diff --git a/.gitignore b/.gitignore index f7fdc59..26d4495 100644 --- a/.gitignore +++ b/.gitignore @@ -2,6 +2,9 @@ Manifest.toml docs/Manifest.toml +# Mathematica (useful for translation of new rules) +*.m + # Documentation build artifacts docs/build/ diff --git a/README.md b/README.md index 585d661..78e3313 100644 --- a/README.md +++ b/README.md @@ -1,6 +1,6 @@ # SymbolicIntegration.jl -[![Build Status](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/CI.yml?query=branch%3Amain) +[![Build Status](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/ci.yml/badge.svg?branch=main)](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/ci.yml?query=branch%3Amain) [![Spell Check](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/spellcheck.yml/badge.svg?branch=main)](https://github.com/JuliaSymbolics/SymbolicIntegration.jl/actions/workflows/spellcheck.yml) [![Rules](https://img.shields.io/badge/dynamic/json?url=https://raw.githubusercontent.com/JuliaSymbolics/SymbolicIntegration.jl/main/.github/badges/rules-count.json&query=$.message&label=Total%20rules&color=blue)](https://github.com/JuliaSymbolics/SymbolicIntegration.jl) @@ -83,7 +83,7 @@ trigonometric functions | ? | sometimes hyperbolic functions | ✅ | sometimes Nonelementary integrals | ❌ | most of them Special functions | ❌ | ❌ -more than one symbolic
variable in the expression | ❌ | ✅ +multiple symbols | ❌ | ✅ More info about them in the [methods documentation](https://docs.sciml.ai/SymbolicIntegration/dev/methods/overview/) diff --git a/docs/src/index.md b/docs/src/index.md index 75b4b24..248d644 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -51,7 +51,7 @@ trigonometric functions | ? | sometimes hyperbolic functions | ✅ | sometimes Nonelementary integrals | ❌ | most of them Special functions | ❌ | ❌ -more than one symbolic
variable in the expression | ❌ | ✅ +multiple symbols | ❌ | ✅ [→ See complete methods documentation](methods/overview.md) @@ -68,7 +68,7 @@ integrate(x^2 + 1, x, RischMethod(use_algebraic_closure=false, catch_errors=true - `use_algebraic_closure` does what? - `catch_errors` does what? -[→ See detailed Risch documentation](risch.md) +[→ See detailed Risch documentation](methods/risch.md) ### RuleBased This method uses a large number of integration rules that specify how to integrate various mathematical expressions. @@ -102,10 +102,7 @@ If you use SymbolicIntegration.jl in your research, please cite: ```@contents Pages = [ - "manual/getting_started.md", - "manual/basic_usage.md", - "manual/rational_functions.md", - "manual/transcendental_functions.md", + "manual/contributing.md", "api.md" ] Depth = 2 diff --git a/docs/src/manual/contributing.md b/docs/src/manual/contributing.md index de9d2c2..ccf3ab0 100644 --- a/docs/src/manual/contributing.md +++ b/docs/src/manual/contributing.md @@ -1,4 +1,10 @@ -- [Contributing to improving RuleBasedMethod](#contributing-to-improving-rulebasedmethod) +# Contributing + +We welcome contributions! + +Below there are detailed info on how to contribute to the translation of new rules from the Mathematica [RUBI](https://rulebasedintegration.org/) package, or translation of solved integrals to use as tests, from the same package. + +- [Contributing to RuleBasedMethod](#contributing-to-rulebasedmethod) - [Common problems when translating rules](#common-problems-when-translating-rules) - [function not translated](#function-not-translated) - [Sum function translation](#sum-function-translation) @@ -13,10 +19,10 @@ - [end](#end) - [Adding Testsuites](#adding-testsuites) -# Contributing to improving RuleBasedMethod +# Contributing to translating new rules for RuleBasedMethod -In this repo there is also some software that serves the sole purpose of helping with the translation of rules from Mathematica syntax, and not for the actual package working. The important ones are: -- translator_of_rules.jl is a script that with regex and other string manipulations translates from Mathematica syntax to julia syntax +In the github repo of the package there is also some software that serves the sole purpose of helping with the translation of rules from Mathematica syntax, and not for the actual package working. The important ones are: +- translator_of_rules.jl is a script that with regex and other string manipulations translates from Mathematica syntax to julia syntax (see "houw to use it" section later) - translator_of_testset.jl is a script that translates the testsets into julia syntax (much simpler than translator_of_rules.jl) - `reload_rules` function in rules_loader.jl. When developing the package using Revise is not enough because rules are defined with a macro. So this function reloads rules from a specific .jl file or from all files if called without arguments. @@ -75,10 +81,11 @@ This script is used to translate integration rules from Mathematica syntax to julia Syntax. ### How to use it +In the branch `rules` of the github repo of the package there are all the Mathematica files containing the untranslated rules already in the correct folders in this repo, so that you can use the translator script like this: ``` bash julia src/translator_of_rules.jl "src/rules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.m" ``` -and will produce the julia file at the path `src/rules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.jl` +this will produce the julia file at the path `src/rules/4 Trig functions/4.1 Sine/4.1.8 trig^m (a+b cos^p+c sin^q)^n.jl` ### How it works internally (useful to know if you have to debug it) It processes line per line, so the integration rule must be all on only one @@ -170,4 +177,4 @@ finally the rule is placed in a tuple (index, rule), and all the tuples are put into a array, ready to be included by load_rules ## Adding Testsuites -There is a test suite of 27585 solved integrals taken from the RUBI package, in the folders `test/test_files/0 Independent test suites` (1796 tests) and `test/test_files/1 Algebraic functions` (25798 tests). But more test can be translated from the [RUBI testsuite](https://rulebasedintegration.org/testProblems.html). In [this](https://github.com/Bumblebee00/SymbolicIntegration.jl?tab=readme-ov-file#testing) repo there are the tests still in Mathematica syntax and a script to translate them to julia. \ No newline at end of file +There is a test suite of 27585 solved integrals taken from the RUBI package, in the folders `test/test_files/0 Independent test suites` (1796 tests) and `test/test_files/1 Algebraic functions` (25798 tests). But more test can be translated from the [RUBI testsuite](https://rulebasedintegration.org/testProblems.html). In the branch `rules` of this repo there are the tests still in Mathematica syntax and a script to translate them to julia. \ No newline at end of file diff --git a/test/methods/rule_based/test_files/2 Exponentials/2.1 u (F^(c (a+b x)))^n.m b/test/methods/rule_based/test_files/2 Exponentials/2.1 u (F^(c (a+b x)))^n.m deleted file mode 100644 index 1c0d1c3..0000000 --- a/test/methods/rule_based/test_files/2 Exponentials/2.1 u (F^(c (a+b x)))^n.m +++ /dev/null @@ -1,177 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (F^(c (a+b x)))^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u F^(c (a+b x))*) - - -(* Note: The optimal antiderivatives in this file are for when the control variable $UseGamma is False. *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m F^(c (a+b x))*) - - -{F^(c*(a + b*x))*(d + e*x)^m, x, 1, (F^(c*(a - (b*d)/e))*(d + e*x)^m*Gamma[1 + m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^m*(b*c*Log[F]))} - -{F^(c*(a + b*x))*(d + e*x)^4, x, 5, (24*e^4*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (24*e^3*F^(c*(a + b*x))*(d + e*x))/(b^4*c^4*Log[F]^4) + (12*e^2*F^(c*(a + b*x))*(d + e*x)^2)/(b^3*c^3*Log[F]^3) - (4*e*F^(c*(a + b*x))*(d + e*x)^3)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^4)/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^3, x, 4, -((6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4)) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/(b^3*c^3*Log[F]^3) - (3*e*F^(c*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^2, x, 3, (2*e^2*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (2*e*F^(c*(a + b*x))*(d + e*x))/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^2)/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^1, x, 2, -((e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2)) + (F^(c*(a + b*x))*(d + e*x))/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^0, x, 1, F^(c*(a + b*x))/(b*c*Log[F])} -{F^(c*(a + b*x))/(d + e*x)^1, x, 1, (F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e])/e} -{F^(c*(a + b*x))/(d + e*x)^2, x, 2, -(F^(c*(a + b*x))/(e*(d + e*x))) + (b*c*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F])/e^2} -{F^(c*(a + b*x))/(d + e*x)^3, x, 3, -(F^(c*(a + b*x))/(2*e*(d + e*x)^2)) - (b*c*F^(c*(a + b*x))*Log[F])/(2*e^2*(d + e*x)) + (b^2*c^2*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^2)/(2*e^3)} -{F^(c*(a + b*x))/(d + e*x)^4, x, 4, -(F^(c*(a + b*x))/(3*e*(d + e*x)^3)) - (b*c*F^(c*(a + b*x))*Log[F])/(6*e^2*(d + e*x)^2) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(6*e^3*(d + e*x)) + (b^3*c^3*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/(6*e^4)} -{F^(c*(a + b*x))/(d + e*x)^5, x, 5, -(F^(c*(a + b*x))/(4*e*(d + e*x)^4)) - (b*c*F^(c*(a + b*x))*Log[F])/(12*e^2*(d + e*x)^3) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(24*e^3*(d + e*x)^2) - (b^3*c^3*F^(c*(a + b*x))*Log[F]^3)/(24*e^4*(d + e*x)) + (b^4*c^4*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/(24*e^5)} - - -{F^(c*(a + b*x))*Expand[(d + e*x)^4], x, 6, (24*e^4*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (24*e^3*F^(c*(a + b*x))*(d + e*x))/(b^4*c^4*Log[F]^4) + (12*e^2*F^(c*(a + b*x))*(d + e*x)^2)/(b^3*c^3*Log[F]^3) - (4*e*F^(c*(a + b*x))*(d + e*x)^3)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^4)/(b*c*Log[F])} -{F^(c*(a + b*x))*Expand[(d + e*x)^3], x, 5, -((6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4)) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/(b^3*c^3*Log[F]^3) - (3*e*F^(c*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])} -{F^(c*(a + b*x))*Expand[(d + e*x)^2], x, 4, (2*e^2*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (2*e*F^(c*(a + b*x))*(d + e*x))/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^2)/(b*c*Log[F])} - -{F^(c*(a + b*x))/Expand[(d + e*x)^2], x, 3, -(F^(c*(a + b*x))/(e*(d + e*x))) + (b*c*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F])/e^2} -{F^(c*(a + b*x))/Expand[(d + e*x)^3], x, 4, -(F^(c*(a + b*x))/(2*e*(d + e*x)^2)) - (b*c*F^(c*(a + b*x))*Log[F])/(2*e^2*(d + e*x)) + (b^2*c^2*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^2)/(2*e^3)} -{F^(c*(a + b*x))/Expand[(d + e*x)^4], x, 5, -(F^(c*(a + b*x))/(3*e*(d + e*x)^3)) - (b*c*F^(c*(a + b*x))*Log[F])/(6*e^2*(d + e*x)^2) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(6*e^3*(d + e*x)) + (b^3*c^3*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^3)/(6*e^4)} -{F^(c*(a + b*x))/Expand[(d + e*x)^5], x, 6, -(F^(c*(a + b*x))/(4*e*(d + e*x)^4)) - (b*c*F^(c*(a + b*x))*Log[F])/(12*e^2*(d + e*x)^3) - (b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(24*e^3*(d + e*x)^2) - (b^3*c^3*F^(c*(a + b*x))*Log[F]^3)/(24*e^4*(d + e*x)) + (b^4*c^4*F^(c*(a - (b*d)/e))*ExpIntegralEi[(b*c*(d + e*x)*Log[F])/e]*Log[F]^4)/(24*e^5)} - - -{F^(c*(a + b*x))*Expand[(d + e*x)^n]^m, x, 2, (F^(c*(a - (b*d)/e))*((d + e*x)^n)^m*Gamma[1 + m*n, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^(m*n)*(b*c*Log[F]))} - -{F^(c*(a + b*x))*Expand[(d + e*x)^4]^m, x, 2, (F^(c*(a - (b*d)/e))*((d + e*x)^4)^m*Gamma[1 + 4*m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^(4*m)*(b*c*Log[F]))} -{F^(c*(a + b*x))*Expand[(d + e*x)^3]^m, x, 2, (F^(c*(a - (b*d)/e))*((d + e*x)^3)^m*Gamma[1 + 3*m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^(3*m)*(b*c*Log[F]))} -{F^(c*(a + b*x))*Expand[(d + e*x)^2]^m, x, 2, (F^(c*(a - (b*d)/e))*((d + e*x)^2)^m*Gamma[1 + 2*m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^(2*m)*(b*c*Log[F]))} -{F^(c*(a + b*x))*Expand[(d + e*x)^1]^m, x, 1, (F^(c*(a - (b*d)/e))*(d + e*x)^m*Gamma[1 + m, -((b*c*(d + e*x)*Log[F])/e)])/((-((b*c*(d + e*x)*Log[F])/e))^m*(b*c*Log[F]))} -{F^(c*(a + b*x))/Expand[(d + e*x)^1]^m, x, 1, (F^(c*(a - (b*d)/e))*Gamma[1 - m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*x)*Log[F])/e))^m)/((d + e*x)^m*(b*c*Log[F]))} -{F^(c*(a + b*x))/Expand[(d + e*x)^2]^m, x, 2, (F^(c*(a - (b*d)/e))*Gamma[1 - 2*m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*x)*Log[F])/e))^(2*m))/(((d + e*x)^2)^m*(b*c*Log[F]))} -{F^(c*(a + b*x))/Expand[(d + e*x)^3]^m, x, 2, (F^(c*(a - (b*d)/e))*Gamma[1 - 3*m, -((b*c*(d + e*x)*Log[F])/e)]*(-((b*c*(d + e*x)*Log[F])/e))^(3*m))/(((d + e*x)^3)^m*(b*c*Log[F]))} - - -{F^(2 + 5*x), x, 1, F^(2 + 5*x)/(5*Log[F])} -{F^(a + b*x), x, 1, F^(a + b*x)/(b*Log[F])} -{10^(2 + 5*x), x, 1, (2^(2 + 5*x)*5^(1 + 5*x))/Log[10]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(m/2) F^(c (a+b x))*) - - -{x^(7/2)*F^(a + b*x), x, 6, (105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(16*b^(9/2)*Log[F]^(9/2)) - (105*F^(a + b*x)*Sqrt[x])/(8*b^4*Log[F]^4) + (35*F^(a + b*x)*x^(3/2))/(4*b^3*Log[F]^3) - (7*F^(a + b*x)*x^(5/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(7/2))/(b*Log[F])} -{x^(5/2)*F^(a + b*x), x, 5, -((15*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(8*b^(7/2)*Log[F]^(7/2))) + (15*F^(a + b*x)*Sqrt[x])/(4*b^3*Log[F]^3) - (5*F^(a + b*x)*x^(3/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(5/2))/(b*Log[F])} -{x^(3/2)*F^(a + b*x), x, 4, (3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(4*b^(5/2)*Log[F]^(5/2)) - (3*F^(a + b*x)*Sqrt[x])/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(3/2))/(b*Log[F])} -{x^(1/2)*F^(a + b*x), x, 3, -((F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(2*b^(3/2)*Log[F]^(3/2))) + (F^(a + b*x)*Sqrt[x])/(b*Log[F])} -{F^(a + b*x)/x^(1/2), x, 2, (F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(Sqrt[b]*Sqrt[Log[F]])} -{F^(a + b*x)/x^(3/2), x, 3, -((2*F^(a + b*x))/Sqrt[x]) + 2*Sqrt[b]*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Sqrt[Log[F]]} -{F^(a + b*x)/x^(5/2), x, 4, -((2*F^(a + b*x))/(3*x^(3/2))) - (4*b*F^(a + b*x)*Log[F])/(3*Sqrt[x]) + (4/3)*b^(3/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(3/2)} -{F^(a + b*x)/x^(7/2), x, 5, -((2*F^(a + b*x))/(5*x^(5/2))) - (4*b*F^(a + b*x)*Log[F])/(15*x^(3/2)) - (8*b^2*F^(a + b*x)*Log[F]^2)/(15*Sqrt[x]) + (8/15)*b^(5/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(5/2)} -{F^(a + b*x)/x^(9/2), x, 6, -((2*F^(a + b*x))/(7*x^(7/2))) - (4*b*F^(a + b*x)*Log[F])/(35*x^(5/2)) - (8*b^2*F^(a + b*x)*Log[F]^2)/(105*x^(3/2)) - (16*b^3*F^(a + b*x)*Log[F]^3)/(105*Sqrt[x]) + (16/105)*b^(7/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Log[F]^(7/2)} - - -{F^(c*(a + b*x))*(d + e*x)^(7/2), x, 6, (105*e^(7/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(16*b^(9/2)*c^(9/2)*Log[F]^(9/2)) - (105*e^3*F^(c*(a + b*x))*Sqrt[d + e*x])/(8*b^4*c^4*Log[F]^4) + (35*e^2*F^(c*(a + b*x))*(d + e*x)^(3/2))/(4*b^3*c^3*Log[F]^3) - (7*e*F^(c*(a + b*x))*(d + e*x)^(5/2))/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(7/2))/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^(5/2), x, 5, -((15*e^(5/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(8*b^(7/2)*c^(7/2)*Log[F]^(7/2))) + (15*e^2*F^(c*(a + b*x))*Sqrt[d + e*x])/(4*b^3*c^3*Log[F]^3) - (5*e*F^(c*(a + b*x))*(d + e*x)^(3/2))/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(5/2))/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^(3/2), x, 4, (3*e^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(4*b^(5/2)*c^(5/2)*Log[F]^(5/2)) - (3*e*F^(c*(a + b*x))*Sqrt[d + e*x])/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(3/2))/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x)^(1/2), x, 3, -((Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(2*b^(3/2)*c^(3/2)*Log[F]^(3/2))) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])} -{F^(c*(a + b*x))/(d + e*x)^(1/2), x, 2, (F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(Sqrt[b]*Sqrt[c]*Sqrt[e]*Sqrt[Log[F]])} -{F^(c*(a + b*x))/(d + e*x)^(3/2), x, 3, -((2*F^(c*(a + b*x)))/(e*Sqrt[d + e*x])) + (2*Sqrt[b]*Sqrt[c]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Sqrt[Log[F]])/e^(3/2)} -{F^(c*(a + b*x))/(d + e*x)^(5/2), x, 4, -((2*F^(c*(a + b*x)))/(3*e*(d + e*x)^(3/2))) - (4*b*c*F^(c*(a + b*x))*Log[F])/(3*e^2*Sqrt[d + e*x]) + (4*b^(3/2)*c^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Log[F]^(3/2))/(3*e^(5/2))} -{F^(c*(a + b*x))/(d + e*x)^(7/2), x, 5, -((2*F^(c*(a + b*x)))/(5*e*(d + e*x)^(5/2))) - (4*b*c*F^(c*(a + b*x))*Log[F])/(15*e^2*(d + e*x)^(3/2)) - (8*b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(15*e^3*Sqrt[d + e*x]) + (8*b^(5/2)*c^(5/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Log[F]^(5/2))/(15*e^(7/2))} -{F^(c*(a + b*x))/(d + e*x)^(9/2), x, 6, -((2*F^(c*(a + b*x)))/(7*e*(d + e*x)^(7/2))) - (4*b*c*F^(c*(a + b*x))*Log[F])/(35*e^2*(d + e*x)^(5/2)) - (8*b^2*c^2*F^(c*(a + b*x))*Log[F]^2)/(105*e^3*(d + e*x)^(3/2)) - (16*b^3*c^3*F^(c*(a + b*x))*Log[F]^3)/(105*e^4*Sqrt[d + e*x]) + (16*b^(7/2)*c^(7/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]*Log[F]^(7/2))/(105*e^(9/2))} - - -{x^(13/2)/E^(b*x), x, 9, -((135135*Sqrt[x])/(E^(b*x)*(64*b^7))) - (45045*x^(3/2))/(E^(b*x)*(32*b^6)) - (9009*x^(5/2))/(E^(b*x)*(16*b^5)) - (1287*x^(7/2))/(E^(b*x)*(8*b^4)) - (143*x^(9/2))/(E^(b*x)*(4*b^3)) - (13*x^(11/2))/(E^(b*x)*(2*b^2)) - x^(13/2)/(E^(b*x)*b) + (135135*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(128*b^(15/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(m/3) (F^(c (a+b x)))^n*) - - -{(d + e*x)^(4/3)*F^(c*(a + b*x)), x, 1, -((e*F^(c*(a - (b*d)/e))*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*(d + e*x)*Log[F])/e)])/(b^2*c^2*Log[F]^2*(-((b*c*(d + e*x)*Log[F])/e))^(1/3)))} - - -{(d + e*x)^(4/3)*(F^(c*(a + b*x)))^n, x, 2, -((e*F^(c*(a - (b*d)/e)*n - c*n*(a + b*x))*(F^(c*(a + b*x)))^n*(d + e*x)^(1/3)*Gamma[7/3, -((b*c*n*(d + e*x)*Log[F])/e)])/(b^2*c^2*n^2*Log[F]^2*(-((b*c*n*(d + e*x)*Log[F])/e))^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Px F^(c (a+b x))*) - - -{F^(c*(a + b*x))*(d + e*x), x, 2, -((e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2)) + (F^(c*(a + b*x))*(d + e*x))/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x + f*x^2), x, 8, (2*f*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) - (e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2) - (2*f*F^(c*(a + b*x))*x)/(b^2*c^2*Log[F]^2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) + (e*F^(c*(a + b*x))*x)/(b*c*Log[F]) + (f*F^(c*(a + b*x))*x^2)/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3), x, 12, -((6*F^(c*(a + b*x))*g)/(b^4*c^4*Log[F]^4)) + (2*f*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (6*F^(c*(a + b*x))*g*x)/(b^3*c^3*Log[F]^3) - (e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2) - (2*f*F^(c*(a + b*x))*x)/(b^2*c^2*Log[F]^2) - (3*F^(c*(a + b*x))*g*x^2)/(b^2*c^2*Log[F]^2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) + (e*F^(c*(a + b*x))*x)/(b*c*Log[F]) + (f*F^(c*(a + b*x))*x^2)/(b*c*Log[F]) + (F^(c*(a + b*x))*g*x^3)/(b*c*Log[F])} -{F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4), x, 17, (24*F^(c*(a + b*x))*h)/(b^5*c^5*Log[F]^5) - (6*F^(c*(a + b*x))*g)/(b^4*c^4*Log[F]^4) - (24*F^(c*(a + b*x))*h*x)/(b^4*c^4*Log[F]^4) + (2*f*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (6*F^(c*(a + b*x))*g*x)/(b^3*c^3*Log[F]^3) + (12*F^(c*(a + b*x))*h*x^2)/(b^3*c^3*Log[F]^3) - (e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2) - (2*f*F^(c*(a + b*x))*x)/(b^2*c^2*Log[F]^2) - (3*F^(c*(a + b*x))*g*x^2)/(b^2*c^2*Log[F]^2) - (4*F^(c*(a + b*x))*h*x^3)/(b^2*c^2*Log[F]^2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) + (e*F^(c*(a + b*x))*x)/(b*c*Log[F]) + (f*F^(c*(a + b*x))*x^2)/(b*c*Log[F]) + (F^(c*(a + b*x))*g*x^3)/(b*c*Log[F]) + (F^(c*(a + b*x))*h*x^4)/(b*c*Log[F])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (e+f x)^n F^(a+b (c+d x))*) - - -{x^m*(a + b*x)^3/E^(a + b*x), x, 6, -((a^3*x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*b)) - (3*a^2*x^m*Gamma[2 + m, b*x])/(E^a*(b*x)^m*b) - (3*a*x^m*Gamma[3 + m, b*x])/(E^a*(b*x)^m*b) - (x^m*Gamma[4 + m, b*x])/(E^a*(b*x)^m*b)} - -{x^3*(a + b*x)^3/E^(a + b*x), x, 24, -((720*E^(-a - b*x))/b^4) - (360*a*E^(-a - b*x))/b^4 - (72*a^2*E^(-a - b*x))/b^4 - (6*a^3*E^(-a - b*x))/b^4 - (720*E^(-a - b*x)*x)/b^3 - (360*a*E^(-a - b*x)*x)/b^3 - (72*a^2*E^(-a - b*x)*x)/b^3 - (6*a^3*E^(-a - b*x)*x)/b^3 - (360*E^(-a - b*x)*x^2)/b^2 - (180*a*E^(-a - b*x)*x^2)/b^2 - (36*a^2*E^(-a - b*x)*x^2)/b^2 - (3*a^3*E^(-a - b*x)*x^2)/b^2 - (120*E^(-a - b*x)*x^3)/b - (60*a*E^(-a - b*x)*x^3)/b - (12*a^2*E^(-a - b*x)*x^3)/b - (a^3*E^(-a - b*x)*x^3)/b - 30*E^(-a - b*x)*x^4 - 15*a*E^(-a - b*x)*x^4 - 3*a^2*E^(-a - b*x)*x^4 - 6*b*E^(-a - b*x)*x^5 - 3*a*b*E^(-a - b*x)*x^5 - b^2*E^(-a - b*x)*x^6} -{x^2*(a + b*x)^3/E^(a + b*x), x, 20, -((120*E^(-a - b*x))/b^3) - (72*a*E^(-a - b*x))/b^3 - (18*a^2*E^(-a - b*x))/b^3 - (2*a^3*E^(-a - b*x))/b^3 - (120*E^(-a - b*x)*x)/b^2 - (72*a*E^(-a - b*x)*x)/b^2 - (18*a^2*E^(-a - b*x)*x)/b^2 - (2*a^3*E^(-a - b*x)*x)/b^2 - (60*E^(-a - b*x)*x^2)/b - (36*a*E^(-a - b*x)*x^2)/b - (9*a^2*E^(-a - b*x)*x^2)/b - (a^3*E^(-a - b*x)*x^2)/b - 20*E^(-a - b*x)*x^3 - 12*a*E^(-a - b*x)*x^3 - 3*a^2*E^(-a - b*x)*x^3 - 5*b*E^(-a - b*x)*x^4 - 3*a*b*E^(-a - b*x)*x^4 - b^2*E^(-a - b*x)*x^5} -{x^1*(a + b*x)^3/E^(a + b*x), x, 11, -((24*E^(-a - b*x))/b^2) + (6*a*E^(-a - b*x))/b^2 - (24*E^(-a - b*x)*(a + b*x))/b^2 + (6*a*E^(-a - b*x)*(a + b*x))/b^2 - (12*E^(-a - b*x)*(a + b*x)^2)/b^2 + (3*a*E^(-a - b*x)*(a + b*x)^2)/b^2 - (4*E^(-a - b*x)*(a + b*x)^3)/b^2 + (a*E^(-a - b*x)*(a + b*x)^3)/b^2 - (E^(-a - b*x)*(a + b*x)^4)/b^2} -{x^0*(a + b*x)^3/E^(a + b*x), x, 4, -((6*E^(-a - b*x))/b) - (6*E^(-a - b*x)*(a + b*x))/b - (3*E^(-a - b*x)*(a + b*x)^2)/b - (E^(-a - b*x)*(a + b*x)^3)/b} -{(a + b*x)^3/(x^1*E^(a + b*x)), x, 9, -2*E^(-a - b*x) - 3*a*E^(-a - b*x) - 3*a^2*E^(-a - b*x) - 2*b*E^(-a - b*x)*x - 3*a*b*E^(-a - b*x)*x - b^2*E^(-a - b*x)*x^2 + (a^3*ExpIntegralEi[(-b)*x])/E^a} -{(a + b*x)^3/(x^2*E^(a + b*x)), x, 8, (-b)*E^(-a - b*x) - 3*a*b*E^(-a - b*x) - (a^3*E^(-a - b*x))/x - b^2*E^(-a - b*x)*x + (3*a^2*b*ExpIntegralEi[(-b)*x])/E^a - (a^3*b*ExpIntegralEi[(-b)*x])/E^a} -{(a + b*x)^3/(x^3*E^(a + b*x)), x, 9, (-b^2)*E^(-a - b*x) - (a^3*E^(-a - b*x))/(2*x^2) - (3*a^2*b*E^(-a - b*x))/x + (a^3*b*E^(-a - b*x))/(2*x) + (3*a*b^2*ExpIntegralEi[(-b)*x])/E^a - (3*a^2*b^2*ExpIntegralEi[(-b)*x])/E^a + ((1/2)*a^3*b^2*ExpIntegralEi[(-b)*x])/E^a} -{(a + b*x)^3/(x^4*E^(a + b*x)), x, 12, -((a^3*E^(-a - b*x))/(3*x^3)) - (3*a^2*b*E^(-a - b*x))/(2*x^2) + (a^3*b*E^(-a - b*x))/(6*x^2) - (3*a*b^2*E^(-a - b*x))/x + (3*a^2*b^2*E^(-a - b*x))/(2*x) - (a^3*b^2*E^(-a - b*x))/(6*x) + (b^3*ExpIntegralEi[(-b)*x])/E^a - (3*a*b^3*ExpIntegralEi[(-b)*x])/E^a + ((3/2)*a^2*b^3*ExpIntegralEi[(-b)*x])/E^a - ((1/6)*a^3*b^3*ExpIntegralEi[(-b)*x])/E^a} - - -{x^m*(e + f*x)^2*F^(a + b*(c + d*x)), x, 5, (f^2*F^(a + b*c)*x^m*Gamma[3 + m, (-b)*d*x*Log[F]])/(((-b)*d*x*Log[F])^m*(b^3*d^3*Log[F]^3)) - (2*e*f*F^(a + b*c)*x^m*Gamma[2 + m, (-b)*d*x*Log[F]])/(((-b)*d*x*Log[F])^m*(b^2*d^2*Log[F]^2)) + (e^2*F^(a + b*c)*x^m*Gamma[1 + m, (-b)*d*x*Log[F]])/(((-b)*d*x*Log[F])^m*(b*d*Log[F]))} - -{x^3*(e + f*x)^2*F^(a + b*(c + d*x)), x, 17, -((120*f^2*F^(a + b*c + b*d*x))/(b^6*d^6*Log[F]^6)) + (48*e*f*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) + (120*f^2*F^(a + b*c + b*d*x)*x)/(b^5*d^5*Log[F]^5) - (6*e^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (48*e*f*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) - (60*f^2*F^(a + b*c + b*d*x)*x^2)/(b^4*d^4*Log[F]^4) + (6*e^2*F^(a + b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) + (24*e*f*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) + (20*f^2*F^(a + b*c + b*d*x)*x^3)/(b^3*d^3*Log[F]^3) - (3*e^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (8*e*f*F^(a + b*c + b*d*x)*x^3)/(b^2*d^2*Log[F]^2) - (5*f^2*F^(a + b*c + b*d*x)*x^4)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^5)/(b*d*Log[F])} -{x^2*(e + f*x)^2*F^(a + b*(c + d*x)), x, 14, (24*f^2*F^(a + b*c + b*d*x))/(b^5*d^5*Log[F]^5) - (12*e*f*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4) - (24*f^2*F^(a + b*c + b*d*x)*x)/(b^4*d^4*Log[F]^4) + (2*e^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (12*e*f*F^(a + b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) + (12*f^2*F^(a + b*c + b*d*x)*x^2)/(b^3*d^3*Log[F]^3) - (2*e^2*F^(a + b*c + b*d*x)*x)/(b^2*d^2*Log[F]^2) - (6*e*f*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) - (4*f^2*F^(a + b*c + b*d*x)*x^3)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x^2)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^4)/(b*d*Log[F])} -{x^1*(e + f*x)^2*F^(a + b*(c + d*x)), x, 11, -((6*f^2*F^(a + b*c + b*d*x))/(b^4*d^4*Log[F]^4)) + (4*e*f*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) + (6*f^2*F^(a + b*c + b*d*x)*x)/(b^3*d^3*Log[F]^3) - (e^2*F^(a + b*c + b*d*x))/(b^2*d^2*Log[F]^2) - (4*e*f*F^(a + b*c + b*d*x)*x)/(b^2*d^2*Log[F]^2) - (3*f^2*F^(a + b*c + b*d*x)*x^2)/(b^2*d^2*Log[F]^2) + (e^2*F^(a + b*c + b*d*x)*x)/(b*d*Log[F]) + (2*e*f*F^(a + b*c + b*d*x)*x^2)/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x^3)/(b*d*Log[F])} -{x^0*(e + f*x)^2*F^(a + b*(c + d*x)), x, 4, (2*f^2*F^(a + b*c + b*d*x))/(b^3*d^3*Log[F]^3) - (2*f*F^(a + b*c + b*d*x)*(e + f*x))/(b^2*d^2*Log[F]^2) + (F^(a + b*c + b*d*x)*(e + f*x)^2)/(b*d*Log[F])} -{(e + f*x)^2*F^(a + b*(c + d*x))/x^1, x, 6, e^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]] - (f^2*F^(a + b*c + b*d*x))/(b^2*d^2*Log[F]^2) + (2*e*f*F^(a + b*c + b*d*x))/(b*d*Log[F]) + (f^2*F^(a + b*c + b*d*x)*x)/(b*d*Log[F])} -{(e + f*x)^2*F^(a + b*(c + d*x))/x^2, x, 6, -((e^2*F^(a + b*c + b*d*x))/x) + 2*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]] + (f^2*F^(a + b*c + b*d*x))/(b*d*Log[F]) + b*d*e^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]} -{(e + f*x)^2*F^(a + b*(c + d*x))/x^3, x, 8, -((e^2*F^(a + b*c + b*d*x))/(2*x^2)) - (2*e*f*F^(a + b*c + b*d*x))/x + f^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]] - (b*d*e^2*F^(a + b*c + b*d*x)*Log[F])/(2*x) + 2*b*d*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F] + (1/2)*b^2*d^2*e^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^2} -{(e + f*x)^2*F^(a + b*(c + d*x))/x^4, x, 11, -((e^2*F^(a + b*c + b*d*x))/(3*x^3)) - (e*f*F^(a + b*c + b*d*x))/x^2 - (f^2*F^(a + b*c + b*d*x))/x - (b*d*e^2*F^(a + b*c + b*d*x)*Log[F])/(6*x^2) - (b*d*e*f*F^(a + b*c + b*d*x)*Log[F])/x + b*d*f^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F] - (b^2*d^2*e^2*F^(a + b*c + b*d*x)*Log[F]^2)/(6*x) + b^2*d^2*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^2 + (1/6)*b^3*d^3*e^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^3} -{(e + f*x)^2*F^(a + b*(c + d*x))/x^5, x, 14, -((e^2*F^(a + b*c + b*d*x))/(4*x^4)) - (2*e*f*F^(a + b*c + b*d*x))/(3*x^3) - (f^2*F^(a + b*c + b*d*x))/(2*x^2) - (b*d*e^2*F^(a + b*c + b*d*x)*Log[F])/(12*x^3) - (b*d*e*f*F^(a + b*c + b*d*x)*Log[F])/(3*x^2) - (b*d*f^2*F^(a + b*c + b*d*x)*Log[F])/(2*x) - (b^2*d^2*e^2*F^(a + b*c + b*d*x)*Log[F]^2)/(24*x^2) - (b^2*d^2*e*f*F^(a + b*c + b*d*x)*Log[F]^2)/(3*x) + (1/2)*b^2*d^2*f^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^2 - (b^3*d^3*e^2*F^(a + b*c + b*d*x)*Log[F]^3)/(24*x) + (1/3)*b^3*d^3*e*f*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^3 + (1/24)*b^4*d^4*e^2*F^(a + b*c)*ExpIntegralEi[b*d*x*Log[F]]*Log[F]^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b x)^n F^(e (a+b x))*) - - -{(c + d*x)^3*(a + b*x)^4/E^(a + b*x), x, 28, -((5040*d^3*E^(-a - b*x))/b^4) - (2160*d^2*(b*c - a*d)*E^(-a - b*x))/b^4 - (360*d*(b*c - a*d)^2*E^(-a - b*x))/b^4 - (24*(b*c - a*d)^3*E^(-a - b*x))/b^4 - (5040*d^3*E^(-a - b*x)*(a + b*x))/b^4 - (2160*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^4 - (360*d*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x))/b^4 - (24*(b*c - a*d)^3*E^(-a - b*x)*(a + b*x))/b^4 - (2520*d^3*E^(-a - b*x)*(a + b*x)^2)/b^4 - (1080*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/b^4 - (180*d*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^2)/b^4 - (12*(b*c - a*d)^3*E^(-a - b*x)*(a + b*x)^2)/b^4 - (840*d^3*E^(-a - b*x)*(a + b*x)^3)/b^4 - (360*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^4 - (60*d*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^3)/b^4 - (4*(b*c - a*d)^3*E^(-a - b*x)*(a + b*x)^3)/b^4 - (210*d^3*E^(-a - b*x)*(a + b*x)^4)/b^4 - (90*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^4 - (15*d*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^4)/b^4 - ((b*c - a*d)^3*E^(-a - b*x)*(a + b*x)^4)/b^4 - (42*d^3*E^(-a - b*x)*(a + b*x)^5)/b^4 - (18*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^5)/b^4 - (3*d*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^5)/b^4 - (7*d^3*E^(-a - b*x)*(a + b*x)^6)/b^4 - (3*d^2*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^6)/b^4 - (d^3*E^(-a - b*x)*(a + b*x)^7)/b^4} -{(c + d*x)^2*(a + b*x)^4/E^(a + b*x), x, 20, -((720*d^2*E^(-a - b*x))/b^3) - (240*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (24*(b*c - a*d)^2*E^(-a - b*x))/b^3 - (720*d^2*E^(-a - b*x)*(a + b*x))/b^3 - (240*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^3 - (24*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x))/b^3 - (360*d^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/b^3 - (12*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (40*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^3 - (4*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (30*d^2*E^(-a - b*x)*(a + b*x)^4)/b^3 - (10*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^3 - ((b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^4)/b^3 - (6*d^2*E^(-a - b*x)*(a + b*x)^5)/b^3 - (2*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^5)/b^3 - (d^2*E^(-a - b*x)*(a + b*x)^6)/b^3} -{(c + d*x)^1*(a + b*x)^4/E^(a + b*x), x, 13, -((120*d*E^(-a - b*x))/b^2) - (24*(b*c - a*d)*E^(-a - b*x))/b^2 - (120*d*E^(-a - b*x)*(a + b*x))/b^2 - (24*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^2 - (60*d*E^(-a - b*x)*(a + b*x)^2)/b^2 - (12*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/b^2 - (20*d*E^(-a - b*x)*(a + b*x)^3)/b^2 - (4*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^2 - (5*d*E^(-a - b*x)*(a + b*x)^4)/b^2 - ((b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^2 - (d*E^(-a - b*x)*(a + b*x)^5)/b^2} -{(c + d*x)^0*(a + b*x)^4/E^(a + b*x), x, 5, -((24*E^(-a - b*x))/b) - (24*E^(-a - b*x)*(a + b*x))/b - (12*E^(-a - b*x)*(a + b*x)^2)/b - (4*E^(-a - b*x)*(a + b*x)^3)/b - (E^(-a - b*x)*(a + b*x)^4)/b} -{(a + b*x)^4/((c + d*x)^1*E^(a + b*x)), x, 13, -((6*E^(-a - b*x))/d) + (2*(b*c - a*d)*E^(-a - b*x))/d^2 - ((b*c - a*d)^2*E^(-a - b*x))/d^3 + ((b*c - a*d)^3*E^(-a - b*x))/d^4 - (6*E^(-a - b*x)*(a + b*x))/d + (2*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/d^2 - ((b*c - a*d)^2*E^(-a - b*x)*(a + b*x))/d^3 - (3*E^(-a - b*x)*(a + b*x)^2)/d + ((b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/d^2 - (E^(-a - b*x)*(a + b*x)^3)/d + ((b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5} -{(a + b*x)^4/((c + d*x)^2*E^(a + b*x)), x, 11, -((2*b*E^(-a - b*x))/d^2) + (4*b*(b*c - a*d)*E^(-a - b*x))/d^3 - (6*b*(b*c - a*d)^2*E^(-a - b*x))/d^4 - ((b*c - a*d)^4*E^(-a - b*x))/(d^5*(c + d*x)) - (2*b^2*E^(-a - b*x)*(c + d*x))/d^3 + (4*b^2*(b*c - a*d)*E^(-a - b*x)*(c + d*x))/d^4 - (b^3*E^(-a - b*x)*(c + d*x)^2)/d^4 - (4*b*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 - (b*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^6} -{(a + b*x)^4/((c + d*x)^3*E^(a + b*x)), x, 11, -((b^2*E^(-a - b*x))/d^3) + (b^2*(3*b*c - 4*a*d)*E^(-a - b*x))/d^4 - (b^3*E^(-a - b*x)*x)/d^3 - ((b*c - a*d)^4*E^(-a - b*x))/(2*d^5*(c + d*x)^2) + (4*b*(b*c - a*d)^3*E^(-a - b*x))/(d^5*(c + d*x)) + (b*(b*c - a*d)^4*E^(-a - b*x))/(2*d^6*(c + d*x)) + (6*b^2*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 + (4*b^2*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^6 + (b^2*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^7)} -{(a + b*x)^4/((c + d*x)^4*E^(a + b*x)), x, 13, -((b^3*E^(-a - b*x))/d^4) - ((b*c - a*d)^4*E^(-a - b*x))/(3*d^5*(c + d*x)^3) + (2*b*(b*c - a*d)^3*E^(-a - b*x))/(d^5*(c + d*x)^2) + (b*(b*c - a*d)^4*E^(-a - b*x))/(6*d^6*(c + d*x)^2) - (6*b^2*(b*c - a*d)^2*E^(-a - b*x))/(d^5*(c + d*x)) - (2*b^2*(b*c - a*d)^3*E^(-a - b*x))/(d^6*(c + d*x)) - (b^2*(b*c - a*d)^4*E^(-a - b*x))/(6*d^7*(c + d*x)) - (4*b^3*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 - (6*b^3*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^6 - (2*b^3*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^7 - (b^3*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^8)} -{(a + b*x)^4/((c + d*x)^5*E^(a + b*x)), x, 17, -(((b*c - a*d)^4*E^(-a - b*x))/(4*d^5*(c + d*x)^4)) + (4*b*(b*c - a*d)^3*E^(-a - b*x))/(3*d^5*(c + d*x)^3) + (b*(b*c - a*d)^4*E^(-a - b*x))/(12*d^6*(c + d*x)^3) - (3*b^2*(b*c - a*d)^2*E^(-a - b*x))/(d^5*(c + d*x)^2) - (2*b^2*(b*c - a*d)^3*E^(-a - b*x))/(3*d^6*(c + d*x)^2) - (b^2*(b*c - a*d)^4*E^(-a - b*x))/(24*d^7*(c + d*x)^2) + (4*b^3*(b*c - a*d)*E^(-a - b*x))/(d^5*(c + d*x)) + (3*b^3*(b*c - a*d)^2*E^(-a - b*x))/(d^6*(c + d*x)) + (2*b^3*(b*c - a*d)^3*E^(-a - b*x))/(3*d^7*(c + d*x)) + (b^3*(b*c - a*d)^4*E^(-a - b*x))/(24*d^8*(c + d*x)) + (b^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 + (4*b^4*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^6 + (3*b^4*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^7 + (2*b^4*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(3*d^8) + (b^4*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(24*d^9)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m F(c (a+b x)) Log[d x] (e+(f+g x) Log[d x])*) - - -{x^m*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 + m + b*c*x*Log[F])*Log[d*x]), x, 1, e*F^(c*(a + b*x))*x^(1 + m)*Log[d*x]^(1 + n)} - -{x^2*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 + 2 + b*c*x*Log[F])*Log[d*x]), x, 1, e*F^(c*(a + b*x))*x^3*Log[d*x]^(1 + n)} -{x^1*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 + 1 + b*c*x*Log[F])*Log[d*x]), x, 1, e*F^(c*(a + b*x))*x^2*Log[d*x]^(1 + n)} -{x^0*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 + 0 + b*c*x*Log[F])*Log[d*x]), x, 1, e*F^(c*(a + b*x))*x*Log[d*x]^(1 + n)} -{1/x^1*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 - 1 + b*c*x*Log[F])*Log[d*x]), x, 1, e*F^(c*(a + b*x))*Log[d*x]^(1 + n)} -{1/x^2*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 - 2 + b*c*x*Log[F])*Log[d*x]), x, 1, (e*F^(c*(a + b*x))*Log[d*x]^(1 + n))/x} -{1/x^3*F^(c*(a + b*x))*Log[d*x]^n*(e + e*n + e*(1 - 3 + b*c*x*Log[F])*Log[d*x]), x, 1, (e*F^(c*(a + b*x))*Log[d*x]^(1 + n))/x^2} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (F^(c (a+b x)))^n*) - - -{Sqrt[E^(a + b*x)]*x^4, x, 5, (768*Sqrt[E^(a + b*x)])/b^5 - (384*Sqrt[E^(a + b*x)]*x)/b^4 + (96*Sqrt[E^(a + b*x)]*x^2)/b^3 - (16*Sqrt[E^(a + b*x)]*x^3)/b^2 + (2*Sqrt[E^(a + b*x)]*x^4)/b} -{Sqrt[E^(a + b*x)]*x^3, x, 4, -((96*Sqrt[E^(a + b*x)])/b^4) + (48*Sqrt[E^(a + b*x)]*x)/b^3 - (12*Sqrt[E^(a + b*x)]*x^2)/b^2 + (2*Sqrt[E^(a + b*x)]*x^3)/b} -{Sqrt[E^(a + b*x)]*x^2, x, 3, (16*Sqrt[E^(a + b*x)])/b^3 - (8*Sqrt[E^(a + b*x)]*x)/b^2 + (2*Sqrt[E^(a + b*x)]*x^2)/b} -{Sqrt[E^(a + b*x)]*x^1, x, 2, -((4*Sqrt[E^(a + b*x)])/b^2) + (2*Sqrt[E^(a + b*x)]*x)/b} -{Sqrt[E^(a + b*x)]*x^0, x, 1, (2*Sqrt[E^(a + b*x)])/b} -{Sqrt[E^(a + b*x)]/x^1, x, 2, (Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/E^((b*x)/2)} -{Sqrt[E^(a + b*x)]/x^2, x, 3, -(Sqrt[E^(a + b*x)]/x) + ((1/2)*b*Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/E^((b*x)/2)} -{Sqrt[E^(a + b*x)]/x^3, x, 4, -(Sqrt[E^(a + b*x)]/(2*x^2)) - (b*Sqrt[E^(a + b*x)])/(4*x) + ((1/8)*b^2*Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/E^((b*x)/2)} -{Sqrt[E^(a + b*x)]/x^4, x, 5, -(Sqrt[E^(a + b*x)]/(3*x^3)) - (b*Sqrt[E^(a + b*x)])/(12*x^2) - (b^2*Sqrt[E^(a + b*x)])/(24*x) + ((1/48)*b^3*Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/E^((b*x)/2)} diff --git a/test/methods/rule_based/test_files/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m b/test/methods/rule_based/test_files/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m deleted file mode 100644 index f316416..0000000 --- a/test/methods/rule_based/test_files/2 Exponentials/2.2 (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p.m +++ /dev/null @@ -1,192 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b (F^(g (e+f x)))^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b F^(e+f x))^p*) - - -(* ::Subsection:: *) -(*p>0*) - - -(* ::Subsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*E^(c + d*x)), x, 6, x^4/(4*a) - (x^3*Log[1 + (b*E^(c + d*x))/a])/(a*d) - (3*x^2*PolyLog[2, -((b*E^(c + d*x))/a)])/(a*d^2) + (6*x*PolyLog[3, -((b*E^(c + d*x))/a)])/(a*d^3) - (6*PolyLog[4, -((b*E^(c + d*x))/a)])/(a*d^4)} -{x^2/(a + b*E^(c + d*x)), x, 5, x^3/(3*a) - (x^2*Log[1 + (b*E^(c + d*x))/a])/(a*d) - (2*x*PolyLog[2, -((b*E^(c + d*x))/a)])/(a*d^2) + (2*PolyLog[3, -((b*E^(c + d*x))/a)])/(a*d^3)} -{x^1/(a + b*E^(c + d*x)), x, 4, x^2/(2*a) - (x*Log[1 + (b*E^(c + d*x))/a])/(a*d) - PolyLog[2, -((b*E^(c + d*x))/a)]/(a*d^2)} -{x^0/(a + b*E^(c + d*x)), x, 4, x/a - Log[a + b*E^(c + d*x)]/(a*d)} -{1/(x^1*(a + b*E^(c + d*x))), x, 0, Unintegrable[1/((a + b*E^(c + d*x))*x), x]} -{1/(x^2*(a + b*E^(c + d*x))), x, 0, Unintegrable[1/((a + b*E^(c + d*x))*x^2), x]} - -{1/(a + b*E^(c - d*x)), x, 4, x/a + Log[a + b*E^(c - d*x)]/(a*d)} -{1/(a + b*E^(-c - d*x)), x, 4, x/a + Log[a + b*E^(-c - d*x)]/(a*d)} - - -{x^3/(a + b*E^(c + d*x))^2, x, 13, -(x^3/(a^2*d)) + x^3/(a*d*(a + b*E^(c + d*x))) + x^4/(4*a^2) + (3*x^2*Log[1 + (b*E^(c + d*x))/a])/(a^2*d^2) - (x^3*Log[1 + (b*E^(c + d*x))/a])/(a^2*d) + (6*x*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^2*d^3) - (3*x^2*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^2*d^2) - (6*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^2*d^4) + (6*x*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^2*d^3) - (6*PolyLog[4, -((b*E^(c + d*x))/a)])/(a^2*d^4)} -{x^2/(a + b*E^(c + d*x))^2, x, 11, -(x^2/(a^2*d)) + x^2/(a*d*(a + b*E^(c + d*x))) + x^3/(3*a^2) + (2*x*Log[1 + (b*E^(c + d*x))/a])/(a^2*d^2) - (x^2*Log[1 + (b*E^(c + d*x))/a])/(a^2*d) + (2*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^2*d^3) - (2*x*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^2*d^2) + (2*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^2*d^3)} -{x^1/(a + b*E^(c + d*x))^2, x, 10, -(x/(a^2*d)) + x/(a*d*(a + b*E^(c + d*x))) + x^2/(2*a^2) + Log[a + b*E^(c + d*x)]/(a^2*d^2) - (x*Log[1 + (b*E^(c + d*x))/a])/(a^2*d) - PolyLog[2, -((b*E^(c + d*x))/a)]/(a^2*d^2)} -{x^0/(a + b*E^(c + d*x))^2, x, 3, 1/(a*d*(a + b*E^(c + d*x))) + x/a^2 - Log[a + b*E^(c + d*x)]/(a^2*d)} -{1/(x^1*(a + b*E^(c + d*x))^2), x, 0, Unintegrable[1/((a + b*E^(c + d*x))^2*x), x]} -{1/(x^2*(a + b*E^(c + d*x))^2), x, 0, Unintegrable[1/((a + b*E^(c + d*x))^2*x^2), x]} - -{1/(a + b*E^(c - d*x))^2, x, 3, -(1/(a*d*(a + b*E^(c - d*x)))) + x/a^2 + Log[a + b*E^(c - d*x)]/(a^2*d)} -{1/(a + b*E^(-c - d*x))^2, x, 3, -(1/(a*d*(a + b*E^(-c - d*x)))) + x/a^2 + Log[a + b*E^(-c - d*x)]/(a^2*d)} - - -{x^3/(a + b*E^(c + d*x))^3, x, 26, (3*x^2)/(2*a^3*d^2) - (3*x^2)/(2*a^2*d^2*(a + b*E^(c + d*x))) - (3*x^3)/(2*a^3*d) + x^3/(2*a*d*(a + b*E^(c + d*x))^2) + x^3/(a^2*d*(a + b*E^(c + d*x))) + x^4/(4*a^3) - (3*x*Log[1 + (b*E^(c + d*x))/a])/(a^3*d^3) + (9*x^2*Log[1 + (b*E^(c + d*x))/a])/(2*a^3*d^2) - (x^3*Log[1 + (b*E^(c + d*x))/a])/(a^3*d) - (3*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^3*d^4) + (9*x*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^3*d^3) - (3*x^2*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^3*d^2) - (9*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^3*d^4) + (6*x*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^3*d^3) - (6*PolyLog[4, -((b*E^(c + d*x))/a)])/(a^3*d^4)} -{x^2/(a + b*E^(c + d*x))^3, x, 23, x/(a^3*d^2) - x/(a^2*d^2*(a + b*E^(c + d*x))) - (3*x^2)/(2*a^3*d) + x^2/(2*a*d*(a + b*E^(c + d*x))^2) + x^2/(a^2*d*(a + b*E^(c + d*x))) + x^3/(3*a^3) - Log[a + b*E^(c + d*x)]/(a^3*d^3) + (3*x*Log[1 + (b*E^(c + d*x))/a])/(a^3*d^2) - (x^2*Log[1 + (b*E^(c + d*x))/a])/(a^3*d) + (3*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^3*d^3) - (2*x*PolyLog[2, -((b*E^(c + d*x))/a)])/(a^3*d^2) + (2*PolyLog[3, -((b*E^(c + d*x))/a)])/(a^3*d^3)} -{x^1/(a + b*E^(c + d*x))^3, x, 15, -(1/(2*a^2*d^2*(a + b*E^(c + d*x)))) - (3*x)/(2*a^3*d) + x/(2*a*d*(a + b*E^(c + d*x))^2) + x/(a^2*d*(a + b*E^(c + d*x))) + x^2/(2*a^3) + (3*Log[a + b*E^(c + d*x)])/(2*a^3*d^2) - (x*Log[1 + (b*E^(c + d*x))/a])/(a^3*d) - PolyLog[2, -((b*E^(c + d*x))/a)]/(a^3*d^2)} -{x^0/(a + b*E^(c + d*x))^3, x, 3, 1/(2*a*d*(a + b*E^(c + d*x))^2) + 1/(a^2*d*(a + b*E^(c + d*x))) + x/a^3 - Log[a + b*E^(c + d*x)]/(a^3*d)} -{1/(x^1*(a + b*E^(c + d*x))^3), x, 0, Unintegrable[1/((a + b*E^(c + d*x))^3*x), x]} -{1/(x^2*(a + b*E^(c + d*x))^3), x, 0, Unintegrable[1/((a + b*E^(c + d*x))^3*x^2), x]} - -{1/(a + b*E^(c - d*x))^3, x, 3, -(1/(2*a*d*(a + b*E^(c - d*x))^2)) - 1/(a^2*d*(a + b*E^(c - d*x))) + x/a^3 + Log[a + b*E^(c - d*x)]/(a^3*d)} -{1/(a + b*E^(-c - d*x))^3, x, 3, -(1/(2*a*d*(a + b*E^(-c - d*x))^2)) - 1/(a^2*d*(a + b*E^(-c - d*x))) + x/a^3 + Log[a + b*E^(-c - d*x)]/(a^3*d)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b (F^(g (e+f x)))^n)^p*) - - -(* ::Subsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^3*(a + b*(F^(g*(e + f*x)))^n), x, 6, (a*(c + d*x)^4)/(4*d) - (6*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) + (6*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) - (3*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F])} -{(c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n), x, 5, (a*(c + d*x)^3)/(3*d) + (2*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) - (2*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F])} -{(c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n), x, 4, (a*(c + d*x)^2)/(2*d) - (b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(e*g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F])} -{(c + d*x)^0*(a + b*(F^(g*(e + f*x)))^n), x, 2, a*x + (b*(F^(g*(e + f*x)))^n)/(f*g*n*Log[F])} -{(a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^1, x, 4, (b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d])/d + (a*Log[c + d*x])/d} -{(a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^2, x, 5, -(a/(d*(c + d*x))) - (b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) + (b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2} -{(a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^3, x, 6, -(a/(2*d*(c + d*x)^2)) - (b*(F^(e*g + f*g*x))^n)/(2*d*(c + d*x)^2) - (b*f*(F^(e*g + f*g*x))^n*g*n*Log[F])/(2*d^2*(c + d*x)) + (b*f^2*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g^2*n^2*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/(2*d^3)} - - -{(c + d*x)^3*(a + b*(F^(g*(e + f*x)))^n)^2, x, 10, (a^2*(c + d*x)^4)/(4*d) - (12*a*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) - (3*b^2*d^3*(F^(e*g + f*g*x))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) + (12*a*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (3*b^2*d^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(4*f^3*g^3*n^3*Log[F]^3) - (6*a*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) - (3*b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(4*f^2*g^2*n^2*Log[F]^2) + (2*a*b*(F^(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^3)/(2*f*g*n*Log[F])} -{(c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n)^2, x, 8, (a^2*(c + d*x)^3)/(3*d) + (4*a*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) + (b^2*d^2*(F^(e*g + f*g*x))^(2*n))/(4*f^3*g^3*n^3*Log[F]^3) - (4*a*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*f^2*g^2*n^2*Log[F]^2) + (2*a*b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(2*f*g*n*Log[F])} -{(c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n)^2, x, 6, (a^2*(c + d*x)^2)/(2*d) - (2*a*b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) - (b^2*d*(F^(e*g + f*g*x))^(2*n))/(4*f^2*g^2*n^2*Log[F]^2) + (2*a*b*(F^(e*g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F]) + (b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*f*g*n*Log[F])} -{(c + d*x)^0*(a + b*(F^(g*(e + f*x)))^n)^2, x, 4, a^2*x + (2*a*b*(F^(g*(e + f*x)))^n)/(f*g*n*Log[F]) + (b^2*(F^(g*(e + f*x)))^(2*n))/(2*f*g*n*Log[F])} -{(a + b*(F^(g*(e + f*x)))^n)^2/(c + d*x)^1, x, 6, (2*a*b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d])/d + (b^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d])/d + (a^2*Log[c + d*x])/d} -{(a + b*(F^(g*(e + f*x)))^n)^2/(c + d*x)^2, x, 8, -(a^2/(d*(c + d*x))) - (2*a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) - (b^2*(F^(e*g + f*g*x))^(2*n))/(d*(c + d*x)) + (2*a*b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2 + (2*b^2*f*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g*n*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2} -{(a + b*(F^(g*(e + f*x)))^n)^2/(c + d*x)^3, x, 10, -(a^2/(2*d*(c + d*x)^2)) - (a*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)^2) - (b^2*(F^(e*g + f*g*x))^(2*n))/(2*d*(c + d*x)^2) - (a*b*f*(F^(e*g + f*g*x))^n*g*n*Log[F])/(d^2*(c + d*x)) - (b^2*f*(F^(e*g + f*g*x))^(2*n)*g*n*Log[F])/(d^2*(c + d*x)) + (a*b*f^2*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g^2*n^2*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3 + (2*b^2*f^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g^2*n^2*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3} - - -{(c + d*x)^3*(a + b*(F^(g*(e + f*x)))^n)^3, x, 14, (a^3*(c + d*x)^4)/(4*d) - (18*a^2*b*d^3*(F^(e*g + f*g*x))^n)/(f^4*g^4*n^4*Log[F]^4) - (9*a*b^2*d^3*(F^(e*g + f*g*x))^(2*n))/(8*f^4*g^4*n^4*Log[F]^4) - (2*b^3*d^3*(F^(e*g + f*g*x))^(3*n))/(27*f^4*g^4*n^4*Log[F]^4) + (18*a^2*b*d^2*(F^(e*g + f*g*x))^n*(c + d*x))/(f^3*g^3*n^3*Log[F]^3) + (9*a*b^2*d^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^3*g^3*n^3*Log[F]^3) - (9*a^2*b*d*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f^2*g^2*n^2*Log[F]^2) - (9*a*b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(4*f^2*g^2*n^2*Log[F]^2) - (b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/(3*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*g + f*g*x))^n*(c + d*x)^3)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^3)/(2*f*g*n*Log[F]) + (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^3)/(3*f*g*n*Log[F])} -{(c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n)^3, x, 11, (a^3*(c + d*x)^3)/(3*d) + (6*a^2*b*d^2*(F^(e*g + f*g*x))^n)/(f^3*g^3*n^3*Log[F]^3) + (3*a*b^2*d^2*(F^(e*g + f*g*x))^(2*n))/(4*f^3*g^3*n^3*Log[F]^3) + (2*b^3*d^2*(F^(e*g + f*g*x))^(3*n))/(27*f^3*g^3*n^3*Log[F]^3) - (6*a^2*b*d*(F^(e*g + f*g*x))^n*(c + d*x))/(f^2*g^2*n^2*Log[F]^2) - (3*a*b^2*d*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*f^2*g^2*n^2*Log[F]^2) - (2*b^3*d*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(9*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*g + f*g*x))^n*(c + d*x)^2)/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^2)/(2*f*g*n*Log[F]) + (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^2)/(3*f*g*n*Log[F])} -{(c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n)^3, x, 8, (a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) - (3*a*b^2*d*(F^(e*g + f*g*x))^(2*n))/(4*f^2*g^2*n^2*Log[F]^2) - (b^3*d*(F^(e*g + f*g*x))^(3*n))/(9*f^2*g^2*n^2*Log[F]^2) + (3*a^2*b*(F^(e*g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F]) + (3*a*b^2*(F^(e*g + f*g*x))^(2*n)*(c + d*x))/(2*f*g*n*Log[F]) + (b^3*(F^(e*g + f*g*x))^(3*n)*(c + d*x))/(3*f*g*n*Log[F])} -{(c + d*x)^0*(a + b*(F^(g*(e + f*x)))^n)^3, x, 4, a^3*x + (3*a^2*b*(F^(g*(e + f*x)))^n)/(f*g*n*Log[F]) + (3*a*b^2*(F^(g*(e + f*x)))^(2*n))/(2*f*g*n*Log[F]) + (b^3*(F^(g*(e + f*x)))^(3*n))/(3*f*g*n*Log[F])} -{(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^1, x, 8, (3*a^2*b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d])/d + (3*a*b^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d])/d + (b^3*F^(3*(e - (c*f)/d)*g*n - 3*g*n*(e + f*x))*(F^(e*g + f*g*x))^(3*n)*ExpIntegralEi[(3*f*g*n*(c + d*x)*Log[F])/d])/d + (a^3*Log[c + d*x])/d} -{(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^2, x, 11, -(a^3/(d*(c + d*x))) - (3*a^2*b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) - (3*a*b^2*(F^(e*g + f*g*x))^(2*n))/(d*(c + d*x)) - (b^3*(F^(e*g + f*g*x))^(3*n))/(d*(c + d*x)) + (3*a^2*b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2 + (6*a*b^2*f*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g*n*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2 + (3*b^3*f*F^(3*(e - (c*f)/d)*g*n - 3*g*n*(e + f*x))*(F^(e*g + f*g*x))^(3*n)*g*n*ExpIntegralEi[(3*f*g*n*(c + d*x)*Log[F])/d]*Log[F])/d^2} -{(a + b*(F^(g*(e + f*x)))^n)^3/(c + d*x)^3, x, 14, -(a^3/(2*d*(c + d*x)^2)) - (3*a^2*b*(F^(e*g + f*g*x))^n)/(2*d*(c + d*x)^2) - (3*a*b^2*(F^(e*g + f*g*x))^(2*n))/(2*d*(c + d*x)^2) - (b^3*(F^(e*g + f*g*x))^(3*n))/(2*d*(c + d*x)^2) - (3*a^2*b*f*(F^(e*g + f*g*x))^n*g*n*Log[F])/(2*d^2*(c + d*x)) - (3*a*b^2*f*(F^(e*g + f*g*x))^(2*n)*g*n*Log[F])/(d^2*(c + d*x)) - (3*b^3*f*(F^(e*g + f*g*x))^(3*n)*g*n*Log[F])/(2*d^2*(c + d*x)) + (3*a^2*b*f^2*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g^2*n^2*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/(2*d^3) + (6*a*b^2*f^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*g^2*n^2*ExpIntegralEi[(2*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/d^3 + (9*b^3*f^2*F^(3*(e - (c*f)/d)*g*n - 3*g*n*(e + f*x))*(F^(e*g + f*g*x))^(3*n)*g^2*n^2*ExpIntegralEi[(3*f*g*n*(c + d*x)*Log[F])/d]*Log[F]^2)/(2*d^3)} - - -(* ::Subsection::Closed:: *) -(*p<0*) - - -{(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n), x, 6, (c + d*x)^4/(4*a*d) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*Log[F]) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*n^2*Log[F]^2) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^4*g^4*n^4*Log[F]^4)} -{(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n), x, 5, (c + d*x)^3/(3*a*d) - ((c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*Log[F]) - (2*d*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*n^2*Log[F]^2) + (2*d^2*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^3*g^3*n^3*Log[F]^3)} -{(c + d*x)^1/(a + b*(F^(g*(e + f*x)))^n), x, 4, (c + d*x)^2/(2*a*d) - ((c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a*f*g*n*Log[F]) - (d*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a*f^2*g^2*n^2*Log[F]^2)} -{(c + d*x)^0/(a + b*(F^(g*(e + f*x)))^n), x, 5, x/a - Log[a + b*(F^(g*(e + f*x)))^n]/(a*f*g*n*Log[F])} -{1/((c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n)), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)*(c + d*x)), x]} -{1/((c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n)), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)*(c + d*x)^2), x]} - - -{(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^2, x, 13, (c + d*x)^4/(4*a^2*d) - (c + d*x)^3/(a^2*f*g*n*Log[F]) + (c + d*x)^3/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (3*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^2*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^2*f*g*n*Log[F]) + (6*d^2*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^2*g^2*n^2*Log[F]^2) - (6*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^4*g^4*n^4*Log[F]^4)} -{(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^2, x, 11, (c + d*x)^3/(3*a^2*d) - (c + d*x)^2/(a^2*f*g*n*Log[F]) + (c + d*x)^2/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (2*d*(c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^2*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^2*f*g*n*Log[F]) + (2*d^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3) - (2*d*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^2*g^2*n^2*Log[F]^2) + (2*d^2*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^3*g^3*n^3*Log[F]^3)} -{(c + d*x)^1/(a + b*(F^(g*(e + f*x)))^n)^2, x, 11, (c + d*x)^2/(2*a^2*d) - (d*x)/(a^2*f*g*n*Log[F]) + (c + d*x)/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (d*Log[a + b*(F^(g*(e + f*x)))^n])/(a^2*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^2*f*g*n*Log[F]) - (d*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^2*f^2*g^2*n^2*Log[F]^2)} -{(c + d*x)^0/(a + b*(F^(g*(e + f*x)))^n)^2, x, 4, x/a^2 + 1/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - Log[a + b*(F^(g*(e + f*x)))^n]/(a^2*f*g*n*Log[F])} -{1/((c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n)^2), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)^2*(c + d*x)), x]} -{1/((c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n)^2), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)^2*(c + d*x)^2), x]} - - -{(c + d*x)^3/(a + b*(F^(g*(e + f*x)))^n)^3, x, 26, (c + d*x)^4/(4*a^3*d) + (3*d*(c + d*x)^2)/(2*a^3*f^2*g^2*n^2*Log[F]^2) - (3*d*(c + d*x)^2)/(2*a^2*f^2*(a + b*(F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*x)^3)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^3/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (c + d*x)^3/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (3*d^2*(c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^3*g^3*n^3*Log[F]^3) + (9*d*(c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (3*d^3*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (9*d^2*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (3*d*(c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^2*g^2*n^2*Log[F]^2) - (9*d^3*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4) + (6*d^2*(c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (6*d^3*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^4*g^4*n^4*Log[F]^4)} -{(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3, x, 24, (c + d*x)^3/(3*a^3*d) + (d^2*x)/(a^3*f^2*g^2*n^2*Log[F]^2) - (d*(c + d*x))/(a^2*f^2*(a + b*(F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*(c + d*x)^2)/(2*a^3*f*g*n*Log[F]) + (c + d*x)^2/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (c + d*x)^2/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - (d^2*Log[a + b*(F^(g*(e + f*x)))^n])/(a^3*f^3*g^3*n^3*Log[F]^3) + (3*d*(c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) + (3*d^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3) - (2*d*(c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^2*g^2*n^2*Log[F]^2) + (2*d^2*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^3*g^3*n^3*Log[F]^3)} -{(c + d*x)^1/(a + b*(F^(g*(e + f*x)))^n)^3, x, 17, (c + d*x)^2/(2*a^3*d) - d/(2*a^2*f^2*(a + b*(F^(g*(e + f*x)))^n)*g^2*n^2*Log[F]^2) - (3*d*x)/(2*a^3*f*g*n*Log[F]) + (c + d*x)/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (c + d*x)/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) + (3*d*Log[a + b*(F^(g*(e + f*x)))^n])/(2*a^3*f^2*g^2*n^2*Log[F]^2) - ((c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(a^3*f*g*n*Log[F]) - (d*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(a^3*f^2*g^2*n^2*Log[F]^2)} -{(c + d*x)^0/(a + b*(F^(g*(e + f*x)))^n)^3, x, 4, x/a^3 + 1/(2*a*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + 1/(a^2*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - Log[a + b*(F^(g*(e + f*x)))^n]/(a^3*f*g*n*Log[F])} -{1/((c + d*x)^1*(a + b*(F^(g*(e + f*x)))^n)^3), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)^3*(c + d*x)), x]} -{1/((c + d*x)^2*(a + b*(F^(g*(e + f*x)))^n)^3), x, 1, Unintegrable[1/((a + b*(F^(e*g + f*g*x))^n)^3*(c + d*x)^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) (a+b (F^(g (e+f x)))^n)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[c + d*x]*(a + b*E^x), x, 5, b*E^x*Sqrt[c + d*x] + (2*a*(c + d*x)^(3/2))/(3*d) - ((1/2)*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d)} - - -{Sqrt[c + d*x]*(a + b*E^x)^2, x, 8, 2*a*b*E^x*Sqrt[c + d*x] + (1/2)*b^2*E^(2*x)*Sqrt[c + d*x] + (2*a^2*(c + d*x)^(3/2))/(3*d) - (a*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d) - ((1/4)*b^2*Sqrt[d]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/E^((2*c)/d)} - - -{Sqrt[c + d*x]*(a + b*E^x)^3, x, 11, 3*a^2*b*E^x*Sqrt[c + d*x] + (3/2)*a*b^2*E^(2*x)*Sqrt[c + d*x] + (1/3)*b^3*E^(3*x)*Sqrt[c + d*x] + (2*a^3*(c + d*x)^(3/2))/(3*d) - ((3/2)*a^2*b*Sqrt[d]*Sqrt[Pi]*Erfi[Sqrt[c + d*x]/Sqrt[d]])/E^(c/d) - ((3/4)*a*b^2*Sqrt[d]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c + d*x])/Sqrt[d]])/E^((2*c)/d) - ((1/6)*b^3*Sqrt[d]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c + d*x])/Sqrt[d]])/E^((3*c)/d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sqrt[c + d*x]/(a + b*E^x), x, 0, Unintegrable[Sqrt[c + d*x]/(a + b*E^x), x]} - - -{Sqrt[c + d*x]/(a + b*E^x)^2, x, 0, Unintegrable[Sqrt[c + d*x]/(a + b*E^x)^2, x]} - - -{Sqrt[c + d*x]/(a + b*E^x)^3, x, 0, Unintegrable[Sqrt[c + d*x]/(a + b*E^x)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b (F^(g (e+f x)))^n)^p with m symbolic*) - - -{(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^3, x, 8, (a^3*(c + d*x)^(1 + m))/(d*(1 + m)) + (3^(-1 - m)*b^3*F^(3*(e - (c*f)/d)*g*n - 3*g*n*(e + f*x))*(F^(e*g + f*g*x))^(3*n)*(c + d*x)^m*Gamma[1 + m, -((3*f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F])) + (3*2^(-1 - m)*a*b^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^m*Gamma[1 + m, -((2*f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F])) + (3*a^2*b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*(c + d*x)^m*Gamma[1 + m, -((f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F]))} -{(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^2, x, 6, (a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (2^(-1 - m)*b^2*F^(2*(e - (c*f)/d)*g*n - 2*g*n*(e + f*x))*(F^(e*g + f*g*x))^(2*n)*(c + d*x)^m*Gamma[1 + m, -((2*f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F])) + (2*a*b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*(c + d*x)^m*Gamma[1 + m, -((f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F]))} -{(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^1, x, 4, (a*(c + d*x)^(1 + m))/(d*(1 + m)) + (b*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*(c + d*x)^m*Gamma[1 + m, -((f*g*n*(c + d*x)*Log[F])/d)])/((-((f*g*n*(c + d*x)*Log[F])/d))^m*(f*g*n*Log[F]))} -{(c + d*x)^m/(a + b*(F^(g*(e + f*x)))^n)^1, x, 1, Unintegrable[(c + d*x)^m/(a + b*(F^(e*g + f*g*x))^n), x]} -{(c + d*x)^m/(a + b*(F^(g*(e + f*x)))^n)^2, x, 1, Unintegrable[(c + d*x)^m/(a + b*(F^(e*g + f*g*x))^n)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b (F^(g (e+f x)))^n)^p with p symbolic*) - - -{(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^p, x, 1, Unintegrable[(a + b*(F^(e*g + f*g*x))^n)^p*(c + d*x)^m, x]} - - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (F^(g (e+f x)))^n (a+b (F^(g (e+f x)))^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(e+f x) (a+b F^(e+f x))^n*) - - -{x^3*F^(c + d*x)/(a + b*F^(c + d*x)), x, 5, (x^3*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + (3*x^2*PolyLog[2, -((b*F^(c + d*x))/a)])/(b*d^2*Log[F]^2) - (6*x*PolyLog[3, -((b*F^(c + d*x))/a)])/(b*d^3*Log[F]^3) + (6*PolyLog[4, -((b*F^(c + d*x))/a)])/(b*d^4*Log[F]^4)} -{x^2*F^(c + d*x)/(a + b*F^(c + d*x)), x, 4, (x^2*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + (2*x*PolyLog[2, -((b*F^(c + d*x))/a)])/(b*d^2*Log[F]^2) - (2*PolyLog[3, -((b*F^(c + d*x))/a)])/(b*d^3*Log[F]^3)} -{x^1*F^(c + d*x)/(a + b*F^(c + d*x)), x, 3, (x*Log[1 + (b*F^(c + d*x))/a])/(b*d*Log[F]) + PolyLog[2, -((b*F^(c + d*x))/a)]/(b*d^2*Log[F]^2)} -{x^0*F^(c + d*x)/(a + b*F^(c + d*x)), x, 2, Log[a + b*F^(c + d*x)]/(b*d*Log[F])} -{1/x^1*F^(c + d*x)/(a + b*F^(c + d*x)), x, 0, Unintegrable[F^(c + d*x)/((a + b*F^(c + d*x))*x), x]} -{1/x^2*F^(c + d*x)/(a + b*F^(c + d*x)), x, 0, Unintegrable[F^(c + d*x)/((a + b*F^(c + d*x))*x^2), x]} - - -{x^3*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 6, x^3/(a*b*d*Log[F]) - x^3/(b*d*(a + b*F^(c + d*x))*Log[F]) - (3*x^2*Log[1 + (b*F^(c + d*x))/a])/(a*b*d^2*Log[F]^2) - (6*x*PolyLog[2, -((b*F^(c + d*x))/a)])/(a*b*d^3*Log[F]^3) + (6*PolyLog[3, -((b*F^(c + d*x))/a)])/(a*b*d^4*Log[F]^4)} -{x^2*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 5, x^2/(a*b*d*Log[F]) - x^2/(b*d*(a + b*F^(c + d*x))*Log[F]) - (2*x*Log[1 + (b*F^(c + d*x))/a])/(a*b*d^2*Log[F]^2) - (2*PolyLog[2, -((b*F^(c + d*x))/a)])/(a*b*d^3*Log[F]^3)} -{x^1*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 5, x/(a*b*d*Log[F]) - x/(b*d*(a + b*F^(c + d*x))*Log[F]) - Log[a + b*F^(c + d*x)]/(a*b*d^2*Log[F]^2)} -{x^0*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 2, -(1/(b*d*(a + b*F^(c + d*x))*Log[F]))} -{1/x^1*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 1, -(1/(b*d*(a + b*F^(c + d*x))*x*Log[F])) - Unintegrable[1/((a + b*F^(c + d*x))*x^2), x]/(b*d*Log[F])} -{1/x^2*F^(c + d*x)/(a + b*F^(c + d*x))^2, x, 1, -(1/(b*d*(a + b*F^(c + d*x))*x^2*Log[F])) - (2*Unintegrable[1/((a + b*F^(c + d*x))*x^3), x])/(b*d*Log[F])} - - -{x^3*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 12, -((3*x^2)/(2*a^2*b*d^2*Log[F]^2)) + (3*x^2)/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x^3/(2*a^2*b*d*Log[F]) - x^3/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) + (3*x*Log[1 + (b*F^(c + d*x))/a])/(a^2*b*d^3*Log[F]^3) - (3*x^2*Log[1 + (b*F^(c + d*x))/a])/(2*a^2*b*d^2*Log[F]^2) + (3*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*Log[F]^4) - (3*x*PolyLog[2, -((b*F^(c + d*x))/a)])/(a^2*b*d^3*Log[F]^3) + (3*PolyLog[3, -((b*F^(c + d*x))/a)])/(a^2*b*d^4*Log[F]^4)} -{x^2*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 11, -(x/(a^2*b*d^2*Log[F]^2)) + x/(a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x^2/(2*a^2*b*d*Log[F]) - x^2/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) + Log[a + b*F^(c + d*x)]/(a^2*b*d^3*Log[F]^3) - (x*Log[1 + (b*F^(c + d*x))/a])/(a^2*b*d^2*Log[F]^2) - PolyLog[2, -((b*F^(c + d*x))/a)]/(a^2*b*d^3*Log[F]^3)} -{x^1*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 4, 1/(2*a*b*d^2*(a + b*F^(c + d*x))*Log[F]^2) + x/(2*a^2*b*d*Log[F]) - x/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]) - Log[a + b*F^(c + d*x)]/(2*a^2*b*d^2*Log[F]^2)} -{x^0*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 2, -(1/(2*b*d*(a + b*F^(c + d*x))^2*Log[F]))} -{1/x^1*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 1, -(1/(2*b*d*(a + b*F^(c + d*x))^2*x*Log[F])) - Unintegrable[1/((a + b*F^(c + d*x))^2*x^2), x]/(2*b*d*Log[F])} -{1/x^2*F^(c + d*x)/(a + b*F^(c + d*x))^3, x, 1, -(1/(2*b*d*(a + b*F^(c + d*x))^2*x^2*Log[F])) - Unintegrable[1/((a + b*F^(c + d*x))^2*x^3), x]/(b*d*Log[F])} diff --git a/test/methods/rule_based/test_files/2 Exponentials/2.3 Exponential functions.m b/test/methods/rule_based/test_files/2 Exponentials/2.3 Exponential functions.m deleted file mode 100644 index f6e1c17..0000000 --- a/test/methods/rule_based/test_files/2 Exponentials/2.3 Exponential functions.m +++ /dev/null @@ -1,1295 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Exponentials*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (G^(h (f+g x)))^m (a+b (F^(e (c+d x)))^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(e+f x) (a+b F^(e+f x))^p*) - - -{E^x/(4 + 6*E^x), x, 2, (1/6)*Log[2 + 3*E^x]} -{E^x/(a + b*E^x), x, 2, Log[a + b*E^x]/b} -{E^(d*x)/(a + b*E^(c + d*x)), x, 3, Log[a + b*E^(c + d*x)]/(E^c*(b*d))} -{E^(c + d*x)/(a + b*E^(c + d*x)), x, 2, Log[a + b*E^(c + d*x)]/(b*d)} - -{E^x*(a + b*E^x)^n, x, 2, (a + b*E^x)^(1 + n)/(b*(1 + n))} -{E^(d*x)*(a + b*E^(c + d*x))^n, x, 3, (a + b*E^(c + d*x))^(1 + n)/(E^c*(b*d*(1 + n)))} -{E^(c + d*x)*(a + b*E^(c + d*x))^n, x, 2, (a + b*E^(c + d*x))^(1 + n)/(b*d*(1 + n))} - - -{F^x/(a + b*F^x), x, 2, Log[a + b*F^x]/(b*Log[F])} -{F^(d*x)/(a + b*F^(c + d*x)), x, 3, Log[a + b*F^(c + d*x)]/(F^c*(b*d*Log[F]))} -{F^(c + d*x)/(a + b*F^(c + d*x)), x, 2, Log[a + b*F^(c + d*x)]/(b*d*Log[F])} - -{F^x*(a + b*F^x)^n, x, 2, (a + b*F^x)^(1 + n)/(b*(1 + n)*Log[F])} -{F^(d*x)*(a + b*F^(c + d*x))^n, x, 3, (a + b*F^(c + d*x))^(1 + n)/(F^c*(b*d*(1 + n)*Log[F]))} -{F^(c + d*x)*(a + b*F^(c + d*x))^n, x, 2, (a + b*F^(c + d*x))^(1 + n)/(b*d*(1 + n)*Log[F])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (G^(h (f+g x)))^m (a+b (F^(e (c+d x)))^n)^p when d e n Log[f]=g h m Log[g]*) - - -{(E^x)^n*(a + b*(E^x)^n)^p, x, 2, (a + b*(E^x)^n)^(1 + p)/(b*n*(1 + p))} -{E^(n*x)*(a + b*(E^x)^n)^p, x, 3, (E^(n*x)*(a + b*(E^x)^n)^(1 + p))/((E^x)^n*(b*n*(1 + p)))} - - -{(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)^p, x, 2, (a + b*(F^(e*(c + d*x)))^n)^(1 + p)/(b*d*e*n*(1 + p)*Log[F])} - - -{(G^(h*(f + g*x)))^(d*e*n*Log[F]/(g*h*Log[G]))*(a + b*(F^(e*(c + d*x)))^n)^p, x, 3, ((a + b*(F^(e*(c + d*x)))^n)^(1 + p)*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])))/((F^(e*(c + d*x)))^n*(b*d*e*n*(1 + p)*Log[F]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form G^(h (f+g x)) (a+b F^(e (c+d x)))^p*) - - -{E^(2*x)/(a + b*E^x), x, 3, E^x/b - (a*Log[a + b*E^x])/b^2} -{E^(2*x)/(a + b*E^x)^2, x, 3, a/(b^2*(a + b*E^x)) + Log[a + b*E^x]/b^2} -{E^(2*x)/(a + b*E^x)^3, x, 2, E^(2*x)/(2*a*(a + b*E^x)^2)} -{E^(2*x)/(a + b*E^x)^4, x, 3, a/(3*b^2*(a + b*E^x)^3) - 1/(2*b^2*(a + b*E^x)^2)} - -{E^(4*x)/(a + b*E^(2*x)), x, 3, E^(2*x)/(2*b) - (a*Log[a + b*E^(2*x)])/(2*b^2)} -{E^(4*x)/(a + b*E^(2*x))^2, x, 3, a/(2*b^2*(a + b*E^(2*x))) + Log[a + b*E^(2*x)]/(2*b^2)} -{E^(4*x)/(a + b*E^(2*x))^3, x, 2, E^(4*x)/(4*a*(a + b*E^(2*x))^2)} -{E^(4*x)/(a + b*E^(2*x))^4, x, 3, a/(6*b^2*(a + b*E^(2*x))^3) - 1/(4*b^2*(a + b*E^(2*x))^2)} - -{E^(4*x)/(a + b*E^(2*x))^(2/3), x, 3, -((3*a*(a + b*E^(2*x))^(1/3))/(2*b^2)) + (3*(a + b*E^(2*x))^(4/3))/(8*b^2)} - - -{E^(-n*x)*(a + b*E^(n*x)), x, 3, -(a/(E^(n*x)*n)) + b*x} -{E^(-n*x)*(a + b*E^(n*x))^2, x, 3, -(a^2/(E^(n*x)*n)) + (b^2*E^(n*x))/n + 2*a*b*x} -{E^(-n*x)*(a + b*E^(n*x))^3, x, 3, -(a^3/(E^(n*x)*n)) + (3*a*b^2*E^(n*x))/n + (b^3*E^(2*n*x))/(2*n) + 3*a^2*b*x} - -{E^(-n*x)/(a + b*E^(n*x)), x, 3, -(1/(E^(n*x)*(a*n))) - (b*x)/a^2 + (b*Log[a + b*E^(n*x)])/(a^2*n)} -{E^(-n*x)/(a + b*E^(n*x))^2, x, 3, -(1/(E^(n*x)*(a^2*n))) - b/(a^2*(a + b*E^(n*x))*n) - (2*b*x)/a^3 + (2*b*Log[a + b*E^(n*x)])/(a^3*n)} -{E^(-n*x)/(a + b*E^(n*x))^3, x, 3, -(1/(E^(n*x)*(a^3*n))) - b/(2*a^2*(a + b*E^(n*x))^2*n) - (2*b)/(a^3*(a + b*E^(n*x))*n) - (3*b*x)/a^4 + (3*b*Log[a + b*E^(n*x)])/(a^4*n)} - - -{f^(a + b*x)/(c + d*f^(e + 2*b*x)), x, 2, (f^(a - e/2)*ArcTan[(Sqrt[d]*f^(e/2 + b*x))/Sqrt[c]])/(b*Sqrt[c]*Sqrt[d]*Log[f])} -{f^(a + 2*b*x)/(c + d*f^(e + 2*b*x)), x, 3, (f^(a - e)*Log[c + d*f^(e + 2*b*x)])/(2*b*d*Log[f])} -{f^(a + 3*b*x)/(c + d*f^(e + 2*b*x)), x, 3, f^((1/2)*(2*a - 3*e) + (1/2)*(e + 2*b*x))/(b*d*Log[f]) - (Sqrt[c]*f^(a - (3*e)/2)*ArcTan[(Sqrt[d]*f^((1/2)*(e + 2*b*x)))/Sqrt[c]])/(b*d^(3/2)*Log[f])} -{f^(a + 4*b*x)/(c + d*f^(e + 2*b*x)), x, 3, f^(a - e + 2*b*x)/(2*b*d*Log[f]) - (c*f^(a - 2*e)*Log[c + d*f^(e + 2*b*x)])/(2*b*d^2*Log[f])} -{f^(a + 5*b*x)/(c + d*f^(e + 2*b*x)), x, 4, -((c*f^((1/2)*(2*a - 5*e) + (1/2)*(e + 2*b*x)))/(b*d^2*Log[f])) + f^((1/2)*(2*a - 5*e) + (3/2)*(e + 2*b*x))/(3*b*d*Log[f]) + (c^(3/2)*f^(a - (5*e)/2)*ArcTan[(Sqrt[d]*f^((1/2)*(e + 2*b*x)))/Sqrt[c]])/(b*d^(5/2)*Log[f])} - - -{x^0*E^x/(1 + E^(2*x)), x, 2, ArcTan[E^x]} -{x^0*E^x/(1 - E^(2*x)), x, 2, ArcTanh[E^x]} -{x^1*E^x/(1 - E^(2*x)), x, 3, x*ArcTanh[E^x] + (1/2)*PolyLog[2, -E^x] - (1/2)*PolyLog[2, E^x]} -{x^2*E^x/(1 - E^(2*x)), x, 8, x^2*ArcTanh[E^x] + x*PolyLog[2, -E^x] - x*PolyLog[2, E^x] - PolyLog[3, -E^x] + PolyLog[3, E^x]} -{x^3*E^x/(1 - E^(2*x)), x, 10, x^3*ArcTanh[E^x] + (3/2)*x^2*PolyLog[2, -E^x] - (3/2)*x^2*PolyLog[2, E^x] - 3*x*PolyLog[3, -E^x] + 3*x*PolyLog[3, E^x] + 3*PolyLog[4, -E^x] - 3*PolyLog[4, E^x]} - - -{x^0*f^x/(a + b*f^(2*x)), x, 2, ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[f])} -{x^1*f^x/(a + b*f^(2*x)), x, 6, (x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2)} -{x^2*f^x/(a + b*f^(2*x)), x, 9, (x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*x*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^3)} -{x^3*f^x/(a + b*f^(2*x)), x, 11, (x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (3*I*x^2*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (3*I*x^2*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (3*I*x*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (3*I*x*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (3*I*PolyLog[4, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(Sqrt[a]*Sqrt[b]*Log[f]^4) + (3*I*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Log[f]^4)} - - -{x^0*f^x/(a + b*f^(2*x))^2, x, 3, f^x/(2*a*(a + b*f^(2*x))*Log[f]) + ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*Log[f])} -{x^1*f^x/(a + b*f^(2*x))^2, x, 8, -(ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*Log[f]^2)) + (f^x*x)/(2*a*(a + b*f^(2*x))*Log[f]) + (x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]) - (I*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(4*a^(3/2)*Sqrt[b]*Log[f]^2) + (I*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b]*Log[f]^2)} -{x^2*f^x/(a + b*f^(2*x))^2, x, 16, -((x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]*Log[f]^2)) + (f^x*x^2)/(2*a*(a + b*f^(2*x))*Log[f]) + (x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]) + (I*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) - (I*x*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^2) - (I*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) + (I*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^3)} -{x^3*f^x/(a + b*f^(2*x))^2, x, 21, -((3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^2)) + (f^x*x^3)/(2*a*(a + b*f^(2*x))*Log[f]) + (x^3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]) + (3*I*x*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) - (3*I*x^2*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(4*a^(3/2)*Sqrt[b]*Log[f]^2) - (3*I*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) + (3*I*x^2*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b]*Log[f]^2) - (3*I*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^4) + (3*I*x*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) + (3*I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^4) - (3*I*x*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^3) - (3*I*PolyLog[4, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(3/2)*Sqrt[b]*Log[f]^4) + (3*I*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(3/2)*Sqrt[b]*Log[f]^4)} - - -{x^0*f^x/(a + b*f^(2*x))^3, x, 4, f^x/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f])} -{x^1*f^x/(a + b*f^(2*x))^3, x, 11, -(f^x/(8*a^2*(a + b*f^(2*x))*Log[f]^2)) - ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(2*a^(5/2)*Sqrt[b]*Log[f]^2) + (f^x*x)/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x*x)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3*x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]) - (3*I*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(16*a^(5/2)*Sqrt[b]*Log[f]^2) + (3*I*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]*Log[f]^2)} -{x^2*f^x/(a + b*f^(2*x))^3, x, 24, ArcTan[(Sqrt[b]*f^x)/Sqrt[a]]/(4*a^(5/2)*Sqrt[b]*Log[f]^3) - (f^x*x)/(4*a^2*(a + b*f^(2*x))*Log[f]^2) - (x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(a^(5/2)*Sqrt[b]*Log[f]^2) + (f^x*x^2)/(4*a*(a + b*f^(2*x))^2*Log[f]) + (3*f^x*x^2)/(8*a^2*(a + b*f^(2*x))*Log[f]) + (3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]) + (I*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(2*a^(5/2)*Sqrt[b]*Log[f]^3) - (3*I*x*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(8*a^(5/2)*Sqrt[b]*Log[f]^2) - (I*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]*Log[f]^3) + (3*I*x*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]^2) + (3*I*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(8*a^(5/2)*Sqrt[b]*Log[f]^3) - (3*I*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]^3)} -(* {x^3*f^x/(a + b*f^(2*x))^3, x, 30, (3*x*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(4*a^(5/2)*Sqrt[b]*Log[f]^3) - (3*f^x*x^2)/(8*a^2*(a + b*f^(2*x))*Log[f]^2) - (3*x^2*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/(2*a^(5/2)*Sqrt[b]*Log[f]^2) + (x^3*((2*a^(3/2)*f^x)/(a + b*f^(2*x))^2 + (3*Sqrt[a]*f^x)/(a + b*f^(2*x)) + (3*ArcTan[(Sqrt[b]*f^x)/Sqrt[a]])/Sqrt[b]))/(8*a^(5/2)*Log[f]) - (3*I*(2 - 8*x*Log[f] + 3*x^2*Log[f]^2)*PolyLog[2, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(16*a^(5/2)*Sqrt[b]*Log[f]^4) + (3*I*(2 - 8*x*Log[f] + 3*x^2*Log[f]^2)*PolyLog[2, (I*Sqrt[b]*f^x)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b]*Log[f]^4) - (3*I*(4 - 3*x*Log[f])*PolyLog[3, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(8*a^(5/2)*Sqrt[b]*Log[f]^4) + (3*I*(4 - 3*x*Log[f])*PolyLog[3, (I*Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]^4) - (9*I*PolyLog[4, -((I*Sqrt[b]*f^x)/Sqrt[a])])/(8*a^(5/2)*Sqrt[b]*Log[f]^4) + (9*I*PolyLog[4, (I*Sqrt[b]*f^x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*Log[f]^4)} *) - - -{x^0/(a*f^x + b*f^(-x)), x, 2, ArcTan[(Sqrt[a]*f^x)/Sqrt[b]]/(Sqrt[a]*Sqrt[b]*Log[f])} -{x^1/(a*f^x + b*f^(-x)), x, 6, (x*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2)} -{x^2/(a*f^x + b*f^(-x)), x, 9, (x^2*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (I*x*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^2) + (I*PolyLog[3, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3)} -{x^3/(a*f^x + b*f^(-x)), x, 11, (x^3*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]) - (3*I*x^2*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (3*I*x^2*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(2*Sqrt[a]*Sqrt[b]*Log[f]^2) + (3*I*x*PolyLog[3, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (3*I*x*PolyLog[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^3) - (3*I*PolyLog[4, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(Sqrt[a]*Sqrt[b]*Log[f]^4) + (3*I*PolyLog[4, (I*Sqrt[a]*f^x)/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*Log[f]^4)} - - -{x^0/(a*f^x + b*f^(-x))^2, x, 2, -(1/(2*a*(b + a*f^(2*x))*Log[f]))} -{x^1/(a*f^x + b*f^(-x))^2, x, 6, x/(2*a*b*Log[f]) - x/(2*a*(b + a*f^(2*x))*Log[f]) - Log[b + a*f^(2*x)]/(4*a*b*Log[f]^2)} -{x^2/(a*f^x + b*f^(-x))^2, x, 6, x^2/(2*a*b*Log[f]) - x^2/(2*a*(b + a*f^(2*x))*Log[f]) - (x*Log[1 + (a*f^(2*x))/b])/(2*a*b*Log[f]^2) - PolyLog[2, -((a*f^(2*x))/b)]/(4*a*b*Log[f]^3)} -{x^3/(a*f^x + b*f^(-x))^2, x, 7, x^3/(2*a*b*Log[f]) - x^3/(2*a*(b + a*f^(2*x))*Log[f]) - (3*x^2*Log[1 + (a*f^(2*x))/b])/(4*a*b*Log[f]^2) - (3*x*PolyLog[2, -((a*f^(2*x))/b)])/(4*a*b*Log[f]^3) + (3*PolyLog[3, -((a*f^(2*x))/b)])/(8*a*b*Log[f]^4)} - - -{x^0/(a*f^x + b*f^(-x))^3, x, 4, -(f^x/(4*a*(b + a*f^(2*x))^2*Log[f])) + f^x/(8*a*b*(b + a*f^(2*x))*Log[f]) + ArcTan[(Sqrt[a]*f^x)/Sqrt[b]]/(8*a^(3/2)*b^(3/2)*Log[f])} -{x^1/(a*f^x + b*f^(-x))^3, x, 22, f^x/(8*a*b*(b + a*f^(2*x))*Log[f]^2) - (f^x*x)/(4*a*(b + a*f^(2*x))^2*Log[f]) + (f^x*x)/(8*a*b*(b + a*f^(2*x))*Log[f]) + (x*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]) - (I*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(16*a^(3/2)*b^(3/2)*Log[f]^2) + (I*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(16*a^(3/2)*b^(3/2)*Log[f]^2)} -{x^2/(a*f^x + b*f^(-x))^3, x, 43, -(ArcTan[(Sqrt[a]*f^x)/Sqrt[b]]/(4*a^(3/2)*b^(3/2)*Log[f]^3)) + (f^x*x)/(4*a*b*(b + a*f^(2*x))*Log[f]^2) - (f^x*x^2)/(4*a*(b + a*f^(2*x))^2*Log[f]) + (f^x*x^2)/(8*a*b*(b + a*f^(2*x))*Log[f]) + (x^2*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]) - (I*x*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(8*a^(3/2)*b^(3/2)*Log[f]^2) + (I*x*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]^2) + (I*PolyLog[3, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(8*a^(3/2)*b^(3/2)*Log[f]^3) - (I*PolyLog[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]^3)} -(* {x^3/(a*f^x + b*f^(-x))^3, x, 49, -((3*x*ArcTan[(Sqrt[a]*f^x)/Sqrt[b]])/(4*a^(3/2)*b^(3/2)*Log[f]^3)) + (3*f^x*x^2)/(8*a*b*(b + a*f^(2*x))*Log[f]^2) - (f^x*(5 + (3*a*f^(2*x))/b)*x^3)/(8*a*(b + a*f^(2*x))^2*Log[f]) + (x^3*((4*Sqrt[a]*Sqrt[b]*f^x)/(b + a*f^(2*x)) + ArcTan[(Sqrt[a]*f^x)/Sqrt[b]]))/(8*a^(3/2)*b^(3/2)*Log[f]) + (3*I*(2 - x^2*Log[f]^2)*PolyLog[2, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(16*a^(3/2)*b^(3/2)*Log[f]^4) - (3*I*(2 - x^2*Log[f]^2)*PolyLog[2, (I*Sqrt[a]*f^x)/Sqrt[b]])/(16*a^(3/2)*b^(3/2)*Log[f]^4) + (3*I*x*PolyLog[3, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(8*a^(3/2)*b^(3/2)*Log[f]^3) - (3*I*x*PolyLog[3, (I*Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]^3) - (3*I*PolyLog[4, -((I*Sqrt[a]*f^x)/Sqrt[b])])/(8*a^(3/2)*b^(3/2)*Log[f]^4) + (3*I*PolyLog[4, (I*Sqrt[a]*f^x)/Sqrt[b]])/(8*a^(3/2)*b^(3/2)*Log[f]^4)} *) - - -{f^(a + b*x + c*x^2)*g^(d + e*x + f*x^2), x, 3, (f^a*g^d*Sqrt[Pi]*Erfi[(b*Log[f] + e*Log[g] + 2*x*(c*Log[f] + f*Log[g]))/(2*Sqrt[c*Log[f] + f*Log[g]])])/(E^((b*Log[f] + e*Log[g])^2/(4*(c*Log[f] + f*Log[g])))*(2*Sqrt[c*Log[f] + f*Log[g]]))} - - -{F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n, x, 2, (F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n*Hypergeometric2F1[-n, (d*e*Log[F])/(g*h*Log[G]), 1 + (d*e*Log[F])/(g*h*Log[G]), -((b*G^(h*(f + g*x)))/a)])/((1 + (b*G^(h*(f + g*x)))/a)^n*(d*e*Log[F]))} - - -{F^(e*(c + d*x))*H^(t*(r + s*x))/(a + b*F^(e*(c + d*x))), x, 2, (H^(t*(r + s*x))*Hypergeometric2F1[1, -((s*t*Log[H])/(d*e*Log[F])), 1 - (s*t*Log[H])/(d*e*Log[F]), -(a/(F^(e*(c + d*x))*b))])/(b*s*t*Log[H])} -{F^(e*(f + d*x))*H^(t*(r + s*x))/(a + b*F^(e*(c + d*x))), x, 2, (H^(t*(r + s*x))*Hypergeometric2F1[1, -((s*t*Log[H])/(d*e*Log[F])), 1 - (s*t*Log[H])/(d*e*Log[F]), -(a/(F^(e*(c + d*x))*b))])/(F^(e*(c - f))*(b*s*t*Log[H]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(a+b x^n)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{f^(a + b*x^2)*x^m, x, 1, (-(1/2))*f^a*x^(1 + m)*Gamma[(1 + m)/2, (-b)*x^2*Log[f]]*((-b)*x^2*Log[f])^((1/2)*(-1 - m))} - -{f^(a + b*x^2)*x^11, x, 1, -((f^(a + b*x^2)*(120 - 120*b*x^2*Log[f] + 60*b^2*x^4*Log[f]^2 - 20*b^3*x^6*Log[f]^3 + 5*b^4*x^8*Log[f]^4 - b^5*x^10*Log[f]^5))/(2*b^6*Log[f]^6))} -{f^(a + b*x^2)*x^9, x, 1, (f^(a + b*x^2)*(24 - 24*b*x^2*Log[f] + 12*b^2*x^4*Log[f]^2 - 4*b^3*x^6*Log[f]^3 + b^4*x^8*Log[f]^4))/(2*b^5*Log[f]^5)} -{f^(a + b*x^2)*x^7, x, 4, -((3*f^(a + b*x^2))/(b^4*Log[f]^4)) + (3*f^(a + b*x^2)*x^2)/(b^3*Log[f]^3) - (3*f^(a + b*x^2)*x^4)/(2*b^2*Log[f]^2) + (f^(a + b*x^2)*x^6)/(2*b*Log[f])} -{f^(a + b*x^2)*x^5, x, 3, f^(a + b*x^2)/(b^3*Log[f]^3) - (f^(a + b*x^2)*x^2)/(b^2*Log[f]^2) + (f^(a + b*x^2)*x^4)/(2*b*Log[f])} -{f^(a + b*x^2)*x^3, x, 2, -(f^(a + b*x^2)/(2*b^2*Log[f]^2)) + (f^(a + b*x^2)*x^2)/(2*b*Log[f])} -{f^(a + b*x^2)*x^1, x, 1, f^(a + b*x^2)/(2*b*Log[f])} -{f^(a + b*x^2)/x^1, x, 1, (1/2)*f^a*ExpIntegralEi[b*x^2*Log[f]]} -{f^(a + b*x^2)/x^3, x, 2, -(f^(a + b*x^2)/(2*x^2)) + (1/2)*b*f^a*ExpIntegralEi[b*x^2*Log[f]]*Log[f]} -{f^(a + b*x^2)/x^5, x, 3, -(f^(a + b*x^2)/(4*x^4)) - (b*f^(a + b*x^2)*Log[f])/(4*x^2) + (1/4)*b^2*f^a*ExpIntegralEi[b*x^2*Log[f]]*Log[f]^2} -{f^(a + b*x^2)/x^7, x, 4, -(f^(a + b*x^2)/(6*x^6)) - (b*f^(a + b*x^2)*Log[f])/(12*x^4) - (b^2*f^(a + b*x^2)*Log[f]^2)/(12*x^2) + (1/12)*b^3*f^a*ExpIntegralEi[b*x^2*Log[f]]*Log[f]^3} -{f^(a + b*x^2)/x^9, x, 1, (-(1/2))*b^4*f^a*Gamma[-4, (-b)*x^2*Log[f]]*Log[f]^4} -{f^(a + b*x^2)/x^11, x, 1, (1/2)*b^5*f^a*Gamma[-5, (-b)*x^2*Log[f]]*Log[f]^5} - -{f^(a + b*x^2)*x^12, x, 1, -((f^a*x^13*Gamma[13/2, (-b)*x^2*Log[f]])/(2*((-b)*x^2*Log[f])^(13/2)))} -{f^(a + b*x^2)*x^10, x, 1, -((f^a*x^11*Gamma[11/2, (-b)*x^2*Log[f]])/(2*((-b)*x^2*Log[f])^(11/2)))} -{f^(a + b*x^2)*x^8, x, 5, (105*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(32*b^(9/2)*Log[f]^(9/2)) - (105*f^(a + b*x^2)*x)/(16*b^4*Log[f]^4) + (35*f^(a + b*x^2)*x^3)/(8*b^3*Log[f]^3) - (7*f^(a + b*x^2)*x^5)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^7)/(2*b*Log[f])} -{f^(a + b*x^2)*x^6, x, 4, -((15*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(16*b^(7/2)*Log[f]^(7/2))) + (15*f^(a + b*x^2)*x)/(8*b^3*Log[f]^3) - (5*f^(a + b*x^2)*x^3)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^5)/(2*b*Log[f])} -{f^(a + b*x^2)*x^4, x, 3, (3*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(8*b^(5/2)*Log[f]^(5/2)) - (3*f^(a + b*x^2)*x)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^3)/(2*b*Log[f])} -{f^(a + b*x^2)*x^2, x, 2, -((f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(4*b^(3/2)*Log[f]^(3/2))) + (f^(a + b*x^2)*x)/(2*b*Log[f])} -{f^(a + b*x^2)*x^0, x, 1, (f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(2*Sqrt[b]*Sqrt[Log[f]])} -{f^(a + b*x^2)/x^2, x, 2, -(f^(a + b*x^2)/x) + Sqrt[b]*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Sqrt[Log[f]]} -{f^(a + b*x^2)/x^4, x, 3, -(f^(a + b*x^2)/(3*x^3)) - (2*b*f^(a + b*x^2)*Log[f])/(3*x) + (2/3)*b^(3/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Log[f]^(3/2)} -{f^(a + b*x^2)/x^6, x, 4, -(f^(a + b*x^2)/(5*x^5)) - (2*b*f^(a + b*x^2)*Log[f])/(15*x^3) - (4*b^2*f^(a + b*x^2)*Log[f]^2)/(15*x) + (4/15)*b^(5/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Log[f]^(5/2)} -{f^(a + b*x^2)/x^8, x, 5, -(f^(a + b*x^2)/(7*x^7)) - (2*b*f^(a + b*x^2)*Log[f])/(35*x^5) - (4*b^2*f^(a + b*x^2)*Log[f]^2)/(105*x^3) - (8*b^3*f^(a + b*x^2)*Log[f]^3)/(105*x) + (8/105)*b^(7/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Log[f]^(7/2)} -{f^(a + b*x^2)/x^10, x, 1, -((f^a*Gamma[-(9/2), (-b)*x^2*Log[f]]*((-b)*x^2*Log[f])^(9/2))/(2*x^9))} -{f^(a + b*x^2)/x^12, x, 1, -((f^a*Gamma[-(11/2), (-b)*x^2*Log[f]]*((-b)*x^2*Log[f])^(11/2))/(2*x^11))} - - -{f^(a + b*x^3)*x^m, x, 1, (-(1/3))*f^a*x^(1 + m)*Gamma[(1 + m)/3, (-b)*x^3*Log[f]]*((-b)*x^3*Log[f])^((1/3)*(-1 - m))} - -{f^(a + b*x^3)*x^17, x, 1, -((f^(a + b*x^3)*(120 - 120*b*x^3*Log[f] + 60*b^2*x^6*Log[f]^2 - 20*b^3*x^9*Log[f]^3 + 5*b^4*x^12*Log[f]^4 - b^5*x^15*Log[f]^5))/(3*b^6*Log[f]^6))} -{f^(a + b*x^3)*x^14, x, 1, (f^(a + b*x^3)*(24 - 24*b*x^3*Log[f] + 12*b^2*x^6*Log[f]^2 - 4*b^3*x^9*Log[f]^3 + b^4*x^12*Log[f]^4))/(3*b^5*Log[f]^5)} -{f^(a + b*x^3)*x^11, x, 4, -((2*f^(a + b*x^3))/(b^4*Log[f]^4)) + (2*f^(a + b*x^3)*x^3)/(b^3*Log[f]^3) - (f^(a + b*x^3)*x^6)/(b^2*Log[f]^2) + (f^(a + b*x^3)*x^9)/(3*b*Log[f])} -{f^(a + b*x^3)*x^8, x, 3, (2*f^(a + b*x^3))/(3*b^3*Log[f]^3) - (2*f^(a + b*x^3)*x^3)/(3*b^2*Log[f]^2) + (f^(a + b*x^3)*x^6)/(3*b*Log[f])} -{f^(a + b*x^3)*x^5, x, 2, -(f^(a + b*x^3)/(3*b^2*Log[f]^2)) + (f^(a + b*x^3)*x^3)/(3*b*Log[f])} -{f^(a + b*x^3)*x^2, x, 1, f^(a + b*x^3)/(3*b*Log[f])} -{f^(a + b*x^3)/x^1, x, 1, (1/3)*f^a*ExpIntegralEi[b*x^3*Log[f]]} -{f^(a + b*x^3)/x^4, x, 2, -(f^(a + b*x^3)/(3*x^3)) + (1/3)*b*f^a*ExpIntegralEi[b*x^3*Log[f]]*Log[f]} -{f^(a + b*x^3)/x^7, x, 3, -(f^(a + b*x^3)/(6*x^6)) - (b*f^(a + b*x^3)*Log[f])/(6*x^3) + (1/6)*b^2*f^a*ExpIntegralEi[b*x^3*Log[f]]*Log[f]^2} -{f^(a + b*x^3)/x^10, x, 4, -(f^(a + b*x^3)/(9*x^9)) - (b*f^(a + b*x^3)*Log[f])/(18*x^6) - (b^2*f^(a + b*x^3)*Log[f]^2)/(18*x^3) + (1/18)*b^3*f^a*ExpIntegralEi[b*x^3*Log[f]]*Log[f]^3} -{f^(a + b*x^3)/x^13, x, 1, (-(1/3))*b^4*f^a*Gamma[-4, (-b)*x^3*Log[f]]*Log[f]^4} -{f^(a + b*x^3)/x^16, x, 1, (1/3)*b^5*f^a*Gamma[-5, (-b)*x^3*Log[f]]*Log[f]^5} - -{f^(a + b*x^3)*x^4, x, 1, -((f^a*x^5*Gamma[5/3, (-b)*x^3*Log[f]])/(3*((-b)*x^3*Log[f])^(5/3)))} -{f^(a + b*x^3)*x^3, x, 1, -((f^a*x^4*Gamma[4/3, (-b)*x^3*Log[f]])/(3*((-b)*x^3*Log[f])^(4/3)))} -{f^(a + b*x^3)*x^1, x, 1, -((f^a*x^2*Gamma[2/3, (-b)*x^3*Log[f]])/(3*((-b)*x^3*Log[f])^(2/3)))} -{f^(a + b*x^3)*x^0, x, 1, -((f^a*x*Gamma[1/3, (-b)*x^3*Log[f]])/(3*((-b)*x^3*Log[f])^(1/3)))} -{f^(a + b*x^3)/x^2, x, 1, -((f^a*Gamma[-(1/3), (-b)*x^3*Log[f]]*((-b)*x^3*Log[f])^(1/3))/(3*x))} -{f^(a + b*x^3)/x^3, x, 1, -((f^a*Gamma[-(2/3), (-b)*x^3*Log[f]]*((-b)*x^3*Log[f])^(2/3))/(3*x^2))} - - -{E^(4*x^3)*x^2, x, 1, E^(4*x^3)/12} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{f^(a + b/x)*x^m, x, 1, f^a*x^(1 + m)*Gamma[-1 - m, -((b*Log[f])/x)]*(-((b*Log[f])/x))^(1 + m)} - -{f^(a + b/x)*x^4, x, 1, (-b^5)*f^a*Gamma[-5, -((b*Log[f])/x)]*Log[f]^5} -{f^(a + b/x)*x^3, x, 1, b^4*f^a*Gamma[-4, -((b*Log[f])/x)]*Log[f]^4} -{f^(a + b/x)*x^2, x, 4, (1/3)*f^(a + b/x)*x^3 + (1/6)*b*f^(a + b/x)*x^2*Log[f] + (1/6)*b^2*f^(a + b/x)*x*Log[f]^2 - (1/6)*b^3*f^a*ExpIntegralEi[(b*Log[f])/x]*Log[f]^3} -{f^(a + b/x)*x^1, x, 3, (1/2)*f^(a + b/x)*x^2 + (1/2)*b*f^(a + b/x)*x*Log[f] - (1/2)*b^2*f^a*ExpIntegralEi[(b*Log[f])/x]*Log[f]^2} -{f^(a + b/x)*x^0, x, 2, f^(a + b/x)*x - b*f^a*ExpIntegralEi[(b*Log[f])/x]*Log[f]} -{f^(a + b/x)/x^1, x, 1, (-f^a)*ExpIntegralEi[(b*Log[f])/x]} -{f^(a + b/x)/x^2, x, 1, -(f^(a + b/x)/(b*Log[f]))} -{f^(a + b/x)/x^3, x, 2, f^(a + b/x)/(b^2*Log[f]^2) - f^(a + b/x)/(b*x*Log[f])} -{f^(a + b/x)/x^4, x, 3, -((2*f^(a + b/x))/(b^3*Log[f]^3)) + (2*f^(a + b/x))/(b^2*x*Log[f]^2) - f^(a + b/x)/(b*x^2*Log[f])} -{f^(a + b/x)/x^5, x, 4, (6*f^(a + b/x))/(b^4*Log[f]^4) - (6*f^(a + b/x))/(b^3*x*Log[f]^3) + (3*f^(a + b/x))/(b^2*x^2*Log[f]^2) - f^(a + b/x)/(b*x^3*Log[f])} -{f^(a + b/x)/x^6, x, 1, -((f^(a + b/x)*(24*x^4 - 24*b*x^3*Log[f] + 12*b^2*x^2*Log[f]^2 - 4*b^3*x*Log[f]^3 + b^4*Log[f]^4))/(b^5*x^4*Log[f]^5))} -{f^(a + b/x)/x^7, x, 1, (f^(a + b/x)*(120*x^5 - 120*b*x^4*Log[f] + 60*b^2*x^3*Log[f]^2 - 20*b^3*x^2*Log[f]^3 + 5*b^4*x*Log[f]^4 - b^5*Log[f]^5))/(b^6*x^5*Log[f]^6)} - - -{f^(a + b/x^2)*x^m, x, 1, (1/2)*f^a*x^(1 + m)*Gamma[(1/2)*(-1 - m), -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^((1 + m)/2)} - -{f^(a + b/x^2)*x^9, x, 1, (-(1/2))*b^5*f^a*Gamma[-5, -((b*Log[f])/x^2)]*Log[f]^5} -{f^(a + b/x^2)*x^7, x, 1, (1/2)*b^4*f^a*Gamma[-4, -((b*Log[f])/x^2)]*Log[f]^4} -{f^(a + b/x^2)*x^5, x, 4, (1/6)*f^(a + b/x^2)*x^6 + (1/12)*b*f^(a + b/x^2)*x^4*Log[f] + (1/12)*b^2*f^(a + b/x^2)*x^2*Log[f]^2 - (1/12)*b^3*f^a*ExpIntegralEi[(b*Log[f])/x^2]*Log[f]^3} -{f^(a + b/x^2)*x^3, x, 3, (1/4)*f^(a + b/x^2)*x^4 + (1/4)*b*f^(a + b/x^2)*x^2*Log[f] - (1/4)*b^2*f^a*ExpIntegralEi[(b*Log[f])/x^2]*Log[f]^2} -{f^(a + b/x^2)*x^1, x, 2, (1/2)*f^(a + b/x^2)*x^2 - (1/2)*b*f^a*ExpIntegralEi[(b*Log[f])/x^2]*Log[f]} -{f^(a + b/x^2)/x^1, x, 1, (-(1/2))*f^a*ExpIntegralEi[(b*Log[f])/x^2]} -{f^(a + b/x^2)/x^3, x, 1, -(f^(a + b/x^2)/(2*b*Log[f]))} -{f^(a + b/x^2)/x^5, x, 2, f^(a + b/x^2)/(2*b^2*Log[f]^2) - f^(a + b/x^2)/(2*b*x^2*Log[f])} -{f^(a + b/x^2)/x^7, x, 3, -(f^(a + b/x^2)/(b^3*Log[f]^3)) + f^(a + b/x^2)/(b^2*x^2*Log[f]^2) - f^(a + b/x^2)/(2*b*x^4*Log[f])} -{f^(a + b/x^2)/x^9, x, 4, (3*f^(a + b/x^2))/(b^4*Log[f]^4) - (3*f^(a + b/x^2))/(b^3*x^2*Log[f]^3) + (3*f^(a + b/x^2))/(2*b^2*x^4*Log[f]^2) - f^(a + b/x^2)/(2*b*x^6*Log[f])} -{f^(a + b/x^2)/x^11, x, 1, -((f^(a + b/x^2)*(24*x^8 - 24*b*x^6*Log[f] + 12*b^2*x^4*Log[f]^2 - 4*b^3*x^2*Log[f]^3 + b^4*Log[f]^4))/(2*b^5*x^8*Log[f]^5))} -{f^(a + b/x^2)/x^13, x, 1, (f^(a + b/x^2)*(120*x^10 - 120*b*x^8*Log[f] + 60*b^2*x^6*Log[f]^2 - 20*b^3*x^4*Log[f]^3 + 5*b^4*x^2*Log[f]^4 - b^5*Log[f]^5))/(2*b^6*x^10*Log[f]^6)} - -{f^(a + b/x^2)*x^10, x, 1, (1/2)*f^a*x^11*Gamma[-(11/2), -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(11/2)} -{f^(a + b/x^2)*x^8, x, 1, (1/2)*f^a*x^9*Gamma[-(9/2), -((b*Log[f])/x^2)]*(-((b*Log[f])/x^2))^(9/2)} -{f^(a + b/x^2)*x^6, x, 6, (1/7)*f^(a + b/x^2)*x^7 + (2/35)*b*f^(a + b/x^2)*x^5*Log[f] + (4/105)*b^2*f^(a + b/x^2)*x^3*Log[f]^2 + (8/105)*b^3*f^(a + b/x^2)*x*Log[f]^3 - (8/105)*b^(7/2)*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(7/2)} -{f^(a + b/x^2)*x^4, x, 5, (1/5)*f^(a + b/x^2)*x^5 + (2/15)*b*f^(a + b/x^2)*x^3*Log[f] + (4/15)*b^2*f^(a + b/x^2)*x*Log[f]^2 - (4/15)*b^(5/2)*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(5/2)} -{f^(a + b/x^2)*x^2, x, 4, (1/3)*f^(a + b/x^2)*x^3 + (2/3)*b*f^(a + b/x^2)*x*Log[f] - (2/3)*b^(3/2)*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Log[f]^(3/2)} -{f^(a + b/x^2)*x^0, x, 3, f^(a + b/x^2)*x - Sqrt[b]*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x]*Sqrt[Log[f]]} -{f^(a + b/x^2)/x^2, x, 2, -((f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(2*Sqrt[b]*Sqrt[Log[f]]))} -{f^(a + b/x^2)/x^4, x, 3, (f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(4*b^(3/2)*Log[f]^(3/2)) - f^(a + b/x^2)/(2*b*x*Log[f])} -{f^(a + b/x^2)/x^6, x, 4, -((3*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(8*b^(5/2)*Log[f]^(5/2))) + (3*f^(a + b/x^2))/(4*b^2*x*Log[f]^2) - f^(a + b/x^2)/(2*b*x^3*Log[f])} -{f^(a + b/x^2)/x^8, x, 5, (15*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(16*b^(7/2)*Log[f]^(7/2)) - (15*f^(a + b/x^2))/(8*b^3*x*Log[f]^3) + (5*f^(a + b/x^2))/(4*b^2*x^3*Log[f]^2) - f^(a + b/x^2)/(2*b*x^5*Log[f])} -{f^(a + b/x^2)/x^10, x, 6, -((105*f^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[f]])/x])/(32*b^(9/2)*Log[f]^(9/2))) + (105*f^(a + b/x^2))/(16*b^4*x*Log[f]^4) - (35*f^(a + b/x^2))/(8*b^3*x^3*Log[f]^3) + (7*f^(a + b/x^2))/(4*b^2*x^5*Log[f]^2) - f^(a + b/x^2)/(2*b*x^7*Log[f])} -{f^(a + b/x^2)/x^12, x, 1, (f^a*Gamma[11/2, -((b*Log[f])/x^2)])/(2*x^11*(-((b*Log[f])/x^2))^(11/2))} -{f^(a + b/x^2)/x^14, x, 1, (f^a*Gamma[13/2, -((b*Log[f])/x^2)])/(2*x^13*(-((b*Log[f])/x^2))^(13/2))} - - -{f^(a + b/x^3)*x^m, x, 1, (1/3)*f^a*x^(1 + m)*Gamma[(1/3)*(-1 - m), -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^((1 + m)/3)} - -{f^(a + b/x^3)*x^14, x, 1, (-(1/3))*b^5*f^a*Gamma[-5, -((b*Log[f])/x^3)]*Log[f]^5} -{f^(a + b/x^3)*x^11, x, 1, (1/3)*b^4*f^a*Gamma[-4, -((b*Log[f])/x^3)]*Log[f]^4} -{f^(a + b/x^3)*x^8, x, 4, (1/9)*f^(a + b/x^3)*x^9 + (1/18)*b*f^(a + b/x^3)*x^6*Log[f] + (1/18)*b^2*f^(a + b/x^3)*x^3*Log[f]^2 - (1/18)*b^3*f^a*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]^3} -{f^(a + b/x^3)*x^5, x, 3, (1/6)*f^(a + b/x^3)*x^6 + (1/6)*b*f^(a + b/x^3)*x^3*Log[f] - (1/6)*b^2*f^a*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]^2} -{f^(a + b/x^3)*x^2, x, 2, (1/3)*f^(a + b/x^3)*x^3 - (1/3)*b*f^a*ExpIntegralEi[(b*Log[f])/x^3]*Log[f]} -{f^(a + b/x^3)/x^1, x, 1, (-(1/3))*f^a*ExpIntegralEi[(b*Log[f])/x^3]} -{f^(a + b/x^3)/x^4, x, 1, -(f^(a + b/x^3)/(3*b*Log[f]))} -{f^(a + b/x^3)/x^7, x, 2, f^(a + b/x^3)/(3*b^2*Log[f]^2) - f^(a + b/x^3)/(3*b*x^3*Log[f])} -{f^(a + b/x^3)/x^10, x, 3, -((2*f^(a + b/x^3))/(3*b^3*Log[f]^3)) + (2*f^(a + b/x^3))/(3*b^2*x^3*Log[f]^2) - f^(a + b/x^3)/(3*b*x^6*Log[f])} -{f^(a + b/x^3)/x^13, x, 4, (2*f^(a + b/x^3))/(b^4*Log[f]^4) - (2*f^(a + b/x^3))/(b^3*x^3*Log[f]^3) + f^(a + b/x^3)/(b^2*x^6*Log[f]^2) - f^(a + b/x^3)/(3*b*x^9*Log[f])} -{f^(a + b/x^3)/x^16, x, 1, -((f^(a + b/x^3)*(24*x^12 - 24*b*x^9*Log[f] + 12*b^2*x^6*Log[f]^2 - 4*b^3*x^3*Log[f]^3 + b^4*Log[f]^4))/(3*b^5*x^12*Log[f]^5))} -{f^(a + b/x^3)/x^19, x, 1, (f^(a + b/x^3)*(120*x^15 - 120*b*x^12*Log[f] + 60*b^2*x^9*Log[f]^2 - 20*b^3*x^6*Log[f]^3 + 5*b^4*x^3*Log[f]^4 - b^5*Log[f]^5))/(3*b^6*x^15*Log[f]^6)} - -{f^(a + b/x^3)*x^4, x, 1, (1/3)*f^a*x^5*Gamma[-(5/3), -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(5/3)} -{f^(a + b/x^3)*x^3, x, 1, (1/3)*f^a*x^4*Gamma[-(4/3), -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(4/3)} -{f^(a + b/x^3)*x^1, x, 1, (1/3)*f^a*x^2*Gamma[-(2/3), -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(2/3)} -{f^(a + b/x^3)*x^0, x, 1, (1/3)*f^a*x*Gamma[-(1/3), -((b*Log[f])/x^3)]*(-((b*Log[f])/x^3))^(1/3)} -{f^(a + b/x^3)/x^2, x, 1, (f^a*Gamma[1/3, -((b*Log[f])/x^3)])/(3*x*(-((b*Log[f])/x^3))^(1/3))} -{f^(a + b/x^3)/x^3, x, 1, (f^a*Gamma[2/3, -((b*Log[f])/x^3)])/(3*x^2*(-((b*Log[f])/x^3))^(2/3))} -{f^(a + b/x^3)/x^5, x, 1, (f^a*Gamma[4/3, -((b*Log[f])/x^3)])/(3*x^4*(-((b*Log[f])/x^3))^(4/3))} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{f^(a + b*x^n)*x^m, x, 1, -((f^a*x^(1 + m)*Gamma[(1 + m)/n, (-b)*x^n*Log[f]])/(((-b)*x^n*Log[f])^((1 + m)/n)*n))} - -{f^(a + b*x^n)*x^3, x, 1, -((f^a*x^4*Gamma[4/n, (-b)*x^n*Log[f]])/(((-b)*x^n*Log[f])^(4/n)*n))} -{f^(a + b*x^n)*x^2, x, 1, -((f^a*x^3*Gamma[3/n, (-b)*x^n*Log[f]])/(((-b)*x^n*Log[f])^(3/n)*n))} -{f^(a + b*x^n)*x^1, x, 1, -((f^a*x^2*Gamma[2/n, (-b)*x^n*Log[f]])/(((-b)*x^n*Log[f])^(2/n)*n))} -{f^(a + b*x^n)*x^0, x, 1, -((f^a*x*Gamma[1/n, (-b)*x^n*Log[f]])/(((-b)*x^n*Log[f])^n^(-1)*n))} -{f^(a + b*x^n)/x^1, x, 1, (f^a*ExpIntegralEi[b*x^n*Log[f]])/n} -{f^(a + b*x^n)/x^2, x, 1, -((f^a*Gamma[-(1/n), (-b)*x^n*Log[f]]*((-b)*x^n*Log[f])^(1/n))/(n*x))} -{f^(a + b*x^n)/x^3, x, 1, -((f^a*Gamma[-(2/n), (-b)*x^n*Log[f]]*((-b)*x^n*Log[f])^(2/n))/(n*x^2))} -{f^(a + b*x^n)/x^4, x, 1, -((f^a*Gamma[-(3/n), (-b)*x^n*Log[f]]*((-b)*x^n*Log[f])^(3/n))/(n*x^3))} - - -{f^(a + b*x^n)*x^(6*n/2-1), x, 3, (2*f^(a + b*x^n))/(b^3*n*Log[f]^3) - (2*f^(a + b*x^n)*x^n)/(b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^(2*n))/(b*n*Log[f])} -{f^(a + b*x^n)*x^(4*n/2-1), x, 2, -(f^(a + b*x^n)/(b^2*n*Log[f]^2)) + (f^(a + b*x^n)*x^n)/(b*n*Log[f])} -{f^(a + b*x^n)*x^(2*n/2-1), x, 1, f^(a + b*x^n)/(b*n*Log[f])} -{f^(a + b*x^n)*x^(0*n/2-1), x, 1, (f^a*ExpIntegralEi[b*x^n*Log[f]])/n} -{f^(a + b*x^n)*x^(-2*n/2-1), x, 2, -(f^(a + b*x^n)/(x^n*n)) + (b*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f])/n} -{f^(a + b*x^n)*x^(-4*n/2-1), x, 3, -(f^(a + b*x^n)/(x^(2*n)*(2*n))) - (b*f^(a + b*x^n)*Log[f])/(x^n*(2*n)) + (b^2*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f]^2)/(2*n)} - -{f^(a + b*x^n)*x^(5*n/2-1), x, 4, (3*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(4*b^(5/2)*n*Log[f]^(5/2)) - (3*f^(a + b*x^n)*x^(n/2))/(2*b^2*n*Log[f]^2) + (f^(a + b*x^n)*x^((3*n)/2))/(b*n*Log[f])} -{f^(a + b*x^n)*x^(3*n/2-1), x, 3, -((f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(2*b^(3/2)*n*Log[f]^(3/2))) + (f^(a + b*x^n)*x^(n/2))/(b*n*Log[f])} -{f^(a + b*x^n)*x^(1*n/2-1), x, 2, (f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]])/(Sqrt[b]*n*Sqrt[Log[f]])} -{f^(a + b*x^n)*x^(-1*n/2-1), x, 3, -((2*f^(a + b*x^n))/(x^(n/2)*n)) + (2*Sqrt[b]*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]]*Sqrt[Log[f]])/n} -{f^(a + b*x^n)*x^(-3*n/2-1), x, 4, -((2*f^(a + b*x^n))/(x^((3*n)/2)*(3*n))) - (4*b*f^(a + b*x^n)*Log[f])/(x^(n/2)*(3*n)) + (4*b^(3/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)*Sqrt[Log[f]]]*Log[f]^(3/2))/(3*n)} - - -{x/E^(0.1*x), x, 2, -100./E^(0.1*x) - (10.*x)/E^(0.1*x)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(c (a+b x)^n)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m F^(c (a+b x)^n)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{f^(c*(a + b*x)^2)*x^3, x, 8, -(f^(c*(a + b*x)^2)/(2*b^4*c^2*Log[f]^2)) + (3*a*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(4*b^4*c^(3/2)*Log[f]^(3/2)) + (3*a^2*f^(c*(a + b*x)^2))/(2*b^4*c*Log[f]) - (3*a*f^(c*(a + b*x)^2)*(a + b*x))/(2*b^4*c*Log[f]) + (f^(c*(a + b*x)^2)*(a + b*x)^2)/(2*b^4*c*Log[f]) - (a^3*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^4*Sqrt[c]*Sqrt[Log[f]])} -{f^(c*(a + b*x)^2)*x^2, x, 6, -((Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(4*b^3*c^(3/2)*Log[f]^(3/2))) - (a*f^(c*(a + b*x)^2))/(b^3*c*Log[f]) + (f^(c*(a + b*x)^2)*(a + b*x))/(2*b^3*c*Log[f]) + (a^2*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^3*Sqrt[c]*Sqrt[Log[f]])} -{f^(c*(a + b*x)^2)*x^1, x, 4, f^(c*(a + b*x)^2)/(2*b^2*c*Log[f]) - (a*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b^2*Sqrt[c]*Sqrt[Log[f]])} -{f^(c*(a + b*x)^2)*x^0, x, 1, (Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b*Sqrt[c]*Sqrt[Log[f]])} -{f^(c*(a + b*x)^2)/x^1, x, 0, Unintegrable[f^(c*(a + b*x)^2)/x, x]} -{f^(c*(a + b*x)^2)/x^2, x, 2, -(f^(c*(a + b*x)^2)/x) + b*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]]*Sqrt[Log[f]] + 2*a*b*c*Log[f]*Unintegrable[f^(c*(a + b*x)^2)/x, x]} -{f^(c*(a + b*x)^2)/x^3, x, 3, -(f^(c*(a + b*x)^2)/(2*x^2)) - (a*b*c*f^(c*(a + b*x)^2)*Log[f])/x + a*b^2*c^(3/2)*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]]*Log[f]^(3/2) + b^2*c*Log[f]*Unintegrable[f^(c*(a + b*x)^2)/x, x] + 2*a^2*b^2*c^2*Log[f]^2*Unintegrable[f^(c*(a + b*x)^2)/x, x]} - - -{f^(c*(a + b*x)^3)*x^2, x, 5, f^(c*(a + b*x)^3)/(3*b^3*c*Log[f]) + (2*a*(a + b*x)^2*Gamma[2/3, (-c)*(a + b*x)^3*Log[f]])/(3*b^3*((-c)*(a + b*x)^3*Log[f])^(2/3)) - (a^2*(a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]])/(3*b^3*((-c)*(a + b*x)^3*Log[f])^(1/3))} -{f^(c*(a + b*x)^3)*x^1, x, 4, -(((a + b*x)^2*Gamma[2/3, (-c)*(a + b*x)^3*Log[f]])/(3*b^2*((-c)*(a + b*x)^3*Log[f])^(2/3))) + (a*(a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]])/(3*b^2*((-c)*(a + b*x)^3*Log[f])^(1/3))} -{f^(c*(a + b*x)^3)*x^0, x, 1, -(((a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]])/(3*b*((-c)*(a + b*x)^3*Log[f])^(1/3)))} -{f^(c*(a + b*x)^3)/x^1, x, 0, Unintegrable[f^(c*(a + b*x)^3)/x, x]} -{f^(c*(a + b*x)^3)/x^2, x, 5, -(f^(c*(a + b*x)^3)/x) - (b*c*(a + b*x)^2*Gamma[2/3, (-c)*(a + b*x)^3*Log[f]]*Log[f])/((-c)*(a + b*x)^3*Log[f])^(2/3) - (a*b*c*(a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]]*Log[f])/((-c)*(a + b*x)^3*Log[f])^(1/3) + 3*a^2*b*c*Log[f]*Unintegrable[f^(c*(a + b*x)^3)/x, x]} -{f^(c*(a + b*x)^3)/x^3, x, 9, -(f^(c*(a + b*x)^3)/(2*x^2)) - (3*a^2*b*c*f^(c*(a + b*x)^3)*Log[f])/(2*x) - (3*a^2*b^2*c^2*(a + b*x)^2*Gamma[2/3, (-c)*(a + b*x)^3*Log[f]]*Log[f]^2)/(2*((-c)*(a + b*x)^3*Log[f])^(2/3)) - (b^2*c*(a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]]*Log[f])/(2*((-c)*(a + b*x)^3*Log[f])^(1/3)) - (3*a^3*b^2*c^2*(a + b*x)*Gamma[1/3, (-c)*(a + b*x)^3*Log[f]]*Log[f]^2)/(2*((-c)*(a + b*x)^3*Log[f])^(1/3)) + 3*a*b^2*c*Log[f]*Unintegrable[f^(c*(a + b*x)^3)/x, x] + (9/2)*a^4*b^2*c^2*Log[f]^2*Unintegrable[f^(c*(a + b*x)^3)/x, x]} - - -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^4, x, 8, (2*a^2*E^(a + b*x)^3)/b^5 - (a^4*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^5*(-(a + b*x)^3)^(1/3)) + (4*a^3*(a + b*x)^2*Gamma[2/3, -(a + b*x)^3])/(3*b^5*(-(a + b*x)^3)^(2/3)) + (4*a*(a + b*x)^4*Gamma[4/3, -(a + b*x)^3])/(3*b^5*(-(a + b*x)^3)^(4/3)) - ((a + b*x)^5*Gamma[5/3, -(a + b*x)^3])/(3*b^5*(-(a + b*x)^3)^(5/3))} -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^3, x, 7, -((a*E^(a + b*x)^3)/b^4) + (a^3*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^4*(-(a + b*x)^3)^(1/3)) - (a^2*(a + b*x)^2*Gamma[2/3, -(a + b*x)^3])/(b^4*(-(a + b*x)^3)^(2/3)) - ((a + b*x)^4*Gamma[4/3, -(a + b*x)^3])/(3*b^4*(-(a + b*x)^3)^(4/3))} -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^2, x, 6, E^(a + b*x)^3/(3*b^3) - (a^2*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^3*(-(a + b*x)^3)^(1/3)) + (2*a*(a + b*x)^2*Gamma[2/3, -(a + b*x)^3])/(3*b^3*(-(a + b*x)^3)^(2/3))} -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^1, x, 5, (a*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(1/3)) - ((a + b*x)^2*Gamma[2/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(2/3))} -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^0, x, 2, -(((a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b*(-(a + b*x)^3)^(1/3)))} -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)/x^1, x, 0, CannotIntegrate[E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)/x, x]} - - -{E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^m, x, 0, CannotIntegrate[E^(a^3 + 3*a^2*b*x + 3*a*b^2*x^2 + b^3*x^3)*x^m, x]} - - -{E^Sqrt[5 + 3*x], x, 3, (-(2/3))*E^Sqrt[5 + 3*x] + (2/3)*E^Sqrt[5 + 3*x]*Sqrt[5 + 3*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{f^(c/(a + b*x))*x^4, x, 13, (a^4*f^(c/(a + b*x))*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x))*(a + b*x)^2)/b^5 + (2*a^2*f^(c/(a + b*x))*(a + b*x)^3)/b^5 - (2*a^3*c*f^(c/(a + b*x))*(a + b*x)*Log[f])/b^5 + (a^2*c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/b^5 - (a^4*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^5 + (a^2*c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/b^5 + (2*a^3*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/b^5 - (a^2*c^3*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^3)/b^5 - (4*a*c^4*Gamma[-4, -((c*Log[f])/(a + b*x))]*Log[f]^4)/b^5 - (c^5*Gamma[-5, -((c*Log[f])/(a + b*x))]*Log[f]^5)/b^5} -{f^(c/(a + b*x))*x^3, x, 12, -((a^3*f^(c/(a + b*x))*(a + b*x))/b^4) + (3*a^2*f^(c/(a + b*x))*(a + b*x)^2)/(2*b^4) - (a*f^(c/(a + b*x))*(a + b*x)^3)/b^4 + (3*a^2*c*f^(c/(a + b*x))*(a + b*x)*Log[f])/(2*b^4) - (a*c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/(2*b^4) + (a^3*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^4 - (a*c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/(2*b^4) - (3*a^2*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/(2*b^4) + (a*c^3*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^3)/(2*b^4) + (c^4*Gamma[-4, -((c*Log[f])/(a + b*x))]*Log[f]^4)/b^4} -{f^(c/(a + b*x))*x^2, x, 11, (a^2*f^(c/(a + b*x))*(a + b*x))/b^3 - (a*f^(c/(a + b*x))*(a + b*x)^2)/b^3 + (f^(c/(a + b*x))*(a + b*x)^3)/(3*b^3) - (a*c*f^(c/(a + b*x))*(a + b*x)*Log[f])/b^3 + (c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/(6*b^3) - (a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^3 + (c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/(6*b^3) + (a*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/b^3 - (c^3*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^3)/(6*b^3)} -{f^(c/(a + b*x))*x^1, x, 7, -((a*f^(c/(a + b*x))*(a + b*x))/b^2) + (f^(c/(a + b*x))*(a + b*x)^2)/(2*b^2) + (c*f^(c/(a + b*x))*(a + b*x)*Log[f])/(2*b^2) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^2 - (c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^2)/(2*b^2)} -{f^(c/(a + b*x))*x^0, x, 2, (f^(c/(a + b*x))*(a + b*x))/b - (c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b} -{f^(c/(a + b*x))/x^1, x, 4, -ExpIntegralEi[(c*Log[f])/(a + b*x)] + f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a*(a + b*x)))]} -{f^(c/(a + b*x))/x^2, x, 9, -((b*f^(c/(a + b*x)))/a) - f^(c/(a + b*x))/x - (b*c*f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a*(a + b*x)))]*Log[f])/a^2} -{f^(c/(a + b*x))/x^3, x, 18, (b^2*f^(c/(a + b*x)))/(2*a^2) - f^(c/(a + b*x))/(2*x^2) + (b^2*c*f^(c/(a + b*x))*Log[f])/(2*a^3) + (b*c*f^(c/(a + b*x))*Log[f])/(2*a^2*x) + (b^2*c*f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a*(a + b*x)))]*Log[f])/a^3 + (b^2*c^2*f^(c/a)*ExpIntegralEi[-((b*c*x*Log[f])/(a*(a + b*x)))]*Log[f]^2)/(2*a^4)} - - -{f^(c/(a + b*x)^2)*x^4, x, 19, (a^4*f^(c/(a + b*x)^2)*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^5 + (2*a^2*f^(c/(a + b*x)^2)*(a + b*x)^3)/b^5 - (a*f^(c/(a + b*x)^2)*(a + b*x)^4)/b^5 + (f^(c/(a + b*x)^2)*(a + b*x)^5)/(5*b^5) - (a^4*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^5 + (4*a^2*c*f^(c/(a + b*x)^2)*(a + b*x)*Log[f])/b^5 - (a*c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/b^5 + (2*c*f^(c/(a + b*x)^2)*(a + b*x)^3*Log[f])/(15*b^5) + (2*a^3*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^5 - (4*a^2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/b^5 + (4*c^2*f^(c/(a + b*x)^2)*(a + b*x)*Log[f]^2)/(15*b^5) + (a*c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/b^5 - (4*c^(5/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(5/2))/(15*b^5)} -{f^(c/(a + b*x)^2)*x^3, x, 14, -((a^3*f^(c/(a + b*x)^2)*(a + b*x))/b^4) + (3*a^2*f^(c/(a + b*x)^2)*(a + b*x)^2)/(2*b^4) - (a*f^(c/(a + b*x)^2)*(a + b*x)^3)/b^4 + (f^(c/(a + b*x)^2)*(a + b*x)^4)/(4*b^4) + (a^3*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^4 - (2*a*c*f^(c/(a + b*x)^2)*(a + b*x)*Log[f])/b^4 + (c*f^(c/(a + b*x)^2)*(a + b*x)^2*Log[f])/(4*b^4) - (3*a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/(2*b^4) + (2*a*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/b^4 - (c^2*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f]^2)/(4*b^4)} -{f^(c/(a + b*x)^2)*x^2, x, 11, (a^2*f^(c/(a + b*x)^2)*(a + b*x))/b^3 - (a*f^(c/(a + b*x)^2)*(a + b*x)^2)/b^3 + (f^(c/(a + b*x)^2)*(a + b*x)^3)/(3*b^3) - (a^2*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^3 + (2*c*f^(c/(a + b*x)^2)*(a + b*x)*Log[f])/(3*b^3) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/b^3 - (2*c^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Log[f]^(3/2))/(3*b^3)} -{f^(c/(a + b*x)^2)*x^1, x, 7, -((a*f^(c/(a + b*x)^2)*(a + b*x))/b^2) + (f^(c/(a + b*x)^2)*(a + b*x)^2)/(2*b^2) + (a*Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b^2 - (c*ExpIntegralEi[(c*Log[f])/(a + b*x)^2]*Log[f])/(2*b^2)} -{f^(c/(a + b*x)^2)*x^0, x, 3, (f^(c/(a + b*x)^2)*(a + b*x))/b - (Sqrt[c]*Sqrt[Pi]*Erfi[(Sqrt[c]*Sqrt[Log[f]])/(a + b*x)]*Sqrt[Log[f]])/b} -{f^(c/(a + b*x)^2)/x^1, x, 0, Unintegrable[f^(c/(a + b*x)^2)/x, x]} -{f^(c/(a + b*x)^2)/x^2, x, 0, CannotIntegrate[f^(c/(a + b*x)^2)/x^2, x]} -{f^(c/(a + b*x)^2)/x^3, x, 0, CannotIntegrate[f^(c/(a + b*x)^2)/x^3, x]} - - -{f^(c/(a + b*x)^3)*x^4, x, 8, (2*a^2*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^5 - (2*a^2*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^5 + (a^4*(a + b*x)*Gamma[-(1/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^5) - (4*a^3*(a + b*x)^2*Gamma[-(2/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/(3*b^5) - (4*a*(a + b*x)^4*Gamma[-(4/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(4/3))/(3*b^5) + ((a + b*x)^5*Gamma[-(5/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(5/3))/(3*b^5)} -{f^(c/(a + b*x)^3)*x^3, x, 7, -((a*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^4) + (a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/b^4 - (a^3*(a + b*x)*Gamma[-(1/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^4) + (a^2*(a + b*x)^2*Gamma[-(2/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/b^4 + ((a + b*x)^4*Gamma[-(4/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(4/3))/(3*b^4)} -{f^(c/(a + b*x)^3)*x^2, x, 6, (f^(c/(a + b*x)^3)*(a + b*x)^3)/(3*b^3) - (c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/(3*b^3) + (a^2*(a + b*x)*Gamma[-(1/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^3) - (2*a*(a + b*x)^2*Gamma[-(2/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/(3*b^3)} -{f^(c/(a + b*x)^3)*x^1, x, 4, -((a*(a + b*x)*Gamma[-(1/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^2)) + ((a + b*x)^2*Gamma[-(2/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/(3*b^2)} -{f^(c/(a + b*x)^3)*x^0, x, 1, ((a + b*x)*Gamma[-(1/3), -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b)} -{f^(c/(a + b*x)^3)/x^1, x, 0, Unintegrable[f^(c/(a + b*x)^3)/x, x]} -{f^(c/(a + b*x)^3)/x^2, x, 0, CannotIntegrate[f^(c/(a + b*x)^3)/x^2, x]} -{f^(c/(a + b*x)^3)/x^3, x, 0, CannotIntegrate[f^(c/(a + b*x)^3)/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m F^(c (a+b x)^n) with m symbolic*) - - -{f^(c*(a + b*x)^3)*x^m, x, 0, CannotIntegrate[f^(c*(a + b*x)^3)*x^m, x]} -{f^(c*(a + b*x)^2)*x^m, x, 1, Unintegrable[f^(a^2*c + 2*a*b*c*x + b^2*c*x^2)*x^m, x]} -{f^(c*(a + b*x)^1)*x^m, x, 1, (f^(a*c)*x^m*Gamma[1 + m, (-b)*c*x*Log[f]])/(((-b)*c*x*Log[f])^m*(b*c*Log[f]))} -{f^(c/(a + b*x)^1)*x^m, x, 0, CannotIntegrate[f^(c/(a + b*x))*x^m, x]} -{f^(c/(a + b*x)^2)*x^m, x, 0, CannotIntegrate[f^(c/(a + b*x)^2)*x^m, x]} -{f^(c/(a + b*x)^3)*x^m, x, 0, CannotIntegrate[f^(c/(a + b*x)^3)*x^m, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m F^(c (a+b x)^n) with n symbolic*) - - -{f^(c*(a + b*x)^n)*x^m, x, 0, CannotIntegrate[f^(c*(a + b*x)^n)*x^m, x]} - - -{f^(c*(a + b*x)^n)*x^3, x, 6, -(((a + b*x)^4*Gamma[4/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(4/n)*(b^4*n))) + (3*a*(a + b*x)^3*Gamma[3/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(3/n)*(b^4*n)) - (3*a^2*(a + b*x)^2*Gamma[2/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(2/n)*(b^4*n)) + (a^3*(a + b*x)*Gamma[1/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^n^(-1)*(b^4*n))} -{f^(c*(a + b*x)^n)*x^2, x, 5, -(((a + b*x)^3*Gamma[3/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(3/n)*(b^3*n))) + (2*a*(a + b*x)^2*Gamma[2/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(2/n)*(b^3*n)) - (a^2*(a + b*x)*Gamma[1/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^n^(-1)*(b^3*n))} -{f^(c*(a + b*x)^n)*x^1, x, 4, -(((a + b*x)^2*Gamma[2/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^(2/n)*(b^2*n))) + (a*(a + b*x)*Gamma[1/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^n^(-1)*(b^2*n))} -{f^(c*(a + b*x)^n)*x^0, x, 1, -(((a + b*x)*Gamma[1/n, (-c)*(a + b*x)^n*Log[f]])/(((-c)*(a + b*x)^n*Log[f])^n^(-1)*(b*n)))} -{f^(c*(a + b*x)^n)/x^1, x, 0, Unintegrable[f^(c*(a + b*x)^n)/x, x]} -{f^(c*(a + b*x)^n)/x^2, x, 0, CannotIntegrate[f^(c*(a + b*x)^n)/x^2, x]} -{f^(c*(a + b*x)^n)/x^3, x, 0, CannotIntegrate[f^(c*(a + b*x)^n)/x^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m F^(a+b (c+d x)^n)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{F^(a + b*(c + d*x)^2)*(c + d*x)^m, x, 1, -((F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/2, (-b)*(c + d*x)^2*Log[F]]*((-b)*(c + d*x)^2*Log[F])^((1/2)*(-1 - m)))/(2*d))} - -{F^(a + b*(c + d*x)^2)*(c + d*x)^11, x, 1, -((F^(a + b*(c + d*x)^2)*(120 - 120*b*(c + d*x)^2*Log[F] + 60*b^2*(c + d*x)^4*Log[F]^2 - 20*b^3*(c + d*x)^6*Log[F]^3 + 5*b^4*(c + d*x)^8*Log[F]^4 - b^5*(c + d*x)^10*Log[F]^5))/(2*b^6*d*Log[F]^6))} -{F^(a + b*(c + d*x)^2)*(c + d*x)^9, x, 1, (F^(a + b*(c + d*x)^2)*(24 - 24*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c + d*x)^6*Log[F]^3 + b^4*(c + d*x)^8*Log[F]^4))/(2*b^5*d*Log[F]^5)} -{F^(a + b*(c + d*x)^2)*(c + d*x)^7, x, 4, -((3*F^(a + b*(c + d*x)^2))/(b^4*d*Log[F]^4)) + (3*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b^3*d*Log[F]^3) - (3*F^(a + b*(c + d*x)^2)*(c + d*x)^4)/(2*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^6)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^5, x, 3, F^(a + b*(c + d*x)^2)/(b^3*d*Log[F]^3) - (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^4)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^3, x, 2, -(F^(a + b*(c + d*x)^2)/(2*b^2*d*Log[F]^2)) + (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^1, x, 1, F^(a + b*(c + d*x)^2)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)/(c + d*x)^1, x, 1, (F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]])/(2*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^3, x, 2, -(F^(a + b*(c + d*x)^2)/(2*d*(c + d*x)^2)) + (b*F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]]*Log[F])/(2*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^5, x, 3, -(F^(a + b*(c + d*x)^2)/(4*d*(c + d*x)^4)) - (b*F^(a + b*(c + d*x)^2)*Log[F])/(4*d*(c + d*x)^2) + (b^2*F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]]*Log[F]^2)/(4*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^7, x, 4, -(F^(a + b*(c + d*x)^2)/(6*d*(c + d*x)^6)) - (b*F^(a + b*(c + d*x)^2)*Log[F])/(12*d*(c + d*x)^4) - (b^2*F^(a + b*(c + d*x)^2)*Log[F]^2)/(12*d*(c + d*x)^2) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^2*Log[F]]*Log[F]^3)/(12*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^9, x, 1, -((b^4*F^a*Gamma[-4, (-b)*(c + d*x)^2*Log[F]]*Log[F]^4)/(2*d))} -{F^(a + b*(c + d*x)^2)/(c + d*x)^11, x, 1, (b^5*F^a*Gamma[-5, (-b)*(c + d*x)^2*Log[F]]*Log[F]^5)/(2*d)} - -{F^(a + b*(c + d*x)^2)*(c + d*x)^12, x, 1, -((F^a*(c + d*x)^13*Gamma[13/2, (-b)*(c + d*x)^2*Log[F]])/(2*d*((-b)*(c + d*x)^2*Log[F])^(13/2)))} -{F^(a + b*(c + d*x)^2)*(c + d*x)^10, x, 1, -((F^a*(c + d*x)^11*Gamma[11/2, (-b)*(c + d*x)^2*Log[F]])/(2*d*((-b)*(c + d*x)^2*Log[F])^(11/2)))} -{F^(a + b*(c + d*x)^2)*(c + d*x)^8, x, 5, (105*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(32*b^(9/2)*d*Log[F]^(9/2)) - (105*F^(a + b*(c + d*x)^2)*(c + d*x))/(16*b^4*d*Log[F]^4) + (35*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(8*b^3*d*Log[F]^3) - (7*F^(a + b*(c + d*x)^2)*(c + d*x)^5)/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^7)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^6, x, 4, -((15*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(16*b^(7/2)*d*Log[F]^(7/2))) + (15*F^(a + b*(c + d*x)^2)*(c + d*x))/(8*b^3*d*Log[F]^3) - (5*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^5)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^4, x, 3, (3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d*Log[F]^(5/2)) - (3*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^2, x, 2, -((F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(4*b^(3/2)*d*Log[F]^(3/2))) + (F^(a + b*(c + d*x)^2)*(c + d*x))/(2*b*d*Log[F])} -{F^(a + b*(c + d*x)^2)*(c + d*x)^0, x, 1, (F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)/(c + d*x)^2, x, 2, -(F^(a + b*(c + d*x)^2)/(d*(c + d*x))) + (Sqrt[b]*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]])/d} -{F^(a + b*(c + d*x)^2)/(c + d*x)^4, x, 3, -(F^(a + b*(c + d*x)^2)/(3*d*(c + d*x)^3)) - (2*b*F^(a + b*(c + d*x)^2)*Log[F])/(3*d*(c + d*x)) + (2*b^(3/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(3/2))/(3*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^6, x, 4, -(F^(a + b*(c + d*x)^2)/(5*d*(c + d*x)^5)) - (2*b*F^(a + b*(c + d*x)^2)*Log[F])/(15*d*(c + d*x)^3) - (4*b^2*F^(a + b*(c + d*x)^2)*Log[F]^2)/(15*d*(c + d*x)) + (4*b^(5/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(5/2))/(15*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^8, x, 5, -(F^(a + b*(c + d*x)^2)/(7*d*(c + d*x)^7)) - (2*b*F^(a + b*(c + d*x)^2)*Log[F])/(35*d*(c + d*x)^5) - (4*b^2*F^(a + b*(c + d*x)^2)*Log[F]^2)/(105*d*(c + d*x)^3) - (8*b^3*F^(a + b*(c + d*x)^2)*Log[F]^3)/(105*d*(c + d*x)) + (8*b^(7/2)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(7/2))/(105*d)} -{F^(a + b*(c + d*x)^2)/(c + d*x)^10, x, 1, -((F^a*Gamma[-(9/2), (-b)*(c + d*x)^2*Log[F]]*((-b)*(c + d*x)^2*Log[F])^(9/2))/(2*d*(c + d*x)^9))} -{F^(a + b*(c + d*x)^2)/(c + d*x)^12, x, 1, -((F^a*Gamma[-(11/2), (-b)*(c + d*x)^2*Log[F]]*((-b)*(c + d*x)^2*Log[F])^(11/2))/(2*d*(c + d*x)^11))} - - -{F^(a + b*(c + d*x)^3)*(c + d*x)^m, x, 1, -((F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/3, (-b)*(c + d*x)^3*Log[F]]*((-b)*(c + d*x)^3*Log[F])^((1/3)*(-1 - m)))/(3*d))} - -{F^(a + b*(c + d*x)^3)*(c + d*x)^17, x, 1, -((F^(a + b*(c + d*x)^3)*(120 - 120*b*(c + d*x)^3*Log[F] + 60*b^2*(c + d*x)^6*Log[F]^2 - 20*b^3*(c + d*x)^9*Log[F]^3 + 5*b^4*(c + d*x)^12*Log[F]^4 - b^5*(c + d*x)^15*Log[F]^5))/(3*b^6*d*Log[F]^6))} -{F^(a + b*(c + d*x)^3)*(c + d*x)^14, x, 1, (F^(a + b*(c + d*x)^3)*(24 - 24*b*(c + d*x)^3*Log[F] + 12*b^2*(c + d*x)^6*Log[F]^2 - 4*b^3*(c + d*x)^9*Log[F]^3 + b^4*(c + d*x)^12*Log[F]^4))/(3*b^5*d*Log[F]^5)} -{F^(a + b*(c + d*x)^3)*(c + d*x)^11, x, 4, -((2*F^(a + b*(c + d*x)^3))/(b^4*d*Log[F]^4)) + (2*F^(a + b*(c + d*x)^3)*(c + d*x)^3)/(b^3*d*Log[F]^3) - (F^(a + b*(c + d*x)^3)*(c + d*x)^6)/(b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^3)*(c + d*x)^9)/(3*b*d*Log[F])} -{F^(a + b*(c + d*x)^3)*(c + d*x)^8, x, 3, (2*F^(a + b*(c + d*x)^3))/(3*b^3*d*Log[F]^3) - (2*F^(a + b*(c + d*x)^3)*(c + d*x)^3)/(3*b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^3)*(c + d*x)^6)/(3*b*d*Log[F])} -{F^(a + b*(c + d*x)^3)*(c + d*x)^5, x, 2, -(F^(a + b*(c + d*x)^3)/(3*b^2*d*Log[F]^2)) + (F^(a + b*(c + d*x)^3)*(c + d*x)^3)/(3*b*d*Log[F])} -{F^(a + b*(c + d*x)^3)*(c + d*x)^2, x, 1, F^(a + b*(c + d*x)^3)/(3*b*d*Log[F])} -{F^(a + b*(c + d*x)^3)/(c + d*x)^1, x, 1, (F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]])/(3*d)} -{F^(a + b*(c + d*x)^3)/(c + d*x)^4, x, 2, -(F^(a + b*(c + d*x)^3)/(3*d*(c + d*x)^3)) + (b*F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F])/(3*d)} -{F^(a + b*(c + d*x)^3)/(c + d*x)^7, x, 3, -(F^(a + b*(c + d*x)^3)/(6*d*(c + d*x)^6)) - (b*F^(a + b*(c + d*x)^3)*Log[F])/(6*d*(c + d*x)^3) + (b^2*F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^2)/(6*d)} -{F^(a + b*(c + d*x)^3)/(c + d*x)^10, x, 4, -(F^(a + b*(c + d*x)^3)/(9*d*(c + d*x)^9)) - (b*F^(a + b*(c + d*x)^3)*Log[F])/(18*d*(c + d*x)^6) - (b^2*F^(a + b*(c + d*x)^3)*Log[F]^2)/(18*d*(c + d*x)^3) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^3)/(18*d)} -{F^(a + b*(c + d*x)^3)/(c + d*x)^13, x, 1, -((b^4*F^a*Gamma[-4, (-b)*(c + d*x)^3*Log[F]]*Log[F]^4)/(3*d))} -{F^(a + b*(c + d*x)^3)/(c + d*x)^16, x, 1, (b^5*F^a*Gamma[-5, (-b)*(c + d*x)^3*Log[F]]*Log[F]^5)/(3*d)} - -{F^(a + b*(c + d*x)^3)*(c + d*x)^3, x, 1, -((F^a*(c + d*x)^4*Gamma[4/3, (-b)*(c + d*x)^3*Log[F]])/(3*d*((-b)*(c + d*x)^3*Log[F])^(4/3)))} -{F^(a + b*(c + d*x)^3)*(c + d*x)^1, x, 1, -((F^a*(c + d*x)^2*Gamma[2/3, (-b)*(c + d*x)^3*Log[F]])/(3*d*((-b)*(c + d*x)^3*Log[F])^(2/3)))} -{F^(a + b*(c + d*x)^3)*(c + d*x)^0, x, 1, -((F^a*(c + d*x)*Gamma[1/3, (-b)*(c + d*x)^3*Log[F]])/(3*d*((-b)*(c + d*x)^3*Log[F])^(1/3)))} -{F^(a + b*(c + d*x)^3)/(c + d*x)^2, x, 1, -((F^a*Gamma[-(1/3), (-b)*(c + d*x)^3*Log[F]]*((-b)*(c + d*x)^3*Log[F])^(1/3))/(3*d*(c + d*x)))} -{F^(a + b*(c + d*x)^3)/(c + d*x)^3, x, 1, -((F^a*Gamma[-(2/3), (-b)*(c + d*x)^3*Log[F]]*((-b)*(c + d*x)^3*Log[F])^(2/3))/(3*d*(c + d*x)^2))} -{F^(a + b*(c + d*x)^3)/(c + d*x)^5, x, 1, -((F^a*Gamma[-(4/3), (-b)*(c + d*x)^3*Log[F]]*((-b)*(c + d*x)^3*Log[F])^(4/3))/(3*d*(c + d*x)^4))} - - -{f^(a + b*(c + d*x)^(1/2)), x, 3, -((2*f^(a + b*Sqrt[c + d*x]))/(b^2*d*Log[f]^2)) + (2*f^(a + b*Sqrt[c + d*x])*Sqrt[c + d*x])/(b*d*Log[f])} -{f^(a + b*(c + d*x)^(1/3)), x, 4, (6*f^(a + b*(c + d*x)^(1/3)))/(b^3*d*Log[f]^3) - (6*f^(a + b*(c + d*x)^(1/3))*(c + d*x)^(1/3))/(b^2*d*Log[f]^2) + (3*f^(a + b*(c + d*x)^(1/3))*(c + d*x)^(2/3))/(b*d*Log[f])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{F^(a + b/(c + d*x))*(c + d*x)^m, x, 1, (F^a*(c + d*x)^(1 + m)*Gamma[-1 - m, -((b*Log[F])/(c + d*x))]*(-((b*Log[F])/(c + d*x)))^(1 + m))/d} - -{F^(a + b/(c + d*x))*(c + d*x)^4, x, 1, -((b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x))]*Log[F]^5)/d)} -{F^(a + b/(c + d*x))*(c + d*x)^3, x, 1, (b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x))]*Log[F]^4)/d} -{F^(a + b/(c + d*x))*(c + d*x)^2, x, 4, (F^(a + b/(c + d*x))*(c + d*x)^3)/(3*d) + (b*F^(a + b/(c + d*x))*(c + d*x)^2*Log[F])/(6*d) + (b^2*F^(a + b/(c + d*x))*(c + d*x)*Log[F]^2)/(6*d) - (b^3*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F]^3)/(6*d)} -{F^(a + b/(c + d*x))*(c + d*x)^1, x, 3, (F^(a + b/(c + d*x))*(c + d*x)^2)/(2*d) + (b*F^(a + b/(c + d*x))*(c + d*x)*Log[F])/(2*d) - (b^2*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F]^2)/(2*d)} -{F^(a + b/(c + d*x))*(c + d*x)^0, x, 2, (F^(a + b/(c + d*x))*(c + d*x))/d - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)]*Log[F])/d} -{F^(a + b/(c + d*x))/(c + d*x)^1, x, 1, -((F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)])/d)} -{F^(a + b/(c + d*x))/(c + d*x)^2, x, 1, -(F^(a + b/(c + d*x))/(b*d*Log[F]))} -{F^(a + b/(c + d*x))/(c + d*x)^3, x, 2, F^(a + b/(c + d*x))/(b^2*d*Log[F]^2) - F^(a + b/(c + d*x))/(b*d*(c + d*x)*Log[F])} -{F^(a + b/(c + d*x))/(c + d*x)^4, x, 3, -((2*F^(a + b/(c + d*x)))/(b^3*d*Log[F]^3)) + (2*F^(a + b/(c + d*x)))/(b^2*d*(c + d*x)*Log[F]^2) - F^(a + b/(c + d*x))/(b*d*(c + d*x)^2*Log[F])} -{F^(a + b/(c + d*x))/(c + d*x)^5, x, 4, (6*F^(a + b/(c + d*x)))/(b^4*d*Log[F]^4) - (6*F^(a + b/(c + d*x)))/(b^3*d*(c + d*x)*Log[F]^3) + (3*F^(a + b/(c + d*x)))/(b^2*d*(c + d*x)^2*Log[F]^2) - F^(a + b/(c + d*x))/(b*d*(c + d*x)^3*Log[F])} -{F^(a + b/(c + d*x))/(c + d*x)^6, x, 1, -((F^(a + b/(c + d*x))*(24*(c + d*x)^4 - 24*b*(c + d*x)^3*Log[F] + 12*b^2*(c + d*x)^2*Log[F]^2 - 4*b^3*(c + d*x)*Log[F]^3 + b^4*Log[F]^4))/(b^5*d*(c + d*x)^4*Log[F]^5))} -{F^(a + b/(c + d*x))/(c + d*x)^7, x, 1, (F^(a + b/(c + d*x))*(120*(c + d*x)^5 - 120*b*(c + d*x)^4*Log[F] + 60*b^2*(c + d*x)^3*Log[F]^2 - 20*b^3*(c + d*x)^2*Log[F]^3 + 5*b^4*(c + d*x)*Log[F]^4 - b^5*Log[F]^5))/(b^6*d*(c + d*x)^5*Log[F]^6)} - - -{F^(a + b/(c + d*x)^2)*(c + d*x)^m, x, 1, (F^a*(c + d*x)^(1 + m)*Gamma[(1/2)*(-1 - m), -((b*Log[F])/(c + d*x)^2)]*(-((b*Log[F])/(c + d*x)^2))^((1 + m)/2))/(2*d)} - -{F^(a + b/(c + d*x)^2)*(c + d*x)^9, x, 1, -((b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^2)]*Log[F]^5)/(2*d))} -{F^(a + b/(c + d*x)^2)*(c + d*x)^7, x, 1, (b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^2)]*Log[F]^4)/(2*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^5, x, 4, (F^(a + b/(c + d*x)^2)*(c + d*x)^6)/(6*d) + (b*F^(a + b/(c + d*x)^2)*(c + d*x)^4*Log[F])/(12*d) + (b^2*F^(a + b/(c + d*x)^2)*(c + d*x)^2*Log[F]^2)/(12*d) - (b^3*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2]*Log[F]^3)/(12*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^3, x, 3, (F^(a + b/(c + d*x)^2)*(c + d*x)^4)/(4*d) + (b*F^(a + b/(c + d*x)^2)*(c + d*x)^2*Log[F])/(4*d) - (b^2*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2]*Log[F]^2)/(4*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^1, x, 2, (F^(a + b/(c + d*x)^2)*(c + d*x)^2)/(2*d) - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2]*Log[F])/(2*d)} -{F^(a + b/(c + d*x)^2)/(c + d*x)^1, x, 1, -((F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^2])/(2*d))} -{F^(a + b/(c + d*x)^2)/(c + d*x)^3, x, 1, -(F^(a + b/(c + d*x)^2)/(2*b*d*Log[F]))} -{F^(a + b/(c + d*x)^2)/(c + d*x)^5, x, 2, F^(a + b/(c + d*x)^2)/(2*b^2*d*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^2*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^7, x, 3, -(F^(a + b/(c + d*x)^2)/(b^3*d*Log[F]^3)) + F^(a + b/(c + d*x)^2)/(b^2*d*(c + d*x)^2*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^4*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^9, x, 4, (3*F^(a + b/(c + d*x)^2))/(b^4*d*Log[F]^4) - (3*F^(a + b/(c + d*x)^2))/(b^3*d*(c + d*x)^2*Log[F]^3) + (3*F^(a + b/(c + d*x)^2))/(2*b^2*d*(c + d*x)^4*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^6*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^11, x, 1, -((F^(a + b/(c + d*x)^2)*(24*(c + d*x)^8 - 24*b*(c + d*x)^6*Log[F] + 12*b^2*(c + d*x)^4*Log[F]^2 - 4*b^3*(c + d*x)^2*Log[F]^3 + b^4*Log[F]^4))/(2*b^5*d*(c + d*x)^8*Log[F]^5))} -{F^(a + b/(c + d*x)^2)/(c + d*x)^13, x, 1, (F^(a + b/(c + d*x)^2)*(120*(c + d*x)^10 - 120*b*(c + d*x)^8*Log[F] + 60*b^2*(c + d*x)^6*Log[F]^2 - 20*b^3*(c + d*x)^4*Log[F]^3 + 5*b^4*(c + d*x)^2*Log[F]^4 - b^5*Log[F]^5))/(2*b^6*d*(c + d*x)^10*Log[F]^6)} - -{F^(a + b/(c + d*x)^2)*(c + d*x)^10, x, 1, (F^a*(c + d*x)^11*Gamma[-(11/2), -((b*Log[F])/(c + d*x)^2)]*(-((b*Log[F])/(c + d*x)^2))^(11/2))/(2*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^8, x, 1, (F^a*(c + d*x)^9*Gamma[-(9/2), -((b*Log[F])/(c + d*x)^2)]*(-((b*Log[F])/(c + d*x)^2))^(9/2))/(2*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^6, x, 6, (F^(a + b/(c + d*x)^2)*(c + d*x)^7)/(7*d) + (2*b*F^(a + b/(c + d*x)^2)*(c + d*x)^5*Log[F])/(35*d) + (4*b^2*F^(a + b/(c + d*x)^2)*(c + d*x)^3*Log[F]^2)/(105*d) + (8*b^3*F^(a + b/(c + d*x)^2)*(c + d*x)*Log[F]^3)/(105*d) - (8*b^(7/2)*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)]*Log[F]^(7/2))/(105*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^4, x, 5, (F^(a + b/(c + d*x)^2)*(c + d*x)^5)/(5*d) + (2*b*F^(a + b/(c + d*x)^2)*(c + d*x)^3*Log[F])/(15*d) + (4*b^2*F^(a + b/(c + d*x)^2)*(c + d*x)*Log[F]^2)/(15*d) - (4*b^(5/2)*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)]*Log[F]^(5/2))/(15*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^2, x, 4, (F^(a + b/(c + d*x)^2)*(c + d*x)^3)/(3*d) + (2*b*F^(a + b/(c + d*x)^2)*(c + d*x)*Log[F])/(3*d) - (2*b^(3/2)*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)]*Log[F]^(3/2))/(3*d)} -{F^(a + b/(c + d*x)^2)*(c + d*x)^0, x, 3, (F^(a + b/(c + d*x)^2)*(c + d*x))/d - (Sqrt[b]*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)]*Sqrt[Log[F]])/d} -{F^(a + b/(c + d*x)^2)/(c + d*x)^2, x, 2, -((F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(2*Sqrt[b]*d*Sqrt[Log[F]]))} -{F^(a + b/(c + d*x)^2)/(c + d*x)^4, x, 3, (F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(4*b^(3/2)*d*Log[F]^(3/2)) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^6, x, 4, -((3*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(8*b^(5/2)*d*Log[F]^(5/2))) + (3*F^(a + b/(c + d*x)^2))/(4*b^2*d*(c + d*x)*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^3*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^8, x, 5, (15*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(16*b^(7/2)*d*Log[F]^(7/2)) - (15*F^(a + b/(c + d*x)^2))/(8*b^3*d*(c + d*x)*Log[F]^3) + (5*F^(a + b/(c + d*x)^2))/(4*b^2*d*(c + d*x)^3*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^5*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^10, x, 6, -((105*F^a*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)])/(32*b^(9/2)*d*Log[F]^(9/2))) + (105*F^(a + b/(c + d*x)^2))/(16*b^4*d*(c + d*x)*Log[F]^4) - (35*F^(a + b/(c + d*x)^2))/(8*b^3*d*(c + d*x)^3*Log[F]^3) + (7*F^(a + b/(c + d*x)^2))/(4*b^2*d*(c + d*x)^5*Log[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^7*Log[F])} -{F^(a + b/(c + d*x)^2)/(c + d*x)^12, x, 1, (F^a*Gamma[11/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^11*(-((b*Log[F])/(c + d*x)^2))^(11/2))} -{F^(a + b/(c + d*x)^2)/(c + d*x)^14, x, 1, (F^a*Gamma[13/2, -((b*Log[F])/(c + d*x)^2)])/(2*d*(c + d*x)^13*(-((b*Log[F])/(c + d*x)^2))^(13/2))} - - -{F^(a + b/(c + d*x)^3)*(c + d*x)^m, x, 1, (F^a*(c + d*x)^(1 + m)*Gamma[(1/3)*(-1 - m), -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^((1 + m)/3))/(3*d)} - -{F^(a + b/(c + d*x)^3)*(c + d*x)^14, x, 1, -((b^5*F^a*Gamma[-5, -((b*Log[F])/(c + d*x)^3)]*Log[F]^5)/(3*d))} -{F^(a + b/(c + d*x)^3)*(c + d*x)^11, x, 1, (b^4*F^a*Gamma[-4, -((b*Log[F])/(c + d*x)^3)]*Log[F]^4)/(3*d)} -{F^(a + b/(c + d*x)^3)*(c + d*x)^8, x, 4, (F^(a + b/(c + d*x)^3)*(c + d*x)^9)/(9*d) + (b*F^(a + b/(c + d*x)^3)*(c + d*x)^6*Log[F])/(18*d) + (b^2*F^(a + b/(c + d*x)^3)*(c + d*x)^3*Log[F]^2)/(18*d) - (b^3*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F]^3)/(18*d)} -{F^(a + b/(c + d*x)^3)*(c + d*x)^5, x, 3, (F^(a + b/(c + d*x)^3)*(c + d*x)^6)/(6*d) + (b*F^(a + b/(c + d*x)^3)*(c + d*x)^3*Log[F])/(6*d) - (b^2*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F]^2)/(6*d)} -{F^(a + b/(c + d*x)^3)*(c + d*x)^2, x, 2, (F^(a + b/(c + d*x)^3)*(c + d*x)^3)/(3*d) - (b*F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^3]*Log[F])/(3*d)} -{F^(a + b/(c + d*x)^3)/(c + d*x)^1, x, 1, -((F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)^3])/(3*d))} -{F^(a + b/(c + d*x)^3)/(c + d*x)^4, x, 1, -(F^(a + b/(c + d*x)^3)/(3*b*d*Log[F]))} -{F^(a + b/(c + d*x)^3)/(c + d*x)^7, x, 2, F^(a + b/(c + d*x)^3)/(3*b^2*d*Log[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^3*Log[F])} -{F^(a + b/(c + d*x)^3)/(c + d*x)^10, x, 3, -((2*F^(a + b/(c + d*x)^3))/(3*b^3*d*Log[F]^3)) + (2*F^(a + b/(c + d*x)^3))/(3*b^2*d*(c + d*x)^3*Log[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^6*Log[F])} -{F^(a + b/(c + d*x)^3)/(c + d*x)^13, x, 4, (2*F^(a + b/(c + d*x)^3))/(b^4*d*Log[F]^4) - (2*F^(a + b/(c + d*x)^3))/(b^3*d*(c + d*x)^3*Log[F]^3) + F^(a + b/(c + d*x)^3)/(b^2*d*(c + d*x)^6*Log[F]^2) - F^(a + b/(c + d*x)^3)/(3*b*d*(c + d*x)^9*Log[F])} -{F^(a + b/(c + d*x)^3)/(c + d*x)^16, x, 1, -((F^(a + b/(c + d*x)^3)*(24*(c + d*x)^12 - 24*b*(c + d*x)^9*Log[F] + 12*b^2*(c + d*x)^6*Log[F]^2 - 4*b^3*(c + d*x)^3*Log[F]^3 + b^4*Log[F]^4))/(3*b^5*d*(c + d*x)^12*Log[F]^5))} -{F^(a + b/(c + d*x)^3)/(c + d*x)^19, x, 1, (F^(a + b/(c + d*x)^3)*(120*(c + d*x)^15 - 120*b*(c + d*x)^12*Log[F] + 60*b^2*(c + d*x)^9*Log[F]^2 - 20*b^3*(c + d*x)^6*Log[F]^3 + 5*b^4*(c + d*x)^3*Log[F]^4 - b^5*Log[F]^5))/(3*b^6*d*(c + d*x)^15*Log[F]^6)} - -{F^(a + b/(c + d*x)^3)*(c + d*x)^3, x, 1, (F^a*(c + d*x)^4*Gamma[-(4/3), -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(4/3))/(3*d)} -{F^(a + b/(c + d*x)^3)*(c + d*x)^1, x, 1, (F^a*(c + d*x)^2*Gamma[-(2/3), -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(2/3))/(3*d)} -{F^(a + b/(c + d*x)^3)*(c + d*x)^0, x, 1, (F^a*(c + d*x)*Gamma[-(1/3), -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)} -{F^(a + b/(c + d*x)^3)/(c + d*x)^2, x, 1, (F^a*Gamma[1/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)*(-((b*Log[F])/(c + d*x)^3))^(1/3))} -{F^(a + b/(c + d*x)^3)/(c + d*x)^3, x, 1, (F^a*Gamma[2/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)^2*(-((b*Log[F])/(c + d*x)^3))^(2/3))} -{F^(a + b/(c + d*x)^3)/(c + d*x)^5, x, 1, (F^a*Gamma[4/3, -((b*Log[F])/(c + d*x)^3)])/(3*d*(c + d*x)^4*(-((b*Log[F])/(c + d*x)^3))^(4/3))} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{F^(a + b*(c + d*x)^n)*(c + d*x)^m, x, 1, -((F^a*(c + d*x)^(1 + m)*Gamma[(1 + m)/n, (-b)*(c + d*x)^n*Log[F]])/(((-b)*(c + d*x)^n*Log[F])^((1 + m)/n)*(d*n)))} - -{F^(a + b*(c + d*x)^n)*(c + d*x)^3, x, 1, -((F^a*(c + d*x)^4*Gamma[4/n, (-b)*(c + d*x)^n*Log[F]])/(((-b)*(c + d*x)^n*Log[F])^(4/n)*(d*n)))} -{F^(a + b*(c + d*x)^n)*(c + d*x)^2, x, 1, -((F^a*(c + d*x)^3*Gamma[3/n, (-b)*(c + d*x)^n*Log[F]])/(((-b)*(c + d*x)^n*Log[F])^(3/n)*(d*n)))} -{F^(a + b*(c + d*x)^n)*(c + d*x)^1, x, 1, -((F^a*(c + d*x)^2*Gamma[2/n, (-b)*(c + d*x)^n*Log[F]])/(((-b)*(c + d*x)^n*Log[F])^(2/n)*(d*n)))} -{F^(a + b*(c + d*x)^n)*(c + d*x)^0, x, 1, -((F^a*(c + d*x)*Gamma[1/n, (-b)*(c + d*x)^n*Log[F]])/(((-b)*(c + d*x)^n*Log[F])^n^(-1)*(d*n)))} -{F^(a + b*(c + d*x)^n)/(c + d*x)^1, x, 1, (F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)} -{F^(a + b*(c + d*x)^n)/(c + d*x)^2, x, 1, -((F^a*Gamma[-(1/n), (-b)*(c + d*x)^n*Log[F]]*((-b)*(c + d*x)^n*Log[F])^(1/n))/(d*n*(c + d*x)))} -{F^(a + b*(c + d*x)^n)/(c + d*x)^3, x, 1, -((F^a*Gamma[-(2/n), (-b)*(c + d*x)^n*Log[F]]*((-b)*(c + d*x)^n*Log[F])^(2/n))/(d*n*(c + d*x)^2))} -{F^(a + b*(c + d*x)^n)/(c + d*x)^4, x, 1, -((F^a*Gamma[-(3/n), (-b)*(c + d*x)^n*Log[F]]*((-b)*(c + d*x)^n*Log[F])^(3/n))/(d*n*(c + d*x)^3))} - - -{F^(a + b*(c + d*x)^n)*(c + d*x)^(6*n - 1), x, 1, -((F^(a + b*(c + d*x)^n)*(120 - 120*b*(c + d*x)^n*Log[F] + 60*b^2*(c + d*x)^(2*n)*Log[F]^2 - 20*b^3*(c + d*x)^(3*n)*Log[F]^3 + 5*b^4*(c + d*x)^(4*n)*Log[F]^4 - b^5*(c + d*x)^(5*n)*Log[F]^5))/(b^6*d*n*Log[F]^6))} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(5*n - 1), x, 1, (F^(a + b*(c + d*x)^n)*(24 - 24*b*(c + d*x)^n*Log[F] + 12*b^2*(c + d*x)^(2*n)*Log[F]^2 - 4*b^3*(c + d*x)^(3*n)*Log[F]^3 + b^4*(c + d*x)^(4*n)*Log[F]^4))/(b^5*d*n*Log[F]^5)} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(4*n - 1), x, 4, -((6*F^(a + b*(c + d*x)^n))/(b^4*d*n*Log[F]^4)) + (6*F^(a + b*(c + d*x)^n)*(c + d*x)^n)/(b^3*d*n*Log[F]^3) - (3*F^(a + b*(c + d*x)^n)*(c + d*x)^(2*n))/(b^2*d*n*Log[F]^2) + (F^(a + b*(c + d*x)^n)*(c + d*x)^(3*n))/(b*d*n*Log[F])} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(3*n - 1), x, 3, (2*F^(a + b*(c + d*x)^n))/(b^3*d*n*Log[F]^3) - (2*F^(a + b*(c + d*x)^n)*(c + d*x)^n)/(b^2*d*n*Log[F]^2) + (F^(a + b*(c + d*x)^n)*(c + d*x)^(2*n))/(b*d*n*Log[F])} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(2*n - 1), x, 2, -(F^(a + b*(c + d*x)^n)/(b^2*d*n*Log[F]^2)) + (F^(a + b*(c + d*x)^n)*(c + d*x)^n)/(b*d*n*Log[F])} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(1*n - 1), x, 1, F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(0*n - 1), x, 1, (F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]])/(d*n)} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(-1*n - 1), x, 2, -(F^(a + b*(c + d*x)^n)/((c + d*x)^n*(d*n))) + (b*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F])/(d*n)} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(-2*n - 1), x, 3, -(F^(a + b*(c + d*x)^n)/((c + d*x)^(2*n)*(2*d*n))) - (b*F^(a + b*(c + d*x)^n)*Log[F])/((c + d*x)^n*(2*d*n)) + (b^2*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^2)/(2*d*n)} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(-3*n - 1), x, 4, -(F^(a + b*(c + d*x)^n)/((c + d*x)^(3*n)*(3*d*n))) - (b*F^(a + b*(c + d*x)^n)*Log[F])/((c + d*x)^(2*n)*(6*d*n)) - (b^2*F^(a + b*(c + d*x)^n)*Log[F]^2)/((c + d*x)^n*(6*d*n)) + (b^3*F^a*ExpIntegralEi[b*(c + d*x)^n*Log[F]]*Log[F]^3)/(6*d*n)} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(-4*n - 1), x, 1, -((b^4*F^a*Gamma[-4, (-b)*(c + d*x)^n*Log[F]]*Log[F]^4)/(d*n))} -{F^(a + b*(c + d*x)^n)*(c + d*x)^(-5*n - 1), x, 1, (b^5*F^a*Gamma[-5, (-b)*(c + d*x)^n*Log[F]]*Log[F]^5)/(d*n)} - - -{(a + b*x)^(n/2 - 1)*F^(c*(a + b*x)^n), x, 2, (Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])} -{(a + b*x)^(n/2 - 1)*F^(-c*(a + b*x)^n), x, 2, (Sqrt[Pi]*Erf[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m F^(a+b (c+d x)^n)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{F^(a + b*(c + d*x)^2)*(e + f*x)^5, x, 14, (f^5*F^(a + b*(c + d*x)^2))/(b^3*d^6*Log[F]^3) + (15*f^4*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d^6*Log[F]^(5/2)) - (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(b^2*d^6*Log[F]^2) - (15*f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^6*Log[F]^2) - (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b^2*d^6*Log[F]^2) - (5*f^2*(d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^6*Log[F]^(3/2)) + (5*f*(d*e - c*f)^4*F^(a + b*(c + d*x)^2))/(2*b*d^6*Log[F]) + (5*f^2*(d*e - c*f)^3*F^(a + b*(c + d*x)^2)*(c + d*x))/(b*d^6*Log[F]) + (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b*d^6*Log[F]) + (5*f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d^6*Log[F]) + (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)^4)/(2*b*d^6*Log[F]) + ((d*e - c*f)^5*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^6*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)*(e + f*x)^4, x, 11, (3*f^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(8*b^(5/2)*d^5*Log[F]^(5/2)) - (2*f^3*(d*e - c*f)*F^(a + b*(c + d*x)^2))/(b^2*d^5*Log[F]^2) - (3*f^4*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^5*Log[F]^2) - (3*f^2*(d*e - c*f)^2*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^5*Log[F]^(3/2)) + (2*f*(d*e - c*f)^3*F^(a + b*(c + d*x)^2))/(b*d^5*Log[F]) + (3*f^2*(d*e - c*f)^2*F^(a + b*(c + d*x)^2)*(c + d*x))/(b*d^5*Log[F]) + (2*f^3*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b*d^5*Log[F]) + (f^4*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d^5*Log[F]) + ((d*e - c*f)^4*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^5*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)*(e + f*x)^3, x, 8, -((f^3*F^(a + b*(c + d*x)^2))/(2*b^2*d^4*Log[F]^2)) - (3*f^2*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(4*b^(3/2)*d^4*Log[F]^(3/2)) + (3*f*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(2*b*d^4*Log[F]) + (3*f^2*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(2*b*d^4*Log[F]) + (f^3*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*d^4*Log[F]) + ((d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^4*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)*(e + f*x)^2, x, 6, -((f^2*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(4*b^(3/2)*d^3*Log[F]^(3/2))) + (f*(d*e - c*f)*F^(a + b*(c + d*x)^2))/(b*d^3*Log[F]) + (f^2*F^(a + b*(c + d*x)^2)*(c + d*x))/(2*b*d^3*Log[F]) + ((d*e - c*f)^2*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^3*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)*(e + f*x)^1, x, 4, (f*F^(a + b*(c + d*x)^2))/(2*b*d^2*Log[F]) + ((d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^2*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)*(e + f*x)^0, x, 1, (F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d*Sqrt[Log[F]])} -{F^(a + b*(c + d*x)^2)/(e + f*x)^1, x, 0, Unintegrable[F^(a + b*(c + d*x)^2)/(e + f*x), x]} -{F^(a + b*(c + d*x)^2)/(e + f*x)^2, x, 2, -(F^(a + b*(c + d*x)^2)/(f*(e + f*x))) + (Sqrt[b]*d*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]])/f^2 - (2*b*d*(d*e - c*f)*Log[F]*Unintegrable[F^(a + b*(c + d*x)^2)/(e + f*x), x])/f^2} -{F^(a + b*(c + d*x)^2)/(e + f*x)^3, x, 3, -(F^(a + b*(c + d*x)^2)/(2*f*(e + f*x)^2)) + (b*d*(d*e - c*f)*F^(a + b*(c + d*x)^2)*Log[F])/(f^3*(e + f*x)) - (b^(3/2)*d^2*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(3/2))/f^4 + (b*d^2*Log[F]*Unintegrable[F^(a + b*(c + d*x)^2)/(e + f*x), x])/f^2 + (2*b^2*d^2*(d*e - c*f)^2*Log[F]^2*Unintegrable[F^(a + b*(c + d*x)^2)/(e + f*x), x])/f^4} - - -{E^(e*(c + d*x)^3)*(a + b*x)^3, x, 6, -((b^2*(b*c - a*d)*E^(e*(c + d*x)^3))/(d^4*e)) + ((b*c - a*d)^3*(c + d*x)*Gamma[1/3, (-e)*(c + d*x)^3])/(3*d^4*((-e)*(c + d*x)^3)^(1/3)) - (b*(b*c - a*d)^2*(c + d*x)^2*Gamma[2/3, (-e)*(c + d*x)^3])/(d^4*((-e)*(c + d*x)^3)^(2/3)) - (b^3*(c + d*x)^4*Gamma[4/3, (-e)*(c + d*x)^3])/(3*d^4*((-e)*(c + d*x)^3)^(4/3))} -{E^(e*(c + d*x)^3)*(a + b*x)^2, x, 5, (b^2*E^(e*(c + d*x)^3))/(3*d^3*e) - ((b*c - a*d)^2*(c + d*x)*Gamma[1/3, (-e)*(c + d*x)^3])/(3*d^3*((-e)*(c + d*x)^3)^(1/3)) + (2*b*(b*c - a*d)*(c + d*x)^2*Gamma[2/3, (-e)*(c + d*x)^3])/(3*d^3*((-e)*(c + d*x)^3)^(2/3))} -{E^(e*(c + d*x)^3)*(a + b*x)^1, x, 4, ((b*c - a*d)*(c + d*x)*Gamma[1/3, -(e*(c + d*x)^3)])/(3*d^2*(-(e*(c + d*x)^3))^(1/3)) - (b*(c + d*x)^2*Gamma[2/3, -(e*(c + d*x)^3)])/(3*d^2*(-(e*(c + d*x)^3))^(2/3))} -{E^(e*(c + d*x)^3)*(a + b*x)^0, x, 1, -(((c + d*x)*Gamma[1/3, (-e)*(c + d*x)^3])/(3*d*((-e)*(c + d*x)^3)^(1/3)))} -{E^(e*(c + d*x)^3)/(a + b*x)^1, x, 0, Unintegrable[E^(e*(c + d*x)^3)/(a + b*x), x]} -{E^(e*(c + d*x)^3)/(a + b*x)^2, x, 5, -(E^(e*(c + d*x)^3)/(b*(a + b*x))) - (d*(b*c - a*d)*e*(c + d*x)*Gamma[1/3, (-e)*(c + d*x)^3])/(b^3*((-e)*(c + d*x)^3)^(1/3)) - (d*e*(c + d*x)^2*Gamma[2/3, (-e)*(c + d*x)^3])/(b^2*((-e)*(c + d*x)^3)^(2/3)) + (3*d*(b*c - a*d)^2*e*Unintegrable[E^(e*(c + d*x)^3)/(a + b*x), x])/b^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{F^(a + b/(c + d*x))/(e + f*x)^1, x, 4, -((F^a*ExpIntegralEi[(b*Log[F])/(c + d*x)])/f) + (F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(d*b*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))])/f} -{F^(a + b/(c + d*x))/(e + f*x)^2, x, 9, (d*F^(a + b/(c + d*x)))/(f*(d*e - c*f)) - F^(a + b/(c + d*x))/(f*(e + f*x)) - (b*d*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^2} -{F^(a + b/(c + d*x))/(e + f*x)^3, x, 18, (d^2*F^(a + b/(c + d*x)))/(2*f*(d*e - c*f)^2) - F^(a + b/(c + d*x))/(2*f*(e + f*x)^2) - (b*d^2*F^(a + b/(c + d*x))*Log[F])/(2*(d*e - c*f)^3) + (b*d*F^(a + b/(c + d*x))*Log[F])/(2*(d*e - c*f)^2*(e + f*x)) - (b*d^2*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^3 + (b^2*d^2*f*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F]^2)/(2*(d*e - c*f)^4)} -{F^(a + b/(c + d*x))/(e + f*x)^4, x, 36, (d^3*F^(a + b/(c + d*x)))/(3*f*(d*e - c*f)^3) - F^(a + b/(c + d*x))/(3*f*(e + f*x)^3) - (5*b*d^3*F^(a + b/(c + d*x))*Log[F])/(6*(d*e - c*f)^4) + (b*d*F^(a + b/(c + d*x))*Log[F])/(6*(d*e - c*f)^2*(e + f*x)^2) + (2*b*d^2*F^(a + b/(c + d*x))*Log[F])/(3*(d*e - c*f)^3*(e + f*x)) - (b*d^3*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^4 + (b^2*d^3*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*(d*e - c*f)^5) - (b^2*d^2*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*(d*e - c*f)^4*(e + f*x)) + (b^2*d^3*f*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F]^2)/(d*e - c*f)^5 - (b^3*d^3*f^2*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F]^3)/(6*(d*e - c*f)^6)} - - -{E^(e/(c + d*x))*(a + b*x)^4, x, 13, ((b*c - a*d)^4*E^(e/(c + d*x))*(c + d*x))/d^5 - (2*b*(b*c - a*d)^3*e*E^(e/(c + d*x))*(c + d*x))/d^5 + (b^2*(b*c - a*d)^2*e^2*E^(e/(c + d*x))*(c + d*x))/d^5 - (2*b*(b*c - a*d)^3*E^(e/(c + d*x))*(c + d*x)^2)/d^5 + (b^2*(b*c - a*d)^2*e*E^(e/(c + d*x))*(c + d*x)^2)/d^5 + (2*b^2*(b*c - a*d)^2*E^(e/(c + d*x))*(c + d*x)^3)/d^5 - ((b*c - a*d)^4*e*ExpIntegralEi[e/(c + d*x)])/d^5 + (2*b*(b*c - a*d)^3*e^2*ExpIntegralEi[e/(c + d*x)])/d^5 - (b^2*(b*c - a*d)^2*e^3*ExpIntegralEi[e/(c + d*x)])/d^5 - (b^4*e^5*Gamma[-5, -(e/(c + d*x))])/d^5 - (4*b^3*(b*c - a*d)*e^4*Gamma[-4, -(e/(c + d*x))])/d^5} -{E^(e/(c + d*x))*(a + b*x)^3, x, 12, -(((b*c - a*d)^3*E^(e/(c + d*x))*(c + d*x))/d^4) + (3*b*(b*c - a*d)^2*e*E^(e/(c + d*x))*(c + d*x))/(2*d^4) - (b^2*(b*c - a*d)*e^2*E^(e/(c + d*x))*(c + d*x))/(2*d^4) + (3*b*(b*c - a*d)^2*E^(e/(c + d*x))*(c + d*x)^2)/(2*d^4) - (b^2*(b*c - a*d)*e*E^(e/(c + d*x))*(c + d*x)^2)/(2*d^4) - (b^2*(b*c - a*d)*E^(e/(c + d*x))*(c + d*x)^3)/d^4 + ((b*c - a*d)^3*e*ExpIntegralEi[e/(c + d*x)])/d^4 - (3*b*(b*c - a*d)^2*e^2*ExpIntegralEi[e/(c + d*x)])/(2*d^4) + (b^2*(b*c - a*d)*e^3*ExpIntegralEi[e/(c + d*x)])/(2*d^4) + (b^3*e^4*Gamma[-4, -(e/(c + d*x))])/d^4} -{E^(e/(c + d*x))*(a + b*x)^2, x, 11, ((b*c - a*d)^2*E^(e/(c + d*x))*(c + d*x))/d^3 - (b*(b*c - a*d)*e*E^(e/(c + d*x))*(c + d*x))/d^3 + (b^2*e^2*E^(e/(c + d*x))*(c + d*x))/(6*d^3) - (b*(b*c - a*d)*E^(e/(c + d*x))*(c + d*x)^2)/d^3 + (b^2*e*E^(e/(c + d*x))*(c + d*x)^2)/(6*d^3) + (b^2*E^(e/(c + d*x))*(c + d*x)^3)/(3*d^3) - ((b*c - a*d)^2*e*ExpIntegralEi[e/(c + d*x)])/d^3 + (b*(b*c - a*d)*e^2*ExpIntegralEi[e/(c + d*x)])/d^3 - (b^2*e^3*ExpIntegralEi[e/(c + d*x)])/(6*d^3)} -{E^(e/(c + d*x))*(a + b*x)^1, x, 7, -(((b*c - a*d)*E^(e/(c + d*x))*(c + d*x))/d^2) + (b*e*E^(e/(c + d*x))*(c + d*x))/(2*d^2) + (b*E^(e/(c + d*x))*(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)])/d^2 - (b*e^2*ExpIntegralEi[e/(c + d*x)])/(2*d^2)} -{E^(e/(c + d*x))*(a + b*x)^0, x, 2, (E^(e/(c + d*x))*(c + d*x))/d - (e*ExpIntegralEi[e/(c + d*x)])/d} -{E^(e/(c + d*x))/(a + b*x)^1, x, 4, -(ExpIntegralEi[e/(c + d*x)]/b) + (E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/b} -{E^(e/(c + d*x))/(a + b*x)^2, x, 9, -((d*E^(e/(c + d*x)))/(b*(b*c - a*d))) - E^(e/(c + d*x))/(b*(a + b*x)) - (d*e*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(b*c - a*d)^2} -{E^(e/(c + d*x))/(a + b*x)^3, x, 18, (d^2*E^(e/(c + d*x)))/(2*b*(b*c - a*d)^2) + (d^2*e*E^(e/(c + d*x)))/(2*(b*c - a*d)^3) - E^(e/(c + d*x))/(2*b*(a + b*x)^2) + (d*e*E^(e/(c + d*x)))/(2*(b*c - a*d)^2*(a + b*x)) + (d^2*e*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(b*c - a*d)^3 + (b*d^2*e^2*E^((b*e)/(b*c - a*d))*ExpIntegralEi[-((d*e*(a + b*x))/((b*c - a*d)*(c + d*x)))])/(2*(b*c - a*d)^4)} - - -{E^(e/(c + d*x)^2)*(a + b*x)^3, x, 14, -(((b*c - a*d)^3*E^(e/(c + d*x)^2)*(c + d*x))/d^4) - (2*b^2*(b*c - a*d)*e*E^(e/(c + d*x)^2)*(c + d*x))/d^4 + (3*b*(b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x)^2)/(2*d^4) + (b^3*e*E^(e/(c + d*x)^2)*(c + d*x)^2)/(4*d^4) - (b^2*(b*c - a*d)*E^(e/(c + d*x)^2)*(c + d*x)^3)/d^4 + (b^3*E^(e/(c + d*x)^2)*(c + d*x)^4)/(4*d^4) + ((b*c - a*d)^3*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 + (2*b^2*(b*c - a*d)*e^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 - (3*b*(b*c - a*d)^2*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^4) - (b^3*e^2*ExpIntegralEi[e/(c + d*x)^2])/(4*d^4)} -{E^(e/(c + d*x)^2)*(a + b*x)^2, x, 11, ((b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x))/d^3 + (2*b^2*e*E^(e/(c + d*x)^2)*(c + d*x))/(3*d^3) - (b*(b*c - a*d)*E^(e/(c + d*x)^2)*(c + d*x)^2)/d^3 + (b^2*E^(e/(c + d*x)^2)*(c + d*x)^3)/(3*d^3) - ((b*c - a*d)^2*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^3 - (2*b^2*e^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/(3*d^3) + (b*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^2])/d^3} -{E^(e/(c + d*x)^2)*(a + b*x)^1, x, 7, -(((b*c - a*d)*E^(e/(c + d*x)^2)*(c + d*x))/d^2) + (b*E^(e/(c + d*x)^2)*(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^2 - (b*e*ExpIntegralEi[e/(c + d*x)^2])/(2*d^2)} -{E^(e/(c + d*x)^2)*(a + b*x)^0, x, 3, (E^(e/(c + d*x)^2)*(c + d*x))/d - (Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d} -{E^(e/(c + d*x)^2)/(a + b*x)^1, x, 0, Unintegrable[E^(e/(c + d*x)^2)/(a + b*x), x]} -{E^(e/(c + d*x)^2)/(a + b*x)^2, x, 0, CannotIntegrate[E^(e/(c + d*x)^2)/(a + b*x)^2, x]} -{E^(e/(c + d*x)^2)/(a + b*x)^3, x, 0, CannotIntegrate[E^(e/(c + d*x)^2)/(a + b*x)^3, x]} - - -{E^(e/(c + d*x)^3)*(a + b*x)^3, x, 7, -((b^2*(b*c - a*d)*E^(e/(c + d*x)^3)*(c + d*x)^3)/d^4) + (b^2*(b*c - a*d)*e*ExpIntegralEi[e/(c + d*x)^3])/d^4 + (b^3*(-(e/(c + d*x)^3))^(4/3)*(c + d*x)^4*Gamma[-(4/3), -(e/(c + d*x)^3)])/(3*d^4) + (b*(b*c - a*d)^2*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-(2/3), -(e/(c + d*x)^3)])/d^4 - ((b*c - a*d)^3*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -(e/(c + d*x)^3)])/(3*d^4)} -{E^(e/(c + d*x)^3)*(a + b*x)^2, x, 6, (b^2*E^(e/(c + d*x)^3)*(c + d*x)^3)/(3*d^3) - (b^2*e*ExpIntegralEi[e/(c + d*x)^3])/(3*d^3) - (2*b*(b*c - a*d)*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-(2/3), -(e/(c + d*x)^3)])/(3*d^3) + ((b*c - a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -(e/(c + d*x)^3)])/(3*d^3)} -{E^(e/(c + d*x)^3)*(a + b*x)^1, x, 4, (b*(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-(2/3), -(e/(c + d*x)^3)])/(3*d^2) - ((b*c - a*d)*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -(e/(c + d*x)^3)])/(3*d^2)} -{E^(e/(c + d*x)^3)*(a + b*x)^0, x, 1, ((-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -(e/(c + d*x)^3)])/(3*d)} -{E^(e/(c + d*x)^3)/(a + b*x)^1, x, 0, Unintegrable[E^(e/(c + d*x)^3)/(a + b*x), x]} -{E^(e/(c + d*x)^3)/(a + b*x)^2, x, 0, CannotIntegrate[E^(e/(c + d*x)^3)/(a + b*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x)^m F^(e+f (a+b x)/(c+d x))*) - - -{F^(e + f*(a + b*x)/(c + d*x))/(g + h*x)^1, x, 5, -((F^(e + (b*f)/d)*ExpIntegralEi[-(((b*c - a*d)*f*Log[F])/(d*(c + d*x)))])/h) + (F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))])/h} -{F^(e + f*(a + b*x)/(c + d*x))/(g + h*x)^2, x, 12, (d*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(h*(d*g - c*h)) - F^(e + (f*(a + b*x))/(c + d*x))/(h*(g + h*x)) + ((b*c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F])/(d*g - c*h)^2} -{F^(e + f*(a + b*x)/(c + d*x))/(g + h*x)^3, x, 24, (d^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(2*h*(d*g - c*h)^2) - F^(e + (f*(a + b*x))/(c + d*x))/(2*h*(g + h*x)^2) + (d*(b*c - a*d)*f*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(2*(d*g - c*h)^3) - ((b*c - a*d)*f*F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(2*(d*g - c*h)^2*(g + h*x)) + (d*(b*c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F])/(d*g - c*h)^3 + ((b*c - a*d)^2*f^2*F^(e + (f*(b*g - a*h))/(d*g - c*h))*h*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F]^2)/(2*(d*g - c*h)^4)} -{F^(e + f*(a + b*x)/(c + d*x))/(g + h*x)^4, x, 48, (d^3*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(3*h*(d*g - c*h)^3) - F^(e + (f*(a + b*x))/(c + d*x))/(3*h*(g + h*x)^3) + (5*d^2*(b*c - a*d)*f*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*Log[F])/(6*(d*g - c*h)^4) - ((b*c - a*d)*f*F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(6*(d*g - c*h)^2*(g + h*x)^2) - (2*d*(b*c - a*d)*f*F^(e + (f*(a + b*x))/(c + d*x))*Log[F])/(3*(d*g - c*h)^3*(g + h*x)) + (d^2*(b*c - a*d)*f*F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F])/(d*g - c*h)^4 + (d*(b*c - a*d)^2*f^2*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x)))*h*Log[F]^2)/(6*(d*g - c*h)^5) - ((b*c - a*d)^2*f^2*F^(e + (f*(a + b*x))/(c + d*x))*h*Log[F]^2)/(6*(d*g - c*h)^4*(g + h*x)) + (d*(b*c - a*d)^2*f^2*F^(e + (f*(b*g - a*h))/(d*g - c*h))*h*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F]^2)/(d*g - c*h)^5 + ((b*c - a*d)^3*f^3*F^(e + (f*(b*g - a*h))/(d*g - c*h))*h^2*ExpIntegralEi[-(((b*c - a*d)*f*(g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))]*Log[F]^3)/(6*(d*g - c*h)^6)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(a+b x+c x^2)*) - - -{x^3*f^(a + b*x + c*x^2), x, 10, -(f^(a + b*x + c*x^2)/(2*c^2*Log[f]^2)) + (3*b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (b^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]) - (b*f^(a + b*x + c*x^2)*x)/(4*c^2*Log[f]) + (f^(a + b*x + c*x^2)*x^2)/(2*c*Log[f]) - (b^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2)*Sqrt[Log[f]])} -{x^2*f^(a + b*x + c*x^2), x, 6, -((f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Log[f]^(3/2))) - (b*f^(a + b*x + c*x^2))/(4*c^2*Log[f]) + (f^(a + b*x + c*x^2)*x)/(2*c*Log[f]) + (b^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Sqrt[Log[f]])} -{x*f^(a + b*x + c*x^2), x, 3, f^(a + b*x + c*x^2)/(2*c*Log[f]) - (b*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])} -{f^(a + b*x + c*x^2), x, 2, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(2*Sqrt[c]*Sqrt[Log[f]])} -{f^(a + b*x + c*x^2)/x, x, 0, Unintegrable[f^(a + b*x + c*x^2)/x, x]} -{f^(a + b*x + c*x^2)/x^2, x, 3, -(f^(a + b*x + c*x^2)/x) + Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]] + b*Unintegrable[f^(a + b*x + c*x^2)/x, x]*Log[f]} - - -{x^3*E^(a + b*x - c*x^2), x, 10, -((b^2*E^(a + b*x - c*x^2))/(8*c^3)) - E^(a + b*x - c*x^2)/(2*c^2) - (b*E^(a + b*x - c*x^2)*x)/(4*c^2) - (E^(a + b*x - c*x^2)*x^2)/(2*c) - (b^3*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(16*c^(7/2)) - (3*b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(5/2))} -{x^2*E^(a + b*x - c*x^2), x, 6, -((b*E^(a + b*x - c*x^2))/(4*c^2)) - (E^(a + b*x - c*x^2)*x)/(2*c) - (b^2*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(5/2)) - (E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))} -{x*E^(a + b*x - c*x^2), x, 3, -(E^(a + b*x - c*x^2)/(2*c)) - (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*c^(3/2))} -{E^(a + b*x - c*x^2), x, 2, -((E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(2*Sqrt[c]))} -{E^(a + b*x - c*x^2)/x, x, 0, Unintegrable[E^(a + b*x - c*x^2)/x, x]} -{E^(a + b*x - c*x^2)/x^2, x, 3, -(E^(a + b*x - c*x^2)/x) + Sqrt[c]*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])] + b*Unintegrable[E^(a + b*x - c*x^2)/x, x]} - - -{x^3*E^((a + b*x)*(c + d*x)), x, 11, -(E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b^2*d^2)) + ((b*c + a*d)^2*E^(a*c + (b*c + a*d)*x + b*d*x^2))/(8*b^3*d^3) - ((b*c + a*d)*E^(a*c + (b*c + a*d)*x + b*d*x^2)*x)/(4*b^2*d^2) + (E^(a*c + (b*c + a*d)*x + b*d*x^2)*x^2)/(2*b*d) + (3*(b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(8*b^(5/2)*d^(5/2))) - ((b*c + a*d)^3*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(16*b^(7/2)*d^(7/2)))} -{x^2*E^((a + b*x)*(c + d*x)), x, 7, -(((b*c + a*d)*E^(a*c + (b*c + a*d)*x + b*d*x^2))/(4*b^2*d^2)) + (E^(a*c + (b*c + a*d)*x + b*d*x^2)*x)/(2*b*d) - (Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(4*b^(3/2)*d^(3/2))) + ((b*c + a*d)^2*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(8*b^(5/2)*d^(5/2)))} -{x*E^((a + b*x)*(c + d*x)), x, 4, E^(a*c + (b*c + a*d)*x + b*d*x^2)/(2*b*d) - ((b*c + a*d)*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(4*b^(3/2)*d^(3/2)))} -{E^((a + b*x)*(c + d*x)), x, 3, (Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/(E^((b*c - a*d)^2/(4*b*d))*(2*Sqrt[b]*Sqrt[d]))} -{E^((a + b*x)*(c + d*x))/x, x, 1, Unintegrable[E^(a*c + (b*c + a*d)*x + b*d*x^2)/x, x]} -{E^((a + b*x)*(c + d*x))/x^2, x, 4, -(E^(a*c + (b*c + a*d)*x + b*d*x^2)/x) + (Sqrt[b]*Sqrt[d]*Sqrt[Pi]*Erfi[(b*c + a*d + 2*b*d*x)/(2*Sqrt[b]*Sqrt[d])])/E^((b*c - a*d)^2/(4*b*d)) + (b*c + a*d)*Unintegrable[E^(a*c + (b*c + a*d)*x + b*d*x^2)/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m F^(a+b x+c x^2)*) - - -{(d + e*x)^3*f^(a + b*x + c*x^2), x, 10, -((e^3*f^(a + b*x + c*x^2))/(2*c^2*Log[f]^2)) - (3*e^2*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (e*(2*c*d - b*e)^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]) + (e*(2*c*d - b*e)*f^(a + b*x + c*x^2)*(d + e*x))/(4*c^2*Log[f]) + (e*f^(a + b*x + c*x^2)*(d + e*x)^2)/(2*c*Log[f]) + ((2*c*d - b*e)^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2)*Sqrt[Log[f]])} -{(d + e*x)^2*f^(a + b*x + c*x^2), x, 6, -((e^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Log[f]^(3/2))) + (e*(2*c*d - b*e)*f^(a + b*x + c*x^2))/(4*c^2*Log[f]) + (e*f^(a + b*x + c*x^2)*(d + e*x))/(2*c*Log[f]) + ((2*c*d - b*e)^2*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Sqrt[Log[f]])} -{(d + e*x)*f^(a + b*x + c*x^2), x, 3, (e*f^(a + b*x + c*x^2))/(2*c*Log[f]) + ((2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*c^(3/2)*Sqrt[Log[f]])} -{f^(a + b*x + c*x^2)/(d + e*x), x, 0, Unintegrable[f^(a + b*x + c*x^2)/(d + e*x), x]} -{f^(a + b*x + c*x^2)/(d + e*x)^2, x, 3, -(f^(a + b*x + c*x^2)/(e*(d + e*x))) + (Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/e^2 - ((2*c*d - b*e)*Unintegrable[f^(a + b*x + c*x^2)/(d + e*x), x]*Log[f])/e^2} -{f^(a + b*x + c*x^2)/(d + e*x)^3, x, 4, -(f^(a + b*x + c*x^2)/(2*e*(d + e*x)^2)) + ((2*c*d - b*e)*f^(a + b*x + c*x^2)*Log[f])/(2*e^3*(d + e*x)) + (c*Unintegrable[f^(a + b*x + c*x^2)/(d + e*x), x]*Log[f])/e^2 - (Sqrt[c]*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Log[f]^(3/2))/(2*e^4) + ((2*c*d - b*e)^2*Unintegrable[f^(a + b*x + c*x^2)/(d + e*x), x]*Log[f]^2)/(2*e^4)} - - -{(b + 2*c*x)^3*f^(a + b*x + c*x^2), x, 2, -((4*c*f^(a + b*x + c*x^2))/Log[f]^2) + (f^(a + b*x + c*x^2)*(b + 2*c*x)^2)/Log[f]} -{(b + 2*c*x)^2*f^(a + b*x + c*x^2), x, 3, -((Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/Log[f]^(3/2)) + (f^(a + b*x + c*x^2)*(b + 2*c*x))/Log[f]} -{(b + 2*c*x)*f^(a + b*x + c*x^2), x, 1, f^(a + b*x + c*x^2)/Log[f]} -{f^(a + b*x + c*x^2)/(b + 2*c*x), x, 1, (f^(a - b^2/(4*c))*ExpIntegralEi[((b + 2*c*x)^2*Log[f])/(4*c)])/(4*c)} -{f^(a + b*x + c*x^2)/(b + 2*c*x)^2, x, 3, -(f^(a + b*x + c*x^2)/(2*c*(b + 2*c*x))) + (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/(4*c^(3/2))} -{f^(a + b*x + c*x^2)/(b + 2*c*x)^3, x, 2, -(f^(a + b*x + c*x^2)/(4*c*(b + 2*c*x)^2)) + (f^(a - b^2/(4*c))*ExpIntegralEi[((b + 2*c*x)^2*Log[f])/(4*c)]*Log[f])/(16*c^2)} - - -{(b + 2*c*x)^3*f^(b*x + c*x^2), x, 2, -((4*c*f^(b*x + c*x^2))/Log[f]^2) + (f^(b*x + c*x^2)*(b + 2*c*x)^2)/Log[f]} -{(b + 2*c*x)^2*f^(b*x + c*x^2), x, 3, -((Sqrt[c]*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(f^(b^2/(4*c))*Log[f]^(3/2))) + (f^(b*x + c*x^2)*(b + 2*c*x))/Log[f]} -{(b + 2*c*x)*f^(b*x + c*x^2), x, 1, f^(b*x + c*x^2)/Log[f]} -{f^(b*x + c*x^2)/(b + 2*c*x), x, 1, ExpIntegralEi[((b + 2*c*x)^2*Log[f])/(4*c)]/(f^(b^2/(4*c))*(4*c))} -{f^(b*x + c*x^2)/(b + 2*c*x)^2, x, 3, -(f^(b*x + c*x^2)/(2*c*(b + 2*c*x))) + (Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]])/(f^(b^2/(4*c))*(4*c^(3/2)))} -{f^(b*x + c*x^2)/(b + 2*c*x)^3, x, 2, -(f^(b*x + c*x^2)/(4*c*(b + 2*c*x)^2)) + (ExpIntegralEi[((b + 2*c*x)^2*Log[f])/(4*c)]*Log[f])/(f^(b^2/(4*c))*(16*c^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(d+e x)^n (a+b x+c x^2)^p*) - - -{E^(a + b*x)/(x^2*(c + d*x^2)), x, 8, -(E^(a + b*x)/(c*x)) + (b*E^a*ExpIntegralEi[b*x])/c + (Sqrt[d]*E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*(-c)^(3/2))} -{E^(a + b*x)/(x^1*(c + d*x^2)), x, 7, (E^a*ExpIntegralEi[b*x])/c - (E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*c) - (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*c)} -{x^0*E^(a + b*x)/(c + d*x^2), x, 4, (E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*Sqrt[-c]*Sqrt[d]) - (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d])} -{x^1*E^(a + b*x)/(c + d*x^2), x, 4, (E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*d) + (E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*d)} -{x^2*E^(a + b*x)/(c + d*x^2), x, 7, E^(a + b*x)/(b*d) + (Sqrt[-c]*E^(a + (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[-((b*(Sqrt[-c] - Sqrt[d]*x))/Sqrt[d])])/(2*d^(3/2)) - (Sqrt[-c]*E^(a - (b*Sqrt[-c])/Sqrt[d])*ExpIntegralEi[(b*(Sqrt[-c] + Sqrt[d]*x))/Sqrt[d]])/(2*d^(3/2))} - - -{E^(d + e*x)/(x^2*(a + b*x + c*x^2)), x, 9, -(E^(d + e*x)/(a*x)) - (b*E^d*ExpIntegralEi[e*x])/a^2 + (e*E^d*ExpIntegralEi[e*x])/a + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*a^2) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*a^2)} -{E^(d + e*x)/(x^1*(a + b*x + c*x^2)), x, 7, (E^d*ExpIntegralEi[e*x])/a - ((1 + b/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*a) - ((1 - b/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*a)} -{x^0*E^(d + e*x)/(a + b*x + c*x^2), x, 4, (E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/Sqrt[b^2 - 4*a*c] - (E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/Sqrt[b^2 - 4*a*c]} -{x^1*E^(d + e*x)/(a + b*x + c*x^2), x, 4, ((1 - b/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c) + ((1 + b/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c)} -{x^2*E^(d + e*x)/(a + b*x + c*x^2), x, 7, E^(d + e*x)/(c*e) - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^2) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^2)} -{x^3*E^(d + e*x)/(a + b*x + c*x^2), x, 9, -(E^(d + e*x)/(c*e^2)) - (b*E^(d + e*x))/(c^2*e) + (E^(d + e*x)*x)/(c*e) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b - Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*E^(d - ((b + Sqrt[b^2 - 4*a*c])*e)/(2*c))*ExpIntegralEi[(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c)])/(2*c^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form f^x / (a+b g^x)*) - - -{4^x/(a + b*2^x), x, 3, 2^x/(b*Log[2]) - (a*Log[a + 2^x*b])/(b^2*Log[2])} -{2^(2*x)/(a + b*2^x), x, 3, 2^x/(b*Log[2]) - (a*Log[a + 2^x*b])/(b^2*Log[2])} - -{4^x/(a - b*2^x), x, 3, -(2^x/(b*Log[2])) - (a*Log[a - 2^x*b])/(b^2*Log[2])} -{2^(2*x)/(a - b*2^x), x, 3, -(2^x/(b*Log[2])) - (a*Log[a - 2^x*b])/(b^2*Log[2])} - - -{4^x/(a + b/2^x), x, 3, (b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) - (2^x*b)/(a^2*Log[2]) + (b^2*Log[a + b/2^x])/(a^3*Log[2])} -{2^(2*x)/(a + b/2^x), x, 3, (b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) - (2^x*b)/(a^2*Log[2]) + (b^2*Log[a + b/2^x])/(a^3*Log[2])} - -{4^x/(a - b/2^x), x, 3, (b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) + (2^x*b)/(a^2*Log[2]) + (b^2*Log[a - b/2^x])/(a^3*Log[2])} -{2^(2*x)/(a - b/2^x), x, 3, (b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) + (2^x*b)/(a^2*Log[2]) + (b^2*Log[a - b/2^x])/(a^3*Log[2])} - - -{2^x/(a + b*4^x), x, 2, ArcTan[(2^x*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[2])} -{2^x/(a + b*2^(2*x)), x, 2, ArcTan[(2^x*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[2])} - -{2^x/(a - b*4^x), x, 2, ArcTanh[(2^x*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[2])} -{2^x/(a - b*2^(2*x)), x, 2, ArcTanh[(2^x*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[2])} - - -{2^x/(a + b/4^x), x, 4, 2^x/(a*Log[2]) - (Sqrt[b]*ArcTan[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])} -{2^x/(a + b/2^(2*x)), x, 4, 2^x/(a*Log[2]) - (Sqrt[b]*ArcTan[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])} - -{2^x/(a - b/4^x), x, 4, 2^x/(a*Log[2]) - (Sqrt[b]*ArcTanh[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])} -{2^x/(a - b/2^(2*x)), x, 4, 2^x/(a*Log[2]) - (Sqrt[b]*ArcTanh[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form f^x / Sqrt[a+b g^x]*) - - -(* Contributed by Robert Israel in sci.math.symbolic *) -{2^x/Sqrt[a + b*4^x], x, 3, ArcTanh[(2^x*Sqrt[b])/Sqrt[a + 4^x*b]]/(Sqrt[b]*Log[2])} -{2^x/Sqrt[a + b*2^(2*x)], x, 3, ArcTanh[(2^x*Sqrt[b])/Sqrt[a + 4^x*b]]/(Sqrt[b]*Log[2])} - -{2^x/Sqrt[a - b*4^x], x, 3, ArcTan[(2^x*Sqrt[b])/Sqrt[a - 4^x*b]]/(Sqrt[b]*Log[2])} -{2^x/Sqrt[a - b*2^(2*x)], x, 3, ArcTan[(2^x*Sqrt[b])/Sqrt[a - 4^x*b]]/(Sqrt[b]*Log[2])} - - -{2^x/Sqrt[a + b/4^x], x, 2, (2^x*Sqrt[a + b/2^(2*x)])/(a*Log[2])} -{2^x/Sqrt[a + b/2^(2*x)], x, 2, (2^x*Sqrt[a + b/2^(2*x)])/(a*Log[2])} - -{2^x/Sqrt[a - b/4^x], x, 2, (2^x*Sqrt[a - b/2^(2*x)])/(a*Log[2])} -{2^x/Sqrt[a - b/2^(2*x)], x, 2, (2^x*Sqrt[a - b/2^(2*x)])/(a*Log[2])} - - -{4^x/Sqrt[a + b*2^x], x, 3, -((2*a*Sqrt[a + 2^x*b])/(b^2*Log[2])) + (2*(a + 2^x*b)^(3/2))/(3*b^2*Log[2])} -{2^(2*x)/Sqrt[a + b*2^x], x, 3, -((2*a*Sqrt[a + 2^x*b])/(b^2*Log[2])) + (2*(a + 2^x*b)^(3/2))/(3*b^2*Log[2])} - -{4^x/Sqrt[a - b*2^x], x, 3, -((2*a*Sqrt[a - 2^x*b])/(b^2*Log[2])) + (2*(a - 2^x*b)^(3/2))/(3*b^2*Log[2])} -{2^(2*x)/Sqrt[a - b*2^x], x, 3, -((2*a*Sqrt[a - 2^x*b])/(b^2*Log[2])) + (2*(a - 2^x*b)^(3/2))/(3*b^2*Log[2])} - - -{4^x/Sqrt[a + b/2^x], x, 5, (2^(-1 + 2*x)*Sqrt[a + b/2^x])/(a*Log[2]) - (3*2^(-2 + x)*b*Sqrt[a + b/2^x])/(a^2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a + b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])} -{2^(2*x)/Sqrt[a + b/2^x], x, 5, (2^(-1 + 2*x)*Sqrt[a + b/2^x])/(a*Log[2]) - (3*2^(-2 + x)*b*Sqrt[a + b/2^x])/(a^2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a + b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])} - -{4^x/Sqrt[a - b/2^x], x, 5, (2^(-1 + 2*x)*Sqrt[a - b/2^x])/(a*Log[2]) + (3*2^(-2 + x)*b*Sqrt[a - b/2^x])/(a^2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a - b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])} -{2^(2*x)/Sqrt[a - b/2^x], x, 5, (2^(-1 + 2*x)*Sqrt[a - b/2^x])/(a*Log[2]) + (3*2^(-2 + x)*b*Sqrt[a - b/2^x])/(a^2*Log[2]) + (3*b^2*ArcTanh[Sqrt[a - b/2^x]/Sqrt[a]])/(4*a^(5/2)*Log[2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b f^(d+e x)+c f^(2 (d+e x)))^n*) - - -{1/(1 + 2*E^x + E^(2*x)), x, 3, 1/(1 + E^x) + x - Log[1 + E^x]} -{1/(2 + 3*E^x + E^(2*x)), x, 6, x/2 - Log[1 + E^x] + (1/2)*Log[2 + E^x]} -{1/(-1 + E^x + E^(2*x)), x, 6, -x + (1/10)*(5 + Sqrt[5])*Log[1 - Sqrt[5] + 2*E^x] + (1/10)*(5 - Sqrt[5])*Log[1 + Sqrt[5] + 2*E^x]} -{1/(3 + 3*E^x + E^(2*x)), x, 7, x/3 - ArcTan[(3 + 2*E^x)/Sqrt[3]]/Sqrt[3] - (1/6)*Log[3 + 3*E^x + E^(2*x)]} -{1/(a + b*E^x + c*E^(2*x)), x, 7, x/a + (b*ArcTanh[(b + 2*c*E^x)/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) - Log[a + b*E^x + c*E^(2*x)]/(2*a)} - -{x/(1 + 2*E^x + E^(2*x)), x, 11, -x + x/(1 + E^x) + x^2/2 + Log[1 + E^x] - x*Log[1 + E^x] - PolyLog[2, -E^x]} -{x/(2 + 3*E^x + E^(2*x)), x, 9, x^2/4 + (1/2)*x*Log[1 + E^x/2] - x*Log[1 + E^x] - PolyLog[2, -E^x] + (1/2)*PolyLog[2, -(E^x/2)]} -{x/(-1 + E^x + E^(2*x)), x, 9, x^2/(Sqrt[5]*(1 - Sqrt[5])) - x^2/(Sqrt[5]*(1 + Sqrt[5])) - (2*x*Log[1 + (2*E^x)/(1 - Sqrt[5])])/(Sqrt[5]*(1 - Sqrt[5])) + (2*x*Log[1 + (2*E^x)/(1 + Sqrt[5])])/(Sqrt[5]*(1 + Sqrt[5])) - (2*PolyLog[2, -((2*E^x)/(1 - Sqrt[5]))])/(Sqrt[5]*(1 - Sqrt[5])) + (2*PolyLog[2, -((2*E^x)/(1 + Sqrt[5]))])/(Sqrt[5]*(1 + Sqrt[5]))} -{x/(3 + 3*E^x + E^(2*x)), x, 9, -(x^2/(Sqrt[3]*(3*I - Sqrt[3]))) + x^2/(Sqrt[3]*(3*I + Sqrt[3])) - (2*x*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*x*Log[1 + (2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (2*PolyLog[2, -((2*E^x)/(3 - I*Sqrt[3]))])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*PolyLog[2, -((2*E^x)/(3 + I*Sqrt[3]))])/(Sqrt[3]*(3*I - Sqrt[3]))} -{x/(a + b*E^x + c*E^(2*x)), x, 9, -((c*x^2)/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (c*x^2)/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) + (2*c*x*Log[1 + (2*c*E^x)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (2*c*x*Log[1 + (2*c*E^x)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) + (2*c*PolyLog[2, -((2*c*E^x)/(b - Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (2*c*PolyLog[2, -((2*c*E^x)/(b + Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])} - -{x^2/(1 + 2*E^x + E^(2*x)), x, 12, -x^2 + x^2/(1 + E^x) + x^3/3 + 2*x*Log[1 + E^x] - x^2*Log[1 + E^x] + 2*PolyLog[2, -E^x] - 2*x*PolyLog[2, -E^x] + 2*PolyLog[3, -E^x]} -{x^2/(2 + 3*E^x + E^(2*x)), x, 11, x^3/6 + (1/2)*x^2*Log[1 + E^x/2] - x^2*Log[1 + E^x] - 2*x*PolyLog[2, -E^x] + x*PolyLog[2, -(E^x/2)] + 2*PolyLog[3, -E^x] - PolyLog[3, -(E^x/2)]} -{x^2/(-1 + E^x + E^(2*x)), x, 11, (2*x^3)/(3*Sqrt[5]*(1 - Sqrt[5])) - (2*x^3)/(3*Sqrt[5]*(1 + Sqrt[5])) - (2*x^2*Log[1 + (2*E^x)/(1 - Sqrt[5])])/(Sqrt[5]*(1 - Sqrt[5])) + (2*x^2*Log[1 + (2*E^x)/(1 + Sqrt[5])])/(Sqrt[5]*(1 + Sqrt[5])) - (4*x*PolyLog[2, -((2*E^x)/(1 - Sqrt[5]))])/(Sqrt[5]*(1 - Sqrt[5])) + (4*x*PolyLog[2, -((2*E^x)/(1 + Sqrt[5]))])/(Sqrt[5]*(1 + Sqrt[5])) + (4*PolyLog[3, -((2*E^x)/(1 - Sqrt[5]))])/(Sqrt[5]*(1 - Sqrt[5])) - (4*PolyLog[3, -((2*E^x)/(1 + Sqrt[5]))])/(Sqrt[5]*(1 + Sqrt[5]))} -{x^2/(3 + 3*E^x + E^(2*x)), x, 11, -((2*x^3)/(3*Sqrt[3]*(3*I - Sqrt[3]))) + (2*x^3)/(3*Sqrt[3]*(3*I + Sqrt[3])) - (2*x^2*Log[1 + (2*E^x)/(3 - I*Sqrt[3])])/(Sqrt[3]*(3*I + Sqrt[3])) + (2*x^2*Log[1 + (2*E^x)/(3 + I*Sqrt[3])])/(Sqrt[3]*(3*I - Sqrt[3])) - (4*x*PolyLog[2, -((2*E^x)/(3 - I*Sqrt[3]))])/(Sqrt[3]*(3*I + Sqrt[3])) + (4*x*PolyLog[2, -((2*E^x)/(3 + I*Sqrt[3]))])/(Sqrt[3]*(3*I - Sqrt[3])) + (4*PolyLog[3, -((2*E^x)/(3 - I*Sqrt[3]))])/(Sqrt[3]*(3*I + Sqrt[3])) - (4*PolyLog[3, -((2*E^x)/(3 + I*Sqrt[3]))])/(Sqrt[3]*(3*I - Sqrt[3]))} -{x^2/(a + b*E^x + c*E^(2*x)), x, 11, -((2*c*x^3)/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]))) - (2*c*x^3)/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])) + (2*c*x^2*Log[1 + (2*c*E^x)/(b - Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (2*c*x^2*Log[1 + (2*c*E^x)/(b + Sqrt[b^2 - 4*a*c])])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) + (4*c*x*PolyLog[2, -((2*c*E^x)/(b - Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + (4*c*x*PolyLog[2, -((2*c*E^x)/(b + Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (4*c*PolyLog[3, -((2*c*E^x)/(b - Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) - (4*c*PolyLog[3, -((2*c*E^x)/(b + Sqrt[b^2 - 4*a*c]))])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])} - - -{1/(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x)), x, 3, x + 1/(d*(1 + f^(c + d*x))*Log[f]) - Log[1 + f^(c + d*x)]/(d*Log[f])} -{1/(a + b*f^(c + d*x) + c*f^(2*c + 2*d*x)), x, 7, x/a + (b*ArcTanh[(b + 2*c*f^(c + d*x))/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*d*Log[f]) - Log[a + b*f^(c + d*x) + c*f^(2*c + 2*d*x)]/(2*a*d*Log[f])} -{1/(a + b*f^(g + h*x) + c*f^(2*(g + h*x))), x, 7, x/a + (b*ArcTanh[(b + 2*c*f^(g + h*x))/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*h*Log[f]) - Log[a + b*f^(g + h*x) + c*f^(2*g + 2*h*x)]/(2*a*h*Log[f])} - -{x/(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x)), x, 11, x^2/2 - x/(d*Log[f]) + x/(d*(1 + f^(c + d*x))*Log[f]) + Log[1 + f^(c + d*x)]/(d^2*Log[f]^2) - (x*Log[1 + f^(c + d*x)])/(d*Log[f]) - PolyLog[2, -f^(c + d*x)]/(d^2*Log[f]^2)} -{x/(a + b*f^(c + d*x) + c*f^(2*c + 2*d*x)), x, 9, -((c*x^2)/(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])) - (c*x^2)/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - (2*c*x*Log[1 + (2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d*Log[f]) + (2*c*x*Log[1 + (2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d*Log[f]) - (2*c*PolyLog[2, -((2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d^2*Log[f]^2) + (2*c*PolyLog[2, -((2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d^2*Log[f]^2)} - -{x^2/(1 + 2*f^(c + d*x) + f^(2*c + 2*d*x)), x, 12, x^3/3 - x^2/(d*Log[f]) + x^2/(d*(1 + f^(c + d*x))*Log[f]) + (2*x*Log[1 + f^(c + d*x)])/(d^2*Log[f]^2) - (x^2*Log[1 + f^(c + d*x)])/(d*Log[f]) + (2*PolyLog[2, -f^(c + d*x)])/(d^3*Log[f]^3) - (2*x*PolyLog[2, -f^(c + d*x)])/(d^2*Log[f]^2) + (2*PolyLog[3, -f^(c + d*x)])/(d^3*Log[f]^3)} -{x^2/(a + b*f^(c + d*x) + c*f^(2*c + 2*d*x)), x, 11, -((2*c*x^3)/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]))) - (2*c*x^3)/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])) - (2*c*x^2*Log[1 + (2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d*Log[f]) + (2*c*x^2*Log[1 + (2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d*Log[f]) - (4*c*x*PolyLog[2, -((2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d^2*Log[f]^2) + (4*c*x*PolyLog[2, -((2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d^2*Log[f]^2) + (4*c*PolyLog[3, -((2*c*f^(c + d*x))/(b - Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*d^3*Log[f]^3) - (4*c*PolyLog[3, -((2*c*f^(c + d*x))/(b + Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*d^3*Log[f]^3)} - - -{(d + e*f^(g + h*x))/(a + b*f^(g + h*x) + c*f^(2*g + 2*h*x)), x, 7, (d*x)/a + ((b*d - 2*a*e)*ArcTanh[(b + 2*c*f^(g + h*x))/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*h*Log[f]) - (d*Log[a + b*f^(g + h*x) + c*f^(2*g + 2*h*x)])/(2*a*h*Log[f])} -{(d + e*f^(g + h*x))/(a + b*f^(g + h*x) + c*f^(2*(g + h*x))), x, 7, (d*x)/a + ((b*d - 2*a*e)*ArcTanh[(b + 2*c*f^(g + h*x))/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*h*Log[f]) - (d*Log[a + b*f^(g + h*x) + c*f^(2*g + 2*h*x)])/(2*a*h*Log[f])} - - -{1/(2 + E^(-x) + E^x), x, 2, -(1/(1 + E^x))} -{x/(2 + E^(-x) + E^x), x, 7, x - x/(1 + E^x) - Log[1 + E^x]} -{x^2/(2 + E^(-x) + E^x), x, 7, x^2 - x^2/(1 + E^x) - 2*x*Log[1 + E^x] - 2*PolyLog[2, -E^x]} - -{1/(2 + f^(-c - d*x) + f^(c + d*x)), x, 2, -(1/(d*(1 + f^(c + d*x))*Log[f]))} -{x/(2 + f^(-c - d*x) + f^(c + d*x)), x, 7, x/(d*Log[f]) - x/(d*(1 + f^(c + d*x))*Log[f]) - Log[1 + f^(c + d*x)]/(d^2*Log[f]^2)} -{x^2/(2 + f^(-c - d*x) + f^(c + d*x)), x, 7, x^2/(d*Log[f]) - x^2/(d*(1 + f^(c + d*x))*Log[f]) - (2*x*Log[1 + f^(c + d*x)])/(d^2*Log[f]^2) - (2*PolyLog[2, -f^(c + d*x)])/(d^3*Log[f]^3)} - - -{1/(2 + 3^(-x) + 3^x), x, 2, -(1/((1 + 3^x)*Log[3]))} -{1/(1 - E^(-x) + 2*E^x), x, 4, (1/3)*Log[1 - 2*E^x] - (1/3)*Log[1 + E^x]} - -{1/(a + b*E^(-x) + c*E^x), x, 4, -((2*ArcTanh[(a + 2*c*E^x)/Sqrt[a^2 - 4*b*c]])/Sqrt[a^2 - 4*b*c])} -{x/(a + b*E^(-x) + c*E^x), x, 8, (x*Log[1 + (2*c*E^x)/(a - Sqrt[a^2 - 4*b*c])])/Sqrt[a^2 - 4*b*c] - (x*Log[1 + (2*c*E^x)/(a + Sqrt[a^2 - 4*b*c])])/Sqrt[a^2 - 4*b*c] + PolyLog[2, -((2*c*E^x)/(a - Sqrt[a^2 - 4*b*c]))]/Sqrt[a^2 - 4*b*c] - PolyLog[2, -((2*c*E^x)/(a + Sqrt[a^2 - 4*b*c]))]/Sqrt[a^2 - 4*b*c]} -{x^2/(a + b*E^(-x) + c*E^x), x, 10, (x^2*Log[1 + (2*c*E^x)/(a - Sqrt[a^2 - 4*b*c])])/Sqrt[a^2 - 4*b*c] - (x^2*Log[1 + (2*c*E^x)/(a + Sqrt[a^2 - 4*b*c])])/Sqrt[a^2 - 4*b*c] + (2*x*PolyLog[2, -((2*c*E^x)/(a - Sqrt[a^2 - 4*b*c]))])/Sqrt[a^2 - 4*b*c] - (2*x*PolyLog[2, -((2*c*E^x)/(a + Sqrt[a^2 - 4*b*c]))])/Sqrt[a^2 - 4*b*c] - (2*PolyLog[3, -((2*c*E^x)/(a - Sqrt[a^2 - 4*b*c]))])/Sqrt[a^2 - 4*b*c] + (2*PolyLog[3, -((2*c*E^x)/(a + Sqrt[a^2 - 4*b*c]))])/Sqrt[a^2 - 4*b*c]} - -{1/(a + b*f^(-c - d*x) + c*f^(c + d*x)), x, 4, -((2*ArcTanh[(a + 2*c*f^(c + d*x))/Sqrt[a^2 - 4*b*c]])/(Sqrt[a^2 - 4*b*c]*d*Log[f]))} -{x/(a + b*f^(-c - d*x) + c*f^(c + d*x)), x, 8, (x*Log[1 + (2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) - (x*Log[1 + (2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) + PolyLog[2, -((2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c]))]/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2) - PolyLog[2, -((2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c]))]/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2)} -{x^2/(a + b*f^(-c - d*x) + c*f^(c + d*x)), x, 10, (x^2*Log[1 + (2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) - (x^2*Log[1 + (2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c])])/(Sqrt[a^2 - 4*b*c]*d*Log[f]) + (2*x*PolyLog[2, -((2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c]))])/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2) - (2*x*PolyLog[2, -((2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c]))])/(Sqrt[a^2 - 4*b*c]*d^2*Log[f]^2) - (2*PolyLog[3, -((2*c*f^(c + d*x))/(a - Sqrt[a^2 - 4*b*c]))])/(Sqrt[a^2 - 4*b*c]*d^3*Log[f]^3) + (2*PolyLog[3, -((2*c*f^(c + d*x))/(a + Sqrt[a^2 - 4*b*c]))])/(Sqrt[a^2 - 4*b*c]*d^3*Log[f]^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b F^(c Sqrt[d+e x]/Sqrt[f+g x]))^n*) - - -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^n/(d*f + (e*f + d*g)*x + e*g*x^2), x, 0, Unintegrable[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))^n/(d*f + (e*f + d*g)*x + e*g*x^2), x]} - - -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^3/(d*f + (e*f + d*g)*x + e*g*x^2), x, 6, (6*a^2*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (6*a*b^2*ExpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (2*b^3*ExpIntegralEi[(3*c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (2*a^3*Log[Sqrt[d + e*x]/Sqrt[f + g*x]])/(e*f - d*g)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^2/(d*f + (e*f + d*g)*x + e*g*x^2), x, 5, (4*a*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (2*b^2*ExpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (2*a^2*Log[Sqrt[d + e*x]/Sqrt[f + g*x]])/(e*f - d*g)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^1/(d*f + (e*f + d*g)*x + e*g*x^2), x, 4, (2*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[f + g*x]])/(e*f - d*g) + (2*a*Log[Sqrt[d + e*x]/Sqrt[f + g*x]])/(e*f - d*g)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^0/(d*f + (e*f + d*g)*x + e*g*x^2), x, 3, Log[d + e*x]/(e*f - d*g) - Log[f + g*x]/(e*f - d*g)} -{1/((a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^1*(d*f + (e*f + d*g)*x + e*g*x^2)), x, 0, Unintegrable[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))*(d*f + (e*f + d*g)*x + e*g*x^2)), x]} -{1/((a + b*F^(c*Sqrt[d + e*x]/Sqrt[f + g*x]))^2*(d*f + (e*f + d*g)*x + e*g*x^2)), x, 0, Unintegrable[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[f + g*x]))^2*(d*f + (e*f + d*g)*x + e*g*x^2)), x]} - - -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^n/(d^2 - e^2*x^2), x, 0, Unintegrable[(a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^n/(d^2 - e^2*x^2), x]} - - -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^3/(d^2 - e^2*x^2), x, 6, (3*a^2*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (3*a*b^2*ExpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (b^3*ExpIntegralEi[(3*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (a^3*Log[Sqrt[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^2/(d^2 - e^2*x^2), x, 5, (2*a*b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (b^2*ExpIntegralEi[(2*c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (a^2*Log[Sqrt[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^1/(d^2 - e^2*x^2), x, 4, (b*ExpIntegralEi[(c*Sqrt[d + e*x]*Log[F])/Sqrt[d*f - e*f*x]])/(d*e) + (a*Log[Sqrt[d + e*x]/Sqrt[d*f - e*f*x]])/(d*e)} -{(a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^0/(d^2 - e^2*x^2), x, 1, ArcTanh[(e*x)/d]/(d*e)} -{1/((a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^1*(d^2 - e^2*x^2)), x, 0, Unintegrable[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))*(d^2 - e^2*x^2)), x]} -{1/((a + b*F^(c*Sqrt[d + e*x]/Sqrt[d*f - e*f*x]))^2*(d^2 - e^2*x^2)), x, 0, Unintegrable[1/((a + b*F^((c*Sqrt[d + e*x])/Sqrt[d*f - e*f*x]))^2*(d^2 - e^2*x^2)), x]} - - -{(F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^n/(1 - a^2*x^2), x, 3, -(((F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^n*ExpIntegralEi[(n*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]])/(F^((n*Sqrt[1 - a*x])/Sqrt[1 + a*x])*a))} - -{(F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^3/(1 - a^2*x^2), x, 2, -(ExpIntegralEi[(3*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)} -{(F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^2/(1 - a^2*x^2), x, 2, -(ExpIntegralEi[(2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)} -{(F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^1/(1 - a^2*x^2), x, 2, -(ExpIntegralEi[(Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x]]/a)} -{1/((F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^1*(1 - a^2*x^2)), x, 2, -(ExpIntegralEi[-((Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x])]/a)} -{1/((F^(Sqrt[1 - a*x]/Sqrt[1 + a*x]))^2*(1 - a^2*x^2)), x, 2, -(ExpIntegralEi[-((2*Sqrt[1 - a*x]*Log[F])/Sqrt[1 + a*x])]/a)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(a+b x) G^(c+d x)*) - - -{x^2*a^x*b^x, x, 4, (2*a^x*b^x)/(Log[a] + Log[b])^3 - (2*a^x*b^x*x)/(Log[a] + Log[b])^2 + (a^x*b^x*x^2)/(Log[a] + Log[b])} -{x*a^x*b^x, x, 3, -((a^x*b^x)/(Log[a] + Log[b])^2) + (a^x*b^x*x)/(Log[a] + Log[b])} -{a^x*b^x, x, 2, (a^x*b^x)/(Log[a] + Log[b])} -{a^x*b^x/x, x, 2, ExpIntegralEi[x*(Log[a] + Log[b])]} -{a^x*b^x/x^2, x, 3, -((a^x*b^x)/x) + ExpIntegralEi[x*(Log[a] + Log[b])]*(Log[a] + Log[b])} -{a^x*b^x/x^3, x, 4, -((a^x*b^x)/(2*x^2)) - (a^x*b^x*(Log[a] + Log[b]))/(2*x) + (1/2)*ExpIntegralEi[x*(Log[a] + Log[b])]*(Log[a] + Log[b])^2} - - -{a^x*b^x*c^x, x, 3, (a^x*b^x*c^x)/(Log[a] + Log[b] + Log[c])} -{a^x/b^x, x, 2, a^x/(b^x*(Log[a] - Log[b]))} - -{(a^x*x^2)/b^x, x, 4, (2*a^x)/(b^x*(Log[a] - Log[b])^3) - (2*a^x*x)/(b^x*(Log[a] - Log[b])^2) + (a^x*x^2)/(b^x*(Log[a] - Log[b]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e F^(h+i x)) (a+b F^(h+i x)+c F^(2 h+2 i x))^n*) - - -{(f + g*x)^3*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 13, ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^4)/(4*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^4)/(4*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^3*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - (3*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)^2*PolyLog[2, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^2) - (3*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)^2*PolyLog[2, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^2) + (6*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^2*(f + g*x)*PolyLog[3, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^3) + (6*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^2*(f + g*x)*PolyLog[3, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^3) - (6*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^3*PolyLog[4, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^4) - (6*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^3*PolyLog[4, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^4)} -{(f + g*x)^2*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 11, ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^3)/(3*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^3)/(3*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - (2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)*PolyLog[2, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^2) - (2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*(f + g*x)*PolyLog[2, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^2) + (2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^2*PolyLog[3, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^3) + (2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g^2*PolyLog[3, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^3)} -{(f + g*x)^1*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 9, ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2)/(2*(b + Sqrt[b^2 - 4*a*c])*g) + ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)^2)/(2*(b - Sqrt[b^2 - 4*a*c])*g) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*i) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f + g*x)*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*i) - ((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, -((2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*i^2) - ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*g*PolyLog[2, -((2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*i^2)} -{(f + g*x)^0*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 7, (d*x)/a + ((b*d - 2*a*e)*ArcTanh[(b + 2*c*E^(h + i*x))/Sqrt[b^2 - 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]*i) - (d*Log[a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)])/(2*a*i)} -{1/(f + g*x)^1*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 2, d*CannotIntegrate[1/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x] + e*CannotIntegrate[E^(h + i*x)/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)), x]} -{1/(f + g*x)^2*(d + e*E^(h + i*x))/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)), x, 2, d*CannotIntegrate[1/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)^2), x] + e*CannotIntegrate[E^(h + i*x)/((a + b*E^(h + i*x) + c*E^(2*h + 2*i*x))*(f + g*x)^2), x]} - - -{x*(b*e - a*e*E^(c + d*x))/(b*e - 2*a*e*E^(c + d*x) - b*e*E^(2*(c + d*x))), x, 9, x^2/2 - (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*d) - (x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*d) - PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))]/(2*d^2) - PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]/(2*d^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m F^(a+b Log[c+d x^n])*) - - -{x^2*F^(a + b*Log[c + d*x^n]), x, 4, ((1/3)*F^a*x^3*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[3/n, (-b)*Log[F], (3 + n)/n, -((d*x^n)/c)])/(1 + (d*x^n)/c)^(b*Log[F])} -{x^1*F^(a + b*Log[c + d*x^n]), x, 4, ((1/2)*F^a*x^2*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[2/n, (-b)*Log[F], (2 + n)/n, -((d*x^n)/c)])/(1 + (d*x^n)/c)^(b*Log[F])} -{x^0*F^(a + b*Log[c + d*x^n]), x, 4, (F^a*x*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[1/n, (-b)*Log[F], 1 + 1/n, -((d*x^n)/c)])/(1 + (d*x^n)/c)^(b*Log[F])} -{F^(a + b*Log[c + d*x^n])/x^1, x, 4, -((F^a*(c + d*x^n)^(1 + b*Log[F])*Hypergeometric2F1[1, 1 + b*Log[F], 2 + b*Log[F], 1 + (d*x^n)/c])/(c*n*(1 + b*Log[F])))} -{F^(a + b*Log[c + d*x^n])/x^2, x, 4, -((F^a*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[-(1/n), (-b)*Log[F], -((1 - n)/n), -((d*x^n)/c)])/((1 + (d*x^n)/c)^(b*Log[F])*x))} -{F^(a + b*Log[c + d*x^n])/x^3, x, 4, -((F^a*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[-(2/n), (-b)*Log[F], -((2 - n)/n), -((d*x^n)/c)])/((1 + (d*x^n)/c)^(b*Log[F])*(2*x^2)))} - - -{(d x)^m*F^(a + b*Log[c + d*x^n]), x, 4, (F^a*(d*x)^(1 + m)*(c + d*x^n)^(b*Log[F])*Hypergeometric2F1[(1 + m)/n, (-b)*Log[F], (1 + m + n)/n, -((d*x^n)/c)])/((1 + (d*x^n)/c)^(b*Log[F])*(d*(1 + m)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n]^2))*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n]^2)) when e g-d h=0*) - - -{(d + e*x)^m*E^Log[(d + e*x)^n]^2, x, 3, (Sqrt[Pi]*(d + e*x)^(1 + m)*Erfi[(1 + m + 2*n*Log[(d + e*x)^n])/(2*n)])/(E^((1 + m)^2/(4*n^2))*((d + e*x)^n)^((1 + m)/n)*(2*e*n))} -{(d*g + e*g*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 3, (F^(a*f)*Sqrt[Pi]*(d*g + e*g*x)^(1 + m)*Erfi[(1 + m + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^((1 + m)^2/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^((1 + m)/n)*(2*Sqrt[b]*e*Sqrt[f]*g*n*Sqrt[Log[F]]))} - - -{(d*g + e*g*x)^2*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 3, (F^(a*f)*g^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(9/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*Sqrt[b]*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{(d*g + e*g*x)^1*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 3, (F^(a*f)*g*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*Sqrt[b]*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{(d*g + e*g*x)^0*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 3, (F^(a*f)*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*Sqrt[b]*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(d*g + e*g*x)^1, x, 2, (F^(a*f)*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[f]*Sqrt[Log[F]]*Log[c*(d + e*x)^n]])/(2*Sqrt[b]*e*Sqrt[f]*g*n*Sqrt[Log[F]])} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(d*g + e*g*x)^2, x, 3, If[$VersionNumber>=8, -((F^(a*f)*Sqrt[Pi]*(c*(d + e*x)^n)^(1/n)*Erfi[(1 - 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(2*Sqrt[b]*e*Sqrt[f]*g^2*n*(d + e*x)*Sqrt[Log[F]]))), (F^(a*f)*Sqrt[Pi]*(c*(d + e*x)^n)^(1/n)*Erfi[-((1 - 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]]))])/(E^(1/(4*b*f*n^2*Log[F]))*(2*Sqrt[b]*e*Sqrt[f]*g^2*n*(d + e*x)*Sqrt[Log[F]]))]} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(d*g + e*g*x)^3, x, 3, If[$VersionNumber>=8, -((F^(a*f)*Sqrt[Pi]*(c*(d + e*x)^n)^(2/n)*Erfi[(1 - b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(b*f*n^2*Log[F]))*(2*Sqrt[b]*e*Sqrt[f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]]))), (F^(a*f)*Sqrt[Pi]*(c*(d + e*x)^n)^(2/n)*Erfi[-((1 - b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]]))])/(E^(1/(b*f*n^2*Log[F]))*(2*Sqrt[b]*e*Sqrt[f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]]))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n]^2))*) - - -{(g + h*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n]^2))*(g + h*x)^m, x]} - - -{(g + h*x)^3*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 14, (3*F^(a*f)*h*(e*g - d*h)^2*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*Sqrt[b]*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + (F^(a*f)*h^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(4/(b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(4/n)*(2*Sqrt[b]*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + (F^(a*f)*(e*g - d*h)^3*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*Sqrt[b]*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + (3*F^(a*f)*h^2*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^3*Erfi[(3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(9/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*Sqrt[b]*e^4*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^2*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 11, (F^(a*f)*h*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(Sqrt[b]*e^3*Sqrt[f]*n*Sqrt[Log[F]])) + (F^(a*f)*(e*g - d*h)^2*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*Sqrt[b]*e^3*Sqrt[f]*n*Sqrt[Log[F]])) + (F^(a*f)*h^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(9/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*Sqrt[b]*e^3*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^1*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 8, (F^(a*f)*h*Sqrt[Pi]*(d + e*x)^2*Erfi[(1 + b*f*n*Log[F]*Log[c*(d + e*x)^n])/(Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(b*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*Sqrt[b]*e^2*Sqrt[f]*n*Sqrt[Log[F]])) + (F^(a*f)*(e*g - d*h)*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*Sqrt[b]*e^2*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^0*F^(f*(a + b*Log[c*(d + e*x)^n]^2)), x, 3, (F^(a*f)*Sqrt[Pi]*(d + e*x)*Erfi[(1 + 2*b*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*Sqrt[b]*Sqrt[f]*n*Sqrt[Log[F]])])/(E^(1/(4*b*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*Sqrt[b]*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x)^1, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x), x]} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x)^2, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x)^2, x]} -{F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x)^3, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n]^2))/(g + h*x)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n])^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n])^2) when e g-d h=0*) - - -{(d*g + e*g*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 4, (F^(a^2*f)*Sqrt[Pi]*(d + e*x)*(d*g + e*g*x)^m*Erfi[(1 + m + 2*a*b*f*n*Log[F] + 2*b^2*f*n*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*n*Sqrt[Log[F]])])/(E^((1 + m + 2*a*b*f*n*Log[F])^2/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^((1 + m)/n)*(2*b*e*Sqrt[f]*n*Sqrt[Log[F]]))} - - -{(d*g + e*g*x)^2*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 4, (g^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*b*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{(d*g + e*g*x)^1*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 4, (g*Sqrt[Pi]*(d + e*x)^2*Erfi[(1/n + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*b*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{(d*g + e*g*x)^0*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 4, (Sqrt[Pi]*(d + e*x)*Erfi[(1/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*b*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(d*g + e*g*x)^1, x, 4, (Sqrt[Pi]*Erfi[a*Sqrt[f]*Sqrt[Log[F]] + b*Sqrt[f]*Sqrt[Log[F]]*Log[c*(d + e*x)^n]])/(2*b*e*Sqrt[f]*g*n*Sqrt[Log[F]])} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(d*g + e*g*x)^2, x, 4, If[$VersionNumber>=8, -((E^(a/(b*n) - 1/(4*b^2*f*n^2*Log[F]))*Sqrt[Pi]*(c*(d + e*x)^n)^(1/n)*Erfi[(1/n - 2*a*b*f*Log[F] - 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(2*b*e*Sqrt[f]*g^2*n*(d + e*x)*Sqrt[Log[F]])), (E^(a/(b*n) - 1/(4*b^2*f*n^2*Log[F]))*Sqrt[Pi]*(c*(d + e*x)^n)^(1/n)*Erfi[-((1/n - 2*a*b*f*Log[F] - 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]]))])/(2*b*e*Sqrt[f]*g^2*n*(d + e*x)*Sqrt[Log[F]])]} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(d*g + e*g*x)^3, x, 4, If[$VersionNumber>=8, -((Sqrt[Pi]*(c*(d + e*x)^n)^(2/n)*Erfi[(1/n - a*b*f*Log[F] - b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 - 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(2*b*e*Sqrt[f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]]))), (Sqrt[Pi]*(c*(d + e*x)^n)^(2/n)*Erfi[-((1/n - a*b*f*Log[F] - b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]]))])/(E^((1 - 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(2*b*e*Sqrt[f]*g^3*n*(d + e*x)^2*Sqrt[Log[F]]))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m F^(f (a+b Log[c (d+e x)^n])^2)*) - - -{(g + h*x)^m*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(g + h*x)^m, x]} - - -{(g + h*x)^3*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 18, (3*h*(e*g - d*h)^2*Sqrt[Pi]*(d + e*x)^2*Erfi[(1/n + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*b*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + (h^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2/n + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((4*(1 + a*b*f*n*Log[F]))/(b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(4/n)*(2*b*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + ((e*g - d*h)^3*Sqrt[Pi]*(d + e*x)*Erfi[(1/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*b*e^4*Sqrt[f]*n*Sqrt[Log[F]])) + (3*h^2*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^3*Erfi[(3/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*b*e^4*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^2*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 14, (h*(e*g - d*h)*Sqrt[Pi]*(d + e*x)^2*Erfi[(1/n + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(b*e^3*Sqrt[f]*n*Sqrt[Log[F]])) + ((e*g - d*h)^2*Sqrt[Pi]*(d + e*x)*Erfi[(1/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*b*e^3*Sqrt[f]*n*Sqrt[Log[F]])) + (h^2*Sqrt[Pi]*(d + e*x)^3*Erfi[(3/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((3*(3 + 4*a*b*f*n*Log[F]))/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(3/n)*(2*b*e^3*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^1*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 10, (h*Sqrt[Pi]*(d + e*x)^2*Erfi[(1/n + a*b*f*Log[F] + b^2*f*Log[F]*Log[c*(d + e*x)^n])/(b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 2*a*b*f*n*Log[F])/(b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^(2/n)*(2*b*e^2*Sqrt[f]*n*Sqrt[Log[F]])) + ((e*g - d*h)*Sqrt[Pi]*(d + e*x)*Erfi[(1/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*b*e^2*Sqrt[f]*n*Sqrt[Log[F]]))} -{(g + h*x)^0*F^(f*(a + b*Log[c*(d + e*x)^n])^2), x, 4, (Sqrt[Pi]*(d + e*x)*Erfi[(1/n + 2*a*b*f*Log[F] + 2*b^2*f*Log[F]*Log[c*(d + e*x)^n])/(2*b*Sqrt[f]*Sqrt[Log[F]])])/(E^((1 + 4*a*b*f*n*Log[F])/(4*b^2*f*n^2*Log[F]))*(c*(d + e*x)^n)^n^(-1)*(2*b*e*Sqrt[f]*n*Sqrt[Log[F]]))} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x)^1, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x), x]} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x)^2, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x)^2, x]} -{F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x)^3, x, 0, Unintegrable[F^(f*(a + b*Log[c*(d + e*x)^n])^2)/(g + h*x)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u F^v D[v,x]*) - - -{F^(a + b*x + c*x^3)*(b + 3*c*x^2), x, 1, F^(a + b*x + c*x^3)/Log[F]} -{F^(1/(a + b*x + c*x^2))*(b + 2*c*x)/(a + b*x + c*x^2)^2, x, 1, -(F^(1/(a + b*x + c*x^2))/Log[F])} - - -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^m*(b + 2*c*x), x, 2, ((a + b*x + c*x^2)^m*Gamma[1 + m, -a - b*x - c*x^2])/(-a - b*x - c*x^2)^m} - -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^3*(b + 2*c*x), x, 5, -6*E^(a + b*x + c*x^2) + 6*E^(a + b*x + c*x^2)*(a + b*x + c*x^2) - 3*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2 + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^3} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2*(b + 2*c*x), x, 4, 2*E^(a + b*x + c*x^2) - 2*E^(a + b*x + c*x^2)*(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^2} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^1*(b + 2*c*x), x, 3, -E^(a + b*x + c*x^2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^0*(b + 2*c*x), x, 1, E^(a + b*x + c*x^2)} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^1*(b + 2*c*x), x, 2, ExpIntegralEi[a + b*x + c*x^2]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^2*(b + 2*c*x), x, 3, -(E^(a + b*x + c*x^2)/(a + b*x + c*x^2)) + ExpIntegralEi[a + b*x + c*x^2]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^3*(b + 2*c*x), x, 4, -(E^(a + b*x + c*x^2)/(2*(a + b*x + c*x^2)^2)) - E^(a + b*x + c*x^2)/(2*(a + b*x + c*x^2)) + (1/2)*ExpIntegralEi[a + b*x + c*x^2]} - - -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(7/2)*(b + 2*c*x), x, 7, (-(105/8))*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] + (35/4)*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2) - (7/2)*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(7/2) + (105/16)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2)*(b + 2*c*x), x, 6, (15/4)*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] - (5/2)*E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2) + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(5/2) - (15/8)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2)*(b + 2*c*x), x, 5, (-(3/2))*E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] + E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(3/2) + (3/4)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)*(a + b*x + c*x^2)^(1/2)*(b + 2*c*x), x, 4, E^(a + b*x + c*x^2)*Sqrt[a + b*x + c*x^2] - (1/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^(1/2)*(b + 2*c*x), x, 3, Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^(3/2)*(b + 2*c*x), x, 4, -((2*E^(a + b*x + c*x^2))/Sqrt[a + b*x + c*x^2]) + 2*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^(5/2)*(b + 2*c*x), x, 5, -((2*E^(a + b*x + c*x^2))/(3*(a + b*x + c*x^2)^(3/2))) - (4*E^(a + b*x + c*x^2))/(3*Sqrt[a + b*x + c*x^2]) + (4/3)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^(7/2)*(b + 2*c*x), x, 6, -((2*E^(a + b*x + c*x^2))/(5*(a + b*x + c*x^2)^(5/2))) - (4*E^(a + b*x + c*x^2))/(15*(a + b*x + c*x^2)^(3/2)) - (8*E^(a + b*x + c*x^2))/(15*Sqrt[a + b*x + c*x^2]) + (8/15)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} -{E^(a + b*x + c*x^2)/(a + b*x + c*x^2)^(9/2)*(b + 2*c*x), x, 7, -((2*E^(a + b*x + c*x^2))/(7*(a + b*x + c*x^2)^(7/2))) - (4*E^(a + b*x + c*x^2))/(35*(a + b*x + c*x^2)^(5/2)) - (8*E^(a + b*x + c*x^2))/(105*(a + b*x + c*x^2)^(3/2)) - (16*E^(a + b*x + c*x^2))/(105*Sqrt[a + b*x + c*x^2]) + (16/105)*Sqrt[Pi]*Erfi[Sqrt[a + b*x + c*x^2]]} - - -(* ::Section::Closed:: *) -(*Problems from Calculus textbooks*) - - -(* ::Subsection::Closed:: *) -(*Anton Calculus, 4th Edition*) - - -{1/(E^x*Sqrt[1 - E^(-2*x)]), x, 2, -ArcSin[E^(-x)]} -{E^x/(4 + E^(2*x)), x, 2, ArcTan[E^x/2]/2} -{E^x/(1 - E^(2*x)), x, 2, ArcTanh[E^x]} -{E^x/(3 - 4*E^(2*x)), x, 2, ArcTanh[(2*E^x)/Sqrt[3]]/(2*Sqrt[3])} -{E^x*Sqrt[3 - 4*E^(2*x)], x, 3, (1/2)*E^x*Sqrt[3 - 4*E^(2*x)] + (3/4)*ArcSin[(2*E^x)/Sqrt[3]]} -{E^x^2*x^3, x, 2, -(E^x^2/2) + (1/2)*E^x^2*x^2} -{E^x*Sqrt[1 - E^(2*x)], x, 3, (1/2)*E^x*Sqrt[1 - E^(2*x)] + ArcSin[E^x]/2} -{E^x/Sqrt[1 + E^x + E^(2*x)], x, 3, ArcSinh[(1 + 2*E^x)/Sqrt[3]]} -{E^x/(-4 + E^(2*x)), x, 2, -ArcTanh[E^x/2]/2} - - -(* ::Subsection::Closed:: *) -(*Ayres Calculus, 1964 edition*) - - -{E^(2 - x^2)*x, x, 1, -E^(2 - x^2)/2} -{E^x - x^E, x, 2, E^x - x^(1 + E)/(1 + E)} -{(-1 + E^(2*x))/(3 + E^(2*x)), x, 3, -(x/3) + (2/3)*Log[3 + E^(2*x)]} -{E^x/Sqrt[1 - E^(2*x)], x, 2, ArcSin[E^x]} -{E^(2*x)/(1 + E^(4*x)), x, 2, ArcTan[E^(2*x)]/2} -{1/(-3*E^x + E^(2*x)), x, 3, 1/(E^x*3) - x/9 + (1/9)*Log[3 - E^x]} -{(E^x*(-2 + E^x))/(1 + E^x), x, 3, E^x - 3*Log[1 + E^x]} - - -(* ::Subsection::Closed:: *) -(*Edwards and Penney Calculus*) - - -{E^x/(-1 + E^(2*x)), x, 2, -ArcTanh[E^x]} -{E^x/(1 + E^(2*x)), x, 2, ArcTan[E^x]} -{(E^(-x) + E^x)/(-E^(-x) + E^x), x, 4, Log[E^(-x) - E^x], -x + Log[1 - E^(2*x)]} -{(-E^(-x) + E^x)/(E^(-x) + E^x), x, 4, Log[E^(-x) + E^x], -x + Log[1 + E^(2*x)]} -{(E^(-2*x) + E^(2*x))/(-E^(-2*x) + E^(2*x)), x, 4, -x + (1/2)*Log[1 - E^(4*x)]} -{E^x/Sqrt[1 + E^(2*x)], x, 2, ArcSinh[E^x]} -{E^Sqrt[4 + x]/Sqrt[4 + x], x, 1, 2*E^Sqrt[4 + x]} -{x/Sqrt[-1 + E^(2*x^2)], x, 4, ArcTan[Sqrt[-1 + E^(2*x^2)]]/2} -{E^x*Sqrt[9 + E^(2*x)], x, 3, (1/2)*E^x*Sqrt[9 + E^(2*x)] + (9/2)*ArcSinh[E^x/3]} -{E^x*Sqrt[1 + E^(2*x)], x, 3, (1/2)*E^x*Sqrt[1 + E^(2*x)] + ArcSinh[E^x]/2} -{(E^x^2*x)/(1 + E^(2*x^2)), x, 3, ArcTan[E^x^2]/2} -{E^x^(3/2)*x^2, x, 3, (-(2/3))*E^x^(3/2) + (2/3)*E^x^(3/2)*x^(3/2)} - - -(* ::Subsection::Closed:: *) -(*Grossman Calculus*) - - -{E^x/Sqrt[-3 + E^(2*x)], x, 3, ArcTanh[E^x/Sqrt[-3 + E^(2*x)]]} -{E^x/(16 - E^(2*x)), x, 2, ArcTanh[E^x/4]/4} -{E^(5*x)/(1 + E^(10*x)), x, 2, ArcTan[E^(5*x)]/5} -{E^(4*x)/Sqrt[16 + E^(8*x)], x, 2, ArcSinh[E^(4*x)/4]/4} -{E^(4*x^3)*x^2*Cos[7*x^3], x, 2, (4/195)*E^(4*x^3)*Cos[7*x^3] + (7/195)*E^(4*x^3)*Sin[7*x^3]} - - -(* ::Subsection::Closed:: *) -(*Hughes, Hallet, Gleason, et al Calculus, 2nd Edition*) - - -{E^(1 + x^2)*x, x, 1, E^(1 + x^2)/2} -{E^(1 + x^3)*x^2, x, 1, E^(1 + x^3)/3} -{E^Sqrt[x]/Sqrt[x], x, 1, 2*E^Sqrt[x]} -{E^x^(1/3)/x^(2/3), x, 1, 3*E^x^(1/3)} -{E^(3*x)*(-8 + 2*x^3 + x^5), x, 13, -((724*E^(3*x))/243) + (76/81)*E^(3*x)*x - (38/27)*E^(3*x)*x^2 + (38/27)*E^(3*x)*x^3 - (5/9)*E^(3*x)*x^4 + (1/3)*E^(3*x)*x^5} -{(E^x + x)^2, x, 5, -2*E^x + E^(2*x)/2 + 2*E^x*x + x^3/3} - - -(* ::Subsection::Closed:: *) -(*Spivak Calculus*) - - -{(E^x + E^(2*x) + E^(3*x))/E^(4*x), x, 3, -(1/3)/E^(3*x) - 1/(E^(2*x)*2) - E^(-x)} -{E^x/(1 + 2*E^x + E^(2*x)), x, 2, -(1 + E^x)^(-1)} - - -(* ::Subsection::Closed:: *) -(*Stewart Calculus*) - - -{Cos[3*x]/E^x, x, 1, ((-(1/10))*Cos[3*x])/E^x + ((3/10)*Sin[3*x])/E^x} -{E^(2*x)/(2 + 3*E^x + E^(2*x)), x, 4, -Log[1 + E^x] + 2*Log[2 + E^x]} -{E^(2*x)/(1 + E^x), x, 3, E^x - Log[1 + E^x]} -{E^(3*x)*Cos[5*x], x, 1, (3/34)*E^(3*x)*Cos[5*x] + (5/34)*E^(3*x)*Sin[5*x]} -{E^x*Sech[E^x], x, 2, ArcTan[Sinh[E^x]]} -{1/(E^x*(1 + 2*E^x)), x, 3, -E^(-x) - 2*x + 2*Log[1 + 2*E^x]} -{E^x*Cos[4 + 3*x], x, 1, (1/10)*E^x*Cos[4 + 3*x] + (3/10)*E^x*Sin[4 + 3*x]} - - -(* ::Subsection::Closed:: *) -(*Thomas Calculus, 8th Edition*) - - -{E^x*Sec[1 - E^x]^3, x, 3, (-(1/2))*ArcTanh[Sin[1 - E^x]] - (1/2)*Sec[1 - E^x]*Tan[1 - E^x]} -{(E^(-x) + E^x)*x, x, 6, -E^(-x) - E^x - x/E^x + E^x*x} -{E^x/(2 + 3*E^x + E^(2*x)), x, 4, Log[1 + E^x] - Log[2 + E^x]} -{E^(2*x)/(1 + E^x)^(1/3), x, 3, (-(3/2))*(1 + E^x)^(2/3) + (3/5)*(1 + E^x)^(5/3)} -{E^(2*x)/(1 + E^x)^(1/4), x, 3, (-(4/3))*(1 + E^x)^(3/4) + (4/7)*(1 + E^x)^(7/4)} -{(-E^x + 2*E^(2*x))/Sqrt[-1 - 6*E^x + 3*E^(2*x)], x, 4, (2/3)*Sqrt[-1 - 6*E^x + 3*E^(2*x)] - ArcTanh[(Sqrt[3]*(1 - E^x))/Sqrt[-1 - 6*E^x + 3*E^(2*x)]]/Sqrt[3]} - - -(* ::Section::Closed:: *) -(*Problems from integration competitions*) - - -(* ::Subsection::Closed:: *) -(*MIT Integration Competition*) - - -{E^x*(-5*x + x^2), x, 8, 7*E^x - 7*E^x*x + E^x*x^2} -{E^(3*x)*(-x + x^2), x, 8, (5*E^(3*x))/27 - (5/9)*E^(3*x)*x + (1/3)*E^(3*x)*x^2} - - -(* ::Subsection::Closed:: *) -(*University of Wisconsin Integration Competition*) - - -{E^x^x*x^(2*x)*(1 + Log[x]), x, -2, E^x^x*(-1 + x^x)} -{(E^(5*x) + E^(7*x))/(E^(-x) + E^x), x, 2, E^(6*x)/6} -{x^(-2 - x^(-1))*(1 - Log[x]), x, -2, -x^(-x^(-1))} - - -(* ::Section::Closed:: *) -(*Miscellaneous problems*) - - -(* Note: Apart should NOT be used to expand integrands of this form! *) -{(a + b*E^x)^2, x, 3, 2*a*b*E^x + (1/2)*b^2*E^(2*x) + a^2*x} -{(a + b*E^x)^3, x, 3, 3*a^2*b*E^x + (3/2)*a*b^2*E^(2*x) + (1/3)*b^3*E^(3*x) + a^3*x} -{(a + b*E^x)^4, x, 3, 4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4/3)*a*b^3*E^(3*x) + (1/4)*b^4*E^(4*x) + a^4*x} - -{1/Sqrt[a + b*E^(c + d*x)], x, 3, -((2*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/(Sqrt[a]*d))} -{1/Sqrt[-a + b*E^(c + d*x)], x, 3, (2*ArcTan[Sqrt[-a + b*E^(c + d*x)]/Sqrt[a]])/(Sqrt[a]*d)} - -{Sqrt[a + b*E^(c + d*x)], x, 4, (2*Sqrt[a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/d} -{Sqrt[-a + b*E^(c + d*x)], x, 4, (2*Sqrt[-a + b*E^(c + d*x)])/d - (2*Sqrt[a]*ArcTan[Sqrt[-a + b*E^(c + d*x)]/Sqrt[a]])/d} - - -{E^(6*x)*Sin[3*x], x, 1, (-(1/15))*E^(6*x)*Cos[3*x] + (2/15)*E^(6*x)*Sin[3*x]} -{E^(3*x)/(1 + E^(2*x)), x, 3, E^x - ArcTan[E^x]} -{E^(3*x)/(-1 + E^(2*x)), x, 3, E^x - ArcTanh[E^x]} -{1/(E^x*Sqrt[1 + E^(2*x)]), x, 2, -(Sqrt[1 + E^(2*x)]/E^x)} - - -{E^x/(-1 - 8*E^x + E^(2*x)), x, 3, ArcTanh[(4 - E^x)/Sqrt[17]]/Sqrt[17]} -{E^(7*x)*x^3, x, 4, -((6*E^(7*x))/2401) + (6/343)*E^(7*x)*x - (3/49)*E^(7*x)*x^2 + (1/7)*E^(7*x)*x^3} -{E^(8 - 2*x)*x^3, x, 4, (-(3/8))*E^(8 - 2*x) - (3/4)*E^(8 - 2*x)*x - (3/4)*E^(8 - 2*x)*x^2 - (1/2)*E^(8 - 2*x)*x^3} -{E^x*Sqrt[9 - E^(2*x)], x, 3, (1/2)*E^x*Sqrt[9 - E^(2*x)] + (9/2)*ArcSin[E^x/3]} -{E^(6*x)*Sqrt[9 - E^(2*x)], x, 3, -27*(9 - E^(2*x))^(3/2) + (18/5)*(9 - E^(2*x))^(5/2) - (1/7)*(9 - E^(2*x))^(7/2)} -{E^(6*x)/(9 - E^x)^(5/2), x, 3, 39366/(9 - E^x)^(3/2) - 65610/Sqrt[9 - E^x] - 14580*Sqrt[9 - E^x] + 540*(9 - E^x)^(3/2) - 18*(9 - E^x)^(5/2) + (2/7)*(9 - E^x)^(7/2)} -{(2 - 7*E^x^4)^5*x^3, x, 4, -140*E^x^4 + 490*E^(2*x^4) - (3430*E^(3*x^4))/3 + (12005*E^(4*x^4))/8 - (16807*E^(5*x^4))/20 + 8*x^4} -{E^x^2*Sqrt[1 - E^(2*x^2)]*x, x, 4, (1/4)*E^x^2*Sqrt[1 - E^(2*x^2)] + (1/4)*ArcSin[E^x^2]} -{E^x^3*(1 - E^(4*x^3))^2*x^2, x, 4, E^x^3/3 - (2*E^(5*x^3))/15 + E^(9*x^3)/27} -{E^(E^x + x), x, 2, E^E^x} -{E^(E^E^x + E^x + x), x, 3, E^E^E^x} - - -{(E^(-x) + E^x)^2, x, 4, -(1/2)/E^(2*x) + E^(2*x)/2 + 2*x} -{1/(E^(-x) + E^x), x, 2, ArcTan[E^x]} -{1/(E^(-x) + E^x)^2, x, 2, -1/(2*(1 + E^(2*x)))} - - -{1/(-E^(-x) + E^x), x, 2, -ArcTanh[E^x]} -{1/(-E^(-x) + E^x)^2, x, 2, 1/(2*(1 - E^(2*x)))} - - -{E^x*(-E^(-x) + E^x)^2, x, 3, -E^(-x) - 2*E^x + E^(3*x)/3} -{E^x*(-E^(-x) + E^x)^3, x, 4, 1/(E^(2*x)*2) - (3*E^(2*x))/2 + E^(4*x)/4 + 3*x} - - -{(1 + 4^x)/(1 + 2^x), x, 3, x + 2^x/Log[2] - (2*Log[1 + 2^x])/Log[2]} -{(1 + 4^x)/(1 + 2^(-x)), x, 3, -(2^x/Log[2]) + 2^(-1 + 2*x)/Log[2] + (2*Log[1 + 2^x])/Log[2]} - -{E^(a + x)^2/x^2 - (2*a*E^(a + x)^2)/x, x, 3, -(E^(a + x)^2/x) + Sqrt[Pi]*Erfi[a + x]} -{(x^4 + x^6 + x^8)/E^x^2, x, 15, ((-(147/16))*x)/E^x^2 - ((49/8)*x^3)/E^x^2 - ((9/4)*x^5)/E^x^2 - ((1/2)*x^7)/E^x^2 + (147/32)*Sqrt[Pi]*Erf[x]} - -{1/(-E^x + E^(3*x)), x, 3, E^(-x) - ArcTanh[E^x]} -{(E^x*(-5 + x + x^2))/(-1 + x)^2, x, 6, E^x - (3*E^x)/(1 - x)} -{(E^x^2*x^3)/(1 + x^2)^2, x, 1, E^x^2/(2*(1 + x^2))} -{E^(3*x)/Sqrt[25 + 16*E^(2*x)], x, 3, (1/32)*E^x*Sqrt[25 + 16*E^(2*x)] - (25/128)*ArcSinh[(4*E^x)/5]} - -(* {E^(a + b*x + c*x^2)/(d + e*x)^2, x, 0} *) -{(1 + E^x)/Sqrt[E^x + x], x, 1, 2*Sqrt[E^x + x]} -{(1 + E^x)/(E^x + x), x, 1, Log[E^x + x]} -{E^x^2/x^2, x, 2, -(E^x^2/x) + Sqrt[Pi]*Erfi[x]} -{(E^x^2*(1 + 4*x^4))/x^2, x, 6, -(E^x^2/x) + 2*E^x^2*x} - -{Sqrt[f^x]*(a + b*x)^2, x, 3, (16*b^2*Sqrt[f^x])/Log[f]^3 - (8*b*Sqrt[f^x]*(a + b*x))/Log[f]^2 + (2*Sqrt[f^x]*(a + b*x)^2)/Log[f]} - -{3^(1 + x^2)*x, x, 1, 3^(1 + x^2)/(2*Log[3])} -{2^Sqrt[x]/Sqrt[x], x, 1, 2^(1 + Sqrt[x])/Log[2]} -{2^x^(-1)/x^2, x, 1, -(2^x^(-1)/Log[2])} -{2^(-x) + 2^x, x, 3, -(1/(2^x*Log[2])) + 2^x/Log[2]} -{(2 - 3*x + x^2)/E^(4*x), x, 8, -(11/32)/E^(4*x) + ((5/8)*x)/E^(4*x) - ((1/4)*x^2)/E^(4*x)} -{k^(x/2) + x^Sqrt[k], x, 2, x^(1 + Sqrt[k])/(1 + Sqrt[k]) + (2*k^(x/2))/Log[k]} -{10^Sqrt[x]/Sqrt[x], x, 1, (2^(1 + Sqrt[x])*5^Sqrt[x])/Log[10]} - - -(* Problems requiring simplification of irreducible integrands *) -{E^x/(E^x + x)^(1/2) + 1/Sqrt[E^x + x], x, 2, 2*Sqrt[E^x + x]} -{((1 + E^x)*x)/Sqrt[E^x + x] + 2*Sqrt[E^x + x], x, 6, 2*x*Sqrt[E^x + x]} -{x/Sqrt[E^x + x] + (E^x*x)/Sqrt[E^x + x] + 2*Sqrt[E^x + x], x, 4, 2*x*Sqrt[E^x + x]} -{((1 + E^x)*x)/Sqrt[E^x + x], x, 5, 2*x*Sqrt[E^x + x] - 2*CannotIntegrate[Sqrt[E^x + x], x]} -{x/Sqrt[E^x + x] + (E^x*x)/Sqrt[E^x + x], x, 4, 2*x*Sqrt[E^x + x] - 2*CannotIntegrate[Sqrt[E^x + x], x]} -{x*E^x/Sqrt[E^x + x], x, 2, 2*Sqrt[E^x + x] + 2*x*Sqrt[E^x + x] - CannotIntegrate[1/Sqrt[E^x + x], x] - 3*CannotIntegrate[Sqrt[E^x + x], x]} - -{(x^2*(5*E^x + 3*x^2))/(5*Sqrt[5*E^x + x^3]) + (4/5)*x*Sqrt[5*E^x + x^3], x, 4, (2/5)*x^2*Sqrt[5*E^x + x^3]} -{x^2*E^x/Sqrt[5*E^x + x^3], x, 1, (2/5)*x^2*Sqrt[5*E^x + x^3] - (3/5)*CannotIntegrate[x^4/Sqrt[5*E^x + x^3], x] - (4/5)*CannotIntegrate[x*Sqrt[5*E^x + x^3], x]} - -{-((1 + E^x)/(E^x + x)^(1/3)), x, 1, (-(3/2))*(E^x + x)^(2/3)} -{-(1/(E^x + x)^(1/3)) + x/(E^x + x)^(1/3) - (E^x + x)^(2/3), x, 2, (-(3/2))*(E^x + x)^(2/3)} -{x/(E^x + x)^(1/3), x, 1, (-(3/2))*(E^x + x)^(2/3) + CannotIntegrate[1/(E^x + x)^(1/3), x] + CannotIntegrate[(E^x + x)^(2/3), x]} - -{(5*x + E^x*(3 + 2*x))/(E^x + x)^(1/3), x, 8, 3*x*(E^x + x)^(2/3)} -{(2*x)/(E^x + x)^(1/3) + (2*E^x*x)/(E^x + x)^(1/3) + 3*(E^x + x)^(2/3), x, 4, 3*x*(E^x + x)^(2/3)} - - -{E^x*(-E^(-x) + E^x)*(E^(-x) + E^x)^2, x, 3, 1/(E^(2*x)*2) + E^(2*x)/2 + E^(4*x)/4 - x} - - -(* Unwise expansion leads to infinite recursion *) -{x/(E^x + x), x, 0, CannotIntegrate[x/(E^x + x), x]} -{x^2/Sqrt[E^x + x], x, 0, CannotIntegrate[x^2/Sqrt[E^x + x], x]} -{E^x/(E^x + x), x, 0, CannotIntegrate[E^x/(E^x + x), x]} -{E^x/(E^x + x^2), x, 0, CannotIntegrate[E^x/(E^x + x^2), x]} - -{F0[x]/(F0[x] + x), x, 2, x - CannotIntegrate[x/(x + F0[x]), x]} -{F0[x]/(F0[x] + x^2), x, 2, x - CannotIntegrate[x^2/(x^2 + F0[x]), x]} -{F0[x]/(F0[x] + x)^2, x, 2, -CannotIntegrate[x/(x + F0[x])^2, x] + CannotIntegrate[1/(x + F0[x]), x]} -{F0[x]/(F0[x] + x^2)^2, x, 2, -CannotIntegrate[x^2/(x^2 + F0[x])^2, x] + CannotIntegrate[1/(x^2 + F0[x]), x]} - - -{(a*F^(c + d*x))^m*(b*F^(e + f*x))^n, x, 4, ((a*F^(c + d*x))^m*(b*F^(e + f*x))^n)/((d*m + f*n)*Log[F])} - - -{E^(a + b*x^n)*E^(c + d*x^n), x, 2, -((E^(a + c)*x*Gamma[1/n, -((b + d)*x^n)])/((-((b + d)*x^n))^n^(-1)*n))} -{f^(a + b*x^n)*g^(c + d*x^n), x, 2, -((f^a*g^c*x*Gamma[1/n, (-x^n)*(b*Log[f] + d*Log[g])])/(((-x^n)*(b*Log[f] + d*Log[g]))^n^(-1)*n))} - - -{x^m*E^(x^n), x, 1, -((x^(1 + m)*Gamma[(1 + m)/n, -x^n])/((-x^n)^((1 + m)/n)*n))} -{x^m*f^(x^n), x, 1, -((x^(1 + m)*Gamma[(1 + m)/n, (-x^n)*Log[f]])/(((-x^n)*Log[f])^((1 + m)/n)*n))} - -{(a + b*x)^m*E^((a + b*x)^n), x, 1, -(((a + b*x)^(1 + m)*Gamma[(1 + m)/n, -(a + b*x)^n])/((-(a + b*x)^n)^((1 + m)/n)*(b*n)))} -{(a + b*x)^m*f^((a + b*x)^n), x, 1, -(((a + b*x)^(1 + m)*Gamma[(1 + m)/n, (-(a + b*x)^n)*Log[f]])/(((-(a + b*x)^n)*Log[f])^((1 + m)/n)*(b*n)))} - - -(* Contributed by Oleg Marichev, Wolfram Research *) -{x*E^(a + b*x)^3, x, 4, (a*(a + b*x)*Gamma[1/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(1/3)) - ((a + b*x)^2*Gamma[2/3, -(a + b*x)^3])/(3*b^2*(-(a + b*x)^3)^(2/3))} - - -(* Problem posted on Maple Primes on 1 June 2017 *) -{(5*x^2 + 3*(x + E^x)^(1/3) + E^x*(2*x^2 + 3*x))/(x*(x + E^x)^(1/3)), x, 8, 3*x*(E^x + x)^(2/3) + 3*Log[x]} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m deleted file mode 100644 index f40cce6..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.1.2 (d x)^m (a+b log(c x^n))^p.m +++ /dev/null @@ -1,324 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b Log[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Log[c x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Log[c*x], x, 1, -x^4/16 + (x^4*Log[c*x])/4} -{x^2*Log[c*x], x, 1, -x^3/9 + (x^3*Log[c*x])/3} -{x^1*Log[c*x], x, 1, -x^2/4 + (x^2*Log[c*x])/2} -{x^0*Log[c*x], x, 1, -x + x*Log[c*x]} -{Log[c*x]/x^1, x, 1, Log[c*x]^2/2} -{Log[c*x]/x^2, x, 1, -x^(-1) - Log[c*x]/x} -{Log[c*x]/x^3, x, 1, -1/(4*x^2) - Log[c*x]/(2*x^2)} - - -{x^3*Log[c*x]^2, x, 2, x^4/32 - (x^4*Log[c*x])/8 + (x^4*Log[c*x]^2)/4} -{x^2*Log[c*x]^2, x, 2, (2*x^3)/27 - (2*x^3*Log[c*x])/9 + (x^3*Log[c*x]^2)/3} -{x^1*Log[c*x]^2, x, 2, x^2/4 - (x^2*Log[c*x])/2 + (x^2*Log[c*x]^2)/2} -{x^0*Log[c*x]^2, x, 2, 2*x - 2*x*Log[c*x] + x*Log[c*x]^2} -{Log[c*x]^2/x^1, x, 2, Log[c*x]^3/3} -{Log[c*x]^2/x^2, x, 2, -2/x - (2*Log[c*x])/x - Log[c*x]^2/x} -{Log[c*x]^2/x^3, x, 2, -1/(4*x^2) - Log[c*x]/(2*x^2) - Log[c*x]^2/(2*x^2)} - - -{x^3*Log[c*x]^3, x, 3, (-3*x^4)/128 + (3*x^4*Log[c*x])/32 - (3*x^4*Log[c*x]^2)/16 + (x^4*Log[c*x]^3)/4} -{x^2*Log[c*x]^3, x, 3, (-2*x^3)/27 + (2*x^3*Log[c*x])/9 - (x^3*Log[c*x]^2)/3 + (x^3*Log[c*x]^3)/3} -{x^1*Log[c*x]^3, x, 3, (-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2} -{x^0*Log[c*x]^3, x, 3, -6*x + 6*x*Log[c*x] - 3*x*Log[c*x]^2 + x*Log[c*x]^3} -{Log[c*x]^3/x^1, x, 2, Log[c*x]^4/4} -{Log[c*x]^3/x^2, x, 3, -6/x - (6*Log[c*x])/x - (3*Log[c*x]^2)/x - Log[c*x]^3/x} -{Log[c*x]^3/x^3, x, 3, -3/(8*x^2) - (3*Log[c*x])/(4*x^2) - (3*Log[c*x]^2)/(4*x^2) - Log[c*x]^3/(2*x^2)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/Log[c*x], x, 2, ExpIntegralEi[4*Log[c*x]]/c^4} -{x^2/Log[c*x], x, 2, ExpIntegralEi[3*Log[c*x]]/c^3} -{x^1/Log[c*x], x, 2, ExpIntegralEi[2*Log[c*x]]/c^2} -{x^0/Log[c*x], x, 1, LogIntegral[c*x]/c} -{1/(x^1*Log[c*x]), x, 2, Log[Log[c*x]]} -{1/(x^2*Log[c*x]), x, 2, c*ExpIntegralEi[-Log[c*x]]} -{1/(x^3*Log[c*x]), x, 2, c^2*ExpIntegralEi[-2*Log[c*x]]} - - -{x^3/Log[c*x]^2, x, 3, (4*ExpIntegralEi[4*Log[c*x]])/c^4 - x^4/Log[c*x]} -{x^2/Log[c*x]^2, x, 3, (3*ExpIntegralEi[3*Log[c*x]])/c^3 - x^3/Log[c*x]} -{x^1/Log[c*x]^2, x, 3, (2*ExpIntegralEi[2*Log[c*x]])/c^2 - x^2/Log[c*x]} -{x^0/Log[c*x]^2, x, 2, -(x/Log[c*x]) + LogIntegral[c*x]/c} -{1/(x^1*Log[c*x]^2), x, 2, -Log[c*x]^(-1)} -{1/(x^2*Log[c*x]^2), x, 3, -(c*ExpIntegralEi[-Log[c*x]]) - 1/(x*Log[c*x])} -{1/(x^3*Log[c*x]^2), x, 3, -2*c^2*ExpIntegralEi[-2*Log[c*x]] - 1/(x^2*Log[c*x])} - - -{x^3/Log[c*x]^3, x, 4, (8*ExpIntegralEi[4*Log[c*x]])/c^4 - x^4/(2*Log[c*x]^2) - (2*x^4)/Log[c*x]} -{x^2/Log[c*x]^3, x, 4, (9*ExpIntegralEi[3*Log[c*x]])/(2*c^3) - x^3/(2*Log[c*x]^2) - (3*x^3)/(2*Log[c*x])} -{x^1/Log[c*x]^3, x, 4, (2*ExpIntegralEi[2*Log[c*x]])/c^2 - x^2/(2*Log[c*x]^2) - x^2/Log[c*x]} -{x^0/Log[c*x]^3, x, 3, -(x/(2*Log[c*x]^2)) - x/(2*Log[c*x]) + LogIntegral[c*x]/(2*c)} -{1/(x^1*Log[c*x]^3), x, 2, -1/(2*Log[c*x]^2)} -{1/(x^2*Log[c*x]^3), x, 4, (1/2)*c*ExpIntegralEi[-Log[c*x]] - 1/(2*x*Log[c*x]^2) + 1/(2*x*Log[c*x])} -{1/(x^3*Log[c*x]^3), x, 4, 2*c^2*ExpIntegralEi[-2*Log[c*x]] - 1/(2*x^2*Log[c*x]^2) + 1/(x^2*Log[c*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b Log[c x^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Log[c*x^n]), x, 1, -(b*n*x^4)/16 + (x^4*(a + b*Log[c*x^n]))/4} -{x^2*(a + b*Log[c*x^n]), x, 1, -(b*n*x^3)/9 + (x^3*(a + b*Log[c*x^n]))/3} -{x^1*(a + b*Log[c*x^n]), x, 1, -(b*n*x^2)/4 + (x^2*(a + b*Log[c*x^n]))/2} -{x^0*(a + b*Log[c*x^n]), x, 2, a*x - b*n*x + b*x*Log[c*x^n]} -{(a + b*Log[c*x^n])/x^1, x, 1, (a + b*Log[c*x^n])^2/(2*b*n)} -{(a + b*Log[c*x^n])/x^2, x, 1, -((b*n)/x) - (a + b*Log[c*x^n])/x} -{(a + b*Log[c*x^n])/x^3, x, 1, -(b*n)/(4*x^2) - (a + b*Log[c*x^n])/(2*x^2)} - - -{x^3*(a + b*Log[c*x^n])^2, x, 2, (b^2*n^2*x^4)/32 - (b*n*x^4*(a + b*Log[c*x^n]))/8 + (x^4*(a + b*Log[c*x^n])^2)/4} -{x^2*(a + b*Log[c*x^n])^2, x, 2, (2*b^2*n^2*x^3)/27 - (2*b*n*x^3*(a + b*Log[c*x^n]))/9 + (x^3*(a + b*Log[c*x^n])^2)/3} -{x^1*(a + b*Log[c*x^n])^2, x, 2, (b^2*n^2*x^2)/4 - (b*n*x^2*(a + b*Log[c*x^n]))/2 + (x^2*(a + b*Log[c*x^n])^2)/2} -{x^0*(a + b*Log[c*x^n])^2, x, 3, -2*a*b*n*x + 2*b^2*n^2*x - 2*b^2*n*x*Log[c*x^n] + x*(a + b*Log[c*x^n])^2} -{(a + b*Log[c*x^n])^2/x^1, x, 2, (a + b*Log[c*x^n])^3/(3*b*n)} -{(a + b*Log[c*x^n])^2/x^2, x, 2, (-2*b^2*n^2)/x - (2*b*n*(a + b*Log[c*x^n]))/x - (a + b*Log[c*x^n])^2/x} -{(a + b*Log[c*x^n])^2/x^3, x, 2, -(b^2*n^2)/(4*x^2) - (b*n*(a + b*Log[c*x^n]))/(2*x^2) - (a + b*Log[c*x^n])^2/(2*x^2)} - - -{x^3*(a + b*Log[c*x^n])^3, x, 3, (-3*b^3*n^3*x^4)/128 + (3*b^2*n^2*x^4*(a + b*Log[c*x^n]))/32 - (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 + (x^4*(a + b*Log[c*x^n])^3)/4} -{x^2*(a + b*Log[c*x^n])^3, x, 3, (-2*b^3*n^3*x^3)/27 + (2*b^2*n^2*x^3*(a + b*Log[c*x^n]))/9 - (b*n*x^3*(a + b*Log[c*x^n])^2)/3 + (x^3*(a + b*Log[c*x^n])^3)/3} -{x^1*(a + b*Log[c*x^n])^3, x, 3, (-3*b^3*n^3*x^2)/8 + (3*b^2*n^2*x^2*(a + b*Log[c*x^n]))/4 - (3*b*n*x^2*(a + b*Log[c*x^n])^2)/4 + (x^2*(a + b*Log[c*x^n])^3)/2} -{x^0*(a + b*Log[c*x^n])^3, x, 4, 6*a*b^2*n^2*x - 6*b^3*n^3*x + 6*b^3*n^2*x*Log[c*x^n] - 3*b*n*x*(a + b*Log[c*x^n])^2 + x*(a + b*Log[c*x^n])^3} -{(a + b*Log[c*x^n])^3/x^1, x, 2, (a + b*Log[c*x^n])^4/(4*b*n)} -{(a + b*Log[c*x^n])^3/x^2, x, 3, (-6*b^3*n^3)/x - (6*b^2*n^2*(a + b*Log[c*x^n]))/x - (3*b*n*(a + b*Log[c*x^n])^2)/x - (a + b*Log[c*x^n])^3/x} -{(a + b*Log[c*x^n])^3/x^3, x, 3, (-3*b^3*n^3)/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n]))/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2)/(4*x^2) - (a + b*Log[c*x^n])^3/(2*x^2)} -{(a + b*Log[c*x^n])^3/x^4, x, 3, (-2*b^3*n^3)/(27*x^3) - (2*b^2*n^2*(a + b*Log[c*x^n]))/(9*x^3) - (b*n*(a + b*Log[c*x^n])^2)/(3*x^3) - (a + b*Log[c*x^n])^3/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Log[c*x^n]), x, 2, (x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((4*a)/(b*n))*n*(c*x^n)^(4/n))} -{x^2/(a + b*Log[c*x^n]), x, 2, (x^3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((3*a)/(b*n))*n*(c*x^n)^(3/n))} -{x^1/(a + b*Log[c*x^n]), x, 2, (x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b*E^((2*a)/(b*n))*n*(c*x^n)^(2/n))} -{x^0/(a + b*Log[c*x^n]), x, 2, (x*ExpIntegralEi[(a + b*Log[c*x^n])/(b*n)])/(b*E^(a/(b*n))*n*(c*x^n)^n^(-1))} -{1/(x^1*(a + b*Log[c*x^n])), x, 2, Log[a + b*Log[c*x^n]]/(b*n)} -{1/(x^2*(a + b*Log[c*x^n])), x, 2, (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b*n*x)} -{1/(x^3*(a + b*Log[c*x^n])), x, 2, (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b*n)])/(b*n*x^2)} -{1/(x^4*(a + b*Log[c*x^n])), x, 2, (E^((3*a)/(b*n))*(c*x^n)^(3/n)*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(b*n)])/(b*n*x^3)} - - -{x^3/(a + b*Log[c*x^n])^2, x, 3, (4*x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b^2*E^((4*a)/(b*n))*n^2*(c*x^n)^(4/n)) - x^4/(b*n*(a + b*Log[c*x^n]))} -{x^2/(a + b*Log[c*x^n])^2, x, 3, (3*x^3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(b^2*E^((3*a)/(b*n))*n^2*(c*x^n)^(3/n)) - x^3/(b*n*(a + b*Log[c*x^n]))} -{x^1/(a + b*Log[c*x^n])^2, x, 3, (2*x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b^2*E^((2*a)/(b*n))*n^2*(c*x^n)^(2/n)) - x^2/(b*n*(a + b*Log[c*x^n]))} -{x^0/(a + b*Log[c*x^n])^2, x, 3, (x*ExpIntegralEi[(a + b*Log[c*x^n])/(b*n)])/(b^2*E^(a/(b*n))*n^2*(c*x^n)^n^(-1)) - x/(b*n*(a + b*Log[c*x^n]))} -{1/(x^1*(a + b*Log[c*x^n])^2), x, 2, -(1/(b*n*(a + b*Log[c*x^n])))} -{1/(x^2*(a + b*Log[c*x^n])^2), x, 3, -((E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/(b^2*n^2*x)) - 1/(b*n*x*(a + b*Log[c*x^n]))} -{1/(x^3*(a + b*Log[c*x^n])^2), x, 3, (-2*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b*n)])/(b^2*n^2*x^2) - 1/(b*n*x^2*(a + b*Log[c*x^n]))} -{1/(x^4*(a + b*Log[c*x^n])^2), x, 3, (-3*E^((3*a)/(b*n))*(c*x^n)^(3/n)*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(b*n)])/(b^2*n^2*x^3) - 1/(b*n*x^3*(a + b*Log[c*x^n]))} - - -{x^3/(a + b*Log[c*x^n])^3, x, 4, (8*x^4*ExpIntegralEi[(4*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((4*a)/(b*n))*n^3*(c*x^n)^(4/n)) - x^4/(2*b*n*(a + b*Log[c*x^n])^2) - (2*x^4)/(b^2*n^2*(a + b*Log[c*x^n]))} -{x^2/(a + b*Log[c*x^n])^3, x, 4, (9*x^3*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(b*n)])/(2*b^3*E^((3*a)/(b*n))*n^3*(c*x^n)^(3/n)) - x^3/(2*b*n*(a + b*Log[c*x^n])^2) - (3*x^3)/(2*b^2*n^2*(a + b*Log[c*x^n]))} -{x^1/(a + b*Log[c*x^n])^3, x, 4, (2*x^2*ExpIntegralEi[(2*(a + b*Log[c*x^n]))/(b*n)])/(b^3*E^((2*a)/(b*n))*n^3*(c*x^n)^(2/n)) - x^2/(2*b*n*(a + b*Log[c*x^n])^2) - x^2/(b^2*n^2*(a + b*Log[c*x^n]))} -{x^0/(a + b*Log[c*x^n])^3, x, 4, (x*ExpIntegralEi[(a + b*Log[c*x^n])/(b*n)])/(2*b^3*E^(a/(b*n))*n^3*(c*x^n)^n^(-1)) - x/(2*b*n*(a + b*Log[c*x^n])^2) - x/(2*b^2*n^2*(a + b*Log[c*x^n]))} -{1/(x^1*(a + b*Log[c*x^n])^3), x, 2, -1/(2*b*n*(a + b*Log[c*x^n])^2)} -{1/(x^2*(a + b*Log[c*x^n])^3), x, 4, (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-((a + b*Log[c*x^n])/(b*n))])/(2*b^3*n^3*x) - 1/(2*b*n*x*(a + b*Log[c*x^n])^2) + 1/(2*b^2*n^2*x*(a + b*Log[c*x^n]))} -{1/(x^3*(a + b*Log[c*x^n])^3), x, 4, (2*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[(-2*(a + b*Log[c*x^n]))/(b*n)])/(b^3*n^3*x^2) - 1/(2*b*n*x^2*(a + b*Log[c*x^n])^2) + 1/(b^2*n^2*x^2*(a + b*Log[c*x^n]))} -{1/(x^4*(a + b*Log[c*x^n])^3), x, 4, (9*E^((3*a)/(b*n))*(c*x^n)^(3/n)*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(b*n)])/(2*b^3*n^3*x^3) - 1/(2*b*n*x^3*(a + b*Log[c*x^n])^2) + 3/(2*b^2*n^2*x^3*(a + b*Log[c*x^n]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b Log[c x^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(d*x)^(5/2)*(a + b*Log[c*x^n]), x, 1, (-4*b*n*(d*x)^(7/2))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(7*d)} -{(d*x)^(3/2)*(a + b*Log[c*x^n]), x, 1, (-4*b*n*(d*x)^(5/2))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(5*d)} -{(d*x)^(1/2)*(a + b*Log[c*x^n]), x, 1, (-4*b*n*(d*x)^(3/2))/(9*d) + (2*(d*x)^(3/2)*(a + b*Log[c*x^n]))/(3*d)} -{(a + b*Log[c*x^n])/(d*x)^(1/2), x, 1, (-4*b*n*Sqrt[d*x])/d + (2*Sqrt[d*x]*(a + b*Log[c*x^n]))/d} -{(a + b*Log[c*x^n])/(d*x)^(3/2), x, 1, (-4*b*n)/(d*Sqrt[d*x]) - (2*(a + b*Log[c*x^n]))/(d*Sqrt[d*x])} -{(a + b*Log[c*x^n])/(d*x)^(5/2), x, 1, (-4*b*n)/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n]))/(3*d*(d*x)^(3/2))} - - -{(d*x)^(5/2)*(a + b*Log[c*x^n])^2, x, 2, (16*b^2*n^2*(d*x)^(7/2))/(343*d) - (8*b*n*(d*x)^(7/2)*(a + b*Log[c*x^n]))/(49*d) + (2*(d*x)^(7/2)*(a + b*Log[c*x^n])^2)/(7*d)} -{(d*x)^(3/2)*(a + b*Log[c*x^n])^2, x, 2, (16*b^2*n^2*(d*x)^(5/2))/(125*d) - (8*b*n*(d*x)^(5/2)*(a + b*Log[c*x^n]))/(25*d) + (2*(d*x)^(5/2)*(a + b*Log[c*x^n])^2)/(5*d)} -{(d*x)^(1/2)*(a + b*Log[c*x^n])^2, x, 2, (16*b^2*n^2*(d*x)^(3/2))/(27*d) - (8*b*n*(d*x)^(3/2)*(a + b*Log[c*x^n]))/(9*d) + (2*(d*x)^(3/2)*(a + b*Log[c*x^n])^2)/(3*d)} -{(a + b*Log[c*x^n])^2/(d*x)^(1/2), x, 2, (16*b^2*n^2*Sqrt[d*x])/d - (8*b*n*Sqrt[d*x]*(a + b*Log[c*x^n]))/d + (2*Sqrt[d*x]*(a + b*Log[c*x^n])^2)/d} -{(a + b*Log[c*x^n])^2/(d*x)^(3/2), x, 2, (-16*b^2*n^2)/(d*Sqrt[d*x]) - (8*b*n*(a + b*Log[c*x^n]))/(d*Sqrt[d*x]) - (2*(a + b*Log[c*x^n])^2)/(d*Sqrt[d*x])} -{(a + b*Log[c*x^n])^2/(d*x)^(5/2), x, 2, (-16*b^2*n^2)/(27*d*(d*x)^(3/2)) - (8*b*n*(a + b*Log[c*x^n]))/(9*d*(d*x)^(3/2)) - (2*(a + b*Log[c*x^n])^2)/(3*d*(d*x)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(d*x)^(5/2)/(a + b*Log[c*x^n]), x, 2, ((d*x)^(7/2)*ExpIntegralEi[(7*(a + b*Log[c*x^n]))/(2*b*n)])/(b*d*E^((7*a)/(2*b*n))*n*(c*x^n)^(7/(2*n)))} -{(d*x)^(3/2)/(a + b*Log[c*x^n]), x, 2, ((d*x)^(5/2)*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(b*d*E^((5*a)/(2*b*n))*n*(c*x^n)^(5/(2*n)))} -{(d*x)^(1/2)/(a + b*Log[c*x^n]), x, 2, ((d*x)^(3/2)*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(2*b*n)])/(b*d*E^((3*a)/(2*b*n))*n*(c*x^n)^(3/(2*n)))} -{1/((d*x)^(1/2)*(a + b*Log[c*x^n])), x, 2, (Sqrt[d*x]*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(b*d*E^(a/(2*b*n))*n*(c*x^n)^(1/(2*n)))} -{1/((d*x)^(3/2)*(a + b*Log[c*x^n])), x, 2, (E^(a/(2*b*n))*(c*x^n)^(1/(2*n))*ExpIntegralEi[-(a + b*Log[c*x^n])/(2*b*n)])/(b*d*n*Sqrt[d*x])} -{1/((d*x)^(5/2)*(a + b*Log[c*x^n])), x, 2, (E^((3*a)/(2*b*n))*(c*x^n)^(3/(2*n))*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(2*b*n)])/(b*d*n*(d*x)^(3/2))} - - -{(d*x)^(5/2)/(a + b*Log[c*x^n])^2, x, 3, (7*(d*x)^(7/2)*ExpIntegralEi[(7*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*E^((7*a)/(2*b*n))*n^2*(c*x^n)^(7/(2*n))) - (d*x)^(7/2)/(b*d*n*(a + b*Log[c*x^n]))} -{(d*x)^(3/2)/(a + b*Log[c*x^n])^2, x, 3, (5*(d*x)^(5/2)*ExpIntegralEi[(5*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*E^((5*a)/(2*b*n))*n^2*(c*x^n)^(5/(2*n))) - (d*x)^(5/2)/(b*d*n*(a + b*Log[c*x^n]))} -{(d*x)^(1/2)/(a + b*Log[c*x^n])^2, x, 3, (3*(d*x)^(3/2)*ExpIntegralEi[(3*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*E^((3*a)/(2*b*n))*n^2*(c*x^n)^(3/(2*n))) - (d*x)^(3/2)/(b*d*n*(a + b*Log[c*x^n]))} -{1/((d*x)^(1/2)*(a + b*Log[c*x^n])^2), x, 3, (Sqrt[d*x]*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(2*b^2*d*E^(a/(2*b*n))*n^2*(c*x^n)^(1/(2*n))) - Sqrt[d*x]/(b*d*n*(a + b*Log[c*x^n]))} -{1/((d*x)^(3/2)*(a + b*Log[c*x^n])^2), x, 3, -(E^(a/(2*b*n))*(c*x^n)^(1/(2*n))*ExpIntegralEi[-(a + b*Log[c*x^n])/(2*b*n)])/(2*b^2*d*n^2*Sqrt[d*x]) - 1/(b*d*n*Sqrt[d*x]*(a + b*Log[c*x^n]))} -{1/((d*x)^(5/2)*(a + b*Log[c*x^n])^2), x, 3, (-3*E^((3*a)/(2*b*n))*(c*x^n)^(3/(2*n))*ExpIntegralEi[(-3*(a + b*Log[c*x^n]))/(2*b*n)])/(2*b^2*d*n^2*(d*x)^(3/2)) - 1/(b*d*n*(d*x)^(3/2)*(a + b*Log[c*x^n]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b Log[c x^n])^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Log[c*x^n])^(1/2), x, 4, ((-(1/2))*Sqrt[b]*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) + x*Sqrt[a + b*Log[c*x^n]]} - - -{x^3*Sqrt[Log[a*x^n]], x, 4, ((-(1/16))*Sqrt[n]*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(4/n) + (1/4)*x^4*Sqrt[Log[a*x^n]]} -{x^2*Sqrt[Log[a*x^n]], x, 4, ((-(1/6))*Sqrt[n]*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(3/n) + (1/3)*x^3*Sqrt[Log[a*x^n]]} -{x^1*Sqrt[Log[a*x^n]], x, 4, ((-(1/4))*Sqrt[n]*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(2/n) + (1/2)*x^2*Sqrt[Log[a*x^n]]} -{x^0*Sqrt[Log[a*x^n]], x, 4, ((-(1/2))*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(a*x^n)^n^(-1) + x*Sqrt[Log[a*x^n]]} -{Sqrt[Log[a*x^n]]/x^1, x, 2, (2*Log[a*x^n]^(3/2))/(3*n)} -{Sqrt[Log[a*x^n]]/x^2, x, 4, (Sqrt[n]*Sqrt[Pi]*(a*x^n)^(1/n)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(2*x) - Sqrt[Log[a*x^n]]/x} -{Sqrt[Log[a*x^n]]/x^3, x, 4, (Sqrt[n]*Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(4*x^2) - Sqrt[Log[a*x^n]]/(2*x^2)} - - -{x^3*Log[a*x^n]^(3/2), x, 5, ((3/128)*n^(3/2)*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(4/n) - (3/32)*n*x^4*Sqrt[Log[a*x^n]] + (1/4)*x^4*Log[a*x^n]^(3/2)} -{x^2*Log[a*x^n]^(3/2), x, 5, ((1/12)*n^(3/2)*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(3/n) - (1/6)*n*x^3*Sqrt[Log[a*x^n]] + (1/3)*x^3*Log[a*x^n]^(3/2)} -{x^1*Log[a*x^n]^(3/2), x, 5, ((3/16)*n^(3/2)*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(a*x^n)^(2/n) - (3/8)*n*x^2*Sqrt[Log[a*x^n]] + (1/2)*x^2*Log[a*x^n]^(3/2)} -{x^0*Log[a*x^n]^(3/2), x, 5, ((3/4)*n^(3/2)*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(a*x^n)^n^(-1) - (3/2)*n*x*Sqrt[Log[a*x^n]] + x*Log[a*x^n]^(3/2)} -{Log[a*x^n]^(3/2)/x^1, x, 2, (2*Log[a*x^n]^(5/2))/(5*n)} -{Log[a*x^n]^(3/2)/x^2, x, 5, (3*n^(3/2)*Sqrt[Pi]*(a*x^n)^(1/n)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(4*x) - (3*n*Sqrt[Log[a*x^n]])/(2*x) - Log[a*x^n]^(3/2)/x} -{Log[a*x^n]^(3/2)/x^3, x, 5, (3*n^(3/2)*Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(16*x^2) - (3*n*Sqrt[Log[a*x^n]])/(8*x^2) - Log[a*x^n]^(3/2)/(2*x^2)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/Sqrt[Log[a*x^n]], x, 3, (Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(2*Sqrt[n]*(a*x^n)^(4/n))} -{x^2/Sqrt[Log[a*x^n]], x, 3, (Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(Sqrt[n]*(a*x^n)^(3/n))} -{x^1/Sqrt[Log[a*x^n]], x, 3, (Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(Sqrt[n]*(a*x^n)^(2/n))} -{x^0/Sqrt[Log[a*x^n]], x, 3, (Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(Sqrt[n]*(a*x^n)^n^(-1))} -{1/(x^1*Sqrt[Log[a*x^n]]), x, 2, (2*Sqrt[Log[a*x^n]])/n} -{1/(x^2*Sqrt[Log[a*x^n]]), x, 3, (Sqrt[Pi]*(a*x^n)^n^(-1)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(Sqrt[n]*x)} -{1/(x^3*Sqrt[Log[a*x^n]]), x, 3, (Sqrt[Pi/2]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(Sqrt[n]*x^2)} - - -{x^3/Log[a*x^n]^(3/2), x, 4, (4*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/(n^(3/2)*(a*x^n)^(4/n)) - (2*x^4)/(n*Sqrt[Log[a*x^n]])} -{x^2/Log[a*x^n]^(3/2), x, 4, (2*Sqrt[3*Pi]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^(3/n)*n^(3/2)) - (2*x^3)/(n*Sqrt[Log[a*x^n]])} -{x^1/Log[a*x^n]^(3/2), x, 4, (2*Sqrt[2*Pi]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^(2/n)*n^(3/2)) - (2*x^2)/(n*Sqrt[Log[a*x^n]])} -{x^0/Log[a*x^n]^(3/2), x, 4, (2*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/(n^(3/2)*(a*x^n)^n^(-1)) - (2*x)/(n*Sqrt[Log[a*x^n]])} -{1/(x^1*Log[a*x^n]^(3/2)), x, 2, -2/(n*Sqrt[Log[a*x^n]])} -{1/(x^2*Log[a*x^n]^(3/2)), x, 4, -((2*Sqrt[Pi]*(a*x^n)^(1/n)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(n^(3/2)*x)) - 2/(n*x*Sqrt[Log[a*x^n]])} -{1/(x^3*Log[a*x^n]^(3/2)), x, 4, -((2*Sqrt[2*Pi]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(n^(3/2)*x^2)) - 2/(n*x^2*Sqrt[Log[a*x^n]])} - - -{x^3/Log[a*x^n]^(5/2), x, 5, (32*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^(4/n)*(3*n^(5/2))) - (2*x^4)/(3*n*Log[a*x^n]^(3/2)) - (16*x^4)/(3*n^2*Sqrt[Log[a*x^n]])} -{x^2/Log[a*x^n]^(5/2), x, 5, (4*Sqrt[3*Pi]*x^3*Erfi[(Sqrt[3]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^(3/n)*n^(5/2)) - (2*x^3)/(3*n*Log[a*x^n]^(3/2)) - (4*x^3)/(n^2*Sqrt[Log[a*x^n]])} -{x^1/Log[a*x^n]^(5/2), x, 5, (8*Sqrt[2*Pi]*x^2*Erfi[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^(2/n)*(3*n^(5/2))) - (2*x^2)/(3*n*Log[a*x^n]^(3/2)) - (8*x^2)/(3*n^2*Sqrt[Log[a*x^n]])} -{x^0/Log[a*x^n]^(5/2), x, 5, (4*Sqrt[Pi]*x*Erfi[Sqrt[Log[a*x^n]]/Sqrt[n]])/((a*x^n)^n^(-1)*(3*n^(5/2))) - (2*x)/(3*n*Log[a*x^n]^(3/2)) - (4*x)/(3*n^2*Sqrt[Log[a*x^n]])} -{1/(x^1*Log[a*x^n]^(5/2)), x, 2, -(2/(3*n*Log[a*x^n]^(3/2)))} -{1/(x^2*Log[a*x^n]^(5/2)), x, 5, (4*Sqrt[Pi]*(a*x^n)^(1/n)*Erf[Sqrt[Log[a*x^n]]/Sqrt[n]])/(3*n^(5/2)*x) - 2/(3*n*x*Log[a*x^n]^(3/2)) + 4/(3*n^2*x*Sqrt[Log[a*x^n]])} -{1/(x^3*Log[a*x^n]^(5/2)), x, 5, (8*Sqrt[2*Pi]*(a*x^n)^(2/n)*Erf[(Sqrt[2]*Sqrt[Log[a*x^n]])/Sqrt[n]])/(3*n^(5/2)*x^2) - 2/(3*n*x^2*Log[a*x^n]^(3/2)) + 8/(3*n^2*x^2*Sqrt[Log[a*x^n]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b Log[c x^n])^p with m symbolic*) - - -{(d*x)^m*(a + a*(m + 1)/n*Log[c*x^n]), x, 1, (a*(d*x)^(1 + m)*Log[c*x^n])/(d*n)} - - -{(d*x)^m*(a + b*Log[c*x^n])^3, x, 3, -((6*b^3*n^3*(d*x)^(1 + m))/(d*(1 + m)^4)) + (6*b^2*n^2*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^3) - (3*b*n*(d*x)^(1 + m)*(a + b*Log[c*x^n])^2)/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])^3)/(d*(1 + m))} -{(d*x)^m*(a + b*Log[c*x^n])^2, x, 2, (2*b^2*n^2*(d*x)^(1 + m))/(d*(1 + m)^3) - (2*b*n*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])^2)/(d*(1 + m))} -{(d*x)^m*(a + b*Log[c*x^n])^1, x, 1, -((b*n*(d*x)^(1 + m))/(d*(1 + m)^2)) + ((d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m))} -{(d*x)^m/(a + b*Log[c*x^n])^1, x, 2, ((d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(b*d*n))} -{(d*x)^m/(a + b*Log[c*x^n])^2, x, 3, ((1 + m)*(d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(b^2*d*n^2)) - (d*x)^(1 + m)/(b*d*n*(a + b*Log[c*x^n]))} -{(d*x)^m/(a + b*Log[c*x^n])^3, x, 4, ((1 + m)^2*(d*x)^(1 + m)*ExpIntegralEi[((1 + m)*(a + b*Log[c*x^n]))/(b*n)])/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(2*b^3*d*n^3)) - (d*x)^(1 + m)/(2*b*d*n*(a + b*Log[c*x^n])^2) - ((1 + m)*(d*x)^(1 + m))/(2*b^2*d*n^2*(a + b*Log[c*x^n]))} - - -{(d*x)^(n - 1)*Log[c*x^n]^3, x, 3, -((6*(d*x)^n)/(d*n)) + (6*(d*x)^n*Log[c*x^n])/(d*n) - (3*(d*x)^n*Log[c*x^n]^2)/(d*n) + ((d*x)^n*Log[c*x^n]^3)/(d*n)} -{(d*x)^(n - 1)*Log[c*x^n]^2, x, 2, (2*(d*x)^n)/(d*n) - (2*(d*x)^n*Log[c*x^n])/(d*n) + ((d*x)^n*Log[c*x^n]^2)/(d*n)} -{(d*x)^(n - 1)*Log[c*x^n]^1, x, 1, -((d*x)^n/(d*n)) + ((d*x)^n*Log[c*x^n])/(d*n)} -{(d*x)^(n - 1)/Log[c*x^n]^1, x, 3, (x^(1 - n)*(d*x)^(-1 + n)*LogIntegral[c*x^n])/(c*n)} -{(d*x)^(n - 1)/Log[c*x^n]^2, x, 4, -((d*x)^n/(d*n*Log[c*x^n])) + (x^(1 - n)*(d*x)^(-1 + n)*LogIntegral[c*x^n])/(c*n)} -{(d*x)^(n - 1)/Log[c*x^n]^3, x, 5, -((d*x)^n/(2*d*n*Log[c*x^n]^2)) - (d*x)^n/(2*d*n*Log[c*x^n]) + (x^(1 - n)*(d*x)^(-1 + n)*LogIntegral[c*x^n])/(2*c*n)} - - -{x^m*Log[a*x^n]^(3/2), x, 5, (3*n^(3/2)*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^((1 + m)/n)*(4*(1 + m)^(5/2))) - (3*n*x^(1 + m)*Sqrt[Log[a*x^n]])/(2*(1 + m)^2) + (x^(1 + m)*Log[a*x^n]^(3/2))/(1 + m)} -{x^m*Log[a*x^n]^(1/2), x, 4, -((Sqrt[n]*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^((1 + m)/n)*(2*(1 + m)^(3/2)))) + (x^(1 + m)*Sqrt[Log[a*x^n]])/(1 + m)} -{x^m/Log[a*x^n]^(1/2), x, 3, (Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^((1 + m)/n)*(Sqrt[1 + m]*Sqrt[n]))} -{x^m/Log[a*x^n]^(3/2), x, 4, (2*Sqrt[1 + m]*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^((1 + m)/n)*n^(3/2)) - (2*x^(1 + m))/(n*Sqrt[Log[a*x^n]])} -{x^m/Log[a*x^n]^(5/2), x, 5, (4*(1 + m)^(3/2)*Sqrt[Pi]*x^(1 + m)*Erfi[(Sqrt[1 + m]*Sqrt[Log[a*x^n]])/Sqrt[n]])/((a*x^n)^((1 + m)/n)*(3*n^(5/2))) - (2*x^(1 + m))/(3*n*Log[a*x^n]^(3/2)) - (4*(1 + m)*x^(1 + m))/(3*n^2*Sqrt[Log[a*x^n]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b Log[a x^n])^p with p symbolic*) - - -{(d*x)^m*(a + b*Log[c*x^n])^p, x, 2, ((d*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(d*(1 + m)))} - -{x^2*(a + b*Log[c*x^n])^p, x, 2, (3^(-1 - p)*x^3*Gamma[1 + p, -((3*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((3*a)/(b*n))*(c*x^n)^(3/n)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{x^1*(a + b*Log[c*x^n])^p, x, 2, (2^(-1 - p)*x^2*Gamma[1 + p, -((2*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((2*a)/(b*n))*(c*x^n)^(2/n)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{x^0*(a + b*Log[c*x^n])^p, x, 2, (x*Gamma[1 + p, -((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{(a + b*Log[c*x^n])^p/x^1, x, 2, (a + b*Log[c*x^n])^(1 + p)/(b*n*(1 + p))} -{(a + b*Log[c*x^n])^p/x^2, x, 2, -((E^(a/(b*n))*(c*x^n)^(1/n)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x))} -{(a + b*Log[c*x^n])^p/x^3, x, 2, -((2^(-1 - p)*E^((2*a)/(b*n))*(c*x^n)^(2/n)*Gamma[1 + p, (2*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x^2))} -{(a + b*Log[c*x^n])^p/x^4, x, 2, -((3^(-1 - p)*E^((3*a)/(b*n))*(c*x^n)^(3/n)*Gamma[1 + p, (3*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x^3))} - - -{(d*x)^m*(a + b*Log[c*x])^p, x, 2, ((c*x)^(-1 - m)*(d*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x]))/b)]*(a + b*Log[c*x])^p)/(E^((a*(1 + m))/b)*(-(((1 + m)*(a + b*Log[c*x]))/b))^p*(d*(1 + m)))} - -{x^2*(a + b*Log[c*x])^p, x, 2, (3^(-1 - p)*Gamma[1 + p, -((3*(a + b*Log[c*x]))/b)]*(a + b*Log[c*x])^p)/(E^((3*a)/b)*(-((a + b*Log[c*x])/b))^p*c^3)} -{x^1*(a + b*Log[c*x])^p, x, 2, (2^(-1 - p)*Gamma[1 + p, -((2*(a + b*Log[c*x]))/b)]*(a + b*Log[c*x])^p)/(E^((2*a)/b)*(-((a + b*Log[c*x])/b))^p*c^2)} -{x^0*(a + b*Log[c*x])^p, x, 2, (Gamma[1 + p, -((a + b*Log[c*x])/b)]*(a + b*Log[c*x])^p)/(E^(a/b)*(-((a + b*Log[c*x])/b))^p*c)} -{(a + b*Log[c*x])^p/x^1, x, 2, (a + b*Log[c*x])^(1 + p)/(b*(1 + p))} -{(a + b*Log[c*x])^p/x^2, x, 2, ((-c)*E^(a/b)*Gamma[1 + p, (a + b*Log[c*x])/b]*(a + b*Log[c*x])^p)/((a + b*Log[c*x])/b)^p} -{(a + b*Log[c*x])^p/x^3, x, 2, ((-2^(-1 - p))*c^2*E^((2*a)/b)*Gamma[1 + p, (2*(a + b*Log[c*x]))/b]*(a + b*Log[c*x])^p)/((a + b*Log[c*x])/b)^p} -{(a + b*Log[c*x])^p/x^4, x, 2, ((-3^(-1 - p))*c^3*E^((3*a)/b)*Gamma[1 + p, (3*(a + b*Log[c*x]))/b]*(a + b*Log[c*x])^p)/((a + b*Log[c*x])/b)^p} - - -{(d*x)^m*(a + b*Log[c*Sqrt[x]])^p, x, 2, ((d*x)^(1 + m)*Gamma[1 + p, -((2*(1 + m)*(a + b*Log[c*Sqrt[x]]))/b)]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*E^((2*a*(1 + m))/b)*(c*Sqrt[x])^(2*(1 + m))*(-(((1 + m)*(a + b*Log[c*Sqrt[x]]))/b))^p*(d*(1 + m)))} - -{x^2*(a + b*Log[c*Sqrt[x]])^p, x, 2, (3^(-1 - p)*Gamma[1 + p, -((6*(a + b*Log[c*Sqrt[x]]))/b)]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*E^((6*a)/b)*(-((a + b*Log[c*Sqrt[x]])/b))^p*c^6)} -{x^1*(a + b*Log[c*Sqrt[x]])^p, x, 2, (2^(-1 - 2*p)*Gamma[1 + p, -((4*(a + b*Log[c*Sqrt[x]]))/b)]*(a + b*Log[c*Sqrt[x]])^p)/(E^((4*a)/b)*(-((a + b*Log[c*Sqrt[x]])/b))^p*c^4)} -{x^0*(a + b*Log[c*Sqrt[x]])^p, x, 2, (Gamma[1 + p, -((2*(a + b*Log[c*Sqrt[x]]))/b)]*(a + b*Log[c*Sqrt[x]])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*Sqrt[x]])/b))^p*c^2)} -{(a + b*Log[c*Sqrt[x]])^p/x^1, x, 2, (2*(a + b*Log[c*Sqrt[x]])^(1 + p))/(b*(1 + p))} -{(a + b*Log[c*Sqrt[x]])^p/x^2, x, 2, ((-2^(-p))*c^2*E^((2*a)/b)*Gamma[1 + p, (2*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/((a + b*Log[c*Sqrt[x]])/b)^p} -{(a + b*Log[c*Sqrt[x]])^p/x^3, x, 2, ((-2^(-1 - 2*p))*c^4*E^((4*a)/b)*Gamma[1 + p, (4*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/((a + b*Log[c*Sqrt[x]])/b)^p} -{(a + b*Log[c*Sqrt[x]])^p/x^4, x, 2, ((-2^(-p))*3^(-1 - p)*c^6*E^((6*a)/b)*Gamma[1 + p, (6*(a + b*Log[c*Sqrt[x]]))/b]*(a + b*Log[c*Sqrt[x]])^p)/((a + b*Log[c*Sqrt[x]])/b)^p} - - -{x^(n - 1)*(a + b*Log[c*x^n])^p, x, 2, (Gamma[1 + p, -((a + b*Log[c*x^n])/b)]*(a + b*Log[c*x^n])^p)/(E^(a/b)*(-((a + b*Log[c*x^n])/b))^p*(c*n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x^q)^m (a+b Log[c x^n])^p*) - - -{(d*x^q)^m*(a + b*Log[c*x^n])^p, x, 3, (x*(d*x^q)^m*Gamma[1 + p, -(((1 + m*q)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a + a*m*q)/(b*n))*(c*x^n)^((1 + m*q)/n)*(-(((1 + m*q)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m*q))} - - -{(d1*x^q1)^m1*(d2*x^q2)^m2*(a + b*Log[c*x^n])^p, x, 4, (x*(d1*x^q1)^m1*(d2*x^q2)^m2*Gamma[1 + p, -(((1 + m1*q1 + m2*q2)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m1*q1 + m2*q2))/(b*n))*(c*x^n)^((1 + m1*q1 + m2*q2)/n)*(-(((1 + m1*q1 + m2*q2)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m1*q1 + m2*q2))} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m deleted file mode 100644 index 23b4ae0..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.1.4 (f x)^m (d+e x^r)^q (a+b log(c x^n))^p.m +++ /dev/null @@ -1,869 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (f x)^m (d+e x^r)^q (a+b Log[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^1)^q (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b Log[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(d + e*x)*(a + b*Log[c*x^n]), x, 4, (-(1/16))*b*d*n*x^4 - (1/25)*b*e*n*x^5 + (1/20)*(5*d*x^4 + 4*e*x^5)*(a + b*Log[c*x^n])} -{x^2*(d + e*x)*(a + b*Log[c*x^n]), x, 4, (-(1/9))*b*d*n*x^3 - (1/16)*b*e*n*x^4 + (1/12)*(4*d*x^3 + 3*e*x^4)*(a + b*Log[c*x^n])} -{x^1*(d + e*x)*(a + b*Log[c*x^n]), x, 4, (-(1/4))*b*d*n*x^2 - (1/9)*b*e*n*x^3 + (1/6)*(3*d*x^2 + 2*e*x^3)*(a + b*Log[c*x^n])} -{x^0*(d + e*x)*(a + b*Log[c*x^n]), x, 2, (-b)*d*n*x - (1/4)*b*e*n*x^2 + d*x*(a + b*Log[c*x^n]) + (1/2)*e*x^2*(a + b*Log[c*x^n])} -{(d + e*x)*(a + b*Log[c*x^n])/x^1, x, 4, a*e*x - b*e*n*x + b*e*x*Log[c*x^n] + (d*(a + b*Log[c*x^n])^2)/(2*b*n)} -{(d + e*x)*(a + b*Log[c*x^n])/x^2, x, 4, -((b*d*n)/x) - (d*(a + b*Log[c*x^n]))/x + (e*(a + b*Log[c*x^n])^2)/(2*b*n), -((b*d*n)/x) - (1/2)*b*e*n*Log[x]^2 - (d*(a + b*Log[c*x^n]))/x + e*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)*(a + b*Log[c*x^n])/x^3, x, 4, -((b*d*n)/(4*x^2)) - (b*e*n)/x + (b*e^2*n*Log[x])/(2*d) - ((d + e*x)^2*(a + b*Log[c*x^n]))/(2*d*x^2)} -{(d + e*x)*(a + b*Log[c*x^n])/x^4, x, 4, -((b*d*n)/(9*x^3)) - (b*e*n)/(4*x^2) - (d*(a + b*Log[c*x^n]))/(3*x^3) - (e*(a + b*Log[c*x^n]))/(2*x^2)} - - -{x^3*(d + e*x)^2*(a + b*Log[c*x^n]), x, 4, (-(1/16))*b*d^2*n*x^4 - (2/25)*b*d*e*n*x^5 - (1/36)*b*e^2*n*x^6 + (1/60)*(15*d^2*x^4 + 24*d*e*x^5 + 10*e^2*x^6)*(a + b*Log[c*x^n])} -{x^2*(d + e*x)^2*(a + b*Log[c*x^n]), x, 4, (-(1/9))*b*d^2*n*x^3 - (1/8)*b*d*e*n*x^4 - (1/25)*b*e^2*n*x^5 + (1/30)*(10*d^2*x^3 + 15*d*e*x^4 + 6*e^2*x^5)*(a + b*Log[c*x^n])} -{x^1*(d + e*x)^2*(a + b*Log[c*x^n]), x, 4, (-(1/4))*b*d^2*n*x^2 - (2/9)*b*d*e*n*x^3 - (1/16)*b*e^2*n*x^4 + (1/12)*(6*d^2*x^2 + 8*d*e*x^3 + 3*e^2*x^4)*(a + b*Log[c*x^n])} -{x^0*(d + e*x)^2*(a + b*Log[c*x^n]), x, 4, (-b)*d^2*n*x - (1/2)*b*d*e*n*x^2 - (1/9)*b*e^2*n*x^3 - (b*d^3*n*Log[x])/(3*e) + ((d + e*x)^3*(a + b*Log[c*x^n]))/(3*e)} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^1, x, 3, (-(1/4))*b*n*(4*d + e*x)^2 - (1/2)*b*d^2*n*Log[x]^2 + 2*d*e*x*(a + b*Log[c*x^n]) + (1/2)*e^2*x^2*(a + b*Log[c*x^n]) + d^2*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^2, x, 3, -((b*d^2*n)/x) - b*e^2*n*x - b*d*e*n*Log[x]^2 - (d^2*(a + b*Log[c*x^n]))/x + e^2*x*(a + b*Log[c*x^n]) + 2*d*e*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^3, x, 6, -((b*n*(d + 4*e*x)^2)/(4*x^2)) - (1/2)*b*e^2*n*Log[x]^2 - (d^2*(a + b*Log[c*x^n]))/(2*x^2) - (2*d*e*(a + b*Log[c*x^n]))/x + e^2*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^4, x, 4, -((b*d^2*n)/(9*x^3)) - (b*d*e*n)/(2*x^2) - (b*e^2*n)/x + (b*e^3*n*Log[x])/(3*d) - ((d + e*x)^3*(a + b*Log[c*x^n]))/(3*d*x^3)} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^5, x, 4, -((b*d^2*n)/(16*x^4)) - (2*b*d*e*n)/(9*x^3) - (b*e^2*n)/(4*x^2) - (d^2*(a + b*Log[c*x^n]))/(4*x^4) - (2*d*e*(a + b*Log[c*x^n]))/(3*x^3) - (e^2*(a + b*Log[c*x^n]))/(2*x^2)} -{(d + e*x)^2*(a + b*Log[c*x^n])/x^6, x, 4, -((b*d^2*n)/(25*x^5)) - (b*d*e*n)/(8*x^4) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(5*x^5) - (d*e*(a + b*Log[c*x^n]))/(2*x^4) - (e^2*(a + b*Log[c*x^n]))/(3*x^3)} - - -{x^3*(d + e*x)^3*(a + b*Log[c*x^n]), x, 4, (-(1/16))*b*d^3*n*x^4 - (3/25)*b*d^2*e*n*x^5 - (1/12)*b*d*e^2*n*x^6 - (1/49)*b*e^3*n*x^7 + (1/140)*(35*d^3*x^4 + 84*d^2*e*x^5 + 70*d*e^2*x^6 + 20*e^3*x^7)*(a + b*Log[c*x^n])} -{x^2*(d + e*x)^3*(a + b*Log[c*x^n]), x, 4, (-(1/9))*b*d^3*n*x^3 - (3/16)*b*d^2*e*n*x^4 - (3/25)*b*d*e^2*n*x^5 - (1/36)*b*e^3*n*x^6 + (1/60)*(20*d^3*x^3 + 45*d^2*e*x^4 + 36*d*e^2*x^5 + 10*e^3*x^6)*(a + b*Log[c*x^n])} -{x^1*(d + e*x)^3*(a + b*Log[c*x^n]), x, 5, (b*d^4*n*x)/(5*e) + (3/20)*b*d^3*n*x^2 + (1/15)*b*d^2*e*n*x^3 + (1/80)*b*d*e^2*n*x^4 - (b*n*(d + e*x)^5)/(25*e^2) + (b*d^5*n*Log[x])/(20*e^2) - (1/20)*((5*d*(d + e*x)^4)/e^2 - (4*(d + e*x)^5)/e^2)*(a + b*Log[c*x^n])} -{x^0*(d + e*x)^3*(a + b*Log[c*x^n]), x, 4, (-b)*d^3*n*x - (3/4)*b*d^2*e*n*x^2 - (1/3)*b*d*e^2*n*x^3 - (1/16)*b*e^3*n*x^4 - (b*d^4*n*Log[x])/(4*e) + ((d + e*x)^4*(a + b*Log[c*x^n]))/(4*e)} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^1, x, 4, -3*b*d^2*e*n*x - (3/4)*b*d*e^2*n*x^2 - (1/9)*b*e^3*n*x^3 - (1/2)*b*d^3*n*Log[x]^2 + 3*d^2*e*x*(a + b*Log[c*x^n]) + (3/2)*d*e^2*x^2*(a + b*Log[c*x^n]) + (1/3)*e^3*x^3*(a + b*Log[c*x^n]) + d^3*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^2, x, 3, -((b*d^3*n)/x) - 3*b*d*e^2*n*x - (1/4)*b*e^3*n*x^2 - (3/2)*b*d^2*e*n*Log[x]^2 - (d^3*(a + b*Log[c*x^n]))/x + 3*d*e^2*x*(a + b*Log[c*x^n]) + (1/2)*e^3*x^2*(a + b*Log[c*x^n]) + 3*d^2*e*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^3, x, 3, -((b*d^3*n)/(4*x^2)) - (3*b*d^2*e*n)/x - b*e^3*n*x - (3/2)*b*d*e^2*n*Log[x]^2 - (d^3*(a + b*Log[c*x^n]))/(2*x^2) - (3*d^2*e*(a + b*Log[c*x^n]))/x + e^3*x*(a + b*Log[c*x^n]) + 3*d*e^2*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^4, x, 7, -((b*d^3*n)/(9*x^3)) - (3*b*d^2*e*n)/(4*x^2) - (3*b*d*e^2*n)/x - (1/2)*b*e^3*n*Log[x]^2 - (d^3*(a + b*Log[c*x^n]))/(3*x^3) - (3*d^2*e*(a + b*Log[c*x^n]))/(2*x^2) - (3*d*e^2*(a + b*Log[c*x^n]))/x + e^3*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^5, x, 4, -((b*d^3*n)/(16*x^4)) - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/(4*x^2) - (b*e^3*n)/x + (b*e^4*n*Log[x])/(4*d) - ((d + e*x)^4*(a + b*Log[c*x^n]))/(4*d*x^4)} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^6, x, 5, (b*d^2*e*n)/(80*x^4) + (b*d*e^2*n)/(15*x^3) + (3*b*e^3*n)/(20*x^2) + (b*e^4*n)/(5*d*x) - (b*n*(d + e*x)^5)/(25*d^2*x^5) - (b*e^5*n*Log[x])/(20*d^2) - ((d + e*x)^4*(a + b*Log[c*x^n]))/(5*d*x^5) + (e*(d + e*x)^4*(a + b*Log[c*x^n]))/(20*d^2*x^4)} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^7, x, 4, -((b*d^3*n)/(36*x^6)) - (3*b*d^2*e*n)/(25*x^5) - (3*b*d*e^2*n)/(16*x^4) - (b*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*x^n]))/(6*x^6) - (3*d^2*e*(a + b*Log[c*x^n]))/(5*x^5) - (3*d*e^2*(a + b*Log[c*x^n]))/(4*x^4) - (e^3*(a + b*Log[c*x^n]))/(3*x^3)} -{(d + e*x)^3*(a + b*Log[c*x^n])/x^8, x, 4, -((b*d^3*n)/(49*x^7)) - (b*d^2*e*n)/(12*x^6) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(16*x^4) - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (d^2*e*(a + b*Log[c*x^n]))/(2*x^6) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c*x^n]))/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3*(a + b*Log[c*x^n])/(d + e*x), x, 8, (a*d^2*x)/e^3 - (b*d^2*n*x)/e^3 + (b*d*n*x^2)/(4*e^2) - (b*n*x^3)/(9*e) + (b*d^2*x*Log[c*x^n])/e^3 - (d*x^2*(a + b*Log[c*x^n]))/(2*e^2) + (x^3*(a + b*Log[c*x^n]))/(3*e) - (d^3*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 - (b*d^3*n*PolyLog[2, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])/(d + e*x), x, 7, -((a*d*x)/e^2) + (b*d*n*x)/e^2 - (b*n*x^2)/(4*e) - (b*d*x*Log[c*x^n])/e^2 + (x^2*(a + b*Log[c*x^n]))/(2*e) + (d^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + (b*d^2*n*PolyLog[2, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])/(d + e*x), x, 6, (a*x)/e - (b*n*x)/e + (b*x*Log[c*x^n])/e - (d*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^2 - (b*d*n*PolyLog[2, -((e*x)/d)])/e^2} -{x^0*(a + b*Log[c*x^n])/(d + e*x), x, 2, ((a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e + (b*n*PolyLog[2, -((e*x)/d)])/e} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)), x, 2, -((Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)), x, 4, -((b*n)/(d*x)) - (a + b*Log[c*x^n])/(d*x) + (e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^2 - (b*e*n*PolyLog[2, -(d/(e*x))])/d^2} -{(a + b*Log[c*x^n])/(x^3*(d + e*x)), x, 6, -((b*n)/(4*d*x^2)) + (b*e*n)/(d^2*x) - (a + b*Log[c*x^n])/(2*d*x^2) + (e*(a + b*Log[c*x^n]))/(d^2*x) - (e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^3 + (b*e^2*n*PolyLog[2, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])/(x^4*(d + e*x)), x, 8, -((b*n)/(9*d*x^3)) + (b*e*n)/(4*d^2*x^2) - (b*e^2*n)/(d^3*x) - (a + b*Log[c*x^n])/(3*d*x^3) + (e*(a + b*Log[c*x^n]))/(2*d^2*x^2) - (e^2*(a + b*Log[c*x^n]))/(d^3*x) + (e^3*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^4 - (b*e^3*n*PolyLog[2, -(d/(e*x))])/d^4} - - -{x^3*(a + b*Log[c*x^n])/(d + e*x)^2, x, 8, (3*b*d*n*x)/e^3 - (d*(3*a + b*n)*x)/e^3 - (3*b*n*x^2)/(4*e^2) - (3*b*d*x*Log[c*x^n])/e^3 - (x^3*(a + b*Log[c*x^n]))/(e*(d + e*x)) + (x^2*(3*a + b*n + 3*b*Log[c*x^n]))/(2*e^2) + (d^2*(3*a + b*n + 3*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + (3*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])/(d + e*x)^2, x, 7, -((b*n*x)/e^2) + (2*x*(a + b*Log[c*x^n]))/e^2 - (x^2*(a + b*Log[c*x^n]))/(e*(d + e*x)) - (d*(2*a + b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 - (2*b*d*n*PolyLog[2, -((e*x)/d)])/e^3, -((2*b*n*x)/e^2) + ((2*a + b*n)*x)/e^2 + (2*b*x*Log[c*x^n])/e^2 - (x^2*(a + b*Log[c*x^n]))/(e*(d + e*x)) - (d*(2*a + b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 - (2*b*d*n*PolyLog[2, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])/(d + e*x)^2, x, 3, -((x*(a + b*Log[c*x^n]))/(e*(d + e*x))) + ((a + b*n + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^2 + (b*n*PolyLog[2, -((e*x)/d)])/e^2} -{x^0*(a + b*Log[c*x^n])/(d + e*x)^2, x, 2, (x*(a + b*Log[c*x^n]))/(d*(d + e*x)) - (b*n*Log[d + e*x])/(d*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)^2), x, 5, -((e*x*(a + b*Log[c*x^n]))/(d^2*(d + e*x))) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^2 + (b*n*Log[d + e*x])/d^2 + (b*n*PolyLog[2, -(d/(e*x))])/d^2} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)^2), x, 7, -((b*n)/(d^2*x)) - (a + b*Log[c*x^n])/(d^2*x) + (e^2*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) + (2*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^3 - (b*e*n*Log[d + e*x])/d^3 - (2*b*e*n*PolyLog[2, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])/(x^3*(d + e*x)^2), x, 8, -((b*n)/(4*d^2*x^2)) + (2*b*e*n)/(d^3*x) - (a + b*Log[c*x^n])/(2*d^2*x^2) + (2*e*(a + b*Log[c*x^n]))/(d^3*x) - (e^3*x*(a + b*Log[c*x^n]))/(d^4*(d + e*x)) - (3*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^4 + (b*e^2*n*Log[d + e*x])/d^4 + (3*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^4} - - -{x^3*(a + b*Log[c*x^n])/(d + e*x)^3, x, 8, -((3*b*n*x)/e^3) + ((6*a + 5*b*n)*x)/(2*e^3) + (3*b*x*Log[c*x^n])/e^3 - (x^3*(a + b*Log[c*x^n]))/(2*e*(d + e*x)^2) - (x^2*(3*a + b*n + 3*b*Log[c*x^n]))/(2*e^2*(d + e*x)) - (d*(6*a + 5*b*n + 6*b*Log[c*x^n])*Log[1 + (e*x)/d])/(2*e^4) - (3*b*d*n*PolyLog[2, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])/(d + e*x)^3, x, 4, -((x^2*(a + b*Log[c*x^n]))/(2*e*(d + e*x)^2)) - (x*(2*a + b*n + 2*b*Log[c*x^n]))/(2*e^2*(d + e*x)) + ((2*a + 3*b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/(2*e^3) + (b*n*PolyLog[2, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])/(d + e*x)^3, x, 3, -((b*n)/(2*e^2*(d + e*x))) + (x^2*(a + b*Log[c*x^n]))/(2*d*(d + e*x)^2) - (b*n*Log[d + e*x])/(2*d*e^2)} -{x^0*(a + b*Log[c*x^n])/(d + e*x)^3, x, 3, (b*n)/(2*d*e*(d + e*x)) + (b*n*Log[x])/(2*d^2*e) - (a + b*Log[c*x^n])/(2*e*(d + e*x)^2) - (b*n*Log[d + e*x])/(2*d^2*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)^3), x, 9, -((b*n)/(2*d^2*(d + e*x))) - (b*n*Log[x])/(2*d^3) + (a + b*Log[c*x^n])/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^3 + (3*b*n*Log[d + e*x])/(2*d^3) + (b*n*PolyLog[2, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)^3), x, 10, -((b*n)/(d^3*x)) + (b*e*n)/(2*d^3*(d + e*x)) + (b*e*n*Log[x])/(2*d^4) - (a + b*Log[c*x^n])/(d^3*x) - (e*(a + b*Log[c*x^n]))/(2*d^2*(d + e*x)^2) + (2*e^2*x*(a + b*Log[c*x^n]))/(d^4*(d + e*x)) + (3*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^4 - (5*b*e*n*Log[d + e*x])/(2*d^4) - (3*b*e*n*PolyLog[2, -(d/(e*x))])/d^4} -{(a + b*Log[c*x^n])/(x^3*(d + e*x)^3), x, 11, -((b*n)/(4*d^3*x^2)) + (3*b*e*n)/(d^4*x) - (b*e^2*n)/(2*d^4*(d + e*x)) - (b*e^2*n*Log[x])/(2*d^5) - (a + b*Log[c*x^n])/(2*d^3*x^2) + (3*e*(a + b*Log[c*x^n]))/(d^4*x) + (e^2*(a + b*Log[c*x^n]))/(2*d^3*(d + e*x)^2) - (3*e^3*x*(a + b*Log[c*x^n]))/(d^5*(d + e*x)) - (6*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^5 + (7*b*e^2*n*Log[d + e*x])/(2*d^5) + (6*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^5} - - -{x^5*(a + b*Log[c*x^n])/(d + e*x)^4, x, 10, (10*b*d*n*x)/e^5 - (d*(60*a + 47*b*n)*x)/(6*e^5) - (5*b*n*x^2)/(2*e^4) - (10*b*d*x*Log[c*x^n])/e^5 - (x^5*(a + b*Log[c*x^n]))/(3*e*(d + e*x)^3) - (x^4*(5*a + b*n + 5*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x^3*(20*a + 9*b*n + 20*b*Log[c*x^n]))/(6*e^3*(d + e*x)) + (x^2*(60*a + 47*b*n + 60*b*Log[c*x^n]))/(12*e^4) + (d^2*(60*a + 47*b*n + 60*b*Log[c*x^n])*Log[1 + (e*x)/d])/(6*e^6) + (10*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^6} -{x^4*(a + b*Log[c*x^n])/(d + e*x)^4, x, 9, -((4*b*n*x)/e^4) + ((12*a + 13*b*n)*x)/(3*e^4) + (4*b*x*Log[c*x^n])/e^4 - (x^4*(a + b*Log[c*x^n]))/(3*e*(d + e*x)^3) - (x^3*(4*a + b*n + 4*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x^2*(12*a + 7*b*n + 12*b*Log[c*x^n]))/(6*e^3*(d + e*x)) - (d*(12*a + 13*b*n + 12*b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*e^5) - (4*b*d*n*PolyLog[2, -((e*x)/d)])/e^5} -{x^3*(a + b*Log[c*x^n])/(d + e*x)^4, x, 5, -((x^3*(a + b*Log[c*x^n]))/(3*e*(d + e*x)^3)) - (x^2*(3*a + b*n + 3*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x*(6*a + 5*b*n + 6*b*Log[c*x^n]))/(6*e^3*(d + e*x)) + ((6*a + 11*b*n + 6*b*Log[c*x^n])*Log[1 + (e*x)/d])/(6*e^4) + (b*n*PolyLog[2, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])/(d + e*x)^4, x, 3, (b*d*n)/(6*e^3*(d + e*x)^2) - (2*b*n)/(3*e^3*(d + e*x)) + (x^3*(a + b*Log[c*x^n]))/(3*d*(d + e*x)^3) - (b*n*Log[d + e*x])/(3*d*e^3)} -{x^1*(a + b*Log[c*x^n])/(d + e*x)^4, x, 4, -((b*n)/(6*e^2*(d + e*x)^2)) + (b*n)/(6*d*e^2*(d + e*x)) + (b*n*Log[x])/(6*d^2*e^2) + (d*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x)^3) - (a + b*Log[c*x^n])/(2*e^2*(d + e*x)^2) - (b*n*Log[d + e*x])/(6*d^2*e^2)} -{x^0*(a + b*Log[c*x^n])/(d + e*x)^4, x, 3, (b*n)/(6*d*e*(d + e*x)^2) + (b*n)/(3*d^2*e*(d + e*x)) + (b*n*Log[x])/(3*d^3*e) - (a + b*Log[c*x^n])/(3*e*(d + e*x)^3) - (b*n*Log[d + e*x])/(3*d^3*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)^4), x, 13, -((b*n)/(6*d^2*(d + e*x)^2)) - (5*b*n)/(6*d^3*(d + e*x)) - (5*b*n*Log[x])/(6*d^4) + (a + b*Log[c*x^n])/(3*d*(d + e*x)^3) + (a + b*Log[c*x^n])/(2*d^2*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n]))/(d^4*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^4 + (11*b*n*Log[d + e*x])/(6*d^4) + (b*n*PolyLog[2, -(d/(e*x))])/d^4} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)^4), x, 13, -((b*n)/(d^4*x)) + (b*e*n)/(6*d^3*(d + e*x)^2) + (4*b*e*n)/(3*d^4*(d + e*x)) + (4*b*e*n*Log[x])/(3*d^5) - (a + b*Log[c*x^n])/(d^4*x) - (e*(a + b*Log[c*x^n]))/(3*d^2*(d + e*x)^3) - (e*(a + b*Log[c*x^n]))/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n]))/(d^5*(d + e*x)) + (4*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^5 - (13*b*e*n*Log[d + e*x])/(3*d^5) - (4*b*e*n*PolyLog[2, -(d/(e*x))])/d^5} -{(a + b*Log[c*x^n])/(x^3*(d + e*x)^4), x, 14, -((b*n)/(4*d^4*x^2)) + (4*b*e*n)/(d^5*x) - (b*e^2*n)/(6*d^4*(d + e*x)^2) - (11*b*e^2*n)/(6*d^5*(d + e*x)) - (11*b*e^2*n*Log[x])/(6*d^6) - (a + b*Log[c*x^n])/(2*d^4*x^2) + (4*e*(a + b*Log[c*x^n]))/(d^5*x) + (e^2*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)^3) + (3*e^2*(a + b*Log[c*x^n]))/(2*d^4*(d + e*x)^2) - (6*e^3*x*(a + b*Log[c*x^n]))/(d^6*(d + e*x)) - (10*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^6 + (47*b*e^2*n*Log[d + e*x])/(6*d^6) + (10*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^6} - - -{x^8*(a + b*Log[c*x^n])/(d + e*x)^7, x, 13, (28*b*d*n*x)/e^8 - (d*(280*a + 341*b*n)*x)/(10*e^8) - (7*b*n*x^2)/e^7 - (28*b*d*x*Log[c*x^n])/e^8 - (x^8*(a + b*Log[c*x^n]))/(6*e*(d + e*x)^6) - (x^7*(8*a + b*n + 8*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^6*(56*a + 15*b*n + 56*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^5*(168*a + 73*b*n + 168*b*Log[c*x^n]))/(180*e^4*(d + e*x)^3) + (x^2*(280*a + 341*b*n + 280*b*Log[c*x^n]))/(20*e^7) - (x^4*(840*a + 533*b*n + 840*b*Log[c*x^n]))/(360*e^5*(d + e*x)^2) - (x^3*(840*a + 743*b*n + 840*b*Log[c*x^n]))/(90*e^6*(d + e*x)) + (d^2*(280*a + 341*b*n + 280*b*Log[c*x^n])*Log[1 + (e*x)/d])/(10*e^9) + (28*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^9} -{x^7*(a + b*Log[c*x^n])/(d + e*x)^7, x, 12, -((7*b*n*x)/e^7) + ((140*a + 223*b*n)*x)/(20*e^7) + (7*b*x*Log[c*x^n])/e^7 - (x^7*(a + b*Log[c*x^n]))/(6*e*(d + e*x)^6) - (x^6*(7*a + b*n + 7*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^5*(42*a + 13*b*n + 42*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^2*(140*a + 153*b*n + 140*b*Log[c*x^n]))/(40*e^6*(d + e*x)) - (x^4*(210*a + 107*b*n + 210*b*Log[c*x^n]))/(360*e^4*(d + e*x)^3) - (x^3*(420*a + 319*b*n + 420*b*Log[c*x^n]))/(360*e^5*(d + e*x)^2) - (d*(140*a + 223*b*n + 140*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^8) - (7*b*d*n*PolyLog[2, -((e*x)/d)])/e^8} -{x^6*(a + b*Log[c*x^n])/(d + e*x)^7, x, 8, -((x^6*(a + b*Log[c*x^n]))/(6*e*(d + e*x)^6)) - (x^5*(6*a + b*n + 6*b*Log[c*x^n]))/(30*e^2*(d + e*x)^5) - (x^2*(20*a + 19*b*n + 20*b*Log[c*x^n]))/(40*e^5*(d + e*x)^2) - (x*(20*a + 29*b*n + 20*b*Log[c*x^n]))/(20*e^6*(d + e*x)) - (x^4*(30*a + 11*b*n + 30*b*Log[c*x^n]))/(120*e^3*(d + e*x)^4) - (x^3*(60*a + 37*b*n + 60*b*Log[c*x^n]))/(180*e^4*(d + e*x)^3) + ((20*a + 49*b*n + 20*b*Log[c*x^n])*Log[1 + (e*x)/d])/(20*e^7) + (b*n*PolyLog[2, -((e*x)/d)])/e^7} -{x^5*(a + b*Log[c*x^n])/(d + e*x)^7, x, 3, -((b*d^4*n)/(30*e^6*(d + e*x)^5)) + (5*b*d^3*n)/(24*e^6*(d + e*x)^4) - (5*b*d^2*n)/(9*e^6*(d + e*x)^3) + (5*b*d*n)/(6*e^6*(d + e*x)^2) - (5*b*n)/(6*e^6*(d + e*x)) + (x^6*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) - (b*n*Log[d + e*x])/(6*d*e^6)} -{x^4*(a + b*Log[c*x^n])/(d + e*x)^7, x, 5, -((b*n*x^5)/(30*d^2*(d + e*x)^5)) + (b*d^2*n)/(120*e^5*(d + e*x)^4) - (2*b*d*n)/(45*e^5*(d + e*x)^3) + (b*n)/(10*e^5*(d + e*x)^2) - (2*b*n)/(15*d*e^5*(d + e*x)) + (x^5*(a + b*Log[c*x^n]))/(6*d*(d + e*x)^6) + (x^5*(a + b*Log[c*x^n]))/(30*d^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^2*e^5)} -{x^3*(a + b*Log[c*x^n])/(d + e*x)^7, x, 4, -((b*d^2*n)/(30*e^4*(d + e*x)^5)) + (13*b*d*n)/(120*e^4*(d + e*x)^4) - (19*b*n)/(180*e^4*(d + e*x)^3) + (b*n)/(120*d*e^4*(d + e*x)^2) + (b*n)/(60*d^2*e^4*(d + e*x)) + (b*n*Log[x])/(60*d^3*e^4) + (d^3*(a + b*Log[c*x^n]))/(6*e^4*(d + e*x)^6) - (3*d^2*(a + b*Log[c*x^n]))/(5*e^4*(d + e*x)^5) + (3*d*(a + b*Log[c*x^n]))/(4*e^4*(d + e*x)^4) - (a + b*Log[c*x^n])/(3*e^4*(d + e*x)^3) - (b*n*Log[d + e*x])/(60*d^3*e^4)} -{x^2*(a + b*Log[c*x^n])/(d + e*x)^7, x, 4, (b*d*n)/(30*e^3*(d + e*x)^5) - (7*b*n)/(120*e^3*(d + e*x)^4) + (b*n)/(180*d*e^3*(d + e*x)^3) + (b*n)/(120*d^2*e^3*(d + e*x)^2) + (b*n)/(60*d^3*e^3*(d + e*x)) + (b*n*Log[x])/(60*d^4*e^3) - (d^2*(a + b*Log[c*x^n]))/(6*e^3*(d + e*x)^6) + (2*d*(a + b*Log[c*x^n]))/(5*e^3*(d + e*x)^5) - (a + b*Log[c*x^n])/(4*e^3*(d + e*x)^4) - (b*n*Log[d + e*x])/(60*d^4*e^3)} -{x^1*(a + b*Log[c*x^n])/(d + e*x)^7, x, 4, -((b*n)/(30*e^2*(d + e*x)^5)) + (b*n)/(120*d*e^2*(d + e*x)^4) + (b*n)/(90*d^2*e^2*(d + e*x)^3) + (b*n)/(60*d^3*e^2*(d + e*x)^2) + (b*n)/(30*d^4*e^2*(d + e*x)) + (b*n*Log[x])/(30*d^5*e^2) + (d*(a + b*Log[c*x^n]))/(6*e^2*(d + e*x)^6) - (a + b*Log[c*x^n])/(5*e^2*(d + e*x)^5) - (b*n*Log[d + e*x])/(30*d^5*e^2)} -{x^0*(a + b*Log[c*x^n])/(d + e*x)^7, x, 3, (b*n)/(30*d*e*(d + e*x)^5) + (b*n)/(24*d^2*e*(d + e*x)^4) + (b*n)/(18*d^3*e*(d + e*x)^3) + (b*n)/(12*d^4*e*(d + e*x)^2) + (b*n)/(6*d^5*e*(d + e*x)) + (b*n*Log[x])/(6*d^6*e) - (a + b*Log[c*x^n])/(6*e*(d + e*x)^6) - (b*n*Log[d + e*x])/(6*d^6*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)^7), x, 25, -((b*n)/(30*d^2*(d + e*x)^5)) - (11*b*n)/(120*d^3*(d + e*x)^4) - (37*b*n)/(180*d^4*(d + e*x)^3) - (19*b*n)/(40*d^5*(d + e*x)^2) - (29*b*n)/(20*d^6*(d + e*x)) - (29*b*n*Log[x])/(20*d^7) + (a + b*Log[c*x^n])/(6*d*(d + e*x)^6) + (a + b*Log[c*x^n])/(5*d^2*(d + e*x)^5) + (a + b*Log[c*x^n])/(4*d^3*(d + e*x)^4) + (a + b*Log[c*x^n])/(3*d^4*(d + e*x)^3) + (a + b*Log[c*x^n])/(2*d^5*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n]))/(d^7*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^7 + (49*b*n*Log[d + e*x])/(20*d^7) + (b*n*PolyLog[2, -(d/(e*x))])/d^7} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)^7), x, 22, -((b*n)/(d^7*x)) + (b*e*n)/(30*d^3*(d + e*x)^5) + (17*b*e*n)/(120*d^4*(d + e*x)^4) + (79*b*e*n)/(180*d^5*(d + e*x)^3) + (53*b*e*n)/(40*d^6*(d + e*x)^2) + (103*b*e*n)/(20*d^7*(d + e*x)) + (103*b*e*n*Log[x])/(20*d^8) - (a + b*Log[c*x^n])/(d^7*x) - (e*(a + b*Log[c*x^n]))/(6*d^2*(d + e*x)^6) - (2*e*(a + b*Log[c*x^n]))/(5*d^3*(d + e*x)^5) - (3*e*(a + b*Log[c*x^n]))/(4*d^4*(d + e*x)^4) - (4*e*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)^3) - (5*e*(a + b*Log[c*x^n]))/(2*d^6*(d + e*x)^2) + (6*e^2*x*(a + b*Log[c*x^n]))/(d^8*(d + e*x)) + (7*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^8 - (223*b*e*n*Log[d + e*x])/(20*d^8) - (7*b*e*n*PolyLog[2, -(d/(e*x))])/d^8} -{(a + b*Log[c*x^n])/(x^3*(d + e*x)^7), x, 23, -((b*n)/(4*d^7*x^2)) + (7*b*e*n)/(d^8*x) - (b*e^2*n)/(30*d^4*(d + e*x)^5) - (23*b*e^2*n)/(120*d^5*(d + e*x)^4) - (34*b*e^2*n)/(45*d^6*(d + e*x)^3) - (14*b*e^2*n)/(5*d^7*(d + e*x)^2) - (131*b*e^2*n)/(10*d^8*(d + e*x)) - (131*b*e^2*n*Log[x])/(10*d^9) - (a + b*Log[c*x^n])/(2*d^7*x^2) + (7*e*(a + b*Log[c*x^n]))/(d^8*x) + (e^2*(a + b*Log[c*x^n]))/(6*d^3*(d + e*x)^6) + (3*e^2*(a + b*Log[c*x^n]))/(5*d^4*(d + e*x)^5) + (3*e^2*(a + b*Log[c*x^n]))/(2*d^5*(d + e*x)^4) + (10*e^2*(a + b*Log[c*x^n]))/(3*d^6*(d + e*x)^3) + (15*e^2*(a + b*Log[c*x^n]))/(2*d^7*(d + e*x)^2) - (21*e^3*x*(a + b*Log[c*x^n]))/(d^9*(d + e*x)) - (28*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^9 + (341*b*e^2*n*Log[d + e*x])/(10*d^9) + (28*b*e^2*n*PolyLog[2, -(d/(e*x))])/d^9} - - -{Log[c*x]/(1 - c*x), x, 1, PolyLog[2, 1 - c*x]/c} -{Log[x/c]/(c - x), x, 1, PolyLog[2, 1 - x/c]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b Log[c x^n])^2*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^2*(d + e*x)*(a + b*Log[c*x^n])^2, x, 6, (2/27)*b^2*d*n^2*x^3 + (1/32)*b^2*e*n^2*x^4 - (2/9)*b*d*n*x^3*(a + b*Log[c*x^n]) - (1/8)*b*e*n*x^4*(a + b*Log[c*x^n]) + (1/3)*d*x^3*(a + b*Log[c*x^n])^2 + (1/4)*e*x^4*(a + b*Log[c*x^n])^2} -{x^1*(d + e*x)*(a + b*Log[c*x^n])^2, x, 6, (1/4)*b^2*d*n^2*x^2 + (2/27)*b^2*e*n^2*x^3 - (1/2)*b*d*n*x^2*(a + b*Log[c*x^n]) - (2/9)*b*e*n*x^3*(a + b*Log[c*x^n]) + (1/2)*d*x^2*(a + b*Log[c*x^n])^2 + (1/3)*e*x^3*(a + b*Log[c*x^n])^2} -{x^0*(d + e*x)*(a + b*Log[c*x^n])^2, x, 7, -2*a*b*d*n*x + 2*b^2*d*n^2*x + (1/4)*b^2*e*n^2*x^2 - 2*b^2*d*n*x*Log[c*x^n] - (1/2)*b*e*n*x^2*(a + b*Log[c*x^n]) + d*x*(a + b*Log[c*x^n])^2 + (1/2)*e*x^2*(a + b*Log[c*x^n])^2} -{(d + e*x)*(a + b*Log[c*x^n])^2/x^1, x, 6, -2*a*b*e*n*x + 2*b^2*e*n^2*x - 2*b^2*e*n*x*Log[c*x^n] + e*x*(a + b*Log[c*x^n])^2 + (d*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x)*(a + b*Log[c*x^n])^2/x^2, x, 6, -((2*b^2*d*n^2)/x) - (2*b*d*n*(a + b*Log[c*x^n]))/x - (d*(a + b*Log[c*x^n])^2)/x + (e*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x)*(a + b*Log[c*x^n])^2/x^3, x, 6, -((b^2*d*n^2)/(4*x^2)) - (2*b^2*e*n^2)/x - (b*d*n*(a + b*Log[c*x^n]))/(2*x^2) - (2*b*e*n*(a + b*Log[c*x^n]))/x - (d*(a + b*Log[c*x^n])^2)/(2*x^2) - (e*(a + b*Log[c*x^n])^2)/x} -{(d + e*x)*(a + b*Log[c*x^n])^2/x^4, x, 6, -((2*b^2*d*n^2)/(27*x^3)) - (b^2*e*n^2)/(4*x^2) - (2*b*d*n*(a + b*Log[c*x^n]))/(9*x^3) - (b*e*n*(a + b*Log[c*x^n]))/(2*x^2) - (d*(a + b*Log[c*x^n])^2)/(3*x^3) - (e*(a + b*Log[c*x^n])^2)/(2*x^2)} -{(d + e*x)*(a + b*Log[c*x^n])^2/x^5, x, 6, -((b^2*d*n^2)/(32*x^4)) - (2*b^2*e*n^2)/(27*x^3) - (b*d*n*(a + b*Log[c*x^n]))/(8*x^4) - (2*b*e*n*(a + b*Log[c*x^n]))/(9*x^3) - (d*(a + b*Log[c*x^n])^2)/(4*x^4) - (e*(a + b*Log[c*x^n])^2)/(3*x^3)} - - -{x^2*(d + e*x)^2*(a + b*Log[c*x^n])^2, x, 8, (2/27)*b^2*d^2*n^2*x^3 + (1/16)*b^2*d*e*n^2*x^4 + (2/125)*b^2*e^2*n^2*x^5 - (2/9)*b*d^2*n*x^3*(a + b*Log[c*x^n]) - (1/4)*b*d*e*n*x^4*(a + b*Log[c*x^n]) - (2/25)*b*e^2*n*x^5*(a + b*Log[c*x^n]) + (1/3)*d^2*x^3*(a + b*Log[c*x^n])^2 + (1/2)*d*e*x^4*(a + b*Log[c*x^n])^2 + (1/5)*e^2*x^5*(a + b*Log[c*x^n])^2} -{x^1*(d + e*x)^2*(a + b*Log[c*x^n])^2, x, 8, (1/4)*b^2*d^2*n^2*x^2 + (4/27)*b^2*d*e*n^2*x^3 + (1/32)*b^2*e^2*n^2*x^4 - (1/2)*b*d^2*n*x^2*(a + b*Log[c*x^n]) - (4/9)*b*d*e*n*x^3*(a + b*Log[c*x^n]) - (1/8)*b*e^2*n*x^4*(a + b*Log[c*x^n]) + (1/2)*d^2*x^2*(a + b*Log[c*x^n])^2 + (2/3)*d*e*x^3*(a + b*Log[c*x^n])^2 + (1/4)*e^2*x^4*(a + b*Log[c*x^n])^2} -{x^0*(d + e*x)^2*(a + b*Log[c*x^n])^2, x, 5, 2*b^2*d^2*n^2*x + (1/2)*b^2*d*e*n^2*x^2 + (2/27)*b^2*e^2*n^2*x^3 + (b^2*d^3*n^2*Log[x]^2)/(3*e) - 2*b*d^2*n*x*(a + b*Log[c*x^n]) - b*d*e*n*x^2*(a + b*Log[c*x^n]) - (2/9)*b*e^2*n*x^3*(a + b*Log[c*x^n]) - (2*b*d^3*n*Log[x]*(a + b*Log[c*x^n]))/(3*e) + ((d + e*x)^3*(a + b*Log[c*x^n])^2)/(3*e)} -{(d + e*x)^2*(a + b*Log[c*x^n])^2/x^1, x, 14, -4*a*b*d*e*n*x + 4*b^2*d*e*n^2*x + (1/4)*b^2*e^2*n^2*x^2 - 4*b^2*d*e*n*x*Log[c*x^n] - (1/2)*b*e^2*n*x^2*(a + b*Log[c*x^n]) + 2*d*e*x*(a + b*Log[c*x^n])^2 + (1/2)*e^2*x^2*(a + b*Log[c*x^n])^2 + (d^2*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x)^2*(a + b*Log[c*x^n])^2/x^2, x, 9, -((2*b^2*d^2*n^2)/x) - 2*a*b*e^2*n*x + 2*b^2*e^2*n^2*x - 2*b^2*e^2*n*x*Log[c*x^n] - (2*b*d^2*n*(a + b*Log[c*x^n]))/x - (d^2*(a + b*Log[c*x^n])^2)/x + e^2*x*(a + b*Log[c*x^n])^2 + (2*d*e*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x)^2*(a + b*Log[c*x^n])^2/x^3, x, 8, -((b^2*d^2*n^2)/(4*x^2)) - (4*b^2*d*e*n^2)/x - (b*d^2*n*(a + b*Log[c*x^n]))/(2*x^2) - (4*b*d*e*n*(a + b*Log[c*x^n]))/x - (d^2*(a + b*Log[c*x^n])^2)/(2*x^2) - (2*d*e*(a + b*Log[c*x^n])^2)/x + (e^2*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x)^2*(a + b*Log[c*x^n])^2/x^4, x, 8, -((2*b^2*d^2*n^2)/(27*x^3)) - (b^2*d*e*n^2)/(2*x^2) - (2*b^2*e^2*n^2)/x - (2*b*d^2*n*(a + b*Log[c*x^n]))/(9*x^3) - (b*d*e*n*(a + b*Log[c*x^n]))/x^2 - (2*b*e^2*n*(a + b*Log[c*x^n]))/x - (d^2*(a + b*Log[c*x^n])^2)/(3*x^3) - (d*e*(a + b*Log[c*x^n])^2)/x^2 - (e^2*(a + b*Log[c*x^n])^2)/x} -{(d + e*x)^2*(a + b*Log[c*x^n])^2/x^5, x, 8, -((b^2*d^2*n^2)/(32*x^4)) - (4*b^2*d*e*n^2)/(27*x^3) - (b^2*e^2*n^2)/(4*x^2) - (b*d^2*n*(a + b*Log[c*x^n]))/(8*x^4) - (4*b*d*e*n*(a + b*Log[c*x^n]))/(9*x^3) - (b*e^2*n*(a + b*Log[c*x^n]))/(2*x^2) - (d^2*(a + b*Log[c*x^n])^2)/(4*x^4) - (2*d*e*(a + b*Log[c*x^n])^2)/(3*x^3) - (e^2*(a + b*Log[c*x^n])^2)/(2*x^2)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3*(a + b*Log[c*x^n])^2/(d + e*x), x, 12, -((2*a*b*d^2*n*x)/e^3) + (2*b^2*d^2*n^2*x)/e^3 - (b^2*d*n^2*x^2)/(4*e^2) + (2*b^2*n^2*x^3)/(27*e) - (2*b^2*d^2*n*x*Log[c*x^n])/e^3 + (b*d*n*x^2*(a + b*Log[c*x^n]))/(2*e^2) - (2*b*n*x^3*(a + b*Log[c*x^n]))/(9*e) + (d^2*x*(a + b*Log[c*x^n])^2)/e^3 - (d*x^2*(a + b*Log[c*x^n])^2)/(2*e^2) + (x^3*(a + b*Log[c*x^n])^2)/(3*e) - (d^3*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 - (2*b*d^3*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 + (2*b^2*d^3*n^2*PolyLog[3, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])^2/(d + e*x), x, 10, (2*a*b*d*n*x)/e^2 - (2*b^2*d*n^2*x)/e^2 + (b^2*n^2*x^2)/(4*e) + (2*b^2*d*n*x*Log[c*x^n])/e^2 - (b*n*x^2*(a + b*Log[c*x^n]))/(2*e) - (d*x*(a + b*Log[c*x^n])^2)/e^2 + (x^2*(a + b*Log[c*x^n])^2)/(2*e) + (d^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^3 + (2*b*d^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^3 - (2*b^2*d^2*n^2*PolyLog[3, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])^2/(d + e*x), x, 8, -((2*a*b*n*x)/e) + (2*b^2*n^2*x)/e - (2*b^2*n*x*Log[c*x^n])/e + (x*(a + b*Log[c*x^n])^2)/e - (d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^2 - (2*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^2 + (2*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^2} -{x^0*(a + b*Log[c*x^n])^2/(d + e*x), x, 3, ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x)), x, 3, -((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d + (2*b^2*n^2*PolyLog[3, -(d/(e*x))])/d} -{(a + b*Log[c*x^n])^2/(x^2*(d + e*x)), x, 6, -((2*b^2*n^2)/(d*x)) - (2*b*n*(a + b*Log[c*x^n]))/(d*x) - (a + b*Log[c*x^n])^2/(d*x) + (e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^2 - (2*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^2 - (2*b^2*e*n^2*PolyLog[3, -(d/(e*x))])/d^2} -{(a + b*Log[c*x^n])^2/(x^3*(d + e*x)), x, 9, -((b^2*n^2)/(4*d*x^2)) + (2*b^2*e*n^2)/(d^2*x) - (b*n*(a + b*Log[c*x^n]))/(2*d*x^2) + (2*b*e*n*(a + b*Log[c*x^n]))/(d^2*x) - (a + b*Log[c*x^n])^2/(2*d*x^2) + (e*(a + b*Log[c*x^n])^2)/(d^2*x) - (e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^3 + (2*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 + (2*b^2*e^2*n^2*PolyLog[3, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])^2/(x^4*(d + e*x)), x, 12, -((2*b^2*n^2)/(27*d*x^3)) + (b^2*e*n^2)/(4*d^2*x^2) - (2*b^2*e^2*n^2)/(d^3*x) - (2*b*n*(a + b*Log[c*x^n]))/(9*d*x^3) + (b*e*n*(a + b*Log[c*x^n]))/(2*d^2*x^2) - (2*b*e^2*n*(a + b*Log[c*x^n]))/(d^3*x) - (a + b*Log[c*x^n])^2/(3*d*x^3) + (e*(a + b*Log[c*x^n])^2)/(2*d^2*x^2) - (e^2*(a + b*Log[c*x^n])^2)/(d^3*x) + (e^3*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^4 - (2*b*e^3*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 - (2*b^2*e^3*n^2*PolyLog[3, -(d/(e*x))])/d^4} - - -{x^3*(a + b*Log[c*x^n])^2/(d + e*x)^2, x, 13, (4*a*b*d*n*x)/e^3 - (4*b^2*d*n^2*x)/e^3 + (b^2*n^2*x^2)/(4*e^2) + (4*b^2*d*n*x*Log[c*x^n])/e^3 - (b*n*x^2*(a + b*Log[c*x^n]))/(2*e^2) - (2*d*x*(a + b*Log[c*x^n])^2)/e^3 + (x^2*(a + b*Log[c*x^n])^2)/(2*e^2) - (d^2*x*(a + b*Log[c*x^n])^2)/(e^3*(d + e*x)) + (2*b*d^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + (3*d^2*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 + (2*b^2*d^2*n^2*PolyLog[2, -((e*x)/d)])/e^4 + (6*b*d^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 - (6*b^2*d^2*n^2*PolyLog[3, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])^2/(d + e*x)^2, x, 11, -((2*a*b*n*x)/e^2) + (2*b^2*n^2*x)/e^2 - (2*b^2*n*x*Log[c*x^n])/e^2 + (x*(a + b*Log[c*x^n])^2)/e^2 + (d*x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) - (2*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 - (2*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^3 - (2*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/e^3 - (4*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^3 + (4*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])^2/(d + e*x)^2, x, 8, -((x*(a + b*Log[c*x^n])^2)/(e*(d + e*x))) + (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^2 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^2 + (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/e^2 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^2 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^2} -{x^0*(a + b*Log[c*x^n])^2/(d + e*x)^2, x, 3, (x*(a + b*Log[c*x^n])^2)/(d*(d + e*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(d*e) - (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/(d*e)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x)^2), x, 7, -((e*x*(a + b*Log[c*x^n])^2)/(d^2*(d + e*x))) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^2 + (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^2 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^2 + (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/d^2 + (2*b^2*n^2*PolyLog[3, -(d/(e*x))])/d^2} -{(a + b*Log[c*x^n])^2/(x^2*(d + e*x)^2), x, 10, -((2*b^2*n^2)/(d^2*x)) - (2*b*n*(a + b*Log[c*x^n]))/(d^2*x) - (a + b*Log[c*x^n])^2/(d^2*x) + (e^2*x*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)) + (2*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^3 - (2*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 - (4*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 - (2*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^3 - (4*b^2*e*n^2*PolyLog[3, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])^2/(x^3*(d + e*x)^2), x, 12, -((b^2*n^2)/(4*d^2*x^2)) + (4*b^2*e*n^2)/(d^3*x) - (b*n*(a + b*Log[c*x^n]))/(2*d^2*x^2) + (4*b*e*n*(a + b*Log[c*x^n]))/(d^3*x) - (a + b*Log[c*x^n])^2/(2*d^2*x^2) + (2*e*(a + b*Log[c*x^n])^2)/(d^3*x) - (e^3*x*(a + b*Log[c*x^n])^2)/(d^4*(d + e*x)) - (3*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^4 + (2*b*e^2*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^4 + (6*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 + (2*b^2*e^2*n^2*PolyLog[2, -((e*x)/d)])/d^4 + (6*b^2*e^2*n^2*PolyLog[3, -(d/(e*x))])/d^4} - - -{x^3*(a + b*Log[c*x^n])^2/(d + e*x)^3, x, 17, -((2*a*b*n*x)/e^3) + (2*b^2*n^2*x)/e^3 - (2*b^2*n*x*Log[c*x^n])/e^3 + (b*d*n*x*(a + b*Log[c*x^n]))/(e^3*(d + e*x)) - (d*(a + b*Log[c*x^n])^2)/(2*e^4) + (x*(a + b*Log[c*x^n])^2)/e^3 + (d^3*(a + b*Log[c*x^n])^2)/(2*e^4*(d + e*x)^2) + (3*d*x*(a + b*Log[c*x^n])^2)/(e^3*(d + e*x)) - (b^2*d*n^2*Log[d + e*x])/e^4 - (5*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 - (3*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 - (5*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/e^4 - (6*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 + (6*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^4, -((2*a*b*n*x)/e^3) + (2*b^2*n^2*x)/e^3 - (2*b^2*n*x*Log[c*x^n])/e^3 + (b*d*n*x*(a + b*Log[c*x^n]))/(e^3*(d + e*x)) + (b*d*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/e^4 + (x*(a + b*Log[c*x^n])^2)/e^3 + (d^3*(a + b*Log[c*x^n])^2)/(2*e^4*(d + e*x)^2) + (3*d*x*(a + b*Log[c*x^n])^2)/(e^3*(d + e*x)) - (b^2*d*n^2*Log[d + e*x])/e^4 - (6*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 - (3*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 - (b^2*d*n^2*PolyLog[2, -(d/(e*x))])/e^4 - (6*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/e^4 - (6*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 + (6*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])^2/(d + e*x)^3, x, 14, -((b*n*x*(a + b*Log[c*x^n]))/(e^2*(d + e*x))) + (a + b*Log[c*x^n])^2/(2*e^3) - (d^2*(a + b*Log[c*x^n])^2)/(2*e^3*(d + e*x)^2) - (2*x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) + (b^2*n^2*Log[d + e*x])/e^3 + (3*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^3 + (3*b^2*n^2*PolyLog[2, -((e*x)/d)])/e^3 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^3 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^3, -((b*n*x*(a + b*Log[c*x^n]))/(e^2*(d + e*x))) - (b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/e^3 - (d^2*(a + b*Log[c*x^n])^2)/(2*e^3*(d + e*x)^2) - (2*x*(a + b*Log[c*x^n])^2)/(e^2*(d + e*x)) + (b^2*n^2*Log[d + e*x])/e^3 + (4*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^3 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^3 + (b^2*n^2*PolyLog[2, -(d/(e*x))])/e^3 + (4*b^2*n^2*PolyLog[2, -((e*x)/d)])/e^3 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^3 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^3} -{x^1*(a + b*Log[c*x^n])^2/(d + e*x)^3, x, 4, (b*n*x*(a + b*Log[c*x^n]))/(d*e*(d + e*x)) + (x^2*(a + b*Log[c*x^n])^2)/(2*d*(d + e*x)^2) - (b*n*(a + b*n + b*Log[c*x^n])*Log[1 + (e*x)/d])/(d*e^2) - (b^2*n^2*PolyLog[2, -((e*x)/d)])/(d*e^2)} -{x^0*(a + b*Log[c*x^n])^2/(d + e*x)^3, x, 6, -((b*n*x*(a + b*Log[c*x^n]))/(d^2*(d + e*x))) - (b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(d^2*e) - (a + b*Log[c*x^n])^2/(2*e*(d + e*x)^2) + (b^2*n^2*Log[d + e*x])/(d^2*e) + (b^2*n^2*PolyLog[2, -(d/(e*x))])/(d^2*e)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x)^3), x, 14, (b*e*n*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) - (a + b*Log[c*x^n])^2/(2*d^3) + (a + b*Log[c*x^n])^2/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)) + (a + b*Log[c*x^n])^3/(3*b*d^3*n) - (b^2*n^2*Log[d + e*x])/d^3 + (3*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 - ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^3 + (3*b^2*n^2*PolyLog[2, -((e*x)/d)])/d^3 - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^3 + (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/d^3, (b*e*n*x*(a + b*Log[c*x^n]))/(d^3*(d + e*x)) + (b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^3 + (a + b*Log[c*x^n])^2/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^3 - (b^2*n^2*Log[d + e*x])/d^3 + (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 - (b^2*n^2*PolyLog[2, -(d/(e*x))])/d^3 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 + (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/d^3 + (2*b^2*n^2*PolyLog[3, -(d/(e*x))])/d^3} -{(a + b*Log[c*x^n])^2/(x^2*(d + e*x)^3), x, 16, -((2*b^2*n^2)/(d^3*x)) - (2*b*n*(a + b*Log[c*x^n]))/(d^3*x) - (b*e^2*n*x*(a + b*Log[c*x^n]))/(d^4*(d + e*x)) + (e*(a + b*Log[c*x^n])^2)/(2*d^4) - (a + b*Log[c*x^n])^2/(d^3*x) - (e*(a + b*Log[c*x^n])^2)/(2*d^2*(d + e*x)^2) + (2*e^2*x*(a + b*Log[c*x^n])^2)/(d^4*(d + e*x)) - (e*(a + b*Log[c*x^n])^3)/(b*d^4*n) + (b^2*e*n^2*Log[d + e*x])/d^4 - (5*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^4 + (3*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^4 - (5*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^4 + (6*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^4 - (6*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^4, -((2*b^2*n^2)/(d^3*x)) - (2*b*n*(a + b*Log[c*x^n]))/(d^3*x) - (b*e^2*n*x*(a + b*Log[c*x^n]))/(d^4*(d + e*x)) - (b*e*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^4 - (a + b*Log[c*x^n])^2/(d^3*x) - (e*(a + b*Log[c*x^n])^2)/(2*d^2*(d + e*x)^2) + (2*e^2*x*(a + b*Log[c*x^n])^2)/(d^4*(d + e*x)) + (3*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^4 + (b^2*e*n^2*Log[d + e*x])/d^4 - (4*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^4 + (b^2*e*n^2*PolyLog[2, -(d/(e*x))])/d^4 - (6*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 - (4*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^4 - (6*b^2*e*n^2*PolyLog[3, -(d/(e*x))])/d^4} - - -{x^4*(a + b*Log[c*x^n])^2/(d + e*x)^4, x, 27, -((2*a*b*n*x)/e^4) + (2*b^2*n^2*x)/e^4 - (b^2*d^2*n^2)/(3*e^5*(d + e*x)) - (b^2*d*n^2*Log[x])/(3*e^5) - (2*b^2*n*x*Log[c*x^n])/e^4 + (b*d^3*n*(a + b*Log[c*x^n]))/(3*e^5*(d + e*x)^2) + (10*b*d*n*x*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x)) - (5*d*(a + b*Log[c*x^n])^2)/(3*e^5) + (x*(a + b*Log[c*x^n])^2)/e^4 - (d^4*(a + b*Log[c*x^n])^2)/(3*e^5*(d + e*x)^3) + (2*d^3*(a + b*Log[c*x^n])^2)/(e^5*(d + e*x)^2) + (6*d*x*(a + b*Log[c*x^n])^2)/(e^4*(d + e*x)) - (3*b^2*d*n^2*Log[d + e*x])/e^5 - (26*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*e^5) - (4*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^5 - (26*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/(3*e^5) - (8*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^5 + (8*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^5, -((2*a*b*n*x)/e^4) + (2*b^2*n^2*x)/e^4 - (b^2*d^2*n^2)/(3*e^5*(d + e*x)) - (b^2*d*n^2*Log[x])/(3*e^5) - (2*b^2*n*x*Log[c*x^n])/e^4 + (b*d^3*n*(a + b*Log[c*x^n]))/(3*e^5*(d + e*x)^2) + (10*b*d*n*x*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x)) + (10*b*d*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*e^5) + (x*(a + b*Log[c*x^n])^2)/e^4 - (d^4*(a + b*Log[c*x^n])^2)/(3*e^5*(d + e*x)^3) + (2*d^3*(a + b*Log[c*x^n])^2)/(e^5*(d + e*x)^2) + (6*d*x*(a + b*Log[c*x^n])^2)/(e^4*(d + e*x)) - (3*b^2*d*n^2*Log[d + e*x])/e^5 - (12*b*d*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^5 - (4*d*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^5 - (10*b^2*d*n^2*PolyLog[2, -(d/(e*x))])/(3*e^5) - (12*b^2*d*n^2*PolyLog[2, -((e*x)/d)])/e^5 - (8*b*d*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^5 + (8*b^2*d*n^2*PolyLog[3, -((e*x)/d)])/e^5} -{x^3*(a + b*Log[c*x^n])^2/(d + e*x)^4, x, 24, (b^2*d*n^2)/(3*e^4*(d + e*x)) + (b^2*n^2*Log[x])/(3*e^4) - (b*d^2*n*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x)^2) - (7*b*n*x*(a + b*Log[c*x^n]))/(3*e^3*(d + e*x)) + (7*(a + b*Log[c*x^n])^2)/(6*e^4) + (d^3*(a + b*Log[c*x^n])^2)/(3*e^4*(d + e*x)^3) - (3*d^2*(a + b*Log[c*x^n])^2)/(2*e^4*(d + e*x)^2) - (3*x*(a + b*Log[c*x^n])^2)/(e^3*(d + e*x)) + (2*b^2*n^2*Log[d + e*x])/e^4 + (11*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*e^4) + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 + (11*b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*e^4) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^4, (b^2*d*n^2)/(3*e^4*(d + e*x)) + (b^2*n^2*Log[x])/(3*e^4) - (b*d^2*n*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x)^2) - (7*b*n*x*(a + b*Log[c*x^n]))/(3*e^3*(d + e*x)) - (7*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*e^4) + (d^3*(a + b*Log[c*x^n])^2)/(3*e^4*(d + e*x)^3) - (3*d^2*(a + b*Log[c*x^n])^2)/(2*e^4*(d + e*x)^2) - (3*x*(a + b*Log[c*x^n])^2)/(e^3*(d + e*x)) + (2*b^2*n^2*Log[d + e*x])/e^4 + (6*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^4 + ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e^4 + (7*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*e^4) + (6*b^2*n^2*PolyLog[2, -((e*x)/d)])/e^4 + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/e^4 - (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/e^4} -{x^2*(a + b*Log[c*x^n])^2/(d + e*x)^4, x, 5, (b*n*x^2*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) + (x^3*(a + b*Log[c*x^n])^2)/(3*d*(d + e*x)^3) + (b*n*x*(2*a + b*n + 2*b*Log[c*x^n]))/(3*d*e^2*(d + e*x)) - (b*n*(2*a + 3*b*n + 2*b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d*e^3) - (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*d*e^3)} -{x^1*(a + b*Log[c*x^n])^2/(d + e*x)^4, x, 8, (b^2*n^2)/(3*d*e^2*(d + e*x)) + (b*n*(a + b*Log[c*x^n]))/(3*d*e^2*(d + e*x)) - (b*n*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x)^2) + (a + b*Log[c*x^n])^2/(6*d^2*e^2) + (d*(a + b*Log[c*x^n])^2)/(3*e^2*(d + e*x)^3) - (a + b*Log[c*x^n])^2/(2*e^2*(d + e*x)^2) - (b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d^2*e^2) - (b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*d^2*e^2), (b^2*n^2)/(3*d*e^2*(d + e*x)) - (b*n*x^2*(a + b*Log[c*x^n]))/(3*d^2*(d + e*x)^2) + (b*n*x*(a + b*Log[c*x^n]))/(3*d^2*e*(d + e*x)) + (x^2*(a + b*Log[c*x^n])^2)/(3*d*(d + e*x)^3) + (x^2*(a + b*Log[c*x^n])^2)/(6*d^2*(d + e*x)^2) + (b^2*n^2*Log[d + e*x])/(3*d^2*e^2) - (b*n*(a + b*n + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d^2*e^2) - (b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*d^2*e^2)} -{x^0*(a + b*Log[c*x^n])^2/(d + e*x)^4, x, 10, -((b^2*n^2)/(3*d^2*e*(d + e*x))) - (b^2*n^2*Log[x])/(3*d^3*e) + (b*n*(a + b*Log[c*x^n]))/(3*d*e*(d + e*x)^2) - (2*b*n*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)) - (2*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^3*e) - (a + b*Log[c*x^n])^2/(3*e*(d + e*x)^3) + (b^2*n^2*Log[d + e*x])/(d^3*e) + (2*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*d^3*e)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x)^4), x, 25, (b^2*n^2)/(3*d^3*(d + e*x)) + (b^2*n^2*Log[x])/(3*d^4) - (b*n*(a + b*Log[c*x^n]))/(3*d^2*(d + e*x)^2) + (5*b*e*n*x*(a + b*Log[c*x^n]))/(3*d^4*(d + e*x)) - (5*(a + b*Log[c*x^n])^2)/(6*d^4) + (a + b*Log[c*x^n])^2/(3*d*(d + e*x)^3) + (a + b*Log[c*x^n])^2/(2*d^2*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^2)/(d^4*(d + e*x)) + (a + b*Log[c*x^n])^3/(3*b*d^4*n) - (2*b^2*n^2*Log[d + e*x])/d^4 + (11*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d^4) - ((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^4 + (11*b^2*n^2*PolyLog[2, -((e*x)/d)])/(3*d^4) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^4 + (2*b^2*n^2*PolyLog[3, -((e*x)/d)])/d^4, (b^2*n^2)/(3*d^3*(d + e*x)) + (b^2*n^2*Log[x])/(3*d^4) - (b*n*(a + b*Log[c*x^n]))/(3*d^2*(d + e*x)^2) + (5*b*e*n*x*(a + b*Log[c*x^n]))/(3*d^4*(d + e*x)) + (5*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^4) + (a + b*Log[c*x^n])^2/(3*d*(d + e*x)^3) + (a + b*Log[c*x^n])^2/(2*d^2*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^2)/(d^4*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^4 - (2*b^2*n^2*Log[d + e*x])/d^4 + (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^4 - (5*b^2*n^2*PolyLog[2, -(d/(e*x))])/(3*d^4) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^4 + (2*b^2*n^2*PolyLog[2, -((e*x)/d)])/d^4 + (2*b^2*n^2*PolyLog[3, -(d/(e*x))])/d^4} -{(a + b*Log[c*x^n])^2/(x^2*(d + e*x)^4), x, 26, -((2*b^2*n^2)/(d^4*x)) - (b^2*e*n^2)/(3*d^4*(d + e*x)) - (b^2*e*n^2*Log[x])/(3*d^5) - (2*b*n*(a + b*Log[c*x^n]))/(d^4*x) + (b*e*n*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)^2) - (8*b*e^2*n*x*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)) + (4*e*(a + b*Log[c*x^n])^2)/(3*d^5) - (a + b*Log[c*x^n])^2/(d^4*x) - (e*(a + b*Log[c*x^n])^2)/(3*d^2*(d + e*x)^3) - (e*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n])^2)/(d^5*(d + e*x)) - (4*e*(a + b*Log[c*x^n])^3)/(3*b*d^5*n) + (3*b^2*e*n^2*Log[d + e*x])/d^5 - (26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*d^5) + (4*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^5 - (26*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/(3*d^5) + (8*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^5 - (8*b^2*e*n^2*PolyLog[3, -((e*x)/d)])/d^5, -((2*b^2*n^2)/(d^4*x)) - (b^2*e*n^2)/(3*d^4*(d + e*x)) - (b^2*e*n^2*Log[x])/(3*d^5) - (2*b*n*(a + b*Log[c*x^n]))/(d^4*x) + (b*e*n*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x)^2) - (8*b*e^2*n*x*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)) - (8*b*e*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^5) - (a + b*Log[c*x^n])^2/(d^4*x) - (e*(a + b*Log[c*x^n])^2)/(3*d^2*(d + e*x)^3) - (e*(a + b*Log[c*x^n])^2)/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n])^2)/(d^5*(d + e*x)) + (4*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^5 + (3*b^2*e*n^2*Log[d + e*x])/d^5 - (6*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^5 + (8*b^2*e*n^2*PolyLog[2, -(d/(e*x))])/(3*d^5) - (8*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^5 - (6*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^5 - (8*b^2*e*n^2*PolyLog[3, -(d/(e*x))])/d^5} - - -{(x*Log[x]^2)/(d + e*x)^4, x, 8, -(x/(3*d^2*e*(d + e*x))) + (x*Log[x])/(3*d*e*(d + e*x)^2) + (x^2*(3*d + e*x)*Log[x]^2)/(6*d^2*(d + e*x)^3) - (Log[x]*Log[1 + (e*x)/d])/(3*d^2*e^2) - PolyLog[2, -((e*x)/d)]/(3*d^2*e^2), 1/(3*d*e^2*(d + e*x)) - (x^2*Log[x])/(3*d^2*(d + e*x)^2) + (x*Log[x])/(3*d^2*e*(d + e*x)) + (x^2*Log[x]^2)/(3*d*(d + e*x)^3) + (x^2*Log[x]^2)/(6*d^2*(d + e*x)^2) + Log[d + e*x]/(3*d^2*e^2) - ((1 + Log[x])*Log[1 + (e*x)/d])/(3*d^2*e^2) - PolyLog[2, -((e*x)/d)]/(3*d^2*e^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b Log[c x^n])^3*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a + b*Log[c*x^n])^3/(x*(d + e*x)), x, 4, -((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))])/d + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])/d + (6*b^3*n^3*PolyLog[4, -(d/(e*x))])/d} - - -{(a + b*Log[c*x^n])^3/(x*(d + e*x)^2), x, 9, -((e*x*(a + b*Log[c*x^n])^3)/(d^2*(d + e*x))) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d^2 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^2 + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))])/d^2 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^2 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])/d^2 - (6*b^3*n^3*PolyLog[3, -((e*x)/d)])/d^2 + (6*b^3*n^3*PolyLog[4, -(d/(e*x))])/d^2} - - -{(a + b*Log[c*x^n])^3/(x*(d + e*x)^3), x, 18, (3*b*e*n*x*(a + b*Log[c*x^n])^2)/(2*d^3*(d + e*x)) - (a + b*Log[c*x^n])^3/(2*d^3) + (a + b*Log[c*x^n])^3/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^3)/(d^3*(d + e*x)) + (a + b*Log[c*x^n])^4/(4*b*d^3*n) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 + (9*b*n*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/(2*d^3) - ((a + b*Log[c*x^n])^3*Log[1 + (e*x)/d])/d^3 - (3*b^3*n^3*PolyLog[2, -((e*x)/d)])/d^3 + (9*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^3 - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((e*x)/d)])/d^3 - (9*b^3*n^3*PolyLog[3, -((e*x)/d)])/d^3 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((e*x)/d)])/d^3 - (6*b^3*n^3*PolyLog[4, -((e*x)/d)])/d^3, (3*b*e*n*x*(a + b*Log[c*x^n])^2)/(2*d^3*(d + e*x)) + (3*b*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/(2*d^3) + (a + b*Log[c*x^n])^3/(2*d*(d + e*x)^2) - (e*x*(a + b*Log[c*x^n])^3)/(d^3*(d + e*x)) - (Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d^3 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^3 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/d^3 - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))])/d^3 + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))])/d^3 - (3*b^3*n^3*PolyLog[2, -((e*x)/d)])/d^3 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/d^3 - (3*b^3*n^3*PolyLog[3, -(d/(e*x))])/d^3 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d/(e*x))])/d^3 - (6*b^3*n^3*PolyLog[3, -((e*x)/d)])/d^3 + (6*b^3*n^3*PolyLog[4, -(d/(e*x))])/d^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b Log[c x^n])^(1/2)*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(d + e*x)^1*(a + b*Log[c*x^n])^(1/2), x, 10, ((-(1/2))*Sqrt[b]*d*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) - ((1/4)*Sqrt[b]*e*Sqrt[n]*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) + d*x*Sqrt[a + b*Log[c*x^n]] + (1/2)*e*x^2*Sqrt[a + b*Log[c*x^n]]} - - -{(d + e*x)^2*(a + b*Log[c*x^n])^(1/2), x, 14, ((-(1/2))*Sqrt[b]*d^2*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) - ((1/2)*Sqrt[b]*d*e*Sqrt[n]*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) - ((1/6)*Sqrt[b]*e^2*Sqrt[n]*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) + d^2*x*Sqrt[a + b*Log[c*x^n]] + d*e*x^2*Sqrt[a + b*Log[c*x^n]] + (1/3)*e^2*x^3*Sqrt[a + b*Log[c*x^n]]} - - -{(d + e*x)^3*(a + b*Log[c*x^n])^(1/2), x, 18, ((-(1/2))*Sqrt[b]*d^3*Sqrt[n]*Sqrt[Pi]*x*Erfi[Sqrt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*x^n)^n^(-1)) - ((1/16)*Sqrt[b]*e^3*Sqrt[n]*Sqrt[Pi]*x^4*Erfi[(2*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((4*a)/(b*n))*(c*x^n)^(4/n)) - ((3/4)*Sqrt[b]*d^2*e*Sqrt[n]*Sqrt[Pi/2]*x^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) - ((1/2)*Sqrt[b]*d*e^2*Sqrt[n]*Sqrt[Pi/3]*x^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) + d^3*x*Sqrt[a + b*Log[c*x^n]] + (3/2)*d^2*e*x^2*Sqrt[a + b*Log[c*x^n]] + d*e^2*x^3*Sqrt[a + b*Log[c*x^n]] + (1/4)*e^3*x^4*Sqrt[a + b*Log[c*x^n]]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a + b*Log[c*x^n])^(1/2)/(d + e*x)^1, x, 0, Unintegrable[Sqrt[a + b*Log[c*x^n]]/(d + e*x), x]} - - -{(a + b*Log[c*x^n])^(1/2)/(d + e*x)^2, x, 1, (x*Sqrt[a + b*Log[c*x^n]])/(d*(d + e*x)) - (b*n*Unintegrable[1/((d + e*x)*Sqrt[a + b*Log[c*x^n]]), x])/(2*d)} - - -{(a + b*Log[c*x^n])^(1/2)/(d + e*x)^3, x, 1, -(Sqrt[a + b*Log[c*x^n]]/(2*e*(d + e*x)^2)) + (b*n*Unintegrable[1/(x*(d + e*x)^2*Sqrt[a + b*Log[c*x^n]]), x])/(4*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^(q/2) (a+b Log[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*Sqrt[d + e*x]*(a + b*Log[c*x^n]), x, If[$VersionNumber>=8, 8, 9], (64*b*d^4*n*Sqrt[d + e*x])/(315*e^4) + (64*b*d^3*n*(d + e*x)^(3/2))/(945*e^4) - (356*b*d^2*n*(d + e*x)^(5/2))/(1575*e^4) + (80*b*d*n*(d + e*x)^(7/2))/(441*e^4) - (4*b*n*(d + e*x)^(9/2))/(81*e^4) - (64*b*d^(9/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(315*e^4) - (2*d^3*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^4) + (6*d^2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^4) - (6*d*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4) + (2*(d + e*x)^(9/2)*(a + b*Log[c*x^n]))/(9*e^4)} -{x^2*Sqrt[d + e*x]*(a + b*Log[c*x^n]), x, If[$VersionNumber>=8, 6, 7], -((32*b*d^3*n*Sqrt[d + e*x])/(105*e^3)) - (32*b*d^2*n*(d + e*x)^(3/2))/(315*e^3) + (36*b*d*n*(d + e*x)^(5/2))/(175*e^3) - (4*b*n*(d + e*x)^(7/2))/(49*e^3) + (32*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(105*e^3) + (2*d^2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^3)} -{x^1*Sqrt[d + e*x]*(a + b*Log[c*x^n]), x, If[$VersionNumber>=8, 7, 8], (8*b*d^2*n*Sqrt[d + e*x])/(15*e^2) + (8*b*d*n*(d + e*x)^(3/2))/(45*e^2) - (4*b*n*(d + e*x)^(5/2))/(25*e^2) - (8*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(15*e^2) - (2*d*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^2)} -{x^0*Sqrt[d + e*x]*(a + b*Log[c*x^n]), x, 5, -((4*b*d*n*Sqrt[d + e*x])/(3*e)) - (4*b*n*(d + e*x)^(3/2))/(9*e) + (4*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3*e) + (2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e)} -{Sqrt[d + e*x]*(a + b*Log[c*x^n])/x^1, x, 12, -4*b*n*Sqrt[d + e*x] + 4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2 + 2*Sqrt[d + e*x]*(a + b*Log[c*x^n]) - 2*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]) - 4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] - 2*b*Sqrt[d]*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]} -{Sqrt[d + e*x]*(a + b*Log[c*x^n])/x^2, x, 11, -((b*n*Sqrt[d + e*x])/x) - (b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/Sqrt[d] - (Sqrt[d + e*x]*(a + b*Log[c*x^n]))/x - (e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[d] - (2*b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/Sqrt[d] - (b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/Sqrt[d]} -{Sqrt[d + e*x]*(a + b*Log[c*x^n])/x^3, x, 16, -((b*n*Sqrt[d + e*x])/(4*x^2)) - (3*b*e*n*Sqrt[d + e*x])/(8*d*x) - (b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(8*d^(3/2)) - (b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/(4*d^(3/2)) - (Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(2*x^2) - (e*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(4*d*x) + (e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/(4*d^(3/2)) + (b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(2*d^(3/2)) + (b*e^2*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(4*d^(3/2))} - - -{x^3*(d + e*x)^(3/2)*(a + b*Log[c*x^n]), x, 9, (64*b*d^5*n*Sqrt[d + e*x])/(1155*e^4) + (64*b*d^4*n*(d + e*x)^(3/2))/(3465*e^4) + (64*b*d^3*n*(d + e*x)^(5/2))/(5775*e^4) - (172*b*d^2*n*(d + e*x)^(7/2))/(1617*e^4) + (32*b*d*n*(d + e*x)^(9/2))/(297*e^4) - (4*b*n*(d + e*x)^(11/2))/(121*e^4) - (64*b*d^(11/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(1155*e^4) - (2*d^3*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^4) + (6*d^2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4) - (2*d*(d + e*x)^(9/2)*(a + b*Log[c*x^n]))/(3*e^4) + (2*(d + e*x)^(11/2)*(a + b*Log[c*x^n]))/(11*e^4)} -{x^2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]), x, 6, -((32*b*d^4*n*Sqrt[d + e*x])/(315*e^3)) - (32*b*d^3*n*(d + e*x)^(3/2))/(945*e^3) - (32*b*d^2*n*(d + e*x)^(5/2))/(1575*e^3) + (44*b*d*n*(d + e*x)^(7/2))/(441*e^3) - (4*b*n*(d + e*x)^(9/2))/(81*e^3) + (32*b*d^(9/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(315*e^3) + (2*d^2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) - (4*d*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^3) + (2*(d + e*x)^(9/2)*(a + b*Log[c*x^n]))/(9*e^3)} -{x^1*(d + e*x)^(3/2)*(a + b*Log[c*x^n]), x, 8, (8*b*d^3*n*Sqrt[d + e*x])/(35*e^2) + (8*b*d^2*n*(d + e*x)^(3/2))/(105*e^2) + (8*b*d*n*(d + e*x)^(5/2))/(175*e^2) - (4*b*n*(d + e*x)^(7/2))/(49*e^2) - (8*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(35*e^2) - (2*d*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^2) + (2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^2)} -{x^0*(d + e*x)^(3/2)*(a + b*Log[c*x^n]), x, 6, -((4*b*d^2*n*Sqrt[d + e*x])/(5*e)) - (4*b*d*n*(d + e*x)^(3/2))/(15*e) - (4*b*n*(d + e*x)^(5/2))/(25*e) + (4*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(5*e) + (2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e)} -{(d + e*x)^(3/2)*(a + b*Log[c*x^n])/x^1, x, 18, (-(16/3))*b*d*n*Sqrt[d + e*x] - (4/9)*b*n*(d + e*x)^(3/2) + (16/3)*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2 + 2*d*Sqrt[d + e*x]*(a + b*Log[c*x^n]) + (2/3)*(d + e*x)^(3/2)*(a + b*Log[c*x^n]) - 2*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]) - 4*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] - 2*b*d^(3/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]} -{(d + e*x)^(3/2)*(a + b*Log[c*x^n])/x^2, x, 14, -4*b*e*n*Sqrt[d + e*x] - (b*d*n*Sqrt[d + e*x])/x + 3*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 3*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2 + 3*e*Sqrt[d + e*x]*(a + b*Log[c*x^n]) - ((d + e*x)^(3/2)*(a + b*Log[c*x^n]))/x - 3*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]) - 6*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])] - 3*b*Sqrt[d]*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])]} -{(d + e*x)^(3/2)*(a + b*Log[c*x^n])/x^3, x, 16, -((b*d*n*Sqrt[d + e*x])/(4*x^2)) - (11*b*e*n*Sqrt[d + e*x])/(8*x) - (9*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(8*Sqrt[d]) + (3*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/(4*Sqrt[d]) - (3*e*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(4*x) - ((d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(2*x^2) - (3*e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/(4*Sqrt[d]) - (3*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(2*Sqrt[d]) - (3*b*e^2*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(4*Sqrt[d])} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3*(a + b*Log[c*x^n])/Sqrt[d + e*x], x, 7, (64*b*d^3*n*Sqrt[d + e*x])/(35*e^4) - (76*b*d^2*n*(d + e*x)^(3/2))/(105*e^4) + (64*b*d*n*(d + e*x)^(5/2))/(175*e^4) - (4*b*n*(d + e*x)^(7/2))/(49*e^4) - (64*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(35*e^4) - (2*d^3*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4)} -{x^2*(a + b*Log[c*x^n])/Sqrt[d + e*x], x, 6, -((32*b*d^2*n*Sqrt[d + e*x])/(15*e^3)) + (28*b*d*n*(d + e*x)^(3/2))/(45*e^3) - (4*b*n*(d + e*x)^(5/2))/(25*e^3) + (32*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(15*e^3) + (2*d^2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3)} -{x^1*(a + b*Log[c*x^n])/Sqrt[d + e*x], x, 6, (8*b*d*n*Sqrt[d + e*x])/(3*e^2) - (4*b*n*(d + e*x)^(3/2))/(9*e^2) - (8*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3*e^2) - (2*d*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^2 + (2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^2)} -{x^0*(a + b*Log[c*x^n])/Sqrt[d + e*x], x, 4, -((4*b*n*Sqrt[d + e*x])/e) + (4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e} -{(a + b*Log[c*x^n])/(x^1*Sqrt[d + e*x]), x, 7, (2*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/Sqrt[d] - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[d] - (4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/Sqrt[d] - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/Sqrt[d]} -{(a + b*Log[c*x^n])/(x^2*Sqrt[d + e*x]), x, 11, -((b*n*Sqrt[d + e*x])/(d*x)) - (b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2) - (b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/d^(3/2) - (Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(d*x) + (e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(3/2) + (2*b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(3/2) + (b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(3/2)} -{(a + b*Log[c*x^n])/(x^3*Sqrt[d + e*x]), x, 16, -((b*n*Sqrt[d + e*x])/(4*d*x^2)) + (5*b*e*n*Sqrt[d + e*x])/(8*d^2*x) + (7*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(8*d^(5/2)) + (3*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/(4*d^(5/2)) - (Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(2*d*x^2) + (3*e*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/(4*d^2*x) - (3*e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/(4*d^(5/2)) - (3*b*e^2*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(2*d^(5/2)) - (3*b*e^2*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/(4*d^(5/2))} - - -{x^3*(a + b*Log[c*x^n])/(d + e*x)^(3/2), x, 6, -((44*b*d^2*n*Sqrt[d + e*x])/(5*e^4)) + (16*b*d*n*(d + e*x)^(3/2))/(15*e^4) - (4*b*n*(d + e*x)^(5/2))/(25*e^4) + (64*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(5*e^4) + (2*d^3*(a + b*Log[c*x^n]))/(e^4*Sqrt[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/e^4 + (2*(d + e*x)^(5/2)*(a + b*Log[c*x^n]))/(5*e^4)} -{x^2*(a + b*Log[c*x^n])/(d + e*x)^(3/2), x, 6, (20*b*d*n*Sqrt[d + e*x])/(3*e^3) - (4*b*n*(d + e*x)^(3/2))/(9*e^3) - (32*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(3*e^3) - (2*d^2*(a + b*Log[c*x^n]))/(e^3*Sqrt[d + e*x]) - (4*d*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^3 + (2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3)} -{x^1*(a + b*Log[c*x^n])/(d + e*x)^(3/2), x, 5, -((4*b*n*Sqrt[d + e*x])/e^2) + (8*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e^2 + (2*d*(a + b*Log[c*x^n]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^2} -{x^0*(a + b*Log[c*x^n])/(d + e*x)^(3/2), x, 3, -((4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(Sqrt[d]*e)) - (2*(a + b*Log[c*x^n]))/(e*Sqrt[d + e*x])} -{(a + b*Log[c*x^n])/(x^1*(d + e*x)^(3/2)), x, 11, (4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2) + (2*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/d^(3/2) + (2*(a + b*Log[c*x^n]))/(d*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(3/2) - (4*b*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(3/2) - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(3/2)} -{(a + b*Log[c*x^n])/(x^2*(d + e*x)^(3/2)), x, If[$VersionNumber<9, 11, 15], -((b*n*Sqrt[d + e*x])/(d^2*x)) - (5*b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(5/2) - (3*b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]^2)/d^(5/2) - (3*e*(a + b*Log[c*x^n]))/(d^2*Sqrt[d + e*x]) - (a + b*Log[c*x^n])/(d*x*Sqrt[d + e*x]) + (3*e*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(5/2) + (6*b*e*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(5/2) + (3*b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x])])/d^(5/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q / (a+b Log[c x^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^2/((d + e*x)*(a + b*Log[c*x^n])), x, 0, Unintegrable[x^2/((d + e*x)*(a + b*Log[c*x^n])), x]} -{x^1/((d + e*x)*(a + b*Log[c*x^n])), x, 0, Unintegrable[x/((d + e*x)*(a + b*Log[c*x^n])), x]} -{x^0/((d + e*x)*(a + b*Log[c*x^n])), x, 0, Unintegrable[1/((d + e*x)*(a + b*Log[c*x^n])), x]} -{1/(x^1*(d + e*x)*(a + b*Log[c*x^n])), x, 0, Unintegrable[1/(x*(d + e*x)*(a + b*Log[c*x^n])), x]} -{1/(x^2*(d + e*x)*(a + b*Log[c*x^n])), x, 0, Unintegrable[1/(x^2*(d + e*x)*(a + b*Log[c*x^n])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^q (a+b Log[c x^n])^p when m symbolic*) - - -{(f*x)^m*(d + e*x)^3*(a + b*Log[c*x^n]), x, 3, -((b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (3*b*d^2*e*n*(f*x)^(2 + m))/(f^2*(2 + m)^2) - (3*b*d*e^2*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (b*e^3*n*(f*x)^(4 + m))/(f^4*(4 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(2 + m)*(a + b*Log[c*x^n]))/(f^2*(2 + m)) + (3*d*e^2*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (e^3*(f*x)^(4 + m)*(a + b*Log[c*x^n]))/(f^4*(4 + m))} -{(f*x)^m*(d + e*x)^2*(a + b*Log[c*x^n]), x, 4, -((b*d^2*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (2*b*d*e*n*(f*x)^(2 + m))/(f^2*(2 + m)^2) - (b*e^2*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(2 + m)*(a + b*Log[c*x^n]))/(f^2*(2 + m)) + (e^2*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m))} -{(f*x)^m*(d + e*x)^1*(a + b*Log[c*x^n]), x, 4, -((b*d*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (b*e*n*(f*x)^(2 + m))/(f^2*(2 + m)^2) + (d*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (e*(f*x)^(2 + m)*(a + b*Log[c*x^n]))/(f^2*(2 + m))} -{(f*x)^m*(d + e*x)^0*(a + b*Log[c*x^n]), x, 1, -((b*n*(f*x)^(1 + m))/(f*(1 + m)^2)) + ((f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x)^1, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x), x]} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x)^2, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^q (a+b Log[c x^n])^p when q symbolic*) - - -{x^1*(a + b*x)^m*Log[c*x^n], x, 0, Unintegrable[x*(a + b*x)^m*Log[c*x^n], x]} -{x^0*(a + b*x)^m*Log[c*x^n], x, 2, (n*(a + b*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (b*x)/a])/(a*b*(2 + 3*m + m^2)) + ((a + b*x)^(1 + m)*Log[c*x^n])/(b*(1 + m))} -{(a + b*x)^m*Log[c*x^n]/x^1, x, 0, Unintegrable[((a + b*x)^m*Log[c*x^n])/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^q (a+b Log[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*(d + e*x^2)*(a + b*Log[c*x^n]), x, 2, (-(1/36))*b*d*n*x^6 - (1/64)*b*e*n*x^8 + (1/24)*(4*d*x^6 + 3*e*x^8)*(a + b*Log[c*x^n])} -{x^3*(d + e*x^2)*(a + b*Log[c*x^n]), x, 2, (-(1/16))*b*d*n*x^4 - (1/36)*b*e*n*x^6 + (1/12)*(3*d*x^4 + 2*e*x^6)*(a + b*Log[c*x^n])} -{x^1*(d + e*x^2)*(a + b*Log[c*x^n]), x, 4, (-(1/4))*b*d*n*x^2 - (1/16)*b*e*n*x^4 + (1/4)*(2*d*x^2 + e*x^4)*(a + b*Log[c*x^n])} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^1, x, 4, -(b*e*n*x^2)/4 + (e*x^2*(a + b*Log[c*x^n]))/2 + (d*(a + b*Log[c*x^n])^2)/(2*b*n)} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^3, x, 3, -(b*d*n)/(4*x^2) - (d*(a + b*Log[c*x^n]))/(2*x^2) + (e*(a + b*Log[c*x^n])^2)/(2*b*n), -((b*d*n)/(4*x^2)) - (1/2)*b*e*n*Log[x]^2 - (d*(a + b*Log[c*x^n]))/(2*x^2) + e*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^5, x, 4, -((b*d*n)/(16*x^4)) - (b*e*n)/(4*x^2) - (d*(a + b*Log[c*x^n]))/(4*x^4) - (e*(a + b*Log[c*x^n]))/(2*x^2)} - -{x^4*(d + e*x^2)*(a + b*Log[c*x^n]), x, 2, (-(1/25))*b*d*n*x^5 - (1/49)*b*e*n*x^7 + (1/35)*(7*d*x^5 + 5*e*x^7)*(a + b*Log[c*x^n])} -{x^2*(d + e*x^2)*(a + b*Log[c*x^n]), x, 2, (-(1/9))*b*d*n*x^3 - (1/25)*b*e*n*x^5 + (1/15)*(5*d*x^3 + 3*e*x^5)*(a + b*Log[c*x^n])} -{x^0*(d + e*x^2)*(a + b*Log[c*x^n]), x, 2, (-b)*d*n*x - (1/9)*b*e*n*x^3 + d*x*(a + b*Log[c*x^n]) + (1/3)*e*x^3*(a + b*Log[c*x^n])} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^2, x, 2, -((b*d*n)/x) - b*e*n*x - (d*(a + b*Log[c*x^n]))/x + e*x*(a + b*Log[c*x^n])} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^4, x, 4, -((b*d*n)/(9*x^3)) - (b*e*n)/x - (d*(a + b*Log[c*x^n]))/(3*x^3) - (e*(a + b*Log[c*x^n]))/x} -{((d + e*x^2)*(a + b*Log[c*x^n]))/x^6, x, 4, -((b*d*n)/(25*x^5)) - (b*e*n)/(9*x^3) - (d*(a + b*Log[c*x^n]))/(5*x^5) - (e*(a + b*Log[c*x^n]))/(3*x^3)} - - -{x^5*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 4, (-(1/36))*b*d^2*n*x^6 - (1/32)*b*d*e*n*x^8 - (1/100)*b*e^2*n*x^10 + (1/60)*(10*d^2*x^6 + 15*d*e*x^8 + 6*e^2*x^10)*(a + b*Log[c*x^n])} -{x^3*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 4, (-(1/16))*b*d^2*n*x^4 - (1/18)*b*d*e*n*x^6 - (1/64)*b*e^2*n*x^8 + (1/24)*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8)*(a + b*Log[c*x^n])} -{x^1*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 5, -(b*d^2*n*x^2)/4 - (b*d*e*n*x^4)/8 - (b*e^2*n*x^6)/36 - (b*d^3*n*Log[x])/(6*e) + ((d + e*x^2)^3*(a + b*Log[c*x^n]))/(6*e)} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^1, x, 3, (-(1/2))*b*d*e*n*x^2 - (1/16)*b*e^2*n*x^4 - (1/2)*b*d^2*n*Log[x]^2 + d*e*x^2*(a + b*Log[c*x^n]) + (1/4)*e^2*x^4*(a + b*Log[c*x^n]) + d^2*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^3, x, 7, -((b*d^2*n)/(4*x^2)) - (1/4)*b*e^2*n*x^2 - b*d*e*n*Log[x]^2 - (d^2*(a + b*Log[c*x^n]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*Log[c*x^n]) + 2*d*e*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^5, x, 7, -((b*d^2*n)/(16*x^4)) - (b*d*e*n)/(2*x^2) - (1/2)*b*e^2*n*Log[x]^2 - (d^2*(a + b*Log[c*x^n]))/(4*x^4) - (d*e*(a + b*Log[c*x^n]))/x^2 + e^2*Log[x]*(a + b*Log[c*x^n])} - -{x^4*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 2, (-(1/25))*b*d^2*n*x^5 - (2/49)*b*d*e*n*x^7 - (1/81)*b*e^2*n*x^9 + (1/315)*(63*d^2*x^5 + 90*d*e*x^7 + 35*e^2*x^9)*(a + b*Log[c*x^n])} -{x^2*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 2, (-(1/9))*b*d^2*n*x^3 - (2/25)*b*d*e*n*x^5 - (1/49)*b*e^2*n*x^7 + (1/105)*(35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x^7)*(a + b*Log[c*x^n])} -{x^0*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 2, (-b)*d^2*n*x - (2/9)*b*d*e*n*x^3 - (1/25)*b*e^2*n*x^5 + d^2*x*(a + b*Log[c*x^n]) + (2/3)*d*e*x^3*(a + b*Log[c*x^n]) + (1/5)*e^2*x^5*(a + b*Log[c*x^n])} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^2, x, 2, -((b*d^2*n)/x) - 2*b*d*e*n*x - (1/9)*b*e^2*n*x^3 - (d^2*(a + b*Log[c*x^n]))/x + 2*d*e*x*(a + b*Log[c*x^n]) + (1/3)*e^2*x^3*(a + b*Log[c*x^n])} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^4, x, 2, -((b*d^2*n)/(9*x^3)) - (2*b*d*e*n)/x - b*e^2*n*x - (d^2*(a + b*Log[c*x^n]))/(3*x^3) - (2*d*e*(a + b*Log[c*x^n]))/x + e^2*x*(a + b*Log[c*x^n])} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^6, x, 4, -((b*d^2*n)/(25*x^5)) - (2*b*d*e*n)/(9*x^3) - (b*e^2*n)/x - (d^2*(a + b*Log[c*x^n]))/(5*x^5) - (2*d*e*(a + b*Log[c*x^n]))/(3*x^3) - (e^2*(a + b*Log[c*x^n]))/x} -{((d + e*x^2)^2*(a + b*Log[c*x^n]))/x^8, x, 4, -((b*d^2*n)/(49*x^7)) - (2*b*d*e*n)/(25*x^5) - (b*e^2*n)/(9*x^3) - (d^2*(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*(a + b*Log[c*x^n]))/(5*x^5) - (e^2*(a + b*Log[c*x^n]))/(3*x^3)} - - -{x^5*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 4, (-(1/36))*b*d^3*n*x^6 - (3/64)*b*d^2*e*n*x^8 - (3/100)*b*d*e^2*n*x^10 - (1/144)*b*e^3*n*x^12 + (1/120)*(20*d^3*x^6 + 45*d^2*e*x^8 + 36*d*e^2*x^10 + 10*e^3*x^12)*(a + b*Log[c*x^n])} -{x^3*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 6, (b*d^4*n*x^2)/(20*e) + (3/80)*b*d^3*n*x^4 + (1/60)*b*d^2*e*n*x^6 + (1/320)*b*d*e^2*n*x^8 - (b*n*(d + e*x^2)^5)/(100*e^2) + (b*d^5*n*Log[x])/(40*e^2) - (1/40)*((5*d*(d + e*x^2)^4)/e^2 - (4*(d + e*x^2)^5)/e^2)*(a + b*Log[c*x^n])} -{x^1*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 5, -(b*d^3*n*x^2)/4 - (3*b*d^2*e*n*x^4)/16 - (b*d*e^2*n*x^6)/12 - (b*e^3*n*x^8)/64 - (b*d^4*n*Log[x])/(8*e) + ((d + e*x^2)^4*(a + b*Log[c*x^n]))/(8*e)} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^1, x, 5, (-(3/4))*b*d^2*e*n*x^2 - (3/16)*b*d*e^2*n*x^4 - (1/36)*b*e^3*n*x^6 - (1/2)*b*d^3*n*Log[x]^2 + (3/2)*d^2*e*x^2*(a + b*Log[c*x^n]) + (3/4)*d*e^2*x^4*(a + b*Log[c*x^n]) + (1/6)*e^3*x^6*(a + b*Log[c*x^n]) + d^3*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^3, x, 7, -((b*d^3*n)/(4*x^2)) - (3/4)*b*d*e^2*n*x^2 - (1/16)*b*e^3*n*x^4 - (3/2)*b*d^2*e*n*Log[x]^2 - (d^3*(a + b*Log[c*x^n]))/(2*x^2) + (3/2)*d*e^2*x^2*(a + b*Log[c*x^n]) + (1/4)*e^3*x^4*(a + b*Log[c*x^n]) + 3*d^2*e*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^5, x, 7, -((b*d^3*n)/(16*x^4)) - (3*b*d^2*e*n)/(4*x^2) - (1/4)*b*e^3*n*x^2 - (3/2)*b*d*e^2*n*Log[x]^2 - (d^3*(a + b*Log[c*x^n]))/(4*x^4) - (3*d^2*e*(a + b*Log[c*x^n]))/(2*x^2) + (1/2)*e^3*x^2*(a + b*Log[c*x^n]) + 3*d*e^2*Log[x]*(a + b*Log[c*x^n])} - -{x^4*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 2, (-(1/25))*b*d^3*n*x^5 - (3/49)*b*d^2*e*n*x^7 - (1/27)*b*d*e^2*n*x^9 - (1/121)*b*e^3*n*x^11 + ((231*d^3*x^5 + 495*d^2*e*x^7 + 385*d*e^2*x^9 + 105*e^3*x^11)*(a + b*Log[c*x^n]))/1155} -{x^2*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 2, (-(1/9))*b*d^3*n*x^3 - (3/25)*b*d^2*e*n*x^5 - (3/49)*b*d*e^2*n*x^7 - (1/81)*b*e^3*n*x^9 + (1/315)*(105*d^3*x^3 + 189*d^2*e*x^5 + 135*d*e^2*x^7 + 35*e^3*x^9)*(a + b*Log[c*x^n])} -{x^0*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 2, (-b)*d^3*n*x - (1/3)*b*d^2*e*n*x^3 - (3/25)*b*d*e^2*n*x^5 - (1/49)*b*e^3*n*x^7 + d^3*x*(a + b*Log[c*x^n]) + d^2*e*x^3*(a + b*Log[c*x^n]) + (3/5)*d*e^2*x^5*(a + b*Log[c*x^n]) + (1/7)*e^3*x^7*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^2, x, 2, -((b*d^3*n)/x) - 3*b*d^2*e*n*x - (1/3)*b*d*e^2*n*x^3 - (1/25)*b*e^3*n*x^5 - (d^3*(a + b*Log[c*x^n]))/x + 3*d^2*e*x*(a + b*Log[c*x^n]) + d*e^2*x^3*(a + b*Log[c*x^n]) + (1/5)*e^3*x^5*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^4, x, 3, -((b*d^3*n)/(9*x^3)) - (3*b*d^2*e*n)/x - 3*b*d*e^2*n*x - (1/9)*b*e^3*n*x^3 - (d^3*(a + b*Log[c*x^n]))/(3*x^3) - (3*d^2*e*(a + b*Log[c*x^n]))/x + 3*d*e^2*x*(a + b*Log[c*x^n]) + (1/3)*e^3*x^3*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^6, x, 2, -((b*d^3*n)/(25*x^5)) - (b*d^2*e*n)/(3*x^3) - (3*b*d*e^2*n)/x - b*e^3*n*x - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (d^2*e*(a + b*Log[c*x^n]))/x^3 - (3*d*e^2*(a + b*Log[c*x^n]))/x + e^3*x*(a + b*Log[c*x^n])} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^8, x, 4, -((b*d^3*n)/(49*x^7)) - (3*b*d^2*e*n)/(25*x^5) - (b*d*e^2*n)/(3*x^3) - (b*e^3*n)/x - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*(a + b*Log[c*x^n]))/(5*x^5) - (d*e^2*(a + b*Log[c*x^n]))/x^3 - (e^3*(a + b*Log[c*x^n]))/x} -{((d + e*x^2)^3*(a + b*Log[c*x^n]))/x^10, x, 4, -((b*d^3*n)/(81*x^9)) - (3*b*d^2*e*n)/(49*x^7) - (3*b*d*e^2*n)/(25*x^5) - (b*e^3*n)/(9*x^3) - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (3*d^2*e*(a + b*Log[c*x^n]))/(7*x^7) - (3*d*e^2*(a + b*Log[c*x^n]))/(5*x^5) - (e^3*(a + b*Log[c*x^n]))/(3*x^3)} - - -(* ::InheritFromParent:: *) -(**) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^5*(a + b*Log[c*x^n])/(d + e*x^2), x, 6, (b*d*n*x^2)/(4*e^2) - (b*n*x^4)/(16*e) - (d*x^2*(a + b*Log[c*x^n]))/(2*e^2) + (x^4*(a + b*Log[c*x^n]))/(4*e) + (d^2*(a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e^3) + (b*d^2*n*PolyLog[2, -((e*x^2)/d)])/(4*e^3)} -{x^3*(a + b*Log[c*x^n])/(d + e*x^2), x, 5, -((b*n*x^2)/(4*e)) + (x^2*(a + b*Log[c*x^n]))/(2*e) - (d*(a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e^2) - (b*d*n*PolyLog[2, -((e*x^2)/d)])/(4*e^2)} -{x^1*(a + b*Log[c*x^n])/(d + e*x^2), x, 2, ((a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e) + (b*n*PolyLog[2, -((e*x^2)/d)])/(4*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^2)), x, 2, -((Log[1 + d/(e*x^2)]*(a + b*Log[c*x^n]))/(2*d)) + (b*n*PolyLog[2, -(d/(e*x^2))])/(4*d)} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^2)), x, 4, -((b*n)/(4*d*x^2)) - (a + b*Log[c*x^n])/(2*d*x^2) + (e*Log[1 + d/(e*x^2)]*(a + b*Log[c*x^n]))/(2*d^2) - (b*e*n*PolyLog[2, -(d/(e*x^2))])/(4*d^2)} -{(a + b*Log[c*x^n])/(x^5*(d + e*x^2)), x, 6, -((b*n)/(16*d*x^4)) + (b*e*n)/(4*d^2*x^2) - (a + b*Log[c*x^n])/(4*d*x^4) + (e*(a + b*Log[c*x^n]))/(2*d^2*x^2) - (e^2*Log[1 + d/(e*x^2)]*(a + b*Log[c*x^n]))/(2*d^3) + (b*e^2*n*PolyLog[2, -(d/(e*x^2))])/(4*d^3)} - -{x^4*(a + b*Log[c*x^n])/(d + e*x^2), x, 10, -((a*d*x)/e^2) + (b*d*n*x)/e^2 - (b*n*x^3)/(9*e) - (b*d*x*Log[c*x^n])/e^2 + (x^3*(a + b*Log[c*x^n]))/(3*e) + (d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/e^(5/2) - (I*b*d^(3/2)*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(2*e^(5/2)) + (I*b*d^(3/2)*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(2*e^(5/2))} -{x^2*(a + b*Log[c*x^n])/(d + e*x^2), x, 9, (a*x)/e - (b*n*x)/e + (b*x*Log[c*x^n])/e - (Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/e^(3/2) + (I*b*Sqrt[d]*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(2*e^(3/2)) - (I*b*Sqrt[d]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(2*e^(3/2))} -{x^0*(a + b*Log[c*x^n])/(d + e*x^2), x, 5, (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[d]*Sqrt[e]) - (I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(2*Sqrt[d]*Sqrt[e]) + (I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*Sqrt[e])} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^2)), x, 7, -((b*n)/(d*x)) - (a + b*Log[c*x^n])/(d*x) - (Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(3/2) + (I*b*Sqrt[e]*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(2*d^(3/2)) - (I*b*Sqrt[e]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2))} -{(a + b*Log[c*x^n])/(x^4*(d + e*x^2)), x, 9, -((b*n)/(9*d*x^3)) + (b*e*n)/(d^2*x) - (a + b*Log[c*x^n])/(3*d*x^3) + (e*(a + b*Log[c*x^n]))/(d^2*x) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/d^(5/2) - (I*b*e^(3/2)*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(2*d^(5/2)) + (I*b*e^(3/2)*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2))} - - -{x^5*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 7, -((b*n*x^2)/(4*e^2)) + (x^2*(a + b*Log[c*x^n]))/(2*e^2) + (d*x^2*(a + b*Log[c*x^n]))/(2*e^2*(d + e*x^2)) - (b*d*n*Log[d + e*x^2])/(4*e^3) - (d*(a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/e^3 - (b*d*n*PolyLog[2, -((e*x^2)/d)])/(2*e^3)} -{x^3*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 6, -((x^2*(a + b*Log[c*x^n]))/(2*e*(d + e*x^2))) + (b*n*Log[d + e*x^2])/(4*e^2) + ((a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e^2) + (b*n*PolyLog[2, -((e*x^2)/d)])/(4*e^2)} -{x^1*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 2, (x^2*(a + b*Log[c*x^n]))/(2*d*(d + e*x^2)) - (b*n*Log[d + e*x^2])/(4*d*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^2)^2), x, 3, (a + b*Log[c*x^n])/(2*d*(d + e*x^2)) - (Log[1 + d/(e*x^2)]*(2*a - b*n + 2*b*Log[c*x^n]))/(4*d^2) + (b*n*PolyLog[2, -(d/(e*x^2))])/(4*d^2)} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^2)^2), x, 5, -((b*n)/(2*d^2*x^2)) + (a + b*Log[c*x^n])/(2*d*x^2*(d + e*x^2)) - (4*a - b*n + 4*b*Log[c*x^n])/(4*d^2*x^2) + (e*Log[1 + d/(e*x^2)]*(4*a - b*n + 4*b*Log[c*x^n]))/(4*d^3) - (b*e*n*PolyLog[2, -(d/(e*x^2))])/(2*d^3)} - -{x^4*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 16, (a*x)/e^2 - (b*n*x)/e^2 - (b*Sqrt[d]*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(5/2)) + (b*x*Log[c*x^n])/e^2 + (d*x*(a + b*Log[c*x^n]))/(2*e^2*(d + e*x^2)) - (3*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(5/2)) + (3*I*b*Sqrt[d]*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(4*e^(5/2)) - (3*I*b*Sqrt[d]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(4*e^(5/2))} -{x^2*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 14, (b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) - (x*(a + b*Log[c*x^n]))/(2*e*(d + e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*Sqrt[d]*e^(3/2)) - (I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(4*Sqrt[d]*e^(3/2)) + (I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(4*Sqrt[d]*e^(3/2))} -{x^0*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 7, -((b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e])) + (x*(a + b*Log[c*x^n]))/(2*d*(d + e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(3/2)*Sqrt[e]) - (I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(4*d^(3/2)*Sqrt[e]) + (I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(4*d^(3/2)*Sqrt[e])} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^2)^2), x, 8, -((3*b*n)/(2*d^2*x)) + (a + b*Log[c*x^n])/(2*d*x*(d + e*x^2)) - (3*a - b*n + 3*b*Log[c*x^n])/(2*d^2*x) - (Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(3*a - b*n + 3*b*Log[c*x^n]))/(2*d^(5/2)) + (3*I*b*Sqrt[e]*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(4*d^(5/2)) - (3*I*b*Sqrt[e]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(4*d^(5/2))} -{(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^2), x, 10, -((5*b*n)/(18*d^2*x^3)) + (5*b*e*n)/(2*d^3*x) + (a + b*Log[c*x^n])/(2*d*x^3*(d + e*x^2)) - (5*a - b*n + 5*b*Log[c*x^n])/(6*d^2*x^3) + (e*(5*a - b*n + 5*b*Log[c*x^n]))/(2*d^3*x) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(5*a - b*n + 5*b*Log[c*x^n]))/(2*d^(7/2)) - (5*I*b*e^(3/2)*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(4*d^(7/2)) + (5*I*b*e^(3/2)*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(4*d^(7/2))} - - -{x^5*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 10, (b*d*n)/(8*e^3*(d + e*x^2)) + (b*n*Log[x])/(4*e^3) - (d^2*(a + b*Log[c*x^n]))/(4*e^3*(d + e*x^2)^2) - (x^2*(a + b*Log[c*x^n]))/(e^2*(d + e*x^2)) + (3*b*n*Log[d + e*x^2])/(8*e^3) + ((a + b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(2*e^3) + (b*n*PolyLog[2, -((e*x^2)/d)])/(4*e^3)} -{x^3*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 4, -((b*n)/(8*e^2*(d + e*x^2))) + (x^4*(a + b*Log[c*x^n]))/(4*d*(d + e*x^2)^2) - (b*n*Log[d + e*x^2])/(8*d*e^2)} -{x^1*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 4, (b*n)/(8*d*e*(d + e*x^2)) + (b*n*Log[x])/(4*d^2*e) - (a + b*Log[c*x^n])/(4*e*(d + e*x^2)^2) - (b*n*Log[d + e*x^2])/(8*d^2*e)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^2)^3), x, 4, (a + b*Log[c*x^n])/(4*d*(d + e*x^2)^2) - (Log[1 + d/(e*x^2)]*(4*a - 3*b*n + 4*b*Log[c*x^n]))/(8*d^3) + (4*a - b*n + 4*b*Log[c*x^n])/(8*d^2*(d + e*x^2)) + (b*n*PolyLog[2, -(d/(e*x^2))])/(4*d^3)} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^2)^3), x, 6, -((3*b*n)/(4*d^3*x^2)) + (a + b*Log[c*x^n])/(4*d*x^2*(d + e*x^2)^2) + (6*a - b*n + 6*b*Log[c*x^n])/(8*d^2*x^2*(d + e*x^2)) - (12*a - 5*b*n + 12*b*Log[c*x^n])/(8*d^3*x^2) + (e*Log[1 + d/(e*x^2)]*(12*a - 5*b*n + 12*b*Log[c*x^n]))/(8*d^4) - (3*b*e*n*PolyLog[2, -(d/(e*x^2))])/(4*d^4)} - -{x^4*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 24, -((b*n*x)/(8*e^2*(d + e*x^2))) + (b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(5/2)) + (d*x*(a + b*Log[c*x^n]))/(4*e^2*(d + e*x^2)^2) - (5*x*(a + b*Log[c*x^n]))/(8*e^2*(d + e*x^2)) + (3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*Sqrt[d]*e^(5/2)) - (3*I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(16*Sqrt[d]*e^(5/2)) + (3*I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(16*Sqrt[d]*e^(5/2))} -{x^2*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 19, (b*n*x)/(8*d*e*(d + e*x^2)) - (x*(a + b*Log[c*x^n]))/(4*e*(d + e*x^2)^2) + (x*(a + b*Log[c*x^n]))/(8*d*e*(d + e*x^2)) + (ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*d^(3/2)*e^(3/2)) - (I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(16*d^(3/2)*e^(3/2)) + (I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(16*d^(3/2)*e^(3/2))} -{x^0*(a + b*Log[c*x^n])/(d + e*x^2)^3, x, 10, -((b*n*x)/(8*d^2*(d + e*x^2))) - (b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*Log[c*x^n]))/(8*d^2*(d + e*x^2)) + (3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*d^(5/2)*Sqrt[e]) - (3*I*b*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(16*d^(5/2)*Sqrt[e]) + (3*I*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(16*d^(5/2)*Sqrt[e])} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^2)^3), x, 9, -((15*b*n)/(8*d^3*x)) + (a + b*Log[c*x^n])/(4*d*x*(d + e*x^2)^2) + (5*a - b*n + 5*b*Log[c*x^n])/(8*d^2*x*(d + e*x^2)) - (15*a - 8*b*n + 15*b*Log[c*x^n])/(8*d^3*x) - (Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(15*a - 8*b*n + 15*b*Log[c*x^n]))/(8*d^(7/2)) + (15*I*b*Sqrt[e]*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(16*d^(7/2)) - (15*I*b*Sqrt[e]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(16*d^(7/2))} -{(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^3), x, 11, -((35*b*n)/(72*d^3*x^3)) + (35*b*e*n)/(8*d^4*x) + (a + b*Log[c*x^n])/(4*d*x^3*(d + e*x^2)^2) + (7*a - b*n + 7*b*Log[c*x^n])/(8*d^2*x^3*(d + e*x^2)) - (35*a - 12*b*n + 35*b*Log[c*x^n])/(24*d^3*x^3) + (e*(35*a - 12*b*n + 35*b*Log[c*x^n]))/(8*d^4*x) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(35*a - 12*b*n + 35*b*Log[c*x^n]))/(8*d^(9/2)) - (35*I*b*e^(3/2)*n*PolyLog[2, -((I*Sqrt[e]*x)/Sqrt[d])])/(16*d^(9/2)) + (35*I*b*e^(3/2)*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(16*d^(9/2))} - - -{x*Log[c*x^2]/(1 - c*x^2), x, 2, PolyLog[2, 1 - c*x^2]/(2*c)} -{x*Log[x^2/c]/(c - x^2), x, 2, (1/2)*PolyLog[2, 1 - x^2/c]} - - -{Log[x]/(1 - x^2), x, 2, ArcTanh[x]*Log[x] + (1/2)*PolyLog[2, -x] - (1/2)*PolyLog[2, x]} -{Log[x]/(1 + x^2), x, 4, ArcTan[x]*Log[x] - (1/2)*I*PolyLog[2, (-I)*x] + (1/2)*I*PolyLog[2, I*x]} - - -{(a + b*Log[c*x])/(1 - e*x^2), x, 3, (ArcTanh[Sqrt[e]*x]*(a + b*Log[c*x]))/Sqrt[e] + (b*PolyLog[2, (-Sqrt[e])*x])/(2*Sqrt[e]) - (b*PolyLog[2, Sqrt[e]*x])/(2*Sqrt[e])} -{(a + b*Log[c*x^n])/(1 - e*x^2), x, 3, (ArcTanh[Sqrt[e]*x]*(a + b*Log[c*x^n]))/Sqrt[e] + (b*n*PolyLog[2, (-Sqrt[e])*x])/(2*Sqrt[e]) - (b*n*PolyLog[2, Sqrt[e]*x])/(2*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^q (a+b Log[c x^n])^2*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a + b*Log[c*x^n])^2/(d + e*x^2)^2, x, 16, (x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*n*(a + b*Log[c*x^n])*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - ((a + b*Log[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (b^2*n^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (b^2*n^2*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^q (a+b Log[c x^n])^3*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a + b*Log[c*x^n])^3/(d + e*x^2)^2, x, 20, (x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^3)/(4*(-d)^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*Log[c*x^n])^3*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^3*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b^3*n^3*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + (3*b^3*n^3*PolyLog[4, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) - (3*b^3*n^3*PolyLog[4, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^q (a+b Log[c x^n])^p*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{1/((d + e*x^2)^2*(a + b*Log[c*x^n])^1), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*Log[c*x^n])), x]} -{1/((d + e*x^2)^2*(a + b*Log[c*x^n])^2), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*Log[c*x^n])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(q/2) (a+b Log[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, If[$VersionNumber>=8, 7, 8], -((8*b*d^3*n*Sqrt[d + e*x^2])/(105*e^3)) - (8*b*d^2*n*(d + e*x^2)^(3/2))/(315*e^3) + (9*b*d*n*(d + e*x^2)^(5/2))/(175*e^3) - (b*n*(d + e*x^2)^(7/2))/(49*e^3) + (8*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(105*e^3) + (d^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*Log[c*x^n]))/(7*e^3)} -{x^3*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, If[$VersionNumber>=8, 8, 9], (2*b*d^2*n*Sqrt[d + e*x^2])/(15*e^2) + (2*b*d*n*(d + e*x^2)^(3/2))/(45*e^2) - (b*n*(d + e*x^2)^(5/2))/(25*e^2) - (2*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(15*e^2) - (d*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^2)} -{x^1*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, 6, -((b*d*n*Sqrt[d + e*x^2])/(3*e)) - (b*n*(d + e*x^2)^(3/2))/(9*e) + (b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*e) + ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e)} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^1, x, 12, (-b)*n*Sqrt[d + e*x^2] + b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + (1/2)*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2 + (Sqrt[d + e*x^2] - Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])*(a + b*Log[c*x^n]) - b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (1/2)*b*Sqrt[d]*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])]} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^3, x, 14, -((b*n*Sqrt[d + e*x^2])/(4*x^2)) - (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(4*Sqrt[d]) + (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(4*Sqrt[d]) - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2*x^2) - (e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*Sqrt[d]) - (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*Sqrt[d]) - (b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(4*Sqrt[d])} - -{x^4*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, 19, (7*b*d^2*n*x*Sqrt[d + e*x^2])/(192*e^2) - (5*b*d*n*x^3*Sqrt[d + e*x^2])/(288*e) - (1/36)*b*n*x^5*Sqrt[d + e*x^2] + (5*b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(192*e^(5/2)*Sqrt[1 + (e*x^2)/d]) + (b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(32*e^(5/2)*Sqrt[1 + (e*x^2)/d]) - (b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(16*e^(5/2)*Sqrt[1 + (e*x^2)/d]) - (d^2*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(16*e^2) + (d*x^3*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(24*e) + (1/6)*x^5*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) + (d^(5/2)*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(16*e^(5/2)*Sqrt[1 + (e*x^2)/d]) - (b*d^(5/2)*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(32*e^(5/2)*Sqrt[1 + (e*x^2)/d])} -{x^2*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, 15, -((3*b*d*n*x*Sqrt[d + e*x^2])/(32*e)) - (1/16)*b*n*x^3*Sqrt[d + e*x^2] - (b*d^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(32*e^(3/2)*Sqrt[1 + (e*x^2)/d]) - (b*d^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(16*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (b*d^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(8*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (d*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(8*e) + (1/4)*x^3*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) - (d^(3/2)*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (b*d^(3/2)*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(16*e^(3/2)*Sqrt[1 + (e*x^2)/d])} -{x^0*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]), x, 11, (-(1/4))*b*n*x*Sqrt[d + e*x^2] + (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(4*Sqrt[e]*Sqrt[d + e*x^2]) - (b*d*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(4*Sqrt[e]) - (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[e]*Sqrt[d + e*x^2]) + (1/2)*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) + (d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*Sqrt[e]*Sqrt[d + e*x^2]) - (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*Sqrt[e]*Sqrt[d + e*x^2])} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^2, x, 11, -((b*n*Sqrt[d + e*x^2])/x) + (b*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[1 + (e*x^2)/d]) + (b*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (b*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/x + (Sqrt[e]*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (b*Sqrt[e]*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[d]*Sqrt[1 + (e*x^2)/d])} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^4, x, 5, -((b*e*n*Sqrt[d + e*x^2])/(3*d*x)) - (b*n*(d + e*x^2)^(3/2))/(9*d*x^3) + (b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*d*x^3)} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^6, x, If[$VersionNumber>=8, 7, 8], If[$VersionNumber>=8, (2*b*e^2*n*Sqrt[d + e*x^2])/(15*d^2*x) + (2*b*e*n*(d + e*x^2)^(3/2))/(45*d^2*x^3) - (b*n*(d + e*x^2)^(5/2))/(25*d^2*x^5) - (2*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(15*d^2) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(15*d^2*x^3), (2*b*e^2*n*Sqrt[d + e*x^2])/(15*d^2*x) + (2*b*e*n*(d + e*x^2)^(3/2))/(45*d^2*x^3) - (b*n*(d + e*x^2)^(5/2))/(25*d^2*x^5) - (2*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(15*d^2) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(15*d^2*x^3)]} -{Sqrt[d + e*x^2]*(a + b*Log[c*x^n])/x^8, x, If[$VersionNumber>=8, 8, 9], If[$VersionNumber>=8, -((8*b*e^3*n*Sqrt[d + e*x^2])/(105*d^3*x)) - (8*b*e^2*n*(d + e*x^2)^(3/2))/(315*d^3*x^3) - (b*n*(d + e*x^2)^(5/2))/(49*d^2*x^7) + (38*b*e*n*(d + e*x^2)^(5/2))/(1225*d^3*x^5) + (8*b*e^(7/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(105*d^3) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(7*d*x^7) + (4*e*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(35*d^2*x^5) - (8*e^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(105*d^3*x^3), -((8*b*e^3*n*Sqrt[d + e*x^2])/(105*d^3*x)) - (b*n*(d + e*x^2)^(3/2))/(49*d*x^7) + (13*b*e*n*(d + e*x^2)^(3/2))/(1225*d^2*x^5) + (62*b*e^2*n*(d + e*x^2)^(3/2))/(11025*d^3*x^3) + (8*b*e^(7/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(105*d^3) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(7*d*x^7) + (4*e*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(35*d^2*x^5) - (8*e^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(105*d^3*x^3)]} - - -{x^5*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]), x, 7, -((8*b*d^4*n*Sqrt[d + e*x^2])/(315*e^3)) - (8*b*d^3*n*(d + e*x^2)^(3/2))/(945*e^3) - (8*b*d^2*n*(d + e*x^2)^(5/2))/(1575*e^3) + (11*b*d*n*(d + e*x^2)^(7/2))/(441*e^3) - (b*n*(d + e*x^2)^(9/2))/(81*e^3) + (8*b*d^(9/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(315*e^3) + (d^2*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3) - (2*d*(d + e*x^2)^(7/2)*(a + b*Log[c*x^n]))/(7*e^3) + ((d + e*x^2)^(9/2)*(a + b*Log[c*x^n]))/(9*e^3)} -{x^3*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]), x, 9, (2*b*d^3*n*Sqrt[d + e*x^2])/(35*e^2) + (2*b*d^2*n*(d + e*x^2)^(3/2))/(105*e^2) + (2*b*d*n*(d + e*x^2)^(5/2))/(175*e^2) - (b*n*(d + e*x^2)^(7/2))/(49*e^2) - (2*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(35*e^2) - (d*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*Log[c*x^n]))/(7*e^2)} -{x^1*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]), x, 7, -((b*d^2*n*Sqrt[d + e*x^2])/(5*e)) - (b*d*n*(d + e*x^2)^(3/2))/(15*e) - (b*n*(d + e*x^2)^(5/2))/(25*e) + (b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(5*e) + ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e)} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^1, x, 17, (-(4/3))*b*d*n*Sqrt[d + e*x^2] - (1/9)*b*n*(d + e*x^2)^(3/2) + (4/3)*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + (1/2)*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2 + (1/3)*(3*d*Sqrt[d + e*x^2] + (d + e*x^2)^(3/2) - 3*d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])*(a + b*Log[c*x^n]) - b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (1/2)*b*d^(3/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])]} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^3, x, 18, (-b)*e*n*Sqrt[d + e*x^2] - (b*d*n*Sqrt[d + e*x^2])/(4*x^2) + (3/4)*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + (3/4)*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2 + (3/2)*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(2*x^2) - (3/2)*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]) - (3/2)*b*Sqrt[d]*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])] - (3/4)*b*Sqrt[d]*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])]} - -{x^2*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]), x, 19, -((11*b*d^2*n*x*Sqrt[d + e*x^2])/(192*e)) - (23/288)*b*d*n*x^3*Sqrt[d + e*x^2] - (1/36)*b*e*n*x^5*Sqrt[d + e*x^2] - (b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(192*e^(3/2)*Sqrt[1 + (e*x^2)/d]) - (b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(32*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (b*d^(5/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(16*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (d^2*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(16*e) + (1/8)*d*x^3*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) + (1/6)*x^3*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]) - (d^(5/2)*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(16*e^(3/2)*Sqrt[1 + (e*x^2)/d]) + (b*d^(5/2)*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(32*e^(3/2)*Sqrt[1 + (e*x^2)/d])} -{x^0*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]), x, 16, (-(9/32))*b*d*n*x*Sqrt[d + e*x^2] - (1/16)*b*n*x*(d + e*x^2)^(3/2) + (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(16*Sqrt[e]*Sqrt[d + e*x^2]) - (9*b*d^2*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(32*Sqrt[e]) - (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(8*Sqrt[e]*Sqrt[d + e*x^2]) + (3/8)*d*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) + (1/4)*x*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]) + (3*d^(5/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(8*Sqrt[e]*Sqrt[d + e*x^2]) - (3*b*d^(5/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(16*Sqrt[e]*Sqrt[d + e*x^2])} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^2, x, 14, -((b*d*n*Sqrt[d + e*x^2])/x) - (1/4)*b*e*n*x*Sqrt[d + e*x^2] + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(4*Sqrt[1 + (e*x^2)/d]) + (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(4*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[1 + (e*x^2)/d]) + (3/2)*e*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]) - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/x + (3*Sqrt[d]*Sqrt[e]*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*Sqrt[1 + (e*x^2)/d]) - (3*b*Sqrt[d]*Sqrt[e]*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*Sqrt[1 + (e*x^2)/d])} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^4, x, 13, -((4*b*e*n*Sqrt[d + e*x^2])/(3*x)) - (b*n*(d + e*x^2)^(3/2))/(9*x^3) + (4*b*e^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(3*Sqrt[d]*Sqrt[1 + (e*x^2)/d]) + (b*e^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (b*e^(3/2)*n*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/x - ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*x^3) + (e^(3/2)*Sqrt[d + e*x^2]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[d]*Sqrt[1 + (e*x^2)/d]) - (b*e^(3/2)*n*Sqrt[d + e*x^2]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[d]*Sqrt[1 + (e*x^2)/d])} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^6, x, 6, -((b*e^2*n*Sqrt[d + e*x^2])/(5*d*x)) - (b*e*n*(d + e*x^2)^(3/2))/(15*d*x^3) - (b*n*(d + e*x^2)^(5/2))/(25*d*x^5) + (b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*d) - ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*d*x^5)} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^8, x, 8, (2*b*e^3*n*Sqrt[d + e*x^2])/(35*d^2*x) + (2*b*e^2*n*(d + e*x^2)^(3/2))/(105*d^2*x^3) + (2*b*e*n*(d + e*x^2)^(5/2))/(175*d^2*x^5) - (b*n*(d + e*x^2)^(7/2))/(49*d^2*x^7) - (2*b*e^(7/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(35*d^2) - ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(35*d^2*x^5)} -{(d + e*x^2)^(3/2)*(a + b*Log[c*x^n])/x^10, x, 9, -((8*b*e^4*n*Sqrt[d + e*x^2])/(315*d^3*x)) - (8*b*e^3*n*(d + e*x^2)^(3/2))/(945*d^3*x^3) - (8*b*e^2*n*(d + e*x^2)^(5/2))/(1575*d^3*x^5) - (b*n*(d + e*x^2)^(7/2))/(81*d^2*x^9) + (50*b*e*n*(d + e*x^2)^(7/2))/(3969*d^3*x^7) + (8*b*e^(9/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(315*d^3) - ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(9*d*x^9) + (4*e*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(63*d^2*x^7) - (8*e^2*(d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(315*d^3*x^5)} - - -{x*Sqrt[4 + x^2]*Log[x], x, 6, (-(4/3))*Sqrt[4 + x^2] - (1/9)*(4 + x^2)^(3/2) + (8/3)*ArcTanh[Sqrt[4 + x^2]/2] + (1/3)*(4 + x^2)^(3/2)*Log[x]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^5*(a + b*Log[c*x^n])/Sqrt[d + e*x^2], x, 7, -((8*b*d^2*n*Sqrt[d + e*x^2])/(15*e^3)) + (7*b*d*n*(d + e*x^2)^(3/2))/(45*e^3) - (b*n*(d + e*x^2)^(5/2))/(25*e^3) + (8*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(15*e^3) + (d^2*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^3)} -{x^3*(a + b*Log[c*x^n])/Sqrt[d + e*x^2], x, 7, (2*b*d*n*Sqrt[d + e*x^2])/(3*e^2) - (b*n*(d + e*x^2)^(3/2))/(9*e^2) - (2*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*e^2) - (d*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^2)} -{x^1*(a + b*Log[c*x^n])/Sqrt[d + e*x^2], x, 5, -((b*n*Sqrt[d + e*x^2])/e) + (b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/e + (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e} -{(a + b*Log[c*x^n])/(x^1*Sqrt[d + e*x^2]), x, 8, (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(2*Sqrt[d]) - (ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]))/Sqrt[d] - (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/Sqrt[d] - (b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*Sqrt[d])} -{(a + b*Log[c*x^n])/(x^3*Sqrt[d + e*x^2]), x, 14, -((b*n*Sqrt[d + e*x^2])/(4*d*x^2)) - (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(4*d^(3/2)) - (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(4*d^(3/2)) - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2*d*x^2) + (e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(3/2)) + (b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*d^(3/2)) + (b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(4*d^(3/2))} - -{x^2*(a + b*Log[c*x^n])/Sqrt[d + e*x^2], x, 12, -((b*n*x*Sqrt[d + e*x^2])/(4*e)) - (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(4*e^(3/2)*Sqrt[d + e*x^2]) - (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(4*e^(3/2)*Sqrt[d + e*x^2]) + (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(3/2)*Sqrt[d + e*x^2]) + (x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2*e) - (d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(3/2)*Sqrt[d + e*x^2]) + (b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*e^(3/2)*Sqrt[d + e*x^2])} -{x^0*(a + b*Log[c*x^n])/Sqrt[d + e*x^2], x, 7, (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[e]*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(Sqrt[e]*Sqrt[d + e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[e]*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[e]*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^2*Sqrt[d + e*x^2]), x, 4, -((b*n*Sqrt[d + e*x^2])/(d*x)) + (b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/d - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(d*x)} -{(a + b*Log[c*x^n])/(x^4*Sqrt[d + e*x^2]), x, 6, (2*b*e*n*Sqrt[d + e*x^2])/(3*d^2*x) - (b*n*(d + e*x^2)^(3/2))/(9*d^2*x^3) - (2*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^2) - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(3*d^2*x)} -{(a + b*Log[c*x^n])/(x^6*Sqrt[d + e*x^2]), x, 7, -((8*b*e^2*n*Sqrt[d + e*x^2])/(15*d^3*x)) - (b*n*(d + e*x^2)^(3/2))/(25*d^2*x^5) + (26*b*e*n*(d + e*x^2)^(3/2))/(225*d^3*x^3) + (8*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(15*d^3) - (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(5*d*x^5) + (4*e*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(15*d^2*x^3) - (8*e^2*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(15*d^3*x)} - - -{x^7*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 7, -((11*b*d^2*n*Sqrt[d + e*x^2])/(5*e^4)) + (4*b*d*n*(d + e*x^2)^(3/2))/(15*e^4) - (b*n*(d + e*x^2)^(5/2))/(25*e^4) + (16*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(5*e^4) + (d^3*(a + b*Log[c*x^n]))/(e^4*Sqrt[d + e*x^2]) + (3*d^2*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^4 - (d*(d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/e^4 + ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*e^4)} -{x^5*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 7, (5*b*d*n*Sqrt[d + e*x^2])/(3*e^3) - (b*n*(d + e*x^2)^(3/2))/(9*e^3) - (8*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*e^3) - (d^2*(a + b*Log[c*x^n]))/(e^3*Sqrt[d + e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^3)} -{x^3*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 6, -((b*n*Sqrt[d + e*x^2])/e^2) + (2*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/e^2 + (d*(a + b*Log[c*x^n]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^2} -{x^1*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 4, -((b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(Sqrt[d]*e)) - (a + b*Log[c*x^n])/(e*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^2)^(3/2)), x, 11, (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) + (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(2*d^(3/2)) + (1/(d*Sqrt[d + e*x^2]) - ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]/d^(3/2))*(a + b*Log[c*x^n]) - (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/d^(3/2) - (b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*d^(3/2))} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^2)^(3/2)), x, 12, -((b*n*Sqrt[d + e*x^2])/(4*d^2*x^2)) - (5*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(4*d^(5/2)) - (3*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(4*d^(5/2)) - (3*e*(a + b*Log[c*x^n]))/(2*d^2*Sqrt[d + e*x^2]) - (a + b*Log[c*x^n])/(2*d*x^2*Sqrt[d + e*x^2]) + (3*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(5/2)) + (3*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*d^(5/2)) + (3*b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(4*d^(5/2))} - -{x^2*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 11, (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*Sqrt[d + e*x^2]) + (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*e^(3/2)*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(e^(3/2)*Sqrt[d + e*x^2]) - (x*(a + b*Log[c*x^n]))/(e*Sqrt[d + e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(e^(3/2)*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(3/2)*Sqrt[d + e*x^2])} -{x^0*(a + b*Log[c*x^n])/(d + e*x^2)^(3/2), x, 3, -((b*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(d*Sqrt[e])) + (x*(a + b*Log[c*x^n]))/(d*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^2)^(3/2)), x, 5, -((b*n*Sqrt[d + e*x^2])/(d^2*x)) + (2*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/d^2 - (a + b*Log[c*x^n])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*Log[c*x^n]))/(d^2*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^(3/2)), x, 6, -((b*n*Sqrt[d + e*x^2])/(9*d^2*x^3)) + (14*b*e*n*Sqrt[d + e*x^2])/(9*d^3*x) - (8*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^3) - (a + b*Log[c*x^n])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*Log[c*x^n]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*Log[c*x^n]))/(3*d^3*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^6*(d + e*x^2)^(3/2)), x, 8, -((b*n*Sqrt[d + e*x^2])/(25*d^2*x^5)) + (14*b*e*n*Sqrt[d + e*x^2])/(75*d^3*x^3) - (148*b*e^2*n*Sqrt[d + e*x^2])/(75*d^4*x) + (16*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*d^4) - (a + b*Log[c*x^n])/(5*d*x^5*Sqrt[d + e*x^2]) + (2*e*(a + b*Log[c*x^n]))/(5*d^2*x^3*Sqrt[d + e*x^2]) - (8*e^2*(a + b*Log[c*x^n]))/(5*d^3*x*Sqrt[d + e*x^2]) - (16*e^3*x*(a + b*Log[c*x^n]))/(5*d^4*Sqrt[d + e*x^2])} - - -{x^7*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 9, -((b*d^2*n)/(3*e^4*Sqrt[d + e*x^2])) + (8*b*d*n*Sqrt[d + e*x^2])/(3*e^4) - (b*n*(d + e*x^2)^(3/2))/(9*e^4) - (16*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*e^4) + (d^3*(a + b*Log[c*x^n]))/(3*e^4*(d + e*x^2)^(3/2)) - (3*d^2*(a + b*Log[c*x^n]))/(e^4*Sqrt[d + e*x^2]) - (3*d*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^4 + ((d + e*x^2)^(3/2)*(a + b*Log[c*x^n]))/(3*e^4)} -{x^5*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 7, (b*d*n)/(3*e^3*Sqrt[d + e*x^2]) - (b*n*Sqrt[d + e*x^2])/e^3 + (8*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*e^3) - (d^2*(a + b*Log[c*x^n]))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*Log[c*x^n]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/e^3} -{x^3*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 6, -((b*n)/(3*e^2*Sqrt[d + e*x^2])) - (2*b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*Sqrt[d]*e^2) + (d*(a + b*Log[c*x^n]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*Log[c*x^n])/(e^2*Sqrt[d + e*x^2])} -{x^1*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 5, (b*n)/(3*d*e*Sqrt[d + e*x^2]) - (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(3/2)*e) - (a + b*Log[c*x^n])/(3*e*(d + e*x^2)^(3/2))} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^2)^(5/2)), x, 15, -((b*n)/(3*d^2*Sqrt[d + e*x^2])) + (4*b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(5/2)) + (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(2*d^(5/2)) + (1/3)*(1/(d*(d + e*x^2)^(3/2)) + 3/(d^2*Sqrt[d + e*x^2]) - (3*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(5/2))*(a + b*Log[c*x^n]) - (b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/d^(5/2) - (b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*d^(5/2))} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^2)^(5/2)), x, 14, (b*e*n)/(3*d^3*Sqrt[d + e*x^2]) - (b*n*Sqrt[d + e*x^2])/(4*d^3*x^2) - (31*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(12*d^(7/2)) - (5*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]^2)/(4*d^(7/2)) - (5*e*(a + b*Log[c*x^n]))/(6*d^2*(d + e*x^2)^(3/2)) - (a + b*Log[c*x^n])/(2*d*x^2*(d + e*x^2)^(3/2)) - (5*e*(a + b*Log[c*x^n]))/(2*d^3*Sqrt[d + e*x^2]) + (5*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(7/2)) + (5*b*e*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(2*d^(7/2)) + (5*b*e*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^2])])/(4*d^(7/2))} - -{x^6*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 24, If[$VersionNumber<11, (5*b*d*n*x)/(6*e^3*Sqrt[d + e*x^2]) + (b*n*x^3)/(2*e^2*Sqrt[d + e*x^2]) - (3*b*n*x*Sqrt[d + e*x^2])/(4*e^3) - (31*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(12*e^(7/2)*Sqrt[d + e*x^2]) - (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(4*e^(7/2)*Sqrt[d + e*x^2]) - (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*ArcTanh[E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 + E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(7/2)*Sqrt[d + e*x^2]) - (x^5*(a + b*Log[c*x^n]))/(3*e*(d + e*x^2)^(3/2)) - (5*x^3*(a + b*Log[c*x^n]))/(3*e^2*Sqrt[d + e*x^2]) + (5*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2*e^3) - (5*d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*e^(7/2)*Sqrt[d + e*x^2]), (b*d*n*x)/(3*e^3*Sqrt[d + e*x^2]) - (b*n*x*Sqrt[d + e*x^2])/(4*e^3) - (31*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(12*e^(7/2)*Sqrt[d + e*x^2]) - (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(4*e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(7/2)*Sqrt[d + e*x^2]) - (x^5*(a + b*Log[c*x^n]))/(3*e*(d + e*x^2)^(3/2)) - (5*x^3*(a + b*Log[c*x^n]))/(3*e^2*Sqrt[d + e*x^2]) + (5*x*Sqrt[d + e*x^2]*(a + b*Log[c*x^n]))/(2*e^3) - (5*d^(3/2)*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*e^(7/2)*Sqrt[d + e*x^2]) + (5*b*d^(3/2)*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(4*e^(7/2)*Sqrt[d + e*x^2])]} -{x^4*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 13, -((b*n*x)/(3*e^2*Sqrt[d + e*x^2])) + (4*b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(5/2)*Sqrt[d + e*x^2]) + (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*e^(5/2)*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(e^(5/2)*Sqrt[d + e*x^2]) - (x^3*(a + b*Log[c*x^n]))/(3*e*(d + e*x^2)^(3/2)) - (x*(a + b*Log[c*x^n]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(e^(5/2)*Sqrt[d + e*x^2]) - (b*Sqrt[d]*n*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*e^(5/2)*Sqrt[d + e*x^2])} -{x^2*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 4, (b*n*x)/(3*d*e*Sqrt[d + e*x^2]) - (b*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d*e^(3/2)) + (x^3*(a + b*Log[c*x^n]))/(3*d*(d + e*x^2)^(3/2))} -{x^0*(a + b*Log[c*x^n])/(d + e*x^2)^(5/2), x, 5, -((b*n*x)/(3*d^2*Sqrt[d + e*x^2])) - (2*b*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^2*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*Log[c*x^n]))/(3*d^2*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^2)^(5/2)), x, 6, -((b*n)/(d^2*x*Sqrt[d + e*x^2])) - (2*b*e*n*x)/(3*d^3*Sqrt[d + e*x^2]) + (8*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^3) - (a + b*Log[c*x^n])/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*Log[c*x^n]))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*Log[c*x^n]))/(3*d^3*Sqrt[d + e*x^2])} -{(a + b*Log[c*x^n])/(x^4*(d + e*x^2)^(5/2)), x, 7, -((b*e^2*n*x)/(3*d^4*Sqrt[d + e*x^2])) - (b*n*Sqrt[d + e*x^2])/(9*d^3*x^3) + (23*b*e*n*Sqrt[d + e*x^2])/(9*d^4*x) - (16*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*d^4) - (a + b*Log[c*x^n])/(3*d*x^3*(d + e*x^2)^(3/2)) + (2*e*(a + b*Log[c*x^n]))/(d^2*x*(d + e*x^2)^(3/2)) + (8*e^2*x*(a + b*Log[c*x^n]))/(3*d^3*(d + e*x^2)^(3/2)) + (16*e^2*x*(a + b*Log[c*x^n]))/(3*d^4*Sqrt[d + e*x^2])} - - -{x^3*(a + b*Log[c*x^n])/(Sqrt[d + e*x]*Sqrt[d - e*x]), x, 8, (2*b*d^2*n*(d^2 - e^2*x^2))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*n*(d^2 - e^2*x^2)^2)/(9*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (2*b*d^4*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) - (d^2*(d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^4*Sqrt[d - e*x]*Sqrt[d + e*x]) + ((d^2 - e^2*x^2)^2*(a + b*Log[c*x^n]))/(3*e^4*Sqrt[d - e*x]*Sqrt[d + e*x])} -{x^1*(a + b*Log[c*x^n])/(Sqrt[d + e*x]*Sqrt[d - e*x]), x, 6, (b*n*(d^2 - e^2*x^2))/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*d^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(e^2*Sqrt[d - e*x]*Sqrt[d + e*x])} -{(a + b*Log[c*x^n])/(x^1*Sqrt[d + e*x]*Sqrt[d - e*x]), x, 8, (b*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]^2)/(2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*(a + b*Log[c*x^n]))/(Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*Log[2/(1 - Sqrt[1 - (e^2*x^2)/d^2])])/(Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2, -((1 + Sqrt[1 - (e^2*x^2)/d^2])/(1 - Sqrt[1 - (e^2*x^2)/d^2]))])/(2*Sqrt[d - e*x]*Sqrt[d + e*x])} -{(a + b*Log[c*x^n])/(x^3*Sqrt[d + e*x]*Sqrt[d - e*x]), x, 13, -((b*n*(d^2 - e^2*x^2))/(4*d^2*x^2*Sqrt[d - e*x]*Sqrt[d + e*x])) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]^2)/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(2*d^2*x^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (e^2*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*(a + b*Log[c*x^n]))/(2*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*Log[2/(1 - Sqrt[1 - (e^2*x^2)/d^2])])/(2*d^2*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*e^2*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2, -((1 + Sqrt[1 - (e^2*x^2)/d^2])/(1 - Sqrt[1 - (e^2*x^2)/d^2]))])/(4*d^2*Sqrt[d - e*x]*Sqrt[d + e*x])} - -{x^2*(a + b*Log[c*x^n])/(Sqrt[d + e*x]*Sqrt[d - e*x]), x, 12, (b*n*x*(d^2 - e^2*x^2))/(4*e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (b*d^3*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d])/(4*e^3*Sqrt[d - e*x]*Sqrt[d + e*x]) + (I*b*d^3*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]^2)/(4*e^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*d^3*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]*Log[1 - E^(2*I*ArcSin[(e*x)/d])])/(2*e^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - (x*(d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(2*e^2*Sqrt[d - e*x]*Sqrt[d + e*x]) + (d^3*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]*(a + b*Log[c*x^n]))/(2*e^3*Sqrt[d - e*x]*Sqrt[d + e*x]) + (I*b*d^3*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2, E^(2*I*ArcSin[(e*x)/d])])/(4*e^3*Sqrt[d - e*x]*Sqrt[d + e*x])} -{x^0*(a + b*Log[c*x^n])/(Sqrt[d + e*x]*Sqrt[d - e*x]), x, 7, (I*b*d*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]^2)/(2*e*Sqrt[d - e*x]*Sqrt[d + e*x]) - (b*d*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]*Log[1 - E^(2*I*ArcSin[(e*x)/d])])/(e*Sqrt[d - e*x]*Sqrt[d + e*x]) + (d*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d]*(a + b*Log[c*x^n]))/(e*Sqrt[d - e*x]*Sqrt[d + e*x]) + (I*b*d*n*Sqrt[1 - (e^2*x^2)/d^2]*PolyLog[2, E^(2*I*ArcSin[(e*x)/d])])/(2*e*Sqrt[d - e*x]*Sqrt[d + e*x])} -{(a + b*Log[c*x^n])/(x^2*Sqrt[d + e*x]*Sqrt[d - e*x]), x, 4, -((b*n*(d^2 - e^2*x^2))/(d^2*x*Sqrt[d - e*x]*Sqrt[d + e*x])) - (b*e*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d])/(d*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(d^2*x*Sqrt[d - e*x]*Sqrt[d + e*x])} -{(a + b*Log[c*x^n])/(x^4*Sqrt[d + e*x]*Sqrt[d - e*x]), x, 6, -((2*b*e^2*n*(d^2 - e^2*x^2))/(3*d^4*x*Sqrt[d - e*x]*Sqrt[d + e*x])) - (b*n*(d^2 - e^2*x^2)^2)/(9*d^4*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - (2*b*e^3*n*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d])/(3*d^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - ((d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(3*d^2*x^3*Sqrt[d - e*x]*Sqrt[d + e*x]) - (2*e^2*(d^2 - e^2*x^2)*(a + b*Log[c*x^n]))/(3*d^4*x*Sqrt[d - e*x]*Sqrt[d + e*x])} - - -{x*Log[x]/Sqrt[-1 + x^2], x, 5, -Sqrt[-1 + x^2] + ArcTan[Sqrt[-1 + x^2]] + Sqrt[-1 + x^2]*Log[x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b Log[c x^n]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*Log[c*x^n]), x, 3, -((b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (3*b*d^2*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (3*b*d*e^2*n*(f*x)^(5 + m))/(f^5*(5 + m)^2) - (b*e^3*n*(f*x)^(7 + m))/(f^7*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*Log[c*x^n]))/(f^7*(7 + m))} -{(f*x)^m*(d + e*x^2)^2*(a + b*Log[c*x^n]), x, 4, -((b*d^2*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (2*b*d*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) - (b*e^2*n*(f*x)^(5 + m))/(f^5*(5 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*Log[c*x^n]))/(f^5*(5 + m))} -{(f*x)^m*(d + e*x^2)^1*(a + b*Log[c*x^n]), x, 4, -((b*d*n*(f*x)^(1 + m))/(f*(1 + m)^2)) - (b*e*n*(f*x)^(3 + m))/(f^3*(3 + m)^2) + (d*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*Log[c*x^n]))/(f^3*(3 + m))} -{(f*x)^m*(d + e*x^2)^0*(a + b*Log[c*x^n]), x, 1, -((b*n*(f*x)^(1 + m))/(f*(1 + m)^2)) + ((f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2), x]} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^3)^q (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^3)^q (a+b Log[c x^n])^p*) - - -{(a + b*Log[c*x^n])^3/(d + e*x^3)^2, x, 26, If[$VersionNumber>=8, (x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^3)/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) - (b*n*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^3*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (3*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*I*Sqrt[3]*(a + b*Log[c*x^n])^3*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^3*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (2*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (6*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (6*I*Sqrt[3]*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (6*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b^3*n^3*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (4*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (6*(-1)^(1/3)*b^3*n^3*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (12*I*Sqrt[3]*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) - (2*(-1)^(1/3)*b^3*n^3*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (12*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (4*b^3*n^3*PolyLog[4, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (12*I*Sqrt[3]*b^3*n^3*PolyLog[4, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (12*b^3*n^3*PolyLog[4, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)), (x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^3)/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^3)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) - (b*n*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^3*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (3*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(3*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])^3*Log[1 - ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])^3*Log[1 - ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (2*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (6*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (4*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) + (2*b^3*n^3*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (4*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (6*(-1)^(1/3)*b^3*n^3*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*(-1)^(1/3)*b^3*n^3*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) + (8*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) + (8*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) + (4*b^3*n^3*PolyLog[4, -((e^(1/3)*x)/d^(1/3))])/(3*d^(5/3)*e^(1/3)) - (8*b^3*n^3*PolyLog[4, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (8*b^3*n^3*PolyLog[4, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(3*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3))]} -{(a + b*Log[c*x^n])^2/(d + e*x^3)^2, x, 20, If[$VersionNumber>=8, (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^2)/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*I*Sqrt[3]*(a + b*Log[c*x^n])^2*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (2*b^2*n^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (4*I*Sqrt[3]*b*n*(a + b*Log[c*x^n])*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*I*Sqrt[3]*b^2*n^2*PolyLog[3, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)), (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n])^2)/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n])^2)/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])^2*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b*n*(a + b*Log[c*x^n])*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])^2*Log[1 - ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])^2*Log[1 - ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) - (2*b^2*n^2*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*(-1)^(1/3)*b^2*n^2*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - (8*b*n*(a + b*Log[c*x^n])*PolyLog[2, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (8*b*n*(a + b*Log[c*x^n])*PolyLog[2, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*b^2*n^2*PolyLog[3, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) + (8*b^2*n^2*PolyLog[3, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) + (8*b^2*n^2*PolyLog[3, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3))]} -{(a + b*Log[c*x^n])^1/(d + e*x^3)^2, x, 14, If[$VersionNumber>=8, (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n]))/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) + ((-1)^(1/3)*b*n*Log[(-(-1)^(2/3))*d^(1/3) - e^(1/3)*x])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (b*n*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) - (2*I*Sqrt[3]*(a + b*Log[c*x^n])*Log[1 - ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + ((-1)^(2/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - (2*I*Sqrt[3]*b*n*PolyLog[2, ((-1)^(1/3)*e^(1/3)*x)/d^(1/3)])/((1 + (-1)^(1/3))^5*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -(((-1)^(2/3)*e^(1/3)*x)/d^(1/3))])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)), (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + e^(1/3)*x)) - ((-1)^(1/3)*x*(a + b*Log[c*x^n]))/((1 + (-1)^(1/3))^4*d^(5/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x)) + (x*(a + b*Log[c*x^n]))/(9*d^(5/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)) + ((-1)^(1/3)*b*n*Log[(-(-1)^(2/3))*d^(1/3) - e^(1/3)*x])/((1 + (-1)^(1/3))^4*d^(5/3)*e^(1/3)) - (b*n*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + ((-1)^(1/3)*b*n*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(9*d^(5/3)*e^(1/3)) + (2*(a + b*Log[c*x^n])*Log[1 + (e^(1/3)*x)/d^(1/3)])/(9*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])*Log[1 - ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*(a + b*Log[c*x^n])*Log[1 - ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3)) + (2*b*n*PolyLog[2, -((e^(1/3)*x)/d^(1/3))])/(9*d^(5/3)*e^(1/3)) - (4*b*n*PolyLog[2, ((1 - I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 - I*Sqrt[3])*d^(5/3)*e^(1/3)) - (4*b*n*PolyLog[2, ((1 + I*Sqrt[3])*e^(1/3)*x)/(2*d^(1/3))])/(9*(1 + I*Sqrt[3])*d^(5/3)*e^(1/3))]} -{1/((d + e*x^3)^2*(a + b*Log[c*x^n])^1), x, 0, Unintegrable[1/((d + e*x^3)^2*(a + b*Log[c*x^n])), x]} -{1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2), x, 0, Unintegrable[1/((d + e*x^3)^2*(a + b*Log[c*x^n])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^r)^q (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e / x)^q (a+b Log[c x^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3*(a + b*Log[c*x^n])/(d + e/x), x, 9, -((a*e^3*x)/d^4) + (b*e^3*n*x)/d^4 - (b*e^2*n*x^2)/(4*d^3) + (b*e*n*x^3)/(9*d^2) - (b*n*x^4)/(16*d) - (b*e^3*x*Log[c*x^n])/d^4 + (e^2*x^2*(a + b*Log[c*x^n]))/(2*d^3) - (e*x^3*(a + b*Log[c*x^n]))/(3*d^2) + (x^4*(a + b*Log[c*x^n]))/(4*d) + (e^4*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^5 + (b*e^4*n*PolyLog[2, -((d*x)/e)])/d^5} -{x^2*(a + b*Log[c*x^n])/(d + e/x), x, 8, (a*e^2*x)/d^3 - (b*e^2*n*x)/d^3 + (b*e*n*x^2)/(4*d^2) - (b*n*x^3)/(9*d) + (b*e^2*x*Log[c*x^n])/d^3 - (e*x^2*(a + b*Log[c*x^n]))/(2*d^2) + (x^3*(a + b*Log[c*x^n]))/(3*d) - (e^3*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^4 - (b*e^3*n*PolyLog[2, -((d*x)/e)])/d^4} -{x^1*(a + b*Log[c*x^n])/(d + e/x), x, 7, -((a*e*x)/d^2) + (b*e*n*x)/d^2 - (b*n*x^2)/(4*d) - (b*e*x*Log[c*x^n])/d^2 + (x^2*(a + b*Log[c*x^n]))/(2*d) + (e^2*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^3 + (b*e^2*n*PolyLog[2, -((d*x)/e)])/d^3} -{x^0*(a + b*Log[c*x^n])/(d + e/x), x, 6, (a*x)/d - (b*n*x)/d + (b*x*Log[c*x^n])/d - (e*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^2 - (b*e*n*PolyLog[2, -((d*x)/e)])/d^2} -{(a + b*Log[c*x^n])/((d + e/x)*x^1), x, 3, ((a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d + (b*n*PolyLog[2, -((d*x)/e)])/d} -{(a + b*Log[c*x^n])/((d + e/x)*x^2), x, 2, -((Log[1 + e/(d*x)]*(a + b*Log[c*x^n]))/e) + (b*n*PolyLog[2, -(e/(d*x))])/e} -{(a + b*Log[c*x^n])/((d + e/x)*x^3), x, 6, -((b*n)/(e*x)) - (a + b*Log[c*x^n])/(e*x) - (d*(a + b*Log[c*x^n])^2)/(2*b*e^2*n) + (d*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/e^2 + (b*d*n*PolyLog[2, -((d*x)/e)])/e^2} -{(a + b*Log[c*x^n])/((d + e/x)*x^4), x, 7, -((b*n)/(4*e*x^2)) + (b*d*n)/(e^2*x) - (a + b*Log[c*x^n])/(2*e*x^2) + (d*(a + b*Log[c*x^n]))/(e^2*x) + (d^2*(a + b*Log[c*x^n])^2)/(2*b*e^3*n) - (d^2*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/e^3 - (b*d^2*n*PolyLog[2, -((d*x)/e)])/e^3} - - -{x^3*(a + b*Log[c*x])/(d + e/x), x, 9, -((a*e^3*x)/d^4) + (b*e^3*x)/d^4 - (b*e^2*x^2)/(4*d^3) + (b*e*x^3)/(9*d^2) - (b*x^4)/(16*d) - (b*e^3*x*Log[c*x])/d^4 + (e^2*x^2*(a + b*Log[c*x]))/(2*d^3) - (e*x^3*(a + b*Log[c*x]))/(3*d^2) + (x^4*(a + b*Log[c*x]))/(4*d) + (e^4*(a + b*Log[c*x])*Log[1 + (d*x)/e])/d^5 + (b*e^4*PolyLog[2, -((d*x)/e)])/d^5} -{x^2*(a + b*Log[c*x])/(d + e/x), x, 8, (a*e^2*x)/d^3 - (b*e^2*x)/d^3 + (b*e*x^2)/(4*d^2) - (b*x^3)/(9*d) + (b*e^2*x*Log[c*x])/d^3 - (e*x^2*(a + b*Log[c*x]))/(2*d^2) + (x^3*(a + b*Log[c*x]))/(3*d) - (e^3*(a + b*Log[c*x])*Log[1 + (d*x)/e])/d^4 - (b*e^3*PolyLog[2, -((d*x)/e)])/d^4} -{x^1*(a + b*Log[c*x])/(d + e/x), x, 7, -((a*e*x)/d^2) + (b*e*x)/d^2 - (b*x^2)/(4*d) - (b*e*x*Log[c*x])/d^2 + (x^2*(a + b*Log[c*x]))/(2*d) + (e^2*(a + b*Log[c*x])*Log[1 + (d*x)/e])/d^3 + (b*e^2*PolyLog[2, -((d*x)/e)])/d^3} -{x^0*(a + b*Log[c*x])/(d + e/x), x, 6, (a*x)/d - (b*x)/d + (b*x*Log[c*x])/d - (e*(a + b*Log[c*x])*Log[1 + (d*x)/e])/d^2 - (b*e*PolyLog[2, -((d*x)/e)])/d^2} -{(a + b*Log[c*x])/((d + e/x)*x^1), x, 3, ((a + b*Log[c*x])*Log[1 + (d*x)/e])/d + (b*PolyLog[2, -((d*x)/e)])/d} -{(a + b*Log[c*x])/((d + e/x)*x^2), x, 2, -((Log[1 + e/(d*x)]*(a + b*Log[c*x]))/e) + (b*PolyLog[2, -(e/(d*x))])/e} -{(a + b*Log[c*x])/((d + e/x)*x^3), x, 6, -(b/(e*x)) - (a + b*Log[c*x])/(e*x) - (d*(a + b*Log[c*x])^2)/(2*b*e^2) + (d*(a + b*Log[c*x])*Log[1 + (d*x)/e])/e^2 + (b*d*PolyLog[2, -((d*x)/e)])/e^2} -{(a + b*Log[c*x])/((d + e/x)*x^4), x, 7, -(b/(4*e*x^2)) + (b*d)/(e^2*x) - (a + b*Log[c*x])/(2*e*x^2) + (d*(a + b*Log[c*x]))/(e^2*x) + (d^2*(a + b*Log[c*x])^2)/(2*b*e^3) - (d^2*(a + b*Log[c*x])*Log[1 + (d*x)/e])/e^3 - (b*d^2*PolyLog[2, -((d*x)/e)])/e^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^n)^q (a+b Log[c x^n])^p when m=n-1 and c=-e/d*) - - -{x^(n - 1)*Log[e*x^n]/(1 - e*x^n), x, 2, PolyLog[2, 1 - e*x^n]/(e*n)} -{x^(n - 1)*Log[x^n/d]/(d - x^n), x, 2, PolyLog[2, 1 - x^n/d]/n} -{x^(n - 1)*Log[-e*x^n/d]/(d + e*x^n), x, 2, -(PolyLog[2, 1 + (e*x^n)/d]/(e*n))} - - -{Log[a/x]/(a*x - x^2), x, 4, PolyLog[2, 1 - a/x]/a} -{Log[a/x^2]/(a*x - x^3), x, 4, PolyLog[2, 1 - a/x^2]/(2*a)} -{Log[a/x^(n - 1)]/(a*x - x^n), x, 3, -(PolyLog[2, 1 - a*x^(1 - n)]/(a*(1 - n)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^r)^q (a+b Log[c x^n])^p when m=r-1*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f*x)^(m - 1)*(d + e*x^m)^3*(a + b*Log[c*x^n]), x, 5, -((b*d^3*n*x*(f*x)^(-1 + m))/m^2) - (3*b*d^2*e*n*x^(1 + m)*(f*x)^(-1 + m))/(4*m^2) - (b*d*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m))/(3*m^2) - (b*e^3*n*x^(1 + 3*m)*(f*x)^(-1 + m))/(16*m^2) - (b*d^4*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(4*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^4*(a + b*Log[c*x^n]))/(4*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^2*(a + b*Log[c*x^n]), x, 5, -((b*d^2*n*x*(f*x)^(-1 + m))/m^2) - (b*d*e*n*x^(1 + m)*(f*x)^(-1 + m))/(2*m^2) - (b*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m))/(9*m^2) - (b*d^3*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(3*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n]))/(3*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^1*(a + b*Log[c*x^n]), x, 5, -((b*d*n*(f*x)^m)/(f*m^2)) - (b*e*n*x^m*(f*x)^m)/(4*f*m^2) + (d*(f*x)^m*(a + b*Log[c*x^n]))/(f*m) + (e*x^m*(f*x)^m*(a + b*Log[c*x^n]))/(2*f*m), -((b*d*n*x*(f*x)^(-1 + m))/m^2) - (b*e*n*x^(1 + m)*(f*x)^(-1 + m))/(4*m^2) - (b*d^2*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(2*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^2*(a + b*Log[c*x^n]))/(2*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^0*(a + b*Log[c*x^n]), x, 1, -((b*n*(f*x)^m)/(f*m^2)) + ((f*x)^m*(a + b*Log[c*x^n]))/(f*m)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])/(d + e*x^m)^1, x, 3, (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])*Log[1 + (e*x^m)/d])/(e*m) + (b*n*x^(1 - m)*(f*x)^(-1 + m)*PolyLog[2, -((e*x^m)/d)])/(e*m^2)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])/(d + e*x^m)^2, x, 3, ((f*x)^m*(a + b*Log[c*x^n]))/(d*f*m*(d + e*x^m)) - (b*n*(f*x)^m*Log[d + e*x^m])/(x^m*(d*e*f*m^2))} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])/(d + e*x^m)^3, x, 5, (b*n*x^(1 - m)*(f*x)^(-1 + m))/(2*d*e*m^2*(d + e*x^m)) + (b*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(2*d^2*e*m) - (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(2*e*m*(d + e*x^m)^2) - (b*n*x^(1 - m)*(f*x)^(-1 + m)*Log[d + e*x^m])/(2*d^2*e*m^2)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])/(d + e*x^m)^4, x, 5, (b*n*x^(1 - m)*(f*x)^(-1 + m))/(6*d*e*m^2*(d + e*x^m)^2) + (b*n*x^(1 - m)*(f*x)^(-1 + m))/(3*d^2*e*m^2*(d + e*x^m)) + (b*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(3*d^3*e*m) - (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(3*e*m*(d + e*x^m)^3) - (b*n*x^(1 - m)*(f*x)^(-1 + m)*Log[d + e*x^m])/(3*d^3*e*m^2)} - - -{(f*x)^(m - 1)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2, x, 7, (2*b^2*d^3*n^2*x*(f*x)^(-1 + m))/m^3 + (3*b^2*d^2*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(4*m^3) + (2*b^2*d*e^2*n^2*x^(1 + 2*m)*(f*x)^(-1 + m))/(9*m^3) + (b^2*e^3*n^2*x^(1 + 3*m)*(f*x)^(-1 + m))/(32*m^3) + (b^2*d^4*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x]^2)/(4*e*m) - (2*b*d^3*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (3*b*d^2*e*n*x^(1 + m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(2*m^2) - (2*b*d*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(3*m^2) - (b*e^3*n*x^(1 + 3*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(8*m^2) - (b*d^4*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x]*(a + b*Log[c*x^n]))/(2*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^4*(a + b*Log[c*x^n])^2)/(4*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^2*(a + b*Log[c*x^n])^2, x, 7, (2*b^2*d^2*n^2*x*(f*x)^(-1 + m))/m^3 + (b^2*d*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(2*m^3) + (2*b^2*e^2*n^2*x^(1 + 2*m)*(f*x)^(-1 + m))/(27*m^3) + (b^2*d^3*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x]^2)/(3*e*m) - (2*b*d^2*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (b*d*e*n*x^(1 + m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (2*b*e^2*n*x^(1 + 2*m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(9*m^2) - (2*b*d^3*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x]*(a + b*Log[c*x^n]))/(3*e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^3*(a + b*Log[c*x^n])^2)/(3*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^1*(a + b*Log[c*x^n])^2, x, 7, (2*b^2*d*n^2*x*(f*x)^(-1 + m))/m^3 + (b^2*e*n^2*x^(1 + m)*(f*x)^(-1 + m))/(4*m^3) + (b^2*d^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x]^2)/(2*e*m) - (2*b*d*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/m^2 - (b*e*n*x^(1 + m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(2*m^2) - (b*d^2*n*x^(1 - m)*(f*x)^(-1 + m)*Log[x]*(a + b*Log[c*x^n]))/(e*m) + (x^(1 - m)*(f*x)^(-1 + m)*(d + e*x^m)^2*(a + b*Log[c*x^n])^2)/(2*e*m)} -{(f*x)^(m - 1)*(d + e*x^m)^0*(a + b*Log[c*x^n])^2, x, 2, (2*b^2*n^2*(f*x)^m)/(f*m^3) - (2*b*n*(f*x)^m*(a + b*Log[c*x^n]))/(f*m^2) + ((f*x)^m*(a + b*Log[c*x^n])^2)/(f*m)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])^2/(d + e*x^m)^1, x, 4, (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])^2*Log[1 + (e*x^m)/d])/(e*m) + (2*b*n*x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])*PolyLog[2, -((e*x^m)/d)])/(e*m^2) - (2*b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*PolyLog[3, -((e*x^m)/d)])/(e*m^3)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])^2/(d + e*x^m)^2, x, 4, -((x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])^2)/(e*m*(d + e*x^m))) - (2*b*n*x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])*Log[1 + d/(x^m*e)])/(d*e*m^2) + (2*b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*PolyLog[2, -(d/(x^m*e))])/(d*e*m^3)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])^2/(d + e*x^m)^3, x, 7, -((b*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(d^2*m^2*(d + e*x^m))) - (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])^2)/(2*e*m*(d + e*x^m)^2) - (b*n*x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])*Log[1 + d/(x^m*e)])/(d^2*e*m^2) + (b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[d + e*x^m])/(d^2*e*m^3) + (b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*PolyLog[2, -(d/(x^m*e))])/(d^2*e*m^3)} -{(f*x)^(m - 1)*(a + b*Log[c*x^n])^2/(d + e*x^m)^4, x, 12, -((b^2*n^2*x^(1 - m)*(f*x)^(-1 + m))/(3*d^2*e*m^3*(d + e*x^m))) - (b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[x])/(3*d^3*e*m^2) + (b*n*x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(3*d*e*m^2*(d + e*x^m)^2) - (2*b*n*x*(f*x)^(-1 + m)*(a + b*Log[c*x^n]))/(3*d^3*m^2*(d + e*x^m)) - (x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])^2)/(3*e*m*(d + e*x^m)^3) - (2*b*n*x^(1 - m)*(f*x)^(-1 + m)*(a + b*Log[c*x^n])*Log[1 + d/(x^m*e)])/(3*d^3*e*m^2) + (b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*Log[d + e*x^m])/(d^3*e*m^3) + (2*b^2*n^2*x^(1 - m)*(f*x)^(-1 + m)*PolyLog[2, -(d/(x^m*e))])/(3*d^3*e*m^3)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^r)^q (a+b Log[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*(d + e*x^r)*(a + b*Log[c*x^n]), x, 4, -(b*d*n*x^6)/36 - (b*e*n*x^(6 + r))/(6 + r)^2 + ((d*x^6 + (6*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6} -{x^3*(d + e*x^r)*(a + b*Log[c*x^n]), x, 4, -(b*d*n*x^4)/16 - (b*e*n*x^(4 + r))/(4 + r)^2 + ((d*x^4 + (4*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4} -{x^1*(d + e*x^r)*(a + b*Log[c*x^n]), x, 4, -(b*d*n*x^2)/4 - (b*e*n*x^(2 + r))/(2 + r)^2 + ((d*x^2 + (2*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^1, x, 4, -((b*e*n*x^r)/r^2) + (e*x^r*(a + b*Log[c*x^n]))/r + (d*(a + b*Log[c*x^n])^2)/(2*b*n)} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^3, x, 4, -((b*d*n)/(4*x^2)) - (b*e*n*x^(-2 + r))/(2 - r)^2 - (d*(a + b*Log[c*x^n]))/(2*x^2) - (e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r)} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^5, x, 4, -((b*d*n)/(16*x^4)) - (b*e*n*x^(-4 + r))/(4 - r)^2 - (d*(a + b*Log[c*x^n]))/(4*x^4) - (e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r)} - -{x^4*(d + e*x^r)*(a + b*Log[c*x^n]), x, 4, -(b*d*n*x^5)/25 - (b*e*n*x^(5 + r))/(5 + r)^2 + ((d*x^5 + (5*e*x^(5 + r))/(5 + r))*(a + b*Log[c*x^n]))/5} -{x^2*(d + e*x^r)*(a + b*Log[c*x^n]), x, 4, -(b*d*n*x^3)/9 - (b*e*n*x^(3 + r))/(3 + r)^2 + ((d*x^3 + (3*e*x^(3 + r))/(3 + r))*(a + b*Log[c*x^n]))/3} -{x^0*(d + e*x^r)*(a + b*Log[c*x^n]), x, 3, (-b)*d*n*x - (b*e*n*x^(1 + r))/(1 + r)^2 + d*x*(a + b*Log[c*x^n]) + (e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r)} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^2, x, 4, -((b*d*n)/x) - (b*e*n*x^(-1 + r))/(1 - r)^2 - (d*(a + b*Log[c*x^n]))/x - (e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r)} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^4, x, 4, -((b*d*n)/(9*x^3)) - (b*e*n*x^(-3 + r))/(3 - r)^2 - (d*(a + b*Log[c*x^n]))/(3*x^3) - (e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r)} -{((d + e*x^r)*(a + b*Log[c*x^n]))/x^6, x, 4, -((b*d*n)/(25*x^5)) - (b*e*n*x^(-5 + r))/(5 - r)^2 - (d*(a + b*Log[c*x^n]))/(5*x^5) - (e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r)} - - -{x^5*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 4, -(b*d^2*n*x^6)/36 - (b*e^2*n*x^(2*(3 + r)))/(4*(3 + r)^2) - (2*b*d*e*n*x^(6 + r))/(6 + r)^2 + ((d^2*x^6 + (3*e^2*x^(2*(3 + r)))/(3 + r) + (12*d*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6} -{x^3*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 4, -(b*d^2*n*x^4)/16 - (b*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (2*b*d*e*n*x^(4 + r))/(4 + r)^2 + ((d^2*x^4 + (2*e^2*x^(2*(2 + r)))/(2 + r) + (8*d*e*x^(4 + r))/(4 + r))*(a + b*Log[c*x^n]))/4} -{x^1*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 4, -(b*d^2*n*x^2)/4 - (b*e^2*n*x^(2*(1 + r)))/(4*(1 + r)^2) - (2*b*d*e*n*x^(2 + r))/(2 + r)^2 + ((d^2*x^2 + (e^2*x^(2*(1 + r)))/(1 + r) + (4*d*e*x^(2 + r))/(2 + r))*(a + b*Log[c*x^n]))/2} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^1, x, 5, -((2*b*d*e*n*x^r)/r^2) - (b*e^2*n*x^(2*r))/(4*r^2) - (1/2)*b*d^2*n*Log[x]^2 + (2*d*e*x^r*(a + b*Log[c*x^n]))/r + (e^2*x^(2*r)*(a + b*Log[c*x^n]))/(2*r) + d^2*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^3, x, 4, -((b*d^2*n)/(4*x^2)) - (b*e^2*n)/(x^(2*(1 - r))*(4*(1 - r)^2)) - (2*b*d*e*n*x^(-2 + r))/(2 - r)^2 - (d^2*(a + b*Log[c*x^n]))/(2*x^2) - (e^2*(a + b*Log[c*x^n]))/(x^(2*(1 - r))*(2*(1 - r))) - (2*d*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r)} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^5, x, 4, -((b*d^2*n)/(16*x^4)) - (b*e^2*n)/(x^(2*(2 - r))*(4*(2 - r)^2)) - (2*b*d*e*n*x^(-4 + r))/(4 - r)^2 - (d^2*(a + b*Log[c*x^n]))/(4*x^4) - (e^2*(a + b*Log[c*x^n]))/(x^(2*(2 - r))*(2*(2 - r))) - (2*d*e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r)} - -{x^4*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 4, -(b*d^2*n*x^5)/25 - (2*b*d*e*n*x^(5 + r))/(5 + r)^2 - (b*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 + ((d^2*x^5 + (10*d*e*x^(5 + r))/(5 + r) + (5*e^2*x^(5 + 2*r))/(5 + 2*r))*(a + b*Log[c*x^n]))/5} -{x^2*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 4, -(b*d^2*n*x^3)/9 - (2*b*d*e*n*x^(3 + r))/(3 + r)^2 - (b*e^2*n*x^(3 + 2*r))/(3 + 2*r)^2 + ((d^2*x^3 + (6*d*e*x^(3 + r))/(3 + r) + (3*e^2*x^(3 + 2*r))/(3 + 2*r))*(a + b*Log[c*x^n]))/3} -{x^0*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 2, (-b)*d^2*n*x - (2*b*d*e*n*x^(1 + r))/(1 + r)^2 - (b*e^2*n*x^(1 + 2*r))/(1 + 2*r)^2 + d^2*x*(a + b*Log[c*x^n]) + (2*d*e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r) + (e^2*x^(1 + 2*r)*(a + b*Log[c*x^n]))/(1 + 2*r)} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^2, x, 3, -((b*d^2*n)/x) - (2*b*d*e*n*x^(-1 + r))/(1 - r)^2 - (b*e^2*n*x^(-1 + 2*r))/(1 - 2*r)^2 - (d^2*(a + b*Log[c*x^n]))/x - (2*d*e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r) - (e^2*x^(-1 + 2*r)*(a + b*Log[c*x^n]))/(1 - 2*r)} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^4, x, 4, -((b*d^2*n)/(9*x^3)) - (2*b*d*e*n*x^(-3 + r))/(3 - r)^2 - (b*e^2*n*x^(-3 + 2*r))/(3 - 2*r)^2 - (d^2*(a + b*Log[c*x^n]))/(3*x^3) - (2*d*e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r) - (e^2*x^(-3 + 2*r)*(a + b*Log[c*x^n]))/(3 - 2*r)} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^6, x, 4, -((b*d^2*n)/(25*x^5)) - (2*b*d*e*n*x^(-5 + r))/(5 - r)^2 - (b*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (d^2*(a + b*Log[c*x^n]))/(5*x^5) - (2*d*e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r) - (e^2*x^(-5 + 2*r)*(a + b*Log[c*x^n]))/(5 - 2*r)} -{((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^8, x, 4, -((b*d^2*n)/(49*x^7)) - (2*b*d*e*n*x^(-7 + r))/(7 - r)^2 - (b*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (d^2*(a + b*Log[c*x^n]))/(7*x^7) - (2*d*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r) - (e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 - 2*r)} - - -{x^5*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 4, -(b*d^3*n*x^6)/36 - (b*e^3*n*x^(3*(2 + r)))/(9*(2 + r)^2) - (3*b*d*e^2*n*x^(2*(3 + r)))/(4*(3 + r)^2) - (3*b*d^2*e*n*x^(6 + r))/(6 + r)^2 + ((d^3*x^6 + (2*e^3*x^(3*(2 + r)))/(2 + r) + (9*d*e^2*x^(2*(3 + r)))/(3 + r) + (18*d^2*e*x^(6 + r))/(6 + r))*(a + b*Log[c*x^n]))/6} -{x^3*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 4, -(b*d^3*n*x^4)/16 - (3*b*d*e^2*n*x^(2*(2 + r)))/(4*(2 + r)^2) - (3*b*d^2*e*n*x^(4 + r))/(4 + r)^2 - (b*e^3*n*x^(4 + 3*r))/(4 + 3*r)^2 + ((d^3*x^4 + (6*d*e^2*x^(2*(2 + r)))/(2 + r) + (12*d^2*e*x^(4 + r))/(4 + r) + (4*e^3*x^(4 + 3*r))/(4 + 3*r))*(a + b*Log[c*x^n]))/4} -{x^1*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 4, -(b*d^3*n*x^2)/4 - (3*b*d*e^2*n*x^(2*(1 + r)))/(4*(1 + r)^2) - (3*b*d^2*e*n*x^(2 + r))/(2 + r)^2 - (b*e^3*n*x^(2 + 3*r))/(2 + 3*r)^2 + ((d^3*x^2 + (3*d*e^2*x^(2*(1 + r)))/(1 + r) + (6*d^2*e*x^(2 + r))/(2 + r) + (2*e^3*x^(2 + 3*r))/(2 + 3*r))*(a + b*Log[c*x^n]))/2} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^1, x, 5, -((3*b*d^2*e*n*x^r)/r^2) - (3*b*d*e^2*n*x^(2*r))/(4*r^2) - (b*e^3*n*x^(3*r))/(9*r^2) - (1/2)*b*d^3*n*Log[x]^2 + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/r + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(3*r) + d^3*Log[x]*(a + b*Log[c*x^n])} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^3, x, 4, -((b*d^3*n)/(4*x^2)) - (3*b*d*e^2*n)/(x^(2*(1 - r))*(4*(1 - r)^2)) - (3*b*d^2*e*n*x^(-2 + r))/(2 - r)^2 - (b*e^3*n*x^(-2 + 3*r))/(2 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(2*x^2) - (3*d*e^2*(a + b*Log[c*x^n]))/(x^(2*(1 - r))*(2*(1 - r))) - (3*d^2*e*x^(-2 + r)*(a + b*Log[c*x^n]))/(2 - r) - (e^3*x^(-2 + 3*r)*(a + b*Log[c*x^n]))/(2 - 3*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^5, x, 4, -((b*d^3*n)/(16*x^4)) - (3*b*d*e^2*n)/(x^(2*(2 - r))*(4*(2 - r)^2)) - (3*b*d^2*e*n*x^(-4 + r))/(4 - r)^2 - (b*e^3*n*x^(-4 + 3*r))/(4 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(4*x^4) - (3*d*e^2*(a + b*Log[c*x^n]))/(x^(2*(2 - r))*(2*(2 - r))) - (3*d^2*e*x^(-4 + r)*(a + b*Log[c*x^n]))/(4 - r) - (e^3*x^(-4 + 3*r)*(a + b*Log[c*x^n]))/(4 - 3*r)} - -{x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 4, -(b*d^3*n*x^5)/25 - (3*b*d^2*e*n*x^(5 + r))/(5 + r)^2 - (3*b*d*e^2*n*x^(5 + 2*r))/(5 + 2*r)^2 - (b*e^3*n*x^(5 + 3*r))/(5 + 3*r)^2 + ((d^3*x^5 + (15*d^2*e*x^(5 + r))/(5 + r) + (15*d*e^2*x^(5 + 2*r))/(5 + 2*r) + (5*e^3*x^(5 + 3*r))/(5 + 3*r))*(a + b*Log[c*x^n]))/5} -{x^2*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 4, -(b*d^3*n*x^3)/9 - (b*e^3*n*x^(3*(1 + r)))/(9*(1 + r)^2) - (3*b*d^2*e*n*x^(3 + r))/(3 + r)^2 - (3*b*d*e^2*n*x^(3 + 2*r))/(3 + 2*r)^2 + ((d^3*x^3 + (e^3*x^(3*(1 + r)))/(1 + r) + (9*d^2*e*x^(3 + r))/(3 + r) + (9*d*e^2*x^(3 + 2*r))/(3 + 2*r))*(a + b*Log[c*x^n]))/3} -{x^0*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 2, (-b)*d^3*n*x - (3*b*d^2*e*n*x^(1 + r))/(1 + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r))/(1 + 2*r)^2 - (b*e^3*n*x^(1 + 3*r))/(1 + 3*r)^2 + d^3*x*(a + b*Log[c*x^n]) + (3*d^2*e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(1 + 2*r)*(a + b*Log[c*x^n]))/(1 + 2*r) + (e^3*x^(1 + 3*r)*(a + b*Log[c*x^n]))/(1 + 3*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^2, x, 3, -((b*d^3*n)/x) - (3*b*d^2*e*n*x^(-1 + r))/(1 - r)^2 - (3*b*d*e^2*n*x^(-1 + 2*r))/(1 - 2*r)^2 - (b*e^3*n*x^(-1 + 3*r))/(1 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/x - (3*d^2*e*x^(-1 + r)*(a + b*Log[c*x^n]))/(1 - r) - (3*d*e^2*x^(-1 + 2*r)*(a + b*Log[c*x^n]))/(1 - 2*r) - (e^3*x^(-1 + 3*r)*(a + b*Log[c*x^n]))/(1 - 3*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^4, x, 4, -((b*d^3*n)/(9*x^3)) - (b*e^3*n)/(x^(3*(1 - r))*(9*(1 - r)^2)) - (3*b*d^2*e*n*x^(-3 + r))/(3 - r)^2 - (3*b*d*e^2*n*x^(-3 + 2*r))/(3 - 2*r)^2 - (d^3*(a + b*Log[c*x^n]))/(3*x^3) - (e^3*(a + b*Log[c*x^n]))/(x^(3*(1 - r))*(3*(1 - r))) - (3*d^2*e*x^(-3 + r)*(a + b*Log[c*x^n]))/(3 - r) - (3*d*e^2*x^(-3 + 2*r)*(a + b*Log[c*x^n]))/(3 - 2*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^6, x, 4, -((b*d^3*n)/(25*x^5)) - (3*b*d^2*e*n*x^(-5 + r))/(5 - r)^2 - (3*b*d*e^2*n*x^(-5 + 2*r))/(5 - 2*r)^2 - (b*e^3*n*x^(-5 + 3*r))/(5 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(5*x^5) - (3*d^2*e*x^(-5 + r)*(a + b*Log[c*x^n]))/(5 - r) - (3*d*e^2*x^(-5 + 2*r)*(a + b*Log[c*x^n]))/(5 - 2*r) - (e^3*x^(-5 + 3*r)*(a + b*Log[c*x^n]))/(5 - 3*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^8, x, 4, -((b*d^3*n)/(49*x^7)) - (3*b*d^2*e*n*x^(-7 + r))/(7 - r)^2 - (3*b*d*e^2*n*x^(-7 + 2*r))/(7 - 2*r)^2 - (b*e^3*n*x^(-7 + 3*r))/(7 - 3*r)^2 - (d^3*(a + b*Log[c*x^n]))/(7*x^7) - (3*d^2*e*x^(-7 + r)*(a + b*Log[c*x^n]))/(7 - r) - (3*d*e^2*x^(-7 + 2*r)*(a + b*Log[c*x^n]))/(7 - 2*r) - (e^3*x^(-7 + 3*r)*(a + b*Log[c*x^n]))/(7 - 3*r)} -{((d + e*x^r)^3*(a + b*Log[c*x^n]))/x^10, x, 4, -((b*d^3*n)/(81*x^9)) - (b*e^3*n)/(x^(3*(3 - r))*(9*(3 - r)^2)) - (3*b*d^2*e*n*x^(-9 + r))/(9 - r)^2 - (3*b*d*e^2*n*x^(-9 + 2*r))/(9 - 2*r)^2 - (d^3*(a + b*Log[c*x^n]))/(9*x^9) - (e^3*(a + b*Log[c*x^n]))/(x^(3*(3 - r))*(3*(3 - r))) - (3*d^2*e*x^(-9 + r)*(a + b*Log[c*x^n]))/(9 - r) - (3*d*e^2*x^(-9 + 2*r)*(a + b*Log[c*x^n]))/(9 - 2*r)} - - -(* ::InheritFromParent:: *) -(**) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(x^3*(a + b*Log[c*x^n]))/(d + e*x^r), x, 0, Unintegrable[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r), x]} -{(x^1*(a + b*Log[c*x^n]))/(d + e*x^r), x, 0, Unintegrable[(x*(a + b*Log[c*x^n]))/(d + e*x^r), x]} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^r)), x, 2, -(((a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d*r)) + (b*n*PolyLog[2, -(d/(e*x^r))])/(d*r^2)} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^r)), x, 0, Unintegrable[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)), x]} - -{x^2*(a + b*Log[c*x^n])/(d + e*x^r), x, 0, Unintegrable[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r), x]} -{x^0*(a + b*Log[c*x^n])/(d + e*x^r), x, 0, Unintegrable[(a + b*Log[c*x^n])/(d + e*x^r), x]} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^r)), x, 0, Unintegrable[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)), x]} - - -{(x^3*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x, 0, Unintegrable[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]} -{(x^1*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x, 0, Unintegrable[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^r)^2), x, 5, -((e*x^r*(a + b*Log[c*x^n]))/(d^2*r*(d + e*x^r))) - ((a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d^2*r) + (b*n*Log[d + e*x^r])/(d^2*r^2) + (b*n*PolyLog[2, -(d/(x^r*e))])/(d^2*r^2)} -{(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2), x, 0, Unintegrable[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2), x]} - -{x^2*(a + b*Log[c*x^n])/(d + e*x^r)^2, x, 0, Unintegrable[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]} -{x^0*(a + b*Log[c*x^n])/(d + e*x^r)^2, x, 0, Unintegrable[(a + b*Log[c*x^n])/(d + e*x^r)^2, x]} -{(a + b*Log[c*x^n])/(x^2*(d + e*x^r)^2), x, 0, Unintegrable[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)^2), x]} - - -{(a + b*Log[c*x^n])/(x*(c - x^(-n))), x, 4, (a*Log[1 - c*x^n])/(c*n) - (b*PolyLog[2, 1 - c*x^n])/(c*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^r)^q (a+b Log[c x^n])^p/x*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(d + e*x^r)^3*(a + b*Log[c*x^n])/x, x, 5, -((3*b*d^2*e*n*x^r)/r^2) - (3*b*d*e^2*n*x^(2*r))/(4*r^2) - (b*e^3*n*x^(3*r))/(9*r^2) - (1/2)*b*d^3*n*Log[x]^2 + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/r + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(3*r) + d^3*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x^r)^2*(a + b*Log[c*x^n])/x, x, 5, -((2*b*d*e*n*x^r)/r^2) - (b*e^2*n*x^(2*r))/(4*r^2) - (1/2)*b*d^2*n*Log[x]^2 + (2*d*e*x^r*(a + b*Log[c*x^n]))/r + (e^2*x^(2*r)*(a + b*Log[c*x^n]))/(2*r) + d^2*Log[x]*(a + b*Log[c*x^n])} -{(d + e*x^r)^1*(a + b*Log[c*x^n])/x, x, 4, -((b*e*n*x^r)/r^2) + (e*x^r*(a + b*Log[c*x^n]))/r + (d*(a + b*Log[c*x^n])^2)/(2*b*n)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^r)^1), x, 2, -(((a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d*r)) + (b*n*PolyLog[2, -(d/(x^r*e))])/(d*r^2)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^r)^2), x, 5, -((e*x^r*(a + b*Log[c*x^n]))/(d^2*r*(d + e*x^r))) - ((a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d^2*r) + (b*n*Log[d + e*x^r])/(d^2*r^2) + (b*n*PolyLog[2, -(d/(x^r*e))])/(d^2*r^2)} -{(a + b*Log[c*x^n])/(x^1*(d + e*x^r)^3), x, 10, -((b*n)/(2*d^2*r^2*(d + e*x^r))) - (b*n*Log[x])/(2*d^3*r) + (a + b*Log[c*x^n])/(2*d*r*(d + e*x^r)^2) - (e*x^r*(a + b*Log[c*x^n]))/(d^3*r*(d + e*x^r)) - ((a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d^3*r) + (3*b*n*Log[d + e*x^r])/(2*d^3*r^2) + (b*n*PolyLog[2, -(d/(x^r*e))])/(d^3*r^2)} - - -{(d + e*x^r)^3*(a + b*Log[c*x^n])^2/x, x, 10, (6*b^2*d^2*e*n^2*x^r)/r^3 + (3*b^2*d*e^2*n^2*x^(2*r))/(4*r^3) + (2*b^2*e^3*n^2*x^(3*r))/(27*r^3) - (6*b*d^2*e*n*x^r*(a + b*Log[c*x^n]))/r^2 - (3*b*d*e^2*n*x^(2*r)*(a + b*Log[c*x^n]))/(2*r^2) - (2*b*e^3*n*x^(3*r)*(a + b*Log[c*x^n]))/(9*r^2) + (3*d^2*e*x^r*(a + b*Log[c*x^n])^2)/r + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n])^2)/(2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n])^2)/(3*r) + (d^3*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x^r)^2*(a + b*Log[c*x^n])^2/x, x, 8, (4*b^2*d*e*n^2*x^r)/r^3 + (b^2*e^2*n^2*x^(2*r))/(4*r^3) - (4*b*d*e*n*x^r*(a + b*Log[c*x^n]))/r^2 - (b*e^2*n*x^(2*r)*(a + b*Log[c*x^n]))/(2*r^2) + (2*d*e*x^r*(a + b*Log[c*x^n])^2)/r + (e^2*x^(2*r)*(a + b*Log[c*x^n])^2)/(2*r) + (d^2*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(d + e*x^r)^1*(a + b*Log[c*x^n])^2/x, x, 6, (2*b^2*e*n^2*x^r)/r^3 - (2*b*e*n*x^r*(a + b*Log[c*x^n]))/r^2 + (e*x^r*(a + b*Log[c*x^n])^2)/r + (d*(a + b*Log[c*x^n])^3)/(3*b*n)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x^r)^1), x, 3, -(((a + b*Log[c*x^n])^2*Log[1 + d/(x^r*e)])/(d*r)) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(x^r*e))])/(d*r^2) + (2*b^2*n^2*PolyLog[3, -(d/(x^r*e))])/(d*r^3)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x^r)^2), x, 7, (a + b*Log[c*x^n])^2/(d*r*(d + e*x^r)) + (2*b*n*(a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d^2*r^2) - ((a + b*Log[c*x^n])^2*Log[1 + d/(x^r*e)])/(d^2*r) - (2*b^2*n^2*PolyLog[2, -(d/(x^r*e))])/(d^2*r^3) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(x^r*e))])/(d^2*r^2) + (2*b^2*n^2*PolyLog[3, -(d/(x^r*e))])/(d^2*r^3)} -{(a + b*Log[c*x^n])^2/(x^1*(d + e*x^r)^3), x, 14, (b*e*n*x^r*(a + b*Log[c*x^n]))/(d^3*r^2*(d + e*x^r)) + (a + b*Log[c*x^n])^2/(2*d*r*(d + e*x^r)^2) + (a + b*Log[c*x^n])^2/(d^2*r*(d + e*x^r)) + (3*b*n*(a + b*Log[c*x^n])*Log[1 + d/(x^r*e)])/(d^3*r^2) - ((a + b*Log[c*x^n])^2*Log[1 + d/(x^r*e)])/(d^3*r) - (b^2*n^2*Log[d + e*x^r])/(d^3*r^3) - (3*b^2*n^2*PolyLog[2, -(d/(x^r*e))])/(d^3*r^3) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(x^r*e))])/(d^3*r^2) + (2*b^2*n^2*PolyLog[3, -(d/(x^r*e))])/(d^3*r^3)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^r)^(q/2) (a+b Log[c x^n])*) - - -{(d + e*x^r)^(5/2)*(a + b*Log[c*x^n])/x, x, 23, -((92*b*d^2*n*Sqrt[d + e*x^r])/(15*r^2)) - (32*b*d*n*(d + e*x^r)^(3/2))/(45*r^2) - (4*b*n*(d + e*x^r)^(5/2))/(25*r^2) + (92*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(15*r^2) + (2*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/r^2 + (2/15)*((15*d^2*Sqrt[d + e*x^r])/r + (5*d*(d + e*x^r)^(3/2))/r + (3*(d + e*x^r)^(5/2))/r - (15*d^(5/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/r)*(a + b*Log[c*x^n]) - (4*b*d^(5/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2 - (2*b*d^(5/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2} -{(d + e*x^r)^(3/2)*(a + b*Log[c*x^n])/x, x, 17, -((16*b*d*n*Sqrt[d + e*x^r])/(3*r^2)) - (4*b*n*(d + e*x^r)^(3/2))/(9*r^2) + (16*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(3*r^2) + (2*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/r^2 + (2/3)*((3*d*Sqrt[d + e*x^r])/r + (d + e*x^r)^(3/2)/r - (3*d^(3/2)*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/r)*(a + b*Log[c*x^n]) - (4*b*d^(3/2)*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2 - (2*b*d^(3/2)*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2} -{(d + e*x^r)^(1/2)*(a + b*Log[c*x^n])/x, x, 12, -((4*b*n*Sqrt[d + e*x^r])/r^2) + (4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/r^2 + (2*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/r^2 + 2*(Sqrt[d + e*x^r]/r - (Sqrt[d]*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/r)*(a + b*Log[c*x^n]) - (4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2 - (2*b*Sqrt[d]*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/r^2} -{(a + b*Log[c*x^n])/(x*(d + e*x^r)^(1/2)), x, 8, (2*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/(Sqrt[d]*r^2) - (2*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*(a + b*Log[c*x^n]))/(Sqrt[d]*r) - (4*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(Sqrt[d]*r^2) - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(Sqrt[d]*r^2)} -{(a + b*Log[c*x^n])/(x*(d + e*x^r)^(3/2)), x, 11, (4*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(d^(3/2)*r^2) + (2*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/(d^(3/2)*r^2) + 2*(1/(d*r*Sqrt[d + e*x^r]) - ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]/(d^(3/2)*r))*(a + b*Log[c*x^n]) - (4*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(3/2)*r^2) - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(3/2)*r^2)} -{(a + b*Log[c*x^n])/(x*(d + e*x^r)^(5/2)), x, 15, -((4*b*n)/(3*d^2*r^2*Sqrt[d + e*x^r])) + (16*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(3*d^(5/2)*r^2) + (2*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/(d^(5/2)*r^2) + (2/3)*(1/(d*r*(d + e*x^r)^(3/2)) + 3/(d^2*r*Sqrt[d + e*x^r]) - (3*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(d^(5/2)*r))*(a + b*Log[c*x^n]) - (4*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(5/2)*r^2) - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(5/2)*r^2)} -{(a + b*Log[c*x^n])/(x*(d + e*x^r)^(7/2)), x, 20, -((4*b*n)/(15*d^2*r^2*(d + e*x^r)^(3/2))) - (32*b*n)/(15*d^3*r^2*Sqrt[d + e*x^r]) + (92*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(15*d^(7/2)*r^2) + (2*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]^2)/(d^(7/2)*r^2) + (2/15)*(3/(d*r*(d + e*x^r)^(5/2)) + 5/(d^2*r*(d + e*x^r)^(3/2)) + 15/(d^3*r*Sqrt[d + e*x^r]) - (15*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]])/(d^(7/2)*r))*(a + b*Log[c*x^n]) - (4*b*n*ArcTanh[Sqrt[d + e*x^r]/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(7/2)*r^2) - (2*b*n*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] - Sqrt[d + e*x^r])])/(d^(7/2)*r^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^r)^q (a+b Log[c x^n]) when m and r symbolic*) - - -{(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n]), x, 9, -((3*b*d^2*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2) - (3*b*d*e^2*n*x^(1 + 2*r)*(f*x)^m)/(1 + m + 2*r)^2 - (b*e^3*n*x^(1 + 3*r)*(f*x)^m)/(1 + m + 3*r)^2 - (b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (3*d^2*e*x^(1 + r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + r) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(1 + 3*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 3*r) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(d + e*x^r)^2*(a + b*Log[c*x^n]), x, 7, -((2*b*d*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2) - (b*e^2*n*x^(1 + 2*r)*(f*x)^m)/(1 + m + 2*r)^2 - (b*d^2*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (2*d*e*x^(1 + r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + r) + (e^2*x^(1 + 2*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (d^2*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(d + e*x^r)^1*(a + b*Log[c*x^n]), x, 6, -((b*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2) - (b*d*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (e*x^(1 + r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + r) + (d*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(d + e*x^r)^0*(a + b*Log[c*x^n]), x, 1, -((b*n*(f*x)^(1 + m))/(f*(1 + m)^2)) + ((f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x^r)^1, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r), x]} -{(f*x)^m*(a + b*Log[c*x^n])/(d + e*x^r)^2, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^r)^q (a+b Log[c x^n])^p when m, p and r symbolic*) - - -{(d + e/x^(1/(q + 1)))^q*(a + b*Log[c*x^n]), x, 3, ((-b)*n*x*(d + e/x^(1/(1 + q)))^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -(e/(x^(1/(1 + q))*d))])/(1 + e/(x^(1/(1 + q))*d))^q + (x*(d + e/x^(1/(1 + q)))^(1 + q)*(a + b*Log[c*x^n]))/d} - - -{(d + e*x^r)^q*(a + b*Log[c*x^n])/(f*x)^(r*(q + 1) + 1), x, 3, -((b*n*(d + e*x^r)^q*Hypergeometric2F1[-1 - q, -1 - q, -q, -((e*x^r)/d)])/((f*x)^((1 + q)*r)*(1 + (e*x^r)/d)^q*(f*(1 + q)^2*r^2))) - ((d + e*x^r)^(1 + q)*(a + b*Log[c*x^n]))/((f*x)^((1 + q)*r)*(d*f*(1 + q)*r))} - - -{(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n])^p, x, 13, (d^3*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(f*(1 + m))) + (3*d^2*e*x^(1 + r)*(f*x)^m*Gamma[1 + p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(c*x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + r)) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*Gamma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + 2*r))/(b*n))*(c*x^n)^((1 + m + 2*r)/n)*(-(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + 2*r)) + (e^3*x^(1 + 3*r)*(f*x)^m*Gamma[1 + p, -(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + 3*r))/(b*n))*(c*x^n)^((1 + m + 3*r)/n)*(-(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + 3*r))} -{(f*x)^m*(d + e*x^r)^2*(a + b*Log[c*x^n])^p, x, 10, (d^2*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(f*(1 + m))) + (2*d*e*x^(1 + r)*(f*x)^m*Gamma[1 + p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(c*x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + r)) + (e^2*x^(1 + 2*r)*(f*x)^m*Gamma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + 2*r))/(b*n))*(c*x^n)^((1 + m + 2*r)/n)*(-(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + 2*r))} -{(f*x)^m*(d + e*x^r)^1*(a + b*Log[c*x^n])^p, x, 7, (d*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(f*(1 + m))) + (e*x^(1 + r)*(f*x)^m*Gamma[1 + p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(c*x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m + r))} -{(f*x)^m*(d + e*x^r)^0*(a + b*Log[c*x^n])^p, x, 2, ((f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(f*(1 + m)))} -{(f*x)^m*(a + b*Log[c*x^n])^p/(d + e*x^r)^1, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n])^p)/(d + e*x^r), x]} -{(f*x)^m*(a + b*Log[c*x^n])^p/(d + e*x^r)^2, x, 0, Unintegrable[((f*x)^m*(a + b*Log[c*x^n])^p)/(d + e*x^r)^2, x]} - - -(* ::Title:: *) -(*Integrands of the form (f+g x)^m (d+e x)^q (a+b Log[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x)^q (a+b Log[c x^n])^p*) - - -{(f + g*x)*(a + b*Log[c*x^n])^1/(d + e*x)^3, x, 3, (b*(e*f - d*g)*n)/(2*d*e^2*(d + e*x)) + (b*f^2*n*Log[x])/(2*d^2*(e*f - d*g)) - ((f + g*x)^2*(a + b*Log[c*x^n]))/(2*(e*f - d*g)*(d + e*x)^2) - (b*(e*f + d*g)*n*Log[d + e*x])/(2*d^2*e^2)} -{(f + g*x)*(a + b*Log[c*x^n])^2/(d + e*x)^3, x, 8, -((b*(e*f - d*g)*n*x*(a + b*Log[c*x^n]))/(d^2*e*(d + e*x))) + (f^2*(a + b*Log[c*x^n])^2)/(2*d^2*(e*f - d*g)) - ((f + g*x)^2*(a + b*Log[c*x^n])^2)/(2*(e*f - d*g)*(d + e*x)^2) + (b^2*(e*f - d*g)*n^2*Log[d + e*x])/(d^2*e^2) - (b*(e*f + d*g)*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(d^2*e^2) - (b^2*(e*f + d*g)*n^2*PolyLog[2, -((e*x)/d)])/(d^2*e^2)} -{(f + g*x)*(a + b*Log[c*x^n])^3/(d + e*x)^3, x, 11, -((3*b*(e*f - d*g)*n*x*(a + b*Log[c*x^n])^2)/(2*d^2*e*(d + e*x))) + (f^2*(a + b*Log[c*x^n])^3)/(2*d^2*(e*f - d*g)) - ((f + g*x)^2*(a + b*Log[c*x^n])^3)/(2*(e*f - d*g)*(d + e*x)^2) + (3*b^2*(e*f - d*g)*n^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/(d^2*e^2) - (3*b*(e*f + d*g)*n*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/(2*d^2*e^2) + (3*b^3*(e*f - d*g)*n^3*PolyLog[2, -((e*x)/d)])/(d^2*e^2) - (3*b^2*(e*f + d*g)*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)])/(d^2*e^2) + (3*b^3*(e*f + d*g)*n^3*PolyLog[3, -((e*x)/d)])/(d^2*e^2)} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m deleted file mode 100644 index d233d62..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.1.5 u (a+b log(c x^n))^p.m +++ /dev/null @@ -1,475 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form AF[x] (a+b Log[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g x)^m (d+e x+f x^2)^q RFx (a+b Log[c x^n])^p*) - - -{(a + b*Log[c*x^n])/(d + e*x + f*x^2), x, 6, ((a + b*Log[c*x^n])*Log[1 + (2*f*x)/(e - Sqrt[e^2 - 4*d*f])])/Sqrt[e^2 - 4*d*f] - ((a + b*Log[c*x^n])*Log[1 + (2*f*x)/(e + Sqrt[e^2 - 4*d*f])])/Sqrt[e^2 - 4*d*f] + (b*n*PolyLog[2, -((2*f*x)/(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (b*n*PolyLog[2, -((2*f*x)/(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f]} - - -(* ::Title::Closed:: *) -(*Integrands of the form EF[x] (a+b Log[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g x)^q Log[d (e+f x^m)] (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^1)] (a+b Log[c x^n])^p when d e=1*) - - -{x^3*(a + b*Log[c*x^n])*Log[1 + e*x], x, 6, -((5*b*n*x)/(16*e^3)) + (3*b*n*x^2)/(32*e^2) - (7*b*n*x^3)/(144*e) + (1/32)*b*n*x^4 + (x*(a + b*Log[c*x^n]))/(4*e^3) - (x^2*(a + b*Log[c*x^n]))/(8*e^2) + (x^3*(a + b*Log[c*x^n]))/(12*e) - (1/16)*x^4*(a + b*Log[c*x^n]) + (b*n*Log[1 + e*x])/(16*e^4) - (1/16)*b*n*x^4*Log[1 + e*x] - ((a + b*Log[c*x^n])*Log[1 + e*x])/(4*e^4) + (1/4)*x^4*(a + b*Log[c*x^n])*Log[1 + e*x] - (b*n*PolyLog[2, (-e)*x])/(4*e^4)} -{x^2*(a + b*Log[c*x^n])*Log[1 + e*x], x, 6, (4*b*n*x)/(9*e^2) - (5*b*n*x^2)/(36*e) + (2/27)*b*n*x^3 - (x*(a + b*Log[c*x^n]))/(3*e^2) + (x^2*(a + b*Log[c*x^n]))/(6*e) - (1/9)*x^3*(a + b*Log[c*x^n]) - (b*n*Log[1 + e*x])/(9*e^3) - (1/9)*b*n*x^3*Log[1 + e*x] + ((a + b*Log[c*x^n])*Log[1 + e*x])/(3*e^3) + (1/3)*x^3*(a + b*Log[c*x^n])*Log[1 + e*x] + (b*n*PolyLog[2, (-e)*x])/(3*e^3)} -{x^1*(a + b*Log[c*x^n])*Log[1 + e*x], x, 6, -((3*b*n*x)/(4*e)) + (1/4)*b*n*x^2 + (x*(a + b*Log[c*x^n]))/(2*e) - (1/4)*x^2*(a + b*Log[c*x^n]) + (b*n*Log[1 + e*x])/(4*e^2) - (1/4)*b*n*x^2*Log[1 + e*x] - ((a + b*Log[c*x^n])*Log[1 + e*x])/(2*e^2) + (1/2)*x^2*(a + b*Log[c*x^n])*Log[1 + e*x] - (b*n*PolyLog[2, (-e)*x])/(2*e^2)} -{x^0*(a + b*Log[c*x^n])*Log[1 + e*x], x, 7, 2*b*n*x - x*(a + b*Log[c*x^n]) - (b*n*(1 + e*x)*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])*Log[1 + e*x])/e + (b*n*PolyLog[2, (-e)*x])/e} -{(a + b*Log[c*x^n])*Log[1 + e*x]/x^1, x, 2, -((a + b*Log[c*x^n])*PolyLog[2, (-e)*x]) + b*n*PolyLog[3, (-e)*x]} -{(a + b*Log[c*x^n])*Log[1 + e*x]/x^2, x, 8, b*e*n*Log[x] - (1/2)*b*e*n*Log[x]^2 + e*Log[x]*(a + b*Log[c*x^n]) - b*e*n*Log[1 + e*x] - (b*n*Log[1 + e*x])/x - e*(a + b*Log[c*x^n])*Log[1 + e*x] - ((a + b*Log[c*x^n])*Log[1 + e*x])/x - b*e*n*PolyLog[2, (-e)*x]} -{(a + b*Log[c*x^n])*Log[1 + e*x]/x^3, x, 7, -((3*b*e*n)/(4*x)) - (1/4)*b*e^2*n*Log[x] + (1/4)*b*e^2*n*Log[x]^2 - (e*(a + b*Log[c*x^n]))/(2*x) - (1/2)*e^2*Log[x]*(a + b*Log[c*x^n]) + (1/4)*b*e^2*n*Log[1 + e*x] - (b*n*Log[1 + e*x])/(4*x^2) + (1/2)*e^2*(a + b*Log[c*x^n])*Log[1 + e*x] - ((a + b*Log[c*x^n])*Log[1 + e*x])/(2*x^2) + (1/2)*b*e^2*n*PolyLog[2, (-e)*x]} -{(a + b*Log[c*x^n])*Log[1 + e*x]/x^4, x, 7, -((5*b*e*n)/(36*x^2)) + (4*b*e^2*n)/(9*x) + (1/9)*b*e^3*n*Log[x] - (1/6)*b*e^3*n*Log[x]^2 - (e*(a + b*Log[c*x^n]))/(6*x^2) + (e^2*(a + b*Log[c*x^n]))/(3*x) + (1/3)*e^3*Log[x]*(a + b*Log[c*x^n]) - (1/9)*b*e^3*n*Log[1 + e*x] - (b*n*Log[1 + e*x])/(9*x^3) - (1/3)*e^3*(a + b*Log[c*x^n])*Log[1 + e*x] - ((a + b*Log[c*x^n])*Log[1 + e*x])/(3*x^3) - (1/3)*b*e^3*n*PolyLog[2, (-e)*x]} - - -{x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x], x, 15, -((a*b*n*x)/(2*e^3)) + (21*b^2*n^2*x)/(32*e^3) - (7*b^2*n^2*x^2)/(64*e^2) + (37*b^2*n^2*x^3)/(864*e) - (3/128)*b^2*n^2*x^4 - (b^2*n*x*Log[c*x^n])/(2*e^3) - (b*n*x*(a + b*Log[c*x^n]))/(8*e^3) + (3*b*n*x^2*(a + b*Log[c*x^n]))/(16*e^2) - (7*b*n*x^3*(a + b*Log[c*x^n]))/(72*e) + (1/16)*b*n*x^4*(a + b*Log[c*x^n]) + (x*(a + b*Log[c*x^n])^2)/(4*e^3) - (x^2*(a + b*Log[c*x^n])^2)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^2)/(12*e) - (1/16)*x^4*(a + b*Log[c*x^n])^2 - (b^2*n^2*Log[1 + e*x])/(32*e^4) + (1/32)*b^2*n^2*x^4*Log[1 + e*x] + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(8*e^4) - (1/8)*b*n*x^4*(a + b*Log[c*x^n])*Log[1 + e*x] - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*e^4) + (1/4)*x^4*(a + b*Log[c*x^n])^2*Log[1 + e*x] + (b^2*n^2*PolyLog[2, (-e)*x])/(8*e^4) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/(2*e^4) + (b^2*n^2*PolyLog[3, (-e)*x])/(2*e^4)} -{x^2*(a + b*Log[c*x^n])^2*Log[1 + e*x], x, 14, (2*a*b*n*x)/(3*e^2) - (26*b^2*n^2*x)/(27*e^2) + (19*b^2*n^2*x^2)/(108*e) - (2/27)*b^2*n^2*x^3 + (2*b^2*n*x*Log[c*x^n])/(3*e^2) + (2*b*n*x*(a + b*Log[c*x^n]))/(9*e^2) - (5*b*n*x^2*(a + b*Log[c*x^n]))/(18*e) + (4/27)*b*n*x^3*(a + b*Log[c*x^n]) - (x*(a + b*Log[c*x^n])^2)/(3*e^2) + (x^2*(a + b*Log[c*x^n])^2)/(6*e) - (1/9)*x^3*(a + b*Log[c*x^n])^2 + (2*b^2*n^2*Log[1 + e*x])/(27*e^3) + (2/27)*b^2*n^2*x^3*Log[1 + e*x] - (2*b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(9*e^3) - (2/9)*b*n*x^3*(a + b*Log[c*x^n])*Log[1 + e*x] + ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(3*e^3) + (1/3)*x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x] - (2*b^2*n^2*PolyLog[2, (-e)*x])/(9*e^3) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/(3*e^3) - (2*b^2*n^2*PolyLog[3, (-e)*x])/(3*e^3)} -{x^1*(a + b*Log[c*x^n])^2*Log[1 + e*x], x, 13, -((a*b*n*x)/e) + (7*b^2*n^2*x)/(4*e) - (3/8)*b^2*n^2*x^2 - (b^2*n*x*Log[c*x^n])/e - (b*n*x*(a + b*Log[c*x^n]))/(2*e) + (1/2)*b*n*x^2*(a + b*Log[c*x^n]) + (x*(a + b*Log[c*x^n])^2)/(2*e) - (1/4)*x^2*(a + b*Log[c*x^n])^2 - (b^2*n^2*Log[1 + e*x])/(4*e^2) + (1/4)*b^2*n^2*x^2*Log[1 + e*x] + (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(2*e^2) - (1/2)*b*n*x^2*(a + b*Log[c*x^n])*Log[1 + e*x] - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(2*e^2) + (1/2)*x^2*(a + b*Log[c*x^n])^2*Log[1 + e*x] + (b^2*n^2*PolyLog[2, (-e)*x])/(2*e^2) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/e^2 + (b^2*n^2*PolyLog[3, (-e)*x])/e^2} -{x^0*(a + b*Log[c*x^n])^2*Log[1 + e*x], x, 14, 2*a*b*n*x - 6*b^2*n^2*x + 2*b^2*n*x*Log[c*x^n] + 2*b*n*x*(a + b*Log[c*x^n]) - x*(a + b*Log[c*x^n])^2 + (2*b^2*n^2*(1 + e*x)*Log[1 + e*x])/e - (2*b*n*(1 + e*x)*(a + b*Log[c*x^n])*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])^2*Log[1 + e*x])/e - (2*b^2*n^2*PolyLog[2, (-e)*x])/e + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/e - (2*b^2*n^2*PolyLog[3, (-e)*x])/e} -{(a + b*Log[c*x^n])^2*Log[1 + e*x]/x^1, x, 3, (-(a + b*Log[c*x^n])^2)*PolyLog[2, (-e)*x] + 2*b*n*(a + b*Log[c*x^n])*PolyLog[3, (-e)*x] - 2*b^2*n^2*PolyLog[4, (-e)*x]} -{(a + b*Log[c*x^n])^2*Log[1 + e*x]/x^2, x, 10, 2*b^2*e*n^2*Log[x] - 2*b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]) - e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 - 2*b^2*e*n^2*Log[1 + e*x] - (2*b^2*n^2*Log[1 + e*x])/x - (2*b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/x - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/x + 2*b^2*e*n^2*PolyLog[2, -(1/(e*x))] + 2*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))] + 2*b^2*e*n^2*PolyLog[3, -(1/(e*x))]} -{(a + b*Log[c*x^n])^2*Log[1 + e*x]/x^3, x, 14, -((7*b^2*e*n^2)/(4*x)) - (1/4)*b^2*e^2*n^2*Log[x] - (3*b*e*n*(a + b*Log[c*x^n]))/(2*x) + (1/2)*b*e^2*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]) - (e*(a + b*Log[c*x^n])^2)/(2*x) + (1/2)*e^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 + (1/4)*b^2*e^2*n^2*Log[1 + e*x] - (b^2*n^2*Log[1 + e*x])/(4*x^2) - (b*n*(a + b*Log[c*x^n])*Log[1 + e*x])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[1 + e*x])/(2*x^2) - (1/2)*b^2*e^2*n^2*PolyLog[2, -(1/(e*x))] - b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))] - b^2*e^2*n^2*PolyLog[3, -(1/(e*x))]} - - -{x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x], x, 29, (15*a*b^2*n^2*x)/(8*e^3) - (255*b^3*n^3*x)/(128*e^3) + (45*b^3*n^3*x^2)/(256*e^2) - (175*b^3*n^3*x^3)/(3456*e) + (3/128)*b^3*n^3*x^4 + (15*b^3*n^2*x*Log[c*x^n])/(8*e^3) + (3*b^2*n^2*x*(a + b*Log[c*x^n]))/(32*e^3) - (21*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(64*e^2) + (37*b^2*n^2*x^3*(a + b*Log[c*x^n]))/(288*e) - (9/128)*b^2*n^2*x^4*(a + b*Log[c*x^n]) - (15*b*n*x*(a + b*Log[c*x^n])^2)/(16*e^3) + (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(32*e^2) - (7*b*n*x^3*(a + b*Log[c*x^n])^2)/(48*e) + (3/32)*b*n*x^4*(a + b*Log[c*x^n])^2 + (x*(a + b*Log[c*x^n])^3)/(4*e^3) - (x^2*(a + b*Log[c*x^n])^3)/(8*e^2) + (x^3*(a + b*Log[c*x^n])^3)/(12*e) - (1/16)*x^4*(a + b*Log[c*x^n])^3 + (3*b^3*n^3*Log[1 + e*x])/(128*e^4) - (3/128)*b^3*n^3*x^4*Log[1 + e*x] - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(32*e^4) + (3/32)*b^2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + e*x] + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(16*e^4) - (3/16)*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + e*x] - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(4*e^4) + (1/4)*x^4*(a + b*Log[c*x^n])^3*Log[1 + e*x] - (3*b^3*n^3*PolyLog[2, (-e)*x])/(32*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/(8*e^4) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-e)*x])/(4*e^4) - (3*b^3*n^3*PolyLog[3, (-e)*x])/(8*e^4) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-e)*x])/(2*e^4) - (3*b^3*n^3*PolyLog[4, (-e)*x])/(2*e^4)} -{x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x], x, 26, -((8*a*b^2*n^2*x)/(3*e^2)) + (80*b^3*n^3*x)/(27*e^2) - (65*b^3*n^3*x^2)/(216*e) + (8/81)*b^3*n^3*x^3 - (8*b^3*n^2*x*Log[c*x^n])/(3*e^2) - (2*b^2*n^2*x*(a + b*Log[c*x^n]))/(9*e^2) + (19*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(36*e) - (2/9)*b^2*n^2*x^3*(a + b*Log[c*x^n]) + (4*b*n*x*(a + b*Log[c*x^n])^2)/(3*e^2) - (5*b*n*x^2*(a + b*Log[c*x^n])^2)/(12*e) + (2/9)*b*n*x^3*(a + b*Log[c*x^n])^2 - (x*(a + b*Log[c*x^n])^3)/(3*e^2) + (x^2*(a + b*Log[c*x^n])^3)/(6*e) - (1/9)*x^3*(a + b*Log[c*x^n])^3 - (2*b^3*n^3*Log[1 + e*x])/(27*e^3) - (2/27)*b^3*n^3*x^3*Log[1 + e*x] + (2*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(9*e^3) + (2/9)*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[1 + e*x] - (b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(3*e^3) - (1/3)*b*n*x^3*(a + b*Log[c*x^n])^2*Log[1 + e*x] + ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(3*e^3) + (1/3)*x^3*(a + b*Log[c*x^n])^3*Log[1 + e*x] + (2*b^3*n^3*PolyLog[2, (-e)*x])/(9*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/(3*e^3) + (b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-e)*x])/e^3 + (2*b^3*n^3*PolyLog[3, (-e)*x])/(3*e^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-e)*x])/e^3 + (2*b^3*n^3*PolyLog[4, (-e)*x])/e^3} -{x^1*(a + b*Log[c*x^n])^3*Log[1 + e*x], x, 23, (9*a*b^2*n^2*x)/(2*e) - (45*b^3*n^3*x)/(8*e) + (3/4)*b^3*n^3*x^2 + (9*b^3*n^2*x*Log[c*x^n])/(2*e) + (3*b^2*n^2*x*(a + b*Log[c*x^n]))/(4*e) - (9/8)*b^2*n^2*x^2*(a + b*Log[c*x^n]) - (9*b*n*x*(a + b*Log[c*x^n])^2)/(4*e) + (3/4)*b*n*x^2*(a + b*Log[c*x^n])^2 + (x*(a + b*Log[c*x^n])^3)/(2*e) - (1/4)*x^2*(a + b*Log[c*x^n])^3 + (3*b^3*n^3*Log[1 + e*x])/(8*e^2) - (3/8)*b^3*n^3*x^2*Log[1 + e*x] - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*e^2) + (3/4)*b^2*n^2*x^2*(a + b*Log[c*x^n])*Log[1 + e*x] + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*e^2) - (3/4)*b*n*x^2*(a + b*Log[c*x^n])^2*Log[1 + e*x] - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*e^2) + (1/2)*x^2*(a + b*Log[c*x^n])^3*Log[1 + e*x] - (3*b^3*n^3*PolyLog[2, (-e)*x])/(4*e^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/(2*e^2) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-e)*x])/(2*e^2) - (3*b^3*n^3*PolyLog[3, (-e)*x])/(2*e^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-e)*x])/e^2 - (3*b^3*n^3*PolyLog[4, (-e)*x])/e^2} -{x^0*(a + b*Log[c*x^n])^3*Log[1 + e*x], x, 24, -12*a*b^2*n^2*x + 24*b^3*n^3*x - 12*b^3*n^2*x*Log[c*x^n] - 6*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b*n*x*(a + b*Log[c*x^n])^2 - x*(a + b*Log[c*x^n])^3 - (6*b^3*n^3*(1 + e*x)*Log[1 + e*x])/e + (6*b^2*n^2*(1 + e*x)*(a + b*Log[c*x^n])*Log[1 + e*x])/e - (3*b*n*(1 + e*x)*(a + b*Log[c*x^n])^2*Log[1 + e*x])/e + ((1 + e*x)*(a + b*Log[c*x^n])^3*Log[1 + e*x])/e + (6*b^3*n^3*PolyLog[2, (-e)*x])/e - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-e)*x])/e + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-e)*x])/e + (6*b^3*n^3*PolyLog[3, (-e)*x])/e - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-e)*x])/e + (6*b^3*n^3*PolyLog[4, (-e)*x])/e} -{(a + b*Log[c*x^n])^3*Log[1 + e*x]/x^1, x, 4, (-(a + b*Log[c*x^n])^3)*PolyLog[2, (-e)*x] + 3*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, (-e)*x] - 6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, (-e)*x] + 6*b^3*n^3*PolyLog[5, (-e)*x]} -{(a + b*Log[c*x^n])^3*Log[1 + e*x]/x^2, x, 14, 6*b^3*e*n^3*Log[x] - 6*b^2*e*n^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]) - 3*b*e*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 - e*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3 - 6*b^3*e*n^3*Log[1 + e*x] - (6*b^3*n^3*Log[1 + e*x])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/x - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/x + 6*b^3*e*n^3*PolyLog[2, -(1/(e*x))] + 6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))] + 3*b*e*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))] + 6*b^3*e*n^3*PolyLog[3, -(1/(e*x))] + 6*b^2*e*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] + 6*b^3*e*n^3*PolyLog[4, -(1/(e*x))]} -{(a + b*Log[c*x^n])^3*Log[1 + e*x]/x^3, x, 22, -((45*b^3*e*n^3)/(8*x)) - (3/8)*b^3*e^2*n^3*Log[x] - (21*b^2*e*n^2*(a + b*Log[c*x^n]))/(4*x) + (3/4)*b^2*e^2*n^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n]) - (9*b*e*n*(a + b*Log[c*x^n])^2)/(4*x) + (3/4)*b*e^2*n*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^2 - (e*(a + b*Log[c*x^n])^3)/(2*x) + (1/2)*e^2*Log[1 + 1/(e*x)]*(a + b*Log[c*x^n])^3 + (3/8)*b^3*e^2*n^3*Log[1 + e*x] - (3*b^3*n^3*Log[1 + e*x])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + e*x])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + e*x])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + e*x])/(2*x^2) - (3/4)*b^3*e^2*n^3*PolyLog[2, -(1/(e*x))] - (3/2)*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(e*x))] - (3/2)*b*e^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(e*x))] - (3/2)*b^3*e^2*n^3*PolyLog[3, -(1/(e*x))] - 3*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(e*x))] - 3*b^3*e^2*n^3*PolyLog[4, -(1/(e*x))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^2)] (a+b Log[c x^n])^p when d e=1*) - - -{x^3*(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)], x, 7, -((3*b*n*x^2)/(16*d*f)) + (1/16)*b*n*x^4 + (x^2*(a + b*Log[c*x^n]))/(4*d*f) - (1/8)*x^4*(a + b*Log[c*x^n]) + (b*n*Log[1 + d*f*x^2])/(16*d^2*f^2) - (1/16)*b*n*x^4*Log[1 + d*f*x^2] - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(4*d^2*f^2) + (1/4)*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] - (b*n*PolyLog[2, (-d)*f*x^2])/(8*d^2*f^2)} -{x^1*(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)], x, 8, (1/2)*b*n*x^2 - (1/2)*x^2*(a + b*Log[c*x^n]) - (b*n*(1 + d*f*x^2)*Log[1 + d*f*x^2])/(4*d*f) + ((1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*d*f) + (b*n*PolyLog[2, (-d)*f*x^2])/(4*d*f)} -{(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)]/x^1, x, 2, (-(1/2))*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^2] + (1/4)*b*n*PolyLog[3, (-d)*f*x^2]} -{(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)]/x^3, x, 9, (1/2)*b*d*f*n*Log[x] - (1/2)*b*d*f*n*Log[x]^2 + d*f*Log[x]*(a + b*Log[c*x^n]) - (1/4)*b*d*f*n*Log[1 + d*f*x^2] - (b*n*Log[1 + d*f*x^2])/(4*x^2) - (1/2)*d*f*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*x^2) - (1/4)*b*d*f*n*PolyLog[2, (-d)*f*x^2]} - -{x^2*(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)], x, 9, -((8*b*n*x)/(9*d*f)) + (4/27)*b*n*x^3 + (2*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (2*x*(a + b*Log[c*x^n]))/(3*d*f) - (2/9)*x^3*(a + b*Log[c*x^n]) - (2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*f^(3/2)) - (1/9)*b*n*x^3*Log[1 + d*f*x^2] + (1/3)*x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] + (I*b*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2)*f^(3/2)) - (I*b*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2)*f^(3/2))} -{x^0*(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)], x, 8, 4*b*n*x - (2*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - 2*x*(a + b*Log[c*x^n]) + (2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(Sqrt[d]*Sqrt[f]) - b*n*x*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] - (I*b*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + (I*b*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f])} -{(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)]/x^2, x, 7, 2*b*Sqrt[d]*Sqrt[f]*n*ArcTan[Sqrt[d]*Sqrt[f]*x] + 2*Sqrt[d]*Sqrt[f]*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]) - (b*n*Log[1 + d*f*x^2])/x - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/x - I*b*Sqrt[d]*Sqrt[f]*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + I*b*Sqrt[d]*Sqrt[f]*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]} -{(a + b*Log[c*x^n])*Log[d*(1/d + f*x^2)]/x^4, x, 8, -((8*b*d*f*n)/(9*x)) - (2/9)*b*d^(3/2)*f^(3/2)*n*ArcTan[Sqrt[d]*Sqrt[f]*x] - (2*d*f*(a + b*Log[c*x^n]))/(3*x) - (2/3)*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]) - (b*n*Log[1 + d*f*x^2])/(9*x^3) - ((a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(3*x^3) + (1/3)*I*b*d^(3/2)*f^(3/2)*n*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - (1/3)*I*b*d^(3/2)*f^(3/2)*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]} - - -{x^3*(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)], x, 13, (7*b^2*n^2*x^2)/(32*d*f) - (3/64)*b^2*n^2*x^4 - (3*b*n*x^2*(a + b*Log[c*x^n]))/(8*d*f) + (1/8)*b*n*x^4*(a + b*Log[c*x^n]) + (x^2*(a + b*Log[c*x^n])^2)/(4*d*f) - (1/8)*x^4*(a + b*Log[c*x^n])^2 - (b^2*n^2*Log[1 + d*f*x^2])/(32*d^2*f^2) + (1/32)*b^2*n^2*x^4*Log[1 + d*f*x^2] + (b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(8*d^2*f^2) - (1/8)*b*n*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*d^2*f^2) + (1/4)*x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + (b^2*n^2*PolyLog[2, (-d)*f*x^2])/(16*d^2*f^2) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^2])/(4*d^2*f^2) + (b^2*n^2*PolyLog[3, (-d)*f*x^2])/(8*d^2*f^2)} -{x^1*(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)], x, 15, (-(3/4))*b^2*n^2*x^2 + b*n*x^2*(a + b*Log[c*x^n]) - (1/2)*x^2*(a + b*Log[c*x^n])^2 + (b^2*n^2*(1 + d*f*x^2)*Log[1 + d*f*x^2])/(4*d*f) - (b*n*(1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*d*f) + ((1 + d*f*x^2)*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(2*d*f) - (b^2*n^2*PolyLog[2, (-d)*f*x^2])/(4*d*f) + (b*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^2])/(2*d*f) - (b^2*n^2*PolyLog[3, (-d)*f*x^2])/(4*d*f)} -{(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)]/x^1, x, 3, (-(1/2))*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*x^2] + (1/2)*b*n*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*x^2] - (1/4)*b^2*n^2*PolyLog[4, (-d)*f*x^2]} -{(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)]/x^3, x, 11, (1/2)*b^2*d*f*n^2*Log[x] - (1/2)*b*d*f*n*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n]) - (1/2)*d*f*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^2 - (1/4)*b^2*d*f*n^2*Log[1 + d*f*x^2] - (b^2*n^2*Log[1 + d*f*x^2])/(4*x^2) - (b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(2*x^2) + (1/4)*b^2*d*f*n^2*PolyLog[2, -(1/(d*f*x^2))] + (1/2)*b*d*f*n*(a + b*Log[c*x^n])*PolyLog[2, -(1/(d*f*x^2))] + (1/4)*b^2*d*f*n^2*PolyLog[3, -(1/(d*f*x^2))]} - -{x^2*(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)], x, 30, -((16*a*b*n*x)/(9*d*f)) + (52*b^2*n^2*x)/(27*d*f) - (4/27)*b^2*n^2*x^3 - (4*b^2*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x])/(27*d^(3/2)*f^(3/2)) - (16*b^2*n*x*Log[c*x^n])/(9*d*f) + (8/27)*b*n*x^3*(a + b*Log[c*x^n]) + (4*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(9*d^(3/2)*f^(3/2)) + (2*x*(a + b*Log[c*x^n])^2)/(3*d*f) - (2/9)*x^3*(a + b*Log[c*x^n])^2 - ((a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2)) + ((a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2)) + (2/27)*b^2*n^2*x^3*Log[1 + d*f*x^2] - (2/9)*b*n*x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] + (1/3)*x^3*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2)) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2)) - (2*I*b^2*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (2*I*b^2*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) - (2*b^2*n^2*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2)) + (2*b^2*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(3*(-d)^(3/2)*f^(3/2))} -{x^0*(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)], x, 26, 4*a*b*n*x - 8*b^2*n^2*x + 4*b*n*(a - b*n)*x - (4*b*n*(a - b*n)*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + 8*b^2*n*x*Log[c*x^n] - (4*b^2*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*Sqrt[f]) - 2*x*(a + b*Log[c*x^n])^2 - ((a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + ((a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - 2*a*b*n*x*Log[1 + d*f*x^2] + 2*b^2*n^2*x*Log[1 + d*f*x^2] - 2*b^2*n*x*Log[c*x^n]*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (2*I*b^2*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - (2*I*b^2*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - (2*b^2*n^2*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (2*b^2*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f])} -{(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)]/x^2, x, 16, 4*b^2*Sqrt[d]*Sqrt[f]*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x] + 4*b*Sqrt[d]*Sqrt[f]*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]) + Sqrt[-d]*Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x] - Sqrt[-d]*Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x] - (2*b^2*n^2*Log[1 + d*f*x^2])/x - (2*b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/x - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/x - 2*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x] + 2*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x] - 2*I*b^2*Sqrt[d]*Sqrt[f]*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 2*I*b^2*Sqrt[d]*Sqrt[f]*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + 2*b^2*Sqrt[-d]*Sqrt[f]*n^2*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x] - 2*b^2*Sqrt[-d]*Sqrt[f]*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x]} -{(a + b*Log[c*x^n])^2*Log[d*(1/d + f*x^2)]/x^4, x, 22, -((52*b^2*d*f*n^2)/(27*x)) - (4/27)*b^2*d^(3/2)*f^(3/2)*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x] - (16*b*d*f*n*(a + b*Log[c*x^n]))/(9*x) - (4/9)*b*d^(3/2)*f^(3/2)*n*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]) - (2*d*f*(a + b*Log[c*x^n])^2)/(3*x) + (1/3)*(-d)^(3/2)*f^(3/2)*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x] - (1/3)*(-d)^(3/2)*f^(3/2)*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x] - (2*b^2*n^2*Log[1 + d*f*x^2])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(3*x^3) - (2/3)*b*(-d)^(3/2)*f^(3/2)*n*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x] + (2/3)*b*(-d)^(3/2)*f^(3/2)*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x] + (2/9)*I*b^2*d^(3/2)*f^(3/2)*n^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - (2/9)*I*b^2*d^(3/2)*f^(3/2)*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + (2/3)*b^2*(-d)^(3/2)*f^(3/2)*n^2*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x] - (2/3)*b^2*(-d)^(3/2)*f^(3/2)*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x]} - - -{x^3*(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)], x, 22, -((45*b^3*n^3*x^2)/(128*d*f)) + (3/64)*b^3*n^3*x^4 + (21*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(32*d*f) - (9/64)*b^2*n^2*x^4*(a + b*Log[c*x^n]) - (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(16*d*f) + (3/16)*b*n*x^4*(a + b*Log[c*x^n])^2 + (x^2*(a + b*Log[c*x^n])^3)/(4*d*f) - (1/8)*x^4*(a + b*Log[c*x^n])^3 + (3*b^3*n^3*Log[1 + d*f*x^2])/(128*d^2*f^2) - (3/128)*b^3*n^3*x^4*Log[1 + d*f*x^2] - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(32*d^2*f^2) + (3/32)*b^2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2] + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(16*d^2*f^2) - (3/16)*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(4*d^2*f^2) + (1/4)*x^4*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2] - (3*b^3*n^3*PolyLog[2, (-d)*f*x^2])/(64*d^2*f^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^2])/(16*d^2*f^2) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*x^2])/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[3, (-d)*f*x^2])/(32*d^2*f^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*x^2])/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[4, (-d)*f*x^2])/(16*d^2*f^2)} -{x^1*(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)], x, 24, (3/2)*b^3*n^3*x^2 - (9/4)*b^2*n^2*x^2*(a + b*Log[c*x^n]) + (3/2)*b*n*x^2*(a + b*Log[c*x^n])^2 - (1/2)*x^2*(a + b*Log[c*x^n])^3 - (3*b^3*n^3*(1 + d*f*x^2)*Log[1 + d*f*x^2])/(8*d*f) + (3*b^2*n^2*(1 + d*f*x^2)*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(4*d*f) - (3*b*n*(1 + d*f*x^2)*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*d*f) + ((1 + d*f*x^2)*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(2*d*f) + (3*b^3*n^3*PolyLog[2, (-d)*f*x^2])/(8*d*f) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^2])/(4*d*f) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*x^2])/(4*d*f) + (3*b^3*n^3*PolyLog[3, (-d)*f*x^2])/(8*d*f) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*x^2])/(4*d*f) + (3*b^3*n^3*PolyLog[4, (-d)*f*x^2])/(8*d*f)} -{(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)]/x^1, x, 4, (-(1/2))*(a + b*Log[c*x^n])^3*PolyLog[2, (-d)*f*x^2] + (3/4)*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, (-d)*f*x^2] - (3/4)*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, (-d)*f*x^2] + (3/8)*b^3*n^3*PolyLog[5, (-d)*f*x^2]} -{(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)]/x^3, x, 15, (3/4)*b^3*d*f*n^3*Log[x] - (3/4)*b^2*d*f*n^2*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n]) - (3/4)*b*d*f*n*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^2 - (1/2)*d*f*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^3 - (3/8)*b^3*d*f*n^3*Log[1 + d*f*x^2] - (3*b^3*n^3*Log[1 + d*f*x^2])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(2*x^2) + (3/8)*b^3*d*f*n^3*PolyLog[2, -(1/(d*f*x^2))] + (3/4)*b^2*d*f*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(1/(d*f*x^2))] + (3/4)*b*d*f*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(d*f*x^2))] + (3/8)*b^3*d*f*n^3*PolyLog[3, -(1/(d*f*x^2))] + (3/4)*b^2*d*f*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(1/(d*f*x^2))] + (3/8)*b^3*d*f*n^3*PolyLog[4, -(1/(d*f*x^2))]} - -{x^0*(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)], x, 42, -24*a*b^2*n^2*x + 36*b^3*n^3*x - 12*b^2*n^2*(a - b*n)*x + (12*b^2*n^2*(a - b*n)*ArcTan[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) - 36*b^3*n^2*x*Log[c*x^n] + (12*b^3*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*Sqrt[f]) + 12*b*n*x*(a + b*Log[c*x^n])^2 - 2*x*(a + b*Log[c*x^n])^3 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - ((a + b*Log[c*x^n])^3*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + ((a + b*Log[c*x^n])^3*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + 6*a*b^2*n^2*x*Log[1 + d*f*x^2] - 6*b^3*n^3*x*Log[1 + d*f*x^2] + 6*b^3*n^2*x*Log[c*x^n]*Log[1 + d*f*x^2] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2] + x*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2] - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (6*I*b^3*n^3*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + (6*I*b^3*n^3*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*Sqrt[f]) + (6*b^3*n^3*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (6*b^3*n^3*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) + (6*b^3*n^3*PolyLog[4, (-Sqrt[-d])*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f]) - (6*b^3*n^3*PolyLog[4, Sqrt[-d]*Sqrt[f]*x])/(Sqrt[-d]*Sqrt[f])} -{(a + b*Log[c*x^n])^3*Log[d*(1/d + f*x^2)]/x^2, x, 26, 12*b^3*Sqrt[d]*Sqrt[f]*n^3*ArcTan[Sqrt[d]*Sqrt[f]*x] + 12*b^2*Sqrt[d]*Sqrt[f]*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]) + 3*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x] + Sqrt[-d]*Sqrt[f]*(a + b*Log[c*x^n])^3*Log[1 - Sqrt[-d]*Sqrt[f]*x] - 3*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x] - Sqrt[-d]*Sqrt[f]*(a + b*Log[c*x^n])^3*Log[1 + Sqrt[-d]*Sqrt[f]*x] - (6*b^3*n^3*Log[1 + d*f*x^2])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/x - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/x - 6*b^2*Sqrt[-d]*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x] - 3*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-Sqrt[-d])*Sqrt[f]*x] + 6*b^2*Sqrt[-d]*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f]*x] + 3*b*Sqrt[-d]*Sqrt[f]*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d]*Sqrt[f]*x] - 6*I*b^3*Sqrt[d]*Sqrt[f]*n^3*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + 6*I*b^3*Sqrt[d]*Sqrt[f]*n^3*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + 6*b^3*Sqrt[-d]*Sqrt[f]*n^3*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x] + 6*b^2*Sqrt[-d]*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-Sqrt[-d])*Sqrt[f]*x] - 6*b^3*Sqrt[-d]*Sqrt[f]*n^3*PolyLog[3, Sqrt[-d]*Sqrt[f]*x] - 6*b^2*Sqrt[-d]*Sqrt[f]*n^2*(a + b*Log[c*x^n])*PolyLog[3, Sqrt[-d]*Sqrt[f]*x] - 6*b^3*Sqrt[-d]*Sqrt[f]*n^3*PolyLog[4, (-Sqrt[-d])*Sqrt[f]*x] + 6*b^3*Sqrt[-d]*Sqrt[f]*n^3*PolyLog[4, Sqrt[-d]*Sqrt[f]*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^(1/2))] (a+b Log[c x^n])^p when d e=1*) - - -{x^2*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]), x, 7, (-7*b*n*Sqrt[x])/(9*d^5*f^5) + (2*b*n*x)/(9*d^4*f^4) - (b*n*x^(3/2))/(9*d^3*f^3) + (5*b*n*x^2)/(72*d^2*f^2) - (11*b*n*x^(5/2))/(225*d*f) + (b*n*x^3)/27 + (b*n*Log[1 + d*f*Sqrt[x]])/(9*d^6*f^6) - (b*n*x^3*Log[1 + d*f*Sqrt[x]])/9 + (Sqrt[x]*(a + b*Log[c*x^n]))/(3*d^5*f^5) - (x*(a + b*Log[c*x^n]))/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n]))/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n]))/(15*d*f) - (x^3*(a + b*Log[c*x^n]))/18 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*d^6*f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/3 - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6)} -{x^1*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]), x, 7, (-5*b*n*Sqrt[x])/(4*d^3*f^3) + (3*b*n*x)/(8*d^2*f^2) - (7*b*n*x^(3/2))/(36*d*f) + (b*n*x^2)/8 + (b*n*Log[1 + d*f*Sqrt[x]])/(4*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]])/4 + (Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) - (x*(a + b*Log[c*x^n]))/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n]))/(6*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*n*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4)} -{x^0*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]), x, 7, (-3*b*n*Sqrt[x])/(d*f) + b*n*x - b*n*x*Log[d*(1/d + f*Sqrt[x])] + (b*n*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) - (x*(a + b*Log[c*x^n]))/2 + x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2)} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])/x^1, x, 2, -2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 4*b*n*PolyLog[3, -(d*f*Sqrt[x])]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])/x^2, x, 8, (-3*b*d*f*n)/Sqrt[x] + b*d^2*f^2*n*Log[1 + d*f*Sqrt[x]] - (b*n*Log[1 + d*f*Sqrt[x]])/x - (b*d^2*f^2*n*Log[x])/2 + (b*d^2*f^2*n*Log[x]^2)/4 - (d*f*(a + b*Log[c*x^n]))/Sqrt[x] + d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/x - (d^2*f^2*Log[x]*(a + b*Log[c*x^n]))/2 + 2*b*d^2*f^2*n*PolyLog[2, -(d*f*Sqrt[x])]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])/x^3, x, 8, (-7*b*d*f*n)/(36*x^(3/2)) + (3*b*d^2*f^2*n)/(8*x) - (5*b*d^3*f^3*n)/(4*Sqrt[x]) + (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]])/4 - (b*n*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (b*d^4*f^4*n*Log[x])/8 + (b*d^4*f^4*n*Log[x]^2)/8 - (d*f*(a + b*Log[c*x^n]))/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n]))/(4*x) - (d^3*f^3*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*x^2) - (d^4*f^4*Log[x]*(a + b*Log[c*x^n]))/4 + b*d^4*f^4*n*PolyLog[2, -(d*f*Sqrt[x])]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])/x^4, x, 8, (-11*b*d*f*n)/(225*x^(5/2)) + (5*b*d^2*f^2*n)/(72*x^2) - (b*d^3*f^3*n)/(9*x^(3/2)) + (2*b*d^4*f^4*n)/(9*x) - (7*b*d^5*f^5*n)/(9*Sqrt[x]) + (b*d^6*f^6*n*Log[1 + d*f*Sqrt[x]])/9 - (b*n*Log[1 + d*f*Sqrt[x]])/(9*x^3) - (b*d^6*f^6*n*Log[x])/18 + (b*d^6*f^6*n*Log[x]^2)/12 - (d*f*(a + b*Log[c*x^n]))/(15*x^(5/2)) + (d^2*f^2*(a + b*Log[c*x^n]))/(12*x^2) - (d^3*f^3*(a + b*Log[c*x^n]))/(9*x^(3/2)) + (d^4*f^4*(a + b*Log[c*x^n]))/(6*x) - (d^5*f^5*(a + b*Log[c*x^n]))/(3*Sqrt[x]) + (d^6*f^6*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*x^3) - (d^6*f^6*Log[x]*(a + b*Log[c*x^n]))/6 + (2*b*d^6*f^6*n*PolyLog[2, -(d*f*Sqrt[x])])/3} - - -{x^2*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 18, (86*b^2*n^2*Sqrt[x])/(27*d^5*f^5) + (a*b*n*x)/(3*d^4*f^4) - (13*b^2*n^2*x)/(27*d^4*f^4) + (14*b^2*n^2*x^(3/2))/(81*d^3*f^3) - (19*b^2*n^2*x^2)/(216*d^2*f^2) + (182*b^2*n^2*x^(5/2))/(3375*d*f) - (1/27)*b^2*n^2*x^3 - (2*b^2*n^2*Log[1 + d*f*Sqrt[x]])/(27*d^6*f^6) + (2/27)*b^2*n^2*x^3*Log[1 + d*f*Sqrt[x]] + (b^2*n*x*Log[c*x^n])/(3*d^4*f^4) - (14*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(9*d^5*f^5) + (b*n*x*(a + b*Log[c*x^n]))/(9*d^4*f^4) - (2*b*n*x^(3/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) + (5*b*n*x^2*(a + b*Log[c*x^n]))/(36*d^2*f^2) - (22*b*n*x^(5/2)*(a + b*Log[c*x^n]))/(225*d*f) + (2/27)*b*n*x^3*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*d^6*f^6) - (2/9)*b*n*x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*d^5*f^5) - (x*(a + b*Log[c*x^n])^2)/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n])^2)/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n])^2)/(15*d*f) - (1/18)*x^3*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(3*d^6*f^6) + (1/3)*x^3*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 + (4*b^2*n^2*PolyLog[2, (-d)*f*Sqrt[x]])/(9*d^6*f^6) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]])/(3*d^6*f^6) + (8*b^2*n^2*PolyLog[3, (-d)*f*Sqrt[x]])/(3*d^6*f^6)} -{x^1*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 16, (21*b^2*n^2*Sqrt[x])/(4*d^3*f^3) + (a*b*n*x)/(2*d^2*f^2) - (7*b^2*n^2*x)/(8*d^2*f^2) + (37*b^2*n^2*x^(3/2))/(108*d*f) - (3/16)*b^2*n^2*x^2 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*d^4*f^4) + (1/4)*b^2*n^2*x^2*Log[1 + d*f*Sqrt[x]] + (b^2*n*x*Log[c*x^n])/(2*d^2*f^2) - (5*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) + (b*n*x*(a + b*Log[c*x^n]))/(4*d^2*f^2) - (7*b*n*x^(3/2)*(a + b*Log[c*x^n]))/(18*d*f) + (1/4)*b*n*x^2*(a + b*Log[c*x^n]) + (b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) - (1/2)*b*n*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^2)/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(6*d*f) - (1/8)*x^2*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*d^4*f^4) + (1/2)*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 + (b^2*n^2*PolyLog[2, (-d)*f*Sqrt[x]])/(d^4*f^4) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]])/(d^4*f^4) + (4*b^2*n^2*PolyLog[3, (-d)*f*Sqrt[x]])/(d^4*f^4)} -{x^0*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 14, (14*b^2*n^2*Sqrt[x])/(d*f) + a*b*n*x - 3*b^2*n^2*x + 2*b^2*n^2*x*Log[d*(1/d + f*Sqrt[x])] - (2*b^2*n^2*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) + b^2*n*x*Log[c*x^n] - (6*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) + b*n*x*(a + b*Log[c*x^n]) - 2*b*n*x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - (x*(a + b*Log[c*x^n])^2)/2 + x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (4*b^2*n^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (8*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2)} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^1, x, 3, -2*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])] + 8*b*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])] - 16*b^2*n^2*PolyLog[4, -(d*f*Sqrt[x])]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^2, x, 17, -((14*b^2*d*f*n^2)/Sqrt[x]) + 2*b^2*d^2*f^2*n^2*Log[1 + d*f*Sqrt[x]] - (2*b^2*n^2*Log[1 + d*f*Sqrt[x]])/x - b^2*d^2*f^2*n^2*Log[x] + (1/2)*b^2*d^2*f^2*n^2*Log[x]^2 - (6*b*d*f*n*(a + b*Log[c*x^n]))/Sqrt[x] + 2*b*d^2*f^2*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/x - b*d^2*f^2*n*Log[x]*(a + b*Log[c*x^n]) - (d*f*(a + b*Log[c*x^n])^2)/Sqrt[x] + d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/x - (d^2*f^2*(a + b*Log[c*x^n])^3)/(6*b*n) + 4*b^2*d^2*f^2*n^2*PolyLog[2, (-d)*f*Sqrt[x]] + 4*b*d^2*f^2*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]] - 8*b^2*d^2*f^2*n^2*PolyLog[3, (-d)*f*Sqrt[x]]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^3, x, 19, -((37*b^2*d*f*n^2)/(108*x^(3/2))) + (7*b^2*d^2*f^2*n^2)/(8*x) - (21*b^2*d^3*f^3*n^2)/(4*Sqrt[x]) + (1/4)*b^2*d^4*f^4*n^2*Log[1 + d*f*Sqrt[x]] - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (1/8)*b^2*d^4*f^4*n^2*Log[x] + (1/8)*b^2*d^4*f^4*n^2*Log[x]^2 - (7*b*d*f*n*(a + b*Log[c*x^n]))/(18*x^(3/2)) + (3*b*d^2*f^2*n*(a + b*Log[c*x^n]))/(4*x) - (5*b*d^3*f^3*n*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (1/2)*b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*x^2) - (1/4)*b*d^4*f^4*n*Log[x]*(a + b*Log[c*x^n]) - (d*f*(a + b*Log[c*x^n])^2)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^2)/(4*x) - (d^3*f^3*(a + b*Log[c*x^n])^2)/(2*Sqrt[x]) + (1/2)*d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*x^2) - (d^4*f^4*(a + b*Log[c*x^n])^3)/(12*b*n) + b^2*d^4*f^4*n^2*PolyLog[2, (-d)*f*Sqrt[x]] + 2*b*d^4*f^4*n*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]] - 4*b^2*d^4*f^4*n^2*PolyLog[3, (-d)*f*Sqrt[x]]} - - -{x^1*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3, x, 30, -((255*b^3*n^3*Sqrt[x])/(8*d^3*f^3)) - (9*a*b^2*n^2*x)/(4*d^2*f^2) + (45*b^3*n^3*x)/(16*d^2*f^2) - (175*b^3*n^3*x^(3/2))/(216*d*f) + (3/8)*b^3*n^3*x^2 + (3*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(8*d^4*f^4) - (3/8)*b^3*n^3*x^2*Log[1 + d*f*Sqrt[x]] - (9*b^3*n^2*x*Log[c*x^n])/(4*d^2*f^2) + (63*b^2*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/(4*d^3*f^3) - (3*b^2*n^2*x*(a + b*Log[c*x^n]))/(8*d^2*f^2) + (37*b^2*n^2*x^(3/2)*(a + b*Log[c*x^n]))/(36*d*f) - (9/16)*b^2*n^2*x^2*(a + b*Log[c*x^n]) - (3*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*d^4*f^4) + (3/4)*b^2*n^2*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (15*b*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/(4*d^3*f^3) + (9*b*n*x*(a + b*Log[c*x^n])^2)/(8*d^2*f^2) - (7*b*n*x^(3/2)*(a + b*Log[c*x^n])^2)/(12*d*f) + (3/8)*b*n*x^2*(a + b*Log[c*x^n])^2 + (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(4*d^4*f^4) - (3/4)*b*n*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 + (Sqrt[x]*(a + b*Log[c*x^n])^3)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^3)/(4*d^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^3)/(6*d*f) - (1/8)*x^2*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(2*d^4*f^4) + (1/2)*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3 - (3*b^3*n^3*PolyLog[2, (-d)*f*Sqrt[x]])/(2*d^4*f^4) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]])/(d^4*f^4) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*Sqrt[x]])/(d^4*f^4) - (6*b^3*n^3*PolyLog[3, (-d)*f*Sqrt[x]])/(d^4*f^4) + (12*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*Sqrt[x]])/(d^4*f^4) - (24*b^3*n^3*PolyLog[4, (-d)*f*Sqrt[x]])/(d^4*f^4)} -{x^0*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3, x, 24, (-90*b^3*n^3*Sqrt[x])/(d*f) - 6*a*b^2*n^2*x + 12*b^3*n^3*x - 6*b^3*n^3*x*Log[d*(1/d + f*Sqrt[x])] + (6*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) - 6*b^3*n^2*x*Log[c*x^n] + (42*b^2*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) - 3*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b^2*n^2*x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (6*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) - (9*b*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) + 3*b*n*x*(a + b*Log[c*x^n])^2 - 3*b*n*x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^3)/(d*f) - (x*(a + b*Log[c*x^n])^3)/2 + x*Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(d^2*f^2) - (12*b^3*n^3*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (12*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (6*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (24*b^3*n^3*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) + (24*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) - (48*b^3*n^3*PolyLog[4, -(d*f*Sqrt[x])])/(d^2*f^2)} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^1, x, 4, -2*(a + b*Log[c*x^n])^3*PolyLog[2, -(d*f*Sqrt[x])] + 12*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -(d*f*Sqrt[x])] - 48*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -(d*f*Sqrt[x])] + 96*b^3*n^3*PolyLog[5, -(d*f*Sqrt[x])]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^2, x, 28, -((90*b^3*d*f*n^3)/Sqrt[x]) + 6*b^3*d^2*f^2*n^3*Log[1 + d*f*Sqrt[x]] - (6*b^3*n^3*Log[1 + d*f*Sqrt[x]])/x - 3*b^3*d^2*f^2*n^3*Log[x] + (3/2)*b^3*d^2*f^2*n^3*Log[x]^2 - (42*b^2*d*f*n^2*(a + b*Log[c*x^n]))/Sqrt[x] + 6*b^2*d^2*f^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (6*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/x - 3*b^2*d^2*f^2*n^2*Log[x]*(a + b*Log[c*x^n]) - (9*b*d*f*n*(a + b*Log[c*x^n])^2)/Sqrt[x] + 3*b*d^2*f^2*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 - (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/x - (1/2)*d^2*f^2*(a + b*Log[c*x^n])^3 - (d*f*(a + b*Log[c*x^n])^3)/Sqrt[x] + d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/x - (d^2*f^2*(a + b*Log[c*x^n])^4)/(8*b*n) + 12*b^3*d^2*f^2*n^3*PolyLog[2, (-d)*f*Sqrt[x]] + 12*b^2*d^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]] + 6*b*d^2*f^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*Sqrt[x]] - 24*b^3*d^2*f^2*n^3*PolyLog[3, (-d)*f*Sqrt[x]] - 24*b^2*d^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*Sqrt[x]] + 48*b^3*d^2*f^2*n^3*PolyLog[4, (-d)*f*Sqrt[x]]} -{Log[d*(1/d + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^3, x, 34, -((175*b^3*d*f*n^3)/(216*x^(3/2))) + (45*b^3*d^2*f^2*n^3)/(16*x) - (255*b^3*d^3*f^3*n^3)/(8*Sqrt[x]) + (3/8)*b^3*d^4*f^4*n^3*Log[1 + d*f*Sqrt[x]] - (3*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(8*x^2) - (3/16)*b^3*d^4*f^4*n^3*Log[x] + (3/16)*b^3*d^4*f^4*n^3*Log[x]^2 - (37*b^2*d*f*n^2*(a + b*Log[c*x^n]))/(36*x^(3/2)) + (21*b^2*d^2*f^2*n^2*(a + b*Log[c*x^n]))/(8*x) - (63*b^2*d^3*f^3*n^2*(a + b*Log[c*x^n]))/(4*Sqrt[x]) + (3/4)*b^2*d^4*f^4*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (3*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*x^2) - (3/8)*b^2*d^4*f^4*n^2*Log[x]*(a + b*Log[c*x^n]) - (7*b*d*f*n*(a + b*Log[c*x^n])^2)/(12*x^(3/2)) + (9*b*d^2*f^2*n*(a + b*Log[c*x^n])^2)/(8*x) - (15*b*d^3*f^3*n*(a + b*Log[c*x^n])^2)/(4*Sqrt[x]) + (3/4)*b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 - (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(4*x^2) - (1/8)*d^4*f^4*(a + b*Log[c*x^n])^3 - (d*f*(a + b*Log[c*x^n])^3)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^3)/(4*x) - (d^3*f^3*(a + b*Log[c*x^n])^3)/(2*Sqrt[x]) + (1/2)*d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(2*x^2) - (d^4*f^4*(a + b*Log[c*x^n])^4)/(16*b*n) + (3/2)*b^3*d^4*f^4*n^3*PolyLog[2, (-d)*f*Sqrt[x]] + 3*b^2*d^4*f^4*n^2*(a + b*Log[c*x^n])*PolyLog[2, (-d)*f*Sqrt[x]] + 3*b*d^4*f^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*Sqrt[x]] - 6*b^3*d^4*f^4*n^3*PolyLog[3, (-d)*f*Sqrt[x]] - 12*b^2*d^4*f^4*n^2*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*Sqrt[x]] + 24*b^3*d^4*f^4*n^3*PolyLog[4, (-d)*f*Sqrt[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^m)] (a+b Log[c x^n])^p when d e=1*) - - -{Log[d*(1/d + f*x^m)]*(a + b*Log[c*x^n])^4/x, x, 5, -(((a + b*Log[c*x^n])^4*PolyLog[2, (-d)*f*x^m])/m) + (4*b*n*(a + b*Log[c*x^n])^3*PolyLog[3, (-d)*f*x^m])/m^2 - (12*b^2*n^2*(a + b*Log[c*x^n])^2*PolyLog[4, (-d)*f*x^m])/m^3 + (24*b^3*n^3*(a + b*Log[c*x^n])*PolyLog[5, (-d)*f*x^m])/m^4 - (24*b^4*n^4*PolyLog[6, (-d)*f*x^m])/m^5} -{Log[d*(1/d + f*x^m)]*(a + b*Log[c*x^n])^3/x, x, 4, -(((a + b*Log[c*x^n])^3*PolyLog[2, (-d)*f*x^m])/m) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, (-d)*f*x^m])/m^2 - (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, (-d)*f*x^m])/m^3 + (6*b^3*n^3*PolyLog[5, (-d)*f*x^m])/m^4} -{Log[d*(1/d + f*x^m)]*(a + b*Log[c*x^n])^2/x, x, 3, -(((a + b*Log[c*x^n])^2*PolyLog[2, (-d)*f*x^m])/m) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[3, (-d)*f*x^m])/m^2 - (2*b^2*n^2*PolyLog[4, (-d)*f*x^m])/m^3} -{Log[d*(1/d + f*x^m)]*(a + b*Log[c*x^n])^1/x, x, 2, -(((a + b*Log[c*x^n])*PolyLog[2, (-d)*f*x^m])/m) + (b*n*PolyLog[3, (-d)*f*x^m])/m^2} -{Log[d*(1/d + f*x^m)]/(x*(a + b*Log[c*x^n])^1), x, 0, Unintegrable[Log[d*(1/d + f*x^m)]/(x*(a + b*Log[c*x^n])), x]} -{Log[d*(1/d + f*x^m)]/(x*(a + b*Log[c*x^n])^2), x, 0, Unintegrable[Log[d*(1/d + f*x^m)]/(x*(a + b*Log[c*x^n])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g x)^q Log[d (e+f x^m)^r] (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^1)^k] (a+b Log[c x^n])^p*) - - -{x^3*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n]), x, 7, -((5*b*e^3*m*n*x)/(16*f^3)) + (3*b*e^2*m*n*x^2)/(32*f^2) - (7*b*e*m*n*x^3)/(144*f) + (1/32)*b*m*n*x^4 + (e^3*m*x*(a + b*Log[c*x^n]))/(4*f^3) - (e^2*m*x^2*(a + b*Log[c*x^n]))/(8*f^2) + (e*m*x^3*(a + b*Log[c*x^n]))/(12*f) - (1/16)*m*x^4*(a + b*Log[c*x^n]) + (b*e^4*m*n*Log[e + f*x])/(16*f^4) + (b*e^4*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(4*f^4) - (e^4*m*(a + b*Log[c*x^n])*Log[e + f*x])/(4*f^4) - (1/16)*b*n*x^4*Log[d*(e + f*x)^m] + (1/4)*x^4*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] + (b*e^4*m*n*PolyLog[2, 1 + (f*x)/e])/(4*f^4)} -{x^2*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n]), x, 7, (4*b*e^2*m*n*x)/(9*f^2) - (5*b*e*m*n*x^2)/(36*f) + (2/27)*b*m*n*x^3 - (e^2*m*x*(a + b*Log[c*x^n]))/(3*f^2) + (e*m*x^2*(a + b*Log[c*x^n]))/(6*f) - (1/9)*m*x^3*(a + b*Log[c*x^n]) - (b*e^3*m*n*Log[e + f*x])/(9*f^3) - (b*e^3*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(3*f^3) + (e^3*m*(a + b*Log[c*x^n])*Log[e + f*x])/(3*f^3) - (1/9)*b*n*x^3*Log[d*(e + f*x)^m] + (1/3)*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] - (b*e^3*m*n*PolyLog[2, 1 + (f*x)/e])/(3*f^3)} -{x^1*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n]), x, 7, -((3*b*e*m*n*x)/(4*f)) + (1/4)*b*m*n*x^2 + (e*m*x*(a + b*Log[c*x^n]))/(2*f) - (1/4)*m*x^2*(a + b*Log[c*x^n]) + (b*e^2*m*n*Log[e + f*x])/(4*f^2) + (b*e^2*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(2*f^2) - (e^2*m*(a + b*Log[c*x^n])*Log[e + f*x])/(2*f^2) - (1/4)*b*n*x^2*Log[d*(e + f*x)^m] + (1/2)*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] + (b*e^2*m*n*PolyLog[2, 1 + (f*x)/e])/(2*f^2)} -{x^0*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n]), x, 8, 2*b*m*n*x - m*x*(a + b*Log[c*x^n]) - (b*n*(e + f*x)*Log[d*(e + f*x)^m])/f - (b*e*n*Log[-((f*x)/e)]*Log[d*(e + f*x)^m])/f + ((e + f*x)*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/f - (b*e*m*n*PolyLog[2, 1 + (f*x)/e])/f} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])/x^1, x, 4, ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(2*b*n) - (m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(2*b*n) - m*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)] + b*m*n*PolyLog[3, -((f*x)/e)]} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])/x^2, x, 9, (b*f*m*n*Log[x])/e - (b*f*m*n*Log[x]^2)/(2*e) + (f*m*Log[x]*(a + b*Log[c*x^n]))/e - (b*f*m*n*Log[e + f*x])/e + (b*f*m*n*Log[-((f*x)/e)]*Log[e + f*x])/e - (f*m*(a + b*Log[c*x^n])*Log[e + f*x])/e - (b*n*Log[d*(e + f*x)^m])/x - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/x + (b*f*m*n*PolyLog[2, 1 + (f*x)/e])/e} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])/x^3, x, 8, -((3*b*f*m*n)/(4*e*x)) - (b*f^2*m*n*Log[x])/(4*e^2) + (b*f^2*m*n*Log[x]^2)/(4*e^2) - (f*m*(a + b*Log[c*x^n]))/(2*e*x) - (f^2*m*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) + (b*f^2*m*n*Log[e + f*x])/(4*e^2) - (b*f^2*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(2*e^2) + (f^2*m*(a + b*Log[c*x^n])*Log[e + f*x])/(2*e^2) - (b*n*Log[d*(e + f*x)^m])/(4*x^2) - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(2*x^2) - (b*f^2*m*n*PolyLog[2, 1 + (f*x)/e])/(2*e^2)} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])/x^4, x, 8, -((5*b*f*m*n)/(36*e*x^2)) + (4*b*f^2*m*n)/(9*e^2*x) + (b*f^3*m*n*Log[x])/(9*e^3) - (b*f^3*m*n*Log[x]^2)/(6*e^3) - (f*m*(a + b*Log[c*x^n]))/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n]))/(3*e^2*x) + (f^3*m*Log[x]*(a + b*Log[c*x^n]))/(3*e^3) - (b*f^3*m*n*Log[e + f*x])/(9*e^3) + (b*f^3*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(3*e^3) - (f^3*m*(a + b*Log[c*x^n])*Log[e + f*x])/(3*e^3) - (b*n*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(3*x^3) + (b*f^3*m*n*PolyLog[2, 1 + (f*x)/e])/(3*e^3)} - - -{x^2*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2, x, 24, (8*a*b*e^2*m*n*x)/(9*f^2) - (26*b^2*e^2*m*n^2*x)/(27*f^2) + (19*b^2*e*m*n^2*x^2)/(108*f) - (2/27)*b^2*m*n^2*x^3 + (8*b^2*e^2*m*n*x*Log[c*x^n])/(9*f^2) - (5*b*e*m*n*x^2*(a + b*Log[c*x^n]))/(18*f) + (4/27)*b*m*n*x^3*(a + b*Log[c*x^n]) - (e^2*m*x*(a + b*Log[c*x^n])^2)/(3*f^2) + (e*m*x^2*(a + b*Log[c*x^n])^2)/(6*f) - (1/9)*m*x^3*(a + b*Log[c*x^n])^2 + (2*b^2*e^3*m*n^2*Log[e + f*x])/(27*f^3) + (2/27)*b^2*n^2*x^3*Log[d*(e + f*x)^m] - (2/9)*b*n*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] + (1/3)*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] - (2*b*e^3*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(9*f^3) + (e^3*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(3*f^3) - (2*b^2*e^3*m*n^2*PolyLog[2, -((f*x)/e)])/(9*f^3) + (2*b*e^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(3*f^3) - (2*b^2*e^3*m*n^2*PolyLog[3, -((f*x)/e)])/(3*f^3)} -{x^1*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2, x, 21, -((3*a*b*e*m*n*x)/(2*f)) + (7*b^2*e*m*n^2*x)/(4*f) - (3/8)*b^2*m*n^2*x^2 - (3*b^2*e*m*n*x*Log[c*x^n])/(2*f) + (1/2)*b*m*n*x^2*(a + b*Log[c*x^n]) + (e*m*x*(a + b*Log[c*x^n])^2)/(2*f) - (1/4)*m*x^2*(a + b*Log[c*x^n])^2 - (b^2*e^2*m*n^2*Log[e + f*x])/(4*f^2) + (1/4)*b^2*n^2*x^2*Log[d*(e + f*x)^m] - (1/2)*b*n*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] + (1/2)*x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] + (b*e^2*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(2*f^2) - (e^2*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(2*f^2) + (b^2*e^2*m*n^2*PolyLog[2, -((f*x)/e)])/(2*f^2) - (b*e^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f^2 + (b^2*e^2*m*n^2*PolyLog[3, -((f*x)/e)])/f^2} -{x^0*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2, x, 18, 2*a*b*m*n*x - 4*b^2*m*n^2*x + 2*b*m*n*(a - b*n)*x + 4*b^2*m*n*x*Log[c*x^n] - m*x*(a + b*Log[c*x^n])^2 - (2*b*e*m*n*(a - b*n)*Log[e + f*x])/f - 2*a*b*n*x*Log[d*(e + f*x)^m] + 2*b^2*n^2*x*Log[d*(e + f*x)^m] - 2*b^2*n*x*Log[c*x^n]*Log[d*(e + f*x)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] - (2*b^2*e*m*n*Log[c*x^n]*Log[1 + (f*x)/e])/f + (e*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f - (2*b^2*e*m*n^2*PolyLog[2, -((f*x)/e)])/f + (2*b*e*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f - (2*b^2*e*m*n^2*PolyLog[3, -((f*x)/e)])/f} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2/x^1, x, 5, ((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/(3*b*n) - (m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/(3*b*n) - m*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)] + 2*b*m*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)] - 2*b^2*m*n^2*PolyLog[4, -((f*x)/e)]} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2/x^2, x, 10, (2*b^2*f*m*n^2*Log[x])/e - (2*b*f*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/e - (f*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/e - (2*b^2*f*m*n^2*Log[e + f*x])/e - (2*b^2*n^2*Log[d*(e + f*x)^m])/x - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/x - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/x + (2*b^2*f*m*n^2*PolyLog[2, -(e/(f*x))])/e + (2*b*f*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/e + (2*b^2*f*m*n^2*PolyLog[3, -(e/(f*x))])/e} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2/x^3, x, 14, -((7*b^2*f*m*n^2)/(4*e*x)) - (b^2*f^2*m*n^2*Log[x])/(4*e^2) - (3*b*f*m*n*(a + b*Log[c*x^n]))/(2*e*x) + (b*f^2*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/(2*e^2) - (f*m*(a + b*Log[c*x^n])^2)/(2*e*x) + (f^2*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(2*e^2) + (b^2*f^2*m*n^2*Log[e + f*x])/(4*e^2) - (b^2*n^2*Log[d*(e + f*x)^m])/(4*x^2) - (b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(2*x^2) - (b^2*f^2*m*n^2*PolyLog[2, -(e/(f*x))])/(2*e^2) - (b*f^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/e^2 - (b^2*f^2*m*n^2*PolyLog[3, -(e/(f*x))])/e^2} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^2/x^4, x, 19, -((19*b^2*f*m*n^2)/(108*e*x^2)) + (26*b^2*f^2*m*n^2)/(27*e^2*x) + (2*b^2*f^3*m*n^2*Log[x])/(27*e^3) - (5*b*f*m*n*(a + b*Log[c*x^n]))/(18*e*x^2) + (8*b*f^2*m*n*(a + b*Log[c*x^n]))/(9*e^2*x) - (2*b*f^3*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/(9*e^3) - (f*m*(a + b*Log[c*x^n])^2)/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n])^2)/(3*e^2*x) - (f^3*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(3*e^3) - (2*b^2*f^3*m*n^2*Log[e + f*x])/(27*e^3) - (2*b^2*n^2*Log[d*(e + f*x)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(3*x^3) + (2*b^2*f^3*m*n^2*PolyLog[2, -(e/(f*x))])/(9*e^3) + (2*b*f^3*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/(3*e^3) + (2*b^2*f^3*m*n^2*PolyLog[3, -(e/(f*x))])/(3*e^3)} - - -{x^1*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^3, x, 34, (21*a*b^2*e*m*n^2*x)/(4*f) - (45*b^3*e*m*n^3*x)/(8*f) + (3/4)*b^3*m*n^3*x^2 + (21*b^3*e*m*n^2*x*Log[c*x^n])/(4*f) - (9/8)*b^2*m*n^2*x^2*(a + b*Log[c*x^n]) - (9*b*e*m*n*x*(a + b*Log[c*x^n])^2)/(4*f) + (3/4)*b*m*n*x^2*(a + b*Log[c*x^n])^2 + (e*m*x*(a + b*Log[c*x^n])^3)/(2*f) - (1/4)*m*x^2*(a + b*Log[c*x^n])^3 + (3*b^3*e^2*m*n^3*Log[e + f*x])/(8*f^2) - (3/8)*b^3*n^3*x^2*Log[d*(e + f*x)^m] + (3/4)*b^2*n^2*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m] - (3/4)*b*n*x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] + (1/2)*x^2*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m] - (3*b^2*e^2*m*n^2*(a + b*Log[c*x^n])*Log[1 + (f*x)/e])/(4*f^2) + (3*b*e^2*m*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/(4*f^2) - (e^2*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/(2*f^2) - (3*b^3*e^2*m*n^3*PolyLog[2, -((f*x)/e)])/(4*f^2) + (3*b^2*e^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/(2*f^2) - (3*b*e^2*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)])/(2*f^2) - (3*b^3*e^2*m*n^3*PolyLog[3, -((f*x)/e)])/(2*f^2) + (3*b^2*e^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/f^2 - (3*b^3*e^2*m*n^3*PolyLog[4, -((f*x)/e)])/f^2} -{x^0*Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^3, x, 28, -12*a*b^2*m*n^2*x + 18*b^3*m*n^3*x - 6*b^2*m*n^2*(a - b*n)*x - 18*b^3*m*n^2*x*Log[c*x^n] + 6*b*m*n*x*(a + b*Log[c*x^n])^2 - m*x*(a + b*Log[c*x^n])^3 + (6*b^2*e*m*n^2*(a - b*n)*Log[e + f*x])/f + 6*a*b^2*n^2*x*Log[d*(e + f*x)^m] - 6*b^3*n^3*x*Log[d*(e + f*x)^m] + 6*b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m] + (6*b^3*e*m*n^2*Log[c*x^n]*Log[1 + (f*x)/e])/f - (3*b*e*m*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x)/e])/f + (e*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x)/e])/f + (6*b^3*e*m*n^3*PolyLog[2, -((f*x)/e)])/f - (6*b^2*e*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x)/e)])/f + (3*b*e*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x)/e)])/f + (6*b^3*e*m*n^3*PolyLog[3, -((f*x)/e)])/f - (6*b^2*e*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x)/e)])/f + (6*b^3*e*m*n^3*PolyLog[4, -((f*x)/e)])/f} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^3/x^1, x, 6, ((a + b*Log[c*x^n])^4*Log[d*(e + f*x)^m])/(4*b*n) - (m*(a + b*Log[c*x^n])^4*Log[1 + (f*x)/e])/(4*b*n) - m*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x)/e)] + 3*b*m*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x)/e)] - 6*b^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*x)/e)] + 6*b^3*m*n^3*PolyLog[5, -((f*x)/e)]} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^3/x^2, x, 14, (6*b^3*f*m*n^3*Log[x])/e - (6*b^2*f*m*n^2*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/e - (3*b*f*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/e - (f*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^3)/e - (6*b^3*f*m*n^3*Log[e + f*x])/e - (6*b^3*n^3*Log[d*(e + f*x)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/x - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/x + (6*b^3*f*m*n^3*PolyLog[2, -(e/(f*x))])/e + (6*b^2*f*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/e + (3*b*f*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e/(f*x))])/e + (6*b^3*f*m*n^3*PolyLog[3, -(e/(f*x))])/e + (6*b^2*f*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x))])/e + (6*b^3*f*m*n^3*PolyLog[4, -(e/(f*x))])/e} -{Log[d*(e + f*x)^m]*(a + b*Log[c*x^n])^3/x^3, x, 22, -((45*b^3*f*m*n^3)/(8*e*x)) - (3*b^3*f^2*m*n^3*Log[x])/(8*e^2) - (21*b^2*f*m*n^2*(a + b*Log[c*x^n]))/(4*e*x) + (3*b^2*f^2*m*n^2*Log[1 + e/(f*x)]*(a + b*Log[c*x^n]))/(4*e^2) - (9*b*f*m*n*(a + b*Log[c*x^n])^2)/(4*e*x) + (3*b*f^2*m*n*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^2)/(4*e^2) - (f*m*(a + b*Log[c*x^n])^3)/(2*e*x) + (f^2*m*Log[1 + e/(f*x)]*(a + b*Log[c*x^n])^3)/(2*e^2) + (3*b^3*f^2*m*n^3*Log[e + f*x])/(8*e^2) - (3*b^3*n^3*Log[d*(e + f*x)^m])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x)^m])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x)^m])/(2*x^2) - (3*b^3*f^2*m*n^3*PolyLog[2, -(e/(f*x))])/(4*e^2) - (3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x))])/(2*e^2) - (3*b*f^2*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e/(f*x))])/(2*e^2) - (3*b^3*f^2*m*n^3*PolyLog[3, -(e/(f*x))])/(2*e^2) - (3*b^2*f^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x))])/e^2 - (3*b^3*f^2*m*n^3*PolyLog[4, -(e/(f*x))])/e^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^2)^k] (a+b Log[c x^n])^p*) - - -{x^3*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n]), x, 9, -((3*b*e*m*n*x^2)/(16*f)) + (1/16)*b*m*n*x^4 + (e*m*x^2*(a + b*Log[c*x^n]))/(4*f) - (1/8)*m*x^4*(a + b*Log[c*x^n]) + (b*e^2*m*n*Log[e + f*x^2])/(16*f^2) + (b*e^2*m*n*Log[-((f*x^2)/e)]*Log[e + f*x^2])/(8*f^2) - (e^2*m*(a + b*Log[c*x^n])*Log[e + f*x^2])/(4*f^2) - (1/16)*b*n*x^4*Log[d*(e + f*x^2)^m] + (1/4)*x^4*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] + (b*e^2*m*n*PolyLog[2, 1 + (f*x^2)/e])/(8*f^2)} -{x^1*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n]), x, 9, (1/2)*b*m*n*x^2 - (1/2)*m*x^2*(a + b*Log[c*x^n]) - (b*n*(e + f*x^2)*Log[d*(e + f*x^2)^m])/(4*f) - (b*e*n*Log[-((f*x^2)/e)]*Log[d*(e + f*x^2)^m])/(4*f) + ((e + f*x^2)*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(2*f) - (b*e*m*n*PolyLog[2, 1 + (f*x^2)/e])/(4*f)} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^1, x, 4, ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(2*b*n) - (m*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(2*b*n) - (1/2)*m*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)] + (1/4)*b*m*n*PolyLog[3, -((f*x^2)/e)]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^3, x, 11, (b*f*m*n*Log[x])/(2*e) - (b*f*m*n*Log[x]^2)/(2*e) + (f*m*Log[x]*(a + b*Log[c*x^n]))/e - (b*f*m*n*Log[e + f*x^2])/(4*e) + (b*f*m*n*Log[-((f*x^2)/e)]*Log[e + f*x^2])/(4*e) - (f*m*(a + b*Log[c*x^n])*Log[e + f*x^2])/(2*e) - (b*n*Log[d*(e + f*x^2)^m])/(4*x^2) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(2*x^2) + (b*f*m*n*PolyLog[2, 1 + (f*x^2)/e])/(4*e)} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^5, x, 10, -((3*b*f*m*n)/(16*e*x^2)) - (b*f^2*m*n*Log[x])/(8*e^2) + (b*f^2*m*n*Log[x]^2)/(4*e^2) - (f*m*(a + b*Log[c*x^n]))/(4*e*x^2) - (f^2*m*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) + (b*f^2*m*n*Log[e + f*x^2])/(16*e^2) - (b*f^2*m*n*Log[-((f*x^2)/e)]*Log[e + f*x^2])/(8*e^2) + (f^2*m*(a + b*Log[c*x^n])*Log[e + f*x^2])/(4*e^2) - (b*n*Log[d*(e + f*x^2)^m])/(16*x^4) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(4*x^4) - (b*f^2*m*n*PolyLog[2, 1 + (f*x^2)/e])/(8*e^2)} - -{x^2*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n]), x, 9, -((8*b*e*m*n*x)/(9*f)) + (4/27)*b*m*n*x^3 + (2*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(9*f^(3/2)) + (2*e*m*x*(a + b*Log[c*x^n]))/(3*f) - (2/9)*m*x^3*(a + b*Log[c*x^n]) - (2*e^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(3*f^(3/2)) - (1/9)*b*n*x^3*Log[d*(e + f*x^2)^m] + (1/3)*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] + (I*b*e^(3/2)*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(3*f^(3/2)) - (I*b*e^(3/2)*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(3*f^(3/2))} -{x^0*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n]), x, 8, 4*b*m*n*x - (2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - 2*m*x*(a + b*Log[c*x^n]) + (2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/Sqrt[f] - b*n*x*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] - (I*b*Sqrt[e]*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[f] + (I*b*Sqrt[e]*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^2, x, 7, (2*b*Sqrt[f]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (2*Sqrt[f]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/Sqrt[e] - (b*n*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - (I*b*Sqrt[f]*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[e] + (I*b*Sqrt[f]*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[e]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^4, x, 8, -((8*b*f*m*n)/(9*e*x)) - (2*b*f^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(9*e^(3/2)) - (2*f*m*(a + b*Log[c*x^n]))/(3*e*x) - (2*f^(3/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(3*e^(3/2)) - (b*n*Log[d*(e + f*x^2)^m])/(9*x^3) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(3*x^3) + (I*b*f^(3/2)*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(3*e^(3/2)) - (I*b*f^(3/2)*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(3*e^(3/2))} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])/x^6, x, 9, -((16*b*f*m*n)/(225*e*x^3)) + (12*b*f^2*m*n)/(25*e^2*x) + (2*b*f^(5/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(25*e^(5/2)) - (2*f*m*(a + b*Log[c*x^n]))/(15*e*x^3) + (2*f^2*m*(a + b*Log[c*x^n]))/(5*e^2*x) + (2*f^(5/2)*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(5*e^(5/2)) - (b*n*Log[d*(e + f*x^2)^m])/(25*x^5) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(5*x^5) - (I*b*f^(5/2)*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(5*e^(5/2)) + (I*b*f^(5/2)*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(5*e^(5/2))} - - -{x^1*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2, x, 17, (-(3/4))*b^2*m*n^2*x^2 + b*m*n*x^2*(a + b*Log[c*x^n]) - (1/2)*m*x^2*(a + b*Log[c*x^n])^2 + (b^2*e*m*n^2*Log[e + f*x^2])/(4*f) + (1/4)*b^2*n^2*x^2*Log[d*(e + f*x^2)^m] - (1/2)*b*n*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] + (1/2)*x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] - (b*e*m*n*(a + b*Log[c*x^n])*Log[1 + (f*x^2)/e])/(2*f) + (e*m*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(2*f) - (b^2*e*m*n^2*PolyLog[2, -((f*x^2)/e)])/(4*f) + (b*e*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/(2*f) - (b^2*e*m*n^2*PolyLog[3, -((f*x^2)/e)])/(4*f)} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2/x^1, x, 5, ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(3*b*n) - (m*(a + b*Log[c*x^n])^3*Log[1 + (f*x^2)/e])/(3*b*n) - (1/2)*m*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x^2)/e)] + (1/2)*b*m*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*x^2)/e)] - (1/4)*b^2*m*n^2*PolyLog[4, -((f*x^2)/e)]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2/x^3, x, 11, (b^2*f*m*n^2*Log[x])/(2*e) - (b*f*m*n*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n]))/(2*e) - (f*m*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(2*e) - (b^2*f*m*n^2*Log[e + f*x^2])/(4*e) - (b^2*n^2*Log[d*(e + f*x^2)^m])/(4*x^2) - (b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(2*x^2) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(2*x^2) + (b^2*f*m*n^2*PolyLog[2, -(e/(f*x^2))])/(4*e) + (b*f*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x^2))])/(2*e) + (b^2*f*m*n^2*PolyLog[3, -(e/(f*x^2))])/(4*e)} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2/x^5, x, 15, -((7*b^2*f*m*n^2)/(32*e*x^2)) - (b^2*f^2*m*n^2*Log[x])/(16*e^2) - (3*b*f*m*n*(a + b*Log[c*x^n]))/(8*e*x^2) + (b*f^2*m*n*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n]))/(8*e^2) - (f*m*(a + b*Log[c*x^n])^2)/(4*e*x^2) + (f^2*m*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(4*e^2) + (b^2*f^2*m*n^2*Log[e + f*x^2])/(32*e^2) - (b^2*n^2*Log[d*(e + f*x^2)^m])/(32*x^4) - (b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(8*x^4) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(4*x^4) - (b^2*f^2*m*n^2*PolyLog[2, -(e/(f*x^2))])/(16*e^2) - (b*f^2*m*n*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x^2))])/(4*e^2) - (b^2*f^2*m*n^2*PolyLog[3, -(e/(f*x^2))])/(8*e^2)} - -{x^2*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2, x, 30, -((16*a*b*e*m*n*x)/(9*f)) + (52*b^2*e*m*n^2*x)/(27*f) - (4/27)*b^2*m*n^2*x^3 - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2)) - (16*b^2*e*m*n*x*Log[c*x^n])/(9*f) + (8/27)*b*m*n*x^3*(a + b*Log[c*x^n]) + (4*b*e^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) + (2*e*m*x*(a + b*Log[c*x^n])^2)/(3*f) - (2/9)*m*x^3*(a + b*Log[c*x^n])^2 - ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2/27)*b^2*n^2*x^3*Log[d*(e + f*x^2)^m] - (2/9)*b*n*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] + (1/3)*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (2*b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (2*I*b^2*e^(3/2)*m*n^2*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(9*f^(3/2)) + (2*I*b^2*e^(3/2)*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(9*f^(3/2)) - (2*b^2*(-e)^(3/2)*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (2*b^2*(-e)^(3/2)*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2))} -{x^0*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2, x, 26, 4*a*b*m*n*x - 8*b^2*m*n^2*x + 4*b*m*n*(a - b*n)*x - (4*b*Sqrt[e]*m*n*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] + 8*b^2*m*n*x*Log[c*x^n] - (4*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n])/Sqrt[f] - 2*m*x*(a + b*Log[c*x^n])^2 - (Sqrt[-e]*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (Sqrt[-e]*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - 2*a*b*n*x*Log[d*(e + f*x^2)^m] + 2*b^2*n^2*x*Log[d*(e + f*x^2)^m] - 2*b^2*n*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (2*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (2*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (2*I*b^2*Sqrt[e]*m*n^2*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[f] - (2*I*b^2*Sqrt[e]*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - (2*b^2*Sqrt[-e]*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] + (2*b^2*Sqrt[-e]*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2/x^2, x, 16, (4*b^2*Sqrt[f]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (4*b*Sqrt[f]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/Sqrt[e] + (Sqrt[f]*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (Sqrt[f]*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (2*b^2*n^2*Log[d*(e + f*x^2)^m])/x - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - (2*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (2*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (2*I*b^2*Sqrt[f]*m*n^2*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[e] + (2*I*b^2*Sqrt[f]*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (2*b^2*Sqrt[f]*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (2*b^2*Sqrt[f]*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^2/x^4, x, 22, -((52*b^2*f*m*n^2)/(27*e*x)) - (4*b^2*f^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)) - (16*b*f*m*n*(a + b*Log[c*x^n]))/(9*e*x) - (4*b*f^(3/2)*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) - (2*f*m*(a + b*Log[c*x^n])^2)/(3*e*x) + (f^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (f^(3/2)*m*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^2*n^2*Log[d*(e + f*x^2)^m])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(3*x^3) - (2*b*f^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) + (2*b*f^(3/2)*m*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) + (2*I*b^2*f^(3/2)*m*n^2*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(9*e^(3/2)) - (2*I*b^2*f^(3/2)*m*n^2*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(9*e^(3/2)) + (2*b^2*f^(3/2)*m*n^2*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (2*b^2*f^(3/2)*m*n^2*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2))} - - -{x^1*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3, x, 26, (3/2)*b^3*m*n^3*x^2 - (9/4)*b^2*m*n^2*x^2*(a + b*Log[c*x^n]) + (3/2)*b*m*n*x^2*(a + b*Log[c*x^n])^2 - (1/2)*m*x^2*(a + b*Log[c*x^n])^3 - (3*b^3*e*m*n^3*Log[e + f*x^2])/(8*f) - (3/8)*b^3*n^3*x^2*Log[d*(e + f*x^2)^m] + (3/4)*b^2*n^2*x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] - (3/4)*b*n*x^2*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (1/2)*x^2*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] + (3*b^2*e*m*n^2*(a + b*Log[c*x^n])*Log[1 + (f*x^2)/e])/(4*f) - (3*b*e*m*n*(a + b*Log[c*x^n])^2*Log[1 + (f*x^2)/e])/(4*f) + (e*m*(a + b*Log[c*x^n])^3*Log[1 + (f*x^2)/e])/(2*f) + (3*b^3*e*m*n^3*PolyLog[2, -((f*x^2)/e)])/(8*f) - (3*b^2*e*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^2)/e)])/(4*f) + (3*b*e*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x^2)/e)])/(4*f) + (3*b^3*e*m*n^3*PolyLog[3, -((f*x^2)/e)])/(8*f) - (3*b^2*e*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*x^2)/e)])/(4*f) + (3*b^3*e*m*n^3*PolyLog[4, -((f*x^2)/e)])/(8*f)} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3/x^1, x, 6, ((a + b*Log[c*x^n])^4*Log[d*(e + f*x^2)^m])/(4*b*n) - (m*(a + b*Log[c*x^n])^4*Log[1 + (f*x^2)/e])/(4*b*n) - (1/2)*m*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^2)/e)] + (3/4)*b*m*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^2)/e)] - (3/4)*b^2*m*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^2)/e)] + (3/8)*b^3*m*n^3*PolyLog[5, -((f*x^2)/e)]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3/x^3, x, 15, (3*b^3*f*m*n^3*Log[x])/(4*e) - (3*b^2*f*m*n^2*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n]))/(4*e) - (3*b*f*m*n*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^2)/(4*e) - (f*m*Log[1 + e/(f*x^2)]*(a + b*Log[c*x^n])^3)/(2*e) - (3*b^3*f*m*n^3*Log[e + f*x^2])/(8*e) - (3*b^3*n^3*Log[d*(e + f*x^2)^m])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(4*x^2) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(2*x^2) + (3*b^3*f*m*n^3*PolyLog[2, -(e/(f*x^2))])/(8*e) + (3*b^2*f*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(e/(f*x^2))])/(4*e) + (3*b*f*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(e/(f*x^2))])/(4*e) + (3*b^3*f*m*n^3*PolyLog[3, -(e/(f*x^2))])/(8*e) + (3*b^2*f*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(e/(f*x^2))])/(4*e) + (3*b^3*f*m*n^3*PolyLog[4, -(e/(f*x^2))])/(8*e)} - -{x^2*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3, x, 49, (52*a*b^2*e*m*n^2*x)/(9*f) - (160*b^3*e*m*n^3*x)/(27*f) + (16/81)*b^3*m*n^3*x^3 + (4*b^3*e^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*f^(3/2)) + (52*b^3*e*m*n^2*x*Log[c*x^n])/(9*f) - (4/9)*b^2*m*n^2*x^3*(a + b*Log[c*x^n]) - (4*b^2*e^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*f^(3/2)) - (8*b*e*m*n*x*(a + b*Log[c*x^n])^2)/(3*f) + (4/9)*b*m*n*x^3*(a + b*Log[c*x^n])^2 + (2*e*m*x*(a + b*Log[c*x^n])^3)/(3*f) - (2/9)*m*x^3*(a + b*Log[c*x^n])^3 + (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - ((-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + ((-e)^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (2/27)*b^3*n^3*x^3*Log[d*(e + f*x^2)^m] + (2/9)*b^2*n^2*x^3*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m] - (1/3)*b*n*x^3*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + (1/3)*x^3*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) + (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) - (b*(-e)^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (2*I*b^3*e^(3/2)*m*n^3*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(9*f^(3/2)) - (2*I*b^3*e^(3/2)*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(9*f^(3/2)) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*f^(3/2)) - (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*f^(3/2)) + (2*b^2*(-e)^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (2*b^3*(-e)^(3/2)*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2)} -{x^0*Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3, x, 42, -24*a*b^2*m*n^2*x + 36*b^3*m*n^3*x - 12*b^2*m*n^2*(a - b*n)*x + (12*b^2*Sqrt[e]*m*n^2*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] - 36*b^3*m*n^2*x*Log[c*x^n] + (12*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n])/Sqrt[f] + 12*b*m*n*x*(a + b*Log[c*x^n])^2 - 2*m*x*(a + b*Log[c*x^n])^3 + (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (Sqrt[-e]*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (Sqrt[-e]*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + 6*a*b^2*n^2*x*Log[d*(e + f*x^2)^m] - 6*b^3*n^3*x*Log[d*(e + f*x^2)^m] + 6*b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] - (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] + (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] + (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (3*b*Sqrt[-e]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] - (6*I*b^3*Sqrt[e]*m*n^3*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[f] + (6*I*b^3*Sqrt[e]*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[f] + (6*b^3*Sqrt[-e]*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^3*Sqrt[-e]*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (6*b^2*Sqrt[-e]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f] + (6*b^3*Sqrt[-e]*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[f] - (6*b^3*Sqrt[-e]*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[f]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3/x^2, x, 26, (12*b^3*Sqrt[f]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (12*b^2*Sqrt[f]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/Sqrt[e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] + (Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (Sqrt[f]*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*b^3*n^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/x - (3*b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/x - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/x - (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] + (3*b*Sqrt[f]*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*I*b^3*Sqrt[f]*m*n^3*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/Sqrt[e] + (6*I*b^3*Sqrt[f]*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/Sqrt[e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*b^2*Sqrt[f]*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e] - (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/Sqrt[-e] + (6*b^3*Sqrt[f]*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/Sqrt[-e]} -{Log[d*(e + f*x^2)^m]*(a + b*Log[c*x^n])^3/x^4, x, 36, -((160*b^3*f*m*n^3)/(27*e*x)) - (4*b^3*f^(3/2)*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(27*e^(3/2)) - (52*b^2*f*m*n^2*(a + b*Log[c*x^n]))/(9*e*x) - (4*b^2*f^(3/2)*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*x^n]))/(9*e^(3/2)) - (8*b*f*m*n*(a + b*Log[c*x^n])^2)/(3*e*x) - (2*f*m*(a + b*Log[c*x^n])^3)/(3*e*x) + (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) + (f^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (f^(3/2)*m*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^3*n^3*Log[d*(e + f*x^2)^m])/(27*x^3) - (2*b^2*n^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^2)^m])/(9*x^3) - (b*n*(a + b*Log[c*x^n])^2*Log[d*(e + f*x^2)^m])/(3*x^3) - ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m])/(3*x^3) - (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) - (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) + (b*f^(3/2)*m*n*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(-e)^(3/2) + (2*I*b^3*f^(3/2)*m*n^3*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(9*e^(3/2)) - (2*I*b^3*f^(3/2)*m*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(9*e^(3/2)) + (2*b^3*f^(3/2)*m*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(3*(-e)^(3/2)) + (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) - (2*b^3*f^(3/2)*m*n^3*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(3*(-e)^(3/2)) - (2*b^2*f^(3/2)*m*n^2*(a + b*Log[c*x^n])*PolyLog[3, (Sqrt[f]*x)/Sqrt[-e]])/(-e)^(3/2) - (2*b^3*f^(3/2)*m*n^3*PolyLog[4, -((Sqrt[f]*x)/Sqrt[-e])])/(-e)^(3/2) + (2*b^3*f^(3/2)*m*n^3*PolyLog[4, (Sqrt[f]*x)/Sqrt[-e]])/(-e)^(3/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^(1/2))^k] (a+b Log[c x^n])^p*) - - -{x^2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]), x, 9, -((7*b*e^5*k*n*Sqrt[x])/(9*f^5)) + (2*b*e^4*k*n*x)/(9*f^4) - (b*e^3*k*n*x^(3/2))/(9*f^3) + (5*b*e^2*k*n*x^2)/(72*f^2) - (11*b*e*k*n*x^(5/2))/(225*f) + (1/27)*b*k*n*x^3 + (b*e^6*k*n*Log[e + f*Sqrt[x]])/(9*f^6) - (1/9)*b*n*x^3*Log[d*(e + f*Sqrt[x])^k] + (2*b*e^6*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*f^6) + (e^5*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^5) - (e^4*k*x*(a + b*Log[c*x^n]))/(6*f^4) + (e^3*k*x^(3/2)*(a + b*Log[c*x^n]))/(9*f^3) - (e^2*k*x^2*(a + b*Log[c*x^n]))/(12*f^2) + (e*k*x^(5/2)*(a + b*Log[c*x^n]))/(15*f) - (1/18)*k*x^3*(a + b*Log[c*x^n]) - (e^6*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^6) + (1/3)*x^3*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) + (2*b*e^6*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^6)} -{x^1*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]), x, 9, -((5*b*e^3*k*n*Sqrt[x])/(4*f^3)) + (3*b*e^2*k*n*x)/(8*f^2) - (7*b*e*k*n*x^(3/2))/(36*f) + (1/8)*b*k*n*x^2 + (b*e^4*k*n*Log[e + f*Sqrt[x]])/(4*f^4) - (1/4)*b*n*x^2*Log[d*(e + f*Sqrt[x])^k] + (b*e^4*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^4 + (e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/(4*f^2) + (e*k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (1/8)*k*x^2*(a + b*Log[c*x^n]) - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*f^4) + (1/2)*x^2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) + (b*e^4*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^4} -{x^0*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]), x, 9, -((3*b*e*k*n*Sqrt[x])/f) + b*k*n*x + (b*e^2*k*n*Log[e + f*Sqrt[x]])/f^2 - b*n*x*Log[d*(e + f*Sqrt[x])^k] + (2*b*e^2*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + (e*k*Sqrt[x]*(a + b*Log[c*x^n]))/f - (1/2)*k*x*(a + b*Log[c*x^n]) - (e^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) + (2*b*e^2*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^1, x, 4, (Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])^2)/(2*b*n) - (k*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(2*b*n) - 2*k*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)] + 4*b*k*n*PolyLog[3, -((f*Sqrt[x])/e)]} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^2, x, 10, -((3*b*f*k*n)/(e*Sqrt[x])) + (b*f^2*k*n*Log[e + f*Sqrt[x]])/e^2 - (b*n*Log[d*(e + f*Sqrt[x])^k])/x - (2*b*f^2*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b*f^2*k*n*Log[x])/(2*e^2) + (b*f^2*k*n*Log[x]^2)/(4*e^2) - (f*k*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (f^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 - (Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x - (f^2*k*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) - (2*b*f^2*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^2} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^3, x, 10, -((7*b*f*k*n)/(36*e*x^(3/2))) + (3*b*f^2*k*n)/(8*e^2*x) - (5*b*f^3*k*n)/(4*e^3*Sqrt[x]) + (b*f^4*k*n*Log[e + f*Sqrt[x]])/(4*e^4) - (b*n*Log[d*(e + f*Sqrt[x])^k])/(4*x^2) - (b*f^4*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^4 - (b*f^4*k*n*Log[x])/(8*e^4) + (b*f^4*k*n*Log[x]^2)/(8*e^4) - (f*k*(a + b*Log[c*x^n]))/(6*e*x^(3/2)) + (f^2*k*(a + b*Log[c*x^n]))/(4*e^2*x) - (f^3*k*(a + b*Log[c*x^n]))/(2*e^3*Sqrt[x]) + (f^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*e^4) - (Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(2*x^2) - (f^4*k*Log[x]*(a + b*Log[c*x^n]))/(4*e^4) - (b*f^4*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^4} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^4, x, 10, -((11*b*f*k*n)/(225*e*x^(5/2))) + (5*b*f^2*k*n)/(72*e^2*x^2) - (b*f^3*k*n)/(9*e^3*x^(3/2)) + (2*b*f^4*k*n)/(9*e^4*x) - (7*b*f^5*k*n)/(9*e^5*Sqrt[x]) + (b*f^6*k*n*Log[e + f*Sqrt[x]])/(9*e^6) - (b*n*Log[d*(e + f*Sqrt[x])^k])/(9*x^3) - (2*b*f^6*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^6) - (b*f^6*k*n*Log[x])/(18*e^6) + (b*f^6*k*n*Log[x]^2)/(12*e^6) - (f*k*(a + b*Log[c*x^n]))/(15*e*x^(5/2)) + (f^2*k*(a + b*Log[c*x^n]))/(12*e^2*x^2) - (f^3*k*(a + b*Log[c*x^n]))/(9*e^3*x^(3/2)) + (f^4*k*(a + b*Log[c*x^n]))/(6*e^4*x) - (f^5*k*(a + b*Log[c*x^n]))/(3*e^5*Sqrt[x]) + (f^6*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^6) - (Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(3*x^3) - (f^6*k*Log[x]*(a + b*Log[c*x^n]))/(6*e^6) - (2*b*f^6*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*e^6)} - - -{x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 22, (86*b^2*e^5*n^2*Sqrt[x])/(27*f^5) + (a*b*e^4*n*x)/(3*f^4) - (13*b^2*e^4*n^2*x)/(27*f^4) + (14*b^2*e^3*n^2*x^(3/2))/(81*f^3) - (19*b^2*e^2*n^2*x^2)/(216*f^2) + (182*b^2*e*n^2*x^(5/2))/(3375*f) - (1/27)*b^2*n^2*x^3 - (2*b^2*e^6*n^2*Log[e + f*Sqrt[x]])/(27*f^6) + (2/27)*b^2*n^2*x^3*Log[d*(e + f*Sqrt[x])] - (4*b^2*e^6*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(9*f^6) + (b^2*e^4*n*x*Log[c*x^n])/(3*f^4) - (14*b*e^5*n*Sqrt[x]*(a + b*Log[c*x^n]))/(9*f^5) + (b*e^4*n*x*(a + b*Log[c*x^n]))/(9*f^4) - (2*b*e^3*n*x^(3/2)*(a + b*Log[c*x^n]))/(9*f^3) + (5*b*e^2*n*x^2*(a + b*Log[c*x^n]))/(36*f^2) - (22*b*e*n*x^(5/2)*(a + b*Log[c*x^n]))/(225*f) + (2/27)*b*n*x^3*(a + b*Log[c*x^n]) + (2*b*e^6*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*f^6) - (2/9)*b*n*x^3*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (e^5*Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*f^5) - (e^4*x*(a + b*Log[c*x^n])^2)/(6*f^4) + (e^3*x^(3/2)*(a + b*Log[c*x^n])^2)/(9*f^3) - (e^2*x^2*(a + b*Log[c*x^n])^2)/(12*f^2) + (e*x^(5/2)*(a + b*Log[c*x^n])^2)/(15*f) - (1/18)*x^3*(a + b*Log[c*x^n])^2 + (1/3)*x^3*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (e^6*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(3*f^6) - (4*b^2*e^6*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/(9*f^6) - (4*b*e^6*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/(3*f^6) + (8*b^2*e^6*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/(3*f^6)} -{x^1*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 20, (21*b^2*e^3*n^2*Sqrt[x])/(4*f^3) + (a*b*e^2*n*x)/(2*f^2) - (7*b^2*e^2*n^2*x)/(8*f^2) + (37*b^2*e*n^2*x^(3/2))/(108*f) - (3/16)*b^2*n^2*x^2 - (b^2*e^4*n^2*Log[e + f*Sqrt[x]])/(4*f^4) + (1/4)*b^2*n^2*x^2*Log[d*(e + f*Sqrt[x])] - (b^2*e^4*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^4 + (b^2*e^2*n*x*Log[c*x^n])/(2*f^2) - (5*b*e^3*n*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) + (b*e^2*n*x*(a + b*Log[c*x^n]))/(4*f^2) - (7*b*e*n*x^(3/2)*(a + b*Log[c*x^n]))/(18*f) + (1/4)*b*n*x^2*(a + b*Log[c*x^n]) + (b*e^4*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*f^4) - (1/2)*b*n*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (e^3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(2*f^3) - (e^2*x*(a + b*Log[c*x^n])^2)/(4*f^2) + (e*x^(3/2)*(a + b*Log[c*x^n])^2)/(6*f) - (1/8)*x^2*(a + b*Log[c*x^n])^2 + (1/2)*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (e^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(2*f^4) - (b^2*e^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^4 - (2*b*e^4*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^4 + (4*b^2*e^4*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/f^4} -{x^0*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2, x, 18, (14*b^2*e*n^2*Sqrt[x])/f + a*b*n*x - 3*b^2*n^2*x - (2*b^2*e^2*n^2*Log[e + f*Sqrt[x]])/f^2 + 2*b^2*n^2*x*Log[d*(e + f*Sqrt[x])] - (4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + b^2*n*x*Log[c*x^n] - (6*b*e*n*Sqrt[x]*(a + b*Log[c*x^n]))/f + b*n*x*(a + b*Log[c*x^n]) + (2*b*e^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 - 2*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (e*Sqrt[x]*(a + b*Log[c*x^n])^2)/f - (1/2)*x*(a + b*Log[c*x^n])^2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 - (4*b^2*e^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 - (4*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 + (8*b^2*e^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/f^2} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^1, x, 5, (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/(3*b*n) - (Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(3*b*n) - 2*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)] + 8*b*n*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)] - 16*b^2*n^2*PolyLog[4, -((f*Sqrt[x])/e)]} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^2, x, 21, -((14*b^2*f*n^2)/(e*Sqrt[x])) + (2*b^2*f^2*n^2*Log[e + f*Sqrt[x]])/e^2 - (2*b^2*n^2*Log[d*(e + f*Sqrt[x])])/x - (4*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b^2*f^2*n^2*Log[x])/e^2 + (b^2*f^2*n^2*Log[x]^2)/(2*e^2) - (6*b*f*n*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (2*b*f^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 - (2*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (b*f^2*n*Log[x]*(a + b*Log[c*x^n]))/e^2 - (f*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(6*b*e^2*n) - (4*b^2*f^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^2 + (4*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (8*b^2*f^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/e^2} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2/x^3, x, 23, -((37*b^2*f*n^2)/(108*e*x^(3/2))) + (7*b^2*f^2*n^2)/(8*e^2*x) - (21*b^2*f^3*n^2)/(4*e^3*Sqrt[x]) + (b^2*f^4*n^2*Log[e + f*Sqrt[x]])/(4*e^4) - (b^2*n^2*Log[d*(e + f*Sqrt[x])])/(4*x^2) - (b^2*f^4*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^4 - (b^2*f^4*n^2*Log[x])/(8*e^4) + (b^2*f^4*n^2*Log[x]^2)/(8*e^4) - (7*b*f*n*(a + b*Log[c*x^n]))/(18*e*x^(3/2)) + (3*b*f^2*n*(a + b*Log[c*x^n]))/(4*e^2*x) - (5*b*f^3*n*(a + b*Log[c*x^n]))/(2*e^3*Sqrt[x]) + (b*f^4*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*e^4) - (b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(2*x^2) - (b*f^4*n*Log[x]*(a + b*Log[c*x^n]))/(4*e^4) - (f*(a + b*Log[c*x^n])^2)/(6*e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^2)/(4*e^2*x) - (f^3*(a + b*Log[c*x^n])^2)/(2*e^3*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(2*x^2) + (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(2*e^4) - (f^4*(a + b*Log[c*x^n])^3)/(12*b*e^4*n) - (b^2*f^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^4 + (2*b*f^4*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 - (4*b^2*f^4*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/e^4} - - -{x^1*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3, x, 36, -((255*b^3*e^3*n^3*Sqrt[x])/(8*f^3)) - (9*a*b^2*e^2*n^2*x)/(4*f^2) + (45*b^3*e^2*n^3*x)/(16*f^2) - (175*b^3*e*n^3*x^(3/2))/(216*f) + (3/8)*b^3*n^3*x^2 + (3*b^3*e^4*n^3*Log[e + f*Sqrt[x]])/(8*f^4) - (3/8)*b^3*n^3*x^2*Log[d*(e + f*Sqrt[x])] + (3*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(2*f^4) - (9*b^3*e^2*n^2*x*Log[c*x^n])/(4*f^2) + (63*b^2*e^3*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/(4*f^3) - (3*b^2*e^2*n^2*x*(a + b*Log[c*x^n]))/(8*f^2) + (37*b^2*e*n^2*x^(3/2)*(a + b*Log[c*x^n]))/(36*f) - (9/16)*b^2*n^2*x^2*(a + b*Log[c*x^n]) - (3*b^2*e^4*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*f^4) + (3/4)*b^2*n^2*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (15*b*e^3*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/(4*f^3) + (9*b*e^2*n*x*(a + b*Log[c*x^n])^2)/(8*f^2) - (7*b*e*n*x^(3/2)*(a + b*Log[c*x^n])^2)/(12*f) + (3/8)*b*n*x^2*(a + b*Log[c*x^n])^2 - (3/4)*b*n*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (3*b*e^4*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*f^4) + (e^3*Sqrt[x]*(a + b*Log[c*x^n])^3)/(2*f^3) - (e^2*x*(a + b*Log[c*x^n])^3)/(4*f^2) + (e*x^(3/2)*(a + b*Log[c*x^n])^3)/(6*f) - (1/8)*x^2*(a + b*Log[c*x^n])^3 + (1/2)*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (e^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(2*f^4) + (3*b^3*e^4*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*f^4) + (3*b^2*e^4*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^4 - (3*b*e^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^4 - (6*b^3*e^4*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/f^4 + (12*b^2*e^4*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^4 - (24*b^3*e^4*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^4} -{x^0*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3, x, 30, -((90*b^3*e*n^3*Sqrt[x])/f) - 6*a*b^2*n^2*x + 12*b^3*n^3*x + (6*b^3*e^2*n^3*Log[e + f*Sqrt[x]])/f^2 - 6*b^3*n^3*x*Log[d*(e + f*Sqrt[x])] + (12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 - 6*b^3*n^2*x*Log[c*x^n] + (42*b^2*e*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/f - 3*b^2*n^2*x*(a + b*Log[c*x^n]) - (6*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + 6*b^2*n^2*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (9*b*e*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/f + 3*b*n*x*(a + b*Log[c*x^n])^2 - 3*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (3*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 + (e*Sqrt[x]*(a + b*Log[c*x^n])^3)/f - (1/2)*x*(a + b*Log[c*x^n])^3 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/f^2 + (12*b^3*e^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 + (12*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 - (6*b*e^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 - (24*b^3*e^2*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 + (24*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 - (48*b^3*e^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^2} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^1, x, 6, (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^4)/(4*b*n) - (Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^4)/(4*b*n) - 2*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*Sqrt[x])/e)] + 12*b*n*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*Sqrt[x])/e)] - 48*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[4, -((f*Sqrt[x])/e)] + 96*b^3*n^3*PolyLog[5, -((f*Sqrt[x])/e)]} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^2, x, 34, -((90*b^3*f*n^3)/(e*Sqrt[x])) + (6*b^3*f^2*n^3*Log[e + f*Sqrt[x]])/e^2 - (6*b^3*n^3*Log[d*(e + f*Sqrt[x])])/x - (12*b^3*f^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (3*b^3*f^2*n^3*Log[x])/e^2 + (3*b^3*f^2*n^3*Log[x]^2)/(2*e^2) - (42*b^2*f*n^2*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (6*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 - (6*b^2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (3*b^2*f^2*n^2*Log[x]*(a + b*Log[c*x^n]))/e^2 - (9*b*f*n*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x + (3*b*f^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(2*e^2) - (f*(a + b*Log[c*x^n])^3)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/e^2 - (f^2*(a + b*Log[c*x^n])^4)/(8*b*e^2*n) - (12*b^3*f^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/e^2 + (12*b^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 + (6*b*f^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (24*b^3*f^2*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 - (24*b^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 + (48*b^3*f^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^2} -{Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3/x^3, x, 40, -((175*b^3*f*n^3)/(216*e*x^(3/2))) + (45*b^3*f^2*n^3)/(16*e^2*x) - (255*b^3*f^3*n^3)/(8*e^3*Sqrt[x]) + (3*b^3*f^4*n^3*Log[e + f*Sqrt[x]])/(8*e^4) - (3*b^3*n^3*Log[d*(e + f*Sqrt[x])])/(8*x^2) - (3*b^3*f^4*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(2*e^4) - (3*b^3*f^4*n^3*Log[x])/(16*e^4) + (3*b^3*f^4*n^3*Log[x]^2)/(16*e^4) - (37*b^2*f*n^2*(a + b*Log[c*x^n]))/(36*e*x^(3/2)) + (21*b^2*f^2*n^2*(a + b*Log[c*x^n]))/(8*e^2*x) - (63*b^2*f^3*n^2*(a + b*Log[c*x^n]))/(4*e^3*Sqrt[x]) + (3*b^2*f^4*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) - (3*b^2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) - (3*b^2*f^4*n^2*Log[x]*(a + b*Log[c*x^n]))/(8*e^4) - (7*b*f*n*(a + b*Log[c*x^n])^2)/(12*e*x^(3/2)) + (9*b*f^2*n*(a + b*Log[c*x^n])^2)/(8*e^2*x) - (15*b*f^3*n*(a + b*Log[c*x^n])^2)/(4*e^3*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) + (3*b*f^4*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) - (f^4*(a + b*Log[c*x^n])^3)/(8*e^4) - (f*(a + b*Log[c*x^n])^3)/(6*e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4*e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/(2*x^2) + (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(2*e^4) - (f^4*(a + b*Log[c*x^n])^4)/(16*b*e^4*n) - (3*b^3*f^4*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) + (3*b^2*f^4*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 + (3*b*f^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 - (6*b^3*f^4*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^4 - (12*b^2*f^4*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^4 + (24*b^3*f^4*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^(q/2) Log[d (e+f x^m)^k] (a+b Log[c x^n])*) - - -{x^(3/2)*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]), x, 9, (24*b*e^4*k*n*Sqrt[x])/(25*f^4) - (7*b*e^3*k*n*x)/(25*f^3) + (32*b*e^2*k*n*x^(3/2))/(225*f^2) - (9*b*e*k*n*x^2)/(100*f) + (8/125)*b*k*n*x^(5/2) - (4*b*e^5*k*n*Log[e + f*Sqrt[x]])/(25*f^5) - (4/25)*b*n*x^(5/2)*Log[d*(e + f*Sqrt[x])^k] - (4*b*e^5*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(5*f^5) - (2*e^4*k*Sqrt[x]*(a + b*Log[c*x^n]))/(5*f^4) + (e^3*k*x*(a + b*Log[c*x^n]))/(5*f^3) - (2*e^2*k*x^(3/2)*(a + b*Log[c*x^n]))/(15*f^2) + (e*k*x^2*(a + b*Log[c*x^n]))/(10*f) - (2/25)*k*x^(5/2)*(a + b*Log[c*x^n]) + (2*e^5*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(5*f^5) + (2/5)*x^(5/2)*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) - (4*b*e^5*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(5*f^5)} -{x^(1/2)*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]), x, 9, (16*b*e^2*k*n*Sqrt[x])/(9*f^2) - (5*b*e*k*n*x)/(9*f) + (8/27)*b*k*n*x^(3/2) - (4*b*e^3*k*n*Log[e + f*Sqrt[x]])/(9*f^3) - (4/9)*b*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] - (4*b*e^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*f^3) - (2*e^2*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^2) + (e*k*x*(a + b*Log[c*x^n]))/(3*f) - (2/9)*k*x^(3/2)*(a + b*Log[c*x^n]) + (2*e^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^3) + (2/3)*x^(3/2)*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) - (4*b*e^3*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^3)} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^(3/2), x, 11, -((4*b*f*k*n*Log[e + f*Sqrt[x]])/e) - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/Sqrt[x] + (4*b*f*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e + (2*b*f*k*n*Log[x])/e - (b*f*k*n*Log[x]^2)/(2*e) - (2*f*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/Sqrt[x] + (f*k*Log[x]*(a + b*Log[c*x^n]))/e + (4*b*f*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/e} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^(5/2), x, 10, -((5*b*f*k*n)/(9*e*x)) + (16*b*f^2*k*n)/(9*e^2*Sqrt[x]) - (4*b*f^3*k*n*Log[e + f*Sqrt[x]])/(9*e^3) - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/(9*x^(3/2)) + (4*b*f^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^3) + (2*b*f^3*k*n*Log[x])/(9*e^3) - (b*f^3*k*n*Log[x]^2)/(6*e^3) - (f*k*(a + b*Log[c*x^n]))/(3*e*x) + (2*f^2*k*(a + b*Log[c*x^n]))/(3*e^2*Sqrt[x]) - (2*f^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^3) - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(3*x^(3/2)) + (f^3*k*Log[x]*(a + b*Log[c*x^n]))/(3*e^3) + (4*b*f^3*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*e^3)} -{Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])/x^(7/2), x, 10, -((9*b*f*k*n)/(100*e*x^2)) + (32*b*f^2*k*n)/(225*e^2*x^(3/2)) - (7*b*f^3*k*n)/(25*e^3*x) + (24*b*f^4*k*n)/(25*e^4*Sqrt[x]) - (4*b*f^5*k*n*Log[e + f*Sqrt[x]])/(25*e^5) - (4*b*n*Log[d*(e + f*Sqrt[x])^k])/(25*x^(5/2)) + (4*b*f^5*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(5*e^5) + (2*b*f^5*k*n*Log[x])/(25*e^5) - (b*f^5*k*n*Log[x]^2)/(10*e^5) - (f*k*(a + b*Log[c*x^n]))/(10*e*x^2) + (2*f^2*k*(a + b*Log[c*x^n]))/(15*e^2*x^(3/2)) - (f^3*k*(a + b*Log[c*x^n]))/(5*e^3*x) + (2*f^4*k*(a + b*Log[c*x^n]))/(5*e^4*Sqrt[x]) - (2*f^5*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(5*e^5) - (2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(5*x^(5/2)) + (f^5*k*Log[x]*(a + b*Log[c*x^n]))/(5*e^5) + (4*b*f^5*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(5*e^5)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[d (e+f x^m)^k] (a+b Log[c x^n]) with m symbolic*) - - -{(g*x)^q*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 0, Unintegrable[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]} - - -{Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^3/x, x, 6, ((a + b*Log[c*x^n])^4*Log[d*(e + f*x^m)^r])/(4*b*n) - (r*(a + b*Log[c*x^n])^4*Log[1 + (f*x^m)/e])/(4*b*n) - (r*(a + b*Log[c*x^n])^3*PolyLog[2, -((f*x^m)/e)])/m + (3*b*n*r*(a + b*Log[c*x^n])^2*PolyLog[3, -((f*x^m)/e)])/m^2 - (6*b^2*n^2*r*(a + b*Log[c*x^n])*PolyLog[4, -((f*x^m)/e)])/m^3 + (6*b^3*n^3*r*PolyLog[5, -((f*x^m)/e)])/m^4} -{Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^2/x, x, 5, ((a + b*Log[c*x^n])^3*Log[d*(e + f*x^m)^r])/(3*b*n) - (r*(a + b*Log[c*x^n])^3*Log[1 + (f*x^m)/e])/(3*b*n) - (r*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*x^m)/e)])/m + (2*b*n*r*(a + b*Log[c*x^n])*PolyLog[3, -((f*x^m)/e)])/m^2 - (2*b^2*n^2*r*PolyLog[4, -((f*x^m)/e)])/m^3} -{Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^1/x, x, 4, ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^m)^r])/(2*b*n) - (r*(a + b*Log[c*x^n])^2*Log[1 + (f*x^m)/e])/(2*b*n) - (r*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^m)/e)])/m + (b*n*r*PolyLog[3, -((f*x^m)/e)])/m^2} -{Log[d*(e + f*x^m)^r]/(x*(a + b*Log[c*x^n])^1), x, 0, Unintegrable[Log[d*(e + f*x^m)^r]/(x*(a + b*Log[c*x^n])), x]} -{Log[d*(e + f*x^m)^r]/(x*(a + b*Log[c*x^n])^2), x, 0, Unintegrable[Log[d*(e + f*x^m)^r]/(x*(a + b*Log[c*x^n])^2), x]} - - -{x^2*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 0, Unintegrable[x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]} -{x^1*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 0, Unintegrable[x*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]} -{x^0*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 0, Unintegrable[(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]} -{Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n])/x^1, x, 4, ((a + b*Log[c*x^n])^2*Log[d*(e + f*x^m)^k])/(2*b*n) - (k*(a + b*Log[c*x^n])^2*Log[1 + (f*x^m)/e])/(2*b*n) - (k*(a + b*Log[c*x^n])*PolyLog[2, -((f*x^m)/e)])/m + (b*k*n*PolyLog[3, -((f*x^m)/e)])/m^2} -{Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n])/x^2, x, 0, Unintegrable[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^2, x]} -{Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n])/x^3, x, 0, Unintegrable[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3, x]} - - -{(g*x)^(3*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 18, (2*b*k*n*(g*x)^(3*m))/(27*g*m^2) + (4*b*e^2*k*n*(g*x)^(3*m))/(x^(2*m)*(9*f^2*g*m^2)) - (5*b*e*k*n*(g*x)^(3*m))/(x^m*(36*f*g*m^2)) - (k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(9*g*m) - (e^2*k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(x^(2*m)*(3*f^2*g*m)) + (e*k*(g*x)^(3*m)*(a + b*Log[c*x^n]))/(x^m*(6*f*g*m)) - (b*e^3*k*n*(g*x)^(3*m)*Log[e + f*x^m])/(x^(3*m)*(9*f^3*g*m^2)) - (b*e^3*k*n*(g*x)^(3*m)*Log[-((f*x^m)/e)]*Log[e + f*x^m])/(x^(3*m)*(3*f^3*g*m^2)) + (e^3*k*(g*x)^(3*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/(x^(3*m)*(3*f^3*g*m)) - (b*n*(g*x)^(3*m)*Log[d*(e + f*x^m)^k])/(9*g*m^2) + ((g*x)^(3*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(3*g*m) - (b*e^3*k*n*(g*x)^(3*m)*PolyLog[2, 1 + (f*x^m)/e])/(x^(3*m)*(3*f^3*g*m^2))} -{(g*x)^(2*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 16, (b*k*n*(g*x)^(2*m))/(4*g*m^2) - (3*b*e*k*n*(g*x)^(2*m))/(x^m*(4*f*g*m^2)) - (k*(g*x)^(2*m)*(a + b*Log[c*x^n]))/(4*g*m) + (e*k*(g*x)^(2*m)*(a + b*Log[c*x^n]))/(x^m*(2*f*g*m)) + (b*e^2*k*n*(g*x)^(2*m)*Log[e + f*x^m])/(x^(2*m)*(4*f^2*g*m^2)) + (b*e^2*k*n*(g*x)^(2*m)*Log[-((f*x^m)/e)]*Log[e + f*x^m])/(x^(2*m)*(2*f^2*g*m^2)) - (e^2*k*(g*x)^(2*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/(x^(2*m)*(2*f^2*g*m)) - (b*n*(g*x)^(2*m)*Log[d*(e + f*x^m)^k])/(4*g*m^2) + ((g*x)^(2*m)*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(2*g*m) + (b*e^2*k*n*(g*x)^(2*m)*PolyLog[2, 1 + (f*x^m)/e])/(x^(2*m)*(2*f^2*g*m^2))} -{(g*x)^(1*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 14, (2*b*k*n*(g*x)^m)/(g*m^2) - (k*(g*x)^m*(a + b*Log[c*x^n]))/(g*m) - (b*e*k*n*(g*x)^m*Log[e + f*x^m])/(x^m*(f*g*m^2)) - (b*e*k*n*(g*x)^m*Log[-((f*x^m)/e)]*Log[e + f*x^m])/(x^m*(f*g*m^2)) + (e*k*(g*x)^m*(a + b*Log[c*x^n])*Log[e + f*x^m])/(x^m*(f*g*m)) - (b*n*(g*x)^m*Log[d*(e + f*x^m)^k])/(g*m^2) + ((g*x)^m*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/(g*m) - (b*e*k*n*(g*x)^m*PolyLog[2, 1 + (f*x^m)/e])/(x^m*(f*g*m^2))} -{(g*x)^(-1*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 15, (b*f*k*n*x^m*Log[x])/((g*x)^m*(e*g*m)) - (b*f*k*n*x^m*Log[x]^2)/((g*x)^m*(2*e*g)) + (f*k*x^m*Log[x]*(a + b*Log[c*x^n]))/((g*x)^m*(e*g)) - (b*f*k*n*x^m*Log[e + f*x^m])/((g*x)^m*(e*g*m^2)) + (b*f*k*n*x^m*Log[-((f*x^m)/e)]*Log[e + f*x^m])/((g*x)^m*(e*g*m^2)) - (f*k*x^m*(a + b*Log[c*x^n])*Log[e + f*x^m])/((g*x)^m*(e*g*m)) - (b*n*Log[d*(e + f*x^m)^k])/((g*x)^m*(g*m^2)) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/((g*x)^m*(g*m)) + (b*f*k*n*x^m*PolyLog[2, 1 + (f*x^m)/e])/((g*x)^m*(e*g*m^2))} -{(g*x)^(-2*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 16, -((3*b*f*k*n*x^m)/((g*x)^(2*m)*(4*e*g*m^2))) - (b*f^2*k*n*x^(2*m)*Log[x])/((g*x)^(2*m)*(4*e^2*g*m)) + (b*f^2*k*n*x^(2*m)*Log[x]^2)/((g*x)^(2*m)*(4*e^2*g)) - (f*k*x^m*(a + b*Log[c*x^n]))/((g*x)^(2*m)*(2*e*g*m)) - (f^2*k*x^(2*m)*Log[x]*(a + b*Log[c*x^n]))/((g*x)^(2*m)*(2*e^2*g)) + (b*f^2*k*n*x^(2*m)*Log[e + f*x^m])/((g*x)^(2*m)*(4*e^2*g*m^2)) - (b*f^2*k*n*x^(2*m)*Log[-((f*x^m)/e)]*Log[e + f*x^m])/((g*x)^(2*m)*(2*e^2*g*m^2)) + (f^2*k*x^(2*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/((g*x)^(2*m)*(2*e^2*g*m)) - (b*n*Log[d*(e + f*x^m)^k])/((g*x)^(2*m)*(4*g*m^2)) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/((g*x)^(2*m)*(2*g*m)) - (b*f^2*k*n*x^(2*m)*PolyLog[2, 1 + (f*x^m)/e])/((g*x)^(2*m)*(2*e^2*g*m^2))} -{(g*x)^(-3*m - 1)*Log[d*(e + f*x^m)^k]*(a + b*Log[c*x^n]), x, 18, -((5*b*f*k*n*x^m)/((g*x)^(3*m)*(36*e*g*m^2))) + (4*b*f^2*k*n*x^(2*m))/((g*x)^(3*m)*(9*e^2*g*m^2)) + (b*f^3*k*n*x^(3*m)*Log[x])/((g*x)^(3*m)*(9*e^3*g*m)) - (b*f^3*k*n*x^(3*m)*Log[x]^2)/((g*x)^(3*m)*(6*e^3*g)) - (f*k*x^m*(a + b*Log[c*x^n]))/((g*x)^(3*m)*(6*e*g*m)) + (f^2*k*x^(2*m)*(a + b*Log[c*x^n]))/((g*x)^(3*m)*(3*e^2*g*m)) + (f^3*k*x^(3*m)*Log[x]*(a + b*Log[c*x^n]))/((g*x)^(3*m)*(3*e^3*g)) - (b*f^3*k*n*x^(3*m)*Log[e + f*x^m])/((g*x)^(3*m)*(9*e^3*g*m^2)) + (b*f^3*k*n*x^(3*m)*Log[-((f*x^m)/e)]*Log[e + f*x^m])/((g*x)^(3*m)*(3*e^3*g*m^2)) - (f^3*k*x^(3*m)*(a + b*Log[c*x^n])*Log[e + f*x^m])/((g*x)^(3*m)*(3*e^3*g*m)) - (b*n*Log[d*(e + f*x^m)^k])/((g*x)^(3*m)*(9*g*m^2)) - ((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/((g*x)^(3*m)*(3*g*m)) + (b*f^3*k*n*x^(3*m)*PolyLog[2, 1 + (f*x^m)/e])/((g*x)^(3*m)*(3*e^3*g*m^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g x)^m (a+b Log[c x^n])^p (d+e Log[f x^r])^q*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g x)^m (a+b Log[c x^n])^p (d+e Log[f x^r])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]), x, 3, (1/27)*b*e*n*r*x^3 - (1/27)*e*r*x^3*(3*a - b*n + 3*b*Log[c*x^n]) - (1/9)*b*n*x^3*(d + e*Log[f*x^r]) + (1/3)*x^3*(a + b*Log[c*x^n])*(d + e*Log[f*x^r])} -{x^1*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]), x, 3, (1/8)*b*e*n*r*x^2 - (1/8)*e*r*x^2*(2*a - b*n + 2*b*Log[c*x^n]) - (1/4)*b*n*x^2*(d + e*Log[f*x^r]) + (1/2)*x^2*(a + b*Log[c*x^n])*(d + e*Log[f*x^r])} -{x^0*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]), x, 3, b*e*n*r*x - e*(a - b*n)*r*x - b*e*r*x*Log[c*x^n] + a*x*(d + e*Log[f*x^r]) - b*n*x*(d + e*Log[f*x^r]) + b*x*Log[c*x^n]*(d + e*Log[f*x^r])} -{(a + b*Log[c*x^n])*(d + e*Log[f*x^r])/x^1, x, 4, -((e*r*(a + b*Log[c*x^n])^3)/(6*b^2*n^2)) + ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/(2*b*n)} -{(a + b*Log[c*x^n])*(d + e*Log[f*x^r])/x^2, x, 2, -((b*e*n*r)/x) - (e*r*(a + b*n + b*Log[c*x^n]))/x - (b*n*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x} -{(a + b*Log[c*x^n])*(d + e*Log[f*x^r])/x^3, x, 3, -((b*e*n*r)/(8*x^2)) - (e*r*(2*a + b*n + 2*b*Log[c*x^n]))/(8*x^2) - (b*n*(d + e*Log[f*x^r]))/(4*x^2) - ((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(2*x^2)} -{(a + b*Log[c*x^n])*(d + e*Log[f*x^r])/x^4, x, 3, -((b*e*n*r)/(27*x^3)) - (e*r*(3*a + b*n + 3*b*Log[c*x^n]))/(27*x^3) - (b*n*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(3*x^3)} - - -{x^2*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]), x, 7, (-(2/81))*b^2*e*n^2*r*x^3 + (2/81)*b*e*n*(3*a - b*n)*r*x^3 - (1/81)*e*(9*a^2 - 6*a*b*n + 2*b^2*n^2)*r*x^3 + (2/27)*b^2*e*n*r*x^3*Log[c*x^n] - (2/27)*b*e*(3*a - b*n)*r*x^3*Log[c*x^n] - (1/9)*b^2*e*r*x^3*Log[c*x^n]^2 + (2/27)*b^2*n^2*x^3*(d + e*Log[f*x^r]) - (2/9)*b*n*x^3*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]) + (1/3)*x^3*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])} -{x^1*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]), x, 7, (-(1/8))*b^2*e*n^2*r*x^2 + (1/8)*b*e*n*(2*a - b*n)*r*x^2 - (1/8)*e*(2*a^2 - 2*a*b*n + b^2*n^2)*r*x^2 + (1/4)*b^2*e*n*r*x^2*Log[c*x^n] - (1/4)*b*e*(2*a - b*n)*r*x^2*Log[c*x^n] - (1/4)*b^2*e*r*x^2*Log[c*x^n]^2 + (1/4)*b^2*n^2*x^2*(d + e*Log[f*x^r]) - (1/2)*b*n*x^2*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]) + (1/2)*x^2*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])} -{x^0*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]), x, 6, 2*a*b*e*n*r*x - 4*b^2*e*n^2*r*x + 2*b*e*n*(a - b*n)*r*x + 4*b^2*e*n*r*x*Log[c*x^n] - e*r*x*(a + b*Log[c*x^n])^2 - 2*a*b*n*x*(d + e*Log[f*x^r]) + 2*b^2*n^2*x*(d + e*Log[f*x^r]) - 2*b^2*n*x*Log[c*x^n]*(d + e*Log[f*x^r]) + x*(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])} -{(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])/x^1, x, 4, -((e*r*(a + b*Log[c*x^n])^4)/(12*b^2*n^2)) + ((a + b*Log[c*x^n])^3*(d + e*Log[f*x^r]))/(3*b*n)} -{(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])/x^2, x, 6, -((2*b^2*e*n^2*r)/x) - (2*b*e*n*(a + b*n)*r)/x - (e*(a^2 + 2*a*b*n + 2*b^2*n^2)*r)/x - (2*b^2*e*n*r*Log[c*x^n])/x - (2*b*e*(a + b*n)*r*Log[c*x^n])/x - (b^2*e*r*Log[c*x^n]^2)/x - (2*b^2*n^2*(d + e*Log[f*x^r]))/x - (2*b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x} -{(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])/x^3, x, 7, -((b^2*e*n^2*r)/(8*x^2)) - (b*e*n*(2*a + b*n)*r)/(8*x^2) - (e*(2*a^2 + 2*a*b*n + b^2*n^2)*r)/(8*x^2) - (b^2*e*n*r*Log[c*x^n])/(4*x^2) - (b*e*(2*a + b*n)*r*Log[c*x^n])/(4*x^2) - (b^2*e*r*Log[c*x^n]^2)/(4*x^2) - (b^2*n^2*(d + e*Log[f*x^r]))/(4*x^2) - (b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(2*x^2) - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/(2*x^2)} -{(a + b*Log[c*x^n])^2*(d + e*Log[f*x^r])/x^4, x, 7, -((2*b^2*e*n^2*r)/(81*x^3)) - (2*b*e*n*(3*a + b*n)*r)/(81*x^3) - (e*(9*a^2 + 6*a*b*n + 2*b^2*n^2)*r)/(81*x^3) - (2*b^2*e*n*r*Log[c*x^n])/(27*x^3) - (2*b*e*(3*a + b*n)*r*Log[c*x^n])/(27*x^3) - (b^2*e*r*Log[c*x^n]^2)/(9*x^3) - (2*b^2*n^2*(d + e*Log[f*x^r]))/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^2*(a + b*Log[c*x^n])/(d + e*Log[f*x^m]), x, 6, (b*n*x^3)/(3*e*m) - (b*n*x^3*ExpIntegralEi[(3*(d + e*Log[f*x^m]))/(e*m)]*(d + e*Log[f*x^m]))/(E^((3*d)/(e*m))*(f*x^m)^(3/m)*(e^2*m^2)) + (x^3*ExpIntegralEi[(3*(d + e*Log[f*x^m]))/(e*m)]*(a + b*Log[c*x^n]))/(E^((3*d)/(e*m))*(f*x^m)^(3/m)*(e*m))} -{x^1*(a + b*Log[c*x^n])/(d + e*Log[f*x^m]), x, 6, (b*n*x^2)/(2*e*m) - (b*n*x^2*ExpIntegralEi[(2*(d + e*Log[f*x^m]))/(e*m)]*(d + e*Log[f*x^m]))/(E^((2*d)/(e*m))*(f*x^m)^(2/m)*(e^2*m^2)) + (x^2*ExpIntegralEi[(2*(d + e*Log[f*x^m]))/(e*m)]*(a + b*Log[c*x^n]))/(E^((2*d)/(e*m))*(f*x^m)^(2/m)*(e*m))} -{x^0*(a + b*Log[c*x^n])/(d + e*Log[f*x^m]), x, 6, (b*n*x)/(e*m) - (b*n*x*ExpIntegralEi[(d + e*Log[f*x^m])/(e*m)]*(d + e*Log[f*x^m]))/(E^(d/(e*m))*(f*x^m)^m^(-1)*(e^2*m^2)) + (x*ExpIntegralEi[(d + e*Log[f*x^m])/(e*m)]*(a + b*Log[c*x^n]))/(E^(d/(e*m))*(f*x^m)^m^(-1)*(e*m))} -{(a + b*Log[c*x^n])/(x^1*(d + e*Log[f*x^m])), x, 5, (b*n*Log[x])/(e*m) - (b*n*(d + e*Log[f*x^m])*Log[d + e*Log[f*x^m]])/(e^2*m^2) + ((a + b*Log[c*x^n])*Log[d + e*Log[f*x^m]])/(e*m)} -{(a + b*Log[c*x^n])/(x^2*(d + e*Log[f*x^m])), x, 6, -((b*n)/(e*m*x)) - (b*E^(d/(e*m))*n*(f*x^m)^(1/m)*ExpIntegralEi[-((d + e*Log[f*x^m])/(e*m))]*(d + e*Log[f*x^m]))/(e^2*m^2*x) + (E^(d/(e*m))*(f*x^m)^(1/m)*ExpIntegralEi[-((d + e*Log[f*x^m])/(e*m))]*(a + b*Log[c*x^n]))/(e*m*x)} -{(a + b*Log[c*x^n])/(x^3*(d + e*Log[f*x^m])), x, 6, -((b*n)/(2*e*m*x^2)) - (b*E^((2*d)/(e*m))*n*(f*x^m)^(2/m)*ExpIntegralEi[-((2*(d + e*Log[f*x^m]))/(e*m))]*(d + e*Log[f*x^m]))/(e^2*m^2*x^2) + (E^((2*d)/(e*m))*(f*x^m)^(2/m)*ExpIntegralEi[-((2*(d + e*Log[f*x^m]))/(e*m))]*(a + b*Log[c*x^n]))/(e*m*x^2)} - - -{(a + b*Log[c*x^n])/(d + e*Log[c*x^n])^2, x, 7, (((-b)*d + a*e + b*e*n)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/(e*n))*(c*x^n)^n^(-1)*(e^3*n^2)) + ((b*d - a*e)*x)/(e^2*n*(d + e*Log[c*x^n])), -(((b*d - a*e)*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/(e*n))*(c*x^n)^n^(-1)*(e^3*n^2))) + (b*x*ExpIntegralEi[(d + e*Log[c*x^n])/(e*n)])/(E^(d/(e*n))*(c*x^n)^n^(-1)*(e^2*n)) + ((b*d - a*e)*x)/(e^2*n*(d + e*Log[c*x^n]))} - - -{(a + b*Log[c*x^n])/(x*Log[x]), x, 2, b*n*Log[x] - b*n*Log[x]*Log[Log[x]] + (a + b*Log[c*x^n])*Log[Log[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g x)^m (a+b Log[c x^n])^p (d+e Log[f x^r]) with p symbolic*) - - -{(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]), x, 8, -((e*r*x*(g*x)^m*Gamma[2 + p, -((a*(1 + m))/(b*n)) - ((1 + m)*Log[c*x^n])/n]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(1 + m)^2)) - (e*r*x*(g*x)^m*Gamma[1 + p, -((a*(1 + m))/(b*n)) - ((1 + m)*Log[c*x^n])/n]*(a + b*Log[c*x^n])^(1 + p))/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(b*(1 + m)*n)) + ((g*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p*(g*(1 + m)))} - - -{x^2*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]), x, 7, ((-3^(-2 - p))*e*r*x^3*Gamma[2 + p, -((3*a)/(b*n)) - (3*Log[c*x^n])/n]*(a + b*Log[c*x^n])^p)/(E^((3*a)/(b*n))*(c*x^n)^(3/n)*(-((a + b*Log[c*x^n])/(b*n)))^p) - (3^(-1 - p)*e*r*x^3*Gamma[1 + p, -((3*a)/(b*n)) - (3*Log[c*x^n])/n]*(a + b*Log[c*x^n])^(1 + p))/(E^((3*a)/(b*n))*(c*x^n)^(3/n)*(-((a + b*Log[c*x^n])/(b*n)))^p*(b*n)) + (3^(-1 - p)*x^3*Gamma[1 + p, -((3*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(E^((3*a)/(b*n))*(c*x^n)^(3/n)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{x^1*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]), x, 7, ((-2^(-2 - p))*e*r*x^2*Gamma[2 + p, -((2*a)/(b*n)) - (2*Log[c*x^n])/n]*(a + b*Log[c*x^n])^p)/(E^((2*a)/(b*n))*(c*x^n)^(2/n)*(-((a + b*Log[c*x^n])/(b*n)))^p) - (2^(-1 - p)*e*r*x^2*Gamma[1 + p, -((2*a)/(b*n)) - (2*Log[c*x^n])/n]*(a + b*Log[c*x^n])^(1 + p))/(E^((2*a)/(b*n))*(c*x^n)^(2/n)*(-((a + b*Log[c*x^n])/(b*n)))^p*(b*n)) + (2^(-1 - p)*x^2*Gamma[1 + p, -((2*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(E^((2*a)/(b*n))*(c*x^n)^(2/n)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{x^0*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]), x, 7, ((-e)*r*x*Gamma[2 + p, -(a/(b*n)) - Log[c*x^n]/n]*(a + b*Log[c*x^n])^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c*x^n])/(b*n)))^p) - (e*r*x*Gamma[1 + p, -(a/(b*n)) - Log[c*x^n]/n]*(a + b*Log[c*x^n])^(1 + p))/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c*x^n])/(b*n)))^p*(b*n)) + (x*Gamma[1 + p, -((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c*x^n])/(b*n)))^p)} -{(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])/x^1, x, 4, -((e*r*(a + b*Log[c*x^n])^(2 + p))/(b^2*n^2*(1 + p)*(2 + p))) + ((a + b*Log[c*x^n])^(1 + p)*(d + e*Log[f*x^r]))/(b*n*(1 + p))} -{(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])/x^2, x, 7, -((e*E^(a/(b*n))*r*(c*x^n)^(1/n)*Gamma[2 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x)) + (e*E^(a/(b*n))*r*(c*x^n)^(1/n)*Gamma[1 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^(1 + p))/(((a + b*Log[c*x^n])/(b*n))^p*(b*n*x)) - (E^(a/(b*n))*(c*x^n)^(1/n)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(((a + b*Log[c*x^n])/(b*n))^p*x)} -{(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])/x^3, x, 7, -((2^(-2 - p)*e*E^((2*a)/(b*n))*r*(c*x^n)^(2/n)*Gamma[2 + p, (2*a)/(b*n) + (2*Log[c*x^n])/n]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x^2)) + (2^(-1 - p)*e*E^((2*a)/(b*n))*r*(c*x^n)^(2/n)*Gamma[1 + p, (2*a)/(b*n) + (2*Log[c*x^n])/n]*(a + b*Log[c*x^n])^(1 + p))/(((a + b*Log[c*x^n])/(b*n))^p*(b*n*x^2)) - (2^(-1 - p)*E^((2*a)/(b*n))*(c*x^n)^(2/n)*Gamma[1 + p, (2*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(((a + b*Log[c*x^n])/(b*n))^p*x^2)} -{(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r])/x^4, x, 7, -((3^(-2 - p)*e*E^((3*a)/(b*n))*r*(c*x^n)^(3/n)*Gamma[2 + p, (3*a)/(b*n) + (3*Log[c*x^n])/n]*(a + b*Log[c*x^n])^p)/(((a + b*Log[c*x^n])/(b*n))^p*x^3)) + (3^(-1 - p)*e*E^((3*a)/(b*n))*r*(c*x^n)^(3/n)*Gamma[1 + p, (3*a)/(b*n) + (3*Log[c*x^n])/n]*(a + b*Log[c*x^n])^(1 + p))/(((a + b*Log[c*x^n])/(b*n))^p*(b*n*x^3)) - (3^(-1 - p)*E^((3*a)/(b*n))*(c*x^n)^(3/n)*Gamma[1 + p, (3*(a + b*Log[c*x^n]))/(b*n)]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(((a + b*Log[c*x^n])/(b*n))^p*x^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Pq[x] ArcTrig[d (e +x)]^m (a+b Log[c x^n])*) - - -{(d + e*x^2)*ArcSin[a*x]*Log[c*x^n], x, 17, -((d*n*Sqrt[1 - a^2*x^2])/a) - ((3*a^2*d + e)*n*Sqrt[1 - a^2*x^2])/(3*a^3) + (2*e*n*(1 - a^2*x^2)^(3/2))/(27*a^3) - d*n*x*ArcSin[a*x] - (1/9)*e*n*x^3*ArcSin[a*x] - (e*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) + ((3*a^2*d + e)*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(3*a^3) + ((3*a^2*d + e)*Sqrt[1 - a^2*x^2]*Log[c*x^n])/(3*a^3) - (e*(1 - a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcSin[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcSin[a*x]*Log[c*x^n]} -{(d + e*x^2)*ArcCos[a*x]*Log[c*x^n], x, 17, (d*n*Sqrt[1 - a^2*x^2])/a + ((3*a^2*d + e)*n*Sqrt[1 - a^2*x^2])/(3*a^3) - (2*e*n*(1 - a^2*x^2)^(3/2))/(27*a^3) - d*n*x*ArcCos[a*x] - (1/9)*e*n*x^3*ArcCos[a*x] + (e*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) - ((3*a^2*d + e)*n*ArcTanh[Sqrt[1 - a^2*x^2]])/(3*a^3) - ((3*a^2*d + e)*Sqrt[1 - a^2*x^2]*Log[c*x^n])/(3*a^3) + (e*(1 - a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcCos[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcCos[a*x]*Log[c*x^n]} -{(d + e*x^2)*ArcTan[a*x]*Log[c*x^n], x, 9, (5*e*n*x^2)/(36*a) - d*n*x*ArcTan[a*x] - (1/9)*e*n*x^3*ArcTan[a*x] - (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcTan[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcTan[a*x]*Log[c*x^n] + (d*n*Log[1 + a^2*x^2])/(2*a) - (e*n*Log[1 + a^2*x^2])/(18*a^3) - ((3*a^2*d - e)*Log[c*x^n]*Log[1 + a^2*x^2])/(6*a^3) - ((3*a^2*d - e)*n*PolyLog[2, (-a^2)*x^2])/(12*a^3)} -{(d + e*x^2)*ArcCot[a*x]*Log[c*x^n], x, 9, -((5*e*n*x^2)/(36*a)) - d*n*x*ArcCot[a*x] - (1/9)*e*n*x^3*ArcCot[a*x] + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCot[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcCot[a*x]*Log[c*x^n] - (d*n*Log[1 + a^2*x^2])/(2*a) + (e*n*Log[1 + a^2*x^2])/(18*a^3) + ((3*a^2*d - e)*Log[c*x^n]*Log[1 + a^2*x^2])/(6*a^3) + ((3*a^2*d - e)*n*PolyLog[2, (-a^2)*x^2])/(12*a^3)} - -{(d + e*x^2)*ArcSinh[a*x]*Log[c*x^n], x, 17, (d*n*Sqrt[1 + a^2*x^2])/a + ((3*a^2*d - e)*n*Sqrt[1 + a^2*x^2])/(3*a^3) + (2*e*n*(1 + a^2*x^2)^(3/2))/(27*a^3) - d*n*x*ArcSinh[a*x] - (1/9)*e*n*x^3*ArcSinh[a*x] - ((3*a^2*d - e)*n*ArcTanh[Sqrt[1 + a^2*x^2]])/(3*a^3) - (e*n*ArcTanh[Sqrt[1 + a^2*x^2]])/(9*a^3) - ((3*a^2*d - e)*Sqrt[1 + a^2*x^2]*Log[c*x^n])/(3*a^3) - (e*(1 + a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcSinh[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcSinh[a*x]*Log[c*x^n]} -{(d + e*x^2)*ArcCosh[a*x]*Log[c*x^n], x, 12, (d*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a + (2*e*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3) + ((9*a^2*d + 2*e)*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) + (e*n*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (e*n*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2))/(27*a^3) - d*n*x*ArcCosh[a*x] - (1/9)*e*n*x^3*ArcCosh[a*x] - ((9*a^2*d + 2*e)*n*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/(9*a^3) - ((9*a^2*d + 2*e)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a^3) - (e*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[c*x^n])/(9*a) + d*x*ArcCosh[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcCosh[a*x]*Log[c*x^n]} -{(d + e*x^2)*ArcTanh[a*x]*Log[c*x^n], x, 9, -((5*e*n*x^2)/(36*a)) - d*n*x*ArcTanh[a*x] - (1/9)*e*n*x^3*ArcTanh[a*x] + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcTanh[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcTanh[a*x]*Log[c*x^n] - (d*n*Log[1 - a^2*x^2])/(2*a) - (e*n*Log[1 - a^2*x^2])/(18*a^3) + ((3*a^2*d + e)*Log[c*x^n]*Log[1 - a^2*x^2])/(6*a^3) + ((3*a^2*d + e)*n*PolyLog[2, a^2*x^2])/(12*a^3)} -{(d + e*x^2)*ArcCoth[a*x]*Log[c*x^n], x, 9, -((5*e*n*x^2)/(36*a)) - d*n*x*ArcCoth[a*x] - (1/9)*e*n*x^3*ArcCoth[a*x] + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCoth[a*x]*Log[c*x^n] + (1/3)*e*x^3*ArcCoth[a*x]*Log[c*x^n] - (d*n*Log[1 - a^2*x^2])/(2*a) - (e*n*Log[1 - a^2*x^2])/(18*a^3) + ((3*a^2*d + e)*Log[c*x^n]*Log[1 - a^2*x^2])/(6*a^3) + ((3*a^2*d + e)*n*PolyLog[2, a^2*x^2])/(12*a^3)} - - -{(d + e*x^2)*ArcSin[a*x]^2*Log[c*x^n], x, 21, 2*d*n*x + (2*e*n*x)/(27*a^2) + (4/9)*(9*d + (2*e)/a^2)*n*x + (2/27)*e*n*x^3 - (2*d*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a - (4*e*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a^3) - (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^3) - (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a) + (2*e*n*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])/(27*a^3) - d*n*x*ArcSin[a*x]^2 - (1/9)*e*n*x^3*ArcSin[a*x]^2 + (4*(9*a^2*d + 2*e)*n*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])])/(9*a^3) - 2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2/27)*e*x^3*Log[c*x^n] + (2*d*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/(9*a^3) + (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/(9*a) + d*x*ArcSin[a*x]^2*Log[c*x^n] + (1/3)*e*x^3*ArcSin[a*x]^2*Log[c*x^n] - (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, -E^(I*ArcSin[a*x])])/(9*a^3) + (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, E^(I*ArcSin[a*x])])/(9*a^3)} -{(d + e*x^2)*ArcCos[a*x]^2*Log[c*x^n], x, 21, 2*d*n*x + (2*e*n*x)/(27*a^2) + (4/9)*(9*d + (2*e)/a^2)*n*x + (2/27)*e*n*x^3 + (2*d*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a + (4*e*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(9*a^3) + (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a) - (2*e*n*(1 - a^2*x^2)^(3/2)*ArcCos[a*x])/(27*a^3) - d*n*x*ArcCos[a*x]^2 - (1/9)*e*n*x^3*ArcCos[a*x]^2 + (4*I*(9*a^2*d + 2*e)*n*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])])/(9*a^3) - 2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2/27)*e*x^3*Log[c*x^n] - (2*d*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/a - (4*e*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a) + d*x*ArcCos[a*x]^2*Log[c*x^n] + (1/3)*e*x^3*ArcCos[a*x]^2*Log[c*x^n] - (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, (-I)*E^(I*ArcCos[a*x])])/(9*a^3) + (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, I*E^(I*ArcCos[a*x])])/(9*a^3)} - -{(d + e*x^2)*ArcSinh[a*x]^2*Log[c*x^n], x, 21, -2*d*n*x + (2*e*n*x)/(27*a^2) - (4/9)*(9*d - (2*e)/a^2)*n*x - (2/27)*e*n*x^3 + (2*d*n*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + (2*(9*a^2*d - 2*e)*n*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^3) - (4*e*n*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(27*a^3) + (2*e*n*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(27*a) + (2*e*n*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x])/(27*a^3) - d*n*x*ArcSinh[a*x]^2 - (1/9)*e*n*x^3*ArcSinh[a*x]^2 - (4*(9*a^2*d - 2*e)*n*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]])/(9*a^3) + 2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) + (2/27)*e*x^3*Log[c*x^n] - (2*d*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[c*x^n])/(9*a) + d*x*ArcSinh[a*x]^2*Log[c*x^n] + (1/3)*e*x^3*ArcSinh[a*x]^2*Log[c*x^n] - (2*(9*a^2*d - 2*e)*n*PolyLog[2, -E^ArcSinh[a*x]])/(9*a^3) + (2*(9*a^2*d - 2*e)*n*PolyLog[2, E^ArcSinh[a*x]])/(9*a^3)} -{(d + e*x^2)*ArcCosh[a*x]^2*Log[c*x^n], x, 22, -2*d*n*x - (2*e*n*x)/(27*a^2) - (4/9)*(9*d + (2*e)/a^2)*n*x - (2/27)*e*n*x^3 + (2*d*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a + (4*e*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a^3) + (2*(9*a^2*d + 2*e)*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^3) + (2*e*n*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a) + (2*e*n*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(27*a^3) - d*n*x*ArcCosh[a*x]^2 - (1/9)*e*n*x^3*ArcCosh[a*x]^2 - (4*(9*a^2*d + 2*e)*n*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/(9*a^3) + 2*d*x*Log[c*x^n] + (4*e*x*Log[c*x^n])/(9*a^2) + (2/27)*e*x^3*Log[c*x^n] - (2*d*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*Log[c*x^n])/a - (4*e*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*Log[c*x^n])/(9*a) + d*x*ArcCosh[a*x]^2*Log[c*x^n] + (1/3)*e*x^3*ArcCosh[a*x]^2*Log[c*x^n] + (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, (-I)*E^ArcCosh[a*x]])/(9*a^3) - (2*I*(9*a^2*d + 2*e)*n*PolyLog[2, I*E^ArcCosh[a*x]])/(9*a^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[k, e x^q] (a+b Log[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form PolyLog[k, e x^q] (a+b Log[c x^n])^p / x*) - - -{PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^p/x, x, 0, Unintegrable[((a + b*Log[c*x^n])^p*PolyLog[k, e*x^q])/x, x]} - - -{PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^3/x, x, 4, ((a + b*Log[c*x^n])^3*PolyLog[1 + k, e*x^q])/q - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2 + k, e*x^q])/q^2 + (6*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3 + k, e*x^q])/q^3 - (6*b^3*n^3*PolyLog[4 + k, e*x^q])/q^4} -{PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^2/x, x, 3, ((a + b*Log[c*x^n])^2*PolyLog[1 + k, e*x^q])/q - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2 + k, e*x^q])/q^2 + (2*b^2*n^2*PolyLog[3 + k, e*x^q])/q^3} -{PolyLog[k, e*x^q]*(a + b*Log[c*x^n])^1/x, x, 2, ((a + b*Log[c*x^n])*PolyLog[1 + k, e*x^q])/q - (b*n*PolyLog[2 + k, e*x^q])/q^2} -{PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^1), x, 0, Unintegrable[PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])), x]} -{PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^2), x, 1, -(PolyLog[k, e*x^q]/(b*n*(a + b*Log[c*x^n]))) + (q*Unintegrable[PolyLog[-1 + k, e*x^q]/(x*(a + b*Log[c*x^n])), x])/(b*n)} -{PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^3), x, 2, -((q*PolyLog[-1 + k, e*x^q])/(2*b^2*n^2*(a + b*Log[c*x^n]))) - PolyLog[k, e*x^q]/(2*b*n*(a + b*Log[c*x^n])^2) + (q^2*Unintegrable[PolyLog[-2 + k, e*x^q]/(x*(a + b*Log[c*x^n])), x])/(2*b^2*n^2)} - - -{(Log[x]*PolyLog[n, a*x])/x, x, 2, Log[x]*PolyLog[1 + n, a*x] - PolyLog[2 + n, a*x]} -{(Log[x]^2*PolyLog[n, a*x])/x, x, 3, Log[x]^2*PolyLog[1 + n, a*x] - 2*Log[x]*PolyLog[2 + n, a*x] + 2*PolyLog[3 + n, a*x]} - - -{q*PolyLog[k - 1, e*x^q]/(b*n*x*(a + b*Log[c*x^n])) - PolyLog[k, e*x^q]/(x*(a + b*Log[c*x^n])^2), x, 2, PolyLog[k, e*x^q]/(b*n*(a + b*Log[c*x^n]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[k, e x^q] (a+b Log[c x^n])*) - - -{x^2*PolyLog[2, e*x]*(a + b*Log[c*x^n]), x, 10, (5*b*n*x)/(27*e^2) + (7*b*n*x^2)/(108*e) + (1/27)*b*n*x^3 - (x*(a + b*Log[c*x^n]))/(9*e^2) - (x^2*(a + b*Log[c*x^n]))/(18*e) - (1/27)*x^3*(a + b*Log[c*x^n]) + (2*b*n*Log[1 - e*x])/(27*e^3) - (2/27)*b*n*x^3*Log[1 - e*x] - ((a + b*Log[c*x^n])*Log[1 - e*x])/(9*e^3) + (1/9)*x^3*(a + b*Log[c*x^n])*Log[1 - e*x] - (b*n*PolyLog[2, e*x])/(9*e^3) - (1/9)*b*n*x^3*PolyLog[2, e*x] + (1/3)*x^3*(a + b*Log[c*x^n])*PolyLog[2, e*x]} -{x^1*PolyLog[2, e*x]*(a + b*Log[c*x^n]), x, 10, (b*n*x)/(2*e) + (3/16)*b*n*x^2 - (x*(a + b*Log[c*x^n]))/(4*e) - (1/8)*x^2*(a + b*Log[c*x^n]) + (b*n*Log[1 - e*x])/(4*e^2) - (1/4)*b*n*x^2*Log[1 - e*x] - ((a + b*Log[c*x^n])*Log[1 - e*x])/(4*e^2) + (1/4)*x^2*(a + b*Log[c*x^n])*Log[1 - e*x] - (b*n*PolyLog[2, e*x])/(4*e^2) - (1/4)*b*n*x^2*PolyLog[2, e*x] + (1/2)*x^2*(a + b*Log[c*x^n])*PolyLog[2, e*x]} -{x^0*PolyLog[2, e*x]*(a + b*Log[c*x^n]), x, 10, 3*b*n*x - x*(a + b*Log[c*x^n]) + (2*b*n*(1 - e*x)*Log[1 - e*x])/e - ((1 - e*x)*(a + b*Log[c*x^n])*Log[1 - e*x])/e - (b*n*PolyLog[2, e*x])/e - b*n*x*PolyLog[2, e*x] + x*(a + b*Log[c*x^n])*PolyLog[2, e*x]} -{PolyLog[2, e*x]*(a + b*Log[c*x^n])/x^1, x, 2, (a + b*Log[c*x^n])*PolyLog[3, e*x] - b*n*PolyLog[4, e*x]} -{PolyLog[2, e*x]*(a + b*Log[c*x^n])/x^2, x, 13, 2*b*e*n*Log[x] - (1/2)*b*e*n*Log[x]^2 + e*Log[x]*(a + b*Log[c*x^n]) - 2*b*e*n*Log[1 - e*x] + (2*b*n*Log[1 - e*x])/x - e*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/x - b*e*n*PolyLog[2, e*x] - (b*n*PolyLog[2, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/x} -{PolyLog[2, e*x]*(a + b*Log[c*x^n])/x^3, x, 11, -((b*e*n)/(2*x)) + (1/4)*b*e^2*n*Log[x] - (1/8)*b*e^2*n*Log[x]^2 - (e*(a + b*Log[c*x^n]))/(4*x) + (1/4)*e^2*Log[x]*(a + b*Log[c*x^n]) - (1/4)*b*e^2*n*Log[1 - e*x] + (b*n*Log[1 - e*x])/(4*x^2) - (1/4)*e^2*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/(4*x^2) - (1/4)*b*e^2*n*PolyLog[2, e*x] - (b*n*PolyLog[2, e*x])/(4*x^2) - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/(2*x^2)} - - -{x^2*PolyLog[3, e*x]*(a + b*Log[c*x^n]), x, 15, -((2*b*n*x)/(27*e^2)) - (b*n*x^2)/(36*e) - (4/243)*b*n*x^3 + (x*(a + b*Log[c*x^n]))/(27*e^2) + (x^2*(a + b*Log[c*x^n]))/(54*e) + (1/81)*x^3*(a + b*Log[c*x^n]) - (b*n*Log[1 - e*x])/(27*e^3) + (1/27)*b*n*x^3*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/(27*e^3) - (1/27)*x^3*(a + b*Log[c*x^n])*Log[1 - e*x] + (b*n*PolyLog[2, e*x])/(27*e^3) + (2/27)*b*n*x^3*PolyLog[2, e*x] - (1/9)*x^3*(a + b*Log[c*x^n])*PolyLog[2, e*x] - (1/9)*b*n*x^3*PolyLog[3, e*x] + (1/3)*x^3*(a + b*Log[c*x^n])*PolyLog[3, e*x]} -{x^1*PolyLog[3, e*x]*(a + b*Log[c*x^n]), x, 15, -((5*b*n*x)/(16*e)) - (1/8)*b*n*x^2 + (x*(a + b*Log[c*x^n]))/(8*e) + (1/16)*x^2*(a + b*Log[c*x^n]) - (3*b*n*Log[1 - e*x])/(16*e^2) + (3/16)*b*n*x^2*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/(8*e^2) - (1/8)*x^2*(a + b*Log[c*x^n])*Log[1 - e*x] + (b*n*PolyLog[2, e*x])/(8*e^2) + (1/4)*b*n*x^2*PolyLog[2, e*x] - (1/4)*x^2*(a + b*Log[c*x^n])*PolyLog[2, e*x] - (1/4)*b*n*x^2*PolyLog[3, e*x] + (1/2)*x^2*(a + b*Log[c*x^n])*PolyLog[3, e*x]} -{x^0*PolyLog[3, e*x]*(a + b*Log[c*x^n]), x, 14, -4*b*n*x + x*(a + b*Log[c*x^n]) - (3*b*n*(1 - e*x)*Log[1 - e*x])/e + ((1 - e*x)*(a + b*Log[c*x^n])*Log[1 - e*x])/e + (b*n*PolyLog[2, e*x])/e + 2*b*n*x*PolyLog[2, e*x] - x*(a + b*Log[c*x^n])*PolyLog[2, e*x] - b*n*x*PolyLog[3, e*x] + x*(a + b*Log[c*x^n])*PolyLog[3, e*x]} -{PolyLog[3, e*x]*(a + b*Log[c*x^n])/x^1, x, 2, (a + b*Log[c*x^n])*PolyLog[4, e*x] - b*n*PolyLog[5, e*x]} -{PolyLog[3, e*x]*(a + b*Log[c*x^n])/x^2, x, 19, 3*b*e*n*Log[x] - (1/2)*b*e*n*Log[x]^2 + e*Log[x]*(a + b*Log[c*x^n]) - 3*b*e*n*Log[1 - e*x] + (3*b*n*Log[1 - e*x])/x - e*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/x - b*e*n*PolyLog[2, e*x] - (2*b*n*PolyLog[2, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/x - (b*n*PolyLog[3, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[3, e*x])/x} -{PolyLog[3, e*x]*(a + b*Log[c*x^n])/x^3, x, 16, -((5*b*e*n)/(16*x)) + (3/16)*b*e^2*n*Log[x] - (1/16)*b*e^2*n*Log[x]^2 - (e*(a + b*Log[c*x^n]))/(8*x) + (1/8)*e^2*Log[x]*(a + b*Log[c*x^n]) - (3/16)*b*e^2*n*Log[1 - e*x] + (3*b*n*Log[1 - e*x])/(16*x^2) - (1/8)*e^2*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/(8*x^2) - (1/8)*b*e^2*n*PolyLog[2, e*x] - (b*n*PolyLog[2, e*x])/(4*x^2) - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/(4*x^2) - (b*n*PolyLog[3, e*x])/(4*x^2) - ((a + b*Log[c*x^n])*PolyLog[3, e*x])/(2*x^2)} - - -{(d*x)^m*PolyLog[1, e*x^q]*(a + b*Log[c*x^n]), x, 0, -Unintegrable[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x]} -{(d*x)^m*PolyLog[2, e*x^q]*(a + b*Log[c*x^n]), x, 4, -((b*e*n*q^2*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^3*(1 + m + q))) - (b*n*q*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^3) - (b*n*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)) + (q*Unintegrable[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x])/(1 + m)} -{(d*x)^m*PolyLog[3, e*x^q]*(a + b*Log[c*x^n]), x, 9, (2*b*e*n*q^3*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^4*(1 + m + q)) + (2*b*n*q^2*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^4) + (2*b*n*q*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*(1 + m)^3) - (q*(d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)^2) - (b*n*(d*x)^(1 + m)*PolyLog[3, e*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[3, e*x^q])/(d*(1 + m)) - (q^2*Unintegrable[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x])/(1 + m)^2} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b Log[c (d x^m)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^q (a+b Log[c (d x^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q (a+b Log[c (d x^m)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Log[c*(b*x^n)^p]*x^2, x, 2, (-(1/9))*n*p*x^3 + (1/3)*x^3*Log[c*(b*x^n)^p]} -{Log[c*(b*x^n)^p]*x^1, x, 2, (-(1/4))*n*p*x^2 + (1/2)*x^2*Log[c*(b*x^n)^p]} -{Log[c*(b*x^n)^p]*x^0, x, 2, (-n)*p*x + x*Log[c*(b*x^n)^p]} -{Log[c*(b*x^n)^p]/x^1, x, 2, Log[c*(b*x^n)^p]^2/(2*n*p)} -{Log[c*(b*x^n)^p]/x^2, x, 2, -((n*p)/x) - Log[c*(b*x^n)^p]/x} -{Log[c*(b*x^n)^p]/x^3, x, 2, -((n*p)/(4*x^2)) - Log[c*(b*x^n)^p]/(2*x^2)} -{Log[c*(b*x^n)^p]/x^4, x, 2, -((n*p)/(9*x^3)) - Log[c*(b*x^n)^p]/(3*x^3)} - - -{Log[c*(b*x^n)^p]^2*x^2, x, 3, (2/27)*n^2*p^2*x^3 - (2/9)*n*p*x^3*Log[c*(b*x^n)^p] + (1/3)*x^3*Log[c*(b*x^n)^p]^2} -{Log[c*(b*x^n)^p]^2*x^1, x, 3, (1/4)*n^2*p^2*x^2 - (1/2)*n*p*x^2*Log[c*(b*x^n)^p] + (1/2)*x^2*Log[c*(b*x^n)^p]^2} -{Log[c*(b*x^n)^p]^2*x^0, x, 3, 2*n^2*p^2*x - 2*n*p*x*Log[c*(b*x^n)^p] + x*Log[c*(b*x^n)^p]^2} -{Log[c*(b*x^n)^p]^2/x^1, x, 3, Log[c*(b*x^n)^p]^3/(3*n*p)} -{Log[c*(b*x^n)^p]^2/x^2, x, 3, -((2*n^2*p^2)/x) - (2*n*p*Log[c*(b*x^n)^p])/x - Log[c*(b*x^n)^p]^2/x} -{Log[c*(b*x^n)^p]^2/x^3, x, 3, -((n^2*p^2)/(4*x^2)) - (n*p*Log[c*(b*x^n)^p])/(2*x^2) - Log[c*(b*x^n)^p]^2/(2*x^2)} -{Log[c*(b*x^n)^p]^2/x^4, x, 3, -((2*n^2*p^2)/(27*x^3)) - (2*n*p*Log[c*(b*x^n)^p])/(9*x^3) - Log[c*(b*x^n)^p]^2/(3*x^3)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^q (a+b Log[c (d x^m)^n])^p with q symbolic*) - - -{(e*x)^q*(a + b*Log[c*(d*x^m)^n])^3, x, 4, -((6*b^3*m^3*n^3*(e*x)^(1 + q))/(e*(1 + q)^4)) + (6*b^2*m^2*n^2*(e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 + q)^3) - (3*b*m*n*(e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n])^2)/(e*(1 + q)^2) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n])^3)/(e*(1 + q))} -{(e*x)^q*(a + b*Log[c*(d*x^m)^n])^2, x, 3, (2*b^2*m^2*n^2*(e*x)^(1 + q))/(e*(1 + q)^3) - (2*b*m*n*(e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 + q)^2) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n])^2)/(e*(1 + q))} -{(e*x)^q*(a + b*Log[c*(d*x^m)^n])^1, x, 2, -((b*m*n*(e*x)^(1 + q))/(e*(1 + q)^2)) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 + q))} -{(e*x)^q/(a + b*Log[c*(d*x^m)^n])^1, x, 3, ((e*x)^(1 + q)*ExpIntegralEi[((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)])/(E^((a*(1 + q))/(b*m*n))*(c*(d*x^m)^n)^((1 + q)/(m*n))*(b*e*m*n))} -{(e*x)^q/(a + b*Log[c*(d*x^m)^n])^2, x, 4, ((1 + q)*(e*x)^(1 + q)*ExpIntegralEi[((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)])/(E^((a*(1 + q))/(b*m*n))*(c*(d*x^m)^n)^((1 + q)/(m*n))*(b^2*e*m^2*n^2)) - (e*x)^(1 + q)/(b*e*m*n*(a + b*Log[c*(d*x^m)^n]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^q (a+b Log[c (d x^m)^n])^p with p symbolic*) - - -{(e*x)^q*(a + b*Log[c*(d*x^m)^n])^p, x, 3, ((e*x)^(1 + q)*Gamma[1 + p, -(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(E^((a*(1 + q))/(b*m*n))*(c*(d*x^m)^n)^((1 + q)/(m*n))*(-(((1 + q)*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)))^p*(e*(1 + q)))} - - -{x^2*(a + b*Log[c*(d*x^m)^n])^p, x, 3, (3^(-1 - p)*x^3*Gamma[1 + p, -((3*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(E^((3*a)/(b*m*n))*(c*(d*x^m)^n)^(3/(m*n))*(-((a + b*Log[c*(d*x^m)^n])/(b*m*n)))^p)} -{x^1*(a + b*Log[c*(d*x^m)^n])^p, x, 3, (2^(-1 - p)*x^2*Gamma[1 + p, -((2*(a + b*Log[c*(d*x^m)^n]))/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(E^((2*a)/(b*m*n))*(c*(d*x^m)^n)^(2/(m*n))*(-((a + b*Log[c*(d*x^m)^n])/(b*m*n)))^p)} -{x^0*(a + b*Log[c*(d*x^m)^n])^p, x, 3, (x*Gamma[1 + p, -((a + b*Log[c*(d*x^m)^n])/(b*m*n))]*(a + b*Log[c*(d*x^m)^n])^p)/(E^(a/(b*m*n))*(c*(d*x^m)^n)^(1/(m*n))*(-((a + b*Log[c*(d*x^m)^n])/(b*m*n)))^p)} -{(a + b*Log[c*(d*x^m)^n])^p/x^1, x, 3, (a + b*Log[c*(d*x^m)^n])^(1 + p)/(b*m*n*(1 + p))} -{(a + b*Log[c*(d*x^m)^n])^p/x^2, x, 3, -((E^(a/(b*m*n))*(c*(d*x^m)^n)^(1/(m*n))*Gamma[1 + p, (a + b*Log[c*(d*x^m)^n])/(b*m*n)]*(a + b*Log[c*(d*x^m)^n])^p)/(((a + b*Log[c*(d*x^m)^n])/(b*m*n))^p*x))} -{(a + b*Log[c*(d*x^m)^n])^p/x^3, x, 3, -((2^(-1 - p)*E^((2*a)/(b*m*n))*(c*(d*x^m)^n)^(2/(m*n))*Gamma[1 + p, (2*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)]*(a + b*Log[c*(d*x^m)^n])^p)/(((a + b*Log[c*(d*x^m)^n])/(b*m*n))^p*x^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x^r)^q (a+b Log[c (d x^m)^n])^p*) - - -{(a + b*Log[c*(d*x^m)^n])/(e + f*x^2), x, 6, (ArcTan[(Sqrt[f]*x)/Sqrt[e]]*(a + b*Log[c*(d*x^m)^n]))/(Sqrt[e]*Sqrt[f]) - (I*b*m*n*PolyLog[2, -((I*Sqrt[f]*x)/Sqrt[e])])/(2*Sqrt[e]*Sqrt[f]) + (I*b*m*n*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/(2*Sqrt[e]*Sqrt[f])} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m deleted file mode 100644 index ecfb8f0..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.2.1 (f+g x)^m (A+B log(e ((a+b x) over (c+d x))^n))^p.m +++ /dev/null @@ -1,584 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p when b f-a g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p and b f-a g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a*g + b*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, (B*(b*c - a*d)^4*g^4*n*x)/(5*d^4) - (B*(b*c - a*d)^3*g^4*n*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*g^4*n*(a + b*x)^3)/(15*b*d^2) - (B*(b*c - a*d)*g^4*n*(a + b*x)^4)/(20*b*d) + (g^4*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b) - (B*(b*c - a*d)^5*g^4*n*Log[c + d*x])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -(B*(b*c - a*d)^3*g^3*n*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*n*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*n*(a + b*x)^3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b) + (B*(b*c - a*d)^4*g^3*n*Log[c + d*x])/(4*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, (B*(b*c - a*d)^2*g^2*n*x)/(3*d^2) - (B*(b*c - a*d)*g^2*n*(a + b*x)^2)/(6*b*d) + (g^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b) - (B*(b*c - a*d)^3*g^2*n*Log[c + d*x])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -(B*(b*c - a*d)*g*n*x)/(2*d) + (g*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b) + (B*(b*c - a*d)^2*g*n*Log[c + d*x])/(2*b*d^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*g)) + (B*n*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2, x, 2, -((B*n)/(b*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*g^2*(a + b*x)), -((B*n*(c + d*x))/((b*c - a*d)*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^3, x, 4, -(B*n)/(4*b*g^3*(a + b*x)^2) + (B*d*n)/(2*b*(b*c - a*d)*g^3*(a + b*x)) + (B*d^2*n*Log[a + b*x])/(2*b*(b*c - a*d)^2*g^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*b*g^3*(a + b*x)^2) - (B*d^2*n*Log[c + d*x])/(2*b*(b*c - a*d)^2*g^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4, x, 4, -(B*n)/(9*b*g^4*(a + b*x)^3) + (B*d*n)/(6*b*(b*c - a*d)*g^4*(a + b*x)^2) - (B*d^2*n)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) - (B*d^3*n*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*b*g^4*(a + b*x)^3) + (B*d^3*n*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^5, x, 4, -(B*n)/(16*b*g^5*(a + b*x)^4) + (B*d*n)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2*n)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B*d^3*n)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*n*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*b*g^5*(a + b*x)^4) - (B*d^4*n*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5)} - - -{(a*g + b*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 8, -((B*(b*c - a*d)*g^4*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b*d)) + (g^4*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b) + (B*(b*c - a*d)^2*g^4*n*(a + b*x)^3*(4*A + B*n + 4*B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b*d^2) - (B*(b*c - a*d)^3*g^4*n*(a + b*x)^2*(12*A + 7*B*n + 12*B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b*d^3) + (B*(b*c - a*d)^4*g^4*n*(a + b*x)*(12*A + 13*B*n + 12*B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b*d^4) + (B*(b*c - a*d)^5*g^4*n*(12*A + 25*B*n + 12*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(30*b*d^5) + (2*B^2*(b*c - a*d)^5*g^4*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 7, -((B*(b*c - a*d)*g^3*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*d)) + (g^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b) + (B*(b*c - a*d)^2*g^3*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(12*b*d^2) - (B*(b*c - a*d)^3*g^3*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(12*b*d^3) - (B*(b*c - a*d)^4*g^3*n*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(12*b*d^4) - (B^2*(b*c - a*d)^4*g^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 6, -((B*(b*c - a*d)*g^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*d)) + (g^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b) + (B*(b*c - a*d)^2*g^2*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*d^2) + (B*(b*c - a*d)^3*g^2*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b*d^3) + (2*B^2*(b*c - a*d)^3*g^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 5, -((B*(b*c - a*d)*g*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*d)) + (g*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b) - (B*(b*c - a*d)^2*g*n*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b*d^2) - (B^2*(b*c - a*d)^2*g*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^1, x, 4, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b*g)) + (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (2*B^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*g)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^2, x, 3, (-2*B^2*n^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (2*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^3, x, 7, (2*B^2*d*n^2*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^2*g^3*(a + b*x)^2) + (2*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^2*g^3*(a + b*x)^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^4, x, 9, (-2*B^2*d^2*n^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B^2*d*n^2*(c + d*x)^2)/(2*(b*c - a*d)^3*g^4*(a + b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) - (2*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^4*(a + b*x)^2) - (2*b^2*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^4*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^3*g^4*(a + b*x)^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5, x, 11, (2*B^2*d^3*n^2*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*n^2*(c + d*x)^4)/(32*(b*c - a*d)^4*g^5*(a + b*x)^4) + (2*B*d^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B*d^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a*g + b*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{(a*g + b*g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{1/((a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 3, (E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n))])/(B*(b*c - a*d)*g^2*n*(a + b*x))} -{1/((a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 7, (b*E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(B*(b*c - a*d)^2*g^3*n*(a + b*x)^2) - (d*E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n))])/(B*(b*c - a*d)^2*g^3*n*(a + b*x))} - - -{(a*g + b*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{(a*g + b*g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{1/((a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 4, -((E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n))])/(B^2*(b*c - a*d)*g^2*n^2*(a + b*x))) - (c + d*x)/(B*(b*c - a*d)*g^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} -{1/((a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 9, (-2*b*E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(B^2*(b*c - a*d)^2*g^3*n^2*(a + b*x)^2) + (d*E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n))])/(B^2*(b*c - a*d)^2*g^3*n^2*(a + b*x)) + (d*(c + d*x))/(B*(b*c - a*d)^2*g^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])) - (b*(c + d*x)^2)/(B*(b*c - a*d)^2*g^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p when d f-c g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p and d f-c g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c*g + d*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -((B*(b*c - a*d)^4*g^4*n*x)/(5*b^4)) - (B*(b*c - a*d)^3*g^4*n*(c + d*x)^2)/(10*b^3*d) - (B*(b*c - a*d)^2*g^4*n*(c + d*x)^3)/(15*b^2*d) - (B*(b*c - a*d)*g^4*n*(c + d*x)^4)/(20*b*d) - (B*(b*c - a*d)^5*g^4*n*Log[a + b*x])/(5*b^5*d) + (g^4*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d)} -{(c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -((B*(b*c - a*d)^3*g^3*n*x)/(4*b^3)) - (B*(b*c - a*d)^2*g^3*n*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*g^3*n*(c + d*x)^3)/(12*b*d) - (B*(b*c - a*d)^4*g^3*n*Log[a + b*x])/(4*b^4*d) + (g^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d)} -{(c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -((B*(b*c - a*d)^2*g^2*n*x)/(3*b^2)) - (B*(b*c - a*d)*g^2*n*(c + d*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*g^2*n*Log[a + b*x])/(3*b^3*d) + (g^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d)} -{(c*g + d*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -((B*(b*c - a*d)*g*n*x)/(2*b)) - (B*(b*c - a*d)^2*g*n*Log[a + b*x])/(2*b^2*d) + (g*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^1, x, 5, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d*g)) - (B*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*g)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^2, x, 3, (A*(a + b*x))/((b*c - a*d)*g^2*(c + d*x)) - (B*n*(a + b*x))/((b*c - a*d)*g^2*(c + d*x)) + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*g^2*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^3, x, 4, (B*n)/(4*d*g^3*(c + d*x)^2) + (b*B*n)/(2*d*(b*c - a*d)*g^3*(c + d*x)) + (b^2*B*n*Log[a + b*x])/(2*d*(b*c - a*d)^2*g^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*d*g^3*(c + d*x)^2) - (b^2*B*n*Log[c + d*x])/(2*d*(b*c - a*d)^2*g^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^4, x, 4, (B*n)/(9*d*g^4*(c + d*x)^3) + (b*B*n)/(6*d*(b*c - a*d)*g^4*(c + d*x)^2) + (b^2*B*n)/(3*d*(b*c - a*d)^2*g^4*(c + d*x)) + (b^3*B*n*Log[a + b*x])/(3*d*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*d*g^4*(c + d*x)^3) - (b^3*B*n*Log[c + d*x])/(3*d*(b*c - a*d)^3*g^4)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^5, x, 4, (B*n)/(16*d*g^5*(c + d*x)^4) + (b*B*n)/(12*d*(b*c - a*d)*g^5*(c + d*x)^3) + (b^2*B*n)/(8*d*(b*c - a*d)^2*g^5*(c + d*x)^2) + (b^3*B*n)/(4*d*(b*c - a*d)^3*g^5*(c + d*x)) + (b^4*B*n*Log[a + b*x])/(4*d*(b*c - a*d)^4*g^5) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*d*g^5*(c + d*x)^4) - (b^4*B*n*Log[c + d*x])/(4*d*(b*c - a*d)^4*g^5)} - - -{(c*g + d*g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 19, (13*B^2*(b*c - a*d)^4*g^4*n^2*x)/(30*b^4) + (7*B^2*(b*c - a*d)^3*g^4*n^2*(c + d*x)^2)/(60*b^3*d) + (B^2*(b*c - a*d)^2*g^4*n^2*(c + d*x)^3)/(30*b^2*d) - (2*B*(b*c - a*d)^4*g^4*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^5) - (B*(b*c - a*d)^3*g^4*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^3*d) - (2*B*(b*c - a*d)^2*g^4*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^2*d) - (B*(b*c - a*d)*g^4*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b*d) + (g^4*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*d) + (13*B^2*(b*c - a*d)^5*g^4*n^2*Log[(a + b*x)/(c + d*x)])/(30*b^5*d) + (5*B^2*(b*c - a*d)^5*g^4*n^2*Log[c + d*x])/(6*b^5*d) + (2*B*(b*c - a*d)^5*g^4*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(5*b^5*d) - (2*B^2*(b*c - a*d)^5*g^4*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(5*b^5*d)} -{(c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 15, (5*B^2*(b*c - a*d)^3*g^3*n^2*x)/(12*b^3) + (B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x)^2)/(12*b^2*d) - (B*(b*c - a*d)^3*g^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^4) - (B*(b*c - a*d)^2*g^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2*d) - (B*(b*c - a*d)*g^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*d) + (g^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*d) + (5*B^2*(b*c - a*d)^4*g^3*n^2*Log[(a + b*x)/(c + d*x)])/(12*b^4*d) + (11*B^2*(b*c - a*d)^4*g^3*n^2*Log[c + d*x])/(12*b^4*d) + (B*(b*c - a*d)^4*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d) - (B^2*(b*c - a*d)^4*g^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d)} -{(c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 11, (B^2*(b*c - a*d)^2*g^2*n^2*x)/(3*b^2) - (2*B*(b*c - a*d)^2*g^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3) - (B*(b*c - a*d)*g^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*d) + (g^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*d) + (B^2*(b*c - a*d)^3*g^2*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d) + (B^2*(b*c - a*d)^3*g^2*n^2*Log[c + d*x])/(b^3*d) + (2*B*(b*c - a*d)^3*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d) - (2*B^2*(b*c - a*d)^3*g^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d)} -{(c*g + d*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 7, -((B*(b*c - a*d)*g*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/b^2) + (g*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d) + (B^2*(b*c - a*d)^2*g*n^2*Log[c + d*x])/(b^2*d) + (B*(b*c - a*d)^2*g*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*d) - (B^2*(b*c - a*d)^2*g*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*d)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^1, x, 4, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d*g)) - (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*g) + (2*B^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d*g)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^2, x, 4, -((2*A*B*n*(a + b*x))/((b*c - a*d)*g^2*(c + d*x))) + (2*B^2*n^2*(a + b*x))/((b*c - a*d)*g^2*(c + d*x)) - (2*B^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*g^2*(c + d*x)) + ((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*g^2*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^3, x, 8, -((B^2*d*n^2*(a + b*x)^2)/(4*(b*c - a*d)^2*g^3*(c + d*x)^2)) - (2*A*b*B*n*(a + b*x))/((b*c - a*d)^2*g^3*(c + d*x)) + (2*b*B^2*n^2*(a + b*x))/((b*c - a*d)^2*g^3*(c + d*x)) - (2*b*B^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^2*g^3*(c + d*x)) + (B*d*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^3*(c + d*x)^2) - (d*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^2*g^3*(c + d*x)^2) + (b*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^2*g^3*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^4, x, 6, (2*B^2*d^2*n^2*(a + b*x)^3)/(27*(b*c - a*d)^3*g^4*(c + d*x)^3) - (b*B^2*d*n^2*(a + b*x)^2)/(2*(b*c - a*d)^3*g^4*(c + d*x)^2) + (2*b^2*B^2*n^2*(a + b*x))/((b*c - a*d)^3*g^4*(c + d*x)) - (2*B*d^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^4*(c + d*x)^3) + (b*B*d*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^4*(c + d*x)^2) - (2*b^2*B*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^4*(c + d*x)) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(3*d*g^4*(c + d*x)^3) + (2*b^3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/(3*d*(b*c - a*d)^3*g^4) - (b^3*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2)/(3*d*(b*c - a*d)^3*g^4)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^5, x, 5, -((B^2*d^3*n^2*(a + b*x)^4)/(32*(b*c - a*d)^4*g^5*(c + d*x)^4)) + (2*b*B^2*d^2*n^2*(a + b*x)^3)/(9*(b*c - a*d)^4*g^5*(c + d*x)^3) - (3*b^2*B^2*d*n^2*(a + b*x)^2)/(4*(b*c - a*d)^4*g^5*(c + d*x)^2) + (2*b^3*B^2*n^2*(a + b*x))/((b*c - a*d)^4*g^5*(c + d*x)) + (B*d^3*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^4*g^5*(c + d*x)^4) - (2*b*B*d^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^4*g^5*(c + d*x)^3) + (3*b^2*B*d*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^5*(c + d*x)^2) - (2*b^3*B*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^5*(c + d*x)) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*d*g^5*(c + d*x)^4) + (b^4*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/(2*d*(b*c - a*d)^4*g^5) - (b^4*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2)/(4*d*(b*c - a*d)^4*g^5)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(c*g + d*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(c*g + d*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{(c*g + d*g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(c*g + d*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{1/((c*g + d*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 0, Unintegrable[1/((c*g + d*g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]} -{1/((c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 3, ((a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n)])/(E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(B*(b*c - a*d)*g^2*n*(c + d*x)))} -{1/((c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 7, (b*(a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n)])/(E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(B*(b*c - a*d)^2*g^3*n*(c + d*x))) - (d*(a + b*x)^2*ExpIntegralEi[(2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(B*(b*c - a*d)^2*g^3*n*(c + d*x)^2))} - - -{(c*g + d*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(c*g + d*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{(c*g + d*g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(c*g + d*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{1/((c*g + d*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 0, Unintegrable[1/((c*g + d*g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]} -{1/((c*g + d*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 4, ((a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n)])/(E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(B^2*(b*c - a*d)*g^2*n^2*(c + d*x))) - (a + b*x)/(B*(b*c - a*d)*g^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} -{1/((c*g + d*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 10, (b*(a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n)])/(E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x))) - (2*d*(a + b*x)^2*ExpIntegralEi[(2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x)^2)) - (a + b*x)/(B*(b*c - a*d)*g^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, (B*(b*c - a*d)*g*(a^3*d^3*g^3 - a^2*b*d^2*g^2*(5*d*f - c*g) + a*b^2*d*g*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2) - b^3*(10*d^3*f^3 - 10*c*d^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*n*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2 - a*b*d*g*(5*d*f - c*g) + b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*n*x^2)/(10*b^3*d^3) - (B*(b*c - a*d)*g^3*(5*b*d*f - b*c*g - a*d*g)*n*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*n*x^4)/(20*b*d) - (B*(b*f - a*g)^5*n*Log[a + b*x])/(5*b^5*g) + ((f + g*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*g) + (B*(d*f - c*g)^5*n*Log[c + d*x])/(5*d^5*g)} -{(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, -(B*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n*x)/(4*b^3*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*n*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*g^3*n*x^3)/(12*b*d) - (B*(b*f - a*g)^4*n*Log[a + b*x])/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*g) + (B*(d*f - c*g)^4*n*Log[c + d*x])/(4*d^4*g)} -{(f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, -(B*(b*c - a*d)*g*(3*b*d*f - b*c*g - a*d*g)*n*x)/(3*b^2*d^2) - (B*(b*c - a*d)*g^2*n*x^2)/(6*b*d) - (B*(b*f - a*g)^3*n*Log[a + b*x])/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*g) + (B*(d*f - c*g)^3*n*Log[c + d*x])/(3*d^3*g)} -{(f + g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, -(B*(b*c - a*d)*g*n*x)/(2*b*d) - (B*(b*f - a*g)^2*n*Log[a + b*x])/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*g) + (B*(d*f - c*g)^2*n*Log[c + d*x])/(2*d^2*g)} -{(f + g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, A*x + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b - (B*(b*c - a*d)*n*Log[c + d*x])/(b*d)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^1, x, 7, -((B*n*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g) + ((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[f + g*x])/g + (B*n*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (B*n*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (B*n*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^2, x, 3, ((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*f - a*g)*(f + g*x)) + (B*(b*c - a*d)*n*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^3, x, 3, -(B*(b*c - a*d)*n)/(2*(b*f - a*g)*(d*f - c*g)*(f + g*x)) + (b^2*B*n*Log[a + b*x])/(2*g*(b*f - a*g)^2) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*g*(f + g*x)^2) - (B*d^2*n*Log[c + d*x])/(2*g*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n*Log[f + g*x])/(2*(b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^4, x, 3, -(B*(b*c - a*d)*n)/(6*(b*f - a*g)*(d*f - c*g)*(f + g*x)^2) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n)/(3*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (b^3*B*n*Log[a + b*x])/(3*g*(b*f - a*g)^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*g*(f + g*x)^3) - (B*d^3*n*Log[c + d*x])/(3*g*(d*f - c*g)^3) + (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n*Log[f + g*x])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^5, x, 3, -(B*(b*c - a*d)*n)/(12*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n)/(8*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n)/(4*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*n*Log[a + b*x])/(4*g*(b*f - a*g)^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4*g*(f + g*x)^4) - (B*d^4*n*Log[c + d*x])/(4*g*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n*Log[f + g*x])/(4*(b*f - a*g)^4*(d*f - c*g)^4)} - - -{(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 15, (B^2*(b*c - a*d)^3*g^3*n^2*x)/(6*b^3*d^3) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*n^2*x)/(4*b^3*d^3) + (B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x)^2)/(12*b^2*d^4) - (B*(b*c - a*d)*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^4*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2*d^4) - (B*(b*c - a*d)*g^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*d^4) - ((b*f - a*g)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*g) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(2*b^4*d^4) + (B^2*(b*c - a*d)^4*g^3*n^2*Log[(a + b*x)/(c + d*x)])/(6*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*n^2*Log[(a + b*x)/(c + d*x)])/(4*b^4*d^4) + (B^2*(b*c - a*d)^4*g^3*n^2*Log[c + d*x])/(6*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*n^2*Log[c + d*x])/(4*b^4*d^4) + (B^2*(b*c - a*d)^2*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*n^2*Log[c + d*x])/(2*b^4*d^4) - (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b^4*d^4)} -{(f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 12, (B^2*(b*c - a*d)^2*g^2*n^2*x)/(3*b^2*d^2) - (2*B*(b*c - a*d)*g*(3*b*d*f - 2*b*c*g - a*d*g)*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*d^2) - (B*(b*c - a*d)*g^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*d^3) - ((b*f - a*g)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*g) + (2*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b^3*d^3) + (B^2*(b*c - a*d)^3*g^2*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d^3) + (B^2*(b*c - a*d)^3*g^2*n^2*Log[c + d*x])/(3*b^3*d^3) + (2*B^2*(b*c - a*d)^2*g*(3*b*d*f - 2*b*c*g - a*d*g)*n^2*Log[c + d*x])/(3*b^3*d^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^3*d^3)} -{(f + g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 9, -((B*(b*c - a*d)*g*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*d)) - ((b*f - a*g)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*g) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^2*d^2) + (B^2*(b*c - a*d)^2*g*n^2*Log[c + d*x])/(b^2*d^2) + (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)} -{(f + g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 6, ((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/b + (2*B*(b*c - a*d)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b*d) + (2*B^2*(b*c - a*d)*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^1, x, 9, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/g) + ((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g - (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/g + (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g + (2*B^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/g - (2*B^2*n^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^2, x, 4, ((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*f - a*g)*(f + g*x)) + (2*B*(b*c - a*d)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g)) + (2*B^2*(b*c - a*d)*n^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^3, x, 9, (B*(b*c - a*d)*g*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*f - a*g)^2*(d*f - c*g)*(f + g*x)) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*g*(b*f - a*g)^2) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(2*g*(f + g*x)^2) + (B^2*(b*c - a*d)^2*g*n^2*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^2*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2) + (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^4, x, 12, (B^2*(b*c - a*d)^2*g^2*n^2*(c + d*x))/(3*(b*f - a*g)^2*(d*f - c*g)^3*(f + g*x)) - (B*(b*c - a*d)*g^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*f - a*g)*(d*f - c*g)^3*(f + g*x)^2) + (2*B*(b*c - a*d)*g*(3*b*d*f - b*c*g - 2*a*d*g)*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*f - a*g)^3*(d*f - c*g)^2*(f + g*x)) + (b^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*g*(b*f - a*g)^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(3*g*(f + g*x)^3) + (B^2*(b*c - a*d)^3*g^2*n^2*Log[(a + b*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) - (B^2*(b*c - a*d)^3*g^2*n^2*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B^2*(b*c - a*d)^2*g*(3*b*d*f - b*c*g - 2*a*d*g)*n^2*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*n^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^5, x, 15, -((B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x)^2)/(12*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2)) - (B^2*(b*c - a*d)^3*g^3*n^2*(c + d*x))/(6*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*n^2*(c + d*x))/(4*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B*(b*c - a*d)*g^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*(b*f - a*g)*(d*f - c*g)^4*(f + g*x)^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2) + (B*(b*c - a*d)*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*f - a*g)^4*(d*f - c*g)^3*(f + g*x)) + (b^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*g*(b*f - a*g)^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*g*(f + g*x)^4) - (B^2*(b*c - a*d)^4*g^3*n^2*Log[(a + b*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*n^2*Log[(a + b*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^4*g^3*n^2*Log[(f + g*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*n^2*Log[(f + g*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^2*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n^2*Log[(f + g*x)/(c + d*x)])/(2*(b*f - a*g)^4*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*f - a*g)^4*(d*f - c*g)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{(f + g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(f + g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]} -{(f + g*x)^0/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 0, Unintegrable[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^(-1), x]} -{1/((f + g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]} -{1/((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]} -{1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x]} - - -{(f + g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{(f + g*x)^1/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(f + g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x]} -{(f + g*x)^0/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 0, Unintegrable[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^(-2), x]} -{1/((f + g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]} -{1/((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]} -{1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]} - - -(* ::Title:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p when b f-a g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^1/(c+d x)^1])^p when b f-a g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 4, (B*(b*c - a*d)^4*g^4*x)/(5*d^4) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) - (B*(b*c - a*d)*g^4*(a + b*x)^4)/(20*b*d) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b) - (B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 4, -(B*(b*c - a*d)^3*g^3*x)/(4*d^3) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*g^3*(a + b*x)^3)/(12*b*d) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + (B*(b*c - a*d)^4*g^3*Log[c + d*x])/(4*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 4, (B*(b*c - a*d)^2*g^2*x)/(3*d^2) - (B*(b*c - a*d)*g^2*(a + b*x)^2)/(6*b*d) + (g^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b) - (B*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 4, -(B*(b*c - a*d)*g*x)/(2*d) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b) + (B*(b*c - a*d)^2*g*Log[c + d*x])/(2*b*d^2)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g)) + (B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^2, x, 2, -(B/(b*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)*g^2*(a + b*x)), -((B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^3, x, 4, -B/(4*b*g^3*(a + b*x)^2) + (B*d)/(2*b*(b*c - a*d)*g^3*(a + b*x)) + (B*d^2*Log[a + b*x])/(2*b*(b*c - a*d)^2*g^3) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(2*b*g^3*(a + b*x)^2) - (B*d^2*Log[c + d*x])/(2*b*(b*c - a*d)^2*g^3)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^4, x, 4, -B/(9*b*g^4*(a + b*x)^3) + (B*d)/(6*b*(b*c - a*d)*g^4*(a + b*x)^2) - (B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) - (B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(3*b*g^4*(a + b*x)^3) + (B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^5, x, 4, -B/(16*b*g^5*(a + b*x)^4) + (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5)} - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 8, -((B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b*d)) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) + (B*(b*c - a*d)^2*g^4*(a + b*x)^3*(4*A + B + 4*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^2) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(12*A + 7*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b*d^3) + (B*(b*c - a*d)^4*g^4*(a + b*x)*(12*A + 13*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^4) + (B*(b*c - a*d)^5*g^4*Log[(b*c - a*d)/(b*(c + d*x))]*(12*A + 25*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^5) + (2*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 7, -((B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b*d)) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b*d^2) - (B*(b*c - a*d)^3*g^3*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b*d^3) - (B*(b*c - a*d)^4*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b*d^4) - (B^2*(b*c - a*d)^4*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 6, -((B*(b*c - a*d)*g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*d)) + (g^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b) + (B*(b*c - a*d)^2*g^2*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*d^2) + (B*(b*c - a*d)^3*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*d^3) + (2*B^2*(b*c - a*d)^3*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 5, -((B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*d)) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b) - (B*(b*c - a*d)^2*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*d^2) - (B^2*(b*c - a*d)^2*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^1, x, 4, -(((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b*g)) + (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (2*B^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^2, x, 3, (-2*B^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (2*B*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^3, x, 7, (2*B^2*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*(c + d*x)^2)/(4*(b*c - a*d)^2*g^3*(a + b*x)^2) + (2*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^2*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^4, x, 9, (-2*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B^2*d*(c + d*x)^2)/(2*(b*c - a*d)^3*g^4*(a + b*x)^2) - (2*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) - (2*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^4*(a + b*x)^2) - (2*b^2*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^5, x, 11, (2*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(32*(b*c - a*d)^4*g^5*(a + b*x)^4) + (2*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^4)} - - -{Log[d*(a + b*x)/(b*(c + d*x))]/(c*f + d*f*x), x, 1, PolyLog[2, (b*c - a*d)/(b*(c + d*x))]/(d*f), PolyLog[2, 1 - (d*(a + b*x))/(b*(c + d*x))]/(d*f)} - - -{Log[1 + 1/(a + b*x)]/(a + b*x), x, 1, PolyLog[2, -(1/(a + b*x))]/b} -{Log[1 - 1/(a + b*x)]/(a + b*x), x, 1, PolyLog[2, 1/(a + b*x)]/b} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 3, (e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/(B*(b*c - a*d)*g^2)} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 7, (b*e^2*E^((2*A)/B)*ExpIntegralEi[(-2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/B])/(B*(b*c - a*d)^2*g^3) - (d*e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/(B*(b*c - a*d)^2*g^3)} - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 4, -((e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/(B^2*(b*c - a*d)*g^2)) - (c + d*x)/(B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 9, (-2*b*e^2*E^((2*A)/B)*ExpIntegralEi[(-2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/B])/(B^2*(b*c - a*d)^2*g^3) + (d*e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/(B^2*(b*c - a*d)^2*g^3) + (d*(c + d*x))/(B*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])) - (b*(c + d*x)^2)/(B*(b*c - a*d)^2*g^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^2/(c+d x)^2])^p when b f-a g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 4, (2*B*(b*c - a*d)^4*g^4*x)/(5*d^4) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(5*b*d^3) + (2*B*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) - (B*(b*c - a*d)*g^4*(a + b*x)^4)/(10*b*d) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*b) - (2*B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 4, -(B*(b*c - a*d)^3*g^3*x)/(2*d^3) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(4*b*d^2) - (B*(b*c - a*d)*g^3*(a + b*x)^3)/(6*b*d) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*b) + (B*(b*c - a*d)^4*g^3*Log[c + d*x])/(2*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 4, (2*B*(b*c - a*d)^2*g^2*x)/(3*d^2) - (B*(b*c - a*d)*g^2*(a + b*x)^2)/(3*b*d) + (g^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b) - (2*B*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 4, -((B*(b*c - a*d)*g*x)/d) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*b) + (B*(b*c - a*d)^2*g*Log[c + d*x])/(b*d^2)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b*g)) + (2*B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^2, x, 2, -((2*B)/(b*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)*g^2*(a + b*x)), -((2*B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x))) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^3, x, 4, -B/(2*b*g^3*(a + b*x)^2) + (B*d)/(b*(b*c - a*d)*g^3*(a + b*x)) + (B*d^2*Log[a + b*x])/(b*(b*c - a*d)^2*g^3) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*b*g^3*(a + b*x)^2) - (B*d^2*Log[c + d*x])/(b*(b*c - a*d)^2*g^3)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^4, x, 4, (-2*B)/(9*b*g^4*(a + b*x)^3) + (B*d)/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) - (2*B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) - (2*B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(3*b*g^4*(a + b*x)^3) + (2*B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(a*g + b*g*x)^5, x, 4, -B/(8*b*g^5*(a + b*x)^4) + (B*d)/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) - (B*d^2)/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B*d^3)/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*Log[a + b*x])/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5)} - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 8, -((B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*b*d)) + (g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(5*b) + (2*B*(b*c - a*d)^2*g^4*(a + b*x)^3*(2*A + B + 2*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(15*b*d^2) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(6*A + 7*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(15*b*d^3) + (2*B*(b*c - a*d)^4*g^4*(a + b*x)*(6*A + 13*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(15*b*d^4) + (2*B*(b*c - a*d)^5*g^4*(6*A + 25*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(15*b*d^5) + (8*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 7, -((B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d)) + (g^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*b) + (B*(b*c - a*d)^2*g^3*(a + b*x)^2*(3*A + 2*B + 3*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(6*b*d^2) - (B*(b*c - a*d)^3*g^3*(a + b*x)*(3*A + 5*B + 3*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d^3) - (B*(b*c - a*d)^4*g^3*(3*A + 11*B + 3*B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b*d^4) - (2*B^2*(b*c - a*d)^4*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 6, -((2*B*(b*c - a*d)*g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d)) + (g^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*b) + (4*B*(b*c - a*d)^2*g^2*(a + b*x)*(A + B + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d^2) + (4*B*(b*c - a*d)^3*g^2*(A + 3*B + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b*d^3) + (8*B^2*(b*c - a*d)^3*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 5, -((2*B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b*d)) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*b) - (2*B*(b*c - a*d)^2*g*(A + 2*B + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(b*d^2) - (4*B^2*(b*c - a*d)^2*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^1, x, 4, -(((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b*g)) + (4*B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (8*B^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^2, x, 3, (-8*B^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (4*B*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^3, x, 7, (8*B^2*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*(c + d*x)^2)/((b*c - a*d)^2*g^3*(a + b*x)^2) + (4*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*(b*c - a*d)^2*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^4, x, 9, (-8*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (2*b*B^2*d*(c + d*x)^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (8*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) - (4*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x)) + (2*b*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x)^2) - (4*b^2*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^3*g^4*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*(b*c - a*d)^3*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^5, x, 11, (8*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/((b*c - a*d)^4*g^5*(a + b*x)^2) + (8*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (4*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)^2) + (4*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 3, (E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*ExpIntegralEi[-(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*B)])/(2*B*(b*c - a*d)*g^2*(a + b*x))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 7, (b*e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/B)])/(2*B*(b*c - a*d)^2*g^3) - (d*E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*ExpIntegralEi[-(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*B)])/(2*B*(b*c - a*d)^2*g^3*(a + b*x))} - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 4, -(E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*ExpIntegralEi[-(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*B)])/(4*B^2*(b*c - a*d)*g^2*(a + b*x)) - (c + d*x)/(2*B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 9, -(b*e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/B)])/(2*B^2*(b*c - a*d)^2*g^3) + (d*E^(A/(2*B))*Sqrt[(e*(a + b*x)^2)/(c + d*x)^2]*(c + d*x)*ExpIntegralEi[-(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*B)])/(4*B^2*(b*c - a*d)^2*g^3*(a + b*x)) + (d*(c + d*x))/(2*B*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])) - (b*(c + d*x)^2)/(2*B*(b*c - a*d)^2*g^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p when b f-a g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(a + b*x)^4, x, 3, (B*(b*c - a*d)^4*n*x)/(5*d^4) - (B*(b*c - a*d)^3*n*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*n*(a + b*x)^3)/(15*b*d^2) - (B*(b*c - a*d)*n*(a + b*x)^4)/(20*b*d) - (B*(b*c - a*d)^5*n*Log[c + d*x])/(5*b*d^5) + ((a + b*x)^5*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(5*b)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(a + b*x)^3, x, 3, -((B*(b*c - a*d)^3*n*x)/(4*d^3)) + (B*(b*c - a*d)^2*n*(a + b*x)^2)/(8*b*d^2) - (B*(b*c - a*d)*n*(a + b*x)^3)/(12*b*d) + (B*(b*c - a*d)^4*n*Log[c + d*x])/(4*b*d^4) + ((a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*b)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(a + b*x)^2, x, 3, (B*(b*c - a*d)^2*n*x)/(3*d^2) - (B*(b*c - a*d)*n*(a + b*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*n*Log[c + d*x])/(3*b*d^3) + ((a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(a + b*x)^1, x, 3, -((B*(b*c - a*d)*n*x)/(2*d)) + (B*(b*c - a*d)^2*n*Log[c + d*x])/(2*b*d^2) + ((a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*b)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a + b*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/b) + (B*n*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a + b*x)^2, x, 3, -((B*n)/(b*(a + b*x))) - (B*d*n*Log[a + b*x])/(b*(b*c - a*d)) + (B*d*n*Log[c + d*x])/(b*(b*c - a*d)) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*(a + b*x))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a + b*x)^3, x, 3, -((B*n)/(4*b*(a + b*x)^2)) + (B*d*n)/(2*b*(b*c - a*d)*(a + b*x)) + (B*d^2*n*Log[a + b*x])/(2*b*(b*c - a*d)^2) - (B*d^2*n*Log[c + d*x])/(2*b*(b*c - a*d)^2) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*(a + b*x)^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a + b*x)^4, x, 3, -((B*n)/(9*b*(a + b*x)^3)) + (B*d*n)/(6*b*(b*c - a*d)*(a + b*x)^2) - (B*d^2*n)/(3*b*(b*c - a*d)^2*(a + b*x)) - (B*d^3*n*Log[a + b*x])/(3*b*(b*c - a*d)^3) + (B*d^3*n*Log[c + d*x])/(3*b*(b*c - a*d)^3) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(a + b*x)^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a + b*x)^5, x, 3, -((B*n)/(16*b*(a + b*x)^4)) + (B*d*n)/(12*b*(b*c - a*d)*(a + b*x)^3) - (B*d^2*n)/(8*b*(b*c - a*d)^2*(a + b*x)^2) + (B*d^3*n)/(4*b*(b*c - a*d)^3*(a + b*x)) + (B*d^4*n*Log[a + b*x])/(4*b*(b*c - a*d)^4) - (B*d^4*n*Log[c + d*x])/(4*b*(b*c - a*d)^4) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*b*(a + b*x)^4)} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(a + b*x)^3, x, 8, -((B*(b*c - a*d)*n*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(6*b*d)) + ((a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*b) + (B*(b*c - a*d)^2*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(12*b*d^2) - (B*(b*c - a*d)^3*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(12*b*d^3) - (B*(b*c - a*d)^4*n*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(12*b*d^4) - (B^2*(b*c - a*d)^4*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(a + b*x)^2, x, 7, -((B*(b*c - a*d)*n*(a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b*d)) + ((a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*b) + (B*(b*c - a*d)^2*n*(a + b*x)*(2*A + B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b*d^2) + (B*(b*c - a*d)^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B*n + 2*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b*d^3) + (2*B^2*(b*c - a*d)^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(a + b*x)^1, x, 6, -((B*(b*c - a*d)*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d)) + ((a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b) - (B*(b*c - a*d)^2*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*n + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d^2) - (B^2*(b*c - a*d)^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a + b*x)^1, x, 5, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (2*B^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/b} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a + b*x)^2, x, 4, -((2*n^2*(c + d*x)*B^2)/((b*c - a*d)*(a + b*x))) - (2*n*(c + d*x)*B*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)*(a + b*x))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a + b*x)^3, x, 8, (2*B^2*d*n^2*(c + d*x))/((b*c - a*d)^2*(a + b*x)) - (b*B^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^2*(a + b*x)^2) + (2*B*d*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^2*(a + b*x)) - (b*B*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*(b*c - a*d)^2*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^2*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*c - a*d)^2*(a + b*x)^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a + b*x)^4, x, 10, -((2*B^2*d^2*n^2*(c + d*x))/((b*c - a*d)^3*(a + b*x))) + (b*B^2*d*n^2*(c + d*x)^2)/(2*(b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*(a + b*x)^3) - (2*B*d^2*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^3*(a + b*x)) + (b*B*d*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(9*(b*c - a*d)^3*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^3*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^3*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*(b*c - a*d)^3*(a + b*x)^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a + b*x)^5, x, 12, (2*B^2*d^3*n^2*(c + d*x))/((b*c - a*d)^4*(a + b*x)) - (3*b*B^2*d^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*(a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3)/(9*(b*c - a*d)^4*(a + b*x)^3) - (b^3*B^2*n^2*(c + d*x)^4)/(32*(b*c - a*d)^4*(a + b*x)^4) + (2*B*d^3*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^4*(a + b*x)) - (3*b*B*d^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*(b*c - a*d)^4*(a + b*x)^2) + (2*b^2*B*d*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*(b*c - a*d)^4*(a + b*x)^3) - (b^3*B*n*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(8*(b*c - a*d)^4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*c - a*d)^4*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*(b*c - a*d)^4*(a + b*x)^4)} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(a + b*x)^3, x, 27, -((B^3*(b*c - a*d)^3*n^3*x)/(4*d^3)) - (B^3*(b*c - a*d)^4*n^3*Log[(a + b*x)/(c + d*x)])/(4*b*d^4) + (3*B^3*(b*c - a*d)^4*n^3*Log[c + d*x])/(2*b*d^4) - (7*B^2*(b*c - a*d)^3*n^2*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*b*d^3) + (b*B^2*(b*c - a*d)^2*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*d^4) - (9*B^2*(b*c - a*d)^4*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*b*d^4) - (9*B*(b*c - a*d)^3*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*b*d^3) + (9*b*B*(b*c - a*d)^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(8*d^4) - (b^2*B*(b*c - a*d)*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*d^4) - (3*B*(b*c - a*d)^4*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*b*d^4) + ((a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(4*b) + (7*B^2*(b*c - a*d)^4*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(4*b*d^4) - (9*B^3*(b*c - a*d)^4*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4) - (3*B^2*(b*c - a*d)^4*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4) - (7*B^3*(b*c - a*d)^4*n^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(4*b*d^4) + (3*B^3*(b*c - a*d)^4*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(a + b*x)^2, x, 17, -((B^3*(b*c - a*d)^3*n^3*Log[c + d*x])/(b*d^3)) + (B^2*(b*c - a*d)^2*n^2*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d^2) + (4*B^2*(b*c - a*d)^3*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d^3) + (2*B*(b*c - a*d)^2*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b*d^2) - (b*B*(b*c - a*d)*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*d^3) + (B*(b*c - a*d)^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b*d^3) + ((a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(3*b) - (B^2*(b*c - a*d)^3*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b*d^3) + (4*B^3*(b*c - a*d)^3*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^3) + (2*B^2*(b*c - a*d)^3*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^3) + (B^3*(b*c - a*d)^3*n^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*d^3) - (2*B^3*(b*c - a*d)^3*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*d^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(a + b*x)^1, x, 11, -((3*B^2*(b*c - a*d)^2*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d^2)) - (3*B*(b*c - a*d)*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b*d) - (3*B*(b*c - a*d)^2*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b*d^2) + ((a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*b) - (3*B^3*(b*c - a*d)^2*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2) - (3*B^2*(b*c - a*d)^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2) + (3*B^3*(b*c - a*d)^2*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a + b*x)^1, x, 6, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/b + (6*B^3*n^3*PolyLog[4, (b*(c + d*x))/(d*(a + b*x))])/b} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a + b*x)^2, x, 5, -((6*B^3*n^3*(c + d*x))/((b*c - a*d)*(a + b*x))) - (6*B^2*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)*(a + b*x)) - (3*B*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)*(a + b*x))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a + b*x)^3, x, 10, (6*B^3*d*n^3*(c + d*x))/((b*c - a*d)^2*(a + b*x)) - (3*b*B^3*n^3*(c + d*x)^2)/(8*(b*c - a*d)^2*(a + b*x)^2) + (6*B^2*d*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^2*(a + b*x)) - (3*b*B^2*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*(b*c - a*d)^2*(a + b*x)^2) + (3*B*d*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^2*(a + b*x)) - (3*b*B*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*(b*c - a*d)^2*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^2*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*(b*c - a*d)^2*(a + b*x)^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a + b*x)^4, x, 13, -((6*B^3*d^2*n^3*(c + d*x))/((b*c - a*d)^3*(a + b*x))) + (3*b*B^3*d*n^3*(c + d*x)^2)/(4*(b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B^3*n^3*(c + d*x)^3)/(27*(b*c - a*d)^3*(a + b*x)^3) - (6*B^2*d^2*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^3*(a + b*x)) + (3*b*B^2*d*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*(b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(9*(b*c - a*d)^3*(a + b*x)^3) - (3*B*d^2*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^3*(a + b*x)) + (3*b*B*d*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*c - a*d)^3*(a + b*x)^2) - (b^2*B*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*(b*c - a*d)^3*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(3*(b*c - a*d)^3*(a + b*x)^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a + b*x)^5, x, 16, (6*B^3*d^3*n^3*(c + d*x))/((b*c - a*d)^4*(a + b*x)) - (9*b*B^3*d^2*n^3*(c + d*x)^2)/(8*(b*c - a*d)^4*(a + b*x)^2) + (2*b^2*B^3*d*n^3*(c + d*x)^3)/(9*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^3*n^3*(c + d*x)^4)/(128*(b*c - a*d)^4*(a + b*x)^4) + (6*B^2*d^3*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^4*(a + b*x)) - (9*b*B^2*d^2*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*(b*c - a*d)^4*(a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^2*n^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(32*(b*c - a*d)^4*(a + b*x)^4) + (3*B*d^3*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)) - (9*b*B*d^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*(b*c - a*d)^4*(a + b*x)^2) + (b^2*B*d*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B*n*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(16*(b*c - a*d)^4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^4*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*(b*c - a*d)^4*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^4*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(4*(b*c - a*d)^4*(a + b*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/((a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])), x, 4, (E^(A/(B*n))*(c + d*x)*((e*(a + b*x)^n)/(c + d*x)^n)^(1/n)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(B*n))])/(B*(b*c - a*d)*g^2*n*(a + b*x))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p when d f-c g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^1/(c+d x)^1])^p when d f-c g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 4, -(B*(b*c - a*d)^4*g^4*x)/(5*d^4) + (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(10*b*d^3) - (B*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^4)/(20*b*d) + (B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(5*b)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 4, (B*(b*c - a*d)^3*g^3*x)/(4*d^3) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(8*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3)/(12*b*d) - (B*(b*c - a*d)^4*g^3*Log[c + d*x])/(4*b*d^4) + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(4*b)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 4, -(B*(b*c - a*d)^2*g^2*x)/(3*d^2) + (B*(b*c - a*d)*g^2*(a + b*x)^2)/(6*b*d) + (B*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b*d^3) + (g^2*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*b)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 4, (B*(b*c - a*d)*g*x)/(2*d) - (B*(b*c - a*d)^2*g*Log[c + d*x])/(2*b*d^2) + (g*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*b)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(b*g)) - (B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^2, x, 3, -((A - B)/(b*g^2*(a + b*x))) - (B*(c + d*x)*Log[(e*(c + d*x))/(a + b*x)])/((b*c - a*d)*g^2*(a + b*x)), -((A*(c + d*x))/((b*c - a*d)*g^2*(a + b*x))) + (B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (B*(c + d*x)*Log[(e*(c + d*x))/(a + b*x)])/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^3, x, 4, B/(4*b*g^3*(a + b*x)^2) - (B*d)/(2*b*(b*c - a*d)*g^3*(a + b*x)) - (B*d^2*Log[a + b*x])/(2*b*(b*c - a*d)^2*g^3) + (B*d^2*Log[c + d*x])/(2*b*(b*c - a*d)^2*g^3) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(2*b*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^4, x, 4, B/(9*b*g^4*(a + b*x)^3) - (B*d)/(6*b*(b*c - a*d)*g^4*(a + b*x)^2) + (B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) + (B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(3*b*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^5, x, 4, B/(16*b*g^5*(a + b*x)^4) - (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)^2) - (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(4*b*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 19, (13*B^2*(b*c - a*d)^4*g^4*x)/(30*d^4) - (7*B^2*(b*c - a*d)^3*g^4*(a + b*x)^2)/(60*b*d^3) + (B^2*(b*c - a*d)^2*g^4*(a + b*x)^3)/(30*b*d^2) - (5*B^2*(b*c - a*d)^5*g^4*Log[a + b*x])/(6*b*d^5) - (13*B^2*(b*c - a*d)^5*g^4*Log[(c + d*x)/(a + b*x)])/(30*b*d^5) + (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(5*b*d^3) - (2*B*(b*c - a*d)^2*g^4*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(10*b*d) - (2*B*(b*c - a*d)^4*g^4*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(5*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(5*b) - (2*B*(b*c - a*d)^5*g^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5) + (2*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 15, -((5*B^2*(b*c - a*d)^3*g^3*x)/(12*d^3)) + (B^2*(b*c - a*d)^2*g^3*(a + b*x)^2)/(12*b*d^2) + (11*B^2*(b*c - a*d)^4*g^3*Log[a + b*x])/(12*b*d^4) + (5*B^2*(b*c - a*d)^4*g^3*Log[(c + d*x)/(a + b*x)])/(12*b*d^4) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(4*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(6*b*d) + (B*(b*c - a*d)^3*g^3*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*d^4) + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(4*b) + (B*(b*c - a*d)^4*g^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4) - (B^2*(b*c - a*d)^4*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 11, (B^2*(b*c - a*d)^2*g^2*x)/(3*d^2) - (B^2*(b*c - a*d)^3*g^2*Log[a + b*x])/(b*d^3) - (B^2*(b*c - a*d)^3*g^2*Log[(c + d*x)/(a + b*x)])/(3*b*d^3) + (B*(b*c - a*d)*g^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*b*d) - (2*B*(b*c - a*d)^2*g^2*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*d^3) + (g^2*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(3*b) - (2*B*(b*c - a*d)^3*g^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3) + (2*B^2*(b*c - a*d)^3*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 7, (B^2*(b*c - a*d)^2*g*Log[a + b*x])/(b*d^2) + (B*(b*c - a*d)*g*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/d^2 + (g*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(2*b) + (B*(b*c - a*d)^2*g*(A + B*Log[(e*(c + d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b*d^2) - (B^2*(b*c - a*d)^2*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^1, x, 4, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(b*g)) - (2*B*(A + B*Log[(e*(c + d*x))/(a + b*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (2*B^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^2, x, 4, (2*A*B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (2*B^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) + (2*B^2*(c + d*x)*Log[(e*(c + d*x))/(a + b*x)])/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^3, x, 8, (-2*A*B*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) + (2*B^2*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*(c + d*x)^2)/(4*(b*c - a*d)^2*g^3*(a + b*x)^2) - (2*B^2*d*(c + d*x)*Log[(e*(c + d*x))/(a + b*x)])/((b*c - a*d)^2*g^3*(a + b*x)) + (b*B*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*(b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2)/(2*(b*c - a*d)^2*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^4, x, 6, -((2*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x))) + (b*B^2*d*(c + d*x)^2)/(2*(b*c - a*d)^3*g^4*(a + b*x)^2) - (2*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) + (B^2*d^3*Log[(c + d*x)/(a + b*x)]^2)/(3*b*(b*c - a*d)^3*g^4) + (2*B*d^2*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^3*g^4*(a + b*x)) - (b*B*d*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^3*g^4*(a + b*x)^2) + (2*b^2*B*(c + d*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (2*B*d^3*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(3*b*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^5, x, 5, (2*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(32*(b*c - a*d)^4*g^5*(a + b*x)^4) - (B^2*d^4*Log[(c + d*x)/(a + b*x)]^2)/(4*b*(b*c - a*d)^4*g^5) - (2*B*d^3*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^4*g^5*(a + b*x)) + (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) - (2*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) + (b^3*B*(c + d*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (B*d^4*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(4*b*g^5*(a + b*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x))/(a + b*x)]), x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(c + d*x))/(a + b*x)]), x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(c + d*x))/(a + b*x)]), x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x))/(a + b*x)])), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])), x, 3, -(ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B]/(B*(b*c - a*d)*e*E^(A/B)*g^2))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])), x, 7, (d*ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B])/(B*(b*c - a*d)^2*e*E^(A/B)*g^3) - (b*ExpIntegralEi[(2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/B])/(B*(b*c - a*d)^2*e^2*E^((2*A)/B)*g^3)} - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(c + d*x))/(a + b*x)])^2, x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2), x, 4, -(ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B]/(B^2*(b*c - a*d)*e*E^(A/B)*g^2)) + (c + d*x)/(B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2), x, 10, (d*ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B])/(E^(A/B)*(B^2*(b*c - a*d)^2*e*g^3)) - (2*b*ExpIntegralEi[(2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/B])/(E^((2*A)/B)*(B^2*(b*c - a*d)^2*e^2*g^3)) + (c + d*x)/(B*(b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^2/(c+d x)^2])^p when d f-c g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 4, (-2*B*(b*c - a*d)^4*g^4*x)/(5*d^4) + (B*(b*c - a*d)^3*g^4*(a + b*x)^2)/(5*b*d^3) - (2*B*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^4)/(10*b*d) + (2*B*(b*c - a*d)^5*g^4*Log[c + d*x])/(5*b*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 4, (B*(b*c - a*d)^3*g^3*x)/(2*d^3) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2)/(4*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3)/(6*b*d) - (B*(b*c - a*d)^4*g^3*Log[c + d*x])/(2*b*d^4) + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(4*b)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 4, (-2*B*(b*c - a*d)^2*g^2*x)/(3*d^2) + (B*(b*c - a*d)*g^2*(a + b*x)^2)/(3*b*d) + (2*B*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b*d^3) + (g^2*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 4, (B*(b*c - a*d)*g*x)/d - (B*(b*c - a*d)^2*g*Log[c + d*x])/(b*d^2) + (g*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(2*b)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^1, x, 5, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(b*g)) - (2*B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^2, x, 3, -((A*(c + d*x))/((b*c - a*d)*g^2*(a + b*x))) + (2*B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (B*(c + d*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2])/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^3, x, 4, B/(2*b*g^3*(a + b*x)^2) - (B*d)/(b*(b*c - a*d)*g^3*(a + b*x)) - (B*d^2*Log[a + b*x])/(b*(b*c - a*d)^2*g^3) + (B*d^2*Log[c + d*x])/(b*(b*c - a*d)^2*g^3) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*b*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^4, x, 4, (2*B)/(9*b*g^4*(a + b*x)^3) - (B*d)/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) + (2*B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) + (2*B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (2*B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(3*b*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(a*g + b*g*x)^5, x, 4, B/(8*b*g^5*(a + b*x)^4) - (B*d)/(6*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2) - (B*d^3)/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(2*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(4*b*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 19, (26*B^2*(b*c - a*d)^4*g^4*x)/(15*d^4) - (7*B^2*(b*c - a*d)^3*g^4*(a + b*x)^2)/(15*b*d^3) + (2*B^2*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) - (10*B^2*(b*c - a*d)^5*g^4*Log[a + b*x])/(3*b*d^5) - (26*B^2*(b*c - a*d)^5*g^4*Log[(c + d*x)/(a + b*x)])/(15*b*d^5) + (2*B*(b*c - a*d)^3*g^4*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b*d^3) - (4*B*(b*c - a*d)^2*g^4*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b*d) - (4*B*(b*c - a*d)^4*g^4*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(5*b) - (4*B*(b*c - a*d)^5*g^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5) + (8*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)} -{(a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 15, -((5*B^2*(b*c - a*d)^3*g^3*x)/(3*d^3)) + (B^2*(b*c - a*d)^2*g^3*(a + b*x)^2)/(3*b*d^2) + (11*B^2*(b*c - a*d)^4*g^3*Log[a + b*x])/(3*b*d^4) + (5*B^2*(b*c - a*d)^4*g^3*Log[(c + d*x)/(a + b*x)])/(3*b*d^4) - (B*(b*c - a*d)^2*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(2*b*d^2) + (B*(b*c - a*d)*g^3*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*d) + (B*(b*c - a*d)^3*g^3*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/d^4 + (g^3*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(4*b) + (B*(b*c - a*d)^4*g^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b*d^4) - (2*B^2*(b*c - a*d)^4*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^4)} -{(a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 11, (4*B^2*(b*c - a*d)^2*g^2*x)/(3*d^2) - (4*B^2*(b*c - a*d)^3*g^2*Log[a + b*x])/(b*d^3) - (4*B^2*(b*c - a*d)^3*g^2*Log[(c + d*x)/(a + b*x)])/(3*b*d^3) + (2*B*(b*c - a*d)*g^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*d) - (4*B*(b*c - a*d)^2*g^2*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*d^3) + (g^2*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(3*b) - (4*B*(b*c - a*d)^3*g^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3) + (8*B^2*(b*c - a*d)^3*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3)} -{(a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 7, (4*B^2*(b*c - a*d)^2*g*Log[a + b*x])/(b*d^2) + (2*B*(b*c - a*d)*g*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/d^2 + (g*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(2*b) + (2*B*(b*c - a*d)^2*g*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b*d^2) - (4*B^2*(b*c - a*d)^2*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^2)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(a*g + b*g*x)^1, x, 4, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(b*g)) - (4*B*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (8*B^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b*g)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(a*g + b*g*x)^2, x, 4, (4*A*B*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) - (8*B^2*(c + d*x))/((b*c - a*d)*g^2*(a + b*x)) + (4*B^2*(c + d*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2])/((b*c - a*d)*g^2*(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/((b*c - a*d)*g^2*(a + b*x))} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(a*g + b*g*x)^3, x, 8, (-4*A*B*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) + (8*B^2*d*(c + d*x))/((b*c - a*d)^2*g^3*(a + b*x)) - (b*B^2*(c + d*x)^2)/((b*c - a*d)^2*g^3*(a + b*x)^2) - (4*B^2*d*(c + d*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2])/((b*c - a*d)^2*g^3*(a + b*x)) + (b*B*(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^2*g^3*(a + b*x)^2) + (d*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/((b*c - a*d)^2*g^3*(a + b*x)) - (b*(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(2*(b*c - a*d)^2*g^3*(a + b*x)^2)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(a*g + b*g*x)^4, x, 6, -((8*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x))) + (2*b*B^2*d*(c + d*x)^2)/((b*c - a*d)^3*g^4*(a + b*x)^2) - (8*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) + (4*B^2*d^3*Log[(c + d*x)/(a + b*x)]^2)/(3*b*(b*c - a*d)^3*g^4) + (4*B*d^2*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x)) - (2*b*B*d*(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x)^2) + (4*b^2*B*(c + d*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (4*B*d^3*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(3*b*g^4*(a + b*x)^3)} -{(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(a*g + b*g*x)^5, x, 5, (8*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/((b*c - a*d)^4*g^5*(a + b*x)^2) + (8*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) - (B^2*d^4*Log[(c + d*x)/(a + b*x)]^2)/(b*(b*c - a*d)^4*g^5) - (4*B*d^3*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)) + (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^4*g^5*(a + b*x)^2) - (4*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) + (b^3*B*(c + d*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(4*(b*c - a*d)^4*g^5*(a + b*x)^4) + (B*d^4*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(4*b*g^5*(a + b*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]), x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])), x, 3, -((c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)])/(2*B*(b*c - a*d)*E^(A/(2*B))*g^2*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2])} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])), x, 7, (d*(c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)])/(2*B*(b*c - a*d)^2*E^(A/(2*B))*g^3*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2]) - (b*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/B])/(2*B*(b*c - a*d)^2*e*E^(A/B)*g^3)} - - -{(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 0, Unintegrable[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x]} -{(a*g + b*g*x)^1/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x, 0, Unintegrable[(a*g + b*g*x)/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x]} -{1/((a*g + b*g*x)^1*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2), x, 0, Unintegrable[1/((a*g + b*g*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2), x]} -{1/((a*g + b*g*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2), x, 4, -((c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)])/(4*B^2*(b*c - a*d)*E^(A/(2*B))*g^2*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2]) + (c + d*x)/(2*B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))} -{1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2), x, 10, (d*(c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)])/(E^(A/(2*B))*(4*B^2*(b*c - a*d)^2*g^3*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2])) - (b*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/B])/(E^(A/B)*(2*B^2*(b*c - a*d)^2*e*g^3)) + (c + d*x)/(2*B*(b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p when d f-c g=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/((a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])), x, 4, (E^(A/(B*n))*(c + d*x)*((e*(a + b*x)^n)/(c + d*x)^n)^(1/n)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(B*n))])/(B*(b*c - a*d)*g^2*n*(a + b*x))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^1/(c+d x)^1])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 3, (B*(b*c - a*d)*g*(a^3*d^3*g^3 - a^2*b*d^2*g^2*(5*d*f - c*g) + a*b^2*d*g*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2) - b^3*(10*d^3*f^3 - 10*c*d^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2 - a*b*d*g*(5*d*f - c*g) + b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*x^2)/(10*b^3*d^3) - (B*(b*c - a*d)*g^3*(5*b*d*f - b*c*g - a*d*g)*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*x^4)/(20*b*d) - (B*(b*f - a*g)^5*Log[a + b*x])/(5*b^5*g) + ((f + g*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*g) + (B*(d*f - c*g)^5*Log[c + d*x])/(5*d^5*g)} -{(f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 3, -(B*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*x)/(4*b^3*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*g^3*x^3)/(12*b*d) - (B*(b*f - a*g)^4*Log[a + b*x])/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*g) + (B*(d*f - c*g)^4*Log[c + d*x])/(4*d^4*g)} -{(f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 3, -(B*(b*c - a*d)*g*(3*b*d*f - b*c*g - a*d*g)*x)/(3*b^2*d^2) - (B*(b*c - a*d)*g^2*x^2)/(6*b*d) - (B*(b*f - a*g)^3*Log[a + b*x])/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*g) + (B*(d*f - c*g)^3*Log[c + d*x])/(3*d^3*g)} -{(f + g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 3, -(B*(b*c - a*d)*g*x)/(2*b*d) - (B*(b*f - a*g)^2*Log[a + b*x])/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*g) + (B*(d*f - c*g)^2*Log[c + d*x])/(2*d^2*g)} -{(f + g*x)^0*(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 3, A*x + (B*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/b - (B*(b*c - a*d)*Log[c + d*x])/(b*d)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^1, x, 7, -((B*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g) + ((A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x])/g + (B*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (B*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (B*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^2, x, 3, ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*f - a*g)*(f + g*x)) + (B*(b*c - a*d)*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^3, x, 3, -(B*(b*c - a*d))/(2*(b*f - a*g)*(d*f - c*g)*(f + g*x)) + (b^2*B*Log[a + b*x])/(2*g*(b*f - a*g)^2) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(2*g*(f + g*x)^2) - (B*d^2*Log[c + d*x])/(2*g*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*Log[f + g*x])/(2*(b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^4, x, 3, -(B*(b*c - a*d))/(6*(b*f - a*g)*(d*f - c*g)*(f + g*x)^2) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g))/(3*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (b^3*B*Log[a + b*x])/(3*g*(b*f - a*g)^3) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(3*g*(f + g*x)^3) - (B*d^3*Log[c + d*x])/(3*g*(d*f - c*g)^3) + (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*Log[f + g*x])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^5, x, 3, -(B*(b*c - a*d))/(12*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g))/(8*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2)))/(4*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*Log[a + b*x])/(4*g*(b*f - a*g)^4) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(4*g*(f + g*x)^4) - (B*d^4*Log[c + d*x])/(4*g*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*Log[f + g*x])/(4*(b*f - a*g)^4*(d*f - c*g)^4)} - - -{(f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 15, (B^2*(b*c - a*d)^3*g^3*x)/(6*b^3*d^3) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*x)/(4*b^3*d^3) + (B^2*(b*c - a*d)^2*g^3*(c + d*x)^2)/(12*b^2*d^4) + (B^2*(b*c - a*d)^4*g^3*Log[(a + b*x)/(c + d*x)])/(6*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*Log[(a + b*x)/(c + d*x)])/(4*b^4*d^4) - (B*(b*c - a*d)*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2*d^4) - (B*(b*c - a*d)*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b*d^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*d^4) - ((b*f - a*g)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*g) + (B^2*(b*c - a*d)^4*g^3*Log[c + d*x])/(6*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*Log[c + d*x])/(4*b^4*d^4) + (B^2*(b*c - a*d)^2*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*Log[c + d*x])/(2*b^4*d^4) - (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(2*b^4*d^4)} -{(f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 12, (B^2*(b*c - a*d)^2*g^2*x)/(3*b^2*d^2) + (B^2*(b*c - a*d)^3*g^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d^3) - (2*B*(b*c - a*d)*g*(3*b*d*f - 2*b*c*g - a*d*g)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3*d^2) - (B*(b*c - a*d)*g^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*d^3) + (2*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3*d^3) - ((b*f - a*g)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*g) + (B^2*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b^3*d^3) + (2*B^2*(b*c - a*d)^2*g*(3*b*d*f - 2*b*c*g - a*d*g)*Log[c + d*x])/(3*b^3*d^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^3*d^3)} -{(f + g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 9, -((B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*d)) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*d^2) - ((b*f - a*g)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*g) + (B^2*(b*c - a*d)^2*g*Log[c + d*x])/(b^2*d^2) + (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)} -{(f + g*x)^0*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 6, (2*B*(b*c - a*d)*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*d) + ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/b + (2*B^2*(b*c - a*d)*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^1, x, 9, -((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/g) + ((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/g + (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g + (2*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/g - (2*B^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^2, x, 4, ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*f - a*g)*(f + g*x)) + (2*B*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g)) + (2*B^2*(b*c - a*d)*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^3, x, 9, (B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*f - a*g)^2*(d*f - c*g)*(f + g*x)) + (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*g*(b*f - a*g)^2) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(2*g*(f + g*x)^2) + (B^2*(b*c - a*d)^2*g*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^2*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2) + (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^4, x, 12, (B^2*(b*c - a*d)^2*g^2*(c + d*x))/(3*(b*f - a*g)^2*(d*f - c*g)^3*(f + g*x)) + (B^2*(b*c - a*d)^3*g^2*Log[(a + b*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) - (B*(b*c - a*d)*g^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*f - a*g)*(d*f - c*g)^3*(f + g*x)^2) + (2*B*(b*c - a*d)*g*(3*b*d*f - b*c*g - 2*a*d*g)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*f - a*g)^3*(d*f - c*g)^2*(f + g*x)) + (b^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*g*(b*f - a*g)^3) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(3*g*(f + g*x)^3) - (B^2*(b*c - a*d)^3*g^2*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B^2*(b*c - a*d)^2*g*(3*b*d*f - b*c*g - 2*a*d*g)*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x)^5, x, 15, -((B^2*(b*c - a*d)^2*g^3*(c + d*x)^2)/(12*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2)) - (B^2*(b*c - a*d)^3*g^3*(c + d*x))/(6*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*(c + d*x))/(4*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) - (B^2*(b*c - a*d)^4*g^3*Log[(a + b*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*Log[(a + b*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B*(b*c - a*d)*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*(b*f - a*g)*(d*f - c*g)^4*(f + g*x)^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2) + (B*(b*c - a*d)*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*f - a*g)^4*(d*f - c*g)^3*(f + g*x)) + (b^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*g*(b*f - a*g)^4) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(4*g*(f + g*x)^4) + (B^2*(b*c - a*d)^4*g^3*Log[(f + g*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*Log[(f + g*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^2*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*Log[(f + g*x)/(c + d*x)])/(2*(b*f - a*g)^4*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*f - a*g)^4*(d*f - c*g)^4)} - - -{Log[(1 + x)/(-1 + x)]/x^2, x, 3, 2*Log[-(x/(1 - x))] - ((1 + x)*Log[-((1 + x)/(1 - x))])/x} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x]} -{(f + g*x)^1/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 0, Unintegrable[(f + g*x)/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x]} -{(f + g*x)^0/(A + B*Log[(e*(a + b*x))/(c + d*x)]), x, 0, Unintegrable[(A + B*Log[(e*(a + b*x))/(c + d*x)])^(-1), x]} -{1/((f + g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x]} -{1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x]} -{1/((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])), x]} - - -{(f + g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x]} -{(f + g*x)^1/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 0, Unintegrable[(f + g*x)/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x]} -{(f + g*x)^0/(A + B*Log[(e*(a + b*x))/(c + d*x)])^2, x, 0, Unintegrable[(A + B*Log[(e*(a + b*x))/(c + d*x)])^(-2), x]} -{1/((f + g*x)^1*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]} -{1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]} -{1/((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^2/(c+d x)^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 3, (2*B*(b*c - a*d)*g*(a^3*d^3*g^3 - a^2*b*d^2*g^2*(5*d*f - c*g) + a*b^2*d*g*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2) - b^3*(10*d^3*f^3 - 10*c*d^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2 - a*b*d*g*(5*d*f - c*g) + b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*x^2)/(5*b^3*d^3) - (2*B*(b*c - a*d)*g^3*(5*b*d*f - b*c*g - a*d*g)*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*x^4)/(10*b*d) - (2*B*(b*f - a*g)^5*Log[a + b*x])/(5*b^5*g) + ((f + g*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*g) + (2*B*(d*f - c*g)^5*Log[c + d*x])/(5*d^5*g)} -{(f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 3, -(B*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*x)/(2*b^3*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*x^2)/(4*b^2*d^2) - (B*(b*c - a*d)*g^3*x^3)/(6*b*d) - (B*(b*f - a*g)^4*Log[a + b*x])/(2*b^4*g) + ((f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*g) + (B*(d*f - c*g)^4*Log[c + d*x])/(2*d^4*g)} -{(f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 3, (-2*B*(b*c - a*d)*g*(3*b*d*f - b*c*g - a*d*g)*x)/(3*b^2*d^2) - (B*(b*c - a*d)*g^2*x^2)/(3*b*d) - (2*B*(b*f - a*g)^3*Log[a + b*x])/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*g) + (2*B*(d*f - c*g)^3*Log[c + d*x])/(3*d^3*g)} -{(f + g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 3, -((B*(b*c - a*d)*g*x)/(b*d)) - (B*(b*f - a*g)^2*Log[a + b*x])/(b^2*g) + ((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*g) + (B*(d*f - c*g)^2*Log[c + d*x])/(d^2*g)} -{(f + g*x)^0*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 3, A*x + (B*(a + b*x)*Log[(e*(a + b*x)^2)/(c + d*x)^2])/b - (2*B*(b*c - a*d)*Log[c + d*x])/(b*d)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^1, x, 7, (-2*B*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g + ((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[f + g*x])/g + (2*B*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (2*B*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (2*B*PolyLog[2, (d*(f + g*x))/(d*f - c*g)])/g} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^2, x, 3, ((a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*f - a*g)*(f + g*x)) + (2*B*(b*c - a*d)*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^3, x, 3, -((B*(b*c - a*d))/((b*f - a*g)*(d*f - c*g)*(f + g*x))) + (b^2*B*Log[a + b*x])/(g*(b*f - a*g)^2) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(2*g*(f + g*x)^2) - (B*d^2*Log[c + d*x])/(g*(d*f - c*g)^2) + (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*Log[f + g*x])/((b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^4, x, 3, -(B*(b*c - a*d))/(3*(b*f - a*g)*(d*f - c*g)*(f + g*x)^2) - (2*B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g))/(3*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (2*b^3*B*Log[a + b*x])/(3*g*(b*f - a*g)^3) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(3*g*(f + g*x)^3) - (2*B*d^3*Log[c + d*x])/(3*g*(d*f - c*g)^3) + (2*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*Log[f + g*x])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^5, x, 3, -(B*(b*c - a*d))/(6*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g))/(4*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2)))/(2*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*Log[a + b*x])/(2*g*(b*f - a*g)^4) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(4*g*(f + g*x)^4) - (B*d^4*Log[c + d*x])/(2*g*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*Log[f + g*x])/(2*(b*f - a*g)^4*(d*f - c*g)^4)} - - -{(f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 15, (2*B^2*(b*c - a*d)^3*g^3*x)/(3*b^3*d^3) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*x)/(b^3*d^3) + (B^2*(b*c - a*d)^2*g^3*(c + d*x)^2)/(3*b^2*d^4) - (B*(b*c - a*d)*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b^4*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*b^2*d^4) - (B*(b*c - a*d)*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d^4) - ((b*f - a*g)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*g) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*d^4) + (2*B^2*(b*c - a*d)^4*g^3*Log[(a + b*x)/(c + d*x)])/(3*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*Log[(a + b*x)/(c + d*x)])/(b^4*d^4) + (2*B^2*(b*c - a*d)^4*g^3*Log[c + d*x])/(3*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*Log[c + d*x])/(b^4*d^4) + (2*B^2*(b*c - a*d)^2*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*Log[c + d*x])/(b^4*d^4) - (2*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*d^4)} -{(f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 12, (4*B^2*(b*c - a*d)^2*g^2*x)/(3*b^2*d^2) - (4*B*(b*c - a*d)*g*(3*b*d*f - 2*b*c*g - a*d*g)*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b^3*d^2) - (2*B*(b*c - a*d)*g^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d^3) - ((b*f - a*g)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*g) + (4*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b^3*d^3) + (4*B^2*(b*c - a*d)^3*g^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d^3) + (4*B^2*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b^3*d^3) + (8*B^2*(b*c - a*d)^2*g*(3*b*d*f - 2*b*c*g - a*d*g)*Log[c + d*x])/(3*b^3*d^3) + (8*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^3*d^3)} -{(f + g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 9, -((2*B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b^2*d)) - ((b*f - a*g)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*b^2*g) + ((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*g) + (2*B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^2*d^2) + (4*B^2*(b*c - a*d)^2*g*Log[c + d*x])/(b^2*d^2) + (4*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)} -{(f + g*x)^0*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 6, ((a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/b + (4*B*(b*c - a*d)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(b*d) + (8*B^2*(b*c - a*d)*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^1, x, 9, -(((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2*Log[(b*c - a*d)/(b*(c + d*x))])/g) + ((A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g - (4*B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/g + (4*B*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g + (8*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/g - (8*B^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^2, x, 4, ((a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/((b*f - a*g)*(f + g*x)) + (4*B*(b*c - a*d)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g)) + (8*B^2*(b*c - a*d)*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)*(d*f - c*g))} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^3, x, 9, (2*B*(b*c - a*d)*g*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*f - a*g)^2*(d*f - c*g)*(f + g*x)) + (b^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(2*g*(b*f - a*g)^2) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(2*g*(f + g*x)^2) + (4*B^2*(b*c - a*d)^2*g*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^2*(d*f - c*g)^2) + (2*B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2) + (4*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^4, x, 12, (4*B^2*(b*c - a*d)^2*g^2*(c + d*x))/(3*(b*f - a*g)^2*(d*f - c*g)^3*(f + g*x)) - (2*B*(b*c - a*d)*g^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*f - a*g)*(d*f - c*g)^3*(f + g*x)^2) + (4*B*(b*c - a*d)*g*(3*b*d*f - b*c*g - 2*a*d*g)*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*f - a*g)^3*(d*f - c*g)^2*(f + g*x)) + (b^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*g*(b*f - a*g)^3) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(3*g*(f + g*x)^3) + (4*B^2*(b*c - a*d)^3*g^2*Log[(a + b*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) - (4*B^2*(b*c - a*d)^3*g^2*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (8*B^2*(b*c - a*d)^2*g*(3*b*d*f - b*c*g - 2*a*d*g)*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (4*B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (8*B^2*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3)} -{(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^5, x, 15, -((B^2*(b*c - a*d)^2*g^3*(c + d*x)^2)/(3*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2)) - (2*B^2*(b*c - a*d)^3*g^3*(c + d*x))/(3*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*(c + d*x))/((b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B*(b*c - a*d)*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*f - a*g)*(d*f - c*g)^4*(f + g*x)^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2) + (B*(b*c - a*d)*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/((b*f - a*g)^4*(d*f - c*g)^3*(f + g*x)) + (b^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*g*(b*f - a*g)^4) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(4*g*(f + g*x)^4) - (2*B^2*(b*c - a*d)^4*g^3*Log[(a + b*x)/(c + d*x)])/(3*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*Log[(a + b*x)/(c + d*x)])/((b*f - a*g)^4*(d*f - c*g)^4) + (2*B^2*(b*c - a*d)^4*g^3*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^4*(d*f - c*g)^4) + (2*B^2*(b*c - a*d)^2*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*Log[(f + g*x)/(c + d*x)])/((b*f - a*g)^4*(d*f - c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^4*(d*f - c*g)^4) - (2*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^4*(d*f - c*g)^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x]} -{(f + g*x)^1/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 0, Unintegrable[(f + g*x)/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x]} -{(f + g*x)^0/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]), x, 0, Unintegrable[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-1), x]} -{1/((f + g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]} -{1/((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]} -{1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]} - - -{(f + g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 0, Unintegrable[(f + g*x)^2/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x]} -{(f + g*x)^1/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 0, Unintegrable[(f + g*x)/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x]} -{(f + g*x)^0/(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2, x, 0, Unintegrable[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2), x]} -{1/((f + g*x)^1*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 0, Unintegrable[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]} -{1/((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 0, Unintegrable[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]} -{1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x, 0, Unintegrable[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (A+B Log[e (a+b x)^n/(c+d x)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(g + h*x)^4, x, 3, (B*(b*c - a*d)*h*(a^3*d^3*h^3 - a^2*b*d^2*h^2*(5*d*g - c*h) + a*b^2*d*h*(10*d^2*g^2 - 5*c*d*g*h + c^2*h^2) - b^3*(10*d^3*g^3 - 10*c*d^2*g^2*h + 5*c^2*d*g*h^2 - c^3*h^3))*n*x)/(5*b^4*d^4) - (B*(b*c - a*d)*h^2*(a^2*d^2*h^2 - a*b*d*h*(5*d*g - c*h) + b^2*(10*d^2*g^2 - 5*c*d*g*h + c^2*h^2))*n*x^2)/(10*b^3*d^3) - (B*(b*c - a*d)*h^3*(5*b*d*g - b*c*h - a*d*h)*n*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*h^4*n*x^4)/(20*b*d) - (B*(b*g - a*h)^5*n*Log[a + b*x])/(5*b^5*h) + (B*(d*g - c*h)^5*n*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(5*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(g + h*x)^3, x, 3, -((B*(b*c - a*d)*h*(a^2*d^2*h^2 - a*b*d*h*(4*d*g - c*h) + b^2*(6*d^2*g^2 - 4*c*d*g*h + c^2*h^2))*n*x)/(4*b^3*d^3)) - (B*(b*c - a*d)*h^2*(4*b*d*g - b*c*h - a*d*h)*n*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*h^3*n*x^3)/(12*b*d) - (B*(b*g - a*h)^4*n*Log[a + b*x])/(4*b^4*h) + (B*(d*g - c*h)^4*n*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(g + h*x)^2, x, 3, -((B*(b*c - a*d)*h*(3*b*d*g - b*c*h - a*d*h)*n*x)/(3*b^2*d^2)) - (B*(b*c - a*d)*h^2*n*x^2)/(6*b*d) - (B*(b*g - a*h)^3*n*Log[a + b*x])/(3*b^3*h) + (B*(d*g - c*h)^3*n*Log[c + d*x])/(3*d^3*h) + ((g + h*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(g + h*x)^1, x, 3, -((B*(b*c - a*d)*h*n*x)/(2*b*d)) - (B*(b*g - a*h)^2*n*Log[a + b*x])/(2*b^2*h) + (B*(d*g - c*h)^2*n*Log[c + d*x])/(2*d^2*h) + ((g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(2*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])*(g + h*x)^0, x, 3, A*x - (B*(b*c - a*d)*n*Log[c + d*x])/(b*d) + (B*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/b} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g + h*x)^1, x, 7, -((B*n*Log[-((h*(a + b*x))/(b*g - a*h))]*Log[g + h*x])/h) + (B*n*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[g + h*x])/h + ((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[g + h*x])/h - (B*n*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/h + (B*n*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/h} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g + h*x)^2, x, 3, (b*B*n*Log[a + b*x])/(h*(b*g - a*h)) - (B*d*n*Log[c + d*x])/(h*(d*g - c*h)) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(h*(g + h*x)) + (B*(b*c - a*d)*n*Log[g + h*x])/((b*g - a*h)*(d*g - c*h))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g + h*x)^3, x, 3, -((B*(b*c - a*d)*n)/(2*(b*g - a*h)*(d*g - c*h)*(g + h*x))) + (b^2*B*n*Log[a + b*x])/(2*h*(b*g - a*h)^2) - (B*d^2*n*Log[c + d*x])/(2*h*(d*g - c*h)^2) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*h*(g + h*x)^2) + (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*Log[g + h*x])/(2*(b*g - a*h)^2*(d*g - c*h)^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g + h*x)^4, x, 3, -((B*(b*c - a*d)*n)/(6*(b*g - a*h)*(d*g - c*h)*(g + h*x)^2)) - (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n)/(3*(b*g - a*h)^2*(d*g - c*h)^2*(g + h*x)) + (b^3*B*n*Log[a + b*x])/(3*h*(b*g - a*h)^3) - (B*d^3*n*Log[c + d*x])/(3*h*(d*g - c*h)^3) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*h*(g + h*x)^3) + (B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[g + h*x])/(3*(b*g - a*h)^3*(d*g - c*h)^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(g + h*x)^5, x, 3, -((B*(b*c - a*d)*n)/(12*(b*g - a*h)*(d*g - c*h)*(g + h*x)^3)) - (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n)/(8*(b*g - a*h)^2*(d*g - c*h)^2*(g + h*x)^2) - (B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n)/(4*(b*g - a*h)^3*(d*g - c*h)^3*(g + h*x)) + (b^4*B*n*Log[a + b*x])/(4*h*(b*g - a*h)^4) - (B*d^4*n*Log[c + d*x])/(4*h*(d*g - c*h)^4) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*h*(g + h*x)^4) - (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*(2*a*b*d^2*g*h - a^2*d^2*h^2 - b^2*(2*d^2*g^2 - 2*c*d*g*h + c^2*h^2))*n*Log[g + h*x])/(4*(b*g - a*h)^4*(d*g - c*h)^4)} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(g + h*x)^2, x, 13, (B^2*(b*c - a*d)^2*h^2*n^2*x)/(3*b^2*d^2) + (B^2*(b*c - a*d)^3*h^2*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d^3) + (B^2*(b*c - a*d)^3*h^2*n^2*Log[c + d*x])/(3*b^3*d^3) + (2*B^2*(b*c - a*d)^2*h*(3*b*d*g - 2*b*c*h - a*d*h)*n^2*Log[c + d*x])/(3*b^3*d^3) - (2*B*(b*c - a*d)*h*(3*b*d*g - 2*b*c*h - a*d*h)*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b^3*d^2) - (B*(b*c - a*d)*h^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b*d^3) + (2*B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*b^3*d^3) - ((b*g - a*h)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*b^3*h) + ((g + h*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*h) + (2*B^2*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^3*d^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(g + h*x)^1, x, 10, (B^2*(b*c - a*d)^2*h*n^2*Log[c + d*x])/(b^2*d^2) - (B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^2*d) + (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^2*d^2) - ((b*g - a*h)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b^2*h) + ((g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*h) + (B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(g + h*x)^0, x, 6, (2*B*(b*c - a*d)*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b*d) + ((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/b + (2*B^2*(b*c - a*d)*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(g + h*x)^1, x, 10, -((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/h) + ((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h - (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/h + (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h + (2*B^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/h - (2*B^2*n^2*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(g + h*x)^2, x, 5, ((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*g - a*h)*(g + h*x)) + (2*B*(b*c - a*d)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) + (2*B^2*(b*c - a*d)*n^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(g + h*x)^3, x, 10, (B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*g - a*h)^2*(d*g - c*h)*(g + h*x)) + (b^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*h*(b*g - a*h)^2) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(2*h*(g + h*x)^2) + (B^2*(b*c - a*d)^2*h*n^2*Log[(g + h*x)/(c + d*x)])/((b*g - a*h)^2*(d*g - c*h)^2) + (B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) + (B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2)} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(g + h*x)^2, x, 19, -((B^3*(b*c - a*d)^3*h^2*n^3*Log[c + d*x])/(b^3*d^3)) + (B^2*(b*c - a*d)^2*h^2*n^2*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^3*d^2) - (2*B^2*(b*c - a*d)^2*h*(3*b*d*g - 2*b*c*h - a*d*h)*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^3*d^3) - (B*(b*c - a*d)*h*(3*b*d*g - 2*b*c*h - a*d*h)*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b^3*d^2) - (B*(b*c - a*d)*h^2*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b*d^3) + (B*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b^3*d^3) - ((b*g - a*h)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(3*b^3*h) + ((g + h*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(3*h) - (B^2*(b*c - a*d)^3*h^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*d^3) - (2*B^3*(b*c - a*d)^2*h*(3*b*d*g - 2*b*c*h - a*d*h)*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) + (2*B^2*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) + (B^3*(b*c - a*d)^3*h^2*n^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*d^3) - (2*B^3*(b*c - a*d)*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(g + h*x)^1, x, 13, -((3*B^2*(b*c - a*d)^2*h*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(b^2*d^2)) - (3*B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b^2*d) + (3*B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*b^2*d^2) - ((b*g - a*h)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*b^2*h) + ((g + h*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*h) - (3*B^3*(b*c - a*d)^2*h*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2) + (3*B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2) - (3*B^3*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(g + h*x)^0, x, 6, (3*B*(b*c - a*d)*n*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(b*d) + ((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/b + (6*B^2*(b*c - a*d)*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d) - (6*B^3*(b*c - a*d)*n^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*d)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(g + h*x)^1, x, 12, -((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/h) + ((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h - (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/h + (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/h - (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h - (6*B^3*n^3*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))])/h + (6*B^3*n^3*PolyLog[4, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/h} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(g + h*x)^2, x, 6, ((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*g - a*h)*(g + h*x)) + (3*B*(b*c - a*d)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) + (6*B^2*(b*c - a*d)*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) - (6*B^3*(b*c - a*d)*n^3*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(g + h*x)^3, x, 13, (3*B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*g - a*h)^2*(d*g - c*h)*(g + h*x)) + (b^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*h*(b*g - a*h)^2) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(2*h*(g + h*x)^2) + (3*B^2*(b*c - a*d)^2*h*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) + (3*B*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/(2*(b*g - a*h)^2*(d*g - c*h)^2) + (3*B^3*(b*c - a*d)^2*h*n^3*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) + (3*B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) - (3*B^3*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^3*PolyLog[3, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2)} - - -(* ::Subsubsection:: *) -(*p<0*) diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m deleted file mode 100644 index 353b59e..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.2.2 (f+g x)^m (h+i x)^q (A+B log(e ((a+b x) over (c+d x))^n))^p.m +++ /dev/null @@ -1,439 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x)/(c+d x))^n])^p when b f-a g=0 and d h-c i=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x) / (c+d x)])^1 when b f-a g=0 and d h-c i=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, -((B*(b*c - a*d)^4*g^3*i*x)/(20*b*d^3)) + (B*(b*c - a*d)^3*g^3*i*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d)^2*g^3*i*(a + b*x)^3)/(60*b^2*d) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^2) + (B*(b*c - a*d)^5*g^3*i*Log[c + d*x])/(20*b^2*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, (B*(b*c - a*d)^3*g^2*i*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*(a + b*x)^2)/(24*b^2*d) + (g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2) - (B*(b*c - a*d)^4*g^2*i*Log[c + d*x])/(12*b^2*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, -((B*(b*c - a*d)^2*g*i*x)/(6*b*d)) + (g*i*(a + b*x)^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b) + ((b*c - a*d)*g*i*(a + b*x)^2*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^2) + (B*(b*c - a*d)^3*g*i*Log[c + d*x])/(6*b^2*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 4, -((B*(b*c - a*d)*i*x)/(2*b)) - (B*(b*c - a*d)^2*i*Log[a + b*x])/(2*b^2*d) + (i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^1, x, 6, (i*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g) - ((b*c - a*d)*i*Log[-((b*c - a*d)/(d*(a + b*x)))]*(A - B + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g) + (B*(b*c - a*d)*i*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^2*g)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^2, x, 5, -((B*i*(c + d*x))/(b*g^2*(a + b*x))) - (i*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g^2*(a + b*x)) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (B*d*i*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^3, x, 2, -((B*i*(c + d*x)^2)/(4*(b*c - a*d)*g^3*(a + b*x)^2)) - (i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)*g^3*(a + b*x)^2)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^4, x, 5, (B*d*i*(c + d*x)^2)/(4*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*B*i*(c + d*x)^3)/(9*(b*c - a*d)^2*g^4*(a + b*x)^3) + (d*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^2*g^4*(a + b*x)^3)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^5, x, 5, -((B*d^2*i*(c + d*x)^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^2)) + (2*b*B*d*i*(c + d*x)^3)/(9*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B*i*(c + d*x)^4)/(16*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^3*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, (B*(b*c - a*d)^5*g^3*i^2*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*(c + d*x)^2)/(120*b*d^4) - (19*B*(b*c - a*d)^3*g^3*i^2*(c + d*x)^3)/(180*d^4) + (13*b*B*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4)/(120*d^4) - (b^2*B*(b*c - a*d)*g^3*i^2*(c + d*x)^5)/(30*d^4) + (B*(b*c - a*d)^6*g^3*i^2*Log[(a + b*x)/(c + d*x)])/(60*b^3*d^4) - ((b*c - a*d)^3*g^3*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^4) + (3*b*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^4) - (3*b^2*(b*c - a*d)*g^3*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^4) + (b^3*g^3*i^2*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^4) + (B*(b*c - a*d)^6*g^3*i^2*Log[c + d*x])/(60*b^3*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, -((B*(b*c - a*d)^4*g^2*i^2*x)/(30*b^2*d^2)) - (B*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2)/(60*b*d^3) + (B*(b*c - a*d)^2*g^2*i^2*(c + d*x)^3)/(10*d^3) - (b*B*(b*c - a*d)*g^2*i^2*(c + d*x)^4)/(20*d^3) - (B*(b*c - a*d)^5*g^2*i^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) + ((b*c - a*d)^2*g^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^3) - (b*(b*c - a*d)*g^2*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^3) + (b^2*g^2*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^3) - (B*(b*c - a*d)^5*g^2*i^2*Log[c + d*x])/(30*b^3*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, (B*(b*c - a*d)^3*g*i^2*x)/(12*b^2*d) + (B*(b*c - a*d)^2*g*i^2*(c + d*x)^2)/(24*b*d^2) - (B*(b*c - a*d)*g*i^2*(c + d*x)^3)/(12*d^2) + (B*(b*c - a*d)^4*g*i^2*Log[(a + b*x)/(c + d*x)])/(12*b^3*d^2) - ((b*c - a*d)*g*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^2) + (b*g*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2) + (B*(b*c - a*d)^4*g*i^2*Log[c + d*x])/(12*b^3*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 4, -((B*(b*c - a*d)^2*i^2*x)/(3*b^2)) - (B*(b*c - a*d)*i^2*(c + d*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*i^2*Log[a + b*x])/(3*b^3*d) + (i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^1, x, 10, -((B*d*(b*c - a*d)*i^2*x)/(2*b^2*g)) - (B*(b*c - a*d)^2*i^2*Log[(a + b*x)/(c + d*x)])/(2*b^3*g) + (d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g) + (i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b*g) - (3*B*(b*c - a*d)^2*i^2*Log[c + d*x])/(2*b^3*g) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (B*(b*c - a*d)^2*i^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^2, x, 8, -((B*(b*c - a*d)*i^2*(c + d*x))/(b^2*g^2*(a + b*x))) + (d^2*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2) - ((b*c - a*d)*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^2*(a + b*x)) - (B*d*(b*c - a*d)*i^2*Log[c + d*x])/(b^3*g^2) - (2*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (2*B*d*(b*c - a*d)*i^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^3, x, 7, -((B*d*i^2*(c + d*x))/(b^2*g^3*(a + b*x))) - (B*i^2*(c + d*x)^2)/(4*b*g^3*(a + b*x)^2) - (d*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^3*(a + b*x)) - (i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b*g^3*(a + b*x)^2) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (B*d^2*i^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^4, x, 2, -((B*i^2*(c + d*x)^3)/(9*(b*c - a*d)*g^4*(a + b*x)^3)) - (i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)*g^4*(a + b*x)^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^5, x, 5, (B*d*i^2*(c + d*x)^3)/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B*i^2*(c + d*x)^4)/(16*(b*c - a*d)^2*g^5*(a + b*x)^4) + (d*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^6, x, 5, -((B*d^2*i^2*(c + d*x)^3)/(9*(b*c - a*d)^3*g^6*(a + b*x)^3)) + (b*B*d*i^2*(c + d*x)^4)/(8*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*B*i^2*(c + d*x)^5)/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, (B*(b*c - a*d)^6*g^3*i^3*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2)/(280*b^2*d^4) + (B*(b*c - a*d)^4*g^3*i^3*(c + d*x)^3)/(420*b*d^4) - (17*B*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4)/(280*d^4) + (b*B*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5)/(14*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*(c + d*x)^6)/(42*d^4) + (B*(b*c - a*d)^7*g^3*i^3*Log[(a + b*x)/(c + d*x)])/(140*b^4*d^4) - ((b*c - a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^4) + (3*b*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^4) - (b^2*(b*c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4) + (b^3*g^3*i^3*(c + d*x)^7*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(7*d^4) + (B*(b*c - a*d)^7*g^3*i^3*Log[c + d*x])/(140*b^4*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, -((B*(b*c - a*d)^5*g^2*i^3*x)/(60*b^3*d^2)) - (B*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*c - a*d)^3*g^2*i^3*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*(c + d*x)^4)/(120*d^3) - (b*B*(b*c - a*d)*g^2*i^3*(c + d*x)^5)/(30*d^3) - (B*(b*c - a*d)^6*g^2*i^3*Log[(a + b*x)/(c + d*x)])/(60*b^4*d^3) + ((b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^3) - (B*(b*c - a*d)^6*g^2*i^3*Log[c + d*x])/(60*b^4*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 5, (B*(b*c - a*d)^4*g*i^3*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4)/(20*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[(a + b*x)/(c + d*x)])/(20*b^4*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^2) + (b*g*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^2) + (B*(b*c - a*d)^5*g*i^3*Log[c + d*x])/(20*b^4*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)]), x, 4, -((B*(b*c - a*d)^3*i^3*x)/(4*b^3)) - (B*(b*c - a*d)^2*i^3*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*i^3*(c + d*x)^3)/(12*b*d) - (B*(b*c - a*d)^4*i^3*Log[a + b*x])/(4*b^4*d) + (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^1, x, 14, -((5*B*d*(b*c - a*d)^2*i^3*x)/(6*b^3*g)) - (B*(b*c - a*d)*i^3*(c + d*x)^2)/(6*b^2*g) - (5*B*(b*c - a*d)^3*i^3*Log[(a + b*x)/(c + d*x)])/(6*b^4*g) + (d*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g) + ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g) + (i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*g) - (11*B*(b*c - a*d)^3*i^3*Log[c + d*x])/(6*b^4*g) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (B*(b*c - a*d)^3*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^2, x, 11, -((B*d^2*(b*c - a*d)*i^3*x)/(2*b^3*g^2)) - (B*(b*c - a*d)^2*i^3*(c + d*x))/(b^3*g^2*(a + b*x)) - (B*d*(b*c - a*d)^2*i^3*Log[(a + b*x)/(c + d*x)])/(2*b^4*g^2) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^2) - ((b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^2) - (5*B*d*(b*c - a*d)^2*i^3*Log[c + d*x])/(2*b^4*g^2) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (3*B*d*(b*c - a*d)^2*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^3, x, 9, -((2*B*d*(b*c - a*d)*i^3*(c + d*x))/(b^3*g^3*(a + b*x))) - (B*(b*c - a*d)*i^3*(c + d*x)^2)/(4*b^2*g^3*(a + b*x)^2) + (d^3*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^3) - (2*d*(b*c - a*d)*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^3*(a + b*x)) - ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^3*(a + b*x)^2) - (B*d^2*(b*c - a*d)*i^3*Log[c + d*x])/(b^4*g^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (3*B*d^2*(b*c - a*d)*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^4, x, 9, -((B*d^2*i^3*(c + d*x))/(b^3*g^4*(a + b*x))) - (B*d*i^3*(c + d*x)^2)/(4*b^2*g^4*(a + b*x)^2) - (B*i^3*(c + d*x)^3)/(9*b*g^4*(a + b*x)^3) - (d^2*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^4*(a + b*x)) - (d*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^4*(a + b*x)^2) - (i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*g^4*(a + b*x)^3) - (d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4) + (B*d^3*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^5, x, 2, -((B*i^3*(c + d*x)^4)/(16*(b*c - a*d)*g^5*(a + b*x)^4)) - (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^6, x, 5, (B*d*i^3*(c + d*x)^4)/(16*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*B*i^3*(c + d*x)^5)/(25*(b*c - a*d)^2*g^6*(a + b*x)^5) + (d*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c - a*d)^2*g^6*(a + b*x)^5)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(a*g + b*g*x)^7, x, 5, -((B*d^2*i^3*(c + d*x)^4)/(16*(b*c - a*d)^3*g^7*(a + b*x)^4)) + (2*b*B*d*i^3*(c + d*x)^5)/(25*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*B*i^3*(c + d*x)^6)/(36*(b*c - a*d)^3*g^7*(a + b*x)^6) - (d^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^3*g^7*(a + b*x)^4) + (2*b*d*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*(b*c - a*d)^3*g^7*(a + b*x)^6)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x), x, 6, (g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d*i) - ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^2*i) + ((b*c - a*d)^2*g^3*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^3*i) + ((b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^4*i) + (B*(b*c - a*d)^3*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x), x, 5, (g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*i) - ((b*c - a*d)*g^2*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^2*i) - ((b*c - a*d)^2*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^3*i) - (B*(b*c - a*d)^2*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x), x, 4, (g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i) + ((b*c - a*d)*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i) + (B*(b*c - a*d)*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x), x, 5, -((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i)) - (B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^1*(c*i + d*i*x)), x, 2, (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(2*B*(b*c - a*d)*g*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^2*(c*i + d*i*x)), x, 5, -((b*B*(c + d*x))/((b*c - a*d)^2*g^2*i*(a + b*x))) + (B*d*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^2*g^2*i) - (b*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (d*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^2*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)), x, 7, -((B*(c + d*x)^2*(b - (4*d*(a + b*x))/(c + d*x))^2)/(4*(b*c - a*d)^3*g^3*i*(a + b*x)^2)) - (B*d^2*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^3*g^3*i) + (2*b*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^3*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^4*(c*i + d*i*x)), x, 8, -((3*b*B*d^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x))) + (3*b^2*B*d*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*B*(c + d*x)^3)/(9*(b*c - a*d)^4*g^4*i*(a + b*x)^3) + (B*d^3*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^4*i) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^2, x, 9, (3*B*(b*c - a*d)^2*g^3*(a + b*x))/(d^3*i^2*(c + d*x)) - ((6*A + 5*B)*(b*c - a*d)^2*g^3*(a + b*x))/(2*d^3*i^2*(c + d*x)) - (3*B*(b*c - a*d)^2*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i^2*(c + d*x)) + (g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*i^2*(c + d*x)) - ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^2*i^2*(c + d*x)) - (b*(b*c - a*d)^2*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4*i^2) - (3*b*B*(b*c - a*d)^2*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^2, x, 8, -((2*B*(b*c - a*d)*g^2*(a + b*x))/(d^2*i^2*(c + d*x))) + ((2*A + B)*(b*c - a*d)*g^2*(a + b*x))/(d^2*i^2*(c + d*x)) + (2*B*(b*c - a*d)*g^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*i^2*(c + d*x)) + (g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i^2*(c + d*x)) + (b*(b*c - a*d)*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^2) + (2*b*B*(b*c - a*d)*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^2, x, 7, -((A*g*(a + b*x))/(d*i^2*(c + d*x))) + (B*g*(a + b*x))/(d*i^2*(c + d*x)) - (B*g*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d*i^2*(c + d*x)) - (b*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i^2) - (b*B*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^2, x, 3, (A*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) - (B*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) + (B*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)*i^2*(c + d*x))} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^1*(c*i + d*i*x)^2), x, 5, -((A*d*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x))) + (B*d*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) - (B*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^2*g*i^2*(c + d*x)) + (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*B*(b*c - a*d)^2*g*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^2*(c*i + d*i*x)^2), x, 4, -((B*d^2*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x))) - (b^2*B*(c + d*x))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) + (b*B*d*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^3*g^2*i^2) + (d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (b^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*b*d*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^2*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)^2), x, 8, (B*d^3*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*B*d*(c + d*x))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B*(c + d*x)^2)/(4*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) - (3*b*B*d^2*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^3*i^2) - (d^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (3*b*d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^4*(c*i + d*i*x)^2), x, 4, -((B*d^4*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x))) - (6*b^2*B*d^2*(c + d*x))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (b^3*B*d*(c + d*x)^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*B*(c + d*x)^3)/(9*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) + (2*b*B*d^3*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^4*i^2) + (d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (6*b^2*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (4*b*d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^3, x, 9, -((3*B*(b*c - a*d)*g^3*(a + b*x)^2)/(4*d^2*i^3*(c + d*x)^2)) - (3*b*B*(b*c - a*d)*g^3*(a + b*x))/(d^3*i^3*(c + d*x)) + (b*(3*A + B)*(b*c - a*d)*g^3*(a + b*x))/(d^3*i^3*(c + d*x)) + (3*b*B*(b*c - a*d)*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i^3*(c + d*x)) + (g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i^3*(c + d*x)^2) + ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^2*i^3*(c + d*x)^2) + (b^2*(b*c - a*d)*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^3) + (3*b^2*B*(b*c - a*d)*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^3, x, 8, (B*g^2*(a + b*x)^2)/(4*d*i^3*(c + d*x)^2) - (A*b*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) + (b*B*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) - (b*B*g^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*i^3*(c + d*x)) - (g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*i^3*(c + d*x)^2) - (b^2*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^3) - (b^2*B*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^3, x, 2, -((B*g*(a + b*x)^2)/(4*(b*c - a*d)*i^3*(c + d*x)^2)) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)*i^3*(c + d*x)^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])/(c*i + d*i*x)^3, x, 4, B/(4*d*i^3*(c + d*x)^2) + (b*B)/(2*d*(b*c - a*d)*i^3*(c + d*x)) + (b^2*B*Log[a + b*x])/(2*d*(b*c - a*d)^2*i^3) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(2*d*i^3*(c + d*x)^2) - (b^2*B*Log[c + d*x])/(2*d*(b*c - a*d)^2*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^1*(c*i + d*i*x)^3), x, 4, -((B*(4*b - (d*(a + b*x))/(c + d*x))^2)/(4*(b*c - a*d)^3*g*i^3)) - (b^2*B*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^3*g*i^3) + (d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) - (2*b*d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g*i^3*(c + d*x)) + (b^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^2*(c*i + d*i*x)^3), x, 4, (B*d^3*(a + b*x)^2)/(4*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (3*b*B*d^2*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*B*(c + d*x))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) + (3*b^2*B*d*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^2*i^3) - (d^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (3*b^2*d*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^2*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)^3), x, 5, -((B*d^4*(a + b*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2)) + (4*b*B*d^3*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*B*d*(c + d*x))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B*(c + d*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) - (3*b^2*B*d^2*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^3*i^3) + (d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (6*b^2*d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])/((a*g + b*g*x)^4*(c*i + d*i*x)^3), x, 8, (B*d^5*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (5*b*B*d^4*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*B*d^2*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*(c + d*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*B*(c + d*x)^3)/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) + (5*b^2*B*d^3*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3) - (d^5*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x) / (c+d x)])^2 when b f-a g=0 and d h-c i=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 11, (3*B^2*(b*c - a*d)^4*g^3*i*x)/(10*b*d^3) - (3*B^2*(b*c - a*d)^3*g^3*i*(c + d*x)^2)/(20*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i*(c + d*x)^3)/(30*d^4) - (B*(b*c - a*d)^2*g^3*i*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^2*d) - (B*(b*c - a*d)*g^3*i*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^2) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^2) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) + (B*(b*c - a*d)^3*g^3*i*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^2*d^2) - (B*(b*c - a*d)^4*g^3*i*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^2*d^3) - (B*(b*c - a*d)^5*g^3*i*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^2*d^4) - (B^2*(b*c - a*d)^5*g^3*i*Log[c + d*x])/(10*b^2*d^4) - (B^2*(b*c - a*d)^5*g^3*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(10*b^2*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 10, -((B^2*(b*c - a*d)^3*g^2*i*x)/(3*b*d^2)) + (B^2*(b*c - a*d)^2*g^2*i*(c + d*x)^2)/(12*d^3) - (B*(b*c - a*d)^2*g^2*i*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d) - (B*(b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^2) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(12*b^2) + (g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b) + (B*(b*c - a*d)^3*g^2*i*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d^2) + (B*(b*c - a*d)^4*g^2*i*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*Log[c + d*x])/(6*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^2*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 9, (B^2*(b*c - a*d)^2*g*i*x)/(3*b*d) - (B*(b*c - a*d)^2*g*i*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^2*d) - (B*(b*c - a*d)*g*i*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^2) + ((b*c - a*d)*g*i*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b^2) + (g*i*(a + b*x)^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b) - (B*(b*c - a*d)^3*g*i*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^2*d^2) - (B^2*(b*c - a*d)^3*g*i*Log[c + d*x])/(3*b^2*d^2) - (B^2*(b*c - a*d)^3*g*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^2*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 7, -((B*(b*c - a*d)*i*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/b^2) + (i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d) + (B^2*(b*c - a*d)^2*i*Log[c + d*x])/(b^2*d) + (B*(b*c - a*d)^2*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*d) - (B^2*(b*c - a*d)^2*i*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*d)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^1, x, 8, (2*B*(b*c - a*d)*i*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g) + (d*i*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*g) + (2*B*(b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^2, x, 7, -((2*B^2*i*(c + d*x))/(b*g^2*(a + b*x))) - (2*B*i*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g^2*(a + b*x)) - (i*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b*g^2*(a + b*x)) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (2*B*d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (2*B^2*d*i*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^3, x, 3, -((B^2*i*(c + d*x)^2)/(4*(b*c - a*d)*g^3*(a + b*x)^2)) - (B*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)*g^3*(a + b*x)^2) - (i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)*g^3*(a + b*x)^2)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^4, x, 7, (B^2*d*i*(c + d*x)^2)/(4*(b*c - a*d)^2*g^4*(a + b*x)^2) - (2*b*B^2*i*(c + d*x)^3)/(27*(b*c - a*d)^2*g^4*(a + b*x)^3) + (B*d*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (2*b*B*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^2*g^4*(a + b*x)^3) + (d*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^2*g^4*(a + b*x)^3)} -{(c*i + d*i*x)*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^5, x, 9, -((B^2*d^2*i*(c + d*x)^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^2)) + (4*b*B^2*d*i*(c + d*x)^3)/(27*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B^2*i*(c + d*x)^4)/(32*(b*c - a*d)^3*g^5*(a + b*x)^4) - (B*d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (4*b*B*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 17, (3*B^2*(b*c - a*d)^5*g^3*i^2*x)/(20*b^2*d^3) + (B^2*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4)/(60*b^3) - (3*B^2*(b*c - a*d)^4*g^3*i^2*(c + d*x)^2)/(40*b*d^4) + (B^2*(b*c - a*d)^3*g^3*i^2*(c + d*x)^3)/(60*d^4) - (B*(b*c - a*d)^3*g^3*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(90*b^3*d) - (B*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^3) - (B*(b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(15*b^2) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(60*b^3) + ((b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(15*b^2) + (g^3*i^2*(a + b*x)^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b) + (B*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^2) - (B*(b*c - a*d)^5*g^3*i^2*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^3) - (B*(b*c - a*d)^6*g^3*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(180*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*Log[c + d*x])/(20*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(30*b^3*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 15, -((B^2*(b*c - a*d)^4*g^2*i^2*x)/(10*b^2*d^2)) - (B^2*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2)/(20*b*d^3) + (B^2*(b*c - a*d)^2*g^2*i^2*(c + d*x)^3)/(30*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) - (B*(b*c - a*d)^3*g^2*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(15*b^3) - (B*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b*d^3) + (4*B*(b*c - a*d)^2*g^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(15*d^3) - (b*B*(b*c - a*d)*g^2*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*d^3) + ((b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(30*b^3) + ((b*c - a*d)*g^2*i^2*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^2) + (g^2*i^2*(a + b*x)^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) + (B*(b*c - a*d)^4*g^2*i^2*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^3*d^2) + (B*(b*c - a*d)^5*g^2*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*Log[c + d*x])/(10*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(15*b^3*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 14, (B^2*(b*c - a*d)^3*g*i^2*x)/(12*b^2*d) + (B^2*(b*c - a*d)^2*g*i^2*(c + d*x)^2)/(12*b*d^2) - (B^2*(b*c - a*d)^4*g*i^2*Log[(a + b*x)/(c + d*x)])/(12*b^3*d^2) - (B*(b*c - a*d)^3*g*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*d) - (B*(b*c - a*d)^2*g*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3) + (B*(b*c - a*d)^2*g*i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b*d^2) - (B*(b*c - a*d)*g*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^2) + ((b*c - a*d)^2*g*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(12*b^3) + ((b*c - a*d)*g*i^2*(a + b*x)^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b^2) + (g*i^2*(a + b*x)^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b) - (B*(b*c - a*d)^4*g*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*Log[c + d*x])/(4*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^3*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 11, (B^2*(b*c - a*d)^2*i^2*x)/(3*b^2) + (B^2*(b*c - a*d)^3*i^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d) - (2*B*(b*c - a*d)^2*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^3) - (B*(b*c - a*d)*i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*d) + (i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*d) + (B^2*(b*c - a*d)^3*i^2*Log[c + d*x])/(b^3*d) + (2*B*(b*c - a*d)^3*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d) - (2*B^2*(b*c - a*d)^3*i^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^1, x, 15, -((B*d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g)) + (2*B*(b*c - a*d)^2*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g) + (d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g) + (i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b*g) + (B^2*(b*c - a*d)^2*i^2*Log[c + d*x])/(b^3*g) + (B*(b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B^2*(b*c - a*d)^2*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*g) - (B^2*(b*c - a*d)^2*i^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B*(b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B^2*(b*c - a*d)^2*i^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^2, x, 11, -((2*B^2*(b*c - a*d)*i^2*(c + d*x))/(b^2*g^2*(a + b*x))) - (2*B*(b*c - a*d)*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^2*(a + b*x)) + (2*B*d*(b*c - a*d)*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2) + (d^2*i^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^2) - ((b*c - a*d)*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g^2*(a + b*x)) - (2*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (2*B^2*d*(b*c - a*d)*i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*g^2) + (4*B*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (4*B^2*d*(b*c - a*d)*i^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^3, x, 10, -((2*B^2*d*i^2*(c + d*x))/(b^2*g^3*(a + b*x))) - (B^2*i^2*(c + d*x)^2)/(4*b*g^3*(a + b*x)^2) - (2*B*d*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^3*(a + b*x)) - (B*i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b*g^3*(a + b*x)^2) - (d*i^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g^3*(a + b*x)) - (i^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b*g^3*(a + b*x)^2) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (2*B*d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (2*B^2*d^2*i^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^4, x, 3, -((2*B^2*i^2*(c + d*x)^3)/(27*(b*c - a*d)*g^4*(a + b*x)^3)) - (2*B*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)*g^4*(a + b*x)^3) - (i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)*g^4*(a + b*x)^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^5, x, 7, (2*B^2*d*i^2*(c + d*x)^3)/(27*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B^2*i^2*(c + d*x)^4)/(32*(b*c - a*d)^2*g^5*(a + b*x)^4) + (2*B*d*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^2*g^5*(a + b*x)^4) + (d*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^6, x, 9, -((2*B^2*d^2*i^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^6*(a + b*x)^3)) + (b*B^2*d*i^2*(c + d*x)^4)/(16*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B^2*i^2*(c + d*x)^5)/(125*(b*c - a*d)^3*g^6*(a + b*x)^5) - (2*B*d^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 22, (5*B^2*(b*c - a*d)^6*g^3*i^3*x)/(84*b^3*d^3) + (B^2*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4)/(140*b^4) - (29*B^2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2)/(840*b^2*d^4) + (47*B^2*(b*c - a*d)^4*g^3*i^3*(c + d*x)^3)/(1260*b*d^4) - (13*B^2*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4)/(420*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5)/(105*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*Log[(a + b*x)/(c + d*x)])/(210*b^4*d^4) - (B*(b*c - a*d)^4*g^3*i^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(210*b^4*d) - (3*B*(b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(140*b^4) - (B*(b*c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(35*b^3) + (2*B*(b*c - a*d)^4*g^3*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(21*b*d^4) - (3*B*(b*c - a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(14*d^4) + (6*b*B*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(35*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(21*d^4) + ((b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(140*b^4) + ((b*c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(35*b^3) + ((b*c - a*d)*g^3*i^3*(a + b*x)^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(14*b^2) + (g^3*i^3*(a + b*x)^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(7*b) + (B*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(420*b^4*d^2) - (B*(b*c - a*d)^6*g^3*i^3*(a + b*x)*(6*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(420*b^4*d^3) - (B*(b*c - a*d)^7*g^3*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)]))/(420*b^4*d^4) - (11*B^2*(b*c - a*d)^7*g^3*i^3*Log[c + d*x])/(420*b^4*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(70*b^4*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 20, -((7*B^2*(b*c - a*d)^5*g^2*i^3*x)/(180*b^3*d^2)) - (7*B^2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2)/(360*b^2*d^3) - (B^2*(b*c - a*d)^3*g^2*i^3*(c + d*x)^3)/(60*b*d^3) + (B^2*(b*c - a*d)^2*g^2*i^3*(c + d*x)^4)/(60*d^3) + (B^2*(b*c - a*d)^6*g^2*i^3*Log[(a + b*x)/(c + d*x)])/(36*b^4*d^3) - (B*(b*c - a*d)^4*g^2*i^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^4*d) - (B*(b*c - a*d)^3*g^2*i^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^4) - (B*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^2*d^3) + (B*(b*c - a*d)^3*g^2*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(45*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(60*d^3) - (b*B*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(15*d^3) + ((b*c - a*d)^3*g^2*i^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(60*b^4) + ((b*c - a*d)^2*g^2*i^3*(a + b*x)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^3) + ((b*c - a*d)*g^2*i^3*(a + b*x)^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^2) + (g^2*i^3*(a + b*x)^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*b) + (B*(b*c - a*d)^5*g^2*i^3*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^4*d^2) + (B*(b*c - a*d)^6*g^2*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b^4*d^3) + (11*B^2*(b*c - a*d)^6*g^2*i^3*Log[c + d*x])/(180*b^4*d^3) + (B^2*(b*c - a*d)^6*g^2*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(30*b^4*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 19, (B^2*(b*c - a*d)^4*g*i^3*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*c - a*d)^2*g*i^3*(c + d*x)^3)/(30*b*d^2) - (B^2*(b*c - a*d)^5*g*i^3*Log[(a + b*x)/(c + d*x)])/(12*b^4*d^2) - (B*(b*c - a*d)^4*g*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d) - (B*(b*c - a*d)^3*g*i^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4) + (3*B*(b*c - a*d)^3*g*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*d^2) + ((b*c - a*d)^3*g*i^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^4) + ((b*c - a*d)^2*g*i^3*(a + b*x)^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^3) + (3*(b*c - a*d)*g*i^3*(a + b*x)^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^2) + (g*i^3*(a + b*x)^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) - (B*(b*c - a*d)^5*g*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d^2) - (11*B^2*(b*c - a*d)^5*g*i^3*Log[c + d*x])/(60*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(10*b^4*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2, x, 15, (5*B^2*(b*c - a*d)^3*i^3*x)/(12*b^3) + (B^2*(b*c - a*d)^2*i^3*(c + d*x)^2)/(12*b^2*d) + (5*B^2*(b*c - a*d)^4*i^3*Log[(a + b*x)/(c + d*x)])/(12*b^4*d) - (B*(b*c - a*d)^3*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4) - (B*(b*c - a*d)^2*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2*d) - (B*(b*c - a*d)*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b*d) + (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*d) + (11*B^2*(b*c - a*d)^4*i^3*Log[c + d*x])/(12*b^4*d) + (B*(b*c - a*d)^4*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d) - (B^2*(b*c - a*d)^4*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^1, x, 26, (B^2*d*(b*c - a*d)^2*i^3*x)/(3*b^3*g) + (B^2*(b*c - a*d)^3*i^3*Log[(a + b*x)/(c + d*x)])/(3*b^4*g) - (5*B*d*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^4*g) - (B*(b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b^2*g) + (2*B*(b*c - a*d)^3*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g) + (d*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g) + ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g) + (i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b*g) + (2*B^2*(b*c - a*d)^3*i^3*Log[c + d*x])/(b^4*g) + (5*B*(b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*b^4*g) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g) - (5*B^2*(b*c - a*d)^3*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^4*g) + (2*B*(b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^2, x, 17, -((2*B^2*(b*c - a*d)^2*i^3*(c + d*x))/(b^3*g^2*(a + b*x))) - (B*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^2) - (2*B*(b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^2*(a + b*x)) + (4*B*d*(b*c - a*d)^2*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^2) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^2) - ((b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g^2) + (B^2*d*(b*c - a*d)^2*i^3*Log[c + d*x])/(b^4*g^2) + (B*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (4*B^2*d*(b*c - a*d)^2*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^2) - (B^2*d*(b*c - a*d)^2*i^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^3, x, 13, -((4*B^2*d*(b*c - a*d)*i^3*(c + d*x))/(b^3*g^3*(a + b*x))) - (B^2*(b*c - a*d)*i^3*(c + d*x)^2)/(4*b^2*g^3*(a + b*x)^2) - (4*B*d*(b*c - a*d)*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^3*(a + b*x)) - (B*(b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^3*(a + b*x)^2) + (2*B*d^2*(b*c - a*d)*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^3) + (d^3*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^3) - (2*d*(b*c - a*d)*i^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^3*g^3*(a + b*x)) - ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g^3*(a + b*x)^2) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (2*B^2*d^2*(b*c - a*d)*i^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^3) + (6*B*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (6*B^2*d^2*(b*c - a*d)*i^3*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^5, x, 3, -((B^2*i^3*(c + d*x)^4)/(32*(b*c - a*d)*g^5*(a + b*x)^4)) - (B*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)*g^5*(a + b*x)^4) - (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^6, x, 7, (B^2*d*i^3*(c + d*x)^4)/(32*(b*c - a*d)^2*g^6*(a + b*x)^4) - (2*b*B^2*i^3*(c + d*x)^5)/(125*(b*c - a*d)^2*g^6*(a + b*x)^5) + (B*d*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^2*g^6*(a + b*x)^4) - (2*b*B*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*(b*c - a*d)^2*g^6*(a + b*x)^5) + (d*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(b*c - a*d)^2*g^6*(a + b*x)^5)} -{(c*i + d*i*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(a*g + b*g*x)^7, x, 9, -((B^2*d^2*i^3*(c + d*x)^4)/(32*(b*c - a*d)^3*g^7*(a + b*x)^4)) + (4*b*B^2*d*i^3*(c + d*x)^5)/(125*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*B^2*i^3*(c + d*x)^6)/(108*(b*c - a*d)^3*g^7*(a + b*x)^6) - (B*d^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^3*g^7*(a + b*x)^4) + (4*b*B*d*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*B*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(18*(b*c - a*d)^3*g^7*(a + b*x)^6) - (d^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^3*g^7*(a + b*x)^4) + (2*b*d*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*(b*c - a*d)^3*g^7*(a + b*x)^6)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x), x, 25, (b*B^2*(b*c - a*d)^2*g^3*x)/(3*d^3*i) + (B^2*(b*c - a*d)^3*g^3*Log[(a + b*x)/(c + d*x)])/(3*d^4*i) + (7*B*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^3*i) - (b^2*B*(b*c - a*d)*g^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^4*i) + (6*B*(b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i) + (3*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i) - (3*b^2*(b*c - a*d)*g^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^4*i) + (b^3*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*d^4*i) + ((b*c - a*d)^3*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*Log[c + d*x])/(d^4*i) - (7*B*(b*c - a*d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*d^4*i) + (6*B^2*(b*c - a*d)^3*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (2*B*(b*c - a*d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (7*B^2*(b*c - a*d)^3*g^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x), x, 15, -((B*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i)) - (4*B*(b*c - a*d)^2*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i) - (2*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i) + (b^2*g^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^3*i) - ((b*c - a*d)^2*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i) + (B^2*(b*c - a*d)^2*g^2*Log[c + d*x])/(d^3*i) + (B*(b*c - a*d)^2*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(d^3*i) - (4*B^2*(b*c - a*d)^2*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i) - (2*B*(b*c - a*d)^2*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(d^3*i) + (2*B^2*(b*c - a*d)^2*g^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x), x, 9, (2*B*(b*c - a*d)*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i) + (g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d*i) + ((b*c - a*d)*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i) + (2*B^2*(b*c - a*d)*g*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i) + (2*B*(b*c - a*d)*g*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i) - (2*B^2*(b*c - a*d)*g*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x), x, 4, -((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d*i)) - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*i) + (2*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^1*(c*i + d*i*x)), x, 3, (A + B*Log[(e*(a + b*x))/(c + d*x)])^3/(3*B*(b*c - a*d)*g*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^2*(c*i + d*i*x)), x, 7, -((2*b*B^2*(c + d*x))/((b*c - a*d)^2*g^2*i*(a + b*x))) - (2*b*B*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (b*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^2*g^2*i*(a + b*x)) - (d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^2*g^2*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^3*(c*i + d*i*x)), x, 9, (4*b*B^2*d*(c + d*x))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*B^2*(c + d*x)^2)/(4*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (4*b*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (2*b*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^3*g^3*i)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)), x, 11, -((6*b*B^2*d^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x))) + (3*b^2*B^2*d*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B^2*(c + d*x)^3)/(27*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (6*b*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^4*g^4*i)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^2, x, 18, (2*A*B*(b*c - a*d)^2*g^3*(a + b*x))/(d^3*i^2*(c + d*x)) - (2*B^2*(b*c - a*d)^2*g^3*(a + b*x))/(d^3*i^2*(c + d*x)) + (2*B^2*(b*c - a*d)^2*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i^2*(c + d*x)) - (b*B*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^2) - (6*b*B*(b*c - a*d)^2*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^2) - (3*b*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^2) - ((b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^2*(c + d*x)) + (b^3*g^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^4*i^2) - (3*b*(b*c - a*d)^2*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i^2) + (b*B^2*(b*c - a*d)^2*g^3*Log[c + d*x])/(d^4*i^2) + (b*B*(b*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(d^4*i^2) - (6*b*B^2*(b*c - a*d)^2*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) - (6*b*B*(b*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) - (b*B^2*(b*c - a*d)^2*g^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(d^4*i^2) + (6*b*B^2*(b*c - a*d)^2*g^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^2, x, 12, -((2*A*B*(b*c - a*d)*g^2*(a + b*x))/(d^2*i^2*(c + d*x))) + (2*B^2*(b*c - a*d)*g^2*(a + b*x))/(d^2*i^2*(c + d*x)) - (2*B^2*(b*c - a*d)*g^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*i^2*(c + d*x)) + (2*b*B*(b*c - a*d)*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^2) + (b*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i^2) + ((b*c - a*d)*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i^2*(c + d*x)) + (2*b*(b*c - a*d)*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^2) + (2*b*B^2*(b*c - a*d)*g^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2) + (4*b*B*(b*c - a*d)*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2) - (4*b*B^2*(b*c - a*d)*g^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^2, x, 9, (2*A*B*g*(a + b*x))/(d*i^2*(c + d*x)) - (2*B^2*g*(a + b*x))/(d*i^2*(c + d*x)) + (2*B^2*g*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d*i^2*(c + d*x)) - (g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d*i^2*(c + d*x)) - (b*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i^2) - (2*b*B*g*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2) + (2*b*B^2*g*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^2, x, 4, -((2*A*B*(a + b*x))/((b*c - a*d)*i^2*(c + d*x))) + (2*B^2*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) - (2*B^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)*i^2*(c + d*x)) + ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)*i^2*(c + d*x))} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^1*(c*i + d*i*x)^2), x, 7, (2*A*B*d*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) - (2*B^2*d*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) + (2*B^2*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^2*g*i^2*(c + d*x)) - (d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^2*g*i^2*(c + d*x)) + (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^2*g*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^2*(c*i + d*i*x)^2), x, 10, -((2*A*B*d^2*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x))) + (2*B^2*d^2*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (2*b^2*B^2*(c + d*x))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*B^2*d^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (2*b^2*B*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) + (d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (b^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*b*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^3*g^2*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^3*(c*i + d*i*x)^2), x, 12, (2*A*B*d^3*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) - (2*B^2*d^3*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (6*b^2*B^2*d*(c + d*x))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (2*B^2*d^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (6*b^2*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) - (d^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (b*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(B*(b*c - a*d)^4*g^3*i^2)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^2), x, 14, -((2*A*B*d^4*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x))) + (2*B^2*d^4*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (12*b^2*B^2*d^2*(c + d*x))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (b^3*B^2*d*(c + d*x)^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (2*b^4*B^2*(c + d*x)^3)/(27*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (2*B^2*d^4*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (12*b^2*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (2*b^4*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) + (d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (6*b^2*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (4*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^5*g^4*i^2)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^3, x, 14, (B^2*(b*c - a*d)*g^3*(a + b*x)^2)/(4*d^2*i^3*(c + d*x)^2) - (4*A*b*B*(b*c - a*d)*g^3*(a + b*x))/(d^3*i^3*(c + d*x)) + (4*b*B^2*(b*c - a*d)*g^3*(a + b*x))/(d^3*i^3*(c + d*x)) - (4*b*B^2*(b*c - a*d)*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i^3*(c + d*x)) - (B*(b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^2*i^3*(c + d*x)^2) + (2*b^2*B*(b*c - a*d)*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^3) + (b^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^3) + ((b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^2*i^3*(c + d*x)^2) + (2*b*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^3*(c + d*x)) + (3*b^2*(b*c - a*d)*g^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i^3) + (2*b^2*B^2*(b*c - a*d)*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3) + (6*b^2*B*(b*c - a*d)*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3) - (6*b^2*B^2*(b*c - a*d)*g^3*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3)} -{(a*g + b*g*x)^2*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^3, x, 11, -((B^2*g^2*(a + b*x)^2)/(4*d*i^3*(c + d*x)^2)) + (2*A*b*B*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) - (2*b*B^2*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) + (2*b*B^2*g^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*i^3*(c + d*x)) + (B*g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*i^3*(c + d*x)^2) - (g^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d*i^3*(c + d*x)^2) - (b*g^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*i^3*(c + d*x)) - (b^2*g^2*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^3) - (2*b^2*B*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3) + (2*b^2*B^2*g^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^3, x, 3, (B^2*g*(a + b*x)^2)/(4*(b*c - a*d)*i^3*(c + d*x)^2) - (B*g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)*i^3*(c + d*x)^2) + (g*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)*i^3*(c + d*x)^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*(a + b*x)/(c + d*x)])^2/(c*i + d*i*x)^3, x, 8, -((B^2*d*(a + b*x)^2)/(4*(b*c - a*d)^2*i^3*(c + d*x)^2)) - (2*A*b*B*(a + b*x))/((b*c - a*d)^2*i^3*(c + d*x)) + (2*b*B^2*(a + b*x))/((b*c - a*d)^2*i^3*(c + d*x)) - (2*b*B^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^2*i^3*(c + d*x)) + (B*d*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*i^3*(c + d*x)^2) - (d*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^2*i^3*(c + d*x)^2) + (b*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^2*i^3*(c + d*x))} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^1*(c*i + d*i*x)^3), x, 15, (B^2*d^2*(a + b*x)^2)/(4*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (4*A*b*B*d*(a + b*x))/((b*c - a*d)^3*g*i^3*(c + d*x)) - (4*b*B^2*d*(a + b*x))/((b*c - a*d)^3*g*i^3*(c + d*x)) + (4*b*B^2*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^3*g*i^3*(c + d*x)) - (B*d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) - (2*b*d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g*i^3*(c + d*x)) + (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^3*g*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^2*(c*i + d*i*x)^3), x, 12, -((B^2*d^3*(a + b*x)^2)/(4*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2)) - (6*A*b*B*d^2*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) + (6*b*B^2*d^2*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (2*b^3*B^2*(c + d*x))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (6*b*B^2*d^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^4*g^2*i^3*(c + d*x)) + (B*d^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (2*b^3*B*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (d^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(B*(b*c - a*d)^4*g^2*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^3*(c*i + d*i*x)^3), x, 14, (B^2*d^4*(a + b*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (8*A*b*B*d^3*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) - (8*b*B^2*d^3*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (8*b^3*B^2*d*(c + d*x))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B^2*(c + d*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (8*b*B^2*d^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^5*g^3*i^3*(c + d*x)) - (B*d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (8*b^3*B*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (d^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (2*b^2*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(B*(b*c - a*d)^5*g^3*i^3)} -{(A + B*Log[e*(a + b*x)/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^3), x, 16, -((B^2*d^5*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2)) - (10*A*b*B*d^4*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) + (10*b*B^2*d^4*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (20*b^3*B^2*d^2*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B^2*d*(c + d*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (2*b^5*B^2*(c + d*x)^3)/(27*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b*B^2*d^4*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/((b*c - a*d)^6*g^4*i^3*(c + d*x)) + (B*d^5*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (20*b^3*B*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (2*b^5*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^6*g^4*i^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x) / (c+d x))^n])^1 when b f-a g=0 and d h-c i=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, -((B*(b*c - a*d)^4*g^3*i*n*x)/(20*b*d^3)) + (B*(b*c - a*d)^3*g^3*i*n*(a + b*x)^2)/(40*b^2*d^2) - (B*(b*c - a*d)^2*g^3*i*n*(a + b*x)^3)/(60*b^2*d) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^2) + (B*(b*c - a*d)^5*g^3*i*n*Log[c + d*x])/(20*b^2*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, (B*(b*c - a*d)^3*g^2*i*n*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*n*(a + b*x)^2)/(24*b^2*d) + (g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]))/(12*b^2) - (B*(b*c - a*d)^4*g^2*i*n*Log[c + d*x])/(12*b^2*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, -((B*(b*c - a*d)^2*g*i*n*x)/(6*b*d)) + (g*i*(a + b*x)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b) + ((b*c - a*d)*g*i*(a + b*x)^2*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^2) + (B*(b*c - a*d)^3*g*i*n*Log[c + d*x])/(6*b^2*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -(B*(b*c - a*d)*i*n*x)/(2*b) - (B*(b*c - a*d)^2*i*n*Log[a + b*x])/(2*b^2*d) + (i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^1, x, 6, (i*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*g) - ((b*c - a*d)*i*Log[-((b*c - a*d)/(d*(a + b*x)))]*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*g) + (B*(b*c - a*d)*i*n*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^2*g)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2, x, 5, -((B*i*n*(c + d*x))/(b*g^2*(a + b*x))) - (i*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*g^2*(a + b*x)) - (d*i*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (B*d*i*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^3, x, 2, -(B*i*n*(c + d*x)^2)/(4*(b*c - a*d)*g^3*(a + b*x)^2) - (i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)*g^3*(a + b*x)^2)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4, x, 5, (B*d*i*n*(c + d*x)^2)/(4*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*B*i*n*(c + d*x)^3)/(9*(b*c - a*d)^2*g^4*(a + b*x)^3) + (d*i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*i*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^2*g^4*(a + b*x)^3)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^5, x, 5, -(B*d^2*i*n*(c + d*x)^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*B*d*i*n*(c + d*x)^3)/(9*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B*i*n*(c + d*x)^4)/(16*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*c - a*d)^3*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, (B*(b*c - a*d)^5*g^3*i^2*n*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*n*(c + d*x)^2)/(120*b*d^4) - (19*B*(b*c - a*d)^3*g^3*i^2*n*(c + d*x)^3)/(180*d^4) + (13*b*B*(b*c - a*d)^2*g^3*i^2*n*(c + d*x)^4)/(120*d^4) - (b^2*B*(b*c - a*d)*g^3*i^2*n*(c + d*x)^5)/(30*d^4) - ((b*c - a*d)^3*g^3*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^4) + (3*b*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^4) - (3*b^2*(b*c - a*d)*g^3*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^4) + (b^3*g^3*i^2*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^4) + (B*(b*c - a*d)^6*g^3*i^2*n*Log[(a + b*x)/(c + d*x)])/(60*b^3*d^4) + (B*(b*c - a*d)^6*g^3*i^2*n*Log[c + d*x])/(60*b^3*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, -((B*(b*c - a*d)^4*g^2*i^2*n*x)/(30*b^2*d^2)) - (B*(b*c - a*d)^3*g^2*i^2*n*(c + d*x)^2)/(60*b*d^3) + (B*(b*c - a*d)^2*g^2*i^2*n*(c + d*x)^3)/(10*d^3) - (b*B*(b*c - a*d)*g^2*i^2*n*(c + d*x)^4)/(20*d^3) + ((b*c - a*d)^2*g^2*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^3) - (b*(b*c - a*d)*g^2*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^3) + (b^2*g^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3) - (B*(b*c - a*d)^5*g^2*i^2*n*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) - (B*(b*c - a*d)^5*g^2*i^2*n*Log[c + d*x])/(30*b^3*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, (B*(b*c - a*d)^3*g*i^2*n*x)/(12*b^2*d) + (B*(b*c - a*d)^2*g*i^2*n*(c + d*x)^2)/(24*b*d^2) - (B*(b*c - a*d)*g*i^2*n*(c + d*x)^3)/(12*d^2) - ((b*c - a*d)*g*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^2) + (b*g*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^2) + (B*(b*c - a*d)^4*g*i^2*n*Log[(a + b*x)/(c + d*x)])/(12*b^3*d^2) + (B*(b*c - a*d)^4*g*i^2*n*Log[c + d*x])/(12*b^3*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -(B*(b*c - a*d)^2*i^2*n*x)/(3*b^2) - (B*(b*c - a*d)*i^2*n*(c + d*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*i^2*n*Log[a + b*x])/(3*b^3*d) + (i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^1, x, 10, -((B*d*(b*c - a*d)*i^2*n*x)/(2*b^2*g)) + (d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g) + (i^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*g) - (B*(b*c - a*d)^2*i^2*n*Log[(a + b*x)/(c + d*x)])/(2*b^3*g) - (3*B*(b*c - a*d)^2*i^2*n*Log[c + d*x])/(2*b^3*g) - ((b*c - a*d)^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (B*(b*c - a*d)^2*i^2*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2, x, 8, -((B*(b*c - a*d)*i^2*n*(c + d*x))/(b^2*g^2*(a + b*x))) + (d^2*i^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^2) - ((b*c - a*d)*i^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*g^2*(a + b*x)) - (B*d*(b*c - a*d)*i^2*n*Log[c + d*x])/(b^3*g^2) - (2*d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (2*B*d*(b*c - a*d)*i^2*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^3, x, 7, -((B*d*i^2*n*(c + d*x))/(b^2*g^3*(a + b*x))) - (B*i^2*n*(c + d*x)^2)/(4*b*g^3*(a + b*x)^2) - (d*i^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*g^3*(a + b*x)) - (i^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*g^3*(a + b*x)^2) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (B*d^2*i^2*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4, x, 2, -(B*i^2*n*(c + d*x)^3)/(9*(b*c - a*d)*g^4*(a + b*x)^3) - (i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)*g^4*(a + b*x)^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^5, x, 5, (B*d*i^2*n*(c + d*x)^3)/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B*i^2*n*(c + d*x)^4)/(16*(b*c - a*d)^2*g^5*(a + b*x)^4) + (d*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^6, x, 5, -(B*d^2*i^2*n*(c + d*x)^3)/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*x)^4)/(8*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*B*i^2*n*(c + d*x)^5)/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, (B*(b*c - a*d)^6*g^3*i^3*n*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*n*(c + d*x)^2)/(280*b^2*d^4) + (B*(b*c - a*d)^4*g^3*i^3*n*(c + d*x)^3)/(420*b*d^4) - (17*B*(b*c - a*d)^3*g^3*i^3*n*(c + d*x)^4)/(280*d^4) + (b*B*(b*c - a*d)^2*g^3*i^3*n*(c + d*x)^5)/(14*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*n*(c + d*x)^6)/(42*d^4) - ((b*c - a*d)^3*g^3*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^4) + (3*b*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^4) - (b^2*(b*c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^4) + (b^3*g^3*i^3*(c + d*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(7*d^4) + (B*(b*c - a*d)^7*g^3*i^3*n*Log[(a + b*x)/(c + d*x)])/(140*b^4*d^4) + (B*(b*c - a*d)^7*g^3*i^3*n*Log[c + d*x])/(140*b^4*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, -((B*(b*c - a*d)^5*g^2*i^3*n*x)/(60*b^3*d^2)) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*c - a*d)^3*g^2*i^3*n*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4)/(120*d^3) - (b*B*(b*c - a*d)*g^2*i^3*n*(c + d*x)^5)/(30*d^3) + ((b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[(a + b*x)/(c + d*x)])/(60*b^4*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[c + d*x])/(60*b^4*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 5, (B*(b*c - a*d)^4*g*i^3*n*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*n*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^4)/(20*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^2) + (b*g*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log[(a + b*x)/(c + d*x)])/(20*b^4*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log[c + d*x])/(20*b^4*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 4, -(B*(b*c - a*d)^3*i^3*n*x)/(4*b^3) - (B*(b*c - a*d)^2*i^3*n*(c + d*x)^2)/(8*b^2*d) - (B*(b*c - a*d)*i^3*n*(c + d*x)^3)/(12*b*d) - (B*(b*c - a*d)^4*i^3*n*Log[a + b*x])/(4*b^4*d) + (i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^1, x, 14, -((5*B*d*(b*c - a*d)^2*i^3*n*x)/(6*b^3*g)) - (B*(b*c - a*d)*i^3*n*(c + d*x)^2)/(6*b^2*g) + (d*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g) + ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g) + (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*g) - (5*B*(b*c - a*d)^3*i^3*n*Log[(a + b*x)/(c + d*x)])/(6*b^4*g) - (11*B*(b*c - a*d)^3*i^3*n*Log[c + d*x])/(6*b^4*g) - ((b*c - a*d)^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (B*(b*c - a*d)^3*i^3*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^2, x, 11, -((B*d^2*(b*c - a*d)*i^3*n*x)/(2*b^3*g^2)) - (B*(b*c - a*d)^2*i^3*n*(c + d*x))/(b^3*g^2*(a + b*x)) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^2) - ((b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^2) - (B*d*(b*c - a*d)^2*i^3*n*Log[(a + b*x)/(c + d*x)])/(2*b^4*g^2) - (5*B*d*(b*c - a*d)^2*i^3*n*Log[c + d*x])/(2*b^4*g^2) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (3*B*d*(b*c - a*d)^2*i^3*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^3, x, 9, -((2*B*d*(b*c - a*d)*i^3*n*(c + d*x))/(b^3*g^3*(a + b*x))) - (B*(b*c - a*d)*i^3*n*(c + d*x)^2)/(4*b^2*g^3*(a + b*x)^2) + (d^3*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^3) - (2*d*(b*c - a*d)*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^3*(a + b*x)) - ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^3*(a + b*x)^2) - (B*d^2*(b*c - a*d)*i^3*n*Log[c + d*x])/(b^4*g^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (3*B*d^2*(b*c - a*d)*i^3*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^4, x, 9, -((B*d^2*i^3*n*(c + d*x))/(b^3*g^4*(a + b*x))) - (B*d*i^3*n*(c + d*x)^2)/(4*b^2*g^4*(a + b*x)^2) - (B*i^3*n*(c + d*x)^3)/(9*b*g^4*(a + b*x)^3) - (d^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^4*(a + b*x)) - (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^4*(a + b*x)^2) - (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*g^4*(a + b*x)^3) - (d^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4) + (B*d^3*i^3*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x), x, 6, (g^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d*i) - ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^2*i) + ((b*c - a*d)^2*g^3*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3*i) + ((b*c - a*d)^3*g^3*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(6*d^4*i) + (B*(b*c - a*d)^3*g^3*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x), x, 5, (g^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i) - ((b*c - a*d)*g^2*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^2*i) - ((b*c - a*d)^2*g^2*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(2*d^3*i) - (B*(b*c - a*d)^2*g^2*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x), x, 4, (g*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i) + ((b*c - a*d)*g*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i) + (B*(b*c - a*d)*g*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x), x, 5, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d*i)) - (B*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*i)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^1*(c*i + d*i*x)), x, 2, (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(2*B*(b*c - a*d)*g*i*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^2*(c*i + d*i*x)), x, 5, -((b*B*n*(c + d*x))/((b*c - a*d)^2*g^2*i*(a + b*x))) - (b*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^2*g^2*i) + (B*d*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^2*g^2*i)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^3*(c*i + d*i*x)), x, 7, -(B*n*(c + d*x)^2*(b - (4*d*(a + b*x))/(c + d*x))^2)/(4*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (2*b*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^3*g^3*i) - (B*d^2*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^3*g^3*i)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^4*(c*i + d*i*x)), x, 8, (-3*b*B*d^2*n*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*n*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*B*n*(c + d*x)^3)/(9*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (3*b*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^4*g^4*i) + (B*d^3*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^4*i)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^2, x, 9, (3*B*(b*c - a*d)^2*g^3*n*(a + b*x))/(d^3*i^2*(c + d*x)) - ((b*c - a*d)^2*g^3*(6*A + 5*B*n)*(a + b*x))/(2*d^3*i^2*(c + d*x)) - (3*B*(b*c - a*d)^2*g^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^3*i^2*(c + d*x)) + (g^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i^2*(c + d*x)) - ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^2*i^2*(c + d*x)) - (b*(b*c - a*d)^2*g^3*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(2*d^4*i^2) - (3*b*B*(b*c - a*d)^2*g^3*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^2, x, 8, (-2*B*(b*c - a*d)*g^2*n*(a + b*x))/(d^2*i^2*(c + d*x)) + ((b*c - a*d)*g^2*(2*A + B*n)*(a + b*x))/(d^2*i^2*(c + d*x)) + (2*B*(b*c - a*d)*g^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^2*i^2*(c + d*x)) + (g^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i^2*(c + d*x)) + (b*(b*c - a*d)*g^2*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^2) + (2*b*B*(b*c - a*d)*g^2*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^2, x, 7, -((A*g*(a + b*x))/(d*i^2*(c + d*x))) + (B*g*n*(a + b*x))/(d*i^2*(c + d*x)) - (B*g*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d*i^2*(c + d*x)) - (b*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i^2) - (b*B*g*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^2, x, 3, (A*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) - (B*n*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*i^2*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^1*(c*i + d*i*x)^2), x, 5, -((A*d*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x))) + (B*d*n*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) - (B*d*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^2*g*i^2*(c + d*x)) + (b*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*B*(b*c - a*d)^2*g*i^2*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^2*(c*i + d*i*x)^2), x, 4, -((B*d^2*n*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x))) - (b^2*B*n*(c + d*x))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) + (d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (b^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*b*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^3*g^2*i^2) + (b*B*d*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^3*g^2*i^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^3*(c*i + d*i*x)^2), x, 8, (B*d^3*n*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*B*d*n*(c + d*x))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B*n*(c + d*x)^2)/(4*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) - (d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (3*b*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^4*g^3*i^2) - (3*b*B*d^2*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^3*i^2)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^4*(c*i + d*i*x)^2), x, 4, -((B*d^4*n*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x))) - (6*b^2*B*d^2*n*(c + d*x))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (b^3*B*d*n*(c + d*x)^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*B*n*(c + d*x)^3)/(9*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) + (d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (6*b^2*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (4*b*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^5*g^4*i^2) + (2*b*B*d^3*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^4*i^2)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3, x, 9, (-3*B*(b*c - a*d)*g^3*n*(a + b*x)^2)/(4*d^2*i^3*(c + d*x)^2) - (3*b*B*(b*c - a*d)*g^3*n*(a + b*x))/(d^3*i^3*(c + d*x)) + (b*(b*c - a*d)*g^3*(3*A + B*n)*(a + b*x))/(d^3*i^3*(c + d*x)) + (3*b*B*(b*c - a*d)*g^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^3*i^3*(c + d*x)) + (g^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i^3*(c + d*x)^2) + ((b*c - a*d)*g^3*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^2*i^3*(c + d*x)^2) + (b^2*(b*c - a*d)*g^3*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i^3) + (3*b^2*B*(b*c - a*d)*g^3*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3, x, 8, (B*g^2*n*(a + b*x)^2)/(4*d*i^3*(c + d*x)^2) - (A*b*g^2*(a + b*x))/(d^2*i^3*(c + d*x)) + (b*B*g^2*n*(a + b*x))/(d^2*i^3*(c + d*x)) - (b*B*g^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^2*i^3*(c + d*x)) - (g^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i^3*(c + d*x)^2) - (b^2*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^3) - (b^2*B*g^2*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3, x, 2, -(B*g*n*(a + b*x)^2)/(4*(b*c - a*d)*i^3*(c + d*x)^2) + (g*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)*i^3*(c + d*x)^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*i + d*i*x)^3, x, 4, (B*n)/(4*d*i^3*(c + d*x)^2) + (b*B*n)/(2*d*(b*c - a*d)*i^3*(c + d*x)) + (b^2*B*n*Log[a + b*x])/(2*d*(b*c - a*d)^2*i^3) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*d*i^3*(c + d*x)^2) - (b^2*B*n*Log[c + d*x])/(2*d*(b*c - a*d)^2*i^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^1*(c*i + d*i*x)^3), x, 4, -(B*n*(4*b - (d*(a + b*x))/(c + d*x))^2)/(4*(b*c - a*d)^3*g*i^3) + (d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) - (2*b*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g*i^3*(c + d*x)) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^3*g*i^3) - (b^2*B*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^3*g*i^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^2*(c*i + d*i*x)^3), x, 4, (B*d^3*n*(a + b*x)^2)/(4*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (3*b*B*d^2*n*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*B*n*(c + d*x))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (d^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (3*b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^4*g^2*i^3) + (3*b^2*B*d*n*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c - a*d)^4*g^2*i^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^3*(c*i + d*i*x)^3), x, 5, -(B*d^4*n*(a + b*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (4*b*B*d^3*n*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*B*d*n*(c + d*x))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B*n*(c + d*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (d^4*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (6*b^2*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^5*g^3*i^3) - (3*b^2*B*d^2*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^3*i^3)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^4*(c*i + d*i*x)^3), x, 8, (B*d^5*n*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (5*b*B*d^4*n*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*B*d^2*n*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*n*(c + d*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*B*n*(c + d*x)^3)/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((b*c - a*d)^6*g^4*i^3) + (5*b^2*B*d^3*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x) / (c+d x))^n])^2 when b f-a g=0 and d h-c i=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 11, (3*B^2*(b*c - a*d)^4*g^3*i*n^2*x)/(10*b*d^3) - (3*B^2*(b*c - a*d)^3*g^3*i*n^2*(c + d*x)^2)/(20*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i*n^2*(c + d*x)^3)/(30*d^4) - (B*(b*c - a*d)^2*g^3*i*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^2*d) - (B*(b*c - a*d)*g^3*i*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^2) + ((b*c - a*d)*g^3*i*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(20*b^2) + (g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b) + (B*(b*c - a*d)^3*g^3*i*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b^2*d^2) - (B*(b*c - a*d)^4*g^3*i*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b^2*d^3) - (B*(b*c - a*d)^5*g^3*i*n*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(60*b^2*d^4) - (B^2*(b*c - a*d)^5*g^3*i*n^2*Log[c + d*x])/(10*b^2*d^4) - (B^2*(b*c - a*d)^5*g^3*i*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(10*b^2*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 10, -(B^2*(b*c - a*d)^3*g^2*i*n^2*x)/(3*b*d^2) + (B^2*(b*c - a*d)^2*g^2*i*n^2*(c + d*x)^2)/(12*d^3) - (B*(b*c - a*d)^2*g^2*i*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(12*b^2*d) - (B*(b*c - a*d)*g^2*i*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^2) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(12*b^2) + (g^2*i*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b) + (B*(b*c - a*d)^3*g^2*i*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(12*b^2*d^2) + (B*(b*c - a*d)^4*g^2*i*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(12*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*n^2*Log[c + d*x])/(6*b^2*d^3) + (B^2*(b*c - a*d)^4*g^2*i*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^2*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 9, (B^2*(b*c - a*d)^2*g*i*n^2*x)/(3*b*d) - (B*(b*c - a*d)^2*g*i*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2*d) - (B*(b*c - a*d)*g*i*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2) + ((b*c - a*d)*g*i*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b^2) + (g*i*(a + b*x)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b) - (B*(b*c - a*d)^3*g*i*n*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(3*b^2*d^2) - (B^2*(b*c - a*d)^3*g*i*n^2*Log[c + d*x])/(3*b^2*d^2) - (B^2*(b*c - a*d)^3*g*i*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b^2*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 7, -((B*(b*c - a*d)*i*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/b^2) + (i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d) + (B^2*(b*c - a*d)^2*i*n^2*Log[c + d*x])/(b^2*d) + (B*(b*c - a*d)^2*i*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*d) - (B^2*(b*c - a*d)^2*i*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*d)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^1, x, 8, (d*i*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^2*g) + (2*B*(b*c - a*d)*i*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^2*g) - ((b*c - a*d)*i*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*g) + (2*B*(b*c - a*d)*i*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^2, x, 7, -((2*B^2*i*n^2*(c + d*x))/(b*g^2*(a + b*x))) - (2*B*i*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*g^2*(a + b*x)) - (i*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b*g^2*(a + b*x)) - (d*i*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (2*B*d*i*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2) + (2*B^2*d*i*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g^2)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^3, x, 3, -(B^2*i*n^2*(c + d*x)^2)/(4*(b*c - a*d)*g^3*(a + b*x)^2) - (B*i*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)*g^3*(a + b*x)^2) - (i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)*g^3*(a + b*x)^2)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^4, x, 7, (B^2*d*i*n^2*(c + d*x)^2)/(4*(b*c - a*d)^2*g^4*(a + b*x)^2) - (2*b*B^2*i*n^2*(c + d*x)^3)/(27*(b*c - a*d)^2*g^4*(a + b*x)^3) + (B*d*i*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (2*b*B*i*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^2*g^4*(a + b*x)^3) + (d*i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*i*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^2*g^4*(a + b*x)^3)} -{(c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5, x, 9, -(B^2*d^2*i*n^2*(c + d*x)^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^2) + (4*b*B^2*d*i*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B^2*i*n^2*(c + d*x)^4)/(32*(b*c - a*d)^3*g^5*(a + b*x)^4) - (B*d^2*i*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (4*b*B*d*i*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B*i*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^4)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 17, (3*B^2*(b*c - a*d)^5*g^3*i^2*n^2*x)/(20*b^2*d^3) + (B^2*(b*c - a*d)^2*g^3*i^2*n^2*(a + b*x)^4)/(60*b^3) - (3*B^2*(b*c - a*d)^4*g^3*i^2*n^2*(c + d*x)^2)/(40*b*d^4) + (B^2*(b*c - a*d)^3*g^3*i^2*n^2*(c + d*x)^3)/(60*d^4) - (B*(b*c - a*d)^3*g^3*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(90*b^3*d) - (B*(b*c - a*d)^2*g^3*i^2*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^3) - (B*(b*c - a*d)*g^3*i^2*n*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^2) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(60*b^3) + ((b*c - a*d)*g^3*i^2*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(15*b^2) + (g^3*i^2*(a + b*x)^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b) + (B*(b*c - a*d)^4*g^3*i^2*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(180*b^3*d^2) - (B*(b*c - a*d)^5*g^3*i^2*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(180*b^3*d^3) - (B*(b*c - a*d)^6*g^3*i^2*n*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(180*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*Log[c + d*x])/(20*b^3*d^4) - (B^2*(b*c - a*d)^6*g^3*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(30*b^3*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 15, -(B^2*(b*c - a*d)^4*g^2*i^2*n^2*x)/(10*b^2*d^2) - (B^2*(b*c - a*d)^3*g^2*i^2*n^2*(c + d*x)^2)/(20*b*d^3) + (B^2*(b*c - a*d)^2*g^2*i^2*n^2*(c + d*x)^3)/(30*d^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i^2*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^3) - (B*(b*c - a*d)^3*g^2*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b*d^3) + (4*B*(b*c - a*d)^2*g^2*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*d^3) - (b*B*(b*c - a*d)*g^2*i^2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*d^3) + ((b*c - a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(30*b^3) + ((b*c - a*d)*g^2*i^2*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(10*b^2) + (g^2*i^2*(a + b*x)^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b) + (B*(b*c - a*d)^4*g^2*i^2*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^3*d^2) + (B*(b*c - a*d)^5*g^2*i^2*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*Log[c + d*x])/(10*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(15*b^3*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 14, (B^2*(b*c - a*d)^3*g*i^2*n^2*x)/(12*b^2*d) + (B^2*(b*c - a*d)^2*g*i^2*n^2*(c + d*x)^2)/(12*b*d^2) - (B*(b*c - a*d)^3*g*i^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^3*d) - (B*(b*c - a*d)^2*g*i^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^3) + (B*(b*c - a*d)^2*g*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b*d^2) - (B*(b*c - a*d)*g*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^2) + ((b*c - a*d)^2*g*i^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(12*b^3) + ((b*c - a*d)*g*i^2*(a + b*x)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b^2) + (g*i^2*(a + b*x)^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b) - (B*(b*c - a*d)^4*g*i^2*n*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(6*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*n^2*Log[(a + b*x)/(c + d*x)])/(12*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*n^2*Log[c + d*x])/(4*b^3*d^2) - (B^2*(b*c - a*d)^4*g*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(6*b^3*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 11, (B^2*(b*c - a*d)^2*i^2*n^2*x)/(3*b^2) - (2*B*(b*c - a*d)^2*i^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3) - (B*(b*c - a*d)*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*d) + (i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*d) + (B^2*(b*c - a*d)^3*i^2*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^3*d) + (B^2*(b*c - a*d)^3*i^2*n^2*Log[c + d*x])/(b^3*d) + (2*B*(b*c - a*d)^3*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d) - (2*B^2*(b*c - a*d)^3*i^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^3*d)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^1, x, 15, -((B*d*(b*c - a*d)*i^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g)) + (d*(b*c - a*d)*i^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g) + (i^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b*g) + (2*B*(b*c - a*d)^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^3*g) + (B^2*(b*c - a*d)^2*i^2*n^2*Log[c + d*x])/(b^3*g) + (B*(b*c - a*d)^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) - ((b*c - a*d)^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B^2*(b*c - a*d)^2*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*g) - (B^2*(b*c - a*d)^2*i^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B*(b*c - a*d)^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g) + (2*B^2*(b*c - a*d)^2*i^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^2, x, 11, -((2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x))/(b^2*g^2*(a + b*x))) - (2*B*(b*c - a*d)*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*g^2*(a + b*x)) + (d^2*i^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^2) - ((b*c - a*d)*i^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^2*g^2*(a + b*x)) + (2*B*d*(b*c - a*d)*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^3*g^2) - (2*d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (2*B^2*d*(b*c - a*d)*i^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*g^2) + (4*B*d*(b*c - a*d)*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2) + (4*B^2*d*(b*c - a*d)*i^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^2)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^3, x, 10, -((2*B^2*d*i^2*n^2*(c + d*x))/(b^2*g^3*(a + b*x))) - (B^2*i^2*n^2*(c + d*x)^2)/(4*b*g^3*(a + b*x)^2) - (2*B*d*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^2*g^3*(a + b*x)) - (B*i^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*g^3*(a + b*x)^2) - (d*i^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^2*g^3*(a + b*x)) - (i^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b*g^3*(a + b*x)^2) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (2*B*d^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3) + (2*B^2*d^2*i^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^3*g^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^4, x, 3, (-2*B^2*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)*g^4*(a + b*x)^3) - (2*B*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)*g^4*(a + b*x)^3) - (i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)*g^4*(a + b*x)^3)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5, x, 7, (2*B^2*d*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B^2*i^2*n^2*(c + d*x)^4)/(32*(b*c - a*d)^2*g^5*(a + b*x)^4) + (2*B*d*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B*i^2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^2*g^5*(a + b*x)^4) + (d*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)} -{(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^6, x, 9, (-2*B^2*d^2*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B^2*d*i^2*n^2*(c + d*x)^4)/(16*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B^2*i^2*n^2*(c + d*x)^5)/(125*(b*c - a*d)^3*g^6*(a + b*x)^5) - (2*B*d^2*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B*i^2*n*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)} - - -{(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 22, (5*B^2*(b*c - a*d)^6*g^3*i^3*n^2*x)/(84*b^3*d^3) + (B^2*(b*c - a*d)^3*g^3*i^3*n^2*(a + b*x)^4)/(140*b^4) - (29*B^2*(b*c - a*d)^5*g^3*i^3*n^2*(c + d*x)^2)/(840*b^2*d^4) + (47*B^2*(b*c - a*d)^4*g^3*i^3*n^2*(c + d*x)^3)/(1260*b*d^4) - (13*B^2*(b*c - a*d)^3*g^3*i^3*n^2*(c + d*x)^4)/(420*d^4) + (b*B^2*(b*c - a*d)^2*g^3*i^3*n^2*(c + d*x)^5)/(105*d^4) - (B*(b*c - a*d)^4*g^3*i^3*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(210*b^4*d) - (3*B*(b*c - a*d)^3*g^3*i^3*n*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(140*b^4) - (B*(b*c - a*d)^2*g^3*i^3*n*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(35*b^3) + (2*B*(b*c - a*d)^4*g^3*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(21*b*d^4) - (3*B*(b*c - a*d)^3*g^3*i^3*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(14*d^4) + (6*b*B*(b*c - a*d)^2*g^3*i^3*n*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(35*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*n*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(21*d^4) + ((b*c - a*d)^3*g^3*i^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(140*b^4) + ((b*c - a*d)^2*g^3*i^3*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(35*b^3) + ((b*c - a*d)*g^3*i^3*(a + b*x)^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(14*b^2) + (g^3*i^3*(a + b*x)^4*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(7*b) + (B*(b*c - a*d)^5*g^3*i^3*n*(a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(420*b^4*d^2) - (B*(b*c - a*d)^6*g^3*i^3*n*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n]))/(420*b^4*d^3) - (B*(b*c - a*d)^7*g^3*i^3*n*(6*A + 11*B*n + 6*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(420*b^4*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(210*b^4*d^4) - (11*B^2*(b*c - a*d)^7*g^3*i^3*n^2*Log[c + d*x])/(420*b^4*d^4) - (B^2*(b*c - a*d)^7*g^3*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(70*b^4*d^4)} -{(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 20, (-7*B^2*(b*c - a*d)^5*g^2*i^3*n^2*x)/(180*b^3*d^2) - (7*B^2*(b*c - a*d)^4*g^2*i^3*n^2*(c + d*x)^2)/(360*b^2*d^3) - (B^2*(b*c - a*d)^3*g^2*i^3*n^2*(c + d*x)^3)/(60*b*d^3) + (B^2*(b*c - a*d)^2*g^2*i^3*n^2*(c + d*x)^4)/(60*d^3) - (B*(b*c - a*d)^4*g^2*i^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b^4*d) - (B*(b*c - a*d)^3*g^2*i^3*n*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^4) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^2*d^3) + (B*(b*c - a*d)^3*g^2*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(45*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*d^3) - (b*B*(b*c - a*d)*g^2*i^3*n*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*d^3) + ((b*c - a*d)^3*g^2*i^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(60*b^4) + ((b*c - a*d)^2*g^2*i^3*(a + b*x)^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(20*b^3) + ((b*c - a*d)*g^2*i^3*(a + b*x)^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(10*b^2) + (g^2*i^3*(a + b*x)^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*b) + (B*(b*c - a*d)^5*g^2*i^3*n*(a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*b^4*d^2) + (B*(b*c - a*d)^6*g^2*i^3*n*(2*A + 3*B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(60*b^4*d^3) + (B^2*(b*c - a*d)^6*g^2*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(36*b^4*d^3) + (11*B^2*(b*c - a*d)^6*g^2*i^3*n^2*Log[c + d*x])/(180*b^4*d^3) + (B^2*(b*c - a*d)^6*g^2*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(30*b^4*d^3)} -{(a*g + b*g*x)^1*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 19, (B^2*(b*c - a*d)^4*g*i^3*n^2*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i^3*n^2*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*c - a*d)^2*g*i^3*n^2*(c + d*x)^3)/(30*b*d^2) - (B*(b*c - a*d)^4*g*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^4*d) - (B*(b*c - a*d)^3*g*i^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^4) + (3*B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*d^2) + ((b*c - a*d)^3*g*i^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(20*b^4) + ((b*c - a*d)^2*g*i^3*(a + b*x)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(10*b^3) + (3*(b*c - a*d)*g*i^3*(a + b*x)^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(20*b^2) + (g*i^3*(a + b*x)^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b) - (B*(b*c - a*d)^5*g*i^3*n*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(10*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(12*b^4*d^2) - (11*B^2*(b*c - a*d)^5*g*i^3*n^2*Log[c + d*x])/(60*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(10*b^4*d^2)} -{(a*g + b*g*x)^0*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2, x, 15, (5*B^2*(b*c - a*d)^3*i^3*n^2*x)/(12*b^3) + (B^2*(b*c - a*d)^2*i^3*n^2*(c + d*x)^2)/(12*b^2*d) - (B*(b*c - a*d)^3*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^4) - (B*(b*c - a*d)^2*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2*d) - (B*(b*c - a*d)*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*d) + (i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*d) + (5*B^2*(b*c - a*d)^4*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(12*b^4*d) + (11*B^2*(b*c - a*d)^4*i^3*n^2*Log[c + d*x])/(12*b^4*d) + (B*(b*c - a*d)^4*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d) - (B^2*(b*c - a*d)^4*i^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(2*b^4*d)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^1, x, 26, (B^2*d*(b*c - a*d)^2*i^3*n^2*x)/(3*b^3*g) - (5*B*d*(b*c - a*d)^2*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^4*g) - (B*(b*c - a*d)*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2*g) + (d*(b*c - a*d)^2*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g) + ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g) + (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b*g) + (2*B*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*g) + (B^2*(b*c - a*d)^3*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(3*b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*n^2*Log[c + d*x])/(b^4*g) + (5*B*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*b^4*g) - ((b*c - a*d)^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g) - (5*B^2*(b*c - a*d)^3*i^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*b^4*g) + (2*B*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g) + (2*B^2*(b*c - a*d)^3*i^3*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^2, x, 17, -((2*B^2*(b*c - a*d)^2*i^3*n^2*(c + d*x))/(b^3*g^2*(a + b*x))) - (B*d^2*(b*c - a*d)*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^2) - (2*B*(b*c - a*d)^2*i^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^2*(a + b*x)) + (2*d^2*(b*c - a*d)*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g^2) - ((b*c - a*d)^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^2*(a + b*x)) + (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g^2) + (4*B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*g^2) + (B^2*d*(b*c - a*d)^2*i^3*n^2*Log[c + d*x])/(b^4*g^2) + (B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (4*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^2) - (B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2) + (6*B^2*d*(b*c - a*d)^2*i^3*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^2)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^3, x, 13, -((4*B^2*d*(b*c - a*d)*i^3*n^2*(c + d*x))/(b^3*g^3*(a + b*x))) - (B^2*(b*c - a*d)*i^3*n^2*(c + d*x)^2)/(4*b^2*g^3*(a + b*x)^2) - (4*B*d*(b*c - a*d)*i^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^3*(a + b*x)) - (B*(b*c - a*d)*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^3*(a + b*x)^2) + (d^3*i^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g^3) - (2*d*(b*c - a*d)*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^3*(a + b*x)) - ((b*c - a*d)*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g^3*(a + b*x)^2) + (2*B*d^2*(b*c - a*d)*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*g^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (2*B^2*d^2*(b*c - a*d)*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*g^3) + (6*B*d^2*(b*c - a*d)*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3) + (6*B^2*d^2*(b*c - a*d)*i^3*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^3)} -{(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^4, x, 13, -((2*B^2*d^2*i^3*n^2*(c + d*x))/(b^3*g^4*(a + b*x))) - (B^2*d*i^3*n^2*(c + d*x)^2)/(4*b^2*g^4*(a + b*x)^2) - (2*B^2*i^3*n^2*(c + d*x)^3)/(27*b*g^4*(a + b*x)^3) - (2*B*d^2*i^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^4*(a + b*x)) - (B*d*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^4*(a + b*x)^2) - (2*B*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*b*g^4*(a + b*x)^3) - (d^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^4*(a + b*x)) - (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b^2*g^4*(a + b*x)^2) - (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b*g^4*(a + b*x)^3) - (d^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4) + (2*B*d^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4) + (2*B^2*d^3*i^3*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x), x, 25, (b*B^2*(b*c - a*d)^2*g^3*n^2*x)/(3*d^3*i) + (7*B*(b*c - a*d)^2*g^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^3*i) - (b^2*B*(b*c - a*d)*g^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^4*i) + (3*(b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*i) - (3*b^2*(b*c - a*d)*g^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d^4*i) + (b^3*g^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*d^4*i) + (6*B*(b*c - a*d)^3*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i) + ((b*c - a*d)^3*g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i) + (B^2*(b*c - a*d)^3*g^3*n^2*Log[(a + b*x)/(c + d*x)])/(3*d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*n^2*Log[c + d*x])/(d^4*i) - (7*B*(b*c - a*d)^3*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*d^4*i) + (6*B^2*(b*c - a*d)^3*g^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (2*B*(b*c - a*d)^3*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i) + (7*B^2*(b*c - a*d)^3*g^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*d^4*i) - (2*B^2*(b*c - a*d)^3*g^3*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x), x, 15, -((B*(b*c - a*d)*g^2*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d^2*i)) - (2*(b*c - a*d)*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^2*i) + (b^2*g^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d^3*i) - (4*B*(b*c - a*d)^2*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i) - ((b*c - a*d)^2*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i) + (B^2*(b*c - a*d)^2*g^2*n^2*Log[c + d*x])/(d^3*i) + (B*(b*c - a*d)^2*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(d^3*i) - (4*B^2*(b*c - a*d)^2*g^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i) - (2*B*(b*c - a*d)^2*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i) - (B^2*(b*c - a*d)^2*g^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(d^3*i) + (2*B^2*(b*c - a*d)^2*g^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x), x, 9, (g*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d*i) + (2*B*(b*c - a*d)*g*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i) + ((b*c - a*d)*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i) + (2*B^2*(b*c - a*d)*g*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i) + (2*B*(b*c - a*d)*g*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i) - (2*B^2*(b*c - a*d)*g*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x), x, 4, -(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d*i)) - (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d*i) + (2*B^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d*i)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^1*(c*i + d*i*x)), x, 3, (A + B*Log[e*((a + b*x)/(c + d*x))^n])^3/(3*B*(b*c - a*d)*g*i*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^2*(c*i + d*i*x)), x, 7, (-2*b*B^2*n^2*(c + d*x))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (2*b*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^2*g^2*i*(a + b*x)) - (b*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^2*g^2*i*(a + b*x)) - (d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^2*g^2*i*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^3*(c*i + d*i*x)), x, 9, (4*b*B^2*d*n^2*(c + d*x))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*B^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (4*b*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (2*b*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^3*i*(a + b*x)) - (b^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2) + (d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^3*g^3*i*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^4*(c*i + d*i*x)), x, 11, (-6*b*B^2*d^2*n^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B^2*d*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (6*b*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (3*b*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^4*g^4*i*n)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^2, x, 18, (2*A*B*(b*c - a*d)^2*g^3*n*(a + b*x))/(d^3*i^2*(c + d*x)) - (2*B^2*(b*c - a*d)^2*g^3*n^2*(a + b*x))/(d^3*i^2*(c + d*x)) + (2*B^2*(b*c - a*d)^2*g^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^3*i^2*(c + d*x)) - (b*B*(b*c - a*d)*g^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d^3*i^2) - (3*b*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*i^2) - ((b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*i^2*(c + d*x)) + (b^3*g^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d^4*i^2) - (6*b*B*(b*c - a*d)^2*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i^2) - (3*b*(b*c - a*d)^2*g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i^2) + (b*B^2*(b*c - a*d)^2*g^3*n^2*Log[c + d*x])/(d^4*i^2) + (b*B*(b*c - a*d)^2*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(d^4*i^2) - (6*b*B^2*(b*c - a*d)^2*g^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) - (6*b*B*(b*c - a*d)^2*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) - (b*B^2*(b*c - a*d)^2*g^3*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(d^4*i^2) + (6*b*B^2*(b*c - a*d)^2*g^3*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^2, x, 12, (-2*A*B*(b*c - a*d)*g^2*n*(a + b*x))/(d^2*i^2*(c + d*x)) + (2*B^2*(b*c - a*d)*g^2*n^2*(a + b*x))/(d^2*i^2*(c + d*x)) - (2*B^2*(b*c - a*d)*g^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^2*i^2*(c + d*x)) + (b*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^2*i^2) + ((b*c - a*d)*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^2*i^2*(c + d*x)) + (2*b*B*(b*c - a*d)*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^2) + (2*b*(b*c - a*d)*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^2) + (2*b*B^2*(b*c - a*d)*g^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2) + (4*b*B*(b*c - a*d)*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2) - (4*b*B^2*(b*c - a*d)*g^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^2)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^2, x, 9, (2*A*B*g*n*(a + b*x))/(d*i^2*(c + d*x)) - (2*B^2*g*n^2*(a + b*x))/(d*i^2*(c + d*x)) + (2*B^2*g*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d*i^2*(c + d*x)) - (g*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d*i^2*(c + d*x)) - (b*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i^2) - (2*b*B*g*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2) + (2*b*B^2*g*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^2*i^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^2, x, 4, (-2*A*B*n*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) + (2*B^2*n^2*(a + b*x))/((b*c - a*d)*i^2*(c + d*x)) - (2*B^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*i^2*(c + d*x)) + ((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*i^2*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^1*(c*i + d*i*x)^2), x, 7, (2*A*B*d*n*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) - (2*B^2*d*n^2*(a + b*x))/((b*c - a*d)^2*g*i^2*(c + d*x)) + (2*B^2*d*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^2*g*i^2*(c + d*x)) - (d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^2*g*i^2*(c + d*x)) + (b*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^2*g*i^2*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^2*(c*i + d*i*x)^2), x, 10, (-2*A*B*d^2*n*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x)) + (2*B^2*d^2*n^2*(a + b*x))/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (2*b^2*B^2*n^2*(c + d*x))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*B^2*d^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (2*b^2*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^3*g^2*i^2*(a + b*x)) + (d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^2*i^2*(c + d*x)) - (b^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^2*i^2*(a + b*x)) - (2*b*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^3*g^2*i^2*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^3*(c*i + d*i*x)^2), x, 12, (2*A*B*d^3*n*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) - (2*B^2*d^3*n^2*(a + b*x))/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (6*b^2*B^2*d*n^2*(c + d*x))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (2*B^2*d^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (6*b^2*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) - (d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^3*i^2*(c + d*x)) + (3*b^2*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^3*i^2*(a + b*x)) - (b^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2) + (b*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(B*(b*c - a*d)^4*g^3*i^2*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^2), x, 14, (-2*A*B*d^4*n*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) + (2*B^2*d^4*n^2*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (12*b^2*B^2*d^2*n^2*(c + d*x))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (b^3*B^2*d*n^2*(c + d*x)^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (2*b^4*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (2*B^2*d^4*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (12*b^2*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (2*b^4*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) + (d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (6*b^2*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^5*g^4*i^2*(a + b*x)^3) - (4*b*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^5*g^4*i^2*n)} - - -{(a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^3, x, 14, (B^2*(b*c - a*d)*g^3*n^2*(a + b*x)^2)/(4*d^2*i^3*(c + d*x)^2) - (4*A*b*B*(b*c - a*d)*g^3*n*(a + b*x))/(d^3*i^3*(c + d*x)) + (4*b*B^2*(b*c - a*d)*g^3*n^2*(a + b*x))/(d^3*i^3*(c + d*x)) - (4*b*B^2*(b*c - a*d)*g^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^3*i^3*(c + d*x)) - (B*(b*c - a*d)*g^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^2*i^3*(c + d*x)^2) + (b^2*g^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*i^3) + ((b*c - a*d)*g^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d^2*i^3*(c + d*x)^2) + (2*b*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*i^3*(c + d*x)) + (2*b^2*B*(b*c - a*d)*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i^3) + (3*b^2*(b*c - a*d)*g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^4*i^3) + (2*b^2*B^2*(b*c - a*d)*g^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3) + (6*b^2*B*(b*c - a*d)*g^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3) - (6*b^2*B^2*(b*c - a*d)*g^3*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^3)} -{(a*g + b*g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^3, x, 11, -(B^2*g^2*n^2*(a + b*x)^2)/(4*d*i^3*(c + d*x)^2) + (2*A*b*B*g^2*n*(a + b*x))/(d^2*i^3*(c + d*x)) - (2*b*B^2*g^2*n^2*(a + b*x))/(d^2*i^3*(c + d*x)) + (2*b*B^2*g^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(d^2*i^3*(c + d*x)) + (B*g^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*i^3*(c + d*x)^2) - (g^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d*i^3*(c + d*x)^2) - (b*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^2*i^3*(c + d*x)) - (b^2*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[(b*c - a*d)/(b*(c + d*x))])/(d^3*i^3) - (2*b^2*B*g^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3) + (2*b^2*B^2*g^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(d^3*i^3)} -{(a*g + b*g*x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^3, x, 3, (B^2*g*n^2*(a + b*x)^2)/(4*(b*c - a*d)*i^3*(c + d*x)^2) - (B*g*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)*i^3*(c + d*x)^2) + (g*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)*i^3*(c + d*x)^2)} -{(a*g + b*g*x)^0*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^3, x, 8, -(B^2*d*n^2*(a + b*x)^2)/(4*(b*c - a*d)^2*i^3*(c + d*x)^2) - (2*A*b*B*n*(a + b*x))/((b*c - a*d)^2*i^3*(c + d*x)) + (2*b*B^2*n^2*(a + b*x))/((b*c - a*d)^2*i^3*(c + d*x)) - (2*b*B^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^2*i^3*(c + d*x)) + (B*d*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*i^3*(c + d*x)^2) - (d*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^2*i^3*(c + d*x)^2) + (b*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^2*i^3*(c + d*x))} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^1*(c*i + d*i*x)^3), x, 15, (B^2*d^2*n^2*(a + b*x)^2)/(4*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (4*A*b*B*d*n*(a + b*x))/((b*c - a*d)^3*g*i^3*(c + d*x)) - (4*b*B^2*d*n^2*(a + b*x))/((b*c - a*d)^3*g*i^3*(c + d*x)) + (4*b*B^2*d*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^3*g*i^3*(c + d*x)) - (B*d^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) - (2*b*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g*i^3*(c + d*x)) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^3*g*i^3*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^2*(c*i + d*i*x)^3), x, 12, -(B^2*d^3*n^2*(a + b*x)^2)/(4*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (6*A*b*B*d^2*n*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) + (6*b*B^2*d^2*n^2*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (2*b^3*B^2*n^2*(c + d*x))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (6*b*B^2*d^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^4*g^2*i^3*(c + d*x)) + (B*d^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (2*b^3*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (d^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(B*(b*c - a*d)^4*g^2*i^3*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^3*(c*i + d*i*x)^3), x, 14, (B^2*d^4*n^2*(a + b*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (8*A*b*B*d^3*n*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) - (8*b*B^2*d^3*n^2*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (8*b^3*B^2*d*n^2*(c + d*x))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (8*b*B^2*d^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^5*g^3*i^3*(c + d*x)) - (B*d^4*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (8*b^3*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (d^4*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^5*g^3*i^3*(a + b*x)^2) + (2*b^2*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(B*(b*c - a*d)^5*g^3*i^3*n)} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^3), x, 16, -(B^2*d^5*n^2*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (10*A*b*B*d^4*n*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) + (10*b*B^2*d^4*n^2*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (20*b^3*B^2*d^2*n^2*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B^2*d*n^2*(c + d*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (2*b^5*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b*B^2*d^4*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^6*g^4*i^3*(c + d*x)) + (B*d^5*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (20*b^3*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (2*b^5*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^6*g^4*i^3*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x) / (c+d x))^n])^p when b f-a g=0 and d h-c i=0 and m+q+2=0*) - - -{(a*g + b*g*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/(c*i + d*i*x)^(m + 2), x, 3, ((a + b*x)*(g*(a + b*x))^m*Gamma[1 + p, -(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p)/(E^((A*(1 + m))/(B*n))*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^m*(-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)))^p*((b*c - a*d)*i^2*(1 + m)*(c + d*x)))} -{(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/(a*g + b*g*x)^(m + 2), x, 3, -((E^((A*(1 + m))/(B*n))*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^(2 + m)*Gamma[1 + p, ((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p)/((((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))^p*((b*c - a*d)*i^2*(1 + m)*(c + d*x))))} - - -{(a*g + b*g*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3/(c*i + d*i*x)^(m + 2), x, 4, -((6*B^3*n^3*(a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^4*(c + d*x)))) + (6*B^2*n^2*(a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^3*(c + d*x))) - (3*B*n*(a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^2*(c + d*x))) + ((a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)*(c + d*x)))} -{(a*g + b*g*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*i + d*i*x)^(m + 2), x, 3, (2*B^2*n^2*(a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^3*(c + d*x))) - (2*B*n*(a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^2*(c + d*x))) + ((a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)*(c + d*x)))} -{(a*g + b*g*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^1/(c*i + d*i*x)^(m + 2), x, 2, -((B*n*(a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)^2*(c + d*x)))) + ((a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((i*(c + d*x))^m*((b*c - a*d)*i^2*(1 + m)*(c + d*x)))} -{(a*g + b*g*x)^m/((c*i + d*i*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^1), x, 3, ((a + b*x)*(g*(a + b*x))^m*ExpIntegralEi[((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(E^((A*(1 + m))/(B*n))*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^m*(B*(b*c - a*d)*i^2*n*(c + d*x)))} -{(a*g + b*g*x)^m/((c*i + d*i*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 4, ((1 + m)*(a + b*x)*(g*(a + b*x))^m*ExpIntegralEi[((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(E^((A*(1 + m))/(B*n))*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^m*(B^2*(b*c - a*d)*i^2*n^2*(c + d*x))) - ((a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*(B*(b*c - a*d)*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))} -{(a*g + b*g*x)^m/((c*i + d*i*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3), x, 5, ((1 + m)^2*(a + b*x)*(g*(a + b*x))^m*ExpIntegralEi[((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(E^((A*(1 + m))/(B*n))*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^m*(2*B^3*(b*c - a*d)*i^2*n^3*(c + d*x))) - ((a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*(2*B*(b*c - a*d)*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)) - ((1 + m)*(a + b*x)*(g*(a + b*x))^m)/((i*(c + d*x))^m*(2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))} - - -{(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3/(a*g + b*g*x)^(m + 2), x, 4, -((6*B^3*n^3*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/((b*c - a*d)*i^2*(1 + m)^4*(c + d*x))) - (6*B^2*n^2*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*i^2*(1 + m)^3*(c + d*x)) - (3*B*n*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*i^2*(1 + m)^2*(c + d*x)) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/((b*c - a*d)*i^2*(1 + m)*(c + d*x))} -{(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^(m + 2), x, 3, -((2*B^2*n^2*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/((b*c - a*d)*i^2*(1 + m)^3*(c + d*x))) - (2*B*n*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*i^2*(1 + m)^2*(c + d*x)) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*i^2*(1 + m)*(c + d*x))} -{(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^1/(a*g + b*g*x)^(m + 2), x, 2, -((B*n*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/((b*c - a*d)*i^2*(1 + m)^2*(c + d*x))) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*i^2*(1 + m)*(c + d*x))} -{(c*i + d*i*x)^m/((a*g + b*g*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^1), x, 3, (E^((A*(1 + m))/(B*n))*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(B*(b*c - a*d)*i^2*n*(c + d*x))} -{(c*i + d*i*x)^m/((a*g + b*g*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x, 4, -((E^((A*(1 + m))/(B*n))*(1 + m)*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(B^2*(b*c - a*d)*i^2*n^2*(c + d*x))) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(B*(b*c - a*d)*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} -{(c*i + d*i*x)^m/((a*g + b*g*x)^(m + 2)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3), x, 5, (E^((A*(1 + m))/(B*n))*(1 + m)^2*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(2*B^3*(b*c - a*d)*i^2*n^3*(c + d*x)) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(2*B*(b*c - a*d)*i^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2) + ((1 + m)*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x) / (c+d x))^n])^p when b f-a g=0 and d h-c i=0 and p symbolic*) - - -{Log[e*((a + b*x)/(c + d*x))^n]^p/((a + b*x)*(c + d*x)), x, 3, Log[e*((a + b*x)/(c + d*x))^n]^(1 + p)/((b*c - a*d)*n*(1 + p))} -{Log[e*((a + b*x)/(c + d*x))^n]^p/(a*c + (b*c + a*d)*x + b*d*x^2), x, 4, Log[e*((a + b*x)/(c + d*x))^n]^(1 + p)/((b*c - a*d)*n*(1 + p))} - - -(* ::Section:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x)/(c+d x))^n])^p when b f-a g=0*) - - -(* ::Section:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Title:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x)^n/(c+d x)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x)^n/(c+d x)^n])^p when b f-a g=0 and d h-c i=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x)^n/(c+d x)^n]) when b f-a g=0 and d h-c i=0*) - - -{(a*g + b*g*x)^m*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/(c*i + d*i*x)^(m + 2), x, 4, ((a + b*x)*(g*(a + b*x))^m*Gamma[1 + p, -(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n))]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p)/(E^((A*(1 + m))/(B*n))*((e*(a + b*x)^n)/(c + d*x)^n)^((1 + m)/n)*(i*(c + d*x))^m*(-(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n)))^p*((b*c - a*d)*i^2*(1 + m)*(c + d*x)))} -{(c*i + d*i*x)^m*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/(a*g + b*g*x)^(m + 2), x, 4, -((1/((b*c - a*d)*i^2*(1 + m)*(c + d*x)))*((E^((A*(1 + m))/(B*n))*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m)*((e*(a + b*x)^n)/(c + d*x)^n)^((1 + m)/n)*Gamma[1 + p, ((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n)]*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p)/(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n))^p))} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/((a + b*x)*(c + d*x)), x, 4, (A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^4/(4*B*(b*c - a*d)*n)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/((a + b*x)*(c + d*x)), x, 4, (A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(3*B*(b*c - a*d)*n)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1/((a + b*x)*(c + d*x)), x, 3, (A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(2*B*(b*c - a*d)*n)} -{1/((A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1*(a + b*x)*(c + d*x)), x, 4, Log[A + B*Log[e*(a + b*x)^n/(c + d*x)^n]]/(B*(b*c - a*d)*n)} -{1/((A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(a + b*x)*(c + d*x)), x, 4, -(1/(B*(b*c - a*d)*n*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])))} -{1/((A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3*(a + b*x)*(c + d*x)), x, 4, -(1/(2*B*(b*c - a*d)*n*(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2))} - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/((a + b*x)*(c + d*x)), x, 4, (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^(1 + p)/(B*(b*c - a*d)*n*(1 + p))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/((a*f + b*f*x)*(c*g + d*g*x)), x, 4, (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^(1 + p)/(B*(b*c - a*d)*f*g*n*(1 + p))} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^p/(a*c*f + (b*c + a*d)*f*x + b*d*f*x^2), x, 5, (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^(1 + p)/(B*(b*c - a*d)*f*n*(1 + p))} - - -{1/(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/((a + b*x)*(c + d*x)), x, 4, Log[A + B*Log[e*(a + b*x)^n/(c + d*x)^n]]/(B*(b*c - a*d)*n)} -{1/(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/((a*f + b*f*x)*(c*g + d*g*x)), x, 4, Log[A + B*Log[e*(a + b*x)^n/(c + d*x)^n]]/(B*(b*c - a*d)*f*g*n)} -{1/(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])/(a*c*f + (b*c + a*d)*f*x + b*d*f*x^2), x, 5, Log[A + B*Log[e*(a + b*x)^n/(c + d*x)^n]]/(B*(b*c - a*d)*f*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q / (A+B Log[e (a+b x)^n/(c+d x)^n]) when b f-a g=0 and d h-c i=0 and m+n+2=0*) - - -{(a + b*x)^m/((c + d*x)^(m + 2)*Log[e*(a + b*x)^n/(c + d*x)^n]), x, 4, ((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*ExpIntegralEi[((1 + m)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/n])/(((e*(a + b*x)^n)/(c + d*x)^n)^((1 + m)/n)*((b*c - a*d)*n))} - - -{(a + b*x)^3/((c + d*x)^5*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((a + b*x)^4*ExpIntegralEi[(4*Log[e*((a + b*x)/(c + d*x))^n])/n])/((e*((a + b*x)/(c + d*x))^n)^(4/n)*((b*c - a*d)*n*(c + d*x)^4))} -{(a + b*x)^2/((c + d*x)^4*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((a + b*x)^3*ExpIntegralEi[(3*Log[e*((a + b*x)/(c + d*x))^n])/n])/((e*((a + b*x)/(c + d*x))^n)^(3/n)*((b*c - a*d)*n*(c + d*x)^3))} -{(a + b*x)^1/((c + d*x)^3*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((a + b*x)^2*ExpIntegralEi[(2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((e*((a + b*x)/(c + d*x))^n)^(2/n)*((b*c - a*d)*n*(c + d*x)^2))} -{(a + b*x)^0/((c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((a + b*x)*ExpIntegralEi[Log[e*((a + b*x)/(c + d*x))^n]/n])/((e*((a + b*x)/(c + d*x))^n)^n^(-1)*((b*c - a*d)*n*(c + d*x)))} -{1/((a + b*x)^1*(c + d*x)^1*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, Log[Log[e*((a + b*x)/(c + d*x))^n]]/((b*c - a*d)*n)} -{1/((a + b*x)^2*(c + d*x)^0*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((e*((a + b*x)/(c + d*x))^n)^(1/n)*(c + d*x)*ExpIntegralEi[-(Log[e*((a + b*x)/(c + d*x))^n]/n)])/((b*c - a*d)*n*(a + b*x))} -{1/((a + b*x)^3*(c + d*x)^(-1)*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[-((2*Log[e*((a + b*x)/(c + d*x))^n])/n)])/((b*c - a*d)*n*(a + b*x)^2)} -{1/((a + b*x)^4*(c + d*x)^(-2)*Log[e*((a + b*x)/(c + d*x))^n]), x, 3, ((e*((a + b*x)/(c + d*x))^n)^(3/n)*(c + d*x)^3*ExpIntegralEi[-((3*Log[e*((a + b*x)/(c + d*x))^n])/n)])/((b*c - a*d)*n*(a + b*x)^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (h+i x)^q (A+B Log[e (a+b x)^n/(c+d x)^n])^p when b f-a g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Log[e (a+b x)^n/(c+d x)^n])^p / ((f+g x) (h+i x)) when b f-a g=0*) - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^4/((a*h + b*h*x)*(f + g*x)), x, 8, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^4*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (4*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (12*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (24*B^3*n^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[4, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (24*B^4*n^4*PolyLog[5, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/((a*h + b*h*x)*(f + g*x)), x, 7, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (6*B^3*n^3*PolyLog[4, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/((a*h + b*h*x)*(f + g*x)), x, 6, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (2*B^2*n^2*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1/((a*h + b*h*x)*(f + g*x)), x, 5, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (B*n*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{1/((A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1*(a*h + b*h*x)*(f + g*x)), x, 1, Defer[Subst][Unintegrable[1/((f + g*x)*(a*h + b*h*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x], e*((a + b*x)/(c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]} -{1/((A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2*(a*h + b*h*x)*(f + g*x)), x, 1, Defer[Subst][Unintegrable[1/((f + g*x)*(a*h + b*h*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x], e*((a + b*x)/(c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]} - - -{Log[(c + d*x)/(a + b*x)]/((a + b*x)*(h*(a - c) + h*(b - d)*x)), x, 2, -(PolyLog[2, 1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*h))} - - -{Log[(a - c*g + (b - d*g)*x)/(a + b*x)]/((a + b*x)*(c + d*x)), x, 2, PolyLog[2, (g*(c + d*x))/(a + b*x)]/(b*c - a*d)} -{Log[1 - (g*(c + d*x))/(a + b*x)]/((a + b*x)*(c + d*x)), x, 3, PolyLog[2, (g*(c + d*x))/(a + b*x)]/(b*c - a*d)} -{Log[(a - c*g + b*x - d*g*x)/(a + b*x)]/((a + b*x)*(c + d*x)), x, 3, PolyLog[2, (g*(c + d*x))/(a + b*x)]/(b*c - a*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Log[e (a+b x)^n/(c+d x)^n])^p / (g+h x+i x^2)*) - - -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^3/(a*f*h + h*(b*f*x + a*g*x) + b*g*h*x^2), x, 8, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (3*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (6*B^3*n^3*PolyLog[4, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a*f*h + h*(b*f*x + a*g*x) + b*g*h*x^2), x, 7, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (2*B^2*n^2*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1/(a*f*h + h*(b*f*x + a*g*x) + b*g*h*x^2), x, 6, -(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)) + (B*n*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)} -{1/(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^1/(a*f*h + h*(b*f*x + a*g*x) + b*g*h*x^2), x, 3, Defer[Subst][Unintegrable[1/((a + b*x)*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])), x], e*((a + b*x)/(c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]/h} -{1/(A + B*Log[e*(a + b*x)^n/(c + d*x)^n])^2/(a*f*h + h*(b*f*x + a*g*x) + b*g*h*x^2), x, 3, Defer[Subst][Unintegrable[1/((a + b*x)*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x], e*((a + b*x)/(c + d*x))^n, (e*(a + b*x)^n)/(c + d*x)^n]/h} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m b/test/methods/rule_based/test_files/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m deleted file mode 100644 index 71068f0..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.2.3 u log(e (f (a+b x)^p (c+d x)^q)^r)^s.m +++ /dev/null @@ -1,221 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x^q)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g/x)^m (A+B Log[e ((a+b x)/(c+d x))^n])^p*) - - -{(f + g/x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 16, -((B*(b*c - a*d)*g^3*n)/(2*a*c*x)) + A*f^3*x - (1/2)*B*(b^2/a^2 - d^2/c^2)*g^3*n*Log[x] + (b^2*B*g^3*n*Log[a + b*x])/(2*a^2) - 3*B*f^2*g*n*Log[x]*Log[1 + (b*x)/a] + (B*f^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b - (g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*x^2) + (3*(b*c - a*d)*f*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*(c + d*x)*(a - (c*(a + b*x))/(c + d*x))) + 3*f^2*g*Log[x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - (B*(b*c - a*d)*f^3*n*Log[c + d*x])/(b*d) - (B*d^2*g^3*n*Log[c + d*x])/(2*c^2) + 3*B*f^2*g*n*Log[x]*Log[1 + (d*x)/c] + (3*B*(b*c - a*d)*f*g^2*n*Log[a - (c*(a + b*x))/(c + d*x)])/(a*c) - 3*B*f^2*g*n*PolyLog[2, -((b*x)/a)] + 3*B*f^2*g*n*PolyLog[2, -((d*x)/c)]} -{(f + g/x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 13, A*f^2*x - 2*B*f*g*n*Log[x]*Log[1 + (b*x)/a] + (B*f^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + ((b*c - a*d)*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*(c + d*x)*(a - (c*(a + b*x))/(c + d*x))) + 2*f*g*Log[x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - (B*(b*c - a*d)*f^2*n*Log[c + d*x])/(b*d) + 2*B*f*g*n*Log[x]*Log[1 + (d*x)/c] + (B*(b*c - a*d)*g^2*n*Log[a - (c*(a + b*x))/(c + d*x)])/(a*c) - 2*B*f*g*n*PolyLog[2, -((b*x)/a)] + 2*B*f*g*n*PolyLog[2, -((d*x)/c)]} -{(f + g/x)^1*(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x, 10, A*f*x - B*g*n*Log[x]*Log[1 + (b*x)/a] + (B*f*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + g*Log[x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - (B*(b*c - a*d)*f*n*Log[c + d*x])/(b*d) + B*g*n*Log[x]*Log[1 + (d*x)/c] - B*g*n*PolyLog[2, -((b*x)/a)] + B*g*n*PolyLog[2, -((d*x)/c)]} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g/x)^1, x, 12, (A*x)/f + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(b*f) - (B*(b*c - a*d)*n*Log[c + d*x])/(b*d*f) + (B*g*n*Log[(f*(a + b*x))/(a*f - b*g)]*Log[g + f*x])/f^2 - (g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[g + f*x])/f^2 - (B*g*n*Log[(f*(c + d*x))/(c*f - d*g)]*Log[g + f*x])/f^2 + (B*g*n*PolyLog[2, -((b*(g + f*x))/(a*f - b*g))])/f^2 - (B*g*n*PolyLog[2, -((d*(g + f*x))/(c*f - d*g))])/f^2} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g/x)^2, x, 15, (A*x)/f^2 + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(b*f^2) - (g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(f^2*(a*f - b*g)*(g + f*x)) - (B*(b*c - a*d)*n*Log[c + d*x])/(b*d*f^2) + (2*B*g*n*Log[(f*(a + b*x))/(a*f - b*g)]*Log[g + f*x])/f^3 - (2*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[g + f*x])/f^3 - (2*B*g*n*Log[(f*(c + d*x))/(c*f - d*g)]*Log[g + f*x])/f^3 + (B*(b*c - a*d)*g^2*n*Log[(g + f*x)/(c + d*x)])/(f^2*(a*f - b*g)*(c*f - d*g)) + (2*B*g*n*PolyLog[2, -((b*(g + f*x))/(a*f - b*g))])/f^3 - (2*B*g*n*PolyLog[2, -((d*(g + f*x))/(c*f - d*g))])/f^3} -{(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g/x)^3, x, 18, (A*x)/f^3 + (B*(b*c - a*d)*g^3*n)/(2*f^3*(a*f - b*g)*(c*f - d*g)*(g + f*x)) - (b^2*B*g^3*n*Log[a + b*x])/(2*f^4*(a*f - b*g)^2) + (B*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(b*f^3) + (g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*f^4*(g + f*x)^2) - (3*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(f^3*(a*f - b*g)*(g + f*x)) - (B*(b*c - a*d)*n*Log[c + d*x])/(b*d*f^3) + (B*d^2*g^3*n*Log[c + d*x])/(2*f^4*(c*f - d*g)^2) + (B*(b*c - a*d)*g^3*(b*c*f + a*d*f - 2*b*d*g)*n*Log[g + f*x])/(2*f^3*(a*f - b*g)^2*(c*f - d*g)^2) + (3*B*g*n*Log[(f*(a + b*x))/(a*f - b*g)]*Log[g + f*x])/f^4 - (3*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[g + f*x])/f^4 - (3*B*g*n*Log[(f*(c + d*x))/(c*f - d*g)]*Log[g + f*x])/f^4 + (3*B*(b*c - a*d)*g^2*n*Log[(g + f*x)/(c + d*x)])/(f^3*(a*f - b*g)*(c*f - d*g)) + (3*B*g*n*PolyLog[2, -((b*(g + f*x))/(a*f - b*g))])/f^4 - (3*B*g*n*PolyLog[2, -((d*(g + f*x))/(c*f - d*g))])/f^4} - - -(* ::Title:: *) -(*Integrands of the form u (A+B Log[e (f (a+b x)^s (c+d x)^t)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x)^m (A+B Log[e (f (a+b x)^s (c+d x)^t)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m (A+B Log[e (f (a+b x)^s (c+d x)^t)^n])^p when b g-a h=0*) - - -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(a + b*x)^4, x, 4, -(((b*c - a*d)^4*q*r*x)/(5*d^4)) + ((b*c - a*d)^3*q*r*(a + b*x)^2)/(10*b*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^3)/(15*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^4)/(20*b*d) - (p*r*(a + b*x)^5)/(25*b) - (q*r*(a + b*x)^5)/(25*b) + ((b*c - a*d)^5*q*r*Log[c + d*x])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(a + b*x)^3, x, 4, ((b*c - a*d)^3*q*r*x)/(4*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^2)/(8*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^3)/(12*b*d) - (p*r*(a + b*x)^4)/(16*b) - (q*r*(a + b*x)^4)/(16*b) - ((b*c - a*d)^4*q*r*Log[c + d*x])/(4*b*d^4) + ((a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(a + b*x)^2, x, 4, -(((b*c - a*d)^2*q*r*x)/(3*d^2)) + ((b*c - a*d)*q*r*(a + b*x)^2)/(6*b*d) - (p*r*(a + b*x)^3)/(9*b) - (q*r*(a + b*x)^3)/(9*b) + ((b*c - a*d)^3*q*r*Log[c + d*x])/(3*b*d^3) + ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(a + b*x)^1, x, 4, (-(1/2))*a*p*r*x + ((b*c - a*d)*q*r*x)/(2*d) - (1/4)*b*p*r*x^2 - (q*r*(a + b*x)^2)/(4*b) - ((b*c - a*d)^2*q*r*Log[c + d*x])/(2*b*d^2) + ((a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^1, x, 6, -((p*r*Log[a + b*x]^2)/(2*b)) - (q*r*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/b + (Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/b - (q*r*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/b} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^2, x, 5, -((p*r)/(b*(a + b*x))) + (d*q*r*Log[a + b*x])/(b*(b*c - a*d)) - (d*q*r*Log[c + d*x])/(b*(b*c - a*d)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(b*(a + b*x))} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^3, x, 4, -((p*r)/(4*b*(a + b*x)^2)) - (d*q*r)/(2*b*(b*c - a*d)*(a + b*x)) - (d^2*q*r*Log[a + b*x])/(2*b*(b*c - a*d)^2) + (d^2*q*r*Log[c + d*x])/(2*b*(b*c - a*d)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(2*b*(a + b*x)^2)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^4, x, 4, -((p*r)/(9*b*(a + b*x)^3)) - (d*q*r)/(6*b*(b*c - a*d)*(a + b*x)^2) + (d^2*q*r)/(3*b*(b*c - a*d)^2*(a + b*x)) + (d^3*q*r*Log[a + b*x])/(3*b*(b*c - a*d)^3) - (d^3*q*r*Log[c + d*x])/(3*b*(b*c - a*d)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(3*b*(a + b*x)^3)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^5, x, 4, -((p*r)/(16*b*(a + b*x)^4)) - (d*q*r)/(12*b*(b*c - a*d)*(a + b*x)^3) + (d^2*q*r)/(8*b*(b*c - a*d)^2*(a + b*x)^2) - (d^3*q*r)/(4*b*(b*c - a*d)^3*(a + b*x)) - (d^4*q*r*Log[a + b*x])/(4*b*(b*c - a*d)^4) + (d^4*q*r*Log[c + d*x])/(4*b*(b*c - a*d)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*b*(a + b*x)^4)} - - -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(a + b*x)^4, x, 32, -((a*(b*c - a*d)^3*p*q*r^2*x)/(5*d^3)) + (2*(b*c - a*d)^4*p*q*r^2*x)/(25*d^4) + (77*(b*c - a*d)^4*q^2*r^2*x)/(150*d^4) + (2*(b*c - a*d)^4*q*(p + q)*r^2*x)/(5*d^4) - (b*(b*c - a*d)^3*p*q*r^2*x^2)/(10*d^3) - ((b*c - a*d)^3*p*q*r^2*(a + b*x)^2)/(25*b*d^3) - (77*(b*c - a*d)^3*q^2*r^2*(a + b*x)^2)/(300*b*d^3) + (16*(b*c - a*d)^2*p*q*r^2*(a + b*x)^3)/(225*b*d^2) + (47*(b*c - a*d)^2*q^2*r^2*(a + b*x)^3)/(450*b*d^2) - (9*(b*c - a*d)*p*q*r^2*(a + b*x)^4)/(200*b*d) - (9*(b*c - a*d)*q^2*r^2*(a + b*x)^4)/(200*b*d) + (2*p^2*r^2*(a + b*x)^5)/(125*b) + (4*p*q*r^2*(a + b*x)^5)/(125*b) + (2*q^2*r^2*(a + b*x)^5)/(125*b) - (2*(b*c - a*d)^5*p*q*r^2*Log[c + d*x])/(25*b*d^5) - (137*(b*c - a*d)^5*q^2*r^2*Log[c + d*x])/(150*b*d^5) - (2*(b*c - a*d)^5*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(5*b*d^5) - ((b*c - a*d)^5*q^2*r^2*Log[c + d*x]^2)/(5*b*d^5) - (2*(b*c - a*d)^4*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(15*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(10*b*d) - (2*p*r*(a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(25*b) - (2*q*r*(a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(25*b) + (2*(b*c - a*d)^5*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(5*b) - (2*(b*c - a*d)^5*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(5*b*d^5)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(a + b*x)^3, x, 28, (a*(b*c - a*d)^2*p*q*r^2*x)/(4*d^2) - ((b*c - a*d)^3*p*q*r^2*x)/(8*d^3) - (13*(b*c - a*d)^3*q^2*r^2*x)/(24*d^3) - ((b*c - a*d)^3*q*(p + q)*r^2*x)/(2*d^3) + (b*(b*c - a*d)^2*p*q*r^2*x^2)/(8*d^2) + ((b*c - a*d)^2*p*q*r^2*(a + b*x)^2)/(16*b*d^2) + (13*(b*c - a*d)^2*q^2*r^2*(a + b*x)^2)/(48*b*d^2) - (7*(b*c - a*d)*p*q*r^2*(a + b*x)^3)/(72*b*d) - (7*(b*c - a*d)*q^2*r^2*(a + b*x)^3)/(72*b*d) + (p^2*r^2*(a + b*x)^4)/(32*b) + (p*q*r^2*(a + b*x)^4)/(16*b) + (q^2*r^2*(a + b*x)^4)/(32*b) + ((b*c - a*d)^4*p*q*r^2*Log[c + d*x])/(8*b*d^4) + (25*(b*c - a*d)^4*q^2*r^2*Log[c + d*x])/(24*b*d^4) + ((b*c - a*d)^4*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b*d^4) + ((b*c - a*d)^4*q^2*r^2*Log[c + d*x]^2)/(4*b*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(6*b*d) - (p*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(8*b) - (q*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(8*b) - ((b*c - a*d)^4*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*d^4) + ((a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(4*b) + ((b*c - a*d)^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*d^4)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(a + b*x)^2, x, 24, -((a*(b*c - a*d)*p*q*r^2*x)/(3*d)) + (2*(b*c - a*d)^2*p*q*r^2*x)/(9*d^2) + (5*(b*c - a*d)^2*q^2*r^2*x)/(9*d^2) + (2*(b*c - a*d)^2*q*(p + q)*r^2*x)/(3*d^2) - (b*(b*c - a*d)*p*q*r^2*x^2)/(6*d) - ((b*c - a*d)*p*q*r^2*(a + b*x)^2)/(9*b*d) - (5*(b*c - a*d)*q^2*r^2*(a + b*x)^2)/(18*b*d) + (2*p^2*r^2*(a + b*x)^3)/(27*b) + (4*p*q*r^2*(a + b*x)^3)/(27*b) + (2*q^2*r^2*(a + b*x)^3)/(27*b) - (2*(b*c - a*d)^3*p*q*r^2*Log[c + d*x])/(9*b*d^3) - (11*(b*c - a*d)^3*q^2*r^2*Log[c + d*x])/(9*b*d^3) - (2*(b*c - a*d)^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b*d^3) - ((b*c - a*d)^3*q^2*r^2*Log[c + d*x]^2)/(3*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d) - (2*p*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(9*b) - (2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(9*b) + (2*(b*c - a*d)^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^3) + ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*b) - (2*(b*c - a*d)^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b*d^3)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(a + b*x)^1, x, 20, (1/2)*a*p^2*r^2*x + (1/2)*a*p*q*r^2*x - ((b*c - a*d)*p*q*r^2*x)/(2*d) - ((b*c - a*d)*q^2*r^2*x)/(2*d) - ((b*c - a*d)*q*(p + q)*r^2*x)/d + (1/4)*b*p^2*r^2*x^2 + (1/4)*b*p*q*r^2*x^2 + (p*q*r^2*(a + b*x)^2)/(4*b) + (q^2*r^2*(a + b*x)^2)/(4*b) + ((b*c - a*d)^2*p*q*r^2*Log[c + d*x])/(2*b*d^2) + (3*(b*c - a*d)^2*q^2*r^2*Log[c + d*x])/(2*b*d^2) + ((b*c - a*d)^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*d^2) + ((b*c - a*d)^2*q^2*r^2*Log[c + d*x]^2)/(2*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d) - (p*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b) - (q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b) - ((b*c - a*d)^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d^2) + ((a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(2*b) + ((b*c - a*d)^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*d^2)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^1, x, 19, Log[(a + b*x)^(p*r)]^3/(3*b*p*r) - (q*Log[(a + b*x)^(p*r)]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(b*p) + (Log[(a + b*x)^(p*r)]^2*Log[(c + d*x)^(q*r)])/(b*p*r) + (Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(q*r)]^2)/b - (2*q*r*Log[(a + b*x)^(p*r)]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/b + (2*q*r*Log[(c + d*x)^(q*r)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/b - (1/4)*(Log[(a + b*x)^(p*r)] + Log[(c + d*x)^(q*r)] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*((Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])^2/(b*p*r) + 8*((Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(q*r)])/b + (q*r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/b)) + (2*p*q*r^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/b - (2*q^2*r^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/b} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^2, x, 20, -((2*p^2*r^2)/(b*(a + b*x))) + (2*d*p*q*r^2*Log[a + b*x])/(b*(b*c - a*d)) - (d*p*q*r^2*Log[a + b*x]^2)/(b*(b*c - a*d)) - (2*d*p*q*r^2*Log[c + d*x])/(b*(b*c - a*d)) + (2*d*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*(b*c - a*d)) + (d*q^2*r^2*Log[c + d*x]^2)/(b*(b*c - a*d)) - (2*d*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)) - (2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(a + b*x)) + (2*d*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)) - (2*d*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(b*(a + b*x)) - (2*d*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)) + (2*d*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d))} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^3, x, 24, -((p^2*r^2)/(4*b*(a + b*x)^2)) - (3*d*p*q*r^2)/(2*b*(b*c - a*d)*(a + b*x)) - (d^2*p*q*r^2*Log[a + b*x])/(2*b*(b*c - a*d)^2) + (d^2*q^2*r^2*Log[a + b*x])/(b*(b*c - a*d)^2) + (d^2*p*q*r^2*Log[a + b*x]^2)/(2*b*(b*c - a*d)^2) + (d^2*p*q*r^2*Log[c + d*x])/(2*b*(b*c - a*d)^2) - (d^2*q^2*r^2*Log[c + d*x])/(b*(b*c - a*d)^2) - (d^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*(b*c - a*d)^2) - (d^2*q^2*r^2*Log[c + d*x]^2)/(2*b*(b*c - a*d)^2) + (d^2*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)^2) - (p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(a + b*x)^2) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)*(a + b*x)) - (d^2*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)^2) + (d^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(2*b*(a + b*x)^2) + (d^2*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)^2) - (d^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)^2)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^4, x, 28, -((2*p^2*r^2)/(27*b*(a + b*x)^3)) - (5*d*p*q*r^2)/(18*b*(b*c - a*d)*(a + b*x)^2) + (8*d^2*p*q*r^2)/(9*b*(b*c - a*d)^2*(a + b*x)) - (d^2*q^2*r^2)/(3*b*(b*c - a*d)^2*(a + b*x)) + (2*d^3*p*q*r^2*Log[a + b*x])/(9*b*(b*c - a*d)^3) - (d^3*q^2*r^2*Log[a + b*x])/(b*(b*c - a*d)^3) - (d^3*p*q*r^2*Log[a + b*x]^2)/(3*b*(b*c - a*d)^3) - (2*d^3*p*q*r^2*Log[c + d*x])/(9*b*(b*c - a*d)^3) + (d^3*q^2*r^2*Log[c + d*x])/(b*(b*c - a*d)^3) + (2*d^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b*(b*c - a*d)^3) + (d^3*q^2*r^2*Log[c + d*x]^2)/(3*b*(b*c - a*d)^3) - (2*d^3*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3) - (2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(9*b*(a + b*x)^3) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)*(a + b*x)^2) + (2*d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^2*(a + b*x)) + (2*d^3*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^3) - (2*d^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(3*b*(a + b*x)^3) - (2*d^3*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3) + (2*d^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^5, x, 32, -((p^2*r^2)/(32*b*(a + b*x)^4)) - (7*d*p*q*r^2)/(72*b*(b*c - a*d)*(a + b*x)^3) + (3*d^2*p*q*r^2)/(16*b*(b*c - a*d)^2*(a + b*x)^2) - (d^2*q^2*r^2)/(12*b*(b*c - a*d)^2*(a + b*x)^2) - (5*d^3*p*q*r^2)/(8*b*(b*c - a*d)^3*(a + b*x)) + (5*d^3*q^2*r^2)/(12*b*(b*c - a*d)^3*(a + b*x)) - (d^4*p*q*r^2*Log[a + b*x])/(8*b*(b*c - a*d)^4) + (11*d^4*q^2*r^2*Log[a + b*x])/(12*b*(b*c - a*d)^4) + (d^4*p*q*r^2*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4) + (d^4*p*q*r^2*Log[c + d*x])/(8*b*(b*c - a*d)^4) - (11*d^4*q^2*r^2*Log[c + d*x])/(12*b*(b*c - a*d)^4) - (d^4*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4) - (d^4*q^2*r^2*Log[c + d*x]^2)/(4*b*(b*c - a*d)^4) + (d^4*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4) - (p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(8*b*(a + b*x)^4) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(6*b*(b*c - a*d)*(a + b*x)^3) + (d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b*(b*c - a*d)^2*(a + b*x)^2) - (d^3*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^3*(a + b*x)) - (d^4*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^4) + (d^4*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(4*b*(a + b*x)^4) + (d^4*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*b*(b*c - a*d)^4) - (d^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^m (A+B Log[e (f (a+b x)^s (c+d x)^t)^n])^p*) - - -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(g + h*x)^4, x, 5, -(((b*g - a*h)^4*p*r*x)/(5*b^4)) - ((d*g - c*h)^4*q*r*x)/(5*d^4) - ((b*g - a*h)^3*p*r*(g + h*x)^2)/(10*b^3*h) - ((d*g - c*h)^3*q*r*(g + h*x)^2)/(10*d^3*h) - ((b*g - a*h)^2*p*r*(g + h*x)^3)/(15*b^2*h) - ((d*g - c*h)^2*q*r*(g + h*x)^3)/(15*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^4)/(20*b*h) - ((d*g - c*h)*q*r*(g + h*x)^4)/(20*d*h) - (p*r*(g + h*x)^5)/(25*h) - (q*r*(g + h*x)^5)/(25*h) - ((b*g - a*h)^5*p*r*Log[a + b*x])/(5*b^5*h) - ((d*g - c*h)^5*q*r*Log[c + d*x])/(5*d^5*h) + ((g + h*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(g + h*x)^3, x, 5, -(((b*g - a*h)^3*p*r*x)/(4*b^3)) - ((d*g - c*h)^3*q*r*x)/(4*d^3) - ((b*g - a*h)^2*p*r*(g + h*x)^2)/(8*b^2*h) - ((d*g - c*h)^2*q*r*(g + h*x)^2)/(8*d^2*h) - ((b*g - a*h)*p*r*(g + h*x)^3)/(12*b*h) - ((d*g - c*h)*q*r*(g + h*x)^3)/(12*d*h) - (p*r*(g + h*x)^4)/(16*h) - (q*r*(g + h*x)^4)/(16*h) - ((b*g - a*h)^4*p*r*Log[a + b*x])/(4*b^4*h) - ((d*g - c*h)^4*q*r*Log[c + d*x])/(4*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(g + h*x)^2, x, 5, -(((b*g - a*h)^2*p*r*x)/(3*b^2)) - ((d*g - c*h)^2*q*r*x)/(3*d^2) - ((b*g - a*h)*p*r*(g + h*x)^2)/(6*b*h) - ((d*g - c*h)*q*r*(g + h*x)^2)/(6*d*h) - (p*r*(g + h*x)^3)/(9*h) - (q*r*(g + h*x)^3)/(9*h) - ((b*g - a*h)^3*p*r*Log[a + b*x])/(3*b^3*h) - ((d*g - c*h)^3*q*r*Log[c + d*x])/(3*d^3*h) + ((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(g + h*x)^1, x, 5, -(((b*g - a*h)*p*r*x)/(2*b)) - ((d*g - c*h)*q*r*x)/(2*d) - (p*r*(g + h*x)^2)/(4*h) - (q*r*(g + h*x)^2)/(4*h) - ((b*g - a*h)^2*p*r*Log[a + b*x])/(2*b^2*h) - ((d*g - c*h)^2*q*r*Log[c + d*x])/(2*d^2*h) + ((g + h*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(g + h*x)^0, x, 3, -((p + q)*r*x) + ((b*c - a*d)*q*r*Log[c + d*x])/(b*d) + ((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/b} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^1, x, 7, -((p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*Log[g + h*x])/h) - (q*r*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[g + h*x])/h + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x])/h - (p*r*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/h - (q*r*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/h} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^2, x, 7, (b*p*r*Log[a + b*x])/(h*(b*g - a*h)) + (d*q*r*Log[c + d*x])/(h*(d*g - c*h)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(g + h*x)) - (b*p*r*Log[g + h*x])/(h*(b*g - a*h)) - (d*q*r*Log[g + h*x])/(h*(d*g - c*h))} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^3, x, 5, (b*p*r)/(2*h*(b*g - a*h)*(g + h*x)) + (d*q*r)/(2*h*(d*g - c*h)*(g + h*x)) + (b^2*p*r*Log[a + b*x])/(2*h*(b*g - a*h)^2) + (d^2*q*r*Log[c + d*x])/(2*h*(d*g - c*h)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(2*h*(g + h*x)^2) - (b^2*p*r*Log[g + h*x])/(2*h*(b*g - a*h)^2) - (d^2*q*r*Log[g + h*x])/(2*h*(d*g - c*h)^2)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^4, x, 5, (b*p*r)/(6*h*(b*g - a*h)*(g + h*x)^2) + (d*q*r)/(6*h*(d*g - c*h)*(g + h*x)^2) + (b^2*p*r)/(3*h*(b*g - a*h)^2*(g + h*x)) + (d^2*q*r)/(3*h*(d*g - c*h)^2*(g + h*x)) + (b^3*p*r*Log[a + b*x])/(3*h*(b*g - a*h)^3) + (d^3*q*r*Log[c + d*x])/(3*h*(d*g - c*h)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(3*h*(g + h*x)^3) - (b^3*p*r*Log[g + h*x])/(3*h*(b*g - a*h)^3) - (d^3*q*r*Log[g + h*x])/(3*h*(d*g - c*h)^3)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^5, x, 5, (b*p*r)/(12*h*(b*g - a*h)*(g + h*x)^3) + (d*q*r)/(12*h*(d*g - c*h)*(g + h*x)^3) + (b^2*p*r)/(8*h*(b*g - a*h)^2*(g + h*x)^2) + (d^2*q*r)/(8*h*(d*g - c*h)^2*(g + h*x)^2) + (b^3*p*r)/(4*h*(b*g - a*h)^3*(g + h*x)) + (d^3*q*r)/(4*h*(d*g - c*h)^3*(g + h*x)) + (b^4*p*r*Log[a + b*x])/(4*h*(b*g - a*h)^4) + (d^4*q*r*Log[c + d*x])/(4*h*(d*g - c*h)^4) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*h*(g + h*x)^4) - (b^4*p*r*Log[g + h*x])/(4*h*(b*g - a*h)^4) - (d^4*q*r*Log[g + h*x])/(4*h*(d*g - c*h)^4)} - - -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(g + h*x)^3, x, 53, (2*(b*g - a*h)^3*p^2*r^2*x)/b^3 + (5*(b*g - a*h)^3*p*q*r^2*x)/(8*b^3) + (5*(b*g - a*h)^2*(d*g - c*h)*p*q*r^2*x)/(12*b^2*d) + (5*(b*g - a*h)*(d*g - c*h)^2*p*q*r^2*x)/(12*b*d^2) + (5*(d*g - c*h)^3*p*q*r^2*x)/(8*d^3) + (2*(d*g - c*h)^3*q^2*r^2*x)/d^3 + (3*h*(b*g - a*h)^2*p^2*r^2*(a + b*x)^2)/(4*b^4) + (2*h^2*(b*g - a*h)*p^2*r^2*(a + b*x)^3)/(9*b^4) + (h^3*p^2*r^2*(a + b*x)^4)/(32*b^4) + (3*h*(d*g - c*h)^2*q^2*r^2*(c + d*x)^2)/(4*d^4) + (2*h^2*(d*g - c*h)*q^2*r^2*(c + d*x)^3)/(9*d^4) + (h^3*q^2*r^2*(c + d*x)^4)/(32*d^4) + (3*(b*g - a*h)^2*p*q*r^2*(g + h*x)^2)/(16*b^2*h) + ((b*g - a*h)*(d*g - c*h)*p*q*r^2*(g + h*x)^2)/(6*b*d*h) + (3*(d*g - c*h)^2*p*q*r^2*(g + h*x)^2)/(16*d^2*h) + (7*(b*g - a*h)*p*q*r^2*(g + h*x)^3)/(72*b*h) + (7*(d*g - c*h)*p*q*r^2*(g + h*x)^3)/(72*d*h) + (p*q*r^2*(g + h*x)^4)/(16*h) + ((b*g - a*h)^4*p*q*r^2*Log[a + b*x])/(8*b^4*h) + ((b*g - a*h)^3*(d*g - c*h)*p*q*r^2*Log[a + b*x])/(6*b^3*d*h) + ((b*g - a*h)^2*(d*g - c*h)^2*p*q*r^2*Log[a + b*x])/(4*b^2*d^2*h) - (2*(b*g - a*h)^3*p^2*r^2*(a + b*x)*Log[a + b*x])/b^4 - ((d*g - c*h)^3*p*q*r^2*(a + b*x)*Log[a + b*x])/(2*b*d^3) - (3*h*(b*g - a*h)^2*p^2*r^2*(a + b*x)^2*Log[a + b*x])/(2*b^4) - (2*h^2*(b*g - a*h)*p^2*r^2*(a + b*x)^3*Log[a + b*x])/(3*b^4) - (h^3*p^2*r^2*(a + b*x)^4*Log[a + b*x])/(8*b^4) - ((d*g - c*h)^2*p*q*r^2*(g + h*x)^2*Log[a + b*x])/(4*d^2*h) - ((d*g - c*h)*p*q*r^2*(g + h*x)^3*Log[a + b*x])/(6*d*h) - (p*q*r^2*(g + h*x)^4*Log[a + b*x])/(8*h) - ((b*g - a*h)^4*p^2*r^2*Log[a + b*x]^2)/(4*b^4*h) + ((b*g - a*h)^2*(d*g - c*h)^2*p*q*r^2*Log[c + d*x])/(4*b^2*d^2*h) + ((b*g - a*h)*(d*g - c*h)^3*p*q*r^2*Log[c + d*x])/(6*b*d^3*h) + ((d*g - c*h)^4*p*q*r^2*Log[c + d*x])/(8*d^4*h) - ((b*g - a*h)^3*p*q*r^2*(c + d*x)*Log[c + d*x])/(2*b^3*d) - (2*(d*g - c*h)^3*q^2*r^2*(c + d*x)*Log[c + d*x])/d^4 - (3*h*(d*g - c*h)^2*q^2*r^2*(c + d*x)^2*Log[c + d*x])/(2*d^4) - (2*h^2*(d*g - c*h)*q^2*r^2*(c + d*x)^3*Log[c + d*x])/(3*d^4) - (h^3*q^2*r^2*(c + d*x)^4*Log[c + d*x])/(8*d^4) - ((b*g - a*h)^2*p*q*r^2*(g + h*x)^2*Log[c + d*x])/(4*b^2*h) - ((b*g - a*h)*p*q*r^2*(g + h*x)^3*Log[c + d*x])/(6*b*h) - (p*q*r^2*(g + h*x)^4*Log[c + d*x])/(8*h) - ((b*g - a*h)^4*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b^4*h) - ((d*g - c*h)^4*q^2*r^2*Log[c + d*x]^2)/(4*d^4*h) - ((d*g - c*h)^4*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*d^4*h) + ((b*g - a*h)^3*p*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*b^3) + ((d*g - c*h)^3*q*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*d^3) + ((b*g - a*h)^2*p*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(4*b^2*h) + ((d*g - c*h)^2*q*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(4*d^2*h) + ((b*g - a*h)*p*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(6*b*h) + ((d*g - c*h)*q*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(6*d*h) + (p*r*(g + h*x)^4*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(8*h) + (q*r*(g + h*x)^4*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(8*h) + ((b*g - a*h)^4*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*b^4*h) + ((d*g - c*h)^4*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(4*h) - ((d*g - c*h)^4*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*d^4*h) - ((b*g - a*h)^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b^4*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(g + h*x)^2, x, 47, (2*(b*g - a*h)^2*p^2*r^2*x)/b^2 + (8*(b*g - a*h)^2*p*q*r^2*x)/(9*b^2) + (2*(b*g - a*h)*(d*g - c*h)*p*q*r^2*x)/(3*b*d) + (8*(d*g - c*h)^2*p*q*r^2*x)/(9*d^2) + (2*(d*g - c*h)^2*q^2*r^2*x)/d^2 + (h*(b*g - a*h)*p^2*r^2*(a + b*x)^2)/(2*b^3) + (2*h^2*p^2*r^2*(a + b*x)^3)/(27*b^3) + (h*(d*g - c*h)*q^2*r^2*(c + d*x)^2)/(2*d^3) + (2*h^2*q^2*r^2*(c + d*x)^3)/(27*d^3) + (5*(b*g - a*h)*p*q*r^2*(g + h*x)^2)/(18*b*h) + (5*(d*g - c*h)*p*q*r^2*(g + h*x)^2)/(18*d*h) + (4*p*q*r^2*(g + h*x)^3)/(27*h) + (2*(b*g - a*h)^3*p*q*r^2*Log[a + b*x])/(9*b^3*h) + ((b*g - a*h)^2*(d*g - c*h)*p*q*r^2*Log[a + b*x])/(3*b^2*d*h) - (2*(b*g - a*h)^2*p^2*r^2*(a + b*x)*Log[a + b*x])/b^3 - (2*(d*g - c*h)^2*p*q*r^2*(a + b*x)*Log[a + b*x])/(3*b*d^2) - (h*(b*g - a*h)*p^2*r^2*(a + b*x)^2*Log[a + b*x])/b^3 - (2*h^2*p^2*r^2*(a + b*x)^3*Log[a + b*x])/(9*b^3) - ((d*g - c*h)*p*q*r^2*(g + h*x)^2*Log[a + b*x])/(3*d*h) - (2*p*q*r^2*(g + h*x)^3*Log[a + b*x])/(9*h) - ((b*g - a*h)^3*p^2*r^2*Log[a + b*x]^2)/(3*b^3*h) + ((b*g - a*h)*(d*g - c*h)^2*p*q*r^2*Log[c + d*x])/(3*b*d^2*h) + (2*(d*g - c*h)^3*p*q*r^2*Log[c + d*x])/(9*d^3*h) - (2*(b*g - a*h)^2*p*q*r^2*(c + d*x)*Log[c + d*x])/(3*b^2*d) - (2*(d*g - c*h)^2*q^2*r^2*(c + d*x)*Log[c + d*x])/d^3 - (h*(d*g - c*h)*q^2*r^2*(c + d*x)^2*Log[c + d*x])/d^3 - (2*h^2*q^2*r^2*(c + d*x)^3*Log[c + d*x])/(9*d^3) - ((b*g - a*h)*p*q*r^2*(g + h*x)^2*Log[c + d*x])/(3*b*h) - (2*p*q*r^2*(g + h*x)^3*Log[c + d*x])/(9*h) - (2*(b*g - a*h)^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b^3*h) - ((d*g - c*h)^3*q^2*r^2*Log[c + d*x]^2)/(3*d^3*h) - (2*(d*g - c*h)^3*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*d^3*h) + (2*(b*g - a*h)^2*p*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*b^2) + (2*(d*g - c*h)^2*q*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*d^2) + ((b*g - a*h)*p*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*b*h) + ((d*g - c*h)*q*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*d*h) + (2*p*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(9*h) + (2*q*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(9*h) + (2*(b*g - a*h)^3*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*b^3*h) + (2*(d*g - c*h)^3*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*d^3*h) + ((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*h) - (2*(d*g - c*h)^3*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*d^3*h) - (2*(b*g - a*h)^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b^3*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(g + h*x)^1, x, 39, (3*(b*g - a*h)*p*q*r^2*x)/(2*b) + (3*(d*g - c*h)*p*q*r^2*x)/(2*d) + (p*q*r^2*(g + h*x)^2)/(2*h) + (p^2*r^2*(4*b*g - 3*a*h + b*h*x)^2)/(4*b^2*h) + (q^2*r^2*(4*d*g - 3*c*h + d*h*x)^2)/(4*d^2*h) + ((b*g - a*h)^2*p*q*r^2*Log[a + b*x])/(2*b^2*h) - (2*(b*g - a*h)*p^2*r^2*(a + b*x)*Log[a + b*x])/b^2 - ((d*g - c*h)*p*q*r^2*(a + b*x)*Log[a + b*x])/(b*d) - (h*p^2*r^2*(a + b*x)^2*Log[a + b*x])/(2*b^2) - (p*q*r^2*(g + h*x)^2*Log[a + b*x])/(2*h) - ((b*g - a*h)^2*p^2*r^2*Log[a + b*x]^2)/(2*b^2*h) + ((d*g - c*h)^2*p*q*r^2*Log[c + d*x])/(2*d^2*h) - ((b*g - a*h)*p*q*r^2*(c + d*x)*Log[c + d*x])/(b*d) - (2*(d*g - c*h)*q^2*r^2*(c + d*x)*Log[c + d*x])/d^2 - (h*q^2*r^2*(c + d*x)^2*Log[c + d*x])/(2*d^2) - (p*q*r^2*(g + h*x)^2*Log[c + d*x])/(2*h) - ((b*g - a*h)^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b^2*h) - ((d*g - c*h)^2*q^2*r^2*Log[c + d*x]^2)/(2*d^2*h) - ((d*g - c*h)^2*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(d^2*h) + ((b*g - a*h)*p*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/b + ((d*g - c*h)*q*r*x*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/d + (p*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*h) + (q*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(2*h) + ((b*g - a*h)^2*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(b^2*h) + ((d*g - c*h)^2*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(d^2*h) + ((g + h*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(2*h) - ((d*g - c*h)^2*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^2*h) - ((b*g - a*h)^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b^2*h)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*(g + h*x)^0, x, 10, 2*(p + q)^2*r^2*x - (2*(b*c - a*d)*q*(p + q)*r^2*Log[c + d*x])/(b*d) - (2*(b*c - a*d)*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*d) - ((b*c - a*d)*q^2*r^2*Log[c + d*x]^2)/(b*d) - (2*(p + q)*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/b + (2*(b*c - a*d)*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d) + ((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/b - (2*(b*c - a*d)*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*d)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^1, x, -29, (p*q*r^2*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]^2)/h + (p^2*r^2*Log[a + b*x]^2*Log[g + h*x])/h + (2*p*q*r^2*Log[a + b*x]*Log[c + d*x]*Log[g + h*x])/h + (q^2*r^2*Log[c + d*x]^2*Log[g + h*x])/h - (2*p*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x])/h - (2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x])/h + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*Log[g + h*x])/h - (p^2*r^2*Log[a + b*x]^2*Log[(b*(g + h*x))/(b*g - a*h)])/h - (2*p*q*r^2*Log[a + b*x]*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[(b*(g + h*x))/(b*g - a*h)])/h + (p*q*r^2*Log[-((h*(c + d*x))/(d*g - c*h))]^2*Log[(b*(g + h*x))/(b*g - a*h)])/h - (2*p*q*r^2*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Log[(b*(g + h*x))/(b*g - a*h)])/h + (p*q*r^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]^2*Log[(b*(g + h*x))/(b*g - a*h)])/h + (2*p*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[(b*(g + h*x))/(b*g - a*h)])/h - (2*p*q*r^2*Log[a + b*x]*Log[c + d*x]*Log[(d*(g + h*x))/(d*g - c*h)])/h - (q^2*r^2*Log[c + d*x]^2*Log[(d*(g + h*x))/(d*g - c*h)])/h + (2*p*q*r^2*Log[a + b*x]*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[(d*(g + h*x))/(d*g - c*h)])/h - (p*q*r^2*Log[-((h*(c + d*x))/(d*g - c*h))]^2*Log[(d*(g + h*x))/(d*g - c*h)])/h + (2*p*q*r^2*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*Log[(d*(g + h*x))/(d*g - c*h)])/h + (2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[(d*(g + h*x))/(d*g - c*h)])/h - (p*q*r^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]^2*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/h - (2*p*r*(q*r*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*PolyLog[2, -((h*(a + b*x))/(b*g - a*h))])/h + (2*q*r*(p*r*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*PolyLog[2, -((h*(c + d*x))/(d*g - c*h))])/h + (2*p*q*r^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/h - (2*p*q*r^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*PolyLog[2, ((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))])/h - (2*p^2*r^2*PolyLog[3, -((h*(a + b*x))/(b*g - a*h))])/h - (2*p*q*r^2*PolyLog[3, -((h*(a + b*x))/(b*g - a*h))])/h - (2*p*q*r^2*PolyLog[3, -((h*(c + d*x))/(d*g - c*h))])/h - (2*q^2*r^2*PolyLog[3, -((h*(c + d*x))/(d*g - c*h))])/h - (2*p*q*r^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/h + (2*p*q*r^2*PolyLog[3, ((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))])/h} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^2, x, 31, (2*b*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(h*(b*g - a*h)) + (2*d*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(h*(d*g - c*h)) - (2*b*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(b*g - a*h)) - (2*d*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(d*g - c*h)) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(h*(g + h*x)) + (2*b*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(h*(b*g - a*h)) + (2*d*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(h*(d*g - c*h)) - (2*d*p*q*r^2*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)])/(h*(d*g - c*h)) - (2*b*p*q*r^2*Log[c + d*x]*Log[(d*(g + h*x))/(d*g - c*h)])/(h*(b*g - a*h)) - (2*b*p^2*r^2*Log[a + b*x]*Log[1 + (b*g - a*h)/(h*(a + b*x))])/(h*(b*g - a*h)) - (2*d*q^2*r^2*Log[c + d*x]*Log[1 + (d*g - c*h)/(h*(c + d*x))])/(h*(d*g - c*h)) + (2*b*p^2*r^2*PolyLog[2, -((b*g - a*h)/(h*(a + b*x)))])/(h*(b*g - a*h)) + (2*d*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(h*(d*g - c*h)) - (2*d*p*q*r^2*PolyLog[2, -((h*(a + b*x))/(b*g - a*h))])/(h*(d*g - c*h)) + (2*d*q^2*r^2*PolyLog[2, -((d*g - c*h)/(h*(c + d*x)))])/(h*(d*g - c*h)) + (2*b*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(h*(b*g - a*h)) - (2*b*p*q*r^2*PolyLog[2, -((h*(c + d*x))/(d*g - c*h))])/(h*(b*g - a*h))} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^3, x, 43, -((b*d*p*q*r^2*Log[a + b*x])/(h*(b*g - a*h)*(d*g - c*h))) + (d*p*q*r^2*Log[a + b*x])/(h*(d*g - c*h)*(g + h*x)) - (b*p^2*r^2*(a + b*x)*Log[a + b*x])/((b*g - a*h)^2*(g + h*x)) - (b*d*p*q*r^2*Log[c + d*x])/(h*(b*g - a*h)*(d*g - c*h)) + (b*p*q*r^2*Log[c + d*x])/(h*(b*g - a*h)*(g + h*x)) - (d*q^2*r^2*(c + d*x)*Log[c + d*x])/((d*g - c*h)^2*(g + h*x)) + (b^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(h*(b*g - a*h)^2) + (d^2*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(h*(d*g - c*h)^2) - (b*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(b*g - a*h)*(g + h*x)) - (d*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(d*g - c*h)*(g + h*x)) - (b^2*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(b*g - a*h)^2) - (d^2*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(h*(d*g - c*h)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(2*h*(g + h*x)^2) + (b^2*p^2*r^2*Log[g + h*x])/(h*(b*g - a*h)^2) + (2*b*d*p*q*r^2*Log[g + h*x])/(h*(b*g - a*h)*(d*g - c*h)) + (d^2*q^2*r^2*Log[g + h*x])/(h*(d*g - c*h)^2) + (b^2*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(h*(b*g - a*h)^2) + (d^2*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(h*(d*g - c*h)^2) - (d^2*p*q*r^2*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)])/(h*(d*g - c*h)^2) - (b^2*p*q*r^2*Log[c + d*x]*Log[(d*(g + h*x))/(d*g - c*h)])/(h*(b*g - a*h)^2) - (b^2*p^2*r^2*Log[a + b*x]*Log[1 + (b*g - a*h)/(h*(a + b*x))])/(h*(b*g - a*h)^2) - (d^2*q^2*r^2*Log[c + d*x]*Log[1 + (d*g - c*h)/(h*(c + d*x))])/(h*(d*g - c*h)^2) + (b^2*p^2*r^2*PolyLog[2, -((b*g - a*h)/(h*(a + b*x)))])/(h*(b*g - a*h)^2) + (d^2*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(h*(d*g - c*h)^2) - (d^2*p*q*r^2*PolyLog[2, -((h*(a + b*x))/(b*g - a*h))])/(h*(d*g - c*h)^2) + (d^2*q^2*r^2*PolyLog[2, -((d*g - c*h)/(h*(c + d*x)))])/(h*(d*g - c*h)^2) + (b^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(h*(b*g - a*h)^2) - (b^2*p*q*r^2*PolyLog[2, -((h*(c + d*x))/(d*g - c*h))])/(h*(b*g - a*h)^2)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^4, x, 57, -((b^2*p^2*r^2)/(3*h*(b*g - a*h)^2*(g + h*x))) - (2*b*d*p*q*r^2)/(3*h*(b*g - a*h)*(d*g - c*h)*(g + h*x)) - (d^2*q^2*r^2)/(3*h*(d*g - c*h)^2*(g + h*x)) - (b^3*p^2*r^2*Log[a + b*x])/(3*h*(b*g - a*h)^3) - (2*b*d^2*p*q*r^2*Log[a + b*x])/(3*h*(b*g - a*h)*(d*g - c*h)^2) - (b^2*d*p*q*r^2*Log[a + b*x])/(3*h*(b*g - a*h)^2*(d*g - c*h)) + (b*p^2*r^2*Log[a + b*x])/(3*h*(b*g - a*h)*(g + h*x)^2) + (d*p*q*r^2*Log[a + b*x])/(3*h*(d*g - c*h)*(g + h*x)^2) + (2*d^2*p*q*r^2*Log[a + b*x])/(3*h*(d*g - c*h)^2*(g + h*x)) - (2*b^2*p^2*r^2*(a + b*x)*Log[a + b*x])/(3*(b*g - a*h)^3*(g + h*x)) - (b*d^2*p*q*r^2*Log[c + d*x])/(3*h*(b*g - a*h)*(d*g - c*h)^2) - (2*b^2*d*p*q*r^2*Log[c + d*x])/(3*h*(b*g - a*h)^2*(d*g - c*h)) - (d^3*q^2*r^2*Log[c + d*x])/(3*h*(d*g - c*h)^3) + (b*p*q*r^2*Log[c + d*x])/(3*h*(b*g - a*h)*(g + h*x)^2) + (d*q^2*r^2*Log[c + d*x])/(3*h*(d*g - c*h)*(g + h*x)^2) + (2*b^2*p*q*r^2*Log[c + d*x])/(3*h*(b*g - a*h)^2*(g + h*x)) - (2*d^2*q^2*r^2*(c + d*x)*Log[c + d*x])/(3*(d*g - c*h)^3*(g + h*x)) + (2*b^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*h*(b*g - a*h)^3) + (2*d^3*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*h*(d*g - c*h)^3) - (b*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(b*g - a*h)*(g + h*x)^2) - (d*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(d*g - c*h)*(g + h*x)^2) - (2*b^2*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(b*g - a*h)^2*(g + h*x)) - (2*d^2*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(d*g - c*h)^2*(g + h*x)) - (2*b^3*p*r*Log[a + b*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(b*g - a*h)^3) - (2*d^3*q*r*Log[c + d*x]*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(3*h*(d*g - c*h)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(3*h*(g + h*x)^3) + (b^3*p^2*r^2*Log[g + h*x])/(h*(b*g - a*h)^3) + (b*d^2*p*q*r^2*Log[g + h*x])/(h*(b*g - a*h)*(d*g - c*h)^2) + (b^2*d*p*q*r^2*Log[g + h*x])/(h*(b*g - a*h)^2*(d*g - c*h)) + (d^3*q^2*r^2*Log[g + h*x])/(h*(d*g - c*h)^3) + (2*b^3*p*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(3*h*(b*g - a*h)^3) + (2*d^3*q*r*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*Log[g + h*x])/(3*h*(d*g - c*h)^3) - (2*d^3*p*q*r^2*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)])/(3*h*(d*g - c*h)^3) - (2*b^3*p*q*r^2*Log[c + d*x]*Log[(d*(g + h*x))/(d*g - c*h)])/(3*h*(b*g - a*h)^3) - (2*b^3*p^2*r^2*Log[a + b*x]*Log[1 + (b*g - a*h)/(h*(a + b*x))])/(3*h*(b*g - a*h)^3) - (2*d^3*q^2*r^2*Log[c + d*x]*Log[1 + (d*g - c*h)/(h*(c + d*x))])/(3*h*(d*g - c*h)^3) + (2*b^3*p^2*r^2*PolyLog[2, -((b*g - a*h)/(h*(a + b*x)))])/(3*h*(b*g - a*h)^3) + (2*d^3*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*h*(d*g - c*h)^3) - (2*d^3*p*q*r^2*PolyLog[2, -((h*(a + b*x))/(b*g - a*h))])/(3*h*(d*g - c*h)^3) + (2*d^3*q^2*r^2*PolyLog[2, -((d*g - c*h)/(h*(c + d*x)))])/(3*h*(d*g - c*h)^3) + (2*b^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*h*(b*g - a*h)^3) - (2*b^3*p*q*r^2*PolyLog[2, -((h*(c + d*x))/(d*g - c*h))])/(3*h*(b*g - a*h)^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x^2)^m (A+B Log[e (f (a+b x)^s (c+d x)^t)^n])^p*) - - -{(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 2, -((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^(1 + n)/(b*c*(1 + n)))} - - -{(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 5, -((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(4*b*c))} -{(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 5, -((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(3*b*c))} -{(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 4, -((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(2*b*c))} -{1/((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 2, -(Log[a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]]]/(b*c))} -{1/((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 2, 1/(b*c*(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]]))} -{1/((a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*(1 - c^2*x^2)), x, 2, 1/(2*b*c*(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2)} - - -{Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1/(1 - a^2*x^2), x, 4, -(Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(2*a))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (s+t Log[i (g+h x)^n])^m Log[e (f (a+b x)^p (c+d x)^q)^r]/(g+h x)*) - - -{(s + t*Log[i*(g + h*x)^n])^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*g + k*h*x), x, 11, -((p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t)) - (q*r*Log[-((h*(c + d*x))/(d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n])^3)/(3*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (q*r*(s + t*Log[i*(g + h*x)^n])^2*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (2*n*p*r*t*(s + t*Log[i*(g + h*x)^n])*PolyLog[3, (b*(g + h*x))/(b*g - a*h)])/(h*k) + (2*n*q*r*t*(s + t*Log[i*(g + h*x)^n])*PolyLog[3, (d*(g + h*x))/(d*g - c*h)])/(h*k) - (2*n^2*p*r*t^2*PolyLog[4, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (2*n^2*q*r*t^2*PolyLog[4, (d*(g + h*x))/(d*g - c*h)])/(h*k)} -{(s + t*Log[i*(g + h*x)^n])^1*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*g + k*h*x), x, 9, -((p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*(s + t*Log[i*(g + h*x)^n])^2)/(2*h*k*n*t)) - (q*r*Log[-((h*(c + d*x))/(d*g - c*h))]*(s + t*Log[i*(g + h*x)^n])^2)/(2*h*k*n*t) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*(s + t*Log[i*(g + h*x)^n])^2)/(2*h*k*n*t) - (p*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (q*r*(s + t*Log[i*(g + h*x)^n])*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k) + (n*p*r*t*PolyLog[3, (b*(g + h*x))/(b*g - a*h)])/(h*k) + (n*q*r*t*PolyLog[3, (d*(g + h*x))/(d*g - c*h)])/(h*k)} -{(s + t*Log[i*(g + h*x)^n])^0*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(k*g + k*h*x), x, 7, -((p*r*Log[-((h*(a + b*x))/(b*g - a*h))]*Log[g*k + h*k*x])/(h*k)) - (q*r*Log[-((h*(c + d*x))/(d*g - c*h))]*Log[g*k + h*k*x])/(h*k) + (Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g*k + h*k*x])/(h*k) - (p*r*PolyLog[2, (b*(g + h*x))/(b*g - a*h)])/(h*k) - (q*r*PolyLog[2, (d*(g + h*x))/(d*g - c*h)])/(h*k)} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((k*g + k*h*x)*(s + t*Log[i*(g + h*x)^n])^1), x, 0, Unintegrable[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])), x]} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((k*g + k*h*x)*(s + t*Log[i*(g + h*x)^n])^2), x, 0, Unintegrable[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/((g*k + h*k*x)*(s + t*Log[i*(g + h*x)^n])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[i (j (h x)^t)^u]^m Log[e (f (a+b x)^p (c+d x)^q)^r]^s/x*) - - -{Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x, x, 13, -((p*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (b*x)/a])/(4*t*u)) + (Log[i*(j*(h*x)^t)^u]^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^4*Log[1 + (d*x)/c])/(4*t*u) - p*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^3*PolyLog[2, -((d*x)/c)] + 3*p*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((b*x)/a)] + 3*q*r*t*u*Log[i*(j*(h*x)^t)^u]^2*PolyLog[3, -((d*x)/c)] - 6*p*r*t^2*u^2*Log[i*(j*(h*x)^t)^u]*PolyLog[4, -((b*x)/a)] - 6*q*r*t^2*u^2*Log[i*(j*(h*x)^t)^u]*PolyLog[4, -((d*x)/c)] + 6*p*r*t^3*u^3*PolyLog[5, -((b*x)/a)] + 6*q*r*t^3*u^3*PolyLog[5, -((d*x)/c)]} -{Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x, x, 11, -((p*r*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (b*x)/a])/(3*t*u)) + (Log[i*(j*(h*x)^t)^u]^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^3*Log[1 + (d*x)/c])/(3*t*u) - p*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]^2*PolyLog[2, -((d*x)/c)] + 2*p*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[3, -((b*x)/a)] + 2*q*r*t*u*Log[i*(j*(h*x)^t)^u]*PolyLog[3, -((d*x)/c)] - 2*p*r*t^2*u^2*PolyLog[4, -((b*x)/a)] - 2*q*r*t^2*u^2*PolyLog[4, -((d*x)/c)]} -{Log[i*(j*(h*x)^t)^u]^1*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x, x, 9, -((p*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (b*x)/a])/(2*t*u)) + (Log[i*(j*(h*x)^t)^u]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*t*u) - (q*r*Log[i*(j*(h*x)^t)^u]^2*Log[1 + (d*x)/c])/(2*t*u) - p*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((b*x)/a)] - q*r*Log[i*(j*(h*x)^t)^u]*PolyLog[2, -((d*x)/c)] + p*r*t*u*PolyLog[3, -((b*x)/a)] + q*r*t*u*PolyLog[3, -((d*x)/c)]} -{Log[i*(j*(h*x)^t)^u]^0*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/x, x, 5, (-p)*r*Log[x]*Log[1 + (b*x)/a] + Log[x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - q*r*Log[x]*Log[1 + (d*x)/c] - p*r*PolyLog[2, -((b*x)/a)] - q*r*PolyLog[2, -((d*x)/c)]} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(x*Log[i*(j*(h*x)^t)^u]^1), x, 0, CannotIntegrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(x*Log[i*(j*(h*x)^t)^u]), x]} -{Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(x*Log[i*(j*(h*x)^t)^u]^2), x, 0, CannotIntegrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(x*Log[i*(j*(h*x)^t)^u]^2), x]} - - -(* Simplification and normalization formerly caused infinite recursion. *) -{Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3/x, x, 0, Unintegrable[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^3)/x, x]} -{Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^2/x, x, 0, Unintegrable[(Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^2)/x, x]} -{Log[x]*Log[(a + b*x)/((b*c - a*d)*x)]^1/x, x, 5, (-(1/2))*Log[1 + a/(b*x)]*Log[x]^2 + (1/2)*Log[b/(b*c - a*d) + a/((b*c - a*d)*x)]*Log[x]^2 + Log[x]*PolyLog[2, -(a/(b*x))] + PolyLog[3, -(a/(b*x))]} -{Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^1), x, 0, Unintegrable[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]), x]} -{Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x, 0, Unintegrable[Log[x]/(x*Log[(a + b*x)/((b*c - a*d)*x)]^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[i (j (g+h x)^t)^u] Log[e (f (a+b x)^p (c+d x)^q)^r]^s / ((a+b x) (c+d x))*) - - -{Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x))*Log[e*((a + b*x)/(c + d*x))^n]^3, x, 14, (m*Log[e*((a + b*x)/(c + d*x))^n]^4*Log[(b*c - a*d)/(b*(c + d*x))])/(4*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c + d*x))^n]^4*Log[h*(f + g*x)^m])/(4*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c + d*x))^n]^4*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(4*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) - (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (3*m*n*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) + (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) - (6*m*n^2*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) - (6*m*n^3*PolyLog[5, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (6*m*n^3*PolyLog[5, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)} -{Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x))*Log[e*((a + b*x)/(c + d*x))^n]^2, x, 12, (m*Log[e*((a + b*x)/(c + d*x))^n]^3*Log[(b*c - a*d)/(b*(c + d*x))])/(3*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c + d*x))^n]^3*Log[h*(f + g*x)^m])/(3*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c + d*x))^n]^3*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) - (2*m*n*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (2*m*n*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) + (2*m*n^2*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) - (2*m*n^2*PolyLog[4, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)} -{Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x))*Log[e*((a + b*x)/(c + d*x))^n]^1, x, 10, (m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[(b*c - a*d)/(b*(c + d*x))])/(2*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c + d*x))^n]^2*Log[h*(f + g*x)^m])/(2*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d) - (m*n*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (m*n*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)} -{Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x))/Log[e*((a + b*x)/(c + d*x))^n]^1, x, 2, (b*Unintegrable[Log[h*(f + g*x)^m]/((a + b*x)*Log[e*((a + b*x)/(c + d*x))^n]), x])/(b*c - a*d) - (d*Unintegrable[Log[h*(f + g*x)^m]/((c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]), x])/(b*c - a*d)} -{Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x))/Log[e*((a + b*x)/(c + d*x))^n]^2, x, 1, -(Log[h*(f + g*x)^m]/((b*c - a*d)*n*Log[e*((a + b*x)/(c + d*x))^n])) + (g*m*Unintegrable[1/((f + g*x)*Log[e*((a + b*x)/(c + d*x))^n]), x])/((b*c - a*d)*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x+h x^2)^m Log[1-f (a+b x)/(c+d x)] Log[e ((a+b x) / (c+d x))^n]^p*) - - -{Log[1 - (a + b*x)/(c + d*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2), x, 2, (b*CannotIntegrate[Log[1 - (a + b*x)/(c + d*x)]/((a + b*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d) - (d*CannotIntegrate[Log[1 - (a + b*x)/(c + d*x)]/((c + d*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d)} -{Log[1 - (c + d*x)/(a + b*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2), x, 2, (b*CannotIntegrate[Log[1 - (c + d*x)/(a + b*x)]/((a + b*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d) - (d*CannotIntegrate[Log[1 - (c + d*x)/(a + b*x)]/((c + d*x)*Log[(a + b*x)/(c + d*x)]^2), x])/(b*c - a*d)} - - -{Log[1 - (a + b*x)/(c + d*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2) + 1/((c + d*x)*(-a + c + (-b + d)*x)*Log[(a + b*x)/(c + d*x)]), x, -3, -(Log[1 - (a + b*x)/(c + d*x)]/((b*c - a*d)*Log[(a + b*x)/(c + d*x)]))} -{Log[1 - (c + d*x)/(a + b*x)]/((a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2) - 1/((a + b*x)*(a - c + (b - d)*x)*Log[(a + b*x)/(c + d*x)]), x, -3, -(Log[1 - (c + d*x)/(a + b*x)]/((b*c - a*d)*Log[(a + b*x)/(c + d*x)]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form RFx Log[e (f (a+b x)^p (c+d x)^q)^r]^s*) - - -{x^3*Log[e*((a + b*x)/(c + d*x))^n]/(f - g*x^2), x, 30, -((a*n*x)/(2*b*g)) + (c*n*x)/(2*d*g) + (a^2*n*Log[a + b*x])/(2*b^2*g) - (n*x^2*Log[a + b*x])/(2*g) - (c^2*n*Log[c + d*x])/(2*d^2*g) + (n*x^2*Log[c + d*x])/(2*g) + (x^2*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(2*g) - (f*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^2) - (f*n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^2) + (f*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g^2) + (f*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f - g*x^2])/(2*g^2) - (f*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^2) - (f*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*g^2) + (f*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^2)} -{x^2*Log[e*((a + b*x)/(c + d*x))^n]/(f - g*x^2), x, 27, -((n*(a + b*x)*Log[a + b*x])/(b*g)) + (n*(c + d*x)*Log[c + d*x])/(d*g) + (x*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/g - (Sqrt[f]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/g^(3/2) - (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*g^(3/2)) + (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^(3/2))} -{x^1*Log[e*((a + b*x)/(c + d*x))^n]/(f - g*x^2), x, 18, -((n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g)) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g) - (n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g) + ((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f - g*x^2])/(2*g) - (n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g) - (n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g) + (n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*g) + (n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g)} -{x^0*Log[e*((a + b*x)/(c + d*x))^n]/(f - g*x^2), x, 7, (Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - ((d*Sqrt[f] - c*Sqrt[g])*(a + b*x))/((b*Sqrt[f] - a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g]) - (Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - ((d*Sqrt[f] + c*Sqrt[g])*(a + b*x))/((b*Sqrt[f] + a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g]) + (n*PolyLog[2, ((d*Sqrt[f] - c*Sqrt[g])*(a + b*x))/((b*Sqrt[f] - a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g]) - (n*PolyLog[2, ((d*Sqrt[f] + c*Sqrt[g])*(a + b*x))/((b*Sqrt[f] + a*Sqrt[g])*(c + d*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[e*((a + b*x)/(c + d*x))^n]/(x^1*(f - g*x^2)), x, 29, (n*Log[-((b*x)/a)]*Log[a + b*x])/f - (n*Log[-((d*x)/c)]*Log[c + d*x])/f - (Log[x]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f - (n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f) - (n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*f) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*f) + ((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f - g*x^2])/(2*f) - (n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*f) - (n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f) + (n*PolyLog[2, 1 + (b*x)/a])/f + (n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*f) + (n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f) - (n*PolyLog[2, 1 + (d*x)/c])/f} -{Log[e*((a + b*x)/(c + d*x))^n]/(x^2*(f - g*x^2)), x, 31, (b*n*Log[x])/(a*f) - (d*n*Log[x])/(c*f) - (b*n*Log[a + b*x])/(a*f) - (n*Log[a + b*x])/(f*x) + (d*n*Log[c + d*x])/(c*f) + (n*Log[c + d*x])/(f*x) + (n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])/(f*x) - (Sqrt[g]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f^(3/2) - (Sqrt[g]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*f^(3/2)) - (Sqrt[g]*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*f^(3/2)) - (Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) - (Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2))} - - -{x^3*Log[e*((a + b*x)/(c + d*x))^n]/(f + g*x + h*x^2), x, 37, (a*n*x)/(2*b*h) - (c*n*x)/(2*d*h) - (a^2*n*Log[a + b*x])/(2*b^2*h) + (n*x^2*Log[a + b*x])/(2*h) - (g*n*(a + b*x)*Log[a + b*x])/(b*h^2) + (c^2*n*Log[c + d*x])/(2*d^2*h) - (n*x^2*Log[c + d*x])/(2*h) + (g*n*(c + d*x)*Log[c + d*x])/(d*h^2) + (g*x*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/h^2 - (x^2*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(2*h) - (g*(g^2 - 3*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(h^3*Sqrt[g^2 - 4*f*h]) + ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) + ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^3) - ((g^2 - f*h)*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*h^3) + ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^3) + ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^3) - ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^3) - ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^3)} -{x^2*Log[e*((a + b*x)/(c + d*x))^n]/(f + g*x + h*x^2), x, 30, (n*(a + b*x)*Log[a + b*x])/(b*h) - (n*(c + d*x)*Log[c + d*x])/(d*h) - (x*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/h + ((g^2 - 2*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(h^2*Sqrt[g^2 - 4*f*h]) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^2) + ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^2) + ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^2) + (g*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*h^2) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^2) + ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^2) + ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^2)} -{x^1*Log[e*((a + b*x)/(c + d*x))^n]/(f + g*x + h*x^2), x, 21, -((g*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(h*Sqrt[g^2 - 4*f*h])) + ((1 - g/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*h) - ((1 - g/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*h) + ((1 + g/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2*h) - ((1 + g/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h) - ((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*h) + ((1 - g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h) + ((1 + g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*h) - ((1 - g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*h) - ((1 + g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*h)} -{x^0*Log[e*((a + b*x)/(c + d*x))^n]/(f + g*x + h*x^2), x, 7, -((Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h - (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h]) + (Log[e*((a + b*x)/(c + d*x))^n]*Log[1 - (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h + (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h] - (n*PolyLog[2, (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h - (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h] + (n*PolyLog[2, (2*(d^2*f - c*d*g + c^2*h)*(a + b*x))/((2*b*d*f - b*c*g - a*d*g + 2*a*c*h + (b*c - a*d)*Sqrt[g^2 - 4*f*h])*(c + d*x))])/Sqrt[g^2 - 4*f*h]} -{Log[e*((a + b*x)/(c + d*x))^n]/(x^1*(f + g*x + h*x^2)), x, 31, (n*Log[-((b*x)/a)]*Log[a + b*x])/f - (n*Log[-((d*x)/c)]*Log[c + d*x])/f - (g*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(f*Sqrt[g^2 - 4*f*h]) - (Log[x]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f - ((1 + g/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*f) + ((1 + g/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*f) - ((1 - g/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2*f) + ((1 - g/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*f) + ((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*f) - ((1 + g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*f) - ((1 - g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*f) + (n*PolyLog[2, 1 + (b*x)/a])/f + ((1 + g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*f) + ((1 - g/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*f) - (n*PolyLog[2, 1 + (d*x)/c])/f} -{Log[e*((a + b*x)/(c + d*x))^n]/(x^2*(f + g*x + h*x^2)), x, 40, (b*n*Log[x])/(a*f) - (d*n*Log[x])/(c*f) - (b*n*Log[a + b*x])/(a*f) - (n*Log[a + b*x])/(f*x) - (g*n*Log[-((b*x)/a)]*Log[a + b*x])/f^2 + (d*n*Log[c + d*x])/(c*f) + (n*Log[c + d*x])/(f*x) + (g*n*Log[-((d*x)/c)]*Log[c + d*x])/f^2 + (n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])/(f*x) + ((g^2 - 2*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f*h]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/(f^2*Sqrt[g^2 - 4*f*h]) + (g*Log[x]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f^2 + ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*f^2) - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h])))])/(2*f^2) + ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[a + b*x]*Log[-((b*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/(2*f^2) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*f^2) - (g*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f + g*x + h*x^2])/(2*f^2) + ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*f^2) + ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))])/(2*f^2) - (g*n*PolyLog[2, 1 + (b*x)/a])/f^2 - ((g + (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^2 - 4*f*h]))])/(2*f^2) - ((g - (g^2 - 2*f*h)/Sqrt[g^2 - 4*f*h])*n*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g + Sqrt[g^2 - 4*f*h]))])/(2*f^2) + (g*n*PolyLog[2, 1 + (d*x)/c])/f^2} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving Log[e (f (a+b x)^p (c+d x)^q)^r]^s*) - - -{Log[(c*x)/(a + b*x)]/(a + b*x), x, 5, -((Log[a/(a + b*x)]*Log[(c*x)/(a + b*x)])/b) - PolyLog[2, 1 - a/(a + b*x)]/b} -{Log[(c*x)/(a + b*x)]^2/(x*(a + b*x)), x, 3, Log[(c*x)/(a + b*x)]^3/(3*a)} - - -{(Log[a/(a + b*x)]*Log[(c*x)/(a + b*x)]^2)/(x*(a + b*x)), x, 3, -((Log[(c*x)/(a + b*x)]^2*PolyLog[2, 1 - a/(a + b*x)])/a) + (2*Log[(c*x)/(a + b*x)]*PolyLog[3, 1 - a/(a + b*x)])/a - (2*PolyLog[4, 1 - a/(a + b*x)])/a} - - -{Log[e*(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*(c + d*x))]/((g*a + g*b*x)*(c + d*x)), x, 3, -((Log[(e*(a + b*x))/(c + d*x)]^2*PolyLog[2, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g)) + (2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g) - (2*PolyLog[4, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g)} -{Log[e*((a + b*x)/(c + d*x))^n]^2*Log[(b*c - a*d)/(b*(c + d*x))]/((g*a + g*b*x)*(c + d*x)), x, 3, -((Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[2, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g)) + (2*n*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g) - (2*n^2*PolyLog[4, 1 - (b*c - a*d)/(b*(c + d*x))])/((b*c - a*d)*g)} - - -{Log[(c*(b + a*x))/x]^1, x, 4, x*Log[a*c + (b*c)/x] + (b*Log[b + a*x])/a} -{Log[(c*(b + a*x))/x]^2, x, 5, ((b + a*x)*Log[a*c + (b*c)/x]^2)/a - (2*b*Log[c*(a + b/x)]*Log[-(b/(a*x))])/a - (2*b*PolyLog[2, 1 + b/(a*x)])/a} -{Log[(c*(b + a*x))/x]^3, x, 7, ((b + a*x)*Log[a*c + (b*c)/x]^3)/a - (3*b*Log[c*(a + b/x)]^2*Log[-(b/(a*x))])/a - (6*b*Log[c*(a + b/x)]*PolyLog[2, 1 + b/(a*x)])/a + (6*b*PolyLog[3, 1 + b/(a*x)])/a} - -{Log[(c*(b + a*x)^2)/x^2], x, 2, (2*b*Log[b + a*x])/a + x*Log[(c*(b + a*x)^2)/x^2]} -{Log[(c*(b + a*x)^2)/x^2]^2, x, 6, -((4*b*Log[b/(b + a*x)]*Log[(c*(b + a*x)^2)/x^2])/a) + x*Log[(c*(b + a*x)^2)/x^2]^2 + (8*b*PolyLog[2, 1 - b/(b + a*x)])/a} -{Log[(c*(b + a*x)^2)/x^2]^3, x, 5, x*Log[(c*(b + a*x)^2)/x^2]^3 - (6*b*Log[(c*(b + a*x)^2)/x^2]^2*Log[1 - (a*x)/(b + a*x)])/a + (24*b*Log[(c*(b + a*x)^2)/x^2]*PolyLog[2, (a*x)/(b + a*x)])/a + (48*b*PolyLog[3, (a*x)/(b + a*x)])/a} - -{Log[(c*x^2)/(b + a*x)^2]^1, x, 2, x*Log[(c*x^2)/(b + a*x)^2] - (2*b*Log[b + a*x])/a} -{Log[(c*x^2)/(b + a*x)^2]^2, x, 6, x*Log[(c*x^2)/(b + a*x)^2]^2 + (4*b*Log[(c*x^2)/(b + a*x)^2]*Log[b/(b + a*x)])/a + (8*b*PolyLog[2, 1 - b/(b + a*x)])/a} -{Log[(c*x^2)/(b + a*x)^2]^3, x, 5, x*Log[(c*x^2)/(b + a*x)^2]^3 + (6*b*Log[(c*x^2)/(b + a*x)^2]^2*Log[b/(b + a*x)])/a + (24*b*Log[(c*x^2)/(b + a*x)^2]*PolyLog[2, (a*x)/(b + a*x)])/a - (48*b*PolyLog[3, (a*x)/(b + a*x)])/a} - - -{PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x, 1, -(PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))]/(b*c - a*d))} - - -{Log[(e*(c + d*x))/(a + b*x)]*Log[((-b)*c + a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x, 2, (Log[(e*(c + d*x))/(a + b*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*c - a*d) - PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))]/(b*c - a*d)} - - -{Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(a + b*x), x, 4, -((Log[((-b)*c + a*d)/(d*(a + b*x))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/b) - (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/b} - - -{Log[(e*(c + d*x))/(a + b*x)]*Log[(((-b)*c + a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))]/((a + b*x)*(c + d*x)), x, 2, (Log[(e*(c + d*x))/(a + b*x)]*PolyLog[2, 1 + ((b*c - a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))])/(b*c - a*d) - PolyLog[3, 1 + ((b*c - a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))]/(b*c - a*d)} - - -{Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/((a + b*x)*(e + f*x)), x, 4, -((Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(b*e - a*f)) - (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(b*e - a*f) + (2*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(b*e - a*f)} - - -{Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2/(e + f*x), x, 9, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/f) + (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f - (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/f + (2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f + (2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/f - (2*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f} - - -{Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(b*(e + f*x))/(b*e - a*f)]/((a + b*x)*(c + d*x)), x, 10, -((Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/(2*(b*c - a*d))) - (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[(b*(e + f*x))/(b*e - a*f)])/(2*(b*c - a*d)) + (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(2*(b*c - a*d)) - (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*c - a*d) + (Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/(b*c - a*d) + PolyLog[3, (b*(c + d*x))/(d*(a + b*x))]/(b*c - a*d) - PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]/(b*c - a*d)} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m deleted file mode 100644 index 37605c1..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.3 u (a+b log(c (d+e x)^n))^p.m +++ /dev/null @@ -1,1221 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form AF[x] (a+b Log[c (d+e x)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[c (d+e x)]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c (d+e x)]^p*) - - -{Log[c*(d + e*x)]^4, x, 5, 24*x - (24*(d + e*x)*Log[c*(d + e*x)])/e + (12*(d + e*x)*Log[c*(d + e*x)]^2)/e - (4*(d + e*x)*Log[c*(d + e*x)]^3)/e + ((d + e*x)*Log[c*(d + e*x)]^4)/e} -{Log[c*(d + e*x)]^3, x, 4, -6*x + (6*(d + e*x)*Log[c*(d + e*x)])/e - (3*(d + e*x)*Log[c*(d + e*x)]^2)/e + ((d + e*x)*Log[c*(d + e*x)]^3)/e} -{Log[c*(d + e*x)]^2, x, 3, 2*x - (2*(d + e*x)*Log[c*(d + e*x)])/e + ((d + e*x)*Log[c*(d + e*x)]^2)/e} -{Log[c*(d + e*x)]^1, x, 2, -x + ((d + e*x)*Log[c*(d + e*x)])/e} -{1/Log[c*(d + e*x)]^1, x, 2, LogIntegral[c*(d + e*x)]/(c*e)} -{1/Log[c*(d + e*x)]^2, x, 3, -((d + e*x)/(e*Log[c*(d + e*x)])) + LogIntegral[c*(d + e*x)]/(c*e)} -{1/Log[c*(d + e*x)]^3, x, 4, -((d + e*x)/(2*e*Log[c*(d + e*x)]^2)) - (d + e*x)/(2*e*Log[c*(d + e*x)]) + LogIntegral[c*(d + e*x)]/(2*c*e)} -{1/Log[c*(d + e*x)]^4, x, 5, -((d + e*x)/(3*e*Log[c*(d + e*x)]^3)) - (d + e*x)/(6*e*Log[c*(d + e*x)]^2) - (d + e*x)/(6*e*Log[c*(d + e*x)]) + LogIntegral[c*(d + e*x)]/(6*c*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c (d+e x)]^(p/2)*) - - -{Log[c*(d + e*x)]^(5/2), x, 7, -((15*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(8*c*e)) + (15*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(4*e) - (5*(d + e*x)*Log[c*(d + e*x)]^(3/2))/(2*e) + ((d + e*x)*Log[c*(d + e*x)]^(5/2))/e} -{Log[c*(d + e*x)]^(3/2), x, 6, (3*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(4*c*e) - (3*(d + e*x)*Sqrt[Log[c*(d + e*x)]])/(2*e) + ((d + e*x)*Log[c*(d + e*x)]^(3/2))/e} -{Log[c*(d + e*x)]^(1/2), x, 5, -((Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(2*c*e)) + ((d + e*x)*Sqrt[Log[c*(d + e*x)]])/e} -{1/Log[c*(d + e*x)]^(1/2), x, 4, (Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e)} -{1/Log[c*(d + e*x)]^(3/2), x, 5, (2*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e) - (2*(d + e*x))/(e*Sqrt[Log[c*(d + e*x)]])} -{1/Log[c*(d + e*x)]^(5/2), x, 6, (4*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(3*c*e) - (2*(d + e*x))/(3*e*Log[c*(d + e*x)]^(3/2)) - (4*(d + e*x))/(3*e*Sqrt[Log[c*(d + e*x)]])} -{1/Log[c*(d + e*x)]^(7/2), x, 7, (8*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(15*c*e) - (2*(d + e*x))/(5*e*Log[c*(d + e*x)]^(5/2)) - (4*(d + e*x))/(15*e*Log[c*(d + e*x)]^(3/2)) - (8*(d + e*x))/(15*e*Sqrt[Log[c*(d + e*x)]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c (d+e x)]^p with p symbolic*) - - -{Log[c*(d + e*x)]^p, x, 3, (Gamma[1 + p, -Log[c*(d + e*x)]]*Log[c*(d + e*x)]^p)/((-Log[c*(d + e*x)])^p*(c*e))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e x)^n])^p*) - - -{(a + b*Log[c*(d + e*x)^n])^4, x, 6, -24*a*b^3*n^3*x + 24*b^4*n^4*x - (24*b^4*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e + (12*b^2*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e - (4*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4)/e} -{(a + b*Log[c*(d + e*x)^n])^3, x, 5, 6*a*b^2*n^2*x - 6*b^3*n^3*x + (6*b^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e} -{(a + b*Log[c*(d + e*x)^n])^2, x, 4, -2*a*b*n*x + 2*b^2*n^2*x - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^n])/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e} -{(a + b*Log[c*(d + e*x)^n])^1, x, 3, a*x - b*n*x + (b*(d + e*x)*Log[c*(d + e*x)^n])/e} -{1/(a + b*Log[c*(d + e*x)^n])^1, x, 3, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e*n))} -{1/(a + b*Log[c*(d + e*x)^n])^2, x, 4, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e*n^2)) - (d + e*x)/(b*e*n*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^3, x, 5, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*b^3*e*n^3)) - (d + e*x)/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - (d + e*x)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e x)^n])^(p/2)*) - - -{(a + b*Log[c*(d + e*x)^n])^(5/2), x, 7, -((15*b^(5/2)*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(8*e))) + (15*b^2*n^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e) - (5*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/e} -{(a + b*Log[c*(d + e*x)^n])^(3/2), x, 6, (3*b^(3/2)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(4*e)) - (3*b*n*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e} -{(a + b*Log[c*(d + e*x)^n])^(1/2), x, 5, -((Sqrt[b]*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*e))) + ((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e} -{1/(a + b*Log[c*(d + e*x)^n])^(1/2), x, 4, (Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(Sqrt[b]*e*Sqrt[n]))} -{1/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 5, (2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^(3/2)*e*n^(3/2))) - (2*(d + e*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{1/(a + b*Log[c*(d + e*x)^n])^(5/2), x, 6, (4*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(3*b^(5/2)*e*n^(5/2))) - (2*(d + e*x))/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) - (4*(d + e*x))/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{1/(a + b*Log[c*(d + e*x)^n])^(7/2), x, 7, (8*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(15*b^(7/2)*e*n^(7/2))) - (2*(d + e*x))/(5*b*e*n*(a + b*Log[c*(d + e*x)^n])^(5/2)) - (4*(d + e*x))/(15*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])^(3/2)) - (8*(d + e*x))/(15*b^3*e*n^3*Sqrt[a + b*Log[c*(d + e*x)^n]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e x)^n])^p with p symbolic*) - - -{(a + b*Log[c*(d + e*x)^n])^p, x, 3, ((d + e*x)*Gamma[1 + p, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^p)/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^p*e)} - - -{(a + b*Log[c*(d + e*x)^(1/2)])^p, x, 3, (Gamma[1 + p, -((2*(a + b*Log[c*Sqrt[d + e*x]]))/b)]*(a + b*Log[c*Sqrt[d + e*x]])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*Sqrt[d + e*x]])/b))^p*(c^2*e))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p when e f-d g=0*) - - -{(e + f*x)^(p - 1)/Log[d*(e + f*x)^p], x, 3, LogIntegral[d*(e + f*x)^p]/(d*f*p)} -{(e*g + f*g*x)^(p - 1)/Log[d*(e + f*x)^p], x, 4, ((e + f*x)^(1 - p)*(g*(e + f*x))^(-1 + p)*LogIntegral[d*(e + f*x)^p])/(d*f*p)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Log[c*(d + e*x)^n])*(f + g*x)^4, x, 3, -((b*(e*f - d*g)^4*n*x)/(5*e^4)) - (b*(e*f - d*g)^3*n*(f + g*x)^2)/(10*e^3*g) - (b*(e*f - d*g)^2*n*(f + g*x)^3)/(15*e^2*g) - (b*(e*f - d*g)*n*(f + g*x)^4)/(20*e*g) - (b*n*(f + g*x)^5)/(25*g) - (b*(e*f - d*g)^5*n*Log[d + e*x])/(5*e^5*g) + ((f + g*x)^5*(a + b*Log[c*(d + e*x)^n]))/(5*g)} -{(a + b*Log[c*(d + e*x)^n])*(f + g*x)^3, x, 3, -((b*(e*f - d*g)^3*n*x)/(4*e^3)) - (b*(e*f - d*g)^2*n*(f + g*x)^2)/(8*e^2*g) - (b*(e*f - d*g)*n*(f + g*x)^3)/(12*e*g) - (b*n*(f + g*x)^4)/(16*g) - (b*(e*f - d*g)^4*n*Log[d + e*x])/(4*e^4*g) + ((f + g*x)^4*(a + b*Log[c*(d + e*x)^n]))/(4*g)} -{(a + b*Log[c*(d + e*x)^n])*(f + g*x)^2, x, 3, -((b*(e*f - d*g)^2*n*x)/(3*e^2)) - (b*(e*f - d*g)*n*(f + g*x)^2)/(6*e*g) - (b*n*(f + g*x)^3)/(9*g) - (b*(e*f - d*g)^3*n*Log[d + e*x])/(3*e^3*g) + ((f + g*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*g)} -{(a + b*Log[c*(d + e*x)^n])*(f + g*x)^1, x, 3, -((b*(e*f - d*g)*n*x)/(2*e)) - (b*n*(f + g*x)^2)/(4*g) - (b*(e*f - d*g)^2*n*Log[d + e*x])/(2*e^2*g) + ((f + g*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*g)} -{(a + b*Log[c*(d + e*x)^n])*(f + g*x)^0, x, 3, a*x - b*n*x + (b*(d + e*x)*Log[c*(d + e*x)^n])/e} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^1, x, 3, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 4, (b*e*n*Log[d + e*x])/(g*(e*f - d*g)) - (a + b*Log[c*(d + e*x)^n])/(g*(f + g*x)) - (b*e*n*Log[f + g*x])/(g*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^3, x, 3, (b*e*n)/(2*g*(e*f - d*g)*(f + g*x)) + (b*e^2*n*Log[d + e*x])/(2*g*(e*f - d*g)^2) - (a + b*Log[c*(d + e*x)^n])/(2*g*(f + g*x)^2) - (b*e^2*n*Log[f + g*x])/(2*g*(e*f - d*g)^2)} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^4, x, 3, (b*e*n)/(6*g*(e*f - d*g)*(f + g*x)^2) + (b*e^2*n)/(3*g*(e*f - d*g)^2*(f + g*x)) + (b*e^3*n*Log[d + e*x])/(3*g*(e*f - d*g)^3) - (a + b*Log[c*(d + e*x)^n])/(3*g*(f + g*x)^3) - (b*e^3*n*Log[f + g*x])/(3*g*(e*f - d*g)^3)} - - -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^3, x, 8, (2*b^2*(e*f - d*g)^3*n^2*x)/e^3 + (3*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2)/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2*(d + e*x)^3)/(9*e^4) + (b^2*g^3*n^2*(d + e*x)^4)/(32*e^4) + (b^2*(e*f - d*g)^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^4 - (3*b*g*(e*f - d*g)^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4) - (2*b*g^2*(e*f - d*g)*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) - (b*g^3*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4) - (b*(e*f - d*g)^4*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + ((f + g*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g)} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^2, x, 8, (2*b^2*(e*f - d*g)^2*n^2*x)/e^2 + (b^2*g*(e*f - d*g)*n^2*(d + e*x)^2)/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3)/(27*e^3) + (b^2*(e*f - d*g)^3*n^2*Log[d + e*x]^2)/(3*e^3*g) - (2*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^3 - (b*g*(e*f - d*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/e^3 - (2*b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*(e*f - d*g)^3*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3*g) + ((f + g*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*g)} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^1, x, 9, -((2*a*b*(e*f - d*g)*n*x)/e) + (2*b^2*(e*f - d*g)*n^2*x)/e + (b^2*g*n^2*(d + e*x)^2)/(4*e^2) - (2*b^2*(e*f - d*g)*n*(d + e*x)*Log[c*(d + e*x)^n])/e^2 - (b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2)} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^0, x, 4, -2*a*b*n*x + 2*b^2*n^2*x - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^n])/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^1, x, 4, ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g - (2*b^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^2, x, 4, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/((e*f - d*g)*(f + g*x)) - (2*b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)) - (2*b^2*e*n^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^3, x, 7, -((b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)^2*(f + g*x))) - (a + b*Log[c*(d + e*x)^n])^2/(2*g*(f + g*x)^2) + (b^2*e^2*n^2*Log[f + g*x])/(g*(e*f - d*g)^2) - (b*e^2*n*(a + b*Log[c*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^2) + (b^2*e^2*n^2*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^2)} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^4, x, 11, -((b^2*e^2*n^2)/(3*g*(e*f - d*g)^2*(f + g*x))) - (b^2*e^3*n^2*Log[d + e*x])/(3*g*(e*f - d*g)^3) + (b*e*n*(a + b*Log[c*(d + e*x)^n]))/(3*g*(e*f - d*g)*(f + g*x)^2) - (2*b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(3*(e*f - d*g)^3*(f + g*x)) - (a + b*Log[c*(d + e*x)^n])^2/(3*g*(f + g*x)^3) + (b^2*e^3*n^2*Log[f + g*x])/(g*(e*f - d*g)^3) - (2*b*e^3*n*(a + b*Log[c*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(3*g*(e*f - d*g)^3) + (2*b^2*e^3*n^2*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(3*g*(e*f - d*g)^3)} - - -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^3, x, 19, (6*a*b^2*(e*f - d*g)^3*n^2*x)/e^3 - (6*b^3*(e*f - d*g)^3*n^3*x)/e^3 - (9*b^3*g*(e*f - d*g)^2*n^3*(d + e*x)^2)/(8*e^4) - (2*b^3*g^2*(e*f - d*g)*n^3*(d + e*x)^3)/(9*e^4) - (3*b^3*g^3*n^3*(d + e*x)^4)/(128*e^4) + (6*b^3*(e*f - d*g)^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^4 + (9*b^2*g*(e*f - d*g)^2*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^4) + (2*b^2*g^2*(e*f - d*g)*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) + (3*b^2*g^3*n^2*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(32*e^4) - (3*b*(e*f - d*g)^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^4 - (9*b*g*(e*f - d*g)^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^4) - (b*g^2*(e*f - d*g)*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/e^4 - (3*b*g^3*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(16*e^4) + ((e*f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (3*g*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^4) + (g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/e^4 + (g^3*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^3)/(4*e^4)} -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^2, x, 15, (6*a*b^2*(e*f - d*g)^2*n^2*x)/e^2 - (6*b^3*(e*f - d*g)^2*n^3*x)/e^2 - (3*b^3*g*(e*f - d*g)*n^3*(d + e*x)^2)/(4*e^3) - (2*b^3*g^2*n^3*(d + e*x)^3)/(27*e^3) + (6*b^3*(e*f - d*g)^2*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^3 + (3*b^2*g*(e*f - d*g)*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (2*b^2*g^2*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (3*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^3 - (3*b*g*(e*f - d*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) - (b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3)} -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^1, x, 11, (6*a*b^2*(e*f - d*g)*n^2*x)/e - (6*b^3*(e*f - d*g)*n^3*x)/e - (3*b^3*g*n^3*(d + e*x)^2)/(8*e^2) + (6*b^3*(e*f - d*g)*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^2 + (3*b^2*g*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) - (3*b*(e*f - d*g)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^2 - (3*b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^2)} -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^0, x, 5, 6*a*b^2*n^2*x - 6*b^3*n^3*x + (6*b^3*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e} -{(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^1, x, 5, ((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g + (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g - (6*b^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g + (6*b^3*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^2, x, 5, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*f - d*g)*(f + g*x)) - (3*b*e*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)) - (6*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)) + (6*b^3*e*n^3*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^3, x, 9, -((3*b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*(e*f - d*g)^2*(f + g*x))) - (a + b*Log[c*(d + e*x)^n])^3/(2*g*(f + g*x)^2) + (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)^2) - (3*b*e^2*n*(a + b*Log[c*(d + e*x)^n])^2*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(2*g*(e*f - d*g)^2) + (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^2) + (3*b^3*e^2*n^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)^2) + (3*b^3*e^2*n^3*PolyLog[3, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^2)} -{(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^4, x, 16, (b^2*e^2*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)^3*(f + g*x)) + (b*e*n*(a + b*Log[c*(d + e*x)^n])^2)/(2*g*(e*f - d*g)*(f + g*x)^2) - (b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/((e*f - d*g)^3*(f + g*x)) - (a + b*Log[c*(d + e*x)^n])^3/(3*g*(f + g*x)^3) - (b^3*e^3*n^3*Log[f + g*x])/(g*(e*f - d*g)^3) + (2*b^2*e^3*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)^3) + (b^2*e^3*n^2*(a + b*Log[c*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b*e^3*n*(a + b*Log[c*(d + e*x)^n])^2*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(g*(e*f - d*g)^3) - (b^3*e^3*n^3*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^3) + (2*b^2*e^3*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^3) + (2*b^3*e^3*n^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)^3) + (2*b^3*e^3*n^3*PolyLog[3, -((e*f - d*g)/(g*(d + e*x)))])/(g*(e*f - d*g)^3)} - - -{(a + b*Log[c*(d + e*x)^n])^4*(f + g*x)^1, x, 13, -((24*a*b^3*(e*f - d*g)*n^3*x)/e) + (24*b^4*(e*f - d*g)*n^4*x)/e + (3*b^4*g*n^4*(d + e*x)^2)/(4*e^2) - (24*b^4*(e*f - d*g)*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e^2 - (3*b^3*g*n^3*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (12*b^2*(e*f - d*g)*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + (3*b^2*g*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (4*b*(e*f - d*g)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e^2 - (b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/e^2 + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^4)/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^4)/(2*e^2)} -{(a + b*Log[c*(d + e*x)^n])^4*(f + g*x)^0, x, 6, -24*a*b^3*n^3*x + 24*b^4*n^4*x - (24*b^4*n^3*(d + e*x)*Log[c*(d + e*x)^n])/e + (12*b^2*n^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e - (4*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4)/e} -{(a + b*Log[c*(d + e*x)^n])^4/(f + g*x)^1, x, 6, ((a + b*Log[c*(d + e*x)^n])^4*Log[(e*(f + g*x))/(e*f - d*g)])/g + (4*b*n*(a + b*Log[c*(d + e*x)^n])^3*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g - (12*b^2*n^2*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g + (24*b^3*n^3*(a + b*Log[c*(d + e*x)^n])*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g - (24*b^4*n^4*PolyLog[5, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])^4/(f + g*x)^2, x, 6, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^4)/((e*f - d*g)*(f + g*x)) - (4*b*e*n*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)) - (12*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)) + (24*b^3*e*n^3*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g)) - (24*b^4*e*n^4*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/(g*(e*f - d*g))} - - -{Log[a + b*x], x, 2, -x + ((a + b*x)*Log[a + b*x])/b} -{Log[a + b*x]^2, x, 3, 2*x - (2*(a + b*x)*Log[a + b*x])/b + ((a + b*x)*Log[a + b*x]^2)/b} -{Log[a + b*x]^3, x, 4, -6*x + (6*(a + b*x)*Log[a + b*x])/b - (3*(a + b*x)*Log[a + b*x]^2)/b + ((a + b*x)*Log[a + b*x]^3)/b} - - -{Log[a + b*x + c*x], x, 3, -x + ((a + (b + c)*x)*Log[a + (b + c)*x])/(b + c)} -{Log[a + b*x + c*x]^2, x, 4, 2*x - (2*(a + (b + c)*x)*Log[a + (b + c)*x])/(b + c) + ((a + (b + c)*x)*Log[a + (b + c)*x]^2)/(b + c)} -{Log[a + b*x + c*x]^3, x, 5, -6*x + (6*(a + (b + c)*x)*Log[a + (b + c)*x])/(b + c) - (3*(a + (b + c)*x)*Log[a + (b + c)*x]^2)/(b + c) + ((a + (b + c)*x)*Log[a + (b + c)*x]^3)/(b + c)} - - -{Log[c*(d + e*x)^n], x, 2, (-n)*x + ((d + e*x)*Log[c*(d + e*x)^n])/e} - - -{Log[-g*(d + e*x)/(e*f - d*g)]/(f + g*x), x, 2, -(PolyLog[2, (e*(f + g*x))/(e*f - d*g)]/g)} - - -{(a + b*Log[c*(1/c + e*x)])/x, x, 2, a Log[x]-b PolyLog[2,-c e x]} - - -{Log[3 + e*x]/x, x, 2, Log[3]*Log[x] - PolyLog[2, -((e*x)/3)]} -{Log[2 + e*x]/x, x, 2, Log[2]*Log[x] - PolyLog[2, -((e*x)/2)]} -{Log[1 + e*x]/x, x, 1, -PolyLog[2, -e*x]} -{Log[0 + e*x]/x, x, 1, (1/2)*Log[e*x]^2} -{Log[-1 + e*x]/x, x, 2, Log[e*x]*Log[-1 + e*x] + PolyLog[2, 1 - e*x]} -{Log[-2 + e*x]/x, x, 2, Log[(e*x)/2]*Log[-2 + e*x] + PolyLog[2, 1 - (e*x)/2]} - - -{(a + b*Log[3 + e*x])/x, x, 2, (a + b*Log[3])*Log[x] - b*PolyLog[2, -((e*x)/3)]} -{(a + b*Log[2 + e*x])/x, x, 2, (a + b*Log[2])*Log[x] - b*PolyLog[2, -((e*x)/2)]} -{(a + b*Log[1 + e*x])/x, x, 2, a*Log[x] - b*PolyLog[2, (-e)*x]} -{(a + b*Log[0 + e*x])/x, x, 1, (a + b*Log[e*x])^2/(2*b)} -{(a + b*Log[-1 + e*x])/x, x, 2, Log[e*x]*(a + b*Log[-1 + e*x]) + b*PolyLog[2, 1 - e*x]} -{(a + b*Log[-2 + e*x])/x, x, 2, Log[(e*x)/2]*(a + b*Log[-2 + e*x]) + b*PolyLog[2, 1 - (e*x)/2]} - - -{Log[c*(a + b*x)^n]^2*x^2, x, 7, (2*a^2*n^2*x)/b^2 - (a*n^2*(a + b*x)^2)/(2*b^3) + (2*n^2*(a + b*x)^3)/(27*b^3) - (a^3*n^2*Log[a + b*x]^2)/(3*b^3) - (2*a^2*n*(a + b*x)*Log[c*(a + b*x)^n])/b^3 + (a*n*(a + b*x)^2*Log[c*(a + b*x)^n])/b^3 - (2*n*(a + b*x)^3*Log[c*(a + b*x)^n])/(9*b^3) + (2*a^3*n*Log[a + b*x]*Log[c*(a + b*x)^n])/(3*b^3) + (1/3)*x^3*Log[c*(a + b*x)^n]^2} -{Log[c*(a + b*x)^n]^2/x^4, x, 11, -((b^2*n^2)/(3*a^2*x)) - (b^3*n^2*Log[x])/a^3 + (b^3*n^2*Log[a + b*x])/(3*a^3) - (b*n*Log[c*(a + b*x)^n])/(3*a*x^2) + (2*b^2*n*(a + b*x)*Log[c*(a + b*x)^n])/(3*a^3*x) - Log[c*(a + b*x)^n]^2/(3*x^3) + (2*b^3*n*Log[c*(a + b*x)^n]*Log[1 - a/(a + b*x)])/(3*a^3) - (2*b^3*n^2*PolyLog[2, a/(a + b*x)])/(3*a^3)} - - -{x^2*Log[c*(a + b*x)^n]^3, x, 14, -((6*a^2*n^3*x)/b^2) + (3*a*n^3*(a + b*x)^2)/(4*b^3) - (2*n^3*(a + b*x)^3)/(27*b^3) + (6*a^2*n^2*(a + b*x)*Log[c*(a + b*x)^n])/b^3 - (3*a*n^2*(a + b*x)^2*Log[c*(a + b*x)^n])/(2*b^3) + (2*n^2*(a + b*x)^3*Log[c*(a + b*x)^n])/(9*b^3) - (3*a^2*n*(a + b*x)*Log[c*(a + b*x)^n]^2)/b^3 + (3*a*n*(a + b*x)^2*Log[c*(a + b*x)^n]^2)/(2*b^3) - (n*(a + b*x)^3*Log[c*(a + b*x)^n]^2)/(3*b^3) + (a^2*(a + b*x)*Log[c*(a + b*x)^n]^3)/b^3 - (a*(a + b*x)^2*Log[c*(a + b*x)^n]^3)/b^3 + ((a + b*x)^3*Log[c*(a + b*x)^n]^3)/(3*b^3)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(a + b*Log[c*(d + e*x)^n])*(f + g*x)^3, x, 14, ((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e^4*n)) + (3*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b*e^4*n)) + (3*g^2*(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b*e^4*n)) + (g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(b*e^4*n))} -{1/(a + b*Log[c*(d + e*x)^n])*(f + g*x)^2, x, 11, ((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e^3*n)) + (2*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b*e^3*n)) + (g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b*e^3*n))} -{1/(a + b*Log[c*(d + e*x)^n])*(f + g*x)^1, x, 8, ((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e^2*n)) + (g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b*e^2*n))} -{1/(a + b*Log[c*(d + e*x)^n])*(f + g*x)^0, x, 3, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e*n))} -{1/(a + b*Log[c*(d + e*x)^n])/(f + g*x)^1, x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]} -{1/(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 0, Unintegrable[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])), x]} - - -{1/(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^3, x, 26, ((e*f - d*g)^3*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e^4*n^2)) + (6*g*(e*f - d*g)^2*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^2*e^4*n^2)) + (9*g^2*(e*f - d*g)*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^2*e^4*n^2)) + (4*g^3*(d + e*x)^4*ExpIntegralEi[(4*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(b^2*e^4*n^2)) - ((d + e*x)*(f + g*x)^3)/(b*e*n*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^2, x, 20, ((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e^3*n^2)) + (4*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^2*e^3*n^2)) + (3*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^2*e^3*n^2)) - ((d + e*x)*(f + g*x)^2)/(b*e*n*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^1, x, 12, ((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e^2*n^2)) + (2*g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^2*e^2*n^2)) - ((d + e*x)*(f + g*x))/(b*e*n*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^0, x, 4, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e*n^2)) - (d + e*x)/(b*e*n*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^1, x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x]} -{1/(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^2, x, 0, Unintegrable[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x]} - - -{1/(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^2, x, 33, ((e*f - d*g)^2*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*b^3*e^3*n^3)) + (4*g*(e*f - d*g)*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^3*e^3*n^3)) + (9*g^2*(d + e*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(2*b^3*e^3*n^3)) - ((d + e*x)*(f + g*x)^2)/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) + ((e*f - d*g)*(d + e*x)*(f + g*x))/(b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - (3*(d + e*x)*(f + g*x)^2)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^1, x, 17, ((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*b^3*e^2*n^3)) + (2*g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^3*e^2*n^3)) - ((d + e*x)*(f + g*x))/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) + ((e*f - d*g)*(d + e*x))/(2*b^2*e^2*n^2*(a + b*Log[c*(d + e*x)^n])) - ((d + e*x)*(f + g*x))/(b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^0, x, 5, ((d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*b^3*e*n^3)) - (d + e*x)/(2*b*e*n*(a + b*Log[c*(d + e*x)^n])^2) - (d + e*x)/(2*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n]))} -{1/(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^1, x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^3), x]} -{1/(a + b*Log[c*(d + e*x)^n])^3/(f + g*x)^2, x, 0, Unintegrable[1/((f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d (e+f x)^p)^q])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]], x, 17, -((Sqrt[b]*(e*f - d*g)^2*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*e^3))) - (Sqrt[b]*g*(e*f - d*g)*Sqrt[n]*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(2*e^3)) - (Sqrt[b]*g^2*Sqrt[n]*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(6*e^3)) + ((e*f - d*g)^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g*(e*f - d*g)*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^3 + (g^2*(d + e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(3*e^3)} -{(f + g*x)^1*Sqrt[a + b*Log[c*(d + e*x)^n]], x, 12, -((Sqrt[b]*(e*f - d*g)*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*e^2))) - (Sqrt[b]*g*Sqrt[n]*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(4*e^2)) + ((e*f - d*g)*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e^2 + (g*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e^2)} -{(f + g*x)^0*Sqrt[a + b*Log[c*(d + e*x)^n]], x, 5, -((Sqrt[b]*Sqrt[n]*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(2*e))) + ((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/e} -{Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^1, x, 0, Unintegrable[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x), x]} -{Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^2, x, 1, ((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/((e*f - d*g)*(f + g*x)) - (b*e*n*Unintegrable[1/((f + g*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x])/(2*(e*f - d*g))} -{Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^3, x, 1, -(Sqrt[a + b*Log[c*(d + e*x)^n]]/(2*g*(f + g*x)^2)) + (b*e*n*Unintegrable[1/((d + e*x)*(f + g*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]]), x])/(4*g)} - - -{(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2), x, 20, (3*b^(3/2)*(e*f - d*g)^2*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(4*e^3)) + (3*b^(3/2)*g*(e*f - d*g)*n^(3/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(8*e^3)) + (b^(3/2)*g^2*n^(3/2)*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(12*e^3)) - (3*b*(e*f - d*g)^2*n*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e^3) - (3*b*g*(e*f - d*g)*n*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e^3) - (b*g^2*n*(d + e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(6*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^(3/2))/(3*e^3)} -{(f + g*x)^1*(a + b*Log[c*(d + e*x)^n])^(3/2), x, 14, (3*b^(3/2)*(e*f - d*g)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(4*e^2)) + (3*b^(3/2)*g*n^(3/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(16*e^2)) - (3*b*(e*f - d*g)*n*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e^2) - (3*b*g*n*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(8*e^2) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e^2)} -{(f + g*x)^0*(a + b*Log[c*(d + e*x)^n])^(3/2), x, 6, (3*b^(3/2)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(4*e)) - (3*b*n*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(2*e) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e} -{(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^1, x, 0, Unintegrable[(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x), x]} -{(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^2, x, 1, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/((e*f - d*g)*(f + g*x)) - (3*b*e*n*Unintegrable[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x), x])/(2*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^3, x, 1, -((a + b*Log[c*(d + e*x)^n])^(3/2)/(2*g*(f + g*x)^2)) + (3*b*e*n*Unintegrable[Sqrt[a + b*Log[c*(d + e*x)^n]]/((d + e*x)*(f + g*x)^2), x])/(4*g)} - - -{(f + g*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2), x, 23, -((15*b^(5/2)*(e*f - d*g)^2*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(8*e^3))) - (15*b^(5/2)*g*(e*f - d*g)*n^(5/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(32*e^3)) - (5*b^(5/2)*g^2*n^(5/2)*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(72*e^3)) + (15*b^2*(e*f - d*g)^2*n^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e^3) + (15*b^2*g*(e*f - d*g)*n^2*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(16*e^3) + (5*b^2*g^2*n^2*(d + e*x)^3*Sqrt[a + b*Log[c*(d + e*x)^n]])/(36*e^3) - (5*b*(e*f - d*g)^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e^3) - (5*b*g*(e*f - d*g)*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(4*e^3) - (5*b*g^2*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^(3/2))/(18*e^3) + ((e*f - d*g)^2*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/e^3 + (g*(e*f - d*g)*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2))/e^3 + (g^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^(5/2))/(3*e^3)} -{(f + g*x)^1*(a + b*Log[c*(d + e*x)^n])^(5/2), x, 16, -((15*b^(5/2)*(e*f - d*g)*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(8*e^2))) - (15*b^(5/2)*g*n^(5/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(64*e^2)) + (15*b^2*(e*f - d*g)*n^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e^2) + (15*b^2*g*n^2*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(32*e^2) - (5*b*(e*f - d*g)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e^2) - (5*b*g*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(8*e^2) + ((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/e^2 + (g*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^(5/2))/(2*e^2)} -{(f + g*x)^0*(a + b*Log[c*(d + e*x)^n])^(5/2), x, 7, -((15*b^(5/2)*n^(5/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(8*e))) + (15*b^2*n^2*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(4*e) - (5*b*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/e} -{(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^1, x, 0, Unintegrable[(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x), x]} -{(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^2, x, 1, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/((e*f - d*g)*(f + g*x)) - (5*b*e*n*Unintegrable[(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x), x])/(2*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^3, x, 1, -((a + b*Log[c*(d + e*x)^n])^(5/2)/(2*g*(f + g*x)^2)) + (5*b*e*n*Unintegrable[(a + b*Log[c*(d + e*x)^n])^(3/2)/((d + e*x)*(f + g*x)^2), x])/(4*g)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x)^3/Sqrt[a + b*Log[c*(d + e*x)^n]], x, 18, ((e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(Sqrt[b]*e^4*Sqrt[n])) + (g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(2*Sqrt[b]*e^4*Sqrt[n])) + (3*g*(e*f - d*g)^2*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(Sqrt[b]*e^4*Sqrt[n])) + (g^2*(e*f - d*g)*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(Sqrt[b]*e^4*Sqrt[n]))} -{(f + g*x)^2/Sqrt[a + b*Log[c*(d + e*x)^n]], x, 14, ((e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(Sqrt[b]*e^3*Sqrt[n])) + (g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(Sqrt[b]*e^3*Sqrt[n])) + (g^2*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(Sqrt[b]*e^3*Sqrt[n]))} -{(f + g*x)^1/Sqrt[a + b*Log[c*(d + e*x)^n]], x, 10, ((e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(Sqrt[b]*e^2*Sqrt[n])) + (g*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(Sqrt[b]*e^2*Sqrt[n]))} -{(f + g*x)^0/Sqrt[a + b*Log[c*(d + e*x)^n]], x, 4, (Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(Sqrt[b]*e*Sqrt[n]))} -{1/((f + g*x)^1*Sqrt[a + b*Log[c*(d + e*x)^n]]), x, 0, Unintegrable[1/((f + g*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]} - - -{(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 33, (2*(e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^(3/2)*e^4*n^(3/2))) + (4*g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(b^(3/2)*e^4*n^(3/2))) + (6*g*(e*f - d*g)^2*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^(3/2)*e^4*n^(3/2))) + (6*g^2*(e*f - d*g)*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^(3/2)*e^4*n^(3/2))) - (2*(d + e*x)*(f + g*x)^3)/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 25, (2*(e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^(3/2)*e^3*n^(3/2))) + (4*g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^(3/2)*e^3*n^(3/2))) + (2*g^2*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^(3/2)*e^3*n^(3/2))) - (2*(d + e*x)*(f + g*x)^2)/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^1/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 15, (2*(e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^(3/2)*e^2*n^(3/2))) + (2*g*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^(3/2)*e^2*n^(3/2))) - (2*(d + e*x)*(f + g*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^0/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 5, (2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^(3/2)*e*n^(3/2))) - (2*(d + e*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{1/((f + g*x)^1*(a + b*Log[c*(d + e*x)^n])^(3/2)), x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(3/2)), x]} - - -{(f + g*x)^3/(a + b*Log[c*(d + e*x)^n])^(5/2), x, 59, (4*(e*f - d*g)^3*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(3*b^(5/2)*e^4*n^(5/2))) + (32*g^3*Sqrt[Pi]*(d + e*x)^4*Erfi[(2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(3*b^(5/2)*e^4*n^(5/2))) + (8*g*(e*f - d*g)^2*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(b^(5/2)*e^4*n^(5/2))) + (12*g^2*(e*f - d*g)*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^(5/2)*e^4*n^(5/2))) - (2*(d + e*x)*(f + g*x)^3)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (4*(e*f - d*g)*(d + e*x)*(f + g*x)^2)/(b^2*e^2*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]) - (16*(d + e*x)*(f + g*x)^3)/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^2/(a + b*Log[c*(d + e*x)^n])^(5/2), x, 41, (4*(e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(3*b^(5/2)*e^3*n^(5/2))) + (16*g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(3*b^(5/2)*e^3*n^(5/2))) + (4*g^2*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(b^(5/2)*e^3*n^(5/2))) - (2*(d + e*x)*(f + g*x)^2)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (8*(e*f - d*g)*(d + e*x)*(f + g*x))/(3*b^2*e^2*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]) - (4*(d + e*x)*(f + g*x)^2)/(b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^1/(a + b*Log[c*(d + e*x)^n])^(5/2), x, 21, (4*(e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(3*b^(5/2)*e^2*n^(5/2))) + (8*g*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(3*b^(5/2)*e^2*n^(5/2))) - (2*(d + e*x)*(f + g*x))/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (4*(e*f - d*g)*(d + e*x))/(3*b^2*e^2*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]]) - (8*(d + e*x)*(f + g*x))/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{(f + g*x)^0/(a + b*Log[c*(d + e*x)^n])^(5/2), x, 6, (4*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(3*b^(5/2)*e*n^(5/2))) - (2*(d + e*x))/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) - (4*(d + e*x))/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])} -{1/((f + g*x)^1*(a + b*Log[c*(d + e*x)^n])^(5/2)), x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^(m/2) (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]), x, 6, -((4*b*(e*f - d*g)^2*n*Sqrt[f + g*x])/(5*e^2*g)) - (4*b*(e*f - d*g)*n*(f + g*x)^(3/2))/(15*e*g) - (4*b*n*(f + g*x)^(5/2))/(25*g) + (4*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(5*e^(5/2)*g) + (2*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(5*g)} -{(f + g*x)^(1/2)*(a + b*Log[c*(d + e*x)^n]), x, 5, -((4*b*(e*f - d*g)*n*Sqrt[f + g*x])/(3*e*g)) - (4*b*n*(f + g*x)^(3/2))/(9*g) + (4*b*(e*f - d*g)^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(3*e^(3/2)*g) + (2*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(3*g)} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(1/2), x, 4, -((4*b*n*Sqrt[f + g*x])/g) + (4*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(Sqrt[e]*g) + (2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/g} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(3/2), x, 3, -((4*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(g*Sqrt[e*f - d*g])) - (2*(a + b*Log[c*(d + e*x)^n]))/(g*Sqrt[f + g*x])} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(5/2), x, 4, (4*b*e*n)/(3*g*(e*f - d*g)*Sqrt[f + g*x]) - (4*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(3*g*(e*f - d*g)^(3/2)) - (2*(a + b*Log[c*(d + e*x)^n]))/(3*g*(f + g*x)^(3/2))} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(7/2), x, 5, (4*b*e*n)/(15*g*(e*f - d*g)*(f + g*x)^(3/2)) + (4*b*e^2*n)/(5*g*(e*f - d*g)^2*Sqrt[f + g*x]) - (4*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(5*g*(e*f - d*g)^(5/2)) - (2*(a + b*Log[c*(d + e*x)^n]))/(5*g*(f + g*x)^(5/2))} -{(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(9/2), x, 6, (4*b*e*n)/(35*g*(e*f - d*g)*(f + g*x)^(5/2)) + (4*b*e^2*n)/(21*g*(e*f - d*g)^2*(f + g*x)^(3/2)) + (4*b*e^3*n)/(7*g*(e*f - d*g)^3*Sqrt[f + g*x]) - (4*b*e^(7/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(7*g*(e*f - d*g)^(7/2)) - (2*(a + b*Log[c*(d + e*x)^n]))/(7*g*(f + g*x)^(7/2))} - - -{(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])^2, x, 28, (368*b^2*(e*f - d*g)^2*n^2*Sqrt[f + g*x])/(75*e^2*g) + (128*b^2*(e*f - d*g)*n^2*(f + g*x)^(3/2))/(225*e*g) + (16*b^2*n^2*(f + g*x)^(5/2))/(125*g) - (368*b^2*(e*f - d*g)^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(75*e^(5/2)*g) - (8*b^2*(e*f - d*g)^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(5*e^(5/2)*g) - (8*b*(e*f - d*g)^2*n*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/(5*e^2*g) - (8*b*(e*f - d*g)*n*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(15*e*g) - (8*b*n*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(25*g) + (8*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(5*e^(5/2)*g) + (2*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n])^2)/(5*g) + (16*b^2*(e*f - d*g)^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(5*e^(5/2)*g) + (8*b^2*(e*f - d*g)^(5/2)*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(5*e^(5/2)*g)} -{(f + g*x)^(1/2)*(a + b*Log[c*(d + e*x)^n])^2, x, 21, (64*b^2*(e*f - d*g)*n^2*Sqrt[f + g*x])/(9*e*g) + (16*b^2*n^2*(f + g*x)^(3/2))/(27*g) - (64*b^2*(e*f - d*g)^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(9*e^(3/2)*g) - (8*b^2*(e*f - d*g)^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(3*e^(3/2)*g) - (8*b*(e*f - d*g)*n*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e*g) - (8*b*n*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(9*g) + (8*b*(e*f - d*g)^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(3*e^(3/2)*g) + (2*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])^2)/(3*g) + (16*b^2*(e*f - d*g)^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(3*e^(3/2)*g) + (8*b^2*(e*f - d*g)^(3/2)*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(3*e^(3/2)*g)} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^(1/2), x, 15, (16*b^2*n^2*Sqrt[f + g*x])/g - (16*b^2*Sqrt[e*f - d*g]*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(Sqrt[e]*g) - (8*b^2*Sqrt[e*f - d*g]*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(Sqrt[e]*g) - (8*b*n*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/g + (8*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[e]*g) + (2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n])^2)/g + (16*b^2*Sqrt[e*f - d*g]*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(Sqrt[e]*g) + (8*b^2*Sqrt[e*f - d*g]*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(Sqrt[e]*g)} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^(3/2), x, 10, (8*b^2*Sqrt[e]*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(g*Sqrt[e*f - d*g]) - (8*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(g*Sqrt[e*f - d*g]) - (2*(a + b*Log[c*(d + e*x)^n])^2)/(g*Sqrt[f + g*x]) - (16*b^2*Sqrt[e]*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(g*Sqrt[e*f - d*g]) - (8*b^2*Sqrt[e]*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(g*Sqrt[e*f - d*g])} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^(5/2), x, 14, (16*b^2*e^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(3*g*(e*f - d*g)^(3/2)) + (8*b^2*e^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(3*g*(e*f - d*g)^(3/2)) + (8*b*e*n*(a + b*Log[c*(d + e*x)^n]))/(3*g*(e*f - d*g)*Sqrt[f + g*x]) - (8*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(3*g*(e*f - d*g)^(3/2)) - (2*(a + b*Log[c*(d + e*x)^n])^2)/(3*g*(f + g*x)^(3/2)) - (16*b^2*e^(3/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(3*g*(e*f - d*g)^(3/2)) - (8*b^2*e^(3/2)*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(3*g*(e*f - d*g)^(3/2))} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^(7/2), x, 19, -((16*b^2*e^2*n^2)/(15*g*(e*f - d*g)^2*Sqrt[f + g*x])) + (64*b^2*e^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(15*g*(e*f - d*g)^(5/2)) + (8*b^2*e^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(5*g*(e*f - d*g)^(5/2)) + (8*b*e*n*(a + b*Log[c*(d + e*x)^n]))/(15*g*(e*f - d*g)*(f + g*x)^(3/2)) + (8*b*e^2*n*(a + b*Log[c*(d + e*x)^n]))/(5*g*(e*f - d*g)^2*Sqrt[f + g*x]) - (8*b*e^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(5*g*(e*f - d*g)^(5/2)) - (2*(a + b*Log[c*(d + e*x)^n])^2)/(5*g*(f + g*x)^(5/2)) - (16*b^2*e^(5/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(5*g*(e*f - d*g)^(5/2)) - (8*b^2*e^(5/2)*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(5*g*(e*f - d*g)^(5/2))} -{(a + b*Log[c*(d + e*x)^n])^2/(f + g*x)^(9/2), x, 25, -((16*b^2*e^2*n^2)/(105*g*(e*f - d*g)^2*(f + g*x)^(3/2))) - (128*b^2*e^3*n^2)/(105*g*(e*f - d*g)^3*Sqrt[f + g*x]) + (368*b^2*e^(7/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(105*g*(e*f - d*g)^(7/2)) + (8*b^2*e^(7/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(7*g*(e*f - d*g)^(7/2)) + (8*b*e*n*(a + b*Log[c*(d + e*x)^n]))/(35*g*(e*f - d*g)*(f + g*x)^(5/2)) + (8*b*e^2*n*(a + b*Log[c*(d + e*x)^n]))/(21*g*(e*f - d*g)^2*(f + g*x)^(3/2)) + (8*b*e^3*n*(a + b*Log[c*(d + e*x)^n]))/(7*g*(e*f - d*g)^3*Sqrt[f + g*x]) - (8*b*e^(7/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(7*g*(e*f - d*g)^(7/2)) - (2*(a + b*Log[c*(d + e*x)^n])^2)/(7*g*(f + g*x)^(7/2)) - (16*b^2*e^(7/2)*n^2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(7*g*(e*f - d*g)^(7/2)) - (8*b^2*e^(7/2)*n^2*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(7*g*(e*f - d*g)^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x)^(3/2)/(a + b*Log[c*(d + e*x)^n]), x, 0, Unintegrable[(f + g*x)^(3/2)/(a + b*Log[c*(d + e*x)^n]), x]} -{(f + g*x)^(1/2)/(a + b*Log[c*(d + e*x)^n]), x, 0, Unintegrable[Sqrt[f + g*x]/(a + b*Log[c*(d + e*x)^n]), x]} -{1/((f + g*x)^(1/2)*(a + b*Log[c*(d + e*x)^n])), x, 0, Unintegrable[1/(Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n])), x]} -{1/((f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])), x, 0, Unintegrable[1/((f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^(m/2) (a+b Log[c (d (e+f x)^p)^q])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]], x, 1, (2*(f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(3*g) - (b*e*n*Unintegrable[(f + g*x)^(3/2)/((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x])/(3*g)} -{Sqrt[a + b*Log[c*(d + e*x)^n]]/Sqrt[f + g*x], x, 1, (2*Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]])/g - (b*e*n*Unintegrable[Sqrt[f + g*x]/((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x])/g} -{Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^(3/2), x, 1, -((2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(g*Sqrt[f + g*x])) + (b*e*n*Unintegrable[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x])/g} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sqrt[f + g*x]/Sqrt[a + b*Log[c*(d + e*x)^n]], x, 0, Unintegrable[Sqrt[f + g*x]/Sqrt[a + b*Log[c*(d + e*x)^n]], x]} -{1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x, 0, Unintegrable[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]} -{1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x, 0, Unintegrable[1/((f + g*x)^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p when m symbolic*) - - -(* {(a + b*Log[c*(d + e*x)^n])^3*(f + g*x)^m, x, 0, (3*a^2*b*f*p*q*(f + g*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(f + g*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + (6*a*b^2*p^2*q^2*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, ((-f)*g + e*h)/(h*(e + f*x))])/(h*(1 + m)^3) - (6*b^3*p^3*q^3*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m, -1 - m}, {-m, -m, -m}, ((-f)*g + e*h)/(h*(e + f*x))])/(h*(1 + m)^4) - (6*a*b^2*p*q*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, ((-f)*g + e*h)/(h*(e + f*x))]*Log[c*(d + e*x)^n])/(h*(1 + m)^2) + (6*b^3*p^2*q^2*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, ((-f)*g + e*h)/(h*(e + f*x))]*Log[c*(d + e*x)^n])/(h*(1 + m)^3) - (3*b^3*p*q*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, ((-f)*g + e*h)/(h*(e + f*x))]*Log[c*(d + e*x)^n]^2)/(h*(1 + m)^2) + ((f + g*x)^(1 + m)*(a + b*Log[c*(d + e*x)^n])^3)/(h*(1 + m))} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*x)^m, x, 0, (a^2*(f + g*x)^(1 + m))/(h*(1 + m)) + (2*a*b*f*p*q*(f + g*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(f + g*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + (2*b^2*p^2*q^2*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, ((-f)*g + e*h)/(h*(e + f*x))])/(h*(1 + m)^3) + (2*a*b*(f + g*x)^(1 + m)*Log[c*(d + e*x)^n])/(h*(1 + m)) - (2*b^2*p*q*(f + g*x)^(1 + m)*((f*(f + g*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, ((-f)*g + e*h)/(h*(e + f*x))]*Log[c*(d + e*x)^n])/(h*(1 + m)^2) + (b^2*(f + g*x)^(1 + m)*Log[c*(d + e*x)^n]^2)/(h*(1 + m))} *) -{(a + b*Log[c*(d + e*x)^n])^1*(f + g*x)^m, x, 2, (b*e*n*(f + g*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (e*(f + g*x))/(e*f - d*g)])/(g*(e*f - d*g)*(1 + m)*(2 + m)) + ((f + g*x)^(1 + m)*(a + b*Log[c*(d + e*x)^n]))/(g*(1 + m))} -{(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^1, x, 0, Unintegrable[(f + g*x)^m/(a + b*Log[c*(d + e*x)^n]), x]} -{(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^2, x, 0, Unintegrable[(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^2, x]} - - -{(f + g*x)^m*(a + b*Log[c*(d + e*x)^n])^(3/2), x, 0, Unintegrable[(f + g*x)^m*(a + b*Log[c*(d + e*x)^n])^(3/2), x]} -{(f + g*x)^m*(a + b*Log[c*(d + e*x)^n])^(1/2), x, 0, Unintegrable[(f + g*x)^m*Sqrt[a + b*Log[c*(d + e*x)^n]], x]} -{(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^(1/2), x, 0, Unintegrable[(f + g*x)^m/Sqrt[a + b*Log[c*(d + e*x)^n]], x]} -{(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^(3/2), x, 0, Unintegrable[(f + g*x)^m/(a + b*Log[c*(d + e*x)^n])^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p when n symbolic*) - - -{(a + b*Log[c*(d + e*x)^n])^n*(f + g*x)^m, x, 0, Unintegrable[(f + g*x)^m*(a + b*Log[c*(d + e*x)^n])^n, x]} - - -{(a + b*Log[c*(d + e*x)^n])^n*(f + g*x)^3, x, 14, (4^(-1 - n)*g^3*(d + e*x)^4*Gamma[1 + n, -((4*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^((4*a)/(b*n))*(c*(d + e*x)^n)^(4/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^4) + (g^2*(e*f - d*g)*(d + e*x)^3*Gamma[1 + n, -((3*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(3^n*E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^4) + (3*2^(-1 - n)*g*(e*f - d*g)^2*(d + e*x)^2*Gamma[1 + n, -((2*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^4) + ((e*f - d*g)^3*(d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^4)} -{(a + b*Log[c*(d + e*x)^n])^n*(f + g*x)^2, x, 11, (3^(-1 - n)*g^2*(d + e*x)^3*Gamma[1 + n, -((3*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^3) + (g*(e*f - d*g)*(d + e*x)^2*Gamma[1 + n, -((2*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(2^n*E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^3) + ((e*f - d*g)^2*(d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^3)} -{(a + b*Log[c*(d + e*x)^n])^n*(f + g*x)^1, x, 8, (2^(-1 - n)*g*(d + e*x)^2*Gamma[1 + n, -((2*(a + b*Log[c*(d + e*x)^n]))/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^2) + ((e*f - d*g)*(d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e^2)} -{(a + b*Log[c*(d + e*x)^n])^n*(f + g*x)^0, x, 3, ((d + e*x)*Gamma[1 + n, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*(a + b*Log[c*(d + e*x)^n])^n)/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^n*e)} -{(a + b*Log[c*(d + e*x)^n])^n/(f + g*x)^1, x, 0, Unintegrable[(a + b*Log[c*(d + e*x)^n])^n/(f + g*x), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h+i x)^q (a+b Log[c (d+e x)^n])^p / (f+g x) when e f-d g=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (h+i x)^q (a+b Log[c (d+e x)^n])^p / (d+e x)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(h + i*x)^4*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 8, -((4*b*i*(f*h - e*i)^3*x)/(d*f^4)) - (3*b*i^2*(f*h - e*i)^2*(e + f*x)^2)/(2*d*f^5) - (4*b*i^3*(f*h - e*i)*(e + f*x)^3)/(9*d*f^5) - (b*i^4*(e + f*x)^4)/(16*d*f^5) - (b*(f*h - e*i)^4*Log[e + f*x]^2)/(2*d*f^5) + (4*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^5) + (3*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(d*f^5) + (4*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(3*d*f^5) + (i^4*(e + f*x)^4*(a + b*Log[c*(e + f*x)]))/(4*d*f^5) + ((f*h - e*i)^4*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(d*f^5)} -{(h + i*x)^3*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 8, -((3*b*i*(f*h - e*i)^2*x)/(d*f^3)) - (3*b*i^2*(f*h - e*i)*(e + f*x)^2)/(4*d*f^4) - (b*i^3*(e + f*x)^3)/(9*d*f^4) - (b*(f*h - e*i)^3*Log[e + f*x]^2)/(2*d*f^4) + (3*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^4) + (3*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^4) + (i^3*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(3*d*f^4) + ((f*h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(d*f^4)} -{(h + i*x)^2*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 7, -((b*(4*f*h - 3*e*i + f*i*x)^2)/(4*d*f^3)) - (b*(f*h - e*i)^2*Log[e + f*x]^2)/(2*d*f^3) + (2*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^3) + (i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^3) + ((f*h - e*i)^2*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(d*f^3)} -{(h + i*x)^1*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 6, (a*i*x)/(d*f) - (b*i*x)/(d*f) + (b*i*(e + f*x)*Log[c*(e + f*x)])/(d*f^2) + ((f*h - e*i)*(a + b*Log[c*(e + f*x)])^2)/(2*b*d*f^2)} -{(h + i*x)^0*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 3, (a + b*Log[c*(e + f*x)])^2/(2*b*d*f)} -{1/(h + i*x)^1*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 4, -(((a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i))) + (b*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i))} -{1/(h + i*x)^2*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 7, -((i*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*(f*h - e*i)^2*(h + i*x))) + (b*f*Log[h + i*x])/(d*(f*h - e*i)^2) - (f*(a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i)^2) + (b*f*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^2)} -{1/(h + i*x)^3*(a + b*Log[c*(e + f*x)])/(d*e + d*f*x), x, 11, -((b*f)/(2*d*(f*h - e*i)^2*(h + i*x))) - (b*f^2*Log[e + f*x])/(2*d*(f*h - e*i)^3) + (a + b*Log[c*(e + f*x)])/(2*d*(f*h - e*i)*(h + i*x)^2) - (f*i*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*(f*h - e*i)^3*(h + i*x)) + (3*b*f^2*Log[h + i*x])/(2*d*(f*h - e*i)^3) - (f^2*(a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i)^3) + (b*f^2*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^3)} - - -{(h + i*x)^4*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 32, -((4*a*b*i*(f*h - e*i)^3*x)/(d*f^4)) + (8*b^2*i*(f*h - e*i)^3*x)/(d*f^4) + (3*b^2*i^2*(f*h - e*i)^2*(e + f*x)^2)/(2*d*f^5) + (8*b^2*i^3*(f*h - e*i)*(e + f*x)^3)/(27*d*f^5) + (b^2*i^4*(e + f*x)^4)/(32*d*f^5) + (7*b^2*(f*h - e*i)^4*Log[e + f*x]^2)/(12*d*f^5) - (4*b^2*i*(f*h - e*i)^3*(e + f*x)*Log[c*(e + f*x)])/(d*f^5) - (4*b*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^5) - (3*b*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(d*f^5) - (8*b*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(9*d*f^5) - (b*i^4*(e + f*x)^4*(a + b*Log[c*(e + f*x)]))/(8*d*f^5) - (7*b*(f*h - e*i)^4*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(6*d*f^5) + (2*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^5) + (i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*d*f^5) + ((f*h - e*i)*(h + i*x)^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f^2) + ((h + i*x)^4*(a + b*Log[c*(e + f*x)])^2)/(4*d*f) + ((f*h - e*i)^4*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^5)} -{(h + i*x)^3*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 24, -((4*a*b*i*(f*h - e*i)^2*x)/(d*f^3)) + (6*b^2*i*(f*h - e*i)^2*x)/(d*f^3) + (3*b^2*i^2*(f*h - e*i)*(e + f*x)^2)/(4*d*f^4) + (2*b^2*i^3*(e + f*x)^3)/(27*d*f^4) + (b^2*(f*h - e*i)^3*Log[e + f*x]^2)/(3*d*f^4) - (4*b^2*i*(f*h - e*i)^2*(e + f*x)*Log[c*(e + f*x)])/(d*f^4) - (2*b*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*f^4) - (3*b*i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^4) - (2*b*i^3*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/(9*d*f^4) - (2*b*(f*h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)]))/(3*d*f^4) + (2*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^4) + (i^2*(f*h - e*i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*d*f^4) + ((h + i*x)^3*(a + b*Log[c*(e + f*x)])^2)/(3*d*f) + ((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^4)} -{(h + i*x)^2*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 16, -((4*a*b*i*(f*h - e*i)*x)/(d*f^2)) + (4*b^2*i*(f*h - e*i)*x)/(d*f^2) + (b^2*i^2*(e + f*x)^2)/(4*d*f^3) - (4*b^2*i*(f*h - e*i)*(e + f*x)*Log[c*(e + f*x)])/(d*f^3) - (b*i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/(2*d*f^3) + (2*i*(f*h - e*i)*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^3) + (i^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2)/(2*d*f^3) + ((f*h - e*i)^2*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^3)} -{(h + i*x)^1*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 8, -((2*a*b*i*x)/(d*f)) + (2*b^2*i*x)/(d*f) - (2*b^2*i*(e + f*x)*Log[c*(e + f*x)])/(d*f^2) + (i*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*f^2) + ((f*h - e*i)*(a + b*Log[c*(e + f*x)])^3)/(3*b*d*f^2)} -{(h + i*x)^0*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 4, (a + b*Log[c*(e + f*x)])^3/(3*b*d*f)} -{1/(h + i*x)^1*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 5, -(((a + b*Log[c*(e + f*x)])^2*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i))) + (2*b*(a + b*Log[c*(e + f*x)])*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)) + (2*b^2*PolyLog[3, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i))} -{1/(h + i*x)^2*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 9, -((i*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*(f*h - e*i)^2*(h + i*x))) + (2*b*f*(a + b*Log[c*(e + f*x)])*Log[(f*(h + i*x))/(f*h - e*i)])/(d*(f*h - e*i)^2) - (f*(a + b*Log[c*(e + f*x)])^2*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i)^2) + (2*b*f*(a + b*Log[c*(e + f*x)])*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^2) + (2*b^2*f*PolyLog[2, -((i*(e + f*x))/(f*h - e*i))])/(d*(f*h - e*i)^2) + (2*b^2*f*PolyLog[3, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^2)} -{1/(h + i*x)^3*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 16, (b*f*i*(e + f*x)*(a + b*Log[c*(e + f*x)]))/(d*(f*h - e*i)^3*(h + i*x)) + (a + b*Log[c*(e + f*x)])^2/(2*d*(f*h - e*i)*(h + i*x)^2) - (f*i*(e + f*x)*(a + b*Log[c*(e + f*x)])^2)/(d*(f*h - e*i)^3*(h + i*x)) - (b^2*f^2*Log[h + i*x])/(d*(f*h - e*i)^3) + (2*b*f^2*(a + b*Log[c*(e + f*x)])*Log[(f*(h + i*x))/(f*h - e*i)])/(d*(f*h - e*i)^3) + (b*f^2*(a + b*Log[c*(e + f*x)])*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i)^3) - (f^2*(a + b*Log[c*(e + f*x)])^2*Log[1 + (f*h - e*i)/(i*(e + f*x))])/(d*(f*h - e*i)^3) - (b^2*f^2*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^3) + (2*b*f^2*(a + b*Log[c*(e + f*x)])*PolyLog[2, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^3) + (2*b^2*f^2*PolyLog[2, -((i*(e + f*x))/(f*h - e*i))])/(d*(f*h - e*i)^3) + (2*b^2*f^2*PolyLog[3, -((f*h - e*i)/(i*(e + f*x)))])/(d*(f*h - e*i)^3)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(h + i*x)^4/((a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 14, (4*i*(f*h - e*i)^3*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(E^(a/b)*(b*c*d*f^5)) + (6*i^2*(f*h - e*i)^2*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(E^((2*a)/b)*(b*c^2*d*f^5)) + (4*i^3*(f*h - e*i)*ExpIntegralEi[(3*(a + b*Log[c*(e + f*x)]))/b])/(E^((3*a)/b)*(b*c^3*d*f^5)) + (i^4*ExpIntegralEi[(4*(a + b*Log[c*(e + f*x)]))/b])/(E^((4*a)/b)*(b*c^4*d*f^5)) + ((f*h - e*i)^4*Log[a + b*Log[c*(e + f*x)]])/(b*d*f^5)} -{(h + i*x)^3/((a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 12, (3*i*(f*h - e*i)^2*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(E^(a/b)*(b*c*d*f^4)) + (3*i^2*(f*h - e*i)*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(E^((2*a)/b)*(b*c^2*d*f^4)) + (i^3*ExpIntegralEi[(3*(a + b*Log[c*(e + f*x)]))/b])/(E^((3*a)/b)*(b*c^3*d*f^4)) + ((f*h - e*i)^3*Log[a + b*Log[c*(e + f*x)]])/(b*d*f^4)} -{(h + i*x)^2/((a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 10, (2*i*(f*h - e*i)*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(E^(a/b)*(b*c*d*f^3)) + (i^2*ExpIntegralEi[(2*(a + b*Log[c*(e + f*x)]))/b])/(E^((2*a)/b)*(b*c^2*d*f^3)) + ((f*h - e*i)^2*Log[a + b*Log[c*(e + f*x)]])/(b*d*f^3)} -{(h + i*x)^1/((a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 8, (i*ExpIntegralEi[(a + b*Log[c*(e + f*x)])/b])/(E^(a/b)*(b*c*d*f^2)) + ((f*h - e*i)*Log[a + b*Log[c*(e + f*x)]])/(b*d*f^2)} -{(h + i*x)^0/((a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 4, Log[a + b*Log[c*(e + f*x)]]/(b*d*f)} -{1/((h + i*x)^1*(a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 5, Log[a + b*Log[c*(e + f*x)]]/(b*d*(f*h - e*i)) - (i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(e + f*x)])), x])/(d*(f*h - e*i))} -{1/((h + i*x)^2*(a + b*Log[c*(e + f*x)])*(d*e + d*f*x)), x, 5, (f*Log[a + b*Log[c*(e + f*x)]])/(b*d*(f*h - e*i)^2) - (i*Unintegrable[1/((h + i*x)^2*(a + b*Log[c*(e + f*x)])), x])/(d*(f*h - e*i)) - (f*i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(e + f*x)])), x])/(d*(f*h - e*i)^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^(m/2) (a+b Log[c (d+e x)^n])^p / (e+f x)*) - - -{(f + g*x)^(5/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 27, -((92*b*(e*f - d*g)^2*n*Sqrt[f + g*x])/(15*e^3)) - (32*b*(e*f - d*g)*n*(f + g*x)^(3/2))/(45*e^2) - (4*b*n*(f + g*x)^(5/2))/(25*e) + (92*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(15*e^(7/2)) + (2*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/e^(7/2) + (2*(e*f - d*g)^2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/e^3 + (2*(e*f - d*g)*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(3*e^2) + (2*(f + g*x)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(5*e) - (2*(e*f - d*g)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/e^(7/2) - (4*b*(e*f - d*g)^(5/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(7/2) - (2*b*(e*f - d*g)^(5/2)*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(7/2)} -{(f + g*x)^(3/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 20, -((16*b*(e*f - d*g)*n*Sqrt[f + g*x])/(3*e^2)) - (4*b*n*(f + g*x)^(3/2))/(9*e) + (16*b*(e*f - d*g)^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(3*e^(5/2)) + (2*b*(e*f - d*g)^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/e^(5/2) + (2*(e*f - d*g)*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/e^2 + (2*(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]))/(3*e) - (2*(e*f - d*g)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/e^(5/2) - (4*b*(e*f - d*g)^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(5/2) - (2*b*(e*f - d*g)^(3/2)*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(5/2)} -{(f + g*x)^(1/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 14, -((4*b*n*Sqrt[f + g*x])/e) + (4*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/e^(3/2) + (2*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/e^(3/2) + (2*Sqrt[f + g*x]*(a + b*Log[c*(d + e*x)^n]))/e - (2*Sqrt[e*f - d*g]*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/e^(3/2) - (4*b*Sqrt[e*f - d*g]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(3/2) - (2*b*Sqrt[e*f - d*g]*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/e^(3/2)} -{1/(f + g*x)^(1/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 9, (2*b*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(Sqrt[e]*Sqrt[e*f - d*g]) - (2*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[e]*Sqrt[e*f - d*g]) - (4*b*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(Sqrt[e]*Sqrt[e*f - d*g]) - (2*b*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(Sqrt[e]*Sqrt[e*f - d*g])} -{1/(f + g*x)^(3/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 13, (4*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e*f - d*g)^(3/2) + (2*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(e*f - d*g)^(3/2) + (2*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)*Sqrt[f + g*x]) - (2*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(e*f - d*g)^(3/2) - (4*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(e*f - d*g)^(3/2) - (2*b*Sqrt[e]*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(e*f - d*g)^(3/2)} -{1/(f + g*x)^(5/2)*((a + b*Log[c*(d + e*x)^n])/(d + e*x)), x, 18, -((4*b*e*n)/(3*(e*f - d*g)^2*Sqrt[f + g*x])) + (16*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(3*(e*f - d*g)^(5/2)) + (2*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]^2)/(e*f - d*g)^(5/2) + (2*(a + b*Log[c*(d + e*x)^n]))/(3*(e*f - d*g)*(f + g*x)^(3/2)) + (2*e*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)^2*Sqrt[f + g*x]) - (2*e^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*(a + b*Log[c*(d + e*x)^n]))/(e*f - d*g)^(5/2) - (4*b*e^(3/2)*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]]*Log[2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(e*f - d*g)^(5/2) - (2*b*e^(3/2)*n*PolyLog[2, 1 - 2/(1 - (Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g])])/(e*f - d*g)^(5/2)} - - -(* These special cases of the above form formerly caused infinite recursion! *) -{(d + e*x)^(3/2)*Log[a + b*x]/(a + b*x), x, 20, -((16*(b*d - a*e)*Sqrt[d + e*x])/(3*b^2)) - (4*(d + e*x)^(3/2))/(9*b) + (16*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(3*b^(5/2)) + (2*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]^2)/b^(5/2) + (2*(b*d - a*e)*Sqrt[d + e*x]*Log[a + b*x])/b^2 + (2*(d + e*x)^(3/2)*Log[a + b*x])/(3*b) - (2*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/b^(5/2) - (4*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(5/2) - (2*(b*d - a*e)^(3/2)*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(5/2)} -{(d + e*x)^(1/2)*Log[a + b*x]/(a + b*x), x, 14, -((4*Sqrt[d + e*x])/b) + (4*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2) + (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]^2)/b^(3/2) + (2*Sqrt[d + e*x]*Log[a + b*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/b^(3/2) - (4*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(3/2) - (2*Sqrt[b*d - a*e]*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/b^(3/2)} -{Log[a + b*x]/((a + b*x)*(d + e*x)^(1/2)), x, 9, (2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]^2)/(Sqrt[b]*Sqrt[b*d - a*e]) - (2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/(Sqrt[b]*Sqrt[b*d - a*e]) - (4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(Sqrt[b]*Sqrt[b*d - a*e]) - (2*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(Sqrt[b]*Sqrt[b*d - a*e])} -{Log[a + b*x]/((a + b*x)*(d + e*x)^(3/2)), x, 13, (4*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2) + (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]^2)/(b*d - a*e)^(3/2) + (2*Log[a + b*x])/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/(b*d - a*e)^(3/2) - (4*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(b*d - a*e)^(3/2) - (2*Sqrt[b]*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(b*d - a*e)^(3/2)} -{Log[a + b*x]/((a + b*x)*(d + e*x)^(5/2)), x, 18, -((4*b)/(3*(b*d - a*e)^2*Sqrt[d + e*x])) + (16*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(3*(b*d - a*e)^(5/2)) + (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]^2)/(b*d - a*e)^(5/2) + (2*Log[a + b*x])/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (2*b*Log[a + b*x])/((b*d - a*e)^2*Sqrt[d + e*x]) - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[a + b*x])/(b*d - a*e)^(5/2) - (4*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]*Log[2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(b*d - a*e)^(5/2) - (2*b^(3/2)*PolyLog[2, 1 - 2/(1 - (Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e])])/(b*d - a*e)^(5/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p / (e+f x) with q symbolic*) - - -(* {(h + i*x)^m*(a + b*Log[c*(e + f*x)])^2/(d*e + d*f*x), x, 0, -((a^2*(h + i*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (f*(h + i*x))/(f*i - e*j)])/(d*(f*i - e*j)*(1 + m))) - (2*a*b*(h + i*x)^m*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, ((-f)*i + e*j)/(j*(e + f*x))])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m^2)) + (2*b^2*(h + i*x)^m*HypergeometricPFQ[{-m, -m, -m, -m}, {1 - m, 1 - m, 1 - m}, ((-f)*i + e*j)/(j*(e + f*x))])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m^3)) + (2*a*b*(h + i*x)^m*Hypergeometric2F1[-m, -m, 1 - m, ((-f)*i + e*j)/(j*(e + f*x))]*Log[c*(e + f*x)])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m)) - (2*b^2*(h + i*x)^m*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, ((-f)*i + e*j)/(j*(e + f*x))]*Log[c*(e + f*x)])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m^2)) + (b^2*(h + i*x)^m*Hypergeometric2F1[-m, -m, 1 - m, ((-f)*i + e*j)/(j*(e + f*x))]*Log[c*(e + f*x)]^2)/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m))} -{(h + i*x)^m*(a + b*Log[c*(e + f*x)])^1/(d*e + d*f*x), x, 0, -((a*(h + i*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (f*(h + i*x))/(f*i - e*j)])/(d*(f*i - e*j)*(1 + m))) - (b*(h + i*x)^m*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, ((-f)*i + e*j)/(j*(e + f*x))])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m^2)) + (b*(h + i*x)^m*Hypergeometric2F1[-m, -m, 1 - m, ((-f)*i + e*j)/(j*(e + f*x))]*Log[c*(e + f*x)])/(((f*(h + i*x))/(j*(e + f*x)))^m*(d*f*m))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p / (e+f x) with p symbolic*) - - -{(h + i*x)^q*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 0, Unintegrable[((h + i*x)^q*(a + b*Log[c*(e + f*x)])^p)/(d*e + d*f*x), x]} - - -{(h + i*x)^3*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 12, ((f*h - e*i)^3*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^4*(1 + p)) + (3^(-1 - p)*i^3*Gamma[1 + p, -((3*(a + b*Log[c*(e + f*x)]))/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c^3*d*f^4)) + (3*2^(-1 - p)*i^2*(f*h - e*i)*Gamma[1 + p, -((2*(a + b*Log[c*(e + f*x)]))/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c^2*d*f^4)) + (3*i*(f*h - e*i)^2*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^(a/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c*d*f^4))} -{(h + i*x)^2*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 10, ((f*h - e*i)^2*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^3*(1 + p)) + (2^(-1 - p)*i^2*Gamma[1 + p, -((2*(a + b*Log[c*(e + f*x)]))/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c^2*d*f^3)) + (2*i*(f*h - e*i)*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^(a/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c*d*f^3))} -{(h + i*x)^1*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 8, ((f*h - e*i)*(a + b*Log[c*(e + f*x)])^(1 + p))/(b*d*f^2*(1 + p)) + (i*Gamma[1 + p, -((a + b*Log[c*(e + f*x)])/b)]*(a + b*Log[c*(e + f*x)])^p)/(E^(a/b)*(-((a + b*Log[c*(e + f*x)])/b))^p*(c*d*f^2))} -{(h + i*x)^0*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 4, (a + b*Log[c*(e + f*x)])^(1 + p)/(b*d*f*(1 + p))} -{1/(h + i*x)^1*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 0, Unintegrable[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)), x]} -{1/(h + i*x)^2*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 0, Unintegrable[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2), x]} -{1/(h + i*x)^3*(a + b*Log[c*(e + f*x)])^p/(d*e + d*f*x), x, 0, Unintegrable[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^q (a+b Log[c (d+e x)^n])^p / (h+i x)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (i+j x)^m (a+b Log[c (d+e x)^n])^p / (g+h x)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(h + i*x)^3*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 14, (a*i*(g*h - f*i)^2*x)/g^3 - (b*i*(e*h - d*i)^2*n*x)/(3*e^2*g) - (b*i*(e*h - d*i)*(g*h - f*i)*n*x)/(2*e*g^2) - (b*i*(g*h - f*i)^2*n*x)/g^3 - (b*(e*h - d*i)*n*(h + i*x)^2)/(6*e*g) - (b*(g*h - f*i)*n*(h + i*x)^2)/(4*g^2) - (b*n*(h + i*x)^3)/(9*g) - (b*(e*h - d*i)^3*n*Log[d + e*x])/(3*e^3*g) - (b*(e*h - d*i)^2*(g*h - f*i)*n*Log[d + e*x])/(2*e^2*g^2) + (b*i*(g*h - f*i)^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^3) + ((g*h - f*i)*(h + i*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^2) + ((h + i*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((g*h - f*i)^3*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^4 + (b*(g*h - f*i)^3*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^4} -{(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 11, (a*i*(g*h - f*i)*x)/g^2 - (b*i*(e*h - d*i)*n*x)/(2*e*g) - (b*i*(g*h - f*i)*n*x)/g^2 - (b*n*(h + i*x)^2)/(4*g) - (b*(e*h - d*i)^2*n*Log[d + e*x])/(2*e^2*g) + (b*i*(g*h - f*i)*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + ((h + i*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (b*(g*h - f*i)^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3} -{(h + i*x)^1*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 8, (a*i*x)/g - (b*i*n*x)/g + (b*i*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + ((g*h - f*i)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 + (b*(g*h - f*i)*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2} -{(h + i*x)^0*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 3, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])/((f + g*x)*(h + i*x)^1), x, 8, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i) + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (b*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)} -{(a + b*Log[c*(d + e*x)^n])/((f + g*x)*(h + i*x)^2), x, 12, -((b*e*n*Log[d + e*x])/((e*h - d*i)*(g*h - f*i))) + (a + b*Log[c*(d + e*x)^n])/((g*h - f*i)*(h + i*x)) + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^2 + (b*e*n*Log[h + i*x])/((e*h - d*i)*(g*h - f*i)) - (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^2 + (b*g*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 - (b*g*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2} -{(a + b*Log[c*(d + e*x)^n])/((f + g*x)*(h + i*x)^3), x, 15, -((b*e*n)/(2*(e*h - d*i)*(g*h - f*i)*(h + i*x))) - (b*e*g*n*Log[d + e*x])/((e*h - d*i)*(g*h - f*i)^2) - (b*e^2*n*Log[d + e*x])/(2*(e*h - d*i)^2*(g*h - f*i)) + (a + b*Log[c*(d + e*x)^n])/(2*(g*h - f*i)*(h + i*x)^2) + (g*(a + b*Log[c*(d + e*x)^n]))/((g*h - f*i)^2*(h + i*x)) + (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^3 + (b*e*g*n*Log[h + i*x])/((e*h - d*i)*(g*h - f*i)^2) + (b*e^2*n*Log[h + i*x])/(2*(e*h - d*i)^2*(g*h - f*i)) - (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^3 + (b*g^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^3 - (b*g^2*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^3} - - -{(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x), x, 19, -((2*a*b*i*(e*h - d*i)*n*x)/(e*g)) - (2*a*b*i*(g*h - f*i)*n*x)/g^2 + (2*b^2*i*(e*h - d*i)*n^2*x)/(e*g) + (2*b^2*i*(g*h - f*i)*n^2*x)/g^2 + (b^2*i^2*n^2*(d + e*x)^2)/(4*e^2*g) - (2*b^2*i*(e*h - d*i)*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g) - (2*b^2*i*(g*h - f*i)*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (b*i^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g) + (i*(e*h - d*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g) + (i*(g*h - f*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) + (i^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (2*b*(g*h - f*i)^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3 - (2*b^2*(g*h - f*i)^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^3} -{(h + i*x)^1*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x), x, 10, -((2*a*b*i*n*x)/g) + (2*b^2*i*n^2*x)/g - (2*b^2*i*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + (i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g) + ((g*h - f*i)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 + (2*b*(g*h - f*i)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2 - (2*b^2*(g*h - f*i)*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^2} -{(h + i*x)^0*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x), x, 4, ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/g + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g - (2*b^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])^2/((f + g*x)*(h + i*x)^1), x, 10, ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i) + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i) - (2*b^2*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) + (2*b^2*n^2*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)} -{(a + b*Log[c*(d + e*x)^n])^2/((f + g*x)*(h + i*x)^2), x, 14, -((i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/((e*h - d*i)*(g*h - f*i)*(h + i*x))) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^2 + (2*b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/((e*h - d*i)*(g*h - f*i)) - (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^2 + (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 + (2*b^2*e*n^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) - (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 - (2*b^2*g*n^2*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 + (2*b^2*g*n^2*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2} - - -{(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^3/(f + g*x), x, 23, (6*a*b^2*i*(e*h - d*i)*n^2*x)/(e*g) + (6*a*b^2*i*(g*h - f*i)*n^2*x)/g^2 - (6*b^3*i*(e*h - d*i)*n^3*x)/(e*g) - (6*b^3*i*(g*h - f*i)*n^3*x)/g^2 - (3*b^3*i^2*n^3*(d + e*x)^2)/(8*e^2*g) + (6*b^3*i*(e*h - d*i)*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g) + (6*b^3*i*(g*h - f*i)*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (3*b^2*i^2*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2*g) - (3*b*i*(e*h - d*i)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g) - (3*b*i*(g*h - f*i)*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) - (3*b*i^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2*g) + (i*(e*h - d*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e^2*g) + (i*(g*h - f*i)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e*g^2) + (i^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^2*g) + ((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (3*b*(g*h - f*i)^2*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3 - (6*b^2*(g*h - f*i)^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^3 + (6*b^3*(g*h - f*i)^2*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g^3} -{(h + i*x)^1*(a + b*Log[c*(d + e*x)^n])^3/(f + g*x), x, 12, (6*a*b^2*i*n^2*x)/g - (6*b^3*i*n^3*x)/g + (6*b^3*i*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) - (3*b*i*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g) + (i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(e*g) + ((g*h - f*i)*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 + (3*b*(g*h - f*i)*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2 - (6*b^2*(g*h - f*i)*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g^2 + (6*b^3*(g*h - f*i)*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g^2} -{(h + i*x)^0*(a + b*Log[c*(d + e*x)^n])^3/(f + g*x), x, 5, ((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/g + (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g - (6*b^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/g + (6*b^3*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])^3/((f + g*x)*(h + i*x)^1), x, 12, ((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i) - ((a + b*Log[c*(d + e*x)^n])^3*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i) + (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (3*b*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i) - (6*b^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) + (6*b^2*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i) + (6*b^3*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i) - (6*b^3*n^3*PolyLog[4, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)} -{(a + b*Log[c*(d + e*x)^n])^3/((f + g*x)*(h + i*x)^2), x, 17, -((i*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/((e*h - d*i)*(g*h - f*i)*(h + i*x))) + (g*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(f + g*x))/(e*f - d*g)])/(g*h - f*i)^2 + (3*b*e*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(h + i*x))/(e*h - d*i)])/((e*h - d*i)*(g*h - f*i)) - (g*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^2 + (3*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 + (6*b^2*e*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) - (3*b*g*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 - (6*b^2*g*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 - (6*b^3*e*n^3*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/((e*h - d*i)*(g*h - f*i)) + (6*b^2*g*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2 + (6*b^3*g*n^3*PolyLog[4, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^2 - (6*b^3*g*n^3*PolyLog[4, -((i*(d + e*x))/(e*h - d*i))])/(g*h - f*i)^2} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(h + i*x)^1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x, 5, (i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e*g*n)) + ((g*h - f*i)*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/g} -{(h + i*x)^0/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x]} -{1/((f + g*x)*(h + i*x)^1*(a + b*Log[c*(d + e*x)^n])), x, 2, (g*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/(g*h - f*i) - (i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(d + e*x)^n])), x])/(g*h - f*i)} -{1/((f + g*x)*(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])), x, 2, (g^2*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])), x])/(g*h - f*i)^2 - (i*Unintegrable[1/((h + i*x)^2*(a + b*Log[c*(d + e*x)^n])), x])/(g*h - f*i) - (g*i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(d + e*x)^n])), x])/(g*h - f*i)^2} - - -{(h + i*x)^1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x, 6, (i*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b^2*e*g*n^2)) - (i*(d + e*x))/(b*e*g*n*(a + b*Log[c*(d + e*x)^n])) + ((g*h - f*i)*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x])/g} -{(h + i*x)^0/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x, 0, Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x]} -{1/((f + g*x)*(h + i*x)^1*(a + b*Log[c*(d + e*x)^n])^2), x, 2, (g*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x])/(g*h - f*i) - (i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(d + e*x)^n])^2), x])/(g*h - f*i)} -{1/((f + g*x)*(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x, 2, (g^2*Unintegrable[1/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2), x])/(g*h - f*i)^2 - (i*Unintegrable[1/((h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x])/(g*h - f*i) - (g*i*Unintegrable[1/((h + i*x)*(a + b*Log[c*(d + e*x)^n])^2), x])/(g*h - f*i)^2} - - -(* ::Subsection:: *) -(*Integrands of the form (i+j x)^(m/2) (a+b Log[c (d+e x)^n])^p / (g+h x)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^m (f+g x^r)^q (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g x^1)^q (a+b Log[c (d+e x)^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 14, (a*f^2*x)/g^3 - (b*f^2*n*x)/g^3 - (b*d*f*n*x)/(2*e*g^2) - (b*d^2*n*x)/(3*e^2*g) + (b*f*n*x^2)/(4*g^2) + (b*d*n*x^2)/(6*e*g) - (b*n*x^3)/(9*g) + (b*d^2*f*n*Log[d + e*x])/(2*e^2*g^2) + (b*d^3*n*Log[d + e*x])/(3*e^3*g) + (b*f^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^3) - (f*x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^2) + (x^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) - (f^3*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^4 - (b*f^3*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^4} -{x^2*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 11, -((a*f*x)/g^2) + (b*f*n*x)/g^2 + (b*d*n*x)/(2*e*g) - (b*n*x^2)/(4*g) - (b*d^2*n*Log[d + e*x])/(2*e^2*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 + (b*f^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3} -{x^1*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 8, (a*x)/g - (b*n*x)/g + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) - (f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 - (b*f*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2} -{x^0*(a + b*Log[c*(d + e*x)^n])/(f + g*x), x, 3, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g} -{(a + b*Log[c*(d + e*x)^n])/(x^1*(f + g*x)), x, 7, (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f - (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f + (b*n*PolyLog[2, 1 + (e*x)/d])/f} -{(a + b*Log[c*(d + e*x)^n])/(x^2*(f + g*x)), x, 11, (b*e*n*Log[x])/(d*f) - (b*e*n*Log[d + e*x])/(d*f) - (a + b*Log[c*(d + e*x)^n])/(f*x) - (g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^2 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^2 + (b*g*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f^2 - (b*g*n*PolyLog[2, 1 + (e*x)/d])/f^2} -{(a + b*Log[c*(d + e*x)^n])/(x^3*(f + g*x)), x, 14, -((b*e*n)/(2*d*f*x)) - (b*e^2*n*Log[x])/(2*d^2*f) - (b*e*g*n*Log[x])/(d*f^2) + (b*e^2*n*Log[d + e*x])/(2*d^2*f) + (b*e*g*n*Log[d + e*x])/(d*f^2) - (a + b*Log[c*(d + e*x)^n])/(2*f*x^2) + (g*(a + b*Log[c*(d + e*x)^n]))/(f^2*x) + (g^2*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^3 - (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^3 - (b*g^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f^3 + (b*g^2*n*PolyLog[2, 1 + (e*x)/d])/f^3} - - -{x^3*(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 15, -((2*a*f*x)/g^3) + (2*b*f*n*x)/g^3 + (b*d*n*x)/(2*e*g^2) - (b*n*x^2)/(4*g^2) - (b*d^2*n*Log[d + e*x])/(2*e^2*g^2) - (b*e*f^3*n*Log[d + e*x])/(g^4*(e*f - d*g)) - (2*b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^3) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^2) + (f^3*(a + b*Log[c*(d + e*x)^n]))/(g^4*(f + g*x)) + (b*e*f^3*n*Log[f + g*x])/(g^4*(e*f - d*g)) + (3*f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^4 + (3*b*f^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^4} -{x^2*(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 12, (a*x)/g^2 - (b*n*x)/g^2 + (b*e*f^2*n*Log[d + e*x])/(g^3*(e*f - d*g)) + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f^2*(a + b*Log[c*(d + e*x)^n]))/(g^3*(f + g*x)) - (b*e*f^2*n*Log[f + g*x])/(g^3*(e*f - d*g)) - (2*f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^3 - (2*b*f*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^3} -{x^1*(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 9, -((b*e*f*n*Log[d + e*x])/(g^2*(e*f - d*g))) + (f*(a + b*Log[c*(d + e*x)^n]))/(g^2*(f + g*x)) + (b*e*f*n*Log[f + g*x])/(g^2*(e*f - d*g)) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^2 + (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2} -{x^0*(a + b*Log[c*(d + e*x)^n])/(f + g*x)^2, x, 4, (b*e*n*Log[d + e*x])/(g*(e*f - d*g)) - (a + b*Log[c*(d + e*x)^n])/(g*(f + g*x)) - (b*e*n*Log[f + g*x])/(g*(e*f - d*g))} -{(a + b*Log[c*(d + e*x)^n])/(x^1*(f + g*x)^2), x, 11, -((b*e*n*Log[d + e*x])/(f*(e*f - d*g))) + (a + b*Log[c*(d + e*x)^n])/(f*(f + g*x)) + (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^2 + (b*e*n*Log[f + g*x])/(f*(e*f - d*g)) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^2 - (b*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f^2 + (b*n*PolyLog[2, 1 + (e*x)/d])/f^2} -{(a + b*Log[c*(d + e*x)^n])/(x^2*(f + g*x)^2), x, 15, (b*e*n*Log[x])/(d*f^2) - (b*e*n*Log[d + e*x])/(d*f^2) + (b*e*g*n*Log[d + e*x])/(f^2*(e*f - d*g)) - (a + b*Log[c*(d + e*x)^n])/(f^2*x) - (g*(a + b*Log[c*(d + e*x)^n]))/(f^2*(f + g*x)) - (2*g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^3 - (b*e*g*n*Log[f + g*x])/(f^2*(e*f - d*g)) + (2*g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^3 + (2*b*g*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f^3 - (2*b*g*n*PolyLog[2, 1 + (e*x)/d])/f^3} -{(a + b*Log[c*(d + e*x)^n])/(x^3*(f + g*x)^2), x, 18, -((b*e*n)/(2*d*f^2*x)) - (b*e^2*n*Log[x])/(2*d^2*f^2) - (2*b*e*g*n*Log[x])/(d*f^3) + (b*e^2*n*Log[d + e*x])/(2*d^2*f^2) + (2*b*e*g*n*Log[d + e*x])/(d*f^3) - (b*e*g^2*n*Log[d + e*x])/(f^3*(e*f - d*g)) - (a + b*Log[c*(d + e*x)^n])/(2*f^2*x^2) + (2*g*(a + b*Log[c*(d + e*x)^n]))/(f^3*x) + (g^2*(a + b*Log[c*(d + e*x)^n]))/(f^3*(f + g*x)) + (3*g^2*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^4 + (b*e*g^2*n*Log[f + g*x])/(f^3*(e*f - d*g)) - (3*g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/f^4 - (3*b*g^2*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/f^4 + (3*b*g^2*n*PolyLog[2, 1 + (e*x)/d])/f^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g x^2)^q (a+b Log[c (d+e x)^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 16, -((b*d*f*n*x)/(2*e*g^2)) + (b*d^3*n*x)/(4*e^3*g) + (b*f*n*x^2)/(4*g^2) - (b*d^2*n*x^2)/(8*e^2*g) + (b*d*n*x^3)/(12*e*g) - (b*n*x^4)/(16*g) + (b*d^2*f*n*Log[d + e*x])/(2*e^2*g^2) - (b*d^4*n*Log[d + e*x])/(4*e^4*g) - (f*x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^2) + (x^4*(a + b*Log[c*(d + e*x)^n]))/(4*g) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3) + (b*f^2*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^3) + (b*f^2*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3)} -{x^3*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 13, (b*d*n*x)/(2*e*g) - (b*n*x^2)/(4*g) - (b*d^2*n*Log[d + e*x])/(2*e^2*g) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g) - (f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) - (f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^2) - (b*f*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^2) - (b*f*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2)} -{x^1*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 8, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g) + (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g)} -{(a + b*Log[c*(d + e*x)^n])/(x^1*(f + g*x^2)), x, 12, (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f) - (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f) - (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f) + (b*n*PolyLog[2, 1 + (e*x)/d])/f} -{(a + b*Log[c*(d + e*x)^n])/(x^3*(f + g*x^2)), x, 15, -((b*e*n)/(2*d*f*x)) - (b*e^2*n*Log[x])/(2*d^2*f) + (b*e^2*n*Log[d + e*x])/(2*d^2*f) - (a + b*Log[c*(d + e*x)^n])/(2*f*x^2) - (g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^2 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) + (b*g*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f^2) + (b*g*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) - (b*g*n*PolyLog[2, 1 + (e*x)/d])/f^2} - -{x^4*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 16, -((a*f*x)/g^2) + (b*f*n*x)/g^2 - (b*d^2*n*x)/(3*e^2*g) + (b*d*n*x^2)/(6*e*g) - (b*n*x^3)/(9*g) + (b*d^3*n*Log[d + e*x])/(3*e^3*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2)) - ((-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(5/2)) - (b*(-f)^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) + (b*(-f)^(3/2)*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2))} -{x^2*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 13, (a*x)/g - (b*n*x)/g + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + (Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(3/2)) - (b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(3/2)) + (b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(3/2))} -{x^0*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2), x, 8, ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*Sqrt[g]) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g])} -{(a + b*Log[c*(d + e*x)^n])/(x^2*(f + g*x^2)), x, 14, (b*e*n*Log[x])/(d*f) - (b*e*n*Log[d + e*x])/(d*f) - (a + b*Log[c*(d + e*x)^n])/(f*x) + (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(3/2)) - (b*Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)) + (b*Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2))} -{(a + b*Log[c*(d + e*x)^n])/(x^4*(f + g*x^2)), x, 17, -((b*e*n)/(6*d*f*x^2)) + (b*e^2*n)/(3*d^2*f*x) + (b*e^3*n*Log[x])/(3*d^3*f) - (b*e*g*n*Log[x])/(d*f^2) - (b*e^3*n*Log[d + e*x])/(3*d^3*f) + (b*e*g*n*Log[d + e*x])/(d*f^2) - (a + b*Log[c*(d + e*x)^n])/(3*f*x^3) + (g*(a + b*Log[c*(d + e*x)^n]))/(f^2*x) + (g^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2)) - (g^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(5/2)) - (b*g^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)) + (b*g^(3/2)*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2))} - - -{x^5*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 19, (b*d*n*x)/(2*e*g^2) - (b*n*x^2)/(4*g^2) + (b*d*e*f^(3/2)*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*g^(5/2)*(e^2*f + d^2*g)) - (b*d^2*n*Log[d + e*x])/(2*e^2*g^2) + (b*e^2*f^2*n*Log[d + e*x])/(2*g^3*(e^2*f + d^2*g)) + (x^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^2) - (f^2*(a + b*Log[c*(d + e*x)^n]))/(2*g^3*(f + g*x^2)) - (f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (f*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/g^3 - (b*e^2*f^2*n*Log[f + g*x^2])/(4*g^3*(e^2*f + d^2*g)) - (b*f*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 - (b*f*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3} -{x^3*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 16, -((b*d*e*Sqrt[f]*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*g^(3/2)*(e^2*f + d^2*g))) - (b*e^2*f*n*Log[d + e*x])/(2*g^2*(e^2*f + d^2*g)) + (f*(a + b*Log[c*(d + e*x)^n]))/(2*g^2*(f + g*x^2)) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^2) + (b*e^2*f*n*Log[f + g*x^2])/(4*g^2*(e^2*f + d^2*g)) + (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^2) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2)} -{x^1*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 6, (b*d*e*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*Sqrt[f]*Sqrt[g]*(e^2*f + d^2*g)) + (b*e^2*n*Log[d + e*x])/(2*g*(e^2*f + d^2*g)) - (a + b*Log[c*(d + e*x)^n])/(2*g*(f + g*x^2)) - (b*e^2*n*Log[f + g*x^2])/(4*g*(e^2*f + d^2*g))} -{(a + b*Log[c*(d + e*x)^n])/(x^1*(f + g*x^2)^2), x, 18, -((b*d*e*Sqrt[g]*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*f^(3/2)*(e^2*f + d^2*g))) - (b*e^2*n*Log[d + e*x])/(2*f*(e^2*f + d^2*g)) + (a + b*Log[c*(d + e*x)^n])/(2*f*(f + g*x^2)) + (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^2 - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) + (b*e^2*n*Log[f + g*x^2])/(4*f*(e^2*f + d^2*g)) - (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f^2) - (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (b*n*PolyLog[2, 1 + (e*x)/d])/f^2} -{(a + b*Log[c*(d + e*x)^n])/(x^3*(f + g*x^2)^2), x, 21, -((b*e*n)/(2*d*f^2*x)) + (b*d*e*g^(3/2)*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*f^(5/2)*(e^2*f + d^2*g)) - (b*e^2*n*Log[x])/(2*d^2*f^2) + (b*e^2*n*Log[d + e*x])/(2*d^2*f^2) + (b*e^2*g*n*Log[d + e*x])/(2*f^2*(e^2*f + d^2*g)) - (a + b*Log[c*(d + e*x)^n])/(2*f^2*x^2) - (g*(a + b*Log[c*(d + e*x)^n]))/(2*f^2*(f + g*x^2)) - (2*g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/f^3 - (b*e^2*g*n*Log[f + g*x^2])/(4*f^2*(e^2*f + d^2*g)) + (b*g*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^3 + (b*g*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (2*b*g*n*PolyLog[2, 1 + (e*x)/d])/f^3} - -{x^4*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 31, (a*x)/g^2 - (b*n*x)/g^2 - (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) + (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f*(a + b*Log[c*(d + e*x)^n]))/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*(a + b*Log[c*(d + e*x)^n]))/(4*g^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*f*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b*e*f*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*g^(5/2)) - (3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*g^(5/2)) + (3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))} -{x^2*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 28, (b*e*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - (b*e*n*Log[d + e*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + (a + b*Log[c*(d + e*x)^n])/(4*g^(3/2)*(Sqrt[-f] - Sqrt[g]*x)) - (a + b*Log[c*(d + e*x)^n])/(4*g^(3/2)*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b*e*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*Sqrt[-f]*g^(3/2)) + (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2))} -{x^0*(a + b*Log[c*(d + e*x)^n])/(f + g*x^2)^2, x, 18, (b*e*n*Log[d + e*x])/(4*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) + (b*e*n*Log[d + e*x])/(4*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) - (a + b*Log[c*(d + e*x)^n])/(4*f*Sqrt[g]*(Sqrt[-f] - Sqrt[g]*x)) + (a + b*Log[c*(d + e*x)^n])/(4*f*Sqrt[g]*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g]) - (b*e*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + ((a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g]) + (b*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*(-f)^(3/2)*Sqrt[g]) - (b*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g])} -{(a + b*Log[c*(d + e*x)^n])/(x^2*(f + g*x^2)^2), x, 32, (b*e*n*Log[x])/(d*f^2) - (b*e*n*Log[d + e*x])/(d*f^2) - (b*e*Sqrt[g]*n*Log[d + e*x])/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (b*e*Sqrt[g]*n*Log[d + e*x])/(4*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) - (a + b*Log[c*(d + e*x)^n])/(f^2*x) + (Sqrt[g]*(a + b*Log[c*(d + e*x)^n]))/(4*f^2*(Sqrt[-f] - Sqrt[g]*x)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n]))/(4*f^2*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*Sqrt[g]*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2)) + (b*e*Sqrt[g]*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(5/2)) + (3*b*Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*(-f)^(5/2)) - (3*b*Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2))} - - -{(a + b*Log[c*(d + e*x)^n])/Sqrt[2 + g*x^2], x, 10, (b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]^2)/(2*Sqrt[g]) - (b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + d^2*g])])/Sqrt[g] - (b*n*ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g])])/Sqrt[g] + (ArcSinh[(Sqrt[g]*x)/Sqrt[2]]*(a + b*Log[c*(d + e*x)^n]))/Sqrt[g] - (b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] - Sqrt[2*e^2 + d^2*g]))])/Sqrt[g] - (b*n*PolyLog[2, -((Sqrt[2]*e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[2]])/(d*Sqrt[g] + Sqrt[2*e^2 + d^2*g]))])/Sqrt[g]} -{(a + b*Log[c*(d + e*x)^n])/Sqrt[f + g*x^2], x, 11, (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]^2)/(2*Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] - Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + (e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g])])/(Sqrt[g]*Sqrt[f + g*x^2]) + (Sqrt[f]*Sqrt[1 + (g*x^2)/f]*ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] - Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*x^2]) - (b*Sqrt[f]*n*Sqrt[1 + (g*x^2)/f]*PolyLog[2, -((e*E^ArcSinh[(Sqrt[g]*x)/Sqrt[f]]*Sqrt[f])/(d*Sqrt[g] + Sqrt[e^2*f + d^2*g]))])/(Sqrt[g]*Sqrt[f + g*x^2])} - -{(a + b*Log[c*(d + e*x)^n])/(Sqrt[2 + g*x]*Sqrt[2 - g*x]), x, 9, (I*b*n*ArcSin[(g*x)/2]^2)/(2*g) - (b*n*ArcSin[(g*x)/2]*Log[1 + (2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g - Sqrt[4*e^2 - d^2*g^2])])/g - (b*n*ArcSin[(g*x)/2]*Log[1 + (2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g + Sqrt[4*e^2 - d^2*g^2])])/g + (ArcSin[(g*x)/2]*(a + b*Log[c*(d + e*x)^n]))/g + (I*b*n*PolyLog[2, -((2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g - Sqrt[4*e^2 - d^2*g^2]))])/g + (I*b*n*PolyLog[2, -((2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g + Sqrt[4*e^2 - d^2*g^2]))])/g} -{(a + b*Log[c*(d + e*x)^n])/(Sqrt[f + g*x]*Sqrt[f - g*x]), x, 11, (I*b*f*n*Sqrt[1 - (g^2*x^2)/f^2]*ArcSin[(g*x)/f]^2)/(2*g*Sqrt[f - g*x]*Sqrt[f + g*x]) - (b*f*n*Sqrt[1 - (g^2*x^2)/f^2]*ArcSin[(g*x)/f]*Log[1 + (e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g - Sqrt[e^2*f^2 - d^2*g^2])])/(g*Sqrt[f - g*x]*Sqrt[f + g*x]) - (b*f*n*Sqrt[1 - (g^2*x^2)/f^2]*ArcSin[(g*x)/f]*Log[1 + (e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g + Sqrt[e^2*f^2 - d^2*g^2])])/(g*Sqrt[f - g*x]*Sqrt[f + g*x]) + (f*Sqrt[1 - (g^2*x^2)/f^2]*ArcSin[(g*x)/f]*(a + b*Log[c*(d + e*x)^n]))/(g*Sqrt[f - g*x]*Sqrt[f + g*x]) + (I*b*f*n*Sqrt[1 - (g^2*x^2)/f^2]*PolyLog[2, -((e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g - Sqrt[e^2*f^2 - d^2*g^2]))])/(g*Sqrt[f - g*x]*Sqrt[f + g*x]) + (I*b*f*n*Sqrt[1 - (g^2*x^2)/f^2]*PolyLog[2, -((e*E^(I*ArcSin[(g*x)/f])*f)/(I*d*g + Sqrt[e^2*f^2 - d^2*g^2]))])/(g*Sqrt[f - g*x]*Sqrt[f + g*x])} - - -{Log[2*e/(e + f*x)]/(e^2 - f^2*x^2), x, 2, PolyLog[2, 1 - (2*e)/(e + f*x)]/(2*e*f)} -{Log[e/(e + f*x)]/(e^2 - f^2*x^2), x, 4, -((ArcTanh[(f*x)/e]*Log[2])/(e*f)) + PolyLog[2, 1 - (2*e)/(e + f*x)]/(2*e*f)} - -{(a + b*Log[2*e/(e + f*x)])/(e^2 - f^2*x^2), x, 4, (a*ArcTanh[(f*x)/e])/(e*f) + (b*PolyLog[2, 1 - (2*e)/(e + f*x)])/(2*e*f)} -{(a + b*Log[e/(e + f*x)])/(e^2 - f^2*x^2), x, 4, (ArcTanh[(f*x)/e]*(a - b*Log[2]))/(e*f) + (b*PolyLog[2, 1 - (2*e)/(e + f*x)])/(2*e*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g x^3)^q (a+b Log[c (d+e x)^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^5*Log[c + d*x]/(a + b*x^3), x, 16, -((c^2*x)/(3*b*d^2)) + (c*x^2)/(6*b*d) - x^3/(9*b) + (c^3*Log[c + d*x])/(3*b*d^3) + (x^3*Log[c + d*x])/(3*b) - (a*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^2)} -{x^2*Log[c + d*x]/(a + b*x^3), x, 11, (Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b) + (Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*b) + (Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*b) + PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*b) + PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]/(3*b) + PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)]/(3*b)} -{Log[c + d*x]/(x^1*(a + b*x^3)), x, 15, (Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a) - (Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a) - (Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*a) + PolyLog[2, 1 + (d*x)/c]/a - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]/(3*a) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)]/(3*a)} -{Log[c + d*x]/(x^4*(a + b*x^3)), x, 18, -(d/(6*a*c*x^2)) + d^2/(3*a*c^2*x) + (d^3*Log[x])/(3*a*c^3) - (d^3*Log[c + d*x])/(3*a*c^3) - Log[c + d*x]/(3*a*x^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*a^2) - (b*PolyLog[2, 1 + (d*x)/c])/a^2} - -{x^4*Log[c + d*x]/(a + b*x^3), x, 16, (c*x)/(2*b*d) - x^2/(4*b) - (c^2*Log[c + d*x])/(2*b*d^2) + (x^2*Log[c + d*x])/(2*b) + (a^(2/3)*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*b^(5/3)) + ((-1)^(2/3)*a^(2/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^(5/3)) + (a^(2/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^(5/3)) + ((-1)^(2/3)*a^(2/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*b^(5/3))} -{x^3*Log[c + d*x]/(a + b*x^3), x, 15, -(x/b) + ((c + d*x)*Log[c + d*x])/(b*d) - (a^(1/3)*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*b^(4/3)) + ((-1)^(1/3)*a^(1/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^(4/3)) - (a^(1/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^(4/3)) + ((-1)^(1/3)*a^(1/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*b^(4/3))} -{x^1*Log[c + d*x]/(a + b*x^3), x, 11, -((Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(1/3)*b^(2/3))) + ((-1)^(1/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(1/3)*b^(2/3)) - PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(1/3)*b^(2/3))} -{x^0*Log[c + d*x]/(a + b*x^3), x, 11, (Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(2/3)*b^(1/3)) - ((-1)^(1/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(2/3)*b^(1/3)) + PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(3*a^(2/3)*b^(1/3)) - ((-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(2/3)*b^(1/3))} -{Log[c + d*x]/(x^2*(a + b*x^3)), x, 17, (d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/3)*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(4/3)) + ((-1)^(2/3)*b^(1/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(4/3)) + (b^(1/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^(4/3)) + ((-1)^(2/3)*b^(1/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(4/3))} -{Log[c + d*x]/(x^3*(a + b*x^3)), x, 16, -(d/(2*a*c*x)) - (d^2*Log[x])/(2*a*c^2) + (d^2*Log[c + d*x])/(2*a*c^2) - Log[c + d*x]/(2*a*x^2) - (b^(2/3)*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(5/3)) - (b^(2/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(5/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g x^4)^q (a+b Log[c (d+e x)^n])*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^7*Log[c + d*x]/(a + b*x^4), x, 23, (c^3*x)/(4*b*d^3) - (c^2*x^2)/(8*b*d^2) + (c*x^3)/(12*b*d) - x^4/(16*b) - (c^4*Log[c + d*x])/(4*b*d^4) + (x^4*Log[c + d*x])/(4*b) - (a*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b^2) - (a*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^2)} -{x^3*Log[c + d*x]/(a + b*x^4), x, 18, (Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b) + (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*b) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*b)} -{Log[c + d*x]/(x^1*(a + b*x^4)), x, 22, (Log[-((d*x)/c)]*Log[c + d*x])/a - (Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*a) - (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*a) - (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*a) - (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*a) + PolyLog[2, 1 + (d*x)/c]/a - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*a) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*a)} - -{x^5*Log[c + d*x]/(a + b*x^4), x, 23, (c*x)/(2*b*d) - x^2/(4*b) - (c^2*Log[c + d*x])/(2*b*d^2) + (x^2*Log[c + d*x])/(2*b) - (Sqrt[-a]*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b^(3/2)) + (Sqrt[-a]*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^(3/2)) - (Sqrt[-a]*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^(3/2)) + (Sqrt[-a]*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^(3/2)) - (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^(3/2)) - (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^(3/2)) + (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^(3/2)) + (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^(3/2))} -{x^1*Log[c + d*x]/(a + b*x^4), x, 18, -((Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b])) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b]) - (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b]) + (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*Sqrt[-a]*Sqrt[b]) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-a]*Sqrt[b]) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-a]*Sqrt[b]) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*Sqrt[-a]*Sqrt[b]) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*Sqrt[-a]*Sqrt[b])} -{Log[c + d*x]/(x^3*(a + b*x^4)), x, 23, -(d/(2*a*c*x)) - (d^2*Log[x])/(2*a*c^2) + (d^2*Log[c + d*x])/(2*a*c^2) - Log[c + d*x]/(2*a*x^2) - (Sqrt[b]*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))} - -{x^4*Log[c + d*x]/(a + b*x^4), x, 22, -(x/b) + ((c + d*x)*Log[c + d*x])/(b*d) + (Sqrt[-Sqrt[-a]]*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b^(5/4)) + ((-a)^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^(5/4)) - (Sqrt[-Sqrt[-a]]*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^(5/4)) - ((-a)^(1/4)*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^(5/4)) - (Sqrt[-Sqrt[-a]]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^(5/4)) + (Sqrt[-Sqrt[-a]]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^(5/4)) - ((-a)^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^(5/4)) + ((-a)^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^(5/4))} -{x^2*Log[c + d*x]/(a + b*x^4), x, 18, (Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*Sqrt[-Sqrt[-a]]*b^(3/4)) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(1/4)*b^(3/4)) - (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*Sqrt[-Sqrt[-a]]*b^(3/4)) - (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(1/4)*b^(3/4)) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-Sqrt[-a]]*b^(3/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*Sqrt[-Sqrt[-a]]*b^(3/4)) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*(-a)^(1/4)*b^(3/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*(-a)^(1/4)*b^(3/4))} -{x^0*Log[c + d*x]/(a + b*x^4), x, 18, (Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) + (Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/4)*b^(1/4)) - (Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) - (Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/4)*b^(1/4)) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)]/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]/(4*(-Sqrt[-a])^(3/2)*b^(1/4)) - PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)]/(4*(-a)^(3/4)*b^(1/4)) + PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)]/(4*(-a)^(3/4)*b^(1/4))} -{Log[c + d*x]/(x^2*(a + b*x^4)), x, 24, (d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/4)*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(5/4)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(5/4))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g / x^1)^q (a+b Log[c (d+e x)^n])*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^1*(f + g/x)^1*(a + b*Log[c*(d + e*x)^n]), x, 4, (b*(d*f - e*g)*n*x)/(2*e) - (b*n*(g + f*x)^2)/(4*f) - (b*(d*f - e*g)^2*n*Log[d + e*x])/(2*e^2*f) + ((g + f*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*f)} - - -{x^2*(f + g/x)^2*(a + b*Log[c*(d + e*x)^n]), x, 4, -((b*(d*f - e*g)^2*n*x)/(3*e^2)) + (b*(d*f - e*g)*n*(g + f*x)^2)/(6*e*f) - (b*n*(g + f*x)^3)/(9*f) + (b*(d*f - e*g)^3*n*Log[d + e*x])/(3*e^3*f) + ((g + f*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*f)} - - -{x^3*(f + g/x)^3*(a + b*Log[c*(d + e*x)^n]), x, 4, (b*(d*f - e*g)^3*n*x)/(4*e^3) - (b*(d*f - e*g)^2*n*(g + f*x)^2)/(8*e^2*f) + (b*(d*f - e*g)*n*(g + f*x)^3)/(12*e*f) - (b*n*(g + f*x)^4)/(16*f) - (b*(d*f - e*g)^4*n*Log[d + e*x])/(4*e^4*f) + ((g + f*x)^4*(a + b*Log[c*(d + e*x)^n]))/(4*f)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(a + b*Log[c*(d + e*x)^n])/(x^1*(f + g/x)^1), x, 4, ((a + b*Log[c*(d + e*x)^n])*Log[-((e*(g + f*x))/(d*f - e*g))])/f + (b*n*PolyLog[2, (f*(d + e*x))/(d*f - e*g)])/f} - - -{(a + b*Log[c*(d + e*x)^n])/(x^2*(f + g/x)^2), x, 5, -((b*e*n*Log[d + e*x])/(f*(d*f - e*g))) - (a + b*Log[c*(d + e*x)^n])/(f*(g + f*x)) + (b*e*n*Log[g + f*x])/(f*(d*f - e*g))} - - -{(a + b*Log[c*(d + e*x)^n])/(x^3*(f + g/x)^3), x, 4, -((b*e*n)/(2*f*(d*f - e*g)*(g + f*x))) + (b*e^2*n*Log[d + e*x])/(2*f*(d*f - e*g)^2) - (a + b*Log[c*(d + e*x)^n])/(2*f*(g + f*x)^2) - (b*e^2*n*Log[g + f*x])/(2*f*(d*f - e*g)^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g / x^2)^q (a+b Log[c (d+e x)^n])*) - - -{Log[a + b*x]/(c + d/x^2), x, 12, -(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]*Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))])/(2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (f+g x^2)^q (a+b Log[c (d+e x)^n])^2*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 28, -((2*a*b*d*f*n*x)/(e*g^2)) + (2*b^2*d*f*n^2*x)/(e*g^2) - (2*b^2*d^3*n^2*x)/(e^3*g) - (b^2*f*n^2*(d + e*x)^2)/(4*e^2*g^2) + (3*b^2*d^2*n^2*(d + e*x)^2)/(4*e^4*g) - (2*b^2*d*n^2*(d + e*x)^3)/(9*e^4*g) + (b^2*n^2*(d + e*x)^4)/(32*e^4*g) + (b^2*d^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b^2*d*f*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) + (2*b*d^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(e^4*g) + (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g^2) - (3*b*d^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + (2*b*d*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4*g) - (b*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4*g) - (b*d^4*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + (x^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) + (d*f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g^2) - (f*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^2) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3) + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (b^2*f^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 - (b^2*f^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3} -{x^3*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 21, (2*a*b*d*n*x)/(e*g) - (2*b^2*d*n^2*x)/(e*g) + (b^2*n^2*(d + e*x)^2)/(4*e^2*g) + (2*b^2*d*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g) - (b*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g) - (d*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g) + ((d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g) - (f*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) - (f*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^2) - (b*f*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^2 - (b*f*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^2 + (b^2*f*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^2 + (b^2*f*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^2} -{x^1*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 10, ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g} -{(a + b*Log[c*(d + e*x)^n])^2/(x^1*(f + g*x^2)), x, 16, (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f - (2*b^2*n^2*PolyLog[3, 1 + (e*x)/d])/f} -{(a + b*Log[c*(d + e*x)^n])^2/(x^3*(f + g*x^2)), x, 23, (b^2*e^2*n^2*Log[x])/(d^2*f) - (b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(d^2*f*x) - (a + b*Log[c*(d + e*x)^n])^2/(2*f*x^2) - (g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^2 + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) - (b*e^2*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(d^2*f) + (b^2*e^2*n^2*PolyLog[2, d/(d + e*x)])/(d^2*f) + (b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 + (b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 - (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f^2 - (b^2*g*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 - (b^2*g*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 + (2*b^2*g*n^2*PolyLog[3, 1 + (e*x)/d])/f^2} - -{x^4*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 23, (2*a*b*f*n*x)/g^2 - (2*b^2*f*n^2*x)/g^2 + (2*b^2*d^2*n^2*x)/(e^2*g) - (b^2*d*n^2*(d + e*x)^2)/(2*e^3*g) + (2*b^2*n^2*(d + e*x)^3)/(27*e^3*g) - (b^2*d^3*n^2*Log[d + e*x]^2)/(3*e^3*g) + (2*b^2*f*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (2*b*d^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(e^3*g) + (b*d*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(e^3*g) - (2*b*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(9*e^3*g) + (2*b*d^3*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3*g) + (x^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*g) - (f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) + ((-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2)) - ((-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(5/2)) - (b*(-f)^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^(5/2) + (b*(-f)^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^(5/2) + (b^2*(-f)^(3/2)*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^(5/2) - (b^2*(-f)^(3/2)*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^(5/2)} -{x^2*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 16, -((2*a*b*n*x)/g) + (2*b^2*n^2*x)/g - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g) + (Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(3/2)) - (b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^(3/2) + (b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^(3/2) + (b^2*Sqrt[-f]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^(3/2) - (b^2*Sqrt[-f]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^(3/2)} -{x^0*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2), x, 10, ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(Sqrt[-f]*Sqrt[g]) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(Sqrt[-f]*Sqrt[g]) + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(Sqrt[-f]*Sqrt[g]) - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(Sqrt[-f]*Sqrt[g])} -{(a + b*Log[c*(d + e*x)^n])^2/(x^2*(f + g*x^2)), x, 15, (2*b*e*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f*x) + (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(3/2)) - (b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(3/2) + (b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(3/2) + (2*b^2*e*n^2*PolyLog[2, 1 + (e*x)/d])/(d*f) + (b^2*Sqrt[g]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(3/2) - (b^2*Sqrt[g]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(3/2)} -{(a + b*Log[c*(d + e*x)^n])^2/(x^4*(f + g*x^2)), x, 26, -((b^2*e^2*n^2)/(3*d^2*f*x)) - (b^2*e^3*n^2*Log[x])/(d^3*f) + (b^2*e^3*n^2*Log[d + e*x])/(3*d^3*f) - (b*e*n*(a + b*Log[c*(d + e*x)^n]))/(3*d*f*x^2) + (2*b*e^2*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(3*d^3*f*x) - (2*b*e*g*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f^2) - (a + b*Log[c*(d + e*x)^n])^2/(3*f*x^3) + (g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f^2*x) + (g^(3/2)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2)) - (g^(3/2)*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(-f)^(5/2)) + (2*b*e^3*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(3*d^3*f) - (2*b^2*e^3*n^2*PolyLog[2, d/(d + e*x)])/(3*d^3*f) - (b*g^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(5/2) + (b*g^(3/2)*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2) - (2*b^2*e*g*n^2*PolyLog[2, 1 + (e*x)/d])/(d*f^2) + (b^2*g^(3/2)*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(5/2) - (b^2*g^(3/2)*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2)} - - -{x^5*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 34, (2*a*b*d*n*x)/(e*g^2) - (2*b^2*d*n^2*x)/(e*g^2) + (b^2*n^2*(d + e*x)^2)/(4*e^2*g^2) + (2*b^2*d*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) - (b*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2*g^2) + (e^2*f^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^3*(e^2*f + d^2*g)) - (d*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e^2*g^2) + ((d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^2) - (f^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^3*(f + g*x^2)) - (b*e*f*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3*(e^2*f + d^2*g)) - (f*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (b*e*(-f)^(3/2)*(e*Sqrt[-f] + d*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^3*(e^2*f + d^2*g)) - (f*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/g^3 - (b^2*e*(-f)^(3/2)*(e*Sqrt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^3*(e^2*f + d^2*g)) - (2*b*f*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 - (b^2*e*(-f)^(3/2)*(e*Sqrt[-f] - d*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3*(e^2*f + d^2*g)) - (2*b*f*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 + (2*b^2*f*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (2*b^2*f*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3} -{x^3*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 25, -((e^2*f*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^2*(e^2*f + d^2*g))) + (f*(a + b*Log[c*(d + e*x)^n])^2)/(2*g^2*(f + g*x^2)) + (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2*(e^2*f + d^2*g)) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2) + (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^2*(e^2*f + d^2*g)) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^2) - (b^2*e*Sqrt[-f]*(e*Sqrt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^2*(e^2*f + d^2*g)) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^2 + (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^2*(e^2*f + d^2*g)) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^2 - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^2 - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^2} -{x^1*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 13, (e^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*g*(e^2*f + d^2*g)) - (a + b*Log[c*(d + e*x)^n])^2/(2*g*(f + g*x^2)) - (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*g*(e^2*f + d^2*g)) - (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f*g*(e^2*f + d^2*g)) - (b^2*e*(e*Sqrt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*g*(e^2*f + d^2*g)) - (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*g*(e^2*f + d^2*g))} -{(a + b*Log[c*(d + e*x)^n])^2/(x^1*(f + g*x^2)^2), x, 29, -((e^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*f*(e^2*f + d^2*g))) + (a + b*Log[c*(d + e*x)^n])^2/(2*f*(f + g*x^2)) + (Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^2 + (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2) + (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2) - (b^2*e*(e*Sqrt[-f] + d*Sqrt[g])*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*(e^2*f + d^2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 + (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e^2*f + d^2*g)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 + (2*b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f^2 + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^2 + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^2 - (2*b^2*n^2*PolyLog[3, 1 + (e*x)/d])/f^2} -{(a + b*Log[c*(d + e*x)^n])^2/(x^3*(f + g*x^2)^2), x, 36, (b^2*e^2*n^2*Log[x])/(d^2*f^2) - (b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(d^2*f^2*x) + (e^2*g*(a + b*Log[c*(d + e*x)^n])^2)/(2*f^2*(e^2*f + d^2*g)) - (a + b*Log[c*(d + e*x)^n])^2/(2*f^2*x^2) - (g*(a + b*Log[c*(d + e*x)^n])^2)/(2*f^2*(f + g*x^2)) - (2*g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^3 - (b*e*(e*f + d*Sqrt[-f]*Sqrt[g])*g*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^3*(e^2*f + d^2*g)) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*g*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^3*(e^2*f + d^2*g)) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/f^3 - (b*e^2*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(d^2*f^2) + (b^2*e^2*n^2*PolyLog[2, d/(d + e*x)])/(d^2*f^2) - (b^2*e*(e*Sqrt[-f] + d*Sqrt[g])*g*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)*(e^2*f + d^2*g)) + (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^3 - (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*g*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^3*(e^2*f + d^2*g)) + (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (4*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/f^3 - (2*b^2*g*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^3 - (2*b^2*g*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 + (4*b^2*g*n^2*PolyLog[3, 1 + (e*x)/d])/f^3} - -{x^4*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 36, -((2*a*b*n*x)/g^2) + (2*b^2*n^2*x)/g^2 - (2*b^2*n*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(e*g^2) - (f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^2*(Sqrt[-f] - Sqrt[g]*x)) - (f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^2*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*f*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b*e*f*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*g^(5/2)) + (b^2*e*f*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) - (b^2*e*f*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*b*Sqrt[-f]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2)) + (3*b^2*Sqrt[-f]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) - (3*b^2*Sqrt[-f]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2))} -{x^2*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 32, ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] + d*Sqrt[g])*g*(Sqrt[-f] - Sqrt[g]*x)) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] - d*Sqrt[g])*g*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b^2*e*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*g^(3/2)) + (b^2*e*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*g^(3/2)) + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*g^(3/2)) - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*g^(3/2))} -{x^0*(a + b*Log[c*(d + e*x)^n])^2/(f + g*x^2)^2, x, 20, -(((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f*(e*Sqrt[-f] + d*Sqrt[g])*(Sqrt[-f] - Sqrt[g]*x))) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f*(e*Sqrt[-f] - d*Sqrt[g])*(Sqrt[-f] + Sqrt[g]*x)) - (b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g]) - (b*e*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(e*(-f)^(3/2) + d*f*Sqrt[g])*Sqrt[g]) + (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*e*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*(e*Sqrt[-f] + d*Sqrt[g])*Sqrt[g]) - (b*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)*Sqrt[g]) - (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(3/2)*Sqrt[g]) + (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(3/2)*Sqrt[g])} -{(a + b*Log[c*(d + e*x)^n])^2/(x^2*(f + g*x^2)^2), x, 35, (2*b*e*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(d*f^2) - ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(d*f^2*x) + (g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])*(Sqrt[-f] - Sqrt[g]*x)) + (g*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*f^2*(e*Sqrt[-f] - d*Sqrt[g])*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2)) + (b*e*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(5/2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)) + (b^2*e*Sqrt[g]*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*b*Sqrt[g]*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2)) + (2*b^2*e*n^2*PolyLog[2, 1 + (e*x)/d])/(d*f^2) - (3*b^2*Sqrt[g]*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)) + (3*b^2*Sqrt[g]*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(-f)^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x^2)^q (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{Log[c*(a + b*x)^n]^3/(d + e*x^2), x, 12, (Log[c*(a + b*x)^n]^3*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a + b*x)^n]^3*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(2*Sqrt[-d]*Sqrt[e]) + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) + (3*n^2*Log[c*(a + b*x)^n]*PolyLog[3, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) - (3*n^2*Log[c*(a + b*x)^n]*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - (3*n^3*PolyLog[4, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (3*n^3*PolyLog[4, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])} -{Log[c*(a + b*x)^n]^2/(d + e*x^2), x, 10, (Log[c*(a + b*x)^n]^2*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a + b*x)^n]^2*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[c*(a + b*x)^n]*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) + (n*Log[c*(a + b*x)^n]*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + (n^2*PolyLog[3, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(Sqrt[-d]*Sqrt[e]) - (n^2*PolyLog[3, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])} -{Log[c*(a + b*x)^n]/(d + e*x^2), x, 8, (Log[c*(a + b*x)^n]*Log[(b*(Sqrt[-d] - Sqrt[e]*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (Log[c*(a + b*x)^n]*Log[(b*(Sqrt[-d] + Sqrt[e]*x))/(b*Sqrt[-d] - a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, -((Sqrt[e]*(a + b*x))/(b*Sqrt[-d] - a*Sqrt[e]))])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (Sqrt[e]*(a + b*x))/(b*Sqrt[-d] + a*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])} -{1/((d + e*x^2)*Log[c*(a + b*x)^n]), x, 2, -(Unintegrable[1/((Sqrt[-d] - Sqrt[e]*x)*Log[c*(a + b*x)^n]), x]/(2*Sqrt[-d])) - Unintegrable[1/((Sqrt[-d] + Sqrt[e]*x)*Log[c*(a + b*x)^n]), x]/(2*Sqrt[-d])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h+i x)^m (f+g x^r)^q (a+b Log[c (d+e x)^n])^p*) - - -{Log[c - a*(1 - c)/(b*x^m)]/(x*(a + b*x^m)), x, 4, PolyLog[2, ((1 - c)*(b + a/x^m))/b]/(a*m)} -{Log[(-a + a*c + b*c*x^m)/(b*x^m)]/(x*(a + b*x^m)), x, 5, PolyLog[2, ((1 - c)*(b + a/x^m))/b]/(a*m)} - -{Log[c*(a - (d - a*c*d)/(c*e*x^m))]/(x*(d + e*x^m)), x, 4, PolyLog[2, ((1 - a*c)*(e + d/x^m))/e]/(d*m)} -{Log[(-d + a*c*d + a*c*e*x^m)/(e*x^m)]/(x*(d + e*x^m)), x, 5, PolyLog[2, ((1 - a*c)*(e + d/x^m))/e]/(d*m)} - - -{Log[(2*a)/(a + b*x)]/(a^2 - b^2*x^2), x, 2, PolyLog[2, 1 - (2*a)/(a + b*x)]/(2*a*b)} -{Log[(2*a)/(a + b*x)]/((a - b*x)*(a + b*x)), x, 4, PolyLog[2, 1 - (2*a)/(a + b*x)]/(2*a*b)} - -{Log[(a*(1 - c) + b*(1 + c)*x)/(a + b*x)]/(a^2 - b^2*x^2), x, 1, PolyLog[2, 1 - (a*(1 - c) + b*(1 + c)*x)/(a + b*x)]/(2*a*b)} -{Log[(a*(1 - c) + b*(1 + c)*x)/(a + b*x)]/((a - b*x)*(a + b*x)), x, 2, PolyLog[2, (c*(a - b*x))/(a + b*x)]/(2*a*b)} - -{Log[1 - (c*(a - b*x))/(a + b*x)]/(a^2 - b^2*x^2), x, 1, PolyLog[2, c*((a - b*x)/(a + b*x))]/(2*a*b)} -{Log[1 - (c*(a - b*x))/(a + b*x)]/((a - b*x)*(a + b*x)), x, 3, PolyLog[2, c*((a - b*x)/(a + b*x))]/(2*a*b)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x+i x^2)^m (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c (a+b x)^n]^p / (d + e x + f x^2)*) - - -{Log[c*(a + b*x)^n]^3/(d*x + e*x^2), x, 13, (Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^3)/d - (Log[c*(a + b*x)^n]^3*Log[(b*(d + e*x))/(b*d - a*e)])/d - (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, 1 + (b*x)/a])/d + (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, 1 + (b*x)/a])/d - (6*n^3*PolyLog[4, -((e*(a + b*x))/(b*d - a*e))])/d + (6*n^3*PolyLog[4, 1 + (b*x)/a])/d} -{Log[c*(a + b*x)^n]^2/(d*x + e*x^2), x, 11, (Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^2)/d - (Log[c*(a + b*x)^n]^2*Log[(b*(d + e*x))/(b*d - a*e)])/d - (2*n*Log[c*(a + b*x)^n]*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (2*n*Log[c*(a + b*x)^n]*PolyLog[2, 1 + (b*x)/a])/d + (2*n^2*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (2*n^2*PolyLog[3, 1 + (b*x)/a])/d} -{Log[c*(a + b*x)^n]/(d*x + e*x^2), x, 8, (Log[-((b*x)/a)]*Log[c*(a + b*x)^n])/d - (Log[c*(a + b*x)^n]*Log[(b*(d + e*x))/(b*d - a*e)])/d + (n*PolyLog[2, 1 + (b*x)/a])/d - (n*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d} -{1/((d*x + e*x^2)*Log[c*(a + b*x)^n]), x, 3, Unintegrable[1/(x*Log[c*(a + b*x)^n]), x]/d - (e*Unintegrable[1/((d + e*x)*Log[c*(a + b*x)^n]), x])/d} - - -{Log[c*(a + b*x)^n]^3/(d + e*x + f*x^2), x, 12, (Log[c*(a + b*x)^n]^3*Log[-((b*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] - (Log[c*(a + b*x)^n]^3*Log[-((b*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] + (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] + (6*n^3*PolyLog[4, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (6*n^3*PolyLog[4, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f]} -{Log[c*(a + b*x)^n]^2/(d + e*x + f*x^2), x, 10, (Log[c*(a + b*x)^n]^2*Log[-((b*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] - (Log[c*(a + b*x)^n]^2*Log[-((b*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] + (2*n*Log[c*(a + b*x)^n]*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (2*n*Log[c*(a + b*x)^n]*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (2*n^2*PolyLog[3, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] + (2*n^2*PolyLog[3, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f]} -{Log[c*(a + b*x)^n]/(d + e*x + f*x^2), x, 8, (Log[c*(a + b*x)^n]*Log[-((b*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] - (Log[c*(a + b*x)^n]*Log[-((b*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] + (n*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e - Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f] - (n*PolyLog[2, (2*f*(a + b*x))/(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))])/Sqrt[e^2 - 4*d*f]} -{1/((d + e*x + f*x^2)*Log[c*(a + b*x)^n]), x, 2, (2*f*Unintegrable[1/((e - Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f] - (2*f*Unintegrable[1/((e + Sqrt[e^2 - 4*d*f] + 2*f*x)*Log[c*(a + b*x)^n]), x])/Sqrt[e^2 - 4*d*f]} - - -{x^3*Log[x]/(a + b*x + c*x^2), x, 10, (b*x)/c^2 - x^2/(4*c) - (b*x*Log[x])/c^2 + (x^2*Log[x])/(2*c) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*c^3) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*c^3) + ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*c^3) + ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*c^3)} -{x^2*Log[x]/(a + b*x + c*x^2), x, 9, -(x/c) + (x*Log[x])/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*c^2) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*c^2) - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*c^2) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*c^2)} -{x^1*Log[x]/(a + b*x + c*x^2), x, 6, ((1 - b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*c) + ((1 + b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*c) + ((1 - b/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*c) + ((1 + b/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*c)} -{x^0*Log[x]/(a + b*x + c*x^2), x, 6, (Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[b^2 - 4*a*c] - (Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[b^2 - 4*a*c] + PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))]/Sqrt[b^2 - 4*a*c] - PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))]/Sqrt[b^2 - 4*a*c]} -{Log[x]/(x^1*(a + b*x + c*x^2)), x, 9, Log[x]^2/(2*a) - ((1 + b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a) - ((1 - b/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a) - ((1 + b/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*a) - ((1 - b/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*a)} -{Log[x]/(x^2*(a + b*x + c*x^2)), x, 10, -(1/(a*x)) - Log[x]/(a*x) - (b*Log[x]^2)/(2*a^2) + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2) + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*a^2) + ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*a^2)} -{Log[x]/(x^3*(a + b*x + c*x^2)), x, 11, -(1/(4*a*x^2)) + b/(a^2*x) - Log[x]/(2*a*x^2) + (b*Log[x])/(a^2*x) + ((b^2 - a*c)*Log[x]^2)/(2*a^3) - ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/(2*a^3) - ((b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/(2*a^3)} - - -(* ::Title::Closed:: *) -(*Integrands of the form EF[x] (a+b Log[c (d+e x)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g x)^q Log[f x^m] (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[f x^m] (a+b Log[c (d+e x)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]), x, 11, -((5*b*d^3*m*n*x)/(16*e^3)) + (3*b*d^2*m*n*x^2)/(32*e^2) - (7*b*d*m*n*x^3)/(144*e) + (1/32)*b*m*n*x^4 + (b*d^3*n*x*Log[f*x^m])/(4*e^3) - (b*d^2*n*x^2*Log[f*x^m])/(8*e^2) + (b*d*n*x^3*Log[f*x^m])/(12*e) - (1/16)*b*n*x^4*Log[f*x^m] + (b*d^4*m*n*Log[d + e*x])/(16*e^4) - (1/16)*(m*x^4 - 4*x^4*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) - (b*d^4*n*Log[f*x^m]*Log[1 + (e*x)/d])/(4*e^4) - (b*d^4*m*n*PolyLog[2, -((e*x)/d)])/(4*e^4)} -{x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]), x, 10, (4*b*d^2*m*n*x)/(9*e^2) - (5*b*d*m*n*x^2)/(36*e) + (2/27)*b*m*n*x^3 - (b*d^2*n*x*Log[f*x^m])/(3*e^2) + (b*d*n*x^2*Log[f*x^m])/(6*e) - (1/9)*b*n*x^3*Log[f*x^m] - (b*d^3*m*n*Log[d + e*x])/(9*e^3) - (1/9)*(m*x^3 - 3*x^3*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + (b*d^3*n*Log[f*x^m]*Log[1 + (e*x)/d])/(3*e^3) + (b*d^3*m*n*PolyLog[2, -((e*x)/d)])/(3*e^3)} -{x^1*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]), x, 9, -((3*b*d*m*n*x)/(4*e)) + (1/4)*b*m*n*x^2 + (b*d*n*x*Log[f*x^m])/(2*e) - (1/4)*b*n*x^2*Log[f*x^m] + (b*d^2*m*n*Log[d + e*x])/(4*e^2) - (1/4)*(m*x^2 - 2*x^2*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) - (b*d^2*n*Log[f*x^m]*Log[1 + (e*x)/d])/(2*e^2) - (b*d^2*m*n*PolyLog[2, -((e*x)/d)])/(2*e^2)} -{x^0*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]), x, 8, 2*b*m*n*x - b*n*x*Log[f*x^m] - (b*d*m*n*Log[d + e*x])/e - x*(m - Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + (b*d*n*Log[f*x^m]*Log[1 + (e*x)/d])/e + (b*d*m*n*PolyLog[2, -((e*x)/d)])/e} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])/x^1, x, 4, (Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m) - (b*n*Log[f*x^m]^2*Log[1 + (e*x)/d])/(2*m) - b*n*Log[f*x^m]*PolyLog[2, -((e*x)/d)] + b*m*n*PolyLog[3, -((e*x)/d)]} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])/x^2, x, 6, (b*e*m*n*Log[x])/d - (b*e*n*Log[1 + d/(e*x)]*Log[f*x^m])/d - (b*e*m*n*Log[d + e*x])/d - (m/x + Log[f*x^m]/x)*(a + b*Log[c*(d + e*x)^n]) + (b*e*m*n*PolyLog[2, -(d/(e*x))])/d} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])/x^3, x, 7, -((3*b*e*m*n)/(4*d*x)) - (b*e^2*m*n*Log[x])/(4*d^2) - (b*e*n*Log[f*x^m])/(2*d*x) + (b*e^2*n*Log[1 + d/(e*x)]*Log[f*x^m])/(2*d^2) + (b*e^2*m*n*Log[d + e*x])/(4*d^2) - (1/4)*(m/x^2 + (2*Log[f*x^m])/x^2)*(a + b*Log[c*(d + e*x)^n]) - (b*e^2*m*n*PolyLog[2, -(d/(e*x))])/(2*d^2)} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])/x^4, x, 9, -((5*b*e*m*n)/(36*d*x^2)) + (4*b*e^2*m*n)/(9*d^2*x) + (b*e^3*m*n*Log[x])/(9*d^3) - (b*e*n*Log[f*x^m])/(6*d*x^2) + (b*e^2*n*Log[f*x^m])/(3*d^2*x) - (b*e^3*n*Log[1 + d/(e*x)]*Log[f*x^m])/(3*d^3) - (b*e^3*m*n*Log[d + e*x])/(9*d^3) - (1/9)*(m/x^3 + (3*Log[f*x^m])/x^3)*(a + b*Log[c*(d + e*x)^n]) + (b*e^3*m*n*PolyLog[2, -(d/(e*x))])/(3*d^3)} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])/x^5, x, 11, -((7*b*e*m*n)/(144*d*x^3)) + (3*b*e^2*m*n)/(32*d^2*x^2) - (5*b*e^3*m*n)/(16*d^3*x) - (b*e^4*m*n*Log[x])/(16*d^4) - (b*e*n*Log[f*x^m])/(12*d*x^3) + (b*e^2*n*Log[f*x^m])/(8*d^2*x^2) - (b*e^3*n*Log[f*x^m])/(4*d^3*x) + (b*e^4*n*Log[1 + d/(e*x)]*Log[f*x^m])/(4*d^4) + (b*e^4*m*n*Log[d + e*x])/(16*d^4) - (1/16)*(m/x^4 + (4*Log[f*x^m])/x^4)*(a + b*Log[c*(d + e*x)^n]) - (b*e^4*m*n*PolyLog[2, -(d/(e*x))])/(4*d^4)} - - -{x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2, x, 52, (2*a*b*d^2*m*n*x)/(9*e^2) - (71*b^2*d^2*m*n^2*x)/(54*e^2) + (b*d^2*m*n*(6*a - 11*b*n)*x)/(9*e^2) + (19*b^2*d*m*n^2*x^2)/(54*e) - (2/27)*b^2*m*n^2*x^3 - (2*a*b*d^2*n*x*Log[f*x^m])/(3*e^2) + (11*b^2*d^2*n^2*x*Log[f*x^m])/(9*e^2) - (5*b^2*d*n^2*x^2*Log[f*x^m])/(18*e) + (2/27)*b^2*n^2*x^3*Log[f*x^m] + (23*b^2*d^3*m*n^2*Log[d + e*x])/(54*e^3) + (5*b^2*d^3*m*n^2*Log[-((e*x)/d)]*Log[d + e*x])/(9*e^3) - (5*b^2*d^3*n^2*Log[f*x^m]*Log[d + e*x])/(9*e^3) + (8*b^2*d^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(9*e^3) + (2*b^2*d^3*m*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/(3*e^3) - (2*b^2*d^2*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/(3*e^3) - (5*b*d*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(18*e) + (4/27)*b*m*n*x^3*(a + b*Log[c*(d + e*x)^n]) + (b*d*n*x^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(3*e) - (2/9)*b*n*x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]) - (d^3*m*(a + b*Log[c*(d + e*x)^n])^2)/(9*e^3) - (1/9)*m*x^3*(a + b*Log[c*(d + e*x)^n])^2 - (d^3*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (d^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (1/3)*x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2 + (11*b^2*d^3*m*n^2*PolyLog[2, 1 + (e*x)/d])/(9*e^3) - (2*b*d^3*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/(3*e^3) + (2*b^2*d^3*m*n^2*PolyLog[3, 1 + (e*x)/d])/(3*e^3), (11*a*b*d^2*m*n*x)/(9*e^2) - (28*b^2*d^2*m*n^2*x)/(9*e^2) + (5*b^2*d*m*n^2*x^2)/(36*e) - (2/81)*b^2*m*n^2*x^3 + (13*b^2*d*m*n^2*(d + e*x)^2)/(36*e^3) - (4*b^2*m*n^2*(d + e*x)^3)/(81*e^3) + (23*b^2*d^3*m*n^2*Log[x])/(54*e^3) + (2*b^2*d^2*n^2*x*Log[f*x^m])/e^2 - (b^2*d*n^2*(d + e*x)^2*Log[f*x^m])/(2*e^3) + (2*b^2*n^2*(d + e*x)^3*Log[f*x^m])/(27*e^3) + (b^2*d^3*m*n^2*Log[d + e*x]^2)/(9*e^3) + (b^2*d^3*m*n^2*Log[x]*Log[d + e*x]^2)/(3*e^3) - (b^2*d^3*n^2*Log[f*x^m]*Log[d + e*x]^2)/(3*e^3) + (11*b^2*d^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(9*e^3) + (2*b*d^2*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) - (13*b*d*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(18*e^3) + (4*b*m*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(27*e^3) + (11*b*d^3*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*d^2*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/e^3 + (b*d*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/e^3 - (2*b*n*(d + e*x)^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*d^3*m*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (2*b*d^3*m*n*Log[x]*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) + (2*b*d^3*n*Log[f*x^m]*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/(3*e^3) - (1/9)*m*x^3*(a + b*Log[c*(d + e*x)^n])^2 + (d^3*m*Log[x]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) - (d^3*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (1/3)*x^3*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2 + (11*b^2*d^3*m*n^2*PolyLog[2, 1 + (e*x)/d])/(9*e^3) - (2*b*d^3*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/(3*e^3) + (2*b^2*d^3*m*n^2*PolyLog[3, 1 + (e*x)/d])/(3*e^3)} -{x^1*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2, x, 38, -((a*b*d*m*n*x)/(2*e)) + (2*b^2*d*m*n^2*x)/e - (2*b*d*m*n*(a - b*n)*x)/e - (1/8)*b^2*m*n^2*x^2 - (b^2*m*n^2*(d + e*x)^2)/(4*e^2) - (b^2*d^2*m*n^2*Log[x])/(4*e^2) + (2*a*b*d*n*x*Log[f*x^m])/e - (2*b^2*d*n^2*x*Log[f*x^m])/e + (b^2*n^2*(d + e*x)^2*Log[f*x^m])/(4*e^2) - (5*b^2*d*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(2*e^2) - (2*b^2*d^2*m*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e^2 + (2*b^2*d*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e^2 + (b*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (b*d^2*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) - (b*n*(d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (d*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) + (d^2*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (d*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e^2 + ((d + e*x)^2*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) - (3*b^2*d^2*m*n^2*PolyLog[2, 1 + (e*x)/d])/(2*e^2) + (b*d^2*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e^2 - (b^2*d^2*m*n^2*PolyLog[3, 1 + (e*x)/d])/e^2} -{x^0*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2, x, 17, 2*a*b*m*n*x - 4*b^2*m*n^2*x + 2*b*m*n*(a - b*n)*x - 2*a*b*n*x*Log[f*x^m] + 2*b^2*n^2*x*Log[f*x^m] + (4*b^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e + (2*b^2*d*m*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e - (2*b^2*n*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e - (m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e - (d*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/e + ((d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e + (2*b^2*d*m*n^2*PolyLog[2, 1 + (e*x)/d])/e - (2*b*d*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e + (2*b^2*d*m*n^2*PolyLog[3, 1 + (e*x)/d])/e} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2/x^1, x, -1, (1/2)*m*Log[x]^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + Log[x]*((-m)*Log[x] + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 2*b*n*((-m)*Log[x] + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + 2*b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((1/2)*Log[x]^2*(Log[d + e*x] - Log[1 + (e*x)/d]) - Log[x]*PolyLog[2, -((e*x)/d)] + PolyLog[3, -((e*x)/d)]) - b^2*n^2*(m*Log[x] - Log[f*x^m])*(Log[-((e*x)/d)]*Log[d + e*x]^2 + 2*Log[d + e*x]*PolyLog[2, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d]) + (1/12)*b^2*m*n^2*(Log[-((e*x)/d)]^4 + 6*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e*x))]^2 - 4*(Log[-((e*x)/d)] + Log[d/(d + e*x)])*Log[-((e*x)/(d + e*x))]^3 + Log[-((e*x)/(d + e*x))]^4 + 6*Log[x]^2*Log[d + e*x]^2 + 4*(2*Log[-((e*x)/d)]^3 - 3*Log[x]^2*Log[d + e*x])*Log[1 + (e*x)/d] + 6*(Log[x] - Log[-((e*x)/d)])*(Log[x] + 3*Log[-((e*x)/d)])*Log[1 + (e*x)/d]^2 - 4*Log[-((e*x)/d)]^2*Log[-((e*x)/(d + e*x))]*(Log[-((e*x)/d)] + 3*Log[1 + (e*x)/d]) + 12*(Log[-((e*x)/d)]^2 - 2*Log[-((e*x)/d)]*(Log[-((e*x)/(d + e*x))] + Log[1 + (e*x)/d]) + 2*Log[x]*(-Log[d + e*x] + Log[1 + (e*x)/d]))*PolyLog[2, -((e*x)/d)] - 12*Log[-((e*x)/(d + e*x))]^2*PolyLog[2, (e*x)/(d + e*x)] + 12*(Log[-((e*x)/d)] - Log[-((e*x)/(d + e*x))])^2*PolyLog[2, 1 + (e*x)/d] + 24*(Log[x] - Log[-((e*x)/d)])*Log[1 + (e*x)/d]*PolyLog[2, 1 + (e*x)/d] + 24*(Log[-((e*x)/(d + e*x))] + Log[d + e*x])*PolyLog[3, -((e*x)/d)] + 24*Log[-((e*x)/(d + e*x))]*PolyLog[3, (e*x)/(d + e*x)] + 24*(-Log[x] + Log[-((e*x)/(d + e*x))])*PolyLog[3, 1 + (e*x)/d] - 24*(PolyLog[4, -((e*x)/d)] + PolyLog[4, (e*x)/(d + e*x)] - PolyLog[4, 1 + (e*x)/d]))} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2/x^2, x, -1, -((b^2*e*m*n^2*Log[x]^2*Log[d + e*x])/d) + (2*b^2*e*m*n^2*Log[-((e*x)/d)]*Log[d + e*x])/d + (2*b^2*e*n^2*Log[x]*Log[f*x^m]*Log[d + e*x])/d - (b^2*e*m*n^2*Log[d + e*x]^2)/d - (b^2*m*n^2*Log[d + e*x]^2)/x + (b^2*e*m*n^2*Log[-((e*x)/d)]*Log[d + e*x]^2)/d - (b^2*e*n^2*Log[f*x^m]*Log[d + e*x]^2)/d - (b^2*n^2*Log[f*x^m]*Log[d + e*x]^2)/x - (2*b*n*(m*Log[x] - Log[f*x^m])*(e*x*Log[-((e*x)/d)] - (d + e*x)*Log[d + e*x])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]))/(d*x) - (m*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/x - ((m - m*Log[x] + Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/x + (b^2*e*m*n^2*Log[x]^2*Log[1 + (e*x)/d])/d - (2*b^2*e*n^2*Log[x]*Log[f*x^m]*Log[1 + (e*x)/d])/d - (2*b^2*e*n^2*Log[f*x^m]*PolyLog[2, -((e*x)/d)])/d + (b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(2*e*x*Log[-((e*x)/d)] - 2*(d + e*x)*Log[d + e*x] - 2*d*Log[x]*Log[d + e*x] + e*x*(Log[x]^2 - 2*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)]))))/(d*x) + (2*b^2*e*m*n^2*(1 + Log[d + e*x])*PolyLog[2, 1 + (e*x)/d])/d + (2*b^2*e*m*n^2*PolyLog[3, -((e*x)/d)])/d - (2*b^2*e*m*n^2*PolyLog[3, 1 + (e*x)/d])/d} -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2/x^3, x, -1, (b^2*e^2*m*n^2*Log[x])/d^2 - (b^2*e^2*m*n^2*Log[x]^2)/(2*d^2) + (b^2*e^2*m*n^2*Log[-((e*x)/d)])/(2*d^2) + (b^2*e^2*n^2*Log[x]*Log[f*x^m])/d^2 - (3*b^2*e^2*m*n^2*Log[d + e*x])/(2*d^2) - (3*b^2*e*m*n^2*Log[d + e*x])/(2*d*x) + (b^2*e^2*m*n^2*Log[x]*Log[d + e*x])/d^2 + (b^2*e^2*m*n^2*Log[x]^2*Log[d + e*x])/(2*d^2) - (b^2*e^2*m*n^2*Log[-((e*x)/d)]*Log[d + e*x])/(2*d^2) - (b^2*e^2*n^2*Log[f*x^m]*Log[d + e*x])/d^2 - (b^2*e*n^2*Log[f*x^m]*Log[d + e*x])/(d*x) - (b^2*e^2*n^2*Log[x]*Log[f*x^m]*Log[d + e*x])/d^2 + (b^2*e^2*m*n^2*Log[d + e*x]^2)/(4*d^2) - (b^2*m*n^2*Log[d + e*x]^2)/(4*x^2) - (b^2*e^2*m*n^2*Log[-((e*x)/d)]*Log[d + e*x]^2)/(2*d^2) + (b^2*e^2*n^2*Log[f*x^m]*Log[d + e*x]^2)/(2*d^2) - (b^2*n^2*Log[f*x^m]*Log[d + e*x]^2)/(2*x^2) + (b*n*(m*Log[x] - Log[f*x^m])*(e^2*x^2*Log[-((e*x)/d)] + (d + e*x)*(e*x + (d - e*x)*Log[d + e*x]))*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]))/(d^2*x^2) - (m*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(2*x^2) - ((m - 2*m*Log[x] + 2*Log[f*x^m])*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(4*x^2) - (b^2*e^2*m*n^2*Log[x]*Log[1 + (e*x)/d])/d^2 - (b^2*e^2*m*n^2*Log[x]^2*Log[1 + (e*x)/d])/(2*d^2) + (b^2*e^2*n^2*Log[x]*Log[f*x^m]*Log[1 + (e*x)/d])/d^2 - (b^2*e^2*n^2*(m - Log[f*x^m])*PolyLog[2, -((e*x)/d)])/d^2 - (1/(2*d^2*x^2))*(b*m*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e*x*(d + e*x) + e^2*x^2*Log[-((e*x)/d)] + (d^2 - e^2*x^2)*Log[d + e*x] + 2*d^2*Log[x]*Log[d + e*x] + e*x*(e*x*Log[x]^2 + 2*d*(1 + Log[x]) - 2*e*x*(Log[x]*Log[1 + (e*x)/d] + PolyLog[2, -((e*x)/d)])))) - (b^2*e^2*m*n^2*(1 + 2*Log[d + e*x])*PolyLog[2, 1 + (e*x)/d])/(2*d^2) - (b^2*e^2*m*n^2*PolyLog[3, -((e*x)/d)])/d^2 + (b^2*e^2*m*n^2*PolyLog[3, 1 + (e*x)/d])/d^2} - - -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3, x, 28, -12*a*b^2*m*n^2*x + 18*b^3*m*n^3*x - 6*b^2*m*n^2*(a - b*n)*x + 6*a*b^2*n^2*x*Log[f*x^m] - 6*b^3*n^3*x*Log[f*x^m] - (18*b^3*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (6*b^3*d*m*n^2*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/e + (6*b^3*n^2*(d + e*x)*Log[f*x^m]*Log[c*(d + e*x)^n])/e + (6*b*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (3*b*d*m*n*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/e - (3*b*n*(d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^2)/e - (m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e - (d*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^3)/e + ((d + e*x)*Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^3)/e - (6*b^3*d*m*n^3*PolyLog[2, 1 + (e*x)/d])/e + (6*b^2*d*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d])/e - (3*b*d*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[3, 1 + (e*x)/d])/e + (6*b^2*d*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, 1 + (e*x)/d])/e - (6*b^3*d*m*n^3*PolyLog[4, 1 + (e*x)/d])/e} - - -{(Log[x]*Log[a + b*x]^2)/x, x, -1, (1/12)*(Log[-((b*x)/a)]^4 + 6*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]^2 - 4*(Log[-((b*x)/a)] + Log[a/(a + b*x)])*Log[-((b*x)/(a + b*x))]^3 + Log[-((b*x)/(a + b*x))]^4 + 6*Log[x]^2*Log[a + b*x]^2 + 4*(2*Log[-((b*x)/a)]^3 - 3*Log[x]^2*Log[a + b*x])*Log[1 + (b*x)/a] + 6*(Log[x] - Log[-((b*x)/a)])*(Log[x] + 3*Log[-((b*x)/a)])*Log[1 + (b*x)/a]^2 - 4*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]*(Log[-((b*x)/a)] + 3*Log[1 + (b*x)/a]) + 12*(Log[-((b*x)/a)]^2 - 2*Log[-((b*x)/a)]*(Log[-((b*x)/(a + b*x))] + Log[1 + (b*x)/a]) + 2*Log[x]*(-Log[a + b*x] + Log[1 + (b*x)/a]))*PolyLog[2, -((b*x)/a)] - 12*Log[-((b*x)/(a + b*x))]^2*PolyLog[2, (b*x)/(a + b*x)] + 12*(Log[-((b*x)/a)] - Log[-((b*x)/(a + b*x))])^2*PolyLog[2, 1 + (b*x)/a] + 24*(Log[x] - Log[-((b*x)/a)])*Log[1 + (b*x)/a]*PolyLog[2, 1 + (b*x)/a] + 24*(Log[-((b*x)/(a + b*x))] + Log[a + b*x])*PolyLog[3, -((b*x)/a)] + 24*Log[-((b*x)/(a + b*x))]*PolyLog[3, (b*x)/(a + b*x)] + 24*(-Log[x] + Log[-((b*x)/(a + b*x))])*PolyLog[3, 1 + (b*x)/a] - 24*(PolyLog[4, -((b*x)/a)] + PolyLog[4, (b*x)/(a + b*x)] - PolyLog[4, 1 + (b*x)/a]))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^1, x, 0, Unintegrable[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n]), x]} -{Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2, x, 0, Unintegrable[Log[f*x^m]/(a + b*Log[c*(d + e*x)^n])^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[f x^m] (a+b Log[c (d+e x)^n])^p with p symbolic*) - - -{Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x, 0, Unintegrable[Log[f*x^m]*(a + b*Log[c*(d + e*x)^n])^p, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (i x)^q Log[f (g+h x)^m] (a+b Log[c (d+e x)^n])^p*) - - -{Log[a + b*x]*Log[c + d*x]/x, x, 1, Log[-((b*x)/a)]*Log[a + b*x]*Log[c + d*x] + (1/2)*(Log[-((b*x)/a)] + Log[(b*c - a*d)/(b*(c + d*x))] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))])*Log[(a*(c + d*x))/(c*(a + b*x))]^2 - (1/2)*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])^2 + (Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (b*x)/a] + Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))] - Log[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] + (Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c] - PolyLog[3, 1 + (b*x)/a] + PolyLog[3, (c*(a + b*x))/(a*(c + d*x))] - PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] - PolyLog[3, 1 + (d*x)/c]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (k x)^r (a+b Log[c (d+e x)^n])^p (f+g Log[h (i+j x)^m])^q*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Log[c (d+e x)^n]) (f+g Log[c (d+e x)^n])*) - - -{x^2*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]), x, 7, (2*b*d^2*g*n^2*x)/e^2 - (b*d*g*n^2*(d + e*x)^2)/(2*e^3) + (2*b*g*n^2*(d + e*x)^3)/(27*e^3) - (b*d^3*g*n^2*Log[d + e*x]^2)/(3*e^3) + (1/3)*x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - (d^2*n*(d + e*x)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/e^3 + (d*n*(d + e*x)^2*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(2*e^3) - (n*(d + e*x)^3*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(9*e^3) + (d^3*n*Log[d + e*x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(3*e^3)} -{x^1*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]), x, 7, -((2*b*d*g*n^2*x)/e) + (b*g*n^2*(d + e*x)^2)/(4*e^2) + (b*d^2*g*n^2*Log[d + e*x]^2)/(2*e^2) + (1/2)*x^2*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) + (d*n*(d + e*x)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/e^2 - (n*(d + e*x)^2*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(4*e^2) - (d^2*n*Log[d + e*x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(2*e^2)} -{x^0*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]), x, 6, -((b*f + a*g)*n*x) + 2*b*g*n^2*x - (2*b*g*n*(d + e*x)*Log[c*(d + e*x)^n])/e + x*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) + (d*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^2)/(4*b*e*g)} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n])/x^1, x, 6, Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]) - (Log[x]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^2)/(4*b*g) + (Log[-((e*x)/d)]*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])^2)/(4*b*g) + n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x)/d] - 2*b*g*n^2*PolyLog[3, 1 + (e*x)/d]} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n])/x^2, x, 4, -(((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x) + (e*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/d - (2*b*e*g*n^2*PolyLog[2, d/(d + e*x)])/d} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n])/x^3, x, 7, (b*e^2*g*n^2*Log[x])/d^2 - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/(2*x^2) - (e*n*(d + e*x)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(2*d^2*x) - (e^2*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(2*d^2) + (b*e^2*g*n^2*PolyLog[2, d/(d + e*x)])/d^2} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n])/x^4, x, 11, -((b*e^2*g*n^2)/(3*d^2*x)) - (b*e^3*g*n^2*Log[x])/d^3 + (b*e^3*g*n^2*Log[d + e*x])/(3*d^3) - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/(3*x^3) - (e*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(6*d*x^2) + (e^2*n*(d + e*x)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(3*d^3*x) + (e^3*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(3*d^3) - (2*b*e^3*g*n^2*PolyLog[2, d/(d + e*x)])/(3*d^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^r (a+b Log[c (d+e x)^n])^p (f+g Log[h (i+j x)^m])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x, 35, (a*g*i^3*m*x)/(4*j^3) + (b*d^3*f*n*x)/(4*e^3) - (5*b*d^3*g*m*n*x)/(16*e^3) - (5*b*g*i^3*m*n*x)/(16*j^3) - (5*b*d*g*i^2*m*n*x)/(24*e*j^2) - (5*b*d^2*g*i*m*n*x)/(24*e^2*j) + (3*b*d^2*g*m*n*x^2)/(32*e^2) + (3*b*g*i^2*m*n*x^2)/(32*j^2) + (b*d*g*i*m*n*x^2)/(12*e*j) - (7*b*d*g*m*n*x^3)/(144*e) - (7*b*g*i*m*n*x^3)/(144*j) + (1/32)*b*g*m*n*x^4 + (b*d^4*g*m*n*Log[d + e*x])/(16*e^4) + (b*d^2*g*i^2*m*n*Log[d + e*x])/(8*e^2*j^2) + (b*d^3*g*i*m*n*Log[d + e*x])/(12*e^3*j) + (b*g*i^3*m*(d + e*x)*Log[c*(d + e*x)^n])/(4*e*j^3) - (g*i^2*m*x^2*(a + b*Log[c*(d + e*x)^n]))/(8*j^2) + (g*i*m*x^3*(a + b*Log[c*(d + e*x)^n]))/(12*j) - (1/16)*g*m*x^4*(a + b*Log[c*(d + e*x)^n]) + (b*g*i^4*m*n*Log[i + j*x])/(16*j^4) + (b*d*g*i^3*m*n*Log[i + j*x])/(12*e*j^3) + (b*d^2*g*i^2*m*n*Log[i + j*x])/(8*e^2*j^2) - (g*i^4*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^4) + (b*d^3*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(4*e^3*j) - (b*d^2*n*x^2*(f + g*Log[h*(i + j*x)^m]))/(8*e^2) + (b*d*n*x^3*(f + g*Log[h*(i + j*x)^m]))/(12*e) - (1/16)*b*n*x^4*(f + g*Log[h*(i + j*x)^m]) - (b*d^4*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(4*e^4) + (1/4)*x^4*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) - (b*g*i^4*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(4*j^4) - (b*d^4*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(4*e^4)} -{x^2*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x, 29, -((a*g*i^2*m*x)/(3*j^2)) - (b*d^2*f*n*x)/(3*e^2) + (4*b*d^2*g*m*n*x)/(9*e^2) + (4*b*g*i^2*m*n*x)/(9*j^2) + (b*d*g*i*m*n*x)/(3*e*j) - (5*b*d*g*m*n*x^2)/(36*e) - (5*b*g*i*m*n*x^2)/(36*j) + (2/27)*b*g*m*n*x^3 - (b*d^3*g*m*n*Log[d + e*x])/(9*e^3) - (b*d^2*g*i*m*n*Log[d + e*x])/(6*e^2*j) - (b*g*i^2*m*(d + e*x)*Log[c*(d + e*x)^n])/(3*e*j^2) + (g*i*m*x^2*(a + b*Log[c*(d + e*x)^n]))/(6*j) - (1/9)*g*m*x^3*(a + b*Log[c*(d + e*x)^n]) - (b*g*i^3*m*n*Log[i + j*x])/(9*j^3) - (b*d*g*i^2*m*n*Log[i + j*x])/(6*e*j^2) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(3*j^3) - (b*d^2*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(3*e^2*j) + (b*d*n*x^2*(f + g*Log[h*(i + j*x)^m]))/(6*e) - (1/9)*b*n*x^3*(f + g*Log[h*(i + j*x)^m]) + (b*d^3*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(3*e^3) + (1/3)*x^3*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) + (b*g*i^3*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) + (b*d^3*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(3*e^3)} -{x^1*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x, 23, (a*g*i*m*x)/(2*j) + (b*d*f*n*x)/(2*e) - (3*b*d*g*m*n*x)/(4*e) - (3*b*g*i*m*n*x)/(4*j) + (1/4)*b*g*m*n*x^2 + (b*d^2*g*m*n*Log[d + e*x])/(4*e^2) + (b*g*i*m*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) - (1/4)*g*m*x^2*(a + b*Log[c*(d + e*x)^n]) + (b*g*i^2*m*n*Log[i + j*x])/(4*j^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j^2) + (b*d*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) - (1/4)*b*n*x^2*(f + g*Log[h*(i + j*x)^m]) - (b*d^2*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(2*e^2) + (1/2)*x^2*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) - (b*g*i^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (b*d^2*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2)} -{x^0*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x, 17, (-a)*g*m*x - b*f*n*x + 2*b*g*m*n*x - (b*g*m*(d + e*x)*Log[c*(d + e*x)^n])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j - (b*g*n*(i + j*x)*Log[h*(i + j*x)^m])/j + (b*d*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/e + x*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]) + (b*g*i*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (b*d*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m])/x^1, x, 13, f*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]) + b*g*m*n*Log[-((e*x)/d)]*Log[d + e*x]*Log[i + j*x] - b*g*m*Log[-((j*x)/i)]*(n*Log[d + e*x] - Log[c*(d + e*x)^n])*Log[i + j*x] + (1/2)*b*g*m*n*(Log[-((e*x)/d)] + Log[(e*i - d*j)/(e*(i + j*x))] - Log[-(((e*i - d*j)*x)/(d*(i + j*x)))])*Log[(d*(i + j*x))/(i*(d + e*x))]^2 - (1/2)*b*g*m*n*(Log[-((e*x)/d)] - Log[-((j*x)/i)])*(Log[d + e*x] + Log[(d*(i + j*x))/(i*(d + e*x))])^2 - b*g*Log[-((e*x)/d)]*Log[c*(d + e*x)^n]*(m*Log[i + j*x] - Log[h*(i + j*x)^m]) + a*g*Log[-((j*x)/i)]*Log[h*(i + j*x)^m] + b*f*n*PolyLog[2, 1 + (e*x)/d] + b*g*m*n*(Log[i + j*x] - Log[(d*(i + j*x))/(i*(d + e*x))])*PolyLog[2, 1 + (e*x)/d] - b*g*n*(m*Log[i + j*x] - Log[h*(i + j*x)^m])*PolyLog[2, 1 + (e*x)/d] + b*g*m*n*Log[(d*(i + j*x))/(i*(d + e*x))]*PolyLog[2, (i*(d + e*x))/(d*(i + j*x))] - b*g*m*n*Log[(d*(i + j*x))/(i*(d + e*x))]*PolyLog[2, (j*(d + e*x))/(e*(i + j*x))] + a*g*m*PolyLog[2, 1 + (j*x)/i] - b*g*m*(n*Log[d + e*x] - Log[c*(d + e*x)^n])*PolyLog[2, 1 + (j*x)/i] + b*g*m*n*(Log[d + e*x] + Log[(d*(i + j*x))/(i*(d + e*x))])*PolyLog[2, 1 + (j*x)/i] - b*g*m*n*PolyLog[3, 1 + (e*x)/d] + b*g*m*n*PolyLog[3, (i*(d + e*x))/(d*(i + j*x))] - b*g*m*n*PolyLog[3, (j*(d + e*x))/(e*(i + j*x))] - b*g*m*n*PolyLog[3, 1 + (j*x)/i]} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m])/x^2, x, 15, (g*j*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/i - (g*j*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/i + (b*e*n*Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m]))/d - (b*e*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/d - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x - (b*g*j*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/i + (b*g*j*m*n*PolyLog[2, 1 + (e*x)/d])/i - (b*e*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/d + (b*e*g*m*n*PolyLog[2, 1 + (j*x)/i])/d} -{(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m])/x^3, x, 23, (b*e*g*j*m*n*Log[x])/(d*i) - (b*e*g*j*m*n*Log[d + e*x])/(2*d*i) - (g*j*m*(a + b*Log[c*(d + e*x)^n]))/(2*i*x) - (g*j^2*m*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/(2*i^2) - (b*e*g*j*m*n*Log[i + j*x])/(2*d*i) + (g*j^2*m*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*i^2) - (b*e*n*(f + g*Log[h*(i + j*x)^m]))/(2*d*x) - (b*e^2*n*Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m]))/(2*d^2) + (b*e^2*n*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m]))/(2*d^2) - ((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/(2*x^2) + (b*g*j^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*i^2) - (b*g*j^2*m*n*PolyLog[2, 1 + (e*x)/d])/(2*i^2) + (b*e^2*g*m*n*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*d^2) - (b*e^2*g*m*n*PolyLog[2, 1 + (j*x)/i])/(2*d^2)} - - -(* {x^3*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]), x, 157, (2*a*b*d^3*f*n*x)/e^3 - (a*b*d^3*g*m*n*x)/e^3 - (5*a*b*g*i^3*m*n*x)/(8*j^3) - (2*a*b*d*g*i^2*m*n*x)/(3*e*j^2) - (3*a*b*d^2*g*i*m*n*x)/(4*e^2*j) - (2*b^2*d^3*f*n^2*x)/e^3 + (649*b^2*d^3*g*m*n^2*x)/(288*e^3) + (21*b^2*g*i^3*m*n^2*x)/(32*j^3) + (119*b^2*d*g*i^2*m*n^2*x)/(144*e*j^2) + (55*b^2*d^2*g*i*m*n^2*x)/(48*e^2*j) - (139*b^2*d^2*g*m*n^2*x^2)/(576*e^2) - (3*b^2*g*i^2*m*n^2*x^2)/(64*j^2) - (b^2*d*g*i*m*n^2*x^2)/(9*e*j) + (53*b^2*d*g*m*n^2*x^3)/(864*e) + (7*b^2*g*i*m*n^2*x^3)/(288*j) - (1/64)*b^2*g*m*n^2*x^4 + (3*b^2*d^2*f*n^2*(d + e*x)^2)/(4*e^4) - (3*b^2*d^2*g*m*n^2*(d + e*x)^2)/(16*e^4) - (b^2*g*i^2*m*n^2*(d + e*x)^2)/(16*e^2*j^2) - (b^2*d*g*i*m*n^2*(d + e*x)^2)/(8*e^3*j) - (2*b^2*d*f*n^2*(d + e*x)^3)/(9*e^4) + (b^2*d*g*m*n^2*(d + e*x)^3)/(18*e^4) + (b^2*g*i*m*n^2*(d + e*x)^3)/(54*e^3*j) + (b^2*f*n^2*(d + e*x)^4)/(32*e^4) - (b^2*g*m*n^2*(d + e*x)^4)/(128*e^4) - (61*b^2*d^4*g*m*n^2*Log[d + e*x])/(288*e^4) - (b^2*d^2*g*i^2*m*n^2*Log[d + e*x])/(16*e^2*j^2) - (b^2*d^3*g*i*m*n^2*Log[d + e*x])/(8*e^3*j) + (2*b^2*d^3*f*n*(d + e*x)*Log[c*(d + e*x)^n])/e^4 - (b^2*d^3*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e^4 - (5*b^2*g*i^3*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(8*e*j^3) - (2*b^2*d*g*i^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(3*e^2*j^2) - (3*b^2*d^2*g*i*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(4*e^3*j) + (b*d^2*g*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(8*e^2) + (b*g*i^2*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(16*j^2) + (b*d*g*i*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(12*e*j) - (b*d*g*m*n*x^3*(a + b*Log[c*(d + e*x)^n]))/(18*e) - (b*g*i*m*n*x^3*(a + b*Log[c*(d + e*x)^n]))/(24*j) + (1/32)*b*g*m*n*x^4*(a + b*Log[c*(d + e*x)^n]) - (3*b*d^2*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4) + (3*b*d^2*g*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(8*e^4) + (b*g*i^2*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(8*e^2*j^2) + (b*d*g*i*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^3*j) + (2*b*d*f*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*e^4) - (b*d*g*m*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(6*e^4) - (b*g*i*m*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(18*e^3*j) - (b*f*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4) + (b*g*m*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(32*e^4) - (d^4*f*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^4) + (d^4*g*m*(a + b*Log[c*(d + e*x)^n])^2)/(16*e^4) + (d^3*g*i*m*(a + b*Log[c*(d + e*x)^n])^2)/(12*e^3*j) + (g*i*m*x^3*(a + b*Log[c*(d + e*x)^n])^2)/(12*j) - (1/16)*g*m*x^4*(a + b*Log[c*(d + e*x)^n])^2 + (g*i^3*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*e*j^3) + (d*g*i^2*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2*j^2) - (g*i^2*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(8*e^2*j^2) - (b^2*g*i^4*m*n^2*Log[i + j*x])/(32*j^4) - (7*b^2*d*g*i^3*m*n^2*Log[i + j*x])/(72*e*j^3) - (13*b^2*d^2*g*i^2*m*n^2*Log[i + j*x])/(48*e^2*j^2) + (b*g*i^4*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(8*j^4) + (b*d*g*i^3*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(6*e*j^3) + (b*d^2*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(4*e^2*j^2) + (b*d^3*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e^3*j) + (d^4*g*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(4*e^4) - (g*i^4*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^4) + (13*b^2*d^2*g*n^2*x^2*Log[h*(i + j*x)^m])/(48*e^2) - (7*b^2*d*g*n^2*x^3*Log[h*(i + j*x)^m])/(72*e) + (1/32)*b^2*g*n^2*x^4*Log[h*(i + j*x)^m] - (25*b^2*d^3*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/(24*e^3*j) + (25*b^2*d^4*g*n^2*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(24*e^4) + (b*d^3*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(2*e^3) - (b*d^2*g*n*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(4*e^2) + (b*d*g*n*x^3*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(6*e) - (1/8)*b*g*n*x^4*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (d^4*g*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(4*e^4) + (1/4)*x^4*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) + (b^2*g*i^4*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(8*j^4) + (b^2*d*g*i^3*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(6*e*j^3) + (b^2*d^2*g*i^2*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(4*e^2*j^2) + (b^2*d^3*g*i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e^3*j) + (b*d^4*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e^4) - (b*g*i^4*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^4) + (25*b^2*d^4*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(24*e^4) - (b^2*d^4*g*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(2*e^4) + (b^2*g*i^4*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(2*j^4)} *) -(* {x^2*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]), x, 108, (2*a*b*d^2*g*m*n*x)/(3*e^2) + (8*a*b*g*i^2*m*n*x)/(9*j^2) + (a*b*d*g*i*m*n*x)/(e*j) + (2*b^2*d^2*f*n^2*x)/e^2 - (151*b^2*d^2*g*m*n^2*x)/(54*e^2) - (26*b^2*g*i^2*m*n^2*x)/(27*j^2) - (25*b^2*d*g*i*m*n^2*x)/(18*e*j) + (7*b^2*d*g*m*n^2*x^2)/(27*e) + (5*b^2*g*i*m*n^2*x^2)/(54*j) - (4/81)*b^2*g*m*n^2*x^3 - (b^2*d*f*n^2*(d + e*x)^2)/(2*e^3) + (b^2*d*g*m*n^2*(d + e*x)^2)/(6*e^3) + (b^2*g*i*m*n^2*(d + e*x)^2)/(12*e^2*j) + (2*b^2*f*n^2*(d + e*x)^3)/(27*e^3) - (2*b^2*g*m*n^2*(d + e*x)^3)/(81*e^3) + (13*b^2*d^3*g*m*n^2*Log[d + e*x])/(54*e^3) + (b^2*d^2*g*i*m*n^2*Log[d + e*x])/(9*e^2*j) - (b^2*d^3*f*n^2*Log[d + e*x]^2)/(3*e^3) + (b^2*d^3*g*m*n^2*Log[d + e*x]^2)/(9*e^3) + (2*b^2*d^2*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(3*e^3) + (8*b^2*g*i^2*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(9*e*j^2) + (b^2*d*g*i*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*j) - (b*d*g*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(6*e) - (b*g*i*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(9*j) + (2/27)*b*g*m*n*x^3*(a + b*Log[c*(d + e*x)^n]) - (b*g*i*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(6*e^2*j) - (1/9)*b*f*n*((18*d^2*(d + e*x))/e^3 - (9*d*(d + e*x)^2)/e^3 + (2*(d + e*x)^3)/e^3 - (6*d^3*Log[d + e*x])/e^3)*(a + b*Log[c*(d + e*x)^n]) + (1/27)*b*g*m*n*((18*d^2*(d + e*x))/e^3 - (9*d*(d + e*x)^2)/e^3 + (2*(d + e*x)^3)/e^3 - (6*d^3*Log[d + e*x])/e^3)*(a + b*Log[c*(d + e*x)^n]) - (1/9)*g*m*x^3*(a + b*Log[c*(d + e*x)^n])^2 - (g*i^2*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(3*e*j^2) - (d*g*i*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^2*j) + (g*i*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(6*e^2*j) + (2*b^2*g*i^3*m*n^2*Log[i + j*x])/(27*j^3) + (5*b^2*d*g*i^2*m*n^2*Log[i + j*x])/(18*e*j^2) - (2*b*g*i^3*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(9*j^3) - (b*d*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(3*e*j^2) - (2*b*d^2*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(3*e^2*j) - (d^3*g*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(3*e^3) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(3*j^3) - (5*b^2*d*g*n^2*x^2*Log[h*(i + j*x)^m])/(18*e) + (2/27)*b^2*g*n^2*x^3*Log[h*(i + j*x)^m] + (11*b^2*d^2*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/(9*e^2*j) - (11*b^2*d^3*g*n^2*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(9*e^3) - (2*b*d^2*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(3*e^2) + (b*d*g*n*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(3*e) - (2/9)*b*g*n*x^3*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] + (d^3*g*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(3*e^3) + (1/3)*x^3*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) - (2*b^2*g*i^3*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(9*j^3) - (b^2*d*g*i^2*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*e*j^2) - (2*b^2*d^2*g*i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*e^2*j) - (2*b*d^3*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*e^3) + (2*b*g*i^3*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) - (11*b^2*d^3*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(9*e^3) + (2*b^2*d^3*g*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(3*e^3) - (2*b^2*g*i^3*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3)} *) -{x^1*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]), x, 73, -((2*a*b*d*g*m*n*x)/e) - (3*a*b*g*i*m*n*x)/(2*j) - (2*b^2*d*f*n^2*x)/e + (15*b^2*d*g*m*n^2*x)/(4*e) + (7*b^2*g*i*m*n^2*x)/(4*j) - (1/4)*b^2*g*m*n^2*x^2 + (b^2*f*n^2*(d + e*x)^2)/(4*e^2) - (b^2*g*m*n^2*(d + e*x)^2)/(8*e^2) - (b^2*d^2*g*m*n^2*Log[d + e*x])/(4*e^2) + (b^2*d^2*f*n^2*Log[d + e*x]^2)/(2*e^2) - (2*b^2*d*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e^2 - (3*b^2*g*i*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) + (1/4)*b*g*m*n*x^2*(a + b*Log[c*(d + e*x)^n]) + (2*b*d*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^2 - (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^2) + (b*g*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) - (b*d^2*f*n*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n]))/e^2 + (d*g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) + (g*i*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e*j) - (g*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) - (b^2*g*i^2*m*n^2*Log[i + j*x])/(4*j^2) + (b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j^2) + (b*d*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(e*j) + (d^2*g*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j^2) + (1/4)*b^2*g*n^2*x^2*Log[h*(i + j*x)^m] - (3*b^2*d*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) + (3*b^2*d^2*g*n^2*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(2*e^2) + (b*d*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/e - (1/2)*b*g*n*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (d^2*g*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(2*e^2) + (1/2)*x^2*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) + (b^2*g*i^2*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) + (b^2*d*g*i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e*j) + (b*d^2*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e^2 - (b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j^2 + (3*b^2*d^2*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2) - (b^2*d^2*g*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e^2 + (b^2*g*i^2*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j^2} -{x^0*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]), x, 41, -2*a*b*f*n*x + 4*a*b*g*m*n*x + 2*b^2*f*n^2*x - 6*b^2*g*m*n^2*x - (2*b^2*f*n*(d + e*x)*Log[c*(d + e*x)^n])/e + (4*b^2*g*m*n*(d + e*x)*Log[c*(d + e*x)^n])/e + (d*f*(a + b*Log[c*(d + e*x)^n])^2)/e - (g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e - (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/j + (2*b^2*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/j - (2*b^2*d*g*n^2*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/e - 2*b*g*n*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) - (2*b^2*g*i*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (2*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e + (2*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (2*b^2*d*g*m*n^2*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e + (2*b^2*d*g*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e - (2*b^2*g*i*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m])/x^1, x, 0, Unintegrable[((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x, x]} -{(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m])/x^2, x, 0, Unintegrable[((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x^2, x]} - - -(* {x^2*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]), x, 262, (6*a*b^2*d^2*f*n^2*x)/e^2 - (28*a*b^2*d^2*g*m*n^2*x)/(3*e^2) - (26*a*b^2*g*i^2*m*n^2*x)/(9*j^2) - (9*a*b^2*d*g*i*m*n^2*x)/(2*e*j) - (6*b^3*d^2*f*n^3*x)/e^2 + (1571*b^3*d^2*g*m*n^3*x)/(108*e^2) + (80*b^3*g*i^2*m*n^3*x)/(27*j^2) + (185*b^3*d*g*i*m*n^3*x)/(36*e*j) - (55*b^3*d*g*m*n^3*x^2)/(108*e) - (5*b^3*g*i*m*n^3*x^2)/(54*j) + (4/81)*b^3*g*m*n^3*x^3 + (3*b^3*d*f*n^3*(d + e*x)^2)/(4*e^3) - (13*b^3*d*g*m*n^3*(d + e*x)^2)/(24*e^3) - (5*b^3*g*i*m*n^3*(d + e*x)^2)/(24*e^2*j) - (2*b^3*f*n^3*(d + e*x)^3)/(27*e^3) + (4*b^3*g*m*n^3*(d + e*x)^3)/(81*e^3) - (53*b^3*d^3*g*m*n^3*Log[d + e*x])/(108*e^3) - (b^3*d^2*g*i*m*n^3*Log[d + e*x])/(9*e^2*j) + (6*b^3*d^2*f*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^3 - (28*b^3*d^2*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(3*e^3) - (26*b^3*g*i^2*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(9*e*j^2) - (9*b^3*d*g*i*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(2*e^2*j) + (5*b^2*d*g*m*n^2*x^2*(a + b*Log[c*(d + e*x)^n]))/(12*e) + (b^2*g*i*m*n^2*x^2*(a + b*Log[c*(d + e*x)^n]))/(9*j) - (2/27)*b^2*g*m*n^2*x^3*(a + b*Log[c*(d + e*x)^n]) - (3*b^2*d*f*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (13*b^2*d*g*m*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(12*e^3) + (5*b^2*g*i*m*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(12*e^2*j) + (2*b^2*f*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(9*e^3) - (4*b^2*g*m*n^2*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/(27*e^3) + (b*d^3*g*m*n*(a + b*Log[c*(d + e*x)^n])^2)/(9*e^3) + (1/9)*b*g*m*n*x^3*(a + b*Log[c*(d + e*x)^n])^2 - (3*b*d^2*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^3 + (5*b*d^2*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) + (4*b*g*i^2*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(3*e*j^2) + (11*b*d*g*i*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(6*e^2*j) + (3*b*d*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) - (3*b*d*g*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^3) - (5*b*g*i*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(12*e^2*j) - (b*f*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(3*e^3) + (b*g*m*n*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2)/(9*e^3) + (d^3*f*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3) - (d^2*g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3) - (g*i^2*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(3*e*j^2) - (d*g*i*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^2*j) + (d*g*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(3*e^3) + (g*i*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(6*e^2*j) - (g*m*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^3)/(9*e^3) - (2*b^3*g*i^3*m*n^3*Log[i + j*x])/(27*j^3) - (19*b^3*d*g*i^2*m*n^3*Log[i + j*x])/(36*e*j^2) + (2*b^2*g*i^3*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(9*j^3) + (5*b^2*d*g*i^2*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(6*e*j^2) + (11*b^2*d^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(3*e^2*j) + (11*b*d^3*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(6*e^3) - (b*g*i^3*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(3*j^3) - (b*d*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e*j^2) - (b*d^2*g*i*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(e^2*j) - (d^3*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/(3*e^3) + (g*i^3*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/(3*j^3) + (19*b^3*d*g*n^3*x^2*Log[h*(i + j*x)^m])/(36*e) - (2/27)*b^3*g*n^3*x^3*Log[h*(i + j*x)^m] - (85*b^3*d^2*g*n^3*(i + j*x)*Log[h*(i + j*x)^m])/(18*e^2*j) + (85*b^3*d^3*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(18*e^3) + (11*b^2*d^2*g*n^2*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(3*e^2) - (5*b^2*d*g*n^2*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(6*e) + (2/9)*b^2*g*n^2*x^3*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (11*b*d^3*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(6*e^3) - (b*d^2*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e^2 + (b*d*g*n*x^2*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(2*e) - (1/3)*b*g*n*x^3*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] + (d^3*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*x)^m])/(3*e^3) + (1/3)*x^3*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) + (2*b^3*g*i^3*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(9*j^3) + (5*b^3*d*g*i^2*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(6*e*j^2) + (11*b^3*d^2*g*i*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*e^2*j) + (11*b^2*d^3*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*e^3) - (2*b^2*g*i^3*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) - (b^2*d*g*i^2*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e*j^2) - (2*b^2*d^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e^2*j) - (b*d^3*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e^3 + (b*g*i^3*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j^3 + (85*b^3*d^3*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(18*e^3) - (11*b^3*d^3*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(3*e^3) + (2*b^3*g*i^3*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(3*j^3) + (b^3*d*g*i^2*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(e*j^2) + (2*b^3*d^2*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(e^2*j) + (2*b^2*d^3*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e^3 - (2*b^2*g*i^3*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j^3 - (2*b^3*d^3*g*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/e^3 + (2*b^3*g*i^3*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j^3} *) -{x^1*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]), x, 148, -((6*a*b^2*d*f*n^2*x)/e) + (12*a*b^2*d*g*m*n^2*x)/e + (21*a*b^2*g*i*m*n^2*x)/(4*j) + (6*b^3*d*f*n^3*x)/e - (141*b^3*d*g*m*n^3*x)/(8*e) - (45*b^3*g*i*m*n^3*x)/(8*j) + (3/8)*b^3*g*m*n^3*x^2 - (3*b^3*f*n^3*(d + e*x)^2)/(8*e^2) + (3*b^3*g*m*n^3*(d + e*x)^2)/(8*e^2) + (3*b^3*d^2*g*m*n^3*Log[d + e*x])/(8*e^2) - (6*b^3*d*f*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^2 + (12*b^3*d*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e^2 + (21*b^3*g*i*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/(4*e*j) - (3/8)*b^2*g*m*n^2*x^2*(a + b*Log[c*(d + e*x)^n]) + (3*b^2*f*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) - (3*b^2*g*m*n^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) + (3*b*d*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e^2 - (15*b*d*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) - (9*b*g*i*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*e*j) - (3*b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) + (3*b*g*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) - (d^2*f*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^2) + (d*g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(2*e^2) + (g*i*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/(2*e*j) - (g*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^3)/(4*e^2) + (3*b^3*g*i^2*m*n^3*Log[i + j*x])/(8*j^2) - (3*b^2*g*i^2*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^2) - (9*b^2*d*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e*j) - (9*b*d^2*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(4*e^2) + (3*b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(4*j^2) + (3*b*d*g*i*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e*j) + (d^2*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j^2) - (3/8)*b^3*g*n^3*x^2*Log[h*(i + j*x)^m] + (21*b^3*d*g*n^3*(i + j*x)*Log[h*(i + j*x)^m])/(4*e*j) - (21*b^3*d^2*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(4*e^2) - (9*b^2*d*g*n^2*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(2*e) + (3/4)*b^2*g*n^2*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] + (9*b*d^2*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(4*e^2) + (3*b*d*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(2*e) - (3/4)*b*g*n*x^2*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] - (d^2*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*x)^m])/(2*e^2) + (1/2)*x^2*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) - (3*b^3*g*i^2*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(4*j^2) - (9*b^3*d*g*i*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e*j) - (9*b^2*d^2*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e^2) + (3*b^2*g*i^2*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) + (3*b^2*d*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e*j) + (3*b*d^2*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e^2) - (3*b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (21*b^3*d^2*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(4*e^2) + (9*b^3*d^2*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(2*e^2) - (3*b^3*g*i^2*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) - (3*b^3*d*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/(e*j) - (3*b^2*d^2*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e^2 + (3*b^2*g*i^2*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j^2 + (3*b^3*d^2*g*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/e^2 - (3*b^3*g*i^2*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j^2} -{x^0*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]), x, 64, 6*a*b^2*f*n^2*x - 18*a*b^2*g*m*n^2*x - 6*b^3*f*n^3*x + 24*b^3*g*m*n^3*x + (6*b^3*f*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (18*b^3*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (6*b*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (d*f*(a + b*Log[c*(d + e*x)^n])^3)/e - (g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^3)/e + (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/j + (3*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e - (3*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/j - (6*b^3*g*n^3*(i + j*x)*Log[h*(i + j*x)^m])/j + (6*b^3*d*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/e + 6*b^2*g*n^2*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (3*b*d*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e - 3*b*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) + (6*b^3*g*i*m*n^3*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e - (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (3*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e + (3*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^3*d*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e - (6*b^3*d*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e - (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j - (6*b^3*d*g*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j} -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m])/x^1, x, 0, Unintegrable[((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x, x]} -{(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m])/x^2, x, 0, Unintegrable[((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x^2, x]} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (i+j x)^q Log[f (g+h x)^m] (a+b Log[c (d+e x)^n])^p*) - - -{((a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/(d + e*x), x, 3, -(((a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/e) + (b*n*PolyLog[3, -((g*(d + e*x))/(e*f - d*g))])/e} - - -{Log[c*(d + e*x)]*(a + b*Log[c*(d + e*x)])/(d + e*x)^2, x, 4, -(b/(e*(d + e*x))) - (b*Log[c*(d + e*x)])/(e*(d + e*x)) - (Log[c*(d + e*x)]*(a + b*Log[c*(d + e*x)]))/(e*(d + e*x)) - (a + b + b*Log[c*(d + e*x)])/(e*(d + e*x))} -{(a + b*Log[c*(d + e*x)])*(f + g*Log[c*(d + e*x)])/(d + e*x)^2, x, 4, -((b*g)/(e*(d + e*x))) - (g*(a + b + b*Log[c*(d + e*x)]))/(e*(d + e*x)) - (b*(f + g*Log[c*(d + e*x)]))/(e*(d + e*x)) - ((a + b*Log[c*(d + e*x)])*(f + g*Log[c*(d + e*x)]))/(e*(d + e*x))} - - -(* ::Title::Closed:: *) -(*Integrands of the form AF[x] (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d (e+f x)^m)^n])^p*) - - -{(a + b*Log[c*(d*(e + f*x)^m)^n])^4, x, 7, -24*a*b^3*m^3*n^3*x + 24*b^4*m^4*n^4*x - (24*b^4*m^3*n^3*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f + (12*b^2*m^2*n^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f - (4*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^3)/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^4)/f} -{(a + b*Log[c*(d*(e + f*x)^m)^n])^3, x, 6, 6*a*b^2*m^2*n^2*x - 6*b^3*m^3*n^3*x + (6*b^3*m^2*n^2*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f - (3*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^3)/f} -{(a + b*Log[c*(d*(e + f*x)^m)^n])^2, x, 5, -2*a*b*m*n*x + 2*b^2*m^2*n^2*x - (2*b^2*m*n*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)/f} -{(a + b*Log[c*(d*(e + f*x)^m)^n])^1, x, 4, a*x - b*m*n*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^m)^n])/f} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^1, x, 4, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(b*f*m*n))} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^2, x, 5, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(b^2*f*m^2*n^2)) - (e + f*x)/(b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n]))} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^3, x, 6, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(2*b^3*f*m^3*n^3)) - (e + f*x)/(2*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^2) - (e + f*x)/(2*b^2*f*m^2*n^2*(a + b*Log[c*(d*(e + f*x)^m)^n]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d (e+f x)^m)^n])^(p/2)*) - - -{(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2), x, 8, -((15*b^(5/2)*m^(5/2)*n^(5/2)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(8*f))) + (15*b^2*m^2*n^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])/(4*f) - (5*b*m*n*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2))/(2*f) + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2))/f} -{(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2), x, 7, (3*b^(3/2)*m^(3/2)*n^(3/2)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(4*f)) - (3*b*m*n*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])/(2*f) + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2))/f} -{(a + b*Log[c*(d*(e + f*x)^m)^n])^(1/2), x, 6, -((Sqrt[b]*Sqrt[m]*Sqrt[n]*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(2*f))) + ((e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])/f} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^(1/2), x, 5, (Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(Sqrt[b]*f*Sqrt[m]*Sqrt[n]))} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2), x, 6, (2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(b^(3/2)*f*m^(3/2)*n^(3/2))) - (2*(e + f*x))/(b*f*m*n*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2), x, 7, (4*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(3*b^(5/2)*f*m^(5/2)*n^(5/2))) - (2*(e + f*x))/(3*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2)) - (4*(e + f*x))/(3*b^2*f*m^2*n^2*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])} -{1/(a + b*Log[c*(d*(e + f*x)^m)^n])^(7/2), x, 8, (8*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(15*b^(7/2)*f*m^(7/2)*n^(7/2))) - (2*(e + f*x))/(5*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2)) - (4*(e + f*x))/(15*b^2*f*m^2*n^2*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2)) - (8*(e + f*x))/(15*b^3*f*m^3*n^3*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d (e+f x)^m)^n])^p with p symbolic*) - - -{(a + b*Log[c*(d*(e + f*x)^m)^n])^p, x, 4, ((e + f*x)*Gamma[1 + p, -((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n))]*(a + b*Log[c*(d*(e + f*x)^m)^n])^p)/(E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(-((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)))^p*f)} - - -{(a + b*Log[c*(d*(e + f*x)^(1/2))^(1/2)])^p, x, 4, (Gamma[1 + p, -((4*(a + b*Log[c*Sqrt[d*Sqrt[e + f*x]]]))/b)]*(a + b*Log[c*Sqrt[d*Sqrt[e + f*x]]])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*Sqrt[d*Sqrt[e + f*x]]])/b))^p*(c^4*d^2*f))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x)^q (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^q (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^3, x, 4, -((b*(f*g - e*h)^3*p*q*x)/(4*f^3)) - (b*(f*g - e*h)^2*p*q*(g + h*x)^2)/(8*f^2*h) - (b*(f*g - e*h)*p*q*(g + h*x)^3)/(12*f*h) - (b*p*q*(g + h*x)^4)/(16*h) - (b*(f*g - e*h)^4*p*q*Log[e + f*x])/(4*f^4*h) + ((g + h*x)^4*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(4*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^2, x, 4, -((b*(f*g - e*h)^2*p*q*x)/(3*f^2)) - (b*(f*g - e*h)*p*q*(g + h*x)^2)/(6*f*h) - (b*p*q*(g + h*x)^3)/(9*h) - (b*(f*g - e*h)^3*p*q*Log[e + f*x])/(3*f^3*h) + ((g + h*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^1, x, 4, -((b*(f*g - e*h)*p*q*x)/(2*f)) - (b*p*q*(g + h*x)^2)/(4*h) - (b*(f*g - e*h)^2*p*q*Log[e + f*x])/(2*f^2*h) + ((g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^0, x, 4, a*x - b*p*q*x + (b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^1, x, 4, ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h + (b*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^2, x, 5, (b*f*p*q*Log[e + f*x])/(h*(f*g - e*h)) - (a + b*Log[c*(d*(e + f*x)^p)^q])/(h*(g + h*x)) - (b*f*p*q*Log[g + h*x])/(h*(f*g - e*h))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^3, x, 4, (b*f*p*q)/(2*h*(f*g - e*h)*(g + h*x)) + (b*f^2*p*q*Log[e + f*x])/(2*h*(f*g - e*h)^2) - (a + b*Log[c*(d*(e + f*x)^p)^q])/(2*h*(g + h*x)^2) - (b*f^2*p*q*Log[g + h*x])/(2*h*(f*g - e*h)^2)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^4, x, 4, (b*f*p*q)/(6*h*(f*g - e*h)*(g + h*x)^2) + (b*f^2*p*q)/(3*h*(f*g - e*h)^2*(g + h*x)) + (b*f^3*p*q*Log[e + f*x])/(3*h*(f*g - e*h)^3) - (a + b*Log[c*(d*(e + f*x)^p)^q])/(3*h*(g + h*x)^3) - (b*f^3*p*q*Log[g + h*x])/(3*h*(f*g - e*h)^3)} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^3, x, 9, (2*b^2*(f*g - e*h)^3*p^2*q^2*x)/f^3 + (3*b^2*h*(f*g - e*h)^2*p^2*q^2*(e + f*x)^2)/(4*f^4) + (2*b^2*h^2*(f*g - e*h)*p^2*q^2*(e + f*x)^3)/(9*f^4) + (b^2*h^3*p^2*q^2*(e + f*x)^4)/(32*f^4) + (b^2*(f*g - e*h)^4*p^2*q^2*Log[e + f*x]^2)/(4*f^4*h) - (2*b*(f*g - e*h)^3*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/f^4 - (3*b*h*(f*g - e*h)^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^4) - (2*b*h^2*(f*g - e*h)*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*f^4) - (b*h^3*p*q*(e + f*x)^4*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(8*f^4) - (b*(f*g - e*h)^4*p*q*Log[e + f*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^4*h) + ((g + h*x)^4*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^2, x, 9, (2*b^2*(f*g - e*h)^2*p^2*q^2*x)/f^2 + (b^2*h*(f*g - e*h)*p^2*q^2*(e + f*x)^2)/(2*f^3) + (2*b^2*h^2*p^2*q^2*(e + f*x)^3)/(27*f^3) + (b^2*(f*g - e*h)^3*p^2*q^2*Log[e + f*x]^2)/(3*f^3*h) - (2*b*(f*g - e*h)^2*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/f^3 - (b*h*(f*g - e*h)*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/f^3 - (2*b*h^2*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*f^3) - (2*b*(f*g - e*h)^3*p*q*Log[e + f*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*f^3*h) + ((g + h*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^1, x, 10, -((2*a*b*(f*g - e*h)*p*q*x)/f) + (2*b^2*(f*g - e*h)*p^2*q^2*x)/f + (b^2*h*p^2*q^2*(e + f*x)^2)/(4*f^2) - (2*b^2*(f*g - e*h)*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f^2 - (b*h*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^2) + ((f*g - e*h)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*f^2)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^0, x, 5, -2*a*b*p*q*x + 2*b^2*p^2*q^2*x - (2*b^2*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^1, x, 5, ((a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/h + (2*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (2*b^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^2, x, 5, ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/((f*g - e*h)*(g + h*x)) - (2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)) - (2*b^2*f*p^2*q^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^3, x, 8, -((b*f*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/((f*g - e*h)^2*(g + h*x))) - (a + b*Log[c*(d*(e + f*x)^p)^q])^2/(2*h*(g + h*x)^2) + (b^2*f^2*p^2*q^2*Log[g + h*x])/(h*(f*g - e*h)^2) - (b*f^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[1 + (f*g - e*h)/(h*(e + f*x))])/(h*(f*g - e*h)^2) + (b^2*f^2*p^2*q^2*PolyLog[2, -((f*g - e*h)/(h*(e + f*x)))])/(h*(f*g - e*h)^2)} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^2, x, 16, (6*a*b^2*(f*g - e*h)^2*p^2*q^2*x)/f^2 - (6*b^3*(f*g - e*h)^2*p^3*q^3*x)/f^2 - (3*b^3*h*(f*g - e*h)*p^3*q^3*(e + f*x)^2)/(4*f^3) - (2*b^3*h^2*p^3*q^3*(e + f*x)^3)/(27*f^3) + (6*b^3*(f*g - e*h)^2*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f^3 + (3*b^2*h*(f*g - e*h)*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^3) + (2*b^2*h^2*p^2*q^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*f^3) - (3*b*(f*g - e*h)^2*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f^3 - (3*b*h*(f*g - e*h)*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*f^3) - (b*h^2*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*f^3) + ((f*g - e*h)^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(3*f^3)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^1, x, 12, (6*a*b^2*(f*g - e*h)*p^2*q^2*x)/f - (6*b^3*(f*g - e*h)*p^3*q^3*x)/f - (3*b^3*h*p^3*q^3*(e + f*x)^2)/(8*f^2) + (6*b^3*(f*g - e*h)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f^2 + (3*b^2*h*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(4*f^2) - (3*b*(f*g - e*h)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f^2 - (3*b*h*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*f^2) + ((f*g - e*h)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(2*f^2)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^0, x, 6, 6*a*b^2*p^2*q^2*x - 6*b^3*p^3*q^3*x + (6*b^3*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f - (3*b*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x)^1, x, 6, ((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h + (3*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (6*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h + (6*b^3*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x)^2, x, 6, ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/((f*g - e*h)*(g + h*x)) - (3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)) - (6*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h)) + (6*b^3*f*p^3*q^3*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x)^3, x, 10, -((3*b*f*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*(f*g - e*h)^2*(g + h*x))) - (a + b*Log[c*(d*(e + f*x)^p)^q])^3/(2*h*(g + h*x)^2) + (3*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)^2) - (3*b*f^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[1 + (f*g - e*h)/(h*(e + f*x))])/(2*h*(f*g - e*h)^2) + (3*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((f*g - e*h)/(h*(e + f*x)))])/(h*(f*g - e*h)^2) + (3*b^3*f^2*p^3*q^3*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h)^2) + (3*b^3*f^2*p^3*q^3*PolyLog[3, -((f*g - e*h)/(h*(e + f*x)))])/(h*(f*g - e*h)^2)} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])^4*(g + h*x)^0, x, 7, -24*a*b^3*p^3*q^3*x + 24*b^4*p^4*q^4*x - (24*b^4*p^3*q^3*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f + (12*b^2*p^2*q^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/f - (4*b*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^4)/f} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^1, x, 7, ((a + b*Log[c*(d*(e + f*x)^p)^q])^4*Log[(f*(g + h*x))/(f*g - e*h)])/h + (4*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (12*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h + (24*b^3*p^3*q^3*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h - (24*b^4*p^4*q^4*PolyLog[5, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^4/(g + h*x)^2, x, 7, ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^4)/((f*g - e*h)*(g + h*x)) - (4*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)) - (12*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h)) + (24*b^3*f*p^3*q^3*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h)) - (24*b^4*f*p^4*q^4*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/(h*(f*g - e*h))} - - -{Log[c*(d*(e + f*x)^p)^q], x, 3, (-p)*q*x + ((e + f*x)*Log[c*(d*(e + f*x)^p)^q])/f} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^2, x, 12, ((f*g - e*h)^2*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b*f^3*p*q)) + (2*h*(f*g - e*h)*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b*f^3*p*q)) + (h^2*(e + f*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(b*f^3*p*q))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^1, x, 9, ((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b*f^2*p*q)) + (h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b*f^2*p*q))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])*(g + h*x)^0, x, 4, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b*f*p*q))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^1, x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^2, x, 0, Unintegrable[1/((g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} - - -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^2, x, 21, ((f*g - e*h)^2*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b^2*f^3*p^2*q^2)) + (4*h*(f*g - e*h)*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b^2*f^3*p^2*q^2)) + (3*h^2*(e + f*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(b^2*f^3*p^2*q^2)) - ((e + f*x)*(g + h*x)^2)/(b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^1, x, 13, ((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b^2*f^2*p^2*q^2)) + (2*h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b^2*f^2*p^2*q^2)) - ((e + f*x)*(g + h*x))/(b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^0, x, 5, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(b^2*f*p^2*q^2)) - (e + f*x)/(b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^1, x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^2, x, 0, Unintegrable[1/((g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} - - -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^2, x, 34, ((f*g - e*h)^2*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(2*b^3*f^3*p^3*q^3)) + (4*h*(f*g - e*h)*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b^3*f^3*p^3*q^3)) + (9*h^2*(e + f*x)^3*ExpIntegralEi[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(2*b^3*f^3*p^3*q^3)) - ((e + f*x)*(g + h*x)^2)/(2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) + ((f*g - e*h)*(e + f*x)*(g + h*x))/(b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])) - (3*(e + f*x)*(g + h*x)^2)/(2*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^1, x, 18, ((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(2*b^3*f^2*p^3*q^3)) + (2*h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(b^3*f^2*p^3*q^3)) - ((e + f*x)*(g + h*x))/(2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) + ((f*g - e*h)*(e + f*x))/(2*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])) - ((e + f*x)*(g + h*x))/(b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^0, x, 6, ((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(2*b^3*f*p^3*q^3)) - (e + f*x)/(2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) - (e + f*x)/(2*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x)^1, x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3), x]} -{1/(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x)^2, x, 0, Unintegrable[1/((g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^q (a+b Log[c (d (e+f x)^m)^n])^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(g + h*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 18, -(Sqrt[b]*(f*g - e*h)^2*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (Sqrt[b]*h*(f*g - e*h)*Sqrt[p]*Sqrt[Pi/2]*Sqrt[q]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*E^((2*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (Sqrt[b]*h^2*Sqrt[p]*Sqrt[Pi/3]*Sqrt[q]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(6*E^((3*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(3/(p*q))) + ((f*g - e*h)^2*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h*(f*g - e*h)*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^3 + (h^2*(e + f*x)^3*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(3*f^3)} -{(g + h*x)^1*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 13, -(Sqrt[b]*(f*g - e*h)*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (Sqrt[b]*h*Sqrt[p]*Sqrt[Pi/2]*Sqrt[q]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(4*E^((2*a)/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + ((f*g - e*h)*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f^2 + (h*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f^2)} -{(g + h*x)^0*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 6, -(Sqrt[b]*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(2*E^(a/(b*p*q))*f*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + ((e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/f} -{Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x)^1, x, 0, Unintegrable[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x), x]} -{Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x)^2, x, 0, Unintegrable[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x)^2, x]} - - -{(g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 21, (3*b^(3/2)*(f*g - e*h)^2*p^(3/2)*Sqrt[Pi]*q^(3/2)*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (3*b^(3/2)*h*(f*g - e*h)*p^(3/2)*Sqrt[Pi/2]*q^(3/2)*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(8*E^((2*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (b^(3/2)*h^2*p^(3/2)*Sqrt[Pi/3]*q^(3/2)*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(12*E^((3*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(3/(p*q))) - (3*b*(f*g - e*h)^2*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f^3) - (3*b*h*(f*g - e*h)*p*q*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(4*f^3) - (b*h^2*p*q*(e + f*x)^3*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(6*f^3) + ((f*g - e*h)^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^3 + (h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^3 + (h^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/(3*f^3)} -{(g + h*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 15, (3*b^(3/2)*(f*g - e*h)*p^(3/2)*Sqrt[Pi]*q^(3/2)*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (3*b^(3/2)*h*p^(3/2)*Sqrt[Pi/2]*q^(3/2)*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(16*E^((2*a)/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (3*b*(f*g - e*h)*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f^2) - (3*b*h*p*q*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(8*f^2) + ((f*g - e*h)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/(2*f^2)} -{(g + h*x)^0*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 7, (3*b^(3/2)*p^(3/2)*Sqrt[Pi]*q^(3/2)*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (3*b*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f) + ((e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)/(g + h*x)^1, x, 0, Unintegrable[(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)/(g + h*x), x]} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)/(g + h*x)^2, x, 0, Unintegrable[(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)/(g + h*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(g + h*x)^2/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 15, ((f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (h^2*Sqrt[Pi/3]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((3*a)/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(3/(p*q)))} -{(g + h*x)^1/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 11, ((f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*Sqrt[Pi/2]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(2/(p*q)))} -{(g + h*x)^0/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 5, (Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a/(b*p*q))*f*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q)))} -{1/((g + h*x)^1*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x, 0, Unintegrable[1/((g + h*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x]} - - -{(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 26, (2*(f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^(a/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (4*h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((2*a)/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (2*h^2*Sqrt[3*Pi]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((3*a)/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(3/(p*q))) - (2*(e + f*x)*(g + h*x)^2)/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{(g + h*x)^1/(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 16, (2*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^(a/(b*p*q))*f^2*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^((2*a)/(b*p*q))*f^2*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*(e + f*x)*(g + h*x))/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{(g + h*x)^0/(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 6, (2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^(a/(b*p*q))*f*p^(3/2)*q^(3/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (2*(e + f*x))/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{1/((g + h*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)), x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)), x]} - - -{(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2), x, 42, (4*(f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^(a/(b*p*q))*f^3*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (16*h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^((2*a)/(b*p*q))*f^3*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (4*h^2*Sqrt[3*Pi]*(e + f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(5/2)*E^((3*a)/(b*p*q))*f^3*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(3/(p*q))) - (2*(e + f*x)*(g + h*x)^2)/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)) + (8*(f*g - e*h)*(e + f*x)*(g + h*x))/(3*b^2*f^2*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]) - (4*(e + f*x)*(g + h*x)^2)/(b^2*f*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{(g + h*x)^1/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2), x, 22, (4*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^(a/(b*p*q))*f^2*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (8*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^((2*a)/(b*p*q))*f^2*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2*(e + f*x)*(g + h*x))/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)) + (4*(f*g - e*h)*(e + f*x))/(3*b^2*f^2*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]) - (8*(e + f*x)*(g + h*x))/(3*b^2*f*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{(g + h*x)^0/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2), x, 7, (4*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(3*b^(5/2)*E^(a/(b*p*q))*f*p^(5/2)*q^(5/2)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - (2*(e + f*x))/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2)) - (4*(e + f*x))/(3*b^2*f*p^2*q^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])} -{1/((g + h*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2)), x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^(q/2) (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]), x, 7, (-4*b*(f*g - e*h)^2*p*q*Sqrt[g + h*x])/(5*f^2*h) - (4*b*(f*g - e*h)*p*q*(g + h*x)^(3/2))/(15*f*h) - (4*b*p*q*(g + h*x)^(5/2))/(25*h) + (4*b*(f*g - e*h)^(5/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(5*f^(5/2)*h) + (2*(g + h*x)^(5/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*h)} -{(g + h*x)^(1/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]), x, 6, (-4*b*(f*g - e*h)*p*q*Sqrt[g + h*x])/(3*f*h) - (4*b*p*q*(g + h*x)^(3/2))/(9*h) + (4*b*(f*g - e*h)^(3/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(3*f^(3/2)*h) + (2*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(1/2), x, 5, (-4*b*p*q*Sqrt[g + h*x])/h + (4*b*Sqrt[f*g - e*h]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(Sqrt[f]*h) + (2*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(3/2), x, 4, (-4*b*Sqrt[f]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(h*Sqrt[f*g - e*h]) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[g + h*x])} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(5/2), x, 5, (4*b*f*p*q)/(3*h*(f*g - e*h)*Sqrt[g + h*x]) - (4*b*f^(3/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(3*h*(f*g - e*h)^(3/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h*(g + h*x)^(3/2))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(7/2), x, 6, (4*b*f*p*q)/(15*h*(f*g - e*h)*(g + h*x)^(3/2)) + (4*b*f^2*p*q)/(5*h*(f*g - e*h)^2*Sqrt[g + h*x]) - (4*b*f^(5/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(5*h*(f*g - e*h)^(5/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*h*(g + h*x)^(5/2))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(9/2), x, 7, (4*b*f*p*q)/(35*h*(f*g - e*h)*(g + h*x)^(5/2)) + (4*b*f^2*p*q)/(21*h*(f*g - e*h)^2*(g + h*x)^(3/2)) + (4*b*f^3*p*q)/(7*h*(f*g - e*h)^3*Sqrt[g + h*x]) - (4*b*f^(7/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(7*h*(f*g - e*h)^(7/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(7*h*(g + h*x)^(7/2))} - - -{(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x, 29, (368*b^2*(f*g - e*h)^2*p^2*q^2*Sqrt[g + h*x])/(75*f^2*h) + (128*b^2*(f*g - e*h)*p^2*q^2*(g + h*x)^(3/2))/(225*f*h) + (16*b^2*p^2*q^2*(g + h*x)^(5/2))/(125*h) - (368*b^2*(f*g - e*h)^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(75*f^(5/2)*h) - (8*b^2*(f*g - e*h)^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(5*f^(5/2)*h) - (8*b*(f*g - e*h)^2*p*q*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*f^2*h) - (8*b*(f*g - e*h)*p*q*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(15*f*h) - (8*b*p*q*(g + h*x)^(5/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(25*h) + (8*b*(f*g - e*h)^(5/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*f^(5/2)*h) + (2*(g + h*x)^(5/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(5*h) + (16*b^2*(f*g - e*h)^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(5*f^(5/2)*h) + (8*b^2*(f*g - e*h)^(5/2)*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(5*f^(5/2)*h)} -{(g + h*x)^(1/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x, 22, (64*b^2*(f*g - e*h)*p^2*q^2*Sqrt[g + h*x])/(9*f*h) + (16*b^2*p^2*q^2*(g + h*x)^(3/2))/(27*h) - (64*b^2*(f*g - e*h)^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(9*f^(3/2)*h) - (8*b^2*(f*g - e*h)^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(3*f^(3/2)*h) - (8*b*(f*g - e*h)*p*q*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*f*h) - (8*b*p*q*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*h) + (8*b*(f*g - e*h)^(3/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*f^(3/2)*h) + (2*(g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*h) + (16*b^2*(f*g - e*h)^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(3*f^(3/2)*h) + (8*b^2*(f*g - e*h)^(3/2)*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(3*f^(3/2)*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(1/2), x, 16, (16*b^2*p^2*q^2*Sqrt[g + h*x])/h - (16*b^2*Sqrt[f*g - e*h]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(Sqrt[f]*h) - (8*b^2*Sqrt[f*g - e*h]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(Sqrt[f]*h) - (8*b*p*q*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h + (8*b*Sqrt[f*g - e*h]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(Sqrt[f]*h) + (2*Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/h + (16*b^2*Sqrt[f*g - e*h]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(Sqrt[f]*h) + (8*b^2*Sqrt[f*g - e*h]*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(Sqrt[f]*h)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(3/2), x, 11, (8*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(h*Sqrt[f*g - e*h]) - (8*b*Sqrt[f]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[f*g - e*h]) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(h*Sqrt[g + h*x]) - (16*b^2*Sqrt[f]*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h]) - (8*b^2*Sqrt[f]*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(h*Sqrt[f*g - e*h])} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(5/2), x, 15, (16*b^2*f^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(3*h*(f*g - e*h)^(3/2)) + (8*b^2*f^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(3*h*(f*g - e*h)^(3/2)) + (8*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h*(f*g - e*h)*Sqrt[g + h*x]) - (8*b*f^(3/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h*(f*g - e*h)^(3/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*h*(g + h*x)^(3/2)) - (16*b^2*f^(3/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(3*h*(f*g - e*h)^(3/2)) - (8*b^2*f^(3/2)*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(3*h*(f*g - e*h)^(3/2))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(7/2), x, 20, -((16*b^2*f^2*p^2*q^2)/(15*h*(f*g - e*h)^2*Sqrt[g + h*x])) + (64*b^2*f^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(15*h*(f*g - e*h)^(5/2)) + (8*b^2*f^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(5*h*(f*g - e*h)^(5/2)) + (8*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(15*h*(f*g - e*h)*(g + h*x)^(3/2)) + (8*b*f^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*h*(f*g - e*h)^2*Sqrt[g + h*x]) - (8*b*f^(5/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(5*h*(f*g - e*h)^(5/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(5*h*(g + h*x)^(5/2)) - (16*b^2*f^(5/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(5*h*(f*g - e*h)^(5/2)) - (8*b^2*f^(5/2)*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(5*h*(f*g - e*h)^(5/2))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x)^(9/2), x, 26, -((16*b^2*f^2*p^2*q^2)/(105*h*(f*g - e*h)^2*(g + h*x)^(3/2))) - (128*b^2*f^3*p^2*q^2)/(105*h*(f*g - e*h)^3*Sqrt[g + h*x]) + (368*b^2*f^(7/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(105*h*(f*g - e*h)^(7/2)) + (8*b^2*f^(7/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]^2)/(7*h*(f*g - e*h)^(7/2)) + (8*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(35*h*(f*g - e*h)*(g + h*x)^(5/2)) + (8*b*f^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(21*h*(f*g - e*h)^2*(g + h*x)^(3/2)) + (8*b*f^3*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(7*h*(f*g - e*h)^3*Sqrt[g + h*x]) - (8*b*f^(7/2)*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(7*h*(f*g - e*h)^(7/2)) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(7*h*(g + h*x)^(7/2)) - (16*b^2*f^(7/2)*p^2*q^2*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]]*Log[2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(7*h*(f*g - e*h)^(7/2)) - (8*b^2*f^(7/2)*p^2*q^2*PolyLog[2, 1 - 2/(1 - (Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h])])/(7*h*(f*g - e*h)^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(g + h*x)^(3/2)/(a + b*Log[c*(d*(e + f*x)^p)^q]), x, 0, Unintegrable[(g + h*x)^(3/2)/(a + b*Log[c*(d*(e + f*x)^p)^q]), x]} -{(g + h*x)^(1/2)/(a + b*Log[c*(d*(e + f*x)^p)^q]), x, 0, Unintegrable[Sqrt[g + h*x]/(a + b*Log[c*(d*(e + f*x)^p)^q]), x]} -{1/((g + h*x)^(1/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[1/(Sqrt[g + h*x]*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} -{1/((g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[1/((g + h*x)^(3/2)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^(q/2) (a+b Log[c (d (e+f x)^m)^n])^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[g + h*x]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 0, Unintegrable[Sqrt[g + h*x]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x]} -{Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/Sqrt[g + h*x], x, 0, Unintegrable[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/Sqrt[g + h*x], x]} -{Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x)^(3/2), x, 0, Unintegrable[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(g + h*x)^(3/2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sqrt[g + h*x]/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x, 0, Unintegrable[Sqrt[g + h*x]/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x]} -{1/(Sqrt[g + h*x]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x, 0, Unintegrable[1/(Sqrt[g + h*x]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x]} -{1/((g + h*x)^(3/2)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x, 0, Unintegrable[1/((g + h*x)^(3/2)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^q (a+b Log[c (d (e+f x)^m)^n])^p when q symbolic*) - - -(* {(a + b*Log[c*(d*(e + f*x)^p)^q])^3*(g + h*x)^m, x, 0, (3*a^2*b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + (6*a*b^2*p^2*q^2*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, -((f*g - e*h)/(h*(e + f*x)))])/(h*(1 + m)^3) - (6*b^3*p^3*q^3*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m, -1 - m}, {-m, -m, -m}, -((f*g - e*h)/(h*(e + f*x)))])/(h*(1 + m)^4) - (6*a*b^2*p*q*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, -((f*g - e*h)/(h*(e + f*x)))]*Log[c*(d*(e + f*x)^p)^q])/(h*(1 + m)^2) + (6*b^3*p^2*q^2*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, -((f*g - e*h)/(h*(e + f*x)))]*Log[c*(d*(e + f*x)^p)^q])/(h*(1 + m)^3) - (3*b^3*p*q*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, -((f*g - e*h)/(h*(e + f*x)))]*Log[c*(d*(e + f*x)^p)^q]^2)/(h*(1 + m)^2) + ((g + h*x)^(1 + m)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(h*(1 + m))} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2*(g + h*x)^m, x, 0, (a^2*(g + h*x)^(1 + m))/(h*(1 + m)) + (2*a*b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + (2*b^2*p^2*q^2*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*HypergeometricPFQ[{-1 - m, -1 - m, -1 - m}, {-m, -m}, -((f*g - e*h)/(h*(e + f*x)))])/(h*(1 + m)^3) + (2*a*b*(g + h*x)^(1 + m)*Log[c*(d*(e + f*x)^p)^q])/(h*(1 + m)) - (2*b^2*p*q*(g + h*x)^(1 + m)*((f*(g + h*x))/(h*(e + f*x)))^(-1 - m)*Hypergeometric2F1[-1 - m, -1 - m, -m, -((f*g - e*h)/(h*(e + f*x)))]*Log[c*(d*(e + f*x)^p)^q])/(h*(1 + m)^2) + (b^2*(g + h*x)^(1 + m)*Log[c*(d*(e + f*x)^p)^q]^2)/(h*(1 + m))} *) -{(a + b*Log[c*(d*(e + f*x)^p)^q])^1*(g + h*x)^m, x, 3, (b*f*p*q*(g + h*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (f*(g + h*x))/(f*g - e*h)])/(h*(f*g - e*h)*(1 + m)*(2 + m)) + ((g + h*x)^(1 + m)*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*(1 + m))} -{(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^1, x, 0, Unintegrable[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q]), x]} -{(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x, 0, Unintegrable[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^2, x]} - - -{(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 0, Unintegrable[(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x]} -{(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^(1/2), x, 0, Unintegrable[(g + h*x)^m*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x]} -{(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^(1/2), x, 0, Unintegrable[(g + h*x)^m/Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]], x]} -{(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x, 0, Unintegrable[(g + h*x)^m/(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x)^q (a+b Log[c (d (e+f x)^m)^n])^p when p symbolic*) - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])^n*(g + h*x)^m, x, 0, Unintegrable[(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^n, x]} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])^n*(g + h*x)^2, x, 12, (3^(-1 - n)*h^2*(e + f*x)^3*Gamma[1 + n, -((3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f^3) + (h*(f*g - e*h)*(e + f*x)^2*Gamma[1 + n, -((2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(2^n*E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f^3) + ((f*g - e*h)^2*(e + f*x)*Gamma[1 + n, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f^3)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^n*(g + h*x)^1, x, 9, (2^(-1 - n)*h*(e + f*x)^2*Gamma[1 + n, -((2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f^2) + ((f*g - e*h)*(e + f*x)*Gamma[1 + n, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f^2)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^n*(g + h*x)^0, x, 4, ((e + f*x)*Gamma[1 + n, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*(a + b*Log[c*(d*(e + f*x)^p)^q])^n)/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*(-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)))^n*f)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^n/(g + h*x)^1, x, 0, Unintegrable[(a + b*Log[c*(d*(e + f*x)^p)^q])^n/(g + h*x), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g+h x^r)^q (a+b Log[c (d (e+f x)^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g+h x^2)^q (a+b Log[c (d (e+f x)^m)^n])*) - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x^2), x, 9, ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] - Sqrt[h]*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h]) - ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(Sqrt[-g] + Sqrt[h]*x))/(f*Sqrt[-g] - e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h]) - (b*p*q*PolyLog[2, -((Sqrt[h]*(e + f*x))/(f*Sqrt[-g] - e*Sqrt[h]))])/(2*Sqrt[-g]*Sqrt[h]) + (b*p*q*PolyLog[2, (Sqrt[h]*(e + f*x))/(f*Sqrt[-g] + e*Sqrt[h])])/(2*Sqrt[-g]*Sqrt[h])} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[2 + h*x^2], x, 11, (b*p*q*ArcSinh[(Sqrt[h]*x)/Sqrt[2]]^2)/(2*Sqrt[h]) - (b*p*q*ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*E^ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*f)/(e*Sqrt[h] - Sqrt[2*f^2 + e^2*h])])/Sqrt[h] - (b*p*q*ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*Log[1 + (Sqrt[2]*E^ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*f)/(e*Sqrt[h] + Sqrt[2*f^2 + e^2*h])])/Sqrt[h] + (ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/Sqrt[h] - (b*p*q*PolyLog[2, -((Sqrt[2]*E^ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*f)/(e*Sqrt[h] - Sqrt[2*f^2 + e^2*h]))])/Sqrt[h] - (b*p*q*PolyLog[2, -((Sqrt[2]*E^ArcSinh[(Sqrt[h]*x)/Sqrt[2]]*f)/(e*Sqrt[h] + Sqrt[2*f^2 + e^2*h]))])/Sqrt[h]} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/Sqrt[g + h*x^2], x, 12, (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]^2)/(2*Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] - Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*Log[1 + (E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h])])/(Sqrt[h]*Sqrt[g + h*x^2]) + (Sqrt[g]*Sqrt[1 + (h*x^2)/g]*ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*PolyLog[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] - Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x^2]) - (b*Sqrt[g]*p*q*Sqrt[1 + (h*x^2)/g]*PolyLog[2, -((E^ArcSinh[(Sqrt[h]*x)/Sqrt[g]]*f*Sqrt[g])/(e*Sqrt[h] + Sqrt[f^2*g + e^2*h]))])/(Sqrt[h]*Sqrt[g + h*x^2])} - - -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[2 + h*x]*Sqrt[2 - h*x]), x, 10, (I*b*p*q*ArcSin[(h*x)/2]^2)/(2*h) - (b*p*q*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2])])/h - (b*p*q*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + Sqrt[4*f^2 - e^2*h^2])])/h + (ArcSin[(h*x)/2]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h + (I*b*p*q*PolyLog[2, -((2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2]))])/h + (I*b*p*q*PolyLog[2, -((2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + Sqrt[4*f^2 - e^2*h^2]))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[g + h*x]*Sqrt[g - h*x]), x, 12, (I*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x)/g]^2)/(2*h*Sqrt[g - h*x]*Sqrt[g + h*x]) - (b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x)/g]*Log[1 + (E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h - Sqrt[f^2*g^2 - e^2*h^2])])/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) - (b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x)/g]*Log[1 + (E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h + Sqrt[f^2*g^2 - e^2*h^2])])/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) + (g*Sqrt[1 - (h^2*x^2)/g^2]*ArcSin[(h*x)/g]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) + (I*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*PolyLog[2, -((E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h - Sqrt[f^2*g^2 - e^2*h^2]))])/(h*Sqrt[g - h*x]*Sqrt[g + h*x]) + (I*b*g*p*q*Sqrt[1 - (h^2*x^2)/g^2]*PolyLog[2, -((E^(I*ArcSin[(h*x)/g])*f*g)/(I*e*h + Sqrt[f^2*g^2 - e^2*h^2]))])/(h*Sqrt[g - h*x]*Sqrt[g + h*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (i+j x)^q (a+b Log[c (d (e+f x)^m)^n])^p / (g+h x)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (i+j x)^q (a+b Log[c (d (e+f x)^m)^n])^p / (g+h x)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(i + j*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x), x, 15, (a*j*(h*i - g*j)^2*x)/h^3 - (b*j*(f*i - e*j)^2*p*q*x)/(3*f^2*h) - (b*j*(f*i - e*j)*(h*i - g*j)*p*q*x)/(2*f*h^2) - (b*j*(h*i - g*j)^2*p*q*x)/h^3 - (b*(f*i - e*j)*p*q*(i + j*x)^2)/(6*f*h) - (b*(h*i - g*j)*p*q*(i + j*x)^2)/(4*h^2) - (b*p*q*(i + j*x)^3)/(9*h) - (b*(f*i - e*j)^3*p*q*Log[e + f*x])/(3*f^3*h) - (b*(f*i - e*j)^2*(h*i - g*j)*p*q*Log[e + f*x])/(2*f^2*h^2) + (b*j*(h*i - g*j)^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^3) + ((h*i - g*j)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*h^2) + ((i + j*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(3*h) + ((h*i - g*j)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h^4 + (b*(h*i - g*j)^3*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^4} -{(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x), x, 12, (a*j*(h*i - g*j)*x)/h^2 - (b*j*(f*i - e*j)*p*q*x)/(2*f*h) - (b*j*(h*i - g*j)*p*q*x)/h^2 - (b*p*q*(i + j*x)^2)/(4*h) - (b*(f*i - e*j)^2*p*q*Log[e + f*x])/(2*f^2*h) + (b*j*(h*i - g*j)*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2) + ((i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h^3 + (b*(h*i - g*j)^2*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^3} -{(i + j*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x), x, 9, (a*j*x)/h - (b*j*p*q*x)/h + (b*j*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h) + ((h*i - g*j)*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h^2 + (b*(h*i - g*j)*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^2} -{(i + j*x)^0*(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x), x, 4, ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/h + (b*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/((g + h*x)*(i + j*x)^1), x, 9, ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j) - ((a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j) + (b*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) - (b*p*q*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/((g + h*x)*(i + j*x)^2), x, 13, -((b*f*p*q*Log[e + f*x])/((f*i - e*j)*(h*i - g*j))) + (a + b*Log[c*(d*(e + f*x)^p)^q])/((h*i - g*j)*(i + j*x)) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j)^2 + (b*f*p*q*Log[i + j*x])/((f*i - e*j)*(h*i - g*j)) - (h*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^2 + (b*h*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 - (b*h*p*q*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2} -{(a + b*Log[c*(d*(e + f*x)^p)^q])/((g + h*x)*(i + j*x)^3), x, 16, -((b*f*p*q)/(2*(f*i - e*j)*(h*i - g*j)*(i + j*x))) - (b*f*h*p*q*Log[e + f*x])/((f*i - e*j)*(h*i - g*j)^2) - (b*f^2*p*q*Log[e + f*x])/(2*(f*i - e*j)^2*(h*i - g*j)) + (a + b*Log[c*(d*(e + f*x)^p)^q])/(2*(h*i - g*j)*(i + j*x)^2) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q]))/((h*i - g*j)^2*(i + j*x)) + (h^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j)^3 + (b*f*h*p*q*Log[i + j*x])/((f*i - e*j)*(h*i - g*j)^2) + (b*f^2*p*q*Log[i + j*x])/(2*(f*i - e*j)^2*(h*i - g*j)) - (h^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^3 + (b*h^2*p*q*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^3 - (b*h^2*p*q*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^3} - - -{(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x), x, 20, -((2*a*b*j*(f*i - e*j)*p*q*x)/(f*h)) - (2*a*b*j*(h*i - g*j)*p*q*x)/h^2 + (2*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) + (2*b^2*j*(h*i - g*j)*p^2*q^2*x)/h^2 + (b^2*j^2*p^2*q^2*(e + f*x)^2)/(4*f^2*h) - (2*b^2*j*(f*i - e*j)*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) - (2*b^2*j*(h*i - g*j)*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2) - (b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^2*h) + (j*(f*i - e*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) + (j^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(2*f^2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/h^3 + (2*b*(h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^3 - (2*b^2*(h*i - g*j)^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3} -{(i + j*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x), x, 11, -((2*a*b*j*p*q*x)/h) + (2*b^2*j*p^2*q^2*x)/h - (2*b^2*j*p*q*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h) + (j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h) + ((h*i - g*j)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/h^2 + (2*b*(h*i - g*j)*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^2 - (2*b^2*(h*i - g*j)*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^2} -{(i + j*x)^0*(a + b*Log[c*(d*(e + f*x)^p)^q])^2/(g + h*x), x, 5, ((a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/h + (2*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (2*b^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/((g + h*x)*(i + j*x)^1), x, 11, ((a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j) - ((a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j) + (2*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) - (2*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j) - (2*b^2*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) + (2*b^2*p^2*q^2*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^2/((g + h*x)*(i + j*x)^2), x, 15, -((j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/((f*i - e*j)*(h*i - g*j)*(i + j*x))) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j)^2 + (2*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*Log[(f*(i + j*x))/(f*i - e*j)])/((f*i - e*j)*(h*i - g*j)) - (h*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^2 + (2*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 + (2*b^2*f*p^2*q^2*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/((f*i - e*j)*(h*i - g*j)) - (2*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2 - (2*b^2*h*p^2*q^2*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 + (2*b^2*h*p^2*q^2*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2} - - -{(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x), x, 24, (6*a*b^2*j*(f*i - e*j)*p^2*q^2*x)/(f*h) + (6*a*b^2*j*(h*i - g*j)*p^2*q^2*x)/h^2 - (6*b^3*j*(f*i - e*j)*p^3*q^3*x)/(f*h) - (6*b^3*j*(h*i - g*j)*p^3*q^3*x)/h^2 - (3*b^3*j^2*p^3*q^3*(e + f*x)^2)/(8*f^2*h) + (6*b^3*j*(f*i - e*j)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f^2*h) + (6*b^3*j*(h*i - g*j)*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h^2) + (3*b^2*j^2*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(4*f^2*h) - (3*b*j*(f*i - e*j)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f^2*h) - (3*b*j*(h*i - g*j)*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h^2) - (3*b*j^2*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*f^2*h) + (j*(f*i - e*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f^2*h) + (j*(h*i - g*j)*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f*h^2) + (j^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(2*f^2*h) + ((h*i - g*j)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^3 + (3*b*(h*i - g*j)^2*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^3 - (6*b^2*(h*i - g*j)^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^3 + (6*b^3*(h*i - g*j)^2*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h^3} -{(i + j*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x), x, 13, (6*a*b^2*j*p^2*q^2*x)/h - (6*b^3*j*p^3*q^3*x)/h + (6*b^3*j*p^2*q^2*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])/(f*h) - (3*b*j*p*q*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(f*h) + (j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(f*h) + ((h*i - g*j)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h^2 + (3*b*(h*i - g*j)*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h^2 - (6*b^2*(h*i - g*j)*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h^2 + (6*b^3*(h*i - g*j)*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h^2} -{(i + j*x)^0*(a + b*Log[c*(d*(e + f*x)^p)^q])^3/(g + h*x), x, 6, ((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/h + (3*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/h - (6*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/h + (6*b^3*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/h} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3/((g + h*x)*(i + j*x)^1), x, 13, ((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j) - ((a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j) + (3*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) - (3*b*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j) - (6*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) + (6*b^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j) + (6*b^3*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j) - (6*b^3*p^3*q^3*PolyLog[4, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)} -{(a + b*Log[c*(d*(e + f*x)^p)^q])^3/((g + h*x)*(i + j*x)^2), x, 18, -((j*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/((f*i - e*j)*(h*i - g*j)*(i + j*x))) + (h*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(g + h*x))/(f*g - e*h)])/(h*i - g*j)^2 + (3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*Log[(f*(i + j*x))/(f*i - e*j)])/((f*i - e*j)*(h*i - g*j)) - (h*(a + b*Log[c*(d*(e + f*x)^p)^q])^3*Log[(f*(i + j*x))/(f*i - e*j)])/(h*i - g*j)^2 + (3*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 + (6*b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/((f*i - e*j)*(h*i - g*j)) - (3*b*h*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2*PolyLog[2, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2 - (6*b^2*h*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 - (6*b^3*f*p^3*q^3*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/((f*i - e*j)*(h*i - g*j)) + (6*b^2*h*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q])*PolyLog[3, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2 + (6*b^3*h*p^3*q^3*PolyLog[4, -((h*(e + f*x))/(f*g - e*h))])/(h*i - g*j)^2 - (6*b^3*h*p^3*q^3*PolyLog[4, -((j*(e + f*x))/(f*i - e*j))])/(h*i - g*j)^2} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(i + j*x)^1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[(i + j*x)/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} -{(i + j*x)^0/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} -{1/((g + h*x)*(i + j*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[1/((g + h*x)*(i + j*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} -{1/((g + h*x)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])), x, 0, Unintegrable[1/((g + h*x)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])), x]} - - -{(i + j*x)^1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x, 0, Unintegrable[(i + j*x)/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} -{(i + j*x)^0/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x, 0, Unintegrable[1/((g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} -{1/((g + h*x)*(i + j*x)^1*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x, 0, Unintegrable[1/((g + h*x)*(i + j*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} -{1/((g + h*x)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x, 0, Unintegrable[1/((g + h*x)*(i + j*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2), x]} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m b/test/methods/rule_based/test_files/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m deleted file mode 100644 index 0b1c96e..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.4 u (a+b log(c (d+e x^m)^n))^p.m +++ /dev/null @@ -1,1225 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form u Log[c (d+e x^m)^n]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^m)^n]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[c (d+e x^m)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Log[c*(a + b*x^2)^p], x, 4, -((2*a^2*p*x)/(5*b^2)) + (2*a*p*x^3)/(15*b) - (2*p*x^5)/25 + (2*a^(5/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(5*b^(5/2)) + (1/5)*x^5*Log[c*(a + b*x^2)^p]} -{x^3*Log[c*(a + b*x^2)^p], x, 4, (a*p*x^2)/(4*b) - (p*x^4)/8 - (a^2*p*Log[a + b*x^2])/(4*b^2) + (1/4)*x^4*Log[c*(a + b*x^2)^p]} -{x^2*Log[c*(a + b*x^2)^p], x, 4, (2*a*p*x)/(3*b) - (2*p*x^3)/9 - (2*a^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(3/2)) + (1/3)*x^3*Log[c*(a + b*x^2)^p]} -{x^1*Log[c*(a + b*x^2)^p], x, 3, -((p*x^2)/2) + ((a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b)} -{x^0*Log[c*(a + b*x^2)^p], x, 3, -2*p*x + (2*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + x*Log[c*(a + b*x^2)^p]} -{Log[c*(a + b*x^2)^p]/x^1, x, 3, (1/2)*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p] + (1/2)*p*PolyLog[2, 1 + (b*x^2)/a]} -{Log[c*(a + b*x^2)^p]/x^2, x, 2, (2*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] - Log[c*(a + b*x^2)^p]/x} -{Log[c*(a + b*x^2)^p]/x^3, x, 5, (b*p*Log[x])/a - ((a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*a*x^2), (b*p*Log[x])/a - (b*p*Log[a + b*x^2])/(2*a) - Log[c*(a + b*x^2)^p]/(2*x^2)} -{Log[c*(a + b*x^2)^p]/x^4, x, 3, -((2*b*p)/(3*a*x)) - (2*b^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*a^(3/2)) - Log[c*(a + b*x^2)^p]/(3*x^3)} -{Log[c*(a + b*x^2)^p]/x^5, x, 4, -((b*p)/(4*a*x^2)) - (b^2*p*Log[x])/(2*a^2) + (b^2*p*Log[a + b*x^2])/(4*a^2) - Log[c*(a + b*x^2)^p]/(4*x^4)} -{Log[c*(a + b*x^2)^p]/x^6, x, 4, -((2*b*p)/(15*a*x^3)) + (2*b^2*p)/(5*a^2*x) + (2*b^(5/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(5*a^(5/2)) - Log[c*(a + b*x^2)^p]/(5*x^5)} -{Log[c*(a + b*x^2)^p]/x^7, x, 4, -((b*p)/(12*a*x^4)) + (b^2*p)/(6*a^2*x^2) + (b^3*p*Log[x])/(3*a^3) - (b^3*p*Log[a + b*x^2])/(6*a^3) - Log[c*(a + b*x^2)^p]/(6*x^6)} - - -{x^5*Log[c*(a + b*x^3)^p], x, 4, (a*p*x^3)/(6*b) - (p*x^6)/12 - (a^2*p*Log[a + b*x^3])/(6*b^2) + (1/6)*x^6*Log[c*(a + b*x^3)^p]} -{x^4*Log[c*(a + b*x^3)^p], x, 9, (3*a*p*x^2)/(10*b) - (3*p*x^5)/25 + (Sqrt[3]*a^(5/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(5*b^(5/3)) + (a^(5/3)*p*Log[a^(1/3) + b^(1/3)*x])/(5*b^(5/3)) - (a^(5/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(10*b^(5/3)) + (1/5)*x^5*Log[c*(a + b*x^3)^p]} -{x^3*Log[c*(a + b*x^3)^p], x, 9, (3*a*p*x)/(4*b) - (3*p*x^4)/16 + (Sqrt[3]*a^(4/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(4*b^(4/3)) - (a^(4/3)*p*Log[a^(1/3) + b^(1/3)*x])/(4*b^(4/3)) + (a^(4/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(8*b^(4/3)) + (1/4)*x^4*Log[c*(a + b*x^3)^p]} -{x^2*Log[c*(a + b*x^3)^p], x, 3, -((p*x^3)/3) + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/(3*b)} -{x^1*Log[c*(a + b*x^3)^p], x, 8, -((3*p*x^2)/4) - (Sqrt[3]*a^(2/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*b^(2/3)) - (a^(2/3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)) + (a^(2/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(2/3)) + (1/2)*x^2*Log[c*(a + b*x^3)^p]} -{x^0*Log[c*(a + b*x^3)^p], x, 8, -3*p*x - (Sqrt[3]*a^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (a^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (a^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)) + x*Log[c*(a + b*x^3)^p]} -{Log[c*(a + b*x^3)^p]/x^1, x, 3, (1/3)*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p] + (1/3)*p*PolyLog[2, 1 + (b*x^3)/a]} -{Log[c*(a + b*x^3)^p]/x^2, x, 7, -((Sqrt[3]*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(1/3)) - (b^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (b^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)) - Log[c*(a + b*x^3)^p]/x} -{Log[c*(a + b*x^3)^p]/x^3, x, 7, -((Sqrt[3]*b^(2/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*a^(2/3))) + (b^(2/3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*a^(2/3)) - (b^(2/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*a^(2/3)) - Log[c*(a + b*x^3)^p]/(2*x^2)} -{Log[c*(a + b*x^3)^p]/x^4, x, 5, (b*p*Log[x])/a - (b*p*Log[a + b*x^3])/(3*a) - Log[c*(a + b*x^3)^p]/(3*x^3)} -{Log[c*(a + b*x^3)^p]/x^5, x, 8, -((3*b*p)/(4*a*x)) + (Sqrt[3]*b^(4/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(4*a^(4/3)) + (b^(4/3)*p*Log[a^(1/3) + b^(1/3)*x])/(4*a^(4/3)) - (b^(4/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(8*a^(4/3)) - Log[c*(a + b*x^3)^p]/(4*x^4)} -{Log[c*(a + b*x^3)^p]/x^6, x, 8, -((3*b*p)/(10*a*x^2)) + (Sqrt[3]*b^(5/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(5*a^(5/3)) - (b^(5/3)*p*Log[a^(1/3) + b^(1/3)*x])/(5*a^(5/3)) + (b^(5/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(10*a^(5/3)) - Log[c*(a + b*x^3)^p]/(5*x^5)} -{Log[c*(a + b*x^3)^p]/x^7, x, 4, -((b*p)/(6*a*x^3)) - (b^2*p*Log[x])/(2*a^2) + (b^2*p*Log[a + b*x^3])/(6*a^2) - Log[c*(a + b*x^3)^p]/(6*x^6)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4*Log[c*(a + b/x)^p], x, 4, -((b^4*p*x)/(5*a^4)) + (b^3*p*x^2)/(10*a^3) - (b^2*p*x^3)/(15*a^2) + (b*p*x^4)/(20*a) + (1/5)*x^5*Log[c*(a + b/x)^p] + (b^5*p*Log[b + a*x])/(5*a^5)} -{x^3*Log[c*(a + b/x)^p], x, 4, (b^3*p*x)/(4*a^3) - (b^2*p*x^2)/(8*a^2) + (b*p*x^3)/(12*a) + (1/4)*x^4*Log[c*(a + b/x)^p] - (b^4*p*Log[b + a*x])/(4*a^4)} -{x^2*Log[c*(a + b/x)^p], x, 4, -((b^2*p*x)/(3*a^2)) + (b*p*x^2)/(6*a) + (1/3)*x^3*Log[c*(a + b/x)^p] + (b^3*p*Log[b + a*x])/(3*a^3)} -{x^1*Log[c*(a + b/x)^p], x, 4, (b*p*x)/(2*a) + (1/2)*x^2*Log[c*(a + b/x)^p] - (b^2*p*Log[b + a*x])/(2*a^2)} -{x^0*Log[c*(a + b/x)^p], x, 3, x*Log[c*(a + b/x)^p] + (b*p*Log[b + a*x])/a} -{Log[c*(a + b/x)^p]/x^1, x, 3, (-Log[c*(a + b/x)^p])*Log[-(b/(a*x))] - p*PolyLog[2, 1 + b/(a*x)]} -{Log[c*(a + b/x)^p]/x^2, x, 3, p/x - ((a + b/x)*Log[c*(a + b/x)^p])/b} -{Log[c*(a + b/x)^p]/x^3, x, 4, p/(4*x^2) - (a*p)/(2*b*x) + (a^2*p*Log[a + b/x])/(2*b^2) - Log[c*(a + b/x)^p]/(2*x^2)} -{Log[c*(a + b/x)^p]/x^4, x, 4, p/(9*x^3) - (a*p)/(6*b*x^2) + (a^2*p)/(3*b^2*x) - (a^3*p*Log[a + b/x])/(3*b^3) - Log[c*(a + b/x)^p]/(3*x^3)} -{Log[c*(a + b/x)^p]/x^5, x, 4, p/(16*x^4) - (a*p)/(12*b*x^3) + (a^2*p)/(8*b^2*x^2) - (a^3*p)/(4*b^3*x) + (a^4*p*Log[a + b/x])/(4*b^4) - Log[c*(a + b/x)^p]/(4*x^4)} - - -{x^4*Log[c*(a + b/x^2)^p], x, 5, -((2*b^2*p*x)/(5*a^2)) + (2*b*p*x^3)/(15*a) + (2*b^(5/2)*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(5*a^(5/2)) + (1/5)*x^5*Log[c*(a + b/x^2)^p]} -{x^3*Log[c*(a + b/x^2)^p], x, 5, (b*p*x^2)/(4*a) + (1/4)*x^4*Log[c*(a + b/x^2)^p] - (b^2*p*Log[b + a*x^2])/(4*a^2)} -{x^2*Log[c*(a + b/x^2)^p], x, 4, (2*b*p*x)/(3*a) - (2*b^(3/2)*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(3*a^(3/2)) + (1/3)*x^3*Log[c*(a + b/x^2)^p]} -{x^1*Log[c*(a + b/x^2)^p], x, 3, (1/2)*x^2*Log[c*(a + b/x^2)^p] + (b*p*Log[b + a*x^2])/(2*a)} -{x^0*Log[c*(a + b/x^2)^p], x, 3, (2*Sqrt[b]*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/Sqrt[a] + x*Log[c*(a + b/x^2)^p]} -{Log[c*(a + b/x^2)^p]/x^1, x, 3, (-(1/2))*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))] - (1/2)*p*PolyLog[2, 1 + b/(a*x^2)]} -{Log[c*(a + b/x^2)^p]/x^2, x, 4, (2*p)/x + (2*Sqrt[a]*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/Sqrt[b] - Log[c*(a + b/x^2)^p]/x} -{Log[c*(a + b/x^2)^p]/x^3, x, 3, p/(2*x^2) - ((a + b/x^2)*Log[c*(a + b/x^2)^p])/(2*b)} -{Log[c*(a + b/x^2)^p]/x^4, x, 5, (2*p)/(9*x^3) - (2*a*p)/(3*b*x) - (2*a^(3/2)*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(3*b^(3/2)) - Log[c*(a + b/x^2)^p]/(3*x^3)} - - -{Log[1 + b/x]/x, x, 1, PolyLog[2, -b/x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[c (d+e x^(m/2))^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Log[c*(a + b*Sqrt[x])^p], x, 4, (a^7*p*Sqrt[x])/(4*b^7) - (a^6*p*x)/(8*b^6) + (a^5*p*x^(3/2))/(12*b^5) - (a^4*p*x^2)/(16*b^4) + (a^3*p*x^(5/2))/(20*b^3) - (a^2*p*x^3)/(24*b^2) + (a*p*x^(7/2))/(28*b) - (p*x^4)/32 - (a^8*p*Log[a + b*Sqrt[x]])/(4*b^8) + (1/4)*x^4*Log[c*(a + b*Sqrt[x])^p]} -{x^2*Log[c*(a + b*Sqrt[x])^p], x, 4, (a^5*p*Sqrt[x])/(3*b^5) - (a^4*p*x)/(6*b^4) + (a^3*p*x^(3/2))/(9*b^3) - (a^2*p*x^2)/(12*b^2) + (a*p*x^(5/2))/(15*b) - (p*x^3)/18 - (a^6*p*Log[a + b*Sqrt[x]])/(3*b^6) + (1/3)*x^3*Log[c*(a + b*Sqrt[x])^p]} -{x^1*Log[c*(a + b*Sqrt[x])^p], x, 4, (a^3*p*Sqrt[x])/(2*b^3) - (a^2*p*x)/(4*b^2) + (a*p*x^(3/2))/(6*b) - (p*x^2)/8 - (a^4*p*Log[a + b*Sqrt[x]])/(2*b^4) + (1/2)*x^2*Log[c*(a + b*Sqrt[x])^p]} -{x^0*Log[c*(a + b*Sqrt[x])^p], x, 4, (a*p*Sqrt[x])/b - (p*x)/2 - (a^2*p*Log[a + b*Sqrt[x]])/b^2 + x*Log[c*(a + b*Sqrt[x])^p]} -{Log[c*(a + b*Sqrt[x])^p]/x^1, x, 3, 2*Log[c*(a + b*Sqrt[x])^p]*Log[-((b*Sqrt[x])/a)] + 2*p*PolyLog[2, 1 + (b*Sqrt[x])/a]} -{Log[c*(a + b*Sqrt[x])^p]/x^2, x, 4, -((b*p)/(a*Sqrt[x])) + (b^2*p*Log[a + b*Sqrt[x]])/a^2 - Log[c*(a + b*Sqrt[x])^p]/x - (b^2*p*Log[x])/(2*a^2)} -{Log[c*(a + b*Sqrt[x])^p]/x^3, x, 4, -((b*p)/(6*a*x^(3/2))) + (b^2*p)/(4*a^2*x) - (b^3*p)/(2*a^3*Sqrt[x]) + (b^4*p*Log[a + b*Sqrt[x]])/(2*a^4) - Log[c*(a + b*Sqrt[x])^p]/(2*x^2) - (b^4*p*Log[x])/(4*a^4)} -{Log[c*(a + b*Sqrt[x])^p]/x^4, x, 4, -((b*p)/(15*a*x^(5/2))) + (b^2*p)/(12*a^2*x^2) - (b^3*p)/(9*a^3*x^(3/2)) + (b^4*p)/(6*a^4*x) - (b^5*p)/(3*a^5*Sqrt[x]) + (b^6*p*Log[a + b*Sqrt[x]])/(3*a^6) - Log[c*(a + b*Sqrt[x])^p]/(3*x^3) - (b^6*p*Log[x])/(6*a^6)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(q/2) Log[c (d+e x^(m/2))^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Log[a + b*Sqrt[x]]/Sqrt[x], x, 3, -2*Sqrt[x] + (2*(a + b*Sqrt[x])*Log[a + b*Sqrt[x]])/b} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^m)^n] when q symbolic*) - - -{(f*x)^m*Log[c*(d + e*x^3)^p], x, 3, -((3*e*p*(f*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/3, (7 + m)/3, -((e*x^3)/d)])/(d*f^4*(1 + m)*(4 + m))) + ((f*x)^(1 + m)*Log[c*(d + e*x^3)^p])/(f*(1 + m))} -{(f*x)^m*Log[c*(d + e*x^2)^p], x, 3, -((2*e*p*(f*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)])/(d*f^3*(1 + m)*(3 + m))) + ((f*x)^(1 + m)*Log[c*(d + e*x^2)^p])/(f*(1 + m))} -{(f*x)^m*Log[c*(d + e*x^1)^p], x, 2, -((e*p*(f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((e*x)/d)])/(d*f^2*(1 + m)*(2 + m))) + ((f*x)^(1 + m)*Log[c*(d + e*x)^p])/(f*(1 + m))} -{(f*x)^m*Log[c*(d + e/x^1)^p], x, 4, (e*p*(f*x)^m*Hypergeometric2F1[1, -m, 1 - m, -(e/(d*x))])/(d*m*(1 + m)) + ((f*x)^(1 + m)*Log[c*(d + e/x)^p])/(f*(1 + m))} -{(f*x)^m*Log[c*(d + e/x^2)^p], x, 4, -((2*e*f*p*(f*x)^(-1 + m)*Hypergeometric2F1[1, (1 - m)/2, (3 - m)/2, -(e/(d*x^2))])/(d*(1 - m^2))) + ((f*x)^(1 + m)*Log[c*(d + e/x^2)^p])/(f*(1 + m))} -{(f*x)^m*Log[c*(d + e/x^3)^p], x, 4, -((3*e*f^2*p*(f*x)^(-2 + m)*Hypergeometric2F1[1, (2 - m)/3, (5 - m)/3, -(e/(d*x^3))])/(d*(2 + m - m^2))) + ((f*x)^(1 + m)*Log[c*(d + e/x^3)^p])/(f*(1 + m))} - - -{(f*x)^m*Log[c*(d + e*Sqrt[x])^p], x, 4, If[$VersionNumber>=8, -((e*p*x^(3/2)*(f*x)^m*Hypergeometric2F1[1, 3 + 2*m, 2*(2 + m), -((e*Sqrt[x])/d)])/(d*(3 + 5*m + 2*m^2))) + ((f*x)^(1 + m)*Log[c*(d + e*Sqrt[x])^p])/(f*(1 + m)), -((e*p*x^(3/2)*(f*x)^m*Hypergeometric2F1[1, 3 + 2*m, 2*(2 + m), -((e*Sqrt[x])/d)])/(d*(1 + m)*(3 + 2*m))) + ((f*x)^(1 + m)*Log[c*(d + e*Sqrt[x])^p])/(f*(1 + m))]} -{(f*x)^m*Log[c*(d + e/Sqrt[x])^p], x, 5, (p*x*(f*x)^m*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((d*Sqrt[x])/e)])/(2*(1 + m)^2) + ((f*x)^(1 + m)*Log[c*(d + e/Sqrt[x])^p])/(f*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^m)^n] when m symbolic*) - - -{(f*x)^m*Log[c*(d + e*x^n)^p], x, 3, -((e*n*p*x^(1 + n)*(f*x)^m*Hypergeometric2F1[1, (1 + m + n)/n, (1 + m + 2*n)/n, -((e*x^n)/d)])/(d*(1 + m)*(1 + m + n))) + ((f*x)^(1 + m)*Log[c*(d + e*x^n)^p])/(f*(1 + m))} - - -{(f*x)^(3*n - 1)*Log[c*(d + e*x^n)^p], x, 5, -((p*(f*x)^(3*n))/(9*f*n)) - (d^2*p*(f*x)^(3*n))/(x^(2*n)*(3*e^2*f*n)) + (d*p*(f*x)^(3*n))/(x^n*(6*e*f*n)) + (d^3*p*(f*x)^(3*n)*Log[d + e*x^n])/(x^(3*n)*(3*e^3*f*n)) + ((f*x)^(3*n)*Log[c*(d + e*x^n)^p])/(3*f*n)} -{(f*x)^(2*n - 1)*Log[c*(d + e*x^n)^p], x, 5, -((p*(f*x)^(2*n))/(4*f*n)) + (d*p*(f*x)^(2*n))/(x^n*(2*e*f*n)) - (d^2*p*(f*x)^(2*n)*Log[d + e*x^n])/(x^(2*n)*(2*e^2*f*n)) + ((f*x)^(2*n)*Log[c*(d + e*x^n)^p])/(2*f*n)} -{(f*x)^(1*n - 1)*Log[c*(d + e*x^n)^p], x, 5, -((p*(f*x)^n)/(f*n)) + (d*p*(f*x)^n*Log[d + e*x^n])/(x^n*(e*f*n)) + ((f*x)^n*Log[c*(d + e*x^n)^p])/(f*n)} -{(f*x)^(0*n - 1)*Log[c*(d + e*x^n)^p], x, 4, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f*n)} -{(f*x)^(-1*n - 1)*Log[c*(d + e*x^n)^p], x, 6, (e*p*x^n*Log[x])/((f*x)^n*(d*f)) - (e*p*x^n*Log[d + e*x^n])/((f*x)^n*(d*f*n)) - Log[c*(d + e*x^n)^p]/((f*x)^n*(f*n))} -{(f*x)^(-2*n - 1)*Log[c*(d + e*x^n)^p], x, 5, -((e*p*x^n)/((f*x)^(2*n)*(2*d*f*n))) - (e^2*p*x^(2*n)*Log[x])/((f*x)^(2*n)*(2*d^2*f)) + (e^2*p*x^(2*n)*Log[d + e*x^n])/((f*x)^(2*n)*(2*d^2*f*n)) - Log[c*(d + e*x^n)^p]/((f*x)^(2*n)*(2*f*n))} - - -{x^2*Log[c*(d + e*x^n)^p], x, 2, -((e*n*p*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(3*d*(3 + n))) + (1/3)*x^3*Log[c*(d + e*x^n)^p]} -{x^1*Log[c*(d + e*x^n)^p], x, 2, -((e*n*p*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + 1/n), -((e*x^n)/d)])/(2*d*(2 + n))) + (1/2)*x^2*Log[c*(d + e*x^n)^p]} -{x^0*Log[c*(d + e*x^n)^p], x, 2, -((e*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + 1/n, 2 + 1/n, -((e*x^n)/d)])/(d*(1 + n))) + x*Log[c*(d + e*x^n)^p]} -{Log[c*(d + e*x^n)^p]/x^1, x, 3, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (p*PolyLog[2, 1 + (e*x^n)/d])/n} -{Log[c*(d + e*x^n)^p]/x^2, x, 2, -((e*n*p*x^(-1 + n)*Hypergeometric2F1[1, -((1 - n)/n), 2 - 1/n, -((e*x^n)/d)])/(d*(1 - n))) - Log[c*(d + e*x^n)^p]/x} -{Log[c*(d + e*x^n)^p]/x^3, x, 2, -((e*n*p*x^(-2 + n)*Hypergeometric2F1[1, -((2 - n)/n), 2*(1 - 1/n), -((e*x^n)/d)])/(2*d*(2 - n))) - Log[c*(d + e*x^n)^p]/(2*x^2)} -{Log[c*(d + e*x^n)^p]/x^4, x, 2, -((e*n*p*x^(-3 + n)*Hypergeometric2F1[1, -((3 - n)/n), 2 - 3/n, -((e*x^n)/d)])/(3*d*(3 - n))) - Log[c*(d + e*x^n)^p]/(3*x^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^m)^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[c (d+e x^2)^n]^p *) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Log[c*(a + b*x^2)^p]^2, x, 8, (a^2*p^2*x^2)/b^2 - (a*p^2*(a + b*x^2)^2)/(4*b^3) + (p^2*(a + b*x^2)^3)/(27*b^3) - (a^3*p^2*Log[a + b*x^2]^2)/(6*b^3) - (a^2*p*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^3 + (a*p*(a + b*x^2)^2*Log[c*(a + b*x^2)^p])/(2*b^3) - (p*(a + b*x^2)^3*Log[c*(a + b*x^2)^p])/(9*b^3) + (a^3*p*Log[a + b*x^2]*Log[c*(a + b*x^2)^p])/(3*b^3) + (1/6)*x^6*Log[c*(a + b*x^2)^p]^2} -{x^3*Log[c*(a + b*x^2)^p]^2, x, 9, -((a*p^2*x^2)/b) + (p^2*(a + b*x^2)^2)/(8*b^2) + (a*p*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^2 - (p*(a + b*x^2)^2*Log[c*(a + b*x^2)^p])/(4*b^2) - (a*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^2) + ((a + b*x^2)^2*Log[c*(a + b*x^2)^p]^2)/(4*b^2)} -{x^1*Log[c*(a + b*x^2)^p]^2, x, 4, p^2*x^2 - (p*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b + ((a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b)} -{Log[c*(a + b*x^2)^p]^2/x^1, x, 5, (1/2)*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p]^2 + p*Log[c*(a + b*x^2)^p]*PolyLog[2, 1 + (b*x^2)/a] - p^2*PolyLog[3, 1 + (b*x^2)/a]} -{Log[c*(a + b*x^2)^p]^2/x^3, x, 4, (b*p*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/a - ((a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*a*x^2) + (b*p^2*PolyLog[2, 1 + (b*x^2)/a])/a} -{Log[c*(a + b*x^2)^p]^2/x^5, x, 8, (b^2*p^2*Log[x])/a^2 - (b*p*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*a^2*x^2) - Log[c*(a + b*x^2)^p]^2/(4*x^4) - (b^2*p*Log[c*(a + b*x^2)^p]*Log[1 - a/(a + b*x^2)])/(2*a^2) + (b^2*p^2*PolyLog[2, a/(a + b*x^2)])/(2*a^2)} -{Log[c*(a + b*x^2)^p]^2/x^7, x, 12, -((b^2*p^2)/(6*a^2*x^2)) - (b^3*p^2*Log[x])/a^3 + (b^3*p^2*Log[a + b*x^2])/(6*a^3) - (b*p*Log[c*(a + b*x^2)^p])/(6*a*x^4) + (b^2*p*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(3*a^3*x^2) - Log[c*(a + b*x^2)^p]^2/(6*x^6) + (b^3*p*Log[c*(a + b*x^2)^p]*Log[1 - a/(a + b*x^2)])/(3*a^3) - (b^3*p^2*PolyLog[2, a/(a + b*x^2)])/(3*a^3)} - -{x^4*Log[c*(a + b*x^2)^p]^2, x, 20, (184*a^2*p^2*x)/(75*b^2) - (64*a*p^2*x^3)/(225*b) + (8*p^2*x^5)/125 - (184*a^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(75*b^(5/2)) + (4*I*a^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(5*b^(5/2)) + (8*a^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(5*b^(5/2)) - (4*a^2*p*x*Log[c*(a + b*x^2)^p])/(5*b^2) + (4*a*p*x^3*Log[c*(a + b*x^2)^p])/(15*b) - (4/25)*p*x^5*Log[c*(a + b*x^2)^p] + (4*a^(5/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(5*b^(5/2)) + (1/5)*x^5*Log[c*(a + b*x^2)^p]^2 + (4*I*a^(5/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(5*b^(5/2))} -{x^2*Log[c*(a + b*x^2)^p]^2, x, 16, -((32*a*p^2*x)/(9*b)) + (8*p^2*x^3)/27 + (32*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(9*b^(3/2)) - (4*I*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(3*b^(3/2)) - (8*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2)) + (4*a*p*x*Log[c*(a + b*x^2)^p])/(3*b) - (4/9)*p*x^3*Log[c*(a + b*x^2)^p] - (4*a^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*b^(3/2)) + (1/3)*x^3*Log[c*(a + b*x^2)^p]^2 - (4*I*a^(3/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2))} -{x^0*Log[c*(a + b*x^2)^p]^2, x, 12, 8*p^2*x - (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + (4*I*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/Sqrt[b] + (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b] - 4*p*x*Log[c*(a + b*x^2)^p] + (4*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b] + x*Log[c*(a + b*x^2)^p]^2 + (4*I*Sqrt[a]*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b]} -{Log[c*(a + b*x^2)^p]^2/x^2, x, 7, (4*I*Sqrt[b]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/Sqrt[a] + (8*Sqrt[b]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[a] + (4*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[a] - Log[c*(a + b*x^2)^p]^2/x + (4*I*Sqrt[b]*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[a]} -{Log[c*(a + b*x^2)^p]^2/x^4, x, 11, (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*a^(3/2)) - (4*I*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(3*a^(3/2)) - (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*a^(3/2)) - (4*b*p*Log[c*(a + b*x^2)^p])/(3*a*x) - (4*b^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*a^(3/2)) - Log[c*(a + b*x^2)^p]^2/(3*x^3) - (4*I*b^(3/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*a^(3/2))} -{Log[c*(a + b*x^2)^p]^2/x^6, x, 14, -((8*b^2*p^2)/(15*a^2*x)) - (32*b^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(15*a^(5/2)) + (4*I*b^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(5*a^(5/2)) + (8*b^(5/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(5*a^(5/2)) - (4*b*p*Log[c*(a + b*x^2)^p])/(15*a*x^3) + (4*b^2*p*Log[c*(a + b*x^2)^p])/(5*a^2*x) + (4*b^(5/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(5*a^(5/2)) - Log[c*(a + b*x^2)^p]^2/(5*x^5) + (4*I*b^(5/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(5*a^(5/2))} -{Log[c*(a + b*x^2)^p]^2/x^8, x, 18, -((8*b^2*p^2)/(105*a^2*x^3)) + (64*b^3*p^2)/(105*a^3*x) + (184*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(105*a^(7/2)) - (4*I*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(7*a^(7/2)) - (8*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(7*a^(7/2)) - (4*b*p*Log[c*(a + b*x^2)^p])/(35*a*x^5) + (4*b^2*p*Log[c*(a + b*x^2)^p])/(21*a^2*x^3) - (4*b^3*p*Log[c*(a + b*x^2)^p])/(7*a^3*x) - (4*b^(7/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(7*a^(7/2)) - Log[c*(a + b*x^2)^p]^2/(7*x^7) - (4*I*b^(7/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(7*a^(7/2))} - - -{x^5*Log[c*(a + b*x^2)^p]^3, x, 15, -((3*a^2*p^3*x^2)/b^2) + (3*a*p^3*(a + b*x^2)^2)/(8*b^3) - (p^3*(a + b*x^2)^3)/(27*b^3) + (3*a^2*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^3 - (3*a*p^2*(a + b*x^2)^2*Log[c*(a + b*x^2)^p])/(4*b^3) + (p^2*(a + b*x^2)^3*Log[c*(a + b*x^2)^p])/(9*b^3) - (3*a^2*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^3) + (3*a*p*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^2)/(4*b^3) - (p*(a + b*x^2)^3*Log[c*(a + b*x^2)^p]^2)/(6*b^3) + (a^2*(a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b^3) - (a*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^3)/(2*b^3) + ((a + b*x^2)^3*Log[c*(a + b*x^2)^p]^3)/(6*b^3)} -{x^3*Log[c*(a + b*x^2)^p]^3, x, 11, (3*a*p^3*x^2)/b - (3*p^3*(a + b*x^2)^2)/(16*b^2) - (3*a*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b^2 + (3*p^2*(a + b*x^2)^2*Log[c*(a + b*x^2)^p])/(8*b^2) + (3*a*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b^2) - (3*p*(a + b*x^2)^2*Log[c*(a + b*x^2)^p]^2)/(8*b^2) - (a*(a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b^2) + ((a + b*x^2)^2*Log[c*(a + b*x^2)^p]^3)/(4*b^2)} -{x^1*Log[c*(a + b*x^2)^p]^3, x, 5, -3*p^3*x^2 + (3*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/b - (3*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*b) + ((a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*b)} -{Log[c*(a + b*x^2)^p]^3/x^1, x, 6, (1/2)*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p]^3 + (3/2)*p*Log[c*(a + b*x^2)^p]^2*PolyLog[2, 1 + (b*x^2)/a] - 3*p^2*Log[c*(a + b*x^2)^p]*PolyLog[3, 1 + (b*x^2)/a] + 3*p^3*PolyLog[4, 1 + (b*x^2)/a]} -{Log[c*(a + b*x^2)^p]^3/x^3, x, 6, (3*b*p*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p]^2)/(2*a) - ((a + b*x^2)*Log[c*(a + b*x^2)^p]^3)/(2*a*x^2) + (3*b*p^2*Log[c*(a + b*x^2)^p]*PolyLog[2, 1 + (b*x^2)/a])/a - (3*b*p^3*PolyLog[3, 1 + (b*x^2)/a])/a} -{Log[c*(a + b*x^2)^p]^3/x^5, x, 10, (3*b^2*p^2*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*a^2) - (3*b*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(4*a^2*x^2) - Log[c*(a + b*x^2)^p]^3/(4*x^4) - (3*b^2*p*Log[c*(a + b*x^2)^p]^2*Log[1 - a/(a + b*x^2)])/(4*a^2) + (3*b^2*p^2*Log[c*(a + b*x^2)^p]*PolyLog[2, a/(a + b*x^2)])/(2*a^2) + (3*b^2*p^3*PolyLog[2, 1 + (b*x^2)/a])/(2*a^2) + (3*b^2*p^3*PolyLog[3, a/(a + b*x^2)])/(2*a^2)} -{Log[c*(a + b*x^2)^p]^3/x^7, x, 17, (b^3*p^3*Log[x])/a^3 - (b^2*p^2*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*a^3*x^2) - (b^3*p^2*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/a^3 - (b*p*Log[c*(a + b*x^2)^p]^2)/(4*a*x^4) + (b^2*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(2*a^3*x^2) - Log[c*(a + b*x^2)^p]^3/(6*x^6) - (b^3*p^2*Log[c*(a + b*x^2)^p]*Log[1 - a/(a + b*x^2)])/(2*a^3) + (b^3*p*Log[c*(a + b*x^2)^p]^2*Log[1 - a/(a + b*x^2)])/(2*a^3) + (b^3*p^3*PolyLog[2, a/(a + b*x^2)])/(2*a^3) - (b^3*p^2*Log[c*(a + b*x^2)^p]*PolyLog[2, a/(a + b*x^2)])/a^3 - (b^3*p^3*PolyLog[2, 1 + (b*x^2)/a])/a^3 - (b^3*p^3*PolyLog[3, a/(a + b*x^2)])/a^3} - -{x^2*Log[c*(a + b*x^2)^p]^3, x, 31, (208*a*p^3*x)/(9*b) - (16*p^3*x^3)/27 - (208*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(9*b^(3/2)) + (32*I*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(3*b^(3/2)) + (64*a^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2)) - (32*a*p^2*x*Log[c*(a + b*x^2)^p])/(3*b) + (8/9)*p^2*x^3*Log[c*(a + b*x^2)^p] + (32*a^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*b^(3/2)) + (2*a*p*x*Log[c*(a + b*x^2)^p]^2)/b - (2/3)*p*x^3*Log[c*(a + b*x^2)^p]^2 + (1/3)*x^3*Log[c*(a + b*x^2)^p]^3 + (32*I*a^(3/2)*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*b^(3/2)) - (2*a^2*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x])/b} -{x^0*Log[c*(a + b*x^2)^p]^3, x, 15, -48*p^3*x + (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - (24*I*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/Sqrt[b] - (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b] + 24*p^2*x*Log[c*(a + b*x^2)^p] - (24*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b] - 6*p*x*Log[c*(a + b*x^2)^p]^2 + x*Log[c*(a + b*x^2)^p]^3 - (24*I*Sqrt[a]*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b] + 6*a*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x]} -{Log[c*(a + b*x^2)^p]^3/x^2, x, 1, -(Log[c*(a + b*x^2)^p]^3/x) + 6*b*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x]} -{Log[c*(a + b*x^2)^p]^3/x^4, x, 10, (8*I*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a^(3/2) + (16*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) + (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/a^(3/2) - (2*b*p*Log[c*(a + b*x^2)^p]^2)/(a*x) - Log[c*(a + b*x^2)^p]^3/(3*x^3) + (8*I*b^(3/2)*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) - (2*b^2*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x])/a} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/Log[c*(a + b*x^2)^p], x, 9, -((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/((c*(a + b*x^2)^p)^p^(-1)*(2*b^2*p))) + ((a + b*x^2)^2*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p])/((c*(a + b*x^2)^p)^(2/p)*(2*b^2*p))} -{x^1/Log[c*(a + b*x^2)^p], x, 4, ((a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/(2*b*p*(c*(a + b*x^2)^p)^p^(-1))} -{1/(x^1*Log[c*(a + b*x^2)^p]), x, 0, Unintegrable[1/(x*Log[c*(a + b*x^2)^p]), x]} -{1/(x^3*Log[c*(a + b*x^2)^p]), x, 0, Unintegrable[1/(x^3*Log[c*(a + b*x^2)^p]), x]} - -{x^2/Log[c*(a + b*x^2)^p], x, 0, Unintegrable[x^2/Log[c*(a + b*x^2)^p], x]} -{x^0/Log[c*(a + b*x^2)^p], x, 0, Unintegrable[1/Log[c*(a + b*x^2)^p], x]} -{1/(x^2*Log[c*(a + b*x^2)^p]), x, 0, Unintegrable[1/(x^2*Log[c*(a + b*x^2)^p]), x]} - - -{x^3/Log[c*(a + b*x^2)^p]^2, x, 13, -((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/((c*(a + b*x^2)^p)^p^(-1)*(2*b^2*p^2))) + ((a + b*x^2)^2*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p])/((c*(a + b*x^2)^p)^(2/p)*(b^2*p^2)) - (x^2*(a + b*x^2))/(2*b*p*Log[c*(a + b*x^2)^p])} -{x^1/Log[c*(a + b*x^2)^p]^2, x, 5, ((a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/((c*(a + b*x^2)^p)^p^(-1)*(2*b*p^2)) - (a + b*x^2)/(2*b*p*Log[c*(a + b*x^2)^p])} -{1/(x^1*Log[c*(a + b*x^2)^p]^2), x, 0, Unintegrable[1/(x*Log[c*(a + b*x^2)^p]^2), x]} -{1/(x^3*Log[c*(a + b*x^2)^p]^2), x, 0, Unintegrable[1/(x^3*Log[c*(a + b*x^2)^p]^2), x]} - -{x^2/Log[c*(a + b*x^2)^p]^2, x, 0, Unintegrable[x^2/Log[c*(a + b*x^2)^p]^2, x]} -{x^0/Log[c*(a + b*x^2)^p]^2, x, 0, Unintegrable[1/Log[c*(a + b*x^2)^p]^2, x]} -{1/(x^2*Log[c*(a + b*x^2)^p]^2), x, 0, Unintegrable[1/(x^2*Log[c*(a + b*x^2)^p]^2), x]} - - -{x^3/Log[c*(a + b*x^2)^p]^3, x, 18, -((a*(a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/((c*(a + b*x^2)^p)^p^(-1)*(4*b^2*p^3))) + ((a + b*x^2)^2*ExpIntegralEi[(2*Log[c*(a + b*x^2)^p])/p])/((c*(a + b*x^2)^p)^(2/p)*(b^2*p^3)) - (x^2*(a + b*x^2))/(4*b*p*Log[c*(a + b*x^2)^p]^2) - (a*(a + b*x^2))/(4*b^2*p^2*Log[c*(a + b*x^2)^p]) - (x^2*(a + b*x^2))/(2*b*p^2*Log[c*(a + b*x^2)^p])} -{x^1/Log[c*(a + b*x^2)^p]^3, x, 6, ((a + b*x^2)*ExpIntegralEi[Log[c*(a + b*x^2)^p]/p])/((c*(a + b*x^2)^p)^p^(-1)*(4*b*p^3)) - (a + b*x^2)/(4*b*p*Log[c*(a + b*x^2)^p]^2) - (a + b*x^2)/(4*b*p^2*Log[c*(a + b*x^2)^p])} -{1/(x^1*Log[c*(a + b*x^2)^p]^3), x, 0, Unintegrable[1/(x*Log[c*(a + b*x^2)^p]^3), x]} -{1/(x^3*Log[c*(a + b*x^2)^p]^3), x, 0, Unintegrable[1/(x^3*Log[c*(a + b*x^2)^p]^3), x]} - -{x^2/Log[c*(a + b*x^2)^p]^3, x, 0, Unintegrable[x^2/Log[c*(a + b*x^2)^p]^3, x]} -{x^0/Log[c*(a + b*x^2)^p]^3, x, 0, Unintegrable[1/Log[c*(a + b*x^2)^p]^3, x]} -{1/(x^2*Log[c*(a + b*x^2)^p]^3), x, 0, Unintegrable[1/(x^2*Log[c*(a + b*x^2)^p]^3), x]} - - -{x^3/Log[c*(a + b*x^2)], x, 8, ExpIntegralEi[2*Log[c*(a + b*x^2)]]/(2*b^2*c^2) - (a*LogIntegral[c*(a + b*x^2)])/(2*b^2*c)} -{x^1/Log[c*(a + b*x^2)], x, 3, LogIntegral[c*(a + b*x^2)]/(2*b*c)} - - -{x^3/Log[c*(a + b*x^2)]^2, x, 11, ExpIntegralEi[2*Log[c*(a + b*x^2)]]/(b^2*c^2) - (x^2*(a + b*x^2))/(2*b*Log[c*(a + b*x^2)]) - (a*LogIntegral[c*(a + b*x^2)])/(2*b^2*c)} -{x^1/Log[c*(a + b*x^2)]^2, x, 4, -((a + b*x^2)/(2*b*Log[c*(a + b*x^2)])) + LogIntegral[c*(a + b*x^2)]/(2*b*c)} - - -{x^3/Log[c*(a + b*x^2)]^3, x, 15, ExpIntegralEi[2*Log[c*(a + b*x^2)]]/(b^2*c^2) - (x^2*(a + b*x^2))/(4*b*Log[c*(a + b*x^2)]^2) - (a*(a + b*x^2))/(4*b^2*Log[c*(a + b*x^2)]) - (x^2*(a + b*x^2))/(2*b*Log[c*(a + b*x^2)]) - (a*LogIntegral[c*(a + b*x^2)])/(4*b^2*c)} -{x^1/Log[c*(a + b*x^2)]^3, x, 5, -((a + b*x^2)/(4*b*Log[c*(a + b*x^2)]^2)) - (a + b*x^2)/(4*b*Log[c*(a + b*x^2)]) + LogIntegral[c*(a + b*x^2)]/(4*b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Log[c (d+e x^3)^n]^p *) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Log[c*(d + e*x^3)^p]^2, x, 9, -((2*d*p^2*x^3)/(3*e)) + (p^2*(d + e*x^3)^2)/(12*e^2) + (2*d*p*(d + e*x^3)*Log[c*(d + e*x^3)^p])/(3*e^2) - (p*(d + e*x^3)^2*Log[c*(d + e*x^3)^p])/(6*e^2) - (d*(d + e*x^3)*Log[c*(d + e*x^3)^p]^2)/(3*e^2) + ((d + e*x^3)^2*Log[c*(d + e*x^3)^p]^2)/(6*e^2)} -{x^2*Log[c*(d + e*x^3)^p]^2, x, 4, (2*p^2*x^3)/3 - (2*p*(d + e*x^3)*Log[c*(d + e*x^3)^p])/(3*e) + ((d + e*x^3)*Log[c*(d + e*x^3)^p]^2)/(3*e)} -{Log[c*(d + e*x^3)^p]^2/x^1, x, 5, (1/3)*Log[-((e*x^3)/d)]*Log[c*(d + e*x^3)^p]^2 + (2/3)*p*Log[c*(d + e*x^3)^p]*PolyLog[2, 1 + (e*x^3)/d] - (2/3)*p^2*PolyLog[3, 1 + (e*x^3)/d]} -{Log[c*(d + e*x^3)^p]^2/x^4, x, 4, (2*e*p*Log[-((e*x^3)/d)]*Log[c*(d + e*x^3)^p])/(3*d) - ((d + e*x^3)*Log[c*(d + e*x^3)^p]^2)/(3*d*x^3) + (2*e*p^2*PolyLog[2, 1 + (e*x^3)/d])/(3*d)} - -{x^1*Log[c*(d + e*x^3)^p]^2, x, If[$VersionNumber>=8, 49, 51], If[$VersionNumber>=8, If[$VersionNumber<11, (9*p^2*x^2)/4 + (3*Sqrt[3]*d^(2/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*e^(2/3)) + (3*d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(2*e^(2/3)) + ((-1)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(2*e^(2/3)) - ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - (3*d^(2/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*e^(2/3)) - (3/2)*p*x^2*Log[c*(d + e*x^3)^p] - (d^(2/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + ((-1)^(1/3)*d^(2/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + (1/2)*x^2*Log[c*(d + e*x^3)^p]^2 + (d^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (d^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3), (9*p^2*x^2)/4 + (3*Sqrt[3]*d^(2/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*e^(2/3)) + (3*d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(2*e^(2/3)) + ((-1)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - (3*d^(2/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*e^(2/3)) - (3/2)*p*x^2*Log[c*(d + e*x^3)^p] - (d^(2/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + ((-1)^(1/3)*d^(2/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + (1/2)*x^2*Log[c*(d + e*x^3)^p]^2 + (d^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) + (d^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3)], (9*p^2*x^2)/4 + (3*Sqrt[3]*d^(2/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*e^(2/3)) + (3*d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(2*e^(2/3)) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(2*e^(2/3)) + ((-1)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/e^(2/3) + ((-1)^(2/3)*d^(2/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(2*e^(2/3)) - ((-1)^(2/3)*d^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + (d^(2/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(2/3) + ((-1)^(1/3)*d^(2/3)*p^2*Log[-(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) - (3*d^(2/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*e^(2/3)) - (3/2)*p*x^2*Log[c*(d + e*x^3)^p] - (d^(2/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + ((-1)^(1/3)*d^(2/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(2/3) + (1/2)*x^2*Log[c*(d + e*x^3)^p]^2 + (d^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3) + (d^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(2/3) - ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) - ((-1)^(2/3)*d^(2/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(2/3) + ((-1)^(1/3)*d^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(2/3)]} -{x^0*Log[c*(d + e*x^3)^p]^2, x, If[$VersionNumber>=8, 49, 51], If[$VersionNumber>=8, If[$VersionNumber<11, 18*p^2*x + (6*Sqrt[3]*d^(1/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/e^(1/3) - (d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/e^(1/3) - (6*d^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/e^(1/3) - ((-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + ((-1)^(1/3)*d^(1/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) + (3*d^(1/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/e^(1/3) - 6*p*x*Log[c*(d + e*x^3)^p] + (2*d^(1/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + (2*(-1)^(2/3)*d^(1/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + x*Log[c*(d + e*x^3)^p]^2 - (2*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*d^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(1/3), 18*p^2*x + (6*Sqrt[3]*d^(1/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/e^(1/3) - (d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/e^(1/3) - (6*d^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/e^(1/3) - ((-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + ((-1)^(1/3)*d^(1/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) + (3*d^(1/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/e^(1/3) - 6*p*x*Log[c*(d + e*x^3)^p] + (2*d^(1/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + (2*(-1)^(2/3)*d^(1/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + x*Log[c*(d + e*x^3)^p]^2 - (2*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*d^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3)], 18*p^2*x + (6*Sqrt[3]*d^(1/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/e^(1/3) - (d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/e^(1/3) - (6*d^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/e^(1/3) - ((-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + (2*(-1)^(1/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/e^(1/3) + ((-1)^(1/3)*d^(1/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*d^(1/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/e^(1/3) + (2*(-1)^(2/3)*d^(1/3)*p^2*Log[-(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) + (3*d^(1/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/e^(1/3) - 6*p*x*Log[c*(d + e*x^3)^p] + (2*d^(1/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + (2*(-1)^(2/3)*d^(1/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/e^(1/3) + x*Log[c*(d + e*x^3)^p]^2 - (2*d^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3) - (2*d^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/e^(1/3) - (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) - (2*(-1)^(1/3)*d^(1/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/e^(1/3) + (2*(-1)^(2/3)*d^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/e^(1/3)]} -{Log[c*(d + e*x^3)^p]^2/x^2, x, If[$VersionNumber>=8, 39, 41], If[$VersionNumber>=8, If[$VersionNumber<11, (e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/d^(1/3) - ((-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + ((-1)^(2/3)*e^(1/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*e^(1/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) + (2*(-1)^(1/3)*e^(1/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - Log[c*(d + e*x^3)^p]^2/x + (2*e^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*e^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(1/3), (e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/d^(1/3) - ((-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + ((-1)^(2/3)*e^(1/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*e^(1/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) + (2*(-1)^(1/3)*e^(1/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - Log[c*(d + e*x^3)^p]^2/x + (2*e^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) + (2*e^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3)], (e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/d^(1/3) - ((-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + (2*(-1)^(2/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/d^(1/3) + ((-1)^(2/3)*e^(1/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*e^(1/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(1/3) + (2*(-1)^(1/3)*e^(1/3)*p^2*Log[-(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) - (2*e^(1/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) + (2*(-1)^(1/3)*e^(1/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(1/3) - Log[c*(d + e*x^3)^p]^2/x + (2*e^(1/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3) + (2*e^(1/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(1/3) - (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) - (2*(-1)^(2/3)*e^(1/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(1/3) + (2*(-1)^(1/3)*e^(1/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(1/3)]} -{Log[c*(d + e*x^3)^p]^2/x^3, x, If[$VersionNumber>=8, 39, 41], If[$VersionNumber>=8, If[$VersionNumber<11, -((e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/(2*d^(2/3))) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/(2*d^(2/3)) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/(2*d^(2/3)) - ((-1)^(1/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) + (e^(2/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) + ((-1)^(2/3)*e^(2/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - Log[c*(d + e*x^3)^p]^2/(2*x^2) - (e^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - (e^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(2/3), -((e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/(2*d^(2/3))) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/(2*d^(2/3)) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/(2*d^(2/3)) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) + (e^(2/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) + ((-1)^(2/3)*e^(2/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - Log[c*(d + e*x^3)^p]^2/(2*x^2) - (e^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - (e^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3)], -((e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]^2)/(2*d^(2/3))) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]^2)/(2*d^(2/3)) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x])/d^(2/3) + ((-1)^(1/3)*e^(2/3)*p^2*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]^2)/(2*d^(2/3)) - ((-1)^(1/3)*e^(2/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - (e^(2/3)*p^2*Log[-d^(1/3) - e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/d^(2/3) + ((-1)^(2/3)*e^(2/3)*p^2*Log[-(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) + (e^(2/3)*p*Log[-d^(1/3) - e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) + ((-1)^(2/3)*e^(2/3)*p*Log[-d^(1/3) + (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p*Log[-d^(1/3) - (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/d^(2/3) - Log[c*(d + e*x^3)^p]^2/(2*x^2) - (e^(2/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3) - (e^(2/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/d^(2/3) - ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) - ((-1)^(1/3)*e^(2/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/d^(2/3) + ((-1)^(2/3)*e^(2/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/d^(2/3)]} -{Log[c*(d + e*x^3)^p]^2/x^5, x, If[$VersionNumber>=8, 48, 50], If[$VersionNumber>=8, If[$VersionNumber<11, -((3*Sqrt[3]*e^(4/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*d^(4/3))) - (3*e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(4*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(4*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(4*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + (3*e^(4/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*d^(4/3)) - (3*e*p*Log[c*(d + e*x^3)^p])/(2*d*x) + (e^(4/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - ((-1)^(1/3)*e^(4/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - Log[c*(d + e*x^3)^p]^2/(4*x^4) - (e^(4/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - (e^(4/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/(2*d^(4/3)), -((3*Sqrt[3]*e^(4/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*d^(4/3))) - (3*e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(4*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(4*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(4*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + (3*e^(4/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*d^(4/3)) - (3*e*p*Log[c*(d + e*x^3)^p])/(2*d*x) + (e^(4/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - ((-1)^(1/3)*e^(4/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - Log[c*(d + e*x^3)^p]^2/(4*x^4) - (e^(4/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) - (e^(4/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, -(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, (d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3))], -((3*Sqrt[3]*e^(4/3)*p^2*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(2*d^(4/3))) - (3*e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x])/(2*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]^2)/(4*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[-(((-1)^(2/3)*d^(1/3) + e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]^2)/(4*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x])/(2*d^(4/3)) - ((-1)^(2/3)*e^(4/3)*p^2*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]^2)/(4*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*Log[((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - (e^(4/3)*p^2*Log[d^(1/3) + e^(1/3)*x]*Log[((-1)^(1/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*Log[-(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 - (-1)^(2/3))*d^(1/3))])/(2*d^(4/3)) - ((-1)^(1/3)*e^(4/3)*p^2*Log[-(((-1)^(1/3)*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[-(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) + (3*e^(4/3)*p^2*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(4*d^(4/3)) - (3*e*p*Log[c*(d + e*x^3)^p])/(2*d*x) + (e^(4/3)*p*Log[d^(1/3) + e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - ((-1)^(1/3)*e^(4/3)*p*Log[d^(1/3) - (-1)^(1/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p*Log[d^(1/3) + (-1)^(2/3)*e^(1/3)*x]*Log[c*(d + e*x^3)^p])/(2*d^(4/3)) - Log[c*(d + e*x^3)^p]^2/(4*x^4) - (e^(4/3)*p^2*PolyLog[2, (d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3)) - (e^(4/3)*p^2*PolyLog[2, (2*(d^(1/3) + e^(1/3)*x))/((3 - I*Sqrt[3])*d^(1/3))])/(2*d^(4/3)) + ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, (d^(1/3) - (-1)^(1/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) + ((-1)^(2/3)*e^(4/3)*p^2*PolyLog[2, ((-1)^(1/3)*(d^(1/3) - (-1)^(1/3)*e^(1/3)*x))/((1 + (-1)^(1/3))*d^(1/3))])/(2*d^(4/3)) - ((-1)^(1/3)*e^(4/3)*p^2*PolyLog[2, -(((-1)^(2/3)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((1 - (-1)^(2/3))*d^(1/3)))])/(2*d^(4/3))]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^8/Log[c*(d + e*x^3)^p], x, 12, (d^2*(d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/((c*(d + e*x^3)^p)^p^(-1)*(3*e^3*p)) - (2*d*(d + e*x^3)^2*ExpIntegralEi[(2*Log[c*(d + e*x^3)^p])/p])/((c*(d + e*x^3)^p)^(2/p)*(3*e^3*p)) + ((d + e*x^3)^3*ExpIntegralEi[(3*Log[c*(d + e*x^3)^p])/p])/((c*(d + e*x^3)^p)^(3/p)*(3*e^3*p))} -{x^5/Log[c*(d + e*x^3)^p], x, 9, -(d*(d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/(3*e^2*p*(c*(d + e*x^3)^p)^p^(-1)) + ((d + e*x^3)^2*ExpIntegralEi[(2*Log[c*(d + e*x^3)^p])/p])/(3*e^2*p*(c*(d + e*x^3)^p)^(2/p))} -{x^2/Log[c*(d + e*x^3)^p], x, 4, ((d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/(3*e*p*(c*(d + e*x^3)^p)^p^(-1))} -{1/(x^1*Log[c*(d + e*x^3)^p]), x, 0, Unintegrable[1/(x*Log[c*(d + e*x^3)^p]), x]} -{1/(x^4*Log[c*(d + e*x^3)^p]), x, 0, Unintegrable[1/(x^4*Log[c*(d + e*x^3)^p]), x]} - -{x^3/Log[c*(d + e*x^3)^p], x, 0, Unintegrable[x^3/Log[c*(d + e*x^3)^p], x]} -{x^1/Log[c*(d + e*x^3)^p], x, 0, Unintegrable[x/Log[c*(d + e*x^3)^p], x]} -{x^0/Log[c*(d + e*x^3)^p], x, 0, Unintegrable[Log[c*(d + e*x^3)^p]^(-1), x]} -{1/(x^2*Log[c*(d + e*x^3)^p]), x, 0, Unintegrable[1/(x^2*Log[c*(d + e*x^3)^p]), x]} -{1/(x^3*Log[c*(d + e*x^3)^p]), x, 0, Unintegrable[1/(x^3*Log[c*(d + e*x^3)^p]), x]} - - -{x^8/Log[c*(d + e*x^3)^p]^2, x, 21, (d^2*(d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/((c*(d + e*x^3)^p)^p^(-1)*(3*e^3*p^2)) - (4*d*(d + e*x^3)^2*ExpIntegralEi[(2*Log[c*(d + e*x^3)^p])/p])/((c*(d + e*x^3)^p)^(2/p)*(3*e^3*p^2)) + ((d + e*x^3)^3*ExpIntegralEi[(3*Log[c*(d + e*x^3)^p])/p])/((c*(d + e*x^3)^p)^(3/p)*(e^3*p^2)) - (x^6*(d + e*x^3))/(3*e*p*Log[c*(d + e*x^3)^p])} -{x^5/Log[c*(d + e*x^3)^p]^2, x, 13, -(d*(d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/(3*e^2*p^2*(c*(d + e*x^3)^p)^p^(-1)) + (2*(d + e*x^3)^2*ExpIntegralEi[(2*Log[c*(d + e*x^3)^p])/p])/(3*e^2*p^2*(c*(d + e*x^3)^p)^(2/p)) - (x^3*(d + e*x^3))/(3*e*p*Log[c*(d + e*x^3)^p])} -{x^2/Log[c*(d + e*x^3)^p]^2, x, 5, ((d + e*x^3)*ExpIntegralEi[Log[c*(d + e*x^3)^p]/p])/(3*e*p^2*(c*(d + e*x^3)^p)^p^(-1)) - (d + e*x^3)/(3*e*p*Log[c*(d + e*x^3)^p])} -{1/(x^1*Log[c*(d + e*x^3)^p]^2), x, 0, Unintegrable[1/(x*Log[c*(d + e*x^3)^p]^2), x]} -{1/(x^4*Log[c*(d + e*x^3)^p]^2), x, 0, Unintegrable[1/(x^4*Log[c*(d + e*x^3)^p]^2), x]} - -{x^3/Log[c*(d + e*x^3)^p]^2, x, 0, Unintegrable[x^3/Log[c*(d + e*x^3)^p]^2, x]} -{x^1/Log[c*(d + e*x^3)^p]^2, x, 0, Unintegrable[x/Log[c*(d + e*x^3)^p]^2, x]} -{x^0/Log[c*(d + e*x^3)^p]^2, x, 0, Unintegrable[Log[c*(d + e*x^3)^p]^(-2), x]} -{1/(x^2*Log[c*(d + e*x^3)^p]^2), x, 0, Unintegrable[1/(x^2*Log[c*(d + e*x^3)^p]^2), x]} -{1/(x^3*Log[c*(d + e*x^3)^p]^2), x, 0, Unintegrable[1/(x^3*Log[c*(d + e*x^3)^p]^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^2)^n]^p when q symbolic*) - - -{(f*x)^m*Log[c*(d + e*x^2)^p]^3, x, 1, ((f*x)^(1 + m)*Log[c*(d + e*x^2)^p]^3)/(f*(1 + m)) - (6*e*p*Unintegrable[((f*x)^(2 + m)*Log[c*(d + e*x^2)^p]^2)/(d + e*x^2), x])/(f^2*(1 + m))} -{(f*x)^m*Log[c*(d + e*x^2)^p]^2, x, 1, ((f*x)^(1 + m)*Log[c*(d + e*x^2)^p]^2)/(f*(1 + m)) - (4*e*p*Unintegrable[((f*x)^(2 + m)*Log[c*(d + e*x^2)^p])/(d + e*x^2), x])/(f^2*(1 + m))} -{(f*x)^m*Log[c*(d + e*x^2)^p]^1, x, 3, -((2*e*p*(f*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2)/d)])/(d*f^3*(1 + m)*(3 + m))) + ((f*x)^(1 + m)*Log[c*(d + e*x^2)^p])/(f*(1 + m))} -{(f*x)^m/Log[c*(d + e*x^2)^p]^1, x, 0, Unintegrable[(f*x)^m/Log[c*(d + e*x^2)^p], x]} -{(f*x)^m/Log[c*(d + e*x^2)^p]^2, x, 0, Unintegrable[(f*x)^m/Log[c*(d + e*x^2)^p]^2, x]} - - -{(f*x)^(3*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 9, (2*d^2*p^2*x^(1 - 2*n)*(f*x)^(-1 + 3*n))/(e^2*n) - (d*p^2*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*(d + e*x^n)^2)/(2*e^3*n) + (2*p^2*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*(d + e*x^n)^3)/(27*e^3*n) - (d^3*p^2*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*Log[d + e*x^n]^2)/(3*e^3*n) - (2*d^2*p*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e^3*n) + (d*p*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*(d + e*x^n)^2*Log[c*(d + e*x^n)^p])/(e^3*n) - (2*p*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*(d + e*x^n)^3*Log[c*(d + e*x^n)^p])/(9*e^3*n) + (2*d^3*p*x^(1 - 3*n)*(f*x)^(-1 + 3*n)*Log[d + e*x^n]*Log[c*(d + e*x^n)^p])/(3*e^3*n) + (x*(f*x)^(-1 + 3*n)*Log[c*(d + e*x^n)^p]^2)/(3*n)} -{(f*x)^(2*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 10, -((2*d*p^2*x^(1 - n)*(f*x)^(-1 + 2*n))/(e*n)) + (p^2*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2)/(4*e^2*n) + (2*d*p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e^2*n) - (p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2*Log[c*(d + e*x^n)^p])/(2*e^2*n) - (d*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p]^2)/(e^2*n) + (x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2*Log[c*(d + e*x^n)^p]^2)/(2*e^2*n)} -{(f*x)^(1*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 5, (2*p^2*x*(f*x)^(-1 + n))/n - (2*p*x^(1 - n)*(f*x)^(-1 + n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e*n) + (x^(1 - n)*(f*x)^(-1 + n)*(d + e*x^n)*Log[c*(d + e*x^n)^p]^2)/(e*n)} -{(f*x)^(0*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 6, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^2)/(f*n) + (2*p*Log[c*(d + e*x^n)^p]*PolyLog[2, 1 + (e*x^n)/d])/(f*n) - (2*p^2*PolyLog[3, 1 + (e*x^n)/d])/(f*n)} -{(f*x)^(-1*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 5, (2*e*p*x^(1 + n)*(f*x)^(-1 - n)*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(d*n) - (x*(f*x)^(-1 - n)*(d + e*x^n)*Log[c*(d + e*x^n)^p]^2)/(d*n) + (2*e*p^2*x^(1 + n)*(f*x)^(-1 - n)*PolyLog[2, 1 + (e*x^n)/d])/(d*n)} -{(f*x)^(-2*n - 1)*Log[c*(d + e*x^n)^p]^2, x, 9, (e^2*p^2*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*Log[x])/d^2 - (e*p*x^(1 + n)*(f*x)^(-1 - 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(d^2*n) - (x*(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]^2)/(2*n) - (e^2*p*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p]*Log[1 - d/(d + e*x^n)])/(d^2*n) + (e^2*p^2*x^(1 + 2*n)*(f*x)^(-1 - 2*n)*PolyLog[2, d/(d + e*x^n)])/(d^2*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^q Log[c (d+e x^m)^n]^p when m symbolic*) - - -{Log[1 + e*x^n]/x, x, 1, -(PolyLog[2, -(e*x^n)]/n)} -{Log[2 + e*x^n]/x, x, 3, Log[2]*Log[x] - PolyLog[2, -((e*x^n)/2)]/n} -{Log[2*(3 + e*x^n)]/x, x, 3, Log[6]*Log[x] - PolyLog[2, -((e*x^n)/3)]/n} -{Log[c*(d + e*x^n)]/x, x, 3, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)])/n + PolyLog[2, 1 + (e*x^n)/d]/n} - - -{Log[c*(d + e*x^n)^p]^1/x, x, 3, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (p*PolyLog[2, 1 + (e*x^n)/d])/n} -{Log[c*(d + e*x^n)^p]^2/x, x, 5, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^2)/n + (2*p*Log[c*(d + e*x^n)^p]*PolyLog[2, 1 + (e*x^n)/d])/n - (2*p^2*PolyLog[3, 1 + (e*x^n)/d])/n} -{Log[c*(d + e*x^n)^p]^3/x, x, 6, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p]^3)/n + (3*p*Log[c*(d + e*x^n)^p]^2*PolyLog[2, 1 + (e*x^n)/d])/n - (6*p^2*Log[c*(d + e*x^n)^p]*PolyLog[3, 1 + (e*x^n)/d])/n + (6*p^3*PolyLog[4, 1 + (e*x^n)/d])/n} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^q Log[c (d+e x^m)^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q Log[c (d+e x^m)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Log[c*(a + b*x)^p]*(d + e*x)^3, x, 3, -(((b*d - a*e)^3*p*x)/(4*b^3)) - ((b*d - a*e)^2*p*(d + e*x)^2)/(8*b^2*e) - ((b*d - a*e)*p*(d + e*x)^3)/(12*b*e) - (p*(d + e*x)^4)/(16*e) - ((b*d - a*e)^4*p*Log[a + b*x])/(4*b^4*e) + ((d + e*x)^4*Log[c*(a + b*x)^p])/(4*e)} -{Log[c*(a + b*x)^p]*(d + e*x)^2, x, 3, -(((b*d - a*e)^2*p*x)/(3*b^2)) - ((b*d - a*e)*p*(d + e*x)^2)/(6*b*e) - (p*(d + e*x)^3)/(9*e) - ((b*d - a*e)^3*p*Log[a + b*x])/(3*b^3*e) + ((d + e*x)^3*Log[c*(a + b*x)^p])/(3*e)} -{Log[c*(a + b*x)^p]*(d + e*x)^1, x, 3, -(((b*d - a*e)*p*x)/(2*b)) - (p*(d + e*x)^2)/(4*e) - ((b*d - a*e)^2*p*Log[a + b*x])/(2*b^2*e) + ((d + e*x)^2*Log[c*(a + b*x)^p])/(2*e)} -{Log[c*(a + b*x)^p]*(d + e*x)^0, x, 2, (-p)*x + ((a + b*x)*Log[c*(a + b*x)^p])/b} -{Log[c*(a + b*x)^p]/(d + e*x)^1, x, 3, (Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e + (p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e} -{Log[c*(a + b*x)^p]/(d + e*x)^2, x, 4, (b*p*Log[a + b*x])/(e*(b*d - a*e)) - Log[c*(a + b*x)^p]/(e*(d + e*x)) - (b*p*Log[d + e*x])/(e*(b*d - a*e))} -{Log[c*(a + b*x)^p]/(d + e*x)^3, x, 3, (b*p)/(2*e*(b*d - a*e)*(d + e*x)) + (b^2*p*Log[a + b*x])/(2*e*(b*d - a*e)^2) - Log[c*(a + b*x)^p]/(2*e*(d + e*x)^2) - (b^2*p*Log[d + e*x])/(2*e*(b*d - a*e)^2)} -{Log[c*(a + b*x)^p]/(d + e*x)^4, x, 3, (b*p)/(6*e*(b*d - a*e)*(d + e*x)^2) + (b^2*p)/(3*e*(b*d - a*e)^2*(d + e*x)) + (b^3*p*Log[a + b*x])/(3*e*(b*d - a*e)^3) - Log[c*(a + b*x)^p]/(3*e*(d + e*x)^3) - (b^3*p*Log[d + e*x])/(3*e*(b*d - a*e)^3)} - - -{Log[c*(a + b*x^2)^p]*(d + e*x)^3, x, 6, -((2*d*(b*d^2 - a*e^2)*p*x)/b) - (e*(6*b*d^2 - a*e^2)*p*x^2)/(4*b) - (2/3)*d*e^2*p*x^3 - (1/8)*e^3*p*x^4 + (2*Sqrt[a]*d*(b*d^2 - a*e^2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) - ((b^2*d^4 - 6*a*b*d^2*e^2 + a^2*e^4)*p*Log[a + b*x^2])/(4*b^2*e) + ((d + e*x)^4*Log[c*(a + b*x^2)^p])/(4*e)} -{Log[c*(a + b*x^2)^p]*(d + e*x)^2, x, 6, -((2*(3*b*d^2 - a*e^2)*p*x)/(3*b)) - d*e*p*x^2 - (2/9)*e^2*p*x^3 + (2*Sqrt[a]*(3*b*d^2 - a*e^2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(3/2)) - (d*(b*d^2 - 3*a*e^2)*p*Log[a + b*x^2])/(3*b*e) + ((d + e*x)^3*Log[c*(a + b*x^2)^p])/(3*e)} -{Log[c*(a + b*x^2)^p]*(d + e*x)^1, x, 6, -2*d*p*x - (1/2)*e*p*x^2 + (2*Sqrt[a]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - ((b*d^2 - a*e^2)*p*Log[a + b*x^2])/(2*b*e) + ((d + e*x)^2*Log[c*(a + b*x^2)^p])/(2*e)} -{Log[c*(a + b*x^2)^p]*(d + e*x)^0, x, 3, -2*p*x + (2*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + x*Log[c*(a + b*x^2)^p]} -{Log[c*(a + b*x^2)^p]/(d + e*x)^1, x, 9, -((p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e) - (p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e + (Log[d + e*x]*Log[c*(a + b*x^2)^p])/e - (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e - (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e} -{Log[c*(a + b*x^2)^p]/(d + e*x)^2, x, 6, (2*Sqrt[a]*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*d^2 + a*e^2) - (2*b*d*p*Log[d + e*x])/(e*(b*d^2 + a*e^2)) + (b*d*p*Log[a + b*x^2])/(e*(b*d^2 + a*e^2)) - Log[c*(a + b*x^2)^p]/(e*(d + e*x))} -{Log[c*(a + b*x^2)^p]/(d + e*x)^3, x, 6, (b*d*p)/(e*(b*d^2 + a*e^2)*(d + e*x)) + (2*Sqrt[a]*b^(3/2)*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*d^2 + a*e^2)^2 - (b*(b*d^2 - a*e^2)*p*Log[d + e*x])/(e*(b*d^2 + a*e^2)^2) + (b*(b*d^2 - a*e^2)*p*Log[a + b*x^2])/(2*e*(b*d^2 + a*e^2)^2) - Log[c*(a + b*x^2)^p]/(2*e*(d + e*x)^2)} - - -{Log[c*(a + b*x^3)^p]*(d + e*x)^3, x, 13, -((3*(4*b*d^3 - a*e^3)*p*x)/(4*b)) - (9/4)*d^2*e*p*x^2 - d*e^2*p*x^3 - (3/16)*e^3*p*x^4 - (Sqrt[3]*a^(1/3)*(4*b*d^3 + 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(4*b^(4/3)) + (a^(1/3)*(4*b*d^3 - 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(1/3) + b^(1/3)*x])/(4*b^(4/3)) - (a^(1/3)*(4*b*d^3 - 6*a^(1/3)*b^(2/3)*d^2*e - a*e^3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(8*b^(4/3)) - (d*(b*d^3 - 4*a*e^3)*p*Log[a + b*x^3])/(4*b*e) + ((d + e*x)^4*Log[c*(a + b*x^3)^p])/(4*e)} -{Log[c*(a + b*x^3)^p]*(d + e*x)^2, x, 12, -3*d^2*p*x - (3/2)*d*e*p*x^2 - (1/3)*e^2*p*x^3 - (Sqrt[3]*a^(1/3)*d*(b^(1/3)*d + a^(1/3)*e)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(2/3) + (a^(1/3)*d*(b^(1/3)*d - a^(1/3)*e)*p*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*d*(b^(1/3)*d - a^(1/3)*e)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(2/3)) - ((b*d^3 - a*e^3)*p*Log[a + b*x^3])/(3*b*e) + ((d + e*x)^3*Log[c*(a + b*x^3)^p])/(3*e)} -{Log[c*(a + b*x^3)^p]*(d + e*x)^1, x, 11, -3*d*p*x - (3/4)*e*p*x^2 - (Sqrt[3]*a^(1/3)*(2*b^(1/3)*d + a^(1/3)*e)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*b^(2/3)) + (a^(1/3)*(2*b^(1/3)*d - a^(1/3)*e)*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)) - (a^(1/3)*(2*b^(1/3)*d - a^(1/3)*e)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(2/3)) - (d^2*p*Log[a + b*x^3])/(2*e) + ((d + e*x)^2*Log[c*(a + b*x^3)^p])/(2*e)} -{Log[c*(a + b*x^3)^p]*(d + e*x)^0, x, 8, -3*p*x - (Sqrt[3]*a^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (a^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (a^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)) + x*Log[c*(a + b*x^3)^p]} -{Log[c*(a + b*x^3)^p]/(d + e*x)^1, x, 12, -((p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e) - (p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e - (p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e + (Log[d + e*x]*Log[c*(a + b*x^3)^p])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e} -{Log[c*(a + b*x^3)^p]/(d + e*x)^2, x, 11, -((Sqrt[3]*a^(1/3)*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(b^(2/3)*d^2 + a^(1/3)*b^(1/3)*d*e + a^(2/3)*e^2)) + (a^(1/3)*b^(1/3)*(b^(1/3)*d + a^(1/3)*e)*p*Log[a^(1/3) + b^(1/3)*x])/(b*d^3 - a*e^3) - (3*b*d^2*p*Log[d + e*x])/(e*(b*d^3 - a*e^3)) - (a^(1/3)*b^(1/3)*(b^(1/3)*d + a^(1/3)*e)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*(b*d^3 - a*e^3)) + (b*d^2*p*Log[a + b*x^3])/(e*(b*d^3 - a*e^3)) - Log[c*(a + b*x^3)^p]/(e*(d + e*x))} -{Log[c*(a + b*x^3)^p]/(d + e*x)^3, x, 11, (3*b*d^2*p)/(2*e*(b*d^3 - a*e^3)*(d + e*x)) - (Sqrt[3]*a^(1/3)*b^(2/3)*(2*b*d^3 - 3*a^(1/3)*b^(2/3)*d^2*e + a*e^3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*(b*d^3 - a*e^3)^2) + (a^(1/3)*b^(2/3)*(2*b*d^3 + 3*a^(1/3)*b^(2/3)*d^2*e + a*e^3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*(b*d^3 - a*e^3)^2) - (3*b*d*(b*d^3 + 2*a*e^3)*p*Log[d + e*x])/(2*e*(b*d^3 - a*e^3)^2) - (a^(1/3)*b^(2/3)*(2*b*d^3 + 3*a^(1/3)*b^(2/3)*d^2*e + a*e^3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*(b*d^3 - a*e^3)^2) + (b*d*(b*d^3 + 2*a*e^3)*p*Log[a + b*x^3])/(2*e*(b*d^3 - a*e^3)^2) - Log[c*(a + b*x^3)^p]/(2*e*(d + e*x)^2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Log[c*(a + b/x)^p]*(d + e*x)^3, x, 4, (b*e*(6*a^2*d^2 - 4*a*b*d*e + b^2*e^2)*p*x)/(4*a^3) + (b*e^2*(4*a*d - b*e)*p*x^2)/(8*a^2) + (b*e^3*p*x^3)/(12*a) + ((d + e*x)^4*Log[c*(a + b/x)^p])/(4*e) + (d^4*p*Log[x])/(4*e) - ((a*d - b*e)^4*p*Log[b + a*x])/(4*a^4*e)} -{Log[c*(a + b/x)^p]*(d + e*x)^2, x, 4, (b*e*(3*a*d - b*e)*p*x)/(3*a^2) + (b*e^2*p*x^2)/(6*a) + ((d + e*x)^3*Log[c*(a + b/x)^p])/(3*e) + (d^3*p*Log[x])/(3*e) - ((a*d - b*e)^3*p*Log[b + a*x])/(3*a^3*e)} -{Log[c*(a + b/x)^p]*(d + e*x)^1, x, 4, (b*e*p*x)/(2*a) + ((d + e*x)^2*Log[c*(a + b/x)^p])/(2*e) + (d^2*p*Log[x])/(2*e) - ((a*d - b*e)^2*p*Log[b + a*x])/(2*a^2*e)} -{Log[c*(a + b/x)^p]/(d + e*x)^1, x, 8, (Log[c*(a + b/x)^p]*Log[d + e*x])/e + (p*Log[-((e*x)/d)]*Log[d + e*x])/e - (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e - (p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e + (p*PolyLog[2, 1 + (e*x)/d])/e} -{Log[c*(a + b/x)^p]/(d + e*x)^2, x, 4, -(Log[c*(a + b/x)^p]/(e*(d + e*x))) - (p*Log[x])/(d*e) + (a*p*Log[b + a*x])/(e*(a*d - b*e)) - (b*p*Log[d + e*x])/(d*(a*d - b*e))} -{Log[c*(a + b/x)^p]/(d + e*x)^3, x, 4, (b*p)/(2*d*(a*d - b*e)*(d + e*x)) - Log[c*(a + b/x)^p]/(2*e*(d + e*x)^2) - (p*Log[x])/(2*d^2*e) + (a^2*p*Log[b + a*x])/(2*e*(a*d - b*e)^2) - (b*(2*a*d - b*e)*p*Log[d + e*x])/(2*d^2*(a*d - b*e)^2)} -{Log[c*(a + b/x)^p]/(d + e*x)^4, x, 4, (b*p)/(6*d*(a*d - b*e)*(d + e*x)^2) + (b*(2*a*d - b*e)*p)/(3*d^2*(a*d - b*e)^2*(d + e*x)) - Log[c*(a + b/x)^p]/(3*e*(d + e*x)^3) - (p*Log[x])/(3*d^3*e) + (a^3*p*Log[b + a*x])/(3*e*(a*d - b*e)^3) - (b*(3*a^2*d^2 - 3*a*b*d*e + b^2*e^2)*p*Log[d + e*x])/(3*d^3*(a*d - b*e)^3)} - - -{Log[a + b/x]/(c + d*x), x, 8, (Log[a + b/x]*Log[c + d*x])/d + (Log[-((d*x)/c)]*Log[c + d*x])/d - (Log[-((d*(b + a*x))/(a*c - b*d))]*Log[c + d*x])/d - PolyLog[2, (a*(c + d*x))/(a*c - b*d)]/d + PolyLog[2, 1 + (d*x)/c]/d} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q Log[c (d+e x^m)^n] when r symbolic*) - - -{(d + e*x)^m*Log[c*(a + b*x^3)^p], x, 6, (b^(1/3)*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/(e*(b^(1/3)*d - a^(1/3)*e)*(1 + m)*(2 + m)) + (b^(1/3)*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/(e*(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)*(1 + m)*(2 + m)) + (b^(1/3)*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/(e*(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1 + m)*Log[c*(a + b*x^3)^p])/(e*(1 + m))} -{(d + e*x)^m*Log[c*(a + b*x^2)^p], x, 5, (Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/(e*(Sqrt[b]*d - Sqrt[-a]*e)*(1 + m)*(2 + m)) + (Sqrt[b]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/(e*(Sqrt[b]*d + Sqrt[-a]*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1 + m)*Log[c*(a + b*x^2)^p])/(e*(1 + m))} -{(d + e*x)^m*Log[c*(a + b*x^1)^p], x, 2, (b*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b*(d + e*x))/(b*d - a*e)])/(e*(b*d - a*e)*(1 + m)*(2 + m)) + ((d + e*x)^(1 + m)*Log[c*(a + b*x)^p])/(e*(1 + m))} -{(d + e*x)^m*Log[c*(a + b/x^1)^p], x, 5, (a*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (a*(d + e*x))/(a*d - b*e)])/(e*(a*d - b*e)*(1 + m)*(2 + m)) - (p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (e*x)/d])/(d*e*(2 + 3*m + m^2)) + ((d + e*x)^(1 + m)*Log[c*(a + b/x)^p])/(e*(1 + m))} -{(d + e*x)^m*Log[c*(a + b/x^2)^p], x, 9, (Sqrt[-a]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/(e*(Sqrt[-a]*d - Sqrt[b]*e)*(1 + m)*(2 + m)) + (Sqrt[-a]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/(e*(Sqrt[-a]*d + Sqrt[b]*e)*(1 + m)*(2 + m)) - (2*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (e*x)/d])/(d*e*(2 + 3*m + m^2)) + ((d + e*x)^(1 + m)*Log[c*(a + b/x^2)^p])/(e*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^q Log[c (d+e x^m)^n] when n symbolic*) - - -{(f + g*x)^m*Log[c*(d + e*x^n)^p], x, 0, Unintegrable[(f + g*x)^m*Log[c*(d + e*x^n)^p], x]} - - -{(f + g*x)^3*Log[c*(d + e*x^n)^p], x, 8, -((e*f^3*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + 1/n, 2 + 1/n, -((e*x^n)/d)])/(d*(1 + n))) - (3*e*f^2*g*n*p*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + 1/n), -((e*x^n)/d)])/(2*d*(2 + n)) - (e*f*g^2*n*p*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(d*(3 + n)) - (e*g^3*n*p*x^(4 + n)*Hypergeometric2F1[1, (4 + n)/n, 2*(1 + 2/n), -((e*x^n)/d)])/(4*d*(4 + n)) - (f^4*p*Log[d + e*x^n])/(4*g) + ((f + g*x)^4*Log[c*(d + e*x^n)^p])/(4*g)} -{(f + g*x)^2*Log[c*(d + e*x^n)^p], x, 7, -((e*f^2*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + 1/n, 2 + 1/n, -((e*x^n)/d)])/(d*(1 + n))) - (e*f*g*n*p*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + 1/n), -((e*x^n)/d)])/(d*(2 + n)) - (e*g^2*n*p*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/n, 2 + 3/n, -((e*x^n)/d)])/(3*d*(3 + n)) - (f^3*p*Log[d + e*x^n])/(3*g) + ((f + g*x)^3*Log[c*(d + e*x^n)^p])/(3*g)} -{(f + g*x)^1*Log[c*(d + e*x^n)^p], x, 6, -((e*f*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + 1/n, 2 + 1/n, -((e*x^n)/d)])/(d*(1 + n))) - (e*g*n*p*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/n, 2*(1 + 1/n), -((e*x^n)/d)])/(2*d*(2 + n)) - (f^2*p*Log[d + e*x^n])/(2*g) + ((f + g*x)^2*Log[c*(d + e*x^n)^p])/(2*g)} -{(f + g*x)^0*Log[c*(d + e*x^n)^p], x, 2, -((e*n*p*x^(1 + n)*Hypergeometric2F1[1, 1 + 1/n, 2 + 1/n, -((e*x^n)/d)])/(d*(1 + n))) + x*Log[c*(d + e*x^n)^p]} -{Log[c*(d + e*x^n)^p]/(f + g*x)^1, x, 0, Unintegrable[Log[c*(d + e*x^n)^p]/(f + g*x), x]} -{Log[c*(d + e*x^n)^p]/(f + g*x)^2, x, 0, Unintegrable[Log[c*(d + e*x^n)^p]/(f + g*x)^2, x]} -{Log[c*(d + e*x^n)^p]/(f + g*x)^3, x, 0, Unintegrable[Log[c*(d + e*x^n)^p]/(f + g*x)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^r (f+g x)^q Log[c (d+e x^m)^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^r / (f+g x) Log[c (d+e x^m)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Log[c*(a + b*x)^p]/(d + e*x), x, 13, -((d^2*p*x)/e^3) - (a*d*p*x)/(2*b*e^2) - (a^2*p*x)/(3*b^2*e) + (d*p*x^2)/(4*e^2) + (a*p*x^2)/(6*b*e) - (p*x^3)/(9*e) + (a^2*d*p*Log[a + b*x])/(2*b^2*e^2) + (a^3*p*Log[a + b*x])/(3*b^3*e) - (d*x^2*Log[c*(a + b*x)^p])/(2*e^2) + (x^3*Log[c*(a + b*x)^p])/(3*e) + (d^2*(a + b*x)*Log[c*(a + b*x)^p])/(b*e^3) - (d^3*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e^4 - (d^3*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^4} -{x^2*Log[c*(a + b*x)^p]/(d + e*x), x, 10, (d*p*x)/e^2 + (a*p*x)/(2*b*e) - (p*x^2)/(4*e) - (a^2*p*Log[a + b*x])/(2*b^2*e) + (x^2*Log[c*(a + b*x)^p])/(2*e) - (d*(a + b*x)*Log[c*(a + b*x)^p])/(b*e^2) + (d^2*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e^3 + (d^2*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^3} -{x^1*Log[c*(a + b*x)^p]/(d + e*x), x, 7, -((p*x)/e) + ((a + b*x)*Log[c*(a + b*x)^p])/(b*e) - (d*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e^2 - (d*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^2} -{x^0*Log[c*(a + b*x)^p]/(d + e*x), x, 3, (Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e + (p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e} -{Log[c*(a + b*x)^p]/(x^1*(d + e*x)), x, 7, (Log[-((b*x)/a)]*Log[c*(a + b*x)^p])/d - (Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/d - (p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (p*PolyLog[2, 1 + (b*x)/a])/d} -{Log[c*(a + b*x)^p]/(x^2*(d + e*x)), x, 11, (b*p*Log[x])/(a*d) - (b*p*Log[a + b*x])/(a*d) - Log[c*(a + b*x)^p]/(d*x) - (e*Log[-((b*x)/a)]*Log[c*(a + b*x)^p])/d^2 + (e*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/d^2 + (e*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d^2 - (e*p*PolyLog[2, 1 + (b*x)/a])/d^2} -{Log[c*(a + b*x)^p]/(x^3*(d + e*x)), x, 14, -((b*p)/(2*a*d*x)) - (b^2*p*Log[x])/(2*a^2*d) - (b*e*p*Log[x])/(a*d^2) + (b^2*p*Log[a + b*x])/(2*a^2*d) + (b*e*p*Log[a + b*x])/(a*d^2) - Log[c*(a + b*x)^p]/(2*d*x^2) + (e*Log[c*(a + b*x)^p])/(d^2*x) + (e^2*Log[-((b*x)/a)]*Log[c*(a + b*x)^p])/d^3 - (e^2*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/d^3 - (e^2*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d^3 + (e^2*p*PolyLog[2, 1 + (b*x)/a])/d^3} - - -{x^3*Log[c*(a + b*x^2)^p]/(d + e*x), x, 21, -((2*d^2*p*x)/e^3) + (2*a*p*x)/(3*b*e) + (d*p*x^2)/(2*e^2) - (2*p*x^3)/(9*e) + (2*Sqrt[a]*d^2*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*e^3) - (2*a^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(3/2)*e) + (d^3*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e^4 + (d^2*x*Log[c*(a + b*x^2)^p])/e^3 + (x^3*Log[c*(a + b*x^2)^p])/(3*e) - (d*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b*e^2) - (d^3*Log[d + e*x]*Log[c*(a + b*x^2)^p])/e^4 + (d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e^4} -{x^2*Log[c*(a + b*x^2)^p]/(d + e*x), x, 17, (2*d*p*x)/e^2 - (p*x^2)/(2*e) - (2*Sqrt[a]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*e^2) - (d^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e^3 - (d*x*Log[c*(a + b*x^2)^p])/e^2 + ((a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b*e) + (d^2*Log[d + e*x]*Log[c*(a + b*x^2)^p])/e^3 - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e^3 - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e^3} -{x^1*Log[c*(a + b*x^2)^p]/(d + e*x), x, 14, -((2*p*x)/e) + (2*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*e) + (d*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e^2 + (d*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e^2 + (x*Log[c*(a + b*x^2)^p])/e - (d*Log[d + e*x]*Log[c*(a + b*x^2)^p])/e^2 + (d*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e^2 + (d*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e^2} -{x^0*Log[c*(a + b*x^2)^p]/(d + e*x), x, 9, -((p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e) - (p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e + (Log[d + e*x]*Log[c*(a + b*x^2)^p])/e - (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e - (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e} -{Log[c*(a + b*x^2)^p]/(x^1*(d + e*x)), x, 14, (p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/d + (p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/d + (Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*d) - (Log[d + e*x]*Log[c*(a + b*x^2)^p])/d + (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/d + (p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d + (p*PolyLog[2, 1 + (b*x^2)/a])/(2*d)} -{Log[c*(a + b*x^2)^p]/(x^2*(d + e*x)), x, 16, (2*Sqrt[b]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*d) - (e*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/d^2 - (e*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/d^2 - Log[c*(a + b*x^2)^p]/(d*x) - (e*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*d^2) + (e*Log[d + e*x]*Log[c*(a + b*x^2)^p])/d^2 - (e*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d^2 - (e*p*PolyLog[2, 1 + (b*x^2)/a])/(2*d^2)} -{Log[c*(a + b*x^2)^p]/(x^3*(d + e*x)), x, 21, -((2*Sqrt[b]*e*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*d^2)) + (b*p*Log[x])/(a*d) + (e^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/d^3 - (b*p*Log[a + b*x^2])/(2*a*d) - Log[c*(a + b*x^2)^p]/(2*d*x^2) + (e*Log[c*(a + b*x^2)^p])/(d^2*x) + (e^2*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*d^3) - (e^2*Log[d + e*x]*Log[c*(a + b*x^2)^p])/d^3 + (e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/d^3 + (e^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/d^3 + (e^2*p*PolyLog[2, 1 + (b*x^2)/a])/(2*d^3)} - - -{x^3*Log[c*(a + b*x^3)^p]/(d + e*x), x, 33, -((3*d^2*p*x)/e^3) + (3*d*p*x^2)/(4*e^2) - (p*x^3)/(3*e) - (Sqrt[3]*a^(1/3)*d^2*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(b^(1/3)*e^3) + (Sqrt[3]*a^(2/3)*d*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*b^(2/3)*e^2) + (a^(1/3)*d^2*p*Log[a^(1/3) + b^(1/3)*x])/(b^(1/3)*e^3) + (a^(2/3)*d*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)*e^2) + (d^3*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e^4 - (a^(1/3)*d^2*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)*e^3) - (a^(2/3)*d*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(2/3)*e^2) + (d^2*x*Log[c*(a + b*x^3)^p])/e^3 - (d*x^2*Log[c*(a + b*x^3)^p])/(2*e^2) + ((a + b*x^3)*Log[c*(a + b*x^3)^p])/(3*b*e) - (d^3*Log[d + e*x]*Log[c*(a + b*x^3)^p])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e^4} -{x^2*Log[c*(a + b*x^3)^p]/(d + e*x), x, 30, (3*d*p*x)/e^2 - (3*p*x^2)/(4*e) + (Sqrt[3]*a^(1/3)*d*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(b^(1/3)*e^2) - (Sqrt[3]*a^(2/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*b^(2/3)*e) - (a^(1/3)*d*p*Log[a^(1/3) + b^(1/3)*x])/(b^(1/3)*e^2) - (a^(2/3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*b^(2/3)*e) - (d^2*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e^3 + (a^(1/3)*d*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)*e^2) + (a^(2/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*b^(2/3)*e) - (d*x*Log[c*(a + b*x^3)^p])/e^2 + (x^2*Log[c*(a + b*x^3)^p])/(2*e) + (d^2*Log[d + e*x]*Log[c*(a + b*x^3)^p])/e^3 - (d^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e^3 - (d^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e^3 - (d^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e^3} -{x^1*Log[c*(a + b*x^3)^p]/(d + e*x), x, 22, -((3*p*x)/e) - (Sqrt[3]*a^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(b^(1/3)*e) + (a^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/(b^(1/3)*e) + (d*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e^2 + (d*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e^2 + (d*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e^2 - (a^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*b^(1/3)*e) + (x*Log[c*(a + b*x^3)^p])/e - (d*Log[d + e*x]*Log[c*(a + b*x^3)^p])/e^2 + (d*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e^2 + (d*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e^2 + (d*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e^2} -{x^0*Log[c*(a + b*x^3)^p]/(d + e*x), x, 12, -((p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/e) - (p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/e - (p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/e + (Log[d + e*x]*Log[c*(a + b*x^3)^p])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/e - (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/e} -{Log[c*(a + b*x^3)^p]/(x^1*(d + e*x)), x, 17, (p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d + (p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/d + (p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/d + (Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d) - (Log[d + e*x]*Log[c*(a + b*x^3)^p])/d + (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/d + (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/d + (p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/d + (p*PolyLog[2, 1 + (b*x^3)/a])/(3*d)} -{Log[c*(a + b*x^3)^p]/(x^2*(d + e*x)), x, 24, -((Sqrt[3]*b^(1/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(a^(1/3)*d)) - (b^(1/3)*p*Log[a^(1/3) + b^(1/3)*x])/(a^(1/3)*d) - (e*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d^2 - (e*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/d^2 - (e*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/d^2 + (b^(1/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)*d) - Log[c*(a + b*x^3)^p]/(d*x) - (e*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d^2) + (e*Log[d + e*x]*Log[c*(a + b*x^3)^p])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/d^2 - (e*p*PolyLog[2, 1 + (b*x^3)/a])/(3*d^2)} -{Log[c*(a + b*x^3)^p]/(x^3*(d + e*x)), x, 31, -((Sqrt[3]*b^(2/3)*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(2*a^(2/3)*d)) + (Sqrt[3]*b^(1/3)*e*p*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(a^(1/3)*d^2) + (b^(2/3)*p*Log[a^(1/3) + b^(1/3)*x])/(2*a^(2/3)*d) + (b^(1/3)*e*p*Log[a^(1/3) + b^(1/3)*x])/(a^(1/3)*d^2) + (e^2*p*Log[-((e*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - a^(1/3)*e))]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e))]*Log[d + e*x])/d^3 + (e^2*p*Log[((-1)^(1/3)*e*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)]*Log[d + e*x])/d^3 - (b^(2/3)*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(4*a^(2/3)*d) - (b^(1/3)*e*p*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(2*a^(1/3)*d^2) - Log[c*(a + b*x^3)^p]/(2*d*x^2) + (e*Log[c*(a + b*x^3)^p])/(d^2*x) + (e^2*Log[-((b*x^3)/a)]*Log[c*(a + b*x^3)^p])/(3*d^3) - (e^2*Log[d + e*x]*Log[c*(a + b*x^3)^p])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, 1 + (b*x^3)/a])/(3*d^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3*Log[c*(a + b/x)^p]/(d + e*x), x, 21, -((b*d*p*x)/(2*a*e^2)) - (b^2*p*x)/(3*a^2*e) + (b*p*x^2)/(6*a*e) + (d^2*x*Log[c*(a + b/x)^p])/e^3 - (d*x^2*Log[c*(a + b/x)^p])/(2*e^2) + (x^3*Log[c*(a + b/x)^p])/(3*e) + (b*d^2*p*Log[b + a*x])/(a*e^3) + (b^2*d*p*Log[b + a*x])/(2*a^2*e^2) + (b^3*p*Log[b + a*x])/(3*a^3*e) - (d^3*Log[c*(a + b/x)^p]*Log[d + e*x])/e^4 - (d^3*p*Log[-((e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e^4 + (d^3*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e^4 - (d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4} -{x^2*Log[c*(a + b/x)^p]/(d + e*x), x, 17, (b*p*x)/(2*a*e) - (d*x*Log[c*(a + b/x)^p])/e^2 + (x^2*Log[c*(a + b/x)^p])/(2*e) - (b*d*p*Log[b + a*x])/(a*e^2) - (b^2*p*Log[b + a*x])/(2*a^2*e) + (d^2*Log[c*(a + b/x)^p]*Log[d + e*x])/e^3 + (d^2*p*Log[-((e*x)/d)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e^3 - (d^2*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e^3 + (d^2*p*PolyLog[2, 1 + (e*x)/d])/e^3} -{x^1*Log[c*(a + b/x)^p]/(d + e*x), x, 13, (x*Log[c*(a + b/x)^p])/e + (b*p*Log[b + a*x])/(a*e) - (d*Log[c*(a + b/x)^p]*Log[d + e*x])/e^2 - (d*p*Log[-((e*x)/d)]*Log[d + e*x])/e^2 + (d*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e^2 + (d*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e^2 - (d*p*PolyLog[2, 1 + (e*x)/d])/e^2} -{x^0*Log[c*(a + b/x)^p]/(d + e*x), x, 8, (Log[c*(a + b/x)^p]*Log[d + e*x])/e + (p*Log[-((e*x)/d)]*Log[d + e*x])/e - (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e - (p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e + (p*PolyLog[2, 1 + (e*x)/d])/e} -{Log[c*(a + b/x)^p]/(x^1*(d + e*x)), x, 13, -((Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d) - (Log[c*(a + b/x)^p]*Log[d + e*x])/d - (p*Log[-((e*x)/d)]*Log[d + e*x])/d + (p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d - (p*PolyLog[2, 1 + b/(a*x)])/d + (p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/d - (p*PolyLog[2, 1 + (e*x)/d])/d} -{Log[c*(a + b/x)^p]/(x^2*(d + e*x)), x, 16, p/(d*x) - ((a + b/x)*Log[c*(a + b/x)^p])/(b*d) + (e*Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d^2 + (e*Log[c*(a + b/x)^p]*Log[d + e*x])/d^2 + (e*p*Log[-((e*x)/d)]*Log[d + e*x])/d^2 - (e*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d^2 + (e*p*PolyLog[2, 1 + b/(a*x)])/d^2 - (e*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/d^2 + (e*p*PolyLog[2, 1 + (e*x)/d])/d^2} -{Log[c*(a + b/x)^p]/(x^3*(d + e*x)), x, 20, p/(4*d*x^2) - (a*p)/(2*b*d*x) - (e*p)/(d^2*x) + (a^2*p*Log[a + b/x])/(2*b^2*d) + (e*(a + b/x)*Log[c*(a + b/x)^p])/(b*d^2) - Log[c*(a + b/x)^p]/(2*d*x^2) - (e^2*Log[c*(a + b/x)^p]*Log[-(b/(a*x))])/d^3 - (e^2*Log[c*(a + b/x)^p]*Log[d + e*x])/d^3 - (e^2*p*Log[-((e*x)/d)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/d^3 - (e^2*p*PolyLog[2, 1 + b/(a*x)])/d^3 + (e^2*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/d^3 - (e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3} - - -{x^3*Log[c*(a + b/x^2)^p]/(d + e*x), x, 25, (2*b*p*x)/(3*a*e) + (2*Sqrt[b]*d^2*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e^3) - (2*b^(3/2)*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(3*a^(3/2)*e) + (d^2*x*Log[c*(a + b/x^2)^p])/e^3 - (d*x^2*Log[c*(a + b/x^2)^p])/(2*e^2) + (x^3*Log[c*(a + b/x^2)^p])/(3*e) - (d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^4 - (2*d^3*p*Log[-((e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^4 - (b*d*p*Log[b + a*x^2])/(2*a*e^2) + (d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/e^4 - (2*d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4} -{x^2*Log[c*(a + b/x^2)^p]/(d + e*x), x, 21, -((2*Sqrt[b]*d*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e^2)) - (d*x*Log[c*(a + b/x^2)^p])/e^2 + (x^2*Log[c*(a + b/x^2)^p])/(2*e) + (d^2*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^3 + (2*d^2*p*Log[-((e*x)/d)]*Log[d + e*x])/e^3 - (d^2*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^3 + (b*p*Log[b + a*x^2])/(2*a*e) - (d^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^3 - (d^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/e^3 + (2*d^2*p*PolyLog[2, 1 + (e*x)/d])/e^3} -{x^1*Log[c*(a + b/x^2)^p]/(d + e*x), x, 18, (2*Sqrt[b]*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e) + (x*Log[c*(a + b/x^2)^p])/e - (d*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^2 - (2*d*p*Log[-((e*x)/d)]*Log[d + e*x])/e^2 + (d*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^2 + (d*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^2 + (d*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^2 + (d*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/e^2 - (2*d*p*PolyLog[2, 1 + (e*x)/d])/e^2} -{x^0*Log[c*(a + b/x^2)^p]/(d + e*x), x, 13, (Log[c*(a + b/x^2)^p]*Log[d + e*x])/e + (2*p*Log[-((e*x)/d)]*Log[d + e*x])/e - (p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e - (p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e - (p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e - (p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/e + (2*p*PolyLog[2, 1 + (e*x)/d])/e} -{Log[c*(a + b/x^2)^p]/(x^1*(d + e*x)), x, 18, -((Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*d)) - (Log[c*(a + b/x^2)^p]*Log[d + e*x])/d - (2*p*Log[-((e*x)/d)]*Log[d + e*x])/d + (p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/d + (p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d - (p*PolyLog[2, 1 + b/(a*x^2)])/(2*d) + (p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/d + (p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/d - (2*p*PolyLog[2, 1 + (e*x)/d])/d} -{Log[c*(a + b/x^2)^p]/(x^2*(d + e*x)), x, 22, (2*p)/(d*x) + (2*Sqrt[a]*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[b]*d) - Log[c*(a + b/x^2)^p]/(d*x) + (e*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*d^2) + (e*Log[c*(a + b/x^2)^p]*Log[d + e*x])/d^2 + (2*e*p*Log[-((e*x)/d)]*Log[d + e*x])/d^2 - (e*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/d^2 - (e*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d^2 + (e*p*PolyLog[2, 1 + b/(a*x^2)])/(2*d^2) - (e*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/d^2 - (e*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/d^2 + (2*e*p*PolyLog[2, 1 + (e*x)/d])/d^2} -{Log[c*(a + b/x^2)^p]/(x^3*(d + e*x)), x, 25, p/(2*d*x^2) - (2*e*p)/(d^2*x) - (2*Sqrt[a]*e*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[b]*d^2) - ((a + b/x^2)*Log[c*(a + b/x^2)^p])/(2*b*d) + (e*Log[c*(a + b/x^2)^p])/(d^2*x) - (e^2*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*d^3) - (e^2*Log[c*(a + b/x^2)^p]*Log[d + e*x])/d^3 - (2*e^2*p*Log[-((e*x)/d)]*Log[d + e*x])/d^3 + (e^2*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d^3 - (e^2*p*PolyLog[2, 1 + b/(a*x^2)])/(2*d^3) + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/d^3 + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/d^3 - (2*e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3} - - -{x^3*Log[c*(a + b/x^3)^p]/(d + e*x), x, 37, -((Sqrt[3]*b^(1/3)*d^2*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(a^(1/3)*e^3)) + (Sqrt[3]*b^(2/3)*d*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(2*a^(2/3)*e^2) + (d^2*x*Log[c*(a + b/x^3)^p])/e^3 - (d*x^2*Log[c*(a + b/x^3)^p])/(2*e^2) + (x^3*Log[c*(a + b/x^3)^p])/(3*e) + (b^(1/3)*d^2*p*Log[b^(1/3) + a^(1/3)*x])/(a^(1/3)*e^3) + (b^(2/3)*d*p*Log[b^(1/3) + a^(1/3)*x])/(2*a^(2/3)*e^2) - (d^3*Log[c*(a + b/x^3)^p]*Log[d + e*x])/e^4 - (3*d^3*p*Log[-((e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e^4 + (d^3*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/e^4 - (b^(1/3)*d^2*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*a^(1/3)*e^3) - (b^(2/3)*d*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(4*a^(2/3)*e^2) + (b*p*Log[b + a*x^3])/(3*a*e) + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/e^4 + (d^3*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e^4 - (3*d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4} -{x^2*Log[c*(a + b/x^3)^p]/(d + e*x), x, 34, (Sqrt[3]*b^(1/3)*d*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(a^(1/3)*e^2) - (Sqrt[3]*b^(2/3)*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(2*a^(2/3)*e) - (d*x*Log[c*(a + b/x^3)^p])/e^2 + (x^2*Log[c*(a + b/x^3)^p])/(2*e) - (b^(1/3)*d*p*Log[b^(1/3) + a^(1/3)*x])/(a^(1/3)*e^2) - (b^(2/3)*p*Log[b^(1/3) + a^(1/3)*x])/(2*a^(2/3)*e) + (d^2*Log[c*(a + b/x^3)^p]*Log[d + e*x])/e^3 + (3*d^2*p*Log[-((e*x)/d)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e^3 - (d^2*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/e^3 + (b^(1/3)*d*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*a^(1/3)*e^2) + (b^(2/3)*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(4*a^(2/3)*e) - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/e^3 - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/e^3 - (d^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e^3 + (3*d^2*p*PolyLog[2, 1 + (e*x)/d])/e^3} -{x^1*Log[c*(a + b/x^3)^p]/(d + e*x), x, 26, -((Sqrt[3]*b^(1/3)*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(a^(1/3)*e)) + (x*Log[c*(a + b/x^3)^p])/e + (b^(1/3)*p*Log[b^(1/3) + a^(1/3)*x])/(a^(1/3)*e) - (d*Log[c*(a + b/x^3)^p]*Log[d + e*x])/e^2 - (3*d*p*Log[-((e*x)/d)]*Log[d + e*x])/e^2 + (d*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e^2 + (d*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e^2 + (d*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/e^2 - (b^(1/3)*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*a^(1/3)*e) + (d*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/e^2 + (d*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/e^2 + (d*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e^2 - (3*d*p*PolyLog[2, 1 + (e*x)/d])/e^2} -{x^0*Log[c*(a + b/x^3)^p]/(d + e*x), x, 16, (Log[c*(a + b/x^3)^p]*Log[d + e*x])/e + (3*p*Log[-((e*x)/d)]*Log[d + e*x])/e - (p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/e - (p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/e - (p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/e - (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/e - (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/e - (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/e + (3*p*PolyLog[2, 1 + (e*x)/d])/e} -{Log[c*(a + b/x^3)^p]/(x^1*(d + e*x)), x, 21, -((Log[c*(a + b/x^3)^p]*Log[-(b/(a*x^3))])/(3*d)) - (Log[c*(a + b/x^3)^p]*Log[d + e*x])/d - (3*p*Log[-((e*x)/d)]*Log[d + e*x])/d + (p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/d + (p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/d + (p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/d - (p*PolyLog[2, 1 + b/(a*x^3)])/(3*d) + (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/d + (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/d + (p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/d - (3*p*PolyLog[2, 1 + (e*x)/d])/d} -{Log[c*(a + b/x^3)^p]/(x^2*(d + e*x)), x, 30, (3*p)/(d*x) - (Sqrt[3]*a^(1/3)*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(b^(1/3)*d) - Log[c*(a + b/x^3)^p]/(d*x) + (e*Log[c*(a + b/x^3)^p]*Log[-(b/(a*x^3))])/(3*d^2) - (a^(1/3)*p*Log[b^(1/3) + a^(1/3)*x])/(b^(1/3)*d) + (e*Log[c*(a + b/x^3)^p]*Log[d + e*x])/d^2 + (3*e*p*Log[-((e*x)/d)]*Log[d + e*x])/d^2 - (e*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/d^2 - (e*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/d^2 - (e*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/d^2 + (a^(1/3)*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*b^(1/3)*d) + (e*p*PolyLog[2, 1 + b/(a*x^3)])/(3*d^2) - (e*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/d^2 - (e*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/d^2 + (3*e*p*PolyLog[2, 1 + (e*x)/d])/d^2} -{Log[c*(a + b/x^3)^p]/(x^3*(d + e*x)), x, 39, (3*p)/(4*d*x^2) - (3*e*p)/(d^2*x) - (Sqrt[3]*a^(2/3)*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(2*b^(2/3)*d) + (Sqrt[3]*a^(1/3)*e*p*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(b^(1/3)*d^2) - Log[c*(a + b/x^3)^p]/(2*d*x^2) + (e*Log[c*(a + b/x^3)^p])/(d^2*x) - (e^2*Log[c*(a + b/x^3)^p]*Log[-(b/(a*x^3))])/(3*d^3) + (a^(2/3)*p*Log[b^(1/3) + a^(1/3)*x])/(2*b^(2/3)*d) + (a^(1/3)*e*p*Log[b^(1/3) + a^(1/3)*x])/(b^(1/3)*d^2) - (e^2*Log[c*(a + b/x^3)^p]*Log[d + e*x])/d^3 - (3*e^2*p*Log[-((e*x)/d)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - b^(1/3)*e))]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*((-1)^(2/3)*b^(1/3) + a^(1/3)*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e))]*Log[d + e*x])/d^3 + (e^2*p*Log[((-1)^(1/3)*e*(b^(1/3) + (-1)^(2/3)*a^(1/3)*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)]*Log[d + e*x])/d^3 - (a^(2/3)*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(4*b^(2/3)*d) - (a^(1/3)*e*p*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(2*b^(1/3)*d^2) - (e^2*p*PolyLog[2, 1 + b/(a*x^3)])/(3*d^3) + (e^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - b^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d + (-1)^(1/3)*b^(1/3)*e)])/d^3 + (e^2*p*PolyLog[2, (a^(1/3)*(d + e*x))/(a^(1/3)*d - (-1)^(2/3)*b^(1/3)*e)])/d^3 - (3*e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x^r)^q Log[c (d+e x^m)^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x^2)^q Log[c (d+e x^m)^n]*) - - -(* ::Subsubsection:: *) -(*q>0*) - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{Log[c*(d + e*x^3)^p]/(f + g*x^2), x, 16, (3*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*I*Sqrt[f]*Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^3)^p])/(Sqrt[f]*Sqrt[g]) - (3*I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(d^(1/3) + e^(1/3)*x))/((I*e^(1/3)*Sqrt[f] + d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*I*Sqrt[f]*Sqrt[g]*((-1)^(2/3)*d^(1/3) + e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(1/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*(-1)^(5/6)*Sqrt[f]*Sqrt[g]*(d^(1/3) + (-1)^(2/3)*e^(1/3)*x))/((e^(1/3)*Sqrt[f] + (-1)^(5/6)*d^(1/3)*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[c*(d + e*x^2)^p]/(f + g*x^2), x, 12, (2*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[c*(d + e*x^1)^p]/(f + g*x^2), x, 8, (Log[c*(d + e*x)^p]*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*x)^p]*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*Sqrt[g])} -{Log[c*(d + e/x^1)^p]/(f + g*x^2), x, 12, (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e/x)^p])/(Sqrt[f]*Sqrt[g]) + (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(e + d*x))/((I*d*Sqrt[f] + e*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, -((I*Sqrt[g]*x)/Sqrt[f])])/(2*Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]])/(2*Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(e + d*x))/((I*d*Sqrt[f] + e*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[c*(d + e/x^2)^p]/(f + g*x^2), x, 18, (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e/x^2)^p])/(Sqrt[f]*Sqrt[g]) + (2*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[e] - Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, -((I*Sqrt[g]*x)/Sqrt[f])])/(Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, (I*Sqrt[g]*x)/Sqrt[f]])/(Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[e] - Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] - Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[e] + Sqrt[-d]*x))/((I*Sqrt[-d]*Sqrt[f] + Sqrt[e]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x^2)^q Log[c (d+e x^(m/2))^n]*) - - -(* {Log[c*(d + e*x^(3/2))^p]/(f + g*x^2), x, 43, (p*Log[-((g^(1/4)*((-1)^(1/3)*d^(1/3) - e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(1/3)*d^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[(g^(1/4)*(d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + d^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[(g^(1/4)*((-1)^(2/3)*d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(2/3)*d^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*((-1)^(1/3)*d^(1/3) - e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - (-1)^(1/3)*d^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[(g^(1/4)*(d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + d^(1/3)*g^(1/4))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[(g^(1/4)*((-1)^(2/3)*d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + (-1)^(2/3)*d^(1/3)*g^(1/4))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[-((g^(1/4)*(d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - d^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[-((g^(1/4)*((-1)^(2/3)*d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(2/3)*d^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[((-1)^(1/3)*g^(1/4)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(1/3)*d^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*(d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - d^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*((-1)^(2/3)*d^(1/3) + e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - (-1)^(2/3)*d^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[((-1)^(1/3)*g^(1/4)*(d^(1/3) + (-1)^(2/3)*e^(1/3)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + (-1)^(1/3)*d^(1/3)*g^(1/4))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]]*Log[c*(d + e*x^(3/2))^p])/(2*Sqrt[-f]*Sqrt[g]) + (Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[c*(d + e*x^(3/2))^p])/(2*Sqrt[-f]*Sqrt[g]) - (Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]]*Log[c*(d + e*x^(3/2))^p])/(2*Sqrt[-f]*Sqrt[g]) + (Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]]*Log[c*(d + e*x^(3/2))^p])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(1/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(2/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - (-1)^(1/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + (-1)^(2/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(1/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (e^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(2/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) + (-1)^(1/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (e^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e^(1/3)*(-f)^(1/4) - (-1)^(2/3)*d^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])} *) -{Log[c*(d + e*x^(1/2))^p]/(f + g*x^2), x, 19, -((Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])} -{Log[c*(d + e/x^(1/2))^p]/(f + g*x^2), x, 20, -((Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])) - (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] - e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - (-f)^(1/4)/Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + (-f)^(1/4)/Sqrt[x]))/(d*(-f)^(1/4) - e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] - e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) - e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])} -(* {Log[c*(d + e/x^(3/2))^p]/(f + g*x^2), x, 0, -((Log[c*(d + e/x^(3/2))^p]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g])) + (p*Log[-((g^(1/4)*((-1)^(1/3)*e^(1/3) - d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(1/3)*e^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[(g^(1/4)*(e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + e^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[(g^(1/4)*((-1)^(2/3)*e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(2/3)*e^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/x^(3/2))^p]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*((-1)^(1/3)*e^(1/3) - d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - (-1)^(1/3)*e^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[(g^(1/4)*(e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + e^(1/3)*g^(1/4))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[(g^(1/4)*((-1)^(2/3)*e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + (-1)^(2/3)*e^(1/3)*g^(1/4))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e/x^(3/2))^p]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[-((g^(1/4)*(e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - e^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[-((g^(1/4)*((-1)^(2/3)*e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(2/3)*e^(1/3)*g^(1/4)))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (p*Log[((-1)^(1/3)*g^(1/4)*(e^(1/3) + (-1)^(2/3)*d^(1/3)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(1/3)*e^(1/3)*g^(1/4))]*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/x^(3/2))^p]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*(e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - e^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[-((g^(1/4)*((-1)^(2/3)*e^(1/3) + d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - (-1)^(2/3)*e^(1/3)*g^(1/4)))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (p*Log[((-1)^(1/3)*g^(1/4)*(e^(1/3) + (-1)^(2/3)*d^(1/3)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + (-1)^(1/3)*e^(1/3)*g^(1/4))]*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]])/(2*Sqrt[-f]*Sqrt[g]) - (3*p*Log[Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]]*Log[-((g^(1/4)*Sqrt[x])/Sqrt[-Sqrt[-f]])])/(2*Sqrt[-f]*Sqrt[g]) - (3*p*Log[Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]]*Log[(g^(1/4)*Sqrt[x])/Sqrt[-Sqrt[-f]]])/(2*Sqrt[-f]*Sqrt[g]) + (3*p*Log[(-f)^(1/4) + g^(1/4)*Sqrt[x]]*Log[-((g^(1/4)*Sqrt[x])/(-f)^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (3*p*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[(g^(1/4)*Sqrt[x])/(-f)^(1/4)])/(2*Sqrt[-f]*Sqrt[g]) - (3*p*PolyLog[2, (Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x])/Sqrt[-Sqrt[-f]]])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(1/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(2/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (3*p*PolyLog[2, ((-f)^(1/4) - g^(1/4)*Sqrt[x])/(-f)^(1/4)])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - (-1)^(1/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + (-1)^(2/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (3*p*PolyLog[2, (Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x])/Sqrt[-Sqrt[-f]]])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] + (-1)^(1/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (d^(1/3)*(Sqrt[-Sqrt[-f]] + g^(1/4)*Sqrt[x]))/(d^(1/3)*Sqrt[-Sqrt[-f]] - (-1)^(2/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (3*p*PolyLog[2, ((-f)^(1/4) + g^(1/4)*Sqrt[x])/(-f)^(1/4)])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) + (-1)^(1/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (d^(1/3)*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(d^(1/3)*(-f)^(1/4) - (-1)^(2/3)*e^(1/3)*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g])} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x^2)^q Log[c (d+e x^2)^n]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x^2)^3*Log[c*(d + e*x^2)^p], x, 17, -2*f^3*p*x + (2*d*f^2*g*p*x)/e - (6*d^2*f*g^2*p*x)/(5*e^2) + (2*d^3*g^3*p*x)/(7*e^3) - (2/3)*f^2*g*p*x^3 + (2*d*f*g^2*p*x^3)/(5*e) - (2*d^2*g^3*p*x^3)/(21*e^2) - (6/25)*f*g^2*p*x^5 + (2*d*g^3*p*x^5)/(35*e) - (2/49)*g^3*p*x^7 + (2*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*f^2*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + (6*d^(5/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (2*d^(7/2)*g^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) + f^3*x*Log[c*(d + e*x^2)^p] + f^2*g*x^3*Log[c*(d + e*x^2)^p] + (3/5)*f*g^2*x^5*Log[c*(d + e*x^2)^p] + (1/7)*g^3*x^7*Log[c*(d + e*x^2)^p]} -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 13, -2*f^2*p*x + (4*d*f*g*p*x)/(3*e) - (2*d^2*g^2*p*x)/(5*e^2) - (4/9)*f*g*p*x^3 + (2*d*g^2*p*x^3)/(15*e) - (2/25)*g^2*p*x^5 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (4*d^(3/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (2*d^(5/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) + f^2*x*Log[c*(d + e*x^2)^p] + (2/3)*f*g*x^3*Log[c*(d + e*x^2)^p] + (1/5)*g^2*x^5*Log[c*(d + e*x^2)^p]} -{(f + g*x^2)^1*Log[c*(d + e*x^2)^p], x, 9, -2*f*p*x + (2*d*g*p*x)/(3*e) - (2/9)*g*p*x^3 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p] + (1/3)*g*x^3*Log[c*(d + e*x^2)^p]} -{Log[c*(d + e*x^2)^p]/(f + g*x^2)^1, x, 12, (2*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[c*(d + e*x^2)^p]/(f + g*x^2)^2, x, 26, (Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(f*(e*f - d*g)) - (e*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) + (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(f^(3/2)*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(2*f^(3/2)*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(3/2)*Sqrt[g]) + (e*p*Log[Sqrt[-f] + Sqrt[g]*x])/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqrt[-f] - Sqrt[g]*x)) + Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqrt[-f] + Sqrt[g]*x)) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*f^(3/2)*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*f^(3/2)*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(3/2)*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(3/2)*Sqrt[g])} - - -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2, x, 50, 8*f^2*p^2*x - (64*d*f*g*p^2*x)/(9*e) + (184*d^2*g^2*p^2*x)/(75*e^2) + (16/27)*f*g*p^2*x^3 - (64*d*g^2*p^2*x^3)/(225*e) + (8/125)*g^2*p^2*x^5 - (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (64*d^(3/2)*f*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - (184*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(75*e^(5/2)) + (4*I*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (8*I*d^(3/2)*f*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(3*e^(3/2)) + (4*I*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(5*e^(5/2)) + (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (16*d^(3/2)*f*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + (8*d^(5/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(5*e^(5/2)) - 4*f^2*p*x*Log[c*(d + e*x^2)^p] + (8*d*f*g*p*x*Log[c*(d + e*x^2)^p])/(3*e) - (4*d^2*g^2*p*x*Log[c*(d + e*x^2)^p])/(5*e^2) - (8/9)*f*g*p*x^3*Log[c*(d + e*x^2)^p] + (4*d*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(15*e) - (4/25)*g^2*p*x^5*Log[c*(d + e*x^2)^p] + (4*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (8*d^(3/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(3*e^(3/2)) + (4*d^(5/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(5*e^(5/2)) + f^2*x*Log[c*(d + e*x^2)^p]^2 + (2/3)*f*g*x^3*Log[c*(d + e*x^2)^p]^2 + (1/5)*g^2*x^5*Log[c*(d + e*x^2)^p]^2 + (4*I*Sqrt[d]*f^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*I*d^(3/2)*f*g*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + (4*I*d^(5/2)*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(5*e^(5/2))} -{(f + g*x^2)^1*Log[c*(d + e*x^2)^p]^2, x, 30, 8*f*p^2*x - (32*d*g*p^2*x)/(9*e) + (8/27)*g*p^2*x^3 - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) + (4*I*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (4*I*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(3*e^(3/2)) + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) - 4*f*p*x*Log[c*(d + e*x^2)^p] + (4*d*g*p*x*Log[c*(d + e*x^2)^p])/(3*e) - (4/9)*g*p*x^3*Log[c*(d + e*x^2)^p] + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (4*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p]^2 + (1/3)*g*x^3*Log[c*(d + e*x^2)^p]^2 + (4*I*Sqrt[d]*f*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (4*I*d^(3/2)*g*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2))} -{Log[c*(d + e*x^2)^p]^2/(f + g*x^2)^1, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^2/(f + g*x^2), x]} -{Log[c*(d + e*x^2)^p]^2/(f + g*x^2)^2, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^2/(f + g*x^2)^2, x]} - - -{(f + g*x^2)^1*Log[c*(d + e*x^2)^p]^3, x, 48, -48*f*p^3*x + (208*d*g*p^3*x)/(9*e) - (16/27)*g*p^3*x^3 + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (208*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - (24*I*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (32*I*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(3*e^(3/2)) - (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (64*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (32*d*g*p^2*x*Log[c*(d + e*x^2)^p])/(3*e) + (8/9)*g*p^2*x^3*Log[c*(d + e*x^2)^p] - (24*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + (32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(3*e^(3/2)) - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (2*d*g*p*x*Log[c*(d + e*x^2)^p]^2)/e - (2/3)*g*p*x^3*Log[c*(d + e*x^2)^p]^2 + f*x*Log[c*(d + e*x^2)^p]^3 + (1/3)*g*x^3*Log[c*(d + e*x^2)^p]^3 - (24*I*Sqrt[d]*f*p^3*PolyLog[2, -((Sqrt[d] - I*Sqrt[e]*x)/(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[e] + (32*I*d^(3/2)*g*p^3*PolyLog[2, -((Sqrt[d] - I*Sqrt[e]*x)/(Sqrt[d] + I*Sqrt[e]*x))])/(3*e^(3/2)) - (2*d*(-3*e*f + d*g)*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x])/e, -48*f*p^3*x + (208*d*g*p^3*x)/(9*e) - (16/27)*g*p^3*x^3 + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (208*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) - (24*I*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (32*I*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(3*e^(3/2)) - (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (64*d^(3/2)*g*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (32*d*g*p^2*x*Log[c*(d + e*x^2)^p])/(3*e) + (8/9)*g*p^2*x^3*Log[c*(d + e*x^2)^p] - (24*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + (32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(3*e^(3/2)) - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (2*d*g*p*x*Log[c*(d + e*x^2)^p]^2)/e - (2/3)*g*p*x^3*Log[c*(d + e*x^2)^p]^2 + f*x*Log[c*(d + e*x^2)^p]^3 + (1/3)*g*x^3*Log[c*(d + e*x^2)^p]^3 - (24*I*Sqrt[d]*f*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (32*I*d^(3/2)*g*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) + 6*d*f*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x] - (2*d^2*g*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x])/e} -{Log[c*(d + e*x^2)^p]^3/(f + g*x^2)^1, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^3/(f + g*x^2), x]} -{Log[c*(d + e*x^2)^p]^3/(f + g*x^2)^2, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^3/(f + g*x^2)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x^2)^2/Log[c*(d + e*x^2)^p], x, 0, Unintegrable[(f + g*x^2)^2/Log[c*(d + e*x^2)^p], x]} -{(f + g*x^2)^1/Log[c*(d + e*x^2)^p], x, 0, Unintegrable[(f + g*x^2)/Log[c*(d + e*x^2)^p], x]} -{1/((f + g*x^2)^1*Log[c*(d + e*x^2)^p]), x, 0, Unintegrable[1/((f + g*x^2)*Log[c*(d + e*x^2)^p]), x]} -{1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]), x, 0, Unintegrable[1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]), x]} - - -{(f + g*x^2)^2/Log[c*(d + e*x^2)^p]^2, x, 0, Unintegrable[(f + g*x^2)^2/Log[c*(d + e*x^2)^p]^2, x]} -{(f + g*x^2)^1/Log[c*(d + e*x^2)^p]^2, x, 0, Unintegrable[(f + g*x^2)/Log[c*(d + e*x^2)^p]^2, x]} -{1/((f + g*x^2)^1*Log[c*(d + e*x^2)^p]^2), x, 0, Unintegrable[1/((f + g*x^2)*Log[c*(d + e*x^2)^p]^2), x]} -{1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2), x, 0, Unintegrable[1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x^3)^q Log[c (d+e x^2)^n]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(f + g*x^3)^3*Log[c*(d + e*x^2)^p], x, 17, -2*f^3*p*x + (6*d^3*f*g^2*p*x)/(7*e^3) + (3*d*f^2*g*p*x^2)/(4*e) - (d^4*g^3*p*x^2)/(10*e^4) - (2*d^2*f*g^2*p*x^3)/(7*e^2) - (3/8)*f^2*g*p*x^4 + (d^3*g^3*p*x^4)/(20*e^3) + (6*d*f*g^2*p*x^5)/(35*e) - (d^2*g^3*p*x^6)/(30*e^2) - (6/49)*f*g^2*p*x^7 + (d*g^3*p*x^8)/(40*e) - (1/50)*g^3*p*x^10 + (2*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (6*d^(7/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (3*d^2*f^2*g*p*Log[d + e*x^2])/(4*e^2) + (d^5*g^3*p*Log[d + e*x^2])/(10*e^5) + f^3*x*Log[c*(d + e*x^2)^p] + (3/4)*f^2*g*x^4*Log[c*(d + e*x^2)^p] + (3/7)*f*g^2*x^7*Log[c*(d + e*x^2)^p] + (1/10)*g^3*x^10*Log[c*(d + e*x^2)^p]} -{(f + g*x^3)^2*Log[c*(d + e*x^2)^p], x, 13, -2*f^2*p*x + (2*d^3*g^2*p*x)/(7*e^3) + (d*f*g*p*x^2)/(2*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (1/4)*f*g*p*x^4 + (2*d*g^2*p*x^5)/(35*e) - (2/49)*g^2*p*x^7 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*Log[d + e*x^2])/(2*e^2) + f^2*x*Log[c*(d + e*x^2)^p] + (1/2)*f*g*x^4*Log[c*(d + e*x^2)^p] + (1/7)*g^2*x^7*Log[c*(d + e*x^2)^p]} -{(f + g*x^3)^1*Log[c*(d + e*x^2)^p], x, 9, -2*f*p*x + (d*g*p*x^2)/(4*e) - (1/8)*g*p*x^4 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (d^2*g*p*Log[d + e*x^2])/(4*e^2) + f*x*Log[c*(d + e*x^2)^p] + (1/4)*g*x^4*Log[c*(d + e*x^2)^p]} -{Log[c*(d + e*x^2)^p]/(f + g*x^3)^1, x, 29, -((p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) - g^(1/3)*x])/(3*f^(2/3)*g^(1/3))) - (p*Log[-((g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3)))]*Log[-f^(1/3) - g^(1/3)*x])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3)*p*Log[-(((-1)^(1/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1/3)))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3)*p*Log[((-1)^(1/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x])/(3*f^(2/3)*g^(1/3)) + ((-1)^(1/3)*p*Log[((-1)^(2/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x])/(3*f^(2/3)*g^(1/3)) + ((-1)^(1/3)*p*Log[-(((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3)))]*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x])/(3*f^(2/3)*g^(1/3)) + (Log[-f^(1/3) - g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(3*f^(2/3)*g^(1/3)) + ((-1)^(2/3)*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(3*f^(2/3)*g^(1/3)) - ((-1)^(1/3)*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(3*f^(2/3)*g^(1/3)) - (p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) - (p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) + ((-1)^(1/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) + ((-1)^(1/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3))} -{Log[c*(d + e*x^2)^p]/(f + g*x^3)^2, x, 47, If[$VersionNumber>=8, (2*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(e*f^(2/3) + d*g^(2/3))) + (2*(-1)^(2/3)*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((1 + (-1)^(1/3))^4*f^(4/3)*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))) + (4*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))) - (2*e*p*Log[f^(1/3) + g^(1/3)*x])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - (2*p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) - (2*p*Log[-((g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3)))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*e*p*Log[f^(1/3) - (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/3)) + (2*I*Sqrt[3]*p*Log[-(((-1)^(1/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1/3)))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + (2*I*Sqrt[3]*p*Log[((-1)^(1/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + (4*(-1)^(1/3)*e*p*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - (2*p*Log[((-1)^(2/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2*p*Log[-(((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3)))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) + (e*p*Log[d + e*x^2])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - ((-1)^(1/3)*e*p*Log[d + e*x^2])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/3)) - (2*(-1)^(1/3)*e*p*Log[d + e*x^2])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - Log[c*(d + e*x^2)^p]/(9*f^(4/3)*g^(1/3)*(f^(1/3) + g^(1/3)*x)) - Log[c*(d + e*x^2)^p]/((1 + (-1)^(1/3))^4*f^(4/3)*g^(1/3)*((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + ((-1)^(1/3)*Log[c*(d + e*x^2)^p])/(9*f^(4/3)*g^(1/3)*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x)) + (2*Log[f^(1/3) + g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*f^(5/3)*g^(1/3)) - (2*I*Sqrt[3]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + (2*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) + (2*I*Sqrt[3]*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + (2*I*Sqrt[3]*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)), (2*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(e*f^(2/3) + d*g^(2/3))) + (2*(-1)^(2/3)*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((1 + (-1)^(1/3))^4*f^(4/3)*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))) + (4*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))) - (2*e*p*Log[f^(1/3) + g^(1/3)*x])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - (2*p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) - (2*p*Log[-((g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3)))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*e*p*Log[f^(1/3) - (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/3)) + (4*(-1)^(1/3)*e*p*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) + (4*p*Log[-(((I + Sqrt[3])*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(2*I*Sqrt[e]*f^(1/3) - (I + Sqrt[3])*Sqrt[-d]*g^(1/3)))]*Log[2*f^(1/3) - (1 - I*Sqrt[3])*g^(1/3)*x])/(9*(1 - I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*Log[((I + Sqrt[3])*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(2*I*Sqrt[e]*f^(1/3) + (I + Sqrt[3])*Sqrt[-d]*g^(1/3))]*Log[2*f^(1/3) - (1 - I*Sqrt[3])*g^(1/3)*x])/(9*(1 - I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*Log[-(((1 + I*Sqrt[3])*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(2*Sqrt[e]*f^(1/3) - (1 + I*Sqrt[3])*Sqrt[-d]*g^(1/3)))]*Log[2*f^(1/3) - (1 + I*Sqrt[3])*g^(1/3)*x])/(9*(1 + I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*Log[((1 + I*Sqrt[3])*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(2*Sqrt[e]*f^(1/3) + (1 + I*Sqrt[3])*Sqrt[-d]*g^(1/3))]*Log[2*f^(1/3) - (1 + I*Sqrt[3])*g^(1/3)*x])/(9*(1 + I*Sqrt[3])*f^(5/3)*g^(1/3)) + (e*p*Log[d + e*x^2])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - ((-1)^(1/3)*e*p*Log[d + e*x^2])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/3)) - (2*(-1)^(1/3)*e*p*Log[d + e*x^2])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - Log[c*(d + e*x^2)^p]/(9*f^(4/3)*g^(1/3)*(f^(1/3) + g^(1/3)*x)) - Log[c*(d + e*x^2)^p]/((1 + (-1)^(1/3))^4*f^(4/3)*g^(1/3)*((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + ((-1)^(1/3)*Log[c*(d + e*x^2)^p])/(9*f^(4/3)*g^(1/3)*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x)) + (2*Log[f^(1/3) + g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*f^(5/3)*g^(1/3)) - (4*Log[2*f^(1/3) - (1 - I*Sqrt[3])*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*(1 - I*Sqrt[3])*f^(5/3)*g^(1/3)) - (4*Log[2*f^(1/3) - (1 + I*Sqrt[3])*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*(1 + I*Sqrt[3])*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) + (4*p*PolyLog[2, (Sqrt[e]*(2*f^(1/3) - (1 - I*Sqrt[3])*g^(1/3)*x))/(2*Sqrt[e]*f^(1/3) + (1 - I*Sqrt[3])*Sqrt[-d]*g^(1/3))])/(9*(1 - I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*PolyLog[2, (Sqrt[e]*(2*f^(1/3) - (1 - I*Sqrt[3])*g^(1/3)*x))/(2*Sqrt[e]*f^(1/3) + I*(I + Sqrt[3])*Sqrt[-d]*g^(1/3))])/(9*(1 - I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*PolyLog[2, (Sqrt[e]*(2*f^(1/3) - (1 + I*Sqrt[3])*g^(1/3)*x))/(2*Sqrt[e]*f^(1/3) - (1 + I*Sqrt[3])*Sqrt[-d]*g^(1/3))])/(9*(1 + I*Sqrt[3])*f^(5/3)*g^(1/3)) + (4*p*PolyLog[2, (Sqrt[e]*(2*f^(1/3) - (1 + I*Sqrt[3])*g^(1/3)*x))/(2*Sqrt[e]*f^(1/3) + (1 + I*Sqrt[3])*Sqrt[-d]*g^(1/3))])/(9*(1 + I*Sqrt[3])*f^(5/3)*g^(1/3))]} - - -{(f + g*x^3)^3*Log[c*(d + e*x^2)^p]^2, x, 55, 8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (3*d*f^2*g*p^2*x^2)/e + (d^4*g^3*p^2*x^2)/e^4 + (568*d^2*f*g^2*p^2*x^3)/(735*e^2) - (288*d*f*g^2*p^2*x^5)/(1225*e) + (24/343)*f*g^2*p^2*x^7 + (3*f^2*g*p^2*(d + e*x^2)^2)/(8*e^2) - (d^3*g^3*p^2*(d + e*x^2)^2)/(2*e^5) + (2*d^2*g^3*p^2*(d + e*x^2)^3)/(9*e^5) - (d*g^3*p^2*(d + e*x^2)^4)/(16*e^5) + (g^3*p^2*(d + e*x^2)^5)/(125*e^5) - (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1408*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(245*e^(7/2)) + (4*I*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (12*I*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(7*e^(7/2)) + (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (24*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - (d^5*g^3*p^2*Log[d + e*x^2]^2)/(10*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*f*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(7*e^2) + (12*d*f*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (12/49)*f*g^2*p*x^7*Log[c*(d + e*x^2)^p] + (3*d*f^2*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (d^4*g^3*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^5 - (3*f^2*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (d^3*g^3*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/e^5 - (2*d^2*g^3*p*(d + e*x^2)^3*Log[c*(d + e*x^2)^p])/(3*e^5) + (d*g^3*p*(d + e*x^2)^4*Log[c*(d + e*x^2)^p])/(4*e^5) - (g^3*p*(d + e*x^2)^5*Log[c*(d + e*x^2)^p])/(25*e^5) + (4*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (12*d^(7/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(7*e^(7/2)) + (d^5*g^3*p*Log[d + e*x^2]*Log[c*(d + e*x^2)^p])/(5*e^5) + f^3*x*Log[c*(d + e*x^2)^p]^2 + (3/7)*f*g^2*x^7*Log[c*(d + e*x^2)^p]^2 + (1/10)*g^3*x^10*Log[c*(d + e*x^2)^p]^2 - (3*d*f^2*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + (3*f^2*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + (4*I*Sqrt[d]*f^3*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (12*I*d^(7/2)*f*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2))} -{(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^2, x, 47, 8*f^2*p^2*x - (1408*d^3*g^2*p^2*x)/(735*e^3) - (2*d*f*g*p^2*x^2)/e + (568*d^2*g^2*p^2*x^3)/(2205*e^2) - (96*d*g^2*p^2*x^5)/(1225*e) + (8/343)*g^2*p^2*x^7 + (f*g*p^2*(d + e*x^2)^2)/(4*e^2) - (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(735*e^(7/2)) + (4*I*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (4*I*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(7*e^(7/2)) + (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - 4*f^2*p*x*Log[c*(d + e*x^2)^p] + (4*d^3*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*g^2*p*x^3*Log[c*(d + e*x^2)^p])/(21*e^2) + (4*d*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (4/49)*g^2*p*x^7*Log[c*(d + e*x^2)^p] + (2*d*f*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(2*e^2) + (4*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (4*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(7*e^(7/2)) + f^2*x*Log[c*(d + e*x^2)^p]^2 + (1/7)*g^2*x^7*Log[c*(d + e*x^2)^p]^2 - (d*f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/e^2 + (f*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + (4*I*Sqrt[d]*f^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (4*I*d^(7/2)*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2))} -{(f + g*x^3)^1*Log[c*(d + e*x^2)^p]^2, x, 23, 8*f*p^2*x - (d*g*p^2*x^2)/e + (g*p^2*(d + e*x^2)^2)/(8*e^2) - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (4*I*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - 4*f*p*x*Log[c*(d + e*x^2)^p] + (d*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + f*x*Log[c*(d + e*x^2)^p]^2 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + (4*I*Sqrt[d]*f*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e]} -{Log[c*(d + e*x^2)^p]^2/(f + g*x^3)^1, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^2/(f + g*x^3), x]} -{Log[c*(d + e*x^2)^p]^2/(f + g*x^3)^2, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^2/(f + g*x^3)^2, x]} - - -{(f + g*x^3)^2*Log[c*(d + e*x^2)^p]^3, x, 103, -48*f^2*p^3*x + (351136*d^3*g^2*p^3*x)/(25725*e^3) + (6*d*f*g*p^3*x^2)/e - (55456*d^2*g^2*p^3*x^3)/(77175*e^2) + (5232*d*g^2*p^3*x^5)/(42875*e) - (48*g^2*p^3*x^7)/2401 - (3*f*g*p^3*(d + e*x^2)^2)/(8*e^2) + (48*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (351136*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(25725*e^(7/2)) - (24*I*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] + (1408*I*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/(245*e^(7/2)) - (48*Sqrt[d]*f^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (2816*d^(7/2)*g^2*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(245*e^(7/2)) + 24*f^2*p^2*x*Log[c*(d + e*x^2)^p] - (1408*d^3*g^2*p^2*x*Log[c*(d + e*x^2)^p])/(245*e^3) + (568*d^2*g^2*p^2*x^3*Log[c*(d + e*x^2)^p])/(735*e^2) - (288*d*g^2*p^2*x^5*Log[c*(d + e*x^2)^p])/(1225*e) + (24/343)*g^2*p^2*x^7*Log[c*(d + e*x^2)^p] - (6*d*f*g*p^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*f*g*p^2*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) - (24*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(245*e^(7/2)) - 6*f^2*p*x*Log[c*(d + e*x^2)^p]^2 + (6*d^3*g^2*p*x*Log[c*(d + e*x^2)^p]^2)/(7*e^3) - (2*d^2*g^2*p*x^3*Log[c*(d + e*x^2)^p]^2)/(7*e^2) + (6*d*g^2*p*x^5*Log[c*(d + e*x^2)^p]^2)/(35*e) - (6/49)*g^2*p*x^7*Log[c*(d + e*x^2)^p]^2 + (3*d*f*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/e^2 - (3*f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + f^2*x*Log[c*(d + e*x^2)^p]^3 + (1/7)*g^2*x^7*Log[c*(d + e*x^2)^p]^3 - (d*f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/e^2 + (f*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(2*e^2) - (24*I*Sqrt[d]*f^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + (1408*I*d^(7/2)*g^2*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(245*e^(7/2)) + 6*d*f^2*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x] - (6*d^4*g^2*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x])/(7*e^3)} -{(f + g*x^3)^1*Log[c*(d + e*x^2)^p]^3, x, 28, -48*f*p^3*x + (3*d*g*p^3*x^2)/e - (3*g*p^3*(d + e*x^2)^2)/(16*e^2) + (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (24*I*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/Sqrt[e] - (48*Sqrt[d]*f*p^3*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 24*f*p^2*x*Log[c*(d + e*x^2)^p] - (3*d*g*p^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 + (3*g*p^2*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(8*e^2) - (24*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/Sqrt[e] - 6*f*p*x*Log[c*(d + e*x^2)^p]^2 + (3*d*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/(2*e^2) - (3*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(8*e^2) + f*x*Log[c*(d + e*x^2)^p]^3 - (d*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^3)/(2*e^2) + (g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^3)/(4*e^2) - (24*I*Sqrt[d]*f*p^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] + 6*d*f*p*Unintegrable[Log[c*(d + e*x^2)^p]^2/(d + e*x^2), x]} -{Log[c*(d + e*x^2)^p]^3/(f + g*x^3)^1, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^3/(f + g*x^3), x]} -{Log[c*(d + e*x^2)^p]^3/(f + g*x^3)^2, x, 0, Unintegrable[Log[c*(d + e*x^2)^p]^3/(f + g*x^3)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(f + g*x^3)^2/Log[c*(d + e*x^2)^p], x, 0, Unintegrable[(f + g*x^3)^2/Log[c*(d + e*x^2)^p], x]} -{(f + g*x^3)^1/Log[c*(d + e*x^2)^p], x, 0, Unintegrable[(f + g*x^3)/Log[c*(d + e*x^2)^p], x]} -{1/((f + g*x^3)^1*Log[c*(d + e*x^2)^p]), x, 0, Unintegrable[1/((f + g*x^3)*Log[c*(d + e*x^2)^p]), x]} -{1/((f + g*x^3)^2*Log[c*(d + e*x^2)^p]), x, 0, Unintegrable[1/((f + g*x^3)^2*Log[c*(d + e*x^2)^p]), x]} - - -{(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2, x, 0, Unintegrable[(f + g*x^3)^2/Log[c*(d + e*x^2)^p]^2, x]} -{(f + g*x^3)^1/Log[c*(d + e*x^2)^p]^2, x, 0, Unintegrable[(f + g*x^3)/Log[c*(d + e*x^2)^p]^2, x]} -{1/((f + g*x^3)^1*Log[c*(d + e*x^2)^p]^2), x, 0, Unintegrable[1/((f + g*x^3)*Log[c*(d + e*x^2)^p]^2), x]} -{1/((f + g*x^3)^2*Log[c*(d + e*x^2)^p]^2), x, 0, Unintegrable[1/((f + g*x^3)^2*Log[c*(d + e*x^2)^p]^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^r (f+g x^s)^q Log[c (d+e x^m)^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^r (f+g x^2)^q Log[c (d+e x^2)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^5*(f + g*x^2)*Log[c*(d + e*x^2)^p], x, 5, -((d^2*(4*e*f - 3*d*g)*p*x^2)/(24*e^3)) + (d*(4*e*f - 3*d*g)*p*x^4)/(48*e^2) - ((4*e*f - 3*d*g)*p*x^6)/(72*e) - (1/32)*g*p*x^8 + (d^3*(4*e*f - 3*d*g)*p*Log[d + e*x^2])/(24*e^4) + (1/6)*f*x^6*Log[c*(d + e*x^2)^p] + (1/8)*g*x^8*Log[c*(d + e*x^2)^p]} -{x^3*(f + g*x^2)*Log[c*(d + e*x^2)^p], x, 5, (d*(3*e*f - 2*d*g)*p*x^2)/(12*e^2) - ((3*e*f - 2*d*g)*p*x^4)/(24*e) - (1/18)*g*p*x^6 - (d^2*(3*e*f - 2*d*g)*p*Log[d + e*x^2])/(12*e^3) + (1/4)*f*x^4*Log[c*(d + e*x^2)^p] + (1/6)*g*x^6*Log[c*(d + e*x^2)^p]} -{x^1*(f + g*x^2)*Log[c*(d + e*x^2)^p], x, 4, -((e*f - d*g)*p*x^2)/(4*e) - (p*(f + g*x^2)^2)/(8*g) - ((e*f - d*g)^2*p*Log[d + e*x^2])/(4*e^2*g) + ((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/(4*g)} -{((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^1, x, 7, (-(1/2))*g*p*x^2 + (g*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e) + (1/2)*f*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + (1/2)*f*p*PolyLog[2, 1 + (e*x^2)/d]} -{((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^3, x, 9, (e*f*p*Log[x])/d - (e*f*p*Log[d + e*x^2])/(2*d) - (f*Log[c*(d + e*x^2)^p])/(2*x^2) + (1/2)*g*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + (1/2)*g*p*PolyLog[2, 1 + (e*x^2)/d]} -{((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^5, x, 5, -((e*f*p)/(4*d*x^2)) - (e*(e*f - 2*d*g)*p*Log[x])/(2*d^2) + ((e*f - d*g)^2*p*Log[d + e*x^2])/(4*d^2*f) - ((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/(4*f*x^4)} -{((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^7, x, 5, -((e*f*p)/(12*d*x^4)) + (e*(2*e*f - 3*d*g)*p)/(12*d^2*x^2) + (e^2*(2*e*f - 3*d*g)*p*Log[x])/(6*d^3) - (e^2*(2*e*f - 3*d*g)*p*Log[d + e*x^2])/(12*d^3) - (f*Log[c*(d + e*x^2)^p])/(6*x^6) - (g*Log[c*(d + e*x^2)^p])/(4*x^4)} -{((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^9, x, 5, -((e*f*p)/(24*d*x^6)) + (e*(3*e*f - 4*d*g)*p)/(48*d^2*x^4) - (e^2*(3*e*f - 4*d*g)*p)/(24*d^3*x^2) - (e^3*(3*e*f - 4*d*g)*p*Log[x])/(12*d^4) + (e^3*(3*e*f - 4*d*g)*p*Log[d + e*x^2])/(24*d^4) - (f*Log[c*(d + e*x^2)^p])/(8*x^8) - (g*Log[c*(d + e*x^2)^p])/(6*x^6)} - -{x^2*(f + g*x^2)*Log[c*(d + e*x^2)^p], x, 10, (2*d*f*p*x)/(3*e) - (2*d^2*g*p*x)/(5*e^2) - (2*f*p*x^3)/9 + (2*d*g*p*x^3)/(15*e) - (2*g*p*x^5)/25 - (2*d^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (2*d^(5/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) + (f*x^3*Log[c*(d + e*x^2)^p])/3 + (g*x^5*Log[c*(d + e*x^2)^p])/5} -{x^0*(f + g*x^2)*Log[c*(d + e*x^2)^p], x, 9, -2*f*p*x + (2*d*g*p*x)/(3*e) - (2*g*p*x^3)/9 + (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + f*x*Log[c*(d + e*x^2)^p] + (g*x^3*Log[c*(d + e*x^2)^p])/3} -{(f + g*x^2)*Log[c*(d + e*x^2)^p]/x^2, x, 7, -2*g*p*x + (2*(e*f + d*g)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) - (f*Log[c*(d + e*x^2)^p])/x + g*x*Log[c*(d + e*x^2)^p], -2*g*p*x + (2*Sqrt[e]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[d]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (f*Log[c*(d + e*x^2)^p])/x + g*x*Log[c*(d + e*x^2)^p]} -{(f + g*x^2)*Log[c*(d + e*x^2)^p]/x^4, x, 7, (-2*e*f*p)/(3*d*x) - (2*e^(3/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (f*Log[c*(d + e*x^2)^p])/(3*x^3) - (g*Log[c*(d + e*x^2)^p])/x} -{(f + g*x^2)*Log[c*(d + e*x^2)^p]/x^6, x, 9, -((2*e*f*p)/(15*d*x^3)) + (2*e^2*f*p)/(5*d^2*x) - (2*e*g*p)/(3*d*x) + (2*e^(5/2)*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*d^(5/2)) - (2*e^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) - (f*Log[c*(d + e*x^2)^p])/(5*x^5) - (g*Log[c*(d + e*x^2)^p])/(3*x^3)} - - -{x^5*(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 5, -((d^2*(e*f - d*g)^2*p*x^2)/(2*e^4)) + (d*(e*f - 2*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(4*e^5) - ((e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2)*p*(d + e*x^2)^3)/(18*e^5) - (g*(e*f - 2*d*g)*p*(d + e*x^2)^4)/(16*e^5) - (g^2*p*(d + e*x^2)^5)/(50*e^5) + (d^3*(10*e^2*f^2 - 15*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2])/(60*e^5) + (1/6)*f^2*x^6*Log[c*(d + e*x^2)^p] + (1/4)*f*g*x^8*Log[c*(d + e*x^2)^p] + (1/10)*g^2*x^10*Log[c*(d + e*x^2)^p]} -{x^3*(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 5, (d*(e*f - d*g)^2*p*x^2)/(2*e^3) - ((e*f - 3*d*g)*(e*f - d*g)*p*(d + e*x^2)^2)/(8*e^4) - (g*(2*e*f - 3*d*g)*p*(d + e*x^2)^3)/(18*e^4) - (g^2*p*(d + e*x^2)^4)/(32*e^4) - (d^2*(6*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2)*p*Log[d + e*x^2])/(24*e^4) + (1/4)*f^2*x^4*Log[c*(d + e*x^2)^p] + (1/3)*f*g*x^6*Log[c*(d + e*x^2)^p] + (1/8)*g^2*x^8*Log[c*(d + e*x^2)^p]} -{x^1*(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 4, -((e*f - d*g)^2*p*x^2)/(6*e^2) - ((e*f - d*g)*p*(f + g*x^2)^2)/(12*e*g) - (p*(f + g*x^2)^3)/(18*g) - ((e*f - d*g)^3*p*Log[d + e*x^2])/(6*e^3*g) + ((f + g*x^2)^3*Log[c*(d + e*x^2)^p])/(6*g)} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^1, x, 10, (-f)*g*p*x^2 + (d*g^2*p*x^2)/(4*e) - (1/8)*g^2*p*x^4 - (d^2*g^2*p*Log[d + e*x^2])/(4*e^2) + (1/4)*g^2*x^4*Log[c*(d + e*x^2)^p] + (f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e + (1/2)*f^2*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + (1/2)*f^2*p*PolyLog[2, 1 + (e*x^2)/d]} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^3, x, 11, (-(1/2))*g^2*p*x^2 + (e*f^2*p*Log[x])/d - (e*f^2*p*Log[d + e*x^2])/(2*d) - (f^2*Log[c*(d + e*x^2)^p])/(2*x^2) + (g^2*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e) + f*g*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + f*g*p*PolyLog[2, 1 + (e*x^2)/d]} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^5, x, 12, -((e*f^2*p)/(4*d*x^2)) - (e^2*f^2*p*Log[x])/(2*d^2) + (2*e*f*g*p*Log[x])/d + (e^2*f^2*p*Log[d + e*x^2])/(4*d^2) - (e*f*g*p*Log[d + e*x^2])/d - (f^2*Log[c*(d + e*x^2)^p])/(4*x^4) - (f*g*Log[c*(d + e*x^2)^p])/x^2 + (1/2)*g^2*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p] + (1/2)*g^2*p*PolyLog[2, 1 + (e*x^2)/d]} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^7, x, 5, -((e*f^2*p)/(12*d*x^4)) + (e*f*(e*f - 3*d*g)*p)/(6*d^2*x^2) + (e*(e^2*f^2 - 3*d*e*f*g + 3*d^2*g^2)*p*Log[x])/(3*d^3) - ((e*f - d*g)^3*p*Log[d + e*x^2])/(6*d^3*f) - ((f + g*x^2)^3*Log[c*(d + e*x^2)^p])/(6*f*x^6)} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^9, x, 5, -((e*f^2*p)/(24*d*x^6)) + (e*f*(3*e*f - 8*d*g)*p)/(48*d^2*x^4) - (e*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p)/(24*d^3*x^2) - (e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*Log[x])/(12*d^4) + (e^2*(3*e^2*f^2 - 8*d*e*f*g + 6*d^2*g^2)*p*Log[d + e*x^2])/(24*d^4) - (f^2*Log[c*(d + e*x^2)^p])/(8*x^8) - (f*g*Log[c*(d + e*x^2)^p])/(3*x^6) - (g^2*Log[c*(d + e*x^2)^p])/(4*x^4)} -{((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/x^11, x, 5, -((e*f^2*p)/(40*d*x^8)) + (e*f*(2*e*f - 5*d*g)*p)/(60*d^2*x^6) - (e*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(120*d^3*x^4) + (e^2*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p)/(60*d^4*x^2) + (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*Log[x])/(30*d^5) - (e^3*(6*e^2*f^2 - 15*d*e*f*g + 10*d^2*g^2)*p*Log[d + e*x^2])/(60*d^5) - (f^2*Log[c*(d + e*x^2)^p])/(10*x^10) - (f*g*Log[c*(d + e*x^2)^p])/(4*x^8) - (g^2*Log[c*(d + e*x^2)^p])/(6*x^6)} - -{x^2*(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 14, (2*d*f^2*p*x)/(3*e) - (4*d^2*f*g*p*x)/(5*e^2) + (2*d^3*g^2*p*x)/(7*e^3) - (2*f^2*p*x^3)/9 + (4*d*f*g*p*x^3)/(15*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (4*f*g*p*x^5)/25 + (2*d*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 - (2*d^(3/2)*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (4*d^(5/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) - (2*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) + (f^2*x^3*Log[c*(d + e*x^2)^p])/3 + (2*f*g*x^5*Log[c*(d + e*x^2)^p])/5 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7} -{x^0*(f + g*x^2)^2*Log[c*(d + e*x^2)^p], x, 13, -2*f^2*p*x + (4*d*f*g*p*x)/(3*e) - (2*d^2*g^2*p*x)/(5*e^2) - (4*f*g*p*x^3)/9 + (2*d*g^2*p*x^3)/(15*e) - (2*g^2*p*x^5)/25 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (4*d^(3/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) + (2*d^(5/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*e^(5/2)) + f^2*x*Log[c*(d + e*x^2)^p] + (2*f*g*x^3*Log[c*(d + e*x^2)^p])/3 + (g^2*x^5*Log[c*(d + e*x^2)^p])/5} -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p]/x^2, x, 11, -4*f*g*p*x + (2*d*g^2*p*x)/(3*e) - (2*g^2*p*x^3)/9 + (2*Sqrt[e]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (4*Sqrt[d]*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(3/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)) - (f^2*Log[c*(d + e*x^2)^p])/x + 2*f*g*x*Log[c*(d + e*x^2)^p] + (g^2*x^3*Log[c*(d + e*x^2)^p])/3} -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p]/x^4, x, 10, (-2*e*f^2*p)/(3*d*x) - 2*g^2*p*x - (2*e^(3/2)*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (4*Sqrt[e]*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[d]*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (f^2*Log[c*(d + e*x^2)^p])/(3*x^3) - (2*f*g*Log[c*(d + e*x^2)^p])/x + g^2*x*Log[c*(d + e*x^2)^p]} -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p]/x^6, x, 11, -((2*e*f^2*p)/(15*d*x^3)) + (2*e^2*f^2*p)/(5*d^2*x) - (4*e*f*g*p)/(3*d*x) + (2*e^(5/2)*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*d^(5/2)) - (4*e^(3/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) + (2*Sqrt[e]*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (f^2*Log[c*(d + e*x^2)^p])/(5*x^5) - (2*f*g*Log[c*(d + e*x^2)^p])/(3*x^3) - (g^2*Log[c*(d + e*x^2)^p])/x} -{(f + g*x^2)^2*Log[c*(d + e*x^2)^p]/x^8, x, 14, -((2*e*f^2*p)/(35*d*x^5)) + (2*e^2*f^2*p)/(21*d^2*x^3) - (4*e*f*g*p)/(15*d*x^3) - (2*e^3*f^2*p)/(7*d^3*x) + (4*e^2*f*g*p)/(5*d^2*x) - (2*e*g^2*p)/(3*d*x) - (2*e^(7/2)*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*d^(7/2)) + (4*e^(5/2)*f*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(5*d^(5/2)) - (2*e^(3/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)) - (f^2*Log[c*(d + e*x^2)^p])/(7*x^7) - (2*f*g*Log[c*(d + e*x^2)^p])/(5*x^5) - (g^2*Log[c*(d + e*x^2)^p])/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2), x, 11, (f*p*x^2)/(2*g^2) + (d*p*x^2)/(4*e*g) - (p*x^4)/(8*g) - (d^2*p*Log[d + e*x^2])/(4*e^2*g) + (x^4*Log[c*(d + e*x^2)^p])/(4*g) - (f*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e*g^2) + (f^2*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*g^3) + (f^2*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g^3)} -{(x^3*Log[c*(d + e*x^2)^p])/(f + g*x^2), x, 8, -(p*x^2)/(2*g) + ((d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e*g) - (f*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*g^2) - (f*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g^2)} -{(x^1*Log[c*(d + e*x^2)^p])/(f + g*x^2), x, 4, (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*g) + (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g)} -{Log[c*(d + e*x^2)^p]/(x^1*(f + g*x^2)), x, 8, (Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/(2*f) - (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*f) - (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*f) + (p*PolyLog[2, 1 + (e*x^2)/d])/(2*f)} -{Log[c*(d + e*x^2)^p]/(x^3*(f + g*x^2)), x, 12, (e*p*Log[x])/(d*f) - (e*p*Log[d + e*x^2])/(2*d*f) - Log[c*(d + e*x^2)^p]/(2*f*x^2) - (g*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/(2*f^2) + (g*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*f^2) + (g*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*f^2) - (g*p*PolyLog[2, 1 + (e*x^2)/d])/(2*f^2)} - -{x^4*Log[c*(d + e*x^2)^p]/(f + g*x^2), x, 21, (2*f*p*x)/g^2 + (2*d*p*x)/(3*e*g) - (2*p*x^3)/(9*g) - (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2) - (2*d^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)*g) + (2*f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) - (f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/g^(5/2) - (f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e*x^2)^p])/(3*g) + (f^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(5/2) - (I*f^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + (I*f^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*g^(5/2)) + (I*f^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*g^(5/2))} -{x^2*Log[c*(d + e*x^2)^p]/(f + g*x^2), x, 17, -((2*p*x)/g) + (2*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g) - (2*Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(3/2) + (Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/g^(3/2) + (Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(3/2) + (x*Log[c*(d + e*x^2)^p])/g - (Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(3/2) + (I*Sqrt[f]*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(3/2) - (I*Sqrt[f]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*g^(3/2)) - (I*Sqrt[f]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*g^(3/2))} -{x^0*Log[c*(d + e*x^2)^p]/(f + g*x^2), x, 12, (2*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(Sqrt[f]*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(Sqrt[f]*Sqrt[g]) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(Sqrt[f]*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*Sqrt[g])} -{Log[c*(d + e*x^2)^p]/(x^2*(f + g*x^2)), x, 16, (2*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f) - (2*Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(3/2) + (Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/f^(3/2) + (Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(3/2) - Log[c*(d + e*x^2)^p]/(f*x) - (Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/f^(3/2) + (I*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(3/2) - (I*Sqrt[g]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(3/2)) - (I*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(3/2))} -{Log[c*(d + e*x^2)^p]/(x^4*(f + g*x^2)), x, 19, -((2*e*p)/(3*d*f*x)) - (2*e^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)*f) - (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f^2) + (2*g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2) - (g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/f^(5/2) - (g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) - Log[c*(d + e*x^2)^p]/(3*f*x^3) + (g*Log[c*(d + e*x^2)^p])/(f^2*x) + (g^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/f^(5/2) - (I*g^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2) + (I*g^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(5/2)) + (I*g^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(5/2))} - - -{(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2)^2, x, 12, -(p*x^2)/(2*g^2) + (e*f^2*p*Log[d + e*x^2])/(2*g^3*(e*f - d*g)) + ((d + e*x^2)*Log[c*(d + e*x^2)^p])/(2*e*g^2) - (f^2*Log[c*(d + e*x^2)^p])/(2*g^3*(f + g*x^2)) - (e*f^2*p*Log[f + g*x^2])/(2*g^3*(e*f - d*g)) - (f*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/g^3 - (f*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/g^3} -{(x^3*Log[c*(d + e*x^2)^p])/(f + g*x^2)^2, x, 10, -(e*f*p*Log[d + e*x^2])/(2*g^2*(e*f - d*g)) + (f*Log[c*(d + e*x^2)^p])/(2*g^2*(f + g*x^2)) + (e*f*p*Log[f + g*x^2])/(2*g^2*(e*f - d*g)) + (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*g^2) + (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*g^2)} -{(x^1*Log[c*(d + e*x^2)^p])/(f + g*x^2)^2, x, 5, (e*p*Log[d + e*x^2])/(2*g*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(2*g*(f + g*x^2)) - (e*p*Log[f + g*x^2])/(2*g*(e*f - d*g))} -{Log[c*(d + e*x^2)^p]/(x^1*(f + g*x^2)^2), x, 12, -((e*p*Log[d + e*x^2])/(2*f*(e*f - d*g))) + Log[c*(d + e*x^2)^p]/(2*f*(f + g*x^2)) + (Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/(2*f^2) + (e*p*Log[f + g*x^2])/(2*f*(e*f - d*g)) - (Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/(2*f^2) - (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/(2*f^2) + (p*PolyLog[2, 1 + (e*x^2)/d])/(2*f^2)} -{Log[c*(d + e*x^2)^p]/(x^3*(f + g*x^2)^2), x, 16, (e*p*Log[x])/(d*f^2) - (e*p*Log[d + e*x^2])/(2*d*f^2) + (e*g*p*Log[d + e*x^2])/(2*f^2*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(2*f^2*x^2) - (g*Log[c*(d + e*x^2)^p])/(2*f^2*(f + g*x^2)) - (g*Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/f^3 - (e*g*p*Log[f + g*x^2])/(2*f^2*(e*f - d*g)) + (g*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/f^3 + (g*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/f^3 - (g*p*PolyLog[2, 1 + (e*x^2)/d])/f^3} - -{x^4*Log[c*(d + e*x^2)^p]/(f + g*x^2)^2, x, 43, -((2*p*x)/g^2) + (2*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2) + (Sqrt[d]*Sqrt[e]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(g^2*(e*f - d*g)) - (e*(-f)^(3/2)*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*g^(5/2)*(e*f - d*g)) - (3*Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + (3*Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(2*g^(5/2)) + (3*Sqrt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*g^(5/2)) + (e*(-f)^(3/2)*p*Log[Sqrt[-f] + Sqrt[g]*x])/(2*g^(5/2)*(e*f - d*g)) + (x*Log[c*(d + e*x^2)^p])/g^2 - (f*Log[c*(d + e*x^2)^p])/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*Log[c*(d + e*x^2)^p])/(4*g^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (3*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*g^(5/2)) + (3*I*Sqrt[f]*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*g^(5/2)) - (3*I*Sqrt[f]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*g^(5/2)) - (3*I*Sqrt[f]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*g^(5/2))} -{x^2*Log[c*(d + e*x^2)^p]/(f + g*x^2)^2, x, 40, -((Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(g*(e*f - d*g))) - (e*Sqrt[-f]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*g^(3/2)*(e*f - d*g)) + (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*g^(3/2)) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(2*Sqrt[f]*g^(3/2)) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*g^(3/2)) + (e*Sqrt[-f]*p*Log[Sqrt[-f] + Sqrt[g]*x])/(2*g^(3/2)*(e*f - d*g)) + Log[c*(d + e*x^2)^p]/(4*g^(3/2)*(Sqrt[-f] - Sqrt[g]*x)) - Log[c*(d + e*x^2)^p]/(4*g^(3/2)*(Sqrt[-f] + Sqrt[g]*x)) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*Sqrt[f]*g^(3/2)) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*Sqrt[f]*g^(3/2)) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*Sqrt[f]*g^(3/2)) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*Sqrt[f]*g^(3/2))} -{x^0*Log[c*(d + e*x^2)^p]/(f + g*x^2)^2, x, 26, (Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(f*(e*f - d*g)) - (e*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) + (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(f^(3/2)*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(2*f^(3/2)*Sqrt[g]) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(3/2)*Sqrt[g]) + (e*p*Log[Sqrt[-f] + Sqrt[g]*x])/(2*Sqrt[-f]*Sqrt[g]*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqrt[-f] - Sqrt[g]*x)) + Log[c*(d + e*x^2)^p]/(4*f*Sqrt[g]*(Sqrt[-f] + Sqrt[g]*x)) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*f^(3/2)*Sqrt[g]) - (I*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*f^(3/2)*Sqrt[g]) + (I*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(3/2)*Sqrt[g]) + (I*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(3/2)*Sqrt[g])} -{Log[c*(d + e*x^2)^p]/(x^2*(f + g*x^2)^2), x, 42, (2*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f^2) - (Sqrt[d]*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(f^2*(e*f - d*g)) - (e*Sqrt[g]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*(-f)^(3/2)*(e*f - d*g)) - (3*Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2) + (3*Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/(2*f^(5/2)) + (3*Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*f^(5/2)) + (e*Sqrt[g]*p*Log[Sqrt[-f] + Sqrt[g]*x])/(2*(-f)^(3/2)*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(f^2*x) + (Sqrt[g]*Log[c*(d + e*x^2)^p])/(4*f^2*(Sqrt[-f] - Sqrt[g]*x)) - (Sqrt[g]*Log[c*(d + e*x^2)^p])/(4*f^2*(Sqrt[-f] + Sqrt[g]*x)) - (3*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*f^(5/2)) + (3*I*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(2*f^(5/2)) - (3*I*Sqrt[g]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(5/2)) - (3*I*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(4*f^(5/2))} - - -{Log[c*(a + b*x^2)^n]/(a + b*x^2), x, 6, (I*n*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(Sqrt[a]*Sqrt[b]) + (2*n*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(Sqrt[a]*Sqrt[b]) + (ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^n])/(Sqrt[a]*Sqrt[b]) + (I*n*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(Sqrt[a]*Sqrt[b])} - - -{Log[1 - x^2]/(2 - x^2), x, 12, Sqrt[2]*ArcTanh[x/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + x)] - (ArcTanh[x/Sqrt[2]]*Log[-((4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + x)))])/Sqrt[2] - (ArcTanh[x/Sqrt[2]]*Log[(4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))])/Sqrt[2] + (ArcTanh[x/Sqrt[2]]*Log[1 - x^2])/Sqrt[2] - PolyLog[2, 1 - (2*Sqrt[2])/(Sqrt[2] + x)]/Sqrt[2] + PolyLog[2, 1 + (4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + x))]/(2*Sqrt[2]) + PolyLog[2, 1 - (4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))]/(2*Sqrt[2])} -{Log[d + e*x^2]/(1 - x^2), x, 11, 2*ArcTanh[x]*Log[2/(1 + x)] - ArcTanh[x]*Log[(2*(Sqrt[-d] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))] - ArcTanh[x]*Log[(2*(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + x))] + ArcTanh[x]*Log[d + e*x^2] - PolyLog[2, 1 - 2/(1 + x)] + (1/2)*PolyLog[2, 1 - (2*(Sqrt[-d] - Sqrt[e]*x))/((Sqrt[-d] - Sqrt[e])*(1 + x))] + (1/2)*PolyLog[2, 1 - (2*(Sqrt[-d] + Sqrt[e]*x))/((Sqrt[-d] + Sqrt[e])*(1 + x))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^r (f+g x^s)^q Log[c (d+e x^m)^n] when s=k n and s symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*r>0*) - - -{(f + g*x^(3*n))*Log[c*(d + e*x^n)^p]/x, x, 8, -((d^2*g*p*x^n)/(3*e^2*n)) + (d*g*p*x^(2*n))/(6*e*n) - (g*p*x^(3*n))/(9*n) + (d^3*g*p*Log[d + e*x^n])/(3*e^3*n) + (g*x^(3*n)*Log[c*(d + e*x^n)^p])/(3*n) + (f*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g*x^(2*n))*Log[c*(d + e*x^n)^p]/x, x, 8, (d*g*p*x^n)/(2*e*n) - (g*p*x^(2*n))/(4*n) - (d^2*g*p*Log[d + e*x^n])/(2*e^2*n) + (g*x^(2*n)*Log[c*(d + e*x^n)^p])/(2*n) + (f*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g*x^(1*n))*Log[c*(d + e*x^n)^p]/x, x, 7, -((g*p*x^n)/n) + (g*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e*n) + (f*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g/x^(1*n))*Log[c*(d + e*x^n)^p]/x, x, 9, (e*g*p*Log[x])/d - (e*g*p*Log[d + e*x^n])/(d*n) - (g*Log[c*(d + e*x^n)^p])/(x^n*n) + (f*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g/x^(2*n))*Log[c*(d + e*x^n)^p]/x, x, 8, -((e*g*p)/(x^n*(2*d*n))) - (e^2*g*p*Log[x])/(2*d^2) + (e^2*g*p*Log[d + e*x^n])/(2*d^2*n) - (g*Log[c*(d + e*x^n)^p])/(x^(2*n)*(2*n)) + (f*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f*p*PolyLog[2, 1 + (e*x^n)/d])/n} - - -{(f + g*x^(3*n))^2*Log[c*(d + e*x^n)^p]/x, x, 11, -((2*d^2*f*g*p*x^n)/(3*e^2*n)) + (d^5*g^2*p*x^n)/(6*e^5*n) + (d*f*g*p*x^(2*n))/(3*e*n) - (d^4*g^2*p*x^(2*n))/(12*e^4*n) - (2*f*g*p*x^(3*n))/(9*n) + (d^3*g^2*p*x^(3*n))/(18*e^3*n) - (d^2*g^2*p*x^(4*n))/(24*e^2*n) + (d*g^2*p*x^(5*n))/(30*e*n) - (g^2*p*x^(6*n))/(36*n) + (2*d^3*f*g*p*Log[d + e*x^n])/(3*e^3*n) - (d^6*g^2*p*Log[d + e*x^n])/(6*e^6*n) + (2*f*g*x^(3*n)*Log[c*(d + e*x^n)^p])/(3*n) + (g^2*x^(6*n)*Log[c*(d + e*x^n)^p])/(6*n) + (f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]/x, x, 11, (d*f*g*p*x^n)/(e*n) + (d^3*g^2*p*x^n)/(4*e^3*n) - (f*g*p*x^(2*n))/(2*n) - (d^2*g^2*p*x^(2*n))/(8*e^2*n) + (d*g^2*p*x^(3*n))/(12*e*n) - (g^2*p*x^(4*n))/(16*n) - (d^2*f*g*p*Log[d + e*x^n])/(e^2*n) - (d^4*g^2*p*Log[d + e*x^n])/(4*e^4*n) + (f*g*x^(2*n)*Log[c*(d + e*x^n)^p])/n + (g^2*x^(4*n)*Log[c*(d + e*x^n)^p])/(4*n) + (f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g*x^(1*n))^2*Log[c*(d + e*x^n)^p]/x, x, 10, -((2*f*g*p*x^n)/n) + (d*g^2*p*x^n)/(2*e*n) - (g^2*p*x^(2*n))/(4*n) - (d^2*g^2*p*Log[d + e*x^n])/(2*e^2*n) + (g^2*x^(2*n)*Log[c*(d + e*x^n)^p])/(2*n) + (2*f*g*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e*n) + (f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g/x^(1*n))^2*Log[c*(d + e*x^n)^p]/x, x, 12, -((e*g^2*p)/(x^n*(2*d*n))) + (2*e*f*g*p*Log[x])/d - (e^2*g^2*p*Log[x])/(2*d^2) - (2*e*f*g*p*Log[d + e*x^n])/(d*n) + (e^2*g^2*p*Log[d + e*x^n])/(2*d^2*n) - (g^2*Log[c*(d + e*x^n)^p])/(x^(2*n)*(2*n)) - (2*f*g*Log[c*(d + e*x^n)^p])/(x^n*n) + (f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n} -{(f + g/x^(2*n))^2*Log[c*(d + e*x^n)^p]/x, x, 11, -((e*g^2*p)/(x^(3*n)*(12*d*n))) + (e^2*g^2*p)/(x^(2*n)*(8*d^2*n)) - (e*f*g*p)/(x^n*(d*n)) - (e^3*g^2*p)/(x^n*(4*d^3*n)) - (e^2*f*g*p*Log[x])/d^2 - (e^4*g^2*p*Log[x])/(4*d^4) + (e^2*f*g*p*Log[d + e*x^n])/(d^2*n) + (e^4*g^2*p*Log[d + e*x^n])/(4*d^4*n) - (g^2*Log[c*(d + e*x^n)^p])/(x^(4*n)*(4*n)) - (f*g*Log[c*(d + e*x^n)^p])/(x^(2*n)*n) + (f^2*Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n} - - -(* ::Subsubsection::Closed:: *) -(*r<0*) - - -{1/(f + g*x^(2*n))*Log[c*(d + e*x^n)^p]/x, x, 13, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] - Sqrt[g]*x^n))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] + Sqrt[g]*x^n))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f*n) - (p*PolyLog[2, -((Sqrt[g]*(d + e*x^n))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f*n) - (p*PolyLog[2, (Sqrt[g]*(d + e*x^n))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f*n)} -{1/(f + g*x^(1*n))*Log[c*(d + e*x^n)^p]/x, x, 8, (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(f + g*x^n))/(e*f - d*g)])/(f*n) - (p*PolyLog[2, -((g*(d + e*x^n))/(e*f - d*g))])/(f*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f*n)} -{1/(f + g/x^(1*n))*Log[c*(d + e*x^n)^p]/x, x, 5, (Log[c*(d + e*x^n)^p]*Log[-((e*(g + f*x^n))/(d*f - e*g))])/(f*n) + (p*PolyLog[2, (f*(d + e*x^n))/(d*f - e*g)])/(f*n)} -{1/(f + g/x^(2*n))*Log[c*(d + e*x^n)^p]/x, x, 9, (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[g] - Sqrt[-f]*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*f*n) + (Log[c*(d + e*x^n)^p]*Log[-((e*(Sqrt[g] + Sqrt[-f]*x^n))/(d*Sqrt[-f] - e*Sqrt[g]))])/(2*f*n) + (p*PolyLog[2, (Sqrt[-f]*(d + e*x^n))/(d*Sqrt[-f] - e*Sqrt[g])])/(2*f*n) + (p*PolyLog[2, (Sqrt[-f]*(d + e*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*f*n)} - - -{1/(f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]/x, x, 19, -((d*e*Sqrt[g]*p*ArcTan[(Sqrt[g]*x^n)/Sqrt[f]])/(2*f^(3/2)*(e^2*f + d^2*g)*n)) - (e^2*p*Log[d + e*x^n])/(2*f*(e^2*f + d^2*g)*n) + Log[c*(d + e*x^n)^p]/(2*f*n*(f + g*x^(2*n))) + (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f^2*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] - Sqrt[g]*x^n))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[-f] + Sqrt[g]*x^n))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*f^2*n) + (e^2*p*Log[f + g*x^(2*n)])/(4*f*(e^2*f + d^2*g)*n) - (p*PolyLog[2, -((Sqrt[g]*(d + e*x^n))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f^2*n) - (p*PolyLog[2, (Sqrt[g]*(d + e*x^n))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^2*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f^2*n)} -{1/(f + g*x^(1*n))^2*Log[c*(d + e*x^n)^p]/x, x, 12, -((e*p*Log[d + e*x^n])/(f*(e*f - d*g)*n)) + Log[c*(d + e*x^n)^p]/(f*n*(f + g*x^n)) + (Log[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/(f^2*n) + (e*p*Log[f + g*x^n])/(f*(e*f - d*g)*n) - (Log[c*(d + e*x^n)^p]*Log[(e*(f + g*x^n))/(e*f - d*g)])/(f^2*n) - (p*PolyLog[2, -((g*(d + e*x^n))/(e*f - d*g))])/(f^2*n) + (p*PolyLog[2, 1 + (e*x^n)/d])/(f^2*n)} -{1/(f + g/x^(1*n))^2*Log[c*(d + e*x^n)^p]/x, x, 10, (e*g*p*Log[d + e*x^n])/(f^2*(d*f - e*g)*n) + (g*Log[c*(d + e*x^n)^p])/(f^2*n*(g + f*x^n)) - (e*g*p*Log[g + f*x^n])/(f^2*(d*f - e*g)*n) + (Log[c*(d + e*x^n)^p]*Log[-((e*(g + f*x^n))/(d*f - e*g))])/(f^2*n) + (p*PolyLog[2, (f*(d + e*x^n))/(d*f - e*g)])/(f^2*n)} -{1/(f + g/x^(2*n))^2*Log[c*(d + e*x^n)^p]/x, x, 17, -((d*e*Sqrt[g]*p*ArcTan[(Sqrt[f]*x^n)/Sqrt[g]])/(2*f^(3/2)*(d^2*f + e^2*g)*n)) - (e^2*g*p*Log[d + e*x^n])/(2*f^2*(d^2*f + e^2*g)*n) + (g*Log[c*(d + e*x^n)^p])/(2*f^2*n*(g + f*x^(2*n))) + (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[g] - Sqrt[-f]*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*f^2*n) + (Log[c*(d + e*x^n)^p]*Log[-((e*(Sqrt[g] + Sqrt[-f]*x^n))/(d*Sqrt[-f] - e*Sqrt[g]))])/(2*f^2*n) + (e^2*g*p*Log[g + f*x^(2*n)])/(4*f^2*(d^2*f + e^2*g)*n) + (p*PolyLog[2, (Sqrt[-f]*(d + e*x^n))/(d*Sqrt[-f] - e*Sqrt[g])])/(2*f^2*n) + (p*PolyLog[2, (Sqrt[-f]*(d + e*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*f^2*n)} - - -{Log[c*(d + e*x^n)]/(x*(c*e - (1 - c*d)/x^n)), x, 4, -(PolyLog[2, 1 - c*(d + e*x^n)]/(c*e*n))} -{(x^(-1 + n)*Log[c*(d + e*x^n)])/(-1 + c*d + c*e*x^n), x, 3, -(PolyLog[2, 1 - c*(d + e*x^n)]/(c*e*n))} - -{Log[c*(d + e/x^n)]/(x*(c*e - (1 - c*d)*x^n)), x, 4, PolyLog[2, 1 - c*(d + e/x^n)]/(c*e*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^r (f+g x^s)^q Log[c (d+e x^m)^n]^q when q symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*r>0*) - - -{(f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q/x, x, 0, (4^(-1 - q)*g^2*(d + e*x^n)^4*Gamma[1 + q, -((4*Log[c*(d + e*x^n)^p])/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^(4/p)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^4*n)) - (d*g^2*(d + e*x^n)^3*Gamma[1 + q, -((3*Log[c*(d + e*x^n)^p])/p)]*Log[c*(d + e*x^n)^p]^q)/(3^q*(c*(d + e*x^n)^p)^(3/p)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^4*n)) + (f*g*(d + e*x^n)^2*Gamma[1 + q, -((2*Log[c*(d + e*x^n)^p])/p)]*Log[c*(d + e*x^n)^p]^q)/(2^q*(c*(d + e*x^n)^p)^(2/p)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^2*n)) + (3*2^(-1 - q)*d^2*g^2*(d + e*x^n)^2*Gamma[1 + q, -((2*Log[c*(d + e*x^n)^p])/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^(2/p)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^4*n)) - (2*d*f*g*(d + e*x^n)*Gamma[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^2*n)) - (d^3*g^2*(d + e*x^n)*Gamma[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^4*n)) + f^2*Unintegrable[Log[c*(d + e*x^n)^p]^q/x, x], Unintegrable[((f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x, x]} -{(f + g*x^(1*n))^2*Log[c*(d + e*x^n)^p]^q/x, x, 0, (2^(-1 - q)*g^2*(d + e*x^n)^2*Gamma[1 + q, -((2*Log[c*(d + e*x^n)^p])/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^(2/p)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^2*n)) + (2*f*g*(d + e*x^n)*Gamma[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e*n)) - (d*g^2*(d + e*x^n)*Gamma[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/((c*(d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n)^p]/p))^q*(e^2*n)) + f^2*Unintegrable[Log[c*(d + e*x^n)^p]^q/x, x], Unintegrable[((f + g*x^n)^2*Log[c*(d + e*x^n)^p]^q)/x, x]} -{(f + g/x^(1*n))^2*Log[c*(d + e*x^n)^p]^q/x, x, 0, Unintegrable[((f + g/x^n)^2*Log[c*(d + e*x^n)^p]^q)/x, x]} -{(f + g/x^(2*n))^2*Log[c*(d + e*x^n)^p]^q/x, x, 0, Unintegrable[((f + g/x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*r<0*) - - -{Log[c*(d + e*x^n)^p]^q/(x*(f + g*x^(2*n))), x, 0, Unintegrable[Log[c*(d + e*x^n)^p]^q/(x*(f + g*x^(2*n))), x]} -{Log[c*(d + e*x^n)^p]^q/(x*(f + g*x^(1*n))), x, 0, Unintegrable[Log[c*(d + e*x^n)^p]^q/(x*(f + g*x^n)), x]} -{Log[c*(d + e*x^n)^p]^q/(x*(f + g/x^(1*n))), x, 0, Unintegrable[Log[c*(d + e*x^n)^p]^q/(x*(f + g/x^n)), x]} -{Log[c*(d + e*x^n)^p]^q/(x*(f + g/x^(2*n))), x, 0, Unintegrable[Log[c*(d + e*x^n)^p]^q/(x*(f + g/x^(2*n))), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^r Log[f x^s]^q Log[c (d+e x^m)^n]^p*) - - -{Log[x]*Log[d + e*x^m]/x, x, 4, (1/2)*Log[x]^2*Log[d + e*x^m] - (1/2)*Log[x]^2*Log[1 + (e*x^m)/d] - (Log[x]*PolyLog[2, -((e*x^m)/d)])/m + PolyLog[3, -((e*x^m)/d)]/m^2} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^q Log[c (d+e x^m)/x^n]*) - - -{Log[(a + x)/x]/x, x, 1, PolyLog[2, -(a/x)], PolyLog[2, 1 - (a + x)/x]} -{Log[(a + x^2)/x^2]/x, x, 2, (1/2)*PolyLog[2, -(a/x^2)]} -{Log[(a + x^n)/x^n]/x, x, 2, PolyLog[2, -a/x^n]/n} - - -{Log[(a + b*x)/x]/x, x, 4, (-Log[b + a/x])*Log[-(a/(b*x))] - PolyLog[2, 1 + a/(b*x)]} -{Log[(a + b*x^2)/x^2]/x, x, 4, (-(1/2))*Log[b + a/x^2]*Log[-(a/(b*x^2))] - (1/2)*PolyLog[2, 1 + a/(b*x^2)]} -{Log[(a + b*x^n)/x^n]/x, x, 4, -((Log[-(a/(x^n*b))]*Log[b + a/x^n])/n) - PolyLog[2, 1 + a/(x^n*b)]/n} - - -{Log[(a + b*x)/x]/(c + d*x), x, 9, (Log[b + a/x]*Log[c + d*x])/d + (Log[-((d*x)/c)]*Log[c + d*x])/d - (Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/d + PolyLog[2, 1 + (d*x)/c]/d - PolyLog[2, (b*(c + d*x))/(b*c - a*d)]/d} -{Log[(a + b*x^2)/x^2]/(c + d*x), x, 14, (Log[b + a/x^2]*Log[c + d*x])/d + (2*Log[-((d*x)/c)]*Log[c + d*x])/d - (Log[(d*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*c + Sqrt[-a]*d)]*Log[c + d*x])/d - (Log[-((d*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*c - Sqrt[-a]*d))]*Log[c + d*x])/d + (2*PolyLog[2, 1 + (d*x)/c])/d - PolyLog[2, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)]/d - PolyLog[2, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)]/d} -{Log[(a + b*x^n)/x^n]/(c + d*x), x, 1, Unintegrable[Log[b + a/x^n]/(c + d*x), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b Log[c (d+e x^m)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^q (a+b Log[c (d+e x^m)^n])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^q (a+b Log[c (d+e x^m)^n]) when q symbolic*) - - -{(f*x)^q*(a + b*Log[c*(d + e*x^m)^n]), x, 3, -((b*e*m*n*x^(1 + m)*(f*x)^q*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m + q)/m, -((e*x^m)/d)])/(d*(1 + q)*(1 + m + q))) + ((f*x)^(1 + q)*(a + b*Log[c*(d + e*x^m)^n]))/(f*(1 + q))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^q (a+b Log[c (d+e x^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q (a+b Log[c (d+e x^(m/2))^n])^p when n symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*m=1*) - - -{x^3*(a + b*Log[c*(d + e*Sqrt[x])^n]), x, 4, (b*d^7*n*Sqrt[x])/(4*e^7) - (b*d^6*n*x)/(8*e^6) + (b*d^5*n*x^(3/2))/(12*e^5) - (b*d^4*n*x^2)/(16*e^4) + (b*d^3*n*x^(5/2))/(20*e^3) - (b*d^2*n*x^3)/(24*e^2) + (b*d*n*x^(7/2))/(28*e) - (1/32)*b*n*x^4 - (b*d^8*n*Log[d + e*Sqrt[x]])/(4*e^8) + (1/4)*x^4*(a + b*Log[c*(d + e*Sqrt[x])^n])} -{x^2*(a + b*Log[c*(d + e*Sqrt[x])^n]), x, 4, (b*d^5*n*Sqrt[x])/(3*e^5) - (b*d^4*n*x)/(6*e^4) + (b*d^3*n*x^(3/2))/(9*e^3) - (b*d^2*n*x^2)/(12*e^2) + (b*d*n*x^(5/2))/(15*e) - (1/18)*b*n*x^3 - (b*d^6*n*Log[d + e*Sqrt[x]])/(3*e^6) + (1/3)*x^3*(a + b*Log[c*(d + e*Sqrt[x])^n])} -{x^1*(a + b*Log[c*(d + e*Sqrt[x])^n]), x, 4, (b*d^3*n*Sqrt[x])/(2*e^3) - (b*d^2*n*x)/(4*e^2) + (b*d*n*x^(3/2))/(6*e) - (1/8)*b*n*x^2 - (b*d^4*n*Log[d + e*Sqrt[x]])/(2*e^4) + (1/2)*x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])} -{x^0*(a + b*Log[c*(d + e*Sqrt[x])^n]), x, 5, (b*d*n*Sqrt[x])/e + a*x - (b*n*x)/2 - (b*d^2*n*Log[d + e*Sqrt[x]])/e^2 + b*x*Log[c*(d + e*Sqrt[x])^n]} -{(a + b*Log[c*(d + e*Sqrt[x])^n])/x^1, x, 3, 2*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)] + 2*b*n*PolyLog[2, 1 + (e*Sqrt[x])/d]} -{(a + b*Log[c*(d + e*Sqrt[x])^n])/x^2, x, 4, -((b*e*n)/(d*Sqrt[x])) + (b*e^2*n*Log[d + e*Sqrt[x]])/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])/x - (b*e^2*n*Log[x])/(2*d^2)} -{(a + b*Log[c*(d + e*Sqrt[x])^n])/x^3, x, 4, -((b*e*n)/(6*d*x^(3/2))) + (b*e^2*n)/(4*d^2*x) - (b*e^3*n)/(2*d^3*Sqrt[x]) + (b*e^4*n*Log[d + e*Sqrt[x]])/(2*d^4) - (a + b*Log[c*(d + e*Sqrt[x])^n])/(2*x^2) - (b*e^4*n*Log[x])/(4*d^4)} -{(a + b*Log[c*(d + e*Sqrt[x])^n])/x^4, x, 4, -((b*e*n)/(15*d*x^(5/2))) + (b*e^2*n)/(12*d^2*x^2) - (b*e^3*n)/(9*d^3*x^(3/2)) + (b*e^4*n)/(6*d^4*x) - (b*e^5*n)/(3*d^5*Sqrt[x]) + (b*e^6*n*Log[d + e*Sqrt[x]])/(3*d^6) - (a + b*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) - (b*e^6*n*Log[x])/(6*d^6)} - - -{x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2, x, 8, (5*b^2*d^4*n^2*(d + e*Sqrt[x])^2)/(2*e^6) - (40*b^2*d^3*n^2*(d + e*Sqrt[x])^3)/(27*e^6) + (5*b^2*d^2*n^2*(d + e*Sqrt[x])^4)/(8*e^6) - (4*b^2*d*n^2*(d + e*Sqrt[x])^5)/(25*e^6) + (b^2*n^2*(d + e*Sqrt[x])^6)/(54*e^6) - (4*b^2*d^5*n^2*Sqrt[x])/e^5 + (b^2*d^6*n^2*Log[d + e*Sqrt[x]]^2)/(3*e^6) + (4*b*d^5*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^6 - (5*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^6 + (40*b*d^3*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) - (5*b*d^2*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^6) + (4*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(5*e^6) - (b*n*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) - (2*b*d^6*n*Log[d + e*Sqrt[x]]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^6) + (1/3)*x^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2} -{x^1*(a + b*Log[c*(d + e*Sqrt[x])^n])^2, x, 8, (3*b^2*d^2*n^2*(d + e*Sqrt[x])^2)/(2*e^4) - (4*b^2*d*n^2*(d + e*Sqrt[x])^3)/(9*e^4) + (b^2*n^2*(d + e*Sqrt[x])^4)/(16*e^4) - (4*b^2*d^3*n^2*Sqrt[x])/e^3 + (b^2*d^4*n^2*Log[d + e*Sqrt[x]]^2)/(2*e^4) + (4*b*d^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^4 - (3*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^4 + (4*b*d*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) - (b*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(4*e^4) - (b*d^4*n*Log[d + e*Sqrt[x]]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^4 + (1/2)*x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2} -{x^0*(a + b*Log[c*(d + e*Sqrt[x])^n])^2, x, 10, (b^2*n^2*(d + e*Sqrt[x])^2)/(2*e^2) + (4*a*b*d*n*Sqrt[x])/e - (4*b^2*d*n^2*Sqrt[x])/e + (4*b^2*d*n*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n])/e^2 - (b*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^2 - (2*d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 + ((d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^1, x, 5, 2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2*Log[-((e*Sqrt[x])/d)] + 4*b*n*(a + b*Log[c*(d + e*Sqrt[x])^n])*PolyLog[2, 1 + (e*Sqrt[x])/d] - 4*b^2*n^2*PolyLog[3, 1 + (e*Sqrt[x])/d]} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^2, x, 8, -((2*b*e*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(d^2*Sqrt[x])) - (2*b*e^2*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/x + (b^2*e^2*n^2*Log[x])/d^2 + (2*b^2*e^2*n^2*PolyLog[2, d/(d + e*Sqrt[x])])/d^2} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^3, x, 16, -((b^2*e^2*n^2)/(6*d^2*x)) + (5*b^2*e^3*n^2)/(6*d^3*Sqrt[x]) - (5*b^2*e^4*n^2*Log[d + e*Sqrt[x]])/(6*d^4) - (b*e*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d*x^(3/2)) + (b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^2*x) - (b*e^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(d^4*Sqrt[x]) - (b*e^4*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/d^4 - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/(2*x^2) + (11*b^2*e^4*n^2*Log[x])/(12*d^4) + (b^2*e^4*n^2*PolyLog[2, d/(d + e*Sqrt[x])])/d^4} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^2/x^4, x, 24, -((b^2*e^2*n^2)/(30*d^2*x^2)) + (b^2*e^3*n^2)/(10*d^3*x^(3/2)) - (47*b^2*e^4*n^2)/(180*d^4*x) + (77*b^2*e^5*n^2)/(90*d^5*Sqrt[x]) - (77*b^2*e^6*n^2*Log[d + e*Sqrt[x]])/(90*d^6) - (2*b*e*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(15*d*x^(5/2)) + (b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(6*d^2*x^2) - (2*b*e^3*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*d^3*x^(3/2)) + (b*e^4*n*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d^4*x) - (2*b*e^5*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d^6*Sqrt[x]) - (2*b*e^6*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*d^6) - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/(3*x^3) + (137*b^2*e^6*n^2*Log[x])/(180*d^6) + (2*b^2*e^6*n^2*PolyLog[2, d/(d + e*Sqrt[x])])/(3*d^6)} - - -{x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3, x, 28, -((15*b^3*d^4*n^3*(d + e*Sqrt[x])^2)/(4*e^6)) + (40*b^3*d^3*n^3*(d + e*Sqrt[x])^3)/(27*e^6) - (15*b^3*d^2*n^3*(d + e*Sqrt[x])^4)/(32*e^6) + (12*b^3*d*n^3*(d + e*Sqrt[x])^5)/(125*e^6) - (b^3*n^3*(d + e*Sqrt[x])^6)/(108*e^6) - (12*a*b^2*d^5*n^2*Sqrt[x])/e^5 + (12*b^3*d^5*n^3*Sqrt[x])/e^5 - (12*b^3*d^5*n^2*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n])/e^6 + (15*b^2*d^4*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^6) - (40*b^2*d^3*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) + (15*b^2*d^2*n^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(8*e^6) - (12*b^2*d*n^2*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(25*e^6) + (b^2*n^2*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(18*e^6) + (6*b*d^5*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^6 - (15*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^6) + (20*b*d^3*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(3*e^6) - (15*b*d^2*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*e^6) + (6*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(5*e^6) - (b*n*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(6*e^6) - (2*d^5*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 + (5*d^4*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 - (20*d^3*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(3*e^6) + (5*d^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 - (2*d*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 + ((d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(3*e^6)} -{x^1*(a + b*Log[c*(d + e*Sqrt[x])^n])^3, x, 20, -((9*b^3*d^2*n^3*(d + e*Sqrt[x])^2)/(4*e^4)) + (4*b^3*d*n^3*(d + e*Sqrt[x])^3)/(9*e^4) - (3*b^3*n^3*(d + e*Sqrt[x])^4)/(64*e^4) - (12*a*b^2*d^3*n^2*Sqrt[x])/e^3 + (12*b^3*d^3*n^3*Sqrt[x])/e^3 - (12*b^3*d^3*n^2*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n])/e^4 + (9*b^2*d^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^4) - (4*b^2*d*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) + (3*b^2*n^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(16*e^4) + (6*b*d^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^4 - (9*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^4) + (2*b*d*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^4 - (3*b*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(8*e^4) - (2*d^3*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + (3*d^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 - (2*d*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + ((d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(2*e^4)} -{x^0*(a + b*Log[c*(d + e*Sqrt[x])^n])^3, x, 12, -((3*b^3*n^3*(d + e*Sqrt[x])^2)/(4*e^2)) - (12*a*b^2*d*n^2*Sqrt[x])/e + (12*b^3*d*n^3*Sqrt[x])/e - (12*b^3*d*n^2*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n])/e^2 + (3*b^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^2) + (6*b*d*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 - (3*b*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^2) - (2*d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^2 + ((d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^2} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^1, x, 6, 2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3*Log[-((e*Sqrt[x])/d)] + 6*b*n*(a + b*Log[c*(d + e*Sqrt[x])^n])^2*PolyLog[2, 1 + (e*Sqrt[x])/d] - 12*b^2*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*PolyLog[3, 1 + (e*Sqrt[x])/d] + 12*b^3*n^3*PolyLog[4, 1 + (e*Sqrt[x])/d]} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^2, x, 10, -((3*b*e*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(d^2*Sqrt[x])) - (3*b*e^2*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])^3/x + (6*b^2*e^2*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)])/d^2 + (6*b^2*e^2*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*PolyLog[2, d/(d + e*Sqrt[x])])/d^2 + (6*b^3*e^2*n^3*PolyLog[2, 1 + (e*Sqrt[x])/d])/d^2 + (6*b^3*e^2*n^3*PolyLog[3, d/(d + e*Sqrt[x])])/d^2} -{(a + b*Log[c*(d + e*Sqrt[x])^n])^3/x^3, x, 28, -((b^3*e^3*n^3)/(2*d^3*Sqrt[x])) + (b^3*e^4*n^3*Log[d + e*Sqrt[x]])/(2*d^4) - (b^2*e^2*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^2*x) + (5*b^2*e^3*n^2*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^4*Sqrt[x]) + (5*b^2*e^4*n^2*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*d^4) - (b*e*n*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*d*x^(3/2)) + (3*b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*d^2*x) - (3*b*e^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*d^4*Sqrt[x]) - (3*b*e^4*n*Log[1 - d/(d + e*Sqrt[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*d^4) - (a + b*Log[c*(d + e*Sqrt[x])^n])^3/(2*x^2) + (3*b^2*e^4*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)])/d^4 - (3*b^3*e^4*n^3*Log[x])/(2*d^4) - (5*b^3*e^4*n^3*PolyLog[2, d/(d + e*Sqrt[x])])/(2*d^4) + (3*b^2*e^4*n^2*(a + b*Log[c*(d + e*Sqrt[x])^n])*PolyLog[2, d/(d + e*Sqrt[x])])/d^4 + (3*b^3*e^4*n^3*PolyLog[2, 1 + (e*Sqrt[x])/d])/d^4 + (3*b^3*e^4*n^3*PolyLog[3, d/(d + e*Sqrt[x])])/d^4} - - -(* ::Subsubsection::Closed:: *) -(*m=-1*) - - -{x^3*(a + b*Log[c*(d + e/Sqrt[x])^n]), x, 4, (b*e^7*n*Sqrt[x])/(4*d^7) - (b*e^6*n*x)/(8*d^6) + (b*e^5*n*x^(3/2))/(12*d^5) - (b*e^4*n*x^2)/(16*d^4) + (b*e^3*n*x^(5/2))/(20*d^3) - (b*e^2*n*x^3)/(24*d^2) + (b*e*n*x^(7/2))/(28*d) - (b*e^8*n*Log[d + e/Sqrt[x]])/(4*d^8) + (x^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/4 - (b*e^8*n*Log[x])/(8*d^8)} -{x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]), x, 4, (b*e^5*n*Sqrt[x])/(3*d^5) - (b*e^4*n*x)/(6*d^4) + (b*e^3*n*x^(3/2))/(9*d^3) - (b*e^2*n*x^2)/(12*d^2) + (b*e*n*x^(5/2))/(15*d) - (b*e^6*n*Log[d + e/Sqrt[x]])/(3*d^6) + (x^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/3 - (b*e^6*n*Log[x])/(6*d^6)} -{x^1*(a + b*Log[c*(d + e/Sqrt[x])^n]), x, 4, (b*e^3*n*Sqrt[x])/(2*d^3) - (b*e^2*n*x)/(4*d^2) + (b*e*n*x^(3/2))/(6*d) - (b*e^4*n*Log[d + e/Sqrt[x]])/(2*d^4) + (x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/2 - (b*e^4*n*Log[x])/(4*d^4)} -{x^0*(a + b*Log[c*(d + e/Sqrt[x])^n]), x, 6, (b*e*n*Sqrt[x])/d + a*x + b*x*Log[c*(d + e/Sqrt[x])^n] - (b*e^2*n*Log[e + d*Sqrt[x]])/d^2} -{(a + b*Log[c*(d + e/Sqrt[x])^n])/x^1, x, 3, -2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))] - 2*b*n*PolyLog[2, 1 + e/(d*Sqrt[x])]} -{(a + b*Log[c*(d + e/Sqrt[x])^n])/x^2, x, 4, (b*n)/(2*x) - (b*d*n)/(e*Sqrt[x]) + (b*d^2*n*Log[d + e/Sqrt[x]])/e^2 - (a + b*Log[c*(d + e/Sqrt[x])^n])/x} -{(a + b*Log[c*(d + e/Sqrt[x])^n])/x^3, x, 4, (b*n)/(8*x^2) - (b*d*n)/(6*e*x^(3/2)) + (b*d^2*n)/(4*e^2*x) - (b*d^3*n)/(2*e^3*Sqrt[x]) + (b*d^4*n*Log[d + e/Sqrt[x]])/(2*e^4) - (a + b*Log[c*(d + e/Sqrt[x])^n])/(2*x^2)} -{(a + b*Log[c*(d + e/Sqrt[x])^n])/x^4, x, 4, (b*n)/(18*x^3) - (b*d*n)/(15*e*x^(5/2)) + (b*d^2*n)/(12*e^2*x^2) - (b*d^3*n)/(9*e^3*x^(3/2)) + (b*d^4*n)/(6*e^4*x) - (b*d^5*n)/(3*e^5*Sqrt[x]) + (b*d^6*n*Log[d + e/Sqrt[x]])/(3*e^6) - (a + b*Log[c*(d + e/Sqrt[x])^n])/(3*x^3)} - - -{x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2, x, 24, -((77*b^2*e^5*n^2*Sqrt[x])/(90*d^5)) + (47*b^2*e^4*n^2*x)/(180*d^4) - (b^2*e^3*n^2*x^(3/2))/(10*d^3) + (b^2*e^2*n^2*x^2)/(30*d^2) + (77*b^2*e^6*n^2*Log[d + e/Sqrt[x]])/(90*d^6) + (2*b*e^5*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*d^6) - (b*e^4*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*d^4) + (2*b*e^3*n*x^(3/2)*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*d^3) - (b*e^2*n*x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(6*d^2) + (2*b*e*n*x^(5/2)*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(15*d) + (2*b*e^6*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*d^6) + (1/3)*x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (2*b^2*e^6*n^2*PolyLog[2, d/(d + e/Sqrt[x])])/(3*d^6)} -{x^1*(a + b*Log[c*(d + e/Sqrt[x])^n])^2, x, 16, -((5*b^2*e^3*n^2*Sqrt[x])/(6*d^3)) + (b^2*e^2*n^2*x)/(6*d^2) + (5*b^2*e^4*n^2*Log[d + e/Sqrt[x]])/(6*d^4) + (b*e^3*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^4 - (b*e^2*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^2) + (b*e*n*x^(3/2)*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*d) + (b*e^4*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^4 + (1/2)*x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (11*b^2*e^4*n^2*Log[x])/(12*d^4) - (b^2*e^4*n^2*PolyLog[2, d/(d + e/Sqrt[x])])/d^4} -{x^0*(a + b*Log[c*(d + e/Sqrt[x])^n])^2, x, 9, (2*b*e*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^2 + (2*b*e^2*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^2 + x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b^2*e^2*n^2*Log[x])/d^2 - (2*b^2*e^2*n^2*PolyLog[2, d/(d + e/Sqrt[x])])/d^2} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^1, x, 5, -2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*Log[-(e/(d*Sqrt[x]))] - 4*b*n*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, 1 + e/(d*Sqrt[x])] + 4*b^2*n^2*PolyLog[3, 1 + e/(d*Sqrt[x])]} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^2, x, 10, -(b^2*n^2*(d + e/Sqrt[x])^2)/(2*e^2) - (4*a*b*d*n)/(e*Sqrt[x]) + (4*b^2*d*n^2)/(e*Sqrt[x]) - (4*b^2*d*n*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^2 + (b*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^2 + (2*d*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^2 - ((d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^2} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^3, x, 8, -((3*b^2*d^2*n^2*(d + e/Sqrt[x])^2)/(2*e^4)) + (4*b^2*d*n^2*(d + e/Sqrt[x])^3)/(9*e^4) - (b^2*n^2*(d + e/Sqrt[x])^4)/(16*e^4) + (4*b^2*d^3*n^2)/(e^3*Sqrt[x]) - (b^2*d^4*n^2*Log[d + e/Sqrt[x]]^2)/(2*e^4) - (4*b*d^3*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^4 + (3*b*d^2*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^4 - (4*b*d*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*e^4) + (b*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(4*e^4) + (b*d^4*n*Log[d + e/Sqrt[x]]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^4 - (a + b*Log[c*(d + e/Sqrt[x])^n])^2/(2*x^2)} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^2/x^4, x, 8, -((5*b^2*d^4*n^2*(d + e/Sqrt[x])^2)/(2*e^6)) + (40*b^2*d^3*n^2*(d + e/Sqrt[x])^3)/(27*e^6) - (5*b^2*d^2*n^2*(d + e/Sqrt[x])^4)/(8*e^6) + (4*b^2*d*n^2*(d + e/Sqrt[x])^5)/(25*e^6) - (b^2*n^2*(d + e/Sqrt[x])^6)/(54*e^6) + (4*b^2*d^5*n^2)/(e^5*Sqrt[x]) - (b^2*d^6*n^2*Log[d + e/Sqrt[x]]^2)/(3*e^6) - (4*b*d^5*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^6 + (5*b*d^4*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^6 - (40*b*d^3*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) + (5*b*d^2*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e^6) - (4*b*d*n*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(5*e^6) + (b*n*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) + (2*b*d^6*n*Log[d + e/Sqrt[x]]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*e^6) - (a + b*Log[c*(d + e/Sqrt[x])^n])^2/(3*x^3)} - - -{x^1*(a + b*Log[c*(d + e/Sqrt[x])^n])^3, x, 28, (b^3*e^3*n^3*Sqrt[x])/(2*d^3) - (b^3*e^4*n^3*Log[d + e/Sqrt[x]])/(2*d^4) - (5*b^2*e^3*n^2*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (b^2*e^2*n^2*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^2) - (5*b^2*e^4*n^2*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (3*b*e^3*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d^4) - (3*b*e^2*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*d^2) + (b*e*n*x^(3/2)*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d) + (3*b*e^4*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d^4) + (1/2)*x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3 - (3*b^2*e^4*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))])/d^4 - (3*b^3*e^4*n^3*Log[x])/(2*d^4) + (5*b^3*e^4*n^3*PolyLog[2, d/(d + e/Sqrt[x])])/(2*d^4) - (3*b^2*e^4*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, d/(d + e/Sqrt[x])])/d^4 - (3*b^3*e^4*n^3*PolyLog[2, 1 + e/(d*Sqrt[x])])/d^4 - (3*b^3*e^4*n^3*PolyLog[3, d/(d + e/Sqrt[x])])/d^4} -{x^0*(a + b*Log[c*(d + e/Sqrt[x])^n])^3, x, 11, (3*b*e*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/d^2 + (3*b*e^2*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/d^2 + x*(a + b*Log[c*(d + e/Sqrt[x])^n])^3 - (6*b^2*e^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))])/d^2 - (6*b^2*e^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, d/(d + e/Sqrt[x])])/d^2 - (6*b^3*e^2*n^3*PolyLog[2, 1 + e/(d*Sqrt[x])])/d^2 - (6*b^3*e^2*n^3*PolyLog[3, d/(d + e/Sqrt[x])])/d^2} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^1, x, 6, -2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3*Log[-(e/(d*Sqrt[x]))] - 6*b*n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*PolyLog[2, 1 + e/(d*Sqrt[x])] + 12*b^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[3, 1 + e/(d*Sqrt[x])] - 12*b^3*n^3*PolyLog[4, 1 + e/(d*Sqrt[x])]} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^2, x, 12, (3*b^3*n^3*(d + e/Sqrt[x])^2)/(4*e^2) + (12*a*b^2*d*n^2)/(e*Sqrt[x]) - (12*b^3*d*n^3)/(e*Sqrt[x]) + (12*b^3*d*n^2*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^2 - (3*b^2*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e^2) - (6*b*d*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^2 + (3*b*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^2) + (2*d*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^2 - ((d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^2} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^3, x, 20, (9*b^3*d^2*n^3*(d + e/Sqrt[x])^2)/(4*e^4) - (4*b^3*d*n^3*(d + e/Sqrt[x])^3)/(9*e^4) + (3*b^3*n^3*(d + e/Sqrt[x])^4)/(64*e^4) + (12*a*b^2*d^3*n^2)/(e^3*Sqrt[x]) - (12*b^3*d^3*n^3)/(e^3*Sqrt[x]) + (12*b^3*d^3*n^2*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^4 - (9*b^2*d^2*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e^4) + (4*b^2*d*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(3*e^4) - (3*b^2*n^2*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(16*e^4) - (6*b*d^3*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^4 + (9*b*d^2*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^4) - (2*b*d*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^4 + (3*b*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(8*e^4) + (2*d^3*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - (3*d^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 + (2*d*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^4 - ((d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*e^4)} -{(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4, x, 28, (15*b^3*d^4*n^3*(d + e/Sqrt[x])^2)/(4*e^6) - (40*b^3*d^3*n^3*(d + e/Sqrt[x])^3)/(27*e^6) + (15*b^3*d^2*n^3*(d + e/Sqrt[x])^4)/(32*e^6) - (12*b^3*d*n^3*(d + e/Sqrt[x])^5)/(125*e^6) + (b^3*n^3*(d + e/Sqrt[x])^6)/(108*e^6) + (12*a*b^2*d^5*n^2)/(e^5*Sqrt[x]) - (12*b^3*d^5*n^3)/(e^5*Sqrt[x]) + (12*b^3*d^5*n^2*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^6 - (15*b^2*d^4*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*e^6) + (40*b^2*d^3*n^2*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(9*e^6) - (15*b^2*d^2*n^2*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(8*e^6) + (12*b^2*d*n^2*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(25*e^6) - (b^2*n^2*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(18*e^6) - (6*b*d^5*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^6 + (15*b*d^4*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^6) - (20*b*d^3*n*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(3*e^6) + (15*b*d^2*n*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*e^6) - (6*b*d*n*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(5*e^6) + (b*n*(d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(6*e^6) + (2*d^5*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 - (5*d^4*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 + (20*d^3*(d + e/Sqrt[x])^3*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(3*e^6) - (5*d^2*(d + e/Sqrt[x])^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 + (2*d*(d + e/Sqrt[x])^5*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^6 - ((d + e/Sqrt[x])^6*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(3*e^6)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q (a+b Log[c (d+e x^(m/3))^n])^p when n symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*m=1*) - - -{x^3*(a + b*Log[c*(d + e*x^(1/3))^n]), x, 4, (b*d^11*n*x^(1/3))/(4*e^11) - (b*d^10*n*x^(2/3))/(8*e^10) + (b*d^9*n*x)/(12*e^9) - (b*d^8*n*x^(4/3))/(16*e^8) + (b*d^7*n*x^(5/3))/(20*e^7) - (b*d^6*n*x^2)/(24*e^6) + (b*d^5*n*x^(7/3))/(28*e^5) - (b*d^4*n*x^(8/3))/(32*e^4) + (b*d^3*n*x^3)/(36*e^3) - (b*d^2*n*x^(10/3))/(40*e^2) + (b*d*n*x^(11/3))/(44*e) - (1/48)*b*n*x^4 - (b*d^12*n*Log[d + e*x^(1/3)])/(4*e^12) + (1/4)*x^4*(a + b*Log[c*(d + e*x^(1/3))^n])} -{x^2*(a + b*Log[c*(d + e*x^(1/3))^n]), x, 4, -((b*d^8*n*x^(1/3))/(3*e^8)) + (b*d^7*n*x^(2/3))/(6*e^7) - (b*d^6*n*x)/(9*e^6) + (b*d^5*n*x^(4/3))/(12*e^5) - (b*d^4*n*x^(5/3))/(15*e^4) + (b*d^3*n*x^2)/(18*e^3) - (b*d^2*n*x^(7/3))/(21*e^2) + (b*d*n*x^(8/3))/(24*e) - (1/27)*b*n*x^3 + (b*d^9*n*Log[d + e*x^(1/3)])/(3*e^9) + (1/3)*x^3*(a + b*Log[c*(d + e*x^(1/3))^n])} -{x^1*(a + b*Log[c*(d + e*x^(1/3))^n]), x, 4, (b*d^5*n*x^(1/3))/(2*e^5) - (b*d^4*n*x^(2/3))/(4*e^4) + (b*d^3*n*x)/(6*e^3) - (b*d^2*n*x^(4/3))/(8*e^2) + (b*d*n*x^(5/3))/(10*e) - (1/12)*b*n*x^2 - (b*d^6*n*Log[d + e*x^(1/3)])/(2*e^6) + (1/2)*x^2*(a + b*Log[c*(d + e*x^(1/3))^n])} -{x^0*(a + b*Log[c*(d + e*x^(1/3))^n]), x, 5, -((b*d^2*n*x^(1/3))/e^2) + (b*d*n*x^(2/3))/(2*e) + a*x - (b*n*x)/3 + (b*d^3*n*Log[d + e*x^(1/3)])/e^3 + b*x*Log[c*(d + e*x^(1/3))^n]} -{(a + b*Log[c*(d + e*x^(1/3))^n])/x^1, x, 3, 3*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)] + 3*b*n*PolyLog[2, 1 + (e*x^(1/3))/d]} -{(a + b*Log[c*(d + e*x^(1/3))^n])/x^2, x, 4, -((b*e*n)/(2*d*x^(2/3))) + (b*e^2*n)/(d^2*x^(1/3)) - (b*e^3*n*Log[d + e*x^(1/3)])/d^3 - (a + b*Log[c*(d + e*x^(1/3))^n])/x + (b*e^3*n*Log[x])/(3*d^3)} -{(a + b*Log[c*(d + e*x^(1/3))^n])/x^3, x, 4, -((b*e*n)/(10*d*x^(5/3))) + (b*e^2*n)/(8*d^2*x^(4/3)) - (b*e^3*n)/(6*d^3*x) + (b*e^4*n)/(4*d^4*x^(2/3)) - (b*e^5*n)/(2*d^5*x^(1/3)) + (b*e^6*n*Log[d + e*x^(1/3)])/(2*d^6) - (a + b*Log[c*(d + e*x^(1/3))^n])/(2*x^2) - (b*e^6*n*Log[x])/(6*d^6)} -{(a + b*Log[c*(d + e*x^(1/3))^n])/x^4, x, 4, -((b*e*n)/(24*d*x^(8/3))) + (b*e^2*n)/(21*d^2*x^(7/3)) - (b*e^3*n)/(18*d^3*x^2) + (b*e^4*n)/(15*d^4*x^(5/3)) - (b*e^5*n)/(12*d^5*x^(4/3)) + (b*e^6*n)/(9*d^6*x) - (b*e^7*n)/(6*d^7*x^(2/3)) + (b*e^8*n)/(3*d^8*x^(1/3)) - (b*e^9*n*Log[d + e*x^(1/3)])/(3*d^9) - (a + b*Log[c*(d + e*x^(1/3))^n])/(3*x^3) + (b*e^9*n*Log[x])/(9*d^9)} - - -{x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2, x, 8, -((6*b^2*d^7*n^2*(d + e*x^(1/3))^2)/e^9) + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3)/(9*e^9) - (21*b^2*d^5*n^2*(d + e*x^(1/3))^4)/(4*e^9) + (84*b^2*d^4*n^2*(d + e*x^(1/3))^5)/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6)/(9*e^9) + (24*b^2*d^2*n^2*(d + e*x^(1/3))^7)/(49*e^9) - (3*b^2*d*n^2*(d + e*x^(1/3))^8)/(32*e^9) + (2*b^2*n^2*(d + e*x^(1/3))^9)/(243*e^9) + (6*b^2*d^8*n^2*x^(1/3))/e^8 - (b^2*d^9*n^2*Log[d + e*x^(1/3)]^2)/(3*e^9) - (6*b*d^8*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 + (12*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 - (56*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (21*b*d^5*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 - (84*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(5*e^9) + (28*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (24*b*d^2*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(7*e^9) + (3*b*d*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^9) - (2*b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(27*e^9) + (2*b*d^9*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (1/3)*x^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2} -{x^1*(a + b*Log[c*(d + e*x^(1/3))^n])^2, x, 8, (15*b^2*d^4*n^2*(d + e*x^(1/3))^2)/(4*e^6) - (20*b^2*d^3*n^2*(d + e*x^(1/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d + e*x^(1/3))^4)/(16*e^6) - (6*b^2*d*n^2*(d + e*x^(1/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(1/3))^6)/(36*e^6) - (6*b^2*d^5*n^2*x^(1/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(1/3)]^2)/(2*e^6) + (6*b*d^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^6 - (15*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^6) + (20*b*d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) - (15*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^6) + (6*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(5*e^6) - (b*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(6*e^6) - (b*d^6*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^6 + (1/2)*x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2} -{x^0*(a + b*Log[c*(d + e*x^(1/3))^n])^2, x, 8, -((3*b^2*d*n^2*(d + e*x^(1/3))^2)/(2*e^3)) + (2*b^2*n^2*(d + e*x^(1/3))^3)/(9*e^3) + (6*b^2*d^2*n^2*x^(1/3))/e^2 - (b^2*d^3*n^2*Log[d + e*x^(1/3)]^2)/e^3 - (6*b*d^2*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 + (3*b*d*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 - (2*b*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^3) + (2*b*d^3*n*Log[d + e*x^(1/3)]*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^3 + x*(a + b*Log[c*(d + e*x^(1/3))^n])^2} -{(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^1, x, 5, 3*(a + b*Log[c*(d + e*x^(1/3))^n])^2*Log[-((e*x^(1/3))/d)] + 6*b*n*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, 1 + (e*x^(1/3))/d] - 6*b^2*n^2*PolyLog[3, 1 + (e*x^(1/3))/d]} -{(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^2, x, 12, -((b^2*e^2*n^2)/(d^2*x^(1/3))) + (b^2*e^3*n^2*Log[d + e*x^(1/3)])/d^3 - (b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d*x^(2/3)) + (2*b*e^2*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d^3*x^(1/3)) + (2*b*e^3*n*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/d^3 - (a + b*Log[c*(d + e*x^(1/3))^n])^2/x - (b^2*e^3*n^2*Log[x])/d^3 - (2*b^2*e^3*n^2*PolyLog[2, d/(d + e*x^(1/3))])/d^3} -{(a + b*Log[c*(d + e*x^(1/3))^n])^2/x^3, x, 24, -((b^2*e^2*n^2)/(20*d^2*x^(4/3))) + (3*b^2*e^3*n^2)/(20*d^3*x) - (47*b^2*e^4*n^2)/(120*d^4*x^(2/3)) + (77*b^2*e^5*n^2)/(60*d^5*x^(1/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(1/3)])/(60*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(5*d*x^(5/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*d^2*x^(4/3)) - (b*e^3*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*d^3*x) + (b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*d^4*x^(2/3)) - (b*e^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d^6*x^(1/3)) - (b*e^6*n*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/d^6 - (a + b*Log[c*(d + e*x^(1/3))^n])^2/(2*x^2) + (137*b^2*e^6*n^2*Log[x])/(180*d^6) + (b^2*e^6*n^2*PolyLog[2, d/(d + e*x^(1/3))])/d^6} - - -{x^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3, x, 52, (-99*b^3*d^10*n^3*(d + e*x^(1/3))^2)/(8*e^12) + (110*b^3*d^9*n^3*(d + e*x^(1/3))^3)/(9*e^12) - (1485*b^3*d^8*n^3*(d + e*x^(1/3))^4)/(128*e^12) + (1188*b^3*d^7*n^3*(d + e*x^(1/3))^5)/(125*e^12) - (77*b^3*d^6*n^3*(d + e*x^(1/3))^6)/(12*e^12) + (1188*b^3*d^5*n^3*(d + e*x^(1/3))^7)/(343*e^12) - (1485*b^3*d^4*n^3*(d + e*x^(1/3))^8)/(1024*e^12) + (110*b^3*d^3*n^3*(d + e*x^(1/3))^9)/(243*e^12) - (99*b^3*d^2*n^3*(d + e*x^(1/3))^10)/(1000*e^12) + (18*b^3*d*n^3*(d + e*x^(1/3))^11)/(1331*e^12) - (b^3*n^3*(d + e*x^(1/3))^12)/(1152*e^12) - (18*a*b^2*d^11*n^2*x^(1/3))/e^11 + (18*b^3*d^11*n^3*x^(1/3))/e^11 - (18*b^3*d^11*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^12 + (99*b^2*d^10*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^12) - (110*b^2*d^9*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^12) + (1485*b^2*d^8*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(32*e^12) - (1188*b^2*d^7*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^12) + (77*b^2*d^6*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^12) - (1188*b^2*d^5*n^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^12) + (1485*b^2*d^4*n^2*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n]))/(128*e^12) - (110*b^2*d^3*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(27*e^12) + (99*b^2*d^2*n^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n]))/(100*e^12) - (18*b^2*d*n^2*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n]))/(121*e^12) + (b^2*n^2*(d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n]))/(96*e^12) + (9*b*d^11*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^12 - (99*b*d^10*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^12) + (55*b*d^9*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^12 - (1485*b*d^8*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) + (594*b*d^7*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^12) - (231*b*d^6*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^12) + (594*b*d^5*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^12) - (1485*b*d^4*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(32*e^12) + (55*b*d^3*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^12) - (99*b*d^2*n*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(20*e^12) + (9*b*d*n*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(11*e^12) - (b*n*(d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) - (3*d^11*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (33*d^10*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (55*d^9*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495*d^8*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12) - (198*d^7*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (231*d^6*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 - (198*d^5*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495*d^4*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12) - (55*d^3*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (33*d^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (3*d*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + ((d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12)} -{x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3, x, 40, (9*b^3*d^7*n^3*(d + e*x^(1/3))^2)/e^9 - (56*b^3*d^6*n^3*(d + e*x^(1/3))^3)/(9*e^9) + (63*b^3*d^5*n^3*(d + e*x^(1/3))^4)/(16*e^9) - (252*b^3*d^4*n^3*(d + e*x^(1/3))^5)/(125*e^9) + (7*b^3*d^3*n^3*(d + e*x^(1/3))^6)/(9*e^9) - (72*b^3*d^2*n^3*(d + e*x^(1/3))^7)/(343*e^9) + (9*b^3*d*n^3*(d + e*x^(1/3))^8)/(256*e^9) - (2*b^3*n^3*(d + e*x^(1/3))^9)/(729*e^9) + (18*a*b^2*d^8*n^2*x^(1/3))/e^8 - (18*b^3*d^8*n^3*x^(1/3))/e^8 + (18*b^3*d^8*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^9 - (18*b^2*d^7*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (63*b^2*d^5*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^9) + (252*b^2*d^4*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (72*b^2*d^2*n^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^9) - (9*b^2*d*n^2*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n]))/(32*e^9) + (2*b^2*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(81*e^9) - (9*b*d^8*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (18*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 - (28*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (63*b*d^5*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^9) - (126*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^9) + (14*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 - (36*b*d^2*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^9) + (9*b*d*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*e^9) - (b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(9*e^9) + (3*d^8*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (12*d^7*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (28*d^6*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (42*d^5*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (42*d^4*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (28*d^3*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (12*d^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (3*d*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + ((d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(3*e^9)} -{x^1*(a + b*Log[c*(d + e*x^(1/3))^n])^3, x, 28, (-45*b^3*d^4*n^3*(d + e*x^(1/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e*x^(1/3))^3)/(9*e^6) - (45*b^3*d^2*n^3*(d + e*x^(1/3))^4)/(64*e^6) + (18*b^3*d*n^3*(d + e*x^(1/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(1/3))^6)/(72*e^6) - (18*a*b^2*d^5*n^2*x^(1/3))/e^5 + (18*b^3*d^5*n^3*x^(1/3))/e^5 - (18*b^3*d^5*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^6 + (45*b^2*d^4*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^6) - (20*b^2*d^3*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^6) + (45*b^2*d^2*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(16*e^6) - (18*b^2*d*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^6) + (b^2*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(12*e^6) + (9*b*d^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^4*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) + (10*b*d^3*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^6 - (45*b*d^2*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*e^6) + (9*b*d*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^6) - (b*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) - (3*d^5*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + (15*d^4*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (10*d^3*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + (15*d^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (3*d*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + ((d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6)} -{x^0*(a + b*Log[c*(d + e*x^(1/3))^n])^3, x, 16, (9*b^3*d*n^3*(d + e*x^(1/3))^2)/(4*e^3) - (2*b^3*n^3*(d + e*x^(1/3))^3)/(9*e^3) + (18*a*b^2*d^2*n^2*x^(1/3))/e^2 - (18*b^3*d^2*n^3*x^(1/3))/e^2 + (18*b^3*d^2*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^3 - (9*b^2*d*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^3) + (2*b^2*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^3) - (9*b*d^2*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^3 + (9*b*d*n*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^3) - (b*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^3 + (3*d^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^3 - (3*d*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^3 + ((d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^3} -{(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^1, x, 6, 3*(a + b*Log[c*(d + e*x^(1/3))^n])^3*Log[-((e*x^(1/3))/d)] + 9*b*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2*PolyLog[2, 1 + (e*x^(1/3))/d] - 18*b^2*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[3, 1 + (e*x^(1/3))/d] + 18*b^3*n^3*PolyLog[4, 1 + (e*x^(1/3))/d]} -{(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^2, x, 17, -((3*b^2*e^2*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(d^3*x^(1/3))) - (3*b^2*e^3*n^2*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/d^3 - (3*b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d*x^(2/3)) + (3*b*e^2*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(d^3*x^(1/3)) + (3*b*e^3*n*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/d^3 - (a + b*Log[c*(d + e*x^(1/3))^n])^3/x - (6*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/d^3 + (b^3*e^3*n^3*Log[x])/d^3 + (3*b^3*e^3*n^3*PolyLog[2, d/(d + e*x^(1/3))])/d^3 - (6*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, d/(d + e*x^(1/3))])/d^3 - (6*b^3*e^3*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/d^3 - (6*b^3*e^3*n^3*PolyLog[3, d/(d + e*x^(1/3))])/d^3} -{(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^3, x, 62, -((b^3*e^3*n^3)/(20*d^3*x)) + (3*b^3*e^4*n^3)/(10*d^4*x^(2/3)) - (71*b^3*e^5*n^3)/(40*d^5*x^(1/3)) + (71*b^3*e^6*n^3*Log[d + e*x^(1/3)])/(40*d^6) - (3*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^2*x^(4/3)) + (9*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^3*x) - (47*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(40*d^4*x^(2/3)) + (77*b^2*e^5*n^2*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6*x^(1/3)) + (77*b^2*e^6*n^2*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n]))/(20*d^6) - (3*b*e*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(10*d*x^(5/3)) + (3*b*e^2*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*d^2*x^(4/3)) - (b*e^3*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^3*x) + (3*b*e^4*n*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*d^4*x^(2/3)) - (3*b*e^5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6*x^(1/3)) - (3*b*e^6*n*Log[1 - d/(d + e*x^(1/3))]*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*d^6) - (a + b*Log[c*(d + e*x^(1/3))^n])^3/(2*x^2) + (3*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*Log[-((e*x^(1/3))/d)])/d^6 - (15*b^3*e^6*n^3*Log[x])/(8*d^6) - (77*b^3*e^6*n^3*PolyLog[2, d/(d + e*x^(1/3))])/(20*d^6) + (3*b^2*e^6*n^2*(a + b*Log[c*(d + e*x^(1/3))^n])*PolyLog[2, d/(d + e*x^(1/3))])/d^6 + (3*b^3*e^6*n^3*PolyLog[2, 1 + (e*x^(1/3))/d])/d^6 + (3*b^3*e^6*n^3*PolyLog[3, d/(d + e*x^(1/3))])/d^6} - - -(* ::Subsubsection::Closed:: *) -(*m=2*) - - -{x^3*(a + b*Log[c*(d + e*x^(2/3))^n]), x, 4, (b*d^5*n*x^(2/3))/(4*e^5) - (b*d^4*n*x^(4/3))/(8*e^4) + (b*d^3*n*x^2)/(12*e^3) - (b*d^2*n*x^(8/3))/(16*e^2) + (b*d*n*x^(10/3))/(20*e) - (1/24)*b*n*x^4 - (b*d^6*n*Log[d + e*x^(2/3)])/(4*e^6) + (1/4)*x^4*(a + b*Log[c*(d + e*x^(2/3))^n])} -{x^2*(a + b*Log[c*(d + e*x^(2/3))^n]), x, 5, -((2*b*d^4*n*x^(1/3))/(3*e^4)) + (2*b*d^3*n*x)/(9*e^3) - (2*b*d^2*n*x^(5/3))/(15*e^2) + (2*b*d*n*x^(7/3))/(21*e) - (2/27)*b*n*x^3 + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(9/2)) + (1/3)*x^3*(a + b*Log[c*(d + e*x^(2/3))^n])} -{x^1*(a + b*Log[c*(d + e*x^(2/3))^n]), x, 4, -((b*d^2*n*x^(2/3))/(2*e^2)) + (b*d*n*x^(4/3))/(4*e) - (1/6)*b*n*x^2 + (b*d^3*n*Log[d + e*x^(2/3)])/(2*e^3) + (1/2)*x^2*(a + b*Log[c*(d + e*x^(2/3))^n])} -{x^0*(a + b*Log[c*(d + e*x^(2/3))^n]), x, 6, (2*b*d*n*x^(1/3))/e + a*x - (2*b*n*x)/3 - (2*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/e^(3/2) + b*x*Log[c*(d + e*x^(2/3))^n]} -{(a + b*Log[c*(d + e*x^(2/3))^n])/x^1, x, 3, (3/2)*(a + b*Log[c*(d + e*x^(2/3))^n])*Log[-((e*x^(2/3))/d)] + (3/2)*b*n*PolyLog[2, 1 + (e*x^(2/3))/d]} -{(a + b*Log[c*(d + e*x^(2/3))^n])/x^2, x, 4, -((2*b*e*n)/(d*x^(1/3))) - (2*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(3/2) - (a + b*Log[c*(d + e*x^(2/3))^n])/x} -{(a + b*Log[c*(d + e*x^(2/3))^n])/x^3, x, 4, -((b*e*n)/(4*d*x^(4/3))) + (b*e^2*n)/(2*d^2*x^(2/3)) - (b*e^3*n*Log[d + e*x^(2/3)])/(2*d^3) - (a + b*Log[c*(d + e*x^(2/3))^n])/(2*x^2) + (b*e^3*n*Log[x])/(3*d^3)} -{(a + b*Log[c*(d + e*x^(2/3))^n])/x^4, x, 7, -((2*b*e*n)/(21*d*x^(7/3))) + (2*b*e^2*n)/(15*d^2*x^(5/3)) - (2*b*e^3*n)/(9*d^3*x) + (2*b*e^4*n)/(3*d^4*x^(1/3)) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*d^(9/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])/(3*x^3)} - - -{x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2, x, 8, (15*b^2*d^4*n^2*(d + e*x^(2/3))^2)/(8*e^6) - (10*b^2*d^3*n^2*(d + e*x^(2/3))^3)/(9*e^6) + (15*b^2*d^2*n^2*(d + e*x^(2/3))^4)/(32*e^6) - (3*b^2*d*n^2*(d + e*x^(2/3))^5)/(25*e^6) + (b^2*n^2*(d + e*x^(2/3))^6)/(72*e^6) - (3*b^2*d^5*n^2*x^(2/3))/e^5 + (b^2*d^6*n^2*Log[d + e*x^(2/3)]^2)/(4*e^6) + (3*b*d^5*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^6 - (15*b*d^4*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*e^6) + (10*b*d^3*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^6) - (15*b*d^2*n*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*e^6) + (3*b*d*n*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*e^6) - (b*n*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(12*e^6) - (b*d^6*n*Log[d + e*x^(2/3)]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*e^6) + (1/4)*x^4*(a + b*Log[c*(d + e*x^(2/3))^n])^2} -{x^1*(a + b*Log[c*(d + e*x^(2/3))^n])^2, x, 8, -((3*b^2*d*n^2*(d + e*x^(2/3))^2)/(4*e^3)) + (b^2*n^2*(d + e*x^(2/3))^3)/(9*e^3) + (3*b^2*d^2*n^2*x^(2/3))/e^2 - (b^2*d^3*n^2*Log[d + e*x^(2/3)]^2)/(2*e^3) - (3*b*d^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3 + (3*b*d*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*e^3) - (b*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^3) + (b*d^3*n*Log[d + e*x^(2/3)]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^3 + (1/2)*x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^1, x, 5, (3/2)*(a + b*Log[c*(d + e*x^(2/3))^n])^2*Log[-((e*x^(2/3))/d)] + 3*b*n*(a + b*Log[c*(d + e*x^(2/3))^n])*PolyLog[2, 1 + (e*x^(2/3))/d] - 3*b^2*n^2*PolyLog[3, 1 + (e*x^(2/3))/d]} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^3, x, 12, -((b^2*e^2*n^2)/(2*d^2*x^(2/3))) + (b^2*e^3*n^2*Log[d + e*x^(2/3)])/(2*d^3) - (b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d*x^(4/3)) + (b*e^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(d^3*x^(2/3)) + (b*e^3*n*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^3 - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(2*x^2) - (b^2*e^3*n^2*Log[x])/d^3 - (b^2*e^3*n^2*PolyLog[2, d/(d + e*x^(2/3))])/d^3} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^5, x, 24, -((b^2*e^2*n^2)/(40*d^2*x^(8/3))) + (3*b^2*e^3*n^2)/(40*d^3*x^2) - (47*b^2*e^4*n^2)/(240*d^4*x^(4/3)) + (77*b^2*e^5*n^2)/(120*d^5*x^(2/3)) - (77*b^2*e^6*n^2*Log[d + e*x^(2/3)])/(120*d^6) - (b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(10*d*x^(10/3)) + (b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*d^2*x^(8/3)) - (b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(6*d^3*x^2) + (b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*d^4*x^(4/3)) - (b*e^5*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^6*x^(2/3)) - (b*e^6*n*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^6) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(4*x^4) + (137*b^2*e^6*n^2*Log[x])/(180*d^6) + (b^2*e^6*n^2*PolyLog[2, d/(d + e*x^(2/3))])/(2*d^6)} - -{x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2, x, 30, -((4*a*b*d^4*n*x^(1/3))/(3*e^4)) + (4504*b^2*d^4*n^2*x^(1/3))/(945*e^4) - (1984*b^2*d^3*n^2*x)/(2835*e^3) + (1144*b^2*d^2*n^2*x^(5/3))/(4725*e^2) - (128*b^2*d*n^2*x^(7/3))/(1323*e) + (8/243)*b^2*n^2*x^3 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(945*e^(9/2)) + (4*I*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/(3*e^(9/2)) + (8*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(3*e^(9/2)) - (4*b^2*d^4*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/(3*e^4) + (4*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(9*e^3) - (4*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*e^2) + (4*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(21*e) - (4/27)*b*n*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]) + (4*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^(9/2)) + (1/3)*x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2 + (4*I*b^2*d^(9/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(3*e^(9/2))} -{x^0*(a + b*Log[c*(d + e*x^(2/3))^n])^2, x, 18, (4*a*b*d*n*x^(1/3))/e - (32*b^2*d*n^2*x^(1/3))/(3*e) + (8/9)*b^2*n^2*x + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(3/2)) - (4*I*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(3/2) - (8*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2) + (4*b^2*d*n*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/e - (4/3)*b*n*x*(a + b*Log[c*(d + e*x^(2/3))^n]) - (4*b*d^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^(3/2) + x*(a + b*Log[c*(d + e*x^(2/3))^n])^2 - (4*I*b^2*d^(3/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2)} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^2, x, 12, (8*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/d^(3/2) - (4*I*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(3/2) - (8*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(3/2) - (4*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(d*x^(1/3)) - (4*b*e^(3/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^(3/2) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/x - (4*I*b^2*e^(3/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(3/2)} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^4, x, 24, -((8*b^2*e^2*n^2)/(105*d^2*x^(5/3))) + (32*b^2*e^3*n^2)/(105*d^3*x) - (568*b^2*e^4*n^2)/(315*d^4*x^(1/3)) - (1408*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(315*d^(9/2)) + (4*I*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/(3*d^(9/2)) + (8*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(3*d^(9/2)) - (4*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(21*d*x^(7/3)) + (4*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*d^2*x^(5/3)) - (4*b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(9*d^3*x) + (4*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*d^4*x^(1/3)) + (4*b*e^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*d^(9/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(3*x^3) + (4*I*b^2*e^(9/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(3*d^(9/2))} -{(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^6, x, 45, -((8*b^2*e^2*n^2)/(715*d^2*x^(11/3))) + (64*b^2*e^3*n^2)/(2145*d^3*x^3) - (2872*b^2*e^4*n^2)/(45045*d^4*x^(7/3)) + (1216*b^2*e^5*n^2)/(9009*d^5*x^(5/3)) - (224072*b^2*e^6*n^2)/(675675*d^6*x) + (344192*b^2*e^7*n^2)/(225225*d^7*x^(1/3)) + (704552*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(225225*d^(15/2)) - (4*I*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/(5*d^(15/2)) - (8*b^2*e^(15/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(5*d^(15/2)) - (4*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(65*d*x^(13/3)) + (4*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(55*d^2*x^(11/3)) - (4*b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(45*d^3*x^3) + (4*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^4*x^(7/3)) - (4*b*e^5*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(25*d^5*x^(5/3)) + (4*b*e^6*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(15*d^6*x) - (4*b*e^7*n*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^7*x^(1/3)) - (4*b*e^(15/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(5*d^(15/2)) - (a + b*Log[c*(d + e*x^(2/3))^n])^2/(5*x^5) - (4*I*b^2*e^(15/2)*n^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(5*d^(15/2))} - - -{x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3, x, 28, (-45*b^3*d^4*n^3*(d + e*x^(2/3))^2)/(16*e^6) + (10*b^3*d^3*n^3*(d + e*x^(2/3))^3)/(9*e^6) - (45*b^3*d^2*n^3*(d + e*x^(2/3))^4)/(128*e^6) + (9*b^3*d*n^3*(d + e*x^(2/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(2/3))^6)/(144*e^6) - (9*a*b^2*d^5*n^2*x^(2/3))/e^5 + (9*b^3*d^5*n^3*x^(2/3))/e^5 - (9*b^3*d^5*n^2*(d + e*x^(2/3))*Log[c*(d + e*x^(2/3))^n])/e^6 + (45*b^2*d^4*n^2*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(8*e^6) - (10*b^2*d^3*n^2*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^6) + (45*b^2*d^2*n^2*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n]))/(32*e^6) - (9*b^2*d*n^2*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n]))/(25*e^6) + (b^2*n^2*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(24*e^6) + (9*b*d^5*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(2*e^6) - (45*b*d^4*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(8*e^6) + (5*b*d^3*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e^6 - (45*b*d^2*n*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(16*e^6) + (9*b*d*n*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(10*e^6) - (b*n*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(8*e^6) - (3*d^5*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^6) + (15*d^4*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(4*e^6) - (5*d^3*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/e^6 + (15*d^2*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(4*e^6) - (3*d*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^6) + ((d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(4*e^6)} -{x^1*(a + b*Log[c*(d + e*x^(2/3))^n])^3, x, 16, (9*b^3*d*n^3*(d + e*x^(2/3))^2)/(8*e^3) - (b^3*n^3*(d + e*x^(2/3))^3)/(9*e^3) + (9*a*b^2*d^2*n^2*x^(2/3))/e^2 - (9*b^3*d^2*n^3*x^(2/3))/e^2 + (9*b^3*d^2*n^2*(d + e*x^(2/3))*Log[c*(d + e*x^(2/3))^n])/e^3 - (9*b^2*d*n^2*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*e^3) + (b^2*n^2*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(3*e^3) - (9*b*d^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(2*e^3) + (9*b*d*n*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(4*e^3) - (b*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(2*e^3) + (3*d^2*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^3) - (3*d*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^3) + ((d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^3)} -{(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^1, x, 6, (3*(a + b*Log[c*(d + e*x^(2/3))^n])^3*Log[-((e*x^(2/3))/d)])/2 + (9*b*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2*PolyLog[2, 1 + (e*x^(2/3))/d])/2 - 9*b^2*n^2*(a + b*Log[c*(d + e*x^(2/3))^n])*PolyLog[3, 1 + (e*x^(2/3))/d] + 9*b^3*n^3*PolyLog[4, 1 + (e*x^(2/3))/d]} -{(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^3, x, 17, -((3*b^2*e^2*n^2*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^3*x^(2/3))) - (3*b^2*e^3*n^2*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(2*d^3) - (3*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(4*d*x^(4/3)) + (3*b*e^2*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(2*d^3*x^(2/3)) + (3*b*e^3*n*Log[1 - d/(d + e*x^(2/3))]*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(2*d^3) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/(2*x^2) - (3*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(2/3))^n])*Log[-((e*x^(2/3))/d)])/d^3 + (b^3*e^3*n^3*Log[x])/d^3 + (3*b^3*e^3*n^3*PolyLog[2, d/(d + e*x^(2/3))])/(2*d^3) - (3*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(2/3))^n])*PolyLog[2, d/(d + e*x^(2/3))])/d^3 - (3*b^3*e^3*n^3*PolyLog[2, 1 + (e*x^(2/3))/d])/d^3 - (3*b^3*e^3*n^3*PolyLog[3, d/(d + e*x^(2/3))])/d^3} - -{x^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3, x, 109, (4504*a*b^2*d^4*n^2*x^(1/3))/(315*e^4) - (3475504*b^3*d^4*n^3*x^(1/3))/(99225*e^4) + (637984*b^3*d^3*n^3*x)/(297675*e^3) - (221344*b^3*d^2*n^3*x^(5/3))/(496125*e^2) + (3088*b^3*d*n^3*x^(7/3))/(27783*e) - (16*b^3*n^3*x^3)/729 + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(99225*e^(9/2)) - (((4504*I)/315)*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(9/2) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(315*e^(9/2)) + (4504*b^3*d^4*n^2*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/(315*e^4) - (1984*b^2*d^3*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/(945*e^3) + (1144*b^2*d^2*n^2*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(1575*e^2) - (128*b^2*d*n^2*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n]))/(441*e) + (8*b^2*n^2*x^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/81 - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(315*e^(9/2)) - (2*b*d^4*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e^4 + (2*b*d^3*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^3) - (2*b*d^2*n*x^(5/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(5*e^2) + (2*b*d*n*x^(7/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*e) - (2*b*n*x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/9 + (x^3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/3 - (((4504*I)/315)*b^3*d^(9/2)*n^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(9/2) + (2*b*d^5*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^n])^2/((d + e*x^(2/3))*x^(2/3)), x])/(3*e^4)} -{x^0*(a + b*Log[c*(d + e*x^(2/3))^n])^3, x, 34, (-32*a*b^2*d*n^2*x^(1/3))/e + (208*b^3*d*n^3*x^(1/3))/(3*e) - (16*b^3*n^3*x)/9 - (208*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(3/2)) + ((32*I)*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/e^(3/2) + (64*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2) - (32*b^3*d*n^2*x^(1/3)*Log[c*(d + e*x^(2/3))^n])/e + (8*b^2*n^2*x*(a + b*Log[c*(d + e*x^(2/3))^n]))/3 + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/e^(3/2) + (6*b*d*n*x^(1/3)*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e - 2*b*n*x*(a + b*Log[c*(d + e*x^(2/3))^n])^2 + x*(a + b*Log[c*(d + e*x^(2/3))^n])^3 + ((32*I)*b^3*d^(3/2)*n^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/e^(3/2) - (2*b*d^2*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^n])^2/((d + e*x^(2/3))*x^(2/3)), x])/e} -{(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2, x, 11, ((24*I)*b^3*e^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/d^(3/2) + (48*b^3*e^(3/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(3/2) + (24*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/d^(3/2) - (6*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(d*x^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/x + ((24*I)*b^3*e^(3/2)*n^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/d^(3/2) - (2*b*e^2*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^n])^2/((d + e*x^(2/3))*x^(2/3)), x])/d} -{(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^4, x, 54, -((16*b^3*e^3*n^3)/(105*d^3*x)) + (16*b^3*e^4*n^3)/(7*d^4*x^(1/3)) + (1376*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(105*d^(9/2)) - (1408*I*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]^2)/(105*d^(9/2)) - (2816*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(105*d^(9/2)) - (8*b^2*e^2*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^2*x^(5/3)) + (32*b^2*e^3*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(35*d^3*x) - (568*b^2*e^4*n^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^4*x^(1/3)) - (1408*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]]*(a + b*Log[c*(d + e*x^(2/3))^n]))/(105*d^(9/2)) - (2*b*e*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(7*d*x^(7/3)) + (2*b*e^2*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(5*d^2*x^(5/3)) - (2*b*e^3*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*d^3*x) + (2*b*e^4*n*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(d^4*x^(1/3)) - (a + b*Log[c*(d + e*x^(2/3))^n])^3/(3*x^3) - (1408*I*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x^(1/3))])/(105*d^(9/2)) + (2*b*e^5*n*Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^n])^2/((d + e*x^(2/3))*x^(2/3)), x])/(3*d^4)} - - -(* ::Subsubsection::Closed:: *) -(*m=-1*) - - -{x^3*(a + b*Log[c*(d + e/x^(1/3))^n]), x, 4, (b*e^11*n*x^(1/3))/(4*d^11) - (b*e^10*n*x^(2/3))/(8*d^10) + (b*e^9*n*x)/(12*d^9) - (b*e^8*n*x^(4/3))/(16*d^8) + (b*e^7*n*x^(5/3))/(20*d^7) - (b*e^6*n*x^2)/(24*d^6) + (b*e^5*n*x^(7/3))/(28*d^5) - (b*e^4*n*x^(8/3))/(32*d^4) + (b*e^3*n*x^3)/(36*d^3) - (b*e^2*n*x^(10/3))/(40*d^2) + (b*e*n*x^(11/3))/(44*d) - (b*e^12*n*Log[d + e/x^(1/3)])/(4*d^12) + (x^4*(a + b*Log[c*(d + e/x^(1/3))^n]))/4 - (b*e^12*n*Log[x])/(12*d^12)} -{x^2*(a + b*Log[c*(d + e/x^(1/3))^n]), x, 4, -(b*e^8*n*x^(1/3))/(3*d^8) + (b*e^7*n*x^(2/3))/(6*d^7) - (b*e^6*n*x)/(9*d^6) + (b*e^5*n*x^(4/3))/(12*d^5) - (b*e^4*n*x^(5/3))/(15*d^4) + (b*e^3*n*x^2)/(18*d^3) - (b*e^2*n*x^(7/3))/(21*d^2) + (b*e*n*x^(8/3))/(24*d) + (b*e^9*n*Log[d + e/x^(1/3)])/(3*d^9) + (x^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/3 + (b*e^9*n*Log[x])/(9*d^9)} -{x^1*(a + b*Log[c*(d + e/x^(1/3))^n]), x, 4, (b*e^5*n*x^(1/3))/(2*d^5) - (b*e^4*n*x^(2/3))/(4*d^4) + (b*e^3*n*x)/(6*d^3) - (b*e^2*n*x^(4/3))/(8*d^2) + (b*e*n*x^(5/3))/(10*d) - (b*e^6*n*Log[d + e/x^(1/3)])/(2*d^6) + (x^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/2 - (b*e^6*n*Log[x])/(6*d^6)} -{x^0*(a + b*Log[c*(d + e/x^(1/3))^n]), x, 6, -((b*e^2*n*x^(1/3))/d^2) + (b*e*n*x^(2/3))/(2*d) + a*x + b*x*Log[c*(d + e/x^(1/3))^n] + (b*e^3*n*Log[e + d*x^(1/3)])/d^3} -{(a + b*Log[c*(d + e/x^(1/3))^n])/x^1, x, 3, -3*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))] - 3*b*n*PolyLog[2, 1 + e/(d*x^(1/3))]} -{(a + b*Log[c*(d + e/x^(1/3))^n])/x^2, x, 4, (b*n)/(3*x) - (b*d*n)/(2*e*x^(2/3)) + (b*d^2*n)/(e^2*x^(1/3)) - (b*d^3*n*Log[d + e/x^(1/3)])/e^3 - (a + b*Log[c*(d + e/x^(1/3))^n])/x} -{(a + b*Log[c*(d + e/x^(1/3))^n])/x^3, x, 4, (b*n)/(12*x^2) - (b*d*n)/(10*e*x^(5/3)) + (b*d^2*n)/(8*e^2*x^(4/3)) - (b*d^3*n)/(6*e^3*x) + (b*d^4*n)/(4*e^4*x^(2/3)) - (b*d^5*n)/(2*e^5*x^(1/3)) + (b*d^6*n*Log[d + e/x^(1/3)])/(2*e^6) - (a + b*Log[c*(d + e/x^(1/3))^n])/(2*x^2)} -{(a + b*Log[c*(d + e/x^(1/3))^n])/x^4, x, 4, (b*n)/(27*x^3) - (b*d*n)/(24*e*x^(8/3)) + (b*d^2*n)/(21*e^2*x^(7/3)) - (b*d^3*n)/(18*e^3*x^2) + (b*d^4*n)/(15*e^4*x^(5/3)) - (b*d^5*n)/(12*e^5*x^(4/3)) + (b*d^6*n)/(9*e^6*x) - (b*d^7*n)/(6*e^7*x^(2/3)) + (b*d^8*n)/(3*e^8*x^(1/3)) - (b*d^9*n*Log[d + e/x^(1/3)])/(3*e^9) - (a + b*Log[c*(d + e/x^(1/3))^n])/(3*x^3)} - - -{x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2, x, 36, (481*b^2*e^8*n^2*x^(1/3))/(420*d^8) - (341*b^2*e^7*n^2*x^(2/3))/(840*d^7) + (743*b^2*e^6*n^2*x)/(3780*d^6) - (533*b^2*e^5*n^2*x^(4/3))/(5040*d^5) + (73*b^2*e^4*n^2*x^(5/3))/(1260*d^4) - (5*b^2*e^3*n^2*x^2)/(168*d^3) + (b^2*e^2*n^2*x^(7/3))/(84*d^2) - (481*b^2*e^9*n^2*Log[d + e/x^(1/3)])/(420*d^9) - (2*b*e^8*n*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9) + (b*e^7*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^7) - (2*b*e^6*n*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*d^6) + (b*e^5*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6*d^5) - (2*b*e^4*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(15*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*d^3) - (2*b*e^2*n*x^(7/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(21*d^2) + (b*e*n*x^(8/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(12*d) - (2*b*e^9*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^9) + (1/3)*x^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2 - (761*b^2*e^9*n^2*Log[x])/(1260*d^9) + (2*b^2*e^9*n^2*PolyLog[2, d/(d + e/x^(1/3))])/(3*d^9)} -{x^1*(a + b*Log[c*(d + e/x^(1/3))^n])^2, x, 24, -((77*b^2*e^5*n^2*x^(1/3))/(60*d^5)) + (47*b^2*e^4*n^2*x^(2/3))/(120*d^4) - (3*b^2*e^3*n^2*x)/(20*d^3) + (b^2*e^2*n^2*x^(4/3))/(20*d^2) + (77*b^2*e^6*n^2*Log[d + e/x^(1/3)])/(60*d^6) + (b*e^5*n*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^6 - (b*e^4*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*d^4) + (b*e^3*n*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*d^3) - (b*e^2*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*d^2) + (b*e*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(5*d) + (b*e^6*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^6 + (1/2)*x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2 + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (b^2*e^6*n^2*PolyLog[2, d/(d + e/x^(1/3))])/d^6} -{x^0*(a + b*Log[c*(d + e/x^(1/3))^n])^2, x, 13, (b^2*e^2*n^2*x^(1/3))/d^2 - (b^2*e^3*n^2*Log[d + e/x^(1/3)])/d^3 - (2*b*e^2*n*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^3 + (b*e*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/d - (2*b*e^3*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^3 + x*(a + b*Log[c*(d + e/x^(1/3))^n])^2 - (b^2*e^3*n^2*Log[x])/d^3 + (2*b^2*e^3*n^2*PolyLog[2, d/(d + e/x^(1/3))])/d^3} -{(a + b*Log[c*(d + e/x^(1/3))^n])^2/x^1, x, 5, -3*(a + b*Log[c*(d + e/x^(1/3))^n])^2*Log[-(e/(d*x^(1/3)))] - 6*b*n*(a + b*Log[c*(d + e/x^(1/3))^n])*PolyLog[2, 1 + e/(d*x^(1/3))] + 6*b^2*n^2*PolyLog[3, 1 + e/(d*x^(1/3))]} -{(a + b*Log[c*(d + e/x^(1/3))^n])^2/x^2, x, 8, (3*b^2*d*n^2*(d + e/x^(1/3))^2)/(2*e^3) - (2*b^2*n^2*(d + e/x^(1/3))^3)/(9*e^3) - (6*b^2*d^2*n^2)/(e^2*x^(1/3)) + (b^2*d^3*n^2*Log[d + e/x^(1/3)]^2)/e^3 + (6*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3 - (3*b*d*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3 + (2*b*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^3) - (2*b*d^3*n*Log[d + e/x^(1/3)]*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^3 - (a + b*Log[c*(d + e/x^(1/3))^n])^2/x} -{(a + b*Log[c*(d + e/x^(1/3))^n])^2/x^3, x, 8, -((15*b^2*d^4*n^2*(d + e/x^(1/3))^2)/(4*e^6)) + (20*b^2*d^3*n^2*(d + e/x^(1/3))^3)/(9*e^6) - (15*b^2*d^2*n^2*(d + e/x^(1/3))^4)/(16*e^6) + (6*b^2*d*n^2*(d + e/x^(1/3))^5)/(25*e^6) - (b^2*n^2*(d + e/x^(1/3))^6)/(36*e^6) + (6*b^2*d^5*n^2)/(e^5*x^(1/3)) - (b^2*d^6*n^2*Log[d + e/x^(1/3)]^2)/(2*e^6) - (6*b*d^5*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^6 + (15*b*d^4*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*e^6) - (20*b*d^3*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^6) + (15*b*d^2*n*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*e^6) - (6*b*d*n*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n]))/(5*e^6) + (b*n*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n]))/(6*e^6) + (b*d^6*n*Log[d + e/x^(1/3)]*(a + b*Log[c*(d + e/x^(1/3))^n]))/e^6 - (a + b*Log[c*(d + e/x^(1/3))^n])^2/(2*x^2)} - - -{x^1*(a + b*Log[c*(d + e/x^(1/3))^n])^3, x, 62, (71*b^3*e^5*n^3*x^(1/3))/(40*d^5) - (3*b^3*e^4*n^3*x^(2/3))/(10*d^4) + (b^3*e^3*n^3*x)/(20*d^3) - (71*b^3*e^6*n^3*Log[d + e/x^(1/3)])/(40*d^6) - (77*b^2*e^5*n^2*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(20*d^6) + (47*b^2*e^4*n^2*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(40*d^4) - (9*b^2*e^3*n^2*x*(a + b*Log[c*(d + e/x^(1/3))^n]))/(20*d^3) + (3*b^2*e^2*n^2*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/(20*d^2) - (77*b^2*e^6*n^2*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/(20*d^6) + (3*b*e^5*n*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*d^6) - (3*b*e^4*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*d^4) + (b*e^3*n*x*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*d^3) - (3*b*e^2*n*x^(4/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(8*d^2) + (3*b*e*n*x^(5/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(10*d) + (3*b*e^6*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*d^6) + (1/2)*x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3 - (3*b^2*e^6*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))])/d^6 - (15*b^3*e^6*n^3*Log[x])/(8*d^6) + (77*b^3*e^6*n^3*PolyLog[2, d/(d + e/x^(1/3))])/(20*d^6) - (3*b^2*e^6*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*PolyLog[2, d/(d + e/x^(1/3))])/d^6 - (3*b^3*e^6*n^3*PolyLog[2, 1 + e/(d*x^(1/3))])/d^6 - (3*b^3*e^6*n^3*PolyLog[3, d/(d + e/x^(1/3))])/d^6} -{x^0*(a + b*Log[c*(d + e/x^(1/3))^n])^3, x, 18, (3*b^2*e^2*n^2*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^3 + (3*b^2*e^3*n^2*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n]))/d^3 - (3*b*e^2*n*(d + e/x^(1/3))*x^(1/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/d^3 + (3*b*e*n*x^(2/3)*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*d) - (3*b*e^3*n*Log[1 - d/(d + e/x^(1/3))]*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/d^3 + x*(a + b*Log[c*(d + e/x^(1/3))^n])^3 + (6*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*Log[-(e/(d*x^(1/3)))])/d^3 + (b^3*e^3*n^3*Log[x])/d^3 - (3*b^3*e^3*n^3*PolyLog[2, d/(d + e/x^(1/3))])/d^3 + (6*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*PolyLog[2, d/(d + e/x^(1/3))])/d^3 + (6*b^3*e^3*n^3*PolyLog[2, 1 + e/(d*x^(1/3))])/d^3 + (6*b^3*e^3*n^3*PolyLog[3, d/(d + e/x^(1/3))])/d^3} -{(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^1, x, 6, -3*(a + b*Log[c*(d + e/x^(1/3))^n])^3*Log[-(e/(d*x^(1/3)))] - 9*b*n*(a + b*Log[c*(d + e/x^(1/3))^n])^2*PolyLog[2, 1 + e/(d*x^(1/3))] + 18*b^2*n^2*(a + b*Log[c*(d + e/x^(1/3))^n])*PolyLog[3, 1 + e/(d*x^(1/3))] - 18*b^3*n^3*PolyLog[4, 1 + e/(d*x^(1/3))]} -{(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2, x, 16, (-9*b^3*d*n^3*(d + e/x^(1/3))^2)/(4*e^3) + (2*b^3*n^3*(d + e/x^(1/3))^3)/(9*e^3) - (18*a*b^2*d^2*n^2)/(e^2*x^(1/3)) + (18*b^3*d^2*n^3)/(e^2*x^(1/3)) - (18*b^3*d^2*n^2*(d + e/x^(1/3))*Log[c*(d + e/x^(1/3))^n])/e^3 + (9*b^2*d*n^2*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*e^3) - (2*b^2*n^2*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 - (9*b*d*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*e^3) + (b*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 - (3*d^2*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 + (3*d*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 - ((d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3} -{(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^3, x, 28, (45*b^3*d^4*n^3*(d + e/x^(1/3))^2)/(8*e^6) - (20*b^3*d^3*n^3*(d + e/x^(1/3))^3)/(9*e^6) + (45*b^3*d^2*n^3*(d + e/x^(1/3))^4)/(64*e^6) - (18*b^3*d*n^3*(d + e/x^(1/3))^5)/(125*e^6) + (b^3*n^3*(d + e/x^(1/3))^6)/(72*e^6) + (18*a*b^2*d^5*n^2)/(e^5*x^(1/3)) - (18*b^3*d^5*n^3)/(e^5*x^(1/3)) + (18*b^3*d^5*n^2*(d + e/x^(1/3))*Log[c*(d + e/x^(1/3))^n])/e^6 - (45*b^2*d^4*n^2*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*e^6) + (20*b^2*d^3*n^2*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(3*e^6) - (45*b^2*d^2*n^2*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n]))/(16*e^6) + (18*b^2*d*n^2*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n]))/(25*e^6) - (b^2*n^2*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n]))/(12*e^6) - (9*b*d^5*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^6 + (45*b*d^4*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*e^6) - (10*b*d^3*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^6 + (45*b*d^2*n*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(8*e^6) - (9*b*d*n*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(5*e^6) + (b*n*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*e^6) + (3*d^5*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - (15*d^4*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6) + (10*d^3*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - (15*d^2*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6) + (3*d*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - ((d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6)} - - -(* ::Subsubsection::Closed:: *) -(*m=-2*) - - -{x^3*(a + b*Log[c*(d + e/x^(2/3))^n]), x, 4, (b*e^5*n*x^(2/3))/(4*d^5) - (b*e^4*n*x^(4/3))/(8*d^4) + (b*e^3*n*x^2)/(12*d^3) - (b*e^2*n*x^(8/3))/(16*d^2) + (b*e*n*x^(10/3))/(20*d) - (b*e^6*n*Log[d + e/x^(2/3)])/(4*d^6) + (x^4*(a + b*Log[c*(d + e/x^(2/3))^n]))/4 - (b*e^6*n*Log[x])/(6*d^6)} -{x^2*(a + b*Log[c*(d + e/x^(2/3))^n]), x, 6, (-2*b*e^4*n*x^(1/3))/(3*d^4) + (2*b*e^3*n*x)/(9*d^3) - (2*b*e^2*n*x^(5/3))/(15*d^2) + (2*b*e*n*x^(7/3))/(21*d) + (2*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/3} -{x^1*(a + b*Log[c*(d + e/x^(2/3))^n]), x, 4, -(b*e^2*n*x^(2/3))/(2*d^2) + (b*e*n*x^(4/3))/(4*d) + (b*e^3*n*Log[d + e/x^(2/3)])/(2*d^3) + (x^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/2 + (b*e^3*n*Log[x])/(3*d^3)} -{x^0*(a + b*Log[c*(d + e/x^(2/3))^n]), x, 6, (2*b*e*n*x^(1/3))/d + a*x - (2*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/d^(3/2) + b*x*Log[c*(d + e/x^(2/3))^n]} -{(a + b*Log[c*(d + e/x^(2/3))^n])/x^1, x, 3, (-3*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))])/2 - (3*b*n*PolyLog[2, 1 + e/(d*x^(2/3))])/2} -{(a + b*Log[c*(d + e/x^(2/3))^n])/x^2, x, 6, (2*b*n)/(3*x) - (2*b*d*n)/(e*x^(1/3)) - (2*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/e^(3/2) - (a + b*Log[c*(d + e/x^(2/3))^n])/x} -{(a + b*Log[c*(d + e/x^(2/3))^n])/x^3, x, 4, (b*n)/(6*x^2) - (b*d*n)/(4*e*x^(4/3)) + (b*d^2*n)/(2*e^2*x^(2/3)) - (b*d^3*n*Log[d + e/x^(2/3)])/(2*e^3) - (a + b*Log[c*(d + e/x^(2/3))^n])/(2*x^2)} -{(a + b*Log[c*(d + e/x^(2/3))^n])/x^4, x, 9, (2*b*n)/(27*x^3) - (2*b*d*n)/(21*e*x^(7/3)) + (2*b*d^2*n)/(15*e^2*x^(5/3)) - (2*b*d^3*n)/(9*e^3*x) + (2*b*d^4*n)/(3*e^4*x^(1/3)) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*e^(9/2)) - (a + b*Log[c*(d + e/x^(2/3))^n])/(3*x^3)} - - -{x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2, x, 24, -((77*b^2*e^5*n^2*x^(2/3))/(120*d^5)) + (47*b^2*e^4*n^2*x^(4/3))/(240*d^4) - (3*b^2*e^3*n^2*x^2)/(40*d^3) + (b^2*e^2*n^2*x^(8/3))/(40*d^2) + (77*b^2*e^6*n^2*Log[d + e/x^(2/3)])/(120*d^6) + (b*e^5*n*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d^6) - (b*e^4*n*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(4*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(6*d^3) - (b*e^2*n*x^(8/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(8*d^2) + (b*e*n*x^(10/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(10*d) + (b*e^6*n*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d^6) + (1/4)*x^4*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + (137*b^2*e^6*n^2*Log[x])/(180*d^6) - (b^2*e^6*n^2*PolyLog[2, d/(d + e/x^(2/3))])/(2*d^6)} -{x^1*(a + b*Log[c*(d + e/x^(2/3))^n])^2, x, 12, (b^2*e^2*n^2*x^(2/3))/(2*d^2) - (b^2*e^3*n^2*Log[d + e/x^(2/3)])/(2*d^3) - (b*e^2*n*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/d^3 + (b*e*n*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d) - (b*e^3*n*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n]))/d^3 + (1/2)*x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2 - (b^2*e^3*n^2*Log[x])/d^3 + (b^2*e^3*n^2*PolyLog[2, d/(d + e/x^(2/3))])/d^3} -{(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^1, x, 5, (-3*(a + b*Log[c*(d + e/x^(2/3))^n])^2*Log[-(e/(d*x^(2/3)))])/2 - 3*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])*PolyLog[2, 1 + e/(d*x^(2/3))] + 3*b^2*n^2*PolyLog[3, 1 + e/(d*x^(2/3))]} -{(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^3, x, 8, (3*b^2*d*n^2*(d + e/x^(2/3))^2)/(4*e^3) - (b^2*n^2*(d + e/x^(2/3))^3)/(9*e^3) - (3*b^2*d^2*n^2)/(e^2*x^(2/3)) + (b^2*d^3*n^2*Log[d + e/x^(2/3)]^2)/(2*e^3) + (3*b*d^2*n*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^3 - (3*b*d*n*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*e^3) + (b*n*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^3) - (b*d^3*n*Log[d + e/x^(2/3)]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^3 - (a + b*Log[c*(d + e/x^(2/3))^n])^2/(2*x^2)} -{(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^5, x, 8, -((15*b^2*d^4*n^2*(d + e/x^(2/3))^2)/(8*e^6)) + (10*b^2*d^3*n^2*(d + e/x^(2/3))^3)/(9*e^6) - (15*b^2*d^2*n^2*(d + e/x^(2/3))^4)/(32*e^6) + (3*b^2*d*n^2*(d + e/x^(2/3))^5)/(25*e^6) - (b^2*n^2*(d + e/x^(2/3))^6)/(72*e^6) + (3*b^2*d^5*n^2)/(e^5*x^(2/3)) - (b^2*d^6*n^2*Log[d + e/x^(2/3)]^2)/(4*e^6) - (3*b*d^5*n*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^6 + (15*b*d^4*n*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(4*e^6) - (10*b*d^3*n*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^6) + (15*b*d^2*n*(d + e/x^(2/3))^4*(a + b*Log[c*(d + e/x^(2/3))^n]))/(8*e^6) - (3*b*d*n*(d + e/x^(2/3))^5*(a + b*Log[c*(d + e/x^(2/3))^n]))/(5*e^6) + (b*n*(d + e/x^(2/3))^6*(a + b*Log[c*(d + e/x^(2/3))^n]))/(12*e^6) + (b*d^6*n*Log[d + e/x^(2/3)]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*e^6) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/(4*x^4)} - -{x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2, x, 28, (-4*a*b*e^4*n*x^(1/3))/(3*d^4) + (568*b^2*e^4*n^2*x^(1/3))/(315*d^4) - (32*b^2*e^3*n^2*x)/(105*d^3) + (8*b^2*e^2*n^2*x^(5/3))/(105*d^2) - (1408*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(315*d^(9/2)) - (((4*I)/3)*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) + (8*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(3*d^(9/2)) - (4*b^2*e^4*n*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/(3*d^4) + (4*b*e^3*n*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(9*d^3) - (4*b*e^2*n*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(15*d^2) + (4*b*e*n*x^(7/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(21*d) + (4*b*e^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*d^(9/2)) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/3 - (((4*I)/3)*b^2*e^(9/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2)} -{x^0*(a + b*Log[c*(d + e/x^(2/3))^n])^2, x, 14, (4*a*b*e*n*x^(1/3))/d + (8*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/d^(3/2) + ((4*I)*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(3/2) - (8*b^2*e^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(3/2) + (4*b^2*e*n*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/d - (4*b*e^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/d^(3/2) + x*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + ((4*I)*b^2*e^(3/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(3/2)} -{(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^2, x, 19, (-8*b^2*n^2)/(9*x) + (32*b^2*d*n^2)/(3*e*x^(1/3)) + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*e^(3/2)) + ((4*I)*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) - (8*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) + (4*b*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*x) - (4*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) - (4*b*d^(3/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/x + ((4*I)*b^2*d^(3/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2)} -(* {(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^4, x, 40, -((8*b^2*n^2)/(243*x^3)) + (128*b^2*d*n^2)/(1323*e*x^(7/3)) - (1144*b^2*d^2*n^2)/(4725*e^2*x^(5/3)) + (1984*b^2*d^3*n^2)/(2835*e^3*x) - (4504*b^2*d^4*n^2)/(945*e^4*x^(1/3)) - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(945*e^(9/2)) - (4*I*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/(3*e^(9/2)) + (8*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(3*e^(9/2)) + (4*b*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(27*x^3) - (4*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(21*e*x^(7/3)) + (4*b*d^2*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(15*e^2*x^(5/3)) - (4*b*d^3*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(9*e^3*x) + (4*b*d^4*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^4*x^(1/3)) + (4*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^(9/2)) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/(3*x^3) - (4*I*b^2*d^(9/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(3*e^(9/2))} *) - - -{x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3, x, 62, (71*b^3*e^5*n^3*x^(2/3))/(80*d^5) - (3*b^3*e^4*n^3*x^(4/3))/(20*d^4) + (b^3*e^3*n^3*x^2)/(40*d^3) - (71*b^3*e^6*n^3*Log[d + e/x^(2/3)])/(80*d^6) - (77*b^2*e^5*n^2*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(40*d^6) + (47*b^2*e^4*n^2*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(80*d^4) - (9*b^2*e^3*n^2*x^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(40*d^3) + (3*b^2*e^2*n^2*x^(8/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(40*d^2) - (77*b^2*e^6*n^2*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(40*d^6) + (3*b*e^5*n*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*d^6) - (3*b*e^4*n*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(8*d^4) + (b*e^3*n*x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*d^3) - (3*b*e^2*n*x^(8/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(16*d^2) + (3*b*e*n*x^(10/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(20*d) + (3*b*e^6*n*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*d^6) + (1/4)*x^4*(a + b*Log[c*(d + e/x^(2/3))^n])^3 - (3*b^2*e^6*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))])/(2*d^6) - (15*b^3*e^6*n^3*Log[x])/(8*d^6) + (77*b^3*e^6*n^3*PolyLog[2, d/(d + e/x^(2/3))])/(40*d^6) - (3*b^2*e^6*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*PolyLog[2, d/(d + e/x^(2/3))])/(2*d^6) - (3*b^3*e^6*n^3*PolyLog[2, 1 + e/(d*x^(2/3))])/(2*d^6) - (3*b^3*e^6*n^3*PolyLog[3, d/(d + e/x^(2/3))])/(2*d^6)} -{x^1*(a + b*Log[c*(d + e/x^(2/3))^n])^3, x, 17, (3*b^2*e^2*n^2*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d^3) + (3*b^2*e^3*n^2*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(2*d^3) - (3*b*e^2*n*(d + e/x^(2/3))*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*d^3) + (3*b*e*n*x^(4/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*d) - (3*b*e^3*n*Log[1 - d/(d + e/x^(2/3))]*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*d^3) + (1/2)*x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3 + (3*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[-(e/(d*x^(2/3)))])/d^3 + (b^3*e^3*n^3*Log[x])/d^3 - (3*b^3*e^3*n^3*PolyLog[2, d/(d + e/x^(2/3))])/(2*d^3) + (3*b^2*e^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*PolyLog[2, d/(d + e/x^(2/3))])/d^3 + (3*b^3*e^3*n^3*PolyLog[2, 1 + e/(d*x^(2/3))])/d^3 + (3*b^3*e^3*n^3*PolyLog[3, d/(d + e/x^(2/3))])/d^3} -{(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^1, x, 6, (-3*(a + b*Log[c*(d + e/x^(2/3))^n])^3*Log[-(e/(d*x^(2/3)))])/2 - (9*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2*PolyLog[2, 1 + e/(d*x^(2/3))])/2 + 9*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*PolyLog[3, 1 + e/(d*x^(2/3))] - 9*b^3*n^3*PolyLog[4, 1 + e/(d*x^(2/3))]} -{(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^3, x, 16, (-9*b^3*d*n^3*(d + e/x^(2/3))^2)/(8*e^3) + (b^3*n^3*(d + e/x^(2/3))^3)/(9*e^3) - (9*a*b^2*d^2*n^2)/(e^2*x^(2/3)) + (9*b^3*d^2*n^3)/(e^2*x^(2/3)) - (9*b^3*d^2*n^2*(d + e/x^(2/3))*Log[c*(d + e/x^(2/3))^n])/e^3 + (9*b^2*d*n^2*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(4*e^3) - (b^2*n^2*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*e^3) - (9*b*d*n*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(4*e^3) + (b*n*(d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(2*e^3) - (3*d^2*(d + e/x^(2/3))*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3) + (3*d*(d + e/x^(2/3))^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3) - ((d + e/x^(2/3))^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/(2*e^3)} - -{x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3, x, 83, (568*a*b^2*e^4*n^2*x^(1/3))/(105*d^4) - (16*b^3*e^4*n^3*x^(1/3))/(7*d^4) + (16*b^3*e^3*n^3*x)/(105*d^3) + (1376*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(105*d^(9/2)) + (((568*I)/105)*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/d^(9/2) - (1136*b^3*e^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(105*d^(9/2)) + (568*b^3*e^4*n^2*x^(1/3)*Log[c*(d + e/x^(2/3))^n])/(105*d^4) - (32*b^2*e^3*n^2*x*(a + b*Log[c*(d + e/x^(2/3))^n]))/(35*d^3) + (8*b^2*e^2*n^2*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n]))/(35*d^2) - (568*b^2*e^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(105*d^(9/2)) - (2*b*e^4*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d^4 + (2*b*e^3*n*x*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*d^3) - (2*b*e^2*n*x^(5/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*d^2) + (2*b*e*n*x^(7/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*d) + (x^3*(a + b*Log[c*(d + e/x^(2/3))^n])^3)/3 + (4*b^2*e^(9/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(9/2) - (2*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) - (4*b^2*e^(9/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)])/(-d)^(9/2) + (2*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(9/2) + (4*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(9/2) + (8*b^3*e^(9/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (((568*I)/105)*b^3*e^(9/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/d^(9/2) + (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) - (4*b^3*e^(9/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(9/2) + (4*b^3*e^(9/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(9/2) - (8*b^3*e^(9/2)*n^3*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(9/2) + (2*b*e^5*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/(3*d^4)} -{x^0*(a + b*Log[c*(d + e/x^(2/3))^n])^3, x, 31, (6*b*e*n*x^(1/3)*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/d + x*(a + b*Log[c*(d + e/x^(2/3))^n])^3 + (12*b^2*e^(3/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)])/(-d)^(3/2) - (6*b^3*e^(3/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]^2)/(-d)^(3/2) - (12*b^2*e^(3/2)*n^2*(a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)])/(-d)^(3/2) + (6*b^3*e^(3/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]^2)/(-d)^(3/2) + (12*b^3*e^(3/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(3/2) - (12*b^3*e^(3/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(3/2) - (24*b^3*e^(3/2)*n^3*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)]*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])])/(-d)^(3/2) + (24*b^3*e^(3/2)*n^3*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) + (24*b^3*e^(3/2)*n^3*PolyLog[2, 1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) - (12*b^3*e^(3/2)*n^3*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])])/(-d)^(3/2) + (12*b^3*e^(3/2)*n^3*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2])/(-d)^(3/2) - (24*b^3*e^(3/2)*n^3*PolyLog[2, 1 + (Sqrt[-d]*x^(1/3))/Sqrt[e]])/(-d)^(3/2) - (2*b*e^2*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/d} -{(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^2, x, 35, (16*b^3*n^3)/(9*x) - (208*b^3*d*n^3)/(3*e*x^(1/3)) - (208*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*e^(3/2)) - ((32*I)*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) + (64*b^3*d^(3/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (8*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(3*x) + (32*b^2*d*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) + (2*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/x - (6*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e*x^(1/3)) - (a + b*Log[c*(d + e/x^(2/3))^n])^3/x - ((32*I)*b^3*d^(3/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) - (2*b*d^2*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/e} -{(a + b*Log[c*(d + e/x^(2/3))^n])^3/x^4, x, 129, (16*b^3*n^3)/(729*x^3) - (3088*b^3*d*n^3)/(27783*e*x^(7/3)) + (221344*b^3*d^2*n^3)/(496125*e^2*x^(5/3)) - (637984*b^3*d^3*n^3)/(297675*e^3*x) + (3475504*b^3*d^4*n^3)/(99225*e^4*x^(1/3)) + (3475504*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(99225*e^(9/2)) + (4504*I*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/(315*e^(9/2)) - (9008*b^3*d^(9/2)*n^3*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(315*e^(9/2)) - (8*b^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(81*x^3) + (128*b^2*d*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(441*e*x^(7/3)) - (1144*b^2*d^2*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(1575*e^2*x^(5/3)) + (1984*b^2*d^3*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(945*e^3*x) - (4504*b^2*d^4*n^2*(a + b*Log[c*(d + e/x^(2/3))^n]))/(315*e^4*x^(1/3)) - (4504*b^2*d^(9/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/(315*e^(9/2)) + (2*b*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(9*x^3) - (2*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(7*e*x^(7/3)) + (2*b*d^2*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(5*e^2*x^(5/3)) - (2*b*d^3*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(3*e^3*x) + (2*b*d^4*n*(a + b*Log[c*(d + e/x^(2/3))^n])^2)/(e^4*x^(1/3)) - (a + b*Log[c*(d + e/x^(2/3))^n])^3/(3*x^3) + (4504*I*b^3*d^(9/2)*n^3*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/(315*e^(9/2)) + (2*b*d^5*n*Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^n])^2/((e + d*x^(2/3))*x^(2/3)), x])/(3*e^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q (a+b Log[c (d+e x^(m/2))^n])^p when p symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*m=1*) - - -{x^3*(a + b*Log[c*(d + e*Sqrt[x])])^p, x, 27, (2^(-2 - 3*p)*Gamma[1 + p, -((8*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^((8*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^8*e^8)) - (2*d*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(7^p*E^((7*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^7*e^8)) + (7*d^2*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(6^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^6*e^8)) - (14*d^3*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^5*e^8)) + (35*2^(-1 - 2*p)*d^4*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^4*e^8)) - (14*d^5*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^3*e^8)) + (7*d^6*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^2*e^8)) - (2*d^7*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c*e^8))} -{x^2*(a + b*Log[c*(d + e*Sqrt[x])])^p, x, 21, (3^(-1 - p)*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^6*e^6)) - (2*d*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^5*e^6)) + (5*d^2*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^4*e^6)) - (20*3^(-1 - p)*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^3*e^6)) + (5*d^4*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^2*e^6)) - (2*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c*e^6))} -{x^1*(a + b*Log[c*(d + e*Sqrt[x])])^p, x, 15, (2^(-1 - 2*p)*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^4*e^4)) - (2*d*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^3*e^4)) + (3*d^2*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^2*e^4)) - (2*d^3*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c*e^4))} -{x^0*(a + b*Log[c*(d + e*Sqrt[x])])^p, x, 9, (Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c^2*e^2)) - (2*d*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])])/b)]*(a + b*Log[c*(d + e*Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])])/b))^p*(c*e^2))} -{(a + b*Log[c*(d + e*Sqrt[x])])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*Sqrt[x])])^p/x, x]} -{(a + b*Log[c*(d + e*Sqrt[x])])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*Sqrt[x])])^p/x^2, x]} - - -{x^3*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x, 27, (Gamma[1 + p, -((4*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(2^(2*(1 + p))*E^((4*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^4*e^8)) - (2^(1 + p)*d*(d + e*Sqrt[x])^7*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(7^p*E^((7*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^8*(c*(d + e*Sqrt[x])^2)^(7/2))) + (7*d^2*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^3*e^8)) - (7*2^(1 + p)*d^3*(d + e*Sqrt[x])^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^8*(c*(d + e*Sqrt[x])^2)^(5/2))) + (35*2^(-1 - p)*d^4*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^2*e^8)) - (7*2^(1 + p)*d^5*(d + e*Sqrt[x])^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^8*(c*(d + e*Sqrt[x])^2)^(3/2))) + (7*d^6*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c*e^8)) - (2^(1 + p)*d^7*(d + e*Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^8*Sqrt[c*(d + e*Sqrt[x])^2]))} -{x^2*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x, 21, (3^(-1 - p)*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^3*e^6)) - (2^(1 + p)*d*(d + e*Sqrt[x])^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^6*(c*(d + e*Sqrt[x])^2)^(5/2))) + (5*d^2*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^2*e^6)) - (5*2^(2 + p)*3^(-1 - p)*d^3*(d + e*Sqrt[x])^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^6*(c*(d + e*Sqrt[x])^2)^(3/2))) + (5*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c*e^6)) - (2^(1 + p)*d^5*(d + e*Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^6*Sqrt[c*(d + e*Sqrt[x])^2]))} -{x^1*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x, 15, (2^(-1 - p)*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c^2*e^4)) - (2^(1 + p)*d*(d + e*Sqrt[x])^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^4*(c*(d + e*Sqrt[x])^2)^(3/2))) + (3*d^2*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c*e^4)) - (2^(1 + p)*d^3*(d + e*Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^4*Sqrt[c*(d + e*Sqrt[x])^2]))} -{x^0*(a + b*Log[c*(d + e*Sqrt[x])^2])^p, x, 9, (Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(c*e^2)) - (2^(1 + p)*d*(d + e*Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e*Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e*Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*Sqrt[x])^2])/b))^p*(e^2*Sqrt[c*(d + e*Sqrt[x])^2]))} -{(a + b*Log[c*(d + e*Sqrt[x])^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*Sqrt[x])^2])^p/x, x]} -{(a + b*Log[c*(d + e*Sqrt[x])^2])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*Sqrt[x])^2])^p/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*m=-1*) - - -{x^1*(a + b*Log[c*(d + e/Sqrt[x])])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/Sqrt[x])])^p, x]} -{x^0*(a + b*Log[c*(d + e/Sqrt[x])])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/Sqrt[x])])^p, x]} -{(a + b*Log[c*(d + e/Sqrt[x])])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/Sqrt[x])])^p/x, x]} -{(a + b*Log[c*(d + e/Sqrt[x])])^p/x^2, x, 9, -((Gamma[1 + p, (-2*(a + b*Log[c*(d + e/Sqrt[x])]))/b]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*c^2*e^2*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)) + (2*d*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])])/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(c*e^2*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p)} -{(a + b*Log[c*(d + e/Sqrt[x])])^p/x^4, x, 21, -((3^(-1 - p)*Gamma[1 + p, -((6*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^6*e^6))) + (2*d*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^5*e^6)) - (5*d^2*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^4*e^6)) + (20*3^(-1 - p)*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^3*e^6)) - (5*d^4*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^2*e^6)) + (2*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])])/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c*e^6))} -{(a + b*Log[c*(d + e/Sqrt[x])])^p/x^6, x, 33, -((5^(-1 - p)*Gamma[1 + p, -((10*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*E^((10*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^10*e^10))) + (2*d*Gamma[1 + p, -((9*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(9^p*E^((9*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^9*e^10)) - (9*d^2*Gamma[1 + p, -((8*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(8^p*E^((8*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^8*e^10)) + (24*d^3*Gamma[1 + p, -((7*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(7^p*E^((7*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^7*e^10)) - (7*6^(1 - p)*d^4*Gamma[1 + p, -((6*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^((6*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^6*e^10)) + (252*5^(-1 - p)*d^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^((5*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^5*e^10)) - (21*2^(1 - 2*p)*d^6*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^4*e^10)) + (8*3^(1 - p)*d^7*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^3*e^10)) - (9*d^8*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/Sqrt[x])]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c^2*e^10)) + (2*d^9*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])])/b)]*(a + b*Log[c*(d + e/Sqrt[x])])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])])/b))^p*(c*e^10))} - - -{x^1*(a + b*Log[c*(d + e/Sqrt[x])^2])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/Sqrt[x])^2])^p, x]} -{x^0*(a + b*Log[c*(d + e/Sqrt[x])^2])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/Sqrt[x])^2])^p, x]} -{(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x, x]} -{(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^2, x, 9, -((Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(c*e^2*E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p)) + (2^(1 + p)*d*(d + e/Sqrt[x])*Gamma[1 + p, -(a + b*Log[c*(d + e/Sqrt[x])^2])/(2*b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(e^2*E^(a/(2*b))*Sqrt[c*(d + e/Sqrt[x])^2]*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p)} -{(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^4, x, 21, -((3^(-1 - p)*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^3*e^6))) + (2^(1 + p)*d*(d + e/Sqrt[x])^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^6*(c*(d + e/Sqrt[x])^2)^(5/2))) - (5*d^2*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^2*e^6)) + (5*2^(2 + p)*3^(-1 - p)*d^3*(d + e/Sqrt[x])^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^6*(c*(d + e/Sqrt[x])^2)^(3/2))) - (5*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c*e^6)) + (2^(1 + p)*d^5*(d + e/Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^6*Sqrt[c*(d + e/Sqrt[x])^2]))} -{(a + b*Log[c*(d + e/Sqrt[x])^2])^p/x^6, x, 33, -((5^(-1 - p)*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((5*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^5*e^10))) + (2^(1 + p)*d*(d + e/Sqrt[x])^9*Gamma[1 + p, -((9*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(9^p*E^((9*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^10*(c*(d + e/Sqrt[x])^2)^(9/2))) - (9*d^2*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^4*e^10)) + (3*2^(3 + p)*d^3*(d + e/Sqrt[x])^7*Gamma[1 + p, -((7*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(7^p*E^((7*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^10*(c*(d + e/Sqrt[x])^2)^(7/2))) - (14*3^(1 - p)*d^4*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^3*e^10)) + (63*2^(2 + p)*5^(-1 - p)*d^5*(d + e/Sqrt[x])^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^10*(c*(d + e/Sqrt[x])^2)^(5/2))) - (21*2^(1 - p)*d^6*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/Sqrt[x])^2]))/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c^2*e^10)) + (2^(3 + p)*3^(1 - p)*d^7*(d + e/Sqrt[x])^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/Sqrt[x])^2]))/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^10*(c*(d + e/Sqrt[x])^2)^(3/2))) - (9*d^8*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/b)]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(c*e^10)) + (2^(1 + p)*d^9*(d + e/Sqrt[x])*Gamma[1 + p, -((a + b*Log[c*(d + e/Sqrt[x])^2])/(2*b))]*(a + b*Log[c*(d + e/Sqrt[x])^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e/Sqrt[x])^2])/b))^p*(e^10*Sqrt[c*(d + e/Sqrt[x])^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q (a+b Log[c (d+e x^(m/3))^n])^p when p symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*m=1*) - - -{x^3*(a + b*Log[c*(d + e*x^(1/3))])^p, x, 39, (4^(-1 - p)*Gamma[1 + p, -((12*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((12*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^12*e^12)) - (3*d*Gamma[1 + p, -((11*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(11^p*E^((11*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^11*e^12)) + (33*2^(-1 - p)*d^2*Gamma[1 + p, -((10*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*E^((10*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^10*e^12)) - (55*d^3*Gamma[1 + p, -((9*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(9^p*E^((9*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^9*e^12)) + (495*2^(-2 - 3*p)*d^4*Gamma[1 + p, -((8*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((8*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^8*e^12)) - (198*d^5*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(7^p*E^((7*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^7*e^12)) + (77*3^(1 - p)*d^6*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(2^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^6*e^12)) - (198*d^7*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^5*e^12)) + (495*4^(-1 - p)*d^8*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^4*e^12)) - (55*d^9*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^3*e^12)) + (33*2^(-1 - p)*d^10*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^2*e^12)) - (3*d^11*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c*e^12))} -{x^2*(a + b*Log[c*(d + e*x^(1/3))])^p, x, 30, (3^(-1 - 2*p)*Gamma[1 + p, -((9*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((9*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^9*e^9)) - (3*d*Gamma[1 + p, -((8*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(8^p*E^((8*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^8*e^9)) + (12*d^2*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(7^p*E^((7*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^7*e^9)) - (7*2^(2 - p)*d^3*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^6*e^9)) + (42*d^4*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^5*e^9)) - (21*2^(1 - 2*p)*d^5*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^4*e^9)) + (28*d^6*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^3*e^9)) - (3*2^(2 - p)*d^7*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^2*e^9)) + (3*d^8*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c*e^9))} -{x^1*(a + b*Log[c*(d + e*x^(1/3))])^p, x, 21, (2^(-1 - p)*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^6*e^6)) - (3*d*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^5*e^6)) + (15*2^(-1 - 2*p)*d^2*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^4*e^6)) - (10*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^3*e^6)) + (15*2^(-1 - p)*d^4*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^2*e^6)) - (3*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c*e^6))} -{x^0*(a + b*Log[c*(d + e*x^(1/3))])^p, x, 12, (Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^3*e^3)) - (3*d*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))]))/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(2^p*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c^2*e^3)) + (3*d^2*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))])/b)]*(a + b*Log[c*(d + e*x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))])/b))^p*(c*e^3))} -{(a + b*Log[c*(d + e*x^(1/3))])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(1/3))])^p/x, x]} -{(a + b*Log[c*(d + e*x^(1/3))])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(1/3))])^p/x^2, x]} - - -{x^3*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x, 39, (2^(-2 - p)*Gamma[1 + p, -((6*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(3^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^6*e^12)) - (3*(2/11)^p*d*(d + e*x^(1/3))^11*Gamma[1 + p, -((11*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((11*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*(c*(d + e*x^(1/3))^2)^(11/2))) + (33*d^2*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(2*c^5*e^12)) - (55*(2/9)^p*d^3*(d + e*x^(1/3))^9*Gamma[1 + p, -((9*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((9*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*(c*(d + e*x^(1/3))^2)^(9/2))) + (495*d^4*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(2^(2*(1 + p))*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^4*e^12)) - (99*2^(1 + p)*d^5*(d + e*x^(1/3))^7*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(7^p*E^((7*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*(c*(d + e*x^(1/3))^2)^(7/2))) + (77*3^(1 - p)*d^6*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^3*e^12)) - (99*2^(1 + p)*d^7*(d + e*x^(1/3))^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*(c*(d + e*x^(1/3))^2)^(5/2))) + (495*2^(-2 - p)*d^8*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^2*e^12)) - (55*(2/3)^p*d^9*(d + e*x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*(c*(d + e*x^(1/3))^2)^(3/2))) + (33*d^10*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(2*c*e^12)) - (3*2^p*d^11*(d + e*x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^12*Sqrt[c*(d + e*x^(1/3))^2]))} -{x^2*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x, 30, (2^p*3^(-1 - 2*p)*(d + e*x^(1/3))^9*Gamma[1 + p, -((9*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((9*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^9*(c*(d + e*x^(1/3))^2)^(9/2))) - (3*d*Gamma[1 + p, -((4*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^4*e^9)) + (3*2^(2 + p)*d^2*(d + e*x^(1/3))^7*Gamma[1 + p, -((7*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(7^p*E^((7*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^9*(c*(d + e*x^(1/3))^2)^(7/2))) - (28*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^3*e^9)) + (21*2^(1 + p)*d^4*(d + e*x^(1/3))^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^9*(c*(d + e*x^(1/3))^2)^(5/2))) - (21*2^(1 - p)*d^5*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^2*e^9)) + (7*2^(2 + p)*d^6*(d + e*x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^9*(c*(d + e*x^(1/3))^2)^(3/2))) - (12*d^7*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c*e^9)) + (3*2^p*d^8*(d + e*x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^9*Sqrt[c*(d + e*x^(1/3))^2]))} -{x^1*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x, 21, (Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(2*c^3*e^6)) - (3*(2/5)^p*d*(d + e*x^(1/3))^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^6*(c*(d + e*x^(1/3))^2)^(5/2))) + (15*2^(-1 - p)*d^2*Gamma[1 + p, -((2*(a + b*Log[c*(d + e*x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c^2*e^6)) - (5*2^(1 + p)*d^3*(d + e*x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^6*(c*(d + e*x^(1/3))^2)^(3/2))) + (15*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(2*c*e^6)) - (3*2^p*d^5*(d + e*x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^6*Sqrt[c*(d + e*x^(1/3))^2]))} -{x^0*(a + b*Log[c*(d + e*x^(1/3))^2])^p, x, 12, ((2/3)^p*(d + e*x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e*x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^3*(c*(d + e*x^(1/3))^2)^(3/2))) - (3*d*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/b)]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(c*e^3)) + (3*2^p*d^2*(d + e*x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e*x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e*x^(1/3))^2])/b))^p*(e^3*Sqrt[c*(d + e*x^(1/3))^2]))} -{(a + b*Log[c*(d + e*x^(1/3))^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(1/3))^2])^p/x, x]} -{(a + b*Log[c*(d + e*x^(1/3))^2])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(1/3))^2])^p/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*m=2*) - - -{x^3*(a + b*Log[c*(d + e*x^(2/3))])^p, x, 21, (2^(-2 - p)*Gamma[1 + p, (-6*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(3^p*c^6*e^6*E^((6*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (3*d*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*5^p*c^5*e^6*E^((5*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) + (15*d^2*Gamma[1 + p, (-4*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2^(2*(1 + p))*c^4*e^6*E^((4*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (5*d^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(3^p*c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) + (15*2^(-2 - p)*d^4*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (3*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3))])/b)]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p)} -{x^1*(a + b*Log[c*(d + e*x^(2/3))])^p, x, 12, (Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*3^p*c^3*e^3*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) - (3*2^(-1 - p)*d*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))]))/b]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(c^2*e^3*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p) + (3*d^2*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3))])/b)]*(a + b*Log[c*(d + e*x^(2/3))])^p)/(2*c*e^3*E^(a/b)*(-((a + b*Log[c*(d + e*x^(2/3))])/b))^p)} -{(a + b*Log[c*(d + e*x^(2/3))])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))])^p/x, x]} -{(a + b*Log[c*(d + e*x^(2/3))])^p/x^3, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))])^p/x^3, x]} - -{x^2*(a + b*Log[c*(d + e*x^(2/3))])^p, x, 1, Unintegrable[x^2*(a + b*Log[c*(d + e*x^(2/3))])^p, x]} -{x^0*(a + b*Log[c*(d + e*x^(2/3))])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))])^p, x]} -{(a + b*Log[c*(d + e*x^(2/3))])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))])^p/x^2, x]} - - -{x^3*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x, 21, (Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))^2]))/b]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(4*3^p*c^3*e^6*E^((3*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (3*2^(-1 + p)*d*(d + e*x^(2/3))^5*Gamma[1 + p, (-5*(a + b*Log[c*(d + e*x^(2/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(5^p*e^6*E^((5*a)/(2*b))*(c*(d + e*x^(2/3))^2)^(5/2)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) + (15*2^(-2 - p)*d^2*Gamma[1 + p, (-2*(a + b*Log[c*(d + e*x^(2/3))^2]))/b]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(c^2*e^6*E^((2*a)/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (5*(2/3)^p*d^3*(d + e*x^(2/3))^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(e^6*E^((3*a)/(2*b))*(c*(d + e*x^(2/3))^2)^(3/2)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) + (15*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3))^2])/b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(4*c*e^6*E^(a/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (3*2^(-1 + p)*d^5*(d + e*x^(2/3))*Gamma[1 + p, -(a + b*Log[c*(d + e*x^(2/3))^2])/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(e^6*E^(a/(2*b))*Sqrt[c*(d + e*x^(2/3))^2]*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p)} -{x^1*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x, 12, (2^(-1 + p)*(d + e*x^(2/3))^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e*x^(2/3))^2]))/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(3^p*e^3*E^((3*a)/(2*b))*(c*(d + e*x^(2/3))^2)^(3/2)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) - (3*d*Gamma[1 + p, -((a + b*Log[c*(d + e*x^(2/3))^2])/b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(2*c*e^3*E^(a/b)*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p) + (3*2^(-1 + p)*d^2*(d + e*x^(2/3))*Gamma[1 + p, -(a + b*Log[c*(d + e*x^(2/3))^2])/(2*b)]*(a + b*Log[c*(d + e*x^(2/3))^2])^p)/(e^3*E^(a/(2*b))*Sqrt[c*(d + e*x^(2/3))^2]*(-((a + b*Log[c*(d + e*x^(2/3))^2])/b))^p)} -{(a + b*Log[c*(d + e*x^(2/3))^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^2])^p/x, x]} -{(a + b*Log[c*(d + e*x^(2/3))^2])^p/x^3, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^2])^p/x^3, x]} - -{x^2*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x, 1, Unintegrable[x^2*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x]} -{x^0*(a + b*Log[c*(d + e*x^(2/3))^2])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^2])^p, x]} -{(a + b*Log[c*(d + e*x^(2/3))^2])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e*x^(2/3))^2])^p/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*m=-1*) - - -{x^1*(a + b*Log[c*(d + e/x^(1/3))])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/x^(1/3))])^p, x]} -{x^0*(a + b*Log[c*(d + e/x^(1/3))])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(1/3))])^p, x]} -{(a + b*Log[c*(d + e/x^(1/3))])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(1/3))])^p/x, x]} -{(a + b*Log[c*(d + e/x^(1/3))])^p/x^2, x, 12, -((Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))]))/b]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(3^p*c^3*e^3*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p)) + (3*d*Gamma[1 + p, (-2*(a + b*Log[c*(d + e/x^(1/3))]))/b]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(2^p*c^2*e^3*E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p) - (3*d^2*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(c*e^3*E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p)} -{(a + b*Log[c*(d + e/x^(1/3))])^p/x^3, x, 21, -((2^(-1 - p)*Gamma[1 + p, -((6*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(3^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^6*e^6))) + (3*d*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^5*e^6)) - (15*2^(-1 - 2*p)*d^2*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^4*e^6)) + (10*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^3*e^6)) - (15*2^(-1 - p)*d^4*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^2*e^6)) + (3*d^5*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c*e^6))} -{(a + b*Log[c*(d + e/x^(1/3))])^p/x^4, x, 30, -((3^(-1 - 2*p)*Gamma[1 + p, -((9*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^((9*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^9*e^9))) + (3*d*Gamma[1 + p, -((8*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(8^p*E^((8*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^8*e^9)) - (12*d^2*Gamma[1 + p, -((7*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(7^p*E^((7*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^7*e^9)) + (7*2^(2 - p)*d^3*Gamma[1 + p, -((6*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(3^p*E^((6*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^6*e^9)) - (42*d^4*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(5^p*E^((5*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^5*e^9)) + (21*2^(1 - 2*p)*d^5*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^4*e^9)) - (28*d^6*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^3*e^9)) + (3*2^(2 - p)*d^7*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/x^(1/3))]))/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c^2*e^9)) - (3*d^8*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))])/b)]*(a + b*Log[c*(d + e/x^(1/3))])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))])/b))^p*(c*e^9))} - - -{x^1*(a + b*Log[c*(d + e/x^(1/3))^2])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/x^(1/3))^2])^p, x]} -{x^0*(a + b*Log[c*(d + e/x^(1/3))^2])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(1/3))^2])^p, x]} -{(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(1/3))^2])^p/x, x]} -{(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^2, x, 12, -(((2/3)^p*(d + e/x^(1/3))^3*Gamma[1 + p, (-3*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^3*E^((3*a)/(2*b))*(c*(d + e/x^(1/3))^2)^(3/2)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p)) + (3*d*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(c*e^3*E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p) - (3*2^p*d^2*(d + e/x^(1/3))*Gamma[1 + p, -(a + b*Log[c*(d + e/x^(1/3))^2])/(2*b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(e^3*E^(a/(2*b))*Sqrt[c*(d + e/x^(1/3))^2]*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p)} -{(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^3, x, 21, -((Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(2*c^3*e^6))) + (3*(2/5)^p*d*(d + e/x^(1/3))^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^6*(c*(d + e/x^(1/3))^2)^(5/2))) - (15*2^(-1 - p)*d^2*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(c^2*e^6)) + (5*2^(1 + p)*d^3*(d + e/x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^6*(c*(d + e/x^(1/3))^2)^(3/2))) - (15*d^4*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(2*c*e^6)) + (3*2^p*d^5*(d + e/x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^6*Sqrt[c*(d + e/x^(1/3))^2]))} -{(a + b*Log[c*(d + e/x^(1/3))^2])^p/x^4, x, 30, -((2^p*3^(-1 - 2*p)*(d + e/x^(1/3))^9*Gamma[1 + p, -((9*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^((9*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^9*(c*(d + e/x^(1/3))^2)^(9/2)))) + (3*d*Gamma[1 + p, -((4*(a + b*Log[c*(d + e/x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(4^p*E^((4*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(c^4*e^9)) - (3*2^(2 + p)*d^2*(d + e/x^(1/3))^7*Gamma[1 + p, -((7*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(7^p*E^((7*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^9*(c*(d + e/x^(1/3))^2)^(7/2))) + (28*d^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*E^((3*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(c^3*e^9)) - (21*2^(1 + p)*d^4*(d + e/x^(1/3))^5*Gamma[1 + p, -((5*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(5^p*E^((5*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^9*(c*(d + e/x^(1/3))^2)^(5/2))) + (21*2^(1 - p)*d^5*Gamma[1 + p, -((2*(a + b*Log[c*(d + e/x^(1/3))^2]))/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^((2*a)/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(c^2*e^9)) - (7*2^(2 + p)*d^6*(d + e/x^(1/3))^3*Gamma[1 + p, -((3*(a + b*Log[c*(d + e/x^(1/3))^2]))/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(3^p*E^((3*a)/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^9*(c*(d + e/x^(1/3))^2)^(3/2))) + (12*d^7*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/b)]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^(a/b)*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(c*e^9)) - (3*2^p*d^8*(d + e/x^(1/3))*Gamma[1 + p, -((a + b*Log[c*(d + e/x^(1/3))^2])/(2*b))]*(a + b*Log[c*(d + e/x^(1/3))^2])^p)/(E^(a/(2*b))*(-((a + b*Log[c*(d + e/x^(1/3))^2])/b))^p*(e^9*Sqrt[c*(d + e/x^(1/3))^2]))} - - -(* ::Subsubsection::Closed:: *) -(*m=-2*) - - -{x^3*(a + b*Log[c*(d + e/x^(2/3))])^p, x, 1, Unintegrable[x^3*(a + b*Log[c*(d + e/x^(2/3))])^p, x]} -{x^2*(a + b*Log[c*(d + e/x^(2/3))])^p, x, 1, Unintegrable[x^2*(a + b*Log[c*(d + e/x^(2/3))])^p, x]} -{x^1*(a + b*Log[c*(d + e/x^(2/3))])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/x^(2/3))])^p, x]} -{x^0*(a + b*Log[c*(d + e/x^(2/3))])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))])^p, x]} -{(a + b*Log[c*(d + e/x^(2/3))])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))])^p/x, x]} -{(a + b*Log[c*(d + e/x^(2/3))])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))])^p/x^2, x]} - - -{x^3*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x, 1, Unintegrable[x^3*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x]} -{x^2*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x, 1, Unintegrable[x^2*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x]} -{x^1*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x, 1, Unintegrable[x*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x]} -{x^0*(a + b*Log[c*(d + e/x^(2/3))^2])^p, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^2])^p, x]} -{(a + b*Log[c*(d + e/x^(2/3))^2])^p/x^1, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^2])^p/x, x]} -{(a + b*Log[c*(d + e/x^(2/3))^2])^p/x^2, x, 1, Unintegrable[(a + b*Log[c*(d + e/x^(2/3))^2])^p/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^r (f+g x)^q (a+b Log[c (d+e x^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (h x)^(r/2) (f+g x)^q (a+b Log[c (d+e x^m)^n])*) - - -(* ::Subsubsection::Closed:: *) -(*r>0*) - - -{(f + g*x)*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(1/2), x, 26, (2*a*f*Sqrt[h*x])/h - (8*b*f*p*Sqrt[h*x])/h - (8*b*g*p*(h*x)^(3/2))/(9*h^2) - (2*Sqrt[2]*b*d^(1/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(3/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(1/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(3/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) + (2*b*f*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h + (2*g*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^2) - (Sqrt[2]*b*d^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*Sqrt[h]) + (Sqrt[2]*b*d^(3/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) + (Sqrt[2]*b*d^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*Sqrt[h]) - (Sqrt[2]*b*d^(3/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h])} -{(f + g*x)*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(3/2), x, 25, (2*a*g*Sqrt[h*x])/h^2 - (8*b*g*p*Sqrt[h*x])/h^2 - (2*Sqrt[2]*b*e^(1/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) - (2*Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*e^(1/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*d^(1/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) + (2*b*g*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h^2 - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(h*Sqrt[h*x]) + (Sqrt[2]*b*e^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(3/2)) - (Sqrt[2]*b*d^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(3/2)) + (Sqrt[2]*b*d^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2))} -{(f + g*x)*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(5/2), x, 23, -((2*Sqrt[2]*b*e^(3/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(5/2))) - (2*Sqrt[2]*b*e^(1/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(5/2)) + (2*Sqrt[2]*b*e^(3/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(5/2)) + (2*Sqrt[2]*b*e^(1/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(5/2)) - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(3*h*(h*x)^(3/2)) - (2*g*(a + b*Log[c*(d + e*x^2)^p]))/(h^2*Sqrt[h*x]) - (Sqrt[2]*b*e^(3/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(5/2)) + (Sqrt[2]*b*e^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(5/2)) + (Sqrt[2]*b*e^(3/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(5/2)) - (Sqrt[2]*b*e^(1/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(5/2))} -{(f + g*x)*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(7/2), x, 24, -((8*b*e*f*p)/(5*d*h^3*Sqrt[h*x])) + (2*Sqrt[2]*b*e^(5/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(3/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(5/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(7/2)) + (2*Sqrt[2]*b*e^(3/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(7/2)) - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(5*h*(h*x)^(5/2)) - (2*g*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^2*(h*x)^(3/2)) - (Sqrt[2]*b*e^(5/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(7/2)) - (Sqrt[2]*b*e^(3/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2)) + (Sqrt[2]*b*e^(5/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(7/2)) + (Sqrt[2]*b*e^(3/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2))} -{(f + g*x)*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(9/2), x, 25, -((8*b*e*f*p)/(21*d*h^3*(h*x)^(3/2))) - (8*b*e*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*f*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*x)^(7/2)) - (2*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) + (Sqrt[2]*b*e^(7/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (Sqrt[2]*b*e^(5/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (Sqrt[2]*b*e^(5/4)*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2))} - - -{(f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(1/2), x, 38, (2*a*f^2*Sqrt[h*x])/h - (8*b*f^2*p*Sqrt[h*x])/h + (8*b*d*g^2*p*Sqrt[h*x])/(5*e*h) - (16*b*f*g*p*(h*x)^(3/2))/(9*h^2) - (8*b*g^2*p*(h*x)^(5/2))/(25*h^3) - (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) - (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*e^(5/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(1/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*Sqrt[h]) + (4*Sqrt[2]*b*d^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(5/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*e^(5/4)*Sqrt[h]) + (2*b*f^2*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h + (4*f*g*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^2) + (2*g^2*(h*x)^(5/2)*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^3) - (Sqrt[2]*b*d^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*Sqrt[h]) + (2*Sqrt[2]*b*d^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) + (Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(5/4)*Sqrt[h]) + (Sqrt[2]*b*d^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*Sqrt[h]) - (2*Sqrt[2]*b*d^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*Sqrt[h]) - (Sqrt[2]*b*d^(5/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*e^(5/4)*Sqrt[h])} -{(f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(3/2), x, 36, (4*a*f*g*Sqrt[h*x])/h^2 - (16*b*f*g*p*Sqrt[h*x])/h^2 - (8*b*g^2*p*(h*x)^(3/2))/(9*h^3) - (2*Sqrt[2]*b*e^(1/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) - (4*Sqrt[2]*b*d^(1/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) - (2*Sqrt[2]*b*d^(3/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*h^(3/2)) + (2*Sqrt[2]*b*e^(1/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(3/2)) + (4*Sqrt[2]*b*d^(1/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*d^(3/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*e^(3/4)*h^(3/2)) + (4*b*f*g*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h^2 - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*Sqrt[h*x]) + (2*g^2*(h*x)^(3/2)*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^3) + (Sqrt[2]*b*e^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(3/2)) - (2*Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) + (Sqrt[2]*b*d^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(3/2)) + (2*Sqrt[2]*b*d^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(3/2)) - (Sqrt[2]*b*d^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*e^(3/4)*h^(3/2))} -{(f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(5/2), x, 35, (2*a*g^2*Sqrt[h*x])/h^3 - (8*b*g^2*p*Sqrt[h*x])/h^3 - (2*Sqrt[2]*b*e^(3/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(5/2)) - (4*Sqrt[2]*b*e^(1/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(5/2)) - (2*Sqrt[2]*b*d^(1/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(5/2)) + (2*Sqrt[2]*b*e^(3/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(5/2)) + (4*Sqrt[2]*b*e^(1/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(5/2)) + (2*Sqrt[2]*b*d^(1/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*h^(5/2)) + (2*b*g^2*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/h^3 - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(3*h*(h*x)^(3/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^p]))/(h^2*Sqrt[h*x]) - (Sqrt[2]*b*e^(3/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(5/2)) + (2*Sqrt[2]*b*e^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(5/2)) - (Sqrt[2]*b*d^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(5/2)) + (Sqrt[2]*b*e^(3/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(5/2)) - (2*Sqrt[2]*b*e^(1/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(5/2)) + (Sqrt[2]*b*d^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*h^(5/2))} -{(f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(7/2), x, 34, -((8*b*e*f^2*p)/(5*d*h^3*Sqrt[h*x])) + (2*Sqrt[2]*b*e^(5/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(7/2)) - (4*Sqrt[2]*b*e^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(1/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(5/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(7/2)) + (4*Sqrt[2]*b*e^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(7/2)) + (2*Sqrt[2]*b*e^(1/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*h^(7/2)) - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*h*(h*x)^(5/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^2*(h*x)^(3/2)) - (2*g^2*(a + b*Log[c*(d + e*x^2)^p]))/(h^3*Sqrt[h*x]) - (Sqrt[2]*b*e^(5/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2)) + (Sqrt[2]*b*e^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(7/2)) + (Sqrt[2]*b*e^(5/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(7/2)) + (2*Sqrt[2]*b*e^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2)) - (Sqrt[2]*b*e^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(7/2))} -{(f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p])/(h*x)^(9/2), x, 35, -((8*b*e*f^2*p)/(21*d*h^3*(h*x)^(3/2))) - (16*b*e*f*g*p)/(5*d*h^4*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) + (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(7/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(7*d^(7/4)*h^(9/2)) - (4*Sqrt[2]*b*e^(5/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(5*d^(5/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(3/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(3*d^(3/4)*h^(9/2)) - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(7*h*(h*x)^(7/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^p]))/(5*h^2*(h*x)^(5/2)) - (2*g^2*(a + b*Log[c*(d + e*x^2)^p]))/(3*h^3*(h*x)^(3/2)) + (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) - (2*Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) - (Sqrt[2]*b*e^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(9/2)) - (Sqrt[2]*b*e^(7/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(7*d^(7/4)*h^(9/2)) + (2*Sqrt[2]*b*e^(5/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(9/2)) + (Sqrt[2]*b*e^(3/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*r<0*) - - -{(h*x)^(1/2)*(a + b*Log[c*(d + e*x^2)^p])/(f + g*x), x, 39, (2*a*Sqrt[h*x])/g - (8*b*p*Sqrt[h*x])/g - (2*Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*g) + (2*Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*g) + (2*b*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/g - (2*Sqrt[f]*Sqrt[h]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2)^p]))/g^(3/2) - (Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*g) + (Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(e^(1/4)*g) - (8*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/g^(3/2) + (2*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (2*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[-((2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])))])/g^(3/2) + (2*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (2*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (4*I*b*Sqrt[f]*Sqrt[h]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/g^(3/2) - (I*b*Sqrt[f]*Sqrt[h]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) - (I*b*Sqrt[f]*Sqrt[h]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) - (I*b*Sqrt[f]*Sqrt[h]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) - (I*b*Sqrt[f]*Sqrt[h]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2)} -{(a + b*Log[c*(d + e*x^2)^p])/((f + g*x)*(h*x)^(1/2)), x, 25, (2*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2)^p]))/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (8*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[-((2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (2*b*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) - (4*I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h]) + (I*b*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(Sqrt[f]*Sqrt[g]*Sqrt[h])} -{(a + b*Log[c*(d + e*x^2)^p])/((f + g*x)*(h*x)^(3/2)), x, 37, -((2*Sqrt[2]*b*e^(1/4)*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2))) + (2*Sqrt[2]*b*e^(1/4)*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(d^(1/4)*f*h^(3/2)) - (2*(a + b*Log[c*(d + e*x^2)^p]))/(f*h*Sqrt[h*x]) - (2*Sqrt[g]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x^2)^p]))/(f^(3/2)*h^(3/2)) + (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (Sqrt[2]*b*e^(1/4)*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*f*h^(3/2)) - (8*b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[-((2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) + (2*b*Sqrt[g]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) + (4*I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x])])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2)) - (I*b*Sqrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/(f^(3/2)*h^(3/2))} - - -(* ::Subsection:: *) -(*Integrands of the form (h x)^(r/2) (f+g x)^q (a+b Log[c (d+e x^2)^n])^2*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (h x)^s Log[f x^r]^q (a+b Log[c (d+e x^m)^n])*) - - -{Log[f*x^p]*Log[1 + e*x^m]/x, x, 2, -((Log[f*x^p]*PolyLog[2, (-e)*x^m])/m) + (p*PolyLog[3, (-e)*x^m])/m^2} - - -{x^(-1 + m)*Log[f*x^p]^2/(d + e*x^m), x, 3, (Log[f*x^p]^2*Log[1 + (e*x^m)/d])/(e*m) + (2*p*Log[f*x^p]*PolyLog[2, -((e*x^m)/d)])/(e*m^2) - (2*p^2*PolyLog[3, -((e*x^m)/d)])/(e*m^3)} - - -{Log[f*x^p]^3*(a + b*Log[c*(d + e*x^m)^n])/x, x, 6, (Log[f*x^p]^4*(a + b*Log[c*(d + e*x^m)^n]))/(4*p) - (b*n*Log[f*x^p]^4*Log[1 + (e*x^m)/d])/(4*p) - (b*n*Log[f*x^p]^3*PolyLog[2, -((e*x^m)/d)])/m + (3*b*n*p*Log[f*x^p]^2*PolyLog[3, -((e*x^m)/d)])/m^2 - (6*b*n*p^2*Log[f*x^p]*PolyLog[4, -((e*x^m)/d)])/m^3 + (6*b*n*p^3*PolyLog[5, -((e*x^m)/d)])/m^4} -{Log[f*x^p]^2*(a + b*Log[c*(d + e*x^m)^n])/x, x, 5, (Log[f*x^p]^3*(a + b*Log[c*(d + e*x^m)^n]))/(3*p) - (b*n*Log[f*x^p]^3*Log[1 + (e*x^m)/d])/(3*p) - (b*n*Log[f*x^p]^2*PolyLog[2, -((e*x^m)/d)])/m + (2*b*n*p*Log[f*x^p]*PolyLog[3, -((e*x^m)/d)])/m^2 - (2*b*n*p^2*PolyLog[4, -((e*x^m)/d)])/m^3} -{Log[f*x^p]^1*(a + b*Log[c*(d + e*x^m)^n])/x, x, 4, (Log[f*x^p]^2*(a + b*Log[c*(d + e*x^m)^n]))/(2*p) - (b*n*Log[f*x^p]^2*Log[1 + (e*x^m)/d])/(2*p) - (b*n*Log[f*x^p]*PolyLog[2, -((e*x^m)/d)])/m + (b*n*p*PolyLog[3, -((e*x^m)/d)])/m^2} -{Log[f*x^p]^0*(a + b*Log[c*(d + e*x^m)^n])/x, x, 3, (Log[-((e*x^m)/d)]*(a + b*Log[c*(d + e*x^m)^n]))/m + (b*n*PolyLog[2, 1 + (e*x^m)/d])/m} -{(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^1), x, 4, (a*Log[Log[f*x^p]])/p + b*Unintegrable[Log[c*(d + e*x^m)^n]/(x*Log[f*x^p]), x]} -{(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^2), x, 1, -((a + b*Log[c*(d + e*x^m)^n])/(p*Log[f*x^p])) + (b*e*m*n*Unintegrable[x^(-1 + m)/((d + e*x^m)*Log[f*x^p]), x])/p} -{(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3), x, 1, -((a + b*Log[c*(d + e*x^m)^n])/(2*p*Log[f*x^p]^2)) + (b*e*m*n*Unintegrable[x^(-1 + m)/((d + e*x^m)*Log[f*x^p]^2), x])/(2*p)} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b Log[c (d+e (f+g x)^m)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e (f+g x)^m)^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c (d+e (f+g x)^m)^n]*) - - -{Log[c*(d + e*(f + g*x)^p)^q], x, 3, -((e*p*q*(f + g*x)^(1 + p)*Hypergeometric2F1[1, 1 + 1/p, 2 + 1/p, -((e*(f + g*x)^p)/d)])/(d*g*(1 + p))) + ((f + g*x)*Log[c*(d + e*(f + g*x)^p)^q])/g} - - -{Log[c*(d + e*(f + g*x)^3)^q], x, 9, -3*q*x - (Sqrt[3]*d^(1/3)*q*ArcTan[(d^(1/3) - 2*e^(1/3)*(f + g*x))/(Sqrt[3]*d^(1/3))])/(e^(1/3)*g) + (d^(1/3)*q*Log[d^(1/3) + e^(1/3)*(f + g*x)])/(e^(1/3)*g) - (d^(1/3)*q*Log[d^(2/3) - d^(1/3)*e^(1/3)*(f + g*x) + e^(2/3)*(f + g*x)^2])/(2*e^(1/3)*g) + ((f + g*x)*Log[c*(d + e*(f + g*x)^3)^q])/g} -{Log[c*(d + e*(f + g*x)^2)^q], x, 4, -2*q*x + (2*Sqrt[d]*q*ArcTan[(Sqrt[e]*(f + g*x))/Sqrt[d]])/(Sqrt[e]*g) + ((f + g*x)*Log[c*(d + e*(f + g*x)^2)^q])/g} -{Log[c*(d + e*(f + g*x)^1)^q], x, 3, (-q)*x + ((d + e*f + e*g*x)*Log[c*(d + e*(f + g*x))^q])/(e*g)} -{Log[c*(d + e/(f + g*x)^1)^q], x, 4, ((f + g*x)*Log[c*(d + e/(f + g*x))^q])/g + (e*q*Log[e + d*(f + g*x)])/(d*g)} -{Log[c*(d + e/(f + g*x)^2)^q], x, 4, (2*Sqrt[e]*q*ArcTan[(Sqrt[d]*(f + g*x))/Sqrt[e]])/(Sqrt[d]*g) + ((f + g*x)*Log[c*(d + e/(f + g*x)^2)^q])/g} -{Log[c*(d + e/(f + g*x)^3)^q], x, 9, -((Sqrt[3]*e^(1/3)*q*ArcTan[(e^(1/3) - 2*d^(1/3)*(f + g*x))/(Sqrt[3]*e^(1/3))])/(d^(1/3)*g)) + ((f + g*x)*Log[c*(d + e/(f + g*x)^3)^q])/g + (e^(1/3)*q*Log[e^(1/3) + d^(1/3)*(f + g*x)])/(d^(1/3)*g) - (e^(1/3)*q*Log[e^(2/3) - d^(1/3)*e^(1/3)*(f + g*x) + d^(2/3)*(f + g*x)^2])/(2*d^(1/3)*g)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Log[c (d+e/(f+g x))^n])^p*) - - -{(a + b*Log[c*(d + e/(f + g*x))^p])^n, x, 0, Unintegrable[(a + b*Log[c*(d + e/(f + g*x))^p])^n, x]} - - -{(a + b*Log[c*(d + e/(f + g*x))^p])^4, x, 8, -((4*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^3)/(d*g)) + ((e + d*(f + g*x))*(a + b*Log[c*(d + e/(f + g*x))^p])^4)/(d*g) - (12*b^2*e*p^2*(a + b*Log[c*(d + e/(f + g*x))^p])^2*PolyLog[2, 1 + e/(d*(f + g*x))])/(d*g) + (24*b^3*e*p^3*(a + b*Log[c*(d + e/(f + g*x))^p])*PolyLog[3, 1 + e/(d*(f + g*x))])/(d*g) - (24*b^4*e*p^4*PolyLog[4, 1 + e/(d*(f + g*x))])/(d*g)} -{(a + b*Log[c*(d + e/(f + g*x))^p])^3, x, 7, -((3*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p])^2)/(d*g)) + ((e + d*(f + g*x))*(a + b*Log[c*(d + e/(f + g*x))^p])^3)/(d*g) - (6*b^2*e*p^2*(a + b*Log[c*(d + e/(f + g*x))^p])*PolyLog[2, 1 + e/(d*(f + g*x))])/(d*g) + (6*b^3*e*p^3*PolyLog[3, 1 + e/(d*(f + g*x))])/(d*g)} -{(a + b*Log[c*(d + e/(f + g*x))^p])^2, x, 5, -((2*b*e*p*Log[-(e/(d*(f + g*x)))]*(a + b*Log[c*(d + e/(f + g*x))^p]))/(d*g)) + ((e + d*(f + g*x))*(a + b*Log[c*(d + e/(f + g*x))^p])^2)/(d*g) - (2*b^2*e*p^2*PolyLog[2, 1 + e/(d*(f + g*x))])/(d*g)} -{(a + b*Log[c*(d + e/(f + g*x))^p])^1, x, 5, a*x + (b*(f + g*x)*Log[c*(d + e/(f + g*x))^p])/g + (b*e*p*Log[e + d*(f + g*x)])/(d*g)} -{1/(a + b*Log[c*(d + e/(f + g*x))^p])^1, x, 0, Unintegrable[1/(a + b*Log[c*(d + e/(f + g*x))^p]), x]} -{1/(a + b*Log[c*(d + e/(f + g*x))^p])^2, x, 0, Unintegrable[1/(a + b*Log[c*(d + e/(f + g*x))^p])^2, x]} diff --git a/test/methods/rule_based/test_files/3 Logarithms/3.5 Logarithm functions.m b/test/methods/rule_based/test_files/3 Logarithms/3.5 Logarithm functions.m deleted file mode 100644 index be3075b..0000000 --- a/test/methods/rule_based/test_files/3 Logarithms/3.5 Logarithm functions.m +++ /dev/null @@ -1,609 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Logarithms*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x^m+e Log[c x^n]^(q-1)) (a x^m+b Log[c x^n]^q)^p / x*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[c x^n]^(q-1) (a x^m+b Log[c x^n]^q)^p / x*) - - -{Log[c*x^n]^(q - 1)*(a*x^m + b*Log[c*x^n]^q)^p/x, x, 1, -((a*m*CannotIntegrate[x^(-1 + m)*(a*x^m + b*Log[c*x^n]^q)^p, x])/(b*n*q)) + (a*x^m + b*Log[c*x^n]^q)^(1 + p)/(b*n*(1 + p)*q)} - - -{Log[c*x^n]^(q - 1)*(a*x^m + b*Log[c*x^n]^q)^3/x, x, 10, (b^3*Log[c*x^n]^(4*q))/(4*n*q) - (3*a*b^2*x^m*Gamma[3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q))/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)*n) - (3*a^2*b*x^(2*m)*Gamma[2*q, -((2*m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q))/(4^q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q)*n) - (a^3*x^(3*m)*Gamma[q, -((3*m*Log[c*x^n])/n)]*Log[c*x^n]^q)/(3^q*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^q*n)} -{Log[c*x^n]^(q - 1)*(a*x^m + b*Log[c*x^n]^q)^2/x, x, 8, (b^2*Log[c*x^n]^(3*q))/(3*n*q) - (2*a*b*x^m*Gamma[2*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q))/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(2*q)*n) - (a^2*x^(2*m)*Gamma[q, -((2*m*Log[c*x^n])/n)]*Log[c*x^n]^q)/(2^q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^q*n)} -{Log[c*x^n]^(q - 1)*(a*x^m + b*Log[c*x^n]^q)^1/x, x, 6, (b*Log[c*x^n]^(2*q))/(2*n*q) - (a*x^m*Gamma[q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^q)/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^q*n)} -{Log[c*x^n]^(q - 1)*(a*x^m + b*Log[c*x^n]^q)^0/x, x, 2, Log[c*x^n]^q/(n*q)} -{Log[c*x^n]^(q - 1)/(x*(a*x^m + b*Log[c*x^n]^q)^1), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q), x])/(b*n*q)) + Log[a*x^m + b*Log[c*x^n]^q]/(b*n*q)} -{Log[c*x^n]^(q - 1)/(x*(a*x^m + b*Log[c*x^n]^q)^2), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^2, x])/(b*n*q)) - 1/(b*n*q*(a*x^m + b*Log[c*x^n]^q))} -{Log[c*x^n]^(q - 1)/(x*(a*x^m + b*Log[c*x^n]^q)^3), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^3, x])/(b*n*q)) - 1/(2*b*n*q*(a*x^m + b*Log[c*x^n]^q)^2)} - - -{Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^3/x, x, 13, -((360*a*b^2*n^5*x^m)/m^6) - (9*a^2*b*n^3*x^(2*m))/(8*m^4) - (a^3*n*x^(3*m))/(9*m^2) + (360*a*b^2*n^4*x^m*Log[c*x^n])/m^5 + (9*a^2*b*n^2*x^(2*m)*Log[c*x^n])/(4*m^3) + (a^3*x^(3*m)*Log[c*x^n])/(3*m) - (180*a*b^2*n^3*x^m*Log[c*x^n]^2)/m^4 - (9*a^2*b*n*x^(2*m)*Log[c*x^n]^2)/(4*m^2) + (60*a*b^2*n^2*x^m*Log[c*x^n]^3)/m^3 + (3*a^2*b*x^(2*m)*Log[c*x^n]^3)/(2*m) - (15*a*b^2*n*x^m*Log[c*x^n]^4)/m^2 + (3*a*b^2*x^m*Log[c*x^n]^5)/m + (b^3*Log[c*x^n]^8)/(8*n)} -{Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^2/x, x, 8, -((12*a*b*n^3*x^m)/m^4) - (a^2*n*x^(2*m))/(4*m^2) + (12*a*b*n^2*x^m*Log[c*x^n])/m^3 + (a^2*x^(2*m)*Log[c*x^n])/(2*m) - (6*a*b*n*x^m*Log[c*x^n]^2)/m^2 + (2*a*b*x^m*Log[c*x^n]^3)/m + (b^2*Log[c*x^n]^6)/(6*n)} -{Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^1/x, x, 5, -((a*n*x^m)/m^2) + (a*x^m*Log[c*x^n])/m + (b*Log[c*x^n]^4)/(4*n)} -{Log[c*x^n]*(a*x^m + b*Log[c*x^n]^2)^0/x, x, 1, Log[c*x^n]^2/(2*n)} -{Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^1), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2), x])/(2*b*n)) + Log[a*x^m + b*Log[c*x^n]^2]/(2*b*n)} -{Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^2), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2)^2, x])/(2*b*n)) - 1/(2*b*n*(a*x^m + b*Log[c*x^n]^2))} -{Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^3), x, 1, -((a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2)^3, x])/(2*b*n)) - 1/(4*b*n*(a*x^m + b*Log[c*x^n]^2)^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x^m+e Log[c x^n]^(q-1)) (a x^m+b Log[c x^n]^q)^p / x when a e m-b d n q=0*) - - -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^p/x, x, 1, (a*x^m + b*Log[c*x^n]^q)^(1 + p)/(1 + p)} - - -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^2/x, x, 1, (1/3)*(a*x^m + b*Log[c*x^n]^q)^3} -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^1/x, x, 1, (1/2)*(a*x^m + b*Log[c*x^n]^q)^2} -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^0/x, x, 4, a*x^m + b*Log[c*x^n]^q} -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^1), x, 1, Log[a*x^m + b*Log[c*x^n]^q]} -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^2), x, 1, -(1/(a*x^m + b*Log[c*x^n]^q))} -{(a*m*x^m + b*n*q*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^3), x, 1, -(1/(2*(a*x^m + b*Log[c*x^n]^q)^2))} - - -{(a*x^2 + b*x*Log[c*x^n]^2)^2*(a/x^2 + 2*b*n/x^3*Log[c*x^n]), x, 3, (1/3)*(a*x + b*Log[c*x^n]^2)^3} -{(a*x^2 + b*x*Log[c*x^n]^2)^1*(a/x^1 + 2*b*n/x^2*Log[c*x^n]), x, 3, (1/2)*(a*x + b*Log[c*x^n]^2)^2} -{(a*x^2 + b*x*Log[c*x^n]^2)^0*(a*x^0 + 2*b*n/x^1*Log[c*x^n]), x, 2, a*x + b*Log[c*x^n]^2} -{(a*x^1 + 2*b*n*x^0*Log[c*x^n])/(a*x^2 + b*x*Log[c*x^n]^2)^1, x, 2, Log[a*x + b*Log[c*x^n]^2]} -{(a*x^2 + 2*b*n*x^1*Log[c*x^n])/(a*x^2 + b*x*Log[c*x^n]^2)^2, x, 3, -(1/(a*x + b*Log[c*x^n]^2))} -{(a*x^3 + 2*b*n*x^2*Log[c*x^n])/(a*x^2 + b*x*Log[c*x^n]^2)^3, x, 3, -(1/(2*(a*x + b*Log[c*x^n]^2)^2))} - - -{(a*(m - 1)*x^(m - 1) + b*n*q*Log[c*x^n]^(q - 1))/(a*x^m + b*x*Log[c*x^n]^q), x, 2, Log[a*x^(m - 1) + b*Log[c*x^n]^q]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x^m+e Log[c x^n]^(q-1)) (a x^m+b Log[c x^n]^q)^p / x*) - - -{(d*x^m + e*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^p/x, x, 1, (d - (a*e*m)/(b*n*q))*CannotIntegrate[x^(-1 + m)*(a*x^m + b*Log[c*x^n]^q)^p, x] + (e*(a*x^m + b*Log[c*x^n]^q)^(1 + p))/(b*n*(1 + p)*q)} - - -{(d*x^m + e*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^3/x, x, 9, -((a^3*(a*e*m - b*d*n*q)*x^(4*m))/(4*b*m*n*q)) - (b^2*(a*e*m - b*d*n*q)*x^m*Gamma[1 + 3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q))/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)*(m*n*q)) - (3*2^(-1 - 2*q)*a*b*(a*e*m - b*d*n*q)*x^(2*m)*Gamma[1 + 2*q, -((2*m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q))/((c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*q)*(m*n*q)) - (a^2*(a*e*m - b*d*n*q)*x^(3*m)*Gamma[1 + q, -((3*m*Log[c*x^n])/n)]*Log[c*x^n]^q)/(3^q*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^q*(m*n*q)) + (e*(a*x^m + b*Log[c*x^n]^q)^4)/(4*b*n*q)} -{(d*x^m + e*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^2/x, x, 7, -((a^2*(a*e*m - b*d*n*q)*x^(3*m))/(3*b*m*n*q)) - (b*(a*e*m - b*d*n*q)*x^m*Gamma[1 + 2*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(2*q))/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(2*q)*(m*n*q)) - (a*(a*e*m - b*d*n*q)*x^(2*m)*Gamma[1 + q, -((2*m*Log[c*x^n])/n)]*Log[c*x^n]^q)/(2^q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^q*(m*n*q)) + (e*(a*x^m + b*Log[c*x^n]^q)^3)/(3*b*n*q)} -{(d*x^m + e*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^1/x, x, 5, -((a*(a*e*m - b*d*n*q)*x^(2*m))/(2*b*m*n*q)) + (((b*d)/m - (a*e)/(n*q))*x^m*Gamma[1 + q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^q)/((c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^q) + (e*(a*x^m + b*Log[c*x^n]^q)^2)/(2*b*n*q)} -{(d*x^m + e*Log[c*x^n]^(q - 1))*(a*x^m + b*Log[c*x^n]^q)^0/x, x, 4, (d*x^m)/m + (e*Log[c*x^n]^q)/(n*q)} -{(d*x^m + e*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^1), x, 1, (d - (a*e*m)/(b*n*q))*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q), x] + (e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q)} -{(d*x^m + e*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^2), x, 1, (d - (a*e*m)/(b*n*q))*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^2, x] - e/(b*n*q*(a*x^m + b*Log[c*x^n]^q))} -{(d*x^m + e*Log[c*x^n]^(q - 1))/(x*(a*x^m + b*Log[c*x^n]^q)^3), x, 1, (d - (a*e*m)/(b*n*q))*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^q)^3, x] - e/(2*b*n*q*(a*x^m + b*Log[c*x^n]^q)^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x^m + e x^m Log[c x^n] + f Log[c x^n]^q) / (x (a x^m + b Log[c x^n]^q)^2)*) - - -{(a*d*n*x^m - a*d*m*x^m*Log[c*x^n] - b*d*n*(q - 1)*Log[c*x^n]^q)/(x*(a*x^m + b*Log[c*x^n]^q)^2), x, 1, (d*Log[c*x^n])/(a*x^m + b*Log[c*x^n]^q)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d + e Log[c x^n])/(a x + b Log[c x^n]^q)^2*) - - -{(n*q - Log[c*x^n])/(a*x + b*Log[c*x^n]^q)^2, x, 1, -((n*(1 - q)*CannotIntegrate[1/(x*(a*x + b*Log[c*x^n]^q)), x])/a) + Log[c*x^n]/(a*(a*x + b*Log[c*x^n]^q))} - - -(* ::Section::Closed:: *) -(*Integrands of the form G[x] Log[F[x]] when C=G[x] (1-F[x]) / D[F[x],x]*) - - -{Log[2*x*(d*Sqrt[-e/d] + e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, -((Sqrt[-(e/d)]*PolyLog[2, 1 - (2*x*(d*Sqrt[-(e/d)] + e*x))/(d + e*x^2)])/(2*e))} -{Log[-2*x*(d*Sqrt[-e/d] - e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, (Sqrt[-(e/d)]*PolyLog[2, 1 + (2*x*(d*Sqrt[-(e/d)] - e*x))/(d + e*x^2)])/(2*e)} - -{Log[2*x*(d*Sqrt[e]/Sqrt[-d] + e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, -(PolyLog[2, 1 + (2*Sqrt[e]*x*(Sqrt[-d] - Sqrt[e]*x))/(d + e*x^2)]/(2*Sqrt[-d]*Sqrt[e]))} -{Log[-2*x*(d*Sqrt[e]/Sqrt[-d] - e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, PolyLog[2, 1 - (2*Sqrt[e]*x*(Sqrt[-d] + Sqrt[e]*x))/(d + e*x^2)]/(2*Sqrt[-d]*Sqrt[e])} - -{Log[2*x*(d*Sqrt[-e]/Sqrt[d] + e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, PolyLog[2, 1 - (2*x*(Sqrt[d]*Sqrt[-e] + e*x))/(d + e*x^2)]/(2*Sqrt[d]*Sqrt[-e])} -{Log[-2*x*(d*Sqrt[-e]/Sqrt[d] - e*x)/(d + e*x^2)]/(d + e*x^2), x, 1, -(PolyLog[2, 1 + (2*x*(Sqrt[d]*Sqrt[-e] - e*x))/(d + e*x^2)]/(2*Sqrt[d]*Sqrt[-e]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Log[c Log[d x^n]^p])*) - - -{(e*x)^m*(a + b*Log[c*Log[d*x]^p]), x, 3, -((b*p*(d*x)^(-1 - m)*(e*x)^(1 + m)*ExpIntegralEi[(1 + m)*Log[d*x]])/(e*(1 + m))) + ((e*x)^(1 + m)*(a + b*Log[c*Log[d*x]^p]))/(e*(1 + m))} -{(e*x)^m*(a + b*Log[c*Log[d*x^n]^p]), x, 3, -((b*p*(e*x)^(1 + m)*ExpIntegralEi[((1 + m)*Log[d*x^n])/n])/((d*x^n)^((1 + m)/n)*(e*(1 + m)))) + ((e*x)^(1 + m)*(a + b*Log[c*Log[d*x^n]^p]))/(e*(1 + m))} - - -{x^2*(a + b*Log[c*Log[d*x^n]^p]), x, 3, ((-(1/3))*b*p*x^3*ExpIntegralEi[(3*Log[d*x^n])/n])/(d*x^n)^(3/n) + (1/3)*x^3*(a + b*Log[c*Log[d*x^n]^p])} -{x^1*(a + b*Log[c*Log[d*x^n]^p]), x, 3, ((-(1/2))*b*p*x^2*ExpIntegralEi[(2*Log[d*x^n])/n])/(d*x^n)^(2/n) + (1/2)*x^2*(a + b*Log[c*Log[d*x^n]^p])} -{x^0*(a + b*Log[c*Log[d*x^n]^p]), x, 4, a*x - (b*p*x*ExpIntegralEi[Log[d*x^n]/n])/(d*x^n)^n^(-1) + b*x*Log[c*Log[d*x^n]^p]} -{(a + b*Log[c*Log[d*x^n]^p])/x^1, x, 1, (-b)*p*Log[x] + (Log[d*x^n]*(a + b*Log[c*Log[d*x^n]^p]))/n} -{(a + b*Log[c*Log[d*x^n]^p])/x^2, x, 3, (b*p*(d*x^n)^(1/n)*ExpIntegralEi[-(Log[d*x^n]/n)])/x - (a + b*Log[c*Log[d*x^n]^p])/x} -{(a + b*Log[c*Log[d*x^n]^p])/x^3, x, 3, (b*p*(d*x^n)^(2/n)*ExpIntegralEi[-((2*Log[d*x^n])/n)])/(2*x^2) - (a + b*Log[c*Log[d*x^n]^p])/(2*x^2)} -{(a + b*Log[c*Log[d*x^n]^p])/x^4, x, 3, (b*p*(d*x^n)^(3/n)*ExpIntegralEi[-((3*Log[d*x^n])/n)])/(3*x^3) - (a + b*Log[c*Log[d*x^n]^p])/(3*x^3)} - - -{Log[c*Log[d*x]^p], x, 2, x*Log[c*Log[d*x]^p] - (p*LogIntegral[d*x])/d} -{Log[c*Log[d*x]^p]/x, x, 1, (-p)*Log[x] + Log[d*x]*Log[c*Log[d*x]^p]} - -{Log[c*Log[d*x^n]^p], x, 3, ((-p)*x*ExpIntegralEi[Log[d*x^n]/n])/(d*x^n)^n^(-1) + x*Log[c*Log[d*x^n]^p]} -{Log[c*Log[d*x^n]^p]/x, x, 1, (-p)*Log[x] + (Log[d*x^n]*Log[c*Log[d*x^n]^p])/n} - - -(* ::Section::Closed:: *) -(*Integrands of the form u Log[d (a+b x+c x^2)^p]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Log[d (a+b x+c x^2)^p]*) - - -{x^m*Log[d*(b*x + c*x^2)^n], x, 3, -((2*n*x^(1 + m))/(1 + m)^2) + (n*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*x)/b)])/(1 + m)^2 + (x^(1 + m)*Log[d*(b*x + c*x^2)^n])/(1 + m)} - -{x^4*Log[d*(b*x + c*x^2)^n], x, 3, -((b^4*n*x)/(5*c^4)) + (b^3*n*x^2)/(10*c^3) - (b^2*n*x^3)/(15*c^2) + (b*n*x^4)/(20*c) - (2*n*x^5)/25 + (b^5*n*Log[b + c*x])/(5*c^5) + (1/5)*x^5*Log[d*(b*x + c*x^2)^n]} -{x^3*Log[d*(b*x + c*x^2)^n], x, 3, (b^3*n*x)/(4*c^3) - (b^2*n*x^2)/(8*c^2) + (b*n*x^3)/(12*c) - (n*x^4)/8 - (b^4*n*Log[b + c*x])/(4*c^4) + (1/4)*x^4*Log[d*(b*x + c*x^2)^n]} -{x^2*Log[d*(b*x + c*x^2)^n], x, 3, -((b^2*n*x)/(3*c^2)) + (b*n*x^2)/(6*c) - (2*n*x^3)/9 + (b^3*n*Log[b + c*x])/(3*c^3) + (1/3)*x^3*Log[d*(b*x + c*x^2)^n]} -{x^1*Log[d*(b*x + c*x^2)^n], x, 3, (b*n*x)/(2*c) - (n*x^2)/2 - (b^2*n*Log[b + c*x])/(2*c^2) + (1/2)*x^2*Log[d*(b*x + c*x^2)^n]} -{x^0*Log[d*(b*x + c*x^2)^n], x, 3, -2*n*x + (b*n*Log[b + c*x])/c + x*Log[d*(b*x + c*x^2)^n]} -{Log[d*(b*x + c*x^2)^n]/x^1, x, 7, (-(1/2))*n*Log[x]^2 - n*Log[x]*Log[1 + (c*x)/b] + Log[x]*Log[d*(b*x + c*x^2)^n] - n*PolyLog[2, -((c*x)/b)]} -{Log[d*(b*x + c*x^2)^n]/x^2, x, 3, -(n/x) + (c*n*Log[x])/b - (c*n*Log[b + c*x])/b - Log[d*(b*x + c*x^2)^n]/x} -{Log[d*(b*x + c*x^2)^n]/x^3, x, 3, -(n/(4*x^2)) - (c*n)/(2*b*x) - (c^2*n*Log[x])/(2*b^2) + (c^2*n*Log[b + c*x])/(2*b^2) - Log[d*(b*x + c*x^2)^n]/(2*x^2)} -{Log[d*(b*x + c*x^2)^n]/x^4, x, 3, -(n/(9*x^3)) - (c*n)/(6*b*x^2) + (c^2*n)/(3*b^2*x) + (c^3*n*Log[x])/(3*b^3) - (c^3*n*Log[b + c*x])/(3*b^3) - Log[d*(b*x + c*x^2)^n]/(3*x^3)} -{Log[d*(b*x + c*x^2)^n]/x^5, x, 3, -(n/(16*x^4)) - (c*n)/(12*b*x^3) + (c^2*n)/(8*b^2*x^2) - (c^3*n)/(4*b^3*x) - (c^4*n*Log[x])/(4*b^4) + (c^4*n*Log[b + c*x])/(4*b^4) - Log[d*(b*x + c*x^2)^n]/(4*x^4)} - - -{x^m*Log[d*(a + b*x + c*x^2)^n], x, 5, -((2*c*n*x^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))])/((b - Sqrt[b^2 - 4*a*c])*(1 + m)*(2 + m))) - (2*c*n*x^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))])/((b + Sqrt[b^2 - 4*a*c])*(1 + m)*(2 + m)) + (x^(1 + m)*Log[d*(a + b*x + c*x^2)^n])/(1 + m)} - -{x^4*Log[d*(a + b*x + c*x^2)^n], x, 7, -(((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x)/(5*c^4)) + (b*(b^2 - 3*a*c)*n*x^2)/(10*c^3) - ((b^2 - 2*a*c)*n*x^3)/(15*c^2) + (b*n*x^4)/(20*c) - (2*n*x^5)/25 + (Sqrt[b^2 - 4*a*c]*(b^4 - 3*a*b^2*c + a^2*c^2)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(5*c^5) + (b*(b^4 - 5*a*b^2*c + 5*a^2*c^2)*n*Log[a + b*x + c*x^2])/(10*c^5) + (1/5)*x^5*Log[d*(a + b*x + c*x^2)^n]} -{x^3*Log[d*(a + b*x + c*x^2)^n], x, 7, (b*(b^2 - 3*a*c)*n*x)/(4*c^3) - ((b^2 - 2*a*c)*n*x^2)/(8*c^2) + (b*n*x^3)/(12*c) - (n*x^4)/8 - (b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*c^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + b*x + c*x^2])/(8*c^4) + (1/4)*x^4*Log[d*(a + b*x + c*x^2)^n]} -{x^2*Log[d*(a + b*x + c*x^2)^n], x, 7, -(((b^2 - 2*a*c)*n*x)/(3*c^2)) + (b*n*x^2)/(6*c) - (2*n*x^3)/9 + (Sqrt[b^2 - 4*a*c]*(b^2 - a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(3*c^3) + (b*(b^2 - 3*a*c)*n*Log[a + b*x + c*x^2])/(6*c^3) + (1/3)*x^3*Log[d*(a + b*x + c*x^2)^n]} -{x^1*Log[d*(a + b*x + c*x^2)^n], x, 7, (b*n*x)/(2*c) - (n*x^2)/2 - (b*Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(2*c^2) - ((b^2 - 2*a*c)*n*Log[a + b*x + c*x^2])/(4*c^2) + (1/2)*x^2*Log[d*(a + b*x + c*x^2)^n]} -{x^0*Log[d*(a + b*x + c*x^2)^n], x, 6, -2*n*x + (Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c + (b*n*Log[a + b*x + c*x^2])/(2*c) + x*Log[d*(a + b*x + c*x^2)^n]} -{Log[d*(a + b*x + c*x^2)^n]/x^1, x, 7, (-n)*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])] - n*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])] + Log[x]*Log[d*(a + b*x + c*x^2)^n] - n*PolyLog[2, -((2*c*x)/(b - Sqrt[b^2 - 4*a*c]))] - n*PolyLog[2, -((2*c*x)/(b + Sqrt[b^2 - 4*a*c]))]} -{Log[d*(a + b*x + c*x^2)^n]/x^2, x, 7, (Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/a + (b*n*Log[x])/a - (b*n*Log[a + b*x + c*x^2])/(2*a) - Log[d*(a + b*x + c*x^2)^n]/x} -{Log[d*(a + b*x + c*x^2)^n]/x^3, x, 7, -((b*n)/(2*a*x)) - (b*Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(2*a^2) - ((b^2 - 2*a*c)*n*Log[x])/(2*a^2) + ((b^2 - 2*a*c)*n*Log[a + b*x + c*x^2])/(4*a^2) - Log[d*(a + b*x + c*x^2)^n]/(2*x^2)} -{Log[d*(a + b*x + c*x^2)^n]/x^4, x, 7, -((b*n)/(6*a*x^2)) + ((b^2 - 2*a*c)*n)/(3*a^2*x) + (Sqrt[b^2 - 4*a*c]*(b^2 - a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(3*a^3) + (b*(b^2 - 3*a*c)*n*Log[x])/(3*a^3) - (b*(b^2 - 3*a*c)*n*Log[a + b*x + c*x^2])/(6*a^3) - Log[d*(a + b*x + c*x^2)^n]/(3*x^3)} -{Log[d*(a + b*x + c*x^2)^n]/x^5, x, 7, -((b*n)/(12*a*x^3)) + ((b^2 - 2*a*c)*n)/(8*a^2*x^2) - (b*(b^2 - 3*a*c)*n)/(4*a^3*x) - (b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*a^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[x])/(4*a^4) + ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + b*x + c*x^2])/(8*a^4) - Log[d*(a + b*x + c*x^2)^n]/(4*x^4)} - - -{Log[1 + x + x^2], x, 6, -2*x + Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + (1/2)*Log[1 + x + x^2] + x*Log[1 + x + x^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Log[d (a+b x+c x^2)^p]*) - - -{Log[d*(a + b*x + c*x^2)^n]*(d + e*x)^4, x, 7, -(((10*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + 2*a*e) - b^2*c*e^3*(5*b*d + 4*a*e) + c^2*e^2*(10*b^2*d^2 + 15*a*b*d*e + 2*a^2*e^2))*n*x)/(5*c^4)) - (e*(20*c^3*d^3 - b^3*e^3 - 10*c^2*d*e*(b*d + a*e) + b*c*e^2*(5*b*d + 3*a*e))*n*x^2)/(10*c^3) - (e^2*(20*c^2*d^2 + b^2*e^2 - c*e*(5*b*d + 2*a*e))*n*x^3)/(15*c^2) - (e^3*(10*c*d - b*e)*n*x^4)/(20*c) - (2/25)*e^4*n*x^5 + (Sqrt[b^2 - 4*a*c]*(5*c^4*d^4 + b^4*e^4 - 10*c^3*d^2*e*(b*d + a*e) - b^2*c*e^3*(5*b*d + 3*a*e) + c^2*e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(5*c^5) - ((2*c*d - b*e)*(c^4*d^4 + b^4*e^4 - 2*c^3*d^2*e*(b*d + 5*a*e) - b^2*c*e^3*(3*b*d + 5*a*e) + c^2*e^2*(4*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2))*n*Log[a + b*x + c*x^2])/(10*c^5*e) + ((d + e*x)^5*Log[d*(a + b*x + c*x^2)^n])/(5*e)} -{Log[d*(a + b*x + c*x^2)^n]*(d + e*x)^3, x, 7, -(((8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*n*x)/(4*c^3)) - (e*(12*c^2*d^2 + b^2*e^2 - 2*c*e*(2*b*d + a*e))*n*x^2)/(8*c^2) - (e^2*(8*c*d - b*e)*n*x^3)/(12*c) - (1/8)*e^3*n*x^4 + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*c^4) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[a + b*x + c*x^2])/(8*c^4*e) + ((d + e*x)^4*Log[d*(a + b*x + c*x^2)^n])/(4*e)} -{Log[d*(a + b*x + c*x^2)^n]*(d + e*x)^2, x, 7, -(((6*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + 2*a*e))*n*x)/(3*c^2)) - (e*(6*c*d - b*e)*n*x^2)/(6*c) - (2/9)*e^2*n*x^3 + (Sqrt[b^2 - 4*a*c]*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(3*c^3) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n*Log[a + b*x + c*x^2])/(6*c^3*e) + ((d + e*x)^3*Log[d*(a + b*x + c*x^2)^n])/(3*e)} -{Log[d*(a + b*x + c*x^2)^n]*(d + e*x)^1, x, 7, (-(1/2))*(4*d - (b*e)/c)*n*x - (1/2)*e*n*x^2 + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(2*c^2) - ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n*Log[a + b*x + c*x^2])/(4*c^2*e) + ((d + e*x)^2*Log[d*(a + b*x + c*x^2)^n])/(2*e)} -{Log[d*(a + b*x + c*x^2)^n]*(d + e*x)^0, x, 6, -2*n*x + (Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c + (b*n*Log[a + b*x + c*x^2])/(2*c) + x*Log[d*(a + b*x + c*x^2)^n]} -{Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^1, x, 9, -((n*Log[-((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))]*Log[d + e*x])/e) - (n*Log[-((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))]*Log[d + e*x])/e + (Log[d + e*x]*Log[d*(a + b*x + c*x^2)^n])/e - (n*PolyLog[2, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)])/e - (n*PolyLog[2, (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/e} -{Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^2, x, 7, (Sqrt[b^2 - 4*a*c]*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*d^2 - b*d*e + a*e^2) - ((2*c*d - b*e)*n*Log[d + e*x])/(e*(c*d^2 - b*d*e + a*e^2)) + ((2*c*d - b*e)*n*Log[a + b*x + c*x^2])/(2*e*(c*d^2 - b*d*e + a*e^2)) - Log[d*(a + b*x + c*x^2)^n]/(e*(d + e*x))} -{Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^3, x, 7, ((2*c*d - b*e)*n)/(2*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(2*(c*d^2 - b*d*e + a*e^2)^2) - ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n*Log[d + e*x])/(2*e*(c*d^2 - b*d*e + a*e^2)^2) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n*Log[a + b*x + c*x^2])/(4*e*(c*d^2 - b*d*e + a*e^2)^2) - Log[d*(a + b*x + c*x^2)^n]/(2*e*(d + e*x)^2)} -{Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^4, x, 7, ((2*c*d - b*e)*n)/(6*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n)/(3*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(3*(c*d^2 - b*d*e + a*e^2)^3) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n*Log[d + e*x])/(3*e*(c*d^2 - b*d*e + a*e^2)^3) + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n*Log[a + b*x + c*x^2])/(6*e*(c*d^2 - b*d*e + a*e^2)^3) - Log[d*(a + b*x + c*x^2)^n]/(3*e*(d + e*x)^3)} -{Log[d*(a + b*x + c*x^2)^n]/(d + e*x)^5, x, 7, ((2*c*d - b*e)*n)/(12*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n)/(8*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*n)/(4*e*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*(c*d^2 - b*d*e + a*e^2)^4) - ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[d + e*x])/(4*e*(c*d^2 - b*d*e + a*e^2)^4) + ((2*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*n*Log[a + b*x + c*x^2])/(8*e*(c*d^2 - b*d*e + a*e^2)^4) - Log[d*(a + b*x + c*x^2)^n]/(4*e*(d + e*x)^4)} - - -(* ::Subsection:: *) -(*Integrands of the form RFx (e+f x)^m Log[d (a+b x+c x^2)^p]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x+g x^2)^m Log[d (a+b x+c x^2)^p]^n*) - - -{Log[d*(a + c*x^2)^n]/(a*e + c*e*x^2), x, 6, (I*n*ArcTan[(Sqrt[c]*x)/Sqrt[a]]^2)/(Sqrt[a]*Sqrt[c]*e) + (2*n*ArcTan[(Sqrt[c]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[c]*x)])/(Sqrt[a]*Sqrt[c]*e) + (ArcTan[(Sqrt[c]*x)/Sqrt[a]]*Log[d*(a + c*x^2)^n])/(Sqrt[a]*Sqrt[c]*e) + (I*n*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[c]*x)])/(Sqrt[a]*Sqrt[c]*e)} -{Log[d*(a + b*x + c*x^2)^n]/(a*e + b*e*x + c*e*x^2), x, 8, (2*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]^2)/(Sqrt[b^2 - 4*a*c]*e) - (4*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]*Log[2/(1 - b/Sqrt[b^2 - 4*a*c] - (2*c*x)/Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*e) - (2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]]*Log[d*(a + b*x + c*x^2)^n])/(Sqrt[b^2 - 4*a*c]*e) - (2*n*PolyLog[2, -((1 + b/Sqrt[b^2 - 4*a*c] + (2*c*x)/Sqrt[b^2 - 4*a*c])/(1 - b/Sqrt[b^2 - 4*a*c] - (2*c*x)/Sqrt[b^2 - 4*a*c]))])/(Sqrt[b^2 - 4*a*c]*e)} - - -{Log[g*(a + b*x + c*x^2)^n]/(d + e*x^2), x, 20, -((n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e])) - (n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (n*Log[-((Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (n*Log[-((Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (Log[Sqrt[-d] - Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2*Sqrt[-d]*Sqrt[e]) - (Log[Sqrt[-d] + Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])} -{Log[g*(a + b*x + c*x^2)^n]/(d + e*x +f*x^2), x, 20, -((n*Log[-((f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*e - b*f + Sqrt[b^2 - 4*a*c]*f - c*Sqrt[e^2 - 4*d*f]))]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f]) - (n*Log[(f*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e - Sqrt[e^2 - 4*d*f]))]*Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f] + (n*Log[(f*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((b - Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f] + (n*Log[(f*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f]))]*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x])/Sqrt[e^2 - 4*d*f] + (Log[e - Sqrt[e^2 - 4*d*f] + 2*f*x]*Log[g*(a + b*x + c*x^2)^n])/Sqrt[e^2 - 4*d*f] - (Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x]*Log[g*(a + b*x + c*x^2)^n])/Sqrt[e^2 - 4*d*f] - (n*PolyLog[2, -((c*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/((b - Sqrt[b^2 - 4*a*c])*f - c*(e - Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] - (n*PolyLog[2, -((c*(e - Sqrt[e^2 - 4*d*f] + 2*f*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e - Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] + (n*PolyLog[2, -((c*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/((b - Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f] + (n*PolyLog[2, -((c*(e + Sqrt[e^2 - 4*d*f] + 2*f*x))/((b + Sqrt[b^2 - 4*a*c])*f - c*(e + Sqrt[e^2 - 4*d*f])))])/Sqrt[e^2 - 4*d*f]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Log[d (a+b x+c x^2)^p]^2*) - - -{Log[d*(b*x + c*x^2)^n]^2, x, 14, 8*n^2*x - (4*b*n^2*Log[b + c*x])/c - (2*b*n^2*Log[-((c*x)/b)]*Log[b + c*x])/c - (b*n^2*Log[b + c*x]^2)/c - 4*n*x*Log[d*(b*x + c*x^2)^n] + (2*b*n*Log[b + c*x]*Log[d*(b*x + c*x^2)^n])/c + x*Log[d*(b*x + c*x^2)^n]^2 - (2*b*n^2*PolyLog[2, 1 + (c*x)/b])/c} -{Log[d*(a + b*x + c*x^2)^n]^2, x, 27, 8*n^2*x - (4*Sqrt[b^2 - 4*a*c]*n^2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c - ((b - Sqrt[b^2 - 4*a*c])*n^2*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]^2)/(2*c) - ((b + Sqrt[b^2 - 4*a*c])*n^2*Log[-((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c]))]*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x])/c - ((b + Sqrt[b^2 - 4*a*c])*n^2*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]^2)/(2*c) - ((b - Sqrt[b^2 - 4*a*c])*n^2*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c - (2*b*n^2*Log[a + b*x + c*x^2])/c - 4*n*x*Log[d*(a + b*x + c*x^2)^n] + ((b - Sqrt[b^2 - 4*a*c])*n*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[d*(a + b*x + c*x^2)^n])/c + ((b + Sqrt[b^2 - 4*a*c])*n*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[d*(a + b*x + c*x^2)^n])/c + x*Log[d*(a + b*x + c*x^2)^n]^2 - ((b - Sqrt[b^2 - 4*a*c])*n^2*PolyLog[2, -((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c]))])/c - ((b + Sqrt[b^2 - 4*a*c])*n^2*PolyLog[2, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c} - - -{x^2*Log[1 + x + x^2]/(2 + 3*x + x^2), x, 28, -2*x + Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - Log[2 + 2*x]*Log[-((1 - I*Sqrt[3] + 2*x)/(1 + I*Sqrt[3]))] + 4*Log[4 + 2*x]*Log[-((1 - I*Sqrt[3] + 2*x)/(3 + I*Sqrt[3]))] - Log[2 + 2*x]*Log[-((1 + I*Sqrt[3] + 2*x)/(1 - I*Sqrt[3]))] + 4*Log[4 + 2*x]*Log[-((1 + I*Sqrt[3] + 2*x)/(3 - I*Sqrt[3]))] + (1/2)*Log[1 + x + x^2] + x*Log[1 + x + x^2] + Log[2 + 2*x]*Log[1 + x + x^2] - 4*Log[4 + 2*x]*Log[1 + x + x^2] - PolyLog[2, (2*(1 + x))/(1 - I*Sqrt[3])] - PolyLog[2, (2*(1 + x))/(1 + I*Sqrt[3])] + 4*PolyLog[2, (2*(2 + x))/(3 - I*Sqrt[3])] + 4*PolyLog[2, (2*(2 + x))/(3 + I*Sqrt[3])]} - - -{Log[1 + x + x^2]^2, x, 27, 8*x - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - (1/2)*(1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]^2 - (1 + I*Sqrt[3])*Log[(I*(1 - I*Sqrt[3] + 2*x))/(2*Sqrt[3])]*Log[1 + I*Sqrt[3] + 2*x] - (1/2)*(1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*x]^2 - (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[-((I*(1 + I*Sqrt[3] + 2*x))/(2*Sqrt[3]))] - 2*Log[1 + x + x^2] - 4*x*Log[1 + x + x^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*x]*Log[1 + x + x^2] + x*Log[1 + x + x^2]^2 - (1 + I*Sqrt[3])*PolyLog[2, -((I - Sqrt[3] + 2*I*x)/(2*Sqrt[3]))] - (1 - I*Sqrt[3])*PolyLog[2, (I + Sqrt[3] + 2*I*x)/(2*Sqrt[3])]} - -{Log[-1 + x + x^2]^2/x^3, x, 34, Log[x] - (1/2)*(1 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x] + 3*Log[(1/2)*(-1 + Sqrt[5])]*Log[1 - Sqrt[5] + 2*x] - (1/4)*(3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x]^2 - (1/2)*(1 - Sqrt[5])*Log[1 + Sqrt[5] + 2*x] - (1/2)*(3 - Sqrt[5])*Log[-((1 - Sqrt[5] + 2*x)/(2*Sqrt[5]))]*Log[1 + Sqrt[5] + 2*x] - (1/4)*(3 - Sqrt[5])*Log[1 + Sqrt[5] + 2*x]^2 - (1/2)*(3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x]*Log[(1 + Sqrt[5] + 2*x)/(2*Sqrt[5])] + 3*Log[x]*Log[1 + (2*x)/(1 + Sqrt[5])] + Log[-1 + x + x^2]/x - 3*Log[x]*Log[-1 + x + x^2] + (1/2)*(3 + Sqrt[5])*Log[1 - Sqrt[5] + 2*x]*Log[-1 + x + x^2] + (1/2)*(3 - Sqrt[5])*Log[1 + Sqrt[5] + 2*x]*Log[-1 + x + x^2] - Log[-1 + x + x^2]^2/(2*x^2) + 3*PolyLog[2, -((2*x)/(1 + Sqrt[5]))] - (1/2)*(3 + Sqrt[5])*PolyLog[2, -((1 - Sqrt[5] + 2*x)/(2*Sqrt[5]))] - (1/2)*(3 - Sqrt[5])*PolyLog[2, (1 + Sqrt[5] + 2*x)/(2*Sqrt[5])] - 3*PolyLog[2, 1 + (2*x)/(1 - Sqrt[5])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u Log[d+e x+f (a+b x+c x^2)^p]*) - - -{x^3*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 25, x/4096 - x^2/1024 + x^3/192 - x^4/32 - (683*Sqrt[-x + x^2])/4096 + (149*(1 - 2*x)*Sqrt[-x + x^2])/2048 - (1/12)*(-x + x^2)^(3/2) - (1/32)*x*(-x + x^2)^(3/2) + ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])]/32768 - (1537*ArcTanh[x/Sqrt[-x + x^2]])/16384 - Log[1 + 8*x]/32768 + (1/4)*x^4*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{x^2*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 20, -(x/384) + x^2/96 - x^3/18 - (85/384)*Sqrt[-x + x^2] + (5/64)*(1 - 2*x)*Sqrt[-x + x^2] - (1/18)*(-x + x^2)^(3/2) - ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])]/3072 - (223*ArcTanh[x/Sqrt[-x + x^2]])/1536 + Log[1 + 8*x]/3072 + (1/3)*x^3*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{x^1*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 16, x/32 - x^2/8 - (11/32)*Sqrt[-x + x^2] + (1/16)*(1 - 2*x)*Sqrt[-x + x^2] + (1/256)*ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])] - (33/128)*ArcTanh[x/Sqrt[-x + x^2]] - (1/256)*Log[1 + 8*x] + (1/2)*x^2*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{x^0*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 13, -(x/2) - (1/2)*Sqrt[-x + x^2] - (1/16)*ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])] - (7/8)*ArcTanh[x/Sqrt[-x + x^2]] + (1/16)*Log[1 + 8*x] + x*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^1, x, 1, CannotIntegrate[Log[-1 + 4*x + 4*Sqrt[-x + x^2]]/x, x]} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^2, x, 19, (4*Sqrt[-x + x^2])/x + 4*ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])] + 4*Log[x] - 4*Log[1 + 8*x] - Log[-1 + 4*x + 4*Sqrt[-x + x^2]]/x} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^3, x, 20, -(2/x) - (10*Sqrt[-x + x^2])/x - (2*(-x + x^2)^(3/2))/(3*x^3) - 16*ArcTanh[(1 - 10*x)/(6*Sqrt[-x + x^2])] - 16*Log[x] + 16*Log[1 + 8*x] - Log[-1 + 4*x + 4*Sqrt[-x + x^2]]/(2*x^2)} - - -{x^(3/2)*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 15, -(Sqrt[x]/160) + x^(3/2)/60 - (2*x^(5/2))/25 - (17*Sqrt[-x + x^2])/(32*Sqrt[x]) - (71*(-x + x^2)^(3/2))/(300*x^(3/2)) - (2*(-x + x^2)^(3/2))/(25*Sqrt[x]) - (Sqrt[-x + x^2]*ArcTan[(2/3)*Sqrt[2]*Sqrt[-1 + x]])/(320*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]) + ArcTan[2*Sqrt[2]*Sqrt[x]]/(320*Sqrt[2]) + (2/5)*x^(5/2)*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{x^(1/2)*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]], x, 13, Sqrt[x]/12 - (2*x^(3/2))/9 - (11*Sqrt[-x + x^2])/(12*Sqrt[x]) - (2*(-x + x^2)^(3/2))/(9*x^(3/2)) + (Sqrt[-x + x^2]*ArcTan[(2/3)*Sqrt[2]*Sqrt[-1 + x]])/(24*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]) - ArcTan[2*Sqrt[2]*Sqrt[x]]/(24*Sqrt[2]) + (2/3)*x^(3/2)*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^(1/2), x, 12, -2*Sqrt[x] - (2*Sqrt[-x + x^2])/Sqrt[x] - (Sqrt[-x + x^2]*ArcTan[(2/3)*Sqrt[2]*Sqrt[-1 + x]])/(Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]) + ArcTan[2*Sqrt[2]*Sqrt[x]]/Sqrt[2] + 2*Sqrt[x]*Log[-1 + 4*x + 4*Sqrt[-x + x^2]]} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^(3/2), x, 15, -((4*Sqrt[2]*Sqrt[-x + x^2]*ArcTan[(2/3)*Sqrt[2]*Sqrt[-1 + x]])/(Sqrt[-1 + x]*Sqrt[x])) + 4*Sqrt[2]*ArcTan[2*Sqrt[2]*Sqrt[x]] - 8*ArcTan[Sqrt[x]/Sqrt[-x + x^2]] - (2*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/Sqrt[x]} -{Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]]/x^(5/2), x, 18, -(16/(3*Sqrt[x])) + (4*Sqrt[-x + x^2])/(3*x^(3/2)) + (32*Sqrt[2]*Sqrt[-x + x^2]*ArcTan[(2/3)*Sqrt[2]*Sqrt[-1 + x]])/(3*Sqrt[-1 + x]*Sqrt[x]) - (32/3)*Sqrt[2]*ArcTan[2*Sqrt[2]*Sqrt[x]] + (44/3)*ArcTan[Sqrt[x]/Sqrt[-x + x^2]] - (2*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/(3*x^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Log[d+e (F^(c (a+b x)))^n]*) - - -{x^3*Log[a + b*E^x], x, 6, (1/4)*x^4*Log[a + b*E^x] - (1/4)*x^4*Log[1 + (b*E^x)/a] - x^3*PolyLog[2, -((b*E^x)/a)] + 3*x^2*PolyLog[3, -((b*E^x)/a)] - 6*x*PolyLog[4, -((b*E^x)/a)] + 6*PolyLog[5, -((b*E^x)/a)]} -{x^2*Log[a + b*E^x], x, 5, (1/3)*x^3*Log[a + b*E^x] - (1/3)*x^3*Log[1 + (b*E^x)/a] - x^2*PolyLog[2, -((b*E^x)/a)] + 2*x*PolyLog[3, -((b*E^x)/a)] - 2*PolyLog[4, -((b*E^x)/a)]} -{x^1*Log[a + b*E^x], x, 4, (1/2)*x^2*Log[a + b*E^x] - (1/2)*x^2*Log[1 + (b*E^x)/a] - x*PolyLog[2, -((b*E^x)/a)] + PolyLog[3, -((b*E^x)/a)]} -{x^0*Log[a + b*E^x], x, 4, x*Log[a + b*E^x] - x*Log[1 + (b*E^x)/a] - PolyLog[2, -((b*E^x)/a)]} -{Log[a + b*E^x]/x^1, x, 0, CannotIntegrate[Log[a + b*E^x]/x, x]} - - -{x^3*Log[1 + e*(f^(c*(a + b*x)))^n], x, 5, -((x^3*PolyLog[2, (-e)*(f^(c*(a + b*x)))^n])/(b*c*n*Log[f])) + (3*x^2*PolyLog[3, (-e)*(f^(c*(a + b*x)))^n])/(b^2*c^2*n^2*Log[f]^2) - (6*x*PolyLog[4, (-e)*(f^(c*(a + b*x)))^n])/(b^3*c^3*n^3*Log[f]^3) + (6*PolyLog[5, (-e)*(f^(c*(a + b*x)))^n])/(b^4*c^4*n^4*Log[f]^4)} -{x^2*Log[1 + e*(f^(c*(a + b*x)))^n], x, 4, -((x^2*PolyLog[2, (-e)*(f^(c*(a + b*x)))^n])/(b*c*n*Log[f])) + (2*x*PolyLog[3, (-e)*(f^(c*(a + b*x)))^n])/(b^2*c^2*n^2*Log[f]^2) - (2*PolyLog[4, (-e)*(f^(c*(a + b*x)))^n])/(b^3*c^3*n^3*Log[f]^3)} -{x^1*Log[1 + e*(f^(c*(a + b*x)))^n], x, 3, -((x*PolyLog[2, (-e)*(f^(c*(a + b*x)))^n])/(b*c*n*Log[f])) + PolyLog[3, (-e)*(f^(c*(a + b*x)))^n]/(b^2*c^2*n^2*Log[f]^2)} -{x^0*Log[1 + e*(f^(c*(a + b*x)))^n], x, 2, -(PolyLog[2, (-e)*(f^(c*(a + b*x)))^n]/(b*c*n*Log[f]))} -{Log[1 + e*(f^(c*(a + b*x)))^n]/x^1, x, 0, CannotIntegrate[Log[1 + e*(f^(c*(a + b*x)))^n]/x, x]} - - -{x^3*Log[d + e*(f^(c*(a + b*x)))^n], x, 6, (1/4)*x^4*Log[d + e*(f^(c*(a + b*x)))^n] - (1/4)*x^4*Log[1 + (e*(f^(c*(a + b*x)))^n)/d] - (x^3*PolyLog[2, -((e*(f^(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + (3*x^2*PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^2*c^2*n^2*Log[f]^2) - (6*x*PolyLog[4, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^3*c^3*n^3*Log[f]^3) + (6*PolyLog[5, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^4*c^4*n^4*Log[f]^4)} -{x^2*Log[d + e*(f^(c*(a + b*x)))^n], x, 5, (1/3)*x^3*Log[d + e*(f^(c*(a + b*x)))^n] - (1/3)*x^3*Log[1 + (e*(f^(c*(a + b*x)))^n)/d] - (x^2*PolyLog[2, -((e*(f^(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + (2*x*PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^2*c^2*n^2*Log[f]^2) - (2*PolyLog[4, -((e*(f^(c*(a + b*x)))^n)/d)])/(b^3*c^3*n^3*Log[f]^3)} -{x^1*Log[d + e*(f^(c*(a + b*x)))^n], x, 4, (1/2)*x^2*Log[d + e*(f^(c*(a + b*x)))^n] - (1/2)*x^2*Log[1 + (e*(f^(c*(a + b*x)))^n)/d] - (x*PolyLog[2, -((e*(f^(c*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)]/(b^2*c^2*n^2*Log[f]^2)} -{x^0*Log[d + e*(f^(c*(a + b*x)))^n], x, 4, x*Log[d + e*(f^(c*(a + b*x)))^n] - x*Log[1 + (e*(f^(c*(a + b*x)))^n)/d] - PolyLog[2, -((e*(f^(c*(a + b*x)))^n)/d)]/(b*c*n*Log[f])} -{Log[d + e*(f^(c*(a + b*x)))^n]/x^1, x, 0, CannotIntegrate[Log[d + e*(f^(c*(a + b*x)))^n]/x, x]} - - -{Log[Pi + b*(F^(e*(c + d*x)))^n], x, 3, x*Log[Pi] - PolyLog[2, -((b*(F^(e*(c + d*x)))^n)/Pi)]/(d*e*n*Log[F])} - - -(* ::Section::Closed:: *) -(*Integrands of the form F[Log[c x^n]]/x*) - - -{1/(x*(3 + Log[x])), x, 2, Log[3 + Log[x]]} -{Sqrt[1 + Log[x]]/x, x, 2, (2*(1 + Log[x])^(3/2))/3} -{(1 + Log[x])^5/x, x, 2, (1 + Log[x])^6/6} -{1/(x*Sqrt[Log[x]]), x, 2, 2*Sqrt[Log[x]]} - -{1/(x*(1 + Log[x]^2)), x, 2, ArcTan[Log[x]]} -{1/(x*Sqrt[-3 + Log[x]^2]), x, 3, ArcTanh[Log[x]/Sqrt[-3 + Log[x]^2]]} -{1/(x*Sqrt[4 - 9*Log[x]^2]), x, 2, ArcSin[(3*Log[x])/2]/3} -{1/(x*Sqrt[4 + Log[x]^2]), x, 2, ArcSinh[Log[x]/2]} -{1/(x*(2 + 3*Log[6*x]^3)), x, 7, -(ArcTan[1/Sqrt[3] - (2^(2/3)*Log[6*x])/3^(1/6)]/(2^(2/3)*3^(5/6))) + Log[2^(1/3) + 3^(1/3)*Log[6*x]]/(3*2^(2/3)*3^(1/3)) - Log[2^(2/3) - 6^(1/3)*Log[6*x] + 3^(2/3)*Log[6*x]^2]/(6*2^(2/3)*3^(1/3))} - -{Log[Log[6*x]]/(x*Log[6*x]), x, 2, Log[Log[6*x]]^2/2} -{2^Log[x]/x, x, 2, 2^Log[x]/Log[2], x^Log[2]/Log[2]} -{Sin[Log[x]]^2/x, x, 3, Log[x]/2 - (1/2)*Cos[Log[x]]*Sin[Log[x]]} -{(7 - Log[x])/(x*(3 + Log[x])), x, 3, -Log[x] + 10*Log[3 + Log[x]]} -{((2 - Log[x])*(3 + Log[x])^2)/x, x, 3, (5/3)*(3 + Log[x])^3 - (1/4)*(3 + Log[x])^4} -{(Log[x]^2*Sqrt[1 + Log[x]^2])/x, x, 4, (-(1/8))*ArcSinh[Log[x]] + (1/8)*Log[x]*Sqrt[1 + Log[x]^2] + (1/4)*Log[x]^3*Sqrt[1 + Log[x]^2]} -{(1 + Log[x])/(x*(3 + 2*Log[x])^2), x, 3, 1/(4*(3 + 2*Log[x])) + (1/4)*Log[3 + 2*Log[x]]} -{Log[x]/(x*Sqrt[1 + Log[x]]), x, 3, -2*Sqrt[1 + Log[x]] + (2/3)*(1 + Log[x])^(3/2)} -{Log[x]/(x*Sqrt[-1 + 4*Log[x]]), x, 3, (1/8)*Sqrt[-1 + 4*Log[x]] + (1/24)*(-1 + 4*Log[x])^(3/2)} -{Sqrt[1 + Log[x]]/(x*Log[x]), x, 4, -2*ArcTanh[Sqrt[1 + Log[x]]] + 2*Sqrt[1 + Log[x]]} -{(1 - 4*Log[x] + Log[x]^2)/(x*(-1 + Log[x])^4), x, 3, -(2/(3*(1 - Log[x])^3)) + 1/(1 - Log[x]) + 1/(-1 + Log[x])^2} - - -{(Log[a*x^n]^2)^p/x, x, 3, (Log[a*x^n]*(Log[a*x^n]^2)^p)/(n*(1 + 2*p))} -{(Log[a*x^n]^m)^p/x, x, 3, (Log[a*x^n]*(Log[a*x^n]^m)^p)/(n*(1 + m*p))} -{Sqrt[Log[a*x^n]^2]/x, x, 3, (Log[a*x^n]*Sqrt[Log[a*x^n]^2])/(2*n)} -{(b*Log[a*x^n]^m)^p/x, x, 3, (Log[a*x^n]*(b*Log[a*x^n]^m)^p)/(n*(1 + m*p))} - - -{1/(x*Log[E^x]), x, 4, -(Log[x]/(x - Log[E^x])) + Log[Log[E^x]]/(x - Log[E^x])} - - -(* ::Section::Closed:: *) -(*Integrands involving logarithms and trig functions*) - - -(* ::Subsection::Closed:: *) -(*Integrands involving products of logarithms and trig functions*) - - -{Log[x]*Sin[a + b*x]^1, x, 5, (Cos[a]*CosIntegral[b*x])/b - (Cos[a + b*x]*Log[x])/b - (Sin[a]*SinIntegral[b*x])/b} -{Log[x]*Sin[a + b*x]^2, x, 5, -(x/2) + (1/2)*x*Log[x] + (CosIntegral[2*b*x]*Sin[2*a])/(4*b) - (Cos[a + b*x]*Log[x]*Sin[a + b*x])/(2*b) + (Cos[2*a]*SinIntegral[2*b*x])/(4*b)} -{Log[x]*Sin[a + b*x]^3, x, 15, (3*Cos[a]*CosIntegral[b*x])/(4*b) - (Cos[3*a]*CosIntegral[3*b*x])/(12*b) - (Cos[a + b*x]*Log[x])/b + (Cos[a + b*x]^3*Log[x])/(3*b) - (3*Sin[a]*SinIntegral[b*x])/(4*b) + (Sin[3*a]*SinIntegral[3*b*x])/(12*b)} - - -{Log[x]*Cos[a + b*x]^1, x, 5, -((CosIntegral[b*x]*Sin[a])/b) + (Log[x]*Sin[a + b*x])/b - (Cos[a]*SinIntegral[b*x])/b} -{Log[x]*Cos[a + b*x]^2, x, 7, -(x/2) + (1/2)*x*Log[x] - (CosIntegral[2*b*x]*Sin[2*a])/(4*b) + (Cos[a + b*x]*Log[x]*Sin[a + b*x])/(2*b) - (Cos[2*a]*SinIntegral[2*b*x])/(4*b)} -{Log[x]*Cos[a + b*x]^3, x, 15, -((3*CosIntegral[b*x]*Sin[a])/(4*b)) - (CosIntegral[3*b*x]*Sin[3*a])/(12*b) + (Log[x]*Sin[a + b*x])/b - (Log[x]*Sin[a + b*x]^3)/(3*b) - (3*Cos[a]*SinIntegral[b*x])/(4*b) - (Cos[3*a]*SinIntegral[3*b*x])/(12*b)} - - -{Cos[x]*Log[x] + Sin[x]/x, x, 4, Log[x]*Sin[x]} - - -(* ::Subsection::Closed:: *) -(*Integrands involving logarithms of trig functions*) - - -{Log[a*Sin[x]], x, 5, (I*x^2)/2 - x*Log[1 - E^(2*I*x)] + x*Log[a*Sin[x]] + (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[a*Sin[x]^2], x, 6, I*x^2 - 2*x*Log[1 - E^(2*I*x)] + x*Log[a*Sin[x]^2] + I*PolyLog[2, E^(2*I*x)]} -{Log[a*Sin[x]^n], x, 6, (1/2)*I*n*x^2 - n*x*Log[1 - E^(2*I*x)] + x*Log[a*Sin[x]^n] + (1/2)*I*n*PolyLog[2, E^(2*I*x)]} - -{Log[a*Cos[x]], x, 5, (I*x^2)/2 - x*Log[1 + E^(2*I*x)] + x*Log[a*Cos[x]] + (1/2)*I*PolyLog[2, -E^(2*I*x)]} -{Log[a*Cos[x]^2], x, 6, I*x^2 - 2*x*Log[1 + E^(2*I*x)] + x*Log[a*Cos[x]^2] + I*PolyLog[2, -E^(2*I*x)]} -{Log[a*Cos[x]^n], x, 6, (1/2)*I*n*x^2 - n*x*Log[1 + E^(2*I*x)] + x*Log[a*Cos[x]^n] + (1/2)*I*n*PolyLog[2, -E^(2*I*x)]} - -{Log[a*Tan[x]], x, 7, 2*x*ArcTanh[E^(2*I*x)] + x*Log[a*Tan[x]] - (1/2)*I*PolyLog[2, -E^(2*I*x)] + (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[a*Tan[x]^2], x, 8, 4*x*ArcTanh[E^(2*I*x)] + x*Log[a*Tan[x]^2] - I*PolyLog[2, -E^(2*I*x)] + I*PolyLog[2, E^(2*I*x)]} -{Log[a*Tan[x]^n], x, 8, 2*n*x*ArcTanh[E^(2*I*x)] + x*Log[a*Tan[x]^n] - (1/2)*I*n*PolyLog[2, -E^(2*I*x)] + (1/2)*I*n*PolyLog[2, E^(2*I*x)]} - -{Log[a*Cot[x]], x, 7, -2*x*ArcTanh[E^(2*I*x)] + x*Log[a*Cot[x]] + (1/2)*I*PolyLog[2, -E^(2*I*x)] - (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[a*Cot[x]^2], x, 8, -4*x*ArcTanh[E^(2*I*x)] + x*Log[a*Cot[x]^2] + I*PolyLog[2, -E^(2*I*x)] - I*PolyLog[2, E^(2*I*x)]} -{Log[a*Cot[x]^n], x, 8, -2*n*x*ArcTanh[E^(2*I*x)] + x*Log[a*Cot[x]^n] + (1/2)*I*n*PolyLog[2, -E^(2*I*x)] - (1/2)*I*n*PolyLog[2, E^(2*I*x)]} - -{Log[a*Sec[x]], x, 5, -((I*x^2)/2) + x*Log[1 + E^(2*I*x)] + x*Log[a*Sec[x]] - (1/2)*I*PolyLog[2, -E^(2*I*x)]} -{Log[a*Sec[x]^2], x, 6, (-I)*x^2 + 2*x*Log[1 + E^(2*I*x)] + x*Log[a*Sec[x]^2] - I*PolyLog[2, -E^(2*I*x)]} -{Log[a*Sec[x]^n], x, 6, (-(1/2))*I*n*x^2 + n*x*Log[1 + E^(2*I*x)] + x*Log[a*Sec[x]^n] - (1/2)*I*n*PolyLog[2, -E^(2*I*x)]} - -{Log[a*Csc[x]], x, 5, -((I*x^2)/2) + x*Log[1 - E^(2*I*x)] + x*Log[a*Csc[x]] - (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[a*Csc[x]^2], x, 6, (-I)*x^2 + 2*x*Log[1 - E^(2*I*x)] + x*Log[a*Csc[x]^2] - I*PolyLog[2, E^(2*I*x)]} -{Log[a*Csc[x]^n], x, 6, (-(1/2))*I*n*x^2 + n*x*Log[1 - E^(2*I*x)] + x*Log[a*Csc[x]^n] - (1/2)*I*n*PolyLog[2, E^(2*I*x)]} - - -{Cos[x]*Log[(1 - Cos[2*x])/2], x, 3, -2*Sin[x] + Log[(1/2)*(1 - Cos[2*x])]*Sin[x]} -{Cot[x]/Log[E*Sin[x]], x, 3, Log[Log[E*Sin[x]]], Log[1 + Log[Sin[x]]]} -{Cot[x]/Log[E^Sin[x]], x, 5, Log[Log[E^Sin[x]]]/(-Log[E^Sin[x]] + Sin[x]) - Log[Sin[x]]/(-Log[E^Sin[x]] + Sin[x])} -{Log[Cos[x]]*Sec[x]^2, x, 3, -x + Tan[x] + Log[Cos[x]]*Tan[x]} -{Cot[x]*Log[Sin[x]], x, 2, Log[Sin[x]]^2/2} -{Cos[x]*Log[Sin[x]]*Sin[x]^2, x, 4, (-(1/9))*Sin[x]^3 + (1/3)*Log[Sin[x]]*Sin[x]^3} -{Log[Sin[a/2 + b*x/2]*Cos[a/2 + b*x/2]]*Cos[a + b*x], x, 2, -(Sin[a + b*x]/b) + (Log[Cos[a/2 + (b*x)/2]*Sin[a/2 + (b*x)/2]]*Sin[a + b*x])/b} -{Tan[x]/Log[Cos[x]], x, 3, -Log[Log[Cos[x]]]} - - -{Log[Cos[x]]*Tan[x], x, 2, -Log[Cos[x]]^2/2} -{Log[Cos[x]]*Sin[x], x, 2, Cos[x] - Cos[x]*Log[Cos[x]]} -{Log[Cos[x]]*Cos[x], x, 4, ArcTanh[Sin[x]] - Sin[x] + Log[Cos[x]]*Sin[x]} - -{Log[Sin[x]]*Cos[x], x, 2, -Sin[x] + Log[Sin[x]]*Sin[x]} -{Log[Sin[x]]*Sin[x]^2, x, 10, x/4 + (I*x^2)/4 - (1/2)*x*Log[1 - E^(2*I*x)] + (1/2)*x*Log[Sin[x]] + (1/4)*I*PolyLog[2, E^(2*I*x)] + (1/4)*Cos[x]*Sin[x] - (1/2)*Cos[x]*Log[Sin[x]]*Sin[x]} -{Log[Sin[x]]*Sin[x]^3, x, 7, (-(2/3))*ArcTanh[Cos[x]] + (2*Cos[x])/3 - Cos[x]^3/9 - Cos[x]*Log[Sin[x]] + (1/3)*Cos[x]^3*Log[Sin[x]]} -{Log[Sin[Sqrt[x]]], x, 8, (I/3)*x^(3/2) - x*Log[1 - E^((2*I)*Sqrt[x])] + x*Log[Sin[Sqrt[x]]] + I*Sqrt[x]*PolyLog[2, E^((2*I)*Sqrt[x])] - PolyLog[3, E^((2*I)*Sqrt[x])]/2} -{Log[Sin[x]]*Csc[x]^2, x, 3, -x - Cot[x] - Cot[x]*Log[Sin[x]]} - - -(* ::Section::Closed:: *) -(*Integrands involving logarithms and hyperbolic functions*) - - -(* ::Subsection::Closed:: *) -(*Integrands involving products of logarithms and hyperbolic functions*) - - -{Log[x]*Sinh[a + b*x]^1, x, 5, -((Cosh[a]*CoshIntegral[b*x])/b) + (Cosh[a + b*x]*Log[x])/b - (Sinh[a]*SinhIntegral[b*x])/b} -{Log[x]*Sinh[a + b*x]^2, x, 7, x/2 - (1/2)*x*Log[x] - (CoshIntegral[2*b*x]*Sinh[2*a])/(4*b) + (Cosh[a + b*x]*Log[x]*Sinh[a + b*x])/(2*b) - (Cosh[2*a]*SinhIntegral[2*b*x])/(4*b)} -{Log[x]*Sinh[a + b*x]^3, x, 15, (3*Cosh[a]*CoshIntegral[b*x])/(4*b) - (Cosh[3*a]*CoshIntegral[3*b*x])/(12*b) - (Cosh[a + b*x]*Log[x])/b + (Cosh[a + b*x]^3*Log[x])/(3*b) + (3*Sinh[a]*SinhIntegral[b*x])/(4*b) - (Sinh[3*a]*SinhIntegral[3*b*x])/(12*b)} - - -{Log[x]*Cosh[a + b*x]^1, x, 5, -((CoshIntegral[b*x]*Sinh[a])/b) + (Log[x]*Sinh[a + b*x])/b - (Cosh[a]*SinhIntegral[b*x])/b} -{Log[x]*Cosh[a + b*x]^2, x, 7, -(x/2) + (1/2)*x*Log[x] - (CoshIntegral[2*b*x]*Sinh[2*a])/(4*b) + (Cosh[a + b*x]*Log[x]*Sinh[a + b*x])/(2*b) - (Cosh[2*a]*SinhIntegral[2*b*x])/(4*b)} -{Log[x]*Cosh[a + b*x]^3, x, 15, -((3*CoshIntegral[b*x]*Sinh[a])/(4*b)) - (CoshIntegral[3*b*x]*Sinh[3*a])/(12*b) + (Log[x]*Sinh[a + b*x])/b + (Log[x]*Sinh[a + b*x]^3)/(3*b) - (3*Cosh[a]*SinhIntegral[b*x])/(4*b) - (Cosh[3*a]*SinhIntegral[3*b*x])/(12*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands involving logarithms of hyperbolic functions*) - - -{Log[a*Sinh[x]], x, 5, x^2/2 - x*Log[1 - E^(2*x)] + x*Log[a*Sinh[x]] - (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Sinh[x]^2], x, 6, x^2 - 2*x*Log[1 - E^(2*x)] + x*Log[a*Sinh[x]^2] - PolyLog[2, E^(2*x)]} -{Log[a*Sinh[x]^n], x, 6, (n*x^2)/2 - n*x*Log[1 - E^(2*x)] + x*Log[a*Sinh[x]^n] - (1/2)*n*PolyLog[2, E^(2*x)]} - -{Log[a*Cosh[x]], x, 5, x^2/2 - x*Log[1 + E^(2*x)] + x*Log[a*Cosh[x]] - (1/2)*PolyLog[2, -E^(2*x)]} -{Log[a*Cosh[x]^2], x, 6, x^2 - 2*x*Log[1 + E^(2*x)] + x*Log[a*Cosh[x]^2] - PolyLog[2, -E^(2*x)]} -{Log[a*Cosh[x]^n], x, 6, (n*x^2)/2 - n*x*Log[1 + E^(2*x)] + x*Log[a*Cosh[x]^n] - (1/2)*n*PolyLog[2, -E^(2*x)]} - -{Log[Tanh[x]], x, 7, 2*x*ArcTanh[E^(2*x)] + x*Log[Tanh[x]] + (1/2)*PolyLog[2, -E^(2*x)] - (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Tanh[x]], x, 7, 2*x*ArcTanh[E^(2*x)] + x*Log[a*Tanh[x]] + (1/2)*PolyLog[2, -E^(2*x)] - (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Tanh[x]^2], x, 8, 4*x*ArcTanh[E^(2*x)] + x*Log[a*Tanh[x]^2] + PolyLog[2, -E^(2*x)] - PolyLog[2, E^(2*x)]} -{Log[a*Tanh[x]^n], x, 8, 2*n*x*ArcTanh[E^(2*x)] + x*Log[a*Tanh[x]^n] + (1/2)*n*PolyLog[2, -E^(2*x)] - (1/2)*n*PolyLog[2, E^(2*x)]} - -{Log[Coth[x]], x, 7, -2*x*ArcTanh[E^(2*x)] + x*Log[Coth[x]] - (1/2)*PolyLog[2, -E^(2*x)] + (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Coth[x]], x, 7, -2*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]] - (1/2)*PolyLog[2, -E^(2*x)] + (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Coth[x]^2], x, 8, -4*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]^2] - PolyLog[2, -E^(2*x)] + PolyLog[2, E^(2*x)]} -{Log[a*Coth[x]^n], x, 8, -2*n*x*ArcTanh[E^(2*x)] + x*Log[a*Coth[x]^n] - (1/2)*n*PolyLog[2, -E^(2*x)] + (1/2)*n*PolyLog[2, E^(2*x)]} - -{Log[a*Sech[x]], x, 5, -(x^2/2) + x*Log[1 + E^(2*x)] + x*Log[a*Sech[x]] + (1/2)*PolyLog[2, -E^(2*x)]} -{Log[a*Sech[x]^2], x, 6, -x^2 + 2*x*Log[1 + E^(2*x)] + x*Log[a*Sech[x]^2] + PolyLog[2, -E^(2*x)]} -{Log[a*Sech[x]^n], x, 6, -((n*x^2)/2) + n*x*Log[1 + E^(2*x)] + x*Log[a*Sech[x]^n] + (1/2)*n*PolyLog[2, -E^(2*x)]} - -{Log[a*Csch[x]], x, 5, -(x^2/2) + x*Log[1 - E^(2*x)] + x*Log[a*Csch[x]] + (1/2)*PolyLog[2, E^(2*x)]} -{Log[a*Csch[x]^2], x, 6, -x^2 + 2*x*Log[1 - E^(2*x)] + x*Log[a*Csch[x]^2] + PolyLog[2, E^(2*x)]} -{Log[a*Csch[x]^n], x, 6, -((n*x^2)/2) + n*x*Log[1 - E^(2*x)] + x*Log[a*Csch[x]^n] + (1/2)*n*PolyLog[2, E^(2*x)]} - - -{Log[Sinh[a/2 + b*x/2]*Cosh[a/2 + b*x/2]]*Cosh[a + b*x], x, 2, -(Sinh[a + b*x]/b) + (Log[Cosh[a/2 + (b*x)/2]*Sinh[a/2 + (b*x)/2]]*Sinh[a + b*x])/b} -{Log[Cosh[x]^2]*Sinh[x], x, 3, -2*Cosh[x] + Cosh[x]*Log[Cosh[x]^2]} - - -(* ::Section::Closed:: *) -(*Problems from Calculus textbooks*) - - -(* ::Subsection::Closed:: *) -(*Anton Calculus, 4th Edition*) - - -{Log[x]/Sqrt[x], x, 1, -4*Sqrt[x] + 2*Sqrt[x]*Log[x]} -{x*Log[2 - 3*x^2], x, 3, -(x^2/2) - (1/6)*(2 - 3*x^2)*Log[2 - 3*x^2]} - - -(* ::Subsection::Closed:: *) -(*Edwards and Penney Calculus*) - - -{1/(x*Sqrt[1 - Log[x]^2]), x, 2, ArcSin[Log[x]]} - - -(* ::Subsection::Closed:: *) -(*Thomas Calculus, 8th Edition*) - - -{16*x^3*Log[x]^2, x, 3, x^4/2 - 2*x^4*Log[x] + 4*x^4*Log[x]^2} -{Log[Sqrt[a + b*x]], x, 2, -(x/2) + ((a + b*x)*Log[Sqrt[a + b*x]])/b} -{x*Log[Sqrt[2 + x]], x, 3, x/2 - x^2/8 + (1/2)*x^2*Log[Sqrt[2 + x]] - Log[2 + x]} -{x*Log[(1 + 3*x)^(1/3)], x, 3, x/18 - x^2/12 + (1/2)*x^2*Log[(1 + 3*x)^(1/3)] - (1/54)*Log[1 + 3*x]} -{x*Log[x + x^3], x, 4, -((3*x^2)/4) + (1/2)*Log[1 + x^2] + (1/2)*x^2*Log[x + x^3]} -{Log[x + Sqrt[1 + x^2]], x, 2, -Sqrt[1 + x^2] + x*Log[x + Sqrt[1 + x^2]]} -{Log[x + Sqrt[-1 + x^2]], x, 2, -Sqrt[-1 + x^2] + x*Log[x + Sqrt[-1 + x^2]]} -{Log[x - Sqrt[-1 + x^2]], x, 2, Sqrt[-1 + x^2] + x*Log[x - Sqrt[-1 + x^2]]} -{Log[Sqrt[x] + Sqrt[1 + x]], x, 6, (-(1/2))*Sqrt[x]*Sqrt[1 + x] + ArcSinh[Sqrt[x]]/2 + x*Log[Sqrt[x] + Sqrt[1 + x]]} - - -(* ::Section::Closed:: *) -(*Problems from integration competitions*) - - -(* ::Subsection::Closed:: *) -(*MIT Integration Competition*) - - -{x^(1/3)*Log[x], x, 1, -((9*x^(4/3))/16) + (3/4)*x^(4/3)*Log[x]} - - -(* ::Subsection::Closed:: *) -(*University of Wisconsin Integration Competition*) - - -{2^Log[x], x, 2, x^(1 + Log[2])/(1 + Log[2])} -{(1 - Log[x])/x^2, x, 1, Log[x]/x} - - -(* ::Section::Closed:: *) -(*Miscellaneous problems*) - - -{Log[1 + x + Sqrt[1 + x]], x, 3, - x + Sqrt[1 + x] + (1/2)*Log[1 + x] + x*Log[1 + x + Sqrt[1 + x]]} -{Log[x + x^3], x, 3, -3*x + 2*ArcTan[x] + x*Log[x + x^3]} -{2^Log[-8 + 7*x], x, 2, (-8 + 7*x)^(1 + Log[2])/(7*(1 + Log[2]))} -{Log[(-11 + 5*x)/(5 + 76*x)], x, 2, (-(1/5))*(11 - 5*x)*Log[-((11 - 5*x)/(5 + 76*x))] - (861/380)*Log[5 + 76*x]} -{Log[1/(13 + x)], x, 2, x + (13 + x)*Log[(13 + x)^(-1)]} -{x*Log[(1 + x)/x^2], x, 4, x/2 + x^2/4 - (1/2)*Log[1 + x] + (1/2)*x^2*Log[(1 + x)/x^2]} -{x^3*Log[(7 + 5*x)/x^2], x, 4, (343*x)/500 - (49*x^2)/200 + (7*x^3)/60 + x^4/16 - (2401*Log[7 + 5*x])/2500 + (1/4)*x^4*Log[(7 + 5*x)/x^2]} -(* {x^3*Log[d + c*x]^4, x, 89, (c*x*(-70140*d^3 + 5190*c*d^2*x - 700*c^2*d*x^2 + 81*c^3*x^3) + 12*(5845*d^4 + 4980*c*d^3*x - 690*c^2*d^2*x^2 + 148*c^3*d*x^3 - 27*c^4*x^4)*Log[d + c*x] - 72*(415*d^4 + 300*c*d^3*x - 78*c^2*d^2*x^2 + 28*c^3*d*x^3 - 9*c^4*x^4)*Log[d + c*x]^2 + 288*(25*d^4 + 12*c*d^3*x - 6*c^2*d^2*x^2 + 4*c^3*d*x^3 - 3*c^4*x^4)*Log[d + c*x]^3 - 864*(d^4 - c^4*x^4)*Log[d + c*x]^4)/(3456*c^4)} *) - - -{(a + b*x)*Log[a + b*x], x, 2, -((a + b*x)^2/(4*b)) + ((a + b*x)^2*Log[a + b*x])/(2*b)} -{(a + b*x)^2*Log[a + b*x], x, 2, -((a + b*x)^3/(9*b)) + ((a + b*x)^3*Log[a + b*x])/(3*b)} -{Log[a + b*x]/(a + b*x), x, 2, Log[a + b*x]^2/(2*b)} -{Log[a + b*x]/(a + b*x)^2, x, 2, -(1/(b*(a + b*x))) - Log[a + b*x]/(b*(a + b*x))} -{(a + b*x)^n*Log[a + b*x], x, 2, -((a + b*x)^(1 + n)/(b*(1 + n)^2)) + ((a + b*x)^(1 + n)*Log[a + b*x])/(b*(1 + n))} - - -{1/(a*x + b*x*Log[c*x^n]), x, 2, Log[a + b*Log[c*x^n]]/(b*n)} -{1/(a*x + b*x*Log[c*x^n]^2), x, 2, ArcTan[(Sqrt[b]*Log[c*x^n])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*n)} -{1/(a*x + b*x*Log[c*x^n]^3), x, 7, -(ArcTan[(a^(1/3) - 2*b^(1/3)*Log[c*x^n])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*n)) + Log[a^(1/3) + b^(1/3)*Log[c*x^n]]/(3*a^(2/3)*b^(1/3)*n) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + b^(2/3)*Log[c*x^n]^2]/(6*a^(2/3)*b^(1/3)*n)} -{1/(a*x + b*x*Log[c*x^n]^4), x, 10, -(ArcTan[1 - (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n)) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Log[c*x^n])/a^(1/4)]/(2*Sqrt[2]*a^(3/4)*b^(1/4)*n) - Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[b]*Log[c*x^n]^2]/(4*Sqrt[2]*a^(3/4)*b^(1/4)*n)} - -{1/(a*x + b*x/Log[c*x^n]), x, 3, Log[x]/a - (b*Log[b + a*Log[c*x^n]])/(a^2*n)} -{1/(a*x + b*x/Log[c*x^n]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[a]*Log[c*x^n])/Sqrt[b]])/(a^(3/2)*n)) + Log[x]/a} -{1/(a*x + b*x/Log[c*x^n]^3), x, 8, (b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Log[c*x^n])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(4/3)*n) + Log[x]/a - (b^(1/3)*Log[b^(1/3) + a^(1/3)*Log[c*x^n]])/(3*a^(4/3)*n) + (b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Log[c*x^n] + a^(2/3)*Log[c*x^n]^2])/(6*a^(4/3)*n)} -{1/(a*x + b*x/Log[c*x^n]^4), x, 11, (b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*Log[c*x^n])/b^(1/4)])/(2*Sqrt[2]*a^(5/4)*n) - (b^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*Log[c*x^n])/b^(1/4)])/(2*Sqrt[2]*a^(5/4)*n) + Log[x]/a + (b^(1/4)*Log[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[a]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(5/4)*n) - (b^(1/4)*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*Log[c*x^n] + Sqrt[a]*Log[c*x^n]^2])/(4*Sqrt[2]*a^(5/4)*n)} - - -{1/(x + x*Log[7*x] + x*Log[7*x]^2), x, 3, (2*ArcTan[(1 + 2*Log[7*x])/Sqrt[3]])/Sqrt[3]} - -{(-1 + Log[3*x])/(x*(1 - Log[3*x] + Log[3*x]^2)), x, 5, ArcTan[(1 - 2*Log[3*x])/Sqrt[3]]/Sqrt[3] + (1/2)*Log[1 - Log[3*x] + Log[3*x]^2]} -{(-1 + Log[3*x]^2)/(x + x*Log[3*x]^3), x, 5, ArcTan[(1 - 2*Log[3*x])/Sqrt[3]]/Sqrt[3] + (1/2)*Log[1 - Log[3*x] + Log[3*x]^2]} -{(-1 + Log[3*x]^2)/(x + x*Log[3*x] + x*Log[3*x]^2), x, 7, -(Sqrt[3]*ArcTan[(1 + 2*Log[3*x])/Sqrt[3]]) + Log[x] - Log[1 + Log[3*x] + Log[3*x]^2]/2} - - -{Log[1/x]^2/x^5, x, 2, -(1/(32*x^4)) + Log[1/x]/(8*x^4) - Log[1/x]^2/(4*x^4)} - -{1/Sqrt[-Log[a*x^2]], x, 3, -((Sqrt[Pi/2]*x*Erf[Sqrt[-Log[a*x^2]]/Sqrt[2]])/Sqrt[a*x^2])} -{1/Sqrt[-Log[a/x^2]], x, 3, Sqrt[Pi/2]*Sqrt[a/x^2]*x*Erfi[Sqrt[-Log[a/x^2]]/Sqrt[2]]} -{1/Sqrt[-Log[a*x^n]], x, 3, -((Sqrt[Pi]*x*Erf[Sqrt[-Log[a*x^n]]/Sqrt[n]])/(Sqrt[n]*(a*x^n)^n^(-1)))} - -{Log[1 + Sqrt[x] - x]/x, x, 8, -2*Log[(1/2)*(1 + Sqrt[5])]*Log[1 + Sqrt[5] - 2*Sqrt[x]] - 2*Log[1 - (2*Sqrt[x])/(1 - Sqrt[5])]*Log[Sqrt[x]] + 2*Log[1 + Sqrt[x] - x]*Log[Sqrt[x]] + 2*PolyLog[2, 1 - (2*Sqrt[x])/(1 + Sqrt[5])] - 2*PolyLog[2, (2*Sqrt[x])/(1 - Sqrt[5])]} - -{(x*Log[c + d*x])/(a + b*x), x, 7, -(x/b) + ((c + d*x)*Log[c + d*x])/(b*d) - (a*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/b^2 - (a*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/b^2} -{Log[x]/(-1 + x), x, 1, -PolyLog[2, 1 - x]} -{(x*Log[1 - a - b*x])/(a + b*x), x, 6, -(x/b) - ((1 - a - b*x)*Log[1 - a - b*x])/b^2 + (a*PolyLog[2, a + b*x])/b^2} -{((b + 2*c*x)*Log[x])/(x*(b + c*x)), x, 5, Log[x]^2/2 + Log[x]*Log[1 + (c*x)/b] + PolyLog[2, -((c*x)/b)]} - -{Sin[x*Log[x]] + Log[x]*Sin[x*Log[x]], x, 2, -Cos[x*Log[x]]} -{Log[(1 - (-1 + x)^2)/(1 + (-1 + x)^2)]/x^2, x, 8, -(1/x) + ArcTan[1 - x] - Log[(1 - (1 - x)^2)/(1 + (-1 + x)^2)]/x + (1/2)*Log[2 - x] + Log[x]/2 - (1/2)*Log[2 - 2*x + x^2]} -{Log[Sqrt[x] + x], x, 4, Sqrt[x] - x - Log[1 + Sqrt[x]] + x*Log[Sqrt[x] + x]} -{Log[-(x/(1 + x))], x, 2, x*Log[-(x/(1 + x))] - Log[1 + x]} -{Log[(-1 + x)/(1 + x)], x, 2, -((1 - x)*Log[-((1 - x)/(1 + x))]) - 2*Log[1 + x]} - -{Log[(1 - x^2)/(1 + x^2)]/(1 + x)^2, x, 8, -(1/(1 + x)) - ArcTan[x] + (1/2)*Log[1 - x^2] - Log[(1 - x^2)/(1 + x^2)]/(1 + x) - (1/2)*Log[1 + x^2]} - - -{Log[c*(1 + x^2)^n]/(1 + x^2), x, 5, I*n*ArcTan[x]^2 + 2*n*ArcTan[x]*Log[2/(1 + I*x)] + ArcTan[x]*Log[c*(1 + x^2)^n] + I*n*PolyLog[2, 1 - 2/(1 + I*x)]} -{Log[x^2/(1 + x^2)]/(1 + x^2), x, 5, I*ArcTan[x]^2 - 2*ArcTan[x]*Log[2 - 2/(1 - I*x)] + ArcTan[x]*Log[x^2/(1 + x^2)] + I*PolyLog[2, -1 + 2/(1 - I*x)]} -{Log[c*x^2/(a + b*x^2)]/(a + b*x^2), x, 5, (I*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/(Sqrt[a]*Sqrt[b]) + (ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(c*x^2)/(a + b*x^2)])/(Sqrt[a]*Sqrt[b]) - (2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[2 - (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b]*x)])/(Sqrt[a]*Sqrt[b]) + (I*PolyLog[2, -1 + (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b]*x)])/(Sqrt[a]*Sqrt[b])} - - -{Log[1 + (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(1 - a^2*x^2), x, 1, PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])]/a} -{Log[1 - (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(1 - a^2*x^2), x, 1, PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/a} - - -{Log[E^(a + b*x)], x, 2, Log[E^(a + b*x)]^2/(2*b)} -{Log[E^(a + b*x^n)], x, 3, -((b*n*x^(1 + n))/(1 + n)) + x*Log[E^(a + b*x^n)]} - - -{E^x*Log[a + b*E^x], x, 5, -E^x + ((a + b*E^x)*Log[a + b*E^x])/b, -E^x + (a*Log[a + b*E^x])/b + E^x*Log[a + b*E^x]} - - -{Log[x]*E^(a + b*x), x, 3, -((E^a*ExpIntegralEi[b*x])/b) + (E^(a + b*x)*Log[x])/b} - - -(* Attempted expansion of these integrands can lead to infinite recursion! *) -{x^2/(x + Log[x]), x, 0, CannotIntegrate[x^2/(x + Log[x]), x]} -{x^1/(x + Log[x]), x, 0, CannotIntegrate[x/(x + Log[x]), x]} -{x^0/(x + Log[x]), x, 0, CannotIntegrate[1/(x + Log[x]), x]} -{1/(x^1*(x + Log[x])), x, 0, CannotIntegrate[1/(x*(x + Log[x])), x]} -{1/(x^2*(x + Log[x])), x, 0, CannotIntegrate[1/(x^2*(x + Log[x])), x]} - - -{Log[x]/(x + 4*x*Log[x]^2), x, 2, Log[1 + 4*Log[x]^2]/8} - - -{(1 - Log[x])/(x*(x + Log[x])), x, 2, Log[1 + Log[x]/x]} -{(1 + x)/(Log[x]*(x + Log[x])), x, 8, Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]} - - -{Log[Sqrt[(x + 1)/x] + 2], x, 5, (-(1/6))*Log[1 - Sqrt[1 + 1/x]] + (1/2)*Log[1 + Sqrt[1 + 1/x]] - (1/3)*Log[2 + Sqrt[1 + 1/x]] + x*Log[2 + Sqrt[(1 + x)/x]]} -{Log[Sqrt[(x + 1)/x] + 1], x, 6, -(1/(2*(1 + Sqrt[1 + 1/x]))) + (1/2)*ArcTanh[Sqrt[(1 + x)/x]] + x*Log[1 + Sqrt[(1 + x)/x]]} -{Log[Sqrt[(x + 1)/x] + 0], x, 4, x*Log[Sqrt[1 + 1/x]] + (1/2)*Log[1 + x]} -{Log[Sqrt[(x + 1)/x] - 1], x, 5, -(1/(2*(1 - Sqrt[1 + 1/x]))) - (1/2)*ArcTanh[Sqrt[1 + 1/x]] + x*Log[-1 + Sqrt[(1 + x)/x]]} -{Log[Sqrt[(x + 1)/x] - 2], x, If[$VersionNumber>=8, 7, 4], (1/2)*Log[1 - Sqrt[1 + 1/x]] - (1/3)*Log[2 - Sqrt[1 + 1/x]] - (1/6)*Log[1 + Sqrt[1 + 1/x]] + x*Log[-2 + Sqrt[(1 + x)/x]]} - - -(* Contributed by Oleg Marichev of Wolfram Research Inc. *) -{x^(a*x) + x^(a*x)*Log[x], x, 2, x^(a*x)/a} - - -{(Log[x]^m)^p, x, 3, (Gamma[1 + m*p, -Log[x]]*(Log[x]^m)^p)/(-Log[x])^(m*p)} - - -(* {Log[a + b*x + c*Sqrt[d + e*x]]/(f + g*x^2), x, 44, (Log[(g^(1/4)*(c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[(g^(1/4)*(c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[a + b*x + c*Sqrt[d + e*x]]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[(g^(1/4)*(c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[(g^(1/4)*(c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[a + b*x + c*Sqrt[d + e*x]]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[-((g^(1/4)*(c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4)))]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[-((g^(1/4)*(c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4)))]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[a + b*x + c*Sqrt[d + e*x]]*Log[Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[-((g^(1/4)*(c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4)))]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) - (Log[-((g^(1/4)*(c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2] + 2*b*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4)))]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + (Log[a + b*x + c*Sqrt[d + e*x]]*Log[Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]])/(2*Sqrt[-f]*Sqrt[g]) + PolyLog[2, (2*b*(Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) + PolyLog[2, (2*b*(Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, (2*b*(Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, (2*b*(Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) + PolyLog[2, (2*b*(Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) + PolyLog[2, (2*b*(Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[(-e)*Sqrt[-f] + d*Sqrt[g]] - (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, (2*b*(Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - (c*e - Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g]) - PolyLog[2, (2*b*(Sqrt[e*Sqrt[-f] + d*Sqrt[g]] + g^(1/4)*Sqrt[d + e*x]))/(2*b*Sqrt[e*Sqrt[-f] + d*Sqrt[g]] - (c*e + Sqrt[4*b^2*d - 4*a*b*e + c^2*e^2])*g^(1/4))]/(2*Sqrt[-f]*Sqrt[g])} *) - - -{Log[x]/Sqrt[a + b*Log[x]], x, 4, -(((2*a + b)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(E^(a/b)*(2*b^(3/2)))) + (x*Sqrt[a + b*Log[x]])/b} -{Log[x]/Sqrt[a - b*Log[x]], x, 4, -(((2*a - b)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*Log[x]]/Sqrt[b]])/(2*b^(3/2))) - (x*Sqrt[a - b*Log[x]])/b} - -{(A + B*Log[x])/Sqrt[a + b*Log[x]], x, 4, ((2*A*b - (2*a + b)*B)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(E^(a/b)*(2*b^(3/2))) + (B*x*Sqrt[a + b*Log[x]])/b} -{(A + B*Log[x])/Sqrt[a - b*Log[x]], x, 4, -(((2*A*b + 2*a*B - b*B)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*Log[x]]/Sqrt[b]])/(2*b^(3/2))) - (B*x*Sqrt[a - b*Log[x]])/b} - - -{x^2*Log[Log[x]*Sin[x]], x, 13, (I*x^4)/12 - (1/3)*ExpIntegralEi[3*Log[x]] - (1/3)*x^3*Log[1 - E^(2*I*x)] + (1/3)*x^3*Log[Log[x]*Sin[x]] + (1/2)*I*x^2*PolyLog[2, E^(2*I*x)] - (1/2)*x*PolyLog[3, E^(2*I*x)] - (1/4)*I*PolyLog[4, E^(2*I*x)]} -{x^1*Log[Log[x]*Sin[x]], x, 12, (I*x^3)/6 - (1/2)*ExpIntegralEi[2*Log[x]] - (1/2)*x^2*Log[1 - E^(2*I*x)] + (1/2)*x^2*Log[Log[x]*Sin[x]] + (1/2)*I*x*PolyLog[2, E^(2*I*x)] - (1/4)*PolyLog[3, E^(2*I*x)]} -{x^0*Log[Log[x]*Sin[x]], x, 7, (I*x^2)/2 - x*Log[1 - E^(2*I*x)] + x*Log[Log[x]*Sin[x]] - LogIntegral[x] + (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[Log[x]*Sin[x]]/x^1, x, 0, CannotIntegrate[Log[Log[x]*Sin[x]]/x, x]} -{Log[Log[x]*Sin[x]]/x^2, x, 5, ExpIntegralEi[-Log[x]] - Log[Log[x]*Sin[x]]/x + Unintegrable[Cot[x]/x, x]} - - -{x^2*Log[E^x*Log[x]*Sin[x]], x, 14, (-(1/12) + I/12)*x^4 - (1/3)*ExpIntegralEi[3*Log[x]] - (1/3)*x^3*Log[1 - E^(2*I*x)] + (1/3)*x^3*Log[E^x*Log[x]*Sin[x]] + (1/2)*I*x^2*PolyLog[2, E^(2*I*x)] - (1/2)*x*PolyLog[3, E^(2*I*x)] - (1/4)*I*PolyLog[4, E^(2*I*x)]} -{x^1*Log[E^x*Log[x]*Sin[x]], x, 13, (-(1/6) + I/6)*x^3 - (1/2)*ExpIntegralEi[2*Log[x]] - (1/2)*x^2*Log[1 - E^(2*I*x)] + (1/2)*x^2*Log[E^x*Log[x]*Sin[x]] + (1/2)*I*x*PolyLog[2, E^(2*I*x)] - (1/4)*PolyLog[3, E^(2*I*x)]} -{x^0*Log[E^x*Log[x]*Sin[x]], x, 7, (-(1/2) + I/2)*x^2 - x*Log[1 - E^(2*I*x)] + x*Log[E^x*Log[x]*Sin[x]] - LogIntegral[x] + (1/2)*I*PolyLog[2, E^(2*I*x)]} -{Log[E^x*Log[x]*Sin[x]]/x^1, x, 0, CannotIntegrate[Log[E^x*Log[x]*Sin[x]]/x, x]} -{Log[E^x*Log[x]*Sin[x]]/x^2, x, 7, ExpIntegralEi[-Log[x]] + Log[x] - Log[E^x*Log[x]*Sin[x]]/x + Unintegrable[Cot[x]/x, x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.0 (a sin)^m (b trg)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.0 (a sin)^m (b trg)^n.m deleted file mode 100644 index 78ac5e8..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.0 (a sin)^m (b trg)^n.m +++ /dev/null @@ -1,913 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Sin[c+d x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[c+d x]^n*) - - -{Sin[a + b*x]^1, x, 1, -(Cos[a + b*x]/b)} -{Sin[a + b*x]^2, x, 2, x/2 - (Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Sin[a + b*x]^3, x, 2, -(Cos[a + b*x]/b) + Cos[a + b*x]^3/(3*b)} -{Sin[a + b*x]^4, x, 3, (3*x)/8 - (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) - (Cos[a + b*x]*Sin[a + b*x]^3)/(4*b)} -{Sin[a + b*x]^5, x, 2, -(Cos[a + b*x]/b) + (2*Cos[a + b*x]^3)/(3*b) - Cos[a + b*x]^5/(5*b)} -{Sin[a + b*x]^6, x, 4, (5*x)/16 - (5*Cos[a + b*x]*Sin[a + b*x])/(16*b) - (5*Cos[a + b*x]*Sin[a + b*x]^3)/(24*b) - (Cos[a + b*x]*Sin[a + b*x]^5)/(6*b)} -{Sin[a + b*x]^7, x, 2, -(Cos[a + b*x]/b) + Cos[a + b*x]^3/b - (3*Cos[a + b*x]^5)/(5*b) + Cos[a + b*x]^7/(7*b)} -{Sin[a + b*x]^8, x, 5, (35*x)/128 - (35*Cos[a + b*x]*Sin[a + b*x])/(128*b) - (35*Cos[a + b*x]*Sin[a + b*x]^3)/(192*b) - (7*Cos[a + b*x]*Sin[a + b*x]^5)/(48*b) - (Cos[a + b*x]*Sin[a + b*x]^7)/(8*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[c+d x])^(n/2)*) - - -{Sin[b*x]^(7/2), x, 3, -((10*EllipticF[Pi/4 - (b*x)/2, 2])/(21*b)) - (10*Cos[b*x]*Sqrt[Sin[b*x]])/(21*b) - (2*Cos[b*x]*Sin[b*x]^(5/2))/(7*b)} -{Sin[b*x]^(5/2), x, 2, -((6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b)) - (2*Cos[b*x]*Sin[b*x]^(3/2))/(5*b)} -{Sin[b*x]^(3/2), x, 2, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b)) - (2*Cos[b*x]*Sqrt[Sin[b*x]])/(3*b)} -{Sin[b*x]^(1/2), x, 1, -((2*EllipticE[Pi/4 - (b*x)/2, 2])/b)} -{1/Sin[b*x]^(1/2), x, 1, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/b)} -{1/Sin[b*x]^(3/2), x, 2, (2*EllipticE[Pi/4 - (b*x)/2, 2])/b - (2*Cos[b*x])/(b*Sqrt[Sin[b*x]])} -{1/Sin[b*x]^(5/2), x, 2, -((2*EllipticF[Pi/4 - (b*x)/2, 2])/(3*b)) - (2*Cos[b*x])/(3*b*Sin[b*x]^(3/2))} -{1/Sin[b*x]^(7/2), x, 3, (6*EllipticE[Pi/4 - (b*x)/2, 2])/(5*b) - (2*Cos[b*x])/(5*b*Sin[b*x]^(5/2)) - (6*Cos[b*x])/(5*b*Sqrt[Sin[b*x]])} - - -{Sin[a + b*x]^(7/2), x, 3, (10*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(21*b) - (10*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(21*b) - (2*Cos[a + b*x]*Sin[a + b*x]^(5/2))/(7*b)} -{Sin[a + b*x]^(5/2), x, 2, (6*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(5*b) - (2*Cos[a + b*x]*Sin[a + b*x]^(3/2))/(5*b)} -{Sin[a + b*x]^(3/2), x, 2, (2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(3*b) - (2*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(3*b)} -{Sin[a + b*x]^(1/2), x, 1, (2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/b} -{1/Sin[a + b*x]^(1/2), x, 1, (2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/b} -{1/Sin[a + b*x]^(3/2), x, 2, -((2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/b) - (2*Cos[a + b*x])/(b*Sqrt[Sin[a + b*x]])} -{1/Sin[a + b*x]^(5/2), x, 2, (2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(3*b) - (2*Cos[a + b*x])/(3*b*Sin[a + b*x]^(3/2))} -{1/Sin[a + b*x]^(7/2), x, 3, -((6*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(5*b)) - (2*Cos[a + b*x])/(5*b*Sin[a + b*x]^(5/2)) - (6*Cos[a + b*x])/(5*b*Sqrt[Sin[a + b*x]])} - - -{(c*Sin[a + b*x])^(7/2), x, 4, (10*c^4*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(21*b*Sqrt[c*Sin[a + b*x]]) - (10*c^3*Cos[a + b*x]*Sqrt[c*Sin[a + b*x]])/(21*b) - (2*c*Cos[a + b*x]*(c*Sin[a + b*x])^(5/2))/(7*b)} -{(c*Sin[a + b*x])^(5/2), x, 3, (6*c^2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[c*Sin[a + b*x]])/(5*b*Sqrt[Sin[a + b*x]]) - (2*c*Cos[a + b*x]*(c*Sin[a + b*x])^(3/2))/(5*b)} -{(c*Sin[a + b*x])^(3/2), x, 3, (2*c^2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b*Sqrt[c*Sin[a + b*x]]) - (2*c*Cos[a + b*x]*Sqrt[c*Sin[a + b*x]])/(3*b)} -{(c*Sin[a + b*x])^(1/2), x, 2, (2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[c*Sin[a + b*x]])/(b*Sqrt[Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2), x, 2, (2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(b*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(3/2), x, 3, -((2*Cos[a + b*x])/(b*c*Sqrt[c*Sin[a + b*x]])) - (2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[c*Sin[a + b*x]])/(b*c^2*Sqrt[Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(5/2), x, 3, -((2*Cos[a + b*x])/(3*b*c*(c*Sin[a + b*x])^(3/2))) + (2*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b*c^2*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(7/2), x, 4, -((2*Cos[a + b*x])/(5*b*c*(c*Sin[a + b*x])^(5/2))) - (6*Cos[a + b*x])/(5*b*c^3*Sqrt[c*Sin[a + b*x]]) - (6*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[c*Sin[a + b*x]])/(5*b*c^4*Sqrt[Sin[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[c+d x])^(n/3)*) - - -{(c*Sin[a + b*x])^(4/3), x, 1, (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/3))/(7*b*c*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x])^(2/3), x, 1, (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/3))/(5*b*c*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x])^(1/3), x, 1, -((3*Sqrt[(3/2)*(3 - I*Sqrt[3])]*c^(1/3)*EllipticE[ArcSin[(Sqrt[2]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)])/Sqrt[3 + I*Sqrt[3]]], (3*I - Sqrt[3])/(3*I + Sqrt[3])]*Sec[a + b*x]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)]*Sqrt[(I + Sqrt[3])/(3*I + Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 - I*Sqrt[3])*c^(2/3))]*Sqrt[(I - Sqrt[3])/(3*I - Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 + I*Sqrt[3])*c^(2/3))])/b) + (3*(1 - I*Sqrt[3])*Sqrt[3 - I*Sqrt[3]]*c^(1/3)*EllipticF[ArcSin[(Sqrt[2]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)])/Sqrt[3 - I*Sqrt[3]]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]*Sec[a + b*x]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)]*Sqrt[(I + Sqrt[3])/(3*I + Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 - I*Sqrt[3])*c^(2/3))]*Sqrt[(I - Sqrt[3])/(3*I - Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 + I*Sqrt[3])*c^(2/3))])/(2*Sqrt[2]*b), (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(4/3))/(4*b*c*Sqrt[Cos[a + b*x]^2])} -{1/(c*Sin[a + b*x])^(1/3), x, 1, -((3*Sqrt[3 - I*Sqrt[3]]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)])/Sqrt[3 - I*Sqrt[3]]], (3*I + Sqrt[3])/(3*I - Sqrt[3])]*Sec[a + b*x]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)]*Sqrt[(I + Sqrt[3])/(3*I + Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 - I*Sqrt[3])*c^(2/3))]*Sqrt[(I - Sqrt[3])/(3*I - Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 + I*Sqrt[3])*c^(2/3))])/(Sqrt[2]*b*c^(1/3))), (3*Cos[a + b*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(2/3))/(2*b*c*Sqrt[Cos[a + b*x]^2])} -{1/(c*Sin[a + b*x])^(2/3), x, 1, (3^(3/4)*EllipticF[ArcCos[(c^(2/3) - (1 - Sqrt[3])*(c*Sin[a + b*x])^(2/3))/(c^(2/3) - (1 + Sqrt[3])*(c*Sin[a + b*x])^(2/3))], (1/4)*(2 + Sqrt[3])]*Sec[a + b*x]*(c*Sin[a + b*x])^(1/3)*(c^(2/3) - (c*Sin[a + b*x])^(2/3))*Sqrt[(c^(4/3)*(1 + (c*Sin[a + b*x])^(2/3)/c^(2/3) + (c*Sin[a + b*x])^(4/3)/c^(4/3)))/(c^(2/3) - (1 + Sqrt[3])*(c*Sin[a + b*x])^(2/3))^2])/(2*b*c^(5/3)*Sqrt[-(((c*Sin[a + b*x])^(2/3)*(c^(2/3) - (c*Sin[a + b*x])^(2/3)))/(c^(2/3) - (1 + Sqrt[3])*(c*Sin[a + b*x])^(2/3))^2)]), (3*Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1/3))/(b*c*Sqrt[Cos[a + b*x]^2])} -{1/(c*Sin[a + b*x])^(4/3), x, 1, -((3*Cos[a + b*x]*Hypergeometric2F1[-(1/6), 1/2, 5/6, Sin[a + b*x]^2])/(b*c*Sqrt[Cos[a + b*x]^2]*(c*Sin[a + b*x])^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[c+d x])^n with n symbolic*) - - -{Sin[a + b*x]^n, x, 1, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sin[a + b*x]^(1 + n))/(b*(1 + n)*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x])^n,x, 1, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + n))/(b*c*(1 + n)*Sqrt[Cos[a + b*x]^2])} - - -(* ::Title:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Trg[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Sin[e+f x])^n with m and n symbolic*) - - -{(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n, x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Sin[e + f*x]^2]*(a*Sin[e + f*x])^(1 + m)*(b*Sin[e + f*x])^n)/(a*f*(1 + m + n)*Sqrt[Cos[e + f*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m Cos[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]^3*Sin[a + b*x], x, 2, -(Cos[a + b*x]^4/(4*b))} -{Cos[a + b*x]^2*Sin[a + b*x], x, 2, -(Cos[a + b*x]^3/(3*b))} -{Cos[a + b*x]^1*Sin[a + b*x], x, 2, Sin[a + b*x]^2/(2*b)} -{Sec[a + b*x]^1*Sin[a + b*x], x, 1, -(Log[Cos[a + b*x]]/b)} -{Sec[a + b*x]^2*Sin[a + b*x], x, 2, Sec[a + b*x]/b} -{Sec[a + b*x]^3*Sin[a + b*x], x, 2, Sec[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4*Sin[a + b*x], x, 2, Sec[a + b*x]^3/(3*b)} - - -{Cos[a + b*x]^7*Sin[a + b*x]^2, x, 3, Sin[a + b*x]^3/(3*b) - (3*Sin[a + b*x]^5)/(5*b) + (3*Sin[a + b*x]^7)/(7*b) - Sin[a + b*x]^9/(9*b)} -{Cos[a + b*x]^5*Sin[a + b*x]^2, x, 3, Sin[a + b*x]^3/(3*b) - (2*Sin[a + b*x]^5)/(5*b) + Sin[a + b*x]^7/(7*b)} -{Cos[a + b*x]^3*Sin[a + b*x]^2, x, 3, Sin[a + b*x]^3/(3*b) - Sin[a + b*x]^5/(5*b)} -{Cos[a + b*x]^1*Sin[a + b*x]^2, x, 2, Sin[a + b*x]^3/(3*b)} - -{Sec[a + b*x]^2*Sin[a + b*x]^2, x, 2, -x + Tan[a + b*x]/b} -{Sec[a + b*x]^4*Sin[a + b*x]^2, x, 2, Tan[a + b*x]^3/(3*b)} -{Sec[a + b*x]^6*Sin[a + b*x]^2, x, 3, Tan[a + b*x]^3/(3*b) + Tan[a + b*x]^5/(5*b)} -{Sec[a + b*x]^8*Sin[a + b*x]^2, x, 3, Tan[a + b*x]^3/(3*b) + (2*Tan[a + b*x]^5)/(5*b) + Tan[a + b*x]^7/(7*b)} -{Sec[a + b*x]^10*Sin[a + b*x]^2, x, 3, Tan[a + b*x]^3/(3*b) + (3*Tan[a + b*x]^5)/(5*b) + (3*Tan[a + b*x]^7)/(7*b) + Tan[a + b*x]^9/(9*b)} - -{Cos[a + b*x]^6*Sin[a + b*x]^2, x, 5, (5*x)/128 + (5*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(192*b) + (Cos[a + b*x]^5*Sin[a + b*x])/(48*b) - (Cos[a + b*x]^7*Sin[a + b*x])/(8*b)} -{Cos[a + b*x]^4*Sin[a + b*x]^2, x, 4, x/16 + (Cos[a + b*x]*Sin[a + b*x])/(16*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(24*b) - (Cos[a + b*x]^5*Sin[a + b*x])/(6*b)} -{Cos[a + b*x]^2*Sin[a + b*x]^2, x, 3, x/8 + (Cos[a + b*x]*Sin[a + b*x])/(8*b) - (Cos[a + b*x]^3*Sin[a + b*x])/(4*b)} -{Cos[a + b*x]^0*Sin[a + b*x]^2, x, 2, x/2 - (Cos[a + b*x]*Sin[a + b*x])/(2*b)} - -{Sec[a + b*x]^1*Sin[a + b*x]^2, x, 3, ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b} -{Sec[a + b*x]^3*Sin[a + b*x]^2, x, 2, -(ArcTanh[Sin[a + b*x]]/(2*b)) + (Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^5*Sin[a + b*x]^2, x, 3, -(ArcTanh[Sin[a + b*x]]/(8*b)) - (Sec[a + b*x]*Tan[a + b*x])/(8*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(4*b)} -{Sec[a + b*x]^7*Sin[a + b*x]^2, x, 4, -(ArcTanh[Sin[a + b*x]]/(16*b)) - (Sec[a + b*x]*Tan[a + b*x])/(16*b) - (Sec[a + b*x]^3*Tan[a + b*x])/(24*b) + (Sec[a + b*x]^5*Tan[a + b*x])/(6*b)} - - -{Cos[a + b*x]^5*Sin[a + b*x]^3, x, 3, -(Cos[a + b*x]^6/(6*b)) + Cos[a + b*x]^8/(8*b)} -{Cos[a + b*x]^4*Sin[a + b*x]^3, x, 3, -(Cos[a + b*x]^5/(5*b)) + Cos[a + b*x]^7/(7*b)} -{Cos[a + b*x]^3*Sin[a + b*x]^3, x, 3, Sin[a + b*x]^4/(4*b) - Sin[a + b*x]^6/(6*b)} -{Cos[a + b*x]^2*Sin[a + b*x]^3, x, 3, -(Cos[a + b*x]^3/(3*b)) + Cos[a + b*x]^5/(5*b)} -{Cos[a + b*x]^1*Sin[a + b*x]^3, x, 2, Sin[a + b*x]^4/(4*b)} -{Sec[a + b*x]^1*Sin[a + b*x]^3, x, 3, Cos[a + b*x]^2/(2*b) - Log[Cos[a + b*x]]/b} -{Sec[a + b*x]^2*Sin[a + b*x]^3, x, 3, Cos[a + b*x]/b + Sec[a + b*x]/b} -{Sec[a + b*x]^3*Sin[a + b*x]^3, x, 2, Log[Cos[a + b*x]]/b + Tan[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4*Sin[a + b*x]^3, x, 2, -(Sec[a + b*x]/b) + Sec[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5*Sin[a + b*x]^3, x, 2, Tan[a + b*x]^4/(4*b)} -{Sec[a + b*x]^6*Sin[a + b*x]^3, x, 3, -(Sec[a + b*x]^3/(3*b)) + Sec[a + b*x]^5/(5*b)} -{Sec[a + b*x]^7*Sin[a + b*x]^3, x, 3, -(Sec[a + b*x]^4/(4*b)) + Sec[a + b*x]^6/(6*b)} -{Sec[a + b*x]^8*Sin[a + b*x]^3, x, 3, -(Sec[a + b*x]^5/(5*b)) + Sec[a + b*x]^7/(7*b)} -{Sec[a + b*x]^9*Sin[a + b*x]^3, x, 3, -(Sec[a + b*x]^6/(6*b)) + Sec[a + b*x]^8/(8*b)} - - -{Cos[a + b*x]^7*Sin[a + b*x]^4, x, 3, Sin[a + b*x]^5/(5*b) - (3*Sin[a + b*x]^7)/(7*b) + Sin[a + b*x]^9/(3*b) - Sin[a + b*x]^11/(11*b)} -{Cos[a + b*x]^5*Sin[a + b*x]^4, x, 3, Sin[a + b*x]^5/(5*b) - (2*Sin[a + b*x]^7)/(7*b) + Sin[a + b*x]^9/(9*b)} -{Cos[a + b*x]^3*Sin[a + b*x]^4, x, 3, Sin[a + b*x]^5/(5*b) - Sin[a + b*x]^7/(7*b)} -{Cos[a + b*x]^1*Sin[a + b*x]^4, x, 2, Sin[a + b*x]^5/(5*b)} - -{Sec[a + b*x]^2*Sin[a + b*x]^4, x, 4, -((3*x)/2) + (3*Tan[a + b*x])/(2*b) - (Sin[a + b*x]^2*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^4*Sin[a + b*x]^4, x, 3, x - Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)} - -{Sec[a + b*x]^6*Sin[a + b*x]^4, x, 2, Tan[a + b*x]^5/(5*b)} -{Sec[a + b*x]^8*Sin[a + b*x]^4, x, 3, Tan[a + b*x]^5/(5*b) + Tan[a + b*x]^7/(7*b)} -{Sec[a + b*x]^10*Sin[a + b*x]^4, x, 3, Tan[a + b*x]^5/(5*b) + (2*Tan[a + b*x]^7)/(7*b) + Tan[a + b*x]^9/(9*b)} - -{Cos[a + b*x]^6*Sin[a + b*x]^4, x, 6, (3*x)/256 + (3*Cos[a + b*x]*Sin[a + b*x])/(256*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^5*Sin[a + b*x])/(160*b) - (3*Cos[a + b*x]^7*Sin[a + b*x])/(80*b) - (Cos[a + b*x]^7*Sin[a + b*x]^3)/(10*b)} -{Cos[a + b*x]^4*Sin[a + b*x]^4, x, 5, (3*x)/128 + (3*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(64*b) - (Cos[a + b*x]^5*Sin[a + b*x])/(16*b) - (Cos[a + b*x]^5*Sin[a + b*x]^3)/(8*b)} -{Cos[a + b*x]^2*Sin[a + b*x]^4, x, 4, x/16 + (Cos[a + b*x]*Sin[a + b*x])/(16*b) - (Cos[a + b*x]^3*Sin[a + b*x])/(8*b) - (Cos[a + b*x]^3*Sin[a + b*x]^3)/(6*b)} -{Cos[a + b*x]^0*Sin[a + b*x]^4, x, 3, (3*x)/8 - (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) - (Cos[a + b*x]*Sin[a + b*x]^3)/(4*b)} - -{Sec[a + b*x]^1*Sin[a + b*x]^4, x, 4, ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b)} -{Sec[a + b*x]^3*Sin[a + b*x]^4, x, 4, -((3*ArcTanh[Sin[a + b*x]])/(2*b)) + (3*Sin[a + b*x])/(2*b) + (Sin[a + b*x]*Tan[a + b*x]^2)/(2*b)} -{Sec[a + b*x]^5*Sin[a + b*x]^4, x, 3, (3*ArcTanh[Sin[a + b*x]])/(8*b) - (3*Sec[a + b*x]*Tan[a + b*x])/(8*b) + (Sec[a + b*x]*Tan[a + b*x]^3)/(4*b)} -{Sec[a + b*x]^7*Sin[a + b*x]^4, x, 4, ArcTanh[Sin[a + b*x]]/(16*b) + (Sec[a + b*x]*Tan[a + b*x])/(16*b) - (Sec[a + b*x]^3*Tan[a + b*x])/(8*b) + (Sec[a + b*x]^3*Tan[a + b*x]^3)/(6*b)} -{Sec[a + b*x]^9*Sin[a + b*x]^4, x, 5, (3*ArcTanh[Sin[a + b*x]])/(128*b) + (3*Sec[a + b*x]*Tan[a + b*x])/(128*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(64*b) - (Sec[a + b*x]^5*Tan[a + b*x])/(16*b) + (Sec[a + b*x]^5*Tan[a + b*x]^3)/(8*b)} - - -{Cos[a + b*x]^7*Sin[a + b*x]^5, x, 4, -(Cos[a + b*x]^8/(8*b)) + Cos[a + b*x]^10/(5*b) - Cos[a + b*x]^12/(12*b)} -{Cos[a + b*x]^6*Sin[a + b*x]^5, x, 3, -(Cos[a + b*x]^7/(7*b)) + (2*Cos[a + b*x]^9)/(9*b) - Cos[a + b*x]^11/(11*b)} -{Cos[a + b*x]^5*Sin[a + b*x]^5, x, 4, Sin[a + b*x]^6/(6*b) - Sin[a + b*x]^8/(4*b) + Sin[a + b*x]^10/(10*b)} -{Cos[a + b*x]^4*Sin[a + b*x]^5, x, 3, -(Cos[a + b*x]^5/(5*b)) + (2*Cos[a + b*x]^7)/(7*b) - Cos[a + b*x]^9/(9*b)} -{Cos[a + b*x]^3*Sin[a + b*x]^5, x, 3, Sin[a + b*x]^6/(6*b) - Sin[a + b*x]^8/(8*b)} -{Cos[a + b*x]^2*Sin[a + b*x]^5, x, 3, -(Cos[a + b*x]^3/(3*b)) + (2*Cos[a + b*x]^5)/(5*b) - Cos[a + b*x]^7/(7*b)} -{Cos[a + b*x]^1*Sin[a + b*x]^5, x, 2, Sin[a + b*x]^6/(6*b)} -{Sec[a + b*x]^1*Sin[a + b*x]^5, x, 4, Cos[a + b*x]^2/b - Cos[a + b*x]^4/(4*b) - Log[Cos[a + b*x]]/b} -{Sec[a + b*x]^2*Sin[a + b*x]^5, x, 3, (2*Cos[a + b*x])/b - Cos[a + b*x]^3/(3*b) + Sec[a + b*x]/b} -{Sec[a + b*x]^3*Sin[a + b*x]^5, x, 4, -(Cos[a + b*x]^2/(2*b)) + (2*Log[Cos[a + b*x]])/b + Sec[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4*Sin[a + b*x]^5, x, 3, -(Cos[a + b*x]/b) - (2*Sec[a + b*x])/b + Sec[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5*Sin[a + b*x]^5, x, 3, -(Log[Cos[a + b*x]]/b) - Tan[a + b*x]^2/(2*b) + Tan[a + b*x]^4/(4*b)} -{Sec[a + b*x]^6*Sin[a + b*x]^5, x, 3, Sec[a + b*x]/b - (2*Sec[a + b*x]^3)/(3*b) + Sec[a + b*x]^5/(5*b)} -{Sec[a + b*x]^7*Sin[a + b*x]^5, x, 2, Tan[a + b*x]^6/(6*b)} -{Sec[a + b*x]^8*Sin[a + b*x]^5, x, 3, Sec[a + b*x]^3/(3*b) - (2*Sec[a + b*x]^5)/(5*b) + Sec[a + b*x]^7/(7*b)} -{Sec[a + b*x]^9*Sin[a + b*x]^5, x, 3, Tan[a + b*x]^6/(6*b) + Tan[a + b*x]^8/(8*b)} -{Sec[a + b*x]^10*Sin[a + b*x]^5, x, 3, Sec[a + b*x]^5/(5*b) - (2*Sec[a + b*x]^7)/(7*b) + Sec[a + b*x]^9/(9*b)} -{Sec[a + b*x]^11*Sin[a + b*x]^5, x, 4, Sec[a + b*x]^6/(6*b) - Sec[a + b*x]^8/(4*b) + Sec[a + b*x]^10/(10*b)} -{Sec[a + b*x]^12*Sin[a + b*x]^5, x, 3, Sec[a + b*x]^7/(7*b) - (2*Sec[a + b*x]^9)/(9*b) + Sec[a + b*x]^11/(11*b)} -{Sec[a + b*x]^13*Sin[a + b*x]^5, x, 4, Sec[a + b*x]^8/(8*b) - Sec[a + b*x]^10/(5*b) + Sec[a + b*x]^12/(12*b)} - - -{Sec[a + b*x]^3*Sin[a + b*x]^6, x, 5, -((5*ArcTanh[Sin[a + b*x]])/(2*b)) + (5*Sin[a + b*x])/(2*b) + (5*Sin[a + b*x]^3)/(6*b) + (Sin[a + b*x]^3*Tan[a + b*x]^2)/(2*b)} - - -{Sec[a + b*x]^6*Sin[a + b*x]^7, x, 3, Cos[a + b*x]/b + (3*Sec[a + b*x])/b - Sec[a + b*x]^3/b + Sec[a + b*x]^5/(5*b)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[a + b*x]^6/Sin[a + b*x], x, 4, -(ArcTanh[Cos[a + b*x]]/b) + Cos[a + b*x]/b + Cos[a + b*x]^3/(3*b) + Cos[a + b*x]^5/(5*b)} -{Cos[a + b*x]^5/Sin[a + b*x], x, 4, Log[Sin[a + b*x]]/b - Sin[a + b*x]^2/b + Sin[a + b*x]^4/(4*b)} -{Cos[a + b*x]^4/Sin[a + b*x], x, 4, -(ArcTanh[Cos[a + b*x]]/b) + Cos[a + b*x]/b + Cos[a + b*x]^3/(3*b)} -{Cos[a + b*x]^3/Sin[a + b*x], x, 3, Log[Sin[a + b*x]]/b - Sin[a + b*x]^2/(2*b)} -{Cos[a + b*x]^2/Sin[a + b*x], x, 3, -(ArcTanh[Cos[a + b*x]]/b) + Cos[a + b*x]/b} -{Cos[a + b*x]^1/Sin[a + b*x], x, 1, Log[Sin[a + b*x]]/b} -{Sec[a + b*x]^1/Sin[a + b*x], x, 2, Log[Tan[a + b*x]]/b} -{Sec[a + b*x]^2/Sin[a + b*x], x, 3, -(ArcTanh[Cos[a + b*x]]/b) + Sec[a + b*x]/b} -{Sec[a + b*x]^3/Sin[a + b*x], x, 3, Log[Tan[a + b*x]]/b + Tan[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4/Sin[a + b*x], x, 4, -(ArcTanh[Cos[a + b*x]]/b) + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5/Sin[a + b*x], x, 4, Log[Tan[a + b*x]]/b + Tan[a + b*x]^2/b + Tan[a + b*x]^4/(4*b)} -{Sec[a + b*x]^6/Sin[a + b*x], x, 4, -(ArcTanh[Cos[a + b*x]]/b) + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b) + Sec[a + b*x]^5/(5*b)} -{Sec[a + b*x]^7/Sin[a + b*x], x, 4, Log[Tan[a + b*x]]/b + (3*Tan[a + b*x]^2)/(2*b) + (3*Tan[a + b*x]^4)/(4*b) + Tan[a + b*x]^6/(6*b)} - - -{Cos[a + b*x]^7/Sin[a + b*x]^2, x, 3, -(Csc[a + b*x]/b) - (3*Sin[a + b*x])/b + Sin[a + b*x]^3/b - Sin[a + b*x]^5/(5*b)} -{Cos[a + b*x]^6/Sin[a + b*x]^2, x, 5, -((15*x)/8) - (15*Cot[a + b*x])/(8*b) + (5*Cos[a + b*x]^2*Cot[a + b*x])/(8*b) + (Cos[a + b*x]^4*Cot[a + b*x])/(4*b)} -{Cos[a + b*x]^5/Sin[a + b*x]^2, x, 3, -(Csc[a + b*x]/b) - (2*Sin[a + b*x])/b + Sin[a + b*x]^3/(3*b)} -{Cos[a + b*x]^4/Sin[a + b*x]^2, x, 4, -((3*x)/2) - (3*Cot[a + b*x])/(2*b) + (Cos[a + b*x]^2*Cot[a + b*x])/(2*b)} -{Cos[a + b*x]^3/Sin[a + b*x]^2, x, 3, -(Csc[a + b*x]/b) - Sin[a + b*x]/b} -{Cos[a + b*x]^2/Sin[a + b*x]^2, x, 2, -x - Cot[a + b*x]/b} -{Cos[a + b*x]^1/Sin[a + b*x]^2, x, 2, -(Csc[a + b*x]/b)} -{Sec[a + b*x]^1/Sin[a + b*x]^2, x, 3, ArcTanh[Sin[a + b*x]]/b - Csc[a + b*x]/b} -{Sec[a + b*x]^2/Sin[a + b*x]^2, x, 3, -(Cot[a + b*x]/b) + Tan[a + b*x]/b} -{Sec[a + b*x]^3/Sin[a + b*x]^2, x, 4, (3*ArcTanh[Sin[a + b*x]])/(2*b) - (3*Csc[a + b*x])/(2*b) + (Csc[a + b*x]*Sec[a + b*x]^2)/(2*b)} -{Sec[a + b*x]^4/Sin[a + b*x]^2, x, 3, -(Cot[a + b*x]/b) + (2*Tan[a + b*x])/b + Tan[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5/Sin[a + b*x]^2, x, 5, (15*ArcTanh[Sin[a + b*x]])/(8*b) - (15*Csc[a + b*x])/(8*b) + (5*Csc[a + b*x]*Sec[a + b*x]^2)/(8*b) + (Csc[a + b*x]*Sec[a + b*x]^4)/(4*b)} - - -{Cos[a + b*x]^7/Sin[a + b*x]^3, x, 4, -(Csc[a + b*x]^2/(2*b)) - (3*Log[Sin[a + b*x]])/b + (3*Sin[a + b*x]^2)/(2*b) - Sin[a + b*x]^4/(4*b)} -{Cos[a + b*x]^6/Sin[a + b*x]^3, x, 5, (5*ArcTanh[Cos[a + b*x]])/(2*b) - (5*Cos[a + b*x])/(2*b) - (5*Cos[a + b*x]^3)/(6*b) - (Cos[a + b*x]^3*Cot[a + b*x]^2)/(2*b)} -{Cos[a + b*x]^5/Sin[a + b*x]^3, x, 4, -(Csc[a + b*x]^2/(2*b)) - (2*Log[Sin[a + b*x]])/b + Sin[a + b*x]^2/(2*b)} -{Cos[a + b*x]^4/Sin[a + b*x]^3, x, 4, (3*ArcTanh[Cos[a + b*x]])/(2*b) - (3*Cos[a + b*x])/(2*b) - (Cos[a + b*x]*Cot[a + b*x]^2)/(2*b)} -{Cos[a + b*x]^3/Sin[a + b*x]^3, x, 2, -(Cot[a + b*x]^2/(2*b)) - Log[Sin[a + b*x]]/b} -{Cos[a + b*x]^2/Sin[a + b*x]^3, x, 2, ArcTanh[Cos[a + b*x]]/(2*b) - (Cot[a + b*x]*Csc[a + b*x])/(2*b)} -{Cos[a + b*x]^1/Sin[a + b*x]^3, x, 2, -(Csc[a + b*x]^2/(2*b))} -{Sec[a + b*x]^1/Sin[a + b*x]^3, x, 3, -(Cot[a + b*x]^2/(2*b)) + Log[Tan[a + b*x]]/b} -{Sec[a + b*x]^2/Sin[a + b*x]^3, x, 4, -((3*ArcTanh[Cos[a + b*x]])/(2*b)) + (3*Sec[a + b*x])/(2*b) - (Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{Sec[a + b*x]^3/Sin[a + b*x]^3, x, 4, -(Cot[a + b*x]^2/(2*b)) + (2*Log[Tan[a + b*x]])/b + Tan[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4/Sin[a + b*x]^3, x, 5, -((5*ArcTanh[Cos[a + b*x]])/(2*b)) + (5*Sec[a + b*x])/(2*b) + (5*Sec[a + b*x]^3)/(6*b) - (Csc[a + b*x]^2*Sec[a + b*x]^3)/(2*b)} -{Sec[a + b*x]^5/Sin[a + b*x]^3, x, 4, -(Cot[a + b*x]^2/(2*b)) + (3*Log[Tan[a + b*x]])/b + (3*Tan[a + b*x]^2)/(2*b) + Tan[a + b*x]^4/(4*b)} - - -{Cos[a + b*x]^9/Sin[a + b*x]^4, x, 3, (4*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + (6*Sin[a + b*x])/b - (4*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)} -{Cos[a + b*x]^8/Sin[a + b*x]^4, x, 6, (35*x)/8 + (35*Cot[a + b*x])/(8*b) - (35*Cot[a + b*x]^3)/(24*b) + (7*Cos[a + b*x]^2*Cot[a + b*x]^3)/(8*b) + (Cos[a + b*x]^4*Cot[a + b*x]^3)/(4*b)} -{Cos[a + b*x]^7/Sin[a + b*x]^4, x, 3, (3*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + (3*Sin[a + b*x])/b - Sin[a + b*x]^3/(3*b)} -{Cos[a + b*x]^6/Sin[a + b*x]^4, x, 5, (5*x)/2 + (5*Cot[a + b*x])/(2*b) - (5*Cot[a + b*x]^3)/(6*b) + (Cos[a + b*x]^2*Cot[a + b*x]^3)/(2*b)} -{Cos[a + b*x]^5/Sin[a + b*x]^4, x, 3, (2*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + Sin[a + b*x]/b} -{Cos[a + b*x]^4/Sin[a + b*x]^4, x, 3, x + Cot[a + b*x]/b - Cot[a + b*x]^3/(3*b)} -{Cos[a + b*x]^3/Sin[a + b*x]^4, x, 2, Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)} -{Cos[a + b*x]^2/Sin[a + b*x]^4, x, 2, -(Cot[a + b*x]^3/(3*b))} -{Cos[a + b*x]^1/Sin[a + b*x]^4, x, 2, -(Csc[a + b*x]^3/(3*b))} -{Sec[a + b*x]^1/Sin[a + b*x]^4, x, 4, ArcTanh[Sin[a + b*x]]/b - Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)} -{Sec[a + b*x]^2/Sin[a + b*x]^4, x, 3, -((2*Cot[a + b*x])/b) - Cot[a + b*x]^3/(3*b) + Tan[a + b*x]/b} -{Sec[a + b*x]^3/Sin[a + b*x]^4, x, 5, (5*ArcTanh[Sin[a + b*x]])/(2*b) - (5*Csc[a + b*x])/(2*b) - (5*Csc[a + b*x]^3)/(6*b) + (Csc[a + b*x]^3*Sec[a + b*x]^2)/(2*b)} -{Sec[a + b*x]^4/Sin[a + b*x]^4, x, 3, -((3*Cot[a + b*x])/b) - Cot[a + b*x]^3/(3*b) + (3*Tan[a + b*x])/b + Tan[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5/Sin[a + b*x]^4, x, 6, (35*ArcTanh[Sin[a + b*x]])/(8*b) - (35*Csc[a + b*x])/(8*b) - (35*Csc[a + b*x]^3)/(24*b) + (7*Csc[a + b*x]^3*Sec[a + b*x]^2)/(8*b) + (Csc[a + b*x]^3*Sec[a + b*x]^4)/(4*b)} - - -{Cos[a + b*x]^9/Sin[a + b*x]^5, x, 4, (2*Csc[a + b*x]^2)/b - Csc[a + b*x]^4/(4*b) + (6*Log[Sin[a + b*x]])/b - (2*Sin[a + b*x]^2)/b + Sin[a + b*x]^4/(4*b)} -{Cos[a + b*x]^8/Sin[a + b*x]^5, x, 6, -((35*ArcTanh[Cos[a + b*x]])/(8*b)) + (35*Cos[a + b*x])/(8*b) + (35*Cos[a + b*x]^3)/(24*b) + (7*Cos[a + b*x]^3*Cot[a + b*x]^2)/(8*b) - (Cos[a + b*x]^3*Cot[a + b*x]^4)/(4*b)} -{Cos[a + b*x]^7/Sin[a + b*x]^5, x, 4, (3*Csc[a + b*x]^2)/(2*b) - Csc[a + b*x]^4/(4*b) + (3*Log[Sin[a + b*x]])/b - Sin[a + b*x]^2/(2*b)} -{Cos[a + b*x]^6/Sin[a + b*x]^5, x, 5, -((15*ArcTanh[Cos[a + b*x]])/(8*b)) + (15*Cos[a + b*x])/(8*b) + (5*Cos[a + b*x]*Cot[a + b*x]^2)/(8*b) - (Cos[a + b*x]*Cot[a + b*x]^4)/(4*b)} -{Cos[a + b*x]^5/Sin[a + b*x]^5, x, 3, Cot[a + b*x]^2/(2*b) - Cot[a + b*x]^4/(4*b) + Log[Sin[a + b*x]]/b} -{Cos[a + b*x]^4/Sin[a + b*x]^5, x, 3, -((3*ArcTanh[Cos[a + b*x]])/(8*b)) + (3*Cot[a + b*x]*Csc[a + b*x])/(8*b) - (Cot[a + b*x]^3*Csc[a + b*x])/(4*b)} -{Cos[a + b*x]^3/Sin[a + b*x]^5, x, 2, -(Cot[a + b*x]^4/(4*b))} -{Cos[a + b*x]^2/Sin[a + b*x]^5, x, 3, ArcTanh[Cos[a + b*x]]/(8*b) + (Cot[a + b*x]*Csc[a + b*x])/(8*b) - (Cot[a + b*x]*Csc[a + b*x]^3)/(4*b)} -{Cos[a + b*x]^1/Sin[a + b*x]^5, x, 2, -(Csc[a + b*x]^4/(4*b))} -{Sec[a + b*x]^1/Sin[a + b*x]^5, x, 4, -(Cot[a + b*x]^2/b) - Cot[a + b*x]^4/(4*b) + Log[Tan[a + b*x]]/b} -{Sec[a + b*x]^2/Sin[a + b*x]^5, x, 5, -((15*ArcTanh[Cos[a + b*x]])/(8*b)) + (15*Sec[a + b*x])/(8*b) - (5*Csc[a + b*x]^2*Sec[a + b*x])/(8*b) - (Csc[a + b*x]^4*Sec[a + b*x])/(4*b)} -{Sec[a + b*x]^3/Sin[a + b*x]^5, x, 4, -((3*Cot[a + b*x]^2)/(2*b)) - Cot[a + b*x]^4/(4*b) + (3*Log[Tan[a + b*x]])/b + Tan[a + b*x]^2/(2*b)} -{Sec[a + b*x]^4/Sin[a + b*x]^5, x, 6, -((35*ArcTanh[Cos[a + b*x]])/(8*b)) + (35*Sec[a + b*x])/(8*b) + (35*Sec[a + b*x]^3)/(24*b) - (7*Csc[a + b*x]^2*Sec[a + b*x]^3)/(8*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(4*b)} -{Sec[a + b*x]^5/Sin[a + b*x]^5, x, 4, -((2*Cot[a + b*x]^2)/b) - Cot[a + b*x]^4/(4*b) + (6*Log[Tan[a + b*x]])/b + (2*Tan[a + b*x]^2)/b + Tan[a + b*x]^4/(4*b)} - - -{Cos[x]^2/Sin[x]^6, x, 3, (-(1/3))*Cot[x]^3 - Cot[x]^5/5} - - -{Cos[x]^3/Sin[x]^7, x, 3, Csc[x]^4/4 - Csc[x]^6/6} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]*(d*Cos[a + b*x])^(3/2), x, 2, -((2*(d*Cos[a + b*x])^(5/2))/(5*b*d))} -{Sin[a + b*x]*(d*Cos[a + b*x])^(1/2), x, 2, -((2*(d*Cos[a + b*x])^(3/2))/(3*b*d))} -{Sin[a + b*x]/(d*Cos[a + b*x])^(1/2), x, 2, -((2*Sqrt[d*Cos[a + b*x]])/(b*d))} -{Sin[a + b*x]/(d*Cos[a + b*x])^(3/2), x, 2, 2/(b*d*Sqrt[d*Cos[a + b*x]])} -{Sin[a + b*x]/(d*Cos[a + b*x])^(5/2), x, 2, 2/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{Sin[a + b*x]/(d*Cos[a + b*x])^(7/2), x, 2, 2/(5*b*d*(d*Cos[a + b*x])^(5/2))} -{Sin[a + b*x]/(d*Cos[a + b*x])^(9/2), x, 2, 2/(7*b*d*(d*Cos[a + b*x])^(7/2))} - - -{Sin[a + b*x]^2*(d*Cos[a + b*x])^(9/2), x, 5, (28*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(195*b*Sqrt[Cos[a + b*x]]) + (28*d^3*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(585*b) + (4*d*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(117*b) - (2*(d*Cos[a + b*x])^(11/2)*Sin[a + b*x])/(13*b*d)} -{Sin[a + b*x]^2*(d*Cos[a + b*x])^(7/2), x, 5, (20*d^4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(231*b*Sqrt[d*Cos[a + b*x]]) + (20*d^3*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(231*b) + (4*d*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(77*b) - (2*(d*Cos[a + b*x])^(9/2)*Sin[a + b*x])/(11*b*d)} -{Sin[a + b*x]^2*(d*Cos[a + b*x])^(5/2), x, 4, (4*d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(15*b*Sqrt[Cos[a + b*x]]) + (4*d*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(45*b) - (2*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(9*b*d)} -{Sin[a + b*x]^2*(d*Cos[a + b*x])^(3/2), x, 4, (4*d^2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(21*b*Sqrt[d*Cos[a + b*x]]) + (4*d*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(21*b) - (2*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(7*b*d)} -{Sin[a + b*x]^2*(d*Cos[a + b*x])^(1/2), x, 3, (4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*Sqrt[Cos[a + b*x]]) - (2*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(5*b*d)} -{Sin[a + b*x]^2/(d*Cos[a + b*x])^(1/2), x, 3, (4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*Sqrt[d*Cos[a + b*x]]) - (2*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(3*b*d)} -{Sin[a + b*x]^2/(d*Cos[a + b*x])^(3/2), x, 3, -((4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*d^2*Sqrt[Cos[a + b*x]])) + (2*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])} -{Sin[a + b*x]^2/(d*Cos[a + b*x])^(5/2), x, 3, -((4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*d^2*Sqrt[d*Cos[a + b*x]])) + (2*Sin[a + b*x])/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{Sin[a + b*x]^2/(d*Cos[a + b*x])^(7/2), x, 4, (4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*d^4*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(5*b*d*(d*Cos[a + b*x])^(5/2)) - (4*Sin[a + b*x])/(5*b*d^3*Sqrt[d*Cos[a + b*x]])} -{Sin[a + b*x]^2/(d*Cos[a + b*x])^(9/2), x, 4, -((4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(21*b*d^4*Sqrt[d*Cos[a + b*x]])) + (2*Sin[a + b*x])/(7*b*d*(d*Cos[a + b*x])^(7/2)) - (4*Sin[a + b*x])/(21*b*d^3*(d*Cos[a + b*x])^(3/2))} - - -{Sin[a + b*x]^3*(d*Cos[a + b*x])^(1/2), x, 3, -((2*(d*Cos[a + b*x])^(3/2))/(3*b*d)) + (2*(d*Cos[a + b*x])^(7/2))/(7*b*d^3)} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(1/2), x, 3, -((2*Sqrt[d*Cos[a + b*x]])/(b*d)) + (2*(d*Cos[a + b*x])^(5/2))/(5*b*d^3)} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(3/2), x, 3, 2/(b*d*Sqrt[d*Cos[a + b*x]]) + (2*(d*Cos[a + b*x])^(3/2))/(3*b*d^3)} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(5/2), x, 3, 2/(3*b*d*(d*Cos[a + b*x])^(3/2)) + (2*Sqrt[d*Cos[a + b*x]])/(b*d^3)} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(7/2), x, 3, 2/(5*b*d*(d*Cos[a + b*x])^(5/2)) - 2/(b*d^3*Sqrt[d*Cos[a + b*x]])} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(9/2), x, 3, 2/(7*b*d*(d*Cos[a + b*x])^(7/2)) - 2/(3*b*d^3*(d*Cos[a + b*x])^(3/2))} -{Sin[a + b*x]^3/(d*Cos[a + b*x])^(11/2), x, 3, 2/(9*b*d*(d*Cos[a + b*x])^(9/2)) - 2/(5*b*d^3*(d*Cos[a + b*x])^(5/2))} - - -{Sin[a + b*x]^4*(d*Cos[a + b*x])^(9/2), x, 6, (56*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(1105*b*Sqrt[Cos[a + b*x]]) + (56*d^3*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(3315*b) + (8*d*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(663*b) - (12*(d*Cos[a + b*x])^(11/2)*Sin[a + b*x])/(221*b*d) - (2*(d*Cos[a + b*x])^(11/2)*Sin[a + b*x]^3)/(17*b*d)} -{Sin[a + b*x]^4*(d*Cos[a + b*x])^(7/2), x, 6, (8*d^4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(231*b*Sqrt[d*Cos[a + b*x]]) + (8*d^3*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(231*b) + (8*d*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(385*b) - (4*(d*Cos[a + b*x])^(9/2)*Sin[a + b*x])/(55*b*d) - (2*(d*Cos[a + b*x])^(9/2)*Sin[a + b*x]^3)/(15*b*d)} -{Sin[a + b*x]^4*(d*Cos[a + b*x])^(5/2), x, 5, (8*d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(65*b*Sqrt[Cos[a + b*x]]) + (8*d*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(195*b) - (4*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x])/(39*b*d) - (2*(d*Cos[a + b*x])^(7/2)*Sin[a + b*x]^3)/(13*b*d)} -{Sin[a + b*x]^4*(d*Cos[a + b*x])^(3/2), x, 5, (8*d^2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(77*b*Sqrt[d*Cos[a + b*x]]) + (8*d*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(77*b) - (12*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(77*b*d) - (2*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x]^3)/(11*b*d)} -{Sin[a + b*x]^4*(d*Cos[a + b*x])^(1/2), x, 4, (8*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(15*b*Sqrt[Cos[a + b*x]]) - (4*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(15*b*d) - (2*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x]^3)/(9*b*d)} -{Sin[a + b*x]^4/(d*Cos[a + b*x])^(1/2), x, 4, (8*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(7*b*Sqrt[d*Cos[a + b*x]]) - (4*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(7*b*d) - (2*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^3)/(7*b*d)} -{Sin[a + b*x]^4/(d*Cos[a + b*x])^(3/2), x, 4, -((24*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*d^2*Sqrt[Cos[a + b*x]])) + (12*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(5*b*d^3) + (2*Sin[a + b*x]^3)/(b*d*Sqrt[d*Cos[a + b*x]])} -{Sin[a + b*x]^4/(d*Cos[a + b*x])^(5/2), x, 4, -((8*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*d^2*Sqrt[d*Cos[a + b*x]])) + (4*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(3*b*d^3) + (2*Sin[a + b*x]^3)/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2), x, 4, (24*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*d^4*Sqrt[Cos[a + b*x]]) - (12*Sin[a + b*x])/(5*b*d^3*Sqrt[d*Cos[a + b*x]]) + (2*Sin[a + b*x]^3)/(5*b*d*(d*Cos[a + b*x])^(5/2))} -{Sin[a + b*x]^4/(d*Cos[a + b*x])^(9/2), x, 4, (8*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(7*b*d^4*Sqrt[d*Cos[a + b*x]]) - (4*Sin[a + b*x])/(7*b*d^3*(d*Cos[a + b*x])^(3/2)) + (2*Sin[a + b*x]^3)/(7*b*d*(d*Cos[a + b*x])^(7/2))} - - -{Sin[a + b*x]^5*Cos[a + b*x]^(3/2), x, 3, -((2*Cos[a + b*x]^(5/2))/(5*b)) + (4*Cos[a + b*x]^(9/2))/(9*b) - (2*Cos[a + b*x]^(13/2))/(13*b)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Csc[a + b*x]*(d*Cos[a + b*x])^(9/2), x, 7, (d^(9/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b - (d^(9/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b + (2*d^3*(d*Cos[a + b*x])^(3/2))/(3*b) + (2*d*(d*Cos[a + b*x])^(7/2))/(7*b)} -{Csc[a + b*x]*(d*Cos[a + b*x])^(7/2), x, 7, -((d^(7/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b) - (d^(7/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b + (2*d^3*Sqrt[d*Cos[a + b*x]])/b + (2*d*(d*Cos[a + b*x])^(5/2))/(5*b)} -{Csc[a + b*x]*(d*Cos[a + b*x])^(5/2), x, 6, (d^(5/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b - (d^(5/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b + (2*d*(d*Cos[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]*(d*Cos[a + b*x])^(3/2), x, 6, -((d^(3/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b) - (d^(3/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b + (2*d*Sqrt[d*Cos[a + b*x]])/b} -{Csc[a + b*x]*(d*Cos[a + b*x])^(1/2), x, 5, (Sqrt[d]*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b - (Sqrt[d]*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/b} -{Csc[a + b*x]/(d*Cos[a + b*x])^(1/2), x, 5, -(ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*Sqrt[d])) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*Sqrt[d])} -{Csc[a + b*x]/(d*Cos[a + b*x])^(3/2), x, 6, ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(3/2)) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(3/2)) + 2/(b*d*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]/(d*Cos[a + b*x])^(5/2), x, 6, -(ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(5/2))) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(5/2)) + 2/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{Csc[a + b*x]/(d*Cos[a + b*x])^(7/2), x, 7, ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(7/2)) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(7/2)) + 2/(5*b*d*(d*Cos[a + b*x])^(5/2)) + 2/(b*d^3*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]/(d*Cos[a + b*x])^(9/2), x, 7, -(ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2))) - ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]]/(b*d^(9/2)) + 2/(7*b*d*(d*Cos[a + b*x])^(7/2)) + 2/(3*b*d^3*(d*Cos[a + b*x])^(3/2))} - - -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(11/2), x, 5, -((d*(d*Cos[a + b*x])^(9/2)*Csc[a + b*x])/b) - (15*d^6*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(7*b*Sqrt[d*Cos[a + b*x]]) - (15*d^5*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(7*b) - (9*d^3*(d*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(7*b)} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(9/2), x, 4, -((d*(d*Cos[a + b*x])^(7/2)*Csc[a + b*x])/b) - (21*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*Sqrt[Cos[a + b*x]]) - (7*d^3*(d*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(5*b)} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(7/2), x, 4, -((d*(d*Cos[a + b*x])^(5/2)*Csc[a + b*x])/b) - (5*d^4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*Sqrt[d*Cos[a + b*x]]) - (5*d^3*Sqrt[d*Cos[a + b*x]]*Sin[a + b*x])/(3*b)} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(5/2), x, 3, -((d*(d*Cos[a + b*x])^(3/2)*Csc[a + b*x])/b) - (3*d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*Sqrt[Cos[a + b*x]])} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(3/2), x, 3, -((d*Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/b) - (d^2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(b*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^(1/2), x, 3, -(((d*Cos[a + b*x])^(3/2)*Csc[a + b*x])/(b*d)) - (Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*Sqrt[Cos[a + b*x]])} -{Csc[a + b*x]^2/(d*Cos[a + b*x])^(1/2), x, 3, -((Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/(b*d)) + (Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(b*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^2/(d*Cos[a + b*x])^(3/2), x, 4, -(Csc[a + b*x]/(b*d*Sqrt[d*Cos[a + b*x]])) - (3*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*d^2*Sqrt[Cos[a + b*x]]) + (3*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^2/(d*Cos[a + b*x])^(5/2), x, 4, -(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(3/2))) + (5*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*d^2*Sqrt[d*Cos[a + b*x]]) + (5*Sin[a + b*x])/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2), x, 5, -(Csc[a + b*x]/(b*d*(d*Cos[a + b*x])^(5/2))) - (21*Sqrt[d*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*d^4*Sqrt[Cos[a + b*x]]) + (7*Sin[a + b*x])/(5*b*d*(d*Cos[a + b*x])^(5/2)) + (21*Sin[a + b*x])/(5*b*d^3*Sqrt[d*Cos[a + b*x]])} - - -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(11/2), x, 8, (9*d^(11/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) + (9*d^(11/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (9*d^5*Sqrt[d*Cos[a + b*x]])/(2*b) - (9*d^3*(d*Cos[a + b*x])^(5/2))/(10*b) - (d*(d*Cos[a + b*x])^(9/2)*Csc[a + b*x]^2)/(2*b)} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(9/2), x, 7, -((7*d^(9/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b)) + (7*d^(9/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (7*d^3*(d*Cos[a + b*x])^(3/2))/(6*b) - (d*(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2)/(2*b)} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(7/2), x, 7, (5*d^(7/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) + (5*d^(7/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (5*d^3*Sqrt[d*Cos[a + b*x]])/(2*b) - (d*(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^2)/(2*b)} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(5/2), x, 6, -((3*d^(5/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b)) + (3*d^(5/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (d*(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2)/(2*b)} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(3/2), x, 6, (d^(3/2)*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) + (d^(3/2)*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (d*Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^2)/(2*b)} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^(1/2), x, 6, (Sqrt[d]*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - (Sqrt[d]*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b) - ((d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2)/(2*b*d)} -{Csc[a + b*x]^3/(d*Cos[a + b*x])^(1/2), x, 6, -((3*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*Sqrt[d])) - (3*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*Sqrt[d]) - (Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^2)/(2*b*d)} -{Csc[a + b*x]^3/(d*Cos[a + b*x])^(3/2), x, 7, (5*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(3/2)) - (5*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(3/2)) + 5/(2*b*d*Sqrt[d*Cos[a + b*x]]) - Csc[a + b*x]^2/(2*b*d*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^3/(d*Cos[a + b*x])^(5/2), x, 7, -((7*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(5/2))) - (7*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(5/2)) + 7/(6*b*d*(d*Cos[a + b*x])^(3/2)) - Csc[a + b*x]^2/(2*b*d*(d*Cos[a + b*x])^(3/2))} -{Csc[a + b*x]^3/(d*Cos[a + b*x])^(7/2), x, 8, (9*ArcTan[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(7/2)) - (9*ArcTanh[Sqrt[d*Cos[a + b*x]]/Sqrt[d]])/(4*b*d^(7/2)) + 9/(10*b*d*(d*Cos[a + b*x])^(5/2)) + 9/(2*b*d^3*Sqrt[d*Cos[a + b*x]]) - Csc[a + b*x]^2/(2*b*d*(d*Cos[a + b*x])^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (b Cos[e+f x])^(n/5)*) - - -{Sin[a + b*x]*(d*Cos[a + b*x])^(1/5), x, 2, -((5*(d*Cos[a + b*x])^(6/5))/(6*b*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) Cos[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[x]^3*Sqrt[Sin[x]], x, 3, (2/3)*Sin[x]^(3/2) - (2/7)*Sin[x]^(7/2)} - - -{Cos[x]^3*Sin[x]^(3/2), x, 3, (2/5)*Sin[x]^(5/2) - (2/9)*Sin[x]^(9/2)} - - -{Cos[x]^3*Sin[x]^(5/2), x, 3, (2/7)*Sin[x]^(7/2) - (2/11)*Sin[x]^(11/2)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[x]^3/Sqrt[Sin[x]], x, 3, 2*Sqrt[Sin[x]] - (2/5)*Sin[x]^(5/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) (b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(9/2), x, 4, (7*d^3*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(30*b*c) + (d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3/2))/(5*b*c) + (7*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(20*b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(5/2), x, 3, (d*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(3*b*c) + (d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(2*b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(1/2), x, 2, (Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(3/2), x, 3, (2*(c*Sin[a + b*x])^(3/2))/(b*c*d*Sqrt[d*Cos[a + b*x]]) - (2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(7/2), x, 4, (2*(c*Sin[a + b*x])^(3/2))/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (4*(c*Sin[a + b*x])^(3/2))/(5*b*c*d^3*Sqrt[d*Cos[a + b*x]]) - (4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(5*b*d^4*Sqrt[Sin[2*a + 2*b*x]])} - -{(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(3/2), x, 11, -((Sqrt[c]*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(4*Sqrt[2]*b)) + (Sqrt[c]*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(4*Sqrt[2]*b) + (Sqrt[c]*d^(3/2)*Log[Sqrt[c] - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(8*Sqrt[2]*b) - (Sqrt[c]*d^(3/2)*Log[Sqrt[c] + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(8*Sqrt[2]*b) + (d*Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2))/(2*b*c)} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(1/2), x, 10, -((Sqrt[c]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*Sqrt[d])) + (Sqrt[c]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*Sqrt[d]) + (Sqrt[c]*Log[Sqrt[c] - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(2*Sqrt[2]*b*Sqrt[d]) - (Sqrt[c]*Log[Sqrt[c] + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(2*Sqrt[2]*b*Sqrt[d])} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(5/2), x, 1, (2*(c*Sin[a + b*x])^(3/2))/(3*b*c*d*(d*Cos[a + b*x])^(3/2))} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(9/2), x, 2, (2*(c*Sin[a + b*x])^(3/2))/(7*b*c*d*(d*Cos[a + b*x])^(7/2)) + (8*(c*Sin[a + b*x])^(3/2))/(21*b*c*d^3*(d*Cos[a + b*x])^(3/2))} -{(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(13/2), x, 3, (2*(c*Sin[a + b*x])^(3/2))/(11*b*c*d*(d*Cos[a + b*x])^(11/2)) + (16*(c*Sin[a + b*x])^(3/2))/(77*b*c*d^3*(d*Cos[a + b*x])^(7/2)) + (64*(c*Sin[a + b*x])^(3/2))/(231*b*c*d^5*(d*Cos[a + b*x])^(3/2))} - - -{(c*Sin[a + b*x])^(3/2)*(d*Cos[a + b*x])^(3/2), x, 4, (c*d*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b) - (c*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3*b*d) + (c^2*d^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(1/2), x, 3, -((c*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(b*d)) + (c^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(5/2), x, 3, (2*c*Sqrt[c*Sin[a + b*x]])/(3*b*d*(d*Cos[a + b*x])^(3/2)) - (c^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(9/2), x, 4, (2*c*Sqrt[c*Sin[a + b*x]])/(7*b*d*(d*Cos[a + b*x])^(7/2)) - (2*c*Sqrt[c*Sin[a + b*x]])/(21*b*d^3*(d*Cos[a + b*x])^(3/2)) - (2*c^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(21*b*d^4*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} - -{(c*Sin[a + b*x])^(3/2)*(d*Cos[a + b*x])^(1/2), x, 11, (c^(3/2)*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(4*Sqrt[2]*b) - (c^(3/2)*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(4*Sqrt[2]*b) - (c^(3/2)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(8*Sqrt[2]*b) + (c^(3/2)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(8*Sqrt[2]*b) - (c*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])/(2*b*d)} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(3/2), x, 11, -((c^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*d^(3/2))) + (c^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*d^(3/2)) + (c^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(2*Sqrt[2]*b*d^(3/2)) - (c^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(2*Sqrt[2]*b*d^(3/2)) + (2*c*Sqrt[c*Sin[a + b*x]])/(b*d*Sqrt[d*Cos[a + b*x]])} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(7/2), x, 1, (2*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*(d*Cos[a + b*x])^(5/2))} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(11/2), x, 3, (2*c*Sqrt[c*Sin[a + b*x]])/(9*b*d*(d*Cos[a + b*x])^(9/2)) - (2*c*Sqrt[c*Sin[a + b*x]])/(45*b*d^3*(d*Cos[a + b*x])^(5/2)) - (8*c*Sqrt[c*Sin[a + b*x]])/(45*b*d^5*Sqrt[d*Cos[a + b*x]])} -{(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(15/2), x, 4, (2*c*Sqrt[c*Sin[a + b*x]])/(13*b*d*(d*Cos[a + b*x])^(13/2)) - (2*c*Sqrt[c*Sin[a + b*x]])/(117*b*d^3*(d*Cos[a + b*x])^(9/2)) - (16*c*Sqrt[c*Sin[a + b*x]])/(585*b*d^5*(d*Cos[a + b*x])^(5/2)) - (64*c*Sqrt[c*Sin[a + b*x]])/(585*b*d^7*Sqrt[d*Cos[a + b*x]])} - - -{(c*Sin[a + b*x])^(5/2)*(d*Cos[a + b*x])^(9/2), x, 5, (c*d^3*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(20*b) + (3*c*d*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3/2))/(70*b) - (c*(d*Cos[a + b*x])^(11/2)*(c*Sin[a + b*x])^(3/2))/(7*b*d) + (3*c^2*d^4*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(40*b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(5/2)*(d*Cos[a + b*x])^(5/2), x, 4, (c*d*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(10*b) - (c*(d*Cos[a + b*x])^(7/2)*(c*Sin[a + b*x])^(3/2))/(5*b*d) + (3*c^2*d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(20*b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(5/2)*(d*Cos[a + b*x])^(1/2), x, 3, -((c*(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2))/(3*b*d)) + (c^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(2*b*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(3/2), x, 3, (2*c*(c*Sin[a + b*x])^(3/2))/(b*d*Sqrt[d*Cos[a + b*x]]) - (3*c^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(7/2), x, 4, (2*c*(c*Sin[a + b*x])^(3/2))/(5*b*d*(d*Cos[a + b*x])^(5/2)) - (6*c*(c*Sin[a + b*x])^(3/2))/(5*b*d^3*Sqrt[d*Cos[a + b*x]]) + (6*c^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(5*b*d^4*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(11/2), x, 5, (2*c*(c*Sin[a + b*x])^(3/2))/(9*b*d*(d*Cos[a + b*x])^(9/2)) - (2*c*(c*Sin[a + b*x])^(3/2))/(15*b*d^3*(d*Cos[a + b*x])^(5/2)) - (4*c*(c*Sin[a + b*x])^(3/2))/(15*b*d^5*Sqrt[d*Cos[a + b*x]]) + (4*c^2*Sqrt[d*Cos[a + b*x]]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[c*Sin[a + b*x]])/(15*b*d^6*Sqrt[Sin[2*a + 2*b*x]])} - -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(1/2), x, 11, -((3*c^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(4*Sqrt[2]*b*Sqrt[d])) + (3*c^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(4*Sqrt[2]*b*Sqrt[d]) + (3*c^(5/2)*Log[Sqrt[c] - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(8*Sqrt[2]*b*Sqrt[d]) - (3*c^(5/2)*Log[Sqrt[c] + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(8*Sqrt[2]*b*Sqrt[d]) - (c*Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2))/(2*b*d)} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(5/2), x, 11, (c^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*d^(5/2)) - (c^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/(Sqrt[c]*Sqrt[d*Cos[a + b*x]])])/(Sqrt[2]*b*d^(5/2)) - (c^(5/2)*Log[Sqrt[c] - (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(2*Sqrt[2]*b*d^(5/2)) + (c^(5/2)*Log[Sqrt[c] + (Sqrt[2]*Sqrt[d]*Sqrt[c*Sin[a + b*x]])/Sqrt[d*Cos[a + b*x]] + Sqrt[c]*Tan[a + b*x]])/(2*Sqrt[2]*b*d^(5/2)) + (2*c*(c*Sin[a + b*x])^(3/2))/(3*b*d*(d*Cos[a + b*x])^(3/2))} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(9/2), x, 1, (2*(c*Sin[a + b*x])^(7/2))/(7*b*c*d*(d*Cos[a + b*x])^(7/2))} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(13/2), x, 3, (2*c*(c*Sin[a + b*x])^(3/2))/(11*b*d*(d*Cos[a + b*x])^(11/2)) - (6*c*(c*Sin[a + b*x])^(3/2))/(77*b*d^3*(d*Cos[a + b*x])^(7/2)) - (8*c*(c*Sin[a + b*x])^(3/2))/(77*b*d^5*(d*Cos[a + b*x])^(3/2))} -{(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(17/2), x, 4, (2*c*(c*Sin[a + b*x])^(3/2))/(15*b*d*(d*Cos[a + b*x])^(15/2)) - (2*c*(c*Sin[a + b*x])^(3/2))/(55*b*d^3*(d*Cos[a + b*x])^(11/2)) - (16*c*(c*Sin[a + b*x])^(3/2))/(385*b*d^5*(d*Cos[a + b*x])^(7/2)) - (64*c*(c*Sin[a + b*x])^(3/2))/(1155*b*d^7*(d*Cos[a + b*x])^(3/2))} - - -{Sin[a + b*x]^(7/2)/Cos[a + b*x]^(7/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b) - ArcTan[1 + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b) - Log[1 + Cot[a + b*x] - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b) + Log[1 + Cot[a + b*x] + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b) - (2*Sqrt[Sin[a + b*x]])/(b*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x]^(5/2))/(5*b*Cos[a + b*x]^(5/2))} - - -{Sin[x]^(3/2)/Cos[x]^(7/2), x, 1, (2*Sin[x]^(5/2))/(5*Cos[x]^(5/2))} -{Sqrt[Sin[x]]/Sqrt[Cos[x]], x, 10, -(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2] + Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2])} -{Sin[x]^(5/2)/Sqrt[Cos[x]], x, 11, -((3*ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]])/(4*Sqrt[2])) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]])/(4*Sqrt[2]) + (3*Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (3*Log[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (1/2)*Sqrt[Cos[x]]*Sin[x]^(3/2)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(7/2), x, 4, (5*d^3*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(6*b*c) + (d*(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]])/(3*b*c) + (5*d^4*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(3/2), x, 3, (d*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])/(b*c) + (d^2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(1/2), x, 2, (EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(5/2), x, 3, (2*Sqrt[c*Sin[a + b*x]])/(3*b*c*d*(d*Cos[a + b*x])^(3/2)) + (2*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(9/2), x, 4, (2*Sqrt[c*Sin[a + b*x]])/(7*b*c*d*(d*Cos[a + b*x])^(7/2)) + (4*Sqrt[c*Sin[a + b*x]])/(7*b*c*d^3*(d*Cos[a + b*x])^(3/2)) + (4*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]])/(7*b*d^4*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])} - -{1/(c*Sin[a + b*x])^(1/2)*(d*Cos[a + b*x])^(1/2), x, 10, (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*Sqrt[c]) - (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/(Sqrt[d]*Sqrt[c*Sin[a + b*x]])])/(Sqrt[2]*b*Sqrt[c]) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] - (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(2*Sqrt[2]*b*Sqrt[c]) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[a + b*x] + (Sqrt[2]*Sqrt[c]*Sqrt[d*Cos[a + b*x]])/Sqrt[c*Sin[a + b*x]]])/(2*Sqrt[2]*b*Sqrt[c])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(3/2), x, 1, (2*Sqrt[c*Sin[a + b*x]])/(b*c*d*Sqrt[d*Cos[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(7/2), x, 2, (2*Sqrt[c*Sin[a + b*x]])/(5*b*c*d*(d*Cos[a + b*x])^(5/2)) + (8*Sqrt[c*Sin[a + b*x]])/(5*b*c*d^3*Sqrt[d*Cos[a + b*x]])} -{1/(c*Sin[a + b*x])^(1/2)/(d*Cos[a + b*x])^(11/2), x, 3, (2*Sqrt[c*Sin[a + b*x]])/(9*b*c*d*(d*Cos[a + b*x])^(9/2)) + (16*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^3*(d*Cos[a + b*x])^(5/2)) + (64*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^5*Sqrt[d*Cos[a + b*x]])} - - -{Cos[a + b*x]^(1/2)/Sin[a + b*x]^(1/2), x, 10, ArcTan[1 - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b) - ArcTan[1 + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b) - Log[1 + Cot[a + b*x] - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b) + Log[1 + Cot[a + b*x] + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b)} - - -{Cos[a + b*x]^(3/2)/Sin[a + b*x]^(3/2), x, 11, ArcTan[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b) - ArcTan[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b) - Log[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]*b) + Log[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]*b) - (2*Sqrt[Cos[a + b*x]])/(b*Sqrt[Sin[a + b*x]])} - - -{Cos[a + b*x]^(5/2)/Sin[a + b*x]^(5/2), x, 11, -(ArcTan[1 - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b)) + ArcTan[1 + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(Sqrt[2]*b) + Log[1 + Cot[a + b*x] - (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b) - Log[1 + Cot[a + b*x] + (Sqrt[2]*Sqrt[Cos[a + b*x]])/Sqrt[Sin[a + b*x]]]/(2*Sqrt[2]*b) - (2*Cos[a + b*x]^(3/2))/(3*b*Sin[a + b*x]^(3/2))} - - -{Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b)) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b) + Log[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]*b) - Log[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]*b) - (2*Cos[a + b*x]^(5/2))/(5*b*Sin[a + b*x]^(5/2)) + (2*Sqrt[Cos[a + b*x]])/(b*Sqrt[Sin[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/3) Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[e + f*x]^4*(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-3/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3))/(4*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^2*(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-1/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3))/(4*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^0*(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3))/(4*b*f*Sqrt[Cos[e + f*x]^2])} -{Sec[e + f*x]^2*(b*Sin[e + f*x])^(1/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[2/3, 3/2, 5/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(4/3))/(4*b*f)} -{Sec[e + f*x]^4*(b*Sin[e + f*x])^(1/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[2/3, 5/2, 5/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(4/3))/(4*b*f)} - - -{Cos[e + f*x]^4*(b*Sin[e + f*x])^(5/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-3/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(8/3))/(8*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^2*(b*Sin[e + f*x])^(5/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(8/3))/(8*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^0*(b*Sin[e + f*x])^(5/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(8/3))/(8*b*f*Sqrt[Cos[e + f*x]^2])} -{Sec[e + f*x]^2*(b*Sin[e + f*x])^(5/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[4/3, 3/2, 7/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(8/3))/(8*b*f)} -{Sec[e + f*x]^4*(b*Sin[e + f*x])^(5/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[4/3, 5/2, 7/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(8/3))/(8*b*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[e + f*x]^4/(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-3/2, 1/3, 4/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))/(2*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^2/(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[-1/2, 1/3, 4/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))/(2*b*f*Sqrt[Cos[e + f*x]^2])} -{Cos[e + f*x]^0/(b*Sin[e + f*x])^(1/3), x, 1, (3*Cos[e + f*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))/(2*b*f*Sqrt[Cos[e + f*x]^2])} -{Sec[e + f*x]^2/(b*Sin[e + f*x])^(1/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 3/2, 4/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(2/3))/(2*b*f)} -{Sec[e + f*x]^4/(b*Sin[e + f*x])^(1/3), x, 1, (3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 5/2, 4/3, Sin[e + f*x]^2]*Sec[e + f*x]*(b*Sin[e + f*x])^(2/3))/(2*b*f)} - - -{Cos[e + f*x]^4/(b*Sin[e + f*x])^(5/3), x, 1, (-3*Cos[e + f*x]*Hypergeometric2F1[-3/2, -1/3, 2/3, Sin[e + f*x]^2])/(2*b*f*Sqrt[Cos[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))} -{Cos[e + f*x]^2/(b*Sin[e + f*x])^(5/3), x, 1, (-3*Cos[e + f*x]*Hypergeometric2F1[-1/2, -1/3, 2/3, Sin[e + f*x]^2])/(2*b*f*Sqrt[Cos[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))} -{Cos[e + f*x]^0/(b*Sin[e + f*x])^(5/3), x, 1, (-3*Cos[e + f*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sin[e + f*x]^2])/(2*b*f*Sqrt[Cos[e + f*x]^2]*(b*Sin[e + f*x])^(2/3))} -{Sec[e + f*x]^2/(b*Sin[e + f*x])^(5/3), x, 1, (-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 3/2, 2/3, Sin[e + f*x]^2]*Sec[e + f*x])/(2*b*f*(b*Sin[e + f*x])^(2/3))} -{Sec[e + f*x]^4/(b*Sin[e + f*x])^(5/3), x, 1, (-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 5/2, 2/3, Sin[e + f*x]^2]*Sec[e + f*x])/(2*b*f*(b*Sin[e + f*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/3) (b Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]^(1/3)/Cos[a + b*x]^(1/3), x, 8, -((Sqrt[3]*ArcTan[(1 - (2*Sin[a + b*x]^(2/3))/Cos[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)]/(2*b) + Log[1 - Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3) + Sin[a + b*x]^(4/3)/Cos[a + b*x]^(4/3)]/(4*b)} - - -{Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3), x, 11, -(ArcTan[Sqrt[3] - (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)]/(2*b)) + ArcTan[Sqrt[3] + (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)]/(2*b) + ArcTan[Sin[a + b*x]^(1/3)/Cos[a + b*x]^(1/3)]/b + (Sqrt[3]*Log[1 - (Sqrt[3]*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)])/(4*b) - (Sqrt[3]*Log[1 + (Sqrt[3]*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)])/(4*b)} - - -{Sin[a + b*x]^(4/3)/Cos[a + b*x]^(4/3), x, 12, -(ArcTan[Sqrt[3] - (2*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)]/(2*b)) + ArcTan[Sqrt[3] + (2*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)]/(2*b) + ArcTan[Cos[a + b*x]^(1/3)/Sin[a + b*x]^(1/3)]/b + (Sqrt[3]*Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3) - (Sqrt[3]*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)])/(4*b) - (Sqrt[3]*Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3) + (Sqrt[3]*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)])/(4*b) + (3*Sin[a + b*x]^(1/3))/(b*Cos[a + b*x]^(1/3))} - - -{Sin[a + b*x]^(5/3)/Cos[a + b*x]^(5/3), x, 9, -((Sqrt[3]*ArcTan[(1 - (2*Cos[a + b*x]^(2/3))/Sin[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) + Log[1 + Cos[a + b*x]^(4/3)/Sin[a + b*x]^(4/3) - Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(4*b) - Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(2*b) + (3*Sin[a + b*x]^(2/3))/(2*b*Cos[a + b*x]^(2/3))} - - -{Sin[a + b*x]^(7/3)/Cos[a + b*x]^(7/3), x, 9, (Sqrt[3]*ArcTan[(1 - (2*Sin[a + b*x]^(2/3))/Cos[a + b*x]^(2/3))/Sqrt[3]])/(2*b) + Log[1 + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)]/(2*b) - Log[1 - Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3) + Sin[a + b*x]^(4/3)/Cos[a + b*x]^(4/3)]/(4*b) + (3*Sin[a + b*x]^(4/3))/(4*b*Cos[a + b*x]^(4/3))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[a + b*x]^(1/3)/Sin[a + b*x]^(1/3), x, 8, (Sqrt[3]*ArcTan[(1 - (2*Cos[a + b*x]^(2/3))/Sin[a + b*x]^(2/3))/Sqrt[3]])/(2*b) - Log[1 + Cos[a + b*x]^(4/3)/Sin[a + b*x]^(4/3) - Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(4*b) + Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(2*b)} - - -{Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3), x, 11, ArcTan[Sqrt[3] - (2*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)]/(2*b) - ArcTan[Sqrt[3] + (2*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)]/(2*b) - ArcTan[Cos[a + b*x]^(1/3)/Sin[a + b*x]^(1/3)]/b - (Sqrt[3]*Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3) - (Sqrt[3]*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)])/(4*b) + (Sqrt[3]*Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3) + (Sqrt[3]*Cos[a + b*x]^(1/3))/Sin[a + b*x]^(1/3)])/(4*b)} - - -{Cos[a + b*x]^(4/3)/Sin[a + b*x]^(4/3), x, 12, ArcTan[Sqrt[3] - (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)]/(2*b) - ArcTan[Sqrt[3] + (2*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3)]/(2*b) - ArcTan[Sin[a + b*x]^(1/3)/Cos[a + b*x]^(1/3)]/b - (Sqrt[3]*Log[1 - (Sqrt[3]*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)])/(4*b) + (Sqrt[3]*Log[1 + (Sqrt[3]*Sin[a + b*x]^(1/3))/Cos[a + b*x]^(1/3) + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)])/(4*b) - (3*Cos[a + b*x]^(1/3))/(b*Sin[a + b*x]^(1/3))} - - -{Cos[a + b*x]^(5/3)/Sin[a + b*x]^(5/3), x, 9, (Sqrt[3]*ArcTan[(1 - (2*Sin[a + b*x]^(2/3))/Cos[a + b*x]^(2/3))/Sqrt[3]])/(2*b) + Log[1 + Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3)]/(2*b) - Log[1 - Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3) + Sin[a + b*x]^(4/3)/Cos[a + b*x]^(4/3)]/(4*b) - (3*Cos[a + b*x]^(2/3))/(2*b*Sin[a + b*x]^(2/3))} - - -{Cos[a + b*x]^(7/3)/Sin[a + b*x]^(7/3), x, 9, -((Sqrt[3]*ArcTan[(1 - (2*Cos[a + b*x]^(2/3))/Sin[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) + Log[1 + Cos[a + b*x]^(4/3)/Sin[a + b*x]^(4/3) - Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(4*b) - Log[1 + Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3)]/(2*b) - (3*Cos[a + b*x]^(4/3))/(4*b*Sin[a + b*x]^(4/3))} - - -{Cos[x]^(2/3)/Sin[x]^(8/3), x, 1, (-3*Cos[x]^(5/3))/(5*Sin[x]^(5/3))} -{Sin[x]^(2/3)/Cos[x]^(8/3), x, 1, (3*Sin[x]^(5/3))/(5*Cos[x]^(5/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n with m symbolic*) - - -{(Sin[e + f*x])^m*(Cos[e + f*x])^n, x, 1, -((Cos[e + f*x]^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x]^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 + n)))} -{(Sin[e + f*x])^m*(d*Cos[e + f*x])^n, x, 1, -(((d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x]^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(d*f*(1 + n)))} -{(b*Sin[e + f*x])^m*(Cos[e + f*x])^n, x, 1, -((b*Cos[e + f*x]^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(b*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 + n)))} -{(b*Sin[e + f*x])^m*(d*Cos[e + f*x])^n, x, 1, -((b*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(b*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(d*f*(1 + n)))} - - -{Cos[a + b*x]^5*(c*Sin[a + b*x])^m, x, 3, (c*Sin[a + b*x])^(1 + m)/(b*c*(1 + m)) - (2*(c*Sin[a + b*x])^(3 + m))/(b*c^3*(3 + m)) + (c*Sin[a + b*x])^(5 + m)/(b*c^5*(5 + m))} -{Cos[a + b*x]^3*(c*Sin[a + b*x])^m, x, 3, (c*Sin[a + b*x])^(1 + m)/(b*c*(1 + m)) - (c*Sin[a + b*x])^(3 + m)/(b*c^3*(3 + m))} -{Cos[a + b*x]^1*(c*Sin[a + b*x])^m, x, 2, (c*Sin[a + b*x])^(1 + m)/(b*c*(1 + m))} -{Sec[a + b*x]^1*(c*Sin[a + b*x])^m, x, 2, (Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} -{Sec[a + b*x]^3*(c*Sin[a + b*x])^m, x, 2, (Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} - -{Cos[a + b*x]^4*(c*Sin[a + b*x])^m, x, 1, (Cos[a + b*x]*Hypergeometric2F1[-(3/2), (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*Sqrt[Cos[a + b*x]^2])} -{Cos[a + b*x]^2*(c*Sin[a + b*x])^m, x, 1, (Cos[a + b*x]*Hypergeometric2F1[-(1/2), (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*Sqrt[Cos[a + b*x]^2])} -{Cos[a + b*x]^0*(c*Sin[a + b*x])^m, x, 1, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*Sqrt[Cos[a + b*x]^2])} -{Sec[a + b*x]^2*(c*Sin[a + b*x])^m, x, 1, (Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*Sec[a + b*x]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} -{Sec[a + b*x]^4*(c*Sin[a + b*x])^m, x, 1, (Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*Sec[a + b*x]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} - - -{(c*Sin[a + b*x])^m*(d*Cos[a + b*x])^(3/2), x, 1, (d*Sqrt[d*Cos[a + b*x]]*Hypergeometric2F1[-(1/4), (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*(Cos[a + b*x]^2)^(1/4))} -{(c*Sin[a + b*x])^m*(d*Cos[a + b*x])^(1/2), x, 1, (d*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*Sqrt[d*Cos[a + b*x]])} -{(c*Sin[a + b*x])^m/(d*Cos[a + b*x])^(1/2), x, 1, (d*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m)*(d*Cos[a + b*x])^(3/2))} -{(c*Sin[a + b*x])^m/(d*Cos[a + b*x])^(3/2), x, 1, ((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*d*(1 + m)*Sqrt[d*Cos[a + b*x]])} -{(c*Sin[a + b*x])^m/(d*Cos[a + b*x])^(5/2), x, 1, ((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*d*(1 + m)*(d*Cos[a + b*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Cos[e+f x])^n with n symbolic*) - - -{Sin[a + b*x]^5*(d*Cos[a + b*x])^n, x, 3, -((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n))) + (2*(d*Cos[a + b*x])^(3 + n))/(b*d^3*(3 + n)) - (d*Cos[a + b*x])^(5 + n)/(b*d^5*(5 + n))} -{Sin[a + b*x]^3*(d*Cos[a + b*x])^n, x, 3, -((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n))) + (d*Cos[a + b*x])^(3 + n)/(b*d^3*(3 + n))} -{Sin[a + b*x]^1*(d*Cos[a + b*x])^n, x, 2, -((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n)))} -{Csc[a + b*x]^1*(d*Cos[a + b*x])^n, x, 2, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2])/(b*d*(1 + n)))} -{Csc[a + b*x]^3*(d*Cos[a + b*x])^n, x, 2, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2])/(b*d*(1 + n)))} -{Csc[a + b*x]^5*(d*Cos[a + b*x])^n, x, 2, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2])/(b*d*(1 + n)))} - -{Sin[a + b*x]^4*(d*Cos[a + b*x])^n, x, 1, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[-(3/2), (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*d*(1 + n)*Sqrt[Sin[a + b*x]^2]))} -{Sin[a + b*x]^2*(d*Cos[a + b*x])^n, x, 1, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[-(1/2), (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*d*(1 + n)*Sqrt[Sin[a + b*x]^2]))} -{Sin[a + b*x]^0*(d*Cos[a + b*x])^n, x, 1, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*d*(1 + n)*Sqrt[Sin[a + b*x]^2]))} -{Csc[a + b*x]^2*(d*Cos[a + b*x])^n, x, 1, -(((d*Cos[a + b*x])^(1 + n)*Csc[a + b*x]*Hypergeometric2F1[3/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2])/(b*d*(1 + n)))} -{Csc[a + b*x]^4*(d*Cos[a + b*x])^n, x, 1, -(((d*Cos[a + b*x])^(1 + n)*Csc[a + b*x]*Hypergeometric2F1[5/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2])/(b*d*(1 + n)))} - - -{(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(5/2), x, 1, -((c*(d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[-(3/4), (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(c*Sin[a + b*x])^(3/2))/(b*d*(1 + n)*(Sin[a + b*x]^2)^(3/4)))} -{(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(3/2), x, 1, -((c*(d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[-(1/4), (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]])/(b*d*(1 + n)*(Sin[a + b*x]^2)^(1/4)))} -{(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(1/2), x, 1, -((c*(d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/4))/(b*d*(1 + n)*Sqrt[c*Sin[a + b*x]]))} -{(d*Cos[a + b*x])^n/(c*Sin[a + b*x])^(1/2), x, 1, -((c*(d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[3/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(3/4))/(b*d*(1 + n)*(c*Sin[a + b*x])^(3/2)))} -{(d*Cos[a + b*x])^n/(c*Sin[a + b*x])^(3/2), x, 1, -(((d*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[5/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/4))/(b*c*d*(1 + n)*Sqrt[c*Sin[a + b*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (b Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^7, x, 3, (2*b^7)/(13*f*(b*Sec[e + f*x])^(13/2)) - (2*b^5)/(3*f*(b*Sec[e + f*x])^(9/2)) + (6*b^3)/(5*f*(b*Sec[e + f*x])^(5/2)) - (2*b)/(f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^5, x, 3, (-2*b^5)/(9*f*(b*Sec[e + f*x])^(9/2)) + (4*b^3)/(5*f*(b*Sec[e + f*x])^(5/2)) - (2*b)/(f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^3, x, 3, (2*b^3)/(5*f*(b*Sec[e + f*x])^(5/2)) - (2*b)/(f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^1, x, 2, (-2*b)/(f*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^1*Sqrt[b*Sec[e + f*x]], x, 5, (Sqrt[b]*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f - (Sqrt[b]*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f} -{Csc[e + f*x]^3*Sqrt[b*Sec[e + f*x]], x, 6, (3*Sqrt[b]*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) - (3*Sqrt[b]*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) - (Cot[e + f*x]^2*(b*Sec[e + f*x])^(3/2))/(2*b*f)} -{Csc[e + f*x]^5*Sqrt[b*Sec[e + f*x]], x, 7, (21*Sqrt[b]*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*f) - (21*Sqrt[b]*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*f) - (7*Cot[e + f*x]^2*(b*Sec[e + f*x])^(3/2))/(16*b*f) - (Cot[e + f*x]^4*(b*Sec[e + f*x])^(7/2))/(4*b^3*f)} - -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^6, x, 5, (80*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(77*f) - (40*b*Sin[e + f*x])/(77*f*Sqrt[b*Sec[e + f*x]]) - (20*b*Sin[e + f*x]^3)/(77*f*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x]^5)/(11*f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^4, x, 4, (8*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(7*f) - (4*b*Sin[e + f*x])/(7*f*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x]^3)/(7*f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^2, x, 3, (4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*f) - (2*b*Sin[e + f*x])/(3*f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^0, x, 2, (2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f} -{Csc[e + f*x]^2*Sqrt[b*Sec[e + f*x]], x, 3, -((b*Csc[e + f*x])/(f*Sqrt[b*Sec[e + f*x]])) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f} -{Csc[e + f*x]^4*Sqrt[b*Sec[e + f*x]], x, 4, (-5*b*Csc[e + f*x])/(6*f*Sqrt[b*Sec[e + f*x]]) - (b*Csc[e + f*x]^3)/(3*f*Sqrt[b*Sec[e + f*x]]) + (5*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(6*f)} -{Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]], x, 5, (-3*b*Csc[e + f*x])/(4*f*Sqrt[b*Sec[e + f*x]]) - (3*b*Csc[e + f*x]^3)/(10*f*Sqrt[b*Sec[e + f*x]]) - (b*Csc[e + f*x]^5)/(5*f*Sqrt[b*Sec[e + f*x]]) + (3*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(4*f)} - - -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^7, x, 3, (2*b^7)/(11*f*(b*Sec[e + f*x])^(11/2)) - (6*b^5)/(7*f*(b*Sec[e + f*x])^(7/2)) + (2*b^3)/(f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]])/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^5, x, 3, (-2*b^5)/(7*f*(b*Sec[e + f*x])^(7/2)) + (4*b^3)/(3*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]])/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^3, x, 3, (2*b^3)/(3*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]])/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^1, x, 2, (2*b*Sqrt[b*Sec[e + f*x]])/f} -{Csc[e + f*x]^1*(b*Sec[e + f*x])^(3/2), x, 6, -((b^(3/2)*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f) - (b^(3/2)*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f + (2*b*Sqrt[b*Sec[e + f*x]])/f} -{Csc[e + f*x]^3*(b*Sec[e + f*x])^(3/2), x, 7, (-5*b^(3/2)*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) - (5*b^(3/2)*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) + (5*b*Sqrt[b*Sec[e + f*x]])/(2*f) - (Cot[e + f*x]^2*(b*Sec[e + f*x])^(5/2))/(2*b*f)} - -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6, x, 5, (-16*b^2*EllipticE[(e + f*x)/2, 2])/(3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (8*b^3*Sin[e + f*x])/(3*f*(b*Sec[e + f*x])^(3/2)) + (20*b^3*Sin[e + f*x]^3)/(9*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^5)/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^4, x, 4, (-24*b^2*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (12*b^3*Sin[e + f*x])/(5*f*(b*Sec[e + f*x])^(3/2)) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^3)/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^2, x, 3, (-4*b^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f} -{(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^0, x, 3, (-2*b^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f} -{Csc[e + f*x]^2*(b*Sec[e + f*x])^(3/2), x, 4, (-3*b^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (b*Csc[e + f*x]*Sqrt[b*Sec[e + f*x]])/f + (3*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f} -{Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2), x, 5, (-7*b^2*EllipticE[(e + f*x)/2, 2])/(2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (7*b*Csc[e + f*x]*Sqrt[b*Sec[e + f*x]])/(6*f) - (b*Csc[e + f*x]^3*Sqrt[b*Sec[e + f*x]])/(3*f) + (7*b*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/(2*f)} - - -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^7, x, 3, (2*b^7)/(9*f*(b*Sec[e + f*x])^(9/2)) - (6*b^5)/(5*f*(b*Sec[e + f*x])^(5/2)) + (6*b^3)/(f*Sqrt[b*Sec[e + f*x]]) + (2*b*(b*Sec[e + f*x])^(3/2))/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^5, x, 3, (-2*b^5)/(5*f*(b*Sec[e + f*x])^(5/2)) + (4*b^3)/(f*Sqrt[b*Sec[e + f*x]]) + (2*b*(b*Sec[e + f*x])^(3/2))/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^3, x, 3, (2*b^3)/(f*Sqrt[b*Sec[e + f*x]]) + (2*b*(b*Sec[e + f*x])^(3/2))/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^1, x, 2, (2*b*(b*Sec[e + f*x])^(3/2))/(3*f)} -{Csc[e + f*x]^1*(b*Sec[e + f*x])^(5/2), x, 6, (b^(5/2)*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f - (b^(5/2)*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f + (2*b*(b*Sec[e + f*x])^(3/2))/(3*f)} -{Csc[e + f*x]^3*(b*Sec[e + f*x])^(5/2), x, 7, (7*b^(5/2)*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) - (7*b^(5/2)*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*f) + (7*b*(b*Sec[e + f*x])^(3/2))/(6*f) - (Cot[e + f*x]^2*(b*Sec[e + f*x])^(7/2))/(2*b*f)} -{Csc[e + f*x]^5*(b*Sec[e + f*x])^(5/2), x, 8, (77*b^(5/2)*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*f) - (77*b^(5/2)*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*f) + (77*b*(b*Sec[e + f*x])^(3/2))/(48*f) - (11*Cot[e + f*x]^2*(b*Sec[e + f*x])^(7/2))/(16*b*f) - (Cot[e + f*x]^4*(b*Sec[e + f*x])^(11/2))/(4*b^3*f)} - -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^6, x, 5, (-80*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(21*f) + (40*b^3*Sin[e + f*x])/(21*f*Sqrt[b*Sec[e + f*x]]) + (20*b^3*Sin[e + f*x]^3)/(21*f*Sqrt[b*Sec[e + f*x]]) + (2*b*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^5)/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^4, x, 4, (-8*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*f) + (4*b^3*Sin[e + f*x])/(3*f*Sqrt[b*Sec[e + f*x]]) + (2*b*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^3)/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^2, x, 3, (-4*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*f) + (2*b*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(3*f)} -{(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^0, x, 3, (2*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*f) + (2*b*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(3*f)} -{Csc[e + f*x]^2*(b*Sec[e + f*x])^(5/2), x, 4, -((5*b^3*Csc[e + f*x])/(3*f*Sqrt[b*Sec[e + f*x]])) + (5*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[b*Sec[e + f*x]])/(3*f) + (2*b*Csc[e + f*x]*(b*Sec[e + f*x])^(3/2))/(3*f)} -{Csc[e + f*x]^4*(b*Sec[e + f*x])^(5/2), x, 5, -((5*b^3*Csc[e + f*x])/(2*f*Sqrt[b*Sec[e + f*x]])) + (5*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[b*Sec[e + f*x]])/(2*f) + (b*Csc[e + f*x]*(b*Sec[e + f*x])^(3/2))/f - (b*Csc[e + f*x]^3*(b*Sec[e + f*x])^(3/2))/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[e + f*x]^7/Sqrt[b*Sec[e + f*x]], x, 3, (2*b^7)/(15*f*(b*Sec[e + f*x])^(15/2)) - (6*b^5)/(11*f*(b*Sec[e + f*x])^(11/2)) + (6*b^3)/(7*f*(b*Sec[e + f*x])^(7/2)) - (2*b)/(3*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^5/Sqrt[b*Sec[e + f*x]], x, 3, (-2*b^5)/(11*f*(b*Sec[e + f*x])^(11/2)) + (4*b^3)/(7*f*(b*Sec[e + f*x])^(7/2)) - (2*b)/(3*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^3/Sqrt[b*Sec[e + f*x]], x, 3, (2*b^3)/(7*f*(b*Sec[e + f*x])^(7/2)) - (2*b)/(3*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^1/Sqrt[b*Sec[e + f*x]], x, 2, (-2*b)/(3*f*(b*Sec[e + f*x])^(3/2))} -{Csc[e + f*x]^1/Sqrt[b*Sec[e + f*x]], x, 5, -(ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(Sqrt[b]*f)) - ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(Sqrt[b]*f)} -{Csc[e + f*x]^3/Sqrt[b*Sec[e + f*x]], x, 6, -ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(4*Sqrt[b]*f) - ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(4*Sqrt[b]*f) - (Cot[e + f*x]^2*Sqrt[b*Sec[e + f*x]])/(2*b*f)} -{Csc[e + f*x]^5/Sqrt[b*Sec[e + f*x]], x, 7, (-5*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*Sqrt[b]*f) - (5*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*Sqrt[b]*f) - (5*Cot[e + f*x]^2*Sqrt[b*Sec[e + f*x]])/(16*b*f) - (Cot[e + f*x]^4*(b*Sec[e + f*x])^(5/2))/(4*b^3*f)} - -{Sin[e + f*x]^6/Sqrt[b*Sec[e + f*x]], x, 5, (16*EllipticE[(e + f*x)/2, 2])/(39*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (8*b*Sin[e + f*x])/(39*f*(b*Sec[e + f*x])^(3/2)) - (20*b*Sin[e + f*x]^3)/(117*f*(b*Sec[e + f*x])^(3/2)) - (2*b*Sin[e + f*x]^5)/(13*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^4/Sqrt[b*Sec[e + f*x]], x, 4, (8*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (4*b*Sin[e + f*x])/(15*f*(b*Sec[e + f*x])^(3/2)) - (2*b*Sin[e + f*x]^3)/(9*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^2/Sqrt[b*Sec[e + f*x]], x, 3, (4*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x])/(5*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^0/Sqrt[b*Sec[e + f*x]], x, 2, (2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^2/Sqrt[b*Sec[e + f*x]], x, 3, -((b*Csc[e + f*x])/(f*(b*Sec[e + f*x])^(3/2))) - EllipticE[(e + f*x)/2, 2]/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^4/Sqrt[b*Sec[e + f*x]], x, 4, -(b*Csc[e + f*x])/(2*f*(b*Sec[e + f*x])^(3/2)) - (b*Csc[e + f*x]^3)/(3*f*(b*Sec[e + f*x])^(3/2)) - EllipticE[(e + f*x)/2, 2]/(2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]], x, 5, (-7*b*Csc[e + f*x])/(20*f*(b*Sec[e + f*x])^(3/2)) - (7*b*Csc[e + f*x]^3)/(30*f*(b*Sec[e + f*x])^(3/2)) - (b*Csc[e + f*x]^5)/(5*f*(b*Sec[e + f*x])^(3/2)) - (7*EllipticE[(e + f*x)/2, 2])/(20*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} - - -{Sin[e + f*x]^7/(b*Sec[e + f*x])^(3/2), x, 3, (2*b^7)/(17*f*(b*Sec[e + f*x])^(17/2)) - (6*b^5)/(13*f*(b*Sec[e + f*x])^(13/2)) + (2*b^3)/(3*f*(b*Sec[e + f*x])^(9/2)) - (2*b)/(5*f*(b*Sec[e + f*x])^(5/2))} -{Sin[e + f*x]^5/(b*Sec[e + f*x])^(3/2), x, 3, (-2*b^5)/(13*f*(b*Sec[e + f*x])^(13/2)) + (4*b^3)/(9*f*(b*Sec[e + f*x])^(9/2)) - (2*b)/(5*f*(b*Sec[e + f*x])^(5/2))} -{Sin[e + f*x]^3/(b*Sec[e + f*x])^(3/2), x, 3, (2*b^3)/(9*f*(b*Sec[e + f*x])^(9/2)) - (2*b)/(5*f*(b*Sec[e + f*x])^(5/2))} -{Sin[e + f*x]^1/(b*Sec[e + f*x])^(3/2), x, 2, (-2*b)/(5*f*(b*Sec[e + f*x])^(5/2))} -{Csc[e + f*x]^1/(b*Sec[e + f*x])^(3/2), x, 6, ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(b^(3/2)*f) - ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(b^(3/2)*f) + 2/(b*f*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^3/(b*Sec[e + f*x])^(3/2), x, 6, -ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(4*b^(3/2)*f) + ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(4*b^(3/2)*f) - (Cot[e + f*x]^2*(b*Sec[e + f*x])^(3/2))/(2*b^3*f)} -{Csc[e + f*x]^5/(b*Sec[e + f*x])^(3/2), x, 7, (-3*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(3/2)*f) + (3*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(3/2)*f) - (3*Cot[e + f*x]^2*(b*Sec[e + f*x])^(3/2))/(16*b^3*f) - (Cot[e + f*x]^4*(b*Sec[e + f*x])^(3/2))/(4*b^3*f)} - -{Sin[e + f*x]^4/(b*Sec[e + f*x])^(3/2), x, 5, (8*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(77*b^2*f) - (12*b*Sin[e + f*x])/(77*f*(b*Sec[e + f*x])^(5/2)) + (8*Sin[e + f*x])/(77*b*f*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x]^3)/(11*f*(b*Sec[e + f*x])^(5/2))} -{Sin[e + f*x]^2/(b*Sec[e + f*x])^(3/2), x, 4, (4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(21*b^2*f) - (2*b*Sin[e + f*x])/(7*f*(b*Sec[e + f*x])^(5/2)) + (4*Sin[e + f*x])/(21*b*f*Sqrt[b*Sec[e + f*x]])} -{Sin[e + f*x]^0/(b*Sec[e + f*x])^(3/2), x, 3, (2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(3*b^2*f) + (2*Sin[e + f*x])/(3*b*f*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^2/(b*Sec[e + f*x])^(3/2), x, 3, -(Csc[e + f*x]/(b*f*Sqrt[b*Sec[e + f*x]])) - (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(b^2*f)} -{Csc[e + f*x]^4/(b*Sec[e + f*x])^(3/2), x, 4, Csc[e + f*x]/(6*b*f*Sqrt[b*Sec[e + f*x]]) - Csc[e + f*x]^3/(3*b*f*Sqrt[b*Sec[e + f*x]]) - (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(6*b^2*f)} -{Csc[e + f*x]^6/(b*Sec[e + f*x])^(3/2), x, 5, Csc[e + f*x]/(12*b*f*Sqrt[b*Sec[e + f*x]]) + Csc[e + f*x]^3/(30*b*f*Sqrt[b*Sec[e + f*x]]) - Csc[e + f*x]^5/(5*b*f*Sqrt[b*Sec[e + f*x]]) - (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(12*b^2*f)} - - -{Sin[e + f*x]^7/(b*Sec[e + f*x])^(5/2), x, 3, (2*b^7)/(19*f*(b*Sec[e + f*x])^(19/2)) - (2*b^5)/(5*f*(b*Sec[e + f*x])^(15/2)) + (6*b^3)/(11*f*(b*Sec[e + f*x])^(11/2)) - (2*b)/(7*f*(b*Sec[e + f*x])^(7/2))} -{Sin[e + f*x]^5/(b*Sec[e + f*x])^(5/2), x, 3, (-2*b^5)/(15*f*(b*Sec[e + f*x])^(15/2)) + (4*b^3)/(11*f*(b*Sec[e + f*x])^(11/2)) - (2*b)/(7*f*(b*Sec[e + f*x])^(7/2))} -{Sin[e + f*x]^3/(b*Sec[e + f*x])^(5/2), x, 3, (2*b^3)/(11*f*(b*Sec[e + f*x])^(11/2)) - (2*b)/(7*f*(b*Sec[e + f*x])^(7/2))} -{Sin[e + f*x]^1/(b*Sec[e + f*x])^(5/2), x, 2, (-2*b)/(7*f*(b*Sec[e + f*x])^(7/2))} -{Csc[e + f*x]^1/(b*Sec[e + f*x])^(5/2), x, 6, -(ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(b^(5/2)*f)) - ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]]/(b^(5/2)*f) + 2/(3*b*f*(b*Sec[e + f*x])^(3/2))} -{Csc[e + f*x]^3/(b*Sec[e + f*x])^(5/2), x, 6, (3*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*b^(5/2)*f) + (3*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(4*b^(5/2)*f) - (Cot[e + f*x]^2*Sqrt[b*Sec[e + f*x]])/(2*b^3*f)} -{Csc[e + f*x]^5/(b*Sec[e + f*x])^(5/2), x, 7, (3*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(5/2)*f) + (3*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/(32*b^(5/2)*f) - (Cot[e + f*x]^2*Sqrt[b*Sec[e + f*x]])/(16*b^3*f) - (Cot[e + f*x]^4*Sqrt[b*Sec[e + f*x]])/(4*b^3*f)} - -{Sin[e + f*x]^4/(b*Sec[e + f*x])^(5/2), x, 5, (8*EllipticE[(e + f*x)/2, 2])/(65*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (4*b*Sin[e + f*x])/(39*f*(b*Sec[e + f*x])^(7/2)) + (8*Sin[e + f*x])/(195*b*f*(b*Sec[e + f*x])^(3/2)) - (2*b*Sin[e + f*x]^3)/(13*f*(b*Sec[e + f*x])^(7/2))} -{Sin[e + f*x]^2/(b*Sec[e + f*x])^(5/2), x, 4, (4*EllipticE[(e + f*x)/2, 2])/(15*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (2*b*Sin[e + f*x])/(9*f*(b*Sec[e + f*x])^(7/2)) + (4*Sin[e + f*x])/(45*b*f*(b*Sec[e + f*x])^(3/2))} -{Sin[e + f*x]^0/(b*Sec[e + f*x])^(5/2), x, 3, (6*EllipticE[(e + f*x)/2, 2])/(5*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (2*Sin[e + f*x])/(5*b*f*(b*Sec[e + f*x])^(3/2))} -{Csc[e + f*x]^2/(b*Sec[e + f*x])^(5/2), x, 3, -(Csc[e + f*x]/(b*f*(b*Sec[e + f*x])^(3/2))) - (3*EllipticE[(e + f*x)/2, 2])/(b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^4/(b*Sec[e + f*x])^(5/2), x, 4, Csc[e + f*x]/(2*b*f*(b*Sec[e + f*x])^(3/2)) - Csc[e + f*x]^3/(3*b*f*(b*Sec[e + f*x])^(3/2)) + EllipticE[(e + f*x)/2, 2]/(2*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{Csc[e + f*x]^6/(b*Sec[e + f*x])^(5/2), x, 5, (3*Csc[e + f*x])/(20*b*f*(b*Sec[e + f*x])^(3/2)) + Csc[e + f*x]^3/(10*b*f*(b*Sec[e + f*x])^(3/2)) - Csc[e + f*x]^5/(5*b*f*(b*Sec[e + f*x])^(3/2)) + (3*EllipticE[(e + f*x)/2, 2])/(20*b^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) (b Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(9/2), x, 13, -((21*a^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(32*Sqrt[2]*Sqrt[b]*f)) + (21*a^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(32*Sqrt[2]*Sqrt[b]*f) + (21*a^(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(64*Sqrt[2]*Sqrt[b]*f) - (21*a^(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(64*Sqrt[2]*Sqrt[b]*f) - (7*a^3*b*(a*Sin[e + f*x])^(3/2))/(16*f*Sqrt[b*Sec[e + f*x]]) - (a*b*(a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2), x, 12, -((3*a^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(4*Sqrt[2]*Sqrt[b]*f)) + (3*a^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(4*Sqrt[2]*Sqrt[b]*f) + (3*a^(5/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(8*Sqrt[2]*Sqrt[b]*f) - (3*a^(5/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(8*Sqrt[2]*Sqrt[b]*f) - (a*b*(a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[b*Sec[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(1/2), x, 11, -((Sqrt[a]*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(Sqrt[2]*Sqrt[b]*f)) + (Sqrt[a]*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(Sqrt[2]*Sqrt[b]*f) + (Sqrt[a]*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(2*Sqrt[2]*Sqrt[b]*f) - (Sqrt[a]*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(2*Sqrt[2]*Sqrt[b]*f)} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(3/2), x, 1, -((2*b)/(a*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]))} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(7/2), x, 2, -((2*b)/(5*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2))) - (8*b)/(5*a^3*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(11/2), x, 3, -((2*b)/(9*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(9/2))) - (16*b)/(45*a^3*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2)) - (64*b)/(45*a^5*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])} - -{Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2), x, 5, -((5*a^3*b*Sqrt[a*Sin[e + f*x]])/(6*f*Sqrt[b*Sec[e + f*x]])) - (a*b*(a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[b*Sec[e + f*x]]) + (5*a^4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(12*f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2), x, 4, -((a*b*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])) + (a^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(1/2), x, 3, (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(5/2), x, 4, -((2*b)/(3*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2))) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^2*f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(9/2), x, 5, -((2*b)/(7*a*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2))) - (4*b)/(7*a^3*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2)) + (4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(7*a^4*f*Sqrt[a*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[e + f*x]^(9/2)/Sqrt[b*Sec[e + f*x]], x, 5, -((7*b*Sin[e + f*x]^(3/2))/(30*f*(b*Sec[e + f*x])^(3/2))) - (b*Sin[e + f*x]^(7/2))/(5*f*(b*Sec[e + f*x])^(3/2)) + (7*b*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(20*b*f*(b*Sec[e + f*x])^(1/2)*Sqrt[Sin[2*e + 2*f*x]])} -{Sin[e + f*x]^(5/2)/Sqrt[b*Sec[e + f*x]], x, 4, -((b*Sin[e + f*x]^(3/2))/(3*f*(b*Sec[e + f*x])^(3/2))) + (EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(2*f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])} -{Sin[e + f*x]^(1/2)/Sqrt[b*Sec[e + f*x]], x, 3, (EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])} -{1/(Sin[e + f*x]^(3/2)*Sqrt[b*Sec[e + f*x]]), x, 4, -((2*b)/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])) - (2*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])} -{1/(Sin[e + f*x]^(7/2)*Sqrt[b*Sec[e + f*x]]), x, 5, -((2*b)/(5*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(5/2))) - (4*b)/(5*f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]) - (4*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[Sin[e + f*x]])/(5*f*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])} - -{Sin[e + f*x]^(3/2)/Sqrt[b*Sec[e + f*x]], x, 12, (Sqrt[b]*ArcTan[1 - (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/(Sqrt[b]*Sqrt[Sin[e + f*x]])])/(4*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (Sqrt[b]*ArcTan[1 + (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/(Sqrt[b]*Sqrt[Sin[e + f*x]])])/(4*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Cot[e + f*x] - (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[e + f*x]]])/(8*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Cot[e + f*x] + (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[e + f*x]]])/(8*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (b*Sqrt[Sin[e + f*x]])/(2*f*(b*Sec[e + f*x])^(3/2))} -{1/(Sin[e + f*x]^(1/2)*Sqrt[b*Sec[e + f*x]]), x, 11, (Sqrt[b]*ArcTan[1 - (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/(Sqrt[b]*Sqrt[Sin[e + f*x]])])/(Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (Sqrt[b]*ArcTan[1 + (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/(Sqrt[b]*Sqrt[Sin[e + f*x]])])/(Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) - (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Cot[e + f*x] - (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[e + f*x]]])/(2*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]]) + (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Cot[e + f*x] + (Sqrt[2]*Sqrt[b*Cos[e + f*x]])/Sqrt[Sin[e + f*x]]])/(2*Sqrt[2]*f*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])} -{1/(Sin[e + f*x]^(5/2)*Sqrt[b*Sec[e + f*x]]), x, 1, -((2*b)/(3*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2)))} -{1/(Sin[e + f*x]^(9/2)*Sqrt[b*Sec[e + f*x]]), x, 2, -((2*b)/(7*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(7/2))) - (8*b)/(21*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))} -{1/(Sin[e + f*x]^(13/2)*Sqrt[b*Sec[e + f*x]]), x, 3, -((2*b)/(11*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(11/2))) - (16*b)/(77*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(7/2)) - (64*b)/(231*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))} -{1/(Sin[e + f*x]^(17/2)*Sqrt[b*Sec[e + f*x]]), x, 4, -((2*b)/(15*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(15/2))) - (8*b)/(55*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(11/2)) - (64*b)/(385*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(7/2)) - (256*b)/(1155*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))} - - -{(a*Sin[e + f*x])^(9/2)/(b*Sec[e + f*x])^(3/2), x, 14, -((7*a^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(128*Sqrt[2]*b^(5/2)*f)) + (7*a^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(128*Sqrt[2]*b^(5/2)*f) + (7*a^(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(256*Sqrt[2]*b^(5/2)*f) - (7*a^(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(256*Sqrt[2]*b^(5/2)*f) - (7*a^3*(a*Sin[e + f*x])^(3/2))/(192*b*f*Sqrt[b*Sec[e + f*x]]) - (a*(a*Sin[e + f*x])^(7/2))/(48*b*f*Sqrt[b*Sec[e + f*x]]) + (a*Sin[e + f*x])^(11/2)/(6*a*b*f*Sqrt[b*Sec[e + f*x]])} -{(a*Sin[e + f*x])^(5/2)/(b*Sec[e + f*x])^(3/2), x, 13, -((3*a^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(32*Sqrt[2]*b^(5/2)*f)) + (3*a^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(32*Sqrt[2]*b^(5/2)*f) + (3*a^(5/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(64*Sqrt[2]*b^(5/2)*f) - (3*a^(5/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(64*Sqrt[2]*b^(5/2)*f) - (a*(a*Sin[e + f*x])^(3/2))/(16*b*f*Sqrt[b*Sec[e + f*x]]) + (a*Sin[e + f*x])^(7/2)/(4*a*b*f*Sqrt[b*Sec[e + f*x]])} -{(a*Sin[e + f*x])^(1/2)/(b*Sec[e + f*x])^(3/2), x, 12, -((Sqrt[a]*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(4*Sqrt[2]*b^(5/2)*f)) + (Sqrt[a]*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(4*Sqrt[2]*b^(5/2)*f) + (Sqrt[a]*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(8*Sqrt[2]*b^(5/2)*f) - (Sqrt[a]*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(8*Sqrt[2]*b^(5/2)*f) + (a*Sin[e + f*x])^(3/2)/(2*a*b*f*Sqrt[b*Sec[e + f*x]])} -{1/((a*Sin[e + f*x])^(3/2)*(b*Sec[e + f*x])^(3/2)), x, 12, (ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(Sqrt[2]*a^(3/2)*b^(5/2)*f) - (ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(Sqrt[2]*a^(3/2)*b^(5/2)*f) - (Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(2*Sqrt[2]*a^(3/2)*b^(5/2)*f) + (Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(2*Sqrt[2]*a^(3/2)*b^(5/2)*f) - 2/(a*b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(7/2)*(b*Sec[e + f*x])^(3/2)), x, 1, -((2*b)/(5*a*f*(b*Sec[e + f*x])^(5/2)*(a*Sin[e + f*x])^(5/2)))} - -{(a*Sin[e + f*x])^(7/2)/(b*Sec[e + f*x])^(3/2), x, 6, -((a^3*Sqrt[a*Sin[e + f*x]])/(12*b*f*Sqrt[b*Sec[e + f*x]])) - (a*(a*Sin[e + f*x])^(5/2))/(30*b*f*Sqrt[b*Sec[e + f*x]]) + (a*Sin[e + f*x])^(9/2)/(5*a*b*f*Sqrt[b*Sec[e + f*x]]) + (a^4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(24*b^2*f*Sqrt[a*Sin[e + f*x]])} -{(a*Sin[e + f*x])^(3/2)/(b*Sec[e + f*x])^(3/2), x, 5, -((a*Sqrt[a*Sin[e + f*x]])/(6*b*f*Sqrt[b*Sec[e + f*x]])) + (a*Sin[e + f*x])^(5/2)/(3*a*b*f*Sqrt[b*Sec[e + f*x]]) + (a^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(12*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(1/2)*(b*Sec[e + f*x])^(3/2)), x, 4, Sqrt[a*Sin[e + f*x]]/(a*b*f*Sqrt[b*Sec[e + f*x]]) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(2*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(5/2)*(b*Sec[e + f*x])^(3/2)), x, 4, -(2/(3*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2))) - (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^2*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(9/2)*(b*Sec[e + f*x])^(3/2)), x, 5, -(2/(7*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2))) + 2/(21*a^3*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2)) - (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(21*a^4*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(13/2)*(b*Sec[e + f*x])^(3/2)), x, 6, -(2/(11*a*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(11/2))) + 2/(77*a^3*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2)) + 4/(77*a^5*b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2)) - (4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(77*a^6*b^2*f*Sqrt[a*Sin[e + f*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (a Sin[e+f x])^(m/3) (b Sec[e+f x])^(n/3)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n with m symbolic*) - - -{(c*Sin[a + b*x])^m*(d*Sec[a + b*x])^(5/2), x, 2, (d*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(d*Sec[a + b*x])^(3/2)*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} -{(c*Sin[a + b*x])^m*(d*Sec[a + b*x])^(3/2), x, 2, (d*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*Sqrt[d*Sec[a + b*x]]*(c*Sin[a + b*x])^(1 + m))/(b*c*(1 + m))} -{(c*Sin[a + b*x])^m*(d*Sec[a + b*x])^(1/2), x, 2, ((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(d*Sec[a + b*x])^(3/2)*(c*Sin[a + b*x])^(1 + m))/(b*c*d*(1 + m))} -{(c*Sin[a + b*x])^m/(d*Sec[a + b*x])^(1/2), x, 2, ((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*Sqrt[d*Sec[a + b*x]]*(c*Sin[a + b*x])^(1 + m))/(b*c*d*(1 + m))} -{(c*Sin[a + b*x])^m/(d*Sec[a + b*x])^(3/2), x, 2, (Hypergeometric2F1[-(1/4), (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m))/(b*c*d*(1 + m)*(Cos[a + b*x]^2)^(1/4)*Sqrt[d*Sec[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Sec[e+f x])^n with n symbolic*) - - -{(Sin[e + f*x])^m*(Sec[e + f*x])^n, x, 2, -((Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x]^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - n)))} -{(a*Sin[e + f*x])^m*(Sec[e + f*x])^n, x, 2, -((a*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*(a*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - n)))} -{(Sin[e + f*x])^m*(b*Sec[e + f*x])^n, x, 2, -((b*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sin[e + f*x]^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - n)))} -{(a*Sin[e + f*x])^m*(b*Sec[e + f*x])^n, x, 2, -((a*b*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*(a*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - n)))} - - -{(b*Sec[e + f*x])^n*Sin[e + f*x]^5, x, 3, -((b^5*(b*Sec[e + f*x])^(-5 + n))/(f*(5 - n))) + (2*b^3*(b*Sec[e + f*x])^(-3 + n))/(f*(3 - n)) - (b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n))} -{(b*Sec[e + f*x])^n*Sin[e + f*x]^3, x, 3, (b^3*(b*Sec[e + f*x])^(-3 + n))/(f*(3 - n)) - (b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n))} -{(b*Sec[e + f*x])^n*Sin[e + f*x]^1, x, 2, -((b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n)))} -{(b*Sec[e + f*x])^n*Csc[e + f*x]^1, x, 2, -((Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(1 + n))/(f*b*(1 + n)))} -{(b*Sec[e + f*x])^n*Csc[e + f*x]^3, x, 2, (Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3 + n))/(f*b^3*(3 + n))} - -{(b*Sec[e + f*x])^n*Sin[e + f*x]^6, x, 2, -((b*Hypergeometric2F1[-(5/2), (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2]))} -{(b*Sec[e + f*x])^n*Sin[e + f*x]^4, x, 2, -((b*Hypergeometric2F1[-(3/2), (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2]))} -{(b*Sec[e + f*x])^n*Sin[e + f*x]^2, x, 2, -((b*Hypergeometric2F1[-(1/2), (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2]))} -{(b*Sec[e + f*x])^n*Sin[e + f*x]^0, x, 2, -((b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2]))} -{(b*Sec[e + f*x])^n*Csc[e + f*x]^2, x, 2, -((b*Csc[e + f*x]*Hypergeometric2F1[3/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sqrt[Sin[e + f*x]^2])/(f*(1 - n)))} -{(b*Sec[e + f*x])^n*Csc[e + f*x]^4, x, 2, -((b*Csc[e + f*x]*Hypergeometric2F1[5/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Sec[e + f*x])^(-1 + n)*Sqrt[Sin[e + f*x]^2])/(f*(1 - n)))} - - -{(b*Sec[a + b*x])^n*(c*Sin[a + b*x])^(3/2), x, 2, -((c*Hypergeometric2F1[-(1/4), (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*(b*Sec[a + b*x])^(-1 + n)*Sqrt[c*Sin[a + b*x]])/((1 - n)*(Sin[a + b*x]^2)^(1/4)))} -{(b*Sec[a + b*x])^n*(c*Sin[a + b*x])^(1/2), x, 2, -((c*Hypergeometric2F1[1/4, (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*(b*Sec[a + b*x])^(-1 + n)*(Sin[a + b*x]^2)^(1/4))/((1 - n)*Sqrt[c*Sin[a + b*x]]))} -{(b*Sec[a + b*x])^n/(c*Sin[a + b*x])^(1/2), x, 2, -((c*Hypergeometric2F1[3/4, (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*(b*Sec[a + b*x])^(-1 + n)*(Sin[a + b*x]^2)^(3/4))/((1 - n)*(c*Sin[a + b*x])^(3/2)))} -{(b*Sec[a + b*x])^n/(c*Sin[a + b*x])^(3/2), x, 2, -((Hypergeometric2F1[5/4, (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*(b*Sec[a + b*x])^(-1 + n)*(Sin[a + b*x]^2)^(1/4))/(c*(1 - n)*Sqrt[c*Sin[a + b*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Csc[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (b Csc[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^4, x, 5, -((2*d^3*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2))) - (10*d*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(21*f)} -{Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^3, x, 4, -((2*d^2*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2))) + (6*d*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^2, x, 4, -((2*d*Cos[e + f*x])/(3*f*Sqrt[d*Csc[e + f*x]])) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*f)} -{Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^1, x, 3, (2*d*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^0, x, 2, (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/f} -{Csc[e + f*x]^1*Sqrt[d*Csc[e + f*x]], x, 4, -((2*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/f) - (2*d*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]], x, 4, -((2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d*f)) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*f)} -{Csc[e + f*x]^3*Sqrt[d*Csc[e + f*x]], x, 5, -((6*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(5*f)) - (2*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*d^2*f) - (6*d*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} - - -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^5, x, 5, -((2*d^4*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2))) - (10*d^2*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(21*f)} -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^4, x, 4, -((2*d^3*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2))) + (6*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^3, x, 4, -((2*d^2*Cos[e + f*x])/(3*f*Sqrt[d*Csc[e + f*x]])) + (2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*f)} -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^2, x, 3, (2*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^1, x, 3, (2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/f} -{(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^0, x, 3, -((2*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/f) - (2*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^1*(d*Csc[e + f*x])^(3/2), x, 4, -((2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*f)) + (2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*f)} -{Csc[e + f*x]^2*(d*Csc[e + f*x])^(3/2), x, 5, -((6*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(5*f)) - (2*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*d*f) - (6*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[e + f*x]^3/Sqrt[d*Csc[e + f*x]], x, 5, -((2*d^2*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2))) - (10*Cos[e + f*x])/(21*f*Sqrt[d*Csc[e + f*x]]) + (10*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(21*d*f)} -{Sin[e + f*x]^2/Sqrt[d*Csc[e + f*x]], x, 4, -((2*d*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2))) + (6*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Sin[e + f*x]^1/Sqrt[d*Csc[e + f*x]], x, 4, -((2*Cos[e + f*x])/(3*f*Sqrt[d*Csc[e + f*x]])) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*d*f)} -{Sin[e + f*x]^0/Sqrt[d*Csc[e + f*x]], x, 2, (2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^1/Sqrt[d*Csc[e + f*x]], x, 3, (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(d*f)} -{Csc[e + f*x]^2/Sqrt[d*Csc[e + f*x]], x, 4, -((2*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(d*f)) - (2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^3/Sqrt[d*Csc[e + f*x]], x, 4, -((2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d^2*f)) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*d*f)} - - -{Sin[e + f*x]^2/(d*Csc[e + f*x])^(3/2), x, 5, -((2*d*Cos[e + f*x])/(7*f*(d*Csc[e + f*x])^(5/2))) - (10*Cos[e + f*x])/(21*d*f*Sqrt[d*Csc[e + f*x]]) + (10*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(21*d^2*f)} -{Sin[e + f*x]^1/(d*Csc[e + f*x])^(3/2), x, 4, -((2*Cos[e + f*x])/(5*f*(d*Csc[e + f*x])^(3/2))) + (6*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*d*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Sin[e + f*x]^0/(d*Csc[e + f*x])^(3/2), x, 3, -((2*Cos[e + f*x])/(3*d*f*Sqrt[d*Csc[e + f*x]])) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*d^2*f)} -{Csc[e + f*x]^1/(d*Csc[e + f*x])^(3/2), x, 3, (2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(d*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^2/(d*Csc[e + f*x])^(3/2), x, 3, (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(d^2*f)} -{Csc[e + f*x]^3/(d*Csc[e + f*x])^(3/2), x, 4, -((2*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(d^2*f)) - (2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(d*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Csc[e + f*x]^4/(d*Csc[e + f*x])^(3/2), x, 4, -((2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d^3*f)) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(3*d^2*f)} -{Csc[e + f*x]^5/(d*Csc[e + f*x])^(3/2), x, 5, -((6*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(5*d^2*f)) - (2*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*d^4*f) - (6*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/(5*d*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) (b Csc[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Csc[e+f x])^n with m and n symbolic*) - - -{(a*Sin[e + f*x])^m*(b*Csc[e + f*x])^n, x, 2, (Cos[e + f*x]*(b*Csc[e + f*x])^n*Hypergeometric2F1[1/2, (1/2)*(1 + m - n), (1/2)*(3 + m - n), Sin[e + f*x]^2]*(a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m - n)*Sqrt[Cos[e + f*x]^2])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.jl b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.jl deleted file mode 100644 index 35a9710..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.jl +++ /dev/null @@ -1,155 +0,0 @@ -# Each tuple is (integrand, result, integration variable, mystery value) -data = [ -# ::Package:: - -# ::Title:: -# Integrands of the form (a+b Sin[c+d x])^n - - -# ::Section::Closed:: -# Integrands of the form (a+a Sin[c+d x])^n - - -# ::Subsection:: -# Integrands of the form (a+a Sin[c+d x])^n - - -# ::Subsection::Closed:: -# Integrands of the form (a+a Sin[c+d x])^(n/2) - - -((a + a*sin(c + d*x))^(7//2), -((256*a^4*cos(c + d*x))/(35*d*sqrt(a + a*sin(c + d*x)))) - (64*a^3*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(35*d) - (24*a^2*cos(c + d*x)*(a + a*sin(c + d*x))^(3//2))/(35*d) - (2*a*cos(c + d*x)*(a + a*sin(c + d*x))^(5//2))/(7*d), x, 4), -((a + a*sin(c + d*x))^(5//2), -((64*a^3*cos(c + d*x))/(15*d*sqrt(a + a*sin(c + d*x)))) - (16*a^2*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(15*d) - (2*a*cos(c + d*x)*(a + a*sin(c + d*x))^(3//2))/(5*d), x, 3), -((a + a*sin(c + d*x))^(3//2), -((8*a^2*cos(c + d*x))/(3*d*sqrt(a + a*sin(c + d*x)))) - (2*a*cos(c + d*x)*sqrt(a + a*sin(c + d*x)))/(3*d), x, 2), -((a + a*sin(c + d*x))^(1//2), -((2*a*cos(c + d*x))/(d*sqrt(a + a*sin(c + d*x)))), x, 1), -(1/(a + a*sin(c + d*x))^(1//2), -((sqrt(2)*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(sqrt(a)*d)), x, 2), -(1/(a + a*sin(c + d*x))^(3//2), -(atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x))))/(2*sqrt(2)*a^(3//2)*d)) - cos(c + d*x)/(2*d*(a + a*sin(c + d*x))^(3//2)), x, 3), -(1/(a + a*sin(c + d*x))^(5//2), -((3*atanh((sqrt(a)*cos(c + d*x))/(sqrt(2)*sqrt(a + a*sin(c + d*x)))))/(16*sqrt(2)*a^(5//2)*d)) - cos(c + d*x)/(4*d*(a + a*sin(c + d*x))^(5//2)) - (3*cos(c + d*x))/(16*a*d*(a + a*sin(c + d*x))^(3//2)), x, 4), - - -# ::Subsection::Closed:: -# Integrands of the form (a+a Sin[c+d x])^(n/3) - - -((a + a*sin(c + d*x))^(4//3), -((2*2^(5//6)*a*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(-(5//6), 1//2, 3//2, (1//2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1//3))/(d*(1 + sin(c + d*x))^(5//6))), x, 2), -((a + a*sin(c + d*x))^(2//3), -((2*2^(1//6)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(-(1//6), 1//2, 3//2, (1//2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(2//3))/(d*(1 + sin(c + d*x))^(7//6))), x, 2), -((a + a*sin(c + d*x))^(1//3), -((2^(5//6)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//6, 1//2, 3//2, (1//2)*(1 - sin(c + d*x)))*(a + a*sin(c + d*x))^(1//3))/(d*(1 + sin(c + d*x))^(5//6))), x, 2), -(1/(a + a*sin(c + d*x))^(1//3), -((2^(1//6)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 5//6, 3//2, (1//2)*(1 - sin(c + d*x))))/(d*(1 + sin(c + d*x))^(1//6)*(a + a*sin(c + d*x))^(1//3))), x, 2), -(1/(a + a*sin(c + d*x))^(2//3), -((cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 7//6, 3//2, (1//2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(1//6))/(2^(1//6)*d*(a + a*sin(c + d*x))^(2//3))), x, 2), -(1/(a + a*sin(c + d*x))^(4//3), -((cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 11//6, 3//2, (1//2)*(1 - sin(c + d*x))))/(2^(5//6)*a*d*(1 + sin(c + d*x))^(1//6)*(a + a*sin(c + d*x))^(1//3))), x, 2), - - -# ::Subsection::Closed:: -# Integrands of the form (a+a Sin[c+d x])^n with n symbolic - - -((a + a*sin(c + d*x))^n, -((2^(1//2 + n)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 1//2 - n, 3//2, (1//2)*(1 - sin(c + d*x)))*(1 + sin(c + d*x))^(-(1//2) - n)*(a + a*sin(c + d*x))^n)/d), x, 2), -((a - a*sin(c + d*x))^n, (2^(1//2 + n)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 1//2 - n, 3//2, (1//2)*(1 + sin(c + d*x)))*(1 - sin(c + d*x))^(-(1//2) - n)*(a - a*sin(c + d*x))^n)/d, x, 2), - - -((2 + 2*sin(c + d*x))^n, -((2^(1//2 + 2*n)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 1//2 - n, 3//2, (1//2)*(1 - sin(c + d*x))))/(d*sqrt(1 + sin(c + d*x)))), x, 1), -((2 - 2*sin(c + d*x))^n, (2^(1//2 + 2*n)*cos(c + d*x)*SymbolicIntegration.hypergeometric2f1(1//2, 1//2 - n, 3//2, (1//2)*(1 + sin(c + d*x))))/(d*sqrt(1 - sin(c + d*x))), x, 1), - - -# ::Section::Closed:: -# Integrands of the form (a+b Sin[c+d x])^n - - -# ::Subsection::Closed:: -# Integrands of the form (a+b Sin[c+d x])^n - - -(1/(5 + 3*sin(c + d*x)), x/4 + atan(cos(c + d*x)/(3 + sin(c + d*x)))/(2*d), x, 1), -(1/(5 + 3*sin(c + d*x))^2, (5*x)/64 + (5*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(32*d) + (3*cos(c + d*x))/(16*d*(5 + 3*sin(c + d*x))), x, 3), -(1/(5 + 3*sin(c + d*x))^3, (59*x)/2048 + (59*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(1024*d) + (3*cos(c + d*x))/(32*d*(5 + 3*sin(c + d*x))^2) + (45*cos(c + d*x))/(512*d*(5 + 3*sin(c + d*x))), x, 4), -(1/(5 + 3*sin(c + d*x))^4, (385*x)/32768 + (385*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(16384*d) + cos(c + d*x)/(16*d*(5 + 3*sin(c + d*x))^3) + (25*cos(c + d*x))/(512*d*(5 + 3*sin(c + d*x))^2) + (311*cos(c + d*x))/(8192*d*(5 + 3*sin(c + d*x))), x, 5), - - -(1/(5 - 3*sin(c + d*x)), x/4 - atan(cos(c + d*x)/(3 - sin(c + d*x)))/(2*d), x, 1), -(1/(5 - 3*sin(c + d*x))^2, (5*x)/64 - (5*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(32*d) - (3*cos(c + d*x))/(16*d*(5 - 3*sin(c + d*x))), x, 3), -(1/(5 - 3*sin(c + d*x))^3, (59*x)/2048 - (59*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(1024*d) - (3*cos(c + d*x))/(32*d*(5 - 3*sin(c + d*x))^2) - (45*cos(c + d*x))/(512*d*(5 - 3*sin(c + d*x))), x, 4), -(1/(5 - 3*sin(c + d*x))^4, (385*x)/32768 - (385*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(16384*d) - cos(c + d*x)/(16*d*(5 - 3*sin(c + d*x))^3) - (25*cos(c + d*x))/(512*d*(5 - 3*sin(c + d*x))^2) - (311*cos(c + d*x))/(8192*d*(5 - 3*sin(c + d*x))), x, 5), - - -(1/(-5 + 3*sin(c + d*x)), -(x/4) + atan(cos(c + d*x)/(3 - sin(c + d*x)))/(2*d), x, 1), -(1/(-5 + 3*sin(c + d*x))^2, (5*x)/64 - (5*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(32*d) - (3*cos(c + d*x))/(16*d*(5 - 3*sin(c + d*x))), x, 3), -(1/(-5 + 3*sin(c + d*x))^3, -((59*x)/2048) + (59*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(1024*d) + (3*cos(c + d*x))/(32*d*(5 - 3*sin(c + d*x))^2) + (45*cos(c + d*x))/(512*d*(5 - 3*sin(c + d*x))), x, 4), -(1/(-5 + 3*sin(c + d*x))^4, (385*x)/32768 - (385*atan(cos(c + d*x)/(3 - sin(c + d*x))))/(16384*d) - cos(c + d*x)/(16*d*(5 - 3*sin(c + d*x))^3) - (25*cos(c + d*x))/(512*d*(5 - 3*sin(c + d*x))^2) - (311*cos(c + d*x))/(8192*d*(5 - 3*sin(c + d*x))), x, 5), - - -(1/(-5 - 3*sin(c + d*x)), -(x/4) - atan(cos(c + d*x)/(3 + sin(c + d*x)))/(2*d), x, 1), -(1/(-5 - 3*sin(c + d*x))^2, (5*x)/64 + (5*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(32*d) + (3*cos(c + d*x))/(16*d*(5 + 3*sin(c + d*x))), x, 3), -(1/(-5 - 3*sin(c + d*x))^3, -((59*x)/2048) - (59*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(1024*d) - (3*cos(c + d*x))/(32*d*(5 + 3*sin(c + d*x))^2) - (45*cos(c + d*x))/(512*d*(5 + 3*sin(c + d*x))), x, 4), -(1/(-5 - 3*sin(c + d*x))^4, (385*x)/32768 + (385*atan(cos(c + d*x)/(3 + sin(c + d*x))))/(16384*d) + cos(c + d*x)/(16*d*(5 + 3*sin(c + d*x))^3) + (25*cos(c + d*x))/(512*d*(5 + 3*sin(c + d*x))^2) + (311*cos(c + d*x))/(8192*d*(5 + 3*sin(c + d*x))), x, 5), - - -(1/(3 + 5*sin(c + d*x)), -(log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x)))/(4*d)) + log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x)))/(4*d), x, 4), -(1/(3 + 5*sin(c + d*x))^2, (3*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(64*d) - (3*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(64*d) - (5*cos(c + d*x))/(16*d*(3 + 5*sin(c + d*x))), x, 6), -(1/(3 + 5*sin(c + d*x))^3, -((43*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(2048*d)) + (43*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(2048*d) - (5*cos(c + d*x))/(32*d*(3 + 5*sin(c + d*x))^2) + (45*cos(c + d*x))/(512*d*(3 + 5*sin(c + d*x))), x, 7), -(1/(3 + 5*sin(c + d*x))^4, (279*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(32768*d) - (279*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(32768*d) - (5*cos(c + d*x))/(48*d*(3 + 5*sin(c + d*x))^3) + (25*cos(c + d*x))/(512*d*(3 + 5*sin(c + d*x))^2) - (995*cos(c + d*x))/(24576*d*(3 + 5*sin(c + d*x))), x, 8), - - -(1/(3 - 5*sin(c + d*x)), -(log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x)))/(4*d)) + log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x)))/(4*d), x, 4), -(1/(3 - 5*sin(c + d*x))^2, (3*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(64*d) - (3*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(64*d) + (5*cos(c + d*x))/(16*d*(3 - 5*sin(c + d*x))), x, 6), -(1/(3 - 5*sin(c + d*x))^3, -((43*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(2048*d)) + (43*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(2048*d) + (5*cos(c + d*x))/(32*d*(3 - 5*sin(c + d*x))^2) - (45*cos(c + d*x))/(512*d*(3 - 5*sin(c + d*x))), x, 7), -(1/(3 - 5*sin(c + d*x))^4, (279*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(32768*d) - (279*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(32768*d) + (5*cos(c + d*x))/(48*d*(3 - 5*sin(c + d*x))^3) - (25*cos(c + d*x))/(512*d*(3 - 5*sin(c + d*x))^2) + (995*cos(c + d*x))/(24576*d*(3 - 5*sin(c + d*x))), x, 8), - - -(1/(-3 + 5*sin(c + d*x)), log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x)))/(4*d) - log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x)))/(4*d), x, 4), -(1/(-3 + 5*sin(c + d*x))^2, (3*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(64*d) - (3*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(64*d) + (5*cos(c + d*x))/(16*d*(3 - 5*sin(c + d*x))), x, 6), -(1/(-3 + 5*sin(c + d*x))^3, (43*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(2048*d) - (43*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(2048*d) - (5*cos(c + d*x))/(32*d*(3 - 5*sin(c + d*x))^2) + (45*cos(c + d*x))/(512*d*(3 - 5*sin(c + d*x))), x, 7), -(1/(-3 + 5*sin(c + d*x))^4, (279*log(cos((1//2)*(c + d*x)) - 3*sin((1//2)*(c + d*x))))/(32768*d) - (279*log(3*cos((1//2)*(c + d*x)) - sin((1//2)*(c + d*x))))/(32768*d) + (5*cos(c + d*x))/(48*d*(3 - 5*sin(c + d*x))^3) - (25*cos(c + d*x))/(512*d*(3 - 5*sin(c + d*x))^2) + (995*cos(c + d*x))/(24576*d*(3 - 5*sin(c + d*x))), x, 8), - - -(1/(-3 - 5*sin(c + d*x)), log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x)))/(4*d) - log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x)))/(4*d), x, 4), -(1/(-3 - 5*sin(c + d*x))^2, (3*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(64*d) - (3*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(64*d) - (5*cos(c + d*x))/(16*d*(3 + 5*sin(c + d*x))), x, 6), -(1/(-3 - 5*sin(c + d*x))^3, (43*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(2048*d) - (43*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(2048*d) + (5*cos(c + d*x))/(32*d*(3 + 5*sin(c + d*x))^2) - (45*cos(c + d*x))/(512*d*(3 + 5*sin(c + d*x))), x, 7), -(1/(-3 - 5*sin(c + d*x))^4, (279*log(3*cos((1//2)*(c + d*x)) + sin((1//2)*(c + d*x))))/(32768*d) - (279*log(cos((1//2)*(c + d*x)) + 3*sin((1//2)*(c + d*x))))/(32768*d) - (5*cos(c + d*x))/(48*d*(3 + 5*sin(c + d*x))^3) + (25*cos(c + d*x))/(512*d*(3 + 5*sin(c + d*x))^2) - (995*cos(c + d*x))/(24576*d*(3 + 5*sin(c + d*x))), x, 8), - - -# ::Subsection::Closed:: -# Integrands of the form (a+b Sin[c+d x])^(n/2) - - -((a + b*sin(c + d*x))^(7//2), -((2*b*(71*a^2 + 25*b^2)*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(105*d)) - (24*a*b*cos(c + d*x)*(a + b*sin(c + d*x))^(3//2))/(35*d) - (2*b*cos(c + d*x)*(a + b*sin(c + d*x))^(5//2))/(7*d) + (32*a*(11*a^2 + 13*b^2)*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(105*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(105*d*sqrt(a + b*sin(c + d*x))), x, 8), -((a + b*sin(c + d*x))^(5//2), -((16*a*b*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(15*d)) - (2*b*cos(c + d*x)*(a + b*sin(c + d*x))^(3//2))/(5*d) + (2*(23*a^2 + 9*b^2)*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (16*a*(a^2 - b^2)*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*d*sqrt(a + b*sin(c + d*x))), x, 7), -((a + b*sin(c + d*x))^(3//2), -((2*b*cos(c + d*x)*sqrt(a + b*sin(c + d*x)))/(3*d)) + (8*a*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*(a^2 - b^2)*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*d*sqrt(a + b*sin(c + d*x))), x, 6), -((a + b*sin(c + d*x))^(1//2), (2*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(d*sqrt((a + b*sin(c + d*x))/(a + b))), x, 2), -(1/(a + b*sin(c + d*x))^(1//2), (2*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(d*sqrt(a + b*sin(c + d*x))), x, 2), -(1/(a + b*sin(c + d*x))^(3//2), (2*b*cos(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))) + (2*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/((a^2 - b^2)*d*sqrt((a + b*sin(c + d*x))/(a + b))), x, 4), -(1/(a + b*sin(c + d*x))^(5//2), (2*b*cos(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(3//2)) + (8*a*b*cos(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))) + (8*a*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(3*(a^2 - b^2)^2*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (2*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(3*(a^2 - b^2)*d*sqrt(a + b*sin(c + d*x))), x, 7), -(1/(a + b*sin(c + d*x))^(7//2), (2*b*cos(c + d*x))/(5*(a^2 - b^2)*d*(a + b*sin(c + d*x))^(5//2)) + (16*a*b*cos(c + d*x))/(15*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))^(3//2)) + (2*b*(23*a^2 + 9*b^2)*cos(c + d*x))/(15*(a^2 - b^2)^3*d*sqrt(a + b*sin(c + d*x))) + (2*(23*a^2 + 9*b^2)*SymbolicIntegration.elliptic_e((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt(a + b*sin(c + d*x)))/(15*(a^2 - b^2)^3*d*sqrt((a + b*sin(c + d*x))/(a + b))) - (16*a*SymbolicIntegration.elliptic_f((1//2)*(c - π/2 + d*x), (2*b)/(a + b))*sqrt((a + b*sin(c + d*x))/(a + b)))/(15*(a^2 - b^2)^2*d*sqrt(a + b*sin(c + d*x))), x, 8), - - -# ::Subsection::Closed:: -# Integrands of the form (a+b Sin[c+d x])^(n/3) - - -((a + b*sin(c + d*x))^(4//3), -((sqrt(2)*(a + b)*SymbolicIntegration.appell_f1(1//2, 1//2, -(4//3), 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^(1//3))/(d*sqrt(1 + sin(c + d*x))*((a + b*sin(c + d*x))/(a + b))^(1//3))), x, 3), -((a + b*sin(c + d*x))^(2//3), -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, -(2//3), 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^(2//3))/(d*sqrt(1 + sin(c + d*x))*((a + b*sin(c + d*x))/(a + b))^(2//3))), x, 3), -((a + b*sin(c + d*x))^(1//3), -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, -(1//3), 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^(1//3))/(d*sqrt(1 + sin(c + d*x))*((a + b*sin(c + d*x))/(a + b))^(1//3))), x, 3), -(1/(a + b*sin(c + d*x))^(1//3), -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, 1//3, 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*((a + b*sin(c + d*x))/(a + b))^(1//3))/(d*sqrt(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1//3))), x, 3), -(1/(a + b*sin(c + d*x))^(2//3), -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, 2//3, 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*((a + b*sin(c + d*x))/(a + b))^(2//3))/(d*sqrt(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(2//3))), x, 3), -(1/(a + b*sin(c + d*x))^(4//3), -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, 4//3, 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*((a + b*sin(c + d*x))/(a + b))^(1//3))/((a + b)*d*sqrt(1 + sin(c + d*x))*(a + b*sin(c + d*x))^(1//3))), x, 3), - - -# ::Subsection::Closed:: -# Integrands of the form (a+b Sin[c+d x])^n with n symbolic - - -((a + b*sin(c + d*x))^n, -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, -n, 3//2, (1//2)*(1 - sin(c + d*x)), (b*(1 - sin(c + d*x)))/(a + b))*cos(c + d*x)*(a + b*sin(c + d*x))^n)/(((a + b*sin(c + d*x))/(a + b))^n*(d*sqrt(1 + sin(c + d*x))))), x, 3), - - -((3 + 4*sin(c + d*x))^n, -((sqrt(2)*7^n*SymbolicIntegration.appell_f1(1//2, 1//2, -n, 3//2, (1//2)*(1 - sin(c + d*x)), (4//7)*(1 - sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 + sin(c + d*x)))), x, 2), -((3 - 4*sin(c + d*x))^n, (sqrt(2)*7^n*SymbolicIntegration.appell_f1(1//2, -n, 1//2, 3//2, (4//7)*(1 + sin(c + d*x)), (1//2)*(1 + sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 - sin(c + d*x))), x, 2), - -((4 + 3*sin(c + d*x))^n, (sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, -n, 3//2, (1//2)*(1 + sin(c + d*x)), -3*(1 + sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 - sin(c + d*x))), x, 2), -((4 - 3*sin(c + d*x))^n, (sqrt(2)*7^n*SymbolicIntegration.appell_f1(1//2, -n, 1//2, 3//2, (3//7)*(1 + sin(c + d*x)), (1//2)*(1 + sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 - sin(c + d*x))), x, 2), - -((-3 + 4*sin(c + d*x))^n, -((sqrt(2)*SymbolicIntegration.appell_f1(1//2, 1//2, -n, 3//2, (1//2)*(1 - sin(c + d*x)), 4*(1 - sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 + sin(c + d*x)))), x, 2), -((-3 - 4*sin(c + d*x))^n, (sqrt(2)*SymbolicIntegration.appell_f1(1//2, -n, 1//2, 3//2, 4*(1 + sin(c + d*x)), (1//2)*(1 + sin(c + d*x)))*cos(c + d*x))/(d*sqrt(1 - sin(c + d*x))), x, 2), - -((-4 + 3*sin(c + d*x))^n, (sqrt(2)*7^n*SymbolicIntegration.appell_f1(1//2, -n, 1//2, 3//2, (3//7)*(1 + sin(c + d*x)), (1//2)*(1 + sin(c + d*x)))*cos(c + d*x)*(-4 + 3*sin(c + d*x))^n)/((4 - 3*sin(c + d*x))^n*(d*sqrt(1 - sin(c + d*x)))), x, 3), -((-4 - 3*sin(c + d*x))^n, -((SymbolicIntegration.appell_f1(1 + n, 1//2, 1//2, 2 + n, 4 + 3*sin(c + d*x), (1//7)*(4 + 3*sin(c + d*x)))*cos(c + d*x)*(-4 - 3*sin(c + d*x))^(1 + n)*sqrt(-1 - sin(c + d*x)))/(sqrt(7)*d*(1 + n)*sqrt(1 - sin(c + d*x))*(1 + sin(c + d*x)))), x, 3), -] -# Total integrals translated: 72 diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m deleted file mode 100644 index 52af0f9..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.1 (a+b sin)^n.m +++ /dev/null @@ -1,151 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Sin[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[c+d x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Sin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[c+d x])^(n/2)*) - - -{(a + a*Sin[c + d*x])^(7/2), x, 4, -((256*a^4*Cos[c + d*x])/(35*d*Sqrt[a + a*Sin[c + d*x]])) - (64*a^3*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) - (24*a^2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*d)} -{(a + a*Sin[c + d*x])^(5/2), x, 3, -((64*a^3*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]])) - (16*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{(a + a*Sin[c + d*x])^(3/2), x, 2, -((8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]])) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{(a + a*Sin[c + d*x])^(1/2), x, 1, -((2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))} -{1/(a + a*Sin[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d))} -{1/(a + a*Sin[c + d*x])^(3/2), x, 3, -(ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d)) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))} -{1/(a + a*Sin[c + d*x])^(5/2), x, 4, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (3*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[c+d x])^(n/3)*) - - -{(a + a*Sin[c + d*x])^(4/3), x, 2, -((2*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6)))} -{(a + a*Sin[c + d*x])^(2/3), x, 2, -((2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6)))} -{(a + a*Sin[c + d*x])^(1/3), x, 2, -((2^(5/6)*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6)))} -{1/(a + a*Sin[c + d*x])^(1/3), x, 2, -((2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} -{1/(a + a*Sin[c + d*x])^(2/3), x, 2, -((Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1/6))/(2^(1/6)*d*(a + a*Sin[c + d*x])^(2/3)))} -{1/(a + a*Sin[c + d*x])^(4/3), x, 2, -((Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[c+d x])^n with n symbolic*) - - -{(a + a*Sin[c + d*x])^n, x, 2, -((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/d)} -{(a - a*Sin[c + d*x])^n, x, 2, (2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(-(1/2) - n)*(a - a*Sin[c + d*x])^n)/d} - - -{(2 + 2*Sin[c + d*x])^n, x, 1, -((2^(1/2 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])])/(d*Sqrt[1 + Sin[c + d*x]]))} -{(2 - 2*Sin[c + d*x])^n, x, 1, (2^(1/2 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 + Sin[c + d*x])])/(d*Sqrt[1 - Sin[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[c+d x])^n*) - - -{1/(5 + 3*Sin[c + d*x]), x, 1, x/4 + ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])]/(2*d)} -{1/(5 + 3*Sin[c + d*x])^2, x, 3, (5*x)/64 + (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(32*d) + (3*Cos[c + d*x])/(16*d*(5 + 3*Sin[c + d*x]))} -{1/(5 + 3*Sin[c + d*x])^3, x, 4, (59*x)/2048 + (59*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(1024*d) + (3*Cos[c + d*x])/(32*d*(5 + 3*Sin[c + d*x])^2) + (45*Cos[c + d*x])/(512*d*(5 + 3*Sin[c + d*x]))} -{1/(5 + 3*Sin[c + d*x])^4, x, 5, (385*x)/32768 + (385*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(16384*d) + Cos[c + d*x]/(16*d*(5 + 3*Sin[c + d*x])^3) + (25*Cos[c + d*x])/(512*d*(5 + 3*Sin[c + d*x])^2) + (311*Cos[c + d*x])/(8192*d*(5 + 3*Sin[c + d*x]))} - - -{1/(5 - 3*Sin[c + d*x]), x, 1, x/4 - ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])]/(2*d)} -{1/(5 - 3*Sin[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(32*d) - (3*Cos[c + d*x])/(16*d*(5 - 3*Sin[c + d*x]))} -{1/(5 - 3*Sin[c + d*x])^3, x, 4, (59*x)/2048 - (59*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(1024*d) - (3*Cos[c + d*x])/(32*d*(5 - 3*Sin[c + d*x])^2) - (45*Cos[c + d*x])/(512*d*(5 - 3*Sin[c + d*x]))} -{1/(5 - 3*Sin[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(16384*d) - Cos[c + d*x]/(16*d*(5 - 3*Sin[c + d*x])^3) - (25*Cos[c + d*x])/(512*d*(5 - 3*Sin[c + d*x])^2) - (311*Cos[c + d*x])/(8192*d*(5 - 3*Sin[c + d*x]))} - - -{1/(-5 + 3*Sin[c + d*x]), x, 1, -(x/4) + ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])]/(2*d)} -{1/(-5 + 3*Sin[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(32*d) - (3*Cos[c + d*x])/(16*d*(5 - 3*Sin[c + d*x]))} -{1/(-5 + 3*Sin[c + d*x])^3, x, 4, -((59*x)/2048) + (59*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(1024*d) + (3*Cos[c + d*x])/(32*d*(5 - 3*Sin[c + d*x])^2) + (45*Cos[c + d*x])/(512*d*(5 - 3*Sin[c + d*x]))} -{1/(-5 + 3*Sin[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])])/(16384*d) - Cos[c + d*x]/(16*d*(5 - 3*Sin[c + d*x])^3) - (25*Cos[c + d*x])/(512*d*(5 - 3*Sin[c + d*x])^2) - (311*Cos[c + d*x])/(8192*d*(5 - 3*Sin[c + d*x]))} - - -{1/(-5 - 3*Sin[c + d*x]), x, 1, -(x/4) - ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])]/(2*d)} -{1/(-5 - 3*Sin[c + d*x])^2, x, 3, (5*x)/64 + (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(32*d) + (3*Cos[c + d*x])/(16*d*(5 + 3*Sin[c + d*x]))} -{1/(-5 - 3*Sin[c + d*x])^3, x, 4, -((59*x)/2048) - (59*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(1024*d) - (3*Cos[c + d*x])/(32*d*(5 + 3*Sin[c + d*x])^2) - (45*Cos[c + d*x])/(512*d*(5 + 3*Sin[c + d*x]))} -{1/(-5 - 3*Sin[c + d*x])^4, x, 5, (385*x)/32768 + (385*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(16384*d) + Cos[c + d*x]/(16*d*(5 + 3*Sin[c + d*x])^3) + (25*Cos[c + d*x])/(512*d*(5 + 3*Sin[c + d*x])^2) + (311*Cos[c + d*x])/(8192*d*(5 + 3*Sin[c + d*x]))} - - -{1/(3 + 5*Sin[c + d*x]), x, 4, -(Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]]/(4*d)) + Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(3 + 5*Sin[c + d*x])^2, x, 6, (3*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(64*d) - (5*Cos[c + d*x])/(16*d*(3 + 5*Sin[c + d*x]))} -{1/(3 + 5*Sin[c + d*x])^3, x, 7, -((43*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(2048*d)) + (43*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(2048*d) - (5*Cos[c + d*x])/(32*d*(3 + 5*Sin[c + d*x])^2) + (45*Cos[c + d*x])/(512*d*(3 + 5*Sin[c + d*x]))} -{1/(3 + 5*Sin[c + d*x])^4, x, 8, (279*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(32768*d) - (5*Cos[c + d*x])/(48*d*(3 + 5*Sin[c + d*x])^3) + (25*Cos[c + d*x])/(512*d*(3 + 5*Sin[c + d*x])^2) - (995*Cos[c + d*x])/(24576*d*(3 + 5*Sin[c + d*x]))} - - -{1/(3 - 5*Sin[c + d*x]), x, 4, -(Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]]/(4*d)) + Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(3 - 5*Sin[c + d*x])^2, x, 6, (3*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(64*d) + (5*Cos[c + d*x])/(16*d*(3 - 5*Sin[c + d*x]))} -{1/(3 - 5*Sin[c + d*x])^3, x, 7, -((43*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(2048*d)) + (43*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(2048*d) + (5*Cos[c + d*x])/(32*d*(3 - 5*Sin[c + d*x])^2) - (45*Cos[c + d*x])/(512*d*(3 - 5*Sin[c + d*x]))} -{1/(3 - 5*Sin[c + d*x])^4, x, 8, (279*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(32768*d) + (5*Cos[c + d*x])/(48*d*(3 - 5*Sin[c + d*x])^3) - (25*Cos[c + d*x])/(512*d*(3 - 5*Sin[c + d*x])^2) + (995*Cos[c + d*x])/(24576*d*(3 - 5*Sin[c + d*x]))} - - -{1/(-3 + 5*Sin[c + d*x]), x, 4, Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]]/(4*d) - Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(-3 + 5*Sin[c + d*x])^2, x, 6, (3*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(64*d) + (5*Cos[c + d*x])/(16*d*(3 - 5*Sin[c + d*x]))} -{1/(-3 + 5*Sin[c + d*x])^3, x, 7, (43*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(2048*d) - (43*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(2048*d) - (5*Cos[c + d*x])/(32*d*(3 - 5*Sin[c + d*x])^2) + (45*Cos[c + d*x])/(512*d*(3 - 5*Sin[c + d*x]))} -{1/(-3 + 5*Sin[c + d*x])^4, x, 8, (279*Log[Cos[(1/2)*(c + d*x)] - 3*Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[3*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(32768*d) + (5*Cos[c + d*x])/(48*d*(3 - 5*Sin[c + d*x])^3) - (25*Cos[c + d*x])/(512*d*(3 - 5*Sin[c + d*x])^2) + (995*Cos[c + d*x])/(24576*d*(3 - 5*Sin[c + d*x]))} - - -{1/(-3 - 5*Sin[c + d*x]), x, 4, Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]]/(4*d) - Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(-3 - 5*Sin[c + d*x])^2, x, 6, (3*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(64*d) - (5*Cos[c + d*x])/(16*d*(3 + 5*Sin[c + d*x]))} -{1/(-3 - 5*Sin[c + d*x])^3, x, 7, (43*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(2048*d) - (43*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(2048*d) + (5*Cos[c + d*x])/(32*d*(3 + 5*Sin[c + d*x])^2) - (45*Cos[c + d*x])/(512*d*(3 + 5*Sin[c + d*x]))} -{1/(-3 - 5*Sin[c + d*x])^4, x, 8, (279*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(32768*d) - (5*Cos[c + d*x])/(48*d*(3 + 5*Sin[c + d*x])^3) + (25*Cos[c + d*x])/(512*d*(3 + 5*Sin[c + d*x])^2) - (995*Cos[c + d*x])/(24576*d*(3 + 5*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[c+d x])^(n/2)*) - - -{(a + b*Sin[c + d*x])^(7/2), x, 8, -((2*b*(71*a^2 + 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(105*d)) - (24*a*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(35*d) - (2*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*d) + (32*a*(11*a^2 + 13*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(105*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(105*d*Sqrt[a + b*Sin[c + d*x]])} -{(a + b*Sin[c + d*x])^(5/2), x, 7, -((16*a*b*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*d)) - (2*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*d) + (2*(23*a^2 + 9*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*d*Sqrt[a + b*Sin[c + d*x]])} -{(a + b*Sin[c + d*x])^(3/2), x, 6, -((2*b*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d)) + (8*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]])} -{(a + b*Sin[c + d*x])^(1/2), x, 2, (2*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])} -{1/(a + b*Sin[c + d*x])^(1/2), x, 2, (2*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{1/(a + b*Sin[c + d*x])^(3/2), x, 4, (2*b*Cos[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])} -{1/(a + b*Sin[c + d*x])^(5/2), x, 7, (2*b*Cos[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (8*a*b*Cos[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) + (8*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]])} -{1/(a + b*Sin[c + d*x])^(7/2), x, 8, (2*b*Cos[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(5/2)) + (16*a*b*Cos[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*b*(23*a^2 + 9*b^2)*Cos[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*(23*a^2 + 9*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*(a^2 - b^2)^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*a*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[c+d x])^(n/3)*) - - -{(a + b*Sin[c + d*x])^(4/3), x, 3, -((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1/3))/(d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^(1/3)))} -{(a + b*Sin[c + d*x])^(2/3), x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(2/3))/(d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^(2/3)))} -{(a + b*Sin[c + d*x])^(1/3), x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1/3))/(d*Sqrt[1 + Sin[c + d*x]]*((a + b*Sin[c + d*x])/(a + b))^(1/3)))} -{1/(a + b*Sin[c + d*x])^(1/3), x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*((a + b*Sin[c + d*x])/(a + b))^(1/3))/(d*Sqrt[1 + Sin[c + d*x]]*(a + b*Sin[c + d*x])^(1/3)))} -{1/(a + b*Sin[c + d*x])^(2/3), x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*((a + b*Sin[c + d*x])/(a + b))^(2/3))/(d*Sqrt[1 + Sin[c + d*x]]*(a + b*Sin[c + d*x])^(2/3)))} -{1/(a + b*Sin[c + d*x])^(4/3), x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, 4/3, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*((a + b*Sin[c + d*x])/(a + b))^(1/3))/((a + b)*d*Sqrt[1 + Sin[c + d*x]]*(a + b*Sin[c + d*x])^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[c+d x])^n with n symbolic*) - - -{(a + b*Sin[c + d*x])^n, x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(d*Sqrt[1 + Sin[c + d*x]])))} - - -{(3 + 4*Sin[c + d*x])^n, x, 2, -((Sqrt[2]*7^n*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (4/7)*(1 - Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 + Sin[c + d*x]]))} -{(3 - 4*Sin[c + d*x])^n, x, 2, (Sqrt[2]*7^n*AppellF1[1/2, -n, 1/2, 3/2, (4/7)*(1 + Sin[c + d*x]), (1/2)*(1 + Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 - Sin[c + d*x]])} - -{(4 + 3*Sin[c + d*x])^n, x, 2, (Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 + Sin[c + d*x]), -3*(1 + Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 - Sin[c + d*x]])} -{(4 - 3*Sin[c + d*x])^n, x, 2, (Sqrt[2]*7^n*AppellF1[1/2, -n, 1/2, 3/2, (3/7)*(1 + Sin[c + d*x]), (1/2)*(1 + Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 - Sin[c + d*x]])} - -{(-3 + 4*Sin[c + d*x])^n, x, 2, -((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), 4*(1 - Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 + Sin[c + d*x]]))} -{(-3 - 4*Sin[c + d*x])^n, x, 2, (Sqrt[2]*AppellF1[1/2, -n, 1/2, 3/2, 4*(1 + Sin[c + d*x]), (1/2)*(1 + Sin[c + d*x])]*Cos[c + d*x])/(d*Sqrt[1 - Sin[c + d*x]])} - -{(-4 + 3*Sin[c + d*x])^n, x, 3, (Sqrt[2]*7^n*AppellF1[1/2, -n, 1/2, 3/2, (3/7)*(1 + Sin[c + d*x]), (1/2)*(1 + Sin[c + d*x])]*Cos[c + d*x]*(-4 + 3*Sin[c + d*x])^n)/((4 - 3*Sin[c + d*x])^n*(d*Sqrt[1 - Sin[c + d*x]]))} -{(-4 - 3*Sin[c + d*x])^n, x, 3, -((AppellF1[1 + n, 1/2, 1/2, 2 + n, 4 + 3*Sin[c + d*x], (1/7)*(4 + 3*Sin[c + d*x])]*Cos[c + d*x]*(-4 - 3*Sin[c + d*x])^(1 + n)*Sqrt[-1 - Sin[c + d*x]])/(Sqrt[7]*d*(1 + n)*Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m deleted file mode 100644 index 7d55e78..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.2 (g cos)^p (a+b sin)^m.m +++ /dev/null @@ -1,1003 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m when a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^p (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x]), x, 3, (8*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(a + a*Sin[c + d*x])^6)/(a^5*d) + (6*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - (a + a*Sin[c + d*x])^8/(8*a^7*d)} -{Cos[c + d*x]^6*(a + a*Sin[c + d*x]), x, 5, (5*a*x)/16 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x]), x, 3, (a + a*Sin[c + d*x])^4/(a^3*d) - (4*(a + a*Sin[c + d*x])^5)/(5*a^4*d) + (a + a*Sin[c + d*x])^6/(6*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, (3*a*x)/8 - (a*Cos[c + d*x]^5)/(5*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x]), x, 3, (2*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - (a + a*Sin[c + d*x])^4/(4*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x]), x, 3, (a*x)/2 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x]), x, 2, (a + a*Sin[c + d*x])^2/(2*a*d), (a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x]), x, 2, -((a*Log[1 - Sin[c + d*x]])/d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x]), x, 3, (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x]), x, 4, (a*ArcTanh[Sin[c + d*x]])/(2*d) + a^2/(2*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x]), x, 3, (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x]), x, 4, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} - - -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 6, (45*a^2*x)/128 - (9*a^2*Cos[c + d*x]^7)/(56*d) + (45*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(8*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 3, (4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(3*a^4*d) + (a + a*Sin[c + d*x])^7/(7*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 5, (7*a^2*x)/16 - (7*a^2*Cos[c + d*x]^5)/(30*d) + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (7*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(6*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 3, (a + a*Sin[c + d*x])^4/(2*a^2*d) - (a + a*Sin[c + d*x])^5/(5*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, (5*a^2*x)/8 - (5*a^2*Cos[c + d*x]^3)/(12*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(4*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 2, (a + a*Sin[c + d*x])^3/(3*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 3, (-2*a^2*Log[1 - Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 3, (-a^2)*x + (2*a^4*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 2, a^3/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 3, (a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2) + (a^4*Cos[c + d*x])/(3*d*(a^2 - a^2*Sin[c + d*x]))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 4, (a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + a^3/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 3, (2*Sec[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(5*d) + (3*a^2*Tan[c + d*x])/(5*d) + (a^2*Tan[c + d*x]^3)/(5*d)} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 4, (a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^5/(12*d*(a - a*Sin[c + d*x])^3) + a^4/(8*d*(a - a*Sin[c + d*x])^2) + (3*a^3)/(16*d*(a - a*Sin[c + d*x])) - a^3/(16*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2, x, 3, (2*Sec[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(7*d) + (5*a^2*Tan[c + d*x])/(7*d) + (10*a^2*Tan[c + d*x]^3)/(21*d) + (a^2*Tan[c + d*x]^5)/(7*d)} - - -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 7, (55*a^3*x)/128 - (11*a^3*Cos[c + d*x]^7)/(56*d) + (55*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (55*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(72*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 3, (2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (a + a*Sin[c + d*x])^8/(8*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 6, (9*a^3*x)/16 - (3*a^3*Cos[c + d*x]^5)/(10*d) + (9*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - (3*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(14*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 3, (2*(a + a*Sin[c + d*x])^5)/(5*a^2*d) - (a + a*Sin[c + d*x])^6/(6*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 5, (7*a^3*x)/8 - (7*a^3*Cos[c + d*x]^3)/(12*d) + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - (7*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(20*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 2, (a + a*Sin[c + d*x])^4/(4*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 3, -((4*a^3*Log[1 - Sin[c + d*x]])/d) - (3*a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, -3*a^3*x + (3*a^3*Cos[c + d*x])/d + (2*a^5*Cos[c + d*x]^3)/(d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 3, (a^3*Log[1 - Sin[c + d*x]])/d + (2*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 2, (a^6*Cos[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^3)} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 2, a^5/(2*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 4, (a^6*Cos[c + d*x])/(5*d*(a - a*Sin[c + d*x])^3) + (2*a^5*Cos[c + d*x])/(15*d*(a - a*Sin[c + d*x])^2) + (2*a^6*Cos[c + d*x])/(15*d*(a^3 - a^3*Sin[c + d*x]))} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 4, (a^3*ArcTanh[Sin[c + d*x]])/(8*d) + a^6/(6*d*(a - a*Sin[c + d*x])^3) + a^5/(8*d*(a - a*Sin[c + d*x])^2) + a^4/(8*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3, x, 4, (3*a^3*Sec[c + d*x]^5)/(35*d) + (2*a*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(7*d) + (3*a^3*Tan[c + d*x])/(7*d) + (2*a^3*Tan[c + d*x]^3)/(7*d) + (3*a^3*Tan[c + d*x]^5)/(35*d)} - - -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^8, x, 3, (4*(a + a*Sin[c + d*x])^11)/(11*a^3*d) - (a + a*Sin[c + d*x])^12/(3*a^4*d) + (a + a*Sin[c + d*x])^13/(13*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8, x, 11, (4199*a^8*x)/1024 - (4199*a^8*Cos[c + d*x]^5)/(1920*d) + (4199*a^8*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (4199*a^8*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) - (323*a^3*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(1320*d) - (19*a^2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^6)/(132*d) - (a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^7)/(12*d) - (4199*a^2*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^3)/(6336*d) - (323*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^4)/(792*d) - (4199*Cos[c + d*x]^5*(a^4 + a^4*Sin[c + d*x])^2)/(4032*d) - (4199*Cos[c + d*x]^5*(a^8 + a^8*Sin[c + d*x]))/(2688*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^8, x, 3, (a + a*Sin[c + d*x])^10/(5*a^2*d) - (a + a*Sin[c + d*x])^11/(11*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^8, x, 10, (2431*a^8*x)/256 - (2431*a^8*Cos[c + d*x]^3)/(384*d) + (2431*a^8*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (17*a^3*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^5)/(48*d) - (17*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^6)/(90*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^7)/(10*d) - (2431*a^2*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^3)/(2016*d) - (221*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^4)/(336*d) - (2431*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x])^2)/(1120*d) - (2431*Cos[c + d*x]^3*(a^8 + a^8*Sin[c + d*x]))/(640*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^8, x, 2, (a + a*Sin[c + d*x])^9/(9*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^8, x, 3, (-128*a^8*Log[1 - Sin[c + d*x]])/d - (64*a^8*Sin[c + d*x])/d - (16*a^5*(a + a*Sin[c + d*x])^3)/(3*d) - (4*a^3*(a + a*Sin[c + d*x])^5)/(5*d) - (a^2*(a + a*Sin[c + d*x])^6)/(3*d) - (a*(a + a*Sin[c + d*x])^7)/(7*d) - (2*(a^2 + a^2*Sin[c + d*x])^4)/d - (16*(a^4 + a^4*Sin[c + d*x])^2)/d} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^8, x, 9, -((3003*a^8*x)/16) + (1001*a^8*Cos[c + d*x]^5)/(10*d) - (3003*a^8*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (1001*a^8*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (2*a^15*Cos[c + d*x]^13)/(d*(a - a*Sin[c + d*x])^7) + (26*a^13*Cos[c + d*x]^11)/(d*(a - a*Sin[c + d*x])^5) + (286*a^14*Cos[c + d*x]^9)/(3*d*(a^2 - a^2*Sin[c + d*x])^3) + (143*a^16*Cos[c + d*x]^7)/(2*d*(a^8 - a^8*Sin[c + d*x]))} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^8, x, 3, (192*a^8*Log[1 - Sin[c + d*x]])/d + (129*a^8*Sin[c + d*x])/d + (36*a^8*Sin[c + d*x]^2)/d + (10*a^8*Sin[c + d*x]^3)/d + (2*a^8*Sin[c + d*x]^4)/d + (a^8*Sin[c + d*x]^5)/(5*d) + (64*a^9)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^8, x, 8, (1155*a^8*x)/8 - (385*a^8*Cos[c + d*x]^3)/(4*d) + (1155*a^8*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (2*a^15*Cos[c + d*x]^11)/(3*d*(a - a*Sin[c + d*x])^7) - (22*a^13*Cos[c + d*x]^9)/(3*d*(a - a*Sin[c + d*x])^5) - (66*a^14*Cos[c + d*x]^7)/(d*(a^2 - a^2*Sin[c + d*x])^3) - (231*a^16*Cos[c + d*x]^5)/(4*d*(a^8 - a^8*Sin[c + d*x]))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^8, x, 3, (-80*a^8*Log[1 - Sin[c + d*x]])/d - (31*a^8*Sin[c + d*x])/d - (4*a^8*Sin[c + d*x]^2)/d - (a^8*Sin[c + d*x]^3)/(3*d) + (16*a^10)/(d*(a - a*Sin[c + d*x])^2) - (80*a^9)/(d*(a - a*Sin[c + d*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^6/(a + a*Sin[c + d*x]), x, 4, (3*x)/(8*a) + Cos[c + d*x]^5/(5*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x]), x, 3, -((2*(a - a*Sin[c + d*x])^3)/(3*a^4*d)) + (a - a*Sin[c + d*x])^4/(4*a^5*d)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x]), x, 3, x/(2*a) + Cos[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 2, Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 2, x/a + Cos[c + d*x]/(a*d)} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x]), x, 2, Log[1 + Sin[c + d*x]]/(a*d)} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, ArcTanh[Sin[c + d*x]]/(2*a*d) - 1/(2*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 3, -(Sec[c + d*x]/(3*d*(a + a*Sin[c + d*x]))) + (2*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, (3*ArcTanh[Sin[c + d*x]])/(8*a*d) + 1/(8*d*(a - a*Sin[c + d*x])) - a/(8*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x]), x, 3, -(Sec[c + d*x]^3/(5*d*(a + a*Sin[c + d*x]))) + (4*Tan[c + d*x])/(5*a*d) + (4*Tan[c + d*x]^3)/(15*a*d)} -{Sec[c + d*x]^5/(a + a*Sin[c + d*x]), x, 4, (5*ArcTanh[Sin[c + d*x]])/(16*a*d) + a/(32*d*(a - a*Sin[c + d*x])^2) + 1/(8*d*(a - a*Sin[c + d*x])) - a^2/(24*d*(a + a*Sin[c + d*x])^3) - (3*a)/(32*d*(a + a*Sin[c + d*x])^2) - 3/(16*d*(a + a*Sin[c + d*x]))} - - -{Cos[c + d*x]^8/(a + a*Sin[c + d*x])^2, x, 5, (7*x)/(16*a^2) + (7*Cos[c + d*x]^5)/(30*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (7*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^2*d) + Cos[c + d*x]^7/(6*d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 3, -((a - a*Sin[c + d*x])^4/(2*a^6*d)) + (a - a*Sin[c + d*x])^5/(5*a^7*d)} -{Cos[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 4, (5*x)/(8*a^2) + (5*Cos[c + d*x]^3)/(12*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + Cos[c + d*x]^5/(4*d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 2, -(a - a*Sin[c + d*x])^3/(3*a^5*d)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 3, (3*x)/(2*a^2) + (3*Cos[c + d*x])/(2*a^2*d) + Cos[c + d*x]^3/(2*d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 3, (2*Log[1 + Sin[c + d*x]])/(a^2*d) - Sin[c + d*x]/(a^2*d)} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 2, -(x/a^2) - (2*Cos[c + d*x])/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 2, -(1/(d*(a^2 + a^2*Sin[c + d*x])))} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(4*a^2*d) - 1/(4*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a^2 + a^2*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 4, -(Sec[c + d*x]/(5*d*(a + a*Sin[c + d*x])^2)) - Sec[c + d*x]/(5*d*(a^2 + a^2*Sin[c + d*x])) + (2*Tan[c + d*x])/(5*a^2*d)} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(4*a^2*d) - a/(12*d*(a + a*Sin[c + d*x])^3) - 1/(8*d*(a + a*Sin[c + d*x])^2) + 1/(16*d*(a^2 - a^2*Sin[c + d*x])) - 3/(16*d*(a^2 + a^2*Sin[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 4, -(Sec[c + d*x]^3/(7*d*(a + a*Sin[c + d*x])^2)) - Sec[c + d*x]^3/(7*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*x])/(7*a^2*d) + (4*Tan[c + d*x]^3)/(21*a^2*d)} -{Sec[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 4, (15*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + 1/(64*d*(a - a*Sin[c + d*x])^2) - a^2/(32*d*(a + a*Sin[c + d*x])^4) - a/(16*d*(a + a*Sin[c + d*x])^3) - 3/(32*d*(a + a*Sin[c + d*x])^2) + 5/(64*d*(a^2 - a^2*Sin[c + d*x])) - 5/(32*d*(a^2 + a^2*Sin[c + d*x]))} - - -{Cos[c + d*x]^8/(a + a*Sin[c + d*x])^3, x, 5, (7*x)/(8*a^3) + (7*Cos[c + d*x]^5)/(15*a^3*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (7*Cos[c + d*x]^3*Sin[c + d*x])/(12*a^3*d) + (2*Cos[c + d*x]^7)/(3*a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^7/(a + a*Sin[c + d*x])^3, x, 2, -((a - a*Sin[c + d*x])^4/(4*a^7*d))} -{Cos[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 4, (5*x)/(2*a^3) + (5*Cos[c + d*x]^3)/(3*a^3*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x]^5)/(a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 3, (4*Log[1 + Sin[c + d*x]])/(a^3*d) - (3*Sin[c + d*x])/(a^3*d) + Sin[c + d*x]^2/(2*a^3*d)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 3, (-3*x)/a^3 - (3*Cos[c + d*x])/(a^3*d) - (2*Cos[c + d*x]^3)/(a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 3, -(Log[1 + Sin[c + d*x]]/(a^3*d)) - 2/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 1, -Cos[c + d*x]^3/(3*d*(a + a*Sin[c + d*x])^3)} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 2, -1/(2*a*d*(a + a*Sin[c + d*x])^2)} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 4, ArcTanh[Sin[c + d*x]]/(8*a^3*d) - 1/(6*d*(a + a*Sin[c + d*x])^3) - 1/(8*a*d*(a + a*Sin[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 5, -(Sec[c + d*x]/(7*d*(a + a*Sin[c + d*x])^3)) - (4*Sec[c + d*x])/(35*a*d*(a + a*Sin[c + d*x])^2) - (4*Sec[c + d*x])/(35*d*(a^3 + a^3*Sin[c + d*x])) + (8*Tan[c + d*x])/(35*a^3*d)} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, (5*ArcTanh[Sin[c + d*x]])/(32*a^3*d) - a/(16*d*(a + a*Sin[c + d*x])^4) - 1/(12*d*(a + a*Sin[c + d*x])^3) - 3/(32*a*d*(a + a*Sin[c + d*x])^2) + 1/(32*d*(a^3 - a^3*Sin[c + d*x])) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 5, -(Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3)) - (2*Sec[c + d*x]^3)/(21*a*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x]^3)/(21*d*(a^3 + a^3*Sin[c + d*x])) + (8*Tan[c + d*x])/(21*a^3*d) + (8*Tan[c + d*x]^3)/(63*a^3*d)} -{Sec[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 4, (21*ArcTanh[Sin[c + d*x]])/(128*a^3*d) + 1/(128*a*d*(a - a*Sin[c + d*x])^2) - a^2/(40*d*(a + a*Sin[c + d*x])^5) - (3*a)/(64*d*(a + a*Sin[c + d*x])^4) - 1/(16*d*(a + a*Sin[c + d*x])^3) - 5/(64*a*d*(a + a*Sin[c + d*x])^2) + 3/(64*d*(a^3 - a^3*Sin[c + d*x])) - 15/(128*d*(a^3 + a^3*Sin[c + d*x]))} - - -{Cos[c + d*x]^8/(a + a*Sin[c + d*x])^8, x, 5, x/a^8 - (2*Cos[c + d*x]^7)/(7*a*d*(a + a*Sin[c + d*x])^7) + (2*Cos[c + d*x]^5)/(5*a^3*d*(a + a*Sin[c + d*x])^5) - (2*Cos[c + d*x]^3)/(3*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) + (2*Cos[c + d*x])/(d*(a^8 + a^8*Sin[c + d*x]))} -{Cos[c + d*x]^7/(a + a*Sin[c + d*x])^8, x, 2, -((a - a*Sin[c + d*x])^4/(8*d*(a^3 + a^3*Sin[c + d*x])^4))} -{Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8, x, 2, -(Cos[c + d*x]^7/(9*d*(a + a*Sin[c + d*x])^8)) - Cos[c + d*x]^7/(63*a*d*(a + a*Sin[c + d*x])^7)} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8, x, 3, -4/(5*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(3*a^5*d*(a + a*Sin[c + d*x])^3) + 1/(d*(a^2 + a^2*Sin[c + d*x])^4)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x])^8, x, 4, -Cos[c + d*x]^5/(11*d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^5/(33*a*d*(a + a*Sin[c + d*x])^7) - (2*Cos[c + d*x]^5)/(231*a^2*d*(a + a*Sin[c + d*x])^6) - (2*Cos[c + d*x]^5)/(1155*a^3*d*(a + a*Sin[c + d*x])^5)} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x])^8, x, 3, -1/(3*a^2*d*(a + a*Sin[c + d*x])^6) + 1/(5*a^3*d*(a + a*Sin[c + d*x])^5)} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x])^8, x, 6, -Cos[c + d*x]^3/(13*d*(a + a*Sin[c + d*x])^8) - (5*Cos[c + d*x]^3)/(143*a*d*(a + a*Sin[c + d*x])^7) - (20*Cos[c + d*x]^3)/(1287*a^2*d*(a + a*Sin[c + d*x])^6) - (20*Cos[c + d*x]^3)/(3003*a^3*d*(a + a*Sin[c + d*x])^5) - (8*Cos[c + d*x]^3)/(3003*d*(a^2 + a^2*Sin[c + d*x])^4) - (8*Cos[c + d*x]^3)/(9009*a^2*d*(a^2 + a^2*Sin[c + d*x])^3)} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x])^8, x, 2, -1/(7*a*d*(a + a*Sin[c + d*x])^7)} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x])^8, x, 4, ArcTanh[Sin[c + d*x]]/(256*a^8*d) - 1/(16*d*(a + a*Sin[c + d*x])^8) - 1/(28*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(80*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(192*a^5*d*(a + a*Sin[c + d*x])^3) - 1/(128*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(256*d*(a^4 + a^4*Sin[c + d*x])^2) - 1/(256*d*(a^8 + a^8*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8, x, 10, -(Sec[c + d*x]/(17*d*(a + a*Sin[c + d*x])^8)) - (3*Sec[c + d*x])/(85*a*d*(a + a*Sin[c + d*x])^7) - (24*Sec[c + d*x])/(1105*a^2*d*(a + a*Sin[c + d*x])^6) - (168*Sec[c + d*x])/(12155*a^3*d*(a + a*Sin[c + d*x])^5) - (112*Sec[c + d*x])/(12155*d*(a^2 + a^2*Sin[c + d*x])^4) - (16*Sec[c + d*x])/(2431*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (64*Sec[c + d*x])/(12155*d*(a^4 + a^4*Sin[c + d*x])^2) - (64*Sec[c + d*x])/(12155*d*(a^8 + a^8*Sin[c + d*x])) + (128*Tan[c + d*x])/(12155*a^8*d)} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x])^8, x, 4, (5*ArcTanh[Sin[c + d*x]])/(512*a^8*d) - a/(36*d*(a + a*Sin[c + d*x])^9) - 1/(32*d*(a + a*Sin[c + d*x])^8) - 3/(112*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(64*a^3*d*(a + a*Sin[c + d*x])^5) - 7/(768*a^5*d*(a + a*Sin[c + d*x])^3) - 3/(256*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(128*d*(a^4 + a^4*Sin[c + d*x])^2) + 1/(1024*d*(a^8 - a^8*Sin[c + d*x])) - 9/(1024*d*(a^8 + a^8*Sin[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x])^8, x, 10, -(Sec[c + d*x]^3/(19*d*(a + a*Sin[c + d*x])^8)) - (11*Sec[c + d*x]^3)/(323*a*d*(a + a*Sin[c + d*x])^7) - (22*Sec[c + d*x]^3)/(969*a^2*d*(a + a*Sin[c + d*x])^6) - (66*Sec[c + d*x]^3)/(4199*a^3*d*(a + a*Sin[c + d*x])^5) - (48*Sec[c + d*x]^3)/(4199*d*(a^2 + a^2*Sin[c + d*x])^4) - (112*Sec[c + d*x]^3)/(12597*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (32*Sec[c + d*x]^3)/(4199*d*(a^4 + a^4*Sin[c + d*x])^2) - (32*Sec[c + d*x]^3)/(4199*d*(a^8 + a^8*Sin[c + d*x])) + (128*Tan[c + d*x])/(4199*a^8*d) + (128*Tan[c + d*x]^3)/(12597*a^8*d)} -{Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8, x, 4, (33*ArcTanh[Sin[c + d*x]])/(2048*a^8*d) - a^2/(80*d*(a + a*Sin[c + d*x])^10) - a/(48*d*(a + a*Sin[c + d*x])^9) - 3/(128*d*(a + a*Sin[c + d*x])^8) - 5/(224*a*d*(a + a*Sin[c + d*x])^7) - 5/(256*a^2*d*(a + a*Sin[c + d*x])^6) - 21/(1280*a^3*d*(a + a*Sin[c + d*x])^5) - 3/(256*a^5*d*(a + a*Sin[c + d*x])^3) - 7/(512*d*(a^2 + a^2*Sin[c + d*x])^4) + 1/(4096*d*(a^4 - a^4*Sin[c + d*x])^2) - 45/(4096*d*(a^4 + a^4*Sin[c + d*x])^2) + 11/(4096*d*(a^8 - a^8*Sin[c + d*x])) - 55/(4096*d*(a^8 + a^8*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^p (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]], x, 3, (16*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) - (24*(a + a*Sin[c + d*x])^(11/2))/(11*a^5*d) + (12*(a + a*Sin[c + d*x])^(13/2))/(13*a^6*d) - (2*(a + a*Sin[c + d*x])^(15/2))/(15*a^7*d)} -{Cos[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]], x, 4, -((256*a^4*Cos[c + d*x]^7)/(3003*d*(a + a*Sin[c + d*x])^(7/2))) - (64*a^3*Cos[c + d*x]^7)/(429*d*(a + a*Sin[c + d*x])^(5/2)) - (24*a^2*Cos[c + d*x]^7)/(143*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]], x, 3, (8*(a + a*Sin[c + d*x])^(7/2))/(7*a^3*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) + (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^5*d)} -{Cos[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 3, (-64*a^3*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2)) - (16*a^2*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^5)/(9*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 3, (4*(a + a*Sin[c + d*x])^(5/2))/(5*a^2*d) - (2*(a + a*Sin[c + d*x])^(7/2))/(7*a^3*d)} -{Cos[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 2, (-8*a^2*Cos[c + d*x]^3)/(15*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^3)/(5*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 2, (2*(a + a*Sin[c + d*x])^(3/2))/(3*a*d)} -{Sec[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 3, (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 3, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[2]*d)) + (Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d} -{Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 5, (3*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*d) - (3*a)/(4*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(2*d)} -{Sec[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 5, -((5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8*Sqrt[2]*d)) - (5*a^2*Cos[c + d*x])/(8*d*(a + a*Sin[c + d*x])^(3/2)) + (5*a*Sec[c + d*x])/(6*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Sec[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]], x, 7, (35*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) - (35*a^2)/(96*d*(a + a*Sin[c + d*x])^(3/2)) - (35*a)/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (7*a*Sec[c + d*x]^2)/(16*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(4*d)} -{Sec[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]], x, 7, -((63*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(128*Sqrt[2]*d)) - (63*a^2*Cos[c + d*x])/(128*d*(a + a*Sin[c + d*x])^(3/2)) - (21*a^2*Sec[c + d*x])/(80*d*(a + a*Sin[c + d*x])^(3/2)) + (21*a*Sec[c + d*x])/(32*d*Sqrt[a + a*Sin[c + d*x]]) + (3*a*Sec[c + d*x]^3)/(10*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(5*d)} - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2), x, 3, (16*(a + a*Sin[c + d*x])^(11/2))/(11*a^4*d) - (24*(a + a*Sin[c + d*x])^(13/2))/(13*a^5*d) + (4*(a + a*Sin[c + d*x])^(15/2))/(5*a^6*d) - (2*(a + a*Sin[c + d*x])^(17/2))/(17*a^7*d)} -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2), x, 5, -((4096*a^5*Cos[c + d*x]^7)/(45045*d*(a + a*Sin[c + d*x])^(7/2))) - (1024*a^4*Cos[c + d*x]^7)/(6435*d*(a + a*Sin[c + d*x])^(5/2)) - (128*a^3*Cos[c + d*x]^7)/(715*d*(a + a*Sin[c + d*x])^(3/2)) - (32*a^2*Cos[c + d*x]^7)/(195*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(15*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2), x, 3, (8*(a + a*Sin[c + d*x])^(9/2))/(9*a^3*d) - (8*(a + a*Sin[c + d*x])^(11/2))/(11*a^4*d) + (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 4, (-256*a^4*Cos[c + d*x]^5)/(1155*d*(a + a*Sin[c + d*x])^(5/2)) - (64*a^3*Cos[c + d*x]^5)/(231*d*(a + a*Sin[c + d*x])^(3/2)) - (8*a^2*Cos[c + d*x]^5)/(33*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 3, (4*(a + a*Sin[c + d*x])^(7/2))/(7*a^2*d) - (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 3, (-64*a^3*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2)) - (16*a^2*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 2, (2*(a + a*Sin[c + d*x])^(5/2))/(5*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (2*a*Sqrt[a + a*Sin[c + d*x]])/d} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 1, (2*a*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 4, (a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d) + (Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(2*d)} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 4, -((a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*d)) + (a*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d)} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2), x, 6, (15*a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*d) - (15*a^2)/(32*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(4*d)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2), x, 6, -((7*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*d)) - (7*a^3*Cos[c + d*x])/(16*d*(a + a*Sin[c + d*x])^(3/2)) + (7*a^2*Sec[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) + (7*a*Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(30*d) + (Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(5*d)} - - -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2), x, 3, (8*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d) - (8*(a + a*Sin[c + d*x])^(13/2))/(13*a^4*d) + (2*(a + a*Sin[c + d*x])^(15/2))/(15*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2), x, 5, (-4096*a^5*Cos[c + d*x]^5)/(15015*d*(a + a*Sin[c + d*x])^(5/2)) - (1024*a^4*Cos[c + d*x]^5)/(3003*d*(a + a*Sin[c + d*x])^(3/2)) - (128*a^3*Cos[c + d*x]^5)/(429*d*Sqrt[a + a*Sin[c + d*x]]) - (32*a^2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2), x, 3, (4*(a + a*Sin[c + d*x])^(9/2))/(9*a^2*d) - (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2), x, 4, (-256*a^4*Cos[c + d*x]^3)/(315*d*(a + a*Sin[c + d*x])^(3/2)) - (64*a^3*Cos[c + d*x]^3)/(105*d*Sqrt[a + a*Sin[c + d*x]]) - (8*a^2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^(5/2), x, 2, (2*(a + a*Sin[c + d*x])^(7/2))/(7*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^(5/2), x, 5, (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (4*a^2*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*(a + a*Sin[c + d*x])^(3/2))/(3*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2), x, 2, (8*a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2), x, 4, -((a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d)) + (a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2), x, 1, (2*a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d)} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2), x, 5, (3*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*d) + (3*a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(16*d) + (Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(4*d)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2), x, 5, -((a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4*Sqrt[2]*d)) + (a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) + (a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(6*d) + (Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(5*d)} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2), x, 7, (35*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*d) - (35*a^3)/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (35*a^2*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(192*d) + (7*a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(48*d) + (Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2))/(6*d)} - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2), x, 3, (16*(a + a*Sin[c + d*x])^(15/2))/(15*a^4*d) - (24*(a + a*Sin[c + d*x])^(17/2))/(17*a^5*d) + (12*(a + a*Sin[c + d*x])^(19/2))/(19*a^6*d) - (2*(a + a*Sin[c + d*x])^(21/2))/(21*a^7*d)} -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2), x, 7, -((131072*a^7*Cos[c + d*x]^7)/(969969*d*(a + a*Sin[c + d*x])^(7/2))) - (32768*a^6*Cos[c + d*x]^7)/(138567*d*(a + a*Sin[c + d*x])^(5/2)) - (12288*a^5*Cos[c + d*x]^7)/(46189*d*(a + a*Sin[c + d*x])^(3/2)) - (1024*a^4*Cos[c + d*x]^7)/(4199*d*Sqrt[a + a*Sin[c + d*x]]) - (64*a^3*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(323*d) - (48*a^2*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(323*d) - (2*a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(19*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2), x, 3, (8*(a + a*Sin[c + d*x])^(13/2))/(13*a^3*d) - (8*(a + a*Sin[c + d*x])^(15/2))/(15*a^4*d) + (2*(a + a*Sin[c + d*x])^(17/2))/(17*a^5*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2), x, 6, -((16384*a^6*Cos[c + d*x]^5)/(45045*d*(a + a*Sin[c + d*x])^(5/2))) - (4096*a^5*Cos[c + d*x]^5)/(9009*d*(a + a*Sin[c + d*x])^(3/2)) - (512*a^4*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) - (128*a^3*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(429*d) - (8*a^2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(39*d) - (2*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(15*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2), x, 3, (4*(a + a*Sin[c + d*x])^(11/2))/(11*a^2*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^3*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2), x, 5, -((4096*a^5*Cos[c + d*x]^3)/(3465*d*(a + a*Sin[c + d*x])^(3/2))) - (1024*a^4*Cos[c + d*x]^3)/(1155*d*Sqrt[a + a*Sin[c + d*x]]) - (128*a^3*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(231*d) - (32*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(99*d) - (2*a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/(11*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^(7/2), x, 2, (2*(a + a*Sin[c + d*x])^(9/2))/(9*a*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^(7/2), x, 6, (8*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*a^3*Sqrt[a + a*Sin[c + d*x]])/d - (4*a^2*(a + a*Sin[c + d*x])^(3/2))/(3*d) - (2*a*(a + a*Sin[c + d*x])^(5/2))/(5*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2), x, 3, (64*a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (16*a^2*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(3*d) - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(3*d)} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2), x, 5, -((3*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d) + (3*a^3*Sqrt[a + a*Sin[c + d*x]])/d + (a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2))/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2), x, 2, -((8*a^2*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d)) + (2*a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/d} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2), x, 5, -((a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*d)) - (a^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(8*d) + (a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(2*d)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2), x, 1, (2*a*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(5*d)} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2), x, 6, (5*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) + (5*a^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(64*d) + (5*a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(48*d) + (Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2))/(6*d)} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])^(7/2), x, 6, -((a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8*Sqrt[2]*d)) + (a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(8*d) + (a^2*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(12*d) + (a*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(10*d) + (Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2))/(7*d)} -{Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2), x, 8, (315*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2048*Sqrt[2]*d) - (315*a^4)/(2048*d*Sqrt[a + a*Sin[c + d*x]]) + (105*a^3*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(1024*d) + (21*a^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(256*d) + (3*a*Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2))/(32*d) + (Sec[c + d*x]^8*(a + a*Sin[c + d*x])^(7/2))/(8*d)} -{Sec[c + d*x]^10*(a + a*Sin[c + d*x])^(7/2), x, 8, -((11*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(64*Sqrt[2]*d)) - (11*a^5*Cos[c + d*x])/(64*d*(a + a*Sin[c + d*x])^(3/2)) + (11*a^4*Sec[c + d*x])/(48*d*Sqrt[a + a*Sin[c + d*x]]) + (11*a^3*Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(120*d) + (11*a^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(140*d) + (11*a*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(126*d) + (Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2))/(9*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^7/Sqrt[a + a*Sin[c + d*x]], x, 3, (16*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(3*a^5*d) + (12*(a + a*Sin[c + d*x])^(11/2))/(11*a^6*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^7*d)} -{Cos[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]], x, 3, -((64*a^3*Cos[c + d*x]^7)/(693*d*(a + a*Sin[c + d*x])^(7/2))) - (16*a^2*Cos[c + d*x]^7)/(99*d*(a + a*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2))} -{Cos[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]], x, 3, (8*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d) - (8*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) + (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^5*d)} -{Cos[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 2, (-8*a^2*Cos[c + d*x]^5)/(35*d*(a + a*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(3/2))} -{Cos[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 3, (4*(a + a*Sin[c + d*x])^(3/2))/(3*a^2*d) - (2*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d)} -{Cos[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 1, (-2*a*Cos[c + d*x]^3)/(3*d*(a + a*Sin[c + d*x])^(3/2))} -{Cos[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 2, (2*Sqrt[a + a*Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 4, ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) - 1/(d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 4, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4*Sqrt[2]*Sqrt[a]*d)) - (3*a*Cos[c + d*x])/(4*d*(a + a*Sin[c + d*x])^(3/2)) + Sec[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 6, (5*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d) - (5*a)/(12*d*(a + a*Sin[c + d*x])^(3/2)) - 5/(8*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^2/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 6, -((35*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(64*Sqrt[2]*Sqrt[a]*d)) - (35*a*Cos[c + d*x])/(64*d*(a + a*Sin[c + d*x])^(3/2)) - (7*a*Sec[c + d*x])/(24*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(48*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^3/(3*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]], x, 8, (63*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*Sqrt[a]*d) - (21*a)/(64*d*(a + a*Sin[c + d*x])^(3/2)) - (9*a*Sec[c + d*x]^2)/(40*d*(a + a*Sin[c + d*x])^(3/2)) - 63/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (63*Sec[c + d*x]^2)/(160*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^4/(4*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]], x, 8, -((231*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d)) - (231*a*Cos[c + d*x])/(512*d*(a + a*Sin[c + d*x])^(3/2)) - (77*a*Sec[c + d*x])/(320*d*(a + a*Sin[c + d*x])^(3/2)) - (11*a*Sec[c + d*x]^3)/(60*d*(a + a*Sin[c + d*x])^(3/2)) + (77*Sec[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^5/(5*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(3/2), x, 3, (16*(a + a*Sin[c + d*x])^(5/2))/(5*a^4*d) - (24*(a + a*Sin[c + d*x])^(7/2))/(7*a^5*d) + (4*(a + a*Sin[c + d*x])^(9/2))/(3*a^6*d) - (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^7*d)} -{Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2), x, 2, -((8*a^2*Cos[c + d*x]^7)/(63*d*(a + a*Sin[c + d*x])^(7/2))) - (2*a*Cos[c + d*x]^7)/(9*d*(a + a*Sin[c + d*x])^(5/2))} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2), x, 3, (8*(a + a*Sin[c + d*x])^(3/2))/(3*a^3*d) - (8*(a + a*Sin[c + d*x])^(5/2))/(5*a^4*d) + (2*(a + a*Sin[c + d*x])^(7/2))/(7*a^5*d)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 1, (-2*a*Cos[c + d*x]^5)/(5*d*(a + a*Sin[c + d*x])^(5/2))} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 3, (4*Sqrt[a + a*Sin[c + d*x]])/(a^2*d) - (2*(a + a*Sin[c + d*x])^(3/2))/(3*a^3*d)} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 3, (-2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (2*Cos[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 2, -2/(a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 5, ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - 1/(3*d*(a + a*Sin[c + d*x])^(3/2)) - 1/(2*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, -((15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(32*Sqrt[2]*a^(3/2)*d)) - (15*Cos[c + d*x])/(32*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(3/2)) + (5*Sec[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 7, (7*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*a^(3/2)*d) - 7/(24*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^2/(5*d*(a + a*Sin[c + d*x])^(3/2)) - 7/(16*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Sec[c + d*x]^2)/(20*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 7, -((105*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d)) - (105*Cos[c + d*x])/(256*d*(a + a*Sin[c + d*x])^(3/2)) - (7*Sec[c + d*x])/(32*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^3/(6*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^3/(4*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2), x, 9, (99*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*a^(3/2)*d) - 33/(128*d*(a + a*Sin[c + d*x])^(3/2)) - (99*Sec[c + d*x]^2)/(560*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^4/(7*d*(a + a*Sin[c + d*x])^(3/2)) - 99/(256*a*d*Sqrt[a + a*Sin[c + d*x]]) + (99*Sec[c + d*x]^2)/(320*a*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^4)/(56*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2), x, 9, -((3003*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8192*Sqrt[2]*a^(3/2)*d)) - (3003*Cos[c + d*x])/(8192*d*(a + a*Sin[c + d*x])^(3/2)) - (1001*Sec[c + d*x])/(5120*d*(a + a*Sin[c + d*x])^(3/2)) - (143*Sec[c + d*x]^3)/(960*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^5/(8*d*(a + a*Sin[c + d*x])^(3/2)) + (1001*Sec[c + d*x])/(2048*a*d*Sqrt[a + a*Sin[c + d*x]]) + (143*Sec[c + d*x]^3)/(640*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Sec[c + d*x]^5)/(80*a*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Cos[c + d*x]^10/(a + a*Sin[c + d*x])^(5/2), x, 3, -((64*a^3*Cos[c + d*x]^11)/(2145*d*(a + a*Sin[c + d*x])^(11/2))) - (16*a^2*Cos[c + d*x]^11)/(195*d*(a + a*Sin[c + d*x])^(9/2)) - (2*a*Cos[c + d*x]^11)/(15*d*(a + a*Sin[c + d*x])^(7/2))} -{Cos[c + d*x]^9/(a + a*Sin[c + d*x])^(5/2), x, 3, (32*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d) - (64*(a + a*Sin[c + d*x])^(7/2))/(7*a^6*d) + (16*(a + a*Sin[c + d*x])^(9/2))/(3*a^7*d) - (16*(a + a*Sin[c + d*x])^(11/2))/(11*a^8*d) + (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^9*d)} -{Cos[c + d*x]^8/(a + a*Sin[c + d*x])^(5/2), x, 2, -((8*a^2*Cos[c + d*x]^9)/(99*d*(a + a*Sin[c + d*x])^(9/2))) - (2*a*Cos[c + d*x]^9)/(11*d*(a + a*Sin[c + d*x])^(7/2))} -{Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(5/2), x, 3, (16*(a + a*Sin[c + d*x])^(3/2))/(3*a^4*d) - (24*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d) + (12*(a + a*Sin[c + d*x])^(7/2))/(7*a^6*d) - (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^7*d)} -{Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(5/2), x, 1, -((2*a*Cos[c + d*x]^7)/(7*d*(a + a*Sin[c + d*x])^(7/2)))} -{Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2), x, 3, (8*Sqrt[a + a*Sin[c + d*x]])/(a^3*d) - (8*(a + a*Sin[c + d*x])^(3/2))/(3*a^4*d) + (2*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d)} -{Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2), x, 4, (-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 3, -4/(a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Sqrt[a + a*Sin[c + d*x]])/(a^3*d)} -{Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 3, ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(Sqrt[2]*a^(5/2)*d) - Cos[c + d*x]/(a*d*(a + a*Sin[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 2, -2/(3*a*d*(a + a*Sin[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 6, ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - 1/(5*d*(a + a*Sin[c + d*x])^(5/2)) - 1/(6*a*d*(a + a*Sin[c + d*x])^(3/2)) - 1/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 6, -((35*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(128*Sqrt[2]*a^(5/2)*d)) - Sec[c + d*x]/(6*d*(a + a*Sin[c + d*x])^(5/2)) - (35*Cos[c + d*x])/(128*a*d*(a + a*Sin[c + d*x])^(3/2)) - (7*Sec[c + d*x])/(48*a*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(96*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 8, (9*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*a^(5/2)*d) - Sec[c + d*x]^2/(7*d*(a + a*Sin[c + d*x])^(5/2)) - 3/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (9*Sec[c + d*x]^2)/(70*a*d*(a + a*Sin[c + d*x])^(3/2)) - 9/(32*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (9*Sec[c + d*x]^2)/(40*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2), x, 8, -((1155*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d)) - Sec[c + d*x]^3/(8*d*(a + a*Sin[c + d*x])^(5/2)) - (1155*Cos[c + d*x])/(4096*a*d*(a + a*Sin[c + d*x])^(3/2)) - (77*Sec[c + d*x])/(512*a*d*(a + a*Sin[c + d*x])^(3/2)) - (11*Sec[c + d*x]^3)/(96*a*d*(a + a*Sin[c + d*x])^(3/2)) + (385*Sec[c + d*x])/(1024*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^3)/(64*a^2*d*Sqrt[a + a*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(p/2) (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x]), x, 5, (-2*a*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x]), x, 4, (-2*a*(e*Cos[c + d*x])^(7/2))/(7*d*e) + (6*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x]), x, 4, (-2*a*(e*Cos[c + d*x])^(5/2))/(5*d*e) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*a*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x]), x, 3, (-2*a*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]], x, 3, (-2*a*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])} -{(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2), x, 4, (2*a)/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2), x, 4, (2*a)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2), x, 5, (2*a)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (6*a*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]])} - - -{(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2, x, 6, (-26*a^2*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (130*a^2*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (130*a^2*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (26*a^2*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*(e*Cos[c + d*x])^(9/2)*(a^2 + a^2*Sin[c + d*x]))/(11*d*e)} -{(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2, x, 5, (-22*a^2*(e*Cos[c + d*x])^(7/2))/(63*d*e) + (22*a^2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (22*a^2*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) - (2*(e*Cos[c + d*x])^(7/2)*(a^2 + a^2*Sin[c + d*x]))/(9*d*e)} -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2, x, 5, (-18*a^2*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (6*a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) + (6*a^2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*d) - (2*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x]))/(7*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2, x, 4, (-14*a^2*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (14*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x]))/(5*d*e)} -{(a + a*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]], x, 4, (-10*a^2*Sqrt[e*Cos[c + d*x]])/(3*d*e) + (10*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x]))/(3*d*e)} -{(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2), x, 4, -((6*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*e^2*Sqrt[Cos[c + d*x]])) + (4*a^4*(e*Cos[c + d*x])^(3/2))/(d*e^3*(a^2 - a^2*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2), x, 4, -((2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]])) + (4*a^4*Sqrt[e*Cos[c + d*x]])/(3*d*e^3*(a^2 - a^2*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2), x, 5, -((2*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^4*Sqrt[Cos[c + d*x]])) + (2*a^4*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a - a*Sin[c + d*x])^2) + (2*a^4*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^2 - a^2*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(9/2), x, 4, (2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(7*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*a^2*Sin[c + d*x])/(7*d*e^3*(e*Cos[c + d*x])^(3/2)) + (4*(a^2 + a^2*Sin[c + d*x]))/(7*d*e*(e*Cos[c + d*x])^(7/2))} -{(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(11/2), x, 5, -((2*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(3*d*e^6*Sqrt[Cos[c + d*x]])) + (2*a^2*Sin[c + d*x])/(9*d*e^3*(e*Cos[c + d*x])^(5/2)) + (2*a^2*Sin[c + d*x])/(3*d*e^5*Sqrt[e*Cos[c + d*x]]) + (4*(a^2 + a^2*Sin[c + d*x]))/(9*d*e*(e*Cos[c + d*x])^(9/2))} - - -{(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^3, x, 7, (-34*a^3*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (170*a^3*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (170*a^3*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (34*a^3*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*a*(e*Cos[c + d*x])^(9/2)*(a + a*Sin[c + d*x])^2)/(13*d*e) - (34*(e*Cos[c + d*x])^(9/2)*(a^3 + a^3*Sin[c + d*x]))/(143*d*e)} -{(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^3, x, 6, (-10*a^3*(e*Cos[c + d*x])^(7/2))/(21*d*e) + (2*a^3*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*a^3*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) - (2*a*(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2)/(11*d*e) - (10*(e*Cos[c + d*x])^(7/2)*(a^3 + a^3*Sin[c + d*x]))/(33*d*e)} -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3, x, 6, (-26*a^3*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (26*a^3*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (26*a^3*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2)/(9*d*e) - (26*(e*Cos[c + d*x])^(5/2)*(a^3 + a^3*Sin[c + d*x]))/(63*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3, x, 5, (-22*a^3*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (22*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2)/(7*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^3 + a^3*Sin[c + d*x]))/(35*d*e)} -{(a + a*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]], x, 5, (-6*a^3*Sqrt[e*Cos[c + d*x]])/(d*e) + (6*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2)/(5*d*e) - (6*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))/(5*d*e)} -{(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2), x, 5, (14*a^3*(e*Cos[c + d*x])^(3/2))/(3*d*e^3) - (14*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(7/2))/(d*e^5*(a - a*Sin[c + d*x])^2)} -{(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2), x, 5, (10*a^3*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) - (10*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(5/2))/(3*d*e^5*(a - a*Sin[c + d*x])^2)} -{(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2), x, 5, (6*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a - a*Sin[c + d*x])^2) - (6*a^6*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^3 - a^3*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2), x, 5, -((2*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]])) + (4*a^5*Sqrt[e*Cos[c + d*x]])/(7*d*e^5*(a - a*Sin[c + d*x])^2) - (2*a^6*Sqrt[e*Cos[c + d*x]])/(21*d*e^5*(a^3 - a^3*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(11/2), x, 6, -((2*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*e^6*Sqrt[Cos[c + d*x]])) + (2*a^6*(e*Cos[c + d*x])^(3/2))/(9*d*e^7*(a - a*Sin[c + d*x])^3) + (2*a^5*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a - a*Sin[c + d*x])^2) + (2*a^6*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^3 - a^3*Sin[c + d*x]))} - - -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4, x, 7, (-442*a^4*(e*Cos[c + d*x])^(5/2))/(385*d*e) + (442*a^4*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (442*a^4*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) - (2*a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^3)/(11*d*e) - (34*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x])^2)/(99*d*e) - (442*(e*Cos[c + d*x])^(5/2)*(a^4 + a^4*Sin[c + d*x]))/(693*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4, x, 6, (-22*a^4*(e*Cos[c + d*x])^(3/2))/(9*d*e) + (22*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*d*Sqrt[Cos[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3)/(9*d*e) - (10*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x])^2)/(21*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^4 + a^4*Sin[c + d*x]))/(21*d*e)} -{(a + a*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]], x, 6, (-78*a^4*Sqrt[e*Cos[c + d*x]])/(7*d*e) + (78*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3)/(7*d*e) - (26*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2)/(35*d*e) - (78*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))/(35*d*e)} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2), x, 6, -((154*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^2*Sqrt[Cos[c + d*x]])) - (154*a^4*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d*e^3) + (4*a^7*(e*Cos[c + d*x])^(11/2))/(d*e^7*(a - a*Sin[c + d*x])^3) + (44*a^8*(e*Cos[c + d*x])^(7/2))/(3*d*e^5*(a^4 - a^4*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2), x, 6, -((10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*e^2*Sqrt[e*Cos[c + d*x]])) - (10*a^4*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(d*e^3) + (4*a^7*(e*Cos[c + d*x])^(9/2))/(3*d*e^7*(a - a*Sin[c + d*x])^3) + (12*a^8*(e*Cos[c + d*x])^(5/2))/(d*e^5*(a^4 - a^4*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2), x, 5, (42*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(7/2))/(5*d*e^7*(a - a*Sin[c + d*x])^3) - (28*a^8*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^4 - a^4*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2), x, 5, (10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(5/2))/(7*d*e^7*(a - a*Sin[c + d*x])^3) - (20*a^8*Sqrt[e*Cos[c + d*x]])/(21*d*e^5*(a^4 - a^4*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2), x, 6, (2*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*e^6*Sqrt[Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(3/2))/(9*d*e^7*(a - a*Sin[c + d*x])^3) - (2*a^8*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^2 - a^2*Sin[c + d*x])^2) - (2*a^8*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^4 - a^4*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(13/2), x, 6, -((2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*d*e^6*Sqrt[e*Cos[c + d*x]])) + (4*a^7*Sqrt[e*Cos[c + d*x]])/(11*d*e^7*(a - a*Sin[c + d*x])^3) - (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^2 - a^2*Sin[c + d*x])^2) - (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^4 - a^4*Sin[c + d*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x]), x, 5, (2*e*(e*Cos[c + d*x])^(9/2))/(9*a*d) + (10*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a*d*Sqrt[e*Cos[c + d*x]]) + (10*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) + (2*e^3*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} -{(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x]), x, 4, (2*e*(e*Cos[c + d*x])^(7/2))/(7*a*d) + (6*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[c + d*x]]) + (2*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x]), x, 4, (2*e*(e*Cos[c + d*x])^(5/2))/(5*a*d) + (2*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a*d*Sqrt[e*Cos[c + d*x]]) + (2*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x]), x, 3, (2*e*(e*Cos[c + d*x])^(3/2))/(3*a*d) + (2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]])} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x]), x, 3, (2*e*Sqrt[e*Cos[c + d*x]])/(a*d) + (2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a*d*Sqrt[e*Cos[c + d*x]])} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]), x, 3, -((2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(a*d*Sqrt[Cos[c + d*x]])) - (2*(e*Cos[c + d*x])^(3/2))/(d*e*(a + a*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(3*d*e*(a + a*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])), x, 4, (-6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*e^2*Sqrt[Cos[c + d*x]]) + (6*Sin[c + d*x])/(5*a*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(5*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])), x, 4, (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a*d*e^2*Sqrt[e*Cos[c + d*x]]) + (10*Sin[c + d*x])/(21*a*d*e*(e*Cos[c + d*x])^(3/2)) - 2/(7*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])), x, 5, (-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*a*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(45*a*d*e*(e*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*a*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x]))} - - -{(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^2, x, 5, (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) + (6*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*a^2*d) + (18*e^3*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^2*d) + (4*e*(e*Cos[c + d*x])^(9/2))/(5*d*(a^2 + a^2*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^2, x, 4, (14*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*Sqrt[Cos[c + d*x]]) + (14*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^2*d) + (4*e*(e*Cos[c + d*x])^(7/2))/(3*d*(a^2 + a^2*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^2, x, 4, (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d*Sqrt[e*Cos[c + d*x]]) + (10*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*e*(e*Cos[c + d*x])^(5/2))/(d*(a^2 + a^2*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^2, x, 3, (-6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a^2*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(d*(a^2 + a^2*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^2, x, 3, (-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a^2*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(3*d*(a^2 + a^2*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^2, x, 4, -((2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d*Sqrt[Cos[c + d*x]])) - (2*(e*Cos[c + d*x])^(3/2))/(5*d*e*(a + a*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(5*d*e*(a^2 + a^2*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(7*d*e*(a + a*Sin[c + d*x])^2) - (2*Sqrt[e*Cos[c + d*x]])/(7*d*e*(a^2 + a^2*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2), x, 5, (-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*a^2*d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*a^2*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(33*a^2*d*e^2*Sqrt[e*Cos[c + d*x]]) + (10*Sin[c + d*x])/(33*a^2*d*e*(e*Cos[c + d*x])^(3/2)) - 2/(11*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2) - 2/(11*d*e*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2), x, 6, (-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*a^2*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(65*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (42*Sin[c + d*x])/(65*a^2*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2) - 2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x]))} - - -{(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^3, x, 6, (26*e^3*(e*Cos[c + d*x])^(9/2))/(45*a^3*d) + (26*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*a^3*d*Sqrt[e*Cos[c + d*x]]) + (26*e^7*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*a^3*d) + (26*e^5*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^3*d) + (4*e*(e*Cos[c + d*x])^(13/2))/(5*a*d*(a + a*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^3, x, 5, (22*e^3*(e*Cos[c + d*x])^(7/2))/(21*a^3*d) + (22*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]) + (22*e^5*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^3*d) + (4*e*(e*Cos[c + d*x])^(11/2))/(3*a*d*(a + a*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^3, x, 5, (18*e^3*(e*Cos[c + d*x])^(5/2))/(5*a^3*d) + (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a^3*d*Sqrt[e*Cos[c + d*x]]) + (6*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(a^3*d) + (4*e*(e*Cos[c + d*x])^(9/2))/(a*d*(a + a*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^3, x, 4, (-14*e^3*(e*Cos[c + d*x])^(3/2))/(3*a^3*d) - (14*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a^3*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(7/2))/(a*d*(a + a*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^3, x, 4, (-10*e^3*Sqrt[e*Cos[c + d*x]])/(3*a^3*d) - (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a^3*d*Sqrt[e*Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(5/2))/(3*a*d*(a + a*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^3, x, 4, (6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(5*a*d*(a + a*Sin[c + d*x])^2) + (6*e*(e*Cos[c + d*x])^(3/2))/(5*d*(a^3 + a^3*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^3, x, 4, (-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^3*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(7*a*d*(a + a*Sin[c + d*x])^2) + (2*e*Sqrt[e*Cos[c + d*x]])/(21*d*(a^3 + a^3*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^3, x, 5, -((2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*a^3*d*Sqrt[Cos[c + d*x]])) - (2*(e*Cos[c + d*x])^(3/2))/(9*d*e*(a + a*Sin[c + d*x])^3) - (2*(e*Cos[c + d*x])^(3/2))/(15*a*d*e*(a + a*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(15*d*e*(a^3 + a^3*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*a^3*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(11*d*e*(a + a*Sin[c + d*x])^3) - (10*Sqrt[e*Cos[c + d*x]])/(77*a*d*e*(a + a*Sin[c + d*x])^2) - (10*Sqrt[e*Cos[c + d*x]])/(77*d*e*(a^3 + a^3*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3), x, 6, (-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^3*d*e^2*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(39*a^3*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(117*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 14/(117*d*e*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))} - - -{(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^4, x, 6, (78*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(7*a^4*d*Sqrt[e*Cos[c + d*x]]) + (78*e^7*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*a^4*d) + (234*e^5*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^4*d) + (4*e*(e*Cos[c + d*x])^(13/2))/(a*d*(a + a*Sin[c + d*x])^3) + (52*e^3*(e*Cos[c + d*x])^(9/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^4, x, 5, -((154*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d*Sqrt[Cos[c + d*x]])) - (154*e^5*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^4*d) - (4*e*(e*Cos[c + d*x])^(11/2))/(a*d*(a + a*Sin[c + d*x])^3) - (44*e^3*(e*Cos[c + d*x])^(7/2))/(3*d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^4, x, 5, -((10*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(a^4*d*Sqrt[e*Cos[c + d*x]])) - (10*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(a^4*d) - (4*e*(e*Cos[c + d*x])^(9/2))/(3*a*d*(a + a*Sin[c + d*x])^3) - (12*e^3*(e*Cos[c + d*x])^(5/2))/(d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^4, x, 4, (42*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(7/2))/(5*a*d*(a + a*Sin[c + d*x])^3) + (28*e^3*(e*Cos[c + d*x])^(3/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^4, x, 4, (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^4*d*Sqrt[e*Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(5/2))/(7*a*d*(a + a*Sin[c + d*x])^3) + (20*e^3*Sqrt[e*Cos[c + d*x]])/(21*d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^4, x, 5, (2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(9*a*d*(a + a*Sin[c + d*x])^3) + (2*e*(e*Cos[c + d*x])^(3/2))/(15*d*(a^2 + a^2*Sin[c + d*x])^2) + (2*e*(e*Cos[c + d*x])^(3/2))/(15*d*(a^4 + a^4*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^4, x, 5, (-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*a^4*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(11*a*d*(a + a*Sin[c + d*x])^3) + (2*e*Sqrt[e*Cos[c + d*x]])/(77*d*(a^2 + a^2*Sin[c + d*x])^2) + (2*e*Sqrt[e*Cos[c + d*x]])/(77*d*(a^4 + a^4*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^4, x, 6, -((2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(39*a^4*d*Sqrt[Cos[c + d*x]])) - (2*(e*Cos[c + d*x])^(3/2))/(13*d*e*(a + a*Sin[c + d*x])^4) - (10*(e*Cos[c + d*x])^(3/2))/(117*a*d*e*(a + a*Sin[c + d*x])^3) - (2*(e*Cos[c + d*x])^(3/2))/(39*d*e*(a^2 + a^2*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(39*d*e*(a^4 + a^4*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4), x, 6, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(33*a^4*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(15*d*e*(a + a*Sin[c + d*x])^4) - (14*Sqrt[e*Cos[c + d*x]])/(165*a*d*e*(a + a*Sin[c + d*x])^3) - (2*Sqrt[e*Cos[c + d*x]])/(33*d*e*(a^2 + a^2*Sin[c + d*x])^2) - (2*Sqrt[e*Cos[c + d*x]])/(33*d*e*(a^4 + a^4*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4), x, 7, (-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(221*a^4*d*e^2*Sqrt[Cos[c + d*x]]) + (42*Sin[c + d*x])/(221*a^4*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) - 18/(221*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(p/2) (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]], x, 8, -((a*(e*Cos[c + d*x])^(5/2))/(2*d*e*Sqrt[a + a*Sin[c + d*x]])) + (3*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (3*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (3*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]], x, 7, -((a*(e*Cos[c + d*x])^(3/2))/(d*e*Sqrt[a + a*Sin[c + d*x]])) + (Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[a + a*Sin[c + d*x]]/Sqrt[e*Cos[c + d*x]], x, 6, -((2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))) + (2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2), x, 1, (2*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[c + d*x]])} -{Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(5/2), x, 2, -((2*Sqrt[a + a*Sin[c + d*x]])/(d*e*(e*Cos[c + d*x])^(3/2))) + (4*(a + a*Sin[c + d*x])^(3/2))/(3*a*d*e*(e*Cos[c + d*x])^(3/2))} -{Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2), x, 3, -((2*Sqrt[a + a*Sin[c + d*x]])/(3*d*e*(e*Cos[c + d*x])^(5/2))) + (8*(a + a*Sin[c + d*x])^(3/2))/(3*a*d*e*(e*Cos[c + d*x])^(5/2)) - (16*(a + a*Sin[c + d*x])^(5/2))/(15*a^2*d*e*(e*Cos[c + d*x])^(5/2))} -{Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2), x, 4, -((2*Sqrt[a + a*Sin[c + d*x]])/(5*d*e*(e*Cos[c + d*x])^(7/2))) - (12*(a + a*Sin[c + d*x])^(3/2))/(5*a*d*e*(e*Cos[c + d*x])^(7/2)) + (16*(a + a*Sin[c + d*x])^(5/2))/(5*a^2*d*e*(e*Cos[c + d*x])^(7/2)) - (32*(a + a*Sin[c + d*x])^(7/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(7/2))} - - -{(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2), x, 10, -((15*a^3*(e*Cos[c + d*x])^(7/2))/(32*d*e*(a + a*Sin[c + d*x])^(3/2))) + (15*a^2*e*(e*Cos[c + d*x])^(3/2))/(64*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a^2*(e*Cos[c + d*x])^(7/2))/(8*d*e*Sqrt[a + a*Sin[c + d*x]]) - (a*(e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) + (45*a*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (45*a*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2), x, 9, -((7*a^2*(e*Cos[c + d*x])^(5/2))/(12*d*e*Sqrt[a + a*Sin[c + d*x]])) + (7*a*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d) - (a*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(3*d*e) - (7*a*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (7*a*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2), x, 8, -((5*a^2*(e*Cos[c + d*x])^(3/2))/(4*d*e*Sqrt[a + a*Sin[c + d*x]])) - (a*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(2*d*e) + (5*a*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*a*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^(3/2)/Sqrt[e*Cos[c + d*x]], x, 7, -((a*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e)) - (3*a*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (3*a*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(3/2), x, 7, (4*a*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) - (2*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))} -{(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2), x, 1, (2*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2), x, 2, (2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(5/2)) - (4*(a + a*Sin[c + d*x])^(5/2))/(5*a*d*e*(e*Cos[c + d*x])^(5/2))} -{(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2), x, 3, -((2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2))) + (8*(a + a*Sin[c + d*x])^(5/2))/(3*a*d*e*(e*Cos[c + d*x])^(7/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(21*a^2*d*e*(e*Cos[c + d*x])^(7/2))} -{(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(11/2), x, 4, -((2*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(9/2))) + (4*(a + a*Sin[c + d*x])^(5/2))/(a*d*e*(e*Cos[c + d*x])^(9/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(5*a^2*d*e*(e*Cos[c + d*x])^(9/2)) + (32*(a + a*Sin[c + d*x])^(9/2))/(45*a^3*d*e*(e*Cos[c + d*x])^(9/2))} - - -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2), x, 10, -((77*a^3*(e*Cos[c + d*x])^(5/2))/(96*d*e*Sqrt[a + a*Sin[c + d*x]])) + (77*a^2*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d) - (11*a^2*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(24*d*e) - (77*a^2*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (77*a^2*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2))/(4*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2), x, 9, -((15*a^3*(e*Cos[c + d*x])^(3/2))/(8*d*e*Sqrt[a + a*Sin[c + d*x]])) - (3*a^2*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) + (15*a^2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (15*a^2*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2))/(3*d*e)} -{(a + a*Sin[c + d*x])^(5/2)/Sqrt[e*Cos[c + d*x]], x, 8, -((7*a^2*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*e)) - (21*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (21*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2))/(2*d*e)} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2), x, 8, (5*a^3*(e*Cos[c + d*x])^(3/2))/(d*e^3*Sqrt[a + a*Sin[c + d*x]]) - (5*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) - (5*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) + (4*a*(a + a*Sin[c + d*x])^(3/2))/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(5/2), x, 7, (2*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(5/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) - (2*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(5/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) + (4*a*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2), x, 1, (2*(a + a*Sin[c + d*x])^(5/2))/(5*d*e*(e*Cos[c + d*x])^(5/2))} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2), x, 2, (2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Cos[c + d*x])^(7/2))} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2), x, 3, (2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(9/2)) - (8*(a + a*Sin[c + d*x])^(7/2))/(5*a*d*e*(e*Cos[c + d*x])^(9/2)) + (16*(a + a*Sin[c + d*x])^(9/2))/(45*a^2*d*e*(e*Cos[c + d*x])^(9/2))} -{(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2), x, 4, -((2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(11/2))) + (4*(a + a*Sin[c + d*x])^(7/2))/(a*d*e*(e*Cos[c + d*x])^(11/2)) - (16*(a + a*Sin[c + d*x])^(9/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(11/2)) + (32*(a + a*Sin[c + d*x])^(11/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(e*Cos[c + d*x])^(5/2)/Sqrt[a + a*Sin[c + d*x]], x, 8, -((a*(e*Cos[c + d*x])^(7/2))/(2*d*e*(a + a*Sin[c + d*x])^(3/2))) + (e*(e*Cos[c + d*x])^(3/2))/(4*d*Sqrt[a + a*Sin[c + d*x]]) + (3*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (3*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/Sqrt[a + a*Sin[c + d*x]], x, 7, (e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) - (e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/Sqrt[a + a*Sin[c + d*x]], x, 6, (2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]), x, 1, (-2*Sqrt[e*Cos[c + d*x]])/(d*e*Sqrt[a + a*Sin[c + d*x]])} -{1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]), x, 2, -(2/(3*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])) + (4*Sqrt[a + a*Sin[c + d*x]])/(3*a*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]), x, 3, -(2/(5*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])) - (8*Sqrt[a + a*Sin[c + d*x]])/(5*a*d*e*(e*Cos[c + d*x])^(3/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(15*a^2*d*e*(e*Cos[c + d*x])^(3/2))} -{1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]]), x, 4, -(2/(7*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])) - (4*Sqrt[a + a*Sin[c + d*x]])/(7*a*d*e*(e*Cos[c + d*x])^(5/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(5/2)) - (32*(a + a*Sin[c + d*x])^(5/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(5/2))} - - -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(3/2), x, 8, (e*(e*Cos[c + d*x])^(5/2))/(2*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*e^3*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d) - (5*e^(7/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*e^(7/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(3/2), x, 7, (e*(e*Cos[c + d*x])^(3/2))/(a*d*Sqrt[a + a*Sin[c + d*x]]) + (3*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^2 + a^2*Cos[c + d*x] + a^2*Sin[c + d*x])) + (3*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^2 + a^2*Cos[c + d*x] + a^2*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(3/2), x, 8, -((2*(e*Cos[c + d*x])^(5/2))/(d*e*(a + a*Sin[c + d*x])^(3/2))) - (2*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d) + (2*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (2*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(3/2), x, 1, (-2*(e*Cos[c + d*x])^(3/2))/(3*d*e*(a + a*Sin[c + d*x])^(3/2))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)), x, 2, (-2*Sqrt[e*Cos[c + d*x]])/(5*d*e*(a + a*Sin[c + d*x])^(3/2)) - (4*Sqrt[e*Cos[c + d*x]])/(5*a*d*e*Sqrt[a + a*Sin[c + d*x]])} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)), x, 3, -(2/(7*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2))) - 8/(21*a*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (16*Sqrt[a + a*Sin[c + d*x]])/(21*a^2*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)), x, 4, -(2/(9*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2))) - 4/(15*a*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (16*Sqrt[a + a*Sin[c + d*x]])/(15*a^2*d*e*(e*Cos[c + d*x])^(3/2)) + (32*(a + a*Sin[c + d*x])^(3/2))/(45*a^3*d*e*(e*Cos[c + d*x])^(3/2))} -{1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)), x, 5, -(2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2))) - 16/(77*a*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (128*(a + a*Sin[c + d*x])^(3/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(5/2)) - (256*(a + a*Sin[c + d*x])^(5/2))/(385*a^4*d*e*(e*Cos[c + d*x])^(5/2))} - - -{(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^(5/2), x, 9, (e*(e*Cos[c + d*x])^(7/2))/(2*a*d*(a + a*Sin[c + d*x])^(3/2)) + (7*e^3*(e*Cos[c + d*x])^(3/2))/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (21*e^(9/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x])) + (21*e^(9/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(5/2), x, 8, -((4*e*(e*Cos[c + d*x])^(5/2))/(a*d*(a + a*Sin[c + d*x])^(3/2))) - (5*e^3*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d) + (5*e^(7/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (5*e^(7/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d*(1 + Cos[c + d*x] + Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(5/2), x, 7, -((4*e*(e*Cos[c + d*x])^(3/2))/(3*a*d*(a + a*Sin[c + d*x])^(3/2))) - (2*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x])) - (2*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(5/2), x, 1, (-2*(e*Cos[c + d*x])^(5/2))/(5*d*e*(a + a*Sin[c + d*x])^(5/2))} -{Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(5/2), x, 2, (-2*(e*Cos[c + d*x])^(3/2))/(7*d*e*(a + a*Sin[c + d*x])^(5/2)) - (4*(e*Cos[c + d*x])^(3/2))/(21*a*d*e*(a + a*Sin[c + d*x])^(3/2))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)), x, 3, (-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) - (8*Sqrt[e*Cos[c + d*x]])/(45*a*d*e*(a + a*Sin[c + d*x])^(3/2)) - (16*Sqrt[e*Cos[c + d*x]])/(45*a^2*d*e*Sqrt[a + a*Sin[c + d*x]])} -{1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)), x, 4, -(2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2))) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^3*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(5/2)), x, 5, -(2/(13*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2))) - 16/(117*a*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)) - 32/(195*a^2*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (128*Sqrt[a + a*Sin[c + d*x]])/(195*a^3*d*e*(e*Cos[c + d*x])^(3/2)) + (256*(a + a*Sin[c + d*x])^(3/2))/(585*a^4*d*e*(e*Cos[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(p/3) (a+a Sin[e+f x])^(m/2)*) - - -{(e*Cos[c + d*x])^(7/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, -((3*2^(1/6)*a*(e*Cos[c + d*x])^(10/3)*Hypergeometric2F1[-(1/6), 5/3, 8/3, (1/2)*(1 - Sin[c + d*x])])/(5*d*e*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(3/2)))} -{(e*Cos[c + d*x])^(5/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, -((3*a*(e*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/6, 4/3, 7/3, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1/6))/(4*2^(1/6)*d*e*(a + a*Sin[c + d*x])^(3/2)))} -{(e*Cos[c + d*x])^(2/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, -((3*2^(1/3)*a*(e*Cos[c + d*x])^(5/3)*Hypergeometric2F1[2/3, 5/6, 11/6, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(2/3))/(5*d*e*(a + a*Sin[c + d*x])^(3/2)))} -{(e*Cos[c + d*x])^(1/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, -((3*a*(e*Cos[c + d*x])^(4/3)*Hypergeometric2F1[2/3, 5/6, 5/3, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(5/6))/(2*2^(5/6)*d*e*(a + a*Sin[c + d*x])^(3/2)))} -{1/(e*Cos[c + d*x])^(1/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, -((3*(e*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 7/6, 4/3, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1/6))/(2*2^(1/6)*d*e*Sqrt[a + a*Sin[c + d*x]]))} -{1/(e*Cos[c + d*x])^(4/3)/Sqrt[a + a*Sin[c + d*x]], x, 3, (3*Hypergeometric2F1[-(1/6), 5/3, 5/6, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(2/3))/(2^(2/3)*d*e*(e*Cos[c + d*x])^(1/3)*Sqrt[a + a*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^p (a+a Sin[e+f x])^m when p symbolic*) - - -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^8, x, 2, -((2^(17/2 + p/2)*a^8*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-15 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(d*e*(1 + p)))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^3, x, 2, -((2^(7/2 + p/2)*a^3*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-5 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(d*e*(1 + p)))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^2, x, 2, -((2^(5/2 + p/2)*a^2*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-3 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(d*e*(1 + p)))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^1, x, 2, -((2^(3/2 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-1 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(d*e*(1 + p)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^1, x, 2, -((2^(-(1/2) + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(3 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(a*d*e*(1 + p)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^2, x, 2, -((2^((1/2)*(-3 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(5 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(a^2*d*e*(1 + p)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^3, x, 2, -((2^((1/2)*(-5 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(7 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(a^3*d*e*(1 + p)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^8, x, 2, -((2^((1/2)*(-15 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(17 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - p)))/(a^8*d*e*(1 + p)))} - - -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(7/2), x, 3, -((2^(4 + p/2)*a^4*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-6 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])])/((1 + Sin[c + d*x])^(p/2)*(d*e*(1 + p)*Sqrt[a + a*Sin[c + d*x]])))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(5/2), x, 3, -((2^(3 + p/2)*a^3*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-4 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])])/((1 + Sin[c + d*x])^(p/2)*(d*e*(1 + p)*Sqrt[a + a*Sin[c + d*x]])))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(3/2), x, 3, -((2^(2 + p/2)*a^2*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-2 - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])])/((1 + Sin[c + d*x])^(p/2)*(d*e*(1 + p)*Sqrt[a + a*Sin[c + d*x]])))} -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(1/2), x, 3, -((2^(1 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[-(p/2), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])])/((1 + Sin[c + d*x])^(p/2)*(d*e*(1 + p)*Sqrt[a + a*Sin[c + d*x]])))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(1/2), x, 3, -((2^(p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(2 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1 - p/2))/(d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(3/2), x, 3, -((2^(-1 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(4 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1 - p/2))/(d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))} -{(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(5/2), x, 3, -((2^(-2 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(6 - p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1 - p/2))/(a*d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^p (a+a Sin[e+f x])^m when m symbolic*) - - -{(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^m, x, 3, -((2^(1/2 + m + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1/2)*(1 - 2*m - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(1 - 2*m - p))*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 + p)))} - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m, x, 3, (8*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) - (12*(a + a*Sin[c + d*x])^(5 + m))/(a^5*d*(5 + m)) + (6*(a + a*Sin[c + d*x])^(6 + m))/(a^6*d*(6 + m)) - (a + a*Sin[c + d*x])^(7 + m)/(a^7*d*(7 + m))} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^m, x, 3, (4*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(5 + m)/(a^5*d*(5 + m))} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^m, x, 3, (2*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) - (a + a*Sin[c + d*x])^(3 + m)/(a^3*d*(3 + m))} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^m, x, 2, (a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^m, x, 2, (Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + Sin[c + d*x])]*(a + a*Sin[c + d*x])^m)/(2*d*m)} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^m, x, 2, -((a*Hypergeometric2F1[2, -1 + m, m, (1/2)*(1 + Sin[c + d*x])]*(a + a*Sin[c + d*x])^(-1 + m))/(4*d*(1 - m)))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^m, x, 2, -((a^2*Hypergeometric2F1[3, -2 + m, -1 + m, (1/2)*(1 + Sin[c + d*x])]*(a + a*Sin[c + d*x])^(-2 + m))/(8*d*(2 - m)))} - -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^m, x, 3, -((2^(5/2 + m)*a^2*Cos[c + d*x]^5*Hypergeometric2F1[5/2, -(3/2) - m, 7/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - m)*(a + a*Sin[c + d*x])^(-2 + m))/(5*d))} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^m, x, 3, -((2^(3/2 + m)*a*Cos[c + d*x]^3*Hypergeometric2F1[3/2, -(1/2) - m, 5/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - m)*(a + a*Sin[c + d*x])^(-1 + m))/(3*d))} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^m, x, 3, (2^(-(1/2) + m)*Hypergeometric2F1[-(1/2), 3/2 - m, 1/2, (1/2)*(1 - Sin[c + d*x])]*Sec[c + d*x]*(1 + Sin[c + d*x])^(1/2 - m)*(a + a*Sin[c + d*x])^m)/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^m, x, 3, (2^(-(3/2) + m)*Hypergeometric2F1[-(3/2), 5/2 - m, -(1/2), (1/2)*(1 - Sin[c + d*x])]*Sec[c + d*x]^3*(1 + Sin[c + d*x])^(1/2 - m)*(a + a*Sin[c + d*x])^(1 + m))/(3*a*d)} - - -{(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^m, x, 3, -((2^(11/4 + m)*a*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, -(3/4) - m, 11/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(3/4) - m)*(a + a*Sin[c + d*x])^(-1 + m))/(7*d*e))} -{(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^m, x, 3, -((2^(9/4 + m)*a*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, -(1/4) - m, 9/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/4) - m)*(a + a*Sin[c + d*x])^(-1 + m))/(5*d*e))} -{(e*Cos[c + d*x])^(1/2)*(a + a*Sin[c + d*x])^m, x, 3, -((2^(7/4 + m)*a*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 1/4 - m, 7/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(3*d*e))} -{(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(1/2), x, 3, -((2^(5/4 + m)*a*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 3/4 - m, 5/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(3/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e))} -{(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2), x, 3, (2^(3/4 + m)*Hypergeometric2F1[-(1/4), 5/4 - m, 3/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(1/4 - m)*(a + a*Sin[c + d*x])^m)/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2), x, 3, (2^(1/4 + m)*Hypergeometric2F1[-(3/4), 7/4 - m, 1/4, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(3/4 - m)*(a + a*Sin[c + d*x])^m)/(3*d*e*(e*Cos[c + d*x])^(3/2))} - - -{(e*Cos[c + d*x])^(-4 - m)*(a + a*Sin[c + d*x])^m, x, 4, -(((e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(3 - m))) - (3*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(1 - m)*(3 - m)) + (6*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*e*(3 - m)*(1 - m^2)) - (6*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*e*(9 - 10*m^2 + m^4))} -{(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^m, x, 3, -(((e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(2 - m))) + (2*(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(2 - m)*m) - (2*(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*e*m*(4 - m^2))} -{(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^m, x, 2, -(((e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(1 - m))) + ((e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(1 - m^2))} -{(e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^m, x, 1, (a + a*Sin[c + d*x])^m/((e*Cos[c + d*x])^m*(d*e*m))} -{(e*Cos[c + d*x])^(0 - m)*(a + a*Sin[c + d*x])^m, x, 3, -((2^(1/2 + m/2)*a*(e*Cos[c + d*x])^(1 - m)*Hypergeometric2F1[(1 - m)/2, (1 - m)/2, (3 - m)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1 - m)/2)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 - m)))} -{(e*Cos[c + d*x])^(1 - m)*(a + a*Sin[c + d*x])^m, x, 3, (2^(1 - m/2)*(e*Cos[c + d*x])^(2 - m)*Hypergeometric2F1[m/2, (2 + m)/2, (4 + m)/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(-1 + m/2)*(a + a*Sin[c + d*x])^m)/(d*e*(2 + m))} -{(e*Cos[c + d*x])^(2 - m)*(a + a*Sin[c + d*x])^m, x, 3, -((2^(3/2 + m/2)*a*(e*Cos[c + d*x])^(3 - m)*Hypergeometric2F1[(1/2)*(-1 - m), (3 - m)/2, (5 - m)/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^((1/2)*(-1 - m))*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(3 - m)))} - - -{(e*Cos[c + d*x])^(5 - 2*m)*(a + a*Sin[c + d*x])^m, x, 3, If[$VersionNumber>=8, -((8*a^3*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-3 + m))/(d*e*(5 - m)*(12 - 7*m + m^2))) - (4*a^2*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(20 - 9*m + m^2)) - (a*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(5 - m)), -((8*a^3*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-3 + m))/(d*e*(60 - 47*m + 12*m^2 - m^3))) - (4*a^2*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(4 - m)*(5 - m)) - (a*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(5 - m))]} -{(e*Cos[c + d*x])^(3 - 2*m)*(a + a*Sin[c + d*x])^m, x, 2, If[$VersionNumber>=8, -((2*a^2*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(6 - 5*m + m^2))) - (a*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(3 - m)), -((2*a^2*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(2 - m)*(3 - m))) - (a*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(3 - m))]} -{(e*Cos[c + d*x])^(1 - 2*m)*(a + a*Sin[c + d*x])^m, x, 1, -((a*(e*Cos[c + d*x])^(2 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 - m)))} -{(e*Cos[c + d*x])^(-1 - 2*m)*(a + a*Sin[c + d*x])^m, x, 3, (Hypergeometric2F1[1, -m, 1 - m, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^m)/((e*Cos[c + d*x])^(2*m)*(2*d*e*m))} -{(e*Cos[c + d*x])^(-3 - 2*m)*(a + a*Sin[c + d*x])^m, x, 3, (Hypergeometric2F1[2, -1 - m, -m, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1 + m))/((e*Cos[c + d*x])^(2*(1 + m))*(4*a*d*e*(1 + m)))} - -{(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^m, x, 4, (2^(5/2 - m)*(e*Cos[c + d*x])^(5 - 2*m)*Hypergeometric2F1[5/2, (1/2)*(-3 + 2*m), 7/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(-(5/2) + m)*(a + a*Sin[c + d*x])^m)/(5*d*e)} -{(e*Cos[c + d*x])^(2 - 2*m)*(a + a*Sin[c + d*x])^m, x, 4, (2^(3/2 - m)*(e*Cos[c + d*x])^(3 - 2*m)*Hypergeometric2F1[3/2, (1/2)*(-1 + 2*m), 5/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(-(3/2) + m)*(a + a*Sin[c + d*x])^m)/(3*d*e)} -{(e*Cos[c + d*x])^(0 - 2*m)*(a + a*Sin[c + d*x])^m, x, 4, (2^(1/2 - m)*(e*Cos[c + d*x])^(1 - 2*m)*Hypergeometric2F1[1/2, (1/2)*(1 + 2*m), 3/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(-(1/2) + m)*(a + a*Sin[c + d*x])^m)/(d*e)} -{(e*Cos[c + d*x])^(-2 - 2*m)*(a + a*Sin[c + d*x])^m, x, 4, -((2^(-(1/2) - m)*(e*Cos[c + d*x])^(-1 - 2*m)*Hypergeometric2F1[-(1/2), (1/2)*(3 + 2*m), 1/2, (1/2)*(1 + Sin[c + d*x])]*(1 - Sin[c + d*x])^(1/2 + m)*(a + a*Sin[c + d*x])^m)/(d*e))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^p (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x]), x, 4, -(b*Cos[c + d*x]^6)/(6*d) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x]), x, 3, -((b*Cos[c + d*x]^4)/(4*d)) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x]), x, 2, (a + b*Sin[c + d*x])^2/(2*b*d), (a*Sin[c + d*x])/d + (b*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x]), x, 4, -((a + b)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)*Log[1 + Sin[c + d*x]])/(2*d)} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x]), x, 3, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x]))/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x]), x, 4, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x]))/(4*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d)} - -{Cos[c + d*x]^4*(a + b*Sin[c + d*x]), x, 4, (3*a*x)/8 - (b*Cos[c + d*x]^5)/(5*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x]), x, 3, (a*x)/2 - (b*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x]), x, 3, (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x]), x, 3, (b*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x]), x, 3, (b*Sec[c + d*x]^5)/(5*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 4, -((a*b*Cos[c + d*x]^6)/(3*d)) + (a^2*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sin[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*d) + (b^2*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 3, -((a^2 - b^2)*(a + b*Sin[c + d*x])^3)/(3*b^3*d) + (a*(a + b*Sin[c + d*x])^4)/(2*b^3*d) - (a + b*Sin[c + d*x])^5/(5*b^3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 2, (a + b*Sin[c + d*x])^3/(3*b*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, -((a + b)^2*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^2*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 3, ((a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 4, ((3*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*(2*a*b + (3*a^2 - b^2)*Sin[c + d*x]))/(8*d)} - -{Cos[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 6, (5/128)*(8*a^2 + b^2)*x - (9*a*b*Cos[c + d*x]^7)/(56*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*a^2 + b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (b*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(8*d)} -{Cos[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 5, (1/16)*(6*a^2 + b^2)*x - (7*a*b*Cos[c + d*x]^5)/(30*d) + ((6*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (b*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(6*d)} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 4, (1/8)*(4*a^2 + b^2)*x - (5*a*b*Cos[c + d*x]^3)/(12*d) + ((4*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 3, (-b^2)*x + (a*b*Cos[c + d*x])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/d} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 4, (a*b*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(3*d) + ((2*a^2 - b^2)*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 4, (a*b*Sec[c + d*x]^3)/(5*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(5*d) + ((4*a^2 - b^2)*Tan[c + d*x])/(5*d) + ((4*a^2 - b^2)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^8*(a + b*Sin[c + d*x])^2, x, 4, (a*b*Sec[c + d*x]^5)/(7*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(7*d) + ((6*a^2 - b^2)*Tan[c + d*x])/(7*d) + (2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(21*d) + ((6*a^2 - b^2)*Tan[c + d*x]^5)/(35*d)} - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 3, ((a^2 - b^2)^2*(a + b*Sin[c + d*x])^4)/(4*b^5*d) - (4*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^5)/(5*b^5*d) + ((3*a^2 - b^2)*(a + b*Sin[c + d*x])^6)/(3*b^5*d) - (4*a*(a + b*Sin[c + d*x])^7)/(7*b^5*d) + (a + b*Sin[c + d*x])^8/(8*b^5*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 3, -((a^2 - b^2)*(a + b*Sin[c + d*x])^4)/(4*b^3*d) + (2*a*(a + b*Sin[c + d*x])^5)/(5*b^3*d) - (a + b*Sin[c + d*x])^6/(6*b^3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 2, (a + b*Sin[c + d*x])^4/(4*b*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 6, -((a + b)^3*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^3*Log[1 + Sin[c + d*x]])/(2*d) - (3*a*b^2*Sin[c + d*x])/d - (b^3*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 6, -((a - 2*b)*(a + b)^2*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^2*(a + 2*b)*Log[1 + Sin[c + d*x]])/(4*d) + (a*b^2*Sin[c + d*x])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 4, (3*a*(a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (3*a*Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)} - -{Cos[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 6, (3/16)*a*(2*a^2 + b^2)*x - (b*(17*a^2 + 4*b^2)*Cos[c + d*x]^5)/(70*d) + (3*a*(2*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(2*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (3*a*b*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(14*d) - (b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(7*d)} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 5, (1/8)*a*(4*a^2 + 3*b^2)*x - (b*(27*a^2 + 8*b^2)*Cos[c + d*x]^3)/(60*d) + (a*(4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (7*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(20*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(5*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 2, -3*a*b^2*x + (2*b*(a^2 + b^2)*Cos[c + d*x])/d + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/d} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 5, (2*b*(a^2 - b^2)*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d) + (2*a*(a^2 - b^2)*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x])^3, x, 5, (2*b*(2*a^2 - b^2)*Sec[c + d*x])/(15*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(5*d) + (2*Sec[c + d*x]^3*(a + b*Sin[c + d*x])*(a*b + (2*a^2 - b^2)*Sin[c + d*x]))/(15*d) + (2*a*(4*a^2 - 3*b^2)*Tan[c + d*x])/(15*d)} -{Sec[c + d*x]^8*(a + b*Sin[c + d*x])^3, x, 5, (2*b*(3*a^2 - b^2)*Sec[c + d*x]^3)/(35*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(7*d) + (2*Sec[c + d*x]^5*(a + b*Sin[c + d*x])*(2*a*b + (3*a^2 - b^2)*Sin[c + d*x]))/(35*d) + (12*a*(2*a^2 - b^2)*Tan[c + d*x])/(35*d) + (4*a*(2*a^2 - b^2)*Tan[c + d*x]^3)/(35*d)} -{Sec[c + d*x]^10*(a + b*Sin[c + d*x])^3, x, 5, (2*b*(4*a^2 - b^2)*Sec[c + d*x]^5)/(63*d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(9*d) + (2*Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(3*a*b + (4*a^2 - b^2)*Sin[c + d*x]))/(63*d) + (2*a*(8*a^2 - 3*b^2)*Tan[c + d*x])/(21*d) + (4*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^3)/(63*d) + (2*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^5)/(105*d)} - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^8, x, 3, ((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^10)/(5*b^5*d) + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/(11*b^5*d) - (a*(a + b*Sin[c + d*x])^12)/(3*b^5*d) + (a + b*Sin[c + d*x])^13/(13*b^5*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^8, x, 3, -((a^2 - b^2)*(a + b*Sin[c + d*x])^9)/(9*b^3*d) + (a*(a + b*Sin[c + d*x])^10)/(5*b^3*d) - (a + b*Sin[c + d*x])^11/(11*b^3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^8, x, 2, (a + b*Sin[c + d*x])^9/(9*b*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^8, x, 6, -((a + b)^8*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*Sin[c + d*x])/d - (4*a*b^3*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2)/d - (b^4*(70*a^4 + 28*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(3*d) - (2*a*b^5*(7*a^2 + b^2)*Sin[c + d*x]^4)/d - (b^6*(28*a^2 + b^2)*Sin[c + d*x]^5)/(5*d) - (4*a*b^7*Sin[c + d*x]^6)/(3*d) - (b^8*Sin[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8, x, 7, -((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]])/(4*d) + (7*b^2*(3*a^6 + 30*a^4*b^2 + 20*a^2*b^4 + b^6)*Sin[c + d*x])/(2*d) + (a*b^3*(35*a^4 + 112*a^2*b^2 + 24*b^4)*Sin[c + d*x]^2)/(2*d) + (7*b^4*(15*a^4 + 20*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(6*d) + (3*a*b^5*(7*a^2 + 4*b^2)*Sin[c + d*x]^4)/(2*d) + (7*b^6*(5*a^2 + b^2)*Sin[c + d*x]^5)/(10*d) + (a*b^7*Sin[c + d*x]^6)/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^8, x, 8, -(((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d)) + ((a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (5*b^2*(6*a^6 - 35*a^4*b^2 - 84*a^2*b^4 - 7*b^6)*Sin[c + d*x])/(8*d) + (a*b^3*(15*a^4 - 77*a^2*b^2 - 48*b^4)*Sin[c + d*x]^2)/(4*d) + (5*b^4*(9*a^4 - 42*a^2*b^2 - 7*b^4)*Sin[c + d*x]^3)/(24*d) - (a*(13 - (3*a^2)/b^2)*b^7*Sin[c + d*x]^4)/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^5*(b*(a^2 + 7*b^2) - a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*d)} - -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^8, x, 10, (1/256)*(128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 9588*a^2*b^4 + 1289*b^6)*Cos[c + d*x]^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(2016*d) - (a*b*(112*a^2 + 109*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(336*d) - (b*(64*a^2 + 21*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^5)/(240*d) - (17*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^6)/(90*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^7)/(10*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8, x, 7, (-(7/16))*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*x + (a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b^4 + 512*b^6)*Cos[c + d*x])/(20*d) + (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 + 175*b^6)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a*b*(40*a^4 + 624*a^2*b^2 + 337*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (b*(120*a^4 + 992*a^2*b^2 + 175*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*d) + (a*b*(30*a^2 + 113*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(30*d) + (b*(6*a^2 + 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(6*d) + (a*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^6)/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/d} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8, x, 7, (35/8)*b^4*(16*a^4 + 16*a^2*b^2 + b^4)*x + (a*b*(8*a^6 - 104*a^4*b^2 - 803*a^2*b^4 - 256*b^6)*Cos[c + d*x])/(6*d) + (b^2*(16*a^6 - 200*a^4*b^2 - 866*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(8*a^4 - 88*a^2*b^2 - 151*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(12*d) + (b*(8*a^4 - 72*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(12*d) + (a*b*(2*a^2 - 13*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(3*d) + (b*(2*a^2 - 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(3*d) - (Sec[c + d*x]*(a + b*Sin[c + d*x])^6*(5*a*b - (2*a^2 - 7*b^2)*Sin[c + d*x]))/(3*d)} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x])^8, x, 7, (-(7/2))*b^6*(8*a^2 + b^2)*x + (2*a*b*(8*a^6 - 48*a^4*b^2 + 163*a^2*b^4 + 192*b^6)*Cos[c + d*x])/(15*d) + (b^2*(16*a^6 - 88*a^4*b^2 + 282*a^2*b^4 + 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + (a*b*(8*a^4 - 32*a^2*b^2 + 87*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(15*d) + (b*(8*a^4 - 16*a^2*b^2 + 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(15*d) + (4*a*b*(2*a^2 + b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(15*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(5*d) - (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^6*(3*a*b - (4*a^2 - 7*b^2)*Sin[c + d*x]))/(15*d) - (4*Sec[c + d*x]*(a + b*Sin[c + d*x])^5*(b*(4*a^2 - 7*b^2) - a*(2*a^2 + b^2)*Sin[c + d*x]))/(15*d)} -{Sec[c + d*x]^8*(a + b*Sin[c + d*x])^8, x, 7, b^8*x + (4*a*b*(24*a^6 - 88*a^4*b^2 + 125*a^2*b^4 - 96*b^6)*Cos[c + d*x])/(105*d) + (b^2*(48*a^6 - 152*a^4*b^2 + 174*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(105*d) + (2*a*b*(24*a^4 - 40*a^2*b^2 + 9*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(105*d) + (2*b*(24*a^4 + 8*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(105*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(7*d) - (2*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^5*(b*(6*a^2 - 7*b^2) - a*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(105*d) - (Sec[c + d*x]^5*(a + b*Sin[c + d*x])^6*(a*b - (6*a^2 - 7*b^2)*Sin[c + d*x]))/(35*d) - (2*Sec[c + d*x]*(a + b*Sin[c + d*x])^4*(3*a*b*(12*a^2 - 11*b^2) - (24*a^4 + 8*a^2*b^2 - 35*b^4)*Sin[c + d*x]))/(105*d)} -{Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8, x, 10, (128*a*b*(a^2 - b^2)^3*Sec[c + d*x])/(315*d) + (64*a*(a^2 - b^2)^2*Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(315*d) + (16*a*(a^2 - b^2)*Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^4)/(105*d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(9*d) + (Sec[c + d*x]^7*(a + b*Sin[c + d*x])^6*(a*b + (8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*d) + (128*a^2*(a^2 - b^2)^3*Tan[c + d*x])/(315*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^5/(a + b*Sin[c + d*x]), x, 3, ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^3*d) - (a*Sin[c + d*x]^3)/(3*b^2*d) + Sin[c + d*x]^4/(4*b*d)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x]), x, 3, -(((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^3*d)) + (a*Sin[c + d*x])/(b^2*d) - Sin[c + d*x]^2/(2*b*d)} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x]), x, 2, Log[a + b*Sin[c + d*x]]/(b*d)} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, -Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, -((a + 2*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a - 2*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^5/(a + b*Sin[c + d*x]), x, 5, -((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b^3 + a*(3*a^2 - 7*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} - -{Cos[c + d*x]^6/(a + b*Sin[c + d*x]), x, 7, (a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*d) + Cos[c + d*x]^5/(5*b*d) - (Cos[c + d*x]^3*(4*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(12*b^3*d) + (Cos[c + d*x]*(8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Sin[c + d*x]))/(8*b^5*d)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x]), x, 6, -((a*(2*a^2 - 3*b^2)*x)/(2*b^4)) + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*d) + Cos[c + d*x]^3/(3*b*d) - (Cos[c + d*x]*(2*(a^2 - b^2) - a*b*Sin[c + d*x]))/(2*b^3*d)} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x]), x, 5, (a*x)/b^2 - (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*d) + Cos[c + d*x]/(b*d)} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x]), x, 5, -((2*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d)) - (Sec[c + d*x]*(b - a*Sin[c + d*x]))/((a^2 - b^2)*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x]), x, 6, (2*b^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*(a^2 - b^2)*d) + (Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^6/(a + b*Sin[c + d*x]), x, 7, -((2*b^6*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) - (Sec[c + d*x]^5*(b - a*Sin[c + d*x]))/(5*(a^2 - b^2)*d) + (Sec[c + d*x]^3*(5*b^3 + a*(4*a^2 - 9*b^2)*Sin[c + d*x]))/(15*(a^2 - b^2)^2*d) - (Sec[c + d*x]*(15*b^5 - a*(8*a^4 - 26*a^2*b^2 + 33*b^4)*Sin[c + d*x]))/(15*(a^2 - b^2)^3*d)} - - -{Cos[c + d*x]^7/(a + b*Sin[c + d*x])^2, x, 3, (6*a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^7*d) - ((5*a^4 - 9*a^2*b^2 + 3*b^4)*Sin[c + d*x])/(b^6*d) + (a*(2*a^2 - 3*b^2)*Sin[c + d*x]^2)/(b^5*d) - ((a^2 - b^2)*Sin[c + d*x]^3)/(b^4*d) + (a*Sin[c + d*x]^4)/(2*b^3*d) - Sin[c + d*x]^5/(5*b^2*d) + (a^2 - b^2)^3/(b^7*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 3, (-4*a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) - (a*Sin[c + d*x]^2)/(b^3*d) + Sin[c + d*x]^3/(3*b^2*d) - (a^2 - b^2)^2/(b^5*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 3, (2*a*Log[a + b*Sin[c + d*x]])/(b^3*d) - Sin[c + d*x]/(b^2*d) + (a^2 - b^2)/(b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 2, -(1/(b*d*(a + b*Sin[c + d*x])))} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 4, -Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) - (2*a*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + b/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 4, -((a + 3*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a - 3*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (4*a*b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (b*(a^2 + 3*b^2))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 5, (-3*(a^2 + 4*a*b + 5*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^4*d) + (3*(a^2 - 4*a*b + 5*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^4*d) - (6*a*b^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (3*b*(a^4 - 4*a^2*b^2 - 5*b^4))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(a^2 + 5*b^2) + 3*a*(a^2 - 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} - -{Cos[c + d*x]^6/(a + b*Sin[c + d*x])^2, x, 7, -((5*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6)) + (10*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*d) + (5*Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(12*b^3*d) - Cos[c + d*x]^5/(b*d*(a + b*Sin[c + d*x])) - (5*Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(8*b^5*d)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 6, (3*(2*a^2 - b^2)*x)/(2*b^4) - (6*a*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*d) + (3*Cos[c + d*x]*(2*a - b*Sin[c + d*x]))/(2*b^3*d) - Cos[c + d*x]^3/(b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 5, -(x/b^2) + (2*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 6, -((6*a*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (b*Sec[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(3*a*b - (a^2 + 2*b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 7, (10*a*b^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (b*Sec[c + d*x]^3)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(5*a*b - (a^2 + 4*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d) + (Sec[c + d*x]*(15*a*b^3 + (2*a^4 - 9*a^2*b^2 - 8*b^4)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d)} - - -{Cos[c + d*x]^7/(a + b*Sin[c + d*x])^3, x, 3, -((3*(5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^7*d)) + (a*(10*a^2 - 9*b^2)*Sin[c + d*x])/(b^6*d) - (3*(2*a^2 - b^2)*Sin[c + d*x]^2)/(2*b^5*d) + (a*Sin[c + d*x]^3)/(b^4*d) - Sin[c + d*x]^4/(4*b^3*d) + (a^2 - b^2)^3/(2*b^7*d*(a + b*Sin[c + d*x])^2) - (6*a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 3, (2*(3*a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) - (3*a*Sin[c + d*x])/(b^4*d) + Sin[c + d*x]^2/(2*b^3*d) - (a^2 - b^2)^2/(2*b^5*d*(a + b*Sin[c + d*x])^2) + (4*a*(a^2 - b^2))/(b^5*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 3, -(Log[a + b*Sin[c + d*x]]/(b^3*d)) + (a^2 - b^2)/(2*b^3*d*(a + b*Sin[c + d*x])^2) - (2*a)/(b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 2, -1/(2*b*d*(a + b*Sin[c + d*x])^2)} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 4, -Log[1 - Sin[c + d*x]]/(2*(a + b)^3*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^3*d) - (b*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + b/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (2*a*b)/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 4, -((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^4*d) + ((a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^4*d) + (2*b^3*(5*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (b*(a^2 + 2*b^2))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*b*(a^2 + 11*b^2))/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 5, (-3*(a^2 + 5*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^5*d) + (3*(a^2 - 5*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^5*d) - (3*b^5*(7*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - (3*b*(a^4 - 5*a^2*b^2 - 4*b^4))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (3*a*b*(a^4 - 6*a^2*b^2 - 27*b^4))/(8*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(2*b*(a^2 + 3*b^2) + a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2)} - -{Cos[c + d*x]^6/(a + b*Sin[c + d*x])^3, x, 7, (5*a*(4*a^2 - 3*b^2)*x)/(2*b^6) - (5*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^5/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*Cos[c + d*x]^3*(4*a + b*Sin[c + d*x]))/(6*b^3*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(2*b^5*d)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 6, -((3*a*x)/b^4) + (3*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^3/(2*b*d*(a + b*Sin[c + d*x])^2) - (3*Cos[c + d*x]*(2*a + b*Sin[c + d*x]))/(2*b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 6, ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]/((a^2 - b^2)^(3/2)*d) - Cos[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*Cos[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 7, -((3*b^2*(4*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) + (b*Sec[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (5*a*b*Sec[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(3*b*(4*a^2 + b^2) - a*(2*a^2 + 13*b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 8, (5*b^4*(6*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (b*Sec[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (7*a*b*Sec[c + d*x]^3)/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(5*b*(6*a^2 + b^2) - a*(2*a^2 + 33*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)^3*d) + (Sec[c + d*x]*(15*b^3*(6*a^2 + b^2) + a*(4*a^4 - 28*a^2*b^2 - 81*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^4*d)} - - -{Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8, x, 3, (a^2 - b^2)^3/(7*b^7*d*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x])^6) + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*Sin[c + d*x])^4) + (5*a^2 - b^2)/(b^7*d*(a + b*Sin[c + d*x])^3) - (3*a)/(b^7*d*(a + b*Sin[c + d*x])^2) + 1/(b^7*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5/(a + b*Sin[c + d*x])^8, x, 3, -(a^2 - b^2)^2/(7*b^5*d*(a + b*Sin[c + d*x])^7) + (2*a*(a^2 - b^2))/(3*b^5*d*(a + b*Sin[c + d*x])^6) - (2*(3*a^2 - b^2))/(5*b^5*d*(a + b*Sin[c + d*x])^5) + a/(b^5*d*(a + b*Sin[c + d*x])^4) - 1/(3*b^5*d*(a + b*Sin[c + d*x])^3)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x])^8, x, 3, (a^2 - b^2)/(7*b^3*d*(a + b*Sin[c + d*x])^7) - a/(3*b^3*d*(a + b*Sin[c + d*x])^6) + 1/(5*b^3*d*(a + b*Sin[c + d*x])^5)} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x])^8, x, 2, -1/(7*b*d*(a + b*Sin[c + d*x])^7)} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x])^8, x, 4, If[$VersionNumber>=8, -Log[1 - Sin[c + d*x]]/(2*(a + b)^8*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*d) - (8*a*b*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^8*d) + b/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (a*b)/(3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(3*a^2 + b^2))/(5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (a*b*(a^2 + b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (a*b*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6))/((a^2 - b^2)^7*d*(a + b*Sin[c + d*x])), -(Log[1 - Sin[c + d*x]]/(2*(a + b)^8*d)) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*d) - (8*a*b*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^8*d) + b/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (a*b)/(3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(3*a^2 + b^2))/(5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (a*b*(a^2 + b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (a*b*(3*a^4 + 10*a^2*b^2 + 3*b^4))/((a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6))/((a^2 - b^2)^7*d*(a + b*Sin[c + d*x]))]} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x])^8, x, 4, -((a + 9*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^9*d) + ((a - 9*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^9*d) + (8*a*b^3*(15*a^6 + 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^9*d) - (b*(7*a^2 + 9*b^2))/(14*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^7) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) - (a*b*(3*a^2 + 13*b^2))/(6*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^6) - (b*(5*a^4 + 50*a^2*b^2 + 9*b^4))/(10*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^5) - (a*b*(a^4 + 20*a^2*b^2 + 11*b^4))/(2*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^4) - (b*(3*a^6 + 115*a^4*b^2 + 129*a^2*b^4 + 9*b^6))/(6*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^3) - (a*b*(a^6 + 77*a^4*b^2 + 147*a^2*b^4 + 31*b^6))/(2*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])^2) - (b*(a^8 + 196*a^6*b^2 + 574*a^4*b^4 + 244*a^2*b^6 + 9*b^8))/(2*(a^2 - b^2)^8*d*(a + b*Sin[c + d*x]))} - -{Cos[c + d*x]^8/(a + b*Sin[c + d*x])^8, x, 11, x/b^8 - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(8*b^8*(a^2 - b^2)^(7/2)*d) - Cos[c + d*x]^7/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]^7)/(6*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) - (a*(6*a^2 - 11*b^2)*Cos[c + d*x]^5)/(24*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) + (a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cos[c + d*x]^3)/(16*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]^5*(6*(a^2 - b^2) + 5*a*b*Sin[c + d*x]))/(30*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (Cos[c + d*x]^3*(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*Sin[c + d*x]))/(24*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (Cos[c + d*x]*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Sin[c + d*x]))/(16*b^7*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8, x, 11, (5*a*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(9/2)*d) - Cos[c + d*x]^5/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*(4*a^2 - b^2)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) + ((4*a^4 - 9*a^2*b^2 + 12*b^4)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (a*(8*a^4 - 30*a^2*b^2 + 57*b^4)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + ((8*a^6 - 38*a^4*b^2 + 87*a^2*b^4 + 48*b^6)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]^3*(2*a + 3*b*Sin[c + d*x]))/(42*b^3*d*(a + b*Sin[c + d*x])^6) - (Cos[c + d*x]*(4*a^2 + 9*b^2 + 10*a*b*Sin[c + d*x]))/(42*b^5*d*(a + b*Sin[c + d*x])^5)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8, x, 11, (3*a*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(11/2)*d) - Cos[c + d*x]^3/(7*b*d*(a + b*Sin[c + d*x])^7) - ((a^2 - 3*b^2)*Cos[c + d*x])/(140*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (a*(2*a^2 - 11*b^2)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) - ((2*a^4 - 15*a^2*b^2 - 8*b^4)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c + d*x])/(560*b^3*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*Cos[c + d*x])/(560*b^3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*(a + 3*b*Sin[c + d*x]))/(28*b^3*d*(a + b*Sin[c + d*x])^6)} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8, x, 11, (a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(13/2)*d) - Cos[c + d*x]/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x])/(42*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) + ((5*a^2 + 6*b^2)*Cos[c + d*x])/(210*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^5) + (a*(20*a^2 + 79*b^2)*Cos[c + d*x])/(840*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(840*b*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x])^8, x, 12, -((9*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(17/2)*d)) + (b*Sec[c + d*x])/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (5*a*b*Sec[c + d*x])/(14*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(49*a^2 + 16*b^2)*Sec[c + d*x])/(70*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (13*a*b*(28*a^2 + 27*b^2)*Sec[c + d*x])/(280*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(700*a^4 + 1317*a^2*b^2 + 128*b^4)*Sec[c + d*x])/(280*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (11*a*b*(280*a^4 + 844*a^2*b^2 + 241*b^4)*Sec[c + d*x])/(560*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9800*a^6 + 41484*a^4*b^2 + 22767*a^2*b^4 + 1024*b^6)*Sec[c + d*x])/(560*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(315*a*b*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6) - (560*a^8 + 42472*a^6*b^2 + 125634*a^4*b^4 + 54511*a^2*b^6 + 2048*b^8)*Sin[c + d*x]))/(560*(a^2 - b^2)^8*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x])^8, x, 13, (165*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(19/2)*d) + (b*Sec[c + d*x]^3)/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (17*a*b*Sec[c + d*x]^3)/(42*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(13*a^2 + 4*b^2)*Sec[c + d*x]^3)/(14*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (a*b*(118*a^2 + 103*b^2)*Sec[c + d*x]^3)/(56*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(882*a^4 + 1421*a^2*b^2 + 128*b^4)*Sec[c + d*x]^3)/(168*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (13*a*b*(140*a^4 + 336*a^2*b^2 + 85*b^4)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9212*a^6 + 28420*a^4*b^2 + 12907*a^2*b^4 + 512*b^6)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(1155*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) - (112*a^8 + 52528*a^6*b^2 + 142902*a^4*b^4 + 57665*a^2*b^6 + 2048*b^8)*Sin[c + d*x]))/(336*(a^2 - b^2)^8*d) + (Sec[c + d*x]*(3465*a*b^3*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) + (224*a^10 - 6048*a^8*b^2 - 207332*a^6*b^4 - 413024*a^4*b^6 - 135489*a^2*b^8 - 4096*b^10)*Sin[c + d*x]))/(336*(a^2 - b^2)^9*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^p (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]], x, 3, (2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) + (2*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d)} -{Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]], x, 3, (-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d) - (2*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Sin[c + d*x]], x, 2, (2*(a + b*Sin[c + d*x])^(3/2))/(3*b*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sin[c + d*x]], x, 5, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d} -{Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]], x, 6, -((2*a - b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) + ((2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a + b]*d) + (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]], x, 7, -((12*a^2 - 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(3/2)*d) + ((12*a^2 + 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(3/2)*d) - (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(a*b - (6*a^2 - 5*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)*d) + (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(4*d)} - -{Cos[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]], x, 8, -((4*a*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(21*b*d)) + (2*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(9*b*d) - (8*(4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*(a^2 - 3*b^2) - 3*b*(a^2 + 7*b^2)*Sin[c + d*x]))/(315*b^3*d)} -{Cos[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]], x, 7, -((4*a*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*b*d)) + (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) + (4*(a^2 + 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]], x, 7, -((EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])) + (a*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]) + (Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]], x, 7, -(((4*a^2 - 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])) + (2*a*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b - (4*a^2 - 3*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)*d) + (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2), x, 3, (2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) + (2*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2), x, 3, (-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d) - (2*(a + b*Sin[c + d*x])^(9/2))/(9*b^3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^(3/2), x, 2, (2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^(3/2), x, 6, -(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + ((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d - (2*b*Sqrt[a + b*Sin[c + d*x]])/d} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2), x, 6, -(Sqrt[a - b]*(2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*d) + ((2*a - b)*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2), x, 7, (-3*(4*a^2 - 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*Sqrt[a - b]*d) + (3*(4*a^2 + 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*Sqrt[a + b]*d) - (Sec[c + d*x]^2*(b - 6*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(4*d)} - -{Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2), x, 8, -((2*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d)) - (32*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^6 - 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1155*b^4*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(a^2 + 3*b^2 + 28*a*b*Sin[c + d*x]))/(231*b*d) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^4 - 21*a^2*b^2 - 15*b^4 - 3*a*b*(a^2 + 31*b^2)*Sin[c + d*x]))/(1155*b^3*d)} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2), x, 7, -((2*b*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(7*d)) + (4*a*(3*a^2 + 29*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(105*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*(3*a^4 + 2*a^2*b^2 - 5*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(105*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(3*a^2 + 5*b^2 + 24*a*b*Sin[c + d*x]))/(105*b*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2), x, 6, (Sec[c + d*x]*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/d - (a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2), x, 7, -((Sec[c + d*x]*(b - 4*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(6*d)) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(3*d) - (2*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(3/2), x, 8, -((Sec[c + d*x]^3*(b - 8*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(30*d)) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(5*d) - (a*(32*a^2 - 29*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(60*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((32*a^2 - 5*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(60*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(8*a^4 - 13*a^2*b^2 + 5*b^4) - a*(32*a^4 - 61*a^2*b^2 + 29*b^4)*Sin[c + d*x]))/(60*(a^2 - b^2)^2*d)} - - -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2), x, 3, (2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d) + (2*(a + b*Sin[c + d*x])^(15/2))/(15*b^5*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2), x, 3, (-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(9/2))/(9*b^3*d) - (2*(a + b*Sin[c + d*x])^(11/2))/(11*b^3*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^(5/2), x, 2, (2*(a + b*Sin[c + d*x])^(7/2))/(7*b*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^(5/2), x, 7, -(((a - b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + ((a + b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d - (4*a*b*Sqrt[a + b*Sin[c + d*x]])/d - (2*b*(a + b*Sin[c + d*x])^(3/2))/(3*d)} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2), x, 7, -((a - b)^(3/2)*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*d) + ((2*a - 3*b)*(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*d) + (a*b*Sqrt[a + b*Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2), x, 7, (-3*Sqrt[a - b]*(4*a^2 + 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*d) + (3*Sqrt[a + b]*(4*a^2 - 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(4*d) + (3*Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(a*b + (2*a^2 - b^2)*Sin[c + d*x]))/(16*d)} - -{Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2), x, 9, -((32*a*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(143*d)) - (2*b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) - (8*(20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*a*(5*a^6 - 45*a^4*b^2 - 53*a^2*b^4 + 93*b^6)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^4*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(a*(5*a^2 + 59*b^2) + 7*b*(53*a^2 + 11*b^2)*Sin[c + d*x]))/(3003*b*d) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*(5*a^4 - 40*a^2*b^2 - 93*b^4) - 3*b*(5*a^4 + 430*a^2*b^2 + 77*b^4)*Sin[c + d*x]))/(15015*b^3*d)} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2), x, 8, -((8*a*b*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(21*d)) - (2*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(9*d) + (4*(5*a^4 + 102*a^2*b^2 + 21*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*a*(5*a^4 + 22*a^2*b^2 - 27*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(5*a^2 + 27*b^2) + 3*b*(25*a^2 + 7*b^2)*Sin[c + d*x]))/(315*b*d)} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2), x, 7, (a*b*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/d - ((a^2 + 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2), x, 7, (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(3*d) - ((4*a^2 - 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b + (4*a^2 - 3*b^2)*Sin[c + d*x]))/(6*d)} -{Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(5/2), x, 8, (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(5*d) - ((32*a^2 - 9*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(60*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(32*a^2 - 17*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(60*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(5*a*b + (8*a^2 - 3*b^2)*Sin[c + d*x]))/(30*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(8*a*b*(a^2 - b^2) - (32*a^4 - 41*a^2*b^2 + 9*b^4)*Sin[c + d*x]))/(60*(a^2 - b^2)*d)} -{Sec[c + d*x]^8*(a + b*Sin[c + d*x])^(5/2), x, 9, (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(7*d) - ((128*a^4 - 144*a^2*b^2 + 21*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(280*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(8*a^2 - 3*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*d*Sqrt[a + b*Sin[c + d*x]]) + (3*Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]]*(3*a*b + (4*a^2 - b^2)*Sin[c + d*x]))/(70*d) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(4*a*b*(a^2 - b^2) - (32*a^4 - 39*a^2*b^2 + 7*b^4)*Sin[c + d*x]))/(140*(a^2 - b^2)*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(32*a^4 - 59*a^2*b^2 + 27*b^4) - (128*a^6 - 272*a^4*b^2 + 165*a^2*b^4 - 21*b^6)*Sin[c + d*x]))/(280*(a^2 - b^2)^2*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]], x, 3, (2*(a^2 - b^2)^2*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (2*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d)} -{Cos[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]], x, 3, (-2*(a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^3*d) + (4*a*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d) - (2*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d)} -{Cos[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]], x, 2, (2*Sqrt[a + b*Sin[c + d*x]])/(b*d)} -{Sec[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]], x, 5, -(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)} -{Sec[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]], x, 6, -((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]], x, 7, (-3*(4*a^2 - 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(5/2)*d) + (3*(4*a^2 + 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(5/2)*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - 7*b^2) - 6*a*(a^2 - 2*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d)} - -{Cos[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]], x, 7, (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(7*b*d) - (32*a*(a^2 - 2*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^4 - 9*a^2*b^2 + 5*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^2 - 5*b^2 - 3*a*b*Sin[c + d*x]))/(35*b^3*d)} -{Cos[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]], x, 6, (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b*d) + (4*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]], x, 6, -((Sec[c + d*x]*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)) - (a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]], x, 7, -((Sec[c + d*x]^3*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)*d)) - (2*a*(a^2 - 2*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - 5*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - 5*b^2) - 4*a*(a^2 - 2*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)^2*d)} - - -{Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2), x, 3, (-2*(a^2 - b^2)^2)/(b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (8*a*(a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) + (2*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2), x, 3, (2*(a^2 - b^2))/(b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (4*a*Sqrt[a + b*Sin[c + d*x]])/(b^3*d) - (2*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d)} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x])^(3/2), x, 2, -2/(b*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x])^(3/2), x, 6, -(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/((a - b)^(3/2)*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2), x, 7, -((2*a - 5*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) - (b*(a^2 + 5*b^2))/(2*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2), x, 8, (-3*(4*a^2 - 14*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(7/2)*d) + (3*(4*a^2 + 14*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(7/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 - 15*b^4))/(16*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^2*(b*(a^2 + 9*b^2) + 2*a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]])} - -{Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(3/2), x, 8, -((2*Cos[c + d*x]^5)/(b*d*Sqrt[a + b*Sin[c + d*x]])) + (20*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63*b^3*d) - (16*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(63*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(63*b^6*d*Sqrt[a + b*Sin[c + d*x]]) - (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*b^5*d)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2), x, 7, -((2*Cos[c + d*x]^3)/(b*d*Sqrt[a + b*Sin[c + d*x]])) + (4*Cos[c + d*x]*(4*a - 3*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(5*b^3*d) + (8*(4*a^2 - 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(5*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (32*a*(a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(5*b^4*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2), x, 6, -((2*Cos[c + d*x])/(b*d*Sqrt[a + b*Sin[c + d*x]])) - (4*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (4*a*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2), x, 7, (2*b*Sec[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - ((a^2 + 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*b - (a^2 + 3*b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2), x, 8, (2*b*Sec[c + d*x]^3)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - ((4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*(a^2 - b^2)^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(a^2 - 3*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a*b - (a^2 + 7*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(a^2 - 33*b^2) - (4*a^4 - 15*a^2*b^2 - 21*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^3*d)} - - -{Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2), x, 3, (-2*(a^2 - b^2)^2)/(3*b^5*d*(a + b*Sin[c + d*x])^(3/2)) + (8*a*(a^2 - b^2))/(b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (4*(3*a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) + (2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2), x, 3, (2*(a^2 - b^2))/(3*b^3*d*(a + b*Sin[c + d*x])^(3/2)) - (4*a)/(b^3*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Sqrt[a + b*Sin[c + d*x]])/(b^3*d)} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x])^(5/2), x, 2, -2/(3*b*d*(a + b*Sin[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x])^(5/2), x, 7, -(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/((a - b)^(5/2)*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/((a + b)^(5/2)*d) + (2*b)/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (4*a*b)/((a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2), x, 8, -((2*a - 7*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(7/2)*d) + ((2*a + 7*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(7/2)*d) - (b*(3*a^2 + 7*b^2))/(6*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^(3/2)) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) - (a*b*(a^2 + 19*b^2))/(2*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2), x, 9, -((12*a^2 - 54*a*b + 77*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(9/2)*d) + ((12*a^2 + 54*a*b + 77*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(9/2)*d) - (b*(18*a^4 - 81*a^2*b^2 - 77*b^4))/(48*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^(3/2)) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) - (a*b*(3*a^4 - 16*a^2*b^2 - 127*b^4))/(8*(a^2 - b^2)^4*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^2*(b*(3*a^2 + 11*b^2) + 2*a*(3*a^2 - 10*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^(3/2))} - -{Cos[c + d*x]^8/(a + b*Sin[c + d*x])^(5/2), x, 9, -((2*Cos[c + d*x]^7)/(3*b*d*(a + b*Sin[c + d*x])^(3/2))) - (128*a*(8*a^2 - 9*b^2)*(4*a^2 - 3*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(99*b^8*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*(128*a^6 - 272*a^4*b^2 + 159*a^2*b^4 - 15*b^6)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(99*b^8*d*Sqrt[a + b*Sin[c + d*x]]) - (28*Cos[c + d*x]^5*(12*a + b*Sin[c + d*x]))/(33*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (40*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 3*b^2 - 28*a*b*Sin[c + d*x]))/(99*b^5*d) - (16*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(128*a^4 - 144*a^2*b^2 + 15*b^4 - 3*a*b*(32*a^2 - 31*b^2)*Sin[c + d*x]))/(99*b^7*d)} -{Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(5/2), x, 8, -((2*Cos[c + d*x]^5)/(3*b*d*(a + b*Sin[c + d*x])^(3/2))) + (16*a*(32*a^2 - 29*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(21*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(21*b^6*d*Sqrt[a + b*Sin[c + d*x]]) - (20*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(21*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d*x]))/(21*b^5*d)} -{Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2), x, 7, -((2*Cos[c + d*x]^3)/(3*b*d*(a + b*Sin[c + d*x])^(3/2))) - (32*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^2 - b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*(4*a + b*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2), x, 7, -((2*Cos[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^(3/2))) + (4*a*Cos[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (4*a*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2), x, 8, (2*b*Sec[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (16*a*b*Sec[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (a*(3*a^2 + 29*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((3*a^2 + 5*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(27*a^2 + 5*b^2) - a*(3*a^2 + 29*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2), x, 9, (2*b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (8*a*b*Sec[c + d*x]^3)/((a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (2*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^4 - 21*a^2*b^2 - 15*b^4)*EllipticF[(1/2)*(c - Pi/2 + d*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(b*(29*a^2 + 3*b^2) - a*(a^2 + 31*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(a^4 - 114*a^2*b^2 - 15*b^4) - 4*a*(a^4 - 6*a^2*b^2 - 27*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(p/2) (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]), x, 5, (-2*b*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]), x, 4, (-2*b*(e*Cos[c + d*x])^(7/2))/(7*d*e) + (6*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]), x, 4, (-2*b*(e*Cos[c + d*x])^(5/2))/(5*d*e) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*a*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]), x, 3, (-2*b*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]], x, 3, (-2*b*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2), x, 4, (2*b)/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2), x, 4, (2*b)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2), x, 5, (2*b)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (6*a*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]])} - - -{(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2, x, 6, (-26*a*b*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (10*(11*a^2 + 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*(11*a^2 + 2*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(11*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(11*d*e)} -{(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2, x, 5, (-22*a*b*(e*Cos[c + d*x])^(7/2))/(63*d*e) + (2*(9*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(9*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(9*d*e)} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2, x, 5, (-18*a*b*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (2*(7*a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (2*(7*a^2 + 2*b^2)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(7*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2, x, 4, (-14*a*b*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (2*(5*a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e)} -{(a + b*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]], x, 4, (-10*a*b*Sqrt[e*Cos[c + d*x]])/(3*d*e) + (2*(3*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e)} -{(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2), x, 4, (2*a*b*(e*Cos[c + d*x])^(3/2))/(d*e^3) - (2*(a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2), x, 4, (2*a*b*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2), x, 5, (2*a*b)/(5*d*e^3*Sqrt[e*Cos[c + d*x]]) - (2*(3*a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(5*d*e*(e*Cos[c + d*x])^(5/2))} - - -{(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3, x, 7, (-2*b*(177*a^2 + 44*b^2)*(e*Cos[c + d*x])^(9/2))/(1287*d*e) + (10*a*(11*a^2 + 6*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*a*(11*a^2 + 6*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(11*a^2 + 6*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (34*a*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(143*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(13*d*e)} -{(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3, x, 6, (-2*b*(43*a^2 + 12*b^2)*(e*Cos[c + d*x])^(7/2))/(231*d*e) + (2*a*(3*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(3*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d) - (10*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(33*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(11*d*e)} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3, x, 6, (-2*b*(89*a^2 + 28*b^2)*(e*Cos[c + d*x])^(5/2))/(315*d*e) + (2*a*(7*a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (2*a*(7*a^2 + 6*b^2)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (26*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(63*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(9*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3, x, 5, (-2*b*(57*a^2 + 20*b^2)*(e*Cos[c + d*x])^(3/2))/(105*d*e) + (2*a*(5*a^2 + 6*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (22*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(35*d*e) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(7*d*e)} -{(a + b*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]], x, 5, (-2*b*(11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]])/(5*d*e) + (2*a*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]) - (6*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(5*d*e) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(5*d*e)} -{(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2), x, 5, (2*b*(3*a^2 + 4*b^2)*(e*Cos[c + d*x])^(3/2))/(3*d*e^3) - (2*a*(a^2 + 6*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2), x, 5, (2*b*(a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*a*(a^2 - 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2), x, 5, (2*b*(3*a^2 - 4*b^2)*(e*Cos[c + d*x])^(3/2))/(5*d*e^5) - (6*a*(a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (2*(a + b*Sin[c + d*x])*(a*b - (3*a^2 - 4*b^2)*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2), x, 5, (2*b*(5*a^2 - 4*b^2)*Sqrt[e*Cos[c + d*x]])/(21*d*e^5) + (2*a*(5*a^2 - 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(7*d*e*(e*Cos[c + d*x])^(7/2)) + (2*(a + b*Sin[c + d*x])*(a*b + (5*a^2 - 4*b^2)*Sin[c + d*x]))/(21*d*e^3*(e*Cos[c + d*x])^(3/2))} - - -{(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^4, x, 8, (-34*a*b*(53*a^2 + 38*b^2)*(e*Cos[c + d*x])^(9/2))/(6435*d*e) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(385*d) - (2*b*(93*a^2 + 26*b^2)*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(715*d*e) - (14*a*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(65*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^3)/(15*d*e)} -{(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4, x, 7, (-10*a*b*(115*a^2 + 94*b^2)*(e*Cos[c + d*x])^(7/2))/(3003*d*e) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*d*Sqrt[Cos[c + d*x]]) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(195*d) - (2*b*(73*a^2 + 22*b^2)*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(429*d*e) - (38*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(143*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3)/(13*d*e)} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4, x, 7, (-26*a*b*(79*a^2 + 74*b^2)*(e*Cos[c + d*x])^(5/2))/(3465*d*e) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) - (2*b*(167*a^2 + 54*b^2)*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(693*d*e) - (34*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(99*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3)/(11*d*e)} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4, x, 6, (-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(105*d*e) - (10*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(21*d*e) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e)} -{(a + b*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]], x, 6, (-6*a*b*(31*a^2 + 34*b^2)*Sqrt[e*Cos[c + d*x]])/(35*d*e) + (2*(7*a^4 + 28*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) - (2*b*(29*a^2 + 10*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(35*d*e) - (26*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(35*d*e) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3)/(7*d*e)} -{(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2), x, 6, (2*a*b*(15*a^2 + 62*b^2)*(e*Cos[c + d*x])^(3/2))/(15*d*e^3) - (2*(5*a^4 + 60*a^2*b^2 + 12*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]) + (2*b*(5*a^2 + 6*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e^3) + (2*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(d*e*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2), x, 6, (2*a*b*(a^2 + 14*b^2)*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*(a^4 - 12*a^2*b^2 - 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*b*(a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e^3) + (2*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(3*d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(3*d*e*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2), x, 6, (2*a*b*(3*a^2 - 10*b^2)*(e*Cos[c + d*x])^(3/2))/(5*d*e^5) - (6*(a^4 - 4*a^2*b^2 - 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (6*b*(a^2 - 2*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e^5) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*(a + b*Sin[c + d*x])^2*(a*b - (a^2 - 2*b^2)*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2), x, 6, (10*a*b*(a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]])/(21*d*e^5) + (2*(5*a^4 - 12*a^2*b^2 + 12*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*b*(5*a^2 - 6*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(21*d*e^5) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(7*d*e*(e*Cos[c + d*x])^(7/2)) - (2*(a + b*Sin[c + d*x])^2*(a*b - (5*a^2 - 6*b^2)*Sin[c + d*x]))/(21*d*e^3*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2), x, 6, (2*a*b*(21*a^2 - 22*b^2)*(e*Cos[c + d*x])^(3/2))/(45*d*e^7) - (2*(7*a^4 - 12*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*e^6*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(9*d*e*(e*Cos[c + d*x])^(9/2)) - (2*(a + b*Sin[c + d*x])*(b*(7*a^2 - 6*b^2) - a*(21*a^2 - 22*b^2)*Sin[c + d*x]))/(45*d*e^5*Sqrt[e*Cos[c + d*x]]) + (2*(a + b*Sin[c + d*x])^2*(a*b + (7*a^2 - 6*b^2)*Sin[c + d*x]))/(45*d*e^3*(e*Cos[c + d*x])^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x]), x, 15, -(((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d)) - ((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + (2*e*(e*Cos[c + d*x])^(9/2))/(9*b*d) + (2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b^6*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*(e*Cos[c + d*x])^(5/2)*(7*(a^2 - b^2) - 5*a*b*Sin[c + d*x]))/(35*b^3*d) + (2*e^5*Sqrt[e*Cos[c + d*x]]*(21*(a^2 - b^2)^2 - a*b*(7*a^2 - 12*b^2)*Sin[c + d*x]))/(21*b^5*d)} -{(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x]), x, 14, ((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) - ((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) + (2*e*(e*Cos[c + d*x])^(7/2))/(7*b*d) - (2*a*(5*a^2 - 8*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*(e*Cos[c + d*x])^(3/2)*(5*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(15*b^3*d)} -{(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x]), x, 14, -(((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d)) - ((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) + (2*e*(e*Cos[c + d*x])^(5/2))/(5*b*d) - (2*a*(3*a^2 - 4*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^4*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*Sqrt[e*Cos[c + d*x]]*(3*(a^2 - b^2) - a*b*Sin[c + d*x]))/(3*b^3*d)} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x]), x, 13, ((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) - ((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) + (2*e*(e*Cos[c + d*x])^(3/2))/(3*b*d) + (2*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) - (a*(a^2 - b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x]), x, 13, -(((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d)) - ((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + (2*e*Sqrt[e*Cos[c + d*x]])/(b*d) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]])} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x]), x, 9, (Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) - (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])), x, 9, -((Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e])) - (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])), x, 13, (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)*d*e^2*Sqrt[Cos[c + d*x]]) - (a*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (a*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])), x, 13, -((b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2))) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Cos[c + d*x]]) - (a*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (a*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2))} -{1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])), x, 14, (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (2*a*(3*a^2 - 8*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Cos[c + d*x]]) + (a*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (a*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)) + (2*(5*b^3 + a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Cos[c + d*x]])} - - -{(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^2, x, 15, (-9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*b^6*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a - 5*b*Sin[c + d*x]))/(35*b^3*d) - (e*(e*Cos[c + d*x])^(9/2))/(b*d*(a + b*Sin[c + d*x])) - (3*e^5*Sqrt[e*Cos[c + d*x]]*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^2)*Sin[c + d*x]))/(7*b^5*d)} -{(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^2, x, 14, (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) - (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) + (7*(5*a^2 - 3*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a - 3*b*Sin[c + d*x]))/(15*b^3*d) - (e*(e*Cos[c + d*x])^(7/2))/(b*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^2, x, 14, (-5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) - (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*(3*a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^4*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a - b*Sin[c + d*x]))/(3*b^3*d) - (e*(e*Cos[c + d*x])^(5/2))/(b*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^2, x, 13, (3*a*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*a*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (3*a^2*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(b*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^2, x, 13, -(a*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (a*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(b*d*(a + b*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^2, x, 13, -(a*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (a*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a^2*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2), x, 13, (3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2), x, 14, (-5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) + (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - ((2*a^2 + 3*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)^2*d*e^2*Sqrt[Cos[c + d*x]]) - (5*a^2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (5*a^2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (5*a*b - (2*a^2 + 3*b^2)*Sin[c + d*x])/((a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2), x, 14, (7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + ((2*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a^2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a^2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])) - (7*a*b - (2*a^2 + 5*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(3/2))} -{1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2), x, 15, (-9*a*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) + (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*(a^2 - b^2)^3*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])) - (9*a*b - (2*a^2 + 7*b^2)*Sin[c + d*x])/(5*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)) + (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Sin[c + d*x]))/(5*(a^2 - b^2)^3*d*e^3*Sqrt[e*Cos[c + d*x]])} - - -{(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^3, x, 15, (-11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*a*(45*a^2 - 37*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(20*b^6*d*Sqrt[Cos[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (11*e^3*(e*Cos[c + d*x])^(7/2)*(9*a + 2*b*Sin[c + d*x]))/(28*b^3*d*(a + b*Sin[c + d*x])) + (11*e^5*(e*Cos[c + d*x])^(3/2)*(5*(9*a^2 - 2*b^2) - 27*a*b*Sin[c + d*x]))/(60*b^5*d)} -{(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^3, x, 15, (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^6*d*Sqrt[e*Cos[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(9/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a + 2*b*Sin[c + d*x]))/(20*b^3*d*(a + b*Sin[c + d*x])) + (3*e^5*Sqrt[e*Cos[c + d*x]]*(3*(7*a^2 - 2*b^2) - 7*a*b*Sin[c + d*x]))/(4*b^5*d)} -{(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^3, x, 14, (7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (35*a*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*b^4*d*Sqrt[Cos[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(7/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a + 2*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^3, x, 14, (-5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^4*d*Sqrt[e*Cos[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(5/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a + 2*b*Sin[c + d*x]))/(4*b^3*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^3, x, 14, (3*(a^2 - 2*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) - (3*(a^2 - 2*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) + (3*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(2*b*d*(a + b*Sin[c + d*x])^2) + (3*a*e*(e*Cos[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^3, x, 14, ((a^2 + 2*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) + ((a^2 + 2*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - (a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*e*Sqrt[e*Cos[c + d*x]])/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^3, x, 14, ((3*a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) - ((3*a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + (5*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/(2*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^2) + (5*a*b*(e*Cos[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3), x, 14, (-3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (7*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*(a^2 - b^2)^2*d*Sqrt[e*Cos[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/(2*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^2) + (7*a*b*Sqrt[e*Cos[c + d*x]])/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3), x, 15, (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (a*(8*a^2 + 37*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*(a^2 - b^2)^3*d*e^2*Sqrt[Cos[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2) + (9*a*b)/(4*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Sin[c + d*x])/(4*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3), x, 15, (-7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) + (a*(8*a^2 + 69*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(12*(a^2 - b^2)^3*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2) + (11*a*b)/(4*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])) - (7*b*(9*a^2 + 2*b^2) - a*(8*a^2 + 69*b^2)*Sin[c + d*x])/(12*(a^2 - b^2)^3*d*e*(e*Cos[c + d*x])^(3/2))} -{1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3), x, 16, (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(20*(a^2 - b^2)^4*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2) + (13*a*b)/(4*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])) - (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*Sin[c + d*x])/(20*(a^2 - b^2)^3*d*e*(e*Cos[c + d*x])^(5/2)) + (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sin[c + d*x]))/(20*(a^2 - b^2)^4*d*e^3*Sqrt[e*Cos[c + d*x]])} - - -{(e*Cos[c + d*x])^(15/2)/(a + b*Sin[c + d*x])^4, x, 16, (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (13*(231*a^4 - 203*a^2*b^2 + 20*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(56*b^8*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^8*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^8*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(13/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (13*e^3*(e*Cos[c + d*x])^(9/2)*(11*a + 4*b*Sin[c + d*x]))/(84*b^3*d*(a + b*Sin[c + d*x])^2) - (39*e^5*(e*Cos[c + d*x])^(5/2)*(77*a^2 - 20*b^2 + 22*a*b*Sin[c + d*x]))/(280*b^5*d*(a + b*Sin[c + d*x])) + (13*e^7*Sqrt[e*Cos[c + d*x]]*(21*a*(11*a^2 - 6*b^2) - b*(77*a^2 - 20*b^2)*Sin[c + d*x]))/(56*b^7*d)} -{(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^4, x, 15, (77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*(15*a^2 - 4*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(40*b^6*d*Sqrt[Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (11*e^3*(e*Cos[c + d*x])^(7/2)*(9*a + 4*b*Sin[c + d*x]))/(60*b^3*d*(a + b*Sin[c + d*x])^2) - (77*e^5*(e*Cos[c + d*x])^(3/2)*(15*a^2 - 4*b^2 + 6*a*b*Sin[c + d*x]))/(120*b^5*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^4, x, 15, -((15*a*(7*a^2 - 6*b^2)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d)) - (15*a*(7*a^2 - 6*b^2)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (5*(21*a^2 - 4*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(8*b^6*d*Sqrt[e*Cos[c + d*x]]) + (15*a^2*(7*a^2 - 6*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (15*a^2*(7*a^2 - 6*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(c + d*x), 2])/(16*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(9/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (e^3*(e*Cos[c + d*x])^(5/2)*(7*a + 4*b*Sin[c + d*x]))/(4*b^3*d*(a + b*Sin[c + d*x])^2) - (5*e^5*Sqrt[e*Cos[c + d*x]]*(21*a^2 - 4*b^2 + 14*a*b*Sin[c + d*x]))/(8*b^5*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^4, x, 15, (7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) - (7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) + (7*(5*a^2 - 4*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*b^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^5*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^5*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(7/2))/(3*b*d*(a + b*Sin[c + d*x])^3) + (7*(5*a^2 - 4*b^2)*e^3*(e*Cos[c + d*x])^(3/2))/(8*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a + 4*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^4, x, 15, (-5*a*(a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) - (5*a*(a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) + (5*(3*a^2 - 4*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*b^4*(a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^4*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^4*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(5/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (5*(3*a^2 - 4*b^2)*e^3*Sqrt[e*Cos[c + d*x]])/(24*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a + 4*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x])^2)} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^4, x, 15, -(a*(a^2 - 6*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + (a*(a^2 - 6*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + ((a^2 + 4*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (a^2*(a^2 - 6*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^3*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (a^2*(a^2 - 6*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^3*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(3*b*d*(a + b*Sin[c + d*x])^3) + (a*e*(e*Cos[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((a^2 + 4*b^2)*e*(e*Cos[c + d*x])^(3/2))/(8*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^4, x, 15, -(a*(a^2 + 6*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - (a*(a^2 + 6*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - ((3*a^2 + 4*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*b^2*(a^2 - b^2)^2*d*Sqrt[e*Cos[c + d*x]]) + (a^2*(a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a^2*(a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(3*b*d*(a + b*Sin[c + d*x])^3) + (a*e*Sqrt[e*Cos[c + d*x]])/(12*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 + 4*b^2)*e*Sqrt[e*Cos[c + d*x]])/(24*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^4, x, 15, (-5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*Sqrt[b]*(-a^2 + b^2)^(13/4)*d) + (5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*Sqrt[b]*(-a^2 + b^2)^(13/4)*d) + ((11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*(a^2 - b^2)^3*d*Sqrt[Cos[c + d*x]]) + (5*a^2*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (5*a^2*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/(3*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^3) + (3*a*b*(e*Cos[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x])^2) + (b*(11*a^2 + 4*b^2)*(e*Cos[c + d*x])^(3/2))/(8*(a^2 - b^2)^3*d*e*(a + b*Sin[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4), x, 15, (7*a*Sqrt[b]*(5*a^2 + 6*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(15/4)*d*Sqrt[e]) + (7*a*Sqrt[b]*(5*a^2 + 6*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(15/4)*d*Sqrt[e]) - ((57*a^2 + 20*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*(a^2 - b^2)^3*d*Sqrt[e*Cos[c + d*x]]) + (7*a^2*(5*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (7*a^2*(5*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/(3*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^3) + (11*a*b*Sqrt[e*Cos[c + d*x]])/(12*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x])^2) + (b*(57*a^2 + 20*b^2)*Sqrt[e*Cos[c + d*x]])/(24*(a^2 - b^2)^3*d*e*(a + b*Sin[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4), x, 16, (-15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) + (15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) - ((16*a^4 + 151*a^2*b^2 + 28*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*(a^2 - b^2)^4*d*e^2*Sqrt[Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(3*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3) + (13*a*b)/(12*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2) + (b*(89*a^2 + 28*b^2))/(24*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (15*a*b*(7*a^2 + 6*b^2) - (16*a^4 + 151*a^2*b^2 + 28*b^4)*Sin[c + d*x])/(8*(a^2 - b^2)^4*d*e*Sqrt[e*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(p/2) (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -(* {(e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[3/2, -(1/4), -(1/4), 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2))/(3*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]], x, 1, (1/(3*b*d*Sqrt[e*Cos[c + d*x]]))*(2*e*AppellF1[3/2, 1/4, 1/4, 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{Sqrt[a + b*Sin[c + d*x]]/Sqrt[e*Cos[c + d*x]], x, 1, (2*e*AppellF1[3/2, 3/4, 3/4, 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(3*b*d*(e*Cos[c + d*x])^(3/2))} -{Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2), x, 1, 0} -{Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(5/2), x, 2, 0} -{Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2), x, 3, 0} -{Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2), x, 4, 0} *) - - -(* {(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(3/2), x, 1, (2*e*AppellF1[5/2, -(3/4), -(3/4), 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2))/(5*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(3/2), x, 1, (2*e*AppellF1[5/2, -(1/4), -(1/4), 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2))/(5*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2), x, 1, (1/(5*b*d*Sqrt[e*Cos[c + d*x]]))*(2*e*AppellF1[5/2, 1/4, 1/4, 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{(a + b*Sin[c + d*x])^(3/2)/Sqrt[e*Cos[c + d*x]], x, 1, (1/(5*b*d*(e*Cos[c + d*x])^(3/2)))*(2*e*AppellF1[5/2, 3/4, 3/4, 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))} -{(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(3/2), x, 8, 0} -{(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2), x, 1, 0} -{(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2), x, 3, 0} -{(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2), x, 4, 0} -{(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(11/2), x, 5, 0} *) - - -(* {(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2), x, 1, (2*e*AppellF1[7/2, -(1/4), -(1/4), 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(7/2))/(7*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2), x, 1, (2*e*AppellF1[7/2, 1/4, 1/4, 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(7*b*d*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^(5/2)/Sqrt[e*Cos[c + d*x]], x, 1, (2*e*AppellF1[7/2, 3/4, 3/4, 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(7*b*d*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2), x, 9, 0} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(5/2), x, 8, 0} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2), x, 1, 0} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2), x, 3, 0} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2), x, 4, 0} -{(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2), x, 5, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x, 2, (2*Sqrt[2]*(-a + b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[ArcSin[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(-a + b)^(1/4)], -1]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))])/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]), (Sqrt[2]*(a - b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[2*ArcTan[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(a - b)^(1/4)], 1/2]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x])*(1 + (Sqrt[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x])))^2)]*(1 + (Sqrt[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x]))))/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x]))])} - - -(* {(e*Cos[c + d*x])^(5/2)/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, -(3/4), -(3/4), 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]])/(b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))} -{(e*Cos[c + d*x])^(3/2)/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, -(1/4), -(1/4), 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]])/(b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{Sqrt[e*Cos[c + d*x]]/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, 1/4, 1/4, 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(b*d*Sqrt[e*Cos[c + d*x]])} -{1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]), x, 1, 0} -{1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]]), x, 2, 0} -{1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + b*Sin[c + d*x]]), x, 3, 0} -{1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + b*Sin[c + d*x]]), x, 4, 0} *) - - -(* {(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(5/4), -(5/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(5/2))/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4)))} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(3/4), -(3/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2))/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4)))} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(1/4), -(1/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]])/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4)))} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*2^(3/4)*e*Hypergeometric2F1[1/4, 3/4, 7/4, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^(1/4))/(3*(a + b)*d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2)), x, 2, 0} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(3/2)), x, 3, 0} -{1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(3/2)), x, 4, 0} -{1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^(3/2)), x, 5, 0} *) - - -(* {(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(7/4), -(7/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(7/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(7/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(7/4)))} -{(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(5/4), -(5/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(5/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4)))} -{(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(3/4), -(3/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4)))} -{(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, 0} -{Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^(5/2), x, 2, 0} -{1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2)), x, 3, 0} -{1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2)), x, 4, 0} -{1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(5/2)), x, 5, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^p (a+b Sin[e+f x])^m when p symbolic*) - - -(* {(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^8, x, 10, -((a*b*(9 + m)*(3*b^6*(5053 + 4443*m + 1163*m^2 + 93*m^3) + a^2*b^4*(101978 + 91503*m + 28373*m^2 + 3753*m^3 + 185*m^4) + a^4*b^2*(95912 + 88506*m + 33953*m^2 + 6723*m^3 + 671*m^4 + 27*m^5) + a^6*(12176 + 11772*m + 6168*m^2 + 1809*m^3 + 303*m^4 + 27*m^5 + m^6))*(e*Cos[c + d*x])^(1 + m))/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(2 + 3*m + m^2))) + (1/(d*e*(6 + m)*(8 + m)*(8 + 6*m + m^2)))*((105*b^8 + 420*a^2*b^6*(8 + m) + 210*a^4*b^4*(48 + 14*m + m^2) + 28*a^6*b^2*(192 + 104*m + 18*m^2 + m^3) + a^8*(384 + 400*m + 140*m^2 + 20*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(Cos[c + d*x]^2)^((1/2)*(-1 - m))*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Sin[c + d*x]^2]*Sin[c + d*x]) - (1/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(6 + 5*m + m^2)))*(b*(105*b^6*(105 + 71*m + 15*m^2 + m^3) + 3*a^2*b^4*(75798 + 51665*m + 12355*m^2 + 1255*m^3 + 47*m^4) + a^4*b^2*(367992 + 253870*m + 72195*m^2 + 10705*m^3 + 813*m^4 + 25*m^5) + a^6*(69264 + 48860*m + 18424*m^2 + 4025*m^3 + 511*m^4 + 35*m^5 + m^6))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])) - (a*b*(11 + m)*(3*b^4*(783 + 308*m + 29*m^2) + 2*a^2*b^2*(3852 + 1529*m + 208*m^2 + 11*m^3) + a^4*(2232 + 902*m + 203*m^2 + 22*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^2)/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(35*b^4*(35 + 12*m + m^2) + 2*a^2*b^2*(6068 + 2091*m + 232*m^2 + 9*m^3) + a^4*(5944 + 2070*m + 355*m^2 + 30*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^3)/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (a*b*(13 + m)*(b^2*(83 + 13*m) + a^2*(82 + 13*m + m^2))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^4)/(d*e*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(7*b^2*(7 + m) + a^2*(146 + 21*m + m^2))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^5)/(d*e*(6 + m)*(7 + m)*(8 + m)) - (a*b*(15 + m)*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^6)/(d*e*(7 + m)*(8 + m)) - (b*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^7)/(d*e*(8 + m))} *) -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^3, x, 4, If[$VersionNumber>=8, -((b*(2*b^2*(2 + p) + a^2*(11 + 6*p + p^2))*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p)*(2 + p)*(3 + p))) - (a*(3*b^2 + a^2*(2 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*(2 + p)*Sqrt[Sin[c + d*x]^2]) - (a*b*(5 + p)*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x]))/(d*e*(2 + p)*(3 + p)) - (b*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x])^2)/(d*e*(3 + p)), -((b*(2*b^2*(2 + p) + a^2*(11 + 6*p + p^2))*(e*Cos[c + d*x])^(1 + p))/(d*e*(3 + p)*(2 + 3*p + p^2))) - (a*(3*b^2 + a^2*(2 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*(2 + p)*Sqrt[Sin[c + d*x]^2]) - (a*b*(5 + p)*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x]))/(d*e*(2 + p)*(3 + p)) - (b*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x])^2)/(d*e*(3 + p))]} -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^2, x, 3, -((a*b*(3 + p)*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p)*(2 + p))) - ((b^2 + a^2*(2 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*(2 + p)*Sqrt[Sin[c + d*x]^2]) - (b*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x]))/(d*e*(2 + p))} -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x]), x, 2, -((b*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p))) - (a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*Sqrt[Sin[c + d*x]^2])} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x]), x, 1, -((e*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(1 - p)))} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^2, x, 1, -((e*AppellF1[2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(2 - p)*(a + b*Sin[c + d*x])))} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^3, x, 1, -((e*AppellF1[3 - p, (1 - p)/2, (1 - p)/2, 4 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(3 - p)*(a + b*Sin[c + d*x])^2))} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8, x, 1, -((e*AppellF1[8 - p, (1 - p)/2, (1 - p)/2, 9 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(8 - p)*(a + b*Sin[c + d*x])^7))} - - -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(5/2), x, 2, (2*e*AppellF1[7/2, (1 - p)/2, (1 - p)/2, 9/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(7*b*d)} -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(3/2), x, 2, (2*e*AppellF1[5/2, (1 - p)/2, (1 - p)/2, 7/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(5*b*d)} -{(e*Cos[c + d*x])^p*Sqrt[a + b*Sin[c + d*x]], x, 2, (2*e*AppellF1[3/2, (1 - p)/2, (1 - p)/2, 5/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(3*b*d)} -{(e*Cos[c + d*x])^p/Sqrt[a + b*Sin[c + d*x]], x, 2, (2*e*AppellF1[1/2, (1 - p)/2, (1 - p)/2, 3/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d)} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(3/2), x, 2, -((2*e*AppellF1[-(1/2), (1 - p)/2, (1 - p)/2, 1/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d*Sqrt[a + b*Sin[c + d*x]]))} -{(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(5/2), x, 2, -((2*e*AppellF1[-(3/2), (1 - p)/2, (1 - p)/2, -(1/2), (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^p (a+b Sin[e+f x])^m when m symbolic*) - - -{(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^m, x, 2, (e*AppellF1[1 + m, (1 - p)/2, (1 - p)/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d*(1 + m))} - - -{Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m, x, 3, -(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(2 + m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*Sin[c + d*x])^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^2 - 3*b^2)*(a + b*Sin[c + d*x])^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/(b^7*d*(5 + m)) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(b^7*d*(6 + m)) - (a + b*Sin[c + d*x])^(7 + m)/(b^7*d*(7 + m))} -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^m, x, 3, ((a^2 - b^2)^2*(a + b*Sin[c + d*x])^(1 + m))/(b^5*d*(1 + m)) - (4*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(b^5*d*(2 + m)) + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(b^5*d*(3 + m)) - (4*a*(a + b*Sin[c + d*x])^(4 + m))/(b^5*d*(4 + m)) + (a + b*Sin[c + d*x])^(5 + m)/(b^5*d*(5 + m))} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^m, x, 3, -(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(b^3*d*(1 + m))) + (2*a*(a + b*Sin[c + d*x])^(2 + m))/(b^3*d*(2 + m)) - (a + b*Sin[c + d*x])^(3 + m)/(b^3*d*(3 + m))} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^m, x, 2, (a + b*Sin[c + d*x])^(1 + m)/(b*d*(1 + m))} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^m, x, 5, -((Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(2*(a - b)*d*(1 + m))) + (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(2*(a + b)*d*(1 + m))} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^m, x, 6, -(((a - b*(1 - m))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(4*(a - b)^2*d*(1 + m))) + ((a + b - b*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(4*(a + b)^2*d*(1 + m)) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^m, x, 7, -(((3*a^2 - 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(16*(a - b)^3*d*(1 + m))) + ((3*a^2 + 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(16*(a + b)^3*d*(1 + m)) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(1 + m)*(b*(b^2*(3 - m) - a^2*(1 + m)) + a*(3*a^2 - b^2*(5 - 2*m))*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} - -{Cos[c + d*x]^4*(a + b*Sin[c + d*x])^m, x, 2, (AppellF1[1 + m, -(3/2), -(3/2), 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/2))} -{Cos[c + d*x]^2*(a + b*Sin[c + d*x])^m, x, 2, (AppellF1[1 + m, -(1/2), -(1/2), 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*Sqrt[1 - (a + b*Sin[c + d*x])/(a - b)]*Sqrt[1 - (a + b*Sin[c + d*x])/(a + b)])} -{Sec[c + d*x]^2*(a + b*Sin[c + d*x])^m, x, 2, (AppellF1[1 + m, 3/2, 3/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/2))/(b*d*(1 + m))} -{Sec[c + d*x]^4*(a + b*Sin[c + d*x])^m, x, 2, (AppellF1[1 + m, 5/2, 5/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/2))/(b*d*(1 + m))} - - -{(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^m, x, 2, (e*AppellF1[1 + m, -(3/4), -(3/4), 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))} -{(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^m, x, 2, (e*AppellF1[1 + m, -(1/4), -(1/4), 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))} -{(e*Cos[c + d*x])^(1/2)*(a + b*Sin[c + d*x])^m, x, 2, (e*AppellF1[1 + m, 1/4, 1/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(b*d*(1 + m)*Sqrt[e*Cos[c + d*x]])} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(1/2), x, 2, (e*AppellF1[1 + m, 3/4, 3/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(3/2))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2), x, 2, (e*AppellF1[1 + m, 5/4, 5/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(5/2))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2), x, 2, (e*AppellF1[1 + m, 7/4, 7/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(7/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(7/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(7/2))} - - -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m + 4), x, 9, If[$VersionNumber>=8, -(((e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(3 + m))) + (2*b*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*d*e^3*(1 + m)*(3 + m)) + (a*(e*Cos[c + d*x])^(-3 - m)*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(3 + m)) + (a*(3*b + a*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^2*d*e*(1 + m)*(3 + m)) - (2^(3/2 - m/2)*a*b*(e*Cos[c + d*x])^(-1 - m)*Hypergeometric2F1[(1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*(a + b)*d*e^3*(1 + m)*(3 + m)) - (2^(-(1/2) - m/2)*a*(2*a*b - b^2 + a^2*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*Hypergeometric2F1[(1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])^2*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((3 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^3*d*e*(1 - m)*(3 + m)), -(((e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(3 + m))) + (2*b*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*d*e^3*(1 + m)*(3 + m)) + (a*(e*Cos[c + d*x])^(-3 - m)*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(3 + m)) + (a*(3*b + a*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^2*d*e*(1 + m)*(3 + m)) - (2^(3/2 - m/2)*a*b*(e*Cos[c + d*x])^(-1 - m)*Hypergeometric2F1[(1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*(a + b)*d*e^3*(3 + 4*m + m^2)) - (2^(-(1/2) - m/2)*a*(2*a*b - b^2 + a^2*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*Hypergeometric2F1[(1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])^2*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((3 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^3*d*e*(1 - m)*(3 + m))]} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m + 3), x, -5, (Sec[c + d*x]^4*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((e*Cos[c + d*x])^m*((a - b)*d*e^3*(2 + m))) + ((-2*b + a*(2 + m))*Sec[c + d*x]^4*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^2*(a + b*Sin[c + d*x])^(1 + m))/((e*Cos[c + d*x])^m*((a - b)^2*d*e^3*m*(2 + m))) - ((-b^2 + a^2*(1 + m))*Hypergeometric2F1[m/2, 1 + m, 2 + m, -((2*(a + b*Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x])))]*Sec[c + d*x]^4*(1 + Sin[c + d*x])^3*(((a + b)*(1 + Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x])))^((1/2)*(-2 + m))*(a + b*Sin[c + d*x])^(1 + m))/((e*Cos[c + d*x])^m*((a - b)^3*d*e^3*m*(1 + m)))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m + 2), x, 3, -(((e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(1 + m))) + (2^(1/2 - m/2)*a*(e*Cos[c + d*x])^(-1 - m)*Hypergeometric2F1[(1/2)*(-1 - m), (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(1 + m))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m + 1), x, 1, (e*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (2*(a + b*Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))]*(1 - Sin[c + d*x])*(-(((a - b)*(1 - Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))))^(m/2)*(a + b*Sin[c + d*x])^(1 + m))/((a + b)*d*(1 + m))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m + 0), x, 2, (e*AppellF1[1 + m, (1 + m)/2, (1 + m)/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 + m)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 + m)/2))/(b*d*(1 + m))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m - 1), x, 2, (e*AppellF1[1 + m, m/2, m/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(m/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(m/2))/((e*Cos[c + d*x])^m*(b*d*(1 + m)))} -{(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(m - 2), x, 2, (e*AppellF1[1 + m, (1/2)*(-1 + m), (1/2)*(-1 + m), 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(1 - m)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1/2)*(-1 + m))*(1 - (a + b*Sin[c + d*x])/(a + b))^((1/2)*(-1 + m)))/(b*d*(1 + m))} - - -(* ::Title:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m when a^2-b^2=0*) - - -(* ::Section:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m deleted file mode 100644 index 7fe2e78..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.1.3 (g tan)^p (a+b sin)^m.m +++ /dev/null @@ -1,351 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Sin[e+f x])^m with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^5*(a + a*Sin[c + d*x]), x, 3, -((23*a*Log[1 - Sin[c + d*x]])/(16*d)) + (7*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Tan[c + d*x]^3*(a + a*Sin[c + d*x]), x, 3, (5*a*Log[1 - Sin[c + d*x]])/(4*d) - (a*Log[1 + Sin[c + d*x]])/(4*d) + (a*Sin[c + d*x])/d + a^2/(2*d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^1*(a + a*Sin[c + d*x]), x, 3, -((a*Log[1 - Sin[c + d*x]])/d) - (a*Sin[c + d*x])/d} -{Cot[c + d*x]^1*(a + a*Sin[c + d*x]), x, 3, (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d} -{Cot[c + d*x]^3*(a + a*Sin[c + d*x]), x, 3, -((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d} -{Cot[c + d*x]^5*(a + a*Sin[c + d*x]), x, 3, (2*a*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d} -{Cot[c + d*x]^7*(a + a*Sin[c + d*x]), x, 3, -((3*a*Csc[c + d*x])/d) - (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d} - -{Tan[c + d*x]^6*(a + a*Sin[c + d*x]), x, 9, (-a)*x + (a*Cos[c + d*x])/d + (3*a*Sec[c + d*x])/d - (a*Sec[c + d*x]^3)/d + (a*Sec[c + d*x]^5)/(5*d) + (a*Tan[c + d*x])/d - (a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^4*(a + a*Sin[c + d*x]), x, 8, a*x - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Tan[c + d*x]^2*(a + a*Sin[c + d*x]), x, 5, (-a)*x + (a*Cos[c + d*x])/d + (a*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} -{Cot[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, (-a)*x - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d - (a*Cot[c + d*x])/d} -{Cot[c + d*x]^4*(a + a*Sin[c + d*x]), x, 9, a*x + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + a*Sin[c + d*x]), x, 11, (-a)*x - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)} - - -{Tan[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 3, -((31*a^2*Log[1 - Sin[c + d*x]])/(8*d)) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (9*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 3, (3*a^2*Log[1 - Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d) + a^3/(d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 3, -((2*a^2*Log[1 - Sin[c + d*x]])/d) - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 2, -((Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4)/(2*a^2*d))} -{Cot[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 3, -((6*a^2*Csc[c + d*x])/d) + (2*a^2*Csc[c + d*x]^3)/d + (a^2*Csc[c + d*x]^4)/(2*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (2*a^2*Log[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d)} - -{Tan[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 14, -((9*a^2*x)/2) + (2*a^2*Cos[c + d*x])/d + (6*a^2*Sec[c + d*x])/d - (2*a^2*Sec[c + d*x]^3)/d + (2*a^2*Sec[c + d*x]^5)/(5*d) + (9*a^2*Tan[c + d*x])/(2*d) - (3*a^2*Tan[c + d*x]^3)/(2*d) + (9*a^2*Tan[c + d*x]^5)/(10*d) - (a^2*Sin[c + d*x]^2*Tan[c + d*x]^5)/(2*d)} -{Tan[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 4, (7*a^2*x)/2 - (16*a^2*Cos[c + d*x])/(3*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (8*a^2*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^2)} -{Tan[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 6, -((5*a^2*x)/2) + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Tan[c + d*x]^0*(a + a*Sin[c + d*x])^2, x, 1, (3*a^2*x)/2 - (2*a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 8, -((a^2*x)/2) - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cot[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 12, -((a^2*x)/2) + (3*a^2*ArcTanh[Cos[c + d*x]])/d - (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} - - -{Tan[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 3, (209*a^3*Log[1 - Sin[c + d*x]])/(16*d) - (a^3*Log[1 + Sin[c + d*x]])/(16*d) + (7*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + a^6/(6*d*(a - a*Sin[c + d*x])^3) - (13*a^5)/(8*d*(a - a*Sin[c + d*x])^2) + (71*a^4)/(8*d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 3, (7*a^3*Log[1 - Sin[c + d*x]])/d + (5*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + (2*a^4)/(d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 3, -((4*a^3*Log[1 - Sin[c + d*x]])/d) - (4*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 3, -((3*a^3*Csc[c + d*x])/d) - (a^3*Csc[c + d*x]^2)/(2*d) + (2*a^3*Log[Sin[c + d*x]])/d - (2*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)} - -{Tan[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 9, -((23*a^3*x)/2) + (136*a^3*Cos[c + d*x])/(5*d) - (136*a^3*Cos[c + d*x]^3)/(15*d) + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^6*Cos[c + d*x]*Sin[c + d*x]^5)/(5*d*(a - a*Sin[c + d*x])^3) - (13*a^5*Cos[c + d*x]*Sin[c + d*x]^4)/(15*d*(a - a*Sin[c + d*x])^2) + (23*a^6*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a^3 - a^3*Sin[c + d*x]))} -{Tan[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 10, (17*a^3*x)/2 - (6*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (25*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Tan[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 8, -((11*a^3*x)/2) + (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Tan[c + d*x]^0*(a + a*Sin[c + d*x])^3, x, 7, (5*a^3*x)/2 - (4*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 10, (a^3*x)/2 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} - - -{Tan[c + d*x]^5*(a + a*Sin[c + d*x])^4, x, 3, -((25*a^4*Log[1 - Sin[c + d*x]])/d) - (16*a^4*Sin[c + d*x])/d - (9*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d) + a^6/(d*(a - a*Sin[c + d*x])^2) - (11*a^5)/(d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^3*(a + a*Sin[c + d*x])^4, x, 3, (16*a^4*Log[1 - Sin[c + d*x]])/d + (12*a^4*Sin[c + d*x])/d + (4*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d) + (4*a^5)/(d*(a - a*Sin[c + d*x]))} -{Tan[c + d*x]^1*(a + a*Sin[c + d*x])^4, x, 3, -((8*a^4*Log[1 - Sin[c + d*x]])/d) - (8*a^4*Sin[c + d*x])/d - (7*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d)} -{Cot[c + d*x]^3*(a + a*Sin[c + d*x])^4, x, 3, -((4*a^4*Csc[c + d*x])/d) - (a^4*Csc[c + d*x]^2)/(2*d) + (5*a^4*Log[Sin[c + d*x]])/d - (5*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*d)} - -{Tan[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 13, (163*a^4*x)/8 - (16*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (56*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Tan[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 11, -((95*a^4*x)/8) + (12*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) + (8*a^4*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (31*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Tan[c + d*x]^0*(a + a*Sin[c + d*x])^4, x, 10, (35*a^4*x)/8 - (8*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cot[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 12, (17*a^4*x)/8 - (4*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x])/d + (23*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cot[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 17, -((61*a^4*x)/8) + (2*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cot[c + d*x]^6*(a + a*Sin[c + d*x])^4, x, 21, (97*a^4*x)/8 + (5*a^4*ArcTanh[Cos[c + d*x]])/(2*d) - (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) + (10*a^4*Cot[c + d*x])/d - (5*a^4*Cot[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x]^5)/(5*d) + (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/d + (15*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^7/(a + a*Sin[c + d*x]), x, 8, -((35*ArcTanh[Sin[c + d*x]])/(128*a*d)) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d*x]^8/(8*a*d)} -{Tan[c + d*x]^5/(a + a*Sin[c + d*x]), x, 7, (5*ArcTanh[Sin[c + d*x]])/(16*a*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(16*a*d) + (5*Sec[c + d*x]*Tan[c + d*x]^3)/(24*a*d) - (Sec[c + d*x]*Tan[c + d*x]^5)/(6*a*d) + Tan[c + d*x]^6/(6*a*d)} -{Tan[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, -((3*ArcTanh[Sin[c + d*x]])/(8*a*d)) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) - (Sec[c + d*x]*Tan[c + d*x]^3)/(4*a*d) + Tan[c + d*x]^4/(4*a*d)} -{Tan[c + d*x]^1/(a + a*Sin[c + d*x]), x, 5, ArcTanh[Sin[c + d*x]]/(2*a*d) + 1/(2*d*(a + a*Sin[c + d*x])), ArcTanh[Sin[c + d*x]]/(2*a*d) + Sec[c + d*x]^2/(2*a*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Cot[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)} -{Cot[c + d*x]^3/(a + a*Sin[c + d*x]), x, 5, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d)} -{Cot[c + d*x]^5/(a + a*Sin[c + d*x]), x, 5, -(Cot[c + d*x]^4/(4*a*d)) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(3*a*d)} -{Cot[c + d*x]^7/(a + a*Sin[c + d*x]), x, 6, -(Cot[c + d*x]^6/(6*a*d)) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)} -{Cot[c + d*x]^9/(a + a*Sin[c + d*x]), x, 6, -(Cot[c + d*x]^8/(8*a*d)) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(a*d) - (3*Csc[c + d*x]^5)/(5*a*d) + Csc[c + d*x]^7/(7*a*d)} - -{Tan[c + d*x]^6/(a + a*Sin[c + d*x]), x, 6, Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^7/(7*a*d) + Tan[c + d*x]^7/(7*a*d)} -{Tan[c + d*x]^4/(a + a*Sin[c + d*x]), x, 6, -(Sec[c + d*x]/(a*d)) + (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^5/(5*a*d)} -{Tan[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]^3/(3*a*d)} -{Tan[c + d*x]^0/(a + a*Sin[c + d*x]), x, 1, -(Cos[c + d*x]/(d*(a + a*Sin[c + d*x])))} -{Cot[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d)} -{Cot[c + d*x]^4/(a + a*Sin[c + d*x]), x, 5, -(ArcTanh[Cos[c + d*x]]/(2*a*d)) - Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cot[c + d*x]^6/(a + a*Sin[c + d*x]), x, 6, (3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]^5/(5*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)} -{Cot[c + d*x]^8/(a + a*Sin[c + d*x]), x, 7, -((5*ArcTanh[Cos[c + d*x]])/(16*a*d)) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)} - - -{Tan[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 4, -((7*ArcTanh[Sin[c + d*x]])/(128*a^2*d)) + a/(192*d*(a - a*Sin[c + d*x])^3) - 1/(32*d*(a - a*Sin[c + d*x])^2) + a^3/(80*d*(a + a*Sin[c + d*x])^5) - (5*a^2)/(64*d*(a + a*Sin[c + d*x])^4) + (19*a)/(96*d*(a + a*Sin[c + d*x])^3) - 1/(4*d*(a + a*Sin[c + d*x])^2) + 21/(256*d*(a^2 - a^2*Sin[c + d*x])) + 35/(256*d*(a^2 + a^2*Sin[c + d*x]))} -{Tan[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 4, (5*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + 1/(64*d*(a - a*Sin[c + d*x])^2) + a^2/(32*d*(a + a*Sin[c + d*x])^4) - (7*a)/(48*d*(a + a*Sin[c + d*x])^3) + 1/(4*d*(a + a*Sin[c + d*x])^2) - 5/(64*d*(a^2 - a^2*Sin[c + d*x])) - 5/(32*d*(a^2 + a^2*Sin[c + d*x]))} -{Tan[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, -(ArcTanh[Sin[c + d*x]]/(8*a^2*d)) + a/(12*d*(a + a*Sin[c + d*x])^3) - 1/(4*d*(a + a*Sin[c + d*x])^2) + 1/(16*d*(a^2 - a^2*Sin[c + d*x])) + 3/(16*d*(a^2 + a^2*Sin[c + d*x]))} -{Tan[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(4*a^2*d) + 1/(4*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a^2 + a^2*Sin[c + d*x]))} -{Cot[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 3, Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cot[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 3, (2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + (2*Log[Sin[c + d*x]])/(a^2*d) - (2*Log[1 + Sin[c + d*x]])/(a^2*d)} -{Cot[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 3, -(Csc[c + d*x]^2/(2*a^2*d)) + (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a^2*d)} -{Cot[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 3, Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a^2*d)} -{Cot[c + d*x]^9/(a + a*Sin[c + d*x])^2, x, 3, -(Csc[c + d*x]^2/(2*a^2*d)) + (2*Csc[c + d*x]^3)/(3*a^2*d) + Csc[c + d*x]^4/(4*a^2*d) - (4*Csc[c + d*x]^5)/(5*a^2*d) + Csc[c + d*x]^6/(6*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d) - Csc[c + d*x]^8/(8*a^2*d)} -{Cot[c + d*x]^11/(a + a*Sin[c + d*x])^2, x, 3, Csc[c + d*x]^2/(2*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(2*a^2*d) + (6*Csc[c + d*x]^5)/(5*a^2*d) - (6*Csc[c + d*x]^7)/(7*a^2*d) + Csc[c + d*x]^8/(4*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d) - Csc[c + d*x]^10/(10*a^2*d)} -{Cot[c + d*x]^13/(a + a*Sin[c + d*x])^2, x, 3, -(Csc[c + d*x]^2/(2*a^2*d)) + (2*Csc[c + d*x]^3)/(3*a^2*d) + (3*Csc[c + d*x]^4)/(4*a^2*d) - (8*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(3*a^2*d) + (12*Csc[c + d*x]^7)/(7*a^2*d) - Csc[c + d*x]^8/(4*a^2*d) - (8*Csc[c + d*x]^9)/(9*a^2*d) + (3*Csc[c + d*x]^10)/(10*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d) - Csc[c + d*x]^12/(12*a^2*d)} - - -{Tan[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 4, ArcTanh[Sin[c + d*x]]/(128*a^3*d) + 1/(128*a*d*(a - a*Sin[c + d*x])^2) + a^2/(40*d*(a + a*Sin[c + d*x])^5) - (7*a)/(64*d*(a + a*Sin[c + d*x])^4) + 1/(6*d*(a + a*Sin[c + d*x])^3) - 5/(64*a*d*(a + a*Sin[c + d*x])^2) - 1/(32*d*(a^3 - a^3*Sin[c + d*x])) - 5/(128*d*(a^3 + a^3*Sin[c + d*x]))} -{Tan[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, -(ArcTanh[Sin[c + d*x]]/(32*a^3*d)) + a/(16*d*(a + a*Sin[c + d*x])^4) - 1/(6*d*(a + a*Sin[c + d*x])^3) + 3/(32*a*d*(a + a*Sin[c + d*x])^2) + 1/(32*d*(a^3 - a^3*Sin[c + d*x])) + 1/(16*d*(a^3 + a^3*Sin[c + d*x]))} -{Tan[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 4, ArcTanh[Sin[c + d*x]]/(8*a^3*d) + 1/(6*d*(a + a*Sin[c + d*x])^3) - 1/(8*a*d*(a + a*Sin[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))} -{Cot[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 3, Log[Sin[c + d*x]]/(a^3*d) - Log[1 + Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 1/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cot[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 3, (3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (5*Log[Sin[c + d*x]])/(a^3*d) - (5*Log[1 + Sin[c + d*x]])/(a^3*d) + 2/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cot[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 3, (4*Csc[c + d*x])/(a^3*d) - (2*Csc[c + d*x]^2)/(a^3*d) + Csc[c + d*x]^3/(a^3*d) - Csc[c + d*x]^4/(4*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)} -{Cot[c + d*x]^7/(a + a*Sin[c + d*x])^3, x, 3, Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + (3*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(6*a^3*d)} -{Cot[c + d*x]^9/(a + a*Sin[c + d*x])^3, x, 3, -(Csc[c + d*x]^3/(3*a^3*d)) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (2*Csc[c + d*x]^5)/(5*a^3*d) - Csc[c + d*x]^6/(3*a^3*d) + (3*Csc[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^8/(8*a^3*d)} -{Cot[c + d*x]^11/(a + a*Sin[c + d*x])^3, x, 3, Csc[c + d*x]^3/(3*a^3*d) - (3*Csc[c + d*x]^4)/(4*a^3*d) + Csc[c + d*x]^5/(5*a^3*d) + (5*Csc[c + d*x]^6)/(6*a^3*d) - (5*Csc[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^8/(8*a^3*d) + Csc[c + d*x]^9/(3*a^3*d) - Csc[c + d*x]^10/(10*a^3*d)} -{Cot[c + d*x]^13/(a + a*Sin[c + d*x])^3, x, 3, -(Csc[c + d*x]^3/(3*a^3*d)) + (3*Csc[c + d*x]^4)/(4*a^3*d) - (4*Csc[c + d*x]^6)/(3*a^3*d) + (6*Csc[c + d*x]^7)/(7*a^3*d) + (3*Csc[c + d*x]^8)/(4*a^3*d) - (8*Csc[c + d*x]^9)/(9*a^3*d) + (3*Csc[c + d*x]^11)/(11*a^3*d) - Csc[c + d*x]^12/(12*a^3*d)} - - -{Tan[c + d*x]^5/(a + a*Sin[c + d*x])^4, x, 4, -(ArcTanh[Sin[c + d*x]]/(128*a^4*d)) + a^2/(48*d*(a + a*Sin[c + d*x])^6) - (7*a)/(80*d*(a + a*Sin[c + d*x])^5) + 1/(8*d*(a + a*Sin[c + d*x])^4) - 5/(96*a*d*(a + a*Sin[c + d*x])^3) + 1/(256*d*(a^2 - a^2*Sin[c + d*x])^2) - 5/(256*d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(256*d*(a^4 - a^4*Sin[c + d*x])) - 1/(256*d*(a^4 + a^4*Sin[c + d*x]))} -{Tan[c + d*x]^3/(a + a*Sin[c + d*x])^4, x, 3, a/(20*d*(a + a*Sin[c + d*x])^5) - 1/(8*d*(a + a*Sin[c + d*x])^4) + 1/(16*a*d*(a + a*Sin[c + d*x])^3) + 1/(32*d*(a^2 + a^2*Sin[c + d*x])^2) + 1/(64*d*(a^4 - a^4*Sin[c + d*x])) + 1/(64*d*(a^4 + a^4*Sin[c + d*x]))} -{Tan[c + d*x]^1/(a + a*Sin[c + d*x])^4, x, 4, ArcTanh[Sin[c + d*x]]/(16*a^4*d) + 1/(8*d*(a + a*Sin[c + d*x])^4) - 1/(12*a*d*(a + a*Sin[c + d*x])^3) - 1/(16*d*(a^2 + a^2*Sin[c + d*x])^2) - 1/(16*d*(a^4 + a^4*Sin[c + d*x]))} -{Cot[c + d*x]^3/(a + a*Sin[c + d*x])^4, x, 3, (4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (9*Log[Sin[c + d*x]])/(a^4*d) - (9*Log[1 + Sin[c + d*x]])/(a^4*d) + 1/(d*(a^2 + a^2*Sin[c + d*x])^2) + 5/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cot[c + d*x]^7/(a + a*Sin[c + d*x])^4, x, 3, (8*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (8*Csc[c + d*x]^3)/(3*a^4*d) - (7*Csc[c + d*x]^4)/(4*a^4*d) + (4*Csc[c + d*x]^5)/(5*a^4*d) - Csc[c + d*x]^6/(6*a^4*d) + (8*Log[Sin[c + d*x]])/(a^4*d) - (8*Log[1 + Sin[c + d*x]])/(a^4*d)} - -{Tan[c + d*x]^2/(a + a*Sin[c + d*x])^4, x, 17, -((4*Sec[c + d*x]^5)/(5*a^4*d)) + (12*Sec[c + d*x]^7)/(7*a^4*d) - (8*Sec[c + d*x]^9)/(9*a^4*d) + Tan[c + d*x]^3/(3*a^4*d) + (9*Tan[c + d*x]^5)/(5*a^4*d) + (16*Tan[c + d*x]^7)/(7*a^4*d) + (8*Tan[c + d*x]^9)/(9*a^4*d)} -{Cot[c + d*x]^2/(a + a*Sin[c + d*x])^4, x, If[$VersionNumber<9, 15, 14], If[$VersionNumber<9, (4*ArcTanh[Cos[c + d*x]])/(a^4*d) - (94*Cot[c + d*x])/(15*a^4*d) + (2*Cot[c + d*x])/(5*a^4*d*(1 + Sin[c + d*x])^3) + (13*Cot[c + d*x])/(15*a^4*d*(1 + Sin[c + d*x])^2) + (4*Cot[c + d*x])/(a^4*d*(1 + Sin[c + d*x])), (4*ArcTanh[Cos[c + d*x]])/(a^4*d) - Cot[c + d*x]/(a^4*d) - (2*Cot[c + d*x])/(5*a^4*d*(1 + Csc[c + d*x])^3) + (31*Cot[c + d*x])/(15*a^4*d*(1 + Csc[c + d*x])^2) - (104*Cot[c + d*x])/(15*a^4*d*(1 + Csc[c + d*x]))]} -{Cot[c + d*x]^4/(a + a*Sin[c + d*x])^4, x, If[$VersionNumber<9, 17, 14], If[$VersionNumber<9, (14*ArcTanh[Cos[c + d*x]])/(a^4*d) - (33*Cot[c + d*x])/(a^4*d) - (11*Cot[c + d*x]^3)/(a^4*d) + (14*Cot[c + d*x]*Csc[c + d*x])/(a^4*d) + (4*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^4*d*(1 + Sin[c + d*x])^2) + (28*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^4*d*(1 + Sin[c + d*x])), (14*ArcTanh[Cos[c + d*x]])/(a^4*d) - (9*Cot[c + d*x])/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + (2*Cot[c + d*x]*Csc[c + d*x])/(a^4*d) + (4*Cot[c + d*x])/(3*a^4*d*(1 + Csc[c + d*x])^2) - (44*Cot[c + d*x])/(3*a^4*d*(1 + Csc[c + d*x]))]} -{Cot[c + d*x]^6/(a + a*Sin[c + d*x])^4, x, 16, If[$VersionNumber<9, (27*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - (40*Cot[c + d*x])/(a^4*d) - (27*Cot[c + d*x]^3)/(a^4*d) - (41*Cot[c + d*x]^5)/(5*a^4*d) + (27*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d) + (9*Cot[c + d*x]*Csc[c + d*x]^3)/(a^4*d) + (8*Cot[c + d*x]*Csc[c + d*x]^4)/(a^4*d*(1 + Sin[c + d*x])), (27*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - (16*Cot[c + d*x])/(a^4*d) - (3*Cot[c + d*x]^3)/(a^4*d) - Cot[c + d*x]^5/(5*a^4*d) + (11*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(a^4*d) - (8*Cot[c + d*x])/(a^4*d*(1 + Csc[c + d*x]))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]], x, 15, (11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(8*Sqrt[2]*f) - (27*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])])/(8*f) - (Sec[e + f*x]^3*Sqrt[a*(1 + Sin[e + f*x])])/(12*f) + (29*Sqrt[a + a*Sin[e + f*x]]*Tan[e + f*x])/(12*f) + (5*Sqrt[a*(1 + Sin[e + f*x])]*Tan[e + f*x]^3)/(12*f), (11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(8*Sqrt[2]*f) + (11*a^2*Cos[e + f*x])/(8*f*(a + a*Sin[e + f*x])^(3/2)) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (11*a*Sec[e + f*x])/(6*f*Sqrt[a + a*Sin[e + f*x]]) - (7*Sec[e + f*x]^3*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*a*f)} -{Tan[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]], x, 4, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*f)) + (5*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(a*f)} -{Cot[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]], x, 4, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (3*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f} -{Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]], x, 7, (11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) + (11*a*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f)} - - -{Tan[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2), x, 14, -((a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*f)) + (2*a^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^(3/2)) - (7*a*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*f) - (4*a^2*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) + (Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f), -((a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*f)) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (a*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*f) - (23*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f) + (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(5/2))/(a*f)} -{Tan[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2), x, 3, (11*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (7*Sec[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(3*f) - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f)} -{Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2), x, 5, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (11*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (5*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (Cot[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/f} -{Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(3/2), x, 8, (37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (29*a^2*Cot[e + f*x])/(24*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(3*f)} - - -{Tan[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2), x, 10, -((2*a^5*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^(5/2))) + (8*a^4*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^(3/2)) - (8*a^2*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f + (2*a*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f) - (12*a^3*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]), -((64*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (46*a^2*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (2*a*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(3/2))/(3*f) + (26*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(5/2))/(3*f) - (4*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(7/2))/(a*f)} -{Tan[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2), x, 4, (124*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) + (31*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (9*Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(5*f) - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(5*a*f)} -{Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2), x, 6, -((5*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (49*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) + (31*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (7*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (Cot[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/f} -{Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2), x, 10, (55*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (9*a^3*Cos[e + f*x])/(40*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) + (17*a^2*Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(24*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (5*a*Cot[e + f*x]*Csc[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(12*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]], x, 17, -((67*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(64*Sqrt[2]*Sqrt[a]*f)) - (Sec[e + f*x]*(53 + 127*Sin[e + f*x]))/(192*f*Sqrt[a + a*Sin[e + f*x]]) + (a*Sin[e + f*x]*Tan[e + f*x])/(24*f*(a + a*Sin[e + f*x])^(3/2)) + Tan[e + f*x]^3/(3*f*Sqrt[a + a*Sin[e + f*x]]), (61*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(64*Sqrt[2]*Sqrt[a]*f) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f) + (61*a*Cos[e + f*x])/(64*f*(a + a*Sin[e + f*x])^(3/2)) + (7*a*Sec[e + f*x])/(24*f*(a + a*Sin[e + f*x])^(3/2)) - (61*Sec[e + f*x])/(48*f*Sqrt[a + a*Sin[e + f*x]]) - (5*Sec[e + f*x]^3)/(6*f*Sqrt[a + a*Sin[e + f*x]]) + (7*Sec[e + f*x]^3*Sqrt[a + a*Sin[e + f*x]])/(12*a*f)} -{Tan[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]], x, 4, (5*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[2]*Sqrt[a]*f) - Sec[e + f*x]/(2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f)} -{Cot[e + f*x]^2/Sqrt[a + a*Sin[e + f*x]], x, 4, ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]]/(Sqrt[a]*f) - Cot[e + f*x]/(f*Sqrt[a + a*Sin[e + f*x]])} -{Cot[e + f*x]^4/Sqrt[a + a*Sin[e + f*x]], x, 11, -((7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*Sqrt[a]*f)) + (9*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) + (Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a + a*Sin[e + f*x]])} - - -{Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2), x, 20, (7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(256*Sqrt[2]*a^(3/2)*f) + (7*Cos[e + f*x])/(256*f*(a + a*Sin[e + f*x])^(3/2)) - (Sec[e + f*x]*(65 + 87*Sin[e + f*x]))/(192*f*(a + a*Sin[e + f*x])^(3/2)) + (a*Sin[e + f*x]*Tan[e + f*x])/(12*f*(a + a*Sin[e + f*x])^(5/2)) + Tan[e + f*x]^3/(3*f*(a + a*Sin[e + f*x])^(3/2)), (7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(256*Sqrt[2]*a^(3/2)*f) + (7*Cos[e + f*x])/(256*f*(a + a*Sin[e + f*x])^(3/2)) + (9*Sec[e + f*x])/(32*f*(a + a*Sin[e + f*x])^(3/2)) - Sec[e + f*x]^3/(6*f*(a + a*Sin[e + f*x])^(3/2)) - (45*Sec[e + f*x])/(64*a*f*Sqrt[a + a*Sin[e + f*x]]) + Sec[e + f*x]^3/(4*a*f*Sqrt[a + a*Sin[e + f*x]])} -{Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2), x, 5, ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(32*Sqrt[2]*a^(3/2)*f) + Cos[e + f*x]/(32*f*(a + a*Sin[e + f*x])^(3/2)) - Sec[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(3/2)) + (5*Sec[e + f*x])/(8*a*f*Sqrt[a + a*Sin[e + f*x]])} -{Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(3/2), x, 6, (3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(a^(3/2)*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*f) - Cot[e + f*x]/(a*f*Sqrt[a + a*Sin[e + f*x]])} -{Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(3/2), x, 10, -(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]]/(8*a^(3/2)*f)) - Cot[e + f*x]/(8*a*f*Sqrt[a + a*Sin[e + f*x]]) + (11*Cot[e + f*x]*Csc[e + f*x])/(12*a*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*a^2*f)} - - -{Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2), x, 23, (317*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(4096*Sqrt[2]*a^(5/2)*f) + (317*Cos[e + f*x])/(3072*f*(a + a*Sin[e + f*x])^(5/2)) + (317*Cos[e + f*x])/(4096*a*f*(a + a*Sin[e + f*x])^(3/2)) - (Sec[e + f*x]*(115 + 129*Sin[e + f*x]))/(384*f*(a + a*Sin[e + f*x])^(5/2)) + (5*a*Sin[e + f*x]*Tan[e + f*x])/(48*f*(a + a*Sin[e + f*x])^(7/2)) + Tan[e + f*x]^3/(3*f*(a + a*Sin[e + f*x])^(5/2)), (317*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(4096*Sqrt[2]*a^(5/2)*f) - Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)) - Sec[e + f*x]^3/(8*f*(a + a*Sin[e + f*x])^(5/2)) + (317*Cos[e + f*x])/(4096*a*f*(a + a*Sin[e + f*x])^(3/2)) + (217*Sec[e + f*x])/(1536*a*f*(a + a*Sin[e + f*x])^(3/2)) + (53*Sec[e + f*x]^3)/(96*a*f*(a + a*Sin[e + f*x])^(3/2)) - (1085*Sec[e + f*x])/(3072*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (31*Sec[e + f*x]^3)/(192*a^2*f*Sqrt[a + a*Sin[e + f*x]])} -{Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2), x, 6, -((11*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(128*Sqrt[2]*a^(5/2)*f)) - Sec[e + f*x]/(6*f*(a + a*Sin[e + f*x])^(5/2)) - (11*Cos[e + f*x])/(128*a*f*(a + a*Sin[e + f*x])^(3/2)) + (17*Sec[e + f*x])/(48*a*f*(a + a*Sin[e + f*x])^(3/2)) + (11*Sec[e + f*x])/(96*a^2*f*Sqrt[a + a*Sin[e + f*x]])} -{Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(5/2), x, 7, (5*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(a^(5/2)*f) - (7*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) - (2*Cos[e + f*x])/(a*f*(a + a*Sin[e + f*x])^(3/2)) - Cot[e + f*x]/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(5/2), x, 16, (45*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*a^(5/2)*f) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*f) - (19*Cot[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (13*Cot[e + f*x]*Csc[e + f*x])/(12*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+a Sin[e+f x])^(m/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[e + f*x]^4*(a + a*Sin[e + f*x])^(1/3), x, 10, -((361*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/(126*f)) + (361*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(1/3))/(63*f) - (Sec[e + f*x]*(65*a^2 - 142*a^2*Sin[e + f*x]))/(42*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3)) + (361*(1 + Sqrt[3])*Sec[e + f*x]*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))/(63*f*(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))) - (361*2^(1/3)*EllipticE[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(21*3^(3/4)*a^(2/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) - (361*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(63*2^(2/3)*3^(1/4)*a^(2/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(2*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3)) - (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(2/3))} -{Tan[e + f*x]^2*(a + a*Sin[e + f*x])^(1/3), x, 4, -((5*a*Cos[e + f*x]*Hypergeometric2F1[1/2, 7/6, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/6))/(3*2^(1/6)*f*(a + a*Sin[e + f*x])^(2/3))) + (7*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1/3))/f - (3*Sec[e + f*x]*(a + a*Sin[e + f*x])^(4/3))/(a*f)} -{Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(1/3), x, 3, (6*Sqrt[2]*AppellF1[11/6, -(1/2), 2, 17/6, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(7/3))/(11*a^2*f)} -{Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(1/3), x, 3, (12*Sqrt[2]*AppellF1[17/6, -(3/2), 4, 23/6, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(10/3))/(17*a^3*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3), x, 8, (973*Sec[e + f*x])/(396*f*(a + a*Sin[e + f*x])^(1/3)) - (973*Sec[e + f*x]*(1 - Sin[e + f*x]))/(495*f*(a + a*Sin[e + f*x])^(1/3)) - (Sec[e + f*x]*(95*a + 356*a*Sin[e + f*x]))/(132*f*(1 - Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3)) + (973*EllipticF[ArcCos[(2^(1/3)*a^(1/3) - (1 - Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3))*Sqrt[(2^(2/3)*a^(2/3) + 2^(1/3)*a^(1/3)*(a + a*Sin[e + f*x])^(1/3) + (a + a*Sin[e + f*x])^(2/3))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2])/(495*2^(1/3)*3^(1/4)*a^(4/3)*f*Sqrt[-(((a + a*Sin[e + f*x])^(1/3)*(2^(1/3)*a^(1/3) - (a + a*Sin[e + f*x])^(1/3)))/(2^(1/3)*a^(1/3) - (1 + Sqrt[3])*(a + a*Sin[e + f*x])^(1/3))^2)]) + (3*a^2*Sin[e + f*x]*Tan[e + f*x])/(4*f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3)) + (3*a^2*Sin[e + f*x]^2*Tan[e + f*x])/(f*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^(4/3))} -{Tan[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3), x, 4, -((3*Sec[e + f*x])/(5*f*(a + a*Sin[e + f*x])^(1/3))) + (11*2^(1/6)*Cos[e + f*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[e + f*x])])/(15*f*(1 + Sin[e + f*x])^(1/6)*(a + a*Sin[e + f*x])^(1/3)) + (4*Sec[e + f*x]*(a + a*Sin[e + f*x])^(2/3))/(5*a*f)} -{Cot[e + f*x]^2/(a + a*Sin[e + f*x])^(1/3), x, 3, (6*Sqrt[2]*AppellF1[7/6, -(1/2), 2, 13/6, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(5/3))/(7*a^2*f)} -{Cot[e + f*x]^4/(a + a*Sin[e + f*x])^(1/3), x, 3, (12*Sqrt[2]*AppellF1[13/6, -(3/2), 4, 19/6, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(8/3))/(13*a^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Sin[e+f x])^m with p symbolic*) - - -{(g*Tan[e + f*x])^p*(a + a*Sin[e + f*x])^3, x, 10, (a^3*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(3*a^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p)) + (1/(f*g*(4 + p)))*(a^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (4 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*(g*Tan[e + f*x])^(1 + p)) + (3*a^3*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))} -{(g*Tan[e + f*x])^p*(a + a*Sin[e + f*x])^2, x, 8, (a^2*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(2*a^2*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p)) + (a^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))} -{(g*Tan[e + f*x])^p*(a + a*Sin[e + f*x])^1, x, 6, (a*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(a*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))} -{(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^1, x, 4, (g*Tan[e + f*x])^(1 + p)/(a*f*g*(1 + p)) - ((Cos[e + f*x]^2)^((3 + p)/2)*Hypergeometric2F1[(2 + p)/2, (3 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(g*Tan[e + f*x])^(2 + p))/(a*f*g^2*(2 + p))} -{(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^2, x, 10, (g*Tan[e + f*x])^(1 + p)/(a^2*f*g*(1 + p)) - (2*(Cos[e + f*x]^2)^((5 + p)/2)*Hypergeometric2F1[(2 + p)/2, (5 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(g*Tan[e + f*x])^(2 + p))/(a^2*f*g^2*(2 + p)) + (2*(g*Tan[e + f*x])^(3 + p))/(a^2*f*g^3*(3 + p))} -{(g*Tan[e + f*x])^p/(a + a*Sin[e + f*x])^3, x, 13, (g*Tan[e + f*x])^(1 + p)/(a^3*f*g*(1 + p)) - (3*(Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(2 + p)/2, (7 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^5*(g*Tan[e + f*x])^(2 + p))/(a^3*f*g^2*(2 + p)) + (5*(g*Tan[e + f*x])^(3 + p))/(a^3*f*g^3*(3 + p)) - ((Cos[e + f*x]^2)^((7 + p)/2)*Hypergeometric2F1[(4 + p)/2, (7 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(g*Tan[e + f*x])^(4 + p))/(a^3*f*g^4*(4 + p)) + (4*(g*Tan[e + f*x])^(5 + p))/(a^3*f*g^5*(5 + p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Sin[e+f x])^m with m symbolic*) - - -{(g*Tan[e + f*x])^p*(a + a*Sin[e + f*x])^m, x, 4, (AppellF1[1 + p, (1 + p)/2, (1/2)*(1 - 2*m + p), 2 + p, Sin[e + f*x], -Sin[e + f*x]]*(1 - Sin[e + f*x])^((1 + p)/2)*(1 + Sin[e + f*x])^((1/2)*(1 - 2*m + p))*(a + a*Sin[e + f*x])^m*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p))} - - -{Tan[e + f*x]^3*(a + a*Sin[e + f*x])^m, x, 4, (a*(4 + m)*Hypergeometric2F1[1, -1 + m, m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^(-1 + m))/(4*f*(1 - m)) - (a^2*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-1 + m))/(f*m*(a - a*Sin[e + f*x])) + ((a + a*Sin[e + f*x])^(-1 + m)*(a*(2 - 3*m - m^2) + 2*a*m*Sin[e + f*x]))/(2*f*(1 - m)*m*(1 - Sin[e + f*x]))} -{Tan[e + f*x]^1*(a + a*Sin[e + f*x])^m, x, 3, -((a + a*Sin[e + f*x])^m/(2*f*m)) + (Hypergeometric2F1[1, 1 + m, 2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(1 + m))} -{Cot[e + f*x]^1*(a + a*Sin[e + f*x])^m, x, 2, -((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)))} -{Cot[e + f*x]^3*(a + a*Sin[e + f*x])^m, x, 3, -((Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(2 + m))/(2*a^2*f)) - ((2 - m)*Hypergeometric2F1[2, 2 + m, 3 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(2 + m))/(2*a^2*f*(2 + m))} -{Cot[e + f*x]^5*(a + a*Sin[e + f*x])^m, x, 4, ((9 - m)*Csc[e + f*x]^3*(a + a*Sin[e + f*x])^(3 + m))/(12*a^3*f) - (Csc[e + f*x]^4*(a + a*Sin[e + f*x])^(3 + m))/(4*a^3*f) - ((12 - 9*m + m^2)*Hypergeometric2F1[3, 3 + m, 4 + m, 1 + Sin[e + f*x]]*(a + a*Sin[e + f*x])^(3 + m))/(12*a^3*f*(3 + m))} - -{Tan[e + f*x]^4*(a + a*Sin[e + f*x])^m, x, 6, (2^(-(3/2) + m)*(9 - 12*m - 7*m^2 + 6*m^3 + m^4)*Hypergeometric2F1[1/2, 5/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(3*f*(1 - m)*m) - (Sec[e + f*x]*(a + a*Sin[e + f*x])^(-1 + m)*(a*(6 - m - 7*m^2 - m^3) - a*(9 - 6*m - 8*m^2 - m^3)*Sin[e + f*x]))/(3*f*(1 - m)*m*(1 - Sin[e + f*x])) + (a^2*Sin[e + f*x]*(a + a*Sin[e + f*x])^(-1 + m)*Tan[e + f*x])/(f*(1 - m)*(a - a*Sin[e + f*x])) - (a^2*Sin[e + f*x]^2*(a + a*Sin[e + f*x])^(-1 + m)*Tan[e + f*x])/(f*m*(a - a*Sin[e + f*x]))} -{Tan[e + f*x]^2*(a + a*Sin[e + f*x])^m, x, 5, (Sec[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 - m)*m) + (2^(-(1/2) + m)*(1 - m - m^2)*Hypergeometric2F1[-(1/2), 3/2 - m, 1/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 - m)*m) - (Sec[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*m)} -{Tan[e + f*x]^0*(a + a*Sin[e + f*x])^m, x, 2, -((2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/f)} -{Cot[e + f*x]^2*(a + a*Sin[e + f*x])^m, x, 3, (2*Sqrt[2]*AppellF1[3/2 + m, -(1/2), 2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(3 + 2*m))} -{Cot[e + f*x]^4*(a + a*Sin[e + f*x])^m, x, 3, (4*Sqrt[2]*AppellF1[5/2 + m, -(3/2), 4, 7/2 + m, (1/2)*(1 + Sin[e + f*x]), 1 + Sin[e + f*x]]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(5 + 2*m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x]), x, 6, ((2*a + 3*b)*Log[1 - Sin[c + d*x]])/(4*d) + ((2*a - 3*b)*Log[1 + Sin[c + d*x]])/(4*d) + (3*b*Sin[c + d*x])/(2*d) + ((a + b*Sin[c + d*x])*Tan[c + d*x]^2)/(2*d)} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x]), x, 5, -(((a + b)*Log[1 - Sin[c + d*x]])/(2*d)) - ((a - b)*Log[1 + Sin[c + d*x]])/(2*d) - (b*Sin[c + d*x])/d} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x]), x, 3, (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x]), x, 3, -((b*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d} -{Cot[c + d*x]^5*(a + b*Sin[c + d*x]), x, 3, (2*b*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (b*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x]), x, 8, a*x - (b*Cos[c + d*x])/d - (2*b*Sec[c + d*x])/d + (b*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, (-a)*x + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, (-a)*x - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d - (a*Cot[c + d*x])/d} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x]), x, 9, a*x + (3*b*ArcTanh[Cos[c + d*x]])/(2*d) - (3*b*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + b*Sin[c + d*x]), x, 11, -(a*x) - (15*b*ArcTanh[Cos[c + d*x]])/(8*d) + (15*b*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*b*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)} - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 7, ((a + b)*(a + 2*b)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - 2*b)*(a - b)*Log[1 + Sin[c + d*x]])/(2*d) + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(2*d)} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, -(((a + b)^2*Log[1 - Sin[c + d*x]])/(2*d)) - ((a - b)^2*Log[1 + Sin[c + d*x]])/(2*d) - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 3, (a^2*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 3, (-2*a*b*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/(2*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 3, (4*a*b*Csc[c + d*x])/d + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) + ((a^2 - 2*b^2)*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 13, a^2*x + (5*b^2*x)/2 - (2*a*b*Cos[c + d*x])/d - (4*a*b*Sec[c + d*x])/d + (2*a*b*Sec[c + d*x]^3)/(3*d) - (a^2*Tan[c + d*x])/d - (5*b^2*Tan[c + d*x])/(2*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (5*b^2*Tan[c + d*x]^3)/(6*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 11, (-a^2)*x - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c + d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 9, (-a^2)*x + (b^2*x)/2 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 13, a^2*x - (3*b^2*x)/2 + (3*a*b*ArcTanh[Cos[c + d*x]])/d - (3*a*b*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (3*b^2*Cot[c + d*x])/(2*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 16, (-a^2)*x + (5*b^2*x)/2 - (15*a*b*ArcTanh[Cos[c + d*x]])/(4*d) + (15*a*b*Cos[c + d*x])/(4*d) - (a^2*Cot[c + d*x])/d + (5*b^2*Cot[c + d*x])/(2*d) + (5*a*b*Cos[c + d*x]*Cot[c + d*x]^2)/(4*d) + (a^2*Cot[c + d*x]^3)/(3*d) - (5*b^2*Cot[c + d*x]^3)/(6*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d)} - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 7, ((a + b)^2*(2*a + 5*b)*Log[1 - Sin[c + d*x]])/(4*d) + ((2*a - 5*b)*(a - b)^2*Log[1 + Sin[c + d*x]])/(4*d) + (b*(6*a^2 + 5*b^2)*Sin[c + d*x])/(2*d) + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(2*d)} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 6, -(((a + b)^3*Log[1 - Sin[c + d*x]])/(2*d)) - ((a - b)^3*Log[1 + Sin[c + d*x]])/(2*d) - (b*(3*a^2 + b^2)*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 3, (a^3*Log[Sin[c + d*x]])/d + (3*a^2*b*Sin[c + d*x])/d + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 3, (-3*a^2*b*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (b*(3*a^2 - b^2)*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 3, (b*(6*a^2 - b^2)*Csc[c + d*x])/d + (a*(2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (a^3*Csc[c + d*x]^4)/(4*d) + (a*(a^2 - 6*b^2)*Log[Sin[c + d*x]])/d + (b*(3*a^2 - 2*b^2)*Sin[c + d*x])/d + (3*a*b^2*Sin[c + d*x]^2)/(2*d) + (b^3*Sin[c + d*x]^3)/(3*d)} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 16, a^3*x + (15*a*b^2*x)/2 - (3*a^2*b*Cos[c + d*x])/d - (3*b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) - (6*a^2*b*Sec[c + d*x])/d - (3*b^3*Sec[c + d*x])/d + (a^2*b*Sec[c + d*x]^3)/d + (b^3*Sec[c + d*x]^3)/(3*d) - (a^3*Tan[c + d*x])/d - (15*a*b^2*Tan[c + d*x])/(2*d) + (a^3*Tan[c + d*x]^3)/(3*d) + (5*a*b^2*Tan[c + d*x]^3)/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 14, (-a^3)*x - (9/2)*a*b^2*x + (3*a^2*b*Cos[c + d*x])/d + (2*b^3*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (9*a*b^2*Tan[c + d*x])/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 11, (-a^3)*x + (3/2)*a*b^2*x - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 17, a^3*x - (9/2)*a*b^2*x + (9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - (9*a^2*b*Cos[c + d*x])/(2*d) + (b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + b*Sin[c + d*x])^3, x, 21, (-a^3)*x + (15/2)*a*b^2*x - (45*a^2*b*ArcTanh[Cos[c + d*x]])/(8*d) + (5*b^3*ArcTanh[Cos[c + d*x]])/(2*d) + (45*a^2*b*Cos[c + d*x])/(8*d) - (5*b^3*Cos[c + d*x])/(2*d) - (5*b^3*Cos[c + d*x]^3)/(6*d) - (a^3*Cot[c + d*x])/d + (15*a*b^2*Cot[c + d*x])/(2*d) + (15*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) - (b^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d) + (a^3*Cot[c + d*x]^3)/(3*d) - (5*a*b^2*Cot[c + d*x]^3)/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^5/(a + b*Sin[c + d*x]), x, 5, -(((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(2*a^2 - b^2) - b*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Tan[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, ((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{Tan[c + d*x]^1/(a + b*Sin[c + d*x]), x, 3, -(Log[1 - Sin[c + d*x]]/(2*(a + b)*d)) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)} -{Cot[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]]/(a*d)} -{Cot[c + d*x]^3/(a + b*Sin[c + d*x]), x, 3, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^3*d)} -{Cot[c + d*x]^5/(a + b*Sin[c + d*x]), x, 3, -((b*(2*a^2 - b^2)*Csc[c + d*x])/(a^4*d)) + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^3*d) + (b*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^5*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^5*d)} - -{Tan[c + d*x]^4/(a + b*Sin[c + d*x]), x, 13, (2*a^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (a^2*b*Sec[c + d*x])/((a^2 - b^2)^2*d) + (b*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^3*Tan[c + d*x])/((a^2 - b^2)^2*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)} -{Tan[c + d*x]^2/(a + b*Sin[c + d*x]), x, 8, -((2*a^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d)) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)} -{Cot[c + d*x]^2/(a + b*Sin[c + d*x]), x, 7, -((2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*d)) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d)} -{Cot[c + d*x]^4/(a + b*Sin[c + d*x]), x, 7, (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*d) - (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)} -{Cot[c + d*x]^6/(a + b*Sin[c + d*x]), x, 9, -((2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d)) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)} - - -{Tan[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 5, -((a*(4*a + b)*Log[1 - Sin[c + d*x]])/(8*(a + b)^4*d)) - (a*(4*a - b)*Log[1 + Sin[c + d*x]])/(8*(a - b)^4*d) + (a^4*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - a^5/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(a^2 + b^2 - 2*a*b*Sin[c + d*x]))/(4*(a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(2*(2*a^4 + 3*a^2*b^2 - b^4) - a*b*(9*a^2 - b^2)*Sin[c + d*x]))/(4*(a^2 - b^2)^3*d)} -{Tan[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 4, (a*Log[1 - Sin[c + d*x]])/(2*(a + b)^3*d) + (a*Log[1 + Sin[c + d*x]])/(2*(a - b)^3*d) - (a^2*(a^2 + 3*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + a^3/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(a^2 + b^2 - 2*a*b*Sin[c + d*x]))/(2*(a^2 - b^2)^2*d)} -{Tan[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 3, -Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) + ((a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - a/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 3, Log[Sin[c + d*x]]/(a^2*d) - Log[a + b*Sin[c + d*x]]/(a^2*d) + 1/(a*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 3, (2*b*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^2*d) - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^2 - 3*b^2)*Log[a + b*Sin[c + d*x]])/(a^4*d) - (a^2 - b^2)/(a^3*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 3, -((4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d)) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))} - -{Tan[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 16, (2*a^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (8*a^3*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(12*(a + b)^2*d*(1 - Sin[c + d*x])^2) + Cos[c + d*x]/(12*(a + b)^2*d*(1 - Sin[c + d*x])) - ((3*a + b)*Cos[c + d*x])/(4*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^2*d*(1 + Sin[c + d*x])^2) - Cos[c + d*x]/(12*(a - b)^2*d*(1 + Sin[c + d*x])) + ((3*a - b)*Cos[c + d*x])/(4*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^4*b*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 12, -((2*a^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) - (4*a*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 8, -((2*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]*d)) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*Cot[c + d*x])/(a^2*d) + Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 8, (2*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) - (b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((7*a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) - ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^3*b*d) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^6/(a + b*Sin[c + d*x])^2, x, 10, -((2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^7*d)) + (b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*ArcTanh[Cos[c + d*x]])/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*Cot[c + d*x])/(15*a^6*d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^3*b^2*d*(a + b*Sin[c + d*x])) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x]))} - - -{Tan[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 5, -(((8*a^2 - 5*a*b - b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^5*d)) - ((8*a^2 + 5*a*b - b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^5*d) + (a^3*(a^4 + 13*a^2*b^2 + 10*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - a^5/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) - (a^4*(a^2 + 5*b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*(a^2 - b^2)^3*d) - (Sec[c + d*x]^2*(8*a^3*(a^2 + 5*b^2) - b*(27*a^4 + 22*a^2*b^2 - b^4)*Sin[c + d*x]))/(8*(a^2 - b^2)^4*d)} -{Tan[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 4, ((2*a - b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^4*d) + ((2*a + b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^4*d) - (a*(a^4 + 8*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) + a^3/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (a^2*(a^2 + 3*b^2))/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)} -{Tan[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 3, -Log[1 - Sin[c + d*x]]/(2*(a + b)^3*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)^3*d) + (a*(a^2 + 3*b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - a/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a^2 + b^2)/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 3, Log[Sin[c + d*x]]/(a^3*d) - Log[a + b*Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + b*Sin[c + d*x])^2) + 1/(a^2*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 3, (3*b*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^3*d) - ((a^2 - 6*b^2)*Log[Sin[c + d*x]])/(a^5*d) + ((a^2 - 6*b^2)*Log[a + b*Sin[c + d*x]])/(a^5*d) - (a^2 - b^2)/(2*a^3*d*(a + b*Sin[c + d*x])^2) - (a^2 - 3*b^2)/(a^4*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 3, (-2*b*(3*a^2 - 5*b^2)*Csc[c + d*x])/(a^6*d) + ((a^2 - 3*b^2)*Csc[c + d*x]^2)/(a^5*d) + (b*Csc[c + d*x]^3)/(a^4*d) - Csc[c + d*x]^4/(4*a^3*d) + ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[Sin[c + d*x]])/(a^7*d) - ((a^4 - 12*a^2*b^2 + 15*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) + (a^2 - b^2)^2/(2*a^5*d*(a + b*Sin[c + d*x])^2) + (a^4 - 6*a^2*b^2 + 5*b^4)/(a^6*d*(a + b*Sin[c + d*x]))} - -{Tan[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 22, (8*a^4*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (12*a^2*b^2*(a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (a^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + Cos[c + d*x]/(12*(a + b)^3*d*(1 - Sin[c + d*x])^2) - (3*a*Cos[c + d*x])/(4*(a + b)^4*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(12*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^3*d*(1 + Sin[c + d*x])^2) + (3*a*Cos[c + d*x])/(4*(a - b)^4*d*(1 + Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^4*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (3*a^5*b*Cos[c + d*x])/(2*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (4*a^3*b^3*Cos[c + d*x])/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 18, -((4*a^2*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) - (a^2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (2*b^2*(3*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^3*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*a*b^3*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 9, -(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)*d)) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x])/(2*a^3*(a^2 - b^2)*d) + Cot[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 9, ((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^3*b*d*(a + b*Sin[c + d*x]))} -{Cot[c + d*x]^6/(a + b*Sin[c + d*x])^3, x, 11, -((Sqrt[a^2 - b^2]*(2*a^4 - 29*a^2*b^2 + 42*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^8*d)) + (b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c + d*x])/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^6*b*d) - ((15*a^4 - 187*a^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + ((5*a^4 - 60*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3)/(20*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^4 - 54*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^4*b^2*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Sin[e+f x])^m with p symbolic*) - - -{(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^3, x, 10, (a^3*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(3*a^2*b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p)) + (1/(f*g*(4 + p)))*(b^3*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (4 + p)/2, (6 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*(g*Tan[e + f*x])^(1 + p)) + (3*a*b^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))} -{(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^2, x, 8, (a^2*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(2*a*b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p)) + (b^2*Hypergeometric2F1[2, (3 + p)/2, (5 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(3 + p))/(f*g^3*(3 + p))} -{(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^1, x, 6, (a*Hypergeometric2F1[1, (1 + p)/2, (3 + p)/2, -Tan[e + f*x]^2]*(g*Tan[e + f*x])^(1 + p))/(f*g*(1 + p)) + (1/(f*g*(2 + p)))*(b*(Cos[e + f*x]^2)^((1 + p)/2)*Hypergeometric2F1[(1 + p)/2, (2 + p)/2, (4 + p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(g*Tan[e + f*x])^(1 + p))} -{(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^1, x, 0, (a*g*AppellF1[(1 - p)/2, (1 - p)/2, 1, (3 - p)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*(Sin[e + f*x]^2)^((1 - p)/2)*(g*Tan[e + f*x])^(-1 + p))/((a^2 - b^2)*f*(-1 + p)) + (b*AppellF1[(1 - p)/2, -(p/2), 1, (3 - p)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]*(g*Tan[e + f*x])^p)/((Sin[e + f*x]^2)^(p/2)*((-a^2 + b^2)*f*(-1 + p))), Unintegrable[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x]), x]} -{(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^2, x, 0, -((a^2*AppellF1[(1 - q)/2, (1 - q)/2, 1, (3 - q)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(-1 - q))*(g*Tan[e + f*x])^q)/((a^2 - b^2)^2*f*(-1 + q))) + (b^2*AppellF1[(1 - q)/2, (1 - q)/2, 1, (3 - q)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(-1 - q))*(g*Tan[e + f*x])^q)/((a^2 - b^2)^2*f*(-1 + q)) + (2*a^2*AppellF1[(1 - q)/2, (1 - q)/2, 2, (3 - q)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(-1 - q))*(g*Tan[e + f*x])^q)/((a^2 - b^2)^2*f*(-1 + q)) - (2*a*b*AppellF1[(1 - q)/2, -(q/2), 2, (3 - q)/2, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]*(g*Tan[e + f*x])^q)/((Sin[e + f*x]^2)^(q/2)*((a^2 - b^2)^2*f*(-1 + q))), Unintegrable[(g*Tan[e + f*x])^p/(a + b*Sin[e + f*x])^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Sin[e+f x])^m with m symbolic*) - - -{(g*Tan[e + f*x])^p*(a + b*Sin[e + f*x])^m, x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m deleted file mode 100644 index eb64da4..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.m +++ /dev/null @@ -1,739 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sin[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sin[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[a + b*x]*(c + d*x)^4, x, 5, -((24*d^4*Cos[a + b*x])/b^5) + (12*d^2*(c + d*x)^2*Cos[a + b*x])/b^3 - ((c + d*x)^4*Cos[a + b*x])/b - (24*d^3*(c + d*x)*Sin[a + b*x])/b^4 + (4*d*(c + d*x)^3*Sin[a + b*x])/b^2} -{Sin[a + b*x]*(c + d*x)^3, x, 4, (6*d^2*(c + d*x)*Cos[a + b*x])/b^3 - ((c + d*x)^3*Cos[a + b*x])/b - (6*d^3*Sin[a + b*x])/b^4 + (3*d*(c + d*x)^2*Sin[a + b*x])/b^2} -{Sin[a + b*x]*(c + d*x)^2, x, 3, (2*d^2*Cos[a + b*x])/b^3 - ((c + d*x)^2*Cos[a + b*x])/b + (2*d*(c + d*x)*Sin[a + b*x])/b^2} -{Sin[a + b*x]*(c + d*x)^1, x, 2, -(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2} -{Sin[a + b*x]/(c + d*x)^1, x, 3, (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{Sin[a + b*x]/(c + d*x)^2, x, 4, (b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2 - Sin[a + b*x]/(d*(c + d*x)) - (b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} -{Sin[a + b*x]/(c + d*x)^3, x, 5, -((b*Cos[a + b*x])/(2*d^2*(c + d*x))) - (b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(2*d^3) - Sin[a + b*x]/(2*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(2*d^3)} - - -{Sin[a + b*x]^2*(c + d*x)^4, x, 6, (3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) + (c + d*x)^5/(10*d) - (3*d^4*Cos[a + b*x]*Sin[a + b*x])/(4*b^5) + (3*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) - ((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x])/(2*b) - (3*d^3*(c + d*x)*Sin[a + b*x]^2)/(2*b^4) + (d*(c + d*x)^3*Sin[a + b*x]^2)/b^2} -{Sin[a + b*x]^2*(c + d*x)^3, x, 4, -((3*c*d^2*x)/(4*b^2)) - (3*d^3*x^2)/(8*b^2) + (c + d*x)^4/(8*d) + (3*d^2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(4*b^3) - ((c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(2*b) - (3*d^3*Sin[a + b*x]^2)/(8*b^4) + (3*d*(c + d*x)^2*Sin[a + b*x]^2)/(4*b^2)} -{Sin[a + b*x]^2*(c + d*x)^2, x, 4, -((d^2*x)/(4*b^2)) + (c + d*x)^3/(6*d) + (d^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^3) - ((c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d*(c + d*x)*Sin[a + b*x]^2)/(2*b^2)} -{Sin[a + b*x]^2*(c + d*x)^1, x, 2, (c*x)/2 + (d*x^2)/4 - ((c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d*Sin[a + b*x]^2)/(4*b^2)} -{Sin[a + b*x]^2/(c + d*x)^1, x, 5, -((Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(2*d)) + Log[c + d*x]/(2*d) + (Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Sin[a + b*x]^2/(c + d*x)^2, x, 5, (b*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^2 - Sin[a + b*x]^2/(d*(c + d*x)) + (b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Sin[a + b*x]^2/(c + d*x)^3, x, 7, (b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^3 - (b*Cos[a + b*x]*Sin[a + b*x])/(d^2*(c + d*x)) - Sin[a + b*x]^2/(2*d*(c + d*x)^2) - (b^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^3} -{Sin[a + b*x]^2/(c + d*x)^4, x, 7, -(b^2/(3*d^3*(c + d*x))) - (2*b^3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(3*d^4) - (b*Cos[a + b*x]*Sin[a + b*x])/(3*d^2*(c + d*x)^2) - Sin[a + b*x]^2/(3*d*(c + d*x)^3) + (2*b^2*Sin[a + b*x]^2)/(3*d^3*(c + d*x)) - (2*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)} - - -{Sin[a + b*x]^3*(c + d*x)^4, x, 12, -((488*d^4*Cos[a + b*x])/(27*b^5)) + (80*d^2*(c + d*x)^2*Cos[a + b*x])/(9*b^3) - (2*(c + d*x)^4*Cos[a + b*x])/(3*b) + (8*d^4*Cos[a + b*x]^3)/(81*b^5) - (160*d^3*(c + d*x)*Sin[a + b*x])/(9*b^4) + (8*d*(c + d*x)^3*Sin[a + b*x])/(3*b^2) + (4*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^3) - ((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) - (8*d^3*(c + d*x)*Sin[a + b*x]^3)/(27*b^4) + (4*d*(c + d*x)^3*Sin[a + b*x]^3)/(9*b^2)} -{Sin[a + b*x]^3*(c + d*x)^3, x, 8, (40*d^2*(c + d*x)*Cos[a + b*x])/(9*b^3) - (2*(c + d*x)^3*Cos[a + b*x])/(3*b) - (40*d^3*Sin[a + b*x])/(9*b^4) + (2*d*(c + d*x)^2*Sin[a + b*x])/b^2 + (2*d^2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^3) - ((c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) - (2*d^3*Sin[a + b*x]^3)/(27*b^4) + (d*(c + d*x)^2*Sin[a + b*x]^3)/(3*b^2)} -{Sin[a + b*x]^3*(c + d*x)^2, x, 6, (14*d^2*Cos[a + b*x])/(9*b^3) - (2*(c + d*x)^2*Cos[a + b*x])/(3*b) - (2*d^2*Cos[a + b*x]^3)/(27*b^3) + (4*d*(c + d*x)*Sin[a + b*x])/(3*b^2) - ((c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (2*d*(c + d*x)*Sin[a + b*x]^3)/(9*b^2)} -{Sin[a + b*x]^3*(c + d*x)^1, x, 3, -((2*(c + d*x)*Cos[a + b*x])/(3*b)) + (2*d*Sin[a + b*x])/(3*b^2) - ((c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (d*Sin[a + b*x]^3)/(9*b^2)} -{Sin[a + b*x]^3/(c + d*x)^1, x, 8, -((CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d)) + (3*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d) + (3*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) - (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{Sin[a + b*x]^3/(c + d*x)^2, x, 8, (3*b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(4*d^2) - (3*b*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(4*d^2) - Sin[a + b*x]^3/(d*(c + d*x)) - (3*b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{Sin[a + b*x]^3/(c + d*x)^3, x, 12, (9*b^2*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(8*d^3) - (3*b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d^3) - (3*b*Cos[a + b*x]*Sin[a + b*x]^2)/(2*d^2*(c + d*x)) - Sin[a + b*x]^3/(2*d*(c + d*x)^2) - (3*b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Csc[a + b*x]*(c + d*x)^3, x, 9, -((2*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b) + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (6*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4} -{Csc[a + b*x]*(c + d*x)^2, x, 7, -((2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b) + (2*I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (2*d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, E^(I*(a + b*x))])/b^3} -{Csc[a + b*x]*(c + d*x)^1, x, 5, -((2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b) + (I*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, E^(I*(a + b*x))])/b^2} -{Csc[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Csc[a + b*x]/(c + d*x), x]} -{Csc[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Csc[a + b*x]/(c + d*x)^2, x]} - - -{Csc[a + b*x]^2*(c + d*x)^3, x, 6, -((I*(c + d*x)^3)/b) - ((c + d*x)^3*Cot[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b^2 - (3*I*d^2*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^4)} -{Csc[a + b*x]^2*(c + d*x)^2, x, 5, -((I*(c + d*x)^2)/b) - ((c + d*x)^2*Cot[a + b*x])/b + (2*d*(c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b^2 - (I*d^2*PolyLog[2, E^(2*I*(a + b*x))])/b^3} -{Csc[a + b*x]^2*(c + d*x)^1, x, 2, -(((c + d*x)*Cot[a + b*x])/b) + (d*Log[Sin[a + b*x]])/b^2} -{Csc[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Csc[a + b*x]^2/(c + d*x), x]} -{Csc[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Csc[a + b*x]^2/(c + d*x)^2, x]} - - -{Csc[a + b*x]^3*(c + d*x)^3, x, 15, -((6*d^2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^3) - ((c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (3*d*(c + d*x)^2*Csc[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]*Csc[a + b*x])/(2*b) + (3*I*d^3*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, E^(I*(a + b*x))])/b^4 - (3*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (3*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (3*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4} -{Csc[a + b*x]^3*(c + d*x)^2, x, 9, -(((c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b) - (d^2*ArcTanh[Cos[a + b*x]])/b^3 - (d*(c + d*x)*Csc[a + b*x])/b^2 - ((c + d*x)^2*Cot[a + b*x]*Csc[a + b*x])/(2*b) + (I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (d^2*PolyLog[3, E^(I*(a + b*x))])/b^3} -{Csc[a + b*x]^3*(c + d*x)^1, x, 6, -(((c + d*x)*ArcTanh[E^(I*(a + b*x))])/b) - (d*Csc[a + b*x])/(2*b^2) - ((c + d*x)*Cot[a + b*x]*Csc[a + b*x])/(2*b) + (I*d*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (I*d*PolyLog[2, E^(I*(a + b*x))])/(2*b^2)} -{Csc[a + b*x]^3/(c + d*x)^1, x, 0, Unintegrable[Csc[a + b*x]^3/(c + d*x), x]} -{Csc[a + b*x]^3/(c + d*x)^2, x, 0, Unintegrable[Csc[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Sin[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[a + b*x]*(c + d*x)^(5/2), x, 8, (15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(4*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/b - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(2*b^2)} -{Sin[a + b*x]*(c + d*x)^(3/2), x, 7, -(((c + d*x)^(3/2)*Cos[a + b*x])/b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(2*b^2)} -{Sin[a + b*x]*(c + d*x)^(1/2), x, 6, -((Sqrt[c + d*x]*Cos[a + b*x])/b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/b^(3/2) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/b^(3/2)} -{Sin[a + b*x]/(c + d*x)^(1/2), x, 5, (Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + (Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])} -{Sin[a + b*x]/(c + d*x)^(3/2), x, 6, (2*Sqrt[b]*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (2*Sqrt[b]*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(3/2) - (2*Sin[a + b*x])/(d*Sqrt[c + d*x])} -{Sin[a + b*x]/(c + d*x)^(5/2), x, 7, -((4*b*Cos[a + b*x])/(3*d^2*Sqrt[c + d*x])) - (4*b^(3/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) - (4*b^(3/2)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(3*d^(5/2)) - (2*Sin[a + b*x])/(3*d*(c + d*x)^(3/2))} -{Sin[a + b*x]/(c + d*x)^(7/2), x, 8, -((4*b*Cos[a + b*x])/(15*d^2*(c + d*x)^(3/2))) - (8*b^(5/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (8*b^(5/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(15*d^(7/2)) - (2*Sin[a + b*x])/(5*d*(c + d*x)^(5/2)) + (8*b^2*Sin[a + b*x])/(15*d^3*Sqrt[c + d*x])} - - -{Sin[a + b*x]^2*(c + d*x)^(5/2), x, 10, -((5*d*(c + d*x)^(3/2))/(16*b^2)) + (c + d*x)^(7/2)/(7*d) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(7/2)) - ((c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x]^2)/(8*b^2) + (15*d^2*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(64*b^3)} -{Sin[a + b*x]^2*(c + d*x)^(3/2), x, 9, -((3*d*Sqrt[c + d*x])/(16*b^2)) + (c + d*x)^(5/2)/(5*d) + (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(32*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(32*b^(5/2)) - ((c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (3*d*Sqrt[c + d*x]*Sin[a + b*x]^2)/(8*b^2)} -{Sin[a + b*x]^2*(c + d*x)^(1/2), x, 8, (c + d*x)^(3/2)/(3*d) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(8*b^(3/2)) + (Sqrt[d]*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(8*b^(3/2)) - (Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(4*b)} -{Sin[a + b*x]^2/(c + d*x)^(1/2), x, 7, Sqrt[c + d*x]/d - (Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2*Sqrt[b]*Sqrt[d]) + (Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2*Sqrt[b]*Sqrt[d])} -{Sin[a + b*x]^2/(c + d*x)^(3/2), x, 7, (2*Sqrt[b]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/d^(3/2) + (2*Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/d^(3/2) - (2*Sin[a + b*x]^2)/(d*Sqrt[c + d*x])} -{Sin[a + b*x]^2/(c + d*x)^(5/2), x, 9, (8*b^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(3*d^(5/2)) - (8*b^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(3*d^(5/2)) - (8*b*Cos[a + b*x]*Sin[a + b*x])/(3*d^2*Sqrt[c + d*x]) - (2*Sin[a + b*x]^2)/(3*d*(c + d*x)^(3/2))} -{Sin[a + b*x]^2/(c + d*x)^(7/2), x, 9, -((16*b^2)/(15*d^3*Sqrt[c + d*x])) - (32*b^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(15*d^(7/2)) - (32*b^(5/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(15*d^(7/2)) - (8*b*Cos[a + b*x]*Sin[a + b*x])/(15*d^2*(c + d*x)^(3/2)) - (2*Sin[a + b*x]^2)/(5*d*(c + d*x)^(5/2)) + (32*b^2*Sin[a + b*x]^2)/(15*d^3*Sqrt[c + d*x])} -{Sin[a + b*x]^2/(c + d*x)^(9/2), x, 11, -((16*b^2)/(105*d^3*(c + d*x)^(3/2))) - (128*b^(7/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(105*d^(9/2)) + (128*b^(7/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(105*d^(9/2)) - (8*b*Cos[a + b*x]*Sin[a + b*x])/(35*d^2*(c + d*x)^(5/2)) + (128*b^3*Cos[a + b*x]*Sin[a + b*x])/(105*d^4*Sqrt[c + d*x]) - (2*Sin[a + b*x]^2)/(7*d*(c + d*x)^(7/2)) + (32*b^2*Sin[a + b*x]^2)/(105*d^3*(c + d*x)^(3/2))} - - -{Sin[a + b*x]^3*(c + d*x)^(5/2), x, 23, (45*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^3) - (2*(c + d*x)^(5/2)*Cos[a + b*x])/(3*b) - (5*d^2*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(144*b^3) - (45*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(3*b^2) - ((c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x]^3)/(18*b^2)} -{Sin[a + b*x]^3*(c + d*x)^(3/2), x, 20, -((2*(c + d*x)^(3/2)*Cos[a + b*x])/(3*b)) - (9*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) - (9*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + (d*Sqrt[c + d*x]*Sin[a + b*x])/b^2 - ((c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (d*Sqrt[c + d*x]*Sin[a + b*x]^3)/(6*b^2)} -{Sin[a + b*x]^3*(c + d*x)^(1/2), x, 14, -((3*Sqrt[c + d*x]*Cos[a + b*x])/(4*b)) + (Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(12*b) + (3*Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (3*Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2))} -{Sin[a + b*x]^3/(c + d*x)^(1/2), x, 12, (3*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(2*Sqrt[b]*Sqrt[d]) + (3*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*Sqrt[b]*Sqrt[d])} -{Sin[a + b*x]^3/(c + d*x)^(3/2), x, 12, (3*Sqrt[b]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/d^(3/2) - (3*Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(3/2) - (2*Sin[a + b*x]^3)/(d*Sqrt[c + d*x])} -{Sin[a + b*x]^3/(c + d*x)^(5/2), x, 18, -((b^(3/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(5/2)) + (b^(3/2)*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(5/2) + (b^(3/2)*Sqrt[6*Pi]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/d^(5/2) - (b^(3/2)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(5/2) - (4*b*Cos[a + b*x]*Sin[a + b*x]^2)/(d^2*Sqrt[c + d*x]) - (2*Sin[a + b*x]^3)/(3*d*(c + d*x)^(3/2))} -{Sin[a + b*x]^3/(c + d*x)^(7/2), x, 19, -((2*b^(5/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2))) + (6*b^(5/2)*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (6*b^(5/2)*Sqrt[6*Pi]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(5*d^(7/2)) + (2*b^(5/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(5*d^(7/2)) - (16*b^2*Sin[a + b*x])/(5*d^3*Sqrt[c + d*x]) - (4*b*Cos[a + b*x]*Sin[a + b*x]^2)/(5*d^2*(c + d*x)^(3/2)) - (2*Sin[a + b*x]^3)/(5*d*(c + d*x)^(5/2)) + (24*b^2*Sin[a + b*x]^3)/(5*d^3*Sqrt[c + d*x])} - - -{Sin[f*x]*(d*x)^(3/2), x, 4, -(((d*x)^(3/2)*Cos[f*x])/f) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(2*f^(5/2)) + (3*d*Sqrt[d*x]*Sin[f*x])/(2*f^2)} -{Sin[f*x]*(d*x)^(1/2), x, 3, -((Sqrt[d*x]*Cos[f*x])/f) + (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/f^(3/2)} -{Sin[f*x]/(d*x)^(1/2), x, 2, (Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[d]*Sqrt[f])} -{Sin[f*x]/(d*x)^(3/2), x, 3, (2*Sqrt[f]*Sqrt[2*Pi]*FresnelC[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Sin[f*x])/(d*Sqrt[d*x])} -{Sin[f*x]/(d*x)^(5/2), x, 4, -((4*f*Cos[f*x])/(3*d^2*Sqrt[d*x])) - (4*f^(3/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[f]*Sqrt[2/Pi]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*Sin[f*x])/(3*d*(d*x)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Csc[a + b*x]*(c + d*x)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x]*Csc[a + b*x], x]} -{Csc[a + b*x]/(c + d*x)^(1/2), x, 0, Unintegrable[Csc[a + b*x]/Sqrt[c + d*x], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/Sin[e + f*x]^(3/2) + x*Sqrt[Sin[e + f*x]], x, 2, -((2*x*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]])) + (4*Sqrt[Sin[e + f*x]])/f^2} -{x^2/Sin[e + f*x]^(3/2) + x^2*Sqrt[Sin[e + f*x]], x, 3, -((16*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/f^3) - (2*x^2*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) + (8*x*Sqrt[Sin[e + f*x]])/f^2} - - -{x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]), x, 2, -((2*x*Cos[e + f*x])/(3*f*Sin[e + f*x]^(3/2))) - 4/(3*f^2*Sqrt[Sin[e + f*x]])} - - -{x/Sin[e + f*x]^(7/2) + (3/5)*x*Sqrt[Sin[e + f*x]], x, 3, -((2*x*Cos[e + f*x])/(5*f*Sin[e + f*x]^(5/2))) - 4/(15*f^2*Sin[e + f*x]^(3/2)) - (6*x*Cos[e + f*x])/(5*f*Sqrt[Sin[e + f*x]]) + (12*Sqrt[Sin[e + f*x]])/(5*f^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sin[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(b*Sin[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(b*Sin[e + f*x])^n, x]} - - -{Sin[a + b*x]^3*(c + d*x)^m, x, 8, -((3*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b))) - (3*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b)) + (3^(-1 - m)*E^(3*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((3*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b)) + (3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, (3*I*b*(c + d*x))/d])/(E^(3*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b))} -{Sin[a + b*x]^2*(c + d*x)^m, x, 5, (c + d*x)^(1 + m)/(2*d*(1 + m)) + (I*2^(-3 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b) - (I*2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b)} -{Sin[a + b*x]^1*(c + d*x)^m, x, 3, -((E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b))) - ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b))} -{Csc[a + b*x]^1*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Csc[a + b*x], x]} -{Csc[a + b*x]^2*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Csc[a + b*x]^2, x]} - - -{x^(m + 3)*Sin[a + b*x], x, 3, (I*E^(I*a)*x^m*Gamma[4 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^4)) - (I*x^m*Gamma[4 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^4))} -{x^(m + 2)*Sin[a + b*x], x, 3, (E^(I*a)*x^m*Gamma[3 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^3)) + (x^m*Gamma[3 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^3))} -{x^(m + 1)*Sin[a + b*x], x, 3, -((I*E^(I*a)*x^m*Gamma[2 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^2))) + (I*x^m*Gamma[2 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^2))} -{x^(m + 0)*Sin[a + b*x], x, 3, -((E^(I*a)*x^m*Gamma[1 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b))) - (x^m*Gamma[1 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b))} -{x^(m - 1)*Sin[a + b*x], x, 3, ((1/2)*I*E^(I*a)*x^m*Gamma[m, (-I)*b*x])/((-I)*b*x)^m - ((1/2)*I*x^m*Gamma[m, I*b*x])/(E^(I*a)*(I*b*x)^m)} -{x^(m - 2)*Sin[a + b*x], x, 3, ((1/2)*b*E^(I*a)*x^m*Gamma[-1 + m, (-I)*b*x])/((-I)*b*x)^m + ((1/2)*b*x^m*Gamma[-1 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)} -{x^(m - 3)*Sin[a + b*x], x, 3, ((-(1/2))*I*b^2*E^(I*a)*x^m*Gamma[-2 + m, (-I)*b*x])/((-I)*b*x)^m + ((1/2)*I*b^2*x^m*Gamma[-2 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)} - - -{x^(m + 3)*Sin[a + b*x]^2, x, 5, x^(4 + m)/(2*(4 + m)) + (2^(-6 - m)*E^(2*I*a)*x^m*Gamma[4 + m, -2*I*b*x])/(((-I)*b*x)^m*b^4) + (2^(-6 - m)*x^m*Gamma[4 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^4)} -{x^(m + 2)*Sin[a + b*x]^2, x, 5, x^(3 + m)/(2*(3 + m)) - (I*2^(-5 - m)*E^(2*I*a)*x^m*Gamma[3 + m, -2*I*b*x])/(((-I)*b*x)^m*b^3) + (I*2^(-5 - m)*x^m*Gamma[3 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^3)} -{x^(m + 1)*Sin[a + b*x]^2, x, 5, x^(2 + m)/(2*(2 + m)) - (2^(-4 - m)*E^(2*I*a)*x^m*Gamma[2 + m, -2*I*b*x])/(((-I)*b*x)^m*b^2) - (2^(-4 - m)*x^m*Gamma[2 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^2)} -{x^(m + 0)*Sin[a + b*x]^2, x, 5, x^(1 + m)/(2*(1 + m)) + (I*2^(-3 - m)*E^(2*I*a)*x^m*Gamma[1 + m, -2*I*b*x])/(((-I)*b*x)^m*b) - (I*2^(-3 - m)*x^m*Gamma[1 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b)} -{x^(m - 1)*Sin[a + b*x]^2, x, 5, x^m/(2*m) + (2^(-2 - m)*E^(2*I*a)*x^m*Gamma[m, -2*I*b*x])/((-I)*b*x)^m + (2^(-2 - m)*x^m*Gamma[m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m)} -{x^(m - 2)*Sin[a + b*x]^2, x, 5, -(x^(-1 + m)/(2*(1 - m))) - (I*2^(-1 - m)*b*E^(2*I*a)*x^m*Gamma[-1 + m, -2*I*b*x])/((-I)*b*x)^m + (I*2^(-1 - m)*b*x^m*Gamma[-1 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m)} -{x^(m - 3)*Sin[a + b*x]^2, x, 5, -(x^(-2 + m)/(2*(2 - m))) - (b^2*E^(2*I*a)*x^m*Gamma[-2 + m, -2*I*b*x])/(2^m*((-I)*b*x)^m) - (b^2*x^m*Gamma[-2 + m, 2*I*b*x])/(2^m*E^(2*I*a)*(I*b*x)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Csc[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Csc[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/Csc[e + f*x]^(3/2) - x*Sqrt[Csc[e + f*x]]/3, x, 4, 4/(9*f^2*Csc[e + f*x]^(3/2)) - (2*x*Cos[e + f*x])/(3*f*Sqrt[Csc[e + f*x]])} -{x^2/Csc[e + f*x]^(3/2) - (1/3)*x^2*Sqrt[Csc[e + f*x]], x, 7, (8*x)/(9*f^2*Csc[e + f*x]^(3/2)) + (16*Cos[e + f*x])/(27*f^3*Sqrt[Csc[e + f*x]]) - (2*x^2*Cos[e + f*x])/(3*f*Sqrt[Csc[e + f*x]]) - (16*Sqrt[Csc[e + f*x]]*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[Sin[e + f*x]])/(27*f^3)} - - -{x/Csc[e + f*x]^(5/2) - 3*x/(5*Sqrt[Csc[e + f*x]]), x, 4, 4/(25*f^2*Csc[e + f*x]^(5/2)) - (2*x*Cos[e + f*x])/(5*f*Csc[e + f*x]^(3/2))} - - -{x/Csc[e + f*x]^(7/2) - (5/21)*x*Sqrt[Csc[e + f*x]], x, 5, 4/(49*f^2*Csc[e + f*x]^(7/2)) - (2*x*Cos[e + f*x])/(7*f*Csc[e + f*x]^(5/2)) + 20/(63*f^2*Csc[e + f*x]^(3/2)) - (10*x*Cos[e + f*x])/(21*f*Sqrt[Csc[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sin[e + f*x])*(c + d*x)^3, x, 6, (a*(c + d*x)^4)/(4*d) + (6*a*d^2*(c + d*x)*Cos[e + f*x])/f^3 - (a*(c + d*x)^3*Cos[e + f*x])/f - (6*a*d^3*Sin[e + f*x])/f^4 + (3*a*d*(c + d*x)^2*Sin[e + f*x])/f^2} -{(a + a*Sin[e + f*x])*(c + d*x)^2, x, 5, (a*(c + d*x)^3)/(3*d) + (2*a*d^2*Cos[e + f*x])/f^3 - (a*(c + d*x)^2*Cos[e + f*x])/f + (2*a*d*(c + d*x)*Sin[e + f*x])/f^2} -{(a + a*Sin[e + f*x])*(c + d*x)^1, x, 4, (a*(c + d*x)^2)/(2*d) - (a*(c + d*x)*Cos[e + f*x])/f + (a*d*Sin[e + f*x])/f^2} -{(a + a*Sin[e + f*x])/(c + d*x)^1, x, 5, (a*Log[c + d*x])/d + (a*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (a*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d} -{(a + a*Sin[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) + (a*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 - (a*Sin[e + f*x])/(d*(c + d*x)) - (a*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2} -{(a + a*Sin[e + f*x])/(c + d*x)^3, x, 7, -(a/(2*d*(c + d*x)^2)) - (a*f*Cos[e + f*x])/(2*d^2*(c + d*x)) - (a*f^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/(2*d^3) - (a*Sin[e + f*x])/(2*d*(c + d*x)^2) - (a*f^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(a + a*Sin[e + f*x])^2*(c + d*x)^3, x, 10, -((3*a^2*c*d^2*x)/(4*f^2)) - (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) + (12*a^2*d^2*(c + d*x)*Cos[e + f*x])/f^3 - (2*a^2*(c + d*x)^3*Cos[e + f*x])/f - (12*a^2*d^3*Sin[e + f*x])/f^4 + (6*a^2*d*(c + d*x)^2*Sin[e + f*x])/f^2 + (3*a^2*d^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*Cos[e + f*x]*Sin[e + f*x])/(2*f) - (3*a^2*d^3*Sin[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sin[e + f*x]^2)/(4*f^2)} -{(a + a*Sin[e + f*x])^2*(c + d*x)^2, x, 9, -((a^2*d^2*x)/(4*f^2)) + (a^2*(c + d*x)^3)/(2*d) + (4*a^2*d^2*Cos[e + f*x])/f^3 - (2*a^2*(c + d*x)^2*Cos[e + f*x])/f + (4*a^2*d*(c + d*x)*Sin[e + f*x])/f^2 + (a^2*d^2*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (a^2*(c + d*x)^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (a^2*d*(c + d*x)*Sin[e + f*x]^2)/(2*f^2)} -{(a + a*Sin[e + f*x])^2*(c + d*x)^1, x, 6, (1/2)*a^2*c*x + (1/4)*a^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (2*a^2*(c + d*x)*Cos[e + f*x])/f + (2*a^2*d*Sin[e + f*x])/f^2 - (a^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (a^2*d*Sin[e + f*x]^2)/(4*f^2)} -{(a + a*Sin[e + f*x])^2/(c + d*x)^1, x, 9, -((a^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*d)) + (3*a^2*Log[c + d*x])/(2*d) + (2*a^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (2*a^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d + (a^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + a*Sin[e + f*x])^2/(c + d*x)^2, x, 9, (2*a^2*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 + (a^2*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d^2 - (4*a^2*Sin[e/2 + Pi/4 + (f*x)/2]^4)/(d*(c + d*x)) - (2*a^2*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2 + (a^2*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + a*Sin[e + f*x])^2/(c + d*x)^3, x, 15, (a^2*f^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/d^3 - (a^2*f^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d^3 - (4*a^2*f*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]^3)/(d^2*(c + d*x)) - (2*a^2*Sin[e/2 + Pi/4 + (f*x)/2]^4)/(d*(c + d*x)^2) - (a^2*f^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^3 - (a^2*f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(a + a*Sin[e + f*x])*(c + d*x)^3, x, 7, -((I*(c + d*x)^3)/(a*f)) - ((c + d*x)^3*Cot[e/2 + Pi/4 + (f*x)/2])/(a*f) + (6*d*(c + d*x)^2*Log[1 - I*E^(I*(e + f*x))])/(a*f^2) - (12*I*d^2*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/(a*f^3) + (12*d^3*PolyLog[3, I*E^(I*(e + f*x))])/(a*f^4)} -{1/(a + a*Sin[e + f*x])*(c + d*x)^2, x, 6, -((I*(c + d*x)^2)/(a*f)) - ((c + d*x)^2*Cot[e/2 + Pi/4 + (f*x)/2])/(a*f) + (4*d*(c + d*x)*Log[1 - I*E^(I*(e + f*x))])/(a*f^2) - (4*I*d^2*PolyLog[2, I*E^(I*(e + f*x))])/(a*f^3)} -{1/(a + a*Sin[e + f*x])*(c + d*x)^1, x, 3, -(((c + d*x)*Cot[e/2 + Pi/4 + (f*x)/2])/(a*f)) + (2*d*Log[Sin[e/2 + Pi/4 + (f*x)/2]])/(a*f^2)} -{1/(a + a*Sin[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Sin[e + f*x])), x]} -{1/(a + a*Sin[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]} - - -{1/(a + a*Sin[e + f*x])^2*(c + d*x)^3, x, 10, -((I*(c + d*x)^3)/(3*a^2*f)) - (2*d^2*(c + d*x)*Cot[e/2 + Pi/4 + (f*x)/2])/(a^2*f^3) - ((c + d*x)^3*Cot[e/2 + Pi/4 + (f*x)/2])/(3*a^2*f) - (d*(c + d*x)^2*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(2*a^2*f^2) - ((c + d*x)^3*Cot[e/2 + Pi/4 + (f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f) + (2*d*(c + d*x)^2*Log[1 - I*E^(I*(e + f*x))])/(a^2*f^2) + (4*d^3*Log[Sin[e/2 + Pi/4 + (f*x)/2]])/(a^2*f^4) - (4*I*d^2*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/(a^2*f^3) + (4*d^3*PolyLog[3, I*E^(I*(e + f*x))])/(a^2*f^4)} -{1/(a + a*Sin[e + f*x])^2*(c + d*x)^2, x, 9, -((I*(c + d*x)^2)/(3*a^2*f)) - (2*d^2*Cot[e/2 + Pi/4 + (f*x)/2])/(3*a^2*f^3) - ((c + d*x)^2*Cot[e/2 + Pi/4 + (f*x)/2])/(3*a^2*f) - (d*(c + d*x)*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(3*a^2*f^2) - ((c + d*x)^2*Cot[e/2 + Pi/4 + (f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f) + (4*d*(c + d*x)*Log[1 - I*E^(I*(e + f*x))])/(3*a^2*f^2) - (4*I*d^2*PolyLog[2, I*E^(I*(e + f*x))])/(3*a^2*f^3)} -{1/(a + a*Sin[e + f*x])^2*(c + d*x)^1, x, 4, -(((c + d*x)*Cot[e/2 + Pi/4 + (f*x)/2])/(3*a^2*f)) - (d*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f^2) - ((c + d*x)*Cot[e/2 + Pi/4 + (f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(6*a^2*f) + (2*d*Log[Sin[e/2 + Pi/4 + (f*x)/2]])/(3*a^2*f^2)} -{1/(a + a*Sin[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Sin[e + f*x])^2), x]} -{1/(a + a*Sin[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Sin[e + f*x])^2), x]} - - -{1/(a - a*Sin[e + f*x])*(c + d*x)^3, x, 7, -((I*(c + d*x)^3)/(a*f)) + (6*d*(c + d*x)^2*Log[1 + I*E^(I*(e + f*x))])/(a*f^2) - (12*I*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/(a*f^3) + (12*d^3*PolyLog[3, (-I)*E^(I*(e + f*x))])/(a*f^4) + ((c + d*x)^3*Tan[e/2 + Pi/4 + (f*x)/2])/(a*f)} -{1/(a - a*Sin[e + f*x])*(c + d*x)^2, x, 6, -((I*(c + d*x)^2)/(a*f)) + (4*d*(c + d*x)*Log[1 + I*E^(I*(e + f*x))])/(a*f^2) - (4*I*d^2*PolyLog[2, (-I)*E^(I*(e + f*x))])/(a*f^3) + ((c + d*x)^2*Tan[e/2 + Pi/4 + (f*x)/2])/(a*f)} -{1/(a - a*Sin[e + f*x])*(c + d*x)^1, x, 3, (2*d*Log[Cos[e/2 + Pi/4 + (f*x)/2]])/(a*f^2) + ((c + d*x)*Tan[e/2 + Pi/4 + (f*x)/2])/(a*f)} -{1/(a - a*Sin[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a - a*Sin[e + f*x])), x]} -{1/(a - a*Sin[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a - a*Sin[e + f*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Sqrt[a + a*Sin[c + d*x]], x, 5, -((96*Sqrt[a + a*Sin[c + d*x]])/d^4) + (12*x^2*Sqrt[a + a*Sin[c + d*x]])/d^2 + (48*x*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/d^3 - (2*x^3*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/d} -{x^2*Sqrt[a + a*Sin[c + d*x]], x, 4, (8*x*Sqrt[a + a*Sin[c + d*x]])/d^2 + (16*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/d^3 - (2*x^2*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/d} -{x^1*Sqrt[a + a*Sin[c + d*x]], x, 3, (4*Sqrt[a + a*Sin[c + d*x]])/d^2 - (2*x*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/d} -{Sqrt[a + a*Sin[c + d*x]]/x^1, x, 4, CosIntegral[(d*x)/2]*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(1/4)*(2*c + Pi)]*Sqrt[a + a*Sin[c + d*x]] + Cos[(1/4)*(2*c + Pi)]*Csc[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]]*SinIntegral[(d*x)/2]} -{Sqrt[a + a*Sin[c + d*x]]/x^2, x, 5, -(Sqrt[a + a*Sin[c + d*x]]/x) - (1/2)*d*CosIntegral[(d*x)/2]*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(1/4)*(2*c - Pi)]*Sqrt[a + a*Sin[c + d*x]] - (1/2)*d*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(1/4)*(2*c + Pi)]*Sqrt[a + a*Sin[c + d*x]]*SinIntegral[(d*x)/2]} -{Sqrt[a + a*Sin[c + d*x]]/x^3, x, 6, -(Sqrt[a + a*Sin[c + d*x]]/(2*x^2)) - (d*Cot[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]])/(4*x) - (1/8)*d^2*CosIntegral[(d*x)/2]*Csc[c/2 + Pi/4 + (d*x)/2]*Sin[(1/4)*(2*c + Pi)]*Sqrt[a + a*Sin[c + d*x]] - (1/8)*d^2*Cos[(1/4)*(2*c + Pi)]*Csc[c/2 + Pi/4 + (d*x)/2]*Sqrt[a + a*Sin[c + d*x]]*SinIntegral[(d*x)/2]} - - -{x^3*(a + a*Sin[e + f*x])^(3/2), x, 9, -((1280*a*Sqrt[a + a*Sin[e + f*x]])/(9*f^4)) + (16*a*x^2*Sqrt[a + a*Sin[e + f*x]])/f^2 + (640*a*x*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (8*a*x^3*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (32*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (4*a*x^3*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (64*a*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(27*f^4) + (8*a*x^2*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f^2)} -{x^2*(a + a*Sin[e + f*x])^(3/2), x, 7, (32*a*x*Sqrt[a + a*Sin[e + f*x]])/(3*f^2) + (224*a*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (8*a*x^2*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (32*a*Cos[e/2 + Pi/4 + (f*x)/2]^2*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(27*f^3) - (4*a*x^2*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (16*a*x*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(9*f^2)} -{x^1*(a + a*Sin[e + f*x])^(3/2), x, 4, (16*a*Sqrt[a + a*Sin[e + f*x]])/(3*f^2) - (8*a*x*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (4*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) + (8*a*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(9*f^2)} -{(a + a*Sin[e + f*x])^(3/2)/x^1, x, 9, (1/2)*a*Cos[(3/4)*(2*e - Pi)]*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]] + (3/2)*a*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(1/4)*(2*e + Pi)]*Sqrt[a + a*Sin[e + f*x]] + (3/2)*a*Cos[(1/4)*(2*e + Pi)]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2] - (1/2)*a*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(3/4)*(2*e - Pi)]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2]} -{(a + a*Sin[e + f*x])^(3/2)/x^2, x, 9, (-(3/4))*a*f*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(1/4)*(2*e - Pi)]*Sqrt[a + a*Sin[e + f*x]] + (3/4)*a*f*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(1/4)*(6*e + Pi)]*Sqrt[a + a*Sin[e + f*x]] - (2*a*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/x - (3/4)*a*f*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(1/4)*(2*e + Pi)]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2] + (3/4)*a*f*Cos[(1/4)*(6*e + Pi)]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2]} -{(a + a*Sin[e + f*x])^(3/2)/x^3, x, 13, (-(9/16))*a*f^2*Cos[(3/4)*(2*e - Pi)]*CosIntegral[(3*f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]] - (3/16)*a*f^2*CosIntegral[(f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(1/4)*(2*e + Pi)]*Sqrt[a + a*Sin[e + f*x]] - (3*a*f*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(2*x) - (a*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/x^2 - (3/16)*a*f^2*Cos[(1/4)*(2*e + Pi)]*Csc[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(f*x)/2] + (9/16)*a*f^2*Csc[e/2 + Pi/4 + (f*x)/2]*Sin[(3/4)*(2*e - Pi)]*Sqrt[a + a*Sin[e + f*x]]*SinIntegral[(3*f*x)/2]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/Sqrt[a + a*Sin[c + d*x]], x, 10, -((4*x^3*ArcTanh[E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d*Sqrt[a + a*Sin[c + d*x]])) + (12*I*x^2*PolyLog[2, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]]) - (12*I*x^2*PolyLog[2, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]]) - (48*x*PolyLog[3, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^3*Sqrt[a + a*Sin[c + d*x]]) + (48*x*PolyLog[3, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^3*Sqrt[a + a*Sin[c + d*x]]) - (96*I*PolyLog[4, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^4*Sqrt[a + a*Sin[c + d*x]]) + (96*I*PolyLog[4, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^4*Sqrt[a + a*Sin[c + d*x]])} -{x^2/Sqrt[a + a*Sin[c + d*x]], x, 8, -((4*x^2*ArcTanh[E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d*Sqrt[a + a*Sin[c + d*x]])) + (8*I*x*PolyLog[2, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]]) - (8*I*x*PolyLog[2, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]]) - (16*PolyLog[3, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^3*Sqrt[a + a*Sin[c + d*x]]) + (16*PolyLog[3, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^3*Sqrt[a + a*Sin[c + d*x]])} -{x^1/Sqrt[a + a*Sin[c + d*x]], x, 6, -((4*x*ArcTanh[E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d*Sqrt[a + a*Sin[c + d*x]])) + (4*I*PolyLog[2, -E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]]) - (4*I*PolyLog[2, E^((1/4)*I*(2*c + Pi + 2*d*x))]*Sin[c/2 + Pi/4 + (d*x)/2])/(d^2*Sqrt[a + a*Sin[c + d*x]])} -{1/(x^1*Sqrt[a + a*Sin[c + d*x]]), x, 0, Unintegrable[1/(x*Sqrt[a + a*Sin[c + d*x]]), x]} -{1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[a + a*Sin[c + d*x]]), x]} - - -{x^3/(a + a*Sin[e + f*x])^(3/2), x, 16, -((3*x^2)/(a*f^2*Sqrt[a + a*Sin[e + f*x]])) - (x^3*Cot[e/2 + Pi/4 + (f*x)/2])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - (24*x*ArcTanh[E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) - (x^3*ArcTanh[E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (24*I*PolyLog[2, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) + (3*I*x^2*PolyLog[2, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (24*I*PolyLog[2, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) - (3*I*x^2*PolyLog[2, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (12*x*PolyLog[3, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) + (12*x*PolyLog[3, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) - (24*I*PolyLog[4, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) + (24*I*PolyLog[4, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]])} -{x^2/(a + a*Sin[e + f*x])^(3/2), x, 10, -((2*x)/(a*f^2*Sqrt[a + a*Sin[e + f*x]])) - (x^2*Cot[e/2 + Pi/4 + (f*x)/2])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - (x^2*ArcTanh[E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - (4*ArcTanh[Cos[e/2 + Pi/4 + (f*x)/2]]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) + (2*I*x*PolyLog[2, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (2*I*x*PolyLog[2, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (4*PolyLog[3, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) + (4*PolyLog[3, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]])} -{x^1/(a + a*Sin[e + f*x])^(3/2), x, 7, -(1/(a*f^2*Sqrt[a + a*Sin[e + f*x]])) - (x*Cot[e/2 + Pi/4 + (f*x)/2])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - (x*ArcTanh[E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (I*PolyLog[2, -E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (I*PolyLog[2, E^((1/4)*I*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]])} -{1/(x^1*(a + a*Sin[e + f*x])^(3/2)), x, 0, Unintegrable[1/(x*(a + a*Sin[e + f*x])^(3/2)), x]} -{1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x, 0, Unintegrable[1/(x^2*(a + a*Sin[e + f*x])^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sin[e+f x])^(n/3)*) - - -(* Used to hang Rubi *) -{(a + a*Sin[c + d*x])^(1/3)/x, x, 0, Unintegrable[(a + a*Sin[c + d*x])^(1/3)/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sin[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + a*Sin[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + a*Sin[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + a*Sin[e + f*x])^3, x, 12, (5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (15*a^3*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(8*f)) - (15*a^3*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(8*f)) + (3*I*2^(-3 - m)*a^3*E^(2*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (3*I*2^(-3 - m)*a^3*(c + d*x)^m*Gamma[1 + m, (2*I*f*(c + d*x))/d])/(E^(2*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f) + (3^(-1 - m)*a^3*E^(3*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((3*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(8*f)) + (3^(-1 - m)*a^3*(c + d*x)^m*Gamma[1 + m, (3*I*f*(c + d*x))/d])/(E^(3*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(8*f))} -{(c + d*x)^m*(a + a*Sin[e + f*x])^2, x, 9, (3*a^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (a^2*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (a^2*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f) + (I*2^(-3 - m)*a^2*E^(2*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (I*2^(-3 - m)*a^2*(c + d*x)^m*Gamma[1 + m, (2*I*f*(c + d*x))/d])/(E^(2*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f)} -{(c + d*x)^m*(a + a*Sin[e + f*x])^1, x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) - (a*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(2*f)) - (a*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(2*f))} -{(c + d*x)^m/(a + a*Sin[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m/(a + a*Sin[e + f*x]), x]} -{(c + d*x)^m/(a + a*Sin[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + a*Sin[e + f*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Sin[e + f*x])*(c + d*x)^3, x, 6, (a*(c + d*x)^4)/(4*d) + (6*b*d^2*(c + d*x)*Cos[e + f*x])/f^3 - (b*(c + d*x)^3*Cos[e + f*x])/f - (6*b*d^3*Sin[e + f*x])/f^4 + (3*b*d*(c + d*x)^2*Sin[e + f*x])/f^2} -{(a + b*Sin[e + f*x])*(c + d*x)^2, x, 5, (a*(c + d*x)^3)/(3*d) + (2*b*d^2*Cos[e + f*x])/f^3 - (b*(c + d*x)^2*Cos[e + f*x])/f + (2*b*d*(c + d*x)*Sin[e + f*x])/f^2} -{(a + b*Sin[e + f*x])*(c + d*x)^1, x, 4, (a*(c + d*x)^2)/(2*d) - (b*(c + d*x)*Cos[e + f*x])/f + (b*d*Sin[e + f*x])/f^2} -{(a + b*Sin[e + f*x])/(c + d*x)^1, x, 5, (a*Log[c + d*x])/d + (b*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (b*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d} -{(a + b*Sin[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) + (b*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 - (b*Sin[e + f*x])/(d*(c + d*x)) - (b*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2} -{(a + b*Sin[e + f*x])/(c + d*x)^3, x, 7, -(a/(2*d*(c + d*x)^2)) - (b*f*Cos[e + f*x])/(2*d^2*(c + d*x)) - (b*f^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/(2*d^3) - (b*Sin[e + f*x])/(2*d*(c + d*x)^2) - (b*f^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(a + b*Sin[e + f*x])^2*(c + d*x)^3, x, 10, -((3*b^2*c*d^2*x)/(4*f^2)) - (3*b^2*d^3*x^2)/(8*f^2) + (a^2*(c + d*x)^4)/(4*d) + (b^2*(c + d*x)^4)/(8*d) + (12*a*b*d^2*(c + d*x)*Cos[e + f*x])/f^3 - (2*a*b*(c + d*x)^3*Cos[e + f*x])/f - (12*a*b*d^3*Sin[e + f*x])/f^4 + (6*a*b*d*(c + d*x)^2*Sin[e + f*x])/f^2 + (3*b^2*d^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (b^2*(c + d*x)^3*Cos[e + f*x]*Sin[e + f*x])/(2*f) - (3*b^2*d^3*Sin[e + f*x]^2)/(8*f^4) + (3*b^2*d*(c + d*x)^2*Sin[e + f*x]^2)/(4*f^2)} -{(a + b*Sin[e + f*x])^2*(c + d*x)^2, x, 9, -((b^2*d^2*x)/(4*f^2)) + (a^2*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(6*d) + (4*a*b*d^2*Cos[e + f*x])/f^3 - (2*a*b*(c + d*x)^2*Cos[e + f*x])/f + (4*a*b*d*(c + d*x)*Sin[e + f*x])/f^2 + (b^2*d^2*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) - (b^2*(c + d*x)^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b^2*d*(c + d*x)*Sin[e + f*x]^2)/(2*f^2)} -{(a + b*Sin[e + f*x])^2*(c + d*x)^1, x, 6, (1/2)*b^2*c*x + (1/4)*b^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (2*a*b*(c + d*x)*Cos[e + f*x])/f + (2*a*b*d*Sin[e + f*x])/f^2 - (b^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b^2*d*Sin[e + f*x]^2)/(4*f^2)} -{(a + b*Sin[e + f*x])^2/(c + d*x)^1, x, 10, -((b^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*d)) + (a^2*Log[c + d*x])/d + (b^2*Log[c + d*x])/(2*d) + (2*a*b*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d + (2*a*b*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d + (b^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + b*Sin[e + f*x])^2/(c + d*x)^2, x, 11, -(a^2/(d*(c + d*x))) + (2*a*b*f*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d^2 - (2*a*b*Sin[e + f*x])/(d*(c + d*x)) - (b^2*Sin[e + f*x]^2)/(d*(c + d*x)) - (2*a*b*f*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + b*Sin[e + f*x])^2/(c + d*x)^3, x, 14, -(a^2/(2*d*(c + d*x)^2)) - (a*b*f*Cos[e + f*x])/(d^2*(c + d*x)) + (b^2*f^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/d^3 - (a*b*f^2*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d^3 - (a*b*Sin[e + f*x])/(d*(c + d*x)^2) - (b^2*f*Cos[e + f*x]*Sin[e + f*x])/(d^2*(c + d*x)) - (b^2*Sin[e + f*x]^2)/(2*d*(c + d*x)^2) - (a*b*f^2*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^3 - (b^2*f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Sin[e + f*x]), x, 12, -((I*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f)) + (I*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - (3*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2) + (3*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2) - (6*I*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^3) + (6*I*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^3) + (6*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^4) - (6*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^4)} -{(c + d*x)^2/(a + b*Sin[e + f*x]), x, 10, -((I*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f)) + (I*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - (2*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2) + (2*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2) - (2*I*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^3) + (2*I*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^3)} -{(c + d*x)^1/(a + b*Sin[e + f*x]), x, 8, -((I*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f)) + (I*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - (d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2) + (d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f^2)} -{1/((c + d*x)^1*(a + b*Sin[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sin[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Sin[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sin[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Sin[e + f*x])^2, x, 22, (I*(c + d*x)^3)/((a^2 - b^2)*f) - (3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) - (I*a*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (3*d*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^3*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (6*I*d^2*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + (6*I*d^2*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) + (3*a*d*(c + d*x)^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^4) - (6*I*a*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) - (6*d^3*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^4) + (6*I*a*d^2*(c + d*x)*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (6*a*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) - (6*a*d^3*PolyLog[4, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^4) + (b*(c + d*x)^3*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{(c + d*x)^2/(a + b*Sin[e + f*x])^2, x, 18, (I*(c + d*x)^2)/((a^2 - b^2)*f) - (2*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) - (I*a*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (2*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (2*I*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + (2*I*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) - (2*I*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (2*I*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (b*(c + d*x)^2*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{(c + d*x)^1/(a + b*Sin[e + f*x])^2, x, 11, -((I*a*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f)) + (I*a*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (d*Log[a + b*Sin[e + f*x]])/((a^2 - b^2)*f^2) - (a*d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + (a*d*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + (b*(c + d*x)*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/((c + d*x)^1*(a + b*Sin[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sin[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Sin[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sin[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sin[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + b*Sin[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + b*Sin[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + b*Sin[e + f*x])^3, x, 18, (a^3*(c + d*x)^(1 + m))/(d*(1 + m)) + (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (3*a^2*b*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(2*f)) - (3*b^3*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(8*f)) - (3*a^2*b*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(2*f)) - (3*b^3*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(8*f)) + (3*I*2^(-3 - m)*a*b^2*E^(2*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (3*I*2^(-3 - m)*a*b^2*(c + d*x)^m*Gamma[1 + m, (2*I*f*(c + d*x))/d])/(E^(2*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f) + (3^(-1 - m)*b^3*E^(3*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((3*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(8*f)) + (3^(-1 - m)*b^3*(c + d*x)^m*Gamma[1 + m, (3*I*f*(c + d*x))/d])/(E^(3*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(8*f))} -{(c + d*x)^m*(a + b*Sin[e + f*x])^2, x, 10, (a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (a*b*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (a*b*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f) + (I*2^(-3 - m)*b^2*E^(2*I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*f) - (I*2^(-3 - m)*b^2*(c + d*x)^m*Gamma[1 + m, (2*I*f*(c + d*x))/d])/(E^(2*I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*f)} -{(c + d*x)^m*(a + b*Sin[e + f*x])^1, x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) - (b*E^(I*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, -((I*f*(c + d*x))/d)])/((-((I*f*(c + d*x))/d))^m*(2*f)) - (b*(c + d*x)^m*Gamma[1 + m, (I*f*(c + d*x))/d])/(E^(I*(e - (c*f)/d))*((I*f*(c + d*x))/d)^m*(2*f))} -{(c + d*x)^m/(a + b*Sin[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m/(a + b*Sin[e + f*x]), x]} -{(c + d*x)^m/(a + b*Sin[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + b*Sin[e + f*x])^2, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+a Sin[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Sin[c + d*x])/(a + a*Sin[c + d*x]), x, 9, (I*(e + f*x)^3)/(a*d) + (e + f*x)^4/(4*a*f) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4)} -{((e + f*x)^2*Sin[c + d*x])/(a + a*Sin[c + d*x]), x, 8, (I*(e + f*x)^2)/(a*d) + (e + f*x)^3/(3*a*f) + ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3)} -{((e + f*x)*Sin[c + d*x])/(a + a*Sin[c + d*x]), x, 5, (e*x)/a + (f*x^2)/(2*a) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2)} -{Sin[c + d*x]/(a + a*Sin[c + d*x]), x, 2, x/a + Cos[c + d*x]/(d*(a + a*Sin[c + d*x]))} -{Sin[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Sin[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{((e + f*x)^3*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 14, ((-I)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) + (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) - ((e + f*x)^3*Cos[c + d*x])/(a*d) - ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) - (6*f^3*Sin[c + d*x])/(a*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2)} -{((e + f*x)^2*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 12, ((-I)*(e + f*x)^2)/(a*d) - (e + f*x)^3/(3*a*f) + (2*f^2*Cos[c + d*x])/(a*d^3) - ((e + f*x)^2*Cos[c + d*x])/(a*d) - ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) - ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + (2*f*(e + f*x)*Sin[c + d*x])/(a*d^2)} -{((e + f*x)*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 8, -((e*x)/a) - (f*x^2)/(2*a) - ((e + f*x)*Cos[c + d*x])/(a*d) - ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2) + (f*Sin[c + d*x])/(a*d^2)} -{Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(x/a) - Cos[c + d*x]/(a*d) - Cos[c + d*x]/(a*d*(1 + Sin[c + d*x]))} -{Sin[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Sin[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{((e + f*x)^3*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 19, (-3*e*f^2*x)/(4*a*d^2) - (3*f^3*x^2)/(8*a*d^2) + (I*(e + f*x)^3)/(a*d) + (3*(e + f*x)^4)/(8*a*f) - (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (6*f^3*Sin[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2) + (3*f^2*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (3*f^3*Sin[c + d*x]^2)/(8*a*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*a*d^2)} -{((e + f*x)^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 17, -(f^2*x)/(4*a*d^2) + (I*(e + f*x)^2)/(a*d) + (e + f*x)^3/(2*a*f) - (2*f^2*Cos[c + d*x])/(a*d^3) + ((e + f*x)^2*Cos[c + d*x])/(a*d) + ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - (2*f*(e + f*x)*Sin[c + d*x])/(a*d^2) + (f^2*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^3) - ((e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + (f*(e + f*x)*Sin[c + d*x]^2)/(2*a*d^2)} -{((e + f*x)*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 11, (3*e*x)/(2*a) + (3*f*x^2)/(4*a) + ((e + f*x)*Cos[c + d*x])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2) - (f*Sin[c + d*x])/(a*d^2) - ((e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + (f*Sin[c + d*x]^2)/(4*a*d^2)} -{Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 2, (3*x)/(2*a) + (2*Cos[c + d*x])/(a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(d*(a + a*Sin[c + d*x]))} -{Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Sin[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Sin[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csc[c + d*x])/(a + a*Sin[c + d*x]), x, 17, (I*(e + f*x)^3)/(a*d) - (2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)} -{((e + f*x)^2*Csc[c + d*x])/(a + a*Sin[c + d*x]), x, 14, (I*(e + f*x)^2)/(a*d) - (2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3)} -{((e + f*x)*Csc[c + d*x])/(a + a*Sin[c + d*x]), x, 9, (-2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2) + (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (I*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2)} -{Csc[c + d*x]/(a + a*Sin[c + d*x]), x, 3, -(ArcTanh[Cos[c + d*x]]/(a*d)) + Cos[c + d*x]/(d*(a + a*Sin[c + d*x]))} -{Csc[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Csc[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{((e + f*x)^3*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 24, ((-2*I)*(e + f*x)^3)/(a*d) + (2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) - ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - ((e + f*x)^3*Cot[c + d*x])/(a*d) + (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) + ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) - ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)} -{((e + f*x)^2*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 20, ((-2*I)*(e + f*x)^2)/(a*d) + (2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) - ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - ((e + f*x)^2*Cot[c + d*x])/(a*d) + (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) + (2*f*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) + ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (I*f^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (2*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3)} -{((e + f*x)*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 12, (2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) - ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - ((e + f*x)*Cot[c + d*x])/(a*d) + (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2) + (f*Log[Sin[c + d*x]])/(a*d^2) - (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + (I*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2)} -{Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, ArcTanh[Cos[c + d*x]]/(a*d) - (2*Cot[c + d*x])/(a*d) + Cot[c + d*x]/(d*(a + a*Sin[c + d*x]))} -{Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Csc[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{((e + f*x)^3*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 40, ((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d^3) - (3*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)^3*Cot[c + d*x])/(a*d) - (3*f*(e + f*x)^2*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (6*f*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) - (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) + ((3*I)*f^3*PolyLog[2, -E^(I*(c + d*x))])/(a*d^4) + (((9*I)/2)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f^3*PolyLog[2, E^(I*(c + d*x))])/(a*d^4) - (((9*I)/2)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (9*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) - ((9*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((9*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)} -{((e + f*x)^2*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 30, ((2*I)*(e + f*x)^2)/(a*d) - (3*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) - (f^2*ArcTanh[Cos[c + d*x]])/(a*d^3) + ((e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)^2*Cot[c + d*x])/(a*d) - (f*(e + f*x)*Csc[c + d*x])/(a*d^2) - ((e + f*x)^2*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (4*f*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d^2) - (2*f*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) + ((3*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + ((4*I)*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (I*f^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) - (3*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (3*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3)} -{((e + f*x)*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]), x, 19, (-3*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)*Cot[c + d*x])/(a*d) - (f*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (2*f*Log[Sin[c/2 + Pi/4 + (d*x)/2]])/(a*d^2) - (f*Log[Sin[c + d*x]])/(a*d^2) + (((3*I)/2)*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (((3*I)/2)*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2)} -{Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, (-3*ArcTanh[Cos[c + d*x]])/(2*a*d) + (2*Cot[c + d*x])/(a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(d*(a + a*Sin[c + d*x]))} -{Csc[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x, 0, Unintegrable[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+a Sin[c+d x]) with m symbolic*) - - -{((e + f*x)^m*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]), x]} -{((e + f*x)^m*Sin[c + d*x])/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sin[c + d*x])/(a + a*Sin[c + d*x]), x]} -{(e + f*x)^m/(a + a*Sin[c + d*x]), x, 0, Unintegrable[(e + f*x)^m/(a + a*Sin[c + d*x]), x]} -{((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x]} -{((e + f*x)^m*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n (a+b Sin[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x]), x, 14, (e + f*x)^4/(4*b*f) + (I*a*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - (I*a*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (3*a*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) + ((6*I)*a*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - ((6*I)*a*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - (6*a*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4) + (6*a*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4)} -{((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x]), x, 12, (e + f*x)^3/(3*b*f) + (I*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - (I*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (2*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (2*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) + ((2*I)*a*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - ((2*I)*a*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3)} -{((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x]), x, 10, (e*x)/b + (f*x^2)/(2*b) + (I*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - (I*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2)} -{Sin[c + d*x]/(a + b*Sin[c + d*x]), x, 4, x/b - (2*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*d)} - - -{((e + f*x)^3*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 19, -(a*(e + f*x)^4)/(4*b^2*f) + (6*f^2*(e + f*x)*Cos[c + d*x])/(b*d^3) - ((e + f*x)^3*Cos[c + d*x])/(b*d) - (I*a^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) + (I*a^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) - (3*a^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) - ((6*I)*a^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^3) + ((6*I)*a^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^3) + (6*a^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^4) - (6*a^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^4) - (6*f^3*Sin[c + d*x])/(b*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x])/(b*d^2)} -{((e + f*x)^2*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 16, -(a*(e + f*x)^3)/(3*b^2*f) + (2*f^2*Cos[c + d*x])/(b*d^3) - ((e + f*x)^2*Cos[c + d*x])/(b*d) - (I*a^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) + (I*a^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) - (2*a^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) + (2*a^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) - ((2*I)*a^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^3) + ((2*I)*a^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^3) + (2*f*(e + f*x)*Sin[c + d*x])/(b*d^2)} -{((e + f*x)*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 13, -((a*e*x)/b^2) - (a*f*x^2)/(2*b^2) - ((e + f*x)*Cos[c + d*x])/(b*d) - (I*a^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) + (I*a^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d) - (a^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) + (a^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*Sqrt[a^2 - b^2]*d^2) + (f*Sin[c + d*x])/(b*d^2)} -{Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 6, -((a*x)/b^2) + (2*a^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]/(b*d)} - - -{((e + f*x)^3*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 24, (-3*e*f^2*x)/(4*b*d^2) - (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*(e + f*x)*Cos[c + d*x])/(b^2*d^3) + (a*(e + f*x)^3*Cos[c + d*x])/(b^2*d) + (I*a^3*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) - (I*a^3*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) + (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) + ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - ((6*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^4) + (6*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^4) + (6*a*f^3*Sin[c + d*x])/(b^2*d^4) - (3*a*f*(e + f*x)^2*Sin[c + d*x])/(b^2*d^2) + (3*f^2*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*b*d) - (3*f^3*Sin[c + d*x]^2)/(8*b*d^4) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*b*d^2)} -{((e + f*x)^2*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 21, -(f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) + (e + f*x)^3/(6*b*f) - (2*a*f^2*Cos[c + d*x])/(b^2*d^3) + (a*(e + f*x)^2*Cos[c + d*x])/(b^2*d) + (I*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) - (I*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) + (2*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) + ((2*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - ((2*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^3) - (2*a*f*(e + f*x)*Sin[c + d*x])/(b^2*d^2) + (f^2*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*d) + (f*(e + f*x)*Sin[c + d*x]^2)/(2*b*d^2)} -{((e + f*x)*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 16, (a^2*e*x)/b^3 + (e*x)/(2*b) + (a^2*f*x^2)/(2*b^3) + (f*x^2)/(4*b) + (a*(e + f*x)*Cos[c + d*x])/(b^2*d) + (I*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) - (I*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d) + (a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*Sqrt[a^2 - b^2]*d^2) - (a*f*Sin[c + d*x])/(b^2*d^2) - ((e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d) + (f*Sin[c + d*x]^2)/(4*b*d^2)} -{Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 6, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + (a*Cos[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csc[c + d*x])/(a + b*Sin[c + d*x]), x, 22, (-2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) + (I*b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (I*b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((6*I)*b*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^3) - ((6*I)*b*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4) - (6*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^4) + (6*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^4)} -{((e + f*x)^2*Csc[c + d*x])/(a + b*Sin[c + d*x]), x, 18, (-2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) + (I*b*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (I*b*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) + ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2) - (2*b*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2) - (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((2*I)*b*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^3) - ((2*I)*b*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^3)} -{((e + f*x)*Csc[c + d*x])/(a + b*Sin[c + d*x]), x, 14, (-2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + (I*b*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (I*b*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) + (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (I*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (b*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2) - (b*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d^2)} -{Csc[c + d*x]/(a + b*Sin[c + d*x]), x, 5, (-2*b*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a*d)} - - -{((e + f*x)^3*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 29, ((-I)*(e + f*x)^3)/(a*d) + (2*b*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a^2*d) - ((e + f*x)^3*Cot[c + d*x])/(a*d) - (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((3*I)*b*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a^2*d^3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^3) + (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) + ((6*I)*b*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a^2*d^4) - ((6*I)*b*f^3*PolyLog[4, E^(I*(c + d*x))])/(a^2*d^4) + (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^4) - (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^4)} -{((e + f*x)^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 24, ((-I)*(e + f*x)^2)/(a*d) + (2*b*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a^2*d) - ((e + f*x)^2*Cot[c + d*x])/(a*d) - (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (2*f*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((2*I)*b*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + ((2*I)*b*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) - (I*f^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (2*b*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (2*b*f^2*PolyLog[3, E^(I*(c + d*x))])/(a^2*d^3) - ((2*I)*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^3) + ((2*I)*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^3)} -{((e + f*x)*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 17, (2*b*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a^2*d) - ((e + f*x)*Cot[c + d*x])/(a*d) - (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d) + (f*Log[Sin[c + d*x]])/(a*d^2) - (I*b*f*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + (I*b*f*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2) + (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*Sqrt[a^2 - b^2]*d^2)} -{Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 7, (2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d)} - - -(* ::Subsubsection::Closed:: *) -(*m symbolic*) - - -{((e + f*x)^m*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sin[c + d*x]^2)/(a + b*Sin[c + d*x]), x]} -{((e + f*x)^m*Sin[c + d*x])/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sin[c + d*x])/(a + b*Sin[c + d*x]), x]} -{(e + f*x)^m/(a + b*Sin[c + d*x]), x, 0, Unintegrable[(e + f*x)^m/(a + b*Sin[c + d*x]), x]} -{((e + f*x)^m*Csc[c + d*x])/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Csc[c + d*x])/(a + b*Sin[c + d*x]), x]} -{((e + f*x)^m*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^2*) - - -{((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x])^2, x, 21, (I*a^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (I*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - (I*a^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (I*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (a*f*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d^2) + (a^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - (f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (a^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + (f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (a*(e + f*x)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^2, x, 30, ((-I)*a*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) + (I*a^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (I*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (2*a*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (I*a^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (I*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - ((2*I)*a*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (2*a^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - (2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - ((2*I)*a*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) - (2*a^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + (2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) + ((2*I)*a^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((2*I)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - ((2*I)*a^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + ((2*I)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - (a*(e + f*x)^2*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x])^2, x, 36, ((-I)*a*(e + f*x)^3)/(b*(a^2 - b^2)*d) + (3*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) + (I*a^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (I*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) + (3*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (I*a^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (I*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d) - ((6*I)*a*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (3*a^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - (3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - ((6*I)*a*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) - (3*a^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + (3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) + (6*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) + ((6*I)*a^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) + (6*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) - ((6*I)*a^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - (6*a^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) + (6*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4) + (6*a^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - (6*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4) - (a*(e + f*x)^3*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[c+d x]^n / (a+b Sin[c+d x])^3*) - - -{((e + f*x)*Sin[c + d*x])/(a + b*Sin[c + d*x])^3, x, 48, (((3*I)/2)*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) - (((3*I)/2)*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (((3*I)/2)*a^3*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*a*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*Log[a + b*Sin[c + d*x]])/(2*b*(a^2 - b^2)^2*d^2) - (f*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d^2) + (3*a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*a^3*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) - (a*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*f)/(2*b*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + ((e + f*x)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{((e + f*x)^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3, x, 73, (((-3*I)/2)*a^2*(e + f*x)^2)/(b*(a^2 - b^2)^2*d) + (I*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d^3) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) + (((3*I)/2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) - (((3*I)/2)*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (((3*I)/2)*a^3*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*a*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - ((3*I)*a^2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - ((3*I)*a^2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((2*I)*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) - (3*a^3*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) + ((3*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) - ((3*I)*a*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*I)*a^3*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) + ((3*I)*a*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (a*(e + f*x)^2*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*f*(e + f*x))/(b*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)^2*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + ((e + f*x)^2*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{((e + f*x)^3*Sin[c + d*x])/(a + b*Sin[c + d*x])^3, x, 92, (((-3*I)/2)*a^2*(e + f*x)^3)/(b*(a^2 - b^2)^2*d) + (I*(e + f*x)^3)/(b*(a^2 - b^2)*d) - ((3*I)*a*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^2*d^2) - (3*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) + (((3*I)/2)*a^3*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) - (((3*I)/2)*a*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) + ((3*I)*a*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^2*d^2) - (3*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^2) - (((3*I)/2)*a^3*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d) + (((3*I)/2)*a*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d) - (3*a*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - ((9*I)*a^2*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (9*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) - (9*a*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) + (3*a*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - ((9*I)*a^2*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) - (9*a^3*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(5/2)*d^2) + (9*a*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) + (9*a^2*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^4) - (6*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) + ((9*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) - ((9*I)*a*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^2*d^4) - (6*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) - ((9*I)*a^3*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^3) + ((9*I)*a*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (9*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^4) + (9*a*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) + (9*a^3*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(5/2)*d^4) - (9*a*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - (a*(e + f*x)^3*Cos[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (3*a*f*(e + f*x)^2)/(2*b*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x])) - (3*a^2*(e + f*x)^3*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + ((e + f*x)^3*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+a Sin[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^3*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 6, -((I*(e + f*x)^4)/(4*a*f)) + (2*(e + f*x)^3*Log[1 - I*E^(I*(c + d*x))])/(a*d) - (6*I*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) + (12*I*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(a*d^4)} -{(e + f*x)^2*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 5, -((I*(e + f*x)^3)/(3*a*f)) + (2*(e + f*x)^2*Log[1 - I*E^(I*(c + d*x))])/(a*d) - (4*I*f*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) + (4*f^2*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3)} -{(e + f*x)^1*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 4, -((I*(e + f*x)^2)/(2*a*f)) + (2*(e + f*x)*Log[1 - I*E^(I*(c + d*x))])/(a*d) - (2*I*f*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2)} -{(e + f*x)^0*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 2, Log[1 + Sin[c + d*x]]/(a*d)} -{1/(e + f*x)^1*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Cos[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{1/(e + f*x)^2*Cos[c + d*x]/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Cos[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{(e + f*x)^3*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 6, (e + f*x)^4/(4*a*f) - (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + (6*f^3*Sin[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2)} -{(e + f*x)^2*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, (e + f*x)^3/(3*a*f) - (2*f^2*Cos[c + d*x])/(a*d^3) + ((e + f*x)^2*Cos[c + d*x])/(a*d) - (2*f*(e + f*x)*Sin[c + d*x])/(a*d^2)} -{(e + f*x)^1*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, (e*x)/a + (f*x^2)/(2*a) + ((e + f*x)*Cos[c + d*x])/(a*d) - (f*Sin[c + d*x])/(a*d^2)} -{(e + f*x)^0*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 2, x/a + Cos[c + d*x]/(a*d)} -{1/(e + f*x)^1*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, Log[e + f*x]/(a*f) - (CosIntegral[(d*e)/f + d*x]*Sin[c - (d*e)/f])/(a*f) - (Cos[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/(a*f)} -{1/(e + f*x)^2*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 6, -(1/(a*f*(e + f*x))) - (d*Cos[c - (d*e)/f]*CosIntegral[(d*e)/f + d*x])/(a*f^2) + Sin[c + d*x]/(a*f*(e + f*x)) + (d*Sin[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/(a*f^2)} - - -{(e + f*x)^3*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 10, -((3*f^3*x)/(8*a*d^3)) + (e + f*x)^3/(4*a*d) - (6*f^3*Cos[c + d*x])/(a*d^4) + (3*f*(e + f*x)^2*Cos[c + d*x])/(a*d^2) - (6*f^2*(e + f*x)*Sin[c + d*x])/(a*d^3) + ((e + f*x)^3*Sin[c + d*x])/(a*d) + (3*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^2) + (3*f^2*(e + f*x)*Sin[c + d*x]^2)/(4*a*d^3) - ((e + f*x)^3*Sin[c + d*x]^2)/(2*a*d)} -{(e + f*x)^2*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 7, (e*f*x)/(2*a*d) + (f^2*x^2)/(4*a*d) + (2*f*(e + f*x)*Cos[c + d*x])/(a*d^2) - (2*f^2*Sin[c + d*x])/(a*d^3) + ((e + f*x)^2*Sin[c + d*x])/(a*d) - (f*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d^2) + (f^2*Sin[c + d*x]^2)/(4*a*d^3) - ((e + f*x)^2*Sin[c + d*x]^2)/(2*a*d)} -{(e + f*x)^1*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, (f*x)/(4*a*d) + (f*Cos[c + d*x])/(a*d^2) + ((e + f*x)*Sin[c + d*x])/(a*d) - (f*Cos[c + d*x]*Sin[c + d*x])/(4*a*d^2) - ((e + f*x)*Sin[c + d*x]^2)/(2*a*d)} -{(e + f*x)^0*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 2, Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)} -{1/(e + f*x)^1*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 9, (Cos[c - (d*e)/f]*CosIntegral[(d*e)/f + d*x])/(a*f) - (CosIntegral[(2*d*e)/f + 2*d*x]*Sin[2*c - (2*d*e)/f])/(2*a*f) - (Sin[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/(a*f) - (Cos[2*c - (2*d*e)/f]*SinIntegral[(2*d*e)/f + 2*d*x])/(2*a*f)} -{1/(e + f*x)^2*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 11, -(Cos[c + d*x]/(a*f*(e + f*x))) - (d*Cos[2*c - (2*d*e)/f]*CosIntegral[(2*d*e)/f + 2*d*x])/(a*f^2) - (d*CosIntegral[(d*e)/f + d*x]*Sin[c - (d*e)/f])/(a*f^2) + Sin[2*c + 2*d*x]/(2*a*f*(e + f*x)) - (d*Cos[c - (d*e)/f]*SinIntegral[(d*e)/f + d*x])/(a*f^2) + (d*Sin[2*c - (2*d*e)/f]*SinIntegral[(2*d*e)/f + 2*d*x])/(a*f^2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^3*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 22, -((3*I*f*(e + f*x)^2)/(2*a*d^2)) - (6*I*f^2*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d^3) - (I*(e + f*x)^3*ArcTan[E^(I*(c + d*x))])/(a*d) + (3*f^2*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/(a*d^3) + (3*I*f^3*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^4) + (3*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(2*a*d^2) - (3*I*f^3*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^4) - (3*I*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/(2*a*d^2) - (3*I*f^3*PolyLog[2, -E^(2*I*(c + d*x))])/(2*a*d^4) - (3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*d^3) + (3*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) - (3*I*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))])/(a*d^4) + (3*I*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(a*d^4) - (3*f*(e + f*x)^2*Sec[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^2)/(2*a*d) + (3*f*(e + f*x)^2*Tan[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{(e + f*x)^2*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 13, -((I*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d)) + (f^2*ArcTanh[Sin[c + d*x]])/(a*d^3) + (f^2*Log[Cos[c + d*x]])/(a*d^3) + (I*f*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) - (I*f*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) - (f^2*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*d^3) + (f^2*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) - (f*(e + f*x)*Sec[c + d*x])/(a*d^2) - ((e + f*x)^2*Sec[c + d*x]^2)/(2*a*d) + (f*(e + f*x)*Tan[c + d*x])/(a*d^2) + ((e + f*x)^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{(e + f*x)^1*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 10, -((I*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d)) + (I*f*PolyLog[2, (-I)*E^(I*(c + d*x))])/(2*a*d^2) - (I*f*PolyLog[2, I*E^(I*(c + d*x))])/(2*a*d^2) - (f*Sec[c + d*x])/(2*a*d^2) - ((e + f*x)*Sec[c + d*x]^2)/(2*a*d) + (f*Tan[c + d*x])/(2*a*d^2) + ((e + f*x)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{(e + f*x)^0*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 4, ArcTanh[Sin[c + d*x]]/(2*a*d) - 1/(2*d*(a + a*Sin[c + d*x]))} -{1/(e + f*x)^1*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{1/(e + f*x)^2*Sec[c + d*x]/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{(e + f*x)^3*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 20, -((2*I*(e + f*x)^3)/(3*a*d)) - (I*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d^2) + (f^3*ArcTanh[Sin[c + d*x]])/(a*d^4) + (2*f*(e + f*x)^2*Log[1 + E^(2*I*(c + d*x))])/(a*d^2) + (f^3*Log[Cos[c + d*x]])/(a*d^4) + (I*f^2*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^3) - (I*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - (2*I*f^2*(e + f*x)*PolyLog[2, -E^(2*I*(c + d*x))])/(a*d^3) - (f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*d^4) + (f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (f^3*PolyLog[3, -E^(2*I*(c + d*x))])/(a*d^4) - (f^2*(e + f*x)*Sec[c + d*x])/(a*d^3) - (f*(e + f*x)^2*Sec[c + d*x]^2)/(2*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^3)/(3*a*d) + (f^2*(e + f*x)*Tan[c + d*x])/(a*d^3) + (2*(e + f*x)^3*Tan[c + d*x])/(3*a*d) + (f*(e + f*x)^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} -{(e + f*x)^2*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 16, -((2*I*(e + f*x)^2)/(3*a*d)) - (2*I*f*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(3*a*d^2) + (4*f*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/(3*a*d^2) + (I*f^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(3*a*d^3) - (I*f^2*PolyLog[2, I*E^(I*(c + d*x))])/(3*a*d^3) - (2*I*f^2*PolyLog[2, -E^(2*I*(c + d*x))])/(3*a*d^3) - (f^2*Sec[c + d*x])/(3*a*d^3) - (f*(e + f*x)*Sec[c + d*x]^2)/(3*a*d^2) - ((e + f*x)^2*Sec[c + d*x]^3)/(3*a*d) + (f^2*Tan[c + d*x])/(3*a*d^3) + (2*(e + f*x)^2*Tan[c + d*x])/(3*a*d) + (f*(e + f*x)*Sec[c + d*x]*Tan[c + d*x])/(3*a*d^2) + ((e + f*x)^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} -{(e + f*x)^1*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 7, (f*ArcTanh[Sin[c + d*x]])/(6*a*d^2) + (2*f*Log[Cos[c + d*x]])/(3*a*d^2) - (f*Sec[c + d*x]^2)/(6*a*d^2) - ((e + f*x)*Sec[c + d*x]^3)/(3*a*d) + (2*(e + f*x)*Tan[c + d*x])/(3*a*d) + (f*Sec[c + d*x]*Tan[c + d*x])/(6*a*d^2) + ((e + f*x)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} -{(e + f*x)^0*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 3, -(Sec[c + d*x]/(3*d*(a + a*Sin[c + d*x]))) + (2*Tan[c + d*x])/(3*a*d)} -{1/(e + f*x)^1*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{1/(e + f*x)^2*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]^2/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -{(e + f*x)^3*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 32, -((I*f*(e + f*x)^2)/(2*a*d^2)) - (5*I*f^2*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d^3) - (3*I*(e + f*x)^3*ArcTan[E^(I*(c + d*x))])/(4*a*d) + (f^2*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/(a*d^3) + (5*I*f^3*PolyLog[2, (-I)*E^(I*(c + d*x))])/(2*a*d^4) + (9*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(8*a*d^2) - (5*I*f^3*PolyLog[2, I*E^(I*(c + d*x))])/(2*a*d^4) - (9*I*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/(8*a*d^2) - (I*f^3*PolyLog[2, -E^(2*I*(c + d*x))])/(2*a*d^4) - (9*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c + d*x))])/(4*a*d^3) + (9*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(4*a*d^3) - (9*I*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))])/(4*a*d^4) + (9*I*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(4*a*d^4) - (f^3*Sec[c + d*x])/(4*a*d^4) - (9*f*(e + f*x)^2*Sec[c + d*x])/(8*a*d^2) - (f^2*(e + f*x)*Sec[c + d*x]^2)/(4*a*d^3) - (f*(e + f*x)^2*Sec[c + d*x]^3)/(4*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^4)/(4*a*d) + (f^3*Tan[c + d*x])/(4*a*d^4) + (f*(e + f*x)^2*Tan[c + d*x])/(2*a*d^2) + (f^2*(e + f*x)*Sec[c + d*x]*Tan[c + d*x])/(4*a*d^3) + (3*(e + f*x)^3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + (f*(e + f*x)^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)} -{(e + f*x)^2*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 17, -((3*I*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(4*a*d)) + (5*f^2*ArcTanh[Sin[c + d*x]])/(6*a*d^3) + (f^2*Log[Cos[c + d*x]])/(3*a*d^3) + (3*I*f*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(4*a*d^2) - (3*I*f*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(4*a*d^2) - (3*f^2*PolyLog[3, (-I)*E^(I*(c + d*x))])/(4*a*d^3) + (3*f^2*PolyLog[3, I*E^(I*(c + d*x))])/(4*a*d^3) - (3*f*(e + f*x)*Sec[c + d*x])/(4*a*d^2) - (f^2*Sec[c + d*x]^2)/(12*a*d^3) - (f*(e + f*x)*Sec[c + d*x]^3)/(6*a*d^2) - ((e + f*x)^2*Sec[c + d*x]^4)/(4*a*d) + (f*(e + f*x)*Tan[c + d*x])/(3*a*d^2) + (f^2*Sec[c + d*x]*Tan[c + d*x])/(12*a*d^3) + (3*(e + f*x)^2*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + (f*(e + f*x)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d^2) + ((e + f*x)^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)} -{(e + f*x)^1*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 11, -((3*I*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(4*a*d)) + (3*I*f*PolyLog[2, (-I)*E^(I*(c + d*x))])/(8*a*d^2) - (3*I*f*PolyLog[2, I*E^(I*(c + d*x))])/(8*a*d^2) - (3*f*Sec[c + d*x])/(8*a*d^2) - (f*Sec[c + d*x]^3)/(12*a*d^2) - ((e + f*x)*Sec[c + d*x]^4)/(4*a*d) + (f*Tan[c + d*x])/(4*a*d^2) + (3*(e + f*x)*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + ((e + f*x)*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d) + (f*Tan[c + d*x]^3)/(12*a*d^2)} -{(e + f*x)^0*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, (3*ArcTanh[Sin[c + d*x]])/(8*a*d) + 1/(8*d*(a - a*Sin[c + d*x])) - a/(8*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a + a*Sin[c + d*x]))} -{1/(e + f*x)^1*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]} -{1/(e + f*x)^2*Sec[c + d*x]^3/(a + a*Sin[c + d*x]), x, 0, Unintegrable[Sec[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+a Sin[c+d x]) with m symbolic*) - - -{(e + f*x)^m*Cos[c + d*x]^4/(a + a*Sin[c + d*x]), x, 14, (e + f*x)^(1 + m)/(2*a*f*(1 + m)) + (E^(I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(8*a*d)) + ((e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(8*a*d)) - (I*2^(-3 - m)*E^(2*I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((2*I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(a*d)) + (I*2^(-3 - m)*(e + f*x)^m*Gamma[1 + m, (2*I*d*(e + f*x))/f])/(E^(2*I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(a*d)) + (3^(-1 - m)*E^(3*I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((3*I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(8*a*d)) + (3^(-1 - m)*(e + f*x)^m*Gamma[1 + m, (3*I*d*(e + f*x))/f])/(E^(3*I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(8*a*d))} -{(e + f*x)^m*Cos[c + d*x]^3/(a + a*Sin[c + d*x]), x, 9, -((I*E^(I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(2*a*d))) + (I*(e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(2*a*d)) + (2^(-3 - m)*E^(2*I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((2*I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(a*d)) + (2^(-3 - m)*(e + f*x)^m*Gamma[1 + m, (2*I*d*(e + f*x))/f])/(E^(2*I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(a*d))} -{(e + f*x)^m*Cos[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, (e + f*x)^(1 + m)/(a*f*(1 + m)) + (E^(I*(c - (d*e)/f))*(e + f*x)^m*Gamma[1 + m, -((I*d*(e + f*x))/f)])/((-((I*d*(e + f*x))/f))^m*(2*a*d)) + ((e + f*x)^m*Gamma[1 + m, (I*d*(e + f*x))/f])/(E^(I*(c - (d*e)/f))*((I*d*(e + f*x))/f)^m*(2*a*d))} -{(e + f*x)^m*Cos[c + d*x]^1/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Cos[c + d*x])/(a + a*Sin[c + d*x]), x]} -{(e + f*x)^m*Cos[c + d*x]^0/(a + a*Sin[c + d*x]), x, 0, Unintegrable[(e + f*x)^m/(a + a*Sin[c + d*x]), x]} -{(e + f*x)^m*Sec[c + d*x]^1/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sec[c + d*x])/(a + a*Sin[c + d*x]), x]} -{(e + f*x)^m*Sec[c + d*x]^2/(a + a*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n (a+b Sin[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cos[c + d*x])/(a + b*Sin[c + d*x]), x, 11, ((-I/4)*(e + f*x)^4)/(b*f) + ((e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) + ((e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - ((3*I)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^2) + (6*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^3) + (6*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^3) + ((6*I)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^4) + ((6*I)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^4)} -{((e + f*x)^2*Cos[c + d*x])/(a + b*Sin[c + d*x]), x, 9, ((-I/3)*(e + f*x)^3)/(b*f) + ((e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) + ((e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - ((2*I)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^2) + (2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^3) + (2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^3)} -{((e + f*x)^1*Cos[c + d*x])/(a + b*Sin[c + d*x]), x, 7, ((-I/2)*(e + f*x)^2)/(b*f) + ((e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) + ((e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (I*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^2) - (I*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^2)} -{Cos[c + d*x]/(a + b*Sin[c + d*x]), x, 2, Log[a + b*Sin[c + d*x]]/(b*d)} - - -{((e + f*x)^3*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 18, (a*(e + f*x)^4)/(4*b^2*f) - (6*f^2*(e + f*x)*Cos[c + d*x])/(b*d^3) + ((e + f*x)^3*Cos[c + d*x])/(b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (3*Sqrt[a^2 - b^2]*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^2) - (3*Sqrt[a^2 - b^2]*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^2) + ((6*I)*Sqrt[a^2 - b^2]*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^3) - ((6*I)*Sqrt[a^2 - b^2]*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^3) - (6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^4) + (6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^4) + (6*f^3*Sin[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(b*d^2)} -{((e + f*x)^2*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 15, (a*(e + f*x)^3)/(3*b^2*f) - (2*f^2*Cos[c + d*x])/(b*d^3) + ((e + f*x)^2*Cos[c + d*x])/(b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (2*Sqrt[a^2 - b^2]*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^2) - (2*Sqrt[a^2 - b^2]*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^2) + ((2*I)*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^3) - ((2*I)*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^3) - (2*f*(e + f*x)*Sin[c + d*x])/(b*d^2)} -{((e + f*x)^1*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 12, (a*e*x)/b^2 + (a*f*x^2)/(2*b^2) + ((e + f*x)*Cos[c + d*x])/(b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d^2) - (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^2) - (f*Sin[c + d*x])/(b*d^2)} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x]), x, 5, (a*x)/b^2 - (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*d) + Cos[c + d*x]/(b*d)} - - -{((e + f*x)^3*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 21, (-3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) + ((I/4)*(a^2 - b^2)*(e + f*x)^4)/(b^3*f) - (6*a*f^3*Cos[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Cos[c + d*x])/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d) - ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^2) + ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^2) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^3) - ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^4) - ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^4) - (6*a*f^2*(e + f*x)*Sin[c + d*x])/(b^2*d^3) + (a*(e + f*x)^3*Sin[c + d*x])/(b^2*d) + (3*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*b*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^2) + (3*f^2*(e + f*x)*Sin[c + d*x]^2)/(4*b*d^3) - ((e + f*x)^3*Sin[c + d*x]^2)/(2*b*d)} -{((e + f*x)^2*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 16, (e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) + ((I/3)*(a^2 - b^2)*(e + f*x)^3)/(b^3*f) + (2*a*f*(e + f*x)*Cos[c + d*x])/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d) - ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^2) + ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^2) - (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^3) - (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^3) - (2*a*f^2*Sin[c + d*x])/(b^2*d^3) + (a*(e + f*x)^2*Sin[c + d*x])/(b^2*d) - (f*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d^2) + (f^2*Sin[c + d*x]^2)/(4*b*d^3) - ((e + f*x)^2*Sin[c + d*x]^2)/(2*b*d)} -{((e + f*x)^1*Cos[c + d*x]^3)/(a + b*Sin[c + d*x]), x, 13, (f*x)/(4*b*d) + ((I/2)*(a^2 - b^2)*(e + f*x)^2)/(b^3*f) + (a*f*Cos[c + d*x])/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^3*d^2) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^3*d^2) + (a*(e + f*x)*Sin[c + d*x])/(b^2*d) - (f*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^2) - ((e + f*x)*Sin[c + d*x]^2)/(2*b*d)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x]), x, 3, -(((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^3*d)) + (a*Sin[c + d*x])/(b^2*d) - Sin[c + d*x]^2/(2*b*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Sec[c + d*x])/(a + b*Sin[c + d*x]), x, 29, -((2*I*a*(e + f*x)^3*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d)) - (b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) - (b*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) + (b*(e + f*x)^3*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d) + (3*I*a*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) - (3*I*a*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (3*I*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) + (3*I*b*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) - (3*I*b*f*(e + f*x)^2*PolyLog[2, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^2) - (6*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (6*a*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^3) - (6*I*a*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) + (6*I*a*f^3*PolyLog[4, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) - (6*I*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^4) - (6*I*b*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^4) + (3*I*b*f^3*PolyLog[4, -E^(2*I*(c + d*x))])/(4*(a^2 - b^2)*d^4)} -{((e + f*x)^2*Sec[c + d*x])/(a + b*Sin[c + d*x]), x, 24, -((2*I*a*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d)) - (b*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) - (b*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) + (b*(e + f*x)^2*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d) + (2*I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) - (2*I*a*f*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (2*I*b*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) + (2*I*b*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) - (I*b*f*(e + f*x)*PolyLog[2, -E^(2*I*(c + d*x))])/((a^2 - b^2)*d^2) - (2*a*f^2*PolyLog[3, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (2*a*f^2*PolyLog[3, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (2*b*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) - (2*b*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) + (b*f^2*PolyLog[3, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^3)} -{((e + f*x)^1*Sec[c + d*x])/(a + b*Sin[c + d*x]), x, 19, -((2*I*a*(e + f*x)*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d)) - (b*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) - (b*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d) + (b*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d) + (I*a*f*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) - (I*a*f*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (I*b*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) + (I*b*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) - (I*b*f*PolyLog[2, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^2)} -{Sec[c + d*x]/(a + b*Sin[c + d*x]), x, 6, -(Log[1 - Sin[c + d*x]]/(2*(a + b)*d)) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)} - - -{((e + f*x)^3*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 29, If[$VersionNumber>=8, -((I*a*(e + f*x)^3)/((a^2 - b^2)*d)) - (6*I*b*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d^2) + (6*I*b*f^2*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*I*b*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (3*I*a*f^2*(e + f*x)*PolyLog[2, -E^(2*I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*b*f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) + (6*b*f^3*PolyLog[3, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) + (6*I*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (6*I*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) + (3*a*f^3*PolyLog[3, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^4) - (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^4) + (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^4) - (b*(e + f*x)^3*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)^3*Tan[c + d*x])/((a^2 - b^2)*d), -((I*a*(e + f*x)^3)/((a^2 - b^2)*d)) - (6*I*b*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d^2) + (6*I*b*f^2*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*I*b*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (3*I*a*f^2*(e + f*x)*PolyLog[2, -E^(2*I*(c + d*x))])/((a^2 - b^2)*d^3) - (6*b*f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) + (6*b*f^3*PolyLog[3, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^4) + (6*I*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (6*I*b^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) + (3*a*f^3*PolyLog[3, -E^(2*I*(c + d*x))])/(2*(a^2 - b^2)*d^4) - (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^4) + (6*b^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^4) - (b*(e + f*x)^3*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)^3*Tan[c + d*x])/((a^2 - b^2)*d)]} -{((e + f*x)^2*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 24, If[$VersionNumber>=8, -((I*a*(e + f*x)^2)/((a^2 - b^2)*d)) - (4*I*b*f*(e + f*x)*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (2*a*f*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d^2) + (2*I*b*f^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (2*I*b*f^2*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (I*a*f^2*PolyLog[2, -E^(2*I*(c + d*x))])/((a^2 - b^2)*d^3) + (2*I*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (2*I*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)^2*Tan[c + d*x])/((a^2 - b^2)*d), -((I*a*(e + f*x)^2)/((a^2 - b^2)*d)) - (4*I*b*f*(e + f*x)*ArcTan[E^(I*(c + d*x))])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (2*a*f*(e + f*x)*Log[1 + E^(2*I*(c + d*x))])/((a^2 - b^2)*d^2) + (2*I*b*f^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) - (2*I*b*f^2*PolyLog[2, I*E^(I*(c + d*x))])/((a^2 - b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (I*a*f^2*PolyLog[2, -E^(2*I*(c + d*x))])/((a^2 - b^2)*d^3) + (2*I*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (2*I*b^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)^2*Tan[c + d*x])/((a^2 - b^2)*d)]} -{((e + f*x)^1*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 15, If[$VersionNumber>=8, (b*f*ArcTanh[Sin[c + d*x]])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (a*f*Log[Cos[c + d*x]])/((a^2 - b^2)*d^2) + (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (b*(e + f*x)*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)*Tan[c + d*x])/((a^2 - b^2)*d), (b*f*ArcTanh[Sin[c + d*x]])/((a^2 - b^2)*d^2) + (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (I*b^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + (a*f*Log[Cos[c + d*x]])/((a^2 - b^2)*d^2) + (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (b^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (b*(e + f*x)*Sec[c + d*x])/((a^2 - b^2)*d) + (a*(e + f*x)*Tan[c + d*x])/((a^2 - b^2)*d)]} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x]), x, 5, -((2*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d)) - (Sec[c + d*x]*(b - a*Sin[c + d*x]))/((a^2 - b^2)*d)} - - -(* ::Subsubsection::Closed:: *) -(*m symbolic*) - - -{(e + f*x)^m*Cos[c + d*x]^2/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Cos[c + d*x]^2)/(a + b*Sin[c + d*x]), x]} -{(e + f*x)^m*Cos[c + d*x]^1/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Cos[c + d*x])/(a + b*Sin[c + d*x]), x]} -{(e + f*x)^m*Cos[c + d*x]^0/(a + b*Sin[c + d*x]), x, 0, Unintegrable[(e + f*x)^m/(a + b*Sin[c + d*x]), x]} -{(e + f*x)^m*Sec[c + d*x]^1/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sec[c + d*x])/(a + b*Sin[c + d*x]), x]} -{(e + f*x)^m*Sec[c + d*x]^2/(a + b*Sin[c + d*x]), x, 0, Unintegrable[((e + f*x)^m*Sec[c + d*x]^2)/(a + b*Sin[c + d*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^2*) - - -{(e + f*x)^1*Cos[c + d*x]/(a + b*Sin[c + d*x])^2, x, 4, (2*f*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*d^2) - (e + f*x)/(b*d*(a + b*Sin[c + d*x]))} -{(e + f*x)^2*Cos[c + d*x]/(a + b*Sin[c + d*x])^2, x, 9, -((2*I*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2)) + (2*I*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) + (2*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - (e + f*x)^2/(b*d*(a + b*Sin[c + d*x]))} -{(e + f*x)^3*Cos[c + d*x]/(a + b*Sin[c + d*x])^2, x, 11, -((3*I*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2)) + (3*I*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^2) - (6*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) + (6*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^3) - (6*I*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4) + (6*I*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*d^4) - (e + f*x)^3/(b*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n / (a+b Sin[c+d x])^3*) - - -{(e + f*x)^1*Cos[c + d*x]/(a + b*Sin[c + d*x])^3, x, 6, (a*f*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d^2) - (e + f*x)/(2*b*d*(a + b*Sin[c + d*x])^2) + (f*Cos[c + d*x])/(2*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x]))} -{(e + f*x)^2*Cos[c + d*x]/(a + b*Sin[c + d*x])^3, x, 12, -((I*a*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2)) + (I*a*f*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^2) - (f^2*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d^3) - (a*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + (a*f^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*Sin[c + d*x])^2) + (f*(e + f*x)*Cos[c + d*x])/((a^2 - b^2)*d^2*(a + b*Sin[c + d*x]))} -{(e + f*x)^3*Cos[c + d*x]/(a + b*Sin[c + d*x])^3, x, 19, (3*I*f*(e + f*x)^2)/(2*b*(a^2 - b^2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) - (3*I*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*f^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^3) + (3*I*a*f*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(2*b*(a^2 - b^2)^(3/2)*d^2) + (3*I*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) + (3*I*f^3*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^3) - (3*I*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) + (3*I*a*f^3*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*(a^2 - b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sin[c + d*x])^2) + (3*f*(e + f*x)^2*Cos[c + d*x])/(2*(a^2 - b^2)*d^2*(a + b*Sin[c + d*x]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n Sin[c+d x]^p (a+b Sin[c+d x])^q*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^p / (a+b Sin[c+d x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^1 / (a+b Sin[c+d x])*) - - -{((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 33, -(e + f*x)^4/(4*b*f) - (2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (3*Sqrt[a^2 - b^2]*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^2) + (3*Sqrt[a^2 - b^2]*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - ((6*I)*Sqrt[a^2 - b^2]*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^3) + ((6*I)*Sqrt[a^2 - b^2]*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4) + (6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^4) - (6*Sqrt[a^2 - b^2]*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^4)} -{((e + f*x)^2*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 27, -(e + f*x)^3/(3*b*f) - (2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d) + ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (2*Sqrt[a^2 - b^2]*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^2) + (2*Sqrt[a^2 - b^2]*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^2) - (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - ((2*I)*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^3) + ((2*I)*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^3)} -{((e + f*x)^1*Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 21, -((e*x)/b) - (f*x^2)/(2*b) - (2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d) + (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (I*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) - (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b*d^2) + (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b*d^2)} -{(Cos[c + d*x]*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 6, -(x/b) + (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b*d) - ArcTanh[Cos[c + d*x]]/(a*d)} - - -{((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 34, ((-I/4)*(e + f*x)^4)/(a*f) - ((I/4)*(a^2 - b^2)*(e + f*x)^4)/(a*b^2*f) + (6*f^3*Cos[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x])/(b*d^2) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d) + ((e + f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a*d) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^2) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^2) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^2) + (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^3) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^4) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^4) + (((3*I)/4)*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a*d^4) + (6*f^2*(e + f*x)*Sin[c + d*x])/(b*d^3) - ((e + f*x)^3*Sin[c + d*x])/(b*d)} -{((e + f*x)^2*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 26, ((-I/3)*(e + f*x)^3)/(a*f) - ((I/3)*(a^2 - b^2)*(e + f*x)^3)/(a*b^2*f) - (2*f*(e + f*x)*Cos[c + d*x])/(b*d^2) + ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d) + ((e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d) - ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^2) - ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^2) - (I*f*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^2) + (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (f^2*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^3) + (2*f^2*Sin[c + d*x])/(b*d^3) - ((e + f*x)^2*Sin[c + d*x])/(b*d)} -{((e + f*x)^1*Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 22, ((-I/2)*(e + f*x)^2)/(a*f) - ((I/2)*(a^2 - b^2)*(e + f*x)^2)/(a*b^2*f) - (f*Cos[c + d*x])/(b*d^2) + ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d) + ((e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d) - (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^2) - (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^2) - ((I/2)*f*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^2) - ((e + f*x)*Sin[c + d*x])/(b*d)} -{(Cos[c + d*x]^2*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a*b^2*d) - Sin[c + d*x]/(b*d)} - - -{((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 53, (3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) - (e + f*x)^4/(8*b*f) + ((a^2 - b^2)*(e + f*x)^4)/(4*b^3*f) - (2*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) - (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*Cos[c + d*x])/(a*b^2*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + ((a^2 - b^2)*(e + f*x)^3*Cos[c + d*x])/(a*b^2*d) + (3*f^3*Cos[c + d*x]^2)/(8*b*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]^2)/(4*b*d^2) + (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*f^3*Sin[c + d*x])/(a*d^4) + (6*(a^2 - b^2)*f^3*Sin[c + d*x])/(a*b^2*d^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*Sin[c + d*x])/(a*b^2*d^2) + (3*f^2*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 41, (f^2*x)/(4*b*d^2) - (e + f*x)^3/(6*b*f) + ((a^2 - b^2)*(e + f*x)^3)/(3*b^3*f) - (2*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d) - (2*f^2*Cos[c + d*x])/(a*d^3) - (2*(a^2 - b^2)*f^2*Cos[c + d*x])/(a*b^2*d^3) + ((e + f*x)^2*Cos[c + d*x])/(a*d) + ((a^2 - b^2)*(e + f*x)^2*Cos[c + d*x])/(a*b^2*d) - (f*(e + f*x)*Cos[c + d*x]^2)/(2*b*d^2) + (I*(a^2 - b^2)^(3/2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d) + ((2*I)*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (2*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) + (2*f^2*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((2*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((2*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^3) - (2*f*(e + f*x)*Sin[c + d*x])/(a*d^2) - (2*(a^2 - b^2)*f*(e + f*x)*Sin[c + d*x])/(a*b^2*d^2) + (f^2*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{((e + f*x)^1*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 31, -(e*x)/(2*b) + ((a^2 - b^2)*e*x)/b^3 - (f*x^2)/(4*b) + ((a^2 - b^2)*f*x^2)/(2*b^3) - (2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cos[c + d*x])/(a*d) + ((a^2 - b^2)*(e + f*x)*Cos[c + d*x])/(a*b^2*d) - (f*Cos[c + d*x]^2)/(4*b*d^2) + (I*(a^2 - b^2)^(3/2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d) + (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (I*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + ((a^2 - b^2)^(3/2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^2) - ((a^2 - b^2)^(3/2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (f*Sin[c + d*x])/(a*d^2) - ((a^2 - b^2)*f*Sin[c + d*x])/(a*b^2*d^2) - ((e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]), x, 6, ((2*a^2 - 3*b^2)*x)/(2*b^3) - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^3*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (a*Cos[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cos[c+d x]^n Cot[c+d x]^2 / (a+b Sin[c+d x])*) - - -{((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 48, ((I/4)*b*(e + f*x)^4)/(a^2*f) + ((I/4)*(a^2 - b^2)*(e + f*x)^4)/(a^2*b*f) - (6*f*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d^2) - ((e + f*x)^3*Csc[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d) - (b*(e + f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((6*I)*f^2*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) + ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^2) + ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^2) + (((3*I)/2)*b*f*(e + f*x)^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) - (6*f^3*PolyLog[3, -E^(I*(c + d*x))])/(a*d^4) + (6*f^3*PolyLog[3, E^(I*(c + d*x))])/(a*d^4) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^3) - (3*b*f^2*(e + f*x)*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a^2*d^3) - ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^4) - ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^4) - (((3*I)/4)*b*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a^2*d^4)} -{((e + f*x)^2*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 37, ((I/3)*b*(e + f*x)^3)/(a^2*f) + ((I/3)*(a^2 - b^2)*(e + f*x)^3)/(a^2*b*f) - (4*f*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d^2) - ((e + f*x)^2*Csc[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d) - (b*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((2*I)*f^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^3) - ((2*I)*f^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) + ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^2) + ((2*I)*(a^2 - b^2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^2) + (I*b*f*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) - (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^3) - (2*(a^2 - b^2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^3) - (b*f^2*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a^2*d^3)} -{((e + f*x)^1*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 28, ((I/2)*b*(e + f*x)^2)/(a^2*f) + ((I/2)*(a^2 - b^2)*(e + f*x)^2)/(a^2*b*f) - (f*ArcTanh[Cos[c + d*x]])/(a*d^2) - ((e + f*x)*Csc[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d) - (b*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b*d^2) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^2) + ((I/2)*b*f*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2)} -{(Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) - ((1 - b^2/a^2)*Log[a + b*Sin[c + d*x]])/(b*d)} - - -{((e + f*x)^3*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 66, ((-I)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) - ((a^2 - b^2)*(e + f*x)^4)/(4*a*b^2*f) + (2*b*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a^2*d) + (6*b*f^2*(e + f*x)*Cos[c + d*x])/(a^2*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*Cos[c + d*x])/(a^2*b*d^3) - (b*(e + f*x)^3*Cos[c + d*x])/(a^2*d) - ((a^2 - b^2)*(e + f*x)^3*Cos[c + d*x])/(a^2*b*d) - ((e + f*x)^3*Cot[c + d*x])/(a*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (I*(a^2 - b^2)^(3/2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((3*I)*b*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + ((3*I)*b*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a^2*d^3) - ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) + ((6*I)*b*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a^2*d^4) - ((6*I)*b*f^3*PolyLog[4, E^(I*(c + d*x))])/(a^2*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^4) - (6*b*f^3*Sin[c + d*x])/(a^2*d^4) - (6*(a^2 - b^2)*f^3*Sin[c + d*x])/(a^2*b*d^4) + (3*b*f*(e + f*x)^2*Sin[c + d*x])/(a^2*d^2) + (3*(a^2 - b^2)*f*(e + f*x)^2*Sin[c + d*x])/(a^2*b*d^2)} -{((e + f*x)^2*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 53, ((-I)*(e + f*x)^2)/(a*d) - (e + f*x)^3/(3*a*f) - ((a^2 - b^2)*(e + f*x)^3)/(3*a*b^2*f) + (2*b*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a^2*d) + (2*b*f^2*Cos[c + d*x])/(a^2*d^3) + (2*(a^2 - b^2)*f^2*Cos[c + d*x])/(a^2*b*d^3) - (b*(e + f*x)^2*Cos[c + d*x])/(a^2*d) - ((a^2 - b^2)*(e + f*x)^2*Cos[c + d*x])/(a^2*b*d) - ((e + f*x)^2*Cot[c + d*x])/(a*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (I*(a^2 - b^2)^(3/2)*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (2*f*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) - ((2*I)*b*f*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + ((2*I)*b*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) + (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) - (I*f^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^3) + (2*b*f^2*PolyLog[3, -E^(I*(c + d*x))])/(a^2*d^3) - (2*b*f^2*PolyLog[3, E^(I*(c + d*x))])/(a^2*d^3) - ((2*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + ((2*I)*(a^2 - b^2)^(3/2)*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^3) + (2*b*f*(e + f*x)*Sin[c + d*x])/(a^2*d^2) + (2*(a^2 - b^2)*f*(e + f*x)*Sin[c + d*x])/(a^2*b*d^2)} -{((e + f*x)^1*Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 38, -((e*x)/a) + ((1 - a^2/b^2)*e*x)/a - (f*x^2)/(2*a) + ((1 - a^2/b^2)*f*x^2)/(2*a) + (2*b*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a^2*d) - (b*(e + f*x)*Cos[c + d*x])/(a^2*d) - ((a^2 - b^2)*(e + f*x)*Cos[c + d*x])/(a^2*b*d) - ((e + f*x)*Cot[c + d*x])/(a*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (I*(a^2 - b^2)^(3/2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d) + (f*Log[Sin[c + d*x]])/(a*d^2) - (I*b*f*PolyLog[2, -E^(I*(c + d*x))])/(a^2*d^2) + (I*b*f*PolyLog[2, E^(I*(c + d*x))])/(a^2*d^2) - ((a^2 - b^2)^(3/2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) + ((a^2 - b^2)^(3/2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^2*d^2) + (b*f*Sin[c + d*x])/(a^2*d^2) + ((a^2 - b^2)*f*Sin[c + d*x])/(a^2*b*d^2)} -{(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 6, -((a*x)/b^2) + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*b^2*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cos[c + d*x]/(b*d) - Cot[c + d*x]/(a*d)} - - -{((e + f*x)^3*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 85, (3*b*f^3*x)/(8*a^2*d^3) + (3*(a^2 - b^2)*f^3*x)/(8*a^2*b*d^3) - (b*(e + f*x)^3)/(4*a^2*d) - ((a^2 - b^2)*(e + f*x)^3)/(4*a^2*b*d) + ((I/4)*b*(e + f*x)^4)/(a^2*f) - ((I/4)*(a^2 - b^2)^2*(e + f*x)^4)/(a^2*b^3*f) - (6*f*(e + f*x)^2*ArcTanh[E^(I*(c + d*x))])/(a*d^2) + (6*f^3*Cos[c + d*x])/(a*d^4) + (6*(a^2 - b^2)*f^3*Cos[c + d*x])/(a*b^2*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x])/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*Cos[c + d*x])/(a*b^2*d^2) - ((e + f*x)^3*Csc[c + d*x])/(a*d) + ((a^2 - b^2)^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d) - (b*(e + f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((6*I)*f^2*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/(a*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) - ((3*I)*(a^2 - b^2)^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) - ((3*I)*(a^2 - b^2)^2*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) + (((3*I)/2)*b*f*(e + f*x)^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) - (6*f^3*PolyLog[3, -E^(I*(c + d*x))])/(a*d^4) + (6*f^3*PolyLog[3, E^(I*(c + d*x))])/(a*d^4) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^3) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^3) - (3*b*f^2*(e + f*x)*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a^2*d^3) + ((6*I)*(a^2 - b^2)^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^4) + ((6*I)*(a^2 - b^2)^2*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^4) - (((3*I)/4)*b*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a^2*d^4) + (6*f^2*(e + f*x)*Sin[c + d*x])/(a*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*Sin[c + d*x])/(a*b^2*d^3) - ((e + f*x)^3*Sin[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)^3*Sin[c + d*x])/(a*b^2*d) - (3*b*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d^4) - (3*(a^2 - b^2)*f^3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*b*d^4) + (3*b*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d^2) + (3*(a^2 - b^2)*f*(e + f*x)^2*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*b*d^2) - (3*b*f^2*(e + f*x)*Sin[c + d*x]^2)/(4*a^2*d^3) - (3*(a^2 - b^2)*f^2*(e + f*x)*Sin[c + d*x]^2)/(4*a^2*b*d^3) + (b*(e + f*x)^3*Sin[c + d*x]^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)^3*Sin[c + d*x]^2)/(2*a^2*b*d)} -{((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 60, -(b*e*f*x)/(2*a^2*d) - ((a^2 - b^2)*e*f*x)/(2*a^2*b*d) - (b*f^2*x^2)/(4*a^2*d) - ((a^2 - b^2)*f^2*x^2)/(4*a^2*b*d) + ((I/3)*b*(e + f*x)^3)/(a^2*f) - ((I/3)*(a^2 - b^2)^2*(e + f*x)^3)/(a^2*b^3*f) - (4*f*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d^2) - (2*f*(e + f*x)*Cos[c + d*x])/(a*d^2) - (2*(a^2 - b^2)*f*(e + f*x)*Cos[c + d*x])/(a*b^2*d^2) - ((e + f*x)^2*Csc[c + d*x])/(a*d) + ((a^2 - b^2)^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d) - (b*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + ((2*I)*f^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^3) - ((2*I)*f^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^3) - ((2*I)*(a^2 - b^2)^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) - ((2*I)*(a^2 - b^2)^2*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) + (I*b*f*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) + (2*(a^2 - b^2)^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^3) + (2*(a^2 - b^2)^2*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^3) - (b*f^2*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a^2*d^3) + (2*f^2*Sin[c + d*x])/(a*d^3) + (2*(a^2 - b^2)*f^2*Sin[c + d*x])/(a*b^2*d^3) - ((e + f*x)^2*Sin[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)^2*Sin[c + d*x])/(a*b^2*d) + (b*f*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d^2) + ((a^2 - b^2)*f*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*b*d^2) - (b*f^2*Sin[c + d*x]^2)/(4*a^2*d^3) - ((a^2 - b^2)*f^2*Sin[c + d*x]^2)/(4*a^2*b*d^3) + (b*(e + f*x)^2*Sin[c + d*x]^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)^2*Sin[c + d*x]^2)/(2*a^2*b*d)} -{((e + f*x)^1*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 45, -(b*f*x)/(4*a^2*d) - ((a^2 - b^2)*f*x)/(4*a^2*b*d) + ((I/2)*b*(e + f*x)^2)/(a^2*f) - ((I/2)*(a^2 - b^2)^2*(e + f*x)^2)/(a^2*b^3*f) - (f*ArcTanh[Cos[c + d*x]])/(a*d^2) - (f*Cos[c + d*x])/(a*d^2) - ((a^2 - b^2)*f*Cos[c + d*x])/(a*b^2*d^2) - ((e + f*x)*Csc[c + d*x])/(a*d) + ((a^2 - b^2)^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d) - (b*(e + f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) - (I*(a^2 - b^2)^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) - (I*(a^2 - b^2)^2*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b^3*d^2) + ((I/2)*b*f*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2) - ((e + f*x)*Sin[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)*Sin[c + d*x])/(a*b^2*d) + (b*f*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d^2) + ((a^2 - b^2)*f*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*b*d^2) + (b*(e + f*x)*Sin[c + d*x]^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)*Sin[c + d*x]^2)/(2*a^2*b*d)} -{(Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^2*b^3*d) - (a*Sin[c + d*x])/(b^2*d) + Sin[c + d*x]^2/(2*b*d)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m deleted file mode 100644 index 8702cbf..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.11 (e x)^m (a+b x^n)^p sin.m +++ /dev/null @@ -1,183 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b x^n)^p Sin[c+d x]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^1)^p Sin[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x)*Sin[c + d*x], x, 11, (-24*b*Cos[c + d*x])/d^5 + (6*a*x*Cos[c + d*x])/d^3 + (12*b*x^2*Cos[c + d*x])/d^3 - (a*x^3*Cos[c + d*x])/d - (b*x^4*Cos[c + d*x])/d - (6*a*Sin[c + d*x])/d^4 - (24*b*x*Sin[c + d*x])/d^4 + (3*a*x^2*Sin[c + d*x])/d^2 + (4*b*x^3*Sin[c + d*x])/d^2} -{x^2*(a + b*x)*Sin[c + d*x], x, 9, (2*a*Cos[c + d*x])/d^3 + (6*b*x*Cos[c + d*x])/d^3 - (a*x^2*Cos[c + d*x])/d - (b*x^3*Cos[c + d*x])/d - (6*b*Sin[c + d*x])/d^4 + (2*a*x*Sin[c + d*x])/d^2 + (3*b*x^2*Sin[c + d*x])/d^2} -{x*(a + b*x)*Sin[c + d*x], x, 7, (2*b*Cos[c + d*x])/d^3 - (a*x*Cos[c + d*x])/d - (b*x^2*Cos[c + d*x])/d + (a*Sin[c + d*x])/d^2 + (2*b*x*Sin[c + d*x])/d^2} -{(a + b*x)*Sin[c + d*x], x, 2, -(((a + b*x)*Cos[c + d*x])/d) + (b*Sin[c + d*x])/d^2} -{((a + b*x)*Sin[c + d*x])/x, x, 6, -((b*Cos[c + d*x])/d) + a*CosIntegral[d*x]*Sin[c] + a*Cos[c]*SinIntegral[d*x]} -{((a + b*x)*Sin[c + d*x])/x^2, x, 9, a*d*Cos[c]*CosIntegral[d*x] + b*CosIntegral[d*x]*Sin[c] - (a*Sin[c + d*x])/x + b*Cos[c]*SinIntegral[d*x] - a*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x)*Sin[c + d*x])/x^3, x, 11, -(a*d*Cos[c + d*x])/(2*x) + b*d*Cos[c]*CosIntegral[d*x] - (a*d^2*CosIntegral[d*x]*Sin[c])/2 - (a*Sin[c + d*x])/(2*x^2) - (b*Sin[c + d*x])/x - (a*d^2*Cos[c]*SinIntegral[d*x])/2 - b*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x)*Sin[c + d*x])/x^4, x, 13, -(a*d*Cos[c + d*x])/(6*x^2) - (b*d*Cos[c + d*x])/(2*x) - (a*d^3*Cos[c]*CosIntegral[d*x])/6 - (b*d^2*CosIntegral[d*x]*Sin[c])/2 - (a*Sin[c + d*x])/(3*x^3) - (b*Sin[c + d*x])/(2*x^2) + (a*d^2*Sin[c + d*x])/(6*x) - (b*d^2*Cos[c]*SinIntegral[d*x])/2 + (a*d^3*Sin[c]*SinIntegral[d*x])/6} -{((a + b*x)*Sin[c + d*x])/x^5, x, 15, -(a*d*Cos[c + d*x])/(12*x^3) - (b*d*Cos[c + d*x])/(6*x^2) + (a*d^3*Cos[c + d*x])/(24*x) - (b*d^3*Cos[c]*CosIntegral[d*x])/6 + (a*d^4*CosIntegral[d*x]*Sin[c])/24 - (a*Sin[c + d*x])/(4*x^4) - (b*Sin[c + d*x])/(3*x^3) + (a*d^2*Sin[c + d*x])/(24*x^2) + (b*d^2*Sin[c + d*x])/(6*x) + (a*d^4*Cos[c]*SinIntegral[d*x])/24 + (b*d^3*Sin[c]*SinIntegral[d*x])/6} - - -{x^2*(a + b*x)^2*Sin[c + d*x], x, 14, (-24*b^2*Cos[c + d*x])/d^5 + (2*a^2*Cos[c + d*x])/d^3 + (12*a*b*x*Cos[c + d*x])/d^3 + (12*b^2*x^2*Cos[c + d*x])/d^3 - (a^2*x^2*Cos[c + d*x])/d - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^4*Cos[c + d*x])/d - (12*a*b*Sin[c + d*x])/d^4 - (24*b^2*x*Sin[c + d*x])/d^4 + (2*a^2*x*Sin[c + d*x])/d^2 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (4*b^2*x^3*Sin[c + d*x])/d^2} -{x*(a + b*x)^2*Sin[c + d*x], x, 11, (4*a*b*Cos[c + d*x])/d^3 + (6*b^2*x*Cos[c + d*x])/d^3 - (a^2*x*Cos[c + d*x])/d - (2*a*b*x^2*Cos[c + d*x])/d - (b^2*x^3*Cos[c + d*x])/d - (6*b^2*Sin[c + d*x])/d^4 + (a^2*Sin[c + d*x])/d^2 + (4*a*b*x*Sin[c + d*x])/d^2 + (3*b^2*x^2*Sin[c + d*x])/d^2} -{(a + b*x)^2*Sin[c + d*x], x, 3, (2*b^2*Cos[c + d*x])/d^3 - ((a + b*x)^2*Cos[c + d*x])/d + (2*b*(a + b*x)*Sin[c + d*x])/d^2} -{((a + b*x)^2*Sin[c + d*x])/x, x, 8, (-2*a*b*Cos[c + d*x])/d - (b^2*x*Cos[c + d*x])/d + a^2*CosIntegral[d*x]*Sin[c] + (b^2*Sin[c + d*x])/d^2 + a^2*Cos[c]*SinIntegral[d*x]} -{((a + b*x)^2*Sin[c + d*x])/x^2, x, 10, -((b^2*Cos[c + d*x])/d) + a^2*d*Cos[c]*CosIntegral[d*x] + 2*a*b*CosIntegral[d*x]*Sin[c] - (a^2*Sin[c + d*x])/x + 2*a*b*Cos[c]*SinIntegral[d*x] - a^2*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x)^2*Sin[c + d*x])/x^3, x, 14, -(a^2*d*Cos[c + d*x])/(2*x) + 2*a*b*d*Cos[c]*CosIntegral[d*x] + b^2*CosIntegral[d*x]*Sin[c] - (a^2*d^2*CosIntegral[d*x]*Sin[c])/2 - (a^2*Sin[c + d*x])/(2*x^2) - (2*a*b*Sin[c + d*x])/x + b^2*Cos[c]*SinIntegral[d*x] - (a^2*d^2*Cos[c]*SinIntegral[d*x])/2 - 2*a*b*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x)^2*Sin[c + d*x])/x^4, x, 17, -(a^2*d*Cos[c + d*x])/(6*x^2) - (a*b*d*Cos[c + d*x])/x + b^2*d*Cos[c]*CosIntegral[d*x] - (a^2*d^3*Cos[c]*CosIntegral[d*x])/6 - a*b*d^2*CosIntegral[d*x]*Sin[c] - (a^2*Sin[c + d*x])/(3*x^3) - (a*b*Sin[c + d*x])/x^2 - (b^2*Sin[c + d*x])/x + (a^2*d^2*Sin[c + d*x])/(6*x) - a*b*d^2*Cos[c]*SinIntegral[d*x] - b^2*d*Sin[c]*SinIntegral[d*x] + (a^2*d^3*Sin[c]*SinIntegral[d*x])/6} -{((a + b*x)^2*Sin[c + d*x])/x^5, x, 20, -(a^2*d*Cos[c + d*x])/(12*x^3) - (a*b*d*Cos[c + d*x])/(3*x^2) - (b^2*d*Cos[c + d*x])/(2*x) + (a^2*d^3*Cos[c + d*x])/(24*x) - (a*b*d^3*Cos[c]*CosIntegral[d*x])/3 - (b^2*d^2*CosIntegral[d*x]*Sin[c])/2 + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4) - (2*a*b*Sin[c + d*x])/(3*x^3) - (b^2*Sin[c + d*x])/(2*x^2) + (a^2*d^2*Sin[c + d*x])/(24*x^2) + (a*b*d^2*Sin[c + d*x])/(3*x) - (b^2*d^2*Cos[c]*SinIntegral[d*x])/2 + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24 + (a*b*d^3*Sin[c]*SinIntegral[d*x])/3} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Sin[c + d*x])/(a + b*x), x, 15, (-2*a*Cos[c + d*x])/(b^2*d^3) + (a^3*Cos[c + d*x])/(b^4*d) + (6*x*Cos[c + d*x])/(b*d^3) - (a^2*x*Cos[c + d*x])/(b^3*d) + (a*x^2*Cos[c + d*x])/(b^2*d) - (x^3*Cos[c + d*x])/(b*d) + (a^4*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^5 - (6*Sin[c + d*x])/(b*d^4) + (a^2*Sin[c + d*x])/(b^3*d^2) - (2*a*x*Sin[c + d*x])/(b^2*d^2) + (3*x^2*Sin[c + d*x])/(b*d^2) + (a^4*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5} -{(x^3*Sin[c + d*x])/(a + b*x), x, 11, (2*Cos[c + d*x])/(b*d^3) - (a^2*Cos[c + d*x])/(b^3*d) + (a*x*Cos[c + d*x])/(b^2*d) - (x^2*Cos[c + d*x])/(b*d) - (a^3*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^4 - (a*Sin[c + d*x])/(b^2*d^2) + (2*x*Sin[c + d*x])/(b*d^2) - (a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4} -{(x^2*Sin[c + d*x])/(a + b*x), x, 8, (a*Cos[c + d*x])/(b^2*d) - (x*Cos[c + d*x])/(b*d) + (a^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^3 + Sin[c + d*x]/(b*d^2) + (a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3} -{(x*Sin[c + d*x])/(a + b*x), x, 6, -(Cos[c + d*x]/(b*d)) - (a*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^2 - (a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2} -{Sin[c + d*x]/(a + b*x), x, 3, (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b + (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b} -{Sin[c + d*x]/(x*(a + b*x)), x, 8, (CosIntegral[d*x]*Sin[c])/a - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a + (Cos[c]*SinIntegral[d*x])/a - (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a} -{Sin[c + d*x]/(x^2*(a + b*x)), x, 12, (d*Cos[c]*CosIntegral[d*x])/a - (b*CosIntegral[d*x]*Sin[c])/a^2 + (b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^2 - Sin[c + d*x]/(a*x) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d*Sin[c]*SinIntegral[d*x])/a + (b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^2} -{Sin[c + d*x]/(x^3*(a + b*x)), x, 17, -(d*Cos[c + d*x])/(2*a*x) - (b*d*Cos[c]*CosIntegral[d*x])/a^2 + (b^2*CosIntegral[d*x]*Sin[c])/a^3 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(2*a*x^2) + (b*Sin[c + d*x])/(a^2*x) + (b^2*Cos[c]*SinIntegral[d*x])/a^3 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) + (b*d*Sin[c]*SinIntegral[d*x])/a^2 - (b^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3} - - -{(x^4*Sin[c + d*x])/(a + b*x)^2, x, 15, (2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^6} -{(x^3*Sin[c + d*x])/(a + b*x)^2, x, 12, (2*a*Cos[c + d*x])/(b^3*d) - (x*Cos[c + d*x])/(b^2*d) - (a^3*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^5 + (3*a^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^4 + Sin[c + d*x]/(b^2*d^2) + (a^3*Sin[c + d*x])/(b^4*(a + b*x)) + (3*a^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4 + (a^3*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5} -{(x^2*Sin[c + d*x])/(a + b*x)^2, x, 10, -(Cos[c + d*x]/(b^2*d)) + (a^2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^4 - (2*a*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^3 - (a^2*Sin[c + d*x])/(b^3*(a + b*x)) - (2*a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3 - (a^2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4} -{(x*Sin[c + d*x])/(a + b*x)^2, x, 9, -((a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^3) + (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^2 + (a*Sin[c + d*x])/(b^2*(a + b*x)) + (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2 + (a*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3} -{Sin[c + d*x]/(a + b*x)^2, x, 4, (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^2 - Sin[c + d*x]/(b*(a + b*x)) - (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2} -{Sin[c + d*x]/(x*(a + b*x)^2), x, 12, -((d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/(a*b)) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^2 + Sin[c + d*x]/(a*(a + b*x)) + (Cos[c]*SinIntegral[d*x])/a^2 - (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^2 + (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(a*b)} -{Sin[c + d*x]/(x^2*(a + b*x)^2), x, 16, (d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/a^2 - (2*b*CosIntegral[d*x]*Sin[c])/a^3 + (2*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(a^2*x) - (b*Sin[c + d*x])/(a^2*(a + b*x)) - (2*b*Cos[c]*SinIntegral[d*x])/a^3 - (d*Sin[c]*SinIntegral[d*x])/a^2 + (2*b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3 - (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^2} - - -{(x^3*Sin[c + d*x])/(a + b*x)^3, x, 15, -(Cos[c + d*x]/(b^3*d)) + (a^3*d*Cos[c + d*x])/(2*b^5*(a + b*x)) + (3*a^2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^5 - (3*a*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^4 + (a^3*d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^6) + (a^3*Sin[c + d*x])/(2*b^4*(a + b*x)^2) - (3*a^2*Sin[c + d*x])/(b^4*(a + b*x)) - (3*a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4 + (a^3*d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^6) - (3*a^2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5} -{(x^2*Sin[c + d*x])/(a + b*x)^3, x, 14, -(a^2*d*Cos[c + d*x])/(2*b^4*(a + b*x)) - (2*a*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^4 + (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^3 - (a^2*d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^5) - (a^2*Sin[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Sin[c + d*x])/(b^3*(a + b*x)) + (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3 - (a^2*d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^5) + (2*a*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^4} -{(x*Sin[c + d*x])/(a + b*x)^3, x, 11, (a*d*Cos[c + d*x])/(2*b^3*(a + b*x)) + (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^3 + (a*d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^4) + (a*Sin[c + d*x])/(2*b^2*(a + b*x)^2) - Sin[c + d*x]/(b^2*(a + b*x)) + (a*d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^4) - (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3} -{Sin[c + d*x]/(a + b*x)^3, x, 5, -(d*Cos[c + d*x])/(2*b^2*(a + b*x)) - (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^3) - Sin[c + d*x]/(2*b*(a + b*x)^2) - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*b^3)} -{Sin[c + d*x]/(x*(a + b*x)^3), x, 17, (d*Cos[c + d*x])/(2*a*b*(a + b*x)) - (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/(a^2*b) + (CosIntegral[d*x]*Sin[c])/a^3 - (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 + (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a*b^2) + Sin[c + d*x]/(2*a*(a + b*x)^2) + Sin[c + d*x]/(a^2*(a + b*x)) + (Cos[c]*SinIntegral[d*x])/a^3 - (Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3 + (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*a*b^2) + (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(a^2*b)} -{Sin[c + d*x]/(x^2*(a + b*x)^3), x, 21, -(d*Cos[c + d*x])/(2*a^2*(a + b*x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (2*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/a^3 - (3*b*CosIntegral[d*x]*Sin[c])/a^4 + (3*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^4 - (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a^2*b) - Sin[c + d*x]/(a^3*x) - (b*Sin[c + d*x])/(2*a^2*(a + b*x)^2) - (2*b*Sin[c + d*x])/(a^3*(a + b*x)) - (3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d*Sin[c]*SinIntegral[d*x])/a^3 + (3*b*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^4 - (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*a^2*b) - (2*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^3} -{Sin[c + d*x]/(x^3*(a + b*x)^3), x, 26, -(d*Cos[c + d*x])/(2*a^3*x) + (b*d*Cos[c + d*x])/(2*a^3*(a + b*x)) - (3*b*d*Cos[c]*CosIntegral[d*x])/a^4 - (3*b*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/a^4 + (6*b^2*CosIntegral[d*x]*Sin[c])/a^5 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^3) - (6*b^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^5 + (d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*a^3) - Sin[c + d*x]/(2*a^3*x^2) + (3*b*Sin[c + d*x])/(a^4*x) + (b^2*Sin[c + d*x])/(2*a^3*(a + b*x)^2) + (3*b^2*Sin[c + d*x])/(a^4*(a + b*x)) + (6*b^2*Cos[c]*SinIntegral[d*x])/a^5 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^3) + (3*b*d*Sin[c]*SinIntegral[d*x])/a^4 - (6*b^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^5 + (d^2*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/(2*a^3) + (3*b*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^4} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^2)^p Sin[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x^2)*Sin[c + d*x], x, 12, (-120*b*x*Cos[c + d*x])/d^5 + (6*a*x*Cos[c + d*x])/d^3 + (20*b*x^3*Cos[c + d*x])/d^3 - (a*x^3*Cos[c + d*x])/d - (b*x^5*Cos[c + d*x])/d + (120*b*Sin[c + d*x])/d^6 - (6*a*Sin[c + d*x])/d^4 - (60*b*x^2*Sin[c + d*x])/d^4 + (3*a*x^2*Sin[c + d*x])/d^2 + (5*b*x^4*Sin[c + d*x])/d^2} -{x^2*(a + b*x^2)*Sin[c + d*x], x, 10, (-24*b*Cos[c + d*x])/d^5 + (2*a*Cos[c + d*x])/d^3 + (12*b*x^2*Cos[c + d*x])/d^3 - (a*x^2*Cos[c + d*x])/d - (b*x^4*Cos[c + d*x])/d - (24*b*x*Sin[c + d*x])/d^4 + (2*a*x*Sin[c + d*x])/d^2 + (4*b*x^3*Sin[c + d*x])/d^2} -{x*(a + b*x^2)*Sin[c + d*x], x, 8, (6*b*x*Cos[c + d*x])/d^3 - (a*x*Cos[c + d*x])/d - (b*x^3*Cos[c + d*x])/d - (6*b*Sin[c + d*x])/d^4 + (a*Sin[c + d*x])/d^2 + (3*b*x^2*Sin[c + d*x])/d^2} -{(a + b*x^2)*Sin[c + d*x], x, 6, (2*b*Cos[c + d*x])/d^3 - (a*Cos[c + d*x])/d - (b*x^2*Cos[c + d*x])/d + (2*b*x*Sin[c + d*x])/d^2} -{((a + b*x^2)*Sin[c + d*x])/x, x, 7, -((b*x*Cos[c + d*x])/d) + a*CosIntegral[d*x]*Sin[c] + (b*Sin[c + d*x])/d^2 + a*Cos[c]*SinIntegral[d*x]} -{((a + b*x^2)*Sin[c + d*x])/x^2, x, 7, -((b*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x^2)*Sin[c + d*x])/x^3, x, 10, -(a*d*Cos[c + d*x])/(2*x) + b*CosIntegral[d*x]*Sin[c] - (a*d^2*CosIntegral[d*x]*Sin[c])/2 - (a*Sin[c + d*x])/(2*x^2) + b*Cos[c]*SinIntegral[d*x] - (a*d^2*Cos[c]*SinIntegral[d*x])/2} -{((a + b*x^2)*Sin[c + d*x])/x^4, x, 12, -(a*d*Cos[c + d*x])/(6*x^2) + b*d*Cos[c]*CosIntegral[d*x] - (a*d^3*Cos[c]*CosIntegral[d*x])/6 - (a*Sin[c + d*x])/(3*x^3) - (b*Sin[c + d*x])/x + (a*d^2*Sin[c + d*x])/(6*x) - b*d*Sin[c]*SinIntegral[d*x] + (a*d^3*Sin[c]*SinIntegral[d*x])/6} -{((a + b*x^2)*Sin[c + d*x])/x^5, x, 14, -(a*d*Cos[c + d*x])/(12*x^3) - (b*d*Cos[c + d*x])/(2*x) + (a*d^3*Cos[c + d*x])/(24*x) - (b*d^2*CosIntegral[d*x]*Sin[c])/2 + (a*d^4*CosIntegral[d*x]*Sin[c])/24 - (a*Sin[c + d*x])/(4*x^4) - (b*Sin[c + d*x])/(2*x^2) + (a*d^2*Sin[c + d*x])/(24*x^2) - (b*d^2*Cos[c]*SinIntegral[d*x])/2 + (a*d^4*Cos[c]*SinIntegral[d*x])/24} - - -{x^2*(a + b*x^2)^2*Sin[c + d*x], x, 17, (720*b^2*Cos[c + d*x])/d^7 - (48*a*b*Cos[c + d*x])/d^5 + (2*a^2*Cos[c + d*x])/d^3 - (360*b^2*x^2*Cos[c + d*x])/d^5 + (24*a*b*x^2*Cos[c + d*x])/d^3 - (a^2*x^2*Cos[c + d*x])/d + (30*b^2*x^4*Cos[c + d*x])/d^3 - (2*a*b*x^4*Cos[c + d*x])/d - (b^2*x^6*Cos[c + d*x])/d + (720*b^2*x*Sin[c + d*x])/d^6 - (48*a*b*x*Sin[c + d*x])/d^4 + (2*a^2*x*Sin[c + d*x])/d^2 - (120*b^2*x^3*Sin[c + d*x])/d^4 + (8*a*b*x^3*Sin[c + d*x])/d^2 + (6*b^2*x^5*Sin[c + d*x])/d^2} -{x*(a + b*x^2)^2*Sin[c + d*x], x, 14, (-120*b^2*x*Cos[c + d*x])/d^5 + (12*a*b*x*Cos[c + d*x])/d^3 - (a^2*x*Cos[c + d*x])/d + (20*b^2*x^3*Cos[c + d*x])/d^3 - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^5*Cos[c + d*x])/d + (120*b^2*Sin[c + d*x])/d^6 - (12*a*b*Sin[c + d*x])/d^4 + (a^2*Sin[c + d*x])/d^2 - (60*b^2*x^2*Sin[c + d*x])/d^4 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (5*b^2*x^4*Sin[c + d*x])/d^2} -{(a + b*x^2)^2*Sin[c + d*x], x, 11, (-24*b^2*Cos[c + d*x])/d^5 + (4*a*b*Cos[c + d*x])/d^3 - (a^2*Cos[c + d*x])/d + (12*b^2*x^2*Cos[c + d*x])/d^3 - (2*a*b*x^2*Cos[c + d*x])/d - (b^2*x^4*Cos[c + d*x])/d - (24*b^2*x*Sin[c + d*x])/d^4 + (4*a*b*x*Sin[c + d*x])/d^2 + (4*b^2*x^3*Sin[c + d*x])/d^2} -{((a + b*x^2)^2*Sin[c + d*x])/x, x, 11, (6*b^2*x*Cos[c + d*x])/d^3 - (2*a*b*x*Cos[c + d*x])/d - (b^2*x^3*Cos[c + d*x])/d + a^2*CosIntegral[d*x]*Sin[c] - (6*b^2*Sin[c + d*x])/d^4 + (2*a*b*Sin[c + d*x])/d^2 + (3*b^2*x^2*Sin[c + d*x])/d^2 + a^2*Cos[c]*SinIntegral[d*x]} -{((a + b*x^2)^2*Sin[c + d*x])/x^2, x, 10, (2*b^2*Cos[c + d*x])/d^3 - (2*a*b*Cos[c + d*x])/d - (b^2*x^2*Cos[c + d*x])/d + a^2*d*Cos[c]*CosIntegral[d*x] - (a^2*Sin[c + d*x])/x + (2*b^2*x*Sin[c + d*x])/d^2 - a^2*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x^2)^2*Sin[c + d*x])/x^3, x, 12, -(a^2*d*Cos[c + d*x])/(2*x) - (b^2*x*Cos[c + d*x])/d + 2*a*b*CosIntegral[d*x]*Sin[c] - (a^2*d^2*CosIntegral[d*x]*Sin[c])/2 + (b^2*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/(2*x^2) + 2*a*b*Cos[c]*SinIntegral[d*x] - (a^2*d^2*Cos[c]*SinIntegral[d*x])/2} -{((a + b*x^2)^2*Sin[c + d*x])/x^4, x, 13, -((b^2*Cos[c + d*x])/d) - (a^2*d*Cos[c + d*x])/(6*x^2) + 2*a*b*d*Cos[c]*CosIntegral[d*x] - (a^2*d^3*Cos[c]*CosIntegral[d*x])/6 - (a^2*Sin[c + d*x])/(3*x^3) - (2*a*b*Sin[c + d*x])/x + (a^2*d^2*Sin[c + d*x])/(6*x) - 2*a*b*d*Sin[c]*SinIntegral[d*x] + (a^2*d^3*Sin[c]*SinIntegral[d*x])/6} -{((a + b*x^2)^2*Sin[c + d*x])/x^5, x, 17, -(a^2*d*Cos[c + d*x])/(12*x^3) - (a*b*d*Cos[c + d*x])/x + (a^2*d^3*Cos[c + d*x])/(24*x) + b^2*CosIntegral[d*x]*Sin[c] - a*b*d^2*CosIntegral[d*x]*Sin[c] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4) - (a*b*Sin[c + d*x])/x^2 + (a^2*d^2*Sin[c + d*x])/(24*x^2) + b^2*Cos[c]*SinIntegral[d*x] - a*b*d^2*Cos[c]*SinIntegral[d*x] + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Sin[c + d*x])/(a + b*x^2), x, 14, (2*Cos[c + d*x])/(b*d^3) + (a*Cos[c + d*x])/(b^2*d) - (x^2*Cos[c + d*x])/(b*d) - ((-a)^(3/2)*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^(5/2)) + ((-a)^(3/2)*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^(5/2)) + (2*x*Sin[c + d*x])/(b*d^2) - ((-a)^(3/2)*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)) - ((-a)^(3/2)*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(5/2))} -{(x^3*Sin[c + d*x])/(a + b*x^2), x, 12, -((x*Cos[c + d*x])/(b*d)) - (a*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) - (a*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + Sin[c + d*x]/(b*d^2) + (a*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2)} -{(x^2*Sin[c + d*x])/(a + b*x^2), x, 11, -(Cos[c + d*x]/(b*d)) - (Sqrt[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^(3/2)) + (Sqrt[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^(3/2)) - (Sqrt[-a]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2))} -{(x*Sin[c + d*x])/(a + b*x^2), x, 8, (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b)} -{Sin[c + d*x]/(a + b*x^2), x, 8, -(CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*Sqrt[-a]*Sqrt[b]) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*Sqrt[-a]*Sqrt[b]) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b])} -{Sin[c + d*x]/(x*(a + b*x^2)), x, 13, (CosIntegral[d*x]*Sin[c])/a - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a) + (Cos[c]*SinIntegral[d*x])/a + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a)} -{Sin[c + d*x]/(x^2*(a + b*x^2)), x, 14, (d*Cos[c]*CosIntegral[d*x])/a - (Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*(-a)^(3/2)) + (Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*(-a)^(3/2)) - Sin[c + d*x]/(a*x) - (d*Sin[c]*SinIntegral[d*x])/a - (Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/2)) - (Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))} -{Sin[c + d*x]/(x^3*(a + b*x^2)), x, 18, -(d*Cos[c + d*x])/(2*a*x) - (b*CosIntegral[d*x]*Sin[c])/a^2 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - Sin[c + d*x]/(2*a*x^2) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) - (b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)} - - -{(x^4*Sin[c + d*x])/(a + b*x^2)^2, x, 24, -(Cos[c + d*x]/(b^2*d)) - (a*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3) - (a*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (3*Sqrt[-a]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (x*Sin[c + d*x])/(2*b^2) - (x^3*Sin[c + d*x])/(2*b*(a + b*x^2)) - (3*Sqrt[-a]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (a*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3) - (3*Sqrt[-a]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (a*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3)} -{(x^3*Sin[c + d*x])/(a + b*x^2)^2, x, 20, (Sqrt[-a]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (Sqrt[-a]*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*b^2) + Sin[c + d*x]/(2*b^2) - (x^2*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) + (Sqrt[-a]*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2))} -{(x^2*Sin[c + d*x])/(a + b*x^2)^2, x, 17, (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) + (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) - (x*Sin[c + d*x])/(2*b*(a + b*x^2)) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2)} -{(x*Sin[c + d*x])/(a + b*x^2)^2, x, 9, (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2)) - Sin[c + d*x]/(2*b*(a + b*x^2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))} -{Sin[c + d*x]/(a + b*x^2)^2, x, 18, -(d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) - Sin[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b)} -{Sin[c + d*x]/(x*(a + b*x^2)^2), x, 22, (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + Sin[c + d*x]/(2*a*(a + b*x^2)) + (Cos[c]*SinIntegral[d*x])/a^2 + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b])} -{Sin[c + d*x]/(x^2*(a + b*x^2)^2), x, 32, (d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a^2) + (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(5/2)) - (3*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(5/2)) - Sin[c + d*x]/(a^2*x) + (Sqrt[b]*Sin[c + d*x])/(4*a^2*(Sqrt[-a] - Sqrt[b]*x)) - (Sqrt[b]*Sin[c + d*x])/(4*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (d*Sin[c]*SinIntegral[d*x])/a^2 + (3*Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(5/2)) + (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a^2) + (3*Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2)) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2)} - - -{(x^3*Sin[c + d*x])/(a + b*x^2)^3, x, 27, -(d*x*Cos[c + d*x])/(8*b^2*(a + b*x^2)) + (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (3*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*b^3) - (x^2*Sin[c + d*x])/(4*b*(a + b*x^2)^2) - Sin[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) + (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) + (3*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2))} -{(x^2*Sin[c + d*x])/(a + b*x^2)^3, x, 28, -(d*Cos[c + d*x])/(8*b^2*(a + b*x^2)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (x*Sin[c + d*x])/(4*b*(a + b*x^2)^2) + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2)) + (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2)} -{(x*Sin[c + d*x])/(a + b*x^2)^3, x, 19, -(d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - Sin[c + d*x]/(4*b*(a + b*x^2)^2) - (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) - (d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2))} -{Sin[c + d*x]/(a + b*x^2)^3, x, 28, (d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b]*x)) - (3*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) + (3*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Sin[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Sin[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (3*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) + (3*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)} -{Sin[c + d*x]/(x*(a + b*x^2)^3), x, 41, (d*Cos[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) - (d*Cos[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (5*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (5*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (CosIntegral[d*x]*Sin[c])/a^3 - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^3) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) - (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^3) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) + Sin[c + d*x]/(4*a*(a + b*x^2)^2) + Sin[c + d*x]/(2*a^2*(a + b*x^2)) + (Cos[c]*SinIntegral[d*x])/a^3 + (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (5*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^3) - (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (5*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b])} -{Sin[c + d*x]/(x^2*(a + b*x^2)^3), x, 60, (d*Cos[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Cos[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)) + (d*Cos[c]*CosIntegral[d*x])/a^3 + (7*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (7*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (15*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (15*Sqrt[b]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - Sin[c + d*x]/(a^3*x) - (Sqrt[b]*Sin[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)^2) + (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*Sin[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)^2) - (7*Sqrt[b]*Sin[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (d*Sin[c]*SinIntegral[d*x])/a^3 - (15*Sqrt[b]*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (7*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) - (15*Sqrt[b]*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (7*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3)} -{Sin[c + d*x]/(x^3*(a + b*x^2)^3), x, 46, -(d*Cos[c + d*x])/(2*a^3*x) - (Sqrt[b]*d*Cos[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Cos[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (9*Sqrt[b]*d*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (9*Sqrt[b]*d*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) - (3*b*CosIntegral[d*x]*Sin[c])/a^4 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^3) + (3*b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (3*b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^4) + (d^2*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) - Sin[c + d*x]/(2*a^3*x^2) - (b*Sin[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Sin[c + d*x])/(a^3*(a + b*x^2)) - (3*b*Cos[c]*SinIntegral[d*x])/a^4 - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^3) - (3*b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (3*b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^4) + (d^2*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (9*Sqrt[b]*d*Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^3)^p Sin[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x^3)*Sin[c + d*x], x, 13, (720*b*Cos[c + d*x])/d^7 + (6*a*x*Cos[c + d*x])/d^3 - (360*b*x^2*Cos[c + d*x])/d^5 - (a*x^3*Cos[c + d*x])/d + (30*b*x^4*Cos[c + d*x])/d^3 - (b*x^6*Cos[c + d*x])/d - (6*a*Sin[c + d*x])/d^4 + (720*b*x*Sin[c + d*x])/d^6 + (3*a*x^2*Sin[c + d*x])/d^2 - (120*b*x^3*Sin[c + d*x])/d^4 + (6*b*x^5*Sin[c + d*x])/d^2} -{x^2*(a + b*x^3)*Sin[c + d*x], x, 11, (2*a*Cos[c + d*x])/d^3 - (120*b*x*Cos[c + d*x])/d^5 - (a*x^2*Cos[c + d*x])/d + (20*b*x^3*Cos[c + d*x])/d^3 - (b*x^5*Cos[c + d*x])/d + (120*b*Sin[c + d*x])/d^6 + (2*a*x*Sin[c + d*x])/d^2 - (60*b*x^2*Sin[c + d*x])/d^4 + (5*b*x^4*Sin[c + d*x])/d^2} -{x*(a + b*x^3)*Sin[c + d*x], x, 9, (-24*b*Cos[c + d*x])/d^5 - (a*x*Cos[c + d*x])/d + (12*b*x^2*Cos[c + d*x])/d^3 - (b*x^4*Cos[c + d*x])/d + (a*Sin[c + d*x])/d^2 - (24*b*x*Sin[c + d*x])/d^4 + (4*b*x^3*Sin[c + d*x])/d^2} -{(a + b*x^3)*Sin[c + d*x], x, 7, -((a*Cos[c + d*x])/d) + (6*b*x*Cos[c + d*x])/d^3 - (b*x^3*Cos[c + d*x])/d - (6*b*Sin[c + d*x])/d^4 + (3*b*x^2*Sin[c + d*x])/d^2} -{((a + b*x^3)*Sin[c + d*x])/x, x, 8, (2*b*Cos[c + d*x])/d^3 - (b*x^2*Cos[c + d*x])/d + a*CosIntegral[d*x]*Sin[c] + (2*b*x*Sin[c + d*x])/d^2 + a*Cos[c]*SinIntegral[d*x]} -{((a + b*x^3)*Sin[c + d*x])/x^2, x, 8, -((b*x*Cos[c + d*x])/d) + a*d*Cos[c]*CosIntegral[d*x] + (b*Sin[c + d*x])/d^2 - (a*Sin[c + d*x])/x - a*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x^3)*Sin[c + d*x])/x^3, x, 8, -((b*Cos[c + d*x])/d) - (a*d*Cos[c + d*x])/(2*x) - (a*d^2*CosIntegral[d*x]*Sin[c])/2 - (a*Sin[c + d*x])/(2*x^2) - (a*d^2*Cos[c]*SinIntegral[d*x])/2} -{((a + b*x^3)*Sin[c + d*x])/x^4, x, 11, -(a*d*Cos[c + d*x])/(6*x^2) - (a*d^3*Cos[c]*CosIntegral[d*x])/6 + b*CosIntegral[d*x]*Sin[c] - (a*Sin[c + d*x])/(3*x^3) + (a*d^2*Sin[c + d*x])/(6*x) + b*Cos[c]*SinIntegral[d*x] + (a*d^3*Sin[c]*SinIntegral[d*x])/6} - - -{x*(a + b*x^3)^2*Sin[c + d*x], x, 17, (-48*a*b*Cos[c + d*x])/d^5 + (5040*b^2*x*Cos[c + d*x])/d^7 - (a^2*x*Cos[c + d*x])/d + (24*a*b*x^2*Cos[c + d*x])/d^3 - (840*b^2*x^3*Cos[c + d*x])/d^5 - (2*a*b*x^4*Cos[c + d*x])/d + (42*b^2*x^5*Cos[c + d*x])/d^3 - (b^2*x^7*Cos[c + d*x])/d - (5040*b^2*Sin[c + d*x])/d^8 + (a^2*Sin[c + d*x])/d^2 - (48*a*b*x*Sin[c + d*x])/d^4 + (2520*b^2*x^2*Sin[c + d*x])/d^6 + (8*a*b*x^3*Sin[c + d*x])/d^2 - (210*b^2*x^4*Sin[c + d*x])/d^4 + (7*b^2*x^6*Sin[c + d*x])/d^2} -{(a + b*x^3)^2*Sin[c + d*x], x, 14, (720*b^2*Cos[c + d*x])/d^7 - (a^2*Cos[c + d*x])/d + (12*a*b*x*Cos[c + d*x])/d^3 - (360*b^2*x^2*Cos[c + d*x])/d^5 - (2*a*b*x^3*Cos[c + d*x])/d + (30*b^2*x^4*Cos[c + d*x])/d^3 - (b^2*x^6*Cos[c + d*x])/d - (12*a*b*Sin[c + d*x])/d^4 + (720*b^2*x*Sin[c + d*x])/d^6 + (6*a*b*x^2*Sin[c + d*x])/d^2 - (120*b^2*x^3*Sin[c + d*x])/d^4 + (6*b^2*x^5*Sin[c + d*x])/d^2} -{((a + b*x^3)^2*Sin[c + d*x])/x, x, 14, (4*a*b*Cos[c + d*x])/d^3 - (120*b^2*x*Cos[c + d*x])/d^5 - (2*a*b*x^2*Cos[c + d*x])/d + (20*b^2*x^3*Cos[c + d*x])/d^3 - (b^2*x^5*Cos[c + d*x])/d + a^2*CosIntegral[d*x]*Sin[c] + (120*b^2*Sin[c + d*x])/d^6 + (4*a*b*x*Sin[c + d*x])/d^2 - (60*b^2*x^2*Sin[c + d*x])/d^4 + (5*b^2*x^4*Sin[c + d*x])/d^2 + a^2*Cos[c]*SinIntegral[d*x]} -{((a + b*x^3)^2*Sin[c + d*x])/x^2, x, 13, (-24*b^2*Cos[c + d*x])/d^5 - (2*a*b*x*Cos[c + d*x])/d + (12*b^2*x^2*Cos[c + d*x])/d^3 - (b^2*x^4*Cos[c + d*x])/d + a^2*d*Cos[c]*CosIntegral[d*x] + (2*a*b*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/x - (24*b^2*x*Sin[c + d*x])/d^4 + (4*b^2*x^3*Sin[c + d*x])/d^2 - a^2*d*Sin[c]*SinIntegral[d*x]} -{((a + b*x^3)^2*Sin[c + d*x])/x^3, x, 12, (-2*a*b*Cos[c + d*x])/d - (a^2*d*Cos[c + d*x])/(2*x) + (6*b^2*x*Cos[c + d*x])/d^3 - (b^2*x^3*Cos[c + d*x])/d - (a^2*d^2*CosIntegral[d*x]*Sin[c])/2 - (6*b^2*Sin[c + d*x])/d^4 - (a^2*Sin[c + d*x])/(2*x^2) + (3*b^2*x^2*Sin[c + d*x])/d^2 - (a^2*d^2*Cos[c]*SinIntegral[d*x])/2} -{((a + b*x^3)^2*Sin[c + d*x])/x^4, x, 14, (2*b^2*Cos[c + d*x])/d^3 - (a^2*d*Cos[c + d*x])/(6*x^2) - (b^2*x^2*Cos[c + d*x])/d - (a^2*d^3*Cos[c]*CosIntegral[d*x])/6 + 2*a*b*CosIntegral[d*x]*Sin[c] - (a^2*Sin[c + d*x])/(3*x^3) + (a^2*d^2*Sin[c + d*x])/(6*x) + (2*b^2*x*Sin[c + d*x])/d^2 + 2*a*b*Cos[c]*SinIntegral[d*x] + (a^2*d^3*Sin[c]*SinIntegral[d*x])/6} -{((a + b*x^3)^2*Sin[c + d*x])/x^5, x, 15, -(a^2*d*Cos[c + d*x])/(12*x^3) + (a^2*d^3*Cos[c + d*x])/(24*x) - (b^2*x*Cos[c + d*x])/d + 2*a*b*d*Cos[c]*CosIntegral[d*x] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 + (b^2*Sin[c + d*x])/d^2 - (a^2*Sin[c + d*x])/(4*x^4) + (a^2*d^2*Sin[c + d*x])/(24*x^2) - (2*a*b*Sin[c + d*x])/x + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24 - 2*a*b*d*Sin[c]*SinIntegral[d*x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Sin[c + d*x])/(a + b*x^3), x, 15, -((x*Cos[c + d*x])/(b*d)) + (a^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*b^(5/3)) + ((-1)^(2/3)*a^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*b^(5/3)) + Sin[c + d*x]/(b*d^2) - ((-1)^(2/3)*a^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(5/3)) + (a^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(5/3))} -{(x^3*Sin[c + d*x])/(a + b*x^3), x, 14, -(Cos[c + d*x]/(b*d)) - (a^(1/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*b^(4/3)) + ((-1)^(1/3)*a^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*b^(4/3)) - ((-1)^(1/3)*a^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - (a^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3))} -{(x^2*Sin[c + d*x])/(a + b*x^3), x, 11, (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*b) + (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*b) + (CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*b) - (Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b) + (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b)} -{(x*Sin[c + d*x])/(a + b*x^3), x, 11, -((CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(1/3)*b^(2/3))) - ((-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(1/3)*b^(2/3)) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3))} -{Sin[c + d*x]/(a + b*x^3), x, 11, (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(2/3)*b^(1/3)) - ((-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(2/3)*b^(1/3)) + ((-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(2/3)*b^(1/3)) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3))} -{Sin[c + d*x]/(x*(a + b*x^3)), x, 16, (CosIntegral[d*x]*Sin[c])/a - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a) - (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a) - (CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a) + (Cos[c]*SinIntegral[d*x])/a + (Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a)} -{Sin[c + d*x]/(x^2*(a + b*x^3)), x, 17, (d*Cos[c]*CosIntegral[d*x])/a + (b^(1/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(4/3)) + ((-1)^(2/3)*b^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(4/3)) - Sin[c + d*x]/(a*x) - (d*Sin[c]*SinIntegral[d*x])/a - ((-1)^(2/3)*b^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(4/3)) + (b^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3))} -{Sin[c + d*x]/(x^3*(a + b*x^3)), x, 18, -((d*Cos[c + d*x])/(2*a*x)) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) - Sin[c + d*x]/(2*a*x^2) - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) - ((-1)^(1/3)*b^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3))} - - -{(x^3*Sin[c + d*x])/(a + b*x^3)^2, x, 23, -(((-1)^(2/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(1/3)*b^(5/3))) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) - (x*Sin[c + d*x])/(3*b*(a + b*x^3)) + ((-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(2/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(1/3)*b^(5/3)) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3))} -{(x^2*Sin[c + d*x])/(a + b*x^3)^2, x, 12, -(((-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3))) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - Sin[c + d*x]/(3*b*(a + b*x^3)) - ((-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3))} -{(x*Sin[c + d*x])/(a + b*x^3)^2, x, 34, -((d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a*b)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b) - (d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b) - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) - ((-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) + ((-1)^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) + Sin[c + d*x]/(3*a*b*x) - Sin[c + d*x]/(3*b*x*(a + b*x^3)) + ((-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) - (d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a*b) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b) + ((-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b)} -{Sin[c + d*x]/(a + b*x^3)^2, x, 36, ((-1)^(2/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) - ((-1)^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) - (2*(-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + (2*(-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + Sin[c + d*x]/(3*a*b*x^2) - Sin[c + d*x]/(3*b*x^2*(a + b*x^3)) + (2*(-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) + ((-1)^(2/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) + (2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (2*(-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) + ((-1)^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3))} -{Sin[c + d*x]/(x*(a + b*x^3)^2), x, 41, ((-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - ((-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^2) - (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^2) - (CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^2) + Sin[c + d*x]/(3*a*b*x^3) - Sin[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^2 + (Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) + ((-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3))} -{Sin[c + d*x]/(x^2*(a + b*x^3)^2), x, 47, (d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^2) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2) + (d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2) + (4*b^(1/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(7/3)) + (4*(-1)^(2/3)*b^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(7/3)) - (4*(-1)^(1/3)*b^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(7/3)) + Sin[c + d*x]/(3*a*b*x^4) - (4*Sin[c + d*x])/(3*a^2*x) - Sin[c + d*x]/(3*b*x^4*(a + b*x^3)) - (d*Sin[c]*SinIntegral[d*x])/a^2 - (4*(-1)^(2/3)*b^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) + (d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^2) + (4*b^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2) - (4*(-1)^(1/3)*b^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2)} -{Sin[c + d*x]/(x^3*(a + b*x^3)^2), x, 51, -((d*Cos[c + d*x])/(2*a^2*x)) - ((-1)^(2/3)*b^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) - (b^(1/3)*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) + ((-1)^(1/3)*b^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^2) - (5*b^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(8/3)) + (5*(-1)^(1/3)*b^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(8/3)) - (5*(-1)^(2/3)*b^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(8/3)) + Sin[c + d*x]/(3*a*b*x^5) - (5*Sin[c + d*x])/(6*a^2*x^2) - Sin[c + d*x]/(3*b*x^5*(a + b*x^3)) - (d^2*Cos[c]*SinIntegral[d*x])/(2*a^2) - (5*(-1)^(1/3)*b^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(8/3)) - ((-1)^(2/3)*b^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) - (5*b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) + (b^(1/3)*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (5*(-1)^(2/3)*b^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) - ((-1)^(1/3)*b^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3))} - - -{(x^3*Sin[c + d*x])/(a + b*x^3)^3, x, 71, (d*Cos[c + d*x])/(18*a*b^2*x) - (d*Cos[c + d*x])/(18*b^2*x*(a + b*x^3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a*b^2) - ((-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a*b^2) + ((-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d^2*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a*b^2) + Sin[c + d*x]/(18*a*b^2*x^2) - (x*Sin[c + d*x])/(6*b*(a + b*x^3)^2) - Sin[c + d*x]/(18*b^2*x^2*(a + b*x^3)) + ((-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a*b^2) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + ((-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2)} -{(x^2*Sin[c + d*x])/(a + b*x^3)^3, x, 37, (d*Cos[c + d*x])/(18*a*b^2*x^2) - (d*Cos[c + d*x])/(18*b^2*x^2*(a + b*x^3)) - ((-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + ((-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^(4/3)*b^(5/3)) - ((-1)^(2/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(4/3)*b^(5/3)) + ((-1)^(1/3)*d^2*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(4/3)*b^(5/3)) - Sin[c + d*x]/(6*b*(a + b*x^3)^2) + ((-1)^(2/3)*d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(4/3)*b^(5/3)) - ((-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(4/3)*b^(5/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + ((-1)^(1/3)*d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(4/3)*b^(5/3)) - ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3))} -{(x*Sin[c + d*x])/(a + b*x^3)^3, x, 89, (d*Cos[c + d*x])/(18*a*b^2*x^3) - (d*Cos[c + d*x])/(18*b^2*x^3*(a + b*x^3)) - (2*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) - (2*(-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) + (2*(-1)^(1/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*d^2*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) - Sin[c + d*x]/(18*a*b^2*x^4) + (2*Sin[c + d*x])/(9*a^2*b*x) - Sin[c + d*x]/(6*b*x*(a + b*x^3)^2) + Sin[c + d*x]/(18*b^2*x^4*(a + b*x^3)) + (2*(-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(1/3)*d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(5/3)*b^(4/3)) - (2*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) + (2*(-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b)} -{Sin[c + d*x]/(a + b*x^3)^3, x, 99, (d*Cos[c + d*x])/(18*a*b^2*x^4) - (d*Cos[c + d*x])/(18*a^2*b*x) - (d*Cos[c + d*x])/(18*b^2*x^4*(a + b*x^3)) + ((-1)^(2/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)*b^(2/3)) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)*b^(2/3)) + (5*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(8/3)*b^(1/3)) - (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^2*b) - (5*(-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(8/3)*b^(1/3)) - (d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^2*b) + (5*(-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(8/3)*b^(1/3)) - (d^2*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a^2*b) - Sin[c + d*x]/(9*a*b^2*x^5) + (5*Sin[c + d*x])/(18*a^2*b*x^2) - Sin[c + d*x]/(6*b*x^2*(a + b*x^3)^2) + Sin[c + d*x]/(9*b^2*x^5*(a + b*x^3)) + (5*(-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(8/3)*b^(1/3)) + (d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^2*b) + ((-1)^(2/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)*b^(2/3)) + (5*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) - (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^2*b) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)*b^(2/3)) + (5*(-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) - (d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^2*b) + ((-1)^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)*b^(2/3))} -{Sin[c + d*x]/(x*(a + b*x^3)^3), x, 110, (d*Cos[c + d*x])/(18*a*b^2*x^5) - (d*Cos[c + d*x])/(18*a^2*b*x^2) - (d*Cos[c + d*x])/(18*b^2*x^5*(a + b*x^3)) + (4*(-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(8/3)*b^(1/3)) - (4*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) - (4*(-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^3 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^3) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^(7/3)*b^(2/3)) - (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^3) + ((-1)^(2/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(7/3)*b^(2/3)) - (CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^3) - ((-1)^(1/3)*d^2*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(7/3)*b^(2/3)) - Sin[c + d*x]/(6*a*b^2*x^6) + Sin[c + d*x]/(3*a^2*b*x^3) - Sin[c + d*x]/(6*b*x^3*(a + b*x^3)^2) + Sin[c + d*x]/(6*b^2*x^6*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^3 + (Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^3) - ((-1)^(2/3)*d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(7/3)*b^(2/3)) + (4*(-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(8/3)*b^(1/3)) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^3) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(7/3)*b^(2/3)) + (4*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^3) - ((-1)^(1/3)*d^2*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(7/3)*b^(2/3)) + (4*(-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m deleted file mode 100644 index 81c2ce5..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.m +++ /dev/null @@ -1,739 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sin[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Sin[c + d*x^2]), x, 6, (a*x^6)/6 + (b*Cos[c + d*x^2])/d^3 - (b*x^4*Cos[c + d*x^2])/(2*d) + (b*x^2*Sin[c + d*x^2])/d^2} -{x^3*(a + b*Sin[c + d*x^2]), x, 5, (a*x^4)/4 - (b*x^2*Cos[c + d*x^2])/(2*d) + (b*Sin[c + d*x^2])/(2*d^2)} -{x^1*(a + b*Sin[c + d*x^2]), x, 4, (a*x^2)/2 - (b*Cos[c + d*x^2])/(2*d)} -{(a + b*Sin[c + d*x^2])/x^1, x, 5, a*Log[x] + (1/2)*b*CosIntegral[d*x^2]*Sin[c] + (1/2)*b*Cos[c]*SinIntegral[d*x^2]} -{(a + b*Sin[c + d*x^2])/x^3, x, 7, -(a/(2*x^2)) + (1/2)*b*d*Cos[c]*CosIntegral[d*x^2] - (b*Sin[c + d*x^2])/(2*x^2) - (1/2)*b*d*Sin[c]*SinIntegral[d*x^2]} -{(a + b*Sin[c + d*x^2])/x^5, x, 8, -(a/(4*x^4)) - (b*d*Cos[c + d*x^2])/(4*x^2) - (1/4)*b*d^2*CosIntegral[d*x^2]*Sin[c] - (b*Sin[c + d*x^2])/(4*x^4) - (1/4)*b*d^2*Cos[c]*SinIntegral[d*x^2]} - -{x^4*(a + b*Sin[c + d*x^2]), x, 7, (a*x^5)/5 - (b*x^3*Cos[c + d*x^2])/(2*d) - (3*b*Sqrt[Pi/2]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/(4*d^(5/2)) - (3*b*Sqrt[Pi/2]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/(4*d^(5/2)) + (3*b*x*Sin[c + d*x^2])/(4*d^2)} -{x^2*(a + b*Sin[c + d*x^2]), x, 6, (a*x^3)/3 - (b*x*Cos[c + d*x^2])/(2*d) + (b*Sqrt[Pi/2]*Cos[c]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x])/(2*d^(3/2)) - (b*Sqrt[Pi/2]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/(2*d^(3/2))} -{x^0*(a + b*Sin[c + d*x^2]), x, 4, a*x + (b*Sqrt[Pi/2]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/Sqrt[d] + (b*Sqrt[Pi/2]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/Sqrt[d]} -{(a + b*Sin[c + d*x^2])/x^2, x, 6, -(a/x) + b*Sqrt[d]*Sqrt[2*Pi]*Cos[c]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x] - b*Sqrt[d]*Sqrt[2*Pi]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c] - (b*Sin[c + d*x^2])/x} -{(a + b*Sin[c + d*x^2])/x^4, x, 7, -(a/(3*x^3)) - (2*b*d*Cos[c + d*x^2])/(3*x) - (2/3)*b*d^(3/2)*Sqrt[2*Pi]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x] - (2/3)*b*d^(3/2)*Sqrt[2*Pi]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c] - (b*Sin[c + d*x^2])/(3*x^3)} - - -{x^5*(a + b*Sin[c + d*x^2])^2, x, 10, -((b^2*x^2)/(8*d^2)) + (a^2*x^6)/6 + (b^2*x^6)/12 + (2*a*b*Cos[c + d*x^2])/d^3 - (a*b*x^4*Cos[c + d*x^2])/d + (2*a*b*x^2*Sin[c + d*x^2])/d^2 + (b^2*Cos[c + d*x^2]*Sin[c + d*x^2])/(8*d^3) - (b^2*x^4*Cos[c + d*x^2]*Sin[c + d*x^2])/(4*d) + (b^2*x^2*Sin[c + d*x^2]^2)/(4*d^2)} -{x^3*(a + b*Sin[c + d*x^2])^2, x, 7, (a^2*x^4)/4 + (b^2*x^4)/8 - (a*b*x^2*Cos[c + d*x^2])/d + (a*b*Sin[c + d*x^2])/d^2 - (b^2*x^2*Cos[c + d*x^2]*Sin[c + d*x^2])/(4*d) + (b^2*Sin[c + d*x^2]^2)/(8*d^2)} -{x^1*(a + b*Sin[c + d*x^2])^2, x, 2, (1/4)*(2*a^2 + b^2)*x^2 - (a*b*Cos[c + d*x^2])/d - (b^2*Cos[c + d*x^2]*Sin[c + d*x^2])/(4*d)} -{(a + b*Sin[c + d*x^2])^2/x^1, x, 9, (-(1/4))*b^2*Cos[2*c]*CosIntegral[2*d*x^2] + (1/2)*(2*a^2 + b^2)*Log[x] + a*b*CosIntegral[d*x^2]*Sin[c] + a*b*Cos[c]*SinIntegral[d*x^2] + (1/4)*b^2*Sin[2*c]*SinIntegral[2*d*x^2]} -{(a + b*Sin[c + d*x^2])^2/x^3, x, 13, -((2*a^2 + b^2)/(4*x^2)) + (b^2*Cos[2*(c + d*x^2)])/(4*x^2) + a*b*d*Cos[c]*CosIntegral[d*x^2] + (1/2)*b^2*d*CosIntegral[2*d*x^2]*Sin[2*c] - (a*b*Sin[c + d*x^2])/x^2 - a*b*d*Sin[c]*SinIntegral[d*x^2] + (1/2)*b^2*d*Cos[2*c]*SinIntegral[2*d*x^2]} -{(a + b*Sin[c + d*x^2])^2/x^5, x, 15, -((2*a^2 + b^2)/(8*x^4)) - (a*b*d*Cos[c + d*x^2])/(2*x^2) + (b^2*Cos[2*(c + d*x^2)])/(8*x^4) + (1/2)*b^2*d^2*Cos[2*c]*CosIntegral[2*d*x^2] - (1/2)*a*b*d^2*CosIntegral[d*x^2]*Sin[c] - (a*b*Sin[c + d*x^2])/(2*x^4) - (b^2*d*Sin[2*(c + d*x^2)])/(4*x^2) - (1/2)*a*b*d^2*Cos[c]*SinIntegral[d*x^2] - (1/2)*b^2*d^2*Sin[2*c]*SinIntegral[2*d*x^2]} - -{x^4*(a + b*Sin[c + d*x^2])^2, x, 13, (1/10)*(2*a^2 + b^2)*x^5 - (a*b*x^3*Cos[c + d*x^2])/d - (3*b^2*x*Cos[2*c + 2*d*x^2])/(32*d^2) + (3*b^2*Sqrt[Pi]*Cos[2*c]*FresnelC[(2*Sqrt[d]*x)/Sqrt[Pi]])/(64*d^(5/2)) - (3*a*b*Sqrt[Pi/2]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/(2*d^(5/2)) - (3*a*b*Sqrt[Pi/2]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/(2*d^(5/2)) - (3*b^2*Sqrt[Pi]*FresnelS[(2*Sqrt[d]*x)/Sqrt[Pi]]*Sin[2*c])/(64*d^(5/2)) + (3*a*b*x*Sin[c + d*x^2])/(2*d^2) - (b^2*x^3*Sin[2*c + 2*d*x^2])/(8*d)} -{x^2*(a + b*Sin[c + d*x^2])^2, x, 11, (1/6)*(2*a^2 + b^2)*x^3 - (a*b*x*Cos[c + d*x^2])/d + (a*b*Sqrt[Pi/2]*Cos[c]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x])/d^(3/2) + (b^2*Sqrt[Pi]*Cos[2*c]*FresnelS[(2*Sqrt[d]*x)/Sqrt[Pi]])/(16*d^(3/2)) - (a*b*Sqrt[Pi/2]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/d^(3/2) + (b^2*Sqrt[Pi]*FresnelC[(2*Sqrt[d]*x)/Sqrt[Pi]]*Sin[2*c])/(16*d^(3/2)) - (b^2*x*Sin[2*c + 2*d*x^2])/(8*d)} -{x^0*(a + b*Sin[c + d*x^2])^2, x, 8, (1/2)*(2*a^2 + b^2)*x - (b^2*Sqrt[Pi]*Cos[2*c]*FresnelC[(2*Sqrt[d]*x)/Sqrt[Pi]])/(4*Sqrt[d]) + (a*b*Sqrt[2*Pi]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x])/Sqrt[d] + (a*b*Sqrt[2*Pi]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c])/Sqrt[d] + (b^2*Sqrt[Pi]*FresnelS[(2*Sqrt[d]*x)/Sqrt[Pi]]*Sin[2*c])/(4*Sqrt[d])} -{(a + b*Sin[c + d*x^2])^2/x^2, x, 11, -((2*a^2 + b^2)/(2*x)) + (b^2*Cos[2*c + 2*d*x^2])/(2*x) + 2*a*b*Sqrt[d]*Sqrt[2*Pi]*Cos[c]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x] + b^2*Sqrt[d]*Sqrt[Pi]*Cos[2*c]*FresnelS[(2*Sqrt[d]*x)/Sqrt[Pi]] - 2*a*b*Sqrt[d]*Sqrt[2*Pi]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c] + b^2*Sqrt[d]*Sqrt[Pi]*FresnelC[(2*Sqrt[d]*x)/Sqrt[Pi]]*Sin[2*c] - (2*a*b*Sin[c + d*x^2])/x} -{(a + b*Sin[c + d*x^2])^2/x^4, x, 13, -((2*a^2 + b^2)/(6*x^3)) - (4*a*b*d*Cos[c + d*x^2])/(3*x) + (b^2*Cos[2*c + 2*d*x^2])/(6*x^3) + (4/3)*b^2*d^(3/2)*Sqrt[Pi]*Cos[2*c]*FresnelC[(2*Sqrt[d]*x)/Sqrt[Pi]] - (4/3)*a*b*d^(3/2)*Sqrt[2*Pi]*Cos[c]*FresnelS[Sqrt[d]*Sqrt[2/Pi]*x] - (4/3)*a*b*d^(3/2)*Sqrt[2*Pi]*FresnelC[Sqrt[d]*Sqrt[2/Pi]*x]*Sin[c] - (4/3)*b^2*d^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[d]*x)/Sqrt[Pi]]*Sin[2*c] - (2*a*b*Sin[c + d*x^2])/(3*x^3) - (2*b^2*d*Sin[2*c + 2*d*x^2])/(3*x)} - - -{x^5*Sin[a + b*x^2]^3, x, 7, (7*Cos[a + b*x^2])/(9*b^3) - (x^4*Cos[a + b*x^2])/(3*b) - Cos[a + b*x^2]^3/(27*b^3) + (2*x^2*Sin[a + b*x^2])/(3*b^2) - (x^4*Cos[a + b*x^2]*Sin[a + b*x^2]^2)/(6*b) + (x^2*Sin[a + b*x^2]^3)/(9*b^2)} -{x^3*Sin[a + b*x^2]^3, x, 4, -((x^2*Cos[a + b*x^2])/(3*b)) + Sin[a + b*x^2]/(3*b^2) - (x^2*Cos[a + b*x^2]*Sin[a + b*x^2]^2)/(6*b) + Sin[a + b*x^2]^3/(18*b^2)} -{x^1*Sin[a + b*x^2]^3, x, 3, -(Cos[a + b*x^2]/(2*b)) + Cos[a + b*x^2]^3/(6*b)} -{Sin[a + b*x^2]^3/x^1, x, 8, (3/8)*CosIntegral[b*x^2]*Sin[a] - (1/8)*CosIntegral[3*b*x^2]*Sin[3*a] + (3/8)*Cos[a]*SinIntegral[b*x^2] - (1/8)*Cos[3*a]*SinIntegral[3*b*x^2]} -{Sin[a + b*x^2]^3/x^3, x, 12, (3/8)*b*Cos[a]*CosIntegral[b*x^2] - (3/8)*b*Cos[3*a]*CosIntegral[3*b*x^2] - (3*Sin[a + b*x^2])/(8*x^2) + Sin[3*(a + b*x^2)]/(8*x^2) - (3/8)*b*Sin[a]*SinIntegral[b*x^2] + (3/8)*b*Sin[3*a]*SinIntegral[3*b*x^2]} - -{x^2*Sin[a + b*x^2]^3, x, 10, -((3*x*Cos[a + b*x^2])/(8*b)) + (x*Cos[3*a + 3*b*x^2])/(24*b) + (3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/(8*b^(3/2)) - (Sqrt[Pi/6]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x])/(24*b^(3/2)) - (3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(8*b^(3/2)) + (Sqrt[Pi/6]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(24*b^(3/2))} -{x^0*Sin[a + b*x^2]^3, x, 8, (3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/(4*Sqrt[b]) - (Sqrt[Pi/6]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x])/(4*Sqrt[b]) + (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(4*Sqrt[b]) - (Sqrt[Pi/6]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(4*Sqrt[b])} -{Sin[a + b*x^2]^3/x^2, x, 9, (3/2)*Sqrt[b]*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x] - (1/2)*Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x] - (3/2)*Sqrt[b]*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a] + (1/2)*Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a] - Sin[a + b*x^2]^3/x} - - -{x^2*Sin[x^2]^3, x, 6, (-(1/2))*x*Cos[x^2] + (1/6)*x*Cos[x^2]^3 + (3/8)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x] - (1/24)*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*x], (-(3/8))*x*Cos[x^2] + (1/24)*x*Cos[3*x^2] + (3/8)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x] - (1/24)*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*x]} -{x^4*Cos[x^2]*Sin[x^2]^2, x, 7, (1/4)*x*Cos[x^2] - (1/12)*x*Cos[x^2]^3 - (3/16)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x] + (1/48)*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*x] + (1/6)*x^3*Sin[x^2]^3, (3/16)*x*Cos[x^2] - (1/48)*x*Cos[3*x^2] - (3/16)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x] + (1/48)*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*x] + (1/6)*x^3*Sin[x^2]^3} - - -{x*Sin[a + b*x^2]^7, x, 3, -(Cos[a + b*x^2]/(2*b)) + Cos[a + b*x^2]^3/(2*b) - (3*Cos[a + b*x^2]^5)/(10*b) + Cos[a + b*x^2]^7/(14*b)} - - -{(1 + Sin[x^2])^2/x^3, x, 8, -(3/(4*x^2)) + Cos[2*x^2]/(4*x^2) + CosIntegral[x^2] - Sin[x^2]/x^2 + (1/2)*SinIntegral[2*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Sin[c + d*x^2]), x, 11, -((I*x^4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(2*Sqrt[a^2 - b^2]*d)) + (I*x^4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(2*Sqrt[a^2 - b^2]*d) - (x^2*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d^2) + (x^2*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d^2) - (I*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d^3) + (I*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d^3)} -{x^3/(a + b*Sin[c + d*x^2]), x, 9, -((I*x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(2*Sqrt[a^2 - b^2]*d)) + (I*x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(2*Sqrt[a^2 - b^2]*d) - PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])]/(2*Sqrt[a^2 - b^2]*d^2) + PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])]/(2*Sqrt[a^2 - b^2]*d^2)} -{x^1/(a + b*Sin[c + d*x^2]), x, 4, ArcTan[(b + a*Tan[(1/2)*(c + d*x^2)])/Sqrt[a^2 - b^2]]/(Sqrt[a^2 - b^2]*d)} -{1/(x^1*(a + b*Sin[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*x^2])), x]} -{1/(x^3*(a + b*Sin[c + d*x^2])), x, 0, Unintegrable[1/(x^3*(a + b*Sin[c + d*x^2])), x]} - -{x^2/(a + b*Sin[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Sin[c + d*x^2]), x]} -{x^0/(a + b*Sin[c + d*x^2]), x, 0, Unintegrable[1/(a + b*Sin[c + d*x^2]), x]} -{1/(x^2*(a + b*Sin[c + d*x^2])), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*x^2])), x]} - - -{x^5/(a + b*Sin[c + d*x^2])^2, x, 19, (I*x^4)/(2*(a^2 - b^2)*d) - (x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) - (I*a*x^4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d) - (x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^2) + (I*a*x^4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d) + (I*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) - (a*x^2*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) + (I*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*d^3) + (a*x^2*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^2) - (I*a*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) + (I*a*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d^3) + (b*x^4*Cos[c + d*x^2])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x^2]))} -{x^3/(a + b*Sin[c + d*x^2])^2, x, 12, -((I*a*x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d)) + (I*a*x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d) - Log[a + b*Sin[c + d*x^2]]/(2*(a^2 - b^2)*d^2) - (a*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d^2) + (a*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(2*(a^2 - b^2)^(3/2)*d^2) + (b*x^2*Cos[c + d*x^2])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x^2]))} -{x^1/(a + b*Sin[c + d*x^2])^2, x, 6, (a*ArcTan[(b + a*Tan[(1/2)*(c + d*x^2)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) + (b*Cos[c + d*x^2])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x^2]))} -{1/(x^1*(a + b*Sin[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*x^2])^2), x]} -{1/(x^3*(a + b*Sin[c + d*x^2])^2), x, 0, Unintegrable[1/(x^3*(a + b*Sin[c + d*x^2])^2), x]} - -{x^2/(a + b*Sin[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Sin[c + d*x^2])^2, x]} -{x^0/(a + b*Sin[c + d*x^2])^2, x, 0, Unintegrable[1/(a + b*Sin[c + d*x^2])^2, x]} -{1/(x^2*(a + b*Sin[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*x^2])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (e x)^(m/2) (a+b Sin[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^2])^p with m symbolic*) - - -{(e*x)^m*(a + b*Sin[c + d*x^2])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^2])^p, x]} - - -{(e*x)^m*(a + b*Sin[c + d*x^2])^3, x, 13, (a*(2*a^2 + 3*b^2)*(e*x)^(1 + m))/(2*e*(1 + m)) + (3*I*b*(4*a^2 + b^2)*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, (-I)*d*x^2])/(16*e) - (3*I*b*(4*a^2 + b^2)*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, I*d*x^2])/(E^(I*c)*(16*e)) + (3*2^(-(7/2) - m/2)*a*b^2*E^(2*I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, -2*I*d*x^2])/e + (3*2^(-(7/2) - m/2)*a*b^2*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, 2*I*d*x^2])/(E^(2*I*c)*e) - (I*3^(-(1/2) - m/2)*b^3*E^(3*I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, -3*I*d*x^2])/(16*e) + (I*3^(-(1/2) - m/2)*b^3*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, 3*I*d*x^2])/(E^(3*I*c)*(16*e))} -{(e*x)^m*(a + b*Sin[c + d*x^2])^2, x, 9, ((2*a^2 + b^2)*(e*x)^(1 + m))/(2*e*(1 + m)) + (I*a*b*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, (-I)*d*x^2])/(2*e) - (I*a*b*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, I*d*x^2])/(E^(I*c)*(2*e)) + (2^(-(7/2) - m/2)*b^2*E^(2*I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, -2*I*d*x^2])/e + (2^(-(7/2) - m/2)*b^2*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, 2*I*d*x^2])/(E^(2*I*c)*e)} -{(e*x)^m*(a + b*Sin[c + d*x^2])^1, x, 5, (a*(e*x)^(1 + m))/(e*(1 + m)) + (I*b*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, (-I)*d*x^2])/(4*e) - (I*b*(e*x)^(1 + m)*(I*d*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, I*d*x^2])/(E^(I*c)*(4*e))} -{(e*x)^m/(a + b*Sin[c + d*x^2])^1, x, 0, Unintegrable[(e*x)^m/(a + b*Sin[c + d*x^2]), x]} -{(e*x)^m/(a + b*Sin[c + d*x^2])^2, x, 0, Unintegrable[(e*x)^m/(a + b*Sin[c + d*x^2])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sin[c+d x^3])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Sin[c + d*x^3]), x, 5, (a*x^6)/6 - (b*x^3*Cos[c + d*x^3])/(3*d) + (b*Sin[c + d*x^3])/(3*d^2)} -{x^2*(a + b*Sin[c + d*x^3]), x, 4, (a*x^3)/3 - (b*Cos[c + d*x^3])/(3*d)} -{(a + b*Sin[c + d*x^3])/x, x, 5, a*Log[x] + (b*CosIntegral[d*x^3]*Sin[c])/3 + (b*Cos[c]*SinIntegral[d*x^3])/3} -{(a + b*Sin[c + d*x^3])/x^4, x, 7, -a/(3*x^3) + (b*d*Cos[c]*CosIntegral[d*x^3])/3 - (b*Sin[c + d*x^3])/(3*x^3) - (b*d*Sin[c]*SinIntegral[d*x^3])/3} - -{x^4*(a + b*Sin[c + d*x^3]), x, 6, (a*x^5)/5 - (b*x^2*Cos[c + d*x^3])/(3*d) - (b*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/(9*d*((-I)*d*x^3)^(2/3)) - (b*x^2*Gamma[2/3, I*d*x^3])/(9*d*E^(I*c)*(I*d*x^3)^(2/3))} -{x*(a + b*Sin[c + d*x^3]), x, 5, (a*x^2)/2 + ((I/6)*b*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^(2/3) - ((I/6)*b*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3))} -{(a + b*Sin[c + d*x^3])/x^2, x, 6, -(a/x) - (b*d*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/(2*((-I)*d*x^3)^(2/3)) - (b*d*x^2*Gamma[2/3, I*d*x^3])/(2*E^(I*c)*(I*d*x^3)^(2/3)) - (b*Sin[c + d*x^3])/x} -{(a + b*Sin[c + d*x^3])/x^5, x, 7, -a/(4*x^4) - (3*b*d*Cos[c + d*x^3])/(4*x) - (((3*I)/8)*b*d^2*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^(2/3) + (((3*I)/8)*b*d^2*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3)) - (b*Sin[c + d*x^3])/(4*x^4)} - -{x^3*(a + b*Sin[c + d*x^3]), x, 6, (a*x^4)/4 - (b*x*Cos[c + d*x^3])/(3*d) - (b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/(18*d*((-I)*d*x^3)^(1/3)) - (b*x*Gamma[1/3, I*d*x^3])/(18*d*E^(I*c)*(I*d*x^3)^(1/3))} -{a + b*Sin[c + d*x^3], x, 4, a*x + ((I/6)*b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) - ((I/6)*b*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3))} -{(a + b*Sin[c + d*x^3])/x^3, x, 6, -a/(2*x^2) - (b*d*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/(4*((-I)*d*x^3)^(1/3)) - (b*d*x*Gamma[1/3, I*d*x^3])/(4*E^(I*c)*(I*d*x^3)^(1/3)) - (b*Sin[c + d*x^3])/(2*x^2)} -{(a + b*Sin[c + d*x^3])/x^6, x, 7, -a/(5*x^5) - (3*b*d*Cos[c + d*x^3])/(10*x^2) - (((3*I)/20)*b*d^2*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) + (((3*I)/20)*b*d^2*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3)) - (b*Sin[c + d*x^3])/(5*x^5)} - - -{x^5*(a + b*Sin[c + d*x^3])^2, x, 7, (a^2*x^6)/6 + (b^2*x^6)/12 - (2*a*b*x^3*Cos[c + d*x^3])/(3*d) + (2*a*b*Sin[c + d*x^3])/(3*d^2) - (b^2*x^3*Cos[c + d*x^3]*Sin[c + d*x^3])/(6*d) + (b^2*Sin[c + d*x^3]^2)/(12*d^2)} -{x^2*(a + b*Sin[c + d*x^3])^2, x, 2, ((2*a^2 + b^2)*x^3)/6 - (2*a*b*Cos[c + d*x^3])/(3*d) - (b^2*Cos[c + d*x^3]*Sin[c + d*x^3])/(6*d)} -{(a + b*Sin[c + d*x^3])^2/x, x, 9, -(b^2*Cos[2*c]*CosIntegral[2*d*x^3])/6 + ((2*a^2 + b^2)*Log[x])/2 + (2*a*b*CosIntegral[d*x^3]*Sin[c])/3 + (2*a*b*Cos[c]*SinIntegral[d*x^3])/3 + (b^2*Sin[2*c]*SinIntegral[2*d*x^3])/6} -{(a + b*Sin[c + d*x^3])^2/x^4, x, 13, -((2*a^2 + b^2)/(6*x^3)) + (b^2*Cos[2*(c + d*x^3)])/(6*x^3) + (2/3)*a*b*d*Cos[c]*CosIntegral[d*x^3] + (1/3)*b^2*d*CosIntegral[2*d*x^3]*Sin[2*c] - (2*a*b*Sin[c + d*x^3])/(3*x^3) - (2/3)*a*b*d*Sin[c]*SinIntegral[d*x^3] + (1/3)*b^2*d*Cos[2*c]*SinIntegral[2*d*x^3]} - -{x^4*(a + b*Sin[c + d*x^3])^2, x, 11, ((2*a^2 + b^2)*x^5)/10 - (2*a*b*x^2*Cos[c + d*x^3])/(3*d) - (2*a*b*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/(9*d*((-I)*d*x^3)^(2/3)) - (2*a*b*x^2*Gamma[2/3, I*d*x^3])/(9*d*E^(I*c)*(I*d*x^3)^(2/3)) + ((I/36)*b^2*E^((2*I)*c)*x^2*Gamma[2/3, (-2*I)*d*x^3])/(2^(2/3)*d*((-I)*d*x^3)^(2/3)) - ((I/36)*b^2*x^2*Gamma[2/3, (2*I)*d*x^3])/(2^(2/3)*d*E^((2*I)*c)*(I*d*x^3)^(2/3)) - (b^2*x^2*Sin[2*c + 2*d*x^3])/(12*d)} -{x*(a + b*Sin[c + d*x^3])^2, x, 9, ((2*a^2 + b^2)*x^2)/4 + ((I/3)*a*b*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^(2/3) - ((I/3)*a*b*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3)) + (b^2*E^((2*I)*c)*x^2*Gamma[2/3, (-2*I)*d*x^3])/(12*2^(2/3)*((-I)*d*x^3)^(2/3)) + (b^2*x^2*Gamma[2/3, (2*I)*d*x^3])/(12*2^(2/3)*E^((2*I)*c)*(I*d*x^3)^(2/3))} -{(a + b*Sin[c + d*x^3])^2/x^2, x, 11, -(2*a^2 + b^2)/(2*x) + (b^2*Cos[2*c + 2*d*x^3])/(2*x) - (a*b*d*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^(2/3) - (a*b*d*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3)) + ((I/2)*b^2*d*E^((2*I)*c)*x^2*Gamma[2/3, (-2*I)*d*x^3])/(2^(2/3)*((-I)*d*x^3)^(2/3)) - ((I/2)*b^2*d*x^2*Gamma[2/3, (2*I)*d*x^3])/(2^(2/3)*E^((2*I)*c)*(I*d*x^3)^(2/3)) - (2*a*b*Sin[c + d*x^3])/x} -{(a + b*Sin[c + d*x^3])^2/x^5, x, 13, -(2*a^2 + b^2)/(8*x^4) - (3*a*b*d*Cos[c + d*x^3])/(2*x) + (b^2*Cos[2*c + 2*d*x^3])/(8*x^4) - (((3*I)/4)*a*b*d^2*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^(2/3) + (((3*I)/4)*a*b*d^2*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3)) - (3*b^2*d^2*E^((2*I)*c)*x^2*Gamma[2/3, (-2*I)*d*x^3])/(4*2^(2/3)*((-I)*d*x^3)^(2/3)) - (3*b^2*d^2*x^2*Gamma[2/3, (2*I)*d*x^3])/(4*2^(2/3)*E^((2*I)*c)*(I*d*x^3)^(2/3)) - (a*b*Sin[c + d*x^3])/(2*x^4) - (3*b^2*d*Sin[2*c + 2*d*x^3])/(4*x)} - -{x^3*(a + b*Sin[c + d*x^3])^2, x, 11, ((2*a^2 + b^2)*x^4)/8 - (2*a*b*x*Cos[c + d*x^3])/(3*d) - (a*b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/(9*d*((-I)*d*x^3)^(1/3)) - (a*b*x*Gamma[1/3, I*d*x^3])/(9*d*E^(I*c)*(I*d*x^3)^(1/3)) + ((I/72)*b^2*E^((2*I)*c)*x*Gamma[1/3, (-2*I)*d*x^3])/(2^(1/3)*d*((-I)*d*x^3)^(1/3)) - ((I/72)*b^2*x*Gamma[1/3, (2*I)*d*x^3])/(2^(1/3)*d*E^((2*I)*c)*(I*d*x^3)^(1/3)) - (b^2*x*Sin[2*c + 2*d*x^3])/(12*d)} -{(a + b*Sin[c + d*x^3])^2, x, 8, ((2*a^2 + b^2)*x)/2 + ((I/3)*a*b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) - ((I/3)*a*b*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3)) + (b^2*E^((2*I)*c)*x*Gamma[1/3, (-2*I)*d*x^3])/(12*2^(1/3)*((-I)*d*x^3)^(1/3)) + (b^2*x*Gamma[1/3, (2*I)*d*x^3])/(12*2^(1/3)*E^((2*I)*c)*(I*d*x^3)^(1/3))} -{(a + b*Sin[c + d*x^3])^2/x^3, x, 11, -(2*a^2 + b^2)/(4*x^2) + (b^2*Cos[2*c + 2*d*x^3])/(4*x^2) - (a*b*d*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/(2*((-I)*d*x^3)^(1/3)) - (a*b*d*x*Gamma[1/3, I*d*x^3])/(2*E^(I*c)*(I*d*x^3)^(1/3)) + ((I/4)*b^2*d*E^((2*I)*c)*x*Gamma[1/3, (-2*I)*d*x^3])/(2^(1/3)*((-I)*d*x^3)^(1/3)) - ((I/4)*b^2*d*x*Gamma[1/3, (2*I)*d*x^3])/(2^(1/3)*E^((2*I)*c)*(I*d*x^3)^(1/3)) - (a*b*Sin[c + d*x^3])/x^2} -{(a + b*Sin[c + d*x^3])^2/x^6, x, 13, -(2*a^2 + b^2)/(10*x^5) - (3*a*b*d*Cos[c + d*x^3])/(5*x^2) + (b^2*Cos[2*c + 2*d*x^3])/(10*x^5) - (((3*I)/10)*a*b*d^2*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) + (((3*I)/10)*a*b*d^2*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3)) - (3*b^2*d^2*E^((2*I)*c)*x*Gamma[1/3, (-2*I)*d*x^3])/(10*2^(1/3)*((-I)*d*x^3)^(1/3)) - (3*b^2*d^2*x*Gamma[1/3, (2*I)*d*x^3])/(10*2^(1/3)*E^((2*I)*c)*(I*d*x^3)^(1/3)) - (2*a*b*Sin[c + d*x^3])/(5*x^5) - (3*b^2*d*Sin[2*c + 2*d*x^3])/(10*x^2)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Sin[c + d*x^3]), x, 9, ((-I/3)*x^3*Log[1 - (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d) + ((I/3)*x^3*Log[1 - (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d) - PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])]/(3*Sqrt[a^2 - b^2]*d^2) + PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])]/(3*Sqrt[a^2 - b^2]*d^2)} -{x^2/(a + b*Sin[c + d*x^3]), x, 4, (2*ArcTan[(b + a*Tan[(c + d*x^3)/2])/Sqrt[a^2 - b^2]])/(3*Sqrt[a^2 - b^2]*d)} -{1/(x*(a + b*Sin[c + d*x^3])), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*x^3])), x]} -{1/(x^4*(a + b*Sin[c + d*x^3])), x, 0, Unintegrable[1/(x^4*(a + b*Sin[c + d*x^3])), x]} - -{x/(a + b*Sin[c + d*x^3]), x, 0, Unintegrable[x/(a + b*Sin[c + d*x^3]), x]} -{1/(x^2*(a + b*Sin[c + d*x^3])), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*x^3])), x]} - -{(a + b*Sin[c + d*x^3])^(-1), x, 0, Unintegrable[(a + b*Sin[c + d*x^3])^(-1), x]} -{1/(x^3*(a + b*Sin[c + d*x^3])), x, 0, Unintegrable[1/(x^3*(a + b*Sin[c + d*x^3])), x]} - - -{x^5/(a + b*Sin[c + d*x^3])^2, x, 12, ((-I/3)*a*x^3*Log[1 - (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) + ((I/3)*a*x^3*Log[1 - (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - Log[a + b*Sin[c + d*x^3]]/(3*(a^2 - b^2)*d^2) - (a*PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a - Sqrt[a^2 - b^2])])/(3*(a^2 - b^2)^(3/2)*d^2) + (a*PolyLog[2, (I*b*E^(I*(c + d*x^3)))/(a + Sqrt[a^2 - b^2])])/(3*(a^2 - b^2)^(3/2)*d^2) + (b*x^3*Cos[c + d*x^3])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x^3]))} -{x^2/(a + b*Sin[c + d*x^3])^2, x, 6, (2*a*ArcTan[(b + a*Tan[(c + d*x^3)/2])/Sqrt[a^2 - b^2]])/(3*(a^2 - b^2)^(3/2)*d) + (b*Cos[c + d*x^3])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x^3]))} -{1/(x*(a + b*Sin[c + d*x^3])^2), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*x^3])^2), x]} -{1/(x^4*(a + b*Sin[c + d*x^3])^2), x, 0, Unintegrable[1/(x^4*(a + b*Sin[c + d*x^3])^2), x]} - -{x/(a + b*Sin[c + d*x^3])^2, x, 0, Unintegrable[x/(a + b*Sin[c + d*x^3])^2, x]} -{1/(x^2*(a + b*Sin[c + d*x^3])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*x^3])^2), x]} - -{(a + b*Sin[c + d*x^3])^(-2), x, 0, Unintegrable[(a + b*Sin[c + d*x^3])^(-2), x]} -{1/(x^3*(a + b*Sin[c + d*x^3])^2), x, 0, Unintegrable[1/(x^3*(a + b*Sin[c + d*x^3])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (e x)^(m/2) (a+b Sin[c+d x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^3])^p with m symbolic*) - - -{(e*x)^m*(a + b*Sin[c + d*x^3])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^3])^p, x]} - - -{(e*x)^m*(a + b*Sin[c + d*x^3])^3, x, 13, (a*(2*a^2 + 3*b^2)*(e*x)^(1 + m))/(2*e*(1 + m)) + ((I/8)*b*(4*a^2 + b^2)*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-I)*d*x^3])/e - ((I/8)*b*(4*a^2 + b^2)*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, I*d*x^3])/(e*E^(I*c)) + (2^(-7/3 - m/3)*a*b^2*E^((2*I)*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-2*I)*d*x^3])/e + (2^(-7/3 - m/3)*a*b^2*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (2*I)*d*x^3])/(e*E^((2*I)*c)) - ((I/8)*3^(-4/3 - m/3)*b^3*E^((3*I)*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-3*I)*d*x^3])/e + ((I/8)*3^(-4/3 - m/3)*b^3*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (3*I)*d*x^3])/(e*E^((3*I)*c))} -{(e*x)^m*(a + b*Sin[c + d*x^3])^2, x, 9, ((2*a^2 + b^2)*(e*x)^(1 + m))/(2*e*(1 + m)) + ((I/3)*a*b*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-I)*d*x^3])/e - ((I/3)*a*b*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, I*d*x^3])/(e*E^(I*c)) + (2^(-7/3 - m/3)*b^2*E^((2*I)*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-2*I)*d*x^3])/(3*e) + (2^(-7/3 - m/3)*b^2*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (2*I)*d*x^3])/(3*e*E^((2*I)*c))} -{(e*x)^m*(a + b*Sin[c + d*x^3])^1, x, 5, (a*(e*x)^(1 + m))/(e*(1 + m)) + ((I/6)*b*E^(I*c)*(e*x)^(1 + m)*((-I)*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, (-I)*d*x^3])/e - ((I/6)*b*(e*x)^(1 + m)*(I*d*x^3)^((-1 - m)/3)*Gamma[(1 + m)/3, I*d*x^3])/(e*E^(I*c))} -{(e*x)^m/(a + b*Sin[c + d*x^3])^1, x, 0, Unintegrable[(e*x)^m/(a + b*Sin[c + d*x^3]), x]} -{(e*x)^m/(a + b*Sin[c + d*x^3])^2, x, 0, Unintegrable[(e*x)^m/(a + b*Sin[c + d*x^3])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d / x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sin[c+d / x^1])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Sin[a + b/x], x, 7, (1/6)*b*x^2*Cos[a + b/x] + (1/6)*b^3*Cos[a]*CosIntegral[b/x] - (1/6)*b^2*x*Sin[a + b/x] + (1/3)*x^3*Sin[a + b/x] - (1/6)*b^3*Sin[a]*SinIntegral[b/x]} -{x^1*Sin[a + b/x], x, 6, (1/2)*b*x*Cos[a + b/x] + (1/2)*b^2*CosIntegral[b/x]*Sin[a] + (1/2)*x^2*Sin[a + b/x] + (1/2)*b^2*Cos[a]*SinIntegral[b/x]} -{x^0*Sin[a + b/x], x, 5, (-b)*Cos[a]*CosIntegral[b/x] + x*Sin[a + b/x] + b*Sin[a]*SinIntegral[b/x]} -{Sin[a + b/x]/x^1, x, 3, (-CosIntegral[b/x])*Sin[a] - Cos[a]*SinIntegral[b/x]} -{Sin[a + b/x]/x^2, x, 2, Cos[a + b/x]/b} -{Sin[a + b/x]/x^3, x, 3, Cos[a + b/x]/(b*x) - Sin[a + b/x]/b^2} -{Sin[a + b/x]/x^4, x, 4, -((2*Cos[a + b/x])/b^3) + Cos[a + b/x]/(b*x^2) - (2*Sin[a + b/x])/(b^2*x)} -{Sin[a + b/x]/x^5, x, 5, Cos[a + b/x]/(b*x^3) - (6*Cos[a + b/x])/(b^3*x) + (6*Sin[a + b/x])/b^4 - (3*Sin[a + b/x])/(b^2*x^2)} - - -{x^2*Sin[a + b/x]^2, x, 9, x^3/6 + (1/3)*b^2*x*Cos[2*(a + b/x)] - (1/6)*x^3*Cos[2*(a + b/x)] + (2/3)*b^3*CosIntegral[(2*b)/x]*Sin[2*a] + (1/6)*b*x^2*Sin[2*(a + b/x)] + (2/3)*b^3*Cos[2*a]*SinIntegral[(2*b)/x]} -{x^1*Sin[a + b/x]^2, x, 8, (-b^2)*Cos[2*a]*CosIntegral[(2*b)/x] + (1/2)*x^2*Sin[a + b/x]^2 + (1/2)*b*x*Sin[2*(a + b/x)] + b^2*Sin[2*a]*SinIntegral[(2*b)/x]} -{x^0*Sin[a + b/x]^2, x, 6, (-b)*CosIntegral[(2*b)/x]*Sin[2*a] + x*Sin[a + b/x]^2 - b*Cos[2*a]*SinIntegral[(2*b)/x]} -{Sin[a + b/x]^2/x^1, x, 5, (1/2)*Cos[2*a]*CosIntegral[(2*b)/x] + Log[x]/2 - (1/2)*Sin[2*a]*SinIntegral[(2*b)/x]} -{Sin[a + b/x]^2/x^2, x, 3, -(1/(2*x)) + (Cos[a + b/x]*Sin[a + b/x])/(2*b)} -{Sin[a + b/x]^2/x^3, x, 3, -(1/(4*x^2)) + (Cos[a + b/x]*Sin[a + b/x])/(2*b*x) - Sin[a + b/x]^2/(4*b^2)} -{Sin[a + b/x]^2/x^4, x, 5, -(1/(6*x^3)) + 1/(4*b^2*x) - (Cos[a + b/x]*Sin[a + b/x])/(4*b^3) + (Cos[a + b/x]*Sin[a + b/x])/(2*b*x^2) - Sin[a + b/x]^2/(2*b^2*x)} -{Sin[a + b/x]^2/x^5, x, 5, -(1/(8*x^4)) + 3/(8*b^2*x^2) + (Cos[a + b/x]*Sin[a + b/x])/(2*b*x^3) - (3*Cos[a + b/x]*Sin[a + b/x])/(4*b^3*x) + (3*Sin[a + b/x]^2)/(8*b^4) - (3*Sin[a + b/x]^2)/(4*b^2*x^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d / x^2])^p*) - - -{Sin[a + b/x^2], x, 5, (-Sqrt[b])*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x] + Sqrt[b]*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a] + x*Sin[a + b/x^2]} -{Sin[a + b/x^2]/x, x, 3, (-(1/2))*CosIntegral[b/x^2]*Sin[a] - (1/2)*Cos[a]*SinIntegral[b/x^2]} -{Sin[a + b/x^2]/x^2, x, 4, -((Sqrt[Pi/2]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x])/Sqrt[b]) - (Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/Sqrt[b]} -{Sin[a + b/x^2]/x^3, x, 2, Cos[a + b/x^2]/(2*b)} -{Sin[a + b/x^2]/x^4, x, 5, Cos[a + b/x^2]/(2*b*x) - (Sqrt[Pi/2]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x])/(2*b^(3/2)) + (Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/(2*b^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^(1/2)])^p*) - - -{Sin[Sqrt[x]]/Sqrt[x], x, 2, -2*Cos[Sqrt[x]]} -{Sin[Sqrt[x]]^3/Sqrt[x], x, 3, -2*Cos[Sqrt[x]] + (2/3)*Cos[Sqrt[x]]^3} -{Sin[Sqrt[x]], x, 3, -2*Sqrt[x]*Cos[Sqrt[x]] + 2*Sin[Sqrt[x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^(1/3)])^p*) - - -{Sin[x^(1/3)]^2, x, 5, -((3*x^(1/3))/4) + x/2 + (3/4)*Cos[x^(1/3)]*Sin[x^(1/3)] - (3/2)*x^(2/3)*Cos[x^(1/3)]*Sin[x^(1/3)] + (3/2)*x^(1/3)*Sin[x^(1/3)]^2} -{Sin[x^(1/3)]^3, x, 7, (14/3)*Cos[x^(1/3)] - 2*x^(2/3)*Cos[x^(1/3)] - (2/9)*Cos[x^(1/3)]^3 + 4*x^(1/3)*Sin[x^(1/3)] - x^(2/3)*Cos[x^(1/3)]*Sin[x^(1/3)]^2 + (2/3)*x^(1/3)*Sin[x^(1/3)]^3} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^n])^p*) - - -{(e*x)^m*(b*Sin[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(b*Sin[c + d*x^n])^p, x]} -{(e*x)^m*(a + b*Sin[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Sin[c + d*x^n])^p, x]} - - -{(e*x)^(n - 1)*(b*Sin[c + d*x^n])^p, x, 3, ((e*x)^n*Cos[c + d*x^n]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[c + d*x^n]^2]*(b*Sin[c + d*x^n])^(1 + p))/(x^n*(b*d*e*n*(1 + p)*Sqrt[Cos[c + d*x^n]^2]))} -{(e*x)^(2*n - 1)*(b*Sin[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(b*Sin[c + d*x^n])^p, x])/(x^(2*n)*e)} - -{(e*x)^(n - 1)*(a + b*Sin[c + d*x^n])^p, x, 5, -((Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1/2)*(1 - Sin[c + d*x^n]), (b*(1 - Sin[c + d*x^n]))/(a + b)]*Cos[c + d*x^n]*(a + b*Sin[c + d*x^n])^p)/(x^n*((a + b*Sin[c + d*x^n])/(a + b))^p*(d*e*n*Sqrt[1 + Sin[c + d*x^n]])))} -{(e*x)^(2*n - 1)*(a + b*Sin[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(a + b*Sin[c + d*x^n])^p, x])/(x^(2*n)*e)} - - -{Sin[a + b*x^n]/x, x, 3, (CosIntegral[b*x^n]*Sin[a])/n + (Cos[a]*SinIntegral[b*x^n])/n} -{Sin[a + b*x^n]^2/x, x, 5, -((Cos[2*a]*CosIntegral[2*b*x^n])/(2*n)) + Log[x]/2 + (Sin[2*a]*SinIntegral[2*b*x^n])/(2*n)} -{Sin[a + b*x^n]^3/x, x, 8, (3*CosIntegral[b*x^n]*Sin[a])/(4*n) - (CosIntegral[3*b*x^n]*Sin[3*a])/(4*n) + (3*Cos[a]*SinIntegral[b*x^n])/(4*n) - (Cos[3*a]*SinIntegral[3*b*x^n])/(4*n)} -{Sin[a + b*x^n]^4/x, x, 8, -((Cos[2*a]*CosIntegral[2*b*x^n])/(2*n)) + (Cos[4*a]*CosIntegral[4*b*x^n])/(8*n) + (3*Log[x])/8 + (Sin[2*a]*SinIntegral[2*b*x^n])/(2*n) - (Sin[4*a]*SinIntegral[4*b*x^n])/(8*n)} - - -{Sin[a + b*x^n], x, 3, (I*E^(I*a)*x*Gamma[1/n, (-I)*b*x^n])/(((-I)*b*x^n)^n^(-1)*(2*n)) - (I*x*Gamma[1/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^n^(-1)*(2*n))} -{Sin[a + b*x^n]^2, x, 5, x/2 + (2^(-2 - 1/n)*E^(2*I*a)*x*Gamma[1/n, -2*I*b*x^n])/(((-I)*b*x^n)^n^(-1)*n) + (2^(-2 - 1/n)*x*Gamma[1/n, 2*I*b*x^n])/(E^(2*I*a)*(I*b*x^n)^n^(-1)*n)} -{Sin[a + b*x^n]^3, x, 8, (3*I*E^(I*a)*x*Gamma[1/n, (-I)*b*x^n])/(((-I)*b*x^n)^n^(-1)*(8*n)) - (3*I*x*Gamma[1/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^n^(-1)*(8*n)) - (I*E^(3*I*a)*x*Gamma[1/n, -3*I*b*x^n])/(3^n^(-1)*((-I)*b*x^n)^n^(-1)*(8*n)) + (I*x*Gamma[1/n, 3*I*b*x^n])/(3^n^(-1)*E^(3*I*a)*(I*b*x^n)^n^(-1)*(8*n))} - -{x^m*Sin[a + b*x^n], x, 3, (I*E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(((-I)*b*x^n)^((1 + m)/n)*(2*n)) - (I*x^(1 + m)*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^((1 + m)/n)*(2*n))} -{x^m*Sin[a + b*x^n]^2, x, 5, x^(1 + m)/(2*(1 + m)) + (E^(2*I*a)*x^(1 + m)*Gamma[(1 + m)/n, -2*I*b*x^n])/(2^((1 + m + 2*n)/n)*((-I)*b*x^n)^((1 + m)/n)*n) + (x^(1 + m)*Gamma[(1 + m)/n, 2*I*b*x^n])/(2^((1 + m + 2*n)/n)*E^(2*I*a)*(I*b*x^n)^((1 + m)/n)*n)} -{x^m*Sin[a + b*x^n]^3, x, 8, (3*I*E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(((-I)*b*x^n)^((1 + m)/n)*(8*n)) - (3*I*x^(1 + m)*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^((1 + m)/n)*(8*n)) - (I*E^(3*I*a)*x^(1 + m)*Gamma[(1 + m)/n, -3*I*b*x^n])/(3^((1 + m)/n)*((-I)*b*x^n)^((1 + m)/n)*(8*n)) + (I*x^(1 + m)*Gamma[(1 + m)/n, 3*I*b*x^n])/(3^((1 + m)/n)*E^(3*I*a)*(I*b*x^n)^((1 + m)/n)*(8*n))} - - -{x^(2*n - 1)*Sin[a + b*x^n], x, 3, -((x^n*Cos[a + b*x^n])/(b*n)) + Sin[a + b*x^n]/(b^2*n)} -{x^(2*n - 1)*Cos[a + b*x^n], x, 3, Cos[a + b*x^n]/(b^2*n) + (x^n*Sin[a + b*x^n])/(b*n)} - - -{Sin[a + b*x^n]/x^(n + 1), x, 5, (b*Cos[a]*CosIntegral[b*x^n])/n - Sin[a + b*x^n]/(x^n*n) - (b*Sin[a]*SinIntegral[b*x^n])/n} -{Sin[a + b*x^n]^2/x^(n + 1), x, 7, -(1/(x^n*(2*n))) + Cos[2*(a + b*x^n)]/(x^n*(2*n)) + (b*CosIntegral[2*b*x^n]*Sin[2*a])/n + (b*Cos[2*a]*SinIntegral[2*b*x^n])/n} -{Sin[a + b*x^n]^3/x^(n + 1), x, 12, (3*b*Cos[a]*CosIntegral[b*x^n])/(4*n) - (3*b*Cos[3*a]*CosIntegral[3*b*x^n])/(4*n) - (3*Sin[a + b*x^n])/(x^n*(4*n)) + Sin[3*(a + b*x^n)]/(x^n*(4*n)) - (3*b*Sin[a]*SinIntegral[b*x^n])/(4*n) + (3*b*Sin[3*a]*SinIntegral[3*b*x^n])/(4*n)} - -{Sin[a + b*x^n]/x^(2*n + 1), x, 6, -((b*Cos[a + b*x^n])/(x^n*(2*n))) - (b^2*CosIntegral[b*x^n]*Sin[a])/(2*n) - Sin[a + b*x^n]/(x^(2*n)*(2*n)) - (b^2*Cos[a]*SinIntegral[b*x^n])/(2*n)} -{Sin[a + b*x^n]^2/x^(2*n + 1), x, 8, -(1/(x^(2*n)*(4*n))) + Cos[2*(a + b*x^n)]/(x^(2*n)*(4*n)) + (b^2*Cos[2*a]*CosIntegral[2*b*x^n])/n - (b*Sin[2*(a + b*x^n)])/(x^n*(2*n)) - (b^2*Sin[2*a]*SinIntegral[2*b*x^n])/n} -{Sin[a + b*x^n]^3/x^(2*n + 1), x, 14, -((3*b*Cos[a + b*x^n])/(x^n*(8*n))) + (3*b*Cos[3*(a + b*x^n)])/(x^n*(8*n)) - (3*b^2*CosIntegral[b*x^n]*Sin[a])/(8*n) + (9*b^2*CosIntegral[3*b*x^n]*Sin[3*a])/(8*n) - (3*Sin[a + b*x^n])/(x^(2*n)*(8*n)) + Sin[3*(a + b*x^n)]/(x^(2*n)*(8*n)) - (3*b^2*Cos[a]*SinIntegral[b*x^n])/(8*n) + (9*b^2*Cos[3*a]*SinIntegral[3*b*x^n])/(8*n)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g+h x)^m (a+b Sin[c+d (e+f x)^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^n]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[b (c+d x)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^3*Sin[b*(c + d*x)^2], x, 10, -((3*f*(d*e - c*f)^2*Cos[b*(c + d*x)^2])/(2*b*d^4)) - (3*f^2*(d*e - c*f)*(c + d*x)*Cos[b*(c + d*x)^2])/(2*b*d^4) - (f^3*(c + d*x)^2*Cos[b*(c + d*x)^2])/(2*b*d^4) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^4) + (f^3*Sin[b*(c + d*x)^2])/(2*b^2*d^4)} -{(e + f*x)^2*Sin[b*(c + d*x)^2], x, 7, -((f*(d*e - c*f)*Cos[b*(c + d*x)^2])/(b*d^3)) - (f^2*(c + d*x)*Cos[b*(c + d*x)^2])/(2*b*d^3) + (f^2*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^3) + ((d*e - c*f)^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^3)} -{(e + f*x)^1*Sin[b*(c + d*x)^2], x, 5, -((f*Cos[b*(c + d*x)^2])/(2*b*d^2)) + ((d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^2)} -{(e + f*x)^0*Sin[b*(c + d*x)^2], x, 1, (Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d)} -{Sin[b*(c + d*x)^2]/(e + f*x)^1, x, 0, Unintegrable[Sin[b*(c + d*x)^2]/(e + f*x), x]} -{Sin[b*(c + d*x)^2]/(e + f*x)^2, x, 0, Unintegrable[Sin[b*(c + d*x)^2]/(e + f*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^3*Sin[b/(c + d*x)^2], x, 16, (2*b*f^2*(d*e - c*f)*(c + d*x)*Cos[b/(c + d*x)^2])/d^4 + (b*f^3*(c + d*x)^2*Cos[b/(c + d*x)^2])/(4*d^4) - (3*b*f*(d*e - c*f)^2*CosIntegral[b/(c + d*x)^2])/(2*d^4) - (Sqrt[b]*(d*e - c*f)^3*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^4 + (2*b^(3/2)*f^2*(d*e - c*f)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^4 + ((d*e - c*f)^3*(c + d*x)*Sin[b/(c + d*x)^2])/d^4 + (3*f*(d*e - c*f)^2*(c + d*x)^2*Sin[b/(c + d*x)^2])/(2*d^4) + (f^2*(d*e - c*f)*(c + d*x)^3*Sin[b/(c + d*x)^2])/d^4 + (f^3*(c + d*x)^4*Sin[b/(c + d*x)^2])/(4*d^4) + (b^2*f^3*SinIntegral[b/(c + d*x)^2])/(4*d^4)} -{(e + f*x)^2*Sin[b/(c + d*x)^2], x, 12, (2*b*f^2*(c + d*x)*Cos[b/(c + d*x)^2])/(3*d^3) - (b*f*(d*e - c*f)*CosIntegral[b/(c + d*x)^2])/d^3 - (Sqrt[b]*(d*e - c*f)^2*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^3 + (2*b^(3/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/(3*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[b/(c + d*x)^2])/d^3 + (f*(d*e - c*f)*(c + d*x)^2*Sin[b/(c + d*x)^2])/d^3 + (f^2*(c + d*x)^3*Sin[b/(c + d*x)^2])/(3*d^3)} -{(e + f*x)^1*Sin[b/(c + d*x)^2], x, 8, -((b*f*CosIntegral[b/(c + d*x)^2])/(2*d^2)) - (Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^2 + ((d*e - c*f)*(c + d*x)*Sin[b/(c + d*x)^2])/d^2 + (f*(c + d*x)^2*Sin[b/(c + d*x)^2])/(2*d^2)} -{(e + f*x)^0*Sin[b/(c + d*x)^2], x, 3, -((Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d) + ((c + d*x)*Sin[b/(c + d*x)^2])/d} -{Sin[b/(c + d*x)^2]/(e + f*x)^1, x, 0, Unintegrable[Sin[b/(c + d*x)^2]/(e + f*x), x]} -{Sin[b/(c + d*x)^2]/(e + f*x)^2, x, 0, Unintegrable[Sin[b/(c + d*x)^2]/(e + f*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^3*Sin[a + b*(c + d*x)^2], x, 14, -((3*f*(d*e - c*f)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4)) - (3*f^2*(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^2])/(2*b*d^4) - (f^3*(c + d*x)^2*Cos[a + b*(c + d*x)^2])/(2*b*d^4) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^4) + ((d*e - c*f)^3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*d^4) - (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)*d^4) + (f^3*Sin[a + b*(c + d*x)^2])/(2*b^2*d^4)} -{(e + f*x)^2*Sin[a + b*(c + d*x)^2], x, 11, -((f*(d*e - c*f)*Cos[a + b*(c + d*x)^2])/(b*d^3)) - (f^2*(c + d*x)*Cos[a + b*(c + d*x)^2])/(2*b*d^3) + (f^2*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)*d^3) + ((d*e - c*f)^2*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^3) + ((d*e - c*f)^2*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*d^3) - (f^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)*d^3)} -{(e + f*x)^1*Sin[a + b*(c + d*x)^2], x, 7, -((f*Cos[a + b*(c + d*x)^2])/(2*b*d^2)) + ((d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d^2) + ((d*e - c*f)*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*d^2)} -{(e + f*x)^0*Sin[a + b*(c + d*x)^2], x, 3, (Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(Sqrt[b]*d) + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(Sqrt[b]*d)} -{Sin[a + b*(c + d*x)^2]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b*(c + d*x)^2]/(e + f*x), x]} -{Sin[a + b*(c + d*x)^2]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b*(c + d*x)^2]/(e + f*x)^2, x]} - - -{(e + f*x)^3*Sin[a + b*(c + d*x)^3], x, 14, -((f^2*(d*e - c*f)*Cos[a + b*(c + d*x)^3])/(b*d^4)) - (f^3*(c + d*x)*Cos[a + b*(c + d*x)^3])/(3*b*d^4) - (E^(I*a)*f^3*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(18*b*d^4*((-I)*b*(c + d*x)^3)^(1/3)) + (I*E^(I*a)*(d*e - c*f)^3*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(6*d^4*((-I)*b*(c + d*x)^3)^(1/3)) - (f^3*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(18*b*d^4*(I*b*(c + d*x)^3)^(1/3))) - (I*(d*e - c*f)^3*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(6*d^4*(I*b*(c + d*x)^3)^(1/3))) + (I*E^(I*a)*f*(d*e - c*f)^2*(c + d*x)^2*Gamma[2/3, (-I)*b*(c + d*x)^3])/(2*d^4*((-I)*b*(c + d*x)^3)^(2/3)) - (I*f*(d*e - c*f)^2*(c + d*x)^2*Gamma[2/3, I*b*(c + d*x)^3])/(E^(I*a)*(2*d^4*(I*b*(c + d*x)^3)^(2/3)))} -{(e + f*x)^2*Sin[a + b*(c + d*x)^3], x, 10, -((f^2*Cos[a + b*(c + d*x)^3])/(3*b*d^3)) + (I*E^(I*a)*(d*e - c*f)^2*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(6*d^3*((-I)*b*(c + d*x)^3)^(1/3)) - (I*(d*e - c*f)^2*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(6*d^3*(I*b*(c + d*x)^3)^(1/3))) + (I*E^(I*a)*f*(d*e - c*f)*(c + d*x)^2*Gamma[2/3, (-I)*b*(c + d*x)^3])/(3*d^3*((-I)*b*(c + d*x)^3)^(2/3)) - (I*f*(d*e - c*f)*(c + d*x)^2*Gamma[2/3, I*b*(c + d*x)^3])/(E^(I*a)*(3*d^3*(I*b*(c + d*x)^3)^(2/3)))} -{(e + f*x)^1*Sin[a + b*(c + d*x)^3], x, 8, (I*E^(I*a)*(d*e - c*f)*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(6*d^2*((-I)*b*(c + d*x)^3)^(1/3)) - (I*(d*e - c*f)*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(6*d^2*(I*b*(c + d*x)^3)^(1/3))) + (I*E^(I*a)*f*(c + d*x)^2*Gamma[2/3, (-I)*b*(c + d*x)^3])/(6*d^2*((-I)*b*(c + d*x)^3)^(2/3)) - (I*f*(c + d*x)^2*Gamma[2/3, I*b*(c + d*x)^3])/(E^(I*a)*(6*d^2*(I*b*(c + d*x)^3)^(2/3)))} -{(e + f*x)^0*Sin[a + b*(c + d*x)^3], x, 3, (I*E^(I*a)*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(6*d*((-I)*b*(c + d*x)^3)^(1/3)) - (I*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(6*d*(I*b*(c + d*x)^3)^(1/3)))} -{Sin[a + b*(c + d*x)^3]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b*(c + d*x)^3]/(e + f*x), x]} -{Sin[a + b*(c + d*x)^3]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b*(c + d*x)^3]/(e + f*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^2*Sin[a + b/(c + d*x)^2], x, 18, (2*b*f^2*(c + d*x)*Cos[a + b/(c + d*x)^2])/(3*d^3) - (b*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^2])/d^3 - (Sqrt[b]*(d*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^3 + (2*b^(3/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/(3*d^3) + (2*b^(3/2)*f^2*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a])/(3*d^3) + (Sqrt[b]*(d*e - c*f)^2*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a])/d^3 + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^2])/d^3 + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^2])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^2])/(3*d^3) + (b*f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^2])/d^3} -{(e + f*x)^1*Sin[a + b/(c + d*x)^2], x, 12, -((b*f*Cos[a]*CosIntegral[b/(c + d*x)^2])/(2*d^2)) - (Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d^2 + (Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a])/d^2 + ((d*e - c*f)*(c + d*x)*Sin[a + b/(c + d*x)^2])/d^2 + (f*(c + d*x)^2*Sin[a + b/(c + d*x)^2])/(2*d^2) + (b*f*Sin[a]*SinIntegral[b/(c + d*x)^2])/(2*d^2)} -{(e + f*x)^0*Sin[a + b/(c + d*x)^2], x, 5, -((Sqrt[b]*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)])/d) + (Sqrt[b]*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)]*Sin[a])/d + ((c + d*x)*Sin[a + b/(c + d*x)^2])/d} -{Sin[a + b/(c + d*x)^2]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b/(c + d*x)^2]/(e + f*x), x]} -{Sin[a + b/(c + d*x)^2]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b/(c + d*x)^2]/(e + f*x)^2, x]} - - -{(e + f*x)^2*Sin[a + b/(c + d*x)^3], x, 13, -((b*f^2*Cos[a]*CosIntegral[b/(c + d*x)^3])/(3*d^3)) - (I*E^(I*a)*f*(d*e - c*f)*(-((I*b)/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-(2/3), -((I*b)/(c + d*x)^3)])/(3*d^3) + (I*f*(d*e - c*f)*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2*Gamma[-(2/3), (I*b)/(c + d*x)^3])/(E^(I*a)*(3*d^3)) - (I*E^(I*a)*(d*e - c*f)^2*(-((I*b)/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -((I*b)/(c + d*x)^3)])/(6*d^3) + (I*(d*e - c*f)^2*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gamma[-(1/3), (I*b)/(c + d*x)^3])/(E^(I*a)*(6*d^3)) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^3])/(3*d^3) + (b*f^2*Sin[a]*SinIntegral[b/(c + d*x)^3])/(3*d^3)} -{(e + f*x)^1*Sin[a + b/(c + d*x)^3], x, 8, -((I*E^(I*a)*f*(-((I*b)/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-(2/3), -((I*b)/(c + d*x)^3)])/(6*d^2)) + (I*f*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2*Gamma[-(2/3), (I*b)/(c + d*x)^3])/(E^(I*a)*(6*d^2)) - (I*E^(I*a)*(d*e - c*f)*(-((I*b)/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -((I*b)/(c + d*x)^3)])/(6*d^2) + (I*(d*e - c*f)*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gamma[-(1/3), (I*b)/(c + d*x)^3])/(E^(I*a)*(6*d^2))} -{(e + f*x)^0*Sin[a + b/(c + d*x)^3], x, 3, -((I*E^(I*a)*(-((I*b)/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-(1/3), -((I*b)/(c + d*x)^3)])/(6*d)) + (I*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gamma[-(1/3), (I*b)/(c + d*x)^3])/(E^(I*a)*(6*d))} -{Sin[a + b/(c + d*x)^3]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b/(c + d*x)^3]/(e + f*x), x]} -{Sin[a + b/(c + d*x)^3]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b/(c + d*x)^3]/(e + f*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^(n/2)]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^2*Sin[a + b*Sqrt[c + d*x]], x, 14, -((240*f^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^5*d^3)) + (24*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (2*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*f^2*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) - (2*f^2*(c + d*x)^(5/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^3) + (240*f^2*Sin[a + b*Sqrt[c + d*x]])/(b^6*d^3) - (24*f*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*(d*e - c*f)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (120*f^2*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*f*(d*e - c*f)*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (10*f^2*(c + d*x)^2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^3)} -{(e + f*x)^1*Sin[a + b*Sqrt[c + d*x]], x, 8, (12*f*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b*d^2) - (2*f*(c + d*x)^(3/2)*Cos[a + b*Sqrt[c + d*x]])/(b*d^2) - (12*f*Sin[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (2*(d*e - c*f)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2) + (6*f*(c + d*x)*Sin[a + b*Sqrt[c + d*x]])/(b^2*d^2)} -{(e + f*x)^0*Sin[a + b*Sqrt[c + d*x]], x, 3, -((2*Sqrt[c + d*x]*Cos[a + b*Sqrt[c + d*x]])/(b*d)) + (2*Sin[a + b*Sqrt[c + d*x]])/(b^2*d)} -{Sin[a + b*Sqrt[c + d*x]]/(e + f*x)^1, x, 8, (CosIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] + b*Sqrt[c + d*x]]*Sin[a - (b*Sqrt[(-d)*e + c*f])/Sqrt[f]])/f + (CosIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] - b*Sqrt[c + d*x]]*Sin[a + (b*Sqrt[(-d)*e + c*f])/Sqrt[f]])/f - (Cos[a + (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] - b*Sqrt[c + d*x]])/f + (Cos[a - (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] + b*Sqrt[c + d*x]])/f} -{Sin[a + b*Sqrt[c + d*x]]/(e + f*x)^2, x, 10, (b*d*Cos[a + (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*CosIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] - b*Sqrt[c + d*x]])/(2*f^(3/2)*Sqrt[(-d)*e + c*f]) - (b*d*Cos[a - (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*CosIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] + b*Sqrt[c + d*x]])/(2*f^(3/2)*Sqrt[(-d)*e + c*f]) - Sin[a + b*Sqrt[c + d*x]]/(f*(e + f*x)) + (b*d*Sin[a + (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] - b*Sqrt[c + d*x]])/(2*f^(3/2)*Sqrt[(-d)*e + c*f]) + (b*d*Sin[a - (b*Sqrt[(-d)*e + c*f])/Sqrt[f]]*SinIntegral[(b*Sqrt[(-d)*e + c*f])/Sqrt[f] + b*Sqrt[c + d*x]])/(2*f^(3/2)*Sqrt[(-d)*e + c*f])} - - -{(e + f*x)^2*Sin[a + b*(c + d*x)^(3/2)], x, 12, -((4*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b*(c + d*x)^(3/2)])/(3*b*d^3)) - (2*f^2*(c + d*x)^(3/2)*Cos[a + b*(c + d*x)^(3/2)])/(3*b*d^3) - (2*E^(I*a)*f*(d*e - c*f)*Sqrt[c + d*x]*Gamma[1/3, (-I)*b*(c + d*x)^(3/2)])/(9*b*d^3*((-I)*b*(c + d*x)^(3/2))^(1/3)) - (2*f*(d*e - c*f)*Sqrt[c + d*x]*Gamma[1/3, I*b*(c + d*x)^(3/2)])/(E^(I*a)*(9*b*d^3*(I*b*(c + d*x)^(3/2))^(1/3))) + (I*E^(I*a)*(d*e - c*f)^2*(c + d*x)*Gamma[2/3, (-I)*b*(c + d*x)^(3/2)])/(3*d^3*((-I)*b*(c + d*x)^(3/2))^(2/3)) - (I*(d*e - c*f)^2*(c + d*x)*Gamma[2/3, I*b*(c + d*x)^(3/2)])/(E^(I*a)*(3*d^3*(I*b*(c + d*x)^(3/2))^(2/3))) + (2*f^2*Sin[a + b*(c + d*x)^(3/2)])/(3*b^2*d^3)} -{(e + f*x)^1*Sin[a + b*(c + d*x)^(3/2)], x, 9, -((2*f*Sqrt[c + d*x]*Cos[a + b*(c + d*x)^(3/2)])/(3*b*d^2)) - (E^(I*a)*f*Sqrt[c + d*x]*Gamma[1/3, (-I)*b*(c + d*x)^(3/2)])/(9*b*d^2*((-I)*b*(c + d*x)^(3/2))^(1/3)) - (f*Sqrt[c + d*x]*Gamma[1/3, I*b*(c + d*x)^(3/2)])/(E^(I*a)*(9*b*d^2*(I*b*(c + d*x)^(3/2))^(1/3))) + (I*E^(I*a)*(d*e - c*f)*(c + d*x)*Gamma[2/3, (-I)*b*(c + d*x)^(3/2)])/(3*d^2*((-I)*b*(c + d*x)^(3/2))^(2/3)) - (I*(d*e - c*f)*(c + d*x)*Gamma[2/3, I*b*(c + d*x)^(3/2)])/(E^(I*a)*(3*d^2*(I*b*(c + d*x)^(3/2))^(2/3)))} -{(e + f*x)^0*Sin[a + b*(c + d*x)^(3/2)], x, 4, (I*E^(I*a)*(c + d*x)*Gamma[2/3, (-I)*b*(c + d*x)^(3/2)])/(3*d*((-I)*b*(c + d*x)^(3/2))^(2/3)) - (I*(c + d*x)*Gamma[2/3, I*b*(c + d*x)^(3/2)])/(E^(I*a)*(3*d*(I*b*(c + d*x)^(3/2))^(2/3)))} -{Sin[a + b*(c + d*x)^(3/2)]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b*(c + d*x)^(3/2)]/(e + f*x), x]} -{Sin[a + b*(c + d*x)^(3/2)]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b*(c + d*x)^(3/2)]/(e + f*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^2*Sin[a + b/Sqrt[c + d*x]], x, 23, (b^5*f^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^3*f*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(6*d^3) + (b*(d*e - c*f)^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d^3 - (b^3*f^2*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(180*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(3*d^3) + (b*f^2*(c + d*x)^(5/2)*Cos[a + b/Sqrt[c + d*x]])/(15*d^3) + (b^6*f^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(360*d^3) - (b^4*f*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(6*d^3) + (b^2*(d*e - c*f)^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/d^3 + (b^4*f^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(360*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(6*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^3 - (b^2*f^2*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/(60*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/Sqrt[c + d*x]])/(3*d^3) + (b^6*f^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(360*d^3) - (b^4*f*(d*e - c*f)*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(6*d^3) + (b^2*(d*e - c*f)^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/d^3} -{(e + f*x)^1*Sin[a + b/Sqrt[c + d*x]], x, 14, -((b^3*f*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(12*d^2)) + (b*(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d^2 + (b*f*(c + d*x)^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(6*d^2) - (b^4*f*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(12*d^2) + (b^2*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/d^2 - (b^2*f*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(12*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d^2 + (f*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/(2*d^2) - (b^4*f*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(12*d^2) + (b^2*(d*e - c*f)*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/d^2} -{(e + f*x)^0*Sin[a + b/Sqrt[c + d*x]], x, 6, (b*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d + (b^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/d + ((c + d*x)*Sin[a + b/Sqrt[c + d*x]])/d + (b^2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/d} -{Sin[a + b/Sqrt[c + d*x]]/(e + f*x)^1, x, 13, -((2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/f) + (CosIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] + b/Sqrt[c + d*x]]*Sin[a - (b*Sqrt[f])/Sqrt[(-d)*e + c*f]])/f + (CosIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] - b/Sqrt[c + d*x]]*Sin[a + (b*Sqrt[f])/Sqrt[(-d)*e + c*f]])/f - (2*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/f - (Cos[a + (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] - b/Sqrt[c + d*x]])/f + (Cos[a - (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] + b/Sqrt[c + d*x]])/f} -{Sin[a + b/Sqrt[c + d*x]]/(e + f*x)^2, x, 10, -((b*d*Cos[a + (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*CosIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] - b/Sqrt[c + d*x]])/(2*Sqrt[f]*((-d)*e + c*f)^(3/2))) + (b*d*Cos[a - (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*CosIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] + b/Sqrt[c + d*x]])/(2*Sqrt[f]*((-d)*e + c*f)^(3/2)) + ((c + d*x)*Sin[a + b/Sqrt[c + d*x]])/((d*e - c*f)*(e + f*x)) - (b*d*Sin[a + (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] - b/Sqrt[c + d*x]])/(2*Sqrt[f]*((-d)*e + c*f)^(3/2)) - (b*d*Sin[a - (b*Sqrt[f])/Sqrt[(-d)*e + c*f]]*SinIntegral[(b*Sqrt[f])/Sqrt[(-d)*e + c*f] + b/Sqrt[c + d*x]])/(2*Sqrt[f]*((-d)*e + c*f)^(3/2))} - - -{(e + f*x)^2*Sin[a + b/(c + d*x)^(3/2)], x, 14, (b*f^2*(c + d*x)^(3/2)*Cos[a + b/(c + d*x)^(3/2)])/(3*d^3) - (2*I*E^(I*a)*f*(d*e - c*f)*(-((I*b)/(c + d*x)^(3/2)))^(4/3)*(c + d*x)^2*Gamma[-(4/3), -((I*b)/(c + d*x)^(3/2))])/(3*d^3) + (2*I*f*(d*e - c*f)*((I*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-(4/3), (I*b)/(c + d*x)^(3/2)])/(E^(I*a)*(3*d^3)) - (I*E^(I*a)*(d*e - c*f)^2*(-((I*b)/(c + d*x)^(3/2)))^(2/3)*(c + d*x)*Gamma[-(2/3), -((I*b)/(c + d*x)^(3/2))])/(3*d^3) + (I*(d*e - c*f)^2*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-(2/3), (I*b)/(c + d*x)^(3/2)])/(E^(I*a)*(3*d^3)) + (b^2*f^2*CosIntegral[b/(c + d*x)^(3/2)]*Sin[a])/(3*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(3/2)])/(3*d^3) + (b^2*f^2*Cos[a]*SinIntegral[b/(c + d*x)^(3/2)])/(3*d^3)} -{(e + f*x)^1*Sin[a + b/(c + d*x)^(3/2)], x, 8, -((I*E^(I*a)*f*(-((I*b)/(c + d*x)^(3/2)))^(4/3)*(c + d*x)^2*Gamma[-(4/3), -((I*b)/(c + d*x)^(3/2))])/(3*d^2)) + (I*f*((I*b)/(c + d*x)^(3/2))^(4/3)*(c + d*x)^2*Gamma[-(4/3), (I*b)/(c + d*x)^(3/2)])/(E^(I*a)*(3*d^2)) - (I*E^(I*a)*(d*e - c*f)*(-((I*b)/(c + d*x)^(3/2)))^(2/3)*(c + d*x)*Gamma[-(2/3), -((I*b)/(c + d*x)^(3/2))])/(3*d^2) + (I*(d*e - c*f)*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-(2/3), (I*b)/(c + d*x)^(3/2)])/(E^(I*a)*(3*d^2))} -{(e + f*x)^0*Sin[a + b/(c + d*x)^(3/2)], x, 4, -((I*E^(I*a)*(-((I*b)/(c + d*x)^(3/2)))^(2/3)*(c + d*x)*Gamma[-(2/3), -((I*b)/(c + d*x)^(3/2))])/(3*d)) + (I*((I*b)/(c + d*x)^(3/2))^(2/3)*(c + d*x)*Gamma[-(2/3), (I*b)/(c + d*x)^(3/2)])/(E^(I*a)*(3*d))} -{Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x), x]} -{Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b/(c + d*x)^(3/2)]/(e + f*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^(n/3)]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^2*Sin[a + b*(c + d*x)^(1/3)], x, 20, -((120960*f^2*Cos[a + b*(c + d*x)^(1/3)])/(b^9*d^3)) + (6*(d*e - c*f)^2*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*f^2*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^7*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*f*(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (5040*f^2*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*f^2*(c + d*x)^2*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (3*f^2*(c + d*x)^(8/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*f*(d*e - c*f)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^8*d^3) + (6*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (360*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (20160*f^2*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30*f*(d*e - c*f)*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*f^2*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (24*f^2*(c + d*x)^(7/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3)} -{(e + f*x)^1*Sin[a + b*(c + d*x)^(1/3)], x, 11, (6*(d*e - c*f)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^2) - (360*f*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d^2) - (3*(d*e - c*f)*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^2) + (60*f*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^2) - (3*f*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^2) + (360*f*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^2) + (6*(d*e - c*f)*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180*f*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d^2) + (15*f*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^2)} -{(e + f*x)^0*Sin[a + b*(c + d*x)^(1/3)], x, 4, (6*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d) - (3*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) + (6*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)} -{Sin[a + b*(c + d*x)^(1/3)]/(e + f*x)^1, x, 11, (CosIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)]*Sin[a - (b*(d*e - c*f)^(1/3))/f^(1/3)])/f + (CosIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)]*Sin[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)])/f + (CosIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)]*Sin[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)])/f - (Cos[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)])/f + (Cos[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/f + (Cos[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/f} -{Sin[a + b*(c + d*x)^(1/3)]/(e + f*x)^2, x, 13, -(((-1)^(1/3)*b*d*Cos[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3))) + (b*d*Cos[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) + ((-1)^(2/3)*b*d*Cos[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - Sin[a + b*(c + d*x)^(1/3)]/(f*(e + f*x)) - ((-1)^(1/3)*b*d*Sin[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - (b*d*Sin[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - ((-1)^(2/3)*b*d*Sin[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3))} - - -{(e + f*x)^2*Sin[a + b*(c + d*x)^(2/3)], x, 17, (6*f*(d*e - c*f)*Cos[a + b*(c + d*x)^(2/3)])/(b^3*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^3) + (105*f^2*(c + d*x)*Cos[a + b*(c + d*x)^(2/3)])/(8*b^3*d^3) - (3*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(2/3)])/(b*d^3) - (3*f^2*(c + d*x)^(7/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^3) + (3*(d*e - c*f)^2*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)*d^3) + (315*f^2*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(16*b^(9/2)*d^3) + (315*f^2*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(16*b^(9/2)*d^3) - (3*(d*e - c*f)^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)*d^3) - (315*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(16*b^4*d^3) + (6*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(b^2*d^3) + (21*f^2*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d^3)} -{(e + f*x)^1*Sin[a + b*(c + d*x)^(2/3)], x, 10, (3*f*Cos[a + b*(c + d*x)^(2/3)])/(b^3*d^2) - (3*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^2) - (3*f*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^2) + (3*(d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)*d^2) - (3*(d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)*d^2) + (3*f*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(b^2*d^2)} -{(e + f*x)^0*Sin[a + b*(c + d*x)^(2/3)], x, 5, -((3*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d)) + (3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)*d) - (3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)*d)} -{Sin[a + b*(c + d*x)^(2/3)]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b*(c + d*x)^(2/3)]/(e + f*x), x]} -{Sin[a + b*(c + d*x)^(2/3)]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b*(c + d*x)^(2/3)]/(e + f*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^2*Sin[a + b/(c + d*x)^(1/3)], x, 29, (b^5*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^7*f^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(120960*d^3) + (b*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d^3) - (b^3*f*(d*e - c*f)*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(60*d^3) + (b^5*f^2*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(1/3)])/(20160*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(5*d^3) - (b^3*f^2*(c + d*x)^2*Cos[a + b/(c + d*x)^(1/3)])/(1008*d^3) + (b*f^2*(c + d*x)^(8/3)*Cos[a + b/(c + d*x)^(1/3)])/(24*d^3) - (b^9*f^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) + (b^3*(d*e - c*f)^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(2*d^3) + (b^6*f*(d*e - c*f)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(120*d^3) + (b^8*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(120960*d^3) - (b^2*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d^3) + (b^4*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^6*f^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/(60480*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f*(d*e - c*f)*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(20*d^3) + (b^4*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(1/3)])/(5040*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f^2*(c + d*x)^(7/3)*Sin[a + b/(c + d*x)^(1/3)])/(168*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(1/3)])/(3*d^3) + (b^6*f*(d*e - c*f)*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(120*d^3) + (b^9*f^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) - (b^3*(d*e - c*f)^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d^3)} -{(e + f*x)^1*Sin[a + b/(c + d*x)^(1/3)], x, 17, (b^5*f*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(240*d^2) + (b*(d*e - c*f)*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d^2) - (b^3*f*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^2) + (b*f*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(10*d^2) + (b^3*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(2*d^2) + (b^6*f*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(240*d^2) - (b^2*(d*e - c*f)*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d^2) + (b^4*f*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(240*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/d^2 - (b^2*f*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(40*d^2) + (f*(c + d*x)^2*Sin[a + b/(c + d*x)^(1/3)])/(2*d^2) + (b^6*f*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(240*d^2) - (b^3*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d^2)} -{(e + f*x)^0*Sin[a + b/(c + d*x)^(1/3)], x, 7, (b*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d) + (b^3*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(2*d) - (b^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d) + ((c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/d - (b^3*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d)} -{Sin[a + b/(c + d*x)^(1/3)]/(e + f*x)^1, x, 16, -((3*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/f) + (CosIntegral[(b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)]*Sin[a - (b*f^(1/3))/(d*e - c*f)^(1/3)])/f + (CosIntegral[((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3) - b/(c + d*x)^(1/3)]*Sin[a + ((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3)])/f + (CosIntegral[((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)]*Sin[a - ((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3)])/f - (3*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/f - (Cos[a + ((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3)]*SinIntegral[((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3) - b/(c + d*x)^(1/3)])/f + (Cos[a - (b*f^(1/3))/(d*e - c*f)^(1/3)]*SinIntegral[(b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)])/f + (Cos[a - ((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3)]*SinIntegral[((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)])/f} -{Sin[a + b/(c + d*x)^(1/3)]/(e + f*x)^2, x, 13, -((b*d*Cos[a + (b*f^(1/3))/((-d)*e + c*f)^(1/3)]*CosIntegral[(b*f^(1/3))/((-d)*e + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3))) - ((-1)^(2/3)*b*d*Cos[a + ((-1)^(2/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3)]*CosIntegral[((-1)^(2/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3)) + ((-1)^(1/3)*b*d*Cos[a - ((-1)^(1/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3)]*CosIntegral[((-1)^(1/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3)) + ((c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/((d*e - c*f)*(e + f*x)) - (b*d*Sin[a + (b*f^(1/3))/((-d)*e + c*f)^(1/3)]*SinIntegral[(b*f^(1/3))/((-d)*e + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3)) - ((-1)^(2/3)*b*d*Sin[a + ((-1)^(2/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3)]*SinIntegral[((-1)^(2/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3) - b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3)) - ((-1)^(1/3)*b*d*Sin[a - ((-1)^(1/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3)]*SinIntegral[((-1)^(1/3)*b*f^(1/3))/((-d)*e + c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f^(2/3)*((-d)*e + c*f)^(4/3))} - - -{(e + f*x)^2*Sin[a + b/(c + d*x)^(2/3)], x, 24, (2*b*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d^3 - (8*b^3*f^2*(c + d*x)*Cos[a + b/(c + d*x)^(2/3)])/(315*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*d^3) + (2*b*f^2*(c + d*x)^(7/3)*Cos[a + b/(c + d*x)^(2/3)])/(21*d^3) + (b^3*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(2*d^3) - (16*b^(9/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(315*d^3) + (2*b^(3/2)*(d*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d^3 + (2*b^(3/2)*(d*e - c*f)^2*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d^3 + (16*b^(9/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(315*d^3) + (16*b^4*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(315*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(2*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/d^3 - (4*b^2*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(2/3)])/(105*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(2/3)])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(2/3)])/(3*d^3) - (b^3*f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(2*d^3)} -{(e + f*x)^1*Sin[a + b/(c + d*x)^(2/3)], x, 15, (2*b*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d^2 + (b*f*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/(4*d^2) + (b^3*f*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(4*d^2) + (2*b^(3/2)*(d*e - c*f)*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d^2 + (2*b^(3/2)*(d*e - c*f)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d^2 - (b^2*f*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(4*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/d^2 + (f*(c + d*x)^2*Sin[a + b/(c + d*x)^(2/3)])/(2*d^2) - (b^3*f*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(4*d^2)} -{(e + f*x)^0*Sin[a + b/(c + d*x)^(2/3)], x, 7, (2*b*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d + (2*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d + (2*b^(3/2)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d + ((c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/d} -{Sin[a + b/(c + d*x)^(2/3)]/(e + f*x)^1, x, 0, Unintegrable[Sin[a + b/(c + d*x)^(2/3)]/(e + f*x), x]} -{Sin[a + b/(c + d*x)^(2/3)]/(e + f*x)^2, x, 0, Unintegrable[Sin[a + b/(c + d*x)^(2/3)]/(e + f*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^(n/3)] when c f-b e=0 *) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)], x, 9, (2160*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^7*d*(c + d*x)^(1/3)) - (1080*e*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d) + (90*e*(c + d*x)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d) - (3*e*(c + d*x)^(5/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) + (2160*e*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d) - (360*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d) + (18*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)} -{(c*e + d*e*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)], x, 7, (36*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d) - (72*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d*(c + d*x)^(2/3)) - (3*(c + d*x)^(2/3)*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) - (72*(e*(c + d*x))^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d*(c + d*x)^(1/3)) + (12*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)} -{(c*e + d*e*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)], x, 6, (18*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d) - (3*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) - (18*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d*(c + d*x)^(1/3)) + (9*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)} -{Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(1/3), x, 4, -((3*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d*(e*(c + d*x))^(1/3))) + (3*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d*(e*(c + d*x))^(1/3))} -{Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(2/3), x, 3, -((3*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d*(e*(c + d*x))^(2/3)))} -{Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3), x, 6, (3*b*(c + d*x)^(1/3)*Cos[a]*CosIntegral[b*(c + d*x)^(1/3)])/(d*e*(e*(c + d*x))^(1/3)) - (3*Sin[a + b*(c + d*x)^(1/3)])/(d*e*(e*(c + d*x))^(1/3)) - (3*b*(c + d*x)^(1/3)*Sin[a]*SinIntegral[b*(c + d*x)^(1/3)])/(d*e*(e*(c + d*x))^(1/3))} -{Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(5/3), x, 7, -((3*b*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(2*d*e*(e*(c + d*x))^(2/3))) - (3*b^2*(c + d*x)^(2/3)*CosIntegral[b*(c + d*x)^(1/3)]*Sin[a])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*Sin[a + b*(c + d*x)^(1/3)])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*b^2*(c + d*x)^(2/3)*Cos[a]*SinIntegral[b*(c + d*x)^(1/3)])/(2*d*e*(e*(c + d*x))^(2/3))} -{Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3), x, 9, (b^3*Cos[a + b*(c + d*x)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3)) - (b*Cos[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*CosIntegral[b*(c + d*x)^(1/3)]*Sin[a])/(8*d*e^2*(e*(c + d*x))^(1/3)) - (3*Sin[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c + d*x)*(e*(c + d*x))^(1/3)) + (b^2*Sin[a + b*(c + d*x)^(1/3)])/(8*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*Cos[a]*SinIntegral[b*(c + d*x)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3))} - - -{(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(2/3)], x, 9, (45*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(8*b^3*d) - (3*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d) - (45*e*Sqrt[Pi]*(e*(c + d*x))^(1/3)*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(8*Sqrt[2]*b^(7/2)*d*(c + d*x)^(1/3)) + (45*e*Sqrt[Pi]*(e*(c + d*x))^(1/3)*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(8*Sqrt[2]*b^(7/2)*d*(c + d*x)^(1/3)) + (15*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d)} -{(c*e + d*e*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)], x, 8, -((3*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d)) - (9*Sqrt[Pi]*(e*(c + d*x))^(2/3)*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) - (9*Sqrt[Pi]*(e*(c + d*x))^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) + (9*(e*(c + d*x))^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d*(c + d*x)^(1/3))} -{(c*e + d*e*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)], x, 5, -((3*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d)) + (3*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(2*b^2*d*(c + d*x)^(1/3))} -{Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(1/3), x, 4, -((3*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d*(e*(c + d*x))^(1/3)))} -{Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(2/3), x, 6, (3*Sqrt[Pi/2]*(c + d*x)^(2/3)*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(Sqrt[b]*d*(e*(c + d*x))^(2/3)) + (3*Sqrt[Pi/2]*(c + d*x)^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(Sqrt[b]*d*(e*(c + d*x))^(2/3))} -{Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3), x, 7, (3*Sqrt[b]*Sqrt[2*Pi]*(c + d*x)^(1/3)*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(d*e*(e*(c + d*x))^(1/3)) - (3*Sqrt[b]*Sqrt[2*Pi]*(c + d*x)^(1/3)*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(d*e*(e*(c + d*x))^(1/3)) - (3*Sin[a + b*(c + d*x)^(2/3)])/(d*e*(e*(c + d*x))^(1/3))} -{Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3), x, 7, (3*b*(c + d*x)^(2/3)*Cos[a]*CosIntegral[b*(c + d*x)^(2/3)])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*Sin[a + b*(c + d*x)^(2/3)])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*b*(c + d*x)^(2/3)*Sin[a]*SinIntegral[b*(c + d*x)^(2/3)])/(2*d*e*(e*(c + d*x))^(2/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)], x, 9, -((b^3*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(8*d)) + (b*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(4*d) - (b^4*(e*(c + d*x))^(1/3)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(8*d*(c + d*x)^(1/3)) - (b^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(8*d) + (3*(c + d*x)*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(4*d) - (b^4*(e*(c + d*x))^(1/3)*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(8*d*(c + d*x)^(1/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(1/3), x, 7, (3*b*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d*(e*(c + d*x))^(1/3)) + (3*b^2*(c + d*x)^(1/3)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(2*d*(e*(c + d*x))^(1/3)) + (3*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/(2*d*(e*(c + d*x))^(1/3)) + (3*b^2*(c + d*x)^(1/3)*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(2/3), x, 6, -((3*b*(c + d*x)^(2/3)*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(d*(e*(c + d*x))^(2/3))) + (3*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/(d*(e*(c + d*x))^(2/3)) + (3*b*(c + d*x)^(2/3)*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(d*(e*(c + d*x))^(2/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3), x, 3, (3*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(5/3), x, 4, (3*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e*(e*(c + d*x))^(2/3)) - (3*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(b^2*d*e*(e*(c + d*x))^(2/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3), x, 6, -((18*Cos[a + b/(c + d*x)^(1/3)])/(b^3*d*e^2*(e*(c + d*x))^(1/3))) + (3*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e^2*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)) - (9*Sin[a + b/(c + d*x)^(1/3)])/(b^2*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) + (18*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(b^4*d*e^2*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(8/3), x, 7, -((36*Cos[a + b/(c + d*x)^(1/3)])/(b^3*d*e^2*(e*(c + d*x))^(2/3))) + (3*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e^2*(c + d*x)^(2/3)*(e*(c + d*x))^(2/3)) + (72*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(b^5*d*e^2*(e*(c + d*x))^(2/3)) - (12*Sin[a + b/(c + d*x)^(1/3)])/(b^2*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)) + (72*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(b^4*d*e^2*(e*(c + d*x))^(2/3))} - - -{(c*e + d*e*x)^(4/3)*Sin[a + b/(c + d*x)^(2/3)], x, 11, -((8*b^3*e*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/(35*d)) + (6*b*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/(35*d) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(35*d*(c + d*x)^(1/3)) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(35*d*(c + d*x)^(1/3)) - (4*b^2*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(35*d) + (3*e*(c + d*x)^2*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(7*d)} -{(c*e + d*e*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)], x, 10, (2*b*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)*Cos[a + b/(c + d*x)^(2/3)])/(5*d) + (4*Sqrt[2]*b^(5/2)*Sqrt[Pi]*(e*(c + d*x))^(2/3)*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(5*d*(c + d*x)^(2/3)) - (4*Sqrt[2]*b^(5/2)*Sqrt[Pi]*(e*(c + d*x))^(2/3)*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(5*d*(c + d*x)^(2/3)) - (4*b^2*(e*(c + d*x))^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(5*d*(c + d*x)^(1/3)) + (3*(c + d*x)*(e*(c + d*x))^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(5*d)} -{(c*e + d*e*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)], x, 8, (3*b*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/(4*d) + (3*b^2*(e*(c + d*x))^(1/3)*CosIntegral[b/(c + d*x)^(2/3)]*Sin[a])/(4*d*(c + d*x)^(1/3)) + (3*(c + d*x)*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(4*d) + (3*b^2*(e*(c + d*x))^(1/3)*Cos[a]*SinIntegral[b/(c + d*x)^(2/3)])/(4*d*(c + d*x)^(1/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(1/3), x, 7, -((3*b*(c + d*x)^(1/3)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(2*d*(e*(c + d*x))^(1/3))) + (3*(c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/(2*d*(e*(c + d*x))^(1/3)) + (3*b*(c + d*x)^(1/3)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(2*d*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(2/3), x, 8, -((3*Sqrt[b]*Sqrt[2*Pi]*(c + d*x)^(2/3)*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(d*(e*(c + d*x))^(2/3))) + (3*Sqrt[b]*Sqrt[2*Pi]*(c + d*x)^(2/3)*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(d*(e*(c + d*x))^(2/3)) + (3*(c + d*x)*Sin[a + b/(c + d*x)^(2/3)])/(d*(e*(c + d*x))^(2/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3), x, 6, -((3*Sqrt[Pi]*(c + d*x)^(1/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(Sqrt[2]*Sqrt[b]*d*e*(e*(c + d*x))^(1/3))) - (3*Sqrt[Pi]*(c + d*x)^(1/3)*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(Sqrt[2]*Sqrt[b]*d*e*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3), x, 4, (3*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*e*(e*(c + d*x))^(2/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(7/3), x, 5, (3*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) - (3*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(2*b^2*d*e^2*(e*(c + d*x))^(1/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(8/3), x, 9, (3*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)) + (9*Sqrt[Pi/2]*(c + d*x)^(2/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(4*b^(5/2)*d*e^2*(e*(c + d*x))^(2/3)) + (9*Sqrt[Pi/2]*(c + d*x)^(2/3)*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(4*b^(5/2)*d*e^2*(e*(c + d*x))^(2/3)) - (9*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(4*b^2*d*e^2*(e*(c + d*x))^(2/3))} -{Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(10/3), x, 10, -((45*Cos[a + b/(c + d*x)^(2/3)])/(8*b^3*d*e^3*(e*(c + d*x))^(1/3))) + (3*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*e^3*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)) + (45*Sqrt[Pi]*(c + d*x)^(1/3)*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(8*Sqrt[2]*b^(7/2)*d*e^3*(e*(c + d*x))^(1/3)) - (45*Sqrt[Pi]*(c + d*x)^(1/3)*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(8*Sqrt[2]*b^(7/2)*d*e^3*(e*(c + d*x))^(1/3)) - (15*Sin[a + b/(c + d*x)^(2/3)])/(4*b^2*d*e^3*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sin[a+b (c+d x)^n] when n symbolic*) - - -{(e*x)^m*Sin[a + b*(c + d*x)^n], x, 0, Unintegrable[(e*x)^m*Sin[a + b*(c + d*x)^n], x]} - - -{x^3*Sin[a + b*(c + d*x)^n], x, 14, -((I*c^3*E^(I*a)*(c + d*x)*Gamma[1/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^n^(-1)*(2*d^4*n))) + (I*c^3*(c + d*x)*Gamma[1/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^n^(-1)*(2*d^4*n)) + (3*I*c^2*E^(I*a)*(c + d*x)^2*Gamma[2/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(2/n)*(2*d^4*n)) - (3*I*c^2*(c + d*x)^2*Gamma[2/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(2/n)*(2*d^4*n)) - (3*I*c*E^(I*a)*(c + d*x)^3*Gamma[3/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(3/n)*(2*d^4*n)) + (3*I*c*(c + d*x)^3*Gamma[3/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(3/n)*(2*d^4*n)) + (I*E^(I*a)*(c + d*x)^4*Gamma[4/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(4/n)*(2*d^4*n)) - (I*(c + d*x)^4*Gamma[4/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(4/n)*(2*d^4*n))} -{x^2*Sin[a + b*(c + d*x)^n], x, 11, (I*c^2*E^(I*a)*(c + d*x)*Gamma[1/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^n^(-1)*(2*d^3*n)) - (I*c^2*(c + d*x)*Gamma[1/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^n^(-1)*(2*d^3*n)) - (I*c*E^(I*a)*(c + d*x)^2*Gamma[2/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(2/n)*(d^3*n)) + (I*c*(c + d*x)^2*Gamma[2/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(2/n)*(d^3*n)) + (I*E^(I*a)*(c + d*x)^3*Gamma[3/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(3/n)*(2*d^3*n)) - (I*(c + d*x)^3*Gamma[3/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(3/n)*(2*d^3*n))} -{x^1*Sin[a + b*(c + d*x)^n], x, 8, -((I*c*E^(I*a)*(c + d*x)*Gamma[1/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^n^(-1)*(2*d^2*n))) + (I*c*(c + d*x)*Gamma[1/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^n^(-1)*(2*d^2*n)) + (I*E^(I*a)*(c + d*x)^2*Gamma[2/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^(2/n)*(2*d^2*n)) - (I*(c + d*x)^2*Gamma[2/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^(2/n)*(2*d^2*n))} -{x^0*Sin[a + b*(c + d*x)^n], x, 3, (I*E^(I*a)*(c + d*x)*Gamma[1/n, (-I)*b*(c + d*x)^n])/(((-I)*b*(c + d*x)^n)^n^(-1)*(2*d*n)) - (I*(c + d*x)*Gamma[1/n, I*b*(c + d*x)^n])/(E^(I*a)*(I*b*(c + d*x)^n)^n^(-1)*(2*d*n))} -{Sin[a + b*(c + d*x)^n]/x^1, x, 0, Unintegrable[Sin[a + b*(c + d*x)^n]/x, x]} -{Sin[a + b*(c + d*x)^n]/x^2, x, 0, Unintegrable[Sin[a + b*(c + d*x)^n]/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d (f+g x)^n])^p when n symbolic*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sin[c+d (f+g x)^n])^p when n symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Sin[c + d*(f + g*x)^n]), x, 16, (a*x^4)/4 - (I*b*E^(I*c)*f^3*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(2*g^4*n)) + (I*b*f^3*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(2*g^4*n)) + (3*I*b*E^(I*c)*f^2*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(2*g^4*n)) - (3*I*b*f^2*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(2/n)*(2*g^4*n)) - (3*I*b*E^(I*c)*f*(f + g*x)^3*Gamma[3/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(3/n)*(2*g^4*n)) + (3*I*b*f*(f + g*x)^3*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(3/n)*(2*g^4*n)) + (I*b*E^(I*c)*(f + g*x)^4*Gamma[4/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(4/n)*(2*g^4*n)) - (I*b*(f + g*x)^4*Gamma[4/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(4/n)*(2*g^4*n))} -{x^2*(a + b*Sin[c + d*(f + g*x)^n]), x, 13, (a*x^3)/3 + (I*b*E^(I*c)*f^2*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(2*g^3*n)) - (I*b*f^2*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(2*g^3*n)) - (I*b*E^(I*c)*f*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(g^3*n)) + (I*b*f*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(2/n)*(g^3*n)) + (I*b*E^(I*c)*(f + g*x)^3*Gamma[3/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(3/n)*(2*g^3*n)) - (I*b*(f + g*x)^3*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(3/n)*(2*g^3*n))} -{x^1*(a + b*Sin[c + d*(f + g*x)^n]), x, 10, (a*x^2)/2 - (I*b*E^(I*c)*f*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(2*g^2*n)) + (I*b*f*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(2*g^2*n)) + (I*b*E^(I*c)*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(2*g^2*n)) - (I*b*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(2/n)*(2*g^2*n))} -{x^0*(a + b*Sin[c + d*(f + g*x)^n]), x, 4, a*x + (I*b*E^(I*c)*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(2*g*n)) - (I*b*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(2*g*n))} -{(a + b*Sin[c + d*(f + g*x)^n])/x^1, x, 2, a*Log[x] + b*Unintegrable[Sin[c + d*(f + g*x)^n]/x, x]} -{(a + b*Sin[c + d*(f + g*x)^n])/x^2, x, 2, -(a/x) + b*Unintegrable[Sin[c + d*(f + g*x)^n]/x^2, x]} - - -{x^2*(a + b*Sin[c + d*(f + g*x)^n])^2, x, 28, ((2*a^2 + b^2)*f^2*x)/(2*g^2) - ((2*a^2 + b^2)*f*(f + g*x)^2)/(2*g^3) + ((2*a^2 + b^2)*(f + g*x)^3)/(6*g^3) + (I*a*b*E^(I*c)*f^2*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g^3*n)) - (I*a*b*f^2*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g^3*n)) + (2^(-2 - 1/n)*b^2*E^(2*I*c)*f^2*(f + g*x)*Gamma[1/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g^3*n)) + (2^(-2 - 1/n)*b^2*f^2*(f + g*x)*Gamma[1/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g^3*n)) - (2*I*a*b*E^(I*c)*f*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(g^3*n)) + (2*I*a*b*f*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(2/n)*(g^3*n)) - (2^(-1 - 2/n)*b^2*E^(2*I*c)*f*(f + g*x)^2*Gamma[2/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(g^3*n)) - (2^(-1 - 2/n)*b^2*f*(f + g*x)^2*Gamma[2/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^(2/n)*(g^3*n)) + (I*a*b*E^(I*c)*(f + g*x)^3*Gamma[3/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(3/n)*(g^3*n)) - (I*a*b*(f + g*x)^3*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(3/n)*(g^3*n)) + (2^(-2 - 3/n)*b^2*E^(2*I*c)*(f + g*x)^3*Gamma[3/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(3/n)*(g^3*n)) + (2^(-2 - 3/n)*b^2*(f + g*x)^3*Gamma[3/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^(3/n)*(g^3*n))} -{x^1*(a + b*Sin[c + d*(f + g*x)^n])^2, x, 19, -(((2*a^2 + b^2)*f*x)/(2*g)) + ((2*a^2 + b^2)*(f + g*x)^2)/(4*g^2) - (I*a*b*E^(I*c)*f*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g^2*n)) + (I*a*b*f*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g^2*n)) - (2^(-2 - 1/n)*b^2*E^(2*I*c)*f*(f + g*x)*Gamma[1/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g^2*n)) - (2^(-2 - 1/n)*b^2*f*(f + g*x)*Gamma[1/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g^2*n)) + (I*a*b*E^(I*c)*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(g^2*n)) - (I*a*b*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^(2/n)*(g^2*n)) + (4^(-1 - 1/n)*b^2*E^(2*I*c)*(f + g*x)^2*Gamma[2/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^(2/n)*(g^2*n)) + (4^(-1 - 1/n)*b^2*(f + g*x)^2*Gamma[2/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^(2/n)*(g^2*n))} -{x^0*(a + b*Sin[c + d*(f + g*x)^n])^2, x, 8, (1/2)*(2*a^2 + b^2)*x + (I*a*b*E^(I*c)*(f + g*x)*Gamma[1/n, (-I)*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g*n)) - (I*a*b*(f + g*x)*Gamma[1/n, I*d*(f + g*x)^n])/(E^(I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g*n)) + (2^(-2 - 1/n)*b^2*E^(2*I*c)*(f + g*x)*Gamma[1/n, -2*I*d*(f + g*x)^n])/(((-I)*d*(f + g*x)^n)^n^(-1)*(g*n)) + (2^(-2 - 1/n)*b^2*(f + g*x)*Gamma[1/n, 2*I*d*(f + g*x)^n])/(E^(2*I*c)*(I*d*(f + g*x)^n)^n^(-1)*(g*n))} -{(a + b*Sin[c + d*(f + g*x)^n])^2/x^1, x, 0, Unintegrable[(a + b*Sin[c + d*(f + g*x)^n])^2/x, x]} -{(a + b*Sin[c + d*(f + g*x)^n])^2/x^2, x, 0, Unintegrable[(a + b*Sin[c + d*(f + g*x)^n])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^2/(a + b*Sin[c + d*(f + g*x)^n]), x, 0, Unintegrable[x^2/(a + b*Sin[c + d*(f + g*x)^n]), x]} -{x^1/(a + b*Sin[c + d*(f + g*x)^n]), x, 0, Unintegrable[x/(a + b*Sin[c + d*(f + g*x)^n]), x]} -{x^0/(a + b*Sin[c + d*(f + g*x)^n]), x, 0, Unintegrable[1/(a + b*Sin[c + d*(f + g*x)^n]), x]} -{1/(x^1*(a + b*Sin[c + d*(f + g*x)^n])), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*(f + g*x)^n])), x]} -{1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])), x]} - - -{x^2/(a + b*Sin[c + d*(f + g*x)^n])^2, x, 0, Unintegrable[x^2/(a + b*Sin[c + d*(f + g*x)^n])^2, x]} -{x^1/(a + b*Sin[c + d*(f + g*x)^n])^2, x, 0, Unintegrable[x/(a + b*Sin[c + d*(f + g*x)^n])^2, x]} -{x^0/(a + b*Sin[c + d*(f + g*x)^n])^2, x, 0, Unintegrable[1/(a + b*Sin[c + d*(f + g*x)^n])^2, x]} -{1/(x^1*(a + b*Sin[c + d*(f + g*x)^n])^2), x, 0, Unintegrable[1/(x*(a + b*Sin[c + d*(f + g*x)^n])^2), x]} -{1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sin[c + d*(f + g*x)^n])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d (f+g x)^n])^p when p symbolic*) - - -{(e*x)^m*(a + b*Sin[c + d*(f + g*x)^n])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Sin[c + d*(f + g*x)^n])^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sin[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sin[c+d x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sin[c+d/x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(e + f*x)^2*(a + b*Sin[c + d/x]), x, 23, a*e^2*x + a*e*f*x^2 + (1/3)*a*f^2*x^3 + b*d*e*f*x*Cos[c + d/x] + (1/6)*b*d*f^2*x^2*Cos[c + d/x] - b*d*e^2*Cos[c]*CosIntegral[d/x] + (1/6)*b*d^3*f^2*Cos[c]*CosIntegral[d/x] + b*d^2*e*f*CosIntegral[d/x]*Sin[c] + b*e^2*x*Sin[c + d/x] - (1/6)*b*d^2*f^2*x*Sin[c + d/x] + b*e*f*x^2*Sin[c + d/x] + (1/3)*b*f^2*x^3*Sin[c + d/x] + b*d^2*e*f*Cos[c]*SinIntegral[d/x] + b*d*e^2*Sin[c]*SinIntegral[d/x] - (1/6)*b*d^3*f^2*Sin[c]*SinIntegral[d/x]} -{(e + f*x)^1*(a + b*Sin[c + d/x]), x, 15, a*e*x + (1/2)*a*f*x^2 + (1/2)*b*d*f*x*Cos[c + d/x] - b*d*e*Cos[c]*CosIntegral[d/x] + (1/2)*b*d^2*f*CosIntegral[d/x]*Sin[c] + b*e*x*Sin[c + d/x] + (1/2)*b*f*x^2*Sin[c + d/x] + (1/2)*b*d^2*f*Cos[c]*SinIntegral[d/x] + b*d*e*Sin[c]*SinIntegral[d/x]} -{(e + f*x)^0*(a + b*Sin[c + d/x]), x, 6, a*x - b*d*Cos[c]*CosIntegral[d/x] + b*x*Sin[c + d/x] + b*d*Sin[c]*SinIntegral[d/x]} -{(a + b*Sin[c + d/x])/(e + f*x)^1, x, 12, (a*Log[f + e/x])/f + (a*Log[x])/f - (b*CosIntegral[d/x]*Sin[c])/f + (b*CosIntegral[d*(f/e + 1/x)]*Sin[c - (d*f)/e])/f + (b*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/f - (b*Cos[c]*SinIntegral[d/x])/f} -{(a + b*Sin[c + d/x])/(e + f*x)^2, x, 7, a/(e*(f + e/x)) - (b*d*Cos[c - (d*f)/e]*CosIntegral[d*(f/e + 1/x)])/e^2 + (b*Sin[c + d/x])/(e*(f + e/x)) + (b*d*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/e^2} -{(a + b*Sin[c + d/x])/(e + f*x)^3, x, 15, -((a*f)/(2*e^2*(f + e/x)^2)) + a/(e^2*(f + e/x)) - (b*d*f*Cos[c + d/x])/(2*e^3*(f + e/x)) - (b*d*Cos[c - (d*f)/e]*CosIntegral[d*(f/e + 1/x)])/e^3 - (b*d^2*f*CosIntegral[d*(f/e + 1/x)]*Sin[c - (d*f)/e])/(2*e^4) - (b*f*Sin[c + d/x])/(2*e^2*(f + e/x)^2) + (b*Sin[c + d/x])/(e^2*(f + e/x)) - (b*d^2*f*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/(2*e^4) + (b*d*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/e^3} - - -{(e + f*x)^1*(a + b*Sin[c + d/x])^2, x, 27, a^2*e*x + (1/2)*a^2*f*x^2 + a*b*d*f*x*Cos[c + d/x] - 2*a*b*d*e*Cos[c]*CosIntegral[d/x] - b^2*d^2*f*Cos[2*c]*CosIntegral[(2*d)/x] + a*b*d^2*f*CosIntegral[d/x]*Sin[c] - b^2*d*e*CosIntegral[(2*d)/x]*Sin[2*c] + 2*a*b*e*x*Sin[c + d/x] + a*b*f*x^2*Sin[c + d/x] + b^2*d*f*x*Cos[c + d/x]*Sin[c + d/x] + b^2*e*x*Sin[c + d/x]^2 + (1/2)*b^2*f*x^2*Sin[c + d/x]^2 + a*b*d^2*f*Cos[c]*SinIntegral[d/x] + 2*a*b*d*e*Sin[c]*SinIntegral[d/x] - b^2*d*e*Cos[2*c]*SinIntegral[(2*d)/x] + b^2*d^2*f*Sin[2*c]*SinIntegral[(2*d)/x]} -{(e + f*x)^0*(a + b*Sin[c + d/x])^2, x, 12, a^2*x - 2*a*b*d*Cos[c]*CosIntegral[d/x] - b^2*d*CosIntegral[(2*d)/x]*Sin[2*c] + 2*a*b*x*Sin[c + d/x] + b^2*x*Sin[c + d/x]^2 + 2*a*b*d*Sin[c]*SinIntegral[d/x] - b^2*d*Cos[2*c]*SinIntegral[(2*d)/x]} -{(a + b*Sin[c + d/x])^2/(e + f*x)^1, x, 22, -((b^2*Cos[2*c - (2*d*f)/e]*CosIntegral[2*d*(f/e + 1/x)])/(2*f)) + (b^2*Cos[2*c]*CosIntegral[(2*d)/x])/(2*f) + (a^2*Log[f + e/x])/f + (b^2*Log[f + e/x])/(2*f) + (a^2*Log[x])/f + (b^2*Log[x])/(2*f) - (2*a*b*CosIntegral[d/x]*Sin[c])/f + (2*a*b*CosIntegral[d*(f/e + 1/x)]*Sin[c - (d*f)/e])/f + (2*a*b*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/f + (b^2*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + 1/x)])/(2*f) - (2*a*b*Cos[c]*SinIntegral[d/x])/f - (b^2*Sin[2*c]*SinIntegral[(2*d)/x])/(2*f)} -{(a + b*Sin[c + d/x])^2/(e + f*x)^2, x, 12, a^2/(e*(f + e/x)) - (2*a*b*d*Cos[c - (d*f)/e]*CosIntegral[d*(f/e + 1/x)])/e^2 - (b^2*d*CosIntegral[2*d*(f/e + 1/x)]*Sin[2*c - (2*d*f)/e])/e^2 + (2*a*b*Sin[c + d/x])/(e*(f + e/x)) + (b^2*Sin[c + d/x]^2)/(e*(f + e/x)) + (2*a*b*d*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/e^2 - (b^2*d*Cos[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + 1/x)])/e^2} -{(a + b*Sin[c + d/x])^2/(e + f*x)^3, x, 27, -((a^2*f)/(2*e^2*(f + e/x)^2)) + a^2/(e^2*(f + e/x)) - (a*b*d*f*Cos[c + d/x])/(e^3*(f + e/x)) - (2*a*b*d*Cos[c - (d*f)/e]*CosIntegral[d*(f/e + 1/x)])/e^3 + (b^2*d^2*f*Cos[2*c - (2*d*f)/e]*CosIntegral[2*d*(f/e + 1/x)])/e^4 - (b^2*d*CosIntegral[2*d*(f/e + 1/x)]*Sin[2*c - (2*d*f)/e])/e^3 - (a*b*d^2*f*CosIntegral[d*(f/e + 1/x)]*Sin[c - (d*f)/e])/e^4 - (a*b*f*Sin[c + d/x])/(e^2*(f + e/x)^2) + (2*a*b*Sin[c + d/x])/(e^2*(f + e/x)) - (b^2*d*f*Cos[c + d/x]*Sin[c + d/x])/(e^3*(f + e/x)) - (b^2*f*Sin[c + d/x]^2)/(2*e^2*(f + e/x)^2) + (b^2*Sin[c + d/x]^2)/(e^2*(f + e/x)) - (a*b*d^2*f*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/e^4 + (2*a*b*d*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + 1/x)])/e^3 - (b^2*d*Cos[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + 1/x)])/e^3 - (b^2*d^2*f*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + 1/x)])/e^4} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(e + f*x)^2/(a + b*Sin[c + d/x]), x, 0, Unintegrable[(e + f*x)^2/(a + b*Sin[c + d/x]), x]} -{(e + f*x)^1/(a + b*Sin[c + d/x]), x, 0, Unintegrable[(e + f*x)/(a + b*Sin[c + d/x]), x]} -{(e + f*x)^0/(a + b*Sin[c + d/x]), x, 0, Unintegrable[1/(a + b*Sin[c + d/x]), x]} -{(e + f*x)^1/(a + b*Sin[c + d/x]), x, 0, Unintegrable[(e + f*x)/(a + b*Sin[c + d/x]), x]} -{(e + f*x)^2/(a + b*Sin[c + d/x]), x, 0, Unintegrable[(e + f*x)^2/(a + b*Sin[c + d/x]), x]} - - -{(e + f*x)^2/(a + b*Sin[c + d/x])^2, x, 0, Unintegrable[(e + f*x)^2/(a + b*Sin[c + d/x])^2, x]} -{(e + f*x)^1/(a + b*Sin[c + d/x])^2, x, 0, Unintegrable[(e + f*x)/(a + b*Sin[c + d/x])^2, x]} -{(e + f*x)^0/(a + b*Sin[c + d/x])^2, x, 0, Unintegrable[1/(a + b*Sin[c + d/x])^2, x]} -{(e + f*x)^1/(a + b*Sin[c + d/x])^2, x, 0, Unintegrable[(e + f*x)/(a + b*Sin[c + d/x])^2, x]} -{(e + f*x)^2/(a + b*Sin[c + d/x])^2, x, 0, Unintegrable[(e + f*x)^2/(a + b*Sin[c + d/x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sin[c+d x^n])^p when p symbolic*) - - -{(e + f*x)^m*(a + b*Sin[c + d/x])^p, x, 0, Unintegrable[(e + f*x)^m*(a + b*Sin[c + d/x])^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sin[c+d x^n]^p)^q*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c Sin[a+b x^n]^p)^(q/3)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c Sin[a+b x^n]^3)^(1/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^m*(c*Sin[a + b*x]^3)^(1/3), x, 4, -(E^(I*a)*x^m*Csc[a + b*x]*Gamma[1 + m, (-I)*b*x]*(c*Sin[a + b*x]^3)^(1/3))/(2*b*((-I)*b*x)^m) - (x^m*Csc[a + b*x]*Gamma[1 + m, I*b*x]*(c*Sin[a + b*x]^3)^(1/3))/(2*b*E^(I*a)*(I*b*x)^m)} -{x^3*(c*Sin[a + b*x]^3)^(1/3), x, 5, (-6*(c*Sin[a + b*x]^3)^(1/3))/b^4 + (3*x^2*(c*Sin[a + b*x]^3)^(1/3))/b^2 + (6*x*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b^3 - (x^3*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b} -{x^2*(c*Sin[a + b*x]^3)^(1/3), x, 4, (2*x*(c*Sin[a + b*x]^3)^(1/3))/b^2 + (2*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b^3 - (x^2*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b} -{x*(c*Sin[a + b*x]^3)^(1/3), x, 3, (c*Sin[a + b*x]^3)^(1/3)/b^2 - (x*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b} -{(c*Sin[a + b*x]^3)^(1/3), x, 2, -((Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b)} -{(c*Sin[a + b*x]^3)^(1/3)/x, x, 4, CosIntegral[b*x]*Csc[a + b*x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3) + Cos[a]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*SinIntegral[b*x]} -{(c*Sin[a + b*x]^3)^(1/3)/x^2, x, 5, -((c*Sin[a + b*x]^3)^(1/3)/x) + b*Cos[a]*CosIntegral[b*x]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3) - b*Csc[a + b*x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3)*SinIntegral[b*x]} -{(c*Sin[a + b*x]^3)^(1/3)/x^3, x, 6, -(c*Sin[a + b*x]^3)^(1/3)/(2*x^2) - (b*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/(2*x) - (b^2*CosIntegral[b*x]*Csc[a + b*x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3))/2 - (b^2*Cos[a]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*SinIntegral[b*x])/2} - - -{x^m*(c*Sin[a + b*x^2]^3)^(1/3), x, 4, (I/4)*E^(I*a)*x^(1 + m)*((-I)*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]*Gamma[(1 + m)/2, (-I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3) - ((I/4)*x^(1 + m)*(I*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]*Gamma[(1 + m)/2, I*b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3))/E^(I*a)} -{x^3*(c*Sin[a + b*x^2]^3)^(1/3), x, 4, (c*Sin[a + b*x^2]^3)^(1/3)/(2*b^2) - (x^2*Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3))/(2*b)} -{x^2*(c*Sin[a + b*x^2]^3)^(1/3), x, 5, -(x*Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3))/(2*b) + (Sqrt[Pi/2]*Cos[a]*Csc[a + b*x^2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*(c*Sin[a + b*x^2]^3)^(1/3))/(2*b^(3/2)) - (Sqrt[Pi/2]*Csc[a + b*x^2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3))/(2*b^(3/2))} -{x*(c*Sin[a + b*x^2]^3)^(1/3), x, 3, -(Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3))/(2*b)} -{(c*Sin[a + b*x^2]^3)^(1/3), x, 4, (Sqrt[Pi/2]*Cos[a]*Csc[a + b*x^2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*(c*Sin[a + b*x^2]^3)^(1/3))/Sqrt[b] + (Sqrt[Pi/2]*Csc[a + b*x^2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3))/Sqrt[b]} -{(c*Sin[a + b*x^2]^3)^(1/3)/x, x, 4, (CosIntegral[b*x^2]*Csc[a + b*x^2]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3))/2 + (Cos[a]*Csc[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3)*SinIntegral[b*x^2])/2} -{(c*Sin[a + b*x^2]^3)^(1/3)/x^2, x, 5, -((c*Sin[a + b*x^2]^3)^(1/3)/x) + Sqrt[b]*Sqrt[2*Pi]*Cos[a]*Csc[a + b*x^2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*(c*Sin[a + b*x^2]^3)^(1/3) - Sqrt[b]*Sqrt[2*Pi]*Csc[a + b*x^2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3)} -{(c*Sin[a + b*x^2]^3)^(1/3)/x^3, x, 6, -(c*Sin[a + b*x^2]^3)^(1/3)/(2*x^2) + (b*Cos[a]*CosIntegral[b*x^2]*Csc[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(1/3))/2 - (b*Csc[a + b*x^2]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3)*SinIntegral[b*x^2])/2} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{x^m*(c*Sin[a + b*x^n]^3)^(1/3), x, 4, ((I/2)*E^(I*a)*x^(1 + m)*Csc[a + b*x^n]*Gamma[(1 + m)/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^((1 + m)/n)) - ((I/2)*x^(1 + m)*Csc[a + b*x^n]*Gamma[(1 + m)/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^((1 + m)/n))} -{x^3*(c*Sin[a + b*x^n]^3)^(1/3), x, 4, ((I/2)*E^(I*a)*x^4*Csc[a + b*x^n]*Gamma[4/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^(4/n)) - ((I/2)*x^4*Csc[a + b*x^n]*Gamma[4/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^(4/n))} -{x^2*(c*Sin[a + b*x^n]^3)^(1/3), x, 4, ((I/2)*E^(I*a)*x^3*Csc[a + b*x^n]*Gamma[3/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^(3/n)) - ((I/2)*x^3*Csc[a + b*x^n]*Gamma[3/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^(3/n))} -{x*(c*Sin[a + b*x^n]^3)^(1/3), x, 4, ((I/2)*E^(I*a)*x^2*Csc[a + b*x^n]*Gamma[2/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^(2/n)) - ((I/2)*x^2*Csc[a + b*x^n]*Gamma[2/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^(2/n))} -{(c*Sin[a + b*x^n]^3)^(1/3), x, 4, ((I/2)*E^(I*a)*x*Csc[a + b*x^n]*Gamma[n^(-1), (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^n^(-1)) - ((I/2)*x*Csc[a + b*x^n]*Gamma[n^(-1), I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^n^(-1))} -{(c*Sin[a + b*x^n]^3)^(1/3)/x, x, 4, (CosIntegral[b*x^n]*Csc[a + b*x^n]*Sin[a]*(c*Sin[a + b*x^n]^3)^(1/3))/n + (Cos[a]*Csc[a + b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3)*SinIntegral[b*x^n])/n} -{(c*Sin[a + b*x^n]^3)^(1/3)/x^2, x, 4, ((I/2)*E^(I*a)*((-I)*b*x^n)^n^(-1)*Csc[a + b*x^n]*Gamma[-n^(-1), (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*x) - ((I/2)*(I*b*x^n)^n^(-1)*Csc[a + b*x^n]*Gamma[-n^(-1), I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*x)} -{(c*Sin[a + b*x^n]^3)^(1/3)/x^3, x, 4, ((I/2)*E^(I*a)*((-I)*b*x^n)^(2/n)*Csc[a + b*x^n]*Gamma[-2/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*x^2) - ((I/2)*(I*b*x^n)^(2/n)*Csc[a + b*x^n]*Gamma[-2/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*x^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c Sin[a+b x^n]^3)^(2/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^m*(c*Sin[a + b*x]^3)^(2/3), x, 6, (x^(1 + m)*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/(2*(1 + m)) + (I*2^(-3 - m)*E^((2*I)*a)*x^m*Csc[a + b*x]^2*Gamma[1 + m, (-2*I)*b*x]*(c*Sin[a + b*x]^3)^(2/3))/(b*((-I)*b*x)^m) - (I*2^(-3 - m)*x^m*Csc[a + b*x]^2*Gamma[1 + m, (2*I)*b*x]*(c*Sin[a + b*x]^3)^(2/3))/(b*E^((2*I)*a)*(I*b*x)^m)} -{x^3*(c*Sin[a + b*x]^3)^(2/3), x, 5, (-3*(c*Sin[a + b*x]^3)^(2/3))/(8*b^4) + (3*x^2*(c*Sin[a + b*x]^3)^(2/3))/(4*b^2) + (3*x*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(4*b^3) - (x^3*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(2*b) - (3*x^2*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/(8*b^2) + (x^4*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/8} -{x^2*(c*Sin[a + b*x]^3)^(2/3), x, 5, (x*(c*Sin[a + b*x]^3)^(2/3))/(2*b^2) + (Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(4*b^3) - (x^2*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(2*b) - (x*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/(4*b^2) + (x^3*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/6} -{x*(c*Sin[a + b*x]^3)^(2/3), x, 3, (c*Sin[a + b*x]^3)^(2/3)/(4*b^2) - (x*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(2*b) + (x^2*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/4} -{(c*Sin[a + b*x]^3)^(2/3), x, 3, -(Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/(2*b) + (x*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/2} -{(c*Sin[a + b*x]^3)^(2/3)/x, x, 6, -(Cos[2*a]*CosIntegral[2*b*x]*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3))/2 + (Csc[a + b*x]^2*Log[x]*(c*Sin[a + b*x]^3)^(2/3))/2 + (Csc[a + b*x]^2*Sin[2*a]*(c*Sin[a + b*x]^3)^(2/3)*SinIntegral[2*b*x])/2} -{(c*Sin[a + b*x]^3)^(2/3)/x^2, x, 6, -((c*Sin[a + b*x]^3)^(2/3)/x) + b*CosIntegral[2*b*x]*Csc[a + b*x]^2*Sin[2*a]*(c*Sin[a + b*x]^3)^(2/3) + b*Cos[2*a]*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*SinIntegral[2*b*x]} -{(c*Sin[a + b*x]^3)^(2/3)/x^3, x, 8, -((c*Sin[a + b*x]^3)^(2/3)/(2*x^2)) - (b*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/x + b^2*Cos[2*a]*CosIntegral[2*b*x]*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3) - b^2*Csc[a + b*x]^2*Sin[2*a]*(c*Sin[a + b*x]^3)^(2/3)*SinIntegral[2*b*x]} - - -{x^m*(c*Sin[a + b*x^2]^3)^(2/3), x, 6, (x^(1 + m)*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(2*(1 + m)) + 2^(-7/2 - m/2)*E^((2*I)*a)*x^(1 + m)*((-I)*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3) + (2^(-7/2 - m/2)*x^(1 + m)*(I*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3))/E^((2*I)*a)} -{x^3*(c*Sin[a + b*x^2]^3)^(2/3), x, 4, (c*Sin[a + b*x^2]^3)^(2/3)/(8*b^2) - (x^2*Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*b) + (x^4*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/8} -{x^2*(c*Sin[a + b*x^2]^3)^(2/3), x, 7, (x^3*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/6 + (Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) + (Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) - (x*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*Sin[2*a + 2*b*x^2])/(8*b)} -{x*(c*Sin[a + b*x^2]^3)^(2/3), x, 4, -(Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*b) + (x^2*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4} -{(c*Sin[a + b*x^2]^3)^(2/3), x, 6, (x*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/2 - (Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*Sqrt[b]) + (Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*Sqrt[b])} -{(c*Sin[a + b*x^2]^3)^(2/3)/x, x, 6, -(Cos[2*a]*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4 + (Csc[a + b*x^2]^2*Log[x]*(c*Sin[a + b*x^2]^3)^(2/3))/2 + (Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2])/4} -{(c*Sin[a + b*x^2]^3)^(2/3)/x^2, x, 7, -((c*Sin[a + b*x^2]^3)^(2/3)/x) + Sqrt[b]*Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3) + Sqrt[b]*Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3)} -{(c*Sin[a + b*x^2]^3)^(2/3)/x^3, x, 8, -((Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(4*x^2)) + (Cos[2*(a + b*x^2)]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(4*x^2) + (1/2)*b*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3) + (1/2)*b*Cos[2*a]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{x^m*(c*Sin[a + b*x^n]^3)^(2/3), x, 6, (x^(1 + m)*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/(2*(1 + m)) + (E^((2*I)*a)*x^(1 + m)*Csc[a + b*x^n]^2*Gamma[(1 + m)/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(2^((1 + m + 2*n)/n)*n*((-I)*b*x^n)^((1 + m)/n)) + (x^(1 + m)*Csc[a + b*x^n]^2*Gamma[(1 + m)/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(2^((1 + m + 2*n)/n)*E^((2*I)*a)*n*(I*b*x^n)^((1 + m)/n))} -{x^3*(c*Sin[a + b*x^n]^3)^(2/3), x, 6, (x^4*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/8 + (4^(-1 - 2/n)*E^((2*I)*a)*x^4*Csc[a + b*x^n]^2*Gamma[4/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^(4/n)) + (4^(-1 - 2/n)*x^4*Csc[a + b*x^n]^2*Gamma[4/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^(4/n))} -{x^2*(c*Sin[a + b*x^n]^3)^(2/3), x, 6, (x^3*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/6 + (2^(-2 - 3/n)*E^((2*I)*a)*x^3*Csc[a + b*x^n]^2*Gamma[3/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^(3/n)) + (2^(-2 - 3/n)*x^3*Csc[a + b*x^n]^2*Gamma[3/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^(3/n))} -{x*(c*Sin[a + b*x^n]^3)^(2/3), x, 6, (x^2*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/4 + (4^(-1 - n^(-1))*E^((2*I)*a)*x^2*Csc[a + b*x^n]^2*Gamma[2/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^(2/n)) + (4^(-1 - n^(-1))*x^2*Csc[a + b*x^n]^2*Gamma[2/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^(2/n))} -{(c*Sin[a + b*x^n]^3)^(2/3), x, 6, (x*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/2 + (2^(-2 - n^(-1))*E^((2*I)*a)*x*Csc[a + b*x^n]^2*Gamma[n^(-1), (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^n^(-1)) + (2^(-2 - n^(-1))*x*Csc[a + b*x^n]^2*Gamma[n^(-1), (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^n^(-1))} -{(c*Sin[a + b*x^n]^3)^(2/3)/x, x, 6, -(Cos[2*a]*CosIntegral[2*b*x^n]*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/(2*n) + (Csc[a + b*x^n]^2*Log[x]*(c*Sin[a + b*x^n]^3)^(2/3))/2 + (Csc[a + b*x^n]^2*Sin[2*a]*(c*Sin[a + b*x^n]^3)^(2/3)*SinIntegral[2*b*x^n])/(2*n)} -{(c*Sin[a + b*x^n]^3)^(2/3)/x^2, x, 6, -(Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/(2*x) + (2^(-2 + n^(-1))*E^((2*I)*a)*((-I)*b*x^n)^n^(-1)*Csc[a + b*x^n]^2*Gamma[-n^(-1), (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*x) + (2^(-2 + n^(-1))*(I*b*x^n)^n^(-1)*Csc[a + b*x^n]^2*Gamma[-n^(-1), (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*x)} -{(c*Sin[a + b*x^n]^3)^(2/3)/x^3, x, 6, -(Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/(4*x^2) + (4^(-1 + n^(-1))*E^((2*I)*a)*((-I)*b*x^n)^(2/n)*Csc[a + b*x^n]^2*Gamma[-2/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*x^2) + (4^(-1 + n^(-1))*(I*b*x^n)^(2/n)*Csc[a + b*x^n]^2*Gamma[-2/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*x^2)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m deleted file mode 100644 index 587796b..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.13 (d+e x)^m sin(a+b x+c x^2)^n.m +++ /dev/null @@ -1,60 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form (d+e x)^m Sin[a+b x+c x^2]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sin[a+b x+c x^2]^n*) - - -{x^2*Sin[a + b*x + c*x^2], x, 8, (b*Cos[a + b*x + c*x^2])/(4*c^2) - (x*Cos[a + b*x + c*x^2])/(2*c) + (Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + (b^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) + (b^2*Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))} -{x*Sin[a + b*x + c*x^2], x, 4, -(Cos[a + b*x + c*x^2]/(2*c)) - (b*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - (b*Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))} -{Sin[a + b*x + c*x^2], x, 3, (Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] + (Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]} -{Sin[a + b*x + c*x^2]/x, x, 0, Unintegrable[Sin[a + b*x + c*x^2]/x, x]} -{Sin[a + b*x + c*x^2]/x^2 - b*Cos[a + b*x + c*x^2]/x, x, 5, Sqrt[c]*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] - Sqrt[c]*Sqrt[2*Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)] - Sin[a + b*x + c*x^2]/x} - -{x^2*Sin[a + b*x - c*x^2], x, 8, (b*Cos[a + b*x - c*x^2])/(4*c^2) + (x*Cos[a + b*x - c*x^2])/(2*c) + (Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + (b^2*Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) - (b^2*Sqrt[Pi/2]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(4*c^(5/2)) + (Sqrt[Pi/2]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(2*c^(3/2))} -{x*Sin[a + b*x - c*x^2], x, 4, Cos[a + b*x - c*x^2]/(2*c) + (b*Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - (b*Sqrt[Pi/2]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(2*c^(3/2))} -{Sin[a + b*x - c*x^2], x, 3, (Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] - (Sqrt[Pi/2]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/Sqrt[c]} -{Sin[a + b*x - c*x^2]/x, x, 0, Unintegrable[Sin[a + b*x - c*x^2]/x, x]} -{Sin[a + b*x - c*x^2]/x^2 - b*Cos[a + b*x - c*x^2]/x, x, 5, Sqrt[c]*Sqrt[2*Pi]*Cos[a + b^2/(4*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] + Sqrt[c]*Sqrt[2*Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)] - Sin[a + b*x - c*x^2]/x} - -{x^2*Sin[1/4 + x + x^2], x, 6, (1/4)*Cos[1/4 + x + x^2] - (1/2)*x*Cos[1/4 + x + x^2] + (1/2)*Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] + (1/4)*Sqrt[Pi/2]*FresnelS[(1 + 2*x)/Sqrt[2*Pi]]} -{x*Sin[1/4 + x + x^2], x, 3, (-(1/2))*Cos[1/4 + x + x^2] - (1/2)*Sqrt[Pi/2]*FresnelS[(1 + 2*x)/Sqrt[2*Pi]]} -{Sin[1/4 + x + x^2], x, 2, Sqrt[Pi/2]*FresnelS[(1 + 2*x)/Sqrt[2*Pi]]} -{Sin[1/4 + x + x^2]/x, x, 0, Unintegrable[Sin[1/4 + x + x^2]/x, x]} -{Sin[1/4 + x + x^2]/x^2, x, 3, Sqrt[2*Pi]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] + Unintegrable[Cos[1/4 + x + x^2]/x, x] - Sin[1/4 + x + x^2]/x} - - -{x^2*Sin[a + b*x + c*x^2]^2, x, 10, x^3/6 - (b^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) + (Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) + (Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) + (b^2*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) + (b*Sin[2*a + 2*b*x + 2*c*x^2])/(16*c^2) - (x*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{x*Sin[a + b*x + c*x^2]^2, x, 6, x^2/4 + (b*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) - (b*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2)) - Sin[2*a + 2*b*x + 2*c*x^2]/(8*c)} -{Sin[a + b*x + c*x^2]^2, x, 5, x/2 - (Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(4*Sqrt[c]) + (Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(4*Sqrt[c])} -{Sin[a + b*x + c*x^2]^2/x, x, 2, (-(1/2))*Unintegrable[Cos[2*a + 2*b*x + 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Sin[a + b*x - c*x^2]^2, x, 10, x^3/6 + (b^2*Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) - (Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) + (Sqrt[Pi]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(16*c^(3/2)) + (b^2*Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(16*c^(5/2)) + (b*Sin[2*a + 2*b*x - 2*c*x^2])/(16*c^2) + (x*Sin[2*a + 2*b*x - 2*c*x^2])/(8*c)} -{x*Sin[a + b*x - c*x^2]^2, x, 6, x^2/4 + (b*Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) + (b*Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(8*c^(3/2)) + Sin[2*a + 2*b*x - 2*c*x^2]/(8*c)} -{Sin[a + b*x - c*x^2]^2, x, 5, x/2 + (Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(4*Sqrt[c]) + (Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(4*Sqrt[c])} -{Sin[a + b*x - c*x^2]^2/x, x, 2, (-(1/2))*Unintegrable[Cos[2*a + 2*b*x - 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Sin[1/4 + x + x^2]^2, x, 8, x^3/6 - (1/16)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]] + (1/16)*Sqrt[Pi]*FresnelS[(1 + 2*x)/Sqrt[Pi]] + (1/16)*Sin[1/2 + 2*x + 2*x^2] - (1/8)*x*Sin[1/2 + 2*x + 2*x^2]} -{x*Sin[1/4 + x + x^2]^2, x, 5, x^2/4 + (1/8)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]] - (1/8)*Sin[1/2 + 2*x + 2*x^2]} -{Sin[1/4 + x + x^2]^2, x, 4, x/2 - (1/4)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]]} -{Sin[1/4 + x + x^2]^2/x, x, 2, (-(1/2))*Unintegrable[Cos[1/2 + 2*x + 2*x^2]/x, x] + Log[x]/2} -{Sin[1/4 + x + x^2]^2/x^2, x, 5, -(1/(2*x)) + Cos[1/2 + 2*x + 2*x^2]/(2*x) + Sqrt[Pi]*FresnelS[(1 + 2*x)/Sqrt[Pi]] + Unintegrable[Sin[1/2 + 2*x + 2*x^2]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m Sin[a+b x+c x^2]^n*) - - -{(d + e*x)^2*Sin[a + b*x + c*x^2], x, 8, -((e*(2*c*d - b*e)*Cos[a + b*x + c*x^2])/(4*c^2)) - (e*(d + e*x)*Cos[a + b*x + c*x^2])/(2*c) + (e^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (e^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))} -{(d + e*x)^1*Sin[a + b*x + c*x^2], x, 4, -((e*Cos[a + b*x + c*x^2])/(2*c)) + ((2*c*d - b*e)*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + ((2*c*d - b*e)*Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))} -{(d + e*x)^0*Sin[a + b*x + c*x^2], x, 3, (Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] + (Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]} -{Sin[a + b*x + c*x^2]/(d + e*x)^1, x, 0, Unintegrable[Sin[a + b*x + c*x^2]/(d + e*x), x]} - - -{(d + e*x)^2*Sin[a + b*x + c*x^2]^2, x, 10, (d + e*x)^3/(6*e) - ((2*c*d - b*e)^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) + (e^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) + (e^2*Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) + ((2*c*d - b*e)^2*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) - (e*(2*c*d - b*e)*Sin[2*a + 2*b*x + 2*c*x^2])/(16*c^2) - (e*(d + e*x)*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{(d + e*x)^1*Sin[a + b*x + c*x^2]^2, x, 6, (d + e*x)^2/(4*e) - ((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) + ((2*c*d - b*e)*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2)) - (e*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{(d + e*x)^0*Sin[a + b*x + c*x^2]^2, x, 5, x/2 - (Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(4*Sqrt[c]) + (Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(4*Sqrt[c])} -{Sin[a + b*x + c*x^2]^2/(d + e*x)^1, x, 2, (-(1/2))*Unintegrable[Cos[2*a + 2*b*x + 2*c*x^2]/(d + e*x), x] + Log[d + e*x]/(2*e)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m deleted file mode 100644 index 578fee9..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.1 (a+b sin)^m (c+d sin)^n.m +++ /dev/null @@ -1,1394 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2, x, 9, (3*a^2*x)/4 - (2*a^2*Cos[e + f*x])/f + (a^2*Cos[e + f*x]^3)/f - (a^2*Cos[e + f*x]^5)/(5*f) - (3*a^2*Cos[e + f*x]*Sin[e + f*x])/(4*f) - (a^2*Cos[e + f*x]*Sin[e + f*x]^3)/(2*f)} - - -{Sin[e + f*x]^3*(a + a*Sin[e + f*x])^3, x, 13, (23*a^3*x)/16 - (4*a^3*Cos[e + f*x])/f + (7*a^3*Cos[e + f*x]^3)/(3*f) - (3*a^3*Cos[e + f*x]^5)/(5*f) - (23*a^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (23*a^3*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (a^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^4/(a + a*Sin[x]), x, 6, -((3*x)/(2*a)) - (4*Cos[x])/a + (4*Cos[x]^3)/(3*a) + (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^3)/(a + a*Sin[x])} -{Sin[x]^3/(a + a*Sin[x]), x, 2, (3*x)/(2*a) + (2*Cos[x])/a - (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^2)/(a + a*Sin[x])} -{Sin[x]^2/(a + a*Sin[x]), x, 4, -(x/a) - Cos[x]/a - Cos[x]/(a*(1 + Sin[x]))} -{Sin[x]^1/(a + a*Sin[x]), x, 2, x/a + Cos[x]/(a + a*Sin[x])} -{Sin[x]^0/(a + a*Sin[x]), x, 1, -(Cos[x]/(a + a*Sin[x]))} -{Csc[x]^1/(a + a*Sin[x]), x, 3, -(ArcTanh[Cos[x]]/a) + Cos[x]/(a + a*Sin[x])} -{Csc[x]^2/(a + a*Sin[x]), x, 5, ArcTanh[Cos[x]]/a - (2*Cot[x])/a + Cot[x]/(a + a*Sin[x])} -{Csc[x]^3/(a + a*Sin[x]), x, 6, -((3*ArcTanh[Cos[x]])/(2*a)) + (2*Cot[x])/a - (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x])/(a + a*Sin[x])} -{Csc[x]^4/(a + a*Sin[x]), x, 6, (3*ArcTanh[Cos[x]])/(2*a) - (4*Cot[x])/a - (4*Cot[x]^3)/(3*a) + (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^2)/(a + a*Sin[x])} - - -{Sin[x]^4/(a + a*Sin[x])^2, x, 3, (7*x)/(2*a^2) + (16*Cos[x])/(3*a^2) - (7*Cos[x]*Sin[x])/(2*a^2) + (8*Cos[x]*Sin[x]^2)/(3*a^2*(1 + Sin[x])) + (Cos[x]*Sin[x]^3)/(3*(a + a*Sin[x])^2)} -{Sin[x]^3/(a + a*Sin[x])^2, x, 6, -((2*x)/a^2) - (4*Cos[x])/(3*a^2) - (2*Cos[x])/(a^2*(1 + Sin[x])) + (Cos[x]*Sin[x]^2)/(3*(a + a*Sin[x])^2)} -{Sin[x]^2/(a + a*Sin[x])^2, x, 3, x/a^2 + (5*Cos[x])/(3*a^2*(1 + Sin[x])) - Cos[x]/(3*(a + a*Sin[x])^2)} -{Sin[x]^1/(a + a*Sin[x])^2, x, 2, Cos[x]/(3*(a + a*Sin[x])^2) - (2*Cos[x])/(3*(a^2 + a^2*Sin[x]))} -{Sin[x]^0/(a + a*Sin[x])^2, x, 2, -(Cos[x]/(3*(a + a*Sin[x])^2)) - Cos[x]/(3*(a^2 + a^2*Sin[x]))} -{Csc[x]^1/(a + a*Sin[x])^2, x, 4, -(ArcTanh[Cos[x]]/a^2) + (4*Cos[x])/(3*a^2*(1 + Sin[x])) + Cos[x]/(3*(a + a*Sin[x])^2)} -{Csc[x]^2/(a + a*Sin[x])^2, x, 6, (2*ArcTanh[Cos[x]])/a^2 - (10*Cot[x])/(3*a^2) + (2*Cot[x])/(a^2*(1 + Sin[x])) + Cot[x]/(3*(a + a*Sin[x])^2)} -{Csc[x]^3/(a + a*Sin[x])^2, x, 7, -((7*ArcTanh[Cos[x]])/(2*a^2)) + (16*Cot[x])/(3*a^2) - (7*Cot[x]*Csc[x])/(2*a^2) + (8*Cot[x]*Csc[x])/(3*a^2*(1 + Sin[x])) + (Cot[x]*Csc[x])/(3*(a + a*Sin[x])^2)} -{Csc[x]^4/(a + a*Sin[x])^2, x, 7, (5*ArcTanh[Cos[x]])/a^2 - (4*Cot[x])/a^2 - Cot[x]^3/(3*a^2) + (Cot[x]*Csc[x])/a^2 - Cos[x]/(3*a^2*(1 + Sin[x])^2) - (13*Cos[x])/(3*a^2*(1 + Sin[x])), (5*ArcTanh[Cos[x]])/a^2 - (12*Cot[x])/a^2 - (4*Cot[x]^3)/a^2 + (5*Cot[x]*Csc[x])/a^2 + (10*Cot[x]*Csc[x]^2)/(3*a^2*(1 + Sin[x])) + (Cot[x]*Csc[x]^2)/(3*(a + a*Sin[x])^2)} - - -{Sin[x]^6/(a + a*Sin[x])^3, x, 8, -((23*x)/(2*a^3)) - (136*Cos[x])/(5*a^3) + (136*Cos[x]^3)/(15*a^3) + (23*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^5)/(5*(a + a*Sin[x])^3) + (13*Cos[x]*Sin[x]^4)/(15*a*(a + a*Sin[x])^2) + (23*Cos[x]*Sin[x]^3)/(3*(a^3 + a^3*Sin[x]))} -{Sin[x]^5/(a + a*Sin[x])^3, x, 4, (13*x)/(2*a^3) + (152*Cos[x])/(15*a^3) - (13*Cos[x]*Sin[x])/(2*a^3) + (Cos[x]*Sin[x]^4)/(5*(a + a*Sin[x])^3) + (11*Cos[x]*Sin[x]^3)/(15*a*(a + a*Sin[x])^2) + (76*Cos[x]*Sin[x]^2)/(15*(a^3 + a^3*Sin[x]))} -{Sin[x]^4/(a + a*Sin[x])^3, x, 7, -((3*x)/a^3) - (9*Cos[x])/(5*a^3) + (Cos[x]*Sin[x]^3)/(5*(a + a*Sin[x])^3) + (3*Cos[x]*Sin[x]^2)/(5*a*(a + a*Sin[x])^2) - (3*Cos[x])/(a^3 + a^3*Sin[x])} -{Sin[x]^3/(a + a*Sin[x])^3, x, 5, x/a^3 + (Cos[x]*Sin[x]^2)/(5*(a + a*Sin[x])^3) - (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (29*Cos[x])/(15*(a^3 + a^3*Sin[x]))} -{Sin[x]^2/(a + a*Sin[x])^3, x, 3, -(Cos[x]/(5*(a + a*Sin[x])^3)) + (8*Cos[x])/(15*a*(a + a*Sin[x])^2) - (7*Cos[x])/(15*(a^3 + a^3*Sin[x]))} -{Sin[x]^1/(a + a*Sin[x])^3, x, 3, Cos[x]/(5*(a + a*Sin[x])^3) - Cos[x]/(5*a*(a + a*Sin[x])^2) - Cos[x]/(5*(a^3 + a^3*Sin[x]))} -{Sin[x]^0/(a + a*Sin[x])^3, x, 3, -(Cos[x]/(5*(a + a*Sin[x])^3)) - (2*Cos[x])/(15*a*(a + a*Sin[x])^2) - (2*Cos[x])/(15*(a^3 + a^3*Sin[x]))} -{Csc[x]^1/(a + a*Sin[x])^3, x, 5, -(ArcTanh[Cos[x]]/a^3) + Cos[x]/(5*(a + a*Sin[x])^3) + (7*Cos[x])/(15*a*(a + a*Sin[x])^2) + (22*Cos[x])/(15*(a^3 + a^3*Sin[x]))} -{Csc[x]^2/(a + a*Sin[x])^3, x, 7, (3*ArcTanh[Cos[x]])/a^3 - (24*Cot[x])/(5*a^3) + Cot[x]/(5*(a + a*Sin[x])^3) + (3*Cot[x])/(5*a*(a + a*Sin[x])^2) + (3*Cot[x])/(a^3 + a^3*Sin[x])} -{Csc[x]^3/(a + a*Sin[x])^3, x, 8, -((13*ArcTanh[Cos[x]])/(2*a^3)) + (152*Cot[x])/(15*a^3) - (13*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x])/(5*(a + a*Sin[x])^3) + (11*Cot[x]*Csc[x])/(15*a*(a + a*Sin[x])^2) + (76*Cot[x]*Csc[x])/(15*(a^3 + a^3*Sin[x]))} -{Csc[x]^4/(a + a*Sin[x])^3, x, 8, (23*ArcTanh[Cos[x]])/(2*a^3) - (136*Cot[x])/(5*a^3) - (136*Cot[x]^3)/(15*a^3) + (23*Cot[x]*Csc[x])/(2*a^3) + (Cot[x]*Csc[x]^2)/(5*(a + a*Sin[x])^3) + (13*Cot[x]*Csc[x]^2)/(15*a*(a + a*Sin[x])^2) + (23*Cot[x]*Csc[x]^2)/(3*(a^3 + a^3*Sin[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 5, -((32*a*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]])) - (16*a*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (64*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (32*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a*d)} -{Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 4, -((4*a*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]])) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d)} -{Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 3, -((14*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]])) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a*d)} -{Sin[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 2, -((2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]])) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Sin[c + d*x]^0*Sqrt[a + a*Sin[c + d*x]], x, 1, -((2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))} -{Csc[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 2, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d)} -{Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 3, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 4, -((3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d)) - (3*a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 5, -((5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d)) - (5*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} - -{Csc[c + d*x]^1*Sqrt[a - a*Sin[c + d*x]], x, 2, -2*(Sqrt[a]/d)*ArcTanh[Sqrt[a]*(Cos[c + d*x]/Sqrt[a - a*Sin[c + d*x]])]} - -{Csc[c + d*x]^1*Sqrt[-a + a*Sin[c + d*x]], x, 2, 2*(Sqrt[a]/d)*ArcTan[Sqrt[a]*(Cos[c + d*x]/Sqrt[-a + a*Sin[c + d*x]])]} - -{Csc[c + d*x]^1*Sqrt[-a - a*Sin[c + d*x]], x, 2, 2*(Sqrt[a]/d)*ArcTan[Sqrt[a]*(Cos[c + d*x]/Sqrt[-a - a*Sin[c + d*x]])]} - - -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 6, -((68*a^2*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]])) - (34*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (136*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (68*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d)} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 4, -((152*a^2*Cos[c + d*x])/(105*d*Sqrt[a + a*Sin[c + d*x]])) - (38*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a*d)} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 3, -((8*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]])) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^(3/2), x, 2, -((8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]])) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 4, -((2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (2*a^2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 4, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a^2*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 5, -((7*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d)) - (7*a^2*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 6, -((11*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d)) - (11*a^2*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2), x, 6, -((284*a^3*Cos[c + d*x])/(99*d*Sqrt[a + a*Sin[c + d*x]])) - (710*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) - (46*a^3*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (568*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*d) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (284*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(231*d)} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2), x, 5, -((832*a^3*Cos[c + d*x])/(315*d*Sqrt[a + a*Sin[c + d*x]])) - (208*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (26*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(63*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2))/(9*a*d)} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(5/2), x, 4, -((64*a^3*Cos[c + d*x])/(21*d*Sqrt[a + a*Sin[c + d*x]])) - (16*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(7*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*d)} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^(5/2), x, 3, -((64*a^3*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]])) - (16*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (2*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(5/2), x, 4, -((2*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (14*a^3*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2), x, 4, -((5*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a^3*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d} -{Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2), x, 4, -((19*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d)) - (9*a^3*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)} -{Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2), x, 5, -((25*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d)) - (25*a^3*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (13*a^3*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2), x, 6, -((163*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d)) - (163*a^3*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) - (163*a^3*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 6, (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (28*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)} -{Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 4, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d)) + (4*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a*d)} -{Sin[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 3, (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Sin[c + d*x]^0/Sqrt[a + a*Sin[c + d*x]], x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d))} -{Csc[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 5, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d)} -{Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 6, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 7, -((7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 7, (15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (31*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (9*Cos[c + d*x]*Sin[c + d*x]^2)/(10*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(10*a^2*d)} -{Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 6, -((11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + (Cos[c + d*x]*Sin[c + d*x]^2)/(2*d*(a + a*Sin[c + d*x])^(3/2)) + (13*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) - (7*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(6*a^2*d)} -{Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 4, (7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 3, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))} -{Sin[c + d*x]^0/(a + a*Sin[c + d*x])^(3/2), x, 3, -(ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d)) - Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))} -{Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 6, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d)) + (5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Cos[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2))} -{Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 7, (3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + Cot[c + d*x]/(2*d*(a + a*Sin[c + d*x])^(3/2)) - (3*Cot[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 8, -((19*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d)) + (13*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*d*(a + a*Sin[c + d*x])^(3/2)) + (7*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Sin[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2), x, 8, (283*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (21*Cos[c + d*x]*Sin[c + d*x]^3)/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (1729*Cos[c + d*x])/(120*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (157*Cos[c + d*x]*Sin[c + d*x]^2)/(80*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (787*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(240*a^3*d)} -{Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2), x, 7, -((163*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (17*Cos[c + d*x]*Sin[c + d*x]^2)/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (197*Cos[c + d*x])/(24*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (95*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(48*a^3*d)} -{Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 6, (75*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (13*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (9*Cos[c + d*x])/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 4, -((19*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (13*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))} -{Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 4, -((5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) + Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (5*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))} -{Sin[c + d*x]^0/(a + a*Sin[c + d*x])^(5/2), x, 4, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) - (3*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))} -{Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 7, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d)) + (43*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Cos[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (11*Cos[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2))} -{Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 8, (5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) - (115*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Cot[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (15*Cot[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (35*Cot[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 9, -((39*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d)) + (219*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(4*d*(a + a*Sin[c + d*x])^(5/2)) + (19*Cot[c + d*x]*Csc[c + d*x])/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) + (63*Cot[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (31*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d*Sqrt[a + a*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^(n/2) (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Sin[e + f*x]]/Sqrt[Sin[e + f*x]], x, 2, -((2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f)} -{Sqrt[a - a*Sin[e + f*x]]/Sqrt[-Sin[e + f*x]], x, 2, (2*Sqrt[a]*ArcSin[(Sqrt[a]*Cos[e + f*x])/Sqrt[a - a*Sin[e + f*x]]])/f} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(Sqrt[1 + Sin[x]]*Sqrt[Sin[x]]), x, 2, (-Sqrt[2])*ArcSin[Cos[x]/(1 + Sin[x])]} -{1/(Sqrt[a + a*Sin[x]]*Sqrt[Sin[x]]), x, 2, -((Sqrt[2]*ArcTan[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a + a*Sin[x]])])/Sqrt[a])} - -{1/(Sqrt[1 - Sin[x]]*Sqrt[Sin[x]]), x, 2, Sqrt[2]*ArcTanh[Cos[x]/(Sqrt[2]*Sqrt[1 - Sin[x]]*Sqrt[Sin[x]])]} -{1/(Sqrt[a - a*Sin[x]]*Sqrt[Sin[x]]), x, 2, (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[x])/(Sqrt[2]*Sqrt[Sin[x]]*Sqrt[a - a*Sin[x]])])/Sqrt[a]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^(n/3) (a+a Sin[e+f x])^m*) - - -{Sin[c + d*x]^(1/3)/(a + a*Sin[c + d*x])^2, x, 5, (4*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[c + d*x]^2]*Sin[c + d*x]^(1/3))/(9*a^2*d*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[c + d*x]^2]*Sin[c + d*x]^(4/3))/(36*a^2*d*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Sin[c + d*x]^(1/3))/(9*a^2*d*(1 + Sin[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x]^(1/3))/(3*d*(a + a*Sin[c + d*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+a Sin[e+f x])^(m/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(2/3), x, 6, -((63*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(220*d)) - (3*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3))/(11*d) - (67*Cos[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(2/3))/(55*2^(5/6)*d*(1 + Sin[c + d*x])^(7/6)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/3))/(44*a*d)} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3), x, 4, (9*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(40*d) - (19*Cos[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(2/3))/(10*2^(5/6)*d*(1 + Sin[c + d*x])^(7/6)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/3))/(8*a*d)} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(2/3), x, 3, -((3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(5*d)) - (4*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(2/3))/(5*d*(1 + Sin[c + d*x])^(7/6))} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^(2/3), x, 2, -((2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6)))} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(2/3), x, 4, -((2*2^(1/6)*AppellF1[1/2, 1, -(1/6), 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6)))} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(2/3), x, 4, -((2*2^(1/6)*AppellF1[1/2, 2, -(1/6), 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(d*(1 + Sin[c + d*x])^(7/6)))} - - -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(4/3), x, 6, -((388*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(455*d*(1 + Sin[c + d*x])^(5/6))) - (72*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(455*d) - (3*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3))/(13*d) - (6*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/3))/(65*a*d)} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3), x, 4, -((37*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(35*d*(1 + Sin[c + d*x])^(5/6))) + (9*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(70*d) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/3))/(10*a*d)} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(4/3), x, 3, -((8*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(7*d*(1 + Sin[c + d*x])^(5/6))) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(4/3))/(7*d)} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^(4/3), x, 2, -((2*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Sin[c + d*x])]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6)))} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(4/3), x, 4, -((2*2^(5/6)*a*AppellF1[1/2, 1, -(5/6), 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6)))} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(4/3), x, 4, -((2*2^(5/6)*a*AppellF1[1/2, 2, -(5/6), 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1/3))/(d*(1 + Sin[c + d*x])^(5/6)))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(1/3), x, 6, -((99*Cos[c + d*x])/(80*d*(a + a*Sin[c + d*x])^(1/3))) - (3*Cos[c + d*x]*Sin[c + d*x]^2)/(8*d*(a + a*Sin[c + d*x])^(1/3)) + (37*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(40*2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)) + (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(40*a*d)} -{Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3), x, 4, (9*Cos[c + d*x])/(10*d*(a + a*Sin[c + d*x])^(1/3)) - (7*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(5*2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)) - (3*Cos[c + d*x]*(a + a*Sin[c + d*x])^(2/3))/(5*a*d)} -{Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(1/3), x, 3, -((3*Cos[c + d*x])/(2*d*(a + a*Sin[c + d*x])^(1/3))) + (Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(2^(5/6)*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))} -{Sin[c + d*x]^0/(a + a*Sin[c + d*x])^(1/3), x, 2, -((2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} -{Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(1/3), x, 4, -((2^(1/6)*AppellF1[1/2, 1, 5/6, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} -{Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(1/3), x, 4, -((2^(1/6)*AppellF1[1/2, 2, 5/6, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x])/(d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} - - -{Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(4/3), x, 6, (6*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3)) - (3*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*(a + a*Sin[c + d*x])^(4/3)) + (6*Cos[c + d*x])/(5*a*d*(a + a*Sin[c + d*x])^(1/3)) - (2*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))} -{Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3), x, 4, -((3*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3))) - (3*Cos[c + d*x])/(2*a*d*(a + a*Sin[c + d*x])^(1/3)) + (13*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(5*2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))} -{Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(4/3), x, 3, (3*Cos[c + d*x])/(5*d*(a + a*Sin[c + d*x])^(4/3)) - (4*2^(1/6)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(5*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3))} -{Sin[c + d*x]^0/(a + a*Sin[c + d*x])^(4/3), x, 2, -((Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 3/2, (1/2)*(1 - Sin[c + d*x])])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} -{Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(4/3), x, 4, -((AppellF1[1/2, 1, 11/6, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} -{Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(4/3), x, 4, -((AppellF1[1/2, 2, 11/6, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+a Sin[e+f x])^m with n symbolic*) - - -{Sin[e + f*x]^n*(1 + Sin[e + f*x])^(3/2), x, 4, -((2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]])) - (2*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]])} -{Sin[e + f*x]^n*(1 + Sin[e + f*x])^(1/2), x, 2, -((2*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[1 + Sin[e + f*x]]))} -{Sin[e + f*x]^n/(1 + Sin[e + f*x])^(1/2), x, 3, -((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]))} -{Sin[e + f*x]^n/(1 + Sin[e + f*x])^(3/2), x, 3, -((AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x])/(2*f*Sqrt[1 + Sin[e + f*x]]))} - - -{Sin[e + f*x]^n*(a + a*Sin[e + f*x])^(3/2), x, 4, -((2*a^2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) - (2*a^2*Cos[e + f*x]*Sin[e + f*x]^(1 + n))/(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])} -{Sin[e + f*x]^n*(a + a*Sin[e + f*x])^(1/2), x, 2, -((2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]))} -{Sin[e + f*x]^n/(a + a*Sin[e + f*x])^(1/2), x, 4, -((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))} -{Sin[e + f*x]^n/(a + a*Sin[e + f*x])^(3/2), x, 4, -((AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]))} - - -{(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(3/2), x, 4, -((2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[1 + Sin[e + f*x]])) + ((5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + n, 2 + n, Sin[e + f*x]]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(3 + 2*n)*Sqrt[1 - Sin[e + f*x]]*Sqrt[1 + Sin[e + f*x]])} -{(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(1/2), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + n, 2 + n, Sin[e + f*x]]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[1 - Sin[e + f*x]]*Sqrt[1 + Sin[e + f*x]])} -{(d*Sin[e + f*x])^n/(1 + Sin[e + f*x])^(1/2), x, 4, -((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{(d*Sin[e + f*x])^n/(1 + Sin[e + f*x])^(3/2), x, 4, -((AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(2*f*Sqrt[1 + Sin[e + f*x]])))} - - -{(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2), x, 5, -((2*a^2*(5 + 4*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))) - (2*a^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])} -{(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(1/2), x, 3, -((2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[a + a*Sin[e + f*x]])))} -{(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(1/2), x, 5, -((AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[a + a*Sin[e + f*x]])))} -{(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2), x, 5, -((AppellF1[1/2, -n, 2, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(2*a*f*Sqrt[a + a*Sin[e + f*x]])))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^n (a+a Sin[e+f x])^m with m symbolic*) - - -{Sin[e + f*x]^n*(1 + Sin[e + f*x])^m, x, 2, -((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x])/(f*Sqrt[1 + Sin[e + f*x]]))} -{(-Sin[e + f*x])^n*(1 - Sin[e + f*x])^m, x, 2, (2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1/2)*(1 + Sin[e + f*x])]*Cos[e + f*x])/(f*Sqrt[1 - Sin[e + f*x]])} - - -{(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^m, x, 3, -((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{(d*Sin[e + f*x])^n*(1 - Sin[e + f*x])^m, x, 3, (2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1/2)*(1 + Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/((-Sin[e + f*x])^n*(f*Sqrt[1 - Sin[e + f*x]]))} - - -{Sin[e + f*x]^n*(a + a*Sin[e + f*x])^m, x, 3, -((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/f)} -{(-Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m, x, 3, (2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1/2)*(1 + Sin[e + f*x])]*Cos[e + f*x]*(1 - Sin[e + f*x])^(-(1/2) - m)*(a - a*Sin[e + f*x])^m)/f} - - -{(d Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m, x, 4, -((2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(Sin[e + f*x]^n*f))} -{(d Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m, x, 4, (2^(1/2 + m)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 + Sin[e + f*x], (1/2)*(1 + Sin[e + f*x])]*Cos[e + f*x]*(1 - Sin[e + f*x])^(-(1/2) - m)*(d*Sin[e + f*x])^n*(a - a*Sin[e + f*x])^m)/((-Sin[e + f*x])^n*f)} - - -{Sin[c + d*x]^4*(a + a*Sin[c + d*x])^n, x, 7, If[$VersionNumber>=8, ((9 - n - n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - (n*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)*(4 + n)) - (Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n)/(d*(4 + n)) - (2^(1/2 + n)*(9 + 12*n + 17*n^2 + 6*n^3 + n^4)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)*(4 + n)) - ((9 + 3*n + n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(2 + n)*(3 + n)*(4 + n)), ((9 - n - n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)) - (n*Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)*(4 + n)) - (Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n)/(d*(4 + n)) - (2^(1/2 + n)*(9 + 12*n + 17*n^2 + 6*n^3 + n^4)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(4 + n)*(6 + 11*n + 6*n^2 + n^3)) - ((9 + 3*n + n^2)*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(4 + n)*(6 + 5*n + n^2))]} -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^n, x, 6, If[$VersionNumber>=8, -(((4 + n)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n))) - (Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)*(3 + n)) - (n*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(6 + 5*n + n^2)), -(((4 + n)*Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(6 + 11*n + 6*n^2 + n^3))) - (Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n)/(d*(3 + n)) - (2^(1/2 + n)*n*(5 + 3*n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(6 + 11*n + 6*n^2 + n^3)) - (n*Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(6 + 5*n + n^2))]} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^n, x, 4, If[$VersionNumber>=8, (Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(2 + 3*n + n^2)) - (2^(1/2 + n)*(1 + n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n)*(2 + n)) - (Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(2 + n)), (Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(2 + 3*n + n^2)) - (2^(1/2 + n)*(1 + n + n^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(2 + 3*n + n^2)) - (Cos[c + d*x]*(a + a*Sin[c + d*x])^(1 + n))/(a*d*(2 + n))]} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^n, x, 3, -((Cos[c + d*x]*(a + a*Sin[c + d*x])^n)/(d*(1 + n))) - (2^(1/2 + n)*n*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/(d*(1 + n))} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^n, x, 2, -((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/d)} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^n, x, 4, -((2^(1/2 + n)*AppellF1[1/2, 1, 1/2 - n, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/d)} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^n, x, 4, -((2^(1/2 + n)*AppellF1[1/2, 2, 1/2 - n, 3/2, 1 - Sin[c + d*x], (1/2)*(1 - Sin[c + d*x])]*Cos[c + d*x]*(1 + Sin[c + d*x])^(-(1/2) - n)*(a + a*Sin[c + d*x])^n)/d)} - - -{(1 + Sin[c + d*x])^n, x, 1, -((2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Sin[c + d*x])])/(d*Sqrt[1 + Sin[c + d*x]]))} -{(1 - Sin[c + d*x])^n, x, 1, (2^(1/2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 + Sin[c + d*x])])/(d*Sqrt[1 - Sin[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[e + f*x]^3*(a + b*Sin[e + f*x]), x, 6, (3*b*x)/8 - (a*Cos[e + f*x])/f + (a*Cos[e + f*x]^3)/(3*f) - (3*b*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)} -{Sin[e + f*x]^2*(a + b*Sin[e + f*x]), x, 5, (a*x)/2 - (b*Cos[e + f*x])/f + (b*Cos[e + f*x]^3)/(3*f) - (a*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Sin[e + f*x]^1*(a + b*Sin[e + f*x]), x, 1, (b*x)/2 - (a*Cos[e + f*x])/f - (b*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Sin[e + f*x]^0*(a + b*Sin[e + f*x]), x, 2, a*x - (b*Cos[e + f*x])/f} -{Csc[e + f*x]^1*(a + b*Sin[e + f*x]), x, 2, b*x - (a*ArcTanh[Cos[e + f*x]])/f} -{Csc[e + f*x]^2*(a + b*Sin[e + f*x]), x, 4, -((b*ArcTanh[Cos[e + f*x]])/f) - (a*Cot[e + f*x])/f} -{Csc[e + f*x]^3*(a + b*Sin[e + f*x]), x, 5, -((a*ArcTanh[Cos[e + f*x]])/(2*f)) - (b*Cot[e + f*x])/f - (a*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^4*(a + b*Sin[e + f*x]), x, 5, -((b*ArcTanh[Cos[e + f*x]])/(2*f)) - (a*Cot[e + f*x])/f - (a*Cot[e + f*x]^3)/(3*f) - (b*Cot[e + f*x]*Csc[e + f*x])/(2*f)} - - -{Sin[e + f*x]^3*(a + b*Sin[e + f*x])^2, x, 7, (3*a*b*x)/4 - ((a^2 + b^2)*Cos[e + f*x])/f + ((a^2 + 2*b^2)*Cos[e + f*x]^3)/(3*f) - (b^2*Cos[e + f*x]^5)/(5*f) - (3*a*b*Cos[e + f*x]*Sin[e + f*x])/(4*f) - (a*b*Cos[e + f*x]*Sin[e + f*x]^3)/(2*f)} -{Sin[e + f*x]^2*(a + b*Sin[e + f*x])^2, x, 6, (1/8)*(4*a^2 + 3*b^2)*x - (2*a*b*Cos[e + f*x])/f + (2*a*b*Cos[e + f*x]^3)/(3*f) - ((4*a^2 + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)} -{Sin[e + f*x]^1*(a + b*Sin[e + f*x])^2, x, 2, a*b*x - (2*(a^2 + b^2)*Cos[e + f*x])/(3*f) - (a*b*Cos[e + f*x]*Sin[e + f*x])/(3*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)} -{Sin[e + f*x]^0*(a + b*Sin[e + f*x])^2, x, 1, (1/2)*(2*a^2 + b^2)*x - (2*a*b*Cos[e + f*x])/f - (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Csc[e + f*x]^1*(a + b*Sin[e + f*x])^2, x, 3, 2*a*b*x - (a^2*ArcTanh[Cos[e + f*x]])/f - (b^2*Cos[e + f*x])/f} -{Csc[e + f*x]^2*(a + b*Sin[e + f*x])^2, x, 4, b^2*x - (2*a*b*ArcTanh[Cos[e + f*x]])/f - (a^2*Cot[e + f*x])/f} -{Csc[e + f*x]^3*(a + b*Sin[e + f*x])^2, x, 5, -(((a^2 + 2*b^2)*ArcTanh[Cos[e + f*x]])/(2*f)) - (2*a*b*Cot[e + f*x])/f - (a^2*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^4*(a + b*Sin[e + f*x])^2, x, 6, -((a*b*ArcTanh[Cos[e + f*x]])/f) - ((2*a^2 + 3*b^2)*Cot[e + f*x])/(3*f) - (a*b*Cot[e + f*x]*Csc[e + f*x])/f - (a^2*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f)} -{Csc[e + f*x]^5*(a + b*Sin[e + f*x])^2, x, 6, -(((3*a^2 + 4*b^2)*ArcTanh[Cos[e + f*x]])/(8*f)) - (2*a*b*Cot[e + f*x])/f - (2*a*b*Cot[e + f*x]^3)/(3*f) - ((3*a^2 + 4*b^2)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)} - - -{Sin[e + f*x]^3*(a + b*Sin[e + f*x])^3, x, 8, (1/16)*b*(18*a^2 + 5*b^2)*x - (a*(a^2 + 3*b^2)*Cos[e + f*x])/f + (a*(a^2 + 6*b^2)*Cos[e + f*x]^3)/(3*f) - (3*a*b^2*Cos[e + f*x]^5)/(5*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (b^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f), (1/16)*b*(18*a^2 + 5*b^2)*x - (a*(5*a^2 + 12*b^2)*Cos[e + f*x])/(5*f) + (a*(5*a^2 + 12*b^2)*Cos[e + f*x]^3)/(15*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (b*(18*a^2 + 5*b^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (13*a*b^2*Cos[e + f*x]*Sin[e + f*x]^4)/(30*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^4*(a + b*Sin[e + f*x]))/(6*f)} -{Sin[e + f*x]^2*(a + b*Sin[e + f*x])^3, x, 4, (1/8)*a*(4*a^2 + 9*b^2)*x - (b*(15*a^2 + 4*b^2)*Cos[e + f*x])/(5*f) + (b*(15*a^2 + 4*b^2)*Cos[e + f*x]^3)/(15*f) - (a*(4*a^2 + 9*b^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (11*a*b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(20*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]))/(5*f), (1/8)*a*(4*a^2 + 9*b^2)*x + ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Cos[e + f*x])/(30*b*f) + (a*(6*a^2 - 71*b^2)*Cos[e + f*x]*Sin[e + f*x])/(120*f) + ((3*a^2 - 16*b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(60*b*f) + (a*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(20*b*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^4)/(5*b*f)} -{Sin[e + f*x]^1*(a + b*Sin[e + f*x])^3, x, 3, (3/8)*b*(4*a^2 + b^2)*x - (a*(a^2 + 4*b^2)*Cos[e + f*x])/(2*f) - (b*(2*a^2 + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(4*f) - (Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)} -{Sin[e + f*x]^0*(a + b*Sin[e + f*x])^3, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x - (2*b*(4*a^2 + b^2)*Cos[e + f*x])/(3*f) - (5*a*b^2*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)} -{Csc[e + f*x]^1*(a + b*Sin[e + f*x])^3, x, 4, (1/2)*b*(6*a^2 + b^2)*x - (a^3*ArcTanh[Cos[e + f*x]])/f - (5*a*b^2*Cos[e + f*x])/(2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*f)} -{Csc[e + f*x]^2*(a + b*Sin[e + f*x])^3, x, 4, 3*a*b^2*x - (3*a^2*b*ArcTanh[Cos[e + f*x]])/f + (b*(a^2 - b^2)*Cos[e + f*x])/f - (a^2*Cot[e + f*x]*(a + b*Sin[e + f*x]))/f} -{Csc[e + f*x]^3*(a + b*Sin[e + f*x])^3, x, 4, b^3*x - (a*(a^2 + 6*b^2)*ArcTanh[Cos[e + f*x]])/(2*f) - (5*a^2*b*Cot[e + f*x])/(2*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]*(a + b*Sin[e + f*x]))/(2*f)} -{Csc[e + f*x]^4*(a + b*Sin[e + f*x])^3, x, 6, -((b*(3*a^2 + 2*b^2)*ArcTanh[Cos[e + f*x]])/(2*f)) - (a*(2*a^2 + 9*b^2)*Cot[e + f*x])/(3*f) - (7*a^2*b*Cot[e + f*x]*Csc[e + f*x])/(6*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^2*(a + b*Sin[e + f*x]))/(3*f)} -{Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3, x, 7, -((3*a*(a^2 + 4*b^2)*ArcTanh[Cos[e + f*x]])/(8*f)) - (b*(2*a^2 + b^2)*Cot[e + f*x])/f - (3*a*(a^2 + 4*b^2)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (3*a^2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(4*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^3*(a + b*Sin[e + f*x]))/(4*f)} - - -{Sin[e + f*x]^0*(a + b*Sin[e + f*x])^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x - (a*b*(19*a^2 + 16*b^2)*Cos[e + f*x])/(6*f) - (b^2*(26*a^2 + 9*b^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (7*a*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^4/(a + b*Sin[x]), x, 7, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]) - ((3*a^2 + 2*b^2)*Cos[x])/(3*b^3) + (a*Cos[x]*Sin[x])/(2*b^2) - (Cos[x]*Sin[x]^2)/(3*b)} -{Sin[x]^3/(a + b*Sin[x]), x, 6, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) + (a*Cos[x])/b^2 - (Cos[x]*Sin[x])/(2*b)} -{Sin[x]^2/(a + b*Sin[x]), x, 6, -((a*x)/b^2) + (2*a^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cos[x]/b} -{Sin[x]^1/(a + b*Sin[x]), x, 4, x/b - (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])} -{Sin[x]^0/(a + b*Sin[x]), x, 3, (2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]} -{Csc[x]^1/(a + b*Sin[x]), x, 5, -((2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])) - ArcTanh[Cos[x]]/a} -{Csc[x]^2/(a + b*Sin[x]), x, 7, (2*b^2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) + (b*ArcTanh[Cos[x]])/a^2 - Cot[x]/a} -{Csc[x]^3/(a + b*Sin[x]), x, 7, -((2*b^3*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2])) - ((a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a^3) + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)} -{Csc[x]^4/(a + b*Sin[x]), x, 8, (2*b^4*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) + (b*(a^2 + 2*b^2)*ArcTanh[Cos[x]])/(2*a^4) - ((2*a^2 + 3*b^2)*Cot[x])/(3*a^3) + (b*Cot[x]*Csc[x])/(2*a^2) - (Cot[x]*Csc[x]^2)/(3*a)} - - -{Sin[x]^4/(a + b*Sin[x])^2, x, 7, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)) + (a*(3*a^2 - 2*b^2)*Cos[x])/(b^3*(a^2 - b^2)) - ((3*a^2 - b^2)*Cos[x]*Sin[x])/(2*b^2*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x]^2)/(b*(a^2 - b^2)*(a + b*Sin[x]))} -{Sin[x]^3/(a + b*Sin[x])^2, x, 6, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)) - ((2*a^2 - b^2)*Cos[x])/(b^2*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x])/(b*(a^2 - b^2)*(a + b*Sin[x]))} -{Sin[x]^2/(a + b*Sin[x])^2, x, 5, x/b^2 - (2*a*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)) + (a^2*Cos[x])/(b*(a^2 - b^2)*(a + b*Sin[x]))} -{Sin[x]^1/(a + b*Sin[x])^2, x, 5, -((2*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (a*Cos[x])/((a^2 - b^2)*(a + b*Sin[x]))} -{Sin[x]^0/(a + b*Sin[x])^2, x, 5, (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*Cos[x])/((a^2 - b^2)*(a + b*Sin[x]))} -{Csc[x]^1/(a + b*Sin[x])^2, x, 6, -((2*b*(2*a^2 - b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2))) - ArcTanh[Cos[x]]/a^2 - (b^2*Cos[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))} -{Csc[x]^2/(a + b*Sin[x])^2, x, 7, (2*b^2*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)) + (2*b*ArcTanh[Cos[x]])/a^3 - ((a^2 - 2*b^2)*Cot[x])/(a^2*(a^2 - b^2)) - (b^2*Cot[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))} -{Csc[x]^3/(a + b*Sin[x])^2, x, 8, -((2*b^3*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2))) - ((a^2 + 6*b^2)*ArcTanh[Cos[x]])/(2*a^4) + (b*(2*a^2 - 3*b^2)*Cot[x])/(a^3*(a^2 - b^2)) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/(2*a^2*(a^2 - b^2)) - (b^2*Cot[x]*Csc[x])/(a*(a^2 - b^2)*(a + b*Sin[x]))} - - -{Sin[x]^5/(a + b*Sin[x])^3, x, 8, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)) + (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[x])/(2*b^4*(a^2 - b^2)^2) - ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[x]*Sin[x])/(2*b^3*(a^2 - b^2)^2) + (a^2*Cos[x]*Sin[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(4*a^2 - 7*b^2)*Cos[x]*Sin[x]^2)/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Sin[x]^4/(a + b*Sin[x])^3, x, 7, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)) - ((3*a^2 - 2*b^2)*Cos[x])/(2*b^3*(a^2 - b^2)) + (a^2*Cos[x]*Sin[x]^2)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) - (3*a^3*(a^2 - 2*b^2)*Cos[x])/(2*b^3*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Sin[x]^3/(a + b*Sin[x])^3, x, 6, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)) + (a^2*Cos[x]*Sin[x])/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (a^2*(2*a^2 - 5*b^2)*Cos[x])/(2*b^2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Sin[x]^2/(a + b*Sin[x])^3, x, 6, ((a^2 + 2*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a^2*Cos[x])/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) - (a*(a^2 - 4*b^2)*Cos[x])/(2*b*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Sin[x]^1/(a + b*Sin[x])^3, x, 6, -((3*a*b*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - (a*Cos[x])/(2*(a^2 - b^2)*(a + b*Sin[x])^2) - ((a^2 + 2*b^2)*Cos[x])/(2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Sin[x]^0/(a + b*Sin[x])^3, x, 6, ((2*a^2 + b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*Cos[x])/(2*(a^2 - b^2)*(a + b*Sin[x])^2) + (3*a*b*Cos[x])/(2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Csc[x]^1/(a + b*Sin[x])^3, x, 7, -((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2))) - ArcTanh[Cos[x]]/a^3 - (b^2*Cos[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (b^2*(5*a^2 - 2*b^2)*Cos[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Csc[x]^2/(a + b*Sin[x])^3, x, 8, (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)) + (3*b*ArcTanh[Cos[x]])/a^4 - ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Cot[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (3*b^2*(2*a^2 - b^2)*Cot[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))} -{Csc[x]^3/(a + b*Sin[x])^3, x, 9, -((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(5/2))) - ((a^2 + 12*b^2)*ArcTanh[Cos[x]])/(2*a^5) + (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Cot[x])/(2*a^4*(a^2 - b^2)^2) - ((a^4 - 10*a^2*b^2 + 6*b^4)*Cot[x]*Csc[x])/(2*a^3*(a^2 - b^2)^2) - (b^2*Cot[x]*Csc[x])/(2*a*(a^2 - b^2)*(a + b*Sin[x])^2) - (b^2*(7*a^2 - 4*b^2)*Cot[x]*Csc[x])/(2*a^2*(a^2 - b^2)^2*(a + b*Sin[x]))} - - -{Sin[c + d*x]^0/(a + b*Sin[c + d*x])^4, x, 7, (a*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (b*Cos[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^3) + (5*a*b*Cos[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (b*(11*a^2 + 4*b^2)*Cos[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]], x, 6, -((2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*f)) + (2*a*EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(3*b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*(a^2 - b^2)*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(3*b*f*Sqrt[a + b*Sin[e + f*x]])} -{Sin[e + f*x]^0*Sqrt[a + b*Sin[e + f*x]], x, 2, (2*EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)])} -{Csc[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]], x, 5, (2*b*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (2*a*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])} -{Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]], x, 9, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/f) - (EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (a*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (b*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]], x, 5, (2*EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(b*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (2*a*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(b*f*Sqrt[a + b*Sin[e + f*x]])} -{Sin[e + f*x]^0/Sqrt[a + b*Sin[e + f*x]], x, 2, (2*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])} -{Csc[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]], x, 2, (2*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])} -{Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]], x, 9, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(a*f)) - (EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(a*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) - (b*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(a*f*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (d Sin[e+f x])^(m/2) (a+b Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^(m/2) (a+b Sin[e+f x])^(n/2)*) - - -{Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]], x, 7, -((Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])) + ((a - b)*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticE[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/d + (a*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(b*d)} -{1/(Sqrt[Sin[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]), x, 1, -((2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n with m symbolic*) - - -{(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^3, x, 5, -((a*b^2*(7 + 2*m)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m))/(d*f*(2 + m)*(3 + m))) + (a*(3*b^2*(1 + m) + a^2*(2 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*(2 + m)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + m) + 3*a^2*(3 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*(3 + m)*Sqrt[Cos[e + f*x]^2]) - (b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(a + b*Sin[e + f*x]))/(d*f*(3 + m))} -{(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^2, x, 4, -((b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m))/(d*f*(2 + m))) + ((b^2*(1 + m) + a^2*(2 + m))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*(2 + m)*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*Sqrt[Cos[e + f*x]^2])} -{(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x])^1, x, 3, (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + m))/(d*f*(1 + m)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + m))/(d^2*f*(2 + m)*Sqrt[Cos[e + f*x]^2])} -{(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^1, x, 5, -((a*d*AppellF1[1/2, (1 - m)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)*f)) + (b*AppellF1[1/2, -(m/2), 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((Sin[e + f*x]^2)^(m/2)*((a^2 - b^2)*f))} -{(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^2, x, 10, -((b^2*AppellF1[1/2, (1/2)*(-1 - m), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(Sin[e + f*x]^2)^((1/2)*(-1 - m)))/((a^2 - b^2)^2*d*f)) - (a^2*d*AppellF1[1/2, (1 - m)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)^2*f) + (2*a*b*AppellF1[1/2, -(m/2), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((Sin[e + f*x]^2)^(m/2)*((a^2 - b^2)^2*f))} -{(d*Sin[e + f*x])^m/(a + b*Sin[e + f*x])^3, x, 13, -((3*a*b^2*AppellF1[1/2, (1/2)*(-1 - m), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + m)*(Sin[e + f*x]^2)^((1/2)*(-1 - m)))/((a^2 - b^2)^3*d*f)) - (a^3*d*AppellF1[1/2, (1 - m)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((a^2 - b^2)^3*f) + (b^3*AppellF1[1/2, (1/2)*(-2 - m), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((Sin[e + f*x]^2)^(m/2)*((a^2 - b^2)^3*f)) + (3*a^2*b*AppellF1[1/2, -(m/2), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Sin[e + f*x])^m)/((Sin[e + f*x]^2)^(m/2)*((a^2 - b^2)^3*f))} - - -{Sin[c + d*x]^(-1 - a^2/(a^2 + b^2))*(a + b*Sin[c + d*x])^2, x, 3, -(((a^2 + b^2)*Cos[c + d*x])/(Sin[c + d*x]^(a^2/(a^2 + b^2))*d)) + (2*a*(a^2 + b^2)*Cos[c + d*x]*Hypergeometric2F1[1/2, b^2/(2*(a^2 + b^2)), (1/2)*(3 - a^2/(a^2 + b^2)), Sin[c + d*x]^2]*Sin[c + d*x]^(b^2/(a^2 + b^2)))/(b*d*Sqrt[Cos[c + d*x]^2])} - - -{(1 + 2*Sin[c + d*x])^2/Sin[c + d*x]^(6/5), x, 3, -((5*Cos[c + d*x])/(d*Sin[c + d*x]^(1/5))) + (5*Cos[c + d*x]*Hypergeometric2F1[2/5, 1/2, 7/5, Sin[c + d*x]^2]*Sin[c + d*x]^(4/5))/(d*Sqrt[Cos[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n with n symbolic*) - - -{Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n, x, 0, Unintegrable[Sin[c + d*x]^m*(a + b*Sin[c + d*x])^n, x]} - - -{Sin[c + d*x]^3*(a + b*Sin[c + d*x])^n, x, 9, (2*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b^2*d*(2 + n)*(3 + n)) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(3 + n)) - (Sqrt[2]*(a + b)*(2*a^2 + b^2*(2 + n)^2)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b^3*d*(2 + n)*(3 + n)*Sqrt[1 + Sin[c + d*x]])) + (Sqrt[2]*a*(2*a^2 + b^2*(4 + 5*n + n^2))*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b^3*d*(2 + n)*(3 + n)*Sqrt[1 + Sin[c + d*x]]))} -{Sin[c + d*x]^2*(a + b*Sin[c + d*x])^n, x, 8, -((Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + n))/(b*d*(2 + n))) + (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b^2*d*(2 + n)*Sqrt[1 + Sin[c + d*x]])) - (Sqrt[2]*(a^2 + b^2*(1 + n))*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b^2*d*(2 + n)*Sqrt[1 + Sin[c + d*x]]))} -{Sin[c + d*x]^1*(a + b*Sin[c + d*x])^n, x, 7, -((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b*d*Sqrt[1 + Sin[c + d*x]]))) + (Sqrt[2]*a*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(b*d*Sqrt[1 + Sin[c + d*x]]))} -{Sin[c + d*x]^0*(a + b*Sin[c + d*x])^n, x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[c + d*x]), (b*(1 - Sin[c + d*x]))/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(((a + b*Sin[c + d*x])/(a + b))^n*(d*Sqrt[1 + Sin[c + d*x]])))} -{Csc[c + d*x]^1*(a + b*Sin[c + d*x])^n, x, 0, Unintegrable[Csc[c + d*x]*(a + b*Sin[c + d*x])^n, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^4, x, 6, (7/8)*a*c^4*x + (7*a*c^4*Cos[e + f*x]^3)/(12*f) + (7*a*c^4*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*(c^2 - c^2*Sin[e + f*x])^2)/(5*f) + (7*a*Cos[e + f*x]^3*(c^4 - c^4*Sin[e + f*x]))/(20*f)} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^3, x, 5, (5/8)*a*c^3*x + (5*a*c^3*Cos[e + f*x]^3)/(12*f) + (5*a*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*(c^3 - c^3*Sin[e + f*x]))/(4*f)} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^2, x, 4, (1/2)*a*c^2*x + (a*c^2*Cos[e + f*x]^3)/(3*f) + (a*c^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^1, x, 1, (a*c*x)/2 + (a*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^1, x, 2, -((a*x)/c) + (2*a*Cos[e + f*x])/(f*(c - c*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^2, x, 2, (a*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3)} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^3, x, 3, (a*c*Cos[e + f*x]^3)/(5*f*(c - c*Sin[e + f*x])^4) + (a*Cos[e + f*x]^3)/(15*f*(c - c*Sin[e + f*x])^3)} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^4, x, 4, (a*c*Cos[e + f*x]^3)/(7*f*(c - c*Sin[e + f*x])^5) + (2*a*Cos[e + f*x]^3)/(35*f*(c - c*Sin[e + f*x])^4) + (2*a*Cos[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^3)} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^5, x, 5, (a*c*Cos[e + f*x]^3)/(9*f*(c - c*Sin[e + f*x])^6) + (a*Cos[e + f*x]^3)/(21*f*(c - c*Sin[e + f*x])^5) + (2*a*Cos[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^4) + (2*a*c*Cos[e + f*x]^3)/(315*f*(c^2 - c^2*Sin[e + f*x])^3)} - - -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5, x, 7, (9/16)*a^2*c^5*x + (3*a^2*c^5*Cos[e + f*x]^5)/(10*f) + (9*a^2*c^5*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (3*a^2*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(8*f) + (a^2*c^3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(7*f) + (3*a^2*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(14*f)} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4, x, 6, (7/16)*a^2*c^4*x + (7*a^2*c^4*Cos[e + f*x]^5)/(30*f) + (7*a^2*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (7*a^2*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^2*Cos[e + f*x]^5*(c^4 - c^4*Sin[e + f*x]))/(6*f)} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3, x, 5, (3/8)*a^2*c^3*x + (a^2*c^3*Cos[e + f*x]^5)/(5*f) + (3*a^2*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2, x, 4, (3/8)*a^2*c^2*x + (3*a^2*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^1, x, 4, (1/2)*a^2*c*x - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^1, x, 4, -((3*a^2*x)/c) + (3*a^2*Cos[e + f*x])/(c*f) + (2*a^2*c*Cos[e + f*x]^3)/(f*(c - c*Sin[e + f*x])^2)} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^2, x, 4, (a^2*x)/c^2 + (2*a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3) - (2*a^2*Cos[e + f*x])/(f*(c^2 - c^2*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^3, x, 2, (a^2*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5)} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4, x, 3, (a^2*c^2*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^6) + (a^2*c*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^5)} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5, x, 4, (a^2*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (2*a^2*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6) + (2*a^2*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^6, x, 5, (a^2*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*c*Cos[e + f*x]^5)/(33*f*(c - c*Sin[e + f*x])^7) + (2*a^2*Cos[e + f*x]^5)/(231*f*(c - c*Sin[e + f*x])^6) + (2*a^2*Cos[e + f*x]^5)/(1155*c*f*(c - c*Sin[e + f*x])^5)} - - -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6, x, 8, (55/128)*a^3*c^6*x + (11*a^3*c^6*Cos[e + f*x]^7)/(56*f) + (55*a^3*c^6*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (55*a^3*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (11*a^3*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) + (a^3*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(9*f) + (11*a^3*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(72*f)} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5, x, 7, (45/128)*a^3*c^5*x + (9*a^3*c^5*Cos[e + f*x]^7)/(56*f) + (45*a^3*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (15*a^3*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (3*a^3*c^5*Cos[e + f*x]^5*Sin[e + f*x])/(16*f) + (a^3*Cos[e + f*x]^7*(c^5 - c^5*Sin[e + f*x]))/(8*f)} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4, x, 6, (5/16)*a^3*c^4*x + (a^3*c^4*Cos[e + f*x]^7)/(7*f) + (5*a^3*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^4*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3, x, 5, (5/16)*a^3*c^3*x + (5*a^3*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*c^3*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2, x, 5, (3/8)*a^3*c^2*x - (a^3*c^2*Cos[e + f*x]^5)/(5*f) + (3*a^3*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^3*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^1, x, 5, (5/8)*a^3*c*x - (5*a^3*c*Cos[e + f*x]^3)/(12*f) + (5*a^3*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (c*Cos[e + f*x]^3*(a^3 + a^3*Sin[e + f*x]))/(4*f)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^1, x, 5, -((15*a^3*x)/(2*c)) + (15*a^3*Cos[e + f*x])/(2*c*f) + (2*a^3*c^2*Cos[e + f*x]^5)/(f*(c - c*Sin[e + f*x])^3) + (5*a^3*Cos[e + f*x]^3)/(2*f*(c - c*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^2, x, 5, (5*a^3*x)/c^2 - (5*a^3*Cos[e + f*x])/(c^2*f) + (2*a^3*c^2*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^4) - (10*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^2)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^3, x, 5, -((a^3*x)/c^3) + (2*a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5) - (2*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3) + (2*a^3*Cos[e + f*x])/(f*(c^3 - c^3*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4, x, 2, (a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5, x, 3, (a^3*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + (a^3*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^7)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6, x, 4, (a^3*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^8) + (2*a^3*c*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^7)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^7, x, 5, (a^3*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (3*a^3*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^9) + (2*a^3*c*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^8) + (2*a^3*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e + f*x])^7)} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^8, x, 6, (a^3*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (4*a^3*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x])^10) + (4*a^3*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (8*a^3*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (8*a^3*Cos[e + f*x]^7)/(45045*c*f*(c - c*Sin[e + f*x])^7)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^4, x, 6, -((35*c^4*x)/(2*a)) - (35*c^4*Cos[e + f*x]^3)/(3*a*f) - (35*c^4*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) - (14*a*c^4*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^3, x, 5, -((15*c^3*x)/(2*a)) - (15*c^3*Cos[e + f*x])/(2*a*f) - (2*a^2*c^3*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^3) - (5*c^3*Cos[e + f*x]^3)/(2*f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^2, x, 4, -((3*c^2*x)/a) - (3*c^2*Cos[e + f*x])/(a*f) - (2*a*c^2*Cos[e + f*x]^3)/(f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^1, x, 2, -((c*x)/a) - (2*c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^1, x, 3, Tan[e + f*x]/(a*c*f)} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^2, x, 4, Sec[e + f*x]/(3*a*f*(c^2 - c^2*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*a*c^2*f)} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^3, x, 5, Sec[e + f*x]/(5*a*c*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]/(5*a*f*(c^3 - c^3*Sin[e + f*x])) + (2*Tan[e + f*x])/(5*a*c^3*f)} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^4, x, 6, Sec[e + f*x]/(7*a*c*f*(c - c*Sin[e + f*x])^3) + (4*Sec[e + f*x])/(35*a*f*(c^2 - c^2*Sin[e + f*x])^2) + (4*Sec[e + f*x])/(35*a*f*(c^4 - c^4*Sin[e + f*x])) + (8*Tan[e + f*x])/(35*a*c^4*f)} - - -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5, x, 7, (105*c^5*x)/(2*a^2) + (35*c^5*Cos[e + f*x]^3)/(a^2*f) + (105*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (2*a^4*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (6*a^2*c^5*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (42*c^5*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4, x, 6, (35*c^4*x)/(2*a^2) + (35*c^4*Cos[e + f*x])/(2*a^2*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(3*f*(a + a*Sin[e + f*x])^5) + (14*a^4*c^4*Cos[e + f*x]^5)/(3*f*(a^2 + a^2*Sin[e + f*x])^3) + (35*c^4*Cos[e + f*x]^3)/(6*f*(a^2 + a^2*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3, x, 5, (5*c^3*x)/a^2 + (5*c^3*Cos[e + f*x])/(a^2*f) - (2*a^2*c^3*Cos[e + f*x]^5)/(3*f*(a + a*Sin[e + f*x])^4) + (10*c^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2, x, 4, (c^2*x)/a^2 - (2*a*c^2*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3) + (2*c^2*Cos[e + f*x])/(f*(a^2 + a^2*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^1, x, 2, -((a*c*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3))} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^1, x, 4, -(Sec[e + f*x]/(3*c*f*(a^2 + a^2*Sin[e + f*x]))) + (2*Tan[e + f*x])/(3*a^2*c*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^2, x, 3, Tan[e + f*x]/(a^2*c^2*f) + Tan[e + f*x]^3/(3*a^2*c^2*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^3, x, 4, Sec[e + f*x]^3/(5*a^2*f*(c^3 - c^3*Sin[e + f*x])) + (4*Tan[e + f*x])/(5*a^2*c^3*f) + (4*Tan[e + f*x]^3)/(15*a^2*c^3*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^4, x, 5, Sec[e + f*x]^3/(7*a^2*f*(c^2 - c^2*Sin[e + f*x])^2) + Sec[e + f*x]^3/(7*a^2*f*(c^4 - c^4*Sin[e + f*x])) + (4*Tan[e + f*x])/(7*a^2*c^4*f) + (4*Tan[e + f*x]^3)/(21*a^2*c^4*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5, x, 6, Sec[e + f*x]^3/(9*a^2*c^2*f*(c - c*Sin[e + f*x])^3) + (2*Sec[e + f*x]^3)/(21*a^2*c^3*f*(c - c*Sin[e + f*x])^2) + (2*Sec[e + f*x]^3)/(21*a^2*f*(c^5 - c^5*Sin[e + f*x])) + (8*Tan[e + f*x])/(21*a^2*c^5*f) + (8*Tan[e + f*x]^3)/(63*a^2*c^5*f)} - - -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5, x, 7, -((63*c^5*x)/(2*a^3)) - (63*c^5*Cos[e + f*x])/(2*a^3*f) - (2*a^4*c^5*Cos[e + f*x]^9)/(5*f*(a + a*Sin[e + f*x])^7) + (6*a^2*c^5*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^5) - (42*c^5*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^3) - (21*c^5*Cos[e + f*x]^3)/(2*f*(a^3 + a^3*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4, x, 6, -((7*c^4*x)/a^3) - (7*c^4*Cos[e + f*x])/(a^3*f) - (2*a^3*c^4*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^6) + (14*a*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^4) - (14*c^4*Cos[e + f*x]^3)/(3*a*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3, x, 5, -((c^3*x)/a^3) - (2*a^2*c^3*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^5) + (2*c^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3) - (2*c^3*Cos[e + f*x])/(f*(a^3 + a^3*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2, x, 2, -((a^2*c^2*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^5))} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^1, x, 3, -((a*c*Cos[e + f*x]^3)/(5*f*(a + a*Sin[e + f*x])^4)) - (c*Cos[e + f*x]^3)/(15*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^1, x, 5, -(Sec[e + f*x]/(5*a*c*f*(a + a*Sin[e + f*x])^2)) - Sec[e + f*x]/(5*c*f*(a^3 + a^3*Sin[e + f*x])) + (2*Tan[e + f*x])/(5*a^3*c*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^2, x, 4, -(Sec[e + f*x]^3/(5*c^2*f*(a^3 + a^3*Sin[e + f*x]))) + (4*Tan[e + f*x])/(5*a^3*c^2*f) + (4*Tan[e + f*x]^3)/(15*a^3*c^2*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^3, x, 3, Tan[e + f*x]/(a^3*c^3*f) + (2*Tan[e + f*x]^3)/(3*a^3*c^3*f) + Tan[e + f*x]^5/(5*a^3*c^3*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4, x, 4, Sec[e + f*x]^5/(7*a^3*f*(c^4 - c^4*Sin[e + f*x])) + (6*Tan[e + f*x])/(7*a^3*c^4*f) + (4*Tan[e + f*x]^3)/(7*a^3*c^4*f) + (6*Tan[e + f*x]^5)/(35*a^3*c^4*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^5, x, 5, Sec[e + f*x]^5/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + Sec[e + f*x]^5/(9*a^3*f*(c^5 - c^5*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*a^3*c^5*f) + (4*Tan[e + f*x]^3)/(9*a^3*c^5*f) + (2*Tan[e + f*x]^5)/(15*a^3*c^5*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^6, x, 6, Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(7/2), x, 5, (256*a*c^5*Cos[e + f*x]^3)/(315*f*(c - c*Sin[e + f*x])^(3/2)) + (64*a*c^4*Cos[e + f*x]^3)/(105*f*Sqrt[c - c*Sin[e + f*x]]) + (8*a*c^3*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(21*f) + (2*a*c^2*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(9*f)} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(5/2), x, 4, (64*a*c^4*Cos[e + f*x]^3)/(105*f*(c - c*Sin[e + f*x])^(3/2)) + (16*a*c^3*Cos[e + f*x]^3)/(35*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^2*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(7*f)} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(3/2), x, 3, (8*a*c^3*Cos[e + f*x]^3)/(15*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a*c^2*Cos[e + f*x]^3)/(5*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(1/2), x, 2, (2*a*c^2*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(1/2), x, 4, (2*Sqrt[2]*a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(3/2), x, 4, -((a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*c^(3/2)*f)) + (a*Cos[e + f*x])/(f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(5/2), x, 5, -((a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(5/2)*f)) + (a*Cos[e + f*x])/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x])/(8*c*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(7/2), x, 6, -((a*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*c^(7/2)*f)) + (a*Cos[e + f*x])/(3*f*(c - c*Sin[e + f*x])^(7/2)) - (a*Cos[e + f*x])/(24*c*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x])/(32*c^2*f*(c - c*Sin[e + f*x])^(3/2))} - - -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2), x, 5, (256*a^2*c^6*Cos[e + f*x]^5)/(1155*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*c^5*Cos[e + f*x]^5)/(231*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*c^4*Cos[e + f*x]^5)/(33*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(11*f)} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2), x, 4, (64*a^2*c^5*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^(5/2)) + (16*a^2*c^4*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^2*c^3*Cos[e + f*x]^5)/(9*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2), x, 3, (8*a^2*c^4*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^2*c^3*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(1/2), x, 2, (2*a^2*c^3*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(1/2), x, 5, (4*Sqrt[2]*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (4*a^2*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(3/2), x, 5, -((3*Sqrt[2]*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f)) + (a^2*c*Cos[e + f*x]^3)/(f*(c - c*Sin[e + f*x])^(5/2)) + (3*a^2*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(5/2), x, 5, (3*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*c^(5/2)*f) + (a^2*c*Cos[e + f*x]^3)/(2*f*(c - c*Sin[e + f*x])^(7/2)) - (3*a^2*Cos[e + f*x])/(4*c*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(7/2), x, 6, (a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*c^(7/2)*f) + (a^2*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*Cos[e + f*x])/(4*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*Cos[e + f*x])/(16*c^2*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(9/2), x, 7, (3*a^2*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f) + (a^2*c*Cos[e + f*x]^3)/(4*f*(c - c*Sin[e + f*x])^(11/2)) - (a^2*Cos[e + f*x])/(8*c*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (3*a^2*Cos[e + f*x])/(256*c^3*f*(c - c*Sin[e + f*x])^(3/2))} - - -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2), x, 5, (256*a^3*c^7*Cos[e + f*x]^7)/(3003*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*c^6*Cos[e + f*x]^7)/(429*f*(c - c*Sin[e + f*x])^(5/2)) + (24*a^3*c^5*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(13*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2), x, 4, (64*a^3*c^6*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^(7/2)) + (16*a^3*c^5*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2), x, 3, (8*a^3*c^5*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(1/2), x, 2, (2*a^3*c^4*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(1/2), x, 6, (8*Sqrt[2]*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2)) - (4*a^3*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (8*a^3*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(3/2), x, 6, -((10*Sqrt[2]*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f)) + (a^3*c^2*Cos[e + f*x]^5)/(f*(c - c*Sin[e + f*x])^(7/2)) + (5*a^3*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) + (10*a^3*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(5/2), x, 6, (15*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(2*Sqrt[2]*c^(5/2)*f) + (a^3*c^2*Cos[e + f*x]^5)/(2*f*(c - c*Sin[e + f*x])^(9/2)) - (5*a^3*Cos[e + f*x]^3)/(4*f*(c - c*Sin[e + f*x])^(5/2)) - (15*a^3*Cos[e + f*x])/(4*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(7/2), x, 6, -((5*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(7/2)*f)) + (a^3*c^2*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^(11/2)) - (5*a^3*Cos[e + f*x]^3)/(12*f*(c - c*Sin[e + f*x])^(7/2)) + (5*a^3*Cos[e + f*x])/(8*c^2*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(9/2), x, 7, -((5*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*c^(9/2)*f)) + (a^3*c^2*Cos[e + f*x]^5)/(4*f*(c - c*Sin[e + f*x])^(13/2)) - (5*a^3*Cos[e + f*x]^3)/(24*f*(c - c*Sin[e + f*x])^(9/2)) + (5*a^3*Cos[e + f*x])/(32*c^2*f*(c - c*Sin[e + f*x])^(5/2)) - (5*a^3*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(11/2), x, 8, -((3*a^3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(512*Sqrt[2]*c^(11/2)*f)) + (a^3*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(15/2)) - (a^3*Cos[e + f*x]^3)/(8*f*(c - c*Sin[e + f*x])^(11/2)) + (a^3*Cos[e + f*x])/(16*c^2*f*(c - c*Sin[e + f*x])^(7/2)) - (a^3*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a^3*Cos[e + f*x])/(512*c^4*f*(c - c*Sin[e + f*x])^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(7/2), x, 5, -((256*c^3*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(5*a*f)) + (64*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(5*a*f) + (8*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(5*a*f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(5*a*f)} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(5/2), x, 4, -((64*c^2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a*f)) + (16*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*a*f) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f)} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(3/2), x, 3, -((8*c*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f)) + (2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f)} -{1/(a + a*Sin[e + f*x])^1*(c - c*Sin[e + f*x])^(1/2), x, 2, -((2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f))} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(1/2), x, 4, ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(Sqrt[2]*a*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*c*f)} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(3/2), x, 5, (3*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*a*c^(3/2)*f) + (3*Cos[e + f*x])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - Sec[e + f*x]/(a*c*f*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^1/(c - c*Sin[e + f*x])^(5/2), x, 6, (15*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*a*c^(5/2)*f) + (15*Cos[e + f*x])/(32*a*c*f*(c - c*Sin[e + f*x])^(3/2)) + Sec[e + f*x]/(4*a*c*f*(c - c*Sin[e + f*x])^(3/2)) - (5*Sec[e + f*x])/(8*a*c^2*f*Sqrt[c - c*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(9/2), x, 6, (4096*c^3*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^2*f) - (1024*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^2*f) + (128*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(5*a^2*f) + (32*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(15*a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(11/2))/(5*a^2*c*f)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2), x, 5, (256*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (64*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*f) + (8*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(3*a^2*c*f)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2), x, 4, (64*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (16*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*f) + (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(a^2*c*f)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2), x, 3, (8*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - (2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(a^2*c*f)} -{1/(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(1/2), x, 2, -((2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*c*f))} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(1/2), x, 5, ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(2*Sqrt[2]*a^2*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(2*a^2*c*f) - (Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*c^2*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(3/2), x, 6, (5*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*a^2*c^(3/2)*f) + (5*Cos[e + f*x])/(8*a^2*f*(c - c*Sin[e + f*x])^(3/2)) - (5*Sec[e + f*x])/(6*a^2*c*f*Sqrt[c - c*Sin[e + f*x]]) - (Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*c^2*f)} -{1/(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^(5/2), x, 7, (35*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(64*Sqrt[2]*a^2*c^(5/2)*f) + (35*Cos[e + f*x])/(64*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) + (7*Sec[e + f*x])/(24*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (35*Sec[e + f*x])/(48*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - Sec[e + f*x]^3/(3*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(9/2), x, 6, -((4096*c^2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*f)) + (1024*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*f) - (128*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*f) + (32*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/(3*a^3*c*f) + (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(13/2))/(3*a^3*c^2*f)} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2), x, 5, -((256*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*f)) + (64*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(a^3*f) - (24*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*c*f) + (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/(a^3*c^2*f)} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2), x, 4, -((64*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*f)) + (16*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*c*f) - (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(a^3*c^2*f)} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2), x, 3, (8*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(15*a^3*c*f) - (2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*a^3*c^2*f)} -{1/(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(1/2), x, 2, -((2*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c^2*f))} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(1/2), x, 6, ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])]/(4*Sqrt[2]*a^3*Sqrt[c]*f) - (Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(4*a^3*c*f) - (Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(6*a^3*c^2*f) - (Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c^3*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(3/2), x, 7, (7*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*a^3*c^(3/2)*f) + (7*Cos[e + f*x])/(16*a^3*f*(c - c*Sin[e + f*x])^(3/2)) - (7*Sec[e + f*x])/(12*a^3*c*f*Sqrt[c - c*Sin[e + f*x]]) - (7*Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(30*a^3*c^2*f) - (Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(5*a^3*c^3*f)} -{1/(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^(5/2), x, 8, (63*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*a^3*c^(5/2)*f) + (63*Cos[e + f*x])/(128*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) + (21*Sec[e + f*x])/(80*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) - (21*Sec[e + f*x])/(32*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - (3*Sec[e + f*x]^3)/(10*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - (Sec[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(5*a^3*c^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(7/2), x, 1, -((a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(5/2), x, 1, -((a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(3/2), x, 1, -((a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, -((a*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(1/2), x, 3, -((a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(3/2), x, 1, (a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(5/2), x, 1, (a*Cos[e + f*x])/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(7/2), x, 1, (a*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))} - - -{(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2), x, 2, -((a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*f*Sqrt[a + a*Sin[e + f*x]])) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*f)} -{(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2), x, 2, -((a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(6*f*Sqrt[a + a*Sin[e + f*x]])) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(4*f)} -{(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2), x, 2, -((a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(3*f)} -{(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(1/2), x, 4, -((2*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(3/2), x, 4, (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(5/2), x, 1, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(7/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(24*c*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(9/2), x, 2, (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*Cos[e + f*x])/(12*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2), x, 2, (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*f*(c - c*Sin[e + f*x])^(11/2)) - (a^2*Cos[e + f*x])/(20*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))} - - -{(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2), x, 3, -((a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(15*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(6*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2), x, 3, -((2*a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(5*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2), x, 2, (c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*f*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(4*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(1/2), x, 5, -((4*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(3/2), x, 5, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(5/2), x, 5, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*(c - c*Sin[e + f*x])^(3/2)) - (a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(7/2), x, 1, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(9/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(48*c*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(11/2), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(40*c*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(240*c^2*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(13/2), x, 3, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(13/2)) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*c*f*(c - c*Sin[e + f*x])^(11/2)) + (a^3*Cos[e + f*x])/(60*c^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))} - - -{(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2), x, 4, -((a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(35*f*Sqrt[a + a*Sin[e + f*x]])) - (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(14*f) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(28*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(8*f)} -{(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2), x, 4, -((2*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(35*f*Sqrt[a + a*Sin[e + f*x]])) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(35*f) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f)} -{(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2), x, 3, (c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(15*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(6*f)} -{(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2), x, 2, (c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f)} -{(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(1/2), x, 6, -((8*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(3/2), x, 6, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(f*(c - c*Sin[e + f*x])^(3/2)) + (12*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(5/2), x, 6, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (6*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(7/2), x, 6, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*(c - c*Sin[e + f*x])^(7/2)) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(9/2), x, 1, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(11/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(13/2), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(60*c*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(480*c^2*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(15/2), x, 4, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(280*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2240*c^3*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)/(c - c*Sin[e + f*x])^(17/2), x, 4, (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(17/2)) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(56*c^2*f*(c - c*Sin[e + f*x])^(13/2)) - (a^4*Cos[e + f*x])/(280*c^3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(5/2), x, 5, (4*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]) + (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(3/2), x, 4, (2*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c - c*Sin[e + f*x])^(1/2), x, 3, (c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(1/2), x, 2, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(3/2), x, 3, Cos[e + f*x]/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)/(c - c*Sin[e + f*x])^(5/2), x, 4, Cos[e + f*x]/(4*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + Cos[e + f*x]/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2), x, 6, -((12*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (6*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - (3*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2), x, 5, -((4*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (2*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2), x, 4, -((c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, -((c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(1/2), x, 3, -(Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(3/2), x, 4, -(Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))) + Cos[e + f*x]/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(5/2), x, 5, -(Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))) + (3*Cos[e + f*x])/(8*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2), x, 7, (24*c^5*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (12*c^4*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*c^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2), x, 6, (6*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (3*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2), x, 5, (c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*(a + a*Sin[e + f*x])^(3/2)) - (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2), x, 1, -((Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(4*f*(a + a*Sin[e + f*x])^(5/2)))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(1/2), x, 1, -((c*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(1/2), x, 4, -(Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])) - Cos[e + f*x]/(4*a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(3/2), x, 5, -(Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))) - (3*Cos[e + f*x])/(8*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + (3*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(5/2)/(c - c*Sin[e + f*x])^(5/2), x, 6, -(Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))) - Cos[e + f*x]/(2*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (3*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m and/or n symbolic*) - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (2^(1/2 + n)*c*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 - 2*n), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*(1 + 2*m))} - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3, x, 4, -((2^(1/2 + m)*a^4*c^3*Cos[e + f*x]^7*Hypergeometric2F1[7/2, 1/2 - m, 9/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-4 + m))/(7*f))} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2, x, 4, -((2^(1/2 + m)*a^3*c^2*Cos[e + f*x]^5*Hypergeometric2F1[5/2, 1/2 - m, 7/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(5*f))} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^1, x, 4, -((2^(1/2 + m)*a^2*c*Cos[e + f*x]^3*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(3*f))} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^1, x, 4, (2^(1/2 + m)*Hypergeometric2F1[-(1/2), 1/2 - m, 1/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(c*f)} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^2, x, 4, (2^(1/2 + m)*Hypergeometric2F1[-(3/2), 1/2 - m, -(1/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*c^2*f)} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^3, x, 4, (2^(1/2 + m)*Hypergeometric2F1[-(5/2), 1/2 - m, -(3/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^5*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(5*a^2*c^3*f)} - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2), x, 3, (64*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(15 + 16*m + 4*m^2)) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m))} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2), x, 2, (8*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m))} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1/2), x, 1, (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(1/2), x, 3, (Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(3/2), x, 3, (Cos[e + f*x]*Hypergeometric2F1[2, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(2*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(5/2), x, 3, (Cos[e + f*x]*Hypergeometric2F1[3, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} - - -(* The same rules should be used to integrate the following two problems: *) -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(1/2), x, 3, (Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(c + c*Sin[e + f*x])^m/(a - a*Sin[e + f*x])^(1/2), x, 3, (Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(c + c*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[a - a*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 3), x, 3, If[$VersionNumber>=8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(f*(5 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*(5 + 2*m)*(3 + 8*m + 4*m^2)), (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(f*(5 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*(15 + 46*m + 36*m^2 + 8*m^3))]} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*(3 + 8*m + 4*m^2))} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 1), x, 1, (Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 0), x, 4, (2^(1/2 - m)*c*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m - 1), x, 4, (2^(3/2 - m)*c^2*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(-1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m - 2), x, 4, (2^(5/2 - m)*c^3*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(-3 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^4, x, 4, (1/8)*a*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*x - (a*(12*c^4 + 95*c^3*d + 112*c^2*d^2 + 80*c*d^3 + 16*d^4)*Cos[e + f*x])/(30*f) - (a*d*(24*c^3 + 130*c^2*d + 116*c*d^2 + 45*d^3)*Cos[e + f*x]*Sin[e + f*x])/(120*f) - (a*(12*c^2 + 35*c*d + 16*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*f) - (a*(4*c + 5*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*f)} -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^3, x, 3, (1/8)*a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*x - (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3)*Cos[e + f*x])/(6*f) - (a*d*(6*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (a*(3*c + 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(12*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f)} -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^2, x, 2, (1/2)*a*(2*c^2 + 2*c*d + d^2)*x - (2*a*(c^2 + 3*c*d + d^2)*Cos[e + f*x])/(3*f) - (a*d*(2*c + 3*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f)} -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^1, x, 1, (1/2)*a*(2*c + d)*x - (a*(c + d)*Cos[e + f*x])/f - (a*d*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^0, x, 2, a*x - (a*Cos[e + f*x])/f} -{(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^1, x, 4, (a*x)/d - (2*a*(c - d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d*Sqrt[c^2 - d^2]*f)} -{(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^2, x, 5, (2*a*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c + d)*Sqrt[c^2 - d^2]*f) - (a*Cos[e + f*x])/((c + d)*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^3, x, 6, (a*(2*c - d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(3/2)*f) - (a*Cos[e + f*x])/(2*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a*(c - 2*d)*Cos[e + f*x])/(2*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^4, x, 7, (a*(2*c^2 - 2*c*d + d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a*(2*c - 3*d)*Cos[e + f*x])/(6*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a*(c - 4*d)*(2*c - d)*Cos[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4, x, 5, (1/16)*a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*x + (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c^2*d^3 - 288*c*d^4 - 64*d^5)*Cos[e + f*x])/(60*d*f) + (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d^4)*Cos[e + f*x]*Sin[e + f*x])/(240*f) + (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d*f) + (a^2*(4*c^2 - 48*c*d - 55*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d*f) + (a^2*(c - 12*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(6*d*f)} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3, x, 4, (3/8)*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*x + (a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*Cos[e + f*x])/(10*d*f) + (a^2*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(40*f) + (a^2*(c^2 - 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(20*d*f) + (a^2*(c - 10*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f)} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2, x, 3, (1/8)*a^2*(12*c^2 + 16*c*d + 7*d^2)*x - (a^2*(12*c^2 + 16*c*d + 7*d^2)*Cos[e + f*x])/(6*f) - (a^2*(12*c^2 + 16*c*d + 7*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((8*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(12*f) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*a*f)} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^1, x, 2, (1/2)*a^2*(3*c + 2*d)*x - (2*a^2*(3*c + 2*d)*Cos[e + f*x])/(3*f) - (a^2*(3*c + 2*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*f)} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^0, x, 1, (3*a^2*x)/2 - (2*a^2*Cos[e + f*x])/f - (a^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^1, x, 5, -((a^2*(c - 2*d)*x)/d^2) + (2*a^2*(c - d)^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*Sqrt[c^2 - d^2]*f) - (a^2*Cos[e + f*x])/(d*f)} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2, x, 5, (a^2*x)/d^2 - (2*a^2*(c - d)^2*(c + 2*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*(c^2 - d^2)^(3/2)*f) + (a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*(c + d*Sin[e + f*x])), (a^2*x)/d^2 - (2*a^2*(c - d)*(c + 2*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*(c + d)*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3, x, 6, (3*a^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c + d)^2*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(c + 4*d)*Cos[e + f*x])/(2*d*(c + d)^2*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4, x, 7, (a^2*(3*c - 2*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c - d)*(c + d)^3*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a^2*(c + 6*d)*Cos[e + f*x])/(6*d*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a^2*(c^2 + 6*c*d - 10*d^2)*Cos[e + f*x])/(6*(c - d)*d*(c + d)^3*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^5, x, 8, (a^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(4*(c - d)^2*(c + d)^4*Sqrt[c^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) - (a^2*(c + 8*d)*Cos[e + f*x])/(12*d*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x])/(24*(c - d)*d*(c + d)^3*f*(c + d*Sin[e + f*x])^2) - (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Cos[e + f*x])/(24*(c - d)^2*d*(c + d)^4*f*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3, x, 6, (1/16)*a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x - (4*a^3*(c + d)^3*Cos[e + f*x])/f + (a^3*(c + d)^2*(c + 7*d)*Cos[e + f*x]^3)/(3*f) - (3*a^3*d^2*(c + d)*Cos[e + f*x]^5)/(5*f) - (a^3*(24*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (a^3*d*(18*c^2 + 54*c*d + 23*d^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (a^3*d^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f), (1/16)*a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*x - (a^3*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5)*Cos[e + f*x])/(60*d^2*f) - (a^3*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d^3 + 345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(240*d*f) - (a^3*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^2*f) - (a^3*(2*c^2 - 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d^2*f) + (a^3*(2*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)} -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2, x, 9, (1/8)*a^3*(20*c^2 + 30*c*d + 13*d^2)*x - (4*a^3*(c + d)^2*Cos[e + f*x])/f + (a^3*(c^2 + 6*c*d + 5*d^2)*Cos[e + f*x]^3)/(3*f) - (a^3*d^2*Cos[e + f*x]^5)/(5*f) - (a^3*(12*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a^3*d*(2*c + 3*d)*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), (1/8)*a^3*(20*c^2 + 30*c*d + 13*d^2)*x - (a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(5*f) + (a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]^3)/(60*f) - (3*a^3*(20*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*Sin[e + f*x])/(40*f) - ((10*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(20*f) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^4)/(5*a*f)} -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^1, x, 8, (5/8)*a^3*(4*c + 3*d)*x - (4*a^3*(c + d)*Cos[e + f*x])/f + (a^3*(c + 3*d)*Cos[e + f*x]^3)/(3*f) - (3*a^3*(4*c + 5*d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a^3*d*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), (5/8)*a^3*(4*c + 3*d)*x - (a^3*(4*c + 3*d)*Cos[e + f*x])/f + (a^3*(4*c + 3*d)*Cos[e + f*x]^3)/(12*f) - (3*a^3*(4*c + 3*d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*f)} -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^0, x, 7, (5*a^3*x)/2 - (4*a^3*Cos[e + f*x])/f + (a^3*Cos[e + f*x]^3)/(3*f) - (3*a^3*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^1, x, 7, (a^3*(2*c^2 - 6*c*d + 7*d^2)*x)/(2*d^3) - (2*a^3*(c - d)^3*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*Sqrt[c^2 - d^2]*f) + (a^3*(2*c - 5*d)*Cos[e + f*x])/(2*d^2*f) - (Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*f)} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2, x, 7, -((a^3*(2*c - 3*d)*x)/d^3) + (2*a^3*(c - d)^2*(2*c + 3*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c + d)*Sqrt[c^2 - d^2]*f) - (2*a^3*c*Cos[e + f*x])/(d^2*(c + d)*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3, x, 7, (a^3*x)/d^3 - (a^3*(c - d)*(2*c^2 + 6*c*d + 7*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c + d)^2*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) + (a^3*(c - d)*(2*c + 5*d)*Cos[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4, x, 8, (5*a^3*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c + d)^3*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^3) + (a^3*(c - d)*(2*c + 7*d)*Cos[e + f*x])/(6*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (a^3*(2*c^2 + 9*c*d + 22*d^2)*Cos[e + f*x])/(6*d^2*(c + d)^3*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^5, x, 9, (5*a^3*(4*c - 3*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(4*(c - d)*(c + d)^4*Sqrt[c^2 - d^2]*f) + ((c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) + (a^3*(c - d)*(2*c + 9*d)*Cos[e + f*x])/(12*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^3*(2*c^2 + 12*c*d + 45*d^2)*Cos[e + f*x])/(24*d^2*(c + d)^3*f*(c + d*Sin[e + f*x])^2) - (a^3*(2*c^3 + 12*c^2*d + 43*c*d^2 - 72*d^3)*Cos[e + f*x])/(24*(c - d)*d^2*(c + d)^4*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^4, x, 3, (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*x)/(2*a) + (2*d*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3)*Cos[e + f*x])/(3*a*f) + (d^2*(6*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a*f) + ((3*c - 4*d)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^3, x, 2, (3*d*(2*c^2 - 2*c*d + d^2)*x)/(2*a) + (2*d*(c^2 - 3*c*d + d^2)*Cos[e + f*x])/(a*f) + ((2*c - 3*d)*d^2*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^2, x, 3, ((2*c - d)*d*x)/a - (d^2*Cos[e + f*x])/(a*f) - ((c - d)^2*Cos[e + f*x])/(a*f*(1 + Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^1, x, 2, (d*x)/a - ((c - d)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^0, x, 1, -(Cos[e + f*x]/(f*(a + a*Sin[e + f*x])))} -{1/(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^1, x, 5, -((2*d*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a*(c - d)*Sqrt[c^2 - d^2]*f)) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^2, x, 6, -((2*d*(2*c + d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(3/2)*f)) - (d*(c + 2*d)*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^3, x, 7, -((3*d*(2*c^2 + 2*c*d + d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*c + 3*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(2*c + d)*(c + 4*d)*Cos[e + f*x])/(2*a*(c - d)^3*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^5, x, 4, (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*x)/(2*a^2) + (2*d*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4)*Cos[e + f*x])/(3*a^2*f) + (d^2*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) + (d*(c^2 + 10*c*d - 12*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f) - ((c - d)*(c + 10*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(3*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4, x, 3, (d^2*(12*c^2 - 16*c*d + 7*d^2)*x)/(2*a^2) + (2*d*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3)*Cos[e + f*x])/(3*a^2*f) + (d^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) - ((c - d)*(c + 8*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3, x, 5, ((3*c - 2*d)*d^2*x)/a^2 + ((c - 4*d)*d^2*Cos[e + f*x])/(3*a^2*f) - ((c - d)^2*(c + 6*d)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2, x, 3, (d^2*x)/a^2 - ((c - d)*(c + 4*d)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(3*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^1, x, 2, -(((c - d)*Cos[e + f*x])/(3*f*(a + a*Sin[e + f*x])^2)) - ((c + 2*d)*Cos[e + f*x])/(3*f*(a^2 + a^2*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^0, x, 2, -(Cos[e + f*x]/(3*f*(a + a*Sin[e + f*x])^2)) - Cos[e + f*x]/(3*f*(a^2 + a^2*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^1, x, 6, (2*d^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^2*Sqrt[c^2 - d^2]*f) - ((c - 4*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2)} -{1/(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2, x, 7, (2*d^2*(3*c + 2*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^3*(c + d)*Sqrt[c^2 - d^2]*f) - (d*(c^2 - 6*c*d - 10*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])) - ((c - 6*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3, x, 8, (d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (d*(2*c^2 - 16*c*d - 21*d^2)*Cos[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((c - 8*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Cos[e + f*x])/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^6, x, 5, (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*x)/(2*a^3) + (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) + (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*a^3*f) - ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^5)/(5*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^5, x, 4, (d^3*(20*c^2 - 30*c*d + 13*d^2)*x)/(2*a^3) + (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*Cos[e + f*x])/(15*a^3*f) + (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Cos[e + f*x]*Sin[e + f*x])/(30*a^3*f) - ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^4, x, 6, ((4*c - 3*d)*d^3*x)/a^3 + (d^2*(2*c^2 + 10*c*d - 27*d^2)*Cos[e + f*x])/(15*a^3*f) - ((c - d)^2*(2*c^2 + 12*c*d + 45*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*(2*c + 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(5*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3, x, 5, (d^3*x)/a^3 - ((c - d)^2*(2*c + 7*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((c - d)*(2*c^2 + 11*c*d + 29*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2, x, 3, -(((c - d)*(2*c + 5*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2)) - ((2*c^2 + 6*c*d + 7*d^2)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(5*f*(a + a*Sin[e + f*x])^3)} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^1, x, 3, -(((c - d)*Cos[e + f*x])/(5*f*(a + a*Sin[e + f*x])^3)) - ((2*c + 3*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((2*c + 3*d)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^0, x, 3, -(Cos[e + f*x]/(5*f*(a + a*Sin[e + f*x])^3)) - (2*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - (2*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^1, x, 7, -((2*d^3*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^3*Sqrt[c^2 - d^2]*f)) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3) - ((2*c - 7*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2) - ((2*c^2 - 9*c*d + 22*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2, x, 8, -((2*d^3*(4*c + 3*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^4*(c + d)*Sqrt[c^2 - d^2]*f)) - (d*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*Cos[e + f*x])/(15*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])) - ((2*c - 9*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])) - ((2*c^2 - 12*c*d + 45*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3, x, 9, -((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*Sqrt[c^2 - d^2]*f)) - (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^2) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*c - 11*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((2*c^2 - 15*c*d + 76*d^2)*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{(A + B*Sin[x])/(1 + Sin[x])^4, x, 4, -(((A - B)*Cos[x])/(7*(1 + Sin[x])^4)) - ((3*A + 4*B)*Cos[x])/(35*(1 + Sin[x])^3) - (2*(3*A + 4*B)*Cos[x])/(105*(1 + Sin[x])^2) - (2*(3*A + 4*B)*Cos[x])/(105*(1 + Sin[x]))} -{(A + B*Sin[x])/(1 - Sin[x])^4, x, 4, ((A + B)*Cos[x])/(7*(1 - Sin[x])^4) + ((3*A - 4*B)*Cos[x])/(35*(1 - Sin[x])^3) + (2*(3*A - 4*B)*Cos[x])/(105*(1 - Sin[x])^2) + (2*(3*A - 4*B)*Cos[x])/(105*(1 - Sin[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2), x, 8, (-2*a*(15*c^2 + 56*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*f) - (2*a*(5*c + 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + (2*a*(15*c^3 + 161*c^2*d + 145*c*d^2 + 63*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c^2 - d^2)*(15*c^2 + 56*c*d + 25*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2), x, 7, (-2*a*(3*c + 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (2*a*(3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(3*c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/2), x, 6, (-2*a*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + (2*a*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/2), x, 5, (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2), x, 6, (-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2), x, 7, (-2*a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - (2*a*(c - 3*d)*Cos[e + f*x])/(3*(c - d)*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d)*d*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2), x, 8, (-2*a*Cos[e + f*x])/(5*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) - (2*a*(3*c - 5*d)*Cos[e + f*x])/(15*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*Cos[e + f*x])/(15*(c - d)^2*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*(3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*(c - d)^2*d*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*(3*c - 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*(c - d)*d*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2), x, 9, (4*a^2*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d*f) + (4*a^2*(5*c*(c - 9*d) - 56*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d*f) + (4*a^2*(c - 9*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) - (4*a^2*(5*c^4 - 45*c^3*d - 381*c^2*d^2 - 435*c*d^3 - 168*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c^2 - d^2)*(5*c^3 - 45*c^2*d - 141*c*d^2 - 75*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2), x, 8, (4*a^2*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(35*d*f) + (4*a^2*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) - (4*a^2*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(35*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c^2 - 7*c*d - 10*d^2)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(35*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1/2), x, 7, (4*a^2*(c - 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f) - (4*a^2*(c^2 - 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(1/2), x, 6, (-2*a^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - (4*a^2*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c - 2*d)*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2), x, 6, (2*a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) + (4*a^2*c*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^2*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2), x, 7, (2*a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c + 3*d)*Cos[e + f*x])/(3*d*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^2*(c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c + 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2), x, 8, (2*a^2*(c - d)*Cos[e + f*x])/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) - (4*a^2*(c + 5*d)*Cos[e + f*x])/(15*d*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*Cos[e + f*x])/(15*(c - d)*d*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^2*(c^2 + 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*(c - d)*d^2*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2), x, 11, (-4*a^3*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(693*d^2*f) - (4*a^3*(4*c^3 - 33*c^2*d + 182*c*d^2 + 231*d^3)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(693*d^2*f) - (4*a^3*(4*c^2 - 33*c*d + 189*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*a^3*(c - 6*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11*d*f) + (4*a^3*(c + 3*d)*(4*c^4 - 45*c^3*d + 309*c^2*d^2 + 525*c*d^3 + 231*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(693*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^4 - 33*c^3*d + 177*c^2*d^2 + 561*c*d^3 + 315*d^4)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(693*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2), x, 10, (-4*a^3*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) - (4*a^3*(4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*a^3*(c - 5*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d*f) + (4*a^3*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(1/2), x, 9, (-4*a^3*(4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f) + (8*a^3*(c - 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d*f) + (4*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c^2 - d^2)*(4*c^2 - 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(1/2), x, 8, (8*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(5*d*f) + (4*a^3*(4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c^2 - 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2), x, 8, (2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(2*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c + d)*f) - (4*a^3*(4*c^2 - 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c - 5*d)*(c - d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2), x, 8, (2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*a^3*(c - d)*(c + 2*d)*Cos[e + f*x])/(3*d^2*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (4*a^3*(4*c^2 + 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2), x, 9, (2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*a^3*(c - d)*(c + 3*d)*Cos[e + f*x])/(15*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*Cos[e + f*x])/(15*d^2*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*(c + d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c^2 + 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2), x, 10, (2*(c - d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(7*d*(c + d)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*a^3*(c - d)*(c + 4*d)*Cos[e + f*x])/(35*d^2*(c + d)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*Cos[e + f*x])/(105*(c - d)*d^2*(c + d)^4*f*Sqrt[c + d*Sin[e + f*x]]) - (4*a^3*(4*c^3 + 21*c^2*d + 62*c*d^2 - 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*(c - d)*d^3*(c + d)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^3*(4*c^2 + 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c + d)^3*f*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x]), x, 7, ((3*c - 5*d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x])) - ((3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x]), x, 6, -(((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + a*Sin[e + f*x]), x, 6, -((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/2)), x, 6, -((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)), x, 7, -((d*(c + 3*d)*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)^2*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)), x, 8, -(d*(3*c + 5*d)*Cos[e + f*x])/(3*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*(c - d)^2*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^2, x, 7, -((c - d)*(c + 5*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(3*f*(a + a*Sin[e + f*x])^2) - ((c^2 + 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^2, x, 7, -((c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f*(a + a*Sin[e + f*x])^2) - ((c + 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(c + 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + a*Sin[e + f*x])^2, x, 7, -(c*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f*(a + a*Sin[e + f*x])^2) - (c*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1/2)), x, 7, -((c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)), x, 8, -(d*(c^2 - 5*c*d - 12*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - ((c - 5*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) - ((c^2 - 5*c*d - 12*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)), x, 9, -(d*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - ((c - 7*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - (d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(3*a^2*(c - d)^4*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((c + 3*d)*(c^2 - 10*c*d - 7*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^4*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c^2 - 7*c*d - 10*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)^3*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(c + d*Sin[e + f*x])^(5/2)/(a + a*Sin[e + f*x])^3, x, 8, (-2*(c - d)*(c + 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 + 15*c*d + 27*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*f*(a^3 + a^3*Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f*(a + a*Sin[e + f*x])^3) - ((4*c^2 + 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(4*c^2 + 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + a*Sin[e + f*x])^3, x, 8, -((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*f*(a + a*Sin[e + f*x])^3) - (2*(c + 2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 + 5*c*d - 3*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 + 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c + d)*(4*c + 5*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + a*Sin[e + f*x])^3, x, 8, -(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*f*(a + a*Sin[e + f*x])^3) - ((2*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 - 5*c*d - 3*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)^2*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 - 5*c*d - 3*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c - 5*d)*(c + d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(1/2)), x, 8, -(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3) - (2*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2) - ((4*c^2 - 15*c*d + 27*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])) - ((4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^2 - 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)), x, 9, -(d*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]]) - (2*(c - 4*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^2 - 21*c*d + 65*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^3*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2)), x, 10, -(d*(4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)) - (2*(c - 5*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) - ((4*c^2 - 27*c*d + 119*d^2)*Cos[e + f*x])/(30*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*c^4 - 27*c^3*d + 111*c^2*d^2 + 579*c*d^3 + 357*d^4)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(30*a^3*(c - d)^5*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*c^3 - 27*c^2*d + 114*c*d^2 + 165*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(30*a^3*(c - d)^4*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^3, x, 4, -((4*a*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(35*f*Sqrt[a + a*Sin[e + f*x]])) - (8*(5*c - d)*d*(c + d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(35*f) - (12*d^2*(c + d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*a*f) - (2*a*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^2, x, 3, -((2*a*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (4*(5*c - d)*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f)} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^1, x, 2, -((2*a*(3*c + d)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]])) - (2*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^0, x, 1, -((2*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^1, x, 2, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[d]*Sqrt[c + d]*f))} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^2, x, 3, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[d]*(c + d)^(3/2)*f)) - (a*Cos[e + f*x])/((c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^3, x, 4, -((3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[d]*(c + d)^(5/2)*f)) - (a*Cos[e + f*x])/(2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*a*Cos[e + f*x])/(4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3, x, 6, (4*a^2*(c - 17*d)*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(315*d*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(c - 17*d)*(5*c - d)*(c + d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) + (4*(c - 17*d)*d*(c + d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) + (2*a^2*(c - 17*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*d*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2, x, 4, -((8*a^2*(35*c^2 + 42*c*d + 19*d^2)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]])) - (2*a*(35*c^2 + 42*c*d + 19*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (4*(7*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*a*f)} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^1, x, 3, -((8*a^2*(5*c + 3*d)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (2*a*(5*c + 3*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^0, x, 2, -((8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]])) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^1, x, 4, (2*a^(3/2)*(c - d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*Sqrt[c + d]*f) - (2*a^2*Cos[e + f*x])/(d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^2, x, 4, -((a^(3/2)*(c + 3*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*(c + d)^(3/2)*f)) + (a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^3, x, 5, -((a^(3/2)*(c + 7*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(3/2)*(c + d)^(5/2)*f)) + (a^2*(c - d)*Cos[e + f*x])/(2*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (a^2*(c + 7*d)*Cos[e + f*x])/(4*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3, x, 6, -((4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x])/(3465*d^2*f*Sqrt[a + a*Sin[e + f*x]])) - (8*a^2*(5*c - d)*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) - (4*a*(c + d)*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) - (2*a^3*(3*c^2 - 38*c*d + 355*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*(3*c - 23*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2, x, 5, -((64*a^3*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]])) - (16*a^2*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) - (4*(9*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(63*f) - (2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^1, x, 4, -((64*a^3*(7*c + 5*d)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]])) - (16*a^2*(7*c + 5*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (2*a*(7*c + 5*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^0, x, 3, -((64*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (16*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^1, x, 4, -((2*a^(5/2)*(c - d)^2*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*Sqrt[c + d]*f)) + (2*a^3*(3*c - 7*d)*Cos[e + f*x])/(3*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d*f)} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^2, x, 4, (a^(5/2)*(c - d)*(3*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*(c + d)^(3/2)*f) - (a^3*(3*c + d)*Cos[e + f*x])/(d^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^3, x, 4, -((a^(5/2)*(3*c^2 + 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(5/2)*(c + d)^(5/2)*f)) + (a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) + (3*a^3*(c - d)*(c + 3*d)*Cos[e + f*x])/(4*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^3, x, 6, -((Sqrt[2]*(c - d)^3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*d*(21*c^2 - 12*c*d + 7*d^2)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*(9*c - d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*a*f) - (2*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^2, x, 4, -((Sqrt[2]*(c - d)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*(3*c - d)*d*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*a*f)} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^1, x, 3, -((Sqrt[2]*(c - d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (2*d*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^0, x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f))} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^1, x, 5, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f)) + (2*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^2, x, 6, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*f)) + (Sqrt[d]*(3*c + d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*(c + d)^(3/2)*f) + (d*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^3, x, 7, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f)) + (Sqrt[d]*(15*c^2 + 10*c*d + 7*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[a]*(c - d)^3*(c + d)^(5/2)*f) + (d*Cos[e + f*x])/(2*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + (d*(7*c + d)*Cos[e + f*x])/(4*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3, x, 6, -(((c - d)^2*(c + 11*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f)) + (d*(3*c^2 - 24*c*d + 13*d^2)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*c - 7*d)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2, x, 4, -(((c - d)*(c + 7*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f)) + ((c - 5*d)*d*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^1, x, 3, -(((c + 3*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f)) - ((c - d)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^0, x, 3, -(ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(2*Sqrt[2]*a^(3/2)*f)) - Cos[e + f*x]/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^1, x, 6, -(((c - 5*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f)) - (2*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^2*Sqrt[c + d]*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^2, x, 7, -(((c - 9*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^3*f)) - (d^(3/2)*(5*c + 3*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^3*(c + d)^(3/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) - (d*(c + 3*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^3, x, 8, -(((c - 13*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^4*f)) - (d^(3/2)*(35*c^2 + 42*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(3/2)*(c - d)^4*(c + d)^(5/2)*f) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(c + 2*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (d*(2*c + d)*(c + 7*d)*Cos[e + f*x])/(4*a*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3, x, 6, -((3*(c - d)*(c^2 + 6*c*d + 25*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f)) - ((c - d)^2*(3*c + 13*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) + ((c - 9*d)*d^2*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2, x, 4, -(((3*c^2 + 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f)) - (3*(c - d)*(c + 3*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^1, x, 4, -(((3*c + 5*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f)) - ((c - d)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c + 5*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^0, x, 4, -((3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f)) - Cos[e + f*x]/(4*f*(a + a*Sin[e + f*x])^(5/2)) - (3*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^1, x, 7, -(((3*c^2 - 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^3*f)) + (2*d^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^3*Sqrt[c + d]*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c - 11*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^2, x, 8, -(((3*c^2 - 22*c*d + 115*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^4*f)) + (d^(5/2)*(7*c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])) - (3*(c - 5*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) - ((c - 7*d)*d*(3*c + 5*d)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^3, x, 9, -((3*(c^2 - 10*c*d + 73*d^2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^5*f)) + (3*d^(5/2)*(21*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*c - 19*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(3*c^2 - 20*c*d - 31*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (3*d*(c + 3*d)*(c^2 - 10*c*d - 7*d^2)*Cos[e + f*x])/(16*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(5/2), x, 5, -((5*Sqrt[a]*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*Sqrt[d]*f)) - (5*a*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (5*a*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(3/2), x, 4, -((3*Sqrt[a]*(c + d)^2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*Sqrt[d]*f)) - (3*a*(c + d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(1/2), x, 3, -((Sqrt[a]*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f)) - (a*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(1/2), x, 2, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[d]*f))} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(3/2), x, 1, -((2*a*Cos[e + f*x])/((c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(5/2), x, 2, -((2*a*Cos[e + f*x])/(3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))) - (4*a*Cos[e + f*x])/(3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(7/2), x, 3, -((2*a*Cos[e + f*x])/(5*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))) - (8*a*Cos[e + f*x])/(15*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a*Cos[e + f*x])/(15*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, (5*a^(3/2)*(c - 15*d)*(c + d)^3*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(64*d^(3/2)*f) + (5*a^2*(c - 15*d)*(c + d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(64*d*f*Sqrt[a + a*Sin[e + f*x]]) + (5*a^2*(c - 15*d)*(c + d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(96*d*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - 15*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(24*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(4*d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2), x, 6, (a^(3/2)*(c - 11*d)*(c + d)^2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*d^(3/2)*f) + (a^2*(c - 11*d)*(c + d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*d*f*Sqrt[a + a*Sin[e + f*x]]) + (a^2*(c - 11*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(3*d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(1/2), x, 5, (a^(3/2)*(c - 7*d)*(c + d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*d^(3/2)*f) + (a^2*(c - 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(1/2), x, 4, (a^(3/2)*(c - 3*d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(3/2)*f) - (a^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2), x, 4, -((2*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(3/2)*f)) + (2*a^2*(c - d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2), x, 3, (2*a^2*(c - d)*Cos[e + f*x])/(3*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*a^2*(c + 5*d)*Cos[e + f*x])/(3*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(7/2), x, 4, (2*a^2*(c - d)*Cos[e + f*x])/(5*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^2*(c + 9*d)*Cos[e + f*x])/(15*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a^2*(c + 9*d)*Cos[e + f*x])/(15*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(9/2), x, 5, (2*a^2*(c - d)*Cos[e + f*x])/(7*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^2*(c + 13*d)*Cos[e + f*x])/(35*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^2*(c + 13*d)*Cos[e + f*x])/(105*d*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^2*(c + 13*d)*Cos[e + f*x])/(105*d*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, -((a^(5/2)*(c + d)^3*(3*c^2 - 34*c*d + 283*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(128*d^(5/2)*f)) - (a^3*(c + d)^2*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(128*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(c + d)*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(192*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 34*c*d + 283*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(240*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*a^3*(c - 7*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(40*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2))/(5*d*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2), x, 6, -((a^(5/2)*(c + d)^2*(3*c^2 - 26*c*d + 163*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(64*d^(5/2)*f)) - (a^3*(c + d)*(3*c^2 - 26*c*d + 163*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(64*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(3*c^2 - 26*c*d + 163*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(96*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a^3*(3*c - 17*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(24*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(4*d*f)} -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(1/2), x, 5, -((a^(5/2)*(c + d)*(c^2 - 6*c*d + 25*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(8*d^(5/2)*f)) - (a^3*(c^2 - 6*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a^3*(3*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*d*f)} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(1/2), x, 4, -((a^(5/2)*(3*c^2 - 10*c*d + 19*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*d^(5/2)*f)) + (3*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*d*f)} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2), x, 4, (a^(5/2)*(3*c - 5*d)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(5/2)*f) + (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (a^3*(3*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2), x, 4, -((2*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(d^(5/2)*f)) + (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*a^3*(c - d)*(3*c + 7*d)*Cos[e + f*x])/(3*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(7/2), x, 3, (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + (2*a^3*(c - d)*(3*c + 11*d)*Cos[e + f*x])/(15*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*a^3*(3*c^2 + 14*c*d + 43*d^2)*Cos[e + f*x])/(15*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(9/2), x, 4, (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(7*d*(c + d)*f*(c + d*Sin[e + f*x])^(7/2)) + (6*a^3*(c - d)*(c + 5*d)*Cos[e + f*x])/(35*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (2*a^3*(3*c^2 + 22*c*d + 115*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (4*a^3*(3*c^2 + 22*c*d + 115*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(11/2), x, 5, (2*a^2*(c - d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(9*d*(c + d)*f*(c + d*Sin[e + f*x])^(9/2)) + (2*a^3*(c - d)*(3*c + 19*d)*Cos[e + f*x])/(63*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2)) - (2*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(105*d^2*(c + d)^3*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)) - (8*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^4*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (16*a^3*(c^2 + 10*c*d + 73*d^2)*Cos[e + f*x])/(315*d^2*(c + d)^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, -((Sqrt[d]*(15*c^2 - 10*c*d + 7*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(4*Sqrt[a]*f)) - (Sqrt[2]*(c - d)^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - ((7*c - d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + a*Sin[e + f*x]]) - (d*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(3/2), x, 6, -(((3*c - d)*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)) - (Sqrt[2]*(c - d)^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f) - (d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(1/2), x, 5, -((2*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)) - (Sqrt[2]*Sqrt[c - d]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(1/2), x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f))} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(3/2), x, 4, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^(3/2)*f)) + (2*d*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(5/2), x, 5, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^(5/2)*f)) + (2*d*Cos[e + f*x])/(3*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + (2*d*(5*c + d)*Cos[e + f*x])/(3*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, -(((5*c - 3*d)*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*f)) - ((c - d)^(3/2)*(c + 9*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + ((c - 3*d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2), x, 6, -((2*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*f)) - (Sqrt[c - d]*(c + 5*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(1/2), x, 4, -(((c + d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*Sqrt[c - d]*f)) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(1/2), x, 4, -(((c - 3*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(3/2)*f)) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2), x, 5, -(((c - 7*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(5/2)*f)) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) - (d*(c + 5*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2), x, 6, -(((c - 11*d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^(7/2)*f)) - Cos[e + f*x]/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c + 7*d)*Cos[e + f*x])/(6*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 38*c*d + 19*d^2)*Cos[e + f*x])/(6*a*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, -((2*d^(5/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(5/2)*f)) - (Sqrt[c - d]*(3*c^2 + 14*c*d + 43*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*(3*c + 11*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2), x, 5, -((3*(c + d)^2*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*Sqrt[c - d]*f)) - ((c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c + 7*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(1/2), x, 5, -(((3*c - 5*d)*(c + d)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(3/2)*f)) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*c - d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(1/2), x, 5, -(((3*c^2 - 10*c*d + 19*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(5/2)*f)) - (Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)) - (3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2))} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2), x, 6, -((3*(c^2 - 6*c*d + 25*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(7/2)*f)) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]]) - ((3*c - 13*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) - ((c - 7*d)*d*(3*c + 7*d)*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2), x, 7, -(((3*c^2 - 26*c*d + 163*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^(9/2)*f)) - Cos[e + f*x]/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2)) - ((3*c - 17*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) - (d*(9*c^2 - 54*c*d - 95*d^2)*Cos[e + f*x])/(48*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (d*(9*c^3 - 57*c^2*d - 493*c*d^2 - 299*d^3)*Cos[e + f*x])/(48*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 4, (Sqrt[2]*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3, x, 6, If[$VersionNumber>=8, -((d*(d^2*(4 + m) - c*d*(5 - 3*m - 2*m^2) + 2*c^2*(8 + 6*m + m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m))) - (2^(1/2 + m)*(d^3*m*(5 + 3*m + m^2) + 3*c^2*d*m*(6 + 5*m + m^2) + 3*c*d^2*(3 + 4*m + 4*m^2 + m^3) + c^3*(6 + 11*m + 6*m^2 + m^3))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m)) - (d^2*(d*m + c*(5 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)*(3 + m)) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2)/(f*(3 + m)), -((d*(d^2*(4 + m) - c*d*(5 - 3*m - 2*m^2) + 2*c^2*(8 + 6*m + m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(3 + m)*(2 + 3*m + m^2))) - (1/(f*(3 + m)*(2 + 3*m + m^2)))*(2^(1/2 + m)*(d^3*m*(5 + 3*m + m^2) + 3*c^2*d*m*(6 + 5*m + m^2) + 3*c*d^2*(3 + 4*m + 4*m^2 + m^3) + c^3*(6 + 11*m + 6*m^2 + m^3))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m) - (d^2*(d*m + c*(5 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)*(3 + m)) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2)/(f*(3 + m))]} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2, x, 4, If[$VersionNumber>=8, (d*(d - 2*c*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)), (d*(d - 2*c*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))]} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^1, x, 3, -((d*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*(c + c*m + d*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m))} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^0, x, 2, -((2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/f)} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^1, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)^2*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)^3*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x, 4, (Sqrt[2]*(c - d)^2*AppellF1[1/2 + m, 1/2, -(5/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2), x, 4, (Sqrt[2]*(c - d)*AppellF1[1/2 + m, 1/2, -(3/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1/2), x, 4, (Sqrt[2]*AppellF1[1/2 + m, 1/2, -(1/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(1/2), x, 4, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x, 4, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2), x, 4, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 5/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)^2*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{(1 + Sin[e + f*x])^m/(3 + 5*Sin[e + f*x])^(m + 1), x, 2, -((4^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (1 - Sin[e + f*x])/(4*(1 + Sin[e + f*x]))])/(f*(1 + Sin[e + f*x]))), -((2^(-1 - 2*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, -((1 - Sin[e + f*x])/(3 + 5*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 5*Sin[e + f*x]))^(1/2 - m))/((3 + 5*Sin[e + f*x])^m*f))} -{(1 + Sin[e + f*x])^m/(3 + 4*Sin[e + f*x])^(m + 1), x, 2, -(((7/2)^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (1 - Sin[e + f*x])/(7*(1 + Sin[e + f*x]))])/(f*(1 + Sin[e + f*x]))), -((2^(1/2 + m)*7^(-(1/2) - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, -((1 - Sin[e + f*x])/(2*(3 + 4*Sin[e + f*x])))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 4*Sin[e + f*x]))^(1/2 - m))/((3 + 4*Sin[e + f*x])^m*f))} -{(1 + Sin[e + f*x])^m/(3 + 3*Sin[e + f*x])^(m + 1), x, 2, -((3^(-1 - m)*Cos[e + f*x])/(f*(1 + Sin[e + f*x])))} -{(1 + Sin[e + f*x])^m/(3 + 2*Sin[e + f*x])^(m + 1), x, 2, -((2^(1/2 + m)*5^(-(1/2) - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/(2*(3 + 2*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + 2*Sin[e + f*x]))^(1/2 - m))/((3 + 2*Sin[e + f*x])^m*f))} -{(1 + Sin[e + f*x])^m/(3 + 1*Sin[e + f*x])^(m + 1), x, 2, -((2^(-(1/2) - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/(3 + Sin[e + f*x])]*(1 + Sin[e + f*x])^(-1 + m)*((1 + Sin[e + f*x])/(3 + Sin[e + f*x]))^(1/2 - m))/((3 + Sin[e + f*x])^m*f))} -{(1 + Sin[e + f*x])^m/(3 + 0*Sin[e + f*x])^(m + 1), x, 2, -((2^(1/2 + m)*3^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])])/(f*Sqrt[1 + Sin[e + f*x]]))} -{(1 + Sin[e + f*x])^m/(3 - 1*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, -((2*(1 - Sin[e + f*x]))/(1 + Sin[e + f*x]))]*(3 - Sin[e + f*x])^(-1 - m)*((3 - Sin[e + f*x])/(1 + Sin[e + f*x]))^(1 + m)*(1 + Sin[e + f*x])^m)/f)} -{(1 + Sin[e + f*x])^m/(3 - 2*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(1 + Sin[e + f*x])^m)/((3 - 2*Sin[e + f*x])^m*(Sqrt[5]*f*m*(1 - Sin[e + f*x])))} -{(1 + Sin[e + f*x])^m/(3 - 3*Sin[e + f*x])^(m + 1), x, 1, (Cos[e + f*x]*(3 - 3*Sin[e + f*x])^(-1 - m)*(1 + Sin[e + f*x])^m)/(f*(1 + 2*m))} -{(1 + Sin[e + f*x])^m/(3 - 4*Sin[e + f*x])^(m + 1), x, 2, (2^(1 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (7*(1 - Sin[e + f*x]))/(1 + Sin[e + f*x])]*(-3 + 4*Sin[e + f*x])^m)/((3 - 4*Sin[e + f*x])^m*(f*(1 + Sin[e + f*x]))), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])^m)/((3 - 4*Sin[e + f*x])^m*(Sqrt[7]*f*m*(1 - Sin[e + f*x])))} -{(1 + Sin[e + f*x])^m/(3 - 5*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (4*(1 - Sin[e + f*x]))/(1 + Sin[e + f*x])]*(-3 + 5*Sin[e + f*x])^m)/((3 - 5*Sin[e + f*x])^m*(f*(1 + Sin[e + f*x]))), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(1 + Sin[e + f*x])^m)/((3 - 5*Sin[e + f*x])^m*(4*f*m*(1 - Sin[e + f*x])))} - - -{(a + a*Sin[e + f*x])^m/(3 + 5*Sin[e + f*x])^(m + 1), x, 2, -((4^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (a - a*Sin[e + f*x])/(4*(a + a*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + 5*Sin[e + f*x])/(4*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((3 + 5*Sin[e + f*x])^m*(4*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(3 + 4*Sin[e + f*x])^(m + 1), x, 2, -(((7/2)^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (a - a*Sin[e + f*x])/(7*(a + a*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 4*Sin[e + f*x]))/(7*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((3 + 4*Sin[e + f*x])^m*(Sqrt[7]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(3 + 3*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*(3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f)} -{(a + a*Sin[e + f*x])^m/(3 + 2*Sin[e + f*x])^(m + 1), x, 2, -(((5/2)^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, -((a - a*Sin[e + f*x])/(5*(a + a*Sin[e + f*x])))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 2*Sin[e + f*x]))/(5*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((3 + 2*Sin[e + f*x])^m*(Sqrt[5]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(3 + 1*Sin[e + f*x])^(m + 1), x, 2, -((2^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, -((a - a*Sin[e + f*x])/(2*(a + a*Sin[e + f*x])))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + Sin[e + f*x])/(2*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((3 + Sin[e + f*x])^m*(2*Sqrt[2]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(3 + 0*Sin[e + f*x])^(m + 1), x, 3, -((2^(1/2 + m)*3^(-1 - m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/f)} -{(a + a*Sin[e + f*x])^m/(3 - 1*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, -((2*(a - a*Sin[e + f*x]))/(a + a*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 - Sin[e + f*x])/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((3 - Sin[e + f*x])^m*(2*Sqrt[2]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(3 - 2*Sin[e + f*x])^(m + 1), x, 2, -((2^(1 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, -((5*(a - a*Sin[e + f*x]))/(a + a*Sin[e + f*x]))]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((3 - 2*Sin[e + f*x])^m*(Sqrt[5]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(3 - 3*Sin[e + f*x])^(m + 1), x, 1, (Cos[e + f*x]*(3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m/(3 - 4*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((3 - 4*Sin[e + f*x])^m*(Sqrt[7]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(3 - 5*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((3 - 5*Sin[e + f*x])^m*(4*f*m*(1 - Sin[e + f*x])))} - - -{(a + a*Sin[e + f*x])^m/(-3 + 5*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (4*(a - a*Sin[e + f*x]))/(a + a*Sin[e + f*x])]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((3 - 5*Sin[e + f*x])/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((-3 + 5*Sin[e + f*x])^m*(4*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 + 4*Sin[e + f*x])^(m + 1), x, 2, -((2^(1 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, (7*(a - a*Sin[e + f*x]))/(a + a*Sin[e + f*x])]*(1 + Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f), -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, -((2*(3 - 4*Sin[e + f*x]))/(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((-3 + 4*Sin[e + f*x])^m*(Sqrt[7]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 + 3*Sin[e + f*x])^(m + 1), x, 1, (Cos[e + f*x]*(-3 + 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m/(-3 + 2*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 - 2*Sin[e + f*x]))/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((-3 + 2*Sin[e + f*x])^m*(Sqrt[5]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 + 1*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 - Sin[e + f*x])/(1 + Sin[e + f*x])]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((-3 + Sin[e + f*x])^m*(2*Sqrt[2]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 + 0*Sin[e + f*x])^(m + 1), x, 3, -(((-3)^(-1 - m)*2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/f)} -{(a + a*Sin[e + f*x])^m/(-3 - 1*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + Sin[e + f*x])/(2*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((-3 - Sin[e + f*x])^m*(2*Sqrt[2]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 - 2*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 2*Sin[e + f*x]))/(5*(1 + Sin[e + f*x]))]*Sqrt[-((1 - Sin[e + f*x])/(1 + Sin[e + f*x]))]*(a + a*Sin[e + f*x])^m)/((-3 - 2*Sin[e + f*x])^m*(Sqrt[5]*f*m*(1 - Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(-3 - 3*Sin[e + f*x])^(m + 1), x, 2, -((Cos[e + f*x]*(-3 - 3*Sin[e + f*x])^(-1 - m)*(a + a*Sin[e + f*x])^m)/f)} -{(a + a*Sin[e + f*x])^m/(-3 - 4*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (2*(3 + 4*Sin[e + f*x]))/(7*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((-3 - 4*Sin[e + f*x])^m*(Sqrt[7]*f*m*(1 - Sin[e + f*x])))} -{(a + a*Sin[e + f*x])^m/(-3 - 5*Sin[e + f*x])^(m + 1), x, 2, (Cos[e + f*x]*Hypergeometric2F1[1/2, -m, 1 - m, (3 + 5*Sin[e + f*x])/(4*(1 + Sin[e + f*x]))]*Sqrt[(1 - Sin[e + f*x])/(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/((-3 - 5*Sin[e + f*x])^m*(4*f*m*(1 - Sin[e + f*x])))} - - -{(a + a*Sin[e + f*x])^m/(d*Sin[e + f*x])^(m + 1), x, 4, -((Cos[e + f*x]*Hypergeometric2F1[1/2 - m, -m, 1 - m, -((2*Sin[e + f*x])/(1 - Sin[e + f*x]))]*((1 + Sin[e + f*x])/(1 - Sin[e + f*x]))^(1/2 - m)*(a + a*Sin[e + f*x])^m)/((d*Sin[e + f*x])^m*(d*f*m*(1 + Sin[e + f*x]))))} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(m + 1), x, 2, -((1/((c + d)*f))*((2^(1/2 + m)*a*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a + a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/(c + d*Sin[e + f*x])^m))} - - -{(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((8*Sqrt[2]*a^3*AppellF1[1/2, -(5/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((4*Sqrt[2]*a^2*AppellF1[1/2, -(3/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((2*Sqrt[2]*a*AppellF1[1/2, -(1/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{(a + a*Sin[e + f*x])^0*(c + d*Sin[e + f*x])^n, x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]])))} -{1/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, 3/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(Sqrt[2]*a*f*Sqrt[1 + Sin[e + f*x]])))} -{1/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, 5/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(2*Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]])))} -{1/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, 7/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(4*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]])))} - - -{(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^n, x, 5, (2*a^3*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(5 + 2*n)) - (2*a^3*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^n, x, 5, -((2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) + (2*a^2*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^n, x, 3, -((2*a*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[a + a*Sin[e + f*x]])))} -{1/(a + a*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, -n, 1, 3/2, (d*(1 - Sin[e + f*x]))/(c + d), (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[a + a*Sin[e + f*x]]))), -((AppellF1[1 + n, 1/2, 1, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)*f*(1 + n)*(1 - Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]]))} -{1/(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, -n, 2, 3/2, (d*(1 - Sin[e + f*x]))/(c + d), (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(2*a*f*Sqrt[a + a*Sin[e + f*x]]))), (d*AppellF1[1 + n, 1/2, 2, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)^2*f*(1 + n)*(a - a*Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]])} -{1/(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[1/2, -n, 3, 3/2, (d*(1 - Sin[e + f*x]))/(c + d), (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]))), -((d^2*AppellF1[1 + n, 1/2, 3, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)^3*f*(1 + n)*Sqrt[a + a*Sin[e + f*x]]*(a^2 - a^2*Sin[e + f*x])))} - - -{(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3), x, 3, -((2*Sqrt[2]*a*AppellF1[1/2, -(1/2), -(1/3), 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3)))} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/3), x, 3, -((2*Sqrt[2]*a*AppellF1[1/2, -(1/2), 1/3, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*((c + d*Sin[e + f*x])/(c + d))^(1/3))/(f*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/3)))} -{(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(4/3), x, 3, -((2*Sqrt[2]*a*AppellF1[1/2, -(1/2), 4/3, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*((c + d*Sin[e + f*x])/(c + d))^(1/3))/((c + d)*f*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/3)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^3, x, 3, (1/8)*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*x - ((4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4*c*d^2))*Cos[e + f*x])/(6*f) - (d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((3*b*c + 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(12*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f)} -{(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^2, x, 2, (1/2)*(2*b*c*d + a*(2*c^2 + d^2))*x - (2*(3*a*c*d + b*(c^2 + d^2))*Cos[e + f*x])/(3*f) - (d*(2*b*c + 3*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f)} -{(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^1, x, 1, (1/2)*(2*a*c + b*d)*x - ((b*c + a*d)*Cos[e + f*x])/f - (b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^0, x, 2, a*x - (b*Cos[e + f*x])/f} -{(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^1, x, 4, (b*x)/d - (2*(b*c - a*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d*Sqrt[c^2 - d^2]*f)} -{(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^2, x, 5, (2*(a*c - b*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(3/2)*f) - ((b*c - a*d)*Cos[e + f*x])/((c^2 - d^2)*f*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^3, x, 6, -(((3*b*c*d - a*(2*c^2 + d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(5/2)*f)) - ((b*c - a*d)*Cos[e + f*x])/(2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((3*a*c*d - b*(c^2 + 2*d^2))*Cos[e + f*x])/(2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} - - -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3, x, 4, (1/8)*(6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*x - ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*Cos[e + f*x])/(30*d*f) - ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*Cos[e + f*x]*Sin[e + f*x])/(120*f) - ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d*f) + (b*(b*c - 10*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f)} -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2, x, 3, (1/8)*(16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*x - ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*Cos[e + f*x])/(6*b*f) - ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (d*(8*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*b*f)} -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^1, x, 2, (1/2)*(2*a^2*c + b^2*c + 2*a*b*d)*x - (2*(3*a*b*c + a^2*d + b^2*d)*Cos[e + f*x])/(3*f) - (b*(3*b*c + 2*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (d*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)} -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^0, x, 1, (1/2)*(2*a^2 + b^2)*x - (2*a*b*Cos[e + f*x])/f - (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^1, x, 5, -((b*(b*c - 2*a*d)*x)/d^2) + (2*(b*c - a*d)^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*Sqrt[c^2 - d^2]*f) - (b^2*Cos[e + f*x])/(d*f)} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2, x, 5, (b^2*x)/d^2 - (2*(b*c - a*d)*(a*c*d + b*(c^2 - 2*d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*(c^2 - d^2)^(3/2)*f) + ((b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(5/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) - ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(2*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4, x, 7, -(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3) - ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Cos[e + f*x])/(6*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) + ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(6*d*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x]))} - - -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3, x, 5, (1/16)*(18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x - ((3*a*b^2*d*(3*c^2 + d^2) + 3*a^2*b*c*(c^2 + 3*d^2) + b^3*c*(c^2 + 3*d^2) + a^3*(3*c^2*d + d^3))*Cos[e + f*x])/f + ((b*c + a*d)*(8*a*b*c*d + a^2*d^2 + b^2*(c^2 + 6*d^2))*Cos[e + f*x]^3)/(3*f) - (3*b^2*d^2*(b*c + a*d)*Cos[e + f*x]^5)/(5*f) - ((24*a^3*c*d^2 + 18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2))*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (5*b^3*d^3*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (3*b*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f) - (b^3*d^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f), (1/16)*(18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x - ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*Cos[e + f*x])/(60*d^2*f) - ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*Cos[e + f*x]*Sin[e + f*x])/(240*d*f) - ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^2*f) + (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d^2*f) + (b^2*(2*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)} -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2, x, 4, (1/8)*(24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*x - ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*Cos[e + f*x])/(30*b*f) - ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*Cos[e + f*x]*Sin[e + f*x])/(120*f) - ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(60*b*f) - (d*(10*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(20*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^4)/(5*b*f)} -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^1, x, 3, (1/8)*(8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*x - ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*Cos[e + f*x])/(6*f) - (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((4*b*c + 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*f) - (d*Cos[e + f*x]*(a + b*Sin[e + f*x])^3)/(4*f)} -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^0, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x - (2*b*(4*a^2 + b^2)*Cos[e + f*x])/(3*f) - (5*a*b^2*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (b*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(3*f)} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^1, x, 6, -((b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*x)/(2*d^3)) - (2*(b*c - a*d)^3*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*Sqrt[c^2 - d^2]*f) + (b^2*(2*b*c - 5*a*d)*Cos[e + f*x])/(2*d^2*f) - (b^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*d*f)} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2, x, 6, -((b^2*(2*b*c - 3*a*d)*x)/d^3) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(3/2)*f) + (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x])/(d^2*(c^2 - d^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3, x, 6, (b^3*x)/d^3 - ((9*a^2*b*c*d^4 - a^3*d^3*(2*c^2 + d^2) - 3*a*b^2*d^3*(c^2 + 2*d^2) + b^3*(2*c^5 - 5*c^3*d^2 + 6*c*d^4))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(5/2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(2*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*Cos[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^4, x, 7, -(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) - ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x]))} - - -{(b*B/a + B*Sin[x])/(a + b*Sin[x]), x, 4, (B*x)/b - (2*Sqrt[a^2 - b^2]*B*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b)} -{(a*B/b + B*Sin[x])/(a + b*Sin[x]), x, 2, (B*x)/b} - -{(a + b*Sin[x])/(b + a*Sin[x])^2, x, 2, -(Cos[x]/(b + a*Sin[x]))} -{(2 - Sin[x])/(2 + Sin[x]), x, 2, -x + (4*x)/Sqrt[3] + (8*ArcTan[Cos[x]/(2 + Sqrt[3] + Sin[x])])/Sqrt[3]} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^4, x, 7, (d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12*c^2 + d^2))*x)/(2*b^4) + (2*(b*c - a*d)^4*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*f) + (d^2*(12*a*b*c*d - 3*a^2*d^2 - b^2*(17*c^2 + 2*d^2))*Cos[e + f*x])/(3*b^3*f) - (d^3*(8*b*c - 3*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*b^2*f) - (d^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*b*f)} -{1/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^3, x, 6, -((d*(6*a*b*c*d - 2*a^2*d^2 - b^2*(6*c^2 + d^2))*x)/(2*b^3)) + (2*(b*c - a*d)^3*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*f) - (d^2*(5*b*c - 2*a*d)*Cos[e + f*x])/(2*b^2*f) - (d^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*b*f)} -{1/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^2, x, 5, (d*(2*b*c - a*d)*x)/b^2 + (2*(b*c - a*d)^2*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*f) - (d^2*Cos[e + f*x])/(b*f)} -{1/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^1, x, 4, (d*x)/b + (2*(b*c - a*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]*f)} -{1/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^0, x, 3, (2*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*f)} -{1/(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^1, x, 7, (2*b*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)*f) - (2*d*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)*Sqrt[c^2 - d^2]*f)} -{1/(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^2, x, 8, (2*b^2*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)^2*f) + (2*d*(a*c*d - b*(2*c^2 - d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*(c^2 - d^2)^(3/2)*f) - (d^2*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^1/(c + d*Sin[e + f*x])^3, x, 9, (2*b^3*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*(c^2 - d^2)^(5/2)*f) - (d^2*Cos[e + f*x])/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) - (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*Cos[e + f*x])/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} - - -{1/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4, x, 7, -((d^2*(16*a*b*c*d - 6*a^2*d^2 - b^2*(12*c^2 + d^2))*x)/(2*b^4)) + (2*(b*c - a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*f) + (d*(2*b*c - a*d)*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(b^3*(a^2 - b^2)*f) + (d^2*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3, x, 6, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (2*(b*c - a*d)^2*(a*b*c + 2*a^2*d - 3*b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*f) + (d*(2*a*b*c*d - 2*a^2*d^2 - b^2*(c^2 - d^2))*Cos[e + f*x])/(b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2, x, 5, (d^2*x)/b^2 + (2*(b*c - a*d)*(a*b*c + a^2*d - 2*b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*f) + ((b*c - a*d)^2*Cos[e + f*x])/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^1, x, 5, (2*(a*c - b*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*f) + ((b*c - a*d)*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^0, x, 5, (2*a*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*f) + (b*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^1, x, 8, (2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^2*f) + (2*d^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]*f) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^2, x, 9, (2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*(c^2 - d^2)^(3/2)*f) + (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^3, x, 10, (2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(5/2)*f) + (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ((3*a^3*c*d^4 - 3*a*b^2*c*d^4 - a^2*b*d^3*(7*c^2 - 4*d^2) - b^3*(2*c^4*d - 11*c^2*d^3 + 6*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} - - -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^5, x, 8, -((d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x)/(2*b^5)) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(5/2)*f) - (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(2*b^4*(a^2 - b^2)^2*f) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)} -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^4, x, 7, (d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*Cos[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)} -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3, x, 6, (d^3*x)/b^3 + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(5/2)*f) + ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*Cos[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x]))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)} -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*Cos[e + f*x])/(2*b*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^1, x, 6, ((2*a^2*c + b^2*c - 3*a*b*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f) + ((b*c - a*d)*Cos[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((3*a*b*c - a^2*d - 2*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^0, x, 6, ((2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*f) + (b*Cos[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (3*a*b*Cos[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^1, x, 9, -((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^3*Sqrt[c^2 - d^2]*f) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^2, x, 10, -((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(3/2)*f) - (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])) + (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{1/(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^3, x, 11, -((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) + (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2), x, 8, (-2*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*f) - (2*(5*b*c + 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*f) - (2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + (2*(15*b*c^3 + 161*a*c^2*d + 145*b*c*d^2 + 63*a*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(15*b*c^2 + 56*a*c*d + 25*b*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2), x, 7, (-2*(3*b*c + 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*f) - (2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + (2*(20*a*c*d + 3*b*(c^2 + 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(3*b*c + 5*a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/2), x, 6, (-2*b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + (2*(b*c + 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/2), x, 5, (2*b*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2), x, 6, (-2*(b*c - a*d)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2), x, 7, (-2*(b*c - a*d)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*Cos[e + f*x])/(3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(4*a*c*d - b*(c^2 + 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2), x, 8, (-2*(b*c - a*d)*Cos[e + f*x])/(5*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*Cos[e + f*x])/(15*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*Cos[e + f*x])/(15*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2), x, 9, (-4*(84*a^2*c*d^2 + 15*a*b*d*(3*c^2 + 5*d^2) - b^2*(5*c^3 - 57*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d*f) - (2*(7*(9*a^2 + 7*b^2)*d^2 - 10*b*c*(b*c - 9*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d*f) + (4*b*(b*c - 9*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) + (2*(21*a^2*d^2*(23*c^2 + 9*d^2) + 30*a*b*d*(3*c^3 + 29*c*d^2) - b^2*(10*c^4 - 279*c^2*d^2 - 147*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*(c^2 - d^2)*(5*b^2*c^3 - 45*a*b*c^2*d - 84*a^2*c*d^2 - 57*b^2*c*d^2 - 75*a*b*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2), x, 8, (-2*(5*(7*a^2 + 5*b^2)*d^2 - 6*b*c*(b*c - 7*a*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d*f) + (4*b*(b*c - 7*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) + (4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(42*a*b*c*d + 35*a^2*d^2 - b^2*(6*c^2 - 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1/2), x, 7, (4*b*(b*c - 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d*f) - (2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f) + (2*(3*(5*a^2 + 3*b^2)*d^2 - 2*b*c*(b*c - 5*a*d))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*b*(b*c - 5*a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(1/2), x, 6, (-2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - (4*b*(b*c - 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*((3*a^2 + b^2)*d^2 + 2*b*c*(b*c - 3*a*d))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(3/2), x, 6, (2*(b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(2*b^2*c^2 - 2*a*b*c*d + (a^2 - b^2)*d^2)*EllipticE[(1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*b*(b*c - a*d)*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(5/2), x, 7, (2*(b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(3*d*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (4*(b*c - a*d)*(2*a*c*d + b*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(2*a*b*c*d - a^2*d^2 + b^2*(2*c^2 - 3*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2), x, 8, (2*(b*c - a*d)^2*Cos[e + f*x])/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*Cos[e + f*x])/(15*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*Cos[e + f*x])/(15*d*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(a^2*d^2*(23*c^2 + 9*d^2) - a*b*(6*c^3*d + 58*c*d^3) - b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*(b*c - a*d)*(4*a*c*d + b*(c^2 - 5*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2), x, 10, (-2*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^2*f) - (2*(1485*a^2*b*c*d^2 + 693*a^3*d^3 - 33*a*b^2*d*(10*c^2 - 49*d^2) + 5*b^3*(8*c^3 + 67*c*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(3465*d^2*f) + (2*b*(66*a*b*c*d - 297*a^2*d^2 - b^2*(8*c^2 + 81*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(693*d^2*f) + (8*b^2*(b*c - 6*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(99*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11*d*f) + (2*(231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) - 33*a*b^2*d*(10*c^4 - 279*c^2*d^2 - 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3465*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(1848*a^3*c*d^3 + 495*a^2*b*d^2*(3*c^2 + 5*d^2) - 66*a*b^2*d*(5*c^3 - 57*c*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3465*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2), x, 9, (-2*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(315*d^2*f) + (2*b*(54*a*b*c*d - 189*a^2*d^2 - b^2*(8*c^2 + 49*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(315*d^2*f) + (8*b^2*(b*c - 5*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(63*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d*f) + (2*(420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3*d^2) - a*b^2*(54*c^3*d - 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(315*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(189*a^2*b*c*d^2 + 105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(315*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(1/2), x, 8, (2*b*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f) + (8*b^2*(b*c - 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d*f) + (2*(105*a^2*b*c*d^2 + 105*a^3*d^3 - 21*a*b^2*d*(2*c^2 - 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*(c^2 - d^2)*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(1/2), x, 7, (8*b^2*(b*c - 3*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])/(5*d*f) - (2*b*(30*a*b*c*d - 45*a^2*d^2 - b^2*(8*c^2 + 9*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(45*a^2*b*c*d^2 - 15*a^3*d^3 - 15*a*b^2*d*(2*c^2 + d^2) + b^3*(8*c^3 + 7*c*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(3/2), x, 7, (2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b*(6*a*b*c*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*(c^2 - d^2)*f) - (2*(9*a^2*b*c*d^2 - 3*a^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*b*(18*a*b*c*d - 9*a^2*d^2 - b^2*(8*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2), x, 7, (2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (8*(b*c - a*d)^2*(a*c*d + b*(c^2 - 2*d^2))*Cos[e + f*x])/(3*d^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15*c^2*d^2 + 3*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^3*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(b*c - a*d)*(2*a*b*c*d - a^2*d^2 + b^2*(8*c^2 - 9*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2), x, 8, (2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + (8*(b*c - a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(3/2)) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[e + f*x])/(15*d^2*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*(a^2*d^2*(23*c^2 + 9*d^2) + 2*a*b*d*(7*c^3 - 39*c*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*(c^2 - d^2)^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 5*d^2) - 3*a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(9/2), x, 9, (2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(7*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(7/2)) + (8*(b*c - a*d)^2*(3*a*c*d + b*(c^2 - 4*d^2))*Cos[e + f*x])/(35*d^2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^(5/2)) - (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*Cos[e + f*x])/(105*d^2*(c^2 - d^2)^4*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(16*a^3*c*d^3*(11*c^2 + 13*d^2) - 6*a*b^2*c*d*(3*c^4 - 62*c^2*d^2 - 133*d^4) - 9*a^2*b*d^2*(5*c^4 + 102*c^2*d^2 + 21*d^4) - b^3*(8*c^6 - 23*c^4*d^2 + 294*c^2*d^4 + 105*d^6))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*(c^2 - d^2)^4*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)*(a^2*d^2*(71*c^2 + 25*d^2) + a*b*(26*c^3*d - 218*c*d^3) + b^2*(8*c^4 - 17*c^2*d^2 + 105*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*d^3*(c^2 - d^2)^3*f*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]), x, 9, (-2*d^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + (2*d*(7*b*c - 3*a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*b^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*d*(6*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*b^3*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)^3*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^3*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x]), x, 8, (2*d*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*d*(b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)^2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + b*Sin[e + f*x]), x, 5, (2*d*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]]) + (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/2)), x, 2, (2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)), x, 7, (-2*d^2*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((b*c - a*d)*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*b*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)), x, 10, (-2*d^2*Cos[e + f*x])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (2*d^2*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*Cos[e + f*x])/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d*(7*b*c^2 - 4*a*c*d - 3*b*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*d*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*(b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*b^2*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^2, x, 10, (d*(6*a*b*c*d - 5*a^2*d^2 - b^2*(3*c^2 - 2*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])) + ((29*a^2*b*c*d^2 - 15*a^3*d^3 + b^3*(3*c^3 - 20*c*d^2) - a*b^2*(9*c^2*d - 12*d^3))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*b^3*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((24*a^3*b*c*d^3 - 15*a^4*d^4 - 12*a*b^3*c*d*(c^2 + 3*d^2) + 2*a^2*b^2*d^2*(c^2 + 8*d^2) + b^4*(3*c^4 + 16*c^2*d^2 + 2*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*b^4*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(2*a*b*c + 5*a^2*d - 7*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^4*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^2, x, 9, ((b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])) - ((2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b^2*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((b*c - a*d)*(2*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 + 4*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^3*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^2*(2*a*b*c + 3*a^2*d - 5*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^3*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^2, x, 9, ((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((2*a*b*c*d + a^2*d^2 - b^2*(c^2 + 2*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b^2*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + b*Sin[e + f*x])^2, x, 9, (b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x])) + (EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((b*c - a*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*a*b*c - a^2*d - b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*b*(a + b)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1/2)), x, 9, (b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])) + (b*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a^2 - b^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*a*b*c - 3*a^2*d + b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)), x, 10, (d*(2*a^2*d^2 + b^2*(c^2 - 3*d^2))*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) + ((2*a^2*d^2 + b^2*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (b*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]]) + (b*(2*a*b*c - 5*a^2*d + 3*b^2*d)*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2)), x, 11, (d*(2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - ((8*a^3*c*d^4 - 8*a*b^2*c*d^4 - 4*a^2*b*d^3*(5*c^2 - 3*d^2) - b^3*(3*c^4*d - 26*c^2*d^3 + 15*d^5))*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^3*c*d^3 - 8*a*b^2*c*d^3 - 4*a^2*b*d^2*(5*c^2 - 3*d^2) - b^3*(3*c^4 - 26*c^2*d^2 + 15*d^4))*EllipticE[(1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((2*a^2*d^2 + b^2*(3*c^2 - 5*d^2))*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(2*a*b*c - 7*a^2*d + 5*b^2*d)*EllipticPi[(2*b)/(a + b), (1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a - b)*(a + b)^2*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])} - - -{(c + d*Sin[e + f*x])^(9/2)/(a + b*Sin[e + f*x])^3, x, 11, (d*(36*a^3*b*c*d^2 - 35*a^4*d^3 + b^4*d*(45*c^2 - 8*d^2) - 18*a*b^3*c*(c^2 + 5*d^2) + a^2*b^2*d*(9*c^2 + 61*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(12*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(6*a*b*c + 7*a^2*d - 13*b^2*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((185*a^4*b*c*d^3 - 105*a^5*d^4 - b^5*c*d*(51*c^2 - 104*d^2) - 15*a^3*b^2*d^2*(3*c^2 - 13*d^2) - a^2*b^3*c*d*(21*c^2 + 361*d^2) + 9*a*b^4*(2*c^4 + 17*c^2*d^2 - 8*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*b^4*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((150*a^5*b*c*d^4 - 105*a^6*d^5 - 12*a^3*b^3*c*d^2*(4*c^2 + 29*d^2) + a^4*b^2*d^3*(26*c^2 + 223*d^2) - b^6*d*(57*c^4 + 136*c^2*d^2 + 8*d^4) + 6*a*b^5*c*(3*c^4 + 38*c^2*d^2 + 48*d^4) - a^2*b^4*d*(33*c^4 + 70*c^2*d^2 + 128*d^4))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(12*b^5*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^3*(20*a^3*b*c*d - 44*a*b^3*c*d + 35*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 43*d^2) + b^4*(4*c^2 + 63*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^5*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(7/2)/(a + b*Sin[e + f*x])^3, x, 10, ((b*c - a*d)^2*(6*a*b*c + 5*a^2*d - 11*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) - ((8*a^3*b*c*d^2 - 15*a^4*d^3 + b^4*d*(13*c^2 - 8*d^2) - 2*a*b^3*c*(3*c^2 + 13*d^2) + a^2*b^2*d*(5*c^2 + 29*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b^3*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (3*(b*c - a*d)*(4*a^3*b*c*d^2 + 5*a^4*d^3 + a^2*b^2*d*(c^2 - 11*d^2) - 2*a*b^3*c*(c^2 + 5*d^2) + b^4*d*(5*c^2 + 8*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^4*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)^2*(12*a^3*b*c*d - 36*a*b^3*c*d + 15*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 19*d^2) + b^4*(4*c^2 + 35*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^4*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x])^3, x, 10, ((b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + (3*(b*c - a*d)*(2*a*b*c + a^2*d - 3*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b^2*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((4*a^3*b*c*d^2 + 3*a^4*d^3 + a^2*b^2*d*(7*c^2 - 5*d^2) + b^4*d*(11*c^2 + 8*d^2) - 2*a*b^3*c*(3*c^2 + 11*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^3*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) + ((b*c - a*d)*(4*a^3*b*c*d - 28*a*b^3*c*d + 3*a^4*d^2 + 2*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^3*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x])^3, x, 10, ((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + ((6*a*b*c - a^2*d - 5*b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((6*a*b*c - a^2*d - 5*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*b*(a^2 - b^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((b*c - a*d)*(6*a*b*c + a^2*d - 7*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b^2*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((4*a^3*b*c*d + 20*a*b^3*c*d + a^4*d^2 - b^4*(4*c^2 + 3*d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b^2*(a + b)^3*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(1/2)/(a + b*Sin[e + f*x])^3, x, 10, (b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2) + (b*(6*a*b*c - 5*a^2*d - b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*(a + b*Sin[e + f*x])) + ((6*a*b*c - 5*a^2*d - b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (3*(2*a*b*c - a^2*d - b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*b*(a^2 - b^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((12*a^3*b*c*d + 12*a*b^3*c*d - 3*a^4*d^2 - b^4*(4*c^2 - d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*b*(a + b)^3*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(1/2)), x, 10, (b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2) + (3*b^2*(2*a*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])) + (3*b*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((6*a*b*c - 7*a^2*d + b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]]) - ((20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])} -{1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(3/2)), x, 11, -(d*(8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]) + (b^2*(6*a*b*c - 11*a^2*d + 5*b^2*d)*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) - ((8*a^4*d^3 + a^2*b^2*d*(13*c^2 - 29*d^2) - b^4*d*(7*c^2 - 15*d^2) - 6*a*b^3*c*(c^2 - d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(4*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (b*(6*a*b*c - 11*a^2*d + 5*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (b*(28*a^3*b*c*d - 4*a*b^3*c*d - 35*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 19*d^2) - b^4*(4*c^2 + 15*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^3*f*Sqrt[c + d*Sin[e + f*x]])} -(* {1/(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2), x, 12, -((d*(8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*Cos[e + f*x])/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2))) + (b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(3/2)) + (b^2*(6*a*b*c - 13*a^2*d + 7*b^2*d)*Cos[e + f*x])/(4*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) + (d*(32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*Cos[e + f*x])/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((32*a^5*c*d^4 - 64*a^3*b^2*c*d^4 - 8*a^4*b*d^3*(13*c^2 - 9*d^2) + 2*a*b^4*c*(9*c^4 - 18*c^2*d^2 + 25*d^4) + b^5*d*(33*c^4 - 170*c^2*d^2 + 105*d^4) - a^2*b^3*d*(51*c^4 - 310*c^2*d^2 + 195*d^4))*EllipticE[Pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(12*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((8*a^4*d^3 + a^2*b^2*d*(45*c^2 - 61*d^2) - b^4*d*(27*c^2 - 35*d^2) - 18*a*b^3*c*(c^2 - d^2))*EllipticF[Pi/4 + (1/2)*(-e - f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(12*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (b^2*(36*a^3*b*c*d - 12*a*b^3*c*d - 63*a^4*d^2 - 2*a^2*b^2*(4*c^2 - 43*d^2) - b^4*(4*c^2 + 35*d^2))*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(4*(a - b)^2*(a + b)^3*(b*c - a*d)^4*f*Sqrt[c + d*Sin[e + f*x]])} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(5/2), x, 8, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*b^2*(b*c - a*d)*f) - (1/(8*b^3*Sqrt[a + b]*d*f))*(Sqrt[c + d]*(5*a^2*b*c*d^2 - a^3*d^3 - a*b^2*d*(15*c^2 + 4*d^2) - 5*b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - ((14*a*b*c*d - 3*a^2*d^2 + b^2*(33*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*b*f*Sqrt[a + b*Sin[e + f*x]]) - (d*(13*b*c - 3*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])/(3*b*f) + ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(2*c + d) + b^2*(33*c^2 + 26*c*d + 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b^3*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(3/2), x, 8, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*(5*b*c + a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b*(b*c - a*d)*f) + (Sqrt[c + d]*(6*a*b*c*d - a^2*d^2 + b^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^2*Sqrt[a + b]*d*f) + ((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*f*Sqrt[a + b*Sin[e + f*x]]) - ((5*b*c + a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*(5*b*c - a*d + 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b^2*Sqrt[c + d]*f) - (b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + b*Sin[e + f*x]])} -{(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(1/2), x, 7, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*Sqrt[a + b]*d*f) - (b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(1/2), x, 1, (2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d*f)} -{(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(3/2), x, 3, -((2*(a - b)*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)) + (2*(a - b)*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)} -{(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(5/2), x, 4, (2*d*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a - b)*Sqrt[a + b]*(4*a*c*d - b*(3*c^2 + d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f) + (2*(a - b)*Sqrt[a + b]*(3*c + d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)} - - -{(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2), x, 9, (1/(192*b^2*d*(b*c - a*d)*f))*(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - (1/(64*b^3*Sqrt[a + b]*d^2*f))*(Sqrt[c + d]*(20*a^3*b*c*d^3 - 3*a^4*d^4 - 60*a*b^3*c*d*(c^2 + 4*d^2) - 6*a^2*b^2*d^2*(15*c^2 + 4*d^2) + b^4*(5*c^4 - 120*c^2*d^2 - 48*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - ((57*a^2*b*c*d^2 - 9*a^3*d^3 + a*b^2*d*(337*c^2 + 156*d^2) + b^3*(15*c^3 + 284*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*b*d*f*Sqrt[a + b*Sin[e + f*x]]) - ((54*a*b*c*d - 9*a^2*d^2 + b^2*(59*c^2 + 36*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(96*b*f) - (d*(17*b*c - 3*a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]])/(24*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]])/(4*b*f) + (1/(192*b^3*d*Sqrt[c + d]*f))*((a + b)^(3/2)*(9*a^3*d^3 - 3*a^2*b*d^2*(17*c + 6*d) + 3*a*b^2*d*(73*c^2 + 36*c*d + 28*d^2) + b^3*(15*c^3 + 118*c^2*d + 284*c*d^2 + 72*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))} -{(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2), x, 8, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*(38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*b*d*(b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*(10*a*b*c*d - a^2*d^2 - b^2*(c^2 - 12*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(8*b^2*Sqrt[a + b]*d^2*f) - ((38*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 + 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*d*f*Sqrt[a + b*Sin[e + f*x]]) - ((3*b*c + 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*f) - ((a + b)^(3/2)*(3*a^2*d^2 - 6*a*b*d*(4*c + d) - b^2*(3*c^2 + 14*c*d + 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b^2*d*Sqrt[c + d]*f) - (b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*f)} -{(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(1/2), x, 7, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c + 5*a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*d*(b*c - a*d)*f) + (Sqrt[c + d]*(6*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 - 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b*Sqrt[a + b]*d^2*f) - (b*(b*c + 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d*f*Sqrt[a + b*Sin[e + f*x]]) - (b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*f) + ((a + b)^(3/2)*(3*a*d + b*(c + 2*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b*d*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(1/2), x, 6, -((b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[c + d*Sin[e + f*x]])) - ((a - b)*b*Sqrt[a + b]*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d*(b*c - a*d)*f) + (Sqrt[a + b]*(b*(c - d) - 2*a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - (Sqrt[a + b]*(b*c - 3*a*d)*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2), x, 5, (2*(a - b)*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*d*Sqrt[c + d]*f) - (1/((c - d)*d^2*Sqrt[c + d]*f))*2*Sqrt[a + b]*(b*(c - 2*d) + a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]) + (2*b*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2), x, 4, -((2*(b*c - a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2))) - (8*(a - b)*Sqrt[a + b]*(a*c - b*d)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f) + (2*(a - b)*Sqrt[a + b]*(a*(3*c + d) - b*(c + 3*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f)} - - -{(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2), x, 10, (1/(1920*b^2*d^2*(b*c - a*d)*f))*(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - (1/(128*b^3*Sqrt[a + b]*d^3*f))*(Sqrt[c + d]*(b*c + a*d)*(28*a^3*b*c*d^3 - 3*a^4*d^4 + 28*a*b^3*c*d*(c^2 - 20*d^2) - 2*a^2*b^2*d^2*(89*c^2 + 20*d^2) - b^4*(3*c^4 + 40*c^2*d^2 + 240*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - ((360*a^3*b*c*d^3 - 45*a^4*d^4 + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) + 8*a*b^3*d*(45*c^3 + 791*c*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(1920*b*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - ((917*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(345*c^2 + 772*d^2) - b^3*(45*c^3 - 516*c*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(960*b*d*f) + (1/(1920*b^3*d^2*Sqrt[c + d]*f))*((a + b)^(3/2)*(45*a^4*d^4 - 30*a^3*b*d^3*(11*c + 3*d) + 30*a^2*b^2*d^2*(64*c^2 + 23*c*d + 22*d^2) + 2*a*b^3*d*(165*c^3 + 917*c^2*d + 2392*c*d^2 + 516*d^3) - b^4*(45*c^4 - 30*c^3*d - 1692*c^2*d^2 - 1544*c*d^3 - 1024*d^4))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])) - ((110*a*b*c*d + 93*a^2*d^2 - b^2*(15*c^2 - 64*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(240*d*f) + (3*b*(b*c - 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(40*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2))/(5*d*f)} -{(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2), x, 9, (1/(192*b*d^2*(b*c - a*d)*f))*(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) + (1/(64*b^2*Sqrt[a + b]*d^3*f))*(Sqrt[c + d]*(60*a^3*b*c*d^3 - 5*a^4*d^4 - 20*a*b^3*c*d*(c^2 - 12*d^2) + 3*b^4*(c^2 + 4*d^2)^2 + 30*a^2*b^2*d^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x])) - ((337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(192*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - ((54*a*b*c*d + 59*a^2*d^2 - 9*b^2*(c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(96*d*f) - (1/(192*b^2*d^2*Sqrt[c + d]*f))*((a + b)^(3/2)*(15*a^3*d^3 - 15*a^2*b*d^2*(11*c + 2*d) - a*b^2*d*(51*c^2 + 172*c*d + 212*d^2) + b^3*(9*c^3 - 6*c^2*d - 156*c*d^2 - 72*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])) + (b*(3*b*c - 17*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(24*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2))/(4*d*f)} -{(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(1/2), x, 8, (Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(24*d^2*(b*c - a*d)*f) + (Sqrt[c + d]*(15*a^2*b*c*d^2 + 5*a^3*d^3 - 5*a*b^2*d*(c^2 - 4*d^2) + b^3*(c^3 + 4*c*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(8*b*Sqrt[a + b]*d^3*f) - (b*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(24*d^2*f*Sqrt[a + b*Sin[e + f*x]]) + (b*(3*b*c - 13*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(12*d*f) + ((a + b)^(3/2)*(15*a^2*d^2 + 6*a*b*d*(2*c + 3*d) - b^2*(3*c^2 - 2*c*d - 16*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(24*b*d^2*Sqrt[c + d]*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*d*f)} -{(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(1/2), x, 7, -((3*b*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - 3*a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*d^2*(b*c - a*d)*f)) - (Sqrt[c + d]*(10*a*b*c*d - 15*a^2*d^2 - b^2*(3*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*Sqrt[a + b]*d^3*f) + (3*b^2*(b*c - 3*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*d*f) - ((a + b)^(3/2)*(3*b*c - 7*a*d - 2*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*d^2*Sqrt[c + d]*f)} -{(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2), x, 7, -((Sqrt[a + b]*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(d^2*Sqrt[c + d]*(b*c - a*d)*f)) - (b*Sqrt[c + d]*(3*b*c - 5*a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*d^3*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (b*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d^2*(c^2 - d^2)*f*Sqrt[a + b*Sin[e + f*x]]) - ((a + b)^(3/2)*(2*a*d - b*(3*c + d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^2*(c + d)^(3/2)*f)} -{(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2), x, 6, (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(a - b)*Sqrt[a + b]*(3*b*c^2 + 4*a*c*d - 7*b*d^2)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*d^2*(c + d)^(3/2)*f) - (1/(3*(c - d)^2*d^3*(c + d)^(3/2)*f))*(2*Sqrt[a + b]*(a^2*d^2*(3*c + d) + a*b*d*(3*c^2 - 4*c*d - 7*d^2) + b^2*(3*c^3 - 6*c^2*d - 2*c*d^2 + 9*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])) + (2*b^2*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(d^3*Sqrt[c + d]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, (3*Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*(3*b*c - a*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^2*(b*c - a*d)*f) - (Sqrt[c + d]*(10*a*b*c*d - 3*a^2*d^2 - b^2*(15*c^2 + 4*d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(4*b^3*Sqrt[a + b]*f) - (3*d*(3*b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(4*b*f*Sqrt[a + b*Sin[e + f*x]]) - (d^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(2*b*f) + (Sqrt[a + b]*(3*a^2*d^2 - a*b*d*(7*c + 3*d) + b^2*(8*c^2 + 9*c*d + 2*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(4*b^3*Sqrt[c + d]*f)} -{1/(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(3/2), x, 6, (Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*(b*c - a*d)*f) + (Sqrt[c + d]*(3*b*c - a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^2*Sqrt[a + b]*f) - (d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[a + b]*(a*d - b*(2*c + d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b^2*Sqrt[c + d]*f)} -{1/(a + b*Sin[e + f*x])^(1/2)*(c + d*Sin[e + f*x])^(1/2), x, 1, (2*Sqrt[a + b]*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)} -{1/(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(1/2), x, 1, (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(3/2), x, 3, (2*(a - b)*Sqrt[a + b]*d*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(1/2)/(c + d*Sin[e + f*x])^(5/2), x, 4, -((2*d^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2))) - (4*(a - b)*Sqrt[a + b]*d*(2*a*c*d - b*(3*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f) - (2*Sqrt[a + b]*(a*d*(3*c + d) - b*(3*c^2 + 3*c*d - 2*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f)} - - -{1/(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2), x, 7, ((c - d)*Sqrt[c + d]*(2*b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2 - b^2*d^2)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b^2*Sqrt[a + b]*(b*c - a*d)*f) + (d*Sqrt[c + d]*(5*b*c - 3*a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + ((4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[a + b]*(3*a^2*d^2 - 2*a*b*d*(c + 3*d) - b^2*(2*c^2 - 6*c*d - d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b^3*Sqrt[c + d]*f)} -{1/(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2), x, 5, (1/((a - b)*b*Sqrt[a + b]*f))*2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]) + (1/(b^2*Sqrt[a + b]*f))*2*d*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]) + (1/((a - b)*b^2*Sqrt[c + d]*f))*2*Sqrt[a + b]*(b*(c - 2*d) + a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])} -{1/(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(1/2), x, 3, (2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)*f) + (2*Sqrt[a + b]*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(1/2), x, 3, (2*b*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(3/2), x, 4, (2*b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (2*(a^2*d^2 + b^2*(c^2 - 2*d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f) + (2*(b*(c - 2*d) - a*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f)} -{1/(a + b*Sin[e + f*x])^(3/2)/(c + d*Sin[e + f*x])^(5/2), x, 5, (2*b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + (2*d*(a^2*d^2 + b^2*(3*c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*(4*a^3*c*d^3 - 4*a*b^2*c*d^3 - a^2*b*d^2*(9*c^2 - 5*d^2) - b^3*(3*c^4 - 15*c^2*d^2 + 8*d^4))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f) + (2*(a^2*d^2*(3*c + d) - 6*a*b*d*(c^2 - d^2) + b^2*(3*c^3 - 9*c^2*d - 6*c*d^2 + 8*d^3))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f)} - - -{1/(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2), x, 6, (2*(c - d)*Sqrt[c + d]*(4*a*b*c + 3*a^2*d - 7*b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*b^2*(a + b)^(3/2)*f) + (2*d^2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a^2*b*(c - 2*d)*d + 3*a^3*d^2 + a*b^2*(3*c^2 - 4*c*d - 2*d^2) + b^3*(c^2 - 7*c*d + 9*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*b^3*Sqrt[a + b]*Sqrt[c + d]*f)} -{1/(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(3/2), x, 4, (8*(c - d)*Sqrt[c + d]*(a*c - b*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)*f) + (2*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(c - d)*(3*a*c + b*c - a*d - 3*b*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(1/2), x, 4, (2*(c - d)*Sqrt[c + d]*(4*a*b*c - 3*a^2*d - b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^2*f) + (2*b*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a + b)*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(1/2), x, 4, (4*b*(c - d)*Sqrt[c + d]*(2*a*b*c - 3*a^2*d + b^2*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(3*(a - b)^2*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)) + (2*(3*a*b*(c - d) - 3*a^2*d + b^2*(c + 2*d))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*(a - b)^2*Sqrt[a + b]*Sqrt[c + d]*(b*c - a*d)^2*f)} -{1/(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2), x, 5, (2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]) + (8*b^2*(a*b*c - 2*a^2*d + b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (2*(3*a^4*d^3 - b^4*d*(5*c^2 - 8*d^2) + 3*a^2*b^2*d*(3*c^2 - 5*d^2) - 4*a*b^3*c*(c^2 - d^2))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^4*f) - (2*(3*a^2*b*(2*c - 3*d)*d - 3*a^3*d^2 - 3*a*b^2*(c^2 - 2*d^2) + b^3*(c^2 - 6*c*d + 8*d^2))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f)} -{1/(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(5/2), x, 6, (2*b^2*Cos[e + f*x])/(3*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)) + (4*b^2*(2*a*b*c - 5*a^2*d + 3*b^2*d)*Cos[e + f*x])/(3*(a^2 - b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) - (2*d*(a^4*d^3 + a^2*b^2*d*(11*c^2 - 13*d^2) - b^4*d*(7*c^2 - 8*d^2) - 4*a*b^3*c*(c^2 - d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - (1/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^5*f))*(8*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])) - (1/(3*Sqrt[a + b]*(a^2 - b^2)*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f))*(2*(a^4*d^3*(3*c + d) - 9*a^3*b*d^2*(c^2 - d^2) + a^2*b^2*d*(9*c^3 - 18*c^2*d - 15*c*d^2 + 16*d^3) + b^4*(c^4 - 9*c^3*d + 16*c^2*d^2 + 12*c*d^3 - 16*d^4) - 3*a*b^3*(c^4 - 5*c^2*d^2 + 4*d^4))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]} - - -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2, x, 8, -((d^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*(a + b)*d*(a*d - 2*b*c*(2 + m))*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]])) - (Sqrt[2]*(a*d*(a*d - 2*b*c*(2 + m)) + b^2*(d^2*(1 + m) + c^2*(2 + m)))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]]))} -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^1, x, 7, -((Sqrt[2]*(a + b)*d*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Sin[e + f*x]]))) - (Sqrt[2]*(b*c - a*d)*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Sin[e + f*x]]))} -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^0, x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sin[e + f*x]])))} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^1, x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x]} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x]} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x]} - - -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]} -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2), x]} -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(1/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/Sqrt[c + d*Sin[e + f*x]], x]} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]} -{(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c (d Sin[e+f x])^p)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c (d Sin[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (d Csc[e+f x])^n with n symbolic*) - - -{(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^3, x, 8, If[$VersionNumber>=8, (a^3*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f*(1 - n)) + (a^3*d^3*(5 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]) + (a^3*d^4*(11 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2]), (a^3*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(2 - 3*n + n^2)) + (d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(a^3 + a^3*Csc[e + f*x]))/(f*(1 - n)) + (a^3*d^3*(5 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - 4*n + n^2)*Sqrt[Cos[e + f*x]^2]) + (a^3*d^4*(11 - 4*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(8 - 6*n + n^2)*Sqrt[Cos[e + f*x]^2])]} -{(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^2, x, 7, If[$VersionNumber>=8, (a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a^2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (a^2*d^3*(3 - 2*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]), (a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a^2*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (a^2*d^3*(3 - 2*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - 4*n + n^2)*Sqrt[Cos[e + f*x]^2])]} -{(d*Csc[e + f*x])^n*(a + a*Sin[e + f*x])^1, x, 6, (a*d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (a*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2])} -{(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^1, x, 7, -((Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(a + a*Csc[e + f*x]))) + (d*n*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(a*f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (Cos[e + f*x]*(d*Csc[e + f*x])^n*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Sin[e + f*x]^2])/(a*f*Sqrt[Cos[e + f*x]^2])} -{(d*Csc[e + f*x])^n/(a + a*Sin[e + f*x])^2, x, 8, -((2*n*Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*a^2*d^2*f*(1 + Csc[e + f*x]))) + (Cot[e + f*x]*(d*Csc[e + f*x])^(2 + n))/(3*d^2*f*(a + a*Csc[e + f*x])^2) + (2*n*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 - n), -(n/2), Sin[e + f*x]^2])/(3*a^2*d^2*f*Sqrt[Cos[e + f*x]^2]) - ((1 + 2*n)*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Sin[e + f*x]^2])/(3*a^2*d*f*Sqrt[Cos[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c (d Sin[e+f x])^p)^n with n and p symbolic*) - - -{(a + a*Sin[e + f*x])^m*(c*(d*Sin[e + f*x])^p)^n, x, 5, -((2^(1/2 + m)*AppellF1[1/2, (-n)*p, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(Sin[e + f*x]^(n*p)*f))} - - -{(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^3, x, 7, -((a^3*(7 + 2*n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a^3*(5 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (a^3*(11 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(a^3 + a^3*Sin[e + f*x]))/(f*(3 + n*p))} -{(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^2, x, 5, -((a^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p))) + (a^2*(3 + 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*a^2*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])} -{(c*(d*Sin[e + f*x])^p)^n*(a + a*Sin[e + f*x])^1, x, 4, (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])} -{(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x])^1, x, 5, (Cos[e + f*x]*Hypergeometric2F1[1/2, (n*p)/2, (1/2)*(2 + n*p), Sin[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*Sqrt[Cos[e + f*x]^2]) - (n*p*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(a*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) - (Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(a + a*Sin[e + f*x]))} -{(c*(d*Sin[e + f*x])^p)^n/(a + a*Sin[e + f*x])^2, x, 6, -((n*p*(1 - 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2])) + (2*(1 - n^2*p^2)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*(1 - n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(1 + Sin[e + f*x])) + (Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*f*(a + a*Sin[e + f*x])^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c (d Sin[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (d Csc[e+f x])^n with n symbolic*) - - -{(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^3, x, 8, If[$VersionNumber>=8, (a^2*b*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(b + a*Csc[e + f*x]))/(f*(1 - n)) + (a*d^3*(3*b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*(4 - n)*Sqrt[Cos[e + f*x]^2]), (a^2*b*d^3*(1 - 2*n)*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n))/(f*(2 - 3*n + n^2)) + (a^2*d^3*Cot[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*(b + a*Csc[e + f*x]))/(f*(1 - n)) + (a*d^3*(3*b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - 4*n + n^2)*Sqrt[Cos[e + f*x]^2]) + (b*d^4*(b^2*(2 - n) + 3*a^2*(3 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-4 + n)*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Sin[e + f*x]^2])/(f*(8 - 6*n + n^2)*Sqrt[Cos[e + f*x]^2])]} -{(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^2, x, 7, If[$VersionNumber>=8, (a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a*b*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (d^3*(b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*(3 - n)*Sqrt[Cos[e + f*x]^2]), (a^2*d^2*Cot[e + f*x]*(d*Csc[e + f*x])^(-2 + n))/(f*(1 - n)) + (2*a*b*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2]) + (d^3*(b^2*(1 - n) + a^2*(2 - n))*Cos[e + f*x]*(d*Csc[e + f*x])^(-3 + n)*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Sin[e + f*x]^2])/(f*(3 - 4*n + n^2)*Sqrt[Cos[e + f*x]^2])]} -{(d*Csc[e + f*x])^n*(a + b*Sin[e + f*x])^1, x, 6, (a*d*Cos[e + f*x]*(d*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2]) + (b*d^2*Cos[e + f*x]*(d*Csc[e + f*x])^(-2 + n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Sin[e + f*x]^2])/(f*(2 - n)*Sqrt[Cos[e + f*x]^2])} -{(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^1, x, 7, (b*AppellF1[1/2, n/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*Sin[e + f*x]*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)*d*f) - (a*AppellF1[1/2, (1 + n)/2, 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(1 + n)*(Sin[e + f*x]^2)^((1 + n)/2))/((a^2 - b^2)*d*f)} -{(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^2, x, 10, -((b^2*AppellF1[1/2, (1/2)*(-1 + n), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((1/2)*(-1 + n)))/((a^2 - b^2)^2*d^2*f)) - (a^2*AppellF1[1/2, (1 + n)/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*Sin[e + f*x]*(Sin[e + f*x]^2)^((1 + n)/2))/((a^2 - b^2)^2*d^2*f) + (2*a*b*AppellF1[1/2, n/2, 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(2 + n)*(Sin[e + f*x]^2)^((2 + n)/2))/((a^2 - b^2)^2*d^2*f)} -{(d*Csc[e + f*x])^n/(a + b*Sin[e + f*x])^3, x, 12, -((3*a*b^2*AppellF1[1/2, (1/2)*(-1 + n), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^4*(Sin[e + f*x]^2)^((1/2)*(-1 + n)))/((a^2 - b^2)^3*d^3*f)) + (b^3*AppellF1[1/2, (1/2)*(-2 + n), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)^3*d^3*f) + (3*a^2*b*AppellF1[1/2, n/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*Sin[e + f*x]^3*(Sin[e + f*x]^2)^(n/2))/((a^2 - b^2)^3*d^3*f) - (a^3*AppellF1[1/2, (1 + n)/2, 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(d*Csc[e + f*x])^(3 + n)*(Sin[e + f*x]^2)^((3 + n)/2))/((a^2 - b^2)^3*d^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c (d Sin[e+f x])^p)^n with n and p symbolic*) - - -{(a + b*Sin[e + f*x])^m*(c*(d*Sin[e + f*x])^p)^n, x, 1, ((c*(d*Sin[e + f*x])^p)^n*Unintegrable[(d*Sin[e + f*x])^(n*p)*(a + b*Sin[e + f*x])^m, x])/(d*Sin[e + f*x])^(n*p)} - - -{(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^3, x, 6, -((a*b^2*(7 + 2*n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p))) + (a*(3*b^2*(1 + n*p) + a^2*(2 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*(b^2*(2 + n*p) + 3*a^2*(3 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*(3 + n*p)*Sqrt[Cos[e + f*x]^2]) - (b^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x]))/(f*(3 + n*p))} -{(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^2, x, 5, -((b^2*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p))) + ((b^2*(1 + n*p) + a^2*(2 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*(2 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])} -{(c*(d*Sin[e + f*x])^p)^n*(a + b*Sin[e + f*x])^1, x, 4, (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (b*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])} -{(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^1, x, 6, (b*AppellF1[1/2, -((n*p)/2), 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((Sin[e + f*x]^2)^((n*p)/2)*((a^2 - b^2)*f)) - (a*AppellF1[1/2, (1/2)*(1 - n*p), 1, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(1 - n*p))*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)*f)} -{(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^2, x, 11, (2*a*b*AppellF1[1/2, -((n*p)/2), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((Sin[e + f*x]^2)^((n*p)/2)*((a^2 - b^2)^2*f)) - (b^2*AppellF1[1/2, (1/2)*(-1 - n*p), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(-1 - n*p))*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^2*f) - (a^2*AppellF1[1/2, (1/2)*(1 - n*p), 2, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(1 - n*p))*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^2*f)} -{(c*(d*Sin[e + f*x])^p)^n/(a + b*Sin[e + f*x])^3, x, 14, (3*a^2*b*AppellF1[1/2, -((n*p)/2), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((Sin[e + f*x]^2)^((n*p)/2)*((a^2 - b^2)^3*f)) + (b^3*AppellF1[1/2, (1/2)*(-2 - n*p), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/((Sin[e + f*x]^2)^((n*p)/2)*((a^2 - b^2)^3*f)) - (3*a*b^2*AppellF1[1/2, (1/2)*(-1 - n*p), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*Sin[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(-1 - n*p))*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f) - (a^3*AppellF1[1/2, (1/2)*(1 - n*p), 3, 3/2, Cos[e + f*x]^2, -((b^2*Cos[e + f*x]^2)/(a^2 - b^2))]*Cot[e + f*x]*(Sin[e + f*x]^2)^((1/2)*(1 - n*p))*(c*(d*Sin[e + f*x])^p)^n)/((a^2 - b^2)^3*f)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m deleted file mode 100644 index ecb2ca0..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.m +++ /dev/null @@ -1,2905 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2), x, 3, -(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(6*c*f)} -{Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2), x, 3, -(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(10*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(5*c*f)} -{Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2), x, 3, -(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(6*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(4*c*f)} -{Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]], x, 3, -(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*c*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(3*c*f)} -{(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]], x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(3/2), x, 5, (-2*a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2), x, 5, (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*a*c*f*(c - c*Sin[e + f*x])^(5/2))} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2), x, 4, (-4*a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(105*c*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(21*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(7*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2), x, 4, -(a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(15*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(6*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2), x, 4, (-2*a^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*c*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(5*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]], x, 3, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(4*a*f)} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]], x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2), x, 6, (-4*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(5/2), x, 6, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(7/2), x, 6, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (a^2*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*a*c*f*(c - c*Sin[e + f*x])^(7/2))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(11/2), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*a*c*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(48*a*c^2*f*(c - c*Sin[e + f*x])^(7/2))} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2), x, 5, -(a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(35*c*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(14*c*f) - (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(28*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(8*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2), x, 5, (-2*a^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(35*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(35*c*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(7*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2), x, 4, (c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(15*a*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(6*a*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]], x, 3, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f)} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]], x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(3/2), x, 7, (-8*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(5/2), x, 7, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (12*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(7/2), x, 7, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (6*a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(9/2), x, 7, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*(c - c*Sin[e + f*x])^(3/2)) + (a^3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*a*c*f*(c - c*Sin[e + f*x])^(9/2))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*a*c*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*a*c^2*f*(c - c*Sin[e + f*x])^(9/2))} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2), x, 6, (-4*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(11/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))/(105*c*f) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(11/2))/(15*c*f) - (4*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(11/2))/(45*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(11/2))/(10*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2), x, 6, (-8*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(63*c*f) - (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(21*c*f) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(9*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*c*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2), x, 5, (c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(35*a*f*Sqrt[c - c*Sin[e + f*x]]) + (c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(14*a*f) + (3*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(3/2))/(28*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(5/2))/(8*a*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2), x, 4, (4*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(105*a*f*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(21*a*f) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a*f)} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]], x, 3, (c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(15*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2)*Sqrt[c - c*Sin[e + f*x]])/(6*a*f)} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]], x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(5*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(3/2), x, 8, (-16*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (8*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2), x, 8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(c*f*(c - c*Sin[e + f*x])^(3/2)) + (32*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (16*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (4*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) + (4*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2), x, 8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) - (24*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (12*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*Sqrt[c - c*Sin[e + f*x]]) - (3*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^3*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2), x, 8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) - (2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(c^3*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (4*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^4*f*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2), x, 8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*c*f*(c - c*Sin[e + f*x])^(9/2)) - (a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c^2*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^4*f*(c - c*Sin[e + f*x])^(3/2)) - (a^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(10*a*c*f*(c - c*Sin[e + f*x])^(11/2))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(12*a*c*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(120*a*c^2*f*(c - c*Sin[e + f*x])^(11/2))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(17/2), x, 4, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(14*a*c*f*(c - c*Sin[e + f*x])^(15/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(84*a*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(840*a*c^3*f*(c - c*Sin[e + f*x])^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]], x, 2, -(Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*c*f*Sqrt[a + a*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]], x, 2, -(Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[a + a*Sin[e + f*x]])} -{(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]], x, 2, -(Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[a + a*Sin[e + f*x]])} -{Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 2, -((Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]))} -{Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 4, -((Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))} -{Cos[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)), x, 2, Cos[e + f*x]/(c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))} - - -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2), x, 8, (16*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (8*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*a*f*Sqrt[a + a*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2), x, 7, (8*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (4*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[a + a*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2), x, 6, (4*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + (Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]])} -{(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2), x, 5, (2*c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]])} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]), x, 4, (Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)), x, 3, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)), x, 4, Cos[e + f*x]/(2*a*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2), x, 9, (-80*c^5*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (40*c^4*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (10*c^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (10*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (5*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2), x, 8, (-32*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (16*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (4*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (4*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2), x, 7, (-12*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (6*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (3*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{(Cos[e + f*x]^2*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2), x, 6, (-4*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) - (Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{(Cos[e + f*x]^2*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2), x, 5, -((c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*(a + a*Sin[e + f*x])^(3/2))} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]), x, 2, -(Cos[e + f*x]/(a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]))} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)), x, 4, -Cos[e + f*x]/(2*a*c*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)), x, 5, -Cos[e + f*x]/(2*a*c*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + Cos[e + f*x]/(2*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m and/or n symbolic*) - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 5, (2^(3/2 + n)*c^2*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(3 + 2*m), (1/2)*(-1 - 2*n), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(f*(3 + 2*m))} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3, x, 4, -((2^(3/2 + m)*a^4*c^3*Cos[e + f*x]^9*Hypergeometric2F1[9/2, -(1/2) - m, 11/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-4 + m))/(9*f))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2, x, 4, -((2^(3/2 + m)*a^3*c^2*Cos[e + f*x]^7*Hypergeometric2F1[7/2, -(1/2) - m, 9/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-3 + m))/(7*f))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^1, x, 4, -((2^(3/2 + m)*a^2*c*Cos[e + f*x]^5*Hypergeometric2F1[5/2, -(1/2) - m, 7/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-2 + m))/(5*f))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^0, x, 3, -((2^(3/2 + m)*a*Cos[e + f*x]^3*Hypergeometric2F1[3/2, -(1/2) - m, 5/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^1, x, 3, -((2^(3/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, -(1/2) - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(c*f))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^2, x, 4, (2^(3/2 + m)*Hypergeometric2F1[-(1/2), -(1/2) - m, 1/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(1 + m))/(a*c^2*f)} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^3, x, 4, (2^(3/2 + m)*Hypergeometric2F1[-(3/2), -(1/2) - m, -(1/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(2 + m))/(3*a^2*c^3*f)} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2), x, 5, If[$VersionNumber>=8, (768*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (192*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(9 + 2*m)*(35 + 24*m + 4*m^2)) + (24*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(63 + 32*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2))/(a*f*(9 + 2*m)), (768*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*(63 + 32*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (192*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(315 + 286*m + 84*m^2 + 8*m^3)) + (24*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(63 + 32*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(5/2))/(a*f*(9 + 2*m))]} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2), x, 4, If[$VersionNumber>=8, (64*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(7 + 2*m)*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(7 + 2*m)), (64*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(105 + 142*m + 60*m^2 + 8*m^3)*Sqrt[c - c*Sin[e + f*x]]) + (16*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(3/2))/(a*f*(7 + 2*m))]} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]], x, 3, (8*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c - c*Sin[e + f*x]])/(a*f*(5 + 2*m))} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]], x, 2, (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2), x, 4, (Cos[e + f*x]*Hypergeometric2F1[1, 3/2 + m, 5/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(a*c*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2), x, 4, (Cos[e + f*x]*Hypergeometric2F1[2, 3/2 + m, 5/2 + m, (1 + Sin[e + f*x])/2]*(a + a*Sin[e + f*x])^(1 + m))/(2*a*c^2*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} - - -(* The same rules should be used to integrate the following two problems: *) -{(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]], x, 2, (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(Cos[e + f*x]^2*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]], x, 2, (2*Cos[e + f*x]*(c + c*Sin[e + f*x])^(1 + m))/(c*f*(3 + 2*m)*Sqrt[a - a*Sin[e + f*x]])} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 5), x, 4, If[$VersionNumber>=8, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^3*f*(7 + 2*m)*(15 + 16*m + 4*m^2)), (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-4 - m))/(a*c*f*(7 + 2*m)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c^2*f*(35 + 24*m + 4*m^2)) + (2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^3*f*(105 + 142*m + 60*m^2 + 8*m^3))]} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 4), x, 3, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-3 - m))/(a*c*f*(5 + 2*m)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c^2*f*(15 + 16*m + 4*m^2))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 3), x, 2, (Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(a*c*f*(3 + 2*m))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 2), x, 5, (2^(-(1/2) - m)*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(3 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 1), x, 5, (2^(1/2 - m)*c*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 0), x, 5, (2^(3/2 - m)*c^2*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(-1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m - 1), x, 5, (2^(5/2 - m)*c^3*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(-3 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2), x, 8, (2*a*c^4*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(7*f*g*Sqrt[a + a*Sin[e + f*x]]) + (10*a*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(11*f*g*Sqrt[a + a*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2), x, 7, (22*a*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(105*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2), x, 6, (2*a*c^2*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]], x, 5, (2*a*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/Sqrt[c - c*Sin[e + f*x]], x, 4, -((2*a*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(3/2), x, 4, (4*a*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(5/2), x, 5, (4*a*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (6*a*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (6*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2), x, 6, (4*a*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2))/(13*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(39*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(65*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2), x, 8, (14*a^2*c^3*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^2*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[a + a*Sin[e + f*x]]) - (14*a^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(11*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2), x, 7, (14*a^2*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^2*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(9*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]], x, 6, -((2*a^2*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (6*a^2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/Sqrt[c - c*Sin[e + f*x]], x, 5, -((14*a^2*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2), x, 5, (4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(3*c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(5/2), x, 5, (4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (42*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(7/2), x, 6, (4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(45*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(117*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(195*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(195*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(195*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(11/2), x, 8, (4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (28*a^2*(g*Cos[e + f*x])^(5/2))/(221*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(1105*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2), x, 9, (154*a^3*c^3*(g*Cos[e + f*x])^(5/2))/(585*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^3*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(195*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(195*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*a^3*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(39*f*g*Sqrt[a + a*Sin[e + f*x]]) - (14*a^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(117*f*g*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(13*f*g) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(13*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2), x, 8, -((14*a^3*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (14*a^3*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(15*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]], x, 7, -((22*a^3*c*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (22*a^3*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(105*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]], x, 6, -((22*a^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (22*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(3/2), x, 6, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(5/2), x, 6, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c*f*g*(c - c*Sin[e + f*x])^(3/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(7/2), x, 6, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(45*c*f*g*(c - c*Sin[e + f*x])^(5/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(45*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(9/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(117*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(585*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(195*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(195*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2), x, 8, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (308*a^3*(g*Cos[e + f*x])^(5/2))/(1989*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(3315*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2), x, 9, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (44*a^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(357*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(1989*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (22*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(3315*c^6*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2), x, 10, -((154*a^4*c^3*(g*Cos[e + f*x])^(5/2))/(585*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (154*a^4*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(195*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^3*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(195*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*f*g*Sqrt[c - c*Sin[e + f*x]]) - (14*a*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(585*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(195*f*g*Sqrt[c - c*Sin[e + f*x]]) + (22*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(195*f*g) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(15*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2), x, 9, -((14*a^4*c^2*(g*Cos[e + f*x])^(5/2))/(39*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (14*a^4*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(13*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(143*f*g*Sqrt[c - c*Sin[e + f*x]]) - (14*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(429*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(143*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(13*f*g)} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]], x, 8, -((2*a^4*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (2*a^4*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(11*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/Sqrt[c - c*Sin[e + f*x]], x, 7, -((22*a^4*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(3/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (66*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*c*f*g*Sqrt[c - c*Sin[e + f*x]]) + (30*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*c*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(5/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (12*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(c*f*g*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*(g*Cos[e + f*x])^(5/2))/(5*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (462*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*c^2*f*g*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (4*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(3*c*f*g*(c - c*Sin[e + f*x])^(5/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(3*c^2*f*g*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(9*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(3*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2), x, 7, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (20*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(39*c^2*f*g*(c - c*Sin[e + f*x])^(5/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(39*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(13*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2), x, 8, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (60*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (220*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(663*c^2*f*g*(c - c*Sin[e + f*x])^(7/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(221*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(221*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2), x, 9, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (20*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(119*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (220*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(1547*c^2*f*g*(c - c*Sin[e + f*x])^(9/2)) - (220*a^4*(g*Cos[e + f*x])^(5/2))/(1989*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(663*c^6*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2), x, 10, (4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(25*f*g*(c - c*Sin[e + f*x])^(15/2)) - (4*a^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(35*c*f*g*(c - c*Sin[e + f*x])^(13/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(595*c^2*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^4*(g*Cos[e + f*x])^(5/2))/(1105*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^6*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5525*c^7*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]], x, 6, (22*c^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*f*g*Sqrt[a + a*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]], x, 5, (14*c^2*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]], x, 4, (2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 3, (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 4, (2*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)), x, 5, (2*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)), x, 6, (2*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (2*(g*Cos[e + f*x])^(5/2))/(15*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(15*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2), x, 7, -((22*c^4*(g*Cos[e + f*x])^(5/2))/(a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (66*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (30*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2))} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2), x, 6, -((154*c^3*(g*Cos[e + f*x])^(5/2))/(15*a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (154*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (22*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(f*g*(a + a*Sin[e + f*x])^(3/2))} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2), x, 5, -((14*c^2*(g*Cos[e + f*x])^(5/2))/(3*a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (14*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(f*g*(a + a*Sin[e + f*x])^(3/2))} -{((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2), x, 4, -((4*c*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])) - (6*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]), x, 4, (-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)), x, 5, (-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + (2*(g*Cos[e + f*x])^(5/2))/(a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)), x, 6, (-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)), x, 7, -((2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))) + (10*(g*Cos[e + f*x])^(5/2))/(9*a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(3*a*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2), x, 8, (418*c^5*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^5*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^4*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (114*c^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (76*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2), x, 7, (154*c^4*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (462*c^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (12*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2), x, 6, (154*c^3*(g*Cos[e + f*x])^(5/2))/(15*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*c^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (44*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))} -{((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2), x, 5, (28*c^2*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (42*c^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*(a + a*Sin[e + f*x])^(5/2))} -{((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(5/2), x, 5, -((4*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])) + (6*c*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + (6*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]), x, 5, (-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)), x, 6, (2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)) - (6*(g*Cos[e + f*x])^(5/2))/(5*c*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*(g*Cos[e + f*x])^(5/2))/(5*a*c*f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)), x, 7, (-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)) - (2*(g*Cos[e + f*x])^(5/2))/(a*f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (6*(g*Cos[e + f*x])^(5/2))/(5*a^2*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)), x, 8, -((2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))) - (14*(g*Cos[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(9*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*a^2*c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m and/or n symbolic*) - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (2^(9/4 + n)*c*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(5 + 4*m), (1/4)*(-1 - 4*n), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(5 + 4*m))} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3, x, 4, -((2^(9/4 + m)*a^4*c^3*(g*Cos[e + f*x])^(17/2)*Hypergeometric2F1[17/4, -(1/4) - m, 21/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-4 + m))/(17*f*g^7))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^2, x, 4, -((2^(9/4 + m)*a^3*c^2*(g*Cos[e + f*x])^(13/2)*Hypergeometric2F1[13/4, -(1/4) - m, 17/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-3 + m))/(13*f*g^5))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^1, x, 4, -((2^(9/4 + m)*a^2*c*(g*Cos[e + f*x])^(9/2)*Hypergeometric2F1[9/4, -(1/4) - m, 13/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-2 + m))/(9*f*g^3))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^0, x, 3, -((2^(9/4 + m)*a*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[5/4, -(1/4) - m, 9/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-1 + m))/(5*f*g))} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^1, x, 4, -((2^(9/4 + m)*g*Sqrt[g*Cos[e + f*x]]*Hypergeometric2F1[1/4, -(1/4) - m, 5/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^m)/(c*f))} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^2, x, 4, (2^(9/4 + m)*g^3*Hypergeometric2F1[-(3/4), -(1/4) - m, 1/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*c^2*f*(g*Cos[e + f*x])^(3/2))} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^3, x, 4, (2^(9/4 + m)*g^5*Hypergeometric2F1[-(7/4), -(1/4) - m, -(3/4), (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(2 + m))/(7*a^2*c^3*f*(g*Cos[e + f*x])^(7/2))} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2), x, 4, -((2^(9/4 + m)*a^3*c^2*(g*Cos[e + f*x])^(15/2)*Hypergeometric2F1[15/4, -(1/4) - m, 19/4, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-3 + m)*Sqrt[c - c*Sin[e + f*x]])/(15*f*g^6))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2), x, 4, -((2^(9/4 + m)*a^2*c*(g*Cos[e + f*x])^(11/2)*Hypergeometric2F1[11/4, -(1/4) - m, 15/4, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-2 + m)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g^4))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1/2), x, 4, -((2^(9/4 + m)*a*(g*Cos[e + f*x])^(7/2)*Hypergeometric2F1[7/4, -(1/4) - m, 11/4, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-1 + m)*Sqrt[c - c*Sin[e + f*x]])/(7*f*g^2))} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(1/2), x, 4, -((2^(9/4 + m)*a*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -(1/4) - m, 7/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[c - c*Sin[e + f*x]]))} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(3/2), x, 4, (2^(9/4 + m)*g^2*Cos[e + f*x]*Hypergeometric2F1[-(1/4), -(1/4) - m, 3/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^m)/(c*f*Sqrt[g*Cos[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/(c - c*Sin[e + f*x])^(5/2), x, 4, (2^(9/4 + m)*g^4*Cos[e + f*x]*Hypergeometric2F1[-(5/4), -(1/4) - m, -(1/4), (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(1 + m))/(5*a*c^2*f*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])} - - -(* The same rules should be used to integrate the following two problems: *) -{((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m)/Sqrt[c - c*Sin[e + f*x]], x, 4, -((2^(9/4 + m)*a*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -(1/4) - m, 7/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[c - c*Sin[e + f*x]]))} -{((g*Cos[e + f*x])^(3/2)*(c + c*Sin[e + f*x])^m)/Sqrt[a - a*Sin[e + f*x]], x, 4, -((2^(9/4 + m)*c*Cos[e + f*x]*(g*Cos[e + f*x])^(3/2)*Hypergeometric2F1[3/4, -(1/4) - m, 7/4, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/4) - m)*(c + c*Sin[e + f*x])^(-1 + m))/(3*f*Sqrt[a - a*Sin[e + f*x]]))} - - -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 3), x, 4, (2^(-(3/4) - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(5 + 4*m), (1/4)*(11 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*g*(5 + 4*m))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 2), x, 4, (2^(1/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(5 + 4*m), (1/4)*(7 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*g*(5 + 4*m))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 1), x, 4, (2^(5/4 - m)*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(3 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m + 0), x, 4, (2^(9/4 - m)*c*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(-1 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m - 1), x, 4, (2^(13/4 - m)*c^2*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(-5 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))} -{(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(m - 2), x, 4, (2^(17/4 - m)*c^3*(g*Cos[e + f*x])^(5/2)*Hypergeometric2F1[(1/4)*(-9 + 4*m), (1/4)*(5 + 4*m), (1/4)*(9 + 4*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(-(1/4) + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*g*(5 + 4*m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m, n and p symbolic*) - - -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (2^(1/2 + n + p/2)*c*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1/2)*(1 - 2*n - p), (1/2)*(1 + 2*m + p), (1/2)*(3 + 2*m + p), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((1/2)*(1 - 2*n - p))*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(1 + 2*m + p))} - - -{(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(m-1), x, 4, -((g*Log[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/((g*Cos[e + f*x])^(2*m)*(c*f)))} - -{(g*Cos[e + f*x])^(5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 3, -((8*a^3*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-3 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n)*(4 - m + n)*(5 - m + n))) - (4*a^2*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(4 - m + n)*(5 - m + n)) - (a*(g*Cos[e + f*x])^(6 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(5 - m + n))} -{(g*Cos[e + f*x])^(3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 2, -((2*a^2*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-2 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(2 - m + n)*(3 - m + n))) - (a*(g*Cos[e + f*x])^(4 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(3 - m + n))} -{(g*Cos[e + f*x])^(1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 1, -((a*(g*Cos[e + f*x])^(2 - 2*m)*(a + a*Sin[e + f*x])^(-1 + m)*(c - c*Sin[e + f*x])^n)/(f*g*(1 - m + n)))} -{(g*Cos[e + f*x])^(-1 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (Hypergeometric2F1[1, -m + n, 1 - m + n, (1/2)*(1 - Sin[e + f*x])]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/((g*Cos[e + f*x])^(2*m)*(2*f*g*(m - n)))} -{(g*Cos[e + f*x])^(-3 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (c*Hypergeometric2F1[2, -1 - m + n, -m + n, (1/2)*(1 - Sin[e + f*x])]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/((g*Cos[e + f*x])^(2*m)*(4*f*g^3*(1 + m - n)))} -{(g*Cos[e + f*x])^(-5 - 2*m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 4, (c^2*Hypergeometric2F1[3, -2 - m + n, -1 - m + n, (1/2)*(1 - Sin[e + f*x])]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/((g*Cos[e + f*x])^(2*m)*(8*f*g^5*(2 + m - n)))} - - -{(g*Cos[e + f*x])^(-1 - m - m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m, x, 3, (ArcTanh[Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^m)/((g*Cos[e + f*x])^(2*m)*(f*g))} - -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n + 3), x, 4, (2^(3 - m/2 + n/2)*c^3*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(1/2)*(-4 + m - n), (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n + 2), x, 4, (2^(2 - m/2 + n/2)*c^2*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(1/2)*(-2 + m - n), (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n + 1), x, 4, (2^(1 - m/2 + n/2)*c*(g*Cos[e + f*x])^(-m - n)*Hypergeometric2F1[(m - n)/2, (m - n)/2, (1/2)*(2 + m - n), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((m - n)/2)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n - 0), x, 1, ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*g*(m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n - 1), x, 2, ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*g*(2 + m - n)) + ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c*f*g*(m - n)*(2 + m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n - 2), x, 3, ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(f*g*(4 + m - n)) + (2*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(c*f*g*(2 + m - n)*(4 + m - n)) + (2*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c^2*f*g*(m - n)*(2 + m - n)*(4 + m - n))} -{(g*Cos[e + f*x])^(-1 - m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(n - 3), x, 4, ((g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 + n))/(f*g*(6 + m - n)) + (3*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 + n))/(c*f*g*(4 + m - n)*(6 + m - n)) + (6*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(c^2*f*g*(2 + m - n)*(4 + m - n)*(6 + m - n)) + (6*(g*Cos[e + f*x])^(-m - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(c^3*f*g*(m - n)*(2 + m - n)*(4 + m - n)*(6 + m - n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) -(**) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m, n and p symbolic*) - - -{(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 5, (2^(1/2 + n - p/2)*c*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m - p), (1/2)*(1 - 2*n + p), (1/2)*(3 + 2*m - p), (1/2)*(1 + Sin[e + f*x])]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1/2)*(1 - 2*n + p))*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n))/(f*(1 + 2*m - p))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^1 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 3, (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d} -{Cos[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 3, -((Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2)/(2*a*d))} -{Cos[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x]^2)/(2*d)) - (a*Csc[c + d*x]^3)/(3*d)} -{Cos[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x]^3)/(3*d)) - (a*Csc[c + d*x]^4)/(4*d)} - - -{Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(2*d) + (a^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Sin[c + d*x]^2)/(2*d) + (2*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 3, (a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x])/d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d} -{Cos[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 4, -((2*a^2*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/(2*d) + (a^2*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 3, -((Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*a*d))} -{Cos[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x]^2)/(2*d)) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d)} -{Cos[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x]^3)/(3*d)) - (a^2*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d)} -{Cos[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x]^4)/(4*d)) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d)} - - -{Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 4, (a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, (a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, -((a + a*Sin[c + d*x])^4/(4*a*d)) + (a + a*Sin[c + d*x])^5/(5*a^2*d)} -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 3, (a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x])/d) + (3*a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 4, -((3*a^3*Csc[c + d*x])/d) - (a^3*Csc[c + d*x]^2)/(2*d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d} -{Cos[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 4, -((3*a^3*Csc[c + d*x])/d) - (3*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) + (a^3*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 3, -((Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(4*a*d))} -{Cos[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 4, (Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4)/(20*a*d) - (Csc[c + d*x]^5*(a + a*Sin[c + d*x])^4)/(5*a*d)} -{Cos[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x]^3)/(3*d)) - (3*a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(6*d)} -{Cos[c + d*x]*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x]^4)/(4*d)) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(2*d) - (a^3*Csc[c + d*x]^7)/(7*d)} - - -{Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 4, (a^4*Sin[c + d*x]^5)/(5*d) + (2*a^4*Sin[c + d*x]^6)/(3*d) + (6*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8)/(2*d) + (a^4*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^4, x, 4, (a^4*Sin[c + d*x]^4)/(4*d) + (4*a^4*Sin[c + d*x]^5)/(5*d) + (a^4*Sin[c + d*x]^6)/d + (4*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8)/(8*d)} -{Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 4, (a + a*Sin[c + d*x])^5/(5*a*d) - (a + a*Sin[c + d*x])^6/(3*a^2*d) + (a + a*Sin[c + d*x])^7/(7*a^3*d)} -{Cos[c + d*x]*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^4, x, 4, -((a + a*Sin[c + d*x])^5/(5*a*d)) + (a + a*Sin[c + d*x])^6/(6*a^2*d)} -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^4, x, 3, (a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (3*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 4, -((a^4*Csc[c + d*x])/d) + (4*a^4*Log[Sin[c + d*x]])/d + (6*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a^4*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^4, x, 4, -((4*a^4*Csc[c + d*x])/d) - (a^4*Csc[c + d*x]^2)/(2*d) + (6*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (a^4*Sin[c + d*x]^2)/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, Log[1 + Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d) + Sin[c + d*x]^4/(4*a*d)} -{Cos[c + d*x]*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, -(Log[1 + Sin[c + d*x]]/(a*d)) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, Log[1 + Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, -(Log[1 + Sin[c + d*x]]/(a*d)) + Sin[c + d*x]/(a*d)} -{Cos[c + d*x]*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) + Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - Log[Sin[c + d*x]]/(a*d) + Log[1 + Sin[c + d*x]]/(a*d)} - - -{Cos[c + d*x]*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 4, -((4*Log[1 + Sin[c + d*x]])/(a^2*d)) + (3*Sin[c + d*x])/(a^2*d) - Sin[c + d*x]^2/(a^2*d) + Sin[c + d*x]^3/(3*a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, (3*Log[1 + Sin[c + d*x]])/(a^2*d) - (2*Sin[c + d*x])/(a^2*d) + Sin[c + d*x]^2/(2*a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 4, -((2*Log[1 + Sin[c + d*x]])/(a^2*d)) + Sin[c + d*x]/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 3, Log[Sin[c + d*x]]/(a^2*d) - Log[1 + Sin[c + d*x]]/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 4, -(Csc[c + d*x]/(a^2*d)) - (2*Log[Sin[c + d*x]])/(a^2*d) + (2*Log[1 + Sin[c + d*x]])/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, (2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + (3*Log[Sin[c + d*x]])/(a^2*d) - (3*Log[1 + Sin[c + d*x]])/(a^2*d) + 1/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 4, -((3*Csc[c + d*x])/(a^2*d)) + Csc[c + d*x]^2/(a^2*d) - Csc[c + d*x]^3/(3*a^2*d) - (4*Log[Sin[c + d*x]])/(a^2*d) + (4*Log[1 + Sin[c + d*x]])/(a^2*d) - 1/(d*(a^2 + a^2*Sin[c + d*x]))} - - -{Cos[c + d*x]*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 4, -((10*Log[1 + Sin[c + d*x]])/(a^3*d)) + (6*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3/(3*a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 4, (6*Log[1 + Sin[c + d*x]])/(a^3*d) - (3*Sin[c + d*x])/(a^3*d) + Sin[c + d*x]^2/(2*a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 4/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, -((3*Log[1 + Sin[c + d*x]])/(a^3*d)) + Sin[c + d*x]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 3/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 4, Log[1 + Sin[c + d*x]]/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 2/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 3, Sin[c + d*x]^2/(2*a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 3, Log[Sin[c + d*x]]/(a^3*d) - Log[1 + Sin[c + d*x]]/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 1/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 4, -(Csc[c + d*x]/(a^3*d)) - (3*Log[Sin[c + d*x]])/(a^3*d) + (3*Log[1 + Sin[c + d*x]])/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 2/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, (3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (6*Log[Sin[c + d*x]])/(a^3*d) - (6*Log[1 + Sin[c + d*x]])/(a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) + 3/(d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 4, -((6*Csc[c + d*x])/(a^3*d)) + (3*Csc[c + d*x]^2)/(2*a^3*d) - Csc[c + d*x]^3/(3*a^3*d) - (10*Log[Sin[c + d*x]])/(a^3*d) + (10*Log[1 + Sin[c + d*x]])/(a^3*d) - 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 4/(d*(a^3 + a^3*Sin[c + d*x]))} - - -{Cos[c + d*x]*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^4, x, 4, (10*Log[1 + Sin[c + d*x]])/(a^4*d) - (4*Sin[c + d*x])/(a^4*d) + Sin[c + d*x]^2/(2*a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 5/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 10/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^4, x, 4, -((4*Log[1 + Sin[c + d*x]])/(a^4*d)) + Sin[c + d*x]/(a^4*d) - 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 2/(d*(a^2 + a^2*Sin[c + d*x])^2) - 6/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^4, x, 4, Log[1 + Sin[c + d*x]]/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 3/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^4, x, 3, Sin[c + d*x]^3/(3*a*d*(a + a*Sin[c + d*x])^3)} -{Cos[c + d*x]*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^4, x, 4, 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 1/(2*d*(a^2 + a^2*Sin[c + d*x])^2)} -{Cos[c + d*x]*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^4, x, 3, Log[Sin[c + d*x]]/(a^4*d) - Log[1 + Sin[c + d*x]]/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 1/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 1/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^4, x, 4, -(Csc[c + d*x]/(a^4*d)) - (4*Log[Sin[c + d*x]])/(a^4*d) + (4*Log[1 + Sin[c + d*x]])/(a^4*d) - 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 1/(d*(a^2 + a^2*Sin[c + d*x])^2) - 3/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^4, x, 4, (4*Csc[c + d*x])/(a^4*d) - Csc[c + d*x]^2/(2*a^4*d) + (10*Log[Sin[c + d*x]])/(a^4*d) - (10*Log[1 + Sin[c + d*x]])/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) + 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 6/(d*(a^4 + a^4*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(1/2), x, 4, -((2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/Sqrt[a]])/d) + (2*Sqrt[a + a*Sin[c + d*x]])/d} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic*) - - -{Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4, x, 3, (a^4*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (4*a^4*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (6*a^4*Sin[c + d*x]^(3 + n))/(d*(3 + n)) + (4*a^4*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a^4*Sin[c + d*x]^(5 + n))/(d*(5 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3, x, 3, (a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (3*a^3*Sin[c + d*x]^(3 + n))/(d*(3 + n)) + (a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2, x, 3, (a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^1, x, 3, (a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^1, x, 2, (Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a*d*(1 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^2, x, 2, (Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^2*d*(1 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^3, x, 2, (Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n))} -{Cos[c + d*x]*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^4, x, 2, (Hypergeometric2F1[4, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 8, (a*x)/16 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, (a*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, (a*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, (a*x)/2 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, (-a)*x - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d - (a*Cot[c + d*x])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 5, (-a)*x + (a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 5, (a*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 6, (a*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + a*Sin[c + d*x]), x, 7, (a*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 12, (a^2*x)/8 - (2*a^2*Cos[c + d*x]^3)/(3*d) + (3*a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 5, (3*a^2*x)/16 - (a^2*Cos[c + d*x]^5)/(10*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 5, (a^2*x)/4 - (2*a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]^5)/(5*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d), (a^2*x)/4 - (a^2*Cos[c + d*x]^3)/(6*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - (Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(10*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 9, a^2*x - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 8, -((a^2*x)/2) - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 7, -2*a^2*x - (a^2*ArcTanh[Cos[c + d*x]])/(2*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 8, (-a^2)*x + (a^2*ArcTanh[Cos[c + d*x]])/d - (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 9, (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 10, (a^2*ArcTanh[Cos[c + d*x]])/(4*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 12, (3*a^2*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 15, (5*a^3*x)/16 - (4*a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/d - (a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x]^3)/(2*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 6, (7*a^3*x)/16 - (4*a^3*Cos[c + d*x]^3)/(3*d) + (3*a^3*Cos[c + d*x]^5)/(5*d) + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (7*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d), (7*a^3*x)/16 - (7*a^3*Cos[c + d*x]^3)/(24*d) + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(10*d) - (Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (7*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(40*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 12, (13*a^3*x)/8 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 10, (a^3*x)/2 - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 10, -((5*a^3*x)/2) - (5*a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 10, -3*a^3*x + (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 11, (-a^3)*x + (13*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (11*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 12, (7*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 14, (7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)} - - -{Cos[c + d*x]^2*Csc[c + d*x]^0*(a + a*Sin[c + d*x])^4, x, 6, (21*a^4*x)/16 - (7*a^4*Cos[c + d*x]^3)/(8*d) + (21*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (3*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^2)/(10*d) - (21*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x]))/(40*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^4, x, 15, (5*a^4*x)/2 - (a^4*ArcTanh[Cos[c + d*x]])/d + (a^4*Cos[c + d*x])/d - (7*a^4*Cos[c + d*x]^3)/(3*d) + (a^4*Cos[c + d*x]^5)/(5*d) + (5*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 12, (17*a^4*x)/8 - (4*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x])/d - (4*a^4*Cos[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x])/d + (23*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 6, (3*x)/(8*a) + Cos[c + d*x]/(a*d) - (2*Cos[c + d*x]^3)/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, -((3*x)/(8*a)) - Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, x/(2*a) + Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, -(x/(2*a)) - Cos[c + d*x]/(a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 3, -(x/a) - ArcTanh[Cos[c + d*x]]/(a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 5, -(ArcTanh[Cos[c + d*x]]/(2*a*d)) + Cot[c + d*x]/(a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 5, ArcTanh[Cos[c + d*x]]/(2*a*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 6, -((3*ArcTanh[Cos[c + d*x]])/(8*a*d)) + Cot[c + d*x]/(a*d) + Cot[c + d*x]^3/(3*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 6, (3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]/(a*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 12, -((27*x)/(8*a^2)) - (4*Cos[c + d*x])/(a^2*d) + (2*Cos[c + d*x]^3)/(3*a^2*d) + (11*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 9, (3*x)/a^2 + (3*Cos[c + d*x])/(a^2*d) - Cos[c + d*x]^3/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 8, -((5*x)/(2*a^2)) - (2*Cos[c + d*x])/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 3, (2*x)/a^2 + Cos[c + d*x]/(a^2*d) + (2*Cos[c + d*x])/(d*(a^2 + a^2*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 5, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, If[$VersionNumber<9, 9, 7], If[$VersionNumber<9, (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - (3*Cot[c + d*x])/(a^2*d) + (2*Cot[c + d*x])/(a^2*d*(1 + Sin[c + d*x])), (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d*(1 + Csc[c + d*x]))]} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 9, -((5*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 11, (3*ArcTanh[Cos[c + d*x]])/(a^2*d) - (3*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d*(1 + Sin[c + d*x]))} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 9, -((11*x)/(2*a^3)) - (3*Cos[c + d*x])/(a^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) - (19*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 4, (3*x)/a^3 + (3*Cos[c + d*x])/(a^3*d) - Cos[c + d*x]^3/(3*d*(a + a*Sin[c + d*x])^3) + (2*Cos[c + d*x]^3)/(a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 3, -(x/a^3) - (7*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])) + (2*Cos[c + d*x])/(3*a*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 7, -(ArcTanh[Cos[c + d*x]]/(a^3*d)) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) + (5*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, If[$VersionNumber<9, 13, 10], If[$VersionNumber<9, (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - (14*Cot[c + d*x])/(3*a^3*d) + (2*Cot[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) + (3*Cot[c + d*x])/(a^3*d*(1 + Sin[c + d*x])), (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) + (2*Cot[c + d*x])/(3*a^3*d*(1 + Csc[c + d*x])^2) - (13*Cot[c + d*x])/(3*a^3*d*(1 + Csc[c + d*x]))]} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 11, -((11*ArcTanh[Cos[c + d*x]])/(2*a^3*d)) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x])^2) + (17*Cos[c + d*x])/(3*a^3*d*(1 + Sin[c + d*x]))} - - -{Cos[e + f*x]^2*Sin[e + f*x]^1/(a + a*Sin[e + f*x])^6, x, 5, (2*Cos[e + f*x])/(9*a*f*(a + a*Sin[e + f*x])^5) - (19*Cos[e + f*x])/(63*a^2*f*(a + a*Sin[e + f*x])^4) + (2*Cos[e + f*x])/(105*f*(a^2 + a^2*Sin[e + f*x])^3) + (4*Cos[e + f*x])/(315*f*(a^3 + a^3*Sin[e + f*x])^2) + (4*Cos[e + f*x])/(315*f*(a^6 + a^6*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 7, -((76*a*Cos[c + d*x])/(495*d*Sqrt[a + a*Sin[c + d*x]])) - (38*a*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (152*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3465*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (76*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(1155*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 4, -((8*a^2*Cos[c + d*x]^3)/(63*d*(a + a*Sin[c + d*x])^(3/2))) - (2*a*Cos[c + d*x]^3)/(21*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 3, -((8*a^2*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2))) - (2*a*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 5, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 4, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (3*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 5, (5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 6, (3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (3*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 8, -((1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]])) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(13*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 5, -((64*a^3*Cos[c + d*x]^3)/(385*d*(a + a*Sin[c + d*x])^(3/2))) - (48*a^2*Cos[c + d*x]^3)/(385*d*Sqrt[a + a*Sin[c + d*x]]) - (6*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(77*d) + (4*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(33*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/(11*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 4, -((64*a^3*Cos[c + d*x]^3)/(315*d*(a + a*Sin[c + d*x])^(3/2))) - (16*a^2*Cos[c + d*x]^3)/(105*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 6, -((2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (2*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 5, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (11*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 6, (a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (13*a^2*Cos[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 6, (13*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (5*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 6, -((4*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]])) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a^2*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 4, -((22*a*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2))) + (12*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*a*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 2, (2*a*Cos[c + d*x]^3)/(15*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^3)/(5*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 4, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d)) + (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 4, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 5, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(4*Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 6, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*Sqrt[a]*d) + Cot[c + d*x]/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 8, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (344*Cos[c + d*x])/(105*a*d*Sqrt[a + a*Sin[c + d*x]]) - (16*Cos[c + d*x]*Sin[c + d*x]^2)/(35*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*a*d*Sqrt[a + a*Sin[c + d*x]]) + (76*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a^2*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, -((2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)) + (18*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^3)/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 6, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 6, (3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - Cot[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 8, -((11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (5*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^2*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 9, (23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (9*Cot[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + a*Sin[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^3 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^3*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^4)/(4*d) + (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, -((a*Cos[c + d*x]^4)/(4*d)) + (a*Sin[c + d*x]^3)/(3*d) - (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^0*(a + a*Sin[c + d*x]), x, 3, (2*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - (a + a*Sin[c + d*x])^4/(4*a^3*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 4, (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 3, -((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^3*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 5, Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^0/(a + a*Sin[c + d*x]), x, 2, Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 5, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 9, (3*a*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (2*a*Cos[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 9, (3*a*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 8, (a*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 7, (a*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 8, (3*a*x)/8 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 9, -((3*a*x)/2) - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 9, -((3*a*x)/2) + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 9, a*x + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 7, a*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x]), x, 6, -((3*a*ArcTanh[Cos[c + d*x]])/(8*d)) - (a*Cot[c + d*x]^5)/(5*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/(16*d)) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + a*Sin[c + d*x]), x, 8, -((a*ArcTanh[Cos[c + d*x]])/(16*d)) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + a*Sin[c + d*x]), x, 9, -((3*a*ArcTanh[Cos[c + d*x]])/(128*d)) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 16, (9*a^2*x)/256 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Cos[c + d*x]^9)/(9*d) + (9*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (3*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 13, (3*a^2*x)/64 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]^7)/(7*d) - (a^2*Cos[c + d*x]^9)/(9*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 14, (11*a^2*x)/128 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (2*a^2*Cos[c + d*x]^7)/(7*d) + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (11*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 6, (a^2*x)/8 - (a^2*Cos[c + d*x]^5)/(15*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - (Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(21*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 11, (3*a^2*x)/4 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 13, -((9*a^2*x)/8) - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 12, -3*a^2*x + (a^2*ArcTanh[Cos[c + d*x]])/(2*d) + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/d} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 12, -((a^2*x)/2) + (3*a^2*ArcTanh[Cos[c + d*x]])/d - (2*a^2*Cos[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 13, 2*a^2*x + (9*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d + (2*a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 10, a^2*x - (3*a^2*ArcTanh[Cos[c + d*x]])/(4*d) + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 11, -((7*a^2*ArcTanh[Cos[c + d*x]])/(16*d)) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + a*Sin[c + d*x])^2, x, 14, -((11*a^2*ArcTanh[Cos[c + d*x]])/(128*d)) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^10*(a + a*Sin[c + d*x])^2, x, 13, -((3*a^2*ArcTanh[Cos[c + d*x]])/(64*d)) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (3*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^11*(a + a*Sin[c + d*x])^2, x, 16, -((9*a^2*ArcTanh[Cos[c + d*x]])/(256*d)) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 19, (15*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (9*a^3*Cos[c + d*x]^7)/(7*d) - (2*a^3*Cos[c + d*x]^9)/(3*d) + (a^3*Cos[c + d*x]^11)/(11*d) + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (3*a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 19, (21*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (a^3*Cos[c + d*x]^7)/d - (a^3*Cos[c + d*x]^9)/(3*d) + (21*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (7*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 17, (17*a^3*x)/128 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (5*a^3*Cos[c + d*x]^7)/(7*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (17*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (17*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (17*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (3*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 7, (27*a^3*x)/128 - (9*a^3*Cos[c + d*x]^5)/(80*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (3*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (9*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(112*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 15, (19*a^3*x)/16 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]^5)/(5*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 15, -((3*a^3*x)/8) - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d + (11*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 15, -((33*a^3*x)/8) - (3*a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 14, -((7*a^3*x)/2) + (7*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (2*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 15, (3*a^3*x)/2 + (33*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (3*a^3*Cos[c + d*x])/d + (2*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (7*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 15, 3*a^3*x + (3*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (a^3*Cos[c + d*x])/d + (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 14, a^3*x - (19*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3, x, 14, -((9*a^3*ArcTanh[Cos[c + d*x]])/(16*d)) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + a*Sin[c + d*x])^3, x, 16, -((27*a^3*ArcTanh[Cos[c + d*x]])/(128*d)) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^10*(a + a*Sin[c + d*x])^3, x, 17, -((17*a^3*ArcTanh[Cos[c + d*x]])/(128*d)) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (5*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^11*(a + a*Sin[c + d*x])^3, x, 19, -((21*a^3*ArcTanh[Cos[c + d*x]])/(256*d)) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/d - (a^3*Cot[c + d*x]^9)/(3*d) - (21*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (29*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 8, (55*a^4*x)/256 - (11*a^4*Cos[c + d*x]^7)/(112*d) + (55*a^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (55*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (11*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(96*d) - (Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(10*a*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x])^2)/(18*d) - (11*Cos[c + d*x]^7*(a^4 + a^4*Sin[c + d*x]))/(144*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 17, -((61*a^4*x)/8) + (2*a^4*ArcTanh[Cos[c + d*x]])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (19*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 8, x/(16*a) + Cos[c + d*x]^3/(3*a*d) - (2*Cos[c + d*x]^5)/(5*a*d) + Cos[c + d*x]^7/(7*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 8, -(x/(16*a)) - Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 7, x/(8*a) + Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 6, -(x/(8*a)) - Cos[c + d*x]^3/(3*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 6, -(x/(2*a)) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 6, -(x/a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cot[c + d*x]/(a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 5, x/a + ArcTanh[Cos[c + d*x]]/(2*a*d) + Cot[c + d*x]/(a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 5, -(ArcTanh[Cos[c + d*x]]/(2*a*d)) - Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 6, ArcTanh[Cos[c + d*x]]/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 7, -(ArcTanh[Cos[c + d*x]]/(8*a*d)) - Cot[c + d*x]^3/(3*a*d) - Cot[c + d*x]^5/(5*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7/(a + a*Sin[c + d*x]), x, 8, ArcTanh[Cos[c + d*x]]/(16*a*d) + Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) + (Cot[c + d*x]*Csc[c + d*x])/(16*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(24*a*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 11, -((5*x)/(8*a^2)) - (2*Cos[c + d*x])/(a^2*d) + (5*Cos[c + d*x]^3)/(3*a^2*d) - (4*Cos[c + d*x]^5)/(5*a^2*d) + Cos[c + d*x]^7/(7*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^5)/(3*a^2*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 12, (11*x)/(16*a^2) + (2*Cos[c + d*x])/(a^2*d) - (4*Cos[c + d*x]^3)/(3*a^2*d) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (11*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*a^2*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 10, -((3*x)/(4*a^2)) - (2*Cos[c + d*x])/(a^2*d) + Cos[c + d*x]^3/(a^2*d) - Cos[c + d*x]^5/(5*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 10, (7*x)/(8*a^2) + (2*Cos[c + d*x])/(a^2*d) - (2*Cos[c + d*x]^3)/(3*a^2*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, -(x/a^2) - (2*Cos[c + d*x]^3)/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - Cos[c + d*x]^5/(d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, -((2*x)/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cos[c + d*x]/(a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 6, x/a^2 + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 8, -((3*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 9, ArcTanh[Cos[c + d*x]]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 10, -((7*ArcTanh[Cos[c + d*x]])/(8*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) - (7*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 10, (3*ArcTanh[Cos[c + d*x]])/(4*a^2*d) - (2*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(a^2*d) - Cot[c + d*x]^5/(5*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(2*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 12, -((11*ArcTanh[Cos[c + d*x]])/(16*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) + (4*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 12, (51*x)/(8*a^3) + (7*Cos[c + d*x])/(a^3*d) - Cos[c + d*x]^3/(a^3*d) - (19*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 9, -((11*x)/(2*a^3)) - (5*Cos[c + d*x])/(a^3*d) + Cos[c + d*x]^3/(3*a^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 4, (9*x)/(2*a^3) + (9*Cos[c + d*x])/(2*a^3*d) + Cos[c + d*x]^5/(d*(a + a*Sin[c + d*x])^3) + (3*Cos[c + d*x]^3)/(2*d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 5, x/a^3 - ArcTanh[Cos[c + d*x]]/(a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 7, (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 9, -((9*ArcTanh[Cos[c + d*x]])/(2*a^3*d)) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, If[$VersionNumber<9, 12, 11], If[$VersionNumber<9, (11*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (13*Cot[c + d*x])/(a^3*d) - (13*Cot[c + d*x]^3)/(3*a^3*d) + (11*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (4*Cot[c + d*x]*Csc[c + d*x]^2)/(a^3*d*(1 + Sin[c + d*x])), (11*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (5*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (4*Cot[c + d*x])/(a^3*d*(1 + Csc[c + d*x]))]} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 14, -((51*ArcTanh[Cos[c + d*x]])/(8*a^3*d)) + (7*Cot[c + d*x])/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (19*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d) + (4*Cos[c + d*x])/(a^3*d*(1 + Sin[c + d*x]))} - - -{Cos[e + f*x]^4*Sin[e + f*x]^1/(a + a*Sin[e + f*x])^6, x, 2, Cos[e + f*x]^5/(7*f*(a + a*Sin[e + f*x])^6) - 6*Cos[e + f*x]^5/(35*a*f*(a + a*Sin[e + f*x])^5)} - - -{Cos[e + f*x]^4*Sin[e + f*x]^2/(a + a*Sin[e + f*x])^7, x, 18, -a*Cos[e + f*x]^7/(18*f*(a + a*Sin[e + f*x])^8) + 25*Cos[e + f*x]^5/(126*a*f*(a + a*Sin[e + f*x])^6) - 47*Cos[e + f*x]^5/(315*a^2*f*(a + a*Sin[e + f*x])^5), -((4*Cos[e + f*x])/(9*a^7*f*(1 + Sin[e + f*x])^5)) + (92*Cos[e + f*x])/(63*a^7*f*(1 + Sin[e + f*x])^4) - (181*Cos[e + f*x])/(105*a^7*f*(1 + Sin[e + f*x])^3) + (268*Cos[e + f*x])/(315*a^7*f*(1 + Sin[e + f*x])^2) - (47*Cos[e + f*x])/(315*a^7*f*(1 + Sin[e + f*x]))} - - -{Cos[e + f*x]^4*Sin[e + f*x]^3/(a + a*Sin[e + f*x])^8, x, 24, (4*Cos[e + f*x])/(11*a^8*f*(1 + Sin[e + f*x])^6) - (52*Cos[e + f*x])/(33*a^8*f*(1 + Sin[e + f*x])^5) + (617*Cos[e + f*x])/(231*a^8*f*(1 + Sin[e + f*x])^4) - (846*Cos[e + f*x])/(385*a^8*f*(1 + Sin[e + f*x])^3) + (1003*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x])^2) - (152*Cos[e + f*x])/(1155*a^8*f*(1 + Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 5, -((1472*a^3*Cos[c + d*x]^5)/(45045*d*(a + a*Sin[c + d*x])^(5/2))) - (368*a^2*Cos[c + d*x]^5)/(9009*d*(a + a*Sin[c + d*x])^(3/2)) - (46*a*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (20*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 4, -((64*a^3*Cos[c + d*x]^5)/(3465*d*(a + a*Sin[c + d*x])^(5/2))) - (16*a^2*Cos[c + d*x]^5)/(693*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^5)/(99*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 9, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (8*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (164*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 8, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (61*a*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*d) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 7, (13*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 7, (11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (2*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (11*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]], x, 9, -((67*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d)) + (61*a*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (61*a*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]], x, 11, -((31*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d)) - (31*a*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x])/(192*d*Sqrt[a + a*Sin[c + d*x]]) + (97*a*Cot[c + d*x]*Csc[c + d*x]^2)/(240*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(5*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]], x, 13, -((55*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d)) - (55*a*Cot[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) - (55*a*Cot[c + d*x]*Csc[c + d*x])/(768*d*Sqrt[a + a*Sin[c + d*x]]) + (329*a*Cot[c + d*x]*Csc[c + d*x]^2)/(960*d*Sqrt[a + a*Sin[c + d*x]]) + (47*a*Cot[c + d*x]*Csc[c + d*x]^3)/(160*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(60*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*Sqrt[a + a*Sin[c + d*x]], x, 15, -((61*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d)) - (61*a*Cot[c + d*x])/(1024*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x])/(1536*d*Sqrt[a + a*Sin[c + d*x]]) - (61*a*Cot[c + d*x]*Csc[c + d*x]^2)/(1920*d*Sqrt[a + a*Sin[c + d*x]]) + (579*a*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a + a*Sin[c + d*x]]) + (193*a*Cot[c + d*x]*Csc[c + d*x]^4)/(840*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(84*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(7*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 6, -((256*a^4*Cos[c + d*x]^5)/(6435*d*(a + a*Sin[c + d*x])^(5/2))) - (64*a^3*Cos[c + d*x]^5)/(1287*d*(a + a*Sin[c + d*x])^(3/2)) - (56*a^2*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) - (14*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(429*d) + (4*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(39*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(15*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 5, -((256*a^4*Cos[c + d*x]^5)/(5005*d*(a + a*Sin[c + d*x])^(5/2))) - (64*a^3*Cos[c + d*x]^5)/(1001*d*(a + a*Sin[c + d*x])^(3/2)) - (8*a^2*Cos[c + d*x]^5)/(143*d*Sqrt[a + a*Sin[c + d*x]]) - (6*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 12, -((2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (14*a^2*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]]) - (34*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (388*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) + (16*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 10, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (171*a^2*Cos[c + d*x])/(35*d*Sqrt[a + a*Sin[c + d*x]]) + (69*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) + (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 9, (9*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (73*a^2*Cos[c + d*x])/(20*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 8, (37*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) - (8*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (29*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2), x, 11, (21*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (2*a^2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (149*a^2*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (19*a^2*Cot[c + d*x]*Csc[c + d*x])/(32*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(8*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2), x, 12, -((165*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*d)) + (91*a^2*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (73*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (31*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(40*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2), x, 14, -((179*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(512*d)) - (179*a^2*Cot[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (111*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d*Sqrt[a + a*Sin[c + d*x]]) + (239*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(320*d*Sqrt[a + a*Sin[c + d*x]]) + (137*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(480*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(20*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^(3/2), x, 16, -((171*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(1024*d)) - (171*a^2*Cot[c + d*x])/(1024*d*Sqrt[a + a*Sin[c + d*x]]) - (57*a^2*Cot[c + d*x]*Csc[c + d*x])/(512*d*Sqrt[a + a*Sin[c + d*x]]) + (199*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(640*d*Sqrt[a + a*Sin[c + d*x]]) + (1237*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*d*Sqrt[a + a*Sin[c + d*x]]) + (9*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(28*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2))/(7*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + a*Sin[c + d*x])^(3/2), x, 18, -((1587*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(16384*d)) - (1587*a^2*Cot[c + d*x])/(16384*d*Sqrt[a + a*Sin[c + d*x]]) - (529*a^2*Cot[c + d*x]*Csc[c + d*x])/(8192*d*Sqrt[a + a*Sin[c + d*x]]) - (529*a^2*Cot[c + d*x]*Csc[c + d*x]^2)/(10240*d*Sqrt[a + a*Sin[c + d*x]]) + (8653*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(35840*d*Sqrt[a + a*Sin[c + d*x]]) + (1957*a^2*Cot[c + d*x]*Csc[c + d*x]^4)/(4480*d*Sqrt[a + a*Sin[c + d*x]]) + (83*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(448*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Csc[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]])/(112*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(8*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 5, -((152*a^2*Cos[c + d*x]^5)/(3465*d*(a + a*Sin[c + d*x])^(5/2))) - (38*a*Cos[c + d*x]^5)/(693*d*(a + a*Sin[c + d*x])^(3/2)) + (20*Cos[c + d*x]^5)/(99*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*a*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 3, (8*a^2*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2)) + (2*a*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(9*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 13, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d)) + (32*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 11, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) + (4*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 11, (9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*Sqrt[a]*d) - (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 11, -((7*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*Sqrt[a]*d)) + (9*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]], x, 15, -((11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*Sqrt[a]*d)) - (11*Cot[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (53*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]], x, 17, -((9*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*Sqrt[a]*d)) - (9*Cot[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) - (3*Cot[c + d*x]*Csc[c + d*x])/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (29*Cot[c + d*x]*Csc[c + d*x]^2)/(80*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*d*Sqrt[a + a*Sin[c + d*x]])} - - -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 12, -((4*Cos[c + d*x])/(165*a*d*Sqrt[a + a*Sin[c + d*x]])) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(231*a*d*Sqrt[a + a*Sin[c + d*x]]) + (14*Cos[c + d*x]*Sin[c + d*x]^4)/(33*a*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(1155*a^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*a^2*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(385*a^3*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 4, -((46*a*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2))) + (20*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(9*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 2, (6*a*Cos[c + d*x]^5)/(35*d*(a + a*Sin[c + d*x])^(5/2)) - (2*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(3/2))} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 6, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d)) + (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 9, (3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - Cos[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 8, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d)) + (7*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 10, -(ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*a^(3/2)*d)) - Cot[c + d*x]/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2), x, 12, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*a^(3/2)*d)) - (3*Cot[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(32*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*Cot[c + d*x]*Csc[c + d*x]^2)/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2), x, 14, -((3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(128*a^(3/2)*d)) - (3*Cot[c + d*x])/(128*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(80*a*d*Sqrt[a + a*Sin[c + d*x]]) + (19*Cot[c + d*x]*Csc[c + d*x]^3)/(40*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2), x, 18, -((4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d)) + (4496*Cos[c + d*x])/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (200*Cos[c + d*x]*Sin[c + d*x]^2)/(231*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (424*Cos[c + d*x]*Sin[c + d*x]^3)/(693*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (46*Cos[c + d*x]*Sin[c + d*x]^4)/(99*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^5)/(11*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (1048*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(693*a^3*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 16, (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2048*Cos[c + d*x])/(315*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (92*Cos[c + d*x]*Sin[c + d*x]^2)/(105*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (38*Cos[c + d*x]*Sin[c + d*x]^3)/(63*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (472*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a^3*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 6, -((4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d)) + (4*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(5/2)) + (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (2*Cos[c + d*x]^5)/(7*a*d*(a + a*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 5, (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2*Cos[c + d*x]^5)/(5*d*(a + a*Sin[c + d*x])^(5/2)) - (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(5/2), x, 9, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d)) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (2*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2), x, 12, (5*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(5/2)*d) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - Cot[c + d*x]/(a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2), x, 14, -((23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(5/2)*d)) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (9*Cot[c + d*x])/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2), x, 16, (45*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(5/2)*d) - (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) - (19*Cot[c + d*x])/(8*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Cot[c + d*x]*Csc[c + d*x])/(12*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d*Sqrt[a + a*Sin[c + d*x]])} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2), x, 18, -((363*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*a^(5/2)*d)) + (4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (149*Cot[c + d*x])/(64*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (107*Cot[c + d*x]*Csc[c + d*x])/(96*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (17*Cot[c + d*x]*Csc[c + d*x]^2)/(24*a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2, x, 5, (a^2*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (a^2*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2])} -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^1, x, 3, (a*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (a*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])} -{Cos[c + d*x]^4*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^1, x, 3, (Cos[c + d*x]*Hypergeometric2F1[-(1/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(a*d*(1 + n)*Sqrt[Cos[c + d*x]^2]) - (Cos[c + d*x]*Hypergeometric2F1[-(1/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(a*d*(2 + n)*Sqrt[Cos[c + d*x]^2])} -{Cos[c + d*x]^4*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^2, x, 5, -((Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(a^2*d*(2 + n))) + ((3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(a^2*d*(1 + n)*(2 + n)*Sqrt[Cos[c + d*x]^2]) - (2*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^6)/(6*d) + (a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) - (2*a*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^6)/(6*d) - (2*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) + (a*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 7, -((a*Cos[c + d*x]^6)/(6*d)) + (a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^5)/(5*d) - (2*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, -((a*Cos[c + d*x]^6)/(6*d)) + (a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, -((a*Cos[c + d*x]^6)/(6*d)) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 4, (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d + (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (2*a*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, (2*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (2*a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 3, (2*a*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (a*Csc[c + d*x]^2)/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (a*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x]), x, 6, -((a*Cot[c + d*x]^6)/(6*d)) - (a*Csc[c + d*x])/d + (2*a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^8*(a + a*Sin[c + d*x]), x, 6, -((a*Cot[c + d*x]^6)/(6*d)) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^9*(a + a*Sin[c + d*x]), x, 7, -((a*Cot[c + d*x]^6)/(6*d)) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^10*(a + a*Sin[c + d*x]), x, 7, -((a*Cot[c + d*x]^6)/(6*d)) - (a*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^5)/(5*d) + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^11*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x]^5)/(5*d)) - (a*Csc[c + d*x]^6)/(6*d) + (2*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^12*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x]^6)/(6*d)) - (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)} - - -{Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Sin[c + d*x]^4)/(4*d) + (2*a^2*Sin[c + d*x]^5)/(5*d) - (a^2*Sin[c + d*x]^6)/(6*d) - (4*a^2*Sin[c + d*x]^7)/(7*d) - (a^2*Sin[c + d*x]^8)/(8*d) + (2*a^2*Sin[c + d*x]^9)/(9*d) + (a^2*Sin[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, (4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(a^4*d) + (13*(a + a*Sin[c + d*x])^7)/(7*a^5*d) - (3*(a + a*Sin[c + d*x])^8)/(4*a^6*d) + (a + a*Sin[c + d*x])^9/(9*a^7*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 4, -((4*(a + a*Sin[c + d*x])^5)/(5*a^3*d)) + (4*(a + a*Sin[c + d*x])^6)/(3*a^4*d) - (5*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (a + a*Sin[c + d*x])^8/(8*a^6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) - (4*a^2*Sin[c + d*x]^3)/(3*d) - (a^2*Sin[c + d*x]^4)/(4*d) + (2*a^2*Sin[c + d*x]^5)/(5*d) + (a^2*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x])/d) + (2*a^2*Log[Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d - (2*a^2*Sin[c + d*x]^2)/d - (a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(2*d) + (a^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 4, -((2*a^2*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/(2*d) - (a^2*Log[Sin[c + d*x]])/d - (4*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + (2*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Csc[c + d*x])/d - (a^2*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (4*a^2*Log[Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/d + (a^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 3, (4*a^2*Csc[c + d*x])/d + (a^2*Csc[c + d*x]^2)/(2*d) - (2*a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) - (a^2*Log[Sin[c + d*x]])/d + (2*a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 4, (a^2*Csc[c + d*x])/d + (2*a^2*Csc[c + d*x]^2)/d + (a^2*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 4, -((2*a^2*Csc[c + d*x])/d) + (a^2*Csc[c + d*x]^2)/(2*d) + (4*a^2*Csc[c + d*x]^3)/(3*d) + (a^2*Csc[c + d*x]^4)/(4*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (a^2*Log[Sin[c + d*x]])/d} - - -{Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, (2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (12*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (13*(a + a*Sin[c + d*x])^8)/(8*a^5*d) - (2*(a + a*Sin[c + d*x])^9)/(3*a^6*d) + (a + a*Sin[c + d*x])^10/(10*a^7*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, -((2*(a + a*Sin[c + d*x])^6)/(3*a^3*d)) + (8*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - (5*(a + a*Sin[c + d*x])^8)/(8*a^5*d) + (a + a*Sin[c + d*x])^9/(9*a^6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, (a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) - (5*a^3*Sin[c + d*x]^4)/(4*d) + (a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x])/d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d - (5*a^3*Sin[c + d*x]^2)/(2*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 4, -((3*a^3*Csc[c + d*x])/d) - (a^3*Csc[c + d*x]^2)/(2*d) + (a^3*Log[Sin[c + d*x]])/d - (5*a^3*Sin[c + d*x])/d - (5*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^4)/(4*d) + (a^3*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x])/d) - (3*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) - (5*a^3*Log[Sin[c + d*x]])/d - (5*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/d + (a^3*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 3, (5*a^3*Csc[c + d*x])/d - (a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/d - (a^3*Csc[c + d*x]^4)/(4*d) - (5*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 4, (5*a^3*Csc[c + d*x])/d + (5*a^3*Csc[c + d*x]^2)/(2*d) - (a^3*Csc[c + d*x]^3)/(3*d) - (3*a^3*Csc[c + d*x]^4)/(4*d) - (a^3*Csc[c + d*x]^5)/(5*d) + (a^3*Log[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x])/d) + (5*a^3*Csc[c + d*x]^2)/(2*d) + (5*a^3*Csc[c + d*x]^3)/(3*d) - (a^3*Csc[c + d*x]^4)/(4*d) - (3*a^3*Csc[c + d*x]^5)/(5*d) - (a^3*Csc[c + d*x]^6)/(6*d) + (3*a^3*Log[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d} - - -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 4, -((4*a^4*Csc[c + d*x])/d) - (2*a^4*Csc[c + d*x]^2)/d - (a^4*Csc[c + d*x]^3)/(3*d) - (4*a^4*Log[Sin[c + d*x]])/d - (10*a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/d + (a^4*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^4, x, 3, (4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/(4*d) - (10*a^4*Log[Sin[c + d*x]])/d - (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^4, x, 4, (10*a^4*Csc[c + d*x])/d + (2*a^4*Csc[c + d*x]^2)/d - (4*a^4*Csc[c + d*x]^3)/(3*d) - (a^4*Csc[c + d*x]^4)/d - (a^4*Csc[c + d*x]^5)/(5*d) - (4*a^4*Log[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (2*a^4*Sin[c + d*x]^2)/d + (a^4*Sin[c + d*x]^3)/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) + Sin[c + d*x]^7/(7*a*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 7, Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d) + Sin[c + d*x]^6/(6*a*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 6, -(Cos[c + d*x]^4/(4*a*d)) - Sin[c + d*x]^3/(3*a*d) + Sin[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - Log[Sin[c + d*x]]/(a*d) + Sin[c + d*x]/(a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) + Log[Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 5, -(Cot[c + d*x]^4/(4*a*d)) - Csc[c + d*x]/(a*d) + Csc[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 6, Cot[c + d*x]^4/(4*a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^7/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]^3/(3*a*d)) + Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^8/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]^4/(4*a*d)) + Csc[c + d*x]^5/(5*a*d) + Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d)} - - -{Cos[c + d*x]^5*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, Sin[c + d*x]^4/(4*a^2*d) - (2*Sin[c + d*x]^5)/(5*a^2*d) + Sin[c + d*x]^6/(6*a^2*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 4, Sin[c + d*x]^3/(3*a^2*d) - Sin[c + d*x]^4/(2*a^2*d) + Sin[c + d*x]^5/(5*a^2*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, Sin[c + d*x]^2/(2*a^2*d) - (2*Sin[c + d*x]^3)/(3*a^2*d) + Sin[c + d*x]^4/(4*a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, Log[Sin[c + d*x]]/(a^2*d) - (2*Sin[c + d*x])/(a^2*d) + Sin[c + d*x]^2/(2*a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 4, -(Csc[c + d*x]/(a^2*d)) - (2*Log[Sin[c + d*x]])/(a^2*d) + Sin[c + d*x]/(a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 4, (2*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a^2*d) + Log[Sin[c + d*x]]/(a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 3, -((Csc[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(3*a^5*d))} -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 3, -(Csc[c + d*x]^2/(2*a^2*d)) + (2*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 4, -(Csc[c + d*x]^3/(3*a^2*d)) + Csc[c + d*x]^4/(2*a^2*d) - Csc[c + d*x]^5/(5*a^2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 4, -(Csc[c + d*x]^4/(4*a^2*d)) + (2*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a^2*d)} - - -{Cos[c + d*x]^5*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, -((4*Log[1 + Sin[c + d*x]])/(a^3*d)) + (4*Sin[c + d*x])/(a^3*d) - (2*Sin[c + d*x]^2)/(a^3*d) + (4*Sin[c + d*x]^3)/(3*a^3*d) - (3*Sin[c + d*x]^4)/(4*a^3*d) + Sin[c + d*x]^5/(5*a^3*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 4, (4*Log[1 + Sin[c + d*x]])/(a^3*d) - (4*Sin[c + d*x])/(a^3*d) + (2*Sin[c + d*x]^2)/(a^3*d) - Sin[c + d*x]^3/(a^3*d) + Sin[c + d*x]^4/(4*a^3*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 4, -((4*Log[1 + Sin[c + d*x]])/(a^3*d)) + (4*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3/(3*a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 4, Log[Sin[c + d*x]]/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d) + Sin[c + d*x]/(a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 4, -(Csc[c + d*x]/(a^3*d)) - (3*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 4, (3*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 4, -((4*Csc[c + d*x])/(a^3*d)) + (3*Csc[c + d*x]^2)/(2*a^3*d) - Csc[c + d*x]^3/(3*a^3*d) - (4*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 3, (4*Csc[c + d*x])/(a^3*d) - (2*Csc[c + d*x]^2)/(a^3*d) + Csc[c + d*x]^3/(a^3*d) - Csc[c + d*x]^4/(4*a^3*d) + (4*Log[Sin[c + d*x]])/(a^3*d) - (4*Log[1 + Sin[c + d*x]])/(a^3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 4, -((4*Csc[c + d*x])/(a^3*d)) + (2*Csc[c + d*x]^2)/(a^3*d) - (4*Csc[c + d*x]^3)/(3*a^3*d) + (3*Csc[c + d*x]^4)/(4*a^3*d) - Csc[c + d*x]^5/(5*a^3*d) - (4*Log[Sin[c + d*x]])/(a^3*d) + (4*Log[1 + Sin[c + d*x]])/(a^3*d)} - - -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^4, x, 3, (12*Csc[c + d*x])/(a^4*d) - (4*Csc[c + d*x]^2)/(a^4*d) + (4*Csc[c + d*x]^3)/(3*a^4*d) - Csc[c + d*x]^4/(4*a^4*d) + (16*Log[Sin[c + d*x]])/(a^4*d) - (16*Log[1 + Sin[c + d*x]])/(a^4*d) + 4/(d*(a^4 + a^4*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^4, x, 4, -((16*Csc[c + d*x])/(a^4*d)) + (6*Csc[c + d*x]^2)/(a^4*d) - (8*Csc[c + d*x]^3)/(3*a^4*d) + Csc[c + d*x]^4/(a^4*d) - Csc[c + d*x]^5/(5*a^4*d) - (20*Log[Sin[c + d*x]])/(a^4*d) + (20*Log[1 + Sin[c + d*x]])/(a^4*d) - 4/(d*(a^4 + a^4*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3, x, 3, (a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) + (a^3*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (5*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (5*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (3*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) + (a^3*Sin[c + d*x]^(8 + n))/(d*(8 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2, x, 3, (a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (4*a^2*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (a^2*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (2*a^2*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (a^2*Sin[c + d*x]^(7 + n))/(d*(7 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^1, x, 3, (a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (2*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a*Sin[c + d*x]^(6 + n))/(d*(6 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^1, x, 3, Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - Sin[c + d*x]^(3 + n)/(a*d*(3 + n)) + Sin[c + d*x]^(4 + n)/(a*d*(4 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^2, x, 3, Sin[c + d*x]^(1 + n)/(a^2*d*(1 + n)) - (2*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)) + Sin[c + d*x]^(3 + n)/(a^2*d*(3 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^3, x, 4, -((3*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n))) + (4*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^3*d*(1 + n)) + Sin[c + d*x]^(2 + n)/(a^3*d*(2 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^4, x, 4, Sin[c + d*x]^(1 + n)/(a^4*d*(1 + n)) - (4*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d) + (4*Sin[c + d*x]^(1 + n))/(d*(a^4 + a^4*Sin[c + d*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 10, (3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (2*a*Cos[c + d*x]^9)/(9*d) - (a*Cos[c + d*x]^11)/(11*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 10, (3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 9, (5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 8, (5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 9, (5*a*x)/16 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 10, -((15*a*x)/8) - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) - (15*a*Cot[c + d*x])/(8*d) + (5*a*Cos[c + d*x]^2*Cot[c + d*x])/(8*d) + (a*Cos[c + d*x]^4*Cot[c + d*x])/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 11, -((15*a*x)/8) + (5*a*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*Cos[c + d*x])/(2*d) - (5*a*Cos[c + d*x]^3)/(6*d) - (15*a*Cot[c + d*x])/(8*d) + (5*a*Cos[c + d*x]^2*Cot[c + d*x])/(8*d) + (a*Cos[c + d*x]^4*Cot[c + d*x])/(4*d) - (a*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 11, (5*a*x)/2 + (5*a*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*Cos[c + d*x])/(2*d) - (5*a*Cos[c + d*x]^3)/(6*d) + (5*a*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*d) - (5*a*Cot[c + d*x]^3)/(6*d) + (a*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 11, (5*a*x)/2 - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) + (5*a*Cot[c + d*x])/(2*d) + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) - (5*a*Cot[c + d*x]^3)/(6*d) + (a*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x]), x, 11, (-a)*x - (15*a*ArcTanh[Cos[c + d*x]])/(8*d) + (15*a*Cos[c + d*x])/(8*d) - (a*Cot[c + d*x])/d + (5*a*Cos[c + d*x]*Cot[c + d*x]^2)/(8*d) + (a*Cot[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x]), x, 9, (-a)*x + (5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x])/d + (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a*Cot[c + d*x]^5*Csc[c + d*x])/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x]), x, 7, (5*a*ArcTanh[Cos[c + d*x]])/(16*d) - (a*Cot[c + d*x]^7)/(7*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a*Cot[c + d*x]^5*Csc[c + d*x])/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^9*(a + a*Sin[c + d*x]), x, 8, (5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^10*(a + a*Sin[c + d*x]), x, 9, (5*a*ArcTanh[Cos[c + d*x]])/(128*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (5*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^11*(a + a*Sin[c + d*x]), x, 10, (3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^12*(a + a*Sin[c + d*x]), x, 10, (3*a*ArcTanh[Cos[c + d*x]])/(256*d) - (a*Cot[c + d*x]^7)/(7*d) - (2*a*Cot[c + d*x]^9)/(9*d) - (a*Cot[c + d*x]^11)/(11*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(256*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)} - - -{Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 18, (17*a^2*x)/1024 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (4*a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Cos[c + d*x]^11)/(11*d) + (17*a^2*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (17*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (17*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(120*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 14, (3*a^2*x)/128 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (a^2*Cos[c + d*x]^9)/(3*d) - (a^2*Cos[c + d*x]^11)/(11*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(80*d) - (3*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(40*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(5*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 16, (13*a^2*x)/256 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (2*a^2*Cos[c + d*x]^9)/(9*d) + (13*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (13*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (13*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (13*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 7, (5*a^2*x)/64 - (a^2*Cos[c + d*x]^7)/(28*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(36*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 12, (5*a^2*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cos[c + d*x]^7)/(7*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 17, -((25*a^2*x)/16) - (2*a^2*ArcTanh[Cos[c + d*x]])/d + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^3)/(3*d) + (2*a^2*Cos[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x])/d - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 16, -((15*a^2*x)/4) + (3*a^2*ArcTanh[Cos[c + d*x]])/(2*d) - (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (9*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 17, (5*a^2*x)/8 + (5*a^2*ArcTanh[Cos[c + d*x]])/d - (4*a^2*Cos[c + d*x])/d - (2*a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/d - (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 16, 5*a^2*x + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (4*a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/d} -{Cos[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^2, x, 15, (3*a^2*x)/2 - (15*a^2*ArcTanh[Cos[c + d*x]])/(4*d) + (2*a^2*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^5)/(5*d) + (9*a^2*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2, x, 17, -2*a^2*x - (25*a^2*ArcTanh[Cos[c + d*x]])/(16*d) + (a^2*Cos[c + d*x])/d - (2*a^2*Cot[c + d*x])/d + (2*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^2, x, 12, (-a^2)*x + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x])/(12*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^9*(a + a*Sin[c + d*x])^2, x, 13, (45*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (35*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^10*(a + a*Sin[c + d*x])^2, x, 12, (5*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^11*(a + a*Sin[c + d*x])^2, x, 16, (13*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) + (13*a^2*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (9*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^12*(a + a*Sin[c + d*x])^2, x, 14, (3*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(3*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^13*(a + a*Sin[c + d*x])^2, x, 18, (17*a^2*ArcTanh[Cos[c + d*x]])/(1024*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (4*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Cot[c + d*x]^11)/(11*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (17*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(1536*d) - (11*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^2*Cot[c + d*x]^3*Csc[c + d*x]^7)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*d)} - - -{Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 21, (27*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (a^3*Cos[c + d*x]^9)/d - (6*a^3*Cos[c + d*x]^11)/(11*d) + (a^3*Cos[c + d*x]^13)/(13*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(512*d) + (9*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(640*d) - (27*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (9*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(40*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(4*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 21, (41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9*d) - (3*a^3*Cos[c + d*x]^11)/(11*d) + (41*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 19, (19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9*d) - (a^3*Cos[c + d*x]^11)/(11*d) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 8, (33*a^3*x)/256 - (33*a^3*Cos[c + d*x]^7)/(560*d) + (33*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(30*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(240*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 17, (125*a^3*x)/128 - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cos[c + d*x]^7)/(7*d) + (125*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (125*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 19, -((15*a^3*x)/16) - (3*a^3*ArcTanh[Cos[c + d*x]])/d + (3*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/d + (3*a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cos[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x])/d + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (11*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 17, -((85*a^3*x)/16) - (a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x]^3)/(3*d) + (3*a^3*Cos[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (43*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 15, -((25*a^3*x)/8) + (13*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^3*Cos[c + d*x])/d - (2*a^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) - (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 16, (45*a^3*x)/8 + (45*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (5*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3, x, 16, (13*a^3*x)/2 - (25*a^3*ArcTanh[Cos[c + d*x]])/(8*d) + (a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (5*a^3*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3, x, 18, -((a^3*x)/2) - (85*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (3*a^3*Cos[c + d*x])/d - (a^3*Cot[c + d*x])/d + (2*a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (43*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^8*(a + a*Sin[c + d*x])^3, x, 18, -3*a^3*x - (15*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cos[c + d*x])/d - (3*a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (11*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^9*(a + a*Sin[c + d*x])^3, x, 17, (-a^3)*x + (125*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Cot[c + d*x]^7)/(7*d) - (115*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(2*d) - (5*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^10*(a + a*Sin[c + d*x])^3, x, 16, (55*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(9*d) - (25*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (15*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^11*(a + a*Sin[c + d*x])^3, x, 18, (33*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/(3*d) + (33*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (29*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^12*(a + a*Sin[c + d*x])^3, x, 19, (19*a^3*ArcTanh[Cos[c + d*x]])/(256*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (5*a^3*Cot[c + d*x]^9)/(9*d) - (a^3*Cot[c + d*x]^11)/(11*d) + (19*a^3*Cot[c + d*x]*Csc[c + d*x])/(256*d) - (7*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) + (5*a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(32*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^13*(a + a*Sin[c + d*x])^3, x, 21, (41*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (7*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Cot[c + d*x]^11)/(11*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (41*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(1536*d) - (35*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(384*d) + (3*a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (3*a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(24*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(12*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^14*(a + a*Sin[c + d*x])^3, x, 21, (27*a^3*ArcTanh[Cos[c + d*x]])/(1024*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) - (a^3*Cot[c + d*x]^9)/d - (6*a^3*Cot[c + d*x]^11)/(11*d) - (a^3*Cot[c + d*x]^13)/(13*d) + (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(1024*d) + (9*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(512*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(128*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x]^7)/(64*d) + (a^3*Cot[c + d*x]^3*Csc[c + d*x]^7)/(8*d) - (a^3*Cot[c + d*x]^5*Csc[c + d*x]^7)/(4*d)} - - -{Cos[c + d*x]^6*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 22, -((135*a^4*x)/16) + (6*a^4*ArcTanh[Cos[c + d*x]])/d - (4*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^5)/(5*d) - (4*a^4*Cot[c + d*x])/d - (a^4*Cot[c + d*x]^3)/(3*d) - (2*a^4*Cot[c + d*x]*Csc[c + d*x])/d - (89*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^4*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^6*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 9, (3*x)/(128*a) + Cos[c + d*x]^5/(5*a*d) - (2*Cos[c + d*x]^7)/(7*a*d) + Cos[c + d*x]^9/(9*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 9, -((3*x)/(128*a)) - Cos[c + d*x]^5/(5*a*d) + Cos[c + d*x]^7/(7*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 8, x/(16*a) + Cos[c + d*x]^5/(5*a*d) - Cos[c + d*x]^7/(7*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 7, -(x/(16*a)) - Cos[c + d*x]^5/(5*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 8, -((3*x)/(8*a)) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 9, -((3*x)/(2*a)) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - (3*Cot[c + d*x])/(2*a*d) + (Cos[c + d*x]^2*Cot[c + d*x])/(2*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 9, (3*x)/(2*a) + (3*ArcTanh[Cos[c + d*x]])/(2*a*d) - (3*Cos[c + d*x])/(2*a*d) + (3*Cot[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Cot[c + d*x])/(2*a*d) - (Cos[c + d*x]*Cot[c + d*x]^2)/(2*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 8, x/a - (3*ArcTanh[Cos[c + d*x]])/(2*a*d) + (3*Cos[c + d*x])/(2*a*d) + Cot[c + d*x]/(a*d) + (Cos[c + d*x]*Cot[c + d*x]^2)/(2*a*d) - Cot[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 7, -(x/a) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]/(a*d) + Cot[c + d*x]^3/(3*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 6, (3*ArcTanh[Cos[c + d*x]])/(8*a*d) - Cot[c + d*x]^5/(5*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d)} - - -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 13, -(x/(8*a^2)) - (2*Cos[c + d*x]^3)/(3*a^2*d) + (3*Cos[c + d*x]^5)/(5*a^2*d) - Cos[c + d*x]^7/(7*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(3*a^2*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 6, (3*x)/(16*a^2) + Cos[c + d*x]^5/(10*a^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*(a - a*Sin[c + d*x])^3)/(6*a^5*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 5, -(x/(4*a^2)) - (2*Cos[c + d*x]^5)/(15*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(6*a^2*d) - Cos[c + d*x]^7/(3*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 10, -(x/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a^2*d) - Cos[c + d*x]^3/(3*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 9, -(x/(2*a^2)) + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 8, (2*x)/a^2 - ArcTanh[Cos[c + d*x]]/(2*a^2*d) + Cos[c + d*x]/(a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 9, -(x/a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 10, (5*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 11, -(ArcTanh[Cos[c + d*x]]/(4*a^2*d)) - (2*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(2*a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 13, (3*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (5*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)} - - -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 14, -((23*x)/(16*a^3)) - (4*Cos[c + d*x])/(a^3*d) + (7*Cos[c + d*x]^3)/(3*a^3*d) - (3*Cos[c + d*x]^5)/(5*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (23*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^3*d) + (Cos[c + d*x]*Sin[c + d*x]^5)/(6*a^3*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 12, (13*x)/(8*a^3) + (4*Cos[c + d*x])/(a^3*d) - (5*Cos[c + d*x]^3)/(3*a^3*d) + Cos[c + d*x]^5/(5*a^3*d) - (13*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 5, -((15*x)/(8*a^3)) - (4*Cos[c + d*x])/(a^3*d) + Cos[c + d*x]^3/(a^3*d) + (15*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^3*d), -((15*x)/(8*a^3)) - (5*Cos[c + d*x]^3)/(4*a^3*d) - (15*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - Cos[c + d*x]^7/(d*(a + a*Sin[c + d*x])^3) - (3*Cos[c + d*x]^5)/(4*d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 7, -((7*x)/(2*a^3)) - ArcTanh[Cos[c + d*x]]/(a^3*d) - (3*Cos[c + d*x])/(a^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 7, (3*x)/a^3 + (3*ArcTanh[Cos[c + d*x]])/(a^3*d) + Cos[c + d*x]/(a^3*d) - Cot[c + d*x]/(a^3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 8, -(x/a^3) - (7*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 10, (5*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (4*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 12, -((15*ArcTanh[Cos[c + d*x]])/(8*a^3*d)) + (4*Cot[c + d*x])/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (15*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 12, (13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) - (4*Cot[c + d*x])/(a^3*d) - (5*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (13*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2) with n symbolic*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3, x, 6, (a^3*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2]) + (a^3*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(4 + n))/(d*(4 + n)*Sqrt[Cos[c + d*x]^2])} -{Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2, x, 5, (a^2*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (a^2*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2])} -{Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x]), x, 3, (a*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (a*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^7 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^7*Sin[c + d*x]^6*(a + a*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^8)/(8*d) - (a*Sin[c + d*x]^9)/(3*d) - (3*a*Sin[c + d*x]^10)/(10*d) + (3*a*Sin[c + d*x]^11)/(11*d) + (a*Sin[c + d*x]^12)/(4*d) - (a*Sin[c + d*x]^13)/(13*d) - (a*Sin[c + d*x]^14)/(14*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^5*(a + a*Sin[c + d*x]), x, 8, -((a*Cos[c + d*x]^8)/(8*d)) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(3*d) + (3*a*Sin[c + d*x]^11)/(11*d) - (a*Sin[c + d*x]^13)/(13*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 8, -((a*Cos[c + d*x]^8)/(8*d)) + (a*Cos[c + d*x]^10)/(5*d) - (a*Cos[c + d*x]^12)/(12*d) + (a*Sin[c + d*x]^5)/(5*d) - (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 7, -((a*Cos[c + d*x]^8)/(8*d)) + (a*Cos[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^5)/(5*d) - (3*a*Sin[c + d*x]^7)/(7*d) + (a*Sin[c + d*x]^9)/(3*d) - (a*Sin[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, -((a*Cos[c + d*x]^8)/(8*d)) + (a*Cos[c + d*x]^10)/(10*d) + (a*Sin[c + d*x]^3)/(3*d) - (3*a*Sin[c + d*x]^5)/(5*d) + (3*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, -((a*Cos[c + d*x]^8)/(8*d)) + (a*Sin[c + d*x]^3)/(3*d) - (3*a*Sin[c + d*x]^5)/(5*d) + (3*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 4, (a*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^4)/(4*d) + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d) - (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (a*Log[Sin[c + d*x]])/d - (3*a*Sin[c + d*x])/d - (3*a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (3*a*Log[Sin[c + d*x]])/d - (3*a*Sin[c + d*x])/d + (3*a*Sin[c + d*x]^2)/(2*d) + (a*Sin[c + d*x]^3)/d - (a*Sin[c + d*x]^4)/(4*d) - (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, (3*a*Csc[c + d*x])/d - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (3*a*Log[Sin[c + d*x]])/d + (3*a*Sin[c + d*x])/d + (3*a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d) - (a*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^5*(a + a*Sin[c + d*x]), x, 4, (3*a*Csc[c + d*x])/d + (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (3*a*Log[Sin[c + d*x]])/d + (3*a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^6*(a + a*Sin[c + d*x]), x, 4, -((3*a*Csc[c + d*x])/d) + (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d - (a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (3*a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^7*(a + a*Sin[c + d*x]), x, 3, -((3*a*Csc[c + d*x])/d) - (3*a*Csc[c + d*x]^2)/(2*d) + (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Log[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d} -{Cos[c + d*x]^7*Csc[c + d*x]^8*(a + a*Sin[c + d*x]), x, 4, (a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]^7*Csc[c + d*x]^9*(a + a*Sin[c + d*x]), x, 6, -((a*Cot[c + d*x]^8)/(8*d)) + (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^10*(a + a*Sin[c + d*x]), x, 6, -((a*Cot[c + d*x]^8)/(8*d)) + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^11*(a + a*Sin[c + d*x]), x, 7, -((a*Cot[c + d*x]^8)/(8*d)) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^3)/(3*d) - (3*a*Csc[c + d*x]^5)/(5*d) + (3*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^12*(a + a*Sin[c + d*x]), x, 7, -((a*Cot[c + d*x]^8)/(8*d)) - (a*Cot[c + d*x]^10)/(10*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^13*(a + a*Sin[c + d*x]), x, 8, -((a*Cot[c + d*x]^8)/(8*d)) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^5)/(5*d) - (3*a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^9)/(3*d) - (a*Csc[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^14*(a + a*Sin[c + d*x]), x, 8, -((a*Cot[c + d*x]^8)/(8*d)) - (a*Cot[c + d*x]^10)/(5*d) - (a*Cot[c + d*x]^12)/(12*d) + (a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(3*d) + (3*a*Csc[c + d*x]^11)/(11*d) - (a*Csc[c + d*x]^13)/(13*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^15*(a + a*Sin[c + d*x]), x, 4, (a*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^9)/(3*d) - (3*a*Csc[c + d*x]^10)/(10*d) + (3*a*Csc[c + d*x]^11)/(11*d) + (a*Csc[c + d*x]^12)/(4*d) - (a*Csc[c + d*x]^13)/(13*d) - (a*Csc[c + d*x]^14)/(14*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^7*Sin[c + d*x]^6/(a + a*Sin[c + d*x]), x, 4, Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(8*a*d) - (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c + d*x]^10/(5*a*d) + Sin[c + d*x]^11/(11*a*d) - Sin[c + d*x]^12/(12*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^5/(a + a*Sin[c + d*x]), x, 4, Sin[c + d*x]^6/(6*a*d) - Sin[c + d*x]^7/(7*a*d) - Sin[c + d*x]^8/(4*a*d) + (2*Sin[c + d*x]^9)/(9*a*d) + Sin[c + d*x]^10/(10*a*d) - Sin[c + d*x]^11/(11*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) - (2*Sin[c + d*x]^7)/(7*a*d) + Sin[c + d*x]^8/(4*a*d) + Sin[c + d*x]^9/(9*a*d) - Sin[c + d*x]^10/(10*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 7, -(Cos[c + d*x]^6/(6*a*d)) + Cos[c + d*x]^8/(8*a*d) - Sin[c + d*x]^5/(5*a*d) + (2*Sin[c + d*x]^7)/(7*a*d) - Sin[c + d*x]^9/(9*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 7, Cos[c + d*x]^6/(6*a*d) - Cos[c + d*x]^8/(8*a*d) + Sin[c + d*x]^3/(3*a*d) - (2*Sin[c + d*x]^5)/(5*a*d) + Sin[c + d*x]^7/(7*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 6, -(Cos[c + d*x]^6/(6*a*d)) - Sin[c + d*x]^3/(3*a*d) + (2*Sin[c + d*x]^5)/(5*a*d) - Sin[c + d*x]^7/(7*a*d)} -{Cos[c + d*x]^7*Sin[c + d*x]^0/(a + a*Sin[c + d*x]), x, 3, -((a - a*Sin[c + d*x])^4/(a^5*d)) + (4*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - (a - a*Sin[c + d*x])^6/(6*a^7*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(a*d) + (2*Sin[c + d*x]^3)/(3*a*d) + Sin[c + d*x]^4/(4*a*d) - Sin[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Log[Sin[c + d*x]]/(a*d) - (2*Sin[c + d*x])/(a*d) + Sin[c + d*x]^2/(a*d) + Sin[c + d*x]^3/(3*a*d) - Sin[c + d*x]^4/(4*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (2*Log[Sin[c + d*x]])/(a*d) + (2*Sin[c + d*x])/(a*d) + Sin[c + d*x]^2/(2*a*d) - Sin[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, (2*Csc[c + d*x])/(a*d) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Log[Sin[c + d*x]])/(a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 4, -((2*Csc[c + d*x])/(a*d)) + Csc[c + d*x]^2/(a*d) + Csc[c + d*x]^3/(3*a*d) - Csc[c + d*x]^4/(4*a*d) + Log[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Csc[c + d*x]^2/(a*d) + (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^4/(4*a*d) - Csc[c + d*x]^5/(5*a*d) - Log[Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^7/(a + a*Sin[c + d*x]), x, 6, -(Cot[c + d*x]^6/(6*a*d)) + Csc[c + d*x]/(a*d) - (2*Csc[c + d*x]^3)/(3*a*d) + Csc[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^8/(a + a*Sin[c + d*x]), x, 6, Cot[c + d*x]^6/(6*a*d) - Csc[c + d*x]^3/(3*a*d) + (2*Csc[c + d*x]^5)/(5*a*d) - Csc[c + d*x]^7/(7*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^9/(a + a*Sin[c + d*x]), x, 7, -(Cot[c + d*x]^6/(6*a*d)) - Cot[c + d*x]^8/(8*a*d) + Csc[c + d*x]^3/(3*a*d) - (2*Csc[c + d*x]^5)/(5*a*d) + Csc[c + d*x]^7/(7*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^10/(a + a*Sin[c + d*x]), x, 7, Cot[c + d*x]^6/(6*a*d) + Cot[c + d*x]^8/(8*a*d) - Csc[c + d*x]^5/(5*a*d) + (2*Csc[c + d*x]^7)/(7*a*d) - Csc[c + d*x]^9/(9*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^11/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*d) - (2*Csc[c + d*x]^7)/(7*a*d) + Csc[c + d*x]^8/(4*a*d) + Csc[c + d*x]^9/(9*a*d) - Csc[c + d*x]^10/(10*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^12/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]^6/(6*a*d) - Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(4*a*d) + (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c + d*x]^10/(10*a*d) - Csc[c + d*x]^11/(11*a*d)} -{Cos[c + d*x]^7*Csc[c + d*x]^13/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]^7/(7*a*d) - Csc[c + d*x]^8/(8*a*d) - (2*Csc[c + d*x]^9)/(9*a*d) + Csc[c + d*x]^10/(5*a*d) + Csc[c + d*x]^11/(11*a*d) - Csc[c + d*x]^12/(12*a*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m win n symbolic*) - - -{Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3, x, 3, (a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (8*a^3*Sin[c + d*x]^(4 + n))/(d*(4 + n)) - (6*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (6*a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (8*a^3*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (3*a^3*Sin[c + d*x]^(9 + n))/(d*(9 + n)) - (a^3*Sin[c + d*x]^(10 + n))/(d*(10 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2, x, 3, (a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a^2*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a^2*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (6*a^2*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (6*a^2*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (2*a^2*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (2*a^2*Sin[c + d*x]^(8 + n))/(d*(8 + n)) - (a^2*Sin[c + d*x]^(9 + n))/(d*(9 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^1, x, 3, (a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (3*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (3*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (3*a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (3*a*Sin[c + d*x]^(6 + n))/(d*(6 + n)) - (a*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (a*Sin[c + d*x]^(8 + n))/(d*(8 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^1, x, 3, Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - (2*Sin[c + d*x]^(3 + n))/(a*d*(3 + n)) + (2*Sin[c + d*x]^(4 + n))/(a*d*(4 + n)) + Sin[c + d*x]^(5 + n)/(a*d*(5 + n)) - Sin[c + d*x]^(6 + n)/(a*d*(6 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^2, x, 3, Sin[c + d*x]^(1 + n)/(a^2*d*(1 + n)) - (2*Sin[c + d*x]^(2 + n))/(a^2*d*(2 + n)) + (2*Sin[c + d*x]^(4 + n))/(a^2*d*(4 + n)) - Sin[c + d*x]^(5 + n)/(a^2*d*(5 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^3, x, 3, Sin[c + d*x]^(1 + n)/(a^3*d*(1 + n)) - (3*Sin[c + d*x]^(2 + n))/(a^3*d*(2 + n)) + (3*Sin[c + d*x]^(3 + n))/(a^3*d*(3 + n)) - Sin[c + d*x]^(4 + n)/(a^3*d*(4 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^4, x, 8, -((7*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n))) + (8*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^4*d*(1 + n)) + (4*Sin[c + d*x]^(2 + n))/(a^4*d*(2 + n)) - Sin[c + d*x]^(3 + n)/(a^4*d*(3 + n))} -{Cos[c + d*x]^7*Sin[c + d*x]^n/(a + a*Sin[c + d*x])^5, x, 4, -((4*(3 + 2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^5*d*(1 + n))) - (Sin[c + d*x]^(1 + n)*(a - a*Sin[c + d*x])^2)/(d*(2 + n)*(a^7 + a^7*Sin[c + d*x])) + (Sin[c + d*x]^(1 + n)*(a*(27 + 30*n + 8*n^2) + a*(7 + 2*n)*Sin[c + d*x]))/(d*(2 + 3*n + n^2)*(a^6 + a^6*Sin[c + d*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^8 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^8 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^8*Sin[c + d*x]^5/(a + a*Sin[c + d*x]), x, 11, (-5*x)/(1024*a) - Cos[c + d*x]^7/(7*a*d) + (2*Cos[c + d*x]^9)/(9*a*d) - Cos[c + d*x]^11/(11*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(1024*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(1536*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(384*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(64*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(24*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*a*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 10, (3*x)/(256*a) + Cos[c + d*x]^7/(7*a*d) - (2*Cos[c + d*x]^9)/(9*a*d) + Cos[c + d*x]^11/(11*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) - (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) - (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 10, (-3*x)/(256*a) - Cos[c + d*x]^7/(7*a*d) + Cos[c + d*x]^9/(9*a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(256*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(128*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(160*a*d) + (3*Cos[c + d*x]^7*Sin[c + d*x])/(80*a*d) + (Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*a*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 9, (5*x)/(128*a) + Cos[c + d*x]^7/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(192*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) - (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 8, (-5*x)/(128*a) - Cos[c + d*x]^7/(7*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(192*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(48*a*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 9, (-5*x)/(16*a) - ArcTanh[Cos[c + d*x]]/(a*d) + Cos[c + d*x]/(a*d) + Cos[c + d*x]^3/(3*a*d) + Cos[c + d*x]^5/(5*a*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(16*a*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 10, (-15*x)/(8*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) - (15*Cot[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) + (Cos[c + d*x]^4*Cot[c + d*x])/(4*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 11, (15*x)/(8*a) + (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (5*Cos[c + d*x])/(2*a*d) - (5*Cos[c + d*x]^3)/(6*a*d) + (15*Cot[c + d*x])/(8*a*d) - (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) - (Cos[c + d*x]^4*Cot[c + d*x])/(4*a*d) - (Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 11, (5*x)/(2*a) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) + (5*Cos[c + d*x])/(2*a*d) + (5*Cos[c + d*x]^3)/(6*a*d) + (5*Cot[c + d*x])/(2*a*d) + (Cos[c + d*x]^3*Cot[c + d*x]^2)/(2*a*d) - (5*Cot[c + d*x]^3)/(6*a*d) + (Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^5/(a + a*Sin[c + d*x]), x, 11, (-5*x)/(2*a) - (15*ArcTanh[Cos[c + d*x]])/(8*a*d) + (15*Cos[c + d*x])/(8*a*d) - (5*Cot[c + d*x])/(2*a*d) + (5*Cos[c + d*x]*Cot[c + d*x]^2)/(8*a*d) + (5*Cot[c + d*x]^3)/(6*a*d) - (Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*a*d) - (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^6/(a + a*Sin[c + d*x]), x, 10, -(x/a) + (15*ArcTanh[Cos[c + d*x]])/(8*a*d) - (15*Cos[c + d*x])/(8*a*d) - Cot[c + d*x]/(a*d) - (5*Cos[c + d*x]*Cot[c + d*x]^2)/(8*a*d) + Cot[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Cot[c + d*x]^4)/(4*a*d) - Cot[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^7/(a + a*Sin[c + d*x]), x, 9, x/a + (5*ArcTanh[Cos[c + d*x]])/(16*a*d) + Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) - (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^8/(a + a*Sin[c + d*x]), x, 7, (-5*ArcTanh[Cos[c + d*x]])/(16*a*d) - Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x])/(24*a*d) + (Cot[c + d*x]^5*Csc[c + d*x])/(6*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^9/(a + a*Sin[c + d*x]), x, 8, (5*ArcTanh[Cos[c + d*x]])/(128*a*d) + Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) - (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) - (Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^10/(a + a*Sin[c + d*x]), x, 9, -((5*ArcTanh[Cos[c + d*x]])/(128*a*d)) - Cot[c + d*x]^7/(7*a*d) - Cot[c + d*x]^9/(9*a*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) + (5*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a*d) - (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) + (Cot[c + d*x]^5*Csc[c + d*x]^3)/(8*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^11/(a + a*Sin[c + d*x]), x, 10, (3*ArcTanh[Cos[c + d*x]])/(256*a*d) + Cot[c + d*x]^7/(7*a*d) + Cot[c + d*x]^9/(9*a*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(256*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(32*a*d) + (Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*a*d) - (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*a*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^12/(a + a*Sin[c + d*x]), x, 10, -((3*ArcTanh[Cos[c + d*x]])/(256*a*d)) - Cot[c + d*x]^7/(7*a*d) - (2*Cot[c + d*x]^9)/(9*a*d) - Cot[c + d*x]^11/(11*a*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(256*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(128*a*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(32*a*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(16*a*d) + (Cot[c + d*x]^5*Csc[c + d*x]^5)/(10*a*d)} - - -{Cos[c + d*x]^8*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 15, (-3*x)/(128*a^2) - (2*Cos[c + d*x]^5)/(5*a^2*d) + (5*Cos[c + d*x]^7)/(7*a^2*d) - (4*Cos[c + d*x]^9)/(9*a^2*d) + Cos[c + d*x]^11/(11*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(64*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(16*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^5)/(5*a^2*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 17, (9*x)/(256*a^2) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (4*Cos[c + d*x]^7)/(7*a^2*d) + (2*Cos[c + d*x]^9)/(9*a^2*d) + (9*Cos[c + d*x]*Sin[c + d*x])/(256*a^2*d) + (3*Cos[c + d*x]^3*Sin[c + d*x])/(128*a^2*d) - (3*Cos[c + d*x]^5*Sin[c + d*x])/(32*a^2*d) - (3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*a^2*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 14, (-3*x)/(64*a^2) - (2*Cos[c + d*x]^5)/(5*a^2*d) + (3*Cos[c + d*x]^7)/(7*a^2*d) - Cos[c + d*x]^9/(9*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(64*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(32*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 15, (11*x)/(128*a^2) + (2*Cos[c + d*x]^5)/(5*a^2*d) - (2*Cos[c + d*x]^7)/(7*a^2*d) + (11*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) + (11*Cos[c + d*x]^3*Sin[c + d*x])/(192*a^2*d) - (11*Cos[c + d*x]^5*Sin[c + d*x])/(48*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 6, -x/(8*a^2) - (2*Cos[c + d*x]^7)/(35*a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(12*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(15*a^2*d) - Cos[c + d*x]^9/(5*d*(a + a*Sin[c + d*x])^2)} -{Cos[c + d*x]^8*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 12, (-3*x)/(4*a^2) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - Cos[c + d*x]^5/(5*a^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(4*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 14, (-9*x)/(8*a^2) + (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*Cos[c + d*x])/(a^2*d) - (2*Cos[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 13, (3*x)/a^2 + ArcTanh[Cos[c + d*x]]/(2*a^2*d) + Cos[c + d*x]^3/(3*a^2*d) + (2*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 13, -x/(2*a^2) - (3*ArcTanh[Cos[c + d*x]])/(a^2*d) + (2*Cos[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 14, (-2*x)/a^2 + (9*ArcTanh[Cos[c + d*x]])/(8*a^2*d) - Cos[c + d*x]/(a^2*d) - (2*Cot[c + d*x])/(a^2*d) + (2*Cot[c + d*x]^3)/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 11, x/a^2 + (3*ArcTanh[Cos[c + d*x]])/(4*a^2*d) + Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 12, -((7*ArcTanh[Cos[c + d*x]])/(16*a^2*d)) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^8/(a + a*Sin[c + d*x])^2, x, 19, ArcTanh[Cos[c + d*x]]/(8*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (7*Cot[c + d*x]*Csc[c + d*x]^3)/(12*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(3*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^9/(a + a*Sin[c + d*x])^2, x, 15, -((11*ArcTanh[Cos[c + d*x]])/(128*a^2*d)) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (2*Cot[c + d*x]^7)/(7*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (7*Cot[c + d*x]*Csc[c + d*x]^3)/(64*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(16*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^10/(a + a*Sin[c + d*x])^2, x, 14, (3*ArcTanh[Cos[c + d*x]])/(64*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (3*Cot[c + d*x]^7)/(7*a^2*d) - Cot[c + d*x]^9/(9*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(32*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(8*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^5)/(4*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^11/(a + a*Sin[c + d*x])^2, x, 17, -((9*ArcTanh[Cos[c + d*x]])/(256*a^2*d)) + (2*Cot[c + d*x]^5)/(5*a^2*d) + (4*Cot[c + d*x]^7)/(7*a^2*d) + (2*Cot[c + d*x]^9)/(9*a^2*d) - (9*Cot[c + d*x]*Csc[c + d*x])/(256*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^3)/(128*a^2*d) + (9*Cot[c + d*x]*Csc[c + d*x]^5)/(160*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x]^7)/(80*a^2*d) - (Cot[c + d*x]^3*Csc[c + d*x]^7)/(10*a^2*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^12/(a + a*Sin[c + d*x])^2, x, 15, (3*ArcTanh[Cos[c + d*x]])/(128*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (4*Cot[c + d*x]^9)/(9*a^2*d) - Cot[c + d*x]^11/(11*a^2*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(128*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(64*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(80*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x]^7)/(40*a^2*d) + (Cot[c + d*x]^3*Csc[c + d*x]^7)/(5*a^2*d)} - - -{Cos[c + d*x]^8*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 18, -((29*x)/(128*a^3)) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (7*Cos[c + d*x]^5)/(5*a^3*d) - (3*Cos[c + d*x]^7)/(7*a^3*d) - (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^3*d) + (29*Cos[c + d*x]^3*Sin[c + d*x]^3)/(48*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a^3*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 16, (5*x)/(16*a^3) + (4*Cos[c + d*x]^3)/(3*a^3*d) - Cos[c + d*x]^5/(a^3*d) + Cos[c + d*x]^7/(7*a^3*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(8*a^3*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(2*a^3*d)} -{Cos[c + d*x]^8*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 6, -((7*x)/(16*a^3)) - (7*Cos[c + d*x]^5)/(30*a^3*d) - (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (7*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) - Cos[c + d*x]^9/(3*d*(a + a*Sin[c + d*x])^3) - Cos[c + d*x]^7/(6*d*(a^3 + a^3*Sin[c + d*x]))} -{Cos[c + d*x]^8*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 13, -((13*x)/(8*a^3)) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(a^3*d) - Cos[c + d*x]^3/(a^3*d) - (13*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 11, x/(2*a^3) + (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - (3*Cos[c + d*x])/(a^3*d) + Cos[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]/(a^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 11, (5*x)/(2*a^3) - (5*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (3*Cos[c + d*x])/(a^3*d) + (3*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 11, -((3*x)/a^3) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) - Cos[c + d*x]/(a^3*d) - (3*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 12, x/a^3 + (13*ArcTanh[Cos[c + d*x]])/(8*a^3*d) + Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(a^3*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 13, -((7*ArcTanh[Cos[c + d*x]])/(8*a^3*d)) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^7/(a + a*Sin[c + d*x])^3, x, 15, (7*ArcTanh[Cos[c + d*x]])/(16*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (3*Cot[c + d*x]^5)/(5*a^3*d) + (7*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) - (17*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^8/(a + a*Sin[c + d*x])^3, x, 17, -((5*ArcTanh[Cos[c + d*x]])/(16*a^3*d)) - (4*Cot[c + d*x]^3)/(3*a^3*d) - Cot[c + d*x]^5/(a^3*d) - Cot[c + d*x]^7/(7*a^3*d) - (5*Cot[c + d*x]*Csc[c + d*x])/(16*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^3*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(2*a^3*d)} -{Cos[c + d*x]^8*Csc[c + d*x]^9/(a + a*Sin[c + d*x])^3, x, 18, (29*ArcTanh[Cos[c + d*x]])/(128*a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (7*Cot[c + d*x]^5)/(5*a^3*d) + (3*Cot[c + d*x]^7)/(7*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x])/(128*a^3*d) + (29*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^3*d) - (23*Cot[c + d*x]*Csc[c + d*x]^5)/(48*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*a^3*d)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^1 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^2*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 8, -((3*a*x)/2) + (2*a*Cos[c + d*x])/d - (a*Cos[c + d*x]^3)/(3*d) + (a*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 8, -((3*a*x)/2) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 5, (-a)*x + (a*Cos[c + d*x])/d + (a*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 5, (-a)*x + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 6, -((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 8, -((3*a*ArcTanh[Cos[c + d*x]])/(2*d)) - (a*Cot[c + d*x])/d + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 8, -((3*a*ArcTanh[Cos[c + d*x]])/(2*d)) - (2*a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a*Tan[c + d*x])/d} - - -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 12, -3*a^2*x + (3*a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]^3)/(3*d) + (2*a^2*Sec[c + d*x])/d + (3*a^2*Tan[c + d*x])/d - (a^2*Sin[c + d*x]^2*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 6, -((5*a^2*x)/2) + (2*a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 3, -2*a^2*x + (2*a^2*Cos[c + d*x])/d + (Sec[c + d*x]*(a + a*Sin[c + d*x])^2)/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 9, -((a^2*ArcTanh[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d + (2*a^2*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 10, -((2*a^2*ArcTanh[Cos[c + d*x]])/d) - (a^2*Cot[c + d*x])/d + (2*a^2*Sec[c + d*x])/d + (2*a^2*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 12, -((5*a^2*ArcTanh[Cos[c + d*x]])/(2*d)) - (2*a^2*Cot[c + d*x])/d + (5*a^2*Sec[c + d*x])/(2*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (2*a^2*Tan[c + d*x])/d} - - -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 11, -((51*a^3*x)/8) + (7*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (19*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 8, -((11*a^3*x)/2) + (5*a^3*Cos[c + d*x])/d - (a^3*Cos[c + d*x]^3)/(3*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 2, -((9*a^3*x)/2) + (6*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + a*Sin[c + d*x])^3)/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, (-a^3)*x - (a^3*ArcTanh[Cos[c + d*x]])/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 6, -((3*a^3*ArcTanh[Cos[c + d*x]])/d) - (a^3*Cot[c + d*x])/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 8, -((9*a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 10, -((11*a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (5*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 7, x/a + Cos[c + d*x]/(a*d) + (2*Sec[c + d*x])/(a*d) - Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, -(x/a) - Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 5, Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 5, Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 7, -(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) - Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 8, ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) + (2*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(3*a*d)} - - -{Sec[c + d*x]^2*Sin[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 15, -((9*x)/(2*a^2)) - (2*Cos[c + d*x])/(a^2*d) - (6*Sec[c + d*x])/(a^2*d) + (2*Sec[c + d*x]^3)/(a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + (9*Tan[c + d*x])/(2*a^2*d) - (3*Tan[c + d*x]^3)/(2*a^2*d) + (9*Tan[c + d*x]^5)/(10*a^2*d) - (Sin[c + d*x]^2*Tan[c + d*x]^5)/(2*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 13, (2*x)/a^2 + Cos[c + d*x]/(a^2*d) + (4*Sec[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^3)/(3*a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x])/(a^2*d) + (2*Tan[c + d*x]^3)/(3*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 12, -(x/a^2) - (2*Sec[c + d*x])/(a^2*d) + (4*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Tan[c + d*x]/(a^2*d) - Tan[c + d*x]^3/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 11, Sec[c + d*x]/(a^2*d) - Sec[c + d*x]^3/(a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 10, (2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, Sec[c + d*x]/(5*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x])/(15*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*x])/(15*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 11, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d) + (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x])/(a^2*d) - (4*Tan[c + d*x]^3)/(3*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 12, (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + (4*Tan[c + d*x])/(a^2*d) + (5*Tan[c + d*x]^3)/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 15, -((9*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) + (9*Sec[c + d*x])/(2*a^2*d) + (3*Sec[c + d*x]^3)/(2*a^2*d) + (9*Sec[c + d*x]^5)/(10*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^5)/(2*a^2*d) - (6*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)} - - -{Sec[c + d*x]^2*Sin[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 16, (3*x)/a^3 + Cos[c + d*x]/(a^3*d) + (7*Sec[c + d*x])/(a^3*d) - (5*Sec[c + d*x]^3)/(a^3*d) + (13*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x])/(a^3*d) + Tan[c + d*x]^3/(a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 16, -(x/a^3) - (3*Sec[c + d*x])/(a^3*d) + (10*Sec[c + d*x]^3)/(3*a^3*d) - (11*Sec[c + d*x]^5)/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]/(a^3*d) - Tan[c + d*x]^3/(3*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 14, Sec[c + d*x]/(a^3*d) - (2*Sec[c + d*x]^3)/(a^3*d) + (9*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 14, Sec[c + d*x]^3/(a^3*d) - (7*Sec[c + d*x]^5)/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 14, -(Sec[c + d*x]^3/(3*a^3*d)) + Sec[c + d*x]^5/(a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) + Tan[c + d*x]^5/(a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 5, Sec[c + d*x]/(7*d*(a + a*Sin[c + d*x])^3) - (3*Sec[c + d*x])/(35*a*d*(a + a*Sin[c + d*x])^2) - (3*Sec[c + d*x])/(35*d*(a^3 + a^3*Sin[c + d*x])) + (6*Tan[c + d*x])/(35*a^3*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 14, -(ArcTanh[Cos[c + d*x]]/(a^3*d)) + Sec[c + d*x]/(a^3*d) + Sec[c + d*x]^3/(3*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) + (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x])/(a^3*d) - (10*Tan[c + d*x]^3)/(3*a^3*d) - (11*Tan[c + d*x]^5)/(5*a^3*d) - (4*Tan[c + d*x]^7)/(7*a^3*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 14, (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^3/(a^3*d) - (3*Sec[c + d*x]^5)/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) + (7*Tan[c + d*x])/(a^3*d) + (5*Tan[c + d*x]^3)/(a^3*d) + (13*Tan[c + d*x]^5)/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -(* {Sec[c + d*x]^2*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 7, -((76*a*Cos[c + d*x])/(495*d*Sqrt[a + a*Sin[c + d*x]])) - (38*a*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (152*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3465*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (76*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(1155*a*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 6, -((2*a*Cos[c + d*x])/(9*d*Sqrt[a + a*Sin[c + d*x]])) + (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(63*d) + (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(9*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(21*a*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 5, -((16*a*Cos[c + d*x])/(105*d*Sqrt[a + a*Sin[c + d*x]])) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) - (18*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 4, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 4, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (3*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 4, (5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 5, (3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (3*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)} *) - - -(* {Sec[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 8, -((1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]])) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(13*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 7, -((62*a^2*Cos[c + d*x])/(165*d*Sqrt[a + a*Sin[c + d*x]])) - (10*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(231*d*Sqrt[a + a*Sin[c + d*x]]) + (124*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(1155*d) + (2*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(33*d) - (62*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(385*d) + (2*Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(11*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 6, -((128*a^2*Cos[c + d*x])/(315*d*Sqrt[a + a*Sin[c + d*x]])) - (32*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d) - (22*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(63*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2))/(9*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 5, -((2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (2*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 5, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (11*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 5, (a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (13*a^2*Cos[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 5, (13*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (5*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)} *) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -(* {Sec[c + d*x]^2*Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 6, -((4*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]])) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 5, -((2*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]])) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 4, (4*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (14*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 3, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d)) + (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2*Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 3, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2*Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 4, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(4*Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2*Csc[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 5, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*Sqrt[a]*d) + Cot[c + d*x]/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} *) - - -(* {Sec[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 6, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (344*Cos[c + d*x])/(105*a*d*Sqrt[a + a*Sin[c + d*x]]) - (16*Cos[c + d*x]*Sin[c + d*x]^2)/(35*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*a*d*Sqrt[a + a*Sin[c + d*x]]) + (76*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, -((2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)) + (16*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 4, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, (3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - Cot[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 6, -((11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (5*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])} -{Sec[c + d*x]^2*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 7, (23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (9*Cot[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + a*Sin[c + d*x]])} *) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^3 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^4*Sin[c + d*x]^6*(a + a*Sin[c + d*x]), x, 9, (5*a*x)/2 - (3*a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (5*a*Tan[c + d*x])/(2*d) + (5*a*Tan[c + d*x]^3)/(6*d) - (a*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^5*(a + a*Sin[c + d*x]), x, 9, (5*a*x)/2 - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (5*a*Tan[c + d*x])/(2*d) + (5*a*Tan[c + d*x]^3)/(6*d) - (a*Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 8, a*x - (a*Cos[c + d*x])/d - (2*a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 6, a*x - (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 5, -((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 5, (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 8, -((a*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) + (2*a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 9, -((5*a*ArcTanh[Cos[c + d*x]])/(2*d)) - (a*Cot[c + d*x])/d + (5*a*Sec[c + d*x])/(2*d) + (5*a*Sec[c + d*x]^3)/(6*d) - (a*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d) + (2*a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} - - -{Sec[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 4, (7*a^2*x)/2 - (2*a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (11*a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), (7*a^2*x)/2 - (16*a^2*Cos[c + d*x])/(3*d) - (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (8*a^2*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 7, 2*a^2*x - (4*a^2*Cos[c + d*x])/(3*d) - (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c + d*x]^2)/(3*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, a^2*x - (5*a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 4, -((2*a^2*Sec[c + d*x])/(3*d)) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(3*d) - (2*a^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 5, -((a^2*ArcTanh[Cos[c + d*x]])/d) + (4*a^2*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 7, -((2*a^2*ArcTanh[Cos[c + d*x]])/d) - (10*a^2*Cot[c + d*x])/(3*d) + (2*a^2*Cot[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^4*Cot[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 8, -((7*a^2*ArcTanh[Cos[c + d*x]])/(2*d)) - (16*a^2*Cot[c + d*x])/(3*d) - (7*a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (8*a^2*Cot[c + d*x]*Csc[c + d*x])/(3*d*(1 - Sin[c + d*x])) + (a^4*Cot[c + d*x]*Csc[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2)} - - -{Sec[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 10, (17*a^3*x)/2 - (6*a^3*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (25*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 8, (11*a^3*x)/2 - (3*a^3*Cos[c + d*x])/d + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (19*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 5, 3*a^3*x - (3*a^3*Cos[c + d*x])/d - (2*a^5*Cos[c + d*x]^3)/(d*(a - a*Sin[c + d*x])^2) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, a^3*x + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d) - (2*a^5*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))} -{Sec[c + d*x]^4*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 6, -((a^3*ArcTanh[Cos[c + d*x]])/d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (5*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^4*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 8, -((3*a^3*ArcTanh[Cos[c + d*x]])/d) - (a^3*Cot[c + d*x])/d + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (11*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 10, -((11*a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (17*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))} -{Sec[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 12, -((17*a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (6*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (23*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x]))} - - -{Sec[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^4, x, 13, (163*a^4*x)/8 - (16*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x]^3)/(3*d) + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (56*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^4*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4, x, 8, (17*a^4*x)/2 - (4*a^4*Cos[c + d*x])/d + (4*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (32*a^4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*Sin[c + d*x]^6/(a + a*Sin[c + d*x]), x, 8, -(x/a) - Cos[c + d*x]/(a*d) - (3*Sec[c + d*x])/(a*d) + Sec[c + d*x]^3/(a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]/(a*d) - Tan[c + d*x]^3/(3*a*d) + Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^5/(a + a*Sin[c + d*x]), x, 8, x/a + Sec[c + d*x]/(a*d) - (2*Sec[c + d*x]^3)/(3*a*d) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d) - Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 6, -(Sec[c + d*x]/(a*d)) + (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 6, -(Sec[c + d*x]^3/(3*a*d)) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 7, Sec[c + d*x]^3/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + Tan[c + d*x]^3/(3*a*d) + Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 6, Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]^3/(3*a*d) - Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 7, -(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + Sec[c + d*x]^5/(5*a*d) - Tan[c + d*x]/(a*d) - (2*Tan[c + d*x]^3)/(3*a*d) - Tan[c + d*x]^5/(5*a*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 8, ArcTanh[Cos[c + d*x]]/(a*d) - Cot[c + d*x]/(a*d) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^3/(3*a*d) - Sec[c + d*x]^5/(5*a*d) + (3*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(a*d) + Tan[c + d*x]^5/(5*a*d)} - - -{Sec[c + d*x]^4*Sin[c + d*x]^7/(a + a*Sin[c + d*x])^2, x, 14, -((2*x)/a^2) - Cos[c + d*x]/(a^2*d) - (5*Sec[c + d*x])/(a^2*d) + (3*Sec[c + d*x]^3)/(a^2*d) - (7*Sec[c + d*x]^5)/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) + (2*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(3*a^2*d) + (2*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^6/(a + a*Sin[c + d*x])^2, x, 13, x/a^2 + (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(a^2*d) + (6*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) - Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d) - Tan[c + d*x]^5/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^2, x, 11, -(Sec[c + d*x]/(a^2*d)) + (4*Sec[c + d*x]^3)/(3*a^2*d) - Sec[c + d*x]^5/(a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^2, x, 10, -((2*Sec[c + d*x]^3)/(3*a^2*d)) + (4*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + Tan[c + d*x]^5/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 12, Sec[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x]^5)/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 12, (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x]^5)/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 4, Sec[c + d*x]^3/(7*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x]^3)/(35*d*(a^2 + a^2*Sin[c + d*x])) + (8*Tan[c + d*x])/(35*a^2*d) + (8*Tan[c + d*x]^3)/(105*a^2*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^2, x, 11, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d) + Sec[c + d*x]^5/(5*a^2*d) + (2*Sec[c + d*x]^7)/(7*a^2*d) - (2*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^3)/(a^2*d) - (6*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^2, x, 12, (2*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a^2*d) - (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) - (2*Sec[c + d*x]^7)/(7*a^2*d) + (5*Tan[c + d*x])/(a^2*d) + (3*Tan[c + d*x]^3)/(a^2*d) + (7*Tan[c + d*x]^5)/(5*a^2*d) + (2*Tan[c + d*x]^7)/(7*a^2*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^2, x, 15, -((11*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) + (2*Cot[c + d*x])/(a^2*d) + (11*Sec[c + d*x])/(2*a^2*d) + (11*Sec[c + d*x]^3)/(6*a^2*d) + (11*Sec[c + d*x]^5)/(10*a^2*d) + (11*Sec[c + d*x]^7)/(14*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^7)/(2*a^2*d) - (8*Tan[c + d*x])/(a^2*d) - (4*Tan[c + d*x]^3)/(a^2*d) - (8*Tan[c + d*x]^5)/(5*a^2*d) - (2*Tan[c + d*x]^7)/(7*a^2*d)} - - -{Sec[c + d*x]^4*Sin[c + d*x]^7/(a + a*Sin[c + d*x])^3, x, 17, x/a^3 + (3*Sec[c + d*x])/(a^3*d) - (13*Sec[c + d*x]^3)/(3*a^3*d) + (21*Sec[c + d*x]^5)/(5*a^3*d) - (15*Sec[c + d*x]^7)/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - Tan[c + d*x]/(a^3*d) + Tan[c + d*x]^3/(3*a^3*d) - Tan[c + d*x]^5/(5*a^3*d) + Tan[c + d*x]^7/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^6/(a + a*Sin[c + d*x])^3, x, 14, -(Sec[c + d*x]/(a^3*d)) + (7*Sec[c + d*x]^3)/(3*a^3*d) - (3*Sec[c + d*x]^5)/(a^3*d) + (13*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^7/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^5/(a + a*Sin[c + d*x])^3, x, 14, -(Sec[c + d*x]^3/(a^3*d)) + (2*Sec[c + d*x]^5)/(a^3*d) - (11*Sec[c + d*x]^7)/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x]^7)/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^3, x, 14, Sec[c + d*x]^3/(3*a^3*d) - (6*Sec[c + d*x]^5)/(5*a^3*d) + (9*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^5/(5*a^3*d) + (5*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^3, x, 15, (3*Sec[c + d*x]^5)/(5*a^3*d) - Sec[c + d*x]^7/(a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d) - Tan[c + d*x]^7/(a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 15, -(Sec[c + d*x]^5/(5*a^3*d)) + (5*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) + (6*Tan[c + d*x]^5)/(5*a^3*d) + (9*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 5, Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3) - Sec[c + d*x]^3/(21*a*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/(21*d*(a^3 + a^3*Sin[c + d*x])) + (4*Tan[c + d*x])/(21*a^3*d) + (4*Tan[c + d*x]^3)/(63*a^3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^3, x, 14, -(ArcTanh[Cos[c + d*x]]/(a^3*d)) + Sec[c + d*x]/(a^3*d) + Sec[c + d*x]^3/(3*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) + Sec[c + d*x]^7/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - (3*Tan[c + d*x])/(a^3*d) - (13*Tan[c + d*x]^3)/(3*a^3*d) - (21*Tan[c + d*x]^5)/(5*a^3*d) - (15*Tan[c + d*x]^7)/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^3, x, 14, (3*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^3*d) - (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^3/(a^3*d) - (3*Sec[c + d*x]^5)/(5*a^3*d) - (3*Sec[c + d*x]^7)/(7*a^3*d) - (4*Sec[c + d*x]^9)/(9*a^3*d) + (8*Tan[c + d*x])/(a^3*d) + (22*Tan[c + d*x]^3)/(3*a^3*d) + (28*Tan[c + d*x]^5)/(5*a^3*d) + (17*Tan[c + d*x]^7)/(7*a^3*d) + (4*Tan[c + d*x]^9)/(9*a^3*d)} - - -{Sec[c + d*x]^4*Sin[c + d*x]^4/(a + a*Sin[c + d*x])^4, x, 17, (4*Sec[c + d*x]^5)/(5*a^4*d) - (16*Sec[c + d*x]^7)/(7*a^4*d) + (20*Sec[c + d*x]^9)/(9*a^4*d) - (8*Sec[c + d*x]^11)/(11*a^4*d) + Tan[c + d*x]^5/(5*a^4*d) + (9*Tan[c + d*x]^7)/(7*a^4*d) + (16*Tan[c + d*x]^9)/(9*a^4*d) + (8*Tan[c + d*x]^11)/(11*a^4*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^4, x, 18, -(Sec[c + d*x]^5/(5*a^4*d)) + (9*Sec[c + d*x]^7)/(7*a^4*d) - (16*Sec[c + d*x]^9)/(9*a^4*d) + (8*Sec[c + d*x]^11)/(11*a^4*d) - (4*Tan[c + d*x]^5)/(5*a^4*d) - (16*Tan[c + d*x]^7)/(7*a^4*d) - (20*Tan[c + d*x]^9)/(9*a^4*d) - (8*Tan[c + d*x]^11)/(11*a^4*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^4, x, 8, -((4*Sec[c + d*x]^7)/(7*a^4*d)) + (4*Sec[c + d*x]^9)/(3*a^4*d) - (8*Sec[c + d*x]^11)/(11*a^4*d) + Tan[c + d*x]^3/(3*a^4*d) + (2*Tan[c + d*x]^5)/(a^4*d) + (25*Tan[c + d*x]^7)/(7*a^4*d) + (8*Tan[c + d*x]^9)/(3*a^4*d) + (8*Tan[c + d*x]^11)/(11*a^4*d), -((a*Sec[c + d*x])/(22*d*(a + a*Sin[c + d*x])^5)) - Sec[c + d*x]/(33*d*(a + a*Sin[c + d*x])^4) - (5*Sec[c + d*x])/(231*a*d*(a + a*Sin[c + d*x])^3) + Sec[c + d*x]^3/(6*a*d*(a + a*Sin[c + d*x])^3) - (4*Sec[c + d*x])/(231*d*(a^2 + a^2*Sin[c + d*x])^2) - (4*Sec[c + d*x])/(231*d*(a^4 + a^4*Sin[c + d*x])) + (8*Tan[c + d*x])/(231*a^4*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^5*Sin[c + d*x]^6*(a + a*Sin[c + d*x]), x, 4, -((39*a*Log[1 - Sin[c + d*x]])/(16*d)) - (9*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/(2*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - (5*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x]), x, 3, -((23*a*Log[1 - Sin[c + d*x]])/(16*d)) + (7*a*Log[1 + Sin[c + d*x]])/(16*d) - (a*Sin[c + d*x])/d + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, -((11*a*Log[1 - Sin[c + d*x]])/(16*d)) - (5*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - (3*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]), x, 5, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(2*d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x]), x, 5, -((a*ArcTanh[Sin[c + d*x]])/(8*d)) + a^3/(8*d*(a - a*Sin[c + d*x])^2) - a^2/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x]), x, 5, -((a*ArcTanh[Sin[c + d*x]])/(8*d)) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x]), x, 4, -((11*a*Log[1 - Sin[c + d*x]])/(16*d)) + (a*Log[Sin[c + d*x]])/d - (5*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(2*d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) - (23*a*Log[1 - Sin[c + d*x]])/(16*d) + (a*Log[Sin[c + d*x]])/d + (7*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (3*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (39*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]])/d - (9*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(d*(a - a*Sin[c + d*x])) + a^2/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x]), x, 4, -((3*a*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (59*a*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*Log[Sin[c + d*x]])/d + (11*a*Log[1 + Sin[c + d*x]])/(16*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + (5*a^2)/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))} - - -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x])^2, x, 3, -((31*a^2*Log[1 - Sin[c + d*x]])/(8*d)) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (2*a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^2)/(2*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (9*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 4, -((17*a^2*Log[1 - Sin[c + d*x]])/(8*d)) + (a^2*Log[1 + Sin[c + d*x]])/(8*d) - (a^2*Sin[c + d*x])/d + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (7*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 4, -((7*a^2*Log[1 - Sin[c + d*x]])/(8*d)) - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 5, (a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - (3*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 5, -((a^2*ArcTanh[Sin[c + d*x]])/(4*d)) + a^4/(4*d*(a - a*Sin[c + d*x])^2) - a^3/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^2, x, 4, -((7*a^2*Log[1 - Sin[c + d*x]])/(8*d)) + (a^2*Log[Sin[c + d*x]])/d - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (3*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x])/d) - (17*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (2*a^2*Log[Sin[c + d*x]])/d + (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (5*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 4, -((2*a^2*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/(2*d) - (31*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (4*a^2*Log[Sin[c + d*x]])/d - (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (7*a^3)/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^2, x, 4, -((4*a^2*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (49*a^2*Log[1 - Sin[c + d*x]])/(8*d) + (6*a^2*Log[Sin[c + d*x]])/d + (a^2*Log[1 + Sin[c + d*x]])/(8*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + (9*a^3)/(4*d*(a - a*Sin[c + d*x]))} - - -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + a*Sin[c + d*x])^3, x, 3, -((10*a^3*Log[1 - Sin[c + d*x]])/d) - (6*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (5*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3, x, 4, -((6*a^3*Log[1 - Sin[c + d*x]])/d) - (3*a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^2)/(2*d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (4*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 4, -((3*a^3*Log[1 - Sin[c + d*x]])/d) - (a^3*Sin[c + d*x])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (3*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Log[1 - Sin[c + d*x]])/d) + a^5/(2*d*(a - a*Sin[c + d*x])^2) - (2*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 3, (a^5*Sin[c + d*x]^2)/(2*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Log[1 - Sin[c + d*x]])/d) + (a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + a^4/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3, x, 4, -((a^3*Csc[c + d*x])/d) - (3*a^3*Log[1 - Sin[c + d*x]])/d + (3*a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + (2*a^4)/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3, x, 4, -((3*a^3*Csc[c + d*x])/d) - (a^3*Csc[c + d*x]^2)/(2*d) - (6*a^3*Log[1 - Sin[c + d*x]])/d + (6*a^3*Log[Sin[c + d*x]])/d + a^5/(2*d*(a - a*Sin[c + d*x])^2) + (3*a^4)/(d*(a - a*Sin[c + d*x]))} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^7 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^7*Sin[c + d*x]^11/(a + a*Sin[c + d*x]), x, 4, (515*Log[1 - Sin[c + d*x]])/(256*a*d) - (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + (5*Sin[c + d*x])/(a*d) - Sin[c + d*x]^2/(2*a*d) + Sin[c + d*x]^3/(3*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (17*a)/(128*d*(a - a*Sin[c + d*x])^2) + 125/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (3*a^2)/(16*d*(a + a*Sin[c + d*x])^3) + (71*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Sin[c + d*x]^10/(a + a*Sin[c + d*x]), x, 4, (325*Log[1 - Sin[c + d*x]])/(256*a*d) + (955*Log[1 + Sin[c + d*x]])/(256*a*d) - Sin[c + d*x]/(a*d) + Sin[c + d*x]^2/(2*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (15*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(6*d*(a + a*Sin[c + d*x])^3) - (55*a)/(64*d*(a + a*Sin[c + d*x])^2) + 105/(32*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Sin[c + d*x]^9/(a + a*Sin[c + d*x]), x, 4, (187*Log[1 - Sin[c + d*x]])/(256*a*d) - (443*Log[1 + Sin[c + d*x]])/(256*a*d) + Sin[c + d*x]/(a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) - (7*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 2/(d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Sin[c + d*x]^8/(a + a*Sin[c + d*x]), x, 4, (93*Log[1 - Sin[c + d*x]])/(256*a*d) + (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) - (11*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(8*d*(a + a*Sin[c + d*x])^3) - (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 35/(32*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Sin[c + d*x]^7/(a + a*Sin[c + d*x]), x, 8, -((35*ArcTanh[Sin[c + d*x]])/(128*a*d)) + (35*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (35*Sec[c + d*x]*Tan[c + d*x]^3)/(192*a*d) + (7*Sec[c + d*x]*Tan[c + d*x]^5)/(48*a*d) - (Sec[c + d*x]*Tan[c + d*x]^7)/(8*a*d) + Tan[c + d*x]^8/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^6/(a + a*Sin[c + d*x]), x, 8, -((5*ArcTanh[Sin[c + d*x]])/(128*a*d)) - (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a*d) + (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a*d) - Tan[c + d*x]^8/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^5/(a + a*Sin[c + d*x]), x, 9, (5*ArcTanh[Sin[c + d*x]])/(128*a*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a*d) - (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a*d) + Tan[c + d*x]^6/(6*a*d) + Tan[c + d*x]^8/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 9, (3*ArcTanh[Sin[c + d*x]])/(128*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(16*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*a*d) - Tan[c + d*x]^6/(6*a*d) - Tan[c + d*x]^8/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 9, -((3*ArcTanh[Sin[c + d*x]])/(128*a*d)) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(8*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(64*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(16*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 9, -((5*ArcTanh[Sin[c + d*x]])/(128*a*d)) + Sec[c + d*x]^6/(6*a*d) - Sec[c + d*x]^8/(8*a*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) - (5*Sec[c + d*x]^3*Tan[c + d*x])/(192*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(48*a*d) + (Sec[c + d*x]^7*Tan[c + d*x])/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 8, (5*ArcTanh[Sin[c + d*x]])/(128*a*d) + Sec[c + d*x]^8/(8*a*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(128*a*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(192*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(48*a*d) - (Sec[c + d*x]^7*Tan[c + d*x])/(8*a*d)} -{Sec[c + d*x]^7*Sin[c + d*x]^0/(a + a*Sin[c + d*x]), x, 4, (35*ArcTanh[Sin[c + d*x]])/(128*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (5*a)/(128*d*(a - a*Sin[c + d*x])^2) + 15/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(24*d*(a + a*Sin[c + d*x])^3) - (5*a)/(64*d*(a + a*Sin[c + d*x])^2) - 5/(32*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, -((93*Log[1 - Sin[c + d*x]])/(256*a*d)) + Log[Sin[c + d*x]]/(a*d) - (163*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (7*a)/(128*d*(a - a*Sin[c + d*x])^2) + 29/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) + a^2/(16*d*(a + a*Sin[c + d*x])^3) + (11*a)/(64*d*(a + a*Sin[c + d*x])^2) + 1/(2*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (187*Log[1 - Sin[c + d*x]])/(256*a*d) - Log[Sin[c + d*x]]/(a*d) + (443*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (9*a)/(128*d*(a - a*Sin[c + d*x])^2) + 47/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(12*d*(a + a*Sin[c + d*x])^3) - (19*a)/(64*d*(a + a*Sin[c + d*x])^2) - 35/(32*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (325*Log[1 - Sin[c + d*x]])/(256*a*d) + (5*Log[Sin[c + d*x]])/(a*d) - (955*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (11*a)/(128*d*(a - a*Sin[c + d*x])^2) + 69/(128*d*(a - a*Sin[c + d*x])) + a^3/(64*d*(a + a*Sin[c + d*x])^4) + (5*a^2)/(48*d*(a + a*Sin[c + d*x])^3) + (29*a)/(64*d*(a + a*Sin[c + d*x])^2) + 2/(d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*Csc[c + d*x]^4/(a + a*Sin[c + d*x]), x, 4, -((5*Csc[c + d*x])/(a*d)) + Csc[c + d*x]^2/(2*a*d) - Csc[c + d*x]^3/(3*a*d) - (515*Log[1 - Sin[c + d*x]])/(256*a*d) - (5*Log[Sin[c + d*x]])/(a*d) + (1795*Log[1 + Sin[c + d*x]])/(256*a*d) + a^2/(96*d*(a - a*Sin[c + d*x])^3) + (13*a)/(128*d*(a - a*Sin[c + d*x])^2) + 95/(128*d*(a - a*Sin[c + d*x])) - a^3/(64*d*(a + a*Sin[c + d*x])^4) - a^2/(8*d*(a + a*Sin[c + d*x])^3) - (41*a)/(64*d*(a + a*Sin[c + d*x])^2) - 105/(32*d*(a + a*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^8 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^8 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^8*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^2, x, 11, (a^2*Sec[c + d*x]^3)/(3*d) - (3*a^2*Sec[c + d*x]^5)/(5*d) + (2*a^2*Sec[c + d*x]^7)/(7*d) + (2*a^2*Tan[c + d*x]^5)/(5*d) + (2*a^2*Tan[c + d*x]^7)/(7*d)} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^9 (d Sin[e+f x])^n (a+a Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^9 Sin[e+f x]^n (a+a Sin[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^9*Sin[c + d*x]^12/(a + a*Sin[c + d*x]), x, 4, -((843*Log[1 - Sin[c + d*x]])/(512*a*d)) - (2229*Log[1 + Sin[c + d*x]])/(512*a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - (3*a^2)/(64*d*(a - a*Sin[c + d*x])^3) + (141*a)/(512*d*(a - a*Sin[c + d*x])^2) - 39/(32*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) + (19*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (53*a^2)/(128*d*(a + a*Sin[c + d*x])^3) + (765*a)/(512*d*(a + a*Sin[c + d*x])^2) - 1155/(256*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Sin[c + d*x]^11/(a + a*Sin[c + d*x]), x, 4, -((437*Log[1 - Sin[c + d*x]])/(512*a*d)) + (949*Log[1 + Sin[c + d*x]])/(512*a*d) - Sin[c + d*x]/(a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - a^2/(24*d*(a - a*Sin[c + d*x])^3) + (109*a)/(512*d*(a - a*Sin[c + d*x])^2) - 203/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) - (17*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (125*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (515*a)/(512*d*(a + a*Sin[c + d*x])^2) + 5/(2*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Sin[c + d*x]^10/(a + a*Sin[c + d*x]), x, 4, -((193*Log[1 - Sin[c + d*x]])/(512*a*d)) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) - (7*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (81*a)/(512*d*(a - a*Sin[c + d*x])^2) - 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) + (15*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (95*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (325*a)/(512*d*(a + a*Sin[c + d*x])^2) - 315/(256*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Sin[c + d*x]^9/(a + a*Sin[c + d*x]), x, 9, (63*ArcTanh[Sin[c + d*x]])/(256*a*d) - (63*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (21*Sec[c + d*x]*Tan[c + d*x]^3)/(128*a*d) - (21*Sec[c + d*x]*Tan[c + d*x]^5)/(160*a*d) + (9*Sec[c + d*x]*Tan[c + d*x]^7)/(80*a*d) - (Sec[c + d*x]*Tan[c + d*x]^9)/(10*a*d) + Tan[c + d*x]^10/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^8/(a + a*Sin[c + d*x]), x, 9, (7*ArcTanh[Sin[c + d*x]])/(256*a*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x]^3)/(96*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]^5)/(80*a*d) + (Sec[c + d*x]^3*Tan[c + d*x]^7)/(10*a*d) - Tan[c + d*x]^10/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^7/(a + a*Sin[c + d*x]), x, 10, -((7*ArcTanh[Sin[c + d*x]])/(256*a*d)) - (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x]^3)/(96*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x]^5)/(80*a*d) - (Sec[c + d*x]^3*Tan[c + d*x]^7)/(10*a*d) + Tan[c + d*x]^8/(8*a*d) + Tan[c + d*x]^10/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^6/(a + a*Sin[c + d*x]), x, 10, -((3*ArcTanh[Sin[c + d*x]])/(256*a*d)) - (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(32*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^3)/(16*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^5)/(10*a*d) - Tan[c + d*x]^8/(8*a*d) - Tan[c + d*x]^10/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^5/(a + a*Sin[c + d*x]), x, 11, (3*ArcTanh[Sin[c + d*x]])/(256*a*d) + Sec[c + d*x]^6/(6*a*d) - Sec[c + d*x]^8/(4*a*d) + Sec[c + d*x]^10/(10*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(32*a*d) + (Sec[c + d*x]^5*Tan[c + d*x]^3)/(16*a*d) - (Sec[c + d*x]^5*Tan[c + d*x]^5)/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^4/(a + a*Sin[c + d*x]), x, 11, (3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(4*a*d) - Sec[c + d*x]^10/(10*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c + d*x])/(160*a*d) - (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^3/(a + a*Sin[c + d*x]), x, 10, -((3*ArcTanh[Sin[c + d*x]])/(256*a*d)) - Sec[c + d*x]^8/(8*a*d) + Sec[c + d*x]^10/(10*a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(160*a*d) + (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) - (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^2/(a + a*Sin[c + d*x]), x, 10, -((7*ArcTanh[Sin[c + d*x]])/(256*a*d)) + Sec[c + d*x]^8/(8*a*d) - Sec[c + d*x]^10/(10*a*d) - (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) - (7*Sec[c + d*x]^3*Tan[c + d*x])/(384*a*d) - (7*Sec[c + d*x]^5*Tan[c + d*x])/(480*a*d) - (Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^9*Tan[c + d*x])/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^1/(a + a*Sin[c + d*x]), x, 9, (7*ArcTanh[Sin[c + d*x]])/(256*a*d) + Sec[c + d*x]^10/(10*a*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(384*a*d) + (7*Sec[c + d*x]^5*Tan[c + d*x])/(480*a*d) + (Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) - (Sec[c + d*x]^9*Tan[c + d*x])/(10*a*d)} -{Sec[c + d*x]^9*Sin[c + d*x]^0/(a + a*Sin[c + d*x]), x, 4, (63*ArcTanh[Sin[c + d*x]])/(256*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(64*d*(a - a*Sin[c + d*x])^3) + (21*a)/(512*d*(a - a*Sin[c + d*x])^2) + 7/(64*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (5*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (5*a^2)/(128*d*(a + a*Sin[c + d*x])^3) - (35*a)/(512*d*(a + a*Sin[c + d*x])^2) - 35/(256*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Csc[c + d*x]^1/(a + a*Sin[c + d*x]), x, 4, -((193*Log[1 - Sin[c + d*x]])/(512*a*d)) + Log[Sin[c + d*x]]/(a*d) - (319*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(48*d*(a - a*Sin[c + d*x])^3) + (37*a)/(512*d*(a - a*Sin[c + d*x])^2) + 65/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (7*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (29*a^2)/(384*d*(a + a*Sin[c + d*x])^3) + (93*a)/(512*d*(a + a*Sin[c + d*x])^2) + 1/(2*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Csc[c + d*x]^2/(a + a*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (437*Log[1 - Sin[c + d*x]])/(512*a*d) - Log[Sin[c + d*x]]/(a*d) + (949*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + (5*a^2)/(192*d*(a - a*Sin[c + d*x])^3) + (57*a)/(512*d*(a - a*Sin[c + d*x])^2) + 61/(128*d*(a - a*Sin[c + d*x])) - a^4/(160*d*(a + a*Sin[c + d*x])^5) - (9*a^3)/(256*d*(a + a*Sin[c + d*x])^4) - (47*a^2)/(384*d*(a + a*Sin[c + d*x])^3) - (187*a)/(512*d*(a + a*Sin[c + d*x])^2) - 315/(256*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^9*Csc[c + d*x]^3/(a + a*Sin[c + d*x]), x, 4, Csc[c + d*x]/(a*d) - Csc[c + d*x]^2/(2*a*d) - (843*Log[1 - Sin[c + d*x]])/(512*a*d) + (6*Log[Sin[c + d*x]])/(a*d) - (2229*Log[1 + Sin[c + d*x]])/(512*a*d) + a^3/(256*d*(a - a*Sin[c + d*x])^4) + a^2/(32*d*(a - a*Sin[c + d*x])^3) + (81*a)/(512*d*(a - a*Sin[c + d*x])^2) + 203/(256*d*(a - a*Sin[c + d*x])) + a^4/(160*d*(a + a*Sin[c + d*x])^5) + (11*a^3)/(256*d*(a + a*Sin[c + d*x])^4) + (23*a^2)/(128*d*(a + a*Sin[c + d*x])^3) + (325*a)/(512*d*(a + a*Sin[c + d*x])^2) + 5/(2*d*(a + a*Sin[c + d*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m with m, n and p symbolic*) - - -{(g*Sec[e + f*x])^p*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^m, x, 5, (AppellF1[1 + n, (1 + p)/2, (1/2)*(1 - 2*m + p), 2 + n, Sin[e + f*x], -Sin[e + f*x]]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 + p)/2)*(d*Sin[e + f*x])^(1 + n)*(1 + Sin[e + f*x])^((1/2)*(1 - 2*m + p))*(a + a*Sin[e + f*x])^m)/(d*f*(1 + n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^1 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x] (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 3, (Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a*f*(1 + m)))} - - -{Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n, x, 3, (a^4*(c - d)^4*(c + d*Sin[e + f*x])^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*Sin[e + f*x])^(2 + n))/(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*Sin[e + f*x])^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*Sin[e + f*x])^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*Sin[e + f*x])^(5 + n))/(d^5*f*(5 + n))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((a^3*(c - d)^3*(c + d*Sin[e + f*x])^(1 + n))/(d^4*f*(1 + n))) + (3*a^3*(c - d)^2*(c + d*Sin[e + f*x])^(2 + n))/(d^4*f*(2 + n)) - (3*a^3*(c - d)*(c + d*Sin[e + f*x])^(3 + n))/(d^4*f*(3 + n)) + (a^3*(c + d*Sin[e + f*x])^(4 + n))/(d^4*f*(4 + n))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, (a^2*(c - d)^2*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(1 + n)) - (2*a^2*(c - d)*(c + d*Sin[e + f*x])^(2 + n))/(d^3*f*(2 + n)) + (a^2*(c + d*Sin[e + f*x])^(3 + n))/(d^3*f*(3 + n))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((a*(c - d)*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(1 + n))) + (a*(c + d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n))} -{Cos[e + f*x]/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 2, -((Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a*(c - d)*f*(1 + n)))} -{Cos[e + f*x]/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 2, (d*Hypergeometric2F1[2, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^2*(c - d)^2*f*(1 + n))} -{Cos[e + f*x]/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 2, -((d^2*Hypergeometric2F1[3, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a^3*(c - d)^3*f*(1 + n)))} - - -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^4, x, 3, ((c - d)^4*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (4*(c - d)^3*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (6*(c - d)^2*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (4*(c - d)*d^3*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) + (d^4*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^3, x, 3, ((c - d)^3*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (3*(c - d)^2*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (3*(c - d)*d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) + (d^3*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2, x, 3, ((c - d)^2*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (2*(c - d)*d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (d^2*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^1, x, 3, ((c - d)*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (d*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^1, x, 2, (Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)*f*(1 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^2, x, 2, (Hypergeometric2F1[2, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)^2*f*(1 + m))} -{Cos[e + f*x]*(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^3, x, 2, (Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(1 + Sin[e + f*x]))/(c - d))]*(a + a*Sin[e + f*x])^(1 + m))/(a*(c - d)^3*f*(1 + m))} - - -{Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^m, x, 3, -((Hypergeometric2F1[1, 2 + n + m, 2 + m, 1 + Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m))), (Hypergeometric2F1[1 + n, -m, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + a*Sin[c + d*x])^m)/((1 + Sin[c + d*x])^m*(d*(1 + n)))} - - -{Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^m, x, 4, (a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (4*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (6*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(5 + m)/(a^5*d*(5 + m))} -{Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^m, x, 4, -((a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))) + (3*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) - (3*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) + (a + a*Sin[c + d*x])^(4 + m)/(a^4*d*(4 + m))} -{Cos[c + d*x]*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^m, x, 4, (a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m)) - (2*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) + (a + a*Sin[c + d*x])^(3 + m)/(a^3*d*(3 + m))} -{Cos[c + d*x]*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^m, x, 4, -((a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))) + (a + a*Sin[c + d*x])^(2 + m)/(a^2*d*(2 + m))} -{Cos[c + d*x]*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^m, x, 2, -((Hypergeometric2F1[1, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))} -{Cos[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m, x, 3, (Hypergeometric2F1[2, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m))} -{Cos[c + d*x]*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^m, x, 3, -((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]), x, 4, (1/8)*a*(4*c + d)*x - (a*(c + d)*Cos[e + f*x]^3)/(3*f) + (a*(4*c + d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*d*Cos[e + f*x]^3*Sin[e + f*x])/(4*f), (1/8)*a*(4*c + d)*x - (a*(4*c + d)*Cos[e + f*x]^3)/(12*f) + (a*(4*c + d)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (d*Cos[e + f*x]^3*(a + a*Sin[e + f*x]))/(4*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^n*) - - -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])), x, 6, -((2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)*f)) + (2*Sqrt[c + d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)*Sqrt[d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)*) - - -{Cos[e + f*x]^2/((a + a*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x, 6, (2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[d]*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[c - d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 4, (2*Sqrt[2]*AppellF1[3/2 + m, -(1/2), -n, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - - -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((16*Sqrt[2]*a^3*AppellF1[3/2, -(7/2), -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(3*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((8*Sqrt[2]*a^2*AppellF1[3/2, -(5/2), -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(3*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((4*Sqrt[2]*a*AppellF1[3/2, -(3/2), -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(3*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 4, -((Sqrt[2]*AppellF1[3/2, 1/2, -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(3*a*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[3/2, 3/2, -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(3*Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[3/2, 5/2, -n, 5/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(6*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]])))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 4, (4*Sqrt[2]*AppellF1[5/2 + m, -(3/2), -n, 7/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(2 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a^2*f*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - - -{Cos[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((16*Sqrt[2]*a^2*AppellF1[5/2, -(7/2), -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(5*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^4*(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((8*Sqrt[2]*a*AppellF1[5/2, -(5/2), -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]^3*(1 - Sin[e + f*x])*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(5*f*(1 + Sin[e + f*x])^(3/2))))} -{Cos[e + f*x]^4/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((2*Sqrt[2]*AppellF1[5/2, -(1/2), -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(5*a*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^4/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 4, -((Sqrt[2]*AppellF1[5/2, 1/2, -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(5*a^2*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^4/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[5/2, 3/2, -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(5*Sqrt[2]*a^3*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^4/(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[5/2, 5/2, -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(10*Sqrt[2]*a^4*f*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^4/(a + a*Sin[e + f*x])^5*(c + d*Sin[e + f*x])^n, x, 3, -((AppellF1[5/2, 7/2, -n, 7/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(1 - Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(20*Sqrt[2]*a^5*f*Sqrt[1 + Sin[e + f*x]])))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (8*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(3*A - 5*B)*(a + a*Sin[c + d*x])^6)/(3*a^5*d) + (6*(A - 3*B)*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^8)/(8*a^7*d) - (B*(a + a*Sin[c + d*x])^9)/(9*a^8*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((A - B)*(a + a*Sin[c + d*x])^4)/(a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^6)/(6*a^5*d) + (B*(a + a*Sin[c + d*x])^7)/(7*a^6*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (2*(A - B)*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^4)/(4*a^3*d) - (B*(a + a*Sin[c + d*x])^5)/(5*a^4*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (a*A*Sin[c + d*x])/d + (a*(A + B)*Sin[c + d*x]^2)/(2*d) + (a*B*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, -((a*(A + B)*Log[1 - Sin[c + d*x]])/d) - (a*B*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 4, (a*(A - B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(A + B))/(2*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 4, (a*(3*A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(A + B))/(8*d*(a - a*Sin[c + d*x])^2) + (a^2*A)/(4*d*(a - a*Sin[c + d*x])) - (a^2*(A - B))/(8*d*(a + a*Sin[c + d*x]))} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 4, (a*(5*A - B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(A + B))/(24*d*(a - a*Sin[c + d*x])^3) + (a^3*(3*A + B))/(32*d*(a - a*Sin[c + d*x])^2) + (3*a^2*A)/(16*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(32*d*(a + a*Sin[c + d*x])^2) - (a^2*(2*A - B))/(16*d*(a + a*Sin[c + d*x]))} - -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 6, (5/128)*a*(8*A + B)*x - (a*(8*A + B)*Cos[c + d*x]^7)/(56*d) + (5*a*(8*A + B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*(8*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*(8*A + B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x]))/(8*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 5, (1/16)*a*(6*A + B)*x - (a*(6*A + B)*Cos[c + d*x]^5)/(30*d) + (a*(6*A + B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x]))/(6*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 4, (1/8)*a*(4*A + B)*x - (a*(4*A + B)*Cos[c + d*x]^3)/(12*d) + (a*(4*A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 2, (-a)*B*x + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x]))/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x]))/(3*d) + (a*(2*A - B)*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x]))/(5*d) + (a*(4*A - B)*Tan[c + d*x])/(5*d) + (a*(4*A - B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x]))/(7*d) + (a*(6*A - B)*Tan[c + d*x])/(7*d) + (2*a*(6*A - B)*Tan[c + d*x]^3)/(21*d) + (a*(6*A - B)*Tan[c + d*x]^5)/(35*d)} -{Sec[c + d*x]^10*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) + (a*(8*A - B)*Tan[c + d*x])/(9*d) + (a*(8*A - B)*Tan[c + d*x]^3)/(9*d) + (a*(8*A - B)*Tan[c + d*x]^5)/(15*d) + (a*(8*A - B)*Tan[c + d*x]^7)/(63*d)} - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, (4*(A - B)*(a + a*Sin[c + d*x])^6)/(3*a^4*d) - (4*(3*A - 5*B)*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (3*(A - 3*B)*(a + a*Sin[c + d*x])^8)/(4*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^9)/(9*a^7*d) - (B*(a + a*Sin[c + d*x])^10)/(10*a^8*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, (4*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(A - 2*B)*(a + a*Sin[c + d*x])^6)/(3*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^7)/(7*a^5*d) + (B*(a + a*Sin[c + d*x])^8)/(8*a^6*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, ((A - B)*(a + a*Sin[c + d*x])^4)/(2*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (B*(a + a*Sin[c + d*x])^6)/(6*a^4*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, ((A - B)*(a + a*Sin[c + d*x])^3)/(3*a*d) + (B*(a + a*Sin[c + d*x])^4)/(4*a^2*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, (-2*a^2*(A + B)*Log[1 - Sin[c + d*x]])/d - (a^2*(A + B)*Sin[c + d*x])/d - (B*(a + a*Sin[c + d*x])^2)/(2*d)} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, (a^2*B*Log[1 - Sin[c + d*x]])/d + (a^3*(A + B))/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(A - B)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^4*(A + B))/(4*d*(a - a*Sin[c + d*x])^2) + (a^3*(A - B))/(4*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(2*A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^5*(A + B))/(12*d*(a - a*Sin[c + d*x])^3) + (a^4*A)/(8*d*(a - a*Sin[c + d*x])^2) + (a^3*(3*A - B))/(16*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(16*d*(a + a*Sin[c + d*x]))} - -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 7, (5/128)*a^2*(9*A + 2*B)*x - (a^2*(9*A + 2*B)*Cos[c + d*x]^7)/(56*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a^2*(9*A + 2*B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - ((9*A + 2*B)*Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(72*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 6, (1/16)*a^2*(7*A + 2*B)*x - (a^2*(7*A + 2*B)*Cos[c + d*x]^5)/(30*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - ((7*A + 2*B)*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(42*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 5, (1/8)*a^2*(5*A + 2*B)*x - (a^2*(5*A + 2*B)*Cos[c + d*x]^3)/(12*d) + (a^2*(5*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - ((5*A + 2*B)*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(20*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, (-a^2)*(A + 2*B)*x + (a^2*(A + 2*B)*Cos[c + d*x])/d + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x])^2)/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(A - 2*B)*Sec[c + d*x])/(3*d) + ((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(3*d) + (a^2*(A - 2*B)*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(3*A - 2*B)*Sec[c + d*x]^3)/(15*d) + ((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(5*d) + (a^2*(3*A - 2*B)*Tan[c + d*x])/(5*d) + (a^2*(3*A - 2*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(5*A - 2*B)*Sec[c + d*x]^5)/(35*d) + ((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(7*d) + (a^2*(5*A - 2*B)*Tan[c + d*x])/(7*d) + (2*a^2*(5*A - 2*B)*Tan[c + d*x]^3)/(21*d) + (a^2*(5*A - 2*B)*Tan[c + d*x]^5)/(35*d)} -{Sec[c + d*x]^10*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(7*A - 2*B)*Sec[c + d*x]^7)/(63*d) + ((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^2)/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x])/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^3)/(9*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^5)/(15*d) + (a^2*(7*A - 2*B)*Tan[c + d*x]^7)/(63*d)} -{Sec[c + d*x]^12*(a + a*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, (a^2*(9*A - 2*B)*Sec[c + d*x]^9)/(99*d) + ((A + B)*Sec[c + d*x]^11*(a + a*Sin[c + d*x])^2)/(11*d) + (a^2*(9*A - 2*B)*Tan[c + d*x])/(11*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^3)/(33*d) + (6*a^2*(9*A - 2*B)*Tan[c + d*x]^5)/(55*d) + (4*a^2*(9*A - 2*B)*Tan[c + d*x]^7)/(77*d) + (a^2*(9*A - 2*B)*Tan[c + d*x]^9)/(99*d)} - - -{Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, (8*(A - B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - ((3*A - 5*B)*(a + a*Sin[c + d*x])^8)/(2*a^5*d) + (2*(A - 3*B)*(a + a*Sin[c + d*x])^9)/(3*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^10)/(10*a^7*d) - (B*(a + a*Sin[c + d*x])^11)/(11*a^8*d)} -{Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, (2*(A - B)*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + ((A - 5*B)*(a + a*Sin[c + d*x])^8)/(8*a^5*d) + (B*(a + a*Sin[c + d*x])^9)/(9*a^6*d)} -{Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, (2*(A - B)*(a + a*Sin[c + d*x])^5)/(5*a^2*d) - ((A - 3*B)*(a + a*Sin[c + d*x])^6)/(6*a^3*d) - (B*(a + a*Sin[c + d*x])^7)/(7*a^4*d)} -{Cos[c + d*x]^1*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, ((A - B)*(a + a*Sin[c + d*x])^4)/(4*a*d) + (B*(a + a*Sin[c + d*x])^5)/(5*a^2*d)} -{Sec[c + d*x]^1*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, -((4*a^3*(A + B)*Log[1 - Sin[c + d*x]])/d) - (3*a^3*(A + B)*Sin[c + d*x])/d - (a^3*(A + B)*Sin[c + d*x]^2)/(2*d) - (B*(a + a*Sin[c + d*x])^3)/(3*d)} -{Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 3, (a^3*(A + 3*B)*Log[1 - Sin[c + d*x]])/d + (a^3*B*Sin[c + d*x])/d + (2*a^4*(A + B))/(d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 2, (a^3*(a*A + a*B*Sin[c + d*x])^2)/(2*(A + B)*d*(a - a*Sin[c + d*x])^2)} -{Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, (a^3*(A - B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^6*(A + B))/(6*d*(a - a*Sin[c + d*x])^3) + (a^5*(A - B))/(8*d*(a - a*Sin[c + d*x])^2) + (a^4*(A - B))/(8*d*(a - a*Sin[c + d*x]))} -{Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, (a^3*(5*A - 3*B)*ArcTanh[Sin[c + d*x]])/(32*d) + (a^7*(A + B))/(16*d*(a - a*Sin[c + d*x])^4) + (a^6*A)/(12*d*(a - a*Sin[c + d*x])^3) + (a^5*(3*A - B))/(32*d*(a - a*Sin[c + d*x])^2) + (a^4*(2*A - B))/(16*d*(a - a*Sin[c + d*x])) - (a^4*(A - B))/(32*d*(a + a*Sin[c + d*x]))} - -{Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 8, (11/256)*a^3*(10*A + 3*B)*x - (11*a^3*(10*A + 3*B)*Cos[c + d*x]^7)/(560*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - (a*(10*A + 3*B)*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(90*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*(10*A + 3*B)*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(720*d)} -{Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 7, (9/128)*a^3*(8*A + 3*B)*x - (3*a^3*(8*A + 3*B)*Cos[c + d*x]^5)/(80*d) + (9*a^3*(8*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (3*a^3*(8*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*(8*A + 3*B)*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(56*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (3*(8*A + 3*B)*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(112*d)} -{Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 6, (7/16)*a^3*(2*A + B)*x - (7*a^3*(2*A + B)*Cos[c + d*x]^3)/(24*d) + (7*a^3*(2*A + B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*(2*A + B)*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(10*d) - (B*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(6*d) - (7*(2*A + B)*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(40*d)} -{Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 2, (-(3/2))*a^3*(2*A + 3*B)*x + (2*a^3*(2*A + 3*B)*Cos[c + d*x])/d + (a^3*(2*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + ((A + B)*Sec[c + d*x]*(a + a*Sin[c + d*x])^3)/d} -{Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, a^3*B*x + ((A + B)*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d) - (2*a^5*B*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))} -{Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, (a^5*(2*A - 3*B)*Cos[c + d*x])/(15*d*(a - a*Sin[c + d*x])^2) + ((A + B)*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(5*d) + (a^5*(2*A - 3*B)*Cos[c + d*x])/(15*d*(a^2 - a^2*Sin[c + d*x]))} -{Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, ((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(7*d) + (2*(4*A - 3*B)*Sec[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(35*d) + (3*a^3*(4*A - 3*B)*Tan[c + d*x])/(35*d) + (a^3*(4*A - 3*B)*Tan[c + d*x]^3)/(35*d)} -{Sec[c + d*x]^10*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]), x, 4, ((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x])^3)/(9*d) + (2*(2*A - B)*Sec[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(21*d) + (5*a^3*(2*A - B)*Tan[c + d*x])/(21*d) + (10*a^3*(2*A - B)*Tan[c + d*x]^3)/(63*d) + (a^3*(2*A - B)*Tan[c + d*x]^5)/(21*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 3, -(((A + B)*(a - a*Sin[c + d*x])^4)/(a^5*d)) + (4*(A + 2*B)*(a - a*Sin[c + d*x])^5)/(5*a^6*d) - ((A + 5*B)*(a - a*Sin[c + d*x])^6)/(6*a^7*d) + (B*(a - a*Sin[c + d*x])^7)/(7*a^8*d)} -{(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 3, (-2*(A + B)*(a - a*Sin[c + d*x])^3)/(3*a^4*d) + ((A + 3*B)*(a - a*Sin[c + d*x])^4)/(4*a^5*d) - (B*(a - a*Sin[c + d*x])^5)/(5*a^6*d)} -{(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 3, (A*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x]^2)/(2*a*d) - (B*Sin[c + d*x]^3)/(3*a*d)} -{(Cos[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 3, ((A - B)*Log[1 + Sin[c + d*x]])/(a*d) + (B*Sin[c + d*x])/(a*d)} -{(Sec[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 4, ((A + B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (A - B)/(2*d*(a + a*Sin[c + d*x]))} -{(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 4, ((3*A + B)*ArcTanh[Sin[c + d*x]])/(8*a*d) + (A + B)/(8*d*(a - a*Sin[c + d*x])) - (a*(A - B))/(8*d*(a + a*Sin[c + d*x])^2) - A/(4*d*(a + a*Sin[c + d*x]))} -{(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 4, ((5*A + B)*ArcTanh[Sin[c + d*x]])/(16*a*d) + (a*(A + B))/(32*d*(a - a*Sin[c + d*x])^2) + (2*A + B)/(16*d*(a - a*Sin[c + d*x])) - (a^2*(A - B))/(24*d*(a + a*Sin[c + d*x])^3) - (a*(3*A - B))/(32*d*(a + a*Sin[c + d*x])^2) - (3*A)/(16*d*(a + a*Sin[c + d*x]))} -{(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x]), x, 4, (5*(7*A + B)*ArcTanh[Sin[c + d*x]])/(128*a*d) + (a^2*(A + B))/(96*d*(a - a*Sin[c + d*x])^3) + (a*(5*A + 3*B))/(128*d*(a - a*Sin[c + d*x])^2) + (5*(3*A + B))/(128*d*(a - a*Sin[c + d*x])) - (a^3*(A - B))/(64*d*(a + a*Sin[c + d*x])^4) - (a^2*(2*A - B))/(48*d*(a + a*Sin[c + d*x])^3) - (a*(5*A - B))/(64*d*(a + a*Sin[c + d*x])^2) - (5*A)/(32*d*(a + a*Sin[c + d*x]))} - - -{(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 3, -((A + B)*(a - a*Sin[c + d*x])^4)/(2*a^6*d) + ((A + 3*B)*(a - a*Sin[c + d*x])^5)/(5*a^7*d) - (B*(a - a*Sin[c + d*x])^6)/(6*a^8*d)} -{(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 3, -((A + B)*(a - a*Sin[c + d*x])^3)/(3*a^5*d) + (B*(a - a*Sin[c + d*x])^4)/(4*a^6*d)} -{(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 3, (2*(A - B)*Log[1 + Sin[c + d*x]])/(a^2*d) - ((A - B)*Sin[c + d*x])/(a^2*d) - (B*(a - a*Sin[c + d*x])^2)/(2*a^4*d)} -{(Cos[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 3, (B*Log[1 + Sin[c + d*x]])/(a^2*d) - (A - B)/(d*(a^2 + a^2*Sin[c + d*x]))} -{(Sec[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 4, ((A + B)*ArcTanh[Sin[c + d*x]])/(4*a^2*d) - (A - B)/(4*d*(a + a*Sin[c + d*x])^2) - (A + B)/(4*d*(a^2 + a^2*Sin[c + d*x]))} -{(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 4, ((2*A + B)*ArcTanh[Sin[c + d*x]])/(8*a^2*d) - (a*(A - B))/(12*d*(a + a*Sin[c + d*x])^3) - A/(8*d*(a + a*Sin[c + d*x])^2) + (A + B)/(16*d*(a^2 - a^2*Sin[c + d*x])) - (3*A + B)/(16*d*(a^2 + a^2*Sin[c + d*x]))} -{(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 4, (5*(3*A + B)*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + (A + B)/(64*d*(a - a*Sin[c + d*x])^2) - (a^2*(A - B))/(32*d*(a + a*Sin[c + d*x])^4) - (a*(3*A - B))/(48*d*(a + a*Sin[c + d*x])^3) - (3*A)/(32*d*(a + a*Sin[c + d*x])^2) + (5*A + 3*B)/(64*d*(a^2 - a^2*Sin[c + d*x])) - (5*A + B)/(32*d*(a^2 + a^2*Sin[c + d*x]))} -{(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + a*Sin[c + d*x])^2, x, 4, (7*(4*A + B)*ArcTanh[Sin[c + d*x]])/(128*a^2*d) + (a*(A + B))/(192*d*(a - a*Sin[c + d*x])^3) + (3*A + 2*B)/(128*d*(a - a*Sin[c + d*x])^2) - (a^3*(A - B))/(80*d*(a + a*Sin[c + d*x])^5) - (a^2*(2*A - B))/(64*d*(a + a*Sin[c + d*x])^4) - (a*(5*A - B))/(96*d*(a + a*Sin[c + d*x])^3) - (5*A)/(64*d*(a + a*Sin[c + d*x])^2) + (3*(7*A + 3*B))/(256*d*(a^2 - a^2*Sin[c + d*x])) - (5*(7*A + B))/(256*d*(a^2 + a^2*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form (e Cos[e+f x])^(m/2) (a+a Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form (e Cos[e+f x])^(m/2) (a+a Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^n (A+B Sin[e+f x]) with n symbolic*) - - -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, -((2^((1/2)*(1 + 2*m + p))*a*(B*m + A*(1 + m + p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1/2)*(1 - 2*m - p), (1 + p)/2, (3 + p)/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^((1/2)*(1 - 2*m - p))*(a + a*Sin[e + f*x])^(-1 + m))/(f*g*(1 + p)*(1 + m + p))) - (B*(g*Cos[e + f*x])^(1 + p)*(a + a*Sin[e + f*x])^m)/(f*g*(1 + m + p))} - -{Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, (8*(A - B)*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) - (4*(3*A - 5*B)*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m)) + (6*(A - 3*B)*(a + a*Sin[e + f*x])^(6 + m))/(a^6*f*(6 + m)) - ((A - 7*B)*(a + a*Sin[e + f*x])^(7 + m))/(a^7*f*(7 + m)) - (B*(a + a*Sin[e + f*x])^(8 + m))/(a^8*f*(8 + m))} -{Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, (4*(A - B)*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) - (4*(A - 2*B)*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m)) + ((A - 5*B)*(a + a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m)) + (B*(a + a*Sin[e + f*x])^(6 + m))/(a^6*f*(6 + m))} -{Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, (2*(A - B)*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m)) - ((A - 3*B)*(a + a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m)) - (B*(a + a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m))} -{Cos[e + f*x]^1*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, ((A - B)*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(1 + m)) + (B*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))} -{Sec[e + f*x]^1*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, ((A - B)*(a + a*Sin[e + f*x])^m)/(2*f*m) + ((A + B)*Hypergeometric2F1[1, 1 + m, 2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(1 + m))} -{Sec[e + f*x]^3*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, -((a*(A*(2 - m) - B*m)*Hypergeometric2F1[1, -1 + m, m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^(-1 + m))/(4*f*(1 - m))) + (a^2*(A + B)*(a + a*Sin[e + f*x])^(-1 + m))/(2*f*(a - a*Sin[e + f*x]))} -{Sec[e + f*x]^5*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 3, -((a^2*(A*(4 - m) - B*m)*Hypergeometric2F1[2, -2 + m, -1 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^(-2 + m))/(16*f*(2 - m))) + (a^4*(A + B)*(a + a*Sin[e + f*x])^(-2 + m))/(4*f*(a - a*Sin[e + f*x])^2)} - -{Cos[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, -((2^(7/2 + m)*a^3*(B*m + A*(7 + m))*Cos[e + f*x]^7*Hypergeometric2F1[7/2, -(5/2) - m, 9/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-3 + m))/(7*f*(7 + m))) - (B*Cos[e + f*x]^7*(a + a*Sin[e + f*x])^m)/(f*(7 + m))} -{Cos[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, -((2^(5/2 + m)*a^2*(B*m + A*(5 + m))*Cos[e + f*x]^5*Hypergeometric2F1[5/2, -(3/2) - m, 7/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-2 + m))/(5*f*(5 + m))) - (B*Cos[e + f*x]^5*(a + a*Sin[e + f*x])^m)/(f*(5 + m))} -{Cos[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, -((2^(3/2 + m)*a*(B*m + A*(3 + m))*Cos[e + f*x]^3*Hypergeometric2F1[3/2, -(1/2) - m, 5/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^(-1 + m))/(3*f*(3 + m))) - (B*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^m)/(f*(3 + m))} -{Sec[e + f*x]^2*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, (B*Sec[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 - m)) + (2^(-(1/2) + m)*(A*(1 - m) - B*m)*Hypergeometric2F1[-(1/2), 3/2 - m, 1/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(f*(1 - m))} -{Sec[e + f*x]^4*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, (B*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^m)/(f*(3 - m)) + (2^(-(3/2) + m)*(A*(3 - m) - B*m)*Hypergeometric2F1[-(3/2), 5/2 - m, -(1/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*f*(3 - m))} -{Sec[e + f*x]^6*(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 4, (B*Sec[e + f*x]^5*(a + a*Sin[e + f*x])^m)/(f*(5 - m)) + (2^(-(5/2) + m)*(A*(5 - m) - B*m)*Hypergeometric2F1[-(5/2), 7/2 - m, -(3/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^5*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(5*a^2*f*(5 - m))} - - -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p + 4), x, 4, If[$VersionNumber>=8, ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(c*f*g*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c^2*f*g*(3 + p)*(5 + p)*(7 + p)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^3*f*g*(1 + p)*(3 + p)*(5 + p)*(7 + p)), ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-4 - p))/(f*g*(7 + p)) + ((3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(c*f*g*(35 + 12*p + p^2)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c^2*f*g*(5 + p)*(21 + 10*p + p^2)) + (2*(3*A - B*(4 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^3*f*g*(5 + 6*p + p^2)*(21 + 10*p + p^2))]} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p + 3), x, 3, If[$VersionNumber>=8, ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(f*g*(5 + p)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c*f*g*(3 + p)*(5 + p)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^2*f*g*(1 + p)*(3 + p)*(5 + p)), ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-3 - p))/(f*g*(5 + p)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(c*f*g*(15 + 8*p + p^2)) + ((2*A - B*(3 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c^2*f*g*(3 + p)*(5 + 6*p + p^2))]} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p + 2), x, 2, If[$VersionNumber>=8, ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(f*g*(3 + p)) + ((A - B*(2 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c*f*g*(1 + p)*(3 + p)), ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-2 - p))/(f*g*(3 + p)) + ((A - B*(2 + p))*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(c*f*g*(3 + 4*p + p^2))]} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p + 1), x, 4, ((A + B)*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (2^(1/2 - p/2)*B*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p))} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p + 0), x, 4, (2^(1/2 - p/2)*c*(A + B*p)*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1 + p)/2, (1 + p)/2, (3 + p)/2, (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p))/((c - c*Sin[e + f*x])^p*(f*g))} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p - 1), x, 4, (2^(1/2 - p/2)*c^2*(2*A - B*(1 - p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-1 + p), (1 + p)/2, (3 + p)/2, (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(1 - p))/(2*f*g)} -{(g*Cos[e + f*x])^p*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(p - 2), x, 4, (2^(5/2 - p/2)*c^3*(3*A - B*(2 - p))*(g*Cos[e + f*x])^(1 + p)*Hypergeometric2F1[(1/2)*(-3 + p), (1 + p)/2, (3 + p)/2, (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^((1 + p)/2)*(c - c*Sin[e + f*x])^(-1 - p))/(3*f*g*(1 + p)) - (B*(g*Cos[e + f*x])^(1 + p)*(c - c*Sin[e + f*x])^(2 - p))/(3*f*g)} - - -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(A*m - A*(m + p + 1)*Sin[e + f*x]), x, 1, (A*(g*Cos[e + f*x])^(1 + p)*(a + a*Sin[e + f*x])^m)/(f*g)} -{(g*Cos[e + f*x])^p*(a - a*Sin[e + f*x])^m*(A*m + A*(m + p + 1)*Sin[e + f*x]), x, 1, -((A*(g*Cos[e + f*x])^(1 + p)*(a - a*Sin[e + f*x])^m)/(f*g))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m, n and p symbolic*) - - -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 4, (1/(a*f*(1 + 2*m + p)))*((2^((1 + p)/2)*g*AppellF1[(1/2)*(1 + 2*m + p), (1 - p)/2, -n, (1/2)*(3 + 2*m + p), (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])^((1 - p)/2)*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/((c + d*Sin[e + f*x])/(c - d))^n)} - - -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((2^(5/2 + p/2)*a^2*AppellF1[(1 + p)/2, (1/2)*(-3 - p), -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^(2 + (1/2)*(-5 - p))*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*g*(1 + p)))), -((2^((5 + p)/2)*a^2*g*AppellF1[(1 + p)/2, (1/2)*(-3 - p), -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*(1 + p))))} -{(g*Cos[e + f*x])^p*(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((2^(3/2 + p/2)*a*AppellF1[(1 + p)/2, (1/2)*(-1 - p), -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^((1/2)*(-1 - p))*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*g*(1 + p)))), -((2^((3 + p)/2)*a*g*AppellF1[(1 + p)/2, (1/2)*(-1 - p), -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*(1 + p))))} -{(g*Cos[e + f*x])^p/(a + a*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 3, -((2^(-(1/2) + p/2)*AppellF1[(1 + p)/2, (3 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^(-1 + (1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a*f*g*(1 + p)))), -((2^(-(1/2) + p/2)*g*AppellF1[(1 + p)/2, (3 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a*f*(1 + p))))} -{(g*Cos[e + f*x])^p/(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 3, -((2^(-(3/2) + p/2)*AppellF1[(1 + p)/2, (5 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^(-2 + (3 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^2*f*g*(1 + p)))), -((2^((1/2)*(-3 + p))*g*AppellF1[(1 + p)/2, (5 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^2*f*(1 + p))))} -{(g*Cos[e + f*x])^p/(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^n, x, 3, -((2^(-(5/2) + p/2)*AppellF1[(1 + p)/2, (7 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^(-3 + (5 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^3*f*g*(1 + p)))), -((2^((1/2)*(-5 + p))*g*AppellF1[(1 + p)/2, (7 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^3*f*(1 + p))))} -{(g*Cos[e + f*x])^p/(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n, x, 3, -((2^(-(7/2) + p/2)*AppellF1[(1 + p)/2, (9 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(1 + p)*(1 + Sin[e + f*x])^(-4 + (7 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^4*f*g*(1 + p)))), -((2^((1/2)*(-7 + p))*g*AppellF1[(1 + p)/2, (9 - p)/2, -n, (3 + p)/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*(g*Cos[e + f*x])^(-1 + p)*(1 - Sin[e + f*x])*(1 + Sin[e + f*x])^((1 - p)/2)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a^4*f*(1 + p))))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) -(**) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m, n and p symbolic*) - - -{(g*Sec[e + f*x])^p*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 5, (1/(a*f*(1 + 2*m - p)))*((2^(1/2 - p/2)*AppellF1[(1/2)*(1 + 2*m - p), (1 + p)/2, -n, (1/2)*(3 + 2*m - p), (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(1 - Sin[e + f*x])^((1 + p)/2)*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/((c + d*Sin[e + f*x])/(c - d))^n)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form Cos[e+f x]^1 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x]), x, 8, (b*x)/16 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (b*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, (a*x)/8 - (b*Cos[c + d*x]^3)/(3*d) + (b*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x]), x, 6, (b*x)/8 - (a*Cos[c + d*x]^3)/(3*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x]), x, 6, (b*x)/2 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, -a*x - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d - (a*Cot[c + d*x])/d} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x]), x, 5, (-b)*x + (a*ArcTanh[Cos[c + d*x]])/(2*d) - (b*Cot[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x]), x, 5, (b*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x]), x, 6, (a*ArcTanh[Cos[c + d*x]])/(8*d) - (b*Cot[c + d*x]^3)/(3*d) + (a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + b*Sin[c + d*x]), x, 7, (b*ArcTanh[Cos[c + d*x]])/(8*d) - (a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 10, (a*b*x)/8 - ((7*a^2 + 4*b^2)*Cos[c + d*x])/(35*d) + ((7*a^2 + 4*b^2)*Cos[c + d*x]^3)/(105*d) - (a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) + ((2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(35*d) + (a*b*Cos[c + d*x]*Sin[c + d*x]^5)/(21*d) + (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(7*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 9, (1/16)*(2*a^2 + b^2)*x - (2*a*b*Cos[c + d*x])/(5*d) + (2*a*b*Cos[c + d*x]^3)/(15*d) - ((2*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a*b*Cos[c + d*x]*Sin[c + d*x]^4)/(15*d) + (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(6*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 5, (a*b*x)/4 - ((a^2 + 4*b^2)*Cos[c + d*x]^3)/(30*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (a*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(10*d) - (Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(5*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, a*b*x - (a^2*ArcTanh[Cos[c + d*x]])/d + ((2*a^2 - b^2)*Cos[c + d*x])/(3*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(3*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 9, (-a^2)*x + (b^2*x)/2 - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d - (a^2*Cot[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 6, -2*a*b*x + ((a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (3*b^2*Cos[c + d*x])/(2*d) - (a*b*Cot[c + d*x])/d - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 6, (-b^2)*x + (a*b*ArcTanh[Cos[c + d*x]])/d + ((a^2 - 2*b^2)*Cot[c + d*x])/(3*d) - (a*b*Cot[c + d*x]*Csc[c + d*x])/(3*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(3*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 8, ((a^2 + 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (2*a*b*Cot[c + d*x])/(3*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^2)/(6*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 9, (a*b*ArcTanh[Cos[c + d*x]])/(4*d) + ((2*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*b*Cot[c + d*x]*Csc[c + d*x])/(4*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(5*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^2, x, 9, ((a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (2*a*b*Cot[c + d*x])/(5*d) + (2*a*b*Cot[c + d*x]^3)/(15*d) + ((a^2 + 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^4)/(15*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(6*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 10, (1/16)*a*(2*a^2 + 3*b^2)*x - (b*(21*a^2 + 4*b^2)*Cos[c + d*x])/(35*d) + (b*(21*a^2 + 4*b^2)*Cos[c + d*x]^3)/(105*d) - (a*(2*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(2*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(56*d) + (b*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(35*d) + (a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(14*d) + (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(7*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 6, (1/16)*b*(6*a^2 + b^2)*x - (a*(2*a^2 + 33*b^2)*Cos[c + d*x]^3)/(120*d) + (b*(6*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((2*a^2 + 5*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(40*d) - (a*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(10*d) - (Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(6*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 7, (1/8)*b*(12*a^2 + b^2)*x - (a^3*ArcTanh[Cos[c + d*x]])/d + (a*(a^2 - 2*b^2)*Cos[c + d*x])/(2*d) + (b*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(4*d) + (Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 11, (-a^3)*x + (3/2)*a*b^2*x - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x])/d + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 7, (-(1/2))*b*(6*a^2 - b^2)*x + (a*(a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (15*a*b^2*Cos[c + d*x])/(2*d) + (5*b^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (3*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 7, -3*a*b^2*x + (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) + (11*b^3*Cos[c + d*x])/(6*d) + (a*(a^2 - 3*b^2)*Cot[c + d*x])/(3*d) - (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(3*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 7, (-b^3)*x + (a*(a^2 + 12*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (b*(2*a^2 - b^2)*Cot[c + d*x])/(2*d) + (a*(a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(4*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^3, x, 9, (b*(3*a^2 + 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + (a*(2*a^2 + 15*b^2)*Cot[c + d*x])/(15*d) + (3*b*(5*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(40*d) + (a*(2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*d) - (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(20*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(5*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^3, x, 10, (a*(a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b*(6*a^2 + 5*b^2)*Cot[c + d*x])/(15*d) + (a*(a^2 + 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (b*(3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) + (a*(5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(120*d) - (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(10*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 9, (a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + ((12*a^2 - b^2)*Cos[c + d*x])/(3*b^4*d) - (2*a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) + (4*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 8, -(((6*a^2 - b^2)*x)/(2*b^4)) + (2*a*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - (3*a*Cos[c + d*x])/(b^3*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^2)/(b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 5, (2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]*(2*a + b*Sin[c + d*x]))/(b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 8, -((2*b*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]*d)) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 8, -((2*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]*d)) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*Cot[c + d*x])/(a^2*d) + Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 9, (2*b*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]*d) + ((a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + (3*b*Cot[c + d*x])/(a^3*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + (Cot[c + d*x]*Csc[c + d*x])/(a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 10, -((2*b^2*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d)) - (b*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) + (2*b*Cot[c + d*x]*Csc[c + d*x])/(a^3*d) - (4*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) + (Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b*Sin[c + d*x]))} - - -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 9, -(((12*a^2 - b^2)*x)/(2*b^5)) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*(a^2 - b^2)^(3/2)*d) - (a*(12*a^2 - 11*b^2)*Cos[c + d*x])/(2*b^4*(a^2 - b^2)*d) + ((6*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*b^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 8, (3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*d) + (3*Cos[c + d*x])/(2*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^2)/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*(3*a^2 - 2*b^2)*Cos[c + d*x])/(2*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 6, -(x/b^3) + (a*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) - (a*Cos[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (Cos[c + d*x]*(2*(a^2 - b^2) + a*b*Sin[c + d*x]))/(2*b^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 8, -((b*(3*a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d)) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((a^2 - 2*b^2)*Cos[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 9, -(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)*d)) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x])/(2*a^3*(a^2 - b^2)*d) + Cot[c + d*x]/(2*a*d*(a + b*Sin[c + d*x])^2) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 10, (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) + ((a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{Cos[e + f*x]^2/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2)), x, 5, (2*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(3*a*d*f*(a + b*Sin[e + f*x])^(3/2)) + (4*b*Cos[e + f*x])/(3*a*(a^2 - b^2)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (4*b*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(3*a^3*Sqrt[a + b]*Sqrt[d]*f) - (4*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(3*a^2*Sqrt[a + b]*Sqrt[d]*f)} - - -(* ::Section:: *) -(*Integrands of the form Cos[e+f x]^3 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^4*(a + b*Sin[c + d*x]), x, 9, (3*a*x)/128 - (b*Cos[c + d*x]^5)/(5*d) + (2*b*Cos[c + d*x]^7)/(7*d) - (b*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (a*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x]), x, 9, (3*b*x)/128 - (a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^7)/(7*d) + (3*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (b*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (b*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x]), x, 8, (a*x)/16 - (b*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + b*Sin[c + d*x]), x, 7, (b*x)/16 - (a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (b*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + b*Sin[c + d*x]), x, 8, (3*b*x)/8 - (a*ArcTanh[Cos[c + d*x]])/d + (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x]), x, 9, -((3*a*x)/2) - (b*ArcTanh[Cos[c + d*x]])/d + (b*Cos[c + d*x])/d + (b*Cos[c + d*x]^3)/(3*d) - (3*a*Cot[c + d*x])/(2*d) + (a*Cos[c + d*x]^2*Cot[c + d*x])/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x]), x, 9, -((3*b*x)/2) + (3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cos[c + d*x])/(2*d) - (3*b*Cot[c + d*x])/(2*d) + (b*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x]), x, 9, a*x + (3*b*ArcTanh[Cos[c + d*x]])/(2*d) - (3*b*Cos[c + d*x])/(2*d) + (a*Cot[c + d*x])/d - (b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x]), x, 7, b*x - (3*a*ArcTanh[Cos[c + d*x]])/(8*d) + (b*Cot[c + d*x])/d - (b*Cot[c + d*x]^3)/(3*d) + (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + b*Sin[c + d*x]), x, 6, -((3*b*ArcTanh[Cos[c + d*x]])/(8*d)) - (a*Cot[c + d*x]^5)/(5*d) + (3*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Cot[c + d*x]^3*Csc[c + d*x])/(4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + b*Sin[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/(16*d)) - (b*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + b*Sin[c + d*x]), x, 8, -((b*ArcTanh[Cos[c + d*x]])/(16*d)) - (a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (b*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (b*Cot[c + d*x]^3*Csc[c + d*x]^3)/(6*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + b*Sin[c + d*x]), x, 9, -((3*a*ArcTanh[Cos[c + d*x]])/(128*d)) - (b*Cot[c + d*x]^5)/(5*d) - (b*Cot[c + d*x]^7)/(7*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 10, (3*a*b*x)/64 - ((9*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + ((9*a^2 + 4*b^2)*Cos[c + d*x]^3)/(315*d) - (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(64*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(32*d) - ((15*a^4 - 44*a^2*b^2 + 6*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(630*b^2*d) - (a*(10*a^2 - 29*b^2)*Cos[c + d*x]*Sin[c + d*x]^5)/(504*b*d) - (5*(3*a^2 - 8*b^2)*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(252*b^2*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(12*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(9*b*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 9, (1/128)*(8*a^2 + 3*b^2)*x - (6*a*b*Cos[c + d*x])/(35*d) + (2*a*b*Cos[c + d*x]^3)/(35*d) - ((8*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) - ((40*a^4 - 140*a^2*b^2 + 21*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(1344*b^2*d) - (a*(20*a^2 - 69*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(840*b*d) - ((20*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(336*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(56*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(8*b*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, (a*b*x)/8 - ((a^2 + 6*b^2)*Cos[c + d*x]^5)/(105*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) - (a*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(21*d) - (Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(7*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, (3*a*b*x)/4 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) - (b^2*Cos[c + d*x]^5)/(5*d) + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(2*d), (3*a*b*x)/4 - (a^2*ArcTanh[Cos[c + d*x]])/d - ((a^4 - 14*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^2*d) - (a*(2*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x])/(60*b*d) - ((a^2 - 12*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(30*b^2*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(10*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^3)/(5*b*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 6, (-(3/8))*(4*a^2 - b^2)*x - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (a*(a^2 + 28*b^2)*Cos[c + d*x])/(6*b*d) + ((2*a^2 + 39*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((a^2 + 12*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(12*a*b*d) - (Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^3)/(a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 6, -3*a*b*x + ((3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - ((4*a^2 - 23*b^2)*Cos[c + d*x])/(6*d) - (b*(a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(3*a*d) - ((2*a^2 - 3*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(6*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 13, a^2*x - (3*b^2*x)/2 + (3*a*b*ArcTanh[Cos[c + d*x]])/d - (3*a*b*Cos[c + d*x])/d + (a^2*Cot[c + d*x])/d - (3*b^2*Cot[c + d*x])/(2*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 6, 2*a*b*x - (3*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*(39*a^2 + 2*b^2)*Cos[c + d*x])/(24*a^2*d) + (17*a*b*Cot[c + d*x])/(12*d) + (5*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(8*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 6, b^2*x - (3*a*b*ArcTanh[Cos[c + d*x]])/(4*d) - ((3*a^4 - 14*a^2*b^2 + b^4)*Cot[c + d*x])/(15*a^2*d) + (b*(27*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(60*a*d) + ((12*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(30*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(10*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(5*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^2, x, 8, -(((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d)) - (2*a*b*Cot[c + d*x])/(5*d) - ((15*a^4 - 80*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(240*a^2*d) + (b*(13*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a*d) + ((35*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(120*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(10*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(6*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + b*Sin[c + d*x])^2, x, 9, -((a*b*ArcTanh[Cos[c + d*x]])/(8*d)) - ((2*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) - ((3*a^4 - 18*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^2*d) + (b*(53*a^2 - 12*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(420*a*d) + (2*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(35*a^2*d) + (2*b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(21*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^3)/(7*a*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 10, (1/128)*a*(8*a^2 + 9*b^2)*x - (b*(27*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + (b*(27*a^2 + 4*b^2)*Cos[c + d*x]^3)/(315*d) - (a*(8*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a*(40*a^4 - 188*a^2*b^2 + 189*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(4032*b^2*d) - ((20*a^4 - 93*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(2520*b*d) - (a*(20*a^2 - 87*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(1008*b^2*d) - (5*(a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(126*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(72*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(9*b*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 7, (3/128)*b*(8*a^2 + b^2)*x - (a*(2*a^2 + 61*b^2)*Cos[c + d*x]^5)/(560*d) + (3*b*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (b*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - ((2*a^2 + 7*b^2)*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(112*d) - (3*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(56*d) - (Cos[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(8*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 7, (1/16)*b*(18*a^2 + b^2)*x - (a^3*ArcTanh[Cos[c + d*x]])/d - (a*(2*a^4 - 43*a^2*b^2 + 36*b^4)*Cos[c + d*x])/(60*b^2*d) - ((4*a^4 - 84*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*b*d) - (a*(2*a^2 - 39*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(120*b^2*d) - ((2*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*b^2*d) + (a*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(15*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^4)/(6*b*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 7, (-(3/8))*a*(4*a^2 - 3*b^2)*x - (3*a^2*b*ArcTanh[Cos[c + d*x]])/d + ((a^4 + 56*a^2*b^2 - 2*b^4)*Cos[c + d*x])/(10*b*d) + (a*(2*a^2 + 83*b^2)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((a^2 + 28*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(20*b*d) + ((a^2 + 20*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(20*a*b*d) - (Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(5*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^4)/(a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 7, (-(3/8))*b*(12*a^2 - b^2)*x + (3*a*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - (a*(a^2 - 17*b^2)*Cos[c + d*x])/(2*d) - (b*(2*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - ((a^2 - 6*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(4*a*d) - ((a^2 - 4*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^4)/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^4)/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 17, a^3*x - (9/2)*a*b^2*x + (9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d) - (b^3*ArcTanh[Cos[c + d*x]])/d - (9*a^2*b*Cos[c + d*x])/(2*d) + (b^3*Cos[c + d*x])/d + (b^3*Cos[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/(2*d) + (3*a*b^2*Cos[c + d*x]^2*Cot[c + d*x])/(2*d) - (3*a^2*b*Cos[c + d*x]*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 7, (3/2)*b*(2*a^2 - b^2)*x - (3*a*(a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*(73*a^2 - 2*b^2)*Cos[c + d*x])/(8*a*d) - (13*b^3*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (17*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^2)/(8*d) + (5*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^3)/(8*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^3, x, 7, 3*a*b^2*x - (3*b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) - (b^3*(83*a^2 + 2*b^2)*Cos[c + d*x])/(40*a^2*d) - (a*(4*a^2 - 29*b^2)*Cot[c + d*x])/(20*d) + (27*b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (2*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3)/(5*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(20*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(5*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^3, x, 7, b^3*x - (a*(a^2 + 18*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) - (b*(36*a^4 - 43*a^2*b^2 + 2*b^4)*Cot[c + d*x])/(60*a^2*d) - ((15*a^4 - 84*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(240*a*d) + (b*(39*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2)/(120*a^2*d) + ((35*a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(120*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(15*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(6*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^8*(a + b*Sin[c + d*x])^3, x, 9, -((3*b*(a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(16*d)) - (a*(2*a^2 + 21*b^2)*Cot[c + d*x])/(35*d) - (b*(105*a^4 - 116*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(560*a^2*d) - ((4*a^4 - 19*a^2*b^2 + 2*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(140*a*d) + (b*(53*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(280*a^2*d) + ((8*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(35*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(7*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^9*(a + b*Sin[c + d*x])^3, x, 10, -((3*a*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d)) - (b*(6*a^2 + 7*b^2)*Cot[c + d*x])/(35*d) - (3*a*(a^2 + 8*b^2)*Cot[c + d*x]*Csc[c + d*x])/(128*d) - (b*(24*a^4 - 25*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(280*a^2*d) - ((35*a^4 - 148*a^2*b^2 + 24*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(2240*a*d) + (3*b*(23*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(560*a^2*d) + ((21*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(112*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^4)/(14*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^4)/(8*a*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 9, -((3*a*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(4*b^7)) + (6*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) - ((30*a^4 - 25*a^2*b^2 + b^4)*Cos[c + d*x])/(5*b^6*d) + (3*a*(4*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*b^5*d) - ((10*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(5*b^4*d) + ((3*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 8, ((40*a^4 - 36*a^2*b^2 + 3*b^4)*x)/(8*b^6) - (2*a*(5*a^4 - 7*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) + (a*(15*a^2 - 11*b^2)*Cos[c + d*x])/(3*b^5*d) - ((20*a^2 - 13*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*a*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 6, -((a*(4*a^2 - 3*b^2)*x)/b^5) + (2*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^3*(4*a + b*Sin[c + d*x]))/(3*b^2*d*(a + b*Sin[c + d*x])) - (Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(b^4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 6, -((2*a*x)/b^3) + (2*Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^3*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) - Cos[c + d*x]/(b^2*d) - ((a^2 - b^2)*Cos[c + d*x])/(a*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 6, x/b^2 - (2*(a^4 + a^2*b^2 - 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^2*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) + ((a^2 - 2*b^2)*Cos[c + d*x])/(a^2*b*d*(a + b*Sin[c + d*x])) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 7, (6*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*d) + (3*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - Cos[c + d*x]/(2*a^2*d*(1 - Cos[c + d*x]^2)) + (2*b*Cot[c + d*x])/(a^3*d) - ((a^2 - b^2)*Cos[c + d*x])/(a^3*d*(a + b*Sin[c + d*x])), (6*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*d) + (3*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) - ((a^2 - 3*b^2)*Cot[c + d*x])/(a^3*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 8, (2*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) - (b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + ((7*a^2 - 12*b^2)*Cot[c + d*x])/(3*a^4*d) - ((a^2 - 2*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^3*b*d) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 9, -((2*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d)) - ((3*a^4 - 36*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - (b*(11*a^2 - 15*b^2)*Cot[c + d*x])/(3*a^5*d) + ((13*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) - ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^3*b*d) + ((4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x]))} - - -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 9, (3*(40*a^4 - 24*a^2*b^2 + b^4)*x)/(8*b^7) - (3*a*(10*a^4 - 11*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (a*(30*a^2 - 13*b^2)*Cos[c + d*x])/(2*b^6*d) - (3*(20*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^5*d) + ((10*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(2*a*b^4*d) - ((15*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 8, (a*(9 - (20*a^2)/b^2)*x)/(2*b^4) + ((20*a^4 - 19*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) - ((60*a^2 - 17*b^2)*Cos[c + d*x])/(6*b^5*d) + ((5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(a*b^4*d) - ((20*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(6*a^2*b^3*d) - ((a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((6*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 6, (3*(4*a^2 - b^2)*x)/(2*b^5) - (3*a*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^3*(2*a + b*Sin[c + d*x]))/(2*b^2*d*(a + b*Sin[c + d*x])^2) + (3*Cos[c + d*x]*(4*a^2 - b^2 + 2*a*b*Sin[c + d*x]))/(2*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 6, x/b^3 - ((2*a^4 - a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^3*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) - ((a^2 - b^2)*Cos[c + d*x])/(2*a*b^2*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 + 2*b^2)*Cos[c + d*x])/(2*a^2*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 7, -((3*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]*d)) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) + ((a^2 - 3*b^2)*Cos[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x])^2) - ((a^2 + 6*b^2)*Cos[c + d*x])/(2*a^3*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 8, (3*b*(3*a^2 - 4*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + (3*(a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) - ((a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*b*d) + ((a^2 - 2*b^2)*Cot[c + d*x])/(2*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^2) - (3*b*Cot[c + d*x])/(a^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 9, ((2*a^4 - 19*a^2*b^2 + 20*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*d) - (b*(9*a^2 - 20*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((17*a^2 - 60*b^2)*Cot[c + d*x])/(6*a^5*d) - ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(a^4*b*d) + ((3*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(6*a^3*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 10, -((3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^7*Sqrt[a^2 - b^2]*d)) - (3*(a^4 - 24*a^2*b^2 + 40*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) - (b*(13*a^2 - 30*b^2)*Cot[c + d*x])/(2*a^6*d) + (3*(7*a^2 - 20*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - ((3*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^4*b*d) + ((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^2*b*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^2 - 15*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(4*a^3*b*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]], x, 10, (16*a*(160*a^4 - 279*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) - (8*(480*a^4 - 937*a^2*b^2 + 231*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) + (8*a*(40*a^2 - 81*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(3003*b^4*d) - (10*(16*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(1287*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(13*b*d) - (8*(320*a^6 - 798*a^4*b^2 + 435*a^2*b^4 - 693*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(160*a^6 - 439*a^4*b^2 + 306*a^2*b^4 - 27*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^4*Sin[c + d*x]^1*Sqrt[a + b*Sin[c + d*x]], x, 8, (-2*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + (8*a*(32*a^4 - 93*a^2*b^2 + 93*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3465*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^2 - 9*b^2 - 7*a*b*Sin[c + d*x]))/(693*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 - 69*a^2*b^2 + 45*b^4 - 24*a*b*(a^2 - 2*b^2)*Sin[c + d*x]))/(3465*b^4*d)} -{Cos[c + d*x]^3*Cot[c + d*x]^1*Sqrt[a + b*Sin[c + d*x]], x, 10, (-2*(8*a^2 - 45*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(105*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(35*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(7*b*d) + (2*a*(8*a^2 - 51*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(105*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^4 - 53*a^2*b^2 - 60*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(105*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^2*Cot[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]], x, 10, ((4*a^2 + 15*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(a*d) - ((4*a^2 + 57*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(4*a^2 + 11*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^1*Cot[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]], x, 10, -((8*a^2 + 3*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^2*d) + (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(2*a*d) + ((8*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(12*a*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^2 + 31*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(12*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((12*a^2 + b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^0*Cot[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]], x, 10, ((32*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(3*a*d) + ((80*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((32*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^1*Sqrt[a + b*Sin[c + d*x]], x, 11, (b*(68*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d) + (5*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(32*a^2*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(4*a*d) + (b*(68*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(196*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 + 24*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^3*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]], x, 12, -((384*a^4 + 332*a^2*b^2 - 105*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1920*a^4*d) + (b*(108*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(960*a^3*d) + ((96*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(240*a^2*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(40*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2))/(5*a*d) - ((384*a^4 + 332*a^2*b^2 - 105*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1920*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((384*a^4 + 116*a^2*b^2 - 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1920*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(48*a^4 - 24*a^2*b^2 + 7*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^4*d*Sqrt[a + b*Sin[c + d*x]])} - - -{Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2), x, 11, (8*(64*a^6 - 174*a^4*b^2 + 81*a^2*b^4 - 195*b^6)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d) + (16*a*(32*a^4 - 47*a^2*b^2 - 27*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(45045*b^5*d) - (8*(160*a^4 - 375*a^2*b^2 + 117*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(45045*b^5*d) + (8*a*(8*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(1287*b^4*d) - (2*(80*a^2 - 221*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(2145*b^3*d) + (4*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(39*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(15*b*d) - (16*a*(32*a^6 - 111*a^4*b^2 + 102*a^2*b^4 - 471*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(64*a^8 - 238*a^6*b^2 + 255*a^4*b^4 - 276*a^2*b^6 + 195*b^8)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^6*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^(3/2), x, 9, (-6*a*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(143*d) - (2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) + (8*(32*a^6 - 137*a^4*b^2 + 258*a^2*b^4 + 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^6 - 145*a^4*b^2 + 290*a^2*b^4 - 177*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(4*a*(2*a^2 - 5*b^2) - 7*b*(a^2 + 11*b^2)*Sin[c + d*x]))/(3003*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^4 - 113*a^2*b^2 + 177*b^4) - 3*b*(8*a^4 - 27*a^2*b^2 - 77*b^4)*Sin[c + d*x]))/(15015*b^4*d)} -{Cos[c + d*x]^3*Cot[c + d*x]^1*(a + b*Sin[c + d*x])^(3/2), x, 11, (-2*a*(8*a^2 - 87*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d) - (2*(8*a^2 - 77*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(315*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(63*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(9*b*d) + (2*(8*a^4 - 93*a^2*b^2 + 84*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(8*a^4 - 95*a^2*b^2 - 228*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2), x, 11, ((4*a^2 + 65*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(35*b*d) + ((4*a^2 + 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(35*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(a*d) - (a*(4*a^2 + 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^4 + 61*a^2*b^2 + 40*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (3*a*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^1*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2), x, 11, -((8*a^2 - 15*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(20*a*d) - ((8*a^2 - 5*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(20*a^2*d) - (b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(2*a*d) + ((8*a^2 - 81*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(20*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(8*a^2 + 37*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(20*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^0*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2), x, 11, -(b*(16*a^2 + b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(8*a^2*d) + ((32*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(24*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(3*a*d) + ((32*a^2 - b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 + 21*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]]) - (b*(36*a^2 + b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^(3/2), x, 11, (b*(68*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + ((20*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(32*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(4*a*d) + (b*(236*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(20*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a*d*Sqrt[a + b*Sin[c + d*x]]) + (3*(16*a^4 - 24*a^2*b^2 + b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2), x, 12, -((128*a^4 - 116*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^3*d) + (3*b*(36*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(320*a^2*d) + ((32*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(80*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(5*a*d) - ((128*a^4 - 116*a^2*b^2 + 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((128*a^4 + 692*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(640*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(48*a^4 + 8*a^2*b^2 - b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^3*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2), x, 13, -(b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d) - ((240*a^4 - 168*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3840*a^3*d) + (b*(156*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(960*a^2*d) + (7*(4*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(96*a^2*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2))/(60*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(6*a*d) - (b*(2064*a^4 + 512*a^2*b^2 - 105*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(7680*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(2544*a^4 + 176*a^2*b^2 - 35*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(7680*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 144*a^4*b^2 - 36*a^2*b^4 + 7*b^6)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^4*d*Sqrt[a + b*Sin[c + d*x]])} - - -{Cos[c + d*x]^4*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^(5/2), x, 10, (-2*(3*a^2 + 13*b^2)*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(429*d) - (2*a*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(39*d) - (2*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2))/(15*d) + (8*a*(32*a^6 - 189*a^4*b^2 + 570*a^2*b^4 + 1635*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(45045*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*(32*a^8 - 197*a^6*b^2 + 615*a^4*b^4 - 255*a^2*b^6 - 195*b^8)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(45045*b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a^4 - 33*a^2*b^2 - 39*b^4 - 7*a*b*(a^2 + 63*b^2)*Sin[c + d*x]))/(9009*b^2*d) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^6 - 165*a^4*b^2 + 450*a^2*b^4 + 195*b^6 - 24*a*b*(a^4 - 5*a^2*b^2 - 60*b^4)*Sin[c + d*x]))/(45045*b^4*d)} -{Cos[c + d*x]^3*Cot[c + d*x]^1*(a + b*Sin[c + d*x])^(5/2), x, 12, (-2*(8*a^4 - 141*a^2*b^2 + 36*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(693*b^2*d) - (2*a*(8*a^2 - 131*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(693*b^2*d) - (2*(8*a^2 - 117*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(693*b^2*d) + (8*a*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(11*b*d) + (2*a*(8*a^4 - 147*a^2*b^2 + 444*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^6 - 149*a^4*b^2 - 516*a^2*b^4 - 36*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(693*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*a^3*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2), x, 12, (a*(20*a^2 + 759*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b*d) + ((20*a^2 + 469*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(315*b*d) + ((4*a^2 + 63*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(63*a*b*d) - (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(9*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(a*d) - ((20*a^4 + 1689*a^2*b^2 - 168*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(20*a^4 + 739*a^2*b^2 + 816*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (5*a^2*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^1*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2), x, 12, -((8*a^2 - 73*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(28*d) - ((8*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(28*a*d) - ((8*a^2 - 21*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(28*a^2*d) - (3*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(2*a*d) + (a*(8*a^2 - 247*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(28*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^4 + 3*a^2*b^2 - 32*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(28*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*a*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*d*Sqrt[a + b*Sin[c + d*x]])} -{Cos[c + d*x]^0*Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2), x, 12, -(b*(96*a^2 - 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(40*a*d) - (b*(208*a^2 - 25*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(120*a^2*d) + ((32*a^2 - 3*b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(24*a^2*d) - (b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(3*a*d) + ((176*a^2 - 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(40*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (a*(96*a^2 + 179*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(40*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*(12*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^(5/2), x, 12, -(b^2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(64*a^2*d) + (5*b*(68*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((60*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) + (b*(492*a^2 - 5*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(64*a*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (b*(148*a^2 + 169*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 360*a^2*b^2 - 5*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2), x, 12, -((128*a^4 - 580*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d) + (b*(36*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(64*a^2*d) + ((32*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(80*a^2*d) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(40*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(5*a*d) - ((128*a^4 - 2476*a^2*b^2 - 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(640*a^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((128*a^4 + 492*a^2*b^2 - 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(640*a*d*Sqrt[a + b*Sin[c + d*x]]) + (3*b*(80*a^4 - 40*a^2*b^2 + b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(128*a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{Cot[c + d*x]^4*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2), x, 13, -(b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d) - ((16*a^4 - 56*a^2*b^2 + 5*b^4)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(256*a^2*d) + (b*(52*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2))/(192*a^2*d) + ((28*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(96*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^(7/2))/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^(7/2))/(6*a*d) - (b*(720*a^4 - 176*a^2*b^2 + 15*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1536*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(816*a^4 + 1696*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1536*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((64*a^6 + 720*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(512*a^3*d*Sqrt[a + b*Sin[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Cos[c + d*x]^4*Sin[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]], x, 10, (64*a*(80*a^4 - 118*a^2*b^2 + 17*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^6*d) - (8*(480*a^4 - 683*a^2*b^2 + 77*b^4)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^5*d) + (4*a*(160*a^2 - 223*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3003*b^4*d) - (10*(8*a^2 - 11*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(429*b^3*d) + (24*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(143*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(13*b*d) + (8*(1280*a^6 - 2048*a^4*b^2 + 453*a^2*b^4 + 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(1280*a^6 - 2368*a^4*b^2 + 875*a^2*b^4 + 213*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^7*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]], x, 9, (-8*(160*a^4 - 247*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^5*d) + (8*a*(120*a^2 - 179*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^4*d) - (2*(80*a^2 - 117*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(693*b^3*d) + (20*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(99*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b*d) - (16*a*(160*a^4 - 267*a^2*b^2 + 69*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3465*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(320*a^6 - 614*a^4*b^2 + 249*a^2*b^4 + 45*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3465*b^6*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^1)/Sqrt[a + b*Sin[c + d*x]], x, 7, (-2*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63*b^2*d) + (8*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(315*b^4*d)} -{(Cos[c + d*x]^3*Cot[c + d*x]^1)/Sqrt[a + b*Sin[c + d*x]], x, 9, (8*a*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d) - (2*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(5*b*d) + (2*(8*a^2 - 21*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*a*(8*a^2 - 23*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^2*Cot[c + d*x]^2)/Sqrt[a + b*Sin[c + d*x]], x, 9, (-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b*d) - (Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d) - ((4*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - 7*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^1*Cot[c + d*x]^3)/Sqrt[a + b*Sin[c + d*x]], x, 9, (3*b*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(2*a*d) + ((8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^2*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((8*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^0*Cot[c + d*x]^4)/Sqrt[a + b*Sin[c + d*x]], x, 10, ((32*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^3*d) + (5*b*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3*a*d) + ((32*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((16*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a^2*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(12*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^3*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cot[c + d*x]^4*Csc[c + d*x]^1)/Sqrt[a + b*Sin[c + d*x]], x, 11, -(b*(188*a^2 - 105*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d) + (5*(12*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(96*a^3*d) + (7*b*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(24*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(4*a*d) - (b*(188*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(192*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (b*(68*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(192*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + ((48*a^4 - 72*a^2*b^2 + 35*b^4)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(64*a^4*d*Sqrt[a + b*Sin[c + d*x]])} - - -{(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2), x, 10, (-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(640*a^4 - 592*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^6*d) + (8*a*(480*a^2 - 419*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^5*d) - (20*(32*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(231*b^4*d) + (2*(40*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(33*a*b^3*d) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b^2*d) - (8*a*(1280*a^4 - 1344*a^2*b^2 + 123*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1155*b^7*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2), x, 9, (-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (8*a*(160*a^2 - 139*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d) - (16*(60*a^2 - 49*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^4*d) + (2*(80*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b^3*d) - (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*b^2*d) + (8*(320*a^4 - 318*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*a*(160*a^4 - 199*a^2*b^2 + 39*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^6*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^1)/(a + b*Sin[c + d*x])^(3/2), x, 7, (-8*a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(7*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d*x]))/(35*b^4*d)} -{(Cos[c + d*x]^3*Cot[c + d*x]^1)/(a + b*Sin[c + d*x])^(3/2), x, 9, (-2*(a^2 - b^2)*Cos[c + d*x])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*d) - (2*(8*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*(8*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(3/2), x, 9, ((2*a^2 - 3*b^2)*Cos[c + d*x])/(a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - Cot[c + d*x]/(a*d*Sqrt[a + b*Sin[c + d*x]]) + ((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(a^2*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (3*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^1*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(3/2), x, 10, ((4*a^2 - 5*b^2)*Cot[c + d*x])/(2*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d) - ((8*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(4*a^3*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((8*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (3*(4*a^2 - 5*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^3*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^0*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^(3/2), x, 11, ((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + (5*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d) - ((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^3*b*d) + (5*(16*a^2 - 21*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(24*a^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(24*a^3*d*Sqrt[a + b*Sin[c + d*x]]) + (b*(36*a^2 - 35*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]])} - - -{(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2), x, 10, (-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(13*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (128*a*(40*a^2 - 19*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^6*d) - (8*(480*a^2 - 203*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(315*b^5*d) + (4*(160*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(63*a*b^4*d) - (10*(8*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(9*a^2*b^3*d) + (8*(1280*a^4 - 768*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^7*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(1280*a^4 - 1088*a^2*b^2 + 123*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^7*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2), x, 9, (-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(11*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (8*(32*a^2 - 11*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(21*b^5*d) + (8*(24*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(21*a*b^4*d) - (2*(80*a^2 - 21*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(21*a^2*b^3*d) - (16*a*(32*a^2 - 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(21*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(64*a^4 - 46*a^2*b^2 + 3*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(21*b^6*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^4*Sin[c + d*x]^1)/(a + b*Sin[c + d*x])^(5/2), x, 7, (8*(32*a^2 - 9*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8*a*(32*a^2 - 17*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*(8*a + 3*b*Sin[c + d*x]))/(15*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x]*(32*a^2 - 9*b^2 + 8*a*b*Sin[c + d*x]))/(15*b^4*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^3*Cot[c + d*x]^1)/(a + b*Sin[c + d*x])^(5/2), x, 9, (-2*(a^2 - b^2)*Cos[c + d*x])/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2*(5*a^2 + 3*b^2)*Cos[c + d*x])/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*(8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a^2*b^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (2*(8*a^2 + b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*a*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (2*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^2*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^2*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^(5/2), x, 10, ((2*a^2 - 5*b^2)*Cos[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x])^(3/2)) - ((4*a^2 + 15*b^2)*Cos[c + d*x])/(3*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((4*a^2 + 15*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*a^3*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) - (5*b*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a^3*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^1*Cot[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2), x, 11, ((4*a^2 - 7*b^2)*Cot[c + d*x])/(6*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^(3/2)) - ((8*a^2 - 105*b^2)*Cos[c + d*x])/(12*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 35*b^2)*Cot[c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(12*a^4*b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((8*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) - ((12*a^2 - 35*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(4*a^4*d*Sqrt[a + b*Sin[c + d*x]])} -{(Cos[c + d*x]^0*Cot[c + d*x]^4)/(a + b*Sin[c + d*x])^(5/2), x, 12, ((2*a^2 - 3*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^(3/2)) + (b*(32*a^2 - 105*b^2)*Cos[c + d*x])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]]) + ((16*a^2 - 35*b^2)*Cot[c + d*x])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) - ((8*a^2 - 21*b^2)*Cot[c + d*x]*Csc[c + d*x])/(12*a^3*b*d*Sqrt[a + b*Sin[c + d*x]]) + ((32*a^2 - 105*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(8*a^5*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((16*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^4*d*Sqrt[a + b*Sin[c + d*x]]) + (15*b*(4*a^2 - 7*b^2)*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(8*a^5*d*Sqrt[a + b*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{Cos[e + f*x]^4/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2)), x, 8, (2*Cos[e + f*x]^3*Sqrt[d*Sin[e + f*x]])/(7*a*d*f*(a + b*Sin[e + f*x])^(7/2)) + (12*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(35*a^2*d*f*(a + b*Sin[e + f*x])^(5/2)) + (8*(a^2 - 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(35*a^3*(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(3/2)) + (32*b*(2*a^2 - b^2)*Cos[e + f*x])/(35*a^3*(a^2 - b^2)^2*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (32*b*(2*a^2 - b^2)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^5*(a - b)*(a + b)^(3/2)*Sqrt[d]*f) - (8*(5*a^2 - 3*a*b - 4*b^2)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(35*a^4*(a - b)*(a + b)^(3/2)*Sqrt[d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^(n/3) (a+b Sin[e+f x])^(m/2)*) - - -{(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x, 0, Unintegrable[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m with n and/or p symbolic*) - - -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x, 0, Unintegrable[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]} - - -{Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x, 0, Unintegrable[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]} -{Cos[c + d*x]^4*Sin[c + d*x]^(-4 - p)*(a + b*Sin[c + d*x])^p, x, 0, Unintegrable[Cos[c + d*x]^4*Sin[c + d*x]^(-4 - p)*(a + b*Sin[c + d*x])^p, x]} - - -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3, x, 8, If[$VersionNumber>=8, -((3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n))) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*Sqrt[Cos[c + d*x]^2]) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2*(7 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*(7 + n)*Sqrt[Cos[c + d*x]^2]) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)*(7 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^4)/(b^2*d*(6 + n)*(7 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^4)/(b*d*(7 + n)), -((3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(5 + n)*(6 + n)*(7 + n)*(8 + 6*n + n^2))) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(6 + n)*(8 + 14*n + 7*n^2 + n^3)*Sqrt[Cos[c + d*x]^2]) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2*(7 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(5 + n)*(7 + n)*(6 + 5*n + n^2)*Sqrt[Cos[c + d*x]^2]) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*(3 + n) - b^2*(6 + n)*(8 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)*(7 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^4)/(b^2*d*(6 + n)*(7 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^4)/(b*d*(7 + n))]} -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2, x, 7, If[$VersionNumber>=8, -(((3*b^4*(5 + n) + 2*a^4*(6 + 5*n + n^2) - 2*a^2*b^2*(40 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n))) + (3*(b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(6 + n)*Sqrt[Cos[c + d*x]^2]) - (2*a*(a^2*(6 + 5*n + n^2) - b^2*(39 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)) + (6*a*b*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*(3 + n)*(5 + n)*Sqrt[Cos[c + d*x]^2]) - ((a^2*(2 + n)*(3 + n) - b^2*(5 + n)*(7 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^3)/(b*d*(6 + n)), -(((3*b^4*(5 + n) + 2*a^4*(6 + 5*n + n^2) - 2*a^2*b^2*(40 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(1 + n))/(b^2*d*(5 + n)*(6 + n)*(8 + 6*n + n^2))) + (3*(b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(4 + n)*(6 + n)*(2 + 3*n + n^2)*Sqrt[Cos[c + d*x]^2]) - (2*a*(a^2*(6 + 5*n + n^2) - b^2*(39 + 13*n + n^2))*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)) + (6*a*b*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(5 + n)*(6 + 5*n + n^2)*Sqrt[Cos[c + d*x]^2]) - ((a^2*(2 + n)*(3 + n) - b^2*(5 + n)*(7 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n)) - (Cos[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^3)/(b*d*(6 + n))]} -{Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^1, x, 3, (a*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*Sqrt[Cos[c + d*x]^2]) + (b*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^6)/(6*d) + (b*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) - (2*b*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*d) + (b*Sin[c + d*x]^11)/(11*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x]), x, 4, (a*Sin[c + d*x]^5)/(5*d) + (b*Sin[c + d*x]^6)/(6*d) - (2*a*Sin[c + d*x]^7)/(7*d) - (b*Sin[c + d*x]^8)/(4*d) + (a*Sin[c + d*x]^9)/(9*d) + (b*Sin[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]), x, 7, -((a*Cos[c + d*x]^6)/(6*d)) + (a*Cos[c + d*x]^8)/(8*d) + (b*Sin[c + d*x]^5)/(5*d) - (2*b*Sin[c + d*x]^7)/(7*d) + (b*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, -((b*Cos[c + d*x]^6)/(6*d)) + (b*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x]^3)/(3*d) - (2*a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1*(a + b*Sin[c + d*x]), x, 6, -((a*Cos[c + d*x]^6)/(6*d)) + (b*Sin[c + d*x]^3)/(3*d) - (2*b*Sin[c + d*x]^5)/(5*d) + (b*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1*(a + b*Sin[c + d*x]), x, 4, (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d - (a*Sin[c + d*x]^2)/d - (2*b*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^4)/(4*d) + (b*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (b*Log[Sin[c + d*x]])/d - (2*a*Sin[c + d*x])/d - (b*Sin[c + d*x]^2)/d + (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x]), x, 4, -((b*Csc[c + d*x])/d) - (a*Csc[c + d*x]^2)/(2*d) - (2*a*Log[Sin[c + d*x]])/d - (2*b*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d) + (b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x]), x, 4, (2*a*Csc[c + d*x])/d - (b*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/(3*d) - (2*b*Log[Sin[c + d*x]])/d + (a*Sin[c + d*x])/d + (b*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + b*Sin[c + d*x]), x, 3, (2*b*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (b*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x]), x, 4, -((a*Csc[c + d*x])/d) + (b*Csc[c + d*x]^2)/d + (2*a*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^4)/(4*d) - (a*Csc[c + d*x]^5)/(5*d) + (b*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^7*(a + b*Sin[c + d*x]), x, 6, -((a*Cot[c + d*x]^6)/(6*d)) - (b*Csc[c + d*x])/d + (2*b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^8*(a + b*Sin[c + d*x]), x, 6, -((b*Cot[c + d*x]^6)/(6*d)) - (a*Csc[c + d*x]^3)/(3*d) + (2*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^9*(a + b*Sin[c + d*x]), x, 7, -((a*Cot[c + d*x]^6)/(6*d)) - (a*Cot[c + d*x]^8)/(8*d) - (b*Csc[c + d*x]^3)/(3*d) + (2*b*Csc[c + d*x]^5)/(5*d) - (b*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^10*(a + b*Sin[c + d*x]), x, 7, -((b*Cot[c + d*x]^6)/(6*d)) - (b*Cot[c + d*x]^8)/(8*d) - (a*Csc[c + d*x]^5)/(5*d) + (2*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^11*(a + b*Sin[c + d*x]), x, 4, -((b*Csc[c + d*x]^5)/(5*d)) - (a*Csc[c + d*x]^6)/(6*d) + (2*b*Csc[c + d*x]^7)/(7*d) + (a*Csc[c + d*x]^8)/(4*d) - (b*Csc[c + d*x]^9)/(9*d) - (a*Csc[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^12*(a + b*Sin[c + d*x]), x, 4, -((b*Csc[c + d*x]^6)/(6*d)) - (a*Csc[c + d*x]^7)/(7*d) + (b*Csc[c + d*x]^8)/(4*d) + (2*a*Csc[c + d*x]^9)/(9*d) - (b*Csc[c + d*x]^10)/(10*d) - (a*Csc[c + d*x]^11)/(11*d)} - - -{Cos[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 4, (a^2*Sin[c + d*x]^3)/(3*d) + (a*b*Sin[c + d*x]^4)/(2*d) - ((2*a^2 - b^2)*Sin[c + d*x]^5)/(5*d) - (2*a*b*Sin[c + d*x]^6)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^7)/(7*d) + (a*b*Sin[c + d*x]^8)/(4*d) + (b^2*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 4, (a^2*Sin[c + d*x]^2)/(2*d) + (2*a*b*Sin[c + d*x]^3)/(3*d) - ((2*a^2 - b^2)*Sin[c + d*x]^4)/(4*d) - (4*a*b*Sin[c + d*x]^5)/(5*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^6)/(6*d) + (2*a*b*Sin[c + d*x]^7)/(7*d) + (b^2*Sin[c + d*x]^8)/(8*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 4, (a^2*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sin[c + d*x]^2)/(2*d) - (4*a*b*Sin[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*d) + (2*a*b*Sin[c + d*x]^5)/(5*d) + (b^2*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 4, -((a^2*Csc[c + d*x])/d) + (2*a*b*Log[Sin[c + d*x]])/d - ((2*a^2 - b^2)*Sin[c + d*x])/d - (2*a*b*Sin[c + d*x]^2)/d + ((a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*d) + (a*b*Sin[c + d*x]^4)/(2*d) + (b^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 4, -((2*a*b*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/(2*d) - ((2*a^2 - b^2)*Log[Sin[c + d*x]])/d - (4*a*b*Sin[c + d*x])/d + ((a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*d) + (2*a*b*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 4, ((2*a^2 - b^2)*Csc[c + d*x])/d - (a*b*Csc[c + d*x]^2)/d - (a^2*Csc[c + d*x]^3)/(3*d) - (4*a*b*Log[Sin[c + d*x]])/d + ((a^2 - 2*b^2)*Sin[c + d*x])/d + (a*b*Sin[c + d*x]^2)/d + (b^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 3, (4*a*b*Csc[c + d*x])/d + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Csc[c + d*x]^4)/(4*d) + ((a^2 - 2*b^2)*Log[Sin[c + d*x]])/d + (2*a*b*Sin[c + d*x])/d + (b^2*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 4, -(((a^2 - 2*b^2)*Csc[c + d*x])/d) + (2*a*b*Csc[c + d*x]^2)/d + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*d) - (a*b*Csc[c + d*x]^4)/(2*d) - (a^2*Csc[c + d*x]^5)/(5*d) + (2*a*b*Log[Sin[c + d*x]])/d + (b^2*Sin[c + d*x])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^2, x, 4, -((2*a*b*Csc[c + d*x])/d) - ((a^2 - 2*b^2)*Csc[c + d*x]^2)/(2*d) + (4*a*b*Csc[c + d*x]^3)/(3*d) + ((2*a^2 - b^2)*Csc[c + d*x]^4)/(4*d) - (2*a*b*Csc[c + d*x]^5)/(5*d) - (a^2*Csc[c + d*x]^6)/(6*d) + (b^2*Log[Sin[c + d*x]])/d} -{Cos[c + d*x]^5*Csc[c + d*x]^8*(a + b*Sin[c + d*x])^2, x, 4, -((b^2*Csc[c + d*x])/d) - (a*b*Csc[c + d*x]^2)/d - ((a^2 - 2*b^2)*Csc[c + d*x]^3)/(3*d) + (a*b*Csc[c + d*x]^4)/d + ((2*a^2 - b^2)*Csc[c + d*x]^5)/(5*d) - (a*b*Csc[c + d*x]^6)/(3*d) - (a^2*Csc[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^9*(a + b*Sin[c + d*x])^2, x, 4, -((b^2*Csc[c + d*x]^2)/(2*d)) - (2*a*b*Csc[c + d*x]^3)/(3*d) - ((a^2 - 2*b^2)*Csc[c + d*x]^4)/(4*d) + (4*a*b*Csc[c + d*x]^5)/(5*d) + ((2*a^2 - b^2)*Csc[c + d*x]^6)/(6*d) - (2*a*b*Csc[c + d*x]^7)/(7*d) - (a^2*Csc[c + d*x]^8)/(8*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 4, (a^2*(7*a^4 - 10*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/(b^8*d) - (2*a*(3*a^4 - 4*a^2*b^2 + b^4)*Sin[c + d*x])/(b^7*d) + ((5*a^4 - 6*a^2*b^2 + b^4)*Sin[c + d*x]^2)/(2*b^6*d) - (4*a*(a^2 - b^2)*Sin[c + d*x]^3)/(3*b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^4*d) - (2*a*Sin[c + d*x]^5)/(5*b^3*d) + Sin[c + d*x]^6/(6*b^2*d) + (a^3*(a^2 - b^2)^2)/(b^8*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 4, -((2*a*(3*a^4 - 4*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^7*d)) + ((5*a^4 - 6*a^2*b^2 + b^4)*Sin[c + d*x])/(b^6*d) - (2*a*(a^2 - b^2)*Sin[c + d*x]^2)/(b^5*d) - ((2 - (3*a^2)/b^2)*Sin[c + d*x]^3)/(3*b^2*d) - (a*Sin[c + d*x]^4)/(2*b^3*d) + Sin[c + d*x]^5/(5*b^2*d) - (a^2*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 4, ((5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^6*d) - (4*a*(a^2 - b^2)*Sin[c + d*x])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^4*d) - (2*a*Sin[c + d*x]^3)/(3*b^3*d) + Sin[c + d*x]^4/(4*b^2*d) + (a*(a^2 - b^2)^2)/(b^6*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 4, Log[Sin[c + d*x]]/(a^2*d) + ((a^2 - b^2)*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/(a^2*b^4*d) - (2*a*Sin[c + d*x])/(b^3*d) + Sin[c + d*x]^2/(2*b^2*d) + (a^2 - b^2)^2/(a*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 4, -(Csc[c + d*x]/(a^2*d)) - (2*b*Log[Sin[c + d*x]])/(a^3*d) - (2*(a^4 - b^4)*Log[a + b*Sin[c + d*x]])/(a^3*b^3*d) + Sin[c + d*x]/(b^2*d) - (a^2 - b^2)^2/(a^2*b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 4, (2*b*Csc[c + d*x])/(a^3*d) - Csc[c + d*x]^2/(2*a^2*d) - ((2*a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^4 + 2*a^2*b^2 - 3*b^4)*Log[a + b*Sin[c + d*x]])/(a^4*b^2*d) + (a^2 - b^2)^2/(a^3*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 4, ((2*a^2 - 3*b^2)*Csc[c + d*x])/(a^4*d) + (b*Csc[c + d*x]^2)/(a^3*d) - Csc[c + d*x]^3/(3*a^2*d) + (4*b*(a^2 - b^2)*Log[Sin[c + d*x]])/(a^5*d) - (4*b*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^5*d) - (a^2 - b^2)^2/(a^4*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 3, (-4*b*(a^2 - b^2)*Csc[c + d*x])/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^2)/(2*a^4*d) + (2*b*Csc[c + d*x]^3)/(3*a^3*d) - Csc[c + d*x]^4/(4*a^2*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[Sin[c + d*x]])/(a^6*d) - ((a^4 - 6*a^2*b^2 + 5*b^4)*Log[a + b*Sin[c + d*x]])/(a^6*d) + (a^2 - b^2)^2/(a^5*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + b*Sin[c + d*x])^2, x, 4, -(((a^4 - 6*a^2*b^2 + 5*b^4)*Csc[c + d*x])/(a^6*d)) - (2*b*(a^2 - b^2)*Csc[c + d*x]^2)/(a^5*d) + ((2*a^2 - 3*b^2)*Csc[c + d*x]^3)/(3*a^4*d) + (b*Csc[c + d*x]^4)/(2*a^3*d) - Csc[c + d*x]^5/(5*a^2*d) - (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[Sin[c + d*x]])/(a^7*d) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[a + b*Sin[c + d*x]])/(a^7*d) - (b*(a^2 - b^2)^2)/(a^6*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with n symbolic*) - - -{Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2, x, 3, (a^2*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (2*a*b*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - ((2*a^2 - b^2)*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (4*a*b*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + ((a^2 - 2*b^2)*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (2*a*b*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (b^2*Sin[c + d*x]^(7 + n))/(d*(7 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^1, x, 3, (a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (b*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a*Sin[c + d*x]^(3 + n))/(d*(3 + n)) - (2*b*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (b*Sin[c + d*x]^(6 + n))/(d*(6 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + b*Sin[c + d*x])^1, x, 5, -((a*(a^2 - 2*b^2)*Sin[c + d*x]^(1 + n))/(b^4*d*(1 + n))) + ((a^2 - b^2)^2*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a*b^4*d*(1 + n)) + ((a^2 - 2*b^2)*Sin[c + d*x]^(2 + n))/(b^3*d*(2 + n)) - (a*Sin[c + d*x]^(3 + n))/(b^2*d*(3 + n)) + Sin[c + d*x]^(4 + n)/(b*d*(4 + n))} -{Cos[c + d*x]^5*Sin[c + d*x]^n/(a + b*Sin[c + d*x])^2, x, 5, ((3*a^2 - 2*b^2)*Sin[c + d*x]^(1 + n))/(b^4*d*(1 + n)) + ((a^2 - b^2)*(b^2*n - a^2*(4 + n))*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a^2*b^4*d*(1 + n)) - (2*a*Sin[c + d*x]^(2 + n))/(b^3*d*(2 + n)) + Sin[c + d*x]^(3 + n)/(b^2*d*(3 + n)) + ((a^2 - b^2)^2*Sin[c + d*x]^(1 + n))/(a*b^4*d*(a + b*Sin[c + d*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 12, (5*a*b*x)/512 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((2*a^2 + 3*b^2)*Cos[c + d*x]^9)/(9*d) - ((a^2 + 3*b^2)*Cos[c + d*x]^11)/(11*d) + (b^2*Cos[c + d*x]^13)/(13*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(512*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(768*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(192*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x])/(32*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^3)/(12*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^5)/(6*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 12, ((12*a^2 + 5*b^2)*x)/1024 - (2*a*b*Cos[c + d*x]^7)/(7*d) + (4*a*b*Cos[c + d*x]^9)/(9*d) - (2*a*b*Cos[c + d*x]^11)/(11*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - ((44*a^2 + 45*b^2)*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) + ((12*a^2 + 25*b^2)*Cos[c + d*x]^9*Sin[c + d*x])/(120*d) - (b^2*Cos[c + d*x]^11*Sin[c + d*x])/(12*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 11, (3*a*b*x)/128 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((a^2 + 2*b^2)*Cos[c + d*x]^9)/(9*d) - (b^2*Cos[c + d*x]^11)/(11*d) + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(80*d) - (3*a*b*Cos[c + d*x]^7*Sin[c + d*x])/(40*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^3)/(5*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 11, (1/256)*(10*a^2 + 3*b^2)*x - (2*a*b*Cos[c + d*x]^7)/(7*d) + (2*a*b*Cos[c + d*x]^9)/(9*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(256*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + ((10*a^2 + 3*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(480*d) - ((10*a^2 + 11*b^2)*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (b^2*Cos[c + d*x]^9*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 7, (5*a*b*x)/64 - ((a^2 + 8*b^2)*Cos[c + d*x]^7)/(252*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (a*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(36*d) - (Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2)/(9*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, If[$VersionNumber<11, 10, 9], (5*a*b*x)/8 - (a^2*ArcTanh[Cos[c + d*x]])/d + (a^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (a^2*Cos[c + d*x]^5)/(5*d) - (b^2*Cos[c + d*x]^7)/(7*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Cos[c + d*x]^3*Sin[c + d*x])/(12*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(3*d)} - -{Cos[c + d*x]^6*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, If[$VersionNumber<11, 11, 12], (1/16)*-5*(6*a^2 - b^2)*x - (2*a*b*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Cos[c + d*x])/d + (2*a*b*Cos[c + d*x]^3)/(3*d) + (2*a*b*Cos[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x])/d - ((14*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((6*a^2 - 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 11, (1/4)*-15*a*b*x + ((5*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - ((2*a^2 - b^2)*Cos[c + d*x])/d - ((a^2 - b^2)*Cos[c + d*x]^3)/(3*d) + (b^2*Cos[c + d*x]^5)/(5*d) - (15*a*b*Cot[c + d*x])/(4*d) + (5*a*b*Cos[c + d*x]^2*Cot[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^4*Cot[c + d*x])/(2*d) - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 12, (5/8)*(4*a^2 - 3*b^2)*x + (5*a*b*ArcTanh[Cos[c + d*x]])/d - (5*a*b*Cos[c + d*x])/d - (5*a*b*Cos[c + d*x]^3)/(3*d) + ((2*a^2 - b^2)*Cot[c + d*x])/d - (a*b*Cos[c + d*x]^3*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) + ((4*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 12, 5*a*b*x - (5*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(8*d) + ((a^2 - 2*b^2)*Cos[c + d*x])/d - (b^2*Cos[c + d*x]^3)/(3*d) + (5*a*b*Cot[c + d*x])/d - (5*a*b*Cot[c + d*x]^3)/(3*d) + (a*b*Cos[c + d*x]^2*Cot[c + d*x]^3)/d + ((9*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 16, -(a^2*x) + (5*b^2*x)/2 - (15*a*b*ArcTanh[Cos[c + d*x]])/(4*d) + (15*a*b*Cos[c + d*x])/(4*d) - (a^2*Cot[c + d*x])/d + (5*b^2*Cot[c + d*x])/(2*d) + (5*a*b*Cos[c + d*x]*Cot[c + d*x]^2)/(4*d) + (a^2*Cot[c + d*x]^3)/(3*d) - (5*b^2*Cot[c + d*x]^3)/(6*d) + (b^2*Cos[c + d*x]^2*Cot[c + d*x]^3)/(2*d) - (a*b*Cos[c + d*x]*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7*(a + b*Sin[c + d*x])^2, x, 11, -2*a*b*x + (5*(a^2 - 6*b^2)*ArcTanh[Cos[c + d*x]])/(16*d) + (b^2*Cos[c + d*x])/d - (2*a*b*Cot[c + d*x])/d + (2*a*b*Cot[c + d*x]^3)/(3*d) - (2*a*b*Cot[c + d*x]^5)/(5*d) - ((11*a^2 - 18*b^2)*Cot[c + d*x]*Csc[c + d*x])/(16*d) + ((13*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^8*(a + b*Sin[c + d*x])^2, x, 9, (-b^2)*x + (5*a*b*ArcTanh[Cos[c + d*x]])/(8*d) - (b^2*Cot[c + d*x])/d + (b^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a*b*Cot[c + d*x]^3*Csc[c + d*x])/(12*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^9*(a + b*Sin[c + d*x])^2, x, 9, (5*(a^2 + 8*b^2)*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) + ((5*a^2 - 88*b^2)*Cot[c + d*x]*Csc[c + d*x])/(128*d) - ((59*a^2 - 104*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*d) + ((17*a^2 - 8*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(48*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(8*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^10*(a + b*Sin[c + d*x])^2, x, 9, (5*a*b*ArcTanh[Cos[c + d*x]])/(64*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - (a^2*Cot[c + d*x]^9)/(9*d) + (5*a*b*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (5*a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a*b*Cot[c + d*x]^3*Csc[c + d*x]^3)/(24*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^11*(a + b*Sin[c + d*x])^2, x, 11, ((3*a^2 + 10*b^2)*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a*b*Cot[c + d*x]^7)/(7*d) - (2*a*b*Cot[c + d*x]^9)/(9*d) + ((3*a^2 + 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(256*d) + ((3*a^2 - 118*b^2)*Cot[c + d*x]*Csc[c + d*x]^3)/(384*d) - ((93*a^2 - 170*b^2)*Cot[c + d*x]*Csc[c + d*x]^5)/(480*d) + ((21*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^9)/(10*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^12*(a + b*Sin[c + d*x])^2, x, 10, (3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[c + d*x]^5*Csc[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 11, (a*(64*a^6 - 120*a^4*b^2 + 60*a^2*b^4 - 5*b^6)*x)/(8*b^9) - (2*a^2*(8*a^2 - 3*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^9*d) + ((840*a^6 - 1435*a^4*b^2 + 588*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^8*d) - (a*(32*a^4 - 52*a^2*b^2 + 19*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b^7*d) + ((280*a^4 - 441*a^2*b^2 + 150*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) - ((24*a^4 - 37*a^2*b^2 + 12*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a*b^5*d) + ((224*a^4 - 340*a^2*b^2 + 105*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])) - (3*b*Cos[c + d*x]*Sin[c + d*x]^5)/(20*a^2*d*(a + b*Sin[c + d*x])) - ((20*a^4 - 30*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(15*a^2*b^3*d*(a + b*Sin[c + d*x])) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(21*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^7)/(7*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 10, -(((112*a^6 - 200*a^4*b^2 + 90*a^2*b^4 - 5*b^6)*x)/(16*b^8)) + (2*a*(7*a^2 - 2*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^8*d) - (a*(105*a^4 - 170*a^2*b^2 + 61*b^4)*Cos[c + d*x])/(15*b^7*d) + ((56*a^4 - 86*a^2*b^2 + 27*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) - ((35*a^4 - 52*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*a*b^5*d) + ((42*a^4 - 61*a^2*b^2 + 16*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*b^4*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(6*a^2*d*(a + b*Sin[c + d*x])) - ((14*a^4 - 20*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(10*a^2*b^3*d*(a + b*Sin[c + d*x])) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(30*b^2*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*Sin[c + d*x]^6)/(6*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 7, (a*(24*a^4 - 40*a^2*b^2 + 15*b^4)*x)/(4*b^7) - (2*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*d) + (Cos[c + d*x]^5*(6*a + b*Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])) - (Cos[c + d*x]^3*(2*(6*a^2 - b^2) - 9*a*b*Sin[c + d*x]))/(6*b^4*d) + (Cos[c + d*x]*(4*(6*a^4 - 7*a^2*b^2 + b^4) - a*b*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(4*b^6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 16, (a*x)/b^3 + (2*a*(2*a^2 - 3*b^2)*x)/b^5 + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*d) - (2*(a^2 - b^2)^(3/2)*(5*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^5*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Cos[c + d*x])/(b^4*d) - Cos[c + d*x]^3/(3*b^2*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 16, -(x/(2*b^2)) - (3*(a^2 - b^2)*x)/b^4 - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b^4*d) + (4*(2*a^6 - 3*a^4*b^2 + b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^4*Sqrt[a^2 - b^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*a*Cos[c + d*x])/(b^3*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(a^2*b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 16, (2*a*x)/b^3 + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^3*d) - (6*(a^2 - b^2)^(3/2)*(a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*b^3*d) - ArcTanh[Cos[c + d*x]]/(2*a^2*d) + (3*(a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b^2*d) + (2*b*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(a^3*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 17, -(x/b^2) - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^2*d) + (4*(a^6 - 3*a^2*b^4 + 2*b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*b^2*Sqrt[a^2 - b^2]*d) + (b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) - Cot[c + d*x]/(a^2*d) + (3*(a^2 - b^2)*Cot[c + d*x])/(a^4*d) - Cot[c + d*x]^3/(3*a^2*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(a^3*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(a^4*b*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + b*Sin[c + d*x])^2, x, 9, -((10*b*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d)) - (5*(3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) + ((3*a^4 - 20*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(3*a^5*b*d) + (5*(5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) - Cot[c + d*x]/(b*d*(a + b*Sin[c + d*x])) - ((6*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^3*d*(a + b*Sin[c + d*x])) + (5*b*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + b*Sin[c + d*x])^2, x, 10, -((2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^7*d)) + (b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*ArcTanh[Cos[c + d*x]])/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*Cot[c + d*x])/(15*a^6*d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(6*a^3*b^2*d*(a + b*Sin[c + d*x])) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^7/(a + b*Sin[c + d*x])^2, x, 11, (2*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^8*d) + ((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Cot[c + d*x])/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^3*b^2*d*(a + b*Sin[c + d*x])) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4)/(30*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d*(a + b*Sin[c + d*x]))} - - -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 11, -(((448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/(16*b^9)) + (a*Sqrt[a^2 - b^2]*(56*a^4 - 47*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(840*a^4 - 985*a^2*b^2 + 213*b^4)*Cos[c + d*x])/(30*b^8*d) + ((224*a^4 - 244*a^2*b^2 + 43*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^7*d) - ((280*a^4 - 291*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a*b^6*d) + ((168*a^4 - 169*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(10*a^2*d*(a + b*Sin[c + d*x])^2) - ((56*a^4 - 60*a^2*b^2 + 9*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(60*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (4*a*Cos[c + d*x]*Sin[c + d*x]^6)/(15*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^7)/(6*b*d*(a + b*Sin[c + d*x])^2) - ((112*a^4 - 110*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(20*a^2*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 10, (a*(168*a^4 - 200*a^2*b^2 + 45*b^4)*x)/(8*b^8) - (Sqrt[a^2 - b^2]*(42*a^4 - 29*a^2*b^2 + 2*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^8*d) + ((630*a^4 - 645*a^2*b^2 + 91*b^4)*Cos[c + d*x])/(30*b^7*d) - ((84*a^4 - 79*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*a*b^6*d) + ((210*a^4 - 187*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(30*a^2*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(12*a^2*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 60*a^2*b^2 + 5*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(60*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(20*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^6)/(5*b*d*(a + b*Sin[c + d*x])^2) - ((63*a^4 - 54*a^2*b^2 + 4*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(12*a^2*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 7, -((15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7)) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^5*(3*a + b*Sin[c + d*x]))/(4*b^2*d*(a + b*Sin[c + d*x])^2) + (5*Cos[c + d*x]^3*(4*a^2 - b^2 + a*b*Sin[c + d*x]))/(4*b^4*d*(a + b*Sin[c + d*x])) - (15*Cos[c + d*x]*(4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*Sin[c + d*x]))/(8*b^6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 20, -(x/(2*b^3)) - (3*(2*a^2 - b^2)*x)/b^5 + (Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b^5*d) - (2*Sqrt[a^2 - b^2]*(5*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b^5*d) + (2*(10*a^6 - 9*a^4*b^2 - b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^5*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) - (3*a*Cos[c + d*x])/(b^4*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(2*a*b^4*d*(a + b*Sin[c + d*x])^2) + (3*(a^2 - b^2)*Cos[c + d*x])/(2*b^4*d*(a + b*Sin[c + d*x])) - ((a^2 - b^2)*(5*a^2 + b^2)*Cos[c + d*x])/(a^2*b^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 20, (3*a*x)/b^4 + (3*Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^4*d) - (6*(2*a^6 - a^4*b^2 - b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*b^4*Sqrt[a^2 - b^2]*d) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b^3*d) - Cot[c + d*x]/(a^3*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(2*a^2*b^3*d*(a + b*Sin[c + d*x])^2) - (3*(a^2 - b^2)*Cos[c + d*x])/(2*a*b^3*d*(a + b*Sin[c + d*x])) + (2*(a^2 - b^2)*(2*a^2 + b^2)*Cos[c + d*x])/(a^3*b^3*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 21, -(x/b^3) - (6*Sqrt[a^2 - b^2]*(a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + (Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + (6*(a^6 + a^2*b^4 - 2*b^6)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*b^3*Sqrt[a^2 - b^2]*d) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) + (3*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + ((a^2 - b^2)^2*Cos[c + d*x])/(2*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*(a^2 - b^2)*Cos[c + d*x])/(2*a^2*b^2*d*(a + b*Sin[c + d*x])) - (3*(a^4 - b^4)*Cos[c + d*x])/(a^4*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 9, (5*(a^2 - 4*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d) - (5*b*(3*a^2 - 4*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^6*d) + ((3*a^4 + 35*a^2*b^2 - 60*b^4)*Cot[c + d*x])/(6*a^5*b^2*d) - Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x])^2) - (a*Cot[c + d*x])/(2*b^2*d*(a + b*Sin[c + d*x])^2) - ((3*a^2 - 5*b^2)*Cot[c + d*x])/(3*a^3*d*(a + b*Sin[c + d*x])^2) + (5*b*Cot[c + d*x]*Csc[c + d*x])/(6*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*(a + b*Sin[c + d*x])^2) - (5*(a^2 - 2*b^2)*Cot[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + b*Sin[c + d*x])^3, x, 10, -((15*b*(a^2 - 2*b^2)*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^7*d)) - (15*(a^4 - 8*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^7*d) + ((a^4 - 25*a^2*b^2 + 30*b^4)*Cot[c + d*x])/(2*a^6*b*d) + (15*(3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^5*d) - Cot[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) - ((4*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x])/(4*a^3*d*(a + b*Sin[c + d*x])^2) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(2*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d*(a + b*Sin[c + d*x])^2) - ((7*a^2 - 10*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^4*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + b*Sin[c + d*x])^3, x, 11, -((Sqrt[a^2 - b^2]*(2*a^4 - 29*a^2*b^2 + 42*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^8*d)) + (b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c + d*x])/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^6*b*d) - ((15*a^4 - 187*a^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + ((5*a^4 - 60*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3)/(20*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^4 - 54*a^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^4*b^2*d*(a + b*Sin[c + d*x]))} -{Cos[c + d*x]^6*Csc[c + d*x]^8/(a + b*Sin[c + d*x])^3, x, 13, -((3*b^2*Sqrt[a^2 - b^2]*(4*a^4 - 23*a^2*b^2 + 24*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^10*d)) - (3*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^10*d) + ((10*a^6 - 889*a^4*b^2 + 3255*a^2*b^4 - 2520*b^6)*Cot[c + d*x])/(70*a^9*d) + (3*b*(27*a^4 - 116*a^2*b^2 + 96*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^8*d) - ((205*a^4 - 973*a^2*b^2 + 840*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(70*a^7*d) + ((16*a^4 - 81*a^2*b^2 + 72*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^6*b*d) - (3*(35*a^4 - 185*a^2*b^2 + 168*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(70*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(5*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(10*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^4 - 35*a^2*b^2 + 30*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(35*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*b*Cot[c + d*x]*Csc[c + d*x]^5)/(14*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d*(a + b*Sin[c + d*x])^2) + ((12*a^4 - 65*a^2*b^2 + 60*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(10*a^4*b^2*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{Cos[e + f*x]^6/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(13/2)), x, 8, (2*Cos[e + f*x]^5*Sqrt[d*Sin[e + f*x]])/(11*a*d*f*(a + b*Sin[e + f*x])^(11/2)) - (20*(a^2 - b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(99*a^2*b^2*d*f*(a + b*Sin[e + f*x])^(9/2)) + (80*(3*a^2 + 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(693*a^3*b^2*d*f*(a + b*Sin[e + f*x])^(7/2)) - (4*(5*a^4 - 17*a^2*b^2 + 16*b^4)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(231*a^4*b^2*(a^2 - b^2)*d*f*(a + b*Sin[e + f*x])^(5/2)) - (8*(5*a^6 - 22*a^4*b^2 + 65*a^2*b^4 - 32*b^6)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(693*a^5*b^2*(a^2 - b^2)^2*d*f*(a + b*Sin[e + f*x])^(3/2)) + (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Cos[e + f*x])/(693*a^5*(a^2 - b^2)^3*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (16*b*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(693*a^7*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f) - (16*(45*a^4 - 48*a^3*b - 69*a^2*b^2 + 24*a*b^3 + 32*b^4)*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(693*a^6*(a - b)^2*(a + b)^(5/2)*Sqrt[d]*f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^2*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]), x, 9, (2*(a^2 + b^2)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) + (4*a*b*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2)) + ((2*a^2 - b^2)*EllipticF[(1/4)*(4*e - Pi) + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*f*g^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]), (2*(a^2 + b^2)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) + (4*a*b*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2)) - (2*(2*a^2 - b^2)*(1 - Csc[e + f*x]^2)^(3/4)*EllipticF[(1/2)*ArcCsc[Sin[e + f*x]], 2]*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2))} - - -{(a + b*Sin[e + f*x])^2/((g*Cos[e + f*x])^(7/2)*Sqrt[d*Sin[e + f*x]]), x, 6, (8*a^2*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^3*Sqrt[g*Cos[e + f*x]]) + (8*a*b*(d*Sin[e + f*x])^(3/2))/(5*d^2*f*g^3*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2)/(5*d*f*g*(g*Cos[e + f*x])^(5/2)) - (8*a*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*d*f*g^4*Sqrt[Sin[2*e + 2*f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n / (a+b Sin[e+f x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^p Sin[e+f x]^n / (a+b Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cos[c + d*x]*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, -((a^3*Log[a + b*Sin[c + d*x]])/(b^4*d)) + (a^2*Sin[c + d*x])/(b^3*d) - (a*Sin[c + d*x]^2)/(2*b^2*d) + Sin[c + d*x]^3/(3*b*d)} -{Cos[c + d*x]*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, (a^2*Log[a + b*Sin[c + d*x]])/(b^3*d) - (a*Sin[c + d*x])/(b^2*d) + Sin[c + d*x]^2/(2*b*d)} -{Cos[c + d*x]*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, -((a*Log[a + b*Sin[c + d*x]])/(b^2*d)) + Sin[c + d*x]/(b*d)} -{Cos[c + d*x]*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]]/(a*d)} -{Cos[c + d*x]*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) + (b*Log[a + b*Sin[c + d*x]])/(a^2*d)} -{Cos[c + d*x]*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) + (b^2*Log[Sin[c + d*x]])/(a^3*d) - (b^2*Log[a + b*Sin[c + d*x]])/(a^3*d)} - - -{Cos[c + d*x]^2*Sin[c + d*x]^4/(a + b*Sin[c + d*x]), x, 10, (a*(8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^6) - (2*a^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*d) + ((15*a^4 - 5*a^2*b^2 - 2*b^4)*Cos[c + d*x])/(15*b^5*d) - (a*(4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + ((5*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 9, -(((8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^5)) + (2*a^3*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*d) - (a*(3*a^2 - b^2)*Cos[c + d*x])/(3*b^4*d) + ((4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 8, (a*(2*a^2 - b^2)*x)/(2*b^4) - (2*a^2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^4*d) + ((3*a^2 - b^2)*Cos[c + d*x])/(3*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d)} -{Cos[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 5, -(((2*a^2 - b^2)*x)/(2*b^3)) + (2*a*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*d) - (Cos[c + d*x]*(2*a - b*Sin[c + d*x]))/(2*b^2*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, -(x/b) + (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b*d) - ArcTanh[Cos[c + d*x]]/(a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 7, -((2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*d)) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 8, (2*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*d) + ((a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^4/(a + b*Sin[c + d*x]), x, 9, -((2*b^2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*d)) - (b*(a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^5/(a + b*Sin[c + d*x]), x, 10, (2*b^3*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*d) + ((a^4 + 4*a^2*b^2 - 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^5*d) - (b*(a^2 - 3*b^2)*Cot[c + d*x])/(3*a^4*d) + ((a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^2*Csc[c + d*x]^6/(a + b*Sin[c + d*x]), x, 11, -((2*b^4*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d)) - (b*(a^4 + 4*a^2*b^2 - 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) + ((2*a^4 + 5*a^2*b^2 - 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (b*(a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)} - - -{Cos[c + d*x]^3*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (a^3*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^6*d) - (a^2*(a^2 - b^2)*Sin[c + d*x])/(b^5*d) + (a*(a^2 - b^2)*Sin[c + d*x]^2)/(2*b^4*d) - ((a^2 - b^2)*Sin[c + d*x]^3)/(3*b^3*d) + (a*Sin[c + d*x]^4)/(4*b^2*d) - Sin[c + d*x]^5/(5*b*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -((a^2*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d)) + (a*(a^2 - b^2)*Sin[c + d*x])/(b^4*d) - ((a^2 - b^2)*Sin[c + d*x]^2)/(2*b^3*d) + (a*Sin[c + d*x]^3)/(3*b^2*d) - Sin[c + d*x]^4/(4*b*d)} -{Cos[c + d*x]^3*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, (a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^4*d) - ((a^2 - b^2)*Sin[c + d*x])/(b^3*d) + (a*Sin[c + d*x]^2)/(2*b^2*d) - Sin[c + d*x]^3/(3*b*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a*b^2*d) - Sin[c + d*x]/(b*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) - ((1 - b^2/a^2)*Log[a + b*Sin[c + d*x]])/(b*d)} -{Cos[c + d*x]^3*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 3, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) + ((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(a^3*d)} - - -{Cos[c + d*x]^4*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 9, ((16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*x)/(16*b^7) - (2*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*d) + (a*(15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^6*d) - ((8*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^5*d) + (a*(5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^4*d) - ((6*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*b*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 8, -((a*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6)) + (2*a^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*d) - ((15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^5*d) + (a*(4*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) - ((5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d)} -{Cos[c + d*x]^4*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, ((8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^5*d) - (Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(12*b^2*d) + (Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(8*b^4*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, ((2*a^2 - 3*b^2)*x)/(2*b^3) - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b^3*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (a*Cos[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 6, -((a*x)/b^2) + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^2*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cos[c + d*x]/(b*d) - Cot[c + d*x]/(a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 6, x/b - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b*d) + ((3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^4/(a + b*Sin[c + d*x]), x, 7, (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*d) - (b*(3*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^4*d) + ((4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^5/(a + b*Sin[c + d*x]), x, 8, -((2*b*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*d)) - ((3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^5*d) - (b*(4*a^2 - 3*b^2)*Cot[c + d*x])/(3*a^4*d) + ((5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^4*Csc[c + d*x]^6/(a + b*Sin[c + d*x]), x, 9, (2*b^2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d) + (b*(3*a^4 - 12*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((3*a^4 - 20*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (b*(5*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((6*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)} - - -{Cos[c + d*x]^5*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, -((a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^8*d)) + (a^2*(a^2 - b^2)^2*Sin[c + d*x])/(b^7*d) - (a*(a^2 - b^2)^2*Sin[c + d*x]^2)/(2*b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*b^3*d) - (a*Sin[c + d*x]^6)/(6*b^2*d) + Sin[c + d*x]^7/(7*b*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, (a^2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^7*d) - (a*(a^2 - b^2)^2*Sin[c + d*x])/(b^6*d) + ((a^2 - b^2)^2*Sin[c + d*x]^2)/(2*b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^4)/(4*b^3*d) - (a*Sin[c + d*x]^5)/(5*b^2*d) + Sin[c + d*x]^6/(6*b*d)} -{Cos[c + d*x]^5*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, -((a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^6*d)) + ((a^2 - b^2)^2*Sin[c + d*x])/(b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^3)/(3*b^3*d) - (a*Sin[c + d*x]^4)/(4*b^2*d) + Sin[c + d*x]^5/(5*b*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, Log[Sin[c + d*x]]/(a*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a*b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x])/(b^3*d) - (a*Sin[c + d*x]^2)/(2*b^2*d) + Sin[c + d*x]^3/(3*b*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^2*b^3*d) - (a*Sin[c + d*x])/(b^2*d) + Sin[c + d*x]^2/(2*b*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((2*a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^3*b^2*d) + Sin[c + d*x]/(b*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^4/(a + b*Sin[c + d*x]), x, 4, ((2*a^2 - b^2)*Csc[c + d*x])/(a^3*d) + (b*Csc[c + d*x]^2)/(2*a^2*d) - Csc[c + d*x]^3/(3*a*d) + (b*(2*a^2 - b^2)*Log[Sin[c + d*x]])/(a^4*d) + ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^4*b*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^5/(a + b*Sin[c + d*x]), x, 3, -((b*(2*a^2 - b^2)*Csc[c + d*x])/(a^4*d)) + ((2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^3*d) + (b*Csc[c + d*x]^3)/(3*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^5*d) - ((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^5*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^6/(a + b*Sin[c + d*x]), x, 4, -(((a^2 - b^2)^2*Csc[c + d*x])/(a^5*d)) - (b*(2*a^2 - b^2)*Csc[c + d*x]^2)/(2*a^4*d) + ((2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^3*d) + (b*Csc[c + d*x]^4)/(4*a^2*d) - Csc[c + d*x]^5/(5*a*d) - (b*(a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^6*d) + (b*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^6*d)} -{Cos[c + d*x]^5*Csc[c + d*x]^7/(a + b*Sin[c + d*x]), x, 4, (b*(a^2 - b^2)^2*Csc[c + d*x])/(a^6*d) - ((a^2 - b^2)^2*Csc[c + d*x]^2)/(2*a^5*d) - (b*(2*a^2 - b^2)*Csc[c + d*x]^3)/(3*a^4*d) + ((2*a^2 - b^2)*Csc[c + d*x]^4)/(4*a^3*d) + (b*Csc[c + d*x]^5)/(5*a^2*d) - Csc[c + d*x]^6/(6*a*d) + (b^2*(a^2 - b^2)^2*Log[Sin[c + d*x]])/(a^7*d) - (b^2*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(a^7*d)} - - -{Cos[c + d*x]^6*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 11, -(((128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*x)/(128*b^9)) + (2*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^9*d) - (a*(105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^8*d) + ((64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x]*Sin[c + d*x])/(128*b^7*d) - (a*(35*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^6*d) + ((48*a^4 - 104*a^2*b^2 + 59*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(192*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d) - ((28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(5*a^2*d) + ((40*a^4 - 85*a^2*b^2 + 48*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(240*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^6)/(7*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^7)/(8*b*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 10, (a*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^8) - (2*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^8*d) + ((105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(105*b^7*d) - (a*(8*a^4 - 18*a^2*b^2 + 11*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^6*d) + ((35*a^4 - 77*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(105*b^5*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d) - ((6*a^4 - 13*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*a*b^4*d) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(4*a^2*d) + ((28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(140*a^2*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^6)/(7*b*d)} -{Cos[c + d*x]^6*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 7, -(((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^7)) + (2*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^7*d) - (Cos[c + d*x]^5*(6*a - 5*b*Sin[c + d*x]))/(30*b^2*d) + (Cos[c + d*x]^3*(8*a*(a^2 - b^2) - b*(6*a^2 - 5*b^2)*Sin[c + d*x]))/(24*b^4*d) - (Cos[c + d*x]*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^2*b^2 + 5*b^4)*Sin[c + d*x]))/(16*b^6*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 14, -((3*x)/(8*b)) - ((a^2 - 3*b^2)*x)/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*x)/b^5 + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*b^5*d) - ArcTanh[Cos[c + d*x]]/(a*d) - (a*Cos[c + d*x])/(b^2*d) - (a*(a^2 - 3*b^2)*Cos[c + d*x])/(b^4*d) + (a*Cos[c + d*x]^3)/(3*b^2*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) + ((a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*d) + (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 13, (a*x)/(2*b^2) + (a*(a^2 - 3*b^2)*x)/b^4 - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*b^4*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) + Cos[c + d*x]/(b*d) + ((a^2 - 3*b^2)*Cos[c + d*x])/(b^3*d) - Cos[c + d*x]^3/(3*b*d) - Cot[c + d*x]/(a*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 6, -(((2*a^2 - 5*b^2)*x)/(2*b^3)) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*b^3*d) + ((5*a^2 - 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (a*Cos[c + d*x])/(b^2*d) + (b*Cot[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^4/(a + b*Sin[c + d*x]), x, 13, (a*x)/b^2 - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*b^2*d) + (b*ArcTanh[Cos[c + d*x]])/(2*a^2*d) - (b*(3*a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + Cos[c + d*x]/(b*d) - Cot[c + d*x]/(a*d) + ((3*a^2 - b^2)*Cot[c + d*x])/(a^3*d) - Cot[c + d*x]^3/(3*a*d) + (b*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^5/(a + b*Sin[c + d*x]), x, 15, -(x/b) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*b*d) - ((15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^5*d) + (b*(-2*a^2 + b^2)*Cot[c + d*x])/(a^4*d) + (b*Cot[c + d*x]^3)/(3*a^2*d) + ((7*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^3*d) - (Cot[c + d*x]^3*Csc[c + d*x])/(4*a*d), -(x/b) + (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*b*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + ((3*a^2 - b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - ((3*a^4 - 3*a^2*b^2 + b^4)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (b*Cot[c + d*x])/(a^2*d) - (b*(3*a^2 - b^2)*Cot[c + d*x])/(a^4*d) + (b*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + ((3*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^6/(a + b*Sin[c + d*x]), x, 9, -((2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d)) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) + (b*(-9*a^2 + 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*d) + ((11*a^2 - 5*b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^3*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d), -((2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^6*d)) + (b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^6*d) - ((23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^5*d) - (Cot[c + d*x]*Csc[c + d*x])/(b*d) + ((8*a^4 - 9*a^2*b^2 + 4*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(2*b^2*d) - ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^3)/(4*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^7/(a + b*Sin[c + d*x]), x, 10, (2*b*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^7*d) + ((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^7*d) + (b*(23*a^4 - 35*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(15*a^6*d) - ((11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(2*b*d) + ((15*a^4 - 22*a^2*b^2 + 10*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^3)/(3*b^2*d) - ((8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^4)/(5*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^8/(a + b*Sin[c + d*x]), x, 11, -((2*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^8*d)) - (b*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + ((15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(105*a^7*d) + (b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) - ((45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(3*b*d) + ((8*a^4 - 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(4*b^2*d) - ((35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)} -{Cos[c + d*x]^6*Csc[c + d*x]^9/(a + b*Sin[c + d*x]), x, 12, (2*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^9*d) + ((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*ArcTanh[Cos[c + d*x]])/(128*a^9*d) - (b*(15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[c + d*x])/(105*a^8*d) + ((5*a^6 - 88*a^4*b^2 + 144*a^2*b^4 - 64*b^6)*Cot[c + d*x]*Csc[c + d*x])/(128*a^7*d) + (b*(45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^6*d) - ((59*a^4 - 104*a^2*b^2 + 48*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(4*b*d) + ((35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(5*b^2*d) - ((48*a^4 - 85*a^2*b^2 + 40*b^4)*Cot[c + d*x]*Csc[c + d*x]^5)/(240*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6)/(7*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*a*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[c + d*x]*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, -Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a^3*Log[a + b*Sin[c + d*x]])/(b^2*(a^2 - b^2)*d) - Sin[c + d*x]/(b*d)} -{Sec[c + d*x]*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (a^2*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)*d)} -{Sec[c + d*x]*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 3, -Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (a*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)} -{Sec[c + d*x]*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, -Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[Sin[c + d*x]]/(a*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (b^2*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)*d)} -{Sec[c + d*x]*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b^3*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d)} -{Sec[c + d*x]*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + ((a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - Log[1 + Sin[c + d*x]]/(2*(a - b)*d) + (b^4*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)*d)} - - -{Sec[c + d*x]^2*Sin[c + d*x]^5/(a + b*Sin[c + d*x]), x, 14, (3*b*x)/(2*(a^2 - b^2)) - (a^2*(2*a^2 + b^2)*x)/(2*b^3*(a^2 - b^2)) + (2*a^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) + (a*Cos[c + d*x])/((a^2 - b^2)*d) - (a^3*Cos[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*Sec[c + d*x])/((a^2 - b^2)*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d) - (3*b*Tan[c + d*x])/(2*(a^2 - b^2)*d) + (b*Sin[c + d*x]^2*Tan[c + d*x])/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + b*Sin[c + d*x]), x, 12, -((a*x)/(a^2 - b^2)) + (a^3*x)/(b^2*(a^2 - b^2)) - (2*a^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(3/2)*d) + (a^2*Cos[c + d*x])/(b*(a^2 - b^2)*d) - (b*Cos[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 9, -((a^2*x)/(b*(a^2 - b^2))) + (b*x)/(a^2 - b^2) + (2*a^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)*d) + (a*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Tan[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 8, -((2*a^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d)) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 5, (2*a*b*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) + (Sec[c + d*x]*(a - b*Sin[c + d*x]))/((a^2 - b^2)*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 10, (2*b^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(3/2)*d) - ArcTanh[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + (b*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a*(a^2 - b^2)*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 13, -((2*b^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d)) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) - (b*Sec[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d), -((2*b^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d)) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) - (b*Sec[c + d*x])/(a^2*d) - (b^2*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a^2*(a^2 - b^2)*d) + Tan[c + d*x]/(a*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 17, (2*b^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d) - ((3*a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^3*d) + (b*Cot[c + d*x])/(a^2*d) + ((3*a^2 - b^2)*Sec[c + d*x])/(2*a*(a^2 - b^2)*d) - (Csc[c + d*x]^2*Sec[c + d*x])/(2*a*d) - (b*Tan[c + d*x])/((a^2 - b^2)*d), (2*b^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(3/2)*d) - (3*ArcTanh[Cos[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Cot[c + d*x])/(a^2*d) + (3*Sec[c + d*x])/(2*a*d) + (b^2*Sec[c + d*x])/(a^3*d) - (Csc[c + d*x]^2*Sec[c + d*x])/(2*a*d) + (b^3*Sec[c + d*x]*(b - a*Sin[c + d*x]))/(a^3*(a^2 - b^2)*d) - (b*Tan[c + d*x])/(a^2*d)} - - -{Sec[c + d*x]^3*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, ((2*a + b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a - b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^3*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 5, (a*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) - (a*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (a^2*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^3*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 5, -((b*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d)) + (b*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (a*b^2*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (Sec[c + d*x]^2*(a - b*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{Sec[c + d*x]^3*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, -((2*a + 3*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + Log[Sin[c + d*x]]/(a*d) - ((2*a - 3*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (b^4*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) + 1/(4*(a - b)*d*(1 + Sin[c + d*x]))} -{Sec[c + d*x]^3*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - ((3*a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((3*a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^5*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))} -{Sec[c + d*x]^3*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((4*a + 5*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((2*a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((4*a - 5*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) - (b^6*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) + 1/(4*(a - b)*d*(1 + Sin[c + d*x]))} - - -{Sec[c + d*x]^4*Sin[c + d*x]^4/(a + b*Sin[c + d*x]), x, 13, (2*a^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (a^2*b*Sec[c + d*x])/((a^2 - b^2)^2*d) + (b*Sec[c + d*x])/((a^2 - b^2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^3*Tan[c + d*x])/((a^2 - b^2)^2*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 10, -((2*a^3*b*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (a*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]*(a - b*Sin[c + d*x]))/((a^2 - b^2)^2*d) - (b*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 10, (2*a^2*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d) + (a^2*Sec[c + d*x]*(b - a*Sin[c + d*x]))/((a^2 - b^2)^2*d) + (a*Tan[c + d*x])/((a^2 - b^2)*d) + (a*Tan[c + d*x]^3)/(3*(a^2 - b^2)*d)} -{Sec[c + d*x]^4*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, -((2*a*b^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) + (Sec[c + d*x]^3*(a - b*Sin[c + d*x]))/(3*(a^2 - b^2)*d) - (Sec[c + d*x]*(3*a*b^2 - b*(a^2 + 2*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 12, -((2*b^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)*d)) - ArcTanh[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^3/(3*a*d) + (b*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a*(a^2 - b^2)*d) - (b*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 15, (2*b^6*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) + (b*(-a^2 + 2*b^2)*Sec[c + d*x])/((a^2 - b^2)^2*d) + (b*Sec[c + d*x]^3*(-a + b*Sin[c + d*x]))/(3*a*(a^2 - b^2)*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*Tan[c + d*x])/(3*a*(a^2 - b^2)^2*d) + Tan[c + d*x]^3/(3*a*d), (2*b^6*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - Cot[c + d*x]/(a*d) - (b*Sec[c + d*x])/(a^2*d) - (b*Sec[c + d*x]^3)/(3*a^2*d) - (b^2*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*d) + (2*Tan[c + d*x])/(a*d) + Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^4*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 20, -((2*b^7*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d)) - (5*ArcTanh[Cos[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Cot[c + d*x])/(a^2*d) + (5*Sec[c + d*x])/(2*a*d) + (b^2*Sec[c + d*x])/(a^3*d) + (5*Sec[c + d*x]^3)/(6*a*d) + (b^2*Sec[c + d*x]^3)/(3*a^3*d) - (Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*a*d) + (b^3*Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*d) - (b^3*Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*d) - (2*b*Tan[c + d*x])/(a^2*d) - (b*Tan[c + d*x]^3)/(3*a^2*d)} - - -{Sec[c + d*x]^5*Sin[c + d*x]^8/(a + b*Sin[c + d*x]), x, 6, -(((35*a^2 + 57*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) + ((35*a^2 - 57*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^8*Log[a + b*Sin[c + d*x]])/(b^3*(a^2 - b^2)^3*d) + (a*Sin[c + d*x])/(b^2*d) - Sin[c + d*x]^2/(2*b*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(4*a^2 - 3*b^2) - a*(13*a^2 - 9*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^7/(a + b*Sin[c + d*x]), x, 6, -(((24*a^2 + 37*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) - ((24*a^2 - 37*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^7*Log[a + b*Sin[c + d*x]])/(b^2*(a^2 - b^2)^3*d) - Sin[c + d*x]/(b*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(3*a^2 - 2*b^2) - b*(13*a^2 - 9*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^6/(a + b*Sin[c + d*x]), x, 6, -(((15*a^2 + 21*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) + ((15*a^2 - 21*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^6*Log[a + b*Sin[c + d*x]])/(b*(a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(3*a^2 - 2*b^2) - a*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^5/(a + b*Sin[c + d*x]), x, 5, -(((8*a^2 + 9*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) - ((8*a^2 - 9*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*(2*a^2 - b^2) - b*(9*a^2 - 5*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^4/(a + b*Sin[c + d*x]), x, 6, -((a*(3*a + b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) + (a*(3*a - b)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^4*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b*(2*a^2 - b^2) - a*(5*a^2 - b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^3/(a + b*Sin[c + d*x]), x, 6, (b*(3*a + b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - ((3*a - b)*b*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a^3*b^2*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a^3 - b*(5*a^2 - b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^2/(a + b*Sin[c + d*x]), x, 6, (a*(a + 3*b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - (a*(a - 3*b)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (a^2*b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (a*Sec[c + d*x]^2*(4*a*b - (a^2 + 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^1/(a + b*Sin[c + d*x]), x, 6, -((b*(a + 3*b)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d)) + ((a - 3*b)*b*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (a*b^4*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + (Sec[c + d*x]^4*(a - b*Sin[c + d*x]))/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*(4*a*b^2 - b*(a^2 + 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^1/(a + b*Sin[c + d*x]), x, 4, -((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + Log[Sin[c + d*x]]/(a*d) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (b^6*Log[a + b*Sin[c + d*x]])/(a*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (5*a + 7*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) + (5*a - 7*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^2/(a + b*Sin[c + d*x]), x, 4, -(Csc[c + d*x]/(a*d)) - ((15*a^2 + 37*a*b + 24*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + ((15*a^2 - 37*a*b + 24*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^7*Log[a + b*Sin[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (7*a + 9*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) - 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (7*a - 9*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))} -{Sec[c + d*x]^5*Csc[c + d*x]^3/(a + b*Sin[c + d*x]), x, 4, (b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) - ((24*a^2 + 57*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2 + b^2)*Log[Sin[c + d*x]])/(a^3*d) - ((24*a^2 - 57*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) + (b^8*Log[a + b*Sin[c + d*x]])/(a^3*(a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Sin[c + d*x])^2) + (9*a + 11*b)/(16*(a + b)^2*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) + (9*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^p (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(p/2) Sin[e+f x]^n / (a+b Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^4)/(a + b*Sin[e + f*x]), x, 21, (a^4*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (a^4*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*(-a^2 + b^2)^(1/4)*f) - (2*a^2*(g*Cos[e + f*x])^(3/2))/(3*b^3*f*g) - (2*(g*Cos[e + f*x])^(3/2))/(3*b*f*g) + (2*(g*Cos[e + f*x])^(7/2))/(7*b*f*g^3) - (2*a^3*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(b^4*f*Sqrt[Cos[e + f*x]]) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*b^2*f*Sqrt[Cos[e + f*x]]) + (a^5*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^5*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (2*a*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^2*f*g)} -{(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 18, -((a^3*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(1/4)*f)) + (a^3*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(1/4)*f) + (2*a*(g*Cos[e + f*x])^(3/2))/(3*b^2*f*g) + (2*a^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^3*f*Sqrt[Cos[e + f*x]]) + (4*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b*f*Sqrt[Cos[e + f*x]]) - (a^4*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^4*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b*f*g)} -{(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 15, (a^2*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (a^2*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(1/4)*f) - (2*(g*Cos[e + f*x])^(3/2))/(3*b*f*g) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^2*f*Sqrt[Cos[e + f*x]]) + (a^3*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^3*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{(Sqrt[g*Cos[e + f*x]]*Sin[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 12, -((a*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(1/4)*f)) + (a*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(1/4)*f) + (2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b*f*Sqrt[Cos[e + f*x]]) - (a^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 16, (Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) - (Sqrt[b]*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(1/4)*f) - (Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + (Sqrt[b]*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(1/4)*f) - (g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 19, -((b*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + (b^(3/2)*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) + (b*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (b^(3/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(1/4)*f) - ((g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a*f*g) - (Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (b*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{(Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 25, (Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) + (b^2*Sqrt[g]*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) - (b^(5/2)*Sqrt[g]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(1/4)*f) - (Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*Sqrt[g]*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(5/2)*Sqrt[g]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(1/4)*f) + (b*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a^2*f*g) - ((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(2*a*f*g) + (b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a^2*f*Sqrt[Cos[e + f*x]]) - (b^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (b^2*g*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} - - -{((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 24, (a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) + (a^3*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) - (2*a^3*g*Sqrt[g*Cos[e + f*x]])/(b^4*f) + (2*a*(g*Cos[e + f*x])^(5/2))/(5*b^2*f*g) - (2*a^4*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(b^5*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*b^3*f*Sqrt[g*Cos[e + f*x]]) + (4*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(21*b*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^3*f) + (4*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(21*b*f) - (2*(g*Cos[e + f*x])^(5/2)*Sin[e + f*x])/(7*b*f*g)} -{((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 20, -((a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f)) - (a^2*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f) + (2*a^2*g*Sqrt[g*Cos[e + f*x]])/(b^3*f) - (2*(g*Cos[e + f*x])^(5/2))/(5*b*f*g) + (2*a^3*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(b^4*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*b^2*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^2*f)} -{((g*Cos[e + f*x])^(3/2)*Sin[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 13, (a*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*f) + (a*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*f) - (2*(3*a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b^3*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]]*(3*a - b*Sin[e + f*x]))/(3*b^2*f)} -{((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 21, -((g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f)) + ((-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*Sqrt[b]*f) - (g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + ((-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*Sqrt[b]*f) - (2*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(b*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 24, (b*g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (Sqrt[b]*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*f) + (b*g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - (Sqrt[b]*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*f) - (g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a*f) + (g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(a*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{((g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 30, (g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(3/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(3/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (b^(3/2)*(-a^2 + b^2)^(1/4)*g^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (b*g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a^2*f) - (g*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(2*a*f) - (b*g^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(a^2*f*Sqrt[g*Cos[e + f*x]]) + (b*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (b*(a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} - - -{((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 24, -((a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(11/2)*f)) + (a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(11/2)*f) - (2*a^3*g*(g*Cos[e + f*x])^(3/2))/(3*b^4*f) + (2*a*(g*Cos[e + f*x])^(7/2))/(7*b^2*f*g) - (2*a^4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(b^5*f*Sqrt[Cos[e + f*x]]) + (6*a^2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*b^3*f*Sqrt[Cos[e + f*x]]) + (4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(15*b*f*Sqrt[Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^6*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^6*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (2*a^2*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^3*f) + (4*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(45*b*f) - (2*(g*Cos[e + f*x])^(7/2)*Sin[e + f*x])/(9*b*f*g)} -{((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 20, (a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) - (a^2*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(9/2)*f) + (2*a^2*g*(g*Cos[e + f*x])^(3/2))/(3*b^3*f) - (2*(g*Cos[e + f*x])^(7/2))/(7*b*f*g) + (2*a^3*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(b^4*f*Sqrt[Cos[e + f*x]]) - (6*a*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(5*b^2*f*Sqrt[Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^3*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^5*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*a*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^2*f)} -{((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 13, -((a*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f)) + (a*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*f) - (2*(5*a^2 - 3*b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*b^3*f*Sqrt[Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^2*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^4*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (2*g*(g*Cos[e + f*x])^(3/2)*(5*a - 3*b*Sin[e + f*x]))/(15*b^2*f)} -{((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^1)/(a + b*Sin[e + f*x]), x, 21, (g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*b^(3/2)*f) - (g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a*f) + ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*b^(3/2)*f) - (2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(b*f*Sqrt[Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^2)/(a + b*Sin[e + f*x]), x, 24, -((b*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f)) + ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*Sqrt[b]*f) + (b*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f) - ((-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*Sqrt[b]*f) - (g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a*f) - (g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*f*Sqrt[Cos[e + f*x]]) - ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*b*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*b*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} -{((g*Cos[e + f*x])^(5/2)*Csc[e + f*x]^3)/(a + b*Sin[e + f*x]), x, 30, -((3*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f)) + (b^2*g^(5/2)*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) - (Sqrt[b]*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (3*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f) - (b^2*g^(5/2)*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f) + (Sqrt[b]*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*f) + (b*g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x])/(a^2*f) - (g*(g*Cos[e + f*x])^(3/2)*Csc[e + f*x]^2)/(2*a*f) + (b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a^2*f*Sqrt[Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a^2*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + ((a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a^2*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^4/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 23, -((a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g])) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(7/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (2*a^2*Sqrt[g*Cos[e + f*x]])/(b^3*f*g) - (2*Sqrt[g*Cos[e + f*x]])/(b*f*g) + (2*(g*Cos[e + f*x])^(5/2))/(5*b*f*g^3) - (2*a^3*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(b^4*f*Sqrt[g*Cos[e + f*x]]) - (4*a*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*b^2*f*Sqrt[g*Cos[e + f*x]]) + (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (2*a*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b^2*f*g)} -{Sin[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 19, (a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (2*a*Sqrt[g*Cos[e + f*x]])/(b^2*f*g) + (2*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^3*f*Sqrt[g*Cos[e + f*x]]) + (4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*b*f*Sqrt[g*Cos[e + f*x]]) - (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(3*b*f*g)} -{Sin[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 15, -((a^2*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g])) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (2*Sqrt[g*Cos[e + f*x]])/(b*f*g) - (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) + (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{Sin[e + f*x]^1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 12, (a*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (a*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*f*Sqrt[g*Cos[e + f*x]]) - (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2 + b*Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) - (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{Csc[e + f*x]^1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 16, -(ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*Sqrt[g])) + (b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*Sqrt[g]) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{Csc[e + f*x]^2/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 19, (b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*Sqrt[g]) - (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*Sqrt[g]) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a*f*g) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a*f*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b^2 + b*Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} -{Csc[e + f*x]^3/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])), x, 25, (-3*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f*Sqrt[g]) - (b^2*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f*Sqrt[g]) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) - (3*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(4*a*f*Sqrt[g]) - (b^2*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^3*f*Sqrt[g]) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^3*(-a^2 + b^2)^(3/4)*f*Sqrt[g]) + (b*Sqrt[g*Cos[e + f*x]]*Csc[e + f*x])/(a^2*f*g) - (Sqrt[g*Cos[e + f*x]]*Csc[e + f*x]^2)/(2*a*f*g) - (b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(a^2*f*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(a^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*Sqrt[g*Cos[e + f*x]])} - - -{Sin[e + f*x]^4/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 22, (a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(5/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a^2*(g*Cos[e + f*x])^(3/2))/(3*b*(a^2 - b^2)*f*g^3) - (2*b*(g*Cos[e + f*x])^(3/2))/(3*(a^2 - b^2)*f*g^3) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (2*a^3*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(b^2*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} -{Sin[e + f*x]^3/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 18, -((a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*a)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^2*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} -{Sin[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 15, (a^2*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (a^2*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} -{Sin[e + f*x]^1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 13, -((a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(5/4)*f*g^(3/2))) + (a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(5/4)*f*g^(3/2)) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (a^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*(a - b*Sin[e + f*x]))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} -{Csc[e + f*x]^1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 21, ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(3/2)) - (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(3/2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + 2/(a*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/((a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (b^2*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*(b - a*Sin[e + f*x]))/(a*(a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} -{Csc[e + f*x]^2/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 25, -((b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(3/2))) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(3/2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(5/4)*f*g^(3/2)) - (2*b)/(a^2*f*g*Sqrt[g*Cos[e + f*x]]) - Csc[e + f*x]/(a*f*g*Sqrt[g*Cos[e + f*x]]) - (3*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*f*g^2*Sqrt[Cos[e + f*x]]) - (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(1/2)*(e + f*x), 2])/(a*(a^2 - b^2)*f*g^2*Sqrt[Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) - (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*f*g*Sqrt[g*Cos[e + f*x]]) + (3*Sin[e + f*x])/(a*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b^2*(b - a*Sin[e + f*x]))/(a^2*(a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]])} - - -{Sin[e + f*x]^4/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 22, -((a^4*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^4*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(3/2)*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) + (2*a^2*Sqrt[g*Cos[e + f*x]])/(b*(a^2 - b^2)*f*g^3) - (2*b*Sqrt[g*Cos[e + f*x]])/((a^2 - b^2)*f*g^3) - (4*a*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a^3*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(b^2*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^5*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} -{Sin[e + f*x]^3/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 18, (a^3*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a^3*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(Sqrt[b]*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (2*a)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) - (2*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(b*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (4*b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (2*b*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} -{Sin[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 15, -((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2))) - (a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2)) + (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (a^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*a*Sin[e + f*x])/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} -{Sin[e + f*x]^1/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 13, (a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) + (a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/((-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^2*b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (a^2*b*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*(a - b*Sin[e + f*x]))/(3*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} -{Csc[e + f*x]^1/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 21, -(ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(5/2))) + (b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]]/(a*f*g^(5/2)) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + 2/(3*a*f*g*(g*Cos[e + f*x])^(3/2)) - (2*b*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) + (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (b^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*b*(b - a*Sin[e + f*x]))/(3*a*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} -{Csc[e + f*x]^2/((g*Cos[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 25, (b*ArcTan[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(5/2)) - (b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) + (b*ArcTanh[Sqrt[g*Cos[e + f*x]]/Sqrt[g]])/(a^2*f*g^(5/2)) - (b^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(a^2*(-a^2 + b^2)^(7/4)*f*g^(5/2)) - (2*b)/(3*a^2*f*g*(g*Cos[e + f*x])^(3/2)) - Csc[e + f*x]/(a*f*g*(g*Cos[e + f*x])^(3/2)) + (5*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*a*f*g^2*Sqrt[g*Cos[e + f*x]]) + (2*b^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2])/(3*a*(a^2 - b^2)*f*g^2*Sqrt[g*Cos[e + f*x]]) - (b^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) - (b^4*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (1/2)*(e + f*x), 2])/(a*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*f*g^2*Sqrt[g*Cos[e + f*x]]) + (5*Sin[e + f*x])/(3*a*f*g*(g*Cos[e + f*x])^(3/2)) - (2*b^2*(b - a*Sin[e + f*x]))/(3*a^2*(a^2 - b^2)*f*g*(g*Cos[e + f*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(p/2) (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(5/2), x, 31, (a^2*d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (d^(5/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) - (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + (a^2*d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) + (d^(5/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (d^2*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f*g) - (a*d^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(3/2), x, 19, -((a*d^(3/2)*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*f)) + (a*d^(3/2)*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*f) + (a*d^(3/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*f) - (a*d^(3/2)*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*f) + (2*Sqrt[2]*a^2*d^2*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*a^2*d^2*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (d*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b*f*Sqrt[Sin[2*e + 2*f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(1/2), x, 16, (Sqrt[d]*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*f) + (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*f) - (2*Sqrt[2]*a*d*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a*d*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(1/2), x, 5, (2*Sqrt[2]*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Sin[e + f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(3/2), x, 9, -((2*(g*Cos[e + f*x])^(3/2))/(a*d*f*g*Sqrt[d*Sin[e + f*x]])) - (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a + b]*Sqrt[a + b]*d*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a + b]*Sqrt[a + b]*d*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*d^2*f*Sqrt[Sin[2*e + 2*f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(5/2), x, 11, -((2*(g*Cos[e + f*x])^(3/2))/(3*a*d*f*g*(d*Sin[e + f*x])^(3/2))) + (2*b*(g*Cos[e + f*x])^(3/2))/(a^2*d^2*f*g*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*Sqrt[-a + b]*Sqrt[a + b]*d^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*Sqrt[-a + b]*Sqrt[a + b]*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*d^3*f*Sqrt[Sin[2*e + 2*f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(7/2), x, 16, -((2*(g*Cos[e + f*x])^(3/2))/(5*a*d*f*g*(d*Sin[e + f*x])^(5/2))) + (2*b*(g*Cos[e + f*x])^(3/2))/(3*a^2*d^2*f*g*(d*Sin[e + f*x])^(3/2)) - (4*(g*Cos[e + f*x])^(3/2))/(5*a*d^3*f*g*Sqrt[d*Sin[e + f*x]]) - (2*b^2*(g*Cos[e + f*x])^(3/2))/(a^3*d^3*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*Sqrt[-a + b]*Sqrt[a + b]*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*Sqrt[-a + b]*Sqrt[a + b]*d^3*f*Sqrt[d*Sin[e + f*x]]) - (4*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a*d^4*f*Sqrt[Sin[2*e + 2*f*x]]) - (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^3*d^4*f*Sqrt[Sin[2*e + 2*f*x]])} -{Sqrt[g*Cos[e + f*x]]/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(9/2), x, 19, -((2*(g*Cos[e + f*x])^(3/2))/(7*a*d*f*g*(d*Sin[e + f*x])^(7/2))) + (2*b*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^2*f*g*(d*Sin[e + f*x])^(5/2)) - (8*(g*Cos[e + f*x])^(3/2))/(21*a*d^3*f*g*(d*Sin[e + f*x])^(3/2)) - (2*b^2*(g*Cos[e + f*x])^(3/2))/(3*a^3*d^3*f*g*(d*Sin[e + f*x])^(3/2)) + (4*b*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^4*f*g*Sqrt[d*Sin[e + f*x]]) + (2*b^3*(g*Cos[e + f*x])^(3/2))/(a^4*d^4*f*g*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^4*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*Sqrt[-a + b]*Sqrt[a + b]*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^4*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*Sqrt[-a + b]*Sqrt[a + b]*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^2*d^5*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^3*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^4*d^5*f*Sqrt[Sin[2*e + 2*f*x]])} - - -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(3/2), x, 31, (3*d^(3/2)*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(4*Sqrt[2]*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^3*f) - (3*d^(3/2)*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(4*Sqrt[2]*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^3*f) + (2*Sqrt[2]*a*Sqrt[-a^2 + b^2]*d^(3/2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^3*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*a*Sqrt[-a^2 + b^2]*d^(3/2)*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^3*f*Sqrt[g*Cos[e + f*x]]) - (3*d^(3/2)*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(8*Sqrt[2]*b*f) - ((a^2 - b^2)*d^(3/2)*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^3*f) + (3*d^(3/2)*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(8*Sqrt[2]*b*f) + ((a^2 - b^2)*d^(3/2)*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^3*f) - (a*d*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b^2*f) + (g*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))/(2*b*f) + (a*d^2*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(1/2), x, 19, -((a*Sqrt[d]*g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f)) + (a*Sqrt[d]*g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f) - (2*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[d]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*Sqrt[d]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*f*Sqrt[g*Cos[e + f*x]]) + (a*Sqrt[d]*g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f) - (a*Sqrt[d]*g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f) + (g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b*f) - (d*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(1/2), x, 18, (g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]*f) - (g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]*f) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(3/2), x, 8, -((2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*d^(3/2)*f*Sqrt[g*Cos[e + f*x]])) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(a*d*f*Sqrt[d*Sin[e + f*x]]) - (b*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^2*d*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(5/2), x, 12, (2*Sqrt[2]*b*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(3*a*d*f*(d*Sin[e + f*x])^(3/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - ((a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^3*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(7/2), x, 15, -((2*Sqrt[2]*b^2*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^4*d^(7/2)*f*Sqrt[g*Cos[e + f*x]])) + (2*Sqrt[2]*b^2*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^4*d^(7/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(5*a*d*f*(d*Sin[e + f*x])^(5/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(3*a^2*d^2*f*(d*Sin[e + f*x])^(3/2)) - (8*g*Sqrt[g*Cos[e + f*x]])/(5*a*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (2*b*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^2*d^3*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (b*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^4*d^3*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{(g*Cos[e + f*x])^(3/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(9/2), x, 20, (2*Sqrt[2]*b^3*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^5*d^(9/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^5*d^(9/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*g*Sqrt[g*Cos[e + f*x]])/(7*a*d*f*(d*Sin[e + f*x])^(7/2)) + (2*b*g*Sqrt[g*Cos[e + f*x]])/(5*a^2*d^2*f*(d*Sin[e + f*x])^(5/2)) - (4*g*Sqrt[g*Cos[e + f*x]])/(7*a*d^3*f*(d*Sin[e + f*x])^(3/2)) + (2*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(3*a^3*d^3*f*(d*Sin[e + f*x])^(3/2)) + (8*b*g*Sqrt[g*Cos[e + f*x]])/(5*a^2*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*b*(a^2 - b^2)*g*Sqrt[g*Cos[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(7*a*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (2*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a^3*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (b^2*(a^2 - b^2)*g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^5*d^4*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} - - -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])*(d*Sin[e + f*x])^(1/2), x, 31, -((Sqrt[d]*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f)) - ((a^2 - b^2)*Sqrt[d]*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (Sqrt[d]*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(4*Sqrt[2]*b*f) + ((a^2 - b^2)*Sqrt[d]*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^3*f) + (Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) + ((a^2 - b^2)*Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(8*Sqrt[2]*b*f) - ((a^2 - b^2)*Sqrt[d]*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^3*f) - (2*Sqrt[2]*a*Sqrt[-a + b]*Sqrt[a + b]*d*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a*Sqrt[-a + b]*Sqrt[a + b]*d*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^3*f*Sqrt[d*Sin[e + f*x]]) + (g*(g*Cos[e + f*x])^(3/2)*Sqrt[d*Sin[e + f*x]])/(2*b*f) + (a*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b^2*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(1/2), x, 19, (a*g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*Sqrt[d]*f) - (a*g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b^2*Sqrt[d]*f) - (a*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*Sqrt[d]*f) + (a*g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b^2*Sqrt[d]*f) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b^2*f*Sqrt[d*Sin[e + f*x]]) - (g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(b*d*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(3/2), x, 20, -((g^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*d^(3/2)*f)) + (g^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*d^(3/2)*f) + (g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*d^(3/2)*f) - (g^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*d^(3/2)*f) - (2*g*(g*Cos[e + f*x])^(3/2))/(a*d*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*b*d*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*b*d*f*Sqrt[d*Sin[e + f*x]]) - (2*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*d^2*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(5/2), x, 10, -((2*g*(g*Cos[e + f*x])^(3/2))/(3*a*d*f*(d*Sin[e + f*x])^(3/2))) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) + (2*b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*d^3*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(7/2), x, 15, -((2*g*(g*Cos[e + f*x])^(3/2))/(5*a*d*f*(d*Sin[e + f*x])^(5/2))) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(3*a^2*d^2*f*(d*Sin[e + f*x])^(3/2)) - (4*g*(g*Cos[e + f*x])^(3/2))/(5*a*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^3*d^3*f*Sqrt[d*Sin[e + f*x]]) - (4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a*d^4*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^3*d^4*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(9/2), x, 18, -((2*g*(g*Cos[e + f*x])^(3/2))/(7*a*d*f*(d*Sin[e + f*x])^(7/2))) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^2*f*(d*Sin[e + f*x])^(5/2)) - (8*g*(g*Cos[e + f*x])^(3/2))/(21*a*d^3*f*(d*Sin[e + f*x])^(3/2)) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(3*a^3*d^3*f*(d*Sin[e + f*x])^(3/2)) + (4*b*g*(g*Cos[e + f*x])^(3/2))/(5*a^2*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*b*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^2*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^4*d^4*f*Sqrt[d*Sin[e + f*x]]) + (4*b*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^2*d^5*f*Sqrt[Sin[2*e + 2*f*x]]) - (2*b*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^4*d^5*f*Sqrt[Sin[2*e + 2*f*x]])} -{(g*Cos[e + f*x])^(5/2)/(a + b*Sin[e + f*x])/(d*Sin[e + f*x])^(11/2), x, 24, -((2*g*(g*Cos[e + f*x])^(3/2))/(9*a*d*f*(d*Sin[e + f*x])^(9/2))) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(7*a^2*d^2*f*(d*Sin[e + f*x])^(7/2)) - (4*g*(g*Cos[e + f*x])^(3/2))/(15*a*d^3*f*(d*Sin[e + f*x])^(5/2)) + (2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(5*a^3*d^3*f*(d*Sin[e + f*x])^(5/2)) + (8*b*g*(g*Cos[e + f*x])^(3/2))/(21*a^2*d^4*f*(d*Sin[e + f*x])^(3/2)) - (2*b*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(3*a^4*d^4*f*(d*Sin[e + f*x])^(3/2)) - (8*g*(g*Cos[e + f*x])^(3/2))/(15*a*d^5*f*Sqrt[d*Sin[e + f*x]]) + (4*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(5*a^3*d^5*f*Sqrt[d*Sin[e + f*x]]) + (2*b^2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (8*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(15*a*d^6*f*Sqrt[Sin[2*e + 2*f*x]]) + (4*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^3*d^6*f*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^2*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^5*d^6*f*Sqrt[Sin[2*e + 2*f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(5/2), x, 19, (a*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f*Sqrt[g]) - (a*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b^2*f*Sqrt[g]) - (2*Sqrt[2]*a^2*d^(5/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) + (2*Sqrt[2]*a^2*d^(5/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b^2*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) - (a*d^(5/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f*Sqrt[g]) + (a*d^(5/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b^2*f*Sqrt[g]) - (d^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(b*f*g) + (d^3*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*b*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(3/2), x, 15, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*f*Sqrt[g])) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*f*Sqrt[g]) + (2*Sqrt[2]*a*d^(3/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*a*d^(3/2)*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]]) + (d^(3/2)*Log[Sqrt[d] - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*f*Sqrt[g]) - (d^(3/2)*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*f*Sqrt[g])} -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(1/2), x, 4, -((2*Sqrt[2]*Sqrt[d]*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]])) + (2*Sqrt[2]*Sqrt[d]*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Cos[e + f*x]])} -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(1/2), x, 7, (2*Sqrt[2]*b*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(3/2), x, 9, -((2*Sqrt[2]*b^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*d^(3/2)*f*Sqrt[g*Cos[e + f*x]])) + (2*Sqrt[2]*b^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*d^(3/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]])/(a*d*f*g*Sqrt[d*Sin[e + f*x]]) - (b*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^2*d*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} -{1/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(5/2), x, 13, (2*Sqrt[2]*b^3*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*Sqrt[-a^2 + b^2]*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]*b^3*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^3*Sqrt[-a^2 + b^2]*d^(5/2)*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]])/(3*a*d*f*g*(d*Sin[e + f*x])^(3/2)) + (2*b*Sqrt[g*Cos[e + f*x]])/(a^2*d^2*f*g*Sqrt[d*Sin[e + f*x]]) + (2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*a*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (b^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a^3*d^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} - - -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(5/2), x, 31, -((a^2*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2))) + (b*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) - (a^2*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) + (b*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) - (2*Sqrt[2]*a^3*d^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*a^3*d^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*d^2*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*d*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*d^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(3/2), x, 10, (2*Sqrt[2]*a^2*d^2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*a^2*d^2*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*a*d*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*d*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))*(d*Sin[e + f*x])^(1/2), x, 11, -((2*Sqrt[2]*a*b*d*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]])) + (2*Sqrt[2]*a*b*d*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(1/2), x, 11, (2*Sqrt[2]*b^2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^2*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/((-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*a*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]) - (2*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]) + (2*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(3/2), x, 16, -((2*a)/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(a*(a^2 - b^2)*d*f*g^3*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]) + (4*a*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*(a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} -{1/((g*Cos[e + f*x])^(3/2)*(a + b*Sin[e + f*x]))/(d*Sin[e + f*x])^(5/2), x, 19, -((2*a)/(3*(a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(3*a*(a^2 - b^2)*d*f*g^3*(d*Sin[e + f*x])^(3/2)) + (2*b)/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) - (2*b^3*(g*Cos[e + f*x])^(3/2))/(a^2*(a^2 - b^2)*d^2*f*g^3*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^4*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^4*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (8*a*Sqrt[d*Sin[e + f*x]])/(3*(a^2 - b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^4*f*g*Sqrt[g*Cos[e + f*x]]) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^3*f*g^2*Sqrt[Sin[2*e + 2*f*x]]) - (2*b^3*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*(a^2 - b^2)*d^3*f*g^2*Sqrt[Sin[2*e + 2*f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n / (a+b Sin[e+f x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^(p/2) (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^2), x, 8, (Sqrt[2]*b*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (Sqrt[2]*b*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a^2*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) + (g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])/(a*d*f*(a + b*Sin[e + f*x])) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(2*a^2*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]])} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^1 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^2*Sin[c + d*x]^4*(a + b*Sin[c + d*x]), x, 8, -((3*a*x)/2) + (2*b*Cos[c + d*x])/d - (b*Cos[c + d*x]^3)/(3*d) + (b*Sec[c + d*x])/d + (3*a*Tan[c + d*x])/(2*d) - (a*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x]), x, 8, -((3*b*x)/2) + (a*Cos[c + d*x])/d + (a*Sec[c + d*x])/d + (3*b*Tan[c + d*x])/(2*d) - (b*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, (-a)*x + (b*Cos[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x]), x, 5, (-b)*x + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x]), x, 6, -((a*ArcTanh[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d + (b*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x]), x, 7, -((b*ArcTanh[Cos[c + d*x]])/d) - (a*Cot[c + d*x])/d + (b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x]), x, 8, -((3*a*ArcTanh[Cos[c + d*x]])/(2*d)) - (b*Cot[c + d*x])/d + (3*a*Sec[c + d*x])/(2*d) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (b*Tan[c + d*x])/d} - - -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 8, -3*a*b*x + ((a^2 + 2*b^2)*Cos[c + d*x])/d - (b^2*Cos[c + d*x]^3)/(3*d) + ((a^2 + b^2)*Sec[c + d*x])/d + (3*a*b*Tan[c + d*x])/d - (a*b*Sin[c + d*x]^2*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 11, (-a^2)*x - (3*b^2*x)/2 + (2*a*b*Cos[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (3*b^2*Tan[c + d*x])/(2*d) - (b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 4, -2*a*b*x + (2*b^2*Cos[c + d*x])/d + (Sec[c + d*x]*(a + b*Sin[c + d*x])^2)/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 8, ((a^2 + b^2)*Sec[c + d*x])/d - (a^2*ArcTanh[Cos[c + d*x]])/d + (2*a*b*Tan[c + d*x])/d, ((a^2 + b^2)*Sec[c + d*x])/d - (a^2*ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x])/d + (2*a*b*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 7, -((2*a*b*ArcTanh[Cos[c + d*x]])/d) - (a^2*Cot[c + d*x])/d + (2*a*b*Sec[c + d*x])/d + ((a^2 + b^2)*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 10, -(((3*a^2 + 2*b^2)*ArcTanh[Cos[c + d*x]])/(2*d)) - (2*a*b*Cot[c + d*x])/d + ((3*a^2 + 2*b^2)*Sec[c + d*x])/(2*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (2*a*b*Tan[c + d*x])/d, -((2*a*b*Cot[c + d*x])/d) + ((3*a^2 + 2*b^2)*Sec[c + d*x])/(2*d) - ((3*a^2 + 2*b^2)*ArcTanh[Sqrt[Cos[c + d*x]^2]]*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x])/(2*d) - (a^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (2*a*b*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 8, -((3*a*b*ArcTanh[Cos[c + d*x]])/d) - ((2*a^2 + b^2)*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (3*a*b*Sec[c + d*x])/d - (a*b*Csc[c + d*x]^2*Sec[c + d*x])/d + ((a^2 + b^2)*Tan[c + d*x])/d} - - -{Sec[c + d*x]^2*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 17, (-(9/2))*a^2*b*x - (15*b^3*x)/8 + (a^3*Cos[c + d*x])/d + (6*a*b^2*Cos[c + d*x])/d - (a*b^2*Cos[c + d*x]^3)/d + (a^3*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d + (9*a^2*b*Tan[c + d*x])/(2*d) + (15*b^3*Tan[c + d*x])/(8*d) - (3*a^2*b*Sin[c + d*x]^2*Tan[c + d*x])/(2*d) - (5*b^3*Sin[c + d*x]^2*Tan[c + d*x])/(8*d) - (b^3*Sin[c + d*x]^4*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 14, (-a^3)*x - (9/2)*a*b^2*x + (3*a^2*b*Cos[c + d*x])/d + (2*b^3*Cos[c + d*x])/d - (b^3*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (9*a*b^2*Tan[c + d*x])/(2*d) - (3*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 3, (-(3/2))*b*(2*a^2 + b^2)*x + (6*a*b^2*Cos[c + d*x])/d + (3*b^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (Sec[c + d*x]*(a + b*Sin[c + d*x])^3)/d} -{Sec[c + d*x]^2*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 11, (-b^3)*x - (a^3*ArcTanh[Cos[c + d*x]])/d + (a^3*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d + (3*a^2*b*Tan[c + d*x])/d + (b^3*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 12, -((3*a^2*b*ArcTanh[Cos[c + d*x]])/d) - (a^3*Cot[c + d*x])/d + (3*a^2*b*Sec[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 14, -((3*a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (3*a*b^2*ArcTanh[Cos[c + d*x]])/d - (3*a^2*b*Cot[c + d*x])/d + (3*a^3*Sec[c + d*x])/(2*d) + (3*a*b^2*Sec[c + d*x])/d - (a^3*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (3*a^2*b*Tan[c + d*x])/d + (b^3*Tan[c + d*x])/d} -{Sec[c + d*x]^2*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 15, -((9*a^2*b*ArcTanh[Cos[c + d*x]])/(2*d)) - (b^3*ArcTanh[Cos[c + d*x]])/d - (2*a^3*Cot[c + d*x])/d - (3*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (9*a^2*b*Sec[c + d*x])/(2*d) + (b^3*Sec[c + d*x])/d - (3*a^2*b*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a^3*Tan[c + d*x])/d + (3*a*b^2*Tan[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + b*Sin[c + d*x])^2, x, 12, -(x/b^2) - (2*a^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)*d) + (4*a^3*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^4*Cos[c + d*x])/(b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 12, (2*a^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)*d) - (2*a^2*(a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (a^3*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 12, -((2*a^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d)) - (4*a*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 6, (2*b*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (a*Sec[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x]))/((a^2 - b^2)^2*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^2, x, 13, (2*b^3*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (2*b^3*(3*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])/(a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^2, x, 15, -((2*b^4*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)*d)) - (4*b^4*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - Cot[c + d*x]/(a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) - (b^5*Cos[c + d*x])/(a^2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^2, x, 17, (2*b^5*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)*d) + (2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(5/2)*d) - ArcTanh[Cos[c + d*x]]/(2*a^2*d) - ((a^2 + 3*b^2)*ArcTanh[Cos[c + d*x]])/(a^4*d) + (2*b*Cot[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + Cos[c + d*x]/(2*(a + b)^2*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^2*d*(1 + Sin[c + d*x])) + (b^6*Cos[c + d*x])/(a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} - - -{Sec[c + d*x]^2*Sin[c + d*x]^4/(a + b*Sin[c + d*x])^3, x, 18, (4*a^4*(a^2 - 2*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) - (a^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) - (2*a^2*(a^4 - 3*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^4*Cos[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^5*Cos[c + d*x])/(2*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (2*a^3*(a^2 - 2*b^2)*Cos[c + d*x])/(b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 18, -((2*a^3*(a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(7/2)*d)) + (a^3*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(7/2)*d) + (2*a*b*(a^2 + 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (a^3*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*a^4*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (a^2*(a^2 - 3*b^2)*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 18, -((4*a^2*b^2*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d)) - (a^2*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (2*b^2*(3*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (a^2*b*Cos[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*a^3*b*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*a*b^3*Cos[c + d*x])/((a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Sin[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 7, (3*a*b*(2*a^2 + 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (a*Sec[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - ((3*a^2 + 2*b^2)*Sec[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]*(3*a*(2*a^2 + 3*b^2) - b*(11*a^2 + 4*b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)} -{Sec[c + d*x]^2*Csc[c + d*x]^1/(a + b*Sin[c + d*x])^3, x, 19, (2*b^3*(3*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (b^3*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(7/2)*d) + (2*b^3*(6*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^4*Cos[c + d*x])/(2*a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^4*Cos[c + d*x])/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^4*(3*a^2 - b^2)*Cos[c + d*x])/(a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^2/(a + b*Sin[c + d*x])^3, x, 21, -((4*b^4*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(7/2)*d)) - (b^4*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(7/2)*d) - (2*b^4*(10*a^4 - 9*a^2*b^2 + 3*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(7/2)*d) + (3*b*ArcTanh[Cos[c + d*x]])/(a^4*d) - Cot[c + d*x]/(a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) - (b^5*Cos[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (3*b^5*Cos[c + d*x])/(2*a*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (2*b^5*(2*a^2 - b^2)*Cos[c + d*x])/(a^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} -{Sec[c + d*x]^2*Csc[c + d*x]^3/(a + b*Sin[c + d*x])^3, x, 23, (2*b^5*(5*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (b^5*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(7/2)*d) + (2*b^5*(15*a^4 - 17*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(7/2)*d) - ArcTanh[Cos[c + d*x]]/(2*a^3*d) - ((a^2 + 6*b^2)*ArcTanh[Cos[c + d*x]])/(a^5*d) + (3*b*Cot[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + Cos[c + d*x]/(2*(a + b)^3*d*(1 - Sin[c + d*x])) + Cos[c + d*x]/(2*(a - b)^3*d*(1 + Sin[c + d*x])) + (b^6*Cos[c + d*x])/(2*a^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) + (3*b^6*Cos[c + d*x])/(2*a^2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) + (b^6*(5*a^2 - 3*b^2)*Cos[c + d*x])/(a^4*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(1/2)/Sqrt[d*Sin[e + f*x]], x, 2, (Sec[e + f*x]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])/(d*f) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f)} - - -{Sec[e + f*x]^2*(a + b*Sin[e + f*x])^(3/2)/Sqrt[d*Sin[e + f*x]], x, -1, (Sec[e + f*x]*(b + a*Sin[e + f*x])*Sqrt[a + b*Sin[e + f*x]])/(f*Sqrt[d*Sin[e + f*x]]) - ((a + b)^(3/2)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(Sqrt[d]*f) - (b*(a + b)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(b + a*Csc[e + f*x])/(-a + b)]*EllipticE[ArcSin[Sqrt[-((b + a*Csc[e + f*x])/(a - b))]], (-a + b)/(a + b)]*(1 + Sin[e + f*x])*Tan[e + f*x])/(f*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^3 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{(Sec[e + f*x]^4*(a + b*Sin[e + f*x])^(5/2))/Sqrt[d*Sin[e + f*x]], x, -1, (5*a*Sec[e + f*x]*(b + a*Sin[e + f*x])*Sqrt[a + b*Sin[e + f*x]])/(6*f*Sqrt[d*Sin[e + f*x]]) + (Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(5/2))/(3*d*f) - (5*a*(a + b)^(3/2)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(6*Sqrt[d]*f) - (5*a*b*(a + b)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(b + a*Csc[e + f*x])/(-a + b)]*EllipticE[ArcSin[Sqrt[-((b + a*Csc[e + f*x])/(a - b))]], (-a + b)/(a + b)]*(1 + Sin[e + f*x])*Tan[e + f*x])/(6*f*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^5*Sin[c + d*x]^7*(a + b*Sin[c + d*x]), x, 11, (35*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Cos[c + d*x]^2)/(2*d) - (3*a*Log[Cos[c + d*x]])/d - (3*a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^4)/(4*d) - (35*b*Sin[c + d*x])/(8*d) - (35*b*Sin[c + d*x]^3)/(24*d) - (7*b*Sin[c + d*x]^3*Tan[c + d*x]^2)/(8*d) + (b*Sin[c + d*x]^3*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^6*(a + b*Sin[c + d*x]), x, 10, (15*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Cos[c + d*x]^2)/(2*d) - (3*b*Log[Cos[c + d*x]])/d - (3*b*Sec[c + d*x]^2)/(2*d) + (b*Sec[c + d*x]^4)/(4*d) - (15*a*Sin[c + d*x])/(8*d) - (5*a*Sin[c + d*x]*Tan[c + d*x]^2)/(8*d) + (a*Sin[c + d*x]*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]), x, 7, -(((8*a + 15*b)*Log[1 - Sin[c + d*x]])/(16*d)) - ((8*a - 15*b)*Log[1 + Sin[c + d*x]])/(16*d) - (15*b*Sin[c + d*x])/(8*d) - ((4*a + 5*b*Sin[c + d*x])*Tan[c + d*x]^2)/(8*d) + ((a + b*Sin[c + d*x])*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x]), x, 7, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) - (b*Log[Cos[c + d*x]])/d - (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (b*Tan[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (b*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x]), x, 6, (3*b*ArcTanh[Sin[c + d*x]])/(8*d) - (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (a*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x]), x, 6, -((a*ArcTanh[Sin[c + d*x]])/(8*d)) - (a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + b*Sin[c + d*x]), x, 6, -((b*ArcTanh[Sin[c + d*x]])/(8*d)) + (a*Sec[c + d*x]^4)/(4*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + b*Sin[c + d*x]), x, 8, (3*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Log[Tan[c + d*x]])/d + (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^2)/d + (a*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x]), x, 10, (15*a*ArcTanh[Sin[c + d*x]])/(8*d) - (15*a*Csc[c + d*x])/(8*d) + (b*Log[Tan[c + d*x]])/d + (5*a*Csc[c + d*x]*Sec[c + d*x]^2)/(8*d) + (a*Csc[c + d*x]*Sec[c + d*x]^4)/(4*d) + (b*Tan[c + d*x]^2)/d + (b*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x]), x, 10, (15*b*ArcTanh[Sin[c + d*x]])/(8*d) - (a*Cot[c + d*x]^2)/(2*d) - (15*b*Csc[c + d*x])/(8*d) + (3*a*Log[Tan[c + d*x]])/d + (5*b*Csc[c + d*x]*Sec[c + d*x]^2)/(8*d) + (b*Csc[c + d*x]*Sec[c + d*x]^4)/(4*d) + (3*a*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^4*(a + b*Sin[c + d*x]), x, 11, (35*a*ArcTanh[Sin[c + d*x]])/(8*d) - (b*Cot[c + d*x]^2)/(2*d) - (35*a*Csc[c + d*x])/(8*d) - (35*a*Csc[c + d*x]^3)/(24*d) + (3*b*Log[Tan[c + d*x]])/d + (7*a*Csc[c + d*x]^3*Sec[c + d*x]^2)/(8*d) + (a*Csc[c + d*x]^3*Sec[c + d*x]^4)/(4*d) + (3*b*Tan[c + d*x]^2)/(2*d) + (b*Tan[c + d*x]^4)/(4*d)} - - -{Sec[c + d*x]^5*Sin[c + d*x]^6*(a + b*Sin[c + d*x])^2, x, 9, -(((15*a^2 + 48*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d)) + ((15*a^2 - 48*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - ((a^2 + 3*b^2)*Sin[c + d*x])/d - (a*b*Sin[c + d*x]^2)/d - (b^2*Sin[c + d*x]^3)/(3*d) - (Sec[c + d*x]^2*(11*b + 9*a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2, x, 8, -(((4*a^2 + 15*a*b + 12*b^2)*Log[1 - Sin[c + d*x]])/(8*d)) + ((15*a*b - 4*(a^2 + 3*b^2))*Log[1 + Sin[c + d*x]])/(8*d) - (2*a*b*Sin[c + d*x])/d - (b^2*Sin[c + d*x]^2)/(2*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(4*a + 5*b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2, x, 9, -(((3*a^2 + 16*a*b + 15*b^2)*Log[1 - Sin[c + d*x]])/(16*d)) + ((3*a^2 - 16*a*b + 15*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (b^2*Sin[c + d*x])/d - (Sec[c + d*x]^2*(7*b + 5*a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 7, -((b*(3*a + 4*b)*Log[1 - Sin[c + d*x]])/(8*d)) + ((3*a - 4*b)*b*Log[1 + Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(2*a + 3*b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 5, -(((a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d)) - (Sec[c + d*x]^2*(4*a*b + (a^2 + 3*b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, -((a*b*ArcTanh[Sin[c + d*x]])/(4*d)) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(4*d) - (Sec[c + d*x]^2*(b^2 + a*b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^2, x, 6, -((a*(4*a + 3*b)*Log[1 - Sin[c + d*x]])/(8*d)) + (a^2*Log[Sin[c + d*x]])/d - (a*(4*a - 3*b)*Log[1 + Sin[c + d*x]])/(8*d) + (a*Sec[c + d*x]^2*(2*a + 3*b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^2, x, 6, -((a^2*Csc[c + d*x])/d) - ((15*a^2 + 16*a*b + 3*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + (2*a*b*Log[Sin[c + d*x]])/d + ((15*a^2 - 16*a*b + 3*b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (b*Sec[c + d*x]^2*(8*a + (3 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d) + (b*Sec[c + d*x]^4*(2*a + ((a^2 + b^2)*Sin[c + d*x])/b))/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^2, x, 6, -((2*a*b*Csc[c + d*x])/d) - (a^2*Csc[c + d*x]^2)/(2*d) - ((12*a^2 + 15*a*b + 4*b^2)*Log[1 - Sin[c + d*x]])/(8*d) + ((3*a^2 + b^2)*Log[Sin[c + d*x]])/d - ((12*a^2 - 15*a*b + 4*b^2)*Log[1 + Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*(2*(2*a^2 + b^2) + 7*a*b*Sin[c + d*x]))/(4*d)} - - -{Sec[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^3, x, 8, -(((a + b)*(8*a^2 + 37*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d)) - ((a - b)*(8*a^2 - 37*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (b*(24*a^2 + 35*b^2)*Sin[c + d*x])/(8*d) - (3*a*b^2*Sin[c + d*x]^2)/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*(8*a + 11*b*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3, x, 9, -((3*(a + b)*(a^2 + 7*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*d)) + (3*(a - b)*(a^2 - 7*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*d) - (29*a*b^2*Sin[c + d*x])/(8*d) - (b^3*Sin[c + d*x]^2)/(2*d) - (Sec[c + d*x]^2*(8*b + 5*a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 8, -((3*b*(a + b)*(3*a + 5*b)*Log[1 - Sin[c + d*x]])/(16*d)) + (3*(3*a - 5*b)*(a - b)*b*Log[1 + Sin[c + d*x]])/(16*d) - (15*b^3*Sin[c + d*x])/(8*d) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2*(4*a + 7*b*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 7, ((a^3 - 9*a*b^2 - 8*b^3)*Log[1 - Sin[c + d*x]])/(16*d) - ((a^3 - 9*a*b^2 + 8*b^3)*Log[1 + Sin[c + d*x]])/(16*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(5*a*b + (a^2 + 4*b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 5, -((3*b*(a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d)) + (Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(4*d) - (3*Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(b^2 + a*b*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^1*(a + b*Sin[c + d*x])^3, x, 6, -(((8*a^3 + 9*a^2*b - b^3)*Log[1 - Sin[c + d*x]])/(16*d)) + (a^3*Log[Sin[c + d*x]])/d - ((8*a^3 - 9*a^2*b + b^3)*Log[1 + Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^2*(4*a^3 + b*(9*a^2 - b^2)*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^4*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^3, x, 6, -((a^3*Csc[c + d*x])/d) - (3*a*(a + b)*(5*a + 3*b)*Log[1 - Sin[c + d*x]])/(16*d) + (3*a^2*b*Log[Sin[c + d*x]])/d + (3*a*(5*a - 3*b)*(a - b)*Log[1 + Sin[c + d*x]])/(16*d) + (b*Sec[c + d*x]^4*(3*a^2 + b^2 + a*(3 + a^2/b^2)*b*Sin[c + d*x]))/(4*d) + (a*b*Sec[c + d*x]^2*(12*a + (9 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^3, x, 6, -((3*a^2*b*Csc[c + d*x])/d) - (a^3*Csc[c + d*x]^2)/(2*d) - (3*(a + b)*(8*a^2 + 7*a*b + b^2)*Log[1 - Sin[c + d*x]])/(16*d) + (3*a*(a^2 + b^2)*Log[Sin[c + d*x]])/d - (3*(a - b)*(8*a^2 - 7*a*b + b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (b^2*Sec[c + d*x]^4*(a*(3 + a^2/b^2) + (1 + (3*a^2)/b^2)*b*Sin[c + d*x]))/(4*d) + (b^2*Sec[c + d*x]^2*(4*a*(3 + (2*a^2)/b^2) + 3*(1 + (7*a^2)/b^2)*b*Sin[c + d*x]))/(8*d)} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with n symbolic*) - - -{Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^4, x, 6, -(((6*a^2*b^2*(1 - n^2) - a^4*(3 - 4*n + n^2) - b^4*(3 + 4*n + n^2))*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(8*d*(1 + n))) - (a*b*n*(a^2*(2 - n) - b^2*(2 + n))*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(2*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a^4 + 6*a^2*b^2 + b^4 + 4*a*b*(a^2 + b^2)*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*Sin[c + d*x]^(1 + n)*(a^4*(3 - n) - 6*a^2*b^2*(1 + n) - b^4*(5 + n) + 4*a*b*(a^2*(2 - n) - b^2*(2 + n))*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^3, x, 5, (a*(a^2*(3 - n) - 3*b^2*(1 + n))*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(4*d*(1 + n)) + (b*(3*a^2*(2 - n) - b^2*(2 + n))*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(4*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^2, x, 5, ((a^2*(3 - n) - b^2*(1 + n))*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(4*d*(1 + n)) + (a*b*(2 - n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(2*d*(2 + n)) + (Sec[c + d*x]^4*Sin[c + d*x]^(1 + n)*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(4*d)} -{Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^1, x, 4, (a*Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (b*Hypergeometric2F1[3, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(2 + n))} -{Sec[c + d*x]^5*Sin[c + d*x]^n/(a + b*Sin[c + d*x])^1, x, 10, ((3*a^2 - 9*a*b + 8*b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a - b)^3*d*(1 + n)) + ((3*a^2 + 9*a*b + 8*b^2)*Hypergeometric2F1[1, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a + b)^3*d*(1 + n)) - (b^6*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Sin[c + d*x])/a)]*Sin[c + d*x]^(1 + n))/(a*(a^2 - b^2)^3*d*(1 + n)) + ((3*a - 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a - b)^2*d*(1 + n)) + ((3*a + 5*b)*Hypergeometric2F1[2, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(16*(a + b)^2*d*(1 + n)) + (Hypergeometric2F1[3, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(8*(a - b)*d*(1 + n)) + (Hypergeometric2F1[3, 1 + n, 2 + n, Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(8*(a + b)*d*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with m symbolic*) - - -{Sec[c + d*x]^5*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x, 17, (1/(16*d*(1 + n)))*((3*AppellF1[1 + n, -p, 1, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1[1 + n, -p, 1, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1[1 + n, -p, 2, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p) + (1/(16*d*(1 + n)))*((3*AppellF1[1 + n, -p, 2, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p) + (1/(8*d*(1 + n)))*((AppellF1[1 + n, -p, 3, 2 + n, -((b*Sin[c + d*x])/a), -Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p) + (1/(8*d*(1 + n)))*((AppellF1[1 + n, -p, 3, 2 + n, -((b*Sin[c + d*x])/a), Sin[c + d*x]]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^p)/(1 + (b*Sin[c + d*x])/a)^p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)*) - - -{(Sec[e + f*x]^6*(a + b*Sin[e + f*x])^(9/2))/Sqrt[d*Sin[e + f*x]], x, -1, -((3*a*b*(-2*a^2 + b^2)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(5*f*Sqrt[d*Sin[e + f*x]])) + (Sec[e + f*x]^5*Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])^(9/2))/(5*d*f) - (3*a*Sec[e + f*x]^3*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*((-a)*(7*a^2 + b^2) + 2*b*(-7*a^2 + b^2)*Sin[e + f*x] + 5*a*(a^2 - b^2)*Sin[e + f*x]^2 + (8*a^2*b - 4*b^3)*Sin[e + f*x]^3))/(20*d*f) - (3*a*(a + b)^(3/2)*(5*a^2 + 3*a*b - 4*b^2)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(20*Sqrt[d]*f) - (3*b*(2*a^4 - 3*a^2*b^2 + b^4)*Sqrt[-((a*(-1 + Csc[e + f*x]))/(a + b))]*EllipticE[ArcSin[Sqrt[-((b + a*Csc[e + f*x])/(a - b))]], 1 - (2*a)/(a + b)]*Sqrt[d*Sin[e + f*x]]*Sqrt[-((a*Csc[e + f*x]^2*(1 + Sin[e + f*x])*(a + b*Sin[e + f*x]))/(a - b)^2)]*Tan[e + f*x])/(5*d*f*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^(n/3)*) - - -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3), x, 11, -((9*(64*a*b*c*d - 26*a^2*d^2 - b^2*(18*c^2 - 13*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(2080*d^3*f)) - (9*b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(208*d^2*f) + (3*Cos[e + f*x]*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(7/3))/(16*d*f) - (3*(c + d)^2*(208*a^2*c*d^2 - 64*a*b*d*(3*c^2 - 5*d^2) + b^2*c*(54*c^2 + d^2))*AppellF1[1/2, 1/2, -(7/3), 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(1040*Sqrt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(192*a*b*c*d - 208*a^2*d^2 - b^2*(54*c^2 + 91*d^2))*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(1040*Sqrt[2]*d^4*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^(4/3), x, 10, -((3*(6*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(130*d^2*f)) + (3*b*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(13*d*f) + (3*(c + d)^2*(6*b*c^2 - 13*a*c*d - 10*b*d^2)*AppellF1[1/2, 1/2, -(7/3), 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(65*Sqrt[2]*d^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3)) - (3*(c - d)*(c + d)^2*(6*b*c - 13*a*d)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(65*Sqrt[2]*d^3*f*Sqrt[1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^0*(c + d*Sin[e + f*x])^(4/3), x, 2, (3*AppellF1[7/3, -(1/2), -(1/2), 10/3, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/3))/(7*d*f*Sqrt[1 - (c + d*Sin[e + f*x])/(c - d)]*Sqrt[1 - (c + d*Sin[e + f*x])/(c + d)])} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^(4/3), x, 0, Unintegrable[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]), x]} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(4/3), x, 0, Unintegrable[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x])^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic*) - - -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x]} - - -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(4/3), x]} - - -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 11, (((2*a^2*d^2*(3 + n) - 4*a*b*c*d*(4 + n) + b^2*(6*c^2 - d^2*(3 + n)))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^3*f*(2 + n)*(3 + n)*(4 + n)) - (b*(3*b*c - 2*a*d)*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + n)*(4 + n)) + (Cos[e + f*x]*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(4 + n)) - (Sqrt[2]*(c + d)*(a^2*c*d^2*(12 + 7*n + n^2) - 2*a*b*d*(4 + n)*(2*c^2 - d^2*(2 + n)) + b^2*c*(6*c^2 - d^2*(3 - n - n^2)))*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^4*f*(2 + n)*(3 + n)*(4 + n)*Sqrt[1 + Sin[e + f*x]])) - (Sqrt[2]*(c^2 - d^2)*(4*a*b*c*d*(4 + n) - a^2*d^2*(12 + 7*n + n^2) - b^2*(6*c^2 + d^2*(3 + 4*n + n^2)))*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^4*f*(2 + n)*(3 + n)*(4 + n)*Sqrt[1 + Sin[e + f*x]])))} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 10, -(((2*b*c - a*d*(3 + n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(2 + n)*(3 + n))) + (b*Cos[e + f*x]*Sin[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + n)) - (Sqrt[2]*(c + d)*(a*c*d*(3 + n) - b*(2*c^2 - d^2*(2 + n)))*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^3*f*(2 + n)*(3 + n)*Sqrt[1 + Sin[e + f*x]])) - (Sqrt[2]*(c^2 - d^2)*(2*b*c - a*d*(3 + n))*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^3*f*(2 + n)*(3 + n)*Sqrt[1 + Sin[e + f*x]]))} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x])^0*(c + d*Sin[e + f*x])^n, x, 2, (AppellF1[1 + n, -(1/2), -(1/2), 2 + n, (c + d*Sin[e + f*x])/(c - d), (c + d*Sin[e + f*x])/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[1 - (c + d*Sin[e + f*x])/(c - d)]*Sqrt[1 - (c + d*Sin[e + f*x])/(c + d)])} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x])^1*(c + d*Sin[e + f*x])^n, x, 0, Unintegrable[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x]), x]} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n, x, 0, Unintegrable[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((3*a*A - b*B)*Sin[c + d*x]^3)/(3*d) - (3*(A*b + a*B)*Sin[c + d*x]^4)/(4*d) + (3*(a*A - b*B)*Sin[c + d*x]^5)/(5*d) + ((A*b + a*B)*Sin[c + d*x]^6)/(2*d) - ((a*A - 3*b*B)*Sin[c + d*x]^7)/(7*d) - ((A*b + a*B)*Sin[c + d*x]^8)/(8*d) - (b*B*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((2*a*A - b*B)*Sin[c + d*x]^3)/(3*d) - ((A*b + a*B)*Sin[c + d*x]^4)/(2*d) + ((a*A - 2*b*B)*Sin[c + d*x]^5)/(5*d) + ((A*b + a*B)*Sin[c + d*x]^6)/(6*d) + (b*B*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) - ((a*A - b*B)*Sin[c + d*x]^3)/(3*d) - ((A*b + a*B)*Sin[c + d*x]^4)/(4*d) - (b*B*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, (a*A*Sin[c + d*x])/d + ((A*b + a*B)*Sin[c + d*x]^2)/(2*d) + (b*B*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 5, -((a + b)*(A + B)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)*(A - B)*Log[1 + Sin[c + d*x]])/(2*d) - (b*B*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 3, ((a*A - b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 4, ((3*a*A - b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(4*d) + ((3*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d)} -{Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(A + B*Sin[c + d*x]), x, 5, ((5*a*A - b*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^6*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(6*d) + ((5*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((5*a*A - b*B)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d)} - - -{Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, -(((a^2 - b^2)^3*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^8*d)) + ((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^8*d) - (3*(a^2 - b^2)*(5*a^2*A*b - A*b^3 - 7*a^3*B + 3*a*b^2*B)*(a + b*Sin[c + d*x])^5)/(5*b^8*d) + ((20*a^3*A*b - 12*a*A*b^3 - 35*a^4*B + 30*a^2*b^2*B - 3*b^4*B)*(a + b*Sin[c + d*x])^6)/(6*b^8*d) - ((15*a^2*A*b - 3*A*b^3 - 35*a^3*B + 15*a*b^2*B)*(a + b*Sin[c + d*x])^7)/(7*b^8*d) + (3*(2*a*A*b - 7*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^8)/(8*b^8*d) - ((A*b - 7*a*B)*(a + b*Sin[c + d*x])^9)/(9*b^8*d) - (B*(a + b*Sin[c + d*x])^10)/(10*b^8*d)} -{Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, ((a^2 - b^2)^2*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^6*d) - ((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^6*d) + (2*(3*a^2*A*b - A*b^3 - 5*a^3*B + 3*a*b^2*B)*(a + b*Sin[c + d*x])^5)/(5*b^6*d) - ((2*a*A*b - 5*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^6)/(3*b^6*d) + ((A*b - 5*a*B)*(a + b*Sin[c + d*x])^7)/(7*b^6*d) + (B*(a + b*Sin[c + d*x])^8)/(8*b^6*d)} -{Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, -((a^2 - b^2)*(A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^4*d) + ((2*a*A*b - 3*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/(4*b^4*d) - ((A*b - 3*a*B)*(a + b*Sin[c + d*x])^5)/(5*b^4*d) - (B*(a + b*Sin[c + d*x])^6)/(6*b^4*d)} -{Cos[c + d*x]^1*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 3, ((A*b - a*B)*(a + b*Sin[c + d*x])^3)/(3*b^2*d) + (B*(a + b*Sin[c + d*x])^4)/(4*b^2*d)} -{Sec[c + d*x]^1*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 6, -((a + b)^2*(A + B)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^2*(A - B)*Log[1 + Sin[c + d*x]])/(2*d) - (b*(A*b + 2*a*B)*Sin[c + d*x])/d - (b^2*B*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 5, -(((a + b)*(a*A - b*(A + 2*B))*Log[1 - Sin[c + d*x]])/(4*d)) + ((a - b)*(a*A + b*(A - 2*B))*Log[1 + Sin[c + d*x]])/(4*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])*(A*b + a*B + (a*A + b*B)*Sin[c + d*x]))/(2*d)} -{Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 4, ((3*a^2*A - A*b^2 - 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(B + A*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(4*d) + (Sec[c + d*x]^2*(2*b*(2*a*A - b*B) + (3*a^2*A + A*b^2 - 2*a*b*B)*Sin[c + d*x]))/(8*d)} -{Sec[c + d*x]^7*(a + b*Sin[c + d*x])^2*(A + B*Sin[c + d*x]), x, 5, ((5*a^2*A - A*b^2 - 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^6*(B + A*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(6*d) + (Sec[c + d*x]^4*(2*b*(4*a*A - b*B) + (5*a^2*A + 3*A*b^2 - 2*a*b*B)*Sin[c + d*x]))/(24*d) + ((5*a^2*A - A*b^2 - 2*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 3, -(((a^2 - b^2)^3*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^8*d)) + ((a^5*A*b - 3*a^3*A*b^3 + 3*a*A*b^5 - a^6*B + 3*a^4*b^2*B - 3*a^2*b^4*B + b^6*B)*Sin[c + d*x])/(b^7*d) - ((a^4 - 3*a^2*b^2 + 3*b^4)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^6*d) + ((a^3*A*b - 3*a*A*b^3 - a^4*B + 3*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^3)/(3*b^5*d) - ((a^2 - 3*b^2)*(A*b - a*B)*Sin[c + d*x]^4)/(4*b^4*d) + ((a*A*b - a^2*B + 3*b^2*B)*Sin[c + d*x]^5)/(5*b^3*d) - ((A*b - a*B)*Sin[c + d*x]^6)/(6*b^2*d) - (B*Sin[c + d*x]^7)/(7*b*d)} -{(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 3, ((a^2 - b^2)^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^6*d) - ((a^3*A*b - 2*a*A*b^3 - a^4*B + 2*a^2*b^2*B - b^4*B)*Sin[c + d*x])/(b^5*d) + ((a^2 - 2*b^2)*(A*b - a*B)*Sin[c + d*x]^2)/(2*b^4*d) - ((a*A*b - a^2*B + 2*b^2*B)*Sin[c + d*x]^3)/(3*b^3*d) + ((A*b - a*B)*Sin[c + d*x]^4)/(4*b^2*d) + (B*Sin[c + d*x]^5)/(5*b*d)} -{(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 3, -(((a^2 - b^2)*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^4*d)) + ((a*A*b - a^2*B + b^2*B)*Sin[c + d*x])/(b^3*d) - ((A*b - a*B)*Sin[c + d*x]^2)/(2*b^2*d) - (B*Sin[c + d*x]^3)/(3*b*d)} -{(Cos[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 3, ((A*b - a*B)*Log[a + b*Sin[c + d*x]])/(b^2*d) + (B*Sin[c + d*x])/(b*d)} -{(Sec[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 3, -((A + B)*Log[1 - Sin[c + d*x]])/(2*(a + b)*d) + ((A - B)*Log[1 + Sin[c + d*x]])/(2*(a - b)*d) - ((A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)} -{(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 4, -((a*A + b*(2*A + B))*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a*A - b*(2*A - B))*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^2*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(2*(a^2 - b^2)*d)} -{(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 5, -((3*a^2*A + a*b*(9*A + B) + b^2*(8*A + 3*B))*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2*A + b^2*(8*A - 3*B) - a*b*(9*A - B))*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^4*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b^2*(A*b - a*B) + (3*a^3*A - 7*a*A*b^2 + a^2*b*B + 3*b^3*B)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)} -{(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x]), x, 6, -((5*a^3*A + a^2*b*(20*A + B) + a*b^2*(29*A + 4*B) + b^3*(16*A + 5*B))*Log[1 - Sin[c + d*x]])/(32*(a + b)^4*d) + ((5*a^3*A - b^3*(16*A - 5*B) + a*b^2*(29*A - 4*B) - a^2*b*(20*A - B))*Log[1 + Sin[c + d*x]])/(32*(a - b)^4*d) + (b^6*(A*b - a*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(6*(a^2 - b^2)*d) + (Sec[c + d*x]^4*(6*b^2*(A*b - a*B) + (5*a^3*A - 11*a*A*b^2 + a^2*b*B + 5*b^3*B)*Sin[c + d*x]))/(24*(a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(8*b^4*(A*b - a*B) - (5*a^5*A - 16*a^3*A*b^2 + 19*a*A*b^4 + a^4*b*B - 4*a^2*b^3*B - 5*b^5*B)*Sin[c + d*x]))/(16*(a^2 - b^2)^3*d)} - - -{(Cos[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 3, ((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^8*d) - ((5*a^4*A*b - 9*a^2*A*b^3 + 3*A*b^5 - 6*a^5*B + 12*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(b^7*d) + ((4*a^3*A*b - 6*a*A*b^3 - 5*a^4*B + 9*a^2*b^2*B - 3*b^4*B)*Sin[c + d*x]^2)/(2*b^6*d) - ((3*a^2*A*b - 3*A*b^3 - 4*a^3*B + 6*a*b^2*B)*Sin[c + d*x]^3)/(3*b^5*d) + ((2*a*A*b - 3*a^2*B + 3*b^2*B)*Sin[c + d*x]^4)/(4*b^4*d) - ((A*b - 2*a*B)*Sin[c + d*x]^5)/(5*b^3*d) - (B*Sin[c + d*x]^6)/(6*b^2*d) + ((a^2 - b^2)^3*(A*b - a*B))/(b^8*d*(a + b*Sin[c + d*x]))} -{(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 3, -(((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^6*d)) + ((3*a^2*A*b - 2*A*b^3 - 4*a^3*B + 4*a*b^2*B)*Sin[c + d*x])/(b^5*d) - ((2*a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x]^2)/(2*b^4*d) + ((A*b - 2*a*B)*Sin[c + d*x]^3)/(3*b^3*d) + (B*Sin[c + d*x]^4)/(4*b^2*d) - ((a^2 - b^2)^2*(A*b - a*B))/(b^6*d*(a + b*Sin[c + d*x]))} -{(Cos[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 3, ((2*a*A*b - 3*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]])/(b^4*d) - ((A*b - 2*a*B)*Sin[c + d*x])/(b^3*d) - (B*Sin[c + d*x]^2)/(2*b^2*d) + ((a^2 - b^2)*(A*b - a*B))/(b^4*d*(a + b*Sin[c + d*x]))} -{(Cos[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 3, (B*Log[a + b*Sin[c + d*x]])/(b^2*d) - (A*b - a*B)/(b^2*d*(a + b*Sin[c + d*x]))} -{(Sec[c + d*x]^1*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 3, -((A + B)*Log[1 - Sin[c + d*x]])/(2*(a + b)^2*d) + ((A - B)*Log[1 + Sin[c + d*x]])/(2*(a - b)^2*d) - ((2*a*A*b - a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + (A*b - a*B)/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{(Sec[c + d*x]^3*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 4, -((a*A + 3*A*b + 2*b*B)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a*A - 3*A*b + 2*b*B)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (b^2*(4*a*A*b - 3*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (b*(a^2*A + 3*A*b^2 - 4*a*b*B))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^2*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))} -{(Sec[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 5, -((3*a^2*A + 2*a*b*(6*A + B) + b^2*(15*A + 8*B))*Log[1 - Sin[c + d*x]])/(16*(a + b)^4*d) + ((3*a^2*A + b^2*(15*A - 8*B) - 2*a*b*(6*A - B))*Log[1 + Sin[c + d*x]])/(16*(a - b)^4*d) - (b^4*(6*a*A*b - 5*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (b*(3*a^4*A - 12*a^2*A*b^2 - 15*A*b^4 + 2*a^3*b*B + 22*a*b^3*B))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^4*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(a^2*A + 5*A*b^2 - 6*a*b*B) + (3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))} -{(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2, x, 6, -((5*a^3*A + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B) + b^3*(35*A + 16*B))*Log[1 - Sin[c + d*x]])/(32*(a + b)^5*d) + ((5*a^3*A - b^3*(35*A - 16*B) + a*b^2*(47*A - 10*B) - a^2*(25*A*b - 2*b*B))*Log[1 + Sin[c + d*x]])/(32*(a - b)^5*d) + (b^6*(8*a*A*b - 7*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - (b*(5*a^6*A - 23*a^4*A*b^2 + 47*a^2*A*b^4 + 35*A*b^6 + 2*a^5*b*B - 12*a^3*b^3*B - 54*a*b^5*B))/(16*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^6*(A*b - a*B - (a*A - b*B)*Sin[c + d*x]))/(6*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^4*(b*(a^2*A + 7*A*b^2 - 8*a*b*B) + (5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B)*Sin[c + d*x]))/(24*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(5*a^4*A - 18*a^2*A*b^2 - 35*A*b^4 + 2*a^3*b*B + 46*a*b^3*B) + 3*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*Sin[c + d*x]))/(48*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Cos[e+f x])^(m/2) (a+b Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Cos[e+f x])^(m/2) (a+b Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic*) - - -{(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x, 0, Unintegrable[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with p symbolic*) - - -{(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^1), x, 4, -((g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 - p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))) + (g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(1 - p))} -{(g*Cos[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2), x, 5, -((b*g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 - p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p))) + (b*g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)^2*f*(1 - p)) + (g*AppellF1[2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*(g*Cos[e + f*x])^(-1 + p)*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 - p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 - p)/2))/((b*c - a*d)*f*(2 - p)*(c + d*Sin[e + f*x]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with p symbolic*) - - -{(g*Sec[e + f*x])^p/((a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])), x, 5, (-((AppellF1[1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (a + b)/(a + b*Sin[e + f*x]), (a - b)/(a + b*Sin[e + f*x])]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(-((b*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])))^((1 + p)/2)*((b*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x]))^((1 + p)/2))/((b*c - a*d)*f*(1 + p))) + (AppellF1[1 + p, (1 + p)/2, (1 + p)/2, 2 + p, (c + d)/(c + d*Sin[e + f*x]), (c - d)/(c + d*Sin[e + f*x])]*Sec[e + f*x]*(g*Sec[e + f*x])^p*(-((d*(1 - Sin[e + f*x]))/(c + d*Sin[e + f*x])))^((1 + p)/2)*((d*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^((1 + p)/2))/((b*c - a*d)*f*(1 + p)))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m deleted file mode 100644 index 528bf9e..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.2.3 (g sin)^p (a+b sin)^m (c+d sin)^n.m +++ /dev/null @@ -1,354 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])*) - - -{Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 13, (1/16)*a^2*c*x - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]^5)/(5*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)} -{Sin[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 11, (1/8)*a^2*c*x - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]^5)/(5*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)} -{Sin[e + f*x]^1*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 10, (1/8)*a^2*c*x - (a^2*c*Cos[e + f*x]^3)/(3*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)} -{Sin[e + f*x]^0*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 4, (1/2)*a^2*c*x - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Csc[e + f*x]^1*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 6, (1/2)*a^2*c*x - (a^2*c*ArcTanh[Cos[e + f*x]])/f + (a^2*c*Cos[e + f*x])/f + (a^2*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Csc[e + f*x]^2*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 8, (-a^2)*c*x - (a^2*c*ArcTanh[Cos[e + f*x]])/f + (a^2*c*Cos[e + f*x])/f - (a^2*c*Cot[e + f*x])/f} -{Csc[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 7, (-a^2)*c*x + (a^2*c*ArcTanh[Cos[e + f*x]])/(2*f) - (a^2*c*Cot[e + f*x])/f - (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 6, (a^2*c*ArcTanh[Cos[e + f*x]])/(2*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^5*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 11, (a^2*c*ArcTanh[Cos[e + f*x]])/(8*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)} -{Csc[e + f*x]^6*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 11, (a^2*c*ArcTanh[Cos[e + f*x]])/(8*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]^5)/(5*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)} -{Csc[e + f*x]^7*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]), x, 13, (a^2*c*ArcTanh[Cos[e + f*x]])/(16*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]^5)/(5*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(16*f) + (a^2*c*Cot[e + f*x]*Csc[e + f*x]^3)/(24*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x]^5)/(6*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])*) - - -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2)*(c - c*Sin[c + d*x]), x, 5, -((8*a^3*c*Cos[c + d*x]^3)/(63*d*(a + a*Sin[c + d*x])^(3/2))) - (2*a^2*c*Cos[c + d*x]^3)/(21*d*Sqrt[a + a*Sin[c + d*x]]) + (4*a*c*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*c*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*d), -((2*a^2*c*Cos[c + d*x])/(9*d*Sqrt[a + a*Sin[c + d*x]])) + (2*a^2*c*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (4*a*c*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(63*d) + (2*a*c*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(9*d) - (2*c*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(21*d)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Sin[e+f x])^m / (c-c Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Sin[e+f x])^(m/2) / (c-c Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]/(Sin[e + f*x]*(c - c*Sin[e + f*x])), x, 5, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(c*f)) + (2*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])), x, 8, -((2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(Sqrt[a]*c*f)) + ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+a Sin[e+f x])^(m/2) / (c-c Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]/(c - c*Sin[e + f*x]), x, 6, (2*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(c*f) + (2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f)} -{Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c - c*Sin[e + f*x])), x, 3, (2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f*g)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])), x, 6, (Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[2]*Sqrt[a]*c*f) + (Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f)} -{1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])), x, 6, -(ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f*Sqrt[g])) + (Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(a*c*f*g)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n / Sin[e+f x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2) / Sin[e+f x]*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]/Sin[e + f*x], x, 2, (Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/f} -{Sqrt[a + a*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c - c*Sin[e + f*x]]), x, 6, -((a*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(c*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[c - c*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]), x, 6, -((c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Log[Sin[e + f*x]]*Sec[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(a*f)} -{1/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 3, (Cos[e + f*x]*Log[Tan[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Sin[e+f x])^m / (c+d Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Sin[e+f x])^(m/2) / (c+d Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]/(Sin[e + f*x]*(c + d*Sin[e + f*x])), x, 5, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(c*f)) + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(c*Sqrt[c + d]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 8, -((2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(Sqrt[a]*c*f)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*c*(c - d)*Sqrt[c + d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+a Sin[e+f x])^(m/2) / (c+d Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x, 5, -((2*Sqrt[a]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(d*f)) + (2*Sqrt[a]*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(d*Sqrt[c + d]*f)} -{Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 2, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[c]*Sqrt[c + d]*f*Sqrt[g]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 5, (Sqrt[2]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[c]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)} -{1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f*Sqrt[g])) + (2*d*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c]*(c - d)*Sqrt[c + d]*f*Sqrt[g])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (c+d Sin[e+f x])^n / (a+a Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (c+d Sin[e+f x])^(m/2) / (a+a Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*(c + c*Sin[e + f*x])), x, 9, (EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - ((a - b)*EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (2*a*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(f*(c + c*Sin[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])), x, 9, (EllipticE[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) - (EllipticF[(1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (2*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/(c*f*Sqrt[a + b*Sin[e + f*x]]) + (Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*f*(c + c*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (c+d Sin[e+f x])^(m/2) / (a+a Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]/(c + c*Sin[e + f*x]), x, 3, (2*Sqrt[g]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])], -((a - b)/(a + b))]*Sec[e + f*x]*Sqrt[(a*(1 - Sin[e + f*x]))/(a + b*Sin[e + f*x])]*Sqrt[(a*(1 + Sin[e + f*x]))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f) + (g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])} -{Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + c*Sin[e + f*x])), x, 1, -((EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/(c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])), x, 3, (g*EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))]) - (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/((a - b)*c*f)} -{1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + c*Sin[e + f*x])), x, 3, -((EllipticE[ArcSin[Cos[e + f*x]/(1 + Sin[e + f*x])], -((a - b)/(a + b))]*Sqrt[Sin[e + f*x]/(1 + Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]])/((a - b)*c*f*Sqrt[g*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a + b)*(1 + Sin[e + f*x]))])) + (2*b*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*(a - b)*c*f*Sqrt[g])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n / Sin[e+f x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2) / Sin[e+f x]*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/Sin[e + f*x], x, 5, -((2*Sqrt[a]*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f) - (2*Sqrt[a]*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/f} -{Sqrt[a + a*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x, 2, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[c]*f))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[c + d*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]), x, 5, -((2*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)) + (Sqrt[2]*Sqrt[c - d]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*f)} -{1/(Sin[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x, 5, -((2*ArcTanh[(Sqrt[a]*Sqrt[c]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c]*f)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Sin[e+f x])^m / (c+d Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+b Sin[e+f x])^m / (c+d Sin[e+f x])*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sin[e + f*x]^2/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])), x, 8, -((2*a*(a^2*c - 2*b^2*c + a*b*d)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*d)^2*f)) + (2*c^2*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]*f) + (a^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+b Sin[e+f x])^(m/2) / (c+d Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -(* {(c + d*Sin[e + f*x])^(5/2)/(Sin[e + f*x]*(a + b*Sin[e + f*x])), x, 0, -((2*d^2*EllipticE[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)])) + (2*d^2*(-2*b*c + a*d)*EllipticF[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b^2*f*Sqrt[c + d*Sin[e + f*x]]) + (2*c^3*EllipticPi[2, -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)^3*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*b^2*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -{(c + d*Sin[e + f*x])^(3/2)/(Sin[e + f*x]*(a + b*Sin[e + f*x])), x, 0, -((2*d^2*EllipticF[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + f*x]])) + (2*c^2*EllipticPi[2, -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)^2*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} *) -{(c + d*Sin[e + f*x])^(1/2)/(Sin[e + f*x]*(a + b*Sin[e + f*x])), x, 5, (2*c*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*(b*c - a*d)*EllipticPi[(2*b)/(a + b), (1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(Sin[e + f*x]*(c + d*Sin[e + f*x])^(1/2)*(a + b*Sin[e + f*x])), x, 5, (2*EllipticPi[2, (1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]]) - (2*b*EllipticPi[(2*b)/(a + b), (1/2)*(e - Pi/2 + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -(* {1/(Sin[e + f*x]*(c + d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 0, (2*d^3*Cos[e + f*x])/(c*(b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d^2*EllipticE[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(c*(b*c - a*d)*(c^2 - d^2)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*EllipticPi[2, -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*c*f*Sqrt[c + d*Sin[e + f*x]]) - (2*b^2*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*(b*c - a*d)*f*Sqrt[c + d*Sin[e + f*x]])} -{1/(Sin[e + f*x]*(c + d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 0, sdx[(2*d^3*Cos[e + f*x])/(3*c*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (2*d^3*(10*b*c^3 - 7*a*c^2*d - 6*b*c*d^2 + 3*a*d^3)*Cos[e + f*x])/(3*c^2*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*d^2*(10*b*c^3 - 7*a*c^2*d - 6*b*c*d^2 + 3*a*d^3)*EllipticE[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*c^2*(b*c - a*d)^2*(c^2 - d^2)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*d^2*EllipticF[(1/4)*(-2*e + Pi - 2*f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*c*(b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (2*EllipticPi[2, -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*c^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*b^3*EllipticPi[(2*b)/(a + b), -(Pi/4) + (1/2)*(e + f*x), (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*(a + b)*(b*c - a*d)^2*f*Sqrt[c + d*Sin[e + f*x]])]} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+b Sin[e+f x])^(m/2) / (c+d Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x, 3, (2*Sqrt[a + b]*Sqrt[g]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(d*f) - (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(d*(c + d)*f*Sqrt[a + b*Sin[e + f*x]])} -{Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 3, -((2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(c*f*Sqrt[g])) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[g*Sin[e + f*x]]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 1, (2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((c + d)*f*Sqrt[a + b*Sin[e + f*x]])} -{1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 3, -((2*Sqrt[a + b]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcSin[(Sqrt[g]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[g*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*c*f*Sqrt[g])) - (2*d*Sqrt[-Cot[e + f*x]^2]*Sqrt[(b + a*Csc[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*a)/(a + b)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(c*(c + d)*f*g*Sqrt[a + b*Sin[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Sin[e+f x])^m / (c+d Sin[e+f x])^(1/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+b Sin[e+f x])^(m/2) / (c+d Sin[e+f x])^(1/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -(* {Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2)/(a + b*Sin[e + f*x]), x, 0, 0} -{Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)/(a + b*Sin[e + f*x]), x, 0, 0} *) -{Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/2)/(a + b*Sin[e + f*x]), x, 3, (2*Sqrt[c + d]*Sqrt[g]*Sqrt[(c*(1 - Csc[e + f*x]))/(c + d)]*Sqrt[(c*(1 + Csc[e + f*x]))/(c - d)]*EllipticPi[(c + d)/d, ArcSin[(Sqrt[g]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x]])], -((c + d)/(c - d))]*Tan[e + f*x])/(b*f) + (2*(b*c - a*d)*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/(b*(a + b)*f*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[g*Sin[e + f*x]]/((c + d*Sin[e + f*x])^(1/2)*(a + b*Sin[e + f*x])), x, 1, (2*Sqrt[-Cot[e + f*x]^2]*Sqrt[(d + c*Csc[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Csc[e + f*x]]/Sqrt[2]], (2*c)/(c + d)]*Sqrt[g*Sin[e + f*x]]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sin[e + f*x]])} -(* {Sqrt[g*Sin[e + f*x]]/((c + d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])), x, 0, 0} -{Sqrt[g*Sin[e + f*x]]/((c + d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])), x, 0, 0} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n / Sin[e+f x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2) / Sin[e+f x]*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]/Sin[e + f*x], x, 3, -((2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*f)) + (2*Sqrt[c + d]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*f)} -{Sqrt[a + b*Sin[e + f*x]]/(Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]), x, 1, -((2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(Sqrt[a + b]*c*f))} - - -(* {(c + d*Sin[e + f*x])^(5/2)/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x, 0, 0} -{(c + d*Sin[e + f*x])^(3/2)/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]), x, 0, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x, 3, -((2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(a*Sqrt[a + b]*c*f)) - (2*b*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(a*Sqrt[c + d]*(b*c - a*d)*f)} - - -(* {1/(Sin[e + f*x]*(c + d*Sin[e + f*x])^(3/2)*Sqrt[a + b*Sin[e + f*x]]), x, 0, 0} -{1/(Sin[e + f*x]*(c + d*Sin[e + f*x])^(5/2)*Sqrt[a + b*Sin[e + f*x]]), x, 0, 0} *) - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (h Sin[e+f x])^q (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (h Sin[e+f x])^q (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n*) - - -(* {(g*Cos[e + f*x])^p*(h*Sin[e + f*x])^q*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n, x, 0, 0} *) - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])^p*) - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + B*Sin[e + f*x])^p, x, 4, (2^(1/2 + n)*AppellF1[1/2 + m, 1/2 - n, -p, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((B*(1 + Sin[e + f*x]))/(A - B))]*Sec[e + f*x]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^(1 + m)*(A + B*Sin[e + f*x])^p*(c - c*Sin[e + f*x])^n)/(((A + B*Sin[e + f*x])/(A - B))^p*(a*f*(1 + 2*m)))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m deleted file mode 100644 index d370fec..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.3.1 (a+b sin)^m (c+d sin)^n (A+B sin).m +++ /dev/null @@ -1,620 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c Sin[e+f x])^n (A+B Sin[e+f x])*) - - -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^3, x, 7, If[$VersionNumber>=8, -((a^3*(B*(27 + 14*n + 2*n^2) + A*(28 + 15*n + 2*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n)*(3 + n)*(4 + n))) + (a^3*(B*(15 + 19*n + 4*n^2) + A*(20 + 21*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*(4 + n)*Sqrt[Cos[e + f*x]^2]) + (a^3*(B*(9 + 4*n) + A*(11 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*(3 + n)*Sqrt[Cos[e + f*x]^2]) - (a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a + a*Sin[e + f*x])^2)/(d*f*(4 + n)) - ((A*(4 + n) + B*(6 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a^3 + a^3*Sin[e + f*x]))/(d*f*(3 + n)*(4 + n)), -((a^3*(B*(27 + 14*n + 2*n^2) + A*(28 + 15*n + 2*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(4 + n)*(6 + 5*n + n^2))) + (a^3*(B*(15 + 19*n + 4*n^2) + A*(20 + 21*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(4 + n)*(2 + 3*n + n^2)*Sqrt[Cos[e + f*x]^2]) + (a^3*(B*(9 + 4*n) + A*(11 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*(3 + n)*Sqrt[Cos[e + f*x]^2]) - (a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a + a*Sin[e + f*x])^2)/(d*f*(4 + n)) - ((A*(4 + n) + B*(6 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a^3 + a^3*Sin[e + f*x]))/(d*f*(3 + n)*(4 + n))]} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^2, x, 6, -((a^2*(A*(3 + n) + B*(4 + n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n)*(3 + n))) + (a^2*(2*B*(1 + n) + A*(3 + 2*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*Sqrt[Cos[e + f*x]^2]) + (a^2*(2*A*(3 + n) + B*(5 + 2*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*(3 + n)*Sqrt[Cos[e + f*x]^2]) - (B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a^2 + a^2*Sin[e + f*x]))/(d*f*(3 + n))} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^1, x, 5, -((a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(2 + n))) + (a*(B*(1 + n) + A*(2 + n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*Sqrt[Cos[e + f*x]^2]) + (a*(A + B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2])} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^1, x, 4, ((B - A*n + B*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(a*d*f*(1 + n)*Sqrt[Cos[e + f*x]^2]) + ((A - B)*(1 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(a*d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2]) + ((A - B)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(a + a*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^2, x, 5, -((n*(A - 2*A*n + 2*B*(1 + n))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(3*a^2*d*f*(1 + n)*Sqrt[Cos[e + f*x]^2])) + ((1 + n)*(B + 2*A*(1 - n) + 2*B*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(3*a^2*d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2]) + ((B + 2*A*(1 - n) + 2*B*n)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(3*a^2*d*f*(1 + Sin[e + f*x])) + ((A - B)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(3*d*f*(a + a*Sin[e + f*x])^2)} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^3, x, 6, -((n*(B*(3 - n - 4*n^2) + A*(2 - 9*n + 4*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(1 + n))/(15*a^3*d*f*(1 + n)*Sqrt[Cos[e + f*x]^2])) + ((1 - n)*(1 + n)*(7*A + 3*B - 4*A*n + 4*B*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^(2 + n))/(15*a^3*d^2*f*(2 + n)*Sqrt[Cos[e + f*x]^2]) + ((A - B)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(5*d*f*(a + a*Sin[e + f*x])^3) + ((A*(5 - 2*n) + 2*B*n)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(15*a*d*f*(a + a*Sin[e + f*x])^2) + ((1 - n)*(7*A + 3*B - 4*A*n + 4*B*n)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(15*d*f*(a^3 + a^3*Sin[e + f*x]))} - - -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(5/2), x, 6, -((2*a^3*(2*B*(115 + 203*n + 104*n^2 + 16*n^3) + A*(301 + 478*n + 224*n^2 + 32*n^3))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))) - (2*a^3*(2*B*(35 + 23*n + 4*n^2) + A*(77 + 50*n + 8*n^2))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*(5 + 2*n)*(7 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(2*B*(5 + n) + A*(7 + 2*n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*Sqrt[a + a*Sin[e + f*x]])/(d*f*(5 + 2*n)*(7 + 2*n)) - (2*a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(a + a*Sin[e + f*x])^(3/2))/(d*f*(7 + 2*n))} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2), x, 5, -((2*a^2*(2*B*(9 + 13*n + 4*n^2) + A*(25 + 30*n + 8*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))) - (2*a^2*(2*B*(3 + n) + A*(5 + 2*n))*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n)*Sqrt[a + a*Sin[e + f*x]])/(d*f*(5 + 2*n))} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(1/2), x, 4, -((2*a*(2*B*(1 + n) + A*(3 + 2*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))) - (2*a*B*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(1/2), x, 9, -(((A - B)*AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[a + a*Sin[e + f*x]]))) - (2*B*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(f*Sqrt[a + a*Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2), x, 10, ((A - B)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n))/(2*d*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - 4*A*n + B*(3 + 4*n))*AppellF1[1/2, -n, 1, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(4*a*f*Sqrt[a + a*Sin[e + f*x]])) - ((A - B)*(1 + 2*n)*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - Sin[e + f*x]]*(d*Sin[e + f*x])^n)/(Sin[e + f*x]^n*(2*a*f*Sqrt[a + a*Sin[e + f*x]]))} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(d*Sin[e + f*x])^n, x, 9, -((2^(3/2 + m)*B*AppellF1[1/2, -n, -(1/2) - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(Sin[e + f*x]^n*f)) - (2^(1/2 + m)*(A - B)*AppellF1[1/2, -n, 1/2 - m, 3/2, 1 - Sin[e + f*x], (1/2)*(1 - Sin[e + f*x])]*Cos[e + f*x]*(d*Sin[e + f*x])^n*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(Sin[e + f*x]^n*f)} - - -{(a + a*Sin[e + f*x])^m*(a - a*Sin[e + f*x])*(d*Sin[e + f*x])^n, x, 4, (AppellF1[1 + n, -(1/2), 1/2 - m, 2 + n, Sin[e + f*x], -Sin[e + f*x]]*Sec[e + f*x]*(d*Sin[e + f*x])^(1 + n)*(1 + Sin[e + f*x])^(1/2 - m)*(a - a*Sin[e + f*x])*(a + a*Sin[e + f*x])^m)/(d*f*(1 + n)*Sqrt[1 - Sin[e + f*x]])} - - -{Sin[c + d*x]^n*(a + a*Sin[c + d*x])^(-2 - n)*(-1 - n - (-2 - n)*Sin[c + d*x]), x, 1, -((Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + a*Sin[c + d*x])^(-2 - n))/d)} -{(a + a*Sin[c + d*x])^m*(1 + m - m*Sin[c + d*x])/Sin[c + d*x]^(m + 2), x, 1, -((Cos[c + d*x]*Sin[c + d*x]^(-1 - m)*(a + a*Sin[c + d*x])^m)/d)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^n (a+b Sin[e+f x])^m (A+B Sin[e+f x])*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sin[e + f*x]^2*(A + B*Sin[e + f*x])/(a + b*Sin[e + f*x])^2, x, 6, ((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTan[(b + a*Tan[(1/2)*(e + f*x)])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*f) - (B*Cos[e + f*x])/(b^2*f) + (a^2*(A*b - a*B)*Cos[e + f*x])/(b^2*(a^2 - b^2)*f*(a + b*Sin[e + f*x]))} - - -(* ::Title:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4, x, 7, (7/16)*a*(2*A - B)*c^4*x + (7*a*(2*A - B)*c^4*Cos[e + f*x]^3)/(24*f) + (7*a*(2*A - B)*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (a*B*c*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^3)/(6*f) + (a*(2*A - B)*Cos[e + f*x]^3*(c^2 - c^2*Sin[e + f*x])^2)/(10*f) + (7*a*(2*A - B)*Cos[e + f*x]^3*(c^4 - c^4*Sin[e + f*x]))/(40*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3, x, 6, (1/8)*a*(5*A - 2*B)*c^3*x + (a*(5*A - 2*B)*c^3*Cos[e + f*x]^3)/(12*f) + (a*(5*A - 2*B)*c^3*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*B*c*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^2)/(5*f) + (a*(5*A - 2*B)*Cos[e + f*x]^3*(c^3 - c^3*Sin[e + f*x]))/(20*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2, x, 5, (1/8)*a*(4*A - B)*c^2*x + (a*(A - B)*c^2*Cos[e + f*x]^3)/(3*f) + (a*(4*A - B)*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*B*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f), (1/8)*a*(4*A - B)*c^2*x + (a*(4*A - B)*c^2*Cos[e + f*x]^3)/(12*f) + (a*(4*A - B)*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*B*Cos[e + f*x]^3*(c^2 - c^2*Sin[e + f*x]))/(4*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1, x, 4, (1/2)*a*A*c*x - (a*B*c*Cos[e + f*x]^3)/(3*f) + (a*A*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^1, x, 4, -((a*(A + 2*B)*x)/c) + (a*B*Cos[e + f*x])/(c*f) + (2*a*(A + B)*Cos[e + f*x])/(f*(c - c*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2, x, 4, (a*B*x)/c^2 - (a*(A + 7*B)*Cos[e + f*x])/(3*c^2*f*(1 - Sin[e + f*x])) + (2*a*(A + B)*Cos[e + f*x])/(3*f*(c - c*Sin[e + f*x])^2)} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3, x, 4, (2*a*(A + B)*Cos[e + f*x])/(5*f*(c - c*Sin[e + f*x])^3) - (a*(A + 11*B)*c*Cos[e + f*x])/(15*f*(c^2 - c^2*Sin[e + f*x])^2) - (a*(A - 4*B)*Cos[e + f*x])/(15*f*(c^3 - c^3*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4, x, 5, (2*a*(A + B)*Cos[e + f*x])/(7*f*(c - c*Sin[e + f*x])^4) - (a*(A + 15*B)*Cos[e + f*x])/(35*c*f*(c - c*Sin[e + f*x])^3) - (a*(2*A - 5*B)*Cos[e + f*x])/(105*f*(c^2 - c^2*Sin[e + f*x])^2) - (a*(2*A - 5*B)*Cos[e + f*x])/(105*f*(c^4 - c^4*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5, x, 6, (2*a*(A + B)*Cos[e + f*x])/(9*f*(c - c*Sin[e + f*x])^5) - (a*(A + 19*B)*Cos[e + f*x])/(63*c*f*(c - c*Sin[e + f*x])^4) - (a*(A - 2*B)*c*Cos[e + f*x])/(105*f*(c^2 - c^2*Sin[e + f*x])^3) - (2*a*(A - 2*B)*c*Cos[e + f*x])/(315*f*(c^3 - c^3*Sin[e + f*x])^2) - (2*a*(A - 2*B)*Cos[e + f*x])/(315*f*(c^5 - c^5*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5, x, 8, (9/128)*a^2*(8*A - 3*B)*c^5*x + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^5)/(80*f) + (9*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (a^2*(8*A - 3*B)*c^3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(56*f) - (a^2*B*c^2*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^3)/(8*f) + (3*a^2*(8*A - 3*B)*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(112*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4, x, 7, (1/16)*a^2*(7*A - 2*B)*c^4*x + (a^2*(7*A - 2*B)*c^4*Cos[e + f*x]^5)/(30*f) + (a^2*(7*A - 2*B)*c^4*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a^2*(7*A - 2*B)*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (a^2*B*Cos[e + f*x]^5*(c^2 - c^2*Sin[e + f*x])^2)/(7*f) + (a^2*(7*A - 2*B)*Cos[e + f*x]^5*(c^4 - c^4*Sin[e + f*x]))/(42*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3, x, 6, (1/16)*a^2*(6*A - B)*c^3*x + (a^2*(6*A - B)*c^3*Cos[e + f*x]^5)/(30*f) + (a^2*(6*A - B)*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a^2*(6*A - B)*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (a^2*B*Cos[e + f*x]^5*(c^3 - c^3*Sin[e + f*x]))/(6*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2, x, 5, (3/8)*a^2*A*c^2*x - (a^2*B*c^2*Cos[e + f*x]^5)/(5*f) + (3*a^2*A*c^2*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a^2*A*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1, x, 5, (1/8)*a^2*(4*A + B)*c*x - (a^2*(4*A + B)*c*Cos[e + f*x]^3)/(12*f) + (a^2*(4*A + B)*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (B*c*Cos[e + f*x]^3*(a^2 + a^2*Sin[e + f*x]))/(4*f)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^1, x, 5, -((3*a^2*(2*A + 3*B)*x)/(2*c)) + (3*a^2*(2*A + 3*B)*Cos[e + f*x])/(2*c*f) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(f*(c - c*Sin[e + f*x])^3) + (a^2*(2*A + 3*B)*Cos[e + f*x]^3)/(2*f*(c - c*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2, x, 5, (a^2*(A + 4*B)*x)/c^2 - (a^2*(A + 4*B)*Cos[e + f*x])/(c^2*f) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^4) - (2*a^2*(A + 4*B)*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^2)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3, x, 5, -((a^2*B*x)/c^3) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5) - (2*a^2*B*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^3) + (2*a^2*B*Cos[e + f*x])/(f*(c^3 - c^3*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4, x, 3, (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^6) + (a^2*(A - 6*B)*c*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^5)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5, x, 4, (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (a^2*(2*A - 7*B)*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6) + (a^2*(2*A - 7*B)*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6, x, 5, (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*(3*A - 8*B)*c*Cos[e + f*x]^5)/(99*f*(c - c*Sin[e + f*x])^7) + (2*a^2*(3*A - 8*B)*Cos[e + f*x]^5)/(693*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(3*A - 8*B)*Cos[e + f*x]^5)/(3465*c*f*(c - c*Sin[e + f*x])^5)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7, x, 6, (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(13*f*(c - c*Sin[e + f*x])^9) + (a^2*(4*A - 9*B)*c*Cos[e + f*x]^5)/(143*f*(c - c*Sin[e + f*x])^8) + (a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(429*f*(c - c*Sin[e + f*x])^7) + (2*a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(3003*c*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(15015*c^2*f*(c - c*Sin[e + f*x])^5)} - - -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^6, x, 9, (11/256)*a^3*(10*A - 3*B)*c^6*x + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^7)/(560*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]*Sin[e + f*x])/(256*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(384*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(480*f) - (a^3*B*Cos[e + f*x]^7*(c^2 - c^2*Sin[e + f*x])^3)/(10*f) + (a^3*(10*A - 3*B)*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(90*f) + (11*a^3*(10*A - 3*B)*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(720*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5, x, 8, (5/128)*a^3*(9*A - 2*B)*c^5*x + (a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^7)/(56*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*c^3*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^2)/(9*f) + (a^3*(9*A - 2*B)*Cos[e + f*x]^7*(c^5 - c^5*Sin[e + f*x]))/(72*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4, x, 7, (5/128)*a^3*(8*A - B)*c^4*x + (a^3*(8*A - B)*c^4*Cos[e + f*x]^7)/(56*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(8*A - B)*c^4*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*Cos[e + f*x]^7*(c^4 - c^4*Sin[e + f*x]))/(8*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3, x, 6, (5/16)*a^3*A*c^3*x - (a^3*B*c^3*Cos[e + f*x]^7)/(7*f) + (5*a^3*A*c^3*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (5*a^3*A*c^3*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a^3*A*c^3*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2, x, 6, (1/16)*a^3*(6*A + B)*c^2*x - (a^3*(6*A + B)*c^2*Cos[e + f*x]^5)/(30*f) + (a^3*(6*A + B)*c^2*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a^3*(6*A + B)*c^2*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (B*c^2*Cos[e + f*x]^5*(a^3 + a^3*Sin[e + f*x]))/(6*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1, x, 6, (1/8)*a^3*(5*A + 2*B)*c*x - (a^3*(5*A + 2*B)*c*Cos[e + f*x]^3)/(12*f) + (a^3*(5*A + 2*B)*c*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (a*B*c*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^2)/(5*f) - ((5*A + 2*B)*c*Cos[e + f*x]^3*(a^3 + a^3*Sin[e + f*x]))/(20*f)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^1, x, 6, -((5*a^3*(3*A + 4*B)*x)/(2*c)) + (5*a^3*(3*A + 4*B)*Cos[e + f*x]^3)/(3*c*f) - (5*a^3*(3*A + 4*B)*Cos[e + f*x]*Sin[e + f*x])/(2*c*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(f*(c - c*Sin[e + f*x])^4) + (2*a^3*(3*A + 4*B)*c^3*Cos[e + f*x]^5)/(f*(c^2 - c^2*Sin[e + f*x])^2)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^2, x, 6, (5*a^3*(2*A + 5*B)*x)/(2*c^2) - (5*a^3*(2*A + 5*B)*Cos[e + f*x])/(2*c^2*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(3*f*(c - c*Sin[e + f*x])^5) - (2*a^3*(2*A + 5*B)*c*Cos[e + f*x]^5)/(3*f*(c - c*Sin[e + f*x])^3) - (5*a^3*(2*A + 5*B)*Cos[e + f*x]^3)/(6*f*(c^2 - c^2*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3, x, 6, -((a^3*(A + 6*B)*x)/c^3) + (a^3*(A + 6*B)*Cos[e + f*x])/(c^3*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(5*f*(c - c*Sin[e + f*x])^6) - (2*a^3*(A + 6*B)*c*Cos[e + f*x]^5)/(15*f*(c - c*Sin[e + f*x])^4) + (2*a^3*(A + 6*B)*c^3*Cos[e + f*x]^3)/(3*f*(c^3 - c^3*Sin[e + f*x])^2)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^4, x, 6, (a^3*B*x)/c^4 + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7) - (2*a^3*B*c*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^5) + (2*a^3*B*c^2*Cos[e + f*x]^3)/(3*f*(c^2 - c^2*Sin[e + f*x])^3) - (2*a^3*B*Cos[e + f*x])/(f*(c^4 - c^4*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5, x, 3, (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + (a^3*(A - 8*B)*c^2*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^7)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6, x, 4, (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (a^3*(2*A - 9*B)*c^2*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^8) + (a^3*(2*A - 9*B)*c*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^7)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7, x, 5, (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (a^3*(3*A - 10*B)*c^2*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^9) + (2*a^3*(3*A - 10*B)*c*Cos[e + f*x]^7)/(1287*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(3*A - 10*B)*Cos[e + f*x]^7)/(9009*f*(c - c*Sin[e + f*x])^7)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8, x, 6, (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (a^3*(4*A - 11*B)*c^2*Cos[e + f*x]^7)/(195*f*(c - c*Sin[e + f*x])^10) + (a^3*(4*A - 11*B)*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (2*a^3*(4*A - 11*B)*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(4*A - 11*B)*Cos[e + f*x]^7)/(45045*c*f*(c - c*Sin[e + f*x])^7)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x]), x, 7, -((35*(4*A - 5*B)*c^4*x)/(8*a)) - (35*(4*A - 5*B)*c^4*Cos[e + f*x]^3)/(12*a*f) - (35*(4*A - 5*B)*c^4*Cos[e + f*x]*Sin[e + f*x])/(8*a*f) - (a^4*(A - B)*c^4*Cos[e + f*x]^9)/(f*(a + a*Sin[e + f*x])^5) - (2*a^2*(4*A - 5*B)*c^4*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^3) - (7*(4*A - 5*B)*c^4*Cos[e + f*x]^5)/(4*f*(a + a*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x]), x, 6, -((5*(3*A - 4*B)*c^3*x)/(2*a)) - (5*(3*A - 4*B)*c^3*Cos[e + f*x]^3)/(3*a*f) - (5*(3*A - 4*B)*c^3*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - (a^3*(A - B)*c^3*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) - (2*a^3*(3*A - 4*B)*c^3*Cos[e + f*x]^5)/(f*(a^2 + a^2*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x]), x, 5, -((3*(2*A - 3*B)*c^2*x)/(2*a)) - (3*(2*A - 3*B)*c^2*Cos[e + f*x])/(2*a*f) - (a^2*(A - B)*c^2*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^3) - ((2*A - 3*B)*c^2*Cos[e + f*x]^3)/(2*f*(a + a*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1)/(a + a*Sin[e + f*x]), x, 4, -(((A - 2*B)*c*x)/a) + (B*c*Cos[e + f*x])/(a*f) - (2*(A - B)*c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^1), x, 4, (B*Sec[e + f*x])/(a*c*f) + (A*Tan[e + f*x])/(a*c*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2), x, 4, ((A + B)*Sec[e + f*x])/(3*a*f*(c^2 - c^2*Sin[e + f*x])) + ((2*A - B)*Tan[e + f*x])/(3*a*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3), x, 5, ((A + B)*Sec[e + f*x])/(5*a*c*f*(c - c*Sin[e + f*x])^2) + ((3*A - 2*B)*Sec[e + f*x])/(15*a*f*(c^3 - c^3*Sin[e + f*x])) + (2*(3*A - 2*B)*Tan[e + f*x])/(15*a*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^4), x, 6, ((A + B)*Sec[e + f*x])/(7*a*c*f*(c - c*Sin[e + f*x])^3) + ((4*A - 3*B)*Sec[e + f*x])/(35*a*f*(c^2 - c^2*Sin[e + f*x])^2) + ((4*A - 3*B)*Sec[e + f*x])/(35*a*f*(c^4 - c^4*Sin[e + f*x])) + (2*(4*A - 3*B)*Tan[e + f*x])/(35*a*c^4*f)} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^2, x, 8, (105*(4*A - 7*B)*c^5*x)/(8*a^2) + (35*(4*A - 7*B)*c^5*Cos[e + f*x]^3)/(4*a^2*f) + (105*(4*A - 7*B)*c^5*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) - (a^5*(A - B)*c^5*Cos[e + f*x]^11)/(3*f*(a + a*Sin[e + f*x])^7) + (2*a^3*(4*A - 7*B)*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^5) + (6*a^4*(4*A - 7*B)*c^5*Cos[e + f*x]^7)/(f*(a^2 + a^2*Sin[e + f*x])^3) + (21*(4*A - 7*B)*c^5*Cos[e + f*x]^5)/(4*f*(a^2 + a^2*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^2, x, 7, (35*(A - 2*B)*c^4*x)/(2*a^2) + (35*(A - 2*B)*c^4*Cos[e + f*x]^3)/(3*a^2*f) + (35*(A - 2*B)*c^4*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (a^4*(A - B)*c^4*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (2*a^2*(A - 2*B)*c^4*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (14*(A - 2*B)*c^4*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^2, x, 6, (5*(2*A - 5*B)*c^3*x)/(2*a^2) + (5*(2*A - 5*B)*c^3*Cos[e + f*x])/(2*a^2*f) - (a^3*(A - B)*c^3*Cos[e + f*x]^7)/(3*f*(a + a*Sin[e + f*x])^5) + (2*a*(2*A - 5*B)*c^3*Cos[e + f*x]^5)/(3*f*(a + a*Sin[e + f*x])^3) + (5*(2*A - 5*B)*c^3*Cos[e + f*x]^3)/(6*f*(a^2 + a^2*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2, x, 5, ((A - 4*B)*c^2*x)/a^2 + ((A - 4*B)*c^2*Cos[e + f*x])/(a^2*f) - (a^2*(A - B)*c^2*Cos[e + f*x]^5)/(3*f*(a + a*Sin[e + f*x])^4) + (2*(A - 4*B)*c^2*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1)/(a + a*Sin[e + f*x])^2, x, 4, -((B*c*x)/a^2) + ((A - 7*B)*c*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - (2*(A - B)*c*Cos[e + f*x])/(3*f*(a + a*Sin[e + f*x])^2)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^1), x, 4, -(((A - B)*Sec[e + f*x])/(3*c*f*(a^2 + a^2*Sin[e + f*x]))) + ((2*A + B)*Tan[e + f*x])/(3*a^2*c*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^2), x, 4, (B*Sec[e + f*x]^3)/(3*a^2*c^2*f) + (A*Tan[e + f*x])/(a^2*c^2*f) + (A*Tan[e + f*x]^3)/(3*a^2*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^3), x, 4, ((A + B)*Sec[e + f*x]^3)/(5*a^2*f*(c^3 - c^3*Sin[e + f*x])) + ((4*A - B)*Tan[e + f*x])/(5*a^2*c^3*f) + ((4*A - B)*Tan[e + f*x]^3)/(15*a^2*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^4), x, 5, ((A + B)*Sec[e + f*x]^3)/(7*a^2*f*(c^2 - c^2*Sin[e + f*x])^2) + ((5*A - 2*B)*Sec[e + f*x]^3)/(35*a^2*f*(c^4 - c^4*Sin[e + f*x])) + (4*(5*A - 2*B)*Tan[e + f*x])/(35*a^2*c^4*f) + (4*(5*A - 2*B)*Tan[e + f*x]^3)/(105*a^2*c^4*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5), x, 6, ((A + B)*Sec[e + f*x]^3)/(9*a^2*c^2*f*(c - c*Sin[e + f*x])^3) + ((2*A - B)*Sec[e + f*x]^3)/(21*a^2*c^3*f*(c - c*Sin[e + f*x])^2) + ((2*A - B)*Sec[e + f*x]^3)/(21*a^2*f*(c^5 - c^5*Sin[e + f*x])) + (4*(2*A - B)*Tan[e + f*x])/(21*a^2*c^5*f) + (4*(2*A - B)*Tan[e + f*x]^3)/(63*a^2*c^5*f)} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^3, x, 8, -((21*(3*A - 8*B)*c^5*x)/(2*a^3)) - (7*(3*A - 8*B)*c^5*Cos[e + f*x]^3)/(a^3*f) - (21*(3*A - 8*B)*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^3*f) - (a^5*(A - B)*c^5*Cos[e + f*x]^11)/(5*f*(a + a*Sin[e + f*x])^8) + (2*a^3*(3*A - 8*B)*c^5*Cos[e + f*x]^9)/(15*f*(a + a*Sin[e + f*x])^6) - (6*a^5*(3*A - 8*B)*c^5*Cos[e + f*x]^7)/(5*f*(a^2 + a^2*Sin[e + f*x])^4) - (42*a^5*(3*A - 8*B)*c^5*Cos[e + f*x]^5)/(5*f*(a^4 + a^4*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^3, x, 7, -((7*(2*A - 7*B)*c^4*x)/(2*a^3)) - (7*(2*A - 7*B)*c^4*Cos[e + f*x])/(2*a^3*f) - (a^4*(A - B)*c^4*Cos[e + f*x]^9)/(5*f*(a + a*Sin[e + f*x])^7) + (2*a^2*(2*A - 7*B)*c^4*Cos[e + f*x]^7)/(15*f*(a + a*Sin[e + f*x])^5) - (14*(2*A - 7*B)*c^4*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^3) - (7*(2*A - 7*B)*c^4*Cos[e + f*x]^3)/(6*f*(a^3 + a^3*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^3, x, 6, -(((A - 6*B)*c^3*x)/a^3) - ((A - 6*B)*c^3*Cos[e + f*x])/(a^3*f) - (a^3*(A - B)*c^3*Cos[e + f*x]^7)/(5*f*(a + a*Sin[e + f*x])^6) + (2*a*(A - 6*B)*c^3*Cos[e + f*x]^5)/(15*f*(a + a*Sin[e + f*x])^4) - (2*a^3*(A - 6*B)*c^3*Cos[e + f*x]^3)/(3*f*(a^3 + a^3*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3, x, 5, (B*c^2*x)/a^3 - (a^2*(A - B)*c^2*Cos[e + f*x]^5)/(5*f*(a + a*Sin[e + f*x])^5) - (2*B*c^2*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^3) + (2*B*c^2*Cos[e + f*x])/(f*(a^3 + a^3*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1)/(a + a*Sin[e + f*x])^3, x, 4, -((2*(A - B)*c*Cos[e + f*x])/(5*f*(a + a*Sin[e + f*x])^3)) + (a*(A - 11*B)*c*Cos[e + f*x])/(15*f*(a^2 + a^2*Sin[e + f*x])^2) + ((A + 4*B)*c*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^1), x, 5, -(((A - B)*Sec[e + f*x])/(5*a*c*f*(a + a*Sin[e + f*x])^2)) - ((3*A + 2*B)*Sec[e + f*x])/(15*c*f*(a^3 + a^3*Sin[e + f*x])) + (2*(3*A + 2*B)*Tan[e + f*x])/(15*a^3*c*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^2), x, 4, -(((A - B)*Sec[e + f*x]^3)/(5*c^2*f*(a^3 + a^3*Sin[e + f*x]))) + ((4*A + B)*Tan[e + f*x])/(5*a^3*c^2*f) + ((4*A + B)*Tan[e + f*x]^3)/(15*a^3*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^3), x, 4, (B*Sec[e + f*x]^5)/(5*a^3*c^3*f) + (A*Tan[e + f*x])/(a^3*c^3*f) + (2*A*Tan[e + f*x]^3)/(3*a^3*c^3*f) + (A*Tan[e + f*x]^5)/(5*a^3*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4), x, 4, ((A + B)*Sec[e + f*x]^5)/(7*a^3*f*(c^4 - c^4*Sin[e + f*x])) + ((6*A - B)*Tan[e + f*x])/(7*a^3*c^4*f) + (2*(6*A - B)*Tan[e + f*x]^3)/(21*a^3*c^4*f) + ((6*A - B)*Tan[e + f*x]^5)/(35*a^3*c^4*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5), x, 5, ((A + B)*Sec[e + f*x]^5)/(9*a^3*c^3*f*(c - c*Sin[e + f*x])^2) + ((7*A - 2*B)*Sec[e + f*x]^5)/(63*a^3*f*(c^5 - c^5*Sin[e + f*x])) + (2*(7*A - 2*B)*Tan[e + f*x])/(21*a^3*c^5*f) + (4*(7*A - 2*B)*Tan[e + f*x]^3)/(63*a^3*c^5*f) + (2*(7*A - 2*B)*Tan[e + f*x]^5)/(105*a^3*c^5*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6), x, 6, ((A + B)*Sec[e + f*x]^5)/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2) + ((8*A - 3*B)*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (2*(8*A - 3*B)*Tan[e + f*x])/(33*a^3*c^6*f) + (4*(8*A - 3*B)*Tan[e + f*x]^3)/(99*a^3*c^6*f) + (2*(8*A - 3*B)*Tan[e + f*x]^5)/(165*a^3*c^6*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 6, (256*a*(11*A - 5*B)*c^5*Cos[e + f*x]^3)/(3465*f*(c - c*Sin[e + f*x])^(3/2)) + (64*a*(11*A - 5*B)*c^4*Cos[e + f*x]^3)/(1155*f*Sqrt[c - c*Sin[e + f*x]]) + (8*a*(11*A - 5*B)*c^3*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(231*f) + (2*a*(11*A - 5*B)*c^2*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(99*f) - (2*a*B*c*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(11*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 5, (64*a*(3*A - B)*c^4*Cos[e + f*x]^3)/(315*f*(c - c*Sin[e + f*x])^(3/2)) + (16*a*(3*A - B)*c^3*Cos[e + f*x]^3)/(105*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a*(3*A - B)*c^2*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(21*f) - (2*a*B*c*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(9*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 4, (8*a*(7*A - B)*c^3*Cos[e + f*x]^3)/(105*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a*(7*A - B)*c^2*Cos[e + f*x]^3)/(35*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a*B*c*Cos[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(7*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, (2*a*(5*A + B)*c^2*Cos[e + f*x]^3)/(15*f*(c - c*Sin[e + f*x])^(3/2)) - (2*a*B*c*Cos[e + f*x]^3)/(5*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(1/2), x, 5, (2*Sqrt[2]*a*(A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a*(3*A + 5*B)*Cos[e + f*x])/(3*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a*B*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*c*f)} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2), x, 5, -((a*(A + 5*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*c^(3/2)*f)) + (a*(A + B)*Cos[e + f*x])/(f*(c - c*Sin[e + f*x])^(3/2)) + (2*a*B*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2), x, 5, -(a*(A - 7*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(5/2)*f) + (a*(A + B)*Cos[e + f*x])/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a*(A + 9*B)*Cos[e + f*x])/(8*c*f*(c - c*Sin[e + f*x])^(3/2))} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2), x, 6, -(a*(A - 3*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*c^(7/2)*f) + (a*(A + B)*Cos[e + f*x])/(3*f*(c - c*Sin[e + f*x])^(7/2)) - (a*(A + 13*B)*Cos[e + f*x])/(24*c*f*(c - c*Sin[e + f*x])^(5/2)) - (a*(A - 3*B)*Cos[e + f*x])/(32*c^2*f*(c - c*Sin[e + f*x])^(3/2))} - - -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 6, (256*a^2*(13*A - 3*B)*c^6*Cos[e + f*x]^5)/(15015*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*(13*A - 3*B)*c^5*Cos[e + f*x]^5)/(3003*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*(13*A - 3*B)*c^4*Cos[e + f*x]^5)/(429*f*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*(13*A - 3*B)*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(143*f) - (2*a^2*B*c^2*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(13*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 5, (64*a^2*(11*A - B)*c^5*Cos[e + f*x]^5)/(3465*f*(c - c*Sin[e + f*x])^(5/2)) + (16*a^2*(11*A - B)*c^4*Cos[e + f*x]^5)/(693*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^2*(11*A - B)*c^3*Cos[e + f*x]^5)/(99*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^2*B*c^2*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(11*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 4, (8*a^2*(9*A + B)*c^4*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^2*(9*A + B)*c^3*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^(3/2)) - (2*a^2*B*c^2*Cos[e + f*x]^5)/(9*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, (2*a^2*(7*A + 3*B)*c^3*Cos[e + f*x]^5)/(35*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a^2*B*c^2*Cos[e + f*x]^5)/(7*f*(c - c*Sin[e + f*x])^(3/2))} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(1/2), x, 6, (4*Sqrt[2]*a^2*(A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^2*B*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a^2*(A + B)*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (4*a^2*(A + B)*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2), x, 6, -((Sqrt[2]*a^2*(3*A + 7*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f)) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(2*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*(3*A + 7*B)*Cos[e + f*x]^3)/(6*f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*(3*A + 7*B)*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2), x, 6, (3*a^2*(A + 9*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*c^(5/2)*f) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(4*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(A + 9*B)*Cos[e + f*x]^3)/(8*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a^2*(A + 9*B)*Cos[e + f*x])/(8*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2), x, 6, (a^2*(A - 11*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*c^(7/2)*f) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(6*f*(c - c*Sin[e + f*x])^(11/2)) + (a^2*(A - 11*B)*Cos[e + f*x]^3)/(24*f*(c - c*Sin[e + f*x])^(7/2)) - (a^2*(A - 11*B)*Cos[e + f*x])/(16*c^2*f*(c - c*Sin[e + f*x])^(3/2))} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2), x, 7, (a^2*(3*A - 13*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f) + (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(8*f*(c - c*Sin[e + f*x])^(13/2)) + (a^2*(3*A - 13*B)*Cos[e + f*x]^3)/(48*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(3*A - 13*B)*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(3*A - 13*B)*Cos[e + f*x])/(256*c^3*f*(c - c*Sin[e + f*x])^(3/2))} - - -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 6, (256*a^3*(15*A - B)*c^7*Cos[e + f*x]^7)/(45045*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*(15*A - B)*c^6*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^(5/2)) + (8*a^3*(15*A - B)*c^5*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^3*(15*A - B)*c^4*Cos[e + f*x]^7)/(195*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*B*c^3*Cos[e + f*x]^7*Sqrt[c - c*Sin[e + f*x]])/(15*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 5, (64*a^3*(13*A + B)*c^6*Cos[e + f*x]^7)/(9009*f*(c - c*Sin[e + f*x])^(7/2)) + (16*a^3*(13*A + B)*c^5*Cos[e + f*x]^7)/(1287*f*(c - c*Sin[e + f*x])^(5/2)) + (2*a^3*(13*A + B)*c^4*Cos[e + f*x]^7)/(143*f*(c - c*Sin[e + f*x])^(3/2)) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(13*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 4, (8*a^3*(11*A + 3*B)*c^5*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^(7/2)) + (2*a^3*(11*A + 3*B)*c^4*Cos[e + f*x]^7)/(99*f*(c - c*Sin[e + f*x])^(5/2)) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^(3/2))} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, (2*a^3*(9*A + 5*B)*c^4*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^(5/2))} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(1/2), x, 7, (8*Sqrt[2]*a^3*(A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[c]*f) - (2*a^3*B*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^(7/2)) - (2*a^3*(A + B)*c^2*Cos[e + f*x]^5)/(5*f*(c - c*Sin[e + f*x])^(5/2)) - (4*a^3*(A + B)*c*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) - (8*a^3*(A + B)*Cos[e + f*x])/(f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(3/2), x, 7, -((2*Sqrt[2]*a^3*(5*A + 9*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(c^(3/2)*f)) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(2*f*(c - c*Sin[e + f*x])^(9/2)) + (a^3*(5*A + 9*B)*c*Cos[e + f*x]^5)/(10*f*(c - c*Sin[e + f*x])^(5/2)) + (a^3*(5*A + 9*B)*Cos[e + f*x]^3)/(3*f*(c - c*Sin[e + f*x])^(3/2)) + (2*a^3*(5*A + 9*B)*Cos[e + f*x])/(c*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(5/2), x, 7, (5*a^3*(3*A + 11*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(2*Sqrt[2]*c^(5/2)*f) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(4*f*(c - c*Sin[e + f*x])^(11/2)) - (a^3*(3*A + 11*B)*c*Cos[e + f*x]^5)/(8*f*(c - c*Sin[e + f*x])^(7/2)) - (5*a^3*(3*A + 11*B)*Cos[e + f*x]^3)/(24*c*f*(c - c*Sin[e + f*x])^(3/2)) - (5*a^3*(3*A + 11*B)*Cos[e + f*x])/(4*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(7/2), x, 7, -((5*a^3*(A + 13*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*c^(7/2)*f)) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(6*f*(c - c*Sin[e + f*x])^(13/2)) - (a^3*(A + 13*B)*c*Cos[e + f*x]^5)/(24*f*(c - c*Sin[e + f*x])^(9/2)) + (5*a^3*(A + 13*B)*Cos[e + f*x]^3)/(48*c*f*(c - c*Sin[e + f*x])^(5/2)) + (5*a^3*(A + 13*B)*Cos[e + f*x])/(16*c^3*f*Sqrt[c - c*Sin[e + f*x]])} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(9/2), x, 7, -((5*a^3*(A - 15*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*c^(9/2)*f)) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(8*f*(c - c*Sin[e + f*x])^(15/2)) + (a^3*(A - 15*B)*c*Cos[e + f*x]^5)/(48*f*(c - c*Sin[e + f*x])^(11/2)) - (5*a^3*(A - 15*B)*Cos[e + f*x]^3)/(192*c*f*(c - c*Sin[e + f*x])^(7/2)) + (5*a^3*(A - 15*B)*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(3/2))} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2), x, 8, -((a^3*(3*A - 17*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(512*Sqrt[2]*c^(11/2)*f)) + (a^3*(A + B)*c^3*Cos[e + f*x]^7)/(10*f*(c - c*Sin[e + f*x])^(17/2)) + (a^3*(3*A - 17*B)*c*Cos[e + f*x]^5)/(80*f*(c - c*Sin[e + f*x])^(13/2)) - (a^3*(3*A - 17*B)*Cos[e + f*x]^3)/(96*c*f*(c - c*Sin[e + f*x])^(9/2)) + (a^3*(3*A - 17*B)*Cos[e + f*x])/(128*c^3*f*(c - c*Sin[e + f*x])^(5/2)) - (a^3*(3*A - 17*B)*Cos[e + f*x])/(512*c^4*f*(c - c*Sin[e + f*x])^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x]), x, 6, -((128*(7*A - 9*B)*c^4*Cos[e + f*x])/(35*a*f*Sqrt[c - c*Sin[e + f*x]])) - (32*(7*A - 9*B)*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(35*a*f) - (12*(7*A - 9*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(35*a*f) - ((7*A - 9*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(7*a*f) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(a*c*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x]), x, 5, -((32*(5*A - 7*B)*c^3*Cos[e + f*x])/(15*a*f*Sqrt[c - c*Sin[e + f*x]])) - (8*(5*A - 7*B)*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(15*a*f) - ((5*A - 7*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(5*a*f) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(a*c*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x]), x, 4, -((4*(3*A - 5*B)*c^2*Cos[e + f*x])/(3*a*f*Sqrt[c - c*Sin[e + f*x]])) - ((3*A - 5*B)*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a*f) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(a*c*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/(a + a*Sin[e + f*x]), x, 3, -(((A - 3*B)*c*Cos[e + f*x])/(a*f*Sqrt[c - c*Sin[e + f*x]])) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a*c*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2)), x, 4, ((A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*a*Sqrt[c]*f) - ((A - B)*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*c*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2)), x, 5, ((3*A - B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*a*c^(3/2)*f) + ((3*A - B)*Cos[e + f*x])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - ((A - B)*Sec[e + f*x])/(a*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2)), x, 6, (3*(5*A - 3*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(32*Sqrt[2]*a*c^(5/2)*f) + (3*(5*A - 3*B)*Cos[e + f*x])/(32*a*c*f*(c - c*Sin[e + f*x])^(3/2)) + ((A + B)*Sec[e + f*x])/(4*a*c*f*(c - c*Sin[e + f*x])^(3/2)) - ((5*A - 3*B)*Sec[e + f*x])/(8*a*c^2*f*Sqrt[c - c*Sin[e + f*x]])} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^2, x, 7, (2048*(7*A - 13*B)*c^4*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(105*a^2*f) - (512*(7*A - 13*B)*c^3*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(105*a^2*f) - (64*(7*A - 13*B)*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(105*a^2*f) - (16*(7*A - 13*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(105*a^2*f) - ((7*A - 13*B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(21*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(13/2))/(3*a^2*c^2*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^2, x, 6, (128*(5*A - 11*B)*c^3*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(15*a^2*f) - (32*(5*A - 11*B)*c^2*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(15*a^2*f) - (4*(5*A - 11*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*a^2*f) - ((5*A - 11*B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(15*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(11/2))/(3*a^2*c^2*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^2, x, 5, (32*(A - 3*B)*c^2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*f) - (8*(A - 3*B)*c*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - ((A - 3*B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(3*a^2*c^2*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^2, x, 4, (4*(A - 7*B)*c*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*f) - ((A - 7*B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(3*a^2*c^2*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/(a + a*Sin[e + f*x])^2, x, 3, -(((A + 5*B)*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*f)) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(1/2)), x, 5, ((A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(2*Sqrt[2]*a^2*Sqrt[c]*f) - ((A + B)*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(2*a^2*c*f) - ((A - B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(3*a^2*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(3/2)), x, 6, ((5*A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(8*Sqrt[2]*a^2*c^(3/2)*f) + ((5*A + B)*Cos[e + f*x])/(8*a^2*f*(c - c*Sin[e + f*x])^(3/2)) - ((5*A + B)*Sec[e + f*x])/(6*a^2*c*f*Sqrt[c - c*Sin[e + f*x]]) - ((A - B)*Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(3*a^2*c^2*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(5/2)), x, 7, (5*(7*A - B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(64*Sqrt[2]*a^2*c^(5/2)*f) + (5*(7*A - B)*Cos[e + f*x])/(64*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) + ((7*A - B)*Sec[e + f*x])/(24*a^2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (5*(7*A - B)*Sec[e + f*x])/(48*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - ((A - B)*Sec[e + f*x]^3)/(3*a^2*c^2*f*Sqrt[c - c*Sin[e + f*x]])} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^3, x, 7, -((2048*(A - 3*B)*c^3*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^3*f)) + (512*(A - 3*B)*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*f) - (64*(A - 3*B)*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(5*a^3*f) - (16*(A - 3*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(15*a^3*f) - ((A - 3*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(11/2))/(5*a^3*c*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(15/2))/(5*a^3*c^3*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^3, x, 6, -((128*(3*A - 13*B)*c^2*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^3*f)) + (32*(3*A - 13*B)*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*f) - (4*(3*A - 13*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(5*a^3*f) - ((3*A - 13*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(9/2))/(15*a^3*c*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(13/2))/(5*a^3*c^3*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^3, x, 5, -((32*(A - 11*B)*c*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^3*f)) + (8*(A - 11*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*f) - ((A - 11*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(7/2))/(5*a^3*c*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/(5*a^3*c^3*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^3, x, 4, (4*(A + 9*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^3*f) - ((A + 9*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/(5*a^3*c^3*f)} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/(a + a*Sin[e + f*x])^3, x, 3, -(((3*A + 7*B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(15*a^3*c*f)) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(5*a^3*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(1/2)), x, 6, ((A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(4*Sqrt[2]*a^3*Sqrt[c]*f) - ((A + B)*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(4*a^3*c*f) - ((A + B)*Sec[e + f*x]^3*(c - c*Sin[e + f*x])^(3/2))/(6*a^3*c^2*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/(5*a^3*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(3/2)), x, 7, ((7*A + 3*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(16*Sqrt[2]*a^3*c^(3/2)*f) + ((7*A + 3*B)*Cos[e + f*x])/(16*a^3*f*(c - c*Sin[e + f*x])^(3/2)) - ((7*A + 3*B)*Sec[e + f*x])/(12*a^3*c*f*Sqrt[c - c*Sin[e + f*x]]) - ((7*A + 3*B)*Sec[e + f*x]^3*Sqrt[c - c*Sin[e + f*x]])/(30*a^3*c^2*f) - ((A - B)*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(5*a^3*c^3*f)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(5/2)), x, 8, (7*(9*A + B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(128*Sqrt[2]*a^3*c^(5/2)*f) + (7*(9*A + B)*Cos[e + f*x])/(128*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) + (7*(9*A + B)*Sec[e + f*x])/(240*a^3*c*f*(c - c*Sin[e + f*x])^(3/2)) - (7*(9*A + B)*Sec[e + f*x])/(96*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - ((9*A + B)*Sec[e + f*x]^3)/(30*a^3*c^2*f*Sqrt[c - c*Sin[e + f*x]]) - ((A - B)*Sec[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(5*a^3*c^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 3, -((a*(A + B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*Sqrt[a + a*Sin[e + f*x]])) + (a*B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(5*c*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 3, -((a*(A + B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])) + (a*B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*c*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 3, -((a*(A + B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])) + (a*B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*c*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, -((a*(A + B)*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])) + (a*B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(1/2), x, 5, -((a*(A + B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (a*B*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2), x, 5, (a*(A + B)*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (a*B*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2), x, 3, (a*(A + B)*Cos[e + f*x])/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (a*B*Cos[e + f*x])/(c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2), x, 3, (a*(A + B)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (a*B*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))} - - -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 3, -(a^2*(3*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(30*f*Sqrt[a + a*Sin[e + f*x]]) - (a*(3*A - B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(15*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(6*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 3, -(a^2*(5*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(30*f*Sqrt[a + a*Sin[e + f*x]]) - (a*(5*A - B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(20*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 3, -((a^2*A*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])) - (a*A*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2))/(3*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2))/(4*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, ((A - B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]]) + (B*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(1/2), x, 5, -((2*a^2*(A + B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (a*(A + B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2), x, 5, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^2*(A + 3*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (a*(A + 3*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2), x, 5, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*f*(c - c*Sin[e + f*x])^(5/2)) - (a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*(c - c*Sin[e + f*x])^(3/2)) - (a^2*B*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2), x, 2, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) + ((A - 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(24*c*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(9/2), x, 3, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(8*f*(c - c*Sin[e + f*x])^(9/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(24*c*f*(c - c*Sin[e + f*x])^(7/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(96*c^2*f*(c - c*Sin[e + f*x])^(5/2))} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(11/2), x, 3, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (a*(3*A - 7*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(40*c*f*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(3*A - 7*B)*Cos[e + f*x])/(120*c^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))} - - -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 4, -(a^3*(7*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^2*(7*A - B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(105*f) - (a*(7*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(42*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 4, -((2*a^3*A*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(15*f*Sqrt[a + a*Sin[e + f*x]])) - (a^2*A*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(5*f) - (a*A*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2))/(5*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(6*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 3, ((5*A + B)*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(30*f*Sqrt[c - c*Sin[e + f*x]]) + ((5*A + B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(20*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(5*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, ((A - B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]]) + (B*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(1/2), x, 6, -((4*a^3*(A + B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (2*a^2*(A + B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*(A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*Sqrt[c - c*Sin[e + f*x]]) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2), x, 6, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^3*(A + 2*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^2*(A + 2*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (a*(A + 2*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2), x, 6, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(4*f*(c - c*Sin[e + f*x])^(5/2)) - (a*(A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*c*f*(c - c*Sin[e + f*x])^(3/2)) - (a^3*(A + 5*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a^2*(A + 5*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2), x, 6, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^3*B*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(9/2), x, 2, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(8*f*(c - c*Sin[e + f*x])^(9/2)) + ((A - 7*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(48*c*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(11/2), x, 3, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 4*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(40*c*f*(c - c*Sin[e + f*x])^(9/2)) + ((A - 4*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(240*c^2*f*(c - c*Sin[e + f*x])^(7/2))} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(13/2), x, 4, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(40*c*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(160*c^2*f*(c - c*Sin[e + f*x])^(9/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(960*c^3*f*(c - c*Sin[e + f*x])^(7/2))} - - -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2), x, 5, -(a^4*(9*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(9*A - B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(126*f) - (a^2*(9*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(84*f) - (a*(9*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(9/2))/(72*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*f)} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2), x, 5, -((2*a^4*A*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(35*f*Sqrt[a + a*Sin[e + f*x]])) - (4*a^3*A*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2))/(35*f) - (a^2*A*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f) - (a*A*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(7*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(7/2))/(8*f)} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 4, ((7*A + B)*c^3*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(105*f*Sqrt[c - c*Sin[e + f*x]]) + (2*(7*A + B)*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(105*f) + ((7*A + B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(42*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2))/(7*f)} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 3, ((3*A + B)*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(30*f*Sqrt[c - c*Sin[e + f*x]]) + ((3*A + B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(15*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(6*f)} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, ((A - B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]]) + (B*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(5*a*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(1/2), x, 7, -((8*a^4*(A + B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (4*a^3*(A + B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]]) - (a^2*(A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*Sqrt[c - c*Sin[e + f*x]]) - (a*(A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[c - c*Sin[e + f*x]]) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2), x, 7, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*f*(c - c*Sin[e + f*x])^(3/2)) + (4*a^4*(3*A + 5*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a^3*(3*A + 5*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c*f*Sqrt[c - c*Sin[e + f*x]]) + (a^2*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[c - c*Sin[e + f*x]]) + (a*(3*A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*c*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2), x, 7, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*(c - c*Sin[e + f*x])^(5/2)) - (a*(A + 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*c*f*(c - c*Sin[e + f*x])^(3/2)) - (6*a^4*(A + 3*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*a^3*(A + 3*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^2*f*Sqrt[c - c*Sin[e + f*x]]) - (3*a^2*(A + 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*c^2*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(7/2), x, 7, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(6*f*(c - c*Sin[e + f*x])^(7/2)) - (a*(A + 7*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(12*c*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(A + 7*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*c^2*f*(c - c*Sin[e + f*x])^(3/2)) + (a^4*(A + 7*B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (a^3*(A + 7*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*c^3*f*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(9/2), x, 7, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8*f*(c - c*Sin[e + f*x])^(9/2)) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*c*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^2*f*(c - c*Sin[e + f*x])^(5/2)) - (a^3*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*(c - c*Sin[e + f*x])^(3/2)) - (a^4*B*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(11/2), x, 2, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 9*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(13/2), x, 3, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(12*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(60*c*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(480*c^2*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(15/2), x, 4, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(168*c*f*(c - c*Sin[e + f*x])^(13/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(840*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(6720*c^3*f*(c - c*Sin[e + f*x])^(9/2))} -{(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(17/2), x, 5, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(16*f*(c - c*Sin[e + f*x])^(17/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(224*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(1120*c^3*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8960*c^4*f*(c - c*Sin[e + f*x])^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/Sqrt[a + a*Sin[e + f*x]], x, 6, (4*(A - B)*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*(A - B)*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]) + ((A - B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]]) - (B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*f*Sqrt[a + a*Sin[e + f*x]])} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/Sqrt[a + a*Sin[e + f*x]], x, 5, (2*(A - B)*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A - B)*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]) - (B*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*Sqrt[a + a*Sin[e + f*x]])} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/Sqrt[a + a*Sin[e + f*x]], x, 5, ((A - B)*c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (B*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(1/2)), x, 7, -(((A + B)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + ((A - B)*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 3, ((A + B)*Cos[e + f*x])/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + ((A - B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)), x, 4, ((A + B)*Cos[e + f*x])/(4*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + ((A - B)*Cos[e + f*x])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + ((A - B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(3/2), x, 7, (-4*(3*A - 5*B)*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*(3*A - 5*B)*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - ((3*A - 5*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((3*A - 5*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(6*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(3/2), x, 6, (-4*(A - 2*B)*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (2*(A - 2*B)*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - 2*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(3/2), x, 5, -(((A - 3*B)*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - ((A - 3*B)*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/(a + a*Sin[e + f*x])^(3/2), x, 5, -(((A - B)*c*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])) + (B*c*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(1/2)), x, 3, -((A - B)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + ((A + B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)), x, 4, -((A - B)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + (A*Cos[e + f*x])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (A*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*a*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)), x, 5, -((A - B)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + ((3*A - B)*Cos[e + f*x])/(8*a*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + ((3*A - B)*Cos[e + f*x])/(8*a*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + ((3*A - B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2), x, 8, (8*(3*A - 7*B)*c^5*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (4*(3*A - 7*B)*c^4*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*A - 7*B)*c^3*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*A - 7*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(3*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*A - 7*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2), x, 7, (6*(A - 3*B)*c^4*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (3*(A - 3*B)*c^3*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (3*(A - 3*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + ((A - 3*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(2*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2))/(a + a*Sin[e + f*x])^(5/2), x, 6, ((A - 5*B)*c^3*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A - 5*B)*c^2*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(2*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + ((A - 5*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(4*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2))/(a + a*Sin[e + f*x])^(5/2), x, 5, -((B*c^2*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - (B*c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2))/(a + a*Sin[e + f*x])^(5/2), x, 3, -(((A - B)*c*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])) - (B*c*Cos[e + f*x])/(a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(1/2)), x, 4, -((A - B)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - ((A + B)*Cos[e + f*x])/(4*a*f*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) + ((A + B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)), x, 5, -(((A - B)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))) - ((3*A + B)*Cos[e + f*x])/(8*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(3/2)) + ((3*A + B)*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + ((3*A + B)*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)), x, 6, -((A - B)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2)) - (A*Cos[e + f*x])/(2*a*f*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2)) + (3*A*Cos[e + f*x])/(8*a^2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (3*A*Cos[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (3*A*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a^2*c^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x]) when m and/or n symbolic*) - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^n, x, 5, (1/(f*(1 + 2*m)*(1 + m + n)))*(2^(1/2 + n)*c*(B*(m - n) + A*(1 + m + n))*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 - 2*n), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*(1 + m + n))} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3, x, 5, (2^(1/2 + m)*a^4*c^3*(B*(3 - m) - A*(4 + m))*Cos[e + f*x]^7*Hypergeometric2F1[7/2, 1/2 - m, 9/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-4 + m))/(7*f*(4 + m)) - (a^3*B*c^3*Cos[e + f*x]^7*(a + a*Sin[e + f*x])^(-3 + m))/(f*(4 + m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^2, x, 5, (2^(1/2 + m)*a^3*c^2*(B*(2 - m) - A*(3 + m))*Cos[e + f*x]^5*Hypergeometric2F1[5/2, 1/2 - m, 7/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-3 + m))/(5*f*(3 + m)) - (a^2*B*c^2*Cos[e + f*x]^5*(a + a*Sin[e + f*x])^(-2 + m))/(f*(3 + m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^1, x, 5, (2^(1/2 + m)*a^2*c*(B*(1 - m) - A*(2 + m))*Cos[e + f*x]^3*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(-2 + m))/(3*f*(2 + m)) - (a*B*c*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^(-1 + m))/(f*(2 + m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^0, x, 3, -((B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*(A + A*m + B*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^1, x, 5, (2^(1/2 + m)*(B + A*m + B*m)*Hypergeometric2F1[-(1/2), 1/2 - m, 1/2, (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^m)/(c*f*m) - (B*Sec[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*c*f*m)} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^2, x, 5, (2^(1/2 + m)*(A*(1 - m) - B*(2 + m))*Hypergeometric2F1[-(3/2), 1/2 - m, -(1/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^3*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(1 + m))/(3*a*c^2*f*(1 - m)) + (B*Sec[e + f*x]^3*(a + a*Sin[e + f*x])^(2 + m))/(a^2*c^2*f*(1 - m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^3, x, 5, (2^(1/2 + m)*(A*(2 - m) - B*(3 + m))*Hypergeometric2F1[-(5/2), 1/2 - m, -(3/2), (1/2)*(1 - Sin[e + f*x])]*Sec[e + f*x]^5*(1 + Sin[e + f*x])^(1/2 - m)*(a + a*Sin[e + f*x])^(2 + m))/(5*a^2*c^3*f*(2 - m)) + (B*Sec[e + f*x]^5*(a + a*Sin[e + f*x])^(3 + m))/(a^3*c^3*f*(2 - m))} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/Sqrt[c - c*Sin[e + f*x]], x, 4, -((2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])) + ((A + B)*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])*(c + c*Sin[e + f*x])^m/Sqrt[a - a*Sin[e + f*x]], x, 4, -((2*B*Cos[e + f*x]*(c + c*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[a - a*Sin[e + f*x]])) + ((A + B)*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(c + c*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[a - a*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(5/2), x, 4, If[$VersionNumber>=8, -((64*c^3*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (16*c^2*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)), -((64*c^3*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (16*c^2*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(35 + 24*m + 4*m^2)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(3/2), x, 3, (4*(A - B)*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) - (2*(A - 3*B)*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) - (2*B*c^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(2 + m))/(a^2*f*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]), If[$VersionNumber>=8, -((8*c^2*(B*(3 - 2*m) - A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (2*c*(B*(3 - 2*m) - A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m)*(5 + 2*m)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)), -((8*c^2*(B*(3 - 2*m) - A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (2*c*(B*(3 - 2*m) - A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(15 + 16*m + 4*m^2)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1/2), x, 3, (2*(A - B)*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*B*c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(1/2), x, 4, -((2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])) + ((A + B)*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(3/2), x, 4, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(2*f*(c - c*Sin[e + f*x])^(3/2)) + ((A*(1 - 2*m) - B*(3 + 2*m))*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c - c*Sin[e + f*x])^(5/2), x, 4, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(4*f*(c - c*Sin[e + f*x])^(5/2)) + ((A*(3 - 2*m) - B*(5 + 2*m))*Cos[e + f*x]*Hypergeometric2F1[2, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(16*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-4 - m), x, 4, If[$VersionNumber>=8, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(7 + 2*m)) + ((3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(c*f*(5 + 2*m)*(7 + 2*m)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c^2*f*(7 + 2*m)*(15 + 16*m + 4*m^2)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^3*f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)), ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-4 - m))/(f*(7 + 2*m)) + ((3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(c*f*(35 + 24*m + 4*m^2)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c^2*f*(105 + 142*m + 60*m^2 + 8*m^3)) + (2*(3*A - 2*B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^3*f*(105 + 352*m + 344*m^2 + 128*m^3 + 16*m^4))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-3 - m), x, 3, If[$VersionNumber>=8, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(f*(5 + 2*m)) + ((2*A - B*(3 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c*f*(3 + 2*m)*(5 + 2*m)) + ((2*A - B*(3 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*(5 + 2*m)*(3 + 8*m + 4*m^2)), ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-3 - m))/(f*(5 + 2*m)) + ((2*A - B*(3 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(c*f*(15 + 16*m + 4*m^2)) + ((2*A - B*(3 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c^2*f*(15 + 46*m + 36*m^2 + 8*m^3))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-2 - m), x, 2, If[$VersionNumber>=8, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m)) + ((A - 2*B*(1 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*(1 + 2*m)*(3 + 2*m)), ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m)) + ((A - 2*B*(1 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(c*f*(3 + 8*m + 4*m^2))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(-1 - m), x, 5, ((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m)) - (2^(1/2 - m)*B*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(0 - m), x, 5, (2^(1/2 - m)*c*(A + 2*B*m)*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(f*(1 + 2*m)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - c*Sin[e + f*x])^m*f)} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(1 - m), x, 5, (1/(f*(1 + 2*m)))*(2^(1/2 - m)*c^2*(2*A - B*(1 - 2*m))*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(-1 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 - m))/(2*f)} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(2 - m), x, 5, (1/(3*f*(1 + 2*m)))*(2^(5/2 - m)*c^3*(3*A - 2*B*(1 - m))*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(-3 + 2*m), (1/2)*(1 + 2*m), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(2 - m))/(3*f)} - - -(* Degenerate special cases *) -{(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^n*(B*(3 - n) - (3 + n + 1)*B*Sin[e + f*x]), x, 2, (a^3*B*c^3*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^(-3 + n))/f} -{(a - a*Sin[e + f*x])^3*(c + c*Sin[e + f*x])^n*(B*(3 - n) + (3 + n + 1)*B*Sin[e + f*x]), x, 2, -((a^3*B*c^3*Cos[e + f*x]^7*(c + c*Sin[e + f*x])^(-3 + n))/f)} - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^3*(B*(m - 3) - (m + 3 + 1)*B*Sin[e + f*x]), x, 2, (a^3*B*c^3*Cos[e + f*x]^7*(a + a*Sin[e + f*x])^(-3 + m))/f} -{(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^3*(B*(m - 3) + (m + 3 + 1)*B*Sin[e + f*x]), x, 2, -((a^3*B*c^3*Cos[e + f*x]^7*(a - a*Sin[e + f*x])^(-3 + m))/f)} - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(B*(m - n) - (m + n + 1)*B*Sin[e + f*x]), x, 1, (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/f} -{(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n*(B*(m - n) + (m + n + 1)*B*Sin[e + f*x]), x, 1, -((B*Cos[e + f*x]*(a - a*Sin[e + f*x])^m*(c + c*Sin[e + f*x])^n)/f)} - - -(* ::InheritFromParent:: *) -(**) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^n (a+a Sin[e+f x])^m (A-A Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 13, (1/8)*a^3*A*x - (2*a^3*A*Cos[c + d*x]^3)/(3*d) + (3*a^3*A*Cos[c + d*x]^5)/(5*d) - (a^3*A*Cos[c + d*x]^7)/(7*d) - (a^3*A*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a^3*A*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) + (a^3*A*Cos[c + d*x]*Sin[c + d*x]^5)/(3*d)} -{Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 12, (3/16)*a^3*A*x - (2*a^3*A*Cos[c + d*x]^3)/(3*d) + (2*a^3*A*Cos[c + d*x]^5)/(5*d) - (3*a^3*A*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*A*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) + (a^3*A*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{Sin[c + d*x]^1*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 10, (1/4)*a^3*A*x - (2*a^3*A*Cos[c + d*x]^3)/(3*d) + (a^3*A*Cos[c + d*x]^5)/(5*d) - (a^3*A*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^3*A*Cos[c + d*x]*Sin[c + d*x]^3)/(2*d)} -{Sin[c + d*x]^0*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 5, (5/8)*a^3*A*x - (5*a^3*A*Cos[c + d*x]^3)/(12*d) + (5*a^3*A*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (A*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(4*d)} -{Csc[c + d*x]^1*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 7, a^3*A*x - (a^3*A*ArcTanh[Cos[c + d*x]])/d + (a^3*A*Cos[c + d*x])/d - (a^3*A*Cos[c + d*x]^3)/(3*d) + (a^3*A*Cos[c + d*x]*Sin[c + d*x])/d} -{Csc[c + d*x]^2*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 9, (-(1/2))*a^3*A*x - (2*a^3*A*ArcTanh[Cos[c + d*x]])/d + (2*a^3*A*Cos[c + d*x])/d - (a^3*A*Cot[c + d*x])/d + (a^3*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 7, -2*a^3*A*x - (a^3*A*ArcTanh[Cos[c + d*x]])/(2*d) + (a^3*A*Cos[c + d*x])/d - (2*a^3*A*Cot[c + d*x])/d - (a^3*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 7, (-a^3)*A*x + (a^3*A*ArcTanh[Cos[c + d*x]])/d - (a^3*A*Cot[c + d*x])/d - (a^3*A*Cot[c + d*x]^3)/(3*d) - (a^3*A*Cot[c + d*x]*Csc[c + d*x])/d} -{Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 10, (5*a^3*A*ArcTanh[Cos[c + d*x]])/(8*d) - (2*a^3*A*Cot[c + d*x]^3)/(3*d) - (3*a^3*A*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^3*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Csc[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 12, (a^3*A*ArcTanh[Cos[c + d*x]])/(4*d) - (2*a^3*A*Cot[c + d*x]^3)/(3*d) - (a^3*A*Cot[c + d*x]^5)/(5*d) + (a^3*A*Cot[c + d*x]*Csc[c + d*x])/(4*d) - (a^3*A*Cot[c + d*x]*Csc[c + d*x]^3)/(2*d)} -{Csc[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A - A*Sin[c + d*x]), x, 12, (3*a^3*A*ArcTanh[Cos[c + d*x]])/(16*d) - (2*a^3*A*Cot[c + d*x]^3)/(3*d) - (2*a^3*A*Cot[c + d*x]^5)/(5*d) + (3*a^3*A*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a^3*A*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*A*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sin[c + d*x]^4*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 11, -((19*A*x)/(2*a^3)) - (4*A*Cos[c + d*x])/(a^3*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (41*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) - (199*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x]))} -{Sin[c + d*x]^3*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 9, (4*A*x)/a^3 + (A*Cos[c + d*x])/(a^3*d) + (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) - (31*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) + (104*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x]))} -{Sin[c + d*x]^2*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 8, -((A*x)/a^3) - (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (7*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^2) - (13*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x]))} -{Sin[c + d*x]^1*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 8, (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) - (11*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) + (4*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x]))} -{Sin[c + d*x]^0*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 3, -((a*A*Cos[c + d*x]^3)/(5*d*(a + a*Sin[c + d*x])^4)) - (A*Cos[c + d*x]^3)/(15*d*(a + a*Sin[c + d*x])^3)} -{Csc[c + d*x]^1*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 9, -((A*ArcTanh[Cos[c + d*x]])/(a^3*d)) + (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (3*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^2) + (8*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x]))} -{Csc[c + d*x]^2*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, If[$VersionNumber<9, 16, 15], If[$VersionNumber<9, (4*A*ArcTanh[Cos[c + d*x]])/(a^3*d) - (94*A*Cot[c + d*x])/(15*a^3*d) + (2*A*Cot[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (13*A*Cot[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) + (4*A*Cot[c + d*x])/(a^3*d*(1 + Sin[c + d*x])), (4*A*ArcTanh[Cos[c + d*x]])/(a^3*d) - (A*Cot[c + d*x])/(a^3*d) - (2*A*Cot[c + d*x])/(5*a^3*d*(1 + Csc[c + d*x])^3) + (31*A*Cot[c + d*x])/(15*a^3*d*(1 + Csc[c + d*x])^2) - (104*A*Cot[c + d*x])/(15*a^3*d*(1 + Csc[c + d*x]))]} -{Csc[c + d*x]^3*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 13, -((19*A*ArcTanh[Cos[c + d*x]])/(2*a^3*d)) + (4*A*Cot[c + d*x])/(a^3*d) - (A*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) + (29*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x])^2) + (164*A*Cos[c + d*x])/(15*a^3*d*(1 + Sin[c + d*x]))} -{Csc[c + d*x]^4*(A - A*Sin[c + d*x])/(a + a*Sin[c + d*x])^3, x, 15, (18*A*ArcTanh[Cos[c + d*x]])/(a^3*d) - (10*A*Cot[c + d*x])/(a^3*d) - (A*Cot[c + d*x]^3)/(3*a^3*d) + (2*A*Cot[c + d*x]*Csc[c + d*x])/(a^3*d) - (2*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^3) - (13*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x])^2) - (93*A*Cos[c + d*x])/(5*a^3*d*(1 + Sin[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 5, (1/8)*a*(B*(4*c^3 + 12*c^2*d + 9*c*d^2 + 3*d^3) + A*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3))*x - (1/(30*d*f))*(a*(5*A*d*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3) - B*(3*c^4 - 15*c^3*d - 52*c^2*d^2 - 60*c*d^3 - 16*d^4))*Cos[e + f*x]) - (a*(5*A*d*(6*c^2 + 20*c*d + 9*d^2) - B*(6*c^3 - 30*c^2*d - 71*c*d^2 - 45*d^3))*Cos[e + f*x]*Sin[e + f*x])/(120*f) - (a*(4*(5*A + 4*B)*d^2 - 3*c*(B*c - 5*(A + B)*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d*f) + (a*(B*c - 5*(A + B)*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d*f) - (a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 4, (1/8)*a*(4*A*(2*c^2 + 2*c*d + d^2) + B*(4*c^2 + 8*c*d + 3*d^2))*x - (a*(4*A*d*(c^2 + 3*c*d + d^2) - B*(c^3 - 4*c^2*d - 8*c*d^2 - 4*d^3))*Cos[e + f*x])/(6*d*f) - (a*(3*(4*A + 3*B)*d^2 - 2*c*(B*c - 4*(A + B)*d))*Cos[e + f*x]*Sin[e + f*x])/(24*f) + (a*(B*c - 4*(A + B)*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(12*d*f) - (a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*d*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^1, x, 3, (a*(B*(c + d) + A*(2*c + d))*x)/2 - (a*(3*A*(c + d) + B*(3*c + d))*Cos[e + f*x])/(3*f) - (a*(3*B*c + 3*A*d - B*d)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*a*f)} -{(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^0, x, 1, (a*(2*A + B)*x)/2 - (a*(A + B)*Cos[e + f*x])/f - (a*B*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^1, x, 6, -((a*(B*c - (A + B)*d)*x)/d^2) + (2*a*(c - d)*(B*c - A*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*Sqrt[c^2 - d^2]*f) - (a*B*Cos[e + f*x])/(d*f)} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 6, (a*B*x)/d^2 + (2*a*((A + B)*(c - d)*d^2 - B*c*(c^2 - d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^2*(c^2 - d^2)^(3/2)*f) + (a*(B*c - A*d)*Cos[e + f*x])/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 7, (a*(2*A*c + B*c - A*d - 2*B*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(3/2)*f) + (a*(B*c - A*d)*Cos[e + f*x])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a*(A*(c - 2*d)*d + B*(c^2 + 2*c*d - 2*d^2))*Cos[e + f*x])/(2*(c - d)*d*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 6, (1/16)*a^2*(6*A*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3) + B*(16*c^3 + 42*c^2*d + 36*c*d^2 + 11*d^3))*x + (a^2*(6*A*d*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4) - B*(2*c^5 - 12*c^4*d + 47*c^3*d^2 + 208*c^2*d^3 + 216*c*d^4 + 64*d^5))*Cos[e + f*x])/(60*d^2*f) + (a^2*(6*A*d*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3) - B*(4*c^4 - 24*c^3*d + 96*c^2*d^2 + 284*c*d^3 + 165*d^4))*Cos[e + f*x]*Sin[e + f*x])/(240*d*f) + (a^2*(6*A*d*(c^2 - 10*c*d - 12*d^2) - B*(2*c^3 - 12*c^2*d + 51*c*d^2 + 64*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^2*f) + (a^2*(6*A*(c - 10*d)*d - B*(2*c^2 - 12*c*d + 55*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(120*d^2*f) + (a^2*(2*B*c - 6*A*d - 7*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (B*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 5, (1/8)*a^2*(12*A*c^2 + 8*B*c^2 + 16*A*c*d + 14*B*c*d + 7*A*d^2 + 6*B*d^2)*x + (a^2*(5*A*d*(c^3 - 8*c^2*d - 20*c*d^2 - 8*d^3) - 2*B*(c^4 - 5*c^3*d + 16*c^2*d^2 + 40*c*d^3 + 18*d^4))*Cos[e + f*x])/(30*d^2*f) + (a^2*(5*A*d*(2*c^2 - 16*c*d - 21*d^2) - B*(4*c^3 - 20*c^2*d + 66*c*d^2 + 90*d^3))*Cos[e + f*x]*Sin[e + f*x])/(120*d*f) + (a^2*(5*A*(c - 8*d)*d - 2*B*(c^2 - 5*c*d + 18*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d^2*f) + (a^2*(2*B*(c - 3*d) - 5*A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d^2*f) - (B*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(5*d*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]), x, 4, (a^2*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*x)/8 - (a^2*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*Cos[e + f*x])/(6*f) - (a^2*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((4*B*c + 4*A*d - B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(12*f) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*a*f)} -{(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]), x, 2, (a^2*(3*A + 2*B)*x)/2 - (2*a^2*(3*A + 2*B)*Cos[e + f*x])/(3*f) - (a^2*(3*A + 2*B)*Cos[e + f*x]*Sin[e + f*x])/(6*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*f)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]), x, 7, -((a^2*(2*A*(c - 2*d)*d - B*(2*c^2 - 4*c*d + 3*d^2))*x)/(2*d^3)) - (2*a^2*(c - d)^2*(B*c - A*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*Sqrt[c^2 - d^2]*f) + (a^2*(2*B*c - 2*A*d - 3*B*d)*Cos[e + f*x])/(2*d^2*f) - (B*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x]))/(2*d*f)} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 7, -((a^2*(2*B*c - A*d - 2*B*d)*x)/d^3) - (2*a^2*(c - d)*(A*d*(c + 2*d) - B*(2*c^2 + 2*c*d - d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c + d)*Sqrt[c^2 - d^2]*f) + (a^2*(A*d - B*(2*c + d))*Cos[e + f*x])/(d^2*(c + d)*f) + ((B*c - A*d)*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 7, (a^2*B*x)/d^3 + (a^2*(3*A*d^3 - B*(2*c^3 + 4*c^2*d + c*d^2 - 4*d^3))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c + d)^2*Sqrt[c^2 - d^2]*f) + ((B*c - A*d)*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x]))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(3*A*d^2 - B*(2*c^2 + 3*c*d - 2*d^2))*Cos[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 7, (1/16)*a^3*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3))*x - (a^3*(7*A*d*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5) - 3*B*(2*c^6 - 14*c^5*d + 51*c^4*d^2 - 189*c^3*d^3 - 920*c^2*d^4 - 952*c*d^5 - 288*d^6))*Cos[e + f*x])/(420*d^3*f) - (a^3*(7*A*d*(4*c^4 - 36*c^3*d + 216*c^2*d^2 + 626*c*d^3 + 345*d^4) - 3*B*(4*c^5 - 28*c^4*d + 104*c^3*d^2 - 392*c^2*d^3 - 1263*c*d^4 - 735*d^5))*Cos[e + f*x]*Sin[e + f*x])/(1680*d^2*f) - (a^3*(7*A*d*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3) - B*(6*c^4 - 42*c^3*d + 165*c^2*d^2 - 651*c*d^3 - 864*d^4))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(840*d^3*f) - (a^3*(7*A*d*(2*c^2 - 18*c*d + 115*d^2) - B*(6*c^3 - 42*c^2*d + 177*c*d^2 - 735*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(840*d^3*f) - (a^3*(6*B*c^2 - 14*A*c*d - 27*B*c*d + 91*A*d^2 + 87*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(210*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4)/(7*d*f) + ((3*B*(c - 3*d) - 7*A*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(42*d^2*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 6, (1/16)*a^3*(B*(30*c^2 + 52*c*d + 23*d^2) + A*(40*c^2 + 60*c*d + 26*d^2))*x - (a^3*(2*A*d*(2*c^4 - 15*c^3*d + 72*c^2*d^2 + 180*c*d^3 + 76*d^4) - B*(2*c^5 - 12*c^4*d + 37*c^3*d^2 - 112*c^2*d^3 - 304*c*d^4 - 136*d^5))*Cos[e + f*x])/(60*d^3*f) - (a^3*(2*A*d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3) - B*(4*c^4 - 24*c^3*d + 76*c^2*d^2 - 236*c*d^3 - 345*d^4))*Cos[e + f*x]*Sin[e + f*x])/(240*d^2*f) - (a^3*(2*A*d*(2*c^2 - 15*c*d + 76*d^2) - B*(2*c^3 - 12*c^2*d + 41*c*d^2 - 136*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(120*d^3*f) + (a^3*(2*A*(2*c - 11*d)*d - B*(2*c^2 - 8*c*d + 21*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(40*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3)/(6*d*f) + ((3*B*c - 6*A*d - 8*B*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(30*d^2*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^1, x, 10, (1/8)*a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*x - (a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x])/(5*f) + (a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]^3)/(60*f) - (3*a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]*Sin[e + f*x])/(40*f) - ((5*B*c + 5*A*d - B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(20*f) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^4)/(5*a*f)} -{(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^0, x, 8, (5*a^3*(4*A + 3*B)*x)/8 - (5*a^3*(4*A + 3*B)*Cos[e + f*x])/(6*f) - (5*a^3*(4*A + 3*B)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - (a*(4*A + 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(12*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*f), (5/8)*a^3*(4*A + 3*B)*x - (a^3*(4*A + 3*B)*Cos[e + f*x])/f + (a^3*(4*A + 3*B)*Cos[e + f*x]^3)/(12*f) - (3*a^3*(4*A + 3*B)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*f)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^1, x, 8, (a^3*(A*d*(2*c^2 - 6*c*d + 7*d^2) - B*(2*c^3 - 6*c^2*d + 7*c*d^2 - 5*d^3))*x)/(2*d^4) + (2*a^3*(c - d)^3*(B*c - A*d)*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^4*Sqrt[c^2 - d^2]*f) + (a^3*(A*(2*c - 5*d)*d - B*(2*c^2 - 5*c*d + 5*d^2))*Cos[e + f*x])/(2*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*d*f) + ((3*B*c - 3*A*d - 5*B*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(6*d^2*f)} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 8, -((a^3*(2*A*(2*c - 3*d)*d - B*(6*c^2 - 12*c*d + 7*d^2))*x)/(2*d^4)) + (2*a^3*(c - d)^2*(A*d*(2*c + 3*d) - B*(3*c^2 + 3*c*d - d^2))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^4*(c + d)*Sqrt[c^2 - d^2]*f) - (a^3*(4*A*c*d - B*(6*c^2 - 3*c*d - 5*d^2))*Cos[e + f*x])/(2*d^3*(c + d)*f) + ((2*A*d - B*(3*c + d))*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d^2*(c + d)*f) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 8, -((a^3*(3*B*c - A*d - 3*B*d)*x)/d^4) - (a^3*(c - d)*(A*d*(2*c^2 + 6*c*d + 7*d^2) - 3*B*(2*c^3 + 4*c^2*d + c*d^2 - 2*d^3))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (a^3*(3*B*c*(2*c + 3*d) - A*d*(2*c + 5*d))*Cos[e + f*x])/(2*d^3*(c + d)^2*f) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A*d*(c + 4*d) - B*(3*c^2 + 4*c*d - 2*d^2))*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x]))/(2*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x]), x, 3, ((3*A*d*(2*c^2 - 2*c*d + d^2) + B*(2*c^3 - 6*c^2*d + 9*c*d^2 - 3*d^3))*x)/(2*a) + (2*d*(3*A*(c^2 - 3*c*d + d^2) - B*(7*c^2 - 9*c*d + 4*d^2))*Cos[e + f*x])/(3*a*f) + (d^2*(6*A*c - 11*B*c - 9*A*d + 9*B*d)*Cos[e + f*x]*Sin[e + f*x])/(6*a*f) + ((3*A - 4*B)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a*f) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(f*(a + a*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x]), x, 2, ((2*A*(2*c - d)*d + B*(2*c^2 - 4*c*d + 3*d^2))*x)/(2*a) + (2*(A*(c - d) - B*(2*c - d))*d*Cos[e + f*x])/(a*f) + ((2*A - 3*B)*d^2*Cos[e + f*x]*Sin[e + f*x])/(2*a*f) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(f*(a + a*Sin[e + f*x]))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x]), x, 4, ((B*(c - d) + A*d)*x)/a - (B*d*Cos[e + f*x])/(a*f) - ((A - B)*(c - d)*Cos[e + f*x])/(a*f*(1 + Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x]), x, 2, (B*x)/a - ((A - B)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])), x, 5, (2*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*Sqrt[c^2 - d^2]*f) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2), x, 6, (-2*(A*d*(2*c + d) - B*(c^2 + c*d + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(3/2)*f) + (d*(B*(2*c + d) - A*(c + 2*d))*Cos[e + f*x])/(a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3), x, 7, -(((3*A*d*(2*c^2 + 2*c*d + d^2) - B*(2*c^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*A*c - 3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(2*A*c^2 - 5*B*c^2 + 9*A*c*d - 6*B*c*d + 4*A*d^2 - 4*B*d^2)*Cos[e + f*x])/(2*a*(c - d)^3*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^2, x, 3, (d*(2*A*(3*c - 2*d)*d + B*(6*c^2 - 12*c*d + 7*d^2))*x)/(2*a^2) + (2*d*(A*(c^2 + 6*c*d - 5*d^2) + B*(2*c^2 - 15*c*d + 8*d^2))*Cos[e + f*x])/(3*a^2*f) + (d^2*(B*(4*c - 21*d) + 2*A*(c + 6*d))*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) - ((2*B*(c - 4*d) + A*(c + 5*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*a^2*f*(1 + Sin[e + f*x])) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*f*(a + a*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^2, x, 5, (d*(2*B*(c - d) + A*d)*x)/a^2 + ((A - 4*B)*d^2*Cos[e + f*x])/(3*a^2*f) - ((c - d)*(2*B*(c - 3*d) + A*(c + 3*d))*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*f*(a + a*Sin[e + f*x])^2)} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^2, x, 4, (B*d*x)/a^2 - ((A*c + 2*B*c + 2*A*d - 5*B*d)*Cos[e + f*x])/(3*a^2*f*(1 + Sin[e + f*x])) - ((A - B)*(c - d)*Cos[e + f*x])/(3*f*(a + a*Sin[e + f*x])^2)} -{(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^2, x, 2, -((A - B)*Cos[e + f*x])/(3*f*(a + a*Sin[e + f*x])^2) - ((A + 2*B)*Cos[e + f*x])/(3*f*(a^2 + a^2*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])), x, 6, (-2*d*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^2*Sqrt[c^2 - d^2]*f) - ((A*(c - 4*d) + B*(2*c + d))*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - ((A - B)*Cos[e + f*x])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2)} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2), x, 7, (2*d*(A*d*(3*c + 2*d) - B*(2*c^2 + 2*c*d + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^3*(c + d)*Sqrt[c^2 - d^2]*f) - (d*(A*(c^2 - 6*c*d - 10*d^2) + B*(2*c^2 + 9*c*d + 4*d^2))*Cos[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])) - ((A*c + 2*B*c - 6*A*d + 3*B*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])) - ((A - B)*Cos[e + f*x])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3), x, 8, (d*(A*d*(12*c^2 + 16*c*d + 7*d^2) - B*(6*c^3 + 12*c^2*d + 13*c*d^2 + 4*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^2*(c - d)^4*(c + d)^2*Sqrt[c^2 - d^2]*f) - (d*(A*(2*c^2 - 16*c*d - 21*d^2) + B*(4*c^2 + 19*c*d + 12*d^2))*Cos[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A*c + 2*B*c - 8*A*d + 5*B*d)*Cos[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - (d*(A*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3) + B*(4*c^3 + 37*c^2*d + 44*c*d^2 + 20*d^3))*Cos[e + f*x])/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^3, x, 6, (d^2*(3*B*(c - d) + A*d)*x)/a^3 + (d^2*(3*B*(c - 9*d) + A*(2*c + 7*d))*Cos[e + f*x])/(15*a^3*f) - ((c - d)*(3*B*(c^2 + 6*c*d - 15*d^2) + A*(2*c^2 + 7*c*d + 15*d^2))*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((3*B*(c - 3*d) + 2*A*(c + 2*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(15*a*f*(a + a*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(5*f*(a + a*Sin[e + f*x])^3)} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^3, x, 5, (B*d^2*x)/a^3 - ((c - d)*(B*(3*c - 7*d) + 2*A*(c + d))*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((B*(3*c^2 + 14*c*d - 29*d^2) + 2*A*(c^2 + 3*c*d + 2*d^2))*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x])) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*(a + a*Sin[e + f*x])^3)} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^3, x, 4, -((A - B)*(c - d)*Cos[e + f*x])/(5*f*(a + a*Sin[e + f*x])^3) - ((2*A*c + 3*B*c + 3*A*d - 8*B*d)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((2*A*c + 3*B*c + 3*A*d + 7*B*d)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^3, x, 3, -((A - B)*Cos[e + f*x])/(5*f*(a + a*Sin[e + f*x])^3) - ((2*A + 3*B)*Cos[e + f*x])/(15*a*f*(a + a*Sin[e + f*x])^2) - ((2*A + 3*B)*Cos[e + f*x])/(15*f*(a^3 + a^3*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])), x, 7, (2*d^2*(B*c - A*d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^3*Sqrt[c^2 - d^2]*f) - ((A - B)*Cos[e + f*x])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3) - ((2*A*c + 3*B*c - 7*A*d + 2*B*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2) - ((B*(3*c^2 - 16*c*d - 2*d^2) + A*(2*c^2 - 9*c*d + 22*d^2))*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2), x, 8, (-2*d^2*(A*d*(4*c + 3*d) - B*(3*c^2 + 3*c*d + d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^4*(c + d)*Sqrt[c^2 - d^2]*f) - (d*(B*(3*c^3 - 23*c^2*d - 63*c*d^2 - 22*d^3) + A*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3))*Cos[e + f*x])/(15*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])) - ((A - B)*Cos[e + f*x])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])) - ((2*A*c + 3*B*c - 9*A*d + 4*B*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])) - ((B*(3*c^2 - 23*c*d - 15*d^2) + A*(2*c^2 - 12*c*d + 45*d^2))*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3), x, 9, -((d^2*(A*d*(20*c^2 + 30*c*d + 13*d^2) - 3*B*(4*c^3 + 8*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(a^3*(c - d)^5*(c + d)^2*Sqrt[c^2 - d^2]*f)) - (d*(3*B*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3) + A*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3))*Cos[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/(5*(c - d)*f*(a + a*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^2) - ((2*A*c + 3*B*c - 11*A*d + 6*B*d)*Cos[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) - ((3*B*(c^2 - 10*c*d - 12*d^2) + A*(2*c^2 - 15*c*d + 76*d^2))*Cos[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - (d*(3*B*(2*c^4 - 20*c^3*d - 119*c^2*d^2 - 130*c*d^3 - 48*d^4) + A*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4))*Cos[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 5, (4*a*(c + d)*(B*c - 9*A*d - 8*B*d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(315*d*f*Sqrt[a + a*Sin[e + f*x]]) + (8*(5*c - d)*(c + d)*(B*c - 9*A*d - 8*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) + (4*d*(c + d)*(B*c - 9*A*d - 8*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*a*f) + (2*a*(B*c - 9*A*d - 8*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*d*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 4, (2*a*(B*c - 7*A*d - 6*B*d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(105*d*f*Sqrt[a + a*Sin[e + f*x]]) + (4*(5*c - d)*(B*c - 7*A*d - 6*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) + (2*d*(B*c - 7*A*d - 6*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*a*f) - (2*a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*d*f*Sqrt[a + a*Sin[e + f*x]])} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]), x, 4, (-2*a*(15*A*c + 5*B*c + 5*A*d + 7*B*d)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*(5*B*c + 5*A*d - 2*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f)} -{Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]), x, 2, (-2*a*(3*A + B)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)} -{(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]), x, 3, (2*Sqrt[a]*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*Sqrt[c + d]*f) - (2*a*B*Cos[e + f*x])/(d*f*Sqrt[a + a*Sin[e + f*x]])} -{(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 3, -((Sqrt[a]*(A*d + B*(c + 2*d))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(3/2)*(c + d)^(3/2)*f)) + (a*(B*c - A*d)*Cos[e + f*x])/(d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 4, -(Sqrt[a]*(3*A*d + B*(c + 4*d))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(3/2)*(c + d)^(5/2)*f) + (a*(B*c - A*d)*Cos[e + f*x])/(2*d*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (a*(3*A*d + B*(c + 4*d))*Cos[e + f*x])/(4*d*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 6, (4*a^2*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x])/(3465*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(5*c - d)*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) + (4*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) + (2*a^2*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(3*B*(c - 4*d) - 11*A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 5, (2*a^2*(15*c^2 + 10*c*d + 7*d^2)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x])/(315*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (4*a*(5*c - d)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*d*f) + (2*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) + (2*a^2*(3*B*c - 9*A*d - 10*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3)/(9*d*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]), x, 5, (-8*a^2*(35*A*c + 21*B*c + 21*A*d + 19*B*d)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(35*A*c + 21*B*c + 21*A*d + 19*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (2*(7*B*c + 7*A*d - 2*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*a*f)} -{(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]), x, 3, (-8*a^2*(5*A + 3*B)*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*(5*A + 3*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f)} -{((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]), x, 4, (-2*a^(3/2)*(c - d)*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*Sqrt[c + d]*f) + (2*a^2*(3*B*c - 3*A*d - 4*B*d)*Cos[e + f*x])/(3*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d*f)} -{((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 4, -((a^(3/2)*(A*d*(c + 3*d) - B*(3*c^2 + 3*c*d - 2*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(5/2)*(c + d)^(3/2)*f)) - (a^2*(3*B*c - A*d + 2*B*d)*Cos[e + f*x])/(d^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]) + (a*(B*c - A*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 4, -(a^(3/2)*(A*d*(c + 7*d) + 3*B*(c^2 + 3*c*d + 4*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(5/2)*(c + d)^(5/2)*f) + (a*(B*c - A*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) + (a^2*(A*(c - 5*d)*d + B*(3*c^2 + 5*c*d - 4*d^2))*Cos[e + f*x])/(4*d^2*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3, x, 7, (-4*a^3*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x])/(45045*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (8*a^2*(5*c - d)*(c + d)*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(45045*d^2*f) - (4*a*(c + d)*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(15015*d*f) - (2*a^3*(13*A*d*(3*c^2 - 38*c*d + 355*d^2) - B*(15*c^3 - 150*c^2*d + 799*c*d^2 - 4184*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(9009*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a^3*(15*B*c^2 - 39*A*c*d - 75*B*c*d + 299*A*d^2 + 280*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(1287*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 13*A*d - 16*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^4)/(143*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^4)/(13*d*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 6, (-2*a^3*(15*c^2 + 10*c*d + 7*d^2)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x])/(3465*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*(5*c - d)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d^2*f) - (2*a*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*d*f) + (2*a^3*(11*A*(3*c - 19*d)*d - B*(15*c^2 - 65*c*d + 194*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 11*A*d - 14*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3)/(99*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3)/(11*d*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]), x, 6, (-64*a^3*(21*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(21*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) - (2*(9*B*c + 9*A*d - 2*B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(63*f) - (2*B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)} -{(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]), x, 4, (-64*a^3*(7*A + 5*B)*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(7*A + 5*B)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*f) - (2*a*(7*A + 5*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(35*f) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(7*f)} -{((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x]), x, 5, (2*a^(5/2)*(c - d)^2*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(7/2)*Sqrt[c + d]*f) + (2*a^3*(5*A*(3*c - 7*d)*d - B*(15*c^2 - 35*c*d + 32*d^2))*Cos[e + f*x])/(15*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 5*A*d - 8*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*d*f)} -{((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2, x, 5, (a^(5/2)*(c - d)*(A*d*(3*c + 5*d) - B*(5*c^2 + 5*c*d - 2*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(d^(7/2)*(c + d)^(3/2)*f) - (a^3*(3*A*d*(3*c + d) - B*(15*c^2 - 5*c*d - 14*d^2))*Cos[e + f*x])/(3*d^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]) - (a^2*(5*B*c - 3*A*d + 2*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*d^2*(c + d)*f) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(d*(c + d)*f*(c + d*Sin[e + f*x]))} -{((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c + d*Sin[e + f*x])^3, x, 5, -(a^(5/2)*(A*d*(3*c^2 + 10*c*d + 19*d^2) - B*(15*c^3 + 30*c^2*d + 7*c*d^2 - 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*d^(7/2)*(c + d)^(5/2)*f) + (a^3*(3*A*d*(c + 3*d) - B*(15*c^2 + 25*c*d + 4*d^2))*Cos[e + f*x])/(4*d^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a*(B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a^2*(A*d*(c + 7*d) - B*(5*c^2 + 7*c*d - 4*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*d^2*(c + d)^2*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/Sqrt[a + a*Sin[e + f*x]], x, 7, -((Sqrt[2]*(A - B)*(c - d)^3*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*(7*A*d*(21*c^2 - 12*c*d + 7*d^2) + B*(36*c^3 - 63*c^2*d + 144*c*d^2 - 37*d^3))*Cos[e + f*x])/(105*f*Sqrt[a + a*Sin[e + f*x]]) - (2*d*(7*A*(9*c - d)*d + B*(24*c^2 - 15*c*d + 31*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(105*a*f) - (2*(6*B*c + 7*A*d - B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(35*f*Sqrt[a + a*Sin[e + f*x]]) - (2*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(7*f*Sqrt[a + a*Sin[e + f*x]])} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/Sqrt[a + a*Sin[e + f*x]], x, 6, -((Sqrt[2]*(A - B)*(c - d)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (4*(5*A*(3*c - d)*d + B*(6*c^2 - 7*c*d + 7*d^2))*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) - (2*d*(4*B*c + 5*A*d - B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*a*f) - (2*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(5*f*Sqrt[a + a*Sin[e + f*x]])} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/Sqrt[a + a*Sin[e + f*x]], x, 5, -((Sqrt[2]*(A - B)*(c - d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (2*(3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*B*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*a*f)} -{(A + B*Sin[e + f*x])/Sqrt[a + a*Sin[e + f*x]], x, 3, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*f)) - (2*B*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x, 5, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*f)) - (2*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[d]*Sqrt[c + d]*f)} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2), x, 6, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*f)) + ((A*d*(3*c + d) - B*(c^2 + c*d + 2*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^2*Sqrt[d]*(c + d)^(3/2)*f) - ((B*c - A*d)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3), x, 7, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*(c - d)^3*f)) + ((A*d*(15*c^2 + 10*c*d + 7*d^2) - B*(3*c^3 + 6*c^2*d + 19*c*d^2 + 4*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*Sqrt[a]*(c - d)^3*Sqrt[d]*(c + d)^(5/2)*f) - ((B*c - A*d)*Cos[e + f*x])/(2*(c^2 - d^2)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + ((A*d*(7*c + d) - B*(3*c^2 + c*d + 4*d^2))*Cos[e + f*x])/(4*(c^2 - d^2)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^(3/2), x, 7, -((c - d)^2*(3*B*(c - 5*d) + A*(c + 11*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + (d*(15*A*c^2 - 99*B*c^2 - 120*A*c*d + 168*B*c*d + 65*A*d^2 - 93*B*d^2)*Cos[e + f*x])/(15*a*f*Sqrt[a + a*Sin[e + f*x]]) + (d^2*(15*A*c - 51*B*c - 35*A*d + 39*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(30*a^2*f) + ((5*A - 9*B)*d*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(10*a*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^(3/2), x, 6, -((c - d)*(A*c + 3*B*c + 7*A*d - 11*B*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) + (d*(3*A*c - 15*B*c - 9*A*d + 13*B*d)*Cos[e + f*x])/(3*a*f*Sqrt[a + a*Sin[e + f*x]]) + ((3*A - 7*B)*d^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(6*a^2*f) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^(3/2), x, 5, -((A*c + 3*B*c + 3*A*d - 7*B*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((A - B)*(c - d)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2)) - (2*B*d*Cos[e + f*x])/(a*f*Sqrt[a + a*Sin[e + f*x]])} -{(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^(3/2), x, 3, -((A + 3*B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((A - B)*Cos[e + f*x])/(2*f*(a + a*Sin[e + f*x])^(3/2))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])), x, 6, -((A*(c - 5*d) + B*(3*c + d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f) + (2*Sqrt[d]*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^2*Sqrt[c + d]*f) - ((A - B)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2), x, 7, -((A*c + 3*B*c - 9*A*d + 5*B*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^3*f) - (Sqrt[d]*(A*d*(5*c + 3*d) - B*(3*c^2 + 3*c*d + 2*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(3/2)*(c - d)^3*(c + d)^(3/2)*f) - ((A - B)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) + (d*(B*(3*c + d) - A*(c + 3*d))*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3), x, 8, -((A*(c - 13*d) + 3*B*(c + 3*d))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*(c - d)^4*f) - (Sqrt[d]*(A*d*(35*c^2 + 42*c*d + 19*d^2) - 3*B*(5*c^3 + 10*c^2*d + 13*c*d^2 + 4*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(3/2)*(c - d)^4*(c + d)^(5/2)*f) - ((A - B)*Cos[e + f*x])/(2*(c - d)*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) + (d*(B*(2*c + d) - A*(c + 2*d))*Cos[e + f*x])/(2*a*(c - d)^2*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) + (d*(3*B*(3*c^2 + 3*c*d + 2*d^2) - A*(2*c^2 + 15*c*d + 7*d^2))*Cos[e + f*x])/(4*a*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^(5/2), x, 7, -((c - d)*(B*(5*c^2 + 62*c*d - 163*d^2) + 3*A*(c^2 + 6*c*d + 25*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) + (d*(A*(9*c^2 + 36*c*d - 93*d^2) + B*(15*c^2 - 228*c*d + 197*d^2))*Cos[e + f*x])/(24*a^2*f*Sqrt[a + a*Sin[e + f*x]]) + (d^2*(9*A*c + 15*B*c + 39*A*d - 95*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(48*a^3*f) - ((3*A*c + 5*B*c + 9*A*d - 17*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2)/(a + a*Sin[e + f*x])^(5/2), x, 6, -((B*(5*c^2 + 38*c*d - 75*d^2) + A*(3*c^2 + 10*c*d + 19*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*(3*A*c + 5*B*c + 5*A*d - 13*B*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2)) + ((A - 9*B)*d^2*Cos[e + f*x])/(4*a^2*f*Sqrt[a + a*Sin[e + f*x]]) - ((A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(4*f*(a + a*Sin[e + f*x])^(5/2))} -{((A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]))/(a + a*Sin[e + f*x])^(5/2), x, 5, -((3*A*c + 5*B*c + 5*A*d + 19*B*d)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((A - B)*(c - d)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*A*c + 5*B*c + 5*A*d - 13*B*d)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))} -{(A + B*Sin[e + f*x])/(a + a*Sin[e + f*x])^(5/2), x, 4, -((3*A + 5*B)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((A - B)*Cos[e + f*x])/(4*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*A + 5*B)*Cos[e + f*x])/(16*a*f*(a + a*Sin[e + f*x])^(3/2))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])), x, 7, -((B*(5*c^2 - 34*c*d - 3*d^2) + A*(3*c^2 - 14*c*d + 43*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^3*f) - (2*d^(3/2)*(B*c - A*d)*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^3*Sqrt[c + d]*f) - ((A - B)*Cos[e + f*x])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)) - ((3*A*c + 5*B*c - 11*A*d + 3*B*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2), x, 8, -((B*(5*c^2 - 58*c*d - 43*d^2) + A*(3*c^2 - 22*c*d + 115*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^4*f) + (d^(3/2)*(A*d*(7*c + 5*d) - B*(5*c^2 + 5*c*d + 2*d^2))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(a^(5/2)*(c - d)^4*(c + d)^(3/2)*f) - ((A - B)*Cos[e + f*x])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])) - ((3*A*c + 5*B*c - 15*A*d + 7*B*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])) - (d*(A*(3*c^2 - 16*c*d - 35*d^2) + B*(5*c^2 + 32*c*d + 11*d^2))*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} -{(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^3), x, 9, -((B*(5*c^2 - 82*c*d - 115*d^2) + 3*A*(c^2 - 10*c*d + 73*d^2))*ArcTanh[(Sqrt[a]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*(c - d)^5*f) + (d^(3/2)*(3*A*d*(21*c^2 + 30*c*d + 13*d^2) - B*(35*c^3 + 70*c^2*d + 67*c*d^2 + 20*d^3))*ArcTanh[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sin[e + f*x]])])/(4*a^(5/2)*(c - d)^5*(c + d)^(5/2)*f) - ((A - B)*Cos[e + f*x])/(4*(c - d)*f*(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2) - ((3*A*c + 5*B*c - 19*A*d + 11*B*d)*Cos[e + f*x])/(16*a*(c - d)^2*f*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^2) - (d*(A*(3*c^2 - 20*c*d - 31*d^2) + B*(5*c^2 + 28*c*d + 15*d^2))*Cos[e + f*x])/(16*a^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^2) - (d*(3*A*(c^3 - 7*c^2*d - 37*c*d^2 - 21*d^3) + B*(5*c^3 + 73*c^2*d + 79*c*d^2 + 35*d^3))*Cos[e + f*x])/(16*a^2*(c - d)^4*(c + d)^2*f*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic*) - - -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^2, x, 7, -((8*Sqrt[2]*a^2*B*AppellF1[1/2, -(5/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]]))) - (4*Sqrt[2]*a^2*(A - B)*AppellF1[1/2, -(3/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^1, x, 8, -((4*Sqrt[2]*a*B*AppellF1[1/2, -(3/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]]))) - (2*Sqrt[2]*a*(A - B)*AppellF1[1/2, -(1/2), -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[1 + Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^1, x, 7, -((Sqrt[2]*B*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(a*f*Sqrt[1 + Sin[e + f*x]]))) - ((A - B)*AppellF1[1/2, 3/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(Sqrt[2]*a*f*Sqrt[1 + Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^2, x, 7, -((B*AppellF1[1/2, 3/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]]))) - ((A - B)*AppellF1[1/2, 5/2, -n, 3/2, (1/2)*(1 - Sin[e + f*x]), (d*(1 - Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(2*Sqrt[2]*a^2*f*Sqrt[1 + Sin[e + f*x]]))} - -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(3/2), x, 11, If[$VersionNumber>=8, -((2*a^2*(A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) + (2*a^2*B*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(5 + 2*n)) + (2*a^2*(A - B)*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) - (2*a^2*B*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]])), -((2*a^2*(A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) + (2*a^2*B*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(15 + 16*n + 4*n^2)*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(5 + 2*n)) + (2*a^2*(A - B)*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) - (2*a^2*B*(3*c^2 - 2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d^2*f*(15 + 16*n + 4*n^2)*Sqrt[a + a*Sin[e + f*x]]))]} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n*(a + a*Sin[e + f*x])^(1/2), x, 4, -((2*a*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]])) - (2*a*(A*d*(3 + 2*n) - B*(c - 2*d*(1 + n)))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(1/2), x, 7, -(((A - B)*AppellF1[1 + n, 1/2, 1, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)*f*(1 + n)*(1 - Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]])) - (2*B*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c + d))^n*(f*Sqrt[a + a*Sin[e + f*x]]))} -{(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n/(a + a*Sin[e + f*x])^(3/2), x, 7, -((B*AppellF1[1 + n, 1/2, 1, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/(a*(c - d)*f*(1 + n)*(1 - Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]])) + ((A - B)*d*AppellF1[1 + n, 1/2, 2, 2 + n, (c + d*Sin[e + f*x])/(c + d), (c + d*Sin[e + f*x])/(c - d)]*Cos[e + f*x]*Sqrt[(d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*Sin[e + f*x])^(1 + n))/((c - d)^2*f*(1 + n)*(a - a*Sin[e + f*x])*Sqrt[a + a*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2, x, 6, If[$VersionNumber>=8, ((d*(A*d*(3 + m) + B*(2*c + d*m)) - 2*(2 + m)*(A*c*d*(3 + m) + B*(c^2 + d^2 + c*d*m)))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m)) - (1/(f*(1 + m)*(2 + m)*(3 + m)))*(2^(1/2 + m)*(A*(3 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2)) + B*(d^2*m*(5 + 3*m + m^2) + c^2*m*(6 + 5*m + m^2) + 2*c*d*(3 + 4*m + 4*m^2 + m^3)))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m) - (d*(A*d*(3 + m) + B*(2*c + d*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)*(3 + m)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2)/(f*(3 + m)), ((d*(A*d*(3 + m) + B*(2*c + d*m)) - 2*(2 + m)*(A*c*d*(3 + m) + B*(c^2 + d^2 + c*d*m)))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)*(3 + m)) - (1/(f*(3 + m)*(2 + 3*m + m^2)))*(2^(1/2 + m)*(A*(3 + m)*(2*c*d*m*(2 + m) + d^2*(1 + m + m^2) + c^2*(2 + 3*m + m^2)) + B*(d^2*m*(5 + 3*m + m^2) + c^2*m*(6 + 5*m + m^2) + 2*c*d*(3 + 4*m + 4*m^2 + m^3)))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m) - (d*(A*d*(3 + m) + B*(2*c + d*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)*(3 + m)) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^2)/(f*(3 + m))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^1, x, 5, If[$VersionNumber>=8, ((B*d - (B*c + A*d)*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(A*(2 + m)*(c + c*m + d*m) + B*(c*m*(2 + m) + d*(1 + m + m^2)))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)), ((B*d - (B*c + A*d)*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(A*(2 + m)*(c + c*m + d*m) + B*(c*m*(2 + m) + d*(1 + m + m^2)))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))]} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^0, x, 3, -((B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*(A + A*m + B*m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^1, x, 6, -((Sqrt[2]*(B*c - A*d)*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)*d*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])) - (2^(1/2 + m)*B*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(d*f)} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^2, x, 7, (Sqrt[2]*(A*d*(c*(1 - m) - d*m) - B*(d^2 - c^2*m - c*d*m))*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c - d)^2*d*(c + d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]) + (2^(1/2 + m)*(B*c - A*d)*m*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)*f) - ((B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/((c^2 - d^2)*f*(c + d*Sin[e + f*x]))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^3, x, 8, (1/(Sqrt[2]*(c - d)^3*d*(c + d)^2*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]))*((B*(2*d^3*m + c^3*(1 - m)*m + 2*c^2*d*(1 - m)*m - c*d^2*(3 - 3*m + m^2)) - A*d*(2*c*d*(2 - m)*m - c^2*(2 - 3*m + m^2) - d^2*(1 - m + m^2)))*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m) - (2^(-(1/2) + m)*m*(A*d*(c*(3 - m) - d*m) - B*(2*d^2 + c^2*(1 - m) - c*d*m))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)^2*f) - ((B*c - A*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + ((A*d*(c*(3 - m) - d*m) - B*(2*d^2 + c^2*(1 - m) - c*d*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(2*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))} - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2), x, 9, (Sqrt[2]*(A - B)*(c - d)*AppellF1[1/2 + m, 1/2, -(3/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (Sqrt[2]*B*(c - d)*AppellF1[3/2 + m, 1/2, -(3/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/2), x, 9, (Sqrt[2]*(A - B)*AppellF1[1/2 + m, 1/2, -(1/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (Sqrt[2]*B*AppellF1[3/2 + m, 1/2, -(1/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^(1/2), x, 9, (Sqrt[2]*(A - B)*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*B*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2), x, 9, (Sqrt[2]*(A - B)*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*B*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*(c - d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x, 9, (Sqrt[2]*(A - B)*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]])) + (Sqrt[2]*B*AppellF1[3/2 + m, 1/2, -n, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^(-(m + 1)), x, 7, -((2^(1/2 + m)*a*(A - B)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a + a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/((c + d*Sin[e + f*x])^m*((c + d)*f))) + (Sqrt[2]*B*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/((c + d*Sin[e + f*x])^m*(a*(c - d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - -{(a + a*Sin[e + f*x])^m*(a - a*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x, 4, (2*Sqrt[2]*AppellF1[1/2 + m, -(1/2), -n, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(f*(1 + 2*m)))} -{(a + a*Sin[e + f*x])^m*(a - a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(-(m + 1)), x, 4, (2*Sqrt[2]*AppellF1[1/2 + m, -(1/2), 1 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Sec[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/((c + d*Sin[e + f*x])^m*((c - d)*f*(1 + 2*m)))} - - -{(a + a*Sin[e + f*x])^m*(d - m*(c - d) + (c + m*(c - d))*Sin[e + f*x])/(c + d*Sin[e + f*x])^(m + 2), x, 1, -((Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/f)} -{(a - a*Sin[e + f*x])^m*(d + m*(c + d) + (c + m*(c + d))*Sin[e + f*x])/(c + d*Sin[e + f*x])^(m + 2), x, 1, -((Cos[e + f*x]*(a - a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sin[e + f*x])^2*(A + B*Sin[e + f*x])/(c + d*Sin[e + f*x])^2, x, 6, -((b*(2*b*B*c - A*b*d - 2*a*B*d)*x)/d^3) - (2*(b*c - a*d)*(a*d^2*(A*c - B*d) - b*(2*B*c^3 - A*c^2*d - 3*B*c*d^2 + 2*A*d^3))*ArcTan[(d + c*Tan[(1/2)*(e + f*x)])/Sqrt[c^2 - d^2]])/(d^3*(c^2 - d^2)^(3/2)*f) - (b^2*B*Cos[e + f*x])/(d^2*f) - ((b*c - a*d)^2*(B*c - A*d)*Cos[e + f*x])/(d^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x]))} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2) (A+B Sin[e+f x])*) - - -{(c + d*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])/(a + b*Sin[e + f*x])^(3/2), x, 7, ((c - d)*Sqrt[c + d]*(2*A*b^2*c - 2*a*b*B*c - 2*a*A*b*d + 3*a^2*B*d - b^2*B*d)*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b^2*Sqrt[a + b]*(b*c - a*d)*f) + (Sqrt[c + d]*(3*b*B*c + 2*A*b*d - 3*a*B*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b^3*Sqrt[a + b]*f) + (2*(A*b - a*B)*(b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) - ((2*A*b*(b*c - a*d) - B*(2*a*b*c - 3*a^2*d + b^2*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(b*(a^2 - b^2)*f*Sqrt[a + b*Sin[e + f*x]]) + (Sqrt[a + b]*(2*A*b*(b*(c - 2*d) + a*d) - B*(3*a^2*d - 6*a*b*d + b^2*(2*c + d)))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b^3*Sqrt[c + d]*f)} -{(c + d*Sin[e + f*x])^(1/2)*(A + B*Sin[e + f*x])/(a + b*Sin[e + f*x])^(3/2), x, 5, (2*(A*b - a*B)*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*b*Sqrt[a + b]*(b*c - a*d)*f) + (2*Sqrt[a + b]*(A*b - a*B)*(c - d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*b*Sqrt[c + d]*(b*c - a*d)*f) + (2*Sqrt[a + b]*B*EllipticPi[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b^2*Sqrt[c + d]*f)} -{(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(1/2)), x, 3, (2*(A*b - a*B)*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)^2*f) + (2*Sqrt[a + b]*(A - B)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((a - b)*Sqrt[c + d]*(b*c - a*d)*f)} -{(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(3/2)), x, 4, (2*b*(A*b - a*B)*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (2*(A*(a^2*d^2 + b^2*(c^2 - 2*d^2)) - B*(a^2*c*d - b^2*c*d + a*b*(c^2 - d^2)))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^3*f) + (2*(A*b*c + b*B*c - a*A*d - 2*A*b*d + a*B*d)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f)} -{(A + B*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^(5/2)), x, 5, (2*b*(A*b - a*B)*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2)) + (2*d*(A*(a^2*d^2 + b^2*(3*c^2 - 4*d^2)) - B*(a^2*c*d - b^2*c*d + 3*a*b*(c^2 - d^2)))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (1/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^4*f))*(2*(B*(2*a^2*b*c*d*(3*c^2 - d^2) - 2*b^3*c*d*(3*c^2 - d^2) - a^3*d^2*(c^2 + 3*d^2) + a*b^2*(3*c^4 - 5*c^2*d^2 + 6*d^4)) + A*(4*a^3*c*d^3 - 4*a*b^2*c*d^3 - a^2*b*d^2*(9*c^2 - 5*d^2) - b^3*(3*c^4 - 15*c^2*d^2 + 8*d^4)))*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x])) - (2*(B*(a^2*d^2*(c + 3*d) - b^2*c*(3*c^2 + 3*c*d - 2*d^2) - 6*a*b*d*(c^2 - d^2)) - A*(a^2*d^2*(3*c + d) - 6*a*b*d*(c^2 - d^2) + b^2*(3*c^3 - 9*c^2*d - 6*c*d^2 + 8*d^3)))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(3*Sqrt[a + b]*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic*) - - -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x]), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^n, x]} - - -(* ::InheritFromParent:: *) -(**) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m deleted file mode 100644 index 3bc3850..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.1 (a+b sin)^m (A+B sin+C sin^2).m +++ /dev/null @@ -1,68 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (b Sin[e+f x])^m (A+C Sin[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sin[e+f x])^m (A+C Sin[e+f x]^2) when A (m+2)+C (m+1)=0*) - - -{Sin[e + f*x]^m*(1 + m - (2 + m)*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^(1 + m))/f} - -{Sin[e + f*x]^5*(6 - 7*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^6)/f} -{Sin[e + f*x]^4*(5 - 6*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^5)/f} -{Sin[e + f*x]^3*(4 - 5*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^4)/f} -{Sin[e + f*x]^2*(3 - 4*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^3)/f} -{Sin[e + f*x]^1*(2 - 3*Sin[e + f*x]^2), x, 1, (Cos[e + f*x]*Sin[e + f*x]^2)/f} -{Sin[e + f*x]^0*(1 - 2*Sin[e + f*x]^2), x, 3, (Cos[e + f*x]*Sin[e + f*x])/f} -{Csc[e + f*x]^1*(0 - 1*Sin[e + f*x]^2), x, 1, Cos[e + f*x]/f} -{Csc[e + f*x]^2*(-1 - 0*Sin[e + f*x]^2), x, 2, Cot[e + f*x]/f} -{Csc[e + f*x]^3*(-2 + 1*Sin[e + f*x]^2), x, 1, (Cot[e + f*x]*Csc[e + f*x])/f} -{Csc[e + f*x]^4*(-3 + 2*Sin[e + f*x]^2), x, 1, (Cot[e + f*x]*Csc[e + f*x]^2)/f} -{Csc[e + f*x]^5*(-4 + 3*Sin[e + f*x]^2), x, 1, (Cot[e + f*x]*Csc[e + f*x]^3)/f} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (A+C Sin[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (A+C Sin[e+f x]^2)*) - - -{(a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)), (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (A+C Sin[e+f x]^2)*) - - -{(a + b*Sin[e + f*x])^m*(A - A*Sin[e + f*x]^2), x, 7, (4*Sqrt[2]*A*AppellF1[1/2, -(3/2), -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sin[e + f*x]])) - (4*Sqrt[2]*A*AppellF1[1/2, -(1/2), -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sin[e + f*x]]))} - - -{(a + b*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2), x, 8, -((C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*a*(a + b)*C*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]])) - (Sqrt[2]*(a^2*C + b^2*(C*(1 + m) + A*(2 + m)))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]]))} - - -{Sin[e + f*x]^5*(A + C*Sin[e + f*x]^2), x, 3, -(((A + C)*Cos[e + f*x])/f) + ((2*A + 3*C)*Cos[e + f*x]^3)/(3*f) - ((A + 3*C)*Cos[e + f*x]^5)/(5*f) + (C*Cos[e + f*x]^7)/(7*f)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, ((C - B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2) + A*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m)), ((C - B*(2 + m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + m)*(2 + m)) - (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2) + A*(2 + 3*m + m^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sin[e + f*x])]*(1 + Sin[e + f*x])^(-(1/2) - m)*(a + a*Sin[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(2 + m))]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{(a + b*Sin[e + f*x])^m*(A + (A + C)*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 7, -((4*Sqrt[2]*C*AppellF1[1/2, -(3/2), -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sin[e + f*x]]))) - (2*Sqrt[2]*(A - C)*AppellF1[1/2, -(1/2), -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sin[e + f*x]]))} - - -{(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 8, -((C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*(a + b)*(a*C - b*B*(2 + m))*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]])) - (Sqrt[2]*(a^2*C + b^2*C*(1 + m) + A*b^2*(2 + m) - a*b*B*(2 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[e + f*x]), (b*(1 - Sin[e + f*x]))/(a + b)]*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(((a + b*Sin[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sin[e + f*x]]))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m deleted file mode 100644 index 90094ce..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.4.2 (a+b sin)^m (c+d sin)^n (A+B sin+C sin^2).m +++ /dev/null @@ -1,179 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2) (A+C Sin[e+f x]^2)*) - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2), x, 5, If[$VersionNumber>=8, (64*c^3*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)) + (2*c*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)*(9 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m)), (64*c^3*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(9 + 2*m)*(105 + 352*m + 344*m^2 + 128*m^3 + 16*m^4)*Sqrt[c - c*Sin[e + f*x]]) + (16*c^2*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(9 + 2*m)*(105 + 142*m + 60*m^2 + 8*m^3)) + (2*c*(C*(39 - 16*m + 4*m^2) + A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(9 + 2*m)*(35 + 24*m + 4*m^2)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(63 + 32*m + 4*m^2)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, (8*c^2*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m)), (8*c^2*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(7 + 2*m)*(15 + 46*m + 36*m^2 + 8*m^3)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(C*(19 - 8*m + 4*m^2) + A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (4*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(35 + 24*m + 4*m^2)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1/2)*(A + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, (2*c*(C - 6*C*m + A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (4*c*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m)), (2*c*(C - 6*C*m + A*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 12*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (4*c*C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m))]} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(1/2)*(A + C*Sin[e + f*x]^2), x, 4, ((A + C)*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) - (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2), x, 5, ((A + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) + ((A + 2*A*m + C*(9 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + ((A*(1 - 2*m) - C*(7 + 2*m))*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2), x, 5, ((A + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(8*a*f*(c - c*Sin[e + f*x])^(5/2)) + ((A*(5 - 2*m) - C*(11 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(16*c*f*(c - c*Sin[e + f*x])^(3/2)) + ((A*(3 - 8*m + 4*m^2) + C*(19 + 24*m + 4*m^2))*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(32*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2) (A+C Sin[e+f x]^2)*) - - -{(A + C*Sin[e + f*x]^2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 8, ((A + C)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - ((A - 3*C)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A + C)*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+C Sin[e+f x]^2) when m symbolic*) - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + C*Sin[e + f*x]^2), x, 6, (1/(f*(1 + 2*m)*(1 + m + n)*(2 + m + n)))*(2^(1/2 + n)*c*(C*(1 + 2*m)*(m - n) + (1 + m + n)*(C*(1 - m + n) + A*(2 + m + n)))*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 - 2*n), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n)) - (C*(1 + 2*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*(1 + m + n)*(2 + m + n)) + (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 + n))/(c*f*(2 + m + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -{(a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2)*(c + d*Sin[e + f*x])^n, x, 10, -((C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(2 + m + n))) + (Sqrt[2]*(c*(C + 2*C*m) + d*(C*(1 - m + n) + A*(2 + m + n)))*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(d*f*(1 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]])) + (Sqrt[2]*C*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -n, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a*d*f*(3 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]))} -{(a + a*Sin[e + f*x])^m*(A + C*Sin[e + f*x]^2)/(c + d*Sin[e + f*x])^(m + 2), x, 8, ((c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/(d*(c^2 - d^2)*f*(1 + m)) - (2^(1/2 + m)*a*(c*(A + C)*d*(1 + m) + d^2*(C - A*m + C*m) - c^2*(C + 2*C*m))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a + a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/((c + d*Sin[e + f*x])^m*((c - d)*d*(c + d)^2*f*(1 + m))) + (Sqrt[2]*C*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/((c + d*Sin[e + f*x])^m*(a*(c - d)*d*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2))/(d*f*(7 + 2*m))) + (Sqrt[2]*(c - d)*(2*c*(C + 2*C*m) + d*(C*(5 - 2*m) + A*(7 + 2*m)))*AppellF1[1/2 + m, 1/2, -(3/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(1 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*C*(c - d)*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -(3/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1/2)*(A + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2))/(d*f*(5 + 2*m))) + (Sqrt[2]*(2*c*(C + 2*C*m) + d*(C*(3 - 2*m) + A*(5 + 2*m)))*AppellF1[1/2 + m, 1/2, -(1/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(1 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*C*(d*m - c*(1 + m))*AppellF1[3/2 + m, 1/2, -(1/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(1/2)*(A + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(3 + 2*m))) + (Sqrt[2]*(2*c*(C + 2*C*m) + d*(C - 2*C*m + A*(3 + 2*m)))*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(d*f*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (2*Sqrt[2]*C*(c + c*m - d*m)*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*f*(3 + 2*m)^2*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2)*(A + C*Sin[e + f*x]^2), x, 10, (2*(c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(c*(A + C)*d - d^2*(A - C + 4*A*m) - 2*c^2*(C + 2*C*m))*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(d*(c^2 - d^2)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(2*c^2*C*(1 + m) + d^2*(A - C + 2*A*m))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*(c^2 - d^2)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2)*(A + C*Sin[e + f*x]^2), x, 10, (2*(c^2*C + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (Sqrt[2]*(3*c*(A + C)*d + d^2*(A + 3*C - 4*A*m) - 2*c^2*(C + 2*C*m))*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*(c - d)^2*d*(c + d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(2*c^2*C*(1 + m) - d^2*(A + 3*C - 2*A*m))*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*a*(c - d)^2*d*(c + d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Section:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+C Sin[e+f x]^2)*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2) (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2) (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 8, ((A + B + C)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) - ((A - B - 3*C)*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + ((A - B + C)*Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2) with m and/or n symbolic*) - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 6, (1/(f*(1 + 2*m)*(1 + m + n)*(2 + m + n)))*(2^(1/2 + n)*c*((1 + m + n)*(C*(1 - m + n) + A*(2 + m + n)) + (m - n)*(C + 2*C*m + B*(2 + m + n)))*Cos[e + f*x]*Hypergeometric2F1[(1/2)*(1 + 2*m), (1/2)*(1 - 2*n), (1/2)*(3 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 - n)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 + n)) - ((C + 2*C*m + B*(2 + m + n))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^n)/(f*(1 + m + n)*(2 + m + n)) + (C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1 + n))/(c*f*(2 + m + n))} - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 5, If[$VersionNumber>=8, -((64*c^3*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (16*c^2*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(9 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)*(9 + 2*m)) - (2*(9*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m)*(9 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m)), -((64*c^3*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(945 + 3378*m + 3800*m^2 + 1840*m^3 + 400*m^4 + 32*m^5)*Sqrt[c - c*Sin[e + f*x]])) - (16*c^2*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(945 + 1488*m + 824*m^2 + 192*m^3 + 16*m^4)) - (2*c*(B*(45 - 8*m - 4*m^2) - C*(39 - 16*m + 4*m^2) - A*(63 + 32*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(315 + 286*m + 84*m^2 + 8*m^3)) - (2*(9*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(f*(63 + 32*m + 4*m^2)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(7/2))/(c*f*(9 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, -((8*c^2*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]])) - (2*c*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) - (2*(7*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m)), -((8*c^2*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(7 + 2*m)*(15 + 46*m + 36*m^2 + 8*m^3)*Sqrt[c - c*Sin[e + f*x]])) - (2*c*(B*(21 - 8*m - 4*m^2) - C*(19 - 8*m + 4*m^2) - A*(35 + 24*m + 4*m^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c - c*Sin[e + f*x]])/(f*(105 + 142*m + 60*m^2 + 8*m^3)) - (2*(7*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(35 + 24*m + 4*m^2)) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(5/2))/(c*f*(7 + 2*m))]} -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(1/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 4, If[$VersionNumber>=8, (2*c*(C - 6*C*m + A*(5 + 2*m) - B*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(5*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*(5 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m)), (2*c*(C - 6*C*m + A*(5 + 2*m) - B*(5 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 12*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(5*B + 2*C + 2*B*m + 4*C*m)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(15 + 16*m + 4*m^2)*Sqrt[c - c*Sin[e + f*x]]) + (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(c*f*(5 + 2*m))]} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(1/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 5, -((2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])) + ((A + B + C)*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) - (2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(a*f*(3 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 5, ((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) + ((A + B + 2*A*m + 2*B*m + C*(9 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]]) + ((A*(1 - 2*m) - B*(3 + 2*m) - C*(7 + 2*m))*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(4*c*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c - c*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 5, ((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m))/(8*a*f*(c - c*Sin[e + f*x])^(5/2)) + ((A*(5 - 2*m) - B*(3 + 2*m) - C*(11 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(16*c*f*(c - c*Sin[e + f*x])^(3/2)) - ((B*(5 - 8*m - 4*m^2) - A*(3 - 8*m + 4*m^2) - C*(19 + 24*m + 4*m^2))*Cos[e + f*x]*Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sin[e + f*x])]*(a + a*Sin[e + f*x])^m)/(32*c^2*f*(1 + 2*m)*Sqrt[c - c*Sin[e + f*x]])} - - -{(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-m - 2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 6, -((2^(-(1/2) - m)*C*Cos[e + f*x]^3*Hypergeometric2F1[(1/2)*(3 + 2*m), (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (1/2)*(1 + Sin[e + f*x])]*(1 - Sin[e + f*x])^(1/2 + m)*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-2 - m))/(f*(3 + 2*m))) + ((A + B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c - c*Sin[e + f*x])^(-2 - m))/(2*a*f*(3 + 2*m)) + ((A - B + C)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(-1 - m))/(2*c*f*(1 + 2*m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, -((C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(2 + m + n))) + (Sqrt[2]*(c*(C + 2*C*m) + d*(C*(1 - m + n) + A*(2 + m + n) - B*(2 + m + n)))*AppellF1[1/2 + m, 1/2, -n, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(d*f*(1 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]])) - (Sqrt[2]*(c*C*(1 + m) - d*(C*m + B*(2 + m + n)))*AppellF1[3/2 + m, 1/2, -n, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*(c + d*Sin[e + f*x])^n)/(((c + d*Sin[e + f*x])/(c - d))^n*(a*d*f*(3 + 2*m)*(2 + m + n)*Sqrt[1 - Sin[e + f*x]]))} - - -{(a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)/(c + d*Sin[e + f*x])^(m + 2), x, 8, ((c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(-1 - m))/(d*(c^2 - d^2)*f*(1 + m)) - (1/((c - d)*d*(c + d)^2*f*(1 + m)))*((2^(1/2 + m)*a*(c*d*(A + C + A*m + B*m + C*m) - c^2*(C + 2*C*m) - d^2*(A*m + B*(1 + m) - C*(1 + m)))*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, ((c - d)*(1 - Sin[e + f*x]))/(2*(c + d*Sin[e + f*x]))]*(a + a*Sin[e + f*x])^(-1 + m)*(((c + d)*(1 + Sin[e + f*x]))/(c + d*Sin[e + f*x]))^(1/2 - m))/(c + d*Sin[e + f*x])^m) + (Sqrt[2]*C*AppellF1[3/2 + m, 1/2, 1 + m, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*((c + d*Sin[e + f*x])/(c - d))^m)/((c + d*Sin[e + f*x])^m*(a*(c - d)*d*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]))} - - -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2))/(d*f*(7 + 2*m))) + (Sqrt[2]*(c - d)*(2*c*(C + 2*C*m) - d*(7*B - 5*C + 2*B*m + 2*C*m - A*(7 + 2*m)))*AppellF1[1/2 + m, 1/2, -(3/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(1 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) - (Sqrt[2]*(c - d)*(2*c*C*(1 + m) - d*(2*C*m + B*(7 + 2*m)))*AppellF1[3/2 + m, 1/2, -(3/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(1/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(3/2))/(d*f*(5 + 2*m))) + (Sqrt[2]*(2*c*(C + 2*C*m) - d*(5*B - 3*C + 2*B*m + 2*C*m - A*(5 + 2*m)))*AppellF1[1/2 + m, 1/2, -(1/2), 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(1 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) - (Sqrt[2]*(2*c*C*(1 + m) - d*(2*C*m + B*(5 + 2*m)))*AppellF1[3/2 + m, 1/2, -(1/2), 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(5 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(1/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, -((2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(3 + 2*m))) + (Sqrt[2]*(2*c*(C + 2*C*m) - d*(3*B - C + 2*B*m + 2*C*m - A*(3 + 2*m)))*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(d*f*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[2]*(2*c*C*(1 + m) - d*(2*C*m + B*(3 + 2*m)))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*f*(3 + 2*m)^2*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, (2*(c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[2]*(d^2*(A + B - C + 4*A*m) - c*d*(A + B + C + 4*B*m) + 2*c^2*(C + 2*C*m))*AppellF1[1/2 + m, 1/2, 1/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(d*(c^2 - d^2)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[2]*(d*(B*c - A*d)*(1 + 2*m) + C*(d^2 - 2*c^2*(1 + m)))*AppellF1[3/2 + m, 1/2, 1/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(a*d*(c^2 - d^2)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} -{(a + a*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 10, (2*(c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + (Sqrt[2]*(d^2*(A - 3*B + 3*C - 4*A*m) + c*d*(3*A - B + 3*C + 4*B*m) - 2*c^2*(C + 2*C*m))*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*(c - d)^2*d*(c + d)*f*(1 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*(B*c*d*(1 - 2*m) + 2*c^2*C*(1 + m) - d^2*(A + 3*C - 2*A*m))*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + m, (1/2)*(1 + Sin[e + f*x]), -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/(3*a*(c - d)^2*d*(c + d)*f*(3 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{(A + B*Sin[c + d*x] + C*Sin[c + d*x]^2)*(a + b*Sin[c + d*x]), x, 2, (1/2)*(b*B + a*(2*A + C))*x - ((A*b + a*B + b*C)*Cos[c + d*x])/d + (b*C*Cos[c + d*x]^3)/(3*d) - ((b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d), (1/2)*(b*B + a*(2*A + C))*x - ((b^2*(3*A + 2*C) + a*(3*b*B - a*C))*Cos[c + d*x])/(3*b*d) - ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (C*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(3*b*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x])^(n/2) (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^(m/2) (a+b Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2)*) - - -{((a + b*Sin[e + f*x])*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/Sin[e + f*x]^(3/2), x, 5, (2*(b*B - a*(A - C))*EllipticE[(1/2)*(e - Pi/2 + f*x), 2])/f + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(1/2)*(e - Pi/2 + f*x), 2])/(3*f) - (2*a*A*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) - (2*b*C*Cos[e + f*x]*Sqrt[Sin[e + f*x]])/(3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n (A+B Sin[e+f x]+C Sin[e+f x]^2) with m and/or n symbolic*) - - -{(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x, 0, Unintegrable[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+a Sin[e+f x])^(n/2) (c-c Sin[e+f x])^(p/2)*) - - -(* {Sin[e + f*x]^4/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 4, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sin[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sin[e + f*x]^3)/(3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^3/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 4, Cos[e + f*x]^3/(2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Log[Cos[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 3, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(a*f*Sqrt[c - c*Sin[e + f*x]]), (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Sin[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 2, -((Cos[e + f*x]*Log[Cos[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))} -{Csc[e + f*x]^1/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 3, (Cos[e + f*x]*Log[Tan[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 3, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - Cot[e + f*x]/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^3/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 4, -((Cos[e + f*x]*Cot[e + f*x]^2)/(2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Cos[e + f*x]*Log[Tan[e + f*x]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^4/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), x, 4, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - Cot[e + f*x]/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} *) - - -(* {Sin[e + f*x]^4/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 8, -((3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) - Cos[e + f*x]^3/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + Sec[e + f*x]/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (3*Cos[e + f*x]*Sin[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]^2*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^3/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 7, -((3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (3*Cos[e + f*x]*Sin[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]^2*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 6, (Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(4*a*f*(c - c*Sin[e + f*x])^(3/2)) + (3*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*Log[1 + Sin[e + f*x]])/(4*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), -((ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + (Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + Tan[e + f*x]/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Sin[e + f*x]^1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 5, Cos[e + f*x]/(2*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]), -((ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) + Sec[e + f*x]/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + Tan[e + f*x]/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^1/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 7, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*Log[Tan[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + Tan[e + f*x]/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^2/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 8, (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*Cot[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*Log[Tan[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Csc[e + f*x]*Sec[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} -{Csc[e + f*x]^3/(Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)), x, 8, (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (3*Cot[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (Cos[e + f*x]*Cot[e + f*x]^2)/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*Cos[e + f*x]*Log[Tan[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Csc[e + f*x]*Sec[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (Sin[e + f*x]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])} *) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m deleted file mode 100644 index 9187424..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.7 (d trig)^m (a+b (c sin)^n)^p.m +++ /dev/null @@ -1,1186 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^n)^(p/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^2)^(p/2)*) - - -{(a*Sin[x]^2)^(5/2), x, 4, (-(8/15))*a^2*Cot[x]*Sqrt[a*Sin[x]^2] - (4/15)*a*Cot[x]*(a*Sin[x]^2)^(3/2) - (1/5)*Cot[x]*(a*Sin[x]^2)^(5/2)} -{(a*Sin[x]^2)^(3/2), x, 3, (-(2/3))*a*Cot[x]*Sqrt[a*Sin[x]^2] - (1/3)*Cot[x]*(a*Sin[x]^2)^(3/2)} -{(a*Sin[x]^2)^(1/2), x, 2, (-Cot[x])*Sqrt[a*Sin[x]^2]} -{1/(a*Sin[x]^2)^(1/2), x, 2, -((ArcTanh[Cos[x]]*Sin[x])/Sqrt[a*Sin[x]^2])} -{1/(a*Sin[x]^2)^(3/2), x, 3, -(Cot[x]/(2*a*Sqrt[a*Sin[x]^2])) - (ArcTanh[Cos[x]]*Sin[x])/(2*a*Sqrt[a*Sin[x]^2])} -{1/(a*Sin[x]^2)^(5/2), x, 4, -(Cot[x]/(4*a*(a*Sin[x]^2)^(3/2))) - (3*Cot[x])/(8*a^2*Sqrt[a*Sin[x]^2]) - (3*ArcTanh[Cos[x]]*Sin[x])/(8*a^2*Sqrt[a*Sin[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^3)^(p/2)*) - - -{(a*Sin[x]^3)^(5/2), x, 6, (-(26/77))*a^2*Cot[x]*Sqrt[a*Sin[x]^3] - (26*a^2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(77*Sin[x]^(3/2)) - (78/385)*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^3] - (26/165)*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^3] - (2/15)*a^2*Cos[x]*Sin[x]^5*Sqrt[a*Sin[x]^3]} -{(a*Sin[x]^3)^(3/2), x, 4, (-(14/45))*a*Cos[x]*Sqrt[a*Sin[x]^3] - (14*a*EllipticE[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(15*Sin[x]^(3/2)) - (2/9)*a*Cos[x]*Sin[x]^2*Sqrt[a*Sin[x]^3]} -{(a*Sin[x]^3)^(1/2), x, 3, (-(2/3))*Cot[x]*Sqrt[a*Sin[x]^3] - (2*EllipticF[Pi/4 - x/2, 2]*Sqrt[a*Sin[x]^3])/(3*Sin[x]^(3/2))} -{1/(a*Sin[x]^3)^(1/2), x, 3, -((2*Cos[x]*Sin[x])/Sqrt[a*Sin[x]^3]) + (2*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2))/Sqrt[a*Sin[x]^3]} -{1/(a*Sin[x]^3)^(3/2), x, 4, -((10*Cos[x])/(21*a*Sqrt[a*Sin[x]^3])) - (2*Cot[x]*Csc[x])/(7*a*Sqrt[a*Sin[x]^3]) - (10*EllipticF[Pi/4 - x/2, 2]*Sin[x]^(3/2))/(21*a*Sqrt[a*Sin[x]^3])} -{1/(a*Sin[x]^3)^(5/2), x, 6, -((154*Cot[x])/(585*a^2*Sqrt[a*Sin[x]^3])) - (22*Cot[x]*Csc[x]^2)/(117*a^2*Sqrt[a*Sin[x]^3]) - (2*Cot[x]*Csc[x]^4)/(13*a^2*Sqrt[a*Sin[x]^3]) - (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Sin[x]^3]) + (154*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2))/(195*a^2*Sqrt[a*Sin[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^4)^(p/2)*) - - -{(a*Sin[x]^4)^(5/2), x, 7, (-(63/256))*a^2*Cot[x]*Sqrt[a*Sin[x]^4] + (63/256)*a^2*x*Csc[x]^2*Sqrt[a*Sin[x]^4] - (21/128)*a^2*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^4] - (21/160)*a^2*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^4] - (9/80)*a^2*Cos[x]*Sin[x]^5*Sqrt[a*Sin[x]^4] - (1/10)*a^2*Cos[x]*Sin[x]^7*Sqrt[a*Sin[x]^4]} -{(a*Sin[x]^4)^(3/2), x, 5, (-(5/16))*a*Cot[x]*Sqrt[a*Sin[x]^4] + (5/16)*a*x*Csc[x]^2*Sqrt[a*Sin[x]^4] - (5/24)*a*Cos[x]*Sin[x]*Sqrt[a*Sin[x]^4] - (1/6)*a*Cos[x]*Sin[x]^3*Sqrt[a*Sin[x]^4]} -{(a*Sin[x]^4)^(1/2), x, 3, (-(1/2))*Cot[x]*Sqrt[a*Sin[x]^4] + (1/2)*x*Csc[x]^2*Sqrt[a*Sin[x]^4]} -{1/(a*Sin[x]^4)^(1/2), x, 3, -((Cos[x]*Sin[x])/Sqrt[a*Sin[x]^4])} -{1/(a*Sin[x]^4)^(3/2), x, 3, -((2*Cos[x]^2*Cot[x])/(3*a*Sqrt[a*Sin[x]^4])) - (Cos[x]^2*Cot[x]^3)/(5*a*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a*Sqrt[a*Sin[x]^4])} -{1/(a*Sin[x]^4)^(5/2), x, 3, -((4*Cos[x]^2*Cot[x])/(3*a^2*Sqrt[a*Sin[x]^4])) - (6*Cos[x]^2*Cot[x]^3)/(5*a^2*Sqrt[a*Sin[x]^4]) - (4*Cos[x]^2*Cot[x]^5)/(7*a^2*Sqrt[a*Sin[x]^4]) - (Cos[x]^2*Cot[x]^7)/(9*a^2*Sqrt[a*Sin[x]^4]) - (Cos[x]*Sin[x])/(a^2*Sqrt[a*Sin[x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^n)^(p/2)*) - - -{(c*Sin[a + b*x]^m)^(5/2), x, 2, (2*c^2*Cos[a + b*x]*Hypergeometric2F1[1/2, (1/4)*(2 + 5*m), (1/4)*(6 + 5*m), Sin[a + b*x]^2]*Sin[a + b*x]^(1 + 2*m)*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + 5*m)*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x]^m)^(3/2), x, 2, (2*c*Cos[a + b*x]*Hypergeometric2F1[1/2, (1/4)*(2 + 3*m), (3*(2 + m))/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 + m)*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + 3*m)*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x]^m)^(1/2), x, 2, (2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Sin[a + b*x]^2]*Sin[a + b*x]*Sqrt[c*Sin[a + b*x]^m])/(b*(2 + m)*Sqrt[Cos[a + b*x]^2])} -{1/(c*Sin[a + b*x]^m)^(1/2), x, 2, (2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 - m)/4, (6 - m)/4, Sin[a + b*x]^2]*Sin[a + b*x])/(b*(2 - m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])} -{1/(c*Sin[a + b*x]^m)^(3/2), x, 2, (2*Cos[a + b*x]*Hypergeometric2F1[1/2, (1/4)*(2 - 3*m), (3*(2 - m))/4, Sin[a + b*x]^2]*Sin[a + b*x]^(1 - m))/(b*c*(2 - 3*m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])} -{1/(c*Sin[a + b*x]^m)^(5/2), x, 2, (2*Cos[a + b*x]*Hypergeometric2F1[1/2, (1/4)*(2 - 5*m), (1/4)*(6 - 5*m), Sin[a + b*x]^2]*Sin[a + b*x]^(1 - 2*m))/(b*c^2*(2 - 5*m)*Sqrt[Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]^m])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sin[e+f x]^n)^p with p symbolic*) - - -{(b*Sin[c + d*x]^n)^p, x, 2, (Cos[c + d*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[c + d*x]^2]*Sin[c + d*x]*(b*Sin[c + d*x]^n)^p)/(d*(1 + n*p)*Sqrt[Cos[c + d*x]^2])} - - -{(c*Sin[a + b*x]^2)^p, x, 2, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1/2)*(1 + 2*p), (1/2)*(3 + 2*p), Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^2)^p)/(b*(1 + 2*p)*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x]^3)^p, x, 2, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1/2)*(1 + 3*p), (3*(1 + p))/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^3)^p)/(b*(1 + 3*p)*Sqrt[Cos[a + b*x]^2])} -{(c*Sin[a + b*x]^4)^p, x, 2, (Cos[a + b*x]*Hypergeometric2F1[1/2, (1/2)*(1 + 4*p), (1/2)*(3 + 4*p), Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x]^4)^p)/(b*(1 + 4*p)*Sqrt[Cos[a + b*x]^2])} - - -{(c*Sin[a + b*x]^n)^(1/n), x, 2, -((Cot[a + b*x]*(c*Sin[a + b*x]^n)^(1/n))/b)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Sin[e+f x])^n)^p with n symbolic*) - - -{(a*(b*Sin[c + d*x])^p)^n, x, 2, (Cos[c + d*x]*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[c + d*x]^2]*Sin[c + d*x]*(a*(b*Sin[c + d*x])^p)^n)/(d*(1 + n*p)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^2)^p when a+b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a - a*Sin[x]^2)^1, x, 3, (a*x)/2 + (1/2)*a*Cos[x]*Sin[x]} -{(a - a*Sin[x]^2)^2, x, 4, (3*a^2*x)/8 + (3/8)*a^2*Cos[x]*Sin[x] + (1/4)*a^2*Cos[x]^3*Sin[x]} -{(a - a*Sin[x]^2)^3, x, 5, (5*a^3*x)/16 + (5/16)*a^3*Cos[x]*Sin[x] + (5/24)*a^3*Cos[x]^3*Sin[x] + (1/6)*a^3*Cos[x]^5*Sin[x]} -{(a - a*Sin[x]^2)^4, x, 6, (35*a^4*x)/128 + (35/128)*a^4*Cos[x]*Sin[x] + (35/192)*a^4*Cos[x]^3*Sin[x] + (7/48)*a^4*Cos[x]^5*Sin[x] + (1/8)*a^4*Cos[x]^7*Sin[x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2), x, 4, (3*Cos[c + d*x])/(a*d) - Cos[c + d*x]^3/(a*d) + Cos[c + d*x]^5/(5*a*d) + Sec[c + d*x]/(a*d)} -{Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2), x, 4, (2*Cos[c + d*x])/(a*d) - Cos[c + d*x]^3/(3*a*d) + Sec[c + d*x]/(a*d)} -{Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2), x, 4, Cos[c + d*x]/(a*d) + Sec[c + d*x]/(a*d)} -{Sin[c + d*x]^1/(a - a*Sin[c + d*x]^2), x, 3, Sec[c + d*x]/(a*d)} -{Csc[c + d*x]^1/(a - a*Sin[c + d*x]^2), x, 4, -(ArcTanh[Cos[c + d*x]]/(a*d)) + Sec[c + d*x]/(a*d)} -{Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2), x, 5, -((3*ArcTanh[Cos[c + d*x]])/(2*a*d)) + (3*Sec[c + d*x])/(2*a*d) - (Csc[c + d*x]^2*Sec[c + d*x])/(2*a*d)} -{Csc[c + d*x]^5/(a - a*Sin[c + d*x]^2), x, 6, -((15*ArcTanh[Cos[c + d*x]])/(8*a*d)) + (15*Sec[c + d*x])/(8*a*d) - (5*Csc[c + d*x]^2*Sec[c + d*x])/(8*a*d) - (Csc[c + d*x]^4*Sec[c + d*x])/(4*a*d)} - -{Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2), x, 6, -((15*x)/(8*a)) + (15*Tan[c + d*x])/(8*a*d) - (5*Sin[c + d*x]^2*Tan[c + d*x])/(8*a*d) - (Sin[c + d*x]^4*Tan[c + d*x])/(4*a*d)} -{Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2), x, 5, -((3*x)/(2*a)) + (3*Tan[c + d*x])/(2*a*d) - (Sin[c + d*x]^2*Tan[c + d*x])/(2*a*d)} -{Sin[c + d*x]^2/(a - a*Sin[c + d*x]^2), x, 4, -(x/a) + Tan[c + d*x]/(a*d)} -{Sin[c + d*x]^0/(a - a*Sin[c + d*x]^2), x, 3, Tan[c + d*x]/(a*d)} -{Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2), x, 4, -(Cot[c + d*x]/(a*d)) + Tan[c + d*x]/(a*d)} -{Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2), x, 4, -((2*Cot[c + d*x])/(a*d)) - Cot[c + d*x]^3/(3*a*d) + Tan[c + d*x]/(a*d)} -{Csc[c + d*x]^6/(a - a*Sin[c + d*x]^2), x, 4, -((3*Cot[c + d*x])/(a*d)) - Cot[c + d*x]^3/(a*d) - Cot[c + d*x]^5/(5*a*d) + Tan[c + d*x]/(a*d)} - - -{Sin[c + d*x]^7/(a - a*Sin[c + d*x]^2)^2, x, 4, -((3*Cos[c + d*x])/(a^2*d)) + Cos[c + d*x]^3/(3*a^2*d) - (3*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)} -{Sin[c + d*x]^5/(a - a*Sin[c + d*x]^2)^2, x, 4, -(Cos[c + d*x]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)} -{Sin[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2, x, 3, -(Sec[c + d*x]/(a^2*d)) + Sec[c + d*x]^3/(3*a^2*d)} -{Sin[c + d*x]^1/(a - a*Sin[c + d*x]^2)^2, x, 3, Sec[c + d*x]^3/(3*a^2*d)} -{Csc[c + d*x]^1/(a - a*Sin[c + d*x]^2)^2, x, 5, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + Sec[c + d*x]/(a^2*d) + Sec[c + d*x]^3/(3*a^2*d)} -{Csc[c + d*x]^3/(a - a*Sin[c + d*x]^2)^2, x, 6, -((5*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) + (5*Sec[c + d*x])/(2*a^2*d) + (5*Sec[c + d*x]^3)/(6*a^2*d) - (Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*a^2*d)} - -{Sin[c + d*x]^6/(a - a*Sin[c + d*x]^2)^2, x, 6, (5*x)/(2*a^2) - (5*Tan[c + d*x])/(2*a^2*d) + (5*Tan[c + d*x]^3)/(6*a^2*d) - (Sin[c + d*x]^2*Tan[c + d*x]^3)/(2*a^2*d)} -{Sin[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2, x, 4, x/a^2 - Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)} -{Sin[c + d*x]^2/(a - a*Sin[c + d*x]^2)^2, x, 3, Tan[c + d*x]^3/(3*a^2*d)} -{Sin[c + d*x]^0/(a - a*Sin[c + d*x]^2)^2, x, 3, Tan[c + d*x]/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)} -{Csc[c + d*x]^2/(a - a*Sin[c + d*x]^2)^2, x, 4, -(Cot[c + d*x]/(a^2*d)) + (2*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)} -{Csc[c + d*x]^4/(a - a*Sin[c + d*x]^2)^2, x, 4, -((3*Cot[c + d*x])/(a^2*d)) - Cot[c + d*x]^3/(3*a^2*d) + (3*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)} - - -{1/(a - a*Sin[x]^2)^3, x, 3, Tan[x]/a^3 + (2*Tan[x]^3)/(3*a^3) + Tan[x]^5/(5*a^3)} -{1/(a - a*Sin[x]^2)^4, x, 3, Tan[x]/a^4 + Tan[x]^3/a^4 + (3*Tan[x]^5)/(5*a^4) + Tan[x]^7/(7*a^4)} -{1/(a - a*Sin[x]^2)^5, x, 3, Tan[x]/a^5 + (4*Tan[x]^3)/(3*a^5) + (6*Tan[x]^5)/(5*a^5) + (4*Tan[x]^7)/(7*a^5) + Tan[x]^9/(9*a^5)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sin[c + d*x]^3*(a + b*Sin[c + d*x]^2), x, 3, -(((a + b)*Cos[c + d*x])/d) + ((a + 2*b)*Cos[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]^5)/(5*d)} -{Sin[c + d*x]^1*(a + b*Sin[c + d*x]^2), x, 2, -(((a + b)*Cos[c + d*x])/d) + (b*Cos[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^1*(a + b*Sin[c + d*x]^2), x, 2, -((a*ArcTanh[Cos[c + d*x]])/d) - (b*Cos[c + d*x])/d} -{Csc[c + d*x]^3*(a + b*Sin[c + d*x]^2), x, 2, -(((a + 2*b)*ArcTanh[Cos[c + d*x]])/(2*d)) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)} - -{Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2), x, 4, (1/16)*(6*a + 5*b)*x - ((6*a + 5*b)*Cos[c + d*x]*Sin[c + d*x])/(16*d) - ((6*a + 5*b)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{Sin[c + d*x]^2*(a + b*Sin[c + d*x]^2), x, 3, (1/8)*(4*a + 3*b)*x - ((4*a + 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sin[c + d*x]^0*(a + b*Sin[c + d*x]^2), x, 3, a*x + (b*x)/2 - (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + b*Sin[c + d*x]^2), x, 2, b*x - (a*Cot[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Sin[c + d*x]^2), x, 3, -(((2*a + 3*b)*Cot[c + d*x])/(3*d)) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d)} -{Csc[c + d*x]^6*(a + b*Sin[c + d*x]^2), x, 3, -(((4*a + 5*b)*Cot[c + d*x])/(5*d)) - ((4*a + 5*b)*Cot[c + d*x]^3)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d)} - - -{(a + b*Sin[x]^2)^1, x, 3, a*x + (b*x)/2 - (1/2)*b*Cos[x]*Sin[x]} -{(a + b*Sin[x]^2)^2, x, 1, (1/8)*(8*a^2 + 8*a*b + 3*b^2)*x - (1/8)*b*(8*a + 3*b)*Cos[x]*Sin[x] - (1/4)*b^2*Cos[x]*Sin[x]^3} -{(a + b*Sin[x]^2)^3, x, 2, (1/16)*(2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*x - (1/48)*b*(64*a^2 + 54*a*b + 15*b^2)*Cos[x]*Sin[x] - (5/24)*b^2*(2*a + b)*Cos[x]*Sin[x]^3 - (1/6)*b*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^2} -{(a + b*Sin[x]^2)^4, x, 3, (1/128)*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*x - (1/384)*b*(608*a^3 + 808*a^2*b + 480*a*b^2 + 105*b^3)*Cos[x]*Sin[x] - (1/192)*b^2*(104*a^2 + 104*a*b + 35*b^2)*Cos[x]*Sin[x]^3 - (7/48)*b*(2*a + b)*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^2 - (1/8)*b*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^3} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2), x, 4, (a^3*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b]*d) - ((a^2 - a*b + b^2)*Cos[c + d*x])/(b^3*d) - ((a - 2*b)*Cos[c + d*x]^3)/(3*b^2*d) - Cos[c + d*x]^5/(5*b*d)} -{Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2), x, 4, -((a^2*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]*d)) + ((a - b)*Cos[c + d*x])/(b^2*d) + Cos[c + d*x]^3/(3*b*d)} -{Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2), x, 3, (a*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b]*d) - Cos[c + d*x]/(b*d)} -{Sin[c + d*x]^1/(a + b*Sin[c + d*x]^2), x, 2, -(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d))} -{Csc[c + d*x]^1/(a + b*Sin[c + d*x]^2), x, 4, -(ArcTanh[Cos[c + d*x]]/(a*d)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)} -{Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2), x, 5, -(((a - 2*b)*ArcTanh[Cos[c + d*x]])/(2*a^2*d)) - (b^(3/2)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Csc[c + d*x]^5/(a + b*Sin[c + d*x]^2), x, 6, -(((3*a^2 - 4*a*b + 8*b^2)*ArcTanh[Cos[c + d*x]])/(8*a^3*d)) + (b^(5/2)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*d) - ((3*a - 4*b)*Cot[c + d*x]*Csc[c + d*x])/(8*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} - -{Sin[c + d*x]^8/(a + b*Sin[c + d*x]^2), x, 7, -(((16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*x)/(16*b^4)) + (a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^4*Sqrt[a + b]*d) - ((8*a^2 - 6*a*b + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*b^3*d) + ((6*a - 5*b)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5)/(6*b*d)} -{Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2), x, 6, ((8*a^2 - 4*a*b + 3*b^2)*x)/(8*b^3) - (a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^3*Sqrt[a + b]*d) + ((4*a - 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)} -{Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2), x, 5, -(((2*a - b)*x)/(2*b^2)) + (a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b^2*Sqrt[a + b]*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2), x, 3, x/b - (Sqrt[a]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(b*Sqrt[a + b]*d)} -{Sin[c + d*x]^0/(a + b*Sin[c + d*x]^2), x, 2, ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]*d)} -{Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2), x, 3, -((b*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]*d)) - Cot[c + d*x]/(a*d)} -{Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2), x, 4, (b^2*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a + b]*d) - ((a - b)*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a*d)} -{Csc[c + d*x]^6/(a + b*Sin[c + d*x]^2), x, 4, -((b^3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(7/2)*Sqrt[a + b]*d)) - ((a^2 - a*b + b^2)*Cot[c + d*x])/(a^3*d) - ((2*a - b)*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a*d)} -{Csc[c + d*x]^8/(a + b*Sin[c + d*x]^2), x, 4, (b^4*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*Sqrt[a + b]*d) - ((a - b)*(a^2 + b^2)*Cot[c + d*x])/(a^4*d) - ((3*a^2 - 2*a*b + b^2)*Cot[c + d*x]^3)/(3*a^3*d) - ((3*a - b)*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a*d)} - - -{Sin[c + d*x]^7/(a + b*Sin[c + d*x]^2)^2, x, 5, -((a^2*(5*a + 6*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(7/2)*(a + b)^(3/2)*d)) + ((2*a - b)*Cos[c + d*x])/(b^3*d) + Cos[c + d*x]^3/(3*b^2*d) + (a^3*Cos[c + d*x])/(2*b^3*(a + b)*d*(a + b - b*Cos[c + d*x]^2))} -{Sin[c + d*x]^5/(a + b*Sin[c + d*x]^2)^2, x, 5, (a*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(5/2)*(a + b)^(3/2)*d) - Cos[c + d*x]/(b^2*d) - (a^2*Cos[c + d*x])/(2*b^2*(a + b)*d*(a + b - b*Cos[c + d*x]^2))} -{Sin[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2, x, 3, -(((a + 2*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)*d)) + (a*Cos[c + d*x])/(2*b*(a + b)*d*(a + b - b*Cos[c + d*x]^2))} -{Sin[c + d*x]^1/(a + b*Sin[c + d*x]^2)^2, x, 3, -(ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]]/(2*Sqrt[b]*(a + b)^(3/2)*d)) - Cos[c + d*x]/(2*(a + b)*d*(a + b - b*Cos[c + d*x]^2))} -{Csc[c + d*x]^1/(a + b*Sin[c + d*x]^2)^2, x, 5, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*d) + (b*Cos[c + d*x])/(2*a*(a + b)*d*(a + b - b*Cos[c + d*x]^2))} -{Csc[c + d*x]^3/(a + b*Sin[c + d*x]^2)^2, x, 6, -(((a - 4*b)*ArcTanh[Cos[c + d*x]])/(2*a^3*d)) - (b^(3/2)*(5*a + 4*b)*ArcTanh[(Sqrt[b]*Cos[c + d*x])/Sqrt[a + b]])/(2*a^3*(a + b)^(3/2)*d) - (b*(a + 2*b)*Cos[c + d*x])/(2*a^2*(a + b)*d*(a + b - b*Cos[c + d*x]^2)) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b - b*Cos[c + d*x]^2))} - -{Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^2, x, 6, -(((4*a - b)*x)/(2*b^3)) + (a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*b^3*(a + b)^(3/2)*d) - (a*(2*a + b)*Tan[c + d*x])/(2*b^2*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)) - (Sin[c + d*x]^2*Tan[c + d*x])/(2*b*d*(a + (a + b)*Tan[c + d*x]^2))} -{Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2, x, 5, x/b^2 - (Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*b^2*(a + b)^(3/2)*d) + (a*Tan[c + d*x])/(2*b*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))} -{Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2, x, 4, ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]]/(2*Sqrt[a]*(a + b)^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*(a + b)*d*(a + b*Sin[c + d*x]^2))} -{Sin[c + d*x]^0/(a + b*Sin[c + d*x]^2)^2, x, 4, ((2*a + b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a + b)*d*(a + b*Sin[c + d*x]^2))} -{Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^2, x, 4, -((b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^(3/2)*d)) - Cot[c + d*x]/(a*d*(a + b*Sin[c + d*x]^2)) - ((2*a*b + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a + b)*d*(a + b*Sin[c + d*x]^2)), -((b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^(3/2)*d)) - Cot[c + d*x]/(a*d*(a + (a + b)*Tan[c + d*x]^2)) - ((2*a^2 + 4*a*b + 3*b^2)*Tan[c + d*x])/(2*a^2*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))} -{Csc[c + d*x]^4/(a + b*Sin[c + d*x]^2)^2, x, 5, (b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a + b)^(3/2)*d) - ((2*a^2 - a*b - 5*b^2)*Cot[c + d*x])/(2*a^3*(a + b)*d) - ((2*a + 5*b)*Cot[c + d*x]^3)/(6*a^2*(a + b)*d) + (b*Csc[c + d*x]^3*Sec[c + d*x])/(2*a*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2))} - - -{Sin[c + d*x]^6/(a + b*Sin[c + d*x]^2)^3, x, 6, x/b^3 - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*b^3*(a + b)^(5/2)*d) + (a*Tan[c + d*x]^3)/(4*b*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) + (a*(4*a + 7*b)*Tan[c + d*x])/(8*b^2*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))} -{Sin[c + d*x]^4/(a + b*Sin[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*(a + b)^(5/2)*d) - Tan[c + d*x]^3/(4*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) - (3*Tan[c + d*x])/(8*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))} -{Sin[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3, x, 5, ((4*a + b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(4*(a + b)*d*(a + b*Sin[c + d*x]^2)^2) - ((2*a - b)*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)^2*d*(a + b*Sin[c + d*x]^2))} -{Sin[c + d*x]^0/(a + b*Sin[c + d*x]^2)^3, x, 5, ((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(5/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(4*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^2) + (3*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2))} -{Csc[c + d*x]^2/(a + b*Sin[c + d*x]^2)^3, x, 5, -((3*b*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a + b)^(5/2)*d)) - ((2*a + 3*b)*(4*a + 5*b)*Cot[c + d*x])/(8*a^3*(a + b)^2*d) + (b*Csc[c + d*x]*Sec[c + d*x]^3)/(4*a*(a + b)*d*(a + (a + b)*Tan[c + d*x]^2)^2) + (b*Cot[c + d*x]*(4*a + 5*b + (4*a + b)*Tan[c + d*x]^2))/(8*a^2*(a + b)^2*d*(a + (a + b)*Tan[c + d*x]^2))} - - -{1/(a + b*Sin[c + d*x]^2)^4, x, 6, ((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(16*a^(7/2)*(a + b)^(7/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(6*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^3) + (5*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(24*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^2) + (b*(44*a^2 + 44*a*b + 15*b^2)*Cos[c + d*x]*Sin[c + d*x])/(48*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2))} -{1/(a + b*Sin[c + d*x]^2)^5, x, 7, ((128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(128*a^(9/2)*(a + b)^(9/2)*d) + (b*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^4) + (7*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(48*a^2*(a + b)^2*d*(a + b*Sin[c + d*x]^2)^3) + (b*(104*a^2 + 104*a*b + 35*b^2)*Cos[c + d*x]*Sin[c + d*x])/(192*a^3*(a + b)^3*d*(a + b*Sin[c + d*x]^2)^2) + (5*b*(2*a + b)*(40*a^2 + 40*a*b + 21*b^2)*Cos[c + d*x]*Sin[c + d*x])/(384*a^4*(a + b)^4*d*(a + b*Sin[c + d*x]^2))} - - -{Sin[x]/Sqrt[1 + Sin[x]^2], x, 2, -ArcSin[Cos[x]/Sqrt[2]]} -{Sin[x]*Sqrt[1 + Sin[x]^2], x, 3, -ArcSin[Cos[x]/Sqrt[2]] - (Cos[x]*Sqrt[2 - Cos[x]^2])/2} -{Sin[7 + 3*x]/Sqrt[3 + Sin[7 + 3*x]^2], x, 2, -ArcSin[Cos[7 + 3*x]/2]/3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^2)^(p/2) when a+b=0*) - - -{(a - a*Sin[x]^2)^(5/2), x, 5, (8/15)*a^2*Sqrt[a*Cos[x]^2]*Tan[x] + (4/15)*a*(a*Cos[x]^2)^(3/2)*Tan[x] + (1/5)*(a*Cos[x]^2)^(5/2)*Tan[x]} -{(a - a*Sin[x]^2)^(3/2), x, 4, (2/3)*a*Sqrt[a*Cos[x]^2]*Tan[x] + (1/3)*(a*Cos[x]^2)^(3/2)*Tan[x]} -{(a - a*Sin[x]^2)^(1/2), x, 3, Sqrt[a*Cos[x]^2]*Tan[x]} -{1/(a - a*Sin[x]^2)^(1/2), x, 3, (ArcTanh[Sin[x]]*Cos[x])/Sqrt[a*Cos[x]^2]} -{1/(a - a*Sin[x]^2)^(3/2), x, 4, (ArcTanh[Sin[x]]*Cos[x])/(2*a*Sqrt[a*Cos[x]^2]) + Tan[x]/(2*a*Sqrt[a*Cos[x]^2])} -{1/(a - a*Sin[x]^2)^(5/2), x, 5, (3*ArcTanh[Sin[x]]*Cos[x])/(8*a^2*Sqrt[a*Cos[x]^2]) + Tan[x]/(4*a*(a*Cos[x]^2)^(3/2)) + (3*Tan[x])/(8*a^2*Sqrt[a*Cos[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2], x, 5, ((a - 3*b)*(a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 3*b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(8*b*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(4*b*f)} -{Sin[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*Sqrt[b]*f) - (Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]^2], x, 6, -((Sqrt[b]*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f) - (Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f} -{Csc[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((a + b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*Sqrt[a]*f) - (Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2], x, 5, -((3*a - b)*(a + b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*a^(3/2)*f) - ((3*a - b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(8*a*f) - ((a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*a*f)} - -{Sin[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2], x, 8, -(((a + 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f)) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(5*f) - ((2*a^2 - 3*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*(a - 2*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2], x, 6, -((Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f)) + ((a + 2*b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^0*Sqrt[a + b*Sin[e + f*x]^2], x, 2, (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2], x, 8, -(((2*a + b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f)) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, ((a - 5*b)*(a + b)^2*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(16*b^(3/2)*f) + ((a - 5*b)*(a + b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(16*b*f) + ((a - 5*b)*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(24*b*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(5/2))/(6*b*f)} -{Sin[e + f*x]^1*(a + b*Sin[e + f*x]^2)^(3/2), x, 5, (-3*(a + b)^2*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*Sqrt[b]*f) - (3*(a + b)*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(8*f) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(3/2))/(4*f)} -{Csc[e + f*x]^1*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -(Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*f) - (a^(3/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f - (b*Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -((b^(3/2)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/f) - (Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*f) - (a*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*f)} -{Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2), x, 5, (-3*(a + b)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(8*Sqrt[a]*f) - (3*(a + b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(8*f) - ((a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f)} -{Csc[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -((5*a - b)*(a + b)^2*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(16*a^(3/2)*f) - ((5*a - b)*(a + b)*Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(16*a*f) - ((5*a - b)*(a + b - b*Cos[e + f*x]^2)^(3/2)*Cot[e + f*x]*Csc[e + f*x]^3)/(24*a*f) - ((a + b - b*Cos[e + f*x]^2)^(5/2)*Cot[e + f*x]*Csc[e + f*x]^5)/(6*a*f)} - -{Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2), x, 9, -((a^2 + 11*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b*f) - (2*(4*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(35*f) - (b*Cos[e + f*x]*Sin[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2])/(7*f) - (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*(2*a^2 - 5*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(35*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -((3*a + 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*f) - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(5*f) + ((3*a^2 + 13*a*b + 8*b^2)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*(3*a + 4*b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^0*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -(b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -((a*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2), x, 8, (-2*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (a*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*(2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{(a + b*Sin[c + d*x]^2)^(5/2), x, 7, -((4*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]^2])/(15*d)) - (b*Cos[c + d*x]*Sin[c + d*x]*(a + b*Sin[c + d*x]^2)^(3/2))/(5*d) + ((23*a^2 + 23*a*b + 8*b^2)*EllipticE[c + d*x, -(b/a)]*Sqrt[a + b*Sin[c + d*x]^2])/(15*d*Sqrt[1 + (b*Sin[c + d*x]^2)/a]) - (4*a*(a + b)*(2*a + b)*EllipticF[c + d*x, -(b/a)]*Sqrt[1 + (b*Sin[c + d*x]^2)/a])/(15*d*Sqrt[a + b*Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, ((a - b)*ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*b^(3/2)*f) - (Cos[e + f*x]*Sqrt[a + b - b*Cos[e + f*x]^2])/(2*b*f)} -{Sin[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, -(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(Sqrt[b]*f))} -{Csc[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, -(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(Sqrt[a]*f))} -{Csc[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((a - b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*a^(3/2)*f) - (Sqrt[a + b - b*Cos[e + f*x]^2]*Cot[e + f*x]*Csc[e + f*x])/(2*a*f)} - -{Sin[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 7, -((Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f)) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 5, (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^0/Sqrt[a + b*Sin[e + f*x]^2], x, 2, (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f)) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((2*(a - b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f)) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, -(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(b^(3/2)*f)) + (a*Cos[e + f*x])/(b*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2])} -{Sin[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 2, -(Cos[e + f*x]/((a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2]))} -{Csc[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(a^(3/2)*f)) + (b*Cos[e + f*x])/(a*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2])} -{Csc[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -((a - 3*b)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]])/(2*a^(5/2)*f) - (b*(a + 3*b)*Cos[e + f*x])/(2*a^2*(a + b)*f*Sqrt[a + b - b*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*Sqrt[a + b - b*Cos[e + f*x]^2])} - -{Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, (a*Cos[e + f*x]*Sin[e + f*x]^3)/(b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((4*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*(a + b)*f) - ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(8*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2), x, 7, (a*Cos[e + f*x]*Sin[e + f*x])/(b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(b^2*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (2*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -((Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])) - (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, (b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, (b*Cot[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*(a + b)*f) - ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Sin[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2), x, 5, -(ArcTan[(Sqrt[b]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(b^(5/2)*f)) + (a*(3*a + 5*b)*Cos[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2]) + (a*Cos[e + f*x]*Sin[e + f*x]^2)/(3*b*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2))} -{Sin[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2), x, 3, (-2*Cos[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2]) - (Cos[e + f*x]*Sin[e + f*x]^2)/(3*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2))} -{Sin[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 3, -Cos[e + f*x]/(3*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2)) - (2*Cos[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2])} -{Csc[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + b - b*Cos[e + f*x]^2]]/(a^(5/2)*f)) + (b*Cos[e + f*x])/(3*a*(a + b)*f*(a + b - b*Cos[e + f*x]^2)^(3/2)) + (b*(5*a + 3*b)*Cos[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b - b*Cos[e + f*x]^2])} - -{Sin[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2), x, 8, (a*Cos[e + f*x]*Sin[e + f*x]^3)/(3*b*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*a*(2*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(8*a + 9*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2), x, 8, (a*Cos[e + f*x]*Sin[e + f*x])/(3*b*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(3*b*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, -((Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))) - ((a - b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, (b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Csc[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 9, (b*Cot[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(3*a + 2*b)*Cot[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 + 13*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f) - ((3*a^2 + 13*a*b + 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((3*a + 4*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x]) (a+b Sin[e+f x]^2)^p when p symbolic*) - - -{(d*Sin[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p, x, 3, -((d*AppellF1[1/2, (1 - m)/2, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*(d*Sin[e + f*x])^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*f))} - - -{Sin[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p, x, 5, If[$VersionNumber>=8, ((3*a - 2*b*(2 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p)) - ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*(b^2*f*(3 + 2*p)*(5 + 2*p))) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p)*Sin[e + f*x]^2)/(b*f*(5 + 2*p)), ((3*a - 2*b*(2 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b^2*f*(15 + 16*p + 4*p^2)) - ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*(b^2*f*(15 + 16*p + 4*p^2))) - (Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p)*Sin[e + f*x]^2)/(b*f*(5 + 2*p))]} -{Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x, 4, -((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p))) + ((a - 2*b*(1 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*(b*f*(3 + 2*p)))} -{Sin[e + f*x]^1*(a + b*Sin[e + f*x]^2)^p, x, 3, -((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*f))} -{Csc[e + f*x]^1*(a + b*Sin[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 1, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*f))} -{Csc[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 2, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*f))} -{Csc[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 3, -p, 3/2, Cos[e + f*x]^2, (b*Cos[e + f*x]^2)/(a + b)]*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p)/((1 - (b*Cos[e + f*x]^2)/(a + b))^p*f))} - -{Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[5/2, 1/2, -p, 7/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Sin[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*(5*f))} -{Sin[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 2 + p, -p, 5/2, -Tan[e + f*x]^2, -(((a + b)*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^p*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x]^3)/((1 + ((a + b)*Tan[e + f*x]^2)/a)^p*(3*f))} -{Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x, 3, -((AppellF1[-(1/2), 1/2, -p, 1/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*f))} -{Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x, 3, -((AppellF1[-(3/2), 1/2, -p, -(1/2), Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*(3*f)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[c + d*x]^7/(a + b*Sin[c + d*x]^3), x, 17, (3*x)/(8*b) + (2*(-1)^(2/3)*a^(5/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(7/3)*d) - (2*a^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(7/3)*d) + (2*(-1)^(1/3)*a^(5/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(7/3)*d) + (a*Cos[c + d*x])/(b^2*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d)} -{Sin[c + d*x]^5/(a + b*Sin[c + d*x]^3), x, 15, x/(2*b) - (2*a*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(5/3)*d) + (2*a*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*b^(5/3)*d) + (2*a*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(5/3)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Sin[c + d*x]^3/(a + b*Sin[c + d*x]^3), x, 13, x/b - (2*a^(1/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b*d) - (2*a^(1/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b*d) + (2*a^(1/3)*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b*d)} -{Sin[c + d*x]^1/(a + b*Sin[c + d*x]^3), x, 11, (2*(-1)^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(1/3)*d) - (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) - b^(2/3)]*b^(1/3)*d) + (2*(-1)^(1/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(1/3)*d)} -{Csc[c + d*x]^1/(a + b*Sin[c + d*x]^3), x, 14, -((2*b^(1/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a*Sqrt[a^(2/3) - b^(2/3)]*d)) - ArcTanh[Cos[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*a*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*d)} -{Csc[c + d*x]^3/(a + b*Sin[c + d*x]^3), x, 15, -((2*b*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) - b^(2/3)]*d)) - (2*b*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) + (2*b*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - ArcTanh[Cos[c + d*x]]/(2*a*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Csc[c + d*x]^5/(a + b*Sin[c + d*x]^3), x, 18, (2*(-1)^(2/3)*b^(5/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - (2*b^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*(-1)^(1/3)*b^(5/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + (b*Cot[c + d*x])/(a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)} - -{Sin[c + d*x]^6/(a + b*Sin[c + d*x]^3), x, 15, -((a*x)/b^2) + (2*a^(4/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^2*d) + (2*a^(4/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^2*d) - (2*a^(4/3)*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^2*d) - Cos[c + d*x]/(b*d) + Cos[c + d*x]^3/(3*b*d)} -{Sin[c + d*x]^4/(a + b*Sin[c + d*x]^3), x, 14, -((2*(-1)^(2/3)*a^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d)) + (2*a^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(4/3)*d) - (2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(4/3)*d) - Cos[c + d*x]/(b*d)} -{Sin[c + d*x]^2/(a + b*Sin[c + d*x]^3), x, 11, (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) - (2*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*b^(2/3)*d) - (2*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d)} -{Sin[c + d*x]^0/(a + b*Sin[c + d*x]^3), x, 11, (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)} -{Csc[c + d*x]^2/(a + b*Sin[c + d*x]^3), x, 15, -((2*(-1)^(2/3)*b^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)) + (2*b^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) - b^(2/3)]*d) - (2*(-1)^(1/3)*b^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - Cot[c + d*x]/(a*d)} -{Csc[c + d*x]^4/(a + b*Sin[c + d*x]^3), x, 16, (2*b^(4/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^2*Sqrt[a^(2/3) - b^(2/3)]*d) + (b*ArcTanh[Cos[c + d*x]])/(a^2*d) - (2*b^(4/3)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*a^2*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*d) - (2*b^(4/3)*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*a^2*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^3)^(p/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4), x, 6, -(a^(3/2)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(9/4)*d) - (a^(3/2)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(9/4)*d) + ((a + b)*Cos[c + d*x])/(b^2*d) - (2*Cos[c + d*x]^3)/(3*b*d) + Cos[c + d*x]^5/(5*b*d)} -{Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4), x, 6, -((a*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(7/4)*d)) + (a*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(7/4)*d) + Cos[c + d*x]/(b*d) - Cos[c + d*x]^3/(3*b*d)} -{Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4), x, 6, -((Sqrt[a]*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)*d)) - (Sqrt[a]*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)*d) + Cos[c + d*x]/(b*d)} -{Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4), x, 4, -(ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)*d)) + ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)*d)} -{Sin[c + d*x]^1/(a - b*Sin[c + d*x]^4), x, 4, -(ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)*d)) - ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)*d)} -{Csc[c + d*x]^1/(a - b*Sin[c + d*x]^4), x, 7, -((b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] - Sqrt[b]]*d)) - ArcTanh[Cos[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + Sqrt[b]]*d)} -{Csc[c + d*x]^3/(a - b*Sin[c + d*x]^4), x, 7, -((b^(3/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d)) - ArcTanh[Cos[c + d*x]]/(2*a*d) - (b^(3/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(4*a*d*(1 + Cos[c + d*x]))} -{Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4), x, 7, -(b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ((3*a + 8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(16*a*d*(1 - Cos[c + d*x])^2) - 3/(16*a*d*(1 - Cos[c + d*x])) + 1/(16*a*d*(1 + Cos[c + d*x])^2) + 3/(16*a*d*(1 + Cos[c + d*x]))} - -{Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4), x, 12, (5*x)/(8*b) - ((a + b)*x)/b^2 + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4), x, 9, -(x/(2*b)) + (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2)*d) - (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/2)*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4), x, 7, -(x/b) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b*d) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b*d)} -{Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4), x, 4, ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*Sqrt[b]*d) - ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]]*Sqrt[b]*d)} -{Sin[c + d*x]^0/(a - b*Sin[c + d*x]^4), x, 4, ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d)} -{Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4), x, 6, (Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*x]/(a*d)} -{Csc[c + d*x]^4/(a - b*Sin[c + d*x]^4), x, 6, (b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*x]/(a*d) - Cot[c + d*x]^3/(3*a*d)} -{Csc[c + d*x]^6/(a - b*Sin[c + d*x]^4), x, 6, (b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (b^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*Cot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d)} -{Csc[c + d*x]^8/(a - b*Sin[c + d*x]^4), x, 6, (b^2*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b^2*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(11/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*Cot[c + d*x])/(a^2*d) - ((3*a + b)*Cot[c + d*x]^3)/(3*a^2*d) - (3*Cot[c + d*x]^5)/(5*a*d) - Cot[c + d*x]^7/(7*a*d)} - - -{Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^2, x, 7, (Sqrt[a]*(5*Sqrt[a] - 6*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(9/4)*d) + (Sqrt[a]*(5*Sqrt[a] + 6*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(9/4)*d) - Cos[c + d*x]/(b^2*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*(a - b)*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^2, x, 5, ((3*Sqrt[a] - 4*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(7/4)*d) - ((3*Sqrt[a] + 4*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2, x, 5, ((Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(5/4)*d) + ((Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^2, x, 5, -(ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/4)*d)) + ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^1/(a - b*Sin[c + d*x]^4)^2, x, 5, -(((3*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(1/4)*d)) - ((3*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(1/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Csc[c + d*x]^1/(a - b*Sin[c + d*x]^4)^2, x, 11, -((b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d)) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^2*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} - -{Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^2, x, 14, x/b^2 - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*(a - b)*b*d) + Tan[c + d*x]^5/(4*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^2, x, 6, -(((2*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d)) + ((2*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*(a - b)*b*d) + (Sec[c + d*x]^2*Tan[c + d*x]^3)/(4*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^2, x, 7, ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - Tan[c + d*x]/(4*a*(a - b)*d) + Tan[c + d*x]^5/(4*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2, x, 5, ((2*Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ((2*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - (Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))/(4*a*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^0/(a - b*Sin[c + d*x]^4)^2, x, 5, ((4*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - (b*Tan[c + d*x]*(1 + 2*Tan[c + d*x]^2))/(4*a*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^2, x, 7, ((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Cot[c + d*x]/(a^2*d) - (b*Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))/(4*a^2*(a - b)*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} - - -{Sin[c + d*x]^9/(a - b*Sin[c + d*x]^4)^3, x, 6, -(((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(9/4)*d)) - ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) - (a*Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a - b)*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(9*a^2 - 11*a*b - 10*b^2 - 2*(2*a - 5*b)*b*Cos[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^7/(a - b*Sin[c + d*x]^4)^3, x, 6, (3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cos[c + d*x]^2))/(32*(a - b)^2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^3, x, 6, ((3*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(5/4)*d) + ((3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(5/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) + (Cos[c + d*x]*(a^2 - 11*a*b - 2*b^2 + 2*b*(2*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*b*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^3/(a - b*Sin[c + d*x]^4)^3, x, 6, -(((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/4)*d)) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Sin[c + d*x]^1/(a - b*Sin[c + d*x]^4)^3, x, 6, -((3*(7*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(1/4)*d)) - (3*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(1/4)*d) - (Cos[c + d*x]*(a + b - b*Cos[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (Cos[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(2*a - b)*b*Cos[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} -{Csc[c + d*x]^1/(a - b*Sin[c + d*x]^4)^3, x, 16, -(((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*d)) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))} - -{Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3, x, 9, -(((2*Sqrt[a] - 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d)) + ((2*Sqrt[a] + 5*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - ((a + 5*b)*Tan[c + d*x])/(32*a*(a - b)^2*b*d) + Tan[c + d*x]^3/(32*a*(a - b)*b*d) + Tan[c + d*x]^9/(8*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Sec[c + d*x]^2*Tan[c + d*x]^5)/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4)^3, x, 6, -(((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d)) + ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - (Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((2*a*(a^2 - a*b - 8*b^2))/(a - b)^3 + ((2*a^2 + 15*a*b + 3*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^4/(a - b*Sin[c + d*x]^4)^3, x, 6, (3*(2*Sqrt[a] - Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d) - (3*(2*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(3*a + b + 4*(a + b)*Tan[c + d*x]^2))/(8*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((9*a^2 - 24*a*b - b^2)/(a - b)^3 + ((17*a + 3*b)*Tan[c + d*x]^2)/(a - b)^2))/(32*a*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3, x, 6, ((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d) - ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 + (5*(2*a^2 + 3*a*b - b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Sin[c + d*x]^0/(a - b*Sin[c + d*x]^4)^3, x, 6, ((32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + ((32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b^2*Tan[c + d*x]*(3*a + b + 4*(a + b)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (b*Tan[c + d*x]*((17*a^2 - 40*a*b + 7*b^2)/(a - b)^3 + ((33*a - 13*b)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} -{Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3, x, 8, (3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) - (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - Cot[c + d*x]/(a^3*d) - (b^2*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*a^2*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (b*Tan[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 + ((18*a^2 + 15*a*b - 13*b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))} - - -{1/(1 - Sin[x]^4), x, 3, ArcTan[Sqrt[2]*Tan[x]]/(2*Sqrt[2]) + Tan[x]/2, x/(2*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(2*Sqrt[2]) + Tan[x]/2} - - -{1/(a + b*Sin[x]^4), x, 10, -(((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Tan[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])) + ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Tan[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3/4)*Tan[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3/4)*Tan[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])} - - -{1/(1 + Sin[x]^4), x, 10, x/(2*Sqrt[-1 + Sqrt[2]]) + ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 - (-2 + Sqrt[2])*Cos[x]*Sin[x])/(2 + Sqrt[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + Sqrt[2]]) - ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]^2 + (-2 + Sqrt[2])*Cos[x]*Sin[x])/(2 + Sqrt[1 + Sqrt[2]] + (-2 + Sqrt[2])*Cos[x]^2 + 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x])]/(4*Sqrt[-1 + Sqrt[2]]) - (1/8)*Sqrt[-1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*Tan[x] + 2*Tan[x]^2] + (1/8)*Sqrt[-1 + Sqrt[2]]*Log[1 + Sqrt[2*(-1 + Sqrt[2])]*Tan[x] + Sqrt[2]*Tan[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sin[e+f x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4], x, 5, -((Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*d)) + (2*Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (2*b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(3*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) + ((a + b)^(3/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(3*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} -{Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4], x, 8, (Sqrt[-a]*ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]])/(2*d) + (Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(4*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4], x, 5, -((Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*b*d)) + (2*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(3*Sqrt[b]*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (2*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(3*b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) + ((a + b)^(1/4)*(a - 2*b + 2*Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(6*b^(5/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} -{Sin[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4], x, 4, (Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(Sqrt[b]*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - ((a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(2*b^(3/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} -{Sin[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]^4], x, 2, -(((a + b)^(1/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(2*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]))} -{Csc[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]^4], x, 4, -(ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]]/(2*Sqrt[-a]*d)) + (b^(1/4)*(a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(2*a*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(4*a*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} -{Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4], x, 7, -(ArcTan[(Sqrt[-a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]]/(4*Sqrt[-a]*d)) - (Sqrt[b]*Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(2*a*Sqrt[a + b]*d*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])) - (Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) + (b^(1/4)*(a + b)^(3/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(2*a*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - (b^(1/4)*(a + b - Sqrt[b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(2*a*(a + b)^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - ((a + b)^(1/4)*(Sqrt[b] - Sqrt[a + b])^2*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1/2)*(1 + Sqrt[b]/Sqrt[a + b])])/(8*a*b^(1/4)*d*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])} - -{Sin[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4], x, 4, -((ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4]]*Cos[c + d*x]^2*Sqrt[a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4])/(2*Sqrt[b]*d*Sqrt[a + b*Sin[c + d*x]^4])) - (a^(1/4)*(Sqrt[a] + Sqrt[a + b])*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*b*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((Sqrt[a] + Sqrt[a + b])^2*Cos[c + d*x]^2*EllipticPi[-((Sqrt[a] - Sqrt[a + b])^2/(4*Sqrt[a]*Sqrt[a + b])), 2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*b*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} -{Sin[c + d*x]^0/Sqrt[a + b*Sin[c + d*x]^4], x, 2, (Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} -{Csc[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4], x, 5, -((Cos[c + d*x]^2*Cot[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4])) + (Sqrt[a + b]*Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((a + b + Sqrt[a]*Sqrt[a + b])*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(3/4)*(a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -{1/(a + b*Sin[x]^5), x, 17, (2*ArcTan[(b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - b^(2/5)]) + (2*ArcTan[((-1)^(2/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]) + (2*ArcTan[((-1)^(4/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]) - (2*ArcTan[((-1)^(3/5)*(b^(1/5) + (-1)^(2/5)*a^(1/5)*Tan[x/2]))/Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]) - (2*ArcTan[((-1)^(1/5)*(b^(1/5) + (-1)^(4/5)*a^(1/5)*Tan[x/2]))/Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)])} -{1/(a + b*Sin[x]^6), x, 7, ArcTan[(Sqrt[a^(1/3) + b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])} -{1/(a + b*Sin[x]^8), x, 9, -(ArcTan[(Sqrt[(-a)^(1/4) - b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)])) - ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcTan[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTan[(Sqrt[(-a)^(1/4) + b^(1/4)]*Tan[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])} - -{1/(a - b*Sin[x]^5), x, 17, -((2*ArcTan[(b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - b^(2/5)])) - (2*ArcTan[((-1)^(2/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(4/5)*b^(2/5)]) - (2*ArcTan[((-1)^(4/5)*b^(1/5) - a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(3/5)*b^(2/5)]) + (2*ArcTan[((-1)^(1/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) - (-1)^(2/5)*b^(2/5)]) + (2*ArcTan[((-1)^(3/5)*b^(1/5) + a^(1/5)*Tan[x/2])/Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + (-1)^(1/5)*b^(2/5)])} -{1/(a - b*Sin[x]^6), x, 7, ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTan[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tan[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])} -{1/(a - b*Sin[x]^8), x, 9, ArcTan[(Sqrt[a^(1/4) - b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) - I*b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + I*b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[a^(1/4) + b^(1/4)]*Tan[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + b^(1/4)])} - - -{1/(1 + Sin[x]^5), x, 15, (2*ArcTan[((-1)^(2/5) + Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTan[((-1)^(4/5) + Tan[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTan[((-1)^(3/5)*(1 + (-1)^(2/5)*Tan[x/2]))/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]) - (2*ArcTan[((-1)^(1/5)*(1 + (-1)^(4/5)*Tan[x/2]))/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]) - Cos[x]/(5*(1 + Sin[x]))} -{1/(1 + Sin[x]^6), x, 7, x/(3*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(3*Sqrt[2]) + ArcTan[Sqrt[1 - (-1)^(1/3)]*Tan[x]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan[Sqrt[1 + (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(2/3)])} -{1/(1 + Sin[x]^8), x, 9, (1/8)*(Sqrt[1 + Sqrt[4 - 2*Sqrt[2]]] + Sqrt[2 + 2*2^(1/4) + 2*Sqrt[1 + Sqrt[2]] + 2*Sqrt[2 + Sqrt[2]]] + Sqrt[1 + Sqrt[4 + 2*Sqrt[2]]])*(x - ArcTan[Tan[x]]) + ArcTan[Sqrt[1 - (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTan[Sqrt[1 + (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(1/4)]) + ArcTan[Sqrt[1 - (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTan[Sqrt[1 + (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(3/4)]), ArcTan[Sqrt[1 - (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTan[Sqrt[1 + (-1)^(1/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(1/4)]) + ArcTan[Sqrt[1 - (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTan[Sqrt[1 + (-1)^(3/4)]*Tan[x]]/(4*Sqrt[1 + (-1)^(3/4)])} - -{1/(1 - Sin[x]^5), x, 15, -((2*ArcTan[((-1)^(2/5) - Tan[x/2])/Sqrt[1 - (-1)^(4/5)]])/(5*Sqrt[1 - (-1)^(4/5)])) - (2*ArcTan[((-1)^(4/5) - Tan[x/2])/Sqrt[1 + (-1)^(3/5)]])/(5*Sqrt[1 + (-1)^(3/5)]) + (2*ArcTan[((-1)^(1/5) + Tan[x/2])/Sqrt[1 - (-1)^(2/5)]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTan[((-1)^(3/5) + Tan[x/2])/Sqrt[1 + (-1)^(1/5)]])/(5*Sqrt[1 + (-1)^(1/5)]) + Cos[x]/(5*(1 - Sin[x]))} -{1/(1 - Sin[x]^6), x, 8, ArcTan[Sqrt[1 + (-1)^(1/3)]*Tan[x]]/(3*Sqrt[1 + (-1)^(1/3)]) + ArcTan[Sqrt[1 - (-1)^(2/3)]*Tan[x]]/(3*Sqrt[1 - (-1)^(2/3)]) + Tan[x]/3} -{1/(1 - Sin[x]^8), x, 10, x/(4*Sqrt[2]) + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/(4*Sqrt[2]) + ArcTan[Sqrt[1 - I]*Tan[x]]/(4*Sqrt[1 - I]) + ArcTan[Sqrt[1 + I]*Tan[x]]/(4*Sqrt[1 + I]) + Tan[x]/4} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^2)^p when a+b=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]^9/(a - a*Sin[x]^2), x, 3, Sin[x]/a - Sin[x]^3/a + (3*Sin[x]^5)/(5*a) - Sin[x]^7/(7*a)} -{Cos[x]^7/(a - a*Sin[x]^2), x, 3, Sin[x]/a - (2*Sin[x]^3)/(3*a) + Sin[x]^5/(5*a)} -{Cos[x]^5/(a - a*Sin[x]^2), x, 3, Sin[x]/a - Sin[x]^3/(3*a)} -{Cos[x]^3/(a - a*Sin[x]^2), x, 2, Sin[x]/a} -{Cos[x]^1/(a - a*Sin[x]^2), x, 2, ArcTanh[Sin[x]]/a} -{Sec[x]^3/(a - a*Sin[x]^2), x, 4, (3*ArcTanh[Sin[x]])/(8*a) + (3*Sec[x]*Tan[x])/(8*a) + (Sec[x]^3*Tan[x])/(4*a)} - -{Cos[x]^6/(a - a*Sin[x]^2), x, 4, (3*x)/(8*a) + (3*Cos[x]*Sin[x])/(8*a) + (Cos[x]^3*Sin[x])/(4*a)} -{Cos[x]^4/(a - a*Sin[x]^2), x, 3, x/(2*a) + (Cos[x]*Sin[x])/(2*a)} -{Cos[x]^2/(a - a*Sin[x]^2), x, 2, x/a} -{Sec[x]^1/(a - a*Sin[x]^2), x, 3, ArcTanh[Sin[x]]/(2*a) + (Sec[x]*Tan[x])/(2*a)} -{Sec[x]^2/(a - a*Sin[x]^2), x, 3, Tan[x]/a + Tan[x]^3/(3*a)} -{Sec[x]^4/(a - a*Sin[x]^2), x, 3, Tan[x]/a + (2*Tan[x]^3)/(3*a) + Tan[x]^5/(5*a)} - - -{Cos[x]^9/(a - a*Sin[x]^2)^2, x, 3, Sin[x]/a^2 - (2*Sin[x]^3)/(3*a^2) + Sin[x]^5/(5*a^2)} -{Cos[x]^7/(a - a*Sin[x]^2)^2, x, 3, Sin[x]/a^2 - Sin[x]^3/(3*a^2)} -{Cos[x]^5/(a - a*Sin[x]^2)^2, x, 2, Sin[x]/a^2} -{Cos[x]^3/(a - a*Sin[x]^2)^2, x, 2, ArcTanh[Sin[x]]/a^2} -{Cos[x]^1/(a - a*Sin[x]^2)^2, x, 3, ArcTanh[Sin[x]]/(2*a^2) + (Sec[x]*Tan[x])/(2*a^2)} -{Sec[x]^1/(a - a*Sin[x]^2)^2, x, 4, (3*ArcTanh[Sin[x]])/(8*a^2) + (3*Sec[x]*Tan[x])/(8*a^2) + (Sec[x]^3*Tan[x])/(4*a^2)} - -{Cos[x]^8/(a - a*Sin[x]^2)^2, x, 4, (3*x)/(8*a^2) + (3*Cos[x]*Sin[x])/(8*a^2) + (Cos[x]^3*Sin[x])/(4*a^2)} -{Cos[x]^6/(a - a*Sin[x]^2)^2, x, 3, x/(2*a^2) + (Cos[x]*Sin[x])/(2*a^2)} -{Cos[x]^4/(a - a*Sin[x]^2)^2, x, 2, x/a^2} -{Cos[x]^2/(a - a*Sin[x]^2)^2, x, 3, Tan[x]/a^2} -{Sec[x]^2/(a - a*Sin[x]^2)^2, x, 3, Tan[x]/a^2 + (2*Tan[x]^3)/(3*a^2) + Tan[x]^5/(5*a^2)} -{Sec[x]^4/(a - a*Sin[x]^2)^2, x, 3, Tan[x]/a^2 + Tan[x]^3/a^2 + (3*Tan[x]^5)/(5*a^2) + Tan[x]^7/(7*a^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cos[e + f*x]^6*(a + b*Sin[e + f*x]^2), x, 6, (5/128)*(8*a + b)*x + (5*(8*a + b)*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (5*(8*a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + ((8*a + b)*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (b*Cos[e + f*x]^7*Sin[e + f*x])/(8*f)} -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2), x, 5, (1/16)*(6*a + b)*x + ((6*a + b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((6*a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (b*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2), x, 4, (1/8)*(4*a + b)*x + ((4*a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^2), x, 3, a*x + (b*x)/2 - (b*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2), x, 3, (-b)*x + ((a + b)*Tan[e + f*x])/f} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2), x, 2, (a*Tan[e + f*x])/f + ((a + b)*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2), x, 3, (a*Tan[e + f*x])/f + ((2*a + b)*Tan[e + f*x]^3)/(3*f) + ((a + b)*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2), x, 3, (a*Tan[e + f*x])/f + ((3*a + b)*Tan[e + f*x]^3)/(3*f) + ((3*a + 2*b)*Tan[e + f*x]^5)/(5*f) + ((a + b)*Tan[e + f*x]^7)/(7*f)} - - -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2, x, 6, (1/128)*(48*a^2 + 16*a*b + 3*b^2)*x + ((48*a^2 + 16*a*b + 3*b^2)*Cos[e + f*x]*Sin[e + f*x])/(128*f) + ((48*a^2 + 16*a*b + 3*b^2)*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) - (b*(10*a + 3*b)*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (b*Cos[e + f*x]^7*Sin[e + f*x]*(a + (a + b)*Tan[e + f*x]^2))/(8*f)} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2, x, 5, (1/16)*(8*a^2 + 4*a*b + b^2)*x + ((8*a^2 + 4*a*b + b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) - (b*(8*a + 3*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (b*Cos[e + f*x]^5*Sin[e + f*x]*(a + (a + b)*Tan[e + f*x]^2))/(6*f)} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^2)^2, x, 1, (1/8)*(8*a^2 + 8*a*b + 3*b^2)*x - (b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - (b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f)} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^2, x, 5, (-(1/2))*b*(4*a + 3*b)*x + (b^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + ((a + b)^2*Tan[e + f*x])/f} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^2, x, 4, b^2*x + ((a^2 - b^2)*Tan[e + f*x])/f + ((a + b)^2*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]^6*(a + b*Sin[e + f*x]^2)^2, x, 3, (a^2*Tan[e + f*x])/f + (2*a*(a + b)*Tan[e + f*x]^3)/(3*f) + ((a + b)^2*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]^8*(a + b*Sin[e + f*x]^2)^2, x, 3, (a^2*Tan[e + f*x])/f + (a*(3*a + 2*b)*Tan[e + f*x]^3)/(3*f) + ((a + b)*(3*a + b)*Tan[e + f*x]^5)/(5*f) + ((a + b)^2*Tan[e + f*x]^7)/(7*f)} -{Sec[e + f*x]^10*(a + b*Sin[e + f*x]^2)^2, x, 3, (a^2*Tan[e + f*x])/f + (2*a*(2*a + b)*Tan[e + f*x]^3)/(3*f) + ((6*a^2 + 6*a*b + b^2)*Tan[e + f*x]^5)/(5*f) + (2*(a + b)*(2*a + b)*Tan[e + f*x]^7)/(7*f) + ((a + b)^2*Tan[e + f*x]^9)/(9*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]^7/(a + b*Sin[x]^2), x, 4, ((a + b)^3*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)) - ((a^2 + 3*a*b + 3*b^2)*Sin[x])/b^3 + ((a + 3*b)*Sin[x]^3)/(3*b^2) - Sin[x]^5/(5*b)} -{Cos[x]^6/(a + b*Sin[x]^2), x, 6, -(((8*a^2 + 20*a*b + 15*b^2)*x)/(8*b^3)) + ((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b^3) - ((4*a + 7*b)*Cos[x]*Sin[x])/(8*b^2) - (Cos[x]^3*Sin[x])/(4*b)} -{Cos[x]^5/(a + b*Sin[x]^2), x, 4, ((a + b)^2*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - ((a + 2*b)*Sin[x])/b^2 + Sin[x]^3/(3*b)} -{Cos[x]^4/(a + b*Sin[x]^2), x, 5, -(((2*a + 3*b)*x)/(2*b^2)) + ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b^2) - (Cos[x]*Sin[x])/(2*b)} -{Cos[x]^3/(a + b*Sin[x]^2), x, 3, ((a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) - Sin[x]/b} -{Cos[x]^2/(a + b*Sin[x]^2), x, 4, -(x/b) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*b)} -{Cos[x]^1/(a + b*Sin[x]^2), x, 2, ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{Sec[x]^1/(a + b*Sin[x]^2), x, 4, (Sqrt[b]*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)) + ArcTanh[Sin[x]]/(a + b)} -{Sec[x]^2/(a + b*Sin[x]^2), x, 3, (b*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2)) + Tan[x]/(a + b)} -{Sec[x]^3/(a + b*Sin[x]^2), x, 5, (b^(3/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^2) + ((a + 3*b)*ArcTanh[Sin[x]])/(2*(a + b)^2) + (Sec[x]*Tan[x])/(2*(a + b))} -{Sec[x]^4/(a + b*Sin[x]^2), x, 4, (b^2*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)) + ((a + 2*b)*Tan[x])/(a + b)^2 + Tan[x]^3/(3*(a + b))} -{Sec[x]^5/(a + b*Sin[x]^2), x, 6, (b^(5/2)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3) + ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[Sin[x]])/(8*(a + b)^3) + ((3*a + 7*b)*Sec[x]*Tan[x])/(8*(a + b)^2) + (Sec[x]^3*Tan[x])/(4*(a + b))} -{Sec[x]^6/(a + b*Sin[x]^2), x, 4, (b^3*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2)) + ((a^2 + 3*a*b + 3*b^2)*Tan[x])/(a + b)^3 + ((2*a + 3*b)*Tan[x]^3)/(3*(a + b)^2) + Tan[x]^5/(5*(a + b))} - - -{Cos[x]^6/(a + b*Sin[x]^2)^2, x, 6, ((4*a + 5*b)*x)/(2*b^3) - ((4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*b^3) - (Cos[x]*Sin[x])/(2*b*(a + (a + b)*Tan[x]^2)) + ((a + b)*(2*a + b)*Tan[x])/(2*a*b^2*(a + (a + b)*Tan[x]^2))} -{Cos[x]^5/(a + b*Sin[x]^2)^2, x, 5, -(((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2))) + Sin[x]/b^2 + ((a + b)^2*Sin[x])/(2*a*b^2*(a + b*Sin[x]^2))} -{Cos[x]^4/(a + b*Sin[x]^2)^2, x, 5, x/b^2 - ((2*a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*b^2) + ((a + b)*Tan[x])/(2*a*b*(a + (a + b)*Tan[x]^2))} -{Cos[x]^3/(a + b*Sin[x]^2)^2, x, 3, -(((a - b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2))) + ((a + b)*Sin[x])/(2*a*b*(a + b*Sin[x]^2))} -{Cos[x]^2/(a + b*Sin[x]^2)^2, x, 3, ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a + b]) + Tan[x]/(2*a*(a + (a + b)*Tan[x]^2))} -{Cos[x]^1/(a + b*Sin[x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sin[x]/(2*a*(a + b*Sin[x]^2))} -{Sec[x]^1/(a + b*Sin[x]^2)^2, x, 5, (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^2) + ArcTanh[Sin[x]]/(a + b)^2 + (b*Sin[x])/(2*a*(a + b)*(a + b*Sin[x]^2))} -{Sec[x]^2/(a + b*Sin[x]^2)^2, x, 5, (b*(4*a + b)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)) + Tan[x]/(a + b)^2 + (b^2*Tan[x])/(2*a*(a + b)^2*(a + (a + b)*Tan[x]^2))} -{Sec[x]^3/(a + b*Sin[x]^2)^2, x, 6, (b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^3) + ((a + 5*b)*ArcTanh[Sin[x]])/(2*(a + b)^3) - ((a - b)*b*Sin[x])/(2*a*(a + b)^2*(a + b*Sin[x]^2)) + (Sec[x]*Tan[x])/(2*(a + b)*(a + b*Sin[x]^2))} -{Sec[x]^4/(a + b*Sin[x]^2)^2, x, 5, (b^2*(6*a + b)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(7/2)) + ((a + 3*b)*Tan[x])/(a + b)^3 + Tan[x]^3/(3*(a + b)^2) + (b^3*Tan[x])/(2*a*(a + b)^3*(a + (a + b)*Tan[x]^2))} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^2)^(p/2) when a+b=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cos[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2], x, 5, (a*(a + 4*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*b^(3/2)*f) + ((a + 4*b)*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(8*b*f) - (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(4*b*f)} -{Cos[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]^2], x, 4, (a*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*Sqrt[b]*f) + (Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*f)} -{Sec[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]^2], x, 6, -((Sqrt[b]*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f) + (Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f} -{Sec[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2], x, 4, (a*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*Sqrt[a + b]*f) + (Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*f)} -{Sec[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2], x, 5, (a*(3*a + 4*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*(a + b)^(3/2)*f) + ((3*a + 4*b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(8*(a + b)*f) + (Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(4*(a + b)*f)} - -{Cos[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2], x, 8, (2*(a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f) - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(5*b*f) - ((2*a^2 + 7*a*b - 3*b^2)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*(a + b)*(a + 3*b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]), (2*(a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f) - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(5*b*f) - ((2*a^2 + 7*a*b - 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (2*a*(a + b)*(a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2], x, 7, (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*f*Sqrt[a + b*Sin[e + f*x]^2]), (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^0*Sqrt[a + b*Sin[e + f*x]^2], x, 2, (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f, -((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f} -{Sec[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2], x, 8, -(((2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (2*a*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f), -(((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (2*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f)} - - -{Cos[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, (a^2*(a + 6*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(16*b^(3/2)*f) + (a*(a + 6*b)*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(16*b*f) + ((a + 6*b)*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(24*b*f) - (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(5/2))/(6*b*f)} -{Cos[e + f*x]^1*(a + b*Sin[e + f*x]^2)^(3/2), x, 5, (3*a^2*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*a*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(8*f) + (Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/(4*f)} -{Sec[e + f*x]^1*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -((Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*f)) + ((a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f - (b*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*f)} -{Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, (b^(3/2)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/f + ((a - 2*b)*Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*f) + ((a + b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*f)} -{Sec[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2), x, 5, (3*a^2*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(8*Sqrt[a + b]*f) + (3*a*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(8*f) + (Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]^7*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, (a^2*(5*a + 6*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(16*(a + b)^(3/2)*f) + (a*(5*a + 6*b)*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(16*(a + b)*f) + ((5*a + 6*b)*Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/(24*(a + b)*f) + (Sec[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(5/2)*Tan[e + f*x])/(6*(a + b)*f)} - -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2), x, 9, -(((a^2 - 9*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b*f)) + (2*(4*a + b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*f) - (b*Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(7*f) - (2*(a - b)*(a^2 + 6*a*b + b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(35*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(a + b)*(2*a^2 + 9*a*b - b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(35*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2), x, 8, (2*(3*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*f) - (b*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(5*f) - ((3*a^2 - 7*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(15*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (a*(3*a - b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(15*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -((b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f)) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -(((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2), x, 8, -((2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*(2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(a - b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f) + ((a + b)*Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, ((a + 2*b)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*b^(3/2)*f) - (Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(2*b*f)} -{Cos[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(Sqrt[b]*f)} -{Sec[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(Sqrt[a + b]*f)} -{Sec[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, ((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(2*(a + b)*f)} - -{Cos[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 7, -((Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f)) - (2*(a + 2*b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*(2*a + 3*b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]), -((Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b*f)) - (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*(2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 6, -((a*EllipticE[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])) + ((a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2]), -((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^0/Sqrt[a + b*Sin[e + f*x]^2], x, 2, (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/((a + b)*f), -((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/((a + b)*f)} -{Sec[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 8, -((2*(a + 2*b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + ((2*a + 3*b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(a + 2*b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)^2*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f), -((2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + ((2*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(a + 2*b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)^2*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f)} - - -{Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(3/2)*f)) + ((a + b)*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 2, Sin[e + f*x]/(a*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/((a + b)^(3/2)*f) + (b*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 6, ((a + 4*b)*ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]])/(2*(a + b)^(5/2)*f) - ((a - 2*b)*b*Sin[e + f*x])/(2*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sec[e + f*x]*Tan[e + f*x])/(2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} - -{Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, ((a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((4*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b^2*f) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*b^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((a + b)*(8*a + 9*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2), x, 7, ((a + b)*Cos[e + f*x]*Sin[e + f*x])/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 7, (Cos[e + f*x]*Sin[e + f*x])/(a*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, (b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, -(((a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])) - ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Cos[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2), x, 5, ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/(b^(5/2)*f) + ((a + b)*Cos[e + f*x]^2*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*(a + b)*Sin[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2), x, 3, (Cos[e + f*x]^2*Sin[e + f*x])/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*Sin[e + f*x])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 3, Sin[e + f*x]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*Sin[e + f*x])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a + b]*Sin[e + f*x])/Sqrt[a + b*Sin[e + f*x]^2]]/((a + b)^(5/2)*f) + (b*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (b*(5*a + 2*b)*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])} - -{Cos[e + f*x]^6/(a + b*Sin[e + f*x]^2)^(5/2), x, 8, ((a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(2*a - b)*(a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((8*a^2 + 3*a*b - 2*b^2)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((8*a - b)*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^3*f*Sqrt[a + b*Sin[e + f*x]^2]), ((a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(2*a - b)*(a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((8*a - b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2), x, 8, ((a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a - b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((2*a - b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^2*f*Sqrt[a + b*Sin[e + f*x]^2]), ((a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*(a - b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((2*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 8, (Cos[e + f*x]*Sin[e + f*x])/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + 2*b)*EllipticE[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b*f*Sqrt[a + b*Sin[e + f*x]^2]), (Cos[e + f*x]*Sin[e + f*x])/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*b*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cos[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, (b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Sec[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 9, -(((3*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2))) - (b*(3*a^2 - 7*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 - 7*a*b - 2*b^2)*EllipticE[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((3*a - b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)), -(((3*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2))) - (b*(3*a^2 - 7*a*b - 2*b^2)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a^2 - 7*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((3*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^2)^p when p symbolic*) - - -{(d*Cos[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p, x, 3, (d*AppellF1[1/2, (1 - m)/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(d*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} - - -{Cos[e + f*x]^5*(a + b*Sin[e + f*x]^2)^p, x, 5, If[$VersionNumber>=8, -(((3*a + b*(7 + 2*p))*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) - (Cos[e + f*x]^2*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 + 2*a*b*(5 + 2*p) + b^2*(15 + 16*p + 4*p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*(b^2*f*(3 + 2*p)*(5 + 2*p))), -(((3*a + b*(7 + 2*p))*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b^2*f*(15 + 16*p + 4*p^2))) - (Cos[e + f*x]^2*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 + 2*a*b*(5 + 2*p) + b^2*(15 + 16*p + 4*p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*(b^2*f*(15 + 16*p + 4*p^2)))]} -{Cos[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x, 4, -((Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p))) + ((a + b*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*(b*f*(3 + 2*p)))} -{Cos[e + f*x]^1*(a + b*Sin[e + f*x]^2)^p, x, 3, (Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Sec[e + f*x]^1*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Sec[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} - -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, -(3/2), -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, -(1/2), -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 3/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 5/2, -p, 3/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos[e + f*x]^2]*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/((1 + (b*Sin[e + f*x]^2)/a)^p*f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3), x, 11, ((a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(5/3)*d) + ((a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*b^(5/3)*d) - ((a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*b^(5/3)*d) - (2*Log[a + b*Sin[c + d*x]^3])/(3*b*d) + Sin[c + d*x]^2/(2*b*d)} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3), x, 9, -(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(6*a^(2/3)*b^(1/3)*d) - Log[a + b*Sin[c + d*x]^3]/(3*b*d)} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x]^3), x, 7, -(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(6*a^(2/3)*b^(1/3)*d)} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x]^3), x, 11, -((b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2)*d)) - Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*(a^2 - b^2)*d) + (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)*d) - (b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)*d)} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3), x, 11, -((b^(5/3)*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2)^2*d)) - ((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)^2*d) + (b*(a^2 + 2*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))} - -{Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3), x, 38, -((2*(-1)^(2/3)*a^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d)) + (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*a^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(4/3)*d) - (4*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(4/3)*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (4*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*b^(2/3)*d) + (4*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d) - Cos[c + d*x]/(b*d)} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3), x, 24, (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) - (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*b^(2/3)*d) + (2*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d)} -{Cos[c + d*x]^0/(a + b*Sin[c + d*x]^3), x, 11, (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3), x, -1, (2*(-1)^(2/3)*b^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(3/2)*d) - (2*b^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*(a^(2/3) - b^(2/3))^(3/2)*d) + (2*(-1)^(1/3)*b^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(3/2)*d) + (Sec[c + d*x]*(b - a*Sin[c + d*x]))/((-a^2 + b^2)*d)} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3), x, -1, -((2*(-1)^(2/3)*a^(2/3)*b^(8/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*(a^2 - b^2)^2*d)) - (2*b^2*(2*a^2 + b^2)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*(a^2 - b^2)^2*d) + (2*a^(2/3)*b^(8/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(Sqrt[a^(2/3) - b^(2/3)]*(a^2 - b^2)^2*d) + (2*b^2*(2*a^2 + b^2)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*(a^2 - b^2)^2*d) + (2*b^(4/3)*(a^2 + 2*b^2)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*(a^2 - b^2)^2*d) - (2*(-1)^(1/3)*a^(2/3)*b^(8/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*(a^2 - b^2)^2*d) + (2*b^2*(2*a^2 + b^2)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*(a^2 - b^2)^2*d) - (2*b^(4/3)*(a^2 + 2*b^2)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) + b^(2/3)]*(a^2 - b^2)^2*d) - (2*b^(4/3)*(a^2 + 2*b^2)*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(1/2)*(c + d*x)])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*(a^2 - b^2)^2*d) + Cos[c + d*x]/(12*(a + b)*d*(1 - Sin[c + d*x])^2) + Cos[c + d*x]/(12*(a + b)*d*(1 - Sin[c + d*x])) + ((a + 4*b)*Cos[c + d*x])/(4*(a + b)^2*d*(1 - Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)*d*(1 + Sin[c + d*x])^2) - ((a - 4*b)*Cos[c + d*x])/(4*(a - b)^2*d*(1 + Sin[c + d*x])) - Cos[c + d*x]/(12*(a - b)*d*(1 + Sin[c + d*x]))} - - -{Cos[c + d*x]^7/(a + b*Sin[c + d*x]^3)^2, x, 10, -((2*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(7/3)*d)) + (2*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(7/3)*d) - ((2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(9*a^(5/3)*b^(7/3)*d) - Sin[c + d*x]/(b^2*d) - (Sin[c + d*x]*(a^2 - b^2 + 3*a*b*Sin[c + d*x] + 3*b^2*Sin[c + d*x]^2))/(3*a*b^2*d*(a + b*Sin[c + d*x]^3))} -{Cos[c + d*x]^5/(a + b*Sin[c + d*x]^3)^2, x, 8, -((2*(a^(4/3) + b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(5/3)*d)) - (2*(a^(4/3) - b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(5/3)*d) + ((a^(4/3) - b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(9*a^(5/3)*b^(5/3)*d) + (Sin[c + d*x]*(b - a*Sin[c + d*x] - 2*b*Sin[c + d*x]^2))/(3*a*b*d*(a + b*Sin[c + d*x]^3))} -{Cos[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2, x, 9, -((2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(1/3)*d)) + (2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(9*a^(5/3)*b^(1/3)*d) + (a + b*Sin[c + d*x])/(3*a*b*d*(a + b*Sin[c + d*x]^3))} -{Cos[c + d*x]^1/(a + b*Sin[c + d*x]^3)^2, x, 8, -((2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(1/3)*d)) + (2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]/(9*a^(5/3)*b^(1/3)*d) + Sin[c + d*x]/(3*a*d*(a + b*Sin[c + d*x]^3))} -{Sec[c + d*x]^1/(a + b*Sin[c + d*x]^3)^2, x, 18, -((b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 - b^2)*d)) - (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2*d) - Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(1/3)*(a^2 - b^2)^2*d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(18*a^(5/3)*(a^2 - b^2)*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2*d) - (2*a*b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2*d) + (b*(a - Sin[c + d*x]*(b - a*Sin[c + d*x])))/(3*a*(a^2 - b^2)*d*(a + b*Sin[c + d*x]^3))} -{Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2, x, 18, -((b^(5/3)*(4*a^2 - 3*a^(4/3)*b^(2/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 - b^2)^2*d)) - (b^(5/3)*(4*a^(8/3) - 9*a^2*b^(2/3) + 8*a^(2/3)*b^2 - 3*b^(8/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^3*d) - ((a + 7*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a - 7*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (b^(5/3)*(4*a^2 + 3*a^(4/3)*b^(2/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)^2*d) + (b^(5/3)*(3*b^(2/3)*(3*a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(1/3)*(a^2 - b^2)^3*d) - (b^(5/3)*(4*a^2 + 3*a^(4/3)*b^(2/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(18*a^(5/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(3*b^(2/3)*(3*a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^3*d) + (2*a*b*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)^2*d*(1 + Sin[c + d*x])) - (b*(a*(a^2 + 2*b^2) - b*Sin[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x])))/(3*a*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]^3))} - -{Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]} -{Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x]} -{Cos[c + d*x]^0/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[1/(a + b*Sin[c + d*x]^3)^2, x]} -{Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2, x]} -{Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x, 0, Unintegrable[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4), x, 6, ((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - ((Sqrt[a] - Sqrt[b])^3*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - (3*Sin[c + d*x])/(b*d) + Sin[c + d*x]^3/(3*b*d)} -{Cos[c + d*x]^5/(a - b*Sin[c + d*x]^4), x, 6, ((Sqrt[a] + Sqrt[b])^2*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/4)*d) + ((a - 2*Sqrt[a]*Sqrt[b] + b)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/4)*d) - Sin[c + d*x]/(b*d)} -{Cos[c + d*x]^3/(a - b*Sin[c + d*x]^4), x, 4, ((Sqrt[a] + Sqrt[b])*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/4)*d) - ((Sqrt[a] - Sqrt[b])*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/4)*d)} -{Cos[c + d*x]^1/(a - b*Sin[c + d*x]^4), x, 4, ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)]/(2*a^(3/4)*b^(1/4)*d) + ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)]/(2*a^(3/4)*b^(1/4)*d)} -{Sec[c + d*x]^1/(a - b*Sin[c + d*x]^4), x, 7, (b^(1/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])*d) + ArcTanh[Sin[c + d*x]]/((a - b)*d) - (b^(1/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])*d)} -{Sec[c + d*x]^3/(a - b*Sin[c + d*x]^4), x, 7, (b^(3/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^2*d) + ((a - 5*b)*ArcTanh[Sin[c + d*x]])/(2*(a - b)^2*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^2*d) + 1/(4*(a - b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))} -{Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4), x, 7, (b^(5/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^3*d) + ((3*a^2 - 6*a*b + 35*b^2)*ArcTanh[Sin[c + d*x]])/(8*(a - b)^3*d) - (b^(5/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^3*d) + 1/(16*(a - b)*d*(1 - Sin[c + d*x])^2) + (3*a - 11*b)/(16*(a - b)^2*d*(1 - Sin[c + d*x])) - 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (3*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))} - -{Cos[c + d*x]^10/(a - b*Sin[c + d*x]^4), x, 16, -((17*x)/(16*b)) - (4*(a + b)*x)/b^2 - ((a + 3*b)*x)/(2*b^2) - ((Sqrt[a] - Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) + ((Sqrt[a] + Sqrt[b])^(9/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(5/2)*d) - (17*Cos[c + d*x]*Sin[c + d*x])/(16*b*d) - ((a + 3*b)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (17*Cos[c + d*x]^3*Sin[c + d*x])/(24*b*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*b*d)} -{Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4), x, 12, -((11*x)/(8*b)) - ((a + 3*b)*x)/b^2 + ((Sqrt[a] - Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^2*d) + ((Sqrt[a] + Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^6/(a - b*Sin[c + d*x]^4), x, 9, -((5*x)/(2*b)) - ((Sqrt[a] - Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/2)*d) + ((Sqrt[a] + Sqrt[b])^(5/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^4/(a - b*Sin[c + d*x]^4), x, 7, -(x/b) + ((Sqrt[a] - Sqrt[b])^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b*d) + ((Sqrt[a] + Sqrt[b])^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*b*d)} -{Cos[c + d*x]^2/(a - b*Sin[c + d*x]^4), x, 4, -((Sqrt[Sqrt[a] - Sqrt[b]]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[b]*d)) + (Sqrt[Sqrt[a] + Sqrt[b]]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[b]*d)} -{Sec[c + d*x]^2/(a - b*Sin[c + d*x]^4), x, 6, -((Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d)) + (Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + Tan[c + d*x]/((a - b)*d)} -{Sec[c + d*x]^4/(a - b*Sin[c + d*x]^4), x, 6, (b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + (b*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) + ((a - 3*b)*Tan[c + d*x])/((a - b)^2*d) + Tan[c + d*x]^3/(3*(a - b)*d)} -{Sec[c + d*x]^6/(a - b*Sin[c + d*x]^4), x, 6, -((b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(7/2)*d)) + (b^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(7/2)*d) + ((a^2 - 3*a*b + 6*b^2)*Tan[c + d*x])/((a - b)^3*d) + (2*(a - 2*b)*Tan[c + d*x]^3)/(3*(a - b)^2*d) + Tan[c + d*x]^5/(5*(a - b)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^4)^p when p symbolic*) - - -{Cos[e + f*x]^m*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^4)^p, x]} - - -{Cos[e + f*x]^5*(a + b*Sin[e + f*x]^4)^p, x, 8, (Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^(1 + p))/(b*f*(5 + 4*p)) - ((a - b*(5 + 4*p))*Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(b*f*(5 + 4*p))) - (2*Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(3*f))} -{Cos[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p, x, 7, (Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*f) - (Hypergeometric2F1[3/4, -p, 7/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(3*f))} -{Cos[e + f*x]^1*(a + b*Sin[e + f*x]^4)^p, x, 3, (Hypergeometric2F1[1/4, -p, 5/4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*f)} -{Sec[e + f*x]^1*(a + b*Sin[e + f*x]^4)^p, x, 7, (AppellF1[1/4, 1, -p, 5/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*f) + (AppellF1[3/4, 1, -p, 7/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(3*f))} -{Sec[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p, x, 9, (AppellF1[1/4, 2, -p, 5/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*f) + (2*AppellF1[3/4, 2, -p, 7/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(3*f)) + (AppellF1[5/4, 2, -p, 9/4, Sin[e + f*x]^4, -((b*Sin[e + f*x]^4)/a)]*Sin[e + f*x]^5*(a + b*Sin[e + f*x]^4)^p)/((1 + (b*Sin[e + f*x]^4)/a)^p*(5*f))} - -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[(a + b*Sin[e + f*x]^4)^p, x]} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^4)^p, x]} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x, 0, Unintegrable[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Sin[e+f x]^n)^p when p symbolic*) - - -{Cos[e + f*x]^m*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Cos[e + f*x]^m*(a + b*Sin[e + f*x]^n)^p, x]} - - -{Cos[e + f*x]^5*(a + b*Sin[e + f*x]^n)^p, x, 9, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*f) - (2*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*(3*f)) + (Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^5*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*(5*f))} -{Cos[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p, x, 7, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*f) - (Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*(3*f))} -{Cos[e + f*x]^1*(a + b*Sin[e + f*x]^n)^p, x, 3, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]*(a + b*Sin[e + f*x]^n)^p)/((1 + (b*Sin[e + f*x]^n)/a)^p*f)} -{Sec[e + f*x]^1*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Sec[e + f*x]*(a + b*Sin[e + f*x]^n)^p, x]} -{Sec[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Sec[e + f*x]^3*(a + b*Sin[e + f*x]^n)^p, x]} - -{Cos[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]} -{Cos[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]} -{Cos[e + f*x]^0*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[e + f*x]^n)^p, x]} -{Sec[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Sec[e + f*x]^2*(a + b*Sin[e + f*x]^n)^p, x]} -{Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x, 0, Unintegrable[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsection:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^2)^p when a+b=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^2)^p*) - - -{Tan[c + d*x]^7/(a + b*Sin[c + d*x]^2), x, 3, (a^3*Log[Cos[c + d*x]])/((a + b)^4*d) - (a^3*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^4*d) + ((3*a^2 + 3*a*b + b^2)*Sec[c + d*x]^2)/(2*(a + b)^3*d) - ((3*a + 2*b)*Sec[c + d*x]^4)/(4*(a + b)^2*d) + Sec[c + d*x]^6/(6*(a + b)*d)} -{Tan[c + d*x]^5/(a + b*Sin[c + d*x]^2), x, 3, -((a^2*Log[Cos[c + d*x]])/((a + b)^3*d)) + (a^2*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^3*d) - ((2*a + b)*Sec[c + d*x]^2)/(2*(a + b)^2*d) + Sec[c + d*x]^4/(4*(a + b)*d)} -{Tan[c + d*x]^3/(a + b*Sin[c + d*x]^2), x, 3, (a*Log[Cos[c + d*x]])/((a + b)^2*d) - (a*Log[a + b*Sin[c + d*x]^2])/(2*(a + b)^2*d) + Sec[c + d*x]^2/(2*(a + b)*d)} -{Tan[c + d*x]^1/(a + b*Sin[c + d*x]^2), x, 4, -(Log[Cos[c + d*x]]/((a + b)*d)) + Log[a + b*Sin[c + d*x]^2]/(2*(a + b)*d)} -{Cot[c + d*x]^1/(a + b*Sin[c + d*x]^2), x, 4, Log[Sin[c + d*x]]/(a*d) - Log[a + b*Sin[c + d*x]^2]/(2*a*d)} -{Cot[c + d*x]^3/(a + b*Sin[c + d*x]^2), x, 3, -(Csc[c + d*x]^2/(2*a*d)) - ((a + b)*Log[Sin[c + d*x]])/(a^2*d) + ((a + b)*Log[a + b*Sin[c + d*x]^2])/(2*a^2*d)} -{Cot[c + d*x]^5/(a + b*Sin[c + d*x]^2), x, 3, ((2*a + b)*Csc[c + d*x]^2)/(2*a^2*d) - Csc[c + d*x]^4/(4*a*d) + ((a + b)^2*Log[Sin[c + d*x]])/(a^3*d) - ((a + b)^2*Log[a + b*Sin[c + d*x]^2])/(2*a^3*d)} -{Cot[c + d*x]^7/(a + b*Sin[c + d*x]^2), x, 3, -(((3*a^2 + 3*a*b + b^2)*Csc[c + d*x]^2)/(2*a^3*d)) + ((3*a + b)*Csc[c + d*x]^4)/(4*a^2*d) - Csc[c + d*x]^6/(6*a*d) - ((a + b)^3*Log[Sin[c + d*x]])/(a^4*d) + ((a + b)^3*Log[a + b*Sin[c + d*x]^2])/(2*a^4*d)} - -{Tan[c + d*x]^8/(a + b*Sin[c + d*x]^2), x, 4, (a^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(9/2)*d) - (a^3*Tan[c + d*x])/((a + b)^4*d) + (a^2*Tan[c + d*x]^3)/(3*(a + b)^3*d) - (a*Tan[c + d*x]^5)/(5*(a + b)^2*d) + Tan[c + d*x]^7/(7*(a + b)*d)} -{Tan[c + d*x]^6/(a + b*Sin[c + d*x]^2), x, 4, -((a^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(7/2)*d)) + (a^2*Tan[c + d*x])/((a + b)^3*d) - (a*Tan[c + d*x]^3)/(3*(a + b)^2*d) + Tan[c + d*x]^5/(5*(a + b)*d)} -{Tan[c + d*x]^4/(a + b*Sin[c + d*x]^2), x, 4, (a^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(5/2)*d) - (a*Tan[c + d*x])/((a + b)^2*d) + Tan[c + d*x]^3/(3*(a + b)*d)} -{Tan[c + d*x]^2/(a + b*Sin[c + d*x]^2), x, 3, -((Sqrt[a]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/((a + b)^(3/2)*d)) + Tan[c + d*x]/((a + b)*d)} -{Cot[c + d*x]^2/(a + b*Sin[c + d*x]^2), x, 3, -((Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(3/2)*d)) - Cot[c + d*x]/(a*d)} -{Cot[c + d*x]^4/(a + b*Sin[c + d*x]^2), x, 4, ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(5/2)*d) + ((a + b)*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^3/(3*a*d)} -{Cot[c + d*x]^6/(a + b*Sin[c + d*x]^2), x, 5, -(((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(7/2)*d)) - ((a + b)^2*Cot[c + d*x])/(a^3*d) + ((a + b)*Cot[c + d*x]^3)/(3*a^2*d) - Cot[c + d*x]^5/(5*a*d)} -{Cot[c + d*x]^8/(a + b*Sin[c + d*x]^2), x, 6, ((a + b)^(7/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(a^(9/2)*d) + ((a + b)^3*Cot[c + d*x])/(a^4*d) - ((a + b)^2*Cot[c + d*x]^3)/(3*a^3*d) + ((a + b)*Cot[c + d*x]^5)/(5*a^2*d) - Cot[c + d*x]^7/(7*a*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^2)^(p/2) when a+b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^5, x, 5, a^2/(3*f*(a*Cos[e + f*x]^2)^(3/2)) - (2*a)/(f*Sqrt[a*Cos[e + f*x]^2]) - Sqrt[a*Cos[e + f*x]^2]/f} -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^3, x, 5, a/(f*Sqrt[a*Cos[e + f*x]^2]) + Sqrt[a*Cos[e + f*x]^2]/f} -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^1, x, 4, -(Sqrt[a*Cos[e + f*x]^2]/f)} -{Cot[e + f*x]^1*Sqrt[a - a*Sin[e + f*x]^2], x, 5, -((Sqrt[a]*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a*Cos[e + f*x]^2]/f} -{Cot[e + f*x]^3*Sqrt[a - a*Sin[e + f*x]^2], x, 7, (3*Sqrt[a]*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]])/(2*f) - (3*Sqrt[a*Cos[e + f*x]^2])/(2*f) - ((a*Cos[e + f*x]^2)^(3/2)*Csc[e + f*x]^2)/(2*a*f)} - -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^6, x, 7, (15*ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/(8*f) - (15*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/(8*f) - (5*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^3)/(8*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^5)/(4*f)} -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^4, x, 6, (-3*ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/(2*f) + (3*Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/(2*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x]^3)/(2*f)} -{Sqrt[a - a*Sin[e + f*x]^2]*Tan[e + f*x]^2, x, 5, (ArcTanh[Sin[e + f*x]]*Sqrt[a*Cos[e + f*x]^2]*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f} -{Cot[e + f*x]^2*Sqrt[a - a*Sin[e + f*x]^2], x, 5, -((Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f) - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f} -{Cot[e + f*x]^4*Sqrt[a - a*Sin[e + f*x]^2], x, 5, (2*Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x])/(3*f) + (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f} -{Cot[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]^2], x, 5, (-3*Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f + (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x])/f - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^5*Sec[e + f*x])/(5*f) - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/Sqrt[a - a*Sin[e + f*x]^2], x, 5, a^2/(5*f*(a*Cos[e + f*x]^2)^(5/2)) - (2*a)/(3*f*(a*Cos[e + f*x]^2)^(3/2)) + 1/(f*Sqrt[a*Cos[e + f*x]^2])} -{Tan[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2], x, 5, a/(3*f*(a*Cos[e + f*x]^2)^(3/2)) - 1/(f*Sqrt[a*Cos[e + f*x]^2])} -{Tan[e + f*x]^1/Sqrt[a - a*Sin[e + f*x]^2], x, 4, 1/(f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^1/Sqrt[a - a*Sin[e + f*x]^2], x, 4, -(ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))} -{Cot[e + f*x]^3/Sqrt[a - a*Sin[e + f*x]^2], x, 6, ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(2*Sqrt[a]*f) - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^2)/(2*a*f)} - -{Tan[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2], x, 5, (3*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*f*Sqrt[a*Cos[e + f*x]^2]) - (3*Tan[e + f*x])/(8*f*Sqrt[a*Cos[e + f*x]^2]) + Tan[e + f*x]^3/(4*f*Sqrt[a*Cos[e + f*x]^2])} -{Tan[e + f*x]^2/Sqrt[a - a*Sin[e + f*x]^2], x, 4, -(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(2*f*Sqrt[a*Cos[e + f*x]^2]) + Tan[e + f*x]/(2*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^2/Sqrt[a - a*Sin[e + f*x]^2], x, 4, -(Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2]))} -{Cot[e + f*x]^4/Sqrt[a - a*Sin[e + f*x]^2], x, 4, Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^6/Sqrt[a - a*Sin[e + f*x]^2], x, 5, -(Cot[e + f*x]/(f*Sqrt[a*Cos[e + f*x]^2])) + (2*Cot[e + f*x]*Csc[e + f*x]^2)/(3*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*f*Sqrt[a*Cos[e + f*x]^2])} - - -{Tan[e + f*x]^5/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, a^2/(7*f*(a*Cos[e + f*x]^2)^(7/2)) - (2*a)/(5*f*(a*Cos[e + f*x]^2)^(5/2)) + 1/(3*f*(a*Cos[e + f*x]^2)^(3/2))} -{Tan[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, a/(5*f*(a*Cos[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cos[e + f*x]^2)^(3/2))} -{Tan[e + f*x]^1/(a - a*Sin[e + f*x]^2)^(3/2), x, 4, 1/(3*f*(a*Cos[e + f*x]^2)^(3/2))} -{Cot[e + f*x]^1/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, -(ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^3/(a - a*Sin[e + f*x]^2)^(3/2), x, 6, -ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]]/(2*a^(3/2)*f) - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^2)/(2*a^2*f)} - -{Tan[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, -(ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(8*a*f*Sqrt[a*Cos[e + f*x]^2]) - Tan[e + f*x]/(8*a*f*Sqrt[a*Cos[e + f*x]^2]) + (Sec[e + f*x]^2*Tan[e + f*x])/(4*a*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^2/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, (ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(a*f*Sqrt[a*Cos[e + f*x]^2]) - Cot[e + f*x]/(a*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^4/(a - a*Sin[e + f*x]^2)^(3/2), x, 4, -(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^6/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, (Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*Cos[e + f*x]^2])} -{Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2), x, 5, -(Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*f*Sqrt[a*Cos[e + f*x]^2]) + (2*Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^6)/(7*a*f*Sqrt[a*Cos[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^5, x, 6, ((8*a^2 + 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(3/2)*f) - ((8*a^2 + 24*a*b + 15*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)^2*f) - ((8*a + 7*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(8*(a + b)^2*f) + (Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2))/(4*(a + b)*f)} -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^3, x, 5, -((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*Sqrt[a + b]*f) + ((2*a + 3*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*(a + b)*f) + (Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(2*(a + b)*f)} -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^1, x, 4, (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/f - Sqrt[a + b*Sin[e + f*x]^2]/f} -{Cot[e + f*x]^1*Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sin[e + f*x]^2]/f} -{Cot[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2], x, 5, ((2*a - b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*Sqrt[a]*f) - ((2*a - b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*a*f) - (Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(2*a*f)} -{Cot[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2], x, 6, -((8*a^2 - 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(3/2)*f) + ((8*a^2 - 8*a*b - b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^2*f) + ((8*a + b)*Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(8*a^2*f) - (Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2))/(4*a*f)} - -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^4, x, 8, ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((3*a + 4*b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^3)/(3*f)} -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^2, x, 7, -((2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f} -{Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x]^0, x, 2, (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Cot[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2], x, 7, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2], x, 8, ((3*a - b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) - (Cot[e + f*x]^3*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) + ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^5, x, 7, ((8*a^2 + 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*Sqrt[a + b]*f) - ((8*a^2 + 40*a*b + 35*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)*f) - ((8*a^2 + 40*a*b + 35*b^2)*(a + b*Sin[e + f*x]^2)^(3/2))/(24*(a + b)^2*f) - ((8*a + 9*b)*Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(8*(a + b)^2*f) + (Sec[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(5/2))/(4*(a + b)*f)} -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^3, x, 6, -(Sqrt[a + b]*(2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*f) + ((2*a + 5*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*f) + ((2*a + 5*b)*(a + b*Sin[e + f*x]^2)^(3/2))/(6*(a + b)*f) + (Sec[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(2*(a + b)*f)} -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^1, x, 5, ((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/f - ((a + b)*Sqrt[a + b*Sin[e + f*x]^2])/f - (a + b*Sin[e + f*x]^2)^(3/2)/(3*f)} -{Cot[e + f*x]^1*(a + b*Sin[e + f*x]^2)^(3/2), x, 5, -((a^(3/2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sin[e + f*x]^2])/f + (a + b*Sin[e + f*x]^2)^(3/2)/(3*f)} -{Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2), x, 6, (Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*f) - ((2*a - 3*b)*Sqrt[a + b*Sin[e + f*x]^2])/(2*f) - ((2*a - 3*b)*(a + b*Sin[e + f*x]^2)^(3/2))/(6*a*f) - (Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(2*a*f)} -{Cot[e + f*x]^5*(a + b*Sin[e + f*x]^2)^(3/2), x, 7, -((8*a^2 - 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((8*a^2 - 24*a*b + 3*b^2)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a*f) + ((8*a^2 - 24*a*b + 3*b^2)*(a + b*Sin[e + f*x]^2)^(3/2))/(24*a^2*f) + ((8*a - b)*Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(5/2))/(8*a^2*f) - (Csc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(5/2))/(4*a*f)} - -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^4, x, 9, -(((3*a + 8*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f)) + (8*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(5*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((a + 2*b)*Sin[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/f + ((a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^3)/(3*f)} -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^2, x, 8, (4*b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x])/f} -{(a + b*Sin[e + f*x]^2)^(3/2)*Tan[e + f*x]^0, x, 6, -((b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f)) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*(a + b)*EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2), x, 8, (4*b*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]*(a + b*Sin[e + f*x]^2)^(3/2))/f - ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*a*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2), x, 9, ((a - b)*Cos[e + f*x]^2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/f + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f) - (Cot[e + f*x]^3*(a + b*Sin[e + f*x]^2)^(3/2))/(3*f) + (8*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a - 3*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/Sqrt[a + b*Sin[e + f*x]^2], x, 5, ((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(5/2)*f) - ((8*a + 5*b)*Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(8*(a + b)^2*f) + (Sec[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2])/(4*(a + b)*f)} -{Tan[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((2*a + b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(3/2)*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(2*(a + b)*f)} -{Tan[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/(Sqrt[a + b]*f)} -{Cot[e + f*x]^1/Sqrt[a + b*Sin[e + f*x]^2], x, 3, -(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))} -{Cot[e + f*x]^3/Sqrt[a + b*Sin[e + f*x]^2], x, 4, ((2*a + b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(3/2)*f) - (Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(2*a*f)} -{Cot[e + f*x]^5/Sqrt[a + b*Sin[e + f*x]^2], x, 5, -((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) + ((8*a + 3*b)*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^2*f) - (Csc[e + f*x]^4*Sqrt[a + b*Sin[e + f*x]^2])/(4*a*f)} - -{Tan[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 8, (2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(2*a + b)*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)^2*f) + (Sec[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/(3*(a + b)*f)} -{Tan[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 4, -((Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (Sqrt[a + b*Sin[e + f*x]^2]*Tan[e + f*x])/((a + b)*f)} -{Tan[e + f*x]^0/Sqrt[a + b*Sin[e + f*x]^2], x, 2, (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^2/Sqrt[a + b*Sin[e + f*x]^2], x, 5, -((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f)) - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Cot[e + f*x]^4/Sqrt[a + b*Sin[e + f*x]^2], x, 8, (2*(2*a + b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*f) + (2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2), x, 6, ((8*a^2 - 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) - (8*a^2 - 8*a*b - b^2)/(8*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((8*a + 3*b)*Sec[e + f*x]^2)/(8*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Sec[e + f*x]^4/(4*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 5, -((2*a - b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) + (2*a - b)/(2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Sec[e + f*x]^2/(2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(3/2)*f) - 1/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(3/2), x, 5, ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(5/2)*f) - (2*a + 3*b)/(2*a^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - Csc[e + f*x]^2/(2*a*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(3/2), x, 6, -((8*a^2 + 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(7/2)*f) + (8*a^2 + 24*a*b + 15*b^2)/(8*a^3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((8*a + 5*b)*Csc[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - Csc[e + f*x]^4/(4*a*f*Sqrt[a + b*Sin[e + f*x]^2])} - -{Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2), x, 9, ((7*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*a*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) - (4*a*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, -((2*b*Cos[e + f*x]*Sin[e + f*x])/((a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/((a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(3/2), x, 4, (b*Cos[e + f*x]*Sin[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) + (EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(a*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])} -{Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(3/2), x, 8, Cot[e + f*x]/(a*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*f) - (2*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(a^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(a*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(3/2), x, 9, ((a + b)*Cot[e + f*x]*Csc[e + f*x]^2)/(a*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + ((7*a + 8*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*f) - ((3*a + 4*b)*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*b*f) + ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (4*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -{Tan[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, ((8*a^2 - 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - (8*a^2 - 24*a*b + 3*b^2)/(24*(a + b)^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) - ((8*a + b)*Sec[e + f*x]^2)/(8*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + Sec[e + f*x]^4/(4*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (8*a^2 - 24*a*b + 3*b^2)/(8*(a + b)^4*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2), x, 6, -((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) + (2*a - 3*b)/(6*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + Sec[e + f*x]^2/(2*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*a - 3*b)/(2*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 5, ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(5/2)*f) - 1/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) - 1/((a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Sin[e + f*x]^2)^(5/2), x, 5, -(ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + 1/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + 1/(a^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Sin[e + f*x]^2)^(5/2), x, 6, ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(2*a^(7/2)*f) - (2*a + 5*b)/(6*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - Csc[e + f*x]^2/(2*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) - (2*a + 5*b)/(2*a^3*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, -((8*a^2 + 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(8*a^(9/2)*f) + (8*a^2 + 40*a*b + 35*b^2)/(24*a^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((8*a + 7*b)*Csc[e + f*x]^2)/(8*a^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) - Csc[e + f*x]^4/(4*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (8*a^2 + 40*a*b + 35*b^2)/(8*a^4*f*Sqrt[a + b*Sin[e + f*x]^2])} - -{Tan[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2), x, 10, ((5*a - 3*b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^3*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (8*(a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^4*f*Sqrt[a + b*Sin[e + f*x]^2]) + (8*(a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*(a + b)^4*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a - 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - (2*(2*a - b)*Tan[e + f*x])/(3*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))} -{Tan[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 9, -((4*b*Cos[e + f*x]*Sin[e + f*x])/(3*(a + b)^2*f*(a + b*Sin[e + f*x]^2)^(3/2))) - ((7*a - b)*b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)^3*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((7*a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a*(a + b)^3*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + Tan[e + f*x]/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2))} -{Tan[e + f*x]^0/(a + b*Sin[e + f*x]^2)^(5/2), x, 7, (b*Cos[e + f*x]*Sin[e + f*x])/(3*a*(a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*b*(2*a + b)*Cos[e + f*x]*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sin[e + f*x]^2]) + (2*(2*a + b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^2*(a + b)^2*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (EllipticF[e + f*x, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2), x, 9, Cot[e + f*x]/(3*a*f*(a + b*Sin[e + f*x]^2)^(3/2)) + ((3*a + 4*b)*Cot[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b*Sin[e + f*x]^2]) - ((7*a + 8*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*f) - ((7*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*(a + b)*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) + (4*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^2*f*Sqrt[a + b*Sin[e + f*x]^2])} -{Cot[e + f*x]^4/(a + b*Sin[e + f*x]^2)^(5/2), x, 10, ((a + b)*Cot[e + f*x]*Csc[e + f*x]^2)/(3*a*b*f*(a + b*Sin[e + f*x]^2)^(3/2)) + (2*(a + 3*b)*Cot[e + f*x]*Csc[e + f*x]^2)/(3*a^2*b*f*Sqrt[a + b*Sin[e + f*x]^2]) + (8*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^4*f) - ((3*a + 8*b)*Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^3*b*f) + (8*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2])/(3*a^4*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - ((5*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sec[e + f*x]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(3*a^3*f*Sqrt[a + b*Sin[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^2)^p when p symbolic*) - - -{(d*Tan[e + f*x])^m*(a + b*Sin[e + f*x]^2)^p, x, 3, (AppellF1[(1 + m)/2, (1 + m)/2, -p, (3 + m)/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(Cos[e + f*x]^2)^((1 + m)/2)*(a + b*Sin[e + f*x]^2)^p*(d*Tan[e + f*x])^(1 + m))/((1 + (b*Sin[e + f*x]^2)/a)^p*(d*f*(1 + m)))} - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x]^2)^p, x, 3, -(((a + b + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)^2*d*(1 + p))) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)*d)} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x]^2)^p, x, 2, (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^2)/(a + b)]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*(a + b)*d*(1 + p))} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x]^2)^p, x, 2, -((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a*d*(1 + p)))} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x]^2)^p, x, 3, -((Csc[c + d*x]^2*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a*d)) + ((a - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^2)/a]*(a + b*Sin[c + d*x]^2)^(1 + p))/(2*a^2*d*(1 + p))} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p, x, 3, (AppellF1[5/2, 5/2, -p, 7/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/((1 + (b*Sin[c + d*x]^2)/a)^p*(5*d))} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x]^2)^p, x, 3, (AppellF1[3/2, 3/2, -p, 5/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^2)^p*Tan[c + d*x])/((1 + (b*Sin[c + d*x]^2)/a)^p*(3*d))} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x]^2)^p, x, 3, -((AppellF1[-(1/2), -(1/2), -p, 1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/((1 + (b*Sin[c + d*x]^2)/a)^p*d))} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p, x, 3, -((AppellF1[-(3/2), -(3/2), -p, -(1/2), Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^3*Sec[c + d*x]*(a + b*Sin[c + d*x]^2)^p)/((1 + (b*Sin[c + d*x]^2)/a)^p*(3*d)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^3)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]^3/(a + b*Sin[x]^3), x, 11, (b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csc[x]^2/(2*a) - Log[Sin[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sin[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[x] + b^(2/3)*Sin[x]^2])/(6*a^(5/3)) + Log[a + b*Sin[x]^3]/(3*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^3)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cot[x]*Sqrt[a + b*Sin[x]^3], x, 5, (-(2/3))*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]] + (2/3)*Sqrt[a + b*Sin[x]^3]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]^1/Sqrt[a + b*Sin[x]^3], x, 4, -((2*ArcTanh[Sqrt[a + b*Sin[x]^3]/Sqrt[a]])/(3*Sqrt[a]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsection:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4], x, 5, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]])/(2*d)) + Sqrt[a + b*Sin[c + d*x]^4]/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4], x, 4, -((a*ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])])/(2*(a + b)^(3/2)*d)) + (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(2*(a + b)*d)} -{Tan[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]^4], x, 3, ArcTanh[(a + b*Sin[c + d*x]^2)/(Sqrt[a + b]*Sqrt[a + b*Sin[c + d*x]^4])]/(2*Sqrt[a + b]*d)} -{Cot[c + d*x]^1/Sqrt[a + b*Sin[c + d*x]^4], x, 4, -(ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]/(2*Sqrt[a]*d))} -{Cot[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4], x, 5, ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]/(2*Sqrt[a]*d) - (Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(2*a*d)} -{Cot[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]^4], x, 6, -(((2*a - b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]])/(4*a^(3/2)*d)) + (Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]^4])/(a*d) - (Csc[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]^4])/(4*a*d)} - -{Tan[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4], x, 4, (Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(Sqrt[a + b]*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - (a^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/((a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + (a^(1/4)*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*(a + b)^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} -{Tan[c + d*x]^0/Sqrt[a + b*Sin[c + d*x]^4], x, 2, (Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(a + b)^(1/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} -{Cot[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]^4], x, 6, -((Cos[c + d*x]^2*Cot[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4])) + (Sqrt[a + b]*Cos[c + d*x]*Sin[c + d*x]*(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4))/(a*d*Sqrt[a + b*Sin[c + d*x]^4]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)) - ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticE[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4]) + ((a + b)^(1/4)*Cos[c + d*x]^2*EllipticF[2*ArcTan[((a + b)^(1/4)*Tan[c + d*x])/a^(1/4)], (1/2)*(1 - Sqrt[a]/Sqrt[a + b])]*(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)*Sqrt[(a + 2*a*Tan[c + d*x]^2 + (a + b)*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[a + b]*Tan[c + d*x]^2)^2])/(2*a^(3/4)*d*Sqrt[a + b*Sin[c + d*x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^4)^p when p symbolic*) - - -{Tan[c + d*x]^m*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^m, x]} - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x]^4)^p, x, 11, -(((a + b + 2*b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^4)/(a + b)]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*(a + b)^2*d*(1 + p))) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x]^4)^(1 + p))/(2*(a + b)*d) - ((a + b + 2*b*p)*AppellF1[1/2, 1, -p, 3/2, Sin[c + d*x]^4, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/((1 + (b*Sin[c + d*x]^4)/a)^p*(2*(a + b)*d)) + (b*(1 + 2*p)*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/((1 + (b*Sin[c + d*x]^4)/a)^p*(2*(a + b)*d))} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x]^4)^p, x, 7, (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sin[c + d*x]^4)/(a + b)]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*(a + b)*d*(1 + p)) + (AppellF1[1/2, 1, -p, 3/2, Sin[c + d*x]^4, -((b*Sin[c + d*x]^4)/a)]*Sin[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p)/((1 + (b*Sin[c + d*x]^4)/a)^p*(2*d))} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x]^4)^p, x, 3, -((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*a*d*(1 + p)))} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x]^4)^p, x, 6, (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^4)/a]*(a + b*Sin[c + d*x]^4)^(1 + p))/(4*a*d*(1 + p)) - (Csc[c + d*x]^2*Hypergeometric2F1[-(1/2), -p, 1/2, -((b*Sin[c + d*x]^4)/a)]*(a + b*Sin[c + d*x]^4)^p)/((1 + (b*Sin[c + d*x]^4)/a)^p*(2*d))} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^4, x]} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^4)^p*Tan[c + d*x]^2, x]} -{Tan[c + d*x]^0*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^4)^p, x]} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x]} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p, x, 0, Unintegrable[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^4)^p, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sin[e+f x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^n)^p*) - - -{Tan[c + d*x]^m*(a + b*Sin[c + d*x]^n)^3, x, 10, (a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (3*a^2*b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n)) + (3*a*b^2*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + 2*n), (1/2)*(3 + m + 2*n), Sin[c + d*x]^2]*Sin[c + d*x]^(2*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 2*n)) + (b^3*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + 3*n), (1/2)*(3 + m + 3*n), Sin[c + d*x]^2]*Sin[c + d*x]^(3*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 3*n))} -{Tan[c + d*x]^m*(a + b*Sin[c + d*x]^n)^2, x, 8, (a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (2*a*b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n)) + (b^2*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + 2*n), (1/2)*(3 + m + 2*n), Sin[c + d*x]^2]*Sin[c + d*x]^(2*n)*Tan[c + d*x]^(1 + m))/(d*(1 + m + 2*n))} -{Tan[c + d*x]^m*(a + b*Sin[c + d*x]^n)^1, x, 6, (a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (b*(Cos[c + d*x]^2)^((1 + m)/2)*Hypergeometric2F1[(1 + m)/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Sin[c + d*x]^2]*Sin[c + d*x]^n*Tan[c + d*x]^(1 + m))/(d*(1 + m + n))} -{Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^1, x, 0, Unintegrable[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n), x]} -{Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^2, x, 0, Unintegrable[Tan[c + d*x]^m/(a + b*Sin[c + d*x]^n)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^n)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cot[x]*Sqrt[a + b*Sin[x]^n], x, 5, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/n) + (2*Sqrt[a + b*Sin[x]^n])/n} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]/Sqrt[a + b*Sin[x]^n], x, 4, -((2*ArcTanh[Sqrt[a + b*Sin[x]^n]/Sqrt[a]])/(Sqrt[a]*n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sin[e+f x]^n)^p when p symbolic*) - - -{Tan[c + d*x]^m*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^m, x]} - - -{Tan[c + d*x]^3*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^3, x]} -{Tan[c + d*x]^1*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x], x]} -{Cot[c + d*x]^1*(a + b*Sin[c + d*x]^n)^p, x, 3, -((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin[c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)))} -{Cot[c + d*x]^3*(a + b*Sin[c + d*x]^n)^p, x, 7, (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin[c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)) - (Csc[c + d*x]^2*Hypergeometric2F1[-(2/n), -p, -((2 - n)/n), -((b*Sin[c + d*x]^n)/a)]*(a + b*Sin[c + d*x]^n)^p)/((1 + (b*Sin[c + d*x]^n)/a)^p*(2*d))} - -{Tan[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^4, x]} -{Tan[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p*Tan[c + d*x]^2, x]} -{Tan[c + d*x]^0*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[(a + b*Sin[c + d*x]^n)^p, x]} -{Cot[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^n)^p, x]} -{Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x, 0, Unintegrable[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (d Sin[e+f x])^n (a+b Sin[e+f x]^r)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (d Sin[e+f x])^n (a+b Sin[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^(m/2) (d Sin[e+f x])^(n/2) (a+b Sin[e+f x]^2)^p*) - - -{(a + b*Sin[e + f*x]^2)/((g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]), x, 7, (2*(a + b)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) + ((2*a - b)*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(3*f*g^2*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]), (2*(a + b)*Sqrt[d*Sin[e + f*x]])/(3*d*f*g*(g*Cos[e + f*x])^(3/2)) - (2*(2*a - b)*(1 - Csc[e + f*x]^2)^(3/4)*EllipticF[(1/2)*ArcCsc[Sin[e + f*x]], 2]*(d*Sin[e + f*x])^(3/2))/(3*d^2*f*g*(g*Cos[e + f*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (d Sin[e+f x])^n (a+b Sin[e+f x]^2)^p when p symbolic*) - - -{(c*Cos[e + f*x])^m*(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x]^2)^p, x, 3, (c*AppellF1[(1 + n)/2, (1 - m)/2, -p, (3 + n)/2, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*(c*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*(d*Sin[e + f*x])^(1 + n)*(a + b*Sin[e + f*x]^2)^p)/((1 + (b*Sin[e + f*x]^2)/a)^p*(d*f*(1 + n)))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b (c Sin[e+f x]+d Cos[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b (c Sin[e+f x]+d Cos[e+f x])^2)^p*) - - -{Sqrt[a + (b*Sin[e+f*x] + c*Cos[e+f*x])^2], x, 3, (EllipticE[e + f*x + ArcTan[b, c], -((b^2 + c^2)/a)]*Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2])/(f*Sqrt[1 + (c*Cos[e + f*x] + b*Sin[e + f*x])^2/a])} - - -{1/Sqrt[a + (b*Sin[e+f*x] + c*Cos[e+f*x])^2], x, 3, (EllipticF[e + f*x + ArcTan[b, c], -((b^2 + c^2)/a)]*Sqrt[1 + (c*Cos[e + f*x] + b*Sin[e + f*x])^2/a])/(f*Sqrt[a + (c*Cos[e + f*x] + b*Sin[e + f*x])^2])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.8 (a+b sin)^m (c+d trig)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.8 (a+b sin)^m (c+d trig)^n.m deleted file mode 100644 index 7644bb8..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.8 (a+b sin)^m (c+d trig)^n.m +++ /dev/null @@ -1,46 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (A+B Trig[c+d x]) (a+b Sin[a+b x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Cos[c+d x]) (a+b Sin[c+d x])^n*) - - -{(A + B*Cos[x])/(a + b*Sin[x]), x, 7, (2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[a + b*Sin[x]])/b} - -{(A + B*Cos[x])/(1 + Sin[x]), x, 5, B*Log[1 + Sin[x]] - (A*Cos[x])/(1 + Sin[x])} -{(A + B*Cos[x])/(1 - Sin[x]), x, 5, (-B)*Log[1 - Sin[x]] + (A*Cos[x])/(1 - Sin[x])} - - -{(b + c + Cos[x])/(a + b*Sin[x]), x, 7, (2*(b + c)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + Log[a + b*Sin[x]]/b} -{(b + c + Cos[x])/(a - b*Sin[x]), x, 7, -((2*(b + c)*ArcTan[(b - a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]) - Log[a - b*Sin[x]]/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Tan[c+d x]) (a+b Sin[c+d x])^n*) - - -{(A + B*Tan[x])/(a + b*Sin[x]), x, 8, (2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (B*Log[1 - Sin[x]])/(2*(a + b)) - (B*Log[1 + Sin[x]])/(2*(a - b)) + (a*B*Log[a + b*Sin[x]])/(a^2 - b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Cot[c+d x]) (a+b Sin[c+d x])^n*) - - -{(A + B*Cot[x])/(a + b*Sin[x]), x, 9, (2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[Sin[x]])/a - (B*Log[a + b*Sin[x]])/a} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Sec[c+d x]) (a+b Sin[c+d x])^n*) - - -{(A + B*Sec[x])/(a + b*Sin[x]), x, 12, (2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (B*Log[1 - Sin[x]])/(2*(a + b)) + (B*Log[1 + Sin[x]])/(2*(a - b)) - (b*B*Log[a + b*Sin[x]])/(a^2 - b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Csc[c+d x]) (a+b Sin[c+d x])^n*) - - -{(A + B*Csc[x])/(a + b*Sin[x]), x, 6, (2*(a*A - b*B)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]) - (B*ArcTanh[Cos[x]])/a} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m deleted file mode 100644 index ebe664f..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.1 Sine/4.1.9 trig^m (a+b sin^n+c sin^(2 n))^p.m +++ /dev/null @@ -1,73 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[d+e x]^m (a+b Sin[d+e x]^n+c Sin[d+e x]^(2 n))^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Sin[d+e x]^n+c Sin[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Sin[d+e x]+c Sin[d+e x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[x]^4/(a + b*Sin[x] + c*Sin[x]^2), x, 12, x/(2*c) + ((b^2 - a*c)*x)/c^3 - (Sqrt[2]*(b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c^3*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c^3*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + (b*Cos[x])/c^2 - (Cos[x]*Sin[x])/(2*c)} -{Sin[x]^3/(a + b*Sin[x] + c*Sin[x]^2), x, 10, -((b*x)/c^2) + (Sqrt[2]*b*(b - (a*c)/b - b^2/Sqrt[b^2 - 4*a*c] + (3*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c^2*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*b*(b - (a*c)/b + b^2/Sqrt[b^2 - 4*a*c] - (3*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c^2*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - Cos[x]/c} -{Sin[x]^2/(a + b*Sin[x] + c*Sin[x]^2), x, 9, x/c - (Sqrt[2]*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])} -{Sin[x]^1/(a + b*Sin[x] + c*Sin[x]^2), x, 8, (Sqrt[2]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + (Sqrt[2]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]} -{Sin[x]^0/(a + b*Sin[x] + c*Sin[x]^2), x, 7, (2*Sqrt[2]*c*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) - (2*Sqrt[2]*c*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])} -{Csc[x]^1/(a + b*Sin[x] + c*Sin[x]^2), x, 10, -((Sqrt[2]*c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - ArcTanh[Cos[x]]/a} -{Csc[x]^2/(a + b*Sin[x] + c*Sin[x]^2), x, 12, (Sqrt[2]*b*c*(1 + (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a^2*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*b*c*(1 - (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a^2*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + (b*ArcTanh[Cos[x]])/a^2 - Cot[x]/a} -{Csc[x]^3/(a + b*Sin[x] + c*Sin[x]^2), x, 14, -((Sqrt[2]*c*(b^3 - 3*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) + (Sqrt[2]*c*(b^3 - 3*a*b*c - Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) - ArcTanh[Cos[x]]/(2*a) - ((b^2 - a*c)*ArcTanh[Cos[x]])/a^3 + (b*Cot[x])/a^2 - (Cot[x]*Csc[x])/(2*a)} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Sin[d+e x]^2+c Sin[d+e x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Sin[d+e x]^n+c Sin[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Sin[d+e x]+c Sin[d+e x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]^3/(a + b*Sin[x] + c*Sin[x]^2), x, 7, ((b^2 - 2*c*(a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + (b*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*c^2) - Sin[x]/c} -{Cos[x]^2/(a + b*Sin[x] + c*Sin[x]^2), x, 9, -(x/c) - (Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 4*a*c]) + (Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/(c*Sqrt[b^2 - 4*a*c])} -{Cos[x]^1/(a + b*Sin[x] + c*Sin[x]^2), x, 3, -((2*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c])} -{Sec[x]^1/(a + b*Sin[x] + c*Sin[x]^2), x, 9, ((b^2 - 2*a*c - 2*c^2)*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 4*a*c]) - Log[1 - Sin[x]]/(2*(a + b + c)) + Log[1 + Sin[x]]/(2*(a - b + c)) - (b*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*(a - b + c)*(a + b + c))} -{Sec[x]^2/(a + b*Sin[x] + c*Sin[x]^2), x, 11, -((Sqrt[2]*b*c*(1 + (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])) - (Sqrt[2]*b*c*(1 - (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]) + Cos[x]/(2*(a + b + c)*(1 - Sin[x])) - Cos[x]/(2*(a - b + c)*(1 + Sin[x]))} -{Sec[x]^3/(a + b*Sin[x] + c*Sin[x]^2), x, 10, -(((b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c))*ArcTanh[(b + 2*c*Sin[x])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)^2)) - ((a + 2*b + 3*c)*Log[1 - Sin[x]])/(4*(a + b + c)^2) + ((a - 2*b + 3*c)*Log[1 + Sin[x]])/(4*(a - b + c)^2) + (b*(b^2 - 2*c*(a + c))*Log[a + b*Sin[x] + c*Sin[x]^2])/(2*(a^2 - b^2 + 2*a*c + c^2)^2) - (Sec[x]^2*(b - (a + c)*Sin[x]))/(2*(a - b + c)*(a + b + c))} - - -{Cos[x]/(-6 + Sin[x] + Sin[x]^2), x, 4, (1/5)*Log[2 - Sin[x]] - (1/5)*Log[3 + Sin[x]]} -{Cos[x]/(2 - 3*Sin[x] + Sin[x]^2), x, 4, -Log[1 - Sin[x]] + Log[2 - Sin[x]]} -{Cos[x]/(-5 + 4*Sin[x] + Sin[x]^2), x, 4, (1/6)*Log[1 - Sin[x]] - (1/6)*Log[5 + Sin[x]]} -{Cos[x]/(10 - 6*Sin[x] + Sin[x]^2), x, 3, -ArcTan[3 - Sin[x]]} -{Cos[x]/(2 + 2*Sin[x] + Sin[x]^2), x, 3, ArcTan[1 + Sin[x]]} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Sin[d+e x]^2+c Sin[d+e x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Tan[d+e x]^m (a+b Sin[d+e x]^n+c Sin[d+e x]^(2 n))^p*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.0 (a cos)^m (b trg)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.0 (a cos)^m (b trg)^n.m deleted file mode 100644 index a20859d..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.0 (a cos)^m (b trg)^n.m +++ /dev/null @@ -1,522 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Cos[c+d x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Cos[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^n*) - - -{Cos[a + b*x]^1, x, 1, Sin[a + b*x]/b} -{Cos[a + b*x]^2, x, 2, x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^3, x, 2, Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b)} -{Cos[a + b*x]^4, x, 3, (3*x)/8 + (3*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(4*b)} -{Cos[a + b*x]^5, x, 2, Sin[a + b*x]/b - (2*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)} -{Cos[a + b*x]^6, x, 4, (5*x)/16 + (5*Cos[a + b*x]*Sin[a + b*x])/(16*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(24*b) + (Cos[a + b*x]^5*Sin[a + b*x])/(6*b)} -{Cos[a + b*x]^7, x, 2, Sin[a + b*x]/b - Sin[a + b*x]^3/b + (3*Sin[a + b*x]^5)/(5*b) - Sin[a + b*x]^7/(7*b)} -{Cos[a + b*x]^8, x, 5, (35*x)/128 + (35*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (35*Cos[a + b*x]^3*Sin[a + b*x])/(192*b) + (7*Cos[a + b*x]^5*Sin[a + b*x])/(48*b) + (Cos[a + b*x]^7*Sin[a + b*x])/(8*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cos[c+d x])^(n/2)*) - - -{Cos[a + b*x]^(7/2), x, 3, (10*EllipticF[(1/2)*(a + b*x), 2])/(21*b) + (10*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(21*b) + (2*Cos[a + b*x]^(5/2)*Sin[a + b*x])/(7*b)} -{Cos[a + b*x]^(5/2), x, 2, (6*EllipticE[(1/2)*(a + b*x), 2])/(5*b) + (2*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(5*b)} -{Cos[a + b*x]^(3/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)} -{Cos[a + b*x]^(1/2), x, 1, (2*EllipticE[(1/2)*(a + b*x), 2])/b} -{1/Cos[a + b*x]^(1/2), x, 1, (2*EllipticF[(1/2)*(a + b*x), 2])/b} -{1/Cos[a + b*x]^(3/2), x, 2, -((2*EllipticE[(1/2)*(a + b*x), 2])/b) + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])} -{1/Cos[a + b*x]^(5/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sin[a + b*x])/(3*b*Cos[a + b*x]^(3/2))} -{1/Cos[a + b*x]^(7/2), x, 3, -((6*EllipticE[(1/2)*(a + b*x), 2])/(5*b)) + (2*Sin[a + b*x])/(5*b*Cos[a + b*x]^(5/2)) + (6*Sin[a + b*x])/(5*b*Sqrt[Cos[a + b*x]])} - - -{(c*Cos[a + b*x])^(7/2), x, 4, (10*c^4*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(21*b*Sqrt[c*Cos[a + b*x]]) + (10*c^3*Sqrt[c*Cos[a + b*x]]*Sin[a + b*x])/(21*b) + (2*c*(c*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(7*b)} -{(c*Cos[a + b*x])^(5/2), x, 3, (6*c^2*Sqrt[c*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*Sqrt[Cos[a + b*x]]) + (2*c*(c*Cos[a + b*x])^(3/2)*Sin[a + b*x])/(5*b)} -{(c*Cos[a + b*x])^(3/2), x, 3, (2*c^2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*Sqrt[c*Cos[a + b*x]]) + (2*c*Sqrt[c*Cos[a + b*x]]*Sin[a + b*x])/(3*b)} -{(c*Cos[a + b*x])^(1/2), x, 2, (2*Sqrt[c*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*Sqrt[Cos[a + b*x]])} -{1/(c*Cos[a + b*x])^(1/2), x, 2, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(b*Sqrt[c*Cos[a + b*x]])} -{1/(c*Cos[a + b*x])^(3/2), x, 3, -((2*Sqrt[c*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(b*c^2*Sqrt[Cos[a + b*x]])) + (2*Sin[a + b*x])/(b*c*Sqrt[c*Cos[a + b*x]])} -{1/(c*Cos[a + b*x])^(5/2), x, 3, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2])/(3*b*c^2*Sqrt[c*Cos[a + b*x]]) + (2*Sin[a + b*x])/(3*b*c*(c*Cos[a + b*x])^(3/2))} -{1/(c*Cos[a + b*x])^(7/2), x, 4, -((6*Sqrt[c*Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2])/(5*b*c^4*Sqrt[Cos[a + b*x]])) + (2*Sin[a + b*x])/(5*b*c*(c*Cos[a + b*x])^(5/2)) + (6*Sin[a + b*x])/(5*b*c^3*Sqrt[c*Cos[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cos[c+d x])^(n/3)*) - - -{Cos[a + b*x]^(4/3), x, 1, -((3*Cos[a + b*x]^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[a + b*x]^2]*Sin[a + b*x])/(7*b*Sqrt[Sin[a + b*x]^2]))} -{Cos[a + b*x]^(2/3), x, 1, -((3*Cos[a + b*x]^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*Sqrt[Sin[a + b*x]^2]))} -{Cos[a + b*x]^(1/3), x, 1, -((3*Cos[a + b*x]^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sin[a + b*x])/(4*b*Sqrt[Sin[a + b*x]^2]))} -{1/Cos[a + b*x]^(1/3), x, 1, -((3*Cos[a + b*x]^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[a + b*x]^2]*Sin[a + b*x])/(2*b*Sqrt[Sin[a + b*x]^2]))} -{1/Cos[a + b*x]^(2/3), x, 1, -((3*Cos[a + b*x]^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*Sqrt[Sin[a + b*x]^2]))} -{1/Cos[a + b*x]^(4/3), x, 1, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*Cos[a + b*x]^(1/3)*Sqrt[Sin[a + b*x]^2])} - - -{(c*Cos[a + b*x])^(4/3), x, 1, -((3*(c*Cos[a + b*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[a + b*x]^2]*Sin[a + b*x])/(7*b*c*Sqrt[Sin[a + b*x]^2]))} -{(c*Cos[a + b*x])^(2/3), x, 1, -((3*(c*Cos[a + b*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*c*Sqrt[Sin[a + b*x]^2]))} -{(c*Cos[a + b*x])^(1/3), x, 1, -((3*(c*Cos[a + b*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sin[a + b*x])/(4*b*c*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x])^(1/3), x, 1, -((3*(c*Cos[a + b*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[a + b*x]^2]*Sin[a + b*x])/(2*b*c*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x])^(2/3), x, 1, -((3*(c*Cos[a + b*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*c*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x])^(4/3), x, 1, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*c*(c*Cos[a + b*x])^(1/3)*Sqrt[Sin[a + b*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cos[c+d x])^n with n symbolic*) - - -{Cos[a + b*x]^n, x, 1, -((Cos[a + b*x]^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(1 + n)*Sqrt[Sin[a + b*x]^2]))} -{(c*Cos[a + b*x])^n, x, 1, -(((c*Cos[a + b*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*c*(1 + n)*Sqrt[Sin[a + b*x]^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Cos[c+d x]^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cos[c+d x]^p)^(n/2) with p positive integer*) - - -{(a*Cos[x]^2)^(5/2),x, 4, (8/15)*a^2*Sqrt[a*Cos[x]^2]*Tan[x] + (4/15)*a*(a*Cos[x]^2)^(3/2)*Tan[x] + (1/5)*(a*Cos[x]^2)^(5/2)*Tan[x]} -{(a*Cos[x]^2)^(3/2),x, 3, (2/3)*a*Sqrt[a*Cos[x]^2]*Tan[x] + (1/3)*(a*Cos[x]^2)^(3/2)*Tan[x]} -{(a*Cos[x]^2)^(1/2), x, 2, Sqrt[a*Cos[x]^2]*Tan[x]} -{1/(a*Cos[x]^2)^(1/2), x, 2, (ArcTanh[Sin[x]]*Cos[x])/Sqrt[a*Cos[x]^2]} -{1/(a*Cos[x]^2)^(3/2), x, 3, (ArcTanh[Sin[x]]*Cos[x])/(2*a*Sqrt[a*Cos[x]^2]) + Tan[x]/(2*a*Sqrt[a*Cos[x]^2])} -{1/(a*Cos[x]^2)^(5/2), x, 4, (3*ArcTanh[Sin[x]]*Cos[x])/(8*a^2*Sqrt[a*Cos[x]^2]) + Tan[x]/(4*a*(a*Cos[x]^2)^(3/2)) + (3*Tan[x])/(8*a^2*Sqrt[a*Cos[x]^2])} - - -{(a*Cos[x]^3)^(5/2),x, 6, (26*a^2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(77*Cos[x]^(3/2)) + (78/385)*a^2*Cos[x]*Sqrt[a*Cos[x]^3]*Sin[x] + (26/165)*a^2*Cos[x]^3*Sqrt[a*Cos[x]^3]*Sin[x] + (2/15)*a^2*Cos[x]^5*Sqrt[a*Cos[x]^3]*Sin[x] + (26/77)*a^2*Sqrt[a*Cos[x]^3]*Tan[x]} -{(a*Cos[x]^3)^(3/2),x, 4, (14*a*Sqrt[a*Cos[x]^3]*EllipticE[x/2, 2])/(15*Cos[x]^(3/2)) + (14/45)*a*Sqrt[a*Cos[x]^3]*Sin[x] + (2/9)*a*Cos[x]^2*Sqrt[a*Cos[x]^3]*Sin[x]} -{(a*Cos[x]^3)^(1/2), x, 3, (2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)) + (2/3)*Sqrt[a*Cos[x]^3]*Tan[x]} -{1/(a*Cos[x]^3)^(1/2), x, 3, -((2*Cos[x]^(3/2)*EllipticE[x/2, 2])/Sqrt[a*Cos[x]^3]) + (2*Cos[x]*Sin[x])/Sqrt[a*Cos[x]^3]} -{1/(a*Cos[x]^3)^(3/2),x, 4, (10*Cos[x]^(3/2)*EllipticF[x/2, 2])/(21*a*Sqrt[a*Cos[x]^3]) + (10*Sin[x])/(21*a*Sqrt[a*Cos[x]^3]) + (2*Sec[x]*Tan[x])/(7*a*Sqrt[a*Cos[x]^3])} -{1/(a*Cos[x]^3)^(5/2),x, 6, -((154*Cos[x]^(3/2)*EllipticE[x/2, 2])/(195*a^2*Sqrt[a*Cos[x]^3])) + (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Cos[x]^3]) + (154*Tan[x])/(585*a^2*Sqrt[a*Cos[x]^3]) + (22*Sec[x]^2*Tan[x])/(117*a^2*Sqrt[a*Cos[x]^3]) + (2*Sec[x]^4*Tan[x])/(13*a^2*Sqrt[a*Cos[x]^3])} - - -{(a*Cos[x]^4)^(5/2),x, 7, (63/256)*a^2*x*Sqrt[a*Cos[x]^4]*Sec[x]^2 + (21/128)*a^2*Cos[x]*Sqrt[a*Cos[x]^4]*Sin[x] + (21/160)*a^2*Cos[x]^3*Sqrt[a*Cos[x]^4]*Sin[x] + (9/80)*a^2*Cos[x]^5*Sqrt[a*Cos[x]^4]*Sin[x] + (1/10)*a^2*Cos[x]^7*Sqrt[a*Cos[x]^4]*Sin[x] + (63/256)*a^2*Sqrt[a*Cos[x]^4]*Tan[x]} -{(a*Cos[x]^4)^(3/2),x, 5, (5/16)*a*x*Sqrt[a*Cos[x]^4]*Sec[x]^2 + (5/24)*a*Cos[x]*Sqrt[a*Cos[x]^4]*Sin[x] + (1/6)*a*Cos[x]^3*Sqrt[a*Cos[x]^4]*Sin[x] + (5/16)*a*Sqrt[a*Cos[x]^4]*Tan[x]} -{(a*Cos[x]^4)^(1/2), x, 3, (1/2)*x*Sqrt[a*Cos[x]^4]*Sec[x]^2 + (1/2)*Sqrt[a*Cos[x]^4]*Tan[x]} -{1/(a*Cos[x]^4)^(1/2), x, 3, (Cos[x]*Sin[x])/Sqrt[a*Cos[x]^4]} -{1/(a*Cos[x]^4)^(3/2),x, 3, (Cos[x]*Sin[x])/(a*Sqrt[a*Cos[x]^4]) + (2*Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Cos[x]^4]) + (Sin[x]^2*Tan[x]^3)/(5*a*Sqrt[a*Cos[x]^4])} -{1/(a*Cos[x]^4)^(5/2),x, 3, (Cos[x]*Sin[x])/(a^2*Sqrt[a*Cos[x]^4]) + (4*Sin[x]^2*Tan[x])/(3*a^2*Sqrt[a*Cos[x]^4]) + (6*Sin[x]^2*Tan[x]^3)/(5*a^2*Sqrt[a*Cos[x]^4]) + (4*Sin[x]^2*Tan[x]^5)/(7*a^2*Sqrt[a*Cos[x]^4]) + (Sin[x]^2*Tan[x]^7)/(9*a^2*Sqrt[a*Cos[x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cos[c+d x]^p)^n*) - - -{(b*Cos[c + d*x]^m)^n, x, 2, -((Cos[c + d*x]*(b*Cos[c + d*x]^m)^n*Hypergeometric2F1[1/2, (1/2)*(1 + m*n), (1/2)*(3 + m*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m*n)*Sqrt[Sin[c + d*x]^2]))} - - -{(c*Cos[a + b*x]^m)^(5/2), x, 2, -((2*c^2*Cos[a + b*x]^(1 + 2*m)*Sqrt[c*Cos[a + b*x]^m]*Hypergeometric2F1[1/2, (1/4)*(2 + 5*m), (1/4)*(6 + 5*m), Cos[a + b*x]^2]*Sin[a + b*x])/(b*(2 + 5*m)*Sqrt[Sin[a + b*x]^2]))} -{(c*Cos[a + b*x]^m)^(3/2), x, 2, -((2*c*Cos[a + b*x]^(1 + m)*Sqrt[c*Cos[a + b*x]^m]*Hypergeometric2F1[1/2, (1/4)*(2 + 3*m), (3*(2 + m))/4, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(2 + 3*m)*Sqrt[Sin[a + b*x]^2]))} -{(c*Cos[a + b*x]^m)^(1/2), x, 2, -((2*Cos[a + b*x]*Sqrt[c*Cos[a + b*x]^m]*Hypergeometric2F1[1/2, (2 + m)/4, (6 + m)/4, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(2 + m)*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x]^m)^(1/2), x, 2, -((2*Cos[a + b*x]*Hypergeometric2F1[1/2, (2 - m)/4, (6 - m)/4, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(2 - m)*Sqrt[c*Cos[a + b*x]^m]*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x]^m)^(3/2), x, 2, -((2*Cos[a + b*x]^(1 - m)*Hypergeometric2F1[1/2, (1/4)*(2 - 3*m), (3*(2 - m))/4, Cos[a + b*x]^2]*Sin[a + b*x])/(b*c*(2 - 3*m)*Sqrt[c*Cos[a + b*x]^m]*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Cos[a + b*x]^m)^(5/2), x, 2, -((2*Cos[a + b*x]^(1 - 2*m)*Hypergeometric2F1[1/2, (1/4)*(2 - 5*m), (1/4)*(6 - 5*m), Cos[a + b*x]^2]*Sin[a + b*x])/(b*c^2*(2 - 5*m)*Sqrt[c*Cos[a + b*x]^m]*Sqrt[Sin[a + b*x]^2]))} - - -{(c*Cos[a + b*x]^m)^(1/m), x, 2, ((c*Cos[a + b*x]^m)^(1/m)*Tan[a + b*x])/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Cos[c+d x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a (b Cos[c+d x])^p)^n*) - - -{(a*(b*Cos[c + d*x])^p)^n, x, 2, -((Cos[c + d*x]*(a*(b*Cos[c + d*x])^p)^n*Hypergeometric2F1[1/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + n*p)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Title:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Trg[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^5*Sqrt[b*Cos[c + d*x]], x, 6, (30*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*d*Sqrt[b*Cos[c + d*x]]) + (30*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b^2*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^4*d)} -{Cos[c + d*x]^4*Sqrt[b*Cos[c + d*x]], x, 5, (14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^3*d)} -{Cos[c + d*x]^3*Sqrt[b*Cos[c + d*x]], x, 5, (10*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d)} -{Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]], x, 4, (6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^1*Sqrt[b*Cos[c + d*x]], x, 4, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*Sqrt[b*Cos[c + d*x]], x, 2, (2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^1, x, 3, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^2, x, 4, -((2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^3, x, 4, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^4, x, 5, -((6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*b*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^5, x, 5, (10*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^4*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*b^2*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^6, x, 6, -((14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]])) + (2*b^5*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*b^3*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*b*Sin[c + d*x])/(15*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(b*Cos[c + d*x])^(3/2), x, 6, (30*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*d*Sqrt[b*Cos[c + d*x]]) + (30*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^3*d)} -{Cos[c + d*x]^3*(b*Cos[c + d*x])^(3/2), x, 5, (14*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^2*d)} -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(3/2), x, 5, (10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(3/2), x, 4, (6*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(3/2), x, 3, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^1, x, 3, (2*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2, x, 3, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3, x, 4, -((2*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4, x, 4, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^3*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^5, x, 5, -((6*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^4*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*b^2*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^6, x, 5, (10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^5*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*b^3*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^7, x, 6, -((14*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]])) + (2*b^6*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*b^4*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*b^2*Sin[c + d*x])/(15*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(b*Cos[c + d*x])^(5/2), x, 6, (30*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*d*Sqrt[b*Cos[c + d*x]]) + (30*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^2*d)} -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(5/2), x, 5, (14*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (14*b*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(5/2), x, 5, (10*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(5/2), x, 3, (6*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^1, x, 4, (2*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2, x, 3, (2*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3, x, 3, (2*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4, x, 4, -((2*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^3*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5, x, 4, (2*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^4*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^6, x, 5, -((6*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^5*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*b^3*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^7, x, 5, (10*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^6*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*b^4*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^8, x, 6, -((14*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]])) + (2*b^7*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*b^5*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*b^3*Sin[c + d*x])/(15*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(7/2), x, 4, (10*b^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b^3*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^6/Sqrt[b*Cos[c + d*x]], x, 6, (30*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*d*Sqrt[b*Cos[c + d*x]]) + (30*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*b*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b^3*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^5*d)} -{Cos[c + d*x]^5/Sqrt[b*Cos[c + d*x]], x, 5, (14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^2*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^4*d)} -{Cos[c + d*x]^4/Sqrt[b*Cos[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d)} -{Cos[c + d*x]^3/Sqrt[b*Cos[c + d*x]], x, 4, (6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d)} -{Cos[c + d*x]^2/Sqrt[b*Cos[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1/Sqrt[b*Cos[c + d*x]], x, 3, (2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^0/Sqrt[b*Cos[c + d*x]], x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1/Sqrt[b*Cos[c + d*x]], x, 4, -((2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2/Sqrt[b*Cos[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^3/Sqrt[b*Cos[c + d*x]], x, 5, -((6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]])) + (2*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^4/Sqrt[b*Cos[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^3*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*b*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^5/Sqrt[b*Cos[c + d*x]], x, 6, -((14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]])) + (2*b^4*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*b^2*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^7/(b*Cos[c + d*x])^(3/2), x, 6, (30*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*b*d*Sqrt[b*Cos[c + d*x]]) + (30*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*b^2*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b^4*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^6*d)} -{Cos[c + d*x]^6/(b*Cos[c + d*x])^(3/2), x, 5, (14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^3*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^5*d)} -{Cos[c + d*x]^5/(b*Cos[c + d*x])^(3/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d)} -{Cos[c + d*x]^4/(b*Cos[c + d*x])^(3/2), x, 4, (6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d)} -{Cos[c + d*x]^3/(b*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^2/(b*Cos[c + d*x])^(3/2), x, 3, (2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^1/(b*Cos[c + d*x])^(3/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(3/2), x, 3, -((2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1/(b*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2/(b*Cos[c + d*x])^(3/2), x, 5, -((6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]])) + (2*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^3/(b*Cos[c + d*x])^(3/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^4/(b*Cos[c + d*x])^(3/2), x, 6, -((14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]])) + (2*b^3*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*b*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*b*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^8/(b*Cos[c + d*x])^(5/2), x, 6, (30*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(77*b^2*d*Sqrt[b*Cos[c + d*x]]) + (30*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(77*b^3*d) + (18*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b^5*d) + (2*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^7*d)} -{Cos[c + d*x]^7/(b*Cos[c + d*x])^(5/2), x, 5, (14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^3*d*Sqrt[Cos[c + d*x]]) + (14*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^4*d) + (2*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^6*d)} -{Cos[c + d*x]^6/(b*Cos[c + d*x])^(5/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)} -{Cos[c + d*x]^5/(b*Cos[c + d*x])^(5/2), x, 4, (6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d)} -{Cos[c + d*x]^4/(b*Cos[c + d*x])^(5/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d)} -{Cos[c + d*x]^3/(b*Cos[c + d*x])^(5/2), x, 3, (2*Sqrt[b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^2/(b*Cos[c + d*x])^(5/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^1/(b*Cos[c + d*x])^(5/2), x, 4, -((2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(5/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(b*Cos[c + d*x])^(5/2), x, 5, -((6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]])) + (2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (6*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2/(b*Cos[c + d*x])^(5/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*b*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (10*Sin[c + d*x])/(21*b*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^3/(b*Cos[c + d*x])^(5/2), x, 6, -((14*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^3*d*Sqrt[Cos[c + d*x]])) + (2*b^2*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (14*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*b^2*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(7/2), x, 4, -((6*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]])) + (2*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (6*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^(m/2) (b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(1/2), x, 4, (3*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (3*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(1/2), x, 3, (x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(1/2)*(b*Cos[c + d*x])^(1/2), x, 2, (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(b*Cos[c + d*x])^(1/2), x, 2, (x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]]} -{Cos[c + d*x]^(-3/2)*(b*Cos[c + d*x])^(1/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(b*Cos[c + d*x])^(1/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-11/2)*(b*Cos[c + d*x])^(1/2), x, 4, (3*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (3*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2), x, 4, (3*b*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (3*b*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2), x, 3, (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(b*Cos[c + d*x])^(3/2), x, 3, (b*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-1/2)*(b*Cos[c + d*x])^(3/2), x, 2, (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(b*Cos[c + d*x])^(3/2), x, 2, (b*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]]} -{Cos[c + d*x]^(-5/2)*(b*Cos[c + d*x])^(3/2), x, 2, (b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(b*Cos[c + d*x])^(3/2), x, 3, (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(b*Cos[c + d*x])^(3/2), x, 3, (b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-11/2)*(b*Cos[c + d*x])^(3/2), x, 3, (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-13/2)*(b*Cos[c + d*x])^(3/2), x, 4, (3*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (3*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2), x, 4, (3*b^2*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (3*b^2*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(1/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(b*Cos[c + d*x])^(5/2), x, 2, (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(b*Cos[c + d*x])^(5/2), x, 2, (b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]]} -{Cos[c + d*x]^(-7/2)*(b*Cos[c + d*x])^(5/2), x, 2, (b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-13/2)*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-15/2)*(b*Cos[c + d*x])^(5/2), x, 4, (3*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (3*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(11/2)/(b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(9/2)/(b*Cos[c + d*x])^(1/2), x, 4, (3*x*Sqrt[Cos[c + d*x]])/(8*Sqrt[b*Cos[c + d*x]]) + (3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)/(b*Cos[c + d*x])^(1/2), x, 3, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)/(b*Cos[c + d*x])^(1/2), x, 3, (x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(b*Cos[c + d*x])^(1/2), x, 2, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(b*Cos[c + d*x])^(1/2), x, 2, (x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]]} -{Cos[c + d*x]^(-1/2)/(b*Cos[c + d*x])^(1/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)/(b*Cos[c + d*x])^(1/2), x, 3, Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(b*Cos[c + d*x])^(1/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)/(b*Cos[c + d*x])^(1/2), x, 3, Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]^3/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)/(b*Cos[c + d*x])^(1/2), x, 4, (3*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(4*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + (3*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^(11/2)/(b*Cos[c + d*x])^(3/2), x, 4, (3*x*Sqrt[Cos[c + d*x]])/(8*b*Sqrt[b*Cos[c + d*x]]) + (3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(9/2)/(b*Cos[c + d*x])^(3/2), x, 3, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)/(b*Cos[c + d*x])^(3/2), x, 3, (x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)/(b*Cos[c + d*x])^(3/2), x, 2, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(b*Cos[c + d*x])^(3/2), x, 2, (x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(b*Cos[c + d*x])^(3/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)/(b*Cos[c + d*x])^(3/2), x, 3, Sin[c + d*x]/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)/(b*Cos[c + d*x])^(3/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(b*Cos[c + d*x])^(3/2), x, 3, Sin[c + d*x]/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]^3/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)/(b*Cos[c + d*x])^(3/2), x, 4, (3*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(4*b*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + (3*Sin[c + d*x])/(8*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^(13/2)/(b*Cos[c + d*x])^(5/2), x, 4, (3*x*Sqrt[Cos[c + d*x]])/(8*b^2*Sqrt[b*Cos[c + d*x]]) + (3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(11/2)/(b*Cos[c + d*x])^(5/2), x, 3, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(9/2)/(b*Cos[c + d*x])^(5/2), x, 3, (x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)/(b*Cos[c + d*x])^(5/2), x, 2, (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)/(b*Cos[c + d*x])^(5/2), x, 2, (x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(b*Cos[c + d*x])^(5/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(b*Cos[c + d*x])^(5/2), x, 3, Sin[c + d*x]/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)/(b*Cos[c + d*x])^(5/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)/(b*Cos[c + d*x])^(5/2), x, 3, Sin[c + d*x]/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]^3/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(b*Cos[c + d*x])^(5/2), x, 4, (3*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(4*b^2*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + (3*Sin[c + d*x])/(8*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3), x, 2, -((3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]))} - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3), x, 2, -((3*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(1/3), x, 2, -((3*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(1/3), x, 1, -((3*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(1/3)*Sec[c + d*x]^1, x, 2, -((3*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(1/3)*Sec[c + d*x]^2, x, 2, (3*b*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*Sec[c + d*x]^3, x, 2, (3*b^2*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3), x, 2, -((3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]))} - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(2/3), x, 2, -((3*(b*Cos[c + d*x])^(11/3)*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2]*Sin[c + d*x])/(11*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(2/3), x, 2, -((3*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(2/3), x, 1, -((3*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(2/3)*Sec[c + d*x]^1, x, 2, -((3*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(2/3)*Sec[c + d*x]^2, x, 2, (3*b*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*Sec[c + d*x]^3, x, 2, (3*b^2*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3), x, 2, -((3*b*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]))} - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3), x, 2, -((3*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(4/3), x, 2, -((3*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(4/3), x, 1, -((3*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(4/3)*Sec[c + d*x]^1, x, 2, -((3*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(4/3)*Sec[c + d*x]^2, x, 2, -((3*b*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]))} -{(b*Cos[c + d*x])^(4/3)*Sec[c + d*x]^3, x, 2, (3*b^2*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^m/(b*Cos[c + d*x])^(1/3), x, 2, -((3*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]))} - -{Cos[c + d*x]^2/(b*Cos[c + d*x])^(1/3), x, 2, -((3*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1/(b*Cos[c + d*x])^(1/3), x, 2, -((3*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(1/3), x, 1, -((3*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b*d*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^1/(b*Cos[c + d*x])^(1/3), x, 2, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^2/(b*Cos[c + d*x])^(1/3), x, 2, (3*b*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^3/(b*Cos[c + d*x])^(1/3), x, 2, (3*b^2*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m/(b*Cos[c + d*x])^(2/3), x, 2, -((3*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(1 + 3*m), (1/6)*(7 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]))} - -{Cos[c + d*x]^2/(b*Cos[c + d*x])^(2/3), x, 2, -((3*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1/(b*Cos[c + d*x])^(2/3), x, 2, -((3*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(2/3), x, 1, -((3*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^1/(b*Cos[c + d*x])^(2/3), x, 2, (3*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^2/(b*Cos[c + d*x])^(2/3), x, 2, (3*b*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^3/(b*Cos[c + d*x])^(2/3), x, 2, (3*b^2*Hypergeometric2F1[-(4/3), 1/2, -(1/3), Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Cos[c + d*x])^(8/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m/(b*Cos[c + d*x])^(4/3), x, 2, (3*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (1/6)*(-1 + 3*m), (1/6)*(5 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - -{Cos[c + d*x]^2/(b*Cos[c + d*x])^(4/3), x, 2, -((3*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1/(b*Cos[c + d*x])^(4/3), x, 2, -((3*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0/(b*Cos[c + d*x])^(4/3), x, 1, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^1/(b*Cos[c + d*x])^(4/3), x, 2, (3*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^2/(b*Cos[c + d*x])^(4/3), x, 2, (3*b*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^3/(b*Cos[c + d*x])^(4/3), x, 2, (3*b^2*Hypergeometric2F1[-(5/3), 1/2, -(2/3), Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Cos[c + d*x])^(10/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n with n symbolic*) - - -{(a*Cos[e + f*x])^m*(b*Cos[e + f*x])^n, x, 2, -(((a*Cos[e + f*x])^(1 + m)*(b*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(1 + m + n)*Sqrt[Sin[e + f*x]^2]))} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^n, x, 2, -(((b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^n, x, 2, -(((b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^n, x, 1, -(((b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^1*(b*Cos[c + d*x])^n, x, 2, -(((b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^2*(b*Cos[c + d*x])^n, x, 2, (b*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^3*(b*Cos[c + d*x])^n, x, 2, (b^2*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^4*(b*Cos[c + d*x])^n, x, 2, (b^3*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (1/2)*(-3 + n), (1/2)*(-1 + n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n, x, 2, -((2*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n, x, 2, -((2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^(1/2)*(b*Cos[c + d*x])^n, x, 2, -((2*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^(-1/2)*(b*Cos[c + d*x])^n, x, 2, -((2*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^(-3/2)*(b*Cos[c + d*x])^n, x, 2, (2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^(-5/2)*(b*Cos[c + d*x])^n, x, 2, (2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^(-7/2)*(b*Cos[c + d*x])^n, x, 2, (2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^(-9/2)*(b*Cos[c + d*x])^n, x, 2, (2*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-7 + 2*n), (1/4)*(-3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 - 2*n)*Cos[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Sec[e+f x])^n*) - - -{(a*Cos[e + f*x])^m*(b*Sec[e + f*x])^n, x, 2, -(((a*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1/2)*(1 + m - n), (1/2)*(3 + m - n), Cos[e + f*x]^2]*(b*Sec[e + f*x])^n*Sin[e + f*x])/(a*f*(1 + m - n)*Sqrt[Sin[e + f*x]^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Csc[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (b Csc[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]^1*Csc[a + b*x]^(1/2), x, 2, 2/(b*Sqrt[Csc[a + b*x]])} -{Cos[a + b*x]^1/Csc[a + b*x]^(1/2), x, 2, 2/(3*b*Csc[a + b*x]^(3/2))} - - -{Cos[a + b*x]^2*Csc[a + b*x]^(1/2), x, 3, (2*Cos[a + b*x])/(3*b*Sqrt[Csc[a + b*x]]) + (4*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b)} -{Cos[a + b*x]^2/Csc[a + b*x]^(1/2), x, 3, (2*Cos[a + b*x])/(5*b*Csc[a + b*x]^(3/2)) + (4*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(5*b)} - - -{Cos[x]^3*Csc[x]^(9/2), x, 3, (2/3)*Csc[x]^(3/2) - (2/7)*Csc[x]^(7/2)} -{Cos[a + b*x]^3*Csc[a + b*x]^(1/2), x, 3, -(2/(5*b*Csc[a + b*x]^(5/2))) + 2/(b*Sqrt[Csc[a + b*x]])} -{Cos[a + b*x]^3/Csc[a + b*x]^(1/2), x, 3, -(2/(7*b*Csc[a + b*x]^(7/2))) + 2/(3*b*Csc[a + b*x]^(3/2))} - - -{Cos[a + b*x]^4*Csc[a + b*x]^(1/2), x, 4, (4*Cos[a + b*x])/(7*b*Sqrt[Csc[a + b*x]]) + (2*Cos[a + b*x]^3)/(7*b*Sqrt[Csc[a + b*x]]) + (8*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(7*b)} -{Cos[a + b*x]^4/Csc[a + b*x]^(1/2), x, 4, (4*Cos[a + b*x])/(15*b*Csc[a + b*x]^(3/2)) + (2*Cos[a + b*x]^3)/(9*b*Csc[a + b*x]^(3/2)) + (8*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(15*b)} - - -{Cos[x]*Csc[x]^(7/3), x, 2, (-3*Csc[x]^(4/3))/4} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[a + b*x]^1*Csc[a + b*x]^(1/2), x, 5, -(ArcTan[Sqrt[Csc[a + b*x]]]/b) + ArcTanh[Sqrt[Csc[a + b*x]]]/b} -{Sec[a + b*x]^1/Csc[a + b*x]^(1/2), x, 5, ArcTan[Sqrt[Csc[a + b*x]]]/b + ArcTanh[Sqrt[Csc[a + b*x]]]/b} - - -{Sec[a + b*x]^2*Csc[a + b*x]^(1/2), x, 3, Sec[a + b*x]/(b*Sqrt[Csc[a + b*x]]) + (Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} -{Sec[a + b*x]^2/Csc[a + b*x]^(1/2), x, 3, Sec[a + b*x]/(b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} - - -{Sec[a + b*x]^3*Csc[a + b*x]^(1/2), x, 6, -((3*ArcTan[Sqrt[Csc[a + b*x]]])/(4*b)) + (3*ArcTanh[Sqrt[Csc[a + b*x]]])/(4*b) + Sec[a + b*x]^2/(2*b*Sqrt[Csc[a + b*x]])} -{Sec[a + b*x]^3/Csc[a + b*x]^(1/2), x, 6, ArcTan[Sqrt[Csc[a + b*x]]]/(4*b) + ArcTanh[Sqrt[Csc[a + b*x]]]/(4*b) + Sec[a + b*x]^2/(2*b*Csc[a + b*x]^(3/2))} - - -{Sec[a + b*x]^4*Csc[a + b*x]^(1/2), x, 4, (5*Sec[a + b*x])/(6*b*Sqrt[Csc[a + b*x]]) + Sec[a + b*x]^3/(3*b*Sqrt[Csc[a + b*x]]) + (5*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(6*b)} -{Sec[a + b*x]^4/Csc[a + b*x]^(1/2), x, 4, Sec[a + b*x]/(2*b*Csc[a + b*x]^(3/2)) + Sec[a + b*x]^3/(3*b*Csc[a + b*x]^(3/2)) - (Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(2*b)} - - -(* ::Subsection:: *) -(*Integrands of the form (a Cos[e+f x])^(m/2) (b Csc[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Csc[e+f x])^n with n symbolic*) - - -{Csc[a + b*x]^p*(d*Cos[a + b*x])^(3/2), x, 2, (d*Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[-(1/4), (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*(1 - p)*(Cos[a + b*x]^2)^(1/4))} -{Csc[a + b*x]^p*(d*Cos[a + b*x])^(1/2), x, 2, (d*(Cos[a + b*x]^2)^(1/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[1/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*(1 - p)*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^p/(d*Cos[a + b*x])^(1/2), x, 2, (d*(Cos[a + b*x]^2)^(3/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[3/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*(1 - p)*(d*Cos[a + b*x])^(3/2))} -{Csc[a + b*x]^p/(d*Cos[a + b*x])^(3/2), x, 2, ((Cos[a + b*x]^2)^(1/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[5/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*d*(1 - p)*Sqrt[d*Cos[a + b*x]])} -{Csc[a + b*x]^p/(d*Cos[a + b*x])^(5/2), x, 2, ((Cos[a + b*x]^2)^(3/4)*Csc[a + b*x]^(-1 + p)*Hypergeometric2F1[7/4, (1 - p)/2, (3 - p)/2, Sin[a + b*x]^2])/(b*d*(1 - p)*(d*Cos[a + b*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Csc[e+f x])^n with m symbolic*) - - -{(Cos[e + f*x])^m*(Csc[e + f*x])^n, x, 2, (Cos[e + f*x]^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Csc[e + f*x]^(-1 + n)*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n))} -{(a*Cos[e + f*x])^m*(Csc[e + f*x])^n, x, 2, (a*(a*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Csc[e + f*x]^(-1 + n)*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n))} -{(Cos[e + f*x])^m*(b*Csc[e + f*x])^n, x, 2, (b*Cos[e + f*x]^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n))} -{(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^n, x, 2, (a*b*(a*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n))} - - -{(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(7/2), x, 2, -((b^3*(a*Cos[e + f*x])^(1 + m)*Sqrt[b*Csc[e + f*x]]*Hypergeometric2F1[9/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(1/4))/(a*f*(1 + m))), -((b*(a*Cos[e + f*x])^(1 + m)*(b*Csc[e + f*x])^(5/2)*Hypergeometric2F1[9/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(5/4))/(a*f*(1 + m)))} -{(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(5/2), x, 2, -((b*(a*Cos[e + f*x])^(1 + m)*(b*Csc[e + f*x])^(3/2)*Hypergeometric2F1[7/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(3/4))/(a*f*(1 + m)))} -{(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(3/2), x, 2, -((b*(a*Cos[e + f*x])^(1 + m)*Sqrt[b*Csc[e + f*x]]*Hypergeometric2F1[5/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(1/4))/(a*f*(1 + m)))} -{(a*Cos[e + f*x])^m*(b*Csc[e + f*x])^(1/2), x, 2, -(((a*Cos[e + f*x])^(1 + m)*(b*Csc[e + f*x])^(3/2)*Hypergeometric2F1[3/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(3/4))/(a*b*f*(1 + m)))} -{(a*Cos[e + f*x])^m/(b*Csc[e + f*x])^(1/2), x, 2, -(((a*Cos[e + f*x])^(1 + m)*Sqrt[b*Csc[e + f*x]]*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(1/4))/(a*b*f*(1 + m)))} -{(a*Cos[e + f*x])^m/(b*Csc[e + f*x])^(3/2), x, 2, -(((a*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[-(1/4), (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*b*f*(1 + m)*Sqrt[b*Csc[e + f*x]]*(Sin[e + f*x]^2)^(1/4)))} -{(a*Cos[e + f*x])^m/(b*Csc[e + f*x])^(5/2), x, 2, -(((a*Cos[e + f*x])^(1 + m)*Sqrt[b*Csc[e + f*x]]*Hypergeometric2F1[-(3/4), (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^(1/4))/(a*b^3*f*(1 + m))), -(((a*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[-(3/4), (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*b*f*(1 + m)*(b*Csc[e + f*x])^(3/2)*(Sin[e + f*x]^2)^(3/4)))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.1 (a+b cos)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.1 (a+b cos)^n.m deleted file mode 100644 index 6841db4..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.1 (a+b cos)^n.m +++ /dev/null @@ -1,139 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Cos[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Cos[c+d x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Cos[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Cos[c+d x])^(n/2)*) - - -{(a + a*Cos[c + d*x])^(7/2), x, 4, (256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2), x, 3, (64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2), x, 2, (8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{1/(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{1/(a + a*Cos[c + d*x])^(3/2), x, 3, ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{1/(a + a*Cos[c + d*x])^(5/2), x, 4, (3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Cos[c+d x])^(n/3)*) - - -{(a + a*Cos[c + d*x])^(4/3), x, 2, (2*2^(5/6)*a*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[-(5/6), 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(5/6))} -{(a + a*Cos[c + d*x])^(2/3), x, 2, (2*2^(1/6)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))} -{(a + a*Cos[c + d*x])^(1/3), x, 2, (2^(5/6)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(5/6))} -{1/(a + a*Cos[c + d*x])^(1/3), x, 2, (2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))} -{1/(a + a*Cos[c + d*x])^(2/3), x, 2, ((1 + Cos[c + d*x])^(1/6)*Hypergeometric2F1[1/2, 7/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2^(1/6)*d*(a + a*Cos[c + d*x])^(2/3))} -{1/(a + a*Cos[c + d*x])^(4/3), x, 2, (Hypergeometric2F1[1/2, 11/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2^(5/6)*a*d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Cos[c+d x])^n with n symbolic*) - - -{(a + a*Cos[c + d*x])^n, x, 2, (2^(1/2 + n)*(1 + Cos[c + d*x])^(-(1/2) - n)*(a + a*Cos[c + d*x])^n*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/d} -{(a - a*Cos[c + d*x])^n, x, 2, -((2^(1/2 + n)*(1 - Cos[c + d*x])^(-(1/2) - n)*(a - a*Cos[c + d*x])^n*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 + Cos[c + d*x])]*Sin[c + d*x])/d)} - - -{(2 + 2*Cos[c + d*x])^n, x, 1, (2^(1/2 + 2*n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])} -{(2 - 2*Cos[c + d*x])^n, x, 1, -((2^(1/2 + 2*n)*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 + Cos[c + d*x])]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^n*) - - -{1/(5 + 3*Cos[c + d*x]), x, 1, x/4 - ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])]/(2*d)} -{1/(5 + 3*Cos[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(32*d) - (3*Sin[c + d*x])/(16*d*(5 + 3*Cos[c + d*x]))} -{1/(5 + 3*Cos[c + d*x])^3, x, 4, (59*x)/2048 - (59*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1024*d) - (3*Sin[c + d*x])/(32*d*(5 + 3*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x]))} -{1/(5 + 3*Cos[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(16384*d) - Sin[c + d*x]/(16*d*(5 + 3*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x])^2) - (311*Sin[c + d*x])/(8192*d*(5 + 3*Cos[c + d*x]))} - - -{1/(5 - 3*Cos[c + d*x]), x, 1, x/4 + ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])]/(2*d)} -{1/(5 - 3*Cos[c + d*x])^2, x, 3, (5*x)/64 + (5*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(32*d) + (3*Sin[c + d*x])/(16*d*(5 - 3*Cos[c + d*x]))} -{1/(5 - 3*Cos[c + d*x])^3, x, 4, (59*x)/2048 + (59*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(1024*d) + (3*Sin[c + d*x])/(32*d*(5 - 3*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(5 - 3*Cos[c + d*x]))} -{1/(5 - 3*Cos[c + d*x])^4, x, 5, (385*x)/32768 + (385*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(16384*d) + Sin[c + d*x]/(16*d*(5 - 3*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(5 - 3*Cos[c + d*x])^2) + (311*Sin[c + d*x])/(8192*d*(5 - 3*Cos[c + d*x]))} - - -{1/(-5 + 3*Cos[c + d*x]), x, 1, -(x/4) - ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])]/(2*d)} -{1/(-5 + 3*Cos[c + d*x])^2, x, 3, (5*x)/64 + (5*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(32*d) + (3*Sin[c + d*x])/(16*d*(5 - 3*Cos[c + d*x]))} -{1/(-5 + 3*Cos[c + d*x])^3, x, 4, -((59*x)/2048) - (59*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(1024*d) - (3*Sin[c + d*x])/(32*d*(5 - 3*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(5 - 3*Cos[c + d*x]))} -{1/(-5 + 3*Cos[c + d*x])^4, x, 5, (385*x)/32768 + (385*ArcTan[Sin[c + d*x]/(3 - Cos[c + d*x])])/(16384*d) + Sin[c + d*x]/(16*d*(5 - 3*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(5 - 3*Cos[c + d*x])^2) + (311*Sin[c + d*x])/(8192*d*(5 - 3*Cos[c + d*x]))} - - -{1/(-5 - 3*Cos[c + d*x]), x, 1, -(x/4) + ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])]/(2*d)} -{1/(-5 - 3*Cos[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(32*d) - (3*Sin[c + d*x])/(16*d*(5 + 3*Cos[c + d*x]))} -{1/(-5 - 3*Cos[c + d*x])^3, x, 4, -((59*x)/2048) + (59*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1024*d) + (3*Sin[c + d*x])/(32*d*(5 + 3*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x]))} -{1/(-5 - 3*Cos[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(16384*d) - Sin[c + d*x]/(16*d*(5 + 3*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(5 + 3*Cos[c + d*x])^2) - (311*Sin[c + d*x])/(8192*d*(5 + 3*Cos[c + d*x]))} - - -{1/(3 + 5*Cos[c + d*x]), x, 2, -(Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]]/(4*d)) + Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(3 + 5*Cos[c + d*x])^2, x, 4, (3*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(64*d) + (5*Sin[c + d*x])/(16*d*(3 + 5*Cos[c + d*x]))} -{1/(3 + 5*Cos[c + d*x])^3, x, 5, -((43*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(2048*d)) + (43*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(2048*d) + (5*Sin[c + d*x])/(32*d*(3 + 5*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x]))} -{1/(3 + 5*Cos[c + d*x])^4, x, 6, (279*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(32768*d) + (5*Sin[c + d*x])/(48*d*(3 + 5*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x])^2) + (995*Sin[c + d*x])/(24576*d*(3 + 5*Cos[c + d*x]))} - - -{1/(3 - 5*Cos[c + d*x]), x, 2, Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]]/(4*d) - Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(3 - 5*Cos[c + d*x])^2, x, 4, -((3*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(64*d)) + (3*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(64*d) - (5*Sin[c + d*x])/(16*d*(3 - 5*Cos[c + d*x]))} -{1/(3 - 5*Cos[c + d*x])^3, x, 5, (43*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(2048*d) - (43*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(2048*d) - (5*Sin[c + d*x])/(32*d*(3 - 5*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x]))} -{1/(3 - 5*Cos[c + d*x])^4, x, 6, -((279*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(32768*d)) + (279*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(32768*d) - (5*Sin[c + d*x])/(48*d*(3 - 5*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x])^2) - (995*Sin[c + d*x])/(24576*d*(3 - 5*Cos[c + d*x]))} - - -{1/(-3 + 5*Cos[c + d*x]), x, 2, -(Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]]/(4*d)) + Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(-3 + 5*Cos[c + d*x])^2, x, 4, -((3*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(64*d)) + (3*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(64*d) - (5*Sin[c + d*x])/(16*d*(3 - 5*Cos[c + d*x]))} -{1/(-3 + 5*Cos[c + d*x])^3, x, 5, -((43*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(2048*d)) + (43*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(2048*d) + (5*Sin[c + d*x])/(32*d*(3 - 5*Cos[c + d*x])^2) - (45*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x]))} -{1/(-3 + 5*Cos[c + d*x])^4, x, 6, -((279*Log[Cos[(1/2)*(c + d*x)] - 2*Sin[(1/2)*(c + d*x)]])/(32768*d)) + (279*Log[Cos[(1/2)*(c + d*x)] + 2*Sin[(1/2)*(c + d*x)]])/(32768*d) - (5*Sin[c + d*x])/(48*d*(3 - 5*Cos[c + d*x])^3) + (25*Sin[c + d*x])/(512*d*(3 - 5*Cos[c + d*x])^2) - (995*Sin[c + d*x])/(24576*d*(3 - 5*Cos[c + d*x]))} - - -{1/(-3 - 5*Cos[c + d*x]), x, 2, Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]]/(4*d) - Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]]/(4*d)} -{1/(-3 - 5*Cos[c + d*x])^2, x, 4, (3*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(64*d) - (3*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(64*d) + (5*Sin[c + d*x])/(16*d*(3 + 5*Cos[c + d*x]))} -{1/(-3 - 5*Cos[c + d*x])^3, x, 5, (43*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(2048*d) - (43*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(2048*d) - (5*Sin[c + d*x])/(32*d*(3 + 5*Cos[c + d*x])^2) + (45*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x]))} -{1/(-3 - 5*Cos[c + d*x])^4, x, 6, (279*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(32768*d) - (279*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(32768*d) + (5*Sin[c + d*x])/(48*d*(3 + 5*Cos[c + d*x])^3) - (25*Sin[c + d*x])/(512*d*(3 + 5*Cos[c + d*x])^2) + (995*Sin[c + d*x])/(24576*d*(3 + 5*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^(n/2)*) - - -{(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (16*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2), x, 6, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])} -{1/(a + b*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{1/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{1/(a + b*Cos[c + d*x])^(5/2), x, 7, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*b*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^(n/3)*) - - -{(a + b*Cos[c + d*x])^(4/3), x, 3, (Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} -{(a + b*Cos[c + d*x])^(2/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))} -{(a + b*Cos[c + d*x])^(1/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} -{1/(a + b*Cos[c + d*x])^(1/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} -{1/(a + b*Cos[c + d*x])^(2/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))} -{1/(a + b*Cos[c + d*x])^(4/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 4/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/((a + b)*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} - - -(* {(a + b*Cos[c + d*x])^(4/3) - (4*a^2 + b^2 + 5*a*b*Cos[c + d*x])/(4*(a + b*Cos[c + d*x])^(2/3)), x, -11, (3*b*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d)} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^n with n symbolic*) - - -{(a + b*Cos[c + d*x])^n, x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^n*Sin[c + d*x])/(((a + b*Cos[c + d*x])/(a + b))^n*(d*Sqrt[1 + Cos[c + d*x]]))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.2 (g sin)^p (a+b cos)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.2 (g sin)^p (a+b cos)^m.m deleted file mode 100644 index 4fe6817..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.2 (g sin)^p (a+b cos)^m.m +++ /dev/null @@ -1,182 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m when a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Cos[e+f x])^m*) - - -{Sin[x]^4/(a + a*Cos[x]), x, 3, x/(2*a) - (Cos[x]*Sin[x])/(2*a) - Sin[x]^3/(3*a)} -{Sin[x]^3/(a + a*Cos[x]), x, 2, -(Cos[x]/a) + Cos[x]^2/(2*a)} -{Sin[x]^2/(a + a*Cos[x]), x, 2, x/a - Sin[x]/a} -{Sin[x]^1/(a + a*Cos[x]), x, 2, -(Log[1 + Cos[x]]/a)} -{Sin[x]^0/(a + a*Cos[x]), x, 1, Sin[x]/(a + a*Cos[x])} -{Csc[x]^1/(a + a*Cos[x]), x, 4, -(ArcTanh[Cos[x]]/(2*a)) + 1/(2*(a + a*Cos[x]))} -{Csc[x]^2/(a + a*Cos[x]), x, 3, -((2*Cot[x])/(3*a)) + Csc[x]/(3*(a + a*Cos[x]))} -{Csc[x]^3/(a + a*Cos[x]), x, 4, -((3*ArcTanh[Cos[x]])/(8*a)) - 1/(8*(a - a*Cos[x])) + a/(8*(a + a*Cos[x])^2) + 1/(4*(a + a*Cos[x]))} -{Csc[x]^4/(a + a*Cos[x]), x, 3, -((4*Cot[x])/(5*a)) - (4*Cot[x]^3)/(15*a) + Csc[x]^3/(5*(a + a*Cos[x]))} - - -{Sin[2*x]/(1 + Cos[2*x]), x, 2, -Log[Cos[x]], (-(1/2))*Log[1 + Cos[2*x]]} -{Sin[2*x]/(1 - Cos[2*x]), x, 2, Log[Sin[x]], (1/2)*Log[1 - Cos[2*x]]} - - -{Sin[x]/(1 + Cos[x])^2, x, 2, 1/(1 + Cos[x])} -{Sin[x]/(1 - Cos[x])^2, x, 2, -(1/(1 - Cos[x]))} -{Sin[x]^2/(1 + Cos[x])^2, x, 2, -x + (2*Sin[x])/(1 + Cos[x])} -{Sin[x]^2/(1 - Cos[x])^2, x, 2, -x - (2*Sin[x])/(1 - Cos[x])} -{Sin[x]^3/(1 + Cos[x])^2, x, 3, Cos[x] - 2*Log[1 + Cos[x]]} -{Sin[x]^3/(1 - Cos[x])^2, x, 3, Cos[x] + 2*Log[1 - Cos[x]]} - - -{Sin[x]/(1 + Cos[x])^3, x, 2, 1/(2*(1 + Cos[x])^2)} -{Sin[x]/(1 - Cos[x])^3, x, 2, -(1/(2*(1 - Cos[x])^2))} -{Sin[x]^2/(1 + Cos[x])^3, x, 1, Sin[x]^3/(3*(1 + Cos[x])^3)} -{Sin[x]^2/(1 - Cos[x])^3, x, 1, -(Sin[x]^3/(3*(1 - Cos[x])^3))} -{Sin[x]^3/(1 + Cos[x])^3, x, 3, 2/(1 + Cos[x]) + Log[1 + Cos[x]]} -{Sin[x]^3/(1 - Cos[x])^3, x, 3, -(2/(1 - Cos[x])) - Log[1 - Cos[x]]} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^p (a+a Cos[e+f x])^(m/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+a Cos[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+a Cos[e+f x])^(m/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^p (a+b Cos[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^4/(a + b*Cos[x]), x, 5, -((a*(2*a^2 - 3*b^2)*x)/(2*b^4)) + (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/b^4 + ((2*(a^2 - b^2) - a*b*Cos[x])*Sin[x])/(2*b^3) - Sin[x]^3/(3*b)} -{Sin[x]^3/(a + b*Cos[x]), x, 3, -((a*Cos[x])/b^2) + Cos[x]^2/(2*b) + ((a^2 - b^2)*Log[a + b*Cos[x]])/b^3} -{Sin[x]^2/(a + b*Cos[x]), x, 4, (a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/b^2 - Sin[x]/b} -{Sin[x]^1/(a + b*Cos[x]), x, 2, -(Log[a + b*Cos[x]]/b)} -{Sin[x]^0/(a + b*Cos[x]), x, 2, (2*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b])} -{Csc[x]^1/(a + b*Cos[x]), x, 6, Log[1 - Cos[x]]/(2*(a + b)) - Log[1 + Cos[x]]/(2*(a - b)) + (b*Log[a + b*Cos[x]])/(a^2 - b^2)} -{Csc[x]^2/(a + b*Cos[x]), x, 4, -((2*b^2*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2))) + ((b - a*Cos[x])*Csc[x])/(a^2 - b^2)} -{Csc[x]^3/(a + b*Cos[x]), x, 4, ((b - a*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2)) + ((a + 2*b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((a - 2*b)*Log[1 + Cos[x]])/(4*(a - b)^2) - (b^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2} -{Csc[x]^4/(a + b*Cos[x]), x, 5, (2*b^4*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) - ((3*b^3 + a*(2*a^2 - 5*b^2)*Cos[x])*Csc[x])/(3*(a^2 - b^2)^2) + ((b - a*Cos[x])*Csc[x]^3)/(3*(a^2 - b^2))} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^p (a+b Cos[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+b Cos[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2), x, 5, (10*a*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (10*a*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*d) + (2*b*(e*Sin[c + d*x])^(9/2))/(9*d*e)} -{(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2), x, 4, (6*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d) + (2*b*(e*Sin[c + d*x])^(7/2))/(7*d*e)} -{(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2), x, 4, (2*a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*a*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d) + (2*b*(e*Sin[c + d*x])^(5/2))/(5*d*e)} -{(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(1/2), x, 3, (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (2*b*(e*Sin[c + d*x])^(3/2))/(3*d*e)} -{(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(1/2), x, 3, (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*Sqrt[e*Sin[c + d*x]])/(d*e)} -{(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(3/2), x, 4, (-2*b)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]])} -{(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(5/2), x, 4, (-2*b)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]])} -{(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(7/2), x, 5, (-2*b)/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x])/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (6*a*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (6*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]])} - - -{(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2), x, 6, (10*(11*a^2 + 2*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) - (10*(11*a^2 + 2*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*(11*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(77*d) + (26*a*b*(e*Sin[c + d*x])^(9/2))/(99*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(11*d*e)} -{(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2), x, 5, (2*(9*a^2 + 2*b^2)*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(15*d*Sqrt[Sin[c + d*x]]) - (2*(9*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(45*d) + (22*a*b*(e*Sin[c + d*x])^(7/2))/(63*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(9*d*e)} -{(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2), x, 5, (2*(7*a^2 + 2*b^2)*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (2*(7*a^2 + 2*b^2)*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) + (18*a*b*(e*Sin[c + d*x])^(5/2))/(35*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(7*d*e)} -{(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(1/2), x, 4, (2*(5*a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) + (14*a*b*(e*Sin[c + d*x])^(3/2))/(15*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(5*d*e)} -{(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(1/2), x, 4, (2*(3*a^2 + 2*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) + (10*a*b*Sqrt[e*Sin[c + d*x]])/(3*d*e) + (2*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*d*e)} -{(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(3/2), x, 4, (-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*(a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) - (2*a*b*(e*Sin[c + d*x])^(3/2))/(d*e^3)} -{(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(5/2), x, 4, (-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*(a^2 - 2*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (2*a*b*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)} -{(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(7/2), x, 5, (-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (2*a*b)/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*(3*a^2 - 2*b^2)*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*(3*a^2 - 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]])} - - -{(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2), x, 7, (10*a*(11*a^2 + 6*b^2)*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) - (10*a*(11*a^2 + 6*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*a*(11*a^2 + 6*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(77*d) + (2*b*(177*a^2 + 44*b^2)*(e*Sin[c + d*x])^(9/2))/(1287*d*e) + (34*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(143*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(9/2))/(13*d*e)} -{(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2), x, 6, (2*a*(3*a^2 + 2*b^2)*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*(3*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(15*d) + (2*b*(43*a^2 + 12*b^2)*(e*Sin[c + d*x])^(7/2))/(231*d*e) + (10*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(33*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2))/(11*d*e)} -{(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2), x, 6, (2*a*(7*a^2 + 6*b^2)*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (2*a*(7*a^2 + 6*b^2)*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) + (2*b*(89*a^2 + 28*b^2)*(e*Sin[c + d*x])^(5/2))/(315*d*e) + (26*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(63*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2))/(9*d*e)} -{(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(1/2), x, 5, (2*a*(5*a^2 + 6*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) + (2*b*(57*a^2 + 20*b^2)*(e*Sin[c + d*x])^(3/2))/(105*d*e) + (22*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(35*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2))/(7*d*e)} -{(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(1/2), x, 5, (2*a*(a^2 + 2*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*(11*a^2 + 4*b^2)*Sqrt[e*Sin[c + d*x]])/(5*d*e) + (6*a*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(5*d*e) + (2*b*(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]])/(5*d*e)} -{(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(3/2), x, 5, -((2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(d*e*Sqrt[e*Sin[c + d*x]])) - (2*a*(a^2 + 6*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) - (2*b*(3*a^2 + 4*b^2)*(e*Sin[c + d*x])^(3/2))/(3*d*e^3) - (2*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(d*e^3)} -{(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(5/2), x, 5, -((2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(3*d*e*(e*Sin[c + d*x])^(3/2))) + (2*a*(a^2 - 6*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (2*b*(a^2 + 4*b^2)*Sqrt[e*Sin[c + d*x]])/(3*d*e^3) - (2*a*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)} -{(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(7/2), x, 5, -((2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(5*d*e*(e*Sin[c + d*x])^(5/2))) + (2*(a + b*Cos[c + d*x])*(a*b - (3*a^2 - 4*b^2)*Cos[c + d*x]))/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (6*a*(a^2 - 2*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]]) - (2*b*(3*a^2 - 4*b^2)*(e*Sin[c + d*x])^(3/2))/(5*d*e^5)} -{(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(9/2), x, 5, -((2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(7*d*e*(e*Sin[c + d*x])^(7/2))) - (2*(a + b*Cos[c + d*x])*(a*b + (5*a^2 - 4*b^2)*Cos[c + d*x]))/(21*d*e^3*(e*Sin[c + d*x])^(3/2)) + (2*a*(5*a^2 - 6*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*d*e^4*Sqrt[e*Sin[c + d*x]]) - (2*b*(5*a^2 - 4*b^2)*Sqrt[e*Sin[c + d*x]])/(21*d*e^5)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x]), x, 15, ((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + ((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + (2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*b^6*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (2*e^5*(21*(a^2 - b^2)^2 - a*b*(7*a^2 - 12*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(21*b^5*d) + (2*e^3*(7*(a^2 - b^2) - 5*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*b^3*d) - (2*e*(e*Sin[c + d*x])^(9/2))/(9*b*d)} -{(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x]), x, 14, -(((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d)) + ((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) + (a*(a^2 - b^2)^2*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*a*(5*a^2 - 8*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*b^4*d*Sqrt[Sin[c + d*x]]) + (2*e^3*(5*(a^2 - b^2) - 3*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*b^3*d) - (2*e*(e*Sin[c + d*x])^(7/2))/(7*b*d)} -{(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x]), x, 14, ((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) + ((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) - (2*a*(3*a^2 - 4*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*b^4*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(3*(a^2 - b^2) - a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*b^3*d) - (2*e*(e*Sin[c + d*x])^(5/2))/(5*b*d)} -{(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x]), x, 13, -(((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d)) + ((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) - (a*(a^2 - b^2)*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(b^2*d*Sqrt[Sin[c + d*x]]) - (2*e*(e*Sin[c + d*x])^(3/2))/(3*b*d)} -{(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x]), x, 13, ((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + ((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + (2*a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (2*e*Sqrt[e*Sin[c + d*x]])/(b*d)} -{(e*Sin[c + d*x])^(1/2)/(a + b*Cos[c + d*x]), x, 9, -((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d)) + (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) + (a*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(1/2)), x, 9, (Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (a*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)), x, 13, -((b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2))) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) + (2*(b - a*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - (a*b*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (a*b*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)), x, 13, (b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (2*(b - a*Cos[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) - (a*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (a*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2)), x, 14, -((b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2))) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) + (2*(b - a*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(5/2)) - (2*(5*b^3 + a*(3*a^2 - 8*b^2)*Cos[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Sin[c + d*x]]) + (a*b^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (a*b^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*a*(3*a^2 - 8*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Sin[c + d*x]])} - - -{(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^2, x, 15, (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) + (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(7*b^6*d*Sqrt[e*Sin[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*e^5*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(7*b^5*d) - (9*e^3*(7*a - 5*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*b^3*d) + (e*(e*Sin[c + d*x])^(9/2))/(b*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^2, x, 14, (-7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) + (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) - (7*a^2*(a^2 - b^2)*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*(5*a^2 - 3*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*b^4*d*Sqrt[Sin[c + d*x]]) - (7*e^3*(5*a - 3*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*b^3*d) + (e*(e*Sin[c + d*x])^(7/2))/(b*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^2, x, 14, (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*(3*a^2 - b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*b^4*d*Sqrt[e*Sin[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (5*e^3*(3*a - b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*b^3*d) + (e*(e*Sin[c + d*x])^(5/2))/(b*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^2, x, 13, (-3*a*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) + (3*a*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) + (3*a^2*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (3*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(b^2*d*Sqrt[Sin[c + d*x]]) + (e*(e*Sin[c + d*x])^(3/2))/(b*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^2, x, 13, (a*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) + (a*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (e*Sqrt[e*Sin[c + d*x]])/(b*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(1/2)/(a + b*Cos[c + d*x])^2, x, 13, (a*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) - (a*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (a^2*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) - (b*(e*Sin[c + d*x])^(3/2))/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x]))} -{1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(1/2)), x, 13, (-3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (b*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x]))} -{1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2)), x, 14, (5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (5*a*b - (2*a^2 + 3*b^2)*Cos[c + d*x])/((a^2 - b^2)^2*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*b*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*b*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - ((2*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^2*d*e^2*Sqrt[Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)), x, 14, (-7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) - (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*a*b - (2*a^2 + 5*b^2)*Cos[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + ((2*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2)), x, 15, (9*a*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)) + (9*a*b - (2*a^2 + 7*b^2)*Cos[c + d*x])/(5*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(5/2)) - (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Cos[c + d*x]))/(5*(a^2 - b^2)^3*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a^2*b^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a^2*b^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^3*d*e^4*Sqrt[Sin[c + d*x]])} - - -{(e*Sin[c + d*x])^(13/2)/(a + b*Cos[c + d*x])^3, x, 15, (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (11*a*(45*a^2 - 37*b^2)*e^6*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(20*b^6*d*Sqrt[Sin[c + d*x]]) - (11*e^5*(5*(9*a^2 - 2*b^2) - 27*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(60*b^5*d) + (11*e^3*(9*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(28*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(11/2))/(2*b*d*(a + b*Cos[c + d*x])^2)} -{(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3, x, 15, (-9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^6*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (3*e^5*(3*(7*a^2 - 2*b^2) - 7*a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(4*b^5*d) + (9*e^3*(7*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(20*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(9/2))/(2*b*d*(a + b*Cos[c + d*x])^2)} -{(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^3, x, 14, (-7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) + (7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) + (7*a*(5*a^2 - 2*b^2)*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (35*a*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*b^4*d*Sqrt[Sin[c + d*x]]) + (7*e^3*(5*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(12*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(7/2))/(2*b*d*(a + b*Cos[c + d*x])^2)} -{(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^3, x, 14, (5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) + (5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^4*d*Sqrt[e*Sin[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (5*e^3*(3*a + 2*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(4*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(5/2))/(2*b*d*(a + b*Cos[c + d*x])^2)} -{(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^3, x, 14, (-3*(a^2 - 2*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) + (3*(a^2 - 2*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) - (3*a*(a^2 - 2*b^2)*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*b^2*(a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) + (e*(e*Sin[c + d*x])^(3/2))/(2*b*d*(a + b*Cos[c + d*x])^2) - (3*a*e*(e*Sin[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^3, x, 14, -((a^2 + 2*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - ((a^2 + 2*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - (a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (e*Sqrt[e*Sin[c + d*x]])/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*e*Sqrt[e*Sin[c + d*x]])/(4*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(1/2)/(a + b*Cos[c + d*x])^3, x, 14, -((3*a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + ((3*a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + (a*(3*a^2 + 2*b^2)*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (5*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*Sqrt[Sin[c + d*x]]) - (b*(e*Sin[c + d*x])^(3/2))/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2) - (5*a*b*(e*Sin[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x]))} -{1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(1/2)), x, 14, (3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) + (3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (7*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*Sqrt[e*Sin[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (b*Sqrt[e*Sin[c + d*x]])/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2) - (7*a*b*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x]))} -{1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2)), x, 15, (-5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) + (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]]) - (9*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Cos[c + d*x])/(4*(a^2 - b^2)^3*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (a*(8*a^2 + 37*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^3*d*e^2*Sqrt[Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2)), x, 15, (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) + (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2)) - (11*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*b*(9*a^2 + 2*b^2) - a*(8*a^2 + 69*b^2)*Cos[c + d*x])/(12*(a^2 - b^2)^3*d*e*(e*Sin[c + d*x])^(3/2)) + (a*(8*a^2 + 69*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(12*(a^2 - b^2)^3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])} -{1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2)), x, 16, (-9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) + (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)) - (13*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)) + (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*Cos[c + d*x])/(20*(a^2 - b^2)^3*d*e*(e*Sin[c + d*x])^(5/2)) - (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Cos[c + d*x]))/(20*(a^2 - b^2)^4*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(20*(a^2 - b^2)^4*d*e^4*Sqrt[Sin[c + d*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (g Sin[e+f x])^(p/2) (a+b Cos[e+f x])^(m/2)*) - - -(* ::Title:: *) -(*Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m when a^2-b^2=0*) - - -(* ::Section:: *) -(*Integrands of the form (g Csc[e+f x])^p (a+b Cos[e+f x])^m*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.3 (g tan)^p (a+b cos)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.3 (g tan)^p (a+b cos)^m.m deleted file mode 100644 index a73c3cd..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.1.3 (g tan)^p (a+b cos)^m.m +++ /dev/null @@ -1,120 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Cos[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Cos[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+a Cos[e+f x])^m*) - - -{Tan[x]^4/(a + a*Cos[x]), x, 5, ArcTanh[Sin[x]]/(2*a) - (Sec[x]*Tan[x])/(2*a) + Tan[x]^3/(3*a)} -{Tan[x]^3/(a + a*Cos[x]), x, 5, -(Sec[x]/a) + Sec[x]^2/(2*a)} -{Tan[x]^2/(a + a*Cos[x]), x, 4, -(ArcTanh[Sin[x]]/a) + Tan[x]/a} -{Tan[x]^1/(a + a*Cos[x]), x, 4, -(Log[Cos[x]]/a) + Log[1 + Cos[x]]/a} -{Cot[x]^1/(a + a*Cos[x]), x, 5, -(ArcTanh[Cos[x]]/(2*a)) + (Cot[x]*Csc[x])/(2*a) - Csc[x]^2/(2*a)} -{Cot[x]^2/(a + a*Cos[x]), x, 5, -(Cot[x]^3/(3*a)) - Csc[x]/a + Csc[x]^3/(3*a)} -{Cot[x]^3/(a + a*Cos[x]), x, 6, (3*ArcTanh[Cos[x]])/(8*a) - Cot[x]^4/(4*a) - (3*Cot[x]*Csc[x])/(8*a) + (Cot[x]^3*Csc[x])/(4*a)} -{Cot[x]^4/(a + a*Cos[x]), x, 6, -(Cot[x]^5/(5*a)) + Csc[x]/a - (2*Csc[x]^3)/(3*a) + Csc[x]^5/(5*a)} - - -{Tan[3*x]/(1 + Cos[3*x])^2, x, 3, -(1/(3*(1 + Cos[3*x]))) - (1/3)*Log[Cos[3*x]] + (1/3)*Log[1 + Cos[3*x]]} - - -(* ::Subsection:: *) -(*Integrands of the form Tan[e+f x]^p (a+a Cos[e+f x])^(m/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Tan[e+f x])^(p/2) (a+a Cos[e+f x])^m*) - - -(* ::Subsection:: *) -(*Integrands of the form (g Tan[e+f x])^(p/2) (a+a Cos[e+f x])^(m/2)*) - - -(* ::Section:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Cos[e+f x])^m (A+B Cos[e+f x])*) - - -(* ::Section:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Title:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Cos[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Cos[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+b Cos[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[x]^4/(a + b*Cos[x]), x, 6, (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/a^4 + (b*(3*a^2 - 2*b^2)*ArcTanh[Sin[x]])/(2*a^4) - ((4*a^2 - 3*b^2)*Tan[x])/(3*a^3) - (b*Sec[x]*Tan[x])/(2*a^2) + (Sec[x]^2*Tan[x])/(3*a)} -{Tan[x]^3/(a + b*Cos[x]), x, 3, ((a^2 - b^2)*Log[Cos[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cos[x]])/a^3 - (b*Sec[x])/a^2 + Sec[x]^2/(2*a)} -{Tan[x]^2/(a + b*Cos[x]), x, 6, -((2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/a^2) - (b*ArcTanh[Sin[x]])/a^2 + Tan[x]/a} -{Tan[x]^1/(a + b*Cos[x]), x, 4, -(Log[Cos[x]]/a) + Log[a + b*Cos[x]]/a} -{Cot[x]^1/(a + b*Cos[x]), x, 3, Log[1 - Cos[x]]/(2*(a + b)) + Log[1 + Cos[x]]/(2*(a - b)) - (a*Log[a + b*Cos[x]])/(a^2 - b^2)} -{Cot[x]^2/(a + b*Cos[x]), x, 7, -((2*a^2*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2))) - (a*Cot[x])/(a^2 - b^2) + (b*Csc[x])/(a^2 - b^2)} -{Cot[x]^3/(a + b*Cos[x]), x, 4, -(((a - b*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2))) - ((2*a + b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((2*a - b)*Log[1 + Cos[x]])/(4*(a - b)^2) + (a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2} -{Cot[x]^4/(a + b*Cos[x]), x, 12, (2*a^4*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) + (a^3*Cot[x])/(a^2 - b^2)^2 - (a*Cot[x]^3)/(3*(a^2 - b^2)) - (a^2*b*Csc[x])/(a^2 - b^2)^2 - (b*Csc[x])/(a^2 - b^2) + (b*Csc[x]^3)/(3*(a^2 - b^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^p (a+b Cos[e+f x])^(m/2)*) - - -{Cot[x]/Sqrt[3 - Cos[x]], x, 5, -ArcTanh[Sqrt[3 - Cos[x]]/2]/2 - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]} - - -{Tan[x]*Sqrt[a + b*Cos[x]], x, 4, 2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]]/Sqrt[a]] - 2*Sqrt[a + b*Cos[x]]} -{Tan[x]/Sqrt[a + b*Cos[x]], x, 3, (2*ArcTanh[Sqrt[a + b*Cos[x]]/Sqrt[a]])/Sqrt[a]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^(p/2) (a+b Cos[e+f x])^m*) - - -{Sqrt[e*Tan[c + d*x]]/(a + b*Cos[c + d*x]), x, 9, -((2*Sqrt[2]*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(Sqrt[-a + b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]])) + (2*Sqrt[2]*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(Sqrt[-a + b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (g Tan[e+f x])^(p/2) (a+b Cos[e+f x])^(m/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Cos[e+f x])^m with p symbolic*) - - -{(g*Tan[e + f*x])^p*(a + b*Cos[e + f*x])^m, x, 1, (g*Cot[e + f*x])^p*(g*Tan[e + f*x])^p*Unintegrable[(a + b*Cos[e + f*x])^m/(g*Cot[e + f*x])^p, x]} - - -(* {(g*Tan[e + f*x])^p*(a + b*Cos[e + f*x])^3, x, 0, 0} -{(g*Tan[e + f*x])^p*(a + b*Cos[e + f*x])^2, x, 0, 0} -{(g*Tan[e + f*x])^p*(a + b*Cos[e + f*x])^1, x, 0, 0} -{(g*Tan[e + f*x])^p/(a + b*Cos[e + f*x])^1, x, 0, -((g*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(b + a*Sec[e + f*x]), (-a + b)/(b + a*Sec[e + f*x])]*(-((a*(1 - Sec[e + f*x]))/(b + a*Sec[e + f*x])))^((1 - p)/2)*((a*(1 + Sec[e + f*x]))/(b + a*Sec[e + f*x]))^((1 - p)/2)*(g*Tan[e + f*x])^(-1 + p)*(-Tan[e + f*x]^2)^((1 - p)/2 + (1/2)*(-1 + p)))/(a*f*(1 - p)))} -{(g*Tan[e + f*x])^p/(a + b*Cos[e + f*x])^2, x, 0, 0} -{(g*Tan[e + f*x])^p/(a + b*Cos[e + f*x])^3, x, 0, 0} *) - - -(* ::Section:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section:: *) -(*Integrands of the form (g Tan[e+f x])^p (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.10 (c+d x)^m (a+b cos)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.10 (c+d x)^m (a+b cos)^n.m deleted file mode 100644 index b445197..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.10 (c+d x)^m (a+b cos)^n.m +++ /dev/null @@ -1,377 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Cos[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[a + b*x]*(c + d*x)^4, x, 5, -((24*d^3*(c + d*x)*Cos[a + b*x])/b^4) + (4*d*(c + d*x)^3*Cos[a + b*x])/b^2 + (24*d^4*Sin[a + b*x])/b^5 - (12*d^2*(c + d*x)^2*Sin[a + b*x])/b^3 + ((c + d*x)^4*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^3, x, 4, -((6*d^3*Cos[a + b*x])/b^4) + (3*d*(c + d*x)^2*Cos[a + b*x])/b^2 - (6*d^2*(c + d*x)*Sin[a + b*x])/b^3 + ((c + d*x)^3*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^2, x, 3, (2*d*(c + d*x)*Cos[a + b*x])/b^2 - (2*d^2*Sin[a + b*x])/b^3 + ((c + d*x)^2*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^1, x, 2, (d*Cos[a + b*x])/b^2 + ((c + d*x)*Sin[a + b*x])/b} -{Cos[a + b*x]/(c + d*x)^1, x, 3, (Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d - (Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{Cos[a + b*x]/(c + d*x)^2, x, 4, -(Cos[a + b*x]/(d*(c + d*x))) - (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^2 - (b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} -{Cos[a + b*x]/(c + d*x)^3, x, 5, -(Cos[a + b*x]/(2*d*(c + d*x)^2)) - (b^2*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(2*d^3) + (b*Sin[a + b*x])/(2*d^2*(c + d*x)) + (b^2*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(2*d^3)} -{Cos[a + b*x]/(c + d*x)^4, x, 6, -(Cos[a + b*x]/(3*d*(c + d*x)^3)) + (b^2*Cos[a + b*x])/(6*d^3*(c + d*x)) + (b^3*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(6*d^4) + (b*Sin[a + b*x])/(6*d^2*(c + d*x)^2) + (b^3*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(6*d^4)} - - -{Cos[a + b*x]^2*(c + d*x)^4, x, 6, (3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) + (c + d*x)^5/(10*d) - (3*d^3*(c + d*x)*Cos[a + b*x]^2)/(2*b^4) + (d*(c + d*x)^3*Cos[a + b*x]^2)/b^2 + (3*d^4*Cos[a + b*x]*Sin[a + b*x])/(4*b^5) - (3*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + ((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^2*(c + d*x)^3, x, 4, -((3*c*d^2*x)/(4*b^2)) - (3*d^3*x^2)/(8*b^2) + (c + d*x)^4/(8*d) - (3*d^3*Cos[a + b*x]^2)/(8*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x]^2)/(4*b^2) - (3*d^2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(4*b^3) + ((c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^2*(c + d*x)^2, x, 4, -((d^2*x)/(4*b^2)) + (c + d*x)^3/(6*d) + (d*(c + d*x)*Cos[a + b*x]^2)/(2*b^2) - (d^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^3) + ((c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^2*(c + d*x)^1, x, 2, (c*x)/2 + (d*x^2)/4 + (d*Cos[a + b*x]^2)/(4*b^2) + ((c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^2/(c + d*x)^1, x, 5, (Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(2*d) + Log[c + d*x]/(2*d) - (Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Cos[a + b*x]^2/(c + d*x)^2, x, 5, -(Cos[a + b*x]^2/(d*(c + d*x))) - (b*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^2 - (b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Cos[a + b*x]^2/(c + d*x)^3, x, 7, -(Cos[a + b*x]^2/(2*d*(c + d*x)^2)) - (b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^3 + (b*Cos[a + b*x]*Sin[a + b*x])/(d^2*(c + d*x)) + (b^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^3} - - -{Cos[a + b*x]^3*(c + d*x)^4, x, 12, -((160*d^3*(c + d*x)*Cos[a + b*x])/(9*b^4)) + (8*d*(c + d*x)^3*Cos[a + b*x])/(3*b^2) - (8*d^3*(c + d*x)*Cos[a + b*x]^3)/(27*b^4) + (4*d*(c + d*x)^3*Cos[a + b*x]^3)/(9*b^2) + (488*d^4*Sin[a + b*x])/(27*b^5) - (80*d^2*(c + d*x)^2*Sin[a + b*x])/(9*b^3) + (2*(c + d*x)^4*Sin[a + b*x])/(3*b) - (4*d^2*(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(9*b^3) + ((c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) - (8*d^4*Sin[a + b*x]^3)/(81*b^5)} -{Cos[a + b*x]^3*(c + d*x)^3, x, 8, -((40*d^3*Cos[a + b*x])/(9*b^4)) + (2*d*(c + d*x)^2*Cos[a + b*x])/b^2 - (2*d^3*Cos[a + b*x]^3)/(27*b^4) + (d*(c + d*x)^2*Cos[a + b*x]^3)/(3*b^2) - (40*d^2*(c + d*x)*Sin[a + b*x])/(9*b^3) + (2*(c + d*x)^3*Sin[a + b*x])/(3*b) - (2*d^2*(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(9*b^3) + ((c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x])/(3*b)} -{Cos[a + b*x]^3*(c + d*x)^2, x, 6, (4*d*(c + d*x)*Cos[a + b*x])/(3*b^2) + (2*d*(c + d*x)*Cos[a + b*x]^3)/(9*b^2) - (14*d^2*Sin[a + b*x])/(9*b^3) + (2*(c + d*x)^2*Sin[a + b*x])/(3*b) + ((c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2*d^2*Sin[a + b*x]^3)/(27*b^3)} -{Cos[a + b*x]^3*(c + d*x)^1, x, 3, (2*d*Cos[a + b*x])/(3*b^2) + (d*Cos[a + b*x]^3)/(9*b^2) + (2*(c + d*x)*Sin[a + b*x])/(3*b) + ((c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(3*b)} -{Cos[a + b*x]^3/(c + d*x)^1, x, 8, (3*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(4*d) + (Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(4*d) - (3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) - (Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{Cos[a + b*x]^3/(c + d*x)^2, x, 8, -(Cos[a + b*x]^3/(d*(c + d*x))) - (3*b*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d^2) - (3*b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d^2) - (3*b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d^2) - (3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{Cos[a + b*x]^3/(c + d*x)^3, x, 12, -(Cos[a + b*x]^3/(2*d*(c + d*x)^2)) - (3*b^2*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(8*d^3) - (9*b^2*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^3) + (3*b*Cos[a + b*x]^2*Sin[a + b*x])/(2*d^2*(c + d*x)) + (3*b^2*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} - - -{x^3*Cos[a + b*x]^4, x, 8, -((45*x^2)/(128*b^2)) + (3*x^4)/32 - (45*Cos[a + b*x]^2)/(128*b^4) + (9*x^2*Cos[a + b*x]^2)/(16*b^2) - (3*Cos[a + b*x]^4)/(128*b^4) + (3*x^2*Cos[a + b*x]^4)/(16*b^2) - (45*x*Cos[a + b*x]*Sin[a + b*x])/(64*b^3) + (3*x^3*Cos[a + b*x]*Sin[a + b*x])/(8*b) - (3*x*Cos[a + b*x]^3*Sin[a + b*x])/(32*b^3) + (x^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b)} -{x^2*Cos[a + b*x]^4, x, 8, -((15*x)/(64*b^2)) + x^3/8 + (3*x*Cos[a + b*x]^2)/(8*b^2) + (x*Cos[a + b*x]^4)/(8*b^2) - (15*Cos[a + b*x]*Sin[a + b*x])/(64*b^3) + (3*x^2*Cos[a + b*x]*Sin[a + b*x])/(8*b) - (Cos[a + b*x]^3*Sin[a + b*x])/(32*b^3) + (x^2*Cos[a + b*x]^3*Sin[a + b*x])/(4*b)} -{x^1*Cos[a + b*x]^4, x, 3, (3*x^2)/16 + (3*Cos[a + b*x]^2)/(16*b^2) + Cos[a + b*x]^4/(16*b^2) + (3*x*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (x*Cos[a + b*x]^3*Sin[a + b*x])/(4*b)} -{Cos[a + b*x]^4/x^1, x, 8, (1/2)*Cos[2*a]*CosIntegral[2*b*x] + (1/8)*Cos[4*a]*CosIntegral[4*b*x] + (3*Log[x])/8 - (1/2)*Sin[2*a]*SinIntegral[2*b*x] - (1/8)*Sin[4*a]*SinIntegral[4*b*x]} -{Cos[a + b*x]^4/x^2, x, 8, -(Cos[a + b*x]^4/x) - b*CosIntegral[2*b*x]*Sin[2*a] - (1/2)*b*CosIntegral[4*b*x]*Sin[4*a] - b*Cos[2*a]*SinIntegral[2*b*x] - (1/2)*b*Cos[4*a]*SinIntegral[4*b*x]} -{Cos[a + b*x]^4/x^3, x, 14, -(Cos[a + b*x]^4/(2*x^2)) - b^2*Cos[2*a]*CosIntegral[2*b*x] - b^2*Cos[4*a]*CosIntegral[4*b*x] + (2*b*Cos[a + b*x]^3*Sin[a + b*x])/x + b^2*Sin[2*a]*SinIntegral[2*b*x] + b^2*Sin[4*a]*SinIntegral[4*b*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[a + b*x]*(c + d*x)^3, x, 9, -((2*I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b) + (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (6*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4} -{Sec[a + b*x]*(c + d*x)^2, x, 7, -((2*I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b) + (2*I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3} -{Sec[a + b*x]*(c + d*x)^1, x, 5, -((2*I*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b) + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2} -{Sec[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Sec[a + b*x]/(c + d*x), x]} - - -{Sec[a + b*x]^2*(c + d*x)^3, x, 6, -((I*(c + d*x)^3)/b) + (3*d*(c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b^2 - (3*I*d^2*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^4) + ((c + d*x)^3*Tan[a + b*x])/b} -{Sec[a + b*x]^2*(c + d*x)^2, x, 5, -((I*(c + d*x)^2)/b) + (2*d*(c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b^2 - (I*d^2*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + ((c + d*x)^2*Tan[a + b*x])/b} -{Sec[a + b*x]^2*(c + d*x)^1, x, 2, (d*Log[Cos[a + b*x]])/b^2 + ((c + d*x)*Tan[a + b*x])/b} -{Sec[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Sec[a + b*x]^2/(c + d*x), x]} - - -{Sec[a + b*x]^3*(c + d*x)^3, x, 15, -((6*I*d^2*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^3) - (I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b + (3*I*d^3*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^4 + (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, I*E^(I*(a + b*x))])/b^4 - (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (3*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + (3*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4 - (3*d*(c + d*x)^2*Sec[a + b*x])/(2*b^2) + ((c + d*x)^3*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^3*(c + d*x)^2, x, 9, -((I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b) + (d^2*ArcTanh[Sin[a + b*x]])/b^3 + (I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (d*(c + d*x)*Sec[a + b*x])/b^2 + ((c + d*x)^2*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^3*(c + d*x)^1, x, 6, -((I*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b) + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Sec[a + b*x]^2/(c + d*x), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Cos[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[a + b*x]*(c + d*x)^(5/2), x, 8, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(2*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(4*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^(3/2), x, 7, (3*d*Sqrt[c + d*x]*Cos[a + b*x])/(2*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^(1/2), x, 6, -((Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/b^(3/2) + (Sqrt[c + d*x]*Sin[a + b*x])/b} -{Cos[a + b*x]/(c + d*x)^(1/2), x, 5, (Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) - (Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])} -{Cos[a + b*x]/(c + d*x)^(3/2), x, 6, -((2*Cos[a + b*x])/(d*Sqrt[c + d*x])) - (2*Sqrt[b]*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (2*Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(3/2)} -{Cos[a + b*x]/(c + d*x)^(5/2), x, 7, -((2*Cos[a + b*x])/(3*d*(c + d*x)^(3/2))) - (4*b^(3/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) + (4*b^(3/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(3*d^(5/2)) + (4*b*Sin[a + b*x])/(3*d^2*Sqrt[c + d*x])} -{Cos[a + b*x]/(c + d*x)^(7/2), x, 8, -((2*Cos[a + b*x])/(5*d*(c + d*x)^(5/2))) + (8*b^2*Cos[a + b*x])/(15*d^3*Sqrt[c + d*x]) + (8*b^(5/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (8*b^(5/2)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(15*d^(7/2)) + (4*b*Sin[a + b*x])/(15*d^2*(c + d*x)^(3/2))} - - -{Cos[a + b*x]^2*(c + d*x)^(5/2), x, 10, -((5*d*(c + d*x)^(3/2))/(16*b^2)) + (c + d*x)^(7/2)/(7*d) + (5*d*(c + d*x)^(3/2)*Cos[a + b*x]^2)/(8*b^2) + (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(7/2)) + ((c + d*x)^(5/2)*Cos[a + b*x]*Sin[a + b*x])/(2*b) - (15*d^2*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(64*b^3)} -{Cos[a + b*x]^2*(c + d*x)^(3/2), x, 9, -((3*d*Sqrt[c + d*x])/(16*b^2)) + (c + d*x)^(5/2)/(5*d) + (3*d*Sqrt[c + d*x]*Cos[a + b*x]^2)/(8*b^2) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(32*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(32*b^(5/2)) + ((c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x])/(2*b)} -{Cos[a + b*x]^2*(c + d*x)^(1/2), x, 8, (c + d*x)^(3/2)/(3*d) - (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(8*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(8*b^(3/2)) + (Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(4*b)} -{Cos[a + b*x]^2/(c + d*x)^(1/2), x, 7, Sqrt[c + d*x]/d + (Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2*Sqrt[b]*Sqrt[d])} -{Cos[a + b*x]^2/(c + d*x)^(3/2), x, 7, -((2*Cos[a + b*x]^2)/(d*Sqrt[c + d*x])) - (2*Sqrt[b]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/d^(3/2) - (2*Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/d^(3/2)} -{Cos[a + b*x]^2/(c + d*x)^(5/2), x, 9, -((2*Cos[a + b*x]^2)/(3*d*(c + d*x)^(3/2))) - (8*b^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(3*d^(5/2)) + (8*b^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(3*d^(5/2)) + (8*b*Cos[a + b*x]*Sin[a + b*x])/(3*d^2*Sqrt[c + d*x])} -{Cos[a + b*x]^2/(c + d*x)^(7/2), x, 9, -((16*b^2)/(15*d^3*Sqrt[c + d*x])) - (2*Cos[a + b*x]^2)/(5*d*(c + d*x)^(5/2)) + (32*b^2*Cos[a + b*x]^2)/(15*d^3*Sqrt[c + d*x]) + (32*b^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(15*d^(7/2)) + (32*b^(5/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(15*d^(7/2)) + (8*b*Cos[a + b*x]*Sin[a + b*x])/(15*d^2*(c + d*x)^(3/2))} -{Cos[a + b*x]^2/(c + d*x)^(9/2), x, 11, -((16*b^2)/(105*d^3*(c + d*x)^(3/2))) - (2*Cos[a + b*x]^2)/(7*d*(c + d*x)^(7/2)) + (32*b^2*Cos[a + b*x]^2)/(105*d^3*(c + d*x)^(3/2)) + (128*b^(7/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(105*d^(9/2)) - (128*b^(7/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(105*d^(9/2)) + (8*b*Cos[a + b*x]*Sin[a + b*x])/(35*d^2*(c + d*x)^(5/2)) - (128*b^3*Cos[a + b*x]*Sin[a + b*x])/(105*d^4*Sqrt[c + d*x])} - - -{Cos[a + b*x]^3*(c + d*x)^(5/2), x, 23, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(3*b^2) + (5*d*(c + d*x)^(3/2)*Cos[a + b*x]^3)/(18*b^2) + (45*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (45*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^3) + (2*(c + d*x)^(5/2)*Sin[a + b*x])/(3*b) + ((c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) - (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(144*b^3)} -{Cos[a + b*x]^3*(c + d*x)^(3/2), x, 20, (d*Sqrt[c + d*x]*Cos[a + b*x])/b^2 + (d*Sqrt[c + d*x]*Cos[a + b*x]^3)/(6*b^2) - (9*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) + (9*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + (2*(c + d*x)^(3/2)*Sin[a + b*x])/(3*b) + ((c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x])/(3*b)} -{Cos[a + b*x]^3*(c + d*x)^(1/2), x, 14, -((3*Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))) - (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (3*Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2)) + (3*Sqrt[c + d*x]*Sin[a + b*x])/(4*b) + (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]^3/(c + d*x)^(1/2), x, 12, (3*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + (Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(2*Sqrt[b]*Sqrt[d]) - (3*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*Sqrt[b]*Sqrt[d])} -{Cos[a + b*x]^3/(c + d*x)^(3/2), x, 12, -((2*Cos[a + b*x]^3)/(d*Sqrt[c + d*x])) - (3*Sqrt[b]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/d^(3/2) - (3*Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(3/2)} -{Cos[a + b*x]^3/(c + d*x)^(5/2), x, 18, -((2*Cos[a + b*x]^3)/(3*d*(c + d*x)^(3/2))) - (b^(3/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(5/2) - (b^(3/2)*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/d^(5/2) + (b^(3/2)*Sqrt[6*Pi]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/d^(5/2) + (b^(3/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/d^(5/2) + (4*b*Cos[a + b*x]^2*Sin[a + b*x])/(d^2*Sqrt[c + d*x])} -{Cos[a + b*x]^3/(c + d*x)^(7/2), x, 19, -((16*b^2*Cos[a + b*x])/(5*d^3*Sqrt[c + d*x])) - (2*Cos[a + b*x]^3)/(5*d*(c + d*x)^(5/2)) + (24*b^2*Cos[a + b*x]^3)/(5*d^3*Sqrt[c + d*x]) + (2*b^(5/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (6*b^(5/2)*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (6*b^(5/2)*Sqrt[6*Pi]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(5*d^(7/2)) + (2*b^(5/2)*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(5*d^(7/2)) + (4*b*Cos[a + b*x]^2*Sin[a + b*x])/(5*d^2*(c + d*x)^(3/2))} - - -{x^(3/2)*Cos[x], x, 4, (3/2)*Sqrt[x]*Cos[x] - (3/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[x]] + x^(3/2)*Sin[x]} -{x^(1/2)*Cos[x], x, 3, (-Sqrt[Pi/2])*FresnelS[Sqrt[2/Pi]*Sqrt[x]] + Sqrt[x]*Sin[x]} -{Cos[x]/x^(1/2), x, 2, Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[x]]} -{Cos[x]/x^(3/2), x, 3, -((2*Cos[x])/Sqrt[x]) - 2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[x]]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/3) Cos[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[a + b*x]*(c + d*x)^(4/3), x, 5, (4*d*(c + d*x)^(1/3)*Cos[a + b*x])/(3*b^2) + (2*I*d^2*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(2/3)*Gamma[1/3, -((I*b*(c + d*x))/d)])/(9*b^3*(c + d*x)^(2/3)) - (2*I*d^2*((I*b*(c + d*x))/d)^(2/3)*Gamma[1/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(9*b^3*(c + d*x)^(2/3))) + ((c + d*x)^(4/3)*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^(2/3), x, 4, (d*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(1/3)*Gamma[2/3, -((I*b*(c + d*x))/d)])/(3*b^2*(c + d*x)^(1/3)) + (d*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(3*b^2*(c + d*x)^(1/3))) + ((c + d*x)^(2/3)*Sin[a + b*x])/b} -{Cos[a + b*x]*(c + d*x)^(1/3), x, 4, (d*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(2/3)*Gamma[1/3, -((I*b*(c + d*x))/d)])/(6*b^2*(c + d*x)^(2/3)) + (d*((I*b*(c + d*x))/d)^(2/3)*Gamma[1/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(6*b^2*(c + d*x)^(2/3))) + ((c + d*x)^(1/3)*Sin[a + b*x])/b} -{Cos[a + b*x]/(c + d*x)^(1/3), x, 3, -((I*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(1/3)*Gamma[2/3, -((I*b*(c + d*x))/d)])/(2*b*(c + d*x)^(1/3))) + (I*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(2*b*(c + d*x)^(1/3)))} -{Cos[a + b*x]/(c + d*x)^(2/3), x, 3, -((I*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(2/3)*Gamma[1/3, -((I*b*(c + d*x))/d)])/(2*b*(c + d*x)^(2/3))) + (I*((I*b*(c + d*x))/d)^(2/3)*Gamma[1/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(2*b*(c + d*x)^(2/3)))} -{Cos[a + b*x]/(c + d*x)^(4/3), x, 4, -((3*Cos[a + b*x])/(d*(c + d*x)^(1/3))) + (3*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(1/3)*Gamma[2/3, -((I*b*(c + d*x))/d)])/(2*d*(c + d*x)^(1/3)) + (3*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(2*d*(c + d*x)^(1/3)))} -{Cos[a + b*x]/(c + d*x)^(5/3), x, 4, -((3*Cos[a + b*x])/(2*d*(c + d*x)^(2/3))) + (3*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(2/3)*Gamma[1/3, -((I*b*(c + d*x))/d)])/(4*d*(c + d*x)^(2/3)) + (3*((I*b*(c + d*x))/d)^(2/3)*Gamma[1/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(4*d*(c + d*x)^(2/3)))} -{Cos[a + b*x]/(c + d*x)^(7/3), x, 5, -((3*Cos[a + b*x])/(4*d*(c + d*x)^(4/3))) + (9*I*b*E^(I*(a - (b*c)/d))*(-((I*b*(c + d*x))/d))^(1/3)*Gamma[2/3, -((I*b*(c + d*x))/d)])/(8*d^2*(c + d*x)^(1/3)) - (9*I*b*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*(8*d^2*(c + d*x)^(1/3))) + (9*b*Sin[a + b*x])/(4*d^2*(c + d*x)^(1/3))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[e+f x]^(n/2)*) - - -{x^1*Cos[a + b*x]^(1/2), x, 0, Unintegrable[x*Sqrt[Cos[a + b*x]], x]} -{x^0*Cos[a + b*x]^(1/2), x, 1, (2*EllipticE[(1/2)*(a + b*x), 2])/b} -{Cos[a + b*x]^(1/2)/x^1, x, 0, Unintegrable[Sqrt[Cos[a + b*x]]/x, x]} - - -{x^1*Cos[a + b*x]^(3/2), x, 1, (4*Cos[a + b*x]^(3/2))/(9*b^2) + (1/3)*Unintegrable[x/Sqrt[Cos[a + b*x]], x] + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)} -{x^0*Cos[a + b*x]^(3/2), x, 2, (2*EllipticF[(1/2)*(a + b*x), 2])/(3*b) + (2*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)} -{Cos[a + b*x]^(3/2)/x^1, x, 0, Unintegrable[Cos[a + b*x]^(3/2)/x, x]} - -{x^1*Cos[a + b*x]^(3/2) - x/(3*Sqrt[Cos[a + b*x]]), x, 2, (4*Cos[a + b*x]^(3/2))/(9*b^2) + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)} - - -{Cos[x]^(3/2)/x^3, x, 1, -(Cos[x]^(3/2)/(2*x^2)) + (3/8)*Unintegrable[1/(x*Sqrt[Cos[x]]), x] - (9/8)*Unintegrable[Cos[x]^(3/2)/x, x] + (3*Sqrt[Cos[x]]*Sin[x])/(4*x)} - - -{x^1/Cos[a + b*x]^(1/2), x, 0, Unintegrable[x/Sqrt[Cos[a + b*x]], x]} -{x^0/Cos[a + b*x]^(1/2), x, 1, (2*EllipticF[(1/2)*(a + b*x), 2])/b} -{1/(x^1*Cos[a + b*x]^(1/2)), x, 0, Unintegrable[1/(x*Sqrt[Cos[a + b*x]]), x]} - - -{x^1/Cos[a + b*x]^(3/2), x, 1, (4*Sqrt[Cos[a + b*x]])/b^2 + (2*x*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]]) - Unintegrable[x*Sqrt[Cos[a + b*x]], x]} -{x^0/Cos[a + b*x]^(3/2), x, 2, -((2*EllipticE[(1/2)*(a + b*x), 2])/b) + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])} -{1/(x^1*Cos[a + b*x]^(3/2)), x, 0, Unintegrable[1/(x*Cos[a + b*x]^(3/2)), x]} - -{x^1/Cos[a + b*x]^(3/2) + x*Sqrt[Cos[a + b*x]], x, 2, (4*Sqrt[Cos[a + b*x]])/b^2 + (2*x*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])} - - -{x/Cos[x]^(3/2) + x*Sqrt[Cos[x]], x, 2, 4*Sqrt[Cos[x]] + (2*x*Sin[x])/Sqrt[Cos[x]]} -{x/Cos[x]^(5/2) - x/(3*Sqrt[Cos[x]]), x, 2, -(4/(3*Sqrt[Cos[x]])) + (2*x*Sin[x])/(3*Cos[x]^(3/2))} -{x/Cos[x]^(7/2) + (3/5)*x*Sqrt[Cos[x]], x, 3, -(4/(15*Cos[x]^(3/2))) + (12*Sqrt[Cos[x]])/5 + (2*x*Sin[x])/(5*Cos[x]^(5/2)) + (6*x*Sin[x])/(5*Sqrt[Cos[x]])} -{x^2/Cos[x]^(3/2) + x^2*Sqrt[Cos[x]], x, 3, 8*x*Sqrt[Cos[x]] - 16*EllipticE[x/2, 2] + (2*x^2*Sin[x])/Sqrt[Cos[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sec[e+f x]^(n/2)*) - - -{x/Sec[x]^(3/2) - (1/3)*x*Sqrt[Sec[x]], x, 4, 4/(9*Sec[x]^(3/2)) + (2*x*Sin[x])/(3*Sqrt[Sec[x]])} -{x/Sec[x]^(5/2) - (3/5)*x/Sqrt[Sec[x]], x, 4, 4/(25*Sec[x]^(5/2)) + (2*x*Sin[x])/(5*Sec[x]^(3/2))} -{x/Sec[x]^(7/2) - (5/21)*x*Sqrt[Sec[x]], x, 5, 4/(49*Sec[x]^(7/2)) + 20/(63*Sec[x]^(3/2)) + (2*x*Sin[x])/(7*Sec[x]^(5/2)) + (10*x*Sin[x])/(21*Sqrt[Sec[x]])} -{x^2/Sec[x]^(3/2) - (1/3)*x^2*Sqrt[Sec[x]], x, 7, (8*x)/(9*Sec[x]^(3/2)) - (16/27)*Sqrt[Cos[x]]*EllipticF[x/2, 2]*Sqrt[Sec[x]] - (16*Sin[x])/(27*Sqrt[Sec[x]]) + (2*x^2*Sin[x])/(3*Sqrt[Sec[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(b*Cos[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(b*Cos[e + f*x])^n, x]} - - -{Cos[a + b*x]^3*(c + d*x)^m, x, 8, -((3*I*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b))) + (3*I*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b)) - (I*3^(-1 - m)*E^(3*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((3*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b)) + (I*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, (3*I*b*(c + d*x))/d])/(E^(3*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b))} -{Cos[a + b*x]^2*(c + d*x)^m, x, 5, (c + d*x)^(1 + m)/(2*d*(1 + m)) - (I*2^(-3 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b) + (I*2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b)} -{Cos[a + b*x]*(c + d*x)^m, x, 3, -((I*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b))) + (I*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b))} -{Sec[a + b*x]*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Sec[a + b*x], x]} -{Sec[a + b*x]^2*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Sec[a + b*x]^2, x]} - - -{x^(m + 3)*Cos[a + b*x], x, 3, -((E^(I*a)*x^m*Gamma[4 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^4))) - (x^m*Gamma[4 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^4))} -{x^(m + 2)*Cos[a + b*x], x, 3, (I*E^(I*a)*x^m*Gamma[3 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^3)) - (I*x^m*Gamma[3 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^3))} -{x^(m + 1)*Cos[a + b*x], x, 3, (E^(I*a)*x^m*Gamma[2 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b^2)) + (x^m*Gamma[2 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b^2))} -{x^(m + 0)*Cos[a + b*x], x, 3, -((I*E^(I*a)*x^m*Gamma[1 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b))) + (I*x^m*Gamma[1 + m, I*b*x])/(E^(I*a)*(I*b*x)^m*(2*b))} -{x^(m - 1)*Cos[a + b*x], x, 3, ((-(1/2))*E^(I*a)*x^m*Gamma[m, (-I)*b*x])/((-I)*b*x)^m - ((1/2)*x^m*Gamma[m, I*b*x])/(E^(I*a)*(I*b*x)^m)} -{x^(m - 2)*Cos[a + b*x], x, 3, ((1/2)*I*b*E^(I*a)*x^m*Gamma[-1 + m, (-I)*b*x])/((-I)*b*x)^m - ((1/2)*I*b*x^m*Gamma[-1 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)} -{x^(m - 3)*Cos[a + b*x], x, 3, ((1/2)*b^2*E^(I*a)*x^m*Gamma[-2 + m, (-I)*b*x])/((-I)*b*x)^m + ((1/2)*b^2*x^m*Gamma[-2 + m, I*b*x])/(E^(I*a)*(I*b*x)^m)} - - -{x^(m + 3)*Cos[a + b*x]^2, x, 5, x^(4 + m)/(2*(4 + m)) - (2^(-6 - m)*E^(2*I*a)*x^m*Gamma[4 + m, -2*I*b*x])/(((-I)*b*x)^m*b^4) - (2^(-6 - m)*x^m*Gamma[4 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^4)} -{x^(m + 2)*Cos[a + b*x]^2, x, 5, x^(3 + m)/(2*(3 + m)) + (I*2^(-5 - m)*E^(2*I*a)*x^m*Gamma[3 + m, -2*I*b*x])/(((-I)*b*x)^m*b^3) - (I*2^(-5 - m)*x^m*Gamma[3 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^3)} -{x^(m + 1)*Cos[a + b*x]^2, x, 5, x^(2 + m)/(2*(2 + m)) + (2^(-4 - m)*E^(2*I*a)*x^m*Gamma[2 + m, -2*I*b*x])/(((-I)*b*x)^m*b^2) + (2^(-4 - m)*x^m*Gamma[2 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b^2)} -{x^(m + 0)*Cos[a + b*x]^2, x, 5, x^(1 + m)/(2*(1 + m)) - (I*2^(-3 - m)*E^(2*I*a)*x^m*Gamma[1 + m, -2*I*b*x])/(((-I)*b*x)^m*b) + (I*2^(-3 - m)*x^m*Gamma[1 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m*b)} -{x^(m - 1)*Cos[a + b*x]^2, x, 5, x^m/(2*m) - (2^(-2 - m)*E^(2*I*a)*x^m*Gamma[m, -2*I*b*x])/((-I)*b*x)^m - (2^(-2 - m)*x^m*Gamma[m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m)} -{x^(m - 2)*Cos[a + b*x]^2, x, 5, -(x^(-1 + m)/(2*(1 - m))) + (I*2^(-1 - m)*b*E^(2*I*a)*x^m*Gamma[-1 + m, -2*I*b*x])/((-I)*b*x)^m - (I*2^(-1 - m)*b*x^m*Gamma[-1 + m, 2*I*b*x])/(E^(2*I*a)*(I*b*x)^m)} -{x^(m - 3)*Cos[a + b*x]^2, x, 5, -(x^(-2 + m)/(2*(2 - m))) + (b^2*E^(2*I*a)*x^m*Gamma[-2 + m, -2*I*b*x])/(2^m*((-I)*b*x)^m) + (b^2*x^m*Gamma[-2 + m, 2*I*b*x])/(2^m*E^(2*I*a)*(I*b*x)^m)} - - -(* ::Section:: *) -(*Integrands of the form (c+d x)^m (b Sec[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Cos[e + f*x])*(c + d*x)^3, x, 6, (a*(c + d*x)^4)/(4*d) - (6*a*d^3*Cos[e + f*x])/f^4 + (3*a*d*(c + d*x)^2*Cos[e + f*x])/f^2 - (6*a*d^2*(c + d*x)*Sin[e + f*x])/f^3 + (a*(c + d*x)^3*Sin[e + f*x])/f} -{(a + a*Cos[e + f*x])*(c + d*x)^2, x, 5, (a*(c + d*x)^3)/(3*d) + (2*a*d*(c + d*x)*Cos[e + f*x])/f^2 - (2*a*d^2*Sin[e + f*x])/f^3 + (a*(c + d*x)^2*Sin[e + f*x])/f} -{(a + a*Cos[e + f*x])*(c + d*x)^1, x, 4, (a*(c + d*x)^2)/(2*d) + (a*d*Cos[e + f*x])/f^2 + (a*(c + d*x)*Sin[e + f*x])/f} -{(a + a*Cos[e + f*x])/(c + d*x)^1, x, 5, (a*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d + (a*Log[c + d*x])/d - (a*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d} -{(a + a*Cos[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) - (a*Cos[e + f*x])/(d*(c + d*x)) - (a*f*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d^2 - (a*f*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2} - - -{(a + a*Cos[e + f*x])^2*(c + d*x)^3, x, 10, -((3*a^2*c*d^2*x)/(4*f^2)) - (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) - (12*a^2*d^3*Cos[e + f*x])/f^4 + (6*a^2*d*(c + d*x)^2*Cos[e + f*x])/f^2 - (3*a^2*d^3*Cos[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Cos[e + f*x]^2)/(4*f^2) - (12*a^2*d^2*(c + d*x)*Sin[e + f*x])/f^3 + (2*a^2*(c + d*x)^3*Sin[e + f*x])/f - (3*a^2*d^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) + (a^2*(c + d*x)^3*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Cos[e + f*x])^2*(c + d*x)^2, x, 9, -((a^2*d^2*x)/(4*f^2)) + (a^2*(c + d*x)^3)/(2*d) + (4*a^2*d*(c + d*x)*Cos[e + f*x])/f^2 + (a^2*d*(c + d*x)*Cos[e + f*x]^2)/(2*f^2) - (4*a^2*d^2*Sin[e + f*x])/f^3 + (2*a^2*(c + d*x)^2*Sin[e + f*x])/f - (a^2*d^2*Cos[e + f*x]*Sin[e + f*x])/(4*f^3) + (a^2*(c + d*x)^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Cos[e + f*x])^2*(c + d*x)^1, x, 6, (1/2)*a^2*c*x + (1/4)*a^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) + (2*a^2*d*Cos[e + f*x])/f^2 + (a^2*d*Cos[e + f*x]^2)/(4*f^2) + (2*a^2*(c + d*x)*Sin[e + f*x])/f + (a^2*(c + d*x)*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{(a + a*Cos[e + f*x])^2/(c + d*x)^1, x, 9, (2*a^2*Cos[e - (c*f)/d]*CosIntegral[(c*f)/d + f*x])/d + (a^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (3*a^2*Log[c + d*x])/(2*d) - (2*a^2*Sin[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d - (a^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + a*Cos[e + f*x])^2/(c + d*x)^2, x, 9, -((4*a^2*Cos[e/2 + (f*x)/2]^4)/(d*(c + d*x))) - (a^2*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d^2 - (2*a^2*f*CosIntegral[(c*f)/d + f*x]*Sin[e - (c*f)/d])/d^2 - (2*a^2*f*Cos[e - (c*f)/d]*SinIntegral[(c*f)/d + f*x])/d^2 - (a^2*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/d^2} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(a + a*Cos[e + f*x])*(c + d*x)^3, x, 7, -((I*(c + d*x)^3)/(a*f)) + (6*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a*f^2) - (12*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a*f^3) + (12*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a*f^4) + ((c + d*x)^3*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Cos[e + f*x])*(c + d*x)^2, x, 6, -((I*(c + d*x)^2)/(a*f)) + (4*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(a*f^2) - (4*I*d^2*PolyLog[2, -E^(I*(e + f*x))])/(a*f^3) + ((c + d*x)^2*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Cos[e + f*x])*(c + d*x)^1, x, 3, (2*d*Log[Cos[e/2 + (f*x)/2]])/(a*f^2) + ((c + d*x)*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Cos[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Cos[e + f*x])), x]} -{1/(a + a*Cos[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Cos[e + f*x])), x]} - - -{1/(a + a*Cos[e + f*x])^2*(c + d*x)^3, x, 10, -((I*(c + d*x)^3)/(3*a^2*f)) + (2*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a^2*f^2) + (4*d^3*Log[Cos[e/2 + (f*x)/2]])/(a^2*f^4) - (4*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a^2*f^3) + (4*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a^2*f^4) - (d*(c + d*x)^2*Sec[e/2 + (f*x)/2]^2)/(2*a^2*f^2) + (2*d^2*(c + d*x)*Tan[e/2 + (f*x)/2])/(a^2*f^3) + ((c + d*x)^3*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Cos[e + f*x])^2*(c + d*x)^2, x, 9, -((I*(c + d*x)^2)/(3*a^2*f)) + (4*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(3*a^2*f^2) - (4*I*d^2*PolyLog[2, -E^(I*(e + f*x))])/(3*a^2*f^3) - (d*(c + d*x)*Sec[e/2 + (f*x)/2]^2)/(3*a^2*f^2) + (2*d^2*Tan[e/2 + (f*x)/2])/(3*a^2*f^3) + ((c + d*x)^2*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Cos[e + f*x])^2*(c + d*x)^1, x, 4, (2*d*Log[Cos[e/2 + (f*x)/2]])/(3*a^2*f^2) - (d*Sec[e/2 + (f*x)/2]^2)/(6*a^2*f^2) + ((c + d*x)*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Cos[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Cos[e + f*x])^2), x]} -{1/(a + a*Cos[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2), x]} - - -{1/(a - a*Cos[e + f*x])*(c + d*x)^3, x, 7, -((I*(c + d*x)^3)/(a*f)) - ((c + d*x)^3*Cot[e/2 + (f*x)/2])/(a*f) + (6*d*(c + d*x)^2*Log[1 - E^(I*(e + f*x))])/(a*f^2) - (12*I*d^2*(c + d*x)*PolyLog[2, E^(I*(e + f*x))])/(a*f^3) + (12*d^3*PolyLog[3, E^(I*(e + f*x))])/(a*f^4)} -{1/(a - a*Cos[e + f*x])*(c + d*x)^2, x, 6, -((I*(c + d*x)^2)/(a*f)) - ((c + d*x)^2*Cot[e/2 + (f*x)/2])/(a*f) + (4*d*(c + d*x)*Log[1 - E^(I*(e + f*x))])/(a*f^2) - (4*I*d^2*PolyLog[2, E^(I*(e + f*x))])/(a*f^3)} -{1/(a - a*Cos[e + f*x])*(c + d*x)^1, x, 3, -(((c + d*x)*Cot[e/2 + (f*x)/2])/(a*f)) + (2*d*Log[Sin[e/2 + (f*x)/2]])/(a*f^2)} -{1/(a - a*Cos[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a - a*Cos[e + f*x])), x]} -{1/(a - a*Cos[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a - a*Cos[e + f*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Sqrt[a + a*Cos[c + d*x]], x, 5, -((96*Sqrt[a + a*Cos[c + d*x]])/d^4) + (12*x^2*Sqrt[a + a*Cos[c + d*x]])/d^2 - (48*x*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d^3 + (2*x^3*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d} -{x^2*Sqrt[a + a*Cos[c + d*x]], x, 4, (8*x*Sqrt[a + a*Cos[c + d*x]])/d^2 - (16*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d^3 + (2*x^2*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d} -{x*Sqrt[a + a*Cos[c + d*x]], x, 3, (4*Sqrt[a + a*Cos[c + d*x]])/d^2 + (2*x*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/d} -{Sqrt[a + a*Cos[c + d*x]], x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]/x, x, 4, Cos[c/2]*Sqrt[a + a*Cos[c + d*x]]*CosIntegral[(d*x)/2]*Sec[c/2 + (d*x)/2] - Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*Sin[c/2]*SinIntegral[(d*x)/2]} -{Sqrt[a + a*Cos[c + d*x]]/x^2, x, 5, -(Sqrt[a + a*Cos[c + d*x]]/x) - (1/2)*d*Sqrt[a + a*Cos[c + d*x]]*CosIntegral[(d*x)/2]*Sec[c/2 + (d*x)/2]*Sin[c/2] - (1/2)*d*Cos[c/2]*Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*SinIntegral[(d*x)/2]} -{Sqrt[a + a*Cos[c + d*x]]/x^3, x, 6, -(Sqrt[a + a*Cos[c + d*x]]/(2*x^2)) - (1/8)*d^2*Cos[c/2]*Sqrt[a + a*Cos[c + d*x]]*CosIntegral[(d*x)/2]*Sec[c/2 + (d*x)/2] + (1/8)*d^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c/2 + (d*x)/2]*Sin[c/2]*SinIntegral[(d*x)/2] + (d*Sqrt[a + a*Cos[c + d*x]]*Tan[c/2 + (d*x)/2])/(4*x)} - - -{x^3*Sqrt[a + a*Cos[x]], x, 5, -96*Sqrt[a + a*Cos[x]] + 12*x^2*Sqrt[a + a*Cos[x]] - 48*x*Sqrt[a + a*Cos[x]]*Tan[x/2] + 2*x^3*Sqrt[a + a*Cos[x]]*Tan[x/2]} -{x^2*Sqrt[a + a*Cos[x]], x, 4, 8*x*Sqrt[a + a*Cos[x]] - 16*Sqrt[a + a*Cos[x]]*Tan[x/2] + 2*x^2*Sqrt[a + a*Cos[x]]*Tan[x/2]} -{x*Sqrt[a + a*Cos[x]], x, 3, 4*Sqrt[a + a*Cos[x]] + 2*x*Sqrt[a + a*Cos[x]]*Tan[x/2]} -{Sqrt[a + a*Cos[x]], x, 1, (2*a*Sin[x])/Sqrt[a + a*Cos[x]]} -{Sqrt[a + a*Cos[x]]/x, x, 2, Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2]} -{Sqrt[a + a*Cos[x]]/x^2, x, 3, -(Sqrt[a + a*Cos[x]]/x) - (1/2)*Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[x/2]} -{Sqrt[a + a*Cos[x]]/x^3, x, 4, -(Sqrt[a + a*Cos[x]]/(2*x^2)) - (1/8)*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2] + (Sqrt[a + a*Cos[x]]*Tan[x/2])/(4*x)} - - -{x^3*Sqrt[a - a*Cos[x]], x, 5, -96*Sqrt[a - a*Cos[x]] + 12*x^2*Sqrt[a - a*Cos[x]] + 48*x*Sqrt[a - a*Cos[x]]*Cot[x/2] - 2*x^3*Sqrt[a - a*Cos[x]]*Cot[x/2]} -{x^2*Sqrt[a - a*Cos[x]], x, 4, 8*x*Sqrt[a - a*Cos[x]] + 16*Sqrt[a - a*Cos[x]]*Cot[x/2] - 2*x^2*Sqrt[a - a*Cos[x]]*Cot[x/2]} -{x*Sqrt[a - a*Cos[x]], x, 3, 4*Sqrt[a - a*Cos[x]] - 2*x*Sqrt[a - a*Cos[x]]*Cot[x/2]} -{Sqrt[a - a*Cos[x]], x, 1, -((2*a*Sin[x])/Sqrt[a - a*Cos[x]])} -{Sqrt[a - a*Cos[x]]/x, x, 2, Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x/2]} -{Sqrt[a - a*Cos[x]]/x^2, x, 3, -(Sqrt[a - a*Cos[x]]/x) + (1/2)*Sqrt[a - a*Cos[x]]*CosIntegral[x/2]*Csc[x/2]} -{Sqrt[a - a*Cos[x]]/x^3, x, 4, -(Sqrt[a - a*Cos[x]]/(2*x^2)) - (Sqrt[a - a*Cos[x]]*Cot[x/2])/(4*x) - (1/8)*Sqrt[a - a*Cos[x]]*Csc[x/2]*SinIntegral[x/2]} - - -{x^3*(a + a*Cos[x])^(3/2), x, 9, (-(1280/9))*a*Sqrt[a + a*Cos[x]] + 16*a*x^2*Sqrt[a + a*Cos[x]] - (64/27)*a*Cos[x/2]^2*Sqrt[a + a*Cos[x]] + (8/3)*a*x^2*Cos[x/2]^2*Sqrt[a + a*Cos[x]] - (32/9)*a*x*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2] + (4/3)*a*x^3*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2] - (640/9)*a*x*Sqrt[a + a*Cos[x]]*Tan[x/2] + (8/3)*a*x^3*Sqrt[a + a*Cos[x]]*Tan[x/2]} -{x^2*(a + a*Cos[x])^(3/2), x, 7, (32/3)*a*x*Sqrt[a + a*Cos[x]] + (16/9)*a*x*Cos[x/2]^2*Sqrt[a + a*Cos[x]] + (4/3)*a*x^2*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2] - (224/9)*a*Sqrt[a + a*Cos[x]]*Tan[x/2] + (8/3)*a*x^2*Sqrt[a + a*Cos[x]]*Tan[x/2] + (32/27)*a*Sqrt[a + a*Cos[x]]*Sin[x/2]^2*Tan[x/2]} -{x*(a + a*Cos[x])^(3/2), x, 4, (16/3)*a*Sqrt[a + a*Cos[x]] + (8/9)*a*Cos[x/2]^2*Sqrt[a + a*Cos[x]] + (4/3)*a*x*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2] + (8/3)*a*x*Sqrt[a + a*Cos[x]]*Tan[x/2]} -{(a + a*Cos[x])^(3/2)/x, x, 5, (3/2)*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2] + (1/2)*a*Sqrt[a + a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2]} -{(a + a*Cos[x])^(3/2)/x^2, x, 5, -((2*a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/x) - (3/4)*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[x/2] - (3/4)*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*SinIntegral[(3*x)/2]} -{(a + a*Cos[x])^(3/2)/x^3, x, 7, -((a*Cos[x/2]^2*Sqrt[a + a*Cos[x]])/x^2) - (3/16)*a*Sqrt[a + a*Cos[x]]*CosIntegral[x/2]*Sec[x/2] - (9/16)*a*Sqrt[a + a*Cos[x]]*CosIntegral[(3*x)/2]*Sec[x/2] + (3*a*Cos[x/2]*Sqrt[a + a*Cos[x]]*Sin[x/2])/(2*x)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/Sqrt[a + a*Cos[c + d*x]], x, 10, -((4*I*x^3*ArcTan[E^((1/2)*I*(c + d*x))]*Cos[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]])) + (12*I*x^2*Cos[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - (12*I*x^2*Cos[c/2 + (d*x)/2]*PolyLog[2, I*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - (48*x*Cos[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^((1/2)*I*(c + d*x))])/(d^3*Sqrt[a + a*Cos[c + d*x]]) + (48*x*Cos[c/2 + (d*x)/2]*PolyLog[3, I*E^((1/2)*I*(c + d*x))])/(d^3*Sqrt[a + a*Cos[c + d*x]]) - (96*I*Cos[c/2 + (d*x)/2]*PolyLog[4, (-I)*E^((1/2)*I*(c + d*x))])/(d^4*Sqrt[a + a*Cos[c + d*x]]) + (96*I*Cos[c/2 + (d*x)/2]*PolyLog[4, I*E^((1/2)*I*(c + d*x))])/(d^4*Sqrt[a + a*Cos[c + d*x]])} -{x^2/Sqrt[a + a*Cos[c + d*x]], x, 8, -((4*I*x^2*ArcTan[E^((1/2)*I*(c + d*x))]*Cos[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]])) + (8*I*x*Cos[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - (8*I*x*Cos[c/2 + (d*x)/2]*PolyLog[2, I*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - (16*Cos[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^((1/2)*I*(c + d*x))])/(d^3*Sqrt[a + a*Cos[c + d*x]]) + (16*Cos[c/2 + (d*x)/2]*PolyLog[3, I*E^((1/2)*I*(c + d*x))])/(d^3*Sqrt[a + a*Cos[c + d*x]])} -{x/Sqrt[a + a*Cos[c + d*x]], x, 6, -((4*I*x*ArcTan[E^((1/2)*I*(c + d*x))]*Cos[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cos[c + d*x]])) + (4*I*Cos[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]]) - (4*I*Cos[c/2 + (d*x)/2]*PolyLog[2, I*E^((1/2)*I*(c + d*x))])/(d^2*Sqrt[a + a*Cos[c + d*x]])} -{1/Sqrt[a + a*Cos[c + d*x]], x, 2, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{1/(x*Sqrt[a + a*Cos[c + d*x]]), x, 0, Unintegrable[1/(x*Sqrt[a + a*Cos[c + d*x]]), x]} - - -{x^3/Sqrt[a - a*Cos[x]], x, 10, -((4*x^3*ArcTanh[E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]) + (12*I*x^2*PolyLog[2, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (12*I*x^2*PolyLog[2, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (48*x*PolyLog[3, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] + (48*x*PolyLog[3, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (96*I*PolyLog[4, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] + (96*I*PolyLog[4, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]} -{x^2/Sqrt[a - a*Cos[x]], x, 8, -((4*x^2*ArcTanh[E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]) + (8*I*x*PolyLog[2, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (8*I*x*PolyLog[2, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (16*PolyLog[3, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] + (16*PolyLog[3, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]} -{x/Sqrt[a - a*Cos[x]], x, 6, -((4*x*ArcTanh[E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]) + (4*I*PolyLog[2, -E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]] - (4*I*PolyLog[2, E^((I*x)/2)]*Sin[x/2])/Sqrt[a - a*Cos[x]]} -{1/Sqrt[a - a*Cos[x]], x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a - a*Cos[x]])])/Sqrt[a])} -{1/(x*Sqrt[a - a*Cos[x]]), x, 0, Unintegrable[1/(x*Sqrt[a - a*Cos[x]]), x]} - - -{x^3/(a + a*Cos[x])^(3/2), x, 16, -((3*x^2)/(a*Sqrt[a + a*Cos[x]])) - (24*I*x*ArcTan[E^((I*x)/2)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) - (I*x^3*ArcTan[E^((I*x)/2)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + (24*I*Cos[x/2]*PolyLog[2, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (3*I*x^2*Cos[x/2]*PolyLog[2, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (24*I*Cos[x/2]*PolyLog[2, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (3*I*x^2*Cos[x/2]*PolyLog[2, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (12*x*Cos[x/2]*PolyLog[3, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (12*x*Cos[x/2]*PolyLog[3, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (24*I*Cos[x/2]*PolyLog[4, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (24*I*Cos[x/2]*PolyLog[4, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (x^3*Tan[x/2])/(2*a*Sqrt[a + a*Cos[x]])} -{x^2/(a + a*Cos[x])^(3/2), x, 10, -((2*x)/(a*Sqrt[a + a*Cos[x]])) - (I*x^2*ArcTan[E^((I*x)/2)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + (4*ArcTanh[Sin[x/2]]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + (2*I*x*Cos[x/2]*PolyLog[2, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (2*I*x*Cos[x/2]*PolyLog[2, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (4*Cos[x/2]*PolyLog[3, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (4*Cos[x/2]*PolyLog[3, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (x^2*Tan[x/2])/(2*a*Sqrt[a + a*Cos[x]])} -{x^1/(a + a*Cos[x])^(3/2), x, 7, -(1/(a*Sqrt[a + a*Cos[x]])) - (I*x*ArcTan[E^((I*x)/2)]*Cos[x/2])/(a*Sqrt[a + a*Cos[x]]) + (I*Cos[x/2]*PolyLog[2, (-I)*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) - (I*Cos[x/2]*PolyLog[2, I*E^((I*x)/2)])/(a*Sqrt[a + a*Cos[x]]) + (x*Tan[x/2])/(2*a*Sqrt[a + a*Cos[x]])} -{1/(x*(a + a*Cos[x])^(3/2)), x, 0, Unintegrable[1/(x*(a + a*Cos[x])^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cos[e+f x])^(n/3)*) - - -(* Used to hang Rubi *) -{(a + a*Cos[c + d*x])^(1/3)/x, x, 0, Unintegrable[(a + a*Cos[c + d*x])^(1/3)/x, x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^m (a+a Cos[e+f x])^n with m symbolic*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n*) - - -(* {x^3/(a + b*Cos[c + d*x]), x, 12, -((I*x^3*Log[1 + (b*E^(I*c + I*d*x))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d)) + (I*x^3*Log[1 + (b*E^(I*c + I*d*x))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d) - (3*x^2*PolyLog[2, -((b*E^(I*c + I*d*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^2) + (3*x^2*PolyLog[2, -((b*E^(I*c + I*d*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^2) - (6*I*x*PolyLog[3, -((b*E^(I*c + I*d*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^3) + (6*I*x*PolyLog[3, -((b*E^(I*c + I*d*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^3) + (6*PolyLog[4, -((b*E^(I*c + I*d*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^4) - (6*PolyLog[4, -((b*E^(I*c + I*d*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^4)} *) -{x^3/(a + b*Cos[x]), x, 12, -((I*x^3*Log[1 + (b*E^(I*x))/(a - Sqrt[a^2 - b^2])])/Sqrt[a^2 - b^2]) + (I*x^3*Log[1 + (b*E^(I*x))/(a + Sqrt[a^2 - b^2])])/Sqrt[a^2 - b^2] - (3*x^2*PolyLog[2, -((b*E^(I*x))/(a - Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2] + (3*x^2*PolyLog[2, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2] - (6*I*x*PolyLog[3, -((b*E^(I*x))/(a - Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2] + (6*I*x*PolyLog[3, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2] + (6*PolyLog[4, -((b*E^(I*x))/(a - Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2] - (6*PolyLog[4, -((b*E^(I*x))/(a + Sqrt[a^2 - b^2]))])/Sqrt[a^2 - b^2]} -{x^2/(a + b*Cos[c + d*x]), x, 10, -((I*x^2*Log[1 + (b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d)) + (I*x^2*Log[1 + (b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d) - (2*x*PolyLog[2, -((b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^2) + (2*x*PolyLog[2, -((b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^2) - (2*I*PolyLog[3, -((b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^3) + (2*I*PolyLog[3, -((b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*d^3)} -{x^1/(a + b*Cos[c + d*x]), x, 8, -((I*x*Log[1 + (b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d)) + (I*x*Log[1 + (b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*d) - PolyLog[2, -((b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]))]/(Sqrt[a^2 - b^2]*d^2) + PolyLog[2, -((b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2]))]/(Sqrt[a^2 - b^2]*d^2)} -{1/(x^1*(a + b*Cos[x])), x, 0, Unintegrable[1/(x*(a + b*Cos[x])), x]} - - -{(e + f*x)/(a + b*Cos[c + d*x])^2, x, 11, -((I*a*(e + f*x)*Log[1 + (b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d)) + (I*a*(e + f*x)*Log[1 + (b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - (f*Log[a + b*Cos[c + d*x]])/((a^2 - b^2)*d^2) - (a*f*PolyLog[2, -((b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*d^2) + (a*f*PolyLog[2, -((b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*d^2) - (b*(e + f*x)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^m (a+b Cos[e+f x])^n with m symbolic*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.12 (e x)^m (a+b cos(c+d x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.12 (e x)^m (a+b cos(c+d x^n))^p.m deleted file mode 100644 index 496e18e..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.12 (e x)^m (a+b cos(c+d x^n))^p.m +++ /dev/null @@ -1,210 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[a+b x^2]^p*) - - -{x^3*Cos[a + b*x^2], x, 3, Cos[a + b*x^2]/(2*b^2) + (x^2*Sin[a + b*x^2])/(2*b)} -{x^2*Cos[a + b*x^2], x, 4, -((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/(2*b^(3/2))) - (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(2*b^(3/2)) + (x*Sin[a + b*x^2])/(2*b)} -{x^1*Cos[a + b*x^2], x, 2, Sin[a + b*x^2]/(2*b)} -{x^0*Cos[a + b*x^2], x, 3, (Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/Sqrt[b] - (Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/Sqrt[b]} -{Cos[a + b*x^2]/x^1, x, 3, (1/2)*Cos[a]*CosIntegral[b*x^2] - (1/2)*Sin[a]*SinIntegral[b*x^2]} -{Cos[a + b*x^2]/x^2, x, 4, -(Cos[a + b*x^2]/x) - Sqrt[b]*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x] - Sqrt[b]*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]} -{Cos[a + b*x^2]/x^3, x, 5, -(Cos[a + b*x^2]/(2*x^2)) - (1/2)*b*CosIntegral[b*x^2]*Sin[a] - (1/2)*b*Cos[a]*SinIntegral[b*x^2]} - - -{x^3*Cos[a + b*x^2]^2, x, 3, x^4/8 + Cos[a + b*x^2]^2/(8*b^2) + (x^2*Cos[a + b*x^2]*Sin[a + b*x^2])/(4*b)} -{x^2*Cos[a + b*x^2]^2, x, 6, x^3/6 - (Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]])/(16*b^(3/2)) - (Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a])/(16*b^(3/2)) + (x*Sin[2*a + 2*b*x^2])/(8*b)} -{x^1*Cos[a + b*x^2]^2, x, 3, x^2/4 + (Cos[a + b*x^2]*Sin[a + b*x^2])/(4*b)} -{x^0*Cos[a + b*x^2]^2, x, 5, x/2 + (Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]])/(4*Sqrt[b]) - (Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a])/(4*Sqrt[b])} -{Cos[a + b*x^2]^2/x^1, x, 5, (1/4)*Cos[2*a]*CosIntegral[2*b*x^2] + Log[x]/2 - (1/4)*Sin[2*a]*SinIntegral[2*b*x^2]} -{Cos[a + b*x^2]^2/x^2, x, 6, -(Cos[a + b*x^2]^2/x) - Sqrt[b]*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]] - Sqrt[b]*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a]} -{Cos[a + b*x^2]^2/x^3, x, 7, -(1/(4*x^2)) - Cos[2*(a + b*x^2)]/(4*x^2) - (1/2)*b*CosIntegral[2*b*x^2]*Sin[2*a] - (1/2)*b*Cos[2*a]*SinIntegral[2*b*x^2]} - - -{x^3*Cos[a + b*x^2]^3, x, 4, Cos[a + b*x^2]/(3*b^2) + Cos[a + b*x^2]^3/(18*b^2) + (x^2*Sin[a + b*x^2])/(3*b) + (x^2*Cos[a + b*x^2]^2*Sin[a + b*x^2])/(6*b)} -{x^2*Cos[a + b*x^2]^3, x, 10, -((3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/(8*b^(3/2))) - (Sqrt[Pi/6]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x])/(24*b^(3/2)) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(8*b^(3/2)) - (Sqrt[Pi/6]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(24*b^(3/2)) + (3*x*Sin[a + b*x^2])/(8*b) + (x*Sin[3*a + 3*b*x^2])/(24*b)} -{x^1*Cos[a + b*x^2]^3, x, 3, Sin[a + b*x^2]/(2*b) - Sin[a + b*x^2]^3/(6*b)} -{x^0*Cos[a + b*x^2]^3, x, 8, (3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/(4*Sqrt[b]) + (Sqrt[Pi/6]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x])/(4*Sqrt[b]) - (3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(4*Sqrt[b]) - (Sqrt[Pi/6]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(4*Sqrt[b])} -{Cos[a + b*x^2]^3/x^1, x, 8, (3/8)*Cos[a]*CosIntegral[b*x^2] + (1/8)*Cos[3*a]*CosIntegral[3*b*x^2] - (3/8)*Sin[a]*SinIntegral[b*x^2] - (1/8)*Sin[3*a]*SinIntegral[3*b*x^2]} -{Cos[a + b*x^2]^3/x^2, x, 9, -(Cos[a + b*x^2]^3/x) - (3/2)*Sqrt[b]*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x] - (1/2)*Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x] - (3/2)*Sqrt[b]*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a] - (1/2)*Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a]} -{Cos[a + b*x^2]^3/x^3, x, 12, -((3*Cos[a + b*x^2])/(8*x^2)) - Cos[3*(a + b*x^2)]/(8*x^2) - (3/8)*b*CosIntegral[b*x^2]*Sin[a] - (3/8)*b*CosIntegral[3*b*x^2]*Sin[3*a] - (3/8)*b*Cos[a]*SinIntegral[b*x^2] - (3/8)*b*Cos[3*a]*SinIntegral[3*b*x^2]} - - -{x*Cos[a + b*x^2]^7, x, 3, Sin[a + b*x^2]/(2*b) - Sin[a + b*x^2]^3/(2*b) + (3*Sin[a + b*x^2]^5)/(10*b) - Sin[a + b*x^2]^7/(14*b)} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Cos[a+b x^2]^(p/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^(m/2) Cos[a+b x^2]^p*) - - -{x^(5/2)*Cos[a + b*x^2], x, 4, -((3*I*E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/(16*b*((-I)*b*x^2)^(3/4))) + (3*I*x^(3/2)*Gamma[3/4, I*b*x^2])/(E^(I*a)*(16*b*(I*b*x^2)^(3/4))) + (x^(3/2)*Sin[a + b*x^2])/(2*b)} -{x^(3/2)*Cos[a + b*x^2], x, 4, -((I*E^(I*a)*Sqrt[x]*Gamma[1/4, (-I)*b*x^2])/(16*b*((-I)*b*x^2)^(1/4))) + (I*Sqrt[x]*Gamma[1/4, I*b*x^2])/(E^(I*a)*(16*b*(I*b*x^2)^(1/4))) + (Sqrt[x]*Sin[a + b*x^2])/(2*b)} -{x^(1/2)*Cos[a + b*x^2], x, 3, -((E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(3/4))) - (x^(3/2)*Gamma[3/4, I*b*x^2])/(E^(I*a)*(4*(I*b*x^2)^(3/4)))} -{Cos[a + b*x^2]/x^(1/2), x, 3, -((E^(I*a)*Sqrt[x]*Gamma[1/4, (-I)*b*x^2])/(4*((-I)*b*x^2)^(1/4))) - (Sqrt[x]*Gamma[1/4, I*b*x^2])/(E^(I*a)*(4*(I*b*x^2)^(1/4)))} -{Cos[a + b*x^2]/x^(3/2), x, 4, -((2*Cos[a + b*x^2])/Sqrt[x]) - (I*b*E^(I*a)*x^(3/2)*Gamma[3/4, (-I)*b*x^2])/((-I)*b*x^2)^(3/4) + (I*b*x^(3/2)*Gamma[3/4, I*b*x^2])/(E^(I*a)*(I*b*x^2)^(3/4))} -{Cos[a + b*x^2]/x^(5/2), x, 4, -((2*Cos[a + b*x^2])/(3*x^(3/2))) - (I*b*E^(I*a)*Sqrt[x]*Gamma[1/4, (-I)*b*x^2])/(3*((-I)*b*x^2)^(1/4)) + (I*b*Sqrt[x]*Gamma[1/4, I*b*x^2])/(E^(I*a)*(3*(I*b*x^2)^(1/4)))} - - -{x^(5/2)*Cos[a + b*x^2]^2, x, 7, x^(7/2)/7 - (3*I*E^(2*I*a)*x^(3/2)*Gamma[3/4, -2*I*b*x^2])/(64*2^(3/4)*b*((-I)*b*x^2)^(3/4)) + (3*I*x^(3/2)*Gamma[3/4, 2*I*b*x^2])/(E^(2*I*a)*(64*2^(3/4)*b*(I*b*x^2)^(3/4))) + (x^(3/2)*Sin[2*(a + b*x^2)])/(8*b)} -{x^(3/2)*Cos[a + b*x^2]^2, x, 7, x^(5/2)/5 - (I*E^(2*I*a)*Sqrt[x]*Gamma[1/4, -2*I*b*x^2])/(64*2^(1/4)*b*((-I)*b*x^2)^(1/4)) + (I*Sqrt[x]*Gamma[1/4, 2*I*b*x^2])/(E^(2*I*a)*(64*2^(1/4)*b*(I*b*x^2)^(1/4))) + (Sqrt[x]*Sin[2*(a + b*x^2)])/(8*b)} -{x^(1/2)*Cos[a + b*x^2]^2, x, 6, x^(3/2)/3 - (E^(2*I*a)*x^(3/2)*Gamma[3/4, -2*I*b*x^2])/(8*2^(3/4)*((-I)*b*x^2)^(3/4)) - (x^(3/2)*Gamma[3/4, 2*I*b*x^2])/(E^(2*I*a)*(8*2^(3/4)*(I*b*x^2)^(3/4)))} -{Cos[a + b*x^2]^2/x^(1/2), x, 6, Sqrt[x] - (E^(2*I*a)*Sqrt[x]*Gamma[1/4, -2*I*b*x^2])/(8*2^(1/4)*((-I)*b*x^2)^(1/4)) - (Sqrt[x]*Gamma[1/4, 2*I*b*x^2])/(E^(2*I*a)*(8*2^(1/4)*(I*b*x^2)^(1/4)))} -{Cos[a + b*x^2]^2/x^(3/2), x, 7, -(1/Sqrt[x]) - Cos[2*(a + b*x^2)]/Sqrt[x] - (I*b*E^(2*I*a)*x^(3/2)*Gamma[3/4, -2*I*b*x^2])/(2^(3/4)*((-I)*b*x^2)^(3/4)) + (I*b*x^(3/2)*Gamma[3/4, 2*I*b*x^2])/(E^(2*I*a)*(2^(3/4)*(I*b*x^2)^(3/4)))} -{Cos[a + b*x^2]^2/x^(5/2), x, 7, -((2*Cos[a + b*x^2]^2)/(3*x^(3/2))) - (I*b*E^(2*I*a)*Sqrt[x]*Gamma[1/4, -2*I*b*x^2])/(3*2^(1/4)*((-I)*b*x^2)^(1/4)) + (I*b*Sqrt[x]*Gamma[1/4, 2*I*b*x^2])/(E^(2*I*a)*(3*2^(1/4)*(I*b*x^2)^(1/4)))} - - -(* ::Subsection:: *) -(*Integrands of the form x^(m/2) Cos[a+b x^2]^(p/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d / x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[a+b / x]^p*) - - -{Cos[a + b/x], x, 5, x*Cos[a + b/x] + b*CosIntegral[b/x]*Sin[a] + b*Cos[a]*SinIntegral[b/x]} -{Cos[a + b/x]/x, x, 3, (-Cos[a])*CosIntegral[b/x] + Sin[a]*SinIntegral[b/x]} -{Cos[a + b/x]/x^2, x, 2, -(Sin[a + b/x]/b)} -{Cos[a + b/x]/x^3, x, 3, -(Cos[a + b/x]/b^2) - Sin[a + b/x]/(b*x)} -{Cos[a + b/x]/x^4, x, 4, -((2*Cos[a + b/x])/(b^2*x)) + (2*Sin[a + b/x])/b^3 - Sin[a + b/x]/(b*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d / x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[a+b / x^2]^p*) - - -{x^0*Cos[a + b/x^2], x, 5, x*Cos[a + b/x^2] + Sqrt[b]*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x] + Sqrt[b]*Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a]} -{Cos[a + b/x^2]/x^1, x, 3, (-(1/2))*Cos[a]*CosIntegral[b/x^2] + (1/2)*Sin[a]*SinIntegral[b/x^2]} -{Cos[a + b/x^2]/x^2, x, 4, -((Sqrt[Pi/2]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x])/Sqrt[b]) + (Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/Sqrt[b]} -{Cos[a + b/x^2]/x^3, x, 2, -(Sin[a + b/x^2]/(2*b))} -{Cos[a + b/x^2]/x^4, x, 5, (Sqrt[Pi/2]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/x])/(2*b^(3/2)) + (Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/x]*Sin[a])/(2*b^(3/2)) - Sin[a + b/x^2]/(2*b*x)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d x^(1/2)])^p*) - - -{Cos[Sqrt[x]]^2/Sqrt[x], x, 3, Sqrt[x] + Cos[Sqrt[x]]*Sin[Sqrt[x]]} -{Cos[Sqrt[x]]/Sqrt[x], x, 2, 2*Sin[Sqrt[x]]} -{Cos[Sqrt[x]], x, 3, 2*Cos[Sqrt[x]] + 2*Sqrt[x]*Sin[Sqrt[x]]} -{Cos[Sqrt[x]]^2, x, 3, x/2 + (1/2)*Cos[Sqrt[x]]^2 + Sqrt[x]*Cos[Sqrt[x]]*Sin[Sqrt[x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d x^(1/3)])^p*) - - -{x^(3/2)*Cos[a + b*x^(1/3)], x, 13, (135135*Sqrt[x]*Cos[a + b*x^(1/3)])/(32*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)])/(8*b^4) + (39*x^(11/6)*Cos[a + b*x^(1/3)])/(2*b^2) + (405405*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(64*b^(15/2)) + (405405*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/(64*b^(15/2)) - (405405*x^(1/6)*Sin[a + b*x^(1/3)])/(64*b^7) + (27027*x^(5/6)*Sin[a + b*x^(1/3)])/(16*b^5) - (429*x^(3/2)*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(13/6)*Sin[a + b*x^(1/3)])/b} -{x^(1/2)*Cos[a + b*x^(1/3)], x, 10, -((315*x^(1/6)*Cos[a + b*x^(1/3)])/(8*b^4)) + (21*x^(5/6)*Cos[a + b*x^(1/3)])/(2*b^2) + (315*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/(8*b^(9/2)) - (315*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/(8*b^(9/2)) - (105*Sqrt[x]*Sin[a + b*x^(1/3)])/(4*b^3) + (3*x^(7/6)*Sin[a + b*x^(1/3)])/b} -{Cos[a + b*x^(1/3)]/x^(1/2), x, 7, -((3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/b^(3/2)) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/b^(3/2) + (3*x^(1/6)*Sin[a + b*x^(1/3)])/b} -{Cos[a + b*x^(1/3)]/x^(3/2), x, 8, -((2*Cos[a + b*x^(1/3)])/Sqrt[x]) - 4*b^(3/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] + 4*b^(3/2)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/x^(1/6)} -{Cos[a + b*x^(1/3)]/x^(5/2), x, 11, -((2*Cos[a + b*x^(1/3)])/(3*x^(3/2))) + (8*b^2*Cos[a + b*x^(1/3)])/(105*x^(5/6)) - (32*b^4*Cos[a + b*x^(1/3)])/(315*x^(1/6)) - (32/315)*b^(9/2)*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] - (32/315)*b^(9/2)*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + (4*b*Sin[a + b*x^(1/3)])/(21*x^(7/6)) - (16*b^3*Sin[a + b*x^(1/3)])/(315*Sqrt[x])} -{Cos[a + b*x^(1/3)]/x^(7/2), x, 14, -((2*Cos[a + b*x^(1/3)])/(5*x^(5/2))) + (8*b^2*Cos[a + b*x^(1/3)])/(715*x^(11/6)) - (32*b^4*Cos[a + b*x^(1/3)])/(45045*x^(7/6)) + (128*b^6*Cos[a + b*x^(1/3)])/(675675*Sqrt[x]) + (256*b^(15/2)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)])/675675 - (256*b^(15/2)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a])/675675 + (4*b*Sin[a + b*x^(1/3)])/(65*x^(13/6)) - (16*b^3*Sin[a + b*x^(1/3)])/(6435*x^(3/2)) + (64*b^5*Sin[a + b*x^(1/3)])/(225225*x^(5/6)) - (256*b^7*Sin[a + b*x^(1/3)])/(675675*x^(1/6))} - - -{x^(3/2)*Cos[a + b*x^(1/3)]^2, x, 15, -((135135*Sqrt[x])/(4096*b^6)) + (3861*x^(7/6))/(256*b^4) - (39*x^(11/6))/(16*b^2) + x^(5/2)/5 + (135135*Sqrt[x]*Cos[a + b*x^(1/3)]^2)/(2048*b^6) - (3861*x^(7/6)*Cos[a + b*x^(1/3)]^2)/(128*b^4) + (39*x^(11/6)*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (405405*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(32768*b^(15/2)) + (405405*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(32768*b^(15/2)) + (27027*x^(5/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(512*b^5) - (429*x^(3/2)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(32*b^3) + (3*x^(13/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b) - (405405*x^(1/6)*Sin[2*(a + b*x^(1/3))])/(16384*b^7)} -{x^(1/2)*Cos[a + b*x^(1/3)]^2, x, 12, (315*x^(1/6))/(256*b^4) - (21*x^(5/6))/(16*b^2) + x^(3/2)/3 - (315*x^(1/6)*Cos[a + b*x^(1/3)]^2)/(128*b^4) + (21*x^(5/6)*Cos[a + b*x^(1/3)]^2)/(8*b^2) + (315*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(512*b^(9/2)) - (315*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(512*b^(9/2)) - (105*Sqrt[x]*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(32*b^3) + (3*x^(7/6)*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(2*b)} -{Cos[a + b*x^(1/3)]^2/x^(1/2), x, 9, Sqrt[x] - (3*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(8*b^(3/2)) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/(8*b^(3/2)) + (3*x^(1/6)*Sin[2*(a + b*x^(1/3))])/(4*b)} -{Cos[a + b*x^(1/3)]^2/x^(3/2), x, 10, -((2*Cos[a + b*x^(1/3)]^2)/Sqrt[x]) - 8*b^(3/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] + 8*b^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/x^(1/6)} -{Cos[a + b*x^(1/3)]^2/x^(5/2), x, 12, -((16*b^2)/(105*x^(5/6))) + (256*b^4)/(315*x^(1/6)) - (2*Cos[a + b*x^(1/3)]^2)/(3*x^(3/2)) + (32*b^2*Cos[a + b*x^(1/3)]^2)/(105*x^(5/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(315*x^(1/6)) - (512/315)*b^(9/2)*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]] - (512/315)*b^(9/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(21*x^(7/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(315*Sqrt[x])} -{Cos[a + b*x^(1/3)]^2/x^(7/2), x, 16, -((16*b^2)/(715*x^(11/6))) + (256*b^4)/(45045*x^(7/6)) - (4096*b^6)/(675675*Sqrt[x]) - (2*Cos[a + b*x^(1/3)]^2)/(5*x^(5/2)) + (32*b^2*Cos[a + b*x^(1/3)]^2)/(715*x^(11/6)) - (512*b^4*Cos[a + b*x^(1/3)]^2)/(45045*x^(7/6)) + (8192*b^6*Cos[a + b*x^(1/3)]^2)/(675675*Sqrt[x]) + (32768*b^(15/2)*Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]])/675675 - (32768*b^(15/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a])/675675 + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(65*x^(13/6)) - (128*b^3*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(6435*x^(3/2)) + (2048*b^5*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(225225*x^(5/6)) - (32768*b^7*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(675675*x^(1/6))} - - -{Cos[x^(1/3)]^3, x, 7, 4*x^(1/3)*Cos[x^(1/3)] + (2/3)*x^(1/3)*Cos[x^(1/3)]^3 - (14/3)*Sin[x^(1/3)] + 2*x^(2/3)*Sin[x^(1/3)] + x^(2/3)*Cos[x^(1/3)]^2*Sin[x^(1/3)] + (2/9)*Sin[x^(1/3)]^3} -{Cos[x^(1/6)]/x^(5/6), x, 2, 6*Sin[x^(1/6)]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cos[c+d x^n])^p*) - - -{(e*x)^m*(b*Cos[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(b*Cos[c + d*x^n])^p, x]} -{(e*x)^m*(a + b*Cos[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Cos[c + d*x^n])^p, x]} - - -{(e*x)^(n - 1)*(b*Cos[c + d*x^n])^p, x, 3, -(((e*x)^n*(b*Cos[c + d*x^n])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x^n]^2]*Sin[c + d*x^n])/(x^n*(b*d*e*n*(1 + p)*Sqrt[Sin[c + d*x^n]^2])))} -{(e*x)^(2*n - 1)*(b*Cos[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(b*Cos[c + d*x^n])^p, x])/(x^(2*n)*e)} - -{(e*x)^(n - 1)*(a + b*Cos[c + d*x^n])^p, x, 5, (Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1/2)*(1 - Cos[c + d*x^n]), (b*(1 - Cos[c + d*x^n]))/(a + b)]*(a + b*Cos[c + d*x^n])^p*Sin[c + d*x^n])/(x^n*((a + b*Cos[c + d*x^n])/(a + b))^p*(d*e*n*Sqrt[1 + Cos[c + d*x^n]]))} -{(e*x)^(2*n - 1)*(a + b*Cos[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(a + b*Cos[c + d*x^n])^p, x])/(x^(2*n)*e)} - - -{Cos[a + b*x^n]/x, x, 3, (Cos[a]*CosIntegral[b*x^n])/n - (Sin[a]*SinIntegral[b*x^n])/n} -{Cos[a + b*x^n]^2/x, x, 5, (Cos[2*a]*CosIntegral[2*b*x^n])/(2*n) + Log[x]/2 - (Sin[2*a]*SinIntegral[2*b*x^n])/(2*n)} -{Cos[a + b*x^n]^3/x, x, 8, (3*Cos[a]*CosIntegral[b*x^n])/(4*n) + (Cos[3*a]*CosIntegral[3*b*x^n])/(4*n) - (3*Sin[a]*SinIntegral[b*x^n])/(4*n) - (Sin[3*a]*SinIntegral[3*b*x^n])/(4*n)} -{Cos[a + b*x^n]^4/x, x, 8, (Cos[2*a]*CosIntegral[2*b*x^n])/(2*n) + (Cos[4*a]*CosIntegral[4*b*x^n])/(8*n) + (3*Log[x])/8 - (Sin[2*a]*SinIntegral[2*b*x^n])/(2*n) - (Sin[4*a]*SinIntegral[4*b*x^n])/(8*n)} - - -{Cos[a + b*x^n], x, 3, -((E^(I*a)*x*Gamma[1/n, (-I)*b*x^n])/(((-I)*b*x^n)^n^(-1)*(2*n))) - (x*Gamma[1/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^n^(-1)*(2*n))} -{Cos[a + b*x^n]^2, x, 5, x/2 - (2^(-2 - 1/n)*E^(2*I*a)*x*Gamma[1/n, -2*I*b*x^n])/(((-I)*b*x^n)^n^(-1)*n) - (2^(-2 - 1/n)*x*Gamma[1/n, 2*I*b*x^n])/(E^(2*I*a)*(I*b*x^n)^n^(-1)*n)} -{Cos[a + b*x^n]^3, x, 8, -((3*E^(I*a)*x*Gamma[1/n, (-I)*b*x^n])/(((-I)*b*x^n)^n^(-1)*(8*n))) - (3*x*Gamma[1/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^n^(-1)*(8*n)) - (E^(3*I*a)*x*Gamma[1/n, -3*I*b*x^n])/(3^n^(-1)*((-I)*b*x^n)^n^(-1)*(8*n)) - (x*Gamma[1/n, 3*I*b*x^n])/(3^n^(-1)*E^(3*I*a)*(I*b*x^n)^n^(-1)*(8*n))} - -{x^m*Cos[a + b*x^n], x, 3, -((E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(((-I)*b*x^n)^((1 + m)/n)*(2*n))) - (x^(1 + m)*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^((1 + m)/n)*(2*n))} -{x^m*Cos[a + b*x^n]^2, x, 5, x^(1 + m)/(2*(1 + m)) - (E^(2*I*a)*x^(1 + m)*Gamma[(1 + m)/n, -2*I*b*x^n])/(2^((1 + m + 2*n)/n)*((-I)*b*x^n)^((1 + m)/n)*n) - (x^(1 + m)*Gamma[(1 + m)/n, 2*I*b*x^n])/(2^((1 + m + 2*n)/n)*E^(2*I*a)*(I*b*x^n)^((1 + m)/n)*n)} -{x^m*Cos[a + b*x^n]^3, x, 8, -((3*E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(((-I)*b*x^n)^((1 + m)/n)*(8*n))) - (3*x^(1 + m)*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*(I*b*x^n)^((1 + m)/n)*(8*n)) - (E^(3*I*a)*x^(1 + m)*Gamma[(1 + m)/n, -3*I*b*x^n])/(3^((1 + m)/n)*((-I)*b*x^n)^((1 + m)/n)*(8*n)) - (x^(1 + m)*Gamma[(1 + m)/n, 3*I*b*x^n])/(3^((1 + m)/n)*E^(3*I*a)*(I*b*x^n)^((1 + m)/n)*(8*n))} - - -{Cos[a + b*x^n]/x^(n + 1), x, 5, -(Cos[a + b*x^n]/(x^n*n)) - (b*CosIntegral[b*x^n]*Sin[a])/n - (b*Cos[a]*SinIntegral[b*x^n])/n} -{Cos[a + b*x^n]^2/x^(n + 1), x, 7, -(1/(x^n*(2*n))) - Cos[2*(a + b*x^n)]/(x^n*(2*n)) - (b*CosIntegral[2*b*x^n]*Sin[2*a])/n - (b*Cos[2*a]*SinIntegral[2*b*x^n])/n} -{Cos[a + b*x^n]^3/x^(n + 1), x, 12, -((3*Cos[a + b*x^n])/(x^n*(4*n))) - Cos[3*(a + b*x^n)]/(x^n*(4*n)) - (3*b*CosIntegral[b*x^n]*Sin[a])/(4*n) - (3*b*CosIntegral[3*b*x^n]*Sin[3*a])/(4*n) - (3*b*Cos[a]*SinIntegral[b*x^n])/(4*n) - (3*b*Cos[3*a]*SinIntegral[3*b*x^n])/(4*n)} - -{Cos[a + b*x^n]/x^(2*n + 1), x, 6, -(Cos[a + b*x^n]/(x^(2*n)*(2*n))) - (b^2*Cos[a]*CosIntegral[b*x^n])/(2*n) + (b*Sin[a + b*x^n])/(x^n*(2*n)) + (b^2*Sin[a]*SinIntegral[b*x^n])/(2*n)} -{Cos[a + b*x^n]^2/x^(2*n + 1), x, 8, -(1/(x^(2*n)*(4*n))) - Cos[2*(a + b*x^n)]/(x^(2*n)*(4*n)) - (b^2*Cos[2*a]*CosIntegral[2*b*x^n])/n + (b*Sin[2*(a + b*x^n)])/(x^n*(2*n)) + (b^2*Sin[2*a]*SinIntegral[2*b*x^n])/n} -{Cos[a + b*x^n]^3/x^(2*n + 1), x, 14, -((3*Cos[a + b*x^n])/(x^(2*n)*(8*n))) - Cos[3*(a + b*x^n)]/(x^(2*n)*(8*n)) - (3*b^2*Cos[a]*CosIntegral[b*x^n])/(8*n) - (9*b^2*Cos[3*a]*CosIntegral[3*b*x^n])/(8*n) + (3*b*Sin[a + b*x^n])/(x^n*(8*n)) + (3*b*Sin[3*(a + b*x^n)])/(x^n*(8*n)) + (3*b^2*Sin[a]*SinIntegral[b*x^n])/(8*n) + (9*b^2*Sin[3*a]*SinIntegral[3*b*x^n])/(8*n)} - - -(* ::Title:: *) -(*Integrands of the form (e x)^m Cos[a+b (c+d x)^n]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[a+b (c+d x)^n]*) - - -{x^2*Cos[(a + b*x)^2], x, 7, (a^2*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b^3 - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*(a + b*x)])/(2*b^3) - (a*Sin[(a + b*x)^2])/b^3 + ((a + b*x)*Sin[(a + b*x)^2])/(2*b^3)} -{x^1*Cos[(a + b*x)^2], x, 5, -((a*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b^2) + Sin[(a + b*x)^2]/(2*b^2)} -{x^0*Cos[(a + b*x)^2], x, 1, (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b} -{Cos[(a + b*x)^2]/x^1, x, 0, Unintegrable[Cos[(a + b*x)^2]/x, x]} -{Cos[(a + b*x)^2]/x^2, x, 0, Unintegrable[Cos[(a + b*x)^2]/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[a+b (c+d x)^(n/2)]*) - - -{Cos[a + b*Sqrt[c + d*x]]*x^2, x, 14, (240*Cos[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (24*c*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (2*c^2*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (120*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (12*c*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (10*(c + d*x)^2*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (240*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b^5*d^3) + (24*c*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b*d^3) - (40*(c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d^3)} -{Cos[a + b*Sqrt[c + d*x]]*x^1, x, 8, -((12*Cos[a + b*Sqrt[c + d*x]])/(b^4*d^2)) - (2*c*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^2) + (6*(c + d*x)*Cos[a + b*Sqrt[c + d*x]])/(b^2*d^2) - (12*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*c*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b*d^2) + (2*(c + d*x)^(3/2)*Sin[a + b*Sqrt[c + d*x]])/(b*d^2)} -{Cos[a + b*Sqrt[c + d*x]]*x^0, x, 3, (2*Cos[a + b*Sqrt[c + d*x]])/(b^2*d) + (2*Sqrt[c + d*x]*Sin[a + b*Sqrt[c + d*x]])/(b*d)} -{Cos[a + b*Sqrt[c + d*x]]/x^1, x, 8, Cos[a - b*Sqrt[c]]*CosIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Cos[a + b*Sqrt[c]]*CosIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]] - Sin[a - b*Sqrt[c]]*SinIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] + Sin[a + b*Sqrt[c]]*SinIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]]} -{Cos[a + b*Sqrt[c + d*x]]/x^2, x, 10, -(Cos[a + b*Sqrt[c + d*x]]/x) + (b*d*CosIntegral[b*(Sqrt[c] + Sqrt[c + d*x])]*Sin[a - b*Sqrt[c]])/(2*Sqrt[c]) - (b*d*CosIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]]*Sin[a + b*Sqrt[c]])/(2*Sqrt[c]) + (b*d*Cos[a - b*Sqrt[c]]*SinIntegral[b*(Sqrt[c] + Sqrt[c + d*x])])/(2*Sqrt[c]) + (b*d*Cos[a + b*Sqrt[c]]*SinIntegral[b*Sqrt[c] - b*Sqrt[c + d*x]])/(2*Sqrt[c])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[a+b (c+d x)^(n/3)]*) - - -{Cos[a + b*(c + d*x)^(1/3)]*x^2, x, 20, -((720*c*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3)) - (120960*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^8*d^3) + (6*c^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (20160*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (30*c*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (24*(c + d*x)^(7/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (120960*Sin[a + b*(c + d*x)^(1/3)])/(b^9*d^3) - (6*c^2*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (60480*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*c*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) - (168*(c + d*x)^2*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3)} -{Cos[a + b*(c + d*x)^(1/3)]*x^1, x, 11, (360*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^2) - (6*c*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^2) + (15*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^2) + (6*c*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (360*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^2) - (3*c*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2) - (60*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (3*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2)} -{Cos[a + b*(c + d*x)^(1/3)]*x^0, x, 4, (6*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d) - (6*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d) + (3*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d)} -{Cos[a + b*(c + d*x)^(1/3)]/x^1, x, 11, Cos[a + b*c^(1/3)]*CosIntegral[b*c^(1/3) - b*(c + d*x)^(1/3)] + Cos[a + (-1)^(2/3)*b*c^(1/3)]*CosIntegral[(-1)^(2/3)*b*c^(1/3) - b*(c + d*x)^(1/3)] + Cos[a - (-1)^(1/3)*b*c^(1/3)]*CosIntegral[(-1)^(1/3)*b*c^(1/3) + b*(c + d*x)^(1/3)] + Sin[a + b*c^(1/3)]*SinIntegral[b*c^(1/3) - b*(c + d*x)^(1/3)] + Sin[a + (-1)^(2/3)*b*c^(1/3)]*SinIntegral[(-1)^(2/3)*b*c^(1/3) - b*(c + d*x)^(1/3)] - Sin[a - (-1)^(1/3)*b*c^(1/3)]*SinIntegral[(-1)^(1/3)*b*c^(1/3) + b*(c + d*x)^(1/3)]} -{Cos[a + b*(c + d*x)^(1/3)]/x^2, x, 13, -(Cos[a + b*(c + d*x)^(1/3)]/x) - (b*d*CosIntegral[b*c^(1/3) - b*(c + d*x)^(1/3)]*Sin[a + b*c^(1/3)])/(3*c^(2/3)) + ((-1)^(1/3)*b*d*CosIntegral[(-1)^(1/3)*b*c^(1/3) + b*(c + d*x)^(1/3)]*Sin[a - (-1)^(1/3)*b*c^(1/3)])/(3*c^(2/3)) - ((-1)^(2/3)*b*d*CosIntegral[(-1)^(2/3)*b*c^(1/3) - b*(c + d*x)^(1/3)]*Sin[a + (-1)^(2/3)*b*c^(1/3)])/(3*c^(2/3)) + (b*d*Cos[a + b*c^(1/3)]*SinIntegral[b*c^(1/3) - b*(c + d*x)^(1/3)])/(3*c^(2/3)) + ((-1)^(2/3)*b*d*Cos[a + (-1)^(2/3)*b*c^(1/3)]*SinIntegral[(-1)^(2/3)*b*c^(1/3) - b*(c + d*x)^(1/3)])/(3*c^(2/3)) + ((-1)^(1/3)*b*d*Cos[a - (-1)^(1/3)*b*c^(1/3)]*SinIntegral[(-1)^(1/3)*b*c^(1/3) + b*(c + d*x)^(1/3)])/(3*c^(2/3))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.13 (d+e x)^m cos(a+b x+c x^2)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.13 (d+e x)^m cos(a+b x+c x^2)^n.m deleted file mode 100644 index 44c1f64..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.13 (d+e x)^m cos(a+b x+c x^2)^n.m +++ /dev/null @@ -1,58 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form (d+e x)^m Cos[a+b x+c x^2]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[a+b x+c x^2]^n*) - - -{x^2*Cos[a + b*x + c*x^2], x, 8, (b^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) - (Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - (Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2)) - (b^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (b*Sin[a + b*x + c*x^2])/(4*c^2) + (x*Sin[a + b*x + c*x^2])/(2*c)} -{x*Cos[a + b*x + c*x^2], x, 4, -((b*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2))) + (b*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2)) + Sin[a + b*x + c*x^2]/(2*c)} -{Cos[a + b*x + c*x^2], x, 3, (Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] - (Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]} -{Cos[a + b*x + c*x^2]/x, x, 0, Unintegrable[Cos[a + b*x + c*x^2]/x, x]} -{Cos[a + b*x + c*x^2]/x^2 + b*Sin[a + b*x + c*x^2]/x, x, 5, -(Cos[a + b*x + c*x^2]/x) - Sqrt[c]*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] - Sqrt[c]*Sqrt[2*Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)]} - -{x^2*Cos[a + b*x - c*x^2], x, 8, -((b^2*Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2))) + (Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - (Sqrt[Pi/2]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(2*c^(3/2)) - (b^2*Sqrt[Pi/2]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(4*c^(5/2)) - (b*Sin[a + b*x - c*x^2])/(4*c^2) - (x*Sin[a + b*x - c*x^2])/(2*c)} -{x*Cos[a + b*x - c*x^2], x, 4, -((b*Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2))) - (b*Sqrt[Pi/2]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/(2*c^(3/2)) - Sin[a + b*x - c*x^2]/(2*c)} -{Cos[a + b*x - c*x^2], x, 3, -((Sqrt[Pi/2]*Cos[a + b^2/(4*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c]) - (Sqrt[Pi/2]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)])/Sqrt[c]} -{Cos[a + b*x - c*x^2]/x, x, 0, Unintegrable[Cos[a + b*x - c*x^2]/x, x]} -{Cos[a + b*x - c*x^2]/x^2 + b*Sin[a + b*x - c*x^2]/x, x, 5, -(Cos[a + b*x - c*x^2]/x) + Sqrt[c]*Sqrt[2*Pi]*Cos[a + b^2/(4*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] - Sqrt[c]*Sqrt[2*Pi]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a + b^2/(4*c)]} - -{x^2*Cos[1/4 + x + x^2], x, 6, (1/4)*Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] - (1/2)*Sqrt[Pi/2]*FresnelS[(1 + 2*x)/Sqrt[2*Pi]] - (1/4)*Sin[1/4 + x + x^2] + (1/2)*x*Sin[1/4 + x + x^2]} -{x*Cos[1/4 + x + x^2], x, 3, (-(1/2))*Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]] + (1/2)*Sin[1/4 + x + x^2]} -{Cos[1/4 + x + x^2], x, 2, Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]]} -{Cos[1/4 + x + x^2]/x, x, 0, Unintegrable[Cos[1/4 + x + x^2]/x, x]} -{Cos[1/4 + x + x^2]/x^2, x, 3, -(Cos[1/4 + x + x^2]/x) - Sqrt[2*Pi]*FresnelS[(1 + 2*x)/Sqrt[2*Pi]] - Unintegrable[Sin[1/4 + x + x^2]/x, x]} - - -{x^2*Cos[a + b*x + c*x^2]^2, x, 10, x^3/6 + (b^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) - (Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) - (Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) - (b^2*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) - (b*Sin[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (x*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{x*Cos[a + b*x + c*x^2]^2, x, 6, x^2/4 - (b*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) + (b*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2)) + Sin[2*a + 2*b*x + 2*c*x^2]/(8*c)} -{Cos[a + b*x + c*x^2]^2, x, 5, x/2 + (Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(4*Sqrt[c]) - (Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(4*Sqrt[c])} -{Cos[a + b*x + c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cos[2*a + 2*b*x + 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Cos[a + b*x - c*x^2]^2, x, 10, x^3/6 - (b^2*Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) + (Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) - (Sqrt[Pi]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(16*c^(3/2)) - (b^2*Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(16*c^(5/2)) - (b*Sin[2*a + 2*b*x - 2*c*x^2])/(16*c^2) - (x*Sin[2*a + 2*b*x - 2*c*x^2])/(8*c)} -{x*Cos[a + b*x - c*x^2]^2, x, 6, x^2/4 - (b*Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) - (b*Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(8*c^(3/2)) - Sin[2*a + 2*b*x - 2*c*x^2]/(8*c)} -{Cos[a + b*x - c*x^2]^2, x, 5, x/2 - (Sqrt[Pi]*Cos[2*a + b^2/(2*c)]*FresnelC[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(4*Sqrt[c]) - (Sqrt[Pi]*FresnelS[(b - 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a + b^2/(2*c)])/(4*Sqrt[c])} -{Cos[a + b*x - c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cos[2*a + 2*b*x - 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Cos[1/4 + x + x^2]^2, x, 8, x^3/6 + (1/16)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]] - (1/16)*Sqrt[Pi]*FresnelS[(1 + 2*x)/Sqrt[Pi]] - (1/16)*Sin[1/2 + 2*x + 2*x^2] + (1/8)*x*Sin[1/2 + 2*x + 2*x^2]} -{x*Cos[1/4 + x + x^2]^2, x, 5, x^2/4 - (1/8)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]] + (1/8)*Sin[1/2 + 2*x + 2*x^2]} -{Cos[1/4 + x + x^2]^2, x, 4, x/2 + (1/4)*Sqrt[Pi]*FresnelC[(1 + 2*x)/Sqrt[Pi]]} -{Cos[1/4 + x + x^2]^2/x, x, 2, (1/2)*Unintegrable[Cos[1/2 + 2*x + 2*x^2]/x, x] + Log[x]/2} -{Cos[1/4 + x + x^2]^2/x^2, x, 5, -(1/(2*x)) - Cos[1/2 + 2*x + 2*x^2]/(2*x) - Sqrt[Pi]*FresnelS[(1 + 2*x)/Sqrt[Pi]] - Unintegrable[Sin[1/2 + 2*x + 2*x^2]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m Cos[a+b x+c x^2]^n*) - - -{(d + e*x)^2*Cos[a + b*x + c*x^2], x, 8, ((2*c*d - b*e)^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) - (e^2*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - (e^2*Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2)) - ((2*c*d - b*e)^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) + (e*(2*c*d - b*e)*Sin[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sin[a + b*x + c*x^2])/(2*c)} -{(d + e*x)*Cos[a + b*x + c*x^2], x, 4, ((2*c*d - b*e)*Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) - ((2*c*d - b*e)*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2)) + (e*Sin[a + b*x + c*x^2])/(2*c)} -{Cos[a + b*x + c*x^2]/(d + e*x), x, 0, Unintegrable[Cos[a + b*x + c*x^2]/(d + e*x), x]} - - -{(d + e*x)^2*Cos[a + b*x + c*x^2]^2, x, 10, (d + e*x)^3/(6*e) + ((2*c*d - b*e)^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(5/2)) - (e^2*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(16*c^(3/2)) - (e^2*Sqrt[Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(3/2)) - ((2*c*d - b*e)^2*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(16*c^(5/2)) + (e*(2*c*d - b*e)*Sin[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (e*(d + e*x)*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{(d + e*x)*Cos[a + b*x + c*x^2]^2, x, 6, (d + e*x)^2/(4*e) + ((2*c*d - b*e)*Sqrt[Pi]*Cos[2*a - b^2/(2*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])])/(8*c^(3/2)) - ((2*c*d - b*e)*Sqrt[Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[Pi])]*Sin[2*a - b^2/(2*c)])/(8*c^(3/2)) + (e*Sin[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{Cos[a + b*x + c*x^2]^2/(d + e*x), x, 2, (1/2)*Unintegrable[Cos[2*a + 2*b*x + 2*c*x^2]/(d + e*x), x] + Log[d + e*x]/(2*e)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.1 (a+b cos)^m (c+d cos)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.1 (a+b cos)^m (c+d cos)^n.m deleted file mode 100644 index ee409df..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.1 (a+b cos)^m (c+d cos)^n.m +++ /dev/null @@ -1,1524 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Cos[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Cos[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^5*(a + a*Cos[c + d*x]), x, 7, (5*a*x)/16 + (a*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^4*(a + a*Cos[c + d*x]), x, 6, (3*a*x)/8 + (a*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a + a*Cos[c + d*x]), x, 6, (3*a*x)/8 + (a*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x]), x, 5, (a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x]), x, 1, (a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x]), x, 2, a*x + (a*Sin[c + d*x])/d} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x]), x, 2, a*x + (a*ArcTanh[Sin[c + d*x]])/d} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x]), x, 4, (a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + a*Cos[c + d*x]), x, 6, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a + a*Cos[c + d*x]), x, 6, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} - - -{Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2, x, 11, (11*a^2*x)/16 + (2*a^2*Sin[c + d*x])/d + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a^2*Sin[c + d*x]^3)/(3*d) + (2*a^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2, x, 9, (3*a^2*x)/4 + (2*a^2*Sin[c + d*x])/d + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - (a^2*Sin[c + d*x]^3)/d + (a^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2, x, 9, (7*a^2*x)/8 + (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^2, x, 2, a^2*x + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d), a^2*x + (4*a^2*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(3*d) + ((a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^2, x, 1, (3*a^2*x)/2 + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^2, x, 3, 2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^2, x, 5, a^2*x + (2*a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^2, x, 7, (3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^2, x, 8, (a^2*ArcTanh[Sin[c + d*x]])/d + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^2, x, 9, (7*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*Tan[c + d*x])/d + (7*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a^2*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3, x, 13, (23*a^3*x)/16 + (4*a^3*Sin[c + d*x])/d + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (7*a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3, x, 11, (13*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^3, x, 8, (15*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/d, (15*a^3*x)/8 + (3*a^3*Sin[c + d*x])/d + (9*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^3, x, 7, (5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^3, x, 6, (7*a^3*x)/2 + (a^3*ArcTanh[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^3, x, 6, 3*a^3*x + (3*a^3*ArcTanh[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d + (a^3*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^3, x, 7, a^3*x + (7*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^3, x, 9, (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (4*a^3*Tan[c + d*x])/d + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^3*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^3, x, 11, (15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d} -{Sec[c + d*x]^6*(a + a*Cos[c + d*x])^3, x, 11, (13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (5*a^3*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4, x, 15, (49*a^4*x)/16 + (8*a^4*Sin[c + d*x])/d + (49*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (41*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^4*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a^4*Sin[c + d*x]^3)/d + (4*a^4*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^4, x, 11, (7*a^4*x)/2 + (8*a^4*Sin[c + d*x])/d + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d - (8*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^5)/(5*d), (7*a^4*x)/2 + (32*a^4*Sin[c + d*x])/(5*d) + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(10*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(5*d) + ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) - (16*a^4*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^4, x, 10, (35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^4, x, 8, 6*a^4*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Sin[c + d*x])/d + (2*a^4*Cos[c + d*x]*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^4, x, 8, (13*a^4*x)/2 + (4*a^4*ArcTanh[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^4, x, 8, 4*a^4*x + (13*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*Sin[c + d*x])/d + (4*a^4*Tan[c + d*x])/d + (a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^4, x, 9, a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Tan[c + d*x])/d + (2*a^4*Sec[c + d*x]*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^4, x, 12, (35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a + a*Cos[c + d*x])^4, x, 13, (7*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (8*a^4*Tan[c + d*x])/d + (7*a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/d + (8*a^4*Tan[c + d*x]^3)/(3*d) + (a^4*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^7*(a + a*Cos[c + d*x])^4, x, 15, (49*a^4*ArcTanh[Sin[c + d*x]])/(16*d) + (8*a^4*Tan[c + d*x])/d + (49*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (41*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*Tan[c + d*x]^3)/d + (4*a^4*Tan[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^5/(a + a*Cos[c + d*x]), x, 7, (15*x)/(8*a) - (4*Sin[c + d*x])/(a*d) + (15*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + (4*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x]), x, 6, -((3*x)/(2*a)) + (4*Sin[c + d*x])/(a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (4*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x]), x, 2, (3*x)/(2*a) - (2*Sin[c + d*x])/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x]), x, 4, -(x/a) + Sin[c + d*x]/(a*d) + Sin[c + d*x]/(a*d*(1 + Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x]), x, 2, x/a - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x]), x, 1, Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x]), x, 3, ArcTanh[Sin[c + d*x]]/(a*d) - Sin[c + d*x]/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x]), x, 5, -(ArcTanh[Sin[c + d*x]]/(a*d)) + (2*Tan[c + d*x])/(a*d) - Tan[c + d*x]/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x]), x, 6, (3*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*Tan[c + d*x])/(a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Cos[c + d*x]), x, 6, -((3*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (4*Tan[c + d*x])/(a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + (4*Tan[c + d*x]^3)/(3*a*d)} - - -{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^2, x, 7, -((5*x)/a^2) + (12*Sin[c + d*x])/(a^2*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (10*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - (4*Sin[c + d*x]^3)/(a^2*d)} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^2, x, 3, (7*x)/(2*a^2) - (16*Sin[c + d*x])/(3*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (8*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^2, x, 6, -((2*x)/a^2) + (4*Sin[c + d*x])/(3*a^2*d) + (2*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^2, x, 3, x/a^2 - (5*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^2, x, 2, -(Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)) + (2*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^2, x, 2, Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(3*d*(a^2 + a^2*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(a^2*d) - (4*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^2, x, 6, -((2*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (10*Tan[c + d*x])/(3*a^2*d) - (2*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - Tan[c + d*x]/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^2, x, 7, (7*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (16*Tan[c + d*x])/(3*a^2*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (8*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^4/(a + a*Cos[c + d*x])^2, x, 7, -((5*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (12*Tan[c + d*x])/(a^2*d) - (5*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (10*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*Tan[c + d*x]^3)/(a^2*d)} - - -{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3, x, 4, (13*x)/(2*a^3) - (152*Sin[c + d*x])/(15*a^3*d) + (13*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3, x, 7, -((3*x)/a^3) + (9*Sin[c + d*x])/(5*a^3*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (3*Cos[c + d*x]^2*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (3*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^3, x, 5, x/a^3 - (Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (7*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (29*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^3, x, 3, Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (7*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^3, x, 3, -(Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3)) + Sin[c + d*x]/(5*a*d*(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(5*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^3, x, 3, Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^3, x, 5, ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (7*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (22*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^3, x, 7, -((3*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (24*Tan[c + d*x])/(5*a^3*d) - Tan[c + d*x]/(5*d*(a + a*Cos[c + d*x])^3) - (3*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (3*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^3, x, 8, (13*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (152*Tan[c + d*x])/(15*a^3*d) + (13*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - (Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (76*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} - - -{Cos[c + d*x]^6/(a + a*Cos[c + d*x])^4, x, 5, (21*x)/(2*a^4) - (576*Sin[c + d*x])/(35*a^4*d) + (21*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - (43*Cos[c + d*x]^3*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) - (288*Cos[c + d*x]^2*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^4, x, 8, -((4*x)/a^4) + (244*Sin[c + d*x])/(105*a^4*d) - (88*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (12*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^4, x, 6, x/a^4 + (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (43*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^4, x, 5, -((18*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2)) + (12*Sin[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (8*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^4, x, 4, Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (11*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (13*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (13*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^4, x, 4, -(Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4)) + (4*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^4, x, 4, Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) + (3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(35*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(35*d*(a^4 + a^4*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^4, x, 6, ArcTanh[Sin[c + d*x]]/(a^4*d) - (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (32*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) - Sin[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sin[c + d*x])/(7*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^4, x, 8, -((4*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (664*Tan[c + d*x])/(105*a^4*d) - (88*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - Tan[c + d*x]/(7*d*(a + a*Cos[c + d*x])^4) - (12*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^4, x, 9, (21*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (576*Tan[c + d*x])/(35*a^4*d) + (21*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - (43*Sec[c + d*x]*Tan[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])^2) - (288*Sec[c + d*x]*Tan[c + d*x])/(35*a^4*d*(1 + Cos[c + d*x])) - (Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} - - -{Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5, x, 6, (31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5, x, 9, -((5*x)/a^5) + (181*Sin[c + d*x])/(63*a^5*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (5*Cos[c + d*x]^4*Sin[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Cos[c + d*x]^2*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5, x, 7, x/a^5 - (Cos[c + d*x]^4*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (13*Cos[c + d*x]^3*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (173*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (661*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5, x, 6, -((Cos[c + d*x]^3*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5)) - (11*Cos[c + d*x]^2*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (67*Sin[c + d*x])/(315*a^2*d*(a + a*Cos[c + d*x])^3) - (142*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) + (83*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^5, x, 6, -((Cos[c + d*x]^2*Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5)) + Sin[c + d*x]/(7*a*d*(a + a*Cos[c + d*x])^4) - (17*Sin[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) + (5*Sin[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) + (5*Sin[c + d*x])/(63*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^5, x, 5, Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (2*Sin[c + d*x])/(9*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(15*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(45*a^3*d*(a + a*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(45*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^5, x, 5, -(Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5)) + (5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + Sin[c + d*x]/(21*a^2*d*(a + a*Cos[c + d*x])^3) + (2*Sin[c + d*x])/(63*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*Sin[c + d*x])/(63*d*(a^5 + a^5*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^5, x, 5, Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(315*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^5, x, 7, ArcTanh[Sin[c + d*x]]/(a^5*d) - Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (13*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (34*Sin[c + d*x])/(105*a^2*d*(a + a*Cos[c + d*x])^3) - (173*Sin[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (488*Sin[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5, x, 9, -((5*ArcTanh[Sin[c + d*x]])/(a^5*d)) + (496*Tan[c + d*x])/(63*a^5*d) - Tan[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) - (5*Tan[c + d*x])/(21*a*d*(a + a*Cos[c + d*x])^4) - (29*Tan[c + d*x])/(63*a^2*d*(a + a*Cos[c + d*x])^3) - (67*Tan[c + d*x])/(63*a^3*d*(a + a*Cos[c + d*x])^2) - (5*Tan[c + d*x])/(d*(a^5 + a^5*Cos[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5, x, 10, (31*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (7664*Tan[c + d*x])/(315*a^5*d) + (31*Sec[c + d*x]*Tan[c + d*x])/(2*a^5*d) - (Sec[c + d*x]*Tan[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Sec[c + d*x]*Tan[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) - (28*Sec[c + d*x]*Tan[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Sec[c + d*x]*Tan[c + d*x])/(315*a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Sec[c + d*x]*Tan[c + d*x])/(315*d*(a^5 + a^5*Cos[c + d*x]))} - - -{Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6, x, 7, (130*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^3) - (268*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])^2) + (146*Sin[c + d*x])/(693*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^4*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6) - (14*Cos[c + d*x]^3*Sin[c + d*x])/(99*a*d*(a + a*Cos[c + d*x])^5) - (118*Cos[c + d*x]^2*Sin[c + d*x])/(693*a^2*d*(a + a*Cos[c + d*x])^4)} -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6, x, 7, -((241*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^3)) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^2) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(33*a*d*(a + a*Cos[c + d*x])^5) + (9*Sin[c + d*x])/(77*a^2*d*(a + a*Cos[c + d*x])^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^4*(a + a*Cos[c + d*x])^(1/2), x, 5, (32*a*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (64*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (32*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(1/2), x, 4, (4*a*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (8*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (12*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(1/2), x, 3, (14*a*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(1/2), x, 4, (3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (3*a*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(1/2), x, 5, (5*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (5*a*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2), x, 6, (68*a^2*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (34*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (136*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (68*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2), x, 4, (152*a^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(3/2), x, 2, (8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2), x, 4, (3*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^2*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2), x, 5, (7*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (7*a^2*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(3/2), x, 6, (11*a^(3/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (11*a^2*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2), x, 6, (284*a^3*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (710*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (568*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a^2*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (284*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d)} -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2), x, 5, (832*a^3*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (26*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(5/2), x, 3, (64*a^3*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2), x, 4, (2*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2), x, 4, (5*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2), x, 4, (19*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (9*a^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + a*Cos[c + d*x])^(5/2), x, 5, (25*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (25*a^3*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(a + a*Cos[c + d*x])^(5/2), x, 6, (163*a^(5/2)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (163*a^3*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(7/2), x, 4, (256*a^4*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (64*a^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d) + (24*a^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(1/2), x, 7, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (148*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (62*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (28*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Sec[c + d*x]^1/Sqrt[a + a*Cos[c + d*x]], x, 5, (2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Sec[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]], x, 6, -(ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + Tan[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]], x, 7, (7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - Tan[c + d*x]/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]], x, 8, -((9*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (7*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(3/2), x, 7, -((15*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - (Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (31*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + (9*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - (13*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2), x, 6, (11*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - (13*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + (7*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2), x, 4, -((7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(3/2), x, 3, (3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(3/2), x, 3, ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2), x, 7, -((3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + (9*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2), x, 8, (19*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - (13*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (7*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (Sec[c + d*x]*Tan[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(5/2), x, 7, (163*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - (197*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (95*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)} -{Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(5/2), x, 6, -((75*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - (Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (13*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (9*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2), x, 4, (19*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + a*Cos[c + d*x])^(5/2), x, 4, (5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^0/(a + a*Cos[c + d*x])^(5/2), x, 4, (3*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2), x, 8, -((5*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + (115*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]), x, 6, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*a*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]), x, 5, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x]), x, 4, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x]), x, 3, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x]), x, 4, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x]), x, 5, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x]), x, 6, -((6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2, x, 10, (32*a^2*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (20*a^2*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (20*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2, x, 9, (12*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(7*d) + (8*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (4*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^2, x, 7, (16*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^2, x, 6, (4*a^2*EllipticE[(1/2)*(c + d*x), 2])/d + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^2, x, 6, (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^2, x, 7, -((4*a^2*EllipticE[(1/2)*(c + d*x), 2])/d) + (8*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^2, x, 9, -((16*a^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (16*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3, x, 12, (68*a^3*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (44*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (44*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (68*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (6*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^3*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^3, x, 10, (28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (52*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (6*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^3, x, 8, (36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^3, x, 8, (4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^3, x, 8, -((4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d) + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^3, x, 10, -((36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a^3*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (36*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^3, x, 12, -((28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (52*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (28*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4, x, 16, (128*a^4*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (904*a^4*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (904*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (128*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (150*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (8*a^4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a^4*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^4, x, 13, (152*a^4*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(7*d) + (32*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (122*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (8*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^4, x, 11, (64*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (136*a^4*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (94*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (8*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^4*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^4, x, 10, (56*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (8*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^4, x, 10, (40*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*a^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^4, x, 11, -((56*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (32*a^4*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*a^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (66*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^4, x, 13, -((64*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (136*a^4*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (94*a^4*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (64*a^4*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x]), x, 6, (21*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x]), x, 5, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x]), x, 4, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x]), x, 4, -(EllipticE[(1/2)*(c + d*x), 2]/(a*d)) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x]), x, 4, EllipticE[(1/2)*(c + d*x), 2]/(a*d) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x]), x, 5, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) + (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x]), x, 6, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} - - -{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^2, x, 7, (56*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d) + (56*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - (3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^2, x, 6, -((7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2, x, 5, (4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2, x, 5, -(EllipticE[(1/2)*(c + d*x), 2]/(a^2*d)) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^2, x, 3, EllipticF[(1/2)*(c + d*x), 2]/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^2, x, 5, EllipticE[(1/2)*(c + d*x), 2]/(a^2*d) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^2, x, 6, -((4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (4*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (5*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^2, x, 7, (7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - (7*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{Cos[c + d*x]^(11/2)/(a + a*Cos[c + d*x])^3, x, 8, (231*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (21*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (63*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^3, x, 7, -((119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (119*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^3, x, 6, (49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (8*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3, x, 6, -((9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) + (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3, x, 6, -(EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (4*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^3, x, 6, EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^3, x, 6, (9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^3, x, 7, -((49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + (49*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^3, x, 8, (119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (5*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (5*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (3*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(35*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (32*a*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (11*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (11*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (11*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (7*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (7*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (3*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (10*a^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (6*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (12*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (208*a^2*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (163*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (163*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (163*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (17*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (25*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (25*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (13*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-1/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (19*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (9*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (5*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (2*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(a + a*Cos[c + d*x])^(5/2), x, 3, (22*a^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (86*a^3*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(a + a*Cos[c + d*x])^(5/2), x, 4, (6*a^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (92*a^3*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-11/2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (38*a^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (1168*a^3*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Cos[c + d*x]^(-5/4)*(a + a*Cos[c + d*x])^(3/2), x, 2, (4*a^2*Sin[c + d*x])/(d*Cos[c + d*x]^(1/4)*Sqrt[a + a*Cos[c + d*x]])} - - -{Sqrt[a + a*Cos[e + f*x]]/Sqrt[Cos[e + f*x]], x, 2, (2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + a*Cos[e + f*x]]])/f} -{Sqrt[a - a*Cos[e + f*x]]/Sqrt[-Cos[e + f*x]], x, 2, -((2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a - a*Cos[e + f*x]]])/f)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(1/2), x, 7, (7*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(1/2), x, 6, -(ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(1/2), x, 5, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(1/2), x, 5, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)/(a + a*Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)/(1 + Cos[c + d*x])^(1/2), x, 7, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (7*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]])/(4*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 + Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(1 + Cos[c + d*x])^(1/2), x, 6, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d - ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(1 + Cos[c + d*x])^(1/2), x, 5, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]])/d} -{Cos[c + d*x]^(-1/2)/(1 + Cos[c + d*x])^(1/2), x, 2, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d} -{Cos[c + d*x]^(-3/2)/(1 + Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(1 + Cos[c + d*x])^(1/2), x, 5, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)/(1 + Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])])/d) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[1 + Cos[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 7, -((3*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + (9*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(3/2), x, 4, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (5*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(3/2), x, 6, (11*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (7*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (19*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2), x, 8, -((5*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + (115*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (15*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (35*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(5/2), x, 5, (19*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (9*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, -((75*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Sin[c + d*x]/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(5/2), x, 7, (163*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (17*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (95*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (299*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(7/2), x, 9, -((7*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d)) + (637*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (7*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (259*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (189*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(7/2), x, 8, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) - (177*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (17*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (49*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + (67*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + (17*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (13*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)/(a + a*Cos[c + d*x])^(7/2), x, 6, (63*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - (103*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)/(a + a*Cos[c + d*x])^(7/2), x, 7, -((363*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d)) - Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - (19*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - (199*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + (691*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + a*Cos[c + d*x])^(7/2), x, 8, (1015*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - (23*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - (109*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (193*Sin[c + d*x])/(64*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (629*Sin[c + d*x])/(64*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2), x, 7, (35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) - (Cos[c + d*x]^(5/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (19*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)) - (187*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (853*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2), x, 7, (45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)) + (33*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)) + (73*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))} - - -{1/(Sqrt[1 + Cos[x]]*Sqrt[Cos[x]]), x, 2, Sqrt[2]*ArcSin[Sin[x]/(1 + Cos[x])]} -{1/(Sqrt[a + a*Cos[x]]*Sqrt[Cos[x]]), x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]])])/Sqrt[a]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a - a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*(a - a*Cos[c + d*x])^(1/2), x, 4, -((3*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*d)) + (3*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) - (a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a - a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d - (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a - a*Cos[c + d*x])^(1/2), x, 2, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/d)} -{Cos[c + d*x]^(-3/2)*(a - a*Cos[c + d*x])^(1/2), x, 1, (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a - a*Cos[c + d*x])^(1/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) - (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a - a*Cos[c + d*x])^(1/2), x, 3, (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) - (8*a*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(1 - Cos[c + d*x])^(1/2), x, 4, -((3*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d)) + (3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(1 - Cos[c + d*x])^(1/2), x, 3, ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(1 - Cos[c + d*x])^(1/2), x, 2, -((2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d)} -{Cos[c + d*x]^(-3/2)*(1 - Cos[c + d*x])^(1/2), x, 1, (2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(1 - Cos[c + d*x])^(1/2), x, 2, (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) - (4*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(1 - Cos[c + d*x])^(1/2), x, 3, (2*Sin[c + d*x])/(5*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2)) - (8*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (16*Sin[c + d*x])/(15*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/(a - a*Cos[c + d*x])^(1/2), x, 7, (7*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a - a*Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a - a*Cos[c + d*x])^(1/2), x, 6, ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])]/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(a - a*Cos[c + d*x])^(1/2), x, 5, (2*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^(-1/2)/(a - a*Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d))} -{Cos[c + d*x]^(-3/2)/(a - a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a - a*Cos[c + d*x])^(1/2), x, 5, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)/(a - a*Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]) + (2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]]) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)/(1 - Cos[c + d*x])^(1/2), x, 7, (7*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/(4*d) - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[1 - Cos[c + d*x]]) + (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[1 - Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(1 - Cos[c + d*x])^(1/2), x, 6, ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])]/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(1 - Cos[c + d*x])^(1/2), x, 5, (2*ArcTanh[Sin[c + d*x]/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d - (Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d} -{Cos[c + d*x]^(-1/2)/(1 - Cos[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d)} -{Cos[c + d*x]^(-3/2)/(1 - Cos[c + d*x])^(1/2), x, 3, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(1 - Cos[c + d*x])^(1/2), x, 5, -((Sqrt[2]*ArcTanh[Sin[c + d*x]/(Sqrt[2]*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])])/d) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)) + (2*Sin[c + d*x])/(3*d*Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/3) (a+a Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3), x, 3, (2^(5/6)*AppellF1[1/2, -(4/3), 1/6, 3/2, 1 - Cos[c + d*x], (1/2)*(1 - Cos[c + d*x])]*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(5/6))} -{Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3), x, 3, (2*2^(1/6)*AppellF1[1/2, -(4/3), -(1/6), 3/2, 1 - Cos[c + d*x], (1/2)*(1 - Cos[c + d*x])]*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))} -{Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3), x, 3, (2*2^(1/6)*AppellF1[1/2, -(5/3), -(1/6), 3/2, 1 - Cos[c + d*x], (1/2)*(1 - Cos[c + d*x])]*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])^(7/6))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((6*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 6, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d} -{(a + a*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 7, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 8, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])/Sec[c + d*x]^(5/2), x, 9, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*a*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2), x, 9, -((16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2), x, 8, (-4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2), x, 5, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(1/2), x, 7, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(1/2), x, 8, (16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(3/2), x, 9, (12*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2), x, 17, -((28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (28*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (52*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (6*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2), x, 15, -((36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (36*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2), x, 13, -((4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2), x, 13, (4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(1/2), x, 13, (36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(1/2), x, 15, (28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (52*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(3/2), x, 17, (68*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (44*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (68*a^3*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (44*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(9/2), x, 19, -((64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (64*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (94*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (8*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(7/2), x, 17, -((56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (66*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(5/2), x, 16, (40*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(3/2), x, 16, (56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(1/2), x, 17, (64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (8*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (94*a^4*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^4/Sec[c + d*x]^(1/2), x, 19, (152*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (122*a^4*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (32*a^4*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x]), x, 9, (3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x]), x, 8, -((3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) - (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x]), x, 7, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(1/2)), x, 7, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 7, (3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 8, -((3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)), x, 9, (21*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (7*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} - - -{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2, x, 10, (7*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (7*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (10*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - (7*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2, x, 9, -((4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^2, x, 8, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(1/2)), x, 5, (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 8, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 8, (4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)), x, 9, -((7*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)), x, 10, (56*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (56*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(a^2*d*Sqrt[Sec[c + d*x]]) - (3*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} - - -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3, x, 10, -((49*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^3, x, 9, (9*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (9*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(1/2)), x, 9, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 9, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 9, -((9*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a^3 + a^3*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 9, (49*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)), x, 10, -((119*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2), x, 5, (32*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (12*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2), x, 4, (16*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2), x, 3, (4*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2), x, 2, (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2), x, 3, (2*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d} -{Sqrt[a + a*Cos[c + d*x]]/Sec[c + d*x]^(1/2), x, 4, (Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]/Sec[c + d*x]^(3/2), x, 5, (3*Sqrt[a]*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (3*a*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2), x, 6, (208*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (104*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (26*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2), x, 5, (12*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2), x, 4, (10*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2), x, 5, (2*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2), x, 5, (3*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(1/2), x, 6, (7*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (7*a^2*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2), x, 7, (11*a^(3/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (11*a^2*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (11*a^2*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2), x, 6, (1168*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (584*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (146*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (38*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2), x, 5, (92*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (46*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2), x, 4, (86*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (22*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2), x, 5, (2*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (14*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2), x, 5, (5*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2), x, 5, (19*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(1/2), x, 6, (25*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (13*a^3*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (25*a^3*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2), x, 7, (163*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (17*a^3*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (163*a^3*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (163*a^3*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)/Sqrt[1 + Cos[c + d*x]], x, 7, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[1 + Cos[c + d*x]])} -{Sec[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]], x, 6, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[1 + Cos[c + d*x]])} -{Sec[c + d*x]^(3/2)/Sqrt[1 + Cos[c + d*x]], x, 4, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Cos[c + d*x]])} -{Sec[c + d*x]^(1/2)/Sqrt[1 + Cos[c + d*x]], x, 3, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d} -{1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 6, -((Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d) + (2*ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d} -{1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 7, (Sqrt[2]*ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (ArcSin[Sin[c + d*x]/Sqrt[1 + Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + Sin[c + d*x]/(d*Sqrt[1 + Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(1/2)/Sqrt[a + a*Cos[c + d*x]], x, 3, (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x]*Sqrt[Sec[c + d*x]])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d), (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 6, (2*ArcTan[(Sqrt[a]*Sin[c + d*x]*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x]*Sqrt[Sec[c + d*x]])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d), (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 7, -((ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 7, (11*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (7*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 6, -((7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 5, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 5, (ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 7, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)), x, 8, -((3*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d)) + (9*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 8, (163*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - (299*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (17*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (95*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 7, -((75*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, (19*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (9*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 6, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 6, (3*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (7*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 8, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (11*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)), x, 9, -((5*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d)) + (115*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (15*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2), x, 9, (1015*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - (629*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (23*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (109*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (193*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2), x, 8, -((363*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - (199*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + (691*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(7/2), x, 7, (63*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (103*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(1/2)), x, 7, (13*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - (5*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)), x, 7, (7*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + (3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (17*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)), x, 7, (5*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (13*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (67*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)), x, 9, (2*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) - (177*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) - (17*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) - (49*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(9/2)), x, 10, -((7*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d)) + (637*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - Sin[c + d*x]/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) - (7*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) - (259*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (189*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)), x, 8, (45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(32*a*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + (33*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (73*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)), x, 8, (35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)) - (19*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (187*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (853*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/4) / (a+a Cos[e+f x])^(n/2)*) - - -{Sec[c + d*x]^(5/4)*(a + a*Cos[c + d*x])^(3/2), x, 3, (4*a^2*Sec[c + d*x]^(1/4)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^n with m and/or n symbolic*) - - -{Cos[c + d*x]^m*(a + a*Cos[c + d*x])^4, x, 7, If[$VersionNumber>=8, (a^4*(55 + 29*m + 4*m^2)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)*(4 + m)) + (Cos[c + d*x]^(1 + m)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) + (2*(5 + m)*Cos[c + d*x]^(1 + m)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)*(4 + m)) - (a^4*(35 + 40*m + 8*m^2)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (4*a^4*(5 + 2*m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]), (a^4*(55 + 29*m + 4*m^2)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + m)*(6 + 5*m + m^2)) + (Cos[c + d*x]^(1 + m)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) + (2*(5 + m)*Cos[c + d*x]^(1 + m)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)*(4 + m)) - (a^4*(35 + 40*m + 8*m^2)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 14*m + 7*m^2 + m^3)*Sqrt[Sin[c + d*x]^2]) - (4*a^4*(5 + 2*m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(a + a*Cos[c + d*x])^3, x, 6, (a^3*(7 + 2*m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (Cos[c + d*x]^(1 + m)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)) - (a^3*(5 + 4*m)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (a^3*(11 + 4*m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(a + a*Cos[c + d*x])^2, x, 4, (a^2*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) - (a^2*(3 + 2*m)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a^2*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(a + a*Cos[c + d*x])^1, x, 3, -((a*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (a*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m/(a + a*Cos[c + d*x])^1, x, 4, (Cos[c + d*x]^m*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (Cos[c + d*x]^m*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*Sqrt[Sin[c + d*x]^2]) + (m*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*(1 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2, x, 5, -((2*(1 - m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x]))) - (Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((1 - 2*m)*m*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(3*a^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (2*(1 - m)*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(3*a^2*d*(2 + m)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Cos[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^7*(a + b*Cos[c + d*x]), x, 8, (35*b*x)/128 + (a*Sin[c + d*x])/d + (35*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*b*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (b*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^6*(a + b*Cos[c + d*x]), x, 7, (5*a*x)/16 + (b*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (b*Sin[c + d*x]^3)/d + (3*b*Sin[c + d*x]^5)/(5*d) - (b*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^5*(a + b*Cos[c + d*x]), x, 7, (5*b*x)/16 + (a*Sin[c + d*x])/d + (5*b*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*b*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^4*(a + b*Cos[c + d*x]), x, 6, (3*a*x)/8 + (b*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*b*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a + b*Cos[c + d*x]), x, 6, (3*b*x)/8 + (a*Sin[c + d*x])/d + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x]), x, 5, (a*x)/2 + (b*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x]), x, 1, (b*x)/2 + (a*Sin[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x]), x, 2, a*x + (b*Sin[c + d*x])/d} -{Sec[c + d*x]^1*(a + b*Cos[c + d*x]), x, 2, b*x + (a*ArcTanh[Sin[c + d*x]])/d} -{Sec[c + d*x]^2*(a + b*Cos[c + d*x]), x, 4, (b*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Cos[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + b*Cos[c + d*x]), x, 5, (b*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^5*(a + b*Cos[c + d*x]), x, 6, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a + b*Cos[c + d*x]), x, 6, (3*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} - - -{Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2, x, 7, (1/16)*(6*a^2 + 5*b^2)*x + (2*a*b*Sin[c + d*x])/d + ((6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a*b*Sin[c + d*x]^3)/(3*d) + (2*a*b*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2, x, 7, (3*a*b*x)/4 + ((a^2 + b^2)*Sin[c + d*x])/d + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - ((a^2 + 2*b^2)*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2, x, 6, (1/8)*(4*a^2 + 3*b^2)*x + (2*a*b*Sin[c + d*x])/d + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^2, x, 2, a*b*x + (2*(a^2 + b^2)*Sin[c + d*x])/(3*d) + (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + ((a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^2, x, 1, (1/2)*(2*a^2 + b^2)*x + (2*a*b*Sin[c + d*x])/d + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(a + b*Cos[c + d*x])^2, x, 3, 2*a*b*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (b^2*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(a + b*Cos[c + d*x])^2, x, 4, b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Cos[c + d*x])^2, x, 5, ((a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + b*Cos[c + d*x])^2, x, 6, (a*b*ArcTanh[Sin[c + d*x]])/d + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(a + b*Cos[c + d*x])^2, x, 6, ((3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Tan[c + d*x])/d + ((3*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*b*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a + b*Cos[c + d*x])^2, x, 7, (3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) + ((4*a^2 + 5*b^2)*Tan[c + d*x])/(5*d) + (3*a*b*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*b*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (a^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*a^2 + 5*b^2)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3, x, 8, (9/8)*a^2*b*x + (5*b^3*x)/16 + (a*(a^2 + 3*b^2)*Sin[c + d*x])/d + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a*(a^2 + 6*b^2)*Sin[c + d*x]^3)/(3*d) + (3*a*b^2*Sin[c + d*x]^5)/(5*d), (1/16)*b*(18*a^2 + 5*b^2)*x + (a*(5*a^2 + 12*b^2)*Sin[c + d*x])/(5*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(18*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (13*a*b^2*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a*(5*a^2 + 12*b^2)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*a*(4*a^2 + 9*b^2)*x - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*b*d) - (a*(6*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) - ((3*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^3, x, 3, (3/8)*b*(4*a^2 + b^2)*x + (a*(a^2 + 4*b^2)*Sin[c + d*x])/(2*d) + (b*(2*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + ((a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^3, x, 2, a^3*x + (3/2)*a*b^2*x + (b*(3*a^2 + b^2)*Sin[c + d*x])/d + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (b^3*Sin[c + d*x]^3)/(3*d), (1/2)*a*(2*a^2 + 3*b^2)*x + (2*b*(4*a^2 + b^2)*Sin[c + d*x])/(3*d) + (5*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(a + b*Cos[c + d*x])^3, x, 4, (1/2)*b*(6*a^2 + b^2)*x + (a^3*ArcTanh[Sin[c + d*x]])/d + (5*a*b^2*Sin[c + d*x])/(2*d) + (b^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(a + b*Cos[c + d*x])^3, x, 4, 3*a*b^2*x + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (b*(a^2 - b^2)*Sin[c + d*x])/d + (a^2*(a + b*Cos[c + d*x])*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Cos[c + d*x])^3, x, 4, b^3*x + (a*(a^2 + 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^2*b*Tan[c + d*x])/(2*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + b*Cos[c + d*x])^3, x, 6, (b*(3*a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2 + 9*b^2)*Tan[c + d*x])/(3*d) + (7*a^2*b*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(a + b*Cos[c + d*x])^3, x, 7, (3*a*(a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(2*a^2 + b^2)*Tan[c + d*x])/d + (3*a*(a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a^2*b*Sec[c + d*x]^2*Tan[c + d*x])/(4*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(a + b*Cos[c + d*x])^3, x, 7, (b*(9*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(4*a^2 + 15*b^2)*Tan[c + d*x])/(5*d) + (b*(9*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (11*a^2*b*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a^2*(a + b*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(4*a^2 + 15*b^2)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4, x, 9, (1/4)*a*b*(6*a^2 + 5*b^2)*x + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(35*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(37*a^2 + 6*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (8*a*b^3*Cos[c + d*x]^5*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(8*a^4 + 36*a^2*b^2 + 5*b^4)*x - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Sin[c + d*x])/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*Cos[c + d*x]*Sin[c + d*x])/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + ((a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^4, x, 4, (1/2)*a*b*(4*a^2 + 3*b^2)*x + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(15*d) + (a*b*(6*a^2 + 29*b^2)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*a^2 + 4*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + ((a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 + 16*b^2)*Sin[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (7*a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(a + b*Cos[c + d*x])^4, x, 5, 2*a*b*(2*a^2 + b^2)*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (b^2*(17*a^2 + 2*b^2)*Sin[c + d*x])/(3*d) + (4*a*b^3*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (b^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(a + b*Cos[c + d*x])^4, x, 5, (1/2)*b^2*(12*a^2 + b^2)*x + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(a^2 - 2*b^2)*Sin[c + d*x])/d - (b^2*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a + b*Cos[c + d*x])^4, x, 5, 4*a*b^3*x + (a^2*(a^2 + 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a^2 - 2*b^2)*Sin[c + d*x])/(2*d) + (3*a^3*b*Tan[c + d*x])/d + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a + b*Cos[c + d*x])^4, x, 5, b^4*x + (2*a*b*(a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2 + 17*b^2)*Tan[c + d*x])/(3*d) + (4*a^3*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(a + b*Cos[c + d*x])^4, x, 7, ((3*a^4 + 24*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a*b*(2*a^2 + 3*b^2)*Tan[c + d*x])/(3*d) + (a^2*(3*a^2 + 22*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (5*a^3*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(a + b*Cos[c + d*x])^4, x, 8, (a*b*(3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + ((8*a^4 + 60*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(15*d) + (a*b*(3*a^2 + 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*(4*a^2 + 27*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (3*a^3*b*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(a + b*Cos[c + d*x])^4, x, 8, ((5*a^4 + 36*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a*b*(4*a^2 + 5*b^2)*Tan[c + d*x])/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*(5*a^2 + 32*b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (7*a^3*b*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (a^2*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a*b*(4*a^2 + 5*b^2)*Tan[c + d*x]^3)/(15*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^5/(a + b*Cos[c + d*x]), x, 7, ((8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a^5*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - (a*(3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^4*d) + ((4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^4/(a + b*Cos[c + d*x]), x, 6, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2 + 2*b^2)*Sin[c + d*x])/(3*b^3*d) - (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x]), x, 5, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*Sin[c + d*x])/(b^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x]), x, 5, -((a*x)/b^2) + (2*a^2*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Sin[c + d*x]/(b*d)} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x]), x, 3, x/b - (2*a*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x]), x, 2, (2*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x]), x, 4, -((2*b*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)) + ArcTanh[Sin[c + d*x]]/(a*d)} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x]), x, 6, (2*b^2*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (b*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a*d)} -{Sec[c + d*x]^3/(a + b*Cos[c + d*x]), x, 6, -((2*b^3*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*Tan[c + d*x])/(a^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4/(a + b*Cos[c + d*x]), x, 7, (2*b^4*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(3*a^3*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2, x, 7, -((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^2, x, 6, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTanh[((a - b)*Sin[c + d*x]/(1 + Cos[c + d*x]))/Sqrt[-a^2 + b^2]])/(b^4*(-a^2 + b^2)^(3/2)*d) - (2*a*Sin[c + d*x])/(b^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (a^4*Sin[c + d*x])/(b^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])), ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^2, x, 5, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + ((2*a^2 - b^2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^2, x, 4, x/b^2 - (2*a*(a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x])^2, x, 4, -((2*b*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d)) + (a*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^2, x, 4, (2*a*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x])^2, x, 5, -((2*b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^2*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x])^2, x, 6, (2*b^2*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - (2*b*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((a^2 - 2*b^2)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Cos[c + d*x])^2, x, 7, -((2*b^3*(4*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ((a^2 + 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (b*(2*a^2 - 3*b^2)*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Cos[c + d*x])^2, x, 8, (2*b^4*(5*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*Sec[c + d*x]*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3, x, 7, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(4*a^2 - 7*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^3, x, 6, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*a^3*(a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^3, x, 5, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(2*a^2 - 5*b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^3, x, 5, ((a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2 - 4*b^2)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x])^3, x, 5, -((3*a*b*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 2*b^2)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^3, x, 5, ((2*a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (3*a*b*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x])^3, x, 6, -((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^3*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x])^3, x, 7, (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*b^2*(2*a^2 - b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Cos[c + d*x])^3, x, 8, -((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((a^2 + 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b^2*(7*a^2 - 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4, x, 7, -((4*a*x)/b^5) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^4, x, 6, x/b^4 - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a^3*(3*a^2 - 8*b^2)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^4, x, 6, -((b*(3*a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a^2*(2*a^2 - 7*b^2)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^4, x, 6, (a*(a^2 + 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2 - 6*b^2)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x])^4, x, 6, -((b*(4*a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2 + 3*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2 + 13*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^4, x, 6, (a*(2*a^2 + 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (5*a*b*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x])^4, x, 7, -((b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^4*d) + (b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b^2*(8*a^2 - 3*b^2)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x])^4, x, 8, (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b^2*(9*a^2 - 4*b^2)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]], x, 8, (2*a*(8*a^2 + 19*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^4 + 17*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]], x, 7, -((2*(2*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Cos[c + d*x]], x, 6, (2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*Sqrt[a + b*Cos[c + d*x]], x, 2, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])} -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^1, x, 5, (2*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2, x, 9, -((Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^3, x, 10, -(b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (3*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} - - -{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^4 + 31*a^2*b^2 - 39*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(8*a^2 + 39*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2), x, 8, -((4*a*(3*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(6*a^4 - 31*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^(3/2), x, 6, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^1, x, 8, (2*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*a*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2, x, 9, -((a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a^2 + 2*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (3*a*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3, x, 10, (-5*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (7*a*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2 + 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} - - -{Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2), x, 10, (2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (8*a*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*a*(3*a^2 + 29*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(21*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (16*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^1, x, 9, (14*a*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*b*(2*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2, x, 9, -(((a^2 - 2*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (a*(a^2 + 4*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (5*a^2*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3, x, 10, (-9*a*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(11*a^2 + 8*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(4*a^2 + 15*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4, x, 11, -((16*a^2 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(16*a^2 + 59*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*(4*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (13*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^(7/2), x, 8, (32*a*(11*a^2 + 13*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(71*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (24*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} - - -{Cos[c + d*x]^3*Sqrt[3 + 4*Cos[c + d*x]], x, 6, (47*EllipticE[(1/2)*(c + d*x), 8/7])/(20*Sqrt[7]*d) + (59*EllipticF[(1/2)*(c + d*x), 8/7])/(60*Sqrt[7]*d) + (59*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (3*(3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(70*d) + (Cos[c + d*x]*(3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(14*d)} -{Cos[c + d*x]^2*Sqrt[3 + 4*Cos[c + d*x]], x, 5, (21*Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(20*d) - (Sqrt[7]*EllipticF[(1/2)*(c + d*x), 8/7])/(20*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + ((3 + 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^1*Sqrt[3 + 4*Cos[c + d*x]], x, 4, (Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(2*d) + (Sqrt[7]*EllipticF[(1/2)*(c + d*x), 8/7])/(6*d) + (2*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*Sqrt[3 + 4*Cos[c + d*x]], x, 1, (2*Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/d} -{Sec[c + d*x]^1*Sqrt[3 + 4*Cos[c + d*x]], x, 3, (8*EllipticF[(1/2)*(c + d*x), 8/7])/(Sqrt[7]*d) + (6*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(Sqrt[7]*d)} -{Sec[c + d*x]^2*Sqrt[3 + 4*Cos[c + d*x]], x, 6, -((Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/d) + (3*EllipticF[(1/2)*(c + d*x), 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*Sqrt[3 + 4*Cos[c + d*x]], x, 7, -((Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(3*d)) + (3*EllipticF[(1/2)*(c + d*x), 8/7])/(Sqrt[7]*d) + (5*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} - - -{Cos[c + d*x]^3*Sqrt[3 - 4*Cos[c + d*x]], x, 6, -((47*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(20*Sqrt[7]*d)) - (59*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(60*Sqrt[7]*d) + (59*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (3*(3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(70*d) - ((3 - 4*Cos[c + d*x])^(3/2)*Cos[c + d*x]*Sin[c + d*x])/(14*d)} -{Cos[c + d*x]^2*Sqrt[3 - 4*Cos[c + d*x]], x, 5, (21*Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(20*d) - (Sqrt[7]*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(20*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^1*Sqrt[3 - 4*Cos[c + d*x]], x, 4, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(2*d)) - (Sqrt[7]*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(6*d) + (2*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*Sqrt[3 - 4*Cos[c + d*x]], x, 1, (2*Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/d} -{Sec[c + d*x]^1*Sqrt[3 - 4*Cos[c + d*x]], x, 3, -((8*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d)) - (6*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d)} -{Sec[c + d*x]^2*Sqrt[3 - 4*Cos[c + d*x]], x, 6, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/d) + (3*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d) + (4*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*Sqrt[3 - 4*Cos[c + d*x]], x, 7, (Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(3*d) - (3*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d) - (5*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(3*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]], x, 7, (2*(8*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 + 7*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]], x, 6, (-4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1/Sqrt[a + b*Cos[c + d*x]], x, 5, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0/Sqrt[a + b*Cos[c + d*x]], x, 2, (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1/Sqrt[a + b*Cos[c + d*x]], x, 2, (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]], x, 9, -((Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]], x, 10, (3*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2 + 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (3*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} - - -{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (8*a*(4*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(8*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*a*(8*a^2 - 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(8*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x])^(3/2), x, 6, (-2*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x])^(3/2), x, 7, (-2*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2), x, 10, -(((a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - (3*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(a^2 - 3*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + Tan[c + d*x]/(a*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2), x, 11, (b*(7*a^2 - 15*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (5*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2 + 15*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b^2*(7*a^2 - 15*b^2)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*b*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^5/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(128*a^6 - 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(128*a^4 - 116*a^2*b^2 - 17*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a^2*(2*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(32*a^4 - 49*a^2*b^2 + 7*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) + (2*(48*a^4 - 71*a^2*b^2 + 3*b^4)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)} -{Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(5/2), x, 8, -((8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(16*a^4 - 16*a^2*b^2 - b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(8*a^2 - 9*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a^2*(a^2 - 2*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2), x, 7, -((4*a*(a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2 - 3*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*a*(a^2 - 3*b^2)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^1/(a + b*Cos[c + d*x])^(5/2), x, 7, (-2*(a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2 + 3*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^(5/2), x, 7, (8*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*b*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1/(a + b*Cos[c + d*x])^(5/2), x, 10, (-2*b*(7*a^2 - 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2), x, 11, -(((3*a^4 - 26*a^2*b^2 + 15*b^4)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*a^2 - 5*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(3*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + Tan[c + d*x]/(a*d*(a + b*Cos[c + d*x])^(3/2))} - - -{Cos[c + d*x]^0/(a + b*Cos[c + d*x])^(7/2), x, 8, (2*(23*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (16*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sin[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) - (16*a*b*Sin[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(23*a^2 + 9*b^2)*Sin[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]], x, 5, (9*Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(20*d) - (23*EllipticF[(1/2)*(c + d*x), 8/7])/(20*Sqrt[7]*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d) + (Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]], x, 4, -((Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(4*d)) + (17*EllipticF[(1/2)*(c + d*x), 8/7])/(12*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^1/Sqrt[3 + 4*Cos[c + d*x]], x, 3, (Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(2*d) - (3*EllipticF[(1/2)*(c + d*x), 8/7])/(2*Sqrt[7]*d)} -{Cos[c + d*x]^0/Sqrt[3 + 4*Cos[c + d*x]], x, 1, (2*EllipticF[(1/2)*(c + d*x), 8/7])/(Sqrt[7]*d)} -{Sec[c + d*x]^1/Sqrt[3 + 4*Cos[c + d*x]], x, 1, (2*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(Sqrt[7]*d)} -{Sec[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]], x, 6, -((Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(3*d)) + EllipticF[(1/2)*(c + d*x), 8/7]/(Sqrt[7]*d) - (4*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]], x, 7, (Sqrt[7]*EllipticE[(1/2)*(c + d*x), 8/7])/(3*d) - EllipticF[(1/2)*(c + d*x), 8/7]/(3*Sqrt[7]*d) + (Sqrt[7]*EllipticPi[2, (1/2)*(c + d*x), 8/7])/(3*d) - (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)} - - -{Cos[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]], x, 5, -((9*Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(20*d)) + (23*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(20*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(10*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]], x, 4, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(4*d)) + (17*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(12*Sqrt[7]*d) - (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^1/Sqrt[3 - 4*Cos[c + d*x]], x, 3, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(2*d)) + (3*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(2*Sqrt[7]*d)} -{Cos[c + d*x]^0/Sqrt[3 - 4*Cos[c + d*x]], x, 1, (2*EllipticF[(1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d)} -{Sec[c + d*x]^1/Sqrt[3 - 4*Cos[c + d*x]], x, 1, -((2*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(Sqrt[7]*d))} -{Sec[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]], x, 6, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(3*d)) + EllipticF[(1/2)*(c + Pi + d*x), 8/7]/(Sqrt[7]*d) - (4*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(3*Sqrt[7]*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]], x, 7, -((Sqrt[7]*EllipticE[(1/2)*(c + Pi + d*x), 8/7])/(3*d)) + EllipticF[(1/2)*(c + Pi + d*x), 8/7]/(3*Sqrt[7]*d) - (Sqrt[7]*EllipticPi[2, (1/2)*(c + Pi + d*x), 8/7])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d) + (Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]), x, 6, (6*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*B*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]), x, 5, (6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*A*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]), x, 4, (2*A*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(A + B*Cos[c + d*x])/Sqrt[Cos[c + d*x]], x, 3, (2*B*EllipticE[(c + d*x)/2, 2])/d + (2*A*EllipticF[(c + d*x)/2, 2])/d} -{(A + B*Cos[c + d*x])/Cos[c + d*x]^(3/2), x, 4, (-2*A*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/Cos[c + d*x]^(5/2), x, 5, (-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*A*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/Cos[c + d*x]^(7/2), x, 6, (-6*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*A*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2, x, 7, (2*(9*a^2 + 7*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (20*a*b*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (20*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2, x, 6, (12*a*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^2, x, 5, (2*(5*a^2 + 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^2, x, 4, (4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^2, x, 4, -((2*(a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^2, x, 5, -((4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2 + 3*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^2, x, 6, -((2*(3*a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3, x, 7, (2*b*(27*a^2 + 7*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*a^2 + 15*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a^2 + 7*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a*b^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b^2*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^3, x, 6, (2*a*(5*a^2 + 9*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*(21*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b^2*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^3, x, 5, (6*b*(5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^3, x, 5, -((2*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*b*(9*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^3, x, 5, -((2*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(a^2 + 9*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (16*a^2*b*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^3, x, 6, -((6*a*(a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a^2*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (6*a*(a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^3, x, 7, -((2*b*(9*a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(5*a^2 + 21*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (32*a^2*b*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*a^2 + 21*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*b*(9*a^2 + 5*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x]), x, 6, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d)) + (2*(3*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) - (2*a^3*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x]), x, 5, (2*EllipticE[(1/2)*(c + d*x), 2])/(b*d) - (2*a*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) + (2*a^2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x]), x, 3, (2*EllipticF[(1/2)*(c + d*x), 2])/(b*d) - (2*a*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x]), x, 1, (2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a + b)*d)} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x]), x, 5, -((2*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*b*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x]), x, 7, (2*b*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b^2*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*b*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^2, x, 7, -((a*(5*a^2 - 4*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d)) + ((15*a^4 - 16*a^2*b^2 - 2*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) - (a^3*(5*a^2 - 7*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) + ((5*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2, x, 6, ((3*a^2 - 2*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2, x, 6, -((a*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^2, x, 6, EllipticE[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d) + (a*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b*(a + b)^2*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^2, x, 6, -((b*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d)) - EllipticF[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d) + ((3*a^2 - b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^2, x, 7, -(((2*a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + (b*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^2, x, 8, (b*(4*a^2 - 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - (b*(4*a^2 - 5*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^(9/2)/(a + b*Cos[c + d*x])^3, x, 8, -((a*(35*a^4 - 65*a^2*b^2 + 24*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d)) + ((105*a^6 - 223*a^4*b^2 + 128*a^2*b^4 + 8*b^6)*EllipticF[(1/2)*(c + d*x), 2])/(12*b^5*(a^2 - b^2)^2*d) - (a^3*(35*a^4 - 86*a^2*b^2 + 63*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((35*a^4 - 61*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - (a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(7*a^2 - 13*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^3, x, 7, ((15*a^4 - 29*a^2*b^2 + 8*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(5*a^4 - 11*a^2*b^2 + 8*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) + (a^2*(15*a^4 - 38*a^2*b^2 + 35*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - (a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(5*a^2 - 11*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3, x, 7, -((3*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3, x, 7, -(((a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^3, x, 7, ((5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^3, x, 7, -((3*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (3*b^2*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^3, x, 8, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^3, x, 9, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d)) + (Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (Sqrt[a + b]*(a^2 - 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(1/2), x, 1, -((2*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + d*x]))/(a + b*Cos[c + d*x])]*(a + b*Cos[c + d*x])*Csc[c + d*x]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]]], -((a - b)/(a + b))])/(Sqrt[a + b]*d))} -{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(1/2), x, 3, (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(1/2), x, 4, (2*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(1/2), x, 5, (2*(a - b)*Sqrt[a + b]*(9*a^2 - 2*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(a - b)*b*Sqrt[a + b]*(19*a^2 + 8*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(25*a^2 + 6*a*b + 8*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2), x, 8, -(((a - b)*Sqrt[a + b]*(3*a^2 + 16*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d)) + (Sqrt[a + b]*(a + 2*b)*(3*a + 8*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) + (a*Sqrt[a + b]*(a^2 - 12*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((3*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(3/2), x, 8, -((5*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d)) + (Sqrt[a + b]*(5*a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (3*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + ((a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(3/2), x, 6, -(((a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)) + (Sqrt[a + b]*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (3*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (2*(a - 2*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (2*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d} -{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(3/2), x, 4, (8*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*(a - 3*b)*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a - b)*Sqrt[a + b]*(3*a^2 + b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d) - (2*(a - b)*(3*a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(3/2), x, 6, (4*(a - b)*b*Sqrt[a + b]*(41*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*(a - b)*Sqrt[a + b]*(25*a^2 - 57*a*b - 6*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (16*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(a - b)*Sqrt[a + b]*(147*a^4 + 33*a^2*b^2 + 8*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) - (2*(a - b)*Sqrt[a + b]*(147*a^3 - 39*a^2*b - 6*a*b^2 - 8*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (20*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) + (8*b*(22*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(5/2), x, 8, -(((a - b)*Sqrt[a + b]*(33*a^2 + 16*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d)) + (Sqrt[a + b]*(33*a^2 + 26*a*b + 16*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (5*a*Sqrt[a + b]*(a^2 + 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (13*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b^2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, -((9*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d)) + (Sqrt[a + b]*(8*a^2 + 9*a*b + 2*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(15*a^2 + 4*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, ((a - b)*Sqrt[a + b]*(2*a^2 - b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a^2 - 6*a*b - b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (5*a*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^(5/2), x, 6, (14*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) + (2*Sqrt[a + b]*(a^2 - 7*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) - (2*b^2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(a + b*Cos[c + d*x])^(5/2), x, 5, (2*(a - b)*Sqrt[a + b]*(9*a^2 + 23*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*(a - b)*Sqrt[a + b]*(9*a^2 - 8*a*b + 15*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (22*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(a - b)*b*Sqrt[a + b]*(29*a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d) + (2*(a - b)*Sqrt[a + b]*(5*a^2 - 24*a*b + 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (38*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-13/2)*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(a - b)*b*Sqrt[a + b]*(741*a^4 + 51*a^2*b^2 + 8*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d) + (2*(a - b)*Sqrt[a + b]*(135*a^4 - 606*a^3*b + 57*a^2*b^2 + 6*a*b^3 + 8*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (46*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*(81*a^2 + 113*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*b*(229*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a*d*Cos[c + d*x]^(5/2)) + (2*(135*a^4 + 205*a^2*b^2 - 4*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(1/2), x, 8, -(((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]), -(((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (a*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(1/2), x, 1, -((2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d))} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(1/2), x, 1, (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(1/2), x, 3, (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(1/2), x, 4, -((4*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d)) + (2*Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2), x, 7, -(((3*a^2 - b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d)) + ((3*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d) + (3*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(3/2), x, 4, -((2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d)) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) + (2*a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*b*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*(a^2 - 2*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, -((2*b*(5*a^2 - 8*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d)) + (2*(a + 2*b)*(a + 4*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d) - (2*(3*a + 4*b)*(a^2 + 4*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 6*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) - (2*b*(3*a^2 - 8*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(3*a^2 - 7*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*(3*a^2 + a*b - 6*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2), x, 5, (8*b*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) + (2*(a - 3*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (8*a*b*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(5/2), x, 5, -((2*(3*a^2 + b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d)) + (2*(3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^2 + b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^(5/2), x, 5, (4*b*(3*a^2 - b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (2*(3*a^2 - 3*a*b - 2*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*b*(3*a^2 - b^2)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^(5/2), x, 5, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d) - (2*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*(a - b)*(a + b)^(3/2)*d)) + (2*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))} - - -{1/(Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]), x, 1, (2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d)} -{1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]), x, 1, (2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 5])/d} - -{1/(Sqrt[Cos[c + d*x]]*Sqrt[2 - 3*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]]))} -{1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 - 3*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/(d*Sqrt[Cos[c + d*x]]))} - -{1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]), x, 1, (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d} -{1/(Sqrt[Cos[c + d*x]]*Sqrt[3 - 2*Cos[c + d*x]]), x, 1, (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -(1/5)]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d)} - -{1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -(1/5)]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d))} -{1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 - 2*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d)} - - -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]), x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])} -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]), x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 5])/(d*Sqrt[-Cos[c + d*x]])} - -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 - 3*Cos[c + d*x]]), x, 1, -((2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 1/5])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]]))} -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 - 3*Cos[c + d*x]]), x, 1, -((2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 5])/(d*Sqrt[-Cos[c + d*x]]))} - -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]), x, 2, (2*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/(d*Sqrt[-Cos[c + d*x]])} -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 - 2*Cos[c + d*x]]), x, 2, (2*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -(1/5)]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])} - -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]), x, 1, -((2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -(1/5)]*Sqrt[-Tan[c + d*x]^2])/(Sqrt[5]*d))} -{1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 - 2*Cos[c + d*x]]), x, 1, -((2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x]^2])/d)} - - -{Sqrt[Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]], x, 1, -((4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d))} -{Sqrt[Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]], x, 1, -((4*Cot[c + d*x]*EllipticPi[1/3, ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d))} - -{Sqrt[Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]], x, 2, -((4*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[1/3, ArcSin[Sqrt[2 - 3*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[-Cos[c + d*x]]))} -{Sqrt[Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]], x, 2, -((4*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d*Sqrt[-Cos[c + d*x]]))} - -{Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]], x, 1, -((3*Cot[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d)} -{Sqrt[Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]], x, 1, (3*Cot[c + d*x]*EllipticPi[-(1/2), ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*d)} - -{Sqrt[Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]], x, 2, (3*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[-(1/2), ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]], x, 2, -((3*Cos[c + d*x]^(3/2)*Csc[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(d*Sqrt[-Cos[c + d*x]]))} - - -{Sqrt[-Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]], x, 2, -((4*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d))} -{Sqrt[-Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]], x, 2, -((4*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[1/3, ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d))} - -{Sqrt[-Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]], x, 1, -((4*Cot[c + d*x]*EllipticPi[1/3, ArcSin[Sqrt[2 - 3*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*Sqrt[5]*d))} -{Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]], x, 1, -((4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d))} - -(* For some reason, Mathematica cannot verify the following two antiderivatives correct. *) -{Sqrt[-Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]], x, 2, -((3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d)} -{Sqrt[-Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]], x, 2, (3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[-(1/2), ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*d)} - -{Sqrt[-Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]], x, 1, (3*Cot[c + d*x]*EllipticPi[-(1/2), ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*d)} -{Sqrt[-Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]], x, 1, -((3*Cot[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/3) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(2/3)/(a + b*Cos[c + d*x]), x, 5, -((b*AppellF1[1/2, -(1/3), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3))) + (a*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(1/3))} -{Cos[c + d*x]^(1/3)/(a + b*Cos[c + d*x]), x, 5, -((b*AppellF1[1/2, -(1/6), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6))) + (a*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(2/3))} -{Cos[c + d*x]^(-1/3)/(a + b*Cos[c + d*x]), x, 5, -((b*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(1/3))) + (a*AppellF1[1/2, 2/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(4/3))} -{Cos[c + d*x]^(-2/3)/(a + b*Cos[c + d*x]), x, 5, -((b*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(2/3))) + (a*AppellF1[1/2, 5/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(5/6)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x]^(5/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/3) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]], x]} -{Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]} -{Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x]} -{Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x]} -{Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x]} -{Cos[c + d*x]^(-1/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x]} -{Cos[c + d*x]^(-2/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x]} -{Cos[c + d*x]^(-4/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[1/(Cos[c + d*x]^(4/3)*Sqrt[a + b*Cos[c + d*x]]), x]} -{Cos[c + d*x]^(-5/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]), x]} -{Cos[c + d*x]^(-7/3)/Sqrt[a + b*Cos[c + d*x]], x, 0, Unintegrable[1/(Cos[c + d*x]^(7/3)*Sqrt[a + b*Cos[c + d*x]]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, (-6*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, (-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 6, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d} -{(A + B*Cos[c + d*x])/Sqrt[Sec[c + d*x]], x, 7, (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 8, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/Sec[c + d*x]^(5/2), x, 9, (6*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*B*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)*(a + b*Cos[c + d*x])^2, x, 10, -((12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (12*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*a^2 + 7*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2, x, 9, -((2*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2, x, 8, -((4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2, x, 7, -((2*(a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^2, x, 7, (4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^2, x, 8, (2*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^2, x, 9, (12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Cos[c + d*x])^2, x, 10, (2*(9*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (20*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (4*a*b*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(9*a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (20*a*b*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)*(a + b*Cos[c + d*x])^3, x, 10, -((2*b*(9*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(9*a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*a^2 + 21*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (32*a^2*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a^2*Sec[c + d*x]^(5/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^3, x, 9, -((6*a*(a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*b*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (6*a*(a^2 + 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^2*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3, x, 8, -((2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*(a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3, x, 8, -((2*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*b*(9*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^3, x, 8, (6*b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (8*a*b^2*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-1/2)*(a + b*Cos[c + d*x])^3, x, 9, (2*a*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (32*a*b^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-3/2)*(a + b*Cos[c + d*x])^3, x, 10, (2*b*(27*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (40*a*b^2*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*b*(27*a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*a^2 + 15*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b^2*(b + a*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x]), x, 11, (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x]), x, 7, -((2*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) - (2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x]), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a + b)*d)} -{Sec[c + d*x]^(-1/2)/(a + b*Cos[c + d*x]), x, 5, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d)} -{Sec[c + d*x]^(-3/2)/(a + b*Cos[c + d*x]), x, 9, (2*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)} -{Sec[c + d*x]^(-5/2)/(a + b*Cos[c + d*x]), x, 10, -((2*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*d)) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2, x, 12, (b*(4*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) - (b*(4*a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2, x, 11, -(((2*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + (b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^2, x, 10, -((b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d)) - (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + ((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^2, x, 10, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + (a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b*(a + b)^2*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^2, x, 10, -((a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^2, x, 10, ((3*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - (a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} - - -{Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3, x, 13, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3, x, 12, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + (b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^3, x, 11, -((3*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (3*b^2*(3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)/(a + b*Cos[c + d*x])^3, x, 11, ((5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b*(a + b)^3*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) - (b*(7*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)/(a + b*Cos[c + d*x])^3, x, 11, -(((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^2*(a + b)^3*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (3*(a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{Sec[c + d*x]^(-5/2)/(a + b*Cos[c + d*x])^3, x, 11, -((3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (a*(a^2 - 7*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a + 2*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2), x, 5, (2*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2), x, 4, (2*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]], x, 2, -((1/(Sqrt[a + b]*d))*(2*Sqrt[Cos[c + d*x]]*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + d*x]))/(a + b*Cos[c + d*x])]*(a + b*Cos[c + d*x])*Csc[c + d*x]*EllipticPi[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]]], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]]))} -{Sqrt[a + b*Cos[c + d*x]]/Sqrt[Sec[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]/Sec[c + d*x]^(3/2), x, 8, -((a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a + 2*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)} - - -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2), x, 7, (4*(a - b)*b*Sqrt[a + b]*(41*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(25*a^2 - 57*a*b - 6*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (16*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2), x, 6, (2*(a - b)*Sqrt[a + b]*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*(3*a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d*Sqrt[Sec[c + d*x]]) + (4*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2), x, 5, (8*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*(a - 3*b)*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2), x, 6, (2*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (2*(a - 2*b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]])} -{(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]], x, 7, -(((a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*a + b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]], x, 9, (-5*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a + 2*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + ((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2), x, 9, -((a - b)*Sqrt[a + b]*(3*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a + 2*b)*(3*a + 8*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*(a^2 - 12*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + ((a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((3*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)} - - -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2), x, 8, (2*(a - b)*Sqrt[a + b]*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (38*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*b*Sqrt[a + b]*(29*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(5*a^2 - 24*a*b + 3*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (6*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2 + 23*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a^2 - 8*a*b + 15*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (22*a*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2), x, 7, (14*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(a^2 - 7*a*b + 9*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a^2 - 6*a*b - b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (5*a*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]], x, 8, (-9*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^2 + 9*a*b + 2*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(15*a^2 + 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (9*a*b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]], x, 9, -((a - b)*Sqrt[a + b]*(33*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(33*a^2 + 26*a*b + 16*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (5*a*Sqrt[a + b]*(a^2 + 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (13*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Sec[c + d*x]]) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d)} -{(a + b*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2), x, 10, -((a - b)*Sqrt[a + b]*(15*a^2 + 284*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3 + 118*a^2*b + 284*a*b^2 + 72*b^3)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4 - 120*a^2*b^2 - 48*b^4)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (17*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + ((59*a^2 + 36*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sqrt[Sec[c + d*x]]) + (a*(15*a^2 + 284*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)/Sqrt[a + b*Cos[c + d*x]], x, 5, (-4*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(a + 2*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^(3/2)/Sqrt[a + b*Cos[c + d*x]], x, 4, (2*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]/Sqrt[a + b*Cos[c + d*x]], x, 2, (2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{1/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]), x, 2, (-2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]])} -{1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 9, -(((a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + Sin[c + d*x]/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])} -{1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)), x, 8, (3*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - ((3*a - 2*b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2 + 4*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)} - - -{Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*b*(5*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + (2*(a + 2*b)*(a + 4*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(a + 2*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*b*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]), x, 5, -((2*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + (2*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 7, (2*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)), x, 8, -(((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + ((3*a + b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (3*a*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} - - -{Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) + (2*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(5/2), x, 6, (4*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*(3*a^2 - 3*a*b - 2*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (4*b*(3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]), x, 6, -((2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) + (2*(3*a - b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(3*a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 6, (8*b*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*(a - 3*b)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (8*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 8, (2*(3*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(3*a^2 + a*b - 6*b^2)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (2*a^2*(3*a^2 - 7*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n with m and/or n symbolic*) - - -{Cos[c + d*x]^m*(a + b*Cos[c + d*x])^4, x, 6, If[$VersionNumber>=8, (b^2*(b^2*(3 + m) + a^2*(22 + 5*m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b^3*(5 + m)*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b^2*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((b^4*(3 + 4*m + m^2) + 6*a^2*b^2*(4 + 5*m + m^2) + a^4*(8 + 6*m + m^2))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (4*a*b*(b^2*(2 + m) + a^2*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]), (b^2*(b^2*(3 + m) + a^2*(22 + 5*m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b^3*(5 + m)*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b^2*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((b^4*(3 + 4*m + m^2) + 6*a^2*b^2*(4 + 5*m + m^2) + a^4*(8 + 6*m + m^2))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + m)*(2 + 3*m + m^2)*Sqrt[Sin[c + d*x]^2]) - (4*a*b*(b^2*(2 + m) + a^2*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(a + b*Cos[c + d*x])^3, x, 5, (a*b^2*(7 + 2*m)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (b^2*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(d*(3 + m)) - (a*(3*b^2*(1 + m) + a^2*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (b*(b^2*(2 + m) + 3*a^2*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2, x, 4, (b^2*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) - ((b^2*(1 + m) + a^2*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(a + b*Cos[c + d*x])^1, x, 3, -((a*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*Sqrt[Sin[c + d*x]^2])) - (b*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m/(a + b*Cos[c + d*x])^1, x, 5, (a*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/((a^2 - b^2)*d) - (b*AppellF1[1/2, -(m/2), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*((a^2 - b^2)*d))} -{Cos[c + d*x]^m/(a + b*Cos[c + d*x])^2, x, 8, (b^2*AppellF1[1/2, (1/2)*(-1 - m), 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(1 + m)*(Cos[c + d*x]^2)^((1/2)*(-1 - m))*Sin[c + d*x])/((a^2 - b^2)^2*d) + (a^2*AppellF1[1/2, (1 - m)/2, 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/((a^2 - b^2)^2*d) - (2*a*b*AppellF1[1/2, -(m/2), 2, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*((a^2 - b^2)^2*d))} - - -{Sec[c + d*x]^m*(a + b*Cos[c + d*x])^3, x, 8, If[$VersionNumber>=8, -((a^2*b*(1 - 2*m)*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(1 - m)*(2 - m))) - (a^2*Sec[c + d*x]^(-2 + m)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(d*(1 - m)) - (b*(b^2*(2 - m) + 3*a^2*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-4 + m)*Sin[c + d*x])/(d*(2 - m)*(4 - m)*Sqrt[Sin[c + d*x]^2]) - (a*(3*b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(1 - m)*(3 - m)*Sqrt[Sin[c + d*x]^2]), -((a^2*b*(1 - 2*m)*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - 3*m + m^2))) - (a^2*Sec[c + d*x]^(-2 + m)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(d*(1 - m)) - (b*(b^2*(2 - m) + 3*a^2*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-4 + m)*Sin[c + d*x])/(d*(8 - 6*m + m^2)*Sqrt[Sin[c + d*x]^2]) - (a*(3*b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(3 - 4*m + m^2)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(a + b*Cos[c + d*x])^2, x, 7, If[$VersionNumber>=8, -((a^2*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m))) - ((b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(1 - m)*(3 - m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - m)*Sqrt[Sin[c + d*x]^2]), -((a^2*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m))) - ((b^2*(1 - m) + a^2*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-3 + m)*Sin[c + d*x])/(d*(3 - 4*m + m^2)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - m)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(a + b*Cos[c + d*x])^1, x, 6, -((b*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(d*(2 - m)*Sqrt[Sin[c + d*x]^2])) - (a*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -(* ::Section:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (c-c Cos[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -{Sqrt[1 - Cos[x]]/Sqrt[a - Cos[x]], x, 2, -2*ArcTan[Sin[x]/(Sqrt[1 - Cos[x]]*Sqrt[a - Cos[x]])]} -{Sqrt[(1 - Cos[x])/(a - Cos[x])], x, 3, -((2*ArcTan[Sin[x]/(Sqrt[1 - Cos[x]]*Sqrt[a - Cos[x]])]*Sqrt[(1 - Cos[x])/(a - Cos[x])]*Sqrt[a - Cos[x]])/Sqrt[1 - Cos[x]])} - - -{(-B*(1/(1 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 1, (a*B*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(-B*(4/(4 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 1, (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{(-B*(n/(n + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^n, x, 1, (B*(a + a*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + n))} - - -{(-B*(-3/(-3 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 1, -((B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^3))} - - -{(-B*(3/2/(3/2 + 1)) + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 1, (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -{(-B*(-1/2/(-1/2 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 2, (2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{(-B*(-5/2/(-5/2 + 1)) + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 1, -((2*B*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^(5/2)))} - - -{(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(2/3), x, 3, (3*B*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d) + (2*2^(1/6)*(5*A + 2*B)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(5*d*(1 + Cos[c + d*x])^(7/6))} -{(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/3), x, 3, (3*B*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d) + ((4*A + B)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2*2^(1/6)*d*(1 + Cos[c + d*x])^(5/6))} - - -{(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/3), x, 3, (3*B*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(1/3)) + ((2*A - B)*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2^(5/6)*d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))} -{(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(2/3), x, 3, (3*(A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(2/3)) - (2^(5/6)*(A - 2*B)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x])^(5/6))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -{(b*B/a + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*x)/b - (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b*d)} -{(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2, x, 2, Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))} -{(3 + Cos[c + d*x])/(2 - Cos[c + d*x]), x, 2, -x + (5*x)/Sqrt[3] + (10*ArcTan[Sin[c + d*x]/(2 + Sqrt[3] - Cos[c + d*x])])/(Sqrt[3]*d)} - - -{(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 3, (2*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^(n/3) (c+d Cos[e+f x])^n*) -(**) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(2/3), x, 7, (Sqrt[2]*(a + b)*B*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))} -{(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/3), x, 7, (Sqrt[2]*(a + b)*B*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3), x, 7, (Sqrt[2]*B*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} -{(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(2/3), x, 7, (Sqrt[2]*B*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(A*b - a*B)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(b*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c Cos[e+f x])^n (d Cos[e+f x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 9, (6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 8, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 6, (2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 6, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 7, -((2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 9, -((6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 9, (6*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 7, (6*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 7, (2*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 6, (2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 7, -((2*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 8, -((2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 9, -((6*A*b*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^3*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b^2*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 8, (6*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (10*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*b*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 8, (6*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 7, (2*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 6, (2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 7, -((2*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 8, -((2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^3*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 9, -((6*A*b^2*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^5*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^4*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*b^3*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 9, (6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 8, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 7, (2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 5, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 7, -((2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 9, -((6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 9, (6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 8, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 7, (2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 6, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 6, -((2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 9, -((6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^5*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 9, (6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*A*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)} -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 8, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 7, (2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 6, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 7, -((2*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 7, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 9, -((6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])} - - -{(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(7/2), x, 8, -((6*A*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b^2*d*(b*Cos[c + d*x])^(3/2)) + (6*A*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^(m/2) (d Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 7, (3*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 6, (A*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 2, (B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 3, (A*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 3, (B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 5, (B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 6, (A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(1/2), x, 6, (B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 7, (3*b*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 6, (A*b*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 2, (b*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 3, (A*b*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 3, (b*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 5, (b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 6, (A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(3/2), x, 6, (b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 7, (3*b^2*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (3*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 6, (A*b^2*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) - (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 2, (b^2*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 3, (A*b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 3, (b^2*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 5, (b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 6, (A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x])*(b*Cos[c + d*x])^(5/2), x, 6, (b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 6, (A*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 2, (B*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 3, (A*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 3, (B*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 5, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 6, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(1/2), x, 6, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 6, (A*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 2, (B*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 3, (A*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 3, (B*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 5, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 6, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 6, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 6, (A*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 2, (B*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 3, (A*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 3, (B*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 5, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 6, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 6, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x]^3)/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]), x, 4, -((3*A*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^3*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^4*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]), x, 4, -((3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^2*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]), x, 3, -((3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x], x, 4, (-3*A*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2, x, 4, (3*A*b*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3, x, 4, (3*A*b^2*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]), x, 4, -((3*A*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(16/3)*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2]*Sin[c + d*x])/(16*b^4*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]), x, 4, -((3*A*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]), x, 3, -((3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x], x, 4, -((3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2, x, 4, (-3*A*b*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3, x, 4, (3*A*b^2*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3), x, 4, -((3*A*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^4*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3), x, 4, -((3*A*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(2/3), x, 3, (-3*A*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(2/3), x, 4, (3*A*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(2/3), x, 4, (3*A*b*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(2/3), x, 4, (3*A*b^2*Hypergeometric2F1[-(4/3), 1/2, -(1/3), Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Cos[c + d*x])^(8/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3), x, 4, -((3*A*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])) - (3*B*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^4*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3), x, 4, (-3*A*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(4/3), x, 3, (3*A*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3), x, 4, (3*A*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3), x, 4, (3*A*b*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(4/3), x, 4, (3*A*b^2*Hypergeometric2F1[-(5/3), 1/2, -(2/3), Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Cos[c + d*x])^(10/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^n) with m and/or n symbolic*) - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, -((A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])) - (B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, -((A*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(4 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^4*d*(4 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, -((A*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 3, -((A*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^1, x, 4, -((A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])) - (B*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^2, x, 4, (A*b*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^3, x, 4, (A*b^2*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) + (b*B*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^4, x, 4, (A*b^3*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]) + (b^2*B*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, (-2*A*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(9/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(9 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, (-2*A*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]), x, 4, (-2*A*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]], x, 4, (-2*A*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2), x, 4, (2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]) - (2*B*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2), x, 4, (2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2), x, 4, (2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2), x, 4, (2*A*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-7 + 2*n)/4, (-3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 - 2*n)*Cos[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]), x, 4, (-3*A*b*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (10 + 3*m)/6, (16 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]), x, 4, (-3*A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (8 + 3*m)/6, (14 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]), x, 4, (-3*A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(1/3), x, 4, (-3*A*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3), x, 4, (-3*A*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3), x, 4, (3*A*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.2 (g sin)^p (a+b cos)^m (c+d cos)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.2 (g sin)^p (a+b cos)^m (c+d cos)^n.m deleted file mode 100644 index ca6fed4..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.2 (g sin)^p (a+b cos)^m (c+d cos)^n.m +++ /dev/null @@ -1,28 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Cos[e+f x])^m (c-c Cos[e+f x])^n*) - - -(* ::Title:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+a Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -(* ::Title:: *) -(*Integrands of the form (g Sin[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^m (a+b Cos[e+f x])^n / (c+d Cos[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sin[e+f x])^(m/2) (d Cos[e+f x])^(n/2) / (a+b Cos[e+f x])*) - - -{Sqrt[g*Sin[e + f*x]]*Sqrt[d*Cos[e + f*x]]/(a + b*Cos[e + f*x]), x, 16, -((Sqrt[d]*Sqrt[g]*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])])/(Sqrt[2]*b*f)) + (Sqrt[d]*Sqrt[g]*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/(Sqrt[g]*Sqrt[d*Cos[e + f*x]])])/(Sqrt[2]*b*f) + (2*Sqrt[2]*a*d*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) - (2*Sqrt[2]*a*d*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(b*Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]]) + (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/Sqrt[d*Cos[e + f*x]] + Sqrt[g]*Tan[e + f*x]])/(2*Sqrt[2]*b*f) - (Sqrt[d]*Sqrt[g]*Log[Sqrt[g] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Sin[e + f*x]])/Sqrt[d*Cos[e + f*x]] + Sqrt[g]*Tan[e + f*x]])/(2*Sqrt[2]*b*f)} -{Sqrt[d*Cos[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(a + b*Cos[e + f*x])), x, 4, (2*Sqrt[2]*Sqrt[d]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Sin[e + f*x]]) - (2*Sqrt[2]*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(Sqrt[-a^2 + b^2]*f*Sqrt[g*Sin[e + f*x]])} - - -{Sqrt[g*Sin[e + f*x]]/(Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])), x, 5, -((2*Sqrt[2]*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]])) + (2*Sqrt[2]*Sqrt[g]*Sqrt[Cos[e + f*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Sin[e + f*x]]/(Sqrt[g]*Sqrt[1 + Cos[e + f*x]])], -1])/(Sqrt[-a + b]*Sqrt[a + b]*f*Sqrt[d*Cos[e + f*x]])} -{1/(Sqrt[g*Sin[e + f*x]]*Sqrt[d*Cos[e + f*x]]*(a + b*Cos[e + f*x])), x, 7, -((2*Sqrt[2]*b*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Sin[e + f*x]])) + (2*Sqrt[2]*b*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Cos[e + f*x]]/(Sqrt[d]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a^2 + b^2]*Sqrt[d]*f*Sqrt[g*Sin[e + f*x]]) + (EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[d*Cos[e + f*x]]*Sqrt[g*Sin[e + f*x]])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.3 (g cos)^p (a+b cos)^m (c+d cos)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.3 (g cos)^p (a+b cos)^m (c+d cos)^n.m deleted file mode 100644 index b2cd1c3..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.2.3 (g cos)^p (a+b cos)^m (c+d cos)^n.m +++ /dev/null @@ -1,11 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration problems of the form (g Cos[e+f x])^p (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Cos[e+f x])^p (a+a Cos[e+f x])^m (c-c Cos[e+f x])^n*) - - -{((a + a*Cos[e + f*x])^2*Sec[e + f*x]^2)/(-c + c*Cos[e + f*x]), x, 6, -((3*a^2*ArcTanh[Sin[e + f*x]])/(c*f)) + (4*a^2*Sin[e + f*x])/(c*f*(1 - Cos[e + f*x])) - (a^2*Tan[e + f*x])/(c*f)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.3.1 (a+b cos)^m (c+d cos)^n (A+B cos).m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.3.1 (a+b cos)^m (c+d cos)^n (A+B cos).m deleted file mode 100644 index ee3fe78..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.3.1 (a+b cos)^m (c+d cos)^n (A+B cos).m +++ /dev/null @@ -1,1030 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 8, (3/8)*a*(A + B)*x + (a*(5*A + 4*B)*Sin[c + d*x])/(5*d) + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 4*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 7, (1/8)*a*(4*A + 3*B)*x + (a*(A + B)*Sin[c + d*x])/d + (a*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 3, (1/2)*a*(A + B)*x + (a*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 1, (1/2)*a*(2*A + B)*x + (a*(A + B)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, a*(A + B)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 4, a*B*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 6, (a*(A + 2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 7, (a*(A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*B)*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x]), x, 7, (a*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(A + B)*Tan[c + d*x])/d + (a*(3*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(A + B)*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 9, (1/16)*a^2*(12*A + 11*B)*x + (a^2*(9*A + 8*B)*Sin[c + d*x])/(5*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(12*A + 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*(6*A + 7*B)*Cos[c + d*x]^4*Sin[c + d*x])/(30*d) + (B*Cos[c + d*x]^4*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(6*d) - (a^2*(9*A + 8*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 8, (1/8)*a^2*(7*A + 6*B)*x + (a^2*(10*A + 9*B)*Sin[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*A + 6*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(10*A + 9*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 4, (1/8)*a^2*(8*A + 7*B)*x + (a^2*(8*A + 7*B)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A - B)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 2, (1/2)*a^2*(3*A + 2*B)*x + (2*a^2*(3*A + 2*B)*Sin[c + d*x])/(3*d) + (a^2*(3*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, (1/2)*a^2*(4*A + 3*B)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B)*Sin[c + d*x])/(2*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, a^2*(A + 2*B)*x + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(A - B)*Sin[c + d*x])/d + (A*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 5, a^2*B*x + (a^2*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*A + 2*B)*Tan[c + d*x])/(2*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 7, (a^2*(2*A + 3*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(5*A + 6*B)*Tan[c + d*x])/(3*d) + (a^2*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^2, x, 8, (a^2*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*A + 5*B)*Tan[c + d*x])/(3*d) + (a^2*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*A + 4*B)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 9, (1/16)*a^3*(26*A + 23*B)*x + (a^3*(19*A + 17*B)*Sin[c + d*x])/(5*d) + (a^3*(26*A + 23*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(22*A + 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*A + 4*B)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(19*A + 17*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 10, (1/8)*a^3*(15*A + 13*B)*x + (a^3*(15*A + 13*B)*Sin[c + d*x])/(5*d) + (3*a^3*(15*A + 13*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*A - B)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(15*A + 13*B)*Sin[c + d*x]^3)/(60*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 8, (5/8)*a^3*(4*A + 3*B)*x + (a^3*(4*A + 3*B)*Sin[c + d*x])/d + (3*a^3*(4*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(4*A + 3*B)*Sin[c + d*x]^3)/(12*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(7*A + 5*B)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, (1/2)*a^3*(6*A + 7*B)*x + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*B*Sin[c + d*x])/(2*d) - ((2*A - B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, a^3*(A + 3*B)*x + (a^3*(7*A + 6*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + ((2*A + B)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/d + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 6, a^3*B*x + (a^3*(5*A + 7*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A + B)*Tan[c + d*x])/(2*d) + ((5*A + 3*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 8, (5*a^3*(3*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(9*A + 11*B)*Tan[c + d*x])/(3*d) + (a^3*(27*A + 28*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A + 2*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^3, x, 9, (a^3*(13*A + 15*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 45*B)*Tan[c + d*x])/(15*d) + (a^3*(13*A + 15*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*A + 45*B)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((7*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 10, (1/16)*a^4*(49*A + 44*B)*x + (a^4*(252*A + 227*B)*Sin[c + d*x])/(35*d) + (a^4*(49*A + 44*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(301*A + 276*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((7*A + 10*B)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (7*(A + B)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(252*A + 227*B)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 13, (7/16)*a^4*(8*A + 7*B)*x + (4*a^4*(8*A + 7*B)*Sin[c + d*x])/(5*d) + (27*a^4*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(8*A + 7*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*A - B)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (B*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(8*A + 7*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 11, (7/8)*a^4*(5*A + 4*B)*x + (8*a^4*(5*A + 4*B)*Sin[c + d*x])/(5*d) + (27*a^4*(5*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^4*(5*A + 4*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) - (4*a^4*(5*A + 4*B)*Sin[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, (1/8)*a^4*(48*A + 35*B)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(8*A + 7*B)*Sin[c + d*x])/(8*d) + (a*B*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + ((32*A + 35*B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(13*A + 12*B)*x + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A + 2*B)*Sin[c + d*x])/(2*d) - ((3*A - B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) - ((3*A - 8*B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, (1/2)*a^4*(8*A + 13*B)*x + (a^4*(13*A + 8*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B)*Sin[c + d*x])/(2*d) - ((6*A + B)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(2*d) + ((5*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, a^4*(A + 4*B)*x + (a^4*(12*A + 13*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B)*Sin[c + d*x])/(2*d) + ((11*A + 9*B)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(3*d) + ((2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 7, a^4*B*x + (a^4*(35*A + 48*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*(7*A + 8*B)*Tan[c + d*x])/(8*d) + ((35*A + 32*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((7*A + 4*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 9, (7*a^4*(4*A + 5*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(83*A + 100*B)*Tan[c + d*x])/(15*d) + (a^4*(244*A + 275*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((26*A + 25*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((8*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^4, x, 10, (7*a^4*(7*A + 8*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B)*Tan[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((73*A + 72*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 7, -((3*(4*A - 5*B)*x)/(8*a)) + (4*(A - B)*Sin[c + d*x])/(a*d) - (3*(4*A - 5*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) - ((4*A - 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - (4*(A - B)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, (3*(A - B)*x)/(2*a) - ((3*A - 4*B)*Sin[c + d*x])/(a*d) + (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 2, -(((A - B)*x)/a) + (B*x)/(2*a) + ((A - B)*Sin[c + d*x])/(a*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x])), -(((2*A - 3*B)*x)/(2*a)) + (2*(A - B)*Sin[c + d*x])/(a*d) - ((2*A - 3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, ((A - B)*x)/a + (B*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 2, (B*x)/a + ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 3, (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 5, -(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B)*Tan[c + d*x])/(a*d) - ((A - B)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, ((3*A - 2*B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(A - B)*Tan[c + d*x])/(a*d) + ((3*A - 2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x]), x, 6, -((3*(A - B)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((4*A - 3*B)*Tan[c + d*x])/(a*d) - (3*(A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A - 3*B)*Tan[c + d*x]^3)/(3*a*d)} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, ((7*A - 10*B)*x)/(2*a^2) - (4*(2*A - 3*B)*Sin[c + d*x])/(a^2*d) + ((7*A - 10*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((7*A - 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(2*A - 3*B)*Sin[c + d*x]^3)/(3*a^2*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 3, -(((4*A - 7*B)*x)/(2*a^2)) + (2*(5*A - 8*B)*Sin[c + d*x])/(3*a^2*d) - ((4*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((5*A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 6, ((A - 2*B)*x)/a^2 - ((A - 4*B)*Sin[c + d*x])/(3*a^2*d) - ((A - 2*B)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 4, (B*x)/a^2 + ((2*A - 5*B)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 2, ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((A + 2*B)*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((4*A - B)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 6, -(((2*A - B)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(5*A - 2*B)*Tan[c + d*x])/(3*a^2*d) - ((2*A - B)*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, ((7*A - 4*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*A - 5*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^2, x, 7, -(((10*A - 7*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + (4*(3*A - 2*B)*Tan[c + d*x])/(a^2*d) - ((10*A - 7*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((10*A - 7*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(3*A - 2*B)*Tan[c + d*x]^3)/(3*a^2*d)} - - -{Cos[c + d*x]^5*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 8, ((13*A - 23*B)*x)/(2*a^3) - (4*(19*A - 34*B)*Sin[c + d*x])/(5*a^3*d) + ((13*A - 23*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((8*A - 13*B)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((13*A - 23*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(19*A - 34*B)*Sin[c + d*x]^3)/(15*a^3*d)} -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 4, -(((6*A - 13*B)*x)/(2*a^3)) + (8*(9*A - 19*B)*Sin[c + d*x])/(15*a^3*d) - ((6*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*A - 19*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 7, ((A - 3*B)*x)/a^3 - ((7*A - 27*B)*Sin[c + d*x])/(15*a^3*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 3*B)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 5, (B*x)/a^3 + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - 7*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((4*A - 29*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 4, -(((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3)) + ((3*A - 8*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A + 7*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 3, ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*A - B)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 7, -(((3*A - B)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A - 11*B)*Tan[c + d*x])/(15*a^3*d) - ((A - B)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*A - B)*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3, x, 8, ((13*A - 6*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*A - 9*B)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*A - 9*B)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} - - -{Cos[c + d*x]^5*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 5, -(((8*A - 21*B)*x)/(2*a^4)) + (8*(83*A - 216*B)*Sin[c + d*x])/(105*a^4*d) - ((8*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) + ((52*A - 129*B)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*(83*A - 216*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((A - 2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 8, ((A - 4*B)*x)/a^4 - ((55*A - 244*B)*Sin[c + d*x])/(105*a^4*d) + ((25*A - 88*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((A - 4*B)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A - 12*B)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 6, (B*x)/a^4 - ((6*A - 55*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((12*A - 215*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 10*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 5, -((2*(A + 27*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2)) + ((13*A + 36*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((A - 8*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 5, -(((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4)) + ((4*A - 11*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((8*A + 13*B)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((8*A + 13*B)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 4, ((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 6, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 6*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (2*(80*A - 3*B)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 8, -(((4*A - B)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (8*(83*A - 20*B)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 25*B)*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((4*A - B)*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4, x, 9, ((21*A - 8*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A - 52*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(216*A - 83*B)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 5, (4*a*(9*A + 8*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*A + 8*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(9*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (4*(9*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*(7*A + 6*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(7*A + 6*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*(3*A + B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*(A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[a]*(3*A + 4*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(3*A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 5, (Sqrt[a]*(5*A + 6*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 6*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(5*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 6, (4*a^2*(187*A + 168*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(187*A + 168*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(11*A + 12*B)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*a*(187*A + 168*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*B*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(187*A + 168*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*(39*A + 34*B)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(9*A + 10*B)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(39*A + 34*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*B*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(39*A + 34*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (8*a^2*(21*A + 19*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*A + 19*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A - 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (8*a^2*(5*A + 3*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(3*A + 4*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^(3/2)*(3*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(A - 2*B)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^(3/2)*(7*A + 12*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(11*A + 14*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(11*A + 14*B)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 6, (a^(3/2)*(75*A + 88*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(75*A + 88*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(9*A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (2*a^3*(803*A + 710*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(209*A + 194*B)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(803*A + 710*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a^2*(11*A + 14*B)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*(803*A + 710*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*a*B*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (64*a^3*(15*A + 13*B)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*A - 2*B)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*B*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (64*a^3*(7*A + 5*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*A + 5*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (2*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(35*A + 32*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(5*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(5*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*(3*A + 14*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(3*A - 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(19*A + 20*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(9*A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(7*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(25*A + 38*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(3*A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(163*A + 200*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(163*A + 200*B)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(95*A + 104*B)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(11*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 7, (a^(5/2)*(283*A + 326*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(283*A + 326*B)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(157*A + 170*B)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 7, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*A - 37*B)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(7*A - 31*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 6, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(5*A - 7*B)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*A - 2*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 6, -(((A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 7, ((7*A - 4*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 8, -(((15*A - 19*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((651*A - 799*B)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((63*A - 67*B)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((7*A - 11*B)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((273*A - 397*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 7, ((11*A - 15*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((65*A - 93*B)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((35*A - 39*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((7*A - 11*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 13*B)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((3*A - 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 5, ((3*A - 7*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 3, ((A + 3*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 7, -(((3*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 8, ((19*A - 12*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((7*A - 6*B)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 8, ((163*A - 283*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((13*A - 21*B)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((985*A - 1729*B)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((85*A - 157*B)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((475*A - 787*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, -(((75*A - 163*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) + ((A - B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((9*A - 17*B)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((93*A - 197*B)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 95*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 6, ((19*A - 75*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 13*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 9*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, ((5*A + 19*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 13*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 4, ((3*A + 5*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 8, -(((5*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 9, ((39*A - 20*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (7*(9*A - 5*B)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]), x, 8, (2*a*(9*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (10*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(A + B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]), x, 7, (6*a*(A + B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]), x, 6, (2*a*(5*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]], x, 5, (2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2), x, 5, (-2*a*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2), x, 6, (-2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2), x, 7, (-2*a*(3*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]), x, 8, (4*a^2*(9*A + 8*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(6*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^2*(9*A + 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(9*A + 11*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*B*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]), x, 7, (4*a^2*(4*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*B*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]], x, 6, (4*a^2*(5*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2), x, 6, (4*a^2*B*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(3*A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(3*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2), x, 6, (-4*a^2*A*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(5*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2), x, 7, (-4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (4*a^2*(4*A + 5*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2), x, 8, (-4*a^2*(3*A + 4*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(9*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (4*a^2*(6*A + 7*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]), x, 9, (4*a^3*(17*A + 15*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(121*A + 105*B)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 105*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(17*A + 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (20*a^3*(22*A + 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(11*A + 15*B)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(99*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]), x, 8, (4*a^3*(21*A + 17*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(13*A + 11*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^3*(24*A + 23*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*A + 13*B)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(63*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]], x, 7, (4*a^3*(9*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(21*A + 13*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*A + 11*B)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2), x, 7, (4*a^3*(5*A + 9*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(5*A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - B)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2), x, 7, (-4*a^3*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (20*a^3*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(4*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(7*A + 3*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2), x, 7, (-4*a^3*(9*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^3*(21*A + 20*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(9*A + 5*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2), x, 8, (-4*a^3*(7*A + 9*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(41*A + 42*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(7*A + 9*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(11*A + 7*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2), x, 9, (-4*a^3*(17*A + 21*B)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(23*A + 24*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^3*(11*A + 13*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(13*A + 9*B)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]), x, 6, (-3*(5*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(A - B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((5*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]), x, 5, (3*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((3*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x]), x, 4, -(((A - 3*B)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])), x, 4, ((A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])), x, 5, -(((3*A - B)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - B)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])), x, 6, (3*(A - B)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*(A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2, x, 7, (-7*(5*A - 8*B)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) + (5*(2*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (5*(2*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*(5*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((2*A - 3*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2, x, 6, ((4*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) - (5*(A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((4*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2, x, 5, -(((A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A - 5*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^2, x, 5, -((B*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((A + 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2), x, 5, (A*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A + B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2), x, 6, -(((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) - ((5*A - 2*B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2), x, 7, ((7*A - 4*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (5*(2*A - B)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (5*(2*A - B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A - 4*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3, x, 8, (-7*(17*A - 33*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A - 21*B)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A - 21*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (7*(17*A - 33*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) + ((A - B)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((7*A - 12*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*(11*A - 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3, x, 7, (7*(7*A - 17*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 33*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) + (7*(7*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3, x, 6, -((9*A - 49*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3, x, 6, -((A + 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3, x, 6, ((A - B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3), x, 6, ((9*A + B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3), x, 7, -((49*A - 9*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - 9*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3), x, 8, (7*(17*A - 7*B)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - (7*(17*A - 7*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 6, (5*Sqrt[a]*(8*A + 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (5*a*(8*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (5*a*(8*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*A + 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 5, (Sqrt[a]*(6*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(6*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*A + 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[a]*(4*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (Sqrt[a]*(2*A + B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*Sqrt[a]*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 2, (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(2*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 6, (a^(3/2)*(88*A + 75*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(88*A + 75*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(88*A + 75*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(14*A + 11*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(14*A + 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(6*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^(3/2)*(12*A + 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*B*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (a^(3/2)*(2*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(2*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^(3/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(4*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 3, (2*a^2*(6*A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(18*A + 25*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 4, (2*a^2*(8*A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^2*(10*A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 7, (a^(5/2)*(326*A + 283*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(326*A + 283*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(326*A + 283*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(170*A + 157*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(10*A + 13*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(200*A + 163*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(200*A + 163*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(104*A + 95*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(8*A + 11*B)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(38*A + 25*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^3*(54*A + 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(20*A + 19*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(4*A - 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(4*A - B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (a^(5/2)*(2*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(14*A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (2*a^(5/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(32*A + 35*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 4, (2*a^3*(10*A + 11*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(230*A + 301*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 5, (2*a^3*(124*A + 135*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x])*(a + a*Cos[c + d*x])^(5/2), x, 6, (2*a^3*(194*A + 209*B)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 7, -(((4*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 6, ((2*A - B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 5, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 7, ((2*A - 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 4, ((3*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 5, -(((7*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, ((11*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 8, ((2*A - 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((7*A - 15*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((11*A - 35*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 11*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, ((5*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 5, ((19*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 6, -(((75*A - 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, ((163*A - 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A - 147*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 9, ((2*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((3*A - 7*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((79*A - 259*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) - (7*(7*A - 27*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 8, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(7/2)*d) + ((5*A - 177*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((5*A - 17*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 49*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 6, ((7*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) + ((17*A + 67*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 6, ((13*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 17*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 6, ((63*A + 13*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((5*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((103*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 7, -((3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d)) - ((A - B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sin[c + d*x])/(48*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sin[c + d*x])/(192*a^2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(7/2), x, 8, ((1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sin[c + d*x])/(48*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sin[c + d*x])/(64*a^2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sin[c + d*x])/(192*a^3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((1887*A - 691*B)*Sin[c + d*x])/(192*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, (1/8)*(4*a*A + 3*b*B)*x + ((A*b + a*B)*Sin[c + d*x])/d + ((4*a*A + 3*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 3, (1/2)*(A*b + a*B)*x + ((3*a*A + 2*b*B)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 1, (1/2)*(2*a*A + b*B)*x + ((A*b + a*B)*Sin[c + d*x])/d + (b*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 4, (A*b + a*B)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (b*B*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 4, b*B*x + ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 6, ((a*A + 2*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((A*b + a*B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*A + 3*b*B)*Tan[c + d*x])/(3*d) + ((A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, ((3*a*A + 4*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((A*b + a*B)*Tan[c + d*x])/d + ((3*a*A + 4*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A*b + a*B)*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 7, (1/8)*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*x + ((4*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x])/(5*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*(5*A*b + 6*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) - ((4*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 4, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B)*x + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*Sin[c + d*x])/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b - a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 2, (1/2)*(2*a^2*A + A*b^2 + 2*a*b*B)*x + (2*(3*a*A*b + a^2*B + b^2*B)*Sin[c + d*x])/(3*d) + (b*(3*A*b + 2*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 4, ((4*a*A*b + 2*a^2*B + b^2*B)*x)/2 + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (b*(2*A*b + 3*a*B)*Sin[c + d*x])/(2*d) + (b*B*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 4, b*(A*b + 2*a*B)*x + (a*(2*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b^2*B*Sin[c + d*x])/d + (a^2*A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 4, b^2*B*x + ((a^2*A + 2*A*b^2 + 4*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A*b + a*B)*Tan[c + d*x])/d + (a^2*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, ((2*a*A*b + a^2*B + 2*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a^2*A + 3*A*b^2 + 6*a*b*B)*Tan[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 7, ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B)*Tan[c + d*x])/(3*d) + ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 8, (1/16)*(8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*x + ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(5*d) + ((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(18*a*A*b + 14*a^2*B + 5*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*(3*A*b + 4*a*B)*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, (1/8)*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*x + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*A*b - a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 3, (1/8)*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*x + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, (1/2)*(6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*A*b + 8*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*d) + (b^2*(3*A*b + 5*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, (1/2)*b*(6*a*A*b + 6*a^2*B + b^2*B)*x + (a^2*(3*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a^2*A - A*b^2 - 3*a*b*B)*Sin[c + d*x])/d - (b^2*(2*a*A - b*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, b^2*(A*b + 3*a*B)*x + (a*(a^2*A + 6*A*b^2 + 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a*A - 2*b*B)*Sin[c + d*x])/(2*d) + (a^2*(2*A*b + a*B)*Tan[c + d*x])/d + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 5, b^3*B*x + ((3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2*A + 8*A*b^2 + 9*a*b*B)*Tan[c + d*x])/(3*d) + (a^2*(5*A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 7, ((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(3*d) + (a*(3*a^2*A + 10*A*b^2 + 12*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(3*A*b + 2*a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 8, ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 15*b^3*B)*Tan[c + d*x])/(15*d) + ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(4*a^2*A + 12*A*b^2 + 15*a*b*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(7*A*b + 5*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 9, (1/16)*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*x + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x])/(35*d) + ((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b*(224*a^2*A*b + 35*A*b^3 + 104*a^3*B + 140*a*b^2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(168*d) + (b^2*(49*a*A*b + 31*a^2*B + 18*b^2*B)*Cos[c + d*x]^4*Sin[c + d*x])/(105*d) + (b*(7*A*b + 10*a*B)*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(42*d) + (b*B*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) - ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, (1/16)*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Sin[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (B*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 4, (1/8)*(8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*A*b + 4*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, (1/8)*(32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (b*(34*a^2*A*b + 4*A*b^3 + 19*a^3*B + 16*a*b^2*B)*Sin[c + d*x])/(6*d) + (b^2*(32*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*B*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, (1/2)*b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*x + (a^3*(4*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(6*a^3*A - 12*a*A*b^2 - 17*a^2*b*B - 2*b^3*B)*Sin[c + d*x])/(3*d) - (b^2*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*a*A - b*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (a*A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, (1/2)*b^2*(8*a*A*b + 12*a^2*B + b^2*B)*x + (a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(13*a^2*A*b - 2*A*b^3 + 4*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/(2*d) - (b^2*(6*a*A*b + 2*a^2*B - b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, b^3*(A*b + 4*a*B)*x + (a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(8*a*A*b + 3*a^2*B - 6*b^2*B)*Sin[c + d*x])/(6*d) + (a^2*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*Tan[c + d*x])/(3*d) + (a*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 6, b^4*B*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(16*a^2*A*b + 19*A*b^3 + 4*a^3*B + 34*a*b^2*B)*Tan[c + d*x])/(6*d) + (a^2*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (a*(7*A*b + 4*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 8, ((12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*Tan[c + d*x])/(15*d) + (a*(60*a^2*A*b + 56*A*b^3 + 15*a^3*B + 110*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a^2*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (a*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^4, x, 9, ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]])/(16*d) + ((32*a^3*A*b + 40*a*A*b^3 + 8*a^4*B + 60*a^2*b^2*B + 15*b^4*B)*Tan[c + d*x])/(15*d) + ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a*(16*a^2*A*b + 13*A*b^3 + 4*a^3*B + 27*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*(3*A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, ((2*a^2 + b^2)*(A*b - a*B)*x)/(2*b^4) - (2*a^3*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*Sin[c + d*x])/(3*b^3*d) + ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, -(((2*a*A*b - 2*a^2*B - b^2*B)*x)/(2*b^3)) + (2*a^2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((A*b - a*B)*Sin[c + d*x])/(b^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, ((A*b - a*B)*x)/b^2 - (2*a*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*x)/b + (2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 4, -((2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)) + (A*ArcTanh[Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, (2*b*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^2*d) + (A*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, -((2*b^2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((a^2*A + 2*A*b^2 - 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((A*b - a*B)*Tan[c + d*x])/(a^2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 7, (2*b^3*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((a^2 + 2*b^2)*(A*b - a*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B)*Tan[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -(((4*a*A*b - 6*a^2*B - b^2*B)*x)/(2*b^4)) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 5, ((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (B*Sin[c + d*x])/(b^2*d) - (a^2*(A*b - a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 5, (B*x)/b^2 - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 4, (2*(a*A - b*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 5, -((2*(2*a^2*A*b - A*b^3 - a^3*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^2*d) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, (2*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((a^2*A - 2*A*b^2 + a*b*B)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, -((2*b^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ((a^2*A + 6*A*b^2 - 4*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B)*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((6*a*A*b - 12*a^2*B - b^2*B)*x)/(2*b^5)) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 6, ((A*b - 3*a*B)*x)/b^4 - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 5, (B*x)/b^3 + ((a^2*A*b^3 + 2*A*b^5 - 2*a^5*B + 5*a^3*b^2*B - 6*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(A*b - a*B)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 6, -(((3*a*A*b - a^2*B - 2*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 5, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 6, -(((6*a^4*A*b - 5*a^2*A*b^3 + 2*A*b^5 - 2*a^5*B - a^3*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, (b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((3*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(6*a^2*A*b - 3*A*b^3 - 4*a^3*B + a*b^2*B)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 8, -((b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((a^2*A + 12*A*b^2 - 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 7, ((A*b - 4*a*B)*x)/b^5 - (a*(2*a^6*A*b - 7*a^4*A*b^3 + 8*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 28*a^5*b^2*B - 35*a^3*b^4*B + 20*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((3*a^3*A*b - 8*a*A*b^3 - 12*a^4*B + 23*a^2*b^2*B - 6*b^4*B)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a^2*(a^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 6, (B*x)/b^4 - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 34*a*b^4*B)*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 6, ((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*(A*b - a*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 7, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 6, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 7, -(((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 8, (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - ((4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4, x, 9, -((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ((a^2*A + 20*A*b^2 - 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*Sin[c + d*x])/d - (B*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*x)/2 + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (B*Sin[c + d*x])/d} -{Cos[c + d*x]^0*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, B*x} -{Sec[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (B*ArcTanh[Sin[c + d*x]])/d} -{Sec[c + d*x]^2*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (B*Tan[c + d*x])/d + (B*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, ((2*a^2 + b^2)*B*x)/(2*b^3) - (2*a^3*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*B*Sin[c + d*x])/(b^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -((a*B*x)/b^2) + (2*a^2*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(b*d)} -{Cos[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 4, (B*x)/b - (2*a*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Cos[c + d*x]^0*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 3, (2*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{Sec[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 5, -((2*b*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)) + (B*ArcTanh[Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^2*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, (2*b^2*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (b*B*ArcTanh[Sin[c + d*x]])/(a^2*d) + (B*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, -((2*b^3*B*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((a^2 + 2*b^2)*B*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*B*Tan[c + d*x])/(a^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 9, (2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2*A*b + 75*A*b^3 - 16*a^3*B - 36*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) - (2*(36*a*A*b - 24*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) + (2*(3*A*b - 2*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*B*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(14*a*A*b - 8*a^2*B - 25*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(14*a*A*b - 8*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(5*a*A*b - 2*a^2*B + 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b - 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, (2*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 9, -((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a*A + 2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 10, -(((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*A*b + 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A - A*b^2 + 4*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 11, -(((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((16*a^2*A - A*b^2 + 18*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(18*a^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(18*a^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) - (2*(18*a*A*b - 8*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(21*a*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(3*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*a*A*b - a^2*B + b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -(((a*A - 2*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a^2*A + 2*A*b^2 + 2*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 10, -(((5*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((7*a*A*b + 4*a^2*B + 8*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 3*A*b^2 + 12*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((5*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 11, -(((16*a^2*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((16*a^2*A + 17*A*b^2 + 42*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((7*A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -((2*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B - 81*b^2*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, (2*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, (2*(35*a*A*b + 23*a^2*B + 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*A*b + 5*A*b^3 - 8*a^3*B + 8*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -(((3*a^2*A - 6*A*b^2 - 14*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*a^3*A + 12*a*A*b^2 + 4*a^2*b*B + 2*b^3*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*(5*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*a*A - 2*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -(((9*a*A*b + 4*a^2*B - 8*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((11*a^2*A*b + 8*A*b^3 + 4*a^3*B + 16*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(4*a^2*A + 15*A*b^2 + 20*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(7*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((16*a^2*A + 33*A*b^2 + 54*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((16*a^3*A + 59*a*A*b^2 + 66*a^2*b*B + 48*b^3*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*A + 33*A*b^2 + 54*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(3*A*b + 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 12, -(((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((356*a^2*A*b + 133*A*b^3 + 128*a^3*B + 472*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*d) + (a*(11*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(56*a^2*A*b + 63*A*b^3 - 48*a^3*B - 44*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*A*b + 49*a*A*b^3 - 48*a^4*B - 32*a^2*b^2*B - 25*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(28*a*A*b - 24*a^2*B - 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*B*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, -((2*(10*a*A*b - 8*a^2*B - 9*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(10*a^2*A*b + 5*A*b^3 - 8*a^3*B - 7*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*B*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(3*A*b - 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a*A*b - 2*a^2*B - b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 5, (2*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b - a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 5, (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 9, -((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 10, ((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 8, -((2*(40*a^3*A*b - 25*a*A*b^3 - 48*a^4*B + 24*a^2*b^2*B + 9*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(40*a^2*A*b + 5*A*b^3 - 48*a^3*B - 12*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*A*b - 5*A*b^3 - 24*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*A*b - 6*a^2*B + b^2*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(6*a*A*b - 8*a^2*B - b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(A*b - a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 10, -(((a^2*A - 3*A*b^2 + 2*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(a^2*A - 3*A*b^2 + 2*a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 11, ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(80*a^5*A*b - 140*a^3*A*b^3 + 40*a*A*b^5 - 128*a^6*B + 212*a^4*b^2*B - 55*a^2*b^4*B - 9*b^6*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(80*a^4*A*b - 80*a^2*A*b^3 - 5*A*b^5 - 128*a^5*B + 116*a^3*b^2*B + 17*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(5*a^2*A*b - 9*A*b^3 - 8*a^3*B + 12*a*b^2*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(40*a^4*A*b - 65*a^2*A*b^3 + 5*A*b^5 - 64*a^5*B + 98*a^3*b^2*B - 14*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*A*b - 50*a*A*b^3 - 48*a^4*B + 71*a^2*b^2*B - 3*b^4*B)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(8*a^4*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*A*b - 9*a*A*b^3 - 16*a^4*B + 16*a^2*b^2*B + b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a^2*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(2*a^3*A*b - 6*a*A*b^3 - 8*a^4*B + 15*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*A*b - 3*A*b^3 - 8*a^3*B + 9*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(A*b - a*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a*A*b + 2*a^2*B - 3*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 10, -((2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(A*b - a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 11, -(((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*a^2*A - 5*A*b^2 + 2*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(3*a^2*A - 5*A*b^2 + 2*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 12, ((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^4*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*(a + b*Cos[c + d*x])^(3/2)) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^(3/2))} - - -{Cos[c + d*x]^0*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 3, (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 3, (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^0*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 5, (2*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*b*B*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*b*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*B*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^(m/2) (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 8, (2*(9*a*A + 7*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*A + 7*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(A*b + a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*B*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, (6*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a*A + 5*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 6, (2*(5*a*A + 3*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 5, (2*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*a*A + b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 5, -((2*(a*A - b*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 6, -((2*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a*A + 3*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x]), x, 7, -((2*(3*a*A + 5*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a*A + 5*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 8, (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*(9*b^2*B + 11*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*B + 11*a*(2*A*b + a*B))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*A*b + 13*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 7, (2*(7*b^2*B + 9*a*(2*A*b + a*B))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*b^2*B + 9*a*(2*A*b + a*B))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*A*b + 11*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, (2*(3*b^2*B + 5*a*(2*A*b + a*B))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(5*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, -((2*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 5, -((2*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^2, x, 6, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 8, (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*A*b + 26*a^2*B + 9*b^2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b^2*(11*A*b + 15*a*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 7, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*A*b + 13*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(6*a^2*A - A*b^2 - 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*a*A - b*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(7*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(a*A - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^3, x, 6, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(9*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*a^2*A + 14*A*b^2 + 15*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 7, -((2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d)) + (2*(3*a^2 + b^2)*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) - (2*a^3*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 6, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) + (2*a^2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d) + (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) - (2*a*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (2*B*EllipticF[(1/2)*(c + d*x), 2])/(b*d) + (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 5, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 7, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b*(A*b - a*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -(((a*A*b - 3*a^2*B + 2*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, ((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -(((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, -(((2*a^2*A - 3*A*b^2 + a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 8, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d)) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 7, -(((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 8, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3, x, 9, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d} -{Cos[c + d*x]^(-1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 2, (2*B*EllipticF[(1/2)*(c + d*x), 2])/d} -{Cos[c + d*x]^(-3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]), x, 3, (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 7, -((2*a*B*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d)) + (2*(3*a^2 + b^2)*B*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) - (2*a^3*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d) - (2*a*B*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) + (2*a^2*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)} -{Cos[c + d*x]^(1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 4, (2*B*EllipticF[(1/2)*(c + d*x), 2])/(b*d) - (2*a*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)} -{Cos[c + d*x]^(-1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 2, (2*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a + b)*d)} -{Cos[c + d*x]^(-3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 6, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*b*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*B*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2, x, 8, (2*b*B*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b^2*B*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*B*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*b*B*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^(m/2) (a+b Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 8, -(((a - b)*Sqrt[a + b]*(6*a*A*b - 3*a^2*B + 16*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d)) + (Sqrt[a + b]*(a + 2*b)*(6*A*b - 3*a*B + 8*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*A*b - a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*(4*A*b + a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d)) + (Sqrt[a + b]*(4*A*b + (a + 2*b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(4*a*A*b - a^2*B + 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, -(((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)) + (Sqrt[a + b]*(2*A + B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*A*b + a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 5, (2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (2*Sqrt[a + b]*(A*b - a*(A - B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 4, (2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*(a - b)*Sqrt[a + b]*(A - 3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 5, (2*(a - b)*Sqrt[a + b]*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b - 5*a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(a - b)*Sqrt[a + b]*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + a^2*(25*A - 63*B) + 2*a*b*(3*A - 7*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 9, -(((a - b)*Sqrt[a + b]*(24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d)) - (Sqrt[a + b]*(9*a^3*B - 6*a^2*b*(4*A + B) - 8*b^3*(16*A + 9*B) - 4*a*b^2*(28*A + 39*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((8*a*A*b - 3*a^2*B + 12*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 8, -(((a - b)*Sqrt[a + b]*(30*a*A*b + 3*a^2*B + 16*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d)) + (Sqrt[a + b]*(30*a*A*b + 12*A*b^2 + 3*a^2*B + 14*a*b*B + 16*b^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((6*A*b + 7*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*B*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, -(((a - b)*Sqrt[a + b]*(4*A*b + 5*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d)) + (Sqrt[a + b]*(8*a*A + 4*A*b + 5*a*B + 2*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(12*a*A*b + 3*a^2*B + 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + ((4*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, ((a - b)*Sqrt[a + b]*(2*a*A - b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a*(A - B) - b*(4*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*A*b + 3*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(a - b)*Sqrt[a + b]*(4*A*b + 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*Sqrt[a + b]*(3*A*b^2 + a^2*(A - 3*B) - a*(4*A*b - 6*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*b*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) - (2*(a - b)*Sqrt[a + b]*(9*a*A - 3*A*b - 5*a*B + 15*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(a - b)*Sqrt[a + b]*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B) + 3*a*b*(19*A - 7*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(8*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(a - b)*Sqrt[a + b]*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 - a^3*(147*A - 75*B) + 3*a^2*b*(13*A - 57*B) + 6*a*b^2*(A - 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 10, -(((a - b)*Sqrt[a + b]*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d)) - (Sqrt[a + b]*(45*a^4*B - 30*a^3*b*(5*A + B) - 16*b^4*(45*A + 64*B) - 8*a*b^3*(355*A + 193*B) - 4*a^2*b^2*(295*A + 423*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*A*b + 120*A*b^3 - 15*a^3*B + 172*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((50*a*A*b - 15*a^2*B + 64*b^2*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 9, -(((a - b)*Sqrt[a + b]*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d)) + (Sqrt[a + b]*(15*a^3*B + 8*b^3*(16*A + 9*B) + 2*a^2*b*(132*A + 59*B) + 4*a*b^2*(52*A + 71*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((24*a*A*b + 5*a^2*B + 12*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, -(((a - b)*Sqrt[a + b]*(54*a*A*b + 33*a^2*B + 16*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d)) + (Sqrt[a + b]*(4*b^2*(3*A + 4*B) + a^2*(48*A + 33*B) + a*(54*A*b + 26*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (Sqrt[a + b]*(30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, ((a - b)*Sqrt[a + b]*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) - (Sqrt[a + b]*(8*a^2*(A - B) - 2*b^2*(2*A + B) - 3*a*b*(8*A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*a*A - b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, ((a - b)*Sqrt[a + b]*(14*a*A*b + 6*a^2*B - 3*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (Sqrt[a + b]*(2*a*b*(7*A - 9*B) - 2*a^2*(A - 3*B) - 3*b^2*(6*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) - (b*Sqrt[a + b]*(2*A*b + 5*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(2*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*Sqrt[a + b]*(15*A*b^3 - a*b^2*(23*A - 45*B) + a^2*b*(17*A - 35*B) - a^3*(9*A - 5*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*b^2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(a - b)*Sqrt[a + b]*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*A - 63*B) + 15*b^2*(A - 7*B) - 8*a*b*(15*A - 7*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d) + (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Cos[c + d*x]^(-11/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 - 6*a^2*b*(19*A - 60*B) + 3*a^3*(49*A - 25*B) + 15*a*b^2*(11*A - 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*a*(4*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{Cos[c + d*x]^(-13/2)*(A + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(a - b)*Sqrt[a + b]*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d) + (1/(3465*a^3*d))*(2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 3*a^4*(225*A - 539*B) - 6*a^3*b*(505*A - 209*B) + 15*a^2*b^2*(19*A - 121*B) + 10*a*b^3*(3*A - 11*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)]) + (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a*d*Cos[c + d*x]^(5/2)) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -{Cos[c + d*x]^(-5/2)*(3*b*(B/(2*a)) + B*Cos[c + d*x])*(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(a - b)*Sqrt[a + b]*(a^2 + 3*b^2)*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - ((a - 3*b)*Sqrt[a + b]*(2*a^2 - a*b + 3*b^2)*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (b*Sqrt[a + b]*(5*a + (3*b^2)/a)*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*(4*A*b - 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d)) + (Sqrt[a + b]*(4*A*b - 3*a*B + 2*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (Sqrt[a + b]*(4*a*A*b - 3*a^2*B - 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) + ((4*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) - (Sqrt[a + b]*(2*A*b - a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (a*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 3, (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 3, (2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 4, -((2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d)) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/2), x, 5, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*A*b^2 + a^2*(9*A - 5*B) - 2*a*b*(A + 5*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 7, ((2*a*A*b - 3*a^2*B + b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) - ((2*A*b - (3*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*A*b - 3*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d)) + (2*(A*b - a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(A + B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*(a^2*A - 2*A*b^2 + a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(2*A*b + a*(A - B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 5, -((2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d)) + (2*(a + 2*b)*(4*A*b + a*(A - 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 8, ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) - ((6*a^2*A*b + 2*a*A*b^2 - 12*A*b^3 - 15*a^3*B - 5*a^2*b*B + 21*a*b^2*B + 3*b^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^3*(a + b)^(3/2)*d) - (Sqrt[a + b]*(2*A*b - 5*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) + (2*a*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*A*b^2 - 3*A*b^3 - 3*a^3*B - a^2*b*B + 6*a*b^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 5, -((2*(3*a^2*A + A*b^2 - 4*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d)) + (2*(3*a*A - A*b + a*B - 3*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 5, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*A*b^2 - 3*a^2*(A + B) + a*b*(3*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 5, (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d) + (2*(8*A*b^3 - 3*a^3*(A - B) + 2*a*b^2*(3*A - B) - 3*a^2*b*(3*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 6, -((2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*(a - b)*(a + b)^(3/2)*d)) - (2*(16*A*b^4 - a^4*(A - 3*B) + 4*a*b^3*(3*A - 2*B) - 9*a^3*b*(A - B) - 2*a^2*b^2*(8*A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 9, -(((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (a*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (a*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 2, -((2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d))} -{Cos[c + d*x]^(-1/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 2, (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{Cos[c + d*x]^(-3/2)*(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 4, (2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} - - -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]), x, 1, -((Cot[c + d*x]*EllipticE[ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/d)} -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]]), x, 1, -((Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[-2 + 3*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d)} - -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 - 3*Cos[c + d*x]]), x, 2, (Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[2 - 3*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], 1/5]*Sqrt[-1 + Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/d} -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 - 3*Cos[c + d*x]]), x, 2, (Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/d} - -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]), x, 1, (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 + 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)} -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 - 2*Cos[c + d*x]]), x, 1, (2*Sqrt[5]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d)} - -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]), x, 2, -((2*Sqrt[5]*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 + 2*Cos[c + d*x]]/Sqrt[-Cos[c + d*x]]], -(1/5)]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d))} -{(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 - 2*Cos[c + d*x]]), x, 2, -((2*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[-3 - 2*Cos[c + d*x]]/(Sqrt[5]*Sqrt[-Cos[c + d*x]])], -5]*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])/(3*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]) with m and/or n symbolic*) - - -{(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]), x, 0, Unintegrable[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]), x]} - - -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^4, x, 7, If[$VersionNumber>=8, (b*(A*b^3*(15 + 8*m + m^2) + 4*a*b^2*B*(15 + 8*m + m^2) + 2*a^3*B*(28 + 10*m + m^2) + a^2*A*b*(110 + 47*m + 5*m^2))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(4 + m)*(5 + m)) + (b^2*(b^2*B*(4 + m)^2 + 2*a*A*b*(5 + m)^2 + a^2*B*(36 + 11*m + m^2))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(3 + m)*(4 + m)*(5 + m)) + (b*(A*b*(5 + m) + a*B*(8 + m))*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^2*Sin[e + f*x])/(c*f*(4 + m)*(5 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^3*Sin[e + f*x])/(c*f*(5 + m)) - ((A*b^4*(3 + 4*m + m^2) + 4*a*b^3*B*(3 + 4*m + m^2) + 6*a^2*A*b^2*(4 + 5*m + m^2) + 4*a^3*b*B*(4 + 5*m + m^2) + a^4*A*(8 + 6*m + m^2))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^4*B*(8 + 6*m + m^2) + 4*a*A*b^3*(10 + 7*m + m^2) + 6*a^2*b^2*B*(10 + 7*m + m^2) + 4*a^3*A*b*(15 + 8*m + m^2) + a^4*B*(15 + 8*m + m^2))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*(5 + m)*Sqrt[Sin[e + f*x]^2]), (b*(A*b^3*(15 + 8*m + m^2) + 4*a*b^2*B*(15 + 8*m + m^2) + 2*a^3*B*(28 + 10*m + m^2) + a^2*A*b*(110 + 47*m + 5*m^2))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(5 + m)*(8 + 6*m + m^2)) + (b^2*(b^2*B*(4 + m)^2 + 2*a*A*b*(5 + m)^2 + a^2*B*(36 + 11*m + m^2))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(3 + m)*(4 + m)*(5 + m)) + (b*(A*b*(5 + m) + a*B*(8 + m))*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^2*Sin[e + f*x])/(c*f*(4 + m)*(5 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^3*Sin[e + f*x])/(c*f*(5 + m)) - ((A*b^4*(3 + 4*m + m^2) + 4*a*b^3*B*(3 + 4*m + m^2) + 6*a^2*A*b^2*(4 + 5*m + m^2) + 4*a^3*b*B*(4 + 5*m + m^2) + a^4*A*(8 + 6*m + m^2))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(4 + m)*(2 + 3*m + m^2)*Sqrt[Sin[e + f*x]^2]) - ((b^4*B*(8 + 6*m + m^2) + 4*a*A*b^3*(10 + 7*m + m^2) + 6*a^2*b^2*B*(10 + 7*m + m^2) + 4*a^3*A*b*(15 + 8*m + m^2) + a^4*B*(15 + 8*m + m^2))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*(5 + m)*Sqrt[Sin[e + f*x]^2])]} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^3, x, 6, (b*(b^2*B*(3 + m) + 3*a*A*b*(4 + m) + 2*a^2*B*(5 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(4 + m)) + (b^2*(A*b*(4 + m) + a*B*(6 + m))*Cos[e + f*x]*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(3 + m)*(4 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])^2*Sin[e + f*x])/(c*f*(4 + m)) - ((a^2*(2 + m)*(b*B*(1 + m) + a*A*(4 + m)) + b*(1 + m)*(b^2*B*(3 + m) + 3*a*A*b*(4 + m) + 2*a^2*B*(5 + m)))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[e + f*x]^2]) - ((A*b^3*(2 + m) + 3*a*b^2*B*(2 + m) + 3*a^2*A*b*(3 + m) + a^3*B*(3 + m))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e + f*x]^2])} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^2, x, 5, (b*(A*b*(3 + m) + a*B*(4 + m))*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)*(3 + m)) + (b*B*(c*Cos[e + f*x])^(1 + m)*(a + b*Cos[e + f*x])*Sin[e + f*x])/(c*f*(3 + m)) - ((A*b^2*(1 + m) + 2*a*b*B*(1 + m) + a^2*A*(2 + m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((b^2*B*(2 + m) + a*(2*A*b + a*B)*(3 + m))*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*(3 + m)*Sqrt[Sin[e + f*x]^2])} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^1, x, 5, (b*B*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(c*f*(2 + m)) - ((b*B*(1 + m) + a*A*(2 + m))*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c*f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - ((A*b + a*B)*(c*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(c^2*f*(2 + m)*Sqrt[Sin[e + f*x]^2])} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^1, x, 7, (a*(A*b - a*B)*c*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(c*Cos[e + f*x])^(-1 + m)*(Cos[e + f*x]^2)^((1 - m)/2)*Sin[e + f*x])/(b*(a^2 - b^2)*f) - ((A*b - a*B)*AppellF1[1/2, -(m/2), 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(c*Cos[e + f*x])^m*Sin[e + f*x])/((Cos[e + f*x]^2)^(m/2)*((a^2 - b^2)*f)) - (B*(c*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(b*c*f*(1 + m)*Sqrt[Sin[e + f*x]^2])} - - -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^(3/2), x, 1, (2*b*B*(c*Cos[e + f*x])^(1 + m)*Sqrt[a + b*Cos[e + f*x]]*Sin[e + f*x])/(c*f*(5 + 2*m)) + (2*Unintegrable[((c*Cos[e + f*x])^m*((1/2)*a*c*(2*b*B*(1 + m) + 2*a*A*(5/2 + m)) + (1/2)*c*(b^2*B*(3 + 2*m) + a*(2*A*b + a*B)*(5 + 2*m))*Cos[e + f*x] + (1/2)*b*c*(2*a*B*(3 + m) + A*b*(5 + 2*m))*Cos[e + f*x]^2))/Sqrt[a + b*Cos[e + f*x]], x])/(c*(5 + 2*m))} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^(1/2), x, 0, Unintegrable[(c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]), x]} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^(1/2), x, 0, Unintegrable[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]} -{(c*Cos[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^(3/2), x, 1, (2*b*(A*b - a*B)*(c*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*(a^2 - b^2)*c*f*Sqrt[a + b*Cos[e + f*x]]) + (2*Unintegrable[((c*Cos[e + f*x])^m*((1/2)*c*(a*(a*A - b*B) + 2*b*(A*b - a*B)*(1/2 + m)) - (1/2)*a*(A*b - a*B)*c*Cos[e + f*x] - (1/2)*b*(A*b - a*B)*c*(3 + 2*m)*Cos[e + f*x]^2))/Sqrt[a + b*Cos[e + f*x]], x])/(a*(a^2 - b^2)*c)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, (-2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, (-2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (-2*a*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 7, (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 8, (2*a*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 9, (6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^2*(A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(7*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, -((4*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (4*a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 8, (4*a^2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(3*A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(3*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 8, (4*a^2*(5*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(5*A + 7*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 9, (4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(7*A + 9*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} - - -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 10, -((4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(7*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(41*A + 42*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*A*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(11*A + 7*B)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((4*a^3*(9*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(21*A + 20*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(9*A + 5*B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 9, -((4*a^3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (20*a^3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(4*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(A - B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 9, (4*a^3*(5*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A - 6*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*A + 9*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 9, (4*a^3*(9*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 10, (4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(24*A + 23*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^3*(13*A + 11*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*A + 13*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]), x, 9, (3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*(A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]), x, 8, -(((3*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]), x, 7, ((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 7, -(((A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 8, (3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A - 5*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 9, -((3*(5*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d)) + (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((5*A - 7*B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2, x, 9, -(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - 2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2, x, 8, (A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 8, -((B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A + 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 8, -(((A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((2*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 9, ((4*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (5*(A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + ((4*A - 7*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3, x, 10, -(((49*A - 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3, x, 9, ((9*A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((6*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((9*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 9, ((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((4*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 9, -(((A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 9, -(((9*A - 49*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A - 8*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A - 13*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 10, (7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((13*A - 33*B)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 6, (32*a*(8*A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(8*A + 9*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(8*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 5, (16*a*(6*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(6*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(6*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 4, (4*a*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(4*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 3, (2*a*(2*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 4, (2*Sqrt[a]*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 4, (Sqrt[a]*(2*A + B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 5, (Sqrt[a]*(4*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*B*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(4*A + 3*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 6, (Sqrt[a]*(6*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*B*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(6*A + 5*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 7, (32*a^2*(168*A + 187*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(168*A + 187*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(168*A + 187*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(168*A + 187*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(12*A + 11*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 6, (16*a^2*(34*A + 39*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 5, (4*a^2*(52*A + 63*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 4, (2*a^2*(18*A + 25*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(6*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 5, (2*a^(3/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 5, (a^(3/2)*(2*A + 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(2*A - B)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 5, (a^(3/2)*(12*A + 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(4*A + 5*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 6, (a^(3/2)*(14*A + 11*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(6*A + 7*B)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a^2*(14*A + 11*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 7, (a^(3/2)*(88*A + 75*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(8*A + 9*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*B*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(15/2), x, 8, (32*a^3*(4184*A + 4615*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(4184*A + 4615*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(280*A + 299*B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(16*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 7, (16*a^3*(710*A + 803*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(194*A + 209*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 6, (4*a^3*(292*A + 345*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(124*A + 135*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 5, (2*a^3*(230*A + 301*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(10*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(10*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 6, (2*a^(5/2)*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 35*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 6, (a^(5/2)*(2*A + 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(14*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(2*A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 6, (a^(5/2)*(20*A + 19*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(4*A - 9*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(4*A - B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 6, (a^(5/2)*(38*A + 25*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(54*A + 49*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 7, (a^(5/2)*(200*A + 163*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(104*A + 95*B)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(8*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^3*(200*A + 163*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 8, (a^(5/2)*(326*A + 283*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(170*A + 157*B)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(10*A + 13*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (a*B*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]], x, 9, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]], x, 8, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]], x, 5, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/Sqrt[a + a*Cos[c + d*x]], x, 6, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 7, ((2*A - B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 8, -(((4*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A - B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - -{((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 7, ((2*A*b + 2*a*B - b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (b*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2), x, 9, ((19*A - 15*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2), x, 8, -((15*A - 11*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((147*A - 95*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2), x, 7, ((11*A - 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2), x, 6, -((7*A - 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(3/2), x, 5, ((3*A + B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 7, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 8, ((2*A - 3*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((A - 3*B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2), x, 9, -((283*A - 163*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A - 1495*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A - 85*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2), x, 8, ((163*A - 75*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A - 147*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2), x, 7, -((75*A - 19*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(5/2), x, 6, ((19*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 6, ((5*A + 3*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + ((A + 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 8, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 9, ((2*A - 5*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - ((11*A - 35*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(7/2), x, 9, ((1015*A - 363*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - ((1887*A - 691*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((23*A - 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((109*A - 41*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((579*A - 199*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(7/2), x, 8, -((3*(121*A - 21*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d)) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)) - ((19*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)) - ((199*A - 43*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)) + ((691*A - 103*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2))/(a + a*Cos[c + d*x])^(7/2), x, 7, ((63*A + 13*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) - ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) - ((5*A - B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((103*A + 5*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(1/2)), x, 7, ((13*A + 7*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]) + ((A + 3*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((5*A - 17*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)), x, 7, ((7*A + 5*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) + ((A - 13*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + ((17*A + 67*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)), x, 9, (2*B*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) + ((5*A - 177*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)) + ((5*A - 17*B)*Sin[c + d*x])/(48*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 49*B)*Sin[c + d*x])/(64*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)), x, 10, ((2*A - 7*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(7/2)*d) - ((177*A - 637*B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*Sqrt[2]*a^(7/2)*d) + ((A - B)*Sin[c + d*x])/(6*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)) + ((3*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((79*A - 259*B)*Sin[c + d*x])/(192*a^2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) - (7*(7*A - 27*B)*Sin[c + d*x])/(64*a^3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^(m/2) (a+b Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((2*(3*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a*A + 5*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(A*b + a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, (-2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*b + a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (-2*(a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 7, (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 8, (2*(5*a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2), x, 9, (6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*a*A + 5*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((2*(3*a^2*A + 5*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a^2*A + 5*b*(A*b + 2*a*B))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 8, -((2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A*b + 3*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 8, -((2*(a^2*A - b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(6*a*A*b + 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 8, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 9, (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*B*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 10, -((2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*a^2*A + 18*A*b^2 + 21*a*b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(11*A*b + 7*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*Sec[c + d*x]^(3/2)*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 9, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*a^2*A + 14*A*b^2 + 15*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(9*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 9, -((2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(9*a*A*b + 3*a^2*B - 2*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a*A - b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 9, -((2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*(5*A*b + 9*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(5*a*A - b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]], x, 9, (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]], x, 10, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(9*A*b + 13*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*b*B*(b + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]), x, 11, (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]), x, 8, (-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]), x, 6, (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d)} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 8, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) - (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 10, (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) + (2*a^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*B*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2, x, 12, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2, x, 11, -(((2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2, x, 10, -(((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 10, ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 10, -(((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 11, ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x]))} - - -{((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3, x, 12, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3, x, 11, -(((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 11, ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 11, -(((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d)) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 11, -(((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d)) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(b + a*Sec[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Sec[c + d*x]))} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 12, ((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*A*b - 99*a^3*A*b^3 + 72*a*A*b^5 - 105*a^6*B + 223*a^4*b^2*B - 128*a^2*b^4*B - 8*b^6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*b^5*(a^2 - b^2)^2*d) + (a^2*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(b + a*Sec[c + d*x]))} - - -{((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]), x, 4, (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]), x, 4, -((2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{((a*B + b*B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]), x, 3, (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d} -{(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 3, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d} -{(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 4, (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 4, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^(m/2) (a+b Cos[e+f x])^(n/2) (A+B Cos[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*Sqrt[a + b]*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + a^2*(25*A - 63*B) + 2*a*b*(3*A - 7*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 5, (2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(A - 3*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 6, (2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(A*b - a*(A - B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]])} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 7, -(((a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A + B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b + a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 8, -(((a - b)*Sqrt[a + b]*(4*A*b + a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(4*A*b + (a + 2*b)*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 9, -(((a - b)*Sqrt[a + b]*(6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(a + 2*b)*(6*A*b - 3*a*B + 8*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)} - - -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 8, (2*(a - b)*Sqrt[a + b]*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 - a^3*(147*A - 75*B) + 3*a^2*b*(13*A - 57*B) + 6*a*b^2*(A - 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*Sqrt[a + b]*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B) + 3*a*b*(19*A - 7*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (2*(8*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A - 3*A*b - 5*a*B + 15*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(4*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(3*A*b^2 + a^2*(A - 3*B) - a*(4*A*b - 6*b*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(2*a*A - b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a*(A - B) - b*(4*A + B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 8, -(((a - b)*Sqrt[a + b]*(4*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(8*a*A + 4*A*b + 5*a*B + 2*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 9, -(((a - b)*Sqrt[a + b]*(30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(30*a*A*b + 12*A*b^2 + 3*a^2*B + 14*a*b*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + ((6*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Sec[c + d*x]]) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 10, -(((a - b)*Sqrt[a + b]*(24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d*Sqrt[Sec[c + d*x]])) - (Sqrt[a + b]*(9*a^3*B - 6*a^2*b*(4*A + B) - 8*b^3*(16*A + 9*B) - 4*a*b^2*(28*A + 39*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) + ((8*a*A*b - 3*a^2*B + 12*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) + ((8*A*b - 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d)} - - -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2), x, 9, (2*(a - b)*Sqrt[a + b]*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d*Sqrt[Sec[c + d*x]]) + (1/(3465*a^3*d*Sqrt[Sec[c + d*x]]))*(2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 3*a^4*(225*A - 539*B) - 6*a^3*b*(505*A - 209*B) + 15*a^2*b^2*(19*A - 121*B) + 10*a*b^3*(3*A - 11*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)]) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2), x, 8, (2*(a - b)*Sqrt[a + b]*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 - 6*a^2*b*(19*A - 60*B) + 3*a^3*(49*A - 25*B) + 15*a*b^2*(11*A - 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*a*(4*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*Sqrt[a + b]*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*A - 63*B) + 15*b^2*(A - 7*B) - 8*a*b*(15*A - 7*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2), x, 8, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(15*A*b^3 - a*b^2*(23*A - 45*B) + a^2*b*(17*A - 35*B) - a^3*(9*A - 5*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2), x, 9, ((a - b)*Sqrt[a + b]*(14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a*b*(7*A - 9*B) - 2*a^2*(A - 3*B) - 3*b^2*(6*A + B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[a + b]*(2*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*a*(2*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2), x, 9, ((a - b)*Sqrt[a + b]*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a^2*(A - B) - 2*b^2*(2*A + B) - 3*a*b*(8*A + 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(4*a*A - b*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*a*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2), x, 9, -(((a - b)*Sqrt[a + b]*(54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(4*b^2*(3*A + 4*B) + a^2*(48*A + 33*B) + a*(54*A*b + 26*b*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])/Sec[c + d*x]^(1/2), x, 10, -(((a - b)*Sqrt[a + b]*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(15*a^3*B + 8*b^3*(16*A + 9*B) + 2*a^2*b*(132*A + 59*B) + 4*a*b^2*(52*A + 71*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + (b*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + ((24*a*A*b + 5*a^2*B + 12*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + ((8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])/Sec[c + d*x]^(3/2), x, 11, -(((a - b)*Sqrt[a + b]*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]])) - (1/(1920*b^2*d*Sqrt[Sec[c + d*x]]))*(Sqrt[a + b]*(45*a^4*B - 30*a^3*b*(5*A + B) - 16*b^4*(45*A + 64*B) - 8*a*b^3*(355*A + 193*B) - 4*a^2*b^2*(295*A + 423*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)]) + (Sqrt[a + b]*(10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a^2*A*b + 120*A*b^3 - 15*a^3*B + 172*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) + ((50*a*A*b - 15*a^2*B + 64*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) + ((10*A*b - 3*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (B*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(8*A*b^2 + a^2*(9*A - 5*B) - 2*a*b*(A + 5*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)} -{Sec[c + d*x]^(5/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 5, -((2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]])) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^(3/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 4, (2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A - B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 4, (1/(a*d*Sqrt[Sec[c + d*x]]))*(2*A*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)]) - (1/(b*d*Sqrt[Sec[c + d*x]]))*(2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])} -{Sec[c + d*x]^(-1/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b - a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^(-3/2)*(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*(4*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(4*A*b - 3*a*B + 2*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) + ((4*A*b - 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)} - - -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + (2*(a + 2*b)*(4*A*b + a*(A - 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b + a*(A - B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A + B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/(Sec[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(3/2)), x, 7, -((2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)), x, 8, ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((2*A*b - (3*a + b)*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} - - -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) - (2*(16*A*b^4 - a^4*(A - 3*B) + 4*a*b^3*(3*A - 2*B) - 9*a^3*b*(A - B) - 2*a^2*b^2*(8*A + 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 - 3*a^3*(A - B) + 2*a*b^2*(3*A - B) - 3*a^2*b*(3*A + B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])*Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(2*A*b^2 - 3*a^2*(A + B) + a*b*(3*A + B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 6, -((2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) + (2*(a*(3*A + B) - b*(A + 3*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 8, (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*(3*A*b^3 + 3*a^3*B + a^2*b*B - a*b^2*(A + 6*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 9, ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + ((3*b^3*(4*A - B) + 15*a^3*B - a*b^2*(2*A + 21*B) - a^2*(6*A*b - 5*b*B))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)} - - -{(a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2), x, 5, (2*(a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{(a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(1/2)/(a + b*Cos[c + d*x])^(3/2), x, 3, (2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])} -{(a*B + b*B*Cos[c + d*x])/(Sec[c + d*x]^(1/2)*(a + b*Cos[c + d*x])^(3/2)), x, 3, -((2*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]))} -{(a*B + b*B*Cos[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)), x, 10, -(((a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]) with m and/or n symbolic*) - - -{(c*Sec[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]), x, 1, (c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[((a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]))/(c*Cos[e + f*x])^m, x]} - - -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^4, x, 10, If[$VersionNumber>=8, -((c^6*(4*a^3*A*b*(15 - 8*m + m^2) + a^4*B*(15 - 8*m + m^2) + 4*a*A*b^3*(10 - 7*m + m^2) + 6*a^2*b^2*B*(10 - 7*m + m^2) + b^4*B*(8 - 6*m + m^2))*Hypergeometric2F1[1/2, (6 - m)/2, (8 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-6 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*(6 - m)*Sqrt[Sin[e + f*x]^2])) - (c^5*(a^4*A*(8 - 6*m + m^2) + 6*a^2*A*b^2*(4 - 5*m + m^2) + 4*a^3*b*B*(4 - 5*m + m^2) + A*b^4*(3 - 4*m + m^2) + 4*a*b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^5*(4*a^2*A*b*(3 - 4*m + m^2) + a^3*B*(3 - 4*m + m^2) + 2*A*b^3*(4 - 2*m + m^2) + a*b^2*B*(8 - 13*m + 5*m^2))*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(4 - m)) - (a^2*c^5*(2*a*b*B*(1 - m)^2 + a^2*A*(2 - m)^2 + A*b^2*(6 - m + m^2))*Sec[e + f*x]*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)*(3 - m)) - (a*c^5*(a*B*(1 - m) - A*b*(2 + m))*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^5*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(1 - m)), -((c^6*(4*a^3*A*b*(15 - 8*m + m^2) + a^4*B*(15 - 8*m + m^2) + 4*a*A*b^3*(10 - 7*m + m^2) + 6*a^2*b^2*B*(10 - 7*m + m^2) + b^4*B*(8 - 6*m + m^2))*Hypergeometric2F1[1/2, (6 - m)/2, (8 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-6 + m)*Sin[e + f*x])/(f*(48 - 44*m + 12*m^2 - m^3)*Sqrt[Sin[e + f*x]^2])) - (c^5*(a^4*A*(8 - 6*m + m^2) + 6*a^2*A*b^2*(4 - 5*m + m^2) + 4*a^3*b*B*(4 - 5*m + m^2) + A*b^4*(3 - 4*m + m^2) + 4*a*b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(15 - 8*m + m^2)*Sqrt[Sin[e + f*x]^2]) - (a*c^5*(4*a^2*A*b*(3 - 4*m + m^2) + a^3*B*(3 - 4*m + m^2) + 2*A*b^3*(4 - 2*m + m^2) + a*b^2*B*(8 - 13*m + 5*m^2))*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(1 - m)*(8 - 6*m + m^2)) - (a^2*c^5*(2*a*b*B*(1 - m)^2 + a^2*A*(2 - m)^2 + A*b^2*(6 - m + m^2))*Sec[e + f*x]*(c*Sec[e + f*x])^(-5 + m)*Tan[e + f*x])/(f*(6 - 11*m + 6*m^2 - m^3)) - (a*c^5*(a*B*(1 - m) - A*b*(2 + m))*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(2 - 3*m + m^2)) - (a*A*c^5*(c*Sec[e + f*x])^(-5 + m)*(b + a*Sec[e + f*x])^3*Tan[e + f*x])/(f*(1 - m))]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^3, x, 9, If[$VersionNumber>=8, -((c^5*(a^3*A*(8 - 6*m + m^2) + 3*a*A*b^2*(4 - 5*m + m^2) + 3*a^2*b*B*(4 - 5*m + m^2) + b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*(5 - m)*Sqrt[Sin[e + f*x]^2])) - (c^4*(A*b^3*(2 - m) + 3*a*b^2*B*(2 - m) + 3*a^2*A*b*(3 - m) + a^3*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^4*(3*a*b*B*(1 - m) + a^2*A*(2 - m) - 2*A*b^2*m)*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(3 - m)) - (a^2*c^4*(a*B*(1 - m) - A*b*(1 + m))*Sec[e + f*x]*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^4*(c*Sec[e + f*x])^(-4 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m)), -((c^5*(a^3*A*(8 - 6*m + m^2) + 3*a*A*b^2*(4 - 5*m + m^2) + 3*a^2*b*B*(4 - 5*m + m^2) + b^3*B*(3 - 4*m + m^2))*Hypergeometric2F1[1/2, (5 - m)/2, (7 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-5 + m)*Sin[e + f*x])/(f*(1 - m)*(15 - 8*m + m^2)*Sqrt[Sin[e + f*x]^2])) - (c^4*(A*b^3*(2 - m) + 3*a*b^2*B*(2 - m) + 3*a^2*A*b*(3 - m) + a^3*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(8 - 6*m + m^2)*Sqrt[Sin[e + f*x]^2]) - (a*c^4*(3*a*b*B*(1 - m) + a^2*A*(2 - m) - 2*A*b^2*m)*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(3 - 4*m + m^2)) - (a^2*c^4*(a*B*(1 - m) - A*b*(1 + m))*Sec[e + f*x]*(c*Sec[e + f*x])^(-4 + m)*Tan[e + f*x])/(f*(2 - 3*m + m^2)) - (a*A*c^4*(c*Sec[e + f*x])^(-4 + m)*(b + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(1 - m))]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^2, x, 8, If[$VersionNumber>=8, -((c^4*(b^2*B*(2 - m) + 2*a*A*b*(3 - m) + a^2*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(2 - m)*(4 - m)*Sqrt[Sin[e + f*x]^2])) - (c^3*(A*b^2*(1 - m) + 2*a*b*B*(1 - m) + a^2*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*Sqrt[Sin[e + f*x]^2]) - (a*c^3*(a*B*(1 - m) - A*b*m)*(c*Sec[e + f*x])^(-3 + m)*Tan[e + f*x])/(f*(1 - m)*(2 - m)) - (a*A*c^3*(c*Sec[e + f*x])^(-3 + m)*(b + a*Sec[e + f*x])*Tan[e + f*x])/(f*(1 - m)), -((c^4*(b^2*B*(2 - m) + 2*a*A*b*(3 - m) + a^2*B*(3 - m))*Hypergeometric2F1[1/2, (4 - m)/2, (6 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-4 + m)*Sin[e + f*x])/(f*(8 - 6*m + m^2)*Sqrt[Sin[e + f*x]^2])) - (c^3*(A*b^2*(1 - m) + 2*a*b*B*(1 - m) + a^2*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(3 - 4*m + m^2)*Sqrt[Sin[e + f*x]^2]) - (a*c^3*(a*B*(1 - m) - A*b*m)*(c*Sec[e + f*x])^(-3 + m)*Tan[e + f*x])/(f*(2 - 3*m + m^2)) - (a*A*c^3*(c*Sec[e + f*x])^(-3 + m)*(b + a*Sec[e + f*x])*Tan[e + f*x])/(f*(1 - m))]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^1, x, 7, If[$VersionNumber>=8, -((c^3*(b*B*(1 - m) + a*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(1 - m)*(3 - m)*Sqrt[Sin[e + f*x]^2])) - ((A*b + a*B)*c^2*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-2 + m)*Sin[e + f*x])/(f*(2 - m)*Sqrt[Sin[e + f*x]^2]) - (a*A*c^2*(c*Sec[e + f*x])^(-2 + m)*Tan[e + f*x])/(f*(1 - m)), -((c^3*(b*B*(1 - m) + a*A*(2 - m))*Hypergeometric2F1[1/2, (3 - m)/2, (5 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-3 + m)*Sin[e + f*x])/(f*(3 - 4*m + m^2)*Sqrt[Sin[e + f*x]^2])) - ((A*b + a*B)*c^2*Hypergeometric2F1[1/2, (2 - m)/2, (4 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-2 + m)*Sin[e + f*x])/(f*(2 - m)*Sqrt[Sin[e + f*x]^2]) - (a*A*c^2*(c*Sec[e + f*x])^(-2 + m)*Tan[e + f*x])/(f*(1 - m))]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^1, x, 10, -(((A*b - a*B)*AppellF1[1/2, m/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*Cos[e + f*x]*(Cos[e + f*x]^2)^(m/2)*(c*Sec[e + f*x])^(1 + m)*Sin[e + f*x])/((a^2 - b^2)*c*f)) + (a*(A*b - a*B)*AppellF1[1/2, (1 + m)/2, 1, 3/2, Sin[e + f*x]^2, -((b^2*Sin[e + f*x]^2)/(a^2 - b^2))]*(Cos[e + f*x]^2)^((1 + m)/2)*(c*Sec[e + f*x])^(1 + m)*Sin[e + f*x])/(b*(a^2 - b^2)*c*f) - (B*c*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[e + f*x]^2]*(c*Sec[e + f*x])^(-1 + m)*Sin[e + f*x])/(b*f*(1 - m)*Sqrt[Sin[e + f*x]^2])} - - -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^(3/2), x, 2, (2*b*B*Cos[e + f*x]*Sqrt[a + b*Cos[e + f*x]]*(c*Sec[e + f*x])^m*Sin[e + f*x])/(f*(5 - 2*m)) + (2*(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[((1/2)*a*c*(2*b*B*(1 - m) + 2*a*A*(5/2 - m)) + (1/2)*c*(b^2*B*(3 - 2*m) + a*(2*A*b + a*B)*(5 - 2*m))*Cos[e + f*x] + (1/2)*b*c*(A*b*(5 - 2*m) + 2*a*B*(3 - m))*Cos[e + f*x]^2)/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x])/(c*(5 - 2*m))} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])*(a + b*Cos[e + f*x])^(1/2), x, 1, (c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[(Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]))/(c*Cos[e + f*x])^m, x]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^(1/2), x, 1, (c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[(A + B*Cos[e + f*x])/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x]} -{(c*Sec[e + f*x])^m*(A + B*Cos[e + f*x])/(a + b*Cos[e + f*x])^(3/2), x, 2, (2*b*(A*b - a*B)*Cos[e + f*x]*(c*Sec[e + f*x])^m*Sin[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Cos[e + f*x]]) + (2*(c*Cos[e + f*x])^m*(c*Sec[e + f*x])^m*Unintegrable[((1/2)*c*(a^2*A + A*b^2*(1 - 2*m) - 2*a*b*B*(1 - m)) - (1/2)*a*(A*b - a*B)*c*Cos[e + f*x] - (1/2)*b*(A*b - a*B)*c*(3 - 2*m)*Cos[e + f*x]^2)/((c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]), x])/(a*(a^2 - b^2)*c)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (c-c Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (c+d Cos[e+f x])^n (A+B Cos[e+f x])*) - - -(* ::Section:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n (A+B Cos[e+f x])*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.1 (a+b cos)^m (A+B cos+C cos^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.1 (a+b cos)^m (A+B cos+C cos^2).m deleted file mode 100644 index 6254eaa..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.1 (a+b cos)^m (A+B cos+C cos^2).m +++ /dev/null @@ -1,838 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (A+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[c+d x])^m (A+C Cos[c+d x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2)*) - - -{Cos[c + d*x]^7*(A + C*Cos[c + d*x]^2), x, 3, ((A + C)*Sin[c + d*x])/d - ((3*A + 4*C)*Sin[c + d*x]^3)/(3*d) + (3*(A + 2*C)*Sin[c + d*x]^5)/(5*d) - ((A + 4*C)*Sin[c + d*x]^7)/(7*d) + (C*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2), x, 3, ((A + C)*Sin[c + d*x])/d - ((2*A + 3*C)*Sin[c + d*x]^3)/(3*d) + ((A + 3*C)*Sin[c + d*x]^5)/(5*d) - (C*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2), x, 3, ((A + C)*Sin[c + d*x])/d - ((A + 2*C)*Sin[c + d*x]^3)/(3*d) + (C*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2), x, 2, ((A + C)*Sin[c + d*x])/d - (C*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2), x, 2, (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2), x, 2, ((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2), x, 3, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^7*(A + C*Cos[c + d*x]^2), x, 4, ((5*A + 6*C)*ArcTanh[Sin[c + d*x]])/(16*d) + ((5*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((5*A + 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - -{Cos[c + d*x]^6*(A + C*Cos[c + d*x]^2), x, 5, (5*(8*A + 7*C)*x)/128 + (5*(8*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*A + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*A + 7*C)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (C*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2), x, 4, ((6*A + 5*C)*x)/16 + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2), x, 3, ((4*A + 3*C)*x)/8 + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2), x, 2, C*x + (A*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2), x, 3, ((2*A + 3*C)*Tan[c + d*x])/(3*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(A + C*Cos[c + d*x]^2), x, 3, ((4*A + 5*C)*Tan[c + d*x])/(5*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*A + 5*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^8*(A + C*Cos[c + d*x]^2), x, 3, ((6*A + 7*C)*Tan[c + d*x])/(7*d) + (A*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + (2*(6*A + 7*C)*Tan[c + d*x]^3)/(21*d) + ((6*A + 7*C)*Tan[c + d*x]^5)/(35*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^(m/2) (A+C Cos[e+f x]^2)*) - - -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 4, (2*b^2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*b*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (2*b^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{(b*Cos[c + d*x])^(1/2)*(A + C*Cos[c + d*x]^2), x, 3, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 3, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 3, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 3, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2))} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2), x, 4, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(9/2), x, 4, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^4*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(7*b*d*(b*Cos[c + d*x])^(7/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b^3*d*(b*Cos[c + d*x])^(3/2))} - - -{(3 - 5*Cos[c + d*x]^2)*Sqrt[Cos[c + d*x]], x, 1, -((2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/d)} -{(1 - 3*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]], x, 1, (-2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/2) (A+C Cos[e+f x]^2)*) - - -{(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(9/2), x, 5, (2*b^4*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b^3*(5*A + 7*C)*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*A*b^2*(b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(7/2), x, 5, -((2*b^4*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b^3*(3*A + 5*C)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*b^2*(b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(5/2), x, 4, (2*b^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*A*b^2*Sqrt[b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(3/2), x, 4, -((2*b^2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*A*b^2*Tan[c + d*x])/(d*Sqrt[b*Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(1/2), x, 4, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b^2*C*Tan[c + d*x])/(3*d*(b*Sec[c + d*x])^(3/2))} -{(A + C*Cos[c + d*x]^2)/(b*Sec[c + d*x])^(1/2), x, 4, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^2*C*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/2))} -{(A + C*Cos[c + d*x]^2)/(b*Sec[c + d*x])^(3/2), x, 5, (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^2*d) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*b*d*Sqrt[b*Sec[c + d*x]]) + (2*b^2*C*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/2))} -{(A + C*Cos[c + d*x]^2)/(b*Sec[c + d*x])^(5/2), x, 5, (2*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*(9*A + 7*C)*Sin[c + d*x])/(45*b*d*(b*Sec[c + d*x])^(3/2)) + (2*b^2*C*Tan[c + d*x])/(9*d*(b*Sec[c + d*x])^(9/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (A+C Cos[e+f x]^2) with m symbolic*) - - -{(b*Cos[c + d*x])^m*(A + C*Cos[c + d*x]^2), x, 2, (C*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(2 + m)) - ((C*(1 + m) + A*(2 + m))*(b*Cos[c + d*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2])} - - -{(b*Cos[c + d*x])^m*(-((C*(1 + m))/(2 + m)) + C*Cos[c + d*x]^2), x, 1, (C*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(2 + m))} -{(b*Cos[c + d*x])^m*(A - (A*(2 + m)*Cos[c + d*x]^2)/(1 + m)), x, 1, -((A*(b*Cos[c + d*x])^(1 + m)*Sin[c + d*x])/(b*d*(1 + m)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[c+d x])^m (b Cos[c+d x])^n (A+C Cos[c+d x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (b Cos[c+d x])^(n/2) (A+C Cos[c+d x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^3*d)} -{Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d)} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 3, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 4, (2*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, (2*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*A*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 5, (2*b*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} - - -{Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 5, (2*b*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^2*d)} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (2*b^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 4, (2*b*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, (2*b^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, -((2*b*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b^2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 4, (2*b^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 5, -((2*b*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*A*b^4*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 5, (2*b^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^5*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^3*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} - - -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 4, (2*b^2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*b*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 5, (2*b^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, (2*b^2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, (2*b^3*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 4, -((2*b^2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*A*b^3*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 4, (2*b^3*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 5, -((2*b^2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*A*b^5*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^3*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 5, (2*b^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^6*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^4*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 6, (10*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(231*d*Sqrt[b*Cos[c + d*x]]) + (10*(11*A + 9*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(231*b*d) + (2*(11*A + 9*C)*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*b^3*d) + (2*C*(b*Cos[c + d*x])^(9/2)*Sin[c + d*x])/(11*b^5*d)} -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 5, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]]) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^2*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^4*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 5, (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 4, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d)} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 3, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 4, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 4, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 5, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]])) + (2*A*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 5, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^5*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/2), x, 6, -((2*(7*A + 9*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]])) + (2*A*b^4*Sin[c + d*x])/(9*d*(b*Cos[c + d*x])^(9/2)) + (2*b^2*(7*A + 9*C)*Sin[c + d*x])/(45*d*(b*Cos[c + d*x])^(5/2)) + (2*(7*A + 9*C)*Sin[c + d*x])/(15*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 5, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]]) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^3*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^5*d)} -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 5, (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 4, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 4, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 3, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 4, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 5, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]])) + (2*A*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 5, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2))} - - -{Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 5, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^3*d*Sqrt[Cos[c + d*x]]) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^4*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^6*d)} -{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 5, (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)} -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 4, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 4, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 4, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 3, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 5, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 5, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b*d*(b*Cos[c + d*x])^(3/2))} - - -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2), x, 4, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]])) + (2*A*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(9/2), x, 4, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^4*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(7*b*d*(b*Cos[c + d*x])^(7/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b^3*d*(b*Cos[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x])^(m/2) (b Cos[c+d x])^(n/2) (A+C Cos[c+d x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 4, ((A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 4, ((4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + ((4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 3, ((A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 4, (A*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 3, (A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 3, (C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + ((2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 4, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + ((3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (b*(A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b*(A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (b*(4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b*(4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 3, (b*(A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 4, (A*b*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b*C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 3, (A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, (b*C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 3, (b*(A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 4, (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + (b*(2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 4, (b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (b*(3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 4, (b^2*(A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b^2*(A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]]) + (b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 4, (b^2*(4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b^2*(4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 3, (b^2*(A + C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 4, (A*b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b^2*C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, (A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 3, (b^2*C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 3, (b^2*(A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2))} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 4, (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + (b^2*(2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 4, (b^2*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (b^2*(3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 4, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 3, ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 4, (A*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (C*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]), x, 3, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]), x, 3, (C*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]), x, 3, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]), x, 4, (A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[b*Cos[c + d*x]]), x, 4, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 4, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*b*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 3, ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 4, (A*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (C*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 3, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)), x, 3, (C*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)), x, 3, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)), x, 4, (A*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)), x, 4, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*b*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(9/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 4, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*b^2*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 3, ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 4, (A*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (C*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 3, (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 3, (C*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)), x, 3, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)), x, 4, (A*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2)), x, 4, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*b^2*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (b Cos[c+d x])^(n/3) (A+C Cos[c+d x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b^3*d) - (3*(13*A + 10*C)*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(130*b^3*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(7/3)*Sin[c + d*x])/(10*b^2*d) - (3*(10*A + 7*C)*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*b^2*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2), x, 2, (3*C*(b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d) - (3*(7*A + 4*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 3, (3*C*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d) - (3*(4*A + C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 3, (3*A*b*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)) + (3*(A - 2*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 3, (3*A*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)) - (3*(2*A + 5*C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(11/3)*Sin[c + d*x])/(14*b^3*d) - (3*(14*A + 11*C)*(b*Cos[c + d*x])^(11/3)*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2]*Sin[c + d*x])/(154*b^3*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x])/(11*b^2*d) - (3*(11*A + 8*C)*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(88*b^2*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2), x, 2, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 3, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 3, (3*A*b*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 3, (3*A*b^2*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(13/3)*Sin[c + d*x])/(16*b^3*d) - (3*(16*A + 13*C)*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(208*b^3*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2), x, 3, (3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b^2*d) - (3*(13*A + 10*C)*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(130*b^2*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2), x, 2, (3*C*(b*Cos[c + d*x])^(7/3)*Sin[c + d*x])/(10*b*d) - (3*(10*A + 7*C)*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 3, (3*C*(b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*d) - (3*(7*A + 4*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 3, (3*b*C*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d) - (3*b*(4*A + C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 3, (3*A*b^2*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)) + (3*(A - 2*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 3, (3*C*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x])/(11*b^3*d) - (3*(11*A + 8*C)*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(88*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 3, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b^2*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/3), x, 2, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(1/3), x, 3, (3*A*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(1/3), x, 3, (3*A*b*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(1/3), x, 3, (3*A*b^2*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)) + (3*(4*A + 7*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3), x, 3, (3*C*(b*Cos[c + d*x])^(7/3)*Sin[c + d*x])/(10*b^3*d) - (3*(10*A + 7*C)*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3), x, 3, (3*C*(b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b^2*d) - (3*(7*A + 4*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(2/3), x, 2, (3*C*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) - (3*(4*A + C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(2/3), x, 3, (3*A*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)) + (3*(A - 2*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(2/3), x, 3, (3*A*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)) - (3*(2*A + 5*C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(2/3), x, 3, (3*A*b^2*Sin[c + d*x])/(8*d*(b*Cos[c + d*x])^(8/3)) + (3*(5*A + 8*C)*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(16*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 3, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b^3*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 3, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b^2*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(4/3), x, 2, (3*A*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3), x, 3, (3*A*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3), x, 3, (3*A*b*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)) + (3*(4*A + 7*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(4/3), x, 3, (3*A*b^2*Sin[c + d*x])/(10*d*(b*Cos[c + d*x])^(10/3)) + (3*(7*A + 10*C)*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(40*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x])^m (b Cos[c+d x])^n (A+C Cos[c+d x]^2) with m symbolic*) - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (3*b*C*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(10 + 3*m)) - (3*b*(C*(7 + 3*m) + A*(10 + 3*m))*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*(10 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*b*C*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(10 + 3*m)) - (3*b*(C*(7 + 3*m) + A*(10 + 3*m))*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(70 + 51*m + 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(8 + 3*m)) - (3*(C*(5 + 3*m) + A*(8 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(8 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(8 + 3*m)) - (3*(C*(5 + 3*m) + A*(8 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(40 + 39*m + 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) - (3*(C*(4 + 3*m) + A*(7 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) - (3*(C*(4 + 3*m) + A*(7 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(28 + 33*m + 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 3, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 21*m + 9*m^2)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3), x, 3, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)) - (3*(C + 3*C*m + A*(4 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(1 + 3*m), (1/6)*(7 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3*m)*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)) - (3*(C + 3*C*m + A*(4 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(1 + 3*m), (1/6)*(7 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 15*m + 9*m^2)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 3, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^m*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(1 - 3*m) - A*(2 + 3*m))*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (1/6)*(-1 + 3*m), (1/6)*(5 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 - 3*m)*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^m*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(1 - 3*m) - A*(2 + 3*m))*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (1/6)*(-1 + 3*m), (1/6)*(5 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 - 3*m - 9*m^2)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x])^m (b Cos[c+d x])^n (A+C Cos[c+d x]^2) with n symbolic*) - - -{(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, (C*(a*Cos[c + d*x])^(1 + m)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(a*d*(2 + m + n)) - ((C*(1 + m + n) + A*(2 + m + n))*(a*Cos[c + d*x])^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*(1 + m + n)*(2 + m + n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, (C*(b*Cos[c + d*x])^(3 + n)*Sin[c + d*x])/(b^3*d*(4 + n)) - ((C*(3 + n) + A*(4 + n))*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*(4 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, (C*(b*Cos[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(3 + n)) - ((C*(2 + n) + A*(3 + n))*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 2, (C*(b*Cos[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(2 + n)) - ((C*(1 + n) + A*(2 + n))*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*(2 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 3, (C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)) - ((A + A*n + C*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 3, (b*C*(b*Cos[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*n) - (b*(C*(1 - n) - A*n)*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*n*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 3, If[$VersionNumber>=8, -((b^2*C*(b*Cos[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(1 - n))) + (b^2*(A*(1 - n) + C*(2 - n))*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*(2 - n)*Sqrt[Sin[c + d*x]^2]), -((b^2*C*(b*Cos[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(1 - n))) + (b^2*(A*(1 - n) + C*(2 - n))*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - 3*n + n^2)*Sqrt[Sin[c + d*x]^2])]} -{(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 3, If[$VersionNumber>=8, -((b^3*C*(b*Cos[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(2 - n))) + (b^3*(A*(2 - n) + C*(3 - n))*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (1/2)*(-3 + n), (1/2)*(-1 + n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2]), -((b^3*C*(b*Cos[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(2 - n))) + (b^3*(A*(2 - n) + C*(3 - n))*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (1/2)*(-3 + n), (1/2)*(-1 + n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(6 - 5*n + n^2)*Sqrt[Sin[c + d*x]^2])]} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (2*C*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(9 + 2*n)) - (2*(C*(7 + 2*n) + A*(9 + 2*n))*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*(9 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(9 + 2*n)) - (2*(C*(7 + 2*n) + A*(9 + 2*n))*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(63 + 32*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (2*C*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) - (2*(C*(5 + 2*n) + A*(7 + 2*n))*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) - (2*(C*(5 + 2*n) + A*(7 + 2*n))*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(35 + 24*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2), x, 3, If[$VersionNumber>=8, (2*C*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) - (2*(C*(3 + 2*n) + A*(5 + 2*n))*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) - (2*(C*(3 + 2*n) + A*(5 + 2*n))*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(15 + 16*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 3, If[$VersionNumber>=8, (2*C*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 8*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 3, (2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Cos[c + d*x]]) + (2*(A - C*(1 - 2*n) + 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 4*n^2)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 3, If[$VersionNumber>=8, -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Cos[c + d*x]^(3/2))) + (2*(A + C*(3 - 2*n) - 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Cos[c + d*x]^(3/2))) + (2*(A + C*(3 - 2*n) - 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 8*n + 4*n^2)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, If[$VersionNumber>=8, -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(5/2))) + (2*(A*(3 - 2*n) + C*(5 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(5/2))) + (2*(A*(3 - 2*n) + C*(5 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(15 - 16*n + 4*n^2)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 3, If[$VersionNumber>=8, -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(7/2))) + (2*(A*(5 - 2*n) + C*(7 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-7 + 2*n), (1/4)*(-3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*(7 - 2*n)*Cos[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(7/2))) + (2*(A*(5 - 2*n) + C*(7 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-7 + 2*n), (1/4)*(-3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(35 - 24*n + 4*n^2)*Cos[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (A+C Cos[e+f x]^2)*) - - -{(a + a*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2), x, 4, If[$VersionNumber>=8, -((C*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(2 + 3*m + m^2))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(1 + m)*(2 + m)), -((C*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(2 + 3*m + m^2))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(2 + 3*m + m^2))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(2/3), x, 4, -((9*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*d)) + (3*C*(a + a*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*a*d) + ((40*A + 19*C)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(10*2^(5/6)*d*(1 + Cos[c + d*x])^(7/6))} -{(A + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/3), x, 4, -((9*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(28*d)) + (3*C*(a + a*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*a*d) + ((28*A + 13*C)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(14*2^(1/6)*d*(1 + Cos[c + d*x])^(5/6))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3), x, 4, -((9*C*Sin[c + d*x])/(10*d*(a + a*Cos[c + d*x])^(1/3))) + (3*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*a*d) + ((10*A + 7*C)*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(5*2^(5/6)*d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))} -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x, 4, (3*(A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(2/3)) + (3*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*a*d) - ((4*A + 7*C)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2*2^(1/6)*a*d*(1 + Cos[c + d*x])^(5/6))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^(n/3) (A+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d) - (3*a*(a + b)*C*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + ((3*a^2*C + b^2*(8*A + 5*C))*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))} -{(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d) - (3*Sqrt[2]*a*(a + b)*C*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(3*a^2*C + b^2*(7*A + 4*C))*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d) - (3*Sqrt[2]*a*C*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(3*a^2*C + b^2*(5*A + 2*C))*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} -{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) - (3*a*C*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + ((3*a^2*C + b^2*(4*A + C))*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+C Cos[e+f x]^2) with m symbolic*) - - -{(a + b*Cos[e + f*x])^m*(A - A*Cos[e + f*x]^2), x, 7, -((4*Sqrt[2]*A*AppellF1[1/2, -(3/2), -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(f*Sqrt[1 + Cos[e + f*x]]))) + (4*Sqrt[2]*A*AppellF1[1/2, -(1/2), -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(f*Sqrt[1 + Cos[e + f*x]]))} - - -{(a + b*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2), x, 8, (C*(a + b*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(b*f*(2 + m)) - (Sqrt[2]*a*(a + b)*C*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]])) + (Sqrt[2]*(a^2*C + b^2*(C*(1 + m) + A*(2 + m)))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -{(a*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 4, -((B*(a*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a^2*f*(2 + m)*Sqrt[Sin[e + f*x]^2])) - (C*(a*Cos[e + f*x])^(3 + m)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a^3*f*(3 + m)*Sqrt[Sin[e + f*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/2) (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^(m/2) (b Cos[e+f x])^(n/2) (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/3) (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])) - (3*C*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(10 + 3*m), (1/6)*(16 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(8 + 3*m), (1/6)*(14 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])) - (3*C*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(11 + 3*m), (1/6)*(17 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(11 + 3*m)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((3*b*B*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(10 + 3*m), (1/6)*(16 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])) - (3*b*C*Cos[c + d*x]^(4 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(13 + 3*m), (1/6)*(19 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(13 + 3*m)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 5, -((3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) - (3*C*Cos[c + d*x]^(3 + m)*Hypergeometric2F1[1/2, (1/6)*(8 + 3*m), (1/6)*(14 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3), x, 5, -((3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) - (3*C*Cos[c + d*x]^(3 + m)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 5, -((3*B*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) - (3*C*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2) with n symbolic*) - - -{(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((B*(a*Cos[c + d*x])^(2 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/2)*(2 + m + n), (1/2)*(4 + m + n), Cos[c + d*x]^2]*Sin[c + d*x])/(a^2*d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])) - (C*(a*Cos[c + d*x])^(3 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/2)*(3 + m + n), (1/2)*(5 + m + n), Cos[c + d*x]^2]*Sin[c + d*x])/(a^3*d*(3 + m + n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((B*(b*Cos[c + d*x])^(4 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^4*d*(4 + n)*Sqrt[Sin[c + d*x]^2])) - (C*(b*Cos[c + d*x])^(5 + n)*Hypergeometric2F1[1/2, (5 + n)/2, (7 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^5*d*(5 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((B*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])) - (C*(b*Cos[c + d*x])^(4 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^4*d*(4 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, -((B*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])) - (C*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 5, -((B*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])) - (C*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, -((B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])) - (C*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, (b*B*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]) - (C*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (b^2*B*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) + (b*C*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((2*B*Cos[c + d*x]^(9/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(9 + 2*n), (1/4)*(13 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(9 + 2*n)*Sqrt[Sin[c + d*x]^2])) - (2*C*Cos[c + d*x]^(11/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(11 + 2*n), (1/4)*(15 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(11 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((2*B*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])) - (2*C*Cos[c + d*x]^(9/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(9 + 2*n), (1/4)*(13 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(9 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, -((2*B*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])) - (2*C*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, -((2*B*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])) - (2*C*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, -((2*B*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])) - (2*C*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]) - (2*C*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) + (2*C*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 5, (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) + (2*C*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -{(a + a*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 4, If[$VersionNumber>=8, -(((C - B*(2 + m))*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(1 + m)*(2 + m))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(1 + m)*(2 + m)), -(((C - B*(2 + m))*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(1 + m)*(2 + m))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(2 + 3*m + m^2))]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -{(a + b*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 8, (C*(a + b*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(b*f*(2 + m)) - (Sqrt[2]*(a + b)*(a*C - b*B*(2 + m))*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]])) + (Sqrt[2]*(a^2*C + b^2*C*(1 + m) - a*b*B*(2 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d) + ((a + b)*(8*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) - ((8*a*b*B - 3*a^2*C - 5*b^2*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d) + (Sqrt[2]*(a + b)*(7*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) - (Sqrt[2]*(7*a*b*B - 3*a^2*C - 4*b^2*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d) + (Sqrt[2]*(5*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*(5*a*b*B - 3*a^2*C - 2*b^2*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) + ((4*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) - ((4*a*b*B - 3*a^2*C - b^2*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -{(a*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 4, (C*(a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) - ((C*(1 + m) + A*(2 + m))*(a*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]) - (B*(a*Cos[e + f*x])^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a^2*f*(2 + m)*Sqrt[Sin[e + f*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/2) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (10*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^3*d)} -{Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^2*d)} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 7, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 7, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 9, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 10, -((6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^3*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2)) + (6*b*B*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, (2*b*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (10*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^2*d)} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (6*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 8, (2*b*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 7, (2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 7, -((2*b*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 8, -((2*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 9, -((2*b*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^3*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 10, -((6*b*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^5*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^4*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^3*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2)) + (6*b^2*B*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (2*b^2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]) + (10*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 9, (6*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 8, (2*b^2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 7, (2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]) + (2*b^3*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 7, -((2*b^2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 8, -((2*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]])) + (2*b^3*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^4*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^3*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 9, -((2*b^2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^3*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^5*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^4*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*b^3*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 10, -((6*b^2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]])) + (2*b^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^6*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^5*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b^4*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2)) + (6*b^3*B*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 10, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^2*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^4*d)} -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 9, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^3*d)} -{(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 8, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^2*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[b*Cos[c + d*x]], x, 6, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]) + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[b*Cos[c + d*x]], x, 7, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[b*Cos[c + d*x]], x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]])) + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[b*Cos[c + d*x]], x, 9, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[b*Cos[c + d*x]], x, 10, -((6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]])) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^3*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b^2*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*b*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 10, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^3*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^5*d)} -{(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 9, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^2*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^4*d)} -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 8, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^3*d)} -{(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 7, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 6, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(3/2), x, 8, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]])) + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(3/2), x, 9, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*d*(b*Cos[c + d*x])^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(3/2), x, 10, -((6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]])) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b^2*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*b*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*d*(b*Cos[c + d*x])^(3/2)) + (6*B*Sin[c + d*x])/(5*b*d*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 10, (2*(9*A + 7*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(15*b^3*d*Sqrt[Cos[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(9*A + 7*C)*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*b^4*d) + (2*B*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d) + (2*C*(b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b^6*d)} -{(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 9, (6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*(7*A + 5*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*B*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d) + (2*C*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)} -{(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 8, (2*(5*A + 3*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d) + (2*C*(b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b^4*d)} -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 7, (2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]]) + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*d)} -{(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 7, -((2*(A - C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 7, -((2*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(b^3*d*Sqrt[Cos[c + d*x]])) + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(5/2), x, 9, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b*d*(b*Cos[c + d*x])^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(5/2), x, 10, -((6*B*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]])) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (2*A*b*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/2)) + (2*B*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b*d*(b*Cos[c + d*x])^(3/2)) + (6*B*Sin[c + d*x])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])} - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2), x, 8, -((2*(3*A + 5*C)*Sqrt[b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d*Sqrt[b*Cos[c + d*x]]) + (2*A*Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (2*B*Sin[c + d*x])/(3*b^2*d*(b*Cos[c + d*x])^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*b^3*d*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^(m/2) (b Cos[e+f x])^(n/2) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (3*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + ((5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (3*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - ((5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(15*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, ((4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + ((3*A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (A*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 4, (B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 4, (C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + ((2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + ((3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (3*b*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (3*b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - (b*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(15*d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (b*(4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b*(4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 3, (b*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*(3*A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (b*C*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (A*b*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b*C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 4, (b*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 4, (b*C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (b*(A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (b*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b*(2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 7, (b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (b*(3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (b*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (3*b^2*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b^2*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (3*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*B*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b^2*C*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) - (b^2*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(15*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (b^2*(4*A + 3*C)*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b^2*(4*A + 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b^2*C*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 3, (b^2*B*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*(3*A + 2*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (b^2*C*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (A*b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b^2*C*x*Sqrt[b*Cos[c + d*x]])/(2*Sqrt[Cos[c + d*x]]) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 4, (b^2*B*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (A*b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (b^2*C*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (b^2*C*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]] + (b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 6, (b^2*(A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 7, (b^2*B*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(2*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)) + (b^2*(2*A + 3*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 7, (b^2*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[b*Cos[c + d*x]])/(8*d*Sqrt[Cos[c + d*x]]) + (A*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)) + (b^2*(3*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (b^2*B*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 7, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 3, (B*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + ((3*A + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]], x, 5, (A*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (C*x*Sqrt[Cos[c + d*x]])/(2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]), x, 4, (B*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]), x, 4, (C*x*Sqrt[Cos[c + d*x]])/Sqrt[b*Cos[c + d*x]] + (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]), x, 6, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]), x, 7, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[b*Cos[c + d*x]]), x, 7, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x]^3)/(3*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 7, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 3, (B*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + ((3*A + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 5, (A*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (C*x*Sqrt[Cos[c + d*x]])/(2*b*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2), x, 4, (B*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)), x, 4, (C*x*Sqrt[Cos[c + d*x]])/(b*Sqrt[b*Cos[c + d*x]]) + (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)), x, 6, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)), x, 7, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)), x, 7, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*b*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x]^3)/(3*b*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 7, ((4*A + 3*C)*x*Sqrt[Cos[c + d*x]])/(8*b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + ((4*A + 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(4*b^2*d*Sqrt[b*Cos[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 3, (B*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + ((3*A + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Cos[c + d*x]]) + (B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 5, (A*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (C*x*Sqrt[Cos[c + d*x]])/(2*b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 4, (B*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (A*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2), x, 4, (C*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)), x, 6, ((A + 2*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)), x, 7, (B*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(2*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(2*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + ((2*A + 3*C)*Sin[c + d*x])/(3*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2)), x, 7, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b^2*d*Sqrt[b*Cos[c + d*x]]) + (A*Sin[c + d*x])/(4*b^2*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]) + ((3*A + 4*C)*Sin[c + d*x])/(8*b^2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]) + (B*Sin[c + d*x]^3)/(3*b^2*d*Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^(n/3) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (3*C*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x])/(11*b^2*d) - (3*(11*A + 8*C)*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(88*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(11/3)*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2]*Sin[c + d*x])/(11*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 5, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (3*A*b*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)) - (3*B*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, (3*A*b^2*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) + (3*b*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (3*A*b^3*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)) + (3*b^2*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*(4*A + 7*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b^2*d) - (3*(13*A + 10*C)*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(130*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(13/3)*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (3*C*(b*Cos[c + d*x])^(7/3)*Sin[c + d*x])/(10*b*d) - (3*(10*A + 7*C)*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(10/3)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 5, (3*C*(b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*d) - (3*(7*A + 4*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(7/3)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (3*b*C*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*d) - (3*b*(4*A + C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, (3*A*b^2*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)) - (3*b*B*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) + (3*(A - 2*C)*(b*Cos[c + d*x])^(4/3)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (3*A*b^3*Sin[c + d*x])/(5*d*(b*Cos[c + d*x])^(5/3)) + (3*b^2*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*(2*A + 5*C)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 5, (3*C*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x])/(11*b^3*d) - (3*(11*A + 8*C)*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(88*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(11/3)*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2]*Sin[c + d*x])/(11*b^4*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 5, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b^2*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/3), x, 4, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(1/3), x, 5, (3*A*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)) - (3*B*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b*d*Sqrt[Sin[c + d*x]^2]) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(1/3), x, 5, (3*A*b*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(1/3), x, 5, (3*A*b^2*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)) + (3*b*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*(4*A + 7*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -{(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 5, (3*C*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x])/(11*b^4*d) - (3*(11*A + 8*C)*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(88*b^4*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(11/3)*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2]*Sin[c + d*x])/(11*b^5*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 5, (3*C*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b^3*d) - (3*(8*A + 5*C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(40*b^3*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^4*d*Sqrt[Sin[c + d*x]^2])} -{(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 5, (3*C*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b^2*d) - (3*(5*A + 2*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2]) - (3*B*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(4/3), x, 4, (3*A*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)) - (3*B*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*(2*A - C)*(b*Cos[c + d*x])^(5/3)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3), x, 5, (3*A*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*(A + 4*C)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3), x, 5, (3*A*b*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*(4*A + 7*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) with m symbolic*) - - -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, If[$VersionNumber>=8, (3*b*C*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(10 + 3*m)) - (3*b*(C*(7 + 3*m) + A*(10 + 3*m))*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*(10 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(10 + 3*m), (1/6)*(16 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*b*C*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(10 + 3*m)) - (3*b*(C*(7 + 3*m) + A*(10 + 3*m))*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(70 + 51*m + 9*m^2)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Cos[c + d*x]^(3 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(10 + 3*m), (1/6)*(16 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(8 + 3*m)) - (3*(C*(5 + 3*m) + A*(8 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(8 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(8 + 3*m), (1/6)*(14 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(d*(8 + 3*m)) - (3*(C*(5 + 3*m) + A*(8 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(40 + 39*m + 9*m^2)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/2, (1/6)*(8 + 3*m), (1/6)*(14 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) - (3*(C*(4 + 3*m) + A*(7 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) - (3*(C*(4 + 3*m) + A*(7 + 3*m))*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(28 + 33*m + 9*m^2)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (1/6)*(7 + 3*m), (1/6)*(13 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3), x, 5, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(10 + 21*m + 9*m^2)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(5 + 3*m), (1/6)*(11 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3), x, 5, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)) - (3*(C + 3*C*m + A*(4 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(1 + 3*m), (1/6)*(7 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 3*m)*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)) - (3*(C + 3*C*m + A*(4 + 3*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(1 + 3*m), (1/6)*(7 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 15*m + 9*m^2)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (1/6)*(4 + 3*m), (1/6)*(10 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])]} -{(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3), x, 5, If[$VersionNumber>=8, (3*C*Cos[c + d*x]^m*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(1 - 3*m) - A*(2 + 3*m))*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (1/6)*(-1 + 3*m), (1/6)*(5 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 - 3*m)*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Cos[c + d*x]^m*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)) - (3*(C*(1 - 3*m) - A*(2 + 3*m))*Cos[c + d*x]^m*Hypergeometric2F1[1/2, (1/6)*(-1 + 3*m), (1/6)*(5 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 - 3*m - 9*m^2)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1/6)*(2 + 3*m), (1/6)*(8 + 3*m), Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) with n symbolic*) - - -{(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (C*(a*Cos[c + d*x])^(1 + m)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(a*d*(2 + m + n)) - ((C*(1 + m + n) + A*(2 + m + n))*(a*Cos[c + d*x])^(1 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Cos[c + d*x]^2]*Sin[c + d*x])/(a*d*(1 + m + n)*(2 + m + n)*Sqrt[Sin[c + d*x]^2]) - (B*(a*Cos[c + d*x])^(2 + m)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/2)*(2 + m + n), (1/2)*(4 + m + n), Cos[c + d*x]^2]*Sin[c + d*x])/(a^2*d*(2 + m + n)*Sqrt[Sin[c + d*x]^2])} - - -{Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (C*(b*Cos[c + d*x])^(3 + n)*Sin[c + d*x])/(b^3*d*(4 + n)) - ((C*(3 + n) + A*(4 + n))*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*(4 + n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^(4 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^4*d*(4 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^1*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (C*(b*Cos[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(3 + n)) - ((C*(2 + n) + A*(3 + n))*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^3*d*(3 + n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^0*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (C*(b*Cos[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(2 + n)) - ((C*(1 + n) + A*(2 + n))*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*(2 + n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 5, (C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)) - ((A + A*n + C*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*(1 + n)*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (b*C*(b*Cos[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*n) - (b*(C*(1 - n) - A*n)*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*n*Sqrt[Sin[c + d*x]^2]) - (B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, If[$VersionNumber>=8, -((b^2*C*(b*Cos[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(1 - n))) + (b^2*(A*(1 - n) + C*(2 - n))*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*(2 - n)*Sqrt[Sin[c + d*x]^2]) + (b*B*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]), -((b^2*C*(b*Cos[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(1 - n))) + (b^2*(A*(1 - n) + C*(2 - n))*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - 3*n + n^2)*Sqrt[Sin[c + d*x]^2]) + (b*B*(b*Cos[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1/2)*(-1 + n), (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])]} -{(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, If[$VersionNumber>=8, -((b^3*C*(b*Cos[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(2 - n))) + (b^3*(A*(2 - n) + C*(3 - n))*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (1/2)*(-3 + n), (1/2)*(-1 + n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2]) + (b^2*B*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]), -((b^3*C*(b*Cos[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(2 - n))) + (b^3*(A*(2 - n) + C*(3 - n))*(b*Cos[c + d*x])^(-3 + n)*Hypergeometric2F1[1/2, (1/2)*(-3 + n), (1/2)*(-1 + n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(6 - 5*n + n^2)*Sqrt[Sin[c + d*x]^2]) + (b^2*B*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(-2 + n), n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])]} - - -{Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, If[$VersionNumber>=8, (2*C*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) - (2*(C*(5 + 2*n) + A*(7 + 2*n))*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) - (2*(C*(5 + 2*n) + A*(7 + 2*n))*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(35 + 24*n + 4*n^2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(7 + 2*n), (1/4)*(11 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, If[$VersionNumber>=8, (2*C*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) - (2*(C*(3 + 2*n) + A*(5 + 2*n))*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) - (2*(C*(3 + 2*n) + A*(5 + 2*n))*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(15 + 16*n + 4*n^2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, If[$VersionNumber>=8, (2*C*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 8*n + 4*n^2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Cos[c + d*x]]) + (2*(A - C*(1 - 2*n) + 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 4*n^2)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]) - (2*B*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, If[$VersionNumber>=8, -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Cos[c + d*x]^(3/2))) + (2*(A + C*(3 - 2*n) - 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Cos[c + d*x]^(3/2))) + (2*(A + C*(3 - 2*n) - 2*A*n)*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 8*n + 4*n^2)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])]} -{((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, If[$VersionNumber>=8, -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(5/2))) + (2*(A*(3 - 2*n) + C*(5 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*(5 - 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Cos[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(5/2))) + (2*(A*(3 - 2*n) + C*(5 - 2*n))*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-5 + 2*n), (1/4)*(-1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(15 - 16*n + 4*n^2)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) + (2*B*(b*Cos[c + d*x])^n*Hypergeometric2F1[1/2, (1/4)*(-3 + 2*n), (1/4)*(1 + 2*n), Cos[c + d*x]^2]*Sin[c + d*x])/(d*(3 - 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -{(a + a*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 4, If[$VersionNumber>=8, -(((C - B*(2 + m))*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(1 + m)*(2 + m))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(1 + m)*(2 + m)), -(((C - B*(2 + m))*(a + a*Cos[e + f*x])^m*Sin[e + f*x])/(f*(1 + m)*(2 + m))) + (C*(a + a*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(a*f*(2 + m)) + (2^(1/2 + m)*(B*m*(2 + m) + C*(1 + m + m^2) + A*(2 + 3*m + m^2))*(1 + Cos[e + f*x])^(-(1/2) - m)*(a + a*Cos[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Cos[e + f*x])]*Sin[e + f*x])/(f*(2 + 3*m + m^2))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(2/3), x, 4, (3*(8*B - 3*C)*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(40*d) + (3*C*(a + a*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*a*d) + ((40*A + 16*B + 19*C)*(a + a*Cos[c + d*x])^(2/3)*Hypergeometric2F1[-(1/6), 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(10*2^(5/6)*d*(1 + Cos[c + d*x])^(7/6))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/3), x, 4, (3*(7*B - 3*C)*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(28*d) + (3*C*(a + a*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*a*d) + ((28*A + 7*B + 13*C)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(14*2^(1/6)*d*(1 + Cos[c + d*x])^(5/6))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3), x, 4, (3*(5*B - 3*C)*Sin[c + d*x])/(10*d*(a + a*Cos[c + d*x])^(1/3)) + (3*C*(a + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*a*d) + ((10*A - 5*B + 7*C)*Hypergeometric2F1[1/2, 5/6, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(5*2^(5/6)*d*(1 + Cos[c + d*x])^(1/6)*(a + a*Cos[c + d*x])^(1/3))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x, 4, (3*(A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(2/3)) + (3*C*(a + a*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*a*d) - ((4*A - 8*B + 7*C)*(a + a*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/6, 1/2, 3/2, (1/2)*(1 - Cos[c + d*x])]*Sin[c + d*x])/(2*2^(1/6)*a*d*(1 + Cos[c + d*x])^(5/6))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^(n/3) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(5/3)*Sin[c + d*x])/(8*b*d) + ((a + b)*(8*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + ((8*A*b^2 - 8*a*b*B + 3*a^2*C + 5*b^2*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(4*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(4/3)*Sin[c + d*x])/(7*b*d) + (Sqrt[2]*(a + b)*(7*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(7*A*b^2 - 7*a*b*B + 3*a^2*C + 4*b^2*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(7*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b*d) + (Sqrt[2]*(5*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(5*A*b^2 - 5*a*b*B + 3*a^2*C + 2*b^2*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(1/3)*Sin[c + d*x])/(5*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(1/3))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3), x, 8, (3*C*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(4*b*d) + ((4*b*B - 3*a*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*(a + b*Cos[c + d*x])^(1/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*((a + b*Cos[c + d*x])/(a + b))^(1/3)) + ((4*A*b^2 - 4*a*b*B + 3*a^2*C + b^2*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Cos[c + d*x]), (b*(1 - Cos[c + d*x]))/(a + b)]*((a + b*Cos[c + d*x])/(a + b))^(2/3)*Sin[c + d*x])/(2*Sqrt[2]*b^2*d*Sqrt[1 + Cos[c + d*x]]*(a + b*Cos[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2) with m symbolic*) - - -{(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 7, (4*Sqrt[2]*C*AppellF1[1/2, -(3/2), -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(f*Sqrt[1 + Cos[e + f*x]])) + (2*Sqrt[2]*(A - C)*AppellF1[1/2, -(1/2), -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(f*Sqrt[1 + Cos[e + f*x]]))} - - -{(a + b*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2), x, 8, (C*(a + b*Cos[e + f*x])^(1 + m)*Sin[e + f*x])/(b*f*(2 + m)) - (Sqrt[2]*(a + b)*(a*C - b*B*(2 + m))*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]])) + (Sqrt[2]*(a^2*C + b^2*C*(1 + m) + A*b^2*(2 + m) - a*b*B*(2 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Cos[e + f*x]), (b*(1 - Cos[e + f*x]))/(a + b)]*(a + b*Cos[e + f*x])^m*Sin[e + f*x])/(((a + b*Cos[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Cos[e + f*x]]))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.2 (a+b cos)^m (c+d cos)^n (A+B cos+C cos^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.2 (a+b cos)^m (c+d cos)^n (A+B cos+C cos^2).m deleted file mode 100644 index ec8f7b7..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.4.2 (a+b cos)^m (c+d cos)^n (A+B cos+C cos^2).m +++ /dev/null @@ -1,2446 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+a Cos[e+f x])^n (A+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 7, (1/8)*a*(4*A + 3*C)*x + (a*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 3, (1/8)*a*(4*A + 3*C)*x + (a*(3*A + 2*C)*Sin[c + d*x])/(3*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 2, (a*(2*A + C)*x)/2 + (a*(3*A + C)*Sin[c + d*x])/(3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 4, (1/2)*a*(2*A + C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, a*C*x + (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, (a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 9, (1/16)*a^2*(14*A + 11*C)*x + (2*a^2*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a^2*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(10*A + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (2*a^2*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 5, (a^2*(4*A + 3*C)*x)/4 + (a^2*(4*A + 3*C)*Sin[c + d*x])/(3*d) + (a^2*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d) + ((10*A + 3*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(30*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(10*a*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 3, (a^2*(12*A + 7*C)*x)/8 + (a^2*(12*A + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 6, a^2*(2*A + C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Sin[c + d*x])/d + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 6, (a^2*(2*A + 3*C)*x)/2 + (2*a^2*A*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 3*C)*Sin[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 6, 2*a^2*C*x + (a^2*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) + (A*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, a^2*C*x + (a^2*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Tan[c + d*x])/d + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 8, (a^2*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a^2*(11*A + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 9, (a^2*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^2*(18*A + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*(9*A + 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(10*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 10, (1/16)*a^3*(26*A + 21*C)*x + (a^3*(133*A + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(154*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(14*a*d) + ((A + C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^3*(133*A + 108*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 11, (1/16)*a^3*(30*A + 23*C)*x + (a^3*(30*A + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d) - (a^3*(30*A + 23*C)*Sin[c + d*x]^3)/(120*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 9, (1/8)*a^3*(20*A + 13*C)*x + (a^3*(20*A + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(20*A + 13*C)*Sin[c + d*x]^3)/(60*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 7, (a^3*(28*A + 15*C)*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 3*C)*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(4*a*d) + ((4*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(8*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 7, (a^3*(6*A + 5*C)*x)/2 + (3*a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d) - ((6*A - 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 7, (a^3*(2*A + 7*C)*x)/2 + (a^3*(7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - ((4*A - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (3*A*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 7, 3*a^3*C*x + (a^3*(5*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*A*Sin[c + d*x])/(2*d) + ((5*A + 3*C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/(3*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, a^3*C*x + (a^3*(15*A + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*C)*Tan[c + d*x])/(8*d) + ((5*A + 4*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 9, (a^3*(13*A + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 55*C)*Tan[c + d*x])/(15*d) + (a^3*(109*A + 140*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((11*A + 10*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (3*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 10, (a^3*(23*A + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 30*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 11, (1/128)*a^4*(392*A + 323*C)*x + (4*a^4*(63*A + 52*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((56*A + 61*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(8*A + 7*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (4*a^4*(63*A + 52*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 14, (1/4)*a^4*(14*A + 11*C)*x + (16*a^4*(14*A + 11*C)*Sin[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(21*a*d) - (8*a^4*(14*A + 11*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 12, (7/16)*a^4*(10*A + 7*C)*x + (4*a^4*(10*A + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 7*C)*Sin[c + d*x]^3)/(15*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 8, (a^4*(12*A + 7*C)*x)/2 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(10*A + 7*C)*Sin[c + d*x])/(2*d) + (a*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 8, (a^4*(52*A + 35*C)*x)/8 + (4*a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 7*C)*Sin[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - ((12*A - 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - ((12*A - 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 8, 2*a^4*(2*A + 3*C)*x + (a^4*(13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - 2*C)*Sin[c + d*x])/(2*d) - ((15*A - 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((9*A - 4*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 8, (a^4*(2*A + 13*C)*x)/2 + (2*a^4*(3*A + 2*C)*ArcTanh[Sin[c + d*x]])/d - (5*a^4*(2*A - C)*Sin[c + d*x])/(2*d) - ((22*A + 3*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + ((8*A + 3*C)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(3*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 8, 4*a^4*C*x + (a^4*(35*A + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 4*C)*Sin[c + d*x])/(8*d) + ((35*A + 36*C)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(12*d) + ((7*A + 4*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 8, a^4*C*x + (a^4*(7*A + 12*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*(7*A + 10*C)*Tan[c + d*x])/(2*d) + ((7*A + 8*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + ((7*A + 5*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 10, (7*a^4*(7*A + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(18*A + 25*C)*Tan[c + d*x])/(15*d) + (a^4*(417*A + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((43*A + 50*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((37*A + 30*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8, x, 11, (a^4*(11*A + 14*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^4*(454*A + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(11*A + 14*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^4*(247*A + 308*C)*Sec[c + d*x]^2*Tan[c + d*x])/(210*d) + ((109*A + 126*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(210*d) + ((8*A + 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (2*a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(21*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 7, (3*(4*A + 5*C)*x)/(8*a) - ((3*A + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((4*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A + 4*C)*Sin[c + d*x]^3)/(3*a*d)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 6, -(((2*A + 3*C)*x)/(2*a)) + ((3*A + 4*C)*Sin[c + d*x])/(a*d) - ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - ((3*A + 4*C)*Sin[c + d*x]^3)/(3*a*d)} -{(Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 2, ((2*A + 3*C)*x)/(2*a) - ((A + 2*C)*Sin[c + d*x])/(a*d) + ((2*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, -((C*x)/a) + (C*Sin[c + d*x])/(a*d) + ((A + C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, -((A*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A + C)*Tan[c + d*x])/(a*d) - ((A + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]), x, 6, ((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A + C)*Tan[c + d*x])/(a*d) + ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]), x, 6, -(((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((4*A + 3*C)*Tan[c + d*x])/(a*d) - ((3*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A + 3*C)*Tan[c + d*x]^3)/(3*a*d)} - - -{(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 8, ((28*A + 55*C)*x)/(8*a^2) - (8*(A + 2*C)*Sin[c + d*x])/(a^2*d) + ((28*A + 55*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + ((28*A + 55*C)*Cos[c + d*x]^3*Sin[c + d*x])/(12*a^2*d) - (2*(A + 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (8*(A + 2*C)*Sin[c + d*x]^3)/(3*a^2*d)} -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 7, -(((2*A + 5*C)*x)/a^2) + ((5*A + 12*C)*Sin[c + d*x])/(a^2*d) - ((2*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (2*(2*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A + 12*C)*Sin[c + d*x]^3)/(3*a^2*d)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 3, ((2*A + 7*C)*x)/(2*a^2) - (4*(A + 4*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*(A + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 6, (-2*C*x)/a^2 + ((A + 4*C)*Sin[c + d*x])/(3*a^2*d) + (2*C*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - (2*(2*A - C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, (-2*A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((10*A + C)*Tan[c + d*x])/(3*a^2*d) - (2*A*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2, x, 7, ((7*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(4*A + C)*Tan[c + d*x])/(3*a^2*d) + ((7*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (2*(4*A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2, x, 7, -(((5*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((12*A + 5*C)*Tan[c + d*x])/(a^2*d) - ((5*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (2*(5*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((12*A + 5*C)*Tan[c + d*x]^3)/(3*a^2*d)} - - -{(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 8, -(((6*A + 23*C)*x)/(2*a^3)) + (4*(9*A + 34*C)*Sin[c + d*x])/(5*a^3*d) - ((6*A + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) - (4*(9*A + 34*C)*Sin[c + d*x]^3)/(15*a^3*d)} -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 4, ((2*A + 13*C)*x)/(2*a^3) - (2*(11*A + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 7, (-3*C*x)/a^3 + ((2*A + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (3*C*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 5, (C*x)/a^3 - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((6*A - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, (-3*A*ArcTanh[Sin[c + d*x]])/(a^3*d) + (2*(36*A + C)*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (3*A*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3, x, 8, ((13*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3, x, 8, -(((23*A + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (4*(34*A + 9*C)*Tan[c + d*x])/(5*a^3*d) - ((23*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(34*A + 9*C)*Tan[c + d*x]^3)/(15*a^3*d)} - - -{(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4, x, 5, ((2*A + 21*C)*x)/(2*a^4) - (32*(5*A + 54*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (16*(5*A + 54*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4, x, 8, (-4*C*x)/a^4 + (2*(3*A + 122*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*C*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(A - 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4, x, 6, (C*x)/a^4 - ((8*A - 55*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((16*A - 215*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + (2*(2*A - 5*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4, x, 5, ((23*A - 54*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + (4*(2*A + 9*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(3*A - 4*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, ((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 13*C)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((6*A + 13*C)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^4, x, 6, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 8*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (8*(20*A - C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(5*A - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 8, (-4*A*ArcTanh[Sin[c + d*x]])/(a^4*d) + (2*(332*A + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 3*C)*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*A*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(6*A - C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4, x, 9, ((21*A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(54*A + 5*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (16*(54*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^4, x, 9, -((2*(11*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (4*(454*A + 83*C)*Tan[c + d*x])/(35*a^4*d) - (2*(11*A + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(a^4*d) - ((178*A + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(11*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^4*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*(8*A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (4*(454*A + 83*C)*Tan[c + d*x]^3)/(105*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 6, (4*a*(99*A + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)} -{Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (2*a*(21*A + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^1*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (2*a*(35*A + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)} -{Cos[c + d*x]^0*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 3, (2*a*(15*A + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (4*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 4, (2*Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, (Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, (Sqrt[a]*(3*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (Sqrt[a]*(5*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 6, (Sqrt[a]*(35*A + 48*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(35*A + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 6, (2*a^2*(143*A + 112*C)*Sin[c + d*x])/(165*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(33*A + 28*C)*Cos[c + d*x]^3*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1155*d) + (2*a*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 6, (8*a^2*(63*A + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 4, (8*a^2*(35*A + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 5, (2*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(5*A + 4*C)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (3*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(3*A - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, (a^(3/2)*(7*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(5*A - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, (a^(3/2)*(11*A + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(19*A + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 6, (a^(3/2)*(75*A + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(13*A + 16*C)*Sec[c + d*x]*Tan[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 7, (a^(3/2)*(133*A + 176*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(133*A + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 7, (2*a^3*(10439*A + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (10*a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)} -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 7, (64*a^3*(33*A + 25*C)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(231*d) + (2*(99*A + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (10*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 5, (64*a^3*(21*A + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) - (4*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 6, (2*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(49*A + 32*C)*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(7*A + 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 6, (5*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*(15*A + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 6, (a^(5/2)*(19*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(27*A - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, (5*a^(5/2)*(5*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(49*A - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 6, (a^(5/2)*(163*A + 304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(299*A + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 7, (a^(5/2)*(283*A + 400*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 8, (a^(5/2)*(1015*A + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1015*A + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(109*A + 136*C)*Sec[c + d*x]^2*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(23*A + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(96*d) + (a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 8, -((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 7, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 6, -((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) - (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 4, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1)/Sqrt[a + a*Cos[c + d*x]], x, 6, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]], x, 6, -((A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]], x, 7, ((7*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (A*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]], x, 8, -(((9*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((7*A + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]], x, 9, ((107*A + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((21*A + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - (A*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 8, ((11*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 7, -(((7*A + 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((15*A + 31*C)*Sin[c + d*x])/(5*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((5*A + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(10*a^2*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, ((3*A + 11*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ((A - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 7, -((3*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((9*A + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2), x, 8, ((19*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A + 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((7*A + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2), x, 9, -(((47*A + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*a^(3/2)*d)) + ((17*A + 9*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (3*(7*A + 4*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 8, -(((75*A + 283*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 7, ((19*A + 163*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + (5*(3*A + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, (5*(A - 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((3*A - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 9*C)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 4, ((3*A + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 8, -((5*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + ((115*A + 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2), x, 9, ((39*A + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A + 43*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((63*A + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 8, (2*a*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*a*(11*A + 9*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*a*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 7, (2*a*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 6, (2*a*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (2*a*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, -((2*a*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, -((2*a*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, -((2*a*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*a*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 9, (4*a^2*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 8, (16*a^2*(3*A + 2*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*C*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (4*a^2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*C*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (16*a^2*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(15*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (-4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (-16*a^2*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(17*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-4*a^2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 9, (-16*a^2*(2*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(19*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (16*a^2*(2*A + 3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 10, (4*a^3*(221*A + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 95*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(143*A + 118*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (12*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(143*A + 145*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 9, (4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (8*a^3*(44*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*C*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(33*A + 35*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d)} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, (4*a^3*(27*A + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(21*A + 16*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*C*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 73*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, (4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^3*(35*A - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(35*A - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, (-4*a^3*(5*A - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (8*a^3*(10*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(35*A - 3*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 8, (-4*a^3*(9*A - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(21*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)) + (2*(11*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(53*A + 70*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (12*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 9, (-4*a^3*(17*A + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(73*A + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 10, (-4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^3*(35*A + 44*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(5*A + 7*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Cos[c + d*x]^(9/2)) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 7, (-3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A + 9*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 6, (3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 5, -(((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])), x, 4, ((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])), x, 5, -(((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])), x, 6, ((3*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])), x, 7, (-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 7, (4*(5*A + 14*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*(5*A + 14*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 6, -(((A + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (2*(A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 5, (4*C*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2), x, 5, ((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2), x, 6, (-4*A*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (4*A*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2), x, 7, ((7*A + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(5*A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(5*A + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 8, (7*(7*A + 33*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A + 63*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(A + 6*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 7, -((9*A + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 11*C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, -((A - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, ((A - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(2*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3), x, 6, ((9*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(3*A - 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3), x, 7, -((49*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3), x, 8, ((119*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A + C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A + C)*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 6, (Sqrt[a]*(48*A + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 5, (Sqrt[a]*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(8*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 4, (Sqrt[a]*(8*A + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 4, (Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 4, (2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 3, (2*a*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (2*a*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (2*a*A*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 7, (a^(3/2)*(176*A + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(176*A + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (3*a*C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 6, (a^(3/2)*(112*A + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(112*A + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (a^(3/2)*(24*A + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(24*A + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (a^(3/2)*(8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(8*A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (3*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, (2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 4, (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(104*A + 175*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (6*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (2*a^2*(52*A + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 6, (2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 8, (a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1304*A + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(136*A + 109*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(24*A + 23*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d) + (a*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 7, (a^(5/2)*(400*A + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(400*A + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 6, (a^(5/2)*(304*A + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(432*A + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (5*a^(5/2)*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(24*A - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (a^(5/2)*(8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(56*A - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, (5*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(64*A + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(32*A + 49*C)*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 5, (2*a^3*(8*A + 11*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(584*A + 903*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 6, (2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 7, (2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 8, -(((8*A + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 7, ((8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]), x, 6, -((C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]), x, 6, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]), x, 5, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]), x, 6, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]), x, 7, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 8, ((8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 7, -((3*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)), x, 6, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)), x, 5, -((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)), x, 6, ((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)), x, 7, -((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((49*A + 25*C)*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 8, -((5*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)), x, 5, ((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)), x, 6, (-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)), x, 7, ((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+a Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*x)/8 + (C*Sin[c + d*x])/d + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*C*Sin[c + d*x]^3)/(3*d) + (C*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*C*x)/8 + (B*Sin[c + d*x])/d + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (B*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (B*x)/2 + (C*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (C*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, B*x + (C*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, C*x + (B*ArcTanh[Sin[c + d*x]])/d} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, (C*ArcTanh[Sin[c + d*x]])/d + (B*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (B*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Tan[c + d*x])/d + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (C*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (B*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + (C*Tan[c + d*x])/d + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (C*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 9, (3/8)*a*(B + C)*x + (a*(5*B + 4*C)*Sin[c + d*x])/(5*d) + (3*a*(B + C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*B + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 8, (1/8)*a*(4*B + 3*C)*x + (a*(B + C)*Sin[c + d*x])/d + (a*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(B + C)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 2, (1/2)*a*(B + C)*x + (a*(3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 2, (1/2)*a*(2*B + C)*x + (a*(B + C)*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, a*(B + C)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 5, a*C*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, (a*(B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 8, (a*(B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*B + 3*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 8, (a*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(3*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (1/8)*a^2*(7*B + 6*C)*x + (a^2*(10*B + 9*C)*Sin[c + d*x])/(5*d) + (a^2*(7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*B + 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(10*B + 9*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, (a^2*(8*B + 7*C)*x)/8 + (a^2*(8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 3, (a^2*(3*B + 2*C)*x)/2 + (2*a^2*(3*B + 2*C)*Sin[c + d*x])/(3*d) + (a^2*(3*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 6, (a^2*(4*B + 3*C)*x)/2 + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*B + 3*C)*Sin[c + d*x])/(2*d) + (C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 6, a^2*(B + 2*C)*x + (a^2*(2*B + C)*ArcTanh[Sin[c + d*x]])/d - (a^2*(B - C)*Sin[c + d*x])/d + (B*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, a^2*C*x + (a^2*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(3*B + 2*C)*Tan[c + d*x])/(2*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 8, (a^2*(2*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(5*B + 6*C)*Tan[c + d*x])/(3*d) + (a^2*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 9, (a^2*(7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*B + 5*C)*Tan[c + d*x])/(3*d) + (a^2*(7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 9, (a^2*(6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(9*B + 10*C)*Tan[c + d*x])/(5*d) + (a^2*(6*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(6*B + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a^2*(9*B + 10*C)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^1*(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, (1/16)*a^3*(26*B + 23*C)*x + (a^3*(19*B + 17*C)*Sin[c + d*x])/(5*d) + (a^3*(26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(22*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*B + 4*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(19*B + 17*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^0*(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (1/8)*a^3*(15*B + 13*C)*x + (a^3*(15*B + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(15*B + 13*C)*Sin[c + d*x]^3)/(60*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 9, (5/8)*a^3*(4*B + 3*C)*x + (a^3*(4*B + 3*C)*Sin[c + d*x])/d + (3*a^3*(4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(4*B + 3*C)*Sin[c + d*x]^3)/(12*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 7, (a^3*(7*B + 5*C)*x)/2 + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) + (a*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 7, (a^3*(6*B + 7*C)*x)/2 + (a^3*(3*B + C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*C*Sin[c + d*x])/(2*d) - ((2*B - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (a*B*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 7, a^3*(B + 3*C)*x + (a^3*(7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*B*Sin[c + d*x])/(2*d) + ((2*B + C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/d + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, a^3*C*x + (a^3*(5*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) + ((5*B + 3*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 9, (5*a^3*(3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(9*B + 11*C)*Tan[c + d*x])/(3*d) + (a^3*(27*B + 28*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*B + 2*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 10, (a^3*(13*B + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(13*B + 15*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*(43*B + 45*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((7*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 7, (3*(B - C)*x)/(2*a) - ((3*B - 4*C)*Sin[c + d*x])/(a*d) + (3*(B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*B - 4*C)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, -((2*B - 3*C)*x)/(2*a) + (2*(B - C)*Sin[c + d*x])/(a*d) - ((2*B - 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, ((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) - ((B - C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 4, (B*ArcTanh[Sin[c + d*x]])/(a*d) - ((B - C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]), x, 6, -(((B - C)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*B - C)*Tan[c + d*x])/(a*d) - ((B - C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]), x, 7, ((3*B - 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*(B - C)*Tan[c + d*x])/(a*d) + ((3*B - 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + a*Cos[c + d*x]), x, 7, -((3*(B - C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((4*B - 3*C)*Tan[c + d*x])/(a*d) - (3*(B - C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*B - 3*C)*Tan[c + d*x]^3)/(3*a*d)} - - -{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 8, ((7*B - 10*C)*x)/(2*a^2) - (4*(2*B - 3*C)*Sin[c + d*x])/(a^2*d) + ((7*B - 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((7*B - 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + (4*(2*B - 3*C)*Sin[c + d*x]^3)/(3*a^2*d)} -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, -((4*B - 7*C)*x)/(2*a^2) + (2*(5*B - 8*C)*Sin[c + d*x])/(3*a^2*d) - ((4*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + ((5*B - 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, ((B - 2*C)*x)/a^2 - ((B - 4*C)*Sin[c + d*x])/(3*a^2*d) - ((B - 2*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 + ((2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1)/(a + a*Cos[c + d*x])^2, x, 3, ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((B + 2*C)*Sin[c + d*x])/(3*d*(a^2 + a^2*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 5, (B*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((4*B - C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2, x, 7, -(((2*B - C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (2*(5*B - 2*C)*Tan[c + d*x])/(3*a^2*d) - ((2*B - C)*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2, x, 8, ((7*B - 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*B - 5*C)*Tan[c + d*x])/(3*a^2*d) + ((7*B - 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*B - 5*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} - - -{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, -((6*B - 13*C)*x)/(2*a^3) + (8*(9*B - 19*C)*Sin[c + d*x])/(15*a^3*d) - ((6*B - 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + ((B - C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*B - 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + (4*(9*B - 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, ((B - 3*C)*x)/a^3 - ((7*B - 27*C)*Sin[c + d*x])/(15*a^3*d) + ((B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, (C*x)/a^3 + ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*B - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((4*B - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, -((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((3*B + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^1)/(a + a*Cos[c + d*x])^3, x, 4, ((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*B + 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*B + 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 6, (B*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((B - C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*B - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (2*(11*B - C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3, x, 8, -(((3*B - C)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*B - 11*C)*Tan[c + d*x])/(15*a^3*d) - ((B - C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*B - 4*C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*B - C)*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3, x, 9, ((13*B - 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (8*(19*B - 9*C)*Tan[c + d*x])/(15*a^3*d) + ((13*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((B - C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*B - 6*C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - (4*(19*B - 9*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*(5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (8*a^2*(21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (64*a^3*(15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 4, -((Sqrt[2]*(B - C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ((3*B - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 4, ((5*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 7, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*C*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 6, (6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 5, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, (2*C*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/d + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 6, -((2*C*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 7, -((6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection:: *) -(*Integrands of the form Cos[e+f x]^(m/2 (B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (1/16)*(6*A + 5*C)*x + (B*Sin[c + d*x])/d + ((6*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*x)/8 + ((5*A + 4*C)*Sin[c + d*x])/(5*d) + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (1/8)*(4*A + 3*C)*x + (B*Sin[c + d*x])/d + ((4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (B*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 2, (B*x)/2 + ((3*A + 2*C)*Sin[c + d*x])/(3*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 4, A*x + (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 3, C*x + (B*ArcTanh[Sin[c + d*x]])/d + (A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 5, ((A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (B*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A + 3*C)*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, ((3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (B*Tan[c + d*x])/d + ((3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (B*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*A + 5*C)*Tan[c + d*x])/(5*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*A + 5*C)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, (1/8)*a*(4*A + 3*(B + C))*x + (a*(5*A + 5*B + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*(B + C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(5*A + 5*B + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 3, (1/8)*a*(4*A + 4*B + 3*C)*x + (a*(3*A + 2*(B + C))*Sin[c + d*x])/(3*d) + (a*(4*A + 4*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 2, (1/2)*a*(2*A + B + C)*x + (a*(3*A + 3*B + C)*Sin[c + d*x])/(3*d) + (a*(3*B - C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, (1/2)*a*(2*A + 2*B + C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*(B + C)*Sin[c + d*x])/d + (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*(B + C)*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 4, a*C*x + (a*(A + 2*(B + C))*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 6, (a*(A + B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*(B + C))*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x]), x, 7, (a*(3*A + 4*(B + C))*ArcTanh[Sin[c + d*x]])/(8*d) - (a*(A + B - 3*(A + B + C))*Tan[c + d*x])/(3*d) + (a*(3*A + 4*(B + C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 9, (1/16)*a^2*(14*A + 12*B + 11*C)*x + (a^2*(10*A + 9*B + 8*C)*Sin[c + d*x])/(5*d) + (a^2*(14*A + 12*B + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(10*A + 12*B + 9*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((3*B + C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^2*(10*A + 9*B + 8*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 5, (1/8)*a^2*(8*A + 7*B + 6*C)*x + (a^2*(8*A + 7*B + 6*C)*Sin[c + d*x])/(6*d) + (a^2*(8*A + 7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((20*A - 5*B + 6*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d) + ((5*B + 2*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 3, (1/8)*a^2*(12*A + 8*B + 7*C)*x + (a^2*(12*A + 8*B + 7*C)*Sin[c + d*x])/(6*d) + (a^2*(12*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*B - C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (1/2)*a^2*(4*A + 3*B + 2*C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B + 2*C)*Sin[c + d*x])/(2*d) + (C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + ((3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(6*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, (1/2)*a^2*(2*A + 4*B + 3*C)*x + (a^2*(2*A + B)*ArcTanh[Sin[c + d*x]])/d - (a^2*(2*A - 2*B - 3*C)*Sin[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(2*d) + (A*(a + a*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, a^2*(B + 2*C)*x + (a^2*(3*A + 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (a^2*(3*A + 2*B - 2*C)*Sin[c + d*x])/(2*d) + ((A + B)*(a^2 + a^2*Cos[c + d*x])*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 6, a^2*C*x + (a^2*(2*A + 3*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B + 2*C)*Tan[c + d*x])/(2*d) + ((2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 8, (a^2*(7*A + 8*B + 12*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(4*A + 5*B + 6*C)*Tan[c + d*x])/(3*d) + (a^2*(11*A + 16*B + 12*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((A + 2*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^2, x, 9, (a^2*(6*A + 7*B + 8*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(18*A + 20*B + 25*C)*Tan[c + d*x])/(15*d) + (a^2*(6*A + 7*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(18*A + 25*B + 20*C)*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((2*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 10, (1/16)*a^3*(26*A + 23*B + 21*C)*x + (a^3*(133*A + 119*B + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 23*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(154*A + 147*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((7*B + 3*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(133*A + 119*B + 108*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 11, (1/16)*a^3*(30*A + 26*B + 23*C)*x + (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(120*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((2*B + C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(10*a*d) - (a^3*(30*A + 26*B + 23*C)*Sin[c + d*x]^3)/(120*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 9, (1/8)*a^3*(20*A + 15*B + 13*C)*x + (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x])/(5*d) + (3*a^3*(20*A + 15*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((5*B - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) - (a^3*(20*A + 15*B + 13*C)*Sin[c + d*x]^3)/(60*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/8)*a^3*(28*A + 20*B + 15*C)*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(4*A + 4*B + 3*C)*Sin[c + d*x])/(8*d) + (C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + ((4*B + 3*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*a*d) + ((12*A + 20*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(24*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/2)*a^3*(6*A + 7*B + 5*C)*x + (a^3*(3*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(3*a*d) - ((6*A - 3*B - 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, (1/2)*a^3*(2*A + 6*B + 7*C)*x + (a^3*(7*A + 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A - C)*Sin[c + d*x])/(2*d) - ((4*A + 2*B - C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(2*d) + ((3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, a^3*(B + 3*C)*x + (a^3*(5*A + 7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + ((5*A + 6*B + 3*C)*(a^3 + a^3*Cos[c + d*x])*Tan[c + d*x])/(3*d) + ((A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 7, a^3*C*x + (a^3*(15*A + 20*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(3*A + 4*(B + C))*Tan[c + d*x])/(8*d) + ((15*A + 20*B + 12*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A + 4*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 9, (a^3*(13*A + 15*B + 20*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(38*A + 45*B + 55*C)*Tan[c + d*x])/(15*d) + (a^3*(109*A + 135*B + 140*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((11*A + 15*B + 10*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((3*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^3, x, 10, (a^3*(23*A + 26*B + 30*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(34*A + 38*B + 45*C)*Tan[c + d*x])/(15*d) + (a^3*(23*A + 26*B + 30*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(73*A + 86*B + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(120*d) + ((31*A + 42*B + 30*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*a*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 11, (1/128)*a^4*(392*A + 352*B + 323*C)*x + (a^4*(252*A + 227*B + 208*C)*Sin[c + d*x])/(35*d) + (a^4*(392*A + 352*B + 323*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2408*A + 2208*B + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*(2*B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(14*d) + (C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((56*A + 80*B + 61*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(8*A + 8*B + 7*C)*Cos[c + d*x]^3*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(252*A + 227*B + 208*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 14, (1/16)*a^4*(56*A + 49*B + 44*C)*x + (4*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]*Sin[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(210*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(42*a*d) - (2*a^4*(56*A + 49*B + 44*C)*Sin[c + d*x]^3)/(105*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 12, (7/16)*a^4*(10*A + 8*B + 7*C)*x + (4*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*B - C)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/8)*a^4*(48*A + 35*B + 28*C)*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(40*A + 35*B + 28*C)*Sin[c + d*x])/(8*d) + (a*(5*B + 4*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/8)*a^4*(52*A + 48*B + 35*C)*x + (a^4*(4*A + B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 8*B + 7*C)*Sin[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - ((12*A - 4*B - 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - ((12*A - 32*B - 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/2)*a^4*(8*A + 13*B + 12*C)*x + (a^4*(13*A + 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A - B - 2*C)*Sin[c + d*x])/(2*d) - ((15*A + 6*B - 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - ((18*A + 3*B - 8*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + (a*(2*A + B)*(a + a*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, (1/2)*a^4*(2*A + 8*B + 13*C)*x + (a^4*(12*A + 13*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(2*A + B - C)*Sin[c + d*x])/(2*d) - ((22*A + 18*B + 3*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d) + ((16*A + 15*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + (a*(4*A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, a^4*(B + 4*C)*x + (a^4*(35*A + 48*B + 52*C)*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^4*(7*A + 8*B + 4*C)*Sin[c + d*x])/(8*d) + ((35*A + 44*B + 36*C)*(a^4 + a^4*Cos[c + d*x])*Tan[c + d*x])/(12*d) + ((7*A + 8*B + 4*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 8, a^4*C*x + (a^4*(28*A + 35*B + 48*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(28*A + 35*B + 40*C)*Tan[c + d*x])/(8*d) + ((28*A + 35*B + 32*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((28*A + 35*B + 20*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + (a*(4*A + 5*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 10, (7*a^4*(7*A + 8*B + 10*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(72*A + 83*B + 100*C)*Tan[c + d*x])/(15*d) + (a^4*(417*A + 488*B + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((43*A + 52*B + 50*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((37*A + 48*B + 30*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (a*(2*A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^8*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^4, x, 11, (a^4*(44*A + 49*B + 56*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^4*(454*A + 504*B + 581*C)*Tan[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Sec[c + d*x]^2*Tan[c + d*x])/(840*d) + ((436*A + 511*B + 504*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(840*d) + ((16*A + 21*B + 14*C)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(70*d) + (a*(4*A + 7*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 7, (3*(4*A - 4*B + 5*C)*x)/(8*a) - ((3*A - 4*B + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, -(((2*A - 3*B + 3*C)*x)/(2*a)) + ((3*A - 3*B + 4*C)*Sin[c + d*x])/(a*d) - ((2*A - 3*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) - ((3*A - 3*B + 4*C)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 2, ((2*A - 2*B + 3*C)*x)/(2*a) - ((A - 2*B + 2*C)*Sin[c + d*x])/(a*d) + ((2*A - 2*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, ((B - C)*x)/a + (C*Sin[c + d*x])/(a*d) + ((A - B + C)*Sin[c + d*x])/(a*d*(1 + Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 3, (C*x)/a + (A*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 5, -(((A - B)*ArcTanh[Sin[c + d*x]])/(a*d)) + ((2*A - B + C)*Tan[c + d*x])/(a*d) - ((A - B + C)*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, ((3*A - 2*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((2*A - 2*B + C)*Tan[c + d*x])/(a*d) + ((3*A - 2*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]), x, 6, -(((3*A - 3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((4*A - 3*B + 3*C)*Tan[c + d*x])/(a*d) - ((3*A - 3*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A - 3*B + 3*C)*Tan[c + d*x]^3)/(3*a*d)} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, -(((4*A - 7*B + 10*C)*x)/(2*a^2)) + ((5*A - 8*B + 12*C)*Sin[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((5*A - 8*B + 12*C)*Sin[c + d*x]^3)/(3*a^2*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, ((2*A - 4*B + 7*C)*x)/(2*a^2) - (2*(2*A - 5*B + 8*C)*Sin[c + d*x])/(3*a^2*d) + ((2*A - 4*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((2*A - 5*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, ((B - 2*C)*x)/a^2 + ((A - B + 4*C)*Sin[c + d*x])/(3*a^2*d) - ((B - 2*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 3, (C*x)/a^2 + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) + ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 4, (A*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((4*A - B - 2*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 6, -(((2*A - B)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((10*A - 4*B + C)*Tan[c + d*x])/(3*a^2*d) - ((2*A - B)*Tan[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, ((7*A - 4*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(8*A - 5*B + 2*C)*Tan[c + d*x])/(3*a^2*d) + ((7*A - 4*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((8*A - 5*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2, x, 7, -(((10*A - 7*B + 4*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + ((12*A - 8*B + 5*C)*Tan[c + d*x])/(a^2*d) - ((10*A - 7*B + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((10*A - 7*B + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) + ((12*A - 8*B + 5*C)*Tan[c + d*x]^3)/(3*a^2*d)} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, -(((6*A - 13*B + 23*C)*x)/(2*a^3)) + (4*(9*A - 19*B + 34*C)*Sin[c + d*x])/(5*a^3*d) - ((6*A - 13*B + 23*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A - 8*B + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((6*A - 13*B + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) - (4*(9*A - 19*B + 34*C)*Sin[c + d*x]^3)/(15*a^3*d)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 4, ((2*A - 6*B + 13*C)*x)/(2*a^3) - (2*(11*A - 36*B + 76*C)*Sin[c + d*x])/(15*a^3*d) + ((2*A - 6*B + 13*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((A - 6*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((11*A - 36*B + 76*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, ((B - 3*C)*x)/a^3 + ((2*A - 7*B + 27*C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A + 4*B - 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((B - 3*C)*Sin[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, (C*x)/a^3 - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((3*A + 2*B - 7*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((6*A + 4*B - 29*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 3, ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((2*A + 3*B + 7*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 5, (A*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((7*A - 2*B - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((22*A - 2*B - 3*C)*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 7, -(((3*A - B)*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (2*(36*A - 11*B + C)*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((9*A - 4*B - C)*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((3*A - B)*Tan[c + d*x])/(d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, ((13*A - 6*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(76*A - 36*B + 11*C)*Tan[c + d*x])/(15*a^3*d) + ((13*A - 6*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((11*A - 6*B + C)*Sec[c + d*x]*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((76*A - 36*B + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3, x, 8, -(((23*A - 13*B + 6*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (4*(34*A - 19*B + 9*C)*Tan[c + d*x])/(5*a^3*d) - ((23*A - 13*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((13*A - 8*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((23*A - 13*B + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a^3 + a^3*Cos[c + d*x])) + (4*(34*A - 19*B + 9*C)*Tan[c + d*x]^3)/(15*a^3*d)} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, ((2*A - 8*B + 21*C)*x)/(2*a^4) - (8*(20*A - 83*B + 216*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((10*A - 52*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(20*A - 83*B + 216*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((B - 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 8, ((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((B - 4*C)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((2*A + 5*B - 12*C)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, (C*x)/a^4 - ((8*A + 6*B - 55*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((16*A + 12*B - 215*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((4*A + 3*B - 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 5, ((23*A - 2*B - 54*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) + ((8*A + 13*B + 36*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((6*A + B - 8*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 4, ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + ((6*A + 8*B + 13*C)*Sin[c + d*x])/(105*d*(a^4 + a^4*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 6, (A*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 6*B - 8*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (2*(80*A - 3*B - 4*C)*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((10*A - 3*B - 4*C)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 8, -(((4*A - B)*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (2*(332*A - 80*B + 3*C)*Tan[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((4*A - B)*Tan[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((12*A - 5*B - 2*C)*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 9, ((21*A - 8*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(216*A - 83*B + 20*C)*Tan[c + d*x])/(105*a^4*d) + ((21*A - 8*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((129*A - 52*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - (4*(216*A - 83*B + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4, x, 9, -(((44*A - 21*B + 8*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d)) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x])/(35*a^4*d) - ((44*A - 21*B + 8*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((178*A - 87*B + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((44*A - 21*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((16*A - 9*B + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3) + (4*(454*A - 216*B + 83*C)*Tan[c + d*x]^3)/(105*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 6, (4*a*(99*A + 88*B + 80*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(11*B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[a + a*Cos[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*C*Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (2*a*(21*A + 18*B + 16*C)*Sin[c + d*x])/(45*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(9*B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (2*a*(35*A + 49*B + 27*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 14*B + 18*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (2*(7*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 3, (2*a*(15*A + 5*B + 7*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (2*Sqrt[a]*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*(3*B + C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[a]*(A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(A - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[a]*(3*A + 4*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(A + 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 5, (Sqrt[a]*(5*A + 6*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(5*A + 6*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(1/2), x, 6, (Sqrt[a]*(35*A + 40*B + 48*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(35*A + 40*B + 48*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*Cos[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (8*a^2*(63*A + 57*B + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 4, (8*a^2*(35*A + 21*B + 19*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(35*A + 21*B + 19*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (2*a^(3/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(15*A + 20*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(3*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(3*A - 6*B - 8*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(7*A + 12*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(5*A + 4*B - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 4*B)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 5, (a^(3/2)*(11*A + 14*B + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(19*A + 30*B + 24*C)*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(A + 2*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 6, (a^(3/2)*(75*A + 88*B + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B + 112*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(39*A + 56*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 8*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(3/2), x, 7, (a^(3/2)*(133*A + 150*B + 176*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(133*A + 150*B + 176*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Sec[c + d*x]*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Sec[c + d*x]^2*Tan[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(3*A + 10*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 7, (2*a^3*(10439*A + 9230*B + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2522*B + 2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(13*B + 5*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 7, (64*a^3*(165*A + 143*B + 125*C)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(165*A + 143*B + 125*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d) + (2*(11*B + 5*C)*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 5, (64*a^3*(21*A + 15*B + 13*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*C*(a + a*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (2*a^(5/2)*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(245*A + 224*B + 160*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(35*A + 56*B + 40*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(5*A + 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (a^3*(15*A + 70*B + 64*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(15*A - 10*B - 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(19*A + 20*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(27*A - 12*B - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(21*A + 12*B - 8*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (a*(5*A + 4*B)*(a + a*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(25*A + 38*B + 40*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(49*A + 54*B - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(31*A + 42*B + 24*C)*Sqrt[a + a*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(5*A + 6*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 6, (a^(5/2)*(163*A + 200*B + 304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(299*A + 392*B + 432*C)*Tan[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(17*A + 24*B + 16*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (a*(5*A + 8*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 7, (a^(5/2)*(283*A + 326*B + 400*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B + 400*C)*Tan[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Sec[c + d*x]*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + (a*(A + 2*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + a*Cos[c + d*x])^(5/2), x, 8, (a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1015*A + 1132*B + 1304*C)*Tan[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Sec[c + d*x]*Tan[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Sec[c + d*x]^2*Tan[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3*Tan[c + d*x])/(480*d) + (a*(5*A + 12*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*Tan[c + d*x])/(60*d) + (A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 8, -((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A - 111*B + 143*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(9*B - C)*Cos[c + d*x]^3*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^4*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 7, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A - 49*B + 37*C)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(7*B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^3*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(35*A - 7*B + 31*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 6, -((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A - 10*B + 14*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 4, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*(3*B - 2*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 6, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 6, -(((A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (A*Tan[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 7, ((7*A - 4*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 8, -(((9*A - 14*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B + 8*C)*Tan[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 6*B)*Sec[c + d*x]*Tan[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/2), x, 9, ((107*A - 72*B + 112*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C)*Tan[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + ((43*A - 8*B + 48*C)*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 8, ((11*A - 15*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((455*A - 651*B + 799*C)*Sin[c + d*x])/(105*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((35*A - 63*B + 67*C)*Cos[c + d*x]^2*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((7*A - 7*B + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((245*A - 273*B + 397*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(210*a^2*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 7, -(((7*A - 11*B + 15*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((45*A - 65*B + 93*C)*Sin[c + d*x])/(15*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((5*A - 5*B + 9*C)*Cos[c + d*x]^2*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((15*A - 35*B + 39*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(30*a^2*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, ((3*A - 7*B + 11*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((3*A - 9*B + 13*C)*Sin[c + d*x])/(3*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((3*A - 3*B + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(6*a^2*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 4, ((A + 3*B - 7*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 6, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) - ((5*A - B - 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 7, -(((3*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B + C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - B + C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 8, ((19*A - 12*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B + 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((7*A - 6*B + 2*C)*Tan[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((2*A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2), x, 9, -(((47*A - 38*B + 24*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*a^(3/2)*d)) + ((17*A - 13*B + 9*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((21*A - 14*B + 12*C)*Tan[c + d*x])/(8*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A - 12*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(12*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 8, -(((75*A - 163*B + 283*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Cos[c + d*x]^3*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Sin[c + d*x])/(120*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((45*A - 85*B + 157*C)*Cos[c + d*x]^2*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*a^3*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 7, ((19*A - 75*B + 163*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((A - 9*B + 17*C)*Cos[c + d*x]^2*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) - ((21*A - 93*B + 197*C)*Sin[c + d*x])/(24*a^2*d*Sqrt[a + a*Cos[c + d*x]]) + ((15*A - 39*B + 95*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(48*a^3*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 6, ((5*A + 19*B - 75*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((3*A + 5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - B + 9*C)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 4, ((3*A + 5*B + 19*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((3*A + 5*B - 13*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*A*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B - 5*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((11*A - 3*B - 5*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 8, -(((5*A - 2*B)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B + 3*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((15*A - 7*B - C)*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((35*A - 11*B + 3*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2), x, 9, ((39*A - 20*B + 8*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B + 43*C)*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((63*A - 35*B + 11*C)*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((19*A - 11*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((31*A - 15*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, -((2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 6, -((2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (2*a*(9*A + 7*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*a*(11*A + 11*B + 9*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*a*(11*A + 11*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*(B + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(11*A + 11*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*(B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (2*a*(9*A + 9*B + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*A + 5*(B + C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A + 9*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (2*a*(5*A + 3*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(7*A + 7*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (2*a*(5*A + 5*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(3*A + B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, -((2*a*(A - B - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(3*A + 3*B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, -((2*a*(A + B - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + 3*(B + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, -((2*a*(3*A + 5*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(A + B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*(B + C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*a*(3*A + 3*B + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(5*A + 7*(B + C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*(B + C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 3*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (4*a^2*(9*A + 8*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(9*A + 8*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(99*A + 121*B + 89*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(11*B + 4*C)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (4*a^2*(12*A + 9*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*B + 4*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (4*a^2*(5*A + 4*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*B + 4*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (4*a^2*(5*B + 4*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(5*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (-4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(4*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (-4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(4*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-4*a^2*(3*A + 4*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(3*A + 4*B + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 9, (-4*a^2*(8*A + 9*B + 12*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(5*A + 6*B + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(8*A + 9*B + 12*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(4*A + 9*B)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, (4*a^3*(221*A + 195*B + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 195*B + 175*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (20*a^3*(286*A + 273*B + 236*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (2*(13*B + 6*C)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(143*A + 195*B + 145*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (4*a^3*(21*A + 17*B + 15*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(11*B + 6*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(99*A + 143*B + 105*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, (4*a^3*(27*A + 21*B + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, (4*a^3*(5*A + 9*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*(7*A - C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) - (2*(35*A - 7*B - 11*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, (-4*a^3*(5*A - 5*B - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(35*A + 15*B - 3*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 8, (-4*a^3*(9*A + 5*B - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(33*A + 35*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-4*a^3*(7*A + 9*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(6*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 9, (-4*a^3*(17*A + 21*B + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(17*A + 21*B + 27*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 10, (-4*a^3*(15*A + 17*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)) + (4*a^3*(105*A + 121*B + 143*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(15*A + 17*B + 21*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (2*(6*A + 11*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d*Cos[c + d*x]^(9/2)) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 7, (-3*(5*A - 7*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(7*A - 7*B + 9*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((5*A - 7*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((7*A - 7*B + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 6, (3*(5*A - 5*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((5*A - 5*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]), x, 5, -(((A - 3*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((3*A - 3*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])), x, 4, ((A - B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((A + B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])), x, 5, -(((3*A - B + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) - ((A - B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((3*A - B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])), x, 6, ((3*A - 3*B + C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((5*A - 3*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((3*A - 3*B + C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])), x, 7, (-3*(7*A - 5*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A - 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((5*A - 5*B + 3*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 7, ((20*A - 35*B + 56*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((20*A - 35*B + 56*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((A - 2*B + 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 6, -(((A - 4*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A - 5*B + 10*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((A - 4*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2, x, 5, -(((B - 4*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((A + 2*B - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2), x, 5, ((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A + B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2), x, 6, -(((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) - ((5*A - 2*B - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((4*A - B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((5*A - 2*B - C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2), x, 7, ((7*A - 4*B + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((10*A - 5*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((7*A - 4*B + C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((7*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 8, (7*(7*A - 17*B + 33*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 33*B + 63*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(7*A - 17*B + 33*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - 7*B + 12*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 33*B + 63*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 7, -((9*A - 49*B + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B + 33*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((B - 2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((9*A - 49*B + 119*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, -((A + 9*B - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3, x, 6, ((A - B - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3), x, 6, ((9*A + B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((9*A + B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3), x, 7, -((49*A - 9*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((49*A - 9*B - C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3), x, 8, ((119*A - 49*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 6, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a*(48*A + 40*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a*(8*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(6*B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 4, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) + (a*(4*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, (Sqrt[a]*(2*B + C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a*(2*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 4, (2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 3, (2*a*(A + 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(8*A + 10*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 4, (2*a*(A + 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (2*a*(A + 9*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 7, (a^(3/2)*(176*A + 150*B + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^2*(176*A + 150*B + 133*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 90*B + 67*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(10*B + 3*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 6, (a^(3/2)*(112*A + 88*B + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^2*(112*A + 88*B + 75*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(48*A + 56*B + 39*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(8*B + 3*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 5, (a^(3/2)*(24*A + 14*B + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) + (a*(2*B + C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 5, (a^(3/2)*(8*A + 12*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) - (a*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 5, (a^(3/2)*(2*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^2*(8*A + 6*B - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 5, (2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 4, (2*a^2*(4*A + 6*B + 5*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(104*A + 126*B + 175*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (2*a^2*(52*A + 72*B + 63*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(13/2), x, 6, (2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 8, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(680*A + 628*B + 545*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 7, (a^(5/2)*(400*A + 326*B + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(128*d) + (a^3*(400*A + 326*B + 283*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]) + (a^3*(1040*A + 950*B + 787*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(2*B + C)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 6, (a^(5/2)*(304*A + 200*B + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(8*B + 5*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 6, (a^(5/2)*(40*A + 38*B + 25*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (a*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 6, (a^(5/2)*(8*A + 20*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2), x, 6, (a^(5/2)*(2*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d - (a^3*(64*A + 70*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2), x, 6, (2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(11/2), x, 5, (2*a^3*(8*A + 10*B + 11*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(584*A + 690*B + 903*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 90*B + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 9*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(13/2), x, 6, (2*a^3*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(15/2), x, 7, (2*a^3*(2224*A + 2522*B + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 8, -(((8*A - 14*B + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((8*A - 2*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]) + ((6*B - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 7, ((8*A - 4*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*Sqrt[a + a*Cos[c + d*x]]), x, 6, ((2*B - C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]), x, 6, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]), x, 5, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]), x, 6, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(13*A - 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]), x, 7, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - (2*(43*A - 91*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - -{(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]], x, 7, ((8*a*A - 4*A*b - 4*a*B + 7*b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(Sqrt[a]*d) + ((4*A*b + 4*a*B - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]) + (b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 8, ((8*A - 12*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) - ((2*A - 6*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((A - B + 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2), x, 7, ((2*B - 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((A - B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(3/2)), x, 6, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)), x, 5, -(((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)), x, 6, ((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sin[c + d*x])/(6*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((19*A - 15*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)), x, 7, -(((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sin[c + d*x])/(10*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sin[c + d*x])/(30*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + ((147*A - 95*B + 75*C)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 8, ((2*B - 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2), x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Cos[c + d*x])^(5/2)), x, 5, ((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)), x, 6, -(((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)), x, 7, ((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sin[c + d*x])/(48*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) - ((299*A - 147*B + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (A+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 7, (1/8)*a*(4*A + 3*C)*x + (b*(5*A + 4*C)*Sin[c + d*x])/(5*d) + (a*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (b*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 3, (1/8)*b*(4*A + 3*C)*x + (a*(3*A + 2*C)*Sin[c + d*x])/(3*d) + (b*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 2, (a*(2*A + C)*x)/2 - ((a^2*C - b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 4, (1/2)*b*(2*A + C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 4, a*C*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 4, b*C*x + (a*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*b*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, (b*(A + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (A*b*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, (a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(2*A + 3*C)*Tan[c + d*x])/(3*d) + (a*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 7, (b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(4*A + 5*C)*Tan[c + d*x])/(5*d) + (b*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(4*A + 5*C)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 8, (1/16)*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*x + (2*a*b*(5*A + 4*C)*Sin[c + d*x])/(5*d) + ((b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((2*a^2*C + b^2*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*b*C*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) - (2*a*b*(5*A + 4*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 4, (1/4)*a*b*(4*A + 3*C)*x + ((5*a^2*(3*A + 2*C) + 2*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + (a*b*(4*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((2*a^2*C + b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (a*b*C*Cos[c + d*x]^3*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 3, (1/8)*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Sin[c + d*x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 5, a*b*(2*A + C)*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 2*(a^2 + b^2)*C)*Sin[c + d*x])/(3*d) + (a*b*C*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 5, (1/2)*(2*A*b^2 + (2*a^2 + b^2)*C)*x + (2*a*A*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(A - C)*Sin[c + d*x])/d - (b^2*(2*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 5, 2*a*b*C*x + ((2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(A - 2*C)*Sin[c + d*x])/(2*d) + (a*A*b*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 5, b^2*C*x + (a*b*(A + 2*C)*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (a*A*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, ((4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*(2*A + 3*C)*Tan[c + d*x])/(3*d) + ((2*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 8, (a*b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]])/(4*d) + ((5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (a*b*(3*A + 4*C)*Sec[c + d*x]*Tan[c + d*x])/(4*d) + ((2*A*b^2 + a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*A*b*Sec[c + d*x]^3*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 5, (1/16)*b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*x + (a*(5*a^2*(3*A + 2*C) + 6*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + (b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(15*A*b^2 + (a^2 + 12*b^2)*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(6*a^2*C + 5*b^2*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 4, (1/8)*a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*x - ((3*a^4*C - 4*b^4*(5*A + 4*C) - 4*a^2*b^2*(20*A + 13*C))*Sin[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) - ((3*a^2*C - 4*b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) - (a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 6, (1/8)*b*(12*a^2*(2*A + C) + b^2*(4*A + 3*C))*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + (a*(6*A*b^2 + (a^2 + 4*b^2)*C)*Sin[c + d*x])/(2*d) + (b*(2*a^2*C + b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(4*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 6, (1/2)*a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*x + (3*a^2*A*b*ArcTanh[Sin[c + d*x]])/d - (b*(a^2*(6*A - 8*C) - b^2*(3*A + 2*C))*Sin[c + d*x])/(3*d) - (a*b^2*(6*A - 5*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 6, (1/2)*b*(2*A*b^2 + (6*a^2 + b^2)*C)*x + (a*(6*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a*b^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) - (b^3*(4*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*A*b*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 6, 3*a*b^2*C*x + (b*(2*A*b^2 + 3*a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^3*(5*A - 6*C)*Sin[c + d*x])/(6*d) + (a*(3*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 6, b^3*C*x + (a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(A*b^2 + a^2*(4*A + 6*C))*Tan[c + d*x])/(2*d) + (a*(2*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 8, (b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(15*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (3*b*(2*A*b^2 + 5*a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a*(3*A*b^2 + 2*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + (3*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 9, (a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (b*(5*b^2*(2*A + 3*C) + 6*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + (a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (b*(A*b^2 + 3*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(6*A*b^2 + 5*a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -{Cos[c + d*x]^1*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 6, (1/4)*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*x + ((35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + (a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a*b*(126*A*b^2 + 6*a^2*C + 103*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(210*d) + ((2*a^2*C + b^2*(7*A + 6*C))*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(21*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^0*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 5, (1/16)*(8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*x - (a*(4*a^4*C - 32*b^4*(5*A + 4*C) - a^2*b^2*(190*A + 121*C))*Sin[c + d*x])/(60*b*d) - ((8*a^4*C - 15*b^4*(6*A + 5*C) - 2*a^2*b^2*(130*A + 89*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) - ((4*a^2*C - 5*b^2*(6*A + 5*C))*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) - (a*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^1, x, 7, (1/2)*a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + ((6*a^4*C + 2*b^4*(5*A + 4*C) + a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(15*d) + (a*b*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*a^2*C + b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + (a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 7, (1/8)*(8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x + (4*a^3*A*b*ArcTanh[Sin[c + d*x]])/d - (a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (b^2*(a^2*(24*A - 26*C) - 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (a*b*(12*A - 7*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 7, 2*a*b*(2*A*b^2 + (2*a^2 + b^2)*C)*x + (a^2*(12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a^2*(39*A - 34*C) - 2*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (a*b^3*(9*A - 4*C)*Cos[c + d*x]*Sin[c + d*x])/(3*d) - (b^2*(15*A - 2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 7, (1/2)*b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*x + (2*a*b*(2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/d - (2*a*b*(b^2*(11*A - 6*C) + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) - (b^2*(3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) + ((6*A*b^2 + a^2*(2*A + 3*C))*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(3*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 7, 4*a*b^3*C*x + ((8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x])/(24*d) + (a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Tan[c + d*x])/(12*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6, x, 7, b^4*C*x + (a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*A*b^4 + 2*a^4*(4*A + 5*C) + a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(15*d) + (a*b*(6*A*b^2 + a^2*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*A*b^2 + a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7, x, 9, ((8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a*b*(5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((12*A*b^2 + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8, x, 10, (a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(4*d) + ((35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[c + d*x])/(105*d) + (a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(4*d) + ((4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sec[c + d*x]^2*Tan[c + d*x])/(105*d) + (a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Sec[c + d*x]^3*Tan[c + d*x])/(210*d) + ((2*A*b^2 + a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(21*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)} - - -{(a + b*Cos[c + d*x])^3*(a^2 - b^2*Cos[c + d*x]^2), x, 5, (a*(8*a^4 + 8*a^2*b^2 - 9*b^4)*x)/8 + (b*(83*a^4 - 32*a^2*b^2 - 16*b^4)*Sin[c + d*x])/(30*d) + (a*b^2*(106*a^2 - 71*b^2)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(23*a^2 - 16*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (a*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) - (b*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^2*(a^2 - b^2*Cos[c + d*x]^2), x, 4, ((8*a^4 - 3*b^4)*x)/8 + (a*b*(13*a^2 - 8*b^2)*Sin[c + d*x])/(6*d) + (b^2*(14*a^2 - 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{(a + b*Cos[c + d*x])*(a^2 - b^2*Cos[c + d*x]^2), x, 3, (a*(2*a^2 - b^2)*x)/2 + (2*b*(2*a^2 - b^2)*Sin[c + d*x])/(3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, ((8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x)/(8*b^5) - (2*a^3*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - (a*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^4*d) + ((4*a^2*C + b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) - (a*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((a*(2*A*b^2 + (2*a^2 + b^2)*C)*x)/(2*b^4)) + (2*a^2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2*C + b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b^3*d) - (a*C*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, ((2*a^2*C + b^2*(2*A + C))*x)/(2*b^3) - (2*a*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*C*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, -((a*C*x)/b^2) + (2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, (C*x)/b - (2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d) + (A*ArcTanh[Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, (2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - (A*b*ArcTanh[Sin[c + d*x]])/(a^2*d) + (A*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((2*b*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (A*b*Tan[c + d*x])/(a^2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*b^2*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^2 + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*a^3*d) - (A*b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -((a*(2*A*b^2 + (4*a^2 + b^2)*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((a^2*b^2*(6*A - 7*C) + 12*a^4*C - b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((2*A*b^2 + (6*a^2 + b^2)*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, -((2*a*C*x)/b^3) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, (C*x)/b^2 + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, -((2*b*(2*a^2*A - A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - (2*A*b*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((2*A*b^2 - a^2*(A - C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -((2*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ((6*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + (b*(3*A*b^2 - a^2*(2*A - C))*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - a^2*(A - 2*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(4*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(a^5*d) - ((12*A*b^4 - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) + (b*(2*A*b^2 - a^2*(A - C))*Sec[c + d*x]*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((4*A*b^2 - a^2*(A - 3*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((2*A*b^2 + (12*a^2 + b^2)*C)*x)/(2*b^5) - (a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (a*(a^2*b^2*(2*A - 21*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*b^2*(A - 10*C) - b^4*(4*A - C) + 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, -((3*a*C*x)/b^4) + ((2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, (C*x)/b^3 - (a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, ((a^2*(2*A + C) + b^2*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, (b*(5*a^2*A*b^2 - 2*A*b^4 - 3*a^4*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((2*A*b^4 - a^4*C - a^2*b^2*(5*A + 2*C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((15*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d)) - (3*A*b*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((11*a^2*A*b^2 - 6*A*b^4 - a^4*(2*A - 3*C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(6*A + C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -((b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 8, ((2*A*b^2 + (20*a^2 + b^2)*C)*x)/(2*b^6) + ((8*a*A*b^8 - a^7*b^2*(2*A - 69*C) + 7*a^5*b^4*(A - 12*C) - 8*a^3*b^6*(A - 5*C) - 20*a^9*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2 - b^2)^3*d) - (a*(a^4*b^2*(6*A - 167*C) - a^2*b^4*(17*A - 146*C) + 2*b^6*(13*A - 12*C) + 60*a^6*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*b^2*(A - 27*C) - a^2*b^4*(2*A - 23*C) + b^6*(6*A - C) + 10*a^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((4*A*b^4 - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, -((4*a*C*x)/b^5) - ((2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (12*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, (C*x)/b^4 + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*b^6*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((4*A*b^6 + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, -((b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b^2*(2*A - 5*C) + 2*a^4*C + b^4*(13*A + 18*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, (a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, -((b*(7*a^2*A*b^4 - 2*A*b^6 - a^4*b^2*(8*A - C) + 4*a^6*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 8, -(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (4*A*b*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 9, ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(12*A - C) - 8*a^8*b*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((20*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((20*A*b^6 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, -(((8*a^4 - 4*a^2*b^2 - b^4)*x)/(8*b^5)) + (2*a^3*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^5*d) + (a*(3*a^2 - b^2)*Sin[c + d*x])/(3*b^4*d) - ((4*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, (a*(2*a^2 - b^2)*x)/(2*b^4) - (2*a^2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^4*d) - ((3*a^2 - b^2)*Sin[c + d*x])/(3*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, -(((2*a^2 - b^2)*x)/(2*b^3)) + (2*a*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^3*d) + (a*Sin[c + d*x])/(b^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^0*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, (a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^2*d) - Sin[c + d*x]/(b*d)} -{Sec[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, -(x/b) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b*d) + ArcTanh[Sin[c + d*x]]/(a*d)} -{Sec[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, -((2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*d)) - (b*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a*d)} -{Sec[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, (2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*d) - ((a^2 - 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (b*Tan[c + d*x])/(a^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, -((2*Sqrt[a - b]*b^2*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*d)) + (b*(a^2 - 2*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - ((a^2 - 3*b^2)*Tan[c + d*x])/(3*a^3*d) - (b*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{Cos[c + d*x]^4*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, -(((40*a^4 - 12*a^2*b^2 - b^4)*x)/(8*b^6)) + (2*a^3*(5*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*d) + (a*(15*a^2 - 2*b^2)*Sin[c + d*x])/(3*b^5*d) - ((20*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*b^4*d) + (5*a*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^3*d) - (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*b^2*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, (a*(4*a^2 - b^2)*x)/b^5 - (2*a^2*(4*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) - ((12*a^2 - b^2)*Sin[c + d*x])/(3*b^4*d) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(b^3*d) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -(((6*a^2 - b^2)*x)/(2*b^4)) + (2*a*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + (3*a*Sin[c + d*x])/(b^3*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(b*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, (2*a*x)/b^3 - (2*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - Sin[c + d*x]/(b^2*d) - (a*Sin[c + d*x])/(b^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, -(x/b^2) + (2*a*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Sin[c + d*x]/(b*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -((2*b*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d)) + ArcTanh[Sin[c + d*x]]/(a^2*d) - Sin[c + d*x]/(a*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -((2*(a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) - (2*b*ArcTanh[Sin[c + d*x]])/(a^3*d) + (2*Tan[c + d*x])/(a^2*d) - Tan[c + d*x]/(a*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, (2*b*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((a^2 - 6*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (3*b*Tan[c + d*x])/(a^3*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^4*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, -((2*b^2*(3*a^2 - 4*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d)) + (b*(a^2 - 4*b^2)*ArcTanh[Sin[c + d*x]])/(a^5*d) - ((a^2 - 12*b^2)*Tan[c + d*x])/(3*a^4*d) - (2*b*Sec[c + d*x]*Tan[c + d*x])/(a^3*d) + (4*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^4*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, (a*(20*a^2 - 3*b^2)*x)/(2*b^6) - (a^2*(20*a^4 - 33*a^2*b^2 + 12*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^6*(a + b)^(3/2)*d) - ((60*a^4 - 59*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)*d) + (a*(10*a^2 - 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((20*a^2 - 17*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^4*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) + ((5*a^2 - 4*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((12*a^2 - b^2)*x)/(2*b^5)) + (a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + (a*(12*a^2 - 11*b^2)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)*d) - ((6*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) + ((4*a^2 - 3*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, (3*a*x)/b^4 - ((6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (3*Sin[c + d*x])/(2*b^3*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*(3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, -(x/b^3) + (a*(2*a^2 - 3*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) - (a*Sin[c + d*x])/(2*b^2*d*(a + b*Cos[c + d*x])^2) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]]/((a - b)^(3/2)*(a + b)^(3/2)*d) + Sin[c + d*x]/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, -((b*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ArcTanh[Sin[c + d*x]]/(a^3*d) - Sin[c + d*x]/(2*a*d*(a + b*Cos[c + d*x])^2) - ((a^2 - 2*b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) - (3*b*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((5*a^2 - 6*b^2)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - Tan[c + d*x]/(2*a*d*(a + b*Cos[c + d*x])^2) - ((2*a^2 - 3*b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((a^2 - 12*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - (b*(11*a^2 - 12*b^2)*Tan[c + d*x])/(2*a^4*(a^2 - b^2)*d) + ((5*a^2 - 6*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^2) - ((3*a^2 - 4*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^4*(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 9, -((b^2*(12*a^4 - 33*a^2*b^2 + 20*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (b*(3*a^2 - 20*b^2)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((2*a^4 - 59*a^2*b^2 + 60*b^4)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)*d) - (b*(9*a^2 - 10*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)*d) + ((17*a^2 - 20*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*a^3*(a^2 - b^2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^2) - ((4*a^2 - 5*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 3, a*x - (b*Sin[c + d*x])/d} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, -x + (4*a*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, (2*(a^2 + b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (2*a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, (2*a*(a^2 + 2*b^2)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a*b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(2*a^2 + b^2)*Sin[c + d*x])/((a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 9, -((2*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(a^2 - b^2)*(21*A*b^2 + 8*a^2*C + 18*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(21*A*b^2 + 8*a^2*C + 18*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(24*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) - (4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 8, (2*a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 + 8*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 7, -((2*(2*a^2*C - 3*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 9, (2*a*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2*C - b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 9, -(((A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 10, -(A*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(3*A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^2 - 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 11, ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^2*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 10, -((4*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(1155*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(1155*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(16*a^4*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1155*b^3*d) - (4*a*(33*A*b^2 + 8*a^2*C + 34*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*b^3*d) + (2*(8*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(231*b^3*d) - (4*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(33*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 9, (2*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(a^2 - b^2)*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 8, (4*a*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*a^2*C - 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 10, (2*(a^2*C + b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*a*(5*A*b^2 - (a^2 - b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 10, -(a*(3*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((a^2*(3*A - 2*C) + 2*b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (3*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 10, -(b*(5*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*b*(7*A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (3*A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 11, -((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((8*a^2*(2*A + 3*C) + b^2*(17*A + 48*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(A*b^2 - 12*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 12, (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (b*(A*b^2 - 4*a^2*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(64*a^2*d) + ((A*b^2 + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*a*d) + (A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(8*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 11, -((2*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (4*a*(a^2 - b^2)*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) - (2*(240*a^4*C - 539*b^4*(13*A + 11*C) + 10*a^2*b^2*(143*A + 124*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) - (4*a*(143*A*b^2 + 24*a^2*C + 166*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(24*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) - (12*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d)} -{Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 10, (2*a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(693*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) - (8*a*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 9, -((2*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - (4*a*(a^2 - b^2)*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) - (4*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x], x, 11, (2*a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(21*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*b^2*(7*A - C) - 3*a^4*C + b^4*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(21*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(3*a^2*C + b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2, x, 11, -((a^2*(15*A - 46*C) - 6*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(a^2*(15*A - 16*C) + 4*b^2*(15*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (5*a^2*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (a*b*(15*A - 16*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3, x, 11, -(a*b*(27*A - 56*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(8*b^2*(3*A + C) + a^2*(33*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - (b^2*(21*A - 8*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4, x, 11, -((3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (a*(8*a^2*(2*A + 3*C) + b^2*(59*A + 96*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + (5*b*(A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5, x, 12, -(b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(4*a^2*(89*A + 132*C) + b^2*(133*A + 384*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^4 - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((5*A*b^2 + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{(a + b*Cos[c + d*x])^(3/2)*(a^2 - b^2*Cos[c + d*x]^2), x, 9, (4*a*(73*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(41*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (4*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + b*Cos[c + d*x]]*(a^2 - b^2*Cos[c + d*x]^2), x, 8, (2*(17*a^2 - 9*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (4*a*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 9, (2*(128*a^4*C + 21*b^4*(9*A + 7*C) + 12*a^2*b^2*(14*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(128*a^4*C + 4*a^2*b^2*(42*A + 19*C) + 3*b^4*(49*A + 37*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(42*A*b^2 + 32*a^2*C + 31*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^4*d) + (2*(48*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) - (16*a*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b*d)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 8, -((4*a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(48*a^4*C + 5*b^4*(7*A + 5*C) + 2*a^2*b^2*(35*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) - (12*a*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 7, (2*(8*a^2*C + 3*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(15*A*b^2 + 8*a^2*C + 7*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (8*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)} -{(A + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]], x, 6, (-4*a*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(2*a^2*C + b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x, 8, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]], x, 9, -((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]], x, 10, (3*A*b*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]], x, 11, -((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(24*a^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((5*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(24*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(8*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^3*d) - (5*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(384*a^4*C + 5*b^4*(7*A + 5*C) + 4*a^2*b^2*(70*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^5*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*a^2*b^2*(70*A - 31*C) + 192*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^4*(a^2 - b^2)*d) - (2*a*(35*A*b^2 + 48*a^2*C - 13*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b^2*(a^2 - b^2)*d)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*(5*A*b^2 + 2*(4*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a*(5*A*b^2 + 8*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 + 6*a^2*C - b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(3*A*b^2 + (8*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x, 9, (-2*(A*b^2 + a^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a*b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 10, ((3*A*b^2 - a^2*(A - 2*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - a^2*(A - 2*C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2), x, 11, -((b*(15*A*b^2 - a^2*(7*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - (5*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b^2*(15*A*b^2 - a^2*(7*A - 8*C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])} - - -{(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(4*a^4*b^2*(10*A - 53*C) - 5*a^2*b^4*(15*A - 11*C) + 128*a^6*C + 3*b^6*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*a*(4*a^2*b^2*(10*A - 29*C) + 128*a^4*C - b^4*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*(3*A*b^4 - a^2*b^2*(A - 6*C) - 4*a^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*(a^2*b^2*(10*A - 49*C) - b^4*(20*A - 7*C) + 32*a^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) + (2*(a^2*b^2*(15*A - 71*C) - b^4*(35*A - 3*C) + 48*a^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)} -{(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 8, -((4*a*(a^2*b^2*(A - 14*C) - b^4*(3*A - 4*C) + 8*a^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*b^2*(A - 8*C) + 16*a^4*C - b^4*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*a*(A*b^2 - 8*a^2*C + 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (4*a*(2*A*b^2 - (a^2 - 3*b^2)*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 10, (2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*a^2*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 + a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 11, ((26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b^2 - a^2*(3*A - 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 - a^2*(3*A - 2*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))} - - -{(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2), x, 8, -((2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - (4*a*(4*A*b^2 - (a^2 - 5*b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) - (4*a*(4*A*b^2 - a^2*C + 5*b^2*C)*Sin[c + d*x])/(15*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])} - - -{(a^2 - b^2*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]], x, 7, (4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (-2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (4*a*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2), x, 8, (2*(5*a^2 + 3*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (4*a*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (4*a*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(5*a^2 + 3*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 8, (2*a*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*b*(11*A + 9*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*b*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(11*A + 9*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*a*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + (2*b*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 7, (2*b*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2), x, 6, (2*a*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (2*b*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, -((2*a*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*b*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, -((2*b*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, -((2*a*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*A*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*b*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*A*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 8, (4*a*b*(9*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a*b*(9*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (8*a*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2), x, 7, (2*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a*b*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (8*a*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 6, (4*a*b*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(4*a^2*C + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (8*a*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, -((2*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a*b*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a*b*(3*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, -((4*a*b*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (8*a*A*b*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, -((2*(5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a*b*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (8*a*A*b*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(4*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((4*a*b*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (8*a*A*b*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(4*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a*b*(3*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2), x, 8, (2*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(165*d) + (2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(231*d) + (4*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(33*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (2*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(63*A*b^2 + 8*a^2*C + 45*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(24*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (4*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, -((2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*a*b^2*(35*A - 11*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, -((2*b*(15*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(3*b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*a*b^2*(5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d - (2*b^3*(35*A - 3*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, -((2*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(8*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (2*b^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (6*b*(8*A*b^2 + 7*a^2*(3*A + 5*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (12*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, -((2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2), x, 9, (2*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(1/2)*(c + d*x), 2])/(195*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(48*a^2*C + 11*b^2*(13*A + 11*C))*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (16*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, (8*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (4*a*b*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(16*a^2*C + 3*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (16*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, -((2*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (8*a*b*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (4*a*b*(a^2*(63*A - 31*C) - 6*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) - (2*b^2*(3*a^2*(105*A - 41*C) - 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) - (2*a*b*(21*A - 5*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, -((8*a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) - (2*b^2*(3*a^2*(49*A - 13*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (4*a*b^3*(175*A - 27*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b^2*(21*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 8, -((2*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (8*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a*b*(2*b^2*(33*A - 5*C) + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(16*A*b^2 + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, -((8*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a*b*(96*A*b^2 + a^2*(101*A + 175*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, -((2*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (8*a*b*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a*b*(32*A*b^2 + a^2*(101*A + 147*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(192*A*b^4 + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(48*A*b^2 + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 9, -((8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (16*A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]), x, 9, (2*(15*a^4*C + 3*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*b^5*d) - (2*a*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*b^6*d) + (2*a^4*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^6*(a + b)*d) - (2*a*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^4*d) + (2*(9*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*b^3*d) - (2*a*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b^2*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*b*d)} -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]), x, 8, -((2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d)) + (2*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^5*(a + b)*d) + (2*(7*a^2*C + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) - (2*a*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]), x, 7, (2*(5*a^2*C + b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d) - (2*a*(3*A*b^2 + (3*a^2 + b^2)*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) - (2*a*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]), x, 6, (-2*a*C*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*(3*a^2*C + b^2*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])), x, 5, (2*C*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*C*EllipticF[(c + d*x)/2, 2])/(b^2*d) + (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d)} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])), x, 6, (-2*A*EllipticE[(c + d*x)/2, 2])/(a*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(b*d) - (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*b*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])), x, 7, (2*A*b*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*A*EllipticF[(c + d*x)/2, 2])/(3*a*d) + (2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*A*b*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])), x, 8, (-2*(5*A*b^2 + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*A*b*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*A*b*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])), x, 9, (2*b*(5*A*b^2 + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^4*d) + (2*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^3*d) + (2*b^2*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*A*b*Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Cos[c + d*x]^(3/2)) - (2*b*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(11/2)*(a + b*Cos[c + d*x])), x, 10, (-2*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*a^5*d) - (2*b*(7*A*b^2 + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^4*d) - (2*b^3*(A*b^2 + a^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^5*(a + b)*d) + (2*A*Sin[c + d*x])/(9*a*d*Cos[c + d*x]^(9/2)) - (2*A*b*Sin[c + d*x])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*(9*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*a^3*d*Cos[c + d*x]^(5/2)) - (2*b*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^4*d*Cos[c + d*x]^(3/2)) + (2*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sin[c + d*x])/(15*a^5*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 8, ((3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*(a^2 - b^2)*d) - (a*(a^2*b^2*(9*A - 20*C) + 21*a^4*C - 4*b^4*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^5*(a + b)^2*d) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 7, -((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d)) + ((a^2*b^2*(3*A - 16*C) + 15*a^4*C - 2*b^4*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 6, ((A*b^2 + 3*a^2*C - 2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((A*b^4 - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2), x, 6, -(((A*b^2 + a^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2), x, 7, ((3*A*b^2 - a^2*(2*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a^2*(2*A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2), x, 8, -((b*(5*A*b^2 - a^2*(4*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - a^2*(2*A - 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + (b*(5*A*b^2 - a^2*(4*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2), x, 9, ((35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*(a^2 - b^2)*d) + (b*(7*A*b^2 - a^2*(4*A - 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^3*(a^2 - b^2)*d) + (b*(7*A*b^4 - 3*a^2*b^2*(3*A - C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a - b)*(a + b)^2*d) - ((7*A*b^2 - a^2*(2*A - 5*C))*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (b*(7*A*b^2 - a^2*(4*A - 3*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((35*A*b^4 - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 8, -((a*(a^2*b^2*(3*A - 65*C) - 3*b^4*(3*A - 8*C) + 35*a^4*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d)) + ((a^4*b^2*(9*A - 223*C) - a^2*b^4*(15*A - 128*C) + 105*a^6*C + 8*b^6*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((a^2*b^2*(3*A - 61*C) - b^4*(21*A - 8*C) + 35*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((5*A*b^4 - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 7, -(((b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) - (a*(15*a^4*C + b^4*(7*A + 24*C) - a^2*b^2*(A + 33*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 7, ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3), x, 7, ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3), x, 8, -(((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3), x, 9, (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(3*a^2*C - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) - (Sqrt[a + b]*(3*a^2*C - 2*a*b*C - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) - (a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) - ((3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) - (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d)) + (Sqrt[a + b]*(8*A*b + (a + 2*b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + (Sqrt[a + b]*(a^2*C - 4*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, ((a - b)*Sqrt[a + b]*(2*A - C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a*A - 2*A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (2*A*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) - (2*Sqrt[a + b]*(A*b - a*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, -((2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d)) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b + 15*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (2*(a - b)*b*Sqrt[a + b]*(8*A*b^2 + a^2*(19*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b + 8*A*b^2 + 5*a^2*(5*A + 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^2 - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2), x, 9, -(((a - b)*Sqrt[a + b]*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d)) - (Sqrt[a + b]*(3*a^3*C - 2*a^2*b*C - 8*b^3*(4*A + 3*C) - 4*a*b^2*(20*A + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) - (Sqrt[a + b]*(3*a^4*C + 24*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(64*b^2*d*Sqrt[Cos[c + d*x]]) - ((3*a^2*C - 4*b^2*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) - (a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*(3*a^2*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d)) + (Sqrt[a + b]*(48*a*A*b + 24*A*b^2 + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (a*Sqrt[a + b]*(24*A*b^2 - a^2*C + 12*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(8*A - 5*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*a*A - 16*A*b - 5*a*C - 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*A*b^2 + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (a*(8*A - 5*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, ((a - b)*b*Sqrt[a + b]*(8*A - 3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A + 3*C) - a*(8*A*b - 3*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (3*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (2*(a - b)*Sqrt[a + b]*(A*b^2 + a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d) - (2*Sqrt[a + b]*(A*b^2 - 2*a*b*(2*A + 5*C) + a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, -((4*(a - b)*b*Sqrt[a + b]*(3*A*b^2 - a^2*(41*A + 70*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d)) + (2*(a - b)*Sqrt[a + b]*(25*a^2*A - 57*a*A*b - 6*A*b^2 + 35*a^2*C - 105*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (6*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(3*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^2 + 8*A*b^3 - 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(3*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) - (4*b*(2*A*b^2 - a^2*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2), x, 10, ((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) - ((45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) - (3*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 9, -(((a - b)*Sqrt[a + b]*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d)) + (Sqrt[a + b]*(15*a^3*C + 24*b^3*(4*A + 3*C) + 2*a^2*b*(192*A + 59*C) + 4*a*b^2*(108*A + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) + (Sqrt[a + b]*(5*a^4*C - 120*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 9, ((a - b)*Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) - (Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C) - 2*a*b*(72*A + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (5*a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) - ((a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) - (a*b*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 9, ((a - b)*b*Sqrt[a + b]*(56*A - 27*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) + (Sqrt[a + b]*(6*b^2*(12*A + C) + 8*a^2*(A + 3*C) - a*(56*A*b - 27*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (a*b*(56*A - 27*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b^2*(8*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 9, ((a - b)*Sqrt[a + b]*(b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (Sqrt[a + b]*(30*A*b^3 - a*b^2*(46*A - 15*C) - 6*a^3*(3*A + 5*C) + a^2*(34*A*b + 90*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (5*a*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - ((b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (2*(a - b)*b*Sqrt[a + b]*(3*A*b^2 + a^2*(29*A + 49*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d) - (2*Sqrt[a + b]*(3*A*b^3 - 9*a*b^2*(3*A + 7*C) - a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(3*A*b^2 + a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, -((2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d)) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 21*a^3*(7*A + 9*C) + 15*a*b^2*(11*A + 21*C) - 6*a^2*b*(19*A + 28*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*(15*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*b*(5*A*b^2 + a^2*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 8, (2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d) + (2*(5*A*b^2 + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*a^2*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} -(* {((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 10, -((2*(a + b)*(17787*a^6*A + 30669*a^4*A*b^2 - 760*a^2*A*b^4 - 240*A*b^6 + 21021*a^6*C + 39897*a^4*b^2*C - 1430*a^2*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))])/(45045*a^4*d*Sqrt[a + b*Cos[c + d*x]])) + (1/(45045*a^3*d*Sqrt[a + b*Cos[c + d*x]]))*(2*(17787*a^6*A + 30831*a^5*A*b + 30669*a^4*A*b^2 + 16685*a^3*A*b^3 - 760*a^2*A*b^4 - 60*a*A*b^5 - 240*A*b^6 + 21021*a^6*C + 37323*a^5*b*C + 39897*a^4*b^2*C + 22165*a^3*b^3*C - 1430*a^2*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]) + (2*(121*a^2*A + 15*A*b^2 + 143*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (2*b*(2209*a^2*A + 15*A*b^2 + 2717*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*a*d*Cos[c + d*x]^(7/2)) + (2*(5929*a^4*A + 8145*a^2*A*b^2 - 90*A*b^4 + 7007*a^4*C + 10725*a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*a^2*d*Cos[c + d*x]^(5/2)) + (2*b*(18973*a^4*A + 395*a^2*A*b^2 + 120*A*b^4 + 23309*a^4*C + 715*a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*a^3*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2)) + (2*(17787*a^6*A + 30669*a^4*A*b^2 - 760*a^2*A*b^4 - 240*A*b^6 + 21021*a^6*C + 39897*a^4*b^2*C - 1430*a^2*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x]/(1 + Cos[c + d*x]))/(45045*a^4*d*Sqrt[Cos[c + d*x]])} *) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*(15*a^2*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d)) + (Sqrt[a + b]*(15*a^2*C - 10*a*b*C + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) + (a*Sqrt[a + b]*(8*A*b^2 + 5*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) + ((15*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) - (5*a*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 7, (3*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - ((3*a - 2*b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - (Sqrt[a + b]*(3*a^2*C + 4*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) - (3*a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]), x, 6, -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*(2*A*b + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d) + (a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]), x, 6, (2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]), x, 4, -((4*A*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d)) + (2*Sqrt[a + b]*(2*A*b + a*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]), x, 5, (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) + (2*Sqrt[a + b]*(2*a*A*b - 8*A*b^2 - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (8*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]), x, 6, -((4*(a - b)*b*Sqrt[a + b]*(24*A*b^2 + a^2*(22*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d)) - (2*Sqrt[a + b]*(12*a*A*b^2 - 48*A*b^3 - 5*a^3*(5*A + 7*C) - a^2*(44*A*b + 70*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (12*A*b*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Cos[c + d*x]^(3/2))} - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 8, ((8*A*b^2 + 15*a^2*C - 7*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - ((8*A*b^2 + (15*a^2 + 5*a*b - 2*b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((4*A*b^2 + 5*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 7, -(((2*A*b^2 + 3*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d)) + ((2*A*b^2 + a*(3*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) + (3*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)), x, 6, (2*(A*b^2 + a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d) + (2*(A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)), x, 4, -((2*(2*A*b^2 - a^2*(A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d)) - (2*(2*A*b + a*(A - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)), x, 5, (2*b*(8*A*b^2 - a^2*(5*A - 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d) + (2*(6*a*A*b + 8*A*b^2 + a^2*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)), x, 6, -((2*(16*A*b^4 - 2*a^2*b^2*(4*A - 5*C) - a^4*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d)) - (2*(12*a*A*b^2 + 16*A*b^3 + 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (2*b*(8*A*b^2 - a^2*(3*A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} - - -{(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 8, ((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) + ((2*a*A*b^3 - 6*A*b^4 + 15*a^4*C + 5*a^3*b*C - 21*a^2*b^2*C - 3*a*b^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) + (5*a*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) - (2*(A*b^2 + a^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d)) + (2*(3*a*A*b^2 - A*b^3 - 3*a^3*C - a^2*b*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)), x, 5, (4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*A*b^2 + 3*a*b*(A + C) - a^2*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*b*(A*b^2 - a^2*(3*A + 2*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)), x, 5, (2*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(6*a*A*b^2 + 8*A*b^3 - 3*a^3*(A - C) - a^2*b*(9*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) - (4*(2*A*b^4 - a^4*C - a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)), x, 6, -((4*b*(8*A*b^4 + a^4*(4*A - 3*C) - a^2*b^2*(14*A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d)) - (2*(12*a*A*b^3 + 16*A*b^4 - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C) - a^3*(9*A*b - 3*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (A+C Cos[e+f x]^2) with m and/or n symbolic*) - - -{Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, If[$VersionNumber>=8, ((2*a^2*C + b^2*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*(C*(2 + m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]), ((2*a^2*C + b^2*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (2*a*b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + m)*(2 + 3*m + m^2)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*(C*(2 + m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^1, x, 5, (a*C*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) + (b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)) - (a*(C*(1 + m) + A*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (b*(C*(2 + m) + A*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^1, x, 8, (a*(A*b^2 + a^2*C)*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*AppellF1[1/2, -(m/2), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*(b*(a^2 - b^2)*d)) + (a*C*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (C*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 9, ((A*b^4*m - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)^2*d) - ((A*b^4*m - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, -(m/2), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*(a*b*(a^2 - b^2)^2*d)) + ((A*b^2 + a^2*C)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((a^2*C*(1 + m) - b^2*(C - A*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) + ((A*b^2 + a^2*C)*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*b*(a^2 - b^2)*d*(2 + m)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 8, (1/8)*(4*a*B + 3*b*C)*x + ((b*B + a*C)*Sin[c + d*x])/d + ((4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((b*B + a*C)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(b*B + a*C)*x + ((3*a*B + 2*b*C)*Sin[c + d*x])/(3*d) + ((b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*C*Cos[c + d*x]^2*Sin[c + d*x])/(3*d), (1/2)*(b*B + a*C)*x + ((3*a*b*B - a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(2*a*B + b*C)*x + ((b*B + a*C)*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, (b*B + a*C)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 5, b*C*x + ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, ((a*B + 2*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 8, ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*B + 3*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 8, ((3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((b*B + a*C)*Tan[c + d*x])/d + ((3*a*B + 4*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((b*B + a*C)*Tan[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 8, (1/8)*(4*a^2*B + 3*b^2*B + 6*a*b*C)*x + ((4*b^2*C + 5*a*(2*b*B + a*C))*Sin[c + d*x])/(5*d) + ((4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*(5*b*B + 6*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (b*C*Cos[c + d*x]^3*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d) - ((4*b^2*C + 5*a*(2*b*B + a*C))*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a*b*B + 4*a^2*C + 3*b^2*C)*x + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/2)*(2*a^2*B + b^2*B + 2*a*b*C)*x + (2*(3*a*b*B + a^2*C + b^2*C)*Sin[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, (1/2)*(4*a*b*B + 2*a^2*C + b^2*C)*x + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (b*(2*b*B + 3*a*C)*Sin[c + d*x])/(2*d) + (b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, b*(b*B + 2*a*C)*x + (a*(2*b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (b^2*C*Sin[c + d*x])/d + (a^2*B*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, b^2*C*x + ((a^2*B + 2*b^2*B + 4*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*b*B + a*C)*Tan[c + d*x])/d + (a^2*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 7, ((2*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a^2*B + 3*b^2*B + 6*a*b*C)*Tan[c + d*x])/(3*d) + (a*(2*b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 8, ((3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*b*B + 2*a^2*C + 3*b^2*C)*Tan[c + d*x])/(3*d) + ((3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(2*b*B + a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a^2*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*x + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sin[c + d*x])/(6*d) + (b*(20*a*b*B + 6*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/2)*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*x + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*b*B + 8*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*d) + (b^2*(3*b*B + 5*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/2)*b*(6*a*b*B + 6*a^2*C + b^2*C)*x + (a^2*(3*b*B + a*C)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a^2*B - b^2*B - 3*a*b*C)*Sin[c + d*x])/d - (b^2*(2*a*B - b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, b^2*(b*B + 3*a*C)*x + (a*(a^2*B + 6*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(a*B - 2*b*C)*Sin[c + d*x])/(2*d) + (a^2*(2*b*B + a*C)*Tan[c + d*x])/d + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, b^3*C*x + ((3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(2*a^2*B + 8*b^2*B + 9*a*b*C)*Tan[c + d*x])/(3*d) + (a^2*(5*b*B + 3*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 8, ((3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*Tan[c + d*x])/(3*d) + (a*(3*a^2*B + 10*b^2*B + 12*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(3*b*B + 2*a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^7*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 9, ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((8*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 15*b^3*C)*Tan[c + d*x])/(15*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(4*a^2*B + 12*b^2*B + 15*a*b*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a^2*(7*b*B + 5*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (a*B*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, ((2*a^2 + b^2)*(b*B - a*C)*x)/(2*b^4) - (2*a^3*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -(((2*a*b*B - 2*a^2*C - b^2*C)*x)/(2*b^3)) + (2*a^2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, ((b*B - a*C)*x)/b^2 - (2*a*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, (C*x)/b + (2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, -((2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)) + (B*ArcTanh[Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*b*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - ((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) + (B*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, -((2*b^2*(b*B - a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((a^2*B + 2*b^2*B - 2*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((b*B - a*C)*Tan[c + d*x])/(a^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -(((4*a*b*B - 6*a^2*C - b^2*C)*x)/(2*b^4)) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((b*B - 2*a*C)*x)/b^3 - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) - (a^2*(b*B - a*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, (C*x)/b^2 - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, (2*(a*B - b*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -((2*(2*a^2*b*B - b^3*B - a^3*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (B*ArcTanh[Sin[c + d*x]])/(a^2*d) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, (2*b*(3*a^2*b*B - 2*b^3*B - 2*a^3*C + a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((a^2*B - 2*b^2*B + a*b*C)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -(((6*a*b*B - 12*a^2*C - b^2*C)*x)/(2*b^5)) + (a^2*(6*a^4*b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*C - 20*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((b*B - 3*a*C)*x)/b^4 - (a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C - 12*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*b*B - 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a^2*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, (C*x)/b^3 + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 6*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(b*B - a*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, -(((3*a*b*B - a^2*C - 2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(b*B - a*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, ((2*a^2*B + b^2*B - 3*a*b*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((b*B - a*C)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*b*B - a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + (B*ArcTanh[Sin[c + d*x]])/(a^3*d) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, (b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((3*b*B - a*C)*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((2*a^4*B - 11*a^2*b^2*B + 6*b^4*B + 5*a^3*b*C - 2*a*b^3*C)*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(b*B - a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, -((2*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - 19*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(14*a*b*B - 8*a^2*C - 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(14*a*b*B - 8*a^2*C - 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(3*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*b*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 10, -((B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a*B + 2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + ((b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 11, -(((b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*b*B + 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B - b^2*B + 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, -((2*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 33*a^2*b^2*C - 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) - (2*(18*a*b*B - 8*a^2*C - 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B + 3*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, (2*(3*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*a*b*B - a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, -(((a*B - 2*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a^2*B + 2*b^2*B + 2*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 11, -(((5*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((7*a*b*B + 4*a^2*C + 8*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B + 3*b^2*B + 12*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((5*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 12, -(((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((16*a^2*B + 17*b^2*B + 42*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((7*b*B + 6*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (a*B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -((2*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 255*a^3*b^2*C - 3705*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 335*a*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) - (2*(22*a*b*B - 8*a^2*C - 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, (2*(35*a*b*B + 23*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C + 8*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*b*B + 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((3*a^2*B - 6*b^2*B - 14*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*a^3*B + 12*a*b^2*B + 4*a^2*b*C + 2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*(5*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*a*B - 2*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((9*a*b*B + 4*a^2*C - 8*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((11*a^2*b*B + 8*b^3*B + 4*a^3*C + 16*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(4*a^2*B + 15*b^2*B + 20*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(7*b*B + 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 12, -(((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((16*a^3*B + 59*a*b^2*B + 66*a^2*b*C + 48*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + (a*(3*b*B + 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^6*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 13, -(((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((356*a^2*b*B + 133*b^3*B + 128*a^3*C + 472*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) + ((48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((36*a^2*B + 59*b^2*B + 104*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*d) + (a*(11*b*B + 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (a*B*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(10*a*b*B - 8*a^2*C - 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - 7*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(3*a*b*B - 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 10, -((B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - ((b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^4*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 11, ((3*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((b*B - 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*B + 3*b^2*B - 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (B*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 12*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*b*B - 6*a^2*C + b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(6*a*b*B - 8*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(b*B - a*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, -((2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 8, -((2*(b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 11, -(((a^2*B - 3*b^2*B + 2*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(a^2*B - 3*b^2*B + 2*a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 16*a^5*C + 28*a^3*b^2*C - 8*a*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*b*B - 9*a*b^3*B - 16*a^4*C + 16*a^2*b^2*C + b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a^2*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Cos[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 8, -((2*(2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*b*B - 3*b^3*B - 8*a^3*C + 9*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*a^2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a*b*B + 2*a^2*C - 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(b*B - a*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(4*a*b*B - a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(b*B - a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*a*b*B - a^2*C - 3*b^2*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 11, -((2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(b*B - a*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 12, -(((3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*a^2*B - 5*b^2*B + 2*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(3*a^2*B - 5*b^2*B + 2*a*b*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (b*(3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (B*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (2*(9*a*B + 7*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(b*B + a*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (6*(b*B + a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(7*a*B + 5*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (2*(5*a*B + 3*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(b*B + a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (2*(b*B + a*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a*B + b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (-2*(a*B - b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/d + (2*a*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (-2*(b*B + a*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-2*(3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a*B + 5*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (10*(9*b^2*C + 11*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(9*b^2*C + 11*a*(2*b*B + a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 13*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*C*Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(11*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (2*(7*b^2*C + 9*a*(2*b*B + a*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*b^2*C + 9*a*(2*b*B + a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 11*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (2*(5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*b^2*C + 7*a*(2*b*B + a*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*b^2*C + 7*a*(2*b*B + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 9*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (2*(3*b^2*C + 5*a*(2*b*B + a*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(5*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, -((2*(a^2*B - b^2*B - 2*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(6*a*b*B + 3*a^2*C + b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, -((2*(2*a*b*B + a^2*C - b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(2*a*b*B + a^2*C + 3*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, -((2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*B*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(2*b*B + a*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (2*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(33*a*b*B + 26*a^2*C + 9*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b^2*(11*b*B + 15*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*b*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, (2*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(27*a*b*B + 22*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*b*B + 13*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (2*(15*a^2*b*B + 3*b^3*B + 5*a^3*C + 9*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(21*a*b*B + 18*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*b*B + 11*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, -((2*(5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(6*a^2*B - b^2*B - 3*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*a*B - b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, -((2*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(7*b*B + 3*a*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(a*B - b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, -((2*(3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*b*B + 3*b^3*B + a^3*C + 9*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(9*b*B + 5*a*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*(3*a^2*B + 14*b^2*B + 15*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, -((2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(11*b*B + 7*a*C)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*a*(5*a^2*B + 18*b^2*B + 21*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 9, -((2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(13*b*B + 9*a*C)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*a*(7*a^2*B + 22*b^2*B + 27*a*b*C)*Sin[c + d*x])/(45*d*Cos[c + d*x]^(5/2)) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 9, (2*(5*a^2 + 3*b^2)*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*C - 5*b^4*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*b^5*d) + (2*a^4*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^5*(a + b)*d) - (2*(7*a*b*B - 7*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, -((2*(5*a*b*B - 5*a^2*C - 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d)) + (2*(3*a^2 + b^2)*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) - (2*a^3*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^(1/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d) - (2*(3*a*b*B - 3*a^2*C - b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) + (2*a^2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(-1/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, (2*C*EllipticE[(1/2)*(c + d*x), 2])/(b*d) + (2*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) - (2*a*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)} -{Cos[c + d*x]^(-3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, (2*C*EllipticF[(1/2)*(c + d*x), 2])/(b*d) + (2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b*(a + b)*d)} -{Cos[c + d*x]^(-5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - (2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d) + (2*B*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, (2*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*b*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*B*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(b*B - a*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 9, -((2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d)) - (2*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (2*b^2*(b*B - a*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a + b)*d) + (2*B*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(b*B - a*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 9, -((25*a^3*b*B - 20*a*b^3*B - 35*a^4*C + 24*a^2*b^2*C + 6*b^4*C)*EllipticE[(c + d*x)/2, 2])/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - 21*a^5*C + 20*a^3*b^2*C + 4*a*b^4*C)*EllipticF[(c + d*x)/2, 2])/(3*b^5*(a^2 - b^2)*d) - (a^3*(5*a^2*b*B - 7*b^3*B - 7*a^3*C + 9*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^5*(a + b)^2*d) + ((5*a^2*b*B - 2*b^3*B - 7*a^3*C + 4*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) - ((5*a*b*B - 7*a^2*C + 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 8, ((3*a^2*b*B - 2*b^3*B - 5*a^3*C + 4*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - 15*a^4*C + 16*a^2*b^2*C + 2*b^4*C)*EllipticF[(c + d*x)/2, 2])/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*b*B - 5*b^3*B - 5*a^3*C + 7*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^4*(a + b)^2*d) - ((3*a*b*B - 5*a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2, x, 7, -(((a*b*B - 3*a^2*C + 2*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d)) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d) - (a*(a^2*b*B - 3*b^3*B - 3*a^3*C + 5*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) + (a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2), x, 7, ((b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((a*b*B + a^2*C - 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d) - ((a^2*b*B + b^3*B + a^3*C - 3*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a - b)*b^2*(a + b)^2*d) - ((b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2), x, 7, -(((b*B - a*C)*EllipticE[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d)) - ((b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(b*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - a^3*C - a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a*(a - b)*b*(a + b)^2*d) + (b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2), x, 8, -(((2*a^2*B - 3*b^2*B + a*b*C)*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d)) + ((b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(a*(a^2 - b^2)*d) - ((5*a^2*b*B - 3*b^3*B - 3*a^3*C + a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*B - 3*b^2*B + a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2), x, 9, ((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*b*B - 5*b^3*B - 5*a^3*C + 3*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) - ((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 9, ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - 35*a^5*C + 65*a^3*b^2*C - 24*a*b^4*C)*EllipticE[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - 105*a^6*C + 223*a^4*b^2*C - 128*a^2*b^4*C - 8*b^6*C)*EllipticF[(c + d*x)/2, 2])/(12*b^5*(a^2 - b^2)^2*d) + (a^2*(15*a^4*b*B - 38*a^2*b^3*B + 35*b^5*B - 35*a^5*C + 86*a^3*b^2*C - 63*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*b*B - 33*a*b^3*B - 35*a^4*C + 61*a^2*b^2*C - 8*b^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(3*a^2*b*B - 9*b^3*B - 7*a^3*C + 13*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 8, -((3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*EllipticF[(c + d*x)/2, 2])/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*b*B - 6*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 38*a^3*b^2*C - 35*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (a*(a^2*b*B - 7*b^3*B - 5*a^3*C + 11*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3, x, 8, -((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*b*B - 10*a^2*b^3*B - 3*b^5*B + 3*a^5*C - 6*a^3*b^2*C + 15*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3), x, 8, ((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*b*B + 3*b^3*B + a^3*C - 7*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*b*B + 10*a^2*b^3*B - b^5*B + a^5*C - 10*a^3*b^2*C - 3*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3), x, 8, -((9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*b*B - b^3*B - 3*a^3*C - 3*a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*b*B - 6*a^2*b^3*B + 3*b^5*B - 3*a^5*C - 10*a^3*b^2*C + a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + (b*(9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3), x, 9, -((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*EllipticF[(c + d*x)/2, 2])/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*b*B - 38*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 6*a^3*b^2*C - 3*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) + (b*(11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^(m/2) (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, -((a - b)*Sqrt[a + b]*(6*a*b*B - 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) + (Sqrt[a + b]*(a + 2*b)*(6*b*B - 3*a*C + 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) + ((6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(a*C + 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(4*a*b*B - a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)) + (Sqrt[a + b]*(2*B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*b*B + a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (2*Sqrt[a + b]*(b*B - a*(B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, (2*(a - b)*Sqrt[a + b]*(b*B + 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*(a - b)*Sqrt[a + b]*(B - 3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*B - 2*b^2*B + 5*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(9*a*B + 2*b*B - 5*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (2*(a - b)*Sqrt[a + b]*(19*a^2*b*B + 8*b^3*B + 63*a^3*C - 14*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*b^2*B + a^2*(25*B - 63*C) + 2*a*b*(3*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B - 4*b^2*B + 7*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, -((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(16*B + 9*C) - 4*a*b^2*(28*B + 39*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*C - 48*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((8*a*b*B - 3*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 9, -(((a - b)*Sqrt[a + b]*(30*a*b*B + 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d)) + (Sqrt[a + b]*(30*a*b*B + 12*b^2*B + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((6*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + (b*C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, -((a - b)*Sqrt[a + b]*(4*b*B + 5*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) + (Sqrt[a + b]*(8*a*B + 4*b*B + 5*a*C + 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(12*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) + ((4*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, ((a - b)*Sqrt[a + b]*(2*a*B - b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*a*(B - C) - b*(4*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (Sqrt[a + b]*(2*b*B + 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*a*B - b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (2*(a - b)*Sqrt[a + b]*(4*b*B + 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (2*Sqrt[a + b]*(3*b^2*B - a*b*(4*B - 6*C) + a^2*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*B + 3*b^2*B + 20*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) - (2*(a - b)*Sqrt[a + b]*(9*a*B - 3*b*B - 5*a*C + 15*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(6*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (2*(a - b)*Sqrt[a + b]*(82*a^2*b*B - 6*b^3*B + 63*a^3*C + 21*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*b^2*B - a^2*(25*B - 63*C) + 3*a*b*(19*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(8*b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B + 3*b^2*B + 42*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 11, -((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(45*B + 64*C) - 8*a*b^3*(355*B + 193*C) - 4*a^2*b^2*(295*B + 423*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*C - 240*a*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 172*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 10, -(((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d)) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(16*B + 9*C) + 2*a^2*b*(132*B + 59*C) + 4*a*b^2*(52*B + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (b*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 9, -((a - b)*Sqrt[a + b]*(54*a*b*B + 33*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) + (Sqrt[a + b]*(4*b^2*(3*B + 4*C) + a*b*(54*B + 26*C) + a^2*(48*B + 33*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((54*a*b*B + 33*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*(2*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 9, ((a - b)*Sqrt[a + b]*(8*a^2*B - 4*b^2*B - 9*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) - (Sqrt[a + b]*(8*a^2*(B - C) - 2*b^2*(2*B + C) - 3*a*b*(8*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(20*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - ((8*a^2*B - 4*b^2*B - 9*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 9, ((a - b)*Sqrt[a + b]*(14*a*b*B + 6*a^2*C - 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (Sqrt[a + b]*(2*a*b*(7*B - 9*C) - 2*a^2*(B - 3*C) - 3*b^2*(6*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*d) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(2*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((14*a*b*B + 6*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (2*(a - b)*Sqrt[a + b]*(9*a^2*B + 23*b^2*B + 35*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (2*Sqrt[a + b]*(15*b^3*B - a*b^2*(23*B - 45*C) + a^2*b*(17*B - 35*C) - a^3*(9*B - 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*a*(8*b*B + 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (2*(a - b)*Sqrt[a + b]*(145*a^2*b*B + 15*b^3*B + 63*a^3*C + 161*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*(a - b)*Sqrt[a + b]*(a^2*(25*B - 63*C) + 15*b^2*(B - 7*C) - 8*a*b*(15*B - 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d) + (2*a*(10*b*B + 7*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(25*a^2*B + 45*b^2*B + 77*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 8, (2*(a - b)*Sqrt[a + b]*(147*a^4*B + 279*a^2*b^2*B - 10*b^4*B + 435*a^3*b*C + 45*a*b^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*b^3*B - 6*a^2*b*(19*B - 60*C) + 3*a^3*(49*B - 25*C) + 15*a*b^2*(11*B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*a*(4*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(49*a^2*B + 75*b^2*B + 135*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(163*a^2*b*B + 5*b^3*B + 75*a^3*C + 135*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 9, (2*(a - b)*Sqrt[a + b]*(3705*a^4*b*B + 255*a^2*b^3*B + 40*b^5*B + 1617*a^5*C + 3069*a^3*b^2*C - 110*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d) + (2*(a - b)*Sqrt[a + b]*(40*b^4*B + 3*a^4*(225*B - 539*C) - 6*a^3*b*(505*B - 209*C) + 15*a^2*b^2*(19*B - 121*C) + 10*a*b^3*(3*B - 11*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d) + (2*a*(14*b*B + 11*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*(81*a^2*B + 113*b^2*B + 209*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(693*d*Cos[c + d*x]^(7/2)) + (2*(1145*a^2*b*B + 15*b^3*B + 539*a^3*C + 825*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a*d*Cos[c + d*x]^(5/2)) + (2*(675*a^4*B + 1025*a^2*b^2*B - 20*b^4*B + 1793*a^3*b*C + 55*a*b^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Cos[c + d*x]^(3/2)) + (2*a*B*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 9, ((a - b)*Sqrt[a + b]*(18*a*b*B - 15*a^2*C - 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d) - (Sqrt[a + b]*(18*a*b*B - 12*b^2*B - 15*a^2*C + 10*a*b*C - 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) - ((18*a*b*B - 15*a^2*C - 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) + ((6*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + (Sqrt[a + b]*(4*a*b*B - 3*a^2*C - 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]), x, 8, -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) - (Sqrt[a + b]*(2*b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]), -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) - (Sqrt[a + b]*(2*b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (a*C*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]), x, 4, (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]), x, 4, (2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]), x, 5, (-2*(a - b)*Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(2*b*B + a*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]), x, 6, (2*(a - b)*Sqrt[a + b]*(9*a^2*B + 8*b^2*B - 10*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*b^2*B + a^2*(9*B - 5*C) - 2*a*b*(B + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))} - - -{(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 9, -((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d) + ((a*b*(12*B - 5*C) - 15*a^2*C + 2*b^2*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a*b*B - 15*a^2*C - 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) + (2*a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((4*a*b*B - 5*a^2*C + b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)} -{(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 8, ((2*a*b*B - 3*a^2*C + b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) - ((2*b*B - 3*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((2*a*b*B - 3*a^2*C + b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)), x, 7, -((2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d)) + (2*(b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (2*a*(b*B - a*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)), x, 5, (2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*Sqrt[a + b]*d) - (2*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)), x, 5, (2*(a^2*B - 2*b^2*B + a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d) - (2*(2*b*B + a*(B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)), x, 6, -((2*(5*a^2*b*B - 8*b^3*B - 3*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d)) + (2*(a + 2*b)*(4*b*B + a*(B - 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(a^2*B - 4*b^2*B + 3*a*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} - - -{(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 9, ((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) - ((a^2*b*(6*B - 5*C) - 3*b^3*(4*B - C) - 15*a^3*C + a*b^2*(2*B + 21*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(2*b*B - 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) + (2*a*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*b^2*B - 3*b^3*B - 3*a^3*C - a^2*b*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) + (2*a*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)), x, 6, -((2*(3*a^2*B + b^2*B - 4*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*(a - b)*(a + b)^(3/2)*d)) + (2*(3*a*B - b*B + a*C - 3*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^2*B + b^2*B - 4*a*b*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)), x, 6, (2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*b^2*B - 3*a^2*(B + C) + a*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)), x, 6, (2*(3*a^4*B - 15*a^2*b^2*B + 8*b^4*B + 6*a^3*b*C - 2*a*b^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*(a - b)*(a + b)^(3/2)*d) + (2*(8*b^3*B - 3*a^3*(B - C) + 2*a*b^2*(3*B - C) - 3*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*b*(b*B - a*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(8*a^2*b*B - 4*b^3*B - 5*a^3*C + a*b^2*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, (1/8)*(4*a*A + 3*b*B + 3*a*C)*x + ((5*A*b + 5*a*B + 4*b*C)*Sin[c + d*x])/(5*d) + ((4*a*A + 3*b*B + 3*a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((5*A*b + 5*a*B + 4*b*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 3, (1/8)*(4*A*b + 4*a*B + 3*b*C)*x + ((3*a*A + 2*b*B + 2*a*C)*Sin[c + d*x])/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + (b*C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 2, (1/2)*(b*B + a*(2*A + C))*x + ((A*b + a*B + b*C)*Sin[c + d*x])/d + ((b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (b*C*Sin[c + d*x]^3)/(3*d), (1/2)*(b*B + a*(2*A + C))*x + ((b^2*(3*A + 2*C) + a*(3*b*B - a*C))*Sin[c + d*x])/(3*b*d) + ((3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 4, (1/2)*(2*A*b + 2*a*B + b*C)*x + (a*A*ArcTanh[Sin[c + d*x]])/d + ((b*B + a*C)*Sin[c + d*x])/d + (b*C*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 4, (b*B + a*C)*x + ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*C*Sin[c + d*x])/d + (a*A*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 4, b*C*x + ((2*b*B + a*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((A*b + a*B)*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 6, ((A*b + a*B + 2*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a*A + 3*b*B + 3*a*C)*Tan[c + d*x])/(3*d) + ((A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, ((3*a*A + 4*b*B + 4*a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((2*A*b + 2*a*B + 3*b*C)*Tan[c + d*x])/(3*d) + ((3*a*A + 4*b*B + 4*a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 7, ((3*A*b + 3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A + 5*b*B + 5*a*C)*Tan[c + d*x])/(5*d) + ((3*A*b + 3*a*B + 4*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*A*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((4*a*A + 5*b*B + 5*a*C)*Tan[c + d*x]^3)/(15*d)} - - -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 4, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*x + ((20*a*b*B + 5*a^2*(3*A + 2*C) + 2*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((5*A*b^2 + 10*a*b*B + 2*a^2*C + 4*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(5*b*B + 2*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 3, (1/8)*(8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*x + ((4*a^2*b*B + 4*b^3*B - a^3*C + 4*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*b*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, (1/2)*(2*a^2*B + b^2*B + 2*a*b*(2*A + C))*x + (a^2*A*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 6*a*b*B + 2*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, (1/2)*(2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*x + (a*(2*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d - (b*(2*a*A - b*B - 2*a*C)*Sin[c + d*x])/d - (b^2*(2*A - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, b*(b*B + 2*a*C)*x + ((2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(A - 2*C)*Sin[c + d*x])/(2*d) + (a*(A*b + a*B)*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 5, b^2*C*x + ((a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + (a*(2*A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 7, ((8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B + 6*a*b*C)*Tan[c + d*x])/(3*d) + ((2*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A*b + 2*a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 8, ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((20*a*b*B + 5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((2*A*b^2 + 10*a*b*B + a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(2*A*b + 5*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 5, (1/16)*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*x + ((30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A + 2*C) + 6*a*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*d) + ((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((12*a^2*b*B + 4*b^3*B + a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((2*b*B + a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 4, (1/8)*(12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*x + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 4*b^4*(5*A + 4*C) + 4*a^2*b^2*(20*A + 13*C))*Sin[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + a*b^2*(100*A + 71*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((4*b^2*(5*A + 4*C) + 3*a*(5*b*B - a*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*b*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/8)*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*x + (a^3*A*ArcTanh[Sin[c + d*x]])/d + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/2)*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*x + (a^2*(3*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*(9*a*b*B - a^2*(6*A - 8*C) + b^2*(3*A + 2*C))*Sin[c + d*x])/(3*d) - (b^2*(6*a*A - 3*b*B - 5*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, (1/2)*b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*x + (a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(9*a*A*b + 4*a^2*B - 2*b^2*B - 6*a*b*C)*Sin[c + d*x])/(2*d) - (b^2*(4*A*b + 2*a*B - b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + ((3*A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, b^2*(b*B + 3*a*C)*x + ((2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*(5*A*b + 3*a*B - 6*b*C)*Sin[c + d*x])/(6*d) + (a*(3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) + ((A*b + a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 6, b^3*C*x + ((12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*A*b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Tan[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 8, ((3*a^3*B + 12*a*b^2*B + 4*b^3*(A + 2*C) + 3*a^2*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((30*a^2*b*B + 15*b^3*B + 15*a*b^2*(2*A + 3*C) + 2*a^3*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((6*A*b^3 + 15*a^3*B + 50*a*b^2*B + 15*a^2*b*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a*(3*A*b^2 + 15*a*b*B + 2*a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(30*d) + ((3*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 9, ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^3*B + 30*a*b^2*B + 5*b^3*(2*A + 3*C) + 6*a^2*b*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(6*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(10*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} - - -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 6, (1/16)*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*x + ((280*a^3*b*B + 224*a*b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/(105*d) + ((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((91*a^3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^2*(126*A + 103*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(70*d) + ((7*b*B + 4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(42*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 5, (1/16)*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*x + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*Sin[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(30*b*d) + (C*(a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, (1/8)*(8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*x + (a^4*A*ArcTanh[Sin[c + d*x]])/d + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Sin[c + d*x])/(30*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, (1/8)*(32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x + (a^3*(4*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) + (b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, (1/2)*b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*x + (a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(12*a^3*B - 24*a*b^2*B + a^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Sin[c + d*x])/(6*d) - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*A*b + a*B)*(a + b*Cos[c + d*x])^3*Tan[c + d*x])/d + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, (1/2)*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*x + (a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(39*a^2*b*B - 6*b^3*B + 4*a*b^2*(11*A - 6*C) + 4*a^3*(2*A + 3*C))*Sin[c + d*x])/(6*d) - (b^2*(18*a*b*B + 3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) + ((12*A*b^2 + 15*a*b*B + a^2*(4*A + 6*C))*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + ((4*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, b^3*(b*B + 4*a*C)*x + ((8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*(32*a*b*B + 2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x])/(24*d) + (a*(12*A*b^3 + 8*a^3*B + 36*a*b^2*B + a^2*b*(23*A + 36*C))*Tan[c + d*x])/(12*d) + ((4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((A*b + a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 7, b^4*C*x + ((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(30*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((12*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((4*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^7*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 9, ((24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^4*B + 60*a^2*b^2*B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*(4*A*b^3 + 16*a^3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((12*A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(120*d) + ((2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^8*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^4, x, 10, ((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((224*a^3*b*B + 280*a*b^3*B + 35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Tan[c + d*x])/(105*d) + ((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sec[c + d*x]^2*Tan[c + d*x])/(105*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Sec[c + d*x]^3*Tan[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(70*d) + ((4*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(42*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d)} - - -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^3, x, 5, (1/8)*(8*a^4*b*B + 24*a^2*b^3*B + 3*b^5*B - 8*a^5*C - 8*a^3*b^2*C + 9*a*b^4*C)*x + (b*(95*a^3*b*B + 80*a*b^3*B - 83*a^4*C + 32*a^2*b^2*C + 16*b^4*C)*Sin[c + d*x])/(30*d) + (b^2*(130*a^2*b*B + 45*b^3*B - 106*a^3*C + 71*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(120*d) + (b*(35*a*b*B - 23*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + (b*(5*b*B - a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (b*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 4, (1/8)*(8*a^3*b*B + 12*a*b^3*B - 8*a^4*C + 3*b^4*C)*x + (b*(16*a^2*b*B + 4*b^3*B - 13*a^3*C + 8*a*b^2*C)*Sin[c + d*x])/(6*d) + (b^2*(20*a*b*B - 14*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (b*(4*b*B - a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x]), x, 3, (1/2)*(2*a^2*b*B + b^3*B - 2*a^3*C + a*b^2*C)*x + (2*b*(3*a*b*B - 2*a^2*C + b^2*C)*Sin[c + d*x])/(3*d) + (b^2*(3*b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (b*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, -(((8*a^3*b*B + 4*a*b^3*B - 8*a^4*C - 4*a^2*b^2*(2*A + C) - b^4*(4*A + 3*C))*x)/(8*b^5)) - (2*a^3*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) + ((3*a^2*b*B + 2*b^3*B - 3*a^3*C - a*b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*d) + ((4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, ((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*(2*A + C))*x)/(2*b^4) + (2*a^2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, ((b^2*(2*A + C) - 2*a*(b*B - a*C))*x)/(2*b^3) - (2*a*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, ((b*B - a*C)*x)/b^2 + (2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 4, (C*x)/b - (2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d) + (A*ArcTanh[Sin[c + d*x]])/(a*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, (2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^2*d) + (A*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((2*b*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d)) + ((2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((A*b - a*B)*Tan[c + d*x])/(a^2*d) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*b^2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*A*b^3 - a^3*B - 2*a*b^2*B + a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, -((2*b^3*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d)) + ((8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*a^5*d) - ((3*A*b^3 - 2*a^3*B - 3*a*b^2*B + a^2*b*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*d) + ((4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d) + (A*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, ((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*x)/(2*b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 2*a*b^2*(A - C) - 4*a^3*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - a*b^2*(A - 2*C) - 3*a^3*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, ((b*B - 2*a*C)*x)/b^3 - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Sin[c + d*x])/(b^2*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, (C*x)/b^2 + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 5, -((2*(2*a^2*A*b - A*b^3 - a^3*B + a^2*b*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, (2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((2*A*b^2 - a*b*B - a^2*(A - C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, -((2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^4*d) + ((3*A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Tan[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Tan[c + d*x])/(3*a^4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*x)/(2*b^5) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 21*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C))*Cos[c + d*x]^2*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, ((b*B - 3*a*C)*x)/b^4 + ((2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, (C*x)/b^3 + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 3*a*b^4*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, -(((3*a*b*B - a^2*(2*A + C) - b^2*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 6, ((5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (A*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((2*A*b^4 + 3*a^3*b*B - a^4*C - a^2*b^2*(5*A + 2*C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((15*a^2*A*b^4 - 6*A*b^6 + 6*a^5*b*B - 5*a^3*b^3*B + 2*a*b^5*B - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d)) - ((3*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((11*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 2*a*b^3*B - a^4*(2*A - 3*C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 + 4*a^3*b*B - a*b^3*B - 2*a^4*C - a^2*b^2*(6*A + C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -((b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^5*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(21*A - 2*C))*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 8, ((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) + (a*(8*A*b^8 + 8*a^7*b*B - 28*a^5*b^3*B + 35*a^3*b^5*B - 20*a*b^7*B - a^6*b^2*(2*A - 69*C) + 7*a^4*b^4*(A - 12*C) - 8*a^2*b^6*(A - 5*C) - 20*a^8*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((24*a^6*b*B - 68*a^4*b^3*B + 65*a^2*b^5*B - 6*b^7*B - a^5*b^2*(6*A - 167*C) + a^3*b^4*(17*A - 146*C) - 2*a*b^6*(13*A - 12*C) - 60*a^7*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) - ((4*a^5*b*B - 11*a^3*b^3*B + 12*a*b^5*B - a^4*b^2*(A - 27*C) + a^2*b^4*(2*A - 23*C) - b^6*(6*A - C) - 10*a^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((4*A*b^4 + 2*a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, ((b*B - 4*a*C)*x)/b^5 - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, (C*x)/b^4 - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((4*A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, ((a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + a^3*b^2*(2*A - 5*C) + 2*a^5*C + a*b^4*(13*A + 18*C))*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, -(((4*a^2*b*B + b^3*B - a^3*(2*A + C) - a*b^2*(3*A + 4*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((2*a^2*b*B + 3*b^3*B + a^3*C - a*b^2*(5*A + 6*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((2*a^3*b*B + 13*a*b^3*B + a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Sin[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, -(((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + (A*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 8, -(((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d)) - ((4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a^5*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Tan[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 9, (b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + ((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} - - -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^1, x, 3, (b*B - a*C)*x + (b*C*Sin[c + d*x])/d} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 4, C*x + (2*(b*B - 2*a*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 5, (2*(a*b*B - a^2*C - b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 6, ((2*a^2*b*B + b^3*B - 2*a^3*C - 4*a*b^2*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (b*(3*a*b*B - 4*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^5, x, 7, ((2*a^3*b*B + 3*a*b^3*B - 2*a^4*C - 7*a^2*b^2*C - b^4*C)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (b*(5*a*b*B - 7*a^2*C - 3*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - (b*(11*a^2*b*B + 4*b^3*B - 13*a^3*C - 17*a*b^2*C)*Sin[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^(n/2) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, (2*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 6*a^2*b^2*(7*A + 4*C) + 21*b^4*(9*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(24*a^2*b*B + 75*b^3*B - 16*a^3*C - 6*a*b^2*(7*A + 6*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(24*a^2*b*B + 75*b^3*B - 16*a^3*C - 6*a*b^2*(7*A + 6*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^3*d) + (2*(63*A*b^2 - 36*a*b*B + 24*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^3*d) + (2*(3*b*B - 2*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, -((2*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - a*b^2*(35*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - (2*(a^2 - b^2)*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(3*b^2*(5*A + 3*C) + a*(5*b*B - 2*a*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, (2*(3*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - (a^2 - b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 9, -(((A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((a*A + 2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 10, -(((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((3*A*b + 4*a*B + 8*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^2 - 4*a*b*B - 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 11, ((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b^2 - 18*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((A*b^3 + 8*a^3*B - 2*a*b^2*B + 4*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, (2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 8, (2*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*b*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, (2*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(5*a^2*b*B - 5*b^3*B + 3*a^3*C - 3*a*b^2*(5*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, -(((3*a*A - 6*b*B - 8*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((6*a*b*B + a^2*(3*A - 2*C) + 2*b^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(3*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A - 2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 10, -(((5*A*b + 4*a*B - 8*b*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((4*a^2*B + 8*b^2*B + a*b*(7*A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 12*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b + 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 11, -(((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((42*a*b*B + 8*a^2*(2*A + 3*C) + b^2*(17*A + 48*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b^3 - 8*a^3*B - 6*a*b^2*B - 12*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a*d) + ((A*b + 2*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 12, ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((3*A*b^3 - 128*a^3*B - 136*a*b^2*B - 12*a^2*b*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(64*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, (2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(45045*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*b^3*d) + (2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45045*b^3*d) + (2*(104*a^2*b*B + 1053*b^3*B - 48*a^3*C - 2*a*b^2*(143*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9009*b^3*d) + (2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 121*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(1287*b^3*d) + (2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(143*b^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 10, -((2*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 15*a^3*b^2*(33*A + 17*C) - 15*a*b^4*(319*A + 247*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(a^2 - b^2)*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3465*b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 5*a*b^2*(99*A + 67*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 - 22*a*b*B + 8*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(99*b^2*d) + (2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 9, (2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(161*A + 93*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(315*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b*d) + (2*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, (2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*b*B - 56*a*b^3*B - 10*a^2*b^2*(7*A - C) + 15*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a^3*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, ((70*a*b*B - a^2*(15*A - 46*C) + 6*b^2*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((20*a^2*b*B + 10*b^3*B + a^3*(15*A - 16*C) + 4*a*b^2*(15*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (a^2*(5*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (b*(15*a*A - 10*b*B - 16*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((12*a^3*B + 48*a*b^2*B + 8*b^3*(3*A + C) + a^2*b*(33*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*d*Sqrt[a + b*Cos[c + d*x]]) + (a*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(21*A*b + 12*a*B - 8*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d) + ((5*A*b + 4*a*B)*(a + b*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(4*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 11, -(((54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((66*a^2*b*B + 48*b^3*B + 8*a^3*(2*A + 3*C) + a*b^2*(59*A + 96*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*d*Sqrt[a + b*Cos[c + d*x]]) + ((5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 20*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*d) + ((5*A*b + 6*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 12, -(((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((128*a^3*B + 472*a*b^2*B + 4*a^2*b*(89*A + 132*C) + b^3*(133*A + 384*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^4 - 160*a^3*b*B - 40*a*b^3*B - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a*d) + ((5*A*b^2 + 24*a*b*B + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + ((5*A*b + 8*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(5/2), x, 13, ((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(1920*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((15*A*b^4 - 3560*a^3*b*B - 1330*a*b^3*B - 256*a^4*(4*A + 5*C) - 4*a^2*b^2*(809*A + 1180*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(1920*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(128*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(1920*a^2*d) + ((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(240*d) + ((A*b + 2*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)} - - -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(3/2), x, 9, (2*(161*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*b*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(20*a*b*B - 17*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(5*b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(5*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 8, (2*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 2*a*b^2*(35*A + 22*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(56*a^3*b*B + 49*a*b^3*B - 48*a^4*C - 5*b^4*(7*A + 5*C) - 2*a^2*b^2*(35*A + 16*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(105*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*b^2*d) + (2*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - a*b^2*(15*A + 7*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^2*d) + (2*C*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 6, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 8, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 9, -((A*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - ((A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(a*d)} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 10, ((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*A*b^2 - 4*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 4*a*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 11, -(((15*A*b^2 - 18*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + ((5*A*b^2 - 6*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(24*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^3 - 8*a^3*B - 6*a*b^2*B + 4*a^2*b*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(8*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 - 18*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(24*a^3*d) - ((5*A*b - 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(12*a^2*d) + (A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)} - - -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 8, -((2*(40*a^3*b*B - 25*a*b^3*B - 6*a^2*b^2*(5*A - 4*C) - 48*a^4*C + 3*b^4*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 6*a*b^2*(5*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 3*a*b^2*(5*A - 3*C) - 24*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(6*a^2*b*B - 3*b^3*B - a*b^2*(3*A - 5*C) - 8*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 - 6*a*b*B + 8*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d)} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 9, -((2*(A*b^2 - a*(b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*b*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 10, ((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]) - ((3*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 11, -(((15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*(a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - ((5*A*b - 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((15*A*b^2 - 12*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^3*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*Sqrt[a + b*Cos[c + d*x]])} - - -{Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 9, -((2*(80*a^5*b*B - 140*a^3*b^3*B + 40*a*b^5*B - 4*a^4*b^2*(10*A - 53*C) + 5*a^2*b^4*(15*A - 11*C) - 128*a^6*C - 3*b^6*(5*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(80*a^4*b*B - 80*a^2*b^3*B - 5*b^5*B - 4*a^3*b^2*(10*A - 29*C) - 128*a^5*C + a*b^4*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^5*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(6*A*b^4 + 5*a^3*b*B - 9*a*b^3*B - 2*a^2*b^2*(A - 6*C) - 8*a^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(40*a^4*b*B - 65*a^2*b^3*B + 5*b^5*B - 2*a^3*b^2*(10*A - 49*C) + 2*a*b^4*(20*A - 7*C) - 64*a^5*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*b*B - 50*a*b^3*B - a^2*b^2*(15*A - 71*C) + b^4*(35*A - 3*C) - 48*a^4*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*b^3*(a^2 - b^2)^2*d)} -{Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 8, (2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 2*a^3*b^2*(A - 14*C) + 2*a*b^4*(3*A - 4*C) - 16*a^5*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(8*a^3*b*B - 9*a*b^3*B - 2*a^2*b^2*(A - 8*C) - 16*a^4*C + b^4*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*a*(4*A*b^4 + a*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C))*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Cos[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(2*a^2*b*B - 3*b^3*B - 8*a^3*C + a*b^2*(A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*a*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Cos[c + d*x]^0*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) - (2*(A*b^2 - a*b*B - 2*a^2*C + 3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 10, (2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*b*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*A*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 11, ((26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b - 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(a^3*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (A*Tan[c + d*x])/(a*d*(a + b*Cos[c + d*x])^(3/2))} -{Sec[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 12, ((105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((35*A*b^2 - 20*a*b*B + 4*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^4*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*(a + b*Cos[c + d*x])^(3/2)) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^(3/2))} - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2), x, 8, -((2*(3*a^3*b*B + 29*a*b^3*B + 2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])) + (2*(3*a^2*b*B + 5*b^3*B + 2*a^3*C - 2*a*b^2*(4*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(15*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(5*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) + (2*(3*a^2*b*B + 5*b^3*B + 2*a^3*C - 2*a*b^2*(4*A + 5*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*a^3*b*B + 29*a*b^3*B + 2*a^4*C - 3*b^4*(3*A + 5*C) - a^2*b^2*(23*A + 19*C))*Sin[c + d*x])/(15*b*(a^2 - b^2)^3*d*Sqrt[a + b*Cos[c + d*x]])} - - -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/2), x, 7, (2*(3*b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*C*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2), x, 7, (2*(b*B - 2*a*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/((a^2 - b^2)*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*(b*B - 2*a*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2), x, 8, (2*(4*a*b*B - 5*a^2*C - 3*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(b*B - 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*b*(b*B - 2*a*C)*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*b*(4*a*b*B - 5*a^2*C - 3*b^2*C)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(m/2) (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (2*(9*A*b + 9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*b*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 6, (2*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (2*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(b*B + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 5, (2*(b*B - a*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 5, (-2*(A*b + a*B - b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*b*B + a*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, (-2*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-2*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(3*A*b + 3*a*B + 5*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(11*A*b^2 + 22*a*b*B + 4*a^2*C + 9*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(77*d) + (2*b*(11*b*B + 4*a*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 7, (2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 4*a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 6, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 4*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a*A - b*B - 2*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(5*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (-2*(a^2*B - b^2*B + 2*a*b*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(4*A*b + 3*a*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 6, (-2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(4*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(4*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, (-2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(4*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*d*Cos[c + d*x]^(5/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, (2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(11*b*B + 6*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*b*B - 6*a^2*(7*A - 3*C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b^2*(35*a*A - 7*b*B - 11*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (-2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(6*a^2*B - b^2*B + 3*a*b*(5*A - C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(35*A*b + 15*a*B - 3*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (-2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(24*A*b^2 + 35*a*b*B + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 7, (-2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(24*A*b^3 + 21*a^3*B + 98*a*b^2*B + 21*a^2*b*(3*A + 5*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (2*(6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, (-2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, (2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(6435*d) + (2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d) + (2*(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d)} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, (2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, (2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(117*a^2*b*B + 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b^2*(162*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) - (2*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, (-2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b*(42*a^3*B - 28*a*b^2*B + 3*a^2*b*(49*A - 13*C) - b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b^2*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b*(21*A*b + 7*a*B - b*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 8, (-2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) - (2*b*(105*a^2*b*B - 5*b^3*B + 4*a*b^2*(33*A - 5*C) + 6*a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(50*a*b*B + b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(16*A*b^2 + 15*a*b*B + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (-2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(192*A*b^3 + 63*a^3*B + 413*a*b^2*B + a^2*(202*A*b + 350*b*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 8, (-2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(192*A*b^4 + 756*a^3*b*B + 1098*a*b^3*B + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2), x, 9, (-2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2])/(231*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2)) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)) + (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (2*(8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, (2*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a*b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*b^5*d) - (2*a^3*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^5*(a + b)*d) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*b^3*d) + (2*(b*B - a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*d) + (2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d)} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d) + (2*(3*a^2*b*B + b^3*B - 3*a^3*C - a*b^2*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^4*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b*d)} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, (2*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d) + (2*(b^2*(3*A + C) - 3*a*(b*B - a*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 5, (2*C*EllipticE[(1/2)*(c + d*x), 2])/(b*d) + (2*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d)} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 6, -((2*A*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*b*(a + b)*d) + (2*A*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 7, (2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 8, -((2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d)) - (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (2*b*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sin[c + d*x])/(5*a^3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]), x, 9, (2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*a^3*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Cos[c + d*x]^(3/2)) - (2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, -(((25*a^3*b*B - 20*a*b^3*B - 3*a^2*b^2*(5*A - 8*C) - 35*a^4*C + 2*b^4*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*(a^2 - b^2)*d)) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - a^3*b^2*(9*A - 20*C) - 21*a^5*C + 4*a*b^4*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 + 5*a^3*b*B - 7*a*b^3*B - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^5*(a + b)^2*d) + ((5*a^2*b*B - 2*b^3*B - a*b^2*(3*A - 4*C) - 7*a^3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - a^2*b^2*(3*A - 16*C) - 15*a^4*C + 2*b^4*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + a*b^2*(A + 4*C))*EllipticF[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 6, -(((A*b^2 - a*(b*B - a*C))*EllipticE[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 7, ((3*A*b^2 - a*b*B - a^2*(2*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*EllipticF[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 8, -(((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -{Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 9, -(((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(20*b^5*(a^2 - b^2)^2*d)) + ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C) - 189*a^7*C - 24*a*b^6*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^6*(a + b)^3*d) + ((35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - a^4*b^2*(9*A - 223*C) + a^2*b^4*(15*A - 128*C) - 105*a^6*C - 8*b^6*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, -(((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C - a*b^4*(7*A + 24*C) + a^3*b^2*(A + 33*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-1/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 7, ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 8, -(((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d)) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x]))} -{Cos[c + d*x]^(-5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3, x, 9, ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(m/2) (a+b Cos[e+f x])^(n/2) (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 8, -((a - b)*Sqrt[a + b]*(8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d) + (Sqrt[a + b]*(24*A*b^2 + (a + 2*b)*(6*b*B - 3*a*C + 8*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d) + ((8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, -((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(8*A*b + a*C + 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, ((a - b)*Sqrt[a + b]*(2*A - C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (Sqrt[a + b]*(2*A*b - a*(2*A - 2*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (Sqrt[a + b]*(2*b*B + a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 6, (2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) - (2*Sqrt[a + b]*(b*(A - 3*B) - a*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 5, (-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) - (2*(a - b)*Sqrt[a + b]*(2*A*b + a*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 2*a*b*(3*A - 7*B) + a^2*(25*A - 63*B + 35*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Cos[c + d*x]^(3/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 9, -((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4*a*b^2*(60*A + 28*B + 39*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + ((4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d) + ((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d) + (Sqrt[a + b]*(3*a^2*C + 4*b^2*(6*A + 3*B + 4*C) + 2*a*b*(24*A + 15*B + 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + ((2*b*B + a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(8*a*A - 4*b*B - 5*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d) - (Sqrt[a + b]*(a*(8*A - 8*B - 5*C) - 2*b*(8*A + 2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - (Sqrt[a + b]*(8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - ((8*a*A - 4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, ((a - b)*Sqrt[a + b]*(8*A*b + 6*a*B - 3*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A - 3*B + 3*C) - a*b*(8*A - 3*(4*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d) - (Sqrt[a + b]*(2*b*B + 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - ((8*A*b + 6*a*B - 3*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 7, (2*(a - b)*Sqrt[a + b]*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d) - (2*Sqrt[a + b]*(3*b^2*(A - 5*B) - 2*a*b*(6*A - 10*B + 15*C) + a^2*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (2*b*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 6, (-2*(a - b)*Sqrt[a + b]*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B + 35*C) + 3*a*b*(19*A - 7*B + 35*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A - 25*B + 63*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x, 10, -((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(260*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d) + ((10*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 9, -((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(12*A + 16*B + 9*C) + 2*a^2*b*(192*A + 132*B + 59*C) + 4*a*b^2*(108*A + 52*B + 71*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 9, -((a - b)*Sqrt[a + b]*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d) - (Sqrt[a + b]*(a^2*(48*A - 48*B - 33*C) - 4*b^2*(6*A + 3*B + 4*C) - 2*a*b*(72*A + 27*B + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d) + ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) - (b*(8*a*A - 2*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) - (b*(6*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 9, ((a - b)*Sqrt[a + b]*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*a*d) - (Sqrt[a + b]*(a*b*(56*A - 72*B - 27*C) - 6*b^2*(12*A + 2*B + C) - 8*a^2*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d) - (Sqrt[a + b]*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d) - ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b*(8*A*b + 4*a*B - b*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2), x, 9, ((a - b)*Sqrt[a + b]*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) + (Sqrt[a + b]*(30*A*b^3 - 2*a^3*(9*A - 5*B + 15*C) + 2*a^2*b*(17*A - 35*B + 45*C) - a*b^2*(46*A - 15*(6*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(5*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(A*b + a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2), x, 8, (2*(a - b)*Sqrt[a + b]*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d) - (2*Sqrt[a + b]*(15*b^3*(A - 7*B) - a^3*(25*A - 63*B + 35*C) + a^2*b*(145*A - 119*B + 245*C) - a*b^2*(135*A - 161*B + 315*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d) - (2*b^2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 7*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2), x, 7, (-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(315*a*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d) + (Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 12*b^2*B + 15*a^2*C - 10*a*b*C + 16*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*(2*A + C))*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(24*b^3*d*Sqrt[Cos[c + d*x]]) + ((6*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d) + (C*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 7, -((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) - (Sqrt[a + b]*(8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]]) + (C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]), x, 6, -(((a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d)) + (Sqrt[a + b]*(2*A*b + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d) - (Sqrt[a + b]*(2*b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]), x, 5, (2*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) - (2*Sqrt[a + b]*(A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]), x, 4, (-2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]), x, 5, (2*(a - b)*Sqrt[a + b]*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d) - (2*Sqrt[a + b]*(8*A*b^2 - 2*a*b*(A + 5*B) + a^2*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]), x, 6, (-2*(a - b)*Sqrt[a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d) + (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2*(3*A + 14*B) + a^3*(25*A - 63*B + 35*C) + 2*a^2*b*(22*A + 7*(B + 5*C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Cos[c + d*x]^(3/2))} - -{(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(4*A*b + a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d) + (Sqrt[a + b]*(4*A*b + (a + 2*b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d) - (Sqrt[a + b]*(4*a*A*b - a^2*B + 4*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d) + ((4*A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d)} -{(a + a*Cos[c + d*x] + 2*b*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]), x, 4, (-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (4*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d + (2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 8, -((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d) - ((8*A*b^2 - a*b*(12*B - 5*C) + 15*a^2*C - 2*b^2*(2*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2), x, 7, -(((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d)) + ((2*A*b^2 - a*(b*(2*B - C) - 3*a*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)), x, 6, (2*(A*b^2 - a*(b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d) + (2*(A*b + b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)), x, 4, -((2*(2*A*b^2 - a*b*B - a^2*(A - C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d)) - (2*(2*A*b + a*(A - B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)), x, 5, (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d) + (2*(8*A*b^2 + 6*a*b*(A - B) + a^2*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)), x, 6, -((2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^5*Sqrt[a + b]*d)) - (2*(48*A*b^3 + 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*Sqrt[a + b]*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2))} - - -{(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 8, ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d) - ((6*A*b^4 - a*b^3*(2*A + 3*(4*B - C)) + a^3*b*(6*B - 5*C) - 15*a^4*C + a^2*b^2*(2*B + 21*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(2*b*B - 5*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d) - (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d)) - (2*(b^3*(A + 3*B) + 3*a^3*C + a^2*b*C - a*b^2*(3*A + B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)), x, 5, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d) - (2*(2*A*b^2 - a^2*(3*A + 3*B + C) + a*b*(3*A + B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)), x, 5, (2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(8*A*b^3 + 2*a*b^2*(3*A - B) - 3*a^3*(A - B - C) - a^2*b*(9*A + 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)), x, 6, -((2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d)) - (2*(16*A*b^4 + 4*a*b^3*(3*A - 2*B) - 3*a^3*b*(3*A - 3*B - C) - 2*a^2*b^2*(8*A + 3*B - C) - a^4*(A - 3*B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) with m and/or n symbolic*) - - -{Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^2, x, 6, If[$VersionNumber>=8, ((2*a^2*C + b^2*C*(3 + m) + A*b^2*(4 + m) + 2*a*b*B*(4 + m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (b*(2*a*C + b*B*(4 + m))*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((2*a*b*B*(4 + 5*m + m^2) + a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]) - ((b^2*B*(2 + m) + a^2*B*(3 + m) + 2*a*b*(C*(2 + m) + A*(3 + m)))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]), ((2*a^2*C + b^2*C*(3 + m) + A*b^2*(4 + m) + 2*a*b*B*(4 + m))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)*(4 + m)) + (b*(2*a*C + b*B*(4 + m))*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (C*Cos[c + d*x]^(1 + m)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(d*(4 + m)) - ((2*a*b*B*(4 + 5*m + m^2) + a^2*(4 + m)*(C*(1 + m) + A*(2 + m)) + b^2*(1 + m)*(C*(3 + m) + A*(4 + m)))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(4 + m)*(2 + 3*m + m^2)*Sqrt[Sin[c + d*x]^2]) - ((b^2*B*(2 + m) + a^2*B*(3 + m) + 2*a*b*(C*(2 + m) + A*(3 + m)))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])]} -{Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(a + b*Cos[c + d*x])^1, x, 5, ((b*B + a*C)*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + m)) + (b*C*Cos[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(3 + m)) - (((b*B + a*C)*(1 + m) + a*A*(2 + m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 + m)*(2 + m)*Sqrt[Sin[c + d*x]^2]) - ((b*C*(2 + m) + A*b*(3 + m) + a*B*(3 + m))*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(2 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^1, x, 8, (a*(A*b^2 - a*(b*B - a*C))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*AppellF1[1/2, -(m/2), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*(b*(a^2 - b^2)*d)) - ((b*B - a*C)*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) - (C*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2, x, 9, ((A*b^4*m + a^3*b*B*m - a*b^3*B*(1 + m) - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, (1 - m)/2, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^(-1 + m)*(Cos[c + d*x]^2)^((1 - m)/2)*Sin[c + d*x])/(b^2*(a^2 - b^2)^2*d) - ((A*b^4*m + a^3*b*B*m - a*b^3*B*(1 + m) - a^4*C*(1 + m) + a^2*b^2*(A - A*m + C*(2 + m)))*AppellF1[1/2, -(m/2), 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^2 - b^2))]*Cos[c + d*x]^m*Sin[c + d*x])/((Cos[c + d*x]^2)^(m/2)*(a*b*(a^2 - b^2)^2*d)) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(1 + m)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) + ((a*b*B*m - a^2*C*(1 + m) + b^2*(C - A*m))*Cos[c + d*x]^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*(1 + m)*Sqrt[Sin[c + d*x]^2]) + ((A*b^2 - a*(b*B - a*C))*(1 + m)*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(a*b*(a^2 - b^2)*d*(2 + m)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a Cos[e+f x])^n (A+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, -((2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, -((2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (2*a*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 10, (-16*a^2*(2*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (16*a^2*(2*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-4*a^2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (-16*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*A*(a^2 + a^2*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (16*a^2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (16*a^2*(3*A + 2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, (4*a^2*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a^2*(33*A + 25*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 11, (-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(105*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(35*A + 44*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 10, (-4*a^3*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a^3*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(73*A + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(53*A + 70*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (12*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 9, (-4*a^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(11*A + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, (-4*a^3*(5*A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (8*a^3*(10*A - 3*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 3*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, (4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*(7*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, (4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, (4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a^3*(44*A + 35*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Sec[c + d*x]^(3/2)) + (2*(33*A + 35*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 11, (4*a^3*(221*A + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (40*a^3*(143*A + 118*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (12*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(5/2)) + (2*(143*A + 145*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]), x, 8, (-3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]), x, 7, ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]), x, 6, -(((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]), x, 5, ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 6, -(((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 7, (3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 8, (-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2, x, 8, ((7*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*(5*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((7*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2, x, 7, (-4*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2, x, 6, ((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 6, (4*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 7, -(((A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A + 7*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(3/2)) + (2*(A + 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 8, (4*(5*A + 14*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A + 3*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (4*(5*A + 14*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3, x, 9, ((119*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((119*A + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((11*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3, x, 8, -((49*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3, x, 7, ((9*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - (2*(3*A - 2*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A - C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 7, ((A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + (2*(2*A - 3*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - 9*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -((A - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A - 13*C)*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 8, -((9*A + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) - (2*C*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A + 119*C)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((A + 11*C)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 9, (7*(7*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - (2*(A + 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A + 63*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (16*a*(16*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 5, (4*a*(24*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 4, (2*a*(8*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 5, (2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 5, (Sqrt[a]*(8*A + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 6, (Sqrt[a]*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 7, (Sqrt[a]*(48*A + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*C*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 7, (16*a^2*(112*A + 143*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(28*A + 33*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (4*a^2*(136*A + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 5, (2*a^2*(104*A + 175*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (6*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (3*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (a^(3/2)*(8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (a^(3/2)*(24*A + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (a^(3/2)*(112*A + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(16*A + 13*C)*Sin[c + d*x])/(32*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (a^(3/2)*(176*A + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (3*a*C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2), x, 8, (16*a^3*(8368*A + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 7, (4*a^3*(568*A + 759*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(232*A + 297*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (2*a^3*(584*A + 903*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8*A + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (5*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (a^(5/2)*(8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (5*a^(5/2)*(8*A + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 7, (a^(5/2)*(304*A + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (5*a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (a^(5/2)*(400*A + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(24*A + 23*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sec[c + d*x]^(5/2)) + (a*C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]], x, 9, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]], x, 8, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]], x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 7, -((C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]), x, 8, ((8*A + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - (C*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 9, -(((8*A + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) - (C*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2), x, 9, ((19*A + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2), x, 8, -((15*A + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((49*A + 25*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((13*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2), x, 7, ((11*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2), x, 6, -((7*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2), x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]), x, 8, -((3*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d)) + ((A + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 9, ((8*A + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2), x, 9, -((283*A + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((2671*A + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2), x, 8, ((163*A + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2), x, 7, (-5*(15*A - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2), x, 6, ((19*A + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]), x, 8, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 9, -((5*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d)) + ((3*A + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 10, ((8*A + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, -((2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (2*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 6, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 7, (6*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 8, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*C*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 5, (2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]], x, 6, (2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 7, (6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, -((2*a*(3*A + 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(3*A + 3*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*(B + C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, -((2*a*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*(B + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, -((2*a*(A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*(A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, -((2*a*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*a*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(B + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(5*A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(B + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 7*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (2*a*(9*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(B + C)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(9*A + 9*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*(B + C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 10, (-4*a^2*(8*A + 9*B + 12*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(8*A + 9*B + 12*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*(4*A + 9*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-4*a^2*(3*A + 4*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(3*A + 4*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(33*A + 49*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(4*A + 7*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (-4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(4*A + 5*B)*(a^2 + a^2*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(5*A + 3*B - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(4*A + 3*B)*(a^2 + a^2*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (4*a^2*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (4*a^2*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*(7*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (4*a^2*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (2*(9*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(3/2)) + (4*a^2*(7*A + 6*B + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, (4*a^2*(9*A + 8*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(99*A + 121*B + 89*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(11*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(5/2)) + (4*a^2*(9*A + 8*B + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(66*A + 55*B + 50*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 11, (-4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*A + 17*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*(6*A + 11*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 10, (-4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(17*A + 21*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(32*A + 41*B + 42*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-4*a^3*(7*A + 9*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(6*A + 7*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 9, (-4*a^3*(9*A + 5*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(33*A + 35*B + 15*C)*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A + 5*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, (-4*a^3*(5*A - 5*B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*(35*A + 15*B - 3*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(2*A + B)*(a^2 + a^2*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, (4*a^3*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*(7*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d*Sqrt[Sec[c + d*x]]) - (2*(35*A - 7*B - 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, (4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*(3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 99*B + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, (4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*(11*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d*Sec[c + d*x]^(3/2)) + (2*(99*A + 143*B + 105*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d*Sec[c + d*x]^(3/2)) + (4*a^3*(143*A + 121*B + 105*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} -{((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 11, (4*a^3*(221*A + 195*B + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(286*A + 273*B + 236*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*(13*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(5/2)) + (2*(143*A + 195*B + 145*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(5/2)) + (4*a^3*(221*A + 195*B + 175*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(121*A + 105*B + 95*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]), x, 8, (-3*(7*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(7*A - 5*B + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((5*A - 5*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]), x, 7, ((3*A - 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((3*A - 3*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((5*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]), x, 6, -(((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) - ((A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]), x, 5, ((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 6, -(((A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 3*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 7, (3*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((3*A - 5*B + 5*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 8, (-3*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + ((7*A - 7*B + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A - 7*B + 7*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A - 7*B + 9*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2, x, 8, ((7*A - 4*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((7*A - 4*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((10*A - 5*B + 2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2, x, 7, -(((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - ((5*A - 2*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2, x, 6, ((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 6, -(((B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 7, -(((A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((A - 4*B + 7*C)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((2*A - 5*B + 10*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 8, ((20*A - 35*B + 56*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((A - 2*B + 3*C)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(A - 2*B + 3*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3, x, 9, ((119*A - 49*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((119*A - 49*B + 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3, x, 8, -((49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((49*A - 9*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3, x, 7, ((9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]) - ((6*A - B - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((9*A + B - C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 7, ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)) + ((4*A + B - 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A - B - 9*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 7, -((A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((A + 3*B - 13*C)*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 8, -((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)) + ((B - 2*C)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A - 13*B + 33*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 9, (7*(7*A - 17*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - ((2*A - 7*B + 12*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A - 17*B + 33*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A - 33*B + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+a Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (16*a*(16*A + 18*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 5, (4*a*(24*A + 28*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 4, (2*a*(8*A + 10*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 5, (2*Sqrt[a]*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 5, (Sqrt[a]*(2*B + C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 5, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*(4*B + C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 6, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*(6*B + C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)) + (a*(8*A + 6*B + 5*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 7, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*(8*B + C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (C*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(5/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 7, (16*a^2*(336*A + 374*B + 429*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(84*A + 110*B + 99*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (4*a^2*(136*A + 156*B + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(52*A + 72*B + 63*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 5, (2*a^2*(104*A + 126*B + 175*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(4*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 7*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (2*a^(3/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (a^(3/2)*(2*B + 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^2*(8*A + 6*B - 3*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*(A + B)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (a^(3/2)*(8*A + 12*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^2*(8*A - 4*B - 5*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*(4*A - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 6, (a^(3/2)*(24*A + 14*B + 11*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*(2*B + C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 7, (a^(3/2)*(112*A + 88*B + 75*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(48*A + 56*B + 39*C)*Sin[c + d*x])/(96*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a*(8*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(24*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)) + (a^2*(112*A + 88*B + 75*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 8, (a^(3/2)*(176*A + 150*B + 133*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a*(10*B + 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(40*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(5/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2), x, 8, (16*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(45045*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2224*A + 2522*B + 2717*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(1287*d) + (2*a*(5*A + 13*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(143*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(13/2)*Sin[c + d*x])/(13*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 7, (4*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(1160*A + 1364*B + 1485*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*a*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 6, (2*a^3*(584*A + 690*B + 903*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8*A + 10*B + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(64*A + 90*B + 63*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*a*(5*A + 9*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (2*a^(5/2)*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (a^(5/2)*(2*B + 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d - (a^3*(64*A + 70*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*(8*A + 10*B + 5*C)*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (a^(5/2)*(8*A + 20*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) - (a^3*(56*A + 12*B - 27*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A + 4*B - C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 3*B)*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (a^(5/2)*(40*A + 38*B + 25*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) - (a^3*(24*A - 54*B - 49*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*(8*A - 2*B - 3*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (a*(6*A - C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2), x, 7, (a^(5/2)*(304*A + 200*B + 163*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (a*(8*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 8, (a^(5/2)*(400*A + 326*B + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) + (a*(2*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2), x, 9, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(680*A + 628*B + 545*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + (a^2*(120*A + 156*B + 115*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(480*d*Sec[c + d*x]^(5/2)) + (a*(12*B + 5*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(60*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*x]^(5/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2)/Sqrt[a + a*Cos[c + d*x]], x, 9, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(19*A - 3*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2)/Sqrt[a + a*Cos[c + d*x]], x, 8, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]], x, 7, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]], x, 6, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]], x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/Sqrt[a + a*Cos[c + d*x]], x, 7, ((2*B - C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(1/2)), x, 8, ((8*A - 4*B + 7*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*B - C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 9, -(((8*A - 14*B + 9*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) + ((6*B - C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((8*A - 2*B + 7*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - -{((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]], x, 7, ((2*A*b + 2*a*B - b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (b*B*Sin[c + d*x])/(d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(3/2), x, 9, ((19*A - 15*B + 11*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((1201*A - 1029*B + 665*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) + ((397*A - 273*B + 245*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(210*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((67*A - 63*B + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(70*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((11*A - 7*B + 7*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(14*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(3/2), x, 8, -(((15*A - 11*B + 7*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) + ((147*A - 95*B + 75*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((39*A - 35*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2), x, 7, ((11*A - 7*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((19*A - 15*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2), x, 6, -(((7*A - 3*B - C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)) + ((5*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(3/2), x, 7, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(1/2)), x, 8, ((2*B - 3*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((A - B + 3*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 9, ((8*A - 12*B + 19*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((A - B + 2*C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((2*A - 6*B + 7*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2), x, 9, -(((283*A - 163*B + 75*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((787*A - 475*B + 195*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((157*A - 85*B + 45*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2), x, 8, ((163*A - 75*B + 19*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((299*A - 147*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]]) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2), x, 7, -(((75*A - 19*B - 5*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(1/2)/(a + a*Cos[c + d*x])^(5/2), x, 6, ((19*A + 5*B + 3*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) - ((9*A - B - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 8, (2*C*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)) + ((5*A + 3*B - 11*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 9, ((2*B - 5*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((A + 7*B - 15*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((3*A - 11*B + 35*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)), x, 10, ((8*A - 20*B + 39*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(5/2)*d) - ((43*A - 115*B + 219*C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)) - ((3*A - 11*B + 19*C)*Sin[c + d*x])/(16*a*d*(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)) + ((7*A - 15*B + 31*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) - ((11*A - 35*B + 63*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Cos[e+f x])^n (A+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*b*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*A*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*b*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (-2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (2*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a*b*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a*b*(5*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (8*a*A*b*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-4*a*b*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a*b*(3*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(4*A*b^2 + a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (8*a*A*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-2*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(4*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (-4*a*b*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (-2*(5*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(5*A - C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) - (4*a*b*(3*A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 7, (4*a*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*C*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(4*a^2*C + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (2*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*C*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(4*a^2*C + b^2*(9*A + 7*C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (4*a*b*(7*A + 5*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (4*a*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a*b*C*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(4*a^2*C + b^2*(11*A + 9*C))*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (4*a*b*(9*A + 7*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (6*b*(8*A*b^2 + 7*a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (12*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (-2*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^3*(9*A - 5*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(8*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*A*b*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-2*b*(15*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^3*(35*A - 3*C)*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*a*b^2*(5*A - C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (-2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*a*b^2*(35*A - 11*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(7*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (2*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(24*a^2*C + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*a*(63*A*b^2 + 8*a^2*C + 45*b^2*C)*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (4*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (2*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C)*Sin[c + d*x])/(165*d*Sec[c + d*x]^(3/2)) + (4*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(33*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, (2*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*a*(11*a^2*(7*A + 5*C) + 15*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(24*a^2*C + 11*b^2*(13*A + 11*C))*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (6*a*(143*A*b^2 + 8*a^2*C + 117*b^2*C)*Sin[c + d*x])/(1001*d*Sec[c + d*x]^(5/2)) + (12*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (2*a*(11*a^2*(7*A + 5*C) + 15*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 10, (-8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(64*A*b^4 + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (4*a*b*(96*A*b^2 + a^2*(673*A + 891*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(192*A*b^4 + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (4*a*b*(32*A*b^2 + a^2*(101*A + 147*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-8*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (4*a*b*(96*A*b^2 + a^2*(101*A + 175*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 9, (-2*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (4*a*b*(2*b^2*(33*A - 5*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (16*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, (-8*a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a*b^3*(175*A - 27*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) - (2*b^2*(3*a^2*(49*A - 13*C) - b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(21*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (16*A*b*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, (-2*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(3*a^2*(105*A - 41*C) - 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) - (4*a*b*(a^2*(63*A - 31*C) - 6*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*a*b*(21*A - 5*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, (8*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a*b*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(16*a^2*C + 3*b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (16*a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, (2*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a*b*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(48*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(3/2)) + (16*a*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (8*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]), x, 9, (-2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) - (2*A*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]), x, 8, (2*A*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*A*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]), x, 7, (-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]), x, 6, (2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 7, (-2*a*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(3*a^2*C + b^2*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 8, (2*(5*a^2*C + b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) - (2*a*(3*A*b^2 + (3*a^2 + b^2)*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^4*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Sec[c + d*x]^(3/2)) - (2*a*C*Sin[c + d*x])/(3*b^2*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 9, (-2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*d) + (2*(21*a^4*C + 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^5*(a + b)*d) + (2*C*Sin[c + d*x])/(7*b*d*Sec[c + d*x]^(5/2)) - (2*a*C*Sin[c + d*x])/(5*b^2*d*Sec[c + d*x]^(3/2)) + (2*(7*a^2*C + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*b^3*d*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2, x, 9, -((b*(5*A*b^2 - a^2*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b*(5*A*b^2 - a^2*(4*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((5*A*b^2 - a^2*(2*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2, x, 8, ((3*A*b^2 - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a^2*(2*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2, x, 7, -(((A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 7, ((A*b^2 + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((A*b^4 - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 8, -((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) + ((a^2*b^2*(3*A - 16*C) + 15*a^4*C - 2*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 9, ((3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*(a^2 - b^2)*d) - (a*(a^2*b^2*(9*A - 20*C) + 21*a^4*C - 4*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3, x, 10, (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(35*A*b^4 + 3*a^4*(8*A - 3*C) - a^2*b^2*(65*A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 - 5*a^4*C - a^2*b^2*(13*A + C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3, x, 9, -((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3, x, 8, ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((3*A*b^4 - a^4*C - a^2*b^2*(9*A + 5*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 8, ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A*b^4 - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 8, -((b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*C + b^4*(7*A + 24*C) - a^2*b^2*(A + 33*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((3*A*b^4 - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 9, -(a*(a^2*b^2*(3*A - 65*C) - 3*b^4*(3*A - 8*C) + 35*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((a^4*b^2*(9*A - 223*C) - a^2*b^4*(15*A - 128*C) + 105*a^6*C + 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + ((5*A*b^4 - 7*a^4*C + a^2*b^2*(A + 13*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((a^2*b^2*(3*A - 61*C) - b^4*(21*A - 8*C) + 35*a^4*C)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 10, -((a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d) + (a*(a^2*b^4*(33*A - 64*C) - 3*a^4*b^2*(5*A - 43*C) - 63*a^6*C - 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) + ((7*A*b^4 - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((a^2*b^2*(15*A - 101*C) - b^4*(45*A - 8*C) + 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + (a*(b^4*(11*A - 8*C) - 5*a^2*b^2*(A - 7*C) - 21*a^4*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (-2*(a - b)*Sqrt[a + b]*(16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^5*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(12*a*A*b^2 + 16*A*b^3 + 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*b*(8*A*b^2 + a^2*(13*A + 21*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a^2*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*b*Sqrt[a + b]*(8*A*b^2 + a^2*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b + 8*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^2 - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(9*a*A + 2*A*b + 15*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (2*A*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A*b - a*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(2*A - C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*a*A - 2*A*b - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*A*b + (a + 2*b)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(a^2*C - 4*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, ((a - b)*Sqrt[a + b]*(3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2*C - 2*a*b*C - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) - (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) - ((3*a^2*C - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, -((a - b)*Sqrt[a + b]*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 10*a^2*b*C + 24*b^3*(4*A + 3*C) + 4*a*b^2*(12*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) - (5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (a*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)} - - -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^2 + 8*A*b^3 - 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (4*b*(2*A*b^2 - a^2*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (-4*(a - b)*b*Sqrt[a + b]*(3*A*b^2 - a^2*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(25*a^2*A - 57*a*A*b - 6*A*b^2 + 35*a^2*C - 105*a*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (6*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (2*(a - b)*Sqrt[a + b]*(A*b^2 + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A*b^2 - 2*a*b*(2*A + 5*C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, ((a - b)*b*Sqrt[a + b]*(8*A - 3*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A + 3*C) - a*(8*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*b*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - (b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, ((a - b)*Sqrt[a + b]*(8*A - 5*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a*A - 16*A*b - 5*a*C - 2*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (b*(4*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - (a*(8*A - 5*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, -((a - b)*Sqrt[a + b]*(3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(48*a*A*b + 24*A*b^2 + 3*a^2*C + 14*a*b*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(24*A*b^2 - a^2*C + 12*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + (a*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((3*a^2*C + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, -((a - b)*Sqrt[a + b]*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^3*C - 2*a^2*b*C - 8*b^3*(4*A + 3*C) - 4*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^4*C + 24*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) - ((3*a^2*C - 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) - (a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*b^2*d)} - - -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 9, (2*(a - b)*b*Sqrt[a + b]*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) - 6*a^3*b*(101*A + 132*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(693*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^4 - 15*a^4*(9*A + 11*C) - a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*a^2*d) + (2*b*(3*A*b^2 + a^2*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(693*a*d) + (2*(5*A*b^2 + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 21*a^3*(7*A + 9*C) + 15*a*b^2*(11*A + 21*C) - 6*a^2*b*(19*A + 28*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*b*(5*A*b^2 + a^2*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (2*(a - b)*b*Sqrt[a + b]*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(3*A*b^3 - 9*a*b^2*(3*A + 7*C) - a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(21*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 10, ((a - b)*Sqrt[a + b]*(b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(30*A*b^3 - a*b^2*(46*A - 15*C) - 6*a^3*(3*A + 5*C) + a^2*(34*A*b + 90*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (5*a*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) - ((b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 10, ((a - b)*b*Sqrt[a + b]*(56*A - 27*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*b^2*(12*A + C) + 8*a^2*(A + 3*C) - a*(56*A*b - 27*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (b^2*(8*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - (a*b*(56*A - 27*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 10, ((a - b)*Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C) - 2*a*b*(72*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (5*a*Sqrt[a + b]*(8*A*b^2 + (a^2 + 4*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) - (a*b*(8*A - 3*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(6*A - C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - ((a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 10, -((a - b)*Sqrt[a + b]*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C + 24*b^3*(4*A + 3*C) + 2*a^2*b*(192*A + 59*C) + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(5*a^4*C - 120*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + ((5*a^2*C + 4*b^2*(4*A + 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + (5*a*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 11, ((a - b)*Sqrt[a + b]*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C) - 8*a*b^3*(260*A + 193*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) - (a*Sqrt[a + b]*(3*a^4*C + 40*a^2*b^2*(2*A + C) + 80*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + (a*(240*A*b^2 - 15*a^2*C + 172*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) - ((15*a^2*C - 16*b^2*(5*A + 4*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) - (3*a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) - ((45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]], x, 7, (-4*(a - b)*b*Sqrt[a + b]*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(12*a*A*b^2 - 48*A*b^3 - 5*a^3*(5*A + 7*C) - a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^3*d) - (12*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]], x, 6, (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*a*A*b - 8*A*b^2 - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (8*A*b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]], x, 5, (-4*A*(a - b)*b*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*A*b + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]], x, 7, (2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*A*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]], x, 7, -(((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A*b + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]]) + (a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{(A + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]), x, 8, (3*(a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - ((3*a - 2*b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a^2*C + 4*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) - (3*a*C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(16*A*b^4 - 2*a^2*b^2*(4*A - 5*C) - a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^5*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) - (2*(12*a*A*b^2 + 16*A*b^3 + 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(5*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*b*(8*A*b^2 - a^2*(3*A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*b*(8*A*b^2 - a^2*(5*A - 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(6*a*A*b + 8*A*b^2 + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2), x, 5, -((2*(2*A*b^2 - a^2*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) - (2*(2*A*b + a*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]), x, 8, -(((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + ((2*A*b^2 + a*(3*a + b)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (3*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 9, ((8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((8*A*b^2 + (15*a^2 + 5*a*b - 2*b^2)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A*b^2 + 5*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d)} - - -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2), x, 7, -((4*b*(8*A*b^4 + a^4*(4*A - 3*C) - a^2*b^2*(14*A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])) - (2*(12*a*A*b^3 + 16*A*b^4 - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C) - a^3*(9*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(6*a*A*b^2 + 8*A*b^3 - 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (4*(2*A*b^4 - a^4*C - a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2), x, 6, -((4*b*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) - (2*(2*A*b^2 + 3*a*b*(A + C) - a^2*(3*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (4*b*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]), x, 8, -((2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])) - (2*(A*b^3 + 3*a^3*C + a^2*b*C - 3*a*b^2*(A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^2*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 9, ((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) - ((6*A*b^4 - a*b^3*(2*A - 3*C) - 15*a^4*C - 5*a^3*b*C + 21*a^2*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*(a - b)*b^3*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]]) + (5*a*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((8*A*b^4 - (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-2*(3*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*a*A + 5*b*B + 5*a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(A*b + a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 6, (-2*(A*b + a*B - b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*b*B + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*b + a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 6, (2*(b*B - a*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A*b + 3*a*B + b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 6, (2*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(b*B + a*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b + 7*a*B + 5*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (2*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*C*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(9*A*b + 9*a*B + 7*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(7*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*a*(4*A*b + 9*a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 7, (-2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(4*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(4*A*b + 5*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (-2*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(A - C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*(4*A*b + 3*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 7, (2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(5*A - C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a*A - b*B - 2*a*C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 7, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(9*b*B + 4*a*C)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(11*b*B + 4*a*C)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(11*A*b^2 + 22*a*b*B + 4*a^2*C + 9*b^2*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 8, (-2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(24*A*b^3 + 21*a^3*B + 98*a*b^2*B + 21*a^2*b*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(6*A*b + 7*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (-2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(24*A*b^2 + 35*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 8, (-2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(35*A*b + 15*a*B - 3*b*C)*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*b*(6*a^2*B - b^2*B + 3*a*b*(5*A - C))*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*(2*A*b + a*B)*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, (2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(35*a*A - 7*b*B - 11*a*C)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*b*(21*a*b*B - 6*a^2*(7*A - 3*C) + b^2*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(7*A - C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (2*(3*b*B + 2*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*C))*Sin[c + d*x])/(495*d*Sec[c + d*x]^(3/2)) + (2*(11*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, (2*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(143*A*b^2 + 195*a*b*B + 24*a^2*C + 121*b^2*C)*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(338*a^2*b*B + 117*b^3*B + 24*a^3*C + 39*a*b^2*(11*A + 9*C))*Sin[c + d*x])/(1001*d*Sec[c + d*x]^(5/2)) + (2*(13*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(143*d*Sec[c + d*x]^(5/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(5/2)) + (2*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (2*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 10, (-2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(8*A*b + 11*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 9, (-2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(192*A*b^4 + 756*a^3*b*B + 1098*a*b^3*B + 21*a^4*(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(8*A*b + 9*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (-2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(192*A*b^3 + 63*a^3*B + 413*a*b^2*B + a^2*(202*A*b + 350*b*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(8*A*b + 7*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 9, (-2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b^2*(50*a*b*B + b^2*(59*A - 3*C) + 3*a^2*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)) - (2*b*(105*a^2*b*B - 5*b^3*B + 4*a*b^2*(33*A - 5*C) + 6*a^3*(3*A + 5*C))*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + 15*a*b*B + a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(8*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, (-2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b^2*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) - (2*b*(42*a^3*B - 28*a*b^2*B + 3*a^2*b*(49*A - 13*C) - b^3*(7*A + 5*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*A*b + 7*a*B - b*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, (2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b^2*(162*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(117*a^2*b*B + 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) - (2*b*(21*a*A - 3*b*B - 5*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) - (2*b*(9*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, (2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(11*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Sqrt[Sec[c + d*x]])} -{((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, (2*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(3/2)) + (2*(13*b*B + 8*a*C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(3/2)) + (2*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]), x, 9, (-2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (2*b*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) - (2*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + (2*A*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]), x, 8, (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*A*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]), x, 7, (-2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]), x, 6, (2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]), x, 7, (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*(b^2*(3*A + C) - 3*a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)), x, 8, (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) + (2*(3*a^2*b*B + b^3*B - 3*a^3*C - a*b^2*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^4*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Sec[c + d*x]^(3/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)), x, 9, (2*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*b^5*d) - (2*a^3*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^5*(a + b)*d) + (2*C*Sin[c + d*x])/(7*b*d*Sec[c + d*x]^(5/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(5*b^2*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*b^3*d*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2, x, 9, -(((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d) - ((5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2, x, 8, ((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2, x, 7, -(((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]), x, 7, ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + a*b^2*(A + 4*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)), x, 8, ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - a^2*b^2*(3*A - 16*C) - 15*a^4*C + 2*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^4*(a^2 - b^2)*d) + (a*(3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)), x, 9, -((25*a^3*b*B - 20*a*b^3*B - 3*a^2*b^2*(5*A - 8*C) - 35*a^4*C + 2*b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - a^3*b^2*(9*A - 20*C) - 21*a^5*C + 4*a*b^4*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^5*(a^2 - b^2)*d) - (a^2*(5*A*b^4 + 5*a^3*b*B - 7*a*b^3*B - 3*a^2*b^2*(A - 3*C) - 7*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*Sin[c + d*x])/(5*b^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + ((5*a^2*b*B - 2*b^3*B - a*b^2*(3*A - 4*C) - 7*a^3*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3, x, 10, ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) - ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3, x, 9, -((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3, x, 8, ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]), x, 8, ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) - ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)), x, 8, -((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C - a*b^4*(7*A + 24*C) + a^3*b^2*(A + 33*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) + ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)), x, 9, ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - a^4*b^2*(9*A - 223*C) + a^2*b^4*(15*A - 128*C) - 105*a^6*C - 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)), x, 10, -((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C - 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d) + ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C) - 189*a^7*C - 24*a*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + ((35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^(m/2) (A+B Cos[e+f x]+C Cos[e+f x]^2) (a+b Cos[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (-2*(a - b)*Sqrt[a + b]*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^5*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(16*A*b^3 + 12*a*b^2*(A - 2*B) + 6*a^2*b*(6*A - 3*B + 7*C) + 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a^2*d) + (2*(A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^2 + 2*a*b*(3*A - 7*B) + a^2*(25*A - 63*B + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 6, (-2*(a - b)*Sqrt[a + b]*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(2*A*b + a*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(b*(A - 3*B) - a*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 8, ((a - b)*Sqrt[a + b]*(2*A - C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*A*b - a*(2*A - 2*B - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((2*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 8, -((a - b)*Sqrt[a + b]*(4*b*B + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*A*b + a*C + 2*b*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) + ((4*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, -((a - b)*Sqrt[a + b]*(8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(24*A*b^2 + (a + 2*b)*(6*b*B - 3*a*C + 8*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^3*d*Sqrt[Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d*Sqrt[Sec[c + d*x]]) + ((8*b^2*(3*A + 2*C) + 3*a*(2*b*B - a*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d)} -{(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 10, ((a - b)*Sqrt[a + b]*(24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C - 2*a^2*b*(12*B + 5*C) + 4*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 16*B + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^3*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*a^3*b*B + 32*a*b^3*B - 5*a^4*C - 8*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(4*b*d*Sec[c + d*x]^(3/2)) + ((16*A*b^2 - 8*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b^2*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b^2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*b*B - 128*b^3*B - 15*a^3*C - 4*a*b^2*(12*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^3*d)} - - -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (2*(a - b)*Sqrt[a + b]*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(8*A*b^3 + 6*a*b^2*(A - 3*B) + 3*a^2*b*(13*A - 57*B + 21*C) - 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(21*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 7, (-2*(a - b)*Sqrt[a + b]*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(6*A*b^2 - a^2*(25*A - 63*B + 35*C) + 3*a*b*(19*A - 7*B + 35*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 8, (2*(a - b)*Sqrt[a + b]*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(3*b^2*(A - 5*B) - 2*a*b*(6*A - 10*B + 15*C) + a^2*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 9, ((a - b)*Sqrt[a + b]*(8*A*b + 6*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(6*A*b^2 + 2*a^2*(A - 3*B + 3*C) - a*b*(8*A - 3*(4*B + C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(A*b + a*B)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - ((8*A*b + 6*a*B - 3*b*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 9, ((a - b)*Sqrt[a + b]*(8*a*A - 4*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a*(8*A - 8*B - 5*C) - 2*b*(8*A + 2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b*d*Sqrt[Sec[c + d*x]]) - (b*(4*A - C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((8*a*A - 4*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (2*A*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 9, -((a - b)*Sqrt[a + b]*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(3*a^2*C + 4*b^2*(6*A + 3*B + 4*C) + 2*a*b*(24*A + 15*B + 7*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^2*d*Sqrt[Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b*d)} -{((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 10, -((a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4*a*b^2*(60*A + 28*B + 39*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^3*d*Sqrt[Sec[c + d*x]]) + ((4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*b*d*Sqrt[Sec[c + d*x]]) + ((8*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d*Sqrt[Sec[c + d*x]]) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d)} - - -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2), x, 9, (2*(a - b)*Sqrt[a + b]*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*(40*A*b^4 + 10*a*b^3*(3*A - 11*B) + 15*a^2*b^2*(19*A - 121*B + 33*C) + 3*a^4*(225*A - 539*B + 275*C) - 6*a^3*b*(505*A - 209*B + 660*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3465*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2), x, 8, (-2*(a - b)*Sqrt[a + b]*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(10*A*b^3 + 15*a*b^2*(11*A - 3*B + 21*C) - 6*a^2*b*(19*A - 60*B + 28*C) + 3*a^3*(49*A - 25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(5*A*b + 9*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2), x, 9, (2*(a - b)*Sqrt[a + b]*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(15*b^3*(A - 7*B) - a^3*(25*A - 63*B + 35*C) + a^2*b*(145*A - 119*B + 245*C) - a*b^2*(135*A - 161*B + 315*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a*d*Sqrt[Sec[c + d*x]]) - (2*b^2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(5*A*b + 7*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2), x, 10, ((a - b)*Sqrt[a + b]*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(30*A*b^3 - 2*a^3*(9*A - 5*B + 15*C) + 2*a^2*b*(17*A - 35*B + 45*C) - a*b^2*(46*A - 15*(6*B + C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a*d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[a + b]*(2*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) - ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(A*b + a*B)*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2), x, 10, ((a - b)*Sqrt[a + b]*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a*b*(56*A - 72*B - 27*C) - 6*b^2*(12*A + 2*B + C) - 8*a^2*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(12*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(8*A*b + 4*a*B - b*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]) - ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (2*(5*A*b + 3*a*B)*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2), x, 10, -((a - b)*Sqrt[a + b]*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(a^2*(48*A - 48*B - 33*C) - 4*b^2*(6*A + 3*B + 4*C) - 2*a*b*(72*A + 27*B + 13*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b*d*Sqrt[Sec[c + d*x]]) - (b*(8*a*A - 2*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) - (b*(6*A - C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]], x, 10, -((a - b)*Sqrt[a + b]*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*a*b*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(15*a^3*C + 8*b^3*(12*A + 16*B + 9*C) + 2*a^2*b*(192*A + 132*B + 59*C) + 4*a*b^2*(108*A + 52*B + 71*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(192*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(64*b^2*d*Sqrt[Sec[c + d*x]]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(32*d*Sqrt[Sec[c + d*x]]) + ((8*b*B + 5*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(192*b*d)} -{((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 11, -((a - b)*Sqrt[a + b]*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(45*a^4*C - 30*a^3*b*(5*B + C) - 16*b^4*(80*A + 45*B + 64*C) - 8*a*b^3*(260*A + 355*B + 193*C) - 4*a^2*b^2*(660*A + 295*B + 423*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(1920*b^2*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(128*b^3*d*Sqrt[Sec[c + d*x]]) + ((50*a^2*b*B + 120*b^3*B - 15*a^3*C + 4*a*b^2*(60*A + 43*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(320*b*d*Sqrt[Sec[c + d*x]]) + ((80*A*b^2 + 50*a*b*B - 15*a^2*C + 64*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(240*b*d*Sqrt[Sec[c + d*x]]) + ((10*b*B - 3*a*C)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(40*b*d*Sqrt[Sec[c + d*x]]) + (C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(5*b*d*Sqrt[Sec[c + d*x]]) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]], x, 7, (-2*(a - b)*Sqrt[a + b]*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^5*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(48*A*b^3 - 4*a*b^2*(3*A + 14*B) + a^3*(25*A - 63*B + 35*C) + 2*a^2*b*(22*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^3*d) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]], x, 6, (2*(a - b)*Sqrt[a + b]*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(8*A*b^2 - 2*a*b*(A + 5*B) + a^2*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^3*d*Sqrt[Sec[c + d*x]]) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*d*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b]*(2*A*b + a*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]], x, 6, (2*A*(a - b)*Sqrt[a + b]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*(A - B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]], x, 7, -(((a - b)*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A*b + a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]), x, 8, -((a - b)*Sqrt[a + b]*(4*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(3*a*C - 2*b*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^2*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d*Sqrt[Sec[c + d*x]]) + ((4*b*B - 3*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)), x, 9, -((a - b)*Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*a*b^3*d*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b]*(24*A*b^2 - 18*a*b*B + 12*b^2*B + 15*a^2*C - 10*a*b*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(24*b^3*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*(2*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(8*b^4*d*Sqrt[Sec[c + d*x]]) + (C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d*Sec[c + d*x]^(3/2)) + ((6*b*B - 5*a*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(12*b^2*d*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*b^3*d)} - -{((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]], x, 8, -(((a - b)*Sqrt[a + b]*B*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d*Sqrt[Sec[c + d*x]])) + (Sqrt[a + b]*(2*A + B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*A*b + a*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d*Sqrt[Sec[c + d*x]]) + (B*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2), x, 7, -((2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^5*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) - (2*(48*A*b^3 + 4*a*b^2*(9*A - 10*B) + 6*a^2*b*(2*A - 5*B + 5*C) + a^3*(9*A - 5*B + 15*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(15*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2), x, 6, (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^2 + 6*a*b*(A - B) + a^2*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2), x, 5, -((2*(2*A*b^2 - a*b*B - a^2*(A - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) - (2*(2*A*b + a*(A - B - C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2), x, 7, (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) + (2*(A*b + b*B - a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (2*Sqrt[a + b]*C*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^2*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]), x, 8, -(((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]])) + ((2*A*b^2 - a*(b*(2*B - C) - 3*a*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*b^2*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(2*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b^3*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} -{(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)), x, 9, -((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*a*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - ((8*A*b^2 - a*b*(12*B - 5*C) + 15*a^2*C - 2*b^2*(2*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^3*Sqrt[a + b]*d*Sqrt[Sec[c + d*x]]) - (Sqrt[a + b]*(8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(4*b^4*d*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((12*a^2*b*B - 4*b^3*B - a*b^2*(8*A - 7*C) - 15*a^3*C)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d)} - - -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2), x, 7, -((2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^5*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])) - (2*(16*A*b^4 + 4*a*b^3*(3*A - 2*B) - 3*a^3*b*(3*A - 3*B - C) - 2*a^2*b^2*(8*A + 3*B - C) - a^4*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2), x, 6, (2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^4*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 + 2*a*b^2*(3*A - B) - 3*a^3*(A - B - C) - a^2*b*(9*A + 3*B + C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -{((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2), x, 6, -((2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^3*(a - b)*(a + b)^(3/2)*d*Sqrt[Sec[c + d*x]])) - (2*(2*A*b^2 - a^2*(3*A + 3*B + C) + a*b*(3*A + B + 3*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]])} -(* {(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(1/2)), x, 8, (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(3*a*(a - b)*b^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(3*a*A*b - a*b*B - 2*a^2*C + 3*a*b*C + b^2*(A - 3*B + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(3*(a - b)*b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + (4*C*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x]/(1 + Cos[c + d*x]))/(3*a*b^2*(a + b)*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} *) -(* {(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)), x, 9, -((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(3*(a - b)^2*b^3*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(a*A*b^2 + 3*A*b^3 + 2*a^2*b*B - 3*a*b^2*B - 3*b^3*B - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(3*(a - b)^2*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) + (2*(2*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], -((a - b)/(a + b))]*Sqrt[Sec[c + d*x]])/(b^3*d*Sqrt[a + b*Cos[c + d*x]]) - (2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + 9*a^2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x]/(1 + Cos[c + d*x]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} *) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.7 (d trig)^m (a+b (c cos)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.7 (d trig)^m (a+b (c cos)^n)^p.m deleted file mode 100644 index 3a06b42..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.7 (d trig)^m (a+b (c cos)^n)^p.m +++ /dev/null @@ -1,392 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^2)^p when a+b=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[x]^6/(a - a*Cos[x]^2), x, 4, (3*x)/(8*a) - (3*Cos[x]*Sin[x])/(8*a) - (Cos[x]*Sin[x]^3)/(4*a)} -{Sin[x]^5/(a - a*Cos[x]^2), x, 3, -(Cos[x]/a) + Cos[x]^3/(3*a)} -{Sin[x]^4/(a - a*Cos[x]^2), x, 3, x/(2*a) - (Cos[x]*Sin[x])/(2*a)} -{Sin[x]^3/(a - a*Cos[x]^2), x, 2, -(Cos[x]/a)} -{Sin[x]^2/(a - a*Cos[x]^2), x, 2, x/a} -{Sin[x]^1/(a - a*Cos[x]^2), x, 2, -(ArcTanh[Cos[x]]/a)} -{Csc[x]^1/(a - a*Cos[x]^2), x, 3, -(ArcTanh[Cos[x]]/(2*a)) - (Cot[x]*Csc[x])/(2*a)} -{Csc[x]^2/(a - a*Cos[x]^2), x, 3, -(Cot[x]/a) - Cot[x]^3/(3*a)} -{Csc[x]^3/(a - a*Cos[x]^2), x, 4, -((3*ArcTanh[Cos[x]])/(8*a)) - (3*Cot[x]*Csc[x])/(8*a) - (Cot[x]*Csc[x]^3)/(4*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[x]^7/(a + b*Cos[x]^2), x, 4, -(((a + b)^3*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) + ((a^2 + 3*a*b + 3*b^2)*Cos[x])/b^3 - ((a + 3*b)*Cos[x]^3)/(3*b^2) + Cos[x]^5/(5*b)} -{Sin[x]^5/(a + b*Cos[x]^2), x, 4, -(((a + b)^2*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2))) + ((a + 2*b)*Cos[x])/b^2 - Cos[x]^3/(3*b)} -{Sin[x]^3/(a + b*Cos[x]^2), x, 3, -(((a + b)*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Cos[x]/b} -{Sin[x]^1/(a + b*Cos[x]^2), x, 2, -(ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]))} -{Csc[x]^1/(a + b*Cos[x]^2), x, 4, -((Sqrt[b]*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*(a + b))) - ArcTanh[Cos[x]]/(a + b)} -{Csc[x]^3/(a + b*Cos[x]^2), x, 5, -((b^(3/2)*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^2)) - ((a + 3*b)*ArcTanh[Cos[x]])/(2*(a + b)^2) - (Cot[x]*Csc[x])/(2*(a + b))} -{Csc[x]^5/(a + b*Cos[x]^2), x, 6, -((b^(5/2)*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3)) - ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[Cos[x]])/(8*(a + b)^3) - ((3*a + 7*b)*Cot[x]*Csc[x])/(8*(a + b)^2) - (Cot[x]*Csc[x]^3)/(4*(a + b))} - -{Sin[x]^6/(a + b*Cos[x]^2), x, 6, -(((8*a^2 + 20*a*b + 15*b^2)*x)/(8*b^3)) - ((a + b)^(5/2)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*b^3) + ((4*a + 7*b)*Cos[x]*Sin[x])/(8*b^2) + (Cos[x]*Sin[x]^3)/(4*b)} -{Sin[x]^4/(a + b*Cos[x]^2), x, 5, -(((2*a + 3*b)*x)/(2*b^2)) - ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*b^2) + (Cos[x]*Sin[x])/(2*b)} -{Sin[x]^2/(a + b*Cos[x]^2), x, 4, -(x/b) - (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*b)} -{Sin[x]^0/(a + b*Cos[x]^2), x, 2, -(ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]))} -{Csc[x]^2/(a + b*Cos[x]^2), x, 3, -((b*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2))) - Cot[x]/(a + b)} -{Csc[x]^4/(a + b*Cos[x]^2), x, 4, -((b^2*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2))) - ((a + 2*b)*Cot[x])/(a + b)^2 - Cot[x]^3/(3*(a + b))} -{Csc[x]^6/(a + b*Cos[x]^2), x, 4, -((b^3*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Cot[x])/(a + b)^3 - ((2*a + 3*b)*Cot[x]^3)/(3*(a + b)^2) - Cot[x]^5/(5*(a + b))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^3)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[x]/(4 - 3*Cos[x]^3), x, 7, -(ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]]/(2*2^(1/3)*3^(5/6))) + Log[2^(2/3) - 3^(1/3)*Cos[x]]/(6*6^(1/3)) - Log[2*2^(1/3) + 2^(2/3)*3^(1/3)*Cos[x] + 3^(2/3)*Cos[x]^2]/(12*6^(1/3))} - - -(* ::Section:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^2)^p when a+b=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(1 - Cos[x]^2), x, 3, -Cot[x]} -{1/(1 - Cos[x]^2)^2, x, 3, -Cot[x] - Cot[x]^3/3} -{1/(1 - Cos[x]^2)^3, x, 3, -Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]^7/(a + b*Cos[x]^2), x, 4, -((a^3*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b])) + ((a^2 - a*b + b^2)*Sin[x])/b^3 + ((a - 2*b)*Sin[x]^3)/(3*b^2) + Sin[x]^5/(5*b)} -{Cos[x]^5/(a + b*Cos[x]^2), x, 4, (a^2*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]) - ((a - b)*Sin[x])/b^2 - Sin[x]^3/(3*b)} -{Cos[x]^3/(a + b*Cos[x]^2), x, 3, -((a*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sin[x]/b} -{Cos[x]^1/(a + b*Cos[x]^2), x, 2, ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])} -{Sec[x]^1/(a + b*Cos[x]^2), x, 4, ArcTanh[Sin[x]]/a - (Sqrt[b]*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(a*Sqrt[a + b])} -{Sec[x]^3/(a + b*Cos[x]^2), x, 5, ((a - 2*b)*ArcTanh[Sin[x]])/(2*a^2) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]) + (Sec[x]*Tan[x])/(2*a)} -{Sec[x]^5/(a + b*Cos[x]^2), x, 6, ((3*a^2 - 4*a*b + 8*b^2)*ArcTanh[Sin[x]])/(8*a^3) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]) + ((3*a - 4*b)*Sec[x]*Tan[x])/(8*a^2) + (Sec[x]^3*Tan[x])/(4*a)} - -{Cos[x]^6/(a + b*Cos[x]^2), x, 6, ((8*a^2 - 4*a*b + 3*b^2)*x)/(8*b^3) + (a^(5/2)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(b^3*Sqrt[a + b]) - ((4*a - 3*b)*Cos[x]*Sin[x])/(8*b^2) + (Cos[x]^3*Sin[x])/(4*b)} -{Cos[x]^4/(a + b*Cos[x]^2), x, 5, -(((2*a - b)*x)/(2*b^2)) - (a^(3/2)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(b^2*Sqrt[a + b]) + (Cos[x]*Sin[x])/(2*b)} -{Cos[x]^2/(a + b*Cos[x]^2), x, 3, x/b + (Sqrt[a]*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(b*Sqrt[a + b])} -{Cos[x]^0/(a + b*Cos[x]^2), x, 2, -(ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]))} -{Sec[x]^2/(a + b*Cos[x]^2), x, 3, (b*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(a^(3/2)*Sqrt[a + b]) + Tan[x]/a} -{Sec[x]^4/(a + b*Cos[x]^2), x, 4, -((b^2*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(a^(5/2)*Sqrt[a + b])) + ((a - b)*Tan[x])/a^2 + Tan[x]^3/(3*a)} -{Sec[x]^6/(a + b*Cos[x]^2), x, 4, (b^3*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(a^(7/2)*Sqrt[a + b]) + ((a^2 - a*b + b^2)*Tan[x])/a^3 + ((2*a - b)*Tan[x]^3)/(3*a^2) + Tan[x]^5/(5*a)} - - -{1/(a + b*Cos[x]^2)^2, x, 4, -(((2*a + b)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2))) - (b*Cos[x]*Sin[x])/(2*a*(a + b)*(a + b*Cos[x]^2))} - - -{1/(a + b*Cos[x]^2)^3, x, 5, -(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(5/2))) - (b*Cos[x]*Sin[x])/(4*a*(a + b)*(a + b*Cos[x]^2)^2) - (3*b*(2*a + b)*Cos[x]*Sin[x])/(8*a^2*(a + b)^2*(a + b*Cos[x]^2))} - - -{1/(1 + Cos[x]^2)^1, x, 2, x/Sqrt[2] - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/Sqrt[2]} -{1/(1 + Cos[x]^2)^2, x, 4, (3*x)/(4*Sqrt[2]) - (3*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(4*Sqrt[2]) - (Cos[x]*Sin[x])/(4*(1 + Cos[x]^2))} -{1/(1 + Cos[x]^2)^3, x, 5, (19*x)/(32*Sqrt[2]) - (19*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(32*Sqrt[2]) - (Cos[x]*Sin[x])/(8*(1 + Cos[x]^2)^2) - (9*Cos[x]*Sin[x])/(32*(1 + Cos[x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^2)^(p/2) when a+b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[1 - Cos[x]^2], x, 3, -(Cot[x]*Sqrt[Sin[x]^2])} -{Sqrt[-1 + Cos[x]^2], x, 3, (-Cot[x])*Sqrt[-Sin[x]^2]} - - -{(1 - Cos[x]^2)^(3/2), x, 4, (-(2/3))*Cot[x]*Sqrt[Sin[x]^2] - (1/3)*Cot[x]*(Sin[x]^2)^(3/2)} -{(-1 + Cos[x]^2)^(3/2), x, 4, (2/3)*Cot[x]*Sqrt[-Sin[x]^2] - (1/3)*Cot[x]*(-Sin[x]^2)^(3/2)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[1 - Cos[x]^2], x, 3, -((ArcTanh[Cos[x]]*Sin[x])/Sqrt[Sin[x]^2])} -{1/Sqrt[-1 + Cos[x]^2], x, 3, -((ArcTanh[Cos[x]]*Sin[x])/Sqrt[-Sin[x]^2])} - - -{1/(1 - Cos[x]^2)^(3/2), x, 4, -(Cot[x]/(2*Sqrt[Sin[x]^2])) - (ArcTanh[Cos[x]]*Sin[x])/(2*Sqrt[Sin[x]^2])} -{1/(-1 + Cos[x]^2)^(3/2), x, 4, Cot[x]/(2*Sqrt[-Sin[x]^2]) + (ArcTanh[Cos[x]]*Sin[x])/(2*Sqrt[-Sin[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[1 + Cos[x]^2], x, 1, EllipticE[Pi/2 + x, -1]} -{Sqrt[-1 - Cos[x]^2], x, 2, (Sqrt[-1 - Cos[x]^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2]} -{Sqrt[a + b*Cos[x]^2], x, 2, (Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/Sqrt[1 + (b*Cos[x]^2)/a]} - - -{(1 + Cos[x]^2)^(3/2), x, 4, 2*EllipticE[Pi/2 + x, -1] - (2/3)*EllipticF[Pi/2 + x, -1] + (1/3)*Cos[x]*Sqrt[1 + Cos[x]^2]*Sin[x]} -{(-1 - Cos[x]^2)^(3/2), x, 6, -((2*Sqrt[-1 - Cos[x]^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2]) - (2*Sqrt[1 + Cos[x]^2]*EllipticF[Pi/2 + x, -1])/(3*Sqrt[-1 - Cos[x]^2]) - (1/3)*Cos[x]*Sqrt[-1 - Cos[x]^2]*Sin[x]} -{(a + b*Cos[x]^2)^(3/2), x, 6, (2*(2*a + b)*Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/(3*Sqrt[1 + (b*Cos[x]^2)/a]) - (a*(a + b)*Sqrt[1 + (b*Cos[x]^2)/a]*EllipticF[Pi/2 + x, -(b/a)])/(3*Sqrt[a + b*Cos[x]^2]) + (1/3)*b*Cos[x]*Sqrt[a + b*Cos[x]^2]*Sin[x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[1 + Cos[x]^2], x, 1, EllipticF[Pi/2 + x, -1]} -{1/Sqrt[-1 - Cos[x]^2], x, 2, (Sqrt[1 + Cos[x]^2]*EllipticF[Pi/2 + x, -1])/Sqrt[-1 - Cos[x]^2]} -{1/Sqrt[a + b*Cos[x]^2], x, 2, (Sqrt[1 + (b*Cos[x]^2)/a]*EllipticF[Pi/2 + x, -(b/a)])/Sqrt[a + b*Cos[x]^2]} - - -{1/(1 + Cos[x]^2)^(3/2), x, 3, (1/2)*EllipticE[Pi/2 + x, -1] - (Cos[x]*Sin[x])/(2*Sqrt[1 + Cos[x]^2])} -{1/(-1 - Cos[x]^2)^(3/2), x, 4, (Sqrt[-1 - Cos[x]^2]*EllipticE[Pi/2 + x, -1])/(2*Sqrt[1 + Cos[x]^2]) + (Cos[x]*Sin[x])/(2*Sqrt[-1 - Cos[x]^2])} -{1/(a + b*Cos[x]^2)^(3/2), x, 4, (Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/(a*(a + b)*Sqrt[1 + (b*Cos[x]^2)/a]) - (b*Cos[x]*Sin[x])/(a*(a + b)*Sqrt[a + b*Cos[x]^2])} - - -{Cos[x]/Sqrt[1 + Cos[x]^2], x, 2, ArcSin[Sin[x]/Sqrt[2]]} -{Cos[5 + 3*x]/Sqrt[3 + Cos[5 + 3*x]^2], x, 2, ArcSin[Sin[5 + 3*x]/2]/3} -{Cos[x]/Sqrt[4 - Cos[x]^2], x, 2, ArcSinh[Sin[x]/Sqrt[3]]} - - -(* ::Section:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^3)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^4)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(a + b*Cos[x]^4), x, 10, ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Cot[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] + Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Cot[x])/(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a + b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[a] - Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Cot[x] + (a + b)^(3/4)*Cot[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])} - - -{1/(a - b*Cos[x]^4), x, 4, -(ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]])) - ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])} - - -{1/(1 + Cos[x]^4), x, 10, x/(2*Sqrt[-1 + Sqrt[2]]) + ArcTan[((-2 + Sqrt[2])*Cos[x]*Sin[x] + Sqrt[-1 + Sqrt[2]]*(1 - 2*Sin[x]^2))/(2 + Sqrt[1 + Sqrt[2]] + 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x] + (-2 + Sqrt[2])*Sin[x]^2)]/(4*Sqrt[-1 + Sqrt[2]]) + ArcTan[((-2 + Sqrt[2])*Cos[x]*Sin[x] + Sqrt[-1 + Sqrt[2]]*(-1 + 2*Sin[x]^2))/(2 + Sqrt[1 + Sqrt[2]] - 2*Sqrt[-1 + Sqrt[2]]*Cos[x]*Sin[x] + (-2 + Sqrt[2])*Sin[x]^2)]/(4*Sqrt[-1 + Sqrt[2]]) + (1/8)*Sqrt[-1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[-1 + Sqrt[2]]*Cot[x] + 2*Cot[x]^2] - (1/8)*Sqrt[-1 + Sqrt[2]]*Log[1 + Sqrt[2*(-1 + Sqrt[2])]*Cot[x] + Sqrt[2]*Cot[x]^2]} - - -{1/(1 - Cos[x]^4), x, 3, x/(2*Sqrt[2]) - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/(2*Sqrt[2]) - Cot[x]/2} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -{1/(a + b*Cos[x]^5), x, 12, (2*ArcTan[(Sqrt[a^(1/5) - b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt[a^(1/5) + b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])} -{1/(a + b*Cos[x]^6), x, 7, -(ArcTan[(Sqrt[a^(1/3) + b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)])) - ArcTan[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) - ArcTan[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])} -{1/(a + b*Cos[x]^8), x, 9, ArcTan[(Sqrt[(-a)^(1/4) - b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)]) + ArcTan[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) + ArcTan[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) + ArcTan[(Sqrt[(-a)^(1/4) + b^(1/4)]*Cot[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])} - -{1/(a - b*Cos[x]^5), x, 12, (2*ArcTan[(Sqrt[a^(1/5) + b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt[a^(1/5) + b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]) + (2*ArcTan[(Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]*Tan[x/2])/Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])} -{1/(a - b*Cos[x]^6), x, 7, -(ArcTan[(Sqrt[a^(1/3) - b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)])) - ArcTan[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) - ArcTan[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Cot[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])} -{1/(a - b*Cos[x]^8), x, 9, -(ArcTan[(Sqrt[a^(1/4) - b^(1/4)]*Cot[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - b^(1/4)])) - ArcTan[(Sqrt[a^(1/4) - I*b^(1/4)]*Cot[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) - I*b^(1/4)]) - ArcTan[(Sqrt[a^(1/4) + I*b^(1/4)]*Cot[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + I*b^(1/4)]) - ArcTan[(Sqrt[a^(1/4) + b^(1/4)]*Cot[x])/a^(1/8)]/(4*a^(7/8)*Sqrt[a^(1/4) + b^(1/4)])} - - -{1/(1 + Cos[x]^5), x, 11, (2*ArcTan[Sqrt[(1 - (-1)^(2/5))/(1 + (-1)^(2/5))]*Tan[x/2]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTan[Sqrt[(1 - (-1)^(4/5))/(1 + (-1)^(4/5))]*Tan[x/2]])/(5*Sqrt[1 + (-1)^(3/5)]) - (2*ArcTanh[Tan[x/2]/Sqrt[-((1 - (-1)^(1/5))/(1 + (-1)^(1/5)))]])/(5*Sqrt[-1 + (-1)^(2/5)]) - (2*Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*ArcTanh[Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*Tan[x/2]])/(5*(1 + (-1)^(3/5))) + Sin[x]/(5*(1 + Cos[x]))} -{1/(1 + Cos[x]^6), x, 7, ArcTan[Tan[x]/Sqrt[2]]/(3*Sqrt[2]) + ArcTan[Tan[x]/Sqrt[1 - (-1)^(1/3)]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTan[Tan[x]/Sqrt[1 + (-1)^(2/3)]]/(3*Sqrt[1 + (-1)^(2/3)]), x/(3*Sqrt[2]) - ArcTan[Sqrt[1 - (-1)^(1/3)]*Cot[x]]/(3*Sqrt[1 - (-1)^(1/3)]) - ArcTan[Sqrt[1 + (-1)^(2/3)]*Cot[x]]/(3*Sqrt[1 + (-1)^(2/3)]) - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/(3*Sqrt[2])} -{1/(1 + Cos[x]^8), x, 9, -(ArcTan[Sqrt[1 - (-1)^(1/4)]*Cot[x]]/(4*Sqrt[1 - (-1)^(1/4)])) - ArcTan[Sqrt[1 + (-1)^(1/4)]*Cot[x]]/(4*Sqrt[1 + (-1)^(1/4)]) - ArcTan[Sqrt[1 - (-1)^(3/4)]*Cot[x]]/(4*Sqrt[1 - (-1)^(3/4)]) - ArcTan[Sqrt[1 + (-1)^(3/4)]*Cot[x]]/(4*Sqrt[1 + (-1)^(3/4)])} - -{1/(1 - Cos[x]^5), x, 11, (2*ArcTan[Sqrt[(1 - (-1)^(1/5))/(1 + (-1)^(1/5))]*Tan[x/2]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTan[Sqrt[(1 - (-1)^(3/5))/(1 + (-1)^(3/5))]*Tan[x/2]])/(5*Sqrt[1 + (-1)^(1/5)]) - (2*ArcTanh[Tan[x/2]/Sqrt[-((1 - (-1)^(2/5))/(1 + (-1)^(2/5)))]])/(5*Sqrt[-1 + (-1)^(4/5)]) + (2*ArcTanh[Sqrt[-((1 + (-1)^(4/5))/(1 - (-1)^(4/5)))]*Tan[x/2]])/(5*Sqrt[-1 - (-1)^(3/5)]) - Sin[x]/(5*(1 - Cos[x]))} -{1/(1 - Cos[x]^6), x, 8, -(ArcTan[Sqrt[1 + (-1)^(1/3)]*Cot[x]]/(3*Sqrt[1 + (-1)^(1/3)])) - ArcTan[Sqrt[1 - (-1)^(2/3)]*Cot[x]]/(3*Sqrt[1 - (-1)^(2/3)]) - Cot[x]/3} -{1/(1 - Cos[x]^8), x, 10, x/(4*Sqrt[2]) - ArcTan[Sqrt[1 - I]*Cot[x]]/(4*Sqrt[1 - I]) - ArcTan[Sqrt[1 + I]*Cot[x]]/(4*Sqrt[1 + I]) - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/(4*Sqrt[2]) - Cot[x]/4} - - -(* ::Title::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^2)^p*) - - -{Tan[x]/(1 + Cos[x]^2), x, 4, -Log[Cos[x]] + (1/2)*Log[1 + Cos[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]*Sqrt[a + b*Cos[x]^2], x, 4, Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^2]/Sqrt[a]] - Sqrt[a + b*Cos[x]^2]} - - -{Tan[x]*Sqrt[1 - Cos[x]^2], x, 5, ArcTanh[Sqrt[Sin[x]^2]] - Sqrt[Sin[x]^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]/Sqrt[a + b*Cos[x]^2], x, 3, ArcTanh[Sqrt[a + b*Cos[x]^2]/Sqrt[a]]/Sqrt[a]} -{Tan[x]/Sqrt[1 + Cos[x]^2], x, 3, ArcTanh[Sqrt[1 + Cos[x]^2]]} - - -{Tan[x]/Sqrt[1 - Cos[x]^2], x, 4, ArcTanh[Sqrt[Sin[x]^2]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^3)^p*) - - -{Tan[x]^3/(a + b*Cos[x]^3), x, 11, -((b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Cos[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))) + Log[Cos[x]]/a + (b^(2/3)*Log[a^(1/3) + b^(1/3)*Cos[x]])/(3*a^(5/3)) - (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Cos[x] + b^(2/3)*Cos[x]^2])/(6*a^(5/3)) - Log[a + b*Cos[x]^3]/(3*a) + Sec[x]^2/(2*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^3)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]*Sqrt[a + b*Cos[x]^3], x, 5, (2/3)*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]] - (2/3)*Sqrt[a + b*Cos[x]^3]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]/Sqrt[a + b*Cos[x]^3], x, 4, (2*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]])/(3*Sqrt[a])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^4)^p*) - - -(* ::Subsection:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]*Sqrt[a + b*Cos[x]^4], x, 5, (1/2)*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]] - (1/2)*Sqrt[a + b*Cos[x]^4]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]/Sqrt[a + b*Cos[x]^4], x, 4, (2*ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]])/(4*Sqrt[a])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Subsection:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cos[e+f x]^n)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]*Sqrt[a + b*Cos[x]^n], x, 5, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^n]/Sqrt[a]])/n - (2*Sqrt[a + b*Cos[x]^n])/n} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]/Sqrt[a + b*Cos[x]^n], x, 4, (2*ArcTanh[Sqrt[a + b*Cos[x]^n]/Sqrt[a]])/(Sqrt[a]*n)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.8 (a+b cos)^m (c+d trig)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.8 (a+b cos)^m (c+d trig)^n.m deleted file mode 100644 index a34c6e7..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.8 (a+b cos)^m (c+d trig)^n.m +++ /dev/null @@ -1,82 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Trig[c+d x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Sin[c+d x])*) - - -{(A + B*Sin[x])/(a + b*Cos[x]), x, 6, (2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - (B*Log[a + b*Cos[x]])/b} - -{(A + B*Sin[x])/(1 + Cos[x]), x, 5, (-B)*Log[1 + Cos[x]] + (A*Sin[x])/(1 + Cos[x])} -{(A + B*Sin[x])/(1 - Cos[x]), x, 5, B*Log[1 - Cos[x]] - (A*Sin[x])/(1 - Cos[x])} - - -{(b + c + Sin[x])/(a + b*Cos[x]), x, 6, (2*(b + c)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - Log[a + b*Cos[x]]/b} -{(b + c + Sin[x])/(a - b*Cos[x]), x, 6, (2*(b + c)*ArcTan[(Sqrt[a + b]*Tan[x/2])/Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[a + b]) + Log[a - b*Cos[x]]/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Tan[c+d x])*) - - -{(A + B*Tan[x])/(a + b*Cos[x]), x, 8, (2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - (B*Log[Cos[x]])/a + (B*Log[a + b*Cos[x]])/a} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Cot[c+d x])*) - - -{(A + B*Cot[x])/(a + b*Cos[x]), x, 7, (2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cos[x]])/(2*(a + b)) + (B*Log[1 + Cos[x]])/(2*(a - b)) - (a*B*Log[a + b*Cos[x]])/(a^2 - b^2)} - - -(* ::Section:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Sec[c+d x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Csc[c+d x])*) - - -{(A + B*Csc[x])/(a + b*Cos[x]), x, 11, (2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cos[x]])/(2*(a + b)) - (B*Log[1 + Cos[x]])/(2*(a - b)) + (b*B*Log[a + b*Cos[x]])/(a^2 - b^2)} - - -(* ::Title:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Trig[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[e+f x])^m (c+d Sec[e+f x])^n*) - - -{(c + d*Sec[e + f*x])^4/(a + b*Cos[e + f*x]), x, 12, (2*(a*c - b*d)^4*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*(4*a*c - b*d)*ArcTanh[Sin[e + f*x]])/(2*a^2*f) + (d*(2*a*c - b*d)*(2*a^2*c^2 - 2*a*b*c*d + b^2*d^2)*ArcTanh[Sin[e + f*x]])/(a^4*f) + (d^4*Tan[e + f*x])/(a*f) + (d^2*(6*a^2*c^2 - 4*a*b*c*d + b^2*d^2)*Tan[e + f*x])/(a^3*f) + (d^3*(4*a*c - b*d)*Sec[e + f*x]*Tan[e + f*x])/(2*a^2*f) + (d^4*Tan[e + f*x]^3)/(3*a*f)} -{(c + d*Sec[e + f*x])^3/(a + b*Cos[e + f*x]), x, 10, (2*(a*c - b*d)^3*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*ArcTanh[Sin[e + f*x]])/(2*a*f) + (d*(3*a^2*c^2 - 3*a*b*c*d + b^2*d^2)*ArcTanh[Sin[e + f*x]])/(a^3*f) + (d^2*(3*a*c - b*d)*Tan[e + f*x])/(a^2*f) + (d^3*Sec[e + f*x]*Tan[e + f*x])/(2*a*f)} -{(c + d*Sec[e + f*x])^2/(a + b*Cos[e + f*x]), x, 8, (2*(a*c - b*d)^2*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*f) + (d*(2*a*c - b*d)*ArcTanh[Sin[e + f*x]])/(a^2*f) + (d^2*Tan[e + f*x])/(a*f)} -{(c + d*Sec[e + f*x])^1/(a + b*Cos[e + f*x]), x, 5, (2*(a*c - b*d)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*f) + (d*ArcTanh[Sin[e + f*x]])/(a*f)} -{1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^1), x, 6, (2*a*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)*f) - (2*d*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(a*c - b*d)*f)} -{1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^2), x, 7, (2*a^2*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)^2*f) - (2*d*(2*a*c^2 - b*c*d - a*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(a*c - b*d)^2*f) + (d^2*Sin[e + f*x])/((a*c - b*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))} -{1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^3), x, 16, (2*a^3*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)^3*f) - (2*d^3*(3*a*c - 2*b*d)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*(a*c - b*d)^2*f) - (d^3*(c^2 + 2*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^2*(c - d)^(5/2)*(c + d)^(5/2)*(a*c - b*d)*f) - (2*d*(3*a^2*c^2 - 3*a*b*c*d + b^2*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^2*Sqrt[c - d]*Sqrt[c + d]*(a*c - b*d)^3*f) - (d^3*Sin[e + f*x])/(2*c*(a*c - b*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) + (3*d^4*Sin[e + f*x])/(2*c*(a*c - b*d)*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])) + (d^2*(3*a*c - 2*b*d)*Sin[e + f*x])/(c*(a*c - b*d)^2*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))} - - -(* {(c + d*Sec[e + f*x])^(5/2)/(a + b*Cos[e + f*x]), x, 0, 0} *) -(* {(c + d*Sec[e + f*x])^(3/2)/(a + b*Cos[e + f*x]), x, 0, 0} *) -{(c + d*Sec[e + f*x])^(1/2)/(a + b*Cos[e + f*x]), x, 4, (2*Sqrt[c + d]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*f) + (2*(a*c - b*d)*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/(c + d)]*Tan[e + f*x])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -{1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^(1/2)), x, 2, (2*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/(c + d)]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -(* {1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^(3/2)), x, 0, 0} *) -(* {1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^(5/2)), x, 0, 0} *) - - -(* ::Title:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Cos[c+d x]+C Sin[c+d x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cos[c+d x])^m (A+B Cos[c+d x]+C Sin[c+d x]^2)*) - - -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^1, x, 6, (B*x)/b + (2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*e) - (C*Log[a + b*Cos[d + e*x]])/(b*e)} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^2, x, 7, (2*(a*A - b*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*e) + C/(b*e*(a + b*Cos[d + e*x])) - ((A*b - a*B)*Sin[d + e*x])/((a^2 - b^2)*e*(a + b*Cos[d + e*x]))} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^3, x, 8, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*e) + C/(2*b*e*(a + b*Cos[d + e*x])^2) - ((A*b - a*B)*Sin[d + e*x])/(2*(a^2 - b^2)*e*(a + b*Cos[d + e*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sin[d + e*x])/(2*(a^2 - b^2)^2*e*(a + b*Cos[d + e*x]))} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4, x, 9, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*e) + C/(3*b*e*(a + b*Cos[d + e*x])^3) - ((A*b - a*B)*Sin[d + e*x])/(3*(a^2 - b^2)*e*(a + b*Cos[d + e*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^2*e*(a + b*Cos[d + e*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^3*e*(a + b*Cos[d + e*x]))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.9 trig^m (a+b cos^n+c cos^(2 n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.9 trig^m (a+b cos^n+c cos^(2 n))^p.m deleted file mode 100644 index 08cf080..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.2 Cosine/4.2.9 trig^m (a+b cos^n+c cos^(2 n))^p.m +++ /dev/null @@ -1,75 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[d+e x]^m (a+b Cos[d+e x]^n+c Cos[d+e x]^(2 n))^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Cos[d+e x]^n+c Cos[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Cos[d+e x]+c Cos[d+e x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[x]^5/(a + b*Cos[x] + c*Cos[x]^2), x, 7, ((b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c))*ArcTanh[(b + 2*c*Cos[x])/Sqrt[b^2 - 4*a*c]])/(c^4*Sqrt[b^2 - 4*a*c]) - ((b^2 - c*(a + 2*c))*Cos[x])/c^3 + (b*Cos[x]^2)/(2*c^2) - Cos[x]^3/(3*c) + (b*(b^2 - 2*c*(a + c))*Log[a + b*Cos[x] + c*Cos[x]^2])/(2*c^4)} -{Sin[x]^3/(a + b*Cos[x] + c*Cos[x]^2), x, 7, -(((b^2 - 2*c*(a + c))*ArcTanh[(b + 2*c*Cos[x])/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c])) + Cos[x]/c - (b*Log[a + b*Cos[x] + c*Cos[x]^2])/(2*c^2)} -{Sin[x]^1/(a + b*Cos[x] + c*Cos[x]^2), x, 3, (2*ArcTanh[(b + 2*c*Cos[x])/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]} -{Csc[x]^1/(a + b*Cos[x] + c*Cos[x]^2), x, 9, -(((b^2 - 2*a*c - 2*c^2)*ArcTanh[(b + 2*c*Cos[x])/Sqrt[b^2 - 4*a*c]])/((a - b + c)*(a + b + c)*Sqrt[b^2 - 4*a*c])) + Log[1 - Cos[x]]/(2*(a + b + c)) - Log[1 + Cos[x]]/(2*(a - b + c)) + (b*Log[a + b*Cos[x] + c*Cos[x]^2])/(2*(a - b + c)*(a + b + c))} -{Csc[x]^3/(a + b*Cos[x] + c*Cos[x]^2), x, 10, ((b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c))*ArcTanh[(b + 2*c*Cos[x])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(a^2 - b^2 + 2*a*c + c^2)^2) + ((b - (a + c)*Cos[x])*Csc[x]^2)/(2*(a - b + c)*(a + b + c)) + ((a + 2*b + 3*c)*Log[1 - Cos[x]])/(4*(a + b + c)^2) - ((a - 2*b + 3*c)*Log[1 + Cos[x]])/(4*(a - b + c)^2) - (b*(b^2 - 2*c*(a + c))*Log[a + b*Cos[x] + c*Cos[x]^2])/(2*(a^2 - b^2 + 2*a*c + c^2)^2)} - -{Sin[x]^4/(a + b*Cos[x] + c*Cos[x]^2), x, 10, If[$VersionNumber<9, x/(2*c) + ((b^2 - c*(a + 2*c))*x)/c^3 - (2*(b*(b^2 - 2*c*(a + c)) - (b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (2*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - (b*Sin[x])/c^2 + (Cos[x]*Sin[x])/(2*c), x/(2*c) + ((b^2 - c*(a + 2*c))*x)/c^3 + (2*(b^2*(b^2 - 2*c*(a + c)) - b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*c*(a + c)) - 2*c*(a*b^2 - c*(a + c)^2))*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (2*(b^4 + 2*c^2*(a + c)^2 - 2*b^2*c*(2*a + c) + b^3*Sqrt[b^2 - 4*a*c] - 2*b*c*(a + c)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - (b*Sin[x])/c^2 + (Cos[x]*Sin[x])/(2*c)]} -{Sin[x]^2/(a + b*Cos[x] + c*Cos[x]^2), x, 7, -(x/c) + (2*(b - (b^2 - 2*c*(a + c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(b + (b^2 - 2*c*(a + c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Csc[x]^2/(a + b*Cos[x] + c*Cos[x]^2), x, 9, -((2*b*c*(1 + (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/((a - b + c)*(a + b + c)*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]])) - (2*b*c*(1 - (b^2 - 2*c*(a + c))/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/((a - b + c)*(a + b + c)*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - Sin[x]/(2*(a + b + c)*(1 - Cos[x])) + Sin[x]/(2*(a - b + c)*(1 + Cos[x]))} - - -{Sin[x]/(-2 + Cos[x] + Cos[x]^2), x, 4, (-(1/3))*Log[1 - Cos[x]] + (1/3)*Log[2 + Cos[x]]} -{Sin[x]/(4 - 5*Cos[x] + Cos[x]^2), x, 4, (1/3)*Log[1 - Cos[x]] - (1/3)*Log[4 - Cos[x]]} -{Sin[x]/(3 - 2*Cos[x] + Cos[x]^2), x, 3, ArcTan[(1 - Cos[x])/Sqrt[2]]/Sqrt[2]} -{Sin[x]/(13 - 4*Cos[x] + Cos[x]^2)^2, x, 4, (-(1/54))*ArcTan[(1/3)*(-2 + Cos[x])] + (2 - Cos[x])/(18*(13 - 4*Cos[x] + Cos[x]^2))} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Cos[d+e x]^2+c Cos[d+e x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Cos[d+e x]^n+c Cos[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Cos[d+e x]+c Cos[d+e x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]^4/(a + b*Cos[x] + c*Cos[x]^2), x, 10, x/(2*c) + ((b^2 - a*c)*x)/c^3 - (2*(b^3 - 2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (2*(b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c^3*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - (b*Sin[x])/c^2 + (Cos[x]*Sin[x])/(2*c)} -{Cos[x]^3/(a + b*Cos[x] + c*Cos[x]^2), x, 8, -((b*x)/c^2) + (2*(b^2 - a*c - b^3/Sqrt[b^2 - 4*a*c] + (3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c^2*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(b^2 - a*c + b^3/Sqrt[b^2 - 4*a*c] - (3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c^2*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) + Sin[x]/c} -{Cos[x]^2/(a + b*Cos[x] + c*Cos[x]^2), x, 7, x/c - (2*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (2*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Cos[x]^1/(a + b*Cos[x] + c*Cos[x]^2), x, 6, (2*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Cos[x]^0/(a + b*Cos[x] + c*Cos[x]^2), x, 5, (4*c*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (4*c*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Sec[x]^1/(a + b*Cos[x] + c*Cos[x]^2), x, 8, -((2*c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]])) - (2*c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(a*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) + ArcTanh[Sin[x]]/a} -{Sec[x]^2/(a + b*Cos[x] + c*Cos[x]^2), x, 10, (2*b*c*(1 + (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(a^2*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*b*c*(1 - (b^2 - 2*a*c)/(b*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(a^2*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) - (b*ArcTanh[Sin[x]])/a^2 + Tan[x]/a} -{Sec[x]^3/(a + b*Cos[x] + c*Cos[x]^2), x, 12, -((2*c*(b^3 - 3*a*b*c + Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]])) + (2*c*(b^3 - 3*a*b*c - Sqrt[b^2 - 4*a*c]*(b^2 - a*c))*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(a^3*Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) + ArcTanh[Sin[x]]/(2*a) + ((b^2 - a*c)*ArcTanh[Sin[x]])/a^3 - (b*Tan[x])/a^2 + (Sec[x]*Tan[x])/(2*a)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Cos[d+e x]^2+c Cos[d+e x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Tan[d+e x]^m (a+b Cos[d+e x]^n+c Cos[d+e x]^(2 n))^p*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.0 (a trg)^m (b tan)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.0 (a trg)^m (b tan)^n.m deleted file mode 100644 index 931a89d..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.0 (a trg)^m (b tan)^n.m +++ /dev/null @@ -1,732 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Tan[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Tan[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^n*) - - -{Tan[c + d*x]^1, x, 1, -(Log[Cos[c + d*x]]/d)} -{Tan[c + d*x]^2, x, 2, -x + Tan[c + d*x]/d} -{Tan[c + d*x]^3, x, 2, Log[Cos[c + d*x]]/d + Tan[c + d*x]^2/(2*d)} -{Tan[c + d*x]^4, x, 3, x - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d)} -{Tan[c + d*x]^5, x, 3, -(Log[Cos[c + d*x]]/d) - Tan[c + d*x]^2/(2*d) + Tan[c + d*x]^4/(4*d)} -{Tan[c + d*x]^6, x, 4, -x + Tan[c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d)} -{Tan[c + d*x]^7, x, 4, Log[Cos[c + d*x]]/d + Tan[c + d*x]^2/(2*d) - Tan[c + d*x]^4/(4*d) + Tan[c + d*x]^6/(6*d)} -{Tan[c + d*x]^8, x, 5, x - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d) - Tan[c + d*x]^5/(5*d) + Tan[c + d*x]^7/(7*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[c+d x])^(n/2)*) - - -{(b*Tan[c + d*x])^(7/2), x, 13, -((b^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d)) + (b^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) - (b^(7/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (b^(7/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*b^3*Sqrt[b*Tan[c + d*x]])/d + (2*b*(b*Tan[c + d*x])^(5/2))/(5*d)} -{(b*Tan[c + d*x])^(5/2), x, 12, (b^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) - (b^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) - (b^(5/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (b^(5/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (2*b*(b*Tan[c + d*x])^(3/2))/(3*d)} -{(b*Tan[c + d*x])^(3/2), x, 12, (b^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) - (b^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) + (b^(3/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (b^(3/2)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (2*b*Sqrt[b*Tan[c + d*x]])/d} -{(b*Tan[c + d*x])^(1/2), x, 11, -((Sqrt[b]*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d)) + (Sqrt[b]*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*d) + (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (Sqrt[b]*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*d)} -{1/(b*Tan[c + d*x])^(1/2), x, 11, -(ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*Sqrt[b]*d)) + ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*Sqrt[b]*d) - Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*Sqrt[b]*d) + Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*Sqrt[b]*d)} -{1/(b*Tan[c + d*x])^(3/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(3/2)*d) - ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(3/2)*d) - Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(3/2)*d) + Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(3/2)*d) - 2/(b*d*Sqrt[b*Tan[c + d*x]])} -{1/(b*Tan[c + d*x])^(5/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(5/2)*d) - ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(5/2)*d) + Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(5/2)*d) - Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(5/2)*d) - 2/(3*b*d*(b*Tan[c + d*x])^(3/2))} -{1/(b*Tan[c + d*x])^(7/2), x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(7/2)*d)) + ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]]/(Sqrt[2]*b^(7/2)*d) + Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(7/2)*d) - Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]]/(2*Sqrt[2]*b^(7/2)*d) - 2/(5*b*d*(b*Tan[c + d*x])^(5/2)) + 2/(b^3*d*Sqrt[b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[c+d x])^(n/3)*) - - -{(b*Tan[c + d*x])^(4/3), x, 13, -((b^(4/3)*ArcTan[(b*Tan[c + d*x])^(1/3)/b^(1/3)])/d) + (b^(4/3)*ArcTan[Sqrt[3] - (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)])/(2*d) - (b^(4/3)*ArcTan[Sqrt[3] + (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)])/(2*d) + (Sqrt[3]*b^(4/3)*Log[b^(2/3) - Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*d) - (Sqrt[3]*b^(4/3)*Log[b^(2/3) + Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*d) + (3*b*(b*Tan[c + d*x])^(1/3))/d} -{(b*Tan[c + d*x])^(2/3), x, 12, (b^(2/3)*ArcTan[(b*Tan[c + d*x])^(1/3)/b^(1/3)])/d - (b^(2/3)*ArcTan[Sqrt[3] - (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)])/(2*d) + (b^(2/3)*ArcTan[Sqrt[3] + (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)])/(2*d) + (Sqrt[3]*b^(2/3)*Log[b^(2/3) - Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*d) - (Sqrt[3]*b^(2/3)*Log[b^(2/3) + Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*d)} -{(b*Tan[c + d*x])^(1/3), x, 9, -((Sqrt[3]*b^(1/3)*ArcTan[(b^(2/3) - 2*(b*Tan[c + d*x])^(2/3))/(Sqrt[3]*b^(2/3))])/(2*d)) - (b^(1/3)*Log[b^(2/3) + (b*Tan[c + d*x])^(2/3)])/(2*d) + (b^(1/3)*Log[b^(4/3) - b^(2/3)*(b*Tan[c + d*x])^(2/3) + (b*Tan[c + d*x])^(4/3)])/(4*d)} -{1/(b*Tan[c + d*x])^(1/3), x, 9, -((Sqrt[3]*ArcTan[(b^(2/3) - 2*(b*Tan[c + d*x])^(2/3))/(Sqrt[3]*b^(2/3))])/(2*b^(1/3)*d)) + Log[b^(2/3) + (b*Tan[c + d*x])^(2/3)]/(2*b^(1/3)*d) - Log[b^(4/3) - b^(2/3)*(b*Tan[c + d*x])^(2/3) + (b*Tan[c + d*x])^(4/3)]/(4*b^(1/3)*d)} -{1/(b*Tan[c + d*x])^(2/3), x, 12, ArcTan[(b*Tan[c + d*x])^(1/3)/b^(1/3)]/(b^(2/3)*d) - ArcTan[Sqrt[3] - (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)]/(2*b^(2/3)*d) + ArcTan[Sqrt[3] + (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)]/(2*b^(2/3)*d) - (Sqrt[3]*Log[b^(2/3) - Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*b^(2/3)*d) + (Sqrt[3]*Log[b^(2/3) + Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*b^(2/3)*d)} -{1/(b*Tan[c + d*x])^(4/3), x, 13, -(ArcTan[(b*Tan[c + d*x])^(1/3)/b^(1/3)]/(b^(4/3)*d)) + ArcTan[Sqrt[3] - (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)]/(2*b^(4/3)*d) - ArcTan[Sqrt[3] + (2*(b*Tan[c + d*x])^(1/3))/b^(1/3)]/(2*b^(4/3)*d) - (Sqrt[3]*Log[b^(2/3) - Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*b^(4/3)*d) + (Sqrt[3]*Log[b^(2/3) + Sqrt[3]*b^(1/3)*(b*Tan[c + d*x])^(1/3) + (b*Tan[c + d*x])^(2/3)])/(4*b^(4/3)*d) - 3/(b*d*(b*Tan[c + d*x])^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[c+d x])^n with n symbolic*) - - -{(b*Tan[c + d*x])^n, x, 2, (Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(1 + n))/(b*d*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Tan[c+d x]^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[c+d x]^p)^(n/2) with p positive integer*) - - -{(b*Tan[c + d*x]^2)^(5/2),x, 4, -((b^2*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[b*Tan[c + d*x]^2])/d) - (b^2*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^2])/(2*d) + (b^2*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^2])/(4*d)} -{(b*Tan[c + d*x]^2)^(3/2),x, 3, (b*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[b*Tan[c + d*x]^2])/d + (b*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^2])/(2*d)} -{(b*Tan[c + d*x]^2)^(1/2), x, 2, -((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[b*Tan[c + d*x]^2])/d)} -{1/(b*Tan[c + d*x]^2)^(1/2), x, 2, (Log[Sin[c + d*x]]*Tan[c + d*x])/(d*Sqrt[b*Tan[c + d*x]^2])} -{1/(b*Tan[c + d*x]^2)^(3/2), x, 3, -(Cot[c + d*x]/(2*b*d*Sqrt[b*Tan[c + d*x]^2])) - (Log[Sin[c + d*x]]*Tan[c + d*x])/(b*d*Sqrt[b*Tan[c + d*x]^2])} -{1/(b*Tan[c + d*x]^2)^(5/2), x, 4, Cot[c + d*x]/(2*b^2*d*Sqrt[b*Tan[c + d*x]^2]) - Cot[c + d*x]^3/(4*b^2*d*Sqrt[b*Tan[c + d*x]^2]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(b^2*d*Sqrt[b*Tan[c + d*x]^2])} - - -{(b*Tan[c + d*x]^3)^(5/2),x, 16, -((2*b^2*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^3])/d) - (b^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) - (b^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (b^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (2*b^2*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^3])/(5*d) - (2*b^2*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^3])/(9*d) + (2*b^2*Tan[c + d*x]^5*Sqrt[b*Tan[c + d*x]^3])/(13*d)} -{(b*Tan[c + d*x]^3)^(3/2),x, 14, -((2*b*Sqrt[b*Tan[c + d*x]^3])/(3*d)) - (b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (b*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2)) - (b*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (2*b*Tan[c + d*x]^2*Sqrt[b*Tan[c + d*x]^3])/(7*d)} -{(b*Tan[c + d*x]^3)^(1/2), x, 13, (2*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^3])/d + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[b*Tan[c + d*x]^3])/(Sqrt[2]*d*Tan[c + d*x]^(3/2)) + (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2)) - (Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[b*Tan[c + d*x]^3])/(2*Sqrt[2]*d*Tan[c + d*x]^(3/2))} -{1/(b*Tan[c + d*x]^3)^(1/2), x, 13, -((2*Tan[c + d*x])/(d*Sqrt[b*Tan[c + d*x]^3])) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*d*Sqrt[b*Tan[c + d*x]^3]) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*d*Sqrt[b*Tan[c + d*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d*Sqrt[b*Tan[c + d*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d*Sqrt[b*Tan[c + d*x]^3])} -{1/(b*Tan[c + d*x]^3)^(3/2),x, 14, 2/(3*b*d*Sqrt[b*Tan[c + d*x]^3]) - (2*Cot[c + d*x]^2)/(7*b*d*Sqrt[b*Tan[c + d*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*b*d*Sqrt[b*Tan[c + d*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*b*d*Sqrt[b*Tan[c + d*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*b*d*Sqrt[b*Tan[c + d*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*b*d*Sqrt[b*Tan[c + d*x]^3])} -{1/(b*Tan[c + d*x]^3)^(5/2),x, 16, -((2*Cot[c + d*x])/(5*b^2*d*Sqrt[b*Tan[c + d*x]^3])) + (2*Cot[c + d*x]^3)/(9*b^2*d*Sqrt[b*Tan[c + d*x]^3]) - (2*Cot[c + d*x]^5)/(13*b^2*d*Sqrt[b*Tan[c + d*x]^3]) + (2*Tan[c + d*x])/(b^2*d*Sqrt[b*Tan[c + d*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*b^2*d*Sqrt[b*Tan[c + d*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Tan[c + d*x]^(3/2))/(Sqrt[2]*b^2*d*Sqrt[b*Tan[c + d*x]^3]) + (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*b^2*d*Sqrt[b*Tan[c + d*x]^3]) - (Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*b^2*d*Sqrt[b*Tan[c + d*x]^3])} - - -{(b*Tan[c + d*x]^4)^(5/2),x, 7, (b^2*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - b^2*x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4] - (b^2*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/(3*d) + (b^2*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^4])/(5*d) - (b^2*Tan[c + d*x]^5*Sqrt[b*Tan[c + d*x]^4])/(7*d) + (b^2*Tan[c + d*x]^7*Sqrt[b*Tan[c + d*x]^4])/(9*d)} -{(b*Tan[c + d*x]^4)^(3/2),x, 5, (b*Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - b*x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4] - (b*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/(3*d) + (b*Tan[c + d*x]^3*Sqrt[b*Tan[c + d*x]^4])/(5*d)} -{(b*Tan[c + d*x]^4)^(1/2), x, 3, (Cot[c + d*x]*Sqrt[b*Tan[c + d*x]^4])/d - x*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4]} -{1/(b*Tan[c + d*x]^4)^(1/2), x, 3, -(Tan[c + d*x]/(d*Sqrt[b*Tan[c + d*x]^4])) - (x*Tan[c + d*x]^2)/Sqrt[b*Tan[c + d*x]^4]} -{1/(b*Tan[c + d*x]^4)^(3/2),x, 5, Cot[c + d*x]/(3*b*d*Sqrt[b*Tan[c + d*x]^4]) - Cot[c + d*x]^3/(5*b*d*Sqrt[b*Tan[c + d*x]^4]) - Tan[c + d*x]/(b*d*Sqrt[b*Tan[c + d*x]^4]) - (x*Tan[c + d*x]^2)/(b*Sqrt[b*Tan[c + d*x]^4])} -{1/(b*Tan[c + d*x]^4)^(5/2),x, 7, Cot[c + d*x]/(3*b^2*d*Sqrt[b*Tan[c + d*x]^4]) - Cot[c + d*x]^3/(5*b^2*d*Sqrt[b*Tan[c + d*x]^4]) + Cot[c + d*x]^5/(7*b^2*d*Sqrt[b*Tan[c + d*x]^4]) - Cot[c + d*x]^7/(9*b^2*d*Sqrt[b*Tan[c + d*x]^4]) - Tan[c + d*x]/(b^2*d*Sqrt[b*Tan[c + d*x]^4]) - (x*Tan[c + d*x]^2)/(b^2*Sqrt[b*Tan[c + d*x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[c+d x]^p)^n with n symbolic*) - - -{(b*Tan[c + d*x]^p)^n, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^p)^n)/(d*(1 + n*p))} - - -{(b*Tan[c + d*x]^2)^n,x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 2*n), (1/2)*(3 + 2*n), -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^2)^n)/(d*(1 + 2*n))} -{(b*Tan[c + d*x]^3)^n,x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 3*n), (3*(1 + n))/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^3)^n)/(d*(1 + 3*n))} -{(b*Tan[c + d*x]^4)^n,x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 4*n), (1/2)*(3 + 4*n), -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^4)^n)/(d*(1 + 4*n))} - - -{(b*Tan[c + d*x]^p)^(5/2), x, 3, (2*b^2*Hypergeometric2F1[1, (1/4)*(2 + 5*p), (1/4)*(6 + 5*p), -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + 2*p)*Sqrt[b*Tan[c + d*x]^p])/(d*(2 + 5*p))} -{(b*Tan[c + d*x]^p)^(3/2), x, 3, (2*b*Hypergeometric2F1[1, (1/4)*(2 + 3*p), (3*(2 + p))/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + p)*Sqrt[b*Tan[c + d*x]^p])/(d*(2 + 3*p))} -{(b*Tan[c + d*x]^p)^(1/2), x, 3, (2*Hypergeometric2F1[1, (2 + p)/4, (6 + p)/4, -Tan[c + d*x]^2]*Tan[c + d*x]*Sqrt[b*Tan[c + d*x]^p])/(d*(2 + p))} -{1/(b*Tan[c + d*x]^p)^(1/2), x, 3, (2*Hypergeometric2F1[1, (2 - p)/4, (6 - p)/4, -Tan[c + d*x]^2]*Tan[c + d*x])/(d*(2 - p)*Sqrt[b*Tan[c + d*x]^p])} -{1/(b*Tan[c + d*x]^p)^(3/2), x, 3, (2*Hypergeometric2F1[1, (1/4)*(2 - 3*p), (3*(2 - p))/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 - p))/(b*d*(2 - 3*p)*Sqrt[b*Tan[c + d*x]^p])} -{1/(b*Tan[c + d*x]^p)^(5/2), x, 3, (2*Hypergeometric2F1[1, (1/4)*(2 - 5*p), (1/4)*(6 - 5*p), -Tan[c + d*x]^2]*Tan[c + d*x]^(1 - 2*p))/(b^2*d*(2 - 5*p)*Sqrt[b*Tan[c + d*x]^p])} - - -{(b*Tan[c + d*x]^p)^(1/p), x, 2, -((Cot[c + d*x]*Log[Cos[c + d*x]]*(b*Tan[c + d*x]^p)^(1/p))/d)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Tan[c+d x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a (b Tan[c+d x])^p)^n with n symbolic*) - - -{(a*(b*Tan[c + d*x])^p)^n, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[c + d*x]^2]*Tan[c + d*x]*(a*(b*Tan[c + d*x])^p)^n)/(d*(1 + n*p))} - - -(* ::Title:: *) -(*Integrands of the form (a Trg[e+f x])^m (b Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (b Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[a + b*x]^4*(d*Tan[a + b*x])^(1/2), x, 13, -((21*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b)) + (21*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) + (21*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) - (21*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) - (7*Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2))/(16*b*d) - (Cos[a + b*x]^4*(d*Tan[a + b*x])^(7/2))/(4*b*d^3)} -{Sin[a + b*x]^2*(d*Tan[a + b*x])^(1/2), x, 12, -((3*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b)) + (3*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) + (3*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) - (3*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) - (Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2))/(2*b*d)} -{Csc[a + b*x]^2*(d*Tan[a + b*x])^(1/2), x, 2, -((2*d)/(b*Sqrt[d*Tan[a + b*x]]))} -{Csc[a + b*x]^4*(d*Tan[a + b*x])^(1/2), x, 3, -((2*d^3)/(5*b*(d*Tan[a + b*x])^(5/2))) - (2*d)/(b*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^6*(d*Tan[a + b*x])^(1/2), x, 3, -((2*d^5)/(9*b*(d*Tan[a + b*x])^(9/2))) - (4*d^3)/(5*b*(d*Tan[a + b*x])^(5/2)) - (2*d)/(b*Sqrt[d*Tan[a + b*x]])} - -{Sin[a + b*x]^3*(d*Tan[a + b*x])^(1/2), x, 5, -((5*d*Sin[a + b*x])/(6*b*Sqrt[d*Tan[a + b*x]])) - (d*Sin[a + b*x]^3)/(3*b*Sqrt[d*Tan[a + b*x]]) + (5*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(12*b)} -{Sin[a + b*x]^1*(d*Tan[a + b*x])^(1/2), x, 4, -((d*Sin[a + b*x])/(b*Sqrt[d*Tan[a + b*x]])) + (Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(2*b)} -{Csc[a + b*x]^1*(d*Tan[a + b*x])^(1/2), x, 3, (Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/b} -{Csc[a + b*x]^3*(d*Tan[a + b*x])^(1/2), x, 4, -((2*d*Csc[a + b*x])/(3*b*Sqrt[d*Tan[a + b*x]])) + (2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(3*b)} -{Csc[a + b*x]^5*(d*Tan[a + b*x])^(1/2), x, 5, -((4*d*Csc[a + b*x])/(7*b*Sqrt[d*Tan[a + b*x]])) - (2*d*Csc[a + b*x]^3)/(7*b*Sqrt[d*Tan[a + b*x]]) + (4*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(7*b)} - - -{Sin[a + b*x]^4*(d*Tan[a + b*x])^(3/2), x, 14, (45*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) - (45*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) + (45*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) - (45*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) + (45*d*Sqrt[d*Tan[a + b*x]])/(16*b) - (9*Cos[a + b*x]^2*(d*Tan[a + b*x])^(5/2))/(16*b*d) - (Cos[a + b*x]^4*(d*Tan[a + b*x])^(9/2))/(4*b*d^3)} -{Sin[a + b*x]^2*(d*Tan[a + b*x])^(3/2), x, 13, (5*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) - (5*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) + (5*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) - (5*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) + (5*d*Sqrt[d*Tan[a + b*x]])/(2*b) - (Cos[a + b*x]^2*(d*Tan[a + b*x])^(5/2))/(2*b*d)} -{Csc[a + b*x]^2*(d*Tan[a + b*x])^(3/2), x, 2, (2*d*Sqrt[d*Tan[a + b*x]])/b} -{Csc[a + b*x]^4*(d*Tan[a + b*x])^(3/2), x, 3, -((2*d^3)/(3*b*(d*Tan[a + b*x])^(3/2))) + (2*d*Sqrt[d*Tan[a + b*x]])/b} -{Csc[a + b*x]^6*(d*Tan[a + b*x])^(3/2), x, 3, -((2*d^5)/(7*b*(d*Tan[a + b*x])^(7/2))) - (4*d^3)/(3*b*(d*Tan[a + b*x])^(3/2)) + (2*d*Sqrt[d*Tan[a + b*x]])/b} - -{Sin[a + b*x]^3*(d*Tan[a + b*x])^(3/2), x, 5, (7*d^3*Sin[a + b*x]^3)/(3*b*(d*Tan[a + b*x])^(3/2)) - (7*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(2*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]]) + (2*d*Sin[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/b} -{Sin[a + b*x]^1*(d*Tan[a + b*x])^(3/2), x, 4, -((3*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])) + (2*d*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/b} -{Csc[a + b*x]^1*(d*Tan[a + b*x])^(3/2), x, 4, -((2*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])) + (2*d*Sin[a + b*x]*Sqrt[d*Tan[a + b*x]])/b} -{Csc[a + b*x]^3*(d*Tan[a + b*x])^(3/2), x, 5, -((4*d^2*Cos[a + b*x])/(b*Sqrt[d*Tan[a + b*x]])) - (4*d^2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]]) + (2*d*Csc[a + b*x]*Sqrt[d*Tan[a + b*x]])/b} - - -{Sin[a + b*x]^4*(d*Tan[a + b*x])^(5/2), x, 14, (77*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) - (77*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b) - (77*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) + (77*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b) + (77*d*(d*Tan[a + b*x])^(3/2))/(48*b) - (11*Cos[a + b*x]^2*(d*Tan[a + b*x])^(7/2))/(16*b*d) - (Cos[a + b*x]^4*(d*Tan[a + b*x])^(11/2))/(4*b*d^3)} -{Sin[a + b*x]^2*(d*Tan[a + b*x])^(5/2), x, 13, (7*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) - (7*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) - (7*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) + (7*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) + (7*d*(d*Tan[a + b*x])^(3/2))/(6*b) - (Cos[a + b*x]^2*(d*Tan[a + b*x])^(7/2))/(2*b*d)} -{Csc[a + b*x]^2*(d*Tan[a + b*x])^(5/2), x, 2, (2*d*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^4*(d*Tan[a + b*x])^(5/2), x, 3, -((2*d^3)/(b*Sqrt[d*Tan[a + b*x]])) + (2*d*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^6*(d*Tan[a + b*x])^(5/2), x, 3, -((2*d^5)/(5*b*(d*Tan[a + b*x])^(5/2))) - (4*d^3)/(b*Sqrt[d*Tan[a + b*x]]) + (2*d*(d*Tan[a + b*x])^(3/2))/(3*b)} - -{Sin[a + b*x]^3*(d*Tan[a + b*x])^(5/2), x, 6, (5*d^3*Sin[a + b*x])/(2*b*Sqrt[d*Tan[a + b*x]]) + (d^3*Sin[a + b*x]^3)/(b*Sqrt[d*Tan[a + b*x]]) - (5*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(4*b) + (2*d*Sin[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Sin[a + b*x]^1*(d*Tan[a + b*x])^(5/2), x, 5, (5*d^3*Sin[a + b*x])/(3*b*Sqrt[d*Tan[a + b*x]]) - (5*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(6*b) + (2*d*Sin[a + b*x]*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^1*(d*Tan[a + b*x])^(5/2), x, 4, -((d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(3*b)) + (2*d*Csc[a + b*x]*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^3*(d*Tan[a + b*x])^(5/2), x, 4, (2*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(3*b) + (2*d*Csc[a + b*x]*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^5*(d*Tan[a + b*x])^(5/2), x, 5, -((4*d^3*Csc[a + b*x])/(3*b*Sqrt[d*Tan[a + b*x]])) + (4*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(3*b) + (2*d*Csc[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(3*b)} -{Csc[a + b*x]^7*(d*Tan[a + b*x])^(5/2), x, 6, -((40*d^3*Csc[a + b*x])/(21*b*Sqrt[d*Tan[a + b*x]])) - (20*d^3*Csc[a + b*x]^3)/(21*b*Sqrt[d*Tan[a + b*x]]) + (40*d^2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(21*b) + (2*d*Csc[a + b*x]^5*(d*Tan[a + b*x])^(3/2))/(3*b)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[a + b*x]^4/(d*Tan[a + b*x])^(1/2), x, 13, -((5*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*Sqrt[d])) + (5*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*Sqrt[d]) - (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*Sqrt[d]) + (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*Sqrt[d]) - (5*Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(16*b*d) - (Cos[a + b*x]^4*(d*Tan[a + b*x])^(5/2))/(4*b*d^3)} -{Sin[a + b*x]^2/(d*Tan[a + b*x])^(1/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(4*Sqrt[2]*b*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(4*Sqrt[2]*b*Sqrt[d]) - Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(8*Sqrt[2]*b*Sqrt[d]) + Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(8*Sqrt[2]*b*Sqrt[d]) - (Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(2*b*d)} -{Csc[a + b*x]^2/(d*Tan[a + b*x])^(1/2), x, 2, -((2*d)/(3*b*(d*Tan[a + b*x])^(3/2)))} -{Csc[a + b*x]^4/(d*Tan[a + b*x])^(1/2), x, 3, -((2*d^3)/(7*b*(d*Tan[a + b*x])^(7/2))) - (2*d)/(3*b*(d*Tan[a + b*x])^(3/2))} -{Csc[a + b*x]^6/(d*Tan[a + b*x])^(1/2), x, 3, -((2*d^5)/(11*b*(d*Tan[a + b*x])^(11/2))) - (4*d^3)/(7*b*(d*Tan[a + b*x])^(7/2)) - (2*d)/(3*b*(d*Tan[a + b*x])^(3/2))} - -{Sin[a + b*x]^5/(d*Tan[a + b*x])^(1/2), x, 5, -((7*d*Sin[a + b*x]^3)/(30*b*(d*Tan[a + b*x])^(3/2))) - (d*Sin[a + b*x]^5)/(5*b*(d*Tan[a + b*x])^(3/2)) + (7*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(20*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Sin[a + b*x]^3/(d*Tan[a + b*x])^(1/2), x, 4, -((d*Sin[a + b*x]^3)/(3*b*(d*Tan[a + b*x])^(3/2))) + (EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(2*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Sin[a + b*x]^1/(d*Tan[a + b*x])^(1/2), x, 3, (EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^1/(d*Tan[a + b*x])^(1/2), x, 4, -((2*Cos[a + b*x])/(b*Sqrt[d*Tan[a + b*x]])) - (2*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^3/(d*Tan[a + b*x])^(1/2), x, 5, -((2*d*Csc[a + b*x])/(5*b*(d*Tan[a + b*x])^(3/2))) - (4*Cos[a + b*x])/(5*b*Sqrt[d*Tan[a + b*x]]) - (4*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(5*b*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} - - -{Sin[a + b*x]^4/(d*Tan[a + b*x])^(3/2), x, 13, -((3*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(3/2))) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(3/2)) + (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*d^(3/2)) - (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*d^(3/2)) + (3*Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2))/(16*b*d^3) - (Cos[a + b*x]^4*(d*Tan[a + b*x])^(3/2))/(4*b*d^3)} -{Sin[a + b*x]^2/(d*Tan[a + b*x])^(3/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(4*Sqrt[2]*b*d^(3/2))) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(4*Sqrt[2]*b*d^(3/2)) + Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(8*Sqrt[2]*b*d^(3/2)) - Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(8*Sqrt[2]*b*d^(3/2)) + (Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2))/(2*b*d^3)} -{Csc[a + b*x]^2/(d*Tan[a + b*x])^(3/2), x, 2, -((2*d)/(5*b*(d*Tan[a + b*x])^(5/2)))} -{Csc[a + b*x]^4/(d*Tan[a + b*x])^(3/2), x, 3, -((2*d^3)/(9*b*(d*Tan[a + b*x])^(9/2))) - (2*d)/(5*b*(d*Tan[a + b*x])^(5/2))} -{Csc[a + b*x]^6/(d*Tan[a + b*x])^(3/2), x, 3, -((2*d^5)/(13*b*(d*Tan[a + b*x])^(13/2))) - (4*d^3)/(9*b*(d*Tan[a + b*x])^(9/2)) - (2*d)/(5*b*(d*Tan[a + b*x])^(5/2))} - -{Sin[a + b*x]^3/(d*Tan[a + b*x])^(3/2), x, 5, -(Sin[a + b*x]/(6*b*d*Sqrt[d*Tan[a + b*x]])) + Sin[a + b*x]^3/(3*b*d*Sqrt[d*Tan[a + b*x]]) + (Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(12*b*d^2)} -{Sin[a + b*x]^1/(d*Tan[a + b*x])^(3/2), x, 4, Sin[a + b*x]/(b*d*Sqrt[d*Tan[a + b*x]]) + (EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*d*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^1/(d*Tan[a + b*x])^(3/2), x, 4, -((2*Csc[a + b*x])/(3*b*d*Sqrt[d*Tan[a + b*x]])) - (Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(3*b*d^2)} -{Csc[a + b*x]^3/(d*Tan[a + b*x])^(3/2), x, 5, (2*Csc[a + b*x])/(21*b*d*Sqrt[d*Tan[a + b*x]]) - (2*Csc[a + b*x]^3)/(7*b*d*Sqrt[d*Tan[a + b*x]]) - (2*Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/(21*b*d^2)} - - -{Sin[a + b*x]^4/(d*Tan[a + b*x])^(5/2), x, 13, -((3*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(5/2))) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(5/2)) - (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*d^(5/2)) + (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(64*Sqrt[2]*b*d^(5/2)) + (Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(16*b*d^3) - (Cos[a + b*x]^4*Sqrt[d*Tan[a + b*x]])/(4*b*d^3)} -{Sin[a + b*x]^2/(d*Tan[a + b*x])^(5/2), x, 12, -((3*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(5/2))) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(5/2)) - (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b*d^(5/2)) + (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b*d^(5/2)) + (Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(2*b*d^3)} -{Csc[a + b*x]^2/(d*Tan[a + b*x])^(5/2), x, 2, -((2*d)/(7*b*(d*Tan[a + b*x])^(7/2)))} -{Csc[a + b*x]^4/(d*Tan[a + b*x])^(5/2), x, 3, -((2*d^3)/(11*b*(d*Tan[a + b*x])^(11/2))) - (2*d)/(7*b*(d*Tan[a + b*x])^(7/2))} -{Csc[a + b*x]^6/(d*Tan[a + b*x])^(5/2), x, 3, -((2*d^5)/(15*b*(d*Tan[a + b*x])^(15/2))) - (4*d^3)/(11*b*(d*Tan[a + b*x])^(11/2)) - (2*d)/(7*b*(d*Tan[a + b*x])^(7/2))} - -{Sin[a + b*x]^7/(d*Tan[a + b*x])^(5/2), x, 6, -(Sin[a + b*x]^3/(20*b*d*(d*Tan[a + b*x])^(3/2))) - (3*Sin[a + b*x]^5)/(70*b*d*(d*Tan[a + b*x])^(3/2)) + Sin[a + b*x]^7/(7*b*d*(d*Tan[a + b*x])^(3/2)) + (3*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(40*b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Sin[a + b*x]^5/(d*Tan[a + b*x])^(5/2), x, 5, -(Sin[a + b*x]^3/(10*b*d*(d*Tan[a + b*x])^(3/2))) + Sin[a + b*x]^5/(5*b*d*(d*Tan[a + b*x])^(3/2)) + (3*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(20*b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Sin[a + b*x]^3/(d*Tan[a + b*x])^(5/2), x, 4, Sin[a + b*x]^3/(3*b*d*(d*Tan[a + b*x])^(3/2)) + (EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(2*b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Sin[a + b*x]^1/(d*Tan[a + b*x])^(5/2), x, 4, -((2*Sin[a + b*x])/(b*d*(d*Tan[a + b*x])^(3/2))) - (3*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^1/(d*Tan[a + b*x])^(5/2), x, 5, -((2*Csc[a + b*x])/(5*b*d*(d*Tan[a + b*x])^(3/2))) + (6*Cos[a + b*x])/(5*b*d^2*Sqrt[d*Tan[a + b*x]]) + (6*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(5*b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} -{Csc[a + b*x]^3/(d*Tan[a + b*x])^(5/2), x, 6, (2*Csc[a + b*x])/(15*b*d*(d*Tan[a + b*x])^(3/2)) - (2*Csc[a + b*x]^3)/(9*b*d*(d*Tan[a + b*x])^(3/2)) + (4*Cos[a + b*x])/(15*b*d^2*Sqrt[d*Tan[a + b*x]]) + (4*EllipticE[a - Pi/4 + b*x, 2]*Sin[a + b*x])/(15*b*d^2*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) (b Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]], x, 2, (-8*a^2*b*Sqrt[a*Sin[e + f*x]])/(5*f*Sqrt[b*Tan[e + f*x]]) - (2*b*(a*Sin[e + f*x])^(5/2))/(5*f*Sqrt[b*Tan[e + f*x]])} -{(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]], x, 3, (-2*b*(a*Sin[e + f*x])^(3/2))/(3*f*Sqrt[b*Tan[e + f*x]]) + (4*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(3*f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]], x, 1, (-2*b*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[b*Tan[e + f*x]])} -{Sqrt[b*Tan[e + f*x]]/Sqrt[a*Sin[e + f*x]], x, 2, (2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(3/2), x, 7, -((ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(a*f*Sqrt[a*Sin[e + f*x]])) - (ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(a*f*Sqrt[a*Sin[e + f*x]])} -{Sqrt[b*Tan[e + f*x]]/(a*Sin[e + f*x])^(5/2), x, 3, -(b/(a^2*f*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])) + (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(a^2*f*Sqrt[a*Sin[e + f*x]])} - - -{(a*Sin[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2), x, 4, (-24*a^2*b^2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) + (12*a^2*b*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(5*f) - (2*b*(a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]])/(5*f)} -{(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2), x, 2, (8*a^2*b*Sqrt[b*Tan[e + f*x]])/(3*f*Sqrt[a*Sin[e + f*x]]) - (2*b*(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])/(3*f)} -{Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(3/2), x, 3, (-4*b^2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) + (2*b*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])/f} -{(b*Tan[e + f*x])^(3/2)/Sqrt[a*Sin[e + f*x]], x, 1, (2*b*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[a*Sin[e + f*x]])} -{(b*Tan[e + f*x])^(3/2)/(a*Sin[e + f*x])^(3/2), x, 3, (-2*b^2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(a^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) + (2*b*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(a^2*f)} -{(b*Tan[e + f*x])^(3/2)/(a*Sin[e + f*x])^(5/2), x, 8, (b^2*ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(a^3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) - (b^2*ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(a^3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) + (2*b*Sqrt[b*Tan[e + f*x]])/(a^2*f*Sqrt[a*Sin[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(a*Sin[e + f*x])^(9/2)/Sqrt[b*Tan[e + f*x]], x, 4, -((4*a^2*b*(a*Sin[e + f*x])^(5/2))/(15*f*(b*Tan[e + f*x])^(3/2))) - (2*b*(a*Sin[e + f*x])^(9/2))/(9*f*(b*Tan[e + f*x])^(3/2)) + (8*a^4*EllipticE[(1/2)*(e + f*x), 2]*Sqrt[a*Sin[e + f*x]])/(15*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} -{(a*Sin[e + f*x])^(7/2)/Sqrt[b*Tan[e + f*x]], x, 2, -((8*a^2*b*(a*Sin[e + f*x])^(3/2))/(21*f*(b*Tan[e + f*x])^(3/2))) - (2*b*(a*Sin[e + f*x])^(7/2))/(7*f*(b*Tan[e + f*x])^(3/2))} -{(a*Sin[e + f*x])^(5/2)/Sqrt[b*Tan[e + f*x]], x, 3, (-2*b*(a*Sin[e + f*x])^(5/2))/(5*f*(b*Tan[e + f*x])^(3/2)) + (4*a^2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} -{(a*Sin[e + f*x])^(3/2)/Sqrt[b*Tan[e + f*x]], x, 1, (-2*b*(a*Sin[e + f*x])^(3/2))/(3*f*(b*Tan[e + f*x])^(3/2))} -{Sqrt[a*Sin[e + f*x]]/Sqrt[b*Tan[e + f*x]], x, 2, (2*EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} -{1/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]]), x, 7, (ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(a*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) - (ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(a*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} -{1/((a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]), x, 3, -((b*Sqrt[a*Sin[e + f*x]])/(a^2*f*(b*Tan[e + f*x])^(3/2))) - (EllipticE[(e + f*x)/2, 2]*Sqrt[a*Sin[e + f*x]])/(a^2*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} -{1/((a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]), x, 8, -(b/(2*a^2*f*Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(3/2))) + (ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(4*a^3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) - (ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[a*Sin[e + f*x]])/(4*a^3*f*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])} - - -{(a*Sin[e + f*x])^(13/2)/(b*Tan[e + f*x])^(3/2), x, 4, -((64*a^6*Sqrt[a*Sin[e + f*x]])/(585*b*f*Sqrt[b*Tan[e + f*x]])) - (16*a^4*(a*Sin[e + f*x])^(5/2))/(585*b*f*Sqrt[b*Tan[e + f*x]]) - (2*a^2*(a*Sin[e + f*x])^(9/2))/(117*b*f*Sqrt[b*Tan[e + f*x]]) + (2*(a*Sin[e + f*x])^(13/2))/(13*b*f*Sqrt[b*Tan[e + f*x]])} -{(a*Sin[e + f*x])^(9/2)/(b*Tan[e + f*x])^(3/2), x, 3, -((8*a^4*Sqrt[a*Sin[e + f*x]])/(45*b*f*Sqrt[b*Tan[e + f*x]])) - (2*a^2*(a*Sin[e + f*x])^(5/2))/(45*b*f*Sqrt[b*Tan[e + f*x]]) + (2*(a*Sin[e + f*x])^(9/2))/(9*b*f*Sqrt[b*Tan[e + f*x]])} -{(a*Sin[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2), x, 1, (-2*b*(a*Sin[e + f*x])^(5/2))/(5*f*(b*Tan[e + f*x])^(5/2))} -{Sqrt[a*Sin[e + f*x]]/(b*Tan[e + f*x])^(3/2), x, 8, (2*Sqrt[a*Sin[e + f*x]])/(b*f*Sqrt[b*Tan[e + f*x]]) - (a*ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(b^2*f*Sqrt[a*Sin[e + f*x]]) - (a*ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)), x, 8, -(1/(2*b*f*(a*Sin[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])) + (ArcTan[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(4*a*b^2*f*Sqrt[a*Sin[e + f*x]]) + (ArcTanh[Sqrt[Cos[e + f*x]]]*Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(4*a*b^2*f*Sqrt[a*Sin[e + f*x]])} - -{(a*Sin[e + f*x])^(11/2)/(b*Tan[e + f*x])^(3/2), x, 5, -((4*a^4*(a*Sin[e + f*x])^(3/2))/(77*b*f*Sqrt[b*Tan[e + f*x]])) - (2*a^2*(a*Sin[e + f*x])^(7/2))/(77*b*f*Sqrt[b*Tan[e + f*x]]) + (2*(a*Sin[e + f*x])^(11/2))/(11*b*f*Sqrt[b*Tan[e + f*x]]) + (8*a^6*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(77*b^2*f*Sqrt[a*Sin[e + f*x]])} -{(a*Sin[e + f*x])^(7/2)/(b*Tan[e + f*x])^(3/2), x, 4, -((2*a^2*(a*Sin[e + f*x])^(3/2))/(21*b*f*Sqrt[b*Tan[e + f*x]])) + (2*(a*Sin[e + f*x])^(7/2))/(7*b*f*Sqrt[b*Tan[e + f*x]]) + (4*a^4*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(21*b^2*f*Sqrt[a*Sin[e + f*x]])} -{(a*Sin[e + f*x])^(3/2)/(b*Tan[e + f*x])^(3/2), x, 3, (2*(a*Sin[e + f*x])^(3/2))/(3*b*f*Sqrt[b*Tan[e + f*x]]) + (2*a^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(3*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/(Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(3/2)), x, 3, -(1/(b*f*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]])) - (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2)), x, 4, -1/(3*b*f*(a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]) + 1/(6*a^2*b*f*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]]) - (Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(6*a^2*b^2*f*Sqrt[a*Sin[e + f*x]])} -{1/((a*Sin[e + f*x])^(9/2)*(b*Tan[e + f*x])^(3/2)), x, 5, -(1/(5*b*f*(a*Sin[e + f*x])^(9/2)*Sqrt[b*Tan[e + f*x]])) + 1/(30*a^2*b*f*(a*Sin[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]) + 1/(12*a^4*b*f*Sqrt[a*Sin[e + f*x]]*Sqrt[b*Tan[e + f*x]]) - (Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(12*a^4*b^2*f*Sqrt[a*Sin[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/3) (b Tan[e+f x])^(n/2)*) - - -{Sqrt[d*Tan[e + f*x]]*(b*Sin[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[3/4, 17/12, 29/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2))/(17*d*f)} -{Sqrt[d*Tan[e + f*x]]*(b*Sin[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[3/4, 11/12, 23/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*(d*Tan[e + f*x])^(3/2))/(11*d*f)} -{Sqrt[d*Tan[e + f*x]]/(b*Sin[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[7/12, 3/4, 19/12, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(7*d*f*(b*Sin[e + f*x])^(1/3))} -{Sqrt[d*Tan[e + f*x]]/(b*Sin[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[1/12, 3/4, 13/12, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(d*f*(b*Sin[e + f*x])^(4/3))} - - -{(d*Tan[e + f*x])^(3/2)*(b*Sin[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[5/4, 23/12, 35/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(4/3)*(d*Tan[e + f*x])^(5/2))/(23*d*f)} -{(d*Tan[e + f*x])^(3/2)*(b*Sin[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[5/4, 17/12, 29/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*(d*Tan[e + f*x])^(5/2))/(17*d*f)} -{(d*Tan[e + f*x])^(3/2)/(b*Sin[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[13/12, 5/4, 25/12, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(13*d*f*(b*Sin[e + f*x])^(1/3))} -{(d*Tan[e + f*x])^(3/2)/(b*Sin[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[7/12, 5/4, 19/12, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(7*d*f*(b*Sin[e + f*x])^(4/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^(m/2) (b Tan[e+f x])^(n/3)*) - - -{Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(7/6)*Hypergeometric2F1[7/6, 17/12, 29/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(7/3))/(17*d*f)} -{Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[2/3, 11/12, 23/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3))/(11*d*f)} -{Sqrt[b*Sin[e + f*x]]/(d*Tan[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 7/12, 19/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(2/3))/(7*d*f)} -{Sqrt[b*Sin[e + f*x]]/(d*Tan[e + f*x])^(4/3), x, 2, (6*Hypergeometric2F1[-(1/6), 1/12, 13/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]])/(d*f*(Cos[e + f*x]^2)^(1/6)*(d*Tan[e + f*x])^(1/3))} - - -{(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3), x, 2, (6*(Cos[e + f*x]^2)^(7/6)*Hypergeometric2F1[7/6, 23/12, 35/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(7/3))/(23*d*f)} -{(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[2/3, 17/12, 29/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3))/(17*d*f)} -{(b*Sin[e + f*x])^(3/2)/(d*Tan[e + f*x])^(1/3), x, 2, (6*(Cos[e + f*x]^2)^(1/3)*Hypergeometric2F1[1/3, 13/12, 25/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(3/2)*(d*Tan[e + f*x])^(2/3))/(13*d*f)} -{(b*Sin[e + f*x])^(3/2)/(d*Tan[e + f*x])^(4/3), x, 2, (6*Hypergeometric2F1[-(1/6), 7/12, 19/12, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(3/2))/(7*d*f*(Cos[e + f*x]^2)^(1/6)*(d*Tan[e + f*x])^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Tan[e+f x])^n with m symbolic*) - - -{(a*Sin[e + f*x])^m*Tan[e + f*x]^3, x, 2, (Hypergeometric2F1[2, (4 + m)/2, (6 + m)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^(4 + m))/(a^4*f*(4 + m))} -{(a*Sin[e + f*x])^m*Tan[e + f*x]^1, x, 2, (Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))} -{(a*Sin[e + f*x])^m*Cot[e + f*x]^1, x, 2, (a*Sin[e + f*x])^m/(f*m)} -{(a*Sin[e + f*x])^m*Cot[e + f*x]^3, x, 3, -((a^2*(a*Sin[e + f*x])^(-2 + m))/(f*(2 - m))) - (a*Sin[e + f*x])^m/(f*m)} -{(a*Sin[e + f*x])^m*Cot[e + f*x]^5, x, 3, -((a^4*(a*Sin[e + f*x])^(-4 + m))/(f*(4 - m))) + (2*a^2*(a*Sin[e + f*x])^(-2 + m))/(f*(2 - m)) + (a*Sin[e + f*x])^m/(f*m)} - -{(a*Sin[e + f*x])^m*Tan[e + f*x]^4, x, 2, (Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[5/2, (5 + m)/2, (7 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^(5 + m))/(a^5*f*(5 + m))} -{(a*Sin[e + f*x])^m*Tan[e + f*x]^2, x, 2, (Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x])^(3 + m))/(a^3*f*(3 + m))} -{(a*Sin[e + f*x])^m*Cot[e + f*x]^2, x, 2, -((a*Cos[e + f*x]*Hypergeometric2F1[-(1/2), (1/2)*(-1 + m), (1 + m)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^(-1 + m))/(f*(1 - m)*Sqrt[Cos[e + f*x]^2]))} -{(a*Sin[e + f*x])^m*Cot[e + f*x]^4, x, 2, -((a^3*Cos[e + f*x]*Hypergeometric2F1[-(3/2), (1/2)*(-3 + m), (1/2)*(-1 + m), Sin[e + f*x]^2]*(a*Sin[e + f*x])^(-3 + m))/(f*(3 - m)*Sqrt[Cos[e + f*x]^2]))} - - -{(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(3/2), x, 2, (2*(Cos[e + f*x]^2)^(5/4)*Hypergeometric2F1[5/4, (1/4)*(5 + 2*m), (1/4)*(9 + 2*m), Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(5/2))/(b*f*(5 + 2*m))} -{(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(1/2), x, 2, (2*(Cos[e + f*x]^2)^(3/4)*Hypergeometric2F1[3/4, (1/4)*(3 + 2*m), (1/4)*(7 + 2*m), Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(3/2))/(b*f*(3 + 2*m))} -{(a*Sin[e + f*x])^m/(b*Tan[e + f*x])^(1/2), x, 2, (2*(Cos[e + f*x]^2)^(1/4)*Hypergeometric2F1[1/4, (1/4)*(1 + 2*m), (1/4)*(5 + 2*m), Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*Sqrt[b*Tan[e + f*x]])/(b*f*(1 + 2*m))} -{(a*Sin[e + f*x])^m/(b*Tan[e + f*x])^(3/2), x, 2, -((2*Hypergeometric2F1[-(1/4), (1/4)*(-1 + 2*m), (1/4)*(3 + 2*m), Sin[e + f*x]^2]*(a*Sin[e + f*x])^m)/(b*f*(1 - 2*m)*(Cos[e + f*x]^2)^(1/4)*Sqrt[b*Tan[e + f*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Tan[e+f x])^n with n symbolic*) - - -{(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^n, x, 2, ((Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 + m + n))} - - -{Sin[e + f*x]^4*(b*Tan[e + f*x])^n, x, 2, (Hypergeometric2F1[3, (5 + n)/2, (7 + n)/2, -Tan[e + f*x]^2]*(b*Tan[e + f*x])^(5 + n))/(b^5*f*(5 + n))} -{Sin[e + f*x]^2*(b*Tan[e + f*x])^n, x, 2, (Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, -Tan[e + f*x]^2]*(b*Tan[e + f*x])^(3 + n))/(b^3*f*(3 + n))} -{Csc[e + f*x]^2*(b*Tan[e + f*x])^n, x, 2, -((b*(b*Tan[e + f*x])^(-1 + n))/(f*(1 - n)))} -{Csc[e + f*x]^4*(b*Tan[e + f*x])^n, x, 3, -((b^3*(b*Tan[e + f*x])^(-3 + n))/(f*(3 - n))) - (b*(b*Tan[e + f*x])^(-1 + n))/(f*(1 - n))} -{Csc[e + f*x]^6*(b*Tan[e + f*x])^n, x, 3, -((b^5*(b*Tan[e + f*x])^(-5 + n))/(f*(5 - n))) - (2*b^3*(b*Tan[e + f*x])^(-3 + n))/(f*(3 - n)) - (b*(b*Tan[e + f*x])^(-1 + n))/(f*(1 - n))} - -{Sin[e + f*x]^3*(b*Tan[e + f*x])^n, x, 2, ((Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (6 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*(b*Tan[e + f*x])^(1 + n))/(b*f*(4 + n))} -{Sin[e + f*x]^1*(b*Tan[e + f*x])^n, x, 2, ((Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (4 + n)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(b*Tan[e + f*x])^(1 + n))/(b*f*(2 + n))} -{Csc[e + f*x]^1*(b*Tan[e + f*x])^n, x, 2, -((Cos[e + f*x]*Hypergeometric2F1[(1 - n)/2, (2 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Tan[e + f*x])^n)/((Sin[e + f*x]^2)^(n/2)*(f*(1 - n))))} -{Csc[e + f*x]^3*(b*Tan[e + f*x])^n, x, 2, -((Cos[e + f*x]*Hypergeometric2F1[(1 - n)/2, (4 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Tan[e + f*x])^n)/((Sin[e + f*x]^2)^(n/2)*(f*(1 - n))))} -{Csc[e + f*x]^5*(b*Tan[e + f*x])^n, x, 2, -((Cos[e + f*x]*Hypergeometric2F1[(1 - n)/2, (6 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(b*Tan[e + f*x])^n)/((Sin[e + f*x]^2)^(n/2)*(f*(1 - n))))} - - -{(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^n, x, 2, (2*(Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1/4)*(5 + 2*n), (1/4)*(9 + 2*n), Sin[e + f*x]^2]*(a*Sin[e + f*x])^(3/2)*(b*Tan[e + f*x])^(1 + n))/(b*f*(5 + 2*n))} -{(a*Sin[e + f*x])^(1/2)*(b*Tan[e + f*x])^n, x, 2, (2*(Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1/4)*(3 + 2*n), (1/4)*(7 + 2*n), Sin[e + f*x]^2]*Sqrt[a*Sin[e + f*x]]*(b*Tan[e + f*x])^(1 + n))/(b*f*(3 + 2*n))} -{(b*Tan[e + f*x])^n/(a*Sin[e + f*x])^(1/2), x, 2, (2*(Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1/4)*(1 + 2*n), (1/4)*(5 + 2*n), Sin[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 + 2*n)*Sqrt[a*Sin[e + f*x]])} -{(b*Tan[e + f*x])^n/(a*Sin[e + f*x])^(3/2), x, 2, -((2*(Cos[e + f*x]^2)^((1 + n)/2)*Hypergeometric2F1[(1 + n)/2, (1/4)*(-1 + 2*n), (1/4)*(3 + 2*n), Sin[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 - 2*n)*(a*Sin[e + f*x])^(3/2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Tan[e+f x])^n*) - - -{(a*Cos[e + f*x])^m*(b*Tan[e + f*x])^n, x, 2, ((a*Cos[e + f*x])^m*(Cos[e + f*x]^2)^((1/2)*(1 - m + n))*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 - m + n), (3 + n)/2, Sin[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Tan[e+f x])^m (b Tan[e+f x])^n*) - - -{(a*Tan[e + f*x])^m*(b*Tan[e + f*x])^n, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + m + n), (1/2)*(3 + m + n), -Tan[e + f*x]^2]*(a*Tan[e + f*x])^(1 + m)*(b*Tan[e + f*x])^n)/(a*f*(1 + m + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cot[e+f x])^m (b Tan[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Cot[e+f x]^m (b Tan[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cot[e+f x])^(m/2) Tan[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[e + f*x]^4*Sqrt[d*Cot[e + f*x]], x, 14, (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^3)/(5*f*(d*Cot[e + f*x])^(5/2)) - (2*d)/(f*Sqrt[d*Cot[e + f*x]]) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^3*Sqrt[d*Cot[e + f*x]], x, 13, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^2)/(3*f*(d*Cot[e + f*x])^(3/2)) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^2*Sqrt[d*Cot[e + f*x]], x, 13, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d)/(f*Sqrt[d*Cot[e + f*x]]) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^1*Sqrt[d*Cot[e + f*x]], x, 12, (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^0*Sqrt[d*Cot[e + f*x]], x, 11, (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Cot[e + f*x]^1*Sqrt[d*Cot[e + f*x]], x, 13, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (2*Sqrt[d*Cot[e + f*x]])/f - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Cot[e + f*x]^2*Sqrt[d*Cot[e + f*x]], x, 13, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*d*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Cot[e + f*x]^3*Sqrt[d*Cot[e + f*x]], x, 14, (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*Sqrt[d*Cot[e + f*x]])/f - (2*(d*Cot[e + f*x])^(5/2))/(5*d^2*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} - - -{Tan[e + f*x]^5*(d*Cot[e + f*x])^(3/2), x, 14, (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^4)/(5*f*(d*Cot[e + f*x])^(5/2)) - (2*d^2)/(f*Sqrt[d*Cot[e + f*x]]) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^4*(d*Cot[e + f*x])^(3/2), x, 13, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^3)/(3*f*(d*Cot[e + f*x])^(3/2)) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^3*(d*Cot[e + f*x])^(3/2), x, 13, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d^2)/(f*Sqrt[d*Cot[e + f*x]]) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^2*(d*Cot[e + f*x])^(3/2), x, 12, (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^1*(d*Cot[e + f*x])^(3/2), x, 12, (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Tan[e + f*x]^0*(d*Cot[e + f*x])^(3/2), x, 12, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (2*d*Sqrt[d*Cot[e + f*x]])/f - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Cot[e + f*x]^1*(d*Cot[e + f*x])^(3/2), x, 13, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} -{Cot[e + f*x]^2*(d*Cot[e + f*x])^(3/2), x, 14, (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (2*d*Sqrt[d*Cot[e + f*x]])/f - (2*(d*Cot[e + f*x])^(5/2))/(5*d*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]])/(2*Sqrt[2]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[e + f*x]^3/Sqrt[d*Cot[e + f*x]], x, 14, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) + (2*d^2)/(5*f*(d*Cot[e + f*x])^(5/2)) - 2/(f*Sqrt[d*Cot[e + f*x]]) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Tan[e + f*x]^2/Sqrt[d*Cot[e + f*x]], x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) + (2*d)/(3*f*(d*Cot[e + f*x])^(3/2)) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Tan[e + f*x]^1/Sqrt[d*Cot[e + f*x]], x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) + 2/(f*Sqrt[d*Cot[e + f*x]]) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Tan[e + f*x]^0/Sqrt[d*Cot[e + f*x]], x, 11, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Cot[e + f*x]^1/Sqrt[d*Cot[e + f*x]], x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Cot[e + f*x]^2/Sqrt[d*Cot[e + f*x]], x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - (2*Sqrt[d*Cot[e + f*x]])/(d*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} -{Cot[e + f*x]^3/Sqrt[d*Cot[e + f*x]], x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*d^2*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*Sqrt[d]*f)} - - -{Tan[e + f*x]^2/(d*Cot[e + f*x])^(3/2), x, 14, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) + (2*d)/(5*f*(d*Cot[e + f*x])^(5/2)) - 2/(d*f*Sqrt[d*Cot[e + f*x]]) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Tan[e + f*x]^1/(d*Cot[e + f*x])^(3/2), x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) + 2/(3*f*(d*Cot[e + f*x])^(3/2)) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Tan[e + f*x]^0/(d*Cot[e + f*x])^(3/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) + 2/(d*f*Sqrt[d*Cot[e + f*x]]) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Cot[e + f*x]^1/(d*Cot[e + f*x])^(3/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Cot[e + f*x]^2/(d*Cot[e + f*x])^(3/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Cot[e + f*x]^3/(d*Cot[e + f*x])^(3/2), x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - (2*Sqrt[d*Cot[e + f*x]])/(d^2*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Cot[e + f*x]^4/(d*Cot[e + f*x])^(3/2), x, 13, -(ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f)) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - (2*(d*Cot[e + f*x])^(3/2))/(3*d^3*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} -{Cot[e + f*x]^5/(d*Cot[e + f*x])^(3/2), x, 14, ArcTan[1 - (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Cot[e + f*x]])/Sqrt[d]]/(Sqrt[2]*d^(3/2)*f) + (2*Sqrt[d*Cot[e + f*x]])/(d^2*f) - (2*(d*Cot[e + f*x])^(5/2))/(5*d^4*f) + Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] - Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Cot[e + f*x] + Sqrt[2]*Sqrt[d*Cot[e + f*x]]]/(2*Sqrt[2]*d^(3/2)*f)} - - -(* ::Subsection:: *) -(*Integrands of the form (a Cot[e+f x])^(m/2) (b Tan[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cot[e+f x])^m (b Tan[e+f x])^n with n symbolic*) - - -{(Cot[e + f*x])^m*(Tan[e + f*x])^n, x, 3, (1/(f*(1 - m + n)))*(Cot[e + f*x]^m*Hypergeometric2F1[1, (1/2)*(1 - m + n), (1/2)*(3 - m + n), -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))} -{(Cot[e + f*x])^m*(b*Tan[e + f*x])^n, x, 3, (1/(b*f*(1 - m + n)))*(Cot[e + f*x]^m*Hypergeometric2F1[1, (1/2)*(1 - m + n), (1/2)*(3 - m + n), -Tan[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))} -{(a*Cot[e + f*x])^m*(Tan[e + f*x])^n, x, 3, (1/(f*(1 - m + n)))*((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1/2)*(1 - m + n), (1/2)*(3 - m + n), -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))} -{(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^n, x, 3, (1/(b*f*(1 - m + n)))*((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1/2)*(1 - m + n), (1/2)*(3 - m + n), -Tan[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (b Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[e + f*x]^6*Sqrt[d*Tan[e + f*x]], x, 3, (2*(d*Tan[e + f*x])^(3/2))/(3*d*f) + (4*(d*Tan[e + f*x])^(7/2))/(7*d^3*f) + (2*(d*Tan[e + f*x])^(11/2))/(11*d^5*f)} -{Sec[e + f*x]^4*Sqrt[d*Tan[e + f*x]], x, 3, (2*(d*Tan[e + f*x])^(3/2))/(3*d*f) + (2*(d*Tan[e + f*x])^(7/2))/(7*d^3*f)} -{Sec[e + f*x]^2*Sqrt[d*Tan[e + f*x]], x, 2, (2*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{Sec[e + f*x]^0*Sqrt[d*Tan[e + f*x]], x, 11, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(2*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(2*Sqrt[2]*f)} -{Cos[e + f*x]^2*Sqrt[d*Tan[e + f*x]], x, 12, -((Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(4*Sqrt[2]*f)) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(4*Sqrt[2]*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(8*Sqrt[2]*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(8*Sqrt[2]*f) + (Cos[e + f*x]^2*(d*Tan[e + f*x])^(3/2))/(2*d*f)} - -{Sec[e + f*x]^3*Sqrt[d*Tan[e + f*x]], x, 5, -((4*Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(5*f*Sqrt[Sin[2*e + 2*f*x]])) + (4*Cos[e + f*x]*(d*Tan[e + f*x])^(3/2))/(5*d*f) + (2*Sec[e + f*x]*(d*Tan[e + f*x])^(3/2))/(5*d*f)} -{Sec[e + f*x]^1*Sqrt[d*Tan[e + f*x]], x, 4, -((2*Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[Sin[2*e + 2*f*x]])) + (2*Cos[e + f*x]*(d*Tan[e + f*x])^(3/2))/(d*f)} -{Cos[e + f*x]^1*Sqrt[d*Tan[e + f*x]], x, 3, (Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(f*Sqrt[Sin[2*e + 2*f*x]])} -{Cos[e + f*x]^3*Sqrt[d*Tan[e + f*x]], x, 4, (Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(2*f*Sqrt[Sin[2*e + 2*f*x]]) + (Cos[e + f*x]^3*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{Cos[e + f*x]^5*Sqrt[d*Tan[e + f*x]], x, 5, (7*Cos[e + f*x]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Tan[e + f*x]])/(20*f*Sqrt[Sin[2*e + 2*f*x]]) + (7*Cos[e + f*x]^3*(d*Tan[e + f*x])^(3/2))/(30*d*f) + (Cos[e + f*x]^5*(d*Tan[e + f*x])^(3/2))/(5*d*f)} - - -{Sec[a + b*x]^6*(d*Tan[a + b*x])^(3/2), x, 3, (2*(d*Tan[a + b*x])^(5/2))/(5*b*d) + (4*(d*Tan[a + b*x])^(9/2))/(9*b*d^3) + (2*(d*Tan[a + b*x])^(13/2))/(13*b*d^5)} -{Sec[a + b*x]^4*(d*Tan[a + b*x])^(3/2), x, 3, (2*(d*Tan[a + b*x])^(5/2))/(5*b*d) + (2*(d*Tan[a + b*x])^(9/2))/(9*b*d^3)} -{Sec[a + b*x]^2*(d*Tan[a + b*x])^(3/2), x, 2, (2*(d*Tan[a + b*x])^(5/2))/(5*b*d)} -{Sec[a + b*x]^0*(d*Tan[a + b*x])^(3/2), x, 12, (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(Sqrt[2]*b) - (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(Sqrt[2]*b) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(2*Sqrt[2]*b) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(2*Sqrt[2]*b) + (2*d*Sqrt[d*Tan[a + b*x]])/b} -{Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2), x, 12, -((d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b)) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b) - (d*Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(2*b)} - -{Sec[a + b*x]^5*(d*Tan[a + b*x])^(3/2), x, 6, -((4*d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(77*b*Sqrt[d*Tan[a + b*x]])) - (4*d*Sec[a + b*x]*Sqrt[d*Tan[a + b*x]])/(77*b) - (2*d*Sec[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/(77*b) + (2*d*Sec[a + b*x]^5*Sqrt[d*Tan[a + b*x]])/(11*b)} -{Sec[a + b*x]^3*(d*Tan[a + b*x])^(3/2), x, 5, -((2*d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(21*b*Sqrt[d*Tan[a + b*x]])) - (2*d*Sec[a + b*x]*Sqrt[d*Tan[a + b*x]])/(21*b) + (2*d*Sec[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/(7*b)} -{Sec[a + b*x]^1*(d*Tan[a + b*x])^(3/2), x, 4, -((d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*Sqrt[d*Tan[a + b*x]])) + (2*d*Sec[a + b*x]*Sqrt[d*Tan[a + b*x]])/(3*b)} -{Cos[a + b*x]^1*(d*Tan[a + b*x])^(3/2), x, 4, (d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*Sqrt[d*Tan[a + b*x]]) - (d*Cos[a + b*x]*Sqrt[d*Tan[a + b*x]])/b} -{Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2), x, 5, (d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*Sqrt[d*Tan[a + b*x]]) + (d*Cos[a + b*x]*Sqrt[d*Tan[a + b*x]])/(6*b) - (d*Cos[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/(3*b)} -{Cos[a + b*x]^5*(d*Tan[a + b*x])^(3/2), x, 6, (d^2*EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(24*b*Sqrt[d*Tan[a + b*x]]) + (d*Cos[a + b*x]*Sqrt[d*Tan[a + b*x]])/(12*b) + (d*Cos[a + b*x]^3*Sqrt[d*Tan[a + b*x]])/(30*b) - (d*Cos[a + b*x]^5*Sqrt[d*Tan[a + b*x]])/(5*b)} - - -{Sec[e + f*x]^6*(d*Tan[e + f*x])^(5/2), x, 3, (2*(d*Tan[e + f*x])^(7/2))/(7*d*f) + (4*(d*Tan[e + f*x])^(11/2))/(11*d^3*f) + (2*(d*Tan[e + f*x])^(15/2))/(15*d^5*f)} -{Sec[e + f*x]^4*(d*Tan[e + f*x])^(5/2), x, 3, (2*(d*Tan[e + f*x])^(7/2))/(7*d*f) + (2*(d*Tan[e + f*x])^(11/2))/(11*d^3*f)} -{Sec[e + f*x]^2*(d*Tan[e + f*x])^(5/2), x, 2, (2*(d*Tan[e + f*x])^(7/2))/(7*d*f)} -{Sec[e + f*x]^0*(d*Tan[e + f*x])^(5/2), x, 12, (d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*f) - (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(2*Sqrt[2]*f) + (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(2*Sqrt[2]*f) + (2*d*(d*Tan[e + f*x])^(3/2))/(3*f)} -{Cos[e + f*x]^2*(d*Tan[e + f*x])^(5/2), x, 12, -((3*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(4*Sqrt[2]*f)) + (3*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(4*Sqrt[2]*f) + (3*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(8*Sqrt[2]*f) - (3*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(8*Sqrt[2]*f) - (d*Cos[e + f*x]^2*(d*Tan[e + f*x])^(3/2))/(2*f)} -{Cos[e + f*x]^4*(d*Tan[e + f*x])^(5/2), x, 13, -((3*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(32*Sqrt[2]*f)) + (3*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(32*Sqrt[2]*f) + (3*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(64*Sqrt[2]*f) - (3*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(64*Sqrt[2]*f) + (3*d*Cos[e + f*x]^2*(d*Tan[e + f*x])^(3/2))/(16*f) - (d*Cos[e + f*x]^4*(d*Tan[e + f*x])^(3/2))/(4*f)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[e + f*x]^5/Sqrt[d*Tan[e + f*x]], x, 5, (4*EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(7*f*Sqrt[d*Tan[e + f*x]]) + (4*Sec[e + f*x]*Sqrt[d*Tan[e + f*x]])/(7*d*f) + (2*Sec[e + f*x]^3*Sqrt[d*Tan[e + f*x]])/(7*d*f)} -{Sec[e + f*x]^3/Sqrt[d*Tan[e + f*x]], x, 4, (2*EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(3*f*Sqrt[d*Tan[e + f*x]]) + (2*Sec[e + f*x]*Sqrt[d*Tan[e + f*x]])/(3*d*f)} -{Sec[e + f*x]^1/Sqrt[d*Tan[e + f*x]], x, 3, (EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(f*Sqrt[d*Tan[e + f*x]])} -{Cos[e + f*x]^1/Sqrt[d*Tan[e + f*x]], x, 4, (EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[d*Tan[e + f*x]]) + (Cos[e + f*x]*Sqrt[d*Tan[e + f*x]])/(d*f)} -{Cos[e + f*x]^3/Sqrt[d*Tan[e + f*x]], x, 5, (5*EllipticF[e - Pi/4 + f*x, 2]*Sec[e + f*x]*Sqrt[Sin[2*e + 2*f*x]])/(12*f*Sqrt[d*Tan[e + f*x]]) + (5*Cos[e + f*x]*Sqrt[d*Tan[e + f*x]])/(6*d*f) + (Cos[e + f*x]^3*Sqrt[d*Tan[e + f*x]])/(3*d*f)} - - -{Sec[a + b*x]^6/(d*Tan[a + b*x])^(3/2), x, 3, -(2/(b*d*Sqrt[d*Tan[a + b*x]])) + (4*(d*Tan[a + b*x])^(3/2))/(3*b*d^3) + (2*(d*Tan[a + b*x])^(7/2))/(7*b*d^5)} -{Sec[a + b*x]^4/(d*Tan[a + b*x])^(3/2), x, 3, -(2/(b*d*Sqrt[d*Tan[a + b*x]])) + (2*(d*Tan[a + b*x])^(3/2))/(3*b*d^3)} -{Sec[a + b*x]^2/(d*Tan[a + b*x])^(3/2), x, 2, -(2/(b*d*Sqrt[d*Tan[a + b*x]]))} -{Sec[a + b*x]^0/(d*Tan[a + b*x])^(3/2), x, 12, ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(Sqrt[2]*b*d^(3/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(Sqrt[2]*b*d^(3/2)) - Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(2*Sqrt[2]*b*d^(3/2)) + Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]]/(2*Sqrt[2]*b*d^(3/2)) - 2/(b*d*Sqrt[d*Tan[a + b*x]])} -{Cos[a + b*x]^2/(d*Tan[a + b*x])^(3/2), x, 13, (5*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(4*Sqrt[2]*b*d^(3/2)) - (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b*d^(3/2)) + (5*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(8*Sqrt[2]*b*d^(3/2)) - 5/(2*b*d*Sqrt[d*Tan[a + b*x]]) + Cos[a + b*x]^2/(2*b*d*Sqrt[d*Tan[a + b*x]])} - -{Sec[a + b*x]^5/(d*Tan[a + b*x])^(3/2), x, 6, -((2*Sec[a + b*x]^3)/(b*d*Sqrt[d*Tan[a + b*x]])) - (24*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(5*b*d^2*Sqrt[Sin[2*a + 2*b*x]]) + (24*Cos[a + b*x]*(d*Tan[a + b*x])^(3/2))/(5*b*d^3) + (12*Sec[a + b*x]*(d*Tan[a + b*x])^(3/2))/(5*b*d^3)} -{Sec[a + b*x]^3/(d*Tan[a + b*x])^(3/2), x, 5, -((2*Sec[a + b*x])/(b*d*Sqrt[d*Tan[a + b*x]])) - (4*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]]) + (4*Cos[a + b*x]*(d*Tan[a + b*x])^(3/2))/(b*d^3)} -{Sec[a + b*x]^1/(d*Tan[a + b*x])^(3/2), x, 4, -((2*Cos[a + b*x])/(b*d*Sqrt[d*Tan[a + b*x]])) - (2*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^1/(d*Tan[a + b*x])^(3/2), x, 4, -((2*Cos[a + b*x])/(b*d*Sqrt[d*Tan[a + b*x]])) - (3*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(b*d^2*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^3/(d*Tan[a + b*x])^(3/2), x, 5, -((2*Cos[a + b*x]^3)/(b*d*Sqrt[d*Tan[a + b*x]])) - (7*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(2*b*d^2*Sqrt[Sin[2*a + 2*b*x]]) - (7*Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(3*b*d^3)} -{Cos[a + b*x]^5/(d*Tan[a + b*x])^(3/2), x, 6, -((2*Cos[a + b*x]^5)/(b*d*Sqrt[d*Tan[a + b*x]])) - (77*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(20*b*d^2*Sqrt[Sin[2*a + 2*b*x]]) - (77*Cos[a + b*x]^3*(d*Tan[a + b*x])^(3/2))/(30*b*d^3) - (11*Cos[a + b*x]^5*(d*Tan[a + b*x])^(3/2))/(5*b*d^3)} - - -{Sec[a + b*x]^1/(d*Tan[a + b*x])^(5/2), x, 4, -((2*Sec[a + b*x])/(3*b*d*(d*Tan[a + b*x])^(3/2))) - (EllipticF[a - Pi/4 + b*x, 2]*Sec[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2*Sqrt[d*Tan[a + b*x]])} - - -{Sec[a + b*x]^3/(d*Tan[a + b*x])^(7/2), x, 5, -((2*Sec[a + b*x])/(5*b*d*(d*Tan[a + b*x])^(5/2))) - (4*Cos[a + b*x])/(5*b*d^3*Sqrt[d*Tan[a + b*x]]) - (4*Cos[a + b*x]*EllipticE[a - Pi/4 + b*x, 2]*Sqrt[d*Tan[a + b*x]])/(5*b*d^4*Sqrt[Sin[2*a + 2*b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/3) (b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[e + f*x]^(4/3)*Tan[e + f*x]^2, x, 2, (3*Hypergeometric2F1[-(7/6), -(1/2), -(1/6), Cos[e + f*x]^2]*Sec[e + f*x]^(7/3)*Sin[e + f*x])/(7*f*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^(2/3)*Tan[e + f*x]^2, x, 2, (3*Hypergeometric2F1[-(5/6), -(1/2), 1/6, Cos[e + f*x]^2]*Sec[e + f*x]^(5/3)*Sin[e + f*x])/(5*f*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^(1/3)*Tan[e + f*x]^2, x, 2, (3*Hypergeometric2F1[-(2/3), -(1/2), 1/3, Cos[e + f*x]^2]*Sec[e + f*x]^(4/3)*Sin[e + f*x])/(4*f*Sqrt[Sin[e + f*x]^2])} -{Tan[e + f*x]^2/Sec[e + f*x]^(1/3), x, 2, (3*Hypergeometric2F1[-(1/2), -(1/3), 2/3, Cos[e + f*x]^2]*Sec[e + f*x]^(2/3)*Sin[e + f*x])/(2*f*Sqrt[Sin[e + f*x]^2])} -{Tan[e + f*x]^2/Sec[e + f*x]^(2/3), x, 2, (3*Hypergeometric2F1[-(1/2), -(1/6), 5/6, Cos[e + f*x]^2]*Sec[e + f*x]^(1/3)*Sin[e + f*x])/(f*Sqrt[Sin[e + f*x]^2])} - - -{Sec[e + f*x]^(4/3)*Tan[e + f*x]^4, x, 2, (3*Hypergeometric2F1[-(13/6), -(3/2), -(7/6), Cos[e + f*x]^2]*Sec[e + f*x]^(13/3)*Sin[e + f*x])/(13*f*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^(2/3)*Tan[e + f*x]^4, x, 2, (3*Hypergeometric2F1[-(11/6), -(3/2), -(5/6), Cos[e + f*x]^2]*Sec[e + f*x]^(11/3)*Sin[e + f*x])/(11*f*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^(1/3)*Tan[e + f*x]^4, x, 2, (3*Hypergeometric2F1[-(5/3), -(3/2), -(2/3), Cos[e + f*x]^2]*Sec[e + f*x]^(10/3)*Sin[e + f*x])/(10*f*Sqrt[Sin[e + f*x]^2])} -{Tan[e + f*x]^4/Sec[e + f*x]^(1/3), x, 2, (3*Hypergeometric2F1[-(3/2), -(4/3), -(1/3), Cos[e + f*x]^2]*Sec[e + f*x]^(8/3)*Sin[e + f*x])/(8*f*Sqrt[Sin[e + f*x]^2])} -{Tan[e + f*x]^4/Sec[e + f*x]^(2/3), x, 2, (3*Hypergeometric2F1[-(3/2), -(7/6), -(1/6), Cos[e + f*x]^2]*Sec[e + f*x]^(7/3)*Sin[e + f*x])/(7*f*Sqrt[Sin[e + f*x]^2])} - - -{(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^2, x, 1, ((Cos[e + f*x]^2)^(13/6)*Hypergeometric2F1[3/2, 13/6, 5/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^3)/(3*f)} -{(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^2, x, 1, ((Cos[e + f*x]^2)^(11/6)*Hypergeometric2F1[3/2, 11/6, 5/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^3)/(3*f)} -{(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^2, x, 1, ((Cos[e + f*x]^2)^(5/3)*Hypergeometric2F1[3/2, 5/3, 5/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^3)/(3*f)} -{Tan[e + f*x]^2/(d*Sec[e + f*x])^(1/3), x, 1, ((Cos[e + f*x]^2)^(4/3)*Hypergeometric2F1[4/3, 3/2, 5/2, Sin[e + f*x]^2]*Tan[e + f*x]^3)/(3*f*(d*Sec[e + f*x])^(1/3))} -{Tan[e + f*x]^2/(d*Sec[e + f*x])^(2/3), x, 1, ((Cos[e + f*x]^2)^(7/6)*Hypergeometric2F1[7/6, 3/2, 5/2, Sin[e + f*x]^2]*Tan[e + f*x]^3)/(3*f*(d*Sec[e + f*x])^(2/3))} - - -{(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^4, x, 1, ((Cos[e + f*x]^2)^(19/6)*Hypergeometric2F1[5/2, 19/6, 7/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(4/3)*Tan[e + f*x]^5)/(5*f)} -{(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^4, x, 1, ((Cos[e + f*x]^2)^(17/6)*Hypergeometric2F1[5/2, 17/6, 7/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(2/3)*Tan[e + f*x]^5)/(5*f)} -{(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^4, x, 1, ((Cos[e + f*x]^2)^(8/3)*Hypergeometric2F1[5/2, 8/3, 7/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^5)/(5*f)} -{Tan[e + f*x]^4/(d*Sec[e + f*x])^(1/3), x, 1, ((Cos[e + f*x]^2)^(7/3)*Hypergeometric2F1[7/3, 5/2, 7/2, Sin[e + f*x]^2]*Tan[e + f*x]^5)/(5*f*(d*Sec[e + f*x])^(1/3))} -{Tan[e + f*x]^4/(d*Sec[e + f*x])^(2/3), x, 1, ((Cos[e + f*x]^2)^(13/6)*Hypergeometric2F1[13/6, 5/2, 7/2, Sin[e + f*x]^2]*Tan[e + f*x]^5)/(5*f*(d*Sec[e + f*x])^(2/3))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/2) (b Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]], x, 7, -((Sqrt[b]*d^3*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])) + (Sqrt[b]*d^3*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) + (d^2*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2))/(2*b*f)} -{(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]], x, 4, -((d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])) + (d^2*(b*Tan[e + f*x])^(3/2))/(b*f*Sqrt[d*Sec[e + f*x]])} -{(d*Sec[e + f*x])^(1/2)*Sqrt[b*Tan[e + f*x]], x, 6, -((Sqrt[b]*d*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])) + (Sqrt[b]*d*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])} -{Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(1/2), x, 3, (2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])} -{Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(3/2), x, 1, (2*(b*Tan[e + f*x])^(3/2))/(3*b*f*(d*Sec[e + f*x])^(3/2))} -{Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(5/2), x, 4, (4*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(5*d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) + (2*(b*Tan[e + f*x])^(3/2))/(5*b*f*(d*Sec[e + f*x])^(5/2))} -{Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(7/2), x, 2, (2*(b*Tan[e + f*x])^(3/2))/(7*b*f*(d*Sec[e + f*x])^(7/2)) + (8*(b*Tan[e + f*x])^(3/2))/(21*b*d^2*f*(d*Sec[e + f*x])^(3/2))} -{Sqrt[b*Tan[e + f*x]]/(d*Sec[e + f*x])^(9/2), x, 5, (8*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(15*d^4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) + (2*(b*Tan[e + f*x])^(3/2))/(9*b*f*(d*Sec[e + f*x])^(9/2)) + (4*(b*Tan[e + f*x])^(3/2))/(15*b*d^2*f*(d*Sec[e + f*x])^(5/2))} - - -{(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2), x, 5, -((b^2*d^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(6*f*Sqrt[b*Tan[e + f*x]])) - (b*d^2*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(6*f) + (b*(d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]])/(3*f)} -{(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2), x, 7, -((b^(3/2)*d*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(4*f*Sqrt[b*Tan[e + f*x]])) - (b^(3/2)*d*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(4*f*Sqrt[b*Tan[e + f*x]]) + (b*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])/(2*f)} -{(d*Sec[e + f*x])^(1/2)*(b*Tan[e + f*x])^(3/2), x, 4, -((b^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Tan[e + f*x]])) + (b*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])/f} -{(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(1/2), x, 7, -((2*d*Csc[e + f*x]*(b*Tan[e + f*x])^(3/2))/(f*(d*Sec[e + f*x])^(3/2))) + (b^(3/2)*d*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*(b*Tan[e + f*x])^(3/2))/(f*(d*Sec[e + f*x])^(3/2)*(b*Sin[e + f*x])^(3/2)) + (b^(3/2)*d*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*(b*Tan[e + f*x])^(3/2))/(f*(d*Sec[e + f*x])^(3/2)*(b*Sin[e + f*x])^(3/2))} -{(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(3/2), x, 4, (2*b^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*d^2*f*Sqrt[b*Tan[e + f*x]]) - (2*b*Sqrt[b*Tan[e + f*x]])/(3*f*(d*Sec[e + f*x])^(3/2))} -{(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(5/2), x, 1, (2*(b*Tan[e + f*x])^(5/2))/(5*b*f*(d*Sec[e + f*x])^(5/2))} -{(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(7/2), x, 5, (4*b^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(21*d^4*f*Sqrt[b*Tan[e + f*x]]) - (2*b*Sqrt[b*Tan[e + f*x]])/(7*f*(d*Sec[e + f*x])^(7/2)) + (2*b*Sqrt[b*Tan[e + f*x]])/(21*d^2*f*(d*Sec[e + f*x])^(3/2))} -{(b*Tan[e + f*x])^(3/2)/(d*Sec[e + f*x])^(9/2), x, 3, -((2*b*Sqrt[b*Tan[e + f*x]])/(9*f*(d*Sec[e + f*x])^(9/2))) + (2*b*Sqrt[b*Tan[e + f*x]])/(45*d^2*f*(d*Sec[e + f*x])^(5/2)) + (8*b*Sqrt[b*Tan[e + f*x]])/(45*d^4*f*Sqrt[d*Sec[e + f*x]])} - - -{(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(5/2), x, 8, (3*b^(5/2)*d^3*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(32*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) - (3*b^(5/2)*d^3*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(32*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) - (3*b*d^2*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2))/(16*f) + (b*(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2))/(4*f)} -{(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2), x, 5, (b^2*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) - (b*d^2*(b*Tan[e + f*x])^(3/2))/(2*f*Sqrt[d*Sec[e + f*x]]) + (b*(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2))/(3*f)} -{(d*Sec[e + f*x])^(1/2)*(b*Tan[e + f*x])^(5/2), x, 7, (3*b^(5/2)*d*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) - (3*b^(5/2)*d*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) + (b*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2))/(2*f)} -{(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(1/2), x, 4, -((3*b^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])) + (b*(b*Tan[e + f*x])^(3/2))/(f*Sqrt[d*Sec[e + f*x]])} -{(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(3/2), x, 7, -((b^(5/2)*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(d*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])) + (b^(5/2)*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(d*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) - (2*b*(b*Tan[e + f*x])^(3/2))/(3*f*(d*Sec[e + f*x])^(3/2))} -{(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(5/2), x, 4, (6*b^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(5*d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) - (2*b*(b*Tan[e + f*x])^(3/2))/(5*f*(d*Sec[e + f*x])^(5/2))} -{(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(7/2), x, 1, (2*(b*Tan[e + f*x])^(7/2))/(7*b*f*(d*Sec[e + f*x])^(7/2))} -{(b*Tan[e + f*x])^(5/2)/(d*Sec[e + f*x])^(9/2), x, 5, (4*b^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(15*d^4*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) - (2*b*(b*Tan[e + f*x])^(3/2))/(9*f*(d*Sec[e + f*x])^(9/2)) + (2*b*(b*Tan[e + f*x])^(3/2))/(15*d^2*f*(d*Sec[e + f*x])^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*Sec[e + f*x])^(7/2)/Sqrt[b*Tan[e + f*x]], x, 7, (3*d^3*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(4*Sqrt[b]*f*Sqrt[b*Tan[e + f*x]]) + (3*d^3*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(4*Sqrt[b]*f*Sqrt[b*Tan[e + f*x]]) + (d^2*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])/(2*b*f)} -{(d*Sec[e + f*x])^(5/2)/Sqrt[b*Tan[e + f*x]], x, 4, (d^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Tan[e + f*x]]) + (d^2*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])/(b*f)} -{(d*Sec[e + f*x])^(3/2)/Sqrt[b*Tan[e + f*x]], x, 6, (d*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(Sqrt[b]*f*Sqrt[b*Tan[e + f*x]]) + (d*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(Sqrt[b]*f*Sqrt[b*Tan[e + f*x]])} -{(d*Sec[e + f*x])^(1/2)/Sqrt[b*Tan[e + f*x]], x, 3, (2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(f*Sqrt[b*Tan[e + f*x]])} -{1/((d*Sec[e + f*x])^(1/2)*Sqrt[b*Tan[e + f*x]]), x, 1, (2*Sqrt[b*Tan[e + f*x]])/(b*f*Sqrt[d*Sec[e + f*x]])} -{1/((d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]), x, 4, (4*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*d^2*f*Sqrt[b*Tan[e + f*x]]) + (2*Sqrt[b*Tan[e + f*x]])/(3*b*f*(d*Sec[e + f*x])^(3/2))} -{1/((d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]]), x, 2, (2*Sqrt[b*Tan[e + f*x]])/(5*b*f*(d*Sec[e + f*x])^(5/2)) + (8*Sqrt[b*Tan[e + f*x]])/(5*b*d^2*f*Sqrt[d*Sec[e + f*x]])} - - -{(d*Sec[e + f*x])^(5/2)/(b*Tan[e + f*x])^(3/2), x, 7, -((2*d^2*Sqrt[d*Sec[e + f*x]])/(b*f*Sqrt[b*Tan[e + f*x]])) - (d^3*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(b^(3/2)*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]]) + (d^3*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[b*Tan[e + f*x]])/(b^(3/2)*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])} -{(d*Sec[e + f*x])^(3/2)/(b*Tan[e + f*x])^(3/2), x, 4, -((2*d^2)/(b*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])) - (2*d^2*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(b^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])} -{(d*Sec[e + f*x])^(1/2)/(b*Tan[e + f*x])^(3/2), x, 1, -((2*Sqrt[d*Sec[e + f*x]])/(b*f*Sqrt[b*Tan[e + f*x]]))} -{1/((d*Sec[e + f*x])^(1/2)*(b*Tan[e + f*x])^(3/2)), x, 4, -(2/(b*f*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]])) - (4*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(b^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])} -{1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2)), x, 2, 2/(3*b*f*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]]) - (8*Sqrt[d*Sec[e + f*x]])/(3*b*d^2*f*Sqrt[b*Tan[e + f*x]]), -(2/(b*f*(d*Sec[e + f*x])^(3/2)*Sqrt[b*Tan[e + f*x]])) - (8*(b*Tan[e + f*x])^(3/2))/(3*b^3*f*(d*Sec[e + f*x])^(3/2))} -{1/((d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2)), x, 5, -(2/(b*f*(d*Sec[e + f*x])^(5/2)*Sqrt[b*Tan[e + f*x]])) - (24*EllipticE[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[b*Tan[e + f*x]])/(5*b^2*d^2*f*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]) - (12*(b*Tan[e + f*x])^(3/2))/(5*b^3*f*(d*Sec[e + f*x])^(5/2))} - - -{(d*Sec[e + f*x])^(7/2)/(b*Tan[e + f*x])^(5/2), x, 7, -((2*d^2*(d*Sec[e + f*x])^(3/2))/(3*b*f*(b*Tan[e + f*x])^(3/2))) + (d^3*ArcTan[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(b^(5/2)*f*Sqrt[b*Tan[e + f*x]]) + (d^3*ArcTanh[Sqrt[b*Sin[e + f*x]]/Sqrt[b]]*Sqrt[d*Sec[e + f*x]]*Sqrt[b*Sin[e + f*x]])/(b^(5/2)*f*Sqrt[b*Tan[e + f*x]])} -{(d*Sec[e + f*x])^(5/2)/(b*Tan[e + f*x])^(5/2), x, 4, -((2*d^2*Sqrt[d*Sec[e + f*x]])/(3*b*f*(b*Tan[e + f*x])^(3/2))) + (2*d^2*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*b^2*f*Sqrt[b*Tan[e + f*x]])} -{(d*Sec[e + f*x])^(3/2)/(b*Tan[e + f*x])^(5/2), x, 1, -((2*(d*Sec[e + f*x])^(3/2))/(3*b*f*(b*Tan[e + f*x])^(3/2)))} -{(d*Sec[e + f*x])^(1/2)/(b*Tan[e + f*x])^(5/2), x, 4, -((2*Sqrt[d*Sec[e + f*x]])/(3*b*f*(b*Tan[e + f*x])^(3/2))) - (4*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*b^2*f*Sqrt[b*Tan[e + f*x]])} -{1/((d*Sec[e + f*x])^(1/2)*(b*Tan[e + f*x])^(5/2)), x, 2, -(2/(3*b*f*Sqrt[d*Sec[e + f*x]]*(b*Tan[e + f*x])^(3/2))) - (8*Sqrt[b*Tan[e + f*x]])/(3*b^3*f*Sqrt[d*Sec[e + f*x]])} -{1/((d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(5/2)), x, 5, -(2/(3*b*f*(d*Sec[e + f*x])^(3/2)*(b*Tan[e + f*x])^(3/2))) - (8*EllipticF[(1/2)*(e - Pi/2 + f*x), 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*b^2*d^2*f*Sqrt[b*Tan[e + f*x]]) - (4*Sqrt[b*Tan[e + f*x]])/(3*b^3*f*(d*Sec[e + f*x])^(3/2))} -{1/((d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(5/2)), x, 3, -(2/(3*b*f*(d*Sec[e + f*x])^(5/2)*(b*Tan[e + f*x])^(3/2))) - (16*Sqrt[b*Tan[e + f*x]])/(15*b^3*f*(d*Sec[e + f*x])^(5/2)) - (64*Sqrt[b*Tan[e + f*x]])/(15*b^3*d^2*f*Sqrt[d*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/3) (b Tan[e+f x])^(n/2)*) - - -{Sqrt[d*Tan[e + f*x]]*(b*Sec[e + f*x])^(4/3), x, 1, (2*(Cos[e + f*x]^2)^(17/12)*Hypergeometric2F1[3/4, 17/12, 7/4, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{Sqrt[d*Tan[e + f*x]]*(b*Sec[e + f*x])^(1/3), x, 1, (2*(Cos[e + f*x]^2)^(11/12)*Hypergeometric2F1[3/4, 11/12, 7/4, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(1/3)*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(1/3), x, 1, (2*(Cos[e + f*x]^2)^(7/12)*Hypergeometric2F1[7/12, 3/4, 7/4, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(3*d*f*(b*Sec[e + f*x])^(1/3))} -{Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(4/3), x, 1, (2*(Cos[e + f*x]^2)^(1/12)*Hypergeometric2F1[1/12, 3/4, 7/4, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(3*d*f*(b*Sec[e + f*x])^(4/3))} - - -{(d*Tan[e + f*x])^(3/2)*(b*Sec[e + f*x])^(4/3), x, 1, (2*(Cos[e + f*x]^2)^(23/12)*Hypergeometric2F1[5/4, 23/12, 9/4, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(4/3)*(d*Tan[e + f*x])^(5/2))/(5*d*f)} -{(d*Tan[e + f*x])^(3/2)*(b*Sec[e + f*x])^(1/3), x, 1, (2*(Cos[e + f*x]^2)^(17/12)*Hypergeometric2F1[5/4, 17/12, 9/4, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(1/3)*(d*Tan[e + f*x])^(5/2))/(5*d*f)} -{(d*Tan[e + f*x])^(3/2)/(b*Sec[e + f*x])^(1/3), x, 1, (2*(Cos[e + f*x]^2)^(13/12)*Hypergeometric2F1[13/12, 5/4, 9/4, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(5*d*f*(b*Sec[e + f*x])^(1/3))} -{(d*Tan[e + f*x])^(3/2)/(b*Sec[e + f*x])^(4/3), x, 1, (2*(Cos[e + f*x]^2)^(7/12)*Hypergeometric2F1[7/12, 5/4, 9/4, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(5*d*f*(b*Sec[e + f*x])^(4/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/2) (b Tan[e+f x])^(n/3)*) - - -{Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(4/3), x, 1, (3*(Cos[e + f*x]^2)^(17/12)*Hypergeometric2F1[7/6, 17/12, 13/6, Sin[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(7/3))/(7*d*f)} -{Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(1/3), x, 1, (3*(Cos[e + f*x]^2)^(11/12)*Hypergeometric2F1[2/3, 11/12, 5/3, Sin[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(4/3))/(4*d*f)} -{Sqrt[b*Sec[e + f*x]]/(d*Tan[e + f*x])^(1/3), x, 1, (3*(Cos[e + f*x]^2)^(7/12)*Hypergeometric2F1[1/3, 7/12, 4/3, Sin[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(d*Tan[e + f*x])^(2/3))/(2*d*f)} -{Sqrt[b*Sec[e + f*x]]/(d*Tan[e + f*x])^(4/3), x, 1, -((3*(Cos[e + f*x]^2)^(1/12)*Hypergeometric2F1[-(1/6), 1/12, 5/6, Sin[e + f*x]^2]*Sqrt[b*Sec[e + f*x]])/(d*f*(d*Tan[e + f*x])^(1/3)))} - - -{(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3), x, 1, (3*(Cos[e + f*x]^2)^(23/12)*Hypergeometric2F1[7/6, 23/12, 13/6, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(7/3))/(7*d*f)} -{(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(1/3), x, 1, (3*(Cos[e + f*x]^2)^(17/12)*Hypergeometric2F1[2/3, 17/12, 5/3, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(4/3))/(4*d*f)} -{(b*Sec[e + f*x])^(3/2)/(d*Tan[e + f*x])^(1/3), x, 1, (3*(Cos[e + f*x]^2)^(13/12)*Hypergeometric2F1[1/3, 13/12, 4/3, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(d*Tan[e + f*x])^(2/3))/(2*d*f)} -{(b*Sec[e + f*x])^(3/2)/(d*Tan[e + f*x])^(4/3), x, 1, -((3*(Cos[e + f*x]^2)^(7/12)*Hypergeometric2F1[-(1/6), 7/12, 5/6, Sin[e + f*x]^2]*(b*Sec[e + f*x])^(3/2))/(d*f*(d*Tan[e + f*x])^(1/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Tan[e+f x])^n with m symbolic*) - - -{(b*Sec[e + f*x])^m*Tan[e + f*x]^5, x, 3, (b*Sec[e + f*x])^m/(f*m) - (2*(b*Sec[e + f*x])^(2 + m))/(b^2*f*(2 + m)) + (b*Sec[e + f*x])^(4 + m)/(b^4*f*(4 + m))} -{(b*Sec[e + f*x])^m*Tan[e + f*x]^3, x, 3, -((b*Sec[e + f*x])^m/(f*m)) + (b*Sec[e + f*x])^(2 + m)/(b^2*f*(2 + m))} -{(b*Sec[e + f*x])^m*Tan[e + f*x]^1, x, 2, (b*Sec[e + f*x])^m/(f*m)} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^1, x, 2, -((Hypergeometric2F1[1, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m))} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^3, x, 2, (Hypergeometric2F1[2, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m)} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^5, x, 2, -((Hypergeometric2F1[3, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m)/(f*m))} - -{(b*Sec[e + f*x])^m*Tan[e + f*x]^4, x, 1, ((Cos[e + f*x]^2)^((5 + m)/2)*Hypergeometric2F1[5/2, (5 + m)/2, 7/2, Sin[e + f*x]^2]*(b*Sec[e + f*x])^m*Tan[e + f*x]^5)/(5*f)} -{(b*Sec[e + f*x])^m*Tan[e + f*x]^2, x, 1, ((Cos[e + f*x]^2)^((3 + m)/2)*Hypergeometric2F1[3/2, (3 + m)/2, 5/2, Sin[e + f*x]^2]*(b*Sec[e + f*x])^m*Tan[e + f*x]^3)/(3*f)} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^2, x, 1, -(((Cos[e + f*x]^2)^((1/2)*(-1 + m))*Cot[e + f*x]*Hypergeometric2F1[-(1/2), (1/2)*(-1 + m), 1/2, Sin[e + f*x]^2]*(b*Sec[e + f*x])^m)/f)} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^4, x, 1, -(((Cos[e + f*x]^2)^((1/2)*(-3 + m))*Cot[e + f*x]^3*Hypergeometric2F1[-(3/2), (1/2)*(-3 + m), -(1/2), Sin[e + f*x]^2]*(b*Sec[e + f*x])^m)/(3*f))} -{(b*Sec[e + f*x])^m*Cot[e + f*x]^6, x, 1, -(((Cos[e + f*x]^2)^((1/2)*(-5 + m))*Cot[e + f*x]^5*Hypergeometric2F1[-(5/2), (1/2)*(-5 + m), -(3/2), Sin[e + f*x]^2]*(b*Sec[e + f*x])^m)/(5*f))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Tan[e+f x])^n with n symbolic*) - - -{(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^n, x, 1, ((Cos[e + f*x]^2)^((1/2)*(1 + m + n))*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 + m + n), (3 + n)/2, Sin[e + f*x]^2]*(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 + n))} - - -{Sec[a + b*x]^6*(d*Tan[a + b*x])^n, x, 3, (d*Tan[a + b*x])^(1 + n)/(b*d*(1 + n)) + (2*(d*Tan[a + b*x])^(3 + n))/(b*d^3*(3 + n)) + (d*Tan[a + b*x])^(5 + n)/(b*d^5*(5 + n))} -{Sec[a + b*x]^4*(d*Tan[a + b*x])^n, x, 3, (d*Tan[a + b*x])^(n + 1)/(b*d*(1 + n)) + (d*Tan[a + b*x])^(n + 3)/(b*d^3*(3 + n))} -{Sec[a + b*x]^2*(d*Tan[a + b*x])^n, x, 2, (d*Tan[a + b*x])^(1 + n)/(b*d*(1 + n))} -{Sec[a + b*x]^0*(d*Tan[a + b*x])^n, x, 2, (Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Cos[a + b*x]^2*(d*Tan[a + b*x])^n, x, 2, (Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, -Tan[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Cos[a + b*x]^4*(d*Tan[a + b*x])^n, x, 2, (Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, -Tan[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} - -{Sec[a + b*x]^5*(d*Tan[a + b*x])^n, x, 1, ((Cos[a + b*x]^2)^((6 + n)/2)*Hypergeometric2F1[(1 + n)/2, (6 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]^5*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Sec[a + b*x]^3*(d*Tan[a + b*x])^n, x, 1, ((Cos[a + b*x]^2)^((4 + n)/2)*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]^3*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Sec[a + b*x]^1*(d*Tan[a + b*x])^n, x, 1, ((Cos[a + b*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Cos[a + b*x]^1*(d*Tan[a + b*x])^n, x, 1, (Cos[a + b*x]*(Cos[a + b*x]^2)^(n/2)*Hypergeometric2F1[n/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} -{Cos[a + b*x]^3*(d*Tan[a + b*x])^n, x, 1, (Cos[a + b*x]^3*(Cos[a + b*x]^2)^((1/2)*(-2 + n))*Hypergeometric2F1[(1/2)*(-2 + n), (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Tan[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (a Csc[e+f x])^(m/2) (b Tan[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Tan[e+f x])^n with m symbolic*) - - -{(b*Csc[e + f*x])^m*Tan[e + f*x]^3, x, 2, -(((b*Csc[e + f*x])^m*Hypergeometric2F1[2, m/2, (2 + m)/2, Csc[e + f*x]^2])/(f*m))} -{(b*Csc[e + f*x])^m*Tan[e + f*x]^1, x, 2, ((b*Csc[e + f*x])^m*Hypergeometric2F1[1, m/2, (2 + m)/2, Csc[e + f*x]^2])/(f*m)} -{(b*Csc[e + f*x])^m*Cot[e + f*x]^1, x, 2, -((b*Csc[e + f*x])^m/(f*m))} -{(b*Csc[e + f*x])^m*Cot[e + f*x]^3, x, 3, (b*Csc[e + f*x])^m/(f*m) - (b*Csc[e + f*x])^(2 + m)/(b^2*f*(2 + m))} -{(b*Csc[e + f*x])^m*Cot[e + f*x]^5, x, 3, -((b*Csc[e + f*x])^m/(f*m)) + (2*(b*Csc[e + f*x])^(2 + m))/(b^2*f*(2 + m)) - (b*Csc[e + f*x])^(4 + m)/(b^4*f*(4 + m))} - -{(b*Csc[e + f*x])^m*Tan[e + f*x]^4, x, 1, ((b*Csc[e + f*x])^m*Hypergeometric2F1[-(3/2), (1/2)*(-3 + m), -(1/2), Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((1/2)*(-3 + m))*Tan[e + f*x]^3)/(3*f)} -{(b*Csc[e + f*x])^m*Tan[e + f*x]^2, x, 1, ((b*Csc[e + f*x])^m*Hypergeometric2F1[-(1/2), (1/2)*(-1 + m), 1/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((1/2)*(-1 + m))*Tan[e + f*x])/f} -{(b*Csc[e + f*x])^m*Cot[e + f*x]^2, x, 1, -((Cot[e + f*x]^3*(b*Csc[e + f*x])^m*Hypergeometric2F1[3/2, (3 + m)/2, 5/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((3 + m)/2))/(3*f))} -{(b*Csc[e + f*x])^m*Cot[e + f*x]^4, x, 1, -((Cot[e + f*x]^5*(b*Csc[e + f*x])^m*Hypergeometric2F1[5/2, (5 + m)/2, 7/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((5 + m)/2))/(5*f))} - - -{(b*Csc[e + f*x])^m*(d*Tan[e + f*x])^(3/2), x, 3, (2*(Cos[e + f*x]^2)^(5/4)*(b*Csc[e + f*x])^m*Hypergeometric2F1[5/4, (1/4)*(5 - 2*m), (1/4)*(9 - 2*m), Sin[e + f*x]^2]*(d*Tan[e + f*x])^(5/2))/(d*f*(5 - 2*m))} -{(b*Csc[e + f*x])^m*(d*Tan[e + f*x])^(1/2), x, 3, (2*(Cos[e + f*x]^2)^(3/4)*(b*Csc[e + f*x])^m*Hypergeometric2F1[3/4, (1/4)*(3 - 2*m), (1/4)*(7 - 2*m), Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(d*f*(3 - 2*m))} -{(b*Csc[e + f*x])^m/(d*Tan[e + f*x])^(1/2), x, 3, (2*(Cos[e + f*x]^2)^(1/4)*(b*Csc[e + f*x])^m*Hypergeometric2F1[1/4, (1/4)*(1 - 2*m), (1/4)*(5 - 2*m), Sin[e + f*x]^2]*Sqrt[d*Tan[e + f*x]])/(d*f*(1 - 2*m))} -{(b*Csc[e + f*x])^m/(d*Tan[e + f*x])^(3/2), x, 3, -((2*(b*Csc[e + f*x])^m*Hypergeometric2F1[-(1/4), (1/4)*(-1 - 2*m), (1/4)*(3 - 2*m), Sin[e + f*x]^2])/(d*f*(1 + 2*m)*(Cos[e + f*x]^2)^(1/4)*Sqrt[d*Tan[e + f*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Tan[e+f x])^n with n symbolic*) - - -{(a*Csc[e + f*x])^m*(b*Tan[e + f*x])^n, x, 3, ((Cos[e + f*x]^2)^((1 + n)/2)*(a*Csc[e + f*x])^m*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 - m + n), (1/2)*(3 - m + n), Sin[e + f*x]^2]*(b*Tan[e + f*x])^(1 + n))/(b*f*(1 - m + n))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m deleted file mode 100644 index f3ba7c8..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.2 (d sec)^m (a+b tan)^n.m +++ /dev/null @@ -1,1063 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n when a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^10*(a + I*a*Tan[c + d*x]), x, 3, (I*a*Sec[c + d*x]^10)/(10*d) + (a*Tan[c + d*x])/d + (4*a*Tan[c + d*x]^3)/(3*d) + (6*a*Tan[c + d*x]^5)/(5*d) + (4*a*Tan[c + d*x]^7)/(7*d) + (a*Tan[c + d*x]^9)/(9*d)} -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x]), x, 3, (I*a*Sec[c + d*x]^8)/(8*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/d + (3*a*Tan[c + d*x]^5)/(5*d) + (a*Tan[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x]), x, 3, (I*a*Sec[c + d*x]^6)/(6*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x]), x, 3, (I*a*Sec[c + d*x]^4)/(4*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x]), x, 3, -((I*(a + I*a*Tan[c + d*x])^2)/(2*a*d)), (I*a*Sec[c + d*x]^2)/(2*d) + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^0*(a + I*a*Tan[c + d*x]), x, 2, a*x - (I*a*Log[Cos[c + d*x]])/d} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x]), x, 3, (a*x)/2 - (I*a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x]), x, 4, (3*a*x)/8 - (I*a*Cos[c + d*x]^4)/(4*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]), x, 5, (5*a*x)/16 - (I*a*Cos[c + d*x]^6)/(6*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^8*(a + I*a*Tan[c + d*x]), x, 6, (35*a*x)/128 - (I*a*Cos[c + d*x]^8)/(8*d) + (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (7*a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} - -{Sec[c + d*x]^7*(a + I*a*Tan[c + d*x]), x, 5, (5*a*ArcTanh[Sin[c + d*x]])/(16*d) + ((I/7)*a*Sec[c + d*x]^7)/d + (5*a*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (5*a*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^5*(a + I*a*Tan[c + d*x]), x, 4, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + ((I/5)*a*Sec[c + d*x]^5)/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x]), x, 3, (a*ArcTanh[Sin[c + d*x]])/(2*d) + ((I/3)*a*Sec[c + d*x]^3)/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x]), x, 2, (a*ArcTanh[Sin[c + d*x]])/d + (I*a*Sec[c + d*x])/d} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x]), x, 2, -((I*a*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x]), x, 3, -((I*a*Cos[c + d*x]^3)/(3*d)) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x]), x, 3, -((I*a*Cos[c + d*x]^5)/(5*d)) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]), x, 3, -((I*a*Cos[c + d*x]^7)/(7*d)) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^2, x, 3, -((4*I*(a + I*a*Tan[c + d*x])^6)/(3*a^4*d)) + (12*I*(a + I*a*Tan[c + d*x])^7)/(7*a^5*d) - (3*I*(a + I*a*Tan[c + d*x])^8)/(4*a^6*d) + (I*(a + I*a*Tan[c + d*x])^9)/(9*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^2, x, 3, -((4*I*(a + I*a*Tan[c + d*x])^5)/(5*a^3*d)) + (2*I*(a + I*a*Tan[c + d*x])^6)/(3*a^4*d) - (I*(a + I*a*Tan[c + d*x])^7)/(7*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^2, x, 3, -((I*(a + I*a*Tan[c + d*x])^4)/(2*a^2*d)) + (I*(a + I*a*Tan[c + d*x])^5)/(5*a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^2, x, 2, -((I*(a + I*a*Tan[c + d*x])^3)/(3*a*d))} -{Sec[c + d*x]^0*(a + I*a*Tan[c + d*x])^2, x, 2, 2*a^2*x - (2*I*a^2*Log[Cos[c + d*x]])/d - (a^2*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^2, x, 2, -((I*a^3)/(d*(a - I*a*Tan[c + d*x])))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^2, x, 4, (a^2*x)/4 - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2) - (I*a^3)/(4*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^2, x, 4, (a^2*x)/4 - (I*a^5)/(12*d*(a - I*a*Tan[c + d*x])^3) - (I*a^4)/(8*d*(a - I*a*Tan[c + d*x])^2) - (3*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])) + (I*a^3)/(16*d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^2, x, 4, (15*a^2*x)/64 - (I*a^6)/(32*d*(a - I*a*Tan[c + d*x])^4) - (I*a^5)/(16*d*(a - I*a*Tan[c + d*x])^3) - (3*I*a^4)/(32*d*(a - I*a*Tan[c + d*x])^2) - (5*I*a^3)/(32*d*(a - I*a*Tan[c + d*x])) + (I*a^4)/(64*d*(a + I*a*Tan[c + d*x])^2) + (5*I*a^3)/(64*d*(a + I*a*Tan[c + d*x]))} - -{Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^2, x, 5, (7*a^2*ArcTanh[Sin[c + d*x]])/(16*d) + (7*I*a^2*Sec[c + d*x]^5)/(30*d) + (7*a^2*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (7*a^2*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (I*Sec[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x]))/(6*d)} -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^2, x, 4, (5*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (5*I*a^2*Sec[c + d*x]^3)/(12*d) + (5*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (I*Sec[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x]))/(4*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^2, x, 3, (3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (3*I*a^2*Sec[c + d*x])/(2*d) + (I*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^2, x, 2, -((a^2*ArcTanh[Sin[c + d*x]])/d) - (2*I*Cos[c + d*x]*(a^2 + I*a^2*Tan[c + d*x]))/d} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^2, x, 2, (a^2*Sin[c + d*x])/(3*d) - (2*I*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x]))/(3*d)} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^2, x, 3, (3*a^2*Sin[c + d*x])/(5*d) - (a^2*Sin[c + d*x]^3)/(5*d) - (2*I*Cos[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x]))/(5*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2, x, 3, (5*a^2*Sin[c + d*x])/(7*d) - (10*a^2*Sin[c + d*x]^3)/(21*d) + (a^2*Sin[c + d*x]^5)/(7*d) - (2*I*Cos[c + d*x]^7*(a^2 + I*a^2*Tan[c + d*x]))/(7*d)} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2, x, 3, (7*a^2*Sin[c + d*x])/(9*d) - (7*a^2*Sin[c + d*x]^3)/(9*d) + (7*a^2*Sin[c + d*x]^5)/(15*d) - (a^2*Sin[c + d*x]^7)/(9*d) - (2*I*Cos[c + d*x]^9*(a^2 + I*a^2*Tan[c + d*x]))/(9*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^3, x, 3, -((8*I*(a + I*a*Tan[c + d*x])^7)/(7*a^4*d)) + (3*I*(a + I*a*Tan[c + d*x])^8)/(2*a^5*d) - (2*I*(a + I*a*Tan[c + d*x])^9)/(3*a^6*d) + (I*(a + I*a*Tan[c + d*x])^10)/(10*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^3, x, 3, -((2*I*(a + I*a*Tan[c + d*x])^6)/(3*a^3*d)) + (4*I*(a + I*a*Tan[c + d*x])^7)/(7*a^4*d) - (I*(a + I*a*Tan[c + d*x])^8)/(8*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^3, x, 3, -((2*I*(a + I*a*Tan[c + d*x])^5)/(5*a^2*d)) + (I*(a + I*a*Tan[c + d*x])^6)/(6*a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^3, x, 2, -((I*(a + I*a*Tan[c + d*x])^4)/(4*a*d))} -{Sec[c + d*x]^0*(a + I*a*Tan[c + d*x])^3, x, 3, 4*a^3*x - (4*I*a^3*Log[Cos[c + d*x]])/d - (2*a^3*Tan[c + d*x])/d + (I*a*(a + I*a*Tan[c + d*x])^2)/(2*d)} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^3, x, 3, (-a^3)*x + (I*a^3*Log[Cos[c + d*x]])/d - (2*I*a^4)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^3, x, 2, -((I*a^5)/(2*d*(a - I*a*Tan[c + d*x])^2))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^3, x, 4, (a^3*x)/8 - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3) - (I*a^5)/(8*d*(a - I*a*Tan[c + d*x])^2) - (I*a^4)/(8*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^3, x, 4, (5*a^3*x)/32 - (I*a^7)/(16*d*(a - I*a*Tan[c + d*x])^4) - (I*a^6)/(12*d*(a - I*a*Tan[c + d*x])^3) - (3*I*a^5)/(32*d*(a - I*a*Tan[c + d*x])^2) - (I*a^4)/(8*d*(a - I*a*Tan[c + d*x])) + (I*a^4)/(32*d*(a + I*a*Tan[c + d*x]))} - -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^3, x, 5, (7*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (7*I*a^3*Sec[c + d*x]^3)/(12*d) + (7*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (I*a*Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^2)/(5*d) + (7*I*Sec[c + d*x]^3*(a^3 + I*a^3*Tan[c + d*x]))/(20*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^3, x, 4, (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (5*I*a^3*Sec[c + d*x])/(2*d) + (I*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^2)/(3*d) + (5*I*Sec[c + d*x]*(a^3 + I*a^3*Tan[c + d*x]))/(6*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^3, x, 3, -((3*a^3*ArcTanh[Sin[c + d*x]])/d) - (3*I*a^3*Sec[c + d*x])/d - (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^2)/d} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3, x, 1, ((-I/3)*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^3, x, 4, -((I*a^3*Cos[c + d*x]^3)/(15*d)) + (a^3*Sin[c + d*x])/(5*d) - (a^3*Sin[c + d*x]^3)/(15*d) - (2*I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^2)/(5*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^3, x, 4, -((3*I*a^3*Cos[c + d*x]^5)/(35*d)) + (3*a^3*Sin[c + d*x])/(7*d) - (2*a^3*Sin[c + d*x]^3)/(7*d) + (3*a^3*Sin[c + d*x]^5)/(35*d) - (2*I*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/(7*d)} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^3, x, 4, -((5*I*a^3*Cos[c + d*x]^7)/(63*d)) + (5*a^3*Sin[c + d*x])/(9*d) - (5*a^3*Sin[c + d*x]^3)/(9*d) + (a^3*Sin[c + d*x]^5)/(3*d) - (5*a^3*Sin[c + d*x]^7)/(63*d) - (2*I*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2)/(9*d)} - - -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^4, x, 6, (21*a^4*ArcTanh[Sin[c + d*x]])/(16*d) + (7*I*a^4*Sec[c + d*x]^3)/(8*d) + (21*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (I*a*Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(6*d) + (3*I*Sec[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) + (21*I*Sec[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]))/(40*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^4, x, 5, (35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (35*I*a^4*Sec[c + d*x])/(8*d) + (I*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^3)/(4*d) + (7*I*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^2)/(12*d) + (35*I*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/(24*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^4, x, 4, -((15*a^4*ArcTanh[Sin[c + d*x]])/(2*d)) - (15*I*a^4*Sec[c + d*x])/(2*d) - (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^3)/d - (5*I*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^4, x, 3, (a^4*ArcTanh[Sin[c + d*x]])/d - (2*I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(3*d) + (2*I*Cos[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^4, x, 2, -((I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(15*d)) - (I*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^4)/(5*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^4, x, 4, (3*a^4*Sin[c + d*x])/(35*d) - (a^4*Sin[c + d*x]^3)/(35*d) - (2*I*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^3)/(7*d) - (2*I*Cos[c + d*x]^5*(a^4 + I*a^4*Tan[c + d*x]))/(35*d)} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^4, x, 4, (5*a^4*Sin[c + d*x])/(21*d) - (10*a^4*Sin[c + d*x]^3)/(63*d) + (a^4*Sin[c + d*x]^5)/(21*d) - (2*I*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^3)/(9*d) - (2*I*Cos[c + d*x]^7*(a^4 + I*a^4*Tan[c + d*x]))/(21*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^5, x, 3, -((8*I*(a + I*a*Tan[c + d*x])^9)/(9*a^4*d)) + (6*I*(a + I*a*Tan[c + d*x])^10)/(5*a^5*d) - (6*I*(a + I*a*Tan[c + d*x])^11)/(11*a^6*d) + (I*(a + I*a*Tan[c + d*x])^12)/(12*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^5, x, 3, -((I*(a + I*a*Tan[c + d*x])^8)/(2*a^3*d)) + (4*I*(a + I*a*Tan[c + d*x])^9)/(9*a^4*d) - (I*(a + I*a*Tan[c + d*x])^10)/(10*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^5, x, 3, -((2*I*(a + I*a*Tan[c + d*x])^7)/(7*a^2*d)) + (I*(a + I*a*Tan[c + d*x])^8)/(8*a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^5, x, 2, -((I*(a + I*a*Tan[c + d*x])^6)/(6*a*d))} -{Sec[c + d*x]^0*(a + I*a*Tan[c + d*x])^5, x, 5, 16*a^5*x - (16*I*a^5*Log[Cos[c + d*x]])/d - (8*a^5*Tan[c + d*x])/d + (2*I*a^2*(a + I*a*Tan[c + d*x])^3)/(3*d) + (I*a*(a + I*a*Tan[c + d*x])^4)/(4*d) + (2*I*a*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^5, x, 3, -12*a^5*x + (12*I*a^5*Log[Cos[c + d*x]])/d + (5*a^5*Tan[c + d*x])/d + (I*a^5*Tan[c + d*x]^2)/(2*d) - (8*I*a^6)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^5, x, 3, a^5*x - (I*a^5*Log[Cos[c + d*x]])/d - (2*I*a^7)/(d*(a - I*a*Tan[c + d*x])^2) + (4*I*a^6)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^5, x, 3, -((2*I*a^8)/(3*d*(a - I*a*Tan[c + d*x])^3)) + (I*a^7)/(2*d*(a - I*a*Tan[c + d*x])^2)} -{Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^5, x, 2, -((I*a^9)/(4*d*(a - I*a*Tan[c + d*x])^4))} -{Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^5, x, 4, (a^5*x)/32 - (I*a^10)/(10*d*(a - I*a*Tan[c + d*x])^5) - (I*a^9)/(16*d*(a - I*a*Tan[c + d*x])^4) - (I*a^8)/(24*d*(a - I*a*Tan[c + d*x])^3) - (I*a^7)/(32*d*(a - I*a*Tan[c + d*x])^2) - (I*a^6)/(32*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^5, x, 4, (7*a^5*x)/128 - (I*a^11)/(24*d*(a - I*a*Tan[c + d*x])^6) - (I*a^10)/(20*d*(a - I*a*Tan[c + d*x])^5) - (3*I*a^9)/(64*d*(a - I*a*Tan[c + d*x])^4) - (I*a^8)/(24*d*(a - I*a*Tan[c + d*x])^3) - (5*I*a^7)/(128*d*(a - I*a*Tan[c + d*x])^2) - (3*I*a^6)/(64*d*(a - I*a*Tan[c + d*x])) + (I*a^6)/(128*d*(a + I*a*Tan[c + d*x]))} - -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^5, x, 6, (63*a^5*ArcTanh[Sin[c + d*x]])/(8*d) + (63*I*a^5*Sec[c + d*x])/(8*d) + (9*I*a^2*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^3)/(20*d) + (I*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^4)/(5*d) + (21*I*a*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^2)/(20*d) + (21*I*Sec[c + d*x]*(a^5 + I*a^5*Tan[c + d*x]))/(8*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^5, x, 5, -((35*a^5*ArcTanh[Sin[c + d*x]])/(2*d)) - (35*I*a^5*Sec[c + d*x])/(2*d) - (7*I*a^3*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^2)/(3*d) - (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^4)/d - (35*I*Sec[c + d*x]*(a^5 + I*a^5*Tan[c + d*x]))/(6*d)} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^5, x, 4, (5*a^5*ArcTanh[Sin[c + d*x]])/d + (5*I*a^5*Sec[c + d*x])/d + (10*I*a^3*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^2)/(3*d) - (2*I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^4)/(3*d)} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5, x, 1, -((I*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/(5*d))} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5, x, 3, -((2*I*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(105*d)) - (2*I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^4)/(35*d) - (I*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/(7*d)} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^5, x, 5, -((I*a^5*Cos[c + d*x]^5)/(105*d)) + (a^5*Sin[c + d*x])/(21*d) - (2*a^5*Sin[c + d*x]^3)/(63*d) + (a^5*Sin[c + d*x]^5)/(105*d) - (2*I*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2)/(63*d) - (2*I*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^4)/(9*d)} -{Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^5, x, 5, -((5*I*a^5*Cos[c + d*x]^7)/(231*d)) + (5*a^5*Sin[c + d*x])/(33*d) - (5*a^5*Sin[c + d*x]^3)/(33*d) + (a^5*Sin[c + d*x]^5)/(11*d) - (5*a^5*Sin[c + d*x]^7)/(231*d) - (2*I*a^3*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2)/(33*d) - (2*I*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^4)/(11*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^8, x, 3, -((2*I*(a + I*a*Tan[c + d*x])^12)/(3*a^4*d)) + (12*I*(a + I*a*Tan[c + d*x])^13)/(13*a^5*d) - (3*I*(a + I*a*Tan[c + d*x])^14)/(7*a^6*d) + (I*(a + I*a*Tan[c + d*x])^15)/(15*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^8, x, 3, -((4*I*(a + I*a*Tan[c + d*x])^11)/(11*a^3*d)) + (I*(a + I*a*Tan[c + d*x])^12)/(3*a^4*d) - (I*(a + I*a*Tan[c + d*x])^13)/(13*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^8, x, 3, -((I*(a + I*a*Tan[c + d*x])^10)/(5*a^2*d)) + (I*(a + I*a*Tan[c + d*x])^11)/(11*a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^8, x, 2, -((I*(a + I*a*Tan[c + d*x])^9)/(9*a*d))} -{Sec[c + d*x]^0*(a + I*a*Tan[c + d*x])^8, x, 8, 128*a^8*x - (128*I*a^8*Log[Cos[c + d*x]])/d - (64*a^8*Tan[c + d*x])/d + (4*I*a^3*(a + I*a*Tan[c + d*x])^5)/(5*d) + (I*a^2*(a + I*a*Tan[c + d*x])^6)/(3*d) + (I*a*(a + I*a*Tan[c + d*x])^7)/(7*d) + (16*I*a^2*(a^2 + I*a^2*Tan[c + d*x])^3)/(3*d) + (2*I*(a^2 + I*a^2*Tan[c + d*x])^4)/d + (16*I*(a^4 + I*a^4*Tan[c + d*x])^2)/d} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^8, x, 3, -192*a^8*x + (192*I*a^8*Log[Cos[c + d*x]])/d + (129*a^8*Tan[c + d*x])/d + (36*I*a^8*Tan[c + d*x]^2)/d - (10*a^8*Tan[c + d*x]^3)/d - (2*I*a^8*Tan[c + d*x]^4)/d + (a^8*Tan[c + d*x]^5)/(5*d) - (64*I*a^9)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^8, x, 3, 80*a^8*x - (80*I*a^8*Log[Cos[c + d*x]])/d - (31*a^8*Tan[c + d*x])/d - (4*I*a^8*Tan[c + d*x]^2)/d + (a^8*Tan[c + d*x]^3)/(3*d) - (16*I*a^10)/(d*(a - I*a*Tan[c + d*x])^2) + (80*I*a^9)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^8, x, 3, -8*a^8*x + (8*I*a^8*Log[Cos[c + d*x]])/d + (a^8*Tan[c + d*x])/d - (16*I*a^11)/(3*d*(a - I*a*Tan[c + d*x])^3) + (16*I*a^10)/(d*(a - I*a*Tan[c + d*x])^2) - (24*I*a^9)/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^8*(a + I*a*Tan[c + d*x])^8, x, 2, -((I*(a^3 + I*a^3*Tan[c + d*x])^4)/(8*d*(a - I*a*Tan[c + d*x])^4))} -{Cos[c + d*x]^10*(a + I*a*Tan[c + d*x])^8, x, 3, -((4*I*a^13)/(5*d*(a - I*a*Tan[c + d*x])^5)) + (I*a^12)/(d*(a - I*a*Tan[c + d*x])^4) - (I*a^11)/(3*d*(a - I*a*Tan[c + d*x])^3)} -{Cos[c + d*x]^12*(a + I*a*Tan[c + d*x])^8, x, 3, -((I*a^14)/(3*d*(a - I*a*Tan[c + d*x])^6)) + (I*a^13)/(5*d*(a - I*a*Tan[c + d*x])^5)} -{Cos[c + d*x]^14*(a + I*a*Tan[c + d*x])^8, x, 2, -((I*a^15)/(7*d*(a - I*a*Tan[c + d*x])^7))} -{Cos[c + d*x]^16*(a + I*a*Tan[c + d*x])^8, x, 4, (a^8*x)/256 - (I*a^16)/(16*d*(a - I*a*Tan[c + d*x])^8) - (I*a^15)/(28*d*(a - I*a*Tan[c + d*x])^7) - (I*a^14)/(48*d*(a - I*a*Tan[c + d*x])^6) - (I*a^13)/(80*d*(a - I*a*Tan[c + d*x])^5) - (I*a^12)/(128*d*(a - I*a*Tan[c + d*x])^4) - (I*a^11)/(192*d*(a - I*a*Tan[c + d*x])^3) - (I*a^10)/(256*d*(a - I*a*Tan[c + d*x])^2) - (I*a^9)/(256*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^18*(a + I*a*Tan[c + d*x])^8, x, 4, (5*a^8*x)/512 - (I*a^17)/(36*d*(a - I*a*Tan[c + d*x])^9) - (I*a^16)/(32*d*(a - I*a*Tan[c + d*x])^8) - (3*I*a^15)/(112*d*(a - I*a*Tan[c + d*x])^7) - (I*a^14)/(48*d*(a - I*a*Tan[c + d*x])^6) - (I*a^13)/(64*d*(a - I*a*Tan[c + d*x])^5) - (3*I*a^12)/(256*d*(a - I*a*Tan[c + d*x])^4) - (7*I*a^11)/(768*d*(a - I*a*Tan[c + d*x])^3) - (I*a^10)/(128*d*(a - I*a*Tan[c + d*x])^2) - (9*I*a^9)/(1024*d*(a - I*a*Tan[c + d*x])) + (I*a^9)/(1024*d*(a + I*a*Tan[c + d*x]))} - -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^8, x, 8, -((3003*a^8*ArcTanh[Sin[c + d*x]])/(16*d)) - (3003*I*a^8*Sec[c + d*x])/(16*d) - (13*I*a^3*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^5)/(6*d) - (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^7)/d - (429*I*a^2*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^3)/(40*d) - (143*I*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^4)/(30*d) - (1001*I*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x])^2)/(40*d) - (1001*I*Sec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/(16*d)} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^8, x, 7, (1155*a^8*ArcTanh[Sin[c + d*x]])/(8*d) + (1155*I*a^8*Sec[c + d*x])/(8*d) + (22*I*a^3*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^5)/(3*d) - (2*I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^7)/(3*d) + (33*I*a^2*Sec[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^3)/(4*d) + (77*I*Sec[c + d*x]*(a^4 + I*a^4*Tan[c + d*x])^2)/(4*d) + (385*I*Sec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/(8*d)} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^8, x, 6, -((63*a^8*ArcTanh[Sin[c + d*x]])/(2*d)) - (63*I*a^8*Sec[c + d*x])/(2*d) + (6*I*a^3*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^5)/(5*d) - (2*I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^7)/(5*d) - (42*I*a^2*Cos[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^3)/(5*d) - (21*I*Sec[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8, x, 5, (a^8*ArcTanh[Sin[c + d*x]])/d + (2*I*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/(5*d) - (2*I*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/(7*d) - (2*I*a^2*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^3)/(3*d) + (2*I*Cos[c + d*x]*(a^8 + I*a^8*Tan[c + d*x]))/d} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8, x, 2, -((I*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/(63*d)) - (I*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8)/(9*d)} -{Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8, x, 4, -((2*I*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/(1155*d)) - (2*I*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^6)/(231*d) - (I*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^7)/(33*d) - (I*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^8)/(11*d)} -{Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8, x, 6, -((20*I*a^3*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/(3003*d)) - (20*I*a^2*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^6)/(1287*d) - (5*I*a*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^7)/(143*d) - (I*Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^8)/(13*d) - (8*I*a^2*Cos[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^3)/(9009*d) - (8*I*Cos[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^4)/(3003*d)} -{Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^8, x, 6, (7*a^8*Sin[c + d*x])/(1287*d) - (7*a^8*Sin[c + d*x]^3)/(1287*d) + (7*a^8*Sin[c + d*x]^5)/(2145*d) - (a^8*Sin[c + d*x]^7)/(1287*d) - (2*I*a^3*Cos[c + d*x]^13*(a + I*a*Tan[c + d*x])^5)/(195*d) - (2*I*a*Cos[c + d*x]^15*(a + I*a*Tan[c + d*x])^7)/(15*d) - (2*I*a^2*Cos[c + d*x]^11*(a^2 + I*a^2*Tan[c + d*x])^3)/(715*d) - (2*I*Cos[c + d*x]^9*(a^8 + I*a^8*Tan[c + d*x]))/(1287*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x]), x, 3, (8*I*(a - I*a*Tan[c + d*x])^5)/(5*a^6*d) - (2*I*(a - I*a*Tan[c + d*x])^6)/(a^7*d) + (6*I*(a - I*a*Tan[c + d*x])^7)/(7*a^8*d) - (I*(a - I*a*Tan[c + d*x])^8)/(8*a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x]), x, 3, (I*(a - I*a*Tan[c + d*x])^4)/(a^5*d) - (4*I*(a - I*a*Tan[c + d*x])^5)/(5*a^6*d) + (I*(a - I*a*Tan[c + d*x])^6)/(6*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x]), x, 3, (2*I*(a - I*a*Tan[c + d*x])^3)/(3*a^4*d) - (I*(a - I*a*Tan[c + d*x])^4)/(4*a^5*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x]), x, 2, Tan[c + d*x]/(a*d) - (I*Tan[c + d*x]^2)/(2*a*d)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x]), x, 2, x/a + (I*Log[Cos[c + d*x]])/(a*d)} -{Sec[c + d*x]^0/(a + I*a*Tan[c + d*x]), x, 2, x/(2*a) + I/(2*d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x]), x, 4, (3*x)/(8*a) - I/(8*d*(a - I*a*Tan[c + d*x])) + (I*a)/(8*d*(a + I*a*Tan[c + d*x])^2) + I/(4*d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x]), x, 4, (5*x)/(16*a) - (I*a)/(32*d*(a - I*a*Tan[c + d*x])^2) - I/(8*d*(a - I*a*Tan[c + d*x])) + (I*a^2)/(24*d*(a + I*a*Tan[c + d*x])^3) + (3*I*a)/(32*d*(a + I*a*Tan[c + d*x])^2) + (3*I)/(16*d*(a + I*a*Tan[c + d*x]))} - -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x]), x, 4, (3*ArcTanh[Sin[c + d*x]])/(8*a*d) - (I*Sec[c + d*x]^5)/(5*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x]), x, 3, ArcTanh[Sin[c + d*x]]/(2*a*d) - ((I/3)*Sec[c + d*x]^3)/(a*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x]), x, 2, ArcTanh[Sin[c + d*x]]/(a*d) - (I*Sec[c + d*x])/(a*d)} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x]), x, 1, (I*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x]), x, 2, (2*Sin[c + d*x])/(3*a*d) + (I*Cos[c + d*x])/(3*d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x]), x, 3, (4*Sin[c + d*x])/(5*a*d) - (4*Sin[c + d*x]^3)/(15*a*d) + (I*Cos[c + d*x]^3)/(5*d*(a + I*a*Tan[c + d*x]))} -{Cos[c + d*x]^5/(a + I*a*Tan[c + d*x]), x, 3, (6*Sin[c + d*x])/(7*a*d) - (4*Sin[c + d*x]^3)/(7*a*d) + (6*Sin[c + d*x]^5)/(35*a*d) + (I*Cos[c + d*x]^5)/(7*d*(a + I*a*Tan[c + d*x]))} - - -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^2, x, 3, (4*I*(a - I*a*Tan[c + d*x])^5)/(5*a^7*d) - (2*I*(a - I*a*Tan[c + d*x])^6)/(3*a^8*d) + (I*(a - I*a*Tan[c + d*x])^7)/(7*a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^2, x, 3, (I*(a - I*a*Tan[c + d*x])^4)/(2*a^6*d) - (I*(a - I*a*Tan[c + d*x])^5)/(5*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^2, x, 2, (I*(a - I*a*Tan[c + d*x])^3)/(3*a^5*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^2, x, 3, (2*x)/a^2 + ((2*I)*Log[Cos[c + d*x]])/(a^2*d) - Tan[c + d*x]/(a^2*d)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^2, x, 2, I/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sec[c + d*x]^0/(a + I*a*Tan[c + d*x])^2, x, 3, x/(4*a^2) + I/(4*d*(a + I*a*Tan[c + d*x])^2) + I/(4*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^2, x, 4, x/(4*a^2) + (I*a)/(12*d*(a + I*a*Tan[c + d*x])^3) + I/(8*d*(a + I*a*Tan[c + d*x])^2) - I/(16*d*(a^2 - I*a^2*Tan[c + d*x])) + (3*I)/(16*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^2, x, 4, (15*x)/(64*a^2) - I/(64*d*(a - I*a*Tan[c + d*x])^2) + (I*a^2)/(32*d*(a + I*a*Tan[c + d*x])^4) + (I*a)/(16*d*(a + I*a*Tan[c + d*x])^3) + (3*I)/(32*d*(a + I*a*Tan[c + d*x])^2) - (5*I)/(64*d*(a^2 - I*a^2*Tan[c + d*x])) + (5*I)/(32*d*(a^2 + I*a^2*Tan[c + d*x]))} - -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^2, x, 5, (7*ArcTanh[Sin[c + d*x]])/(16*a^2*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(24*a^2*d) + (7*Sec[c + d*x]^5*Tan[c + d*x])/(30*a^2*d) - (2*I*Sec[c + d*x]^7)/(5*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^2, x, 4, (5*ArcTanh[Sin[c + d*x]])/(8*a^2*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(8*a^2*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(12*a^2*d) - (2*I*Sec[c + d*x]^5)/(3*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^2, x, 3, (3*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (2*I*Sec[c + d*x]^3)/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^2, x, 2, -(ArcTanh[Sin[c + d*x]]/(a^2*d)) + ((2*I)*Sec[c + d*x])/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^2, x, 2, (I*Sec[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^2) + (I*Sec[c + d*x])/(3*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^2, x, 3, (3*Sin[c + d*x])/(5*a^2*d) - Sin[c + d*x]^3/(5*a^2*d) + (2*I*Cos[c + d*x]^3)/(5*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^2, x, 3, (5*Sin[c + d*x])/(7*a^2*d) - (10*Sin[c + d*x]^3)/(21*a^2*d) + Sin[c + d*x]^5/(7*a^2*d) + (2*I*Cos[c + d*x]^5)/(7*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^2, x, 3, (7*Sin[c + d*x])/(9*a^2*d) - (7*Sin[c + d*x]^3)/(9*a^2*d) + (7*Sin[c + d*x]^5)/(15*a^2*d) - Sin[c + d*x]^7/(9*a^2*d) + (2*I*Cos[c + d*x]^7)/(9*d*(a^2 + I*a^2*Tan[c + d*x]))} - - -{Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^3, x, 3, (8*I*(a - I*a*Tan[c + d*x])^7)/(7*a^10*d) - (3*I*(a - I*a*Tan[c + d*x])^8)/(2*a^11*d) + (2*I*(a - I*a*Tan[c + d*x])^9)/(3*a^12*d) - (I*(a - I*a*Tan[c + d*x])^10)/(10*a^13*d)} -{Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^3, x, 3, (2*I*(a - I*a*Tan[c + d*x])^6)/(3*a^9*d) - (4*I*(a - I*a*Tan[c + d*x])^7)/(7*a^10*d) + (I*(a - I*a*Tan[c + d*x])^8)/(8*a^11*d)} -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^3, x, 3, (2*I*(a - I*a*Tan[c + d*x])^5)/(5*a^8*d) - (I*(a - I*a*Tan[c + d*x])^6)/(6*a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^3, x, 2, (I*(a - I*a*Tan[c + d*x])^4)/(4*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^3, x, 3, (4*x)/a^3 + (4*I*Log[Cos[c + d*x]])/(a^3*d) - (3*Tan[c + d*x])/(a^3*d) + (I*Tan[c + d*x]^2)/(2*a^3*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^3, x, 3, -(x/a^3) - (I*Log[Cos[c + d*x]])/(a^3*d) + (2*I)/(d*(a^3 + I*a^3*Tan[c + d*x]))} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^3, x, 2, I/(2*a*d*(a + I*a*Tan[c + d*x])^2)} -{Sec[c + d*x]^0/(a + I*a*Tan[c + d*x])^3, x, 4, x/(8*a^3) + I/(6*d*(a + I*a*Tan[c + d*x])^3) + I/(8*a*d*(a + I*a*Tan[c + d*x])^2) + I/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^3, x, 4, (5*x)/(32*a^3) + (I*a)/(16*d*(a + I*a*Tan[c + d*x])^4) + I/(12*d*(a + I*a*Tan[c + d*x])^3) + (3*I)/(32*a*d*(a + I*a*Tan[c + d*x])^2) - I/(32*d*(a^3 - I*a^3*Tan[c + d*x])) + I/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^3, x, 4, (21*x)/(128*a^3) - I/(128*a*d*(a - I*a*Tan[c + d*x])^2) + (I*a^2)/(40*d*(a + I*a*Tan[c + d*x])^5) + (3*I*a)/(64*d*(a + I*a*Tan[c + d*x])^4) + I/(16*d*(a + I*a*Tan[c + d*x])^3) + (5*I)/(64*a*d*(a + I*a*Tan[c + d*x])^2) - (3*I)/(64*d*(a^3 - I*a^3*Tan[c + d*x])) + (15*I)/(128*d*(a^3 + I*a^3*Tan[c + d*x]))} - -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^3, x, 5, (7*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - (7*I*Sec[c + d*x]^5)/(15*a^3*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) + (7*Sec[c + d*x]^3*Tan[c + d*x])/(12*a^3*d) - (2*I*Sec[c + d*x]^7)/(3*a*d*(a + I*a*Tan[c + d*x])^2)} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^3, x, 4, (5*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (((5*I)/3)*Sec[c + d*x]^3)/(a^3*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((2*I)*Sec[c + d*x]^5)/(a*d*(a + I*a*Tan[c + d*x])^2)} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^3, x, 3, (-3*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((3*I)*Sec[c + d*x])/(a^3*d) + ((2*I)*Sec[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^2)} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^3, x, 1, ((I/3)*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^3)} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^3, x, 3, (I*Sec[c + d*x])/(5*d*(a + I*a*Tan[c + d*x])^3) + (2*I*Sec[c + d*x])/(15*a*d*(a + I*a*Tan[c + d*x])^2) + (2*I*Sec[c + d*x])/(15*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^3, x, 4, (12*Sin[c + d*x])/(35*a^3*d) - (4*Sin[c + d*x]^3)/(35*a^3*d) + (I*Cos[c + d*x])/(7*d*(a + I*a*Tan[c + d*x])^3) + (8*I*Cos[c + d*x]^3)/(35*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^3, x, 4, (10*Sin[c + d*x])/(21*a^3*d) - (20*Sin[c + d*x]^3)/(63*a^3*d) + (2*Sin[c + d*x]^5)/(21*a^3*d) + (I*Cos[c + d*x]^3)/(9*d*(a + I*a*Tan[c + d*x])^3) + (4*I*Cos[c + d*x]^5)/(21*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^3, x, 4, (56*Sin[c + d*x])/(99*a^3*d) - (56*Sin[c + d*x]^3)/(99*a^3*d) + (56*Sin[c + d*x]^5)/(165*a^3*d) - (8*Sin[c + d*x]^7)/(99*a^3*d) + (I*Cos[c + d*x]^5)/(11*d*(a + I*a*Tan[c + d*x])^3) + (16*I*Cos[c + d*x]^7)/(99*d*(a^3 + I*a^3*Tan[c + d*x]))} - - -{Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^4, x, 3, (4*I*(a - I*a*Tan[c + d*x])^7)/(7*a^11*d) - (I*(a - I*a*Tan[c + d*x])^8)/(2*a^12*d) + (I*(a - I*a*Tan[c + d*x])^9)/(9*a^13*d)} -{Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^4, x, 3, (I*(a - I*a*Tan[c + d*x])^6)/(3*a^10*d) - (I*(a - I*a*Tan[c + d*x])^7)/(7*a^11*d)} -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^4, x, 2, (I*(a - I*a*Tan[c + d*x])^5)/(5*a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^4, x, 3, (8*x)/a^4 + (8*I*Log[Cos[c + d*x]])/(a^4*d) - (4*Tan[c + d*x])/(a^4*d) - (I*(a - I*a*Tan[c + d*x])^2)/(a^6*d) - (I*(a - I*a*Tan[c + d*x])^3)/(3*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^4, x, 3, -((4*x)/a^4) - (4*I*Log[Cos[c + d*x]])/(a^4*d) + Tan[c + d*x]/(a^4*d) + (4*I)/(d*(a^4 + I*a^4*Tan[c + d*x]))} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^4, x, 2, Tan[c + d*x]/(d*(a^2 + I*a^2*Tan[c + d*x])^2)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^4, x, 2, I/(3*a*d*(a + I*a*Tan[c + d*x])^3)} -{Sec[c + d*x]^0/(a + I*a*Tan[c + d*x])^4, x, 5, x/(16*a^4) + I/(8*d*(a + I*a*Tan[c + d*x])^4) + I/(12*a*d*(a + I*a*Tan[c + d*x])^3) + I/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) + I/(16*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^4, x, 4, (3*x)/(32*a^4) + (I*a)/(20*d*(a + I*a*Tan[c + d*x])^5) + I/(16*d*(a + I*a*Tan[c + d*x])^4) + I/(16*a*d*(a + I*a*Tan[c + d*x])^3) + I/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) - I/(64*d*(a^4 - I*a^4*Tan[c + d*x])) + (5*I)/(64*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^4, x, 4, (7*x)/(64*a^4) + (I*a^2)/(48*d*(a + I*a*Tan[c + d*x])^6) + (3*I*a)/(80*d*(a + I*a*Tan[c + d*x])^5) + (3*I)/(64*d*(a + I*a*Tan[c + d*x])^4) + (5*I)/(96*a*d*(a + I*a*Tan[c + d*x])^3) - I/(256*d*(a^2 - I*a^2*Tan[c + d*x])^2) + (15*I)/(256*d*(a^2 + I*a^2*Tan[c + d*x])^2) - (7*I)/(256*d*(a^4 - I*a^4*Tan[c + d*x])) + (21*I)/(256*d*(a^4 + I*a^4*Tan[c + d*x]))} - -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^4, x, 5, (35*ArcTanh[Sin[c + d*x]])/(8*a^4*d) + (35*Sec[c + d*x]*Tan[c + d*x])/(8*a^4*d) + (35*Sec[c + d*x]^3*Tan[c + d*x])/(12*a^4*d) - (2*I*Sec[c + d*x]^7)/(a*d*(a + I*a*Tan[c + d*x])^3) - (14*I*Sec[c + d*x]^5)/(3*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^4, x, 4, -((15*ArcTanh[Sin[c + d*x]])/(2*a^4*d)) - (15*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) + (2*I*Sec[c + d*x]^5)/(a*d*(a + I*a*Tan[c + d*x])^3) + (10*I*Sec[c + d*x]^3)/(d*(a^4 + I*a^4*Tan[c + d*x]))} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^4, x, 3, ArcTanh[Sin[c + d*x]]/(a^4*d) + (2*I*Sec[c + d*x]^3)/(3*a*d*(a + I*a*Tan[c + d*x])^3) - (2*I*Sec[c + d*x])/(d*(a^4 + I*a^4*Tan[c + d*x]))} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^4, x, 2, (I*Sec[c + d*x]^3)/(5*d*(a + I*a*Tan[c + d*x])^4) + (I*Sec[c + d*x]^3)/(15*a*d*(a + I*a*Tan[c + d*x])^3)} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^4, x, 4, (I*Sec[c + d*x])/(7*d*(a + I*a*Tan[c + d*x])^4) + (3*I*Sec[c + d*x])/(35*a*d*(a + I*a*Tan[c + d*x])^3) + (2*I*Sec[c + d*x])/(35*d*(a^2 + I*a^2*Tan[c + d*x])^2) + (2*I*Sec[c + d*x])/(35*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^4, x, 5, (4*Sin[c + d*x])/(21*a^4*d) - (4*Sin[c + d*x]^3)/(63*a^4*d) + (I*Cos[c + d*x])/(9*d*(a + I*a*Tan[c + d*x])^4) + (5*I*Cos[c + d*x])/(63*a*d*(a + I*a*Tan[c + d*x])^3) + (8*I*Cos[c + d*x]^3)/(63*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^4, x, 5, (10*Sin[c + d*x])/(33*a^4*d) - (20*Sin[c + d*x]^3)/(99*a^4*d) + (2*Sin[c + d*x]^5)/(33*a^4*d) + (I*Cos[c + d*x]^3)/(11*d*(a + I*a*Tan[c + d*x])^4) + (7*I*Cos[c + d*x]^3)/(99*a*d*(a + I*a*Tan[c + d*x])^3) + (4*I*Cos[c + d*x]^5)/(33*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cos[c + d*x]^5/(a + I*a*Tan[c + d*x])^4, x, 5, (56*Sin[c + d*x])/(143*a^4*d) - (56*Sin[c + d*x]^3)/(143*a^4*d) + (168*Sin[c + d*x]^5)/(715*a^4*d) - (8*Sin[c + d*x]^7)/(143*a^4*d) + (I*Cos[c + d*x]^5)/(13*d*(a + I*a*Tan[c + d*x])^4) + (9*I*Cos[c + d*x]^5)/(143*a*d*(a + I*a*Tan[c + d*x])^3) + (16*I*Cos[c + d*x]^7)/(143*d*(a^4 + I*a^4*Tan[c + d*x]))} - - -{Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^8, x, 3, -((192*x)/a^8) - (192*I*Log[Cos[c + d*x]])/(a^8*d) + (129*Tan[c + d*x])/(a^8*d) - (36*I*Tan[c + d*x]^2)/(a^8*d) - (10*Tan[c + d*x]^3)/(a^8*d) + (2*I*Tan[c + d*x]^4)/(a^8*d) + Tan[c + d*x]^5/(5*a^8*d) + (64*I)/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^12/(a + I*a*Tan[c + d*x])^8, x, 3, (80*x)/a^8 + (80*I*Log[Cos[c + d*x]])/(a^8*d) - (31*Tan[c + d*x])/(a^8*d) + (4*I*Tan[c + d*x]^2)/(a^8*d) + Tan[c + d*x]^3/(3*a^8*d) + (16*I)/(d*(a^4 + I*a^4*Tan[c + d*x])^2) - (80*I)/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^8, x, 3, -((8*x)/a^8) - (8*I*Log[Cos[c + d*x]])/(a^8*d) + Tan[c + d*x]/(a^8*d) + (16*I)/(3*a^5*d*(a + I*a*Tan[c + d*x])^3) - (16*I)/(d*(a^4 + I*a^4*Tan[c + d*x])^2) + (24*I)/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^8, x, 2, (I*(a - I*a*Tan[c + d*x])^4)/(8*d*(a^3 + I*a^3*Tan[c + d*x])^4)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^8, x, 3, (4*I)/(5*a^3*d*(a + I*a*Tan[c + d*x])^5) + I/(3*a^5*d*(a + I*a*Tan[c + d*x])^3) - I/(d*(a^2 + I*a^2*Tan[c + d*x])^4)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^8, x, 3, I/(3*a^2*d*(a + I*a*Tan[c + d*x])^6) - I/(5*a^3*d*(a + I*a*Tan[c + d*x])^5)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^8, x, 2, I/(7*a*d*(a + I*a*Tan[c + d*x])^7)} -{Sec[c + d*x]^0/(a + I*a*Tan[c + d*x])^8, x, 9, x/(256*a^8) + I/(16*d*(a + I*a*Tan[c + d*x])^8) + I/(28*a*d*(a + I*a*Tan[c + d*x])^7) + I/(48*a^2*d*(a + I*a*Tan[c + d*x])^6) + I/(80*a^3*d*(a + I*a*Tan[c + d*x])^5) + I/(128*d*(a^2 + I*a^2*Tan[c + d*x])^4) + I/(192*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + I/(256*d*(a^4 + I*a^4*Tan[c + d*x])^2) + I/(256*d*(a^8 + I*a^8*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^8, x, 4, (5*x)/(512*a^8) + (I*a)/(36*d*(a + I*a*Tan[c + d*x])^9) + I/(32*d*(a + I*a*Tan[c + d*x])^8) + (3*I)/(112*a*d*(a + I*a*Tan[c + d*x])^7) + I/(48*a^2*d*(a + I*a*Tan[c + d*x])^6) + I/(64*a^3*d*(a + I*a*Tan[c + d*x])^5) + (7*I)/(768*a^5*d*(a + I*a*Tan[c + d*x])^3) + (3*I)/(256*d*(a^2 + I*a^2*Tan[c + d*x])^4) + I/(128*d*(a^4 + I*a^4*Tan[c + d*x])^2) - I/(1024*d*(a^8 - I*a^8*Tan[c + d*x])) + (9*I)/(1024*d*(a^8 + I*a^8*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^8, x, 4, (33*x)/(2048*a^8) + (I*a^2)/(80*d*(a + I*a*Tan[c + d*x])^10) + (I*a)/(48*d*(a + I*a*Tan[c + d*x])^9) + (3*I)/(128*d*(a + I*a*Tan[c + d*x])^8) + (5*I)/(224*a*d*(a + I*a*Tan[c + d*x])^7) + (5*I)/(256*a^2*d*(a + I*a*Tan[c + d*x])^6) + (21*I)/(1280*a^3*d*(a + I*a*Tan[c + d*x])^5) + (3*I)/(256*a^5*d*(a + I*a*Tan[c + d*x])^3) + (7*I)/(512*d*(a^2 + I*a^2*Tan[c + d*x])^4) - I/(4096*d*(a^4 - I*a^4*Tan[c + d*x])^2) + (45*I)/(4096*d*(a^4 + I*a^4*Tan[c + d*x])^2) - (11*I)/(4096*d*(a^8 - I*a^8*Tan[c + d*x])) + (55*I)/(4096*d*(a^8 + I*a^8*Tan[c + d*x]))} - -{Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^8, x, 7, (1155*ArcTanh[Sin[c + d*x]])/(8*a^8*d) + (1155*Sec[c + d*x]*Tan[c + d*x])/(8*a^8*d) + (385*Sec[c + d*x]^3*Tan[c + d*x])/(4*a^8*d) + (2*I*Sec[c + d*x]^11)/(3*a*d*(a + I*a*Tan[c + d*x])^7) - (22*I*Sec[c + d*x]^9)/(3*a^3*d*(a + I*a*Tan[c + d*x])^5) - (66*I*Sec[c + d*x]^7)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) - (154*I*Sec[c + d*x]^5)/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^8, x, 6, -((63*ArcTanh[Sin[c + d*x]])/(2*a^8*d)) - (63*Sec[c + d*x]*Tan[c + d*x])/(2*a^8*d) + (2*I*Sec[c + d*x]^9)/(5*a*d*(a + I*a*Tan[c + d*x])^7) - (6*I*Sec[c + d*x]^7)/(5*a^3*d*(a + I*a*Tan[c + d*x])^5) + (42*I*Sec[c + d*x]^5)/(5*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (42*I*Sec[c + d*x]^3)/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^8, x, 5, ArcTanh[Sin[c + d*x]]/(a^8*d) + (2*I*Sec[c + d*x]^7)/(7*a*d*(a + I*a*Tan[c + d*x])^7) - (2*I*Sec[c + d*x]^5)/(5*a^3*d*(a + I*a*Tan[c + d*x])^5) + (2*I*Sec[c + d*x]^3)/(3*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) - (2*I*Sec[c + d*x])/(d*(a^8 + I*a^8*Tan[c + d*x]))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^8, x, 2, (I*Sec[c + d*x]^7)/(9*d*(a + I*a*Tan[c + d*x])^8) + (I*Sec[c + d*x]^7)/(63*a*d*(a + I*a*Tan[c + d*x])^7)} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^8, x, 4, (I*Sec[c + d*x]^5)/(11*d*(a + I*a*Tan[c + d*x])^8) + (I*Sec[c + d*x]^5)/(33*a*d*(a + I*a*Tan[c + d*x])^7) + (2*I*Sec[c + d*x]^5)/(231*a^2*d*(a + I*a*Tan[c + d*x])^6) + (2*I*Sec[c + d*x]^5)/(1155*a^3*d*(a + I*a*Tan[c + d*x])^5)} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^8, x, 6, (I*Sec[c + d*x]^3)/(13*d*(a + I*a*Tan[c + d*x])^8) + (5*I*Sec[c + d*x]^3)/(143*a*d*(a + I*a*Tan[c + d*x])^7) + (20*I*Sec[c + d*x]^3)/(1287*a^2*d*(a + I*a*Tan[c + d*x])^6) + (20*I*Sec[c + d*x]^3)/(3003*a^3*d*(a + I*a*Tan[c + d*x])^5) + (8*I*Sec[c + d*x]^3)/(3003*d*(a^2 + I*a^2*Tan[c + d*x])^4) + (8*I*Sec[c + d*x]^3)/(9009*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3)} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^8, x, 8, (I*Sec[c + d*x])/(15*d*(a + I*a*Tan[c + d*x])^8) + (7*I*Sec[c + d*x])/(195*a*d*(a + I*a*Tan[c + d*x])^7) + (14*I*Sec[c + d*x])/(715*a^2*d*(a + I*a*Tan[c + d*x])^6) + (14*I*Sec[c + d*x])/(1287*a^3*d*(a + I*a*Tan[c + d*x])^5) + (8*I*Sec[c + d*x])/(1287*d*(a^2 + I*a^2*Tan[c + d*x])^4) + (8*I*Sec[c + d*x])/(2145*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (16*I*Sec[c + d*x])/(6435*d*(a^4 + I*a^4*Tan[c + d*x])^2) + (16*I*Sec[c + d*x])/(6435*d*(a^8 + I*a^8*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^8, x, 9, (192*Sin[c + d*x])/(12155*a^8*d) - (64*Sin[c + d*x]^3)/(12155*a^8*d) + (I*Cos[c + d*x])/(17*d*(a + I*a*Tan[c + d*x])^8) + (3*I*Cos[c + d*x])/(85*a*d*(a + I*a*Tan[c + d*x])^7) + (24*I*Cos[c + d*x])/(1105*a^2*d*(a + I*a*Tan[c + d*x])^6) + (168*I*Cos[c + d*x])/(12155*a^3*d*(a + I*a*Tan[c + d*x])^5) + (112*I*Cos[c + d*x])/(12155*d*(a^2 + I*a^2*Tan[c + d*x])^4) + (16*I*Cos[c + d*x])/(2431*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (128*I*Cos[c + d*x]^3)/(12155*d*(a^8 + I*a^8*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^8, x, 9, (160*Sin[c + d*x])/(4199*a^8*d) - (320*Sin[c + d*x]^3)/(12597*a^8*d) + (32*Sin[c + d*x]^5)/(4199*a^8*d) + (I*Cos[c + d*x]^3)/(19*d*(a + I*a*Tan[c + d*x])^8) + (11*I*Cos[c + d*x]^3)/(323*a*d*(a + I*a*Tan[c + d*x])^7) + (22*I*Cos[c + d*x]^3)/(969*a^2*d*(a + I*a*Tan[c + d*x])^6) + (66*I*Cos[c + d*x]^3)/(4199*a^3*d*(a + I*a*Tan[c + d*x])^5) + (48*I*Cos[c + d*x]^3)/(4199*d*(a^2 + I*a^2*Tan[c + d*x])^4) + (112*I*Cos[c + d*x]^3)/(12597*a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (64*I*Cos[c + d*x]^5)/(4199*d*(a^8 + I*a^8*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/2) (a+a I Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]), x, 5, (-6*a*e^4*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((2*I)/7)*a*(e*Sec[c + d*x])^(7/2))/d + (6*a*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*e*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]), x, 4, (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d) + (((2*I)/5)*a*(e*Sec[c + d*x])^(5/2))/d + (2*a*e*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]), x, 4, (-2*a*e^2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((2*I)/3)*a*(e*Sec[c + d*x])^(3/2))/d + (2*a*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/d} -{(e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x]), x, 3, ((2*I)*a*Sqrt[e*Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/d} -{(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(1/2), x, 3, ((-2*I)*a)/(d*Sqrt[e*Sec[c + d*x]]) + (2*a*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(3/2), x, 4, -((2*I*a)/(3*d*(e*Sec[c + d*x])^(3/2))) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(3*d*e^2) + (2*a*Sin[c + d*x])/(3*d*e*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(5/2), x, 4, -((2*I*a)/(5*d*(e*Sec[c + d*x])^(5/2))) + (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*a*Sin[c + d*x])/(5*d*e*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])/(e*Sec[c + d*x])^(7/2), x, 5, -((2*I*a)/(7*d*(e*Sec[c + d*x])^(7/2))) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(21*d*e^4) + (2*a*Sin[c + d*x])/(7*d*e*(e*Sec[c + d*x])^(5/2)) + (10*a*Sin[c + d*x])/(21*d*e^3*Sqrt[e*Sec[c + d*x]])} - - -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2, x, 5, -((14*a^2*e^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (14*I*a^2*(e*Sec[c + d*x])^(3/2))/(15*d) + (14*a^2*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*I*(e*Sec[c + d*x])^(3/2)*(a^2 + I*a^2*Tan[c + d*x]))/(5*d)} -{(e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^2, x, 4, (10*I*a^2*Sqrt[e*Sec[c + d*x]])/(3*d) + (10*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(3*d) + (2*I*Sqrt[e*Sec[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x]))/(3*d)} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(1/2), x, 4, (6*a^2*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (6*a^2*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(d*e) - (4*I*(a^2 + I*a^2*Tan[c + d*x]))/(d*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(3/2), x, 3, (-2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d*e^2) - (((4*I)/3)*(a^2 + I*a^2*Tan[c + d*x]))/(d*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(5/2), x, 3, (2*a^2*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((4*I)/5)*(a^2 + I*a^2*Tan[c + d*x]))/(d*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(7/2), x, 4, (2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(7*d*e^4) + (2*a^2*Sin[c + d*x])/(7*d*e^3*Sqrt[e*Sec[c + d*x]]) - (((4*I)/7)*(a^2 + I*a^2*Tan[c + d*x]))/(d*(e*Sec[c + d*x])^(7/2))} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(9/2), x, 4, (2*a^2*EllipticE[(1/2)*(c + d*x), 2])/(3*d*e^4*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*a^2*Sin[c + d*x])/(9*d*e^3*(e*Sec[c + d*x])^(3/2)) - (4*I*(a^2 + I*a^2*Tan[c + d*x]))/(9*d*(e*Sec[c + d*x])^(9/2))} -{(a + I*a*Tan[c + d*x])^2/(e*Sec[c + d*x])^(11/2), x, 5, (10*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(33*d*e^6) + (2*a^2*Sin[c + d*x])/(11*d*e^3*(e*Sec[c + d*x])^(5/2)) + (10*a^2*Sin[c + d*x])/(33*d*e^5*Sqrt[e*Sec[c + d*x]]) - (4*I*(a^2 + I*a^2*Tan[c + d*x]))/(11*d*(e*Sec[c + d*x])^(11/2))} - - -{(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^3, x, 7, -((2*a^3*e^4*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (10*I*a^3*(e*Sec[c + d*x])^(7/2))/(21*d) + (2*a^3*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^3*e*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d) + (2*I*a*(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2)/(11*d) + (10*I*(e*Sec[c + d*x])^(7/2)*(a^3 + I*a^3*Tan[c + d*x]))/(33*d)} -{(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^3, x, 6, (26*a^3*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(21*d) + (26*I*a^3*(e*Sec[c + d*x])^(5/2))/(35*d) + (26*a^3*e*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*I*a*(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2)/(9*d) + (26*I*(e*Sec[c + d*x])^(5/2)*(a^3 + I*a^3*Tan[c + d*x]))/(63*d)} -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3, x, 6, -((22*a^3*e^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (22*I*a^3*(e*Sec[c + d*x])^(3/2))/(15*d) + (22*a^3*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*I*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2)/(7*d) + (22*I*(e*Sec[c + d*x])^(3/2)*(a^3 + I*a^3*Tan[c + d*x]))/(35*d)} -{(e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^3, x, 5, (6*I*a^3*Sqrt[e*Sec[c + d*x]])/d + (6*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/d + (2*I*a*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^2)/(5*d) + (6*I*Sqrt[e*Sec[c + d*x]]*(a^3 + I*a^3*Tan[c + d*x]))/(5*d)} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(1/2), x, 5, -((26*I*a^3)/(3*d*Sqrt[e*Sec[c + d*x]])) + (14*a^3*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (6*a^3*Tan[c + d*x])/(d*Sqrt[e*Sec[c + d*x]]) - (2*I*a^3*Tan[c + d*x]^2)/(3*d*Sqrt[e*Sec[c + d*x]]), (14*a^3*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (14*a^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(d*e) + (2*I*a*(a + I*a*Tan[c + d*x])^2)/(3*d*Sqrt[e*Sec[c + d*x]]) - (28*I*(a^3 + I*a^3*Tan[c + d*x]))/(3*d*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(3/2), x, 4, (((-10*I)/3)*a^3*Sqrt[e*Sec[c + d*x]])/(d*e^2) - (10*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*d*e^2) - (((4*I)/3)*a*(a + I*a*Tan[c + d*x])^2)/(d*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(5/2), x, 4, (((6*I)/5)*a^3)/(d*e^2*Sqrt[e*Sec[c + d*x]]) - (6*a^3*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((4*I)/5)*a*(a + I*a*Tan[c + d*x])^2)/(d*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(7/2), x, 4, -((2*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(21*d*e^4)) - (2*I*(a + I*a*Tan[c + d*x])^3)/(7*d*(e*Sec[c + d*x])^(7/2)) - (4*I*(a^3 + I*a^3*Tan[c + d*x]))/(21*d*e^2*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(9/2), x, 4, (2*a^3*EllipticE[(1/2)*(c + d*x), 2])/(15*d*e^4*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (2*I*(a + I*a*Tan[c + d*x])^3)/(9*d*(e*Sec[c + d*x])^(9/2)) - (4*I*(a^3 + I*a^3*Tan[c + d*x]))/(15*d*e^2*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(11/2), x, 5, (10*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(77*d*e^6) + (10*a^3*Sin[c + d*x])/(77*d*e^5*Sqrt[e*Sec[c + d*x]]) - (2*I*(a + I*a*Tan[c + d*x])^3)/(11*d*(e*Sec[c + d*x])^(11/2)) - (20*I*(a^3 + I*a^3*Tan[c + d*x]))/(77*d*e^2*(e*Sec[c + d*x])^(7/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(13/2), x, 5, (14*a^3*EllipticE[(1/2)*(c + d*x), 2])/(39*d*e^6*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (14*a^3*Sin[c + d*x])/(117*d*e^5*(e*Sec[c + d*x])^(3/2)) - (2*I*(a + I*a*Tan[c + d*x])^3)/(13*d*(e*Sec[c + d*x])^(13/2)) - (28*I*(a^3 + I*a^3*Tan[c + d*x]))/(117*d*e^2*(e*Sec[c + d*x])^(9/2))} -{(a + I*a*Tan[c + d*x])^3/(e*Sec[c + d*x])^(15/2), x, 6, (2*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(11*d*e^8) + (6*a^3*Sin[c + d*x])/(55*d*e^5*(e*Sec[c + d*x])^(5/2)) + (2*a^3*Sin[c + d*x])/(11*d*e^7*Sqrt[e*Sec[c + d*x]]) - (2*I*(a + I*a*Tan[c + d*x])^3)/(15*d*(e*Sec[c + d*x])^(15/2)) - (12*I*(a^3 + I*a^3*Tan[c + d*x]))/(55*d*e^2*(e*Sec[c + d*x])^(11/2))} - - -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^4, x, 7, -((22*a^4*e^2*EllipticE[(1/2)*(c + d*x), 2])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (22*I*a^4*(e*Sec[c + d*x])^(3/2))/(9*d) + (22*a^4*e*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*I*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3)/(9*d) + (10*I*(e*Sec[c + d*x])^(3/2)*(a^2 + I*a^2*Tan[c + d*x])^2)/(21*d) + (22*I*(e*Sec[c + d*x])^(3/2)*(a^4 + I*a^4*Tan[c + d*x]))/(21*d)} -{(e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^4, x, 6, (78*I*a^4*Sqrt[e*Sec[c + d*x]])/(7*d) + (78*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(7*d) + (2*I*a*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3)/(7*d) + (26*I*Sqrt[e*Sec[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x])^2)/(35*d) + (78*I*Sqrt[e*Sec[c + d*x]]*(a^4 + I*a^4*Tan[c + d*x]))/(35*d)} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(1/2), x, 6, (154*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (154*I*a^4*(e*Sec[c + d*x])^(3/2))/(15*d*e^2) - (154*a^4*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d*e) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(d*Sqrt[e*Sec[c + d*x]]) - (22*I*(e*Sec[c + d*x])^(3/2)*(a^4 + I*a^4*Tan[c + d*x]))/(5*d*e^2)} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(3/2), x, 5, -((10*I*a^4*Sqrt[e*Sec[c + d*x]])/(d*e^2)) - (10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(d*e^2) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(3*d*(e*Sec[c + d*x])^(3/2)) - (2*I*Sqrt[e*Sec[c + d*x]]*(a^4 + I*a^4*Tan[c + d*x]))/(d*e^2)} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(5/2), x, 5, -((42*a^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (42*a^4*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*d*e^3) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(5*d*(e*Sec[c + d*x])^(5/2)) + (28*I*(a^4 + I*a^4*Tan[c + d*x]))/(5*d*e^2*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(7/2), x, 4, (10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(21*d*e^4) - (((4*I)/7)*a*(a + I*a*Tan[c + d*x])^3)/(d*(e*Sec[c + d*x])^(7/2)) + (((20*I)/21)*(a^4 + I*a^4*Tan[c + d*x]))/(d*e^2*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(9/2), x, 4, -((2*a^4*EllipticE[(1/2)*(c + d*x), 2])/(15*d*e^4*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(9*d*(e*Sec[c + d*x])^(9/2)) + (4*I*(a^4 + I*a^4*Tan[c + d*x]))/(15*d*e^2*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(11/2), x, 5, -((2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(77*d*e^6)) - (2*a^4*Sin[c + d*x])/(77*d*e^5*Sqrt[e*Sec[c + d*x]]) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(11*d*(e*Sec[c + d*x])^(11/2)) + (4*I*(a^4 + I*a^4*Tan[c + d*x]))/(77*d*e^2*(e*Sec[c + d*x])^(7/2))} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(13/2), x, 5, (2*a^4*EllipticE[(1/2)*(c + d*x), 2])/(39*d*e^6*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*a^4*Sin[c + d*x])/(117*d*e^5*(e*Sec[c + d*x])^(3/2)) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(13*d*(e*Sec[c + d*x])^(13/2)) - (4*I*(a^4 + I*a^4*Tan[c + d*x]))/(117*d*e^2*(e*Sec[c + d*x])^(9/2))} -{(a + I*a*Tan[c + d*x])^4/(e*Sec[c + d*x])^(15/2), x, 6, (2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(33*d*e^8) + (2*a^4*Sin[c + d*x])/(55*d*e^5*(e*Sec[c + d*x])^(5/2)) + (2*a^4*Sin[c + d*x])/(33*d*e^7*Sqrt[e*Sec[c + d*x]]) - (4*I*a*(a + I*a*Tan[c + d*x])^3)/(15*d*(e*Sec[c + d*x])^(15/2)) - (4*I*(a^4 + I*a^4*Tan[c + d*x]))/(55*d*e^2*(e*Sec[c + d*x])^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x]), x, 5, (-6*e^6*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((2*I)/7)*e^2*(e*Sec[c + d*x])^(7/2))/(a*d) + (6*e^5*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) + (2*e^3*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*a*d)} -{(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x]), x, 4, (2*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*a*d) - (((2*I)/5)*e^2*(e*Sec[c + d*x])^(5/2))/(a*d) + (2*e^3*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*a*d)} -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x]), x, 4, (-2*e^4*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (((2*I)/3)*e^2*(e*Sec[c + d*x])^(3/2))/(a*d) + (2*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x]), x, 3, ((-2*I)*e^2*Sqrt[e*Sec[c + d*x]])/(a*d) + (2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(a*d)} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x]), x, 3, ((2*I)*e^2)/(a*d*Sqrt[e*Sec[c + d*x]]) + (2*e^2*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])} -{(e*Sec[c + d*x])^(1/2)/(a + I*a*Tan[c + d*x]), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*a*d) + (((2*I)/3)*Sqrt[e*Sec[c + d*x]])/(d*(a + I*a*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])), x, 3, (6*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + ((2*I)/5)/(d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])), x, 4, (10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(21*a*d*e^2) + (10*Sin[c + d*x])/(21*a*d*e*Sqrt[e*Sec[c + d*x]]) + ((2*I)/7)/(d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])), x, 4, (14*EllipticE[(c + d*x)/2, 2])/(15*a*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (14*Sin[c + d*x])/(45*a*d*e*(e*Sec[c + d*x])^(3/2)) + ((2*I)/9)/(d*(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])), x, 5, (30*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(77*a*d*e^4) + (18*Sin[c + d*x])/(77*a*d*e*(e*Sec[c + d*x])^(5/2)) + (30*Sin[c + d*x])/(77*a*d*e^3*Sqrt[e*Sec[c + d*x]]) + ((2*I)/11)/(d*(e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]))} - - -{(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^2, x, 6, -((22*e^8*EllipticE[(1/2)*(c + d*x), 2])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (22*e^7*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (22*e^5*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(45*a^2*d) + (22*e^3*(e*Sec[c + d*x])^(9/2)*Sin[c + d*x])/(63*a^2*d) - (4*I*e^2*(e*Sec[c + d*x])^(11/2))/(7*d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^2, x, 5, (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(7*a^2*d) + (6*e^5*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*a^2*d) + (18*e^3*(e*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(35*a^2*d) - (4*I*e^2*(e*Sec[c + d*x])^(9/2))/(5*d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^2, x, 5, (-14*e^6*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (14*e^5*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*d) + (14*e^3*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(15*a^2*d) - (((4*I)/3)*e^2*(e*Sec[c + d*x])^(7/2))/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^2, x, 4, (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*a^2*d) + (10*e^3*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((4*I)*e^2*(e*Sec[c + d*x])^(5/2))/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2, x, 4, (6*e^4*EllipticE[(c + d*x)/2, 2])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (6*e^3*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((4*I)*e^2*(e*Sec[c + d*x])^(3/2))/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^2, x, 3, (-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*a^2*d) + (((4*I)/3)*e^2*Sqrt[e*Sec[c + d*x]])/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2, x, 3, (2*e^2*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/5)*e^2)/(d*Sqrt[e*Sec[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(1/2)/(a + I*a*Tan[c + d*x])^2, x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(7*a^2*d) + (2*e*Sin[c + d*x])/(7*a^2*d*Sqrt[e*Sec[c + d*x]]) + (4*I*e^2)/(7*d*(e*Sec[c + d*x])^(3/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^2), x, 4, (2*EllipticE[(1/2)*(c + d*x), 2])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e*Sin[c + d*x])/(9*a^2*d*(e*Sec[c + d*x])^(3/2)) + (4*I*e^2)/(9*d*(e*Sec[c + d*x])^(5/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(33*a^2*d*e^2) + (2*e*Sin[c + d*x])/(11*a^2*d*(e*Sec[c + d*x])^(5/2)) + (10*Sin[c + d*x])/(33*a^2*d*e*Sqrt[e*Sec[c + d*x]]) + (4*I*e^2)/(11*d*(e*Sec[c + d*x])^(7/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2), x, 5, (42*EllipticE[(1/2)*(c + d*x), 2])/(65*a^2*d*e^2*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e*Sin[c + d*x])/(13*a^2*d*(e*Sec[c + d*x])^(7/2)) + (14*Sin[c + d*x])/(65*a^2*d*e*(e*Sec[c + d*x])^(3/2)) + (4*I*e^2)/(13*d*(e*Sec[c + d*x])^(9/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2), x, 6, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(7*a^2*d*e^4) + (2*e*Sin[c + d*x])/(15*a^2*d*(e*Sec[c + d*x])^(9/2)) + (6*Sin[c + d*x])/(35*a^2*d*e*(e*Sec[c + d*x])^(5/2)) + (2*Sin[c + d*x])/(7*a^2*d*e^3*Sqrt[e*Sec[c + d*x]]) + (4*I*e^2)/(15*d*(e*Sec[c + d*x])^(11/2)*(a^2 + I*a^2*Tan[c + d*x]))} - - -{(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^3, x, 6, -((22*e^8*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) - (22*I*e^4*(e*Sec[c + d*x])^(7/2))/(21*a^3*d) + (22*e^7*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*d) + (22*e^5*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(15*a^3*d) - (4*I*e^2*(e*Sec[c + d*x])^(11/2))/(3*a*d*(a + I*a*Tan[c + d*x])^2)} -{(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^3, x, 5, (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(a^3*d) - (((18*I)/5)*e^4*(e*Sec[c + d*x])^(5/2))/(a^3*d) + (6*e^5*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(a^3*d) - ((4*I)*e^2*(e*Sec[c + d*x])^(9/2))/(a*d*(a + I*a*Tan[c + d*x])^2)} -{(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^3, x, 5, (14*e^6*EllipticE[(c + d*x)/2, 2])/(a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((14*I)/3)*e^4*(e*Sec[c + d*x])^(3/2))/(a^3*d) - (14*e^5*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(a^3*d) + ((4*I)*e^2*(e*Sec[c + d*x])^(7/2))/(a*d*(a + I*a*Tan[c + d*x])^2)} -{(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^3, x, 4, (((10*I)/3)*e^4*Sqrt[e*Sec[c + d*x]])/(a^3*d) - (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(3*a^3*d) + (((4*I)/3)*e^2*(e*Sec[c + d*x])^(5/2))/(a*d*(a + I*a*Tan[c + d*x])^2)} -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^3, x, 4, (((-6*I)/5)*e^4)/(a^3*d*Sqrt[e*Sec[c + d*x]]) - (6*e^4*EllipticE[(c + d*x)/2, 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/5)*e^2*(e*Sec[c + d*x])^(3/2))/(a*d*(a + I*a*Tan[c + d*x])^2)} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^3, x, 4, (-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[e*Sec[c + d*x]])/(21*a^3*d) + (((4*I)/7)*e^2*Sqrt[e*Sec[c + d*x]])/(a*d*(a + I*a*Tan[c + d*x])^2) - (((2*I)/21)*e^2*Sqrt[e*Sec[c + d*x]])/(d*(a^3 + I*a^3*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^3, x, 4, (2*e^2*EllipticE[(c + d*x)/2, 2])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/9)*e^2)/(a*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^2) + (((2*I)/45)*e^2)/(d*Sqrt[e*Sec[c + d*x]]*(a^3 + I*a^3*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(1/2)/(a + I*a*Tan[c + d*x])^3, x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(77*a^3*d) + (10*e*Sin[c + d*x])/(77*a^3*d*Sqrt[e*Sec[c + d*x]]) + (2*I*Sqrt[e*Sec[c + d*x]])/(11*d*(a + I*a*Tan[c + d*x])^3) + (20*I*e^2)/(77*d*(e*Sec[c + d*x])^(3/2)*(a^3 + I*a^3*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^3), x, 5, (14*EllipticE[(1/2)*(c + d*x), 2])/(39*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (14*e*Sin[c + d*x])/(117*a^3*d*(e*Sec[c + d*x])^(3/2)) + (2*I)/(13*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3) + (28*I*e^2)/(117*d*(e*Sec[c + d*x])^(5/2)*(a^3 + I*a^3*Tan[c + d*x]))} -{1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3), x, 6, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(11*a^3*d*e^2) + (6*e*Sin[c + d*x])/(55*a^3*d*(e*Sec[c + d*x])^(5/2)) + (2*Sin[c + d*x])/(11*a^3*d*e*Sqrt[e*Sec[c + d*x]]) + (2*I)/(15*d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^3) + (12*I*e^2)/(55*d*(e*Sec[c + d*x])^(7/2)*(a^3 + I*a^3*Tan[c + d*x]))} - - -{(e*Sec[c + d*x])^(15/2)/(a + I*a*Tan[c + d*x])^4, x, 6, (154*e^8*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) - (154*e^7*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a^4*d) - (154*e^5*(e*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(15*a^4*d) + (4*I*e^2*(e*Sec[c + d*x])^(11/2))/(a*d*(a + I*a*Tan[c + d*x])^3) + (44*I*e^4*(e*Sec[c + d*x])^(7/2))/(3*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(13/2)/(a + I*a*Tan[c + d*x])^4, x, 5, -((10*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(a^4*d)) - (10*e^5*(e*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(a^4*d) + (4*I*e^2*(e*Sec[c + d*x])^(9/2))/(3*a*d*(a + I*a*Tan[c + d*x])^3) + (12*I*e^4*(e*Sec[c + d*x])^(5/2))/(d*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(11/2)/(a + I*a*Tan[c + d*x])^4, x, 5, -((42*e^6*EllipticE[(1/2)*(c + d*x), 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]])) + (42*e^5*Sqrt[e*Sec[c + d*x]]*Sin[c + d*x])/(5*a^4*d) + (4*I*e^2*(e*Sec[c + d*x])^(7/2))/(5*a*d*(a + I*a*Tan[c + d*x])^3) - (28*I*e^4*(e*Sec[c + d*x])^(3/2))/(5*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^4, x, 4, (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(21*a^4*d) + (4*I*e^2*(e*Sec[c + d*x])^(5/2))/(7*a*d*(a + I*a*Tan[c + d*x])^3) - (20*I*e^4*Sqrt[e*Sec[c + d*x]])/(21*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^4, x, 4, (-2*e^4*EllipticE[(c + d*x)/2, 2])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (((4*I)/9)*e^2*(e*Sec[c + d*x])^(3/2))/(a*d*(a + I*a*Tan[c + d*x])^3) - (((4*I)/15)*e^4)/(d*Sqrt[e*Sec[c + d*x]]*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^4, x, 5, -((2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(77*a^4*d)) - (2*e^3*Sin[c + d*x])/(77*a^4*d*Sqrt[e*Sec[c + d*x]]) + (4*I*e^2*Sqrt[e*Sec[c + d*x]])/(11*a*d*(a + I*a*Tan[c + d*x])^3) - (4*I*e^4)/(77*d*(e*Sec[c + d*x])^(3/2)*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^4, x, 5, (2*e^2*EllipticE[(1/2)*(c + d*x), 2])/(39*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[e*Sec[c + d*x]]) + (2*e^3*Sin[c + d*x])/(117*a^4*d*(e*Sec[c + d*x])^(3/2)) + (4*I*e^2)/(13*a*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^3) + (4*I*e^4)/(117*d*(e*Sec[c + d*x])^(5/2)*(a^4 + I*a^4*Tan[c + d*x]))} -{(e*Sec[c + d*x])^(1/2)/(a + I*a*Tan[c + d*x])^4, x, 6, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[e*Sec[c + d*x]])/(33*a^4*d) + (2*e*Sin[c + d*x])/(33*a^4*d*Sqrt[e*Sec[c + d*x]]) + (2*I*Sqrt[e*Sec[c + d*x]])/(15*d*(a + I*a*Tan[c + d*x])^4) + (14*I*Sqrt[e*Sec[c + d*x]])/(165*a*d*(a + I*a*Tan[c + d*x])^3) + (4*I*e^2)/(33*d*(e*Sec[c + d*x])^(3/2)*(a^4 + I*a^4*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/3) (a+a I Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x]), x, 4, (((6*I)/5)*2^(5/6)*a*Hypergeometric2F1[-5/6, 5/6, 11/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(5/3))/(f*(1 + I*Tan[e + f*x])^(5/6))} -{(d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x]), x, 4, ((6*I)*2^(1/6)*a*Hypergeometric2F1[-1/6, 1/6, 7/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(1/3))/(f*(1 + I*Tan[e + f*x])^(1/6))} -{(a + I*a*Tan[e + f*x])/(d*Sec[e + f*x])^(1/3), x, 4, ((-3*I)*2^(5/6)*a*Hypergeometric2F1[-1/6, 1/6, 5/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(1/6))/(f*(d*Sec[e + f*x])^(1/3))} -{(a + I*a*Tan[e + f*x])/(d*Sec[e + f*x])^(5/3), x, 4, (((-3*I)/5)*2^(1/6)*a*Hypergeometric2F1[-5/6, 5/6, 1/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(5/6))/(f*(d*Sec[e + f*x])^(5/3))} - - -{(d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])^2, x, 4, (((12*I)/5)*2^(5/6)*a^2*Hypergeometric2F1[-11/6, 5/6, 11/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(5/3))/(f*(1 + I*Tan[e + f*x])^(5/6))} -{(d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^2, x, 4, ((12*I)*2^(1/6)*a^2*Hypergeometric2F1[-7/6, 1/6, 7/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(1/3))/(f*(1 + I*Tan[e + f*x])^(1/6))} -{(a + I*a*Tan[e + f*x])^2/(d*Sec[e + f*x])^(1/3), x, 4, ((-6*I)*2^(5/6)*Hypergeometric2F1[-5/6, -1/6, 5/6, (1 - I*Tan[e + f*x])/2]*(a^2 + I*a^2*Tan[e + f*x]))/(f*(d*Sec[e + f*x])^(1/3)*(1 + I*Tan[e + f*x])^(5/6))} -{(a + I*a*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/3), x, 4, (((-6*I)/5)*2^(1/6)*Hypergeometric2F1[-5/6, -1/6, 1/6, (1 - I*Tan[e + f*x])/2]*(a^2 + I*a^2*Tan[e + f*x]))/(f*(d*Sec[e + f*x])^(5/3)*(1 + I*Tan[e + f*x])^(1/6))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*Sec[e + f*x])^(5/3)/(a + I*a*Tan[e + f*x]), x, 4, (((3*I)/5)*Hypergeometric2F1[5/6, 7/6, 11/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(5/3)*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*f*(a + I*a*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(1/3)/(a + I*a*Tan[e + f*x]), x, 4, ((3*I)*Hypergeometric2F1[1/6, 11/6, 7/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(1/3)*(1 + I*Tan[e + f*x])^(5/6))/(2^(5/6)*f*(a + I*a*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])), x, 4, (((-3*I)/2)*Hypergeometric2F1[-1/6, 13/6, 5/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*a*f*(d*Sec[e + f*x])^(1/3))} -{1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])), x, 4, (((-3*I)/10)*Hypergeometric2F1[-5/6, 17/6, 1/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(5/6))/(2^(5/6)*a*f*(d*Sec[e + f*x])^(5/3))} - - -{(d*Sec[e + f*x])^(5/3)/(a + I*a*Tan[e + f*x])^2, x, 4, (((3*I)/10)*Hypergeometric2F1[5/6, 13/6, 11/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(5/3)*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*f*(a^2 + I*a^2*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(1/3)/(a + I*a*Tan[e + f*x])^2, x, 4, (((3*I)/2)*Hypergeometric2F1[1/6, 17/6, 7/6, (1 - I*Tan[e + f*x])/2]*(d*Sec[e + f*x])^(1/3)*(1 + I*Tan[e + f*x])^(5/6))/(2^(5/6)*f*(a^2 + I*a^2*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^2), x, 4, (((-3*I)/4)*Hypergeometric2F1[-1/6, 19/6, 5/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(1/6))/(2^(1/6)*a^2*f*(d*Sec[e + f*x])^(1/3))} -{1/((d*Sec[e + f*x])^(5/3)*(a + I*a*Tan[e + f*x])^2), x, 4, (((-3*I)/20)*Hypergeometric2F1[-5/6, 23/6, 1/6, (1 - I*Tan[e + f*x])/2]*(1 + I*Tan[e + f*x])^(5/6))/(2^(5/6)*a^2*f*(d*Sec[e + f*x])^(5/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^8*Sqrt[a + I*a*Tan[c + d*x]], x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(9/2))/(9*a^4*d)) + (24*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^5*d) - (12*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(15/2))/(15*a^7*d)} -{Sec[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (((-8*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^3*d) + (((8*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^4*d) - (((2*I)/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^5*d)} -{Sec[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (((-4*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^2*d) + (((2*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^3*d)} -{Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]], x, 2, (((-2*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a*d)} -{Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]], x, 5, -((3*I*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*d)) + (3*I*a)/(4*d*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^2)/(2*d*(a - I*a*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]], x, 7, -((35*I*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d)) + (35*I*a^2)/(96*d*(a + I*a*Tan[c + d*x])^(3/2)) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(3/2)) - (7*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2)) + (35*I*a)/(64*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^6*Sqrt[a + I*a*Tan[c + d*x]], x, 9, -((231*I*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(512*Sqrt[2]*d)) + (231*I*a^3)/(640*d*(a + I*a*Tan[c + d*x])^(5/2)) - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3*(a + I*a*Tan[c + d*x])^(5/2)) - (11*I*a^5)/(48*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(5/2)) - (33*I*a^4)/(64*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2)) + (77*I*a^2)/(256*d*(a + I*a*Tan[c + d*x])^(3/2)) + (231*I*a)/(512*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^7*Sqrt[a + I*a*Tan[c + d*x]], x, 4, (256*I*a^4*Sec[c + d*x]^7)/(3003*d*(a + I*a*Tan[c + d*x])^(7/2)) + (64*I*a^3*Sec[c + d*x]^7)/(429*d*(a + I*a*Tan[c + d*x])^(5/2)) + (24*I*a^2*Sec[c + d*x]^7)/(143*d*(a + I*a*Tan[c + d*x])^(3/2)) + (2*I*a*Sec[c + d*x]^7)/(13*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (64*I*a^3*Sec[c + d*x]^5)/(315*d*(a + I*a*Tan[c + d*x])^(5/2)) + (16*I*a^2*Sec[c + d*x]^5)/(63*d*(a + I*a*Tan[c + d*x])^(3/2)) + (2*I*a*Sec[c + d*x]^5)/(9*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]], x, 2, (8*I*a^2*Sec[c + d*x]^3)/(15*d*(a + I*a*Tan[c + d*x])^(3/2)) + (2*I*a*Sec[c + d*x]^3)/(5*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^1*Sqrt[a + I*a*Tan[c + d*x]], x, 1, ((2*I)*a*Sec[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^1*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*d) - (I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]], x, 5, (5*I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(8*Sqrt[2]*d) + (5*I*a*Cos[c + d*x])/(12*d*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]], x, 7, (63*I*Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(128*Sqrt[2]*d) + (21*I*a*Cos[c + d*x])/(64*d*Sqrt[a + I*a*Tan[c + d*x]]) + (9*I*a*Cos[c + d*x]^3)/(40*d*Sqrt[a + I*a*Tan[c + d*x]]) - (63*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(128*d) - (21*I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(80*d) - (I*Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/(5*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(3/2), x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^4*d)) + (24*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^5*d) - (4*I*(a + I*a*Tan[c + d*x])^(15/2))/(5*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(17/2))/(17*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2), x, 3, (((-8*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^3*d) + (((8*I)/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^4*d) - (((2*I)/13)*(a + I*a*Tan[c + d*x])^(13/2))/(a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2), x, 3, (((-4*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^2*d) + (((2*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2), x, 2, (((-2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a*d)} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2), x, 4, -((I*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d)) - (I*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2), x, 6, -((15*I*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*d)) + (15*I*a^2)/(32*d*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(3/2), x, 8, -((105*I*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*d)) + (35*I*a^3)/(128*d*(a + I*a*Tan[c + d*x])^(3/2)) - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3*(a + I*a*Tan[c + d*x])^(3/2)) - (3*I*a^5)/(16*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(3/2)) - (21*I*a^4)/(64*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2)) + (105*I*a^2)/(256*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2), x, 4, (256*I*a^4*Sec[c + d*x]^5)/(1155*d*(a + I*a*Tan[c + d*x])^(5/2)) + (64*I*a^3*Sec[c + d*x]^5)/(231*d*(a + I*a*Tan[c + d*x])^(3/2)) + (8*I*a^2*Sec[c + d*x]^5)/(33*d*Sqrt[a + I*a*Tan[c + d*x]]) + (2*I*a*Sec[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/(11*d)} -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2), x, 3, (64*I*a^3*Sec[c + d*x]^3)/(105*d*(a + I*a*Tan[c + d*x])^(3/2)) + (16*I*a^2*Sec[c + d*x]^3)/(35*d*Sqrt[a + I*a*Tan[c + d*x]]) + (2*I*a*Sec[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(7*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^(3/2), x, 2, (8*I*a^2*Sec[c + d*x])/(3*d*Sqrt[a + I*a*Tan[c + d*x]]) + (2*I*a*Sec[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^(3/2), x, 1, ((-2*I)*a*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2), x, 4, (I*a^(3/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(2*Sqrt[2]*d) - (I*a*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(2*d) - (I*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2), x, 6, (7*I*a^(3/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(16*Sqrt[2]*d) + (7*I*a^2*Cos[c + d*x])/(24*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*I*a*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(16*d) - (7*I*a*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(30*d) - (I*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2))/(5*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(5/2), x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^4*d)) + (8*I*(a + I*a*Tan[c + d*x])^(15/2))/(5*a^5*d) - (12*I*(a + I*a*Tan[c + d*x])^(17/2))/(17*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(19/2))/(19*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(5/2), x, 3, -((8*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^3*d)) + (8*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^4*d) - (2*I*(a + I*a*Tan[c + d*x])^(15/2))/(15*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2), x, 3, (((-4*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^2*d) + (((2*I)/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2), x, 2, (((-2*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a*d)} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2), x, 4, (I*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) - (I*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2), x, 5, -((3*I*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*d)) - (I*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*(a - I*a*Tan[c + d*x])^2) - (3*I*a^3*Sqrt[a + I*a*Tan[c + d*x]])/(16*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(5/2), x, 7, -((35*I*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*d)) + (35*I*a^3)/(128*d*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3*Sqrt[a + I*a*Tan[c + d*x]]) - (7*I*a^5)/(48*d*(a - I*a*Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]]) - (35*I*a^4)/(192*d*(a - I*a*Tan[c + d*x])*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2), x, 4, (256*I*a^4*Sec[c + d*x]^3)/(315*d*(a + I*a*Tan[c + d*x])^(3/2)) + (64*I*a^3*Sec[c + d*x]^3)/(105*d*Sqrt[a + I*a*Tan[c + d*x]]) + (8*I*a^2*Sec[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) + (2*I*a*Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(9*d)} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^(5/2), x, 3, (64*I*a^3*Sec[c + d*x])/(15*d*Sqrt[a + I*a*Tan[c + d*x]]) + (16*I*a^2*Sec[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) + (2*I*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2))/(5*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^(5/2), x, 2, -((8*I*a^2*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d) + (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2), x, 1, (((-2*I)/3)*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2), x, 5, (I*a^(5/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(4*Sqrt[2]*d) - (I*a^2*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(6*d) - (I*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2))/(5*d)} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2), x, 7, (9*I*a^(5/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(32*Sqrt[2]*d) + (3*I*a^3*Cos[c + d*x])/(16*d*Sqrt[a + I*a*Tan[c + d*x]]) - (9*I*a^2*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(32*d) - (3*I*a^2*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(20*d) - (9*I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2))/(70*d) - (I*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2))/(7*d)} - - -{Sec[c + d*x]^8*(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(15/2))/(15*a^4*d)) + (24*I*(a + I*a*Tan[c + d*x])^(17/2))/(17*a^5*d) - (12*I*(a + I*a*Tan[c + d*x])^(19/2))/(19*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(21/2))/(21*a^7*d)} -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((8*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^3*d)) + (8*I*(a + I*a*Tan[c + d*x])^(15/2))/(15*a^4*d) - (2*I*(a + I*a*Tan[c + d*x])^(17/2))/(17*a^5*d)} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((4*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^2*d)) + (2*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^3*d)} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^(7/2), x, 2, -((2*I*(a + I*a*Tan[c + d*x])^(9/2))/(9*a*d))} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^(7/2), x, 5, (3*I*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (3*I*a^3*Sqrt[a + I*a*Tan[c + d*x]])/d - (I*a^3*(a + I*a*Tan[c + d*x])^(3/2))/(d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^(7/2), x, 5, (I*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*d) - (I*a^5*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*(a - I*a*Tan[c + d*x])^2) + (I*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(8*d*(a - I*a*Tan[c + d*x]))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^(7/2), x, 6, -((5*I*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d)) - (I*a^6*Sqrt[a + I*a*Tan[c + d*x]])/(6*d*(a - I*a*Tan[c + d*x])^3) - (5*I*a^5*Sqrt[a + I*a*Tan[c + d*x]])/(48*d*(a - I*a*Tan[c + d*x])^2) - (5*I*a^4*Sqrt[a + I*a*Tan[c + d*x]])/(64*d*(a - I*a*Tan[c + d*x]))} - -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^(7/2), x, 4, (256*I*a^4*Sec[c + d*x])/(35*d*Sqrt[a + I*a*Tan[c + d*x]]) + (64*I*a^3*Sec[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (24*I*a^2*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2))/(35*d) + (2*I*a*Sec[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2))/(7*d)} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((64*I*a^3*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)) + (16*I*a^2*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) + (2*I*a*Cos[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2))/(3*d)} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(7/2), x, 2, (8*I*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) - (2*I*a*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2))/d} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(7/2), x, 1, -((2*I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2))/(5*d))} -{Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2), x, 6, (I*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(8*Sqrt[2]*d) - (I*a^3*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (I*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(12*d) - (I*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2))/(10*d) - (I*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(7/2))/(7*d)} -{Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^(7/2), x, 8, (11*I*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(64*Sqrt[2]*d) + (11*I*a^4*Cos[c + d*x])/(96*d*Sqrt[a + I*a*Tan[c + d*x]]) - (11*I*a^3*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(64*d) - (11*I*a^3*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(120*d) - (11*I*a^2*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^(3/2))/(140*d) - (11*I*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(5/2))/(126*d) - (I*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^(7/2))/(9*d)} -{Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2), x, 10, (195*I*a^(7/2)*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(1024*Sqrt[2]*d) + (65*I*a^4*Cos[c + d*x])/(512*d*Sqrt[a + I*a*Tan[c + d*x]]) + (39*I*a^4*Cos[c + d*x]^3)/(448*d*Sqrt[a + I*a*Tan[c + d*x]]) - (195*I*a^3*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(1024*d) - (13*I*a^3*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(128*d) - (13*I*a^3*Cos[c + d*x]^5*Sqrt[a + I*a*Tan[c + d*x]])/(168*d) - (65*I*a^2*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^(3/2))/(924*d) - (5*I*a*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^(5/2))/(66*d) - (I*Cos[c + d*x]^11*(a + I*a*Tan[c + d*x])^(7/2))/(11*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^8/Sqrt[a + I*a*Tan[c + d*x]], x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(7/2))/(7*a^4*d)) + (8*I*(a + I*a*Tan[c + d*x])^(9/2))/(3*a^5*d) - (12*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(13/2))/(13*a^7*d)} -{Sec[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]], x, 3, (((-8*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^3*d) + (((8*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^4*d) - (((2*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^5*d)} -{Sec[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]], x, 3, (((-4*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a^2*d) + (((2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^3*d)} -{Sec[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]], x, 2, ((-2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{Cos[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]], x, 6, -((5*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d)) + (5*I*a)/(12*d*(a + I*a*Tan[c + d*x])^(3/2)) - (I*a^2)/(2*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(3/2)) + (5*I)/(8*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]], x, 8, -((63*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*Sqrt[a]*d)) + (63*I*a^2)/(160*d*(a + I*a*Tan[c + d*x])^(5/2)) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(5/2)) - (9*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2)) + (21*I*a)/(64*d*(a + I*a*Tan[c + d*x])^(3/2)) + (63*I)/(128*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^6/Sqrt[a + I*a*Tan[c + d*x]], x, 10, -((429*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(1024*Sqrt[2]*Sqrt[a]*d)) + (429*I*a^3)/(896*d*(a + I*a*Tan[c + d*x])^(7/2)) - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3*(a + I*a*Tan[c + d*x])^(7/2)) - (13*I*a^5)/(48*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(7/2)) - (143*I*a^4)/(192*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(7/2)) + (429*I*a^2)/(1280*d*(a + I*a*Tan[c + d*x])^(5/2)) + (143*I*a)/(512*d*(a + I*a*Tan[c + d*x])^(3/2)) + (429*I)/(1024*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^9/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (256*I*a^4*Sec[c + d*x]^9)/(6435*d*(a + I*a*Tan[c + d*x])^(9/2)) + (64*I*a^3*Sec[c + d*x]^9)/(715*d*(a + I*a*Tan[c + d*x])^(7/2)) + (8*I*a^2*Sec[c + d*x]^9)/(65*d*(a + I*a*Tan[c + d*x])^(5/2)) + (2*I*a*Sec[c + d*x]^9)/(15*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^7/Sqrt[a + I*a*Tan[c + d*x]], x, 3, (64*I*a^3*Sec[c + d*x]^7)/(693*d*(a + I*a*Tan[c + d*x])^(7/2)) + (16*I*a^2*Sec[c + d*x]^7)/(99*d*(a + I*a*Tan[c + d*x])^(5/2)) + (2*I*a*Sec[c + d*x]^7)/(11*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^5/Sqrt[a + I*a*Tan[c + d*x]], x, 2, (8*I*a^2*Sec[c + d*x]^5)/(35*d*(a + I*a*Tan[c + d*x])^(5/2)) + (2*I*a*Sec[c + d*x]^5)/(7*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]], x, 1, (((2*I)/3)*a*Sec[c + d*x]^3)/(d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^1/Sqrt[a + I*a*Tan[c + d*x]], x, 2, (I*Sqrt[2]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^1/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(4*Sqrt[2]*Sqrt[a]*d) + (I*Cos[c + d*x])/(2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (3*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a*d)} -{Cos[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]], x, 6, (35*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(64*Sqrt[2]*Sqrt[a]*d) + (35*I*Cos[c + d*x])/(96*d*Sqrt[a + I*a*Tan[c + d*x]]) + (I*Cos[c + d*x]^3)/(4*d*Sqrt[a + I*a*Tan[c + d*x]]) - (35*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(64*a*d) - (7*I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*a*d)} - - -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(3/2), x, 3, -((16*I*(a + I*a*Tan[c + d*x])^(5/2))/(5*a^4*d)) + (24*I*(a + I*a*Tan[c + d*x])^(7/2))/(7*a^5*d) - (4*I*(a + I*a*Tan[c + d*x])^(9/2))/(3*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2), x, 3, (((-8*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a^3*d) + (((8*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^4*d) - (((2*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^5*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2), x, 3, ((-4*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) + (((2*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a^3*d)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2), x, 2, (2*I)/(a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2), x, 7, -((7*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*a^(3/2)*d)) + (7*I*a)/(20*d*(a + I*a*Tan[c + d*x])^(5/2)) - (I*a^2)/(2*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(5/2)) + (7*I)/(24*d*(a + I*a*Tan[c + d*x])^(3/2)) + (7*I)/(16*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2), x, 9, -((99*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*a^(3/2)*d)) + (99*I*a^2)/(224*d*(a + I*a*Tan[c + d*x])^(7/2)) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(7/2)) - (11*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(7/2)) + (99*I*a)/(320*d*(a + I*a*Tan[c + d*x])^(5/2)) + (33*I)/(128*d*(a + I*a*Tan[c + d*x])^(3/2)) + (99*I)/(256*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^6/(a + I*a*Tan[c + d*x])^(3/2), x, 11, -((715*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2048*Sqrt[2]*a^(3/2)*d)) + (715*I*a^3)/(1152*d*(a + I*a*Tan[c + d*x])^(9/2)) - (I*a^6)/(6*d*(a - I*a*Tan[c + d*x])^3*(a + I*a*Tan[c + d*x])^(9/2)) - (5*I*a^5)/(16*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(9/2)) - (65*I*a^4)/(64*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(9/2)) + (715*I*a^2)/(1792*d*(a + I*a*Tan[c + d*x])^(7/2)) + (143*I*a)/(512*d*(a + I*a*Tan[c + d*x])^(5/2)) + (715*I)/(3072*d*(a + I*a*Tan[c + d*x])^(3/2)) + (715*I)/(2048*a*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (256*I*a^4*Sec[c + d*x]^11)/(12155*d*(a + I*a*Tan[c + d*x])^(11/2)) + (64*I*a^3*Sec[c + d*x]^11)/(1105*d*(a + I*a*Tan[c + d*x])^(9/2)) + (8*I*a^2*Sec[c + d*x]^11)/(85*d*(a + I*a*Tan[c + d*x])^(7/2)) + (2*I*a*Sec[c + d*x]^11)/(17*d*(a + I*a*Tan[c + d*x])^(5/2))} -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(3/2), x, 3, (64*I*a^3*Sec[c + d*x]^9)/(1287*d*(a + I*a*Tan[c + d*x])^(9/2)) + (16*I*a^2*Sec[c + d*x]^9)/(143*d*(a + I*a*Tan[c + d*x])^(7/2)) + (2*I*a*Sec[c + d*x]^9)/(13*d*(a + I*a*Tan[c + d*x])^(5/2))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(3/2), x, 2, (8*I*a^2*Sec[c + d*x]^7)/(63*d*(a + I*a*Tan[c + d*x])^(7/2)) + (2*I*a*Sec[c + d*x]^7)/(9*d*(a + I*a*Tan[c + d*x])^(5/2))} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(3/2), x, 1, (((2*I)/5)*a*Sec[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^(5/2))} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2), x, 3, (2*I*Sqrt[2]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(a^(3/2)*d) - (2*I*Sec[c + d*x])/(a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^(3/2), x, 3, (I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (I*Sec[c + d*x])/(2*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^(3/2), x, 5, (15*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(32*Sqrt[2]*a^(3/2)*d) + (I*Cos[c + d*x])/(4*d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*I*Cos[c + d*x])/(16*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (15*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(32*a^2*d)} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2), x, 7, (105*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d) + (I*Cos[c + d*x]^3)/(6*d*(a + I*a*Tan[c + d*x])^(3/2)) + (35*I*Cos[c + d*x])/(128*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*Cos[c + d*x]^3)/(16*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (105*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(256*a^2*d) - (7*I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(32*a^2*d)} - - -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^(5/2), x, 3, (((-32*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^5*d) + (((64*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^6*d) - (((16*I)/3)*(a + I*a*Tan[c + d*x])^(9/2))/(a^7*d) + (((16*I)/11)*(a + I*a*Tan[c + d*x])^(11/2))/(a^8*d) - (((2*I)/13)*(a + I*a*Tan[c + d*x])^(13/2))/(a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(5/2), x, 3, (((-16*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a^4*d) + (((24*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^5*d) - (((12*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a^6*d) + (((2*I)/9)*(a + I*a*Tan[c + d*x])^(9/2))/(a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(5/2), x, 3, ((-8*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d) + (((8*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a^4*d) - (((2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a^5*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2), x, 3, (4*I)/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d)} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2), x, 2, ((2*I)/3)/(a*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2), x, 8, -((9*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*a^(5/2)*d)) + (9*I*a)/(28*d*(a + I*a*Tan[c + d*x])^(7/2)) - (I*a^2)/(2*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(7/2)) + (9*I)/(40*d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*I)/(16*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (9*I)/(32*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2), x, 10, -((143*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(512*Sqrt[2]*a^(5/2)*d)) + (143*I*a^2)/(288*d*(a + I*a*Tan[c + d*x])^(9/2)) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(9/2)) - (13*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(9/2)) + (143*I*a)/(448*d*(a + I*a*Tan[c + d*x])^(7/2)) + (143*I)/(640*d*(a + I*a*Tan[c + d*x])^(5/2)) + (143*I)/(768*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (143*I)/(512*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(5/2), x, 4, (256*I*a^4*Sec[c + d*x]^13)/(20995*d*(a + I*a*Tan[c + d*x])^(13/2)) + (64*I*a^3*Sec[c + d*x]^13)/(1615*d*(a + I*a*Tan[c + d*x])^(11/2)) + (24*I*a^2*Sec[c + d*x]^13)/(323*d*(a + I*a*Tan[c + d*x])^(9/2)) + (2*I*a*Sec[c + d*x]^13)/(19*d*(a + I*a*Tan[c + d*x])^(7/2))} -{Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(5/2), x, 3, (64*I*a^3*Sec[c + d*x]^11)/(2145*d*(a + I*a*Tan[c + d*x])^(11/2)) + (16*I*a^2*Sec[c + d*x]^11)/(195*d*(a + I*a*Tan[c + d*x])^(9/2)) + (2*I*a*Sec[c + d*x]^11)/(15*d*(a + I*a*Tan[c + d*x])^(7/2))} -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(5/2), x, 2, (8*I*a^2*Sec[c + d*x]^9)/(99*d*(a + I*a*Tan[c + d*x])^(9/2)) + (2*I*a*Sec[c + d*x]^9)/(11*d*(a + I*a*Tan[c + d*x])^(7/2))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(5/2), x, 1, (((2*I)/7)*a*Sec[c + d*x]^7)/(d*(a + I*a*Tan[c + d*x])^(7/2))} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(5/2), x, 4, (4*I*Sqrt[2]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(a^(5/2)*d) - (2*I*Sec[c + d*x]^3)/(3*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - (4*I*Sec[c + d*x])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2), x, 4, -((I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(5/2)*d)) + (I*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^(5/2), x, 4, (3*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (I*Sec[c + d*x])/(4*d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*I*Sec[c + d*x])/(16*a*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^(5/2), x, 6, (35*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(128*Sqrt[2]*a^(5/2)*d) + (I*Cos[c + d*x])/(6*d*(a + I*a*Tan[c + d*x])^(5/2)) + (7*I*Cos[c + d*x])/(48*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (35*I*Cos[c + d*x])/(192*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (35*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(128*a^3*d)} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2), x, 8, (1155*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d) + (I*Cos[c + d*x]^3)/(8*d*(a + I*a*Tan[c + d*x])^(5/2)) + (11*I*Cos[c + d*x]^3)/(96*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (385*I*Cos[c + d*x])/(2048*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (33*I*Cos[c + d*x]^3)/(256*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (1155*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4096*a^3*d) - (77*I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(512*a^3*d)} - - -{Sec[c + d*x]^10/(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((32*I*(a + I*a*Tan[c + d*x])^(3/2))/(3*a^5*d)) + (64*I*(a + I*a*Tan[c + d*x])^(5/2))/(5*a^6*d) - (48*I*(a + I*a*Tan[c + d*x])^(7/2))/(7*a^7*d) + (16*I*(a + I*a*Tan[c + d*x])^(9/2))/(9*a^8*d) - (2*I*(a + I*a*Tan[c + d*x])^(11/2))/(11*a^9*d)} -{Sec[c + d*x]^8/(a + I*a*Tan[c + d*x])^(7/2), x, 3, -((16*I*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d)) + (8*I*(a + I*a*Tan[c + d*x])^(3/2))/(a^5*d) - (12*I*(a + I*a*Tan[c + d*x])^(5/2))/(5*a^6*d) + (2*I*(a + I*a*Tan[c + d*x])^(7/2))/(7*a^7*d)} -{Sec[c + d*x]^6/(a + I*a*Tan[c + d*x])^(7/2), x, 3, (8*I)/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) + (8*I*Sqrt[a + I*a*Tan[c + d*x]])/(a^4*d) - (2*I*(a + I*a*Tan[c + d*x])^(3/2))/(3*a^5*d)} -{Sec[c + d*x]^4/(a + I*a*Tan[c + d*x])^(7/2), x, 3, (4*I)/(3*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) - (2*I)/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^2/(a + I*a*Tan[c + d*x])^(7/2), x, 2, (2*I)/(5*a*d*(a + I*a*Tan[c + d*x])^(5/2))} -{Cos[c + d*x]^2/(a + I*a*Tan[c + d*x])^(7/2), x, 9, -((11*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*a^(7/2)*d)) + (11*I*a)/(36*d*(a + I*a*Tan[c + d*x])^(9/2)) - (I*a^2)/(2*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(9/2)) + (11*I)/(56*d*(a + I*a*Tan[c + d*x])^(7/2)) + (11*I)/(80*a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (11*I)/(96*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (11*I)/(64*a^3*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cos[c + d*x]^4/(a + I*a*Tan[c + d*x])^(7/2), x, 11, -((195*I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(1024*Sqrt[2]*a^(7/2)*d)) + (195*I*a^2)/(352*d*(a + I*a*Tan[c + d*x])^(11/2)) - (I*a^4)/(4*d*(a - I*a*Tan[c + d*x])^2*(a + I*a*Tan[c + d*x])^(11/2)) - (15*I*a^3)/(16*d*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(11/2)) + (65*I*a)/(192*d*(a + I*a*Tan[c + d*x])^(9/2)) + (195*I)/(896*d*(a + I*a*Tan[c + d*x])^(7/2)) + (39*I)/(256*a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (65*I)/(512*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (195*I)/(1024*a^3*d*Sqrt[a + I*a*Tan[c + d*x]])} - -{Sec[c + d*x]^13/(a + I*a*Tan[c + d*x])^(7/2), x, 3, (64*I*a^3*Sec[c + d*x]^13)/(3315*d*(a + I*a*Tan[c + d*x])^(13/2)) + (16*I*a^2*Sec[c + d*x]^13)/(255*d*(a + I*a*Tan[c + d*x])^(11/2)) + (2*I*a*Sec[c + d*x]^13)/(17*d*(a + I*a*Tan[c + d*x])^(9/2))} -{Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^(7/2), x, 2, (8*I*a^2*Sec[c + d*x]^11)/(143*d*(a + I*a*Tan[c + d*x])^(11/2)) + (2*I*a*Sec[c + d*x]^11)/(13*d*(a + I*a*Tan[c + d*x])^(9/2))} -{Sec[c + d*x]^9/(a + I*a*Tan[c + d*x])^(7/2), x, 1, (2*I*a*Sec[c + d*x]^9)/(9*d*(a + I*a*Tan[c + d*x])^(9/2))} -{Sec[c + d*x]^7/(a + I*a*Tan[c + d*x])^(7/2), x, 5, (8*I*Sqrt[2]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(a^(7/2)*d) - (2*I*Sec[c + d*x]^5)/(5*a*d*(a + I*a*Tan[c + d*x])^(5/2)) - (4*I*Sec[c + d*x]^3)/(3*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) - (8*I*Sec[c + d*x])/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^(7/2), x, 5, -((3*I*Sqrt[2]*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(a^(7/2)*d)) - (2*I*Sec[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (6*I*Sec[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2), x, 5, -((I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(8*Sqrt[2]*a^(7/2)*d)) + (I*Sec[c + d*x])/(2*a*d*(a + I*a*Tan[c + d*x])^(5/2)) - (I*Sec[c + d*x])/(8*a^2*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + I*a*Tan[c + d*x])^(7/2), x, 5, (5*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(64*Sqrt[2]*a^(7/2)*d) + (I*Sec[c + d*x])/(6*d*(a + I*a*Tan[c + d*x])^(7/2)) + (5*I*Sec[c + d*x])/(48*a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (5*I*Sec[c + d*x])/(64*a^2*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + I*a*Tan[c + d*x])^(7/2), x, 7, (315*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(2048*Sqrt[2]*a^(7/2)*d) + (I*Cos[c + d*x])/(8*d*(a + I*a*Tan[c + d*x])^(7/2)) + (3*I*Cos[c + d*x])/(32*a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (21*I*Cos[c + d*x])/(256*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (105*I*Cos[c + d*x])/(1024*a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (315*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(2048*a^4*d)} -{Cos[c + d*x]^3/(a + I*a*Tan[c + d*x])^(7/2), x, 9, (3003*I*ArcTanh[(Sqrt[a]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(16384*Sqrt[2]*a^(7/2)*d) + (I*Cos[c + d*x]^3)/(10*d*(a + I*a*Tan[c + d*x])^(7/2)) + (13*I*Cos[c + d*x]^3)/(160*a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (143*I*Cos[c + d*x]^3)/(1920*a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (1001*I*Cos[c + d*x])/(8192*a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) + (429*I*Cos[c + d*x]^3)/(5120*a^3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (3003*I*Cos[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(16384*a^4*d) - (1001*I*Cos[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(10240*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/2) (a+a I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 12, (I*a*(e*Sec[c + d*x])^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]], x, 10, (I*Sqrt[2]*Sqrt[a]*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/d - (I*Sqrt[2]*Sqrt[a]*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/d - (I*Sqrt[a]*Sqrt[e]*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d) + (I*Sqrt[a]*Sqrt[e]*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d)} -{Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[e*Sec[c + d*x]], x, 1, ((-2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]])} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(3/2), x, 2, (((4*I)/3)*a*Sqrt[e*Sec[c + d*x]])/(d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/3)*Sqrt[a + I*a*Tan[c + d*x]])/(d*(e*Sec[c + d*x])^(3/2))} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(5/2), x, 3, (8*I*a)/(15*d*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*(e*Sec[c + d*x])^(5/2)) - (16*I*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*e^2*Sqrt[e*Sec[c + d*x]])} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Sec[c + d*x])^(7/2), x, 4, (12*I*a)/(35*d*e^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (32*I*a*Sqrt[e*Sec[c + d*x]])/(35*d*e^4*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*(e*Sec[c + d*x])^(7/2)) - (16*I*Sqrt[a + I*a*Tan[c + d*x]])/(35*d*e^2*(e*Sec[c + d*x])^(3/2))} - - -{(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 13, (7*I*a^(3/2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(8*Sqrt[2]*d) - (7*I*a^(3/2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(8*Sqrt[2]*d) - (7*I*a^(3/2)*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(16*Sqrt[2]*d) + (7*I*a^(3/2)*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(16*Sqrt[2]*d) + (7*I*a^2*(e*Sec[c + d*x])^(5/2))/(12*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*I*a*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) + (I*a*(e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 13, (5*I*a^2*(e*Sec[c + d*x])^(3/2))/(4*d*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(5/2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(4*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a^(5/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(4*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a^(5/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(8*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(5/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(8*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2), x, 11, (3*I*a^(3/2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*d) - (3*I*a^(3/2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*d) - (3*I*a^(3/2)*Sqrt[e]*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*d) + (3*I*a^(3/2)*Sqrt[e]*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*d) + (I*a*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[e*Sec[c + d*x]], x, 12, (I*Sqrt[2]*a^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*Sqrt[2]*a^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (4*I*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(3/2), x, 1, (((-2*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(5/2), x, 2, (((-4*I)/5)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*e^2*Sqrt[e*Sec[c + d*x]]) - (((2*I)/5)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(7/2), x, 3, (((16*I)/21)*a^2*Sqrt[e*Sec[c + d*x]])/(d*e^4*Sqrt[a + I*a*Tan[c + d*x]]) - (((8*I)/21)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*e^2*(e*Sec[c + d*x])^(3/2)) - (((2*I)/7)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(e*Sec[c + d*x])^(7/2))} -{(a + I*a*Tan[c + d*x])^(3/2)/(e*Sec[c + d*x])^(9/2), x, 4, (16*I*a^2)/(45*d*e^4*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (4*I*a*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*e^2*(e*Sec[c + d*x])^(5/2)) - (32*I*a*Sqrt[a + I*a*Tan[c + d*x]])/(45*d*e^4*Sqrt[e*Sec[c + d*x]]) - (2*I*(a + I*a*Tan[c + d*x])^(3/2))/(9*d*(e*Sec[c + d*x])^(9/2))} - - -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 14, (15*I*a^3*(e*Sec[c + d*x])^(3/2))/(8*d*Sqrt[a + I*a*Tan[c + d*x]]) - (15*I*a^(7/2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(8*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (15*I*a^(7/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(8*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (15*I*a^(7/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(16*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (15*I*a^(7/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(16*Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*a^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (I*a*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2), x, 12, (21*I*a^(5/2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(4*Sqrt[2]*d) - (21*I*a^(5/2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(4*Sqrt[2]*d) - (21*I*a^(5/2)*Sqrt[e]*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(8*Sqrt[2]*d) + (21*I*a^(5/2)*Sqrt[e]*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(8*Sqrt[2]*d) + (7*I*a^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (I*a*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2))/(2*d)} -{(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[e*Sec[c + d*x]], x, 13, (5*I*a^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (10*I*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[e*Sec[c + d*x]]) + (I*a*(a + I*a*Tan[c + d*x])^(3/2))/(d*Sqrt[e*Sec[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(3/2), x, 11, -((I*Sqrt[2]*a^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(d*e^(3/2))) + (I*Sqrt[2]*a^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(d*e^(3/2)) + (I*a^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*e^(3/2)) - (I*a^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*e^(3/2)) - (4*I*a*(a + I*a*Tan[c + d*x])^(3/2))/(3*d*(e*Sec[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(5/2), x, 1, (((-2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(7/2), x, 2, (((-4*I)/21)*a*(a + I*a*Tan[c + d*x])^(3/2))/(d*e^2*(e*Sec[c + d*x])^(3/2)) - (((2*I)/7)*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(7/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(9/2), x, 3, (((-16*I)/45)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*e^4*Sqrt[e*Sec[c + d*x]]) - (((8*I)/45)*a*(a + I*a*Tan[c + d*x])^(3/2))/(d*e^2*(e*Sec[c + d*x])^(5/2)) - (((2*I)/9)*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(9/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/(e*Sec[c + d*x])^(11/2), x, 4, (((32*I)/77)*a^3*Sqrt[e*Sec[c + d*x]])/(d*e^6*Sqrt[a + I*a*Tan[c + d*x]]) - (((16*I)/77)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*e^4*(e*Sec[c + d*x])^(3/2)) - (((12*I)/77)*a*(a + I*a*Tan[c + d*x])^(3/2))/(d*e^2*(e*Sec[c + d*x])^(7/2)) - (((2*I)/11)*(a + I*a*Tan[c + d*x])^(5/2))/(d*(e*Sec[c + d*x])^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sec[c + d*x])^(5/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 11, (I*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]*d) - (I*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]*d) - (I*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*Sqrt[a]*d) + (I*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*Sqrt[a]*d) - (I*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{(e*Sec[c + d*x])^(3/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 11, -((I*Sqrt[2]*Sqrt[a]*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])) + (I*Sqrt[2]*Sqrt[a]*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*Sqrt[a]*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*Sqrt[a]*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{Sqrt[e*Sec[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]], x, 1, ((2*I)*Sqrt[e*Sec[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), x, 2, (2*I)/(3*d*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (4*I*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d*Sqrt[e*Sec[c + d*x]])} -{1/((e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 3, (2*I)/(5*d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (16*I*Sqrt[e*Sec[c + d*x]])/(15*d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (8*I*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d*(e*Sec[c + d*x])^(3/2))} -{1/((e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, (2*I)/(7*d*(e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (16*I)/(35*d*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (12*I*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d*(e*Sec[c + d*x])^(5/2)) - (32*I*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d*e^2*Sqrt[e*Sec[c + d*x]])} -{1/((e*Sec[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 5, (2*I)/(9*d*(e*Sec[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (32*I)/(105*d*e^2*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (256*I*Sqrt[e*Sec[c + d*x]])/(315*d*e^4*Sqrt[a + I*a*Tan[c + d*x]]) - (16*I*Sqrt[a + I*a*Tan[c + d*x]])/(63*a*d*(e*Sec[c + d*x])^(7/2)) - (128*I*Sqrt[a + I*a*Tan[c + d*x]])/(315*a*d*e^2*(e*Sec[c + d*x])^(3/2))} - - -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 13, -((I*e^2*(e*Sec[c + d*x])^(3/2))/(a*d*Sqrt[a + I*a*Tan[c + d*x]])) - (3*I*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*e^(7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*Sqrt[a]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (3*I*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*Sqrt[a]*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 11, -((I*Sqrt[2]*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(a^(3/2)*d)) + (I*Sqrt[2]*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(a^(3/2)*d) + (I*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*a^(3/2)*d) - (I*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*a^(3/2)*d) + (4*I*e^2*Sqrt[e*Sec[c + d*x]])/(a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 1, (((2*I)/3)*(e*Sec[c + d*x])^(3/2))/(d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2), x, 2, (2*I*Sqrt[e*Sec[c + d*x]])/(5*d*(a + I*a*Tan[c + d*x])^(3/2)) + (4*I*Sqrt[e*Sec[c + d*x]])/(5*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)), x, 3, (2*I)/(7*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (8*I)/(21*a*d*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (16*I*Sqrt[a + I*a*Tan[c + d*x]])/(21*a^2*d*Sqrt[e*Sec[c + d*x]])} -{1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 4, (2*I)/(9*d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (4*I)/(15*a*d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (32*I*Sqrt[e*Sec[c + d*x]])/(45*a*d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (16*I*Sqrt[a + I*a*Tan[c + d*x]])/(45*a^2*d*(e*Sec[c + d*x])^(3/2))} -{1/((e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 5, (2*I)/(11*d*(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (16*I)/(77*a*d*(e*Sec[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (128*I)/(385*a*d*e^2*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (96*I*Sqrt[a + I*a*Tan[c + d*x]])/(385*a^2*d*(e*Sec[c + d*x])^(5/2)) - (256*I*Sqrt[a + I*a*Tan[c + d*x]])/(385*a^2*d*e^2*Sqrt[e*Sec[c + d*x]])} - - -{(e*Sec[c + d*x])^(9/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 12, -((5*I*e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*a^(5/2)*d)) + (5*I*e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + (5*I*e^(9/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*a^(5/2)*d) - (5*I*e^(9/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*a^(5/2)*d) + (4*I*e^2*(e*Sec[c + d*x])^(5/2))/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*I*e^4*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d)} -{(e*Sec[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 12, (4*I*e^2*(e*Sec[c + d*x])^(3/2))/(3*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I*Sqrt[2]*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(a^(3/2)*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*Sqrt[2]*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(a^(3/2)*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*e^(7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*a^(3/2)*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*a^(3/2)*d*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 1, (((2*I)/5)*(e*Sec[c + d*x])^(5/2))/(d*(a + I*a*Tan[c + d*x])^(5/2))} -{(e*Sec[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 2, (2*I*(e*Sec[c + d*x])^(3/2))/(7*d*(a + I*a*Tan[c + d*x])^(5/2)) + (4*I*(e*Sec[c + d*x])^(3/2))/(21*a*d*(a + I*a*Tan[c + d*x])^(3/2))} -{Sqrt[e*Sec[c + d*x]]/(a + I*a*Tan[c + d*x])^(5/2), x, 3, (2*I*Sqrt[e*Sec[c + d*x]])/(9*d*(a + I*a*Tan[c + d*x])^(5/2)) + (8*I*Sqrt[e*Sec[c + d*x]])/(45*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (16*I*Sqrt[e*Sec[c + d*x]])/(45*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)), x, 4, (2*I)/(11*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + (12*I)/(77*a*d*Sqrt[e*Sec[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (16*I)/(77*a^2*d*Sqrt[e*Sec[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (32*I*Sqrt[a + I*a*Tan[c + d*x]])/(77*a^3*d*Sqrt[e*Sec[c + d*x]])} -{1/((e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 5, (2*I)/(13*d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + (16*I)/(117*a*d*(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (32*I)/(195*a^2*d*(e*Sec[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (256*I*Sqrt[e*Sec[c + d*x]])/(585*a^2*d*e^2*Sqrt[a + I*a*Tan[c + d*x]]) - (128*I*Sqrt[a + I*a*Tan[c + d*x]])/(585*a^3*d*(e*Sec[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/3) (a+a I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sec[c + d*x])^(7/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*I*2^(2/3)*a*Hypergeometric2F1[1/3, 7/6, 13/6, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(7/3)*(1 + I*Tan[c + d*x])^(1/3))/(7*d*(a + I*a*Tan[c + d*x])^(3/2))} -{(e*Sec[c + d*x])^(5/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*I*2^(1/3)*a*Hypergeometric2F1[2/3, 5/6, 11/6, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(5/3)*(1 + I*Tan[c + d*x])^(2/3))/(5*d*(a + I*a*Tan[c + d*x])^(3/2))} -{(e*Sec[c + d*x])^(2/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*I*Hypergeometric2F1[1/3, 7/6, 4/3, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(2/3)*(1 + I*Tan[c + d*x])^(1/6))/(2*2^(1/6)*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^(1/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, ((3*I)*Hypergeometric2F1[1/6, 4/3, 7/6, (1 - I*Tan[c + d*x])/2]*(e*Sec[c + d*x])^(1/3)*(1 + I*Tan[c + d*x])^(1/3))/(2^(1/3)*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/((e*Sec[c + d*x])^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, ((-3*I)*Hypergeometric2F1[-1/6, 5/3, 5/6, (1 - I*Tan[c + d*x])/2]*(1 + I*Tan[c + d*x])^(2/3))/(2^(2/3)*d*(e*Sec[c + d*x])^(1/3)*Sqrt[a + I*a*Tan[c + d*x]])} -{1/((e*Sec[c + d*x])^(4/3)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, -((3*I*Hypergeometric2F1[-(2/3), 13/6, 1/3, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/6)*Sqrt[a + I*a*Tan[c + d*x]])/(8*2^(1/6)*a*d*(e*Sec[c + d*x])^(4/3)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/3) (a+a I Tan[e+f x])^(n/3)*) - - -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 + 2), x, 9, (I*(d*Sec[e + f*x])^(2/3))/(4*f*(a + I*a*Tan[e + f*x])^(7/3)) - (5*x*(d*Sec[e + f*x])^(2/3))/(72*2^(2/3)*a^(5/3)*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) + (5*I*ArcTan[(a^(1/3) + 2^(2/3)*(a - I*a*Tan[e + f*x])^(1/3))/(Sqrt[3]*a^(1/3))]*(d*Sec[e + f*x])^(2/3))/(12*2^(2/3)*Sqrt[3]*a^(5/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (5*I*Log[Cos[e + f*x]]*(d*Sec[e + f*x])^(2/3))/(72*2^(2/3)*a^(5/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (5*I*Log[2^(1/3)*a^(1/3) - (a - I*a*Tan[e + f*x])^(1/3)]*(d*Sec[e + f*x])^(2/3))/(24*2^(2/3)*a^(5/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) + (5*I*(d*Sec[e + f*x])^(2/3))/(24*f*(a + I*a*Tan[e + f*x])^(1/3)*(a^2 + I*a^2*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 + 1), x, 8, (I*(d*Sec[e + f*x])^(2/3))/(2*f*(a + I*a*Tan[e + f*x])^(4/3)) - (x*(d*Sec[e + f*x])^(2/3))/(6*2^(2/3)*a^(2/3)*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) + (I*ArcTan[(a^(1/3) + 2^(2/3)*(a - I*a*Tan[e + f*x])^(1/3))/(Sqrt[3]*a^(1/3))]*(d*Sec[e + f*x])^(2/3))/(2^(2/3)*Sqrt[3]*a^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (I*Log[Cos[e + f*x]]*(d*Sec[e + f*x])^(2/3))/(6*2^(2/3)*a^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (I*Log[2^(1/3)*a^(1/3) - (a - I*a*Tan[e + f*x])^(1/3)]*(d*Sec[e + f*x])^(2/3))/(2*2^(2/3)*a^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3))} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 + 0), x, 6, -((a^(1/3)*x*(d*Sec[e + f*x])^(2/3))/(2*2^(2/3)*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3))) + (I*Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a - I*a*Tan[e + f*x])^(1/3))/(Sqrt[3]*a^(1/3))]*(d*Sec[e + f*x])^(2/3))/(2^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (I*a^(1/3)*Log[Cos[e + f*x]]*(d*Sec[e + f*x])^(2/3))/(2*2^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3)) - (3*I*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a - I*a*Tan[e + f*x])^(1/3)]*(d*Sec[e + f*x])^(2/3))/(2*2^(2/3)*f*(a - I*a*Tan[e + f*x])^(1/3)*(a + I*a*Tan[e + f*x])^(1/3))} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 - 1), x, 1, (3*I*a*(d*Sec[e + f*x])^(2/3))/(f*(a + I*a*Tan[e + f*x])^(1/3))} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 - 2), x, 2, (9*I*a^2*(d*Sec[e + f*x])^(2/3))/(2*f*(a + I*a*Tan[e + f*x])^(1/3)) + (3*I*a*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3))/(4*f)} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 - 3), x, 3, (54*I*a^3*(d*Sec[e + f*x])^(2/3))/(7*f*(a + I*a*Tan[e + f*x])^(1/3)) + (9*I*a^2*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3))/(7*f) + (3*I*a*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(5/3))/(7*f)} -{(d*Sec[e + f*x])^(2/3)/(a + a*I*Tan[e + f*x])^(1/3 - 4), x, 4, (486*I*a^4*(d*Sec[e + f*x])^(2/3))/(35*f*(a + I*a*Tan[e + f*x])^(1/3)) + (81*I*a^3*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(2/3))/(35*f) + (27*I*a^2*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(5/3))/(35*f) + (3*I*a*(d*Sec[e + f*x])^(2/3)*(a + I*a*Tan[e + f*x])^(8/3))/(10*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^n with m symbolic*) - - -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^5, x, 4, (1/(d*m))*((I*2^(5 + m/2)*a^5*Hypergeometric2F1[-4 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/(1 + I*Tan[c + d*x])^(m/2))} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^3, x, 4, (I*2^(3 + m/2)*a^3*Hypergeometric2F1[-2 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/((1 + I*Tan[c + d*x])^(m/2)*(d*m))} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^2, x, 4, (I*2^(2 + m/2)*a^2*Hypergeometric2F1[-1 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/((1 + I*Tan[c + d*x])^(m/2)*(d*m))} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^1, x, 4, (I*2^(1 + m/2)*a*Hypergeometric2F1[-(m/2), m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/((1 + I*Tan[c + d*x])^(m/2)*(d*m))} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^1, x, 4, (1/(a*d*m))*((I*2^(-1 + m/2)*Hypergeometric2F1[2 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/(1 + I*Tan[c + d*x])^(m/2))} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^2, x, 4, (1/(a^2*d*m))*((I*2^(-2 + m/2)*Hypergeometric2F1[3 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/(1 + I*Tan[c + d*x])^(m/2))} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^3, x, 4, (1/(a^3*d*m))*((I*2^(-3 + m/2)*Hypergeometric2F1[4 - m/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m)/(1 + I*Tan[c + d*x])^(m/2))} - - -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2), x, 4, (1/(d*m))*(I*2^((7 + m)/2)*a^3*Hypergeometric2F1[(1/2)*(-5 - m), m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1/2)*(-1 - m))*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(5/2), x, 4, (1/(d*m))*(I*2^((5 + m)/2)*a^2*Hypergeometric2F1[(1/2)*(-3 - m), m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1/2)*(-1 - m))*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(3/2), x, 4, (1/(d*m))*(I*2^((3 + m)/2)*a*Hypergeometric2F1[(1/2)*(-1 - m), m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1/2)*(-1 - m))*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(1/2), x, 4, (I*2^((1 + m)/2)*a*Hypergeometric2F1[(1 - m)/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1 - m)/2))/(d*m*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^(1/2), x, 4, (I*2^((1/2)*(-1 + m))*Hypergeometric2F1[(3 - m)/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1 - m)/2))/(d*m*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (I*2^((1/2)*(-3 + m))*Hypergeometric2F1[(5 - m)/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1 - m)/2))/(a*d*m*Sqrt[a + I*a*Tan[c + d*x]])} -{(e*Sec[c + d*x])^m/(a + I*a*Tan[c + d*x])^(5/2), x, 4, (I*2^((1/2)*(-5 + m))*Hypergeometric2F1[(7 - m)/2, m/2, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((1 - m)/2))/(a^2*d*m*Sqrt[a + I*a*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+a I Tan[e+f x])^n with n symbolic*) - - -{(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(d*m))*(I*2^(m/2 + n)*Hypergeometric2F1[m/2, 1 - m/2 - n, (2 + m)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^(-(m/2) - n)*(a + I*a*Tan[c + d*x])^n)} - - -{Sec[c + d*x]^6*(a + I*a*Tan[c + d*x])^n, x, 3, -((4*I*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(3 + n))) + (4*I*(a + I*a*Tan[c + d*x])^(4 + n))/(a^4*d*(4 + n)) - (I*(a + I*a*Tan[c + d*x])^(5 + n))/(a^5*d*(5 + n))} -{Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^n, x, 3, ((-2*I)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^2*d*(2 + n)) + (I*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(3 + n))} -{Sec[c + d*x]^2*(a + I*a*Tan[c + d*x])^n, x, 2, ((-I)*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Cos[c + d*x]^2*(a + I*a*Tan[c + d*x])^n, x, 2, (I*a*Hypergeometric2F1[2, -1 + n, n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^(-1 + n))/(4*d*(1 - n))} -{Cos[c + d*x]^4*(a + I*a*Tan[c + d*x])^n, x, 2, (I*a^2*Hypergeometric2F1[3, -2 + n, -1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^(-2 + n))/(8*d*(2 - n))} -{Cos[c + d*x]^6*(a + I*a*Tan[c + d*x])^n, x, 2, (I*a^3*Hypergeometric2F1[4, -3 + n, -2 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^(-3 + n))/(16*d*(3 - n))} - -{Sec[c + d*x]^5*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(5*d))*(I*2^(5/2 + n)*a^2*Hypergeometric2F1[5/2, -(3/2) - n, 7/2, (1/2)*(1 - I*Tan[c + d*x])]*Sec[c + d*x]^5*(1 + I*Tan[c + d*x])^(-(1/2) - n)*(a + I*a*Tan[c + d*x])^(-2 + n))} -{Sec[c + d*x]^3*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(3*d))*(I*2^(3/2 + n)*a*Hypergeometric2F1[3/2, -(1/2) - n, 5/2, (1/2)*(1 - I*Tan[c + d*x])]*Sec[c + d*x]^3*(1 + I*Tan[c + d*x])^(-(1/2) - n)*(a + I*a*Tan[c + d*x])^(-1 + n))} -{Sec[c + d*x]^1*(a + I*a*Tan[c + d*x])^n, x, 4, (1/d)*(I*2^(1/2 + n)*a*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - I*Tan[c + d*x])]*Sec[c + d*x]*(1 + I*Tan[c + d*x])^(1/2 - n)*(a + I*a*Tan[c + d*x])^(-1 + n))} -{Cos[c + d*x]^1*(a + I*a*Tan[c + d*x])^n, x, 4, -((1/d)*(I*2^(-(1/2) + n)*Cos[c + d*x]*Hypergeometric2F1[-(1/2), 3/2 - n, 1/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/2 - n)*(a + I*a*Tan[c + d*x])^n))} -{Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^n, x, 4, -((1/(3*a*d))*(I*2^(-(3/2) + n)*Cos[c + d*x]^3*Hypergeometric2F1[-(3/2), 5/2 - n, -(1/2), (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/2 - n)*(a + I*a*Tan[c + d*x])^(1 + n)))} -{Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^n, x, 4, -((1/(5*a^2*d))*(I*2^(-(5/2) + n)*Cos[c + d*x]^5*Hypergeometric2F1[-(5/2), 7/2 - n, -(3/2), (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/2 - n)*(a + I*a*Tan[c + d*x])^(2 + n)))} - - -{(e*Sec[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(5*d))*(I*2^(9/4 + n)*a*Hypergeometric2F1[5/4, -(1/4) - n, 9/4, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(5/2)*(1 + I*Tan[c + d*x])^(-(1/4) - n)*(a + I*a*Tan[c + d*x])^(-1 + n))} -{(e*Sec[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(3*d))*(I*2^(7/4 + n)*a*Hypergeometric2F1[3/4, 1/4 - n, 7/4, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(3/2)*(1 + I*Tan[c + d*x])^(1/4 - n)*(a + I*a*Tan[c + d*x])^(-1 + n))} -{(e*Sec[c + d*x])^(1/2)*(a + I*a*Tan[c + d*x])^n, x, 4, (1/d)*(I*2^(5/4 + n)*a*Hypergeometric2F1[1/4, 3/4 - n, 5/4, (1/2)*(1 - I*Tan[c + d*x])]*Sqrt[e*Sec[c + d*x]]*(1 + I*Tan[c + d*x])^(3/4 - n)*(a + I*a*Tan[c + d*x])^(-1 + n))} -{(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(1/2), x, 4, -((1/(d*Sqrt[e*Sec[c + d*x]]))*(I*2^(3/4 + n)*Hypergeometric2F1[-(1/4), 5/4 - n, 3/4, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/4 - n)*(a + I*a*Tan[c + d*x])^n))} -{(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(3/2), x, 4, -((1/(3*d*(e*Sec[c + d*x])^(3/2)))*(I*2^(1/4 + n)*Hypergeometric2F1[-(3/4), 7/4 - n, 1/4, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(3/4 - n)*(a + I*a*Tan[c + d*x])^n))} -{(a + I*a*Tan[c + d*x])^n/(e*Sec[c + d*x])^(5/2), x, 4, -((1/(5*a*d*(e*Sec[c + d*x])^(5/2)))*(I*2^(-(1/4) + n)*Hypergeometric2F1[-(5/4), 9/4 - n, -(1/4), (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(1/4 - n)*(a + I*a*Tan[c + d*x])^(1 + n)))} - - -{(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^n, x, 5, (I*(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(4 - n)) + (4*I*(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(8 - 6*n + n^2)) - (12*I*(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^2*d*(2 - n)*(4 - n)*n) + (24*I*(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(4 - n)*n*(4 - n^2)) - (24*I*(e*Sec[c + d*x])^(-4 - n)*(a + I*a*Tan[c + d*x])^(4 + n))/(a^4*d*n*(64 - 20*n^2 + n^4))} -{(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n, x, 4, (I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(3 - n)) + (3*I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(3 - 4*n + n^2)) - (6*I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^2*d*(3 - n)*(1 - n^2)) + (6*I*(e*Sec[c + d*x])^(-3 - n)*(a + I*a*Tan[c + d*x])^(3 + n))/(a^3*d*(9 - 10*n^2 + n^4))} -{(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n, x, 3, (I*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(2 - n)) - (2*I*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(2 - n)*n) + (2*I*(e*Sec[c + d*x])^(-2 - n)*(a + I*a*Tan[c + d*x])^(2 + n))/(a^2*d*n*(4 - n^2))} -{(e*Sec[c + d*x])^(-1 - n)*(a + I*a*Tan[c + d*x])^n, x, 2, (I*(e*Sec[c + d*x])^(-1 - n)*(a + I*a*Tan[c + d*x])^n)/(d*(1 - n)) - (I*(e*Sec[c + d*x])^(-1 - n)*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 - n^2))} -{(e*Sec[c + d*x])^(0 - n)*(a + I*a*Tan[c + d*x])^n, x, 1, ((-I)*(a + I*a*Tan[c + d*x])^n)/(d*n*(e*Sec[c + d*x])^n)} -{(e*Sec[c + d*x])^(1 - n)*(a + I*a*Tan[c + d*x])^n, x, 4, (1/(d*(1 - n)))*(I*2^((1 + n)/2)*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(1 - n)*(1 + I*Tan[c + d*x])^((1/2)*(-1 - n))*(a + I*a*Tan[c + d*x])^n)} -{(e*Sec[c + d*x])^(2 - n)*(a + I*a*Tan[c + d*x])^n, x, 4, (I*2^(1 + n/2)*a*Hypergeometric2F1[(2 - n)/2, -(n/2), (4 - n)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(2 - n)*(a + I*a*Tan[c + d*x])^(-1 + n))/((1 + I*Tan[c + d*x])^(n/2)*(d*(2 - n)))} -{(e*Sec[c + d*x])^(3 - n)*(a + I*a*Tan[c + d*x])^n, x, 4, (I*2^((3 + n)/2)*a*Hypergeometric2F1[(1/2)*(-1 - n), (3 - n)/2, (5 - n)/2, (1/2)*(1 - I*Tan[c + d*x])]*(e*Sec[c + d*x])^(3 - n)*(1 + I*Tan[c + d*x])^((1/2)*(-1 - n))*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(3 - n))} - - -{(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 3, If[$VersionNumber>=8, (8*I*a^3*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-3 + n))/(d*(5 - n)*(12 - 7*n + n^2)) + (4*I*a^2*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(20 - 9*n + n^2)) + (I*a*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(5 - n)), (8*I*a^3*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-3 + n))/(d*(60 - 47*n + 12*n^2 - n^3)) + (4*I*a^2*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(20 - 9*n + n^2)) + (I*a*(e*Sec[c + d*x])^(6 - 2*n)*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(5 - n))]} -{(e*Sec[c + d*x])^(5 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 5, -((I*2^(5/2 - n)*Hypergeometric2F1[5/2, (1/2)*(-3 + 2*n), 7/2, (1/2)*(1 + I*Tan[c + d*x])]*(e*Sec[c + d*x])^(5 - 2*n)*(1 - I*Tan[c + d*x])^(-(5/2) + n)*(a + I*a*Tan[c + d*x])^n)/(5*d))} -{(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 2, If[$VersionNumber>=8, (2*I*a^2*(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(6 - 5*n + n^2)) + (I*a*(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(3 - n)), (2*I*a^2*(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^(-2 + n))/(d*(6 - 5*n + n^2)) + (I*a*(e*Sec[c + d*x])^(4 - 2*n)*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(3 - n))]} -{(e*Sec[c + d*x])^(3 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 5, -((I*2^(3/2 - n)*Hypergeometric2F1[3/2, (1/2)*(-1 + 2*n), 5/2, (1/2)*(1 + I*Tan[c + d*x])]*(e*Sec[c + d*x])^(3 - 2*n)*(1 - I*Tan[c + d*x])^(-(3/2) + n)*(a + I*a*Tan[c + d*x])^n)/(3*d))} -{(e*Sec[c + d*x])^(2 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 1, (I*a*(e*Sec[c + d*x])^(2 - 2*n)*(a + I*a*Tan[c + d*x])^(-1 + n))/(d*(1 - n))} -{(e*Sec[c + d*x])^(1 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 5, -((I*2^(1/2 - n)*Hypergeometric2F1[1/2, (1/2)*(1 + 2*n), 3/2, (1/2)*(1 + I*Tan[c + d*x])]*(e*Sec[c + d*x])^(1 - 2*n)*(1 - I*Tan[c + d*x])^(-(1/2) + n)*(a + I*a*Tan[c + d*x])^n)/d)} -{(e*Sec[c + d*x])^(0 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 3, ((-I/2)*Hypergeometric2F1[1, -n, 1 - n, (1 - I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^n)/(d*n*(e*Sec[c + d*x])^(2*n))} -{(e*Sec[c + d*x])^(-1 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 5, (I*2^(-(1/2) - n)*Hypergeometric2F1[-(1/2), (1/2)*(3 + 2*n), 1/2, (1/2)*(1 + I*Tan[c + d*x])]*(e*Sec[c + d*x])^(-1 - 2*n)*(1 - I*Tan[c + d*x])^(1/2 + n)*(a + I*a*Tan[c + d*x])^n)/d} -{(e*Sec[c + d*x])^(-2 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 4, -((I*Hypergeometric2F1[2, -1 - n, -n, (1/2)*(1 - I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^(1 + n))/((e*Sec[c + d*x])^(2*(1 + n))*(4*a*d*(1 + n))))} -{(e*Sec[c + d*x])^(-3 - 2*n)*(a + I*a*Tan[c + d*x])^n, x, 5, (I*2^(-(3/2) - n)*Hypergeometric2F1[-(3/2), (1/2)*(5 + 2*n), -(1/2), (1/2)*(1 + I*Tan[c + d*x])]*(e*Sec[c + d*x])^(-3 - 2*n)*(1 - I*Tan[c + d*x])^(3/2 + n)*(a + I*a*Tan[c + d*x])^n)/(3*d)} - - -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n + 2), x, 4, (I*Hypergeometric2F1[3, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(8*a^2*f*n))} -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n + 1), x, 4, (I*Hypergeometric2F1[2, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(4*a*f*n))} -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n + 0), x, 3, (I*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(2*f*n))} -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n - 1), x, 1, (I*a*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(f*n))} -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n - 2), x, 2, (I*a*(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(1 - n))/(f*(1 + n)) + (2*I*a^2*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(f*n*(1 + n)))} -{(d*Sec[e + f*x])^(2*n)/(a + a*I*Tan[e + f*x])^(n - 3), x, 3, (4*I*a^2*(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(1 - n))/(f*(2 + 3*n + n^2)) + (I*a*(d*Sec[e + f*x])^(2*n)*(a + I*a*Tan[e + f*x])^(2 - n))/(f*(2 + n)) + (8*I*a^3*(d*Sec[e + f*x])^(2*n))/((a + I*a*Tan[e + f*x])^n*(f*n*(2 + 3*n + n^2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^6*(a + b*Tan[c + d*x]), x, 3, (b*Sec[c + d*x]^6)/(6*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^5*(a + b*Tan[c + d*x]), x, 4, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Sec[c + d*x]^5)/(5*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x]), x, 3, (b*Sec[c + d*x]^4)/(4*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^3*(a + b*Tan[c + d*x]), x, 3, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x]), x, 3, (b*Sec[c + d*x]^2)/(2*d) + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x]), x, 2, (a*ArcTanh[Sin[c + d*x]])/d + (b*Sec[c + d*x])/d} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x]), x, 2, -((b*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x]), x, 3, (a*x)/2 - (b*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x]), x, 3, -((b*Cos[c + d*x]^3)/(3*d)) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x]), x, 4, (3*a*x)/8 - (b*Cos[c + d*x]^4)/(4*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^8*(a + b*Tan[c + d*x])^2, x, 4, (a*b*Sec[c + d*x]^8)/(4*d) + (a^2*Tan[c + d*x])/d + ((3*a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + (3*(a^2 + b^2)*Tan[c + d*x]^5)/(5*d) + ((a^2 + 3*b^2)*Tan[c + d*x]^7)/(7*d) + (b^2*Tan[c + d*x]^9)/(9*d)} -{Sec[c + d*x]^6*(a + b*Tan[c + d*x])^2, x, 4, (a*b*Sec[c + d*x]^6)/(3*d) + (a^2*Tan[c + d*x])/d + ((2*a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + ((a^2 + 2*b^2)*Tan[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 3, ((a^2 + b^2)*(a + b*Tan[c + d*x])^3)/(3*b^3*d) - (a*(a + b*Tan[c + d*x])^4)/(2*b^3*d) + (a + b*Tan[c + d*x])^5/(5*b^3*d)} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 2, (a + b*Tan[c + d*x])^3/(3*b*d)} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 3, (1/2)*(a^2 + b^2)*x - (Cos[c + d*x]^2*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 4, (1/8)*(3*a^2 + b^2)*x - (Cos[c + d*x]^4*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x]))/(4*d) - (Cos[c + d*x]^2*(2*a*b - (3*a^2 + b^2)*Tan[c + d*x]))/(8*d)} - -{Sec[c + d*x]^7*(a + b*Tan[c + d*x])^2, x, 6, (5*(8*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(128*d) + (9*a*b*Sec[c + d*x]^7)/(56*d) + (5*(8*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*(8*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(192*d) + ((8*a^2 - b^2)*Sec[c + d*x]^5*Tan[c + d*x])/(48*d) + (b*Sec[c + d*x]^7*(a + b*Tan[c + d*x]))/(8*d)} -{Sec[c + d*x]^5*(a + b*Tan[c + d*x])^2, x, 5, ((6*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(16*d) + (7*a*b*Sec[c + d*x]^5)/(30*d) + ((6*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((6*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b*Sec[c + d*x]^5*(a + b*Tan[c + d*x]))/(6*d)} -{Sec[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 4, ((4*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a*b*Sec[c + d*x]^3)/(12*d) + ((4*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*(a + b*Tan[c + d*x]))/(4*d)} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 3, ((2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b*Sec[c + d*x])/(2*d) + (b*Sec[c + d*x]*(a + b*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 1, (b^2*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Cos[c + d*x])/d + ((a^2 - b^2)*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 4, -((a*b*Cos[c + d*x]^3)/(6*d)) + ((2*a^2 + b^2)*Sin[c + d*x])/(2*d) - ((2*a^2 + b^2)*Sin[c + d*x]^3)/(6*d) - (b*Cos[c + d*x]^3*(a + b*Tan[c + d*x]))/(2*d)} -{Cos[c + d*x]^5*(a + b*Tan[c + d*x])^2, x, 4, -((3*a*b*Cos[c + d*x]^5)/(20*d)) + ((4*a^2 + b^2)*Sin[c + d*x])/(4*d) - ((4*a^2 + b^2)*Sin[c + d*x]^3)/(6*d) + ((4*a^2 + b^2)*Sin[c + d*x]^5)/(20*d) - (b*Cos[c + d*x]^5*(a + b*Tan[c + d*x]))/(4*d)} -{Cos[c + d*x]^7*(a + b*Tan[c + d*x])^2, x, 4, -((5*a*b*Cos[c + d*x]^7)/(42*d)) + ((6*a^2 + b^2)*Sin[c + d*x])/(6*d) - ((6*a^2 + b^2)*Sin[c + d*x]^3)/(6*d) + ((6*a^2 + b^2)*Sin[c + d*x]^5)/(10*d) - ((6*a^2 + b^2)*Sin[c + d*x]^7)/(42*d) - (b*Cos[c + d*x]^7*(a + b*Tan[c + d*x]))/(6*d)} - - -{Sec[c + d*x]^8*(a + b*Tan[c + d*x])^3, x, 4, (3*a^2*b*Sec[c + d*x]^8)/(8*d) + (a^3*Tan[c + d*x])/d + (a*(a^2 + b^2)*Tan[c + d*x]^3)/d + (b^3*Tan[c + d*x]^4)/(4*d) + (3*a*(a^2 + 3*b^2)*Tan[c + d*x]^5)/(5*d) + (b^3*Tan[c + d*x]^6)/(2*d) + (a*(a^2 + 9*b^2)*Tan[c + d*x]^7)/(7*d) + (3*b^3*Tan[c + d*x]^8)/(8*d) + (a*b^2*Tan[c + d*x]^9)/(3*d) + (b^3*Tan[c + d*x]^10)/(10*d)} -{Sec[c + d*x]^6*(a + b*Tan[c + d*x])^3, x, 3, ((a^2 + b^2)^2*(a + b*Tan[c + d*x])^4)/(4*b^5*d) - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^5)/(5*b^5*d) + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/(3*b^5*d) - (4*a*(a + b*Tan[c + d*x])^7)/(7*b^5*d) + (a + b*Tan[c + d*x])^8/(8*b^5*d)} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x])^3, x, 3, ((a^2 + b^2)*(a + b*Tan[c + d*x])^4)/(4*b^3*d) - (2*a*(a + b*Tan[c + d*x])^5)/(5*b^3*d) + (a + b*Tan[c + d*x])^6/(6*b^3*d)} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 2, (a + b*Tan[c + d*x])^4/(4*b*d)} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 6, (1/2)*a*(a^2 + 3*b^2)*x - (b^3*Log[Cos[c + d*x]])/d - (a*b^2*Tan[c + d*x])/(2*d) - (Cos[c + d*x]^2*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x])^3, x, 4, (3/8)*a*(a^2 + b^2)*x - (3*a*Cos[c + d*x]^2*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*d)} - -{Sec[c + d*x]^5*(a + b*Tan[c + d*x])^3, x, 6, (3*a*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(16*d) + (3*a*(2*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a*(2*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(7*d) + (b*Sec[c + d*x]^5*(4*(8*a^2 - b^2) + 15*a*b*Tan[c + d*x]))/(70*d), (3*a*(2*a^2 - b^2)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(16*d*Sqrt[Sec[c + d*x]^2]) + (3*a*(2*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a*(2*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(7*d) + (b*Sec[c + d*x]^5*(4*(8*a^2 - b^2) + 15*a*b*Tan[c + d*x]))/(70*d)} -{Sec[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 5, (a*(4*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(4*a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(5*d) + (b*Sec[c + d*x]^3*(8*(6*a^2 - b^2) + 21*a*b*Tan[c + d*x]))/(60*d), (a*(4*a^2 - 3*b^2)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(8*d*Sqrt[Sec[c + d*x]^2]) + (a*(4*a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(5*d) + (b*Sec[c + d*x]^3*(8*(6*a^2 - b^2) + 21*a*b*Tan[c + d*x]))/(60*d)} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 4, (a*(2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]*(a + b*Tan[c + d*x])^2)/(3*d) + (b*Sec[c + d*x]*(4*(4*a^2 - b^2) + 5*a*b*Tan[c + d*x]))/(6*d), (a*(2*a^2 - 3*b^2)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(2*d*Sqrt[Sec[c + d*x]^2]) + (b*Sec[c + d*x]*(a + b*Tan[c + d*x])^2)/(3*d) + (b*Sec[c + d*x]*(4*(4*a^2 - b^2) + 5*a*b*Tan[c + d*x]))/(6*d)} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 4, (3*a*b^2*ArcTanh[Sin[c + d*x]])/d - (Cos[c + d*x]*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/d - (b*Sec[c + d*x]*(2*(a^2 - b^2) + a*b*Tan[c + d*x]))/d, (3*a*b^2*ArcSinh[Tan[c + d*x]]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/d - (Cos[c + d*x]*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/d - (b*Sec[c + d*x]*(2*(a^2 - b^2) + a*b*Tan[c + d*x]))/d} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 3, -((2*(a^2 + b^2)*Cos[c + d*x]*(b - a*Tan[c + d*x]))/(3*d)) - (Cos[c + d*x]^3*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/(3*d)} -{Cos[c + d*x]^5*(a + b*Tan[c + d*x])^3, x, 4, -((2*(4*a^2 + b^2)*Cos[c + d*x]*(b - a*Tan[c + d*x]))/(15*d)) - (Cos[c + d*x]^3*(b - 4*a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/(15*d) + (Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(5*d)} -{Cos[c + d*x]^7*(a + b*Tan[c + d*x])^3, x, 5, (8*a*(2*a^2 + b^2)*Sin[c + d*x])/(35*d) - (3*Cos[c + d*x]^5*(b - 2*a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/(35*d) + (Cos[c + d*x]^6*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(7*d) - (2*Cos[c + d*x]^3*(b*(6*a^2 + b^2) - a*(4*a^2 - b^2)*Tan[c + d*x]))/(35*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^6/(a + b*Tan[c + d*x]), x, 3, ((a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])/(b^5*d) - (a*(a^2 + 2*b^2)*Tan[c + d*x])/(b^4*d) + ((a^2 + 2*b^2)*Tan[c + d*x]^2)/(2*b^3*d) - (a*Tan[c + d*x]^3)/(3*b^2*d) + Tan[c + d*x]^4/(4*b*d)} -{Sec[c + d*x]^4/(a + b*Tan[c + d*x]), x, 3, ((a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(b^3*d) - (a*Tan[c + d*x])/(b^2*d) + Tan[c + d*x]^2/(2*b*d)} -{Sec[c + d*x]^2/(a + b*Tan[c + d*x]), x, 2, Log[a + b*Tan[c + d*x]]/(b*d)} -{Cos[c + d*x]^2/(a + b*Tan[c + d*x]), x, 7, (a*(a^2 + 3*b^2)*x)/(2*(a^2 + b^2)^2) + (b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(2*(a^2 + b^2)*d)} -{Cos[c + d*x]^4/(a + b*Tan[c + d*x]), x, 8, (a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/(8*(a^2 + b^2)^3) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d) + (Cos[c + d*x]^2*(4*b^3 + a*(3*a^2 + 7*b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)} - -{Sec[c + d*x]^5/(a + b*Tan[c + d*x]), x, 9, -((a*(2*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d)) - ((a^2 + b^2)^(3/2)*ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]])/(b^4*d) + ((a^2 + b^2)*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^3/(3*b*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d), -((a*ArcTanh[Sin[c + d*x]])/(2*b^2*d)) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((a^2 + b^2)^(3/2)*ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]])/(b^4*d) + ((a^2 + b^2)*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^3/(3*b*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d)} -{Sec[c + d*x]^3/(a + b*Tan[c + d*x]), x, 5, -((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d)} -{Sec[c + d*x]^1/(a + b*Tan[c + d*x]), x, 2, -(ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]]/(Sqrt[a^2 + b^2]*d))} -{Cos[c + d*x]^1/(a + b*Tan[c + d*x]), x, 5, -((b^2*ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) + (b*Cos[c + d*x])/((a^2 + b^2)*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d)} -{Cos[c + d*x]^3/(a + b*Tan[c + d*x]), x, 9, -((b^4*ArcTanh[(Cos[c + d*x]*(b - a*Tan[c + d*x]))/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) + (b^3*Cos[c + d*x])/((a^2 + b^2)^2*d) + (b*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a*b^2*Sin[c + d*x])/((a^2 + b^2)^2*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d) - (a*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d)} - - -{Sec[c + d*x]^8/(a + b*Tan[c + d*x])^2, x, 3, -((6*a*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])/(b^7*d)) + ((5*a^4 + 9*a^2*b^2 + 3*b^4)*Tan[c + d*x])/(b^6*d) - (a*(2*a^2 + 3*b^2)*Tan[c + d*x]^2)/(b^5*d) + ((a^2 + b^2)*Tan[c + d*x]^3)/(b^4*d) - (a*Tan[c + d*x]^4)/(2*b^3*d) + Tan[c + d*x]^5/(5*b^2*d) - (a^2 + b^2)^3/(b^7*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^6/(a + b*Tan[c + d*x])^2, x, 3, -((4*a*(a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(b^5*d)) + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(b^4*d) - (a*Tan[c + d*x]^2)/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d) - (a^2 + b^2)^2/(b^5*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Tan[c + d*x])^2, x, 3, -((2*a*Log[a + b*Tan[c + d*x]])/(b^3*d)) + Tan[c + d*x]/(b^2*d) - (a^2 + b^2)/(b^3*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 2, -(1/(b*d*(a + b*Tan[c + d*x])))} -{Cos[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 7, ((a^4 + 6*a^2*b^2 - 3*b^4)*x)/(2*(a^2 + b^2)^3) + (4*a*b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (b*(a^2 - 3*b^2))/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + b*Tan[c + d*x])^2, x, 8, (3*(a^6 + 5*a^4*b^2 + 15*a^2*b^4 - 5*b^6)*x)/(8*(a^2 + b^2)^4) + (6*a*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) + (3*b*(a^2 - b^2)*(a^2 + 5*b^2))/(8*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(a^2 - 5*b^2) - 3*a*(a^2 + 3*b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} - -{Sec[c + d*x]^7/(a + b*Tan[c + d*x])^2, x, 8, (5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(8*b^6*d*Sqrt[Sec[c + d*x]^2]) + (5*a*(a^2 + b^2)^(3/2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(b^6*d*Sqrt[Sec[c + d*x]^2]) - (5*Sec[c + d*x]^3*(4*a - 3*b*Tan[c + d*x]))/(12*b^3*d) - Sec[c + d*x]^5/(b*d*(a + b*Tan[c + d*x])) - (5*Sec[c + d*x]*(8*a*(a^2 + b^2) - b*(4*a^2 + 3*b^2)*Tan[c + d*x]))/(8*b^5*d)} -{Sec[c + d*x]^5/(a + b*Tan[c + d*x])^2, x, 7, (3*(2*a^2 + b^2)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(2*b^4*d*Sqrt[Sec[c + d*x]^2]) + (3*a*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(b^4*d*Sqrt[Sec[c + d*x]^2]) - (3*Sec[c + d*x]*(2*a - b*Tan[c + d*x]))/(2*b^3*d) - Sec[c + d*x]^3/(b*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 6, ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]*d) - Sec[c + d*x]/(b*d*(a + b*Tan[c + d*x])), (ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(b^2*d*Sqrt[Sec[c + d*x]^2]) + (a*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(b^2*Sqrt[a^2 + b^2]*d*Sqrt[Sec[c + d*x]^2]) - Sec[c + d*x]/(b*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 4, -((a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) - (b*Sec[c + d*x])/((a^2 + b^2)*d*(a + b*Tan[c + d*x])), -((a*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/((a^2 + b^2)^(3/2)*d*Sqrt[Sec[c + d*x]^2])) - (b*Sec[c + d*x])/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 5, -((3*a*b^2*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/((a^2 + b^2)^(5/2)*d)) + (b*(a^2 - 2*b^2)*Sec[c + d*x])/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]*(b + a*Tan[c + d*x]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 6, -((5*a*b^4*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/((a^2 + b^2)^(7/2)*d)) + (b*(2*a^4 + 9*a^2*b^2 - 8*b^4)*Sec[c + d*x])/(3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^3*(b + a*Tan[c + d*x]))/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]*(b*(a^2 - 4*b^2) - a*(2*a^2 + 7*b^2)*Tan[c + d*x]))/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} - - -{Sec[c + d*x]^8/(a + b*Tan[c + d*x])^3, x, 3, (3*(a^2 + b^2)*(5*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(b^7*d) - (a*(10*a^2 + 9*b^2)*Tan[c + d*x])/(b^6*d) + (3*(2*a^2 + b^2)*Tan[c + d*x]^2)/(2*b^5*d) - (a*Tan[c + d*x]^3)/(b^4*d) + Tan[c + d*x]^4/(4*b^3*d) - (a^2 + b^2)^3/(2*b^7*d*(a + b*Tan[c + d*x])^2) + (6*a*(a^2 + b^2)^2)/(b^7*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^6/(a + b*Tan[c + d*x])^3, x, 3, (2*(3*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(b^5*d) - (3*a*Tan[c + d*x])/(b^4*d) + Tan[c + d*x]^2/(2*b^3*d) - (a^2 + b^2)^2/(2*b^5*d*(a + b*Tan[c + d*x])^2) + (4*a*(a^2 + b^2))/(b^5*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Tan[c + d*x])^3, x, 3, Log[a + b*Tan[c + d*x]]/(b^3*d) - (a^2 + b^2)/(2*b^3*d*(a + b*Tan[c + d*x])^2) + (2*a)/(b^3*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 2, -(1/(2*b*d*(a + b*Tan[c + d*x])^2))} -{Cos[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 7, (a*(a^4 + 10*a^2*b^2 - 15*b^4)*x)/(2*(a^2 + b^2)^4) + (2*b^3*(5*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) + (b*(a^2 - 2*b^2))/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*b*(a^2 - 11*b^2))/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^4/(a + b*Tan[c + d*x])^3, x, 8, (3*a*(a^6 + 7*a^4*b^2 + 35*a^2*b^4 - 35*b^6)*x)/(8*(a^2 + b^2)^5) + (3*b^5*(7*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) + (3*b*(a^4 + 5*a^2*b^2 - 4*b^4))/(8*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (3*a*b*(a^4 + 6*a^2*b^2 - 27*b^4))/(8*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(2*b*(a^2 - 3*b^2) - a*(3*a^2 + 11*b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2)} - -{Sec[c + d*x]^7/(a + b*Tan[c + d*x])^3, x, 8, -((5*a*(4*a^2 + 3*b^2)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(2*b^6*d*Sqrt[Sec[c + d*x]^2])) - (5*Sqrt[a^2 + b^2]*(4*a^2 + b^2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(2*b^6*d*Sqrt[Sec[c + d*x]^2]) - Sec[c + d*x]^5/(2*b*d*(a + b*Tan[c + d*x])^2) + (5*Sec[c + d*x]^3*(4*a + b*Tan[c + d*x]))/(6*b^3*d*(a + b*Tan[c + d*x])) + (5*Sec[c + d*x]*(4*a^2 + b^2 - 2*a*b*Tan[c + d*x]))/(2*b^5*d)} -{Sec[c + d*x]^5/(a + b*Tan[c + d*x])^3, x, 7, -((3*a*ArcTanh[Sin[c + d*x]])/(b^4*d)) - (3*(2*a^2 + b^2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^4*Sqrt[a^2 + b^2]*d) - Sec[c + d*x]^3/(2*b*d*(a + b*Tan[c + d*x])^2) + (3*Sec[c + d*x]*(2*a + b*Tan[c + d*x]))/(2*b^3*d*(a + b*Tan[c + d*x])), -((3*a*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(b^4*d*Sqrt[Sec[c + d*x]^2])) - (3*(2*a^2 + b^2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(2*b^4*Sqrt[a^2 + b^2]*d*Sqrt[Sec[c + d*x]^2]) - Sec[c + d*x]^3/(2*b*d*(a + b*Tan[c + d*x])^2) + (3*Sec[c + d*x]*(2*a + b*Tan[c + d*x]))/(2*b^3*d*(a + b*Tan[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Tan[c + d*x])^3, x, 4, -(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2)*d)) - (Sec[c + d*x]*(b - a*Tan[c + d*x]))/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2), -((ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(2*(a^2 + b^2)^(3/2)*d*Sqrt[Sec[c + d*x]^2])) - (Sec[c + d*x]*(b - a*Tan[c + d*x]))/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)} -{Sec[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 5, -(((2*a^2 - b^2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(2*(a^2 + b^2)^(5/2)*d*Sqrt[Sec[c + d*x]^2])) - (b*Sec[c + d*x])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (3*a*b*Sec[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 6, -((3*b^2*(4*a^2 - b^2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/(2*(a^2 + b^2)^(7/2)*d)) + (b*(2*a^2 - 3*b^2)*Sec[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (Cos[c + d*x]*(b + a*Tan[c + d*x]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*b*(2*a^2 - 13*b^2)*Sec[c + d*x])/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Tan[c + d*x])^3, x, 7, -((5*b^4*(6*a^2 - b^2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/(2*(a^2 + b^2)^(9/2)*d)) + (b*(4*a^4 + 24*a^2*b^2 - 15*b^4)*Sec[c + d*x])/(6*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) + (Cos[c + d*x]^3*(b + a*Tan[c + d*x]))/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*b*(4*a^4 + 28*a^2*b^2 - 81*b^4)*Sec[c + d*x])/(6*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]*(b*(2*a^2 - 5*b^2) - a*(2*a^2 + 9*b^2)*Tan[c + d*x]))/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/2) (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d*Sec[e + f*x])^(7/2)*(a + b*Tan[e + f*x]), x, 5, -((6*a*d^4*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])) + (2*b*(d*Sec[e + f*x])^(7/2))/(7*f) + (6*a*d^3*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x])/(5*f) + (2*a*d*(d*Sec[e + f*x])^(5/2)*Sin[e + f*x])/(5*f)} -{(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x]), x, 4, (2*a*d^2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[d*Sec[e + f*x]])/(3*f) + (2*b*(d*Sec[e + f*x])^(5/2))/(5*f) + (2*a*d*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(3*f)} -{(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]), x, 4, (-2*a*d^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*b*(d*Sec[e + f*x])^(3/2))/(3*f) + (2*a*d*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x])/f} -{(d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x]), x, 3, (2*b*Sqrt[d*Sec[e + f*x]])/f + (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[d*Sec[e + f*x]])/f} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(1/2), x, 3, (-2*b)/(f*Sqrt[d*Sec[e + f*x]]) + (2*a*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(3/2), x, 4, (-2*b)/(3*f*(d*Sec[e + f*x])^(3/2)) + (2*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[d*Sec[e + f*x]])/(3*d^2*f) + (2*a*Sin[e + f*x])/(3*d*f*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(5/2), x, 4, (-2*b)/(5*f*(d*Sec[e + f*x])^(5/2)) + (6*a*EllipticE[(e + f*x)/2, 2])/(5*d^2*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*a*Sin[e + f*x])/(5*d*f*(d*Sec[e + f*x])^(3/2))} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(7/2), x, 5, (-2*b)/(7*f*(d*Sec[e + f*x])^(7/2)) + (10*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[d*Sec[e + f*x]])/(21*d^4*f) + (2*a*Sin[e + f*x])/(7*d*f*(d*Sec[e + f*x])^(5/2)) + (10*a*Sin[e + f*x])/(21*d^3*f*Sqrt[d*Sec[e + f*x]])} - - -{(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2, x, 5, (2*(7*a^2 - 2*b^2)*d^2*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[d*Sec[e + f*x]])/(21*f) + (18*a*b*(d*Sec[e + f*x])^(5/2))/(35*f) + (2*(7*a^2 - 2*b^2)*d*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(21*f) + (2*b*(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x]))/(7*f)} -{(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2, x, 5, -((2*(5*a^2 - 2*b^2)*d^2*EllipticE[(1/2)*(e + f*x), 2])/(5*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]])) + (14*a*b*(d*Sec[e + f*x])^(3/2))/(15*f) + (2*(5*a^2 - 2*b^2)*d*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x])/(5*f) + (2*b*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]))/(5*f)} -{(d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x])^2, x, 4, (10*a*b*Sqrt[d*Sec[e + f*x]])/(3*f) + (2*(3*a^2 - 2*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[d*Sec[e + f*x]])/(3*f) + (2*b*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))/(3*f)} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(1/2), x, 4, -((6*a*b)/(f*Sqrt[d*Sec[e + f*x]])) + (2*(a^2 - 2*b^2)*EllipticE[(1/2)*(e + f*x), 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*b*(a + b*Tan[e + f*x]))/(f*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(3/2), x, 5, (2*a*b)/(3*f*(d*Sec[e + f*x])^(3/2)) + (2*(a^2 + 2*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[d*Sec[e + f*x]])/(3*d^2*f) + (2*(a^2 + 2*b^2)*Sin[e + f*x])/(3*d*f*Sqrt[d*Sec[e + f*x]]) - (2*b*(a + b*Tan[e + f*x]))/(f*(d*Sec[e + f*x])^(3/2))} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/2), x, 5, -((2*a*b)/(15*f*(d*Sec[e + f*x])^(5/2))) + (2*(3*a^2 + 2*b^2)*EllipticE[(1/2)*(e + f*x), 2])/(5*d^2*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*(3*a^2 + 2*b^2)*Sin[e + f*x])/(15*d*f*(d*Sec[e + f*x])^(3/2)) - (2*b*(a + b*Tan[e + f*x]))/(3*f*(d*Sec[e + f*x])^(5/2))} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(7/2), x, 6, -((6*a*b)/(35*f*(d*Sec[e + f*x])^(7/2))) + (2*(5*a^2 + 2*b^2)*Sqrt[Cos[e + f*x]]*EllipticF[(1/2)*(e + f*x), 2]*Sqrt[d*Sec[e + f*x]])/(21*d^4*f) + (2*(5*a^2 + 2*b^2)*Sin[e + f*x])/(35*d*f*(d*Sec[e + f*x])^(5/2)) + (2*(5*a^2 + 2*b^2)*Sin[e + f*x])/(21*d^3*f*Sqrt[d*Sec[e + f*x]]) - (2*b*(a + b*Tan[e + f*x]))/(5*f*(d*Sec[e + f*x])^(7/2))} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(9/2), x, 6, -((10*a*b)/(63*f*(d*Sec[e + f*x])^(9/2))) + (2*(7*a^2 + 2*b^2)*EllipticE[(1/2)*(e + f*x), 2])/(15*d^4*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*(7*a^2 + 2*b^2)*Sin[e + f*x])/(63*d*f*(d*Sec[e + f*x])^(7/2)) + (2*(7*a^2 + 2*b^2)*Sin[e + f*x])/(45*d^3*f*(d*Sec[e + f*x])^(3/2)) - (2*b*(a + b*Tan[e + f*x]))/(7*f*(d*Sec[e + f*x])^(9/2))} - - -{(d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3, x, 5, (2*a*(7*a^2 - 6*b^2)*d^2*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*Sqrt[d*Sec[e + f*x]])/(21*f*(Sec[e + f*x]^2)^(1/4)) + (2*a*(7*a^2 - 6*b^2)*d^2*Sqrt[d*Sec[e + f*x]]*Tan[e + f*x])/(21*f) + (2*b*d^2*Sec[e + f*x]^2*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2)/(9*f) + (2*b*d^2*Sec[e + f*x]^2*Sqrt[d*Sec[e + f*x]]*(14*(11*a^2 - 2*b^2) + 65*a*b*Tan[e + f*x]))/(315*f)} -{(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3, x, 5, (-2*a*(5*a^2 - 6*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(d*Sec[e + f*x])^(3/2))/(5*f*(Sec[e + f*x]^2)^(3/4)) + (2*a*(5*a^2 - 6*b^2)*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(5*f) + (2*b*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2)/(7*f) + (2*b*(d*Sec[e + f*x])^(3/2)*(10*(9*a^2 - 2*b^2) + 33*a*b*Tan[e + f*x]))/(105*f)} -{(d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x])^3, x, 4, (2*a*(a^2 - 2*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*Sqrt[d*Sec[e + f*x]])/(f*(Sec[e + f*x]^2)^(1/4)) + (2*b*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2)/(5*f) + (2*b*Sqrt[d*Sec[e + f*x]]*(2*(7*a^2 - 2*b^2) + 3*a*b*Tan[e + f*x]))/(5*f)} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(1/2), x, 5, (2*a*(a^2 - 6*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/(f*Sqrt[d*Sec[e + f*x]]) - (2*a*(a^2 - 6*b^2)*Tan[e + f*x])/(f*Sqrt[d*Sec[e + f*x]]) - (2*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(f*Sqrt[d*Sec[e + f*x]]) - (2*b*Sec[e + f*x]^2*(2*(3*a^2 - 2*b^2) + 3*a*b*Tan[e + f*x]))/(3*f*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(3/2), x, 4, (2*a*(a^2 + 6*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(3*f*(d*Sec[e + f*x])^(3/2)) - (2*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(3*f*(d*Sec[e + f*x])^(3/2)) - (2*b*Sec[e + f*x]^2*(2*(a^2 - 2*b^2) + a*b*Tan[e + f*x]))/(3*f*(d*Sec[e + f*x])^(3/2))} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(5/2), x, 5, (6*a*(a^2 + 2*b^2)*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(1/4))/(5*d^2*f*Sqrt[d*Sec[e + f*x]]) - (6*a*(a^2 + 2*b^2)*Tan[e + f*x])/(5*d^2*f*Sqrt[d*Sec[e + f*x]]) - (2*Cos[e + f*x]^2*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(5*d^2*f*Sqrt[d*Sec[e + f*x]]) - (2*(2*b*(a^2 + 2*b^2) - a*(3*a^2 + 5*b^2)*Tan[e + f*x]))/(5*d^2*f*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(7/2), x, 4, (2*a*(5*a^2 + 6*b^2)*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(3/4))/(21*d^2*f*(d*Sec[e + f*x])^(3/2)) - (2*Cos[e + f*x]^2*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(7*d^2*f*(d*Sec[e + f*x])^(3/2)) - (2*(2*b*(3*a^2 + 2*b^2) - a*(5*a^2 + 3*b^2)*Tan[e + f*x]))/(21*d^2*f*(d*Sec[e + f*x])^(3/2))} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(9/2), x, 4, (2*a*(7*a^2 + 6*b^2)*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(1/4))/(15*d^4*f*Sqrt[d*Sec[e + f*x]]) - (2*Cos[e + f*x]^4*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(9*d^4*f*Sqrt[d*Sec[e + f*x]]) - (2*Cos[e + f*x]^2*(2*b*(5*a^2 + 2*b^2) - a*(7*a^2 + b^2)*Tan[e + f*x]))/(45*d^4*f*Sqrt[d*Sec[e + f*x]])} -{(a + b*Tan[e + f*x])^3/(d*Sec[e + f*x])^(11/2), x, 5, (10*a*(3*a^2 + 2*b^2)*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(3/4))/(77*d^4*f*(d*Sec[e + f*x])^(3/2)) + (10*a*(3*a^2 + 2*b^2)*Tan[e + f*x])/(77*d^4*f*(d*Sec[e + f*x])^(3/2)) - (2*Cos[e + f*x]^4*(b - a*Tan[e + f*x])*(a + b*Tan[e + f*x])^2)/(11*d^4*f*(d*Sec[e + f*x])^(3/2)) - (2*Cos[e + f*x]^2*(2*b*(7*a^2 + 2*b^2) - a*(9*a^2 - b^2)*Tan[e + f*x]))/(77*d^4*f*(d*Sec[e + f*x])^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x]), x, 17, (2*d^2*(d*Sec[e + f*x])^(3/2))/(3*b*f) + ((a^2 + b^2)^(3/4)*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(b^(5/2)*f*(Sec[e + f*x]^2)^(3/4)) - ((a^2 + b^2)^(3/4)*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(b^(5/2)*f*(Sec[e + f*x]^2)^(3/4)) + (2*a*d^2*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(d*Sec[e + f*x])^(3/2))/(b^2*f*(Sec[e + f*x]^2)^(3/4)) - (2*a*d^2*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(b^2*f) - (a*Sqrt[a^2 + b^2]*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(b^3*f*(Sec[e + f*x]^2)^(3/4)) + (a*Sqrt[a^2 + b^2]*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(b^3*f*(Sec[e + f*x]^2)^(3/4))} -{(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x]), x, 17, (2*d^2*Sqrt[d*Sec[e + f*x]])/(b*f) - ((a^2 + b^2)^(1/4)*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(b^(3/2)*f*(Sec[e + f*x]^2)^(1/4)) - ((a^2 + b^2)^(1/4)*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(b^(3/2)*f*(Sec[e + f*x]^2)^(1/4)) - (2*a*d^2*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*Sqrt[d*Sec[e + f*x]])/(b^2*f*(Sec[e + f*x]^2)^(1/4)) + (a*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(b^2*f*(Sec[e + f*x]^2)^(1/4)) + (a*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(b^2*f*(Sec[e + f*x]^2)^(1/4))} -{(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x]), x, 13, (ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(Sqrt[b]*(a^2 + b^2)^(1/4)*f*(Sec[e + f*x]^2)^(3/4)) - (ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(Sqrt[b]*(a^2 + b^2)^(1/4)*f*(Sec[e + f*x]^2)^(3/4)) - (a*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(b*Sqrt[a^2 + b^2]*f*(Sec[e + f*x]^2)^(3/4)) + (a*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(b*Sqrt[a^2 + b^2]*f*(Sec[e + f*x]^2)^(3/4))} -{(d*Sec[e + f*x])^(1/2)/(a + b*Tan[e + f*x]), x, 14, -((Sqrt[b]*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/((a^2 + b^2)^(3/4)*f*(Sec[e + f*x]^2)^(1/4))) - (Sqrt[b]*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/((a^2 + b^2)^(3/4)*f*(Sec[e + f*x]^2)^(1/4)) + (a*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4)) + (a*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4))} -{1/((d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x])), x, 17, (b^(3/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)^(5/4)*f*Sqrt[d*Sec[e + f*x]]) - (b^(3/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)^(5/4)*f*Sqrt[d*Sec[e + f*x]]) + (2*a*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]) - (2*a*Tan[e + f*x])/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]) - (a*b*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^(3/2)*f*Sqrt[d*Sec[e + f*x]]) + (a*b*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^(3/2)*f*Sqrt[d*Sec[e + f*x]]) + (2*(b + a*Tan[e + f*x]))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]])} -{1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])), x, 17, -((b^(5/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/((a^2 + b^2)^(7/4)*f*(d*Sec[e + f*x])^(3/2))) - (b^(5/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/((a^2 + b^2)^(7/4)*f*(d*Sec[e + f*x])^(3/2)) + (2*a*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(3*(a^2 + b^2)*f*(d*Sec[e + f*x])^(3/2)) + (a*b^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)) + (a*b^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)) + (2*(b + a*Tan[e + f*x]))/(3*(a^2 + b^2)*f*(d*Sec[e + f*x])^(3/2))} -{1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])), x, 18, (b^(7/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)^(9/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) - (b^(7/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)^(9/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (2*a*(3*a^2 + 8*b^2)*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(1/4))/(5*(a^2 + b^2)^2*d^2*f*Sqrt[d*Sec[e + f*x]]) - (2*a*(3*a^2 + 8*b^2)*Tan[e + f*x])/(5*(a^2 + b^2)^2*d^2*f*Sqrt[d*Sec[e + f*x]]) - (a*b^3*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^(5/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (a*b^3*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/((a^2 + b^2)^(5/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (2*Cos[e + f*x]^2*(b + a*Tan[e + f*x]))/(5*(a^2 + b^2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (2*(5*b^3 + a*(3*a^2 + 8*b^2)*Tan[e + f*x]))/(5*(a^2 + b^2)^2*d^2*f*Sqrt[d*Sec[e + f*x]])} - - -{(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x])^2, x, 17, -((3*a*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(2*b^(5/2)*(a^2 + b^2)^(1/4)*f*(Sec[e + f*x]^2)^(3/4))) + (3*a*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(2*b^(5/2)*(a^2 + b^2)^(1/4)*f*(Sec[e + f*x]^2)^(3/4)) - (3*d^2*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(d*Sec[e + f*x])^(3/2))/(b^2*f*(Sec[e + f*x]^2)^(3/4)) + (3*d^2*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(b^2*f) + (3*a^2*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(2*b^3*Sqrt[a^2 + b^2]*f*(Sec[e + f*x]^2)^(3/4)) - (3*a^2*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(2*b^3*Sqrt[a^2 + b^2]*f*(Sec[e + f*x]^2)^(3/4)) - (d^2*(d*Sec[e + f*x])^(3/2))/(b*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^2, x, 17, (a*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(2*b^(3/2)*(a^2 + b^2)^(3/4)*f*(Sec[e + f*x]^2)^(1/4)) + (a*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(2*b^(3/2)*(a^2 + b^2)^(3/4)*f*(Sec[e + f*x]^2)^(1/4)) + (d^2*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*Sqrt[d*Sec[e + f*x]])/(b^2*f*(Sec[e + f*x]^2)^(1/4)) - (a^2*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(2*b^2*(a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4)) - (a^2*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(2*b^2*(a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4)) - (d^2*Sqrt[d*Sec[e + f*x]])/(b*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^2, x, 17, (a*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(2*Sqrt[b]*(a^2 + b^2)^(5/4)*f*(Sec[e + f*x]^2)^(3/4)) - (a*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(2*Sqrt[b]*(a^2 + b^2)^(5/4)*f*(Sec[e + f*x]^2)^(3/4)) - (EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(d*Sec[e + f*x])^(3/2))/((a^2 + b^2)*f*(Sec[e + f*x]^2)^(3/4)) + (Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/((a^2 + b^2)*f) - (a^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(2*b*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) + (a^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(2*b*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) - (b*(d*Sec[e + f*x])^(3/2))/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(1/2)/(a + b*Tan[e + f*x])^2, x, 17, (-3*a*Sqrt[b]*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(2*(a^2 + b^2)^(7/4)*f*(Sec[e + f*x]^2)^(1/4)) - (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(2*(a^2 + b^2)^(7/4)*f*(Sec[e + f*x]^2)^(1/4)) - (EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*Sqrt[d*Sec[e + f*x]])/((a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4)) + (3*a^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(1/4)) + (3*a^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(1/4)) - (b*Sqrt[d*Sec[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x])^2), x, 18, (5*a*b^(3/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)^(9/4)*f*Sqrt[d*Sec[e + f*x]]) - (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)^(9/4)*f*Sqrt[d*Sec[e + f*x]]) + ((2*a^2 - 3*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/((a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]) - ((2*a^2 - 3*b^2)*Tan[e + f*x])/((a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]) - (5*a^2*b*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^(5/2)*f*Sqrt[d*Sec[e + f*x]]) + (5*a^2*b*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^(5/2)*f*Sqrt[d*Sec[e + f*x]]) + (b*(2*a^2 - 3*b^2)*Sec[e + f*x]^2)/((a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])) + (2*(b + a*Tan[e + f*x]))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2), x, 18, (-7*a*b^(5/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/(2*(a^2 + b^2)^(11/4)*f*(d*Sec[e + f*x])^(3/2)) - (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/(2*(a^2 + b^2)^(11/4)*f*(d*Sec[e + f*x])^(3/2)) + ((2*a^2 - 5*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(3*(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)) + (7*a^2*b^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^3*f*(d*Sec[e + f*x])^(3/2)) + (7*a^2*b^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^3*f*(d*Sec[e + f*x])^(3/2)) + (b*(2*a^2 - 5*b^2)*Sec[e + f*x]^2)/(3*(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])) + (2*(b + a*Tan[e + f*x]))/(3*(a^2 + b^2)*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2), x, 19, (9*a*b^(7/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)^(13/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)^(13/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (3*(2*a^4 + 10*a^2*b^2 - 7*b^4)*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(1/4))/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]) - (3*(2*a^4 + 10*a^2*b^2 - 7*b^4)*Tan[e + f*x])/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*a^2*b^3*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^(7/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (9*a^2*b^3*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^(7/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (3*b*(2*a^4 + 10*a^2*b^2 - 7*b^4)*Sec[e + f*x]^2)/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])) + (2*Cos[e + f*x]^2*(b + a*Tan[e + f*x]))/(5*(a^2 + b^2)*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])) - (2*(b*(2*a^2 - 7*b^2) - 3*a*(a^2 + 4*b^2)*Tan[e + f*x]))/(5*(a^2 + b^2)^2*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))} - - -{(d*Sec[e + f*x])^(7/2)/(a + b*Tan[e + f*x])^3, x, 18, (3*(a^2 + 2*b^2)*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(8*b^(5/2)*(a^2 + b^2)^(5/4)*f*(Sec[e + f*x]^2)^(3/4)) - (3*(a^2 + 2*b^2)*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(8*b^(5/2)*(a^2 + b^2)^(5/4)*f*(Sec[e + f*x]^2)^(3/4)) + (3*a*d^2*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(d*Sec[e + f*x])^(3/2))/(4*b^2*(a^2 + b^2)*f*(Sec[e + f*x]^2)^(3/4)) - (3*a*d^2*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(4*b^2*(a^2 + b^2)*f) - (3*a*(a^2 + 2*b^2)*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(8*b^3*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) + (3*a*(a^2 + 2*b^2)*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(8*b^3*(a^2 + b^2)^(3/2)*f*(Sec[e + f*x]^2)^(3/4)) - (d^2*(d*Sec[e + f*x])^(3/2))/(2*b*f*(a + b*Tan[e + f*x])^2) + (3*a*d^2*(d*Sec[e + f*x])^(3/2))/(4*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3, x, 18, ((a^2 - 2*b^2)*d^2*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(8*b^(3/2)*(a^2 + b^2)^(7/4)*f*(Sec[e + f*x]^2)^(1/4)) + ((a^2 - 2*b^2)*d^2*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(8*b^(3/2)*(a^2 + b^2)^(7/4)*f*(Sec[e + f*x]^2)^(1/4)) + (a*d^2*EllipticF[(1/2)*ArcTan[Tan[e + f*x]], 2]*Sqrt[d*Sec[e + f*x]])/(4*b^2*(a^2 + b^2)*f*(Sec[e + f*x]^2)^(1/4)) - (a*(a^2 - 2*b^2)*d^2*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(8*b^2*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(1/4)) - (a*(a^2 - 2*b^2)*d^2*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(8*b^2*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(1/4)) - (d^2*Sqrt[d*Sec[e + f*x]])/(2*b*f*(a + b*Tan[e + f*x])^2) + (a*d^2*Sqrt[d*Sec[e + f*x]])/(4*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3, x, 18, ((3*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(8*Sqrt[b]*(a^2 + b^2)^(9/4)*f*(Sec[e + f*x]^2)^(3/4)) - ((3*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(d*Sec[e + f*x])^(3/2))/(8*Sqrt[b]*(a^2 + b^2)^(9/4)*f*(Sec[e + f*x]^2)^(3/4)) - (5*a*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(d*Sec[e + f*x])^(3/2))/(4*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(3/4)) + (5*a*Cos[e + f*x]*(d*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(4*(a^2 + b^2)^2*f) - (a*(3*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(8*b*(a^2 + b^2)^(5/2)*f*(Sec[e + f*x]^2)^(3/4)) + (a*(3*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(d*Sec[e + f*x])^(3/2)*Sqrt[-Tan[e + f*x]^2])/(8*b*(a^2 + b^2)^(5/2)*f*(Sec[e + f*x]^2)^(3/4)) - (b*(d*Sec[e + f*x])^(3/2))/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (5*a*b*(d*Sec[e + f*x])^(3/2))/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} -{(d*Sec[e + f*x])^(1/2)/(a + b*Tan[e + f*x])^3, x, 18, (-3*Sqrt[b]*(5*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(8*(a^2 + b^2)^(11/4)*f*(Sec[e + f*x]^2)^(1/4)) - (3*Sqrt[b]*(5*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*Sqrt[d*Sec[e + f*x]])/(8*(a^2 + b^2)^(11/4)*f*(Sec[e + f*x]^2)^(1/4)) - (7*a*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*Sqrt[d*Sec[e + f*x]])/(4*(a^2 + b^2)^2*f*(Sec[e + f*x]^2)^(1/4)) + (3*a*(5*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^3*f*(Sec[e + f*x]^2)^(1/4)) + (3*a*(5*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*Sqrt[d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^3*f*(Sec[e + f*x]^2)^(1/4)) - (b*Sqrt[d*Sec[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (7*a*b*Sqrt[d*Sec[e + f*x]])/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(1/2)*(a + b*Tan[e + f*x])^3), x, 19, (5*b^(3/2)*(7*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) - (5*b^(3/2)*(7*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) + (a*(8*a^2 - 37*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) - (a*(8*a^2 - 37*b^2)*Tan[e + f*x])/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) - (5*a*b*(7*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(7/2)*f*Sqrt[d*Sec[e + f*x]]) + (5*a*b*(7*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(7/2)*f*Sqrt[d*Sec[e + f*x]]) + (b*(4*a^2 - 5*b^2)*Sec[e + f*x]^2)/(2*(a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (2*(b + a*Tan[e + f*x]))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (a*b*(8*a^2 - 37*b^2)*Sec[e + f*x]^2)/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3), x, 19, (-7*b^(5/2)*(9*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/(8*(a^2 + b^2)^(15/4)*f*(d*Sec[e + f*x])^(3/2)) - (7*b^(5/2)*(9*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/(8*(a^2 + b^2)^(15/4)*f*(d*Sec[e + f*x])^(3/2)) + (a*(8*a^2 - 69*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(12*(a^2 + b^2)^3*f*(d*Sec[e + f*x])^(3/2)) + (7*a*b^2*(9*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^4*f*(d*Sec[e + f*x])^(3/2)) + (7*a*b^2*(9*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(3/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^4*f*(d*Sec[e + f*x])^(3/2)) + (b*(4*a^2 - 7*b^2)*Sec[e + f*x]^2)/(6*(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2) + (2*(b + a*Tan[e + f*x]))/(3*(a^2 + b^2)*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2) + (a*b*(8*a^2 - 69*b^2)*Sec[e + f*x]^2)/(12*(a^2 + b^2)^3*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x]))} -{1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^3), x, 20, (9*b^(7/2)*(11*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(8*(a^2 + b^2)^(17/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*b^(7/2)*(11*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(8*(a^2 + b^2)^(17/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (3*a*(8*a^4 + 64*a^2*b^2 - 139*b^4)*EllipticE[(1/2)*ArcTan[Tan[e + f*x]], 2]*(Sec[e + f*x]^2)^(1/4))/(20*(a^2 + b^2)^4*d^2*f*Sqrt[d*Sec[e + f*x]]) - (3*a*(8*a^4 + 64*a^2*b^2 - 139*b^4)*Tan[e + f*x])/(20*(a^2 + b^2)^4*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*a*b^3*(11*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(9/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (9*a*b^3*(11*a^2 - 2*b^2)*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(9/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (3*b*(4*a^4 + 28*a^2*b^2 - 15*b^4)*Sec[e + f*x]^2)/(10*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (2*Cos[e + f*x]^2*(b + a*Tan[e + f*x]))/(5*(a^2 + b^2)*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (3*a*b*(8*a^4 + 64*a^2*b^2 - 139*b^4)*Sec[e + f*x]^2)/(20*(a^2 + b^2)^4*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])) - (2*(b*(4*a^2 - 9*b^2) - a*(3*a^2 + 16*b^2)*Tan[e + f*x]))/(5*(a^2 + b^2)^2*d^2*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(m/3) (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Tan[e + f*x])*(d*Sec[e + f*x])^(5/3), x, 3, (3*b*(d*Sec[e + f*x])^(5/3))/(5*f) + (3*a*d*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(2/3)*Sin[e + f*x])/(2*f*Sqrt[Sin[e + f*x]^2])} -{(a + b*Tan[e + f*x])*(d*Sec[e + f*x])^(1/3), x, 3, (3*b*(d*Sec[e + f*x])^(1/3))/f - (3*a*d*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[e + f*x]^2]*Sin[e + f*x])/(2*f*(d*Sec[e + f*x])^(2/3)*Sqrt[Sin[e + f*x]^2])} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(1/3), x, 3, -((3*b)/(f*(d*Sec[e + f*x])^(1/3))) - (3*a*d*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[e + f*x]^2]*Sin[e + f*x])/(4*f*(d*Sec[e + f*x])^(4/3)*Sqrt[Sin[e + f*x]^2])} -{(a + b*Tan[e + f*x])/(d*Sec[e + f*x])^(5/3), x, 3, -((3*b)/(5*f*(d*Sec[e + f*x])^(5/3))) - (3*a*d*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[e + f*x]^2]*Sin[e + f*x])/(8*f*(d*Sec[e + f*x])^(8/3)*Sqrt[Sin[e + f*x]^2])} - - -{(a + b*Tan[e + f*x])^2*(d*Sec[e + f*x])^(5/3), x, 4, (33*a*b*(d*Sec[e + f*x])^(5/3))/(40*f) + (3*(8*a^2 - 3*b^2)*d*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(2/3)*Sin[e + f*x])/(16*f*Sqrt[Sin[e + f*x]^2]) + (3*b*(d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x]))/(8*f)} -{(a + b*Tan[e + f*x])^2*(d*Sec[e + f*x])^(1/3), x, 4, (21*a*b*(d*Sec[e + f*x])^(1/3))/(4*f) - (3*(4*a^2 - 3*b^2)*d*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[e + f*x]^2]*Sin[e + f*x])/(8*f*(d*Sec[e + f*x])^(2/3)*Sqrt[Sin[e + f*x]^2]) + (3*b*(d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x]))/(4*f)} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(1/3), x, 4, -((15*a*b)/(2*f*(d*Sec[e + f*x])^(1/3))) - (3*(2*a^2 - 3*b^2)*d*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[e + f*x]^2]*Sin[e + f*x])/(8*f*(d*Sec[e + f*x])^(4/3)*Sqrt[Sin[e + f*x]^2]) + (3*b*(a + b*Tan[e + f*x]))/(2*f*(d*Sec[e + f*x])^(1/3))} -{(a + b*Tan[e + f*x])^2/(d*Sec[e + f*x])^(5/3), x, 4, (3*a*b)/(10*f*(d*Sec[e + f*x])^(5/3)) - (3*(2*a^2 + 3*b^2)*d*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[e + f*x]^2]*Sin[e + f*x])/(16*f*(d*Sec[e + f*x])^(8/3)*Sqrt[Sin[e + f*x]^2]) - (3*b*(a + b*Tan[e + f*x]))/(2*f*(d*Sec[e + f*x])^(5/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x]), x, 16, -((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(5/3))/(2*b^(2/3)*(a^2 + b^2)^(1/6)*f*(Sec[e + f*x]^2)^(5/6))) + (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(5/3))/(2*b^(2/3)*(a^2 + b^2)^(1/6)*f*(Sec[e + f*x]^2)^(5/6)) - (ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e + f*x])^(5/3))/(b^(2/3)*(a^2 + b^2)^(1/6)*f*(Sec[e + f*x]^2)^(5/6)) + (Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(5/3))/(4*b^(2/3)*(a^2 + b^2)^(1/6)*f*(Sec[e + f*x]^2)^(5/6)) - (Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(5/3))/(4*b^(2/3)*(a^2 + b^2)^(1/6)*f*(Sec[e + f*x]^2)^(5/6)) + (AppellF1[1/2, 1, 1/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(5/3)*Tan[e + f*x])/(a*f*(Sec[e + f*x]^2)^(5/6))} -{(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x]), x, 16, (Sqrt[3]*b^(2/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*(a^2 + b^2)^(5/6)*f*(Sec[e + f*x]^2)^(1/6)) - (Sqrt[3]*b^(2/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*(a^2 + b^2)^(5/6)*f*(Sec[e + f*x]^2)^(1/6)) - (b^(2/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e + f*x])^(1/3))/((a^2 + b^2)^(5/6)*f*(Sec[e + f*x]^2)^(1/6)) + (b^(2/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(4*(a^2 + b^2)^(5/6)*f*(Sec[e + f*x]^2)^(1/6)) - (b^(2/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(4*(a^2 + b^2)^(5/6)*f*(Sec[e + f*x]^2)^(1/6)) + (AppellF1[1/2, 1, 5/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x])/(a*f*(Sec[e + f*x]^2)^(1/6))} -{1/((d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x])), x, 17, (3*b)/((a^2 + b^2)*f*(d*Sec[e + f*x])^(1/3)) - (Sqrt[3]*b^(4/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(1/6))/(2*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) + (Sqrt[3]*b^(4/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(1/6))/(2*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) - (b^(4/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(Sec[e + f*x]^2)^(1/6))/((a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) + (b^(4/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(1/6))/(4*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) - (b^(4/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(1/6))/(4*(a^2 + b^2)^(7/6)*f*(d*Sec[e + f*x])^(1/3)) + (AppellF1[1/2, 1, 7/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/6)*Tan[e + f*x])/(a*f*(d*Sec[e + f*x])^(1/3))} -{1/((d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x])), x, 17, (3*b)/(5*(a^2 + b^2)*f*(d*Sec[e + f*x])^(5/3)) + (Sqrt[3]*b^(8/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(5/6))/(2*(a^2 + b^2)^(11/6)*f*(d*Sec[e + f*x])^(5/3)) - (Sqrt[3]*b^(8/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(5/6))/(2*(a^2 + b^2)^(11/6)*f*(d*Sec[e + f*x])^(5/3)) - (b^(8/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(Sec[e + f*x]^2)^(5/6))/((a^2 + b^2)^(11/6)*f*(d*Sec[e + f*x])^(5/3)) + (b^(8/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(5/6))/(4*(a^2 + b^2)^(11/6)*f*(d*Sec[e + f*x])^(5/3)) - (b^(8/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(5/6))/(4*(a^2 + b^2)^(11/6)*f*(d*Sec[e + f*x])^(5/3)) + (AppellF1[1/2, 1, 11/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(5/6)*Tan[e + f*x])/(a*f*(d*Sec[e + f*x])^(5/3))} - - -{(d*Sec[e + f*x])^(5/3)/(a + b*Tan[e + f*x])^2, x, 18, -((a*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(5/3))/(2*Sqrt[3]*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6))) + (a*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(5/3))/(2*Sqrt[3]*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) - (a*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e + f*x])^(5/3))/(3*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) + (a*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(5/3))/(12*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) - (a*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(5/3))/(12*b^(2/3)*(a^2 + b^2)^(7/6)*f*(Sec[e + f*x]^2)^(5/6)) + (AppellF1[1/2, 2, 1/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(5/3)*Tan[e + f*x])/(a^2*f*(Sec[e + f*x]^2)^(5/6)) + (b^2*AppellF1[3/2, 2, 1/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(5/3)*Tan[e + f*x]^3)/(3*a^4*f*(Sec[e + f*x]^2)^(5/6)) - (a*b*(d*Sec[e + f*x])^(5/3))/((a^2 + b^2)*f*(a^2 - b^2*Tan[e + f*x]^2))} -{(d*Sec[e + f*x])^(1/3)/(a + b*Tan[e + f*x])^2, x, 18, (5*a*b^(2/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(d*Sec[e + f*x])^(1/3))/(2*Sqrt[3]*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(d*Sec[e + f*x])^(1/3))/(3*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) + (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) - (5*a*b^(2/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(d*Sec[e + f*x])^(1/3))/(12*(a^2 + b^2)^(11/6)*f*(Sec[e + f*x]^2)^(1/6)) + (AppellF1[1/2, 2, 5/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x])/(a^2*f*(Sec[e + f*x]^2)^(1/6)) + (b^2*AppellF1[3/2, 2, 5/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^(1/3)*Tan[e + f*x]^3)/(3*a^4*f*(Sec[e + f*x]^2)^(1/6)) - (a*b*(d*Sec[e + f*x])^(1/3))/((a^2 + b^2)*f*(a^2 - b^2*Tan[e + f*x]^2))} -{1/((d*Sec[e + f*x])^(1/3)*(a + b*Tan[e + f*x])^2), x, 19, (7*a*b)/((a^2 + b^2)^2*f*(d*Sec[e + f*x])^(1/3)) - (7*a*b^(4/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(1/6))/(2*Sqrt[3]*(a^2 + b^2)^(13/6)*f*(d*Sec[e + f*x])^(1/3)) + (7*a*b^(4/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(1/6))/(2*Sqrt[3]*(a^2 + b^2)^(13/6)*f*(d*Sec[e + f*x])^(1/3)) - (7*a*b^(4/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(Sec[e + f*x]^2)^(1/6))/(3*(a^2 + b^2)^(13/6)*f*(d*Sec[e + f*x])^(1/3)) + (7*a*b^(4/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(1/6))/(12*(a^2 + b^2)^(13/6)*f*(d*Sec[e + f*x])^(1/3)) - (7*a*b^(4/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(1/6))/(12*(a^2 + b^2)^(13/6)*f*(d*Sec[e + f*x])^(1/3)) + (AppellF1[1/2, 2, 7/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/6)*Tan[e + f*x])/(a^2*f*(d*Sec[e + f*x])^(1/3)) + (b^2*AppellF1[3/2, 2, 7/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/6)*Tan[e + f*x]^3)/(3*a^4*f*(d*Sec[e + f*x])^(1/3)) - (a*b)/((a^2 + b^2)*f*(d*Sec[e + f*x])^(1/3)*(a^2 - b^2*Tan[e + f*x]^2))} -{1/((d*Sec[e + f*x])^(5/3)*(a + b*Tan[e + f*x])^2), x, 19, (11*a*b)/(5*(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(5/3)) + (11*a*b^(8/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(5/6))/(2*Sqrt[3]*(a^2 + b^2)^(17/6)*f*(d*Sec[e + f*x])^(5/3)) - (11*a*b^(8/3)*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(Sqrt[3]*(a^2 + b^2)^(1/6))]*(Sec[e + f*x]^2)^(5/6))/(2*Sqrt[3]*(a^2 + b^2)^(17/6)*f*(d*Sec[e + f*x])^(5/3)) - (11*a*b^(8/3)*ArcTanh[(b^(1/3)*(Sec[e + f*x]^2)^(1/6))/(a^2 + b^2)^(1/6)]*(Sec[e + f*x]^2)^(5/6))/(3*(a^2 + b^2)^(17/6)*f*(d*Sec[e + f*x])^(5/3)) + (11*a*b^(8/3)*Log[(a^2 + b^2)^(1/3) - b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(5/6))/(12*(a^2 + b^2)^(17/6)*f*(d*Sec[e + f*x])^(5/3)) - (11*a*b^(8/3)*Log[(a^2 + b^2)^(1/3) + b^(1/3)*(a^2 + b^2)^(1/6)*(Sec[e + f*x]^2)^(1/6) + b^(2/3)*(Sec[e + f*x]^2)^(1/3)]*(Sec[e + f*x]^2)^(5/6))/(12*(a^2 + b^2)^(17/6)*f*(d*Sec[e + f*x])^(5/3)) + (AppellF1[1/2, 2, 11/6, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(5/6)*Tan[e + f*x])/(a^2*f*(d*Sec[e + f*x])^(5/3)) + (b^2*AppellF1[3/2, 2, 11/6, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(5/6)*Tan[e + f*x]^3)/(3*a^4*f*(d*Sec[e + f*x])^(5/3)) - (a*b)/((a^2 + b^2)*f*(d*Sec[e + f*x])^(5/3)*(a^2 - b^2*Tan[e + f*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n with m symbolic*) - - -{(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^3, x, 4, -((a*(3*b^2 - a^2*(1 + m))*Hypergeometric2F1[1/2, 1 - m/2, 3/2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x])/((Sec[e + f*x]^2)^(m/2)*(f*(1 + m)))) + (b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^2)/(f*(2 + m)) - (b*(d*Sec[e + f*x])^m*(2*(1 + m)*(b^2 - a^2*(3 + m)) - a*b*m*(4 + m)*Tan[e + f*x]))/(f*m*(2 + 3*m + m^2))} -{(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^2, x, 4, If[$VersionNumber>=8, (a*b*(2 + m)*(d*Sec[e + f*x])^m)/(f*m*(1 + m)) + (d*(b^2 - a^2*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + m)*Sin[e + f*x])/(f*(1 - m)*(1 + m)*Sqrt[Sin[e + f*x]^2]) + (b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(1 + m)), (a*b*(2 + m)*(d*Sec[e + f*x])^m)/(f*m*(1 + m)) + (d*(b^2 - a^2*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + m)*Sin[e + f*x])/(f*(1 - m^2)*Sqrt[Sin[e + f*x]^2]) + (b*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(1 + m))]} -{(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^1, x, 3, (b*(d*Sec[e + f*x])^m)/(f*m) - (a*d*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + m)*Sin[e + f*x])/(f*(1 - m)*Sqrt[Sin[e + f*x]^2])} -{(d*Sec[e + f*x])^m/(a + b*Tan[e + f*x])^1, x, 6, -((b*Hypergeometric2F1[1, m/2, (2 + m)/2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*(d*Sec[e + f*x])^m)/((a^2 + b^2)*f*m)) + (AppellF1[1/2, 1, 1 - m/2, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x])/((Sec[e + f*x]^2)^(m/2)*(a*f))} -{(d*Sec[e + f*x])^m/(a + b*Tan[e + f*x])^2, x, 7, -((2*a*b*Hypergeometric2F1[2, m/2, (2 + m)/2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)]*(d*Sec[e + f*x])^m)/((a^2 + b^2)^2*f*m)) + (AppellF1[1/2, 2, 1 - m/2, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x])/((Sec[e + f*x]^2)^(m/2)*(a^2*f)) + (b^2*AppellF1[3/2, 2, 1 - m/2, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]^3)/((Sec[e + f*x]^2)^(m/2)*(3*a^4*f))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x])^n with n symbolic*) - - -{(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n, x, 3, (b*AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a + Sqrt[-b^2]), (a + b*Tan[e + f*x])/(a - Sqrt[-b^2])]*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n))/((1 + (a + b*Tan[e + f*x])/(-a + Sqrt[-b^2]))^(m/2)*(1 - (a + b*Tan[e + f*x])/(a + Sqrt[-b^2]))^(m/2)*((a^2 + b^2)*f*(1 + n))), (AppellF1[1 + n, 1 - m/2, 1 - m/2, 2 + n, (a + b*Tan[e + f*x])/(a - Sqrt[-b^2]), (a + b*Tan[e + f*x])/(a + Sqrt[-b^2])]*Cos[e + f*x]^2*(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n)*(1 - (a + b*Tan[e + f*x])/(a - Sqrt[-b^2]))^(1 - m/2)*(1 - (a + b*Tan[e + f*x])/(a + Sqrt[-b^2]))^(1 - m/2))/(b*f*(1 + n))} - - -{Sec[c + d*x]^6*(a + b*Tan[c + d*x])^n, x, 3, ((a^2 + b^2)^2*(a + b*Tan[c + d*x])^(1 + n))/(b^5*d*(1 + n)) - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^(2 + n))/(b^5*d*(2 + n)) + (2*(3*a^2 + b^2)*(a + b*Tan[c + d*x])^(3 + n))/(b^5*d*(3 + n)) - (4*a*(a + b*Tan[c + d*x])^(4 + n))/(b^5*d*(4 + n)) + (a + b*Tan[c + d*x])^(5 + n)/(b^5*d*(5 + n))} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x])^n, x, 3, ((a^2 + b^2)*(a + b*Tan[c + d*x])^(1 + n))/(b^3*d*(1 + n)) - (2*a*(a + b*Tan[c + d*x])^(2 + n))/(b^3*d*(2 + n)) + (a + b*Tan[c + d*x])^(3 + n)/(b^3*d*(3 + n))} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 2, (a + b*Tan[c + d*x])^(1 + n)/(b*d*(1 + n))} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 6, -(((Sqrt[-b^2]*(1 + a^2/b^2 - n) - a*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*(1 + a^2/b^2)*b*(a - Sqrt[-b^2])*d*(1 + n))) + (b*(Sqrt[-b^2]*(1 + a^2/b^2 - n) + a*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*(a^2 + b^2)*(a + Sqrt[-b^2])*d*(1 + n)) + (Cos[c + d*x]^2*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(2*(a^2 + b^2)*d)} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x])^n, x, 7, (b*((a*(5 + (3*a^2)/b^2 - 2*n)*n)/b^2 - (Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))/b^6)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*(1 + a^2/b^2)^2*(a - Sqrt[-b^2])*d*(1 + n)) + (b*((a*(5 + (3*a^2)/b^2 - 2*n)*n)/b^2 + (Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))/b^6)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*(1 + a^2/b^2)^2*(a + Sqrt[-b^2])*d*(1 + n)) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(4*(a^2 + b^2)*d) + (b*Cos[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)*(b^2*(3 - n) + a^2*(1 + n) + a*b*(5 + (3*a^2)/b^2 - 2*n)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)} - -{Sec[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 3, (AppellF1[1 + n, -(1/2), -(1/2), 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]), (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*Sec[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(1 + n)*Sqrt[1 - (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*Sqrt[1 - (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])])} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 3, (AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]), (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*Cos[c + d*x]*(a + b*Tan[c + d*x])^(1 + n)*Sqrt[1 - (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*Sqrt[1 - (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])])/(b*d*(1 + n))} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 3, (AppellF1[1 + n, 3/2, 3/2, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]), (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^(1 + n)*(1 - (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]))^(3/2)*(1 - (a + b*Tan[c + d*x])/(a + Sqrt[-b^2]))^(3/2))/(b*d*(1 + n))} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 3, (AppellF1[1 + n, 5/2, 5/2, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]), (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^(1 + n)*(1 - (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]))^(5/2)*(1 - (a + b*Tan[c + d*x])/(a + Sqrt[-b^2]))^(5/2))/(b*d*(1 + n))} - - -(* ::Title:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n when a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(m/2) (a+a I Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Cos[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x]), x, 6, -((2*I*a*(e*Cos[c + d*x])^(7/2))/(7*d)) + (10*a*(e*Cos[c + d*x])^(7/2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d*Cos[c + d*x]^(7/2)) + (2*a*(e*Cos[c + d*x])^(7/2)*Tan[c + d*x])/(7*d) + (10*a*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2*Tan[c + d*x])/(21*d)} -{(e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x]), x, 5, -((2*I*a*(e*Cos[c + d*x])^(5/2))/(5*d)) + (6*a*(e*Cos[c + d*x])^(5/2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(e*Cos[c + d*x])^(5/2)*Tan[c + d*x])/(5*d)} -{(e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x]), x, 5, -((2*I*a*(e*Cos[c + d*x])^(3/2))/(3*d)) + (2*a*(e*Cos[c + d*x])^(3/2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(e*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(3*d)} -{Sqrt[e*Cos[c + d*x]]*(a + I*a*Tan[c + d*x]), x, 4, ((-2*I)*a*Sqrt[e*Cos[c + d*x]])/d + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])} -{(a + I*a*Tan[c + d*x])/Sqrt[e*Cos[c + d*x]], x, 4, ((2*I)*a)/(d*Sqrt[e*Cos[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])} -{(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(3/2), x, 5, (2*I*a)/(3*d*(e*Cos[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^(3/2)*EllipticE[(1/2)*(c + d*x), 2])/(d*(e*Cos[c + d*x])^(3/2)) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]]), (((2*I)/3)*a)/(d*(e*Cos[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2])/(d*(e*Cos[c + d*x])^(3/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(d*(e*Cos[c + d*x])^(3/2))} -{(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(5/2), x, 5, (((2*I)/5)*a)/(d*(e*Cos[c + d*x])^(5/2)) + (2*a*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2])/(3*d*(e*Cos[c + d*x])^(5/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(3*d*(e*Cos[c + d*x])^(5/2))} -{(a + I*a*Tan[c + d*x])/(e*Cos[c + d*x])^(7/2), x, 6, (((2*I)/7)*a)/(d*(e*Cos[c + d*x])^(7/2)) - (6*a*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(5*d*(e*Cos[c + d*x])^(7/2)) + (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2)) + (6*a*Cos[c + d*x]^3*Sin[c + d*x])/(5*d*(e*Cos[c + d*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Cos[c + d*x])^(7/2)/(a + I*a*Tan[c + d*x])^2, x, 7, (2*(e*Cos[c + d*x])^(7/2)*EllipticF[(1/2)*(c + d*x), 2])/(7*a^2*d*Cos[c + d*x]^(7/2)) + (2*Cos[c + d*x]*(e*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(15*a^2*d) + (6*(e*Cos[c + d*x])^(7/2)*Tan[c + d*x])/(35*a^2*d) + (2*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2*Tan[c + d*x])/(7*a^2*d) + (4*I*Cos[c + d*x]^2*(e*Cos[c + d*x])^(7/2))/(15*d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Cos[c + d*x])^(5/2)/(a + I*a*Tan[c + d*x])^2, x, 6, (42*(e*Cos[c + d*x])^(5/2)*EllipticE[(1/2)*(c + d*x), 2])/(65*a^2*d*Cos[c + d*x]^(5/2)) + (2*Cos[c + d*x]*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*a^2*d) + (14*(e*Cos[c + d*x])^(5/2)*Tan[c + d*x])/(65*a^2*d) + (4*I*Cos[c + d*x]^2*(e*Cos[c + d*x])^(5/2))/(13*d*(a^2 + I*a^2*Tan[c + d*x]))} -{(e*Cos[c + d*x])^(3/2)/(a + I*a*Tan[c + d*x])^2, x, 6, (10*(e*Cos[c + d*x])^(3/2)*EllipticF[(1/2)*(c + d*x), 2])/(33*a^2*d*Cos[c + d*x]^(3/2)) + (2*Cos[c + d*x]*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*a^2*d) + (10*(e*Cos[c + d*x])^(3/2)*Tan[c + d*x])/(33*a^2*d) + (4*I*Cos[c + d*x]^2*(e*Cos[c + d*x])^(3/2))/(11*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Sqrt[e*Cos[c + d*x]]/(a + I*a*Tan[c + d*x])^2, x, 5, (2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(3*a^2*d*Sqrt[Cos[c + d*x]]) + (2*I*Sqrt[e*Cos[c + d*x]])/(9*d*(a + I*a*Tan[c + d*x])^2) + (2*I*Sqrt[e*Cos[c + d*x]])/(9*d*(a^2 + I*a^2*Tan[c + d*x])), (2*Sqrt[e*Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2])/(3*a^2*d*Sqrt[Cos[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(9*a^2*d) + (4*I*Cos[c + d*x]^2*Sqrt[e*Cos[c + d*x]])/(9*d*(a^2 + I*a^2*Tan[c + d*x]))} -{1/(Sqrt[e*Cos[c + d*x]]*(a + I*a*Tan[c + d*x])^2), x, 5, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) + (2*I)/(7*d*Sqrt[e*Cos[c + d*x]]*(a + I*a*Tan[c + d*x])^2) + (2*I)/(7*d*Sqrt[e*Cos[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x])), (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) + (4*I*Cos[c + d*x]^2)/(7*d*Sqrt[e*Cos[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Cos[c + d*x])^(3/2)*(a + I*a*Tan[c + d*x])^2), x, 4, (2*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*(e*Cos[c + d*x])^(3/2)) + (((4*I)/5)*Cos[c + d*x]^2)/(d*(e*Cos[c + d*x])^(3/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Cos[c + d*x])^(5/2)*(a + I*a*Tan[c + d*x])^2), x, 4, (-2*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d*(e*Cos[c + d*x])^(5/2)) + (((4*I)/3)*Cos[c + d*x]^2)/(d*(e*Cos[c + d*x])^(5/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Cos[c + d*x])^(7/2)*(a + I*a*Tan[c + d*x])^2), x, 5, (6*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2])/(a^2*d*(e*Cos[c + d*x])^(7/2)) - (6*Cos[c + d*x]^3*Sin[c + d*x])/(a^2*d*(e*Cos[c + d*x])^(7/2)) + ((4*I)*Cos[c + d*x]^2)/(d*(e*Cos[c + d*x])^(7/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Cos[c + d*x])^(9/2)*(a + I*a*Tan[c + d*x])^2), x, 5, (10*Cos[c + d*x]^(9/2)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d*(e*Cos[c + d*x])^(9/2)) + (10*Cos[c + d*x]^3*Sin[c + d*x])/(3*a^2*d*(e*Cos[c + d*x])^(9/2)) - (4*I*Cos[c + d*x]^2)/(d*(e*Cos[c + d*x])^(9/2)*(a^2 + I*a^2*Tan[c + d*x]))} -{1/((e*Cos[c + d*x])^(11/2)*(a + I*a*Tan[c + d*x])^2), x, 6, -((14*Cos[c + d*x]^(11/2)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d*(e*Cos[c + d*x])^(11/2))) + (14*Cos[c + d*x]^3*Sin[c + d*x])/(15*a^2*d*(e*Cos[c + d*x])^(11/2)) + (14*Cos[c + d*x]^5*Sin[c + d*x])/(5*a^2*d*(e*Cos[c + d*x])^(11/2)) - (4*I*Cos[c + d*x]^2)/(3*d*(e*Cos[c + d*x])^(11/2)*(a^2 + I*a^2*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(m/2) (a+a I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 5, (12*I*a*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2)/(35*d*Sqrt[a + I*a*Tan[c + d*x]]) + (32*I*a*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^4)/(35*d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d) - (16*I*(e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*d)} -{(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 4, (8*I*a*(e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2)/(15*d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) - (16*I*(e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(15*d)} -{(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (4*I*a*e*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x])/(3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d), (((4*I)/3)*a*(e*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/3)*(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]], x, 2, ((-2*I)*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[e*Cos[c + d*x]], x, 10, (I*Sqrt[2]*Sqrt[a]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(d*Sqrt[e]) - (I*Sqrt[2]*Sqrt[a]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(d*Sqrt[e]) - (I*Sqrt[a]*Log[a*Sqrt[e] - Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]] + Sqrt[e]*Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*Sqrt[e]) + (I*Sqrt[a]*Log[a*Sqrt[e] + Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]] + Sqrt[e]*Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(Sqrt[2]*d*Sqrt[e])} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(3/2), x, 13, (I*a)/(d*(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])]*Sec[c + d*x])/(Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])]*Sec[c + d*x])/(Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), (I*a)/(d*(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^(3/2)*e^(3/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*a^(3/2)*e^(3/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(2*Sqrt[2]*d*(e*Cos[c + d*x])^(3/2)*(e*Sec[c + d*x])^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(5/2), x, 13, (3*I*Sqrt[a]*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(4*Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (3*I*Sqrt[a]*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(4*Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (3*I*Sqrt[a]*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) + (3*I*Sqrt[a]*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) + (I*a)/(2*d*(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (3*I*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*(e*Cos[c + d*x])^(5/2))} -{Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(7/2), x, 15, (I*a)/(3*d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a*Cos[c + d*x]^2)/(8*d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(3/2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a^(3/2)*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (5*I*a^(3/2)*e^(7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(16*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*a^(3/2)*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(16*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (5*I*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*d*(e*Cos[c + d*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Cos[c + d*x])^(5/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 5, (2*I*(e*Cos[c + d*x])^(5/2))/(7*d*Sqrt[a + I*a*Tan[c + d*x]]) + (16*I*(e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2)/(35*d*Sqrt[a + I*a*Tan[c + d*x]]) - (12*I*(e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d) - (32*I*(e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d)} -{(e*Cos[c + d*x])^(3/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (2*I*(e*Cos[c + d*x])^(3/2))/(5*d*Sqrt[a + I*a*Tan[c + d*x]]) + (16*I*(e*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2)/(15*d*Sqrt[a + I*a*Tan[c + d*x]]) - (8*I*(e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d)} -{Sqrt[e*Cos[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]], x, 3, (2*I*Sqrt[e*Cos[c + d*x]])/(3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (4*I*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d)} -{1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), x, 2, (2*I)/(d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 11, -((I*Sqrt[2]*Sqrt[a]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])]*Sec[c + d*x])/(d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])) + (I*Sqrt[2]*Sqrt[a]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e])]*Sec[c + d*x])/(d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (I*Sqrt[a]*Log[a*Sqrt[e] - Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]] + Sqrt[e]*Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*Sqrt[a]*Log[a*Sqrt[e] + Sqrt[2]*Sqrt[a]*Sqrt[e*Cos[c + d*x]]*Sqrt[a - I*a*Tan[c + d*x]] + Sqrt[e]*Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*e^(3/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 12, (I*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (I*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])])/(Sqrt[2]*Sqrt[a]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (I*e^(5/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*Sqrt[a]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) + (I*e^(5/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a + I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a + I*a*Tan[c + d*x])])/(2*Sqrt[2]*Sqrt[a]*d*(e*Cos[c + d*x])^(5/2)*(e*Sec[c + d*x])^(5/2)) - (I*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(a*d*(e*Cos[c + d*x])^(5/2))} -{1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 14, (3*I*Cos[c + d*x]^2)/(4*d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (3*I*Sqrt[a]*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(4*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*Sqrt[a]*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(4*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (3*I*Sqrt[a]*e^(7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (3*I*Sqrt[a]*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(8*Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (I*Cos[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d*(e*Cos[c + d*x])^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+a I Tan[e+f x])^n with m symbolic*) - - -{(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^n, x, 5, -((I*2^(-(m/2) + n)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), (1/2)*(2 + m - 2*n), 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^((1/2)*(m - 2*n))*(a + I*a*Tan[c + d*x])^n)/(d*m))} - - -{(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^2, x, 5, -((I*2^(2 - m/2)*a^2*(e*Cos[c + d*x])^m*Hypergeometric2F1[(1/2)*(-2 + m), -(m/2), 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(m/2))/(d*m))} -{(e*Cos[c + d*x])^m*(a + I*a*Tan[c + d*x])^1, x, 5, -((I*2^(1 - m/2)*a*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), m/2, 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(m/2))/(d*m))} -{(e*Cos[c + d*x])^m/(a + I*a*Tan[c + d*x])^1, x, 5, -((I*2^(-1 - m/2)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), (4 + m)/2, 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(m/2))/(a*d*m))} -{(e*Cos[c + d*x])^m/(a + I*a*Tan[c + d*x])^2, x, 5, -((I*2^(-2 - m/2)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), (6 + m)/2, 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^(m/2))/(a^2*d*m))} - - -{(e*Cos[c + d*x])^m*Sqrt[a + I*a*Tan[c + d*x]], x, 5, -((I*2^(1/2 - m/2)*a*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), (1 + m)/2, 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^((1 + m)/2))/(d*m*Sqrt[a + I*a*Tan[c + d*x]]))} -{(e*Cos[c + d*x])^m/Sqrt[a + I*a*Tan[c + d*x]], x, 5, -((I*2^(-(1/2) - m/2)*(e*Cos[c + d*x])^m*Hypergeometric2F1[-(m/2), (3 + m)/2, 1 - m/2, (1/2)*(1 - I*Tan[c + d*x])]*(1 + I*Tan[c + d*x])^((1 + m)/2))/(d*m*Sqrt[a + I*a*Tan[c + d*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n with m symbolic*) - - -{(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^3, x, 5, If[$VersionNumber>=8, -((a*(3*b^2 - a^2*(1 - m))*(d*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (2 + m)/2, 3/2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/(f*(1 - m))) + (b*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^2)/(f*(2 - m)) + (b*(d*Cos[e + f*x])^m*(2*(b^2 - a^2*(3 - m))*(1 - m) + a*b*(4 - m)*m*Tan[e + f*x]))/(f*m*(2 - 3*m + m^2)), -((1/(f*(1 - m)))*(a*(3*b^2 - a^2*(1 - m))*(d*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (2 + m)/2, 3/2, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])) + (b*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^2)/(f*(2 - m)) + (b*(d*Cos[e + f*x])^m*(2*(b^2 - a^2*(3 - m))*(1 - m) + a*b*(4 - m)*m*Tan[e + f*x]))/(f*(1 - m)*(2 - m)*m)]} -{(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^2, x, 5, If[$VersionNumber>=8, -((a*b*(2 - m)*(d*Cos[e + f*x])^m)/(f*(1 - m)*m)) + ((b^2 - a^2*(1 - m))*Cos[e + f*x]*(d*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - m)*(1 + m)*Sqrt[Sin[e + f*x]^2]) + (b*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(1 - m)), -((a*b*(2 - m)*(d*Cos[e + f*x])^m)/(f*(1 - m)*m)) + ((b^2 - a^2*(1 - m))*Cos[e + f*x]*(d*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - m^2)*Sqrt[Sin[e + f*x]^2]) + (b*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]))/(f*(1 - m))]} -{(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^1, x, 4, -((b*(d*Cos[e + f*x])^m)/(f*m)) - (a*(d*Cos[e + f*x])^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*(1 + m)*Sqrt[Sin[e + f*x]^2]), -((b*(d*Cos[e + f*x])^m)/(f*m)) - (a*Cos[e + f*x]*(d*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 + m)*Sqrt[Sin[e + f*x]^2])} -{(d*Cos[e + f*x])^m/(a + b*Tan[e + f*x])^1, x, 7, (b*(d*Cos[e + f*x])^m*Hypergeometric2F1[1, -(m/2), 1 - m/2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])/((a^2 + b^2)*f*m) + (AppellF1[1/2, 1, (2 + m)/2, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/(a*f)} -{(d*Cos[e + f*x])^m/(a + b*Tan[e + f*x])^2, x, 8, (2*a*b*(d*Cos[e + f*x])^m*Hypergeometric2F1[2, -(m/2), 1 - m/2, (b^2*Sec[e + f*x]^2)/(a^2 + b^2)])/((a^2 + b^2)^2*f*m) + (AppellF1[1/2, 2, (2 + m)/2, 3/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x])/(a^2*f) + (b^2*AppellF1[3/2, 2, (2 + m)/2, 5/2, (b^2*Tan[e + f*x]^2)/a^2, -Tan[e + f*x]^2]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]^3)/(3*a^4*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x])^n with n symbolic*) - - -{(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^n, x, 4, (AppellF1[1 + n, (2 + m)/2, (2 + m)/2, 2 + n, (a + b*Tan[e + f*x])/(a - Sqrt[-b^2]), (a + b*Tan[e + f*x])/(a + Sqrt[-b^2])]*Cos[e + f*x]^2*(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x])^(1 + n)*(1 - (a + b*Tan[e + f*x])/(a - Sqrt[-b^2]))^((2 + m)/2)*(1 - (a + b*Tan[e + f*x])/(a + Sqrt[-b^2]))^((2 + m)/2))/(b*f*(1 + n))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m deleted file mode 100644 index 6dc3fb4..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.1.3 (d sin)^m (a+b tan)^n.m +++ /dev/null @@ -1,203 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+a I Tan[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^4/(I + Tan[x]), x, 5, -((I*x)/16) - 1/(32*(I - Tan[x])^2) - I/(8*(I - Tan[x])) + I/(24*(I + Tan[x])^3) - 5/(32*(I + Tan[x])^2) - (3*I)/(16*(I + Tan[x]))} -{Sin[x]^3/(I + Tan[x]), x, 9, (1/3)*I*Cos[x]^3 - (1/5)*I*Cos[x]^5 + Sin[x]^5/5} -{Sin[x]^2/(I + Tan[x]), x, 5, -((I*x)/8) - I/(8*(I - Tan[x])) - 1/(8*(I + Tan[x])^2) - I/(4*(I + Tan[x]))} -{Sin[x]^1/(I + Tan[x]), x, 8, (1/3)*I*Cos[x]^3 + Sin[x]^3/3} -{Csc[x]^1/(I + Tan[x]), x, 8, I*ArcTanh[Cos[x]] - I*Cos[x] + Sin[x]} -{Csc[x]^2/(I + Tan[x]), x, 3, I*x + I*Cot[x] + Log[Cos[x]] + Log[Tan[x]]} -{Csc[x]^3/(I + Tan[x]), x, 8, (-(1/2))*I*ArcTanh[Cos[x]] - Csc[x] + (1/2)*I*Cot[x]*Csc[x]} -{Csc[x]^4/(I + Tan[x]), x, 4, (-(1/2))*Cot[x]^2 + (1/3)*I*Cot[x]^3} -{Csc[x]^5/(I + Tan[x]), x, 9, (-(1/8))*I*ArcTanh[Cos[x]] - (1/8)*I*Cot[x]*Csc[x] - Csc[x]^3/3 + (1/4)*I*Cot[x]*Csc[x]^3} -{Csc[x]^6/(I + Tan[x]), x, 4, (-(1/2))*Cot[x]^2 + (1/3)*I*Cot[x]^3 - Cot[x]^4/4 + (1/5)*I*Cot[x]^5} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[c + d*x]^5*(a + b*Tan[c + d*x]), x, 8, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d + (2*a*Cos[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]^5)/(5*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) - (b*Sin[c + d*x]^5)/(5*d)} -{Sin[c + d*x]^4*(a + b*Tan[c + d*x]), x, 6, (3*a*x)/8 - (b*Log[Cos[c + d*x]])/d - (Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Tan[c + d*x]))/(4*d) - (Cos[c + d*x]*Sin[c + d*x]*(3*a + 4*b*Tan[c + d*x]))/(8*d)} -{Sin[c + d*x]^3*(a + b*Tan[c + d*x]), x, 8, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d)} -{Sin[c + d*x]^2*(a + b*Tan[c + d*x]), x, 5, (a*x)/2 - (b*Log[Cos[c + d*x]])/d - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x]))/(2*d)} -{Sin[c + d*x]^1*(a + b*Tan[c + d*x]), x, 6, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c + d*x])/d - (b*Sin[c + d*x])/d} -{Csc[c + d*x]^1*(a + b*Tan[c + d*x]), x, 4, -((a*ArcTanh[Cos[c + d*x]])/d) + (b*ArcTanh[Sin[c + d*x]])/d} -{Csc[c + d*x]^2*(a + b*Tan[c + d*x]), x, 3, -((a*Cot[c + d*x])/d) + (b*Log[Tan[c + d*x]])/d} -{Csc[c + d*x]^3*(a + b*Tan[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/(2*d)) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{Csc[c + d*x]^4*(a + b*Tan[c + d*x]), x, 3, -((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d) + (b*Log[Tan[c + d*x]])/d} -{Csc[c + d*x]^5*(a + b*Tan[c + d*x]), x, 9, -((3*a*ArcTanh[Cos[c + d*x]])/(8*d)) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (b*Csc[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{Csc[c + d*x]^6*(a + b*Tan[c + d*x]), x, 3, -((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/d - (2*a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d) + (b*Log[Tan[c + d*x]])/d} - - -{Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 8, (3/8)*(a^2 - 5*b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (Cos[c + d*x]^2*(7*b - 5*a*Tan[c + d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(4*d)} -{Sin[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 11, (2*a*b*ArcTanh[Sin[c + d*x]])/d - (a^2*Cos[c + d*x])/d + (2*b^2*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) - (b^2*Cos[c + d*x]^3)/(3*d) + (b^2*Sec[c + d*x])/d - (2*a*b*Sin[c + d*x])/d - (2*a*b*Sin[c + d*x]^3)/(3*d)} -{Sin[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 6, (1/2)*(a^2 - 3*b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (3*b^2*Tan[c + d*x])/(2*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(2*d)} -{Sin[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 9, (2*a*b*ArcTanh[Sin[c + d*x]])/d - (a^2*Cos[c + d*x])/d + (b^2*Cos[c + d*x])/d + (b^2*Sec[c + d*x])/d - (2*a*b*Sin[c + d*x])/d} -{Csc[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 6, -((a^2*ArcTanh[Cos[c + d*x]])/d) + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (b^2*Sec[c + d*x])/d} -{Csc[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 3, -((a^2*Cot[c + d*x])/d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 10, -((a^2*ArcTanh[Cos[c + d*x]])/(2*d)) - (b^2*ArcTanh[Cos[c + d*x]])/d + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Csc[c + d*x])/d - (a^2*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (b^2*Sec[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 3, -(((a^2 + b^2)*Cot[c + d*x])/d) - (a*b*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^5*(a + b*Tan[c + d*x])^2, x, 13, -((3*a^2*ArcTanh[Cos[c + d*x]])/(8*d)) - (3*b^2*ArcTanh[Cos[c + d*x]])/(2*d) + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Csc[c + d*x])/d - (3*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (2*a*b*Csc[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (3*b^2*Sec[c + d*x])/(2*d) - (b^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d)} -{Csc[c + d*x]^6*(a + b*Tan[c + d*x])^2, x, 3, -(((a^2 + 2*b^2)*Cot[c + d*x])/d) - (2*a*b*Cot[c + d*x]^2)/d - ((2*a^2 + b^2)*Cot[c + d*x]^3)/(3*d) - (a*b*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (2*a*b*Log[Tan[c + d*x]])/d + (b^2*Tan[c + d*x])/d} - - -{Sin[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 16, (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cos[c + d*x])/d + (6*a*b^2*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^3)/(3*d) - (a*b^2*Cos[c + d*x]^3)/d + (3*a*b^2*Sec[c + d*x])/d - (3*a^2*b*Sin[c + d*x])/d + (5*b^3*Sin[c + d*x])/(2*d) - (a^2*b*Sin[c + d*x]^3)/d + (5*b^3*Sin[c + d*x]^3)/(6*d) + (b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(2*d)} -{Sin[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 7, (1/2)*a*(a^2 - 9*b^2)*x - (b*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (9*a*b^2*Tan[c + d*x])/(2*d) + (b^3*Tan[c + d*x]^2)/d - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(2*d)} -{Sin[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 13, (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cos[c + d*x])/d + (3*a*b^2*Cos[c + d*x])/d + (3*a*b^2*Sec[c + d*x])/d - (3*a^2*b*Sin[c + d*x])/d + (3*b^3*Sin[c + d*x])/(2*d) + (b^3*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 8, -((a^3*ArcTanh[Cos[c + d*x]])/d) + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (b^3*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 3, -((a^3*Cot[c + d*x])/d) + (3*a^2*b*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 12, -((a^3*ArcTanh[Cos[c + d*x]])/(2*d)) - (3*a*b^2*ArcTanh[Cos[c + d*x]])/d + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Csc[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^4*(a + b*Tan[c + d*x])^3, x, 3, -((a*(a^2 + 3*b^2)*Cot[c + d*x])/d) - (3*a^2*b*Cot[c + d*x]^2)/(2*d) - (a^3*Cot[c + d*x]^3)/(3*d) + (b*(3*a^2 + b^2)*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^5*(a + b*Tan[c + d*x])^3, x, 17, -((3*a^3*ArcTanh[Cos[c + d*x]])/(8*d)) - (9*a*b^2*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Csc[c + d*x])/d - (3*b^3*Csc[c + d*x])/(2*d) - (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*b*Csc[c + d*x]^3)/d - (a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (9*a*b^2*Sec[c + d*x])/(2*d) - (3*a*b^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (b^3*Csc[c + d*x]*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^6*(a + b*Tan[c + d*x])^3, x, 3, -((a*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (b*(6*a^2 + b^2)*Cot[c + d*x]^2)/(2*d) - (a*(2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - (3*a^2*b*Cot[c + d*x]^4)/(4*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Log[Tan[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b^3*Tan[c + d*x]^2)/(2*d)} - - -{Sin[c + d*x]^3*(a + b*Tan[c + d*x])^4, x, 19, (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (12*a^2*b^2*Cos[c + d*x])/d - (3*b^4*Cos[c + d*x])/d + (a^4*Cos[c + d*x]^3)/(3*d) - (2*a^2*b^2*Cos[c + d*x]^3)/d + (b^4*Cos[c + d*x]^3)/(3*d) + (6*a^2*b^2*Sec[c + d*x])/d - (3*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*b*Sin[c + d*x])/d + (10*a*b^3*Sin[c + d*x])/d - (4*a^3*b*Sin[c + d*x]^3)/(3*d) + (10*a*b^3*Sin[c + d*x]^3)/(3*d) + (2*a*b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/d} -{Sin[c + d*x]^2*(a + b*Tan[c + d*x])^4, x, 7, (1/2)*(a^4 - 18*a^2*b^2 + 5*b^4)*x - (4*a*b*(a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (b^2*(18*a^2 - 5*b^2)*Tan[c + d*x])/(2*d) + (4*a*b^3*Tan[c + d*x]^2)/d + (5*b^4*Tan[c + d*x]^3)/(6*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan[c + d*x])^4)/(2*d)} -{Sin[c + d*x]^1*(a + b*Tan[c + d*x])^4, x, 16, (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (6*a*b^3*ArcTanh[Sin[c + d*x]])/d - (a^4*Cos[c + d*x])/d + (6*a^2*b^2*Cos[c + d*x])/d - (b^4*Cos[c + d*x])/d + (6*a^2*b^2*Sec[c + d*x])/d - (2*b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) - (4*a^3*b*Sin[c + d*x])/d + (6*a*b^3*Sin[c + d*x])/d + (2*a*b^3*Sin[c + d*x]*Tan[c + d*x]^2)/d} -{Csc[c + d*x]^1*(a + b*Tan[c + d*x])^4, x, 10, -((a^4*ArcTanh[Cos[c + d*x]])/d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b^3*ArcTanh[Sin[c + d*x]])/d + (6*a^2*b^2*Sec[c + d*x])/d - (b^4*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) + (2*a*b^3*Sec[c + d*x]*Tan[c + d*x])/d} -{Csc[c + d*x]^2*(a + b*Tan[c + d*x])^4, x, 3, -((a^4*Cot[c + d*x])/d) + (4*a^3*b*Log[Tan[c + d*x]])/d + (6*a^2*b^2*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^3*(a + b*Tan[c + d*x])^4, x, 14, -((a^4*ArcTanh[Cos[c + d*x]])/(2*d)) - (6*a^2*b^2*ArcTanh[Cos[c + d*x]])/d + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (2*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (a^4*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (6*a^2*b^2*Sec[c + d*x])/d + (b^4*Sec[c + d*x]^3)/(3*d) + (2*a*b^3*Sec[c + d*x]*Tan[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Tan[c + d*x])^4, x, 3, -((a^2*(a^2 + 6*b^2)*Cot[c + d*x])/d) - (2*a^3*b*Cot[c + d*x]^2)/d - (a^4*Cot[c + d*x]^3)/(3*d) + (4*a*b*(a^2 + b^2)*Log[Tan[c + d*x]])/d + (b^2*(6*a^2 + b^2)*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4, x, 21, -((3*a^4*ArcTanh[Cos[c + d*x]])/(8*d)) - (9*a^2*b^2*ArcTanh[Cos[c + d*x]])/d - (b^4*ArcTanh[Cos[c + d*x]])/d + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (6*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (6*a*b^3*Csc[c + d*x])/d - (3*a^4*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (9*a^2*b^2*Sec[c + d*x])/d + (b^4*Sec[c + d*x])/d - (3*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*x])/d + (2*a*b^3*Csc[c + d*x]*Sec[c + d*x]^2)/d + (b^4*Sec[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^6*(a + b*Tan[c + d*x])^4, x, 3, -(((a^4 + 12*a^2*b^2 + b^4)*Cot[c + d*x])/d) - (2*a*b*(2*a^2 + b^2)*Cot[c + d*x]^2)/d - (2*a^2*(a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - (a^3*b*Cot[c + d*x]^4)/d - (a^4*Cot[c + d*x]^5)/(5*d) + (4*a*b*(a^2 + 2*b^2)*Log[Tan[c + d*x]])/d + (2*b^2*(3*a^2 + b^2)*Tan[c + d*x])/d + (2*a*b^3*Tan[c + d*x]^2)/d + (b^4*Tan[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^7*(a + b*Tan[c + d*x])^4, x, 25, -((5*a^4*ArcTanh[Cos[c + d*x]])/(16*d)) - (45*a^2*b^2*ArcTanh[Cos[c + d*x]])/(4*d) - (5*b^4*ArcTanh[Cos[c + d*x]])/(2*d) + (4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (10*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (10*a*b^3*Csc[c + d*x])/d - (5*a^4*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (10*a*b^3*Csc[c + d*x]^3)/(3*d) - (5*a^4*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (4*a^3*b*Csc[c + d*x]^5)/(5*d) - (a^4*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (45*a^2*b^2*Sec[c + d*x])/(4*d) + (5*b^4*Sec[c + d*x])/(2*d) - (15*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*x])/(4*d) - (3*a^2*b^2*Csc[c + d*x]^4*Sec[c + d*x])/(2*d) + (2*a*b^3*Csc[c + d*x]^3*Sec[c + d*x]^2)/d + (5*b^4*Sec[c + d*x]^3)/(6*d) - (b^4*Csc[c + d*x]^2*Sec[c + d*x]^3)/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[c + d*x]^5/(a + b*Tan[c + d*x]), x, 13, (a^5*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d) + (a^3*b^2*Cos[c + d*x])/((a^2 + b^2)^3*d) + (a*b^2*Cos[c + d*x])/((a^2 + b^2)^2*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) - (a*b^2*Cos[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (2*a*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) - (a*Cos[c + d*x]^5)/(5*(a^2 + b^2)*d) + (a^4*b*Sin[c + d*x])/((a^2 + b^2)^3*d) + (a^2*b*Sin[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (b*Sin[c + d*x]^5)/(5*(a^2 + b^2)*d)} -{Sin[c + d*x]^4/(a + b*Tan[c + d*x]), x, 8, (a*(3*a^4 - 6*a^2*b^2 - b^4)*x)/(8*(a^2 + b^2)^3) + (a^4*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x]))/(4*(a^2 + b^2)*d) - (Cos[c + d*x]^2*(4*b*(2*a^2 + b^2) + a*(5*a^2 + b^2)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)} -{Sin[c + d*x]^3/(a + b*Tan[c + d*x]), x, 10, (a^3*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d) + (a*b^2*Cos[c + d*x])/((a^2 + b^2)^2*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) + (a*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a^2*b*Sin[c + d*x])/((a^2 + b^2)^2*d) + (b*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d)} -{Sin[c + d*x]^2/(a + b*Tan[c + d*x]), x, 7, (a*(a^2 - b^2)*x)/(2*(a^2 + b^2)^2) + (a^2*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(b + a*Tan[c + d*x]))/(2*(a^2 + b^2)*d)} -{Sin[c + d*x]^1/(a + b*Tan[c + d*x]), x, 6, (a*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) - (a*Cos[c + d*x])/((a^2 + b^2)*d) + (b*Sin[c + d*x])/((a^2 + b^2)*d)} -{Csc[c + d*x]^1/(a + b*Tan[c + d*x]), x, 6, -(ArcTanh[Cos[c + d*x]]/(a*d)) + (b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)} -{Csc[c + d*x]^2/(a + b*Tan[c + d*x]), x, 3, -(Cot[c + d*x]/(a*d)) - (b*Log[Tan[c + d*x]])/(a^2*d) + (b*Log[a + b*Tan[c + d*x]])/(a^2*d)} -{Csc[c + d*x]^3/(a + b*Tan[c + d*x]), x, 15, -(ArcTanh[Cos[c + d*x]]/(2*a*d)) - (b^2*ArcTanh[Cos[c + d*x]])/(a^3*d) + (b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^3*d) + (b*Csc[c + d*x])/(a^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d)} -{Csc[c + d*x]^4/(a + b*Tan[c + d*x]), x, 3, -(((a^2 + b^2)*Cot[c + d*x])/(a^3*d)) + (b*Cot[c + d*x]^2)/(2*a^2*d) - Cot[c + d*x]^3/(3*a*d) - (b*(a^2 + b^2)*Log[Tan[c + d*x]])/(a^4*d) + (b*(a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a^4*d)} -(* {Csc[c + d*x]^5/(a + b*Tan[c + d*x]), x, 25, -((3*ArcTanh[Cos[c + d*x]])/(8*a*d)) - (3*b^2*ArcTanh[Cos[c + d*x]])/(2*a^3*d) - (b^4*ArcTanh[Cos[c + d*x]])/(a^5*d) - (b*ArcTanh[Sin[c + d*x]])/(a^2*d) - (b^3*ArcTanh[Sin[c + d*x]])/(a^4*d) + (b*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(a^4*d) + (b*(a^2 + b^2)^(3/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^5*d) + (b*Csc[c + d*x])/(a^2*d) + (3*b^3*Csc[c + d*x])/(2*a^4*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (b*Csc[c + d*x]^3)/(3*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d) + (3*b^2*Sec[c + d*x])/(2*a^3*d) + (b^4*Sec[c + d*x])/(a^5*d) - (b^2*(a^2 + b^2)*Sec[c + d*x])/(a^5*d) - (b^2*Csc[c + d*x]^2*Sec[c + d*x])/(2*a^3*d) - (b^3*Csc[c + d*x]*Sec[c + d*x]^2)/(2*a^4*d) + (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d)} *) -{Csc[c + d*x]^6/(a + b*Tan[c + d*x]), x, 3, -(((a^2 + b^2)^2*Cot[c + d*x])/(a^5*d)) + (b*(2*a^2 + b^2)*Cot[c + d*x]^2)/(2*a^4*d) - ((2*a^2 + b^2)*Cot[c + d*x]^3)/(3*a^3*d) + (b*Cot[c + d*x]^4)/(4*a^2*d) - Cot[c + d*x]^5/(5*a*d) - (b*(a^2 + b^2)^2*Log[Tan[c + d*x]])/(a^6*d) + (b*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])/(a^6*d)} - - -{Sin[c + d*x]^6/(a + b*Tan[c + d*x])^2, x, 9, ((5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*x)/(16*(a^2 + b^2)^5) + (2*a^5*b*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^6*b)/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^6*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^2*d) + (Cos[c + d*x]^4*(12*a*b*(3*a^2 + b^2) + (13*a^4 - 18*a^2*b^2 - 7*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^3*d) - (Cos[c + d*x]^2*(48*a^5*b + (11*a^6 - 43*a^4*b^2 - 7*a^2*b^4 - b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^4*d)} -{Sin[c + d*x]^4/(a + b*Tan[c + d*x])^2, x, 8, ((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*x)/(8*(a^2 + b^2)^4) + (2*a^3*b*(a^2 - 2*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^4*b)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(16*a^3*b + (5*a^4 - 12*a^2*b^2 - b^4)*Tan[c + d*x]))/(8*(a^2 + b^2)^3*d)} -{Sin[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 7, ((a^4 - 6*a^2*b^2 + b^4)*x)/(2*(a^2 + b^2)^3) + (2*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (a^2*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(2*(a^2 + b^2)^2*d)} -{Csc[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 3, -(Cot[c + d*x]/(a^2*d)) - (2*b*Log[Tan[c + d*x]])/(a^3*d) + (2*b*Log[a + b*Tan[c + d*x]])/(a^3*d) - b/(a^2*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^4/(a + b*Tan[c + d*x])^2, x, 3, -(((a^2 + 3*b^2)*Cot[c + d*x])/(a^4*d)) + (b*Cot[c + d*x]^2)/(a^3*d) - Cot[c + d*x]^3/(3*a^2*d) - (2*b*(a^2 + 2*b^2)*Log[Tan[c + d*x]])/(a^5*d) + (2*b*(a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]])/(a^5*d) - (b*(a^2 + b^2))/(a^4*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^6/(a + b*Tan[c + d*x])^2, x, 3, -(((a^2 + b^2)*(a^2 + 5*b^2)*Cot[c + d*x])/(a^6*d)) + (2*b*(a^2 + b^2)*Cot[c + d*x]^2)/(a^5*d) - ((2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*a^4*d) + (b*Cot[c + d*x]^4)/(2*a^3*d) - Cot[c + d*x]^5/(5*a^2*d) - (2*b*(a^2 + b^2)*(a^2 + 3*b^2)*Log[Tan[c + d*x]])/(a^7*d) + (2*b*(a^2 + b^2)*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^7*d) - (b*(a^2 + b^2)^2)/(a^6*d*(a + b*Tan[c + d*x]))} -(* -{Sin[c + d*x]^5/(a + b*Tan[c + d*x])^2, x, 0, (5*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(a*(a^2 + b^2) + 2*b*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])))/(64*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (1/(64*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2))*(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(6*b*(-3*a^2 + b^2)*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x]) - (a^2 + b^2)*(3*a*(a^2 - 3*b^2) + 2*a*(a^2 + b^2)*Cos[2*(c + d*x)] - 2*b*(a^2 + b^2)*Sin[2*(c + d*x)]))) - (3*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*((10*b*(5*a^4 - 10*a^2*b^2 + b^4)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (8*(a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x])/(a^2 + b^2)^3 - (4*(a^2 - b^2)*Cos[3*(c + d*x)])/(3*(a^2 + b^2)^2) - (32*a*b*(a^2 - b^2)*Sin[c + d*x])/(a^2 + b^2)^3 + (a*(a^4 - 10*a^2*b^2 + 5*b^4))/((a^2 + b^2)^3*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (8*a*b*Sin[3*(c + d*x)])/(3*(a^2 + b^2)^2)))/(64*d*(a + b*Tan[c + d*x])^2) + (1/(64*d*(a + b*Tan[c + d*x])^2))*(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*((14*b*(-7*a^6 + 35*a^4*b^2 - 21*a^2*b^4 + b^6)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(9/2) - (12*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Cos[c + d*x])/(a^2 + b^2)^4 + (8*(a^4 - 6*a^2*b^2 + b^4)*Cos[3*(c + d*x)])/(3*(a^2 + b^2)^3) - (4*(a^2 - b^2)*Cos[5*(c + d*x)])/(5*(a^2 + b^2)^2) + (24*a*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Sin[c + d*x])/(a^2 + b^2)^4 - (a*(a^6 - 21*a^4*b^2 + 35*a^2*b^4 - 7*b^6))/((a^2 + b^2)^4*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (32*a*b*(a^2 - b^2)*Sin[3*(c + d*x)])/(3*(a^2 + b^2)^3) + (8*a*b*Sin[5*(c + d*x)])/(5*(a^2 + b^2)^2)))} -{Sin[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 31, (2*a^4*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d) - (3*a^2*b^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d) + (4*a^2*b^2*Cos[c + d*x])/((a^2 + b^2)^3*d) - (a^2*Cos[c + d*x])/((a^2 + b^2)^2*d) + (a^2*Cos[c + d*x]^3)/(3*(a^2 + b^2)^2*d) - (b^2*Cos[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (2*a^3*b*Sin[c + d*x])/((a^2 + b^2)^3*d) - (2*a*b^3*Sin[c + d*x])/((a^2 + b^2)^3*d) + (2*a*b*Sin[c + d*x]^3)/(3*(a^2 + b^2)^2*d) + (a^3*b^2)/((a^2 + b^2)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sin[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 14, (2*a^2*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d) - (b^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d) - (a^2*Cos[c + d*x])/((a^2 + b^2)^2*d) + (b^2*Cos[c + d*x])/((a^2 + b^2)^2*d) + (2*a*b*Sin[c + d*x])/((a^2 + b^2)^2*d) + (a*b^2)/((a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Csc[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 9, -(ArcTanh[Cos[c + d*x]]/(a^2*d)) + (b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) + (b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]*d) + b^2/(a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Csc[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 17, -(ArcTanh[Cos[c + d*x]]/(2*a^2*d)) - (3*b^2*ArcTanh[Cos[c + d*x]])/(a^4*d) - (b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]*d) + (3*b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^4*d) + (2*b*Csc[c + d*x])/(a^3*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^2*d) + b^2/(a^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -*) - - -{Sin[c + d*x]^6/(a + b*Tan[c + d*x])^3, x, 9, (a*(5*a^8 - 180*a^6*b^2 + 390*a^4*b^4 - 68*a^2*b^6 - 3*b^8)*x)/(16*(a^2 + b^2)^6) + (a^4*b*(3*a^4 - 22*a^2*b^2 + 15*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^6*b)/(2*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^2) - (2*a^5*b*(a^2 - 3*b^2))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^6*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(6*b*(9*a^4 - 4*a^2*b^2 - b^4) + a*(13*a^4 - 62*a^2*b^2 - 3*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^4*d) - (a*Cos[c + d*x]^2*(24*a^3*b*(3*a^2 - 5*b^2) + (11*a^6 - 119*a^4*b^2 + 65*a^2*b^4 + 3*b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^5*d)} -{Sin[c + d*x]^4/(a + b*Tan[c + d*x])^3, x, 8, (3*a*(a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(8*(a^2 + b^2)^5) + (3*a^2*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^4*b)/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (2*a^3*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^3*d) - (a*Cos[c + d*x]^2*(24*a*b*(a^2 - b^2) + (5*a^4 - 34*a^2*b^2 + 9*b^4)*Tan[c + d*x]))/(8*(a^2 + b^2)^4*d)} -{Sin[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 7, (a*(a^4 - 14*a^2*b^2 + 9*b^4)*x)/(2*(a^2 + b^2)^4) + (b*(3*a^4 - 8*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^2*b)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a*b*(a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(2*(a^2 + b^2)^3*d)} -{Csc[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 3, -(Cot[c + d*x]/(a^3*d)) - (3*b*Log[Tan[c + d*x]])/(a^4*d) + (3*b*Log[a + b*Tan[c + d*x]])/(a^4*d) - b/(2*a^2*d*(a + b*Tan[c + d*x])^2) - (2*b)/(a^3*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^4/(a + b*Tan[c + d*x])^3, x, 3, -(((a^2 + 6*b^2)*Cot[c + d*x])/(a^5*d)) + (3*b*Cot[c + d*x]^2)/(2*a^4*d) - Cot[c + d*x]^3/(3*a^3*d) - (b*(3*a^2 + 10*b^2)*Log[Tan[c + d*x]])/(a^6*d) + (b*(3*a^2 + 10*b^2)*Log[a + b*Tan[c + d*x]])/(a^6*d) - (b*(a^2 + b^2))/(2*a^4*d*(a + b*Tan[c + d*x])^2) - (2*b*(a^2 + 2*b^2))/(a^5*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^6/(a + b*Tan[c + d*x])^3, x, 3, -(((a^4 + 12*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a^7*d)) + (b*(3*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^6*d) - (2*(a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*a^5*d) + (3*b*Cot[c + d*x]^4)/(4*a^4*d) - Cot[c + d*x]^5/(5*a^3*d) - (b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[Tan[c + d*x]])/(a^8*d) + (b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[a + b*Tan[c + d*x]])/(a^8*d) - (b*(a^2 + b^2)^2)/(2*a^6*d*(a + b*Tan[c + d*x])^2) - (2*b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*d*(a + b*Tan[c + d*x]))} -(* -{Sin[c + d*x]^5/(a + b*Tan[c + d*x])^3, x, 0, (3*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(6*a*b*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (a^2 + b^2)*(a*(2*a^2 - b^2)*Cos[c + d*x] + b*(a^2 - 2*b^2)*Sin[c + d*x])))/(64*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3) + (1/(64*d*(a + b*Tan[c + d*x])^3))*(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*(-((70*a*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(9/2)) - (24*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cos[c + d*x])/(a^2 + b^2)^4 + (8*a*(a^2 - 3*b^2)*Cos[3*(c + d*x)])/(3*(a^2 + b^2)^3) + (24*b*(5*a^4 - 10*a^2*b^2 + b^4)*Sin[c + d*x])/(a^2 + b^2)^4 - (b*(-3*a^4 + 10*a^2*b^2 - 3*b^4)*Sin[c + d*x])/((a^2 + b^2)^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (6*a^6 - 87*a^4*b^2 + 80*a^2*b^4 - 3*b^6)/((a^2 + b^2)^4*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (8*b*(-3*a^2 + b^2)*Sin[3*(c + d*x)])/(3*(a^2 + b^2)^3))) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3*((60*a*b*(a^2 - b^2)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (2*(5*a^3*(a^2 - 5*b^2)*Cos[c + d*x] + a*(a^2 + b^2)^2*Cos[3*(c + d*x)] - b*(-5*b^2*(-5*a^2 + b^2)*Sin[c + d*x] + (a^2 + b^2)^2*Sin[3*(c + d*x)])))/((a^2 + b^2)^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)))/(64*d*(a + b*Tan[c + d*x])^3) - (1/(320*(a^2 + b^2)^6*d*(a + b*Tan[c + d*x])^3))*(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(1260*a*b*Sqrt[a^2 + b^2]*(a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (a^2 + b^2)*(105*a*(a^8 - 26*a^6*b^2 + 56*a^4*b^4 - 14*a^2*b^6 - b^8)*Cos[c + d*x] + 21*a*(a^2 + b^2)^2*(a^4 - 10*a^2*b^2 + 5*b^4)*Cos[3*(c + d*x)] - 3*a^9*Cos[5*(c + d*x)] + 18*a^5*b^4*Cos[5*(c + d*x)] + 24*a^3*b^6*Cos[5*(c + d*x)] + 9*a*b^8*Cos[5*(c + d*x)] + a^9*Cos[7*(c + d*x)] + 4*a^7*b^2*Cos[7*(c + d*x)] + 6*a^5*b^4*Cos[7*(c + d*x)] + 4*a^3*b^6*Cos[7*(c + d*x)] + a*b^8*Cos[7*(c + d*x)] - 105*a^8*b*Sin[c + d*x] - 1470*a^6*b^3*Sin[c + d*x] + 5880*a^4*b^5*Sin[c + d*x] - 2730*a^2*b^7*Sin[c + d*x] + 105*b^9*Sin[c + d*x] - 105*a^8*b*Sin[3*(c + d*x)] + 294*a^4*b^5*Sin[3*(c + d*x)] + 168*a^2*b^7*Sin[3*(c + d*x)] - 21*b^9*Sin[3*(c + d*x)] + 9*a^8*b*Sin[5*(c + d*x)] + 24*a^6*b^3*Sin[5*(c + d*x)] + 18*a^4*b^5*Sin[5*(c + d*x)] - 3*b^9*Sin[5*(c + d*x)] - a^8*b*Sin[7*(c + d*x)] - 4*a^6*b^3*Sin[7*(c + d*x)] - 6*a^4*b^5*Sin[7*(c + d*x)] - 4*a^2*b^7*Sin[7*(c + d*x)] - b^9*Sin[7*(c + d*x)])))} -{Sin[c + d*x]^3/(a + b*Tan[c + d*x])^3, x, 99, (3*a^5*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(9/2)*d) - (23*a^3*b^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*(a^2 + b^2)^(9/2)*d) + (3*a*b^5*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(9/2)*d) + (9*a^3*b^2*Cos[c + d*x])/((a^2 + b^2)^4*d) - (3*a*b^4*Cos[c + d*x])/((a^2 + b^2)^4*d) - (a^3*Cos[c + d*x])/((a^2 + b^2)^3*d) + (a^3*Cos[c + d*x]^3)/(3*(a^2 + b^2)^3*d) - (a*b^2*Cos[c + d*x]^3)/((a^2 + b^2)^3*d) + (3*a^4*b*Sin[c + d*x])/((a^2 + b^2)^4*d) - (9*a^2*b^3*Sin[c + d*x])/((a^2 + b^2)^4*d) + (b^3*Sin[c + d*x])/((a^2 + b^2)^3*d) + (a^2*b*Sin[c + d*x]^3)/((a^2 + b^2)^3*d) - (b^3*Sin[c + d*x]^3)/(3*(a^2 + b^2)^3*d) + (a^3*b^3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*(a^2 + b^2)^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (3*a^4*b^2)/((a^2 + b^2)^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (3*a^2*b^4)/((a^2 + b^2)^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sin[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 0, -((1/(4*(a^2 + b^2)^(7/2)*d*(a + b*Tan[c + d*x])^3))*(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(12*a*b*(2*a^2 - 3*b^2)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + Sqrt[a^2 + b^2]*(a*(3*a^4 - 26*a^2*b^2 + b^4)*Cos[c + d*x] + a*(a^2 + b^2)^2*Cos[3*(c + d*x)] - 2*b*(a^4 + 13*a^2*b^2 - 3*b^4 + (a^2 + b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x]))))} -{Csc[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 0, (1/(2*a^3*d*(a + b*Tan[c + d*x])^3))*(Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*((3*a*b^2*(2*a^2 + b^2)*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(a^2 + b^2)^2 - (2*b*(6*a^4 + 5*a^2*b^2 + 2*b^4)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]]*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^2 + b^2)^(5/2) - 2*Log[Cos[(1/2)*(c + d*x)]]*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 2*Log[Sin[(1/2)*(c + d*x)]]*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - (a*b^3*Tan[c + d*x])/(a^2 + b^2)))} -{Csc[c + d*x]^3/(a + b*Tan[c + d*x])^3, x, 20, -(ArcTanh[Cos[c + d*x]]/(2*a^3*d)) - (6*b^2*ArcTanh[Cos[c + d*x]])/(a^5*d) + (b^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*a^3*(a^2 + b^2)^(3/2)*d) - (3*b*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]*d) + (6*b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(a^5*d) + (3*b*Csc[c + d*x])/(a^4*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*a^3*d) + (b^3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*a^3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (3*b^2)/(a^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -*) - - -{Sin[c + d*x]^4/(a + b*Tan[c + d*x])^4, x, 8, ((3*a^8 - 132*a^6*b^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*x)/(8*(a^2 + b^2)^6) + (4*a*b*(a^2 - b^2)*(a^4 - 8*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^4*b)/(3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^3) - (a^3*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^2) - (3*a^2*b*(a^4 - 5*a^2*b^2 + 2*b^4))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(4*(a^2 + b^2)^4*d) - (Cos[c + d*x]^2*(16*a*b*(2*a^4 - 5*a^2*b^2 + b^4) + (5*a^6 - 65*a^4*b^2 + 55*a^2*b^4 - 3*b^6)*Tan[c + d*x]))/(8*(a^2 + b^2)^5*d)} -{Sin[c + d*x]^2/(a + b*Tan[c + d*x])^4, x, 7, ((a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(2*(a^2 + b^2)^5) + (4*a*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^2*b)/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^3) - (a*b*(a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 - 8*a^2*b^2 + b^4))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(2*(a^2 + b^2)^4*d)} -{Csc[c + d*x]^2/(a + b*Tan[c + d*x])^4, x, 3, -(Cot[c + d*x]/(a^4*d)) - (4*b*Log[Tan[c + d*x]])/(a^5*d) + (4*b*Log[a + b*Tan[c + d*x]])/(a^5*d) - b/(3*a^2*d*(a + b*Tan[c + d*x])^3) - b/(a^3*d*(a + b*Tan[c + d*x])^2) - (3*b)/(a^4*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^4/(a + b*Tan[c + d*x])^4, x, 3, -(((a^2 + 10*b^2)*Cot[c + d*x])/(a^6*d)) + (2*b*Cot[c + d*x]^2)/(a^5*d) - Cot[c + d*x]^3/(3*a^4*d) - (4*b*(a^2 + 5*b^2)*Log[Tan[c + d*x]])/(a^7*d) + (4*b*(a^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/(a^7*d) - (b*(a^2 + b^2))/(3*a^4*d*(a + b*Tan[c + d*x])^3) - (b*(a^2 + 2*b^2))/(a^5*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 + 10*b^2))/(a^6*d*(a + b*Tan[c + d*x]))} -{Csc[c + d*x]^6/(a + b*Tan[c + d*x])^4, x, 3, -(((a^4 + 20*a^2*b^2 + 35*b^4)*Cot[c + d*x])/(a^8*d)) + (2*b*(2*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^7*d) - (2*(a^2 + 5*b^2)*Cot[c + d*x]^3)/(3*a^6*d) + (b*Cot[c + d*x]^4)/(a^5*d) - Cot[c + d*x]^5/(5*a^4*d) - (4*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[Tan[c + d*x]])/(a^9*d) + (4*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[a + b*Tan[c + d*x]])/(a^9*d) - (b*(a^2 + b^2)^2)/(3*a^6*d*(a + b*Tan[c + d*x])^3) - (b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 + 20*a^2*b^2 + 21*b^4))/(a^8*d*(a + b*Tan[c + d*x]))} - - -(* Hangs Mathematica 6 & 7: *) -{Csc[x]/(1 + Tan[x]), x, 6, -ArcTanh[Cos[x]] + ArcTanh[(Cos[x] - Sin[x])/Sqrt[2]]/Sqrt[2]} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (d Sin[e+f x])^(m/2) (a+b Tan[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (d Sin[e+f x])^(m/2) (a+b Tan[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x])^n with m symbolic*) - - -{Sin[c + d*x]^m*(a + b*Tan[c + d*x])^3, x, 8, (a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^2*b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m)) + (3*a*b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(3 + m))/(d*(3 + m)) + (b^3*Hypergeometric2F1[2, (4 + m)/2, (6 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(4 + m))/(d*(4 + m))} -{Sin[c + d*x]^m*(a + b*Tan[c + d*x])^2, x, 6, (a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a*b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m)) + (b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(3 + m))/(d*(3 + m))} -{Sin[c + d*x]^m*(a + b*Tan[c + d*x])^1, x, 5, (a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m))} -{Sin[c + d*x]^m/(a + b*Tan[c + d*x])^1, x, 14, (2^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*d*(1 + m)) + (2^(1 + m)*b*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2])*d*(2 + m)) - (2^(1 + m)*b*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d*(2 + m)) + (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(Sqrt[a^2 + b^2]*(b - Sqrt[a^2 + b^2])^2*d*(3 + m)) - (2^(1 + m)*a*b*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])^2*d*(3 + m))} -(* {Sin[c + d*x]^m/(a + b*Tan[c + d*x])^2, x, 52, -((2^(1 + m)*b^2*AppellF1[(1 + m)/2, 1 + m, 1, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b - Sqrt[a^2 + b^2])*d*(1 + m))) - (2^(2 + m)*b*(1 - (2*b)/Sqrt[a^2 + b^2])*AppellF1[(1 + m)/2, 1 + m, 1, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a^2*(b - Sqrt[a^2 + b^2])*d*(1 + m)) + (2^(1 + m)*b^2*AppellF1[(1 + m)/2, 1 + m, 1, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b + Sqrt[a^2 + b^2])*d*(1 + m)) - (2^(2 + m)*b*(1 + (2*b)/Sqrt[a^2 + b^2])*AppellF1[(1 + m)/2, 1 + m, 1, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a^2*(b + Sqrt[a^2 + b^2])*d*(1 + m)) + (2^(1 + m)*b^2*(b - Sqrt[a^2 + b^2])^2*AppellF1[(1 + m)/2, 1 + m, 2, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(1 + m)) + (2^(1 + m)*b^2*(b + Sqrt[a^2 + b^2])^2*AppellF1[(1 + m)/2, 1 + m, 2, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(1 + m)) + (2^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(1/2)*(c + d*x)]^2]*Tan[(1/2)*(c + d*x)]*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a^2*d*(1 + m)) - (2^(1 + m)*a*b^2*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b - Sqrt[a^2 + b^2])^2*d*(2 + m)) - (2^(2 + m)*b*(1 - (2*b)/Sqrt[a^2 + b^2])*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(b - Sqrt[a^2 + b^2])^2*d*(2 + m)) - (2^(2 + m)*b^3*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(a^2 + b^2)^(3/2)*(b - Sqrt[a^2 + b^2])*d*(2 + m)) + (2^(1 + m)*a*b^2*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b + Sqrt[a^2 + b^2])^2*d*(2 + m)) - (2^(2 + m)*b*(1 + (2*b)/Sqrt[a^2 + b^2])*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(b + Sqrt[a^2 + b^2])^2*d*(2 + m)) + (2^(2 + m)*b^3*AppellF1[(2 + m)/2, 1 + m, 1, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(a^2 + b^2)^(3/2)*(b + Sqrt[a^2 + b^2])*d*(2 + m)) + (2^(2 + m)*a*b^2*(b - Sqrt[a^2 + b^2])*AppellF1[(2 + m)/2, 1 + m, 2, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(2 + m)) + (2^(2 + m)*b^3*(b - Sqrt[a^2 + b^2])^2*AppellF1[(2 + m)/2, 1 + m, 2, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(2 + m)) + (2^(2 + m)*a*b^2*(b + Sqrt[a^2 + b^2])*AppellF1[(2 + m)/2, 1 + m, 2, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(2 + m)) + (2^(2 + m)*b^3*(b + Sqrt[a^2 + b^2])^2*AppellF1[(2 + m)/2, 1 + m, 2, (4 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^2*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/(a*(a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(2 + m)) - (2^(2 + m)*b^3*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b - Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b - Sqrt[a^2 + b^2])^2*d*(3 + m)) + (2^(2 + m)*b^3*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(b + Sqrt[a^2 + b^2])^2]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)^(3/2)*(b + Sqrt[a^2 + b^2])^2*d*(3 + m)) + (2^(1 + m)*a^2*b^2*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(3 + m)) + (2^(3 + m)*b^3*(b - Sqrt[a^2 + b^2])*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(3 + m)) + (2^(1 + m)*a^2*b^2*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(3 + m)) + (2^(3 + m)*b^3*(b + Sqrt[a^2 + b^2])*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^3*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(3 + m)) + (2^(2 + m)*a*b^3*AppellF1[(4 + m)/2, 1 + m, 2, (6 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^4*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b - Sqrt[a^2 + b^2]))^2*d*(4 + m)) + (2^(2 + m)*a*b^3*AppellF1[(4 + m)/2, 1 + m, 2, (6 + m)/2, -Tan[(1/2)*(c + d*x)]^2, (a^2*Tan[(1/2)*(c + d*x)]^2)/(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))]*Tan[(1/2)*(c + d*x)]^4*(Tan[(1/2)*(c + d*x)]/(1 + Tan[(1/2)*(c + d*x)]^2))^m*(1 + Tan[(1/2)*(c + d*x)]^2)^m)/((a^2 + b^2)*(a^2 + 2*b*(b + Sqrt[a^2 + b^2]))^2*d*(4 + m))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x])^n with n symbolic*) - - -{Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x, 0, CannotIntegrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x]} - - -{Sin[c + d*x]^4*(a + b*Tan[c + d*x])^n, x, 7, -(((a*b^2*n*(5*a^2 + b^2*(3 + 2*n)) + Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 + 6*n - n^2) + b^4*(3 + 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*b*(a^2 + b^2)^2*(a - Sqrt[-b^2])*d*(1 + n))) - ((a*b^2*n*(5*a^2 + b^2*(3 + 2*n)) - Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 + 6*n - n^2) + b^4*(3 + 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*b*(a^2 + b^2)^2*(a + Sqrt[-b^2])*d*(1 + n)) + (Cos[c + d*x]^4*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(4*(a^2 + b^2)*d) - (Cos[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)*(b*(a^2*(7 - n) + b^2*(5 + n)) + a*(5*a^2 + b^2*(3 + 2*n))*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)} -{Sin[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 6, -(((a*b^2*n + Sqrt[-b^2]*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(a - Sqrt[-b^2])*d*(1 + n))) - ((a*b^2*n - Sqrt[-b^2]*(a^2 + b^2*(1 + n)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(a + Sqrt[-b^2])*d*(1 + n)) - (Cos[c + d*x]^2*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(2*(a^2 + b^2)*d)} -{Csc[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 2, (b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))} -{Csc[c + d*x]^4*(a + b*Tan[c + d*x])^n, x, 4, (b*(2 - n)*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(6*a^2*d) - (Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(1 + n))/(3*a*d) + (b*(6*a^2 + b^2*(2 - 3*n + n^2))*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(6*a^4*d*(1 + n))} - -{Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 0, CannotIntegrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]} -{Sin[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 0, CannotIntegrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^n, x]} -{Csc[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 0, CannotIntegrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^n, x]} -{Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 0, CannotIntegrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^(m/2) (a+b Tan[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^(m/2) (a+b Tan[e+f x])^(n/2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m deleted file mode 100644 index 864d4bf..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.10 (c+d x)^m (a+b tan)^n.m +++ /dev/null @@ -1,141 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tan[e+f x])^n*) - - -{x^3*Tan[a + b*x], x, 6, (I*x^4)/4 - (x^3*Log[1 + E^(2*I*(a + b*x))])/b + (3*I*x^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) - (3*I*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4)} -{x^2*Tan[a + b*x], x, 5, (I*x^3)/3 - (x^2*Log[1 + E^(2*I*(a + b*x))])/b + (I*x*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - PolyLog[3, -E^(2*I*(a + b*x))]/(2*b^3)} -{x^1*Tan[a + b*x], x, 4, (I*x^2)/2 - (x*Log[1 + E^(2*I*(a + b*x))])/b + (I*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2)} -{Tan[a + b*x]/x^1, x, 0, Unintegrable[Tan[a + b*x]/x, x]} -{Tan[a + b*x]/x^2, x, 0, Unintegrable[Tan[a + b*x]/x^2, x]} - - -{x^3*Tan[a + b*x]^2, x, 7, -((I*x^3)/b) - x^4/4 + (3*x^2*Log[1 + E^(2*I*(a + b*x))])/b^2 - (3*I*x*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + (3*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^4) + (x^3*Tan[a + b*x])/b} -{x^2*Tan[a + b*x]^2, x, 6, -((I*x^2)/b) - x^3/3 + (2*x*Log[1 + E^(2*I*(a + b*x))])/b^2 - (I*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + (x^2*Tan[a + b*x])/b} -{x^1*Tan[a + b*x]^2, x, 3, -x^2/2 + Log[Cos[a + b*x]]/b^2 + (x*Tan[a + b*x])/b} -{Tan[a + b*x]^2/x^1, x, 0, Unintegrable[Tan[a + b*x]^2/x, x]} -{Tan[a + b*x]^2/x^2, x, 0, Unintegrable[Tan[a + b*x]^2/x^2, x]} - - -{x^3*Tan[a + b*x]^3, x, 13, (3*I*x^2)/(2*b^2) + x^3/(2*b) - (I*x^4)/4 - (3*x*Log[1 + E^(2*I*(a + b*x))])/b^3 + (x^3*Log[1 + E^(2*I*(a + b*x))])/b + (3*I*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^4) - (3*I*x^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) + (3*x*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*I*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) - (3*x^2*Tan[a + b*x])/(2*b^2) + (x^3*Tan[a + b*x]^2)/(2*b)} -{x^2*Tan[a + b*x]^3, x, 9, x^2/(2*b) - (I*x^3)/3 + (x^2*Log[1 + E^(2*I*(a + b*x))])/b - Log[Cos[a + b*x]]/b^3 - (I*x*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 + PolyLog[3, -E^(2*I*(a + b*x))]/(2*b^3) - (x*Tan[a + b*x])/b^2 + (x^2*Tan[a + b*x]^2)/(2*b)} -{x^1*Tan[a + b*x]^3, x, 7, x/(2*b) - (I*x^2)/2 + (x*Log[1 + E^(2*I*(a + b*x))])/b - (I*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - Tan[a + b*x]/(2*b^2) + (x*Tan[a + b*x]^2)/(2*b)} -{Tan[a + b*x]^3/x^1, x, 0, Unintegrable[Tan[a + b*x]^3/x, x]} -{Tan[a + b*x]^3/x^2, x, 0, Unintegrable[Tan[a + b*x]^3/x^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tan[e+f x])^(n/2)*) - - -{x^2/Tan[a + b*x]^(3/2) - (4*x)/(b*Sqrt[Tan[a + b*x]]) + x^2*Sqrt[Tan[a + b*x]], x, 76, -((2*x^2)/(b*Sqrt[Tan[a + b*x]]))} -{x^2*Tan[a + b*x^2]^(3/2) + x^2/Sqrt[Tan[a + b*x^2]] + Sqrt[Tan[a + b*x^2]]/b, x, -1, (x*Sqrt[Tan[a + b*x^2]])/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tan[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Tan[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + I*a*Tan[e + f*x]), x, 5, (((3*I)/8)*d^3*x)/(a*f^3) - (3*d*(c + d*x)^2)/(8*a*f^2) - ((I/4)*(c + d*x)^3)/(a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4*(a + I*a*Tan[e + f*x])) - (((3*I)/4)*d^2*(c + d*x))/(f^3*(a + I*a*Tan[e + f*x])) + (3*d*(c + d*x)^2)/(4*f^2*(a + I*a*Tan[e + f*x])) + ((I/2)*(c + d*x)^3)/(f*(a + I*a*Tan[e + f*x]))} -{(c + d*x)^2/(a + I*a*Tan[e + f*x]), x, 4, -(d^2*x)/(4*a*f^2) - ((I/4)*(c + d*x)^2)/(a*f) + (c + d*x)^3/(6*a*d) - ((I/4)*d^2)/(f^3*(a + I*a*Tan[e + f*x])) + (d*(c + d*x))/(2*f^2*(a + I*a*Tan[e + f*x])) + ((I/2)*(c + d*x)^2)/(f*(a + I*a*Tan[e + f*x]))} -{(c + d*x)^1/(a + I*a*Tan[e + f*x]), x, 3, ((-I/4)*d*x)/(a*f) + (c + d*x)^2/(4*a*d) + d/(4*f^2*(a + I*a*Tan[e + f*x])) + ((I/2)*(c + d*x))/(f*(a + I*a*Tan[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Tan[e + f*x])), x, 7, (Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) - ((I/2)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d) - ((I/2)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d) - (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)} -{1/((c + d*x)^2*(a + I*a*Tan[e + f*x])), x, 7, ((-I)*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - (f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d^2) - (f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) + (I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - 1/(d*(c + d*x)*(a + I*a*Tan[e + f*x]))} -{1/((c + d*x)^3*(a + I*a*Tan[e + f*x])), x, 8, ((-I/2)*f)/(a*d^2*(c + d*x)) - (f^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) + (I*f^2*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d^3) + (I*f^2*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) + (f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x)^2*(a + I*a*Tan[e + f*x])) + (I*f)/(d^2*(c + d*x)*(a + I*a*Tan[e + f*x]))} - - -{(c + d*x)^3/(a + I*a*Tan[e + f*x])^2, x, 10, (-3*d^3*E^((-2*I)*e - (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((-4*I)*e - (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^2*f^3) - (((3*I)/128)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^2*f^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(64*a^2*f^2) + ((I/4)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^3)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3)/(a^2*f) + (c + d*x)^4/(16*a^2*d)} -{(c + d*x)^2/(a + I*a*Tan[e + f*x])^2, x, 8, ((-I/8)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^2*f^3) - ((I/128)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^2*f^3) + (d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(4*a^2*f^2) + (d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(32*a^2*f^2) + ((I/4)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a^2*f) + (c + d*x)^3/(12*a^2*d)} -{(c + d*x)^1/(a + I*a*Tan[e + f*x])^2, x, 7, (((-3*I)/16)*d*x)/(a^2*f) - (d*x^2)/(8*a^2) + (x*(c + d*x))/(4*a^2) + d/(16*f^2*(a + I*a*Tan[e + f*x])^2) + ((I/4)*(c + d*x))/(f*(a + I*a*Tan[e + f*x])^2) + (3*d)/(16*f^2*(a^2 + I*a^2*Tan[e + f*x])) + ((I/4)*(c + d*x))/(f*(a^2 + I*a^2*Tan[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Tan[e + f*x])^2), x, 21, (Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + (Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + Log[c + d*x]/(4*a^2*d) - ((I/4)*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^2*d) - ((I/2)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^2*d) - ((I/2)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d) - (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) - ((I/4)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d) - (Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d)} -{1/((c + d*x)^2*(a + I*a*Tan[e + f*x])^2), x, 24, -(1/(4*a^2*d*(c + d*x))) - Cos[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cos[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) - (I*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (I*f*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^2*d^2) - (f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^2*d^2) + (I*Sin[2*e + 2*f*x])/(2*a^2*d*(c + d*x)) + Sin[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + (I*Sin[4*e + 4*f*x])/(4*a^2*d*(c + d*x)) - (f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) + (I*f*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2)} - - -{(c + d*x)^3/(a + I*a*Tan[e + f*x])^3, x, 14, (-9*d^3*E^((-2*I)*e - (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((-4*I)*e - (4*I)*f*x))/(1024*a^3*f^4) - (d^3*E^((-6*I)*e - (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^3*f^3) - (((9*I)/256)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(a^3*f^3) + (9*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(128*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^3)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^3)/(a^3*f) + (c + d*x)^4/(32*a^3*d)} -{(c + d*x)^2/(a + I*a*Tan[e + f*x])^3, x, 11, (((-3*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^3*f^3) - (((3*I)/256)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^3*f^3) - ((I/864)*d^2*E^((-6*I)*e - (6*I)*f*x))/(a^3*f^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + (((3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)} -{(c + d*x)^1/(a + I*a*Tan[e + f*x])^3, x, 11, (((-11*I)/96)*d*x)/(a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) + d/(36*f^2*(a + I*a*Tan[e + f*x])^3) + ((I/6)*(c + d*x))/(f*(a + I*a*Tan[e + f*x])^3) + (5*d)/(96*a*f^2*(a + I*a*Tan[e + f*x])^2) + ((I/8)*(c + d*x))/(a*f*(a + I*a*Tan[e + f*x])^2) + (11*d)/(96*f^2*(a^3 + I*a^3*Tan[e + f*x])) + ((I/8)*(c + d*x))/(f*(a^3 + I*a^3*Tan[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Tan[e + f*x])^3), x, 53, (3*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^3*d) - ((I/8)*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^3*d) - (((3*I)/8)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^3*d) - (3*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (((3*I)/8)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^3*d) - (3*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - ((I/8)*Cos[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(a^3*d) - (Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)} -{1/((c + d*x)^2*(a + I*a*Tan[e + f*x])^3), x, 60, -(1/(8*a^3*d*(c + d*x))) - (9*Cos[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cos[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - Cos[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) - (3*Cos[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*I*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*I*f*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*I*f*Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) - (3*f*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(4*a^3*d^2) - (3*f*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(2*a^3*d^2) - (3*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(4*a^3*d^2) + (15*I*Sin[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) + (3*Sin[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - (I*Sin[2*e + 2*f*x]^3)/(8*a^3*d*(c + d*x)) + (3*I*Sin[4*e + 4*f*x])/(8*a^3*d*(c + d*x)) + (3*I*Sin[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) + (3*I*f*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Cos[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (3*I*f*Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Tan[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + I*a*Tan[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m*(a + I*a*Tan[e + f*x])^2, x]} -{(c + d*x)^m*(a + I*a*Tan[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m*(a + I*a*Tan[e + f*x]), x]} -{(c + d*x)^m/(a + I*a*Tan[e + f*x])^1, x, 2, (c + d*x)^(1 + m)/(2*a*d*(1 + m)) + (I*2^(-2 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(a*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + I*a*Tan[e + f*x])^2, x, 4, (c + d*x)^(1 + m)/(4*a^2*d*(1 + m)) + (I*2^(-2 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(a^2*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (I*4^(-2 - m)*(c + d*x)^m*Gamma[1 + m, ((4*I)*f*(c + d*x))/d])/(a^2*E^((4*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + I*a*Tan[e + f*x])^3, x, 5, (c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) + ((3*I)*2^(-4 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*f*(c + d*x))/d])/(a^3*E^((2*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + ((3*I)*2^(-5 - 2*m)*(c + d*x)^m*Gamma[1 + m, ((4*I)*f*(c + d*x))/d])/(a^3*E^((4*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m) + (I*2^(-4 - m)*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((6*I)*f*(c + d*x))/d])/(a^3*E^((6*I)*(e - (c*f)/d))*f*((I*f*(c + d*x))/d)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Tan[e + f*x]), x, 8, (a*(c + d*x)^4)/(4*d) + ((I/4)*b*(c + d*x)^4)/d - (b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f + (((3*I)/2)*b*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (3*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) - (((3*I)/4)*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4} -{(c + d*x)^2*(a + b*Tan[e + f*x]), x, 7, (a*(c + d*x)^3)/(3*d) + ((I/3)*b*(c + d*x)^3)/d - (b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (I*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Tan[e + f*x]), x, 6, (a*(c + d*x)^2)/(2*d) + ((I/2)*b*(c + d*x)^2)/d - (b*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f + ((I/2)*b*d*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2} -{(a + b*Tan[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tan[e + f*x])/(c + d*x), x]} -{(a + b*Tan[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tan[e + f*x])/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Tan[e + f*x])^2, x, 15, ((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) + ((I/2)*a*b*(c + d*x)^4)/d - (b^2*(c + d*x)^4)/(4*d) + (3*b^2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (2*a*b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a*b*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 + (3*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/f^3 - (((3*I)/2)*a*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 + (b^2*(c + d*x)^3*Tan[e + f*x])/f} -{(c + d*x)^2*(a + b*Tan[e + f*x])^2, x, 13, ((-I)*b^2*(c + d*x)^2)/f + (a^2*(c + d*x)^3)/(3*d) + (((2*I)/3)*a*b*(c + d*x)^3)/d - (b^2*(c + d*x)^3)/(3*d) + (2*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (2*a*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f - (I*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((2*I)*a*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (a*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/f^3 + (b^2*(c + d*x)^2*Tan[e + f*x])/f} -{(c + d*x)^1*(a + b*Tan[e + f*x])^2, x, 9, -(b^2*c*x) - (b^2*d*x^2)/2 + (a^2*(c + d*x)^2)/(2*d) + (I*a*b*(c + d*x)^2)/d - (2*a*b*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f + (b^2*d*Log[Cos[e + f*x]])/f^2 + (I*a*b*d*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 + (b^2*(c + d*x)*Tan[e + f*x])/f} -{(a + b*Tan[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tan[e + f*x])^2/(c + d*x), x]} -{(a + b*Tan[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tan[e + f*x])^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Tan[e + f*x])^3, x, 28, (((3*I)/2)*b^3*d*(c + d*x)^2)/f^2 - ((3*I)*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) + (((3*I)/4)*a^2*b*(c + d*x)^4)/d - (3*a*b^2*(c + d*x)^4)/(4*d) - ((I/4)*b^3*(c + d*x)^4)/d - (3*b^3*d^2*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (3*a^2*b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f + (((3*I)/2)*b^3*d^3*PolyLog[2, -E^((2*I)*(e + f*x))])/f^4 - ((9*I)*a*b^2*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + (((9*I)/2)*a^2*b*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (((3*I)/2)*b^3*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 + (9*a*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) + (3*b^3*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) - (((9*I)/4)*a^2*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 + (((3*I)/4)*b^3*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 - (3*b^3*d*(c + d*x)^2*Tan[e + f*x])/(2*f^2) + (3*a*b^2*(c + d*x)^3*Tan[e + f*x])/f + (b^3*(c + d*x)^3*Tan[e + f*x]^2)/(2*f)} -{(c + d*x)^2*(a + b*Tan[e + f*x])^3, x, 22, (b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) + (I*a^2*b*(c + d*x)^3)/d - (a*b^2*(c + d*x)^3)/d - ((I/3)*b^3*(c + d*x)^3)/d + (6*a*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (3*a^2*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f - (b^3*d^2*Log[Cos[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a^2*b*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (I*b^3*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) + (b^3*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) - (b^3*d*(c + d*x)*Tan[e + f*x])/f^2 + (3*a*b^2*(c + d*x)^2*Tan[e + f*x])/f + (b^3*(c + d*x)^2*Tan[e + f*x]^2)/(2*f)} -{(c + d*x)^1*(a + b*Tan[e + f*x])^3, x, 16, -3*a*b^2*c*x + (b^3*d*x)/(2*f) - (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) + (((3*I)/2)*a^2*b*(c + d*x)^2)/d - ((I/2)*b^3*(c + d*x)^2)/d - (3*a^2*b*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f + (3*a*b^2*d*Log[Cos[e + f*x]])/f^2 + (((3*I)/2)*a^2*b*d*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - ((I/2)*b^3*d*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (b^3*d*Tan[e + f*x])/(2*f^2) + (3*a*b^2*(c + d*x)*Tan[e + f*x])/f + (b^3*(c + d*x)*Tan[e + f*x]^2)/(2*f)} -{(a + b*Tan[e + f*x])^3/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tan[e + f*x])^3/(c + d*x), x]} -{(a + b*Tan[e + f*x])^3/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tan[e + f*x])^3/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Tan[e + f*x]), x, 6, (c + d*x)^4/(4*(a + I*b)*d) + (b*(c + d*x)^3*Log[1 + ((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2])/((a^2 + b^2)*f) - (3*I*b*d*(c + d*x)^2*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/(2*(a^2 + b^2)*f^2) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/(2*(a^2 + b^2)*f^3) + (3*I*b*d^3*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/(4*(a^2 + b^2)*f^4)} -{(c + d*x)^2/(a + b*Tan[e + f*x]), x, 5, (c + d*x)^3/(3*(a + I*b)*d) + (b*(c + d*x)^2*Log[1 + ((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2])/((a^2 + b^2)*f) - (I*b*d*(c + d*x)*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/((a^2 + b^2)*f^2) + (b*d^2*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/(2*(a^2 + b^2)*f^3)} -{(c + d*x)^1/(a + b*Tan[e + f*x]), x, 4, (c + d*x)^2/(2*(a + I*b)*d) + (b*(c + d*x)*Log[1 + ((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2])/((a^2 + b^2)*f) - (I*b*d*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/(2*(a^2 + b^2)*f^2)} -{1/((c + d*x)^1*(a + b*Tan[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Tan[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Tan[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Tan[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Tan[e + f*x])^2, x, 21, -((2*I*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f)) + (2*b^2*(c + d*x)^3)/((a + I*b)*(I*a + b)^2*(I*a - b + (I*a + b)*E^(2*I*e + 2*I*f*x))*f) + (c + d*x)^4/(4*(a - I*b)^2*d) + (b*(c + d*x)^4)/((I*a - b)*(a - I*b)^2*d) - (b^2*(c + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a^2 + b^2)^2*f^2) + (2*b*(c + d*x)^3*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a - I*b)^2*(a + I*b)*f) - (2*I*b^2*(c + d*x)^3*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a^2 + b^2)^2*f) - (3*I*b^2*d^2*(c + d*x)*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) + (3*b*d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((I*a - b)*(a - I*b)^2*f^2) - (3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^2) + (3*b^2*d^3*PolyLog[3, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a - I*b)^2*(a + I*b)*f^3) - (3*I*b^2*d^2*(c + d*x)*PolyLog[3, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) - (3*b*d^3*PolyLog[4, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/(2*(I*a - b)*(a - I*b)^2*f^4) + (3*b^2*d^3*PolyLog[4, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4)} -{(c + d*x)^2/(a + b*Tan[e + f*x])^2, x, 18, -((2*I*b^2*(c + d*x)^2)/((a^2 + b^2)^2*f)) + (2*b^2*(c + d*x)^2)/((a + I*b)*(I*a + b)^2*(I*a - b + (I*a + b)*E^(2*I*e + 2*I*f*x))*f) + (c + d*x)^3/(3*(a - I*b)^2*d) + (4*b*(c + d*x)^3)/(3*(I*a - b)*(a - I*b)^2*d) - (4*b^2*(c + d*x)^3)/(3*(a^2 + b^2)^2*d) + (2*b^2*d*(c + d*x)*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a^2 + b^2)^2*f^2) + (2*b*(c + d*x)^2*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a - I*b)^2*(a + I*b)*f) - (2*I*b^2*(c + d*x)^2*Log[1 + ((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b)])/((a^2 + b^2)^2*f) - (I*b^2*d^2*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) + (2*b*d*(c + d*x)*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((I*a - b)*(a - I*b)^2*f^2) - (2*b^2*d*(c + d*x)*PolyLog[2, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^2) + (b*d^2*PolyLog[3, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a - I*b)^2*(a + I*b)*f^3) - (I*b^2*d^2*PolyLog[3, -(((a - I*b)*E^(2*I*e + 2*I*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3)} -{(c + d*x)^1/(a + b*Tan[e + f*x])^2, x, 5, -((c + d*x)^2/(2*(a^2 + b^2)*d)) + (b*d + 2*a*c*f + 2*a*d*f*x)^2/(4*a*(a + I*b)*(a^2 + b^2)*d*f^2) + (b*(b*d + 2*a*c*f + 2*a*d*f*x)*Log[1 + ((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2])/((a^2 + b^2)^2*f^2) - (I*a*b*d*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(e + f*x)))/(a + I*b)^2)])/((a^2 + b^2)^2*f^2) - (b*(c + d*x))/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{1/((c + d*x)^1*(a + b*Tan[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Tan[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Tan[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Tan[e + f*x])^2), x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m deleted file mode 100644 index 9d690d9..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.11 (e x)^m (a+b tan(c+d x^n))^p.m +++ /dev/null @@ -1,130 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Tan[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Tan[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Tan[c + d*x^2]), x, 7, (a*x^4)/4 + (I/4)*b*x^4 - (b*x^2*Log[1 + E^((2*I)*(c + d*x^2))])/(2*d) + ((I/4)*b*PolyLog[2, -E^((2*I)*(c + d*x^2))])/d^2} -{x^2*(a + b*Tan[c + d*x^2]), x, 2, (a*x^3)/3 + b*Unintegrable[x^2*Tan[c + d*x^2], x]} -{x^1*(a + b*Tan[c + d*x^2]), x, 4, (a*x^2)/2 - (b*Log[Cos[c + d*x^2]])/(2*d)} -{a + b*Tan[c + d*x^2], x, 1, a*x + b*Unintegrable[Tan[c + d*x^2], x]} -{(a + b*Tan[c + d*x^2])/x^1, x, 2, a*Log[x] + b*Unintegrable[Tan[c + d*x^2]/x, x]} -{(a + b*Tan[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Tan[c + d*x^2]/x^2, x]} - - -{x^3*(a + b*Tan[c + d*x^2])^2, x, 10, (a^2*x^4)/4 + (I/2)*a*b*x^4 - (b^2*x^4)/4 - (a*b*x^2*Log[1 + E^((2*I)*(c + d*x^2))])/d + (b^2*Log[Cos[c + d*x^2]])/(2*d^2) + ((I/2)*a*b*PolyLog[2, -E^((2*I)*(c + d*x^2))])/d^2 + (b^2*x^2*Tan[c + d*x^2])/(2*d)} -{x^2*(a + b*Tan[c + d*x^2])^2, x, 0, Unintegrable[x^2*(a + b*Tan[c + d*x^2])^2, x]} -{x^1*(a + b*Tan[c + d*x^2])^2, x, 3, ((a^2 - b^2)*x^2)/2 - (a*b*Log[Cos[c + d*x^2]])/d + (b^2*Tan[c + d*x^2])/(2*d)} -{x^0*(a + b*Tan[c + d*x^2])^2, x, 0, Unintegrable[(a + b*Tan[c + d*x^2])^2, x]} -{(a + b*Tan[c + d*x^2])^2/x^1, x, 0, Unintegrable[(a + b*Tan[c + d*x^2])^2/x, x]} -{(a + b*Tan[c + d*x^2])^2/x^2, x, 0, Unintegrable[(a + b*Tan[c + d*x^2])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Tan[c + d*x^2]), x, 5, x^4/(4*(a + I*b)) + (b*x^2*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*x^2)))/(a + I*b)^2])/(2*(a^2 + b^2)*d) - (I*b*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*x^2)))/(a + I*b)^2)])/(4*(a^2 + b^2)*d^2)} -{x^2/(a + b*Tan[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Tan[c + d*x^2]), x]} -{x^1/(a + b*Tan[c + d*x^2]), x, 3, (a*x^2)/(2*(a^2 + b^2)) + (b*Log[a*Cos[c + d*x^2] + b*Sin[c + d*x^2]])/(2*(a^2 + b^2)*d)} -{x^0/(a + b*Tan[c + d*x^2]), x, 0, Unintegrable[(a + b*Tan[c + d*x^2])^(-1), x]} -{1/(x*(a + b*Tan[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*x^2])), x]} -{1/(x^2*(a + b*Tan[c + d*x^2])), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*x^2])), x]} - - -{x^3/(a + b*Tan[c + d*x^2])^2, x, 6, -(x^4/(4*(a^2 + b^2))) + (b + 2*a*d*x^2)^2/(8*a*(a + I*b)*(a^2 + b^2)*d^2) + (b*(b + 2*a*d*x^2)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*x^2)))/(a + I*b)^2])/(2*(a^2 + b^2)^2*d^2) - (I*a*b*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*x^2)))/(a + I*b)^2)])/(2*(a^2 + b^2)^2*d^2) - (b*x^2)/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x^2]))} -{x^2/(a + b*Tan[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Tan[c + d*x^2])^2, x]} -{x^1/(a + b*Tan[c + d*x^2])^2, x, 4, ((a^2 - b^2)*x^2)/(2*(a^2 + b^2)^2) + (a*b*Log[a*Cos[c + d*x^2] + b*Sin[c + d*x^2]])/((a^2 + b^2)^2*d) - b/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x^2]))} -{x^0/(a + b*Tan[c + d*x^2])^2, x, 0, Unintegrable[(a + b*Tan[c + d*x^2])^(-2), x]} -{1/(x*(a + b*Tan[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*x^2])^2), x]} -{1/(x^2*(a + b*Tan[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*x^2])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Tan[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Tan[c + d*Sqrt[x]]), x, 13, (a*x^4)/4 + (I/4)*b*x^4 - (2*b*x^(7/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d + ((7*I)*b*x^3*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 - (21*b*x^(5/2)*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (((105*I)/2)*b*x^2*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 + (105*b*x^(3/2)*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (((315*I)/2)*b*x*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 - (315*b*Sqrt[x]*PolyLog[7, -E^((2*I)*(c + d*Sqrt[x]))])/(2*d^7) - (((315*I)/4)*b*PolyLog[8, -E^((2*I)*(c + d*Sqrt[x]))])/d^8} -{x^2*(a + b*Tan[c + d*Sqrt[x]]), x, 11, (a*x^3)/3 + (I/3)*b*x^3 - (2*b*x^(5/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d + ((5*I)*b*x^2*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 - (10*b*x^(3/2)*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - ((15*I)*b*x*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 + (15*b*Sqrt[x]*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (((15*I)/2)*b*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))])/d^6} -{x^1*(a + b*Tan[c + d*Sqrt[x]]), x, 9, (a*x^2)/2 + (I/2)*b*x^2 - (2*b*x^(3/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d + ((3*I)*b*x*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 - (3*b*Sqrt[x]*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (((3*I)/2)*b*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^4} -{a + b*Tan[c + d*Sqrt[x]], x, 6, a*x + I*b*x - (2*b*Sqrt[x]*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d + (I*b*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2} -{(a + b*Tan[c + d*Sqrt[x]])/x^1, x, 2, a*Log[x] + b*Unintegrable[Tan[c + d*Sqrt[x]]/x, x]} -{(a + b*Tan[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Tan[c + d*Sqrt[x]]/x^2, x]} - - -{x^2*(a + b*Tan[c + d*Sqrt[x]])^2, x, 20, ((-2*I)*b^2*x^(5/2))/d + (a^2*x^3)/3 + ((2*I)/3)*a*b*x^3 - (b^2*x^3)/3 + (10*b^2*x^2*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 - (4*a*b*x^(5/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d - ((20*I)*b^2*x^(3/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 + ((10*I)*a*b*x^2*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 + (30*b^2*x*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - (20*a*b*x^(3/2)*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 + ((30*I)*b^2*Sqrt[x]*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 - ((30*I)*a*b*x*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - (15*b^2*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 + (30*a*b*Sqrt[x]*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + ((15*I)*a*b*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 + (2*b^2*x^(5/2)*Tan[c + d*Sqrt[x]])/d} -{x^1*(a + b*Tan[c + d*Sqrt[x]])^2, x, 16, ((-2*I)*b^2*x^(3/2))/d + (a^2*x^2)/2 + I*a*b*x^2 - (b^2*x^2)/2 + (6*b^2*x*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 - (4*a*b*x^(3/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 + ((6*I)*a*b*x*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 + (3*b^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - (6*a*b*Sqrt[x]*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - ((3*I)*a*b*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 + (2*b^2*x^(3/2)*Tan[c + d*Sqrt[x]])/d} -{x^0*(a + b*Tan[c + d*Sqrt[x]])^2, x, 10, a^2*x + (2*I)*a*b*x - b^2*x - (4*a*b*Sqrt[x]*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d + (2*b^2*Log[Cos[c + d*Sqrt[x]]])/d^2 + ((2*I)*a*b*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^2 + (2*b^2*Sqrt[x]*Tan[c + d*Sqrt[x]])/d} -{(a + b*Tan[c + d*Sqrt[x]])^2/x^1, x, 0, Unintegrable[(a + b*Tan[c + d*Sqrt[x]])^2/x, x]} -{(a + b*Tan[c + d*Sqrt[x]])^2/x^2, x, 0, Unintegrable[(a + b*Tan[c + d*Sqrt[x]])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Tan[c + d*Sqrt[x]]), x, 11, x^4/(4*(a + I*b)) + (2*b*x^(7/2)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2])/((a^2 + b^2)*d) - (7*I*b*x^3*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (21*b*x^(5/2)*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + (105*I*b*x^2*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^4) - (105*b*x^(3/2)*PolyLog[5, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^5) - (315*I*b*x*PolyLog[6, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^6) + (315*b*Sqrt[x]*PolyLog[7, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^7) + (315*I*b*PolyLog[8, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(4*(a^2 + b^2)*d^8)} -{x^2/(a + b*Tan[c + d*Sqrt[x]]), x, 9, x^3/(3*(a + I*b)) + (2*b*x^(5/2)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2])/((a^2 + b^2)*d) - (5*I*b*x^2*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (10*b*x^(3/2)*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + (15*I*b*x*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^4) - (15*b*Sqrt[x]*PolyLog[5, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^5) - (15*I*b*PolyLog[6, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^6)} -{x^1/(a + b*Tan[c + d*Sqrt[x]]), x, 7, x^2/(2*(a + I*b)) + (2*b*x^(3/2)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2])/((a^2 + b^2)*d) - (3*I*b*x*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (3*b*Sqrt[x]*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + (3*I*b*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^4)} -{x^0/(a + b*Tan[c + d*Sqrt[x]]), x, 5, x/(a + I*b) + (2*b*Sqrt[x]*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2])/((a^2 + b^2)*d) - (I*b*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)*d^2)} -{1/(x*(a + b*Tan[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*Sqrt[x]])), x]} -{1/(x^2*(a + b*Tan[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*Sqrt[x]])), x]} - - -{x^2/(a + b*Tan[c + d*Sqrt[x]])^2, x, 28, -((4*I*b^2*x^(5/2))/((a^2 + b^2)^2*d)) + (4*b^2*x^(5/2))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^(2*I*(c + d*Sqrt[x])))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^2) + (10*b^2*x^2*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (4*b*x^(5/2)*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - (4*I*b^2*x^(5/2)*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d) - (20*I*b^2*x^(3/2)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (10*b*x^2*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (10*b^2*x^2*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (30*b^2*x*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (20*b*x^(3/2)*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - (20*I*b^2*x^(3/2)*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (30*I*b^2*Sqrt[x]*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^5) - (30*b*x*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (30*b^2*x*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4) - (15*b^2*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^6) - (30*b*Sqrt[x]*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^5) + (30*I*b^2*Sqrt[x]*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^5) + (15*b*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6) - (15*b^2*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^6)} -{x^1/(a + b*Tan[c + d*Sqrt[x]])^2, x, 22, -((4*I*b^2*x^(3/2))/((a^2 + b^2)^2*d)) + (4*b^2*x^(3/2))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^(2*I*(c + d*Sqrt[x])))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (6*b^2*x*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (4*b*x^(3/2)*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - (4*I*b^2*x^(3/2)*Log[1 + ((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)])/((a^2 + b^2)^2*d) - (6*I*b^2*Sqrt[x]*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (6*b*x*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (6*b^2*x*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (3*b^2*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (6*b*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - (6*I*b^2*Sqrt[x]*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^3) - (3*b*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (3*b^2*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b))])/((a^2 + b^2)^2*d^4)} -{x^0/(a + b*Tan[c + d*Sqrt[x]])^2, x, 6, (b + 2*a*d*Sqrt[x])^2/(2*a*(a + I*b)*(a^2 + b^2)*d^2) - x/(a^2 + b^2) + (2*b*(b + 2*a*d*Sqrt[x])*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2])/((a^2 + b^2)^2*d^2) - (2*I*a*b*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*Sqrt[x])))/(a + I*b)^2)])/((a^2 + b^2)^2*d^2) - (2*b*Sqrt[x])/((a^2 + b^2)*d*(a + b*Tan[c + d*Sqrt[x]]))} -{1/(x*(a + b*Tan[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*Sqrt[x]])^2), x]} -{1/(x^2*(a + b*Tan[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*Sqrt[x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Tan[c+d x^(1/3)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*(a + b*Tan[c + d*x^(1/3)]), x, 14, (a*x^3)/3 + (I/3)*b*x^3 - (3*b*x^(8/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d + ((12*I)*b*x^(7/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 - (42*b*x^2*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^3 - ((126*I)*b*x^(5/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^4 + (315*b*x^(4/3)*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/d^5 + ((630*I)*b*x*PolyLog[6, -E^((2*I)*(c + d*x^(1/3)))])/d^6 - (945*b*x^(2/3)*PolyLog[7, -E^((2*I)*(c + d*x^(1/3)))])/d^7 - ((945*I)*b*x^(1/3)*PolyLog[8, -E^((2*I)*(c + d*x^(1/3)))])/d^8 + (945*b*PolyLog[9, -E^((2*I)*(c + d*x^(1/3)))])/(2*d^9)} -{x^1*(a + b*Tan[c + d*x^(1/3)]), x, 11, (a*x^2)/2 + (I/2)*b*x^2 - (3*b*x^(5/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d + (((15*I)/2)*b*x^(4/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 - (15*b*x*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^3 - (((45*I)/2)*b*x^(2/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^4 + (45*b*x^(1/3)*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/(2*d^5) + (((45*I)/4)*b*PolyLog[6, -E^((2*I)*(c + d*x^(1/3)))])/d^6} -{a + b*Tan[c + d*x^(1/3)], x, 7, a*x + I*b*x - (3*b*x^(2/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d + ((3*I)*b*x^(1/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 - (3*b*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/(2*d^3)} -{(a + b*Tan[c + d*x^(1/3)])/x^1, x, 2, a*Log[x] + b*Unintegrable[Tan[c + d*x^(1/3)]/x, x]} -{(a + b*Tan[c + d*x^(1/3)])/x^2, x, 2, -(a/x) + b*Unintegrable[Tan[c + d*x^(1/3)]/x^2, x]} - - -{x^2*(a + b*Tan[c + d*x^(1/3)])^2, x, 26, ((-3*I)*b^2*x^(8/3))/d + (a^2*x^3)/3 + ((2*I)/3)*a*b*x^3 - (b^2*x^3)/3 + (24*b^2*x^(7/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d^2 - (6*a*b*x^(8/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d - ((84*I)*b^2*x^2*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + ((24*I)*a*b*x^(7/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 + (252*b^2*x^(5/3)*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^4 - (84*a*b*x^2*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + ((630*I)*b^2*x^(4/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^5 - ((252*I)*a*b*x^(5/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^4 - (1260*b^2*x*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/d^6 + (630*a*b*x^(4/3)*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/d^5 - ((1890*I)*b^2*x^(2/3)*PolyLog[6, -E^((2*I)*(c + d*x^(1/3)))])/d^7 + ((1260*I)*a*b*x*PolyLog[6, -E^((2*I)*(c + d*x^(1/3)))])/d^6 + (1890*b^2*x^(1/3)*PolyLog[7, -E^((2*I)*(c + d*x^(1/3)))])/d^8 - (1890*a*b*x^(2/3)*PolyLog[7, -E^((2*I)*(c + d*x^(1/3)))])/d^7 + ((945*I)*b^2*PolyLog[8, -E^((2*I)*(c + d*x^(1/3)))])/d^9 - ((1890*I)*a*b*x^(1/3)*PolyLog[8, -E^((2*I)*(c + d*x^(1/3)))])/d^8 + (945*a*b*PolyLog[9, -E^((2*I)*(c + d*x^(1/3)))])/d^9 + (3*b^2*x^(8/3)*Tan[c + d*x^(1/3)])/d} -{x^1*(a + b*Tan[c + d*x^(1/3)])^2, x, 20, ((-3*I)*b^2*x^(5/3))/d + (a^2*x^2)/2 + I*a*b*x^2 - (b^2*x^2)/2 + (15*b^2*x^(4/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d^2 - (6*a*b*x^(5/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d - ((30*I)*b^2*x*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + ((15*I)*a*b*x^(4/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 + (45*b^2*x^(2/3)*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^4 - (30*a*b*x*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + ((45*I)*b^2*x^(1/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^5 - ((45*I)*a*b*x^(2/3)*PolyLog[4, -E^((2*I)*(c + d*x^(1/3)))])/d^4 - (45*b^2*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/(2*d^6) + (45*a*b*x^(1/3)*PolyLog[5, -E^((2*I)*(c + d*x^(1/3)))])/d^5 + (((45*I)/2)*a*b*PolyLog[6, -E^((2*I)*(c + d*x^(1/3)))])/d^6 + (3*b^2*x^(5/3)*Tan[c + d*x^(1/3)])/d} -{x^0*(a + b*Tan[c + d*x^(1/3)])^2, x, 14, ((-3*I)*b^2*x^(2/3))/d + a^2*x + (2*I)*a*b*x - b^2*x + (6*b^2*x^(1/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d^2 - (6*a*b*x^(2/3)*Log[1 + E^((2*I)*(c + d*x^(1/3)))])/d - ((3*I)*b^2*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + ((6*I)*a*b*x^(1/3)*PolyLog[2, -E^((2*I)*(c + d*x^(1/3)))])/d^2 - (3*a*b*PolyLog[3, -E^((2*I)*(c + d*x^(1/3)))])/d^3 + (3*b^2*x^(2/3)*Tan[c + d*x^(1/3)])/d} -{(a + b*Tan[c + d*x^(1/3)])^2/x^1, x, 0, Unintegrable[(a + b*Tan[c + d*x^(1/3)])^2/x, x]} -{(a + b*Tan[c + d*x^(1/3)])^2/x^2, x, 0, Unintegrable[(a + b*Tan[c + d*x^(1/3)])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^2/(a + b*Tan[c + d*x^(1/3)]), x, 12, x^3/(3*(a + I*b)) + (3*b*x^(8/3)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d) - (12*I*b*x^(7/3)*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (42*b*x^2*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + (126*I*b*x^(5/3)*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^4) - (315*b*x^(4/3)*PolyLog[5, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^5) - (630*I*b*x*PolyLog[6, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^6) + (945*b*x^(2/3)*PolyLog[7, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^7) + (945*I*b*x^(1/3)*PolyLog[8, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^8) - (945*b*PolyLog[9, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^9)} -{x^1/(a + b*Tan[c + d*x^(1/3)]), x, 9, x^2/(2*(a + I*b)) + (3*b*x^(5/3)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d) - (15*I*b*x^(4/3)*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^2) + (15*b*x*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^3) + (45*I*b*x^(2/3)*PolyLog[4, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^4) - (45*b*x^(1/3)*PolyLog[5, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^5) - (45*I*b*PolyLog[6, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(4*(a^2 + b^2)*d^6)} -{x^0/(a + b*Tan[c + d*x^(1/3)]), x, 6, x/(a + I*b) + (3*b*x^(2/3)*Log[1 + ((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2])/((a^2 + b^2)*d) - (3*I*b*x^(1/3)*PolyLog[2, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/((a^2 + b^2)*d^2) + (3*b*PolyLog[3, -(((a^2 + b^2)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)^2)])/(2*(a^2 + b^2)*d^3)} -{1/(x*(a + b*Tan[c + d*x^(1/3)])), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*x^(1/3)])), x]} -{1/(x^2*(a + b*Tan[c + d*x^(1/3)])), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*x^(1/3)])), x]} - - -{x^2/(a + b*Tan[c + d*x^(1/3)])^2, x, 37, -((6*I*b^2*x^(8/3))/((a^2 + b^2)^2*d)) + (6*b^2*x^(8/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^(2*I*(c + d*x^(1/3))))) + x^3/(3*(a - I*b)^2) + (4*b*x^3)/(3*(I*a - b)*(a - I*b)^2) - (4*b^2*x^3)/(3*(a^2 + b^2)^2) + (24*b^2*x^(7/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(8/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - (6*I*b^2*x^(8/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - (84*I*b^2*x^2*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (24*b*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (24*b^2*x^(7/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (252*b^2*x^(5/3)*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (84*b*x^2*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - (84*I*b^2*x^2*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (630*I*b^2*x^(4/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) - (252*b*x^(5/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (252*b^2*x^(5/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) - (1260*b^2*x*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^6) - (630*b*x^(4/3)*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^5) + (630*I*b^2*x^(4/3)*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) - (1890*I*b^2*x^(2/3)*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^7) + (1260*b*x*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^6) - (1260*b^2*x*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^6) + (1890*b^2*x^(1/3)*PolyLog[7, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) + (1890*b*x^(2/3)*PolyLog[7, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^7) - (1890*I*b^2*x^(2/3)*PolyLog[7, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^7) + (945*I*b^2*PolyLog[8, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9) - (1890*b*x^(1/3)*PolyLog[8, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^8) + (1890*b^2*x^(1/3)*PolyLog[8, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^8) - (945*b*PolyLog[9, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^9) + (945*I*b^2*PolyLog[9, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^9)} -{x^1/(a + b*Tan[c + d*x^(1/3)])^2, x, 28, -((6*I*b^2*x^(5/3))/((a^2 + b^2)^2*d)) + (6*b^2*x^(5/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^(2*I*(c + d*x^(1/3))))) + x^2/(2*(a - I*b)^2) + (2*b*x^2)/((I*a - b)*(a - I*b)^2) - (2*b^2*x^2)/(a^2 + b^2)^2 + (15*b^2*x^(4/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(5/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - (6*I*b^2*x^(5/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - (30*I*b^2*x*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (15*b*x^(4/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (15*b^2*x^(4/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (45*b^2*x^(2/3)*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) + (30*b*x*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - (30*I*b^2*x*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (45*I*b^2*x^(1/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) - (45*b*x^(2/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^4) + (45*b^2*x^(2/3)*PolyLog[4, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^4) - (45*b^2*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6) - (45*b*x^(1/3)*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^5) + (45*I*b^2*x^(1/3)*PolyLog[5, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^5) + (45*b*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/(2*(I*a - b)*(a - I*b)^2*d^6) - (45*b^2*PolyLog[6, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/(2*(a^2 + b^2)^2*d^6)} -{x^0/(a + b*Tan[c + d*x^(1/3)])^2, x, 19, -((6*I*b^2*x^(2/3))/((a^2 + b^2)^2*d)) + (6*b^2*x^(2/3))/((a + I*b)*(I*a + b)^2*d*(I*a - b + (I*a + b)*E^(2*I*(c + d*x^(1/3))))) + x/(a - I*b)^2 + (4*b*x)/((I*a - b)*(a - I*b)^2) - (4*b^2*x)/(a^2 + b^2)^2 + (6*b^2*x^(1/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d^2) + (6*b*x^(2/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a - I*b)^2*(a + I*b)*d) - (6*I*b^2*x^(2/3)*Log[1 + ((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b)])/((a^2 + b^2)^2*d) - (3*I*b^2*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3) + (6*b*x^(1/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((I*a - b)*(a - I*b)^2*d^2) - (6*b^2*x^(1/3)*PolyLog[2, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^2) + (3*b*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a - I*b)^2*(a + I*b)*d^3) - (3*I*b^2*PolyLog[3, -(((a - I*b)*E^(2*I*(c + d*x^(1/3))))/(a + I*b))])/((a^2 + b^2)^2*d^3)} -{1/(x*(a + b*Tan[c + d*x^(1/3)])^2), x, 0, Unintegrable[1/(x*(a + b*Tan[c + d*x^(1/3)])^2), x]} -{1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2), x, 0, Unintegrable[1/(x^2*(a + b*Tan[c + d*x^(1/3)])^2), x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m deleted file mode 100644 index 8471e2f..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.2.1 (a+b tan)^m (c+d tan)^n.m +++ /dev/null @@ -1,2220 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^5*(a + I*a*Tan[c + d*x]), x, 6, (-I)*a*x - (a*Log[Cos[c + d*x]])/d + (I*a*Tan[c + d*x])/d - (a*Tan[c + d*x]^2)/(2*d) - ((I/3)*a*Tan[c + d*x]^3)/d + (a*Tan[c + d*x]^4)/(4*d) + ((I/5)*a*Tan[c + d*x]^5)/d} -{Tan[c + d*x]^4*(a + I*a*Tan[c + d*x]), x, 5, a*x - (I*a*Log[Cos[c + d*x]])/d - (a*Tan[c + d*x])/d - ((I/2)*a*Tan[c + d*x]^2)/d + (a*Tan[c + d*x]^3)/(3*d) + ((I/4)*a*Tan[c + d*x]^4)/d} -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x]), x, 4, I*a*x + (a*Log[Cos[c + d*x]])/d - (I*a*Tan[c + d*x])/d + (a*Tan[c + d*x]^2)/(2*d) + ((I/3)*a*Tan[c + d*x]^3)/d} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x]), x, 3, -(a*x) + (I*a*Log[Cos[c + d*x]])/d + (a*Tan[c + d*x])/d + ((I/2)*a*Tan[c + d*x]^2)/d} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x]), x, 2, (-I)*a*x - (a*Log[Cos[c + d*x]])/d + (I*a*Tan[c + d*x])/d} -{a + I*a*Tan[c + d*x], x, 2, a*x - (I*a*Log[Cos[c + d*x]])/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x]), x, 2, I*a*x + (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x]), x, 3, -(a*x) - (a*Cot[c + d*x])/d + (I*a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x]), x, 4, (-I)*a*x - (I*a*Cot[c + d*x])/d - (a*Cot[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x]), x, 5, a*x + (a*Cot[c + d*x])/d - ((I/2)*a*Cot[c + d*x]^2)/d - (a*Cot[c + d*x]^3)/(3*d) - (I*a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x]), x, 6, I*a*x + (I*a*Cot[c + d*x])/d + (a*Cot[c + d*x]^2)/(2*d) - ((I/3)*a*Cot[c + d*x]^3)/d - (a*Cot[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x]), x, 7, -(a*x) - (a*Cot[c + d*x])/d + ((I/2)*a*Cot[c + d*x]^2)/d + (a*Cot[c + d*x]^3)/(3*d) - ((I/4)*a*Cot[c + d*x]^4)/d - (a*Cot[c + d*x]^5)/(5*d) + (I*a*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^4*(a + I*a*Tan[c + d*x])^2, x, 6, 2*a^2*x - ((2*I)*a^2*Log[Cos[c + d*x]])/d - (2*a^2*Tan[c + d*x])/d - (I*a^2*Tan[c + d*x]^2)/d + (2*a^2*Tan[c + d*x]^3)/(3*d) + ((I/2)*a^2*Tan[c + d*x]^4)/d - (a^2*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^2, x, 5, (2*I)*a^2*x + (2*a^2*Log[Cos[c + d*x]])/d - ((2*I)*a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^2)/d + (((2*I)/3)*a^2*Tan[c + d*x]^3)/d - (a^2*Tan[c + d*x]^4)/(4*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^2, x, 3, -2*a^2*x + (2*I*a^2*Log[Cos[c + d*x]])/d + (a^2*Tan[c + d*x])/d - (I*(a + I*a*Tan[c + d*x])^3)/(3*a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^2, x, 3, -2*I*a^2*x - (2*a^2*Log[Cos[c + d*x]])/d + (I*a^2*Tan[c + d*x])/d + (a + I*a*Tan[c + d*x])^2/(2*d)} -{(a + I*a*Tan[c + d*x])^2, x, 2, 2*a^2*x - ((2*I)*a^2*Log[Cos[c + d*x]])/d - (a^2*Tan[c + d*x])/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^2, x, 3, (2*I)*a^2*x + (a^2*Log[Cos[c + d*x]])/d + (a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2, x, 3, -2*a^2*x - (a^2*Cot[c + d*x])/d + ((2*I)*a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^2, x, 4, (-2*I)*a^2*x - ((2*I)*a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^2)/(2*d) - (2*a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2, x, 5, 2*a^2*x + (2*a^2*Cot[c + d*x])/d - (I*a^2*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) - ((2*I)*a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2, x, 6, (2*I)*a^2*x + ((2*I)*a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^2)/d - (((2*I)/3)*a^2*Cot[c + d*x]^3)/d - (a^2*Cot[c + d*x]^4)/(4*d) + (2*a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^2, x, 7, -2*a^2*x - (2*a^2*Cot[c + d*x])/d + (I*a^2*Cot[c + d*x]^2)/d + (2*a^2*Cot[c + d*x]^3)/(3*d) - ((I/2)*a^2*Cot[c + d*x]^4)/d - (a^2*Cot[c + d*x]^5)/(5*d) + ((2*I)*a^2*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^3, x, 6, (4*I)*a^3*x + (4*a^3*Log[Cos[c + d*x]])/d - ((4*I)*a^3*Tan[c + d*x])/d + (2*a^3*Tan[c + d*x]^2)/d + (((4*I)/3)*a^3*Tan[c + d*x]^3)/d - (11*a^3*Tan[c + d*x]^4)/(20*d) - (Tan[c + d*x]^4*(a^3 + I*a^3*Tan[c + d*x]))/(5*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^3, x, 4, -4*a^3*x + ((4*I)*a^3*Log[Cos[c + d*x]])/d + (2*a^3*Tan[c + d*x])/d - ((I/2)*a*(a + I*a*Tan[c + d*x])^2)/d - ((I/4)*(a + I*a*Tan[c + d*x])^4)/(a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3, x, 4, (-4*I)*a^3*x - (4*a^3*Log[Cos[c + d*x]])/d + ((2*I)*a^3*Tan[c + d*x])/d + (a*(a + I*a*Tan[c + d*x])^2)/(2*d) + (a + I*a*Tan[c + d*x])^3/(3*d)} -{(a + I*a*Tan[c + d*x])^3, x, 3, 4*a^3*x - ((4*I)*a^3*Log[Cos[c + d*x]])/d - (2*a^3*Tan[c + d*x])/d + ((I/2)*a*(a + I*a*Tan[c + d*x])^2)/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3, x, 5, (4*I)*a^3*x + (3*a^3*Log[Cos[c + d*x]])/d + (a^3*Log[Sin[c + d*x]])/d - (a^3 + I*a^3*Tan[c + d*x])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^3, x, 5, -4*a^3*x + (I*a^3*Log[Cos[c + d*x]])/d + ((3*I)*a^3*Log[Sin[c + d*x]])/d - (Cot[c + d*x]*(a^3 + I*a^3*Tan[c + d*x]))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3, x, 4, (-4*I)*a^3*x - ((2*I)*a^3*Cot[c + d*x])/d - (4*a^3*Log[Sin[c + d*x]])/d - (a*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3, x, 5, 4*a^3*x + (2*a^3*Cot[c + d*x])/d - ((4*I)*a^3*Log[Sin[c + d*x]])/d - ((I/2)*a*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2)/d - (Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3, x, 6, (4*I)*a^3*x + ((4*I)*a^3*Cot[c + d*x])/d + (2*a^3*Cot[c + d*x]^2)/d - (((3*I)/4)*a^3*Cot[c + d*x]^3)/d + (4*a^3*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^4*(a^3 + I*a^3*Tan[c + d*x]))/(4*d)} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3, x, 7, -4*a^3*x - (4*a^3*Cot[c + d*x])/d + ((2*I)*a^3*Cot[c + d*x]^2)/d + (4*a^3*Cot[c + d*x]^3)/(3*d) - (((11*I)/20)*a^3*Cot[c + d*x]^4)/d + ((4*I)*a^3*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^5*(a^3 + I*a^3*Tan[c + d*x]))/(5*d)} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^4, x, 7, (8*I)*a^4*x + (8*a^4*Log[Cos[c + d*x]])/d - ((8*I)*a^4*Tan[c + d*x])/d + (4*a^4*Tan[c + d*x]^2)/d + (((8*I)/3)*a^4*Tan[c + d*x]^3)/d - (67*a^4*Tan[c + d*x]^4)/(60*d) - (Tan[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x])^2)/(6*d) - (7*Tan[c + d*x]^4*(a^4 + I*a^4*Tan[c + d*x]))/(15*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4, x, 5, -8*a^4*x + ((8*I)*a^4*Log[Cos[c + d*x]])/d + (4*a^4*Tan[c + d*x])/d - ((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d - ((I/5)*(a + I*a*Tan[c + d*x])^5)/(a*d) - (I*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4, x, 5, (-8*I)*a^4*x - (8*a^4*Log[Cos[c + d*x]])/d + ((4*I)*a^4*Tan[c + d*x])/d + (a*(a + I*a*Tan[c + d*x])^3)/(3*d) + (a + I*a*Tan[c + d*x])^4/(4*d) + (a^2 + I*a^2*Tan[c + d*x])^2/d} -{(a + I*a*Tan[c + d*x])^4, x, 4, 8*a^4*x - ((8*I)*a^4*Log[Cos[c + d*x]])/d - (4*a^4*Tan[c + d*x])/d + ((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d + (I*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4, x, 6, (8*I)*a^4*x + (7*a^4*Log[Cos[c + d*x]])/d + (a^4*Log[Sin[c + d*x]])/d - (a^2 + I*a^2*Tan[c + d*x])^2/(2*d) - (3*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4, x, 5, -8*a^4*x + ((4*I)*a^4*Log[Cos[c + d*x]])/d + ((4*I)*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4, x, 6, (-8*I)*a^4*x - (a^4*Log[Cos[c + d*x]])/d - (7*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*I)*Cot[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4, x, 5, 8*a^4*x + (4*a^4*Cot[c + d*x])/d - ((8*I)*a^4*Log[Sin[c + d*x]])/d - (a*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(3*d) - (I*Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4, x, 6, (8*I)*a^4*x + ((4*I)*a^4*Cot[c + d*x])/d + (8*a^4*Log[Sin[c + d*x]])/d - ((I/3)*a*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - (Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4)/(4*d) + (Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4, x, 7, -8*a^4*x - (8*a^4*Cot[c + d*x])/d + ((4*I)*a^4*Cot[c + d*x]^2)/d + (23*a^4*Cot[c + d*x]^3)/(15*d) + ((8*I)*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/(5*d) - (((3*I)/5)*Cot[c + d*x]^4*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4, x, 8, (-8*I)*a^4*x - ((8*I)*a^4*Cot[c + d*x])/d - (4*a^4*Cot[c + d*x]^2)/d + (((8*I)/3)*a^4*Cot[c + d*x]^3)/d + (67*a^4*Cot[c + d*x]^4)/(60*d) - (8*a^4*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^6*(a^2 + I*a^2*Tan[c + d*x])^2)/(6*d) - (((7*I)/15)*Cot[c + d*x]^5*(a^4 + I*a^4*Tan[c + d*x]))/d} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^6/(a + I*a*Tan[c + d*x]), x, 6, (5*x)/(2*a) + ((3*I)*Log[Cos[c + d*x]])/(a*d) - (5*Tan[c + d*x])/(2*a*d) + (((3*I)/2)*Tan[c + d*x]^2)/(a*d) + (5*Tan[c + d*x]^3)/(6*a*d) - (((3*I)/4)*Tan[c + d*x]^4)/(a*d) - Tan[c + d*x]^5/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x]), x, 5, (((-5*I)/2)*x)/a + (2*Log[Cos[c + d*x]])/(a*d) + (((5*I)/2)*Tan[c + d*x])/(a*d) + Tan[c + d*x]^2/(a*d) - (((5*I)/6)*Tan[c + d*x]^3)/(a*d) - Tan[c + d*x]^4/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x]), x, 4, (-3*x)/(2*a) - ((2*I)*Log[Cos[c + d*x]])/(a*d) + (3*Tan[c + d*x])/(2*a*d) - (I*Tan[c + d*x]^2)/(a*d) - Tan[c + d*x]^3/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x]), x, 3, (((3*I)/2)*x)/a - Log[Cos[c + d*x]]/(a*d) - (((3*I)/2)*Tan[c + d*x])/(a*d) - Tan[c + d*x]^2/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x]), x, 3, x/(2*a) + (I*Log[Cos[c + d*x]])/(a*d) - I/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x]), x, 2, -((I*x)/(2*a)) - 1/(2*d*(a + I*a*Tan[c + d*x]))} -{(a + I*a*Tan[c + d*x])^(-1), x, 2, x/(2*a) + (I/2)/(d*(a + I*a*Tan[c + d*x]))} -{Cot[c + d*x]/(a + I*a*Tan[c + d*x]), x, 4, ((-I/2)*x)/a + Log[Sin[c + d*x]]/(a*d) + 1/(2*d*(a + I*a*Tan[c + d*x]))} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x]), x, 4, (-3*x)/(2*a) - (3*Cot[c + d*x])/(2*a*d) - (I*Log[Sin[c + d*x]])/(a*d) + Cot[c + d*x]/(2*d*(a + I*a*Tan[c + d*x]))} -{Cot[c + d*x]^3/(a + I*a*Tan[c + d*x]), x, 5, (((3*I)/2)*x)/a + (((3*I)/2)*Cot[c + d*x])/(a*d) - Cot[c + d*x]^2/(a*d) - (2*Log[Sin[c + d*x]])/(a*d) + Cot[c + d*x]^2/(2*d*(a + I*a*Tan[c + d*x]))} -{Cot[c + d*x]^4/(a + I*a*Tan[c + d*x]), x, 6, (5*x)/(2*a) + (5*Cot[c + d*x])/(2*a*d) + (I*Cot[c + d*x]^2)/(a*d) - (5*Cot[c + d*x]^3)/(6*a*d) + ((2*I)*Log[Sin[c + d*x]])/(a*d) + Cot[c + d*x]^3/(2*d*(a + I*a*Tan[c + d*x]))} - - -{Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^2, x, 6, (-25*x)/(4*a^2) - ((6*I)*Log[Cos[c + d*x]])/(a^2*d) + (25*Tan[c + d*x])/(4*a^2*d) - ((3*I)*Tan[c + d*x]^2)/(a^2*d) - (25*Tan[c + d*x]^3)/(12*a^2*d) + (((3*I)/2)*Tan[c + d*x]^4)/(a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^5/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^2, x, 5, (((15*I)/4)*x)/a^2 - (4*Log[Cos[c + d*x]])/(a^2*d) - (((15*I)/4)*Tan[c + d*x])/(a^2*d) - (2*Tan[c + d*x]^2)/(a^2*d) + (((5*I)/4)*Tan[c + d*x]^3)/(a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^4/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^2, x, 4, (9*x)/(4*a^2) + ((2*I)*Log[Cos[c + d*x]])/(a^2*d) - (9*Tan[c + d*x])/(4*a^2*d) + (I*Tan[c + d*x]^2)/(a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^3/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^2, x, 6, -((3*I*x)/(4*a^2)) + Log[Cos[c + d*x]]/(a^2*d) - 3/(4*a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^2/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^2, x, 3, -(x/(4*a^2)) + (3*I)/(4*a^2*d*(1 + I*Tan[c + d*x])) - I/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x])^2, x, 3, ((-I/4)*x)/a^2 - 1/(4*d*(a + I*a*Tan[c + d*x])^2) + 1/(4*d*(a^2 + I*a^2*Tan[c + d*x]))} -{Tan[c + d*x]^0/(a + I*a*Tan[c + d*x])^2, x, 3, x/(4*a^2) + (I/4)/(d*(a + I*a*Tan[c + d*x])^2) + (I/4)/(d*(a^2 + I*a^2*Tan[c + d*x]))} -{Cot[c + d*x]^1/(a + I*a*Tan[c + d*x])^2, x, 4, (((-3*I)/4)*x)/a^2 + Log[Sin[c + d*x]]/(a^2*d) + 3/(4*a^2*d*(1 + I*Tan[c + d*x])) + 1/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^2, x, 5, (-9*x)/(4*a^2) - (9*Cot[c + d*x])/(4*a^2*d) - ((2*I)*Log[Sin[c + d*x]])/(a^2*d) + Cot[c + d*x]/(a^2*d*(1 + I*Tan[c + d*x])) + Cot[c + d*x]/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Cot[c + d*x]^3/(a + I*a*Tan[c + d*x])^2, x, 6, (((15*I)/4)*x)/a^2 + (((15*I)/4)*Cot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^2)/(a^2*d) - (4*Log[Sin[c + d*x]])/(a^2*d) + (5*Cot[c + d*x]^2)/(4*a^2*d*(1 + I*Tan[c + d*x])) + Cot[c + d*x]^2/(4*d*(a + I*a*Tan[c + d*x])^2)} - - -{Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^3, x, 6, (55*x)/(8*a^3) + ((7*I)*Log[Cos[c + d*x]])/(a^3*d) - (55*Tan[c + d*x])/(8*a^3*d) + (((7*I)/2)*Tan[c + d*x]^2)/(a^3*d) - Tan[c + d*x]^5/(6*d*(a + I*a*Tan[c + d*x])^3) + (((13*I)/24)*Tan[c + d*x]^4)/(a*d*(a + I*a*Tan[c + d*x])^2) + (55*Tan[c + d*x]^3)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^3, x, 5, (((-25*I)/8)*x)/a^3 + (3*Log[Cos[c + d*x]])/(a^3*d) + (((25*I)/8)*Tan[c + d*x])/(a^3*d) - Tan[c + d*x]^4/(6*d*(a + I*a*Tan[c + d*x])^3) + (((11*I)/24)*Tan[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^2) + (3*Tan[c + d*x]^2)/(2*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^3, x, 7, -((7*x)/(8*a^3)) - (I*Log[Cos[c + d*x]])/(a^3*d) - Tan[c + d*x]^3/(6*d*(a + I*a*Tan[c + d*x])^3) + (3*I*Tan[c + d*x]^2)/(8*a*d*(a + I*a*Tan[c + d*x])^2) + (7*I)/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^3, x, 4, (I*x)/(8*a^3) + 3/(8*a^3*d*(1 + I*Tan[c + d*x])) + (I*Tan[c + d*x]^3)/(6*d*(a + I*a*Tan[c + d*x])^3) - 1/(8*a*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^3, x, 4, -(x/(8*a^3)) - I/(6*d*(a + I*a*Tan[c + d*x])^3) + (3*I)/(8*a*d*(a + I*a*Tan[c + d*x])^2) - I/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Tan[c + d*x]/(a + I*a*Tan[c + d*x])^3, x, 4, ((-I/8)*x)/a^3 - 1/(6*d*(a + I*a*Tan[c + d*x])^3) + 1/(8*a*d*(a + I*a*Tan[c + d*x])^2) + 1/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(a + I*a*Tan[c + d*x])^(-3), x, 4, x/(8*a^3) + (I/6)/(d*(a + I*a*Tan[c + d*x])^3) + (I/8)/(a*d*(a + I*a*Tan[c + d*x])^2) + (I/8)/(d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cot[c + d*x]/(a + I*a*Tan[c + d*x])^3, x, 5, (((-7*I)/8)*x)/a^3 + Log[Sin[c + d*x]]/(a^3*d) + 1/(6*d*(a + I*a*Tan[c + d*x])^3) + 3/(8*a*d*(a + I*a*Tan[c + d*x])^2) + 7/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^3, x, 6, (-25*x)/(8*a^3) - (25*Cot[c + d*x])/(8*a^3*d) - ((3*I)*Log[Sin[c + d*x]])/(a^3*d) + Cot[c + d*x]/(6*d*(a + I*a*Tan[c + d*x])^3) + (11*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (3*Cot[c + d*x])/(2*d*(a^3 + I*a^3*Tan[c + d*x]))} - - -{Tan[c + d*x]^6/(a + I*a*Tan[c + d*x])^4, x, 6, (-65*x)/(16*a^4) - ((4*I)*Log[Cos[c + d*x]])/(a^4*d) + (65*Tan[c + d*x])/(16*a^4*d) - ((2*I)*Tan[c + d*x]^2)/(a^4*d*(1 + I*Tan[c + d*x])) + (31*Tan[c + d*x]^3)/(48*a^4*d*(1 + I*Tan[c + d*x])^2) - Tan[c + d*x]^5/(8*d*(a + I*a*Tan[c + d*x])^4) + (((7*I)/24)*Tan[c + d*x]^4)/(a*d*(a + I*a*Tan[c + d*x])^3)} -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^4, x, 8, (15*I*x)/(16*a^4) - Log[Cos[c + d*x]]/(a^4*d) + 15/(16*a^4*d*(1 + I*Tan[c + d*x])) + (7*Tan[c + d*x]^2)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) - Tan[c + d*x]^4/(8*d*(a + I*a*Tan[c + d*x])^4) + (I*Tan[c + d*x]^3)/(4*a*d*(a + I*a*Tan[c + d*x])^3)} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^4, x, 5, x/(16*a^4) - (3*I)/(16*a^4*d*(1 + I*Tan[c + d*x])) + (I*Tan[c + d*x]^4)/(8*d*(a + I*a*Tan[c + d*x])^4) + Tan[c + d*x]^3/(12*a*d*(a + I*a*Tan[c + d*x])^3) + I/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^4, x, 5, (I*x)/(16*a^4) + 3/(16*a^4*d*(1 + I*Tan[c + d*x])) + Tan[c + d*x]^4/(8*d*(a + I*a*Tan[c + d*x])^4) + (I*Tan[c + d*x]^3)/(12*a*d*(a + I*a*Tan[c + d*x])^3) - 1/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2)} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^4, x, 5, -(x/(16*a^4)) - I/(8*d*(a + I*a*Tan[c + d*x])^4) + I/(4*a*d*(a + I*a*Tan[c + d*x])^3) - I/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) - I/(16*d*(a^4 + I*a^4*Tan[c + d*x]))} -{Tan[c + d*x]/(a + I*a*Tan[c + d*x])^4, x, 5, ((-I/16)*x)/a^4 - 1/(8*d*(a + I*a*Tan[c + d*x])^4) + 1/(12*a*d*(a + I*a*Tan[c + d*x])^3) + 1/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) + 1/(16*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(a + I*a*Tan[c + d*x])^(-4), x, 5, x/(16*a^4) + (I/8)/(d*(a + I*a*Tan[c + d*x])^4) + (I/12)/(a*d*(a + I*a*Tan[c + d*x])^3) + (I/16)/(d*(a^2 + I*a^2*Tan[c + d*x])^2) + (I/16)/(d*(a^4 + I*a^4*Tan[c + d*x]))} -{Cot[c + d*x]/(a + I*a*Tan[c + d*x])^4, x, 6, (((-15*I)/16)*x)/a^4 + Log[Sin[c + d*x]]/(a^4*d) + 7/(16*a^4*d*(1 + I*Tan[c + d*x])^2) + 15/(16*a^4*d*(1 + I*Tan[c + d*x])) + 1/(8*d*(a + I*a*Tan[c + d*x])^4) + 1/(4*a*d*(a + I*a*Tan[c + d*x])^3)} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^4, x, 7, (-65*x)/(16*a^4) - (65*Cot[c + d*x])/(16*a^4*d) - ((4*I)*Log[Sin[c + d*x]])/(a^4*d) + (31*Cot[c + d*x])/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + (2*Cot[c + d*x])/(a^4*d*(1 + I*Tan[c + d*x])) + Cot[c + d*x]/(8*d*(a + I*a*Tan[c + d*x])^4) + (7*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]], x, 6, -((I*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d) + (8*I*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*I*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (2*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(7*d) + (62*I*(a + I*a*Tan[c + d*x])^(3/2))/(105*a*d)} -{Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]], x, 5, (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) + (2*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) - (2*(a + I*a*Tan[c + d*x])^(3/2))/(15*a*d)} -{Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]], x, 3, (I*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (((2*I)/3)*(a + I*a*Tan[c + d*x])^(3/2))/(a*d)} -{Tan[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]], x, 3, -((Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d) + (2*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[a + I*a*Tan[c + d*x]], x, 2, ((-I)*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]], x, 6, (-2*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]], x, 8, ((-I)*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (I*Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]], x, 8, (7*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - ((I/4)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - (Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2), x, 7, (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (((2*I)/7)*a^2*Tan[c + d*x]^3)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*a^2*Tan[c + d*x]^4)/(7*d*Sqrt[a + I*a*Tan[c + d*x]]) - (64*a*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (16*a*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (76*(a + I*a*Tan[c + d*x])^(3/2))/(105*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2), x, 4, ((2*I)*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - ((2*I)*a*Sqrt[a + I*a*Tan[c + d*x]])/d - (((2*I)/5)*(a + I*a*Tan[c + d*x])^(5/2))/(a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2), x, 4, (-2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (2*a*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{(a + I*a*Tan[c + d*x])^(3/2), x, 3, ((-2*I)*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + ((2*I)*a*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2), x, 6, (-2*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2), x, 8, ((-3*I)*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + ((2*I)*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (I*a^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (a^2*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2), x, 9, (11*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - ((I/2)*a^2*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (a^2*Cot[c + d*x]^2)/(2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/4)*a*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2), x, 7, (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (368*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) + (92*a^2*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) + (((38*I)/63)*a^2*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a^2*Tan[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]])/(9*d) - (472*a*(a + I*a*Tan[c + d*x])^(3/2))/(315*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((4*I)*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - ((4*I)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/d - (((2*I)/3)*a*(a + I*a*Tan[c + d*x])^(3/2))/d - (((2*I)/7)*(a + I*a*Tan[c + d*x])^(7/2))/(a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2), x, 5, (-4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (4*a^2*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*a*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) + (2*(a + I*a*Tan[c + d*x])^(5/2))/(5*d)} -{(a + I*a*Tan[c + d*x])^(5/2), x, 4, ((-4*I)*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + ((4*I)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/d + (((2*I)/3)*a*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2), x, 7, (-2*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (2*a^2*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2), x, 7, ((-5*I)*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + ((4*I)*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a^2*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2), x, 8, (23*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (((9*I)/4)*a^2*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d - (a^2*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2), x, 9, (((45*I)/8)*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d - ((4*I)*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (19*a^2*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (((13*I)/12)*a^2*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/d - (a^2*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} - - -{(a + I*a*Tan[c + d*x])^(7/2), x, 5, ((-8*I)*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + ((8*I)*a^3*Sqrt[a + I*a*Tan[c + d*x]])/d + (((4*I)/3)*a^2*(a + I*a*Tan[c + d*x])^(3/2))/d + (((2*I)/5)*a*(a + I*a*Tan[c + d*x])^(5/2))/d} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^5/Sqrt[a + I*a*Tan[c + d*x]], x, 7, -(ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d)) - Tan[c + d*x]^4/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (188*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d) + (47*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*a*d) - (9*I*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(7*a*d) + (223*(a + I*a*Tan[c + d*x])^(3/2))/(105*a^2*d)} -{Tan[c + d*x]^4/Sqrt[a + I*a*Tan[c + d*x]], x, 6, -((I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d)) - Tan[c + d*x]^3/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (28*I*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d) - (7*I*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d) - (23*I*(a + I*a*Tan[c + d*x])^(3/2))/(15*a^2*d)} -{Tan[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]], x, 5, ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) - Tan[c + d*x]^2/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (4*Sqrt[a + I*a*Tan[c + d*x]])/(a*d) - (5*(a + I*a*Tan[c + d*x])^(3/2))/(3*a^2*d)} -{Tan[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) - I/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{Tan[c + d*x]^1/Sqrt[a + I*a*Tan[c + d*x]], x, 3, -(ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d)) - 1/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^0/Sqrt[a + I*a*Tan[c + d*x]], x, 3, ((-I)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + I/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^1/Sqrt[a + I*a*Tan[c + d*x]], x, 7, (-2*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) + 1/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]], x, 8, (I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + Cot[c + d*x]/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{Cot[c + d*x]^3/Sqrt[a + I*a*Tan[c + d*x]], x, 9, (11*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) + Cot[c + d*x]^2/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*I*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a*d) - (3*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d)} - - -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^(3/2), x, 7, -ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]^4/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((19*I)/6)*Tan[c + d*x]^3)/(a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (78*Sqrt[a + I*a*Tan[c + d*x]])/(5*a^2*d) - (39*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(10*a^2*d) - (151*(a + I*a*Tan[c + d*x])^(3/2))/(30*a^3*d)} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(3/2), x, 6, ((-I/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]^3/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (((5*I)/2)*Tan[c + d*x]^2)/(a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((10*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) + (((7*I)/2)*(a + I*a*Tan[c + d*x])^(3/2))/(a^3*d)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2), x, 5, ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]^2/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) - 11/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(3*a^2*d)} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) - I/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (3*I)/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x])^(3/2), x, 4, -ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - 1/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + 1/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^0/(a + I*a*Tan[c + d*x])^(3/2), x, 4, ((-I/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(3/2)*d) + (I/3)/(d*(a + I*a*Tan[c + d*x])^(3/2)) + (I/2)/(a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^1/(a + I*a*Tan[c + d*x])^(3/2), x, 8, (-2*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) + 1/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + 3/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(3/2), x, 9, ((3*I)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I/2)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(3/2)*d) + Cot[c + d*x]/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (13*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d)} -{Cot[c + d*x]^3/(a + I*a*Tan[c + d*x])^(3/2), x, 10, (23*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) + Cot[c + d*x]^2/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (17*Cot[c + d*x]^2)/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((21*I)/4)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) - (11*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(3*a^2*d)} - - -{Tan[c + d*x]^5/(a + I*a*Tan[c + d*x])^(5/2), x, 7, -ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]^4/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (((7*I)/10)*Tan[c + d*x]^3)/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (89*Tan[c + d*x]^2)/(20*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (89*Sqrt[a + I*a*Tan[c + d*x]])/(5*a^3*d) + (361*(a + I*a*Tan[c + d*x])^(3/2))/(60*a^4*d)} -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(5/2), x, 6, ((-I/4)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]^3/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (((17*I)/30)*Tan[c + d*x]^2)/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((151*I)/60)/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((83*I)/30)*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(5/2), x, 5, ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]^2/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) - 13/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + 31/(20*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2), x, 5, (I*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) - I/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + I/(2*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - I/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x])^(5/2), x, 5, -ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - 1/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + 1/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + 1/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^0/(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((-I/4)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(5/2)*d) + (I/5)/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (I/6)/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I/4)/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^1/(a + I*a*Tan[c + d*x])^(5/2), x, 9, (-2*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) + 1/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + 1/(2*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + 7/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(5/2), x, 10, ((5*I)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((I/4)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(5/2)*d) + Cot[c + d*x]/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (19*Cot[c + d*x])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (41*Cot[c + d*x])/(12*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (21*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^3*d)} - - -{(a + I*a*Tan[c + d*x])^(-7/2), x, 6, ((-I/8)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*a^(7/2)*d) + (I/7)/(d*(a + I*a*Tan[c + d*x])^(7/2)) + (I/10)/(a*d*(a + I*a*Tan[c + d*x])^(5/2)) + (I/12)/(a^2*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I/8)/(a^3*d*Sqrt[a + I*a*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x]), x, 5, (-2*(-1)^(3/4)*a*d^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - ((2*I)*a*d^2*Sqrt[d*Tan[e + f*x]])/f + (2*a*d*(d*Tan[e + f*x])^(3/2))/(3*f) + (((2*I)/5)*a*(d*Tan[e + f*x])^(5/2))/f} -{(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x]), x, 4, (2*(-1)^(1/4)*a*d^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (2*a*d*Sqrt[d*Tan[e + f*x]])/f + (((2*I)/3)*a*(d*Tan[e + f*x])^(3/2))/f} -{Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x]), x, 3, (2*(-1)^(3/4)*a*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + ((2*I)*a*Sqrt[d*Tan[e + f*x]])/f} -{(a + I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]], x, 2, (-2*(-1)^(1/4)*a*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f)} -{(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2), x, 3, (-2*(-1)^(3/4)*a*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f) - (2*a)/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2), x, 4, (2*(-1)^(1/4)*a*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (2*a)/(3*d*f*(d*Tan[e + f*x])^(3/2)) - ((2*I)*a)/(d^2*f*Sqrt[d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2), x, 5, (2*(-1)^(3/4)*a*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(7/2)*f) - (2*a)/(5*d*f*(d*Tan[e + f*x])^(5/2)) - (((2*I)/3)*a)/(d^2*f*(d*Tan[e + f*x])^(3/2)) + (2*a)/(d^3*f*Sqrt[d*Tan[e + f*x]])} - -{(d*Tan[e + f*x])^(5/2)*(a - I*a*Tan[e + f*x]), x, 5, (2*(-1)^(3/4)*a*d^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (2*I*a*d^2*Sqrt[d*Tan[e + f*x]])/f + (2*a*d*(d*Tan[e + f*x])^(3/2))/(3*f) - (2*I*a*(d*Tan[e + f*x])^(5/2))/(5*f)} -{(d*Tan[e + f*x])^(3/2)*(a - I*a*Tan[e + f*x]), x, 4, (2*(-1)^(1/4)*a*d^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (2*a*d*Sqrt[d*Tan[e + f*x]])/f - (2*I*a*(d*Tan[e + f*x])^(3/2))/(3*f)} -{Sqrt[d*Tan[e + f*x]]*(a - I*a*Tan[e + f*x]), x, 3, -((2*(-1)^(3/4)*a*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f) - (2*I*a*Sqrt[d*Tan[e + f*x]])/f} -{(a - I*a*Tan[e + f*x])/Sqrt[d*Tan[e + f*x]], x, 2, -((2*(-1)^(1/4)*a*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f))} -{(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2), x, 3, (2*(-1)^(3/4)*a*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f) - (2*a)/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2), x, 4, (2*(-1)^(1/4)*a*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (2*a)/(3*d*f*(d*Tan[e + f*x])^(3/2)) + (2*I*a)/(d^2*f*Sqrt[d*Tan[e + f*x]])} -{(a - I*a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2), x, 5, -((2*(-1)^(3/4)*a*ArcTanh[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(7/2)*f)) - (2*a)/(5*d*f*(d*Tan[e + f*x])^(5/2)) + (2*I*a)/(3*d^2*f*(d*Tan[e + f*x])^(3/2)) + (2*a)/(d^3*f*Sqrt[d*Tan[e + f*x]])} - - -{(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^2, x, 6, (-4*(-1)^(3/4)*a^2*d^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - ((4*I)*a^2*d^2*Sqrt[d*Tan[e + f*x]])/f + (4*a^2*d*(d*Tan[e + f*x])^(3/2))/(3*f) + (((4*I)/5)*a^2*(d*Tan[e + f*x])^(5/2))/f - (2*a^2*(d*Tan[e + f*x])^(7/2))/(7*d*f)} -{(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^2, x, 5, (4*(-1)^(1/4)*a^2*d^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (4*a^2*d*Sqrt[d*Tan[e + f*x]])/f + (((4*I)/3)*a^2*(d*Tan[e + f*x])^(3/2))/f - (2*a^2*(d*Tan[e + f*x])^(5/2))/(5*d*f)} -{Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2, x, 4, (4*(-1)^(3/4)*a^2*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + ((4*I)*a^2*Sqrt[d*Tan[e + f*x]])/f - (2*a^2*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{(a + I*a*Tan[e + f*x])^2/Sqrt[d*Tan[e + f*x]], x, 3, (-4*(-1)^(1/4)*a^2*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f) - (2*a^2*Sqrt[d*Tan[e + f*x]])/(d*f)} -{(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2), x, 3, (-4*(-1)^(3/4)*a^2*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f) - (2*a^2)/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2), x, 4, (4*(-1)^(1/4)*a^2*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (2*a^2)/(3*d*f*(d*Tan[e + f*x])^(3/2)) - ((4*I)*a^2)/(d^2*f*Sqrt[d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(7/2), x, 5, (4*(-1)^(3/4)*a^2*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(7/2)*f) - (2*a^2)/(5*d*f*(d*Tan[e + f*x])^(5/2)) - (((4*I)/3)*a^2)/(d^2*f*(d*Tan[e + f*x])^(3/2)) + (4*a^2)/(d^3*f*Sqrt[d*Tan[e + f*x]])} - - -{(d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^3, x, 7, (-8*(-1)^(3/4)*a^3*d^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - ((8*I)*a^3*d^2*Sqrt[d*Tan[e + f*x]])/f + (8*a^3*d*(d*Tan[e + f*x])^(3/2))/(3*f) + (((8*I)/5)*a^3*(d*Tan[e + f*x])^(5/2))/f - (40*a^3*(d*Tan[e + f*x])^(7/2))/(63*d*f) - (2*(d*Tan[e + f*x])^(7/2)*(a^3 + I*a^3*Tan[e + f*x]))/(9*d*f)} -{(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^3, x, 6, (8*(-1)^(1/4)*a^3*d^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (8*a^3*d*Sqrt[d*Tan[e + f*x]])/f + (((8*I)/3)*a^3*(d*Tan[e + f*x])^(3/2))/f - (32*a^3*(d*Tan[e + f*x])^(5/2))/(35*d*f) - (2*(d*Tan[e + f*x])^(5/2)*(a^3 + I*a^3*Tan[e + f*x]))/(7*d*f)} -{Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^3, x, 5, (8*(-1)^(3/4)*a^3*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + ((8*I)*a^3*Sqrt[d*Tan[e + f*x]])/f - (8*a^3*(d*Tan[e + f*x])^(3/2))/(5*d*f) - (2*(d*Tan[e + f*x])^(3/2)*(a^3 + I*a^3*Tan[e + f*x]))/(5*d*f)} -{(a + I*a*Tan[e + f*x])^3/Sqrt[d*Tan[e + f*x]], x, 4, (-8*(-1)^(1/4)*a^3*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f) - (16*a^3*Sqrt[d*Tan[e + f*x]])/(3*d*f) - (2*Sqrt[d*Tan[e + f*x]]*(a^3 + I*a^3*Tan[e + f*x]))/(3*d*f)} -{(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(3/2), x, 4, (-8*(-1)^(3/4)*a^3*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f) - (2*(a^3 + I*a^3*Tan[e + f*x]))/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(5/2), x, 4, (8*(-1)^(1/4)*a^3*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (((16*I)/3)*a^3)/(d^2*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + I*a^3*Tan[e + f*x]))/(3*d*f*(d*Tan[e + f*x])^(3/2))} -{(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(7/2), x, 5, (8*(-1)^(3/4)*a^3*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(7/2)*f) - (((8*I)/5)*a^3)/(d^2*f*(d*Tan[e + f*x])^(3/2)) + (8*a^3)/(d^3*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + I*a^3*Tan[e + f*x]))/(5*d*f*(d*Tan[e + f*x])^(5/2))} -{(a + I*a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(9/2), x, 6, (-8*(-1)^(1/4)*a^3*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(9/2)*f) - (((32*I)/35)*a^3)/(d^2*f*(d*Tan[e + f*x])^(5/2)) + (8*a^3)/(3*d^3*f*(d*Tan[e + f*x])^(3/2)) + ((8*I)*a^3)/(d^4*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + I*a^3*Tan[e + f*x]))/(7*d*f*(d*Tan[e + f*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x]), x, 13, ((5/4 - (7*I)/4)*d^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f) - ((5/4 - (7*I)/4)*d^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f) + ((5/8 + (7*I)/8)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) - ((5/8 + (7*I)/8)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) + (5*d^3*Sqrt[d*Tan[e + f*x]])/(2*a*f) - (7*I*d^2*(d*Tan[e + f*x])^(3/2))/(6*a*f) - (d*(d*Tan[e + f*x])^(5/2))/(2*f*(a + I*a*Tan[e + f*x]))} -{(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]), x, 12, -(((3/4 + (5*I)/4)*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f)) + ((3/4 + (5*I)/4)*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f) + ((3/8 - (5*I)/8)*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) - ((3/8 - (5*I)/8)*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) - (5*I*d^2*Sqrt[d*Tan[e + f*x]])/(2*a*f) - (d*(d*Tan[e + f*x])^(3/2))/(2*f*(a + I*a*Tan[e + f*x]))} -{(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x]), x, 11, -(((1/4 - (3*I)/4)*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f)) + ((1/4 - (3*I)/4)*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*f) - ((1/8 + (3*I)/8)*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) + ((1/8 + (3*I)/8)*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*f) - (d*Sqrt[d*Tan[e + f*x]])/(2*f*(a + I*a*Tan[e + f*x]))} -{Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x]), x, 3, ((-1)^(3/4)*Sqrt[d]*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*a*f) + ((I/2)*Sqrt[d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x]))} -{1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])), x, 11, -(((3/4 - I/4)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*Sqrt[d]*f)) + ((3/4 - I/4)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*Sqrt[d]*f) - ((3/8 + I/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*Sqrt[d]*f) + ((3/8 + I/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*Sqrt[d]*f) + Sqrt[d*Tan[e + f*x]]/(2*d*f*(a + I*a*Tan[e + f*x]))} -{1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])), x, 12, ((5/4 + (3*I)/4)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*d^(3/2)*f) - ((5/4 + (3*I)/4)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*d^(3/2)*f) - ((5/8 - (3*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*d^(3/2)*f) + ((5/8 - (3*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*d^(3/2)*f) - 5/(2*a*d*f*Sqrt[d*Tan[e + f*x]]) + 1/(2*d*f*Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x]))} -{1/((d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])), x, 13, ((7/4 - (5*I)/4)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*d^(5/2)*f) - ((7/4 - (5*I)/4)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a*d^(5/2)*f) + ((7/8 + (5*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*d^(5/2)*f) - ((7/8 + (5*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a*d^(5/2)*f) - 7/(6*a*d*f*(d*Tan[e + f*x])^(3/2)) + (5*I)/(2*a*d^2*f*Sqrt[d*Tan[e + f*x]]) + 1/(2*d*f*(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x]))} - - -{(d*Tan[e + f*x])^(9/2)/(a + I*a*Tan[e + f*x])^2, x, 14, -(((49/16 + (45*I)/16)*d^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f)) + ((49/16 + (45*I)/16)*d^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) + ((49/32 - (45*I)/32)*d^(9/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - ((49/32 - (45*I)/32)*d^(9/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - (45*I*d^4*Sqrt[d*Tan[e + f*x]])/(8*a^2*f) - (49*d^3*(d*Tan[e + f*x])^(3/2))/(24*a^2*f) + (9*I*d^2*(d*Tan[e + f*x])^(5/2))/(8*a^2*f*(1 + I*Tan[e + f*x])) - (d*(d*Tan[e + f*x])^(7/2))/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x])^2, x, 13, -(((25/16 - (21*I)/16)*d^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f)) + ((25/16 - (21*I)/16)*d^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) - ((25/32 + (21*I)/32)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + ((25/32 + (21*I)/32)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - (25*d^3*Sqrt[d*Tan[e + f*x]])/(8*a^2*f) + (7*I*d^2*(d*Tan[e + f*x])^(3/2))/(8*a^2*f*(1 + I*Tan[e + f*x])) - (d*(d*Tan[e + f*x])^(5/2))/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2, x, 12, ((9/16 + (5*I)/16)*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) - ((9/16 + (5*I)/16)*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) - ((9/32 - (5*I)/32)*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + ((9/32 - (5*I)/32)*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + (5*I*d^2*Sqrt[d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) - (d*(d*Tan[e + f*x])^(3/2))/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2, x, 12, ((1/16 + (3*I)/16)*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) - ((1/16 + (3*I)/16)*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) + ((1/32 - (3*I)/32)*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - ((1/32 - (3*I)/32)*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + (3*d*Sqrt[d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) - (d*Sqrt[d*Tan[e + f*x]])/(4*f*(a + I*a*Tan[e + f*x])^2)} -{Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2, x, 12, -(((1/16 - (3*I)/16)*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f)) + ((1/16 - (3*I)/16)*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*f) + ((1/32 + (3*I)/32)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) - ((1/32 + (3*I)/32)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*f) + (I*Sqrt[d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) + (I*Sqrt[d*Tan[e + f*x]])/(4*f*(a + I*a*Tan[e + f*x])^2)} -{1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2), x, 12, -(((9/16 - (5*I)/16)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*Sqrt[d]*f)) + ((9/16 - (5*I)/16)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*Sqrt[d]*f) - ((9/32 + (5*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*Sqrt[d]*f) + ((9/32 + (5*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*Sqrt[d]*f) + (5*Sqrt[d*Tan[e + f*x]])/(8*a^2*d*f*(1 + I*Tan[e + f*x])) + Sqrt[d*Tan[e + f*x]]/(4*d*f*(a + I*a*Tan[e + f*x])^2)} -{1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^2), x, 13, ((25/16 + (21*I)/16)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*d^(3/2)*f) - ((25/16 + (21*I)/16)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*d^(3/2)*f) - ((25/32 - (21*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*d^(3/2)*f) + ((25/32 - (21*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*d^(3/2)*f) - 25/(8*a^2*d*f*Sqrt[d*Tan[e + f*x]]) + 7/(8*a^2*d*f*(1 + I*Tan[e + f*x])*Sqrt[d*Tan[e + f*x]]) + 1/(4*d*f*Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2)} -{1/((d*Tan[e + f*x])^(5/2)*(a + I*a*Tan[e + f*x])^2), x, 14, ((49/16 - (45*I)/16)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*d^(5/2)*f) - ((49/16 - (45*I)/16)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^2*d^(5/2)*f) + ((49/32 + (45*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*d^(5/2)*f) - ((49/32 + (45*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^2*d^(5/2)*f) - 49/(24*a^2*d*f*(d*Tan[e + f*x])^(3/2)) + 9/(8*a^2*d*f*(1 + I*Tan[e + f*x])*(d*Tan[e + f*x])^(3/2)) + (45*I)/(8*a^2*d^2*f*Sqrt[d*Tan[e + f*x]]) + 1/(4*d*f*(d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^2)} - - -{(d*Tan[e + f*x])^(9/2)/(a + I*a*Tan[e + f*x])^3, x, 14, ((7/4 + (15*I)/8)*d^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) - ((7/4 + (15*I)/8)*d^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) - ((7/8 - (15*I)/16)*d^(9/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) + ((7/8 - (15*I)/16)*d^(9/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) + (15*I*d^4*Sqrt[d*Tan[e + f*x]])/(4*a^3*f) - (d*(d*Tan[e + f*x])^(7/2))/(6*f*(a + I*a*Tan[e + f*x])^3) + (5*I*d^2*(d*Tan[e + f*x])^(5/2))/(12*a*f*(a + I*a*Tan[e + f*x])^2) + (7*d^3*(d*Tan[e + f*x])^(3/2))/(6*f*(a^3 + I*a^3*Tan[e + f*x]))} -{(d*Tan[e + f*x])^(7/2)/(a + I*a*Tan[e + f*x])^3, x, 13, ((5/16 - (7*I)/16)*d^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) - ((5/16 - (7*I)/16)*d^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) + ((5/32 + (7*I)/32)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) - ((5/32 + (7*I)/32)*d^(7/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) - (d*(d*Tan[e + f*x])^(5/2))/(6*f*(a + I*a*Tan[e + f*x])^3) + (I*d^2*(d*Tan[e + f*x])^(3/2))/(3*a*f*(a + I*a*Tan[e + f*x])^2) + (5*d^3*Sqrt[d*Tan[e + f*x]])/(8*f*(a^3 + I*a^3*Tan[e + f*x]))} -{(d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3, x, 16, (d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(8*Sqrt[2]*a^3*f) - (d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(8*Sqrt[2]*a^3*f) - (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(16*Sqrt[2]*a^3*f) + (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(16*Sqrt[2]*a^3*f) - (d*(d*Tan[e + f*x])^(3/2))/(6*f*(a + I*a*Tan[e + f*x])^3) + ((I/4)*d^2*Sqrt[d*Tan[e + f*x]])/(a*f*(a + I*a*Tan[e + f*x])^2) - ((I/4)*d^2*Sqrt[d*Tan[e + f*x]])/(f*(a^3 + I*a^3*Tan[e + f*x]))} -{(d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3, x, 7, ((-1)^(1/4)*d^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(8*a^3*f) - (d*Sqrt[d*Tan[e + f*x]])/(6*f*(a + I*a*Tan[e + f*x])^3) + (d*Sqrt[d*Tan[e + f*x]])/(6*a*f*(a + I*a*Tan[e + f*x])^2) + (d*Sqrt[d*Tan[e + f*x]])/(8*f*(a^3 + I*a^3*Tan[e + f*x]))} -{Sqrt[d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3, x, 14, ((I/8)*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) - ((I/8)*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*f) + ((I/16)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) - ((I/16)*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*f) + ((I/6)*Sqrt[d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^3) + ((I/12)*Sqrt[d*Tan[e + f*x]])/(a*f*(a + I*a*Tan[e + f*x])^2)} -{1/(Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^3), x, 13, -(((7/16 - (5*I)/16)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*Sqrt[d]*f)) + ((7/16 - (5*I)/16)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*Sqrt[d]*f) - ((7/32 + (5*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*Sqrt[d]*f) + ((7/32 + (5*I)/32)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*Sqrt[d]*f) + Sqrt[d*Tan[e + f*x]]/(6*d*f*(a + I*a*Tan[e + f*x])^3) + Sqrt[d*Tan[e + f*x]]/(3*a*d*f*(a + I*a*Tan[e + f*x])^2) + (5*Sqrt[d*Tan[e + f*x]])/(8*d*f*(a^3 + I*a^3*Tan[e + f*x]))} -{1/((d*Tan[e + f*x])^(3/2)*(a + I*a*Tan[e + f*x])^3), x, 14, ((15/8 + (7*I)/4)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*d^(3/2)*f) - ((15/8 + (7*I)/4)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[2]*a^3*d^(3/2)*f) - ((15/16 - (7*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*d^(3/2)*f) + ((15/16 - (7*I)/8)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*a^3*d^(3/2)*f) - 15/(4*a^3*d*f*Sqrt[d*Tan[e + f*x]]) + 1/(6*d*f*Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^3) + 5/(12*a*d*f*Sqrt[d*Tan[e + f*x]]*(a + I*a*Tan[e + f*x])^2) + 7/(6*d*f*Sqrt[d*Tan[e + f*x]]*(a^3 + I*a^3*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 9, (7*(-1)^(3/4)*Sqrt[a]*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d) + ((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (I*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 9, -(((-1)^(1/4)*Sqrt[a]*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]], x, 7, -((2*(-1)^(3/4)*Sqrt[a]*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d} -{Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Tan[c + d*x]], x, 2, ((1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d} -{Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(3/2), x, 3, ((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(5/2), x, 5, ((-1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (((2*I)/3)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{Sqrt[a + I*a*Tan[c + d*x]]/Tan[c + d*x]^(7/2), x, 6, ((-1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (((2*I)/15)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) + (26*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 11, (23*(-1)^(3/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(8*d) + ((2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (I*a^2*Tan[c + d*x]^(5/2))/(3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (a^2*Tan[c + d*x]^(7/2))/(3*d*Sqrt[a + I*a*Tan[c + d*x]]) - (9*I*a*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) + (7*a*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(12*d)} -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 10, -((11*(-1)^(1/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d)) - ((2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (I*a^2*Tan[c + d*x]^(3/2))/(2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (a^2*Tan[c + d*x]^(5/2))/(2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (5*a*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2), x, 9, -((3*(-1)^(3/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (I*a^2*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (a^2*Tan[c + d*x]^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[Tan[c + d*x]], x, 7, (2*(-1)^(1/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(3/2), x, 3, ((2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(5/2), x, 4, ((-2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((2*I)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (2*(a + I*a*Tan[c + d*x])^(3/2))/(3*d*Tan[c + d*x]^(3/2))} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(7/2), x, 7, ((-2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2)/(5*d*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/5)*a^2)/(d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((4*I)/5)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) + (12*a*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*Sqrt[Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(9/2), x, 8, ((2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2)/(7*d*Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((2*I)/7)*a^2)/(d*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((16*I)/35)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(5/2)) + (76*a*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(3/2)) + (((268*I)/105)*a*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 11, (363*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(64*d) + ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (149*I*a^2*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(64*d) + (107*a^2*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) + (17*I*a^2*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a^2*Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d)} -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 10, -((45*(-1)^(1/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(8*d)) - ((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (19*a^2*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) + (13*I*a^2*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) - (a^2*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2), x, 9, -((23*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d)) - ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (9*I*a^2*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a^2*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[Tan[c + d*x]], x, 8, (5*(-1)^(1/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (a^2*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(3/2), x, 8, (2*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(5/2), x, 4, ((-4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - ((4*I)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (2*a*(a + I*a*Tan[c + d*x])^(3/2))/(3*d*Tan[c + d*x]^(3/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(7/2), x, 5, ((-4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (4*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (((2*I)/3)*a*(a + I*a*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(3/2)) - (2*(a + I*a*Tan[c + d*x])^(5/2))/(5*d*Tan[c + d*x]^(5/2))} -{(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(9/2), x, 7, ((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (((6*I)/7)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(5/2)) + (32*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(3/2)) + (((104*I)/21)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(11/2), x, 8, ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(9*d*Tan[c + d*x]^(9/2)) - (((38*I)/63)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(7/2)) + (92*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(5/2)) + (((472*I)/315)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(3/2)) - (1576*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(7/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 10, (11*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*Sqrt[a]*d) + ((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) - Tan[c + d*x]^(5/2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a*d) - (3*I*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d)} -{Tan[c + d*x]^(5/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 9, -(((-1)^(3/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d)) + ((1/2 + I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) - Tan[c + d*x]^(3/2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{Tan[c + d*x]^(3/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 8, -((2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d)) - ((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) - Sqrt[Tan[c + d*x]]/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(1/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 3, ((-1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + (I*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(1/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, ((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + Sqrt[Tan[c + d*x]]/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 5, ((1/2 + I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + 1/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (3*Sqrt[a + I*a*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 6, -(((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d)) + 1/(d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (5*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d*Tan[c + d*x]^(3/2)) + (7*I*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 7, -(((1/2 + I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d)) + 1/(d*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d*Tan[c + d*x]^(5/2)) + (23*I*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d*Tan[c + d*x]^(3/2)) + (61*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(7/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 10, -((3*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d)) + ((1/4 - I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) - Tan[c + d*x]^(5/2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (13*I*Tan[c + d*x]^(3/2))/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d)} -{Tan[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 9, (2*(-1)^(3/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + ((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) - Tan[c + d*x]^(3/2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (3*I*Sqrt[Tan[c + d*x]])/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 4, ((-1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + ((I/3)*Tan[c + d*x]^(3/2))/(d*(a + I*a*Tan[c + d*x])^(3/2)) + Sqrt[Tan[c + d*x]]/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(1/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 4, -(((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d)) + Tan[c + d*x]^(3/2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I*Sqrt[Tan[c + d*x]])/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 5, ((1/4 - I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + Sqrt[Tan[c + d*x]]/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (7*Sqrt[Tan[c + d*x]])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 6, ((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + 1/(3*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + 11/(6*a*d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (25*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 7, ((-1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + 1/(3*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + 5/(2*a*d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d*Tan[c + d*x]^(3/2)) + (((13*I)/2)*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(9/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 11, (5*(-1)^(3/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) - ((1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) - Tan[c + d*x]^(7/2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (19*I*Tan[c + d*x]^(5/2))/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (41*Tan[c + d*x]^(3/2))/(12*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (21*I*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^3*d)} -{Tan[c + d*x]^(7/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 10, (2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) - Tan[c + d*x]^(5/2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (I*Tan[c + d*x]^(3/2))/(2*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (7*Sqrt[Tan[c + d*x]])/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((I/5)*Tan[c + d*x]^(5/2))/(d*(a + I*a*Tan[c + d*x])^(5/2)) + Tan[c + d*x]^(3/2)/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I/4)*Sqrt[Tan[c + d*x]])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((-1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + Tan[c + d*x]^(5/2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((I/6)*Tan[c + d*x]^(3/2))/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) + Sqrt[Tan[c + d*x]]/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(1/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 6, ((-1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((I/5)*Sqrt[Tan[c + d*x]])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + ((I/10)*Sqrt[Tan[c + d*x]])/(a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I/20)*Sqrt[Tan[c + d*x]])/(a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 6, ((1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + Sqrt[Tan[c + d*x]]/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (13*Sqrt[Tan[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (67*Sqrt[Tan[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 7, ((1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + 1/(5*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + 17/(30*a*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + 151/(60*a^2*d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (317*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 8, ((-1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + 1/(5*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + 7/(10*a*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + 89/(20*a^2*d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (361*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d*Tan[c + d*x]^(3/2)) + (((707*I)/60)*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^(n/3)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(10/3)/(a + I*a*Tan[c + d*x]), x, 25, (7*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (7*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (5*I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*Sqrt[3]*a*d) - (7*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) - (5*I*Log[1 + Tan[c + d*x]^(2/3)])/(6*a*d) + (7*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - (7*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) + (5*I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(12*a*d) + (7*Tan[c + d*x]^(1/3))/(2*a*d) - (5*I*Tan[c + d*x]^(4/3))/(4*a*d) - Tan[c + d*x]^(7/3)/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x]), x, 24, -((5*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d)) + (5*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (2*I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a*d) + (5*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) + (2*I*Log[1 + Tan[c + d*x]^(2/3)])/(3*a*d) + (5*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - (5*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(3*a*d) - (2*I*Tan[c + d*x]^(2/3))/(a*d) - Tan[c + d*x]^(5/3)/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x]), x, 23, -(ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)]/(12*a*d)) + ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)]/(12*a*d) + (I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a*d) + ArcTan[Tan[c + d*x]^(1/3)]/(6*a*d) + (I*Log[1 + Tan[c + d*x]^(2/3)])/(3*a*d) - Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(8*Sqrt[3]*a*d) + Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(8*Sqrt[3]*a*d) - (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(6*a*d) - Tan[c + d*x]^(1/3)/(2*d*(a + I*a*Tan[c + d*x]))} -{Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x]), x, 23, -(ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)]/(12*a*d)) + ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)]/(12*a*d) + (I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*Sqrt[3]*a*d) + ArcTan[Tan[c + d*x]^(1/3)]/(6*a*d) - (I*Log[1 + Tan[c + d*x]^(2/3)])/(6*a*d) + Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(8*Sqrt[3]*a*d) - Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(8*Sqrt[3]*a*d) + (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(12*a*d) + (I*Tan[c + d*x]^(2/3))/(2*d*(a + I*a*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])), x, 23, (I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) - ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/(Sqrt[3]*a*d) - (I*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) + Log[1 + Tan[c + d*x]^(2/3)]/(3*a*d) - (I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) + (I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]/(6*a*d) + Tan[c + d*x]^(2/3)/(2*d*(a + I*a*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])), x, 24, (5*I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d) - (5*I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) + (2*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(Sqrt[3]*a*d) - (5*I*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) + (2*Log[1 + Tan[c + d*x]^(2/3)])/(3*a*d) + (5*I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - (5*I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]/(3*a*d) - 2/(a*d*Tan[c + d*x]^(2/3)) + 1/(2*d*Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(7/3)*(a + I*a*Tan[c + d*x])), x, 25, -((7*I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(12*a*d)) + (7*I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(12*a*d) + (5*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*Sqrt[3]*a*d) + (7*I*ArcTan[Tan[c + d*x]^(1/3)])/(6*a*d) - (5*Log[1 + Tan[c + d*x]^(2/3)])/(6*a*d) + (7*I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) - (7*I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(8*Sqrt[3]*a*d) + (5*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(12*a*d) - 5/(4*a*d*Tan[c + d*x]^(4/3)) + (7*I)/(2*a*d*Tan[c + d*x]^(1/3)) + 1/(2*d*Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x]))} - - -{Tan[c + d*x]^(14/3)/(a + I*a*Tan[c + d*x])^2, x, 26, -((121*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d)) + (121*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) - (14*I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) + (121*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) + (14*I*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) + (121*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (121*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (7*I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(9*a^2*d) - (14*I*Tan[c + d*x]^(2/3))/(3*a^2*d) - (121*Tan[c + d*x]^(5/3))/(60*a^2*d) + (7*I*Tan[c + d*x]^(8/3))/(6*a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^(11/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^(10/3)/(a + I*a*Tan[c + d*x])^2, x, 25, -((49*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d)) + (49*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) + (5*I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) + (49*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) + (5*I*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) - (49*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) + (49*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (5*I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(18*a^2*d) - (49*Tan[c + d*x]^(1/3))/(12*a^2*d) + (5*I*Tan[c + d*x]^(4/3))/(6*a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^(7/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^(8/3)/(a + I*a*Tan[c + d*x])^2, x, 24, (25*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) - (25*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) + (2*I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) - (25*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) - (2*I*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) - (25*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) + (25*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) + (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(9*a^2*d) + (2*I*Tan[c + d*x]^(2/3))/(3*a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^(5/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^2, x, 24, ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)]/(72*a^2*d) - ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)]/(72*a^2*d) + (I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) - ArcTan[Tan[c + d*x]^(1/3)]/(36*a^2*d) + (I*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) + Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(48*Sqrt[3]*a^2*d) - Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(48*Sqrt[3]*a^2*d) - (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(18*a^2*d) + Tan[c + d*x]^(1/3)/(3*a^2*d*(1 + I*Tan[c + d*x])) - Tan[c + d*x]^(1/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x])^2, x, 24, -(ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)]/(72*a^2*d)) + ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)]/(72*a^2*d) + (I*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) + ArcTan[Tan[c + d*x]^(1/3)]/(36*a^2*d) - (I*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) + Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(48*Sqrt[3]*a^2*d) - Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)]/(48*Sqrt[3]*a^2*d) + (I*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(18*a^2*d) + (I*Tan[c + d*x]^(2/3))/(3*a^2*d*(1 + I*Tan[c + d*x])) + Tan[c + d*x]^(5/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^2), x, 24, (7*I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) - (7*I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) - (2*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) - (7*I*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) + (2*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) - (7*I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) + (7*I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]/(9*a^2*d) + (7*Tan[c + d*x]^(2/3))/(12*a^2*d*(1 + I*Tan[c + d*x])) + Tan[c + d*x]^(2/3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{1/(Tan[c + d*x]^(5/3)*(a + I*a*Tan[c + d*x])^2), x, 25, (55*I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) - (55*I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) + (8*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(3*Sqrt[3]*a^2*d) - (55*I*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) + (8*Log[1 + Tan[c + d*x]^(2/3)])/(9*a^2*d) + (55*I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (55*I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (4*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(9*a^2*d) - 8/(3*a^2*d*Tan[c + d*x]^(2/3)) + 11/(12*a^2*d*(1 + I*Tan[c + d*x])*Tan[c + d*x]^(2/3)) + 1/(4*d*Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^2)} -{1/(Tan[c + d*x]^(7/3)*(a + I*a*Tan[c + d*x])^2), x, 26, -((91*I*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(72*a^2*d)) + (91*I*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(72*a^2*d) + (25*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(6*Sqrt[3]*a^2*d) + (91*I*ArcTan[Tan[c + d*x]^(1/3)])/(36*a^2*d) - (25*Log[1 + Tan[c + d*x]^(2/3)])/(18*a^2*d) + (91*I*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) - (91*I*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(48*Sqrt[3]*a^2*d) + (25*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(36*a^2*d) - 25/(12*a^2*d*Tan[c + d*x]^(4/3)) + 13/(12*a^2*d*(1 + I*Tan[c + d*x])*Tan[c + d*x]^(4/3)) + (91*I)/(12*a^2*d*Tan[c + d*x]^(1/3)) + 1/(4*d*Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (d Tan[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + I*a*Tan[c + d*x])^(1/2)*Tan[c + d*x]^(4/3), x, 4, (3*a*AppellF1[7/3, 1/2, 1, 10/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(7/3))/(7*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(1/2)*Tan[c + d*x]^(2/3), x, 4, (3*a*AppellF1[5/3, 1/2, 1, 8/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(5/3))/(5*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(1/2)*Tan[c + d*x]^(1/3), x, 4, (3*a*AppellF1[4/3, 1/2, 1, 7/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(4/3))/(4*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(1/2)/Tan[c + d*x]^(1/3), x, 4, (3*a*AppellF1[2/3, 1/2, 1, 5/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(2/3))/(2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(1/2)/Tan[c + d*x]^(2/3), x, 4, (3*a*AppellF1[1/3, 1/2, 1, 4/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1/3))/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(1/2)/Tan[c + d*x]^(4/3), x, 4, -((3*a*AppellF1[-(1/3), 1/2, 1, 2/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]])/(d*Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]))} - - -{(a + I*a*Tan[c + d*x])^(3/2)*Tan[c + d*x]^(4/3), x, 4, (3*a*AppellF1[7/3, -(1/2), 1, 10/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(7/3)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*Sqrt[1 + I*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)*Tan[c + d*x]^(2/3), x, 4, (3*a*AppellF1[5/3, -(1/2), 1, 8/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(5/3)*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*Sqrt[1 + I*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)*Tan[c + d*x]^(1/3), x, 4, (3*a*AppellF1[4/3, -(1/2), 1, 7/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[1 + I*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(1/3), x, 4, (3*a*AppellF1[2/3, -(1/2), 1, 5/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*Sqrt[1 + I*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(2/3), x, 4, (3*a*AppellF1[1/3, -(1/2), 1, 4/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(3/2)/Tan[c + d*x]^(4/3), x, 4, -((3*a*AppellF1[-(1/3), -(1/2), 1, 2/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1/3)))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(4/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*AppellF1[7/3, 3/2, 1, 10/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(7/3))/(7*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(2/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*AppellF1[5/3, 3/2, 1, 8/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(5/3))/(5*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(1/3)/Sqrt[a + I*a*Tan[c + d*x]], x, 4, (3*AppellF1[4/3, 3/2, 1, 7/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(4/3))/(4*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, (3*AppellF1[2/3, 3/2, 1, 5/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(2/3))/(2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(2/3)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, (3*AppellF1[1/3, 3/2, 1, 4/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1/3))/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(4/3)*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, -((3*AppellF1[-(1/3), 3/2, 1, 2/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]])/(d*Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]))} - - -{Tan[c + d*x]^(4/3)/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (3*AppellF1[7/3, 5/2, 1, 10/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(7/3))/(7*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(2/3)/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (3*AppellF1[5/3, 5/2, 1, 8/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(5/3))/(5*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{Tan[c + d*x]^(1/3)/(a + I*a*Tan[c + d*x])^(3/2), x, 4, (3*AppellF1[4/3, 5/2, 1, 7/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(4/3))/(4*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(1/3)*(a + I*a*Tan[c + d*x])^(3/2)), x, 4, (3*AppellF1[2/3, 5/2, 1, 5/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(2/3))/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(2/3)*(a + I*a*Tan[c + d*x])^(3/2)), x, 4, (3*AppellF1[1/3, 5/2, 1, 4/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1/3))/(a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Tan[c + d*x]^(4/3)*(a + I*a*Tan[c + d*x])^(3/2)), x, 4, -((3*AppellF1[-(1/3), 5/2, 1, 2/3, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]])/(a*d*Tan[c + d*x]^(1/3)*Sqrt[a + I*a*Tan[c + d*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/3) (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3), x, 8, -((I*a^(1/3)*x)/(2*2^(2/3))) + (Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) - (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) - (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) - (18*(a + I*a*Tan[c + d*x])^(1/3))/(7*d) + (3*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3))/(7*d) - (3*(a + I*a*Tan[c + d*x])^(4/3))/(28*a*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3), x, 6, (a^(1/3)*x)/(2*2^(2/3)) + (I*Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) - (I*a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) - (3*I*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) - (3*I*(a + I*a*Tan[c + d*x])^(4/3))/(4*a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3), x, 6, (I*a^(1/3)*x)/(2*2^(2/3)) - (Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) + (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) + (3*(a + I*a*Tan[c + d*x])^(1/3))/d} -{(a + I*a*Tan[c + d*x])^(1/3), x, 5, -((a^(1/3)*x)/(2*2^(2/3))) - (I*Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) + (I*a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (3*I*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d)} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3), x, 11, -((I*a^(1/3)*x)/(2*2^(2/3))) - (Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) - (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) - (a^(1/3)*Log[Tan[c + d*x]])/(2*d) + (3*a^(1/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d)} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3), x, 12, (a^(1/3)*x)/(2*2^(2/3)) - (I*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*d) + (I*Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) - (I*a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) - (I*a^(1/3)*Log[Tan[c + d*x]])/(6*d) + (I*a^(1/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*I*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) - (Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(1/3), x, 13, (I*a^(1/3)*x)/(2*2^(2/3)) + (8*a^(1/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*d) - (Sqrt[3]*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) + (a^(1/3)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (4*a^(1/3)*Log[Tan[c + d*x]])/(9*d) - (4*a^(1/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(3*d) + (3*a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) - (I*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/(6*d) - (Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(1/3))/(2*d)} - - -{(a + I*a*Tan[c + d*x])^(2/3), x, 5, -((a^(2/3)*x)/(2*2^(1/3))) + (I*Sqrt[3]*a^(2/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) + (I*a^(2/3)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) + (3*I*a^(2/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d)} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(4/3), x, 9, -((I*a^(4/3)*x)/2^(2/3)) + (2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d - (a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (3*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) - (3*a*(a + I*a*Tan[c + d*x])^(1/3))/d - (9*(a + I*a*Tan[c + d*x])^(4/3))/(20*d) + (3*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3))/(10*d) - (6*(a + I*a*Tan[c + d*x])^(7/3))/(35*a*d)} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3), x, 7, (a^(4/3)*x)/2^(2/3) + (I*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d - (I*a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (3*I*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) - (3*I*a*(a + I*a*Tan[c + d*x])^(1/3))/d - (3*I*(a + I*a*Tan[c + d*x])^(7/3))/(7*a*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3), x, 7, (I*a^(4/3)*x)/2^(2/3) - (2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) + (3*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) + (3*a*(a + I*a*Tan[c + d*x])^(1/3))/d + (3*(a + I*a*Tan[c + d*x])^(4/3))/(4*d)} -{(a + I*a*Tan[c + d*x])^(4/3), x, 6, -((a^(4/3)*x)/2^(2/3)) - (I*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (I*a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) + (3*I*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) + (3*I*a*(a + I*a*Tan[c + d*x])^(1/3))/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3), x, 13, -((I*a^(4/3)*x)/2^(2/3)) - (Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d - (a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (a^(4/3)*Log[Tan[c + d*x]])/(2*d) + (3*a^(4/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d)} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3), x, 13, (a^(4/3)*x)/2^(2/3) - (4*I*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*d) + (I*2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d - (I*a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) - (2*I*a^(4/3)*Log[Tan[c + d*x]])/(3*d) + (2*I*a^(4/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/d - (3*I*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) + (I*a*(a + I*a*Tan[c + d*x])^(1/3))/d - (Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(4/3))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(4/3), x, 13, (I*a^(4/3)*x)/2^(2/3) + (11*a^(4/3)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*d) - (2^(1/3)*Sqrt[3]*a^(4/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (a^(4/3)*Log[Cos[c + d*x]])/(2^(2/3)*d) + (11*a^(4/3)*Log[Tan[c + d*x]])/(18*d) - (11*a^(4/3)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(6*d) + (3*a^(4/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(2/3)*d) - (2*I*a*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(1/3))/(3*d) - (Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(4/3))/(2*d)} - - -{(a + I*a*Tan[c + d*x])^(5/3), x, 6, -((a^(5/3)*x)/2^(1/3)) + (I*2^(2/3)*Sqrt[3]*a^(5/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d + (I*a^(5/3)*Log[Cos[c + d*x]])/(2^(1/3)*d) + (3*I*a^(5/3)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2^(1/3)*d) + (3*I*a*(a + I*a*Tan[c + d*x])^(2/3))/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(1/3), x, 3, (AppellF1[1 + m, 4/3, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(1 + I*Tan[c + d*x])^(1/3)*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^(1/2)/(a + I*a*Tan[c + d*x])^(1/3), x, 4, (2*AppellF1[3/2, 4/3, 1, 5/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(1 + I*Tan[c + d*x])^(1/3)*Tan[c + d*x]^(3/2))/(3*d*(a + I*a*Tan[c + d*x])^(1/3))} - -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(1/3), x, 9, -(x/(4*2^(1/3)*a^(1/3))) + (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) + (I*Log[Cos[c + d*x]])/(4*2^(1/3)*a^(1/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) - (15*I*Tan[c + d*x]^2)/(8*d*(a + I*a*Tan[c + d*x])^(1/3)) + (3*Tan[c + d*x]^3)/(8*d*(a + I*a*Tan[c + d*x])^(1/3)) + (45*I*(a + I*a*Tan[c + d*x])^(2/3))/(8*a*d) - (39*I*(a + I*a*Tan[c + d*x])^(5/3))/(20*a^2*d)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(1/3), x, 8, -((I*x)/(4*2^(1/3)*a^(1/3))) - (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) - Log[Cos[c + d*x]]/(4*2^(1/3)*a^(1/3)*d) - (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) + 21/(10*d*(a + I*a*Tan[c + d*x])^(1/3)) + (3*Tan[c + d*x]^2)/(5*d*(a + I*a*Tan[c + d*x])^(1/3)) + (3*(a + I*a*Tan[c + d*x])^(2/3))/(10*a*d)} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3), x, 7, x/(4*2^(1/3)*a^(1/3)) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) - (I*Log[Cos[c + d*x]])/(4*2^(1/3)*a^(1/3)*d) - (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) - (3*I)/(2*d*(a + I*a*Tan[c + d*x])^(1/3)) - (3*I*(a + I*a*Tan[c + d*x])^(2/3))/(2*a*d)} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x])^(1/3), x, 6, (I*x)/(4*2^(1/3)*a^(1/3)) + (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) + Log[Cos[c + d*x]]/(4*2^(1/3)*a^(1/3)*d) + (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) - 3/(2*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^0/(a + I*a*Tan[c + d*x])^(1/3), x, 6, -(x/(4*2^(1/3)*a^(1/3))) + (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) + (I*Log[Cos[c + d*x]])/(4*2^(1/3)*a^(1/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) + (3*I)/(2*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Cot[c + d*x]^1/(a + I*a*Tan[c + d*x])^(1/3), x, 13, -((I*x)/(4*2^(1/3)*a^(1/3))) + (Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(a^(1/3)*d) - (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) - Log[Cos[c + d*x]]/(4*2^(1/3)*a^(1/3)*d) - Log[Tan[c + d*x]]/(2*a^(1/3)*d) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(1/3)*d) - (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) + 3/(2*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(1/3), x, 13, x/(4*2^(1/3)*a^(1/3)) - (I*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*d) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) - (I*Log[Cos[c + d*x]])/(4*2^(1/3)*a^(1/3)*d) + (I*Log[Tan[c + d*x]])/(6*a^(1/3)*d) - (I*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(1/3)*d) - (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) - (5*I)/(2*d*(a + I*a*Tan[c + d*x])^(1/3)) - Cot[c + d*x]/(d*(a + I*a*Tan[c + d*x])^(1/3))} - - -{1/(a + I*a*Tan[c + d*x])^(2/3), x, 6, -(x/(4*2^(2/3)*a^(2/3))) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(2/3)*a^(2/3)*d) + (I*Log[Cos[c + d*x]])/(4*2^(2/3)*a^(2/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(2/3)*a^(2/3)*d) + (3*I)/(4*d*(a + I*a*Tan[c + d*x])^(2/3))} - - -{Tan[c + d*x]^m/(a + I*a*Tan[c + d*x])^(4/3), x, 3, (AppellF1[1 + m, 7/3, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(1 + I*Tan[c + d*x])^(1/3)*Tan[c + d*x]^(1 + m))/(a*d*(1 + m)*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^(1/2)/(a + I*a*Tan[c + d*x])^(4/3), x, 4, (2*AppellF1[3/2, 7/3, 1, 5/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(1 + I*Tan[c + d*x])^(1/3)*Tan[c + d*x]^(3/2))/(3*a*d*(a + I*a*Tan[c + d*x])^(1/3))} - -{Tan[c + d*x]^4/(a + I*a*Tan[c + d*x])^(4/3), x, 9, -(x/(8*2^(1/3)*a^(4/3))) + (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) + (I*Log[Cos[c + d*x]])/(8*2^(1/3)*a^(4/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) - (39*I*Tan[c + d*x]^2)/(40*d*(a + I*a*Tan[c + d*x])^(4/3)) + (3*Tan[c + d*x]^3)/(5*d*(a + I*a*Tan[c + d*x])^(4/3)) - (51*I)/(10*a*d*(a + I*a*Tan[c + d*x])^(1/3)) - (87*I*(a + I*a*Tan[c + d*x])^(2/3))/(40*a^2*d)} -{Tan[c + d*x]^3/(a + I*a*Tan[c + d*x])^(4/3), x, 8, -((I*x)/(8*2^(1/3)*a^(4/3))) - (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) - Log[Cos[c + d*x]]/(8*2^(1/3)*a^(4/3)*d) - (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) + 15/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) + (3*Tan[c + d*x]^2)/(2*d*(a + I*a*Tan[c + d*x])^(4/3)) - 27/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^2/(a + I*a*Tan[c + d*x])^(4/3), x, 7, x/(8*2^(1/3)*a^(4/3)) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) - (I*Log[Cos[c + d*x]])/(8*2^(1/3)*a^(4/3)*d) - (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) - (3*I)/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) + (9*I)/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^1/(a + I*a*Tan[c + d*x])^(4/3), x, 7, (I*x)/(8*2^(1/3)*a^(4/3)) + (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) + Log[Cos[c + d*x]]/(8*2^(1/3)*a^(4/3)*d) + (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) - 3/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) + 3/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Tan[c + d*x]^0/(a + I*a*Tan[c + d*x])^(4/3), x, 7, -(x/(8*2^(1/3)*a^(4/3))) + (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) + (I*Log[Cos[c + d*x]])/(8*2^(1/3)*a^(4/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) + (3*I)/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) + (3*I)/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Cot[c + d*x]^1/(a + I*a*Tan[c + d*x])^(4/3), x, 15, -((I*x)/(8*2^(1/3)*a^(4/3))) + (Sqrt[3]*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(a^(4/3)*d) - (Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) - Log[Cos[c + d*x]]/(8*2^(1/3)*a^(4/3)*d) - Log[Tan[c + d*x]]/(2*a^(4/3)*d) + (3*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*a^(4/3)*d) - (3*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) + 3/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) + 9/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} -{Cot[c + d*x]^2/(a + I*a*Tan[c + d*x])^(4/3), x, 14, x/(8*2^(1/3)*a^(4/3)) - (4*I*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(1/3)*a^(4/3)*d) - (I*Log[Cos[c + d*x]])/(8*2^(1/3)*a^(4/3)*d) + (2*I*Log[Tan[c + d*x]])/(3*a^(4/3)*d) - (2*I*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(a^(4/3)*d) - (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(1/3)*a^(4/3)*d) - (11*I)/(8*d*(a + I*a*Tan[c + d*x])^(4/3)) - Cot[c + d*x]/(d*(a + I*a*Tan[c + d*x])^(4/3)) - (19*I)/(4*a*d*(a + I*a*Tan[c + d*x])^(1/3))} - - -{(a + I*a*Tan[c + d*x])^(-5/3), x, 7, -(x/(8*2^(2/3)*a^(5/3))) - (I*Sqrt[3]*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(4*2^(2/3)*a^(5/3)*d) + (I*Log[Cos[c + d*x]])/(8*2^(2/3)*a^(5/3)*d) + (3*I*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(8*2^(2/3)*a^(5/3)*d) + (3*I)/(10*d*(a + I*a*Tan[c + d*x])^(5/3)) + (3*I)/(8*a*d*(a + I*a*Tan[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n with n synbolic*) - - -{(e*Tan[c + d*x])^m*(a + I*a*Tan[c + d*x]), x, 2, (a*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m*(a - I*a*Tan[c + d*x]), x, 2, (a*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*Tan[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} - - -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^4, x, 6, If[$VersionNumber>=8, -((2*a^4*(16 + 11*n + 2*n^2)*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)*(3 + n))) + (8*a^4*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) - ((d*Tan[e + f*x])^(1 + n)*(a^2 + I*a^2*Tan[e + f*x])^2)/(d*f*(3 + n)) - (2*(4 + n)*(d*Tan[e + f*x])^(1 + n)*(a^4 + I*a^4*Tan[e + f*x]))/(d*f*(2 + n)*(3 + n)), -((2*a^4*(16 + 11*n + 2*n^2)*(d*Tan[e + f*x])^(1 + n))/(d*f*(3 + n)*(2 + 3*n + n^2))) + (8*a^4*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) - ((d*Tan[e + f*x])^(1 + n)*(a^2 + I*a^2*Tan[e + f*x])^2)/(d*f*(3 + n)) - (2*(4 + n)*(d*Tan[e + f*x])^(1 + n)*(a^4 + I*a^4*Tan[e + f*x]))/(d*f*(2 + n)*(3 + n))]} -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^3, x, 5, -((a^3*(5 + 2*n)*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n))) + (4*a^3*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) - ((d*Tan[e + f*x])^(1 + n)*(a^3 + I*a^3*Tan[e + f*x]))/(d*f*(2 + n))} -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^2, x, 4, -((a^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))) + (2*a^2*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^1, x, 2, (a*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^1, x, 6, ((1 - n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(2*a*d*f*(1 + n)) + (I*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(2*a*d^2*f*(2 + n)) + (d*Tan[e + f*x])^(1 + n)/(2*d*f*(a + I*a*Tan[e + f*x]))} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x, 7, ((1 - n)^2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(4*a^2*d*f*(1 + n)) + ((2 - n)*(d*Tan[e + f*x])^(1 + n))/(4*a^2*d*f*(1 + I*Tan[e + f*x])) + (I*(2 - n)*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(4*a^2*d^2*f*(2 + n)) + (d*Tan[e + f*x])^(1 + n)/(4*d*f*(a + I*a*Tan[e + f*x])^2)} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x, 8, ((1 - 2*n)*(1 - n)*(3 - n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(24*a^3*d*f*(1 + n)) + (I*(5 - 2*n)*(2 - n)*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(24*a^3*d^2*f*(2 + n)) + (d*Tan[e + f*x])^(1 + n)/(6*d*f*(a + I*a*Tan[e + f*x])^3) + ((7 - 2*n)*(d*Tan[e + f*x])^(1 + n))/(24*a*d*f*(a + I*a*Tan[e + f*x])^2) + ((5 - 2*n)*(2 - n)*(d*Tan[e + f*x])^(1 + n))/(24*d*f*(a^3 + I*a^3*Tan[e + f*x]))} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^4, x, 9, ((1 - n)*(3 - n)*(1 - 4*n + n^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(48*a^4*d*f*(1 + n)) + ((13 - 7*n + n^2)*(d*Tan[e + f*x])^(1 + n))/(48*a^4*d*f*(1 + I*Tan[e + f*x])^2) + ((2 - n)^2*(4 - n)*(d*Tan[e + f*x])^(1 + n))/(48*a^4*d*f*(1 + I*Tan[e + f*x])) + (I*(2 - n)^2*(4 - n)*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(48*a^4*d^2*f*(2 + n)) + (d*Tan[e + f*x])^(1 + n)/(8*d*f*(a + I*a*Tan[e + f*x])^4) + ((5 - n)*(d*Tan[e + f*x])^(1 + n))/(24*a*d*f*(a + I*a*Tan[e + f*x])^3)} - -{(d*Tan[e + f*x])^n*(a - I*a*Tan[e + f*x])^1, x, 2, (a*Hypergeometric2F1[1, 1 + n, 2 + n, (-I)*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n/(a - I*a*Tan[e + f*x])^1, x, 6, ((1 - n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(2*a*d*f*(1 + n)) - (I*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(2*a*d^2*f*(2 + n)) + (d*Tan[e + f*x])^(1 + n)/(2*d*f*(a - I*a*Tan[e + f*x]))} - - -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^(3/2), x, 3, (a*AppellF1[1 + n, -(1/2), 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*Sqrt[a + I*a*Tan[e + f*x]])/(d*f*(1 + n)*Sqrt[1 + I*Tan[e + f*x]])} -{(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x])^(1/2), x, 3, (a*AppellF1[1 + n, 1/2, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*Sqrt[1 + I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[a + I*a*Tan[e + f*x]])} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(1/2), x, 3, (AppellF1[1 + n, 3/2, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*Sqrt[1 + I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*Sqrt[a + I*a*Tan[e + f*x]])} -{(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2), x, 3, (AppellF1[1 + n, 5/2, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*Sqrt[1 + I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(a*d*f*(1 + n)*Sqrt[a + I*a*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Tan[e+f x])^n with m synbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(d*Tan[e + f*x])^n, x, 3, (AppellF1[1 + n, 1 - m, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + I*a*Tan[e + f*x])^m)/((1 + I*Tan[e + f*x])^m*(d*f*(1 + n)))} - - -{Tan[c + d*x]^4*(a + I*a*Tan[c + d*x])^m, x, 6, If[$VersionNumber>=8, (2*I*(a + I*a*Tan[c + d*x])^m)/(d*(6 + 5*m + m^2)) - (I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) - (I*m*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m)/(d*(6 + 5*m + m^2)) + (Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^m)/(d*(3 + m)) + (I*(6 + 3*m + m^2)*(a + I*a*Tan[c + d*x])^(1 + m))/(a*d*(1 + m)*(2 + m)*(3 + m)), (2*I*(a + I*a*Tan[c + d*x])^m)/(d*(6 + 5*m + m^2)) - (I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) - (I*m*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m)/(d*(6 + 5*m + m^2)) + (Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^m)/(d*(3 + m)) + (I*(6 + 3*m + m^2)*(a + I*a*Tan[c + d*x])^(1 + m))/(a*d*(3 + m)*(2 + 3*m + m^2))]} -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^m, x, 5, -((2*(a + I*a*Tan[c + d*x])^m)/(d*m*(2 + m))) + (Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) + (Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m)/(d*(2 + m)) - (m*(a + I*a*Tan[c + d*x])^(1 + m))/(a*d*(2 + 3*m + m^2))} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m, x, 3, (I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) - (I*(a + I*a*Tan[c + d*x])^(1 + m))/(a*d*(1 + m))} -{Tan[c + d*x]^1*(a + I*a*Tan[c + d*x])^m, x, 3, (a + I*a*Tan[c + d*x])^m/(d*m) - (Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m)} -{Tan[c + d*x]^0*(a + I*a*Tan[c + d*x])^m, x, 2, -((I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m))} -{Cot[c + d*x]^1*(a + I*a*Tan[c + d*x])^m, x, 5, (Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) - (Hypergeometric2F1[1, m, 1 + m, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m)/(d*m)} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^m, x, 6, -((Cot[c + d*x]*(a + I*a*Tan[c + d*x])^m)/d) + (I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^m)/(2*d*m) - (I*Hypergeometric2F1[1, m, 1 + m, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m)/d} - - -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^m, x, 4, (2*AppellF1[5/2, 1 - m, 1, 7/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^m)/((1 + I*Tan[c + d*x])^m*(5*d))} -{Tan[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^m, x, 4, (2*AppellF1[3/2, 1 - m, 1, 5/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^m)/((1 + I*Tan[c + d*x])^m*(3*d))} -{(a + I*a*Tan[c + d*x])^m/Tan[c + d*x]^(1/2), x, 4, (2*AppellF1[1/2, 1 - m, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m)/((1 + I*Tan[c + d*x])^m*d)} -{(a + I*a*Tan[c + d*x])^m/Tan[c + d*x]^(3/2), x, 4, -((2*AppellF1[-(1/2), 1 - m, 1, 1/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^m)/((1 + I*Tan[c + d*x])^m*(d*Sqrt[Tan[c + d*x]])))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Tan[e+f x])^m (d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(5/2), x, 5, (Sqrt[2]*a*d^(5/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f - (2*a*d^2*Sqrt[d*Tan[e + f*x]])/f + (2*a*d*(d*Tan[e + f*x])^(3/2))/(3*f) + (2*a*(d*Tan[e + f*x])^(5/2))/(5*f)} -{(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(3/2), x, 4, (Sqrt[2]*a*d^(3/2)*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f + (2*a*d*Sqrt[d*Tan[e + f*x]])/f + (2*a*(d*Tan[e + f*x])^(3/2))/(3*f)} -{(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(1/2), x, 3, -((Sqrt[2]*a*Sqrt[d]*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f) + (2*a*Sqrt[d*Tan[e + f*x]])/f} -{(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(1/2), x, 2, -((Sqrt[2]*a*ArcTan[(Sqrt[d]*(1 - Tan[e + f*x]))/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[d]*f))} -{(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2), x, 3, (Sqrt[2]*a*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(3/2)*f) - (2*a)/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2), x, 4, (Sqrt[2]*a*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*a)/(3*d*f*(d*Tan[e + f*x])^(3/2)) - (2*a)/(d^2*f*Sqrt[d*Tan[e + f*x]])} -{(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(7/2), x, 5, -((Sqrt[2]*a*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(7/2)*f)) - (2*a)/(5*d*f*(d*Tan[e + f*x])^(5/2)) - (2*a)/(3*d^2*f*(d*Tan[e + f*x])^(3/2)) + (2*a)/(d^3*f*Sqrt[d*Tan[e + f*x]])} - - -{(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(5/2), x, 16, -((Sqrt[2]*a^2*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f) + (Sqrt[2]*a^2*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (a^2*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) + (a^2*d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) - (4*a^2*d^2*Sqrt[d*Tan[e + f*x]])/f + (4*a^2*(d*Tan[e + f*x])^(5/2))/(5*f) + (2*a^2*(d*Tan[e + f*x])^(7/2))/(7*d*f)} -{(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(3/2), x, 15, (Sqrt[2]*a^2*d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (Sqrt[2]*a^2*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (a^2*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) + (a^2*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) + (4*a^2*(d*Tan[e + f*x])^(3/2))/(3*f) + (2*a^2*(d*Tan[e + f*x])^(5/2))/(5*d*f)} -{(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(1/2), x, 15, (Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) - (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*f) + (4*a^2*Sqrt[d*Tan[e + f*x]])/f + (2*a^2*(d*Tan[e + f*x])^(3/2))/(3*d*f)} -{(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(1/2), x, 14, -((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f)) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(Sqrt[d]*f) + (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*Sqrt[d]*f) - (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*Sqrt[d]*f) + (2*a^2*Sqrt[d*Tan[e + f*x]])/(d*f)} -{(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2), x, 13, -((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f)) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(3/2)*f) - (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*d^(3/2)*f) + (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*d^(3/2)*f) - (2*a^2)/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2), x, 14, (Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(d^(5/2)*f) - (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*d^(5/2)*f) + (a^2*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(Sqrt[2]*d^(5/2)*f) - (2*a^2)/(3*d*f*(d*Tan[e + f*x])^(3/2)) - (4*a^2)/(d^2*f*Sqrt[d*Tan[e + f*x]])} - - -{(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(7/2), x, 8, -((2*Sqrt[2]*a^3*d^(7/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f) + (4*a^3*d^3*Sqrt[d*Tan[e + f*x]])/f - (4*a^3*d^2*(d*Tan[e + f*x])^(3/2))/(3*f) - (4*a^3*d*(d*Tan[e + f*x])^(5/2))/(5*f) + (4*a^3*(d*Tan[e + f*x])^(7/2))/(7*f) + (16*a^3*(d*Tan[e + f*x])^(9/2))/(33*d*f) + (2*(d*Tan[e + f*x])^(9/2)*(a^3 + a^3*Tan[e + f*x]))/(11*d*f)} -{(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(5/2), x, 7, -((2*Sqrt[2]*a^3*d^(5/2)*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f) - (4*a^3*d^2*Sqrt[d*Tan[e + f*x]])/f - (4*a^3*d*(d*Tan[e + f*x])^(3/2))/(3*f) + (4*a^3*(d*Tan[e + f*x])^(5/2))/(5*f) + (40*a^3*(d*Tan[e + f*x])^(7/2))/(63*d*f) + (2*(d*Tan[e + f*x])^(7/2)*(a^3 + a^3*Tan[e + f*x]))/(9*d*f)} -{(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(3/2), x, 6, (2*Sqrt[2]*a^3*d^(3/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f - (4*a^3*d*Sqrt[d*Tan[e + f*x]])/f + (4*a^3*(d*Tan[e + f*x])^(3/2))/(3*f) + (32*a^3*(d*Tan[e + f*x])^(5/2))/(35*d*f) + (2*(d*Tan[e + f*x])^(5/2)*(a^3 + a^3*Tan[e + f*x]))/(7*d*f)} -{(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(1/2), x, 5, (2*Sqrt[2]*a^3*Sqrt[d]*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/f + (4*a^3*Sqrt[d*Tan[e + f*x]])/f + (8*a^3*(d*Tan[e + f*x])^(3/2))/(5*d*f) + (2*(d*Tan[e + f*x])^(3/2)*(a^3 + a^3*Tan[e + f*x]))/(5*d*f)} -{(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(1/2), x, 4, -((2*Sqrt[2]*a^3*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[d]*f)) + (16*a^3*Sqrt[d*Tan[e + f*x]])/(3*d*f) + (2*Sqrt[d*Tan[e + f*x]]*(a^3 + a^3*Tan[e + f*x]))/(3*d*f)} -{(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(3/2), x, 4, -((2*Sqrt[2]*a^3*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(3/2)*f)) + (4*a^3*Sqrt[d*Tan[e + f*x]])/(d^2*f) - (2*(a^3 + a^3*Tan[e + f*x]))/(d*f*Sqrt[d*Tan[e + f*x]])} -{(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(5/2), x, 4, (2*Sqrt[2]*a^3*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(5/2)*f) - (16*a^3)/(3*d^2*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + a^3*Tan[e + f*x]))/(3*d*f*(d*Tan[e + f*x])^(3/2))} -{(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(7/2), x, 5, (2*Sqrt[2]*a^3*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(7/2)*f) - (8*a^3)/(5*d^2*f*(d*Tan[e + f*x])^(3/2)) - (4*a^3)/(d^3*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + a^3*Tan[e + f*x]))/(5*d*f*(d*Tan[e + f*x])^(5/2))} -{(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(9/2), x, 6, -((2*Sqrt[2]*a^3*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(9/2)*f)) - (32*a^3)/(35*d^2*f*(d*Tan[e + f*x])^(5/2)) - (4*a^3)/(3*d^3*f*(d*Tan[e + f*x])^(3/2)) + (4*a^3)/(d^4*f*Sqrt[d*Tan[e + f*x]]) - (2*(a^3 + a^3*Tan[e + f*x]))/(7*d*f*(d*Tan[e + f*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(5/2), x, 7, -((d^(5/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(a*f)) + (d^(5/2)*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[2]*a*f) + (2*d^2*Sqrt[d*Tan[e + f*x]])/(a*f)} -{1/(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(3/2), x, 6, (d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(a*f) - (d^(3/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[2]*a*f)} -{1/(a + a*Tan[e + f*x])*(d*Tan[e + f*x])^(1/2), x, 6, -((Sqrt[d]*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(a*f)) - (Sqrt[d]*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[2]*a*f)} -{1/(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(1/2), x, 6, ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]]/(a*Sqrt[d]*f) + ArcTanh[(Sqrt[d]*(1 + Tan[e + f*x]))/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(Sqrt[2]*a*Sqrt[d]*f)} -{1/(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2), x, 7, -(ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]]/(a*d^(3/2)*f)) + ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(Sqrt[2]*a*d^(3/2)*f) - 2/(a*d*f*Sqrt[d*Tan[e + f*x]])} -{1/(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(5/2), x, 10, ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]]/(a*d^(5/2)*f) - ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(Sqrt[2]*a*d^(5/2)*f) - 2/(3*a*d*f*(d*Tan[e + f*x])^(3/2)) + 2/(a*d^2*f*Sqrt[d*Tan[e + f*x]])} - - -{1/(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(5/2), x, 17, (3*d^(5/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*f) + (d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) - (d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) + (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) - (d^(5/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) - (d^2*Sqrt[d*Tan[e + f*x]])/(2*f*(a^2 + a^2*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(3/2), x, 18, -((d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*f)) - (d^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) + (d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) + (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) - (d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) + (d*Sqrt[d*Tan[e + f*x]])/(2*f*(a^2 + a^2*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^2*(d*Tan[e + f*x])^(1/2), x, 17, -((Sqrt[d]*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*f)) - (Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) + (Sqrt[d]*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/(2*Sqrt[2]*a^2*f) - (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) + (Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(4*Sqrt[2]*a^2*f) - Sqrt[d*Tan[e + f*x]]/(2*f*(a^2 + a^2*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(1/2), x, 18, (3*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*Sqrt[d]*f) + ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*Sqrt[d]*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*Sqrt[d]*f) - Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*Sqrt[d]*f) + Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*Sqrt[d]*f) + Sqrt[d*Tan[e + f*x]]/(2*d*f*(a^2 + a^2*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(3/2), x, 18, -((5*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*d^(3/2)*f)) + ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*d^(3/2)*f) - ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*d^(3/2)*f) + Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*d^(3/2)*f) - Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*d^(3/2)*f) - 5/(2*a^2*d*f*Sqrt[d*Tan[e + f*x]]) + 1/(2*d*f*Sqrt[d*Tan[e + f*x]]*(a^2 + a^2*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^2/(d*Tan[e + f*x])^(5/2), x, 20, (7*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(2*a^2*d^(5/2)*f) - ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*d^(5/2)*f) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(2*Sqrt[2]*a^2*d^(5/2)*f) + Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*d^(5/2)*f) - Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]]/(4*Sqrt[2]*a^2*d^(5/2)*f) - 7/(6*a^2*d*f*(d*Tan[e + f*x])^(3/2)) + 9/(2*a^2*d^2*f*Sqrt[d*Tan[e + f*x]]) + 1/(2*d*f*(d*Tan[e + f*x])^(3/2)*(a^2 + a^2*Tan[e + f*x]))} - - -{1/(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(9/2), x, 9, -((31*d^(9/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*f)) + (d^(9/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(2*Sqrt[2]*a^3*f) + (27*d^4*Sqrt[d*Tan[e + f*x]])/(8*a^3*f) - (9*d^3*(d*Tan[e + f*x])^(3/2))/(8*a^3*f*(1 + Tan[e + f*x])) - (d^2*(d*Tan[e + f*x])^(5/2))/(4*a*f*(a + a*Tan[e + f*x])^2)} -{1/(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(7/2), x, 8, (11*d^(7/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*f) + (d^(7/2)*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(2*Sqrt[2]*a^3*f) - (7*d^3*Sqrt[d*Tan[e + f*x]])/(8*a^3*f*(1 + Tan[e + f*x])) - (d^2*(d*Tan[e + f*x])^(3/2))/(4*a*f*(a + a*Tan[e + f*x])^2)} -{1/(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(5/2), x, 8, (d^(5/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*f) - (d^(5/2)*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(2*Sqrt[2]*a^3*f) + (5*d^2*Sqrt[d*Tan[e + f*x]])/(8*a^3*f*(1 + Tan[e + f*x])) - (d^2*Sqrt[d*Tan[e + f*x]])/(4*a*f*(a + a*Tan[e + f*x])^2)} -{1/(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(3/2), x, 8, -((5*d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*f)) - (d^(3/2)*ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(2*Sqrt[2]*a^3*f) + (d*Sqrt[d*Tan[e + f*x]])/(4*a*f*(a + a*Tan[e + f*x])^2) - (d*Sqrt[d*Tan[e + f*x]])/(8*f*(a^3 + a^3*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^3*(d*Tan[e + f*x])^(1/2), x, 8, (Sqrt[d]*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*f) + (Sqrt[d]*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(2*Sqrt[2]*a^3*f) - Sqrt[d*Tan[e + f*x]]/(4*a*f*(a + a*Tan[e + f*x])^2) - (3*Sqrt[d*Tan[e + f*x]])/(8*f*(a^3 + a^3*Tan[e + f*x]))} -{1/(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(1/2), x, 8, (11*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*Sqrt[d]*f) + ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(2*Sqrt[2]*a^3*Sqrt[d]*f) + (7*Sqrt[d*Tan[e + f*x]])/(8*a^3*d*f*(1 + Tan[e + f*x])) + Sqrt[d*Tan[e + f*x]]/(4*a*d*f*(a + a*Tan[e + f*x])^2)} -{1/(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(3/2), x, 9, -((31*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*d^(3/2)*f)) - ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(2*Sqrt[2]*a^3*d^(3/2)*f) - 27/(8*a^3*d*f*Sqrt[d*Tan[e + f*x]]) + 9/(8*a^3*d*f*Sqrt[d*Tan[e + f*x]]*(1 + Tan[e + f*x])) + 1/(4*a*d*f*Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^2)} -{1/(a + a*Tan[e + f*x])^3/(d*Tan[e + f*x])^(5/2), x, 10, (59*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(8*a^3*d^(5/2)*f) - ArcTan[(Sqrt[d] - Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])]/(2*Sqrt[2]*a^3*d^(5/2)*f) - 55/(24*a^3*d*f*(d*Tan[e + f*x])^(3/2)) + 63/(8*a^3*d^2*f*Sqrt[d*Tan[e + f*x]]) + 11/(8*a^3*d*f*(d*Tan[e + f*x])^(3/2)*(1 + Tan[e + f*x])) + 1/(4*a*d*f*(d*Tan[e + f*x])^(3/2)*(a + a*Tan[e + f*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Tan[e+f x])^(m/2) Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[e + f*x]^5*Sqrt[1 + Tan[e + f*x]], x, 11, -((Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f) - (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f + (2*Sqrt[1 + Tan[e + f*x]])/f + (52*(1 + Tan[e + f*x])^(3/2))/(315*f) - (26*Tan[e + f*x]*(1 + Tan[e + f*x])^(3/2))/(105*f) - (4*Tan[e + f*x]^2*(1 + Tan[e + f*x])^(3/2))/(21*f) + (2*Tan[e + f*x]^3*(1 + Tan[e + f*x])^(3/2))/(9*f)} -{Tan[e + f*x]^3*Sqrt[1 + Tan[e + f*x]], x, 9, (Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f - (2*Sqrt[1 + Tan[e + f*x]])/f - (4*(1 + Tan[e + f*x])^(3/2))/(15*f) + (2*Tan[e + f*x]*(1 + Tan[e + f*x])^(3/2))/(5*f)} -{Tan[e + f*x]^1*Sqrt[1 + Tan[e + f*x]], x, 6, -((Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f) - (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f + (2*Sqrt[1 + Tan[e + f*x]])/f} -{Cot[e + f*x]^1*Sqrt[1 + Tan[e + f*x]], x, 9, (Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f - (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f} -{Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]], x, 11, -((Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f) + (9*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(4*f) - (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(4*f) - (Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(2*f)} -{Cot[e + f*x]^5*Sqrt[1 + Tan[e + f*x]], x, 13, (Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f - (139*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f + (11*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(64*f) + (53*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(96*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)} - -{Tan[e + f*x]^4*Sqrt[1 + Tan[e + f*x]], x, 14, -((Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - (18*(1 + Tan[e + f*x])^(3/2))/(35*f) - (8*Tan[e + f*x]*(1 + Tan[e + f*x])^(3/2))/(35*f) + (2*Tan[e + f*x]^2*(1 + Tan[e + f*x])^(3/2))/(7*f)} -{Tan[e + f*x]^2*Sqrt[1 + Tan[e + f*x]], x, 12, (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) + (2*(1 + Tan[e + f*x])^(3/2))/(3*f)} -{Tan[e + f*x]^0*Sqrt[1 + Tan[e + f*x]], x, 11, -((Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f)} -{Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]], x, 16, (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - ArcTanh[Sqrt[1 + Tan[e + f*x]]]/f - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f} -{Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]], x, 19, -((Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + (7*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) + (9*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(8*f) - (Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(3*f)} - - -{Tan[e + f*x]^5*(1 + Tan[e + f*x])^(3/2), x, 19, (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + (2*Sqrt[1 + Tan[e + f*x]])/f + (2*(1 + Tan[e + f*x])^(3/2))/(3*f) + (20*(1 + Tan[e + f*x])^(5/2))/(231*f) - (50*Tan[e + f*x]*(1 + Tan[e + f*x])^(5/2))/(231*f) - (4*Tan[e + f*x]^2*(1 + Tan[e + f*x])^(5/2))/(33*f) + (2*Tan[e + f*x]^3*(1 + Tan[e + f*x])^(5/2))/(11*f)} -{Tan[e + f*x]^3*(1 + Tan[e + f*x])^(3/2), x, 17, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) - (2*Sqrt[1 + Tan[e + f*x]])/f - (2*(1 + Tan[e + f*x])^(3/2))/(3*f) - (4*(1 + Tan[e + f*x])^(5/2))/(35*f) + (2*Tan[e + f*x]*(1 + Tan[e + f*x])^(5/2))/(7*f)} -{Tan[e + f*x]^1*(1 + Tan[e + f*x])^(3/2), x, 14, (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + (2*Sqrt[1 + Tan[e + f*x]])/f + (2*(1 + Tan[e + f*x])^(3/2))/(3*f)} -{Cot[e + f*x]^1*(1 + Tan[e + f*x])^(3/2), x, 16, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f)} -{Cot[e + f*x]^3*(1 + Tan[e + f*x])^(3/2), x, 18, (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f + (5*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(4*f) + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) - (5*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(4*f) - (Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(2*f)} -{Cot[e + f*x]^5*(1 + Tan[e + f*x])^(3/2), x, 20, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f - (83*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(2*Sqrt[1 + Sqrt[2]]*f) + (83*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(64*f) + (15*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(32*f) - (3*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(8*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)} - -{Tan[e + f*x]^4*(1 + Tan[e + f*x])^(3/2), x, 10, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (2*Sqrt[1 + Tan[e + f*x]])/f - (22*(1 + Tan[e + f*x])^(5/2))/(63*f) - (8*Tan[e + f*x]*(1 + Tan[e + f*x])^(5/2))/(63*f) + (2*Tan[e + f*x]^2*(1 + Tan[e + f*x])^(5/2))/(9*f)} -{Tan[e + f*x]^2*(1 + Tan[e + f*x])^(3/2), x, 8, (Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f - (2*Sqrt[1 + Tan[e + f*x]])/f + (2*(1 + Tan[e + f*x])^(5/2))/(5*f)} -{Tan[e + f*x]^0*(1 + Tan[e + f*x])^(3/2), x, 7, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (2*Sqrt[1 + Tan[e + f*x]])/f} -{Cot[e + f*x]^2*(1 + Tan[e + f*x])^(3/2), x, 11, (Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f - (3*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f} -{Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2), x, 13, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f) + (25*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (7*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(8*f) - (7*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[e + f*x]^5/Sqrt[1 + Tan[e + f*x]], x, 10, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f)) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) + (44*Sqrt[1 + Tan[e + f*x]])/(105*f) - (22*Tan[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(105*f) - (12*Tan[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(35*f) + (2*Tan[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(7*f)} -{Tan[e + f*x]^3/Sqrt[1 + Tan[e + f*x]], x, 8, (Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (4*Sqrt[1 + Tan[e + f*x]])/(3*f) + (2*Tan[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(3*f)} -{Tan[e + f*x]^1/Sqrt[1 + Tan[e + f*x]], x, 5, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f)) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f)} -{Cot[e + f*x]^1/Sqrt[1 + Tan[e + f*x]], x, 9, (Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f)} -{Cot[e + f*x]^3/Sqrt[1 + Tan[e + f*x]], x, 12, -((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f)) + (5*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(4*f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) + (3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(4*f) - (Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(2*f)} -{Cot[e + f*x]^5/Sqrt[1 + Tan[e + f*x]], x, 14, (Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (115*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/(2*f) - (13*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(64*f) + (13*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(96*f) + (7*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)} - -{Tan[e + f*x]^4/Sqrt[1 + Tan[e + f*x]], x, 14, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f)) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) - (14*Sqrt[1 + Tan[e + f*x]])/(15*f) - (8*Tan[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(15*f) + (2*Tan[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(5*f)} -{Tan[e + f*x]^2/Sqrt[1 + Tan[e + f*x]], x, 12, (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + (2*Sqrt[1 + Tan[e + f*x]])/f} -{Tan[e + f*x]^0/Sqrt[1 + Tan[e + f*x]], x, 11, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f)) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f)} -{Cot[e + f*x]^2/Sqrt[1 + Tan[e + f*x]], x, 19, (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) - (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) + ArcTanh[Sqrt[1 + Tan[e + f*x]]]/f + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f} -{Cot[e + f*x]^4/Sqrt[1 + Tan[e + f*x]], x, 19, -((Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f)) + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) - (3*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + (3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(8*f) + (5*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(3*f)} - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Tan[e+f x])^m (d Tan[e+f x])^n with n symbolic*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Tan[e+f x])^m (d Tan[e+f x])^n with m symbolic*) - - -{(a + a*Tan[e + f*x])^m*(d*Tan[e + f*x])^n, x, 7, (AppellF1[1 + n, -m, 1, 2 + n, -Tan[e + f*x], (-I)*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + a*Tan[e + f*x])^m)/((1 + Tan[e + f*x])^m*(2*d*f*(1 + n))) + (AppellF1[1 + n, -m, 1, 2 + n, -Tan[e + f*x], I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + a*Tan[e + f*x])^m)/((1 + Tan[e + f*x])^m*(2*d*f*(1 + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^5*(a + b*Tan[c + d*x]), x, 6, (-b)*x - (a*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x])/d - (a*Tan[c + d*x]^2)/(2*d) - (b*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^4)/(4*d) + (b*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^4*(a + b*Tan[c + d*x]), x, 5, a*x - (b*Log[Cos[c + d*x]])/d - (a*Tan[c + d*x])/d - (b*Tan[c + d*x]^2)/(2*d) + (a*Tan[c + d*x]^3)/(3*d) + (b*Tan[c + d*x]^4)/(4*d)} -{Tan[c + d*x]^3*(a + b*Tan[c + d*x]), x, 4, b*x + (a*Log[Cos[c + d*x]])/d - (b*Tan[c + d*x])/d + (a*Tan[c + d*x]^2)/(2*d) + (b*Tan[c + d*x]^3)/(3*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x]), x, 3, (-a)*x + (b*Log[Cos[c + d*x]])/d + (a*Tan[c + d*x])/d + (b*Tan[c + d*x]^2)/(2*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x]), x, 2, (-b)*x - (a*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x])/d} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x]), x, 2, a*x - (b*Log[Cos[c + d*x]])/d} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x]), x, 2, b*x + (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x]), x, 3, (-a)*x - (a*Cot[c + d*x])/d + (b*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x]), x, 4, (-b)*x - (b*Cot[c + d*x])/d - (a*Cot[c + d*x]^2)/(2*d) - (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x]), x, 5, a*x + (a*Cot[c + d*x])/d - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d) - (b*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x]), x, 6, b*x + (b*Cot[c + d*x])/d + (a*Cot[c + d*x]^2)/(2*d) - (b*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x]), x, 7, (-a)*x - (a*Cot[c + d*x])/d + (b*Cot[c + d*x]^2)/(2*d) + (a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5)/(5*d) + (b*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 6, (a^2 - b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d - ((a^2 - b^2)*Tan[c + d*x])/d - (a*b*Tan[c + d*x]^2)/d + ((a^2 - b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*Tan[c + d*x]^4)/(2*d) + (b^2*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 5, 2*a*b*x + ((a^2 - b^2)*Log[Cos[c + d*x]])/d - (2*a*b*Tan[c + d*x])/d + ((a^2 - b^2)*Tan[c + d*x]^2)/(2*d) + (2*a*b*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^4)/(4*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 3, -((a^2 - b^2)*x) + (2*a*b*Log[Cos[c + d*x]])/d - (b^2*Tan[c + d*x])/d + (a + b*Tan[c + d*x])^3/(3*b*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 3, -2*a*b*x - ((a^2 - b^2)*Log[Cos[c + d*x]])/d + (a*b*Tan[c + d*x])/d + (a + b*Tan[c + d*x])^2/(2*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^2, x, 2, (a^2 - b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^2, x, 3, 2*a*b*x - (b^2*Log[Cos[c + d*x]])/d + (a^2*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2, x, 3, -((a^2 - b^2)*x) - (a^2*Cot[c + d*x])/d + (2*a*b*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2, x, 4, -2*a*b*x - (2*a*b*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^2)/(2*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2, x, 5, (a^2 - b^2)*x + ((a^2 - b^2)*Cot[c + d*x])/d - (a*b*Cot[c + d*x]^2)/d - (a^2*Cot[c + d*x]^3)/(3*d) - (2*a*b*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2, x, 6, 2*a*b*x + (2*a*b*Cot[c + d*x])/d + ((a^2 - b^2)*Cot[c + d*x]^2)/(2*d) - (2*a*b*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^4)/(4*d) + ((a^2 - b^2)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2, x, 7, -((a^2 - b^2)*x) - ((a^2 - b^2)*Cot[c + d*x])/d + (a*b*Cot[c + d*x]^2)/d + ((a^2 - b^2)*Cot[c + d*x]^3)/(3*d) - (a*b*Cot[c + d*x]^4)/(2*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (2*a*b*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 7, b*(3*a^2 - b^2)*x + (a*(a^2 - 3*b^2)*Log[Cos[c + d*x]])/d - (b*(a^2 - b^2)*Tan[c + d*x])/d - (a*(a + b*Tan[c + d*x])^2)/(2*d) - (a + b*Tan[c + d*x])^3/(3*d) - (a*(a + b*Tan[c + d*x])^4)/(20*b^2*d) + (Tan[c + d*x]*(a + b*Tan[c + d*x])^4)/(5*b*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 4, (-a)*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d - (2*a*b^2*Tan[c + d*x])/d - (b*(a + b*Tan[c + d*x])^2)/(2*d) + (a + b*Tan[c + d*x])^4/(4*b*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 4, (-b)*(3*a^2 - b^2)*x - (a*(a^2 - 3*b^2)*Log[Cos[c + d*x]])/d + (b*(a^2 - b^2)*Tan[c + d*x])/d + (a*(a + b*Tan[c + d*x])^2)/(2*d) + (a + b*Tan[c + d*x])^3/(3*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^3, x, 3, a*(a^2 - 3*b^2)*x - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (2*a*b^2*Tan[c + d*x])/d + (b*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^3, x, 4, b*(3*a^2 - b^2)*x - (3*a*b^2*Log[Cos[c + d*x]])/d + (a^3*Log[Sin[c + d*x]])/d + (b^2*(a + b*Tan[c + d*x]))/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3, x, 4, (-a)*(a^2 - 3*b^2)*x - (b^3*Log[Cos[c + d*x]])/d + (3*a^2*b*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]*(a + b*Tan[c + d*x]))/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3, x, 4, (-b)*(3*a^2 - b^2)*x - (5*a^2*b*Cot[c + d*x])/(2*d) - (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^2*(a + b*Tan[c + d*x]))/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3, x, 5, a*(a^2 - 3*b^2)*x + (a*(a^2 - 3*b^2)*Cot[c + d*x])/d - (7*a^2*b*Cot[c + d*x]^2)/(6*d) - (b*(3*a^2 - b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^3*(a + b*Tan[c + d*x]))/(3*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3, x, 6, b*(3*a^2 - b^2)*x + (b*(3*a^2 - b^2)*Cot[c + d*x])/d + (a*(a^2 - 3*b^2)*Cot[c + d*x]^2)/(2*d) - (3*a^2*b*Cot[c + d*x]^3)/(4*d) + (a*(a^2 - 3*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c + d*x]))/(4*d)} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3, x, 7, (-a)*(a^2 - 3*b^2)*x - (a*(a^2 - 3*b^2)*Cot[c + d*x])/d + (b*(3*a^2 - b^2)*Cot[c + d*x]^2)/(2*d) + (a*(a^2 - 3*b^2)*Cot[c + d*x]^3)/(3*d) - (11*a^2*b*Cot[c + d*x]^4)/(20*d) + (b*(3*a^2 - b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^5*(a + b*Tan[c + d*x]))/(5*d)} - - -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^4, x, 8, 4*a*b*(a^2 - b^2)*x + ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d - (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d - ((a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) - (a*(a + b*Tan[c + d*x])^3)/(3*d) - (a + b*Tan[c + d*x])^4/(4*d) - (a*(a + b*Tan[c + d*x])^5)/(30*b^2*d) + (Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4, x, 5, -((a^4 - 6*a^2*b^2 + b^4)*x) + (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d - (b^2*(3*a^2 - b^2)*Tan[c + d*x])/d - (a*b*(a + b*Tan[c + d*x])^2)/d - (b*(a + b*Tan[c + d*x])^3)/(3*d) + (a + b*Tan[c + d*x])^5/(5*b*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^4, x, 5, -4*a*b*(a^2 - b^2)*x - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d + ((a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) + (a*(a + b*Tan[c + d*x])^3)/(3*d) + (a + b*Tan[c + d*x])^4/(4*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^4, x, 4, (a^4 - 6*a^2*b^2 + b^4)*x - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(3*a^2 - b^2)*Tan[c + d*x])/d + (a*b*(a + b*Tan[c + d*x])^2)/d + (b*(a + b*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^4, x, 5, 4*a*b*(a^2 - b^2)*x - (b^2*(6*a^2 - b^2)*Log[Cos[c + d*x]])/d + (a^4*Log[Sin[c + d*x]])/d + (3*a*b^3*Tan[c + d*x])/d + (b^2*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4, x, 5, -((a^4 - 6*a^2*b^2 + b^4)*x) - (4*a*b^3*Log[Cos[c + d*x]])/d + (4*a^3*b*Log[Sin[c + d*x]])/d + (b^2*(a^2 + b^2)*Tan[c + d*x])/d - (a^2*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4, x, 5, -4*a*b*(a^2 - b^2)*x - (3*a^3*b*Cot[c + d*x])/d - (b^4*Log[Cos[c + d*x]])/d - (a^2*(a^2 - 6*b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4, x, 5, (a^4 - 6*a^2*b^2 + b^4)*x + (a^2*(3*a^2 - 17*b^2)*Cot[c + d*x])/(3*d) - (4*a^3*b*Cot[c + d*x]^2)/(3*d) - (4*a*b*(a^2 - b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(3*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4, x, 6, 4*a*b*(a^2 - b^2)*x + (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d + (a^2*(2*a^2 - 11*b^2)*Cot[c + d*x]^2)/(4*d) - (5*a^3*b*Cot[c + d*x]^3)/(6*d) + ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2)/(4*d)} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^4, x, 7, -((a^4 - 6*a^2*b^2 + b^4)*x) - ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x])/d + (2*a*b*(a^2 - b^2)*Cot[c + d*x]^2)/d + (a^2*(5*a^2 - 27*b^2)*Cot[c + d*x]^3)/(15*d) - (3*a^3*b*Cot[c + d*x]^4)/(5*d) + (4*a*b*(a^2 - b^2)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*d)} -{Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4, x, 8, -4*a*b*(a^2 - b^2)*x - (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d - ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/(2*d) + (4*a*b*(a^2 - b^2)*Cot[c + d*x]^3)/(3*d) + (a^2*(3*a^2 - 16*b^2)*Cot[c + d*x]^4)/(12*d) - (7*a^3*b*Cot[c + d*x]^5)/(15*d) - ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2)/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^6/(a + b*Tan[c + d*x]), x, 8, -((a*x)/(a^2 + b^2)) - (b*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^6*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)*d) - (a*(a^2 - b^2)*Tan[c + d*x])/(b^4*d) + ((a^2 - b^2)*Tan[c + d*x]^2)/(2*b^3*d) - (a*Tan[c + d*x]^3)/(3*b^2*d) + Tan[c + d*x]^4/(4*b*d)} -{Tan[c + d*x]^5/(a + b*Tan[c + d*x]), x, 7, (b*x)/(a^2 + b^2) - (a^5*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)*d) - (a*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + ((a^2 - b^2)*Tan[c + d*x])/(b^3*d) - (a*Tan[c + d*x]^2)/(2*b^2*d) + Tan[c + d*x]^3/(3*b*d)} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x]), x, 6, (a*x)/(a^2 + b^2) + (a^4*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) + (b*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a*Tan[c + d*x])/(b^2*d) + Tan[c + d*x]^2/(2*b*d)} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x]), x, 5, -((b*x)/(a^2 + b^2)) - (a^3*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) + (a*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + Tan[c + d*x]/(b*d)} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x]), x, 4, -((a*x)/(a^2 + b^2)) - Log[Cos[c + d*x]]/(b*d) + (a^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b*(a^2 + b^2)*d), -((a*x)/b^2) + (a^3*x)/(b^2*(a^2 + b^2)) - Log[Cos[c + d*x]]/(b*d) + (a^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b*(a^2 + b^2)*d)} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x]), x, 2, (b*x)/(a^2 + b^2) - (a*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x]), x, 2, (a*x)/(a^2 + b^2) + (b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x]), x, 3, -((b*x)/(a^2 + b^2)) + Log[Sin[c + d*x]]/(a*d) - (b^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x]), x, 4, -((a*x)/(a^2 + b^2)) - Cot[c + d*x]/(a*d) - (b*Log[Sin[c + d*x]])/(a^2*d) + (b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)*d)} -{Cot[c + d*x]^3/(a + b*Tan[c + d*x]), x, 5, (b*x)/(a^2 + b^2) + (b*Cot[c + d*x])/(a^2*d) - Cot[c + d*x]^2/(2*a*d) - ((a^2 - b^2)*Log[Sin[c + d*x]])/(a^3*d) - (b^4*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)*d)} -{Cot[c + d*x]^4/(a + b*Tan[c + d*x]), x, 6, (a*x)/(a^2 + b^2) + ((a^2 - b^2)*Cot[c + d*x])/(a^3*d) + (b*Cot[c + d*x]^2)/(2*a^2*d) - Cot[c + d*x]^3/(3*a*d) + (b*(a^2 - b^2)*Log[Sin[c + d*x]])/(a^4*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)*d)} - - -{Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2, x, 8, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*a*b*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (2*a^5*(2*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^2*d) + ((4*a^4 + 2*a^2*b^2 - b^4)*Tan[c + d*x])/(b^4*(a^2 + b^2)*d) - (a*(2*a^2 + b^2)*Tan[c + d*x]^2)/(b^3*(a^2 + b^2)*d) + ((4*a^2 + b^2)*Tan[c + d*x]^3)/(3*b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^4)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^5/(a + b*Tan[c + d*x])^2, x, 7, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^4*(3*a^2 + 5*b^2)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^2*d) - (a*(3*a^2 + 2*b^2)*Tan[c + d*x])/(b^3*(a^2 + b^2)*d) + ((3*a^2 + b^2)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2, x, 6, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (2*a^3*(a^2 + 2*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) + ((2*a^2 + b^2)*Tan[c + d*x])/(b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 5, -((2*a*b*x)/(a^2 + b^2)^2) + (a^2*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d) + ((a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + a^3/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])), -((2*a*b*x)/(a^2 + b^2)^2) + ((a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^2*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 3, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - a^2/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 3, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + a/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x])^2, x, 3, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - b/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x])^2, x, 4, -((2*a*b*x)/(a^2 + b^2)^2) + Log[Sin[c + d*x]]/(a^2*d) - (b^2*(3*a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)^2*d) + b^2/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x])^2, x, 5, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*b*Log[Sin[c + d*x]])/(a^3*d) + (2*b^3*(2*a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (b*(a^2 + 2*b^2))/(a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2, x, 6, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) - (b^4*(5*a^2 + 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b^2*(2*a^2 + 3*b^2))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (3*b*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - Cot[c + d*x]^2/(2*a*d*(a + b*Tan[c + d*x]))} - - -{Tan[c + d*x]^6/(a + b*Tan[c + d*x])^3, x, 8, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a^4*(6*a^4 + 17*a^2*b^2 + 15*b^4)*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^3*d) - (a*(6*a^4 + 11*a^2*b^2 + 3*b^4)*Tan[c + d*x])/(b^4*(a^2 + b^2)^2*d) + ((6*a^4 + 11*a^2*b^2 + b^4)*Tan[c + d*x]^2)/(2*b^3*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x]^4)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a^2*(a^2 + 2*b^2)*Tan[c + d*x]^3)/(b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^5/(a + b*Tan[c + d*x])^3, x, 7, (b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3 - (a*(a^2 - 3*b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) - (a^3*(3*a^4 + 9*a^2*b^2 + 10*b^4)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3*d) + ((3*a^4 + 6*a^2*b^2 + b^4)*Tan[c + d*x])/(b^3*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x]^3)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(3*a^2 + 7*b^2)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x])^3, x, 6, (a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a^2*(a^4 + 3*a^2*b^2 + 6*b^4)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^2)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a^3*(a^2 + 3*b^2))/(b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x])^3, x, 4, -((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + (a*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (a^2*Tan[c + d*x])/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(a^2 + 5*b^2))/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 4, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - a^2/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (2*a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 4, (b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3 - (a*(a^2 - 3*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + a/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a^2 - b^2)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x])^3, x, 4, (a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - b/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x])^3, x, 5, -((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + Log[Sin[c + d*x]]/(a^3*d) - (b^2*(6*a^4 + 3*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^3*d) + b^2/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^2*(3*a^2 + b^2))/(a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x])^3, x, 6, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (3*b*Log[Sin[c + d*x]])/(a^4*d) + (b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2 + 3*b^2))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} - - -{Tan[c + d*x]^6/(a + b*Tan[c + d*x])^4, x, 8, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) - (4*a^3*(a^6 + 4*a^4*b^2 + 6*a^2*b^4 + 5*b^6)*Log[a + b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^4*d) + ((4*a^6 + 12*a^4*b^2 + 13*a^2*b^4 + b^6)*Tan[c + d*x])/(b^4*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^4)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(2*a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a^2*(a^4 + 3*a^2*b^2 + 4*b^4)*Tan[c + d*x]^2)/(b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^5/(a + b*Tan[c + d*x])^4, x, 7, (4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a^2*(a^6 + 4*a^4*b^2 + 5*a^2*b^4 + 10*b^6)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) - (a^2*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 3*b^2)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (a^3*(a^4 + 3*a^2*b^2 + 6*b^4))/(b^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x])^4, x, 5, ((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^2*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^3*(a^2 + 4*b^2))/(3*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a^2*(2*a^4 + 7*a^2*b^2 + 17*b^4))/(3*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x])^4, x, 5, -((4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4) + ((a^4 - 6*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 7*b^2))/(6*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a*(a^2 - 3*b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x])^4, x, 5, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - a^2/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x])^4, x, 5, (4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 - ((a^4 - 6*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) + a/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^2 - b^2)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (a*(a^2 - 3*b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x])^4, x, 5, ((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x])^4, x, 6, -((4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4) + Log[Sin[c + d*x]]/(a^4*d) - (b^2*(10*a^6 + 5*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^4*d) + b^2/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b^2*(3*a^2 + b^2))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b^2*(6*a^4 + 3*a^2*b^2 + b^4))/(a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4, x, 7, -(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*b*Log[Sin[c + d*x]])/(a^5*d) + (4*b^3*(5*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^5*(a^2 + b^2)^4*d) - (b*(3*a^2 + 4*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^3) - (b*(a^4 + 4*a^2*b^2 + 2*b^4))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(a^6 + 13*a^4*b^2 + 12*a^2*b^4 + 4*b^6))/(a^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} - - -{1/(3 + 5*Tan[c + d*x]), x, 2, (3*x)/34 + (5*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(34*d)} -{1/(3 + 5*Tan[c + d*x])^2, x, 3, -((4*x)/289) + (15*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(578*d) - 5/(34*d*(3 + 5*Tan[c + d*x]))} -{1/(3 + 5*Tan[c + d*x])^3, x, 4, -((99*x)/19652) + (5*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(19652*d) - 5/(68*d*(3 + 5*Tan[c + d*x])^2) - 15/(578*d*(3 + 5*Tan[c + d*x]))} -{1/(3 + 5*Tan[c + d*x])^4, x, 5, -((161*x)/334084) - (60*Log[3*Cos[c + d*x] + 5*Sin[c + d*x]])/(83521*d) - 5/(102*d*(3 + 5*Tan[c + d*x])^3) - 15/(1156*d*(3 + 5*Tan[c + d*x])^2) - 5/(19652*d*(3 + 5*Tan[c + d*x]))} - -{1/(5 + 3*Tan[c + d*x]), x, 2, (5*x)/34 + (3*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(34*d)} -{1/(5 + 3*Tan[c + d*x])^2, x, 3, (4*x)/289 + (15*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(578*d) - 3/(34*d*(5 + 3*Tan[c + d*x]))} -{1/(5 + 3*Tan[c + d*x])^3, x, 4, -((5*x)/19652) + (99*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(19652*d) - 3/(68*d*(5 + 3*Tan[c + d*x])^2) - 15/(578*d*(5 + 3*Tan[c + d*x]))} -{1/(5 + 3*Tan[c + d*x])^4, x, 5, -((161*x)/334084) + (60*Log[5*Cos[c + d*x] + 3*Sin[c + d*x]])/(83521*d) - 1/(34*d*(5 + 3*Tan[c + d*x])^3) - 15/(1156*d*(5 + 3*Tan[c + d*x])^2) - 99/(19652*d*(5 + 3*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]], x, 14, (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*(8*a^2 - 35*b^2)*(a + b*Tan[c + d*x])^(3/2))/(105*b^3*d) - (8*a*Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/(35*b^2*d) + (2*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2))/(7*b*d)} -{Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]], x, 11, (Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*Sqrt[a + b*Tan[c + d*x]])/d - (4*a*(a + b*Tan[c + d*x])^(3/2))/(15*b^2*d) + (2*Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/(5*b*d)} -{Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]], x, 12, -((b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*(a + b*Tan[c + d*x])^(3/2))/(3*b*d)} -{Tan[c + d*x]^1*Sqrt[a + b*Tan[c + d*x]], x, 8, -((Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - (Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*Sqrt[a + b*Tan[c + d*x]])/d} -{Tan[c + d*x]^0*Sqrt[a + b*Tan[c + d*x]], x, 11, (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} -{Sqrt[a + b*Tan[c + d*x]]*Cot[c + d*x]^1, x, 11, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + (Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d} -{Sqrt[a + b*Tan[c + d*x]]*Cot[c + d*x]^2, x, 16, -((b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[a + b*Tan[c + d*x]]*Cot[c + d*x]^3, x, 13, ((8*a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - (Sqrt[a - I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*a*d) - (Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d)} - - -{Tan[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2), x, 11, -((I*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + (I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*Sqrt[a + b*Tan[c + d*x]])/d + (2*(8*a^2 - 63*b^2)*(a + b*Tan[c + d*x])^(5/2))/(315*b^3*d) - (8*a*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2))/(63*b^2*d) + (2*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2))/(9*b*d)} -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2), x, 12, ((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/d - (2*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (4*a*(a + b*Tan[c + d*x])^(5/2))/(35*b^2*d) + (2*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2))/(7*b*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2), x, 9, (I*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*b*Sqrt[a + b*Tan[c + d*x]])/d + (2*(a + b*Tan[c + d*x])^(5/2))/(5*b*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^(3/2), x, 9, -(((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*a*Sqrt[a + b*Tan[c + d*x]])/d + (2*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^(3/2), x, 8, -((I*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + (I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*Sqrt[a + b*Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^(3/2)*Cot[c + d*x]^1, x, 11, -((2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + ((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d} -{(a + b*Tan[c + d*x])^(3/2)*Cot[c + d*x]^2, x, 12, -((3*Sqrt[a]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + (I*(a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (a*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^(3/2)*Cot[c + d*x]^3, x, 13, ((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (5*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d)} - - -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2), x, 13, ((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*(a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d - (2*a*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (2*(a + b*Tan[c + d*x])^(5/2))/(5*d) - (4*a*(a + b*Tan[c + d*x])^(7/2))/(63*b^2*d) + (2*Tan[c + d*x]*(a + b*Tan[c + d*x])^(7/2))/(9*b*d)} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2), x, 10, (I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (4*a*b*Sqrt[a + b*Tan[c + d*x]])/d - (2*b*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*(a + b*Tan[c + d*x])^(7/2))/(7*b*d)} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^(5/2), x, 10, -(((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (2*a*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*(a + b*Tan[c + d*x])^(5/2))/(5*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^(5/2), x, 9, -((I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (4*a*b*Sqrt[a + b*Tan[c + d*x]])/d + (2*b*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{(a + b*Tan[c + d*x])^(5/2)*Cot[c + d*x]^1, x, 12, -((2*a^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + ((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b^2*Sqrt[a + b*Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^(5/2)*Cot[c + d*x]^2, x, 12, -((5*a^(3/2)*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + (I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (a^2*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^(5/2)*Cot[c + d*x]^3, x, 13, (Sqrt[a]*(8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*d) - ((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (9*a*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a^2*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d)} -{(a + b*Tan[c + d*x])^(5/2)*Cot[c + d*x]^4, x, 14, (5*b*(8*a^2 - b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*Sqrt[a]*d) - (I*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2 - 11*b^2)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*d) - (13*a*b*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*d) - (a^2*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(3*d)} - - -{(a + b*Tan[c + d*x])^(7/2), x, 10, -((I*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + (I*(a + I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(3*a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (4*a*b*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*b*(a + b*Tan[c + d*x])^(5/2))/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^5/Sqrt[a + b*Tan[c + d*x]], x, 12, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(Sqrt[a - I*b]*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(Sqrt[a + I*b]*d) - (4*a*(24*a^2 - 35*b^2)*Sqrt[a + b*Tan[c + d*x]])/(105*b^4*d) + (2*(24*a^2 - 35*b^2)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(105*b^3*d) - (12*a*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(35*b^2*d) + (2*Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(7*b*d)} -{Tan[c + d*x]^4/Sqrt[a + b*Tan[c + d*x]], x, 14, (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*(8*a^2 - 15*b^2)*Sqrt[a + b*Tan[c + d*x]])/(15*b^3*d) - (8*a*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(15*b^2*d) + (2*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(5*b*d)} -{Tan[c + d*x]^3/Sqrt[a + b*Tan[c + d*x]], x, 10, ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(Sqrt[a - I*b]*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(Sqrt[a + I*b]*d) - (4*a*Sqrt[a + b*Tan[c + d*x]])/(3*b^2*d) + (2*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b*d)} -{Tan[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]], x, 12, -((b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*Sqrt[a + b*Tan[c + d*x]])/(b*d)} -{Tan[c + d*x]^1/Sqrt[a + b*Tan[c + d*x]], x, 7, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(Sqrt[a - I*b]*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(Sqrt[a + I*b]*d)} -{Tan[c + d*x]^0/Sqrt[a + b*Tan[c + d*x]], x, 11, (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} -{Cot[c + d*x]^1/Sqrt[a + b*Tan[c + d*x]], x, 11, -((2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(Sqrt[a - I*b]*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(Sqrt[a + I*b]*d)} -{Cot[c + d*x]^2/Sqrt[a + b*Tan[c + d*x]], x, 17, (b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*d)} -{Cot[c + d*x]^3/Sqrt[a + b*Tan[c + d*x]], x, 14, ((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(5/2)*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/(Sqrt[a - I*b]*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(Sqrt[a + I*b]*d) + (3*b*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*a^2*d) - (Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*a*d)} - - -{Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(3/2), x, 11, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^3)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(16*a^4 + 6*a^2*b^2 - 5*b^4)*Sqrt[a + b*Tan[c + d*x]])/(5*b^4*(a^2 + b^2)*d) - (2*a*(8*a^2 + 3*b^2)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(5*b^3*(a^2 + b^2)*d) + (2*(6*a^2 + b^2)*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(5*b^2*(a^2 + b^2)*d)} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(3/2), x, 10, -((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) - (2*a*(8*a^2 + 5*b^2)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d) + (2*(4*a^2 + b^2)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b^2*(a^2 + b^2)*d)} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2), x, 9, ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) - (2*a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(2*a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d)} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x])^(3/2), x, 8, (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x])^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) + (2*a)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x])^(3/2), x, 8, -((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x])^(3/2), x, 12, -((2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) + (2*b^2)/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(3/2), x, 13, (3*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (b*(a^2 + 3*b^2))/(a^2*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) - Cot[c + d*x]/(a*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2), x, 14, ((8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(7/2)*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(3/2)*d) + (b^2*(7*a^2 + 15*b^2))/(4*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (5*b*Cot[c + d*x])/(4*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - Cot[c + d*x]^2/(2*a*d*Sqrt[a + b*Tan[c + d*x]])} - - -{Tan[c + d*x]^5/(a + b*Tan[c + d*x])^(5/2), x, 11, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^2*(a^2 + 2*b^2)*Tan[c + d*x]^2)/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) - (4*a*(8*a^4 + 15*a^2*b^2 + 4*b^4)*Sqrt[a + b*Tan[c + d*x]])/(3*b^4*(a^2 + b^2)^2*d) + (2*(8*a^4 + 15*a^2*b^2 + b^4)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)^2*d)} -{Tan[c + d*x]^4/(a + b*Tan[c + d*x])^(5/2), x, 10, -((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (4*a^3*(2*a^2 + 5*b^2))/(3*b^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*a^2 + 3*b^2)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d)} -{Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2), x, 9, ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^2*(a^2 + 4*b^2))/(3*b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2), x, 9, (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*a^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (4*a*b)/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^1/(a + b*Tan[c + d*x])^(5/2), x, 9, -(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(5/2)*d) + (2*a)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*(a^2 - b^2))/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^0/(a + b*Tan[c + d*x])^(5/2), x, 9, -((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a*b)/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^1/(a + b*Tan[c + d*x])^(5/2), x, 13, -((2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d)) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/((a + I*b)^(5/2)*d) + (2*b^2)/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b^2*(3*a^2 + b^2))/(a^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2), x, 14, (5*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(7/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (b*(3*a^2 + 5*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (b*(a^4 + 10*a^2*b^2 + 5*b^4))/(a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} - - -{1/(a + b*Tan[c + d*x])^(7/2), x, 10, -((I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(7/2)*d)) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(7/2)*d) - (2*b)/(5*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(5/2)) - (4*a*b)/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*Sqrt[a + b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]), x, 13, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*b*Sqrt[Tan[c + d*x]])/d + (2*a*Tan[c + d*x]^(3/2))/(3*d) + (2*b*Tan[c + d*x]^(5/2))/(5*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]), x, 12, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*a*Sqrt[Tan[c + d*x]])/d + (2*b*Tan[c + d*x]^(3/2))/(3*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]), x, 11, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*Sqrt[Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])/Sqrt[Tan[c + d*x]], x, 10, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Tan[c + d*x]^(3/2), x, 11, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a)/(d*Sqrt[Tan[c + d*x]])} -{(a + b*Tan[c + d*x])/Tan[c + d*x]^(5/2), x, 12, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a)/(3*d*Tan[c + d*x]^(3/2)) - (2*b)/(d*Sqrt[Tan[c + d*x]])} -{(a + b*Tan[c + d*x])/Tan[c + d*x]^(7/2), x, 13, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a)/(5*d*Tan[c + d*x]^(5/2)) - (2*b)/(3*d*Tan[c + d*x]^(3/2)) + (2*a)/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2, x, 14, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (4*a*b*Sqrt[Tan[c + d*x]])/d + (2*(a^2 - b^2)*Tan[c + d*x]^(3/2))/(3*d) + (4*a*b*Tan[c + d*x]^(5/2))/(5*d) + (2*b^2*Tan[c + d*x]^(7/2))/(7*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2, x, 13, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(a^2 - b^2)*Sqrt[Tan[c + d*x]])/d + (4*a*b*Tan[c + d*x]^(3/2))/(3*d) + (2*b^2*Tan[c + d*x]^(5/2))/(5*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2, x, 12, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (4*a*b*Sqrt[Tan[c + d*x]])/d + (2*b^2*Tan[c + d*x]^(3/2))/(3*d)} -{(a + b*Tan[c + d*x])^2/Sqrt[Tan[c + d*x]], x, 11, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b^2*Sqrt[Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(3/2), x, 11, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2)/(d*Sqrt[Tan[c + d*x]])} -{(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(5/2), x, 12, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2)/(3*d*Tan[c + d*x]^(3/2)) - (4*a*b)/(d*Sqrt[Tan[c + d*x]])} -{(a + b*Tan[c + d*x])^2/Tan[c + d*x]^(7/2), x, 13, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2)/(5*d*Tan[c + d*x]^(5/2)) - (4*a*b)/(3*d*Tan[c + d*x]^(3/2)) + (2*(a^2 - b^2))/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3, x, 15, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*b*(3*a^2 - b^2)*Sqrt[Tan[c + d*x]])/d + (2*a*(a^2 - 3*b^2)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*(3*a^2 - b^2)*Tan[c + d*x]^(5/2))/(5*d) + (40*a*b^2*Tan[c + d*x]^(7/2))/(63*d) + (2*b^2*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3, x, 14, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*a*(a^2 - 3*b^2)*Sqrt[Tan[c + d*x]])/d + (2*b*(3*a^2 - b^2)*Tan[c + d*x]^(3/2))/(3*d) + (32*a*b^2*Tan[c + d*x]^(5/2))/(35*d) + (2*b^2*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/(7*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3, x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*(3*a^2 - b^2)*Sqrt[Tan[c + d*x]])/d + (8*a*b^2*Tan[c + d*x]^(3/2))/(5*d) + (2*b^2*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))/(5*d)} -{(a + b*Tan[c + d*x])^3/Sqrt[Tan[c + d*x]], x, 12, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (16*a*b^2*Sqrt[Tan[c + d*x]])/(3*d) + (2*b^2*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(3*d)} -{(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(3/2), x, 12, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*(a^2 + b^2)*Sqrt[Tan[c + d*x]])/d - (2*a^2*(a + b*Tan[c + d*x]))/(d*Sqrt[Tan[c + d*x]])} -{(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(5/2), x, 12, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (16*a^2*b)/(3*d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(3*d*Tan[c + d*x]^(3/2))} -{(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(7/2), x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (8*a^2*b)/(5*d*Tan[c + d*x]^(3/2)) + (2*a*(a^2 - 3*b^2))/(d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(5*d*Tan[c + d*x]^(5/2))} -{(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(9/2), x, 14, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (32*a^2*b)/(35*d*Tan[c + d*x]^(5/2)) + (2*a*(a^2 - 3*b^2))/(3*d*Tan[c + d*x]^(3/2)) + (2*b*(3*a^2 - b^2))/(d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(7*d*Tan[c + d*x]^(7/2))} -{(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(11/2), x, 15, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (40*a^2*b)/(63*d*Tan[c + d*x]^(7/2)) + (2*a*(a^2 - 3*b^2))/(5*d*Tan[c + d*x]^(5/2)) + (2*b*(3*a^2 - b^2))/(3*d*Tan[c + d*x]^(3/2)) - (2*a*(a^2 - 3*b^2))/(d*Sqrt[Tan[c + d*x]]) - (2*a^2*(a + b*Tan[c + d*x]))/(9*d*Tan[c + d*x]^(9/2))} - - -{(a + b*Tan[c + d*x])/Sqrt[Tan[c + d*x]], x, 10, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Sqrt[-Tan[c + d*x]], x, 10, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[-Tan[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[-Tan[c + d*x]]])/(Sqrt[2]*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[-Tan[c + d*x]] - Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[-Tan[c + d*x]] - Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Sqrt[e*Tan[c + d*x]], x, 10, -(((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e])) + ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])} -{(a + b*Tan[c + d*x])/Sqrt[-e*Tan[c + d*x]], x, 10, ((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[(-e)*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[(-e)*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) + ((a + b)*Log[Sqrt[e] - Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[(-e)*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a + b)*Log[Sqrt[e] - Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[(-e)*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x]), x, 17, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(7/2)*(a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*(a^2 - b^2)*Sqrt[Tan[c + d*x]])/(b^3*d) - (2*a*Tan[c + d*x]^(3/2))/(3*b^2*d) + (2*Tan[c + d*x]^(5/2))/(5*b*d)} -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x]), x, 16, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*a*Sqrt[Tan[c + d*x]])/(b^2*d) + (2*Tan[c + d*x]^(3/2))/(3*b*d)} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x]), x, 15, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*Sqrt[Tan[c + d*x]])/(b*d)} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x]), x, 14, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x]), x, 14, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/((a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])), x, 14, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 15, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(a*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 16, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(3*a*d*Tan[c + d*x]^(3/2)) + (2*b)/(a^2*d*Sqrt[Tan[c + d*x]])} -{1/(Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x])), x, 17, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(5*a*d*Tan[c + d*x]^(5/2)) + (2*b)/(3*a^2*d*Tan[c + d*x]^(3/2)) + (2*(a^2 - b^2))/(a^3*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^2, x, 17, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(7/2)*(5*a^2 + 9*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(7/2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (a*(5*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) + ((5*a^2 + 2*b^2)*Tan[c + d*x]^(3/2))/(3*b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^(5/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^2, x, 16, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (a^(5/2)*(3*a^2 + 7*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((3*a^2 + 2*b^2)*Sqrt[Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^(3/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2, x, 15, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(3/2)*(a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (a^2*Sqrt[Tan[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^2, x, 15, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (Sqrt[a]*(a^2 - 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (a*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^2, x, 15, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2), x, 15, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (b^2*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2), x, 16, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + b^2/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2), x, 17, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(7/2)*(9*a^2 + 5*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2 + 5*b^2)/(3*a^2*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)) + (b*(4*a^2 + 5*b^2))/(a^3*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + b^2/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))} - - -{Tan[c + d*x]^(11/2)/(a + b*Tan[c + d*x])^3, x, 18, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^(7/2)*(35*a^4 + 102*a^2*b^2 + 99*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(9/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (a*(35*a^4 + 67*a^2*b^2 + 24*b^4)*Sqrt[Tan[c + d*x]])/(4*b^4*(a^2 + b^2)^2*d) + ((35*a^4 + 67*a^2*b^2 + 8*b^4)*Tan[c + d*x]^(3/2))/(12*b^3*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x]^(7/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(7*a^2 + 15*b^2)*Tan[c + d*x]^(5/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^3, x, 17, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(7/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((15*a^4 + 31*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x]^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(5*a^2 + 13*b^2)*Tan[c + d*x]^(3/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^3, x, 16, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(5/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (a^2*Tan[c + d*x]^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(3*a^2 + 11*b^2)*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^3, x, 16, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(3/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (a^2*Sqrt[Tan[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(a^2 + 9*b^2)*Sqrt[Tan[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3, x, 16, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((3*a^4 - 26*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*Sqrt[a]*Sqrt[b]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (a*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((3*a^2 - 5*b^2)*Sqrt[Tan[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^3, x, 16, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[b]*(15*a^4 - 18*a^2*b^2 - b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(3/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (b*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (b*(7*a^2 - b^2)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3), x, 16, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(5/2)*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (b^2*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^2*(11*a^2 + 3*b^2)*Sqrt[Tan[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3), x, 17, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4 + 31*a^2*b^2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + b^2/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2) + (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))} -{1/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3), x, 18, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b^(7/2)*(99*a^4 + 102*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(9/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4 + 67*a^2*b^2 + 35*b^4)/(12*a^3*(a^2 + b^2)^2*d*Tan[c + d*x]^(3/2)) + (b*(24*a^4 + 67*a^2*b^2 + 35*b^4))/(4*a^4*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + b^2/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2) + (b^2*(15*a^2 + 7*b^2))/(4*a^2*(a^2 + b^2)^2*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(1/2), x, 14, -((Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - ((a^2 + 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*b^(3/2)*d) + (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*b*d) + (Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(2*b*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(1/2), x, 13, (I*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[b]*d) + (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Tan[c + d*x]^(1/2)*(a + b*Tan[c + d*x])^(1/2), x, 11, (Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d} -{Tan[c + d*x]^(-1/2)*(a + b*Tan[c + d*x])^(1/2), x, 7, -((I*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d} -{Tan[c + d*x]^(-3/2)*(a + b*Tan[c + d*x])^(1/2), x, 8, -((Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)*(a + b*Tan[c + d*x])^(1/2), x, 10, (I*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (2*b*Sqrt[a + b*Tan[c + d*x]])/(3*a*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-7/2)*(a + b*Tan[c + d*x])^(1/2), x, 10, (Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*b*Sqrt[a + b*Tan[c + d*x]])/(15*a*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2 + 2*b^2)*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2), x, 15, (I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (a*(a^2 + 24*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(8*b^(3/2)*d) - (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((a^2 + 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(8*b*d) - (a*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(12*b*d) + (Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2))/(3*b*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2), x, 14, ((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((3*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*Sqrt[b]*d) + ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (3*a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*d) + (Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(2*d)} -{Tan[c + d*x]^(1/2)*(a + b*Tan[c + d*x])^(3/2), x, 13, -((I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Tan[c + d*x]^(-1/2)*(a + b*Tan[c + d*x])^(3/2), x, 12, -(((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d} -{Tan[c + d*x]^(-3/2)*(a + b*Tan[c + d*x])^(3/2), x, 8, (I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)*(a + b*Tan[c + d*x])^(3/2), x, 9, ((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (8*b*Sqrt[a + b*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-7/2)*(a + b*Tan[c + d*x])^(3/2), x, 10, -((I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (4*b*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(3/2)) + (2*(5*a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/(5*a*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-9/2)*(a + b*Tan[c + d*x])^(3/2), x, 11, -(((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (16*b*Sqrt[a + b*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (2*(35*a^2 - 3*b^2)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d*Tan[c + d*x]^(3/2)) + (4*b*(70*a^2 + 3*b^2)*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2), x, 16, ((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((5*a^4 + 240*a^2*b^2 - 128*b^4)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(64*b^(3/2)*d) - ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (a*(5*a^2 + 112*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(64*b*d) - ((5*a^2 + 48*b^2)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(96*b*d) - (a*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2))/(24*b*d) + (Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2))/(4*b*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2), x, 15, -((I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (5*a*(a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(8*Sqrt[b]*d) - (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((11*a^2 - 8*b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(8*d) + (13*a*b*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(12*d) + (b^2*Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Tan[c + d*x]^(1/2)*(a + b*Tan[c + d*x])^(5/2), x, 14, -(((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (Sqrt[b]*(15*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*d) + ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (9*a*b*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*d) + (b^2*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(2*d)} -{Tan[c + d*x]^(-1/2)*(a + b*Tan[c + d*x])^(5/2), x, 13, (I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b^2*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Tan[c + d*x]^(-3/2)*(a + b*Tan[c + d*x])^(5/2), x, 13, ((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)*(a + b*Tan[c + d*x])^(5/2), x, 9, -((I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (14*a*b*Sqrt[a + b*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-7/2)*(a + b*Tan[c + d*x])^(5/2), x, 10, -(((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (22*a*b*Sqrt[a + b*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2 - 23*b^2)*Sqrt[a + b*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-9/2)*(a + b*Tan[c + d*x])^(5/2), x, 11, (I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (6*a*b*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(5/2)) + (2*(7*a^2 - 9*b^2)*Sqrt[a + b*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(3/2)) + (2*b*(49*a^2 - 3*b^2)*Sqrt[a + b*Tan[c + d*x]])/(21*a*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-11/2)*(a + b*Tan[c + d*x])^(5/2), x, 12, ((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a + b*Tan[c + d*x]])/(9*d*Tan[c + d*x]^(9/2)) - (38*a*b*Sqrt[a + b*Tan[c + d*x]])/(63*d*Tan[c + d*x]^(7/2)) + (2*(21*a^2 - 25*b^2)*Sqrt[a + b*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(5/2)) + (2*b*(231*a^2 - 5*b^2)*Sqrt[a + b*Tan[c + d*x]])/(315*a*d*Tan[c + d*x]^(3/2)) - (2*(315*a^4 - 483*a^2*b^2 - 10*b^4)*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^(1/2), x, 14, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a - b]*d) + ((3*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*b^(5/2)*d) + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a + b]*d) - (3*a*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*b^2*d) + (Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(2*b*d)} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(1/2), x, 13, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) - (a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(3/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) + (Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(b*d)} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(1/2), x, 12, -(ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a - b]*d)) + (2*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[b]*d) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a + b]*d)} -{Tan[c + d*x]^(1/2)/(a + b*Tan[c + d*x])^(1/2), x, 7, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{Tan[c + d*x]^(-1/2)/(a + b*Tan[c + d*x])^(1/2), x, 7, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a - b]*d) + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a + b]*d)} -{Tan[c + d*x]^(-3/2)/(a + b*Tan[c + d*x])^(1/2), x, 9, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*Sqrt[a + b*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)/(a + b*Tan[c + d*x])^(1/2), x, 10, -(ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a - b]*d)) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/(Sqrt[I*a + b]*d) - (2*Sqrt[a + b*Tan[c + d*x]])/(3*a*d*Tan[c + d*x]^(3/2)) + (4*b*Sqrt[a + b*Tan[c + d*x]])/(3*a^2*d*Sqrt[Tan[c + d*x]])} -{Tan[c + d*x]^(-7/2)/(a + b*Tan[c + d*x])^(1/2), x, 11, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*Sqrt[a + b*Tan[c + d*x]])/(5*a*d*Tan[c + d*x]^(5/2)) + (8*b*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2 - 8*b^2)*Sqrt[a + b*Tan[c + d*x]])/(15*a^3*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^(3/2), x, 14, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) - (3*a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(5/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - (2*a^2*Tan[c + d*x]^(3/2))/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + ((3*a^2 + b^2)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d)} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2), x, 13, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(3/2)*d) + (2*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(3/2)*d) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(3/2)*d) - (2*a^2*Sqrt[Tan[c + d*x]])/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(3/2), x, 8, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d)) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) + (2*a*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(1/2)/(a + b*Tan[c + d*x])^(3/2), x, 8, -(ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(3/2)*d)) + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(3/2)*d) - (2*b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-1/2)/(a + b*Tan[c + d*x])^(3/2), x, 8, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) + (2*b^2*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-3/2)/(a + b*Tan[c + d*x])^(3/2), x, 9, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(3/2)*d) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(3/2)*d) - 2/(a*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - (2*b*(a^2 + 2*b^2)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)/(a + b*Tan[c + d*x])^(3/2), x, 10, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d)) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - 2/(3*a*d*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]) + (8*b)/(3*a^2*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*b^2*(5*a^2 + 8*b^2)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} - - -{Tan[c + d*x]^(9/2)/(a + b*Tan[c + d*x])^(5/2), x, 15, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d)) - (5*a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(7/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^(5/2))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*a^2*(5*a^2 + 11*b^2)*Tan[c + d*x]^(3/2))/(3*b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) + ((5*a^4 + 10*a^2*b^2 + b^4)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d)} -{Tan[c + d*x]^(7/2)/(a + b*Tan[c + d*x])^(5/2), x, 14, -(ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(5/2)*d)) + (2*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(5/2)*d) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(5/2)*d) - (2*a^2*Tan[c + d*x]^(3/2))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*a^2*(a^2 + 3*b^2)*Sqrt[Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(5/2), x, 9, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*a^2*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(a^2 + 7*b^2)*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(5/2), x, 9, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(5/2)*d) + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(5/2)*d) + (2*a*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (4*(a^2 - 2*b^2)*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(1/2)/(a + b*Tan[c + d*x])^(5/2), x, 9, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d)) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*b*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2)*Sqrt[Tan[c + d*x]])/(3*a*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-1/2)/(a + b*Tan[c + d*x])^(5/2), x, 9, -(ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(5/2)*d)) - ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(5/2)*d) + (2*b^2*Sqrt[Tan[c + d*x]])/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^2 + b^2)*Sqrt[Tan[c + d*x]])/(3*a^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-3/2)/(a + b*Tan[c + d*x])^(5/2), x, 10, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - 2/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2 + 4*b^2)*Sqrt[Tan[c + d*x]])/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-5/2)/(a + b*Tan[c + d*x])^(5/2), x, 11, ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a - b)^(5/2)*d) + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/((I*a + b)^(5/2)*d) - 2/(3*a*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) + (4*b)/(a^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*b^2*(7*a^2 + 8*b^2)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^4 + 15*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]])/(3*a^4*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} - - -{1/(Sqrt[Tan[c + d*x]]*Sqrt[2 + 3*Tan[c + d*x]]), x, 7, ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]]/(Sqrt[3 - 2*I]*d) + ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)} -{1/(Sqrt[Tan[c + d*x]]*Sqrt[-2 + 3*Tan[c + d*x]]), x, 7, ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]]/(Sqrt[3 - 2*I]*d) + ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)} - -{1/(Sqrt[Tan[c + d*x]]*Sqrt[2 - 3*Tan[c + d*x]]), x, 7, ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]]/(Sqrt[3 - 2*I]*d) + ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)} -{1/(Sqrt[Tan[c + d*x]]*Sqrt[-2 - 3*Tan[c + d*x]]), x, 7, ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]]/(Sqrt[3 - 2*I]*d) + ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]]/(Sqrt[3 + 2*I]*d)} - -{1/(Sqrt[Tan[c + d*x]]*Sqrt[3 + 2*Tan[c + d*x]]), x, 7, ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)} -{1/(Sqrt[Tan[c + d*x]]*Sqrt[3 - 2*Tan[c + d*x]]), x, 7, ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTan[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)} - -{1/(Sqrt[Tan[c + d*x]]*Sqrt[-3 + 2*Tan[c + d*x]]), x, 7, ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)} -{1/(Sqrt[Tan[c + d*x]]*Sqrt[-3 - 2*Tan[c + d*x]]), x, 7, ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]]/(Sqrt[2 - 3*I]*d) + ArcTan[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]]/(Sqrt[2 + 3*I]*d)} - - -{Sqrt[Tan[c + d*x]]/Sqrt[2 + 3*Tan[c + d*x]], x, 7, (I*ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/(Sqrt[3 - 2*I]*d) - (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)} -{Sqrt[Tan[c + d*x]]/Sqrt[-2 + 3*Tan[c + d*x]], x, 7, -((I*ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]])/(Sqrt[3 - 2*I]*d)) + (I*ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 + 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)} - -{Sqrt[Tan[c + d*x]]/Sqrt[2 - 3*Tan[c + d*x]], x, 7, -((I*ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]])/(Sqrt[3 - 2*I]*d)) + (I*ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)} -{Sqrt[Tan[c + d*x]]/Sqrt[-2 - 3*Tan[c + d*x]], x, 7, (I*ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]])/(Sqrt[3 - 2*I]*d) - (I*ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]])/(Sqrt[3 + 2*I]*d)} - -{Sqrt[Tan[c + d*x]]/Sqrt[3 + 2*Tan[c + d*x]], x, 7, (I*ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]])/(Sqrt[2 - 3*I]*d) - (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 + 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)} -{Sqrt[Tan[c + d*x]]/Sqrt[3 - 2*Tan[c + d*x]], x, 7, -((I*ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2*Tan[c + d*x]]])/(Sqrt[2 - 3*I]*d)) + (I*ArcTan[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)} - -{Sqrt[Tan[c + d*x]]/Sqrt[-3 + 2*Tan[c + d*x]], x, 7, -((I*ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]])/(Sqrt[2 - 3*I]*d)) + (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)} -{Sqrt[Tan[c + d*x]]/Sqrt[-3 - 2*Tan[c + d*x]], x, 7, (I*ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]])/(Sqrt[2 - 3*I]*d) - (I*ArcTan[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 - 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^(n/3)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(5/3)/(a + b*Tan[c + d*x]), x, 32, -((b*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d)) + (b*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) - (Sqrt[3]*a^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x]^(1/3))/(Sqrt[3]*a^(1/3))])/(b^(2/3)*(a^2 + b^2)*d) + (Sqrt[3]*a*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*(a^2 + b^2)*d) + (b*ArcTan[Tan[c + d*x]^(1/3)])/((a^2 + b^2)*d) - (3*a^(5/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]^(1/3)])/(2*b^(2/3)*(a^2 + b^2)*d) - (a*Log[1 + Tan[c + d*x]^(2/3)])/(2*(a^2 + b^2)*d) + (Sqrt[3]*b*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) - (Sqrt[3]*b*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (a^(5/3)*Log[a + b*Tan[c + d*x]])/(2*b^(2/3)*(a^2 + b^2)*d) + (a*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(4*(a^2 + b^2)*d)} -{Tan[c + d*x]^(1/3)/(a + b*Tan[c + d*x]), x, 30, -((b*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d)) + (b*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) + (Sqrt[3]*a^(1/3)*b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x]^(1/3))/(Sqrt[3]*a^(1/3))])/((a^2 + b^2)*d) - (Sqrt[3]*a*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*(a^2 + b^2)*d) + (b*ArcTan[Tan[c + d*x]^(1/3)])/((a^2 + b^2)*d) - (3*a^(1/3)*b^(2/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) - (a*Log[1 + Tan[c + d*x]^(2/3)])/(2*(a^2 + b^2)*d) - (Sqrt[3]*b*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (Sqrt[3]*b*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (a^(1/3)*b^(2/3)*Log[a + b*Tan[c + d*x]])/(2*(a^2 + b^2)*d) + (a*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(4*(a^2 + b^2)*d)} -{Tan[c + d*x]^(-1/3)/(a + b*Tan[c + d*x]), x, 28, (b*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) - (b*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) - (Sqrt[3]*b^(4/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x]^(1/3))/(Sqrt[3]*a^(1/3))])/(a^(1/3)*(a^2 + b^2)*d) - (Sqrt[3]*a*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*(a^2 + b^2)*d) - (b*ArcTan[Tan[c + d*x]^(1/3)])/((a^2 + b^2)*d) - (3*b^(4/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]^(1/3)])/(2*a^(1/3)*(a^2 + b^2)*d) + (a*Log[1 + Tan[c + d*x]^(2/3)])/(2*(a^2 + b^2)*d) - (Sqrt[3]*b*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (Sqrt[3]*b*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (b^(4/3)*Log[a + b*Tan[c + d*x]])/(2*a^(1/3)*(a^2 + b^2)*d) - (a*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(4*(a^2 + b^2)*d)} -{Tan[c + d*x]^(-5/3)/(a + b*Tan[c + d*x]), x, 30, (b*ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) - (b*ArcTan[Sqrt[3] + 2*Tan[c + d*x]^(1/3)])/(2*(a^2 + b^2)*d) + (Sqrt[3]*b^(8/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x]^(1/3))/(Sqrt[3]*a^(1/3))])/(a^(5/3)*(a^2 + b^2)*d) + (Sqrt[3]*a*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]])/(2*(a^2 + b^2)*d) - (b*ArcTan[Tan[c + d*x]^(1/3)])/((a^2 + b^2)*d) - (3*b^(8/3)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]^(1/3)])/(2*a^(5/3)*(a^2 + b^2)*d) + (a*Log[1 + Tan[c + d*x]^(2/3)])/(2*(a^2 + b^2)*d) + (Sqrt[3]*b*Log[1 - Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) - (Sqrt[3]*b*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/(4*(a^2 + b^2)*d) + (b^(8/3)*Log[a + b*Tan[c + d*x]])/(2*a^(5/3)*(a^2 + b^2)*d) - (a*Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)])/(4*(a^2 + b^2)*d) - (3*a)/(2*(a^2 + b^2)*d*Tan[c + d*x]^(2/3)) - (3*b^2)/(2*a*(a^2 + b^2)*d*Tan[c + d*x]^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (d Tan[e+f x])^(n/3)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^(4/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, (3*AppellF1[7/3, 1, 1/2, 10/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(7/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(14*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[7/3, 1, 1/2, 10/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(7/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(14*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(2/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, (3*AppellF1[5/3, 1, 1/2, 8/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(10*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[5/3, 1, 1/2, 8/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(10*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(1/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, (3*AppellF1[4/3, 1, 1/2, 7/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(4/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(8*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[4/3, 1, 1/2, 7/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(4/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(8*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-1/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, (3*AppellF1[2/3, 1, 1/2, 5/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(2/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(4*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[2/3, 1, 1/2, 5/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(2/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(4*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-2/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, (3*AppellF1[1/3, 1, 1/2, 4/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*Sqrt[a + b*Tan[c + d*x]]) + (3*AppellF1[1/3, 1, 1/2, 4/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1/3)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^(-4/3)/Sqrt[a + b*Tan[c + d*x]], x, 9, -((3*AppellF1[-(1/3), 1, 1/2, 2/3, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*Tan[c + d*x]^(1/3)*Sqrt[a + b*Tan[c + d*x]])) - (3*AppellF1[-(1/3), 1, 1/2, 2/3, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*Tan[c + d*x]^(1/3)*Sqrt[a + b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/3) (d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[e + f*x]^4*(c + d*Tan[e + f*x])^(1/3), x, 16, (-(1/4))*(c - Sqrt[-d^2])^(1/3)*x - (1/4)*(c + Sqrt[-d^2])^(1/3)*x - (Sqrt[3]*Sqrt[-d^2]*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*d*f) + (Sqrt[3]*Sqrt[-d^2]*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*d*f) + (Sqrt[-d^2]*(c - Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*d*f) - (Sqrt[-d^2]*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*d*f) + (3*Sqrt[-d^2]*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*d*f) - (3*Sqrt[-d^2]*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*d*f) + (3*(9*c^2 - 35*d^2)*(c + d*Tan[e + f*x])^(4/3))/(140*d^3*f) - (9*c*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(35*d^2*f) + (3*Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(4/3))/(10*d*f)} -{Tan[e + f*x]^3*(c + d*Tan[e + f*x])^(1/3), x, 15, (-(1/4))*I*(c - I*d)^(1/3)*x + (1/4)*I*(c + I*d)^(1/3)*x + (Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/(2*f) + (Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(2*f) - ((c - I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - ((c + I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - (3*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) - (3*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) - (3*(c + d*Tan[e + f*x])^(1/3))/f - (9*c*(c + d*Tan[e + f*x])^(4/3))/(28*d^2*f) + (3*Tan[e + f*x]*(c + d*Tan[e + f*x])^(4/3))/(7*d*f)} -{Tan[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3), x, 14, (1/4)*(c - Sqrt[-d^2])^(1/3)*x + (1/4)*(c + Sqrt[-d^2])^(1/3)*x - (Sqrt[3]*d*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (Sqrt[3]*d*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (d*(c - Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) + (3*d*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (3*d*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) + (3*(c + d*Tan[e + f*x])^(4/3))/(4*d*f)} -{Tan[e + f*x]^1*(c + d*Tan[e + f*x])^(1/3), x, 12, (1/4)*I*(c - I*d)^(1/3)*x - (1/4)*I*(c + I*d)^(1/3)*x - (Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/(2*f) - (Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(2*f) + ((c - I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) + ((c + I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) + (3*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) + (3*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) + (3*(c + d*Tan[e + f*x])^(1/3))/f} -{Tan[e + f*x]^0*(c + d*Tan[e + f*x])^(1/3), x, 13, (-(1/4))*(c - Sqrt[-d^2])^(1/3)*x - (1/4)*(c + Sqrt[-d^2])^(1/3)*x + (Sqrt[3]*d*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) - (Sqrt[3]*d*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) - (d*(c - Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) + (d*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (3*d*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) + (3*d*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f)} -{Cot[e + f*x]^1*(c + d*Tan[e + f*x])^(1/3), x, 19, (-(1/4))*I*(c - I*d)^(1/3)*x + (1/4)*I*(c + I*d)^(1/3)*x - (Sqrt[3]*c^(1/3)*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3))/(Sqrt[3]*c^(1/3))])/f + (Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/(2*f) + (Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(2*f) - ((c - I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - ((c + I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - (c^(1/3)*Log[Tan[e + f*x]])/(2*f) + (3*c^(1/3)*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*f) - (3*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) - (3*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f)} -{Cot[e + f*x]^2*(c + d*Tan[e + f*x])^(1/3), x, 20, (1/4)*(c - Sqrt[-d^2])^(1/3)*x + (1/4)*(c + Sqrt[-d^2])^(1/3)*x - (d*ArcTan[(c^(1/3) + 2*(c + d*Tan[e + f*x])^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*f) - (Sqrt[3]*d*(c - Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (Sqrt[3]*d*(c + Sqrt[-d^2])^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + Sqrt[-d^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-d^2]*f) + (d*(c - Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*(c + Sqrt[-d^2])^(1/3)*Log[Cos[e + f*x]])/(4*Sqrt[-d^2]*f) - (d*Log[Tan[e + f*x]])/(6*c^(2/3)*f) + (d*Log[c^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*c^(2/3)*f) + (3*d*(c - Sqrt[-d^2])^(1/3)*Log[(c - Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (3*d*(c + Sqrt[-d^2])^(1/3)*Log[(c + Sqrt[-d^2])^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*Sqrt[-d^2]*f) - (Cot[e + f*x]*(c + d*Tan[e + f*x])^(1/3))/f} - - -{(a + b*Tan[c + d*x])^(5/3), x, 12, (-(1/4))*(a - I*b)^(5/3)*x - (1/4)*(a + I*b)^(5/3)*x + (I*Sqrt[3]*(a - I*b)^(5/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*d) - (I*Sqrt[3]*(a + I*b)^(5/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*d) + (I*(a - I*b)^(5/3)*Log[Cos[c + d*x]])/(4*d) - (I*(a + I*b)^(5/3)*Log[Cos[c + d*x]])/(4*d) + (3*I*(a - I*b)^(5/3)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) - (3*I*(a + I*b)^(5/3)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) + (3*b*(a + b*Tan[c + d*x])^(2/3))/(2*d)} -{(a + b*Tan[c + d*x])^(4/3), x, 12, (-(1/4))*(a - I*b)^(4/3)*x - (1/4)*(a + I*b)^(4/3)*x - (I*Sqrt[3]*(a - I*b)^(4/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*d) + (I*Sqrt[3]*(a + I*b)^(4/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*d) + (I*(a - I*b)^(4/3)*Log[Cos[c + d*x]])/(4*d) - (I*(a + I*b)^(4/3)*Log[Cos[c + d*x]])/(4*d) + (3*I*(a - I*b)^(4/3)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) - (3*I*(a + I*b)^(4/3)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) + (3*b*(a + b*Tan[c + d*x])^(1/3))/d} -{(a + b*Tan[c + d*x])^(2/3), x, 13, (-(1/4))*(a - Sqrt[-b^2])^(2/3)*x - (1/4)*(a + Sqrt[-b^2])^(2/3)*x - (Sqrt[3]*b*(a - Sqrt[-b^2])^(2/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*d) + (Sqrt[3]*b*(a + Sqrt[-b^2])^(2/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*d) - (b*(a - Sqrt[-b^2])^(2/3)*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*d) + (b*(a + Sqrt[-b^2])^(2/3)*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*d) - (3*b*(a - Sqrt[-b^2])^(2/3)*Log[(a - Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*d) + (3*b*(a + Sqrt[-b^2])^(2/3)*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*d)} -{(a + b*Tan[c + d*x])^(1/3), x, 13, (-(1/4))*(a - Sqrt[-b^2])^(1/3)*x - (1/4)*(a + Sqrt[-b^2])^(1/3)*x + (Sqrt[3]*b*(a - Sqrt[-b^2])^(1/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*d) - (Sqrt[3]*b*(a + Sqrt[-b^2])^(1/3)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*d) - (b*(a - Sqrt[-b^2])^(1/3)*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*d) + (b*(a + Sqrt[-b^2])^(1/3)*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*d) - (3*b*(a - Sqrt[-b^2])^(1/3)*Log[(a - Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*d) + (3*b*(a + Sqrt[-b^2])^(1/3)*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + b*Tan[c + d*x])^(1/3), x, 11, -(x/(4*(a - Sqrt[-b^2])^(1/3))) - x/(4*(a + Sqrt[-b^2])^(1/3)) - (Sqrt[3]*b*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])^(1/3)*d) + (Sqrt[3]*b*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])^(1/3)*d) - (b*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*(a - Sqrt[-b^2])^(1/3)*d) + (b*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*(a + Sqrt[-b^2])^(1/3)*d) - (3*b*Log[(a - Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*(a - Sqrt[-b^2])^(1/3)*d) + (3*b*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*(a + Sqrt[-b^2])^(1/3)*d)} -{1/(a + b*Tan[c + d*x])^(2/3), x, 11, -(x/(4*(a - Sqrt[-b^2])^(2/3))) - x/(4*(a + Sqrt[-b^2])^(2/3)) + (Sqrt[3]*b*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])^(2/3)*d) - (Sqrt[3]*b*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + Sqrt[-b^2])^(1/3))/Sqrt[3]])/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])^(2/3)*d) - (b*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*(a - Sqrt[-b^2])^(2/3)*d) + (b*Log[Cos[c + d*x]])/(4*Sqrt[-b^2]*(a + Sqrt[-b^2])^(2/3)*d) - (3*b*Log[(a - Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*(a - Sqrt[-b^2])^(2/3)*d) + (3*b*Log[(a + Sqrt[-b^2])^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*Sqrt[-b^2]*(a + Sqrt[-b^2])^(2/3)*d)} -{1/(a + b*Tan[c + d*x])^(4/3), x, 12, -(x/(4*(a - I*b)^(4/3))) - x/(4*(a + I*b)^(4/3)) + (I*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*(a - I*b)^(4/3)*d) - (I*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*(a + I*b)^(4/3)*d) + (I*Log[Cos[c + d*x]])/(4*(a - I*b)^(4/3)*d) - (I*Log[Cos[c + d*x]])/(4*(a + I*b)^(4/3)*d) + (3*I*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a - I*b)^(4/3)*d) - (3*I*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a + I*b)^(4/3)*d) - (3*b)/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^(1/3))} -{1/(a + b*Tan[c + d*x])^(5/3), x, 12, -(x/(4*(a - I*b)^(5/3))) - x/(4*(a + I*b)^(5/3)) - (I*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*(a - I*b)^(5/3)*d) + (I*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*(a + I*b)^(5/3)*d) + (I*Log[Cos[c + d*x]])/(4*(a - I*b)^(5/3)*d) - (I*Log[Cos[c + d*x]])/(4*(a + I*b)^(5/3)*d) + (3*I*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a - I*b)^(5/3)*d) - (3*I*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a + I*b)^(5/3)*d) - (3*b)/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n with n symbolic*) - - -{(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^4, x, 8, -((b^2*(b^2*(3 + n) - a^2*(17 + 5*n))*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*(3 + n))) + ((a^4 - 6*a^2*b^2 + b^4)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (2*a*b^3*(4 + n)*Tan[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(2 + n)*(3 + n)) + (4*a*b*(a^2 - b^2)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(d^2*f*(2 + n)) + (b^2*(d*Tan[e + f*x])^(1 + n)*(a + b*Tan[e + f*x])^2)/(d*f*(3 + n))} -{(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^3, x, 7, (a*b^2*(5 + 2*n)*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)*(2 + n)) + (a*(a^2 - 3*b^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b*(3*a^2 - b^2)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(d^2*f*(2 + n)) + (b^2*(d*Tan[e + f*x])^(1 + n)*(a + b*Tan[e + f*x]))/(d*f*(2 + n))} -{(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^2, x, 6, (b^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + ((a^2 - b^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (2*a*b*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(d^2*f*(2 + n))} -{(d*Tan[e + f*x])^n*(a + b*Tan[e + f*x])^1, x, 5, (a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(d^2*f*(2 + n))} -{(d*Tan[e + f*x])^n/(a + b*Tan[e + f*x])^1, x, 8, (a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/((a^2 + b^2)*d*f*(1 + n)) + (b^2*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Tan[e + f*x])/a)]*(d*Tan[e + f*x])^(1 + n))/(a*(a^2 + b^2)*d*f*(1 + n)) - (b*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/((a^2 + b^2)*d^2*f*(2 + n))} -{(d*Tan[e + f*x])^n/(a + b*Tan[e + f*x])^2, x, 9, ((a^2 - b^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/((a^2 + b^2)^2*d*f*(1 + n)) + (b^2*(a^2*(2 - n) - b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -((b*Tan[e + f*x])/a)]*(d*Tan[e + f*x])^(1 + n))/(a^2*(a^2 + b^2)^2*d*f*(1 + n)) - (2*a*b*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/((a^2 + b^2)^2*d^2*f*(2 + n)) + (b^2*(d*Tan[e + f*x])^(1 + n))/(a*(a^2 + b^2)*d*f*(a + b*Tan[e + f*x]))} - - -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2), x, 7, (a*AppellF1[1 + m, -(3/2), 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a]) + (a*AppellF1[1 + m, -(3/2), 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(1/2), x, 7, (AppellF1[1 + m, -(1/2), 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a]) + (AppellF1[1 + m, -(1/2), 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])} -{Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(1/2), x, 7, (AppellF1[1 + m, 1/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + (AppellF1[1 + m, 1/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]])} -{Tan[c + d*x]^m/(a + b*Tan[c + d*x])^(3/2), x, 7, (AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + (AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Tan[e+f x])^n with m symbolic*) - - -{(a + b*Tan[e + f*x])^m*(d*Tan[e + f*x])^n, x, 7, (AppellF1[1 + n, -m, 1, 2 + n, -((b*Tan[e + f*x])/a), (-I)*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*d*f*(1 + n))) + (AppellF1[1 + n, -m, 1, 2 + n, -((b*Tan[e + f*x])/a), I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n)*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*d*f*(1 + n)))} - - -{Tan[c + d*x]^4*(a + b*Tan[c + d*x])^n, x, 8, ((2*a^2 - b^2*(2 + n)*(3 + n))*(a + b*Tan[c + d*x])^(1 + n))/(b^3*d*(1 + n)*(2 + n)*(3 + n)) - (Sqrt[-b^2]*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*b*(a - Sqrt[-b^2])*d*(1 + n)) + (Sqrt[-b^2]*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*b*(a + Sqrt[-b^2])*d*(1 + n)) - (2*a*Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b^2*d*(2 + n)*(3 + n)) + (Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(3 + n))} -{Tan[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 8, -((a*(a + b*Tan[c + d*x])^(1 + n))/(b^2*d*(1 + n)*(2 + n))) + (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) + (Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(2 + n))} -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 6, (a + b*Tan[c + d*x])^(1 + n)/(b*d*(1 + n)) - (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*d*(1 + n)) + (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*d*(1 + n))} -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 5, -((Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n))) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n))} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^n, x, 5, (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*d*(1 + n)) - (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*d*(1 + n))} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^n, x, 8, (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^n, x, 10, -((Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(a*d)) - (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*d*(1 + n)) + (b*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*d*(1 + n)) - (b*n*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^n, x, 11, (b*(1 - n)*Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^2*d) - (Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(2*a*d) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) + ((2*a^2 + b^2*(1 - n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^3*d*(1 + n))} - - -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n, x, 9, (AppellF1[5/2, 1, -n, 7/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d)) + (AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d))} -{Tan[c + d*x]^(1/2)*(a + b*Tan[c + d*x])^n, x, 9, (AppellF1[3/2, 1, -n, 5/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d)) + (AppellF1[3/2, 1, -n, 5/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d))} -{Tan[c + d*x]^(-1/2)*(a + b*Tan[c + d*x])^n, x, 9, (AppellF1[1/2, 1, -n, 3/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d) + (AppellF1[1/2, 1, -n, 3/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)} -{Tan[c + d*x]^(-3/2)*(a + b*Tan[c + d*x])^n, x, 9, -((AppellF1[-(1/2), 1, -n, 1/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Tan[c + d*x]]))) - (AppellF1[-(1/2), 1, -n, 1/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Tan[c + d*x]]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x]), x, 5, (-2*(-1)^(3/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - ((2*I)*a*Sqrt[Cot[c + d*x]])/d - (2*a*Cot[c + d*x]^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x]), x, 4, (-2*(-1)^(1/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a*Sqrt[Cot[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x]), x, 3, (2*(-1)^(3/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d} -{(a + I*a*Tan[c + d*x])/Sqrt[Cot[c + d*x]], x, 4, (2*(-1)^(1/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + ((2*I)*a)/(d*Sqrt[Cot[c + d*x]])} -{(a + I*a*Tan[c + d*x])/Cot[c + d*x]^(3/2), x, 5, (-2*(-1)^(3/4)*a*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (((2*I)/3)*a)/(d*Cot[c + d*x]^(3/2)) + (2*a)/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2, x, 6, (4*(-1)^(1/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (4*a^2*Sqrt[Cot[c + d*x]])/d - (4*I*a^2*Cot[c + d*x]^(3/2))/(3*d) - (2*a^2*Cot[c + d*x]^(5/2))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2, x, 5, -((4*(-1)^(3/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (4*I*a^2*Sqrt[Cot[c + d*x]])/d - (2*a^2*Cot[c + d*x]^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2, x, 4, -((4*(-1)^(1/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a^2*Sqrt[Cot[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2, x, 4, (4*(-1)^(3/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a^2)/(d*Sqrt[Cot[c + d*x]])} -{(a + I*a*Tan[c + d*x])^2/Sqrt[Cot[c + d*x]], x, 5, (4*(-1)^(1/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a^2)/(3*d*Cot[c + d*x]^(3/2)) + (4*I*a^2)/(d*Sqrt[Cot[c + d*x]])} -{(a + I*a*Tan[c + d*x])^2/Cot[c + d*x]^(3/2), x, 6, -((4*(-1)^(3/4)*a^2*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a^2)/(5*d*Cot[c + d*x]^(5/2)) + (4*I*a^2)/(3*d*Cot[c + d*x]^(3/2)) + (4*a^2)/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3, x, 6, (8*(-1)^(1/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (8*a^3*Sqrt[Cot[c + d*x]])/d - (8*I*a^3*Cot[c + d*x]^(3/2))/(5*d) - (2*Cot[c + d*x]^(3/2)*(I*a^3 + a^3*Cot[c + d*x]))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3, x, 5, -((8*(-1)^(3/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (16*I*a^3*Sqrt[Cot[c + d*x]])/(3*d) - (2*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3, x, 5, (-8*(-1)^(1/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*(I*a^3 + a^3*Cot[c + d*x]))/(d*Sqrt[Cot[c + d*x]])} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3, x, 5, (8*(-1)^(3/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (16*a^3)/(3*d*Sqrt[Cot[c + d*x]]) - (2*(I*a^3 + a^3*Cot[c + d*x]))/(3*d*Cot[c + d*x]^(3/2))} -{(a + I*a*Tan[c + d*x])^3/Sqrt[Cot[c + d*x]], x, 6, (8*(-1)^(1/4)*a^3*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (8*a^3)/(5*d*Cot[c + d*x]^(3/2)) + (8*I*a^3)/(d*Sqrt[Cot[c + d*x]]) - (2*(I*a^3 + a^3*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x]), x, 13, -(((5/4 + (3*I)/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d)) + ((5/4 + (3*I)/4)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - (5*Sqrt[Cot[c + d*x]])/(2*a*d) + Cot[c + d*x]^(3/2)/(2*d*(I*a + a*Cot[c + d*x])) - ((5/8 - (3*I)/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) + ((5/8 - (3*I)/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} -{Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x]), x, 12, ((3/4 - I/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((3/4 - I/4)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) + Sqrt[Cot[c + d*x]]/(2*d*(I*a + a*Cot[c + d*x])) - ((3/8 + I/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) + ((3/8 + I/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])), x, 4, ((-1)^(1/4)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/(2*a*d) + (I*Sqrt[Cot[c + d*x]])/(2*d*(I*a + a*Cot[c + d*x]))} -{1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])), x, 12, ((1/4 - (3*I)/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((1/4 - (3*I)/4)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - Sqrt[Cot[c + d*x]]/(2*d*(I*a + a*Cot[c + d*x])) - ((1/8 + (3*I)/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) + ((1/8 + (3*I)/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} -{1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])), x, 13, ((3/4 + (5*I)/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((3/4 + (5*I)/4)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - (5*I)/(2*a*d*Sqrt[Cot[c + d*x]]) - 1/(2*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])) + ((3/8 - (5*I)/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) - ((3/8 - (5*I)/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} - - -{Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^2, x, 14, -(((25/16 + (21*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + ((25/16 + (21*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) - (25*Sqrt[Cot[c + d*x]])/(8*a^2*d) + (7*Cot[c + d*x]^(3/2))/(8*a^2*d*(I + Cot[c + d*x])) + Cot[c + d*x]^(5/2)/(4*d*(I*a + a*Cot[c + d*x])^2) - ((25/32 - (21*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((25/32 - (21*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^2, x, 13, ((9/16 - (5*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) - ((9/16 - (5*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + (5*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) + Cot[c + d*x]^(3/2)/(4*d*(I*a + a*Cot[c + d*x])^2) - ((9/32 + (5*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((9/32 + (5*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2), x, 13, ((1/16 - (3*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) - ((1/16 - (3*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + (3*I*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) + Sqrt[Cot[c + d*x]]/(4*d*(I*a + a*Cot[c + d*x])^2) + ((1/32 + (3*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) - ((1/32 + (3*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2), x, 13, -(((1/16 + (3*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + ((1/16 + (3*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + Sqrt[Cot[c + d*x]]/(8*a^2*d*(I + Cot[c + d*x])) + (I*Sqrt[Cot[c + d*x]])/(4*d*(I*a + a*Cot[c + d*x])^2) + ((1/32 - (3*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) - ((1/32 - (3*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2), x, 13, -(((9/16 + (5*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + ((9/16 + (5*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + (5*I*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) - Sqrt[Cot[c + d*x]]/(4*d*(I*a + a*Cot[c + d*x])^2) - ((9/32 - (5*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((9/32 - (5*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2), x, 14, ((25/16 - (21*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) - ((25/16 - (21*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) - 25/(8*a^2*d*Sqrt[Cot[c + d*x]]) + (7*I)/(8*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])) - 1/(4*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - ((25/32 + (21*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((25/32 + (21*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} - - -{Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3, x, 14, ((7/16 - (5*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - ((7/16 - (5*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + Cot[c + d*x]^(5/2)/(6*d*(I*a + a*Cot[c + d*x])^3) + Cot[c + d*x]^(3/2)/(3*a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - ((7/32 + (5*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) + ((7/32 + (5*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3), x, 16, -((I*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(8*Sqrt[2]*a^3*d)) + (I*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(8*Sqrt[2]*a^3*d) + Cot[c + d*x]^(3/2)/(6*d*(I*a + a*Cot[c + d*x])^3) + Sqrt[Cot[c + d*x]]/(4*a*d*(I*a + a*Cot[c + d*x])^2) + (I*Sqrt[Cot[c + d*x]])/(4*d*(I*a^3 + a^3*Cot[c + d*x])) + (I*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(16*Sqrt[2]*a^3*d) - (I*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(16*Sqrt[2]*a^3*d)} -{1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3), x, 7, -(((-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/(8*a^3*d)) + Sqrt[Cot[c + d*x]]/(6*d*(I*a + a*Cot[c + d*x])^3) + (I*Sqrt[Cot[c + d*x]])/(6*a*d*(I*a + a*Cot[c + d*x])^2) + Sqrt[Cot[c + d*x]]/(8*d*(I*a^3 + a^3*Cot[c + d*x]))} -{1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3), x, 15, -(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*a^3*d)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*a^3*d) + (I*Sqrt[Cot[c + d*x]])/(6*d*(I*a + a*Cot[c + d*x])^3) + Sqrt[Cot[c + d*x]]/(12*a*d*(I*a + a*Cot[c + d*x])^2) - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(16*Sqrt[2]*a^3*d) + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(16*Sqrt[2]*a^3*d)} -{1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3), x, 14, -(((5/16 - (7*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d)) + ((5/16 - (7*I)/16)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - Sqrt[Cot[c + d*x]]/(6*d*(I*a + a*Cot[c + d*x])^3) + (I*Sqrt[Cot[c + d*x]])/(3*a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) + ((5/32 + (7*I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) - ((5/32 + (7*I)/32)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (d Cot[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 7, ((-1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (26*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (((2*I)/15)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 6, ((-1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (((2*I)/3)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]], x, 4, ((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]], x, 3, ((1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[Cot[c + d*x]], x, 8, -((2*(-1)^(3/4)*Sqrt[a]*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{Sqrt[a + I*a*Tan[c + d*x]]/Cot[c + d*x]^(3/2), x, 10, -(((-1)^(1/4)*Sqrt[a]*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((1 - I)*Sqrt[a]*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + Sqrt[a + I*a*Tan[c + d*x]]/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 8, ((-2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (((2*I)/5)*a^2*Cot[c + d*x]^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*a^2*Cot[c + d*x]^(5/2))/(5*d*Sqrt[a + I*a*Tan[c + d*x]]) + (12*a*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) - (((4*I)/5)*a*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 5, ((-2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((2*I)*a*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2), x, 4, ((2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2), x, 8, (2*(-1)^(1/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((2 - 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{(a + I*a*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]], x, 10, -((3*(-1)^(3/4)*a^(3/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((2 + 2*I)*a^(3/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - a^2/(d*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (I*a^2)/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} - - -{Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 8, ((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (((104*I)/21)*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d + (32*a^2*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (((6*I)/7)*a^2*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a^2*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 6, ((-4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (((2*I)/3)*a*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/d - (2*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((-4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((4*I)*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2), x, 9, (2*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2), x, 9, (5*(-1)^(1/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{(a + I*a*Tan[c + d*x])^(5/2)/Sqrt[Cot[c + d*x]], x, 10, -((23*(-1)^(3/4)*a^(5/2)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d)) - ((4 + 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (a^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*Cot[c + d*x]^(3/2)) + (9*I*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cot[c + d*x]^(5/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 7, -(((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d)) + Cot[c + d*x]^(3/2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*I*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d) - (5*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d)} -{Cot[c + d*x]^(3/2)/Sqrt[a + I*a*Tan[c + d*x]], x, 6, ((1/2 + I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + Sqrt[Cot[c + d*x]]/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (3*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{Sqrt[Cot[c + d*x]]/Sqrt[a + I*a*Tan[c + d*x]], x, 5, ((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + 1/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, ((-1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + I/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 9, -((2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d)) - ((1/2 - I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) - 1/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 10, -(((-1)^(3/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d)) + ((1/2 + I/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) - 1/(d*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (2*I*Sqrt[a + I*a*Tan[c + d*x]])/(a*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 8, ((-1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + Cot[c + d*x]^(3/2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*Cot[c + d*x]^(3/2))/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((13*I)/2)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) - (7*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d)} -{Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(3/2), x, 7, ((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + Sqrt[Cot[c + d*x]]/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (11*Sqrt[Cot[c + d*x]])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (25*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d)} -{Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^(3/2), x, 6, ((1/4 - I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + 1/(3*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + 7/(6*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)), x, 5, -(((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d)) + 1/(3*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + I/(2*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 5, ((-1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + (I/3)/(d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + 1/(2*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 10, (2*(-1)^(3/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + ((1/4 + I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) - 1/(3*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (3*I)/(2*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 11, -((3*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d)) + ((1/4 - I/4)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) - 1/(3*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (13*I)/(6*a*d*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (7*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(5/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 9, ((-1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + Cot[c + d*x]^(3/2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (7*Cot[c + d*x]^(3/2))/(10*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (89*Cot[c + d*x]^(3/2))/(20*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (((707*I)/60)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a^3*d) - (361*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d)} -{Cot[c + d*x]^(3/2)/(a + I*a*Tan[c + d*x])^(5/2), x, 8, ((1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + Sqrt[Cot[c + d*x]]/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (17*Sqrt[Cot[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (151*Sqrt[Cot[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (317*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d)} -{Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^(5/2), x, 7, ((1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + 1/(5*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + 13/(30*a*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + 67/(60*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)), x, 7, ((-1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + (I/5)/(d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + (I/10)/(a*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) - (I/20)/(a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 6, ((-1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + 1/(5*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)) + (I/6)/(a*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + 1/(4*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 6, ((1/8 + I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + (I/5)/(d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)) + 1/(6*a*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) - (I/4)/(a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 11, (2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + ((1/8 - I/8)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) - 1/(5*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)) + I/(2*a*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + 7/(4*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^n with n symbolic*) - - -{(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^3, x, 6, If[$VersionNumber>=8, (I*a^3*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(1 - n)*(2 - n)) + (d^2*(d*Cot[e + f*x])^(-2 + n)*(I*a^3 + a^3*Cot[e + f*x]))/(f*(1 - n)) - (4*I*a^3*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, -2 + n, -1 + n, (-I)*Cot[e + f*x]])/(f*(2 - n)), (I*a^3*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(2 - 3*n + n^2)) + (d^2*(d*Cot[e + f*x])^(-2 + n)*(I*a^3 + a^3*Cot[e + f*x]))/(f*(1 - n)) - (4*I*a^3*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, -2 + n, -1 + n, (-I)*Cot[e + f*x]])/(f*(2 - n))]} -{(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^2, x, 5, (a^2*d*(d*Cot[e + f*x])^(-1 + n))/(f*(1 - n)) - (2*a^2*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1, -1 + n, n, (-I)*Cot[e + f*x]])/(f*(1 - n))} -{(d*Cot[e + f*x])^n*(a + I*a*Tan[e + f*x])^1, x, 3, -((I*a*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n, 1 + n, (-I)*Cot[e + f*x]])/(f*n))} -{(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x])^1, x, 7, -((d*Cot[e + f*x])^(2 + n)/(2*d^2*f*(I*a + a*Cot[e + f*x]))) - (I*n*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2])/(2*a*d^2*f*(2 + n)) + ((1 + n)*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/(2*a*d^3*f*(3 + n))} -{(d*Cot[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x, 8, -((I*n*(d*Cot[e + f*x])^(3 + n))/(4*a^2*d^3*f*(I + Cot[e + f*x]))) - (d*Cot[e + f*x])^(3 + n)/(4*d^3*f*(I*a + a*Cot[e + f*x])^2) + ((1 + n)^2*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/(4*a^2*d^3*f*(3 + n)) + (I*n*(2 + n)*(d*Cot[e + f*x])^(4 + n)*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Cot[e + f*x]^2])/(4*a^2*d^4*f*(4 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (d Cot[e+f x])^n with m symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(d*Cot[e + f*x])^n, x, 4, (AppellF1[1 - n, 1 - m, 1, 2 - n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*(d*Cot[e + f*x])^n*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^m)/((1 + I*Tan[e + f*x])^m*(f*(1 - n)))} - - -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n, x, 5, -((2*AppellF1[-(1/2), 1 - n, 1, 1/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d))} -{Cot[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^n, x, 5, (2*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]]))} -{(a + I*a*Tan[c + d*x])^n/Cot[c + d*x]^(1/2), x, 5, (2*AppellF1[3/2, 1 - n, 1, 5/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(3*d*Cot[c + d*x]^(3/2)))} -{(a + I*a*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x, 5, (2*AppellF1[5/2, 1 - n, 1, 7/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(5*d*Cot[c + d*x]^(5/2)))} - - -(* ::Section:: *) -(*Integrands of the form (a+a Tan[e+f x])^m (d Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x]), x, 14, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*a*Sqrt[Cot[c + d*x]])/d - (2*b*Cot[c + d*x]^(3/2))/(3*d) - (2*a*Cot[c + d*x]^(5/2))/(5*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x]), x, 13, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*b*Sqrt[Cot[c + d*x]])/d - (2*a*Cot[c + d*x]^(3/2))/(3*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x]), x, 12, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a*Sqrt[Cot[c + d*x]])/d - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x]), x, 11, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Sqrt[Cot[c + d*x]], x, 12, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b)/(d*Sqrt[Cot[c + d*x]]) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Cot[c + d*x]^(3/2), x, 13, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b)/(3*d*Cot[c + d*x]^(3/2)) + (2*a)/(d*Sqrt[Cot[c + d*x]]) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])/Cot[c + d*x]^(5/2), x, 14, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b)/(5*d*Cot[c + d*x]^(5/2)) + (2*a)/(3*d*Cot[c + d*x]^(3/2)) - (2*b)/(d*Sqrt[Cot[c + d*x]]) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^2, x, 15, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (4*a*b*Sqrt[Cot[c + d*x]])/d + (2*(a^2 - b^2)*Cot[c + d*x]^(3/2))/(3*d) - (4*a*b*Cot[c + d*x]^(5/2))/(5*d) - (2*a^2*Cot[c + d*x]^(7/2))/(7*d) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2, x, 14, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*(a^2 - b^2)*Sqrt[Cot[c + d*x]])/d - (4*a*b*Cot[c + d*x]^(3/2))/(3*d) - (2*a^2*Cot[c + d*x]^(5/2))/(5*d) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2, x, 13, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (4*a*b*Sqrt[Cot[c + d*x]])/d - (2*a^2*Cot[c + d*x]^(3/2))/(3*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2, x, 12, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a^2*Sqrt[Cot[c + d*x]])/d - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2, x, 12, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2)/(d*Sqrt[Cot[c + d*x]]) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])^2/Sqrt[Cot[c + d*x]], x, 13, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2)/(3*d*Cot[c + d*x]^(3/2)) + (4*a*b)/(d*Sqrt[Cot[c + d*x]]) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(3/2), x, 14, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2)/(5*d*Cot[c + d*x]^(5/2)) + (4*a*b)/(3*d*Cot[c + d*x]^(3/2)) + (2*(a^2 - b^2))/(d*Sqrt[Cot[c + d*x]]) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])^2/Cot[c + d*x]^(5/2), x, 15, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2)/(7*d*Cot[c + d*x]^(7/2)) + (4*a*b)/(5*d*Cot[c + d*x]^(5/2)) + (2*(a^2 - b^2))/(3*d*Cot[c + d*x]^(3/2)) - (4*a*b)/(d*Sqrt[Cot[c + d*x]]) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3, x, 15, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]])/d + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*d) - (32*a^2*b*Cot[c + d*x]^(5/2))/(35*d) - (2*a^2*Cot[c + d*x]^(5/2)*(b + a*Cot[c + d*x]))/(7*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3, x, 14, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*a*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/d - (8*a^2*b*Cot[c + d*x]^(3/2))/(5*d) - (2*a^2*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x]))/(5*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3, x, 13, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (16*a^2*b*Sqrt[Cot[c + d*x]])/(3*d) - (2*a^2*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x]))/(3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3, x, 13, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a*(a^2 + b^2)*Sqrt[Cot[c + d*x]])/d + (2*b^2*(b + a*Cot[c + d*x]))/(d*Sqrt[Cot[c + d*x]]) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3, x, 13, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (16*a*b^2)/(3*d*Sqrt[Cot[c + d*x]]) + (2*b^2*(b + a*Cot[c + d*x]))/(3*d*Cot[c + d*x]^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])^3/Sqrt[Cot[c + d*x]], x, 14, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (8*a*b^2)/(5*d*Cot[c + d*x]^(3/2)) + (2*b*(3*a^2 - b^2))/(d*Sqrt[Cot[c + d*x]]) + (2*b^2*(b + a*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a + b*Tan[c + d*x])^3/Cot[c + d*x]^(3/2), x, 15, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (32*a*b^2)/(35*d*Cot[c + d*x]^(5/2)) + (2*b*(3*a^2 - b^2))/(3*d*Cot[c + d*x]^(3/2)) + (2*a*(a^2 - 3*b^2))/(d*Sqrt[Cot[c + d*x]]) + (2*b^2*(b + a*Cot[c + d*x]))/(7*d*Cot[c + d*x]^(7/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x]), x, 17, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(a^2 + b^2)*d) + (2*b*Sqrt[Cot[c + d*x]])/(a^2*d) - (2*Cot[c + d*x]^(3/2))/(3*a*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x]), x, 16, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(5/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(3/2)*(a^2 + b^2)*d) - (2*Sqrt[Cot[c + d*x]])/(a*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x]), x, 15, ((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])), x, 15, ((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/((a^2 + b^2)*d) + ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 15, -(((a - b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(3/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)*d) + ((a + b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 16, -(((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(3/2)*(a^2 + b^2)*d) + 2/(b*d*Sqrt[Cot[c + d*x]]) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} - - -{Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^2, x, 18, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (b^(7/2)*(9*a^2 + 5*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(7/2)*(a^2 + b^2)^2*d) + (b*(4*a^2 + 5*b^2)*Sqrt[Cot[c + d*x]])/(a^3*(a^2 + b^2)*d) - ((2*a^2 + 5*b^2)*Cot[c + d*x]^(3/2))/(3*a^2*(a^2 + b^2)*d) + (b^2*Cot[c + d*x]^(5/2))/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^2, x, 17, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(a^2 + b^2)^2*d) - ((2*a^2 + 3*b^2)*Sqrt[Cot[c + d*x]])/(a^2*(a^2 + b^2)*d) + (b^2*Cot[c + d*x]^(3/2))/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^2, x, 16, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(3/2)*(a^2 + b^2)^2*d) + (b^2*Sqrt[Cot[c + d*x]])/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2), x, 16, ((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)^2*d) - (b*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2), x, 16, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[a]*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)^2*d) + (a*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2), x, 16, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (a^(3/2)*(a^2 + 5*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(3/2)*(a^2 + b^2)^2*d) - (a^2*Sqrt[Cot[c + d*x]])/(b*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) - ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2), x, 17, ((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(5/2)*(3*a^2 + 7*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(5/2)*(a^2 + b^2)^2*d) + (3*a^2 + 2*b^2)/(b^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]) - a^2/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} - - -{Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^3, x, 19, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(7/2)*(99*a^4 + 102*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(9/2)*(a^2 + b^2)^3*d) + (b*(24*a^4 + 67*a^2*b^2 + 35*b^4)*Sqrt[Cot[c + d*x]])/(4*a^4*(a^2 + b^2)^2*d) - ((8*a^4 + 67*a^2*b^2 + 35*b^4)*Cot[c + d*x]^(3/2))/(12*a^3*(a^2 + b^2)^2*d) + (b^2*Cot[c + d*x]^(7/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (b^2*(15*a^2 + 7*b^2)*Cot[c + d*x]^(5/2))/(4*a^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^3, x, 18, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((8*a^4 + 31*a^2*b^2 + 15*b^4)*Sqrt[Cot[c + d*x]])/(4*a^3*(a^2 + b^2)^2*d) + (b^2*Cot[c + d*x]^(5/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (b^2*(13*a^2 + 5*b^2)*Cot[c + d*x]^(3/2))/(4*a^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^3, x, 17, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(5/2)*(a^2 + b^2)^3*d) + (b^2*Cot[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (b^2*(11*a^2 + 3*b^2)*Sqrt[Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3), x, 17, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (Sqrt[b]*(15*a^4 - 18*a^2*b^2 - b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(3/2)*(a^2 + b^2)^3*d) + (b^2*Sqrt[Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (b*(9*a^2 + b^2)*Sqrt[Cot[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3), x, 17, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^4 - 26*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*Sqrt[a]*Sqrt[b]*(a^2 + b^2)^3*d) - (b*Sqrt[Cot[c + d*x]])/(2*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + ((5*a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/(4*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3), x, 17, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*(a^2 + b^2)^3*d) + (a*Sqrt[Cot[c + d*x]])/(2*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (a*(a^2 - 7*b^2)*Sqrt[Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3), x, 17, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*b^(5/2)*(a^2 + b^2)^3*d) - (a^2*Sqrt[Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (a^2*(3*a^2 + 11*b^2)*Sqrt[Cot[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (d Cot[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]], x, 11, (Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2 + 2*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d) - (2*b*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*a*d) - (2*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]], x, 11, (I*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*b*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*a*d) - (2*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]], x, 9, -((Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]], x, 8, ((-I)*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{Sqrt[a + b*Tan[c + d*x]]/Sqrt[Cot[c + d*x]], x, 12, (Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{Sqrt[a + b*Tan[c + d*x]]/Cot[c + d*x]^(3/2), x, 14, (I*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[b]*d) + (I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + Sqrt[a + b*Tan[c + d*x]]/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2), x, 12, -(((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*b*(70*a^2 + 3*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d) + (2*(35*a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d) - (16*b*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(35*d) - (2*a*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2), x, 11, -((I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(5*a^2 - b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(5*a*d) - (4*b*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d) - (2*a*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2), x, 10, ((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (8*b*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*d) - (2*a*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2), x, 9, (I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2), x, 13, -(((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{(a + b*Tan[c + d*x])^(3/2)/Sqrt[Cot[c + d*x]], x, 14, -((I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*(I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{(a + b*Tan[c + d*x])^(3/2)/Cot[c + d*x]^(3/2), x, 15, ((I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((3*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*Sqrt[b]*d) + ((I*a + b)^(3/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (3*a*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]) + (a + b*Tan[c + d*x])^(3/2)/(2*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2), x, 13, ((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(315*a^4 - 483*a^2*b^2 - 10*b^4)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d) + (2*b*(231*a^2 - 5*b^2)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(315*a*d) + (2*(21*a^2 - 25*b^2)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(105*d) - (38*a*b*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(63*d) - (2*a^2*Cot[c + d*x]^(9/2)*Sqrt[a + b*Tan[c + d*x]])/(9*d)} -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2), x, 12, (I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*b*(49*a^2 - 3*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(21*a*d) + (2*(7*a^2 - 9*b^2)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(21*d) - (6*a*b*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d) - (2*a^2*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2), x, 11, -(((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2 - 23*b^2)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (22*a*b*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (2*a^2*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2), x, 10, -((I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (14*a*b*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*d) - (2*a^2*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2), x, 14, ((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a^2*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2), x, 14, (I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b^2*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{(a + b*Tan[c + d*x])^(5/2)/Sqrt[Cot[c + d*x]], x, 15, -(((I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (Sqrt[b]*(15*a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d) + ((I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b^2*Sqrt[a + b*Tan[c + d*x]])/(2*d*Cot[c + d*x]^(3/2)) + (9*a*b*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]])} -{(a + b*Tan[c + d*x])^(5/2)/Cot[c + d*x]^(3/2), x, 16, -((I*(I*a - b)^(5/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (5*a*(a^2 - 8*b^2)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(8*Sqrt[b]*d) - (I*(I*a + b)^(5/2)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b^2*Sqrt[a + b*Tan[c + d*x]])/(3*d*Cot[c + d*x]^(5/2)) + (13*a*b*Sqrt[a + b*Tan[c + d*x]])/(12*d*Cot[c + d*x]^(3/2)) + ((11*a^2 - 8*b^2)*Sqrt[a + b*Tan[c + d*x]])/(8*d*Sqrt[Cot[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cot[c + d*x]^(5/2)/Sqrt[a + b*Tan[c + d*x]], x, 11, -((ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) - (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) + (4*b*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*a^2*d) - (2*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*a*d)} -{Cot[c + d*x]^(3/2)/Sqrt[a + b*Tan[c + d*x]], x, 10, ((-I)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) - (2*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(a*d)} -{Sqrt[Cot[c + d*x]]/Sqrt[a + b*Tan[c + d*x]], x, 8, (ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{1/(Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]), x, 8, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{1/(Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]), x, 13, -((ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) + (2*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[b]*d) - (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{1/(Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]), x, 14, ((-I)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) - (a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(3/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) + Sqrt[a + b*Tan[c + d*x]]/(b*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(3/2), x, 11, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b^2*(5*a^2 + 8*b^2))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (8*b*Sqrt[Cot[c + d*x]])/(3*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - (2*Cot[c + d*x]^(3/2))/(3*a*d*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(3/2), x, 10, (ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) - (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*b*(a^2 + 2*b^2))/(a^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - (2*Sqrt[Cot[c + d*x]])/(a*d*Sqrt[a + b*Tan[c + d*x]])} -{Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^(3/2), x, 9, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)), x, 9, -((ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) + (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*b)/((a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)), x, 9, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*a)/((a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)), x, 14, (ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) + (2*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(3/2)*d) - (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2)), x, 15, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) - (3*a*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(5/2)*d) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*a^2)/(b*(a^2 + b^2)*d*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]) + ((3*a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(5/2)/(a + b*Tan[c + d*x])^(5/2), x, 12, (ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) + (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*b^2*(7*a^2 + 8*b^2))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (4*b*Sqrt[Cot[c + d*x]])/(a^2*d*(a + b*Tan[c + d*x])^(3/2)) - (2*Cot[c + d*x]^(3/2))/(3*a*d*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^4 + 15*a^2*b^2 + 8*b^4))/(3*a^4*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{Cot[c + d*x]^(3/2)/(a + b*Tan[c + d*x])^(5/2), x, 11, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) - (2*b*(3*a^2 + 4*b^2))/(3*a^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*Sqrt[Cot[c + d*x]])/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^4 + 17*a^2*b^2 + 8*b^4))/(3*a^3*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{Sqrt[Cot[c + d*x]]/(a + b*Tan[c + d*x])^(5/2), x, 10, -((ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d)) - (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*b^2)/(3*a*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (4*b^2*(4*a^2 + b^2))/(3*a^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)), x, 10, -((I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d)) + (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) - (2*b)/(3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2))/(3*a*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)), x, 10, (ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) + (ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*a)/(3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (4*(a^2 - 2*b^2))/(3*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{1/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)), x, 10, (I*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) - (I*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) - (2*a^2)/(3*b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(a^2 + 7*b^2))/(3*b*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n with n symbolic*) - - -{(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^3, x, 8, If[$VersionNumber>=8, (a^2*b*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(1 - n)*(2 - n)) + (a^2*d^2*(d*Cot[e + f*x])^(-2 + n)*(b + a*Cot[e + f*x]))/(f*(1 - n)) - (b*(3*a^2 - b^2)*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, (1/2)*(-2 + n), n/2, -Cot[e + f*x]^2])/(f*(2 - n)) - (a*(a^2 - 3*b^2)*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, -Cot[e + f*x]^2])/(f*(1 - n)), (a^2*b*d^2*(1 - 2*n)*(d*Cot[e + f*x])^(-2 + n))/(f*(2 - 3*n + n^2)) + (a^2*d^2*(d*Cot[e + f*x])^(-2 + n)*(b + a*Cot[e + f*x]))/(f*(1 - n)) - (b*(3*a^2 - b^2)*d^2*(d*Cot[e + f*x])^(-2 + n)*Hypergeometric2F1[1, (1/2)*(-2 + n), n/2, -Cot[e + f*x]^2])/(f*(2 - n)) - (a*(a^2 - 3*b^2)*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, -Cot[e + f*x]^2])/(f*(1 - n))]} -{(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^2, x, 7, (a^2*d*(d*Cot[e + f*x])^(-1 + n))/(f*(1 - n)) - ((a^2 - b^2)*d*(d*Cot[e + f*x])^(-1 + n)*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, -Cot[e + f*x]^2])/(f*(1 - n)) - (2*a*b*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n/2, (2 + n)/2, -Cot[e + f*x]^2])/(f*n)} -{(d*Cot[e + f*x])^n*(a + b*Tan[e + f*x])^1, x, 6, -((b*(d*Cot[e + f*x])^n*Hypergeometric2F1[1, n/2, (2 + n)/2, -Cot[e + f*x]^2])/(f*n)) - (a*(d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[e + f*x]^2])/(d*f*(1 + n))} -{(d*Cot[e + f*x])^n/(a + b*Tan[e + f*x])^1, x, 9, -((b*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)*d^2*f*(2 + n))) - (a^2*(d*Cot[e + f*x])^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, -((a*Cot[e + f*x])/b)])/(b*(a^2 + b^2)*d^2*f*(2 + n)) + (a*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)*d^3*f*(3 + n))} -{(d*Cot[e + f*x])^n/(a + b*Tan[e + f*x])^2, x, 10, -((a^2*(d*Cot[e + f*x])^(3 + n))/(b*(a^2 + b^2)*d^3*f*(b + a*Cot[e + f*x]))) + ((a^2 - b^2)*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)^2*d^3*f*(3 + n)) + (a^2*(b^2*n + a^2*(2 + n))*(d*Cot[e + f*x])^(3 + n)*Hypergeometric2F1[1, 3 + n, 4 + n, -((a*Cot[e + f*x])/b)])/(b^2*(a^2 + b^2)^2*d^3*f*(3 + n)) + (2*a*b*(d*Cot[e + f*x])^(4 + n)*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Cot[e + f*x]^2])/((a^2 + b^2)^2*d^4*f*(4 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (d Cot[e+f x])^n with m symbolic*) - - -{(a + b*Tan[e + f*x])^m*(d*Cot[e + f*x])^n, x, 8, (AppellF1[1 - n, -m, 1, 2 - n, -((b*Tan[e + f*x])/a), (-I)*Tan[e + f*x]]*(d*Cot[e + f*x])^n*Tan[e + f*x]*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*f*(1 - n))) + (AppellF1[1 - n, -m, 1, 2 - n, -((b*Tan[e + f*x])/a), I*Tan[e + f*x]]*(d*Cot[e + f*x])^n*Tan[e + f*x]*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*f*(1 - n)))} - - -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n, x, 10, -((AppellF1[-(1/2), 1, -n, 1/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)) - (AppellF1[-(1/2), 1, -n, 1/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n, x, 10, (AppellF1[1/2, 1, -n, 3/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Cot[c + d*x]])) + (AppellF1[1/2, 1, -n, 3/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Cot[c + d*x]]))} -{(a + b*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]], x, 10, (AppellF1[3/2, 1, -n, 5/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d*Cot[c + d*x]^(3/2))) + (AppellF1[3/2, 1, -n, 5/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d*Cot[c + d*x]^(3/2)))} -{(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x, 10, (AppellF1[5/2, 1, -n, 7/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d*Cot[c + d*x]^(5/2))) + (AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d*Cot[c + d*x]^(5/2)))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x]), x, 3, -((I*c*(a + I*a*Tan[e + f*x])^3)/(3*f))} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x]), x, 4, -((I*c*(a + I*a*Tan[e + f*x])^2)/(2*f)), (I*a^2*c*Sec[e + f*x]^2)/(2*f) + (a^2*c*Tan[e + f*x])/f} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x]), x, 3, (a*c*Tan[e + f*x])/f} -{(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^1, x, 3, (I*c)/(f*(a + I*a*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2, x, 3, (I*c)/(2*f*(a + I*a*Tan[e + f*x])^2)} -{(c - I*c*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3, x, 3, (I*c)/(3*f*(a + I*a*Tan[e + f*x])^3)} - - -{(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^2, x, 4, -((I*c^2*(a + I*a*Tan[e + f*x])^4)/(2*f)) + (I*c^2*(a + I*a*Tan[e + f*x])^5)/(5*a*f)} -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2, x, 4, (I*a^3*c^2*Sec[e + f*x]^4)/(4*f) + (a^3*c^2*Tan[e + f*x])/f + (a^3*c^2*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2, x, 3, (a^2*c^2*Tan[e + f*x])/f + (a^2*c^2*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^2, x, 4, (I*a*(c - I*c*Tan[e + f*x])^2)/(2*f), -((I*a*c^2*Sec[e + f*x]^2)/(2*f)) + (a*c^2*Tan[e + f*x])/f} -{(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^1, x, 4, -((c^2*x)/a) - (I*c^2*Log[Cos[e + f*x]])/(a*f) + (2*I*c^2)/(f*(a + I*a*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^2, x, 3, (c^2*Tan[e + f*x])/(f*(a + I*a*Tan[e + f*x])^2)} -{(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3, x, 4, (2*I*c^2)/(3*f*(a + I*a*Tan[e + f*x])^3) - (I*c^2)/(2*a*f*(a + I*a*Tan[e + f*x])^2)} -{(c - I*c*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^4, x, 4, (I*c^2)/(2*f*(a + I*a*Tan[e + f*x])^4) - (I*a^2*c^2)/(3*f*(a^2 + I*a^2*Tan[e + f*x])^3)} - - -{(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^3, x, 4, -((4*I*c^3*(a + I*a*Tan[e + f*x])^5)/(5*f)) + (2*I*c^3*(a + I*a*Tan[e + f*x])^6)/(3*a*f) - (I*c^3*(a + I*a*Tan[e + f*x])^7)/(7*a^2*f)} -{(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^3, x, 4, (I*a^4*c^3*Sec[e + f*x]^6)/(6*f) + (a^4*c^3*Tan[e + f*x])/f + (2*a^4*c^3*Tan[e + f*x]^3)/(3*f) + (a^4*c^3*Tan[e + f*x]^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3, x, 3, (a^3*c^3*Tan[e + f*x])/f + (2*a^3*c^3*Tan[e + f*x]^3)/(3*f) + (a^3*c^3*Tan[e + f*x]^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3, x, 4, -((I*a^2*c^3*Sec[e + f*x]^4)/(4*f)) + (a^2*c^3*Tan[e + f*x])/f + (a^2*c^3*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^3, x, 3, (I*a*(c - I*c*Tan[e + f*x])^3)/(3*f)} -{(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^1, x, 4, -((4*c^3*x)/a) - (4*I*c^3*Log[Cos[e + f*x]])/(a*f) + (c^3*Tan[e + f*x])/(a*f) + (4*I*c^3)/(f*(a + I*a*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2, x, 4, (c^3*x)/a^2 + (I*c^3*Log[Cos[e + f*x]])/(a^2*f) + (2*I*c^3)/(f*(a + I*a*Tan[e + f*x])^2) - (4*I*c^3)/(f*(a^2 + I*a^2*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^3, x, 3, (I*c^3*(a^2 - I*a^2*Tan[e + f*x])^3)/(6*f*(a^3 + I*a^3*Tan[e + f*x])^3)} -{(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^4, x, 4, (I*c^3)/(f*(a + I*a*Tan[e + f*x])^4) - (4*I*c^3)/(3*a*f*(a + I*a*Tan[e + f*x])^3) + (I*c^3)/(2*f*(a^2 + I*a^2*Tan[e + f*x])^2)} -{(c - I*c*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^5, x, 4, (4*I*c^3)/(5*f*(a + I*a*Tan[e + f*x])^5) + (I*c^3)/(3*a^2*f*(a + I*a*Tan[e + f*x])^3) - (I*a^3*c^3)/(f*(a^2 + I*a^2*Tan[e + f*x])^4)} - - -{(a + I*a*Tan[e + f*x])^5*(c - I*c*Tan[e + f*x])^4, x, 4, (I*a^5*c^4*Sec[e + f*x]^8)/(8*f) + (a^5*c^4*Tan[e + f*x])/f + (a^5*c^4*Tan[e + f*x]^3)/f + (3*a^5*c^4*Tan[e + f*x]^5)/(5*f) + (a^5*c^4*Tan[e + f*x]^7)/(7*f)} -{(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^4, x, 3, (a^4*c^4*Tan[e + f*x])/f + (a^4*c^4*Tan[e + f*x]^3)/f + (3*a^4*c^4*Tan[e + f*x]^5)/(5*f) + (a^4*c^4*Tan[e + f*x]^7)/(7*f)} -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4, x, 4, -((I*a^3*c^4*Sec[e + f*x]^6)/(6*f)) + (a^3*c^4*Tan[e + f*x])/f + (2*a^3*c^4*Tan[e + f*x]^3)/(3*f) + (a^3*c^4*Tan[e + f*x]^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4, x, 4, (I*a^2*(c - I*c*Tan[e + f*x])^4)/(2*f) - (I*a^2*(c - I*c*Tan[e + f*x])^5)/(5*c*f)} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^4, x, 3, (I*a*(c - I*c*Tan[e + f*x])^4)/(4*f)} -{(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^1, x, 4, -((12*c^4*x)/a) - (12*I*c^4*Log[Cos[e + f*x]])/(a*f) + (5*c^4*Tan[e + f*x])/(a*f) - (I*c^4*Tan[e + f*x]^2)/(2*a*f) + (8*I*c^4)/(f*(a + I*a*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^2, x, 4, (6*c^4*x)/a^2 + (6*I*c^4*Log[Cos[e + f*x]])/(a^2*f) - (c^4*Tan[e + f*x])/(a^2*f) + (4*I*c^4)/(f*(a + I*a*Tan[e + f*x])^2) - (12*I*c^4)/(f*(a^2 + I*a^2*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^3, x, 4, -((c^4*x)/a^3) - (I*c^4*Log[Cos[e + f*x]])/(a^3*f) + (8*I*c^4)/(3*f*(a + I*a*Tan[e + f*x])^3) - (6*I*c^4)/(a*f*(a + I*a*Tan[e + f*x])^2) + (6*I*c^4)/(f*(a^3 + I*a^3*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^4, x, 3, (I*c^4*(a^2 - I*a^2*Tan[e + f*x])^4)/(8*f*(a^3 + I*a^3*Tan[e + f*x])^4)} -{(c - I*c*Tan[e + f*x])^4/(a + I*a*Tan[e + f*x])^5, x, 4, (I*c^4*(1 - I*Tan[e + f*x])^4)/(10*f*(a + I*a*Tan[e + f*x])^5) + (I*c^4*(a - I*a*Tan[e + f*x])^4)/(80*a^5*f*(a + I*a*Tan[e + f*x])^4)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x]), x, 4, -((12*a^4*x)/c) + (12*I*a^4*Log[Cos[e + f*x]])/(c*f) + (5*a^4*Tan[e + f*x])/(c*f) + (I*a^4*Tan[e + f*x]^2)/(2*c*f) - (8*I*a^4)/(f*(c - I*c*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x]), x, 4, -((4*a^3*x)/c) + (4*I*a^3*Log[Cos[e + f*x]])/(c*f) + (a^3*Tan[e + f*x])/(c*f) - (4*I*a^3)/(f*(c - I*c*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x]), x, 4, -((a^2*x)/c) + (I*a^2*Log[Cos[e + f*x]])/(c*f) - (2*I*a^2)/(f*(c - I*c*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x]), x, 3, -((I*a)/(f*(c - I*c*Tan[e + f*x])))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])), x, 3, x/(2*a*c) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*c*f)} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])), x, 5, (3*x)/(8*a^2*c) + (I*Cos[e + f*x]^4)/(4*a^2*c*f) + (3*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*c*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a^2*c*f)} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])), x, 5, x/(4*a^3*c) - I/(16*a^3*f*(c - I*c*Tan[e + f*x])) + (I*c^2)/(12*a^3*f*(c + I*c*Tan[e + f*x])^3) + (I*c)/(8*a^3*f*(c + I*c*Tan[e + f*x])^2) + (3*I)/(16*a^3*f*(c + I*c*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^2, x, 4, (6*a^4*x)/c^2 - (6*I*a^4*Log[Cos[e + f*x]])/(c^2*f) - (a^4*Tan[e + f*x])/(c^2*f) - (4*I*a^4)/(f*(c - I*c*Tan[e + f*x])^2) + (12*I*a^4)/(f*(c^2 - I*c^2*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^2, x, 4, (a^3*x)/c^2 - (I*a^3*Log[Cos[e + f*x]])/(c^2*f) - (2*I*a^3)/(f*(c - I*c*Tan[e + f*x])^2) + (4*I*a^3)/(f*(c^2 - I*c^2*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^2, x, 3, (a^2*Tan[e + f*x])/(f*(c - I*c*Tan[e + f*x])^2)} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x])^2, x, 3, -((I*a)/(2*f*(c - I*c*Tan[e + f*x])^2))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^2), x, 5, (3*x)/(8*a*c^2) - I/(8*a*f*(c - I*c*Tan[e + f*x])^2) - I/(4*a*f*(c^2 - I*c^2*Tan[e + f*x])) + I/(8*a*f*(c^2 + I*c^2*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2), x, 4, (3*x)/(8*a^2*c^2) + (3*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*c^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a^2*c^2*f)} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2), x, 6, (5*x)/(16*a^3*c^2) + (I*Cos[e + f*x]^6)/(6*a^3*c^2*f) + (5*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*c^2*f) + (5*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^3*c^2*f) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a^3*c^2*f)} - - -{(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3, x, 4, -((40*a^6*x)/c^3) + (40*I*a^6*Log[Cos[e + f*x]])/(c^3*f) + (9*a^6*Tan[e + f*x])/(c^3*f) + (I*a^6*Tan[e + f*x]^2)/(2*c^3*f) - (32*I*a^6)/(3*f*(c - I*c*Tan[e + f*x])^3) + (40*I*a^6)/(c*f*(c - I*c*Tan[e + f*x])^2) - (80*I*a^6)/(f*(c^3 - I*c^3*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^5/(c - I*c*Tan[e + f*x])^3, x, 4, -((8*a^5*x)/c^3) + (8*I*a^5*Log[Cos[e + f*x]])/(c^3*f) + (a^5*Tan[e + f*x])/(c^3*f) - (16*I*a^5)/(3*f*(c - I*c*Tan[e + f*x])^3) - (24*I*a^5)/(f*(c^3 - I*c^3*Tan[e + f*x])) + (16*I*a^5*c^5)/(f*(c^4 - I*c^4*Tan[e + f*x])^2)} -{(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^3, x, 4, -((a^4*x)/c^3) + (I*a^4*Log[Cos[e + f*x]])/(c^3*f) - (8*I*a^4)/(3*f*(c - I*c*Tan[e + f*x])^3) + (6*I*a^4)/(c*f*(c - I*c*Tan[e + f*x])^2) - (6*I*a^4)/(f*(c^3 - I*c^3*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^3, x, 3, -((I*a^3*(c^2 + I*c^2*Tan[e + f*x])^3)/(6*f*(c^3 - I*c^3*Tan[e + f*x])^3))} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^3, x, 4, -((2*I*a^2)/(3*f*(c - I*c*Tan[e + f*x])^3)) + (I*a^2)/(2*c*f*(c - I*c*Tan[e + f*x])^2)} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x])^3, x, 3, -((I*a)/(3*f*(c - I*c*Tan[e + f*x])^3))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^3), x, 5, x/(4*a*c^3) - I/(12*a*f*(c - I*c*Tan[e + f*x])^3) - I/(8*a*c*f*(c - I*c*Tan[e + f*x])^2) - (3*I)/(16*a*f*(c^3 - I*c^3*Tan[e + f*x])) + I/(16*a*f*(c^3 + I*c^3*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3), x, 5, (5*x)/(16*a^2*c^3) - I/(24*a^2*f*(c - I*c*Tan[e + f*x])^3) - (3*I)/(32*a^2*c*f*(c - I*c*Tan[e + f*x])^2) + I/(32*a^2*c*f*(c + I*c*Tan[e + f*x])^2) - (3*I)/(16*a^2*f*(c^3 - I*c^3*Tan[e + f*x])) + I/(8*a^2*f*(c^3 + I*c^3*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3), x, 5, (5*x)/(16*a^3*c^3) + (5*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*c^3*f) + (5*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^3*c^3*f) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a^3*c^3*f)} - - -{(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^4, x, 4, (10*a^6*x)/c^4 - (10*I*a^6*Log[Cos[e + f*x]])/(c^4*f) - (a^6*Tan[e + f*x])/(c^4*f) - (8*I*a^6)/(f*(c - I*c*Tan[e + f*x])^4) + (80*I*a^6)/(3*c*f*(c - I*c*Tan[e + f*x])^3) - (40*I*a^6)/(f*(c^2 - I*c^2*Tan[e + f*x])^2) + (40*I*a^6)/(f*(c^4 - I*c^4*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^5/(c - I*c*Tan[e + f*x])^4, x, 4, (a^5*x)/c^4 - (I*a^5*Log[Cos[e + f*x]])/(c^4*f) - (4*I*a^5)/(f*(c - I*c*Tan[e + f*x])^4) - (12*I*a^5)/(f*(c^2 - I*c^2*Tan[e + f*x])^2) + (32*I*a^5*c^5)/(3*f*(c^3 - I*c^3*Tan[e + f*x])^3) + (8*I*a^5)/(f*(c^4 - I*c^4*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^4/(c - I*c*Tan[e + f*x])^4, x, 3, -((I*a^4*(c^2 + I*c^2*Tan[e + f*x])^4)/(8*f*(c^3 - I*c^3*Tan[e + f*x])^4))} -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^4, x, 4, -((I*a^3)/(f*(c - I*c*Tan[e + f*x])^4)) + (4*I*a^3)/(3*c*f*(c - I*c*Tan[e + f*x])^3) - (I*a^3)/(2*f*(c^2 - I*c^2*Tan[e + f*x])^2)} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^4, x, 4, -((I*a^2)/(2*f*(c - I*c*Tan[e + f*x])^4)) + (I*a^2*c^2)/(3*f*(c^2 - I*c^2*Tan[e + f*x])^3)} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x])^4, x, 3, -((I*a)/(4*f*(c - I*c*Tan[e + f*x])^4))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^4), x, 5, (5*x)/(32*a*c^4) - I/(16*a*f*(c - I*c*Tan[e + f*x])^4) - I/(12*a*c*f*(c - I*c*Tan[e + f*x])^3) - (3*I)/(32*a*f*(c^2 - I*c^2*Tan[e + f*x])^2) - I/(8*a*f*(c^4 - I*c^4*Tan[e + f*x])) + I/(32*a*f*(c^4 + I*c^4*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4), x, 5, (15*x)/(64*a^2*c^4) - I/(32*a^2*f*(c - I*c*Tan[e + f*x])^4) - I/(16*a^2*c*f*(c - I*c*Tan[e + f*x])^3) - (3*I)/(32*a^2*f*(c^2 - I*c^2*Tan[e + f*x])^2) + I/(64*a^2*f*(c^2 + I*c^2*Tan[e + f*x])^2) - (5*I)/(32*a^2*f*(c^4 - I*c^4*Tan[e + f*x])) + (5*I)/(64*a^2*f*(c^4 + I*c^4*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4), x, 5, (35*x)/(128*a^3*c^4) - I/(64*a^3*f*(c - I*c*Tan[e + f*x])^4) - I/(24*a^3*c*f*(c - I*c*Tan[e + f*x])^3) + I/(96*a^3*c*f*(c + I*c*Tan[e + f*x])^3) - (5*I)/(64*a^3*f*(c^2 - I*c^2*Tan[e + f*x])^2) + (5*I)/(128*a^3*f*(c^2 + I*c^2*Tan[e + f*x])^2) - (5*I)/(32*a^3*f*(c^4 - I*c^4*Tan[e + f*x])) + (15*I)/(128*a^3*f*(c^4 + I*c^4*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]], x, 4, (8*I*a^3*Sqrt[c - I*c*Tan[e + f*x]])/f - (8*I*a^3*(c - I*c*Tan[e + f*x])^(3/2))/(3*c*f) + (2*I*a^3*(c - I*c*Tan[e + f*x])^(5/2))/(5*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]], x, 4, ((4*I)*a^2*Sqrt[c - I*c*Tan[e + f*x]])/f - (((2*I)/3)*a^2*(c - I*c*Tan[e + f*x])^(3/2))/(c*f)} -{(a + I*a*Tan[e + f*x])^1*Sqrt[c - I*c*Tan[e + f*x]], x, 3, ((2*I)*a*Sqrt[c - I*c*Tan[e + f*x]])/f} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^1, x, 5, (I*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(2*Sqrt[2]*a*f) + (I*Sqrt[c - I*c*Tan[e + f*x]])/(2*a*f*(1 + I*Tan[e + f*x]))} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2, x, 6, (3*I*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a^2*f) + (I*Sqrt[c - I*c*Tan[e + f*x]])/(4*a^2*f*(1 + I*Tan[e + f*x])^2) + (3*I*Sqrt[c - I*c*Tan[e + f*x]])/(16*a^2*f*(1 + I*Tan[e + f*x]))} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3, x, 7, (5*I*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(64*Sqrt[2]*a^3*f) + (I*Sqrt[c - I*c*Tan[e + f*x]])/(6*a^3*f*(1 + I*Tan[e + f*x])^3) + (5*I*Sqrt[c - I*c*Tan[e + f*x]])/(48*a^3*f*(1 + I*Tan[e + f*x])^2) + (5*I*Sqrt[c - I*c*Tan[e + f*x]])/(64*a^3*f*(1 + I*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2), x, 4, (8*I*a^3*(c - I*c*Tan[e + f*x])^(3/2))/(3*f) - (8*I*a^3*(c - I*c*Tan[e + f*x])^(5/2))/(5*c*f) + (2*I*a^3*(c - I*c*Tan[e + f*x])^(7/2))/(7*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2), x, 4, (((4*I)/3)*a^2*(c - I*c*Tan[e + f*x])^(3/2))/f - (((2*I)/5)*a^2*(c - I*c*Tan[e + f*x])^(5/2))/(c*f)} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^(3/2), x, 3, (((2*I)/3)*a*(c - I*c*Tan[e + f*x])^(3/2))/f} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^1, x, 5, -((I*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a*f)) + (I*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(a*f*(c + I*c*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2, x, 6, -((I*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[2]*a^2*f)) + (I*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^2*f*(c + I*c*Tan[e + f*x])^2) - (I*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(8*a^2*f*(c + I*c*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3, x, 7, -((I*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(32*Sqrt[2]*a^3*f)) + (I*c^4*Sqrt[c - I*c*Tan[e + f*x]])/(3*a^3*f*(c + I*c*Tan[e + f*x])^3) - (I*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(24*a^3*f*(c + I*c*Tan[e + f*x])^2) - (I*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(32*a^3*f*(c + I*c*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2), x, 4, (8*I*a^3*(c - I*c*Tan[e + f*x])^(5/2))/(5*f) - (8*I*a^3*(c - I*c*Tan[e + f*x])^(7/2))/(7*c*f) + (2*I*a^3*(c - I*c*Tan[e + f*x])^(9/2))/(9*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2), x, 4, (((4*I)/5)*a^2*(c - I*c*Tan[e + f*x])^(5/2))/f - (((2*I)/7)*a^2*(c - I*c*Tan[e + f*x])^(7/2))/(c*f)} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^(5/2), x, 3, (((2*I)/5)*a*(c - I*c*Tan[e + f*x])^(5/2))/f} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^1, x, 6, -((3*I*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(a*f)) + (3*I*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(a*f) + (I*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(a*f*(c + I*c*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2, x, 6, (3*I*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(4*Sqrt[2]*a^2*f) + (I*c^3*(c - I*c*Tan[e + f*x])^(3/2))/(2*a^2*f*(c + I*c*Tan[e + f*x])^2) - (3*I*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(4*a^2*f*(c + I*c*Tan[e + f*x]))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3, x, 7, (I*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a^3*f) + (I*c^4*(c - I*c*Tan[e + f*x])^(3/2))/(3*a^3*f*(c + I*c*Tan[e + f*x])^3) - (I*c^4*Sqrt[c - I*c*Tan[e + f*x]])/(4*a^3*f*(c + I*c*Tan[e + f*x])^2) + (I*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(16*a^3*f*(c + I*c*Tan[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^3/Sqrt[c - I*c*Tan[e + f*x]], x, 4, -((8*I*a^3)/(f*Sqrt[c - I*c*Tan[e + f*x]])) - (8*I*a^3*Sqrt[c - I*c*Tan[e + f*x]])/(c*f) + (2*I*a^3*(c - I*c*Tan[e + f*x])^(3/2))/(3*c^2*f)} -{(a + I*a*Tan[e + f*x])^2/Sqrt[c - I*c*Tan[e + f*x]], x, 4, ((-4*I)*a^2)/(f*Sqrt[c - I*c*Tan[e + f*x]]) - ((2*I)*a^2*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)} -{(a + I*a*Tan[e + f*x])^1/Sqrt[c - I*c*Tan[e + f*x]], x, 3, ((-2*I)*a)/(f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^1*Sqrt[c - I*c*Tan[e + f*x]]), x, 6, (3*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(4*Sqrt[2]*a*Sqrt[c]*f) - (3*I)/(4*a*f*Sqrt[c - I*c*Tan[e + f*x]]) + I/(2*a*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]), x, 7, (15*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(32*Sqrt[2]*a^2*Sqrt[c]*f) - (15*I)/(32*a^2*f*Sqrt[c - I*c*Tan[e + f*x]]) + I/(4*a^2*f*(1 + I*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]) + (5*I)/(16*a^2*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]), x, 8, (35*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(128*Sqrt[2]*a^3*Sqrt[c]*f) - (35*I)/(128*a^3*f*Sqrt[c - I*c*Tan[e + f*x]]) + I/(6*a^3*f*(1 + I*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]) + (7*I)/(48*a^3*f*(1 + I*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]) + (35*I)/(192*a^3*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^(3/2), x, 4, -((8*I*a^3)/(3*f*(c - I*c*Tan[e + f*x])^(3/2))) + (8*I*a^3)/(c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (2*I*a^3*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f)} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^(3/2), x, 4, (((-4*I)/3)*a^2)/(f*(c - I*c*Tan[e + f*x])^(3/2)) + ((2*I)*a^2)/(c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x])^(3/2), x, 3, (((-2*I)/3)*a)/(f*(c - I*c*Tan[e + f*x])^(3/2))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^(3/2)), x, 7, (5*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[2]*a*c^(3/2)*f) - (5*I)/(12*a*f*(c - I*c*Tan[e + f*x])^(3/2)) + I/(2*a*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - (5*I)/(8*a*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)), x, 8, (35*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(64*Sqrt[2]*a^2*c^(3/2)*f) - (35*I)/(96*a^2*f*(c - I*c*Tan[e + f*x])^(3/2)) + I/(4*a^2*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)) + (7*I)/(16*a^2*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - (35*I)/(64*a^2*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)), x, 9, (105*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(256*Sqrt[2]*a^3*c^(3/2)*f) - (35*I)/(128*a^3*f*(c - I*c*Tan[e + f*x])^(3/2)) + I/(6*a^3*f*(1 + I*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)) + (3*I)/(16*a^3*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)) + (21*I)/(64*a^3*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - (105*I)/(256*a^3*c*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^3/(c - I*c*Tan[e + f*x])^(5/2), x, 4, -((8*I*a^3)/(5*f*(c - I*c*Tan[e + f*x])^(5/2))) + (8*I*a^3)/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*I*a^3)/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^2/(c - I*c*Tan[e + f*x])^(5/2), x, 4, (((-4*I)/5)*a^2)/(f*(c - I*c*Tan[e + f*x])^(5/2)) + (((2*I)/3)*a^2)/(c*f*(c - I*c*Tan[e + f*x])^(3/2))} -{(a + I*a*Tan[e + f*x])^1/(c - I*c*Tan[e + f*x])^(5/2), x, 3, (((-2*I)/5)*a)/(f*(c - I*c*Tan[e + f*x])^(5/2))} -{1/((a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^(5/2)), x, 8, (7*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a*c^(5/2)*f) - (7*I)/(20*a*f*(c - I*c*Tan[e + f*x])^(5/2)) + I/(2*a*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - (7*I)/(24*a*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (7*I)/(16*a*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)), x, 9, (63*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(128*Sqrt[2]*a^2*c^(5/2)*f) - (63*I)/(160*a^2*f*(c - I*c*Tan[e + f*x])^(5/2)) + I/(4*a^2*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)) + (9*I)/(16*a^2*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - (21*I)/(64*a^2*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (63*I)/(128*a^2*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)), x, 10, (231*I*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(512*Sqrt[2]*a^3*c^(5/2)*f) - (231*I)/(640*a^3*f*(c - I*c*Tan[e + f*x])^(5/2)) + I/(6*a^3*f*(1 + I*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)) + (11*I)/(48*a^3*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)) + (33*I)/(64*a^3*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - (77*I)/(256*a^3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (231*I)/(512*a^3*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (c-c I Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]], x, 6, ((-3*I)*a^(5/2)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (((3*I)/2)*a^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/f + ((I/2)*a*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/f} -{(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]], x, 5, ((-2*I)*a^(3/2)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (I*a*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/f} -{(a + I*a*Tan[e + f*x])^(1/2)*Sqrt[c - I*c*Tan[e + f*x]], x, 4, ((-2*I)*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(1/2), x, 2, (I*Sqrt[c - I*c*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(3/2), x, 3, ((I/3)*Sqrt[c - I*c*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I/3)*Sqrt[c - I*c*Tan[e + f*x]])/(a*f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2), x, 4, ((I/5)*Sqrt[c - I*c*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^(5/2)) + (((2*I)/15)*Sqrt[c - I*c*Tan[e + f*x]])/(a*f*(a + I*a*Tan[e + f*x])^(3/2)) + (((2*I)/15)*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[c - I*c*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(7/2), x, 5, (I*Sqrt[c - I*c*Tan[e + f*x]])/(7*f*(a + I*a*Tan[e + f*x])^(7/2)) + (3*I*Sqrt[c - I*c*Tan[e + f*x]])/(35*a*f*(a + I*a*Tan[e + f*x])^(5/2)) + (2*I*Sqrt[c - I*c*Tan[e + f*x]])/(35*a^2*f*(a + I*a*Tan[e + f*x])^(3/2)) + (2*I*Sqrt[c - I*c*Tan[e + f*x]])/(35*a^3*f*Sqrt[a + I*a*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2), x, 6, -((I*a^(5/2)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) + (a^2*c*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) + (I*a*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2), x, 5, -((I*a^(3/2)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) + (a*c*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f)} -{(a + I*a*Tan[e + f*x])^(1/2)*(c - I*c*Tan[e + f*x])^(3/2), x, 5, ((-2*I)*Sqrt[a]*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f - (I*c*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/f} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(1/2), x, 5, ((2*I)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[a]*f) + ((2*I)*c*Sqrt[c - I*c*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(3/2), x, 2, ((I/3)*(c - I*c*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(5/2), x, 3, ((I/5)*(c - I*c*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(5/2)) + ((I/15)*(c - I*c*Tan[e + f*x])^(3/2))/(a*f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(7/2), x, 4, (I*(c - I*c*Tan[e + f*x])^(3/2))/(7*f*(a + I*a*Tan[e + f*x])^(7/2)) + (2*I*(c - I*c*Tan[e + f*x])^(3/2))/(35*a*f*(a + I*a*Tan[e + f*x])^(5/2)) + (2*I*(c - I*c*Tan[e + f*x])^(3/2))/(105*a^2*f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(9/2), x, 5, (I*(c - I*c*Tan[e + f*x])^(3/2))/(9*f*(a + I*a*Tan[e + f*x])^(9/2)) + (I*(c - I*c*Tan[e + f*x])^(3/2))/(21*a*f*(a + I*a*Tan[e + f*x])^(7/2)) + (2*I*(c - I*c*Tan[e + f*x])^(3/2))/(105*a^2*f*(a + I*a*Tan[e + f*x])^(5/2)) + (2*I*(c - I*c*Tan[e + f*x])^(3/2))/(315*a^3*f*(a + I*a*Tan[e + f*x])^(3/2))} - - -{(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2), x, 6, -((3*I*a^(5/2)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f)) + (3*a^2*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (a*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(4*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2), x, 6, -((I*a^(3/2)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) + (a*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) - (I*c*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f)} -{(a + I*a*Tan[e + f*x])^(1/2)*(c - I*c*Tan[e + f*x])^(5/2), x, 6, ((-3*I)*Sqrt[a]*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f - (((3*I)/2)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/f - ((I/2)*c*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/f} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(1/2), x, 6, (6*I*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[a]*f) + (3*I*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(a*f) + (2*I*c*(c - I*c*Tan[e + f*x])^(3/2))/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2), x, 6, ((-2*I)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(3/2)*f) - ((2*I)*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(a*f*Sqrt[a + I*a*Tan[e + f*x]]) + (((2*I)/3)*c*(c - I*c*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2), x, 2, ((I/5)*(c - I*c*Tan[e + f*x])^(5/2))/(f*(a + I*a*Tan[e + f*x])^(5/2))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(7/2), x, 3, (I*(c - I*c*Tan[e + f*x])^(5/2))/(7*f*(a + I*a*Tan[e + f*x])^(7/2)) + (I*(c - I*c*Tan[e + f*x])^(5/2))/(35*a*f*(a + I*a*Tan[e + f*x])^(5/2))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(9/2), x, 4, (I*(c - I*c*Tan[e + f*x])^(5/2))/(9*f*(a + I*a*Tan[e + f*x])^(9/2)) + (2*I*(c - I*c*Tan[e + f*x])^(5/2))/(63*a*f*(a + I*a*Tan[e + f*x])^(7/2)) + (2*I*(c - I*c*Tan[e + f*x])^(5/2))/(315*a^2*f*(a + I*a*Tan[e + f*x])^(5/2))} -{(c - I*c*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(11/2), x, 5, (I*(c - I*c*Tan[e + f*x])^(5/2))/(11*f*(a + I*a*Tan[e + f*x])^(11/2)) + (I*(c - I*c*Tan[e + f*x])^(5/2))/(33*a*f*(a + I*a*Tan[e + f*x])^(9/2)) + (2*I*(c - I*c*Tan[e + f*x])^(5/2))/(231*a^2*f*(a + I*a*Tan[e + f*x])^(7/2)) + (2*I*(c - I*c*Tan[e + f*x])^(5/2))/(1155*a^3*f*(a + I*a*Tan[e + f*x])^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^(7/2)/Sqrt[c - I*c*Tan[e + f*x]], x, 7, (15*I*a^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - (2*I*a*(a + I*a*Tan[e + f*x])^(5/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (15*I*a^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c*f) - (5*I*a^2*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(2*c*f)} -{(a + I*a*Tan[e + f*x])^(5/2)/Sqrt[c - I*c*Tan[e + f*x]], x, 6, (6*I*a^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - (2*I*a*(a + I*a*Tan[e + f*x])^(3/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (3*I*a^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)} -{(a + I*a*Tan[e + f*x])^(3/2)/Sqrt[c - I*c*Tan[e + f*x]], x, 5, ((2*I)*a^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - ((2*I)*a*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^(1/2)/Sqrt[c - I*c*Tan[e + f*x]], x, 2, ((-I)*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(1/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 2, Tan[e + f*x]/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 3, I/(3*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (2*Tan[e + f*x])/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 4, I/(5*f*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]) + I/(5*a*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (2*Tan[e + f*x])/(5*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 5, I/(7*f*(a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (4*I)/(35*a*f*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (4*I)/(35*a^2*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (8*Tan[e + f*x])/(35*a^3*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(9/2)/(c - I*c*Tan[e + f*x])^(3/2), x, 8, -((35*I*a^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f)) - (2*I*a*(a + I*a*Tan[e + f*x])^(7/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (14*I*a^2*(a + I*a*Tan[e + f*x])^(5/2))/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (35*I*a^4*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c^2*f) + (35*I*a^3*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(6*c^2*f)} -{(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(3/2), x, 7, -((10*I*a^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f)) - (2*I*a*(a + I*a*Tan[e + f*x])^(5/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (10*I*a^2*(a + I*a*Tan[e + f*x])^(3/2))/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (5*I*a^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f)} -{(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(3/2), x, 6, ((-2*I)*a^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f) - (((2*I)/3)*a*(a + I*a*Tan[e + f*x])^(3/2))/(f*(c - I*c*Tan[e + f*x])^(3/2)) + ((2*I)*a^2*Sqrt[a + I*a*Tan[e + f*x]])/(c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^(3/2)/(c - I*c*Tan[e + f*x])^(3/2), x, 2, ((-I/3)*(a + I*a*Tan[e + f*x])^(3/2))/(f*(c - I*c*Tan[e + f*x])^(3/2))} -{(a + I*a*Tan[e + f*x])^(1/2)/(c - I*c*Tan[e + f*x])^(3/2), x, 3, ((-I/3)*Sqrt[a + I*a*Tan[e + f*x]])/(f*(c - I*c*Tan[e + f*x])^(3/2)) - ((I/3)*Sqrt[a + I*a*Tan[e + f*x]])/(c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(1/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 4, I/(f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)) - (2*I*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*I*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 3, Tan[e + f*x]/(3*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (2*Tan[e + f*x])/(3*a*c*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 4, I/(5*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (4*Tan[e + f*x])/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (8*Tan[e + f*x])/(15*a^2*c*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 5, I/(7*f*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(3/2)) + I/(7*a*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (4*Tan[e + f*x])/(21*a^2*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (8*Tan[e + f*x])/(21*a^3*c*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(11/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 9, (63*I*a^(11/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(5/2)*f) - (2*I*a*(a + I*a*Tan[e + f*x])^(9/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (6*I*a^2*(a + I*a*Tan[e + f*x])^(7/2))/(5*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (42*I*a^3*(a + I*a*Tan[e + f*x])^(5/2))/(5*c^2*f*Sqrt[c - I*c*Tan[e + f*x]]) - (63*I*a^5*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c^3*f) - (21*I*a^4*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(2*c^3*f)} -{(a + I*a*Tan[e + f*x])^(9/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 8, (14*I*a^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(5/2)*f) - (2*I*a*(a + I*a*Tan[e + f*x])^(7/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (14*I*a^2*(a + I*a*Tan[e + f*x])^(5/2))/(15*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (14*I*a^3*(a + I*a*Tan[e + f*x])^(3/2))/(3*c^2*f*Sqrt[c - I*c*Tan[e + f*x]]) - (7*I*a^4*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c^3*f)} -{(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 7, (2*I*a^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(5/2)*f) - (2*I*a*(a + I*a*Tan[e + f*x])^(5/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (2*I*a^2*(a + I*a*Tan[e + f*x])^(3/2))/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*I*a^3*Sqrt[a + I*a*Tan[e + f*x]])/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^(5/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 2, ((-I/5)*(a + I*a*Tan[e + f*x])^(5/2))/(f*(c - I*c*Tan[e + f*x])^(5/2))} -{(a + I*a*Tan[e + f*x])^(3/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 3, ((-I/5)*(a + I*a*Tan[e + f*x])^(3/2))/(f*(c - I*c*Tan[e + f*x])^(5/2)) - ((I/15)*(a + I*a*Tan[e + f*x])^(3/2))/(c*f*(c - I*c*Tan[e + f*x])^(3/2))} -{(a + I*a*Tan[e + f*x])^(1/2)/(c - I*c*Tan[e + f*x])^(5/2), x, 4, ((-I/5)*Sqrt[a + I*a*Tan[e + f*x]])/(f*(c - I*c*Tan[e + f*x])^(5/2)) - (((2*I)/15)*Sqrt[a + I*a*Tan[e + f*x]])/(c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (((2*I)/15)*Sqrt[a + I*a*Tan[e + f*x]])/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(1/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 5, I/(f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)) - (3*I*Sqrt[a + I*a*Tan[e + f*x]])/(5*a*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*I*Sqrt[a + I*a*Tan[e + f*x]])/(5*a*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*I*Sqrt[a + I*a*Tan[e + f*x]])/(5*a*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 6, I/(3*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (4*I)/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)) - (4*I*Sqrt[a + I*a*Tan[e + f*x]])/(5*a^2*f*(c - I*c*Tan[e + f*x])^(5/2)) - (8*I*Sqrt[a + I*a*Tan[e + f*x]])/(15*a^2*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (8*I*Sqrt[a + I*a*Tan[e + f*x]])/(15*a^2*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 4, Tan[e + f*x]/(5*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (4*Tan[e + f*x])/(15*a*c*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (8*Tan[e + f*x])/(15*a^2*c^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 5, I/(7*f*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (6*Tan[e + f*x])/(35*a*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (8*Tan[e + f*x])/(35*a^2*c*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (16*Tan[e + f*x])/(35*a^3*c^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n with n symbolic*) - - -{(a + I*a*Tan[e + f*x])^4*(c - I*c*Tan[e + f*x])^n, x, 4, (8*I*a^4*(c - I*c*Tan[e + f*x])^n)/(f*n) - (12*I*a^4*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n)) + (6*I*a^4*(c - I*c*Tan[e + f*x])^(2 + n))/(c^2*f*(2 + n)) - (I*a^4*(c - I*c*Tan[e + f*x])^(3 + n))/(c^3*f*(3 + n))} -{(a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^n, x, 4, (4*I*a^3*(c - I*c*Tan[e + f*x])^n)/(f*n) - (4*I*a^3*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n)) + (I*a^3*(c - I*c*Tan[e + f*x])^(2 + n))/(c^2*f*(2 + n))} -{(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^n, x, 4, ((2*I)*a^2*(c - I*c*Tan[e + f*x])^n)/(f*n) - (I*a^2*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n))} -{(a + I*a*Tan[e + f*x])^1*(c - I*c*Tan[e + f*x])^n, x, 3, (I*a*(c - I*c*Tan[e + f*x])^n)/(f*n)} -{(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^1, x, 3, (I*Hypergeometric2F1[2, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(4*a*f*n)} -{(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x, 3, (I*Hypergeometric2F1[3, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(8*a^2*f*n)} -{(c - I*c*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x, 3, (I*Hypergeometric2F1[4, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(16*a^3*f*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c-c I Tan[e+f x])^n with m symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n, x, 3, ((I/2)*Hypergeometric2F1[1, m + n, 1 + n, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(f*n), -((1/(f*m))*((I*2^(-1 + n)*Hypergeometric2F1[m, 1 - n, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(1 - I*Tan[e + f*x])^n))} - - -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^4, x, 4, -((8*I*c^4*(a + I*a*Tan[e + f*x])^m)/(f*m)) + (12*I*c^4*(a + I*a*Tan[e + f*x])^(1 + m))/(a*f*(1 + m)) - (6*I*c^4*(a + I*a*Tan[e + f*x])^(2 + m))/(a^2*f*(2 + m)) + (I*c^4*(a + I*a*Tan[e + f*x])^(3 + m))/(a^3*f*(3 + m))} -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^3, x, 4, ((-4*I)*c^3*(a + I*a*Tan[e + f*x])^m)/(f*m) + ((4*I)*c^3*(a + I*a*Tan[e + f*x])^(1 + m))/(a*f*(1 + m)) - (I*c^3*(a + I*a*Tan[e + f*x])^(2 + m))/(a^2*f*(2 + m))} -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^2, x, 4, ((-2*I)*c^2*(a + I*a*Tan[e + f*x])^m)/(f*m) + (I*c^2*(a + I*a*Tan[e + f*x])^(1 + m))/(a*f*(1 + m))} -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^1, x, 3, ((-I)*c*(a + I*a*Tan[e + f*x])^m)/(f*m)} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^1, x, 3, -((I*Hypergeometric2F1[2, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(4*c*f*m))} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^2, x, 3, -((I*Hypergeometric2F1[3, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(8*c^2*f*m))} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^3, x, 3, -((I*Hypergeometric2F1[4, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(16*c^3*f*m))} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^4, x, 3, -((I*Hypergeometric2F1[5, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(32*c^4*f*m))} - - -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2), x, 3, ((I/5)*Hypergeometric2F1[1, 5/2 + m, 7/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2))/f, (1/(5*f))*((I*2^m*Hypergeometric2F1[5/2, 1 - m, 7/2, (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(5/2))/(1 + I*Tan[e + f*x])^m)} -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(3/2), x, 3, ((I/3)*Hypergeometric2F1[1, 3/2 + m, 5/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(3/2))/f, (1/(3*f))*((I*2^m*Hypergeometric2F1[3/2, 1 - m, 5/2, (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(3/2))/(1 + I*Tan[e + f*x])^m)} -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^(1/2), x, 3, (I*Hypergeometric2F1[1, 1/2 + m, 3/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m*Sqrt[c - I*c*Tan[e + f*x]])/f, (1/f)*((I*2^m*Hypergeometric2F1[1/2, 1 - m, 3/2, (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[c - I*c*Tan[e + f*x]])/(1 + I*Tan[e + f*x])^m)} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^(1/2), x, 3, ((-I)*Hypergeometric2F1[1, -1/2 + m, 1/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(f*Sqrt[c - I*c*Tan[e + f*x]]), -((1/(f*Sqrt[c - I*c*Tan[e + f*x]]))*((I*2^m*Hypergeometric2F1[-(1/2), 1 - m, 1/2, (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(1 + I*Tan[e + f*x])^m))} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^(3/2), x, 3, ((-I/3)*Hypergeometric2F1[1, -3/2 + m, -1/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(f*(c - I*c*Tan[e + f*x])^(3/2)), -((I*2^m*Hypergeometric2F1[-(3/2), 1 - m, -(1/2), (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(1 + I*Tan[e + f*x])^m/(3*f*(c - I*c*Tan[e + f*x])^(3/2)))} -{(a + I*a*Tan[e + f*x])^m/(c - I*c*Tan[e + f*x])^(5/2), x, 3, ((-I/5)*Hypergeometric2F1[1, -5/2 + m, -3/2, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(f*(c - I*c*Tan[e + f*x])^(5/2)), -((I*2^m*Hypergeometric2F1[-(5/2), 1 - m, -(3/2), (1/2)*(1 - I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(1 + I*Tan[e + f*x])^m/(5*f*(c - I*c*Tan[e + f*x])^(5/2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x]), x, 4, 4*a^3*(c - I*d)*x - (4*a^3*(I*c + d)*Log[Cos[e + f*x]])/f - (2*a^3*(c - I*d)*Tan[e + f*x])/f + (a*(I*c + d)*(a + I*a*Tan[e + f*x])^2)/(2*f) + (d*(a + I*a*Tan[e + f*x])^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]), x, 3, 2*a^2*(c - I*d)*x - (2*a^2*(I*c + d)*Log[Cos[e + f*x]])/f - (a^2*(c - I*d)*Tan[e + f*x])/f + (d*(a + I*a*Tan[e + f*x])^2)/(2*f)} -{(a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x]), x, 2, a*(c - I*d)*x - (a*(I*c + d)*Log[Cos[e + f*x]])/f + (I*a*d*Tan[e + f*x])/f} -{(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^1, x, 2, ((c - I*d)*x)/(2*a) + (I*c - d)/(2*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2, x, 3, ((c - I*d)*x)/(4*a^2) + (I*c - d)/(4*f*(a + I*a*Tan[e + f*x])^2) + (I*c + d)/(4*f*(a^2 + I*a^2*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3, x, 4, ((c - I*d)*x)/(8*a^3) + (I*c - d)/(6*f*(a + I*a*Tan[e + f*x])^3) + (I*c + d)/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (I*c + d)/(8*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2, x, 5, 4*a^3*(c - I*d)^2*x - ((4*I)*a^3*(c - I*d)^2*Log[Cos[e + f*x]])/f - (2*a^3*(c - I*d)^2*Tan[e + f*x])/f + ((I/2)*a*(c - I*d)^2*(a + I*a*Tan[e + f*x])^2)/f + (2*c*d*(a + I*a*Tan[e + f*x])^3)/(3*f) - ((I/4)*d^2*(a + I*a*Tan[e + f*x])^4)/(a*f)} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2, x, 4, 2*a^2*(c - I*d)^2*x - ((2*I)*a^2*(c - I*d)^2*Log[Cos[e + f*x]])/f - (a^2*(c - I*d)^2*Tan[e + f*x])/f + (c*d*(a + I*a*Tan[e + f*x])^2)/f - ((I/3)*d^2*(a + I*a*Tan[e + f*x])^3)/(a*f)} -{(a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^2, x, 3, a*(c - I*d)^2*x - (I*a*(c - I*d)^2*Log[Cos[e + f*x]])/f + (a*d*(I*c + d)*Tan[e + f*x])/f + ((I/2)*a*(c + d*Tan[e + f*x])^2)/f} -{(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^1, x, 3, ((c^2 - 2*I*c*d + d^2)*x)/(2*a) + (I*d^2*Log[Cos[e + f*x]])/(a*f) + (I*(c + I*d)^2)/(2*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^2, x, 3, ((c - I*d)^2*x)/(4*a^2) + ((c + I*d)*(I*c + 3*d))/(4*a^2*f*(1 + I*Tan[e + f*x])) + (I*(c + I*d)^2)/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(c + d*Tan[e + f*x])^2/(a + I*a*Tan[e + f*x])^3, x, 4, ((c - I*d)^2*x)/(8*a^3) + (I*(c + I*d)^2)/(6*f*(a + I*a*Tan[e + f*x])^3) + ((c + I*d)*(I*c + 3*d))/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (I*(c - I*d)^2)/(8*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3, x, 6, 4*a^3*(c - I*d)^3*x + (4*a^3*(I*c + d)^3*Log[Cos[e + f*x]])/f + ((4*I)*a^3*(c - I*d)^2*d*Tan[e + f*x])/f + (2*a^3*(I*c + d)*(c + d*Tan[e + f*x])^2)/f + (((4*I)/3)*a^3*(c + d*Tan[e + f*x])^3)/f + (a^3*(I*c - 11*d)*(c + d*Tan[e + f*x])^4)/(20*d^2*f) - ((a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^4)/(5*d*f)} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3, x, 5, 2*a^2*(c - I*d)^3*x + (2*a^2*(I*c + d)^3*Log[Cos[e + f*x]])/f + ((2*I)*a^2*(c - I*d)^2*d*Tan[e + f*x])/f + (a^2*(I*c + d)*(c + d*Tan[e + f*x])^2)/f + (((2*I)/3)*a^2*(c + d*Tan[e + f*x])^3)/f - (a^2*(c + d*Tan[e + f*x])^4)/(4*d*f)} -{(a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^3, x, 4, a*(c - I*d)^3*x + (a*(I*c + d)^3*Log[Cos[e + f*x]])/f + (I*a*(c - I*d)^2*d*Tan[e + f*x])/f + (a*(I*c + d)*(c + d*Tan[e + f*x])^2)/(2*f) + ((I/3)*a*(c + d*Tan[e + f*x])^3)/f} -{(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^1, x, 3, ((c^3 - (3*I)*c^2*d + 3*c*d^2 + (3*I)*d^3)*x)/(2*a) + (((3*I)*c - d)*d^2*Log[Cos[e + f*x]])/(a*f) - ((c + (3*I)*d)*d^2*Tan[e + f*x])/(2*a*f) + ((I*c - d)*(c + d*Tan[e + f*x])^2)/(2*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2, x, 5, ((c^3 - (3*I)*c^2*d - 3*c*d^2 - (3*I)*d^3)*x)/(4*a^2) + (d^3*Log[Cos[e + f*x]])/(a^2*f) + ((c + I*d)^2*(I*c + 3*d))/(4*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^2)/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^3, x, 4, ((c - I*d)^3*x)/(8*a^3) + ((c + I*d)*(c - 3*I*d)*(I*c + d))/(8*a^3*f*(1 + I*Tan[e + f*x])) + ((c + I*d)^2*(I*c + d))/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (I*(c + d*Tan[e + f*x])^3)/(6*f*(a + I*a*Tan[e + f*x])^3)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x]), x, 5, (4*a^3*x)/(c - I*d) - (a^3*(I*c - 3*d)*Log[Cos[e + f*x]])/(d^2*f) - (a^3*(c + I*d)^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(d^2*(I*c + d)*f) - (a^3 + I*a^3*Tan[e + f*x])/(d*f)} -{(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x]), x, 4, -((a^2*c*(c + I*d)*x)/((c - I*d)*d^2)) + (a^2*(c + (2*I)*d)*x)/d^2 + (a^2*Log[Cos[e + f*x]])/(d*f) - (a^2*(I*c - d)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(d*(I*c + d)*f)} -{(a + I*a*Tan[e + f*x])^1/(c + d*Tan[e + f*x]), x, 2, (a*x)/(c - I*d) + (a*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((I*c + d)*f)} -{1/((a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])), x, 5, x/(2*a*(c + I*d)) - (c*d*x)/(a*(I*c - d)*(c^2 + d^2)) - (d^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a*(I*c - d)*(c^2 + d^2)*f) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])), x, 4, ((c^3 + (3*I)*c^2*d - 3*c*d^2 + (3*I)*d^3)*x)/(4*a^2*(c - I*d)*(c + I*d)^3) - (d^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^2*(c - I*d)*(c + I*d)^3*f) + (I*c - 3*d)/(4*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2)} -{1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])), x, 5, ((c^4 + (4*I)*c^3*d - 6*c^2*d^2 - (4*I)*c*d^3 - 7*d^4)*x)/(8*a^3*(c - I*d)*(c + I*d)^4) + (d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^4*(I*c + d)*f) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (I*c - 3*d)/(8*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2) + (c^2 + (4*I)*c*d - 7*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^2, x, 5, (4*a^3*x)/(c - I*d)^2 + (I*a^3*Log[Cos[e + f*x]])/(d^2*f) - (a^3*(I*c - d)*(c - (3*I)*d)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c - I*d)^2*d^2*f) + ((c + I*d)*(a^3 + I*a^3*Tan[e + f*x]))/((c - I*d)*d*f*(c + d*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^2, x, 3, (2*a^2*x)/(c - I*d)^2 - ((2*I)*a^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c - I*d)^2*f) + (a^2*(I*c - d))/(d*(I*c + d)*f*(c + d*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^1/(c + d*Tan[e + f*x])^2, x, 3, (a*x)/(c - I*d)^2 - (I*a*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c - I*d)^2*f) - a/((I*c + d)*f*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^2), x, 4, ((c^3 + (3*I)*c^2*d + 3*c*d^2 - (3*I)*d^3)*x)/(2*a*(c - I*d)^2*(c + I*d)^3) + ((3*c - I*d)*d^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a*(I*c - d)^3*(c - I*d)^2*f) + ((c - (3*I)*d)*d)/(2*a*(c - I*d)*(c + I*d)^2*f*(c + d*Tan[e + f*x])) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2), x, 5, ((c^4 + (4*I)*c^3*d - 6*c^2*d^2 + (12*I)*c*d^3 + 9*d^4)*x)/(4*a^2*(c - I*d)^2*(c + I*d)^4) - (2*(2*c - I*d)*d^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^2*(c - I*d)^2*(c + I*d)^4*f) + (d*(c^2 + (4*I)*c*d + 9*d^2))/(4*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])) + (I*c - 4*d)/(4*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x])) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2), x, 6, ((c^5 + (5*I)*c^4*d - 10*c^3*d^2 - (10*I)*c^2*d^3 - 35*c*d^4 + (25*I)*d^5)*x)/(8*a^3*(c - I*d)^2*(c + I*d)^5) + ((5*c - (3*I)*d)*d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(I*c - d)^5*(c - I*d)^2*f) + (d*(c^3 + (5*I)*c^2*d - 11*c*d^2 + (25*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])) + ((3*I)*c - 11*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) + (c^2 + (5*I)*c*d - 12*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^3, x, 4, (4*a^3*x)/(c - I*d)^3 - (4*a^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((I*c + d)^3*f) - (a*(a + I*a*Tan[e + f*x])^2)/(2*(I*c + d)*f*(c + d*Tan[e + f*x])^2) + (2*a^3*(c + I*d))/((c - I*d)^2*d*f*(c + d*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^3, x, 4, (2*a^2*x)/(c - I*d)^3 - (2*a^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((I*c + d)^3*f) + (a^2*(I*c - d))/(2*d*(I*c + d)*f*(c + d*Tan[e + f*x])^2) + ((2*I)*a^2)/((c - I*d)^2*f*(c + d*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^1/(c + d*Tan[e + f*x])^3, x, 4, (a*x)/(c - I*d)^3 - (a*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((I*c + d)^3*f) - a/(2*(I*c + d)*f*(c + d*Tan[e + f*x])^2) + (I*a)/((c - I*d)^2*f*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^3), x, 5, ((c^4 + (4*I)*c^3*d + 6*c^2*d^2 - (12*I)*c*d^3 - 3*d^4)*x)/(2*a*(c - I*d)^3*(c + I*d)^4) + (2*d^2*(3*c^2 - (2*I)*c*d - d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a*(c + I*d)^4*(I*c + d)^3*f) + ((c - (2*I)*d)*d)/(2*a*(c - I*d)*(c + I*d)^2*f*(c + d*Tan[e + f*x])^2) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + (d*(c^2 - (8*I)*c*d - 3*d^2))/(2*a*(c - I*d)^2*(c + I*d)^3*f*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3), x, 6, ((c^5 + (5*I)*c^4*d - 10*c^3*d^2 + (30*I)*c^2*d^3 + 45*c*d^4 - (15*I)*d^5)*x)/(4*a^2*(c - I*d)^3*(c + I*d)^5) - (2*d^3*(5*c^2 - (5*I)*c*d - 2*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^2*(I*c - d)^5*(I*c + d)^3*f) + (d*(c^2 + (5*I)*c*d + 8*d^2))/(4*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^2) + (I*c - 5*d)/(4*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) + ((c - (3*I)*d)*d*(c^2 + (8*I)*c*d + 5*d^2))/(4*a^2*(c - I*d)^2*(c + I*d)^4*f*(c + d*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3), x, 7, ((c^6 + (6*I)*c^5*d - 15*c^4*d^2 - (20*I)*c^3*d^3 - 105*c^2*d^4 + (150*I)*c*d^5 + 55*d^6)*x)/(8*a^3*(c - I*d)^3*(c + I*d)^6) - (d^4*(15*c^2 - (18*I)*c*d - 7*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^6*(I*c + d)^3*f) + (d*(c^3 + (6*I)*c^2*d - 17*c*d^2 + (28*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^2) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((3*I)*c - 13*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) + (3*c^2 + (18*I)*c*d - 55*d^2)/(24*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + (d*(c^4 + (6*I)*c^3*d - 16*c^2*d^2 + (94*I)*c*d^3 + 55*d^4))/(8*a^3*(c - I*d)^2*(c + I*d)^5*f*(c + d*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]], x, 6, ((-8*I)*a^3*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((8*I)*a^3*Sqrt[c + d*Tan[e + f*x]])/f + (4*a^3*(I*c - 6*d)*(c + d*Tan[e + f*x])^(3/2))/(15*d^2*f) - (2*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2))/(5*d*f)} -{(a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]], x, 5, ((-4*I)*a^2*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((4*I)*a^2*Sqrt[c + d*Tan[e + f*x]])/f - (2*a^2*(c + d*Tan[e + f*x])^(3/2))/(3*d*f)} -{(a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]], x, 4, ((-2*I)*a*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((2*I)*a*Sqrt[c + d*Tan[e + f*x]])/f} -{Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x]), x, 8, ((-I/2)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*f) + ((I/2)*c*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(a*Sqrt[c + I*d]*f) + ((I/2)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x]))} -{Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^2, x, 9, ((-I/4)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*f) - ((2*c*d - I*(2*c^2 + d^2))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(3/2)*f) + (((2*I)*c - d)*Sqrt[c + d*Tan[e + f*x]])/(8*a^2*(c + I*d)*f*(1 + I*Tan[e + f*x])) + ((I/4)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^2)} -{Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3, x, 10, ((-I/8)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + (((2*I)*c^3 - 4*c^2*d - I*c*d^2 - 2*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(5/2)*f) + ((I/6)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 2*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*d)*f*(a + I*a*Tan[e + f*x])^2) + (c*((2*I)*c - 3*d)*Sqrt[c + d*Tan[e + f*x]])/(16*(c + I*d)^2*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2), x, 7, ((-8*I)*a^3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (8*a^3*(I*c + d)*Sqrt[c + d*Tan[e + f*x]])/f + (((8*I)/3)*a^3*(c + d*Tan[e + f*x])^(3/2))/f + (4*a^3*(I*c - 8*d)*(c + d*Tan[e + f*x])^(5/2))/(35*d^2*f) - (2*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2))/(7*d*f)} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2), x, 6, ((-4*I)*a^2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (4*a^2*(I*c + d)*Sqrt[c + d*Tan[e + f*x]])/f + (((4*I)/3)*a^2*(c + d*Tan[e + f*x])^(3/2))/f - (2*a^2*(c + d*Tan[e + f*x])^(5/2))/(5*d*f)} -{(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2), x, 5, ((-2*I)*a*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (2*a*(I*c + d)*Sqrt[c + d*Tan[e + f*x]])/f + (((2*I)/3)*a*(c + d*Tan[e + f*x])^(3/2))/f} -{(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x]), x, 8, ((-I/2)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*f) + (Sqrt[c + I*d]*(I*c + 2*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*f) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]])/(2*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^2, x, 9, ((-I/4)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*f) + ((2*c*d + I*(2*c^2 + d^2))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*Sqrt[c + I*d]*f) + (((2*I)*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]])/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^3, x, 10, ((-I/8)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + ((I/16)*c*(2*c^2 + 3*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(a^3*(c + I*d)^(3/2)*f) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]])/(6*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c + 4*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*f*(a + I*a*Tan[e + f*x])^2) - ((2*c^2 - I*c*d + 2*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2), x, 8, ((-8*I)*a^3*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((8*I)*a^3*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/f + (8*a^3*(I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (((8*I)/5)*a^3*(c + d*Tan[e + f*x])^(5/2))/f + (4*a^3*(I*c - 10*d)*(c + d*Tan[e + f*x])^(7/2))/(63*d^2*f) - (2*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^(7/2))/(9*d*f)} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2), x, 7, ((-4*I)*a^2*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((4*I)*a^2*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/f + (4*a^2*(I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (((4*I)/5)*a^2*(c + d*Tan[e + f*x])^(5/2))/f - (2*a^2*(c + d*Tan[e + f*x])^(7/2))/(7*d*f)} -{(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2), x, 6, ((-2*I)*a*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((2*I)*a*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/f + (2*a*(I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (((2*I)/5)*a*(c + d*Tan[e + f*x])^(5/2))/f} -{(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]), x, 9, ((-I/2)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*f) + ((c + I*d)^(3/2)*(I*c + 4*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*f) - ((c + (5*I)*d)*d*Sqrt[c + d*Tan[e + f*x]])/(2*a*f) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(2*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^2, x, 9, ((-I/4)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*f) + (Sqrt[c + I*d]*((2*I)*c^2 + 6*c*d - (7*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*f) + ((c + I*d)*((2*I)*c + 5*d)*Sqrt[c + d*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(4*f*(a + I*a*Tan[e + f*x])^2)} -{(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3, x, 10, ((-I/8)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + (((2*I)*c^3 + 4*c^2*d - I*c*d^2 + 2*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*Sqrt[c + I*d]*f) + ((c + I*d)*(I*c + 2*d)*Sqrt[c + d*Tan[e + f*x]])/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (((2*I)*c^2 + 5*c*d - (4*I)*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*f*(a^3 + I*a^3*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(6*f*(a + I*a*Tan[e + f*x])^3)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]], x, 5, ((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (4*a^3*(I*c - 4*d)*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*f) - (2*(a^3 + I*a^3*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*d*f)} -{(a + I*a*Tan[e + f*x])^2/Sqrt[c + d*Tan[e + f*x]], x, 4, ((-4*I)*a^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - (2*a^2*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(a + I*a*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]], x, 3, ((-2*I)*a*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)} -{1/((a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]), x, 8, ((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*Sqrt[c - I*d]*f) + ((I*c - 2*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(3/2)*f) - Sqrt[c + d*Tan[e + f*x]]/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x]))} -{1/((a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]), x, 9, ((-I/4)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*Sqrt[c - I*d]*f) + (((2*I)*c^2 - 6*c*d - (7*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(5/2)*f) + (((2*I)*c - 5*d)*Sqrt[c + d*Tan[e + f*x]])/(8*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])) - Sqrt[c + d*Tan[e + f*x]]/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2)} -{1/((a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]), x, 10, ((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*Sqrt[c - I*d]*f) + (((2*I)*c^3 - 8*c^2*d - (13*I)*c*d^2 + 12*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(7/2)*f) - Sqrt[c + d*Tan[e + f*x]]/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 8*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2) + ((2*c^2 + (7*I)*c*d - 10*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2), x, 5, ((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) + (2*(c + I*d)*(a^3 + I*a^3*Tan[e + f*x]))/((c - I*d)*d*f*Sqrt[c + d*Tan[e + f*x]]) + (4*a^3*c*Sqrt[c + d*Tan[e + f*x]])/(d^2*(I*c + d)*f)} -{(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2), x, 4, ((-4*I)*a^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) + (2*a^2*(I*c - d))/(d*(I*c + d)*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2), x, 4, ((-2*I)*a*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) - (2*a)/((I*c + d)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)), x, 9, ((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*(c - I*d)^(3/2)*f) + ((I*c - 4*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(5/2)*f) + ((c - (5*I)*d)*d)/(2*a*(c - I*d)*(c + I*d)^2*f*Sqrt[c + d*Tan[e + f*x]]) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)), x, 10, ((-I/4)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(3/2)*f) + (((2*I)*c^2 - 10*c*d - (23*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(7/2)*f) + (d*(2*c^2 + (7*I)*c*d + 25*d^2))/(8*a^2*(c - I*d)*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]]) + ((2*I)*c - 7*d)/(8*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)), x, 11, ((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(3/2)*f) + (((2*I)*c^3 - 12*c^2*d - (33*I)*c*d^2 + 58*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(9/2)*f) + (d*(2*c^3 + (9*I)*c^2*d - 17*c*d^2 + (60*I)*d^3))/(16*a^3*(c - I*d)*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]]) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]) + ((3*I)*c - 10*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]) + (6*c^2 + (27*I)*c*d - 56*d^2)/(48*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2), x, 5, ((-8*I)*a^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) + (2*(c + I*d)*(a^3 + I*a^3*Tan[e + f*x]))/(3*(c - I*d)*d*f*(c + d*Tan[e + f*x])^(3/2)) + (4*a^3*(I*c - d)*(c - (4*I)*d))/(3*(c - I*d)^2*d^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(5/2), x, 5, ((-4*I)*a^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) + (2*a^2*(I*c - d))/(3*d*(I*c + d)*f*(c + d*Tan[e + f*x])^(3/2)) + ((4*I)*a^2)/((c - I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2), x, 5, ((-2*I)*a*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) - (2*a)/(3*(I*c + d)*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*a)/((c - I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)), x, 10, ((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*(c - I*d)^(5/2)*f) + ((I*c - 6*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(7/2)*f) + (d*((3*I)*c + 7*d))/(6*a*(I*c - d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(c^2 - (14*I)*c*d - 5*d^2))/(2*a*(c - I*d)^2*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)), x, 11, ((-I/4)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(5/2)*f) + (((2*I)*c^2 - 14*c*d - (47*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(9/2)*f) + (d*(6*c^2 + (27*I)*c*d + 49*d^2))/(24*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*c - 9*d)/(8*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (d*(2*c^3 + (9*I)*c^2*d + 88*c*d^2 - (45*I)*d^3))/(8*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2)), x, 12, ((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(5/2)*f) + (((2*I)*c^3 - 16*c^2*d - (61*I)*c*d^2 + 152*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(11/2)*f) + (d*(6*c^3 + (33*I)*c^2*d - 83*c*d^2 + (154*I)*d^3))/(48*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^(3/2)) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)) + (I*c - 4*d)/(8*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)) + (2*c^2 + (11*I)*c*d - 30*d^2)/(16*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(2*c^4 + (11*I)*c^3*d - 26*c^2*d^2 + (253*I)*c*d^3 + 150*d^4))/(16*a^3*(c - I*d)^2*(c + I*d)^5*f*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^(m/2) (c+d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]], x, 9, -((-1)^(1/4)*a^(5/2)*(c^2 + (10*I)*c*d + 23*d^2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(4*d^(3/2)*f) - ((4*I)*Sqrt[2]*a^(5/2)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a^2*(c + (9*I)*d)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d*f) - (a^2*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*d*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]], x, 9, -(((-1)^(1/4)*a^(3/2)*(I*c + 3*d)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[d]*f)) - ((2*I)*Sqrt[2]*a^(3/2)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a^2*(c + I*d)*Sqrt[c + d*Tan[e + f*x]])/(d*f*Sqrt[a + I*a*Tan[e + f*x]]) - (a^2*(c + d*Tan[e + f*x])^(3/2))/(d*f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]], x, 7, (-2*(-1)^(1/4)*Sqrt[a]*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f - (I*Sqrt[2]*Sqrt[a]*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f} -{Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + I*a*Tan[e + f*x]], x, 3, ((-I)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*f) + (I*Sqrt[c + d*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(3/2), x, 4, ((-I/2)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((I/2)*Sqrt[c + d*Tan[e + f*x]])/(a*f*Sqrt[a + I*a*Tan[e + f*x]]) - (c + d*Tan[e + f*x])^(3/2)/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2))} -{Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^(5/2), x, 6, ((-I/4)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) + ((I/5)*Sqrt[c + d*Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^(5/2)) + (((5*I)*c - 3*d)*Sqrt[c + d*Tan[e + f*x]])/(30*a*(c + I*d)*f*(a + I*a*Tan[e + f*x])^(3/2)) - ((20*c*d - I*(15*c^2 + 3*d^2))*Sqrt[c + d*Tan[e + f*x]])/(60*a^2*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2), x, 10, -((-1)^(1/4)*a^(5/2)*(c - (3*I)*d)*(c^2 + (18*I)*c*d + 15*d^2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(8*d^(3/2)*f) - ((4*I)*Sqrt[2]*a^(5/2)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a^2*(c^2 + (14*I)*c*d + 19*d^2)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*d*f) + (a^2*(c + (13*I)*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(12*d*f) - (a^2*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*d*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2), x, 10, -((-1)^(1/4)*a^(3/2)*((3*I)*c^2 + 18*c*d - (11*I)*d^2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(4*Sqrt[d]*f) - ((2*I)*Sqrt[2]*a^(3/2)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a*((3*I)*c + 5*d)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*f) + (a^2*(c + I*d)*(c + d*Tan[e + f*x])^(3/2))/(2*d*f*Sqrt[a + I*a*Tan[e + f*x]]) - (a^2*(c + d*Tan[e + f*x])^(5/2))/(2*d*f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2), x, 8, -(((-1)^(1/4)*Sqrt[a]*(3*c - I*d)*Sqrt[d]*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f) - (I*Sqrt[2]*Sqrt[a]*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (d*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f} -{(c + d*Tan[e + f*x])^(3/2)/Sqrt[a + I*a*Tan[e + f*x]], x, 8, (2*(-1)^(3/4)*d^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a]*f) - (I*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*f) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(3/2), x, 4, ((-I/2)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((I*c + d)*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I/3)*(c + d*Tan[e + f*x])^(3/2))/(f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c + d*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(5/2), x, 5, ((-I/4)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) + ((I*c + d)*Sqrt[c + d*Tan[e + f*x]])/(4*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I/6)*(c + d*Tan[e + f*x])^(3/2))/(a*f*(a + I*a*Tan[e + f*x])^(3/2)) - (c + d*Tan[e + f*x])^(5/2)/(5*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(5/2))} - - -{(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2), x, 11, -((-1)^(1/4)*a^(5/2)*(5*c^4 + (100*I)*c^3*d + 690*c^2*d^2 - (900*I)*c*d^3 - 363*d^4)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(64*d^(3/2)*f) - ((4*I)*Sqrt[2]*a^(5/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a^2*(5*c^3 + (95*I)*c^2*d + 273*c*d^2 - (149*I)*d^3)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*d*f) + (a^2*(5*c^2 + (90*I)*c*d + 107*d^2)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*d*f) + (a^2*(c + (17*I)*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*d*f) - (a^2*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2), x, 11, -((-1)^(1/4)*a^(3/2)*((5*I)*c^3 + 45*c^2*d - (55*I)*c*d^2 - 23*d^3)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(8*Sqrt[d]*f) - ((2*I)*Sqrt[2]*a^(3/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + (a*(c - (3*I)*d)*((5*I)*c + 3*d)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*f) + (a*((5*I)*c + 7*d)*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(12*f) + (a^2*(c + I*d)*(c + d*Tan[e + f*x])^(5/2))/(3*d*f*Sqrt[a + I*a*Tan[e + f*x]]) - (a^2*(c + d*Tan[e + f*x])^(7/2))/(3*d*f*Sqrt[a + I*a*Tan[e + f*x]])} -{Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2), x, 9, -((-1)^(1/4)*Sqrt[a]*Sqrt[d]*(15*c^2 - (10*I)*c*d - 7*d^2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(4*f) - (I*Sqrt[2]*Sqrt[a]*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f + ((7*c - I*d)*d*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*f) + (d*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*f)} -{(c + d*Tan[e + f*x])^(5/2)/Sqrt[a + I*a*Tan[e + f*x]], x, 9, ((-1)^(1/4)*(5*I*c - d)*d^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a]*f) - (I*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*f) - ((c + 2*I*d)*d*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(a*f) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2), x, 9, (2*(-1)^(1/4)*d^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(a^(3/2)*f) - ((I/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((c + I*d)*(I*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))} -{(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(5/2), x, 5, ((-I/4)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) + ((I/4)*(c - I*d)^2*Sqrt[c + d*Tan[e + f*x]])/(a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I*c + d)*(c + d*Tan[e + f*x])^(3/2))/(6*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I/5)*(c + d*Tan[e + f*x])^(5/2))/(f*(a + I*a*Tan[e + f*x])^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + I*a*Tan[e + f*x])^(5/2)/Sqrt[c + d*Tan[e + f*x]], x, 8, -(((-1)^(1/4)*a^(5/2)*(c + (5*I)*d)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(d^(3/2)*f)) - ((4*I)*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[c - I*d]*f) - (a^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(a + I*a*Tan[e + f*x])^(3/2)/Sqrt[c + d*Tan[e + f*x]], x, 7, (-2*(-1)^(3/4)*a^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[d]*f) - ((2*I)*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)} -{Sqrt[a + I*a*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x, 2, ((-I)*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)} -{1/(Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x, 4, ((-I)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*Sqrt[c - I*d]*f) - Sqrt[c + d*Tan[e + f*x]]/((I*c + d)*f*Sqrt[a + I*a*Tan[e + f*x]]) + (2*d*Sqrt[c + d*Tan[e + f*x]])/((c^2 + d^2)*f*Sqrt[a + I*a*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]), x, 5, ((-I/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*Sqrt[c - I*d]*f) - Sqrt[c + d*Tan[e + f*x]]/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2)) + (((3*I)*c - 7*d)*Sqrt[c + d*Tan[e + f*x]])/(6*a*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]), x, 6, ((-I/4)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*Sqrt[c - I*d]*f) - Sqrt[c + d*Tan[e + f*x]]/(5*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(5/2)) + (((5*I)*c - 13*d)*Sqrt[c + d*Tan[e + f*x]])/(30*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((15*c^2 + (50*I)*c*d - 67*d^2)*Sqrt[c + d*Tan[e + f*x]])/(60*a^2*(I*c - d)^3*f*Sqrt[a + I*a*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(3/2), x, 8, (2*(-1)^(1/4)*a^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/(d^(3/2)*f) - ((4*I)*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(3/2)*f) + (2*a^2*(c + I*d)*Sqrt[a + I*a*Tan[e + f*x]])/((c - I*d)*d*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(3/2), x, 3, ((-2*I)*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(3/2)*f) - (2*a*Sqrt[a + I*a*Tan[e + f*x]])/((I*c + d)*f*Sqrt[c + d*Tan[e + f*x]])} -{Sqrt[a + I*a*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(3/2), x, 3, ((-I)*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(3/2)*f) - (2*d*Sqrt[a + I*a*Tan[e + f*x]])/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)), x, 5, -((I*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*(c - I*d)^(3/2)*f)) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + ((c - 3*I*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(a*(c - I*d)*(c + I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)), x, 6, ((-I/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*(c - I*d)^(3/2)*f) - 1/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) + ((3*I)*c - 11*d)/(6*a*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + ((3*c - (5*I)*d)*(c + (5*I)*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)), x, 7, ((-I/4)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*(c - I*d)^(3/2)*f) - 1/(5*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]) + ((5*I)*c - 17*d)/(30*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) + (15*c^2 + (70*I)*c*d - 151*d^2)/(60*a^2*(I*c - d)^3*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + (d*(15*c^3 + (65*I)*c^2*d - 117*c*d^2 + (317*I)*d^3)*Sqrt[a + I*a*Tan[e + f*x]])/(60*a^3*(c - I*d)*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(5/2), x, 4, ((-4*I)*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(5/2)*f) - (2*a*(a + I*a*Tan[e + f*x])^(3/2))/(3*(I*c + d)*f*(c + d*Tan[e + f*x])^(3/2)) + ((4*I)*a^2*Sqrt[a + I*a*Tan[e + f*x]])/((c - I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(5/2), x, 4, ((-2*I)*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(5/2)*f) - (2*d*(a + I*a*Tan[e + f*x])^(3/2))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((2*I)*a*Sqrt[a + I*a*Tan[e + f*x]])/((c - I*d)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{Sqrt[a + I*a*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(5/2), x, 5, ((-I)*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/((c - I*d)^(5/2)*f) - (2*d*Sqrt[a + I*a*Tan[e + f*x]])/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(5*c + I*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/(Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)), x, 6, -((I*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*(c - I*d)^(5/2)*f)) - 1/((I*c - d)*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*(3*I*c + 5*d)*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(I*c - d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + ((3*c - I*d)*(c - 7*I*d)*d*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*(c - I*d)^2*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)), x, 7, ((-I/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*(c - I*d)^(5/2)*f) - 1/(3*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) + (I*c - 5*d)/(2*a*(c + I*d)^2*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*(3*c^2 + (14*I)*c*d + 21*d^2)*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)*(c + I*d)^3*f*(c + d*Tan[e + f*x])^(3/2)) + ((c - (3*I)*d)*d*(3*c^2 + (22*I)*c*d + 13*d^2)*Sqrt[a + I*a*Tan[e + f*x]])/(6*a^2*(c - I*d)^2*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)), x, 8, ((-I/4)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(5/2)*(c - I*d)^(5/2)*f) - 1/(5*(I*c - d)*f*(a + I*a*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)) + ((5*I)*c - 21*d)/(30*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) + (5*c^2 + (30*I)*c*d - 89*d^2)/(20*a^2*(I*c - d)^3*f*Sqrt[a + I*a*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (d*(15*c^3 + (85*I)*c^2*d - 221*c*d^2 + (361*I)*d^3)*Sqrt[a + I*a*Tan[e + f*x]])/(60*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])^(3/2)) + (d*(15*c^4 + (80*I)*c^3*d - 182*c^2*d^2 + (1224*I)*c*d^3 + 707*d^4)*Sqrt[a + I*a*Tan[e + f*x]])/(60*a^3*(c - I*d)^2*(c + I*d)^5*f*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^n with n symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x, 3, -((1/(2*f*m))*((I*AppellF1[m, -n, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n)/((c + d*Tan[e + f*x])/(c + I*d))^n))} - - -{(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^n, x, 4, (a^3*(I*c - d*(5 + 2*n))*(c + d*Tan[e + f*x])^(1 + n))/(d^2*f*(1 + n)*(2 + n)) + (4*a^3*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/((I*c + d)*f*(1 + n)) - ((a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^(1 + n))/(d*f*(2 + n))} -{(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^n, x, 3, -((a^2*(c + d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))) + (2*a^2*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/((I*c + d)*f*(1 + n))} -{(a + I*a*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^n, x, 2, (a*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/((I*c + d)*f*(1 + n))} -{(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^1, x, 6, (Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(4*a*(I*c + d)*f*(1 + n)) + ((I*c - d + 2*d*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c + I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(4*a*(c + I*d)^2*f*(1 + n)) - (c + d*Tan[e + f*x])^(1 + n)/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^2, x, 7, (Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(8*a^2*(I*c + d)*f*(1 + n)) + ((c^2 + 2*I*c*d*(1 - n) - d^2*(1 - 4*n + 2*n^2))*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c + I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(8*a^2*(I*c - d)^3*f*(1 + n)) + ((I*c - d*(2 - n))*(c + d*Tan[e + f*x])^(1 + n))/(4*a^2*(c + I*d)^2*f*(1 + I*Tan[e + f*x])) - (c + d*Tan[e + f*x])^(1 + n)/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2)} -{(c + d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^3, x, 8, (Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c - I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(16*a^3*(I*c + d)*f*(1 + n)) + ((3*I*c^3 - c^2*d*(9 - 6*n) - 3*I*c*d^2*(3 - 6*n + 2*n^2) + d^3*(3 - 20*n + 18*n^2 - 4*n^3))*Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Tan[e + f*x])/(c + I*d)]*(c + d*Tan[e + f*x])^(1 + n))/(48*a^3*(c + I*d)^4*f*(1 + n)) - (c + d*Tan[e + f*x])^(1 + n)/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + ((3*I*c - d*(7 - 2*n))*(c + d*Tan[e + f*x])^(1 + n))/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2) + ((3*I*c^2 - 3*c*d*(3 - n) - I*d^2*(10 - 9*n + 2*n^2))*(c + d*Tan[e + f*x])^(1 + n))/(24*(c + I*d)^3*f*(a^3 + I*a^3*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c+d Tan[e+f x])^n with m symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3, x, 5, (-2*d*(d^2 + I*c*d*m - c^2*(3 + m))*(a + I*a*Tan[e + f*x])^m)/(f*m*(2 + m)) + ((I*c + d)^3*Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(2*f*m) - (d^2*(d*m + I*c*(4 + m))*(a + I*a*Tan[e + f*x])^(1 + m))/(a*f*(1 + m)*(2 + m)) + (d*(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2)/(f*(2 + m))} -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2, x, 4, (2*c*d*(a + I*a*Tan[e + f*x])^m)/(f*m) - ((I/2)*(c - I*d)^2*Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(f*m) - (I*d^2*(a + I*a*Tan[e + f*x])^(1 + m))/(a*f*(1 + m))} -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^1, x, 3, (d*(a + I*a*Tan[e + f*x])^m)/(f*m) - ((I*c + d)*Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(2*f*m)} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^1, x, 5, (Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(2*(I*c + d)*f*m) - (d*Hypergeometric2F1[1, m, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d))]*(a + I*a*Tan[e + f*x])^m)/((c^2 + d^2)*f*m)} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^2, x, 6, -((I*Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(2*(c - I*d)^2*f*m)) - (d*(c*(2 - m) + I*d*m)*Hypergeometric2F1[1, m, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d))]*(a + I*a*Tan[e + f*x])^m)/((c^2 + d^2)^2*f*m) - (d*(a + I*a*Tan[e + f*x])^m)/((c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^3, x, 7, -((Hypergeometric2F1[1, m, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m)/(2*(I*c + d)^3*f*m)) - (d*(2*I*c*d*(3 - m)*m + c^2*(6 - 5*m + m^2) - d^2*(2 - m + m^2))*Hypergeometric2F1[1, m, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d))]*(a + I*a*Tan[e + f*x])^m)/(2*(c^2 + d^2)^3*f*m) - (d*(a + I*a*Tan[e + f*x])^m)/(2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (d*(c*(4 - m) + I*d*m)*(a + I*a*Tan[e + f*x])^m)/(2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} - - -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2), x, 3, -(((I*c - d)*AppellF1[m, -(3/2), 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(2*f*m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)]))} -{(a + I*a*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(1/2), x, 3, -((I*AppellF1[m, -(1/2), 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(2*f*m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)]))} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(1/2), x, 3, -((I*AppellF1[m, 1/2, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(2*f*m*Sqrt[c + d*Tan[e + f*x]]))} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2), x, 3, (AppellF1[m, 3/2, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(2*(I*c - d)*f*m*Sqrt[c + d*Tan[e + f*x]])} -{(a + I*a*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2), x, 3, -((I*AppellF1[m, 5/2, 1, 1 + m, -((d*(1 + I*Tan[e + f*x]))/(I*c - d)), (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*Sqrt[(c + d*Tan[e + f*x])/(c + I*d)])/(2*(c + I*d)^2*f*m*Sqrt[c + d*Tan[e + f*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x]), x, 4, (a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d)*x - ((3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*Log[Cos[e + f*x]])/f + (b*(2*a*b*c + a^2*d - b^2*d)*Tan[e + f*x])/f + ((b*c + a*d)*(a + b*Tan[e + f*x])^2)/(2*f) + (d*(a + b*Tan[e + f*x])^3)/(3*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x]), x, 3, (a^2*c - b^2*c - 2*a*b*d)*x - ((2*a*b*c + a^2*d - b^2*d)*Log[Cos[e + f*x]])/f + (b*(b*c + a*d)*Tan[e + f*x])/f + (d*(a + b*Tan[e + f*x])^2)/(2*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]), x, 2, (a*c - b*d)*x - ((b*c + a*d)*Log[Cos[e + f*x]])/f + (b*d*Tan[e + f*x])/f} -{(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x]), x, 2, ((a*c + b*d)*x)/(a^2 + b^2) + ((b*c - a*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*f)} -{(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x])^2, x, 3, ((a^2*c - b^2*c + 2*a*b*d)*x)/(a^2 + b^2)^2 + ((2*a*b*c - a^2*d + b^2*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*f) - (b*c - a*d)/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])/(a + b*Tan[e + f*x])^3, x, 4, ((a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d)*x)/(a^2 + b^2)^3 + ((3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - (b*c - a*d)/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (2*a*b*c - a^2*d + b^2*d)/((a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2, x, 5, -((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x) - ((2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*Log[Cos[e + f*x]])/f + (2*b*(b*c + a*d)*(a*c - b*d)*Tan[e + f*x])/f + ((2*a*c*d + b*(c^2 - d^2))*(a + b*Tan[e + f*x])^2)/(2*f) + (2*c*d*(a + b*Tan[e + f*x])^3)/(3*f) + (d^2*(a + b*Tan[e + f*x])^4)/(4*b*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2, x, 4, (a*c - b*c - a*d - b*d)*(a*c + b*c + a*d - b*d)*x - (2*(b*c + a*d)*(a*c - b*d)*Log[Cos[e + f*x]])/f + (b*(2*a*c*d + b*(c^2 - d^2))*Tan[e + f*x])/f + (c*d*(a + b*Tan[e + f*x])^2)/f + (d^2*(a + b*Tan[e + f*x])^3)/(3*b*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2, x, 3, -((2*b*c*d - a*(c^2 - d^2))*x) - ((2*a*c*d + b*(c^2 - d^2))*Log[Cos[e + f*x]])/f + (d*(b*c + a*d)*Tan[e + f*x])/f + (b*(c + d*Tan[e + f*x])^2)/(2*f)} -{(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x]), x, 4, (a*(b*c - a*d)^2*x)/(b^2*(a^2 + b^2)) + (d*(2*b*c - a*d)*x)/b^2 - (d^2*Log[Cos[e + f*x]])/(b*f) + ((b*c - a*d)^2*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/(b*(a^2 + b^2)*f)} -{(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x])^2, x, 3, -(((b*(c - d) - a*(c + d))*(a*(c - d) + b*(c + d))*x)/(a^2 + b^2)^2) + (2*(b*c - a*d)*(a*c + b*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*f) - (b*c - a*d)^2/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^2/(a + b*Tan[e + f*x])^3, x, 4, ((6*a^2*b*c*d - 2*b^3*c*d + a^3*(c^2 - d^2) - 3*a*b^2*(c^2 - d^2))*x)/(a^2 + b^2)^3 - ((2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*(c^2 - d^2) + b^3*(c^2 - d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - (b*c - a*d)^2/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (2*(b*c - a*d)*(a*c + b*d))/((a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3, x, 6, -((a*c - b*d)*(8*a*b*c*d - a^2*(c^2 - 3*d^2) + b^2*(3*c^2 - d^2))*x) + ((b*c + a*d)*(8*a*b*c*d + b^2*(c^2 - 3*d^2) - a^2*(3*c^2 - d^2))*Log[Cos[e + f*x]])/f + (d*(2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*Tan[e + f*x])/f + ((3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*(c + d*Tan[e + f*x])^2)/(2*f) + (b*(3*a^2 - b^2)*(c + d*Tan[e + f*x])^3)/(3*f) - (b^2*(b*c - 11*a*d)*(c + d*Tan[e + f*x])^4)/(20*d^2*f) + (b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^4)/(5*d*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3, x, 5, -((b^2*c*(c^2 - 3*d^2) + 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x) - ((2*a*b*c*(c^2 - 3*d^2) - b^2*d*(3*c^2 - d^2) + a^2*(3*c^2*d - d^3))*Log[Cos[e + f*x]])/f + (2*d*(b*c + a*d)*(a*c - b*d)*Tan[e + f*x])/f + ((2*a*b*c + a^2*d - b^2*d)*(c + d*Tan[e + f*x])^2)/(2*f) + (2*a*b*(c + d*Tan[e + f*x])^3)/(3*f) + (b^2*(c + d*Tan[e + f*x])^4)/(4*d*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3, x, 4, -((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x) - ((b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3)*Log[Cos[e + f*x]])/f + (d*(2*a*c*d + b*(c^2 - d^2))*Tan[e + f*x])/f + ((b*c + a*d)*(c + d*Tan[e + f*x])^2)/(2*f) + (b*(c + d*Tan[e + f*x])^3)/(3*f)} -{(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x]), x, 5, ((a*c^3 + 3*b*c^2*d - 3*a*c*d^2 - b*d^3)*x)/(a^2 + b^2) + ((b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3)*Log[Cos[e + f*x]])/((a^2 + b^2)*f) + ((b*c - a*d)^3*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)*f) + (d^2*(c + d*Tan[e + f*x]))/(b*f)} -{(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^2, x, 5, -(((b^2*c*(c^2 - 3*d^2) - 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x)/(a^2 + b^2)^2) + ((2*a*b*c*(c^2 - 3*d^2) + b^2*d*(3*c^2 - d^2) - a^2*(3*c^2*d - d^3))*Log[Cos[e + f*x]])/((a^2 + b^2)^2*f) + ((b*c - a*d)^2*(2*a*b*c + a^2*d + 3*b^2*d)*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)^2*f) - ((b*c - a*d)^2*(c + d*Tan[e + f*x]))/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^3, x, 4, ((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(a^2 + b^2)^3 + ((b*c - a*d)*(8*a*b*c*d - b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((b*c - a*d)^2*(4*a*b*c + a^2*d + 5*b^2*d))/(2*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((b*c - a*d)^2*(c + d*Tan[e + f*x]))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x]), x, 6, ((a^4*c - 6*a^2*b^2*c + b^4*c + 4*a^3*b*d - 4*a*b^3*d)*x)/(c^2 + d^2) - ((4*a^3*b*c - 4*a*b^3*c - a^4*d + 6*a^2*b^2*d - b^4*d)*Log[Cos[e + f*x]])/((c^2 + d^2)*f) + ((b*c - a*d)^4*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)*f) - (b^3*(b*c - 3*a*d)*Tan[e + f*x])/(d^2*f) + (b^2*(a + b*Tan[e + f*x])^2)/(2*d*f)} -{(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x]), x, 5, ((a^3*c - 3*a*b^2*c + 3*a^2*b*d - b^3*d)*x)/(c^2 + d^2) - ((3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d)*Log[Cos[e + f*x]])/((c^2 + d^2)*f) - ((b*c - a*d)^3*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f) + (b^2*(a + b*Tan[e + f*x]))/(d*f)} -{(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x]), x, 4, -((b*(b*c - 2*a*d)*x)/d^2) + (c*(b*c - a*d)^2*x)/(d^2*(c^2 + d^2)) - (b^2*Log[Cos[e + f*x]])/(d*f) + ((b*c - a*d)^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(d*(c^2 + d^2)*f)} -{(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x]), x, 2, ((a*c + b*d)*x)/(c^2 + d^2) - ((b*c - a*d)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)*f)} -{1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])), x, 3, ((a*c - b*d)*x)/((a^2 + b^2)*(c^2 + d^2)) + (b^2*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f) - (d^2*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f)} -{1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])), x, 4, ((a^2*c - b^2*c - 2*a*b*d)*x)/((a^2 + b^2)^2*(c^2 + d^2)) + (b^2*(2*a*b*c - 3*a^2*d - b^2*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^2*f) + (d^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)*f) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])), x, 5, ((a^3*c - 3*a*b^2*c - 3*a^2*b*d + b^3*d)*x)/((a^2 + b^2)^3*(c^2 + d^2)) - (b^2*(8*a^3*b*c*d - 6*a^4*d^2 + b^4*(c^2 - d^2) - 3*a^2*b^2*(c^2 + d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*(b*c - a*d)^3*f) - (d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)*f) - b^2/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (b^2*(2*a*b*c - 3*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^2, x, 6, ((8*a^3*b*c*d - 8*a*b^3*c*d + a^4*(c^2 - d^2) - 6*a^2*b^2*(c^2 - d^2) + b^4*(c^2 - d^2))*x)/(c^2 + d^2)^2 - (2*(a^2*c - b^2*c + 2*a*b*d)*(2*a*b*c - a^2*d + b^2*d)*Log[Cos[e + f*x]])/((c^2 + d^2)^2*f) - (2*(b*c - a*d)^3*(a*c*d + b*(c^2 + 2*d^2))*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^2*f) - (b^2*(a*d*(2*b*c - a*d) - b^2*(2*c^2 + d^2))*Tan[e + f*x])/(d^2*(c^2 + d^2)*f) - ((b*c - a*d)^2*(a + b*Tan[e + f*x])^2)/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^2, x, 5, ((6*a^2*b*c*d - 2*b^3*c*d + a^3*(c^2 - d^2) - 3*a*b^2*(c^2 - d^2))*x)/(c^2 + d^2)^2 + ((2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*(c^2 - d^2) + b^3*(c^2 - d^2))*Log[Cos[e + f*x]])/((c^2 + d^2)^2*f) + ((b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)^2*f) - ((b*c - a*d)^2*(a + b*Tan[e + f*x]))/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^2, x, 3, -(((b*(c - d) - a*(c + d))*(a*(c - d) + b*(c + d))*x)/(c^2 + d^2)^2) - (2*(b*c - a*d)*(a*c + b*d)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^2*f) - (b*c - a*d)^2/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^2, x, 3, ((2*b*c*d + a*(c^2 - d^2))*x)/(c^2 + d^2)^2 + ((2*a*c*d - b*(c^2 - d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^2*f) + (b*c - a*d)/((c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2), x, 4, -(((2*b*c*d - a*(c^2 - d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^2)) + (b^3*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^2*f) + (d^2*(2*a*c*d - b*(3*c^2 + d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)^2*f) + d^2/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2), x, 5, ((b*(c - d) + a*(c + d))*(a*(c - d) - b*(c + d))*x)/((a^2 + b^2)^2*(c^2 + d^2)^2) + (2*b^3*(a*b*c - 2*a^2*d - b^2*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^3*f) - (2*d^3*(a*c*d - b*(2*c^2 + d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^2*f) - (d*(a^2*d^2 + b^2*(c^2 + 2*d^2)))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2), x, 6, -(((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x)/((a^2 + b^2)^3*(c^2 + d^2)^2)) - (b^3*(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 + b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*(b*c - a*d)^4*f) - (d^4*(5*b*c^2 - 2*a*c*d + 3*b*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^4*(c^2 + d^2)^2*f) + (d*(a^4*d^3 - 2*a*b^3*c*(c^2 + d^2) + 2*a^2*b^2*d*(2*c^2 + 3*d^2) + b^4*d*(2*c^2 + 3*d^2)))/((a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - b^2/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (b^2*(4*a*b*c - 7*a^2*d - 3*b^2*d))/(2*(a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^3, x, 6, -(((6*a^2*b^2*c*(c^2 - 3*d^2) - b^4*c*(c^2 - 3*d^2) - 4*a^3*b*d*(3*c^2 - d^2) + 4*a*b^3*d*(3*c^2 - d^2) - a^4*(c^3 - 3*c*d^2))*x)/(c^2 + d^2)^3) - ((4*a^3*b*c*(c^2 - 3*d^2) - 4*a*b^3*c*(c^2 - 3*d^2) + 6*a^2*b^2*d*(3*c^2 - d^2) - b^4*d*(3*c^2 - d^2) - a^4*(3*c^2*d - d^3))*Log[Cos[e + f*x]])/((c^2 + d^2)^3*f) + ((b*c - a*d)^2*(a^2*d^2*(3*c^2 - d^2) + 2*a*b*c*d*(c^2 + 5*d^2) + b^2*(c^4 + 3*c^2*d^2 + 6*d^4))*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^3*f) - ((b*c - a*d)^2*(a + b*Tan[e + f*x])^2)/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + ((b*c - a*d)^3*(2*a*c*d + b*(c^2 + 3*d^2)))/(d^3*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^3, x, 4, ((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(c^2 + d^2)^3 - ((b*c - a*d)*(8*a*b*c*d - b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^3*f) - ((b*c - a*d)^2*(a + b*Tan[e + f*x]))/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - ((b*c - a*d)^2*(4*a*c*d + b*(c^2 + 5*d^2)))/(2*d^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^3, x, 4, -(((b^2*c*(c^2 - 3*d^2) - 2*a*b*d*(3*c^2 - d^2) - a^2*(c^3 - 3*c*d^2))*x)/(c^2 + d^2)^3) - ((2*a*b*c*(c^2 - 3*d^2) + b^2*d*(3*c^2 - d^2) - a^2*(3*c^2*d - d^3))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^3*f) - (b*c - a*d)^2/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (2*(b*c - a*d)*(a*c + b*d))/((c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^3, x, 4, ((a*c^3 + 3*b*c^2*d - 3*a*c*d^2 - b*d^3)*x)/(c^2 + d^2)^3 + ((a*d*(3*c^2 - d^2) - b*(c^3 - 3*c*d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^3*f) + (b*c - a*d)/(2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (2*a*c*d - b*(c^2 - d^2))/((c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3), x, 5, -(((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^4*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^3*f) + (d^2*(8*a*b*c^3*d - a^2*d^2*(3*c^2 - d^2) - b^2*(6*c^4 + 3*c^2*d^2 + d^4))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + d^2/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3), x, 6, -(((b^2*c*(c^2 - 3*d^2) - a^2*(c^3 - 3*c*d^2) + a*b*(6*c^2*d - 2*d^3))*x)/((a^2 + b^2)^2*(c^2 + d^2)^3)) + (b^4*(2*a*b*c - 5*a^2*d - 3*b^2*d)*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^4*f) + (d^3*(a^2*d^2*(3*c^2 - d^2) - 2*a*b*c*d*(5*c^2 + d^2) + b^2*(10*c^4 + 9*c^2*d^2 + 3*d^4))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^4*(c^2 + d^2)^3*f) - (d*(a^2*d^2 + b^2*(2*c^2 + 3*d^2)))/(2*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + (d*(2*a^3*c*d^3 + 2*a*b^2*c*d^3 - 2*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 6*c^2*d^2 + 3*d^4)))/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]], x, 10, ((I*a + b)^3*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f - ((I*a - b)^3*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*b*(3*a^2 - b^2)*Sqrt[c + d*Tan[e + f*x]])/f - (4*b^2*(b*c - 6*a*d)*(c + d*Tan[e + f*x])^(3/2))/(15*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2))/(5*d*f)} -{(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]], x, 9, ((-I)*(a - I*b)^2*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (4*a*b*Sqrt[c + d*Tan[e + f*x]])/f + (2*b^2*(c + d*Tan[e + f*x])^(3/2))/(3*d*f)} -{(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]], x, 8, -(((I*a + b)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*b*Sqrt[c + d*Tan[e + f*x]])/f} -{Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x]), x, 11, (Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*f) - (Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*Sqrt[b]*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*f)} -{Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^2, x, 12, ((-I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*Sqrt[b*c - a*d]*f) - (b*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^3, x, 13, -((Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + (Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b]*(40*a^3*b*c*d - 24*a*b^3*c*d - 15*a^4*d^2 - 6*a^2*b^2*(4*c^2 - 3*d^2) + b^4*(8*c^2 + d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - (b*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (b*(8*a*b*c - 7*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2), x, 11, ((I*a + b)^3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f - ((I*a - b)^3*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*b*(3*a^2 - b^2)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - (4*b^2*(b*c - 8*a*d)*(c + d*Tan[e + f*x])^(5/2))/(35*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2))/(7*d*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2), x, 10, ((-I)*(a - I*b)^2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*b*c + a^2*d - b^2*d)*Sqrt[c + d*Tan[e + f*x]])/f + (4*a*b*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*b^2*(c + d*Tan[e + f*x])^(5/2))/(5*d*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2), x, 9, -(((I*a + b)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(b*c + a*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*b*(c + d*Tan[e + f*x])^(3/2))/(3*f)} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x]), x, 11, ((c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*f) - ((c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*f)} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^2, x, 12, ((-I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b*c - a*d]*(4*a*b*c - a^2*d + 3*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)^2*f) - ((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^3, x, 13, -(((c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((24*a^3*b*c*d - 40*a*b^3*c*d - 3*a^4*d^2 - 2*a^2*b^2*(12*c^2 - 13*d^2) + b^4*(8*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*Sqrt[b]*(a^2 + b^2)^3*Sqrt[b*c - a*d]*f) - ((b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - ((8*a*b*c - 3*a^2*d + 5*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2), x, 12, ((I*a + b)^3*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f - ((I*a - b)^3*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a^3*c*d - 6*a*b^2*c*d + 3*a^2*b*(c^2 - d^2) - b^3*(c^2 - d^2))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(3*a^2*b*c - b^3*c + a^3*d - 3*a*b^2*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*b*(3*a^2 - b^2)*(c + d*Tan[e + f*x])^(5/2))/(5*f) - (4*b^2*(b*c - 10*a*d)*(c + d*Tan[e + f*x])^(7/2))/(63*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(7/2))/(9*d*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2), x, 11, ((-I)*(a - I*b)^2*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (4*(b*c + a*d)*(a*c - b*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*(2*a*b*c + a^2*d - b^2*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (4*a*b*(c + d*Tan[e + f*x])^(5/2))/(5*f) + (2*b^2*(c + d*Tan[e + f*x])^(7/2))/(7*d*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2), x, 10, -(((I*a + b)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*c*d + b*(c^2 - d^2))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(b*c + a*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*b*(c + d*Tan[e + f*x])^(5/2))/(5*f)} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x]), x, 12, ((c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*f) - ((c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)*f) + (2*d^2*Sqrt[c + d*Tan[e + f*x]])/(b*f)} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^2, x, 12, ((-I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - ((b*c - a*d)^(3/2)*(4*a*b*c + a^2*d + 5*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)^2*f) - ((b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3, x, 13, -(((c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b*c - a*d]*(8*a^3*b*c*d - 56*a*b^3*c*d + a^4*d^2 + b^4*(8*c^2 - 15*d^2) - 6*a^2*b^2*(4*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(a^2 + b^2)^3*f) - ((b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - ((b*c - a*d)*(8*a*b*c + a^2*d + 9*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*b*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + b*Tan[e + f*x])^4/Sqrt[c + d*Tan[e + f*x]], x, 10, -((I*(a - I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) + (I*(a + I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) - (2*b^2*(40*a*b*c*d - 87*a^2*d^2 - b^2*(8*c^2 - 15*d^2))*Sqrt[c + d*Tan[e + f*x]])/(15*d^3*f) - (4*b^3*(2*b*c - 7*a*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(15*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d*f)} -{(a + b*Tan[e + f*x])^3/Sqrt[c + d*Tan[e + f*x]], x, 9, ((I*a + b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - ((I*a - b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) - (4*b^2*(b*c - 4*a*d)*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/(3*d*f)} -{(a + b*Tan[e + f*x])^2/Sqrt[c + d*Tan[e + f*x]], x, 8, ((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + (I*(a + I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*b^2*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(a + b*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]], x, 7, -(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) + ((I*a - b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f)} -{1/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]), x, 11, ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/((I*a + b)*Sqrt[c - I*d]*f) - ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]/((I*a - b)*Sqrt[c + I*d]*f) - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]*f)} -{1/((a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]), x, 12, ((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*Sqrt[c - I*d]*f) + (I*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*Sqrt[c + I*d]*f) - (b^(3/2)*(4*a*b*c - 5*a^2*d - b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(3/2)*f) - (b^2*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(3/2), x, 10, -((I*(a - I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + (I*(a + I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^2)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (2*b*(15*a^2*b*c*d^2 - 6*a^3*d^3 - 12*a*b^2*d*(2*c^2 + d^2) + b^3*(8*c^3 + 5*c*d^2))*Sqrt[c + d*Tan[e + f*x]])/(3*d^3*(c^2 + d^2)*f) - (2*b^2*(3*a*d*(2*b*c - a*d) - b^2*(4*c^2 + d^2))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*(c^2 + d^2)*f)} -{(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2), x, 9, ((I*a + b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) - ((I*a - b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x]))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (2*b*(a*d*(2*b*c - a*d) - b^2*(2*c^2 + d^2))*Sqrt[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f)} -{(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2), x, 8, ((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f) + (I*(a + I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(b*c - a*d)^2)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2), x, 8, -(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + ((I*a - b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) + (2*(b*c - a*d))/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)), x, 12, ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/((I*a + b)*(c - I*d)^(3/2)*f) - ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]/((I*a - b)*(c + I*d)^(3/2)*f) - (2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*(b*c - a*d)^(3/2)*f) + (2*d^2)/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)), x, 13, ((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*(c - I*d)^(3/2)*f) + (I*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*(c + I*d)^(3/2)*f) - (b^(5/2)*(4*a*b*c - 7*a^2*d - 3*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(5/2)*f) - (d*(2*a^2*d^2 + b^2*(c^2 + 3*d^2)))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(5/2), x, 10, -((I*(a - I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) + (I*(a + I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (4*(b*c - a*d)^3*(2*b*c^2 + 3*a*c*d + 5*b*d^2))/(3*d^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) - (2*b^2*(a*d*(2*b*c - a*d) - b^2*(4*c^2 + 3*d^2))*Sqrt[c + d*Tan[e + f*x]])/(3*d^3*(c^2 + d^2)*f)} -{(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(5/2), x, 9, ((I*a + b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) - ((I*a - b)^3*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x]))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (4*(b*c - a*d)^2*(3*a*c*d + b*(c^2 + 4*d^2)))/(3*d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(5/2), x, 9, ((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) + (I*(a + I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (4*(b*c - a*d)*(a*c + b*d))/((c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2), x, 9, -(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) + ((I*a - b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) + (2*(b*c - a*d))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(2*a*c*d - b*(c^2 - d^2)))/((c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)), x, 13, ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/((I*a + b)*(c - I*d)^(5/2)*f) - ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]/((I*a - b)*(c + I*d)^(5/2)*f) - (2*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*(b*c - a*d)^(5/2)*f) + (2*d^2)/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)), x, 14, ((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*(c - I*d)^(5/2)*f) + (I*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*(c + I*d)^(5/2)*f) - (b^(7/2)*(4*a*b*c - 9*a^2*d - 5*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(7/2)*f) - (d*(2*a^2*d^2 + b^2*(3*c^2 + 5*d^2)))/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) + (d*(4*a^3*c*d^3 + 4*a*b^2*c*d^3 - 4*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 10*c^2*d^2 + 5*d^4)))/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (c+d Tan[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]], x, 14, -((I*(a - I*b)^(5/2)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*(a + I*b)^(5/2)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + (Sqrt[b]*(10*a*b*c*d + 15*a^2*d^2 - b^2*(c^2 + 8*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*d^(3/2)*f) - (b*(b*c - 9*a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d*f) + (b^2*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*d*f)} -{(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]], x, 13, -((I*(a - I*b)^(3/2)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*(a + I*b)^(3/2)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + (Sqrt[b]*(b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[d]*f) + (b*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f} -{Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]], x, 11, -((I*Sqrt[a - I*b]*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*Sqrt[a + I*b]*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + (2*Sqrt[b]*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/f} -{Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]], x, 7, -((I*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) + (I*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f)} -{Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^(3/2), x, 8, ((-I)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f) + (I*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) - (2*b*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^(5/2), x, 9, ((-I)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f) + (I*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) - (2*b*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*b*(6*a*b*c - 5*a^2*d + b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2), x, 14, -((I*(a - I*b)^(3/2)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*(a + I*b)^(3/2)*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((18*a*b*c*d + 3*a^2*d^2 + b^2*(3*c^2 - 8*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*Sqrt[b]*Sqrt[d]*f) + ((3*b*c + 5*a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*f) + (b*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*f)} -{Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2), x, 13, -((I*Sqrt[a - I*b]*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*Sqrt[a + I*b]*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + (Sqrt[d]*(3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*f) + (d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f} -{(c + d*Tan[e + f*x])^(3/2)/Sqrt[a + b*Tan[e + f*x]], x, 12, -((I*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) + (I*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f) + (2*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*f)} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(3/2), x, 8, ((-I)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f) + (I*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) - (2*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(5/2), x, 9, ((-I)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f) + (I*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) - (2*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) - (4*(3*a*b*c - a^2*d + 2*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)^2*f*Sqrt[a + b*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(7/2), x, 10, -((I*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) + (I*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2)) - (4*(5*a*b*c - 2*a^2*d + 3*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(15*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) + (2*(50*a^3*b*c*d - 70*a*b^3*c*d - 8*a^4*d^2 - a^2*b^2*(45*c^2 - 49*d^2) + 3*b^4*(5*c^2 - d^2))*Sqrt[c + d*Tan[e + f*x]])/(15*(a^2 + b^2)^3*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2), x, 15, -((I*(a - I*b)^(3/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*(a + I*b)^(3/2)*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((15*a^2*b*c*d^2 - a^3*d^3 + 3*a*b^2*d*(15*c^2 - 8*d^2) + 5*b^3*(c^3 - 8*c*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*b^(3/2)*Sqrt[d]*f) + ((14*a*b*c*d - a^2*d^2 + b^2*(11*c^2 - 8*d^2))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*b*f) + (d*(13*b*c - a*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(12*b*f) + (d^2*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*b*f)} -{Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2), x, 14, -((I*Sqrt[a - I*b]*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (I*Sqrt[a + I*b]*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + (Sqrt[d]*(10*a*b*c*d - a^2*d^2 + b^2*(15*c^2 - 8*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*b^(3/2)*f) + (d*(9*b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + (d^2*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*b*f)} -{(c + d*Tan[e + f*x])^(5/2)/Sqrt[a + b*Tan[e + f*x]], x, 13, -((I*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) + (I*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f) + (d^(3/2)*(5*b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(3/2)*f) + (d^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f)} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(3/2), x, 13, -((I*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f)) + (I*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) + (2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(3/2)*f) - (2*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(5/2), x, 9, ((-I)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f) + (I*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) - (2*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(b*c - a*d)*(6*a*b*c + a^2*d + 7*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)^2*f*Sqrt[a + b*Tan[e + f*x]])} -{(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(7/2), x, 10, -((I*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) + (I*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*(b*c - a*d)*(10*a*b*c + a^2*d + 11*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) + (2*(20*a^3*b*c*d - 100*a*b^3*c*d + 2*a^4*d^2 + b^4*(15*c^2 - 23*d^2) - 3*a^2*b^2*(15*c^2 - 13*d^2))*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^3*f*Sqrt[a + b*Tan[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{(a + b*Tan[e + f*x])^(5/2)/Sqrt[c + d*Tan[e + f*x]], x, 13, -((I*(a - I*b)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) + (I*(a + I*b)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f) - (b^(3/2)*(b*c - 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(3/2)*f) + (b^2*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(a + b*Tan[e + f*x])^(3/2)/Sqrt[c + d*Tan[e + f*x]], x, 12, -((I*(a - I*b)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) + (I*(a + I*b)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f) + (2*b^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[d]*f)} -{Sqrt[a + b*Tan[e + f*x]]/Sqrt[c + d*Tan[e + f*x]], x, 7, -((I*Sqrt[a - I*b]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) + (I*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f)} -{1/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x, 7, -((I*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*Sqrt[c - I*d]*f)) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]*f)} -{1/((a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]), x, 8, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*Sqrt[c - I*d]*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*Sqrt[c + I*d]*f) - (2*b^2*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]), x, 9, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*Sqrt[c - I*d]*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*Sqrt[c + I*d]*f) - (2*b^2*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (4*b^2*(3*a*b*c - 4*a^2*d - b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(3/2), x, 14, -((I*(a - I*b)^(7/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) + (I*(a + I*b)^(7/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) - (b^(5/2)*(3*b*c - 7*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(3/2))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (b*(2*a*d*(2*b*c - a*d) - b^2*(3*c^2 + d^2))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f)} -{(a + b*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(3/2), x, 13, -((I*(a - I*b)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) + (I*(a + I*b)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (2*b^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(3/2)*f) - (2*(b*c - a*d)^2*Sqrt[a + b*Tan[e + f*x]])/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(3/2), x, 8, ((-I)*(a - I*b)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f) + (I*(a + I*b)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]])/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(3/2), x, 8, ((-I)*Sqrt[a - I*b]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f) + (I*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) - (2*d*Sqrt[a + b*Tan[e + f*x]])/((c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)), x, 8, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*(c - I*d)^(3/2)*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*(c + I*d)^(3/2)*f) + (2*d^2*Sqrt[a + b*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)), x, 9, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*(c - I*d)^(3/2)*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*(c + I*d)^(3/2)*f) - (2*b^2)/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*d*(a^2*d^2 + b^2*(c^2 + 2*d^2))*Sqrt[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)), x, 10, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*(c - I*d)^(3/2)*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*(c + I*d)^(3/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) - (4*b^2*(3*a*b*c - 5*a^2*d - 2*b^2*d))/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) + (2*d*(3*a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(5*c^2 + 8*d^2) + a^2*b^2*d*(11*c^2 + 17*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^(9/2)/(c + d*Tan[e + f*x])^(5/2), x, 15, -((I*(a - I*b)^(9/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) + (I*(a + I*b)^(9/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) - (b^(7/2)*(5*b*c - 9*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(7/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(5/2))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*c - a*d)^2*(5*b*c^2 + 6*a*c*d + 11*b*d^2)*(a + b*Tan[e + f*x])^(3/2))/(3*d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (b*(4*a^3*c*d^3 - 4*a^2*b*d^2*(c^2 - 2*d^2) - 4*a*b^2*c*d*(c^2 + 4*d^2) + b^3*(5*c^4 + 10*c^2*d^2 + d^4))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^2*f)} -{(a + b*Tan[e + f*x])^(7/2)/(c + d*Tan[e + f*x])^(5/2), x, 14, -((I*(a - I*b)^(7/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) + (I*(a + I*b)^(7/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) + (2*b^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(3/2))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*c - a*d)^2*(2*a*c*d + b*(c^2 + 3*d^2))*Sqrt[a + b*Tan[e + f*x]])/(d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^(5/2)/(c + d*Tan[e + f*x])^(5/2), x, 9, ((-I)*(a - I*b)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f) + (I*(a + I*b)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2*Sqrt[a + b*Tan[e + f*x]])/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*c - a*d)*(6*a*c*d + b*(c^2 + 7*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*d*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^(3/2)/(c + d*Tan[e + f*x])^(5/2), x, 9, -((I*(a - I*b)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) + (I*(a + I*b)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) + (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]])/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (4*(b*c^2 - 3*a*c*d - 2*b*d^2)*Sqrt[a + b*Tan[e + f*x]])/(3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{Sqrt[a + b*Tan[e + f*x]]/(c + d*Tan[e + f*x])^(5/2), x, 9, ((-I)*Sqrt[a - I*b]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f) + (I*Sqrt[a + I*b]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) - (2*d*Sqrt[a + b*Tan[e + f*x]])/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*d*(6*a*c*d - b*(5*c^2 - d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(b*c - a*d)*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)), x, 9, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*(c - I*d)^(5/2)*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*(c + I*d)^(5/2)*f) + (2*d^2*Sqrt[a + b*Tan[e + f*x]])/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (4*d^2*(3*a*c*d - b*(4*c^2 + d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(b*c - a*d)^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)), x, 10, -((I*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*(c - I*d)^(5/2)*f)) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*(c + I*d)^(5/2)*f) - (2*b^2)/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) - (2*d*(a^2*d^2 + b^2*(3*c^2 + 4*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(6*a^3*c*d^4 + 6*a*b^2*c*d^4 - a^2*b*d^3*(11*c^2 + 5*d^2) - b^3*(3*c^4*d + 17*c^2*d^3 + 8*d^5))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(5/2)), x, 11, ((-I)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*(c - I*d)^(5/2)*f) + (I*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*(c + I*d)^(5/2)*f) - (2*b^2)/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)) - (4*b^2*(a*b*c - 2*a^2*d - b^2*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) + (2*d*(a^4*d^3 - 6*a*b^3*c*(c^2 + d^2) + b^4*d*(7*c^2 + 8*d^2) + a^2*b^2*d*(13*c^2 + 15*d^2))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (4*d*(3*a^5*c*d^4 + 6*a^3*b^2*c*d^4 - a^4*b*d^3*(7*c^2 + 4*d^2) + 3*a*b^4*c*(c^4 + 2*c^2*d^2 + 2*d^4) - b^5*d*(4*c^4 + 15*c^2*d^2 + 8*d^4) - a^2*b^3*d*(7*c^4 + 28*c^2*d^2 + 15*d^4))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^4*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n with m and/or n symbolic*) - - -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x, 7, (AppellF1[1 + m, -n, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^n)/(((b*(c + d*Tan[e + f*x]))/(b*c - a*d))^n*(2*(I*a + b)*f*(1 + m))) - (AppellF1[1 + m, -n, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^n)/(((b*(c + d*Tan[e + f*x]))/(b*c - a*d))^n*(2*(I*a - b)*f*(1 + m)))} - - -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3, x, 7, (d^2*(3*b*c - a*d)*(a + b*Tan[e + f*x])^(1 + m))/(b^2*f*(1 + m)) + ((I*c + d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a - I*b)*f*(1 + m)) - ((I*c - d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m)) + (d^3*(a + b*Tan[e + f*x])^(2 + m))/(b^2*f*(2 + m)), -((d^2*(a*d - b*c*(5 + 2*m))*(a + b*Tan[e + f*x])^(1 + m))/(b^2*f*(1 + m)*(2 + m))) + ((c - I*d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) - ((I*c - d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m)) + (d^2*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x]))/(b*f*(2 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2, x, 6, (d^2*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) + ((c - I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) - ((c + I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*f*(1 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^1, x, 5, ((c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) + ((I*c - d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^0, x, 5, (b*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - Sqrt[-b^2])]*(a + b*Tan[e + f*x])^(1 + m))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*f*(1 + m)) - (b*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + Sqrt[-b^2])]*(a + b*Tan[e + f*x])^(1 + m))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*f*(1 + m))} -{(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^1, x, 8, If[$VersionNumber>=8, (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)*f*(1 + m)) - (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*(I*c - d)*f*(1 + m)) + (d^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m)), (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)*f*(1 + m)) - (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*(c + I*d)*f*(1 + m)) + (d^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m))]} -{(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^2, x, 9, (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)^2*f*(1 + m)) - (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*(c + I*d)^2*f*(1 + m)) - (d^2*(2*a*c*d - b*(c^2*(2 - m) - d^2*m))*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(1 + m)) + (d^2*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^3, x, 10, (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)^3*f*(1 + m)) + (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*(I*c - d)^3*f*(1 + m)) + (d^2*(2*a^2*d^2*(3*c^2 - d^2) - 4*a*b*c*d*(c^2*(3 - m) - d^2*(1 + m)) - b^2*(d^4*(1 - m)*m + 2*c^2*d^2*(1 + 3*m - m^2) - c^4*(6 - 5*m + m^2)))*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)^3*(c^2 + d^2)^3*f*(1 + m)) + (d^2*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (d^2*(4*a*c*d - b*(d^2*(1 - m) + c^2*(5 - m)))*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2), x, 7, ((b*c - a*d)*AppellF1[1 + m, -(3/2), 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(2*b*(I*a + b)*f*(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]) - ((b*c - a*d)*AppellF1[1 + m, -(3/2), 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(2*(I*a - b)*b*f*(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])} -{(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]], x, 7, (AppellF1[1 + m, -(1/2), 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(2*(I*a + b)*f*(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]) - (AppellF1[1 + m, -(1/2), 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(2*(I*a - b)*f*(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])} -{(a + b*Tan[e + f*x])^m/Sqrt[c + d*Tan[e + f*x]], x, 7, (AppellF1[1 + m, 1/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a + b)*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]]) - (AppellF1[1 + m, 1/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a - b)*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2), x, 7, (b*AppellF1[1 + m, 3/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a + b)*(b*c - a*d)*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]]) - (b*AppellF1[1 + m, 3/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a - b)*(b*c - a*d)*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]])} -{(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(5/2), x, 7, (b^2*AppellF1[1 + m, 5/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a + b)*(b*c - a*d)^2*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]]) - (b^2*AppellF1[1 + m, 5/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a - b)*(b*c - a*d)^2*f*(1 + m)*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c (d Tan[e+f x])^p)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c (d Tan[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Tan[e+f x])^m (c (d Tan[e+f x])^p)^n with n and p symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(c*(d*Tan[e + f*x])^p)^n, x, 4, (AppellF1[1 + n*p, 1 - m, 1, 2 + n*p, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^m)/((1 + I*Tan[e + f*x])^m*(f*(1 + n*p)))} - - -{(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^3, x, 8, -((3*a^3*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p))) + (4*a^3*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) - (I*a^3*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^2, x, 5, -((a^2*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p))) + (2*a^2*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n*(a + I*a*Tan[e + f*x])^1, x, 4, (a*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x])^1, x, 8, (Hypergeometric2F1[2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(a*f*(1 + n*p)) - (I*Hypergeometric2F1[2, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(a*f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n/(a + I*a*Tan[e + f*x])^2, x, 8, ((1 - 4*n*p + 2*n^2*p^2)*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, (-I)*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(8*a^2*f*(1 + n*p)) + (Hypergeometric2F1[1, 1 + n*p, 2 + n*p, I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(8*a^2*f*(1 + n*p)) + (Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(4*a^2*f*(1 + I*Tan[e + f*x])^2) + ((2 - n*p)*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(4*a^2*f*(1 + I*Tan[e + f*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c (d Tan[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c (d Tan[e+f x])^p)^n with n and p symbolic*) - - -{(a + b*Tan[e + f*x])^m*(c*(d*Tan[e + f*x])^p)^n, x, 8, (AppellF1[1 + n*p, -m, 1, 2 + n*p, -((b*Tan[e + f*x])/a), (-I)*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*f*(1 + n*p))) + (AppellF1[1 + n*p, -m, 1, 2 + n*p, -((b*Tan[e + f*x])/a), I*Tan[e + f*x]]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^m)/((1 + (b*Tan[e + f*x])/a)^m*(2*f*(1 + n*p)))} - - -{(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^3, x, 7, (3*a*b^2*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) + (a*(a^2 - 3*b^2)*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) + (b^3*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(f*(2 + n*p)) + (b*(3*a^2 - b^2)*Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^2, x, 7, (b^2*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) + ((a^2 - b^2)*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) + (2*a*b*Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n*(a + b*Tan[e + f*x])^1, x, 5, (a*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p)) + (b*Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/(f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n/(a + b*Tan[e + f*x])^1, x, 8, (a*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/((a^2 + b^2)*f*(1 + n*p)) + (b^2*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, -((b*Tan[e + f*x])/a)]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(a*(a^2 + b^2)*f*(1 + n*p)) - (b*Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/((a^2 + b^2)*f*(2 + n*p))} -{(c*(d*Tan[e + f*x])^p)^n/(a + b*Tan[e + f*x])^2, x, 9, ((a^2 - b^2)*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/((a^2 + b^2)^2*f*(1 + n*p)) + (2*b^2*Hypergeometric2F1[1, 1 + n*p, 2 + n*p, -((b*Tan[e + f*x])/a)]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/((a^2 + b^2)^2*f*(1 + n*p)) + (b^2*Hypergeometric2F1[2, 1 + n*p, 2 + n*p, -((b*Tan[e + f*x])/a)]*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(a^2*(a^2 + b^2)*f*(1 + n*p)) - (2*a*b*Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(c*(d*Tan[e + f*x])^p)^n)/((a^2 + b^2)^2*f*(2 + n*p))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m deleted file mode 100644 index ce33e64..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.3.1 (a+b tan)^m (c+d tan)^n (A+B tan).m +++ /dev/null @@ -1,1420 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+a I Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, -(a*(A - I*B)*x) + (a*(I*A + B)*Log[Cos[c + d*x]])/d + (a*(A - I*B)*Tan[c + d*x])/d + (a*(I*A + B)*Tan[c + d*x]^2)/(2*d) + ((I/3)*a*B*Tan[c + d*x]^3)/d} -{Tan[c + d*x]^1*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 3, -(a*(I*A + B)*x) - (a*(A - I*B)*Log[Cos[c + d*x]])/d + (a*(I*A + B)*Tan[c + d*x])/d + ((I/2)*a*B*Tan[c + d*x]^2)/d} -{Tan[c + d*x]^0*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 2, a*(A - I*B)*x - (a*(I*A + B)*Log[Cos[c + d*x]])/d + I*a*B*Tan[c + d*x]/d} -{Cot[c + d*x]^1*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, a*(I*A + B)*x - (I*a*B*Log[Cos[c + d*x]])/d + (a*A*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 3, -(a*(A - I*B)*x) - (a*A*Cot[c + d*x])/d + (a*(I*A + B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, -(a*(I*A + B)*x) - (a*(I*A + B)*Cot[c + d*x])/d - (a*A*Cot[c + d*x]^2)/(2*d) - (a*(A - I*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 5, a*(A - I*B)*x + (a*(A - I*B)*Cot[c + d*x])/d - (a*(I*A + B)*Cot[c + d*x]^2)/(2*d) - (a*A*Cot[c + d*x]^3)/(3*d) - (a*(I*A + B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 6, a*(I*A + B)*x + (a*(I*A + B)*Cot[c + d*x])/d + (a*(A - I*B)*Cot[c + d*x]^2)/(2*d) - (a*(I*A + B)*Cot[c + d*x]^3)/(3*d) - (a*A*Cot[c + d*x]^4)/(4*d) + (a*(A - I*B)*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, -2*a^2*(A - I*B)*x + (2*a^2*(I*A + B)*Log[Cos[c + d*x]])/d + (2*a^2*(A - I*B)*Tan[c + d*x])/d + (a^2*(I*A + B)*Tan[c + d*x]^2)/d - (a^2*(4*A - (5*I)*B)*Tan[c + d*x]^3)/(12*d) + ((I/4)*B*Tan[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x]))/d} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, -2*a^2*(I*A + B)*x - (2*a^2*(A - I*B)*Log[Cos[c + d*x]])/d + (a^2*(I*A + B)*Tan[c + d*x])/d + (A*(a + I*a*Tan[c + d*x])^2)/(2*d) - ((I/3)*B*(a + I*a*Tan[c + d*x])^3)/(a*d)} -{(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 3, 2*a^2*(A - I*B)*x - (2*a^2*(I*A + B)*Log[Cos[c + d*x]])/d - (a^2*(A - I*B)*Tan[c + d*x])/d + (B*(a + I*a*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, 2*a^2*(I*A + B)*x + (a^2*(A - (2*I)*B)*Log[Cos[c + d*x]])/d + (a^2*A*Log[Sin[c + d*x]])/d + (I*B*(a^2 + I*a^2*Tan[c + d*x]))/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, -2*a^2*(A - I*B)*x + (a^2*B*Log[Cos[c + d*x]])/d + (a^2*((2*I)*A + B)*Log[Sin[c + d*x]])/d - (A*Cot[c + d*x]*(a^2 + I*a^2*Tan[c + d*x]))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, -2*a^2*(I*A + B)*x - (a^2*((3*I)*A + 2*B)*Cot[c + d*x])/(2*d) - (2*a^2*(A - I*B)*Log[Sin[c + d*x]])/d - (A*Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x]))/(2*d)} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, 2*a^2*(A - I*B)*x + (2*a^2*(A - I*B)*Cot[c + d*x])/d - (a^2*((4*I)*A + 3*B)*Cot[c + d*x]^2)/(6*d) - (2*a^2*(I*A + B)*Log[Sin[c + d*x]])/d - (A*Cot[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x]))/(3*d)} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 6, 2*a^2*(I*A + B)*x + (2*a^2*(I*A + B)*Cot[c + d*x])/d + (a^2*(A - I*B)*Cot[c + d*x]^2)/d - (a^2*((5*I)*A + 4*B)*Cot[c + d*x]^3)/(12*d) + (2*a^2*(A - I*B)*Log[Sin[c + d*x]])/d - (A*Cot[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x]))/(4*d)} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -4*a^3*(A - I*B)*x + (4*a^3*(I*A + B)*Log[Cos[c + d*x]])/d + (4*a^3*(A - I*B)*Tan[c + d*x])/d + (2*a^3*(I*A + B)*Tan[c + d*x]^2)/d - (a^3*(45*A - (47*I)*B)*Tan[c + d*x]^3)/(60*d) + ((I/5)*a*B*Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^2)/d - ((5*A - (7*I)*B)*Tan[c + d*x]^3*(a^3 + I*a^3*Tan[c + d*x]))/(20*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, -4*a^3*(I*A + B)*x - (4*a^3*(A - I*B)*Log[Cos[c + d*x]])/d + (2*a^3*(I*A + B)*Tan[c + d*x])/d + (a*(A - I*B)*(a + I*a*Tan[c + d*x])^2)/(2*d) + (A*(a + I*a*Tan[c + d*x])^3)/(3*d) - ((I/4)*B*(a + I*a*Tan[c + d*x])^4)/(a*d)} -{(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 4, 4*a^3*(A - I*B)*x - (4*a^3*(I*A + B)*Log[Cos[c + d*x]])/d - (2*a^3*(A - I*B)*Tan[c + d*x])/d + (a*(I*A + B)*(a + I*a*Tan[c + d*x])^2)/(2*d) + (B*(a + I*a*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, 4*a^3*(I*A + B)*x + (a^3*(3*A - (4*I)*B)*Log[Cos[c + d*x]])/d + (a^3*A*Log[Sin[c + d*x]])/d + ((I/2)*a*B*(a + I*a*Tan[c + d*x])^2)/d - ((A - (2*I)*B)*(a^3 + I*a^3*Tan[c + d*x]))/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -4*a^3*(A - I*B)*x + (a^3*(I*A + 3*B)*Log[Cos[c + d*x]])/d + (a^3*(3*I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^2)/d + ((I*A - B)*(a^3 + I*a^3*Tan[c + d*x]))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -4*a^3*(I*A + B)*x + (I*a^3*B*Log[Cos[c + d*x]])/d - (a^3*(4*A - (3*I)*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^2)/(2*d) - (((2*I)*A + B)*Cot[c + d*x]*(a^3 + I*a^3*Tan[c + d*x]))/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, 4*a^3*(A - I*B)*x + (a^3*(17*A - (15*I)*B)*Cot[c + d*x])/(6*d) - (4*a^3*(I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^2)/(3*d) - (((5*I)*A + 3*B)*Cot[c + d*x]^2*(a^3 + I*a^3*Tan[c + d*x]))/(6*d)} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, 4*a^3*(I*A + B)*x + (4*a^3*(I*A + B)*Cot[c + d*x])/d + (a^3*(15*A - (14*I)*B)*Cot[c + d*x]^2)/(12*d) + (4*a^3*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^2)/(4*d) - (((3*I)*A + 2*B)*Cot[c + d*x]^3*(a^3 + I*a^3*Tan[c + d*x]))/(6*d)} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 7, -4*a^3*(A - I*B)*x - (4*a^3*(A - I*B)*Cot[c + d*x])/d + (2*a^3*(I*A + B)*Cot[c + d*x]^2)/d + (a^3*(47*A - (45*I)*B)*Cot[c + d*x]^3)/(60*d) + (4*a^3*(I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^2)/(5*d) - (((7*I)*A + 5*B)*Cot[c + d*x]^4*(a^3 + I*a^3*Tan[c + d*x]))/(20*d)} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -8*a^4*(A - I*B)*x + (8*a^4*(I*A + B)*Log[Cos[c + d*x]])/d + (8*a^4*(A - I*B)*Tan[c + d*x])/d + (4*a^4*(I*A + B)*Tan[c + d*x]^2)/d - (a^4*(92*A - (93*I)*B)*Tan[c + d*x]^3)/(60*d) + ((I/6)*a*B*Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - ((2*A - (3*I)*B)*Tan[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) - ((12*A - (13*I)*B)*Tan[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]))/(20*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, -8*a^4*(I*A + B)*x - (8*a^4*(A - I*B)*Log[Cos[c + d*x]])/d + (4*a^4*(I*A + B)*Tan[c + d*x])/d + (a*(A - I*B)*(a + I*a*Tan[c + d*x])^3)/(3*d) + (A*(a + I*a*Tan[c + d*x])^4)/(4*d) - ((I/5)*B*(a + I*a*Tan[c + d*x])^5)/(a*d) + ((A - I*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 5, 8*a^4*(A - I*B)*x - (8*a^4*(I*A + B)*Log[Cos[c + d*x]])/d - (4*a^4*(A - I*B)*Tan[c + d*x])/d + (a*(I*A + B)*(a + I*a*Tan[c + d*x])^3)/(3*d) + (B*(a + I*a*Tan[c + d*x])^4)/(4*d) + ((I*A + B)*(a^2 + I*a^2*Tan[c + d*x])^2)/d} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, 8*a^4*(I*A + B)*x + (a^4*(7*A - (8*I)*B)*Log[Cos[c + d*x]])/d + (a^4*A*Log[Sin[c + d*x]])/d + ((I/3)*a*B*(a + I*a*Tan[c + d*x])^3)/d - ((A - (2*I)*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*A - (4*I)*B)*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -8*a^4*(A - I*B)*x + (a^4*(4*I*A + 7*B)*Log[Cos[c + d*x]])/d + (a^4*(4*I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3)/d + ((2*I*A - B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - (3*B*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -8*a^4*(I*A + B)*x - (a^4*(A - (4*I)*B)*Log[Cos[c + d*x]])/d - (a^4*(7*A - (4*I)*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^3)/(2*d) - (((5*I)*A + 2*B)*Cot[c + d*x]*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - (3*A*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, 8*a^4*(A - I*B)*x - (a^4*B*Log[Cos[c + d*x]])/d - (a^4*((8*I)*A + 7*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/(3*d) - (((2*I)*A + B)*Cot[c + d*x]^2*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) + ((4*A - (3*I)*B)*Cot[c + d*x]*(a^4 + I*a^4*Tan[c + d*x]))/d} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, 8*a^4*(I*A + B)*x + (a^4*((67*I)*A + 64*B)*Cot[c + d*x])/(12*d) + (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3)/(4*d) - (((7*I)*A + 4*B)*Cot[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2)/(12*d) + ((19*A - (16*I)*B)*Cot[c + d*x]^2*(a^4 + I*a^4*Tan[c + d*x]))/(12*d)} -{Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -8*a^4*(A - I*B)*x - (8*a^4*(A - I*B)*Cot[c + d*x])/d + (a^4*((148*I)*A + 145*B)*Cot[c + d*x]^2)/(60*d) + (8*a^4*(I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3)/(5*d) - (((8*I)*A + 5*B)*Cot[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x])^2)/(20*d) + ((28*A - (25*I)*B)*Cot[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]))/(30*d)} -{Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 8, -8*a^4*(I*A + B)*x - (8*a^4*(I*A + B)*Cot[c + d*x])/d - (4*a^4*(A - I*B)*Cot[c + d*x]^2)/d + (a^4*((93*I)*A + 92*B)*Cot[c + d*x]^3)/(60*d) - (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3)/(6*d) - (((3*I)*A + 2*B)*Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) + ((13*A - (12*I)*B)*Cot[c + d*x]^4*(a^4 + I*a^4*Tan[c + d*x]))/(20*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 4, (3*(I*A - B)*x)/(2*a) - ((A + 2*I*B)*Log[Cos[c + d*x]])/(a*d) - (3*(I*A - B)*Tan[c + d*x])/(2*a*d) - ((A + 2*I*B)*Tan[c + d*x]^2)/(2*a*d) + ((I*A - B)*Tan[c + d*x]^3)/(2*d*(a + I*a*Tan[c + d*x]))} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 3, ((A + 3*I*B)*x)/(2*a) + ((I*A - B)*Log[Cos[c + d*x]])/(a*d) - ((A + 3*I*B)*Tan[c + d*x])/(2*a*d) + ((I*A - B)*Tan[c + d*x]^2)/(2*d*(a + I*a*Tan[c + d*x]))} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 5, -(((I*A - B)*x)/(2*a)) + (I*B*Log[Cos[c + d*x]])/(a*d) - (A + I*B)/(2*a*d*(1 + I*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x]), x, 2, ((A - I*B)*x)/(2*a) + (I*A - B)/(2*d*(a + I*a*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 3, -(((I*A - B)*x)/(2*a)) + (A*Log[Sin[c + d*x]])/(a*d) + (A + I*B)/(2*d*(a + I*a*Tan[c + d*x]))} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 4, -(((3*A + I*B)*x)/(2*a)) - ((3*A + I*B)*Cot[c + d*x])/(2*a*d) - ((I*A - B)*Log[Sin[c + d*x]])/(a*d) + ((A + I*B)*Cot[c + d*x])/(2*d*(a + I*a*Tan[c + d*x]))} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 5, (3*(I*A - B)*x)/(2*a) + (3*(I*A - B)*Cot[c + d*x])/(2*a*d) - ((2*A + I*B)*Cot[c + d*x]^2)/(2*a*d) - ((2*A + I*B)*Log[Sin[c + d*x]])/(a*d) + ((A + I*B)*Cot[c + d*x]^2)/(2*d*(a + I*a*Tan[c + d*x]))} -{(Cot[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 6, ((5*A + 3*I*B)*x)/(2*a) + ((5*A + 3*I*B)*Cot[c + d*x])/(2*a*d) + ((I*A - B)*Cot[c + d*x]^2)/(a*d) - ((5*A + 3*I*B)*Cot[c + d*x]^3)/(6*a*d) + (2*(I*A - B)*Log[Sin[c + d*x]])/(a*d) + ((A + I*B)*Cot[c + d*x]^3)/(2*d*(a + I*a*Tan[c + d*x]))} - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 4, -((3*(I*A - 3*B)*x)/(4*a^2)) + ((A + 2*I*B)*Log[Cos[c + d*x]])/(a^2*d) + (3*(I*A - 3*B)*Tan[c + d*x])/(4*a^2*d) + ((A + 2*I*B)*Tan[c + d*x]^2)/(2*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^3)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 6, -(((A + 3*I*B)*x)/(4*a^2)) + (B*Log[Cos[c + d*x]])/(a^2*d) + (I*A - 3*B)/(4*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^2)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 3, -(((I*A + B)*x)/(4*a^2)) + (A + 3*I*B)/(4*a^2*d*(1 + I*Tan[c + d*x])) - (A + I*B)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^2, x, 3, ((A - I*B)*x)/(4*a^2) + (I*A - B)/(4*d*(a + I*a*Tan[c + d*x])^2) + (I*A + B)/(4*d*(a^2 + I*a^2*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 4, -(((3*I*A - B)*x)/(4*a^2)) + (A*Log[Sin[c + d*x]])/(a^2*d) + (3*A + I*B)/(4*a^2*d*(1 + I*Tan[c + d*x])) + (A + I*B)/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 5, -((3*(3*A + I*B)*x)/(4*a^2)) - (3*(3*A + I*B)*Cot[c + d*x])/(4*a^2*d) - ((2*I*A - B)*Log[Sin[c + d*x]])/(a^2*d) + ((2*A + I*B)*Cot[c + d*x])/(2*a^2*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x])/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 6, (3*(5*I*A - 3*B)*x)/(4*a^2) + (3*(5*I*A - 3*B)*Cot[c + d*x])/(4*a^2*d) - ((2*A + I*B)*Cot[c + d*x]^2)/(a^2*d) - (2*(2*A + I*B)*Log[Sin[c + d*x]])/(a^2*d) + ((5*A + 3*I*B)*Cot[c + d*x]^2)/(4*a^2*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x]^2)/(4*d*(a + I*a*Tan[c + d*x])^2)} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 5, -(((7*A + 25*I*B)*x)/(8*a^3)) - ((I*A - 3*B)*Log[Cos[c + d*x]])/(a^3*d) + ((7*A + 25*I*B)*Tan[c + d*x])/(8*a^3*d) + ((I*A - B)*Tan[c + d*x]^4)/(6*d*(a + I*a*Tan[c + d*x])^3) + ((5*A + 11*I*B)*Tan[c + d*x]^3)/(24*a*d*(a + I*a*Tan[c + d*x])^2) - ((I*A - 3*B)*Tan[c + d*x]^2)/(2*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 7, ((I*A - 7*B)*x)/(8*a^3) - (I*B*Log[Cos[c + d*x]])/(a^3*d) + ((I*A - B)*Tan[c + d*x]^3)/(6*d*(a + I*a*Tan[c + d*x])^3) + ((A + 3*I*B)*Tan[c + d*x]^2)/(8*a*d*(a + I*a*Tan[c + d*x])^2) + (A + 7*I*B)/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 4, -(((A - I*B)*x)/(8*a^3)) + ((I*A - B)*Tan[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d*x])^3) + (I*A - 7*B)/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (I*A + 17*B)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 4, -(((I*A + B)*x)/(8*a^3)) - (A + I*B)/(6*d*(a + I*a*Tan[c + d*x])^3) + (A + 3*I*B)/(8*a*d*(a + I*a*Tan[c + d*x])^2) + (A - I*B)/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^3, x, 4, ((A - I*B)*x)/(8*a^3) + (I*A - B)/(6*d*(a + I*a*Tan[c + d*x])^3) + (I*A + B)/(8*a*d*(a + I*a*Tan[c + d*x])^2) + (I*A + B)/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 5, -(((7*I*A - B)*x)/(8*a^3)) + (A*Log[Sin[c + d*x]])/(a^3*d) + (A + I*B)/(6*d*(a + I*a*Tan[c + d*x])^3) + (3*A + I*B)/(8*a*d*(a + I*a*Tan[c + d*x])^2) + (7*A + I*B)/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 6, -(((25*A + 7*I*B)*x)/(8*a^3)) - ((25*A + 7*I*B)*Cot[c + d*x])/(8*a^3*d) - ((3*I*A - B)*Log[Sin[c + d*x]])/(a^3*d) + ((A + I*B)*Cot[c + d*x])/(6*d*(a + I*a*Tan[c + d*x])^3) + ((11*A + 5*I*B)*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^2) + ((3*A + I*B)*Cot[c + d*x])/(2*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 7, (5*(11*I*A - 5*B)*x)/(8*a^3) + (5*(11*I*A - 5*B)*Cot[c + d*x])/(8*a^3*d) - ((7*A + 3*I*B)*Cot[c + d*x]^2)/(2*a^3*d) - ((7*A + 3*I*B)*Log[Sin[c + d*x]])/(a^3*d) + ((A + I*B)*Cot[c + d*x]^2)/(6*d*(a + I*a*Tan[c + d*x])^3) + ((13*A + 7*I*B)*Cot[c + d*x]^2)/(24*a*d*(a + I*a*Tan[c + d*x])^2) + (5*(11*A + 5*I*B)*Cot[c + d*x]^2)/(24*d*(a^3 + I*a^3*Tan[c + d*x]))} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 8, ((A + 15*I*B)*x)/(16*a^4) - (B*Log[Cos[c + d*x]])/(a^4*d) - (I*A - 15*B)/(16*a^4*d*(1 + I*Tan[c + d*x])) - ((I*A - 7*B)*Tan[c + d*x]^2)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) + ((I*A - B)*Tan[c + d*x]^4)/(8*d*(a + I*a*Tan[c + d*x])^4) + ((A + 3*I*B)*Tan[c + d*x]^3)/(12*a*d*(a + I*a*Tan[c + d*x])^3)} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 5, ((I*A + B)*x)/(16*a^4) - (A - 13*I*B)/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + (5*A - 29*I*B)/(48*a^4*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^3)/(8*d*(a + I*a*Tan[c + d*x])^4) + ((A + 5*I*B)*Tan[c + d*x]^2)/(24*a*d*(a + I*a*Tan[c + d*x])^3)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 5, -(((A - I*B)*x)/(16*a^4)) + (I*A + 5*B)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) - (I*A + B)/(16*a^4*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^2)/(8*d*(a + I*a*Tan[c + d*x])^4) - B/(6*a*d*(a + I*a*Tan[c + d*x])^3)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 5, -(((I*A + B)*x)/(16*a^4)) - (A + I*B)/(8*d*(a + I*a*Tan[c + d*x])^4) + (A + 3*I*B)/(12*a*d*(a + I*a*Tan[c + d*x])^3) + (A - I*B)/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) + (A - I*B)/(16*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^4, x, 5, ((A - I*B)*x)/(16*a^4) + (I*A - B)/(8*d*(a + I*a*Tan[c + d*x])^4) + (I*A + B)/(12*a*d*(a + I*a*Tan[c + d*x])^3) + (I*A + B)/(16*d*(a^2 + I*a^2*Tan[c + d*x])^2) + (I*A + B)/(16*d*(a^4 + I*a^4*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 6, -(((15*I*A - B)*x)/(16*a^4)) + (A*Log[Sin[c + d*x]])/(a^4*d) + (7*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])^2) + (15*A + I*B)/(16*a^4*d*(1 + I*Tan[c + d*x])) + (A + I*B)/(8*d*(a + I*a*Tan[c + d*x])^4) + (3*A + I*B)/(12*a*d*(a + I*a*Tan[c + d*x])^3)} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 7, -((5*(13*A + 3*I*B)*x)/(16*a^4)) - (5*(13*A + 3*I*B)*Cot[c + d*x])/(16*a^4*d) - ((4*I*A - B)*Log[Sin[c + d*x]])/(a^4*d) + ((31*A + 9*I*B)*Cot[c + d*x])/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + ((4*A + I*B)*Cot[c + d*x])/(2*a^4*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x])/(8*d*(a + I*a*Tan[c + d*x])^4) + ((7*A + 3*I*B)*Cot[c + d*x])/(24*a*d*(a + I*a*Tan[c + d*x])^3)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 8, (5*(35*I*A - 13*B)*x)/(16*a^4) + (5*(35*I*A - 13*B)*Cot[c + d*x])/(16*a^4*d) - ((11*A + 4*I*B)*Cot[c + d*x]^2)/(2*a^4*d) - ((11*A + 4*I*B)*Log[Sin[c + d*x]])/(a^4*d) + ((43*A + 17*I*B)*Cot[c + d*x]^2)/(48*a^4*d*(1 + I*Tan[c + d*x])^2) + (5*(35*A + 13*I*B)*Cot[c + d*x]^2)/(48*a^4*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Cot[c + d*x]^2)/(8*d*(a + I*a*Tan[c + d*x])^4) + ((2*A + I*B)*Cot[c + d*x]^2)/(6*a*d*(a + I*a*Tan[c + d*x])^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 6, (Sqrt[2]*Sqrt[a]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*(7*A - I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (2*(7*A - I*B)*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (2*B*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(7*d) - (2*(7*A - (31*I)*B)*(a + I*a*Tan[c + d*x])^(3/2))/(105*a*d)} -{Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 5, (Sqrt[2]*Sqrt[a]*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*B*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) + (2*B*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(5*d) - (2*((5*I)*A + B)*(a + I*a*Tan[c + d*x])^(3/2))/(15*a*d)} -{Tan[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 4, -((Sqrt[2]*Sqrt[a]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d) + (2*A*Sqrt[a + I*a*Tan[c + d*x]])/d - (((2*I)/3)*B*(a + I*a*Tan[c + d*x])^(3/2))/(a*d)} -{Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 3, -((Sqrt[2]*Sqrt[a]*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d) + (2*B*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 6, (-2*Sqrt[a]*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (Sqrt[2]*Sqrt[a]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 7, -((Sqrt[a]*(I*A + 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d) + (Sqrt[2]*Sqrt[a]*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (A*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 8, (Sqrt[a]*(7*A - (4*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (Sqrt[2]*Sqrt[a]*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - ((I*A + 4*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (A*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Cot[c + d*x]^4*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 9, (Sqrt[a]*((9*I)*A + 14*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (Sqrt[2]*Sqrt[a]*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + ((7*A - (2*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - ((I*A + 6*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) - (A*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 6, (2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*a*((7*I)*A + 8*B)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (2*a*((7*I)*A + 8*B)*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) + (((2*I)/7)*a*B*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/d - (4*((21*I)*A + 19*B)*(a + I*a*Tan[c + d*x])^(3/2))/(105*d)} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 5, (-2*Sqrt[2]*a^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (2*a*(A - I*B)*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*A*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) - (((2*I)/5)*B*(a + I*a*Tan[c + d*x])^(5/2))/(a*d)} -{(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 4, (-2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (2*a*(I*A + B)*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*B*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 7, (-2*a^(3/2)*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[2]*a^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + ((2*I)*a*B*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 7, -((a^(3/2)*((3*I)*A + 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d) + (2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a*A*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 8, (a^(3/2)*(11*A - (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (2*Sqrt[2]*a^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a*((5*I)*A + 4*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 9, (a^(3/2)*((23*I)*A + 22*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (2*Sqrt[2]*a^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a*(9*A - (10*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (a*((7*I)*A + 6*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) - (a*A*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 7, (4*Sqrt[2]*a^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*a^2*((45*I)*A + 46*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) + (2*a^2*((45*I)*A + 46*B)*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*(3*A - (4*I)*B)*Tan[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (8*a*((60*I)*A + 59*B)*(a + I*a*Tan[c + d*x])^(3/2))/(315*d) + (((2*I)/9)*a*B*Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Tan[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 6, (-4*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (4*a^2*(A - I*B)*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*a*(A - I*B)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) + (2*A*(a + I*a*Tan[c + d*x])^(5/2))/(5*d) - (((2*I)/7)*B*(a + I*a*Tan[c + d*x])^(7/2))/(a*d)} -{(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 5, (-4*Sqrt[2]*a^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (4*a^2*(I*A + B)*Sqrt[a + I*a*Tan[c + d*x]])/d + (2*a*(I*A + B)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d) + (2*B*(a + I*a*Tan[c + d*x])^(5/2))/(5*d)} -{Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 8, (-2*a^(5/2)*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d + (4*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (2*a^2*(A - (2*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/d + (((2*I)/3)*a*B*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 8, -((a^(5/2)*(5*I*A + 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/d) + (4*Sqrt[2]*a^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*(I*A - 2*B)*Sqrt[a + I*a*Tan[c + d*x]])/d - (a*A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 8, (a^(5/2)*(23*A - (20*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*d) - (4*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (a^2*((7*I)*A + 4*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(3/2))/(2*d)} -{Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 9, (a^(5/2)*((45*I)*A + 46*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (4*Sqrt[2]*a^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*(19*A - (18*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (a^2*((3*I)*A + 2*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, (-3*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(64*d) + (4*Sqrt[2]*a^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(64*d) + (a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2*((11*I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(3/2))/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 6, ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((I*A - B)*Tan[c + d*x]^3)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (4*(5*A + 7*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d) - ((5*A + 7*I*B)*Tan[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d) - ((25*A + 23*I*B)*(a + I*a*Tan[c + d*x])^(3/2))/(15*a^2*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 5, ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((I*A - B)*Tan[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - (4*(I*A - B)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d) + ((3*I*A - 5*B)*(a + I*a*Tan[c + d*x])^(3/2))/(3*a^2*d)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 4, -(((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d)) - (A + I*B)/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((2*I)*B*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{(A + B*Tan[c + d*x])/Sqrt[a + I*a*Tan[c + d*x]], x, 3, -(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d)) + (I*A - B)/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 7, (-2*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + (A + I*B)/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 8, ((I*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((2*A + I*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 9, ((11*A + (4*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x]^2)/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((7*I)*A - 8*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a*d) - ((3*A + (2*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(2*a*d)} - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 6, ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((I*A - B)*Tan[c + d*x]^3)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((3*A + 5*I*B)*Tan[c + d*x]^2)/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - (2*(3*A + 5*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(a^2*d) + ((11*A + 21*I*B)*(a + I*a*Tan[c + d*x])^(3/2))/(6*a^3*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 5, ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((I*A - B)*Tan[c + d*x]^2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (5*I*A - 11*B)/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((I*A - 7*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3*a^2*d)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 4, -(((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d)) - (A + I*B)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (A + 3*I*B)/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(3/2), x, 4, -(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d)) + (I*A - B)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I*A + B)/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 8, (-2*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + (A + I*B)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + (3*A + I*B)/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 9, (((3*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x])/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((13*A + (7*I)*B)*Cot[c + d*x])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (3*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(2*a^2*d)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 10, ((23*A + (12*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*d) + ((A + I*B)*Cot[c + d*x]^2)/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((17*A + (11*I)*B)*Cot[c + d*x]^2)/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + (7*((3*I)*A - 2*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^2*d) - ((22*A + (13*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d)} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 7, -(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d)) + ((I*A - B)*Tan[c + d*x]^4)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((11*A + 21*I*B)*Tan[c + d*x]^3)/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((39*I*A - 89*B)*Tan[c + d*x]^2)/(20*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((39*I*A - 89*B)*Sqrt[a + I*a*Tan[c + d*x]])/(5*a^3*d) - ((151*I*A - 361*B)*(a + I*a*Tan[c + d*x])^(3/2))/(60*a^4*d)} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 6, ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((I*A - B)*Tan[c + d*x]^3)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((7*A + 17*I*B)*Tan[c + d*x]^2)/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (41*A + 151*I*B)/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((13*A + 83*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(30*a^3*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 5, ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((I*A - B)*Tan[c + d*x]^2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*I*A - 13*B)/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - (I*A - 31*B)/(20*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 5, -(((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d)) - (A + I*B)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (A + 3*I*B)/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (A - I*B)/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(5/2), x, 5, -(((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d)) + (I*A - B)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (I*A + B)/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (I*A + B)/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 9, (-2*A*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + (A + I*B)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*A + I*B)/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + (7*A + I*B)/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 10, (((5*I)*A - 2*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((A + I*B)*Cot[c + d*x])/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((19*A + (9*I)*B)*Cot[c + d*x])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((41*A + (15*I)*B)*Cot[c + d*x])/(12*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - (7*(3*A + I*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^3*d)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 11, ((43*A + (20*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(4*a^(5/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*a^(5/2)*d) + ((A + I*B)*Cot[c + d*x]^2)/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((23*A + (13*I)*B)*Cot[c + d*x]^2)/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((337*A + (167*I)*B)*Cot[c + d*x]^2)/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + (21*((2*I)*A - B)*Cot[c + d*x]*Sqrt[a + I*a*Tan[c + d*x]])/(4*a^3*d) - ((85*A + (41*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(12*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 6, -((2*(-1)^(1/4)*a*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (2*a*(I*A + B)*Sqrt[Tan[c + d*x]])/d + (2*a*(A - I*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*a*(I*A + B)*Tan[c + d*x]^(5/2))/(5*d) + (2*I*a*B*Tan[c + d*x]^(7/2))/(7*d)} -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 5, (2*(-1)^(1/4)*a*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (2*a*(A - I*B)*Sqrt[Tan[c + d*x]])/d + (2*a*(I*A + B)*Tan[c + d*x]^(3/2))/(3*d) + (2*I*a*B*Tan[c + d*x]^(5/2))/(5*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, (2*(-1)^(1/4)*a*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (2*a*(I*A + B)*Sqrt[Tan[c + d*x]])/d + (2*I*a*B*Tan[c + d*x]^(3/2))/(3*d)} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 3, -((2*(-1)^(1/4)*a*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) + (2*I*a*B*Sqrt[Tan[c + d*x]])/d} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 3, -((2*(-1)^(1/4)*a*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (2*a*A)/(d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 4, (2*(-1)^(1/4)*a*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (2*a*A)/(3*d*Tan[c + d*x]^(3/2)) - (2*a*(I*A + B))/(d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 5, (2*(-1)^(1/4)*a*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (2*a*A)/(5*d*Tan[c + d*x]^(5/2)) - (2*a*(I*A + B))/(3*d*Tan[c + d*x]^(3/2)) + (2*a*(A - I*B))/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 7, -((4*(-1)^(1/4)*a^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (4*a^2*(I*A + B)*Sqrt[Tan[c + d*x]])/d + (4*a^2*(A - I*B)*Tan[c + d*x]^(3/2))/(3*d) + (4*a^2*(I*A + B)*Tan[c + d*x]^(5/2))/(5*d) - (2*a^2*(9*A - 11*I*B)*Tan[c + d*x]^(7/2))/(63*d) + (2*I*B*Tan[c + d*x]^(7/2)*(a^2 + I*a^2*Tan[c + d*x]))/(9*d)} -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 6, (4*(-1)^(1/4)*a^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (4*a^2*(A - I*B)*Sqrt[Tan[c + d*x]])/d + (4*a^2*(I*A + B)*Tan[c + d*x]^(3/2))/(3*d) - (2*a^2*(7*A - 9*I*B)*Tan[c + d*x]^(5/2))/(35*d) + (2*I*B*Tan[c + d*x]^(5/2)*(a^2 + I*a^2*Tan[c + d*x]))/(7*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, (4*(-1)^(1/4)*a^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (4*a^2*(I*A + B)*Sqrt[Tan[c + d*x]])/d - (2*a^2*(5*A - 7*I*B)*Tan[c + d*x]^(3/2))/(15*d) + (2*I*B*Tan[c + d*x]^(3/2)*(a^2 + I*a^2*Tan[c + d*x]))/(5*d)} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 4, -((4*(-1)^(1/4)*a^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (2*a^2*(3*A - 5*I*B)*Sqrt[Tan[c + d*x]])/(3*d) + (2*I*B*Sqrt[Tan[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x]))/(3*d)} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 4, -((4*(-1)^(1/4)*a^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) + (2*a^2*(I*A - B)*Sqrt[Tan[c + d*x]])/d - (2*A*(a^2 + I*a^2*Tan[c + d*x]))/(d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 4, (4*(-1)^(1/4)*a^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (2*a^2*(5*I*A + 3*B))/(3*d*Sqrt[Tan[c + d*x]]) - (2*A*(a^2 + I*a^2*Tan[c + d*x]))/(3*d*Tan[c + d*x]^(3/2))} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 5, (4*(-1)^(1/4)*a^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (2*a^2*(7*I*A + 5*B))/(15*d*Tan[c + d*x]^(3/2)) + (4*a^2*(A - I*B))/(d*Sqrt[Tan[c + d*x]]) - (2*A*(a^2 + I*a^2*Tan[c + d*x]))/(5*d*Tan[c + d*x]^(5/2))} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 6, -((4*(-1)^(1/4)*a^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (2*a^2*(9*I*A + 7*B))/(35*d*Tan[c + d*x]^(5/2)) + (4*a^2*(A - I*B))/(3*d*Tan[c + d*x]^(3/2)) + (4*a^2*(I*A + B))/(d*Sqrt[Tan[c + d*x]]) - (2*A*(a^2 + I*a^2*Tan[c + d*x]))/(7*d*Tan[c + d*x]^(7/2))} - - -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 7, (8*(-1)^(1/4)*a^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (8*a^3*(A - I*B)*Sqrt[Tan[c + d*x]])/d + (8*a^3*(I*A + B)*Tan[c + d*x]^(3/2))/(3*d) - (16*a^3*(18*A - 19*I*B)*Tan[c + d*x]^(5/2))/(315*d) + (2*I*a*B*Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2)/(9*d) - (2*(9*A - 13*I*B)*Tan[c + d*x]^(5/2)*(a^3 + I*a^3*Tan[c + d*x]))/(63*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, (8*(-1)^(1/4)*a^3*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (8*a^3*(I*A + B)*Sqrt[Tan[c + d*x]])/d - (8*a^3*(21*A - 23*I*B)*Tan[c + d*x]^(3/2))/(105*d) + (2*I*a*B*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2)/(7*d) - (2*(7*A - 11*I*B)*Tan[c + d*x]^(3/2)*(a^3 + I*a^3*Tan[c + d*x]))/(35*d)} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 5, -((8*(-1)^(1/4)*a^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (16*a^3*(5*A - 6*I*B)*Sqrt[Tan[c + d*x]])/(15*d) + (2*I*a*B*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)/(5*d) - (2*(5*A - 9*I*B)*Sqrt[Tan[c + d*x]]*(a^3 + I*a^3*Tan[c + d*x]))/(15*d)} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 5, -((8*(-1)^(1/4)*a^3*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) - (16*a^3*B*Sqrt[Tan[c + d*x]])/(3*d) - (2*a*A*(a + I*a*Tan[c + d*x])^2)/(d*Sqrt[Tan[c + d*x]]) + (2*(3*I*A - B)*Sqrt[Tan[c + d*x]]*(a^3 + I*a^3*Tan[c + d*x]))/(3*d)} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 5, (8*(-1)^(1/4)*a^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (16*a^3*A*Sqrt[Tan[c + d*x]])/(3*d) - (2*a*A*(a + I*a*Tan[c + d*x])^2)/(3*d*Tan[c + d*x]^(3/2)) - (2*(7*I*A + 3*B)*(a^3 + I*a^3*Tan[c + d*x]))/(3*d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 5, (8*(-1)^(1/4)*a^3*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d + (16*a^3*(6*A - 5*I*B))/(15*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^2)/(5*d*Tan[c + d*x]^(5/2)) - (2*(9*I*A + 5*B)*(a^3 + I*a^3*Tan[c + d*x]))/(15*d*Tan[c + d*x]^(3/2))} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 6, -((8*(-1)^(1/4)*a^3*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d) + (8*a^3*(23*A - 21*I*B))/(105*d*Tan[c + d*x]^(3/2)) + (8*a^3*(I*A + B))/(d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^2)/(7*d*Tan[c + d*x]^(7/2)) - (2*(11*I*A + 7*B)*(a^3 + I*a^3*Tan[c + d*x]))/(35*d*Tan[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 13, -(((1/4 + I/4)*((4 + I)*A + (1 + 6*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d)) + ((1/4 + I/4)*((4 + I)*A + (1 + 6*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d) - ((1/8 + I/8)*((1 + 4*I)*A - (6 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) - (((3 - 5*I)*A + (5 + 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) - (5*(I*A - B)*Sqrt[Tan[c + d*x]])/(2*a*d) - ((3*A + 7*I*B)*Tan[c + d*x]^(3/2))/(6*a*d) + ((I*A - B)*Tan[c + d*x]^(5/2))/(2*d*(a + I*a*Tan[c + d*x]))} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 12, -((((1 - 3*I)*A + (3 + 5*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(4*Sqrt[2]*a*d)) - ((1/4 + I/4)*((1 + 2*I)*A - (4 + I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d) - ((1/8 + I/8)*((2 + I)*A + (1 + 4*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) + ((1/8 + I/8)*((2 + I)*A + (1 + 4*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) - ((A + 5*I*B)*Sqrt[Tan[c + d*x]])/(2*a*d) + ((I*A - B)*Tan[c + d*x]^(3/2))/(2*d*(a + I*a*Tan[c + d*x]))} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 11, -(((1/4 - I/4)*(A + (2 - I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d)) + ((1/4 - I/4)*(A + (2 - I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d) + ((1/8 + I/8)*(A - (2 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) - ((1/8 + I/8)*(A - (2 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(2*d*(a + I*a*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])), x, 11, -(((1/4 - I/4)*((2 + I)*A + B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d)) + ((1/4 - I/4)*((2 + I)*A + B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d) - (((3 + I)*A - (1 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) + (((3 + I)*A - (1 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(2*d*(a + I*a*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])), x, 12, (((5 + 3*I)*A - (3 - I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(4*Sqrt[2]*a*d) + (((-5 - 3*I)*A + (3 - I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(4*Sqrt[2]*a*d) - ((1/8 - I/8)*((4 + I)*A + (1 + 2*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a*d) + (((5 - 3*I)*A + (3 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) - (5*A + I*B)/(2*a*d*Sqrt[Tan[c + d*x]]) + (A + I*B)/(2*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])), x, 13, (((7 - 5*I)*A + (5 + 3*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(4*Sqrt[2]*a*d) - ((1/4 - I/4)*((6 + I)*A + (1 + 4*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a*d) + (((7 + 5*I)*A - (5 - 3*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) + (((-7 - 5*I)*A + (5 - 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(8*Sqrt[2]*a*d) - (7*A + 3*I*B)/(6*a*d*Tan[c + d*x]^(3/2)) + (5*(I*A - B))/(2*a*d*Sqrt[Tan[c + d*x]]) + (A + I*B)/(2*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x]))} - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 13, (((9 + 5*I)*A - (25 - 21*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - (((9 + 5*I)*A - (25 - 21*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + (5*(I*A - 5*B)*Sqrt[Tan[c + d*x]])/(8*a^2*d) + ((3*A + 7*I*B)*Tan[c + d*x]^(3/2))/(8*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^(5/2))/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 12, (((1 + 3*I)*A + (9 + 5*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - (((1 + 3*I)*A + (9 + 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) + (((1 - 3*I)*A - (9 - 5*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) - (((1 - 3*I)*A - (9 - 5*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) + ((A + 5*I*B)*Sqrt[Tan[c + d*x]])/(8*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*x]^(3/2))/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 12, (((-1 + 3*I)*A + (1 + 3*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - (((-1 + 3*I)*A + (1 + 3*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) + (((1 + 3*I)*A + (1 - 3*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) - (((1 + 3*I)*A + (1 - 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) + ((I*A + 3*B)*Sqrt[Tan[c + d*x]])/(8*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2), x, 12, ((1/16 + I/16)*((-2 + 7*I)*A + (1 + 2*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^2*d) + (((9 - 5*I)*A + (1 - 3*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) + ((1/32 + I/32)*((-7 + 2*I)*A + (2 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + (((9 + 5*I)*A - (1 + 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) + ((5*A + I*B)*Sqrt[Tan[c + d*x]])/(8*a^2*d*(1 + I*Tan[c + d*x])) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2), x, 13, (((25 + 21*I)*A - (9 - 5*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - ((1/16 - I/16)*((2 + 23*I)*A - (7 + 2*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^2*d) - ((1/32 - I/32)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) - (5*(5*A + I*B))/(8*a^2*d*Sqrt[Tan[c + d*x]]) + (7*A + 3*I*B)/(8*a^2*d*(1 + I*Tan[c + d*x])*Sqrt[Tan[c + d*x]]) + (A + I*B)/(4*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2), x, 14, ((1/16 - I/16)*((47 + 2*I)*A + (2 + 23*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^2*d) - ((1/16 - I/16)*((47 + 2*I)*A + (2 + 23*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^2*d) + (((49 + 45*I)*A - (25 - 21*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^2*d) - ((1/32 - I/32)*((2 + 47*I)*A - (23 + 2*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) - (7*(7*A + 3*I*B))/(24*a^2*d*Tan[c + d*x]^(3/2)) + (9*A + 5*I*B)/(8*a^2*d*(1 + I*Tan[c + d*x])*Tan[c + d*x]^(3/2)) + (5*(9*I*A - 5*B))/(8*a^2*d*Sqrt[Tan[c + d*x]]) + (A + I*B)/(4*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2)} - - -{(Tan[c + d*x]^(9/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 15, ((1/16 + I/16)*((29 + I)*A + (1 + 76*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) - ((1/16 + I/16)*((29 + I)*A + (1 + 76*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) - (((28 - 30*I)*A + (75 + 77*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) - ((1/32 + I/32)*((1 + 29*I)*A - (76 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) + (15*(2*I*A - 5*B)*Sqrt[Tan[c + d*x]])/(8*a^3*d) + (7*(4*A + 11*I*B)*Tan[c + d*x]^(3/2))/(24*a^3*d) + ((I*A - B)*Tan[c + d*x]^(9/2))/(6*d*(a + I*a*Tan[c + d*x])^3) + ((A + 2*I*B)*Tan[c + d*x]^(7/2))/(4*a*d*(a + I*a*Tan[c + d*x])^2) - (3*(2*I*A - 5*B)*Tan[c + d*x]^(5/2))/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 14, -(((1/16 + I/16)*((1 + 6*I)*A - (29 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d)) - (((5 - 7*I)*A + (28 + 30*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) + ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) - ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) + (5*(A + 6*I*B)*Sqrt[Tan[c + d*x]])/(8*a^3*d) + ((I*A - B)*Tan[c + d*x]^(7/2))/(6*d*(a + I*a*Tan[c + d*x])^3) + ((2*A + 5*I*B)*Tan[c + d*x]^(5/2))/(12*a*d*(a + I*a*Tan[c + d*x])^2) - (7*(I*A - 4*B)*Tan[c + d*x]^(3/2))/(24*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 13, ((2*A + (5 - 7*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - ((2*A + (5 - 7*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - ((2*A - (5 + 7*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((2*A - (5 + 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((I*A - B)*Tan[c + d*x]^(5/2))/(6*d*(a + I*a*Tan[c + d*x])^3) + ((A + 4*I*B)*Tan[c + d*x]^(3/2))/(12*a*d*(a + I*a*Tan[c + d*x])^2) + (5*B*Sqrt[Tan[c + d*x]])/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 13, (((1 + I)*A + 2*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (((1 + I)*A + 2*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (((-1 + I)*A + 2*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + (((-1 + I)*A + 2*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((I*A - B)*Tan[c + d*x]^(3/2))/(6*d*(a + I*a*Tan[c + d*x])^3) + (I*B*Sqrt[Tan[c + d*x]])/(4*a*d*(a + I*a*Tan[c + d*x])^2) + ((A - 2*I*B)*Sqrt[Tan[c + d*x]])/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 13, ((1/16 + I/16)*((1 + I)*A + B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) - ((1/16 + I/16)*((1 + I)*A + B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) + ((2*I*A + (1 - I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) - ((2*I*A + (1 - I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(6*d*(a + I*a*Tan[c + d*x])^3) + ((I*A + 2*B)*Sqrt[Tan[c + d*x]])/(12*a*d*(a + I*a*Tan[c + d*x])^2) + (B*Sqrt[Tan[c + d*x]])/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3), x, 13, -((((7 - 5*I)*A - 2*I*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d)) + (((7 - 5*I)*A - 2*I*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (((7 + 5*I)*A - 2*I*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + (((7 + 5*I)*A - 2*I*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(6*d*(a + I*a*Tan[c + d*x])^3) + ((4*A + I*B)*Sqrt[Tan[c + d*x]])/(12*a*d*(a + I*a*Tan[c + d*x])^2) + (5*A*Sqrt[Tan[c + d*x]])/(8*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3), x, 14, (((30 + 28*I)*A - (7 - 5*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^3*d) - ((1/16 - I/16)*((1 + 29*I)*A - (6 + I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) - ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) + ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) - (5*(6*A + I*B))/(8*a^3*d*Sqrt[Tan[c + d*x]]) + (A + I*B)/(6*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^3) + (5*A + 2*I*B)/(12*a*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2) + (7*(4*A + I*B))/(24*d*Sqrt[Tan[c + d*x]]*(a^3 + I*a^3*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3), x, 15, ((1/16 - I/16)*((76 + I)*A + (1 + 29*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) - ((1/16 - I/16)*((76 + I)*A + (1 + 29*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*a^3*d) + (((77 + 75*I)*A - (30 - 28*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(32*Sqrt[2]*a^3*d) - ((1/32 - I/32)*((1 + 76*I)*A - (29 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^3*d) - (7*(11*A + 4*I*B))/(24*a^3*d*Tan[c + d*x]^(3/2)) + (15*(5*I*A - 2*B))/(8*a^3*d*Sqrt[Tan[c + d*x]]) + (A + I*B)/(6*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3) + (2*A + I*B)/(4*a*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2) + (3*(5*A + 2*I*B))/(8*d*Tan[c + d*x]^(3/2)*(a^3 + I*a^3*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 9, ((-1)^(3/4)*Sqrt[a]*(4*I*A + 7*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d) + ((1 + I)*Sqrt[a]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((4*A - I*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (B*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 8, -(((-1)^(3/4)*Sqrt[a]*(2*A - I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((1 + I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (B*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 7, -((2*(-1)^(3/4)*Sqrt[a]*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((1 + I)*Sqrt[a]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 4, ((1 + I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*A*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 5, ((1 + I)*Sqrt[a]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*A*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (2*(I*A + 3*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 6, ((-1 - I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*A*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*(I*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (2*(13*A - (5*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 7, ((1 - I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*A*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*(I*A + 7*B)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (2*(31*A - (7*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(3/2)) + (2*((43*I)*A + 91*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 10, ((-1)^(3/4)*a^(3/2)*(22*I*A + 23*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(8*d) + ((2 + 2*I)*a^(3/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (a*(10*A - 9*I*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) + (a*(6*I*A + 7*B)*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(12*d) + (I*a*B*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 9, -(((-1)^(3/4)*a^(3/2)*(12*A - 11*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d)) - ((2 + 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (a*(4*I*A + 5*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (I*a*B*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(2*d)} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 8, -(((-1)^(3/4)*a^(3/2)*(2*I*A + 3*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d) - ((2 + 2*I)*a^(3/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (I*a*B*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 8, (2*(-1)^(1/4)*a^(3/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((2 + 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 5, ((2 + 2*I)*a^(3/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (2*a*(4*I*A + 3*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 6, ((-2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + I*a*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*a*((6*I)*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (4*a*(9*A - (10*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 7, ((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*a*(8*I*A + 7*B)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (4*a*(19*A - 21*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(3/2)) + (4*a*(67*I*A + 63*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2), x, 8, ((2 + 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + I*a*Tan[c + d*x]])/(9*d*Tan[c + d*x]^(9/2)) - (2*a*((10*I)*A + 9*B)*Sqrt[a + I*a*Tan[c + d*x]])/(63*d*Tan[c + d*x]^(7/2)) + (4*a*(11*A - (12*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(5/2)) + (4*a*((61*I)*A + 57*B)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*Tan[c + d*x]^(3/2)) - (4*a*(193*A - (201*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 11, (3*(-1)^(3/4)*a^(5/2)*(120*I*A + 121*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(64*d) + ((4 + 4*I)*a^(5/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (a^2*(152*A - 149*I*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(64*d) + (a^2*(104*I*A + 107*B)*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2*(8*A - 11*I*B)*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) + (I*a*B*Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2))/(4*d)} -{Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, -(((-1)^(3/4)*a^(5/2)*(46*A - 45*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(8*d)) - ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (a^2*(18*I*A + 19*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(8*d) - (a^2*(2*A - 3*I*B)*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (I*a*B*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 9, -(((-1)^(3/4)*a^(5/2)*(20*I*A + 23*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(4*d)) + ((4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (a^2*(4*A - 7*I*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) + (I*a*B*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2))/(2*d)} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 9, ((-1)^(3/4)*a^(5/2)*(2*A - 5*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + (a^2*(2*I*A - B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(d*Sqrt[Tan[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 9, (2*(-1)^(3/4)*a^(5/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((4 + 4*I)*a^(5/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*(2*I*A + B)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(3*d*Tan[c + d*x]^(3/2))} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 6, ((-4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*((8*I)*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (2*a^2*(38*A - (35*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(5*d*Tan[c + d*x]^(5/2))} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 7, ((4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*(10*I*A + 7*B)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (2*a^2*(80*A - 77*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(3/2)) + (4*a^2*(130*I*A + 133*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(7*d*Tan[c + d*x]^(7/2))} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2), x, 8, ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*((4*I)*A + 3*B)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(7/2)) + (2*a^2*(46*A - (45*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(5/2)) + (8*a^2*((59*I)*A + 60*B)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*Tan[c + d*x]^(3/2)) - (8*a^2*(197*A - (195*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(9*d*Tan[c + d*x]^(9/2))} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(13/2), x, 9, ((4 + 4*I)*a^(5/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*(14*I*A + 11*B)*Sqrt[a + I*a*Tan[c + d*x]])/(99*d*Tan[c + d*x]^(9/2)) + (2*a^2*(212*A - 209*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(693*d*Tan[c + d*x]^(7/2)) + (4*a^2*(250*I*A + 253*B)*Sqrt[a + I*a*Tan[c + d*x]])/(1155*d*Tan[c + d*x]^(5/2)) - (8*a^2*(655*A - 649*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3465*d*Tan[c + d*x]^(3/2)) - (8*a^2*(2155*I*A + 2167*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3465*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + I*a*Tan[c + d*x])^(3/2))/(11*d*Tan[c + d*x]^(11/2))} - - -{((a + I*a*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 9, (2*(-1)^(3/4)*a^(5/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d + ((2 + 2*I)*a^(3/2)*(2*a + 3*I*b)*B*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a*(a + 3*I*b)*B*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (b*B*(a + I*a*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 9, ((-1)^(3/4)*(2*I*A - B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) - ((1/2 - I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + ((I*A - B)*Tan[c + d*x]^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((A + 2*I*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 8, -((2*(-1)^(1/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d)) - ((1/2 + I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), x, 4, ((1/2 - I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 5, ((1/2 + I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + (A + I*B)/(d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - ((3*A + I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 6, ((1/2 + I/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + (A + I*B)/(d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - ((5*A + 3*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d*Tan[c + d*x]^(3/2)) + ((7*I*A - 9*B)*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]), x, 7, ((-1/2 - I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(Sqrt[a]*d) + (A + I*B)/(d*Tan[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]) - ((7*A + (5*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(5*a*d*Tan[c + d*x]^(5/2)) + (((23*I)*A - 25*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d*Tan[c + d*x]^(3/2)) + ((61*A + (35*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(15*a*d*Sqrt[Tan[c + d*x]])} - - -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 9, (2*(-1)^(3/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) - ((1/4 - I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + ((I*A - B)*Tan[c + d*x]^(3/2))/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((A + 3*I*B)*Sqrt[Tan[c + d*x]])/(2*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 5, -(((1/4 + I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d)) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((I*A + 5*B)*Sqrt[Tan[c + d*x]])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)), x, 5, ((1/4 - I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((7*A + I*B)*Sqrt[Tan[c + d*x]])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 6, ((1/4 + I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + (A + I*B)/(3*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (11*A + (5*I)*B)/(6*a*d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - ((25*A + (7*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 7, ((1/4 + I/4)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(3/2)*d) + (A + I*B)/(3*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (5*A + 3*I*B)/(2*a*d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - ((21*A + 11*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d*Tan[c + d*x]^(3/2)) + ((39*I*A - 25*B)*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d*Sqrt[Tan[c + d*x]])} - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 10, (2*(-1)^(1/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((I*A - B)*Tan[c + d*x]^(5/2))/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((A + 3*I*B)*Tan[c + d*x]^(3/2))/(6*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I*A - 7*B)*Sqrt[Tan[c + d*x]])/(4*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 6, -(((1/8 - I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d)) + ((I*A - B)*Tan[c + d*x]^(3/2))/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((A + 11*I*B)*Sqrt[Tan[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((13*A - 37*I*B)*Sqrt[Tan[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 6, -(((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d)) + ((I*A - B)*Sqrt[Tan[c + d*x]])/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((3*I*A + 7*B)*Sqrt[Tan[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((3*I*A - 13*B)*Sqrt[Tan[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)), x, 6, ((1/8 - I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + ((A + I*B)*Sqrt[Tan[c + d*x]])/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((13*A + 3*I*B)*Sqrt[Tan[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((67*A - 3*I*B)*Sqrt[Tan[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 7, ((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + (A + I*B)/(5*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + (17*A + (7*I)*B)/(30*a*d*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (151*A + (41*I)*B)/(60*a^2*d*Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - ((317*A + (67*I)*B)*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 8, ((1/8 + I/8)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/(a^(5/2)*d) + (A + I*B)/(5*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + (21*A + 11*I*B)/(30*a*d*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (89*A + 39*I*B)/(20*a^2*d*Tan[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]) - ((361*A + 151*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d*Tan[c + d*x]^(3/2)) + ((707*I*A - 317*B)*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^0*(a + I*a*Tan[c + d*x])^(1/3)*(A + B*Tan[c + d*x]), x, 6, -((a^(1/3)*(A - I*B)*x)/(2*2^(2/3))) - (Sqrt[3]*a^(1/3)*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(2/3)*d) + (a^(1/3)*(I*A + B)*Log[Cos[c + d*x]])/(2*2^(2/3)*d) + (3*a^(1/3)*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(2/3)*d) + (3*B*(a + I*a*Tan[c + d*x])^(1/3))/d} - - -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 8, (a^(2/3)*(A - I*B)*x)/(2*2^(1/3)) - (Sqrt[3]*a^(2/3)*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) - (a^(2/3)*(I*A + B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) - (3*a^(2/3)*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d) - (9*B*(a + I*a*Tan[c + d*x])^(2/3))/(8*d) + (3*B*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3))/(8*d) - (3*(4*I*A + B)*(a + I*a*Tan[c + d*x])^(5/3))/(20*a*d)} -{Tan[c + d*x]^1*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 7, (a^(2/3)*(I*A + B)*x)/(2*2^(1/3)) + (Sqrt[3]*a^(2/3)*(A - I*B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) + (a^(2/3)*(A - I*B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) + (3*a^(2/3)*(A - I*B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d) + (3*A*(a + I*a*Tan[c + d*x])^(2/3))/(2*d) - (3*I*B*(a + I*a*Tan[c + d*x])^(5/3))/(5*a*d)} -{Tan[c + d*x]^0*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 6, -((a^(2/3)*(A - I*B)*x)/(2*2^(1/3))) + (Sqrt[3]*a^(2/3)*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) + (a^(2/3)*(I*A + B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) + (3*a^(2/3)*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d) + (3*B*(a + I*a*Tan[c + d*x])^(2/3))/(2*d)} -{Cot[c + d*x]^1*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 11, -((a^(2/3)*(I*A + B)*x)/(2*2^(1/3))) + (Sqrt[3]*a^(2/3)*A*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/d - (Sqrt[3]*a^(2/3)*(A - I*B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) - (a^(2/3)*(A - I*B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) - (a^(2/3)*A*Log[Tan[c + d*x]])/(2*d) + (3*a^(2/3)*A*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*a^(2/3)*(A - I*B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d)} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 12, (a^(2/3)*(A - I*B)*x)/(2*2^(1/3)) + (a^(2/3)*(2*I*A + 3*B)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*d) - (Sqrt[3]*a^(2/3)*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) - (a^(2/3)*(I*A + B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) - (a^(2/3)*(2*I*A + 3*B)*Log[Tan[c + d*x]])/(6*d) + (a^(2/3)*(2*I*A + 3*B)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*d) - (3*a^(2/3)*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d) - (A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^(2/3))/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(1/3), x, 6, -(((A - I*B)*x)/(4*2^(1/3)*a^(1/3))) + (Sqrt[3]*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(1/3)*a^(1/3)*d) + ((I*A + B)*Log[Cos[c + d*x]])/(4*2^(1/3)*a^(1/3)*d) + (3*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(1/3)*a^(1/3)*d) + (3*(I*A - B))/(2*d*(a + I*a*Tan[c + d*x])^(1/3))} -{(A + B*Tan[c + d*x])/(a + I*a*Tan[c + d*x])^(2/3), x, 6, -(((A - I*B)*x)/(4*2^(2/3)*a^(2/3))) - (Sqrt[3]*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])^(1/3))/(Sqrt[3]*a^(1/3))])/(2*2^(2/3)*a^(2/3)*d) + ((I*A + B)*Log[Cos[c + d*x]])/(4*2^(2/3)*a^(2/3)*d) + (3*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(4*2^(2/3)*a^(2/3)*d) + (3*(I*A - B))/(4*d*(a + I*a*Tan[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with m symbolic*) - - -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, If[$VersionNumber>=8, -((2*a^4*(A*(64 + 60*m + 19*m^2 + 2*m^3) - I*B*(67 + 60*m + 19*m^2 + 2*m^3))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(2 + m)*(3 + m)*(4 + m))) + (8*a^4*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (I*a*B*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^3)/(d*(4 + m)) - ((A*(4 + m) - I*B*(7 + m))*Tan[c + d*x]^(1 + m)*(a^2 + I*a^2*Tan[c + d*x])^2)/(d*(3 + m)*(4 + m)) - (2*(A*(4 + m)^2 - I*B*(19 + 8*m + m^2))*Tan[c + d*x]^(1 + m)*(a^4 + I*a^4*Tan[c + d*x]))/(d*(2 + m)*(3 + m)*(4 + m)), -((2*a^4*(A*(64 + 60*m + 19*m^2 + 2*m^3) - I*B*(67 + 60*m + 19*m^2 + 2*m^3))*Tan[c + d*x]^(1 + m))/(d*(3 + m)*(4 + m)*(2 + 3*m + m^2))) + (8*a^4*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (I*a*B*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^3)/(d*(4 + m)) - ((A*(4 + m) - I*B*(7 + m))*Tan[c + d*x]^(1 + m)*(a^2 + I*a^2*Tan[c + d*x])^2)/(d*(3 + m)*(4 + m)) - (2*(A*(4 + m)^2 - I*B*(19 + 8*m + m^2))*Tan[c + d*x]^(1 + m)*(a^4 + I*a^4*Tan[c + d*x]))/(d*(4 + m)*(6 + 5*m + m^2))]} -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, If[$VersionNumber>=8, -((a^3*(A*(15 + 11*m + 2*m^2) - I*B*(17 + 11*m + 2*m^2))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(2 + m)*(3 + m))) + (4*a^3*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (I*a*B*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^2)/(d*(3 + m)) - ((A*(3 + m) - I*B*(5 + m))*Tan[c + d*x]^(1 + m)*(a^3 + I*a^3*Tan[c + d*x]))/(d*(2 + m)*(3 + m)), -((a^3*(A*(15 + 11*m + 2*m^2) - I*B*(17 + 11*m + 2*m^2))*Tan[c + d*x]^(1 + m))/(d*(3 + m)*(2 + 3*m + m^2))) + (4*a^3*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (I*a*B*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^2)/(d*(3 + m)) - ((A*(3 + m) - I*B*(5 + m))*Tan[c + d*x]^(1 + m)*(a^3 + I*a^3*Tan[c + d*x]))/(d*(2 + m)*(3 + m))]} -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, (I*a^2*(B + (I*A + B)*(2 + m))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(2 + m)) + (2*a^2*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (I*B*Tan[c + d*x]^(1 + m)*(a^2 + I*a^2*Tan[c + d*x]))/(d*(2 + m))} -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^1*(A + B*Tan[c + d*x]), x, 3, (I*a*B*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (a*(A - I*B)*Hypergeometric2F1[1, 1 + m, 2 + m, I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^1, x, 6, ((A*(1 - m) - I*B*(1 + m))*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(2*a*d*(1 + m)) + ((I*A - B)*m*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(2*a*d*(2 + m)) + ((A + I*B)*Tan[c + d*x]^(1 + m))/(2*d*(a + I*a*Tan[c + d*x]))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 7, ((1 - m)*(A*(1 - m) - I*B*(1 + m))*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(4*a^2*d*(1 + m)) + ((A*(2 - m) - I*B*m)*Tan[c + d*x]^(1 + m))/(4*a^2*d*(1 + I*Tan[c + d*x])) + (m*(I*A*(2 - m) + B*m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(4*a^2*d*(2 + m)) + ((A + I*B)*Tan[c + d*x]^(1 + m))/(4*d*(a + I*a*Tan[c + d*x])^2)} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 8, -(((1 - m)*(I*B*(3 + m - 2*m^2) - A*(3 - 7*m + 2*m^2))*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(24*a^3*d*(1 + m))) + ((2 - m)*m*(B + I*A*(5 - 2*m) + 2*B*m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(24*a^3*d*(2 + m)) + ((A + I*B)*Tan[c + d*x]^(1 + m))/(6*d*(a + I*a*Tan[c + d*x])^3) + ((I*B*(1 - 2*m) + A*(7 - 2*m))*Tan[c + d*x]^(1 + m))/(24*a*d*(a + I*a*Tan[c + d*x])^2) + ((2 - m)*(A*(5 - 2*m) - I*(B + 2*B*m))*Tan[c + d*x]^(1 + m))/(24*d*(a^3 + I*a^3*Tan[c + d*x]))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^4, x, 9, -(((3 - 4*m + m^2)*(I*B*(1 - m^2) - A*(1 - 4*m + m^2))*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(48*a^4*d*(1 + m))) - ((I*B*(1 + 3*m - m^2) - A*(13 - 7*m + m^2))*Tan[c + d*x]^(1 + m))/(48*a^4*d*(1 + I*Tan[c + d*x])^2) - ((2 - m)*(I*B*(2 + 2*m - m^2) - A*(8 - 6*m + m^2))*Tan[c + d*x]^(1 + m))/(48*a^4*d*(1 + I*Tan[c + d*x])) + ((2 - m)*m*(B*(2 + 2*m - m^2) + I*A*(8 - 6*m + m^2))*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(48*a^4*d*(2 + m)) + ((A + I*B)*Tan[c + d*x]^(1 + m))/(8*d*(a + I*a*Tan[c + d*x])^4) + ((I*B*(1 - m) + A*(5 - m))*Tan[c + d*x]^(1 + m))/(24*a*d*(a + I*a*Tan[c + d*x])^3)} - - -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 9, (4*a^3*(A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + (2*a^2*(2*B*(19 + 17*m + 4*m^2) + I*A*(35 + 34*m + 8*m^2))*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*(d*(3 + 2*m)*(5 + 2*m))) + (2*a^2*(2*I*B*(4 + m) - A*(5 + 2*m))*Tan[c + d*x]^(1 + m)*Sqrt[a + I*a*Tan[c + d*x]])/(d*(3 + 2*m)*(5 + 2*m)) + (2*I*a*B*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^(3/2))/(d*(5 + 2*m))} -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 8, (2*a^2*(A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + (2*a*(B + (I*A + B)*(3 + 2*m))*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*(d*(3 + 2*m))) + (2*I*a*B*Tan[c + d*x]^(1 + m)*Sqrt[a + I*a*Tan[c + d*x]])/(d*(3 + 2*m))} -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^(1/2)*(A + B*Tan[c + d*x]), x, 7, (a*(A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + (2*B*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*d)} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(1/2), x, 8, ((A + I*B)*Tan[c + d*x]^(1 + m))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(2*d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + ((I*A - B)*(1 + 2*m)*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*(a*d))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 9, ((A + I*B)*Tan[c + d*x]^(1 + m))/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((A*(5 - 4*m) - I*(B + 4*B*m))*Tan[c + d*x]^(1 + m))/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(4*a*d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + ((1 + 2*m)*(B + I*A*(5 - 4*m) + 4*B*m)*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*(6*a^2*d))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 10, ((A + I*B)*Tan[c + d*x]^(1 + m))/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((I*B*(1 - 4*m) + A*(11 - 4*m))*Tan[c + d*x]^(1 + m))/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) - ((I*B*(13 + 12*m - 16*m^2) - A*(37 - 52*m + 16*m^2))*Tan[c + d*x]^(1 + m))/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[1 + I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m))/(8*a^2*d*(1 + m)*Sqrt[a + I*a*Tan[c + d*x]]) + ((1 + 2*m)*(B*(13 + 12*m - 16*m^2) + I*A*(37 - 52*m + 16*m^2))*Hypergeometric2F1[1/2, -m, 3/2, 1 + I*Tan[c + d*x]]*Tan[c + d*x]^m*Sqrt[a + I*a*Tan[c + d*x]])/(((-I)*Tan[c + d*x])^m*(60*a^3*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with n symbolic*) - - -{Tan[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 7, (1/(d*(1 + m)))*(((A - I*B)*AppellF1[1 + m, 1 - n, 1, 2 + m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^n)/(1 + I*Tan[c + d*x])^n) + (1/(d*(1 + m)))*((I*B*Hypergeometric2F1[1 + m, 1 - n, 2 + m, (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*(a + I*a*Tan[c + d*x])^n)/(1 + I*Tan[c + d*x])^n)} - - -{Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 6, If[$VersionNumber>=8, (2*(I*B*n - A*(3 + n))*(a + I*a*Tan[c + d*x])^n)/(d*n*(2 + n)*(3 + n)) + ((A - I*B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - ((I*B*n - A*(3 + n))*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n)/(d*(2 + n)*(3 + n)) + (B*Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^n)/(d*(3 + n)) - ((A*n*(3 + n) - I*B*(6 + 3*n + n^2))*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n)*(2 + n)*(3 + n)), (2*(I*B*n - A*(3 + n))*(a + I*a*Tan[c + d*x])^n)/(d*n*(2 + n)*(3 + n)) + ((A - I*B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - ((I*B*n - A*(3 + n))*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n)/(d*(2 + n)*(3 + n)) + (B*Tan[c + d*x]^3*(a + I*a*Tan[c + d*x])^n)/(d*(3 + n)) - ((A*n*(3 + n) - I*B*(6 + 3*n + n^2))*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(3 + n)*(2 + 3*n + n^2))]} -{Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 5, -((2*B*(a + I*a*Tan[c + d*x])^n)/(d*n*(2 + n))) + ((I*A + B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) + (B*Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^n)/(d*(2 + n)) - ((B*n + I*A*(2 + n))*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n)*(2 + n))} -{Tan[c + d*x]^1*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 4, (A*(a + I*a*Tan[c + d*x])^n)/(d*n) - ((A - I*B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - (I*B*(a + I*a*Tan[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Tan[c + d*x]^0*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 3, (B*(a + I*a*Tan[c + d*x])^n)/(d*n) - ((I*A + B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n)} -{Cot[c + d*x]^1*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 5, ((A - I*B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - (A*Hypergeometric2F1[1, n, 1 + n, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*n)} -{Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 6, -((A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^n)/d) + ((I*A + B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - ((B + I*A*n)*Hypergeometric2F1[1, n, 1 + n, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*n)} -{Cot[c + d*x]^3*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 7, -(((2*B + I*A*n)*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^n)/(2*d)) - (A*Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^n)/(2*d) - ((A - I*B)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a + I*a*Tan[c + d*x])^n)/(2*d*n) - ((2*I*B*n - A*(2 - n + n^2))*Hypergeometric2F1[1, n, 1 + n, 1 + I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(2*d*n)} - - -{Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 11, If[$VersionNumber>=8, -((2*(2*I*A*n*(5 + 2*n) + B*(15 + 10*n + 4*n^2))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) - (2*(4*B*n*(9 + 8*n + 2*n^2) + I*A*(15 + 36*n + 32*n^2 + 8*n^3))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*(3 + 2*n)*(5 + 2*n))) - (2*(2*I*B*n - A*(5 + 2*n))*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*(5 + 2*n)) + (2*B*Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n)), -((2*(2*I*A*n*(5 + 2*n) + B*(15 + 10*n + 4*n^2))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n)*(3 + 8*n + 4*n^2))) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) - (2*(4*B*n*(9 + 8*n + 2*n^2) + I*A*(15 + 36*n + 32*n^2 + 8*n^3))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(5 + 2*n)*(3 + 8*n + 4*n^2))) - (2*(2*I*B*n - A*(5 + 2*n))*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*(5 + 2*n)) + (2*B*Tan[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n))]} -{Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 10, If[$VersionNumber>=8, -((2*(2*I*B*n - A*(3 + 2*n))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*(3 + 2*n))) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) + (2*(2*A*n*(3 + 2*n) - I*B*(3 + 6*n + 4*n^2))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*(3 + 2*n))) + (2*B*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)), -((2*(2*I*B*n - A*(3 + 2*n))*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 8*n + 4*n^2))) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) + (2*(2*A*n*(3 + 2*n) - I*B*(3 + 6*n + 4*n^2))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(3 + 8*n + 4*n^2))) + (2*B*Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n))]} -{Tan[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 9, (2*B*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)) - (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) + (2*(2*B*n + I*A*(1 + 2*n))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)))} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(1/2), x, 8, (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) + (2*I*B*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d)} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 9, -((2*A*(a + I*a*Tan[c + d*x])^n)/(d*Sqrt[Tan[c + d*x]])) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) - (2*I*A*(1 - 2*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d)} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 10, -((2*A*(a + I*a*Tan[c + d*x])^n)/(3*d*Tan[c + d*x]^(3/2))) - (2*(3*B + 2*I*A*n)*(a + I*a*Tan[c + d*x])^n)/(3*d*Sqrt[Tan[c + d*x]]) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*d) - (2*(1 - 2*n)*(3*I*B - 2*A*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(3*d))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, -((a*A - b*B)*x) + ((A*b + a*B)*Log[Cos[c + d*x]])/d + ((a*A - b*B)*Tan[c + d*x])/d + ((A*b + a*B)*Tan[c + d*x]^2)/(2*d) + (b*B*Tan[c + d*x]^3)/(3*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 3, -((A*b + a*B)*x) - ((a*A - b*B)*Log[Cos[c + d*x]])/d + ((A*b + a*B)*Tan[c + d*x])/d + (b*B*Tan[c + d*x]^2)/(2*d)} -{(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 2, (a*A - b*B)*x - ((A*b + a*B)*Log[Cos[c + d*x]])/d + (b*B*Tan[c + d*x])/d} -{Cot[c + d*x]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, (A*b + a*B)*x - (b*B*Log[Cos[c + d*x]])/d + (a*A*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 3, -((a*A - b*B)*x) - (a*A*Cot[c + d*x])/d + ((A*b + a*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, -((A*b + a*B)*x) - ((A*b + a*B)*Cot[c + d*x])/d - (a*A*Cot[c + d*x]^2)/(2*d) - ((a*A - b*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 5, (a*A - b*B)*x + ((a*A - b*B)*Cot[c + d*x])/d - ((A*b + a*B)*Cot[c + d*x]^2)/(2*d) - (a*A*Cot[c + d*x]^3)/(3*d) - ((A*b + a*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 6, (A*b + a*B)*x + ((A*b + a*B)*Cot[c + d*x])/d + ((a*A - b*B)*Cot[c + d*x]^2)/(2*d) - ((A*b + a*B)*Cot[c + d*x]^3)/(3*d) - (a*A*Cot[c + d*x]^4)/(4*d) + ((a*A - b*B)*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, -((a^2*A - A*b^2 - 2*a*b*B)*x) + ((2*a*A*b + a^2*B - b^2*B)*Log[Cos[c + d*x]])/d - (b*(A*b + a*B)*Tan[c + d*x])/d - (B*(a + b*Tan[c + d*x])^2)/(2*d) + ((4*A*b - a*B)*(a + b*Tan[c + d*x])^3)/(12*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*b*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, -((2*a*A*b + a^2*B - b^2*B)*x) - ((a^2*A - A*b^2 - 2*a*b*B)*Log[Cos[c + d*x]])/d + (b*(a*A - b*B)*Tan[c + d*x])/d + (A*(a + b*Tan[c + d*x])^2)/(2*d) + (B*(a + b*Tan[c + d*x])^3)/(3*b*d)} -{(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 3, (a^2*A - A*b^2 - 2*a*b*B)*x - ((2*a*A*b + a^2*B - b^2*B)*Log[Cos[c + d*x]])/d + (b*(A*b + a*B)*Tan[c + d*x])/d + (B*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, (2*a*A*b + a^2*B - b^2*B)*x - (b*(A*b + 2*a*B)*Log[Cos[c + d*x]])/d + (a^2*A*Log[Sin[c + d*x]])/d + (b^2*B*Tan[c + d*x])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, -((a^2*A - A*b^2 - 2*a*b*B)*x) - (a^2*A*Cot[c + d*x])/d - (b^2*B*Log[Cos[c + d*x]])/d + (a*(2*A*b + a*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 4, (b^2*B - a*(2*A*b + a*B))*x - (a*(2*A*b + a*B)*Cot[c + d*x])/d - (a^2*A*Cot[c + d*x]^2)/(2*d) - ((a^2*A - A*b^2 - 2*a*b*B)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, (a^2*A - A*b^2 - 2*a*b*B)*x + ((a^2*A - A*b^2 - 2*a*b*B)*Cot[c + d*x])/d - (a*(2*A*b + a*B)*Cot[c + d*x]^2)/(2*d) - (a^2*A*Cot[c + d*x]^3)/(3*d) + ((b^2*B - a*(2*A*b + a*B))*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 6, (2*a*A*b + a^2*B - b^2*B)*x - ((b^2*B - a*(2*A*b + a*B))*Cot[c + d*x])/d + ((a^2*A - A*b^2 - 2*a*b*B)*Cot[c + d*x]^2)/(2*d) - (a*(2*A*b + a*B)*Cot[c + d*x]^3)/(3*d) - (a^2*A*Cot[c + d*x]^4)/(4*d) + ((a^2*A - A*b^2 - 2*a*b*B)*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Cos[c + d*x]])/d - (b*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x])/d - ((A*b + a*B)*(a + b*Tan[c + d*x])^2)/(2*d) - (B*(a + b*Tan[c + d*x])^3)/(3*d) + ((5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/(20*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4)/(5*b*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, -((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*x) - ((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Log[Cos[c + d*x]])/d + (b*(a^2*A - A*b^2 - 2*a*b*B)*Tan[c + d*x])/d + ((a*A - b*B)*(a + b*Tan[c + d*x])^2)/(2*d) + (A*(a + b*Tan[c + d*x])^3)/(3*d) + (B*(a + b*Tan[c + d*x])^4)/(4*b*d)} -{(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 4, (a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x - ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Cos[c + d*x]])/d + (b*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x])/d + ((A*b + a*B)*(a + b*Tan[c + d*x])^2)/(2*d) + (B*(a + b*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*x - (b*(3*a*A*b + 3*a^2*B - b^2*B)*Log[Cos[c + d*x]])/d + (a^3*A*Log[Sin[c + d*x]])/d + (b^2*(A*b + 2*a*B)*Tan[c + d*x])/d + (b*B*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, -((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) - (b^2*(A*b + 3*a*B)*Log[Cos[c + d*x]])/d + (a^2*(3*A*b + a*B)*Log[Sin[c + d*x]])/d + (b^2*(a*A + b*B)*Tan[c + d*x])/d - (a*A*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, -((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*x) - (a^2*(2*A*b + a*B)*Cot[c + d*x])/d - (b^3*B*Log[Cos[c + d*x]])/d - (a*(a^2*A - 3*A*b^2 - 3*a*b*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 5, (a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x + (a*(3*a^2*A - 8*A*b^2 - 9*a*b*B)*Cot[c + d*x])/(3*d) - (a^2*(5*A*b + 3*a*B)*Cot[c + d*x]^2)/(6*d) - ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(3*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*x + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Cot[c + d*x])/d + (a*(2*a^2*A - 5*A*b^2 - 6*a*b*B)*Cot[c + d*x]^2)/(4*d) - (a^2*(3*A*b + 2*a*B)*Cot[c + d*x]^3)/(6*d) + ((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2)/(4*d)} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 7, -((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) - ((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Cot[c + d*x])/d + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Cot[c + d*x]^2)/(2*d) + (a*(5*a^2*A - 12*A*b^2 - 15*a*b*B)*Cot[c + d*x]^3)/(15*d) - (a^2*(7*A*b + 5*a*B)*Cot[c + d*x]^4)/(20*d) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*d)} - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x) + ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Log[Cos[c + d*x]])/d - (b*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x])/d - ((2*a*A*b + a^2*B - b^2*B)*(a + b*Tan[c + d*x])^2)/(2*d) - ((A*b + a*B)*(a + b*Tan[c + d*x])^3)/(3*d) - (B*(a + b*Tan[c + d*x])^4)/(4*d) + ((6*A*b - a*B)*(a + b*Tan[c + d*x])^5)/(30*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, -((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x) - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Cos[c + d*x]])/d + (b*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Tan[c + d*x])/d + ((a^2*A - A*b^2 - 2*a*b*B)*(a + b*Tan[c + d*x])^2)/(2*d) + ((a*A - b*B)*(a + b*Tan[c + d*x])^3)/(3*d) + (A*(a + b*Tan[c + d*x])^4)/(4*d) + (B*(a + b*Tan[c + d*x])^5)/(5*b*d)} -{(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 5, (a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Log[Cos[c + d*x]])/d + (b*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x])/d + ((2*a*A*b + a^2*B - b^2*B)*(a + b*Tan[c + d*x])^2)/(2*d) + ((A*b + a*B)*(a + b*Tan[c + d*x])^3)/(3*d) + (B*(a + b*Tan[c + d*x])^4)/(4*d)} -{Cot[c + d*x]*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, (4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x - (b*(6*a^2*A*b - A*b^3 + 4*a^3*B - 4*a*b^2*B)*Log[Cos[c + d*x]])/d + (a^4*A*Log[Sin[c + d*x]])/d + (b^2*(3*a*A*b + 3*a^2*B - b^2*B)*Tan[c + d*x])/d + (b*(A*b + 2*a*B)*(a + b*Tan[c + d*x])^2)/(2*d) + (b*B*(a + b*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, -((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x) - (b^2*(4*a*A*b + 6*a^2*B - b^2*B)*Log[Cos[c + d*x]])/d + (a^3*(4*A*b + a*B)*Log[Sin[c + d*x]])/d + (b^2*(a^2*A + A*b^2 + 3*a*b*B)*Tan[c + d*x])/d + (b*(2*a*A + b*B)*(a + b*Tan[c + d*x])^2)/(2*d) - (a*A*Cot[c + d*x]*(a + b*Tan[c + d*x])^3)/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, -((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x) - (b^3*(A*b + 4*a*B)*Log[Cos[c + d*x]])/d - (a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Log[Sin[c + d*x]])/d + (b^2*(3*a*A*b + a^2*B + b^2*B)*Tan[c + d*x])/d - (a*(5*A*b + 2*a*B)*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/(2*d) - (a*A*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3)/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, (a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x + (a^2*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x])/d - (b^4*B*Log[Cos[c + d*x]])/d - (a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Log[Sin[c + d*x]])/d - (a*(2*A*b + a*B)*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d) - (a*A*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 6, (4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x + (a*(24*a^2*A*b - 19*A*b^3 + 6*a^3*B - 34*a*b^2*B)*Cot[c + d*x])/(6*d) + (a^2*(6*a^2*A - 13*A*b^2 - 16*a*b*B)*Cot[c + d*x]^2)/(12*d) + ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Sin[c + d*x]])/d - (a*(7*A*b + 4*a*B)*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(12*d) - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(4*d)} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 7, -((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x) - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x])/d + (a*(40*a^2*A*b - 28*A*b^3 + 10*a^3*B - 55*a*b^2*B)*Cot[c + d*x]^2)/(20*d) + (a^2*(10*a^2*A - 18*A*b^2 - 25*a*b*B)*Cot[c + d*x]^3)/(30*d) + ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Log[Sin[c + d*x]])/d - (a*(8*A*b + 5*a*B)*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2)/(20*d) - (a*A*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3)/(5*d)} -{Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 8, -((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x) - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Cot[c + d*x])/d - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2)/(2*d) + (a*(20*a^2*A*b - 13*A*b^3 + 5*a^3*B - 27*a*b^2*B)*Cot[c + d*x]^3)/(15*d) + (a^2*(5*a^2*A - 8*A*b^2 - 12*a*b*B)*Cot[c + d*x]^4)/(20*d) - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Sin[c + d*x]])/d - (a*(3*A*b + 2*a*B)*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(10*d) - (a*A*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3)/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 6, -(((A*b - a*B)*x)/(a^2 + b^2)) + ((a*A + b*B)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a^3*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) + ((A*b - a*B)*Tan[c + d*x])/(b^2*d) + (B*Tan[c + d*x]^2)/(2*b*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 5, -(((a*A + b*B)*x)/(a^2 + b^2)) - ((A*b - a*B)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^2*(A*b - a*B)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) + (B*Tan[c + d*x])/(b*d)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 5, ((A*b - a*B)*x)/(a^2 + b^2) - (B*Log[Cos[c + d*x]])/(b*d) - (a*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x]), x, 2, ((a*A + b*B)*x)/(a^2 + b^2) + ((A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 3, -(((A*b - a*B)*x)/(a^2 + b^2)) + (A*Log[Sin[c + d*x]])/(a*d) - (b*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 4, -(((a*A + b*B)*x)/(a^2 + b^2)) - (A*Cot[c + d*x])/(a*d) - ((A*b - a*B)*Log[Sin[c + d*x]])/(a^2*d) + (b^2*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)*d)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 5, ((A*b - a*B)*x)/(a^2 + b^2) + ((A*b - a*B)*Cot[c + d*x])/(a^2*d) - (A*Cot[c + d*x]^2)/(2*a*d) - ((a^2*A - A*b^2 + a*b*B)*Log[Sin[c + d*x]])/(a^3*d) - (b^3*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)*d)} -{(Cot[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 6, ((a*A + b*B)*x)/(a^2 + b^2) + ((a^2*A - A*b^2 + a*b*B)*Cot[c + d*x])/(a^3*d) + ((A*b - a*B)*Cot[c + d*x]^2)/(2*a^2*d) - (A*Cot[c + d*x]^3)/(3*a*d) + ((a^2 - b^2)*(A*b - a*B)*Log[Sin[c + d*x]])/(a^4*d) + (b^4*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)*d)} - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 6, -(((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2) + ((a^2*A - A*b^2 + 2*a*b*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^2*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) - ((a*A*b - 2*a^2*B - b^2*B)*Tan[c + d*x])/(b^2*(a^2 + b^2)*d) + (a*(A*b - a*B)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 5, -(((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2) - ((2*a*A*b - a^2*B + b^2*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d) - (a^2*(A*b - a*B))/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 3, ((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2 - ((a^2*A - A*b^2 + 2*a*b*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + (a*(A*b - a*B))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2, x, 3, ((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2 + ((2*a*A*b - a^2*B + b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - (A*b - a*B)/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 4, -(((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2) + (A*Log[Sin[c + d*x]])/(a^2*d) - (b*(3*a^2*A*b + A*b^3 - 2*a^3*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)^2*d) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 5, -(((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2) - ((2*A*b - a*B)*Log[Sin[c + d*x]])/(a^3*d) + (b^2*(4*a^2*A*b + 2*A*b^3 - 3*a^3*B - a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (b*(a^2*A + 2*A*b^2 - a*b*B))/(a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 6, ((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2 - ((a^2*A - 3*A*b^2 + 2*a*b*B)*Log[Sin[c + d*x]])/(a^4*d) - (b^3*(5*a^2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b*(2*a^2*A*b + 3*A*b^3 - a^3*B - 2*a*b^2*B))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + ((3*A*b - 2*a*B)*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - (A*Cot[c + d*x]^2)/(2*a*d*(a + b*Tan[c + d*x]))} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 7, ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a^2*(a^4*A*b + 3*a^2*A*b^3 + 6*A*b^5 - 3*a^5*B - 9*a^3*b^2*B - 10*a*b^4*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3*d) - ((a^3*A*b + 3*a*A*b^3 - 3*a^4*B - 6*a^2*b^2*B - b^4*B)*Tan[c + d*x])/(b^3*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(a^2*A*b + 5*A*b^3 - 3*a^3*B - 7*a*b^2*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 6, -(((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3) + ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^3*d) + (a*(A*b - a*B)*Tan[c + d*x]^2)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 4, -(((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (a^2*(A*b - a*B))/(2*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 4, ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3 - ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (a*(A*b - a*B))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a^2*A - A*b^2 + 2*a*b*B)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^3, x, 4, ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (A*b - a*B)/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a*A*b - a^2*B + b^2*B)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 5, -(((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3) + (A*Log[Sin[c + d*x]])/(a^3*d) - (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^3*d) + (b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 6, -(((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3) - ((3*A*b - a*B)*Log[Sin[c + d*x]])/(a^4*d) + (b^2*(10*a^4*A*b + 9*a^2*A*b^3 + 3*A*b^5 - 6*a^5*B - 3*a^3*b^2*B - a*b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2*A + 3*A*b^2 - a*b*B))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4*A + 6*a^2*A*b^2 + 3*A*b^4 - 3*a^3*b*B - a*b^3*B))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 7, ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3 - ((a^2*A - 6*A*b^2 + 3*a*b*B)*Log[Sin[c + d*x]])/(a^5*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b*(5*a^2*A*b + 6*A*b^3 - 2*a^3*B - 3*a*b^2*B))/(2*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((2*A*b - a*B)*Cot[c + d*x])/(a^2*d*(a + b*Tan[c + d*x])^2) - (A*Cot[c + d*x]^2)/(2*a*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^4*A*b + 11*a^2*A*b^3 + 6*A*b^5 - a^5*B - 6*a^3*b^2*B - 3*a*b^4*B))/(a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 7, ((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Log[Cos[c + d*x]])/((a^2 + b^2)^4*d) + (a*(4*a^2*A*b^5 - 4*A*b^7 + a^7*B + 4*a^5*b^2*B + 5*a^3*b^4*B + 10*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^4*d) + (a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (a^2*(a^2*A*b^3 - 3*A*b^5 + a^5*B + 3*a^3*b^2*B + 6*a*b^4*B))/(b^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 5, -(((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4) + ((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) + (a*(A*b - a*B)*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^2*(a^2*A*b - 5*A*b^3 + 2*a^3*B + 8*a*b^2*B))/(6*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a*(a^4*A*b + 5*a^2*A*b^3 - 8*A*b^5 + 2*a^5*B + 7*a^3*b^2*B + 17*a*b^4*B))/(3*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 5, -(((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4) - ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^2*(A*b - a*B))/(3*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 5, ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4 - ((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) + (a*(A*b - a*B))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^2*A - A*b^2 + 2*a*b*B)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^4, x, 5, ((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (A*b - a*B)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (2*a*A*b - a^2*B + b^2*B)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 6, -(((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4) + (A*Log[Sin[c + d*x]])/(a^4*d) - (b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^4*d) + (b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B))/(a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 7, -(((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4) - ((4*A*b - a*B)*Log[Sin[c + d*x]])/(a^5*d) + (b^2*(20*a^6*A*b + 24*a^4*A*b^3 + 16*a^2*A*b^5 + 4*A*b^7 - 10*a^7*B - 5*a^5*b^2*B - 4*a^3*b^4*B - a*b^6*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^5*(a^2 + b^2)^4*d) - (b*(3*a^2*A + 4*A*b^2 - a*b*B))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^3) - (b*(2*a^4*A + 8*a^2*A*b^2 + 4*A*b^4 - 3*a^3*b*B - a*b^3*B))/(2*a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(a^6*A + 13*a^4*A*b^2 + 12*a^2*A*b^4 + 4*A*b^6 - 6*a^5*b*B - 3*a^3*b^3*B - a*b^5*B))/(a^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 8, ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4 - ((a^2*A - 10*A*b^2 + 4*a*b*B)*Log[Sin[c + d*x]])/(a^6*d) - (b^3*(35*a^6*A*b + 56*a^4*A*b^3 + 39*a^2*A*b^5 + 10*A*b^7 - 20*a^7*B - 24*a^5*b^2*B - 16*a^3*b^4*B - 4*a*b^6*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^6*(a^2 + b^2)^4*d) + (b*(9*a^2*A*b + 10*A*b^3 - 3*a^3*B - 4*a*b^2*B))/(3*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((5*A*b - 2*a*B)*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])^3) - (A*Cot[c + d*x]^2)/(2*a*d*(a + b*Tan[c + d*x])^3) + (b*(7*a^4*A*b + 19*a^2*A*b^3 + 10*A*b^5 - 2*a^5*B - 8*a^3*b^2*B - 4*a*b^4*B))/(2*a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(4*a^6*A*b + 27*a^4*A*b^3 + 29*a^2*A*b^5 + 10*A*b^7 - a^7*B - 13*a^5*b^2*B - 12*a^3*b^4*B - 4*a*b^6*B))/(a^5*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} - - -{(Tan[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 3, (B*Log[Cos[c + d*x]])/d + (B*Tan[c + d*x]^2)/(2*d)} -{(Tan[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 3, -(B*x) + (B*Tan[c + d*x])/d} -{(Tan[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 2, -((B*Log[Cos[c + d*x]])/d)} -{(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x]), x, 2, B*x} -{(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 2, (B*Log[Sin[c + d*x]])/d} -{(Cot[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 3, -(B*x) - (B*Cot[c + d*x])/d} -{(Cot[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 3, -(B*Cot[c + d*x]^2)/(2*d) - (B*Log[Sin[c + d*x]])/d} -{(Cot[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 4, B*x + (B*Cot[c + d*x])/d - (B*Cot[c + d*x]^3)/(3*d)} - - -{(Tan[c + d*x]^4*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 7, (a*B*x)/(a^2 + b^2) + (b*B*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^4*B*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) - (a*B*Tan[c + d*x])/(b^2*d) + (B*Tan[c + d*x]^2)/(2*b*d)} -{(Tan[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 6, -((b*B*x)/(a^2 + b^2)) + (a*B*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a^3*B*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) + (B*Tan[c + d*x])/(b*d)} -{(Tan[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 5, -((a*B*x)/b^2) + (a^3*B*x)/(b^2*(a^2 + b^2)) - (B*Log[Cos[c + d*x]])/(b*d) + (a^2*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b*(a^2 + b^2)*d)} -{(Tan[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 3, (b*B*x)/(a^2 + b^2) - (a*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2, x, 3, (a*B*x)/(a^2 + b^2) + (b*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 4, -((b*B*x)/(a^2 + b^2)) + (B*Log[Sin[c + d*x]])/(a*d) - (b^2*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)} -{(Cot[c + d*x]^2*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 5, -((a*B*x)/(a^2 + b^2)) - (B*Cot[c + d*x])/(a*d) - (b*B*Log[Sin[c + d*x]])/(a^2*d) + (b^3*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)*d)} -{(Cot[c + d*x]^3*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 6, (b*B*x)/(a^2 + b^2) + (b*B*Cot[c + d*x])/(a^2*d) - (B*Cot[c + d*x]^2)/(2*a*d) - ((a^2 - b^2)*B*Log[Sin[c + d*x]])/(a^3*d) - (b^4*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)*d)} - - -{(3 + Tan[c + d*x])/(2 - Tan[c + d*x]), x, 2, x - Log[2*Cos[c + d*x] - Sin[c + d*x]]/d} -{((b*B)/a + B*Tan[c + d*x])/(a + b*Tan[c + d*x]), x, 2, (2*b*B*x)/(a^2 + b^2) - ((a - b^2/a)*B*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} - -{(a + b*Tan[c + d*x])/(b + a*Tan[c + d*x])^2, x, 3, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^2) + (b*(3*a^2 - b^2)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^2*d) - (a^2 - b^2)/((a^2 + b^2)*d*(b + a*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 11, (Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*A*Sqrt[a + b*Tan[c + d*x]])/d - (2*(14*a*A*b - 8*a^2*B + 35*b^2*B)*(a + b*Tan[c + d*x])^(3/2))/(105*b^3*d) + (2*(7*A*b - 4*a*B)*Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/(35*b^2*d) + (2*B*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2))/(7*b*d)} -{Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 10, (Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*B*Sqrt[a + b*Tan[c + d*x]])/d + (2*(5*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(3/2))/(15*b^2*d) + (2*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/(5*b*d)} -{Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 9, -((Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*A*Sqrt[a + b*Tan[c + d*x]])/d + (2*B*(a + b*Tan[c + d*x])^(3/2))/(3*b*d)} -{Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 8, -((Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*B*Sqrt[a + b*Tan[c + d*x]])/d} -{Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 11, (-2*Sqrt[a]*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + (Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d} -{Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 12, -(((A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) + (Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 13, ((8*a^2*A + A*b^2 - 4*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(3/2)*d) - (Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - ((A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*a*d) - (A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d)} -{Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 14, ((8*a^2*A*b - A*b^3 + 16*a^3*B + 2*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(5/2)*d) - (Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A + A*b^2 - 2*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*a^2*d) - ((A*b + 6*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*a*d) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(3*d)} - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 11, ((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*(A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/d - (2*B*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*(7*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(5/2))/(35*b^2*d) + (2*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2))/(7*b*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 10, -(((a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - ((a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(a*A - b*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*A*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*B*(a + b*Tan[c + d*x])^(5/2))/(5*b*d)} -{(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 9, -(((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*B*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 12, (-2*a^(3/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + ((a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*B*Sqrt[a + b*Tan[c + d*x]])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 12, -((Sqrt[a]*(3*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + ((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (a*A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 13, ((8*a^2*A - 3*A*b^2 - 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*Sqrt[a]*d) - ((a - I*b)^(3/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - ((5*A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 14, ((24*a^2*A*b + A*b^3 + 16*a^3*B - 6*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(3/2)*d) - ((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A - A*b^2 - 10*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*a*d) - ((7*A*b + 6*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*d) - (a*A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(3*d)} - - -{Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 12, ((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Tan[c + d*x]])/d - (2*(A*b + a*B)*(a + b*Tan[c + d*x])^(3/2))/(3*d) - (2*B*(a + b*Tan[c + d*x])^(5/2))/(5*d) + (2*(9*A*b - 2*a*B)*(a + b*Tan[c + d*x])^(7/2))/(63*b^2*d) + (2*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(7/2))/(9*b*d)} -{Tan[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 11, -(((a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - ((a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(a^2*A - A*b^2 - 2*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*(a*A - b*B)*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*A*(a + b*Tan[c + d*x])^(5/2))/(5*d) + (2*B*(a + b*Tan[c + d*x])^(7/2))/(7*b*d)} -{(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, -(((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) + ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*(A*b + a*B)*(a + b*Tan[c + d*x])^(3/2))/(3*d) + (2*B*(a + b*Tan[c + d*x])^(5/2))/(5*d)} -{Cot[c + d*x]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 13, (-2*a^(5/2)*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d + ((a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(A*b + 2*a*B)*Sqrt[a + b*Tan[c + d*x]])/d + (2*b*B*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 13, -((a^(3/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d) + ((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (b*(a*A + 2*b*B)*Sqrt[a + b*Tan[c + d*x]])/d - (a*A*Cot[c + d*x]*(a + b*Tan[c + d*x])^(3/2))/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 13, (Sqrt[a]*(8*a^2*A - 15*A*b^2 - 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*d) - ((a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (a*(7*A*b + 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(3/2))/(2*d)} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 14, ((40*a^2*A*b - 5*A*b^3 + 16*a^3*B - 30*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*Sqrt[a]*d) - ((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A - 11*A*b^2 - 18*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*d) - (a*(3*A*b + 2*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 15, -((128*a^4*A - 240*a^2*A*b^2 - 5*A*b^4 - 320*a^3*b*B + 40*a*b^3*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(64*a^(3/2)*d) + ((a - I*b)^(5/2)*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + ((a + I*b)^(5/2)*(A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((144*a^2*A*b - 5*A*b^3 + 64*a^3*B - 88*a*b^2*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(64*a*d) + ((48*a^2*A - 59*A*b^2 - 104*a*b*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(96*d) - (a*(11*A*b + 8*a*B)*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(3/2))/(4*d)} - - -{(-a + b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(5/2), x, 10, ((I*a - b)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d - (2*b*(a^2 + b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (2*b*(a + b*Tan[c + d*x])^(5/2))/(5*d)} -{(-a + b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(3/2), x, 13, -((b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*b*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{(-a + b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1/2), x, 13, -((b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*b*Sqrt[a + b*Tan[c + d*x]])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 10, ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) - (2*(10*a*A*b - 8*a^2*B + 15*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(15*b^3*d) + (2*(5*A*b - 4*a*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(15*b^2*d) + (2*B*Tan[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(5*b*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 9, ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) + (2*(3*A*b - 2*a*B)*Sqrt[a + b*Tan[c + d*x]])/(3*b^2*d) + (2*B*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b*d)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 8, -(((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) + (2*B*Sqrt[a + b*Tan[c + d*x]])/(b*d)} -{(A + B*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]], x, 7, -(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 11, (-2*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 12, ((A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) - (A*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*d)} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 13, ((8*a^2*A - 3*A*b^2 + 4*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(5/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) + ((3*A*b - 4*a*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(4*a^2*d) - (A*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(2*a*d)} - - -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 10, ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(6*a^2*A*b + 3*A*b^3 - 8*a^3*B - 5*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d) - (2*(3*a*A*b - 4*a^2*B - b^2*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b^2*(a^2 + b^2)*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 9, ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*a^2*(A*b - a*B))/(b^2*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + (2*B*Sqrt[a + b*Tan[c + d*x]])/(b^2*d)} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 8, -(((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*a*(A*b - a*B))/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2), x, 8, -(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*(A*b - a*B))/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 12, (-2*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*b*(A*b - a*B))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 13, ((3*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (b*(a^2*A + 3*A*b^2 - 2*a*b*B))/(a^2*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) - (A*Cot[c + d*x])/(a*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 14, ((8*a^2*A - 15*A*b^2 + 12*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(7/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (b*(7*a^2*A*b + 15*A*b^3 - 4*a^3*B - 12*a*b^2*B))/(4*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + ((5*A*b - 4*a*B)*Cot[c + d*x])/(4*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - (A*Cot[c + d*x]^2)/(2*a*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(Tan[c + d*x]^4*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 11, -(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(a^2*A*b + 3*A*b^3 - 2*a^3*B - 4*a*b^2*B)*Tan[c + d*x]^2)/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) + (2*(8*a^4*A*b + 17*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B - 30*a^3*b^2*B - 8*a*b^4*B)*Sqrt[a + b*Tan[c + d*x]])/(3*b^4*(a^2 + b^2)^2*d) - (2*(4*a^3*A*b + 10*a*A*b^3 - 8*a^4*B - 15*a^2*b^2*B - b^4*B)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)^2*d)} -{(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 10, ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x]^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*a^2*(a^2*A*b + 7*A*b^3 - 4*a^3*B - 10*a*b^2*B))/(3*b^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]]) - (2*(a*A*b - 4*a^2*B - 3*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(3*b^3*(a^2 + b^2)*d)} -{(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 9, ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*a^2*(A*b - a*B))/(3*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Tan[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 9, -(((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*a*(A*b - a*B))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*(a^2*A - A*b^2 + 2*a*b*B))/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2), x, 9, -(((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (2*(A*b - a*B))/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*(2*a*A*b - a^2*B + b^2*B))/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 13, (-2*A*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) + ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 14, ((5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(7/2)*d) + ((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (b*(3*a^2*A + 5*A*b^2 - 2*a*b*B))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (b*(a^4*A + 10*a^2*A*b^2 + 5*A*b^4 - 6*a^3*b*B - 2*a*b^3*B))/(a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 15, ((8*a^2*A - 35*A*b^2 + 20*a*b*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(4*a^(9/2)*d) - ((A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((A + I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (b*(27*a^2*A*b + 35*A*b^3 - 12*a^3*B - 20*a*b^2*B))/(12*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + ((7*A*b - 4*a*B)*Cot[c + d*x])/(4*a^2*d*(a + b*Tan[c + d*x])^(3/2)) - (A*Cot[c + d*x]^2)/(2*a*d*(a + b*Tan[c + d*x])^(3/2)) + (b*(11*a^4*A*b + 62*a^2*A*b^3 + 35*A*b^5 - 4*a^5*B - 40*a^3*b^2*B - 20*a*b^4*B))/(4*a^4*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(a*B + b*B*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]], x, 12, (b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*B*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*B*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} -{(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2), x, 12, (b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*B*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*B*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*B*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a^2 + b^2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} -{(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 12, (-2*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(a*B + b*B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2), x, 9, ((-I)*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + (I*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (2*b*B)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 13, (-2*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*b^2*B)/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(1/2), x, 7, ((I*a - b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(3/2), x, 8, ((I*a - b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(-a + b*Tan[c + d*x])/(a + b*Tan[c + d*x])^(5/2), x, 9, ((I*a - b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (4*a*b)/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(3*a^2 - b^2))/((a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]], x, 3, -((2*I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d))} -{(1 - I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]], x, 3, (2*I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} - - -{(3 + Tan[x])/Sqrt[4 + 3*Tan[x]], x, 2, (-Sqrt[2])*ArcTan[(1 - 3*Tan[x])/(Sqrt[2]*Sqrt[4 + 3*Tan[x]])]} -{(1 - 3*Tan[x])/Sqrt[4 + 3*Tan[x]], x, 2, Sqrt[2]*ArcTanh[(3 + Tan[x])/(Sqrt[2]*Sqrt[4 + 3*Tan[x]])]} - - -{(4 - 3*Tan[a + b*x])/Sqrt[4 + 3*Tan[a + b*x]], x, 5, -((9*ArcTan[(1 - 3*Tan[a + b*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[a + b*x]])])/(5*Sqrt[2]*b)) + (13*ArcTanh[(3 + Tan[a + b*x])/(Sqrt[2]*Sqrt[4 + 3*Tan[a + b*x]])])/(5*Sqrt[2]*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 14, ((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*(A*b + a*B)*Sqrt[Tan[c + d*x]])/d + (2*(a*A - b*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*(A*b + a*B)*Tan[c + d*x]^(5/2))/(5*d) + (2*b*B*Tan[c + d*x]^(7/2))/(7*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 13, ((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(a*A - b*B)*Sqrt[Tan[c + d*x]])/d + (2*(A*b + a*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*B*Tan[c + d*x]^(5/2))/(5*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 12, -(((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(A*b + a*B)*Sqrt[Tan[c + d*x]])/d + (2*b*B*Tan[c + d*x]^(3/2))/(3*d)} -{((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 11, -(((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((b*(A - B) + a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*B*Sqrt[Tan[c + d*x]])/d} -{((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 11, ((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a*A)/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 12, ((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a*A)/(3*d*Tan[c + d*x]^(3/2)) - (2*(A*b + a*B))/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 13, -(((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a*A)/(5*d*Tan[c + d*x]^(5/2)) - (2*(A*b + a*B))/(3*d*Tan[c + d*x]^(3/2)) + (2*(a*A - b*B))/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 15, ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]])/d + (2*(a^2*A - A*b^2 - 2*a*b*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x]^(5/2))/(5*d) + (2*b*(9*A*b + 11*a*B)*Tan[c + d*x]^(7/2))/(63*d) + (2*b*B*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d)} -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 14, ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(a^2*A - A*b^2 - 2*a*b*B)*Sqrt[Tan[c + d*x]])/d + (2*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*(7*A*b + 9*a*B)*Tan[c + d*x]^(5/2))/(35*d) + (2*b*B*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x]))/(7*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 13, -(((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]])/d + (2*b*(5*A*b + 7*a*B)*Tan[c + d*x]^(3/2))/(15*d) + (2*b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))/(5*d)} -{((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 12, -(((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*(3*A*b + 5*a*B)*Sqrt[Tan[c + d*x]])/(3*d) + (2*b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(3*d)} -{((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 12, ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2*A)/(d*Sqrt[Tan[c + d*x]]) + (2*b^2*B*Sqrt[Tan[c + d*x]])/d} -{((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 12, ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2*A)/(3*d*Tan[c + d*x]^(3/2)) - (2*a*(2*A*b + a*B))/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 13, -(((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2*A)/(5*d*Tan[c + d*x]^(5/2)) - (2*a*(2*A*b + a*B))/(3*d*Tan[c + d*x]^(3/2)) + (2*(a^2*A - A*b^2 - 2*a*b*B))/(d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 15, ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Sqrt[Tan[c + d*x]])/d + (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*b*(27*a*A*b + 22*a^2*B - 9*b^2*B)*Tan[c + d*x]^(5/2))/(45*d) + (2*b^2*(9*A*b + 13*a*B)*Tan[c + d*x]^(7/2))/(63*d) + (2*b*B*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2)/(9*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 14, -(((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Tan[c + d*x]])/d + (2*b*(21*a*A*b + 18*a^2*B - 7*b^2*B)*Tan[c + d*x]^(3/2))/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Tan[c + d*x]^(5/2))/(35*d) + (2*b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2)/(7*d)} -{((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 13, -(((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*(15*a*A*b + 14*a^2*B - 5*b^2*B)*Sqrt[Tan[c + d*x]])/(5*d) + (2*b^2*(5*A*b + 9*a*B)*Tan[c + d*x]^(3/2))/(15*d) + (2*b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2)/(5*d)} -{((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 13, ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*b*(2*a^2*A + A*b^2 + 3*a*b*B)*Sqrt[Tan[c + d*x]])/d + (2*b^2*(3*a*A + b*B)*Tan[c + d*x]^(3/2))/(3*d) - (2*a*A*(a + b*Tan[c + d*x])^2)/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 13, ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2*(7*A*b + 3*a*B))/(3*d*Sqrt[Tan[c + d*x]]) + (2*b^2*(a*A + 3*b*B)*Sqrt[Tan[c + d*x]])/(3*d) - (2*a*A*(a + b*Tan[c + d*x])^2)/(3*d*Tan[c + d*x]^(3/2))} -{((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 13, -(((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*a^2*(9*A*b + 5*a*B))/(15*d*Tan[c + d*x]^(3/2)) + (2*a*(5*a^2*A - 14*A*b^2 - 15*a*b*B))/(5*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^2)/(5*d*Tan[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 16, ((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(b^2*d) + (2*B*Tan[c + d*x]^(3/2))/(3*b*d)} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 15, -(((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)*d) + ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*B*Sqrt[Tan[c + d*x]])/(b*d)} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 14, -(((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])), x, 14, ((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 15, ((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*A)/(a*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 16, -(((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)*d) + ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*A)/(3*a*d*Tan[c + d*x]^(3/2)) + (2*(A*b - a*B))/(a^2*d*Sqrt[Tan[c + d*x]])} - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 16, ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^(3/2)*(a^2*A*b + 5*A*b^3 - 3*a^3*B - 7*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a*A*b - 3*a^2*B - 2*b^2*B)*Sqrt[Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) + (a*(A*b - a*B)*Tan[c + d*x]^(3/2))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 15, -(((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (Sqrt[a]*(a^2*A*b - 3*A*b^3 + a^3*B + 5*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 15, -(((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((A*b - a*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2), x, 15, ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (Sqrt[b]*(5*a^2*A*b + A*b^3 - 3*a^3*B + a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2), x, 16, ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (b^(3/2)*(7*a^2*A*b + 3*A*b^3 - 5*a^3*B - a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(5/2)*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2*A + 3*A*b^2 - a*b*B)/(a^2*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2), x, 17, -(((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(5/2)*(9*a^2*A*b + 5*A*b^3 - 7*a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(7/2)*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (2*a^2*A + 5*A*b^2 - 3*a*b*B)/(3*a^2*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)) + (4*a^2*A*b + 5*A*b^3 - 2*a^3*B - 3*a*b^2*B)/(a^3*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x]))} - - -{(Tan[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 17, ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^(3/2)*(3*a^4*A*b + 6*a^2*A*b^3 + 35*A*b^5 - 15*a^5*B - 46*a^3*b^2*B - 63*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(7/2)*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^3*A*b + 11*a*A*b^3 - 15*a^4*B - 31*a^2*b^2*B - 8*b^4*B)*Sqrt[Tan[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Tan[c + d*x]^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(a^2*A*b + 9*A*b^3 - 5*a^3*B - 13*a*b^2*B)*Tan[c + d*x]^(3/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 16, ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (Sqrt[a]*(a^4*A*b + 18*a^2*A*b^3 - 15*A*b^5 + 3*a^5*B + 6*a^3*b^2*B + 35*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*b^(5/2)*(a^2 + b^2)^3*d) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (a*(A*b - a*B)*Tan[c + d*x]^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a*(a^2*A*b - 7*A*b^3 + 3*a^3*B + 11*a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 16, -(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((3*a^4*A*b - 26*a^2*A*b^3 + 3*A*b^5 + a^5*B + 18*a^3*b^2*B - 15*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*Sqrt[a]*b^(3/2)*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (a*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((3*a^2*A*b - 5*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 16, -(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((15*a^4*A*b - 18*a^2*A*b^3 - A*b^5 - 3*a^5*B + 26*a^3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(3/2)*Sqrt[b]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B + 5*a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^3), x, 16, ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (Sqrt[b]*(35*a^4*A*b + 6*a^2*A*b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3*b^2*B + a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(5/2)*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + (b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3), x, 17, ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (b^(3/2)*(63*a^4*A*b + 46*a^2*A*b^3 + 15*A*b^5 - 35*a^5*B - 6*a^3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(7/2)*(a^2 + b^2)^3*d) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B)/(4*a^3*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]) + (b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2) + (b*(13*a^2*A*b + 5*A*b^3 - 9*a^3*B - a*b^2*B))/(4*a^2*(a^2 + b^2)^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x]))} - - -{(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 13, (B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*B*Tan[c + d*x]^(3/2))/(3*d)} -{(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 13, (B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*B*Sqrt[Tan[c + d*x]])/d} -{(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 12, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])), x, 12, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 13, (B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*B)/(d*Sqrt[Tan[c + d*x]])} -{(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 13, (B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) + (B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*B)/(3*d*Tan[c + d*x]^(3/2))} - - -{(Tan[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 16, ((a + b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(5/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)*d) - ((a - b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + (2*B*Sqrt[Tan[c + d*x]])/(b*d)} -{(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 15, ((a - b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(3/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2 + b^2)*d) + ((a + b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 15, -(((a + b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a + b)*B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*Sqrt[a]*Sqrt[b]*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/((a^2 + b^2)*d) + ((a - b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2), x, 15, -(((a - b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a - b)*B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(3/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a + b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2), x, 16, ((a + b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*B*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(5/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)*d) - ((a - b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*B)/(a*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 14, (Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((4*a*A*b - a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*b^(3/2)*d) + (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((4*A*b - a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*b*d) + (B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(2*b*d)} -{Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 13, (Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((2*A*b + a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[b]*d) - (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 12, -((Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (2*Sqrt[b]*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 8, -((Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*A*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 9, (Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*A*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (2*(A*b + 3*a*B)*Sqrt[a + b*Tan[c + d*x]])/(3*a*d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 10, (Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*A*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*(A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(15*a*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2*A + 2*A*b^2 - 5*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d*Sqrt[Tan[c + d*x]])} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 11, -((Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*A*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*(A*b + 7*a*B)*Sqrt[a + b*Tan[c + d*x]])/(35*a*d*Tan[c + d*x]^(5/2)) + (2*(35*a^2*A + 4*A*b^2 - 7*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d*Tan[c + d*x]^(3/2)) + (2*(35*a^2*A*b - 8*A*b^3 + 105*a^3*B + 14*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a^3*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 15, ((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((6*a^2*A*b - 16*A*b^3 - a^3*B - 24*a*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(8*b^(3/2)*d) + ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((6*a*A*b - a^2*B - 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(8*b*d) + ((6*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(12*b*d) + (B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2))/(3*b*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 14, ((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + ((12*a*A*b + 3*a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*Sqrt[b]*d) + ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((4*A*b + 5*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*d) + (b*B*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(2*d)} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 13, -(((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (Sqrt[b]*(2*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b*B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 13, -(((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + (2*b^(3/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 9, ((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + b*Tan[c + d*x]])/(3*d*Tan[c + d*x]^(3/2)) - (2*(4*A*b + 3*a*B)*Sqrt[a + b*Tan[c + d*x]])/(3*d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 10, ((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + b*Tan[c + d*x]])/(5*d*Tan[c + d*x]^(5/2)) - (2*(6*A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2*A - 3*A*b^2 - 20*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(15*a*d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 11, -(((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + b*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (2*(8*A*b + 7*a*B)*Sqrt[a + b*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (2*(35*a^2*A - 3*A*b^2 - 42*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d*Tan[c + d*x]^(3/2)) + (2*(140*a^2*A*b + 6*A*b^3 + 105*a^3*B - 21*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2), x, 12, ((I*a - b)^(3/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*A*Sqrt[a + b*Tan[c + d*x]])/(9*d*Tan[c + d*x]^(9/2)) - (2*(10*A*b + 9*a*B)*Sqrt[a + b*Tan[c + d*x]])/(63*d*Tan[c + d*x]^(7/2)) + (2*(21*a^2*A - A*b^2 - 24*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d*Tan[c + d*x]^(5/2)) + (2*(126*a^2*A*b + 4*A*b^3 + 105*a^3*B - 9*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d*Tan[c + d*x]^(3/2)) - (2*(315*a^4*A - 63*a^2*A*b^2 + 8*A*b^4 - 420*a^3*b*B - 18*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(315*a^3*d*Sqrt[Tan[c + d*x]])} - - -{Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 16, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + ((40*a^3*A*b - 320*a*A*b^3 - 5*a^4*B - 240*a^2*b^2*B + 128*b^4*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(64*b^(3/2)*d) - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((40*a^2*A*b - 64*A*b^3 - 5*a^3*B - 112*a*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(64*b*d) + ((40*a*A*b - 5*a^2*B - 48*b^2*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(96*b*d) + ((8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2))/(24*b*d) + (B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2))/(4*b*d)} -{Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 15, -(((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + ((30*a^2*A*b - 16*A*b^3 + 5*a^3*B - 40*a*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(8*Sqrt[b]*d) + ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((14*a*A*b + 5*a^2*B - 8*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(8*d) + ((2*A*b + 3*a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(4*d) + (b*B*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Tan[c + d*x]], x, 14, ((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (Sqrt[b]*(20*a*A*b + 15*a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(4*d) + ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b*(4*A*b + 7*a*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(4*d) + (b*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/(2*d)} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(3/2), x, 14, ((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b^(3/2)*(2*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + (b*(2*a*A + b*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(d*Sqrt[Tan[c + d*x]])} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 14, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*(2*A*b + a*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(3*d*Tan[c + d*x]^(3/2))} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(7/2), x, 10, -(((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) + ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(15*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(15*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(5*d*Tan[c + d*x]^(5/2))} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(9/2), x, 11, ((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d + ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Tan[c + d*x]])/(35*d*Tan[c + d*x]^(5/2)) + (2*(35*a^2*A - 45*A*b^2 - 77*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(3/2)) + (2*(245*a^2*A*b - 15*A*b^3 + 105*a^3*B - 161*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(7*d*Tan[c + d*x]^(7/2))} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(11/2), x, 12, ((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*(4*A*b + 3*a*B)*Sqrt[a + b*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(7/2)) + (2*(21*a^2*A - 25*A*b^2 - 45*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(105*d*Tan[c + d*x]^(5/2)) + (2*(231*a^2*A*b - 5*A*b^3 + 105*a^3*B - 135*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(315*a*d*Tan[c + d*x]^(3/2)) - (2*(315*a^4*A - 483*a^2*A*b^2 - 10*A*b^4 - 735*a^3*b*B + 45*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(9*d*Tan[c + d*x]^(9/2))} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Tan[c + d*x]^(13/2), x, 13, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d) - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Tan[c + d*x]])/(99*d*Tan[c + d*x]^(9/2)) + (2*(99*a^2*A - 113*A*b^2 - 209*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(693*d*Tan[c + d*x]^(7/2)) + (2*(495*a^2*A*b - 5*A*b^3 + 231*a^3*B - 275*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(1155*a*d*Tan[c + d*x]^(5/2)) - (2*(1155*a^4*A - 1485*a^2*A*b^2 - 20*A*b^4 - 2541*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Tan[c + d*x]])/(3465*a^2*d*Tan[c + d*x]^(3/2)) - (2*(8085*a^4*A*b - 495*a^2*A*b^3 + 40*A*b^5 + 3465*a^5*B - 5313*a^3*b^2*B - 110*a*b^4*B)*Sqrt[a + b*Tan[c + d*x]])/(3465*a^3*d*Sqrt[Tan[c + d*x]]) - (2*a*A*(a + b*Tan[c + d*x])^(3/2))/(11*d*Tan[c + d*x]^(11/2))} - - -{((a + b*Tan[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Tan[c + d*x]))/Tan[c + d*x]^(5/2), x, 14, ((I*a - b)^(5/2)*(2*a - 3*I*b)*B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(2*a*d) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - ((2*a + 3*I*b)*(I*a + b)^(5/2)*B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(2*a*d) - (2*(a^2 + 3*b^2)*B*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]]) - (b*B*(a + b*Tan[c + d*x])^(3/2))/(d*Tan[c + d*x]^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 13, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + ((2*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(3/2)*d) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) + (B*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(b*d)} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 12, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[b]*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]), x, 7, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]), x, 8, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*A*Sqrt[a + b*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]), x, 9, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*A*Sqrt[a + b*Tan[c + d*x]])/(3*a*d*Tan[c + d*x]^(3/2)) + (2*(2*A*b - 3*a*B)*Sqrt[a + b*Tan[c + d*x]])/(3*a^2*d*Sqrt[Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]]), x, 10, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*A*Sqrt[a + b*Tan[c + d*x]])/(5*a*d*Tan[c + d*x]^(5/2)) + (2*(4*A*b - 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d*Tan[c + d*x]^(3/2)) + (2*(15*a^2*A - 8*A*b^2 + 10*a*b*B)*Sqrt[a + b*Tan[c + d*x]])/(15*a^3*d*Sqrt[Tan[c + d*x]])} - - -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 13, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d)) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(3/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) + (2*a*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 8, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d)) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - (2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)), x, 8, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)), x, 9, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - (2*A)/(a*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - (2*b*(a^2*A + 2*A*b^2 - a*b*B)*Sqrt[Tan[c + d*x]])/(a^2*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)), x, 10, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d)) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) - (2*A)/(3*a*d*Tan[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*A*b - 3*a*B))/(3*a^2*d*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*B)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 14, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(b^(5/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x]^(3/2))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(2*A*b^3 - a*(a^2 + 3*b^2)*B)*Sqrt[Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 9, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) + (2*a*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*(2*a^2*A*b - 4*A*b^3 + a^3*B + 7*a*b^2*B)*Sqrt[Tan[c + d*x]])/(3*b*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 9, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d)) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B + 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(3*a*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)), x, 9, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d)) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b + 2*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Tan[c + d*x]])/(3*a^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)), x, 10, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*A)/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^2*A + 4*A*b^2 - a*b*B)*Sqrt[Tan[c + d*x]])/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^4*A + 17*a^2*A*b^2 + 8*A*b^4 - 8*a^3*b*B - 2*a*b^3*B)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)), x, 11, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(5/2)*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(5/2)*d) - (2*A)/(3*a*d*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) + (2*(2*A*b - a*B))/(a^2*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(7*a^2*A*b + 8*A*b^3 - 3*a^3*B - 4*a*b^2*B)*Sqrt[Tan[c + d*x]])/(3*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^4*A*b + 30*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B - 17*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Tan[c + d*x]])/(3*a^4*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])} - - -{(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 13, -((B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d)) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[b]*d) - (B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{(Sqrt[Tan[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 8, (I*B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) - (I*B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{(a*B + b*B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)), x, 8, (B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)} -{(a*B + b*B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)), x, 10, ((-I)*B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (I*B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d) - (2*B*Sqrt[a + b*Tan[c + d*x]])/(a*d*Sqrt[Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Tan[c + d*x])^(2/3)*(A + B*Tan[c + d*x]), x, 12, (-(1/4))*(a - I*b)^(2/3)*(A - I*B)*x - (1/4)*(a + I*b)^(2/3)*(A + I*B)*x + (Sqrt[3]*(a - I*b)^(2/3)*(I*A + B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*d) - (Sqrt[3]*(a + I*b)^(2/3)*(I*A - B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*d) - ((a + I*b)^(2/3)*(I*A - B)*Log[Cos[c + d*x]])/(4*d) + ((a - I*b)^(2/3)*(I*A + B)*Log[Cos[c + d*x]])/(4*d) + (3*(a - I*b)^(2/3)*(I*A + B)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) - (3*(a + I*b)^(2/3)*(I*A - B)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) + (3*B*(a + b*Tan[c + d*x])^(2/3))/(2*d)} -{(a + b*Tan[c + d*x])^(1/3)*(A + B*Tan[c + d*x]), x, 12, (-(1/4))*(a - I*b)^(1/3)*(A - I*B)*x - (1/4)*(a + I*b)^(1/3)*(A + I*B)*x - (Sqrt[3]*(a - I*b)^(1/3)*(I*A + B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*d) + (Sqrt[3]*(a + I*b)^(1/3)*(I*A - B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*d) - ((a + I*b)^(1/3)*(I*A - B)*Log[Cos[c + d*x]])/(4*d) + ((a - I*b)^(1/3)*(I*A + B)*Log[Cos[c + d*x]])/(4*d) + (3*(a - I*b)^(1/3)*(I*A + B)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) - (3*(a + I*b)^(1/3)*(I*A - B)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*d) + (3*B*(a + b*Tan[c + d*x])^(1/3))/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(1/3), x, 11, -(((A - I*B)*x)/(4*(a - I*b)^(1/3))) - ((A + I*B)*x)/(4*(a + I*b)^(1/3)) + (Sqrt[3]*(I*A + B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*(a - I*b)^(1/3)*d) - (Sqrt[3]*(I*A - B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*(a + I*b)^(1/3)*d) - ((I*A - B)*Log[Cos[c + d*x]])/(4*(a + I*b)^(1/3)*d) + ((I*A + B)*Log[Cos[c + d*x]])/(4*(a - I*b)^(1/3)*d) + (3*(I*A + B)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a - I*b)^(1/3)*d) - (3*(I*A - B)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a + I*b)^(1/3)*d)} -{(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^(2/3), x, 11, -(((A - I*B)*x)/(4*(a - I*b)^(2/3))) - ((A + I*B)*x)/(4*(a + I*b)^(2/3)) - (Sqrt[3]*(I*A + B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a - I*b)^(1/3))/Sqrt[3]])/(2*(a - I*b)^(2/3)*d) + (Sqrt[3]*(I*A - B)*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I*b)^(1/3))/Sqrt[3]])/(2*(a + I*b)^(2/3)*d) - ((I*A - B)*Log[Cos[c + d*x]])/(4*(a + I*b)^(2/3)*d) + ((I*A + B)*Log[Cos[c + d*x]])/(4*(a - I*b)^(2/3)*d) + (3*(I*A + B)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a - I*b)^(2/3)*d) - (3*(I*A - B)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/(4*(a + I*b)^(2/3)*d)} - - -{(I - Tan[e + f*x])/(c + d*Tan[e + f*x])^(1/3), x, 5, -((I*x)/(2*(c - I*d)^(1/3))) - (Sqrt[3]*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/((c - I*d)^(1/3)*f) - Log[Cos[e + f*x]]/(2*(c - I*d)^(1/3)*f) - (3*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(2*(c - I*d)^(1/3)*f)} - - -{(d - c*Tan[e + f*x])/(c + d*Tan[e + f*x])^(2/3), x, 11, (-(1/4))*I*(c - I*d)^(1/3)*x + (1/4)*I*(c + I*d)^(1/3)*x + (Sqrt[3]*(c - I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c - I*d)^(1/3))/Sqrt[3]])/(2*f) + (Sqrt[3]*(c + I*d)^(1/3)*ArcTan[(1 + (2*(c + d*Tan[e + f*x])^(1/3))/(c + I*d)^(1/3))/Sqrt[3]])/(2*f) - ((c - I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - ((c + I*d)^(1/3)*Log[Cos[e + f*x]])/(4*f) - (3*(c - I*d)^(1/3)*Log[(c - I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f) - (3*(c + I*d)^(1/3)*Log[(c + I*d)^(1/3) - (c + d*Tan[e + f*x])^(1/3)])/(4*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with m symbolic*) - - -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]), x, 9, If[$VersionNumber>=8, -((b*(A*b^3*(12 + 7*m + m^2) + 4*a*b^2*B*(12 + 7*m + m^2) - 2*a^3*B*(19 + 8*m + m^2) - a^2*A*b*(68 + 37*m + 5*m^2))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(3 + m)*(4 + m))) + ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (b^2*(2*a*A*b*(4 + m)^2 - b^2*B*(12 + 7*m + m^2) + a^2*B*(26 + 9*m + m^2))*Tan[c + d*x]^(2 + m))/(d*(2 + m)*(3 + m)*(4 + m)) + ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (b*(A*b*(4 + m) + a*B*(7 + m))*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^2)/(d*(3 + m)*(4 + m)) + (b*B*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^3)/(d*(4 + m)), -((b*(A*b^3*(12 + 7*m + m^2) + 4*a*b^2*B*(12 + 7*m + m^2) - 2*a^3*B*(19 + 8*m + m^2) - a^2*A*b*(68 + 37*m + 5*m^2))*Tan[c + d*x]^(1 + m))/(d*(4 + m)*(3 + 4*m + m^2))) + ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (b^2*(2*a*A*b*(4 + m)^2 - b^2*B*(12 + 7*m + m^2) + a^2*B*(26 + 9*m + m^2))*Tan[c + d*x]^(2 + m))/(d*(2 + m)*(3 + m)*(4 + m)) + ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (b*(A*b*(4 + m) + a*B*(7 + m))*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^2)/(d*(3 + m)*(4 + m)) + (b*B*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^3)/(d*(4 + m))]} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 8, (b*(3*a*A*b*(3 + m) - b^2*B*(3 + m) + 2*a^2*B*(4 + m))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(3 + m)) + ((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + (b^2*(A*b*(3 + m) + a*B*(5 + m))*Tan[c + d*x]^(2 + m))/(d*(2 + m)*(3 + m)) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (b*B*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^2)/(d*(3 + m))} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 7, (b*(A*b*(2 + m) + a*B*(3 + m))*Tan[c + d*x]^(1 + m))/(d*(1 + m)*(2 + m)) + ((a^2*A - A*b^2 - 2*a*b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + ((2*a*A*b + a^2*B - b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m)) + (b*B*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x]))/(d*(2 + m))} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^1*(A + B*Tan[c + d*x]), x, 6, (b*B*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + ((a*A - b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + m)) + ((A*b + a*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(2 + m))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^1, x, 8, ((a*A + b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/((a^2 + b^2)*d*(1 + m)) + (b*(A*b - a*B)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(1 + m)) - ((A*b - a*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/((a^2 + b^2)*d*(2 + m))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 9, ((a^2*A - A*b^2 + 2*a*b*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/((a^2 + b^2)^2*d*(1 + m)) + (1/(a^2*(a^2 + b^2)^2*d*(1 + m)))*(b*(a^2*A*b*(2 - m) - A*b^3*m + a*b^2*B*(1 + m) - a^3*(B - B*m))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m)) - ((2*a*A*b - a^2*B + b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/((a^2 + b^2)^2*d*(2 + m)) + (b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 10, ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/((a^2 + b^2)^3*d*(1 + m)) - (b*(A*b^5*(1 - m)*m + a*b^4*B*m*(1 + m) - 2*a^3*b^2*B*(3 + m - m^2) + 2*a^2*A*b^3*(1 + 3*m - m^2) - a^4*A*b*(6 - 5*m + m^2) + a^5*B*(2 - 3*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m))/(2*a^3*(a^2 + b^2)^3*d*(1 + m)) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/((a^2 + b^2)^3*d*(2 + m)) + (b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b*(A*b^3*(1 - m) - a^3*B*(3 - m) + a^2*A*b*(5 - m) + a*b^2*B*(1 + m))*Tan[c + d*x]^(1 + m))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4, x, 11, ((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/((a^2 + b^2)^4*d*(1 + m)) - (1/(6*a^4*(a^2 + b^2)^4*d*(1 + m)))*(b*(a*b^6*B*m*(1 - m^2) + 3*a^2*A*b^5*m*(2 - 5*m + m^2) + A*b^7*m*(2 - 3*m + m^2) + 3*a^3*b^4*B*(2 + 5*m + 2*m^2 - m^3) + a^7*B*(6 - 11*m + 6*m^2 - m^3) - a^6*A*b*(24 - 26*m + 9*m^2 - m^3) + 3*a^4*A*b^3*(8 + 10*m - 7*m^2 + m^3) - 3*a^5*b^2*B*(12 - m - 4*m^2 + m^3))*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(1 + m)) - ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/((a^2 + b^2)^4*d*(2 + m)) + (b*(A*b - a*B)*Tan[c + d*x]^(1 + m))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b*(A*b^3*(2 - m) - a^3*B*(5 - m) + a^2*A*b*(8 - m) + a*b^2*B*(1 + m))*Tan[c + d*x]^(1 + m))/(6*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(a*b^4*B*(1 - m^2) + 2*a^3*b^2*B*(7 + 3*m - m^2) + a^4*A*b*(26 - 9*m + m^2) + 2*a^2*A*b^3*(2 - 6*m + m^2) - a^5*B*(11 - 6*m + m^2) + A*b^5*(2 - 3*m + m^2))*Tan[c + d*x]^(1 + m))/(6*a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} - - -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 7, (a^2*(A + I*B)*AppellF1[1 + m, -(5/2), 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a]) + (a^2*(A - I*B)*AppellF1[1 + m, -(5/2), 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 7, (a*(A + I*B)*AppellF1[1 + m, -(3/2), 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a]) + (a*(A - I*B)*AppellF1[1 + m, -(3/2), 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])} -{Tan[c + d*x]^m*(a + b*Tan[c + d*x])^(1/2)*(A + B*Tan[c + d*x]), x, 7, ((A + I*B)*AppellF1[1 + m, -(1/2), 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a]) + ((A - I*B)*AppellF1[1 + m, -(1/2), 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[a + b*Tan[c + d*x]])/(2*d*(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(1/2), x, 7, ((A + I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 1/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]])} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 7, ((A + I*B)*AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 3/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]])} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 7, ((A + I*B)*AppellF1[1 + m, 5/2, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a^2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]]) + ((A - I*B)*AppellF1[1 + m, 5/2, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*Sqrt[1 + (b*Tan[c + d*x])/a])/(2*a^2*d*(1 + m)*Sqrt[a + b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with n symbolic*) - - -{Tan[c + d*x]^m*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 7, (1/(2*d*(1 + m)))*(((A + I*B)*AppellF1[1 + m, -n, 1, 2 + m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^n)/(1 + (b*Tan[c + d*x])/a)^n) + (1/(2*d*(1 + m)))*(((A - I*B)*AppellF1[1 + m, -n, 1, 2 + m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Tan[c + d*x]^(1 + m)*(a + b*Tan[c + d*x])^n)/(1 + (b*Tan[c + d*x])/a)^n)} - - -{Tan[c + d*x]^4*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, -(((A*b^3*(2 + n)*(3 + n)*(4 + n) - a*(b^2*B*(3 + n)*(4 + n) - 2*a*(3*a*B - A*b*(4 + n))))*(a + b*Tan[c + d*x])^(1 + n))/(b^4*d*(1 + n)*(2 + n)*(3 + n)*(4 + n))) + ((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) - ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a - b)*d*(1 + n)) - ((b^2*B*(3 + n)*(4 + n) - 2*a*(3*a*B - A*b*(4 + n)))*Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b^3*d*(2 + n)*(3 + n)*(4 + n)) - ((3*a*B - A*b*(4 + n))*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(b^2*d*(3 + n)*(4 + n)) + (B*Tan[c + d*x]^3*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(4 + n))} -{Tan[c + d*x]^3*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 8, ((2*a^2*B - a*A*b*(3 + n) - b^2*B*(6 + 5*n + n^2))*(a + b*Tan[c + d*x])^(1 + n))/(b^3*d*(1 + n)*(2 + n)*(3 + n)) + ((I*A + B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) + ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) - ((2*a*B - A*b*(3 + n))*Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b^2*d*(2 + n)*(3 + n)) + (B*Tan[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(3 + n))} -{Tan[c + d*x]^2*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 7, -(((a*B - A*b*(2 + n))*(a + b*Tan[c + d*x])^(1 + n))/(b^2*d*(1 + n)*(2 + n))) + ((I*A + B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n)) + ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a - b)*d*(1 + n)) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(2 + n))} -{Tan[c + d*x]^1*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 6, (B*(a + b*Tan[c + d*x])^(1 + n))/(b*d*(1 + n)) - ((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a - I*b)*d*(1 + n)) - ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n))} -{Tan[c + d*x]^0*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 5, ((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) + ((I*A - B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n))} -{Cot[c + d*x]^1*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 8, ((I*A + B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) + ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) - (A*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Cot[c + d*x]^2*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, -((A*Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(a*d)) - ((A - I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) + ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a - b)*d*(1 + n)) - ((a*B + A*b*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))} -{Cot[c + d*x]^3*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 10, -(((2*a*B - A*b*(1 - n))*Cot[c + d*x]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^2*d)) - (A*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n))/(2*a*d) - ((I*A + B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(I*a + b)*d*(1 + n)) - ((A + I*B)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x])^(1 + n))/(2*(a + I*b)*d*(1 + n)) + ((2*a^2*A - 2*a*b*B*n + A*b^2*(1 - n)*n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(2*a^3*d*(1 + n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+a I Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 6, (2*(-1)^(1/4)*a*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (2*a*(A - I*B)*Sqrt[Cot[c + d*x]])/d - (2*a*(I*A + B)*Cot[c + d*x]^(3/2))/(3*d) - (2*a*A*Cot[c + d*x]^(5/2))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 5, -((2*(-1)^(1/4)*a*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a*(I*A + B)*Sqrt[Cot[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, -((2*(-1)^(1/4)*a*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a*A*Sqrt[Cot[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 4, (2*(-1)^(1/4)*a*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (2*I*a*B)/(d*Sqrt[Cot[c + d*x]])} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 5, (2*(-1)^(1/4)*a*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (2*I*a*B)/(3*d*Cot[c + d*x]^(3/2)) + (2*a*(I*A + B))/(d*Sqrt[Cot[c + d*x]])} -{((a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x, 6, -((2*(-1)^(1/4)*a*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) + (2*I*a*B)/(5*d*Cot[c + d*x]^(5/2)) + (2*a*(I*A + B))/(3*d*Cot[c + d*x]^(3/2)) + (2*a*(A - I*B))/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 6, (4*(-1)^(1/4)*a^2*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (4*a^2*(A - I*B)*Sqrt[Cot[c + d*x]])/d - (2*a^2*(7*I*A + 5*B)*Cot[c + d*x]^(3/2))/(15*d) - (2*A*Cot[c + d*x]^(3/2)*(I*a^2 + a^2*Cot[c + d*x]))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, -((4*(-1)^(1/4)*a^2*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a^2*(5*I*A + 3*B)*Sqrt[Cot[c + d*x]])/(3*d) - (2*A*Sqrt[Cot[c + d*x]]*(I*a^2 + a^2*Cot[c + d*x]))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, -((4*(-1)^(1/4)*a^2*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (2*a^2*(A + I*B)*Sqrt[Cot[c + d*x]])/d + (2*I*B*(I*a^2 + a^2*Cot[c + d*x]))/(d*Sqrt[Cot[c + d*x]])} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 5, (4*(-1)^(1/4)*a^2*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a^2*(3*A - 5*I*B))/(3*d*Sqrt[Cot[c + d*x]]) + (2*I*B*(I*a^2 + a^2*Cot[c + d*x]))/(3*d*Cot[c + d*x]^(3/2))} -{((a + I*a*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 6, (4*(-1)^(1/4)*a^2*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (2*a^2*(5*A - 7*I*B))/(15*d*Cot[c + d*x]^(3/2)) + (4*a^2*(I*A + B))/(d*Sqrt[Cot[c + d*x]]) + (2*I*B*(I*a^2 + a^2*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2))} - - -{Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 7, (8*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (8*a^3*(I*A + B)*Sqrt[Cot[c + d*x]])/d + (8*a^3*(23*A - 21*I*B)*Cot[c + d*x]^(3/2))/(105*d) - (2*a*A*Cot[c + d*x]^(3/2)*(I*a + a*Cot[c + d*x])^2)/(7*d) - (2*(11*I*A + 7*B)*Cot[c + d*x]^(3/2)*(I*a^3 + a^3*Cot[c + d*x]))/(35*d)} -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, (8*(-1)^(1/4)*a^3*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d + (16*a^3*(6*A - 5*I*B)*Sqrt[Cot[c + d*x]])/(15*d) - (2*a*A*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2)/(5*d) - (2*(9*I*A + 5*B)*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]))/(15*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -((8*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (16*I*a^3*A*Sqrt[Cot[c + d*x]])/(3*d) + (2*I*a*B*(I*a + a*Cot[c + d*x])^2)/(d*Sqrt[Cot[c + d*x]]) - (2*(A + 3*I*B)*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, -((8*(-1)^(1/4)*a^3*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d) - (16*I*a^3*B*Sqrt[Cot[c + d*x]])/(3*d) + (2*I*a*B*(I*a + a*Cot[c + d*x])^2)/(3*d*Cot[c + d*x]^(3/2)) - (2*(3*A - 7*I*B)*(I*a^3 + a^3*Cot[c + d*x]))/(3*d*Sqrt[Cot[c + d*x]])} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 6, (8*(-1)^(1/4)*a^3*(I*A + B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (16*a^3*(5*A - 6*I*B))/(15*d*Sqrt[Cot[c + d*x]]) + (2*I*a*B*(I*a + a*Cot[c + d*x])^2)/(5*d*Cot[c + d*x]^(5/2)) - (2*(5*A - 9*I*B)*(I*a^3 + a^3*Cot[c + d*x]))/(15*d*Cot[c + d*x]^(3/2))} -{((a + I*a*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 7, (8*(-1)^(1/4)*a^3*(A - I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Cot[c + d*x]]])/d - (8*a^3*(21*A - 23*I*B))/(105*d*Cot[c + d*x]^(3/2)) + (8*a^3*(I*A + B))/(d*Sqrt[Cot[c + d*x]]) + (2*I*a*B*(I*a + a*Cot[c + d*x])^2)/(7*d*Cot[c + d*x]^(7/2)) - (2*(7*A - 11*I*B)*(I*a^3 + a^3*Cot[c + d*x]))/(35*d*Cot[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 14, -(((1/4 - I/4)*((6 + I)*A + (1 + 4*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d)) + (((7 - 5*I)*A + (5 + 3*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(4*Sqrt[2]*a*d) + (5*(I*A - B)*Sqrt[Cot[c + d*x]])/(2*a*d) - ((7*A + 3*I*B)*Cot[c + d*x]^(3/2))/(6*a*d) + ((A + I*B)*Cot[c + d*x]^(5/2))/(2*d*(I*a + a*Cot[c + d*x])) + (((7 + 5*I)*A - (5 - 3*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*d) + (((-7 - 5*I)*A + (5 - 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*d)} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 13, (1/(4*Sqrt[2]*a*d))*(((-5 - 3*I)*A + (3 - I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]) + (1/(4*Sqrt[2]*a*d))*(((5 + 3*I)*A - (3 - I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) - ((5*A + I*B)*Sqrt[Cot[c + d*x]])/(2*a*d) + ((A + I*B)*Cot[c + d*x]^(3/2))/(2*d*(I*a + a*Cot[c + d*x])) - (1/(Sqrt[2]*a*d))*((1/8 - I/8)*((4 + I)*A + (1 + 2*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]) + (1/(8*Sqrt[2]*a*d))*(((5 - 3*I)*A + (3 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x]), x, 12, ((1/4 - I/4)*((2 + I)*A + B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((1/4 - I/4)*((2 + I)*A + B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(2*d*(I*a + a*Cot[c + d*x])) - (((3 + I)*A - (1 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*d) + (((3 + I)*A - (1 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])), x, 12, ((1/4 - I/4)*(A + (2 - I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((1/4 - I/4)*(A + (2 - I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) + ((I*A - B)*Sqrt[Cot[c + d*x]])/(2*d*(I*a + a*Cot[c + d*x])) + ((1/8 + I/8)*(A - (2 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) - ((1/8 + I/8)*(A - (2 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])), x, 13, (((1 - 3*I)*A + (3 + 5*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(4*Sqrt[2]*a*d) + ((1/4 + I/4)*((1 + 2*I)*A - (4 + I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - (A + 5*I*B)/(2*a*d*Sqrt[Cot[c + d*x]]) + (I*A - B)/(2*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])) - ((1/8 + I/8)*((2 + I)*A + (1 + 4*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) + ((1/8 + I/8)*((2 + I)*A + (1 + 4*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])), x, 14, ((1/4 + I/4)*((4 + I)*A + (1 + 6*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((1/4 + I/4)*((4 + I)*A + (1 + 6*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - (3*A + 7*I*B)/(6*a*d*Cot[c + d*x]^(3/2)) - (5*(I*A - B))/(2*a*d*Sqrt[Cot[c + d*x]]) + (I*A - B)/(2*d*Cot[c + d*x]^(3/2)*(I*a + a*Cot[c + d*x])) + (((3 - 5*I)*A + (5 + 7*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(8*Sqrt[2]*a*d) + ((1/8 + I/8)*((1 + 4*I)*A - (6 + I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 14, -(((1/16 - I/16)*((2 + 23*I)*A - (7 + 2*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + (((25 + 21*I)*A - (9 - 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) - (5*(5*A + I*B)*Sqrt[Cot[c + d*x]])/(8*a^2*d) + ((7*A + 3*I*B)*Cot[c + d*x]^(3/2))/(8*a^2*d*(I + Cot[c + d*x])) + ((A + I*B)*Cot[c + d*x]^(5/2))/(4*d*(I*a + a*Cot[c + d*x])^2) - ((1/32 - I/32)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2, x, 13, (((9 - 5*I)*A + (1 - 3*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) + ((1/16 + I/16)*((-2 + 7*I)*A + (1 + 2*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d) + ((5*A + I*B)*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) + ((A + I*B)*Cot[c + d*x]^(3/2))/(4*d*(I*a + a*Cot[c + d*x])^2) + ((1/32 + I/32)*((-7 + 2*I)*A + (2 + I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + (((9 + 5*I)*A - (1 + 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^2*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^2), x, 13, -((((-1 + 3*I)*A + (1 + 3*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d)) + (((-1 + 3*I)*A + (1 + 3*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) + ((3*I*A + B)*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(4*d*(I*a + a*Cot[c + d*x])^2) + (((1 + 3*I)*A + (1 - 3*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^2*d) - (((1 + 3*I)*A + (1 - 3*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^2*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2), x, 13, -(((1/16 + I/16)*((2 + I)*A + (7 - 2*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + (((1 + 3*I)*A + (9 + 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) + ((A + 5*I*B)*Sqrt[Cot[c + d*x]])/(8*a^2*d*(I + Cot[c + d*x])) + ((I*A - B)*Sqrt[Cot[c + d*x]])/(4*d*(I*a + a*Cot[c + d*x])^2) + (((1 - 3*I)*A - (9 - 5*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^2*d) + ((1/32 + I/32)*((1 + 2*I)*A + (2 - 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^2), x, 14, -(((1/16 - I/16)*((2 + 7*I)*A - (23 + 2*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^2*d)) + (((9 + 5*I)*A - (25 - 21*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^2*d) + (5*(I*A - 5*B))/(8*a^2*d*Sqrt[Cot[c + d*x]]) + (3*A + 7*I*B)/(8*a^2*d*Sqrt[Cot[c + d*x]]*(I + Cot[c + d*x])) + (I*A - B)/(4*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((7 + 2*I)*A + (2 + 23*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^2*d)} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 15, -(((1/16 - I/16)*((1 + 29*I)*A - (6 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d)) + (((30 + 28*I)*A - (7 - 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (5*(6*A + I*B)*Sqrt[Cot[c + d*x]])/(8*a^3*d) + ((A + I*B)*Cot[c + d*x]^(7/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + ((5*A + 2*I*B)*Cot[c + d*x]^(5/2))/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (7*(4*A + I*B)*Cot[c + d*x]^(3/2))/(24*d*(I*a^3 + a^3*Cot[c + d*x])) - ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) + ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3, x, 14, -((((-7 + 5*I)*A + 2*I*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d)) + (((-7 + 5*I)*A + 2*I*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + ((A + I*B)*Cot[c + d*x]^(5/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + ((4*A + I*B)*Cot[c + d*x]^(3/2))/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (5*A*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - (((7 + 5*I)*A - 2*I*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d) + (((7 + 5*I)*A - 2*I*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^3), x, 14, -(((1/16 + I/16)*((1 + I)*A + B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d)) + ((1/16 + I/16)*((1 + I)*A + B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + ((A + I*B)*Cot[c + d*x]^(3/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + (A*Sqrt[Cot[c + d*x]])/(4*a*d*(I*a + a*Cot[c + d*x])^2) + ((2*I*A + B)*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) + ((2*I*A + (1 - I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d) - ((2*I*A + (1 - I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^3), x, 14, -((((1 + I)*A + 2*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d)) + (((1 + I)*A + 2*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(6*d*(I*a + a*Cot[c + d*x])^3) + ((2*I*A + B)*Sqrt[Cot[c + d*x]])/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (A*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - (((-1 + I)*A + 2*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d) + (((-1 + I)*A + 2*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3), x, 14, -(((2*A + (5 - 7*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d)) + ((2*A + (5 - 7*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + ((I*A - B)*Sqrt[Cot[c + d*x]])/(6*d*(I*a + a*Cot[c + d*x])^3) + ((A + 4*I*B)*Sqrt[Cot[c + d*x]])/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (5*B*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - ((2*A - (5 + 7*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d) + ((2*A - (5 + 7*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(32*Sqrt[2]*a^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3), x, 15, ((1/16 + I/16)*((1 + 6*I)*A - (29 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((5 - 7*I)*A + (28 + 30*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + (5*(A + 6*I*B))/(8*a^3*d*Sqrt[Cot[c + d*x]]) + (I*A - B)/(6*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^3) + (2*A + 5*I*B)/(12*a*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - (7*(I*A - 4*B))/(24*d*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x])) + ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) - ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 7, ((-1 - I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(13*A - (5*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*(I*A + 5*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*A*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 6, ((1 + I)*Sqrt[a]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(I*A + 3*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(3*d) - (2*A*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 5, ((1 + I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*A*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 8, -((2*(-1)^(3/4)*Sqrt[a]*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((1 - I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{(Sqrt[a + I*a*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 9, -(((-1)^(3/4)*Sqrt[a]*(2*A - I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((1 + I)*Sqrt[a]*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (B*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 8, ((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a*((67*I)*A + 63*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) + (4*a*(19*A - (21*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a*((8*I)*A + 7*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 7, ((-2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a*(9*A - (10*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a*((6*I)*A + 5*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 6, ((2 + 2*I)*a^(3/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*((4*I)*A + 3*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(3*d) - (2*a*A*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 9, (2*(-1)^(1/4)*a^(3/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((2 + 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*A*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 9, -(((-1)^(3/4)*a^(3/2)*(2*I*A + 3*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((2 - 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*a*B*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 10, -(((-1)^(3/4)*a^(3/2)*(12*A - 11*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d)) - ((2 + 2*I)*a^(3/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (I*a*B*Sqrt[a + I*a*Tan[c + d*x]])/(2*d*Cot[c + d*x]^(3/2)) + (a*(4*I*A + 5*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(11/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 9, ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (8*a^2*(197*A - (195*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(315*d) + (8*a^2*((59*I)*A + 60*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(315*d) + (2*a^2*(46*A - (45*I)*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((4*I)*A + 3*B)*Cot[c + d*x]^(7/2)*Sqrt[a + I*a*Tan[c + d*x]])/(21*d) - (2*a*A*Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(3/2))/(9*d)} -{Cot[c + d*x]^(9/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 8, ((4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (4*a^2*((130*I)*A + 133*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) + (2*a^2*(80*A - (77*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(105*d) - (2*a^2*((10*I)*A + 7*B)*Cot[c + d*x]^(5/2)*Sqrt[a + I*a*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(3/2))/(7*d)} -{Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 7, ((-4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*a^2*(38*A - (35*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a^2*((8*I)*A + 5*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(3/2))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, (2*(-1)^(3/4)*a^(5/2)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4 + 4*I)*a^(5/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a^2*(2*I*A + B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, ((-1)^(3/4)*a^(5/2)*(2*A - 5*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (a^2*(2*I*A - B)*Sqrt[a + I*a*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]]) - (2*a*A*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2))/d} -{Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 10, -(((-1)^(3/4)*a^(5/2)*(20*I*A + 23*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d)) + ((4 - 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (a^2*(4*A - 7*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]) + (I*a*B*(a + I*a*Tan[c + d*x])^(3/2))/(2*d*Sqrt[Cot[c + d*x]])} -{((a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 11, -(((-1)^(3/4)*a^(5/2)*(46*A - 45*I*B)*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(8*d)) - ((4 + 4*I)*a^(5/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (a^2*(2*A - 3*I*B)*Sqrt[a + I*a*Tan[c + d*x]])/(4*d*Cot[c + d*x]^(3/2)) + (a^2*(18*I*A + 19*B)*Sqrt[a + I*a*Tan[c + d*x]])/(8*d*Sqrt[Cot[c + d*x]]) + (I*a*B*(a + I*a*Tan[c + d*x])^(3/2))/(3*d*Cot[c + d*x]^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 7, ((1/2 + I/2)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + ((A + I*B)*Cot[c + d*x]^(3/2))/(d*Sqrt[a + I*a*Tan[c + d*x]]) + (((7*I)*A - 9*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d) - ((5*A + (3*I)*B)*Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]])/(3*a*d)} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 6, ((1/2 + I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]) - ((3*A + I*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + I*a*Tan[c + d*x]], x, 5, ((1/2 - I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + (A + I*B)/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]), x, 9, -((2*(-1)^(1/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d)) - ((1/2 + I/2)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[a]*d) + (I*A - B)/(d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 7, ((1/4 + I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(3*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((11*A + (5*I)*B)*Sqrt[Cot[c + d*x]])/(6*a*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((25*A + (7*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(6*a^2*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(3/2), x, 6, ((1/4 - I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + (A + I*B)/(3*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (7*A + I*B)/(6*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)), x, 6, -(((1/4 + I/4)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d)) + (I*A - B)/(3*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (I*A + 5*B)/(6*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)), x, 10, (2*(-1)^(3/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + ((1/4 + I/4)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(3/2)*d) + (I*A - B)/(3*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) + (A + 3*I*B)/(2*a*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 8, ((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + ((A + I*B)*Sqrt[Cot[c + d*x]])/(5*d*(a + I*a*Tan[c + d*x])^(5/2)) + ((17*A + (7*I)*B)*Sqrt[Cot[c + d*x]])/(30*a*d*(a + I*a*Tan[c + d*x])^(3/2)) + ((151*A + (41*I)*B)*Sqrt[Cot[c + d*x]])/(60*a^2*d*Sqrt[a + I*a*Tan[c + d*x]]) - ((317*A + (67*I)*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])/(60*a^3*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^(5/2), x, 7, ((1/8 - I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + (A + I*B)/(5*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + (13*A + (3*I)*B)/(30*a*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (67*A - (3*I)*B)/(60*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)), x, 7, -(((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d)) + (I*A - B)/(5*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(5/2)) + (3*I*A + 7*B)/(30*a*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) - (3*I*A - 13*B)/(60*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 7, ((1/8 + I/8)*(I*A + B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + (I*A - B)/(5*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(5/2)) + (A + 11*I*B)/(30*a*d*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^(3/2)) + (13*A - 37*I*B)/(60*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)), x, 11, (2*(-1)^(1/4)*B*ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + ((1/8 + I/8)*(A - I*B)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*d) + (I*A - B)/(5*d*Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^(5/2)) + (A + 3*I*B)/(6*a*d*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^(3/2)) - (I*A - 7*B)/(4*a^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+i a Tan[c+d x])^n with n symbolic*) - - -{Cot[c + d*x]^m*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 8, (1/(d*(1 - m)))*(((A - I*B)*AppellF1[1 - m, 1 - n, 1, 2 - m, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*Cot[c + d*x]^(-1 + m)*(a + I*a*Tan[c + d*x])^n)/(1 + I*Tan[c + d*x])^n) + (1/(d*(1 - m)))*((I*B*Cot[c + d*x]^(-1 + m)*Hypergeometric2F1[1 - m, 1 - n, 2 - m, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(1 + I*Tan[c + d*x])^n)} - - -{Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 11, -((2*(3*B + 2*I*A*n)*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/(3*d)) - (2*A*Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n)/(3*d) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) - (2*(1 - 2*n)*(3*I*B - 2*A*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(3*d*Sqrt[Cot[c + d*x]]))} -{Cot[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 10, -((2*A*Sqrt[Cot[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/d) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) - (2*I*A*(1 - 2*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]]))} -{Cot[c + d*x]^(1/2)*(a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]), x, 9, (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) + (2*I*B*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]]))} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(1/2), x, 10, If[$VersionNumber>=8, (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*Sqrt[Cot[c + d*x]]) - (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) + (2*(2*B*n + I*A*(1 + 2*n))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*Sqrt[Cot[c + d*x]])), (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*Sqrt[Cot[c + d*x]]) - (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) + (2*(2*B*n + I*A*(1 + 2*n))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*Sqrt[Cot[c + d*x]]))]} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x, 11, If[$VersionNumber>=8, (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*Cot[c + d*x]^(3/2)) - (2*(2*I*B*n - A*(3 + 2*n))*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Cot[c + d*x]]) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) + (2*(2*A*n*(3 + 2*n) - I*B*(3 + 6*n + 4*n^2))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Cot[c + d*x]])), (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*Cot[c + d*x]^(3/2)) - (2*(2*I*B*n - A*(3 + 2*n))*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 8*n + 4*n^2)*Sqrt[Cot[c + d*x]]) - (2*(A - I*B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) + (2*(2*A*n*(3 + 2*n) - I*B*(3 + 6*n + 4*n^2))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(3 + 8*n + 4*n^2)*Sqrt[Cot[c + d*x]]))]} -{((a + I*a*Tan[c + d*x])^n*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(5/2), x, 12, If[$VersionNumber>=8, (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n)*Cot[c + d*x]^(5/2)) - (2*(2*I*B*n - A*(5 + 2*n))*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*(5 + 2*n)*Cot[c + d*x]^(3/2)) - (2*(2*I*A*n*(5 + 2*n) + B*(15 + 10*n + 4*n^2))*(a + I*a*Tan[c + d*x])^n)/(d*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*Sqrt[Cot[c + d*x]]) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) - (2*(4*B*n*(9 + 8*n + 2*n^2) + I*A*(15 + 36*n + 32*n^2 + 8*n^3))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(1 + 2*n)*(3 + 2*n)*(5 + 2*n)*Sqrt[Cot[c + d*x]])), (2*B*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n)*Cot[c + d*x]^(5/2)) - (2*(2*I*B*n - A*(5 + 2*n))*(a + I*a*Tan[c + d*x])^n)/(d*(3 + 2*n)*(5 + 2*n)*Cot[c + d*x]^(3/2)) - (2*(2*I*A*n*(5 + 2*n) + B*(15 + 10*n + 4*n^2))*(a + I*a*Tan[c + d*x])^n)/(d*(5 + 2*n)*(3 + 8*n + 4*n^2)*Sqrt[Cot[c + d*x]]) + (2*(I*A + B)*AppellF1[1/2, 1 - n, 1, 3/2, (-I)*Tan[c + d*x], I*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*Sqrt[Cot[c + d*x]])) - (2*(4*B*n*(9 + 8*n + 2*n^2) + I*A*(15 + 36*n + 32*n^2 + 8*n^3))*Hypergeometric2F1[1/2, 1 - n, 3/2, (-I)*Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^n)/((1 + I*Tan[c + d*x])^n*(d*(5 + 2*n)*(3 + 8*n + 4*n^2)*Sqrt[Cot[c + d*x]]))]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 13, -(((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((b*(A - B) + a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*(A*b + a*B)*Sqrt[Cot[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2))/(3*d) + ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 12, -(((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a*A*Sqrt[Cot[c + d*x]])/d - ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]), x, 12, ((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b*B)/(d*Sqrt[Cot[c + d*x]]) - ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{((a + b*Tan[c + d*x])*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 13, ((a*(A - B) - b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b*B)/(3*d*Cot[c + d*x]^(3/2)) + (2*(A*b + a*B))/(d*Sqrt[Cot[c + d*x]]) + ((b*(A - B) + a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((b*(A - B) + a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 14, ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*(a^2*A - A*b^2 - 2*a*b*B)*Sqrt[Cot[c + d*x]])/d - (2*a*(7*A*b + 5*a*B)*Cot[c + d*x]^(3/2))/(15*d) - (2*a*A*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x]))/(5*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 13, -(((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a*(5*A*b + 3*a*B)*Sqrt[Cot[c + d*x]])/(3*d) - (2*a*A*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x]))/(3*d) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 13, -(((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*B)/(d*Sqrt[Cot[c + d*x]]) - (2*a^2*A*Sqrt[Cot[c + d*x]])/d - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]), x, 13, ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*B)/(3*d*Cot[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B))/(d*Sqrt[Cot[c + d*x]]) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{((a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 14, ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*B)/(5*d*Cot[c + d*x]^(5/2)) + (2*b*(A*b + 2*a*B))/(3*d*Cot[c + d*x]^(3/2)) + (2*(2*a*A*b + a^2*B - b^2*B))/(d*Sqrt[Cot[c + d*x]]) + ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 15, ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cot[c + d*x]])/d + (2*a*(7*a^2*A - 18*A*b^2 - 21*a*b*B)*Cot[c + d*x]^(3/2))/(21*d) - (2*a^2*(11*A*b + 7*a*B)*Cot[c + d*x]^(5/2))/(35*d) - (2*a*A*Cot[c + d*x]^(3/2)*(b + a*Cot[c + d*x])^2)/(7*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 14, ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*a*(5*a^2*A - 14*A*b^2 - 15*a*b*B)*Sqrt[Cot[c + d*x]])/(5*d) - (2*a^2*(9*A*b + 5*a*B)*Cot[c + d*x]^(3/2))/(15*d) - (2*a*A*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])^2)/(5*d) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 14, -(((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*a*(3*a*A*b + a^2*B + 2*b^2*B)*Sqrt[Cot[c + d*x]])/d - (2*a^2*(a*A + 3*b*B)*Cot[c + d*x]^(3/2))/(3*d) + (2*b*B*(b + a*Cot[c + d*x])^2)/(d*Sqrt[Cot[c + d*x]]) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 14, -(((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*(3*A*b + 7*a*B))/(3*d*Sqrt[Cot[c + d*x]]) - (2*a^2*(3*a*A + b*B)*Sqrt[Cot[c + d*x]])/(3*d) + (2*b*B*(b + a*Cot[c + d*x])^2)/(3*d*Cot[c + d*x]^(3/2)) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]), x, 14, ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*(5*A*b + 9*a*B))/(15*d*Cot[c + d*x]^(3/2)) + (2*b*(15*a*A*b + 14*a^2*B - 5*b^2*B))/(5*d*Sqrt[Cot[c + d*x]]) + (2*b*B*(b + a*Cot[c + d*x])^2)/(5*d*Cot[c + d*x]^(5/2)) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{((a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 15, ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*b^2*(7*A*b + 11*a*B))/(35*d*Cot[c + d*x]^(5/2)) + (2*b*(21*a*A*b + 18*a^2*B - 7*b^2*B))/(21*d*Cot[c + d*x]^(3/2)) + (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B))/(d*Sqrt[Cot[c + d*x]]) + (2*b*B*(b + a*Cot[c + d*x])^2)/(7*d*Cot[c + d*x]^(7/2)) + ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 17, ((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(a^2 + b^2)*d) + (2*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(a^2*d) - (2*A*Cot[c + d*x]^(3/2))/(3*a*d) + ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 16, -(((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(3/2)*(a^2 + b^2)*d) - (2*A*Sqrt[Cot[c + d*x]])/(a*d) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 15, -(((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*Sqrt[b]*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])), x, 15, ((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[b]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 16, ((b*(A - B) - a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*a^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(3/2)*(a^2 + b^2)*d) + (2*B)/(b*d*Sqrt[Cot[c + d*x]]) + ((a*(A - B) + b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 17, -(((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d)) + ((a*(A - B) + b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(5/2)*(a^2 + b^2)*d) + (2*B)/(3*b*d*Cot[c + d*x]^(3/2)) + (2*(A*b - a*B))/(b^2*d*Sqrt[Cot[c + d*x]]) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 17, -(((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(3/2)*(7*a^2*A*b + 3*A*b^3 - 5*a^3*B - a*b^2*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(5/2)*(a^2 + b^2)^2*d) - ((2*a^2*A + 3*A*b^2 - a*b*B)*Sqrt[Cot[c + d*x]])/(a^2*(a^2 + b^2)*d) + (b*(A*b - a*B)*Cot[c + d*x]^(3/2))/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2, x, 16, -(((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[b]*(5*a^2*A*b + A*b^3 - 3*a^3*B + a*b^2*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(a^(3/2)*(a^2 + b^2)^2*d) + (b*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^2), x, 16, ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(Sqrt[a]*Sqrt[b]*(a^2 + b^2)^2*d) - ((A*b - a*B)*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x])) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2), x, 16, ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[a]*(a^2*A*b - 3*A*b^3 + a^3*B + 5*a*b^2*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(3/2)*(a^2 + b^2)^2*d) + (a*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(b*(a^2 + b^2)*d*(b + a*Cot[c + d*x])) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2), x, 17, -(((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (a^(3/2)*(a^2*A*b + 5*A*b^3 - 3*a^3*B - 7*a*b^2*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(b^(5/2)*(a^2 + b^2)^2*d) - (a*A*b - 3*a^2*B - 2*b^2*B)/(b^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]) + (a*(A*b - a*B))/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} - - -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 18, -(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b^(3/2)*(63*a^4*A*b + 46*a^2*A*b^3 + 15*A*b^5 - 35*a^5*B - 6*a^3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(7/2)*(a^2 + b^2)^3*d) - ((8*a^4*A + 31*a^2*A*b^2 + 15*A*b^4 - 11*a^3*b*B - 3*a*b^3*B)*Sqrt[Cot[c + d*x]])/(4*a^3*(a^2 + b^2)^2*d) + (b*(A*b - a*B)*Cot[c + d*x]^(5/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (b*(13*a^2*A*b + 5*A*b^3 - 9*a^3*B - a*b^2*B)*Cot[c + d*x]^(3/2))/(4*a^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3, x, 17, -(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[b]*(35*a^4*A*b + 6*a^2*A*b^3 + 3*A*b^5 - 15*a^5*B + 18*a^3*b^2*B + a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(5/2)*(a^2 + b^2)^3*d) + (b*(A*b - a*B)*Cot[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (b*(11*a^2*A*b + 3*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3), x, 17, ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((15*a^4*A*b - 18*a^2*A*b^3 - A*b^5 - 3*a^5*B + 26*a^3*b^2*B - 3*a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*a^(3/2)*Sqrt[b]*(a^2 + b^2)^3*d) + (b*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - ((9*a^2*A*b + A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cot[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3), x, 17, ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^4*A*b - 26*a^2*A*b^3 + 3*A*b^5 + a^5*B + 18*a^3*b^2*B - 15*a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*Sqrt[a]*b^(3/2)*(a^2 + b^2)^3*d) - ((A*b - a*B)*Sqrt[Cot[c + d*x]])/(2*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + ((5*a^2*A*b - 3*A*b^3 - a^3*B + 7*a*b^2*B)*Sqrt[Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3), x, 17, -(((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[a]*(a^4*A*b + 18*a^2*A*b^3 - 15*A*b^5 + 3*a^5*B + 6*a^3*b^2*B + 35*a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*b^(5/2)*(a^2 + b^2)^3*d) + (a*(A*b - a*B)*Sqrt[Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) - (a*(a^2*A*b - 7*A*b^3 + 3*a^3*B + 11*a*b^2*B)*Sqrt[Cot[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^3), x, 18, -(((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((3*a^2*b*(A - B) - b^3*(A - B) - a^3*(A + B) + 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a^(3/2)*(3*a^4*A*b + 6*a^2*A*b^3 + 35*A*b^5 - 15*a^5*B - 46*a^3*b^2*B - 63*a*b^4*B)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/(4*b^(7/2)*(a^2 + b^2)^3*d) - (3*a^3*A*b + 11*a*A*b^3 - 15*a^4*B - 31*a^2*b^2*B - 8*b^4*B)/(4*b^3*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]) + (a*(A*b - a*B))/(2*b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])^2) + (a*(a^2*A*b + 9*A*b^3 - 5*a^3*B - 13*a*b^2*B))/(4*b^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*(b + a*Cot[c + d*x])) - ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) + 3*a^2*b*(A + B) - b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} - - -{(Cot[c + d*x]^(5/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 13, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*B*Cot[c + d*x]^(3/2))/(3*d) + (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(Cot[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 13, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (2*B*Sqrt[Cot[c + d*x]])/d - (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(Sqrt[Cot[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x]), x, 12, (B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a*B + b*B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])), x, 12, (B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) - (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])), x, 13, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*B)/(d*Sqrt[Cot[c + d*x]]) + (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) - (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} -{(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])), x, 13, -((B*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d)) + (B*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*d) + (2*B)/(3*d*Cot[c + d*x]^(3/2)) - (B*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + (B*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cot[c + d*x]^(9/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 12, -((Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(35*a^2*A*b - 8*A*b^3 + 105*a^3*B + 14*a*b^2*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(105*a^3*d) + (2*(35*a^2*A + 4*A*b^2 - 7*a*b*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d) - (2*(A*b + 7*a*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(35*a*d) - (2*A*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 11, (Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2*A + 2*A*b^2 - 5*a*b*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d) - (2*(A*b + 5*a*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*a*d) - (2*A*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 10, (Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(A*b + 3*a*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*a*d) - (2*A*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 9, -((Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*A*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]), x, 13, -((Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (2*Sqrt[b]*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 14, (Sqrt[I*a - b]*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((2*A*b + a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[b]*d) - (Sqrt[I*a + b]*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (B*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{(Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x, 15, (Sqrt[I*a - b]*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4*a*A*b - a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*b^(3/2)*d) + (Sqrt[I*a + b]*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((4*A*b - a*B)*Sqrt[a + b*Tan[c + d*x]])/(4*b*d*Sqrt[Cot[c + d*x]]) + (B*(a + b*Tan[c + d*x])^(3/2))/(2*b*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 13, ((I*a - b)^(3/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(315*a^4*A - 63*a^2*A*b^2 + 8*A*b^4 - 420*a^3*b*B - 18*a*b^3*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(315*a^3*d) + (2*(126*a^2*A*b + 4*A*b^3 + 105*a^3*B - 9*a*b^2*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d) + (2*(21*a^2*A - A*b^2 - 24*a*b*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d) - (2*(10*A*b + 9*a*B)*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(63*d) - (2*a*A*Cot[c + d*x]^(9/2)*Sqrt[a + b*Tan[c + d*x]])/(9*d)} -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 12, -(((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(140*a^2*A*b + 6*A*b^3 + 105*a^3*B - 21*a*b^2*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(105*a^2*d) + (2*(35*a^2*A - 3*A*b^2 - 42*a*b*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(105*a*d) - (2*(8*A*b + 7*a*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(7*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 11, ((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2*A - 3*A*b^2 - 20*a*b*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*a*d) - (2*(6*A*b + 5*a*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 10, ((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(4*A*b + 3*a*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*d) - (2*a*A*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 14, -(((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) + (2*b^(3/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*A*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]), x, 14, -(((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (Sqrt[b]*(2*A*b + 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b*B*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 15, ((a + I*b)^2*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + ((12*a*A*b + 3*a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*Sqrt[b]*d) + ((I*a + b)^(3/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b*B*Sqrt[a + b*Tan[c + d*x]])/(2*d*Cot[c + d*x]^(3/2)) + ((4*A*b + 5*a*B)*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]])} -{((a + b*Tan[c + d*x])^(3/2)*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x, 16, ((I*a - b)^(3/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((6*a^2*A*b - 16*A*b^3 - a^3*B - 24*a*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(8*b^(3/2)*d) + ((I*a + b)^(3/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((6*a*A*b - a^2*B - 8*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(8*b*d*Sqrt[Cot[c + d*x]]) + ((6*A*b - a*B)*(a + b*Tan[c + d*x])^(3/2))/(12*b*d*Sqrt[Cot[c + d*x]]) + (B*(a + b*Tan[c + d*x])^(5/2))/(3*b*d*Sqrt[Cot[c + d*x]])} - - -{Cot[c + d*x]^(13/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 14, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(8085*a^4*A*b - 495*a^2*A*b^3 + 40*A*b^5 + 3465*a^5*B - 5313*a^3*b^2*B - 110*a*b^4*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3465*a^3*d) - (2*(1155*a^4*A - 1485*a^2*A*b^2 - 20*A*b^4 - 2541*a^3*b*B + 55*a*b^3*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3465*a^2*d) + (2*(495*a^2*A*b - 5*A*b^3 + 231*a^3*B - 275*a*b^2*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(1155*a*d) + (2*(99*a^2*A - 113*A*b^2 - 209*a*b*B)*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(693*d) - (2*a*(14*A*b + 11*a*B)*Cot[c + d*x]^(9/2)*Sqrt[a + b*Tan[c + d*x]])/(99*d) - (2*a*A*Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(3/2))/(11*d)} -{Cot[c + d*x]^(11/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 13, ((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*(315*a^4*A - 483*a^2*A*b^2 - 10*A*b^4 - 735*a^3*b*B + 45*a*b^3*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(315*a^2*d) + (2*(231*a^2*A*b - 5*A*b^3 + 105*a^3*B - 135*a*b^2*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(315*a*d) + (2*(21*a^2*A - 25*A*b^2 - 45*a*b*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(105*d) - (2*a*(4*A*b + 3*a*B)*Cot[c + d*x]^(7/2)*Sqrt[a + b*Tan[c + d*x]])/(21*d) - (2*a*A*Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(3/2))/(9*d)} -{Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 12, ((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(245*a^2*A*b - 15*A*b^3 + 105*a^3*B - 161*a*b^2*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(105*a*d) + (2*(35*a^2*A - 45*A*b^2 - 77*a*b*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(105*d) - (2*a*(10*A*b + 7*a*B)*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(35*d) - (2*a*A*Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(3/2))/(7*d)} -{Cot[c + d*x]^(7/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 11, -(((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (2*(15*a^2*A - 23*A*b^2 - 35*a*b*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (2*a*(8*A*b + 5*a*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*d) - (2*a*A*Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(3/2))/(5*d)} -{Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 15, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + (2*b^(5/2)*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - (2*a*(2*A*b + a*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d - (2*a*A*Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 15, ((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b^(3/2)*(2*A*b + 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d - ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b*(2*a*A + b*B)*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Cot[c + d*x]]) - (2*a*A*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/d} -{Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]), x, 15, ((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (Sqrt[b]*(20*a*A*b + 15*a^2*B - 8*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(4*d) + ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + (b*(4*A*b + 7*a*B)*Sqrt[a + b*Tan[c + d*x]])/(4*d*Sqrt[Cot[c + d*x]]) + (b*B*(a + b*Tan[c + d*x])^(3/2))/(2*d*Sqrt[Cot[c + d*x]])} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[Cot[c + d*x]], x, 16, -(((I*a - b)^(5/2)*(A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((30*a^2*A*b - 16*A*b^3 + 5*a^3*B - 40*a*b^2*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(8*Sqrt[b]*d) + ((I*a + b)^(5/2)*(A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((14*a*A*b + 5*a^2*B - 8*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(8*d*Sqrt[Cot[c + d*x]]) + (b*B*(a + b*Tan[c + d*x])^(3/2))/(3*d*Cot[c + d*x]^(3/2)) + ((2*A*b + 3*a*B)*(a + b*Tan[c + d*x])^(3/2))/(4*d*Sqrt[Cot[c + d*x]])} -{((a + b*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/Cot[c + d*x]^(3/2), x, 17, -(((I*a - b)^(5/2)*(I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d) + ((40*a^3*A*b - 320*a*A*b^3 - 5*a^4*B - 240*a^2*b^2*B + 128*b^4*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(64*b^(3/2)*d) - ((I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/d + ((40*a^2*A*b - 64*A*b^3 - 5*a^3*B - 112*a*b^2*B)*Sqrt[a + b*Tan[c + d*x]])/(64*b*d*Sqrt[Cot[c + d*x]]) + ((40*a*A*b - 5*a^2*B - 48*b^2*B)*(a + b*Tan[c + d*x])^(3/2))/(96*b*d*Sqrt[Cot[c + d*x]]) + ((8*A*b - a*B)*(a + b*Tan[c + d*x])^(5/2))/(24*b*d*Sqrt[Cot[c + d*x]]) + (B*(a + b*Tan[c + d*x])^(7/2))/(4*b*d*Sqrt[Cot[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cot[c + d*x]^(7/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 11, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) + (2*(15*a^2*A - 8*A*b^2 + 10*a*b*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(15*a^3*d) + (2*(4*A*b - 5*a*B)*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(15*a^2*d) - (2*A*Cot[c + d*x]^(5/2)*Sqrt[a + b*Tan[c + d*x]])/(5*a*d)} -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 10, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) + (2*(2*A*b - 3*a*B)*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(3*a^2*d) - (2*A*Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]])/(3*a*d)} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 9, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) - (2*A*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/(a*d)} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]], x, 8, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]), x, 13, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[b]*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*Sqrt[a + b*Tan[c + d*x]]), x, 14, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) + ((2*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(3/2)*d) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d) + (B*Sqrt[a + b*Tan[c + d*x]])/(b*d*Sqrt[Cot[c + d*x]])} - - -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 11, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b*(5*a^2*A*b + 8*A*b^3 - 3*a^3*B - 6*a*b^2*B))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) + (2*(4*A*b - 3*a*B)*Sqrt[Cot[c + d*x]])/(3*a^2*d*Sqrt[a + b*Tan[c + d*x]]) - (2*A*Cot[c + d*x]^(3/2))/(3*a*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 10, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*b*(a^2*A + 2*A*b^2 - a*b*B))/(a^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]]) - (2*A*Sqrt[Cot[c + d*x]])/(a*d*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^(1/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 9, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*b*(A*b - a*B))/(a*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(1/2)*(a + b*Tan[c + d*x])^(3/2)), x, 9, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) - (2*(A*b - a*B))/((a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)), x, 14, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(3/2)*d)) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(3/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(3/2)*d) + (2*a*(A*b - a*B))/(b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} - - -{(Cot[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 12, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*b*(7*a^2*A*b + 8*A*b^3 - 3*a^3*B - 4*a*b^2*B))/(3*a^3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*(2*A*b - a*B)*Sqrt[Cot[c + d*x]])/(a^2*d*(a + b*Tan[c + d*x])^(3/2)) - (2*A*Cot[c + d*x]^(3/2))/(3*a*d*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^4*A*b + 30*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B - 17*a^3*b^2*B - 8*a*b^4*B))/(3*a^4*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 11, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) - (2*b*(3*a^2*A + 4*A*b^2 - a*b*B))/(3*a^2*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*A*Sqrt[Cot[c + d*x]])/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (2*b*(3*a^4*A + 17*a^2*A*b^2 + 8*A*b^4 - 8*a^3*b*B - 2*a*b^3*B))/(3*a^3*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2), x, 10, -(((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d)) - ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b + 2*A*b^3 - 5*a^3*B + a*b^2*B))/(3*a^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(5/2)), x, 10, -(((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d)) + ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) - (2*(A*b - a*B))/(3*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B + 4*a*b^2*B))/(3*a*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(5/2)), x, 10, ((A + I*B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) + ((A - I*B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*a*(A*b - a*B))/(3*b*(a^2 + b^2)*d*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) + (2*(2*a^2*A*b - 4*A*b^3 + a^3*B + 7*a*b^2*B))/(3*b*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} -{(A + B*Tan[c + d*x])/(Cot[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^(5/2)), x, 15, ((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a - b)^(5/2)*d) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(b^(5/2)*d) - ((I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/((I*a + b)^(5/2)*d) + (2*a*(A*b - a*B))/(3*b*(a^2 + b^2)*d*Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)) + (2*a*(2*A*b^3 - a*(a^2 + 3*b^2)*B))/(b^2*(a^2 + b^2)^2*d*Sqrt[Cot[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])} - - -{(Sqrt[Cot[c + d*x]]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(3/2), x, 9, (B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) + (B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{(a*B + b*B*Tan[c + d*x])/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)), x, 9, (I*B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d) - (I*B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} -{(a*B + b*B*Tan[c + d*x])/(Cot[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^(3/2)), x, 14, -((B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a - b]*d)) + (2*B*ArcTanh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[b]*d) - (B*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[I*a + b]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^(m/2) (A+B Tan[c+d x]) (a+b Tan[c+d x])^n with n symbolic*) - - -{Cot[c + d*x]^m*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 8, (1/(2*d*(1 - m)))*(((A + I*B)*AppellF1[1 - m, -n, 1, 2 - m, -((b*Tan[c + d*x])/a), (-I)*Tan[c + d*x]]*Cot[c + d*x]^(-1 + m)*(a + b*Tan[c + d*x])^n)/(1 + (b*Tan[c + d*x])/a)^n) + (1/(2*d*(1 - m)))*(((A - I*B)*AppellF1[1 - m, -n, 1, 2 - m, -((b*Tan[c + d*x])/a), I*Tan[c + d*x]]*Cot[c + d*x]^(-1 + m)*(a + b*Tan[c + d*x])^n)/(1 + (b*Tan[c + d*x])/a)^n)} - - -{Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 10, -(((A + I*B)*AppellF1[-(1/2), 1, -n, 1/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)) - ((A - I*B)*AppellF1[-(1/2), 1, -n, 1/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)} -{Sqrt[Cot[c + d*x]]*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 10, ((A + I*B)*AppellF1[1/2, 1, -n, 3/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Cot[c + d*x]])) + ((A - I*B)*AppellF1[1/2, 1, -n, 3/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Cot[c + d*x]]))} -{(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n/Sqrt[Cot[c + d*x]], x, 10, ((A + I*B)*AppellF1[3/2, 1, -n, 5/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d*Cot[c + d*x]^(3/2))) + ((A - I*B)*AppellF1[3/2, 1, -n, 5/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d*Cot[c + d*x]^(3/2)))} -{(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n/Cot[c + d*x]^(3/2), x, 10, ((A + I*B)*AppellF1[5/2, 1, -n, 7/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d*Cot[c + d*x]^(5/2))) + ((A - I*B)*AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d*Cot[c + d*x]^(5/2)))} - - -{Tan[c + d*x]^(3/2)*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, ((A + I*B)*AppellF1[5/2, 1, -n, 7/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d)) + ((A - I*B)*AppellF1[5/2, 1, -n, 7/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(5*d))} -{Tan[c + d*x]^(1/2)*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, ((A + I*B)*AppellF1[3/2, 1, -n, 5/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d)) + ((A - I*B)*AppellF1[3/2, 1, -n, 5/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(3*d))} -{Tan[c + d*x]^(-1/2)*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, ((A + I*B)*AppellF1[1/2, 1, -n, 3/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d) + ((A - I*B)*AppellF1[1/2, 1, -n, 3/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*d)} -{Tan[c + d*x]^(-3/2)*(A + B*Tan[c + d*x])*(a + b*Tan[c + d*x])^n, x, 9, -(((A + I*B)*AppellF1[-(1/2), 1, -n, 1/2, (-I)*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Tan[c + d*x]]))) - ((A - I*B)*AppellF1[-(1/2), 1, -n, 1/2, I*Tan[c + d*x], -((b*Tan[c + d*x])/a)]*(a + b*Tan[c + d*x])^n)/((1 + (b*Tan[c + d*x])/a)^n*(d*Sqrt[Tan[c + d*x]]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n, x, 3, (a*(I*A + B)*(c - I*c*Tan[e + f*x])^n)/(f*n) - (a*B*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n))} - -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4, x, 3, (a*(I*A + B)*c^4*(1 - I*Tan[e + f*x])^4)/(4*f) - (a*B*c^4*(1 - I*Tan[e + f*x])^5)/(5*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3, x, 3, (a*(I*A + B)*c^3*(1 - I*Tan[e + f*x])^3)/(3*f) - (a*B*c^3*(1 - I*Tan[e + f*x])^4)/(4*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2, x, 3, (a*A*c^2*Tan[e + f*x])/f - (a*(I*A - B)*c^2*Tan[e + f*x]^2)/(2*f) - (I*a*B*c^2*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]), x, 2, (a*A*c*Tan[e + f*x])/f + (a*B*c*Tan[e + f*x]^2)/(2*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]), x, 2, a*(A - I*B)*x - (a*(I*A + B)*Log[Cos[e + f*x]])/f + (I*a*B*Tan[e + f*x])/f} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]), x, 3, (I*a*B*x)/c + (a*B*Log[Cos[e + f*x]])/(c*f) + (a*(A - I*B))/(c*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2, x, 2, (a*(A + B*Tan[e + f*x])^2)/(2*(I*A + B)*c^2*f*(1 - I*Tan[e + f*x])^2)} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3, x, 3, -((a*(A - I*B))/(3*c^3*f*(I + Tan[e + f*x])^3)) - (a*B)/(2*c^3*f*(I + Tan[e + f*x])^2)} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4, x, 3, -((a*(I*A + B))/(4*c^4*f*(I + Tan[e + f*x])^4)) - (I*a*B)/(3*c^4*f*(I + Tan[e + f*x])^3)} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5, x, 3, (a*(A - I*B))/(5*c^5*f*(I + Tan[e + f*x])^5) + (a*B)/(4*c^5*f*(I + Tan[e + f*x])^4)} - - -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n, x, 3, (2*a^2*(I*A + B)*(c - I*c*Tan[e + f*x])^n)/(f*n) - (a^2*(I*A + 3*B)*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n)) + (a^2*B*(c - I*c*Tan[e + f*x])^(2 + n))/(c^2*f*(2 + n))} - -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5, x, 3, (2*a^2*(I*A + B)*c^5*(1 - I*Tan[e + f*x])^5)/(5*f) - (a^2*(I*A + 3*B)*c^5*(1 - I*Tan[e + f*x])^6)/(6*f) + (a^2*B*c^5*(1 - I*Tan[e + f*x])^7)/(7*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4, x, 3, (a^2*(I*A + B)*c^4*(1 - I*Tan[e + f*x])^4)/(2*f) - (a^2*(I*A + 3*B)*c^4*(1 - I*Tan[e + f*x])^5)/(5*f) + (a^2*B*c^4*(1 - I*Tan[e + f*x])^6)/(6*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3, x, 3, (2*a^2*(I*A + B)*c^3*(1 - I*Tan[e + f*x])^3)/(3*f) - (a^2*(I*A + 3*B)*c^3*(1 - I*Tan[e + f*x])^4)/(4*f) + (a^2*B*c^3*(1 - I*Tan[e + f*x])^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2, x, 4, (a^2*B*c^2*Sec[e + f*x]^4)/(4*f) + (a^2*A*c^2*Tan[e + f*x])/f + (a^2*A*c^2*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]), x, 3, (a^2*A*c*Tan[e + f*x])/f + (a^2*(I*A + B)*c*Tan[e + f*x]^2)/(2*f) + (I*a^2*B*c*Tan[e + f*x]^3)/(3*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]), x, 3, 2*a^2*(A - I*B)*x - (2*a^2*(I*A + B)*Log[Cos[e + f*x]])/f - (a^2*(A - I*B)*Tan[e + f*x])/f + (B*(a + I*a*Tan[e + f*x])^2)/(2*f)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]), x, 3, -((a^2*(A - (3*I)*B)*x)/c) + (a^2*(I*A + 3*B)*Log[Cos[e + f*x]])/(c*f) - (I*a^2*B*Tan[e + f*x])/(c*f) + (2*a^2*(A - I*B))/(c*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2, x, 3, ((-I)*a^2*B*x)/c^2 - (a^2*B*Log[Cos[e + f*x]])/(c^2*f) + (a^2*(I*A + B))/(c^2*f*(I + Tan[e + f*x])^2) - (a^2*(A - (3*I)*B))/(c^2*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3, x, 3, (-2*a^2*(A - I*B))/(3*c^3*f*(I + Tan[e + f*x])^3) - (a^2*(I*A + 3*B))/(2*c^3*f*(I + Tan[e + f*x])^2) - (I*a^2*B)/(c^3*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4, x, 3, -(a^2*(I*A + B))/(2*c^4*f*(I + Tan[e + f*x])^4) + (a^2*(A - (3*I)*B))/(3*c^4*f*(I + Tan[e + f*x])^3) + (a^2*B)/(2*c^4*f*(I + Tan[e + f*x])^2)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5, x, 3, (2*a^2*(A - I*B))/(5*c^5*f*(I + Tan[e + f*x])^5) + (a^2*(I*A + 3*B))/(4*c^5*f*(I + Tan[e + f*x])^4) + ((I/3)*a^2*B)/(c^5*f*(I + Tan[e + f*x])^3)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6, x, 3, (a^2*(I*A + B))/(3*c^6*f*(I + Tan[e + f*x])^6) - (a^2*(A - (3*I)*B))/(5*c^6*f*(I + Tan[e + f*x])^5) - (a^2*B)/(4*c^6*f*(I + Tan[e + f*x])^4)} - - -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n, x, 3, (4*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^n)/(f*n) - (4*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n)) + (a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(2 + n))/(c^2*f*(2 + n)) - (a^3*B*(c - I*c*Tan[e + f*x])^(3 + n))/(c^3*f*(3 + n))} - -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^6, x, 3, (2*a^3*(I*A + B)*c^6*(1 - I*Tan[e + f*x])^6)/(3*f) - (4*a^3*(I*A + 2*B)*c^6*(1 - I*Tan[e + f*x])^7)/(7*f) + (a^3*(I*A + 5*B)*c^6*(1 - I*Tan[e + f*x])^8)/(8*f) - (a^3*B*c^6*(1 - I*Tan[e + f*x])^9)/(9*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5, x, 3, (4*a^3*(I*A + B)*c^5*(1 - I*Tan[e + f*x])^5)/(5*f) - (2*a^3*(I*A + 2*B)*c^5*(1 - I*Tan[e + f*x])^6)/(3*f) + (a^3*(I*A + 5*B)*c^5*(1 - I*Tan[e + f*x])^7)/(7*f) - (a^3*B*c^5*(1 - I*Tan[e + f*x])^8)/(8*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4, x, 3, (a^3*(I*A + B)*c^4*(1 - I*Tan[e + f*x])^4)/f - (4*a^3*(I*A + 2*B)*c^4*(1 - I*Tan[e + f*x])^5)/(5*f) + (a^3*(I*A + 5*B)*c^4*(1 - I*Tan[e + f*x])^6)/(6*f) - (a^3*B*c^4*(1 - I*Tan[e + f*x])^7)/(7*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3, x, 5, (a^3*B*c^3*Sec[e + f*x]^6)/(6*f) + (a^3*A*c^3*Tan[e + f*x])/f + (2*a^3*A*c^3*Tan[e + f*x]^3)/(3*f) + (a^3*A*c^3*Tan[e + f*x]^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2, x, 3, -((2*a^3*(I*A - B)*c^2*(1 + I*Tan[e + f*x])^3)/(3*f)) + (a^3*(I*A - 3*B)*c^2*(1 + I*Tan[e + f*x])^4)/(4*f) + (a^3*B*c^2*(1 + I*Tan[e + f*x])^5)/(5*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]), x, 3, -((a^3*(I*A - B)*c*(1 + I*Tan[e + f*x])^3)/(3*f)) - (a^3*B*c*(1 + I*Tan[e + f*x])^4)/(4*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]), x, 4, 4*a^3*(A - I*B)*x - (4*a^3*(I*A + B)*Log[Cos[e + f*x]])/f - (2*a^3*(A - I*B)*Tan[e + f*x])/f + (a*(I*A + B)*(a + I*a*Tan[e + f*x])^2)/(2*f) + (B*(a + I*a*Tan[e + f*x])^3)/(3*f)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]), x, 3, (-4*a^3*(A - (2*I)*B)*x)/c + (4*a^3*(I*A + 2*B)*Log[Cos[e + f*x]])/(c*f) + (a^3*(A - (4*I)*B)*Tan[e + f*x])/(c*f) + (a^3*B*Tan[e + f*x]^2)/(2*c*f) + (4*a^3*(A - I*B))/(c*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^2, x, 3, (a^3*(A - (5*I)*B)*x)/c^2 - (a^3*(I*A + 5*B)*Log[Cos[e + f*x]])/(c^2*f) + (I*a^3*B*Tan[e + f*x])/(c^2*f) + (2*a^3*(I*A + B))/(c^2*f*(I + Tan[e + f*x])^2) - (4*a^3*(A - (2*I)*B))/(c^2*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^3, x, 4, (I*a^3*B*x)/c^3 + (a^3*B*Log[Cos[e + f*x]])/(c^3*f) - (a^3*(I*A + B)*(1 + I*Tan[e + f*x])^3)/(6*c^3*f*(1 - I*Tan[e + f*x])^3) - (2*a^3*B)/(c^3*f*(I + Tan[e + f*x])^2) - ((4*I)*a^3*B)/(c^3*f*(I + Tan[e + f*x]))} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^4, x, 3, -(a^3*(I*A + B)*(1 + I*Tan[e + f*x])^3)/(8*c^4*f*(1 - I*Tan[e + f*x])^4) - (a^3*(I*A - 7*B)*(1 + I*Tan[e + f*x])^3)/(48*c^4*f*(1 - I*Tan[e + f*x])^3)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^5, x, 3, (4*a^3*(A - I*B))/(5*c^5*f*(I + Tan[e + f*x])^5) + (a^3*(I*A + 2*B))/(c^5*f*(I + Tan[e + f*x])^4) - (a^3*(A - (5*I)*B))/(3*c^5*f*(I + Tan[e + f*x])^3) - (a^3*B)/(2*c^5*f*(I + Tan[e + f*x])^2)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6, x, 3, (2*a^3*(I*A + B))/(3*c^6*f*(I + Tan[e + f*x])^6) - (4*a^3*(A - (2*I)*B))/(5*c^6*f*(I + Tan[e + f*x])^5) - (a^3*(I*A + 5*B))/(4*c^6*f*(I + Tan[e + f*x])^4) - ((I/3)*a^3*B)/(c^6*f*(I + Tan[e + f*x])^3)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^7, x, 3, (-4*a^3*(A - I*B))/(7*c^7*f*(I + Tan[e + f*x])^7) - (2*a^3*(I*A + 2*B))/(3*c^7*f*(I + Tan[e + f*x])^6) + (a^3*(A - (5*I)*B))/(5*c^7*f*(I + Tan[e + f*x])^5) + (a^3*B)/(4*c^7*f*(I + Tan[e + f*x])^4)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^8, x, 3, -(a^3*(I*A + B))/(2*c^8*f*(I + Tan[e + f*x])^8) + (4*a^3*(A - (2*I)*B))/(7*c^8*f*(I + Tan[e + f*x])^7) + (a^3*(I*A + 5*B))/(6*c^8*f*(I + Tan[e + f*x])^6) + ((I/5)*a^3*B)/(c^8*f*(I + Tan[e + f*x])^5)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x]), x, 3, ((I*A*(1 - n) + B*(1 + n))*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(4*a*f*n) + ((I*A - B)*(c - I*c*Tan[e + f*x])^n)/(2*a*f*(1 + I*Tan[e + f*x]))} - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x]), x, 3, (-4*(3*A + (5*I)*B)*c^4*x)/a - (4*((3*I)*A - 5*B)*c^4*Log[Cos[e + f*x]])/(a*f) - (8*(A + I*B)*c^4)/(a*f*(I - Tan[e + f*x])) + ((5*A + (12*I)*B)*c^4*Tan[e + f*x])/(a*f) - ((I*A - 5*B)*c^4*Tan[e + f*x]^2)/(2*a*f) - ((I/3)*B*c^4*Tan[e + f*x]^3)/(a*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x]), x, 3, (-4*(A + (2*I)*B)*c^3*x)/a - (4*(I*A - 2*B)*c^3*Log[Cos[e + f*x]])/(a*f) - (4*(A + I*B)*c^3)/(a*f*(I - Tan[e + f*x])) + ((A + (4*I)*B)*c^3*Tan[e + f*x])/(a*f) + (B*c^3*Tan[e + f*x]^2)/(2*a*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x]), x, 3, -(((A + (3*I)*B)*c^2*x)/a) - ((I*A - 3*B)*c^2*Log[Cos[e + f*x]])/(a*f) - (2*(A + I*B)*c^2)/(a*f*(I - Tan[e + f*x])) + (I*B*c^2*Tan[e + f*x])/(a*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x]), x, 3, -((I*B*c*x)/a) + (B*c*Log[Cos[e + f*x]])/(a*f) - ((A + I*B)*c)/(a*f*(I - Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x]), x, 2, ((A - I*B)*x)/(2*a) + (I*A - B)/(2*f*(a + I*a*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])), x, 4, (A*x)/(2*a*c) - (Cos[e + f*x]^2*(B - A*Tan[e + f*x]))/(2*a*c*f)} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2), x, 4, ((3*A + I*B)*x)/(8*a*c^2) - (A + I*B)/(8*a*c^2*f*(I - Tan[e + f*x])) + (I*A + B)/(8*a*c^2*f*(I + Tan[e + f*x])^2) + A/(4*a*c^2*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3), x, 4, ((2*A + I*B)*x)/(8*a*c^3) - (A + I*B)/(16*a*c^3*f*(I - Tan[e + f*x])) - (A - I*B)/(12*a*c^3*f*(I + Tan[e + f*x])^3) + ((I/8)*A)/(a*c^3*f*(I + Tan[e + f*x])^2) + (3*A + I*B)/(16*a*c^3*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4), x, 4, ((5*A + (3*I)*B)*x)/(32*a*c^4) - (A + I*B)/(32*a*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(16*a*c^4*f*(I + Tan[e + f*x])^4) - A/(12*a*c^4*f*(I + Tan[e + f*x])^3) + ((3*I)*A - B)/(32*a*c^4*f*(I + Tan[e + f*x])^2) + (2*A + I*B)/(16*a*c^4*f*(I + Tan[e + f*x]))} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^2, x, 3, ((I*A*(2 - n) + B*(2 + n))*Hypergeometric2F1[2, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(16*a^2*f*n) + ((I*A - B)*(c - I*c*Tan[e + f*x])^n)/(4*a^2*f*(1 + I*Tan[e + f*x])^2)} - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^2, x, 3, (8*(3*A + (7*I)*B)*c^5*x)/a^2 + (8*((3*I)*A - 7*B)*c^5*Log[Cos[e + f*x]])/(a^2*f) - (8*(I*A - B)*c^5)/(a^2*f*(I - Tan[e + f*x])^2) + (16*(2*A + (3*I)*B)*c^5)/(a^2*f*(I - Tan[e + f*x])) - ((7*A + (24*I)*B)*c^5*Tan[e + f*x])/(a^2*f) + ((I*A - 7*B)*c^5*Tan[e + f*x]^2)/(2*a^2*f) + ((I/3)*B*c^5*Tan[e + f*x]^3)/(a^2*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^2, x, 3, (6*(A + (3*I)*B)*c^4*x)/a^2 + (6*(I*A - 3*B)*c^4*Log[Cos[e + f*x]])/(a^2*f) - (4*(I*A - B)*c^4)/(a^2*f*(I - Tan[e + f*x])^2) + (4*(3*A + (5*I)*B)*c^4)/(a^2*f*(I - Tan[e + f*x])) - ((A + (6*I)*B)*c^4*Tan[e + f*x])/(a^2*f) - (B*c^4*Tan[e + f*x]^2)/(2*a^2*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x])^2, x, 3, ((A + (5*I)*B)*c^3*x)/a^2 + ((I*A - 5*B)*c^3*Log[Cos[e + f*x]])/(a^2*f) - (2*(I*A - B)*c^3)/(a^2*f*(I - Tan[e + f*x])^2) + (4*(A + (2*I)*B)*c^3)/(a^2*f*(I - Tan[e + f*x])) - (I*B*c^3*Tan[e + f*x])/(a^2*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x])^2, x, 3, (I*B*c^2*x)/a^2 - (B*c^2*Log[Cos[e + f*x]])/(a^2*f) - ((I*A - B)*c^2)/(a^2*f*(I - Tan[e + f*x])^2) + ((A + (3*I)*B)*c^2)/(a^2*f*(I - Tan[e + f*x]))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^2, x, 2, -((c*(A + B*Tan[e + f*x])^2)/(2*a^2*(I*A - B)*f*(1 + I*Tan[e + f*x])^2))} -{(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^2, x, 3, ((A - I*B)*x)/(4*a^2) + (I*A - B)/(4*f*(a + I*a*Tan[e + f*x])^2) + (I*A + B)/(4*f*(a^2 + I*a^2*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])), x, 4, ((3*A - I*B)*x)/(8*a^2*c) - (I*A - B)/(8*a^2*c*f*(I - Tan[e + f*x])^2) - A/(4*a^2*c*f*(I - Tan[e + f*x])) + (A - I*B)/(8*a^2*c*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^2), x, 5, (3*A*x)/(8*a^2*c^2) + (3*A*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*c^2*f) - (Cos[e + f*x]^4*(B - A*Tan[e + f*x]))/(4*a^2*c^2*f)} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^3), x, 4, ((5*A + I*B)*x)/(16*a^2*c^3) - (I*A - B)/(32*a^2*c^3*f*(I - Tan[e + f*x])^2) - (2*A + I*B)/(16*a^2*c^3*f*(I - Tan[e + f*x])) - (A - I*B)/(24*a^2*c^3*f*(I + Tan[e + f*x])^3) + ((3*I)*A + B)/(32*a^2*c^3*f*(I + Tan[e + f*x])^2) + (3*A)/(16*a^2*c^3*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4), x, 4, (5*(3*A + I*B)*x)/(64*a^2*c^4) - (I*A - B)/(64*a^2*c^4*f*(I - Tan[e + f*x])^2) - (5*A + (3*I)*B)/(64*a^2*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(32*a^2*c^4*f*(I + Tan[e + f*x])^4) - (3*A - I*B)/(48*a^2*c^4*f*(I + Tan[e + f*x])^3) + (((3*I)/32)*A)/(a^2*c^4*f*(I + Tan[e + f*x])^2) + (5*A + I*B)/(32*a^2*c^4*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^5), x, 4, (3*(7*A + (3*I)*B)*x)/(128*a^2*c^5) - (I*A - B)/(128*a^2*c^5*f*(I - Tan[e + f*x])^2) - (3*A + (2*I)*B)/(64*a^2*c^5*f*(I - Tan[e + f*x])) + (A - I*B)/(40*a^2*c^5*f*(I + Tan[e + f*x])^5) - ((3*I)*A + B)/(64*a^2*c^5*f*(I + Tan[e + f*x])^4) - A/(16*a^2*c^5*f*(I + Tan[e + f*x])^3) + ((5*I)*A - B)/(64*a^2*c^5*f*(I + Tan[e + f*x])^2) + (5*(3*A + I*B))/(128*a^2*c^5*f*(I + Tan[e + f*x]))} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^n)/(a + I*a*Tan[e + f*x])^3, x, 3, ((I*A*(3 - n) + B*(3 + n))*Hypergeometric2F1[3, n, 1 + n, (1/2)*(1 - I*Tan[e + f*x])]*(c - I*c*Tan[e + f*x])^n)/(48*a^3*f*n) + ((I*A - B)*(c - I*c*Tan[e + f*x])^n)/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^5)/(a + I*a*Tan[e + f*x])^3, x, 3, (-8*(A + (4*I)*B)*c^5*x)/a^3 - (8*(I*A - 4*B)*c^5*Log[Cos[e + f*x]])/(a^3*f) + (16*(A + I*B)*c^5)/(3*a^3*f*(I - Tan[e + f*x])^3) + (8*((2*I)*A - 3*B)*c^5)/(a^3*f*(I - Tan[e + f*x])^2) - (8*(3*A + (7*I)*B)*c^5)/(a^3*f*(I - Tan[e + f*x])) + ((A + (8*I)*B)*c^5*Tan[e + f*x])/(a^3*f) + (B*c^5*Tan[e + f*x]^2)/(2*a^3*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4)/(a + I*a*Tan[e + f*x])^3, x, 3, -(((A + (7*I)*B)*c^4*x)/a^3) - ((I*A - 7*B)*c^4*Log[Cos[e + f*x]])/(a^3*f) + (8*(A + I*B)*c^4)/(3*a^3*f*(I - Tan[e + f*x])^3) + (2*((3*I)*A - 5*B)*c^4)/(a^3*f*(I - Tan[e + f*x])^2) - (6*(A + (3*I)*B)*c^4)/(a^3*f*(I - Tan[e + f*x])) + (I*B*c^4*Tan[e + f*x])/(a^3*f)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^3)/(a + I*a*Tan[e + f*x])^3, x, 4, ((-I)*B*c^3*x)/a^3 + (B*c^3*Log[Cos[e + f*x]])/(a^3*f) - (2*B*c^3)/(a^3*f*(I - Tan[e + f*x])^2) - ((4*I)*B*c^3)/(a^3*f*(I - Tan[e + f*x])) + ((I*A - B)*c^3*(1 - I*Tan[e + f*x])^3)/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^2)/(a + I*a*Tan[e + f*x])^3, x, 3, (2*(A + I*B)*c^2)/(3*a^3*f*(I - Tan[e + f*x])^3) + ((I*A - 3*B)*c^2)/(2*a^3*f*(I - Tan[e + f*x])^2) - (I*B*c^2)/(a^3*f*(I - Tan[e + f*x]))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^3, x, 3, ((A + I*B)*c)/(3*a^3*f*(I - Tan[e + f*x])^3) - (B*c)/(2*a^3*f*(I - Tan[e + f*x])^2)} -{(A + B*Tan[e + f*x])/(a + I*a*Tan[e + f*x])^3, x, 4, ((A - I*B)*x)/(8*a^3) + (I*A - B)/(6*f*(a + I*a*Tan[e + f*x])^3) + (I*A + B)/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (I*A + B)/(8*f*(a^3 + I*a^3*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])), x, 4, ((2*A - I*B)*x)/(8*a^3*c) + (A + I*B)/(12*a^3*c*f*(I - Tan[e + f*x])^3) - ((I/8)*A)/(a^3*c*f*(I - Tan[e + f*x])^2) - (3*A - I*B)/(16*a^3*c*f*(I - Tan[e + f*x])) + (A - I*B)/(16*a^3*c*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2), x, 4, ((5*A - I*B)*x)/(16*a^3*c^2) + (A + I*B)/(24*a^3*c^2*f*(I - Tan[e + f*x])^3) - ((3*I)*A - B)/(32*a^3*c^2*f*(I - Tan[e + f*x])^2) - (3*A)/(16*a^3*c^2*f*(I - Tan[e + f*x])) + (I*A + B)/(32*a^3*c^2*f*(I + Tan[e + f*x])^2) + (2*A - I*B)/(16*a^3*c^2*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^3), x, 6, (5*A*x)/(16*a^3*c^3) + (5*A*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*c^3*f) + (5*A*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^3*c^3*f) - (Cos[e + f*x]^6*(B - A*Tan[e + f*x]))/(6*a^3*c^3*f)} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4), x, 4, (5*(7*A + I*B)*x)/(128*a^3*c^4) + (A + I*B)/(96*a^3*c^4*f*(I - Tan[e + f*x])^3) - ((5*I)*A - 3*B)/(128*a^3*c^4*f*(I - Tan[e + f*x])^2) - (5*(3*A + I*B))/(128*a^3*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(64*a^3*c^4*f*(I + Tan[e + f*x])^4) - (2*A - I*B)/(48*a^3*c^4*f*(I + Tan[e + f*x])^3) + ((5*I)*A + B)/(64*a^3*c^4*f*(I + Tan[e + f*x])^2) + (5*A)/(32*a^3*c^4*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5), x, 4, (7*(4*A + I*B)*x)/(128*a^3*c^5) + (A + I*B)/(192*a^3*c^5*f*(I - Tan[e + f*x])^3) - ((3*I)*A - 2*B)/(128*a^3*c^5*f*(I - Tan[e + f*x])^2) - (3*(7*A + (3*I)*B))/(256*a^3*c^5*f*(I - Tan[e + f*x])) + (A - I*B)/(80*a^3*c^5*f*(I + Tan[e + f*x])^5) - ((2*I)*A + B)/(64*a^3*c^5*f*(I + Tan[e + f*x])^4) - (5*A - I*B)/(96*a^3*c^5*f*(I + Tan[e + f*x])^3) + (((5*I)/64)*A)/(a^3*c^5*f*(I + Tan[e + f*x])^2) + (5*(7*A + I*B))/(256*a^3*c^5*f*(I + Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^6), x, 4, (7*(3*A + I*B)*x)/(128*a^3*c^6) + (A + I*B)/(384*a^3*c^6*f*(I - Tan[e + f*x])^3) - ((7*I)*A - 5*B)/(512*a^3*c^6*f*(I - Tan[e + f*x])^2) - (7*(2*A + I*B))/(256*a^3*c^6*f*(I - Tan[e + f*x])) + (I*A + B)/(96*a^3*c^6*f*(I + Tan[e + f*x])^6) + (2*A - I*B)/(80*a^3*c^6*f*(I + Tan[e + f*x])^5) - ((5*I)*A + B)/(128*a^3*c^6*f*(I + Tan[e + f*x])^4) - (5*A)/(96*a^3*c^6*f*(I + Tan[e + f*x])^3) + (5*((7*I)*A - B))/(512*a^3*c^6*f*(I + Tan[e + f*x])^2) + (7*(4*A + I*B))/(256*a^3*c^6*f*(I + Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^(n/2) (A+B Tan[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 3, (2*a*(I*A + B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*f) - (2*a*B*(c - I*c*Tan[e + f*x])^(9/2))/(9*c*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 3, (2*a*(I*A + B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f) - (2*a*B*(c - I*c*Tan[e + f*x])^(7/2))/(7*c*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 3, (2*a*(I*A + B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f) - (2*a*B*(c - I*c*Tan[e + f*x])^(5/2))/(5*c*f)} -{(a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 3, (2*a*(I*A + B)*Sqrt[c - I*c*Tan[e + f*x]])/f - (2*a*B*(c - I*c*Tan[e + f*x])^(3/2))/(3*c*f)} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 3, (-2*a*(I*A + B))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (2*a*B*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 3, (-2*a*(I*A + B))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a*B)/(c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 3, (-2*a*(I*A + B))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (2*a*B)/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2))} -{((a + I*a*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 3, (-2*a*(I*A + B))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) + (2*a*B)/(5*c*f*(c - I*c*Tan[e + f*x])^(5/2))} - - -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 3, (4*a^2*(I*A + B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*f) - (2*a^2*(I*A + 3*B)*(c - I*c*Tan[e + f*x])^(9/2))/(9*c*f) + (2*a^2*B*(c - I*c*Tan[e + f*x])^(11/2))/(11*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 3, (4*a^2*(I*A + B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f) - (2*a^2*(I*A + 3*B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*c*f) + (2*a^2*B*(c - I*c*Tan[e + f*x])^(9/2))/(9*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 3, (4*a^2*(I*A + B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f) - (2*a^2*(I*A + 3*B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*c*f) + (2*a^2*B*(c - I*c*Tan[e + f*x])^(7/2))/(7*c^2*f)} -{(a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 3, (4*a^2*(I*A + B)*Sqrt[c - I*c*Tan[e + f*x]])/f - (2*a^2*(I*A + 3*B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*c*f) + (2*a^2*B*(c - I*c*Tan[e + f*x])^(5/2))/(5*c^2*f)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 3, (-4*a^2*(I*A + B))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (2*a^2*(I*A + 3*B)*Sqrt[c - I*c*Tan[e + f*x]])/(c*f) + (2*a^2*B*(c - I*c*Tan[e + f*x])^(3/2))/(3*c^2*f)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 3, (-4*a^2*(I*A + B))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a^2*(I*A + 3*B))/(c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (2*a^2*B*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f)} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 3, (-4*a^2*(I*A + B))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (2*a^2*(I*A + 3*B))/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*a^2*B)/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 3, (-4*a^2*(I*A + B))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) + (2*a^2*(I*A + 3*B))/(5*c*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*a^2*B)/(3*c^2*f*(c - I*c*Tan[e + f*x])^(3/2))} - - -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 3, (8*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*f) - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(9/2))/(9*c*f) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(11/2))/(11*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(13/2))/(13*c^3*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 3, (8*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f) - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*c*f) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(9/2))/(9*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(11/2))/(11*c^3*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 3, (8*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f) - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*c*f) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(9/2))/(9*c^3*f)} -{(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 3, (8*a^3*(I*A + B)*Sqrt[c - I*c*Tan[e + f*x]])/f - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*c*f) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(7/2))/(7*c^3*f)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 3, (-8*a^3*(I*A + B))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (8*a^3*(I*A + 2*B)*Sqrt[c - I*c*Tan[e + f*x]])/(c*f) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(5/2))/(5*c^3*f)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 3, (-8*a^3*(I*A + B))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (8*a^3*(I*A + 2*B))/(c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (2*a^3*(I*A + 5*B)*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(3/2))/(3*c^3*f)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 3, (-8*a^3*(I*A + B))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (8*a^3*(I*A + 2*B))/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*a^3*(I*A + 5*B))/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]]) - (2*a^3*B*Sqrt[c - I*c*Tan[e + f*x]])/(c^3*f)} -{((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 3, (-8*a^3*(I*A + B))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) + (8*a^3*(I*A + 2*B))/(5*c*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*a^3*(I*A + 5*B))/(3*c^2*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a^3*B)/(c^3*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x]), x, 7, (-2*Sqrt[2]*((5*I)*A - 9*B)*c^(7/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(a*f) + (2*((5*I)*A - 9*B)*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(a*f) + (((5*I)*A - 9*B)*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(3*a*f) + (((5*I)*A - 9*B)*c*(c - I*c*Tan[e + f*x])^(5/2))/(10*a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(2*a*f*(1 + I*Tan[e + f*x]))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x]), x, 6, -((Sqrt[2]*((3*I)*A - 7*B)*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(a*f)) + (((3*I)*A - 7*B)*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(a*f) + (((3*I)*A - 7*B)*c*(c - I*c*Tan[e + f*x])^(3/2))/(6*a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(2*a*f*(1 + I*Tan[e + f*x]))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x]), x, 5, -(((I*A - 5*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a*f)) + ((I*A - 5*B)*c*Sqrt[c - I*c*Tan[e + f*x]])/(2*a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(2*a*f*(1 + I*Tan[e + f*x]))} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x]), x, 4, ((I*A + 3*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(2*Sqrt[2]*a*f) + ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(2*a*f*(1 + I*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]), x, 5, (((3*I)*A + B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(4*Sqrt[2]*a*Sqrt[c]*f) - ((3*I)*A + B)/(4*a*f*Sqrt[c - I*c*Tan[e + f*x]]) + (I*A - B)/(2*a*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)), x, 6, (((5*I)*A - B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[2]*a*c^(3/2)*f) - ((5*I)*A - B)/(12*a*f*(c - I*c*Tan[e + f*x])^(3/2)) + (I*A - B)/(2*a*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - ((5*I)*A - B)/(8*a*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)), x, 7, (((7*I)*A - 3*B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a*c^(5/2)*f) - ((7*I)*A - 3*B)/(20*a*f*(c - I*c*Tan[e + f*x])^(5/2)) + (I*A - B)/(2*a*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - ((7*I)*A - 3*B)/(24*a*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - ((7*I)*A - 3*B)/(16*a*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^2, x, 8, (7*(5*I*A - 13*B)*c^(9/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a^2*f) - (7*(5*I*A - 13*B)*c^4*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^2*f) - (7*(5*I*A - 13*B)*c^3*(c - I*c*Tan[e + f*x])^(3/2))/(12*a^2*f) - (7*(5*I*A - 13*B)*c^2*(c - I*c*Tan[e + f*x])^(5/2))/(40*a^2*f) - ((5*I*A - 13*B)*c*(c - I*c*Tan[e + f*x])^(7/2))/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(9/2))/(4*a^2*f*(1 + I*Tan[e + f*x])^2)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^2, x, 7, (5*((3*I)*A - 11*B)*c^(7/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(2*Sqrt[2]*a^2*f) - (5*((3*I)*A - 11*B)*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(4*a^2*f) - (5*((3*I)*A - 11*B)*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(24*a^2*f) - (((3*I)*A - 11*B)*c*(c - I*c*Tan[e + f*x])^(5/2))/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(4*a^2*f*(1 + I*Tan[e + f*x])^2)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^2, x, 6, (3*(I*A - 9*B)*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(4*Sqrt[2]*a^2*f) - (3*(I*A - 9*B)*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(8*a^2*f) - ((I*A - 9*B)*c*(c - I*c*Tan[e + f*x])^(3/2))/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(4*a^2*f*(1 + I*Tan[e + f*x])^2)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^2, x, 5, -((I*A + 7*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[2]*a^2*f) + ((I*A + 7*B)*c*Sqrt[c - I*c*Tan[e + f*x]])/(8*a^2*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(4*a^2*f*(1 + I*Tan[e + f*x])^2)} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^2, x, 5, (((3*I)*A + 5*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a^2*f) + ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(4*a^2*f*(1 + I*Tan[e + f*x])^2) + (((3*I)*A + 5*B)*Sqrt[c - I*c*Tan[e + f*x]])/(16*a^2*f*(1 + I*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]), x, 6, (3*((5*I)*A + 3*B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(32*Sqrt[2]*a^2*Sqrt[c]*f) - (3*((5*I)*A + 3*B))/(32*a^2*f*Sqrt[c - I*c*Tan[e + f*x]]) + (I*A - B)/(4*a^2*f*(1 + I*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]) + ((5*I)*A + 3*B)/(16*a^2*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)), x, 7, (5*((7*I)*A + B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(64*Sqrt[2]*a^2*c^(3/2)*f) - (5*((7*I)*A + B))/(96*a^2*f*(c - I*c*Tan[e + f*x])^(3/2)) + (I*A - B)/(4*a^2*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)) + ((7*I)*A + B)/(16*a^2*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - (5*((7*I)*A + B))/(64*a^2*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)), x, 8, (7*((9*I)*A - B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(128*Sqrt[2]*a^2*c^(5/2)*f) - (7*((9*I)*A - B))/(160*a^2*f*(c - I*c*Tan[e + f*x])^(5/2)) + (I*A - B)/(4*a^2*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)) + ((9*I)*A - B)/(16*a^2*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - (7*((9*I)*A - B))/(192*a^2*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (7*((9*I)*A - B))/(128*a^2*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^3, x, 8, -((35*(I*A - 5*B)*c^(9/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(4*Sqrt[2]*a^3*f)) + (35*(I*A - 5*B)*c^4*Sqrt[c - I*c*Tan[e + f*x]])/(8*a^3*f) + (35*(I*A - 5*B)*c^3*(c - I*c*Tan[e + f*x])^(3/2))/(48*a^3*f) + (7*(I*A - 5*B)*c^2*(c - I*c*Tan[e + f*x])^(5/2))/(16*a^3*f*(1 + I*Tan[e + f*x])) - ((I*A - 5*B)*c*(c - I*c*Tan[e + f*x])^(7/2))/(8*a^3*f*(1 + I*Tan[e + f*x])^2) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(9/2))/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^3, x, 7, (-5*(I*A - 13*B)*c^(7/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[2]*a^3*f) + (5*(I*A - 13*B)*c^3*Sqrt[c - I*c*Tan[e + f*x]])/(16*a^3*f) + (5*(I*A - 13*B)*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(48*a^3*f*(1 + I*Tan[e + f*x])) - ((I*A - 13*B)*c*(c - I*c*Tan[e + f*x])^(5/2))/(24*a^3*f*(1 + I*Tan[e + f*x])^2) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^3, x, 6, ((I*A + 11*B)*c^(5/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(16*Sqrt[2]*a^3*f) - ((I*A + 11*B)*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(16*a^3*f*(1 + I*Tan[e + f*x])) + ((I*A + 11*B)*c*(c - I*c*Tan[e + f*x])^(3/2))/(24*a^3*f*(1 + I*Tan[e + f*x])^2) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^3, x, 6, -((I*A + 3*B)*c^(3/2)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(32*Sqrt[2]*a^3*f) + ((I*A + 3*B)*c*Sqrt[c - I*c*Tan[e + f*x]])/(8*a^3*f*(1 + I*Tan[e + f*x])^2) - ((I*A + 3*B)*c*Sqrt[c - I*c*Tan[e + f*x]])/(32*a^3*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(6*a^3*f*(1 + I*Tan[e + f*x])^3)} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^3, x, 6, (((5*I)*A + 7*B)*Sqrt[c]*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(64*Sqrt[2]*a^3*f) + ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(6*a^3*f*(1 + I*Tan[e + f*x])^3) + (((5*I)*A + 7*B)*Sqrt[c - I*c*Tan[e + f*x]])/(48*a^3*f*(1 + I*Tan[e + f*x])^2) + (((5*I)*A + 7*B)*Sqrt[c - I*c*Tan[e + f*x]])/(64*a^3*f*(1 + I*Tan[e + f*x]))} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]), x, 7, (5*((7*I)*A + 5*B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(128*Sqrt[2]*a^3*Sqrt[c]*f) - (5*((7*I)*A + 5*B))/(128*a^3*f*Sqrt[c - I*c*Tan[e + f*x]]) + (I*A - B)/(6*a^3*f*(1 + I*Tan[e + f*x])^3*Sqrt[c - I*c*Tan[e + f*x]]) + ((7*I)*A + 5*B)/(48*a^3*f*(1 + I*Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]) + (5*((7*I)*A + 5*B))/(192*a^3*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)), x, 8, (35*((3*I)*A + B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(256*Sqrt[2]*a^3*c^(3/2)*f) - (35*((3*I)*A + B))/(384*a^3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (I*A - B)/(6*a^3*f*(1 + I*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(3/2)) + ((3*I)*A + B)/(16*a^3*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(3/2)) + (7*((3*I)*A + B))/(64*a^3*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2)) - (35*((3*I)*A + B))/(256*a^3*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)), x, 9, (21*((11*I)*A + B)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(512*Sqrt[2]*a^3*c^(5/2)*f) - (21*((11*I)*A + B))/(640*a^3*f*(c - I*c*Tan[e + f*x])^(5/2)) + (I*A - B)/(6*a^3*f*(1 + I*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^(5/2)) + ((11*I)*A + B)/(48*a^3*f*(1 + I*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^(5/2)) + (3*((11*I)*A + B))/(64*a^3*f*(1 + I*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2)) - (7*((11*I)*A + B))/(256*a^3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (21*((11*I)*A + B))/(512*a^3*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a I Tan[e+f x])^(m/2) (c-c I Tan[e+f x])^(n/2) (A+B Tan[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 8, (-5*Sqrt[a]*((4*I)*A - 3*B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) - (5*((4*I)*A - 3*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) - (5*((4*I)*A - 3*B)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) - (((4*I)*A - 3*B)*c*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2))/(12*f) + (B*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(7/2))/(4*f)} -{Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 7, -((Sqrt[a]*((3*I)*A - 2*B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) - (((3*I)*A - 2*B)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) - (((3*I)*A - 2*B)*c*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(6*f) + (B*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2))/(3*f)} -{Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 6, -((Sqrt[a]*((2*I)*A - B)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) - (((2*I)*A - B)*c*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) + (B*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(2*f)} -{Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 5, ((-2*I)*Sqrt[a]*A*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (B*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/f} -{(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 5, (2*Sqrt[a]*B*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - ((I*A + B)*Sqrt[a + I*a*Tan[e + f*x]])/(f*Sqrt[c - I*c*Tan[e + f*x]])} -{(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 3, -((I*A + B)*Sqrt[a + I*a*Tan[e + f*x]])/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) - ((I*A - 2*B)*Sqrt[a + I*a*Tan[e + f*x]])/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 4, -((I*A + B)*Sqrt[a + I*a*Tan[e + f*x]])/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) - (((2*I)*A - 3*B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (((2*I)*A - 3*B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(Sqrt[a + I*a*Tan[e + f*x]]*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 5, -((I*A + B)*Sqrt[a + I*a*Tan[e + f*x]])/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) - (((3*I)*A - 4*B)*Sqrt[a + I*a*Tan[e + f*x]])/(35*c*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*((3*I)*A - 4*B)*Sqrt[a + I*a*Tan[e + f*x]])/(105*c^2*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*((3*I)*A - 4*B)*Sqrt[a + I*a*Tan[e + f*x]])/(105*c^3*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 8, -(a^(3/2)*((5*I)*A - 2*B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) + (a*(5*A + (2*I)*B)*c^3*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) - (((5*I)*A - 2*B)*c^2*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(12*f) - (((5*I)*A - 2*B)*c*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2))/(20*f) + (B*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(7/2))/(5*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 7, -(a^(3/2)*((4*I)*A - B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) + (a*(4*A + I*B)*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) - (((4*I)*A - B)*c*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(12*f) + (B*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2))/(4*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 6, ((-I)*a^(3/2)*A*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (a*A*c*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) + (B*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f)} -{(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 6, -((a^(3/2)*((2*I)*A + B)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) + (a*((2*I)*A + B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) + (B*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(2*f)} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 6, (2*a^(3/2)*(I*A + 2*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (a*(I*A + 2*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c*f)} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 6, (-2*a^(3/2)*B*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a*B*Sqrt[a + I*a*Tan[e + f*x]])/(c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 3, -((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) - ((I*A - 4*B)*(a + I*a*Tan[e + f*x])^(3/2))/(15*c*f*(c - I*c*Tan[e + f*x])^(3/2))} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 4, -((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) - (((2*I)*A - 5*B)*(a + I*a*Tan[e + f*x])^(3/2))/(35*c*f*(c - I*c*Tan[e + f*x])^(5/2)) - (((2*I)*A - 5*B)*(a + I*a*Tan[e + f*x])^(3/2))/(105*c^2*f*(c - I*c*Tan[e + f*x])^(3/2))} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2), x, 5, -((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(9*f*(c - I*c*Tan[e + f*x])^(9/2)) - ((I*A - 2*B)*(a + I*a*Tan[e + f*x])^(3/2))/(21*c*f*(c - I*c*Tan[e + f*x])^(7/2)) - (2*(I*A - 2*B)*(a + I*a*Tan[e + f*x])^(3/2))/(105*c^2*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*(I*A - 2*B)*(a + I*a*Tan[e + f*x])^(3/2))/(315*c^3*f*(c - I*c*Tan[e + f*x])^(3/2))} -{((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2), x, 6, -((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(11*f*(c - I*c*Tan[e + f*x])^(11/2)) - (((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(99*c*f*(c - I*c*Tan[e + f*x])^(9/2)) - (((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(231*c^2*f*(c - I*c*Tan[e + f*x])^(7/2)) - (2*((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(1155*c^3*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(3465*c^4*f*(c - I*c*Tan[e + f*x])^(3/2))} - - -{(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 8, -(a^(5/2)*((6*I)*A - B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(8*f) + (a^2*(6*A + I*B)*c^3*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) + (a*(6*A + I*B)*c^2*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) - (((6*I)*A - B)*c*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(30*f) + (B*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(7/2))/(6*f)} -{(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 7, (((-3*I)/4)*a^(5/2)*A*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (3*a^2*A*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (a*A*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(4*f) + (B*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f)} -{(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 7, -(a^(5/2)*((4*I)*A + B)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) + (a^2*(4*A - I*B)*c*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (a*((4*I)*A + B)*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(12*f) + (B*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2))/(4*f)} -{(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 7, -((a^(5/2)*((3*I)*A + 2*B)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f) + (a^2*((3*I)*A + 2*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*f) + (a*((3*I)*A + 2*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(6*f) + (B*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])/(3*f)} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 7, (3*a^(5/2)*((2*I)*A + 3*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (3*a^2*((2*I)*A + 3*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c*f) - (a*((2*I)*A + 3*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(2*c*f)} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 7, (-2*a^(5/2)*(I*A + 4*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a*(I*A + 4*B)*(a + I*a*Tan[e + f*x])^(3/2))/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (a^2*(I*A + 4*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c^2*f)} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 7, (2*a^(5/2)*B*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(5/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (2*a*B*(a + I*a*Tan[e + f*x])^(3/2))/(3*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*a^2*B*Sqrt[a + I*a*Tan[e + f*x]])/(c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 3, -((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) - ((I*A - 6*B)*(a + I*a*Tan[e + f*x])^(5/2))/(35*c*f*(c - I*c*Tan[e + f*x])^(5/2))} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2), x, 4, -((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(9*f*(c - I*c*Tan[e + f*x])^(9/2)) - (((2*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(5/2))/(63*c*f*(c - I*c*Tan[e + f*x])^(7/2)) - (((2*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(5/2))/(315*c^2*f*(c - I*c*Tan[e + f*x])^(5/2))} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2), x, 5, -((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(11*f*(c - I*c*Tan[e + f*x])^(11/2)) - (((3*I)*A - 8*B)*(a + I*a*Tan[e + f*x])^(5/2))/(99*c*f*(c - I*c*Tan[e + f*x])^(9/2)) - (2*((3*I)*A - 8*B)*(a + I*a*Tan[e + f*x])^(5/2))/(693*c^2*f*(c - I*c*Tan[e + f*x])^(7/2)) - (2*((3*I)*A - 8*B)*(a + I*a*Tan[e + f*x])^(5/2))/(3465*c^3*f*(c - I*c*Tan[e + f*x])^(5/2))} -{((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(13/2), x, 6, -((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(13*f*(c - I*c*Tan[e + f*x])^(13/2)) - (((4*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(5/2))/(143*c*f*(c - I*c*Tan[e + f*x])^(11/2)) - (((4*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(5/2))/(429*c^2*f*(c - I*c*Tan[e + f*x])^(9/2)) - (2*((4*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(5/2))/(3003*c^3*f*(c - I*c*Tan[e + f*x])^(7/2)) - (2*((4*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(5/2))/(15015*c^4*f*(c - I*c*Tan[e + f*x])^(5/2))} - - -{(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2), x, 9, (-5*a^(7/2)*((8*I)*A - B)*c^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(64*f) + (5*a^3*(8*A + I*B)*c^4*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(128*f) + (5*a^2*(8*A + I*B)*c^3*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(192*f) + (a*(8*A + I*B)*c^2*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(48*f) - (((8*I)*A - B)*c*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(7/2))/(56*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(9/2))/(8*f)} -{(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2), x, 8, (((-5*I)/8)*a^(7/2)*A*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/f + (5*a^3*A*c^3*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) + (5*a^2*A*c^2*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) + (a*A*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(6*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(7/2))/(7*f)} -{(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2), x, 8, -(a^(7/2)*((6*I)*A + B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(8*f) + (a^3*(6*A - I*B)*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) + (a^2*(6*A - I*B)*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) + (a*((6*I)*A + B)*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(30*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2))/(6*f)} -{(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2), x, 8, -(a^(7/2)*((5*I)*A + 2*B)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) + (a^3*(5*A - (2*I)*B)*c*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (a^2*((5*I)*A + 2*B)*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(12*f) + (a*((5*I)*A + 2*B)*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2))/(20*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(3/2))/(5*f)} -{(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]], x, 8, (-5*a^(7/2)*((4*I)*A + 3*B)*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(4*f) + (5*a^3*((4*I)*A + 3*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f) + (5*a^2*((4*I)*A + 3*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(24*f) + (a*((4*I)*A + 3*B)*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])/(12*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c - I*c*Tan[e + f*x]])/(4*f)} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/Sqrt[c - I*c*Tan[e + f*x]], x, 8, (5*a^(7/2)*((3*I)*A + 4*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[c]*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) - (5*a^3*((3*I)*A + 4*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c*f) - (5*a^2*((3*I)*A + 4*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(6*c*f) - (a*((3*I)*A + 4*B)*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])/(3*c*f)} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(3/2), x, 8, (-5*a^(7/2)*((2*I)*A + 5*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a*((2*I)*A + 5*B)*(a + I*a*Tan[e + f*x])^(5/2))/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (5*a^3*((2*I)*A + 5*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*c^2*f) + (5*a^2*((2*I)*A + 5*B)*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])/(6*c^2*f)} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(5/2), x, 8, (2*a^(7/2)*(I*A + 6*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(5/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(5*f*(c - I*c*Tan[e + f*x])^(5/2)) + (2*a*(I*A + 6*B)*(a + I*a*Tan[e + f*x])^(5/2))/(15*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*a^2*(I*A + 6*B)*(a + I*a*Tan[e + f*x])^(3/2))/(3*c^2*f*Sqrt[c - I*c*Tan[e + f*x]]) - (a^3*(I*A + 6*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(c^3*f)} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(7/2), x, 8, (-2*a^(7/2)*B*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(7/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(7*f*(c - I*c*Tan[e + f*x])^(7/2)) + (2*a*B*(a + I*a*Tan[e + f*x])^(5/2))/(5*c*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*a^2*B*(a + I*a*Tan[e + f*x])^(3/2))/(3*c^2*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a^3*B*Sqrt[a + I*a*Tan[e + f*x]])/(c^3*f*Sqrt[c - I*c*Tan[e + f*x]])} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2), x, 3, -((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(9*f*(c - I*c*Tan[e + f*x])^(9/2)) - ((I*A - 8*B)*(a + I*a*Tan[e + f*x])^(7/2))/(63*c*f*(c - I*c*Tan[e + f*x])^(7/2))} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2), x, 4, -((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(11*f*(c - I*c*Tan[e + f*x])^(11/2)) - (((2*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(7/2))/(99*c*f*(c - I*c*Tan[e + f*x])^(9/2)) - (((2*I)*A - 9*B)*(a + I*a*Tan[e + f*x])^(7/2))/(693*c^2*f*(c - I*c*Tan[e + f*x])^(7/2))} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(13/2), x, 5, -((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(13*f*(c - I*c*Tan[e + f*x])^(13/2)) - (((3*I)*A - 10*B)*(a + I*a*Tan[e + f*x])^(7/2))/(143*c*f*(c - I*c*Tan[e + f*x])^(11/2)) - (2*((3*I)*A - 10*B)*(a + I*a*Tan[e + f*x])^(7/2))/(1287*c^2*f*(c - I*c*Tan[e + f*x])^(9/2)) - (2*((3*I)*A - 10*B)*(a + I*a*Tan[e + f*x])^(7/2))/(9009*c^3*f*(c - I*c*Tan[e + f*x])^(7/2))} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(15/2), x, 6, -((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(15*f*(c - I*c*Tan[e + f*x])^(15/2)) - (((4*I)*A - 11*B)*(a + I*a*Tan[e + f*x])^(7/2))/(195*c*f*(c - I*c*Tan[e + f*x])^(13/2)) - (((4*I)*A - 11*B)*(a + I*a*Tan[e + f*x])^(7/2))/(715*c^2*f*(c - I*c*Tan[e + f*x])^(11/2)) - (2*((4*I)*A - 11*B)*(a + I*a*Tan[e + f*x])^(7/2))/(6435*c^3*f*(c - I*c*Tan[e + f*x])^(9/2)) - (2*((4*I)*A - 11*B)*(a + I*a*Tan[e + f*x])^(7/2))/(45045*c^4*f*(c - I*c*Tan[e + f*x])^(7/2))} -{((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(17/2), x, 7, -((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(17*f*(c - I*c*Tan[e + f*x])^(17/2)) - (((5*I)*A - 12*B)*(a + I*a*Tan[e + f*x])^(7/2))/(255*c*f*(c - I*c*Tan[e + f*x])^(15/2)) - (4*((5*I)*A - 12*B)*(a + I*a*Tan[e + f*x])^(7/2))/(3315*c^2*f*(c - I*c*Tan[e + f*x])^(13/2)) - (4*((5*I)*A - 12*B)*(a + I*a*Tan[e + f*x])^(7/2))/(12155*c^3*f*(c - I*c*Tan[e + f*x])^(11/2)) - (8*((5*I)*A - 12*B)*(a + I*a*Tan[e + f*x])^(7/2))/(109395*c^4*f*(c - I*c*Tan[e + f*x])^(9/2)) - (8*((5*I)*A - 12*B)*(a + I*a*Tan[e + f*x])^(7/2))/(765765*c^5*f*(c - I*c*Tan[e + f*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/Sqrt[a + I*a*Tan[e + f*x]], x, 7, (3*((2*I)*A - 3*B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[a]*f) + (3*((2*I)*A - 3*B)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*a*f) + (((2*I)*A - 3*B)*c*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(2*a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/Sqrt[a + I*a*Tan[e + f*x]], x, 6, (2*(I*A - 2*B)*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[a]*f) + ((I*A - 2*B)*c*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(a*f) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/Sqrt[a + I*a*Tan[e + f*x]], x, 5, (-2*B*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(Sqrt[a]*f) + ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]]), x, 3, -((I*A + B)/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])) + (I*A*Sqrt[c - I*c*Tan[e + f*x]])/(c*f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)), x, 4, (I*A - B)/(f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2)) - (((2*I)*A - B)*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*f*(c - I*c*Tan[e + f*x])^(3/2)) - (((2*I)*A - B)*Sqrt[a + I*a*Tan[e + f*x]])/(3*a*c*f*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/(Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)), x, 5, (I*A - B)/(f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)) - (((3*I)*A - 2*B)*Sqrt[a + I*a*Tan[e + f*x]])/(5*a*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*((3*I)*A - 2*B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*a*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*((3*I)*A - 2*B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*a*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^(3/2), x, 8, (-5*((2*I)*A - 5*B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(3/2)*f) - (5*((2*I)*A - 5*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^2*f) - (5*((2*I)*A - 5*B)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(6*a^2*f) - (2*((2*I)*A - 5*B)*c*(c - I*c*Tan[e + f*x])^(5/2))/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^(3/2), x, 7, (-2*(I*A - 4*B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(3/2)*f) - ((I*A - 4*B)*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*f) - (2*(I*A - 4*B)*c*(c - I*c*Tan[e + f*x])^(3/2))/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^(3/2), x, 6, (2*B*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(3/2)*f) + (2*B*c*Sqrt[c - I*c*Tan[e + f*x]])/(a*f*Sqrt[a + I*a*Tan[e + f*x]]) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^(3/2), x, 3, ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(3*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A + 2*B)*Sqrt[c - I*c*Tan[e + f*x]])/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 4, -((I*A + B)/(f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]])) + (((2*I)*A + B)*Sqrt[c - I*c*Tan[e + f*x]])/(3*c*f*(a + I*a*Tan[e + f*x])^(3/2)) + (((2*I)*A + B)*Sqrt[c - I*c*Tan[e + f*x]])/(3*a*c*f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 4, -(I*A + B)/(3*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + ((I/3)*A)/(c*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (2*A*Tan[e + f*x])/(3*a*c*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 6, (I*A - B)/(3*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (4*I*A - B)/(3*a*f*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(5/2)) - ((4*I*A - B)*Sqrt[a + I*a*Tan[e + f*x]])/(5*a^2*f*(c - I*c*Tan[e + f*x])^(5/2)) - (2*(4*I*A - B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*a^2*c*f*(c - I*c*Tan[e + f*x])^(3/2)) - (2*(4*I*A - B)*Sqrt[a + I*a*Tan[e + f*x]])/(15*a^2*c^2*f*Sqrt[c - I*c*Tan[e + f*x]])} - - -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(9/2))/(a + I*a*Tan[e + f*x])^(5/2), x, 9, (7*((2*I)*A - 7*B)*c^(9/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(5/2)*f) + (7*((2*I)*A - 7*B)*c^4*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(2*a^3*f) + (7*((2*I)*A - 7*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*(c - I*c*Tan[e + f*x])^(3/2))/(6*a^3*f) + (14*((2*I)*A - 7*B)*c^2*(c - I*c*Tan[e + f*x])^(5/2))/(15*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) - (2*((2*I)*A - 7*B)*c*(c - I*c*Tan[e + f*x])^(7/2))/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(9/2))/(5*f*(a + I*a*Tan[e + f*x])^(5/2))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2))/(a + I*a*Tan[e + f*x])^(5/2), x, 8, (2*(I*A - 6*B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(5/2)*f) + ((I*A - 6*B)*c^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(a^3*f) + (2*(I*A - 6*B)*c^2*(c - I*c*Tan[e + f*x])^(3/2))/(3*a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) - (2*(I*A - 6*B)*c*(c - I*c*Tan[e + f*x])^(5/2))/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(7/2))/(5*f*(a + I*a*Tan[e + f*x])^(5/2))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2))/(a + I*a*Tan[e + f*x])^(5/2), x, 7, (-2*B*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(a^(5/2)*f) - (2*B*c^2*Sqrt[c - I*c*Tan[e + f*x]])/(a^2*f*Sqrt[a + I*a*Tan[e + f*x]]) + (2*B*c*(c - I*c*Tan[e + f*x])^(3/2))/(3*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + ((I*A - B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*f*(a + I*a*Tan[e + f*x])^(5/2))} -{((A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2))/(a + I*a*Tan[e + f*x])^(5/2), x, 3, ((I*A - B)*(c - I*c*Tan[e + f*x])^(3/2))/(5*f*(a + I*a*Tan[e + f*x])^(5/2)) + ((I*A + 4*B)*(c - I*c*Tan[e + f*x])^(3/2))/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2))} -{((A + B*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])/(a + I*a*Tan[e + f*x])^(5/2), x, 4, ((I*A - B)*Sqrt[c - I*c*Tan[e + f*x]])/(5*f*(a + I*a*Tan[e + f*x])^(5/2)) + (((2*I)*A + 3*B)*Sqrt[c - I*c*Tan[e + f*x]])/(15*a*f*(a + I*a*Tan[e + f*x])^(3/2)) + (((2*I)*A + 3*B)*Sqrt[c - I*c*Tan[e + f*x]])/(15*a^2*f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]), x, 5, -((I*A + B)/(f*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]])) + (((3*I)*A + 2*B)*Sqrt[c - I*c*Tan[e + f*x]])/(5*c*f*(a + I*a*Tan[e + f*x])^(5/2)) + (2*((3*I)*A + 2*B)*Sqrt[c - I*c*Tan[e + f*x]])/(15*a*c*f*(a + I*a*Tan[e + f*x])^(3/2)) + (2*((3*I)*A + 2*B)*Sqrt[c - I*c*Tan[e + f*x]])/(15*a^2*c*f*Sqrt[a + I*a*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)), x, 5, -(I*A + B)/(3*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)) + ((4*I)*A + B)/(15*c*f*(a + I*a*Tan[e + f*x])^(5/2)*Sqrt[c - I*c*Tan[e + f*x]]) + ((4*I)*A + B)/(15*a*c*f*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Tan[e + f*x]]) + (2*(4*A - I*B)*Tan[e + f*x])/(15*a^2*c*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} -{(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)), x, 5, -(I*A + B)/(5*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)) + ((I/5)*A)/(c*f*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (4*A*Tan[e + f*x])/(15*a*c*f*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2)) + (8*A*Tan[e + f*x])/(15*a^2*c^2*f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a I Tan[e+f x])^m (c-c I Tan[e+f x])^n (A+B Tan[e+f x]) when m and n symbolic*) - - -{(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n*(A + B*Tan[e + f*x]), x, 4, ((I*A + B)*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(2*f*n) - (1/(f*m*n))*((2^(-1 + n)*(B*(m - n) + I*A*(m + n))*Hypergeometric2F1[m, -n, 1 + m, (1/2)*(1 + I*Tan[e + f*x])]*(a + I*a*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^n)/(1 - I*Tan[e + f*x])^n)} - - -{(a + I*a*Tan[e + f*x])^(1 + m)*(c - I*c*Tan[e + f*x])^(-1 - m)*(A + B*Tan[e + f*x]), x, 4, -((I*A + B)*(a + I*a*Tan[e + f*x])^(1 + m)*(c - I*c*Tan[e + f*x])^(-1 - m))/(2*f*(1 + m)) + (2^m*a*B*Hypergeometric2F1[-m, -m, 1 - m, (1 - I*Tan[e + f*x])/2]*(a + I*a*Tan[e + f*x])^m)/(c*f*m*(1 + I*Tan[e + f*x])^m*(c - I*c*Tan[e + f*x])^m)} - - -{(c - I*c*Tan[e + f*x])^n*((-I)*(2 + n) + (-2 + n)*Tan[e + f*x])/(-I + Tan[e + f*x])^2, x, 2, (c - I*c*Tan[e + f*x])^n/(f*(I - Tan[e + f*x])^2)} - - -(* ::Section:: *) -(*Integrands of the form (a+a I Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x])*) - - -{((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^2, x, 3, ((A - I*B)*(c - I*d)*x)/(4*a^2) + (B*(c + 3*I*d) + A*(I*c + d))/(4*a^2*f*(1 + I*Tan[e + f*x])) + ((I*A - B)*(c + I*d))/(4*f*(a + I*a*Tan[e + f*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^(n/2) (A+B Tan[e+f x])*) - - -{((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x]))/(a + I*a*Tan[e + f*x])^(3/2), x, 4, -(((I*A + B)*(c - I*d)*ArcTanh[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*a^(3/2)*f)) + ((I*A - B)*(c + I*d))/(3*f*(a + I*a*Tan[e + f*x])^(3/2)) + (B*(c + 3*I*d) + A*(I*c + d))/(2*a*f*Sqrt[a + I*a*Tan[e + f*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (c+d Tan[e+f x])^(n/2) (A+B Tan[e+f x])*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m deleted file mode 100644 index 6b0f918..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.4.2 (a+b tan)^m (c+d tan)^n (A+B tan+C tan^2).m +++ /dev/null @@ -1,313 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m Tan[e+f x]^n (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (B Tan[e+f x]+C Tan[e+f x]^2) (a+b Tan[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, -((a*B - b*C)*x) + ((b*B + a*C)*Log[Cos[c + d*x]])/d + ((a*B - b*C)*Tan[c + d*x])/d + ((b*B + a*C)*Tan[c + d*x]^2)/(2*d) + (b*C*Tan[c + d*x]^3)/(3*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 3, -((b*B + a*C)*x) - ((a*B - b*C)*Log[Cos[c + d*x]])/d + (b*B*Tan[c + d*x])/d + (C*(a + b*Tan[c + d*x])^2)/(2*b*d)} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 3, (a*B - b*C)*x - ((b*B + a*C)*Log[Cos[c + d*x]])/d + (b*C*Tan[c + d*x])/d} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, (b*B + a*C)*x - (b*C*Log[Cos[c + d*x]])/d + (a*B*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 4, -((a*B - b*C)*x) - (a*B*Cot[c + d*x])/d + ((b*B + a*C)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, -((b*B + a*C)*x) - ((b*B + a*C)*Cot[c + d*x])/d - (a*B*Cot[c + d*x]^2)/(2*d) - ((a*B - b*C)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, (a*B - b*C)*x + ((a*B - b*C)*Cot[c + d*x])/d - ((b*B + a*C)*Cot[c + d*x]^2)/(2*d) - (a*B*Cot[c + d*x]^3)/(3*d) - ((b*B + a*C)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 7, (b*B + a*C)*x + ((b*B + a*C)*Cot[c + d*x])/d + ((a*B - b*C)*Cot[c + d*x]^2)/(2*d) - ((b*B + a*C)*Cot[c + d*x]^3)/(3*d) - (a*B*Cot[c + d*x]^4)/(4*d) + ((a*B - b*C)*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^1*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, -((a^2*B - b^2*B - 2*a*b*C)*x) + ((2*a*b*B + a^2*C - b^2*C)*Log[Cos[c + d*x]])/d - (b*(b*B + a*C)*Tan[c + d*x])/d - (C*(a + b*Tan[c + d*x])^2)/(2*d) + ((4*b*B - a*C)*(a + b*Tan[c + d*x])^3)/(12*b^2*d) + (C*Tan[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*b*d)} -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 4, -((2*a*b*B + a^2*C - b^2*C)*x) - ((a^2*B - b^2*B - 2*a*b*C)*Log[Cos[c + d*x]])/d + (b*(a*B - b*C)*Tan[c + d*x])/d + (B*(a + b*Tan[c + d*x])^2)/(2*d) + (C*(a + b*Tan[c + d*x])^3)/(3*b*d)} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 4, (a^2*B - b^2*B - 2*a*b*C)*x - ((2*a*b*B + a^2*C - b^2*C)*Log[Cos[c + d*x]])/d + (b*(b*B + a*C)*Tan[c + d*x])/d + (C*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, (2*a*b*B + a^2*C - b^2*C)*x - (b*(b*B + 2*a*C)*Log[Cos[c + d*x]])/d + (a^2*B*Log[Sin[c + d*x]])/d + (b^2*C*Tan[c + d*x])/d} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, -((a^2*B - b^2*B - 2*a*b*C)*x) - (a^2*B*Cot[c + d*x])/d - (b^2*C*Log[Cos[c + d*x]])/d + (a*(2*b*B + a*C)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, (b^2*C - a*(2*b*B + a*C))*x - (a*(2*b*B + a*C)*Cot[c + d*x])/d - (a^2*B*Cot[c + d*x]^2)/(2*d) - ((a^2*B - b^2*B - 2*a*b*C)*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, (a^2*B - b^2*B - 2*a*b*C)*x + ((a^2*B - b^2*B - 2*a*b*C)*Cot[c + d*x])/d - (a*(2*b*B + a*C)*Cot[c + d*x]^2)/(2*d) - (a^2*B*Cot[c + d*x]^3)/(3*d) + ((b^2*C - a*(2*b*B + a*C))*Log[Sin[c + d*x]])/d} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 7, (2*a*b*B + a^2*C - b^2*C)*x - ((b^2*C - a*(2*b*B + a*C))*Cot[c + d*x])/d + ((a^2*B - b^2*B - 2*a*b*C)*Cot[c + d*x]^2)/(2*d) - (a*(2*b*B + a*C)*Cot[c + d*x]^3)/(3*d) - (a^2*B*Cot[c + d*x]^4)/(4*d) + ((a^2*B - b^2*B - 2*a*b*C)*Log[Sin[c + d*x]])/d} - - -{Tan[c + d*x]^0*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, -((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x) - ((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Log[Cos[c + d*x]])/d + (b*(a^2*B - b^2*B - 2*a*b*C)*Tan[c + d*x])/d + ((a*B - b*C)*(a + b*Tan[c + d*x])^2)/(2*d) + (B*(a + b*Tan[c + d*x])^3)/(3*d) + (C*(a + b*Tan[c + d*x])^4)/(4*b*d)} -{Cot[c + d*x]^1*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 5, (a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x - ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Cos[c + d*x]])/d + (b*(2*a*b*B + a^2*C - b^2*C)*Tan[c + d*x])/d + ((b*B + a*C)*(a + b*Tan[c + d*x])^2)/(2*d) + (C*(a + b*Tan[c + d*x])^3)/(3*d)} -{Cot[c + d*x]^2*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, (3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x - (b*(3*a*b*B + 3*a^2*C - b^2*C)*Log[Cos[c + d*x]])/d + (a^3*B*Log[Sin[c + d*x]])/d + (b^2*(b*B + 2*a*C)*Tan[c + d*x])/d + (b*C*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, -((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x) - (b^2*(b*B + 3*a*C)*Log[Cos[c + d*x]])/d + (a^2*(3*b*B + a*C)*Log[Sin[c + d*x]])/d + (b^2*(a*B + b*C)*Tan[c + d*x])/d - (a*B*Cot[c + d*x]*(a + b*Tan[c + d*x])^2)/d} -{Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, -((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x) - (a^2*(2*b*B + a*C)*Cot[c + d*x])/d - (b^3*C*Log[Cos[c + d*x]])/d - (a*(a^2*B - 3*b^2*B - 3*a*b*C)*Log[Sin[c + d*x]])/d - (a*B*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d)} -{Cot[c + d*x]^5*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 6, (a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x + (a*(3*a^2*B - 8*b^2*B - 9*a*b*C)*Cot[c + d*x])/(3*d) - (a^2*(5*b*B + 3*a*C)*Cot[c + d*x]^2)/(6*d) - ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Sin[c + d*x]])/d - (a*B*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(3*d)} -{Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 7, (3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*x + ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x])/d + (a*(2*a^2*B - 5*b^2*B - 6*a*b*C)*Cot[c + d*x]^2)/(4*d) - (a^2*(3*b*B + 2*a*C)*Cot[c + d*x]^3)/(6*d) + ((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Log[Sin[c + d*x]])/d - (a*B*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^2)/(4*d)} -{Cot[c + d*x]^7*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2), x, 8, -((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x) - ((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Cot[c + d*x])/d + ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x]^2)/(2*d) + (a*(5*a^2*B - 12*b^2*B - 15*a*b*C)*Cot[c + d*x]^3)/(15*d) - (a^2*(7*b*B + 5*a*C)*Cot[c + d*x]^4)/(20*d) + ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Sin[c + d*x]])/d - (a*B*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 7, -(((b*B - a*C)*x)/(a^2 + b^2)) + ((a*B + b*C)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a^3*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Tan[c + d*x]^2)/(2*b*d)} -{Tan[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 6, -(((a*B + b*C)*x)/(a^2 + b^2)) - ((b*B - a*C)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) + (a^2*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)*d) + (C*Tan[c + d*x])/(b*d)} -{Tan[c + d*x]^0*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 6, ((b*B - a*C)*x)/(a^2 + b^2) - ((a*B + b*C)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a*(b*B - a*C)*Log[a + b*Tan[c + d*x]])/(b*(a^2 + b^2)*d)} -{Cot[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 3, ((a*B + b*C)*x)/(a^2 + b^2) + ((b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 4, -(((b*B - a*C)*x)/(a^2 + b^2)) + (B*Log[Sin[c + d*x]])/(a*d) - (b*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d)} -{Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 5, -(((a*B + b*C)*x)/(a^2 + b^2)) - (B*Cot[c + d*x])/(a*d) - ((b*B - a*C)*Log[Sin[c + d*x]])/(a^2*d) + (b^2*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)*d)} -{Cot[c + d*x]^4*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x]), x, 6, ((b*B - a*C)*x)/(a^2 + b^2) + ((b*B - a*C)*Cot[c + d*x])/(a^2*d) - (B*Cot[c + d*x]^2)/(2*a*d) - ((a^2*B - b^2*B + a*b*C)*Log[Sin[c + d*x]])/(a^3*d) - (b^3*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)*d)} - - -{Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 7, -(((2*a*b*B - a^2*C + b^2*C)*x)/(a^2 + b^2)^2) + ((a^2*B - b^2*B + 2*a*b*C)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) - ((a*b*B - 2*a^2*C - b^2*C)*Tan[c + d*x])/(b^2*(a^2 + b^2)*d) + (a*(b*B - a*C)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 6, -(((a^2*B - b^2*B + 2*a*b*C)*x)/(a^2 + b^2)^2) - ((2*a*b*B - a^2*C + b^2*C)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (a*(2*b^3*B - a^3*C - 3*a*b^2*C)*Log[a + b*Tan[c + d*x]])/(b^2*(a^2 + b^2)^2*d) - (a^2*(b*B - a*C))/(b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^0*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 3, ((2*a*b*B - a^2*C + b^2*C)*x)/(a^2 + b^2)^2 - ((a^2*B - b^2*B + 2*a*b*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + (a*(b*B - a*C))/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 4, ((a^2*B - b^2*B + 2*a*b*C)*x)/(a^2 + b^2)^2 + ((2*a*b*B - a^2*C + b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - (b*B - a*C)/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 5, -(((2*a*b*B - a^2*C + b^2*C)*x)/(a^2 + b^2)^2) + (B*Log[Sin[c + d*x]])/(a^2*d) - (b*(3*a^2*b*B + b^3*B - 2*a^3*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)^2*d) + (b*(b*B - a*C))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^2, x, 6, -(((a^2*B - b^2*B + 2*a*b*C)*x)/(a^2 + b^2)^2) - ((2*b*B - a*C)*Log[Sin[c + d*x]])/(a^3*d) + (b^2*(4*a^2*b*B + 2*b^3*B - 3*a^3*C - a*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (b*(a^2*B + 2*b^2*B - a*b*C))/(a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (B*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x]))} - - -{Tan[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 8, ((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*x)/(a^2 + b^2)^3 + ((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a^2*(a^4*b*B + 3*a^2*b^3*B + 6*b^5*B - 3*a^5*C - 9*a^3*b^2*C - 10*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b^4*(a^2 + b^2)^3*d) - ((a^3*b*B + 3*a*b^3*B - 3*a^4*C - 6*a^2*b^2*C - b^4*C)*Tan[c + d*x])/(b^3*(a^2 + b^2)^2*d) + (a*(b*B - a*C)*Tan[c + d*x]^3)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(a^2*b*B + 5*b^3*B - 3*a^3*C - 7*a*b^2*C)*Tan[c + d*x]^2)/(2*b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 7, -(((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*x)/(a^2 + b^2)^3) + ((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*Log[Cos[c + d*x]])/((a^2 + b^2)^3*d) + (a*(a^2*b^3*B - 3*b^5*B + a^5*C + 3*a^3*b^2*C + 6*a*b^4*C)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^3*d) + (a*(b*B - a*C)*Tan[c + d*x]^2)/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (a^2*(2*b^3*B - a^3*C - 3*a*b^2*C))/(b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 5, -(((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*x)/(a^2 + b^2)^3) - ((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (a^2*(b*B - a*C))/(2*b^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a*(2*b^3*B - a^3*C - 3*a*b^2*C))/(b^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Tan[c + d*x]^0*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 4, ((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*x)/(a^2 + b^2)^3 - ((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (a*(b*B - a*C))/(2*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (a^2*B - b^2*B + 2*a*b*C)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^1*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 5, ((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*x)/(a^2 + b^2)^3 + ((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - (b*B - a*C)/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a*b*B - a^2*C + b^2*C)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 6, -(((3*a^2*b*B - b^3*B - a^3*C + 3*a*b^2*C)*x)/(a^2 + b^2)^3) + (B*Log[Sin[c + d*x]])/(a^3*d) - (b*(6*a^4*b*B + 3*a^2*b^3*B + b^5*B - 3*a^5*C + a^3*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^3*d) + (b*(b*B - a*C))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^2*b*B + b^3*B - 2*a^3*C))/(a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cot[c + d*x]^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2)/(a + b*Tan[c + d*x])^3, x, 7, -(((a^3*B - 3*a*b^2*B + 3*a^2*b*C - b^3*C)*x)/(a^2 + b^2)^3) - ((3*b*B - a*C)*Log[Sin[c + d*x]])/(a^4*d) + (b^2*(10*a^4*b*B + 9*a^2*b^3*B + 3*b^5*B - 6*a^5*C - 3*a^3*b^2*C - a*b^4*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2*B + 3*b^2*B - a*b*C))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (B*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4*B + 6*a^2*b^2*B + 3*b^4*B - 3*a^3*b*C - a*b^3*C))/(a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]+C Tan[c+d x]^2) (b Tan[c+d x])^n*) - - -{Tan[c + d*x]^2*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2)*(b*Tan[c + d*x])^n, x, 7, (C*(b*Tan[c + d*x])^(3 + n))/(b^3*d*(3 + n)) + ((A - C)*Hypergeometric2F1[1, (3 + n)/2, (5 + n)/2, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(3 + n))/(b^3*d*(3 + n)) + (B*Hypergeometric2F1[1, (4 + n)/2, (6 + n)/2, -Tan[c + d*x]^2]*(b*Tan[c + d*x])^(4 + n))/(b^4*d*(4 + n))} -{Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2)*(b*Tan[c + d*x])^n, x, 7, (C*Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n)/(d*(1 + m + n)) + ((A - C)*Hypergeometric2F1[1, (1/2)*(1 + m + n), (1/2)*(3 + m + n), -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m)*(b*Tan[c + d*x])^n)/(d*(1 + m + n)) + (B*Hypergeometric2F1[1, (1/2)*(2 + m + n), (1/2)*(4 + m + n), -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m)*(b*Tan[c + d*x])^n)/(d*(2 + m + n))} -{Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2)*(b*Tan[c + d*x])^(1/2), x, 7, (2*C*Tan[c + d*x]^(1 + m)*Sqrt[b*Tan[c + d*x]])/(d*(3 + 2*m)) + (2*(A - C)*Hypergeometric2F1[1, (1/4)*(3 + 2*m), (1/4)*(7 + 2*m), -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m)*Sqrt[b*Tan[c + d*x]])/(d*(3 + 2*m)) + (2*B*Hypergeometric2F1[1, (1/4)*(5 + 2*m), (1/4)*(9 + 2*m), -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m)*Sqrt[b*Tan[c + d*x]])/(d*(5 + 2*m))} -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(b*Tan[c + d*x])^(1/2), x, 7, (2*C*Tan[c + d*x]^(1 + m))/(d*(1 + 2*m)*Sqrt[b*Tan[c + d*x]]) + (2*(A - C)*Hypergeometric2F1[1, (1/4)*(1 + 2*m), (1/4)*(5 + 2*m), -Tan[c + d*x]^2]*Tan[c + d*x]^(1 + m))/(d*(1 + 2*m)*Sqrt[b*Tan[c + d*x]]) + (2*B*Hypergeometric2F1[1, (1/4)*(3 + 2*m), (1/4)*(7 + 2*m), -Tan[c + d*x]^2]*Tan[c + d*x]^(2 + m))/(d*(3 + 2*m)*Sqrt[b*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (A+B Tan[c+d x]+C Tan[c+d x]^2) (a+b Tan[c+d x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Tan[c + d*x]^m*(A + B*Tan[c + d*x] + C*Tan[c + d*x]^2))/Sqrt[a + b*Tan[c + d*x]], x, 13, -(((b*B + Sqrt[-b^2]*(A - C))*AppellF1[1/2, 1, -m, 3/2, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2]), 1 + (b*Tan[c + d*x])/a]*Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]])/((-((b*Tan[c + d*x])/a))^m*(b*(a - Sqrt[-b^2])*d))) - ((b*B - Sqrt[-b^2]*(A - C))*AppellF1[1/2, 1, -m, 3/2, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2]), 1 + (b*Tan[c + d*x])/a]*Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]])/((-((b*Tan[c + d*x])/a))^m*(b*(a + Sqrt[-b^2])*d)) + (2*C*Hypergeometric2F1[1/2, -m, 3/2, 1 + (b*Tan[c + d*x])/a]*Tan[c + d*x]^m*Sqrt[a + b*Tan[c + d*x]])/((-((b*Tan[c + d*x])/a))^m*(b*d))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 6, (a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c + (A - C)*d) + b^3*(B*c + (A - C)*d))*x - ((3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Log[Cos[e + f*x]])/f + (b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2)/(2*f) + ((B*c + (A - C)*d)*(a + b*Tan[e + f*x])^3)/(3*f) - ((a*C*d - 5*b*(c*C + B*d))*(a + b*Tan[e + f*x])^4)/(20*b^2*f) + (C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^4)/(5*b*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 5, (a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d))*x - ((2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Log[Cos[e + f*x]])/f + (b*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*Tan[e + f*x])/f + ((B*c + (A - C)*d)*(a + b*Tan[e + f*x])^2)/(2*f) - ((a*C*d - 4*b*(c*C + B*d))*(a + b*Tan[e + f*x])^3)/(12*b^2*f) + (C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^3)/(4*b*f)} -{(a + b*Tan[e + f*x])^1*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 4, (a*(A*c - c*C - B*d) - b*(B*c + (A - C)*d))*x - ((A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*Log[Cos[e + f*x]])/f + ((A*b + a*B - b*C)*d*Tan[e + f*x])/f - ((b*c*C - 3*b*B*d - 3*a*C*d)*(c + d*Tan[e + f*x])^2)/(6*d^2*f) + (b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^2)/(3*d*f)} -{(a + b*Tan[e + f*x])^0*(c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 3, (A*c - c*C - B*d)*x - ((B*c + (A - C)*d)*Log[Cos[e + f*x]])/f + (B*d*Tan[e + f*x])/f + (C*(c + d*Tan[e + f*x])^2)/(2*d*f)} -{((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^1, x, 5, ((a*(A*c - c*C - B*d) + b*(B*c + (A - C)*d))*x)/(a^2 + b^2) + ((A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)*Log[Cos[e + f*x]])/((a^2 + b^2)*f) + ((A*b^2 - a*(b*B - a*C))*(b*c - a*d)*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)*f) + (C*d*Tan[e + f*x])/(b*f)} -{((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 5, ((a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) + 2*a*b*(B*c + (A - C)*d))*x)/(a^2 + b^2)^2 + ((2*a*b*(A*c - c*C - B*d) - a^2*(B*c + (A - C)*d) + b^2*(B*c + (A - C)*d))*Log[Cos[e + f*x]])/((a^2 + b^2)^2*f) + ((a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2)^2*f) - ((A*b^2 - a*(b*B - a*C))*(b*c - a*d))/(b^2*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 4, ((a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) + 3*a^2*b*(B*c + (A - C)*d) - b^3*(B*c + (A - C)*d))*x)/(a^2 + b^2)^3 + ((3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) - a^3*(B*c + (A - C)*d) + 3*a*b^2*(B*c + (A - C)*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((A*b^2 - a*(b*B - a*C))*(b*c - a*d))/(2*b^2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))/(b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 7, -((a^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 3*a^2*b*(2*c*(A - C)*d + B*(c^2 - d^2)) - b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))*x) + ((3*a^2*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*(c + d*Tan[e + f*x])^2)/(2*f) + ((4*a^3*C*d^3 - 3*a^2*b*d^2*(3*c*C - 16*B*d) + 3*a*b^2*d*(2*c^2*C - 5*B*c*d + 20*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 5*c*(A - C)*d^2 + 20*B*d^3))*(c + d*Tan[e + f*x])^3)/(60*d^4*f) + (b*(5*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(20*d^3*f) - ((b*c*C - 2*b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(10*d^2*f) + (C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^3)/(6*d*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 6, -((a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x) + ((2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f + ((a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^2)/(2*f) + ((8*a^2*C*d^2 - 10*a*b*d*(c*C - 4*B*d) + b^2*(2*c^2*C - 5*B*c*d + 20*(A - C)*d^2))*(c + d*Tan[e + f*x])^3)/(60*d^3*f) - (b*(2*b*c*C - 5*b*B*d - 2*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(20*d^2*f) + (C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3)/(5*d*f)} -{(a + b*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 5, -((a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x) - ((a*(B*c^2 - 2*c*C*d - B*d^2) - b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d + b*(c^2 - d^2)))*Log[Cos[e + f*x]])/f + (d*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*Tan[e + f*x])/f + ((A*b + a*B - b*C)*(c + d*Tan[e + f*x])^2)/(2*f) - ((b*c*C - 4*b*B*d - 4*a*C*d)*(c + d*Tan[e + f*x])^3)/(12*d^2*f) + (b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^3)/(4*d*f)} -{(a + b*Tan[e + f*x])^0*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 4, -((c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2))*x) - ((2*c*(A - C)*d + B*(c^2 - d^2))*Log[Cos[e + f*x]])/f + (d*(B*c + (A - C)*d)*Tan[e + f*x])/f + (B*(c + d*Tan[e + f*x])^2)/(2*f) + (C*(c + d*Tan[e + f*x])^3)/(3*d*f)} -{((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^1, x, 6, -(((a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2)) - ((a*(B*c^2 - 2*c*C*d - B*d^2) + b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d - b*(c^2 - d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)*f) + ((A*b^2 - a*(b*B - a*C))*(b*c - a*d)^2*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^2)*f) + (d*(b*c*C + b*B*d - a*C*d)*Tan[e + f*x])/(b^2*f) + (C*(c + d*Tan[e + f*x])^2)/(2*b*f)} -{((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 6, -(((a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2)^2) - ((2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) - b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)^2*f) - ((b*c - a*d)*(a^3*b*B*d - 2*a^4*C*d - b^4*(B*c + 2*A*d) - a*b^3*(2*A*c - 2*c*C - 3*B*d) + a^2*b^2*(B*c - 4*C*d))*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^2)^2*f) + ((A*b^2 - a*b*B + 2*a^2*C + b^2*C)*d^2*Tan[e + f*x])/(b^2*(a^2 + b^2)*f) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^2)/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 6, -(((a^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a^2*b*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^3*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2)^3) - ((3*a^2*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^3*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + a^3*(2*c*(A - C)*d + B*(c^2 - d^2)) - 3*a*b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)^3*f) + ((a^6*C*d^2 + 3*a^4*b^2*C*d^2 - 3*a^2*b^4*(c^2*C + 2*B*c*d - 2*C*d^2 - A*(c^2 - d^2)) + b^6*(c*(c*C + 2*B*d) - A*(c^2 - d^2)) - a^3*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a*b^5*(2*c*(A - C)*d + B*(c^2 - d^2)))*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^2)^3*f) - ((b*c - a*d)*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d)))/(b^3*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^2)/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)} - - -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 7, (a^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + b^2*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x + ((2*a*b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - a^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/f - (d*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Tan[e + f*x])/f + ((2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^2)/(2*f) + ((a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^3)/(3*f) + ((5*a^2*C*d^2 - 6*a*b*d*(c*C - 5*B*d) + b^2*(c^2*C - 3*B*c*d + 15*(A - C)*d^2))*(c + d*Tan[e + f*x])^4)/(60*d^3*f) - (b*(b*c*C - 3*b*B*d - a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(15*d^2*f) + (C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^4)/(6*d*f)} -{(a + b*Tan[e + f*x])^1*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 6, (a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) - b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x - ((A*(b*c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*d^3))*Log[Cos[e + f*x]])/f + (d*(a*(B*c^2 - 2*c*C*d - B*d^2) - b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d + b*(c^2 - d^2)))*Tan[e + f*x])/f + ((A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*(c + d*Tan[e + f*x])^2)/(2*f) + ((A*b + a*B - b*C)*(c + d*Tan[e + f*x])^3)/(3*f) - ((b*c*C - 5*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^4)/(20*d^2*f) + (b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f)} -{(a + b*Tan[e + f*x])^0*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 5, -((c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2))*x) - (((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2))*Log[Cos[e + f*x]])/f + (d*(2*c*(A - C)*d + B*(c^2 - d^2))*Tan[e + f*x])/f + ((B*c + (A - C)*d)*(c + d*Tan[e + f*x])^2)/(2*f) + (B*(c + d*Tan[e + f*x])^3)/(3*f) + (C*(c + d*Tan[e + f*x])^4)/(4*d*f)} -{((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^1, x, 7, -(((a*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x)/(a^2 + b^2)) - ((b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*d^2 + C*d^3) + A*(a*d*(3*c^2 - d^2) - b*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)*f) + ((A*b^2 - a*(b*B - a*C))*(b*c - a*d)^3*Log[a + b*Tan[e + f*x]])/(b^4*(a^2 + b^2)*f) + (d*(b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(b*c*C + b*B*d - a*C*d))*Tan[e + f*x])/(b^3*f) + ((b*c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^2)/(2*b^2*f) + (C*(c + d*Tan[e + f*x])^3)/(3*b*f)} -{((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 7, -(((b^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + a^2*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x)/(a^2 + b^2)^2) + ((2*a*b*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) - a^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)^2*f) - ((b*c - a*d)^2*(2*a^3*b*B*d - 3*a^4*C*d - b^4*(B*c + 3*A*d) - 2*a*b^3*(A*c - c*C - 2*B*d) + a^2*b^2*(B*c - (A + 5*C)*d))*Log[a + b*Tan[e + f*x]])/(b^4*(a^2 + b^2)^2*f) - (d^2*(3*a^3*C*d - A*b^2*(b*c - a*d) - b^3*(2*c*C + B*d) - a^2*b*(3*c*C + 2*B*d) + a*b^2*(B*c + 2*C*d))*Tan[e + f*x])/(b^3*(a^2 + b^2)*f) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C + b^2*C)*d*(c + d*Tan[e + f*x])^2)/(2*b^2*(a^2 + b^2)*f) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^3)/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 7, -(((3*a*b^2*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + a^3*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) - 3*a^2*b*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*x)/(a^2 + b^2)^3) - ((b^3*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3) + 3*a^2*b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3 - A*(c^3 - 3*c*d^2)) + a^3*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)) - 3*a*b^2*((A - C)*d*(3*c^2 - d^2) + B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((a^2 + b^2)^3*f) - ((b*c - a*d)*(a^5*b*B*d^2 - 3*a^6*C*d^2 + a^4*b^2*d*(B*c - 9*C*d) + a^3*b^3*B*(c^2 + 3*d^2) - b^6*(c*(c*C + 3*B*d) - A*(c^2 - 3*d^2)) - a*b^5*(8*c*(A - C)*d + 3*B*(c^2 - 2*d^2)) + a^2*b^4*(3*c^2*C + 6*B*c*d - 10*C*d^2 - A*(3*c^2 - d^2)))*Log[a + b*Tan[e + f*x]])/(b^4*(a^2 + b^2)^3*f) - (d^2*(a^3*b*B*d - 3*a^4*C*d - a*b^3*(2*A*c - 2*c*C - 3*B*d) + a^2*b^2*(B*c - 6*C*d) - b^4*(B*c + (2*A + C)*d))*Tan[e + f*x])/(b^3*(a^2 + b^2)^2*f) + ((a^3*b*B*d - 3*a^4*C*d - b^4*(2*B*c + 3*A*d) - a*b^3*(4*A*c - 4*c*C - 5*B*d) + a^2*b^2*(2*B*c + (A - 7*C)*d))*(c + d*Tan[e + f*x])^2)/(2*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^3)/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]), x, 7, ((a^3*(A*c - c*C + B*d) - 3*a*b^2*(A*c - c*C + B*d) - 3*a^2*b*(B*c - (A - C)*d) + b^3*(B*c - (A - C)*d))*x)/(c^2 + d^2) - ((3*a^2*b*(A*c - c*C + B*d) - b^3*(A*c - c*C + B*d) + a^3*(B*c - (A - C)*d) - 3*a*b^2*(B*c - (A - C)*d))*Log[Cos[e + f*x]])/((c^2 + d^2)*f) - ((b*c - a*d)^3*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^4*(c^2 + d^2)*f) + (b*(b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - b*B*d - a*C*d))*Tan[e + f*x])/(d^3*f) - ((b*c*C - b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2)/(2*d^2*f) + (C*(a + b*Tan[e + f*x])^3)/(3*d*f)} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]), x, 6, ((a^2*(A*c - c*C + B*d) - b^2*(A*c - c*C + B*d) - 2*a*b*(B*c - (A - C)*d))*x)/(c^2 + d^2) - ((2*a*b*(A*c - c*C + B*d) + a^2*(B*c - (A - C)*d) - b^2*(B*c - (A - C)*d))*Log[Cos[e + f*x]])/((c^2 + d^2)*f) + ((b*c - a*d)^2*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)*f) - (b*(b*c*C - b*B*d - a*C*d)*Tan[e + f*x])/(d^2*f) + (C*(a + b*Tan[e + f*x])^2)/(2*d*f)} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x]), x, 5, ((a*(A*c - c*C + B*d) - b*(B*c - (A - C)*d))*x)/(c^2 + d^2) - ((A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)*Log[Cos[e + f*x]])/((c^2 + d^2)*f) - ((b*c - a*d)*(c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f) + (b*C*Tan[e + f*x])/(d*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x]), x, 4, ((A*c - c*C + B*d)*x)/(c^2 + d^2) - ((B*c - (A - C)*d)*Log[Cos[e + f*x]])/((c^2 + d^2)*f) + ((c^2*C - B*c*d + A*d^2)*Log[c + d*Tan[e + f*x]])/(d*(c^2 + d^2)*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])), x, 3, ((a*(A*c - c*C + B*d) + b*(B*c - (A - C)*d))*x)/((a^2 + b^2)*(c^2 + d^2)) + ((A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f) - ((c^2*C - B*c*d + A*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])), x, 4, ((a^2*(A*c - c*C + B*d) - b^2*(A*c - c*C + B*d) + 2*a*b*(B*c - (A - C)*d))*x)/((a^2 + b^2)^2*(c^2 + d^2)) + ((2*a*b^3*c*(A - C) + 2*a^3*b*B*d - a^4*C*d + b^4*(B*c - A*d) - a^2*b^2*(B*c + 3*A*d - C*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^2*f) + (d*(c^2*C - B*c*d + A*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)*f) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])), x, 5, ((a^3*(A*c - c*C + B*d) - 3*a*b^2*(A*c - c*C + B*d) + 3*a^2*b*(B*c - (A - C)*d) - b^3*(B*c - (A - C)*d))*x)/((a^2 + b^2)^3*(c^2 + d^2)) + ((3*a*b^5*B*c^2 - 3*a^5*b*B*d^2 + a^6*C*d^2 + 3*a^4*b^2*d*(B*c + 2*A*d - C*d) + b^6*(c*(c*C - B*d) - A*(c^2 - d^2)) - a^3*b^3*(8*c*(A - C)*d + B*(c^2 - d^2)) - 3*a^2*b^4*(c*(c*C + 2*B*d) - A*(c^2 + d^2)))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*(b*c - a*d)^3*f) - (d^2*(c^2*C - B*c*d + A*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)*f) - (A*b^2 - a*(b*B - a*C))/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (2*a*b^3*c*(A - C) + 2*a^3*b*B*d - a^4*C*d + b^4*(B*c - A*d) - a^2*b^2*(B*c + 3*A*d - C*d))/((a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*Tan[e + f*x]))} - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2, x, 7, -(((a^3*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a^2*b*(2*c*(A - C)*d - B*(c^2 - d^2)) + b^3*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/(c^2 + d^2)^2) + ((3*a^2*b*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^3*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + a^3*(2*c*(A - C)*d - B*(c^2 - d^2)) - 3*a*b^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[Cos[e + f*x]])/((c^2 + d^2)^2*f) + ((b*c - a*d)^2*(b*(3*c^4*C - 2*B*c^3*d + c^2*(A + 5*C)*d^2 - 4*B*c*d^3 + 3*A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c + d*Tan[e + f*x]])/(d^4*(c^2 + d^2)^2*f) + (b^2*(a*d*(3*c^2*C - B*c*d + (A + 2*C)*d^2) - b*(3*c^3*C - 2*B*c^2*d + c*(A + 2*C)*d^2 - B*d^3))*Tan[e + f*x])/(d^3*(c^2 + d^2)*f) + (b*(3*c^2*C - 2*B*c*d + (2*A + C)*d^2)*(a + b*Tan[e + f*x])^2)/(2*d^2*(c^2 + d^2)*f) - ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2, x, 6, -(((a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 2*a*b*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/(c^2 + d^2)^2) + ((2*a*b*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + a^2*(2*c*(A - C)*d - B*(c^2 - d^2)) - b^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[Cos[e + f*x]])/((c^2 + d^2)^2*f) - ((b*c - a*d)*(b*(2*c^4*C - B*c^3*d + 4*c^2*C*d^2 - 3*B*c*d^3 + 2*A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^2*f) + (b^2*(2*c^2*C - B*c*d + (A + C)*d^2)*Tan[e + f*x])/(d^2*(c^2 + d^2)*f) - ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2, x, 5, -(((a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/(c^2 + d^2)^2) - ((a*(B*c^2 + 2*c*C*d - B*d^2) - b*(c^2*C - 2*B*c*d - C*d^2) - A*(2*a*c*d - b*(c^2 - d^2)))*Log[Cos[e + f*x]])/((c^2 + d^2)^2*f) + ((b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)^2*f) + ((b*c - a*d)*(c^2*C - B*c*d + A*d^2))/(d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^2, x, 3, -(((c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2))*x)/(c^2 + d^2)^2) + ((2*c*(A - C)*d - B*(c^2 - d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^2*f) - (c^2*C - B*c*d + A*d^2)/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2), x, 4, -(((a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)*(c^2 + d^2)^2)) + (b*(A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^2*f) - ((b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)^2*f) + (c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2), x, 5, -(((a^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)^2*(c^2 + d^2)^2)) + (b*(3*a^3*b*B*d - 2*a^4*C*d + b^4*(B*c - 2*A*d) - a^2*b^2*(B*c + 4*A*d) + a*b^3*(2*A*c - 2*c*C + B*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^3*f) + (d*(b*(2*c^4*C - 3*B*c^3*d + 4*A*c^2*d^2 - B*c*d^3 + 2*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^2*f) - (d*(b^2*c*(c*C - B*d) - a*b*B*(c^2 + d^2) + a^2*(2*c^2*C - B*c*d + C*d^2) + A*(a^2*d^2 + b^2*(c^2 + 2*d^2))))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2), x, 6, -(((a^3*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - 3*a*b^2*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 3*a^2*b*(2*c*(A - C)*d - B*(c^2 - d^2)) - b^3*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)^3*(c^2 + d^2)^2)) - (b*(6*a^5*b*B*d^2 - 3*a^6*C*d^2 - a^4*b^2*d*(4*B*c + (10*A - C)*d) - b^6*(c*(c*C - 2*B*d) - A*(c^2 - 3*d^2)) + a*b^5*(2*c*(A - C)*d - B*(3*c^2 - d^2)) + 3*a^2*b^4*(c*(c*C + 2*B*d) - A*(c^2 + 3*d^2)) + a^3*b^3*(10*c*(A - C)*d + B*(c^2 + 3*d^2)))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*(b*c - a*d)^4*f) - (d^2*(b*(3*c^4*C - 4*B*c^3*d + c^2*(5*A + C)*d^2 - 2*B*c*d^3 + 3*A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^4*(c^2 + d^2)^2*f) - (d*(3*a^3*b*B*d*(c^2 + d^2) + a*b^3*(2*A*c - 2*c*C + B*d)*(c^2 + d^2) - a^4*d*(3*c^2*C - B*c*d + (A + 2*C)*d^2) - a^2*b^2*(B*c^3 + 4*A*c^2*d + 2*c^2*C*d - B*c*d^2 + 6*A*d^3) - b^4*(d*(2*A*c^2 + c^2*C + 3*A*d^2) - B*(c^3 + 2*c*d^2))))/((a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - (A*b^2 - a*(b*B - a*C))/(2*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (5*a^3*b*B*d - 3*a^4*C*d + b^4*(2*B*c - 3*A*d) + a*b^3*(4*A*c - 4*c*C + B*d) - a^2*b^2*(2*B*c + (7*A - C)*d))/(2*(a^2 + b^2)^2*(b*c - a*d)^2*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x]))} - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3, x, 7, -(((3*a*b^2*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) + a^3*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) - 3*a^2*b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) + b^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*x)/(c^2 + d^2)^3) - ((3*a^2*b*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) - b^3*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) - a^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) + 3*a*b^2*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((c^2 + d^2)^3*f) - ((b*c - a*d)*(b^2*(3*c^6*C - B*c^5*d + 9*c^4*C*d^2 - 3*B*c^3*d^3 - c^2*(A - 10*C)*d^4 - 6*B*c*d^5 + 3*A*d^6) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) + a*b*d^2*(8*c*(A - C)*d^3 - B*(c^4 + 6*c^2*d^2 - 3*d^4)))*Log[c + d*Tan[e + f*x]])/(d^4*(c^2 + d^2)^3*f) + (b^2*(b*(3*c^4*C - B*c^3*d + 6*c^2*C*d^2 - 3*B*c*d^3 + (2*A + C)*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Tan[e + f*x])/(d^3*(c^2 + d^2)^2*f) - ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - ((b*(3*c^4*C - B*c^3*d - c^2*(A - 7*C)*d^2 - 5*B*c*d^3 + 3*A*d^4) + 2*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*(a + b*Tan[e + f*x])^2)/(2*d^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3, x, 6, -(((b^2*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) + a^2*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) - 2*a*b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*x)/(c^2 + d^2)^3) - ((2*a*b*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) - a^2*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) + b^2*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*Log[Cos[e + f*x]])/((c^2 + d^2)^3*f) - ((2*a*b*d^3*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) - b^2*(c^6*C + 3*c^4*C*d^2 + B*c^3*d^3 - 3*c^2*(A - 2*C)*d^4 - 3*B*c*d^5 + A*d^6) - a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*Log[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^3*f) - ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + ((b*c - a*d)*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/(d^3*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3, x, 4, -(((a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) - b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*x)/(c^2 + d^2)^3) + ((b*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3) - a*(B*c^3 + 3*c^2*C*d - 3*B*c*d^2 - C*d^3) + A*(a*d*(3*c^2 - d^2) - b*(c^3 - 3*c*d^2)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^3*f) + ((b*c - a*d)*(c^2*C - B*c*d + A*d^2))/(2*d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))/(d^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^3, x, 4, -(((c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2))*x)/(c^2 + d^2)^3) + (((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((c^2 + d^2)^3*f) - (c^2*C - B*c*d + A*d^2)/(2*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (2*c*(A - C)*d - B*(c^2 - d^2))/((c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3), x, 5, -(((a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^2*(A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^3*f) - ((b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 - d^4)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + (c^2*C - B*c*d + A*d^2)/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^3), x, 6, -(((b^2*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3) + a^2*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + 2*a*b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)))*x)/((a^2 + b^2)^2*(c^2 + d^2)^3)) + (b^2*(4*a^3*b*B*d - 3*a^4*C*d + b^4*(B*c - 3*A*d) + 2*a*b^3*(A*c - c*C + B*d) - a^2*b^2*(B*c + (5*A + C)*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^4*f) + (1/((b*c - a*d)^4*(c^2 + d^2)^3*f))*(d*(b^2*(3*c^6*C - 6*B*c^5*d + c^4*(10*A - C)*d^2 - 3*B*c^3*d^3 + 9*A*c^2*d^4 - B*c*d^5 + 3*A*d^6) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - 2*a*b*d^2*(c*(A - C)*d*(5*c^2 + d^2) - B*(2*c^4 - 3*c^2*d^2 - d^4)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]]) - (d*(b^2*c*(c*C - B*d) - 2*a*b*B*(c^2 + d^2) + a^2*(3*c^2*C - B*c*d + 2*C*d^2) + A*(a^2*d^2 + b^2*(2*c^2 + 3*d^2))))/(2*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) - (d*(b^3*c*(2*c^3*C - 3*B*c^2*d - B*d^3) + a^2*b*(3*c^4*C - 3*B*c^3*d + 2*c^2*C*d^2 - B*c*d^3 + C*d^4) + a^3*d^2*(2*c*C*d + B*(c^2 - d^2)) + a*b^2*(2*c*C*d^3 - B*(c^4 + c^2*d^2 + 2*d^4)) - A*(2*a^3*c*d^3 + 2*a*b^2*c*d^3 - 2*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 6*c^2*d^2 + 3*d^4))))/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^(n/2) (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 12, -(((a - I*b)^3*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b)^3*(I*A - B - I*C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*Sqrt[c + d*Tan[e + f*x]])/f + (1/(315*d^4*f))*(2*(40*a^3*C*d^3 - 6*a^2*b*d^2*(16*c*C - 45*B*d) + 9*a*b^2*d*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2) - b^3*(16*c^3*C - 24*B*c^2*d + 42*c*(A - C)*d^2 + 105*B*d^3))*(c + d*Tan[e + f*x])^(3/2)) + (2*b*(21*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(2*b*c*C - 3*b*B*d - 2*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(105*d^3*f) - (2*(2*b*c*C - 3*b*B*d - 2*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(21*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2))/(9*d*f)} -{(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 11, -(((a - I*b)^2*(B + I*(A - C))*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) - ((a + I*b)^2*(B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(20*a^2*C*d^2 - 14*a*b*d*(2*c*C - 5*B*d) + b^2*(8*c^2*C - 14*B*c*d + 35*(A - C)*d^2))*(c + d*Tan[e + f*x])^(3/2))/(105*d^3*f) - (2*b*(4*b*c*C - 7*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(35*d^2*f) + (2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2))/(7*d*f)} -{(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 10, -(((I*a + b)*(A - I*B - C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A + I*B - C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(A*b + a*B - b*C)*Sqrt[c + d*Tan[e + f*x]])/f - (2*(2*b*c*C - 5*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(15*d^2*f) + (2*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/2))/(5*d*f)} -{Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 9, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*B*Sqrt[c + d*Tan[e + f*x]])/f + (2*C*(c + d*Tan[e + f*x])^(3/2))/(3*d*f)} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x, 12, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)*f)) + ((I*A - B - I*C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)*f) + (2*C*Sqrt[c + d*Tan[e + f*x]])/(b*f)} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 12, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f)) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - ((a^3*b*B*d + a^4*C*d + b^4*(2*B*c + A*d) + a*b^3*(4*A*c - 4*c*C - 3*B*d) - a^2*b^2*(2*B*c + 3*A*d - 5*C*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)^2*Sqrt[b*c - a*d]*f) - ((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 13, -(((A - I*B - C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((A + I*B - C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + ((3*a^5*b*B*d^2 + a^6*C*d^2 - 3*a^4*b^2*d*(4*B*c + 5*A*d - 6*C*d) - 3*a^2*b^4*(8*A*c^2 - 8*c^2*C - 16*B*c*d - 6*A*d^2 + 5*C*d^2) + 2*a^3*b^3*(20*c*(A - C)*d + B*(4*c^2 - 13*d^2)) - 3*a*b^5*(8*c*(A - C)*d + B*(8*c^2 - d^2)) - b^6*(4*c*(2*c*C + B*d) - A*(8*c^2 + d^2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(a^2 + b^2)^3*(b*c - a*d)^(3/2)*f) - ((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - ((3*a^3*b*B*d + a^4*C*d + b^4*(4*B*c + A*d) + a*b^3*(8*A*c - 8*c*C - 5*B*d) - a^2*b^2*(4*B*c + 7*A*d - 9*C*d))*Sqrt[c + d*Tan[e + f*x]])/(4*b*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} - - -{(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 13, ((I*a + b)^3*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + ((a + I*b)^3*(I*A - B - I*C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(a^3*B - 3*a*b^2*B + 3*a^2*b*(A - C) - b^3*(A - C))*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(168*a^3*C*d^3 - 2*a^2*b*d^2*(192*c*C - 847*B*d) + 33*a*b^2*d*(8*c^2*C - 18*B*c*d + 63*(A - C)*d^2) - b^3*(48*c^3*C - 88*B*c^2*d + 198*c*(A - C)*d^2 + 693*B*d^3))*(c + d*Tan[e + f*x])^(5/2))/(3465*d^4*f) + (2*b*(99*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 11*b*B*d - 6*a*C*d))*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(693*d^3*f) - (2*(6*b*c*C - 11*b*B*d - 6*a*C*d)*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2))/(99*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^(5/2))/(11*d*f)} -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 12, -(((a - I*b)^2*(B + I*(A - C))*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b)^2*(I*A - B - I*C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(28*a^2*C*d^2 - 18*a*b*d*(2*c*C - 7*B*d) + b^2*(8*c^2*C - 18*B*c*d + 63*(A - C)*d^2))*(c + d*Tan[e + f*x])^(5/2))/(315*d^3*f) - (2*b*(4*b*c*C - 9*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(63*d^2*f) + (2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2))/(9*d*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 11, -(((I*a + b)*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A + I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*(A*b + a*B - b*C)*(c + d*Tan[e + f*x])^(3/2))/(3*f) - (2*(2*b*c*C - 7*b*B*d - 7*a*C*d)*(c + d*Tan[e + f*x])^(5/2))/(35*d^2*f) + (2*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(5/2))/(7*d*f)} -{(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 10, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(B*c + (A - C)*d)*Sqrt[c + d*Tan[e + f*x]])/f + (2*B*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*C*(c + d*Tan[e + f*x])^(5/2))/(5*d*f)} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x, 13, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)*f)) - ((A + I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*(A*b^2 - a*(b*B - a*C))*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(5/2)*(a^2 + b^2)*f) + (2*(b*c*C + b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(b^2*f) + (2*C*(c + d*Tan[e + f*x])^(3/2))/(3*b*f)} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 13, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) + (Sqrt[b*c - a*d]*(a^3*b*B*d - 3*a^4*C*d - b^4*(2*B*c + 3*A*d) - a*b^3*(4*A*c - 4*c*C - 5*B*d) + a^2*b^2*(2*B*c + (A - 7*C)*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(5/2)*(a^2 + b^2)^2*f) + ((A*b^2 - a*b*B + 3*a^2*C + 2*b^2*C)*d*Sqrt[c + d*Tan[e + f*x]])/(b^2*(a^2 + b^2)*f) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 13, -(((A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((A + I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) - ((a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(4*B*c + 3*(A + 2*C)*d) - b^6*(8*A*c^2 - 8*c^2*C - 12*B*c*d - 3*A*d^2) + a^2*b^4*(24*A*c^2 - 24*c^2*C - 48*B*c*d - 26*A*d^2 + 35*C*d^2) - 2*a^3*b^3*(12*c*(A - C)*d + B*(4*c^2 - 9*d^2)) + a*b^5*(40*c*(A - C)*d + 3*B*(8*c^2 - 5*d^2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(4*b^(5/2)*(a^2 + b^2)^3*Sqrt[b*c - a*d]*f) - ((a^3*b*B*d + 3*a^4*C*d + b^4*(4*B*c + 3*A*d) + a*b^3*(8*A*c - 8*c*C - 7*B*d) - a^2*b^2*(4*B*c + 5*A*d - 11*C*d))*Sqrt[c + d*Tan[e + f*x]])/(4*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)} - - -{(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 13, -(((a - I*b)^2*(I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((a + I*b)^2*(I*A - B - I*C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f - (2*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(a^2*B - b^2*B + 2*a*b*(A - C))*(c + d*Tan[e + f*x])^(5/2))/(5*f) + (2*(36*a^2*C*d^2 - 22*a*b*d*(2*c*C - 9*B*d) + b^2*(8*c^2*C - 22*B*c*d + 99*(A - C)*d^2))*(c + d*Tan[e + f*x])^(7/2))/(693*d^3*f) - (2*b*(4*b*c*C - 11*b*B*d - 4*a*C*d)*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(99*d^2*f) + (2*C*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(7/2))/(11*d*f)} -{(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 12, -(((I*a + b)*(A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A + I*B - C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(a*(B*c^2 - 2*c*C*d - B*d^2) - b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d + b*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*(A*b + a*B - b*C)*(c + d*Tan[e + f*x])^(5/2))/(5*f) - (2*(2*b*c*C - 9*b*B*d - 9*a*C*d)*(c + d*Tan[e + f*x])^(7/2))/(63*d^2*f) + (2*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(7/2))/(9*d*f)} -{(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 11, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*c*(A - C)*d + B*(c^2 - d^2))*Sqrt[c + d*Tan[e + f*x]])/f + (2*(B*c + (A - C)*d)*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*B*(c + d*Tan[e + f*x])^(5/2))/(5*f) + (2*C*(c + d*Tan[e + f*x])^(7/2))/(7*d*f)} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x, 14, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)*f)) + ((I*A - B - I*C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)*f) - (2*(A*b^2 - a*(b*B - a*C))*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(7/2)*(a^2 + b^2)*f) + (2*(b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(b*c*C + b*B*d - a*C*d))*Sqrt[c + d*Tan[e + f*x]])/(b^3*f) + (2*(b*c*C + b*B*d - a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*f) + (2*C*(c + d*Tan[e + f*x])^(5/2))/(5*b*f)} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2, x, 14, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) + ((b*c - a*d)^(3/2)*(3*a^3*b*B*d - 5*a^4*C*d - b^4*(2*B*c + 5*A*d) - a*b^3*(4*A*c - 4*c*C - 7*B*d) + a^2*b^2*(2*B*c - (A + 9*C)*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(7/2)*(a^2 + b^2)^2*f) - (d*(5*a^3*C*d - A*b^2*(b*c - a*d) - 2*b^3*(2*c*C + B*d) - a^2*b*(5*c*C + 3*B*d) + a*b^2*(B*c + 4*C*d))*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)*f) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C + 2*b^2*C)*d*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*(a^2 + b^2)*f) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(b*(a^2 + b^2)*f*(a + b*Tan[e + f*x]))} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3, x, 14, -(((A - I*B - C)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((A + I*B - C)*(c + I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (1/(4*b^(7/2)*(a^2 + b^2)^3*f))*(Sqrt[b*c - a*d]*(3*a^5*b*B*d^2 - 15*a^6*C*d^2 + a^4*b^2*d*(4*B*c + (A - 46*C)*d) - 3*a^2*b^4*(8*A*c^2 - 8*c^2*C - 16*B*c*d - 6*A*d^2 + 21*C*d^2) - a*b^5*(56*c*(A - C)*d + B*(24*c^2 - 35*d^2)) - b^6*(4*c*(2*c*C + 5*B*d) - A*(8*c^2 - 15*d^2)) + 2*a^3*b^3*(4*c*(A - C)*d + B*(4*c^2 + 3*d^2)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]) - (d*(3*a^3*b*B*d - 15*a^4*C*d - a*b^3*(8*A*c - 8*c*C - 11*B*d) + a^2*b^2*(4*B*c + (A - 31*C)*d) - b^4*(4*B*c + 7*A*d + 8*C*d))*Sqrt[c + d*Tan[e + f*x]])/(4*b^3*(a^2 + b^2)^2*f) + ((a^3*b*B*d - 5*a^4*C*d - b^4*(4*B*c + 5*A*d) - a*b^3*(8*A*c - 8*c*C - 9*B*d) + a^2*b^2*(4*B*c + 3*A*d - 13*C*d))*(c + d*Tan[e + f*x])^(3/2))/(4*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 11, ((I*a + b)^3*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) - ((I*a - b)^3*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*(72*a^3*C*d^3 - 6*a^2*b*d^2*(32*c*C - 49*B*d) + 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) - b^3*(48*c^3*C - 56*B*c^2*d + 70*c*(A - C)*d^2 + 105*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/(105*d^4*f) + (2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(105*d^3*f) - (2*(6*b*c*C - 7*b*B*d - 6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(35*d^2*f) + (2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]])/(7*d*f)} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 10, -(((a - I*b)^2*(B + I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) + ((a + I*b)^2*(I*A - B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*(12*a^2*C*d^2 - 10*a*b*d*(2*c*C - 3*B*d) + b^2*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2))*Sqrt[c + d*Tan[e + f*x]])/(15*d^3*f) - (2*b*(4*b*c*C - 5*b*B*d - 4*a*C*d)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(15*d^2*f) + (2*C*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d*f)} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 9, -(((I*a + b)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) + ((I*a - b)*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) - (2*(2*b*c*C - 3*b*B*d - 3*a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*f) + (2*b*C*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/Sqrt[c + d*Tan[e + f*x]], x, 8, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*C*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]), x, 11, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)*Sqrt[c - I*d]*f)) - ((A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*Sqrt[c + I*d]*f) - (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*Sqrt[b*c - a*d]*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]), x, 12, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*Sqrt[c + I*d]*f) - ((3*a^3*b*B*d - a^4*C*d + b^4*(2*B*c - A*d) + a*b^3*(4*A*c - 4*c*C - B*d) - a^2*b^2*(2*B*c + 5*A*d - 3*C*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)^2*(b*c - a*d)^(3/2)*f) - ((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x]))} - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 11, -(((a - I*b)^3*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) - ((I*a - b)^3*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b*(6*a^2*d^2*(12*c^2*C - 5*B*c*d + (5*A + 7*C)*d^2) - 15*a*b*d*(8*c^3*C - 6*B*c^2*d + c*(3*A + 5*C)*d^2 - 3*B*d^3) + b^2*(48*c^4*C - 40*B*c^3*d + 6*c^2*(5*A + 3*C)*d^2 - 25*B*c*d^3 + 15*(A - C)*d^4))*Sqrt[c + d*Tan[e + f*x]])/(15*d^4*(c^2 + d^2)*f) - (2*b^2*(4*(b*c - a*d)*(6*c^2*C - 5*B*c*d + (5*A + C)*d^2) - 5*d^2*((A - C)*(b*c - a*d) + B*(a*c + b*d)))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(15*d^3*(c^2 + d^2)*f) + (2*b*(6*c^2*C - 5*B*c*d + (5*A + C)*d^2)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/(5*d^2*(c^2 + d^2)*f)} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 10, -(((a - I*b)^2*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) - ((a + I*b)^2*(B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b*(6*a*d*(2*c^2*C - B*c*d + (A + C)*d^2) - b*(8*c^3*C - 6*B*c^2*d + c*(3*A + 5*C)*d^2 - 3*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/(3*d^3*(c^2 + d^2)*f) + (2*b^2*(4*c^2*C - 3*B*c*d + (3*A + C)*d^2)*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d^2*(c^2 + d^2)*f)} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 9, -(((I*a + b)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + ((I*a - b)*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) + (2*(b*c - a*d)*(c^2*C - B*c*d + A*d^2))/(d^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b*C*Sqrt[c + d*Tan[e + f*x]])/(d^2*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2), x, 8, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(c^2*C - B*c*d + A*d^2))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)), x, 12, ((A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*(c - I*d)^(3/2)*f) + ((I*A - B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)*(c + I*d)^(3/2)*f) - (2*Sqrt[b]*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*(b*c - a*d)^(3/2)*f) + (2*(c^2*C - B*c*d + A*d^2))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^(3/2)), x, 13, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*(c - I*d)^(3/2)*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*(c + I*d)^(3/2)*f) - (Sqrt[b]*(5*a^3*b*B*d - 3*a^4*C*d + b^4*(2*B*c - 3*A*d) + a*b^3*(4*A*c - 4*c*C + B*d) - a^2*b^2*(2*B*c + (7*A - C)*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(5/2)*f) - (d*(2*b^2*c*(c*C - B*d) - a*b*B*(c^2 + d^2) + a^2*(3*c^2*C - 2*B*c*d + C*d^2) + A*(2*a^2*d^2 + b^2*(c^2 + 3*d^2))))/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]])} - - -{((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 11, -(((a - I*b)^3*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) - ((I*a - b)^3*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^3)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*(2*c^4*C - B*c^3*d + 4*c^2*C*d^2 - 3*B*c*d^3 + 2*A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*(a + b*Tan[e + f*x])^2)/(d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b*(3*a*b*d*(8*c^4*C - 2*B*c^3*d - c^2*(A - 17*C)*d^2 - 8*B*c*d^3 + (5*A + 3*C)*d^4) - b^2*(16*c^5*C - 8*B*c^4*d + 2*c^3*(A + 15*C)*d^2 - 17*B*c^2*d^3 + 8*c*(A + C)*d^4 - 3*B*d^5) + 6*a^2*d^3*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(3*d^4*(c^2 + d^2)^2*f) + (2*b^2*(b*(8*c^4*C - 4*B*c^3*d + c^2*(A + 15*C)*d^2 - 10*B*c*d^3 + (7*A + C)*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d^3*(c^2 + d^2)^2*f)} -{((a + b*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 10, -(((a - I*b)^2*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) - ((a + I*b)^2*(B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^2)/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*c - a*d)*(b*(4*c^4*C - B*c^3*d - 2*c^2*(A - 5*C)*d^2 - 7*B*c*d^3 + 4*A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/(3*d^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b^2*(4*c^2*C - B*c*d + (A + 3*C)*d^2)*Sqrt[c + d*Tan[e + f*x]])/(3*d^3*(c^2 + d^2)*f)} -{((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 9, -(((a - I*b)*(I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) + ((I*a - b)*(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) + (2*(b*c - a*d)*(c^2*C - B*c*d + A*d^2))/(3*d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/(d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(5/2), x, 9, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(2*c*(A - C)*d - B*(c^2 - d^2)))/((c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(5/2)), x, 13, ((A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*(c - I*d)^(5/2)*f) + ((I*A - B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)*(c + I*d)^(5/2)*f) - (2*b^(3/2)*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*(b*c - a*d)^(5/2)*f) + (2*(c^2*C - B*c*d + A*d^2))/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2))))/((b*c - a*d)^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{1/(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 14, -(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*(c - I*d)^(5/2)*f)) - ((B - I*(A - C))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*(c + I*d)^(5/2)*f) - (b^(3/2)*(7*a^3*b*B*d - 5*a^4*C*d + b^4*(2*B*c - 5*A*d) + a*b^3*(4*A*c - 4*c*C + 3*B*d) - a^2*b^2*(2*B*c + (9*A + C)*d))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(7/2)*f) - (d*(2*b^2*c*(c*C - B*d) - 3*a*b*B*(c^2 + d^2) + a^2*(5*c^2*C - 2*B*c*d + 3*C*d^2) + A*(2*a^2*d^2 + b^2*(3*c^2 + 5*d^2))))/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (A*b^2 - a*(b*B - a*C))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)) - (1/((a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]))*(d*(2*a^3*d^2*(B*c^2 + 2*c*C*d - B*d^2) + 2*b^3*c*(2*c^3*C - 3*B*c^2*d - B*d^3) - a*b^2*(B*c^4 - 4*c*C*d^3 + 3*B*d^4) + a^2*b*(5*c^4*C - 6*B*c^3*d + 2*c^2*C*d^2 - 2*B*c*d^3 + C*d^4) - A*(4*a^3*c*d^3 + 4*a*b^2*c*d^3 - 4*a^2*b*d^2*(2*c^2 + d^2) - b^3*(c^4 + 10*c^2*d^2 + 5*d^4))))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^(m/2) (c+d Tan[e+f x])^(n/2) (A+B Tan[e+f x]+C Tan[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n > 0*) - - -{(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 16, -(((a - I*b)^(5/2)*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - ((a + I*b)^(5/2)*(B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f - ((5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*B*c*d^3 + 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(64*b^(3/2)*d^(7/2)*f) + ((64*b*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 - (b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b*d^3*f) + ((16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(32*d^3*f) - ((5*b*c*C - 8*b*B*d - 5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2))/(4*d*f)} -{(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 15, -(((a - I*b)^(3/2)*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + ((a + I*b)^(3/2)*(I*A - B - I*C)*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f - ((a^3*C*d^3 - 3*a^2*b*d^2*(c*C + 2*B*d) + 3*a*b^2*d*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*b^(3/2)*d^(5/2)*f) + ((8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*b*d^2*f) - ((b*c*C - 2*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*d^2*f) + (C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f)} -{Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 14, -((Sqrt[a - I*b]*(I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - (Sqrt[a + I*b]*(B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f - ((a^2*C*d^2 - 2*a*b*d*(c*C + 2*B*d) + b^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*b^(3/2)*d^(3/2)*f) - ((b*c*C - 4*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*d*f) + (C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*d*f)} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]], x, 13, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f) + ((b*c*C + 2*b*B*d - a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(3/2)*Sqrt[d]*f) + (C*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f)} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2), x, 13, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f)) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) + (2*C*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(3/2)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2), x, 9, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f)) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(2*a^3*b*B*d + a^4*C*d + b^4*(3*B*c + A*d) + 2*a*b^3*(3*A*c - 3*c*C - 2*B*d) - a^2*b^2*(3*B*c + 5*A*d - 7*C*d))*Sqrt[c + d*Tan[e + f*x]])/(3*b*(a^2 + b^2)^2*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])} -{(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2), x, 10, -(((I*A + B - I*C)*Sqrt[c - I*d]*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*(4*a^3*b*B*d + a^4*C*d + b^4*(5*B*c + A*d) + 2*a*b^3*(5*A*c - 5*c*C - 3*B*d) - a^2*b^2*(5*B*c + 9*A*d - 11*C*d))*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) + (2*(8*a^5*b*B*d^2 + 2*a^6*C*d^2 - a^4*b^2*d*(25*B*c + 33*A*d - 39*C*d) - a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 29*A*d^2 + 23*C*d^2) + a^3*b^3*(80*c*(A - C)*d + B*(15*c^2 - 49*d^2)) - a*b^5*(40*c*(A - C)*d + B*(45*c^2 - 3*d^2)) - b^6*(5*c*(3*c*C + B*d) - A*(15*c^2 + 2*d^2)))*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^3*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]])} - - -{(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 16, -(((a - I*b)^(3/2)*(B + I*(A - C))*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - ((a + I*b)^(3/2)*(B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((3*a^4*C*d^4 - 4*a^3*b*d^3*(3*c*C + 2*B*d) + 6*a^2*b^2*d^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - 12*a*b^3*d*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 16*B*d^3) + b^4*(3*c^4*C - 8*B*c^3*d + 48*c^2*(A - C)*d^2 - 192*B*c*d^3 - 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(64*b^(5/2)*d^(5/2)*f) + ((64*b*(a^2*B - b^2*B + 2*a*b*(A - C))*d^3 + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^2*d^2*f) + ((48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b*d^2*f) - ((3*b*c*C - 8*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2))/(4*d*f)} -{Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 15, -((Sqrt[a - I*b]*(I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - (Sqrt[a + I*b]*(B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((a^3*C*d^3 - a^2*b*d^2*(3*c*C + 2*B*d) + a*b^2*d*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - b^3*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 16*B*d^3))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*b^(5/2)*d^(3/2)*f) + ((8*b*(A*b + a*B - b*C)*d^2 - (b*c - a*d)*(b*c*C - 6*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*b^2*d*f) - ((b*c*C - 6*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(12*b*d*f) + (C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*d*f)} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]], x, 14, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) + ((I*A - B - I*C)*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f) + ((3*a^2*C*d^2 - 2*a*b*d*(3*c*C + 2*B*d) + b^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*b^(5/2)*Sqrt[d]*f) + ((3*b*c*C + 4*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b^2*f) + (C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b*f)} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2), x, 14, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) + (Sqrt[d]*(3*b*c*C + 2*b*B*d - 3*a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(5/2)*f) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C + b^2*C)*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b^2*(a^2 + b^2)*f) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(b*(a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2), x, 14, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) + (2*C*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(5/2)*f) - (2*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*Sqrt[c + d*Tan[e + f*x]])/(b^2*(a^2 + b^2)^2*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2))} -{((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2), x, 10, -(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) - (2*(2*a^3*b*B*d + 3*a^4*C*d + b^4*(5*B*c + 3*A*d) + 2*a*b^3*(5*A*c - 5*c*C - 4*B*d) - a^2*b^2*(5*B*c + 7*A*d - 13*C*d))*Sqrt[c + d*Tan[e + f*x]])/(15*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(2*a^5*b*B*d^2 + 3*a^6*C*d^2 + a^4*b^2*d*(10*B*c + (8*A + C)*d) + a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 49*A*d^2 + 58*C*d^2) - a^3*b^3*(50*c*(A - C)*d + B*(15*c^2 - 39*d^2)) + a*b^5*(70*c*(A - C)*d + B*(45*c^2 - 23*d^2)) + b^6*(5*c*(3*c*C + 4*B*d) - 3*A*(5*c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(15*b^2*(a^2 + b^2)^3*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(3/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))} - - -{Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 16, -((Sqrt[a - I*b]*(I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) + (Sqrt[a + I*b]*(I*A - B - I*C)*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f - ((5*a^4*C*d^4 - 4*a^3*b*d^3*(5*c*C + 2*B*d) + 2*a^2*b^2*d^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C)*d^2 + 320*B*c*d^3 + 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(64*b^(7/2)*d^(3/2)*f) + ((64*b^2*d^2*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^3*d*f) + ((48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b^2*d*f) - ((b*c*C - 8*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*b*d*f) + (C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f)} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[a + b*Tan[e + f*x]], x, 15, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*f) - ((5*a^3*C*d^3 - 3*a^2*b*d^2*(5*c*C + 2*B*d) + a*b^2*d*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - b^3*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*b^(7/2)*Sqrt[d]*f) + ((8*b^2*d*(B*c + (A - C)*d) + (b*c - a*d)*(5*b*c*C + 6*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*b^3*f) + ((5*b*c*C + 6*b*B*d - 5*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(12*b^2*f) + (C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*b*f)} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(3/2), x, 15, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*f) + (Sqrt[d]*(15*a^2*C*d^2 - 6*a*b*d*(5*c*C + 2*B*d) + b^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*b^(7/2)*f) - (d*(15*a^3*C*d - 8*A*b^2*(b*c - a*d) - 3*a^2*b*(5*c*C + 4*B*d) - b^3*(7*c*C + 4*B*d) + a*b^2*(8*B*c + 7*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b^3*(a^2 + b^2)*f) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C + b^2*C)*d*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b^2*(a^2 + b^2)*f) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(b*(a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]])} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(5/2), x, 15, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) + (d^(3/2)*(5*b*c*C + 2*b*B*d - 5*a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(7/2)*f) - (d*(2*a^3*b*B*d - 5*a^4*C*d - 2*a*b^3*(2*A*c - 2*c*C - 3*B*d) + 2*a^2*b^2*(B*c - 5*C*d) - b^4*(2*B*c + (4*A + C)*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)^2*f) + (2*(2*a^3*b*B*d - 5*a^4*C*d - b^4*(3*B*c + 5*A*d) - 2*a*b^3*(3*A*c - 3*c*C - 4*B*d) + a^2*b^2*(3*B*c + (A - 11*C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*(a^2 + b^2)^2*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2))} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2), x, 15, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) + (2*C*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(7/2)*f) - (2*(a^6*C*d^2 + 3*a^4*b^2*C*d^2 - 3*a^2*b^4*(c^2*C + 2*B*c*d - 2*C*d^2 - A*(c^2 - d^2)) + b^6*(c*(c*C + 2*B*d) - A*(c^2 - d^2)) - a^3*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a*b^5*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)^3*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))} -{((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(9/2), x, 11, -(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(9/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(9/2)*f) - (2*(6*a^5*b*B*d^2 + 15*a^6*C*d^2 + a^4*b^2*d*(14*B*c + 8*A*d + 37*C*d) + 3*a^2*b^4*(35*A*c^2 - 35*c^2*C - 70*B*c*d - 39*A*d^2 + 54*C*d^2) - a^3*b^3*(98*c*(A - C)*d + B*(35*c^2 - 75*d^2)) + a*b^5*(182*c*(A - C)*d + B*(105*c^2 - 71*d^2)) + b^6*(7*c*(5*c*C + 8*B*d) - 5*A*(7*c^2 - 3*d^2)))*Sqrt[c + d*Tan[e + f*x]])/(105*b^3*(a^2 + b^2)^3*f*(a + b*Tan[e + f*x])^(3/2)) - (1/(105*b^3*(a^2 + b^2)^4*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]))*(2*(6*a^7*b*B*d^3 + 15*a^8*C*d^3 + 2*a^6*b^2*d^2*(7*B*c + 4*A*d + 26*C*d) - 2*a*b^7*(210*A*c^3 - 210*c^3*C - 525*B*c^2*d - 406*A*c*d^2 + 406*c*C*d^2 + 88*B*d^3) - a^4*b^4*(105*B*c^3 + 525*A*c^2*d - 525*c^2*C*d - 749*B*c*d^2 - 311*A*d^3 + 221*C*d^3) + 2*a^2*b^6*(315*B*c^3 + 875*A*c^2*d - 875*c^2*C*d - 812*B*c*d^2 - 261*A*d^3 + 291*C*d^3) + 2*a^5*b^3*d*(56*c*(A - C)*d + B*(35*c^2 - 12*d^2)) - b^8*(5*d*(49*A*c^2 - 49*c^2*C - 3*A*d^2) + 7*B*(15*c^3 - 23*c*d^2)) - 2*a^3*b^5*(210*c^3*C + 700*B*c^2*d - 798*c*C*d^2 - 317*B*d^3 - 42*A*(5*c^3 - 19*c*d^2)))*Sqrt[c + d*Tan[e + f*x]]) - (2*(2*a^3*b*B*d + 5*a^4*C*d + b^4*(7*B*c + 5*A*d) + 2*a*b^3*(7*A*c - 7*c*C - 6*B*d) - a^2*b^2*(7*B*c + 9*A*d - 19*C*d))*(c + d*Tan[e + f*x])^(3/2))/(35*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(7*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n < 0*) - - -{((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 15, -(((a - I*b)^(5/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) - ((a + I*b)^(5/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f) + ((5*a^3*C*d^3 - 15*a^2*b*d^2*(c*C - 2*B*d) + 5*a*b^2*d*(3*c^2*C - 4*B*c*d + 8*(A - C)*d^2) - b^3*(5*c^3*C - 6*B*c^2*d + 8*c*(A - C)*d^2 + 16*B*d^3))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*Sqrt[b]*d^(7/2)*f) + ((8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 6*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*d^3*f) - ((5*b*c*C - 6*b*B*d - 5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(12*d^2*f) + (C*(a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]])/(3*d*f)} -{((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 14, -(((a - I*b)^(3/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) + ((a + I*b)^(3/2)*(I*A - B - I*C)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f) + ((3*a^2*C*d^2 - 6*a*b*d*(c*C - 2*B*d) + b^2*(3*c^2*C - 4*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*Sqrt[b]*d^(5/2)*f) - ((3*b*c*C - 4*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d^2*f) + (C*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*d*f)} -{(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[c + d*Tan[e + f*x]], x, 13, -((Sqrt[a - I*b]*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c - I*d]*f)) + (Sqrt[a + I*b]*(I*A - B - I*C)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[c + I*d]*f) - ((b*c*C - 2*b*B*d - a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*d^(3/2)*f) + (C*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]), x, 12, -(((B + I*(A - C))*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*Sqrt[c - I*d]*f)) + ((I*A - B - I*C)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*Sqrt[c + I*d]*f) + (2*C*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*Sqrt[d]*f)} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]), x, 8, -(((I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*Sqrt[c + I*d]*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(5/2)*Sqrt[c + d*Tan[e + f*x]]), x, 9, -(((I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*Sqrt[c + I*d]*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(5*a^3*b*B*d - 2*a^4*C*d + b^4*(3*B*c - 2*A*d) + a*b^3*(6*A*c - 6*c*C - B*d) - a^2*b^2*(3*B*c + 8*A*d - 4*C*d))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]])} - - -{((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 15, -(((a - I*b)^(5/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) - ((a + I*b)^(5/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (Sqrt[b]*(15*a^2*C*d^2 - 10*a*b*d*(3*c*C - 2*B*d) + b^2*(15*c^2*C - 12*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*d^(7/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(5/2))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (b*(3*(b*c - a*d)*(5*c^2*C - 4*B*c*d + (4*A + C)*d^2) - 4*d^2*((A - C)*(b*c - a*d) + B*(a*c + b*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d^3*(c^2 + d^2)*f) + (b*(5*c^2*C - 4*B*c*d + (4*A + C)*d^2)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*d^2*(c^2 + d^2)*f)} -{((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 14, -(((a - I*b)^(3/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) - ((a + I*b)^(3/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) - (Sqrt[b]*(3*b*c*C - 2*b*B*d - 3*a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(3/2))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (b*(3*c^2*C - 2*B*c*d + (2*A + C)*d^2)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d^2*(c^2 + d^2)*f)} -{(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 13, -((Sqrt[a - I*b]*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) - (Sqrt[a + I*b]*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (2*Sqrt[b]*C*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(3/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*Tan[e + f*x]])/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)), x, 8, -(((B + I*(A - C))*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*(c - I*d)^(3/2)*f)) + ((I*A - B - I*C)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*(c + I*d)^(3/2)*f) + (2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*Tan[e + f*x]])/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2)), x, 9, -(((I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*(c - I*d)^(3/2)*f)) - ((B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*(c + I*d)^(3/2)*f) - (2*(A*b^2 - a*(b*B - a*C)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*d*(b^2*c*(c*C - B*d) - a*b*B*(c^2 + d^2) + a^2*(2*c^2*C - B*c*d + C*d^2) + A*(a^2*d^2 + b^2*(c^2 + 2*d^2)))*Sqrt[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2)), x, 10, -(((I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(5/2)*(c - I*d)^(3/2)*f)) - ((B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*(c + I*d)^(3/2)*f) - (2*(A*b^2 - a*(b*B - a*C)))/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]) - (2*(7*a^3*b*B*d - 4*a^4*C*d + b^4*(3*B*c - 4*A*d) + a*b^3*(6*A*c - 6*c*C + B*d) - a^2*b^2*(3*B*c + 2*(5*A - C)*d)))/(3*(a^2 + b^2)^2*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) - (2*d*(8*a^3*b*B*d*(c^2 + d^2) + 2*a*b^3*(3*A*c - 3*c*C + B*d)*(c^2 + d^2) - a^4*d*(8*c^2*C - 3*B*c*d + (3*A + 5*C)*d^2) - a^2*b^2*(3*B*c^3 + 11*A*c^2*d + 5*c^2*C*d - 3*B*c*d^2 + 17*A*d^3 - C*d^3) - b^4*(d*(5*A*c^2 + 3*c^2*C + 8*A*d^2) - 3*B*(c^3 + 2*c*d^2)))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]])} - - -{((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 15, -(((a - I*b)^(5/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) - ((a + I*b)^(5/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) - (b^(3/2)*(5*b*c*C - 2*b*B*d - 5*a*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(7/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(5/2))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*(5*c^4*C - 2*B*c^3*d - c^2*(A - 11*C)*d^2 - 8*B*c*d^3 + 5*A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*(a + b*Tan[e + f*x])^(3/2))/(3*d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]]) + (b*(b*(5*c^4*C - 2*B*c^3*d + 10*c^2*C*d^2 - 6*B*c*d^3 + (4*A + C)*d^4) + 2*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(d^3*(c^2 + d^2)^2*f)} -{((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 14, -(((a - I*b)^(3/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) - ((a + I*b)^(3/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) + (2*b^(3/2)*C*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(d^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(3/2))/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3 + A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[a + b*Tan[e + f*x]])/(d^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(Sqrt[a + b*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(5/2), x, 9, -((Sqrt[a - I*b]*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c - I*d)^(5/2)*f)) - (Sqrt[a + I*b]*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(5/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*Tan[e + f*x]])/(3*d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*(c^4*C + 2*B*c^3*d - c^2*(5*A - 7*C)*d^2 - 4*B*c*d^3 + A*d^4) + 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[a + b*Tan[e + f*x]])/(3*d*(b*c - a*d)*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2)), x, 9, -(((B + I*(A - C))*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - I*b]*(c - I*d)^(5/2)*f)) + ((I*A - B - I*C)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + I*b]*(c + I*d)^(5/2)*f) + (2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*Tan[e + f*x]])/(3*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (2*(b*(2*c^4*C - 5*B*c^3*d + 4*c^2*(2*A - C)*d^2 + B*c*d^3 + 2*A*d^4) - 3*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Sqrt[a + b*Tan[e + f*x]])/(3*(b*c - a*d)^2*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} -{(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/((a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2)), x, 10, -(((I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a - I*b)^(3/2)*(c - I*d)^(5/2)*f)) - ((B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(3/2)*(c + I*d)^(5/2)*f) - (2*(A*b^2 - a*(b*B - a*C)))/((a^2 + b^2)*(b*c - a*d)*f*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) - (2*d*(b^2*c*(c*C - B*d) - 3*a*b*B*(c^2 + d^2) + a^2*(4*c^2*C - B*c*d + 3*C*d^2) + A*(a^2*d^2 + b^2*(3*c^2 + 4*d^2)))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*d*(b^3*c*(5*c^3*C - 8*B*c^2*d - c*C*d^2 - 2*B*d^3) + a^2*b*(8*c^4*C - 8*B*c^3*d + 5*c^2*C*d^2 - 2*B*c*d^3 + 3*C*d^4) + 3*a^3*d^2*(2*c*C*d + B*(c^2 - d^2)) + 3*a*b^2*(2*c*C*d^3 - B*(c^4 + c^2*d^2 + 2*d^4)) - A*(6*a^3*c*d^3 + 6*a*b^2*c*d^3 - a^2*b*d^2*(11*c^2 + 5*d^2) - b^3*(3*c^4 + 17*c^2*d^2 + 8*d^4)))*Sqrt[a + b*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)^3*(c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x])^m (c+d Tan[e+f x])^n (A+B Tan[e+f x]+C Tan[e+f x]^2) with m and/or n symbolic*) - - -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 9, -(((B + I*(A - C))*AppellF1[1 + m, -n, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^n)/(((b*(c + d*Tan[e + f*x]))/(b*c - a*d))^n*(2*(a - I*b)*f*(1 + m)))) - ((A + I*B - C)*AppellF1[1 + m, -n, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^n)/(((b*(c + d*Tan[e + f*x]))/(b*c - a*d))^n*(2*(I*a - b)*f*(1 + m))) + (C*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^n)/(((b*(c + d*Tan[e + f*x]))/(b*c - a*d))^n*(b*f*(1 + m)))} - - -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 9, ((b*c*(2 + m)*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) - 2*(b*c - a*d)*(3*a*C*d - b*(3*c*C + B*d*(4 + m)))) + d*(b^3*(2*c*(A - C)*d + B*(c^2 - d^2))*(2 + m)*(3 + m)*(4 + m) - a*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) - 2*(b*c - a*d)*(3*a*C*d - b*(3*c*C + B*d*(4 + m))))))*(a + b*Tan[e + f*x])^(1 + m))/(b^4*f*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + ((A - I*B - C)*(c - I*d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) - ((A + I*B - C)*(c + I*d)^3*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*f*(1 + m)) + (d*(b^2*d*(B*c + (A - C)*d)*(3 + m)*(4 + m) - 2*(b*c - a*d)*(3*a*C*d - b*(3*c*C + B*d*(4 + m))))*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b^3*f*(2 + m)*(3 + m)*(4 + m)) - ((3*a*C*d - b*(3*c*C + B*d*(4 + m)))*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^2)/(b^2*f*(3 + m)*(4 + m)) + (C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^3)/(b*f*(4 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 8, ((2*a^2*C*d^2 - a*b*d*(2*c*C + B*d)*(3 + m) + b^2*(2 + m)*(2*c^2*C + 2*B*c*d*(3 + m) + (A - C)*d^2*(3 + m)))*(a + b*Tan[e + f*x])^(1 + m))/(b^3*f*(1 + m)*(2 + m)*(3 + m)) + ((A - I*B - C)*(c - I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) + ((I*A - B - I*C)*(c + I*d)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m)) - (d*(2*a*C*d - b*(2*c*C + B*d*(3 + m)))*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b^2*f*(2 + m)*(3 + m)) + (C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^2)/(b*f*(3 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^1*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 7, -(((a*C*d - b*(c*C + B*d)*(2 + m))*(a + b*Tan[e + f*x])^(1 + m))/(b^2*f*(1 + m)*(2 + m))) + ((A - I*B - C)*(c - I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) - ((A + I*B - C)*(c + I*d)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*f*(1 + m)) + (C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(2 + m))} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^0*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 6, (C*(a + b*Tan[e + f*x])^(1 + m))/(b*f*(1 + m)) + ((A - I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*f*(1 + m)) + ((I*A - B - I*C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*f*(1 + m))} -{((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^1, x, 8, -(((I*A + B - I*C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a - I*b)*(c - I*d)*f*(1 + m))) - ((A + I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a - b)*(c + I*d)*f*(1 + m)) + ((c^2*C - B*c*d + A*d^2)*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m))} -{((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^2, x, 9, ((A - I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)^2*f*(1 + m)) + ((I*A - B - I*C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*(c + I*d)^2*f*(1 + m)) - ((a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)) - b*(A*d^2*(c^2*(2 - m) - d^2*m) - B*c*d*(c^2*(1 - m) - d^2*(1 + m)) - c^2*C*(c^2*m + d^2*(2 + m))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(1 + m)) + ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x]))} -{((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^3, x, 10, ((A - I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)*(c - I*d)^3*f*(1 + m)) + ((A + I*B - C)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(a + I*b)*(I*c - d)^3*f*(1 + m)) + (1/(2*(b*c - a*d)^3*(c^2 + d^2)^3*f*(1 + m)))*((2*a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - 2*a*b*d^2*(B*(6*c^2*d^2 - c^4*(2 - m) - d^4*m) + 2*c*(A - C)*d*(c^2*(3 - m) - d^2*(1 + m))) - b^2*(A*d^2*(d^4*(1 - m)*m + 2*c^2*d^2*(1 + 3*m - m^2) - c^4*(6 - 5*m + m^2)) + B*c*d*(d^4*m*(1 + m) - 2*c^2*d^2*(3 + m - m^2) + c^4*(2 - 3*m + m^2)) + c^2*C*(c^4*(1 - m)*m + 2*c^2*d^2*(3 - m - m^2) - d^4*(2 + 3*m + m^2))))*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m)) + ((c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - ((2*a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)) - b*(c^4*C*(1 - m) + A*d^4*(1 - m) - B*c^3*d*(3 - m) + B*c*d^3*(1 + m) + c^2*d^2*(A*(5 - m) - C*(3 + m))))*(a + b*Tan[e + f*x])^(1 + m))/(2*(b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*x]))} - - -(* {(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 18, -((2*(3*a*C*d - b*(3*c*C + B*d*(5 + 2*m)))*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(b^2*f*(3 + 2*m)*(5 + 2*m))) + (2*C*(a + b*Tan[e + f*x])^(1 + m)*(c + d*Tan[e + f*x])^(3/2))/(b*f*(5 + 2*m)) - ((B*d*(c^2 - d^2 - 2*c*Sqrt[-d^2]) + (A - C)*(2*c*d^2 + c^2*Sqrt[-d^2] + (-d^2)^(3/2)))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f)) - ((B*d*(c^2 - d^2 + 2*c*Sqrt[-d^2]) + (A - C)*(2*c*d^2 - c^2*Sqrt[-d^2] - (-d^2)^(3/2)))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) + (1/(b^2*d*f*(3 + 2*m)*(5 + 2*m)))*((2*(3*a^2*C*d^2 - a*b*d*(6*c*C + B*d*(5 + 2*m)) + b^2*(3*c^2*C + 2*B*c*d*(10 + 9*m + 2*m^2) + (A - C)*d^2*(15 + 16*m + 4*m^2)))*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m)} -{(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(1/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2), x, 16, (2*C*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[c + d*Tan[e + f*x]])/(b*f*(3 + 2*m)) - ((B*d*(c - Sqrt[-d^2]) + (A - C)*(d^2 + c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f)) - ((B*d*(c + Sqrt[-d^2]) + (A - C)*(d^2 - c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) - (2*(a*C*d - b*(c*C + B*d*(3 + 2*m)))*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(b*d*f*(3 + 2*m)))} -{((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(1/2), x, 15, -(((B*d + (A - C)*Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c - Sqrt[-d^2])*f))) - ((B*d - (A - C)*Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c + Sqrt[-d^2])*f)) + (2*C*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*f))} -{((a + b*Tan[e + f*x])^m*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2), x, 16, If[$VersionNumber<9, (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - ((B*c - (A - C)*d - (d*(A*c - c*C + B*d))/Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*((c^2 + d^2)*(c - Sqrt[-d^2])*f)) - ((B*c - (A - C)*d + (d*(A*c - c*C + B*d))/Sqrt[-d^2])*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*((c^2 + d^2)*(c + Sqrt[-d^2])*f)) - (2*b*(c^2*C - B*c*d + A*d^2)*(1 + 2*m)*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(b*c - a*d)*(c^2 + d^2)*f)), (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - ((B*d*(c + Sqrt[-d^2]) - (A - C)*(d^2 - c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c - Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c^2 + d^2)*(c - Sqrt[-d^2])*f)) - ((B*d*(c - Sqrt[-d^2]) - (A - C)*(d^2 + c*Sqrt[-d^2]))*AppellF1[1/2, -m, 1, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d), (c + d*Tan[e + f*x])/(c + Sqrt[-d^2])]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(c^2 + d^2)*(c + Sqrt[-d^2])*f)) - (2*b*(c^2*C - B*c*d + A*d^2)*(1 + 2*m)*Hypergeometric2F1[1/2, -m, 3/2, (b*c + b*d*Tan[e + f*x])/(b*c - a*d)]*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/((1 - (b*c + b*d*Tan[e + f*x])/(b*c - a*d))^m*(d*(b*c - a*d)*(c^2 + d^2)*f))]} *) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m deleted file mode 100644 index 9444798..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.7 (d trig)^m (a+b (c tan)^n)^p.m +++ /dev/null @@ -1,1027 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b (c Tan[e+f x])^n)^(p/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[e+f x]^2)^(p/2)*) - - -{(b*Tan[e + f*x]^2)^(5/2), x, 4, -((b^2*Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f) - (b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^2])/(2*f) + (b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^2])/(4*f)} -{(b*Tan[e + f*x]^2)^(3/2), x, 3, (b*Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f + (b*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^2])/(2*f)} -{(b*Tan[e + f*x]^2)^(1/2), x, 2, -((Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f)} -{1/(b*Tan[e + f*x]^2)^(1/2), x, 2, (Log[Sin[e + f*x]]*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^2])} -{1/(b*Tan[e + f*x]^2)^(3/2), x, 3, -(Cot[e + f*x]/(2*b*f*Sqrt[b*Tan[e + f*x]^2])) - (Log[Sin[e + f*x]]*Tan[e + f*x])/(b*f*Sqrt[b*Tan[e + f*x]^2])} -{1/(b*Tan[e + f*x]^2)^(5/2), x, 4, Cot[e + f*x]/(2*b^2*f*Sqrt[b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(4*b^2*f*Sqrt[b*Tan[e + f*x]^2]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(b^2*f*Sqrt[b*Tan[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[e+f x]^3)^(p/2)*) - - -{(b*Tan[e + f*x]^3)^(5/2), x, 16, -((2*b^2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/f) - (b^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (b^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (2*b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/(5*f) - (2*b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^3])/(9*f) + (2*b^2*Tan[e + f*x]^5*Sqrt[b*Tan[e + f*x]^3])/(13*f)} -{(b*Tan[e + f*x]^3)^(3/2), x, 14, -((2*b*Sqrt[b*Tan[e + f*x]^3])/(3*f)) - (b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (b*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (2*b*Tan[e + f*x]^2*Sqrt[b*Tan[e + f*x]^3])/(7*f)} -{(b*Tan[e + f*x]^3)^(1/2), x, 13, (2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/f + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2))} -{1/(b*Tan[e + f*x]^3)^(1/2), x, 13, -((2*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^3])) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3])} -{1/(b*Tan[e + f*x]^3)^(3/2), x, 14, 2/(3*b*f*Sqrt[b*Tan[e + f*x]^3]) - (2*Cot[e + f*x]^2)/(7*b*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3])} -{1/(b*Tan[e + f*x]^3)^(5/2), x, 16, -((2*Cot[e + f*x])/(5*b^2*f*Sqrt[b*Tan[e + f*x]^3])) + (2*Cot[e + f*x]^3)/(9*b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (2*Cot[e + f*x]^5)/(13*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (2*Tan[e + f*x])/(b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tan[e+f x]^4)^(p/2)*) - - -{(b*Tan[e + f*x]^4)^(5/2), x, 7, (b^2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - b^2*x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4] - (b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/(3*f) + (b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^4])/(5*f) - (b^2*Tan[e + f*x]^5*Sqrt[b*Tan[e + f*x]^4])/(7*f) + (b^2*Tan[e + f*x]^7*Sqrt[b*Tan[e + f*x]^4])/(9*f)} -{(b*Tan[e + f*x]^4)^(3/2), x, 5, (b*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - b*x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4] - (b*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/(3*f) + (b*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^4])/(5*f)} -{(b*Tan[e + f*x]^4)^(1/2), x, 3, (Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4]} -{1/(b*Tan[e + f*x]^4)^(1/2), x, 3, -(Tan[e + f*x]/(f*Sqrt[b*Tan[e + f*x]^4])) - (x*Tan[e + f*x]^2)/Sqrt[b*Tan[e + f*x]^4]} -{1/(b*Tan[e + f*x]^4)^(3/2), x, 5, Cot[e + f*x]/(3*b*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^3/(5*b*f*Sqrt[b*Tan[e + f*x]^4]) - Tan[e + f*x]/(b*f*Sqrt[b*Tan[e + f*x]^4]) - (x*Tan[e + f*x]^2)/(b*Sqrt[b*Tan[e + f*x]^4])} -{1/(b*Tan[e + f*x]^4)^(5/2), x, 7, Cot[e + f*x]/(3*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^3/(5*b^2*f*Sqrt[b*Tan[e + f*x]^4]) + Cot[e + f*x]^5/(7*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^7/(9*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Tan[e + f*x]/(b^2*f*Sqrt[b*Tan[e + f*x]^4]) - (x*Tan[e + f*x]^2)/(b^2*Sqrt[b*Tan[e + f*x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b (c Tan[e+f x])^n)^(p/2)*) - - -{(b*Tan[e + f*x]^n)^(5/2), x, 3, (2*b^2*Hypergeometric2F1[1, (1/4)*(2 + 5*n), (1/4)*(6 + 5*n), -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + 2*n)*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + 5*n))} -{(b*Tan[e + f*x]^n)^(3/2), x, 3, (2*b*Hypergeometric2F1[1, (1/4)*(2 + 3*n), (3*(2 + n))/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n)*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + 3*n))} -{(b*Tan[e + f*x]^n)^(1/2), x, 3, (2*Hypergeometric2F1[1, (2 + n)/4, (6 + n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + n))} -{1/(b*Tan[e + f*x]^n)^(1/2), x, 3, (2*Hypergeometric2F1[1, (2 - n)/4, (6 - n)/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(2 - n)*Sqrt[b*Tan[e + f*x]^n])} -{1/(b*Tan[e + f*x]^n)^(3/2), x, 3, (2*Hypergeometric2F1[1, (1/4)*(2 - 3*n), (3*(2 - n))/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 - n))/(b*f*(2 - 3*n)*Sqrt[b*Tan[e + f*x]^n])} -{1/(b*Tan[e + f*x]^n)^(5/2), x, 3, (2*Hypergeometric2F1[1, (1/4)*(2 - 5*n), (1/4)*(6 - 5*n), -Tan[e + f*x]^2]*Tan[e + f*x]^(1 - 2*n))/(b^2*f*(2 - 5*n)*Sqrt[b*Tan[e + f*x]^n])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(b*Tan[e + f*x]^n)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^n)^p)/(f*(1 + n*p))} - - -{(b*Tan[e + f*x]^2)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 2*p), (1/2)*(3 + 2*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))} -{(b*Tan[e + f*x]^3)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 3*p), (3*(1 + p))/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^3)^p)/(f*(1 + 3*p))} -{(b*Tan[e + f*x]^4)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + 4*p), (1/2)*(3 + 4*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^4)^p)/(f*(1 + 4*p))} - - -{(b*Tan[e + f*x]^n)^(1/n), x, 2, -((Cot[e + f*x]*Log[Cos[e + f*x]]*(b*Tan[e + f*x]^n)^(1/n))/f)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2), x, 3, -(((a - 3*b)*Cos[e + f*x])/f) + ((2*a - 3*b)*Cos[e + f*x]^3)/(3*f) - ((a - b)*Cos[e + f*x]^5)/(5*f) + (b*Sec[e + f*x])/f} -{Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2), x, 3, -(((a - 2*b)*Cos[e + f*x])/f) + ((a - b)*Cos[e + f*x]^3)/(3*f) + (b*Sec[e + f*x])/f} -{Sin[e + f*x]^1*(a + b*Tan[e + f*x]^2), x, 3, -(((a - b)*Cos[e + f*x])/f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^1*(a + b*Tan[e + f*x]^2), x, 3, -((a*ArcTanh[Cos[e + f*x]])/f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2), x, 4, -((a + 2*b)*ArcTanh[Cos[e + f*x]])/(2*f) - (a*Cot[e + f*x]*Csc[e + f*x])/(2*f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2), x, 5, (-3*(a + 4*b)*ArcTanh[Cos[e + f*x]])/(8*f) - ((5*a + 4*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a*Cot[e + f*x]^3*Csc[e + f*x])/(4*f) + (b*Sec[e + f*x])/f} - -{Sin[e + f*x]^6*(a + b*Tan[e + f*x]^2), x, 6, (5*(a - 7*b)*x)/16 - ((11*a - 29*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 19*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - ((a - b)*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f} -{Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2), x, 5, (3*(a - 5*b)*x)/8 - ((5*a - 9*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + ((a - b)*Cos[e + f*x]^3*Sin[e + f*x])/(4*f) + (b*Tan[e + f*x])/f} -{Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2), x, 4, ((a - 3*b)*x)/2 - ((a - b)*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b*Tan[e + f*x])/f} -{Sin[e + f*x]^0*(a + b*Tan[e + f*x]^2), x, 3, a*x - b*x + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2), x, 3, -((a*Cot[e + f*x])/f) + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2), x, 3, -(((a + b)*Cot[e + f*x])/f) - (a*Cot[e + f*x]^3)/(3*f) + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2), x, 3, -(((a + 2*b)*Cot[e + f*x])/f) - ((2*a + b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f) + (b*Tan[e + f*x])/f} - - -{Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2, x, 3, -(((a^2 - 6*a*b + 6*b^2)*Cos[e + f*x])/f) + (2*(a - 2*b)*(a - b)*Cos[e + f*x]^3)/(3*f) - ((a - b)^2*Cos[e + f*x]^5)/(5*f) + (2*(a - 2*b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2, x, 3, -(((a - 3*b)*(a - b)*Cos[e + f*x])/f) + ((a - b)^2*Cos[e + f*x]^3)/(3*f) + ((2*a - 3*b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Sin[e + f*x]^1*(a + b*Tan[e + f*x]^2)^2, x, 3, -(((a - b)^2*Cos[e + f*x])/f) + (2*(a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^1*(a + b*Tan[e + f*x]^2)^2, x, 4, -((a^2*ArcTanh[Cos[e + f*x]])/f) + ((2*a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2, x, 5, -(a*(a + 4*b)*ArcTanh[Cos[e + f*x]])/(2*f) + (a*(a + 4*b)*Sec[e + f*x])/(2*f) - (a^2*Csc[e + f*x]^2*Sec[e + f*x])/(2*f) + (b^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2, x, 6, -((3*a^2 + 24*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - (a*(a + 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) + ((a^2 + 8*a*b + 4*b^2)*Sec[e + f*x])/(4*f) - (a^2*Csc[e + f*x]^4*Sec[e + f*x])/(4*f) + (b^2*Sec[e + f*x]^3)/(3*f)} - -{Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2, x, 6, ((3*a^2 - 30*a*b + 35*b^2)*x)/8 - ((a - 9*b)*(a - b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - ((a^2 - 10*a*b + 13*b^2)*Tan[e + f*x])/(4*f) + ((a - b)^2*Sin[e + f*x]^4*Tan[e + f*x])/(4*f) + (b^2*Tan[e + f*x]^3)/(3*f)} -{Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2, x, 5, ((a - 5*b)*(a - b)*x)/2 - ((a - 5*b)*(a - b)*Tan[e + f*x])/(2*f) + ((a - b)^2*Sin[e + f*x]^2*Tan[e + f*x])/(2*f) + (b^2*Tan[e + f*x]^3)/(3*f)} -{Sin[e + f*x]^0*(a + b*Tan[e + f*x]^2)^2, x, 4, (a - b)^2*x + ((2*a - b)*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2, x, 3, -((a^2*Cot[e + f*x])/f) + (2*a*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2, x, 3, -((a*(a + 2*b)*Cot[e + f*x])/f) - (a^2*Cot[e + f*x]^3)/(3*f) + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2, x, 3, -(((a^2 + 4*a*b + b^2)*Cot[e + f*x])/f) - (2*a*(a + b)*Cot[e + f*x]^3)/(3*f) - (a^2*Cot[e + f*x]^5)/(5*f) + (2*b*(a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2), x, 4, -((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(7/2)*f)) - (a^2*Cos[e + f*x])/((a - b)^3*f) + ((2*a - b)*Cos[e + f*x]^3)/(3*(a - b)^2*f) - Cos[e + f*x]^5/(5*(a - b)*f)} -{Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2), x, 4, -((a*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(5/2)*f)) - (a*Cos[e + f*x])/((a - b)^2*f) + Cos[e + f*x]^3/(3*(a - b)*f)} -{Sin[e + f*x]^1/(a + b*Tan[e + f*x]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(3/2)*f)) - Cos[e + f*x]/((a - b)*f)} -{Csc[e + f*x]^1/(a + b*Tan[e + f*x]^2), x, 4, -((Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a*Sqrt[a - b]*f)) - ArcTanh[Cos[e + f*x]]/(a*f)} -{Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2), x, 5, -((Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a^2*f)) - ((a - 2*b)*ArcTanh[Cos[e + f*x]])/(2*a^2*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f)} -{Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2), x, 6, -(((a - b)^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a^3*f)) - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^3*f) - ((5*a - 4*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f)} - -{Sin[e + f*x]^6/(a + b*Tan[e + f*x]^2), x, 7, ((5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*x)/(16*(a - b)^4) - (a^(5/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^4*f) - ((11*a^2 - 4*a*b + b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*(a - b)^3*f) + ((3*a - b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*(a - b)*f)} -{Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2), x, 6, ((3*a^2 + 6*a*b - b^2)*x)/(8*(a - b)^3) - (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^3*f) - ((5*a - b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f)} -{Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2), x, 5, ((a + b)*x)/(2*(a - b)^2) - (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^2*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f)} -{Sin[e + f*x]^0/(a + b*Tan[e + f*x]^2), x, 3, x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*f)} -{Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*f)) - Cot[e + f*x]/(a*f)} -{Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2), x, 4, -(((a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*f)) - ((a - b)*Cot[e + f*x])/(a^2*f) - Cot[e + f*x]^3/(3*a*f)} -{Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2), x, 4, -(((a - b)^2*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*f)) - ((a - b)^2*Cot[e + f*x])/(a^3*f) - ((2*a - b)*Cot[e + f*x]^3)/(3*a^2*f) - Cot[e + f*x]^5/(5*a*f)} - - -{Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2, x, 6, -(a*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(9/2)*f) - ((5*a^2 + 10*a*b - b^2)*Cos[e + f*x])/(5*(a - b)^4*f) + ((10*a - 3*b)*Cos[e + f*x]^3)/(15*(a - b)^3*f) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)) - (b*(5*a^2 + 2*b^2)*Sec[e + f*x])/(10*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2))} -{Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2, x, 5, -(Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(7/2)*f) - ((a + b)*Cos[e + f*x])/((a - b)^3*f) + Cos[e + f*x]^3/(3*(a - b)^2*f) - (a*b*Sec[e + f*x])/(2*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2))} -{Sin[e + f*x]^1/(a + b*Tan[e + f*x]^2)^2, x, 4, (-3*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(5/2)*f) - (3*Cos[e + f*x])/(2*(a - b)^2*f) + Cos[e + f*x]/(2*(a - b)*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^1/(a + b*Tan[e + f*x]^2)^2, x, 5, -((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^2*(a - b)^(3/2)*f) - ArcTanh[Cos[e + f*x]]/(a^2*f) - (b*Sec[e + f*x])/(2*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2, x, 6, -((3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^3*Sqrt[a - b]*f) - ((a - 4*b)*ArcTanh[Cos[e + f*x]])/(2*a^3*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)) - (b*Sec[e + f*x])/(a^2*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2, x, 7, (-3*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^4*f) - (3*(a^2 - 8*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^4*f) - ((5*a - 6*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)) - (3*(3*a - 4*b)*b*Sec[e + f*x])/(8*a^3*f*(a - b + b*Sec[e + f*x]^2))} - -{Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2, x, 7, (3*(a^2 + 6*a*b + b^2)*x)/(8*(a - b)^4) - (3*Sqrt[a]*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^4*f) - ((5*a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)) - (3*b*(3*a + b)*Tan[e + f*x])/(8*(a - b)^3*f*(a + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2, x, 6, ((a + 3*b)*x)/(2*(a - b)^3) - (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*Sqrt[a]*(a - b)^3*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)) - (b*Tan[e + f*x])/((a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^0/(a + b*Tan[e + f*x]^2)^2, x, 5, x/(a - b)^2 - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*f) - (b*Tan[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2, x, 4, (-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(5/2)*f) - (3*Cot[e + f*x])/(2*a^2*f) + Cot[e + f*x]/(2*a*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2, x, 5, -((3*a - 5*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(7/2)*f) - ((a - 2*b)*Cot[e + f*x])/(a^3*f) - Cot[e + f*x]^3/(3*a^2*f) - ((a - b)*b*Tan[e + f*x])/(2*a^3*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2, x, 6, -(((3*a - 7*b)*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*f)) - ((5*a^2 - 20*a*b + 14*b^2)*Cot[e + f*x])/(5*a^4*f) - ((10*a - 7*b)*Cot[e + f*x]^3)/(15*a^3*f) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)) - (b*(5*a^2 - 10*a*b + 7*b^2)*Tan[e + f*x])/(10*a^4*f*(a + b*Tan[e + f*x]^2))} - - -{Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3, x, 7, -(Sqrt[b]*(15*a^2 + 40*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(11/2)*f) - ((5*a^2 + 20*a*b + 2*b^2)*Cos[e + f*x])/(5*(a - b)^5*f) + ((10*a - b)*Cos[e + f*x]^3)/(15*(a - b)^4*f) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(5*a^2 + 4*b^2)*Sec[e + f*x])/(20*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(35*a^2 + 40*a*b + 24*b^2)*Sec[e + f*x])/(40*(a - b)^5*f*(a - b + b*Sec[e + f*x]^2))} -{Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3, x, 6, (-5*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(9/2)*f) - ((a + 2*b)*Cos[e + f*x])/((a - b)^4*f) + Cos[e + f*x]^3/(3*(a - b)^3*f) - (a*b*Sec[e + f*x])/(4*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(7*a + 4*b)*Sec[e + f*x])/(8*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2))} -{Sin[e + f*x]^1/(a + b*Tan[e + f*x]^2)^3, x, 5, (-15*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(7/2)*f) - (15*Cos[e + f*x])/(8*(a - b)^3*f) + Cos[e + f*x]/(4*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) + (5*Cos[e + f*x])/(8*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^1/(a + b*Tan[e + f*x]^2)^3, x, 6, -(Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^3*(a - b)^(5/2)*f) - ArcTanh[Cos[e + f*x]]/(a^3*f) - (b*Sec[e + f*x])/(4*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) - ((7*a - 4*b)*b*Sec[e + f*x])/(8*a^2*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3, x, 7, -(Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(3/2)*f) - ((a - 6*b)*ArcTanh[Cos[e + f*x]])/(2*a^4*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^2) - (3*b*Sec[e + f*x])/(4*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - ((11*a - 12*b)*b*Sec[e + f*x])/(8*a^3*(a - b)*f*(a - b + b*Sec[e + f*x]^2))} -{Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3, x, 8, (-3*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^5*Sqrt[a - b]*f) - (3*(a^2 - 12*a*b + 16*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^5*f) - ((5*a - 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^2) - ((7*a - 12*b)*b*Sec[e + f*x])/(8*a^3*f*(a - b + b*Sec[e + f*x]^2)^2) - (3*(a - 2*b)*b*Sec[e + f*x])/(2*a^4*f*(a - b + b*Sec[e + f*x]^2))} - -{Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3, x, 8, (3*(a^2 + 10*a*b + 5*b^2)*x)/(8*(a - b)^5) - (3*Sqrt[b]*(5*a^2 + 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^5*f) - ((5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - (b*(7*a + 5*b)*Tan[e + f*x])/(8*(a - b)^3*f*(a + b*Tan[e + f*x]^2)^2) - (3*b*(a + b)*Tan[e + f*x])/(2*(a - b)^4*f*(a + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3, x, 7, ((a + 5*b)*x)/(2*(a - b)^4) - (Sqrt[b]*(15*a^2 + 10*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^4*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^2) - (b*(11*a + b)*Tan[e + f*x])/(8*a*(a - b)^3*f*(a + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^0/(a + b*Tan[e + f*x]^2)^3, x, 6, x/(a - b)^3 - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*f) - (b*Tan[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 3*b)*b*Tan[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3, x, 5, (-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(7/2)*f) - (15*Cot[e + f*x])/(8*a^3*f) + Cot[e + f*x]/(4*a*f*(a + b*Tan[e + f*x]^2)^2) + (5*Cot[e + f*x])/(8*a^2*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3, x, 6, (-5*(3*a - 7*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*f) - ((a - 3*b)*Cot[e + f*x])/(a^4*f) - Cot[e + f*x]^3/(3*a^3*f) - ((a - b)*b*Tan[e + f*x])/(4*a^3*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 11*b)*b*Tan[e + f*x])/(8*a^4*f*(a + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3, x, 7, -(Sqrt[b]*(15*a^2 - 70*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(11/2)*f) - ((5*a^2 - 30*a*b + 27*b^2)*Cot[e + f*x])/(5*a^5*f) - ((10*a - 9*b)*Cot[e + f*x]^3)/(15*a^4*f) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)^2) - (b*(5*a^2 - 10*a*b + 9*b^2)*Tan[e + f*x])/(20*a^4*f*(a + b*Tan[e + f*x]^2)^2) - (b*(35*a^2 - 110*a*b + 99*b^2)*Tan[e + f*x])/(40*a^5*f*(a + b*Tan[e + f*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sin[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f + (2*(5*a - 4*b)*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(15*(a - b)^2*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(3/2))/(5*(a - b)*f)} -{Sin[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2], x, 5, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f + (Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f)} -{Sin[e + f*x]^1*Sqrt[a + b*Tan[e + f*x]^2], x, 4, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f} -{Csc[e + f*x]^1*Sqrt[a + b*Tan[e + f*x]^2], x, 6, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f} -{Csc[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2], x, 7, -((a + b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*Sqrt[a]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2], x, 8, -((3*a^2 + 6*a*b - b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - ((3*a + b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*f)} - -{Sin[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2], x, 8, ((3*a^2 - 12*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*(a - b)*f) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)} -{Sin[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2], x, 7, ((a - 2*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*Sqrt[a - b]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Sin[e + f*x]^0*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f} -{Csc[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2], x, 4, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f} -{Csc[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2], x, 5, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f)} -{Csc[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (2*(5*a - b)*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(15*a^2*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2))/(5*a*f)} - - -{Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ((3*a - 7*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 7*b)*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(a - b)*f) - ((3*a - 7*b)*Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f) + (2*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(5/2))/(3*(a - b)*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(5/2))/(5*(a - b)*f)} -{Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2), x, 6, ((3*a - 5*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 5*b)*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(a - b)*f) - ((3*a - 5*b)*Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f) + (Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(5/2))/(3*(a - b)*f)} -{Sin[e + f*x]^1*(a + b*Tan[e + f*x]^2)^(3/2), x, 5, (3*(a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (3*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f) - (Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/f} -{Csc[e + f*x]^1*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, -((a^(3/2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f) + ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f - (Cot[e + f*x]*Csc[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(2*f)} -{Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2), x, 9, (-3*(a^2 + 6*a*b + b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (3*(a + 3*b)*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*f) - (3*(a + b)*Csc[e + f*x]^2*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*f) - (Cot[e + f*x]*Csc[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(4*f)} - -{Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2), x, 9, (3*(a^2 - 8*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*Sqrt[a - b]*f) + (3*(a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) - (3*(a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) + (3*(a - 2*b)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) - (Cos[e + f*x]*Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(4*f)} -{Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, ((a - 4*b)*Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + ((3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(2*f)} -{Sin[e + f*x]^0*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2), x, 5, (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (3*b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f) - (Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/f} -{Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2), x, 6, (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 2*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f) - ((3*a + 2*b)*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f) - (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/(3*a*f)} -{Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f) - ((3*a + 4*b)*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f) - (2*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/(3*a*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(5/2))/(5*a*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2], x, 4, -((15*a^2 - 10*a*b + 3*b^2)*Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(15*(a - b)^3*f) + (2*(5*a - 3*b)*Cos[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(15*(a - b)^2*f) - (Cos[e + f*x]^5*Sqrt[a - b + b*Sec[e + f*x]^2])/(5*(a - b)*f)} -{Sin[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2], x, 3, -((3*a - b)*Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(3*(a - b)^2*f) + (Cos[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(3*(a - b)*f)} -{Sin[e + f*x]^1/Sqrt[a + b*Tan[e + f*x]^2], x, 2, -((Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/((a - b)*f))} -{Csc[e + f*x]^1/Sqrt[a + b*Tan[e + f*x]^2], x, 3, -(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(Sqrt[a]*f))} -{Csc[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2], x, 5, -((a - b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(3/2)*f) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*a*f)} -{Csc[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2], x, 6, (-3*(a - b)^2*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(5/2)*f) - ((5*a - 3*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*a^2*f) - (Cot[e + f*x]^3*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*a*f)} - -{Sin[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2], x, 6, (3*a^2*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(5/2)*f) - ((5*a - 2*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(4*(a - b)*f)} -{Sin[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2], x, 5, (a*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*(a - b)*f)} -{Sin[e + f*x]^0/Sqrt[a + b*Tan[e + f*x]^2], x, 3, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)} -{Csc[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2], x, 2, -((Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a*f))} -{Csc[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2], x, 3, -((3*a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f)} -{Csc[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2], x, 4, -((15*a^2 - 20*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*f) - (2*(5*a - 2*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a*f)} - - -{Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2), x, 5, -((15*a^2 + 10*a*b - b^2)*Cos[e + f*x])/(15*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2]) + (2*(5*a - 2*b)*Cos[e + f*x]^3)/(15*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - Cos[e + f*x]^5/(5*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (2*b*(15*a^2 + 10*a*b - b^2)*Sec[e + f*x])/(15*(a - b)^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, -((3*a + b)*Cos[e + f*x])/(3*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) + Cos[e + f*x]^3/(3*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (2*b*(3*a + b)*Sec[e + f*x])/(3*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(3/2), x, 3, -(Cos[e + f*x]/((a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])) - (2*b*Sec[e + f*x])/((a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(a^(3/2)*f)) - (b*Sec[e + f*x])/(a*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2), x, 6, -((a - 3*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(5/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (3*b*Sec[e + f*x])/(2*a^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2), x, 7, (-3*(a - 5*b)*(a - b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(7/2)*f) - (5*(a - b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - ((13*a - 15*b)*b*Sec[e + f*x])/(8*a^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])} - -{Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2), x, 7, (3*a*(a + 4*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(7/2)*f) - (5*a*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - (b*(13*a + 2*b)*Tan[e + f*x])/(8*(a - b)^3*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2), x, 6, ((a + 2*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(5/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - (3*b*Tan[e + f*x])/(2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Tan[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2), x, 3, -(Cot[e + f*x]/(a*f*Sqrt[a + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f*x])/(a^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, -((3*a - 4*b)*Cot[e + f*x])/(3*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(3*a*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*(3*a - 4*b)*b*Tan[e + f*x])/(3*a^3*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2), x, 5, -((15*a^2 - 40*a*b + 24*b^2)*Cot[e + f*x])/(15*a^3*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*(5*a - 3*b)*Cot[e + f*x]^3)/(15*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^5/(5*a*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*b*(15*a^2 - 40*a*b + 24*b^2)*Tan[e + f*x])/(15*a^4*f*Sqrt[a + b*Tan[e + f*x]^2])} - - -{Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -((5*a^2 + 10*a*b + b^2)*Cos[e + f*x])/(5*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) + (2*(5*a - b)*Cos[e + f*x]^3)/(15*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 + 10*a*b + b^2)*Sec[e + f*x])/(15*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 + 10*a*b + b^2)*Sec[e + f*x])/(15*(a - b)^5*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2), x, 5, -(((a + b)*Cos[e + f*x])/((a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(a + b)*Sec[e + f*x])/(3*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + b)*Sec[e + f*x])/(3*(a - b)^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(5/2), x, 4, -(Cos[e + f*x]/((a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2))) - (4*b*Sec[e + f*x])/(3*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*Sec[e + f*x])/(3*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(a^(5/2)*f)) - (b*Sec[e + f*x])/(3*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((5*a - 3*b)*b*Sec[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2), x, 7, -((a - 5*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(7/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (5*b*Sec[e + f*x])/(6*a^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((13*a - 15*b)*b*Sec[e + f*x])/(6*a^3*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2), x, 8, -((3*a^2 - 30*a*b + 35*b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(9/2)*f) - ((5*a - 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((23*a - 35*b)*b*Sec[e + f*x])/(24*a^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (5*(11*a - 21*b)*b*Sec[e + f*x])/(24*a^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])} - -{Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2), x, 8, ((3*a^2 + 24*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(9/2)*f) - ((5*a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(23*a + 12*b)*Tan[e + f*x])/(24*(a - b)^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (5*b*(11*a + 10*b)*Tan[e + f*x])/(24*(a - b)^4*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2), x, 7, ((a + 4*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(7/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (5*b*Tan[e + f*x])/(6*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(13*a + 2*b)*Tan[e + f*x])/(6*a*(a - b)^3*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Tan[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2), x, 4, -(Cot[e + f*x]/(a*f*(a + b*Tan[e + f*x]^2)^(3/2))) - (4*b*Tan[e + f*x])/(3*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*b*Tan[e + f*x])/(3*a^3*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2), x, 5, -(((a - 2*b)*Cot[e + f*x])/(a^2*f*(a + b*Tan[e + f*x]^2)^(3/2))) - Cot[e + f*x]^3/(3*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (4*(a - 2*b)*b*Tan[e + f*x])/(3*a^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*(a - 2*b)*b*Tan[e + f*x])/(3*a^4*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -((5*a^2 - 20*a*b + 16*b^2)*Cot[e + f*x])/(5*a^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (2*(5*a - 4*b)*Cot[e + f*x]^3)/(15*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 - 20*a*b + 16*b^2)*Tan[e + f*x])/(15*a^4*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 - 20*a*b + 16*b^2)*Tan[e + f*x])/(15*a^5*f*Sqrt[a + b*Tan[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Sin[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 3, ((Cos[e + f*x]^2)^(1/2 + p)*Hypergeometric2F1[(1/2)*(1 + 2*p), (1/2)*(1 + m + 2*p), (1/2)*(3 + m + 2*p), Sin[e + f*x]^2]*(d*Sin[e + f*x])^m*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p))} - - -{(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(m/2)*(d*Sin[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/a)^p)} - - -{Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p, x, 5, ((10*a - 7*b - 2*b*p)*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(15*(a - b)^2*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(5*(a - b)*f) - ((15*a^2 - 20*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(15*(a - b)^2*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)} -{Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p, x, 4, (Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(3*(a - b)*f) - ((3*a - 2*b*(1 + p))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(3*(a - b)*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)} -{Sin[e + f*x]^1*(a + b*Tan[e + f*x]^2)^p, x, 3, -((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p))} -{Csc[e + f*x]^1*(a + b*Tan[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 1, -p, 3/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/(a - b))]*Sec[e + f*x]*(a - b + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p))} -{Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 2, -p, 5/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/(a - b))]*Sec[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^p)/(3*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)} - -{Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Sin[e + f*x]^0*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p, x, 3, -((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))} -{Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p, x, 4, -(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(1 + p))/(3*a*f) - ((3*a - b*(1 - 2*p))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(3*a*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x, 5, -((10*a - b*(3 - 2*p))*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(1 + p))/(15*a^2*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(1 + p))/(5*a*f) - ((15*a^2 - b*(10*a - b*(3 - 2*p))*(1 - 2*p))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(15*a^2*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} - - -(* ::Section:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Sin[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 3, ((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(1 + m + n*p), (1/2)*(3 + m + n*p), Sin[e + f*x]^2]*(d*Sin[e + f*x])^m*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))} - - -{Sin[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[2, (1/2)*(3 + n*p), (1/2)*(5 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))} -{Sin[e + f*x]^0*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Csc[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 3, -((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p)))} -{Csc[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p))) - (Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p))} -{Csc[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p))) - (2*Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p)) - (Cot[e + f*x]^5*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 - n*p))} - -{Sin[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 3, ((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(4 + n*p), (1/2)*(6 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]^3*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(4 + n*p))} -{Sin[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 3, ((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(2 + n*p), (1/2)*(4 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p))} -{Csc[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 3, ((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*Hypergeometric2F1[(n*p)/2, (1/2)*(1 + n*p), (1/2)*(2 + n*p), Sin[e + f*x]^2]*Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)} -{Csc[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 3, -(((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*Csc[e + f*x]^2*Hypergeometric2F1[(1/2)*(-2 + n*p), (1/2)*(1 + n*p), (n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 - n*p)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p, x, 0, Unintegrable[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Cos[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 3, ((d*Cos[e + f*x])^m*(Cos[e + f*x]^2)^((1/2)*(1 - m + 2*p))*Hypergeometric2F1[(1/2)*(1 + 2*p), (1/2)*(1 - m + 2*p), (1/2)*(3 + 2*p), Sin[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))} - - -{(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 4, (AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/((1 + (b*Tan[e + f*x]^2)/a)^p*f)} - - -(* ::Section:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic*) - - -{(d*Cos[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 3, ((d*Cos[e + f*x])^m*(Cos[e + f*x]^2)^((1/2)*(1 - m + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(1 - m + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Cos[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x, 1, (d*Cos[e + f*x])^m*(Sec[e + f*x]/d)^m*Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p/(Sec[e + f*x]/d)^m, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^p when a-b=0*) - - -{(a + a*Tan[c + d*x]^2)^4, x, 4, (a^4*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/d + (3*a^4*Tan[c + d*x]^5)/(5*d) + (a^4*Tan[c + d*x]^7)/(7*d)} -{(a + a*Tan[c + d*x]^2)^3, x, 4, (a^3*Tan[c + d*x])/d + (2*a^3*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d)} -{(a + a*Tan[c + d*x]^2)^2, x, 4, (a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)} -{1/(a + a*Tan[c + d*x]^2)^1, x, 4, x/(2*a) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{1/(a + a*Tan[c + d*x]^2)^2, x, 5, (3*x)/(8*a^2) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d)} -{1/(a + a*Tan[c + d*x]^2)^3, x, 6, (5*x)/(16*a^3) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2), x, 4, -(((a - b)*Log[Cos[e + f*x]])/f) - ((a - b)*Tan[e + f*x]^2)/(2*f) + ((a - b)*Tan[e + f*x]^4)/(4*f) + (b*Tan[e + f*x]^6)/(6*f)} -{Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2), x, 3, ((a - b)*Log[Cos[e + f*x]])/f + ((a - b)*Tan[e + f*x]^2)/(2*f) + (b*Tan[e + f*x]^4)/(4*f)} -{Tan[e + f*x]^1*(a + b*Tan[e + f*x]^2), x, 2, -(((a - b)*Log[Cos[e + f*x]])/f) + (b*Tan[e + f*x]^2)/(2*f)} -{Cot[e + f*x]^1*(a + b*Tan[e + f*x]^2), x, 3, -((b*Log[Cos[e + f*x]])/f) + (a*Log[Sin[e + f*x]])/f} -{Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2), x, 3, -(a*Cot[e + f*x]^2)/(2*f) - ((a - b)*Log[Sin[e + f*x]])/f} -{Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2), x, 4, ((a - b)*Cot[e + f*x]^2)/(2*f) - (a*Cot[e + f*x]^4)/(4*f) + ((a - b)*Log[Sin[e + f*x]])/f} - -{Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2), x, 5, -((a - b)*x) + ((a - b)*Tan[e + f*x])/f - ((a - b)*Tan[e + f*x]^3)/(3*f) + ((a - b)*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)} -{Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2), x, 4, (a - b)*x - ((a - b)*Tan[e + f*x])/f + ((a - b)*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)} -{Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2), x, 3, -((a - b)*x) + ((a - b)*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)} -{Tan[e + f*x]^0*(a + b*Tan[e + f*x]^2), x, 3, a*x - b*x + (b*Tan[e + f*x])/f} -{Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2), x, 2, -((a - b)*x) - (a*Cot[e + f*x])/f} -{Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2), x, 4, (a - b)*x + ((a - b)*Cot[e + f*x])/f - (a*Cot[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2), x, 5, -((a - b)*x) - ((a - b)*Cot[e + f*x])/f + ((a - b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f)} - - -{Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2, x, 4, -(((a - b)^2*Log[Cos[e + f*x]])/f) - ((a - b)^2*Tan[e + f*x]^2)/(2*f) + ((a - b)^2*Tan[e + f*x]^4)/(4*f) + ((2*a - b)*b*Tan[e + f*x]^6)/(6*f) + (b^2*Tan[e + f*x]^8)/(8*f)} -{Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2, x, 4, ((a - b)^2*Log[Cos[e + f*x]])/f + ((a - b)^2*Tan[e + f*x]^2)/(2*f) + ((2*a - b)*b*Tan[e + f*x]^4)/(4*f) + (b^2*Tan[e + f*x]^6)/(6*f)} -{Tan[e + f*x]^1*(a + b*Tan[e + f*x]^2)^2, x, 4, -(((a - b)^2*Log[Cos[e + f*x]])/f) + ((a - b)*b*Tan[e + f*x]^2)/(2*f) + (a + b*Tan[e + f*x]^2)^2/(4*f)} -{Cot[e + f*x]^1*(a + b*Tan[e + f*x]^2)^2, x, 4, ((a - b)^2*Log[Cos[e + f*x]])/f + (a^2*Log[Tan[e + f*x]])/f + (b^2*Tan[e + f*x]^2)/(2*f)} -{Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2, x, 4, -(a^2*Cot[e + f*x]^2)/(2*f) - ((a - b)^2*Log[Cos[e + f*x]])/f - (a*(a - 2*b)*Log[Tan[e + f*x]])/f} -{Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2, x, 4, (a*(a - 2*b)*Cot[e + f*x]^2)/(2*f) - (a^2*Cot[e + f*x]^4)/(4*f) + ((a - b)^2*Log[Cos[e + f*x]])/f + ((a - b)^2*Log[Tan[e + f*x]])/f} - -{Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2, x, 4, -((a - b)^2*x) + ((a - b)^2*Tan[e + f*x])/f - ((a - b)^2*Tan[e + f*x]^3)/(3*f) + ((a - b)^2*Tan[e + f*x]^5)/(5*f) + ((2*a - b)*b*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)} -{Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2, x, 4, (a - b)^2*x - ((a - b)^2*Tan[e + f*x])/f + ((a - b)^2*Tan[e + f*x]^3)/(3*f) + ((2*a - b)*b*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)} -{Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2, x, 4, -((a - b)^2*x) + ((a - b)^2*Tan[e + f*x])/f + ((2*a - b)*b*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)} -{Tan[e + f*x]^0*(a + b*Tan[e + f*x]^2)^2, x, 4, (a - b)^2*x + ((2*a - b)*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2, x, 4, -((a - b)^2*x) - (a^2*Cot[e + f*x])/f + (b^2*Tan[e + f*x])/f} -{Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2, x, 4, (a - b)^2*x + (a*(a - 2*b)*Cot[e + f*x])/f - (a^2*Cot[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2, x, 4, -((a - b)^2*x) - ((a - b)^2*Cot[e + f*x])/f + (a*(a - 2*b)*Cot[e + f*x]^3)/(3*f) - (a^2*Cot[e + f*x]^5)/(5*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2), x, 4, -(Log[Cos[e + f*x]]/((a - b)*f)) - (a^2*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)*b^2*f) + Tan[e + f*x]^2/(2*b*f)} -{Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2), x, 4, Log[Cos[e + f*x]]/((a - b)*f) + (a*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)*b*f)} -{Tan[e + f*x]^1/(a + b*Tan[e + f*x]^2), x, 5, -Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)*f)} -{Cot[e + f*x]^1/(a + b*Tan[e + f*x]^2), x, 4, Log[Cos[e + f*x]]/((a - b)*f) + Log[Tan[e + f*x]]/(a*f) + (b*Log[a + b*Tan[e + f*x]^2])/(2*a*(a - b)*f)} -{Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2), x, 4, -Cot[e + f*x]^2/(2*a*f) - Log[Cos[e + f*x]]/((a - b)*f) - ((a + b)*Log[Tan[e + f*x]])/(a^2*f) - (b^2*Log[a + b*Tan[e + f*x]^2])/(2*a^2*(a - b)*f)} -{Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2), x, 4, ((a + b)*Cot[e + f*x]^2)/(2*a^2*f) - Cot[e + f*x]^4/(4*a*f) + Log[Cos[e + f*x]]/((a - b)*f) + ((a^2 + a*b + b^2)*Log[Tan[e + f*x]])/(a^3*f) + (b^3*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)*f)} - -{Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2), x, 6, -(x/(a - b)) + (a^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*b^(5/2)*f) - ((a + b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)} -{Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2), x, 5, x/(a - b) - (a^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*b^(3/2)*f) + Tan[e + f*x]/(b*f)} -{Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2), x, 4, -(x/(a - b)) + (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*Sqrt[b]*f)} -{Tan[e + f*x]^0/(a + b*Tan[e + f*x]^2), x, 3, x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*f)} -{Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2), x, 5, -(x/(a - b)) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*(a - b)*f) - Cot[e + f*x]/(a*f)} -{Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2), x, 6, x/(a - b) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*(a - b)*f) + ((a + b)*Cot[e + f*x])/(a^2*f) - Cot[e + f*x]^3/(3*a*f)} -{Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2), x, 7, -(x/(a - b)) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*(a - b)*f) - ((a^2 + a*b + b^2)*Cot[e + f*x])/(a^3*f) + ((a + b)*Cot[e + f*x]^3)/(3*a^2*f) - Cot[e + f*x]^5/(5*a*f)} - - -{Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2, x, 4, -(Log[Cos[e + f*x]]/((a - b)^2*f)) + (a*(a - 2*b)*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)^2*b^2*f) + a^2/(2*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2, x, 4, Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^2*f) - a/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^1/(a + b*Tan[e + f*x]^2)^2, x, 4, -Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^2*f) + 1/(2*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^1/(a + b*Tan[e + f*x]^2)^2, x, 4, Log[Cos[e + f*x]]/((a - b)^2*f) + Log[Tan[e + f*x]]/(a^2*f) + ((2*a - b)*b*Log[a + b*Tan[e + f*x]^2])/(2*a^2*(a - b)^2*f) - b/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2, x, 4, -Cot[e + f*x]^2/(2*a^2*f) - Log[Cos[e + f*x]]/((a - b)^2*f) - ((a + 2*b)*Log[Tan[e + f*x]])/(a^3*f) - ((3*a - 2*b)*b^2*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)^2*f) + b^2/(2*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2, x, 4, ((a + 2*b)*Cot[e + f*x]^2)/(2*a^3*f) - Cot[e + f*x]^4/(4*a^2*f) + Log[Cos[e + f*x]]/((a - b)^2*f) + ((a^2 + 2*a*b + 3*b^2)*Log[Tan[e + f*x]])/(a^4*f) + ((4*a - 3*b)*b^3*Log[a + b*Tan[e + f*x]^2])/(2*a^4*(a - b)^2*f) - b^3/(2*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2))} - -{Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2, x, 6, -(x/(a - b)^2) - (a^(3/2)*(3*a - 5*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^2*b^(5/2)*f) + ((3*a - 2*b)*Tan[e + f*x])/(2*(a - b)*b^2*f) - (a*Tan[e + f*x]^3)/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2, x, 5, x/(a - b)^2 + (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^2*b^(3/2)*f) - (a*Tan[e + f*x])/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2, x, 5, -(x/(a - b)^2) + ((a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*Sqrt[a]*(a - b)^2*Sqrt[b]*f) + Tan[e + f*x]/(2*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^0/(a + b*Tan[e + f*x]^2)^2, x, 5, x/(a - b)^2 - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*f) - (b*Tan[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2, x, 6, -(x/(a - b)^2) + ((5*a - 3*b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(5/2)*(a - b)^2*f) - ((2*a - 3*b)*Cot[e + f*x])/(2*a^2*(a - b)*f) - (b*Cot[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2, x, 7, x/(a - b)^2 - ((7*a - 5*b)*b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^2*f) + ((2*a^2 + 2*a*b - 5*b^2)*Cot[e + f*x])/(2*a^3*(a - b)*f) - ((2*a - 5*b)*Cot[e + f*x]^3)/(6*a^2*(a - b)*f) - (b*Cot[e + f*x]^3)/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2, x, 8, -(x/(a - b)^2) + ((9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*(a - b)^2*f) - ((2*a^3 + 2*a^2*b + 2*a*b^2 - 7*b^3)*Cot[e + f*x])/(2*a^4*(a - b)*f) + ((2*a^2 + 2*a*b - 7*b^2)*Cot[e + f*x]^3)/(6*a^3*(a - b)*f) - ((2*a - 7*b)*Cot[e + f*x]^5)/(10*a^2*(a - b)*f) - (b*Cot[e + f*x]^5)/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))} - - -{Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3, x, 4, -Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) + a^2/(4*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2)^2) - (a*(a - 2*b))/(2*(a - b)^2*b^2*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3, x, 4, Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) - a/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) - 1/(2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^1/(a + b*Tan[e + f*x]^2)^3, x, 4, -Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) + 1/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + 1/(2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^1/(a + b*Tan[e + f*x]^2)^3, x, 4, Log[Cos[e + f*x]]/((a - b)^3*f) + Log[Tan[e + f*x]]/(a^3*f) + (b*(3*a^2 - 3*a*b + b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)^3*f) - b/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((2*a - b)*b)/(2*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3, x, 4, -Cot[e + f*x]^2/(2*a^3*f) - Log[Cos[e + f*x]]/((a - b)^3*f) - ((a + 3*b)*Log[Tan[e + f*x]])/(a^4*f) - (b^2*(6*a^2 - 8*a*b + 3*b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^4*(a - b)^3*f) + b^2/(4*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + ((3*a - 2*b)*b^2)/(2*a^3*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3, x, 4, ((a + 3*b)*Cot[e + f*x]^2)/(2*a^4*f) - Cot[e + f*x]^4/(4*a^3*f) + Log[Cos[e + f*x]]/((a - b)^3*f) + ((a^2 + 3*a*b + 6*b^2)*Log[Tan[e + f*x]])/(a^5*f) + (b^3*(10*a^2 - 15*a*b + 6*b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^5*(a - b)^3*f) - b^3/(4*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((4*a - 3*b)*b^3)/(2*a^4*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} - -{Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3, x, 6, -(x/(a - b)^3) + (Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*(a - b)^3*b^(5/2)*f) - (a*Tan[e + f*x]^3)/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) - (a*(3*a - 7*b)*Tan[e + f*x])/(8*(a - b)^2*b^2*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3, x, 6, x/(a - b)^3 + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^3*b^(3/2)*f) - (a*Tan[e + f*x])/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) + ((a - 5*b)*Tan[e + f*x])/(8*(a - b)^2*b*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3, x, 6, -(x/(a - b)^3) + ((3*a^2 + 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^3*Sqrt[b]*f) + Tan[e + f*x]/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + ((3*a + b)*Tan[e + f*x])/(8*a*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^0/(a + b*Tan[e + f*x]^2)^3, x, 6, x/(a - b)^3 - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*f) - (b*Tan[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 3*b)*b*Tan[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3, x, 7, -(x/(a - b)^3) + (b^(3/2)*(35*a^2 - 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^3*f) - ((8*a^2 - 27*a*b + 15*b^2)*Cot[e + f*x])/(8*a^3*(a - b)^2*f) - (b*Cot[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((9*a - 5*b)*b*Cot[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3, x, 8, x/(a - b)^3 - (b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^3*f) + ((8*a^3 + 8*a^2*b - 55*a*b^2 + 35*b^3)*Cot[e + f*x])/(8*a^4*(a - b)^2*f) - ((8*a^2 - 55*a*b + 35*b^2)*Cot[e + f*x]^3)/(24*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^3)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((11*a - 7*b)*b*Cot[e + f*x]^3)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3, x, 9, -(x/(a - b)^3) + (b^(7/2)*(99*a^2 - 154*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(11/2)*(a - b)^3*f) - ((8*a^4 + 8*a^3*b + 8*a^2*b^2 - 91*a*b^3 + 63*b^4)*Cot[e + f*x])/(8*a^5*(a - b)^2*f) + ((8*a^3 + 8*a^2*b - 91*a*b^2 + 63*b^3)*Cot[e + f*x]^3)/(24*a^4*(a - b)^2*f) - ((8*a^2 - 91*a*b + 63*b^2)*Cot[e + f*x]^5)/(40*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^5)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((13*a - 9*b)*b*Cot[e + f*x]^5)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))} - - -{(a + b*Tan[c + d*x]^2)^4, x, 4, ((a - b)^4*d*x)/d + ((2*a - b)*b*(2*a^2 - 2*a*b + b^2)*Tan[c + d*x])/d + (b^2*(6*a^2 - 4*a*b + b^2)*Tan[c + d*x]^3)/(3*d) + ((4*a - b)*b^3*Tan[c + d*x]^5)/(5*d) + (b^4*Tan[c + d*x]^7)/(7*d)} -{(a + b*Tan[c + d*x]^2)^3, x, 4, ((a - b)^3*d*x)/d + (b*(3*a^2 - 3*a*b + b^2)*Tan[c + d*x])/d + ((3*a - b)*b^2*Tan[c + d*x]^3)/(3*d) + (b^3*Tan[c + d*x]^5)/(5*d)} -{(a + b*Tan[c + d*x]^2)^2, x, 4, ((a - b)^2*d*x)/d + ((2*a - b)*b*Tan[c + d*x])/d + (b^2*Tan[c + d*x]^3)/(3*d)} -{(a + b*Tan[c + d*x]^2)^1, x, 3, a*x - b*x + (b*Tan[c + d*x])/d} -{1/(a + b*Tan[c + d*x]^2)^1, x, 3, (d*x)/((a - b)*d) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)} -{1/(a + b*Tan[c + d*x]^2)^2, x, 5, (d*x)/((a - b)^2*d) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) - (b*Tan[c + d*x])/(2*a*(a - b)*d*(a + b*Tan[c + d*x]^2))} -{1/(a + b*Tan[c + d*x]^2)^3, x, 6, (d*x)/((a - b)^3*d) - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*d) - (b*Tan[c + d*x])/(4*a*(a - b)*d*(a + b*Tan[c + d*x]^2)^2) - ((7*a - 3*b)*b*Tan[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Tan[c + d*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^(p/2) when a-b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]^4*Sqrt[a + a*Tan[x]^2], x, 5, (3/8)*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] - (3/8)*Sqrt[a*Sec[x]^2]*Tan[x] + (1/4)*Sqrt[a*Sec[x]^2]*Tan[x]^3} -{Tan[x]^3*Sqrt[a + a*Tan[x]^2], x, 4, -Sqrt[a*Sec[x]^2] + (a*Sec[x]^2)^(3/2)/(3*a)} -{Tan[x]^2*Sqrt[a + a*Tan[x]^2], x, 4, (-(1/2))*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] + (1/2)*Sqrt[a*Sec[x]^2]*Tan[x]} -{Tan[x]^1*Sqrt[a + a*Tan[x]^2], x, 3, Sqrt[a*Sec[x]^2]} -{Cot[x]^1*Sqrt[a + a*Tan[x]^2], x, 4, (-Sqrt[a])*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]} -{Cot[x]^2*Sqrt[a + a*Tan[x]^2], x, 4, (-Cot[x])*Sqrt[a*Sec[x]^2]} -{Cot[x]^3*Sqrt[a + a*Tan[x]^2], x, 5, (1/2)*Sqrt[a]*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]] - (1/2)*Cot[x]^2*Sqrt[a*Sec[x]^2]} -{Cot[x]^4*Sqrt[a + a*Tan[x]^2], x, 4, Cot[x]*Sqrt[a*Sec[x]^2] - (1/3)*Cot[x]*Csc[x]^2*Sqrt[a*Sec[x]^2]} - - -{(a + a*Tan[c + d*x]^2)^(1/2), x, 4, (Sqrt[a]*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/d} - - -{Tan[x]^3*(a + a*Tan[x]^2)^(3/2), x, 4, (-(1/3))*(a*Sec[x]^2)^(3/2) + (a*Sec[x]^2)^(5/2)/(5*a)} -{Tan[x]^2*(a + a*Tan[x]^2)^(3/2), x, 5, (-(1/8))*a*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] - (1/8)*a*Sqrt[a*Sec[x]^2]*Tan[x] + (1/4)*a*Sec[x]^2*Sqrt[a*Sec[x]^2]*Tan[x]} -{Tan[x]^1*(a + a*Tan[x]^2)^(3/2), x, 3, (1/3)*(a*Sec[x]^2)^(3/2)} -{Cot[x]^1*(a + a*Tan[x]^2)^(3/2), x, 5, (-a^(3/2))*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]] + a*Sqrt[a*Sec[x]^2]} -{Cot[x]^2*(a + a*Tan[x]^2)^(3/2), x, 5, a*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] - a*Cot[x]*Sqrt[a*Sec[x]^2]} - - -{(a + a*Tan[c + d*x]^2)^(3/2), x, 5, (a^(3/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(2*d) + (a*Sqrt[a*Sec[c + d*x]^2]*Tan[c + d*x])/(2*d)} - - -{(a + a*Tan[c + d*x]^2)^(5/2), x, 6, (3*a^(5/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(8*d) + (3*a^2*Sqrt[a*Sec[c + d*x]^2]*Tan[c + d*x])/(8*d) + (a*(a*Sec[c + d*x]^2)^(3/2)*Tan[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]^3/Sqrt[a + a*Tan[x]^2], x, 4, 1/Sqrt[a*Sec[x]^2] + Sqrt[a*Sec[x]^2]/a} -{Tan[x]^2/Sqrt[a + a*Tan[x]^2], x, 5, (ArcTanh[Sin[x]]*Sec[x])/Sqrt[a*Sec[x]^2] - Tan[x]/Sqrt[a*Sec[x]^2]} -{Tan[x]^1/Sqrt[a + a*Tan[x]^2], x, 3, -(1/Sqrt[a*Sec[x]^2])} -{Cot[x]^1/Sqrt[a + a*Tan[x]^2], x, 5, -(ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]/Sqrt[a]) + 1/Sqrt[a*Sec[x]^2]} -{Cot[x]^2/Sqrt[a + a*Tan[x]^2], x, 5, -((Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2]) - Tan[x]/Sqrt[a*Sec[x]^2]} - - -{Tan[x]^3/(a + a*Tan[x]^2)^(3/2), x, 4, 1/(3*(a*Sec[x]^2)^(3/2)) - 1/(a*Sqrt[a*Sec[x]^2])} -{Tan[x]^2/(a + a*Tan[x]^2)^(3/2), x, 4, (Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])} -{Tan[x]^1/(a + a*Tan[x]^2)^(3/2), x, 3, -(1/(3*(a*Sec[x]^2)^(3/2)))} -{Cot[x]^1/(a + a*Tan[x]^2)^(3/2), x, 6, -(ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]/a^(3/2)) + 1/(3*(a*Sec[x]^2)^(3/2)) + 1/(a*Sqrt[a*Sec[x]^2])} -{Cot[x]^2/(a + a*Tan[x]^2)^(3/2), x, 5, -((Csc[x]*Sec[x])/(a*Sqrt[a*Sec[x]^2])) - (2*Tan[x])/(a*Sqrt[a*Sec[x]^2]) + (Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])} - - -{1/(a + a*Tan[c + d*x]^2)^(1/2), x, 3, Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])} -{1/(a + a*Tan[c + d*x]^2)^(3/2), x, 4, Tan[c + d*x]/(3*d*(a*Sec[c + d*x]^2)^(3/2)) + (2*Tan[c + d*x])/(3*a*d*Sqrt[a*Sec[c + d*x]^2])} -{1/(a + a*Tan[c + d*x]^2)^(5/2), x, 5, Tan[c + d*x]/(5*d*(a*Sec[c + d*x]^2)^(5/2)) + (4*Tan[c + d*x])/(15*a*d*(a*Sec[c + d*x]^2)^(3/2)) + (8*Tan[c + d*x])/(15*a^2*d*Sqrt[a*Sec[c + d*x]^2])} -{1/(a + a*Tan[c + d*x]^2)^(7/2), x, 6, Tan[c + d*x]/(7*d*(a*Sec[c + d*x]^2)^(7/2)) + (6*Tan[c + d*x])/(35*a*d*(a*Sec[c + d*x]^2)^(5/2)) + (8*Tan[c + d*x])/(35*a^2*d*(a*Sec[c + d*x]^2)^(3/2)) + (16*Tan[c + d*x])/(35*a^3*d*Sqrt[a*Sec[c + d*x]^2])} - - -{(1 + Tan[x]^2)^(3/2), x, 4, (1/2)*ArcSinh[Tan[x]] + (1/2)*Sqrt[Sec[x]^2]*Tan[x]} -{Sqrt[1 + Tan[x]^2], x, 3, ArcSinh[Tan[x]]} -{1/Sqrt[1 + Tan[x]^2], x, 3, Tan[x]/Sqrt[Sec[x]^2]} - - -{(-1 - Tan[x]^2)^(3/2), x, 5, (1/2)*ArcTan[Tan[x]/Sqrt[-Sec[x]^2]] - (1/2)*Sqrt[-Sec[x]^2]*Tan[x]} -{Sqrt[-1 - Tan[x]^2], x, 4, -ArcTan[Tan[x]/Sqrt[-Sec[x]^2]]} -{1/Sqrt[-1 - Tan[x]^2], x, 3, Tan[x]/Sqrt[-Sec[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2], x, 7, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + Sqrt[a + b*Tan[e + f*x]^2]/f - ((a + b)*(a + b*Tan[e + f*x]^2)^(3/2))/(3*b^2*f) + (a + b*Tan[e + f*x]^2)^(5/2)/(5*b^2*f)} -{Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - Sqrt[a + b*Tan[e + f*x]^2]/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*b*f)} -{Tan[e + f*x]^1*Sqrt[a + b*Tan[e + f*x]^2], x, 5, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + Sqrt[a + b*Tan[e + f*x]^2]/f} -{Cot[e + f*x]^1*Sqrt[a + b*Tan[e + f*x]^2], x, 7, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/f) + (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f} -{Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2], x, 8, ((2*a - b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*Sqrt[a]*f) - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - (Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2], x, 9, -((8*a^2 - 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(3/2)*f) + (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + ((4*a - b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)} - -{Tan[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2], x, 9, -((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) - ((a - 2*b)*(a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(16*b^2*f) + ((a - 6*b)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(24*b*f) + (Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(6*f)} -{Tan[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2], x, 8, (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((a^2 + 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)} -{Tan[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2], x, 7, -((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((a - 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Tan[e + f*x]^0*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f} -{Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2], x, 5, -((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f} -{Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2], x, 6, (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*f)} -{Cot[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2], x, 7, -((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - ((15*a^2 - 5*a*b - 2*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) + ((5*a - b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*f)} - - -{Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*f) - ((a + b)*(a + b*Tan[e + f*x]^2)^(5/2))/(5*b^2*f) + (a + b*Tan[e + f*x]^2)^(7/2)/(7*b^2*f)} -{Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f - (a + b*Tan[e + f*x]^2)^(3/2)/(3*f) + (a + b*Tan[e + f*x]^2)^(5/2)/(5*b*f)} -{Tan[e + f*x]^1*(a + b*Tan[e + f*x]^2)^(3/2), x, 6, -(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*f)} -{Cot[e + f*x]^1*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -((a^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/f) + ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + (b*Sqrt[a + b*Tan[e + f*x]^2])/f} -{Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, (Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*f) - ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - (a*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2), x, 9, -((8*a^2 - 12*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + ((4*a - 5*b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) - (a*Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)} - -{Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2), x, 10, -(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) - ((3*a^3 + 8*a^2*b - 80*a*b^2 + 64*b^3)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 56*a*b + 48*b^2)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(192*b*f) + ((9*a - 8*b)*Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(48*f) + (b*Tan[e + f*x]^7*Sqrt[a + b*Tan[e + f*x]^2])/(8*f)} -{Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2), x, 9, ((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) + ((a^2 - 10*a*b + 8*b^2)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(16*b*f) + ((7*a - 6*b)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(24*f) + (b*Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(6*f)} -{Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + ((5*a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) + (b*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)} -{Tan[e + f*x]^0*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, -(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (a*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f} -{Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2), x, 6, ((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 4*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*f) - (a*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*f)} -{Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2), x, 7, -(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - ((15*a^2 - 20*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a*f) + ((5*a - 6*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*f) - (a*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*f)} - - -{(a + b*Tan[c + d*x]^2)^(5/2), x, 8, ((a - b)^(5/2)*ArcTan[(Sqrt[a - b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^2]])/d + (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^2]])/(8*d) + ((7*a - 4*b)*b*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^2])/(8*d) + (b*Tan[c + d*x]*(a + b*Tan[c + d*x]^2)^(3/2))/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2], x, 6, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f)) - ((a + b)*Sqrt[a + b*Tan[e + f*x]^2])/(b^2*f) + (a + b*Tan[e + f*x]^2)^(3/2)/(3*b^2*f)} -{Tan[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2], x, 5, ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) + Sqrt[a + b*Tan[e + f*x]^2]/(b*f)} -{Tan[e + f*x]^1/Sqrt[a + b*Tan[e + f*x]^2], x, 4, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f))} -{Cot[e + f*x]^1/Sqrt[a + b*Tan[e + f*x]^2], x, 7, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f)} -{Cot[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2], x, 8, ((2*a + b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(3/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) - (Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f)} -{Cot[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2], x, 9, -((8*a^2 + 4*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) + ((4*a + 3*b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*a^2*f) - (Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*a*f)} - -{Tan[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2], x, 8, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) + ((3*a^2 + 4*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - ((3*a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*b^2*f) + (Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*b*f)} -{Tan[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2], x, 7, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f) - ((a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*b*f)} -{Tan[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2], x, 6, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)} -{Tan[e + f*x]^0/Sqrt[a + b*Tan[e + f*x]^2], x, 3, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)} -{Cot[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2], x, 5, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a*f)} -{Cot[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2], x, 6, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f) + ((3*a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f)} -{Cot[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2], x, 7, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) - ((15*a^2 + 10*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*f) + ((5*a + 4*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a*f)} - - -{Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2), x, 6, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f)) + a^2/((a - b)*b^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + Sqrt[a + b*Tan[e + f*x]^2]/(b^2*f)} -{Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - a/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(3/2), x, 5, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f)) + 1/((a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - b/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2), x, 9, ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(5/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - ((a - 3*b)*b)/(2*a^2*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^2/(2*a*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2), x, 10, -((8*a^2 + 12*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(7/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) + (b*(4*a^2 + 3*a*b - 15*b^2))/(8*a^3*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((4*a + 5*b)*Cot[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^4/(4*a*f*Sqrt[a + b*Tan[e + f*x]^2])} - -{Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - ((3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - (a*Tan[e + f*x]^3)/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((3*a - b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*(a - b)*b^2*f)} -{Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - (a*Tan[e + f*x])/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) + Tan[e + f*x]/((a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^0/(a + b*Tan[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Tan[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2), x, 6, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - (b*Cot[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a^2*(a - b)*f)} -{Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2), x, 7, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Cot[e + f*x]^3)/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((3*a - 4*b)*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)*f) - ((a - 4*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*(a - b)*f)} -{Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2), x, 8, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - (b*Cot[e + f*x]^5)/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((15*a^3 + 10*a^2*b + 8*a*b^2 - 48*b^3)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^4*(a - b)*f) + ((5*a^2 + 4*a*b - 24*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*(a - b)*f) - ((a - 6*b)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a^2*(a - b)*f)} - - -{Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f)) + a^2/(3*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (a*(a - 2*b))/((a - b)^2*b^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - a/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) - 1/((a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f)) + 1/(3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + 1/((a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Tan[e + f*x]^2)^(5/2), x, 9, -(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - b/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((2*a - b)*b)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2), x, 10, ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(7/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - ((3*a - 5*b)*b)/(6*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^2/(2*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(a^2 - 8*a*b + 5*b^2))/(2*a^3*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2), x, 11, -((8*a^2 + 20*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(9/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) + (b*(12*a^2 + 15*a*b - 35*b^2))/(24*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((4*a + 7*b)*Cot[e + f*x]^2)/(8*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^4/(4*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) + (b*(4*a^3 + 3*a^2*b - 50*a*b^2 + 35*b^3))/(8*a^4*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} - -{Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2), x, 8, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - (a*Tan[e + f*x]^3)/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (a*(a - 2*b)*Tan[e + f*x])/((a - b)^2*b^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (a*Tan[e + f*x])/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((a - 4*b)*Tan[e + f*x])/(3*(a - b)^2*b*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) + Tan[e + f*x]/(3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((2*a + b)*Tan[e + f*x])/(3*a*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^0/(a + b*Tan[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Tan[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2), x, 7, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) - (b*Cot[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((7*a - 4*b)*b*Cot[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((a - 4*b)*(3*a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)} -{Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2), x, 8, ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*b*Cot[e + f*x]^3)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((a - 2*b)*(3*a^2 + 8*a*b - 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^4*(a - b)^2*f) - ((a^2 - 12*a*b + 8*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)} -{Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2), x, 9, -(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) - (b*Cot[e + f*x]^5)/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((11*a - 8*b)*b*Cot[e + f*x]^5)/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((15*a^4 + 10*a^3*b + 8*a^2*b^2 - 176*a*b^3 + 128*b^4)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^5*(a - b)^2*f) + ((5*a^3 + 4*a^2*b - 88*a*b^2 + 64*b^3)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^4*(a - b)^2*f) - ((a^2 - 22*a*b + 16*b^2)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a^3*(a - b)^2*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 4, (Hypergeometric2F1[1, (1/2)*(1 + m + 2*p), (1/2)*(3 + m + 2*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p))} - - -{(d*Tan[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[(1 + m)/2, 1, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Tan[e + f*x])^(1 + m)*(a + b*Tan[e + f*x]^2)^p)/(d*f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/a)^p)} - - -{Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p, x, 5, -((a + b)*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*b^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) + (a + b*Tan[e + f*x]^2)^(2 + p)/(2*b^2*f*(2 + p))} -{Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p, x, 4, (a + b*Tan[e + f*x]^2)^(1 + p)/(2*b*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p))} -{Tan[e + f*x]^1*(a + b*Tan[e + f*x]^2)^p, x, 3, -(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p))} -{Cot[e + f*x]^1*(a + b*Tan[e + f*x]^2)^p, x, 5, (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))} -{Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p, x, 6, -(Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a*f) - (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) + ((a - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a^2*f*(1 + p))} -{Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p, x, 7, ((2*a + b - b*p)*Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a^2*f) - (Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a*f) + (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) - ((2*a^2 - 2*a*b*p - b^2*(1 - p)*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a^3*f*(1 + p))} - -{Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[7/2, 1, -p, 9/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^7*(a + b*Tan[e + f*x]^2)^p)/(7*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[5/2, 1, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 1, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Tan[e + f*x]^0*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p, x, 3, -((AppellF1[-1/2, 1, -p, 1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))} -{Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p, x, 3, -(AppellF1[-3/2, 1, -p, -1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} -{Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p, x, 3, -(AppellF1[-5/2, 1, -p, -3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/a)^p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x]^3)^p*) - - -{(a + b*Tan[c + d*x]^3)^4, x, 6, (a^4 - 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x])/d + (2*a*b*(a^2 - b^2)*Tan[c + d*x]^2)/d - (b^2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(3*d) + (a*b^3*Tan[c + d*x]^4)/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x]^5)/(5*d) - (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d) + (a*b^3*Tan[c + d*x]^8)/(2*d) - (b^4*Tan[c + d*x]^9)/(9*d) + (b^4*Tan[c + d*x]^11)/(11*d)} -{(a + b*Tan[c + d*x]^3)^3, x, 6, a*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*Tan[c + d*x]^2)/(2*d) - (a*b^2*Tan[c + d*x]^3)/d + (b^3*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) - (b^3*Tan[c + d*x]^6)/(6*d) + (b^3*Tan[c + d*x]^8)/(8*d)} -{(a + b*Tan[c + d*x]^3)^2, x, 6, (a^2 - b^2)*x + (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d - (b^2*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^5)/(5*d)} -{(a + b*Tan[c + d*x]^3)^1, x, 3, a*x + (b*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x]^2)/(2*d)} -{1/(a + b*Tan[c + d*x]^3)^1, x, 14, (a*x)/(a^2 + b^2) + (b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 + b^2)*d) - (b*Log[a*Cos[c + d*x]^3 + b*Sin[c + d*x]^3])/(3*(a^2 + b^2)*d) + (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(2/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(6*a^(2/3)*(a^2 + b^2)*d)} -{1/(a + b*Tan[c + d*x]^3)^2, x, 21, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 + b^2)*d) - (2*a*b*Log[a*Cos[c + d*x]^3 + b*Sin[c + d*x]^3])/(3*(a^2 + b^2)^2*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9*a^(5/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(6*a^(1/3)*(a^2 + b^2)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(18*a^(5/3)*(a^2 + b^2)*d) + (b*(a + Tan[c + d*x]*(b - a*Tan[c + d*x])))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3))} - - -{1/(1 + Tan[x]^3), x, 7, x/2 - (1/2)*Log[Cos[x]] + (1/6)*Log[1 + Tan[x]] - (1/3)*Log[1 - Tan[x] + Tan[x]^2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x]^4)^p*) - - -{(a + b*Tan[c + d*x]^4)^4, x, 4, (a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^7)/(7*d) - (b^3*(4*a + b)*Tan[c + d*x]^9)/(9*d) + (b^3*(4*a + b)*Tan[c + d*x]^11)/(11*d) - (b^4*Tan[c + d*x]^13)/(13*d) + (b^4*Tan[c + d*x]^15)/(15*d)} -{(a + b*Tan[c + d*x]^4)^3, x, 4, (a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*(3*a + b)*Tan[c + d*x]^7)/(7*d) - (b^3*Tan[c + d*x]^9)/(9*d) + (b^3*Tan[c + d*x]^11)/(11*d)} -{(a + b*Tan[c + d*x]^4)^2, x, 4, (a + b)^2*x - (b*(2*a + b)*Tan[c + d*x])/d + (b*(2*a + b)*Tan[c + d*x]^3)/(3*d) - (b^2*Tan[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^7)/(7*d)} -{(a + b*Tan[c + d*x]^4)^1, x, 4, a*x + b*x - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)} -{1/(a + b*Tan[c + d*x]^4)^1, x, 13, x/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d)} -{1/(a + b*Tan[c + d*x]^4)^2, x, 23, x/(a + b)^2 + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)^2*d) + ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^2*d) + ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + (b*Tan[c + d*x]*(1 - Tan[c + d*x]^2))/(4*a*(a + b)*d*(a + b*Tan[c + d*x]^4))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tan[e+f x]^4)^(p/2)*) - - -{(a + b*Tan[c + d*x]^4)^(1/2), x, 8, (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^4]])/(2*d) + (Sqrt[b]*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^4])/(d*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)) - (a^(1/4)*b^(1/4)*EllipticE[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4]) - (b^(1/4)*(a + b)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] + Sqrt[b])*(a + b)*EllipticPi[-((Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b])), 2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4])} -{1/(a + b*Tan[c + d*x]^4)^(1/2), x, 4, ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^4]]/(2*Sqrt[a + b]*d) - (b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] + Sqrt[b])*EllipticPi[-((Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b])), 2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[x]^3*Sqrt[a + b*Tan[x]^4], x, 8, ((a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/(4*Sqrt[b]) + (1/2)*Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - (1/4)*(2 - Tan[x]^2)*Sqrt[a + b*Tan[x]^4]} -{Tan[x]^1*Sqrt[a + b*Tan[x]^4], x, 8, (-(1/2))*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] - (1/2)*Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + (1/2)*Sqrt[a + b*Tan[x]^4]} -{Cot[x]^1*Sqrt[a + b*Tan[x]^4], x, 11, (1/2)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + (1/2)*Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - (1/2)*Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]} - -{Tan[x]^2*Sqrt[a + b*Tan[x]^4], x, 12, (-(1/2))*Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a + b*Tan[x]^4]] + (1/3)*Tan[x]*Sqrt[a + b*Tan[x]^4] - (Sqrt[b]*Tan[x]*Sqrt[a + b*Tan[x]^4])/(Sqrt[a] + Sqrt[b]*Tan[x]^2) + (a^(1/4)*b^(1/4)*EllipticE[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/Sqrt[a + b*Tan[x]^4] + (a^(3/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(3*b^(1/4)*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*a^(1/4)*Sqrt[a + b*Tan[x]^4]) + (b^(1/4)*(a + b)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] + Sqrt[b])*(a + b)*EllipticPi[-((Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b])), 2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4])} - - -{Tan[x]^3*(a + b*Tan[x]^4)^(3/2), x, 9, ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/(16*Sqrt[b]) + (1/2)*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - (1/16)*(8*(a + b) - (3*a + 4*b)*Tan[x]^2)*Sqrt[a + b*Tan[x]^4] - (1/24)*(4 - 3*Tan[x]^2)*(a + b*Tan[x]^4)^(3/2)} -{Tan[x]^1*(a + b*Tan[x]^4)^(3/2), x, 9, (-(1/4))*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] - (1/2)*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + (1/4)*(2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4] + (1/6)*(a + b*Tan[x]^4)^(3/2)} -{Cot[x]^1*(a + b*Tan[x]^4)^(3/2), x, 13, (1/4)*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + (1/2)*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - (1/2)*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]] + (1/2)*a*Sqrt[a + b*Tan[x]^4] - (1/4)*(2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[x]^3/Sqrt[a + b*Tan[x]^4], x, 7, ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]/(2*Sqrt[b]) + ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b])} -{Tan[x]^1/Sqrt[a + b*Tan[x]^4], x, 4, -(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b]))} -{Cot[x]^1/Sqrt[a + b*Tan[x]^4], x, 9, ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b]) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*Sqrt[a])} - -{Tan[x]^2/Sqrt[a + b*Tan[x]^4], x, 4, -(ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a + b*Tan[x]^4]]/(2*Sqrt[a + b])) + (a^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] + Sqrt[b])*EllipticPi[-((Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b])), 2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4])} - - -{Tan[x]^3/(a + b*Tan[x]^4)^(3/2), x, 6, ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2)) - (1 - Tan[x]^2)/(2*(a + b)*Sqrt[a + b*Tan[x]^4])} -{Tan[x]^1/(a + b*Tan[x]^4)^(3/2), x, 6, -(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2))) + (a + b*Tan[x]^2)/(2*a*(a + b)*Sqrt[a + b*Tan[x]^4])} -{Cot[x]^1/(a + b*Tan[x]^4)^(3/2), x, 12, ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2)) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*a^(3/2)) + 1/(2*a*Sqrt[a + b*Tan[x]^4]) - (a + b*Tan[x]^2)/(2*a*(a + b)*Sqrt[a + b*Tan[x]^4])} - - -{Tan[x]^3/(a + b*Tan[x]^4)^(5/2), x, 7, ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2)) - (1 - Tan[x]^2)/(6*(a + b)*(a + b*Tan[x]^4)^(3/2)) - (3*a + (-2*a + b)*Tan[x]^2)/(6*a*(a + b)^2*Sqrt[a + b*Tan[x]^4])} -{Tan[x]^1/(a + b*Tan[x]^4)^(5/2), x, 7, -(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2))) + (a + b*Tan[x]^2)/(6*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Tan[x]^4])} -{Cot[x]^1/(a + b*Tan[x]^4)^(5/2), x, 14, ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2)) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*a^(5/2)) + 1/(6*a*(a + b*Tan[x]^4)^(3/2)) - (a + b*Tan[x]^2)/(6*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + 1/(2*a^2*Sqrt[a + b*Tan[x]^4]) - (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Tan[x]^4])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Tan[e+f x]^(n/2))^p*) - - -{(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]])^2, x, 9, ((a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*f*(1 + m)) + ((a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*f*(1 + m)) + (4*a*b*Hypergeometric2F1[1, (1/4)*(3 + 2*m), (1/4)*(7 + 2*m), -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*f*(3 + 2*m))} -{(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]])^1, x, 7, (a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(f*(1 + m)) + (2*b*Hypergeometric2F1[1, (1/4)*(3 + 2*m), (1/4)*(7 + 2*m), -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*f*(3 + 2*m))} -{(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]])^1, x, 14, (a*(a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)*f*(1 + m)) + (a*(a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)*f*(1 + m)) + (b^4*c^2*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(a*(a^4 + b^4*c^2)*f*(1 + m)) - (b*(a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), -((c*Tan[e + f*x])/Sqrt[-c^2])]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)*f*(3 + 2*m)) - (b*(a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (c*Tan[e + f*x])/Sqrt[-c^2]]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)*f*(3 + 2*m))} -{(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]])^2, x, 15, ((a^6 - 3*a^2*b^4*c^2 - 3*a^4*b^2*Sqrt[-c^2] - b^6*(-c^2)^(3/2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)^2*f*(1 + m)) + ((a^6 - 3*a^2*b^4*c^2 + 3*a^4*b^2*Sqrt[-c^2] + b^6*(-c^2)^(3/2))*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)^2*f*(1 + m)) + (4*a^2*b^4*c^2*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/((a^4 + b^4*c^2)^2*f*(1 + m)) + (b^4*c^2*Hypergeometric2F1[2, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(a^2*(a^4 + b^4*c^2)*f*(1 + m)) - (2*a*b*(a^4 - b^4*c^2 - 2*a^2*b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), -((c*Tan[e + f*x])/Sqrt[-c^2])]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)^2*f*(3 + 2*m)) - (2*a*b*(a^4 - b^4*c^2 + 2*a^2*b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (1/2)*(3 + 2*m), (1/2)*(5 + 2*m), (c*Tan[e + f*x])/Sqrt[-c^2]]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)^2*f*(3 + 2*m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Hypergeometric2F1[1, (1/2)*(1 + m + n*p), (1/2)*(3 + m + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))} - - -{Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Hypergeometric2F1[1, (1/2)*(3 + n*p), (1/2)*(5 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))} -{Tan[e + f*x]^0*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Cot[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]*Hypergeometric2F1[1, (1/2)*(-1 + n*p), (1/2)*(1 + n*p), -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p)))} -{Cot[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]^3*Hypergeometric2F1[1, (1/2)*(-3 + n*p), (1/2)*(-1 + n*p), -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p)))} -{Cot[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]^5*Hypergeometric2F1[1, (1/2)*(-5 + n*p), (1/2)*(-3 + n*p), -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 - n*p)))} - -{Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Hypergeometric2F1[1, (1/2)*(4 + n*p), (1/2)*(6 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p)/(f*(4 + n*p))} -{Tan[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Hypergeometric2F1[1, (1/2)*(2 + n*p), (1/2)*(4 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p))} -{Cot[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Hypergeometric2F1[1, (n*p)/2, 1 + (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)} -{Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 4, -((Cot[e + f*x]^2*Hypergeometric2F1[1, (1/2)*(-2 + n*p), (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 - n*p)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Cot[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 4, ((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1/2)*(1 - m + 2*p), (1/2)*(3 - m + 2*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 - m + 2*p))} - - -{(d*Cot[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 4, (AppellF1[(1 - m)/2, 1, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cot[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/((1 + (b*Tan[e + f*x]^2)/a)^p*(f*(1 - m)))} - - -(* ::Section:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic*) - - -{(d*Cot[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 4, ((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1/2)*(1 - m + n*p), (1/2)*(3 - m + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x, 1, (d*Cot[e + f*x])^m*(Tan[e + f*x]/d)^m*Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2), x, 4, ((4*a - b)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a - b)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x]^2), x, 3, ((2*a - b)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x]^2), x, 3, (b*ArcTanh[Sin[c + d*x]])/d + ((a - b)*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2), x, 2, (a*Sin[c + d*x])/d - ((a - b)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2), x, 3, (a*Sin[c + d*x])/d - ((2*a - b)*Sin[c + d*x]^3)/(3*d) + ((a - b)*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2), x, 3, (a*Sin[c + d*x])/d - ((3*a - b)*Sin[c + d*x]^3)/(3*d) + ((3*a - 2*b)*Sin[c + d*x]^5)/(5*d) - ((a - b)*Sin[c + d*x]^7)/(7*d)} - -{Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2), x, 3, (a*Tan[c + d*x])/d + ((2*a + b)*Tan[c + d*x]^3)/(3*d) + ((a + 2*b)*Tan[c + d*x]^5)/(5*d) + (b*Tan[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2), x, 3, (a*Tan[c + d*x])/d + ((a + b)*Tan[c + d*x]^3)/(3*d) + (b*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2), x, 2, (a*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2), x, 3, (1/2)*(a + b)*x + ((a - b)*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2), x, 4, (1/8)*(3*a + b)*x + ((3*a + b)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a - b)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2), x, 5, (1/16)*(5*a + b)*x + ((5*a + b)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*a + b)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} - - -{Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2, x, 5, ((8*a^2 - 4*a*b + b^2)*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^2 - 4*a*b + b^2)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((8*a - 3*b)*b*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b*Sec[c + d*x]^5*(a - (a - b)*Sin[c + d*x]^2)*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^1*(a + b*Tan[c + d*x]^2)^2, x, 4, ((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (3*(2*a - b)*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*(a - (a - b)*Sin[c + d*x]^2)*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(a + b*Tan[c + d*x]^2)^2, x, 5, ((4*a - 3*b)*b*ArcTanh[Sin[c + d*x]])/(2*d) + ((a - b)^2*Sin[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2, x, 4, (b^2*ArcTanh[Sin[c + d*x]])/d + ((a^2 - b^2)*Sin[c + d*x])/d - ((a - b)^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Sin[c + d*x])/d - (2*a*(a - b)*Sin[c + d*x]^3)/(3*d) + ((a - b)^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Sin[c + d*x])/d - (a*(3*a - 2*b)*Sin[c + d*x]^3)/(3*d) + ((a - b)*(3*a - b)*Sin[c + d*x]^5)/(5*d) - ((a - b)^2*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^9*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Sin[c + d*x])/d - (2*a*(2*a - b)*Sin[c + d*x]^3)/(3*d) + ((6*a^2 - 6*a*b + b^2)*Sin[c + d*x]^5)/(5*d) - (2*(a - b)*(2*a - b)*Sin[c + d*x]^7)/(7*d) + ((a - b)^2*Sin[c + d*x]^9)/(9*d)} - -{Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Tan[c + d*x])/d + (2*a*(a + b)*Tan[c + d*x]^3)/(3*d) + ((a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (2*b*(a + b)*Tan[c + d*x]^7)/(7*d) + (b^2*Tan[c + d*x]^9)/(9*d)} -{Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Tan[c + d*x])/d + (a*(a + 2*b)*Tan[c + d*x]^3)/(3*d) + (b*(2*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2, x, 3, (a^2*Tan[c + d*x])/d + (2*a*b*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2, x, 5, (1/2)*(a - b)*(a + 3*b)*x + ((a - b)^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d} -{Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2, x, 4, (1/8)*(3*a^2 + 2*a*b + 3*b^2)*x + (3*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a - b)*Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(4*d)} -{Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2, x, 5, (1/16)*(5*a^2 + 2*a*b + b^2)*x + ((5*a^2 + 2*a*b + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((a - b)*(5*a + 3*b)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2), x, 5, -(((2*a - 3*b)*ArcTanh[Sin[c + d*x]])/(2*b^2*d)) + ((a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2), x, 4, ArcTanh[Sin[c + d*x]]/(b*d) - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)} -{Sec[c + d*x]^1/(a + b*Tan[c + d*x]^2), x, 2, ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)} -{Cos[c + d*x]^1/(a + b*Tan[c + d*x]^2), x, 3, -((b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(3/2)*d)) + Sin[c + d*x]/((a - b)*d)} -{Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)*d) + ((a - 2*b)*Sin[c + d*x])/((a - b)^2*d) - Sin[c + d*x]^3/(3*(a - b)*d)} -{Cos[c + d*x]^5/(a + b*Tan[c + d*x]^2), x, 4, -((b^3*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Sin[c + d*x])/((a - b)^3*d) - ((2*a - 3*b)*Sin[c + d*x]^3)/(3*(a - b)^2*d) + Sin[c + d*x]^5/(5*(a - b)*d)} - -{Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2), x, 4, -(((a - b)^3*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Tan[c + d*x])/(b^3*d) - ((a - 3*b)*Tan[c + d*x]^3)/(3*b^2*d) + Tan[c + d*x]^5/(5*b*d)} -{Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2), x, 4, ((a - b)^2*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Tan[c + d*x])/(b^2*d) + Tan[c + d*x]^3/(3*b*d)} -{Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2), x, 3, -(((a - b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d)) + Tan[c + d*x]/(b*d)} -{Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)} -{Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2), x, 5, ((a - 3*b)*x)/(2*(a - b)^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*(a - b)*d)} -{Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2), x, 6, ((3*a^2 - 10*a*b + 15*b^2)*x)/(8*(a - b)^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Cos[c + d*x]*Sin[c + d*x])/(8*(a - b)^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*(a - b)*d)} - - -{Sec[c + d*x]^7/(a + b*Tan[c + d*x]^2)^2, x, 6, -(((4*a - 5*b)*ArcTanh[Sin[c + d*x]])/(2*b^3*d)) + ((a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^3*d) + ((a - b)*(2*a - b)*Sin[c + d*x])/(2*a*b^2*d*(a - (a - b)*Sin[c + d*x]^2)) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d*(a - (a - b)*Sin[c + d*x]^2))} -{Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2)^2, x, 5, ArcTanh[Sin[c + d*x]]/(b^2*d) - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) - ((a - b)*Sin[c + d*x])/(2*a*b*d*(a - (a - b)*Sin[c + d*x]^2))} -{Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2, x, 3, ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a - b]*d) + Sin[c + d*x]/(2*a*d*(a - (a - b)*Sin[c + d*x]^2))} -{Sec[c + d*x]^1/(a + b*Tan[c + d*x]^2)^2, x, 3, ((2*a - b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(3/2)*d) - (b*Sin[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Sin[c + d*x]^2))} -{Cos[c + d*x]^1/(a + b*Tan[c + d*x]^2)^2, x, 5, -(((4*a - b)*b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(5/2)*d)) + Sin[c + d*x]/((a - b)^2*d) + (b^2*Sin[c + d*x])/(2*a*(a - b)^2*d*(a - (a - b)*Sin[c + d*x]^2))} -{Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2, x, 5, ((6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(7/2)*d) + ((a - 3*b)*Sin[c + d*x])/((a - b)^3*d) - Sin[c + d*x]^3/(3*(a - b)^2*d) - (b^3*Sin[c + d*x])/(2*a*(a - b)^3*d*(a - (a - b)*Sin[c + d*x]^2))} - -{Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2)^2, x, 5, ((a - b)^2*(5*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(7/2)*d) - ((2*a - 3*b)*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d) - ((a - b)^3*Tan[c + d*x])/(2*a*b^3*d*(a + b*Tan[c + d*x]^2))} -{Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2)^2, x, 5, -(((3*a^2 - 2*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d)) + Tan[c + d*x]/(b^2*d) + ((a - b)^2*Tan[c + d*x])/(2*a*b^2*d*(a + b*Tan[c + d*x]^2))} -{Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2, x, 3, ((a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)*d) - ((a - b)*Tan[c + d*x])/(2*a*b*d*(a + b*Tan[c + d*x]^2))} -{Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Tan[c + d*x]/(2*a*d*(a + b*Tan[c + d*x]^2))} -{Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2, x, 6, ((a - 5*b)*x)/(2*(a - b)^3) + ((5*a - b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*(a - b)*d*(a + b*Tan[c + d*x]^2)) + (b*(a + b)*Tan[c + d*x])/(2*a*(a - b)^2*d*(a + b*Tan[c + d*x]^2))} -{Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2, x, 7, ((3*a^2 - 14*a*b + 35*b^2)*x)/(8*(a - b)^4) - ((7*a - b)*b^(5/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^4*d) + (3*(a - 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*(a - b)^2*d*(a + b*Tan[c + d*x]^2)) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*(a - b)*d*(a + b*Tan[c + d*x]^2)) + ((a - 4*b)*b*(3*a + b)*Tan[c + d*x])/(8*a*(a - b)^3*d*(a + b*Tan[c + d*x]^2))} - - -(* ::Subsection:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Tan[e+f x]^2)^(p/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Sec[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 2, ((Cos[e + f*x]^2)^((1/2)*(1 + m + 2*p))*Hypergeometric2F1[(1/2)*(1 + 2*p), (1/2)*(1 + m + 2*p), (1/2)*(3 + 2*p), Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))} - - -{(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/((Sec[e + f*x]^2)^(m/2)*(1 + (b*Tan[e + f*x]^2)/a)^p*f)} - - -(* ::Section:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic*) - - -{(d*Sec[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 2, ((Cos[e + f*x]^2)^((1/2)*(1 + m + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(1 + m + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} - - -{Sec[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)) + (2*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p)) + (Tan[e + f*x]^5*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 + n*p))} -{Sec[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p, x, 4, (Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)) + (Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))} -{Sec[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Cos[e + f*x]^0*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Cos[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} - -{Sec[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 2, ((Cos[e + f*x]^2)^((1/2)*(4 + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(4 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sec[e + f*x]^3*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Sec[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 2, ((Cos[e + f*x]^2)^((1/2)*(2 + n*p))*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(2 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Cos[e + f*x]^1*(b*(c*Tan[e + f*x])^n)^p, x, 2, ((Cos[e + f*x]^2)^((n*p)/2)*Hypergeometric2F1[(n*p)/2, (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} -{Cos[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p, x, 2, ((Cos[e + f*x]^2)^((n*p)/2)*Hypergeometric2F1[(1/2)*(-2 + n*p), (1/2)*(1 + n*p), (1/2)*(3 + n*p), Sin[e + f*x]^2]*Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]} - - -{Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]} -{Sec[e + f*x]^1*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[Sec[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]} -{Cos[e + f*x]^1*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]} -{Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]} - -{Sec[e + f*x]^6*(a + b*(c*Tan[e + f*x])^n)^p, x, 9, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*f) + (2*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*(3*f)) + (Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^5*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*(5*f))} -{Sec[e + f*x]^4*(a + b*(c*Tan[e + f*x])^n)^p, x, 7, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*f) + (Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*(3*f))} -{Sec[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x, 3, (Hypergeometric2F1[1/n, -p, 1 + 1/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/((1 + (b*(c*Tan[e + f*x])^n)/a)^p*f)} -{Sec[e + f*x]^0*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p, x]} -{Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^2)^p when p symbolic*) - - -{(d*Csc[e + f*x])^m*(b*Tan[e + f*x]^2)^p, x, 4, ((Cos[e + f*x]^2)^(1/2 + p)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1/2)*(1 + 2*p), (1/2)*(1 - m + 2*p), (1/2)*(3 - m + 2*p), Sin[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 - m + 2*p))} - - -{(d*Csc[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p, x, 4, (AppellF1[(1 - m)/2, 1 - m/2, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Csc[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/((Sec[e + f*x]^2)^(m/2)*(1 + (b*Tan[e + f*x]^2)/a)^p*(f*(1 - m)))} - - -(* ::Section:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Tan[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (b (cTan[e+f x])^n)^p when p symbolic*) - - -{(d*Csc[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p, x, 4, ((Cos[e + f*x]^2)^((1/2)*(1 + n*p))*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1/2)*(1 + n*p), (1/2)*(1 - m + n*p), (1/2)*(3 - m + n*p), Sin[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b (c Tan[e+f x])^n)^p when p symbolic*) - - -{(d*Csc[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x, 1, (d*Csc[e + f*x])^m*(Sin[e + f*x]/d)^m*Unintegrable[(a + b*(c*Tan[e + f*x])^n)^p/(Sin[e + f*x]/d)^m, x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m deleted file mode 100644 index 09636b3..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.3 Tangent/4.3.9 trig^m (a+b tan^n+c tan^(2 n))^p.m +++ /dev/null @@ -1,107 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[d+e x]^m (a+b Tan[d+e x]^n+c Tan[d+e x]^(2 n))^p*) - - -(* ::Section:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Tan[d+e x]^n+c Tan[d+e x]^(2 n))^p*) - - -(* ::Section:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Tan[d+e x]^n+c Tan[d+e x]^(2 n))^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[d+e x]^m (a+b Tan[d+e x]^n+c Tan[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[d+e x]^m (a+b Tan[d+e x]+c Tan[d+e x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 21, (1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e))*(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) - (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c^(5/2)*e) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(256*c^(9/2)*e) - (1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e))*(Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e + (b*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) - (b*(7*b^2 - 12*a*c)*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(128*c^4*e) - (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e) + (Tan[d + e*x]^2*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(5*c*e) + ((35*b^2 - 32*a*c - 42*b*c*Tan[d + e*x])*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(240*c^3*e)} -{Tan[d + e*x]^4*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 19, -((1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e))*(Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])) + (Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/e + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*c^(3/2)*e) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(128*c^(7/2)*e) - (1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e))*(Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]) - ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c*e) + ((5*b^2 - 4*a*c)*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(64*c^3*e) - (5*b*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(24*c^2*e) + (Tan[d + e*x]*(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2))/(4*c*e)} -{Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 16, -((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) + (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c^(5/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e - (b*(b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(8*c^2*e) + (a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2)/(3*c*e)} -{Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 10, (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - ((b^2 - 4*(a - 2*c)*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*c^(3/2)*e) + (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + ((b + 2*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c*e)} -{Tan[d + e*x]^1*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 10, (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[c]*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/e} -{Tan[d + e*x]^0*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 9, -((Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) + (Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/e - (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)} -{Cot[d + e*x]^1*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 18, -((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/e + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)} -{Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 17, (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] - (b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*Sqrt[a]*e) + (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b*Sqrt[a^2 + b^2 - 2*a*c + c^2] + (b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/e} -{Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 21, (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/e + ((b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*a^(3/2)*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Tan[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - (Cot[d + e*x]^2*(2*a + b*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a*e)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 15, (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*c^(3/2)*e) - (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c^(7/2)*e) - Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/(c*e) + (Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(3*c*e) + ((15*b^2 - 16*a*c - 10*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(24*c^3*e)} -{Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 14, (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(Sqrt[c]*e) + ((3*b^2 - 4*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*c^(5/2)*e) - (3*b*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*c^2*e) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*c*e)} -{Tan[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 11, -((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*c^(3/2)*e) + Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]/(c*e)} -{Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 9, -((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(Sqrt[c]*e)} -{Tan[d + e*x]^1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 6, (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)} -{Tan[d + e*x]^0/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 6, (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)} -{Cot[d + e*x]^1/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 10, -(ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(Sqrt[a]*e)) - (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)} -{Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 11, -((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b - (a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(3/2)*e) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a*e)} -{Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2], x, 14, ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(Sqrt[a]*e) - ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*a^(5/2)*e) + (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (3*b*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a^2*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a*e)} - - -{Tan[d + e*x]^7/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 20, (3*b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*c^(5/2)*e) - (5*b*(7*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(16*c^(9/2)*e) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*Tan[d + e*x]^2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*Tan[d + e*x]^4*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + ((7*b^2 - 16*a*c)*Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(3*c^2*(b^2 - 4*a*c)*e) - (2*b*Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c*(b^2 - 4*a*c)*e) - ((3*b^2 - 8*a*c - 2*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e) + ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2 - 2*b*c*(35*b^2 - 116*a*c)*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(24*c^4*(b^2 - 4*a*c)*e)} -{Tan[d + e*x]^5/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 14, -((3*b*ArcTanh[(b + 2*c*Tan[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*c^(5/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*Tan[d + e*x]^2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + ((3*b^2 - 8*a*c - 2*b*c*Tan[d + e*x])*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e)} -{Tan[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 10, -((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Tan[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])} -{Tan[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 7, -((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])} -{Tan[d + e*x]^1/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 7, (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])} -{Cot[d + e*x]^1/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 13, -(ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(a^(3/2)*e)) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])} -{Cot[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 13, -((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (3*b*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(5/2)*e) + (2*Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(b*(b^2 - (3*a - c)*c) + c*(b^2 - 2*(a - c)*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a^2*(b^2 - 4*a*c)*e)} -{Cot[d + e*x]^3/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2), x, 18, ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])]/(a^(3/2)*e) - (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*a^(7/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*Cot[d + e*x]^2*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (b*(15*b^2 - 52*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a^3*(b^2 - 4*a*c)*e) - ((5*b^2 - 12*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a^2*(b^2 - 4*a*c)*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[d+e x]^m (a+b Tan[d+e x]^2+c Tan[d+e x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[d + e*x]^5*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 9, -((Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e)) + ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(32*c^(5/2)*e) - (((b - 2*c)*(b + 4*c) + 2*c*(b + 2*c)*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*c^2*e) + (a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2)/(6*c*e)} -{Tan[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 8, (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) - ((b^2 + 4*b*c - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(16*c^(3/2)*e) + ((b - 4*c + 2*c*Tan[d + e*x]^2)*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(8*c*e)} -{Tan[d + e*x]^1*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 8, -((Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e)) + ((b - 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*Sqrt[c]*e) + Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]/(2*e)} -{Cot[d + e*x]^1*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 10, -((Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e)) + (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) + (Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e)} -{Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 22, (Sqrt[a]*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) - (b*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*Sqrt[a]*e) - (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) - (b*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*Sqrt[c]*e) + ((b - 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*Sqrt[c]*e) + (Sqrt[c]*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(2*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*e)} - -{Tan[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 14, -((Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e)) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*e) + (b*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*Sqrt[c]*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*b*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(3*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(6*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((b + Sqrt[a]*Sqrt[c] - c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^0*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 8, (Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((b + Sqrt[a]*Sqrt[c] - c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 9, -((Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e)) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/e + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Cot[d + e*x]^4*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 10, (Sqrt[a - b + c]*ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]])/(2*e) + ((3*a - b)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*a*e) - (Cot[d + e*x]^3*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*e) - ((3*a - b)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(3*a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + ((3*a - b)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(3*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((3*a - b + Sqrt[a]*Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(6*a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (c^(1/4)*(a - b + c)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*(a - b + c)*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[d + e*x]^5/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 8, -(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)) - ((b + 2*c)*ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*c^(3/2)*e) + Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]/(2*c*e)} -{Tan[d + e*x]^3/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 7, ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) + ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[c]*e)} -{Tan[d + e*x]^1/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 4, -(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a - b + c]*e))} -{Cot[d + e*x]^1/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 8, -(ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a]*e)) + ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)} -{Cot[d + e*x]^3/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 11, ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a]*e) + (b*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*a^(3/2)*e) - ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) - (Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*a*e)} - -{Tan[d + e*x]^4/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 5, ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) + (Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(Sqrt[c]*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (a^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (a^(1/4)*(Sqrt[a] - 2*Sqrt[c])*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*(Sqrt[a] - Sqrt[c])*c^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 4, -(ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e)) + (a^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^0/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 4, ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e) - (c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((Sqrt[a] + Sqrt[c])*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Cot[d + e*x]^2/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4], x, 7, -(ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*Sqrt[a - b + c]*e)) - (Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a*e) + (Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) - (c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(a^(3/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((2*Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(3/4)*(Sqrt[a] - Sqrt[c])*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} - - -{Tan[d + e*x]^7/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 8, ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) + ArcTanh[(b + 2*c*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*c^(3/2)*e) + (a*(b^2 - a*(b + 2*c)) + (b^3 + 2*a^2*c - a*b*(b + 3*c))*Tan[d + e*x]^2)/(c*(a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^5/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 6, -(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e)) + (a*(2*a - b) + ((a - b)*b + 2*a*c)*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 6, ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (a*(b - 2*c) + (2*a - b)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Tan[d + e*x]^1/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 6, -(ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e)) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Cot[d + e*x]^1/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 12, -(ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*a^(3/2)*e)) + ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) + (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -{Cot[d + e*x]^3/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 16, ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*a^(3/2)*e) + (3*b*ArcTanh[(2*a + b*Tan[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])])/(4*a^(5/2)*e) - ArcTanh[(2*a - b + (b - 2*c)*Tan[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c + b*c*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (Cot[d + e*x]^2*(b^2 - 2*a*c + b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(2*a^2*(b^2 - 4*a*c)*e)} - -{Tan[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 9, -(ArcTan[(Sqrt[a - b + c]*Tan[d + e*x])/Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]]/(2*(a - b + c)^(3/2)*e)) + (Tan[d + e*x]*(b^2 - 2*a*c - b*c + (b - 2*c)*c*Tan[d + e*x]^2))/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((b - 2*c)*Sqrt[c]*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/((a - b + c)*(b^2 - 4*a*c)*e*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)) + (a^(1/4)*(b - 2*c)*c^(1/4)*EllipticE[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + (c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] - Sqrt[c])*c^(1/4)*EllipticF[2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(2*a^(1/4)*(b - 2*Sqrt[a]*Sqrt[c])*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - ((Sqrt[a] + Sqrt[c])*EllipticPi[-((Sqrt[a] - Sqrt[c])^2/(4*Sqrt[a]*Sqrt[c])), 2*ArcTan[(c^(1/4)*Tan[d + e*x])/a^(1/4)], (1/4)*(2 - b/(Sqrt[a]*Sqrt[c]))]*(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)*Sqrt[(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)/(Sqrt[a] + Sqrt[c]*Tan[d + e*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[c])*c^(1/4)*(a - b + c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])} -(* {Cot[d + e*x]^2/(a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4)^(3/2), x, 21, (Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]^2))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) + ((b^2 - 3*a*c)*(b - Sqrt[b^2 - 4*a*c])*Sqrt[b + Sqrt[b^2 - 4*a*c]]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*a^2*Sqrt[-c]*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (Sqrt[b + Sqrt[b^2 - 4*a*c]]*(b^3 - 4*a*b*c - Sqrt[b^2 - 4*a*c]*(b^2 - 3*a*c))*EllipticF[ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*a^2*Sqrt[-c]*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4]) - (2*(b^2 - 3*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a^2*(b^2 - 4*a*c)*e) - ((b - Sqrt[b^2 - 4*a*c])^2*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(4*a^2*(a - b + c)*Sqrt[b^2 - 4*a*c]*e*(1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c]))) + (b*c*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a^2*(b^2 - 4*a*c)*(1 - (2*c)/(b - Sqrt[b^2 - 4*a*c]))*e*(1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c]))) + (c*(1 + b/Sqrt[b^2 - 4*a*c])*Tan[d + e*x]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(a^2*(b - 2*c - Sqrt[b^2 - 4*a*c])*e*(1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c]))*(1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c]))) - (Sqrt[-c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(Sqrt[2]*a*(a - b + c)*Sqrt[b^2 - 4*a*c]*e*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]) + (b*(b - Sqrt[b^2 - 4*a*c])^3*(b + Sqrt[b^2 - 4*a*c])^(5/2)*EllipticE[ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*Sqrt[2]*a^4*(-c)^(3/2)*(b^2 - 4*a*c)*(b - 2*c - Sqrt[b^2 - 4*a*c])*e*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]) - ((b - Sqrt[b^2 - 4*a*c])^2*(b + Sqrt[b^2 - 4*a*c])^(5/2)*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*EllipticF[ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(16*Sqrt[2]*a^4*(-c)^(3/2)*(b^2 - 4*a*c)*(b - 2*c - Sqrt[b^2 - 4*a*c])*e*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])]) - (Sqrt[b - Sqrt[b^2 - 4*a*c]]*EllipticPi[(b - Sqrt[b^2 - 4*a*c])/(2*c), ArcSin[(Sqrt[2]*Sqrt[-c]*Tan[d + e*x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*Tan[d + e*x]^2 + c*Tan[d + e*x]^4])/(Sqrt[2]*a*Sqrt[-c]*(a - b + c)*e*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*Tan[d + e*x]^2)/(b + Sqrt[b^2 - 4*a*c])])} *) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.0 (a trg)^m (b cot)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.0 (a trg)^m (b cot)^n.m deleted file mode 100644 index fa9ed8d..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.0 (a trg)^m (b cot)^n.m +++ /dev/null @@ -1,177 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Cot[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Cot[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[c+d x]^n*) - - -{Cot[a + b*x]^1, x, 1, Log[Sin[a + b*x]]/b} -{Cot[a + b*x]^2, x, 2, -x - Cot[a + b*x]/b} -{Cot[a + b*x]^3, x, 2, -(Cot[a + b*x]^2/(2*b)) - Log[Sin[a + b*x]]/b} -{Cot[a + b*x]^4, x, 3, x + Cot[a + b*x]/b - Cot[a + b*x]^3/(3*b)} -{Cot[a + b*x]^5, x, 3, Cot[a + b*x]^2/(2*b) - Cot[a + b*x]^4/(4*b) + Log[Sin[a + b*x]]/b} -{Cot[a + b*x]^6, x, 4, -x - Cot[a + b*x]/b + Cot[a + b*x]^3/(3*b) - Cot[a + b*x]^5/(5*b)} -{Cot[a + b*x]^7, x, 4, -(Cot[a + b*x]^2/(2*b)) + Cot[a + b*x]^4/(4*b) - Cot[a + b*x]^6/(6*b) - Log[Sin[a + b*x]]/b} -{Cot[a + b*x]^8, x, 5, x + Cot[a + b*x]/b - Cot[a + b*x]^3/(3*b) + Cot[a + b*x]^5/(5*b) - Cot[a + b*x]^7/(7*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x])^(n/2)*) - - -{(c*Cot[a + b*x])^(7/2), x, 13, (c^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (c^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) + (2*c^3*Sqrt[c*Cot[a + b*x]])/b - (2*c*(c*Cot[a + b*x])^(5/2))/(5*b) + (c^(7/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b) - (c^(7/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b)} -{(c*Cot[a + b*x])^(5/2), x, 12, -((c^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b)) + (c^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (2*c*(c*Cot[a + b*x])^(3/2))/(3*b) + (c^(5/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b) - (c^(5/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b)} -{(c*Cot[a + b*x])^(3/2), x, 12, -((c^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b)) + (c^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (2*c*Sqrt[c*Cot[a + b*x]])/b - (c^(3/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b) + (c^(3/2)*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b)} -{(c*Cot[a + b*x])^(1/2), x, 11, (Sqrt[c]*ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (Sqrt[c]*ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]])/(Sqrt[2]*b) - (Sqrt[c]*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b) + (Sqrt[c]*Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]])/(2*Sqrt[2]*b)} -{1/(c*Cot[a + b*x])^(1/2), x, 11, ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*Sqrt[c]) - ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*Sqrt[c]) + Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*Sqrt[c]) - Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*Sqrt[c])} -{1/(c*Cot[a + b*x])^(3/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(3/2))) + ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(3/2)) + 2/(b*c*Sqrt[c*Cot[a + b*x]]) + Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(3/2)) - Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(3/2))} -{1/(c*Cot[a + b*x])^(5/2), x, 12, -(ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(5/2))) + ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(5/2)) + 2/(3*b*c*(c*Cot[a + b*x])^(3/2)) - Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(5/2)) + Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(5/2))} -{1/(c*Cot[a + b*x])^(7/2), x, 13, ArcTan[1 - (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(7/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[c*Cot[a + b*x]])/Sqrt[c]]/(Sqrt[2]*b*c^(7/2)) + 2/(5*b*c*(c*Cot[a + b*x])^(5/2)) - 2/(b*c^3*Sqrt[c*Cot[a + b*x]]) - Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] - Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(7/2)) + Log[Sqrt[c] + Sqrt[c]*Cot[a + b*x] + Sqrt[2]*Sqrt[c*Cot[a + b*x]]]/(2*Sqrt[2]*b*c^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x])^(n/3)*) - - -{(c*Cot[a + b*x])^(4/3), x, 13, (c^(4/3)*ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)])/b - (c^(4/3)*ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) + (c^(4/3)*ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) - (3*c*(c*Cot[a + b*x])^(1/3))/b - (Sqrt[3]*c^(4/3)*Log[c^(2/3) - Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b) + (Sqrt[3]*c^(4/3)*Log[c^(2/3) + Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b)} -{(c*Cot[a + b*x])^(2/3), x, 12, -((c^(2/3)*ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)])/b) + (c^(2/3)*ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) - (c^(2/3)*ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)])/(2*b) - (Sqrt[3]*c^(2/3)*Log[c^(2/3) - Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b) + (Sqrt[3]*c^(2/3)*Log[c^(2/3) + Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b)} -{(c*Cot[a + b*x])^(1/3), x, 9, (Sqrt[3]*c^(1/3)*ArcTan[(c^(2/3) - 2*(c*Cot[a + b*x])^(2/3))/(Sqrt[3]*c^(2/3))])/(2*b) + (c^(1/3)*Log[c^(2/3) + (c*Cot[a + b*x])^(2/3)])/(2*b) - (c^(1/3)*Log[c^(4/3) - c^(2/3)*(c*Cot[a + b*x])^(2/3) + (c*Cot[a + b*x])^(4/3)])/(4*b)} -{1/(c*Cot[a + b*x])^(1/3), x, 9, (Sqrt[3]*ArcTan[(c^(2/3) - 2*(c*Cot[a + b*x])^(2/3))/(Sqrt[3]*c^(2/3))])/(2*b*c^(1/3)) - Log[c^(2/3) + (c*Cot[a + b*x])^(2/3)]/(2*b*c^(1/3)) + Log[c^(4/3) - c^(2/3)*(c*Cot[a + b*x])^(2/3) + (c*Cot[a + b*x])^(4/3)]/(4*b*c^(1/3))} -{1/(c*Cot[a + b*x])^(2/3), x, 12, -(ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)]/(b*c^(2/3))) + ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(2/3)) - ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(2/3)) + (Sqrt[3]*Log[c^(2/3) - Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3)) - (Sqrt[3]*Log[c^(2/3) + Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(2/3))} -{1/(c*Cot[a + b*x])^(4/3), x, 13, ArcTan[(c*Cot[a + b*x])^(1/3)/c^(1/3)]/(b*c^(4/3)) - ArcTan[Sqrt[3] - (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(4/3)) + ArcTan[Sqrt[3] + (2*(c*Cot[a + b*x])^(1/3))/c^(1/3)]/(2*b*c^(4/3)) + 3/(b*c*(c*Cot[a + b*x])^(1/3)) + (Sqrt[3]*Log[c^(2/3) - Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(4/3)) - (Sqrt[3]*Log[c^(2/3) + Sqrt[3]*c^(1/3)*(c*Cot[a + b*x])^(1/3) + (c*Cot[a + b*x])^(2/3)])/(4*b*c^(4/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x])^n with n symbolic*) - - -{Cot[a + b*x]^n, x, 2, -((Cot[a + b*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[a + b*x]^2])/(b*(1 + n)))} - - -{(b*Cot[c + d*x])^n, x, 2, -(((b*Cot[c + d*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Cot[c + d*x]^2])/(b*d*(1 + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Cot[c+d x]^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x]^2)^(n/2)*) - - -{(a*Cot[x]^2)^(3/2),x, 3, (-(1/2))*a*Cot[x]*Sqrt[a*Cot[x]^2] - a*Sqrt[a*Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{Sqrt[a*Cot[x]^2], x, 2, Sqrt[a*Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{1/Sqrt[a*Cot[x]^2], x, 2, -((Cot[x]*Log[Cos[x]])/Sqrt[a*Cot[x]^2])} -{1/(a*Cot[x]^2)^(3/2),x, 3, (Cot[x]*Log[Cos[x]])/(a*Sqrt[a*Cot[x]^2]) + Tan[x]/(2*a*Sqrt[a*Cot[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x]^3)^(n/2)*) - - -{(a*Cot[x]^3)^(3/2), x, 14, (2/3)*a*Sqrt[a*Cot[x]^3] + (a*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (2/7)*a*Cot[x]^2*Sqrt[a*Cot[x]^3] - (a*Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) + (a*Sqrt[a*Cot[x]^3]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2))} -{Sqrt[a*Cot[x]^3], x, 13, -((ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2))) + (ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Sqrt[a*Cot[x]^3])/(Sqrt[2]*Cot[x]^(3/2)) - (Sqrt[a*Cot[x]^3]*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) + (Sqrt[a*Cot[x]^3]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Cot[x]^(3/2)) - 2*Sqrt[a*Cot[x]^3]*Tan[x]} -{1/Sqrt[a*Cot[x]^3], x, 13, (2*Cot[x])/Sqrt[a*Cot[x]^3] - (ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*Sqrt[a*Cot[x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*Sqrt[a*Cot[x]^3]) + (Cot[x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Sqrt[a*Cot[x]^3]) - (Cot[x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*Sqrt[a*Cot[x]^3])} -{1/(a*Cot[x]^3)^(3/2),x, 14, -(2/(3*a*Sqrt[a*Cot[x]^3])) + (ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*a*Sqrt[a*Cot[x]^3]) - (ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]*Cot[x]^(3/2))/(Sqrt[2]*a*Sqrt[a*Cot[x]^3]) + (Cot[x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*a*Sqrt[a*Cot[x]^3]) - (Cot[x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot[x]])/(2*Sqrt[2]*a*Sqrt[a*Cot[x]^3]) + (2*Tan[x]^2)/(7*a*Sqrt[a*Cot[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x]^4)^(n/2)*) - - -{(a*Cot[x]^4)^(3/2),x, 5, (1/3)*a*Cot[x]*Sqrt[a*Cot[x]^4] - (1/5)*a*Cot[x]^3*Sqrt[a*Cot[x]^4] - a*Sqrt[a*Cot[x]^4]*Tan[x] - a*x*Sqrt[a*Cot[x]^4]*Tan[x]^2} -{Sqrt[a*Cot[x]^4], x, 3, (-Sqrt[a*Cot[x]^4])*Tan[x] - x*Sqrt[a*Cot[x]^4]*Tan[x]^2} -{1/Sqrt[a*Cot[x]^4], x, 3, Cot[x]/Sqrt[a*Cot[x]^4] - (x*Cot[x]^2)/Sqrt[a*Cot[x]^4]} -{1/(a*Cot[x]^4)^(3/2),x, 5, Cot[x]/(a*Sqrt[a*Cot[x]^4]) - (x*Cot[x]^2)/(a*Sqrt[a*Cot[x]^4]) - Tan[x]/(3*a*Sqrt[a*Cot[x]^4]) + Tan[x]^3/(5*a*Sqrt[a*Cot[x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Cot[c+d x]^p)^n with n symbolic*) - - -{(b*Cot[c + d*x]^p)^n, x, 3, -((Cot[c + d*x]*(b*Cot[c + d*x]^p)^n*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Cot[c + d*x]^2])/(d*(1 + n*p)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Cot[c+d x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a (b Cot[c+d x])^p)^n with n symbolic*) - - -{(a*(b*Cot[c + d*x])^p)^n, x, 3, -((Cot[c + d*x]*(a*(b*Cot[c + d*x])^p)^n*Hypergeometric2F1[1, (1/2)*(1 + n*p), (1/2)*(3 + n*p), -Cot[c + d*x]^2])/(d*(1 + n*p)))} - - -(* ::Title:: *) -(*Integrands of the form (a Trg[e+f x])^m (b Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sin[e+f x])^m (b Cot[e+f x])^n*) - - -{(a*Sin[e + f*x])^m*(b*Cot[e + f*x])^n, x, 2, -(((b*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 - m + n), (3 + n)/2, Cos[e + f*x]^2]*(a*Sin[e + f*x])^m*(Sin[e + f*x]^2)^((1/2)*(1 - m + n)))/(b*f*(1 + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cos[e+f x])^m (b Cot[e+f x])^n*) - - -{(a*Cos[e + f*x])^m*(b*Cot[e + f*x])^n, x, 2, -(((a*Cos[e + f*x])^m*(b*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 + m + n), (1/2)*(3 + m + n), Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((1 + n)/2))/(b*f*(1 + m + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Cot[e+f x])^m (b Cot[e+f x])^n*) - - -{(a*Cot[e + f*x])^m*(b*Cot[e + f*x])^n, x, 3, -(((a*Cot[e + f*x])^(1 + m)*(b*Cot[e + f*x])^n*Hypergeometric2F1[1, (1/2)*(1 + m + n), (1/2)*(3 + m + n), -Cot[e + f*x]^2])/(a*f*(1 + m + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Cot[e+f x])^n*) - - -{(a*Sec[e + f*x])^m*(b*Cot[e + f*x])^n, x, 3, -(((b*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 - m + n), (1/2)*(3 - m + n), Cos[e + f*x]^2]*(a*Sec[e + f*x])^m*(Sin[e + f*x]^2)^((1 + n)/2))/(b*f*(1 - m + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Cot[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^m (b Cot[e+f x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (a Csc[e+f x])^(m/2) (b Cot[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Cot[e+f x])^n with n symbolic*) - - -{Csc[e + f*x]^6*(d*Cot[e + f*x])^n, x, 3, -((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n))) - (2*(d*Cot[e + f*x])^(3 + n))/(d^3*f*(3 + n)) - (d*Cot[e + f*x])^(5 + n)/(d^5*f*(5 + n))} -{Csc[e + f*x]^4*(d*Cot[e + f*x])^n, x, 3, -((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n))) - (d*Cot[e + f*x])^(3 + n)/(d^3*f*(3 + n))} -{Csc[e + f*x]^2*(d*Cot[e + f*x])^n, x, 2, -((d*Cot[e + f*x])^(1 + n)/(d*f*(1 + n)))} -{Sin[e + f*x]^2*(d*Cot[e + f*x])^n, x, 2, -(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[2, (1 + n)/2, (3 + n)/2, -Cot[e + f*x]^2])/(d*f*(1 + n)))} -{Sin[e + f*x]^4*(d*Cot[e + f*x])^n, x, 2, -(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, -Cot[e + f*x]^2])/(d*f*(1 + n)))} - -{Csc[e + f*x]^3*(d*Cot[e + f*x])^n, x, 1, -(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]^3*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((4 + n)/2))/(d*f*(1 + n)))} -{Csc[e + f*x]^1*(d*Cot[e + f*x])^n, x, 1, -(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((2 + n)/2))/(d*f*(1 + n)))} -{Sin[e + f*x]^1*(d*Cot[e + f*x])^n, x, 1, -(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[n/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x]*(Sin[e + f*x]^2)^(n/2))/(d*f*(1 + n)))} -{Sin[e + f*x]^3*(d*Cot[e + f*x])^n, x, 1, -(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(1/2)*(-2 + n), (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((1/2)*(-2 + n)))/(d*f*(1 + n)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Cot[e+f x])^n with m and n symbolic*) - - -{(a*Csc[e + f*x])^m*(b*Cot[e + f*x])^n, x, 1, -(((b*Cot[e + f*x])^(1 + n)*(a*Csc[e + f*x])^m*Hypergeometric2F1[(1 + n)/2, (1/2)*(1 + m + n), (3 + n)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((1/2)*(1 + m + n)))/(b*f*(1 + n)))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.2 (d csc)^m (a+b cot)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.2 (d csc)^m (a+b cot)^n.m deleted file mode 100644 index d444fa4..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.2 (d csc)^m (a+b cot)^n.m +++ /dev/null @@ -1,93 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[e+f x]^m (a+a I Cot[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^4/(I + Cot[x]), x, 4, -((5*I*x)/16) + 1/(32*(I - Cot[x])^2) - I/(8*(I - Cot[x])) - I/(24*(I + Cot[x])^3) - 3/(32*(I + Cot[x])^2) + (3*I)/(16*(I + Cot[x]))} -{Sin[x]^3/(I + Cot[x]), x, 3, (4/5)*I*Cos[x] - (4/15)*I*Cos[x]^3 + (I*Sin[x]^3)/(5*(I + Cot[x]))} -{Sin[x]^2/(I + Cot[x]), x, 4, -((3*I*x)/8) - I/(8*(I - Cot[x])) - 1/(8*(I + Cot[x])^2) + I/(4*(I + Cot[x]))} -{Sin[x]^1/(I + Cot[x]), x, 2, (2/3)*I*Cos[x] + (I*Sin[x])/(3*(I + Cot[x]))} -{Csc[x]^1/(I + Cot[x]), x, 1, (I*Csc[x])/(I + Cot[x])} -{Csc[x]^2/(I + Cot[x]), x, 2, (-I)*x + Log[Sin[x]]} -{Csc[x]^3/(I + Cot[x]), x, 2, I*ArcTanh[Cos[x]] - Csc[x]} -{Csc[x]^4/(I + Cot[x]), x, 2, I*Cot[x] - Cot[x]^2/2} -{Csc[x]^5/(I + Cot[x]), x, 3, (1/2)*I*ArcTanh[Cos[x]] + (1/2)*I*Cot[x]*Csc[x] - Csc[x]^3/3} -{Csc[x]^6/(I + Cot[x]), x, 3, I*Cot[x] - Cot[x]^2/2 + (1/3)*I*Cot[x]^3 - Cot[x]^4/4} -{Csc[x]^7/(I + Cot[x]), x, 4, (3/8)*I*ArcTanh[Cos[x]] + (3/8)*I*Cot[x]*Csc[x] + (1/4)*I*Cot[x]*Csc[x]^3 - Csc[x]^5/5} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[e+f x]^m (a+b Cot[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Csc[x]^6/(a + b*Cot[x]), x, 3, (a*(a^2 + 2*b^2)*Cot[x])/b^4 - ((a^2 + 2*b^2)*Cot[x]^2)/(2*b^3) + (a*Cot[x]^3)/(3*b^2) - Cot[x]^4/(4*b) - ((a^2 + b^2)^2*Log[a + b*Cot[x]])/b^5} -{Csc[x]^4/(a + b*Cot[x]), x, 3, (a*Cot[x])/b^2 - Cot[x]^2/(2*b) - ((a^2 + b^2)*Log[a + b*Cot[x]])/b^3} -{Csc[x]^2/(a + b*Cot[x]), x, 2, -(Log[a + b*Cot[x]]/b)} -{Sin[x]^2/(a + b*Cot[x]), x, 7, (a*(a^2 + 3*b^2)*x)/(2*(a^2 + b^2)^2) - (b^3*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^2 - ((b + a*Cot[x])*Sin[x]^2)/(2*(a^2 + b^2))} -{Sin[x]^4/(a + b*Cot[x]), x, 8, (a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/(8*(a^2 + b^2)^3) - (b^5*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 - ((4*b^3 + a*(3*a^2 + 7*b^2)*Cot[x])*Sin[x]^2)/(8*(a^2 + b^2)^2) - ((b + a*Cot[x])*Sin[x]^4)/(4*(a^2 + b^2))} - -{Csc[x]^5/(a + b*Cot[x]), x, 9, (a*ArcTanh[Cos[x]])/(2*b^2) + (a*(a^2 + b^2)*ArcTanh[Cos[x]])/b^4 + ((a^2 + b^2)^(3/2)*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^4 - ((a^2 + b^2)*Csc[x])/b^3 + (a*Cot[x]*Csc[x])/(2*b^2) - Csc[x]^3/(3*b)} -{Csc[x]^3/(a + b*Cot[x]), x, 5, (a*ArcTanh[Cos[x]])/b^2 + (Sqrt[a^2 + b^2]*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^2 - Csc[x]/b} -{Csc[x]^1/(a + b*Cot[x]), x, 2, -(ArcTanh[((-b + a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])} -{Sin[x]^1/(a + b*Cot[x]), x, 5, (b^2*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) - (b*Sin[x])/(a^2 + b^2)} -{Sin[x]^3/(a + b*Cot[x]), x, 9, (b^4*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x])/(a^2 + b^2) + (a*Cos[x]^3)/(3*(a^2 + b^2)) - (b^3*Sin[x])/(a^2 + b^2)^2 - (b*Sin[x]^3)/(3*(a^2 + b^2))} - - -{Csc[x]^2/(a + b*Cot[x])^2, x, 2, 1/(b*(a + b*Cot[x]))} - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^m (a+b Cot[e+f x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^(m/2) (a+b Cot[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^(m/2) (a+b Cot[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n with n symbolic*) -(**) - - -{Csc[x]^2*(a + b*Cot[x])^n, x, 2, -((a + b*Cot[x])^(1 + n)/(b*(1 + n)))} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^(m/2) (a+b Cot[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^(m/2) (a+b Cot[e+f x])^(n/2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.3 (d cos)^m (a+b cot)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.3 (d cos)^m (a+b cot)^n.m deleted file mode 100644 index 4ac84b5..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.1.3 (d cos)^m (a+b cot)^n.m +++ /dev/null @@ -1,62 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cot[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cot[e+f x])^n when a^2+b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[x]^4/(I + Cot[x]), x, 5, -((I*x)/16) + 1/(32*(I - Cot[x])^2) + I/(8*(I - Cot[x])) - I/(24*(I + Cot[x])^3) + 5/(32*(I + Cot[x])^2) + (3*I)/(16*(I + Cot[x]))} -{Cos[x]^3/(I + Cot[x]), x, 9, (-(1/5))*Cos[x]^5 - (1/3)*I*Sin[x]^3 + (1/5)*I*Sin[x]^5} -{Cos[x]^2/(I + Cot[x]), x, 5, -((I*x)/8) + I/(8*(I - Cot[x])) + 1/(8*(I + Cot[x])^2) + I/(4*(I + Cot[x]))} -{Cos[x]^1/(I + Cot[x]), x, 8, (-(1/3))*Cos[x]^3 - (1/3)*I*Sin[x]^3} -{Sec[x]^1/(I + Cot[x]), x, 8, (-I)*ArcTanh[Sin[x]] - Cos[x] + I*Sin[x]} -{Sec[x]^2/(I + Cot[x]), x, 3, I*x - Log[Sin[x]] + Log[Tan[x]] - I*Tan[x]} -{Sec[x]^3/(I + Cot[x]), x, 8, (1/2)*I*ArcTanh[Sin[x]] + Sec[x] - (1/2)*I*Sec[x]*Tan[x]} -{Sec[x]^4/(I + Cot[x]), x, 4, Tan[x]^2/2 - (1/3)*I*Tan[x]^3} -{Sec[x]^5/(I + Cot[x]), x, 9, (1/8)*I*ArcTanh[Sin[x]] + Sec[x]^3/3 + (1/8)*I*Sec[x]*Tan[x] - (1/4)*I*Sec[x]^3*Tan[x]} -{Sec[x]^6/(I + Cot[x]), x, 4, Tan[x]^2/2 - (1/3)*I*Tan[x]^3 + Tan[x]^4/4 - (1/5)*I*Tan[x]^5} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^m (a+b Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cot[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[x]^4/(a + b*Cot[x]), x, 8, (a*(3*a^4 - 6*a^2*b^2 - b^4)*x)/(8*(a^2 + b^2)^3) - (a^4*b*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^3 + ((4*b*(2*a^2 + b^2) + a*(5*a^2 + b^2)*Cot[x])*Sin[x]^2)/(8*(a^2 + b^2)^2) - ((b + a*Cot[x])*Sin[x]^4)/(4*(a^2 + b^2))} -{Cos[x]^3/(a + b*Cot[x]), x, 10, (a^3*b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^2*b*Cos[x])/(a^2 + b^2)^2 - (b*Cos[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Sin[x])/(a^2 + b^2)^2 + (a*Sin[x])/(a^2 + b^2) - (a*Sin[x]^3)/(3*(a^2 + b^2))} -{Cos[x]^2/(a + b*Cot[x]), x, 7, (a*(a^2 - b^2)*x)/(2*(a^2 + b^2)^2) - (a^2*b*Log[b*Cos[x] + a*Sin[x]])/(a^2 + b^2)^2 + ((b + a*Cot[x])*Sin[x]^2)/(2*(a^2 + b^2))} -{Cos[x]^1/(a + b*Cot[x]), x, 6, (a*b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Cos[x])/(a^2 + b^2) + (a*Sin[x])/(a^2 + b^2)} -{Sec[x]^1/(a + b*Cot[x]), x, 6, ArcTanh[Sin[x]]/a + (b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])} -{Sec[x]^2/(a + b*Cot[x]), x, 3, -((b*Log[a + b*Cot[x]])/a^2) - (b*Log[Tan[x]])/a^2 + Tan[x]/a} -{Sec[x]^3/(a + b*Cot[x]), x, 9, ArcTanh[Sin[x]]/(2*a) + (b^2*ArcTanh[Sin[x]])/a^3 + (b*Sqrt[a^2 + b^2]*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/a^3 - (b*Sec[x])/a^2 + (Sec[x]*Tan[x])/(2*a)} -{Sec[x]^4/(a + b*Cot[x]), x, 3, -((b*(a^2 + b^2)*Log[a + b*Cot[x]])/a^4) - (b*(a^2 + b^2)*Log[Tan[x]])/a^4 + ((a^2 + b^2)*Tan[x])/a^3 - (b*Tan[x]^2)/(2*a^2) + Tan[x]^3/(3*a)} - - -(* Following hangs Mathematica 6 & 7: *) -{Sec[x]/(1 + 2*Cot[x]), x, 6, (2*ArcTanh[(Cos[x] - 2*Sin[x])/Sqrt[5]])/Sqrt[5] + ArcTanh[Sin[x]]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.10 (c+d x)^m (a+b cot)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.10 (c+d x)^m (a+b cot)^n.m deleted file mode 100644 index 9fd722a..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.10 (c+d x)^m (a+b cot)^n.m +++ /dev/null @@ -1,137 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cot[a+b x]^n*) - - -{x^3*Cot[a + b*x], x, 6, -((I*x^4)/4) + (x^3*Log[1 - E^(2*I*(a + b*x))])/b - (3*I*x^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) + (3*x*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) + (3*I*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{x^2*Cot[a + b*x], x, 5, -((I*x^3)/3) + (x^2*Log[1 - E^(2*I*(a + b*x))])/b - (I*x*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + PolyLog[3, E^(2*I*(a + b*x))]/(2*b^3)} -{x*Cot[a + b*x], x, 4, -((I*x^2)/2) + (x*Log[1 - E^(2*I*(a + b*x))])/b - (I*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Cot[a + b*x]/x, x, 0, Unintegrable[Cot[a + b*x]/x, x]} -{Cot[a + b*x]/x^2, x, 0, Unintegrable[Cot[a + b*x]/x^2, x]} - - -{x^3*Cot[a + b*x]^2, x, 7, -((I*x^3)/b) - x^4/4 - (x^3*Cot[a + b*x])/b + (3*x^2*Log[1 - E^(2*I*(a + b*x))])/b^2 - (3*I*x*PolyLog[2, E^(2*I*(a + b*x))])/b^3 + (3*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^4)} -{x^2*Cot[a + b*x]^2, x, 6, -((I*x^2)/b) - x^3/3 - (x^2*Cot[a + b*x])/b + (2*x*Log[1 - E^(2*I*(a + b*x))])/b^2 - (I*PolyLog[2, E^(2*I*(a + b*x))])/b^3} -{x*Cot[a + b*x]^2, x, 3, -x^2/2 - (x*Cot[a + b*x])/b + Log[Sin[a + b*x]]/b^2} -{Cot[a + b*x]^2/x, x, 0, Unintegrable[Cot[a + b*x]^2/x, x]} -{Cot[a + b*x]^2/x^2, x, 0, Unintegrable[Cot[a + b*x]^2/x^2, x]} - - -{x^3*Cot[a + b*x]^3, x, 13, -((3*I*x^2)/(2*b^2)) - x^3/(2*b) + (I*x^4)/4 - (3*x^2*Cot[a + b*x])/(2*b^2) - (x^3*Cot[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*I*(a + b*x))])/b^3 - (x^3*Log[1 - E^(2*I*(a + b*x))])/b - (3*I*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^4) + (3*I*x^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (3*I*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{x^2*Cot[a + b*x]^3, x, 9, -(x^2/(2*b)) + (I*x^3)/3 - (x*Cot[a + b*x])/b^2 - (x^2*Cot[a + b*x]^2)/(2*b) - (x^2*Log[1 - E^(2*I*(a + b*x))])/b + Log[Sin[a + b*x]]/b^3 + (I*x*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - PolyLog[3, E^(2*I*(a + b*x))]/(2*b^3)} -{x*Cot[a + b*x]^3, x, 7, -(x/(2*b)) + (I*x^2)/2 - Cot[a + b*x]/(2*b^2) - (x*Cot[a + b*x]^2)/(2*b) - (x*Log[1 - E^(2*I*(a + b*x))])/b + (I*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Cot[a + b*x]^3/x, x, 0, Unintegrable[Cot[a + b*x]^3/x, x]} -{Cot[a + b*x]^3/x^2, x, 0, Unintegrable[Cot[a + b*x]^3/x^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Cot[a+b x]^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cot[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Cot[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + I*a*Cot[e + f*x]), x, 5, (((-3*I)/8)*d^3*x)/(a*f^3) - (3*d*(c + d*x)^2)/(8*a*f^2) + ((I/4)*(c + d*x)^3)/(a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4*(a + I*a*Cot[e + f*x])) + (((3*I)/4)*d^2*(c + d*x))/(f^3*(a + I*a*Cot[e + f*x])) + (3*d*(c + d*x)^2)/(4*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^3)/(f*(a + I*a*Cot[e + f*x]))} -{(c + d*x)^2/(a + I*a*Cot[e + f*x]), x, 4, -(d^2*x)/(4*a*f^2) + ((I/4)*(c + d*x)^2)/(a*f) + (c + d*x)^3/(6*a*d) + ((I/4)*d^2)/(f^3*(a + I*a*Cot[e + f*x])) + (d*(c + d*x))/(2*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^2)/(f*(a + I*a*Cot[e + f*x]))} -{(c + d*x)^1/(a + I*a*Cot[e + f*x]), x, 3, ((I/4)*d*x)/(a*f) + (c + d*x)^2/(4*a*d) + d/(4*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x))/(f*(a + I*a*Cot[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Cot[e + f*x])), x, 7, -(Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) - ((I/2)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d) - ((I/2)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d) + (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)} -{1/((c + d*x)^2*(a + I*a*Cot[e + f*x])), x, 7, ((-I)*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - 1/(d*(c + d*x)*(a + I*a*Cot[e + f*x])) + (f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d^2) + (f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) + (I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^2)} -{1/((c + d*x)^3*(a + I*a*Cot[e + f*x])), x, 8, ((I/2)*f)/(a*d^2*(c + d*x)) + (f^2*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x)^2*(a + I*a*Cot[e + f*x])) - (I*f)/(d^2*(c + d*x)*(a + I*a*Cot[e + f*x])) + (I*f^2*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a*d^3) + (I*f^2*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)} - - -{(c + d*x)^3/(a + I*a*Cot[e + f*x])^2, x, 10, (3*d^3*E^((2*I)*e + (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((4*I)*e + (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d^2*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^2*f^3) + (((3*I)/128)*d^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^2*f^3) - (3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(64*a^2*f^2) + ((I/4)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^2*f) + (c + d*x)^4/(16*a^2*d)} -{(c + d*x)^2/(a + I*a*Cot[e + f*x])^2, x, 8, ((-I/8)*d^2*E^((2*I)*e + (2*I)*f*x))/(a^2*f^3) + ((I/128)*d^2*E^((4*I)*e + (4*I)*f*x))/(a^2*f^3) - (d*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(4*a^2*f^2) + (d*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(32*a^2*f^2) + ((I/4)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(a^2*f) - ((I/16)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(a^2*f) + (c + d*x)^3/(12*a^2*d)} -{(c + d*x)^1/(a + I*a*Cot[e + f*x])^2, x, 7, (((3*I)/16)*d*x)/(a^2*f) - (d*x^2)/(8*a^2) + (x*(c + d*x))/(4*a^2) + d/(16*f^2*(a + I*a*Cot[e + f*x])^2) - ((I/4)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^2) + (3*d)/(16*f^2*(a^2 + I*a^2*Cot[e + f*x])) - ((I/4)*(c + d*x))/(f*(a^2 + I*a^2*Cot[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Cot[e + f*x])^2), x, 21, -(Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + (Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + Log[c + d*x]/(4*a^2*d) + ((I/4)*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^2*d) - ((I/2)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^2*d) - ((I/2)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d) + (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + ((I/4)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d) - (Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d)} -{1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^2), x, 24, -(1/(4*a^2*d*(c + d*x))) + Cos[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cos[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) - (I*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (I*f*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^2*d^2) + (f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^2*d^2) + (I*Sin[2*e + 2*f*x])/(2*a^2*d*(c + d*x)) + Sin[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) - (I*Sin[4*e + 4*f*x])/(4*a^2*d*(c + d*x)) + (f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (I*f*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2)} - - -{(c + d*x)^3/(a + I*a*Cot[e + f*x])^3, x, 14, (9*d^3*E^((2*I)*e + (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((4*I)*e + (4*I)*f*x))/(1024*a^3*f^4) + (d^3*E^((6*I)*e + (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(a^3*f^3) + (((9*I)/256)*d^2*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(a^3*f^3) - (9*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(128*a^3*f^2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^3)/(a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^3)/(a^3*f) + (c + d*x)^4/(32*a^3*d)} -{(c + d*x)^2/(a + I*a*Cot[e + f*x])^3, x, 11, (((-3*I)/32)*d^2*E^((2*I)*e + (2*I)*f*x))/(a^3*f^3) + (((3*I)/256)*d^2*E^((4*I)*e + (4*I)*f*x))/(a^3*f^3) - ((I/864)*d^2*E^((6*I)*e + (6*I)*f*x))/(a^3*f^3) - (3*d*E^((2*I)*e + (2*I)*f*x)*(c + d*x))/(16*a^3*f^2) + (3*d*E^((4*I)*e + (4*I)*f*x)*(c + d*x))/(64*a^3*f^2) - (d*E^((6*I)*e + (6*I)*f*x)*(c + d*x))/(144*a^3*f^2) + (((3*I)/16)*E^((2*I)*e + (2*I)*f*x)*(c + d*x)^2)/(a^3*f) - (((3*I)/32)*E^((4*I)*e + (4*I)*f*x)*(c + d*x)^2)/(a^3*f) + ((I/48)*E^((6*I)*e + (6*I)*f*x)*(c + d*x)^2)/(a^3*f) + (c + d*x)^3/(24*a^3*d)} -{(c + d*x)^1/(a + I*a*Cot[e + f*x])^3, x, 11, (((11*I)/96)*d*x)/(a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) + d/(36*f^2*(a + I*a*Cot[e + f*x])^3) - ((I/6)*(c + d*x))/(f*(a + I*a*Cot[e + f*x])^3) + (5*d)/(96*a*f^2*(a + I*a*Cot[e + f*x])^2) - ((I/8)*(c + d*x))/(a*f*(a + I*a*Cot[e + f*x])^2) + (11*d)/(96*f^2*(a^3 + I*a^3*Cot[e + f*x])) - ((I/8)*(c + d*x))/(f*(a^3 + I*a^3*Cot[e + f*x]))} -{1/((c + d*x)^1*(a + I*a*Cot[e + f*x])^3), x, 53, (-3*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - (Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^3*d) - ((I/8)*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(a^3*d) + (((3*I)/8)*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^3*d) - (((3*I)/8)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^3*d) + (3*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (((3*I)/8)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^3*d) - (3*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - ((I/8)*Cos[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(a^3*d) + (Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)} -{1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^3), x, 60, -(1/(8*a^3*d*(c + d*x))) + (9*Cos[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cos[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) + Cos[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Cos[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*I*f*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*I*f*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*I*f*Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (3*f*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(4*a^3*d^2) - (3*f*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(2*a^3*d^2) + (3*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(4*a^3*d^2) + (15*I*Sin[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) + (3*Sin[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - (I*Sin[2*e + 2*f*x]^3)/(8*a^3*d*(c + d*x)) - (3*I*Sin[4*e + 4*f*x])/(8*a^3*d*(c + d*x)) + (3*I*Sin[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) + (3*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*I*f*Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*I*f*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) + (3*f*Cos[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (3*I*f*Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Cot[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + I*a*Cot[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m*(a + I*a*Cot[e + f*x])^2, x]} -{(c + d*x)^m*(a + I*a*Cot[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m*(a + I*a*Cot[e + f*x]), x]} -{(c + d*x)^m/(a + I*a*Cot[e + f*x])^1, x, 2, (c + d*x)^(1 + m)/(2*a*d*(1 + m)) + (I*2^(-2 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(a*f*(((-I)*f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + I*a*Cot[e + f*x])^2, x, 4, (c + d*x)^(1 + m)/(4*a^2*d*(1 + m)) + (I*2^(-2 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(a^2*f*(((-I)*f*(c + d*x))/d)^m) - (I*4^(-2 - m)*E^((4*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(a^2*f*(((-I)*f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + I*a*Cot[e + f*x])^3, x, 5, (c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) + ((3*I)*2^(-4 - m)*E^((2*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) - ((3*I)*2^(-5 - 2*m)*E^((4*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-4*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m) + (I*2^(-4 - m)*3^(-1 - m)*E^((6*I)*(e - (c*f)/d))*(c + d*x)^m*Gamma[1 + m, ((-6*I)*f*(c + d*x))/d])/(a^3*f*(((-I)*f*(c + d*x))/d)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cot[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Cot[e + f*x]), x, 8, (a*(c + d*x)^4)/(4*d) - ((I/4)*b*(c + d*x)^4)/d + (b*(c + d*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - (((3*I)/2)*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (3*b*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) + (((3*I)/4)*b*d^3*PolyLog[4, E^((2*I)*(e + f*x))])/f^4} -{(c + d*x)^2*(a + b*Cot[e + f*x]), x, 7, (a*(c + d*x)^3)/(3*d) - ((I/3)*b*(c + d*x)^3)/d + (b*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f - (I*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Cot[e + f*x]), x, 6, (a*(c + d*x)^2)/(2*d) - ((I/2)*b*(c + d*x)^2)/d + (b*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f - ((I/2)*b*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2} -{(a + b*Cot[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + b*Cot[e + f*x])/(c + d*x), x]} -{(a + b*Cot[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + b*Cot[e + f*x])/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Cot[e + f*x])^2, x, 15, ((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) - ((I/2)*a*b*(c + d*x)^4)/d - (b^2*(c + d*x)^4)/(4*d) - (b^2*(c + d*x)^3*Cot[e + f*x])/f + (3*b^2*d*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((3*I)*a*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (3*b^2*d^3*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^4) + (3*a*b*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/f^3 + (((3*I)/2)*a*b*d^3*PolyLog[4, E^((2*I)*(e + f*x))])/f^4} -{(c + d*x)^2*(a + b*Cot[e + f*x])^2, x, 13, ((-I)*b^2*(c + d*x)^2)/f + (a^2*(c + d*x)^3)/(3*d) - (((2*I)/3)*a*b*(c + d*x)^3)/d - (b^2*(c + d*x)^3)/(3*d) - (b^2*(c + d*x)^2*Cot[e + f*x])/f + (2*b^2*d*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f - (I*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((2*I)*a*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (a*b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/f^3} -{(c + d*x)^1*(a + b*Cot[e + f*x])^2, x, 9, -(b^2*c*x) - (b^2*d*x^2)/2 + (a^2*(c + d*x)^2)/(2*d) - (I*a*b*(c + d*x)^2)/d - (b^2*(c + d*x)*Cot[e + f*x])/f + (2*a*b*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f + (b^2*d*Log[Sin[e + f*x]])/f^2 - (I*a*b*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2} -{(a + b*Cot[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + b*Cot[e + f*x])^2/(c + d*x), x]} -{(a + b*Cot[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + b*Cot[e + f*x])^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Cot[e + f*x])^3, x, 28, (((-3*I)/2)*b^3*d*(c + d*x)^2)/f^2 - ((3*I)*a*b^2*(c + d*x)^3)/f - (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) - (((3*I)/4)*a^2*b*(c + d*x)^4)/d - (3*a*b^2*(c + d*x)^4)/(4*d) + ((I/4)*b^3*(c + d*x)^4)/d - (3*b^3*d*(c + d*x)^2*Cot[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Cot[e + f*x])/f - (b^3*(c + d*x)^3*Cot[e + f*x]^2)/(2*f) + (3*b^3*d^2*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - (((3*I)/2)*b^3*d^3*PolyLog[2, E^((2*I)*(e + f*x))])/f^4 - ((9*I)*a*b^2*d^2*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - (((9*I)/2)*a^2*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (((3*I)/2)*b^3*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (9*a*b^2*d^3*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^4) + (9*a^2*b*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) - (3*b^3*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) + (((9*I)/4)*a^2*b*d^3*PolyLog[4, E^((2*I)*(e + f*x))])/f^4 - (((3*I)/4)*b^3*d^3*PolyLog[4, E^((2*I)*(e + f*x))])/f^4} -{(c + d*x)^2*(a + b*Cot[e + f*x])^3, x, 22, -((b^3*c*d*x)/f) - (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) - (I*a^2*b*(c + d*x)^3)/d - (a*b^2*(c + d*x)^3)/d + ((I/3)*b^3*(c + d*x)^3)/d - (b^3*d*(c + d*x)*Cot[e + f*x])/f^2 - (3*a*b^2*(c + d*x)^2*Cot[e + f*x])/f - (b^3*(c + d*x)^2*Cot[e + f*x]^2)/(2*f) + (6*a*b^2*d*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f + (b^3*d^2*Log[Sin[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((3*I)*a^2*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (I*b^3*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (3*a^2*b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Cot[e + f*x])^3, x, 16, -3*a*b^2*c*x - (b^3*d*x)/(2*f) - (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (((3*I)/2)*a^2*b*(c + d*x)^2)/d + ((I/2)*b^3*(c + d*x)^2)/d - (b^3*d*Cot[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Cot[e + f*x])/f - (b^3*(c + d*x)*Cot[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)*Log[1 - E^((2*I)*(e + f*x))])/f + (3*a*b^2*d*Log[Sin[e + f*x]])/f^2 - (((3*I)/2)*a^2*b*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + ((I/2)*b^3*d*PolyLog[2, E^((2*I)*(e + f*x))])/f^2} -{(a + b*Cot[e + f*x])^3/(c + d*x)^1, x, 0, Unintegrable[(a + b*Cot[e + f*x])^3/(c + d*x), x]} -{(a + b*Cot[e + f*x])^3/(c + d*x)^2, x, 0, Unintegrable[(a + b*Cot[e + f*x])^3/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Cot[e + f*x]), x, 6, (c + d*x)^4/(4*(a - I*b)*d) - (b*(c + d*x)^3*Log[1 - ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f) + (3*I*b*d*(c + d*x)^2*PolyLog[2, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/(2*(a^2 + b^2)*f^2) - (3*b*d^2*(c + d*x)*PolyLog[3, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/(2*(a^2 + b^2)*f^3) - (3*I*b*d^3*PolyLog[4, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/(4*(a^2 + b^2)*f^4)} -{(c + d*x)^2/(a + b*Cot[e + f*x]), x, 5, (c + d*x)^3/(3*(a - I*b)*d) - (b*(c + d*x)^2*Log[1 - ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f) + (I*b*d*(c + d*x)*PolyLog[2, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f^2) - (b*d^2*PolyLog[3, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/(2*(a^2 + b^2)*f^3)} -{(c + d*x)^1/(a + b*Cot[e + f*x]), x, 4, (c + d*x)^2/(2*(a - I*b)*d) - (b*(c + d*x)*Log[1 - ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)*f) + (I*b*d*PolyLog[2, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/(2*(a^2 + b^2)*f^2)} -{1/((c + d*x)^1*(a + b*Cot[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Cot[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Cot[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Cot[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Cot[e + f*x])^2, x, 21, -((2*I*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f)) - (2*b^2*(c + d*x)^3)/((a - I*b)*(a + I*b)^2*(I*a + b - (I*a - b)*E^(2*I*e + 2*I*f*x))*f) + (c + d*x)^4/(4*(a + I*b)^2*d) - (b*(c + d*x)^4)/((a + I*b)^2*(I*a + b)*d) - (b^2*(c + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) - (2*b*(c + d*x)^3*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a - I*b)*(a + I*b)^2*f) - (2*I*b^2*(c + d*x)^3*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f) - (3*I*b^2*d^2*(c + d*x)*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3) - (3*b*d*(c + d*x)^2*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a + I*b)^2*(I*a + b)*f^2) - (3*b^2*d*(c + d*x)^2*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) + (3*b^2*d^3*PolyLog[3, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/(2*(a^2 + b^2)^2*f^4) - (3*b*d^2*(c + d*x)*PolyLog[3, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a - I*b)*(a + I*b)^2*f^3) - (3*I*b^2*d^2*(c + d*x)*PolyLog[3, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3) + (3*b*d^3*PolyLog[4, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/(2*(a + I*b)^2*(I*a + b)*f^4) + (3*b^2*d^3*PolyLog[4, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/(2*(a^2 + b^2)^2*f^4)} -{(c + d*x)^2/(a + b*Cot[e + f*x])^2, x, 18, -((2*I*b^2*(c + d*x)^2)/((a^2 + b^2)^2*f)) - (2*b^2*(c + d*x)^2)/((a - I*b)*(a + I*b)^2*(I*a + b - (I*a - b)*E^(2*I*e + 2*I*f*x))*f) + (c + d*x)^3/(3*(a + I*b)^2*d) - (4*b*(c + d*x)^3)/(3*(a + I*b)^2*(I*a + b)*d) - (4*b^2*(c + d*x)^3)/(3*(a^2 + b^2)^2*d) + (2*b^2*d*(c + d*x)*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) - (2*b*(c + d*x)^2*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a - I*b)*(a + I*b)^2*f) - (2*I*b^2*(c + d*x)^2*Log[1 - ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f) - (I*b^2*d^2*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3) - (2*b*d*(c + d*x)*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a + I*b)^2*(I*a + b)*f^2) - (2*b^2*d*(c + d*x)*PolyLog[2, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^2) - (b*d^2*PolyLog[3, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a - I*b)*(a + I*b)^2*f^3) - (I*b^2*d^2*PolyLog[3, ((a + I*b)*E^(2*I*e + 2*I*f*x))/(a - I*b)])/((a^2 + b^2)^2*f^3)} -{(c + d*x)^1/(a + b*Cot[e + f*x])^2, x, 5, -((c + d*x)^2/(2*(a^2 + b^2)*d)) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - I*b)^2*(a + I*b)*d*f^2) + (b*(c + d*x))/((a^2 + b^2)*f*(a + b*Cot[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)^2*f^2) + (I*a*b*d*PolyLog[2, ((a + I*b)*E^(2*I*(e + f*x)))/(a - I*b)])/((a^2 + b^2)^2*f^2)} -{1/((c + d*x)^1*(a + b*Cot[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Cot[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Cot[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Cot[e + f*x])^2), x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.2.1 (a+b cot)^m (c+d cot)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.2.1 (a+b cot)^m (c+d cot)^n.m deleted file mode 100644 index 0337d24..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.2.1 (a+b cot)^m (c+d cot)^n.m +++ /dev/null @@ -1,286 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+b Cot[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+I a Cot[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+I a Cot[e+f x])^m (d Cot[e+f x])^n with m symbolic*) - - -{(a + a*I*Cot[c + d*x])^n, x, 2, (I*(a + I*a*Cot[c + d*x])^n*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Cot[c + d*x])])/(2*d*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+a Cot[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^(n/2) (a+a Cot[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(5/2), x, 5, -((Sqrt[2]*a*e^(5/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) + (2*a*e^2*Sqrt[e*Cot[c + d*x]])/d - (2*a*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*a*(e*Cot[c + d*x])^(5/2))/(5*d)} -{(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(3/2), x, 4, -((Sqrt[2]*a*e^(3/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) - (2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*a*(e*Cot[c + d*x])^(3/2))/(3*d)} -{(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(1/2), x, 3, (Sqrt[2]*a*Sqrt[e]*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (2*a*Sqrt[e*Cot[c + d*x]])/d} -{(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(1/2), x, 2, (Sqrt[2]*a*ArcTan[(Sqrt[e]*(1 - Cot[c + d*x]))/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e])} -{(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2), x, 3, -((Sqrt[2]*a*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2))) + (2*a)/(d*e*Sqrt[e*Cot[c + d*x]])} -{(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2), x, 4, -((Sqrt[2]*a*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(5/2))) + (2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (2*a)/(d*e^2*Sqrt[e*Cot[c + d*x]])} - - -{(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(5/2), x, 16, (Sqrt[2]*a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (Sqrt[2]*a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d + (4*a^2*e^2*Sqrt[e*Cot[c + d*x]])/d - (4*a^2*(e*Cot[c + d*x])^(5/2))/(5*d) - (2*a^2*(e*Cot[c + d*x])^(7/2))/(7*d*e) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)} -{(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(3/2), x, 15, -((Sqrt[2]*a^2*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d) + (Sqrt[2]*a^2*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (4*a^2*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*a^2*(e*Cot[c + d*x])^(5/2))/(5*d*e) + (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) - (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)} -{(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(1/2), x, 15, -((Sqrt[2]*a^2*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d) + (Sqrt[2]*a^2*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/d - (4*a^2*Sqrt[e*Cot[c + d*x]])/d - (2*a^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) - (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d) + (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d)} -{(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(1/2), x, 14, (Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*Sqrt[e]) - (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*Sqrt[e]) - (2*a^2*Sqrt[e*Cot[c + d*x]])/(d*e) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*Sqrt[e]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*Sqrt[e])} -{(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2), x, 13, (Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(3/2)) - (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(3/2)) + (2*a^2)/(d*e*Sqrt[e*Cot[c + d*x]]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(3/2))} -{(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2), x, 14, -((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(5/2))) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(5/2)) + (2*a^2)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (4*a^2)/(d*e^2*Sqrt[e*Cot[c + d*x]]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(5/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(5/2))} -{(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2), x, 14, -((Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(7/2))) + (Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (4*a^2)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(7/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(Sqrt[2]*d*e^(7/2))} - - -{(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(5/2), x, 7, (2*Sqrt[2]*a^3*e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d + (4*a^3*e^2*Sqrt[e*Cot[c + d*x]])/d + (4*a^3*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (4*a^3*(e*Cot[c + d*x])^(5/2))/(5*d) - (40*a^3*(e*Cot[c + d*x])^(7/2))/(63*d*e) - (2*(e*Cot[c + d*x])^(7/2)*(a^3 + a^3*Cot[c + d*x]))/(9*d*e)} -{(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(3/2), x, 6, -((2*Sqrt[2]*a^3*e^(3/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) + (4*a^3*e*Sqrt[e*Cot[c + d*x]])/d - (4*a^3*(e*Cot[c + d*x])^(3/2))/(3*d) - (32*a^3*(e*Cot[c + d*x])^(5/2))/(35*d*e) - (2*(e*Cot[c + d*x])^(5/2)*(a^3 + a^3*Cot[c + d*x]))/(7*d*e)} -{(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(1/2), x, 5, -((2*Sqrt[2]*a^3*Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d) - (4*a^3*Sqrt[e*Cot[c + d*x]])/d - (8*a^3*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*(e*Cot[c + d*x])^(3/2)*(a^3 + a^3*Cot[c + d*x]))/(5*d*e)} -{(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(1/2), x, 4, (2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e]) - (16*a^3*Sqrt[e*Cot[c + d*x]])/(3*d*e) - (2*Sqrt[e*Cot[c + d*x]]*(a^3 + a^3*Cot[c + d*x]))/(3*d*e)} -{(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2)) - (4*a^3*Sqrt[e*Cot[c + d*x]])/(d*e^2) + (2*(a^3 + a^3*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]])} -{(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2), x, 4, -((2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(5/2))) + (16*a^3)/(3*d*e^2*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(3*d*e*(e*Cot[c + d*x])^(3/2))} -{(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2), x, 5, -((2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(7/2))) + (8*a^3)/(5*d*e^2*(e*Cot[c + d*x])^(3/2)) + (4*a^3)/(d*e^3*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(5*d*e*(e*Cot[c + d*x])^(5/2))} -{(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2), x, 6, (2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(9/2)) + (32*a^3)/(35*d*e^2*(e*Cot[c + d*x])^(5/2)) + (4*a^3)/(3*d*e^3*(e*Cot[c + d*x])^(3/2)) - (4*a^3)/(d*e^4*Sqrt[e*Cot[c + d*x]]) + (2*(a^3 + a^3*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(5/2), x, 7, (e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d) - (e^(5/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d) - (2*e^2*Sqrt[e*Cot[c + d*x]])/(a*d)} -{1/(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(3/2), x, 6, -((e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d)) + (e^(3/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d)} -{1/(a + a*Cot[c + d*x])*(e*Cot[c + d*x])^(1/2), x, 6, (Sqrt[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(a*d) + (Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(Sqrt[2]*a*d)} -{1/(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(1/2), x, 6, -(ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*Sqrt[e])) - ArcTanh[(Sqrt[e]*(1 + Cot[c + d*x]))/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*Sqrt[e])} -{1/(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2), x, 7, ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*e^(3/2)) - ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*e^(3/2)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]])} -{1/(a + a*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2), x, 10, -(ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]]/(a*d*e^(5/2))) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(Sqrt[2]*a*d*e^(5/2)) + 2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - 2/(a*d*e^2*Sqrt[e*Cot[c + d*x]])} - - -{1/(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(5/2), x, 17, -((3*e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d)) - (e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + (e^2*Sqrt[e*Cot[c + d*x]])/(2*d*(a^2 + a^2*Cot[c + d*x])) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)} -{1/(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(3/2), x, 18, (e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d) + (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (e*Sqrt[e*Cot[c + d*x]])/(2*d*(a^2 + a^2*Cot[c + d*x])) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)} -{1/(a + a*Cot[c + d*x])^2*(e*Cot[c + d*x])^(1/2), x, 17, (Sqrt[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d) + (Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) - (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(2*Sqrt[2]*a^2*d) + Sqrt[e*Cot[c + d*x]]/(2*d*(a^2 + a^2*Cot[c + d*x])) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(4*Sqrt[2]*a^2*d)} -{1/(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(1/2), x, 18, -((3*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*Sqrt[e])) - ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) - Sqrt[e*Cot[c + d*x]]/(2*d*e*(a^2 + a^2*Cot[c + d*x])) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*Sqrt[e])} -{1/(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2), x, 18, (5*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*e^(3/2)) - ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(3/2)) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(3/2)) + 5/(2*a^2*d*e*Sqrt[e*Cot[c + d*x]]) - 1/(2*d*e*Sqrt[e*Cot[c + d*x]]*(a^2 + a^2*Cot[c + d*x])) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(3/2)) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(3/2))} -{1/(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2), x, 20, -((7*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(2*a^2*d*e^(5/2))) + ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(5/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(2*Sqrt[2]*a^2*d*e^(5/2)) + 7/(6*a^2*d*e*(e*Cot[c + d*x])^(3/2)) - 9/(2*a^2*d*e^2*Sqrt[e*Cot[c + d*x]]) - 1/(2*d*e*(e*Cot[c + d*x])^(3/2)*(a^2 + a^2*Cot[c + d*x])) - Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(5/2)) + Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]]/(4*Sqrt[2]*a^2*d*e^(5/2))} - - -{1/(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(5/2), x, 8, -((e^(5/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d)) + (e^(5/2)*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) - (5*e^2*Sqrt[e*Cot[c + d*x]])/(8*a^3*d*(1 + Cot[c + d*x])) + (e^2*Sqrt[e*Cot[c + d*x]])/(4*a*d*(a + a*Cot[c + d*x])^2)} -{1/(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(3/2), x, 8, (5*e^(3/2)*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d) + (e^(3/2)*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) - (e*Sqrt[e*Cot[c + d*x]])/(4*a*d*(a + a*Cot[c + d*x])^2) + (e*Sqrt[e*Cot[c + d*x]])/(8*d*(a^3 + a^3*Cot[c + d*x]))} -{1/(a + a*Cot[c + d*x])^3*(e*Cot[c + d*x])^(1/2), x, 8, -((Sqrt[e]*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d)) - (Sqrt[e]*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(2*Sqrt[2]*a^3*d) + Sqrt[e*Cot[c + d*x]]/(4*a*d*(a + a*Cot[c + d*x])^2) + (3*Sqrt[e*Cot[c + d*x]])/(8*d*(a^3 + a^3*Cot[c + d*x]))} -{1/(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(1/2), x, 8, -((11*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*Sqrt[e])) - ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*Sqrt[e]) - (7*Sqrt[e*Cot[c + d*x]])/(8*a^3*d*e*(1 + Cot[c + d*x])) - Sqrt[e*Cot[c + d*x]]/(4*a*d*e*(a + a*Cot[c + d*x])^2)} -{1/(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2), x, 9, (31*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(3/2)) + ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(3/2)) + 27/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]) - 9/(8*a^3*d*e*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])) - 1/(4*a*d*e*Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2)} -{1/(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2), x, 10, -((59*ArcTan[Sqrt[e*Cot[c + d*x]]/Sqrt[e]])/(8*a^3*d*e^(5/2))) + ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])]/(2*Sqrt[2]*a^3*d*e^(5/2)) + 55/(24*a^3*d*e*(e*Cot[c + d*x])^(3/2)) - 63/(8*a^3*d*e^2*Sqrt[e*Cot[c + d*x]]) - 11/(8*a^3*d*e*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])) - 1/(4*a*d*e*(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+a Cot[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cot[x]^2*Sqrt[1 + Cot[x]], x, 12, (-Sqrt[(1/2)*(1 + Sqrt[2])])*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] + Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (2/3)*(1 + Cot[x])^(3/2) + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])])} -{Cot[x]^1*Sqrt[1 + Cot[x]], x, 6, Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] + Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] - 2*Sqrt[1 + Cot[x]]} - - -{Cot[x]^2*(1 + Cot[x])^(3/2), x, 8, (-Sqrt[-1 + Sqrt[2]])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] - Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] + 2*Sqrt[1 + Cot[x]] - (2/5)*(1 + Cot[x])^(5/2)} -{Cot[x]^1*(1 + Cot[x])^(3/2), x, 14, (-Sqrt[1 + Sqrt[2]])*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] + Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 2*Sqrt[1 + Cot[x]] - (2/3)*(1 + Cot[x])^(3/2) - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Cot[x]^2/Sqrt[1 + Cot[x]], x, 12, (-(1/2))*Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] + (1/2)*Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 2*Sqrt[1 + Cot[x]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]])} -{Cot[x]^1/Sqrt[1 + Cot[x]], x, 5, (1/2)*Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] + (1/2)*Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])]} - - -{Cot[x]^2/(1 + Cot[x])^(3/2), x, 6, (1/2)*Sqrt[(1/2)*(-1 + Sqrt[2])]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] + (1/2)*Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])] + 1/Sqrt[1 + Cot[x]]} -{Cot[x]^1/(1 + Cot[x])^(3/2), x, 13, (1/2)*Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (1/2)*Sqrt[(1/2)*(1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 1/Sqrt[1 + Cot[x]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])])} - - -{Cot[x]^2/(1 + Cot[x])^(5/2), x, 8, (1/4)*Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] + (1/4)*Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])] + 1/(3*(1 + Cot[x])^(3/2)) - 1/Sqrt[1 + Cot[x]]} -{Cot[x]^1/(1 + Cot[x])^(5/2), x, 13, (1/4)*Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (1/4)*Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 1/(3*(1 + Cot[x])^(3/2)) + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]]) - Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(8*Sqrt[1 + Sqrt[2]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+b Cot[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^(n/2) (a+b Cot[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Cot[c + d*x])*(e*Cot[c + d*x])^(3/2), x, 12, -(((a + b)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a + b)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*a*e*Sqrt[e*Cot[c + d*x]])/d - (2*b*(e*Cot[c + d*x])^(3/2))/(3*d) - ((a - b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a - b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])*(e*Cot[c + d*x])^(1/2), x, 11, ((a - b)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a - b)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*b*Sqrt[e*Cot[c + d*x]])/d - ((a + b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a + b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(1/2), x, 10, ((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])} -{(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2), x, 11, -(((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) + (2*a)/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))} -{(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2), x, 12, -(((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (2*a)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (2*b)/(d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))} - - -{(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(3/2), x, 13, -(((a^2 + 2*a*b - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a^2 + 2*a*b - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*(a^2 - b^2)*e*Sqrt[e*Cot[c + d*x]])/d - (4*a*b*(e*Cot[c + d*x])^(3/2))/(3*d) - (2*b^2*(e*Cot[c + d*x])^(5/2))/(5*d*e) - ((a^2 - 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a^2 - 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(1/2), x, 12, ((a^2 - 2*a*b - b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a^2 - 2*a*b - b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (4*a*b*Sqrt[e*Cot[c + d*x]])/d - (2*b^2*(e*Cot[c + d*x])^(3/2))/(3*d*e) - ((a^2 + 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a^2 + 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(1/2), x, 11, ((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (2*b^2*Sqrt[e*Cot[c + d*x]])/(d*e) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])} -{(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2), x, 11, -(((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) + (2*a^2)/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))} -{(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(5/2), x, 12, -(((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (2*a^2)/(3*d*e*(e*Cot[c + d*x])^(3/2)) + (4*a*b)/(d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))} -{(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2), x, 13, ((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (4*a*b)/(3*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*(a^2 - b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))} - - -{(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(3/2), x, 14, -(((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*a*(a^2 - 3*b^2)*e*Sqrt[e*Cot[c + d*x]])/d - (2*b*(3*a^2 - b^2)*(e*Cot[c + d*x])^(3/2))/(3*d) - (32*a*b^2*(e*Cot[c + d*x])^(5/2))/(35*d*e) - (2*b^2*(e*Cot[c + d*x])^(5/2)*(a + b*Cot[c + d*x]))/(7*d*e) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(1/2), x, 13, ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (2*b*(3*a^2 - b^2)*Sqrt[e*Cot[c + d*x]])/d - (8*a*b^2*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*b^2*(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x]))/(5*d*e) - ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d)} -{(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(1/2), x, 12, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (16*a*b^2*Sqrt[e*Cot[c + d*x]])/(3*d*e) - (2*b^2*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]))/(3*d*e) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])} -{(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2), x, 12, -(((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2))) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (2*b*(a^2 + b^2)*Sqrt[e*Cot[c + d*x]])/(d*e^2) + (2*a^2*(a + b*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]]) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2))} -{(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(5/2), x, 12, -(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2))) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (16*a^2*b)/(3*d*e^2*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(3*d*e*(e*Cot[c + d*x])^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2))} -{(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(7/2), x, 13, ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (8*a^2*b)/(5*d*e^2*(e*Cot[c + d*x])^(3/2)) - (2*a*(a^2 - 3*b^2))/(d*e^3*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(5*d*e*(e*Cot[c + d*x])^(5/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2))} -{(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(9/2), x, 14, ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(9/2)) + (32*a^2*b)/(35*d*e^2*(e*Cot[c + d*x])^(5/2)) - (2*a*(a^2 - 3*b^2))/(3*d*e^3*(e*Cot[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/(d*e^4*Sqrt[e*Cot[c + d*x]]) + (2*a^2*(a + b*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2)) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + b*Cot[c + d*x])*(e*Cot[c + d*x])^(5/2), x, 15, (2*a^(5/2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(3/2)*(a^2 + b^2)*d) - ((a + b)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*e^2*Sqrt[e*Cot[c + d*x]])/(b*d) + ((a - b)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(a + b*Cot[c + d*x])*(e*Cot[c + d*x])^(3/2), x, 14, -((2*a^(3/2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d)) - ((a - b)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a + b)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(a + b*Cot[c + d*x])*(e*Cot[c + d*x])^(1/2), x, 14, (2*Sqrt[a]*Sqrt[b]*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/((a^2 + b^2)*d) + ((a + b)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d)} -{1/(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(1/2), x, 14, -((2*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)*d*Sqrt[e])) + ((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) - ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*Sqrt[e]) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*Sqrt[e])} -{1/(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(3/2), x, 15, (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*(a^2 + b^2)*d*e^(3/2)) - ((a + b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + ((a + b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) + 2/(a*d*e*Sqrt[e*Cot[c + d*x]]) + ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2)) - ((a - b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(3/2))} -{1/(a + b*Cot[c + d*x])/(e*Cot[c + d*x])^(5/2), x, 16, -((2*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*(a^2 + b^2)*d*e^(5/2))) - ((a - b)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + ((a - b)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + 2/(3*a*d*e*(e*Cot[c + d*x])^(3/2)) - (2*b)/(a^2*d*e^2*Sqrt[e*Cot[c + d*x]]) - ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(5/2)) + ((a + b)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)*d*e^(5/2))} - - -{1/(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(7/2), x, 16, (a^(5/2)*(3*a^2 + 7*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(5/2)*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((3*a^2 + 2*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(b^2*(a^2 + b^2)*d) + (a^2*e^2*(e*Cot[c + d*x])^(3/2))/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(5/2), x, 15, -((a^(3/2)*(a^2 + 5*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(b^(3/2)*(a^2 + b^2)^2*d)) - ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(3/2), x, 15, -((Sqrt[a]*(a^2 - 3*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)^2*d)) - ((a^2 - 2*a*b - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (a*e*Sqrt[e*Cot[c + d*x]])/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((a^2 + 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(a + b*Cot[c + d*x])^2*(e*Cot[c + d*x])^(1/2), x, 15, (Sqrt[b]*(3*a^2 - b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b*Sqrt[e*Cot[c + d*x]])/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((a^2 - 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d)} -{1/(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(1/2), x, 15, -((b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*(a^2 + b^2)^2*d*Sqrt[e])) + ((a^2 - 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - ((a^2 - 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])) + ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e]) - ((a^2 + 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*Sqrt[e])} -{1/(a + b*Cot[c + d*x])^2/(e*Cot[c + d*x])^(3/2), x, 16, (b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*(a^2 + b^2)^2*d*e^(3/2)) - ((a^2 + 2*a*b - b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) + ((a^2 + 2*a*b - b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) + (2*a^2 + 3*b^2)/(a^2*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]) - b^2/(a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])) + ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2)) - ((a^2 - 2*a*b - b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^2*d*e^(3/2))} - - -{1/(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(9/2), x, 17, (a^(5/2)*(15*a^4 + 46*a^2*b^2 + 63*b^4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(7/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((15*a^4 + 31*a^2*b^2 + 8*b^4)*e^4*Sqrt[e*Cot[c + d*x]])/(4*b^3*(a^2 + b^2)^2*d) + (a^2*e^2*(e*Cot[c + d*x])^(5/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (a^2*(5*a^2 + 13*b^2)*e^3*(e*Cot[c + d*x])^(3/2))/(4*b^2*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(7/2), x, 16, -((a^(3/2)*(3*a^4 + 6*a^2*b^2 + 35*b^4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(5/2)*(a^2 + b^2)^3*d)) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^2*e^2*(e*Cot[c + d*x])^(3/2))/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (a^2*(3*a^2 + 11*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(4*b^2*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(5/2), x, 16, -((Sqrt[a]*(a^4 + 18*a^2*b^2 - 15*b^4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*b^(3/2)*(a^2 + b^2)^3*d)) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (a^2*e^2*Sqrt[e*Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) - (a*(a^2 + 9*b^2)*e^2*Sqrt[e*Cot[c + d*x]])/(4*b*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(3/2), x, 16, -(((3*a^4 - 26*a^2*b^2 + 3*b^4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*Sqrt[a]*Sqrt[b]*(a^2 + b^2)^3*d)) - ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (a*e*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) - ((3*a^2 - 5*b^2)*e*Sqrt[e*Cot[c + d*x]])/(4*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(a + b*Cot[c + d*x])^3*(e*Cot[c + d*x])^(1/2), x, 16, (Sqrt[b]*(15*a^4 - 18*a^2*b^2 - b^4)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(3/2)*(a^2 + b^2)^3*d) + ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) - ((a - b)*(a^2 + 4*a*b + b^2)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d) + (b*Sqrt[e*Cot[c + d*x]])/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (b*(7*a^2 - b^2)*Sqrt[e*Cot[c + d*x]])/(4*a*(a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*b + b^2)*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d)} -{1/(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(1/2), x, 16, -((b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(5/2)*(a^2 + b^2)^3*d*Sqrt[e])) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])^2) - (b^2*(11*a^2 + 3*b^2)*Sqrt[e*Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*e*(a + b*Cot[c + d*x])) + ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e])} -{1/(a + b*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2), x, 17, (b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(4*a^(7/2)*(a^2 + b^2)^3*d*e^(3/2)) - ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) + ((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) + (8*a^4 + 31*a^2*b^2 + 15*b^4)/(4*a^3*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]]) - b^2/(2*a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2) - (b^2*(13*a^2 + 5*b^2))/(4*a^2*(a^2 + b^2)^2*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])) + ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2)) - ((a + b)*(a^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*d*e^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+b Cot[e+f x])^m with m symbolic*) - - -{(a + b*Cot[c + d*x])^n, x, 5, -((b*(a + b*Cot[c + d*x])^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a - Sqrt[-b^2])])/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*d*(1 + n))) + (b*(a + b*Cot[c + d*x])^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Cot[c + d*x])/(a + Sqrt[-b^2])])/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*d*(1 + n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Cot[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+I a Cot[e+f x])^m*) - - -(* ::Section:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+a Cot[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Cot[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Cot[e+f x])^m with m symbolic*) - - -{(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n, x, 8, -((AppellF1[1 - n, -m, 1, 2 - n, -((b*Cot[e + f*x])/a), (-I)*Cot[e + f*x]]*Cot[e + f*x]*(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n)/((1 + (b*Cot[e + f*x])/a)^m*(2*f*(1 - n)))) - (AppellF1[1 - n, -m, 1, 2 - n, -((b*Cot[e + f*x])/a), I*Cot[e + f*x]]*Cot[e + f*x]*(a + b*Cot[e + f*x])^m*(d*Tan[e + f*x])^n)/((1 + (b*Cot[e + f*x])/a)^m*(2*f*(1 - n)))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Cot[e+f x])^m (c+d Cot[e+f x])^n*) - - -(* ::Section:: *) -(*Integrands of the form (a+a I Cot[e+f x])^m (c-c I Cot[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a I Cot[e+f x])^m (c+d Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a I Cot[e+f x])^m (c+d Cot[e+f x])^(n/2)*) - - -{(1 + I*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]], x, 3, (2*I*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)} -{(1 - I*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]], x, 3, -((2*I*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Cot[e+f x])^m (c+d Cot[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cot[e+f x])^m (c+d Cot[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x]), x, 2, ((a*A + b*B)*x)/(a^2 + b^2) - ((A*b - a*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)*d)} - - -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^2, x, 3, ((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2 + (A*b - a*B)/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((2*a*A*b - a^2*B + b^2*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^2*d)} - - -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^3, x, 4, ((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + (A*b - a*B)/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (2*a*A*b - a^2*B + b^2*B)/((a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cot[e+f x])^m (c+d Cot[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2), x, 10, ((a - I*b)^(5/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(5/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*(2*a*A*b + a^2*B - b^2*B)*Sqrt[a + b*Cot[c + d*x]])/d - (2*(A*b + a*B)*(a + b*Cot[c + d*x])^(3/2))/(3*d) - (2*B*(a + b*Cot[c + d*x])^(5/2))/(5*d)} -{(A + B*Cot[c + d*x])*(a + b*Cot[c + d*x])^(3/2), x, 9, ((a - I*b)^(3/2)*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - ((a + I*b)^(3/2)*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*(A*b + a*B)*Sqrt[a + b*Cot[c + d*x]])/d - (2*B*(a + b*Cot[c + d*x])^(3/2))/(3*d)} -{(A + B*Cot[c + d*x])*(a + b*Cot[c + d*x])^(1/2), x, 8, (Sqrt[a - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d - (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d - (2*B*Sqrt[a + b*Cot[c + d*x]])/d} - - -{(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(5/2), x, 10, -(((I*a - b)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/d) + ((a + I*b)^(5/2)*(I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(a^2 + b^2)*Sqrt[a + b*Cot[c + d*x]])/d - (2*b*(a + b*Cot[c + d*x])^(5/2))/(5*d)} -{(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(3/2), x, 13, (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (2*b*(a + b*Cot[c + d*x])^(3/2))/(3*d) + (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*(a^2 + b^2)*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} -{(-a + b*Cot[c + d*x])*(a + b*Cot[c + d*x])^(1/2), x, 13, (b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[a + b*Cot[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) - (2*b*Sqrt[a + b*Cot[c + d*x]])/d - (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Cot[c + d*x] + Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Cot[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(1/2), x, 7, ((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2), x, 8, ((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*(A*b - a*B))/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])} -{(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2), x, 9, ((I*A + B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) + (2*(A*b - a*B))/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3/2)) + (2*(2*a*A*b - a^2*B + b^2*B))/((a^2 + b^2)^2*d*Sqrt[a + b*Cot[c + d*x]])} - - -{(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(1/2), x, 7, -(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)} -{(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(3/2), x, 8, -(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(3/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) - (4*a*b)/((a^2 + b^2)*d*Sqrt[a + b*Cot[c + d*x]])} -{(-a + b*Cot[c + d*x])/(a + b*Cot[c + d*x])^(5/2), x, 9, -(((I*a - b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a - I*b]])/((a - I*b)^(5/2)*d)) + ((I*a + b)*ArcTanh[Sqrt[a + b*Cot[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (4*a*b)/(3*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^(3/2)) - (2*b*(3*a^2 - b^2))/((a^2 + b^2)^2*d*Sqrt[a + b*Cot[c + d*x]])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.7 (d trig)^m (a+b (c cot)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.7 (d trig)^m (a+b (c cot)^n)^p.m deleted file mode 100644 index 96a26b5..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.7 (d trig)^m (a+b (c cot)^n)^p.m +++ /dev/null @@ -1,205 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Cot[e+f x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Cot[e+f x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Cos[e+f x]^m (a+b Cot[e+f x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cot[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Cot[e+f x]^2)^p*) - - -{(A + C*Cot[c + d*x]^2)/Sqrt[b*Tan[c + d*x]], x, 15, -(((A - C)*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*Sqrt[b]*d)) + ((A - C)*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*Sqrt[b]*d) - ((A - C)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[b]*d) + ((A - C)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[b]*d) - (2*b*C)/(3*d*(b*Tan[c + d*x])^(3/2))} - - -(* ::Title::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Cot[c + d*x]^2)^1, x, 3, a*x - b*x - (b*Cot[c + d*x])/d} - - -{(a + b*Cot[c + d*x]^2)^2, x, 4, (a - b)^2*x - ((2*a - b)*b*Cot[c + d*x])/d - (b^2*Cot[c + d*x]^3)/(3*d)} - - -{(a + b*Cot[c + d*x]^2)^3, x, 4, (a - b)^3*x - (b*(3*a^2 - 3*a*b + b^2)*Cot[c + d*x])/d - ((3*a - b)*b^2*Cot[c + d*x]^3)/(3*d) - (b^3*Cot[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(a + b*Cot[c + d*x]^2)^1, x, 3, x/(a - b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)} - - -{1/(a + b*Cot[c + d*x]^2)^2, x, 5, x/(a - b)^2 + ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) + (b*Cot[c + d*x])/(2*a*(a - b)*d*(a + b*Cot[c + d*x]^2))} - - -{1/(a + b*Cot[c + d*x]^2)^3, x, 6, x/(a - b)^3 + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*d) + (b*Cot[c + d*x])/(4*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^2) + ((7*a - 3*b)*b*Cot[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^2)^(p/2) with a-b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(1 + Cot[x]^2)^(3/2), x, 4, (-(1/2))*ArcSinh[Cot[x]] - (1/2)*Cot[x]*Sqrt[Csc[x]^2]} -{Sqrt[1 + Cot[x]^2], x, 3, -ArcSinh[Cot[x]]} -{1/Sqrt[1 + Cot[x]^2], x, 3, -(Cot[x]/Sqrt[Csc[x]^2])} - - -{(-1 - Cot[x]^2)^(3/2), x, 5, (-(1/2))*ArcTan[Cot[x]/Sqrt[-Csc[x]^2]] + (1/2)*Cot[x]*Sqrt[-Csc[x]^2]} -{Sqrt[-1 - Cot[x]^2], x, 4, ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]} -{1/Sqrt[-1 - Cot[x]^2], x, 3, -(Cot[x]/Sqrt[-Csc[x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]^3/Sqrt[a + a*Cot[x]^2], x, 4, -(1/Sqrt[a*Csc[x]^2]) - Sqrt[a*Csc[x]^2]/a} -{Cot[x]^2/Sqrt[a + a*Cot[x]^2], x, 5, Cot[x]/Sqrt[a*Csc[x]^2] - (ArcTanh[Cos[x]]*Csc[x])/Sqrt[a*Csc[x]^2]} -{Cot[x]^1/Sqrt[a + a*Cot[x]^2], x, 3, 1/Sqrt[a*Csc[x]^2]} -{Tan[x]^1/Sqrt[a + a*Cot[x]^2], x, 5, ArcTanh[Sqrt[a*Csc[x]^2]/Sqrt[a]]/Sqrt[a] - 1/Sqrt[a*Csc[x]^2]} -{Tan[x]^2/Sqrt[a + a*Cot[x]^2], x, 5, Cot[x]/Sqrt[a*Csc[x]^2] + (Csc[x]*Sec[x])/Sqrt[a*Csc[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cot[x]^3*Sqrt[a + b*Cot[x]^2], x, 6, (-Sqrt[a - b])*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] + Sqrt[a + b*Cot[x]^2] - (a + b*Cot[x]^2)^(3/2)/(3*b)} -{Cot[x]^1*Sqrt[a + b*Cot[x]^2], x, 5, Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - Sqrt[a + b*Cot[x]^2]} -{Tan[x]^1*Sqrt[a + b*Cot[x]^2], x, 7, Sqrt[a]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]} - -{Cot[x]^2*Sqrt[a + b*Cot[x]^2], x, 7, Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((a - 2*b)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]])/(2*Sqrt[b]) - (1/2)*Cot[x]*Sqrt[a + b*Cot[x]^2]} -{Cot[x]^0*Sqrt[a + b*Cot[x]^2], x, 6, (-Sqrt[a - b])*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]} -{Tan[x]^2*Sqrt[a + b*Cot[x]^2], x, 5, Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] + Sqrt[a + b*Cot[x]^2]*Tan[x]} -{Tan[x]^4*Sqrt[a + b*Cot[x]^2], x, 6, (-Sqrt[a - b])*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a - b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(3*a) + (1/3)*Sqrt[a + b*Cot[x]^2]*Tan[x]^3} - - -{Cot[x]^3*(a + b*Cot[x]^2)^(3/2), x, 7, (-(a - b)^(3/2))*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] + (a - b)*Sqrt[a + b*Cot[x]^2] + (1/3)*(a + b*Cot[x]^2)^(3/2) - (a + b*Cot[x]^2)^(5/2)/(5*b)} -{Cot[x]^2*(a + b*Cot[x]^2)^(3/2), x, 8, (a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]])/(8*Sqrt[b]) - (1/8)*(5*a - 4*b)*Cot[x]*Sqrt[a + b*Cot[x]^2] - (1/4)*b*Cot[x]^3*Sqrt[a + b*Cot[x]^2]} -{Cot[x]^1*(a + b*Cot[x]^2)^(3/2), x, 6, (a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - (a - b)*Sqrt[a + b*Cot[x]^2] - (1/3)*(a + b*Cot[x]^2)^(3/2)} -{Tan[x]^1*(a + b*Cot[x]^2)^(3/2), x, 8, a^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - (a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - b*Sqrt[a + b*Cot[x]^2]} -{Tan[x]^2*(a + b*Cot[x]^2)^(3/2), x, 7, (a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - b^(3/2)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] + a*Sqrt[a + b*Cot[x]^2]*Tan[x]} - - -{(a + b*Cot[c + d*x]^2)^(5/2), x, 8, -(((a - b)^(5/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(8*d) - ((7*a - 4*b)*b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/(8*d) - (b*Cot[c + d*x]*(a + b*Cot[c + d*x]^2)^(3/2))/(4*d)} -{(a + b*Cot[c + d*x]^2)^(3/2), x, 7, -(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(2*d) - (b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/(2*d)} -{(a + b*Cot[c + d*x]^2)^(1/2), x, 6, -((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d} -{1/(a + b*Cot[c + d*x]^2)^(1/2), x, 3, -(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/(Sqrt[a - b]*d))} -{1/(a + b*Cot[c + d*x]^2)^(3/2), x, 4, -(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a - b)*d*Sqrt[a + b*Cot[c + d*x]^2])} -{1/(a + b*Cot[c + d*x]^2)^(5/2), x, 6, -(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(5/2)*d)) + (b*Cot[c + d*x])/(3*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^(3/2)) + ((5*a - 2*b)*b*Cot[c + d*x])/(3*a^2*(a - b)^2*d*Sqrt[a + b*Cot[c + d*x]^2])} -{1/(a + b*Cot[c + d*x]^2)^(7/2), x, 7, -(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^(5/2)) + ((9*a - 4*b)*b*Cot[c + d*x])/(15*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2)^(3/2)) + (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a + b*Cot[c + d*x]^2])} - - -{(1 - Cot[x]^2)^(3/2), x, 6, (5/2)*ArcSin[Cot[x]] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]] + (1/2)*Cot[x]*Sqrt[1 - Cot[x]^2]} -{Sqrt[1 - Cot[x]^2], x, 5, ArcSin[Cot[x]] - Sqrt[2]*ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]]} -{1/Sqrt[1 - Cot[x]^2], x, 3, -(ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]]/Sqrt[2])} - - -{(-1 + Cot[x]^2)^(3/2), x, 7, (5/2)*ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]] - (1/2)*Cot[x]*Sqrt[-1 + Cot[x]^2]} -{Sqrt[-1 + Cot[x]^2], x, 6, -ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]} -{1/Sqrt[-1 + Cot[x]^2], x, 3, -(ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]/Sqrt[2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]^3/Sqrt[a + b*Cot[x]^2], x, 5, -(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]) - Sqrt[a + b*Cot[x]^2]/b} -{Cot[x]^2/Sqrt[a + b*Cot[x]^2], x, 6, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[a - b] - ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[b]} -{Cot[x]^1/Sqrt[a + b*Cot[x]^2], x, 4, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]} -{Tan[x]^1/Sqrt[a + b*Cot[x]^2], x, 7, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]} -{Tan[x]^2/Sqrt[a + b*Cot[x]^2], x, 5, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[a - b] + (Sqrt[a + b*Cot[x]^2]*Tan[x])/a} - - -{Cot[x]^3/(a + b*Cot[x]^2)^(3/2), x, 5, -(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2)) + a/((a - b)*b*Sqrt[a + b*Cot[x]^2])} -{Cot[x]^2/(a + b*Cot[x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) - Cot[x]/((a - b)*Sqrt[a + b*Cot[x]^2])} -{Cot[x]^1/(a + b*Cot[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2) - 1/((a - b)*Sqrt[a + b*Cot[x]^2])} -{Tan[x]^1/(a + b*Cot[x]^2)^(3/2), x, 8, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/a^(3/2) - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2) + b/(a*(a - b)*Sqrt[a + b*Cot[x]^2])} -{Tan[x]^2/(a + b*Cot[x]^2)^(3/2), x, 6, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) + (b*Tan[x])/(a*(a - b)*Sqrt[a + b*Cot[x]^2]) + ((a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(a^2*(a - b))} - - -{Cot[x]^3/(a + b*Cot[x]^2)^(5/2), x, 6, -(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2)) + a/(3*(a - b)*b*(a + b*Cot[x]^2)^(3/2)) + 1/((a - b)^2*Sqrt[a + b*Cot[x]^2])} -{Cot[x]^2/(a + b*Cot[x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) - Cot[x]/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) - ((2*a + b)*Cot[x])/(3*a*(a - b)^2*Sqrt[a + b*Cot[x]^2])} -{Cot[x]^1/(a + b*Cot[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2) - 1/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) - 1/((a - b)^2*Sqrt[a + b*Cot[x]^2])} -{Tan[x]^1/(a + b*Cot[x]^2)^(5/2), x, 9, ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/a^(5/2) - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2) + b/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/2)) + ((2*a - b)*b)/(a^2*(a - b)^2*Sqrt[a + b*Cot[x]^2])} -{Tan[x]^2/(a + b*Cot[x]^2)^(5/2), x, 7, ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) + (b*Tan[x])/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/2)) + ((7*a - 4*b)*b*Tan[x])/(3*a^2*(a - b)^2*Sqrt[a + b*Cot[x]^2]) + ((a - 4*b)*(3*a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(3*a^3*(a - b)^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Cot[e+f x]^3)^n*) - - -{1/(1 + Cot[x]^3), x, 7, x/2 - (1/6)*Log[1 + Cot[x]] + (1/3)*Log[1 - Cot[x] + Cot[x]^2] + (1/2)*Log[Sin[x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^4)^p*) - - -(* ::Subsection:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[e+f x]^m (a+b Cot[e+f x]^4)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cot[x]*Sqrt[a + b*Cot[x]^4], x, 8, (1/2)*Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + (1/2)*Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - (1/2)*Sqrt[a + b*Cot[x]^4]} - - -{Cot[x]*(a + b*Cot[x]^4)^(3/2), x, 9, (1/4)*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + (1/2)*(a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - (1/4)*(2*(a + b) - b*Cot[x]^2)*Sqrt[a + b*Cot[x]^4] - (1/6)*(a + b*Cot[x]^4)^(3/2)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cot[x]/Sqrt[a + b*Cot[x]^4], x, 4, ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])} - - -{Cot[x]/(a + b*Cot[x]^4)^(3/2), x, 6, ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(3/2)) - (a + b*Cot[x]^2)/(2*a*(a + b)*Sqrt[a + b*Cot[x]^4])} - - -{Cot[x]/(a + b*Cot[x]^4)^(5/2), x, 7, ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a + b*Cot[x]^2)/(6*a*(a + b)*(a + b*Cot[x]^4)^(3/2)) - (3*a^2 + b*(5*a + 2*b)*Cot[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Cot[x]^4])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.9 trig^m (a+b cot^n+c cot^(2 n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.9 trig^m (a+b cot^n+c cot^(2 n))^p.m deleted file mode 100644 index 7b7c9db..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.4 Cotangent/4.4.9 trig^m (a+b cot^n+c cot^(2 n))^p.m +++ /dev/null @@ -1,68 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[d+e x]^m (a+b Cot[d+e x]^n+c Cot[d+e x]^(2 n))^p*) - - -(* ::Section:: *) -(*Integrands of the form Sin[d+e x]^m (a+b Cot[d+e x]^n+c Cot[d+e x]^(2 n))^p*) - - -(* ::Section:: *) -(*Integrands of the form Cos[d+e x]^m (a+b Cot[d+e x]^n+c Cot[d+e x]^(2 n))^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cot[d+e x]^m (a+b Cot[d+e x]^n+c Cot[d+e x]^(2 n))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[d+e x]^m (a+b Cot[d+e x]+c Cot[d+e x]^2)^(p/2)*) - - -{Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 15, -((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) + (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(7/2)*e) + Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/(c*e) - (Cot[d + e*x]^2*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(3*c*e) - ((15*b^2 - 16*a*c - 10*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^3*e)} -{Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 11, (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/(c*e)} -{Cot[d + e*x]^1/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 6, -((Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)} -{Tan[d + e*x]^1/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 10, ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(Sqrt[a]*e) + (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e)} -{Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 14, -(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(Sqrt[a]*e)) + ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(5/2)*e) - (Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) + (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]*e) - (3*b*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^2*e) + (Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(2*a*e)} - - -{Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 21, -((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) + (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(5/2)*e) - (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(256*c^(9/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e - (b*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) + (b*(7*b^2 - 12*a*c)*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(128*c^4*e) + (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*c*e) - (Cot[d + e*x]^2*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(5*c*e) - ((35*b^2 - 32*a*c - 42*b*c*Cot[d + e*x])*(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2))/(240*c^3*e)} -{Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 16, (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) - (b*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(5/2)*e) - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e + (b*(b + 2*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(8*c^2*e) - (a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2)/(3*c*e)} -{Cot[d + e*x]^1*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 10, -((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*Sqrt[c]*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) - Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]/e} -{Tan[d + e*x]^1*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 18, (Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)} -{Tan[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2], x, 21, -((Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTan[(b^2 + (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e)) - (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/e - ((b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(3/2)*e) + (Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*ArcTanh[(b^2 + (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + b*Sqrt[a^2 + b^2 - 2*a*c + c^2]*Cot[d + e*x])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*Sqrt[a^2 + b^2 + c*(c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - a*(2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(1/4)*e) + ((2*a + b*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(4*a*e)} - - -{Cot[d + e*x]^7/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 20, -((3*b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(5/2)*e)) + (5*b*(7*b^2 - 12*a*c)*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(16*c^(9/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*Cot[d + e*x]^2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*Cot[d + e*x]^4*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - ((7*b^2 - 16*a*c)*Cot[d + e*x]^2*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(3*c^2*(b^2 - 4*a*c)*e) + (2*b*Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c*(b^2 - 4*a*c)*e) + ((3*b^2 - 8*a*c - 2*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e) - ((105*b^4 - 460*a*b^2*c + 256*a^2*c^2 - 2*b*c*(35*b^2 - 116*a*c)*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(24*c^4*(b^2 - 4*a*c)*e)} -{Cot[d + e*x]^5/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 14, (3*b*ArcTanh[(b + 2*c*Cot[d + e*x])/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(2*c^(5/2)*e) - (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) + (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*Cot[d + e*x]^2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - ((3*b^2 - 8*a*c - 2*b*c*Cot[d + e*x])*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])/(c^2*(b^2 - 4*a*c)*e)} -{Cot[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 10, (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(2*a + b*Cot[d + e*x]))/((b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])} -{Cot[d + e*x]^1/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 7, -((Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])} -{Tan[d + e*x]^1/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 13, ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(a^(3/2)*e) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])} -{Tan[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2), x, 18, -(ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]/(a^(3/2)*e)) + (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(7/2)*e) - (1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e))*(Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]) + (1/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e))*(Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])]) + (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (2*(a*(b^2 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) - (b*(15*b^2 - 52*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^3*(b^2 - 4*a*c)*e) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x])*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + ((5*b^2 - 12*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x]^2)/(2*a^2*(b^2 - 4*a*c)*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cot[d+e x]^m (a+b Cot[d+e x]^2+c Cot[d+e x]^4)^(p/2)*) - - -{Cot[d + e*x]^5/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 8, ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) + ((b + 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*c^(3/2)*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*c*e)} -{Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 7, -(ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)) - ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[c]*e)} -{Cot[d + e*x]^1/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 4, ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)} -{Tan[d + e*x]^1/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 8, ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a]*e) - ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e)} -{Tan[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 11, -(ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a]*e)) - (b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*a^(3/2)*e) + ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*Sqrt[a - b + c]*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a*e)} - - -{Cot[d + e*x]^5*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 9, (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - ((b^3 + 2*b^2*c - 4*b*(a - 2*c)*c - 8*c^2*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(32*c^(5/2)*e) + (((b - 2*c)*(b + 4*c) + 2*c*(b + 2*c)*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(16*c^2*e) - (a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2)/(6*c*e)} -{Cot[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 8, -((Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e)) + ((b^2 + 4*b*c - 4*c*(a + 2*c))*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(16*c^(3/2)*e) - ((b - 4*c + 2*c*Cot[d + e*x]^2)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])/(8*c*e)} -{Cot[d + e*x]^1*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 8, (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]/(2*e)} -{Tan[d + e*x]^1*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 10, (Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e)} -{Tan[d + e*x]^3*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4], x, 22, -((Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e)) + (b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[a]*e) + (Sqrt[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (b*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*Sqrt[c]*e) - (Sqrt[c]*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*e) + (Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*e)} - - -{Cot[d + e*x]^7/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 8, -(ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e)) - ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*c^(3/2)*e) - (a*(b^2 - a*(b + 2*c)) + (b^3 + 2*a^2*c - a*b*(b + 3*c))*Cot[d + e*x]^2)/(c*(a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])} -{Cot[d + e*x]^5/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 6, ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (a*(2*a - b) + ((a - b)*b + 2*a*c)*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])} -{Cot[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 6, -(ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e)) + (a*(b - 2*c) + (2*a - b)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])} -{Cot[d + e*x]^1/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 6, ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])} -{Tan[d + e*x]^1/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 12, ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*a^(3/2)*e) - ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) - (b^2 - 2*a*c + b*c*Cot[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) + (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])} -{Tan[d + e*x]^3/(a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4)^(3/2), x, 16, -(ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*a^(3/2)*e)) - (3*b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(4*a^(5/2)*e) + ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])]/(2*(a - b + c)^(3/2)*e) + (b^2 - 2*a*c + b*c*Cot[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) - (b^2 - 2*a*c - b*c + (b - 2*c)*c*Cot[d + e*x]^2)/((a - b + c)*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) - ((b^2 - 2*a*c + b*c*Cot[d + e*x]^2)*Tan[d + e*x]^2)/(a*(b^2 - 4*a*c)*e*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]) + ((3*b^2 - 8*a*c)*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(2*a^2*(b^2 - 4*a*c)*e)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.0 (a sec)^m (b trg)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.0 (a sec)^m (b trg)^n.m deleted file mode 100644 index 84462a8..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.0 (a sec)^m (b trg)^n.m +++ /dev/null @@ -1,520 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Sec[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sec[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[c+d x])^n*) - - -{Sec[a + b*x]^1, x, 1, ArcTanh[Sin[a + b*x]]/b} -{Sec[a + b*x]^2, x, 2, Tan[a + b*x]/b} -{Sec[a + b*x]^3, x, 2, ArcTanh[Sin[a + b*x]]/(2*b) + (Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{Sec[a + b*x]^4, x, 2, Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)} -{Sec[a + b*x]^5, x, 3, (3*ArcTanh[Sin[a + b*x]])/(8*b) + (3*Sec[a + b*x]*Tan[a + b*x])/(8*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(4*b)} -{Sec[a + b*x]^6, x, 2, Tan[a + b*x]/b + (2*Tan[a + b*x]^3)/(3*b) + Tan[a + b*x]^5/(5*b)} -{Sec[a + b*x]^7, x, 4, (5*ArcTanh[Sin[a + b*x]])/(16*b) + (5*Sec[a + b*x]*Tan[a + b*x])/(16*b) + (5*Sec[a + b*x]^3*Tan[a + b*x])/(24*b) + (Sec[a + b*x]^5*Tan[a + b*x])/(6*b)} -{Sec[a + b*x]^8, x, 2, Tan[a + b*x]/b + Tan[a + b*x]^3/b + (3*Tan[a + b*x]^5)/(5*b) + Tan[a + b*x]^7/(7*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[c+d x])^(n/2)*) - - -{Sec[a + b*x]^(7/2), x, 4, -((6*Sqrt[Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/(5*b)) + (6*Sqrt[Sec[a + b*x]]*Sin[a + b*x])/(5*b) + (2*Sec[a + b*x]^(5/2)*Sin[a + b*x])/(5*b)} -{Sec[a + b*x]^(5/2), x, 3, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/(3*b) + (2*Sec[a + b*x]^(3/2)*Sin[a + b*x])/(3*b)} -{Sec[a + b*x]^(3/2), x, 3, -((2*Sqrt[Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/b) + (2*Sqrt[Sec[a + b*x]]*Sin[a + b*x])/b} -{Sec[a + b*x]^(1/2), x, 2, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/b} -{1/Sec[a + b*x]^(1/2), x, 2, (2*Sqrt[Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/b} -{1/Sec[a + b*x]^(3/2), x, 3, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/(3*b) + (2*Sin[a + b*x])/(3*b*Sqrt[Sec[a + b*x]])} -{1/Sec[a + b*x]^(5/2), x, 3, (6*Sqrt[Cos[a + b*x]]*EllipticE[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/(5*b) + (2*Sin[a + b*x])/(5*b*Sec[a + b*x]^(3/2))} -{1/Sec[a + b*x]^(7/2), x, 4, (10*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[Sec[a + b*x]])/(21*b) + (2*Sin[a + b*x])/(7*b*Sec[a + b*x]^(5/2)) + (10*Sin[a + b*x])/(21*b*Sqrt[Sec[a + b*x]])} - - -{(c*Sec[a + b*x])^(7/2), x, 4, -((6*c^4*EllipticE[(1/2)*(a + b*x), 2])/(5*b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])) + (6*c^3*Sqrt[c*Sec[a + b*x]]*Sin[a + b*x])/(5*b) + (2*c*(c*Sec[a + b*x])^(5/2)*Sin[a + b*x])/(5*b)} -{(c*Sec[a + b*x])^(5/2), x, 3, (2*c^2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[c*Sec[a + b*x]])/(3*b) + (2*c*(c*Sec[a + b*x])^(3/2)*Sin[a + b*x])/(3*b)} -{(c*Sec[a + b*x])^(3/2), x, 3, -((2*c^2*EllipticE[(1/2)*(a + b*x), 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])) + (2*c*Sqrt[c*Sec[a + b*x]]*Sin[a + b*x])/b} -{(c*Sec[a + b*x])^(1/2), x, 2, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[c*Sec[a + b*x]])/b} -{1/(c*Sec[a + b*x])^(1/2), x, 2, (2*EllipticE[(1/2)*(a + b*x), 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(3/2), x, 3, (2*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[c*Sec[a + b*x]])/(3*b*c^2) + (2*Sin[a + b*x])/(3*b*c*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(5/2), x, 3, (6*EllipticE[(1/2)*(a + b*x), 2])/(5*b*c^2*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (2*Sin[a + b*x])/(5*b*c*(c*Sec[a + b*x])^(3/2))} -{1/(c*Sec[a + b*x])^(7/2), x, 4, (10*Sqrt[Cos[a + b*x]]*EllipticF[(1/2)*(a + b*x), 2]*Sqrt[c*Sec[a + b*x]])/(21*b*c^4) + (2*Sin[a + b*x])/(7*b*c*(c*Sec[a + b*x])^(5/2)) + (10*Sin[a + b*x])/(21*b*c^3*Sqrt[c*Sec[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[c+d x])^(n/3)*) - - -{Sec[a + b*x]^(4/3), x, 2, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[a + b*x]^2]*Sec[a + b*x]^(1/3)*Sin[a + b*x])/(b*Sqrt[Sin[a + b*x]^2])} -{Sec[a + b*x]^(2/3), x, 2, -((3*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*Sec[a + b*x]^(1/3)*Sqrt[Sin[a + b*x]^2]))} -{Sec[a + b*x]^(1/3), x, 2, -((3*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[a + b*x]^2]*Sin[a + b*x])/(2*b*Sec[a + b*x]^(2/3)*Sqrt[Sin[a + b*x]^2]))} -{1/Sec[a + b*x]^(1/3), x, 2, -((3*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sin[a + b*x])/(4*b*Sec[a + b*x]^(4/3)*Sqrt[Sin[a + b*x]^2]))} -{1/Sec[a + b*x]^(2/3), x, 2, -((3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*Sec[a + b*x]^(5/3)*Sqrt[Sin[a + b*x]^2]))} -{1/Sec[a + b*x]^(4/3), x, 2, -((3*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[a + b*x]^2]*Sin[a + b*x])/(7*b*Sec[a + b*x]^(7/3)*Sqrt[Sin[a + b*x]^2]))} - - -{(c*Sec[a + b*x])^(4/3), x, 2, (3*c*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[a + b*x]^2]*(c*Sec[a + b*x])^(1/3)*Sin[a + b*x])/(b*Sqrt[Sin[a + b*x]^2])} -{(c*Sec[a + b*x])^(2/3), x, 2, -((3*c*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(c*Sec[a + b*x])^(1/3)*Sqrt[Sin[a + b*x]^2]))} -{(c*Sec[a + b*x])^(1/3), x, 2, -((3*c*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[a + b*x]^2]*Sin[a + b*x])/(2*b*(c*Sec[a + b*x])^(2/3)*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Sec[a + b*x])^(1/3), x, 2, -((3*c*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[a + b*x]^2]*Sin[a + b*x])/(4*b*(c*Sec[a + b*x])^(4/3)*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Sec[a + b*x])^(2/3), x, 2, -((3*c*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[a + b*x]^2]*Sin[a + b*x])/(5*b*(c*Sec[a + b*x])^(5/3)*Sqrt[Sin[a + b*x]^2]))} -{1/(c*Sec[a + b*x])^(4/3), x, 2, -((3*c*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[a + b*x]^2]*Sin[a + b*x])/(7*b*(c*Sec[a + b*x])^(7/3)*Sqrt[Sin[a + b*x]^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[c+d x])^n with n symbolic*) - - -{Sec[a + b*x]^n, x, 2, -((Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*Sec[a + b*x]^(-1 + n)*Sin[a + b*x])/(b*(1 - n)*Sqrt[Sin[a + b*x]^2]))} - - -{(c*Sec[a + b*x])^n, x, 2, -((c*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[a + b*x]^2]*(c*Sec[a + b*x])^(-1 + n)*Sin[a + b*x])/(b*(1 - n)*Sqrt[Sin[a + b*x]^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sec[c+d x]^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[c+d x]^p)^(n/2) with p positive integer*) - - -{(Sec[x]^2)^(7/2), x, 5, (5/16)*ArcSinh[Tan[x]] + (5/16)*Sqrt[Sec[x]^2]*Tan[x] + (5/24)*(Sec[x]^2)^(3/2)*Tan[x] + (1/6)*(Sec[x]^2)^(5/2)*Tan[x]} -{(Sec[x]^2)^(5/2), x, 4, (3/8)*ArcSinh[Tan[x]] + (3/8)*Sqrt[Sec[x]^2]*Tan[x] + (1/4)*(Sec[x]^2)^(3/2)*Tan[x]} -{(Sec[x]^2)^(3/2), x, 3, (1/2)*ArcSinh[Tan[x]] + (1/2)*Sqrt[Sec[x]^2]*Tan[x]} -{(Sec[x]^2)^(1/2), x, 2, ArcSinh[Tan[x]]} -{1/(Sec[x]^2)^(1/2), x, 2, Tan[x]/Sqrt[Sec[x]^2]} -{1/(Sec[x]^2)^(3/2), x, 3, Tan[x]/(3*(Sec[x]^2)^(3/2)) + (2*Tan[x])/(3*Sqrt[Sec[x]^2])} -{1/(Sec[x]^2)^(5/2), x, 4, Tan[x]/(5*(Sec[x]^2)^(5/2)) + (4*Tan[x])/(15*(Sec[x]^2)^(3/2)) + (8*Tan[x])/(15*Sqrt[Sec[x]^2])} -{1/(Sec[x]^2)^(7/2), x, 5, Tan[x]/(7*(Sec[x]^2)^(7/2)) + (6*Tan[x])/(35*(Sec[x]^2)^(5/2)) + (8*Tan[x])/(35*(Sec[x]^2)^(3/2)) + (16*Tan[x])/(35*Sqrt[Sec[x]^2])} - - -{(a*Sec[x]^2)^(7/2), x, 6, (5/16)*a^(7/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]] + (5/16)*a^3*Sqrt[a*Sec[x]^2]*Tan[x] + (5/24)*a^2*(a*Sec[x]^2)^(3/2)*Tan[x] + (1/6)*a*(a*Sec[x]^2)^(5/2)*Tan[x]} -{(a*Sec[x]^2)^(5/2), x, 5, (3/8)*a^(5/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]] + (3/8)*a^2*Sqrt[a*Sec[x]^2]*Tan[x] + (1/4)*a*(a*Sec[x]^2)^(3/2)*Tan[x]} -{(a*Sec[x]^2)^(3/2), x, 4, (1/2)*a^(3/2)*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]] + (1/2)*a*Sqrt[a*Sec[x]^2]*Tan[x]} -{(a*Sec[x]^2)^(1/2), x, 3, Sqrt[a]*ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[a*Sec[x]^2]]} -{1/(a*Sec[x]^2)^(1/2), x, 2, Tan[x]/Sqrt[a*Sec[x]^2]} -{1/(a*Sec[x]^2)^(3/2), x, 3, Tan[x]/(3*(a*Sec[x]^2)^(3/2)) + (2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])} -{1/(a*Sec[x]^2)^(5/2), x, 4, Tan[x]/(5*(a*Sec[x]^2)^(5/2)) + (4*Tan[x])/(15*a*(a*Sec[x]^2)^(3/2)) + (8*Tan[x])/(15*a^2*Sqrt[a*Sec[x]^2])} -{1/(a*Sec[x]^2)^(7/2), x, 5, Tan[x]/(7*(a*Sec[x]^2)^(7/2)) + (6*Tan[x])/(35*a*(a*Sec[x]^2)^(5/2)) + (8*Tan[x])/(35*a^2*(a*Sec[x]^2)^(3/2)) + (16*Tan[x])/(35*a^3*Sqrt[a*Sec[x]^2])} - - -{(a*Sec[x]^3)^(5/2), x, 7, (-(154/195))*a^2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3] + (154/195)*a^2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x] + (154/585)*a^2*Sqrt[a*Sec[x]^3]*Tan[x] + (22/117)*a^2*Sec[x]^2*Sqrt[a*Sec[x]^3]*Tan[x] + (2/13)*a^2*Sec[x]^4*Sqrt[a*Sec[x]^3]*Tan[x]} -{(a*Sec[x]^3)^(3/2), x, 5, (10/21)*a*Cos[x]^(3/2)*EllipticF[x/2, 2]*Sqrt[a*Sec[x]^3] + (10/21)*a*Sqrt[a*Sec[x]^3]*Sin[x] + (2/7)*a*Sec[x]*Sqrt[a*Sec[x]^3]*Tan[x]} -{(a*Sec[x]^3)^(1/2), x, 4, -2*Cos[x]^(3/2)*EllipticE[x/2, 2]*Sqrt[a*Sec[x]^3] + 2*Cos[x]*Sqrt[a*Sec[x]^3]*Sin[x]} -{1/(a*Sec[x]^3)^(1/2), x, 4, (2*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (2*Tan[x])/(3*Sqrt[a*Sec[x]^3])} -{1/(a*Sec[x]^3)^(3/2), x, 5, (14*EllipticE[x/2, 2])/(15*a*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (14*Sin[x])/(45*a*Sqrt[a*Sec[x]^3]) + (2*Cos[x]^2*Sin[x])/(9*a*Sqrt[a*Sec[x]^3])} -{1/(a*Sec[x]^3)^(5/2), x, 7, (26*EllipticF[x/2, 2])/(77*a^2*Cos[x]^(3/2)*Sqrt[a*Sec[x]^3]) + (78*Cos[x]*Sin[x])/(385*a^2*Sqrt[a*Sec[x]^3]) + (26*Cos[x]^3*Sin[x])/(165*a^2*Sqrt[a*Sec[x]^3]) + (2*Cos[x]^5*Sin[x])/(15*a^2*Sqrt[a*Sec[x]^3]) + (26*Tan[x])/(77*a^2*Sqrt[a*Sec[x]^3])} - - -{(a*Sec[x]^4)^(7/2), x, 3, a^3*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + 2*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x] + 3*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^3 + (20/7)*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^5 + (5/3)*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^7 + (6/11)*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^9 + (1/13)*a^3*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^11} -{(a*Sec[x]^4)^(5/2), x, 3, a^2*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (4/3)*a^2*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x] + (6/5)*a^2*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^3 + (4/7)*a^2*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^5 + (1/9)*a^2*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^7} -{(a*Sec[x]^4)^(3/2), x, 3, a*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (2/3)*a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x] + (1/5)*a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x]^3} -{(a*Sec[x]^4)^(1/2), x, 3, Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x]} -{1/(a*Sec[x]^4)^(1/2), x, 3, (x*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4]) + Tan[x]/(2*Sqrt[a*Sec[x]^4])} -{1/(a*Sec[x]^4)^(3/2), x, 5, (5*x*Sec[x]^2)/(16*a*Sqrt[a*Sec[x]^4]) + (5*Cos[x]*Sin[x])/(24*a*Sqrt[a*Sec[x]^4]) + (Cos[x]^3*Sin[x])/(6*a*Sqrt[a*Sec[x]^4]) + (5*Tan[x])/(16*a*Sqrt[a*Sec[x]^4])} -{1/(a*Sec[x]^4)^(5/2), x, 7, (63*x*Sec[x]^2)/(256*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]*Sin[x])/(128*a^2*Sqrt[a*Sec[x]^4]) + (21*Cos[x]^3*Sin[x])/(160*a^2*Sqrt[a*Sec[x]^4]) + (9*Cos[x]^5*Sin[x])/(80*a^2*Sqrt[a*Sec[x]^4]) + (Cos[x]^7*Sin[x])/(10*a^2*Sqrt[a*Sec[x]^4]) + (63*Tan[x])/(256*a^2*Sqrt[a*Sec[x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ((b Sec[c+d x])^p)^n with n symbolic*) - - -{((b*Sec[c + d*x])^p)^n, x, 3, -((Cos[c + d*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[c + d*x]^2]*((b*Sec[c + d*x])^p)^n*Sin[c + d*x])/(d*(1 - n*p)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Sec[c+d x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a (b Sec[c+d x])^p)^n with p symbolic*) - - -{(a*(b*Sec[c + d*x])^p)^n, x, 3, -((Cos[c + d*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[c + d*x]^2]*(a*(b*Sec[c + d*x])^p)^n*Sin[c + d*x])/(d*(1 - n*p)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Title:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Trg[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[c+d x])^m (b Sec[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (b Sec[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*Sqrt[b*Sec[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (10*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*b*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b^3*d)} -{Sec[c + d*x]^3*Sqrt[b*Sec[c + d*x]], x, 5, -((6*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b^2*d)} -{Sec[c + d*x]^2*Sqrt[b*Sec[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*Sqrt[b*Sec[c + d*x]], x, 4, -((2*b*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^0*Sqrt[b*Sec[c + d*x]], x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/d} -{Cos[c + d*x]^1*Sqrt[b*Sec[c + d*x]], x, 3, (2*b*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^2*Sqrt[b*Sec[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b*Sin[c + d*x])/(3*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^3*Sqrt[b*Sec[c + d*x]], x, 4, (6*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^4*Sqrt[b*Sec[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b^3*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*b*Sin[c + d*x])/(21*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^5*Sqrt[b*Sec[c + d*x]], x, 5, (14*b*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^4*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*b^2*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))} - - -{Sec[c + d*x]^3*(b*Sec[c + d*x])^(3/2), x, 5, (10*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (10*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b^2*d)} -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(3/2), x, 5, -((6*b^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^(3/2), x, 4, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^(3/2), x, 3, -((2*b^2*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^(3/2), x, 3, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/d} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(3/2), x, 3, (2*b^2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^(3/2), x, 4, (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b^2*Sin[c + d*x])/(3*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^4*(b*Sec[c + d*x])^(3/2), x, 4, (6*b^2*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^3*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^5*(b*Sec[c + d*x])^(3/2), x, 5, (10*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b^4*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*b^2*Sin[c + d*x])/(21*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^6*(b*Sec[c + d*x])^(3/2), x, 5, (14*b^2*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^5*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*b^3*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(5/2), x, 5, (10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (10*b*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^(5/2), x, 5, -((6*b^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^(5/2), x, 3, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^(5/2), x, 4, -((2*b^3*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(5/2), x, 3, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/d} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^(5/2), x, 3, (2*b^3*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^4*(b*Sec[c + d*x])^(5/2), x, 4, (2*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b^3*Sin[c + d*x])/(3*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^5*(b*Sec[c + d*x])^(5/2), x, 4, (6*b^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^4*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^6*(b*Sec[c + d*x])^(5/2), x, 5, (10*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b^5*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*b^3*Sin[c + d*x])/(21*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^7*(b*Sec[c + d*x])^(5/2), x, 5, (14*b^3*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^6*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*b^4*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))} - - -{(b*Sec[c + d*x])^(7/2), x, 4, -((6*b^4*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*b^3*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^5/Sqrt[b*Sec[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b*d) + (10*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^2*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b^4*d)} -{Sec[c + d*x]^4/Sqrt[b*Sec[c + d*x]], x, 5, -((6*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*b*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b^3*d)} -{Sec[c + d*x]^3/Sqrt[b*Sec[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b*d) + (2*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*b^2*d)} -{Sec[c + d*x]^2/Sqrt[b*Sec[c + d*x]], x, 4, -((2*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^1/Sqrt[b*Sec[c + d*x]], x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(b*d)} -{Sec[c + d*x]^0/Sqrt[b*Sec[c + d*x]], x, 2, (2*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^1/Sqrt[b*Sec[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b*d) + (2*Sin[c + d*x])/(3*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^2/Sqrt[b*Sec[c + d*x]], x, 4, (6*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^3/Sqrt[b*Sec[c + d*x]], x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b*d) + (2*b^2*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*Sin[c + d*x])/(21*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^4/Sqrt[b*Sec[c + d*x]], x, 5, (14*EllipticE[(1/2)*(c + d*x), 2])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^3*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*b*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))} - - -{Sec[c + d*x]^6/(b*Sec[c + d*x])^(3/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^2*d) + (10*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^3*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b^5*d)} -{Sec[c + d*x]^5/(b*Sec[c + d*x])^(3/2), x, 5, -((6*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*b^2*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b^4*d)} -{Sec[c + d*x]^4/(b*Sec[c + d*x])^(3/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*b^3*d)} -{Sec[c + d*x]^3/(b*Sec[c + d*x])^(3/2), x, 4, -((2*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*d)} -{Sec[c + d*x]^2/(b*Sec[c + d*x])^(3/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(b^2*d)} -{Sec[c + d*x]^1/(b*Sec[c + d*x])^(3/2), x, 3, (2*EllipticE[(c + d*x)/2, 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^0/(b*Sec[c + d*x])^(3/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*Sin[c + d*x])/(3*b*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^1/(b*Sec[c + d*x])^(3/2), x, 4, (6*EllipticE[(1/2)*(c + d*x), 2])/(5*b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^2/(b*Sec[c + d*x])^(3/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^2*d) + (2*b*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*Sin[c + d*x])/(21*b*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^3/(b*Sec[c + d*x])^(3/2), x, 5, (14*EllipticE[(1/2)*(c + d*x), 2])/(15*b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*Sin[c + d*x])/(45*d*(b*Sec[c + d*x])^(3/2))} - - -{Sec[c + d*x]^7/(b*Sec[c + d*x])^(5/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^3*d) + (10*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*b^4*d) + (2*(b*Sec[c + d*x])^(7/2)*Sin[c + d*x])/(7*b^6*d)} -{Sec[c + d*x]^6/(b*Sec[c + d*x])^(5/2), x, 5, -((6*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (6*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*b^3*d) + (2*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b^5*d)} -{Sec[c + d*x]^5/(b*Sec[c + d*x])^(5/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*b^4*d)} -{Sec[c + d*x]^4/(b*Sec[c + d*x])^(5/2), x, 4, -((2*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(b^3*d)} -{Sec[c + d*x]^3/(b*Sec[c + d*x])^(5/2), x, 3, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[b*Sec[c + d*x]])/(b^3*d)} -{Sec[c + d*x]^2/(b*Sec[c + d*x])^(5/2), x, 3, (2*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^1/(b*Sec[c + d*x])^(5/2), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^0/(b*Sec[c + d*x])^(5/2), x, 3, (6*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(b*Sec[c + d*x])^(5/2), x, 5, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^3*d) + (2*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(5/2)) + (10*Sin[c + d*x])/(21*b^2*d*Sqrt[b*Sec[c + d*x]])} -{Cos[c + d*x]^2/(b*Sec[c + d*x])^(5/2), x, 5, (14*EllipticE[(1/2)*(c + d*x), 2])/(15*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b*Sin[c + d*x])/(9*d*(b*Sec[c + d*x])^(7/2)) + (14*Sin[c + d*x])/(45*b*d*(b*Sec[c + d*x])^(3/2))} - - -{(b*Sec[c + d*x])^(-7/2), x, 4, (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^4*d) + (2*Sin[c + d*x])/(7*b*d*(b*Sec[c + d*x])^(5/2)) + (10*Sin[c + d*x])/(21*b^3*d*Sqrt[b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[c+d x])^(m/2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[b*Sec[c + d*x]]*Sec[c + d*x]^(9/2), x, 4, (3*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(8*d*Sqrt[Sec[c + d*x]]) + (3*Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (Sec[c + d*x]^(7/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sqrt[b*Sec[c + d*x]]*Sec[c + d*x]^(7/2), x, 3, (Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d)} -{Sqrt[b*Sec[c + d*x]]*Sec[c + d*x]^(5/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(2*d*Sqrt[Sec[c + d*x]]) + (Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Sqrt[b*Sec[c + d*x]]*Sec[c + d*x]^(3/2), x, 3, (Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Sqrt[b*Sec[c + d*x]]*Sec[c + d*x]^(1/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(d*Sqrt[Sec[c + d*x]])} -{Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(1/2), x, 2, (x*Sqrt[b*Sec[c + d*x]])/Sqrt[Sec[c + d*x]]} -{Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(3/2), x, 2, (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])} -{Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(5/2), x, 3, (x*Sqrt[b*Sec[c + d*x]])/(2*Sqrt[Sec[c + d*x]]) + (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2))} -{Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(7/2), x, 3, (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) - (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Sec[c + d*x]])} -{Sqrt[b*Sec[c + d*x]]/Sec[c + d*x]^(9/2), x, 4, (3*x*Sqrt[b*Sec[c + d*x]])/(8*Sqrt[Sec[c + d*x]]) + (Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(7/2)) + (3*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2))} - - -{(b*Sec[c + d*x])^(3/2)*Sec[c + d*x]^(7/2), x, 4, (3*b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(8*d*Sqrt[Sec[c + d*x]]) + (3*b*Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (b*Sec[c + d*x]^(7/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{(b*Sec[c + d*x])^(3/2)*Sec[c + d*x]^(5/2), x, 3, (b*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (b*Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d)} -{(b*Sec[c + d*x])^(3/2)*Sec[c + d*x]^(3/2), x, 3, (b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(2*d*Sqrt[Sec[c + d*x]]) + (b*Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(b*Sec[c + d*x])^(3/2)*Sec[c + d*x]^(1/2), x, 3, (b*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/2), x, 2, (b*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(d*Sqrt[Sec[c + d*x]])} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2), x, 2, (b*x*Sqrt[b*Sec[c + d*x]])/Sqrt[Sec[c + d*x]]} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2), x, 2, (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2), x, 3, (b*x*Sqrt[b*Sec[c + d*x]])/(2*Sqrt[Sec[c + d*x]]) + (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2))} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2), x, 3, (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) - (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Sec[c + d*x]])} -{(b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(11/2), x, 4, (3*b*x*Sqrt[b*Sec[c + d*x]])/(8*Sqrt[Sec[c + d*x]]) + (b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sec[c + d*x]^(7/2)) + (3*b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2))} - - -{(b*Sec[c + d*x])^(5/2)*Sec[c + d*x]^(7/2), x, 3, (b^2*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*b^2*Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d) + (b^2*Sec[c + d*x]^(9/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^5)/(5*d)} -{(b*Sec[c + d*x])^(5/2)*Sec[c + d*x]^(3/2), x, 3, (b^2*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (b^2*Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d)} -{(b*Sec[c + d*x])^(5/2)*Sec[c + d*x]^(1/2), x, 3, (b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(2*d*Sqrt[Sec[c + d*x]]) + (b^2*Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/2), x, 3, (b^2*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2), x, 2, (b^2*ArcTanh[Sin[c + d*x]]*Sqrt[b*Sec[c + d*x]])/(d*Sqrt[Sec[c + d*x]])} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2), x, 2, (b^2*x*Sqrt[b*Sec[c + d*x]])/Sqrt[Sec[c + d*x]]} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2), x, 2, (b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2), x, 3, (b^2*x*Sqrt[b*Sec[c + d*x]])/(2*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2))} -{(b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(11/2), x, 3, (b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]) - (b^2*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)/Sqrt[b*Sec[c + d*x]], x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*d*Sqrt[b*Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/Sqrt[b*Sec[c + d*x]], x, 3, (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/Sqrt[b*Sec[c + d*x]], x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/Sqrt[b*Sec[c + d*x]], x, 2, (x*Sqrt[Sec[c + d*x]])/Sqrt[b*Sec[c + d*x]]} -{1/(Sec[c + d*x]^(1/2)*Sqrt[b*Sec[c + d*x]]), x, 2, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*Sqrt[b*Sec[c + d*x]]), x, 3, (x*Sqrt[Sec[c + d*x]])/(2*Sqrt[b*Sec[c + d*x]]) + Sin[c + d*x]/(2*d*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]), x, 3, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Sec[c + d*x]]) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)/(b*Sec[c + d*x])^(3/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*b*d*Sqrt[b*Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(7/2)/(b*Sec[c + d*x])^(3/2), x, 3, (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/(b*Sec[c + d*x])^(3/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/(b*Sec[c + d*x])^(3/2), x, 2, (x*Sqrt[Sec[c + d*x]])/(b*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/(b*Sec[c + d*x])^(3/2), x, 2, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(1/2)*(b*Sec[c + d*x])^(3/2)), x, 3, (x*Sqrt[Sec[c + d*x]])/(2*b*Sqrt[b*Sec[c + d*x]]) + Sin[c + d*x]/(2*b*d*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^(3/2)), x, 3, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[b*Sec[c + d*x]]) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x]^3)/(3*b*d*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^(3/2)), x, 4, (3*x*Sqrt[Sec[c + d*x]])/(8*b*Sqrt[b*Sec[c + d*x]]) + Sin[c + d*x]/(4*b*d*Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]) + (3*Sin[c + d*x])/(8*b*d*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]])} - - -{Sec[c + d*x]^(11/2)/(b*Sec[c + d*x])^(5/2), x, 3, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*b^2*d*Sqrt[b*Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b^2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(9/2)/(b*Sec[c + d*x])^(5/2), x, 3, (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b^2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(7/2)/(b*Sec[c + d*x])^(5/2), x, 2, (ArcTanh[Sin[c + d*x]]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/(b*Sec[c + d*x])^(5/2), x, 2, (x*Sqrt[Sec[c + d*x]])/(b^2*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/(b*Sec[c + d*x])^(5/2), x, 2, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/(b*Sec[c + d*x])^(5/2), x, 3, (x*Sqrt[Sec[c + d*x]])/(2*b^2*Sqrt[b*Sec[c + d*x]]) + Sin[c + d*x]/(2*b^2*d*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(1/2)*(b*Sec[c + d*x])^(5/2)), x, 3, (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*Sec[c + d*x]]) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x]^3)/(3*b^2*d*Sqrt[b*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^(5/2)), x, 4, (3*x*Sqrt[Sec[c + d*x]])/(8*b^2*Sqrt[b*Sec[c + d*x]]) + Sin[c + d*x]/(4*b^2*d*Sec[c + d*x]^(5/2)*Sqrt[b*Sec[c + d*x]]) + (3*Sin[c + d*x])/(8*b^2*d*Sqrt[Sec[c + d*x]]*Sqrt[b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (b Sec[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(1/3), x, 3, (3*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b*d*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]*(b*Sec[c + d*x])^(1/3), x, 3, (3*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])} -{(b*Sec[c + d*x])^(1/3), x, 2, -((3*b*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]*(b*Sec[c + d*x])^(1/3), x, 3, -((3*b^2*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(1/3), x, 3, -((3*b^3*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(8/3)*Sqrt[Sin[c + d*x]^2]))} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3), x, 3, (3*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]*(b*Sec[c + d*x])^(4/3), x, 3, (3*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{(b*Sec[c + d*x])^(4/3), x, 2, (3*b*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Sec[c + d*x])^(4/3), x, 3, -((3*b^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3), x, 3, -((3*b^3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^2/(b*Sec[c + d*x])^(1/3), x, 3, (3*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b*d*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]/(b*Sec[c + d*x])^(1/3), x, 3, -((3*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]))} -{(b*Sec[c + d*x])^(-1/3), x, 2, -((3*b*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]/(b*Sec[c + d*x])^(1/3), x, 3, -((3*b^2*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^2/(b*Sec[c + d*x])^(1/3), x, 3, -((3*b^3*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3)*Sqrt[Sin[c + d*x]^2]))} - - -{Sec[c + d*x]^2/(b*Sec[c + d*x])^(4/3), x, 3, -((3*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]/(b*Sec[c + d*x])^(4/3), x, 3, -((3*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]))} -{(b*Sec[c + d*x])^(-4/3), x, 2, -((3*b*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]/(b*Sec[c + d*x])^(4/3), x, 3, -((3*b^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^2/(b*Sec[c + d*x])^(4/3), x, 3, -((3*b^3*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2]*Sin[c + d*x])/(13*d*(b*Sec[c + d*x])^(13/3)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (b Sec[c+d x])^n with m symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3), x, 3, (3*b*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3), x, 3, -((3*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3), x, 3, -((3*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^m/(b*Sec[c + d*x])^(1/3), x, 3, -((3*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^m/(b*Sec[c + d*x])^(2/3), x, 3, -((3*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]))} -{Sec[c + d*x]^m/(b*Sec[c + d*x])^(4/3), x, 3, -((3*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (b Sec[c+d x])^n with n symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^n, x, 3, -((Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*Sqrt[Sin[c + d*x]^2]))} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^n, x, 3, (Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^n, x, 3, (Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^n, x, 2, -((b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^n, x, 3, -((b^2*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^n, x, 3, -((b^3*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]))} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^n, x, 3, -((b^4*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(4 - n)*Sqrt[Sin[c + d*x]^2]))} - - -{Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n, x, 3, (2*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n, x, 3, (2*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n, x, 3, -((2*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]))} -{(b*Sec[c + d*x])^n/Sqrt[Sec[c + d*x]], x, 3, -((2*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]))} -{(b*Sec[c + d*x])^n/Sec[c + d*x]^(3/2), x, 3, -((2*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]))} -{(b*Sec[c + d*x])^n/Sec[c + d*x]^(5/2), x, 3, -((2*Hypergeometric2F1[1/2, (1/4)*(7 - 2*n), (1/4)*(11 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 - 2*n)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]))} - - -(* ::Section:: *) -(*Integrands of the form (a Sec[c+d x])^m (b Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^(m/2) (b Csc[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]*(d*Sec[a + b*x])^(7/2), x, 2, (2*d*(d*Sec[a + b*x])^(5/2))/(5*b)} -{Sin[a + b*x]*(d*Sec[a + b*x])^(5/2), x, 2, (2*d*(d*Sec[a + b*x])^(3/2))/(3*b)} -{Sin[a + b*x]*(d*Sec[a + b*x])^(3/2), x, 2, (2*d*Sqrt[d*Sec[a + b*x]])/b} -{Sin[a + b*x]*(d*Sec[a + b*x])^(1/2), x, 2, -((2*d)/(b*Sqrt[d*Sec[a + b*x]]))} -{Sin[a + b*x]/(d*Sec[a + b*x])^(1/2), x, 2, -((2*d)/(3*b*(d*Sec[a + b*x])^(3/2)))} - - -{Sin[a + b*x]^3*(d*Sec[a + b*x])^(5/2), x, 3, 2*d^3/(b*Sqrt[(d*Sec[a + b*x])]) + (2/3)*d*(d*Sec[a + b*x])^(3/2)/b} -{Sin[a + b*x]^3*(d*Sec[a + b*x])^(9/2), x, 3, (-(2/3))*d^3*(d*Sec[a + b*x])^(3/2)/b + (2/7)*d*(d*Sec[a + b*x])^(7/2)/b} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (a Sec[e+f x])^(m/2) (b Csc[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -{(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(9/2), x, 5, -((4*c*d^3*(d*Csc[a + b*x])^(3/2))/(7*b*Sqrt[c*Sec[a + b*x]])) - (2*c*d*(d*Csc[a + b*x])^(7/2))/(7*b*Sqrt[c*Sec[a + b*x]]) + (4*d^4*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(7*b)} -{(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(7/2), x, 2, -((8*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*Sqrt[c*Sec[a + b*x]])) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*Sqrt[c*Sec[a + b*x]])} -{(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(5/2), x, 4, -((2*c*d*(d*Csc[a + b*x])^(3/2))/(3*b*Sqrt[c*Sec[a + b*x]])) + (2*d^2*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b)} -{(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(3/2), x, 1, -((2*c*d*Sqrt[d*Csc[a + b*x]])/(b*Sqrt[c*Sec[a + b*x]]))} -{(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(1/2), x, 3, (Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/b} -{(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(1/2), x, 12, -((ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} -{(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(3/2), x, 4, -(c/(b*d*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]])) + (Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*d^2)} -{(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(5/2), x, 13, -(c/(2*b*d*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]])) - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(4*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(4*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (3*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(8*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (3*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(8*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} - - -{(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(9/2), x, 3, (64*c*d^5*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]]) - (16*c*d^3*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]])/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]])/(7*b)} -{(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(7/2), x, 6, (24*c*d^5*Sqrt[c*Sec[a + b*x]])/(5*b*(d*Csc[a + b*x])^(3/2)) - (12*c*d^3*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(5*b) - (2*c*d*(d*Csc[a + b*x])^(5/2)*Sqrt[c*Sec[a + b*x]])/(5*b) - (24*c^2*d^4*EllipticE[a - Pi/4 + b*x, 2])/(5*b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(5/2), x, 2, (8*c*d^3*Sqrt[c*Sec[a + b*x]])/(3*b*Sqrt[d*Csc[a + b*x]]) - (2*c*d*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]])/(3*b)} -{(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(3/2), x, 5, (4*c*d^3*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2)) - (2*c*d*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]])/b - (4*c^2*d^2*EllipticE[a - Pi/4 + b*x, 2])/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(1/2), x, 1, (2*c*d*Sqrt[c*Sec[a + b*x]])/(b*Sqrt[d*Csc[a + b*x]])} -{(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(1/2), x, 4, (2*c*d*Sqrt[c*Sec[a + b*x]])/(b*(d*Csc[a + b*x])^(3/2)) - (2*c^2*EllipticE[a - Pi/4 + b*x, 2])/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(3/2), x, 13, (2*c*Sqrt[c*Sec[a + b*x]])/(b*d*Sqrt[d*Csc[a + b*x]]) + (c^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) - (c^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) + (c^2*Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) - (c^2*Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]])} -{(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(5/2), x, 4, (2*c*Sqrt[c*Sec[a + b*x]])/(b*d*(d*Csc[a + b*x])^(3/2)) - (3*c^2*EllipticE[a - Pi/4 + b*x, 2])/(b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} - - -{(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(9/2), x, 6, (40*c*d^5*(c*Sec[a + b*x])^(3/2))/(21*b*Sqrt[d*Csc[a + b*x]]) - (20*c*d^3*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2))/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*(c*Sec[a + b*x])^(3/2))/(7*b) + (40*c^2*d^4*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(21*b)} -{(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(7/2), x, 3, -((64*c^3*d^3*Sqrt[d*Csc[a + b*x]])/(15*b*Sqrt[c*Sec[a + b*x]])) + (16*c*d^3*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2))/(15*b) - (2*c*d*(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2))/(5*b)} -{(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(5/2), x, 5, (4*c*d^3*(c*Sec[a + b*x])^(3/2))/(3*b*Sqrt[d*Csc[a + b*x]]) - (2*c*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2))/(3*b) + (4*c^2*d^2*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b)} -{(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(3/2), x, 2, -((8*c^3*d*Sqrt[d*Csc[a + b*x]])/(3*b*Sqrt[c*Sec[a + b*x]])) + (2*c*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2))/(3*b)} -{(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(1/2), x, 4, (2*c*d*(c*Sec[a + b*x])^(3/2))/(3*b*Sqrt[d*Csc[a + b*x]]) + (2*c^2*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b)} -{(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(1/2), x, 1, (2*c*d*(c*Sec[a + b*x])^(3/2))/(3*b*(d*Csc[a + b*x])^(3/2))} -{(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(3/2), x, 4, (2*c*(c*Sec[a + b*x])^(3/2))/(3*b*d*Sqrt[d*Csc[a + b*x]]) - (c^2*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*d^2)} -{(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(5/2), x, 13, (2*c*(c*Sec[a + b*x])^(3/2))/(3*b*d*(d*Csc[a + b*x])^(3/2)) + (c^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (c^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (c^2*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (c^2*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(9/2), x, 2, -((8*c*d^3*(d*Csc[a + b*x])^(3/2))/(21*b*(c*Sec[a + b*x])^(3/2))) - (2*c*d*(d*Csc[a + b*x])^(7/2))/(7*b*(c*Sec[a + b*x])^(3/2))} -{1/(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(7/2), x, 5, -((4*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*(c*Sec[a + b*x])^(3/2))) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*(c*Sec[a + b*x])^(3/2)) - (4*d^4*EllipticE[a - Pi/4 + b*x, 2])/(5*b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(5/2), x, 1, -((2*c*d*(d*Csc[a + b*x])^(3/2))/(3*b*(c*Sec[a + b*x])^(3/2)))} -{1/(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(3/2), x, 4, -((2*c*d*Sqrt[d*Csc[a + b*x]])/(b*(c*Sec[a + b*x])^(3/2))) - (2*d^2*EllipticE[a - Pi/4 + b*x, 2])/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(1/2)*(d*Csc[a + b*x])^(1/2), x, 12, -((ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*Sqrt[c*Sec[a + b*x]])) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*Sqrt[c*Sec[a + b*x]]) - (Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*Sqrt[c*Sec[a + b*x]]) + (Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(1/2), x, 3, EllipticE[a - Pi/4 + b*x, 2]/(b*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(3/2), x, 13, -(c/(2*b*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2))) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(4*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(4*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) - (Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(8*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]]) + (Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(8*Sqrt[2]*b*d^2*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(1/2)/(d*Csc[a + b*x])^(5/2), x, 4, -(c/(3*b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2))) + EllipticE[a - Pi/4 + b*x, 2]/(2*b*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} - - -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(11/2), x, 3, (8*d^5*Sqrt[d*Csc[a + b*x]])/(45*b*c*Sqrt[c*Sec[a + b*x]]) + (2*d^3*(d*Csc[a + b*x])^(5/2))/(45*b*c*Sqrt[c*Sec[a + b*x]]) - (2*d*(d*Csc[a + b*x])^(9/2))/(9*b*c*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(9/2), x, 5, (2*d^3*(d*Csc[a + b*x])^(3/2))/(21*b*c*Sqrt[c*Sec[a + b*x]]) - (2*d*(d*Csc[a + b*x])^(7/2))/(7*b*c*Sqrt[c*Sec[a + b*x]]) - (2*d^4*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(21*b*c^2)} -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(7/2), x, 1, -((2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*(c*Sec[a + b*x])^(5/2)))} -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(5/2), x, 4, -((2*d*(d*Csc[a + b*x])^(3/2))/(3*b*c*Sqrt[c*Sec[a + b*x]])) - (d^2*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(3*b*c^2)} -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(3/2), x, 13, -((2*d*Sqrt[d*Csc[a + b*x]])/(b*c*Sqrt[c*Sec[a + b*x]])) + (d^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (d^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (d^2*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (d^2*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(2*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} -{1/(c*Sec[a + b*x])^(3/2)*(d*Csc[a + b*x])^(1/2), x, 4, d/(b*c*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(2*b*c^2)} -{1/(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(1/2), x, 13, d/(2*b*c*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(4*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(4*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(8*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(8*Sqrt[2]*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} -{1/(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(3/2), x, 5, -(c/(3*b*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(5/2))) + 1/(6*b*c*d*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])/(12*b*c^2*d^2)} -{1/(c*Sec[a + b*x])^(3/2)/(d*Csc[a + b*x])^(5/2), x, 14, -(c/(4*b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(5/2))) + 3/(16*b*c*d*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]]) - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(32*Sqrt[2]*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[c*Sec[a + b*x]])/(32*Sqrt[2]*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) + (3*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(64*Sqrt[2]*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]]) - (3*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[c*Sec[a + b*x]])/(64*Sqrt[2]*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])} - - -{1/(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(9/2), x, 1, -((2*c*d*(d*Csc[a + b*x])^(7/2))/(7*b*(c*Sec[a + b*x])^(7/2)))} -{1/(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(7/2), x, 5, (6*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*c*(c*Sec[a + b*x])^(3/2)) - (2*d*(d*Csc[a + b*x])^(5/2))/(5*b*c*(c*Sec[a + b*x])^(3/2)) + (6*d^4*EllipticE[a - Pi/4 + b*x, 2])/(5*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(5/2), x, 13, -((2*d*(d*Csc[a + b*x])^(3/2))/(3*b*c*(c*Sec[a + b*x])^(3/2))) + (d^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) - (d^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) + (d^2*Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) - (d^2*Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(2*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(3/2), x, 4, -((2*d*Sqrt[d*Csc[a + b*x]])/(b*c*(c*Sec[a + b*x])^(3/2))) - (3*d^2*EllipticE[a - Pi/4 + b*x, 2])/(b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(5/2)*(d*Csc[a + b*x])^(1/2), x, 13, d/(2*b*c*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(4*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(4*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) - (3*Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(8*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]]) + (3*Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(8*Sqrt[2]*b*c^2*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(1/2), x, 4, d/(3*b*c*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2)) + EllipticE[a - Pi/4 + b*x, 2]/(2*b*c^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(3/2), x, 14, -(c/(4*b*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(7/2))) + 1/(16*b*c*d*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(32*Sqrt[2]*b*c^2*d^2*Sqrt[c*Sec[a + b*x]]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(32*Sqrt[2]*b*c^2*d^2*Sqrt[c*Sec[a + b*x]]) - (3*Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(64*Sqrt[2]*b*c^2*d^2*Sqrt[c*Sec[a + b*x]]) + (3*Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(64*Sqrt[2]*b*c^2*d^2*Sqrt[c*Sec[a + b*x]])} -{1/(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(5/2), x, 5, -(c/(5*b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(7/2))) + 1/(10*b*c*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(3/2)) + (3*EllipticE[a - Pi/4 + b*x, 2])/(20*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*x]])} -{1/(c*Sec[a + b*x])^(5/2)/(d*Csc[a + b*x])^(7/2), x, 15, -(c/(6*b*d*(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(7/2))) - (5*c)/(48*b*d^3*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(7/2)) + 5/(192*b*c*d^3*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2)) - (5*ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(128*Sqrt[2]*b*c^2*d^4*Sqrt[c*Sec[a + b*x]]) + (5*ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*x]]]*Sqrt[d*Csc[a + b*x]]*Sqrt[Tan[a + b*x]])/(128*Sqrt[2]*b*c^2*d^4*Sqrt[c*Sec[a + b*x]]) - (5*Sqrt[d*Csc[a + b*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(256*Sqrt[2]*b*c^2*d^4*Sqrt[c*Sec[a + b*x]]) + (5*Sqrt[d*Csc[a + b*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[a + b*x]] + Tan[a + b*x]]*Sqrt[Tan[a + b*x]])/(256*Sqrt[2]*b*c^2*d^4*Sqrt[c*Sec[a + b*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Csc[e+f x])^n with n symbolic*) - - -{(Sec[e + f*x])^m*(Csc[e + f*x])^n, x, 2, ((Cos[e + f*x]^2)^((1 + m)/2)*Csc[e + f*x]^(-1 + n)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sec[e + f*x]^(1 + m))/(f*(1 - n))} -{(a*Sec[e + f*x])^m*(Csc[e + f*x])^n, x, 2, ((Cos[e + f*x]^2)^((1 + m)/2)*Csc[e + f*x]^(-1 + n)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*(a*Sec[e + f*x])^(m + 1))/(a*f*(1 - n))} -{(Sec[e + f*x])^m*(b*Csc[e + f*x])^n, x, 2, (b*(Cos[e + f*x]^2)^((1 + m)/2)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sec[e + f*x]^(1 + m))/(f*(1 - n))} -{(a*Sec[e + f*x])^m*(b*Csc[e + f*x])^n, x, 2, (b*(Cos[e + f*x]^2)^((1 + m)/2)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*(a*Sec[e + f*x])^(1 + m))/(a*f*(1 - n))} - - -{(b*Csc[e + f*x])^n*Sec[e + f*x]^5, x, 2, ((b*Csc[e + f*x])^(5 + n)*Hypergeometric2F1[3, (5 + n)/2, (7 + n)/2, Csc[e + f*x]^2])/(b^5*f*(5 + n))} -{(b*Csc[e + f*x])^n*Sec[e + f*x]^3, x, 2, -(((b*Csc[e + f*x])^(3 + n)*Hypergeometric2F1[2, (3 + n)/2, (5 + n)/2, Csc[e + f*x]^2])/(b^3*f*(3 + n)))} -{(b*Csc[e + f*x])^n*Sec[e + f*x]^1, x, 2, ((b*Csc[e + f*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Csc[e + f*x]^2])/(b*f*(1 + n))} -{(b*Csc[e + f*x])^n*Cos[e + f*x]^1, x, 2, (b*(b*Csc[e + f*x])^(-1 + n))/(f*(1 - n))} -{(b*Csc[e + f*x])^n*Cos[e + f*x]^3, x, 3, -((b^3*(b*Csc[e + f*x])^(-3 + n))/(f*(3 - n))) + (b*(b*Csc[e + f*x])^(-1 + n))/(f*(1 - n))} -{(b*Csc[e + f*x])^n*Cos[e + f*x]^5, x, 3, (b^5*(b*Csc[e + f*x])^(-5 + n))/(f*(5 - n)) - (2*b^3*(b*Csc[e + f*x])^(-3 + n))/(f*(3 - n)) + (b*(b*Csc[e + f*x])^(-1 + n))/(f*(1 - n))} - -{(b*Csc[e + f*x])^n*Sec[e + f*x]^6, x, 2, (b*Sqrt[Cos[e + f*x]^2]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[7/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sec[e + f*x])/(f*(1 - n))} -{(b*Csc[e + f*x])^n*Sec[e + f*x]^4, x, 2, (b*Sqrt[Cos[e + f*x]^2]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[5/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sec[e + f*x])/(f*(1 - n))} -{(b*Csc[e + f*x])^n*Sec[e + f*x]^2, x, 2, (b*Sqrt[Cos[e + f*x]^2]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[3/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sec[e + f*x])/(f*(1 - n))} -{(b*Csc[e + f*x])^n*Sec[e + f*x]^0, x, 2, (b*Cos[e + f*x]*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2])} -{(b*Csc[e + f*x])^n*Cos[e + f*x]^2, x, 2, (b*Cos[e + f*x]*(b*Csc[e + f*x])^(n - 1)*Hypergeometric2F1[-(1/2), (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2])} -{(b*Csc[e + f*x])^n*Cos[e + f*x]^4, x, 2, (b*Cos[e + f*x]*(b*Csc[e + f*x])^(n - 1)*Hypergeometric2F1[-(3/2), (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(f*(1 - n)*Sqrt[Cos[e + f*x]^2])} - - -{(b*Csc[e + f*x])^n*(c*Sec[e + f*x])^(3/2), x, 2, (b*(Cos[e + f*x]^2)^(5/4)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[5/4, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*(c*Sec[e + f*x])^(5/2))/(c*f*(1 - n))} -{(b*Csc[e + f*x])^n*(c*Sec[e + f*x])^(1/2), x, 2, (b*(Cos[e + f*x]^2)^(3/4)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[3/4, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*(c*Sec[e + f*x])^(3/2))/(c*f*(1 - n))} -{(b*Csc[e + f*x])^n/(c*Sec[e + f*x])^(1/2), x, 2, (b*(Cos[e + f*x]^2)^(1/4)*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[1/4, (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2]*Sqrt[c*Sec[e + f*x]])/(c*f*(1 - n))} -{(b*Csc[e + f*x])^n/(c*Sec[e + f*x])^(3/2), x, 2, (b*(b*Csc[e + f*x])^(-1 + n)*Hypergeometric2F1[-(1/4), (1 - n)/2, (3 - n)/2, Sin[e + f*x]^2])/(c*f*(1 - n)*(Cos[e + f*x]^2)^(1/4)*Sqrt[c*Sec[e + f*x]])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m deleted file mode 100644 index ba6d75b..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.2 (d sec)^n (a+b sec)^m.m +++ /dev/null @@ -1,1403 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^4*(a + a*Sec[c + d*x]), x, 6, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^3*(a + a*Sec[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x]), x, 4, (a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x]), x, 2, a*x + (a*ArcTanh[Sin[c + d*x]])/d} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x]), x, 3, a*x + (a*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x]), x, 4, (a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x]), x, 5, (a*x)/2 + (a*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x]), x, 6, (3*a*x)/8 + (a*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*Sin[c + d*x]^3)/(3*d)} - - -{Sec[c + d*x]^4*(a + a*Sec[c + d*x])^2, x, 7, (3*a^2*ArcTanh[Sin[c + d*x]])/(4*d) + (9*a^2*Tan[c + d*x])/(5*d) + (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (a^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (3*a^2*Tan[c + d*x]^3)/(5*d)} -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2, x, 6, (7*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a^2*Tan[c + d*x])/d + (7*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a^2*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2, x, 6, (a^2*ArcTanh[Sin[c + d*x]])/d + (5*a^2*Tan[c + d*x])/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^2, x, 5, (3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^2, x, 4, a^2*x + (2*a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^2, x, 4, 2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2, x, 4, (3*a^2*x)/2 + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2, x, 6, a^2*x + (2*a^2*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2, x, 6, (7*a^2*x)/8 + (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2, x, 8, (3*a^2*x)/4 + (2*a^2*Sin[c + d*x])/d + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - (a^2*Sin[c + d*x]^3)/d + (a^2*Sin[c + d*x]^5)/(5*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3, x, 11, (13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (5*a^3*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3, x, 11, (15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^3, x, 9, (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (4*a^3*Tan[c + d*x])/d + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^3*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^3, x, 5, a^3*x + (7*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*Tan[c + d*x])/(2*d) + ((a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^3, x, 6, 3*a^3*x + (3*a^3*ArcTanh[Sin[c + d*x]])/d + (a^3*Sin[c + d*x])/d + (a^3*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3, x, 6, (7*a^3*x)/2 + (a^3*ArcTanh[Sin[c + d*x]])/d + (3*a^3*Sin[c + d*x])/d + (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3, x, 7, (5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3, x, 10, (15*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/d} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3, x, 11, (13*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3, x, 13, (23*a^3*x)/16 + (4*a^3*Sin[c + d*x])/d + (23*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (7*a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x]^5)/(5*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^4, x, 15, (49*a^4*ArcTanh[Sin[c + d*x]])/(16*d) + (8*a^4*Tan[c + d*x])/d + (49*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (41*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*Tan[c + d*x]^3)/d + (4*a^4*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4, x, 13, (7*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (8*a^4*Tan[c + d*x])/d + (7*a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/d + (8*a^4*Tan[c + d*x]^3)/(3*d) + (a^4*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^4, x, 12, (35*a^4*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*Tan[c + d*x])/d + (27*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (4*a^4*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^4, x, 6, a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (5*a^4*Tan[c + d*x])/d + ((a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + (4*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^4, x, 8, 4*a^4*x + (13*a^4*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*Sin[c + d*x])/d + (4*a^4*Tan[c + d*x])/d + (a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4, x, 8, (13*a^4*x)/2 + (4*a^4*ArcTanh[Sin[c + d*x]])/d + (4*a^4*Sin[c + d*x])/d + (a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Tan[c + d*x])/d} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4, x, 8, 6*a^4*x + (a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Sin[c + d*x])/d + (2*a^4*Cos[c + d*x]*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4, x, 10, (35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4, x, 12, (7*a^4*x)/2 + (8*a^4*Sin[c + d*x])/d + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/d - (8*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4, x, 15, (49*a^4*x)/16 + (8*a^4*Sin[c + d*x])/d + (49*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (41*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^4*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (4*a^4*Sin[c + d*x]^3)/d + (4*a^4*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4, x, 15, (11*a^4*x)/4 + (8*a^4*Sin[c + d*x])/d + (11*a^4*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (11*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (2*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(3*d) - (16*a^4*Sin[c + d*x]^3)/(3*d) + (9*a^4*Sin[c + d*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/(7*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^5, x, 17, (93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (16*a^5*Tan[c + d*x])/d + (93*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (85*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (28*a^5*Tan[c + d*x]^3)/(3*d) + (13*a^5*Tan[c + d*x]^5)/(5*d) + (a^5*Tan[c + d*x]^7)/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^5/(a + a*Sec[c + d*x]), x, 6, -((3*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (4*Tan[c + d*x])/(a*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + (4*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x]), x, 6, (3*ArcTanh[Sin[c + d*x]])/(2*a*d) - (2*Tan[c + d*x])/(a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x]), x, 4, -(ArcTanh[Sin[c + d*x]]/(a*d)) + Tan[c + d*x]/(d*(a + a*Sec[c + d*x])) + Tan[c + d*x]/(a*d)} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x]), x, 3, ArcTanh[Sin[c + d*x]]/(a*d) - Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x]), x, 1, Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x]), x, 2, x/a - Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x]), x, 4, -(x/a) + (2*Sin[c + d*x])/(a*d) - Sin[c + d*x]/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x]), x, 5, (3*x)/(2*a) - (2*Sin[c + d*x])/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^3/(a + a*Sec[c + d*x]), x, 6, -((3*x)/(2*a)) + (4*Sin[c + d*x])/(a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - (4*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^4/(a + a*Sec[c + d*x]), x, 7, (15*x)/(8*a) - (4*Sin[c + d*x])/(a*d) + (15*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + (4*Sin[c + d*x]^3)/(3*a*d)} - - -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^2, x, 7, (7*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (16*Tan[c + d*x])/(3*a^2*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (8*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^2, x, 6, -((2*ArcTanh[Sin[c + d*x]])/(a^2*d)) + (4*Tan[c + d*x])/(3*a^2*d) + (2*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(a^2*d) - (5*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 2, -(Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)) + (2*Tan[c + d*x])/(3*d*(a^2 + a^2*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 2, Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2) + Tan[c + d*x]/(3*d*(a^2 + a^2*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^2, x, 3, x/a^2 - (4*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - Tan[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 5, -((2*x)/a^2) + (10*Sin[c + d*x])/(3*a^2*d) - (2*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 6, (7*x)/(2*a^2) - (16*Sin[c + d*x])/(3*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (8*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 7, -((5*x)/a^2) + (12*Sin[c + d*x])/(a^2*d) - (5*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (10*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*Sin[c + d*x]^3)/(a^2*d)} - - -{Sec[c + d*x]^6/(a + a*Sec[c + d*x])^3, x, 8, (13*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (152*Tan[c + d*x])/(15*a^3*d) + (13*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - (Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (11*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (76*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^3, x, 7, -((3*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (9*Tan[c + d*x])/(5*a^3*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (3*Sec[c + d*x]^2*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) + (3*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^3, x, 5, ArcTanh[Sin[c + d*x]]/(a^3*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (7*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (29*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^3, x, 3, Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (7*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 3, -(Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3)) + Tan[c + d*x]/(5*a*d*(a + a*Sec[c + d*x])^2) + Tan[c + d*x]/(5*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^3, x, 3, Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^3, x, 4, x/a^3 - Tan[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (7*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (22*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^3, x, 6, -((3*x)/a^3) + (24*Sin[c + d*x])/(5*a^3*d) - Sin[c + d*x]/(5*d*(a + a*Sec[c + d*x])^3) - (3*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (3*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 7, (13*x)/(2*a^3) - (152*Sin[c + d*x])/(15*a^3*d) + (13*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (11*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (76*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} - - -{Sec[c + d*x]^7/(a + a*Sec[c + d*x])^4, x, 9, (21*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (576*Tan[c + d*x])/(35*a^4*d) + (21*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - (43*Sec[c + d*x]^3*Tan[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])^2) - (288*Sec[c + d*x]^2*Tan[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^6/(a + a*Sec[c + d*x])^4, x, 8, -((4*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (244*Tan[c + d*x])/(105*a^4*d) - (88*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (12*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^4, x, 6, ArcTanh[Sin[c + d*x]]/(a^4*d) + (11*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])^2) - (43*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Sec[c + d*x]^2*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^4, x, 4, (Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(35*d*(a^2 + a^2*Sec[c + d*x])^2) + Tan[c + d*x]/(5*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^4, x, 4, Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (11*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (13*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (13*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^4, x, 4, -(Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4)) + (4*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (8*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (8*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^4, x, 4, Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) + (3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(35*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(35*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^4, x, 5, x/a^4 - (11*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])^2) - (32*Tan[c + d*x])/(21*a^4*d*(1 + Sec[c + d*x])) - Tan[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (2*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^4, x, 7, -((4*x)/a^4) + (664*Sin[c + d*x])/(105*a^4*d) - (88*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(7*d*(a + a*Sec[c + d*x])^4) - (12*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x])^4, x, 8, (21*x)/(2*a^4) - (576*Sin[c + d*x])/(35*a^4*d) + (21*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - (43*Cos[c + d*x]*Sin[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])^2) - (288*Cos[c + d*x]*Sin[c + d*x])/(35*a^4*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} - - -{Sec[c + d*x]^7/(a + a*Sec[c + d*x])^5, x, 9, -((5*ArcTanh[Sin[c + d*x]])/(a^5*d)) + (181*Tan[c + d*x])/(63*a^5*d) - (Sec[c + d*x]^5*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sec[c + d*x]^4*Tan[c + d*x])/(21*a*d*(a + a*Sec[c + d*x])^4) - (29*Sec[c + d*x]^3*Tan[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sec[c + d*x]^2*Tan[c + d*x])/(63*a^3*d*(a + a*Sec[c + d*x])^2) + (5*Tan[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^6/(a + a*Sec[c + d*x])^5, x, 7, ArcTanh[Sin[c + d*x]]/(a^5*d) - (Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (13*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) + (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (661*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^5, x, 5, (Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) + (4*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + (4*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (32*Tan[c + d*x])/(315*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (4*Tan[c + d*x])/(45*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^5, x, 5, -((Sec[c + d*x]^4*Tan[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5)) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(21*a^2*d*(a + a*Sec[c + d*x])^3) - (8*Tan[c + d*x])/(63*a*d*(a^2 + a^2*Sec[c + d*x])^2) + Tan[c + d*x]/(9*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^5, x, 5, Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (2*Tan[c + d*x])/(9*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(15*a^2*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(45*a^3*d*(a + a*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(45*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^5, x, 5, -(Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5)) + (5*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + Tan[c + d*x]/(21*a^2*d*(a + a*Sec[c + d*x])^3) + (2*Tan[c + d*x])/(63*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*Tan[c + d*x])/(63*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^5, x, 5, Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) + (4*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) + (4*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) + (8*Tan[c + d*x])/(315*a*d*(a^2 + a^2*Sec[c + d*x])^2) + (8*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^5, x, 6, x/a^5 - Tan[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (13*Tan[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (34*Tan[c + d*x])/(105*a^2*d*(a + a*Sec[c + d*x])^3) - (173*Tan[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (488*Tan[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^5, x, 8, -((5*x)/a^5) + (496*Sin[c + d*x])/(63*a^5*d) - Sin[c + d*x]/(9*d*(a + a*Sec[c + d*x])^5) - (5*Sin[c + d*x])/(21*a*d*(a + a*Sec[c + d*x])^4) - (29*Sin[c + d*x])/(63*a^2*d*(a + a*Sec[c + d*x])^3) - (67*Sin[c + d*x])/(63*a^3*d*(a + a*Sec[c + d*x])^2) - (5*Sin[c + d*x])/(d*(a^5 + a^5*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x])^5, x, 9, (31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]*Sin[c + d*x])/(9*d*(a + a*Sec[c + d*x])^5) - (17*Cos[c + d*x]*Sin[c + d*x])/(63*a*d*(a + a*Sec[c + d*x])^4) - (28*Cos[c + d*x]*Sin[c + d*x])/(45*a^2*d*(a + a*Sec[c + d*x])^3) - (577*Cos[c + d*x]*Sin[c + d*x])/(315*a^3*d*(a + a*Sec[c + d*x])^2) - (3832*Cos[c + d*x]*Sin[c + d*x])/(315*d*(a^5 + a^5*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]], x, 4, (4*a*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (8*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(35*d) + (12*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)} -{Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]], x, 3, (14*a*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)} -{Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]], x, 2, (2*a*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]], x, 1, (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^0*Sqrt[a + a*Sec[c + d*x]], x, 2, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d} -{Cos[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]], x, 3, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]], x, 4, (3*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (3*a*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]], x, 5, (5*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (5*a*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]], x, 6, (35*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (35*a*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (35*a*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (7*a*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2), x, 6, (68*a^2*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (34*a^2*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (136*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (68*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)} -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2), x, 4, (152*a^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (38*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) - (4*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2), x, 3, (8*a^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2), x, 2, (8*a^2*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(3/2), x, 4, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2), x, 5, (3*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2), x, 5, (7*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (7*a^2*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2), x, 6, (11*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (11*a^2*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2), x, 6, (284*a^3*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (710*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (46*a^3*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (568*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(693*d) + (2*a^2*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (284*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(231*d)} -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2), x, 5, (832*a^3*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (208*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (26*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) - (4*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2), x, 4, (64*a^3*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2), x, 3, (64*a^3*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(5/2), x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (14*a^3*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2), x, 4, (5*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2), x, 4, (19*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2), x, 5, (25*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (25*a^3*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2), x, 6, (163*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (163*a^3*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (17*a^3*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^1*Sqrt[a - a*Sec[c + d*x]], x, 1, -((2*a*Tan[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]]))} -{Sec[c + d*x]^0*Sqrt[a - a*Sec[c + d*x]], x, 2, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d} -{Cos[c + d*x]^1*Sqrt[a - a*Sec[c + d*x]], x, 3, -((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/d) + (a*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]], x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (28*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)} -{Sec[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]], x, 4, (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]], x, 3, -((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^1/Sqrt[a + a*Sec[c + d*x]], x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Sec[c + d*x]^0/Sqrt[a + a*Sec[c + d*x]], x, 5, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^1/Sqrt[a + a*Sec[c + d*x]], x, 6, -(ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]], x, 7, (7*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - Sin[c + d*x]/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2), x, 6, -((15*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - (Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (31*Tan[c + d*x])/(5*a*d*Sqrt[a + a*Sec[c + d*x]]) + (9*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - (13*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(10*a^2*d)} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2), x, 5, (11*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (13*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + (7*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2), x, 4, -((7*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2), x, 3, (3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^(3/2), x, 3, ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (5*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tan[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^(3/2), x, 7, -((3*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + (9*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2), x, 8, (19*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (13*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (7*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2), x, 6, (163*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (17*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (197*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + (95*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)} -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2), x, 5, -((75*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - (Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (13*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (9*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2), x, 4, (19*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (13*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2), x, 4, (5*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (5*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^(5/2), x, 4, (3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (43*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Tan[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (11*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^(5/2), x, 8, -((5*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (115*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (15*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]/Sqrt[a - a*Sec[c + d*x]], x, 2, -((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d))} -{1/Sqrt[a - a*Sec[c + d*x]], x, 5, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(2/3), x, 7, -((9*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*d)) + (57*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(80*d*(1 + Sec[c + d*x])) + (3*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*a*d) - (19*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(80*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(2/3), x, 6, (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d*(1 + Sec[c + d*x])) - (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(2/3), x, 5, (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])) - (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(2/3), x, 3, (3*Sqrt[2]*AppellF1[7/6, 1/2, 1, 13/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]])} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(2/3), x, 3, -((3*Sqrt[2]*AppellF1[7/6, 1/2, 2, 13/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]))} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/3), x, 8, (147*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(440*d) + (1029*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(880*d*(1 + Sec[c + d*x])) - (9*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(88*d) + (3*(a + a*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*a*d) - (343*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(880*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/3), x, 7, (3*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*d) + (21*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(16*d*(1 + Sec[c + d*x])) + (3*(a + a*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*d) - (7*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(16*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(5/3), x, 6, (3*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (21*a*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(10*d*(1 + Sec[c + d*x])) - (7*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(5/3), x, 3, (3*Sqrt[2]*a*AppellF1[13/6, 1/2, 1, 19/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(13*d*Sqrt[1 - Sec[c + d*x]])} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(5/3), x, 3, -((3*Sqrt[2]*a*AppellF1[13/6, 1/2, 2, 19/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(13*d*Sqrt[1 - Sec[c + d*x]]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(1/3), x, 7, (99*Tan[c + d*x])/(80*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sec[c + d*x]^2*Tan[c + d*x])/(8*d*(a + a*Sec[c + d*x])^(1/3)) - (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*a*d) + (37*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(80*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(1/3), x, 6, -((9*Tan[c + d*x])/(10*d*(a + a*Sec[c + d*x])^(1/3))) + (3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*a*d) - (7*3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(1/3), x, 5, (3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^(1/3), x, 4, -((3^(3/4)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]))} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^(1/3), x, 3, (3*Sqrt[2]*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^(1/3), x, 3, -((3*Sqrt[2]*AppellF1[1/6, 1/2, 2, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)))} - - -{Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/3), x, 9, -((33*Tan[c + d*x])/(28*d*(a + a*Sec[c + d*x])^(5/3))) + (3*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/3)) + (135*Tan[c + d*x])/(14*a*d*(a + a*Sec[c + d*x])^(2/3)) + (375*(1 + Sqrt[3])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*a^2*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (375*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(14*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (125*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(28*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/3), x, 8, (3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) - (36*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (57*(1 + Sqrt[3])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a^2*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (57*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (19*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a^2*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/3), x, 8, -((3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3))) + (15*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) + (15*(1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (15*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (5*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^1/(a + a*Sec[c + d*x])^(5/3), x, 8, (6*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) + (3*Tan[c + d*x])/(7*a*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)) + (6*(1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (6*2^(1/3)*3^(1/4)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (2^(1/3)*3^(3/4)*(1 - Sqrt[3])*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{Sec[c + d*x]^0/(a + a*Sec[c + d*x])^(5/3), x, 3, -((3*Sqrt[2]*AppellF1[-(7/6), 1/2, 1, -(1/6), (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(7*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)))} -{Cos[c + d*x]^1/(a + a*Sec[c + d*x])^(5/3), x, 3, (3*Sqrt[2]*AppellF1[-(7/6), 1/2, 2, -(1/6), (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(7*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sec[c + d*x])*Sec[c + d*x]^(5/2), x, 8, (-6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Sec[c + d*x])*Sec[c + d*x]^(3/2), x, 7, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Sec[c + d*x])*Sec[c + d*x]^(1/2), x, 6, (-2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])/Sec[c + d*x]^(1/2), x, 5, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d} -{(a + a*Sec[c + d*x])/Sec[c + d*x]^(3/2), x, 6, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])/Sec[c + d*x]^(5/2), x, 7, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])/Sec[c + d*x]^(7/2), x, 8, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*a*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Sec[c + d*x])^2*Sec[c + d*x]^(5/2), x, 9, (-12*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (12*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(7*d) + (4*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Sec[c + d*x])^2*Sec[c + d*x]^(3/2), x, 8, (-16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Sec[c + d*x])^2*Sec[c + d*x]^(1/2), x, 7, (-4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(1/2), x, 4, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(3/2), x, 6, (4*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(5/2), x, 7, (16*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(7/2), x, 8, (12*a^2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^2*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Sec[c + d*x])^3*Sec[c + d*x]^(3/2), x, 16, -((28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (28*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (52*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (6*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Sec[c + d*x])^3*Sec[c + d*x]^(1/2), x, 14, -((36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (36*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(1/2), x, 12, -((4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(3/2), x, 12, (4*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(5/2), x, 12, (36*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(7/2), x, 14, (28*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (52*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*a^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (52*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(9/2), x, 16, (68*a^3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (44*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (68*a^3*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (44*a^3*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{(a + a*Sec[c + d*x])^4*Sec[c + d*x]^(3/2), x, 21, -((152*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(7*d) + (152*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (32*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(7*d) + (122*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (8*a^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + (2*a^4*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{(a + a*Sec[c + d*x])^4*Sec[c + d*x]^(1/2), x, 18, -((64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (64*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (94*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (8*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a^4*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(1/2), x, 16, -((56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (66*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^4*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(3/2), x, 15, (40*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (8*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a^4*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(5/2), x, 15, (56*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*a^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a^4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(7/2), x, 16, (64*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (136*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (8*a^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (94*a^4*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(9/2), x, 18, (152*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (32*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(7*d) + (2*a^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (8*a^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (122*a^4*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (32*a^4*Sin[c + d*x])/(7*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(11/2), x, 21, (128*a^4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (904*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (8*a^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (150*a^4*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (128*a^4*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (904*a^4*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x]), x, 8, (3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x]), x, 7, -((3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) - (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x]), x, 6, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + a*Sec[c + d*x]), x, 6, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])), x, 6, (3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, -((3*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d)) + (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, (21*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (7*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} - - -{Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^2, x, 9, (7*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (7*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (10*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - (7*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^2, x, 8, -((4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^2, x, 7, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^2, x, 4, (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^(1/2)/(a + a*Sec[c + d*x])^2, x, 7, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/(Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^2), x, 7, (4*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 8, -((7*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (10*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 9, (56*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (56*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*Sin[c + d*x])/(a^2*d*Sqrt[Sec[c + d*x]]) - (3*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} - - -{Sec[c + d*x]^(11/2)/(a + a*Sec[c + d*x])^3, x, 10, (119*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (119*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + (11*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - (Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^3, x, 9, -((49*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^3, x, 8, (9*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (9*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^3, x, 8, (Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (4*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^3, x, 8, -((Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)/(a + a*Sec[c + d*x])^3, x, 8, -((9*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) + (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a^3 + a^3*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^3), x, 8, (49*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (13*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (8*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 9, -((119*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d)) + (11*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 10, (231*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - (21*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (77*Sin[c + d*x])/(10*a^3*d*Sec[c + d*x]^(3/2)) - (21*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - Sin[c + d*x]/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (4*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (63*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]], x, 4, (3*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (3*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]], x, 3, (Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]], x, 2, (2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d} -{Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(1/2), x, 1, (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(3/2), x, 2, (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(5/2), x, 3, (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(7/2), x, 4, (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (12*a*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (32*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2), x, 6, (11*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (11*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2), x, 5, (7*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (7*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2), x, 4, (3*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/2), x, 4, (2*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2), x, 2, (8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2), x, 3, (8*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2), x, 5, (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (26*a^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (104*a^2*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (208*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2), x, 6, (2*a^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (34*a^2*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (68*a^2*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (272*a^2*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (544*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2), x, 6, (163*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (163*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (17*a^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2), x, 5, (25*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (25*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (13*a^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2), x, 4, (19*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (9*a^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/2), x, 4, (5*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2), x, 4, (2*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (14*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2), x, 3, (64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2), x, 4, (64*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2), x, 5, (38*a^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (146*a^3*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (584*a^3*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (1168*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(11/2), x, 6, (46*a^3*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (710*a^3*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (284*a^3*Sin[c + d*x])/(231*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (1136*a^3*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2272*a^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -{Sec[c + d*x]^(-1/4)*(a + a*Sec[c + d*x])^(3/2), x, 2, (4*a^2*Sec[c + d*x]^(3/4)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sqrt[Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]], x, 2, (2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f} -{Sqrt[-Sec[e + f*x]]*Sqrt[a - a*Sec[e + f*x]], x, 2, (2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a - a*Sec[e + f*x]]])/f} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]], x, 6, -(ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]], x, 5, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Sqrt[Sec[c + d*x]]/Sqrt[a + a*Sec[c + d*x]], x, 2, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 3, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 5, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(3/2), x, 7, -((3*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + (9*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2), x, 3, ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(3/2), x, 3, (3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 4, -((7*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (5*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 5, (11*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (7*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (19*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 6, -((15*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - Sin[c + d*x]/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + (9*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (13*Sin[c + d*x])/(10*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^(5/2), x, 8, -((5*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (115*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (15*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (35*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (11*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(5/2), x, 4, (3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2), x, 4, (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + (5*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(5/2), x, 4, (19*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (9*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 5, -((75*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - (Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (13*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + (49*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 6, (163*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - (17*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (95*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (299*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)/Sqrt[1 + Sec[c + d*x]], x, 7, -((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (7*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/(4*d) - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]) + (Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/Sqrt[1 + Sec[c + d*x]], x, 6, (Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d - ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]]/d + (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/Sqrt[1 + Sec[c + d*x]], x, 5, -((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*ArcSinh[Tan[c + d*x]/Sqrt[1 + Sec[c + d*x]]])/d} -{Sqrt[Sec[c + d*x]]/Sqrt[1 + Sec[c + d*x]], x, 2, (Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]), x, 3, -((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]])} -{1/(Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]]), x, 4, (Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]) - (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[1 + Sec[c + d*x]])} -{1/(Sec[c + d*x]^(5/2)*Sqrt[1 + Sec[c + d*x]]), x, 5, -((Sqrt[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]]) - (2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]) + (26*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[1 + Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/3) (a+a Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Sec[c + d*x]]*(e*Sec[c + d*x])^(4/3), x, 4, (6*a*e*(e*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*e*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3), x, 3, (2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(2/3), x, 4, (3*a*Tan[c + d*x])/(2*d*(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]]) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*d*e*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} - -{Sqrt[a + a*Sec[c + d*x]]*(e*Sec[c + d*x])^(8/3), x, 7, (60*a*e^2*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(91*d*Sqrt[a + a*Sec[c + d*x]]) + (6*a*e*(e*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(13*d*Sqrt[a + a*Sec[c + d*x]]) - (240*a*e^3*Tan[c + d*x])/(91*d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) + (120*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(7/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(91*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (80*Sqrt[2]*3^(3/4)*a^2*e^(7/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(91*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]*(e*Sec[c + d*x])^(5/3), x, 6, (6*a*e*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (24*a*e^2*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) + (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(4/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (8*Sqrt[2]*3^(3/4)*a^2*e^(4/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3), x, 5, -((6*a*e*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3)))) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*e^(1/3)*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) - (2*Sqrt[2]*3^(3/4)*a^2*e^(1/3)*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(1/3), x, 6, (3*a*Tan[c + d*x])/(d*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*d*e^(2/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) + (Sqrt[2]*3^(3/4)*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*e^(2/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} -{Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(4/3), x, 7, (3*a*Tan[c + d*x])/(4*d*(e*Sec[c + d*x])^(4/3)*Sqrt[a + a*Sec[c + d*x]]) + (15*a*Tan[c + d*x])/(8*d*e*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]) + (15*a*Tan[c + d*x])/(8*d*e*Sqrt[a + a*Sec[c + d*x]]*((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))) - (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*EllipticE[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(16*d*e^(5/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]) + (5*3^(3/4)*a^2*EllipticF[ArcSin[((1 - Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))], -7 - 4*Sqrt[3]]*(e^(1/3) - (e*Sec[c + d*x])^(1/3))*Sqrt[(e^(2/3) + e^(1/3)*(e*Sec[c + d*x])^(1/3) + (e*Sec[c + d*x])^(2/3))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*Sqrt[2]*d*e^(5/3)*(a - a*Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]*Sqrt[(e^(1/3)*(e^(1/3) - (e*Sec[c + d*x])^(1/3)))/((1 + Sqrt[3])*e^(1/3) - (e*Sec[c + d*x])^(1/3))^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sec[c + d*x])^(2/3)/Sqrt[a + a*Sec[c + d*x]], x, 4, -((3*AppellF1[2/3, 1/2, 1, 5/3, Sec[c + d*x], -Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]))} -{(e*Sec[c + d*x])^(1/3)/Sqrt[a + a*Sec[c + d*x]], x, 4, -((3*AppellF1[1/3, 1/2, 1, 4/3, Sec[c + d*x], -Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]))} -{1/((e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (3*AppellF1[-(1/3), 1/2, 1, 2/3, Sec[c + d*x], -Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]])} -{1/((e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (3*AppellF1[-(2/3), 1/2, 1, 1/3, Sec[c + d*x], -Sec[c + d*x]]*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/3) (a+a Sec[e+f x])^(m/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(1/3), x, 3, (2^(5/6)*AppellF1[1/2, -(1/3), 1/6, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(5/6))} - - -{Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(2/3), x, 3, (2*2^(1/6)*AppellF1[1/2, -(1/3), -(1/6), 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(7/6))} -{Sec[c + d*x]^(5/3)*(a + a*Sec[c + d*x])^(2/3), x, -3, (9*Sec[c + d*x]^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*Sin[c + d*x])/(4*d) - (3*a*Sec[c + d*x]^(5/3)*Sin[c + d*x])/(2*d*(a*(1 + Sec[c + d*x]))^(1/3)) - (9*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[c + d*x])/(4*d*(1/(1 + Cos[c + d*x]))^(1/3)*(1 + Sec[c + d*x])^(7/3)) + (Hypergeometric2F1[1/4, 1/3, 5/4, Tan[(1/2)*(c + d*x)]^4]*(Cos[c + d*x]*Sec[(1/2)*(c + d*x)]^4)^(1/3)*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[c + d*x])/(8*d*(1/(1 + Cos[c + d*x]))^(1/3)*(1 + Sec[c + d*x])^(4/3)) - (5*Hypergeometric2F1[1/3, 3/4, 7/4, Tan[(1/2)*(c + d*x)]^4]*(Cos[c + d*x]*Sec[(1/2)*(c + d*x)]^4)^(1/3)*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[c + d*x]^3)/(8*d*(1/(1 + Cos[c + d*x]))^(1/3)*(1 + Sec[c + d*x])^(10/3))} - - -{(a + a*Sec[c + d*x])^(4/3)/Sec[c + d*x]^(1/3), x, 3, (2*2^(5/6)*a*AppellF1[1/2, 4/3, -(5/6), 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(5/6))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m with n symbolic*) - - -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^4, x, 8, If[$VersionNumber>=8, (a^4*(30 + 21*n + 4*n^2)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)*(2 + n)*(3 + n)) + (Sec[e + f*x]^(1 + n)*(a^2 + a^2*Sec[e + f*x])^2*Sin[e + f*x])/(f*(3 + n)) + (2*(4 + n)*Sec[e + f*x]^(1 + n)*(a^4 + a^4*Sec[e + f*x])*Sin[e + f*x])/(f*(2 + n)*(3 + n)) - (a^4*(3 + 24*n + 8*n^2)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*(1 + n)*(3 + n)*Sqrt[Sin[e + f*x]^2]) + (4*a^4*(3 + 2*n)*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2]), (a^4*(30 + 21*n + 4*n^2)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(6 + 11*n + 6*n^2 + n^3)) + (Sec[e + f*x]^(1 + n)*(a^2 + a^2*Sec[e + f*x])^2*Sin[e + f*x])/(f*(3 + n)) + (2*(4 + n)*Sec[e + f*x]^(1 + n)*(a^4 + a^4*Sec[e + f*x])*Sin[e + f*x])/(f*(2 + n)*(3 + n)) - (a^4*(3 + 24*n + 8*n^2)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(3 + n)*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (4*a^4*(3 + 2*n)*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2])]} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^3, x, 7, (a^3*(5 + 2*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)*(2 + n)) + (Sec[e + f*x]^(1 + n)*(a^3 + a^3*Sec[e + f*x])*Sin[e + f*x])/(f*(2 + n)) - (a^3*(1 + 4*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(7 + 4*n)*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^2, x, 6, (a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)) - (a^2*(1 + 2*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (2*a^2*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^1, x, 5, -((a*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (a*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^n/(a + a*Sec[e + f*x])^1, x, 6, (Sec[e + f*x]^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) + ((1 - n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-2 + n)*Sin[e + f*x])/(a*f*(2 - n)*Sqrt[Sin[e + f*x]^2]) - (Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2])} -{Sec[e + f*x]^n/(a + a*Sec[e + f*x])^2, x, 7, -((2*(2 - n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))) - (Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - ((3 - 2*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) + (2*(2 - n)*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2])} - - -{Sec[e + f*x]^n*(1 + Sec[e + f*x])^(5/2), x, 4, If[$VersionNumber>=8, (2*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[1 + Sec[e + f*x]]) + (2*Sec[e + f*x]^(1 + n)*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[1 + Sec[e + f*x]]), (2*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[1 + Sec[e + f*x]]) + (2*Sec[e + f*x]^(1 + n)*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[1 + Sec[e + f*x]])]} -{Sec[e + f*x]^n*(1 + Sec[e + f*x])^(3/2), x, 4, (2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) + (2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]])} -{Sec[e + f*x]^n*(1 + Sec[e + f*x])^(1/2), x, 2, (2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])} -{Sec[e + f*x]^n/(1 + Sec[e + f*x])^(1/2), x, 3, (AppellF1[1/2, 1 - n, 1, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])} -{Sec[e + f*x]^n/(1 + Sec[e + f*x])^(3/2), x, 3, (AppellF1[1/2, 1 - n, 2, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*Tan[e + f*x])/(2*f*Sqrt[1 + Sec[e + f*x]])} - -{(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2), x, 4, (2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) - ((1 + 4*n)*Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*(1 + 2*n)*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^(1/2), x, 2, -((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(-Sec[e + f*x])^n/(1 + Sec[e + f*x])^(1/2), x, 2, -((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(-Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2), x, 2, -((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} - -{(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2), x, 4, (2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]]) - ((1 + 4*n)*Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*(1 + 2*n)*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])} -{(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(1/2), x, 2, -((Hypergeometric2F1[1/2, n, 1 + n, Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n/(1 + Sec[e + f*x])^(1/2), x, 2, -((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2), x, 2, -((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} - - -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(5/2), x, 4, If[$VersionNumber>=8, (2*a^3*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Sec[e + f*x]^(1 + n)*Sqrt[a + a*Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]]), (2*a^3*(7 + 4*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Sec[e + f*x]^(1 + n)*Sqrt[a + a*Sec[e + f*x]]*Sin[e + f*x])/(f*(3 + 2*n)) + (2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[a + a*Sec[e + f*x]])]} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(3/2), x, 4, (2*a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(1/2), x, 2, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]^n/(a + a*Sec[e + f*x])^(1/2), x, 4, (AppellF1[1/2, 1 - n, 1, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]^n/(a + a*Sec[e + f*x])^(3/2), x, 4, (AppellF1[1/2, 1 - n, 2, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*Tan[e + f*x])/(2*a*f*Sqrt[a + a*Sec[e + f*x]])} - -{(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2), x, 5, (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(1/2), x, 3, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{(-Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(1/2), x, 3, -((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))} -{(-Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2), x, 3, -((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(-Sec[e + f*x])^n*Tan[e + f*x])/(a*f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))} - -{(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2), x, 5, (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(1/2), x, 3, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{(d*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(1/2), x, 3, -((AppellF1[n, 1/2, 1, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2), x, 3, -((AppellF1[n, 1/2, 2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(a*f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]))} - - -{(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(5/2), x, 4, If[$VersionNumber>=8, (2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^3*(7 + 4*n)*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*(3 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 2*n)), (2*a^3*(3 + 24*n + 16*n^2)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^3*(7 + 4*n)*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(3 + 8*n + 4*n^2)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 2*n))]} -{(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2), x, 4, (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(-Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(1/2), x, 2, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]])} -{(-Sec[e + f*x])^n/(a - a*Sec[e + f*x])^(1/2), x, 4, (AppellF1[1/2, 1 - n, 1, 3/2, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]])} -{(-Sec[e + f*x])^n/(a - a*Sec[e + f*x])^(3/2), x, 4, (AppellF1[1/2, 1 - n, 2, 3/2, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*Tan[e + f*x])/(2*a*f*Sqrt[a - a*Sec[e + f*x]])} - -{Sec[e + f*x]^n*(a - a*Sec[e + f*x])^(3/2), x, 5, (2*a^2*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/((-Sec[e + f*x])^n*(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]))} -{Sec[e + f*x]^n*(a - a*Sec[e + f*x])^(1/2), x, 3, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/((-Sec[e + f*x])^n*(f*Sqrt[a - a*Sec[e + f*x]]))} - -{(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2), x, 5, (2*a^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]) + (2*a^2*(1 + 4*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/((-Sec[e + f*x])^n*(f*(1 + 2*n)*Sqrt[a - a*Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(1/2), x, 3, (2*a*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/((-Sec[e + f*x])^n*(f*Sqrt[a - a*Sec[e + f*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m with m symbolic*) - - -{Sec[e + f*x]^n*(1 + Sec[e + f*x])^m, x, 2, (2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])} -{Sec[e + f*x]^n*(1 - Sec[e + f*x])^m, x, 2, (Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*(1 - Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 + Sec[e + f*x]])} -{Sec[e + f*x]^n*(a + a*Sec[e + f*x])^m, x, 3, (2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/f} -{Sec[e + f*x]^n*(a - a*Sec[e + f*x])^m, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 - Sec[e + f*x], (1/2)*(1 - Sec[e + f*x])]*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 + Sec[e + f*x]])} - -{(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^m, x, 2, (Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*(1 + Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(1 - Sec[e + f*x])^m, x, 2, (2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1 - n, 1/2, 3/2 + m, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])} -{(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m, x, 3, (2^(1/2 + m)*AppellF1[1/2, 1 - n, 1/2 - m, 3/2, 1 + Sec[e + f*x], (1/2)*(1 + Sec[e + f*x])]*(1 - Sec[e + f*x])^(-(1/2) - m)*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/f} - -{(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^m, x, 2, -((AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n*(1 - Sec[e + f*x])^m, x, 2, -((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m, x, 3, -((AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]]))} -{(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m, x, 3, -((AppellF1[n, 1/2 - m, 1/2, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(1 - Sec[e + f*x])^(-(1/2) - m)*(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 + Sec[e + f*x]]))} - - -{Sec[e + f*x]^4*(a + a*Sec[e + f*x])^m, x, 6, If[$VersionNumber>=8, ((4 + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)*(3 + m)) + (Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(3 + m)) + (2^(1/2 + m)*m*(5 + 3*m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)*(3 + m)) + (m*(a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(6 + 5*m + m^2)), ((4 + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(6 + 11*m + 6*m^2 + m^3)) + (Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(3 + m)) + (2^(1/2 + m)*m*(5 + 3*m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(6 + 11*m + 6*m^2 + m^3)) + (m*(a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(6 + 5*m + m^2))]} -{Sec[e + f*x]^3*(a + a*Sec[e + f*x])^m, x, 5, If[$VersionNumber>=8, -(((a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(2 + 3*m + m^2))) + (2^(1/2 + m)*(1 + m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)*(2 + m)) + ((a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(2 + m)), -(((a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(2 + 3*m + m^2))) + (2^(1/2 + m)*(1 + m + m^2)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(2 + 3*m + m^2)) + ((a + a*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(a*f*(2 + m))]} -{Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m, x, 4, ((a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m)) + (2^(1/2 + m)*m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + m))} -{Sec[e + f*x]^1*(a + a*Sec[e + f*x])^m, x, 3, (2^(1/2 + m)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/f} -{Sec[e + f*x]^0*(a + a*Sec[e + f*x])^m, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]])} -{Cos[e + f*x]^1*(a + a*Sec[e + f*x])^m, x, 3, -((Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[1 - Sec[e + f*x]]))} - - -{(a + a*Sec[e + f*x])^m*(d*Sec[e + f*x])^(3/2), x, 3, -((2*AppellF1[3/2, 1/2, 1/2 - m, 5/2, Sec[e + f*x], -Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2)*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(3*f*Sqrt[1 - Sec[e + f*x]]))} -{(a + a*Sec[e + f*x])^m*(d*Sec[e + f*x])^(1/2), x, 3, -((2*AppellF1[1/2, 1/2, 1/2 - m, 3/2, Sec[e + f*x], -Sec[e + f*x]]*Sqrt[d*Sec[e + f*x]]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]]))} -{(a + a*Sec[e + f*x])^m/(d*Sec[e + f*x])^(1/2), x, 3, (2*AppellF1[-(1/2), 1/2, 1/2 - m, 1/2, Sec[e + f*x], -Sec[e + f*x]]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*Sqrt[1 - Sec[e + f*x]]*Sqrt[d*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^m/(d*Sec[e + f*x])^(3/2), x, 3, (2*AppellF1[-(3/2), 1/2, 1/2 - m, -(1/2), Sec[e + f*x], -Sec[e + f*x]]*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(3*f*Sqrt[1 - Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x]), x, 7, (6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x]), x, 6, (6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x]), x, 5, (2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x]), x, 4, (2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/d} -{(a + a*Sec[c + d*x])/Cos[c + d*x]^(1/2), x, 5, (-2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/d + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])/Cos[c + d*x]^(3/2), x, 6, (-2*a*EllipticE[(c + d*x)/2, 2])/d + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])/Cos[c + d*x]^(5/2), x, 7, (-6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])/Cos[c + d*x]^(7/2), x, 8, (-6*a*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*a*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (10*a*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (6*a*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2, x, 10, (32*a^2*EllipticE[(c + d*x)/2, 2])/(15*d) + (20*a^2*EllipticF[(c + d*x)/2, 2])/(21*d) + (20*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2, x, 9, (12*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*EllipticF[(c + d*x)/2, 2])/(7*d) + (8*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d) + (4*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2, x, 8, (16*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2, x, 7, (4*a^2*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^2, x, 5, (4*a^2*EllipticF[(c + d*x)/2, 2])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^2/Cos[c + d*x]^(1/2), x, 8, (-4*a^2*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^2/Cos[c + d*x]^(3/2), x, 9, (-16*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (16*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^2/Cos[c + d*x]^(5/2), x, 10, (-12*a^2*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*EllipticF[(c + d*x)/2, 2])/(7*d) + (2*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (4*a^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*a^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(3/2)) + (12*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3, x, 17, (68*a^3*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (44*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (44*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (68*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (6*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^3*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3, x, 15, (28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (52*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (6*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3, x, 13, (36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d + (2*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3, x, 13, (4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^3, x, 13, -((4*a^3*EllipticE[(1/2)*(c + d*x), 2])/d) + (20*a^3*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^3/Cos[c + d*x]^(1/2), x, 15, -((36*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*EllipticF[(1/2)*(c + d*x), 2])/d + (2*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a^3*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (36*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^3/Cos[c + d*x]^(3/2), x, 17, -((28*a^3*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (52*a^3*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (6*a^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (52*a^3*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (28*a^3*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x]), x, 9, (21*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + (7*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x]), x, 8, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + a*Sec[c + d*x]), x, 7, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])), x, 7, -(EllipticE[(1/2)*(c + d*x), 2]/(a*d)) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) + Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, EllipticE[(1/2)*(c + d*x), 2]/(a*d) + EllipticF[(1/2)*(c + d*x), 2]/(a*d) - Sin[c + d*x]/(d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, -((3*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - EllipticF[(1/2)*(c + d*x), 2]/(a*d) + (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])), x, 9, (3*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + (5*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - (3*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))} - - -{Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^2, x, 10, (56*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(a^2*d) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d) + (56*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - (3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^2, x, 9, -((7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^(1/2)/(a + a*Sec[c + d*x])^2, x, 8, (4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{1/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^2), x, 8, -(EllipticE[(1/2)*(c + d*x), 2]/(a^2*d)) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + Sin[c + d*x]/(a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} -{1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 5, EllipticF[(1/2)*(c + d*x), 2]/(3*a^2*d) + Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} -{1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 8, EllipticE[(1/2)*(c + d*x), 2]/(a^2*d) + (2*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - Sin[c + d*x]/(a^2*d*Sqrt[Cos[c + d*x]]*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} -{1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2), x, 9, -((4*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - (5*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (4*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (5*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2)} -{1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2), x, 10, (7*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (10*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (10*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - (7*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - (7*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(5/2)*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2)} - - -{Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^3, x, 11, (231*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (21*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (4*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^2) - (63*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^3, x, 10, -((119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^(1/2)/(a + a*Sec[c + d*x])^3, x, 9, (49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - Sin[c + d*x]/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^3), x, 9, -((9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) + (2*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 9, -(EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d)) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) + Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - Sin[c + d*x]/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 9, EllipticE[(1/2)*(c + d*x), 2]/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(6*a^3*d) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (4*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2) + Sin[c + d*x]/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3), x, 9, (9*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + EllipticF[(1/2)*(c + d*x), 2]/(2*a^3*d) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (9*Sin[c + d*x])/(10*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3), x, 10, -((49*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) - (13*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + (49*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3) - (8*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3), x, 11, (119*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + (11*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + (11*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - (119*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3) - (2*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2) - (119*Sin[c + d*x])/(30*d*Cos[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]], x, 5, (32*a*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (12*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]], x, 4, (16*a*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]], x, 3, (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]], x, 2, (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[a + a*Sec[c + d*x]]/Sqrt[Cos[c + d*x]], x, 3, (2*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d} -{Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(3/2), x, 4, (Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(5/2), x, 5, (3*Sqrt[a]*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2), x, 6, (208*a^2*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (104*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (26*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2), x, 4, (8*a^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2), x, 3, (8*a^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2), x, 5, (2*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]], x, 5, (3*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2), x, 6, (7*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (7*a^2*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(5/2), x, 7, (11*a^(3/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (11*a^2*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2), x, 6, (1168*a^3*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (584*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (146*a^3*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (38*a^3*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2), x, 5, (64*a^3*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2), x, 4, (64*a^3*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2), x, 5, (2*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (14*a^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2), x, 5, (5*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]], x, 5, (19*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (9*a^3*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2), x, 6, (25*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (13*a^3*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (25*a^3*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))} -{(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(5/2), x, 7, (163*a^(5/2)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (17*a^3*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (163*a^3*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]], x, 6, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (26*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]], x, 5, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Cos[c + d*x]]/Sqrt[a + a*Sec[c + d*x]], x, 4, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 3, (Sqrt[2]*ArcTanh[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d), (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{1/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 6, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{1/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 7, -((ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]), x, 8, (7*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + Sin[c + d*x]/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2), x, 7, -((15*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (13*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + (9*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2), x, 6, (11*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - (19*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (7*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(3/2), x, 5, -((7*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - Sin[c + d*x]/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (5*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 4, (3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 4, (ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 7, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)), x, 8, -((3*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d)) + (9*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - Sin[c + d*x]/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + (3*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2), x, 7, (163*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - (17*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (299*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (95*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(5/2), x, 6, -((75*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - Sin[c + d*x]/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - (13*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (49*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 5, (19*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - (9*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 5, (5*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + (5*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 5, (3*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + (3*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)), x, 8, (2*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - (43*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) - (11*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)), x, 9, -((5*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d)) + (115*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - Sin[c + d*x]/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) - (15*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + (35*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m with n symbolic*) - - -{(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3, x, 8, If[$VersionNumber>=8, -((a^3*(7 - 4*n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a^3*(1 - 4*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + ((d*Cos[e + f*x])^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n)), -((a^3*(7 - 4*n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a^3*(1 - 4*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(2 - 3*n + n^2)) + ((d*Cos[e + f*x])^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n))]} -{(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^2, x, 7, If[$VersionNumber>=8, -((2*a^2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a^2*(1 - 2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)), -((2*a^2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a^2*(1 - 2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n))]} -{(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^1, x, 5, -((a*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*(1 + n)*Sqrt[Sin[e + f*x]^2])} -{(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x])^1, x, 7, ((d*Cos[e + f*x])^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 + n)*Cos[e + f*x]^2*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(a*f*(2 + n)*Sqrt[Sin[e + f*x]^2])} -{(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x])^2, x, 8, (2*(2 + n)*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - ((3 + 2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - (2*(2 + n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - ((d*Cos[e + f*x])^n*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x]), x, 6, (3*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a*Tan[c + d*x])/d + (3*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x]), x, 5, (b*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x]), x, 4, (a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x]), x, 2, a*x + (b*ArcTanh[Sin[c + d*x]])/d} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x]), x, 3, b*x + (a*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x]), x, 4, (a*x)/2 + (b*Sin[c + d*x])/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x]), x, 5, (b*x)/2 + (a*Sin[c + d*x])/d + (b*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x]), x, 6, (3*a*x)/8 + (b*Sin[c + d*x])/d + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x]), x, 6, (3*b*x)/8 + (a*Sin[c + d*x])/d + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x])^2, x, 7, (3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) + ((5*a^2 + 4*b^2)*Tan[c + d*x])/(5*d) + (3*a*b*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a*b*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (b^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*a^2 + 4*b^2)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2, x, 6, ((4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Tan[c + d*x])/d + ((4*a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (2*a*b*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2, x, 6, (a*b*ArcTanh[Sin[c + d*x]])/d + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (b^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^2, x, 5, ((2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Tan[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^2, x, 4, a^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d + (b^2*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^2, x, 4, 2*a*b*x + (b^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2, x, 4, (1/2)*(a^2 + 2*b^2)*x + (2*a*b*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2, x, 6, a*b*x + ((a^2 + b^2)*Sin[c + d*x])/d + (a*b*Cos[c + d*x]*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2, x, 6, (1/8)*(3*a^2 + 4*b^2)*x + (2*a*b*Sin[c + d*x])/d + ((3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a*b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2, x, 8, (3*a*b*x)/4 + ((a^2 + b^2)*Sin[c + d*x])/d + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(2*d) - ((2*a^2 + b^2)*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^5)/(5*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^3, x, 8, (a*(4*a^2 + 9*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*Tan[c + d*x])/(30*b*d) - (a*(6*a^2 - 71*b^2)*Sec[c + d*x]*Tan[c + d*x])/(120*d) - ((3*a^2 - 16*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) - (a*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + ((a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3, x, 7, (3*b*(4*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(a^2 + 4*b^2)*Tan[c + d*x])/(2*d) + (b*(2*a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + ((a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^3, x, 6, (a*(2*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*b*(4*a^2 + b^2)*Tan[c + d*x])/(3*d) + (5*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^3, x, 5, a^3*x + (b*(6*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a*b^2*Tan[c + d*x])/(2*d) + (b^2*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^3, x, 5, 3*a^2*b*x + (3*a*b^2*ArcTanh[Sin[c + d*x]])/d + (a*(a^2 - b^2)*Sin[c + d*x])/d + (b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3, x, 5, (1/2)*a*(a^2 + 6*b^2)*x + (b^3*ArcTanh[Sin[c + d*x]])/d + (5*a^2*b*Sin[c + d*x])/(2*d) + (a^2*Cos[c + d*x]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3, x, 5, (1/2)*b*(3*a^2 + 2*b^2)*x + (a*(2*a^2 + 9*b^2)*Sin[c + d*x])/(3*d) + (7*a^2*b*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3, x, 7, (3/8)*a*(a^2 + 4*b^2)*x + (b*(11*a^2 + 4*b^2)*Sin[c + d*x])/(4*d) + (3*a*(a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])*Sin[c + d*x])/(4*d) - (3*a^2*b*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*b*(9*a^2 + 4*b^2)*x + (a*(4*a^2 + 15*b^2)*Sin[c + d*x])/(5*d) + (b*(9*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (11*a^2*b*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a^2*Cos[c + d*x]^4*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a*(4*a^2 + 15*b^2)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3, x, 9, (1/16)*a*(5*a^2 + 18*b^2)*x + (b*(17*a^2 + 6*b^2)*Sin[c + d*x])/(6*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(5*a^2 + 18*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])*Sin[c + d*x])/(6*d) - (b*(5*a^2 + b^2)*Sin[c + d*x]^3)/(3*d) + (13*a^2*b*Sin[c + d*x]^5)/(30*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^4, x, 9, ((8*a^4 + 36*a^2*b^2 + 5*b^4)*ArcTanh[Sin[c + d*x]])/(16*d) - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*Tan[c + d*x])/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*Sec[c + d*x]*Tan[c + d*x])/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) - (a*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + ((a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4, x, 8, (a*b*(4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*Tan[c + d*x])/(15*d) + (a*b*(6*a^2 + 29*b^2)*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*a^2 + 4*b^2)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + ((a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^4, x, 7, ((8*a^4 + 24*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b*(19*a^2 + 16*b^2)*Tan[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (7*a*b*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^4, x, 6, a^4*x + (2*a*b*(2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/d + (b^2*(17*a^2 + 2*b^2)*Tan[c + d*x])/(3*d) + (4*a*b^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^4, x, 6, 4*a^3*b*x + (b^2*(12*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*a^2 - b^2)*Sin[c + d*x])/(2*d) + (b^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (3*a*b^3*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4, x, 6, (1/2)*a^2*(a^2 + 12*b^2)*x + (4*a*b^3*ArcTanh[Sin[c + d*x]])/d + (3*a^3*b*Sin[c + d*x])/d + (a^2*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a^2 - 2*b^2)*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4, x, 6, 2*a*b*(a^2 + 2*b^2)*x + (b^4*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2 + 17*b^2)*Sin[c + d*x])/(3*d) + (4*a^3*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4, x, 6, (1/8)*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x + (4*a*b*(2*a^2 + 3*b^2)*Sin[c + d*x])/(3*d) + (a^2*(3*a^2 + 22*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a^3*b*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4, x, 8, (1/2)*a*b*(3*a^2 + 4*b^2)*x + ((4*a^4 + 29*a^2*b^2 + 5*b^4)*Sin[c + d*x])/(5*d) + (a*b*(3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a^3*b*Cos[c + d*x]^3*Sin[c + d*x])/(5*d) + (a^2*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a^2*(4*a^2 + 27*b^2)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4, x, 8, (1/16)*(5*a^4 + 36*a^2*b^2 + 8*b^4)*x + (4*a*b*(4*a^2 + 5*b^2)*Sin[c + d*x])/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(5*a^2 + 32*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (7*a^3*b*Cos[c + d*x]^4*Sin[c + d*x])/(15*d) + (a^2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) - (4*a*b*(4*a^2 + 5*b^2)*Sin[c + d*x]^3)/(15*d)} - - -{(a + b*Sec[c + d*x])^5, x, 7, a^5*x + (b*(40*a^4 + 40*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b^2*(53*a^2 + 20*b^2)*Tan[c + d*x])/(6*d) + (b^3*(58*a^2 + 9*b^2)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (11*a*b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b^2*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^5/(a + b*Sec[c + d*x]), x, 8, -((a*(2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d)) + (2*a^4*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(3*b^3*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x]), x, 7, ((2*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*Tan[c + d*x])/(b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x]), x, 6, -((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) + (2*a^2*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + Tan[c + d*x]/(b*d)} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x]), x, 5, ArcTanh[Sin[c + d*x]]/(b*d) - (2*a*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x]), x, 3, (2*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x]), x, 3, x/a - (2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x]), x, 5, -((b*x)/a^2) + (2*b^2*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + Sin[c + d*x]/(a*d)} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x]), x, 6, ((a^2 + 2*b^2)*x)/(2*a^3) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - (b*Sin[c + d*x])/(a^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3/(a + b*Sec[c + d*x]), x, 7, -((b*(a^2 + 2*b^2)*x)/(2*a^4)) + (2*b^4*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^3*d) - (b*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^4/(a + b*Sec[c + d*x]), x, 8, ((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(2*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^4*d) + ((3*a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (b*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} - - -{Sec[c + d*x]^5/(a + b*Sec[c + d*x])^2, x, 8, ((6*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(3*a^2 - 4*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^2, x, 7, -((2*a*ArcTanh[Sin[c + d*x]])/(b^3*d)) + (2*a^2*(2*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + ((2*a^2 - b^2)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 6, ArcTanh[Sin[c + d*x]]/(b^2*d) - (2*a*(a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 5, -((2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d)) + (a*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 5, (2*a*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^2, x, 5, x/a^2 - (2*b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 6, -((2*b*x)/a^3) + (2*b^2*(3*a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2 - 2*b^2)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 7, ((a^2 + 6*b^2)*x)/(2*a^4) - (2*b^3*(4*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(2*a^2 - 3*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 8, -((b*(a^2 + 4*b^2)*x)/a^5) + (2*b^4*(5*a^2 - 4*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^5/(a + b*Sec[c + d*x])^3, x, 8, -((3*a*ArcTanh[Sin[c + d*x]])/(b^4*d)) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((3*a^2 - 2*b^2)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (3*a^3*(a^2 - 2*b^2)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^3, x, 7, ArcTanh[Sin[c + d*x]]/(b^3*d) - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(2*a^2 - 5*b^2)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^3, x, 6, ((a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2 - 4*b^2)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^3, x, 6, -((3*a*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2 + 2*b^2)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^3, x, 6, ((2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (3*a*b*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^3, x, 6, x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x])^3, x, 7, -((3*b*x)/a^4) + (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (3*b^2*(2*a^2 - b^2)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x])^3, x, 8, ((a^2 + 12*b^2)*x)/(2*a^5) - (b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(7*a^2 - 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^6/(a + b*Sec[c + d*x])^4, x, 9, -((4*a*ArcTanh[Sin[c + d*x]])/(b^5*d)) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a^2*(4*a^2 - 9*b^2)*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^5/(a + b*Sec[c + d*x])^4, x, 8, ArcTanh[Sin[c + d*x]]/(b^4*d) - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a^3*(3*a^2 - 8*b^2)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^4, x, 7, -((b*(3*a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*Sec[c + d*x]*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a^2*(2*a^2 - 7*b^2)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^4, x, 7, (a*(a^2 + 4*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2 - 6*b^2)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^4, x, 7, -((b*(4*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2 + 3*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^2 + 13*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^4, x, 7, (a*(2*a^2 + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (5*a*b*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^4, x, 7, x/a^4 - (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2 - 3*b^2)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x])^4, x, 8, -((4*b*x)/a^5) + (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(9*a^2 - 4*b^2)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x])^4, x, 9, ((a^2 + 20*b^2)*x)/(2*a^6) - (b^3*(40*a^6 - 84*a^4*b^2 + 69*a^2*b^4 - 20*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - (b*(24*a^6 - 146*a^4*b^2 + 167*a^2*b^4 - 60*b^6)*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6 - 23*a^4*b^2 + 27*a^2*b^4 - 10*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (5*b^2*(2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(48*a^4 - 53*a^2*b^2 + 20*b^4)*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - - -{1/(3 + 5*Sec[c + d*x]), x, 2, -(x/12) + (5*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(6*d)} -{1/(3 + 5*Sec[c + d*x])^2, x, 4, (29*x)/576 + (35*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(288*d) - (25*Tan[c + d*x])/(48*d*(3 + 5*Sec[c + d*x]))} -{1/(3 + 5*Sec[c + d*x])^3, x, 5, -((1007*x)/55296) + (3055*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(27648*d) - (25*Tan[c + d*x])/(96*d*(3 + 5*Sec[c + d*x])^2) - (125*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x]))} -{1/(3 + 5*Sec[c + d*x])^4, x, 6, (21553*x)/2654208 + (11215*ArcTan[Sin[c + d*x]/(3 + Cos[c + d*x])])/(1327104*d) - (25*Tan[c + d*x])/(144*d*(3 + 5*Sec[c + d*x])^3) - (25*Tan[c + d*x])/(4608*d*(3 + 5*Sec[c + d*x])^2) - (16925*Tan[c + d*x])/(221184*d*(3 + 5*Sec[c + d*x]))} - -{1/(5 + 3*Sec[c + d*x]), x, 3, x/5 + (3*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(20*d) - (3*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(20*d)} -{1/(5 + 3*Sec[c + d*x])^2, x, 5, x/25 + (123*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(1600*d) - (123*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(1600*d) + (9*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x]))} -{1/(5 + 3*Sec[c + d*x])^3, x, 6, x/125 + (8361*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(256000*d) - (8361*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(256000*d) + (9*Tan[c + d*x])/(160*d*(5 + 3*Sec[c + d*x])^2) + (963*Tan[c + d*x])/(12800*d*(5 + 3*Sec[c + d*x]))} -{1/(5 + 3*Sec[c + d*x])^4, x, 7, x/625 + (278151*Log[2*Cos[(1/2)*(c + d*x)] - Sin[(1/2)*(c + d*x)]])/(20480000*d) - (278151*Log[2*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(20480000*d) + (3*Tan[c + d*x])/(80*d*(5 + 3*Sec[c + d*x])^3) + (519*Tan[c + d*x])/(12800*d*(5 + 3*Sec[c + d*x])^2) + (38733*Tan[c + d*x])/(1024000*d*(5 + 3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]], x, 5, (2*(a - b)*Sqrt[a + b]*(2*a^2 - 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(2*a + 9*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) - (4*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]], x, 4, (-2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) - (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]], x, 3, (-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)} -{Sec[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]], x, 1, (-2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]], x, 6, ((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]], x, 7, ((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2 - b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2), x, 7, (-2*(a - b)*Sqrt[a + b]*(8*a^4 + 33*a^2*b^2 + 147*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^3 + 6*a^2*b + 39*a*b^2 - 147*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*a*(8*a^2 + 39*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) - (8*a*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2), x, 6, (4*a*(a - b)*Sqrt[a + b]*(3*a^2 - 41*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2 + 57*a*b - 25*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) - (2*(6*a^2 - 25*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) - (4*a*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2), x, 5, (-2*(a - b)*Sqrt[a + b]*(a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^2*d) - (2*(a - 3*b)*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b*d) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2), x, 4, (-8*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*(a - b)*(3*a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(3/2), x, 5, (-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(2*a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2), x, 6, (a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (3*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2), x, 7, (5*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a + 5*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2 + 3*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (5*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2), x, 8, (-2*a*(a - b)*Sqrt[a + b]*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^4 + 6*a^3*b + 57*a^2*b^2 - 606*a*b^3 + 135*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^3*d) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) - (8*a*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*(10*a^4 - 279*a^2*b^2 - 147*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3 + 165*a^2*b - 114*a*b^2 + 147*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) - (4*a*(5*a^2 - 57*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2), x, 6, (-2*a*(a - b)*Sqrt[a + b]*(3*a^2 + 29*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b^2*d) - (2*(a - b)*Sqrt[a + b]*(3*a^2 - 24*a*b + 5*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b*d) + (2*(3*a^2 + 5*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2), x, 5, (-2*(a - b)*Sqrt[a + b]*(23*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(a - b)*Sqrt[a + b]*(15*a^2 - 8*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (16*a*b*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/2), x, 6, (-14*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) + (2*Sqrt[a + b]*(9*a^2 - 7*a*b + b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a^2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2), x, 6, ((a - b)*Sqrt[a + b]*(a^2 - 2*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(a^2 + 6*a*b - 2*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (5*a*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2), x, 7, (9*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a^2 + 9*a*b + 8*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2 + 15*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (9*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a^2*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2), x, 8, ((a - b)*Sqrt[a + b]*(16*a^2 + 33*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2 + 26*a*b + 33*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (5*b*Sqrt[a + b]*(4*a^2 + b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2 + 33*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (13*a*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a^2*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2), x, 9, ((a - b)*Sqrt[a + b]*(284*a^2 + 15*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(72*a^3 + 284*a^2*b + 118*a*b^2 + 15*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4 + 120*a^2*b^2 - 5*b^4)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (b*(284*a^2 + 15*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2 + 59*b^2)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (17*a*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{(a + b*Sec[c + d*x])^(7/2), x, 7, (-2*(a - b)*Sqrt[a + b]*(58*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) + (2*Sqrt[a + b]*(60*a^3 - 58*a^2*b + 22*a*b^2 - 9*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*a^3*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (26*a*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b^2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]], x, 6, (8*a*(a - b)*Sqrt[a + b]*(12*a^2 + 11*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + b]*(48*a^3 - 12*a^2*b + 44*a*b^2 + 25*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(24*a^2 + 25*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) - (12*a*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*(8*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(8*a^2 - 2*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (8*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]], x, 4, (4*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]], x, 3, (-2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)} -{Sec[c + d*x]^1/Sqrt[a + b*Sec[c + d*x]], x, 1, (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)} -{Sec[c + d*x]^0/Sqrt[a + b*Sec[c + d*x]], x, 1, (-2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{Cos[c + d*x]^1/Sqrt[a + b*Sec[c + d*x]], x, 6, ((a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]], x, 7, (-3*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((2*a - 3*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(4*a^2 + 3*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - (3*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)} - - -{Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2), x, 6, (-2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^5*Sqrt[a + b]*d) - (2*(4*a + 3*b)*(4*a^2 + b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^4*Sqrt[a + b]*d) - (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a*(8*a^2 - 3*b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2), x, 5, (2*a*(8*a^2 - 5*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(2*a + b)*(4*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d)} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2), x, 4, (-2*(2*a^2 - b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) - (2*(2*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*a^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2), x, 4, (2*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*a*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(3/2), x, 5, (-2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*b*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x])^(3/2), x, 7, ((a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((a + 3*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (3*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + Sin[c + d*x]/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2 - 3*b^2)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2), x, 8, -((7*a^2 - 15*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) + ((2*a^2 - 5*a*b - 15*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(4*a^2 + 15*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - (5*b*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) - (b^2*(7*a^2 - 15*b^2)*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(5/2), x, 6, (8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) + (2*(16*a^4 + 12*a^3*b - 16*a^2*b^2 - 9*a*b^3 - b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(2*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/2), x, 5, (-2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) - (2*(8*a^3 + 6*a^2*b - 9*a*b^2 - 3*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (8*a^2*(a^2 - 2*b^2)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/2), x, 5, (4*a*(a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2 + 3*a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a*(a^2 - 3*b^2)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2), x, 5, (2*(a^2 + 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a - 3*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*a*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2 + 3*b^2)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(5/2), x, 5, (-8*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) + (2*(3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*b*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (8*a*b*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(5/2), x, 7, (2*(7*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*(6*a^2 - a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^1/(a + b*Sec[c + d*x])^(5/2), x, 8, ((3*a^4 - 26*a^2*b^2 + 15*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + ((3*a^3 + 21*a^2*b - 5*a*b^2 - 15*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (5*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + Sin[c + d*x]/(a*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^2 - 5*b^2)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2), x, 9, -((33*a^4 - 170*a^2*b^2 + 105*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*(a + b)^(3/2)*d) + ((a + 3*b)*(6*a^3 - 45*a^2*b + 35*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*(a + b)^(3/2)*d) - (Sqrt[a + b]*(4*a^2 + 35*b^2)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - (7*b*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(27*a^2 - 35*b^2)*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(33*a^4 - 170*a^2*b^2 + 105*b^4)*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(a + b*Sec[c + d*x])^(-7/2), x, 8, (2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*(45*a^4 - 13*a^3*b - 36*a^2*b^2 + 5*a*b^3 + 15*b^4)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*(a - b)^2*(a + b)^(5/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (2*b^2*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) + (2*b^2*(13*a^2 - 5*b^2)*Tan[c + d*x])/(15*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(58*a^4 - 41*a^2*b^2 + 15*b^4)*Tan[c + d*x])/(15*a^3*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x]), x, 8, -((6*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (6*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x]), x, 7, -((2*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x]), x, 6, -((2*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x]), x, 5, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x]), x, 6, (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x]), x, 7, (6*a*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x]), x, 8, (6*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (10*a*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (10*a*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2, x, 9, -((12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (12*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*b^2*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2, x, 8, -((2*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*a^2 + 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a*b*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^2, x, 7, -((4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^2, x, 6, (2*(a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^2, x, 6, (4*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^2, x, 7, (2*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^2, x, 8, (12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3, x, 9, -((2*a*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(5*a^2 + 9*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(21*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (32*a*b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b^2*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^3, x, 8, -((6*b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (6*b*(5*a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (8*a*b^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*b^2*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^3, x, 7, (2*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*b*(9*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (16*a*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b^2*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^3, x, 7, (2*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(a^2 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^3, x, 7, (6*a*(a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*b*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*b*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^3, x, 8, (2*b*(9*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (32*a^2*b*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a*(5*a^2 + 21*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(a + b*Sec[c + d*x])^3, x, 9, (2*a*(7*a^2 + 27*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*b*(15*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (40*a^2*b*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a*(7*a^2 + 27*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*b*(15*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4, x, 10, -((2*(15*a^4 + 54*a^2*b^2 + 7*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (8*a*b*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^4 + 54*a^2*b^2 + 7*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a*b*(7*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (14*b^2*(7*a^2 + b^2)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (44*a*b^3*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*b^2*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^4, x, 9, -((8*a*b*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(21*a^4 + 42*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a*b*(5*a^2 + 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*(39*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (36*a*b^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b^2*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^4, x, 8, (2*(5*a^4 - 30*a^2*b^2 - 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b^2*(29*a^2 + 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (28*a*b^3*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b^2*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^4, x, 8, (8*a*b*(a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^4 + 18*a^2*b^2 + b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a*b*(a^2 - 6*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(a^2 - b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^4, x, 8, (2*(3*a^4 + 30*a^2*b^2 - 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a*b*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (28*a^3*b*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^4, x, 8, (8*a*b*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^4 + 42*a^2*b^2 + 21*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (36*a^3*b*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a^2*(5*a^2 + 39*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(a + b*Sec[c + d*x])^4, x, 9, (2*(7*a^4 + 54*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a*b*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (44*a^3*b*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (14*a^2*(a^2 + 7*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a*b*(5*a^2 + 7*b^2)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{Sec[c + d*x]^(-11/2)*(a + b*Sec[c + d*x])^4, x, 10, (8*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (52*a^3*b*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a^2*(9*a^2 + 59*b^2)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (8*a*b*(7*a^2 + 9*b^2)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x]), x, 10, (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b*d) + (2*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) - (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x]), x, 6, (-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) - (2*a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d) + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x]), x, 2, (2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d)} -{Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x]), x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)} -{1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])), x, 8, (2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)} -{1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])), x, 9, (-2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^2, x, 11, (a*(5*a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) + (a^2*(5*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) - (a*(5*a^2 - 4*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((5*a^2 - 2*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^2, x, 10, -(((3*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d)) - (a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^2, x, 9, (a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b*(a + b)^2*d) - (a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2, x, 9, -((Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*d)) - (b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*(a + b)^2*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^2, x, 9, (b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2), x, 9, ((2*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - (b*(4*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2), x, 10, -((b*(4*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) + ((2*a^4 + 16*a^2*b^2 - 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^3, x, 11, -((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) - (a*(5*a^2 - 11*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) - (a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) - (a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(5*a^2 - 11*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^3, x, 10, (3*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d) + (3*(a^4 - 2*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (3*a^2*(a^2 - 3*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^3, x, 10, ((a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 - 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2 - 7*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3, x, 10, -(((5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a^2 - b^2)^2*d)) - (b*(7*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4 + 10*a^2*b^2 - b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (3*(a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^3, x, 10, (3*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (b*(7*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3), x, 10, ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3), x, 11, -(b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(1/2), x, 12, (b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(1/2), x, 7, (2*a*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[(b + a*Cos[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[(b + a*Cos[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^(1/2), x, 3, (2*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^(1/2), x, 8, (2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^(1/2), x, 9, -((4*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(9*a^2 - 2*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^(1/2), x, 10, (2*(25*a^4 - 17*a^2*b^2 - 8*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(19*a^2 + 8*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2 - 4*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2), x, 13, (7*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - (5*a*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (5*a*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(3/2), x, 12, ((2*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (3*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^(3/2), x, 11, (2*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^(3/2), x, 8, (2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (8*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^(3/2), x, 9, (2*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(5*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^(3/2), x, 10, (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(41*a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (16*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2 + 3*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2), x, 14, (b*(59*a^2 + 16*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + (5*a*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((33*a^2 + 16*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((33*a^2 + 16*b^2)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (13*a*b*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (b^2*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(5/2), x, 13, (a*(8*a^2 + 11*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - (9*a*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (9*a*b*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b^2*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Sec[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^(5/2), x, 12, (b*(4*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (5*a*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b^2*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^(5/2), x, 12, (2*a*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (14*a*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^(5/2), x, 9, (16*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 23*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (22*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(a + b*Sec[c + d*x])^(5/2), x, 10, (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(21*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(29*a^2 + 3*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(21*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*(5*a^2 + 9*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-9/2)*(a + b*Sec[c + d*x])^(5/2), x, 11, (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (38*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2 + 75*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*b*(163*a^2 + 5*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(1/2), x, 13, -((a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]])) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (3*a*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (3*a*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(1/2), x, 12, (Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(1/2), x, 3, (2*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[(b + a*Cos[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + b*Sec[c + d*x])^(1/2), x, 3, (2*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[(b + a*Cos[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-1/2)/(a + b*Sec[c + d*x])^(1/2), x, 7, -((2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]])) + (2*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)/(a + b*Sec[c + d*x])^(1/2), x, 8, (2*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (4*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)/(a + b*Sec[c + d*x])^(1/2), x, 9, -((2*b*(7*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(9*a^2 + 8*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (8*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(3/2), x, 13, (Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (3*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(3/2), x, 9, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2), x, 5, -((2*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/((a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + b*Sec[c + d*x])^(3/2), x, 8, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-1/2)/(a + b*Sec[c + d*x])^(3/2), x, 8, -((4*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(a^2 - 2*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)/(a + b*Sec[c + d*x])^(3/2), x, 9, (2*(a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(5*a^2 - 8*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)/(a + b*Sec[c + d*x])^(3/2), x, 10, -((8*b*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(5*a^4*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 6*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (2*b*(3*a^2 - 8*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^(5/2), x, 14, ((5*a^2 - 3*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (5*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((15*a^4 - 26*a^2*b^2 + 3*b^4)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(5*a^2 - 9*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + ((15*a^4 - 26*a^2*b^2 + 3*b^4)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)} -{Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(5/2), x, 13, -((2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) + (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*a*(3*a^2 - 7*b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(5/2), x, 9, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (8*b*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*a^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(a^2 - 5*b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2), x, 9, -((2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) - (2*(3*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*(a^2 + b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(1/2)/(a + b*Sec[c + d*x])^(5/2), x, 9, (2*(3*a^2 - 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-1/2)/(a + b*Sec[c + d*x])^(5/2), x, 9, -((2*b*(9*a^2 - 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^(-3/2)/(a + b*Sec[c + d*x])^(5/2), x, 10, (2*(a^4 + 16*a^2*b^2 - 16*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)/(a + b*Sec[c + d*x])^(5/2), x, 11, -((2*b*(17*a^4 + 116*a^2*b^2 - 128*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) + (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (8*b^2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4 - 71*a^2*b^2 + 48*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} - - -{1/(Sqrt[Sec[c + d*x]]*Sqrt[2 + 3*Sec[c + d*x]]), x, 5, -((3*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]])) + (Sqrt[5]*EllipticE[(1/2)*(c + d*x), 4/5]*Sqrt[2 + 3*Sec[c + d*x]])/(d*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[-2 + 3*Sec[c + d*x]]), x, 5, (3*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]]) - (EllipticE[(1/2)*(c + d*x), -4]*Sqrt[-2 + 3*Sec[c + d*x]])/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} - -{1/(Sqrt[Sec[c + d*x]]*Sqrt[2 - 3*Sec[c + d*x]]), x, 7, (EllipticE[(1/2)*(c + d*x), -4]*Sqrt[2 - 3*Sec[c + d*x]])/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]])} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[-2 - 3*Sec[c + d*x]]), x, 7, -((Sqrt[5]*EllipticE[(1/2)*(c + d*x), 4/5]*Sqrt[-2 - 3*Sec[c + d*x]])/(d*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) - (3*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])} - -{1/(Sqrt[Sec[c + d*x]]*Sqrt[3 + 2*Sec[c + d*x]]), x, 5, -((4*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 6/5]*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]])) + (2*Sqrt[5]*EllipticE[(1/2)*(c + d*x), 6/5]*Sqrt[3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[3 - 2*Sec[c + d*x]]), x, 5, (2*EllipticE[(1/2)*(c + d*x), 6]*Sqrt[3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 + 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[3 - 2*Sec[c + d*x]])} - -{1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 + 2*Sec[c + d*x]]), x, 5, (4*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + Pi + d*x), 6/5]*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[-3 + 2*Sec[c + d*x]]) - (2*Sqrt[5]*EllipticE[(1/2)*(c + Pi + d*x), 6/5]*Sqrt[-3 + 2*Sec[c + d*x]])/(3*d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])} -{1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 - 2*Sec[c + d*x]]), x, 5, -((2*EllipticE[(1/2)*(c + Pi + d*x), 6]*Sqrt[-3 - 2*Sec[c + d*x]])/(3*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) - (4*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + Pi + d*x), 6]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 - 2*Sec[c + d*x]])} - - -{Sqrt[Sec[c + d*x]]/Sqrt[2 + 3*Sec[c + d*x]], x, 2, (2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]/Sqrt[-2 + 3*Sec[c + d*x]], x, 2, (2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]])} - -{Sqrt[Sec[c + d*x]]/Sqrt[2 - 3*Sec[c + d*x]], x, 3, (2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]/Sqrt[-2 - 3*Sec[c + d*x]], x, 3, (2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])} - -{Sqrt[Sec[c + d*x]]/Sqrt[3 + 2*Sec[c + d*x]], x, 2, (2*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]/Sqrt[3 - 2*Sec[c + d*x]], x, 2, (2*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[3 - 2*Sec[c + d*x]])} - -{Sqrt[Sec[c + d*x]]/Sqrt[-3 + 2*Sec[c + d*x]], x, 2, (2*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + Pi + d*x), 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-3 + 2*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]/Sqrt[-3 - 2*Sec[c + d*x]], x, 2, (2*Sqrt[-2 - 3*Cos[c + d*x]]*EllipticF[(1/2)*(c + Pi + d*x), 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-3 - 2*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(1/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(1/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(1/3), x]} - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(2/3), x, 10, (3*(9*a^2 + 32*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(220*b^2*d) - (9*a*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(44*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(11*b*d) + (a*(18*a^2 + 49*b^2)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(110*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - ((9*a^4 + 23*a^2*b^2 - 32*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(55*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(2/3), x, 9, -((9*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(40*b*d)) + (3*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b*d) - ((6*a^2 - 25*b^2)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (3*a*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(10*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(2/3), x, 8, (3*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (2*Sqrt[2]*a*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (2*Sqrt[2]*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(5*b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(2/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(2/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(2/3), x]} - - -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(4/3), x, 3, (Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(4/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(4/3), x]} - - -{Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/3), x, 11, (3*a*(18*a^2 + 97*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(1232*b^2*d) + (3*(18*a^2 + 121*b^2)*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(1232*b^2*d) - (9*a*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(77*b^2*d) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(14*b*d) + ((36*a^4 + 164*a^2*b^2 + 605*b^4)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(616*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(18*a^4 + 79*a^2*b^2 - 97*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(308*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/3), x, 10, -((3*(15*a^2 - 64*b^2)*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(440*b*d)) - (9*a*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(88*b*d) + (3*(a + b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*b*d) - (a*(30*a^2 - 373*b^2)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(220*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + ((15*a^4 - 79*a^2*b^2 + 64*b^4)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(110*Sqrt[2]*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/3), x, 9, (3*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*d) + (3*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*d) + ((2*a^2 + 5*b^2)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(4*Sqrt[2]*b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(a^2 - b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(2*Sqrt[2]*b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/3), x, 3, (Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/3), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(1/3), x, 9, -((9*a*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*b^2*d)) + (3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(8*b*d) + ((18*a^2 + 25*b^2)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(20*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (a*(9*a^2 + 11*b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(10*Sqrt[2]*b^3*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(1/3), x, 8, (3*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d) - (3*Sqrt[2]*a*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(3*a^2 + 2*b^2)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(5*b^2*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(1/3), x, 7, (Sqrt[2]*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(1/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(1/3), x, 0, Unintegrable[1/(a + b*Sec[c + d*x])^(1/3), x]} - - -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(2/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(2/3), x, 0, Unintegrable[1/(a + b*Sec[c + d*x])^(2/3), x]} - - -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(4/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 4/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/((a + b)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(4/3), x, 0, Unintegrable[1/(a + b*Sec[c + d*x])^(4/3), x]} - - -{Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/3), x, 9, -((3*a^2*Sec[c + d*x]*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3))) + (3*(3*a^2 - b^2)*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b^2*(a^2 - b^2)*d) - (a*(9*a^2 - 7*b^2)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(2*Sqrt[2]*b^3*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + ((9*a^4 - 10*a^2*b^2 - b^4)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(2*Sqrt[2]*b^3*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} -{Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/3), x, 8, -((3*a^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3))) + ((3*a^2 - 2*b^2)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(Sqrt[2]*b^2*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (a*(3*a^2 - 4*b^2)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(Sqrt[2]*b^2*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} -{Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/3), x, 8, (3*a*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(2/3)) - (a*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(Sqrt[2]*b*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + ((a^2 - 2*b^2)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(Sqrt[2]*b*(a^2 - b^2)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} -{Sec[c + d*x]^1/(a + b*Sec[c + d*x])^(5/3), x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, 5/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/((a + b)*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} -{Sec[c + d*x]^0/(a + b*Sec[c + d*x])^(5/3), x, 0, Unintegrable[1/(a + b*Sec[c + d*x])^(5/3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/3) (a+b Sec[e+f x])^m*) - - -{Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x]), x, 6, (a*AppellF1[1/2, -(1/6), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)) - (b*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x]), x, 6, (a*AppellF1[1/2, -(1/3), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)) - (b*AppellF1[1/2, 1/6, 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^(-1/3)/(a + b*Sec[c + d*x]), x, 6, -((b*AppellF1[1/2, -(1/6), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3))) + (a*AppellF1[1/2, -(2/3), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3)*Sin[c + d*x])/((a^2 - b^2)*d)} -{Sec[c + d*x]^(-2/3)/(a + b*Sec[c + d*x]), x, 6, -((b*AppellF1[1/2, -(1/3), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3)*Sec[c + d*x]^(2/3))) + (a*AppellF1[1/2, -(5/6), 1, 3/2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*(Cos[c + d*x]^2)^(1/6)*Sec[c + d*x]^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/3) (a+b Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]], x]} -{Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3), x]} -{Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3), x]} -{Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3), x]} -{Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3), x]} -{Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x]} - - -{Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2), x]} -{(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3), x]} -{(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3), x]} -{(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3), x]} -{(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3), x]} -{(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x]} - - -{Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2), x]} -{(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3), x]} -{(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3), x]} -{(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x]} -{(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3), x]} -{(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]], x]} -{Sec[c + d*x]^(-1/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[1/(Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]]), x]} -{Sec[c + d*x]^(-2/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[1/(Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]]), x]} -{Sec[c + d*x]^(-4/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]} -{Sec[c + d*x]^(-5/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[1/(Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]]), x]} -{Sec[c + d*x]^(-7/3)/Sqrt[a + b*Sec[c + d*x]], x, 0, Unintegrable[1/(Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]]), x]} - - -{Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2), x]} -{Sec[c + d*x]^(-1/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)), x]} -{Sec[c + d*x]^(-2/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2)), x]} -{Sec[c + d*x]^(-4/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2)), x]} -{Sec[c + d*x]^(-5/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2)), x]} -{Sec[c + d*x]^(-7/3)/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2)), x]} - - -{Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2), x]} -{Sec[c + d*x]^(-1/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2)), x]} -{Sec[c + d*x]^(-2/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2)), x]} -{Sec[c + d*x]^(-4/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]} -{Sec[c + d*x]^(-5/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2)), x]} -{Sec[c + d*x]^(-7/3)/(a + b*Sec[c + d*x])^(5/2), x, 0, Unintegrable[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m with n symbolic*) - - -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^3, x, 7, -((a*d*(3*b^2*n + a^2*(1 + n))*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2])) + (b*(b^2*(1 + n) + 3*a^2*(2 + n))*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 + 2*n)*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + n)*(2 + n)) + (b^2*(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n))} -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^2, x, 6, -((d*(b^2*n + a^2*(1 + n))*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2])) + (2*a*b*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2]) + (b^2*(d*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + n))} -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^1, x, 5, -((a*d*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^(-1 + n)*Sin[e + f*x])/(f*(1 - n)*Sqrt[Sin[e + f*x]^2])) + (b*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[e + f*x]^2]*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])} -{(d* Sec[e + f*x])^n/(a + b*Sec[e + f*x])^1, x, 6, (a*AppellF1[1/2, (1/2)*(-1 + n), 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n))*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, n/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)*f)} -{(d* Sec[e + f*x])^n/(a + b*Sec[e + f*x])^2, x, 9, (a^2*AppellF1[1/2, (1/2)*(-3 + n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n))*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (1/2)*(-1 + n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n))*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f) - (2*a*b*AppellF1[1/2, (1/2)*(-2 + n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^(n/2)*(d*Sec[e + f*x])^n*Sin[e + f*x])/((a^2 - b^2)^2*f)} - - -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x, 0, Unintegrable[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]} -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(1/2), x, 0, Unintegrable[(d*Sec[e + f*x])^n*Sqrt[a + b*Sec[e + f*x]], x]} -{(d* Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(1/2), x, 0, Unintegrable[(d*Sec[e + f*x])^n/Sqrt[a + b*Sec[e + f*x]], x]} -{(d* Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2), x, 0, Unintegrable[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m with m symbolic*) - - -{Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m, x, 0, Unintegrable[Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m, x]} -{(d* Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m, x, 0, Unintegrable[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m, x]} - - -{Sec[e + f*x]^3*(a + b*Sec[e + f*x])^m, x, 8, ((a + b*Sec[e + f*x])^(1 + m)*Tan[e + f*x])/(b*f*(2 + m)) - (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sec[e + f*x]), (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(((a + b*Sec[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sec[e + f*x]])) + (Sqrt[2]*(a^2 + b^2*(1 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sec[e + f*x]), (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(((a + b*Sec[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Sec[e + f*x]]))} -{Sec[e + f*x]^2*(a + b*Sec[e + f*x])^m, x, 7, (Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sec[e + f*x]), (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(((a + b*Sec[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Sec[e + f*x]])) - (Sqrt[2]*a*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sec[e + f*x]), (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(((a + b*Sec[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Sec[e + f*x]]))} -{Sec[e + f*x]^1*(a + b*Sec[e + f*x])^m, x, 3, (Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sec[e + f*x]), (b*(1 - Sec[e + f*x]))/(a + b)]*(a + b*Sec[e + f*x])^m*Tan[e + f*x])/(((a + b*Sec[e + f*x])/(a + b))^m*(f*Sqrt[1 + Sec[e + f*x]]))} -{Sec[e + f*x]^0*(a + b*Sec[e + f*x])^m, x, 0, Unintegrable[(a + b*Sec[e + f*x])^m, x]} -{Cos[e + f*x]^1*(a + b*Sec[e + f*x])^m, x, 0, Unintegrable[Cos[e + f*x]*(a + b*Sec[e + f*x])^m, x]} -{Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m, x, 0, Unintegrable[Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x]), x, 8, (14*a*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (10*b*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (14*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x]), x, 7, (6*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*a*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x]), x, 6, (6*a*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]), x, 5, (2*b*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x]), x, 4, (2*a*EllipticE[(1/2)*(c + d*x), 2])/d + (2*b*EllipticF[(1/2)*(c + d*x), 2])/d} -{Cos[c + d*x]^(-1/2)*(a + b*Sec[c + d*x]), x, 5, -((2*b*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/d + (2*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(a + b*Sec[c + d*x]), x, 6, -((2*a*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + b*Sec[c + d*x]), x, 7, -((6*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*b*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2, x, 10, (2*(7*a^2 + 9*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (20*a*b*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (20*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*a^2 + 9*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (4*a*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2, x, 9, (12*a*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*a^2 + 7*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2, x, 8, (2*(3*a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (4*a*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2, x, 7, (4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(a^2 + 3*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^2, x, 7, (2*(a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/d + (2*b^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^2, x, 8, -((4*a*b*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(3*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*a*b*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^2, x, 9, -((2*(5*a^2 + 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a*b*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*a*b*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*a^2 + 3*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-5/2)*(a + b*Sec[c + d*x])^2, x, 10, -((12*a*b*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(7*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (4*a*b*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (12*a*b*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3, x, 10, (2*a*(7*a^2 + 27*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*b*(15*a^2 + 7*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(15*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*a^2 + 27*b^2)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (40*a^2*b*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3, x, 9, (2*b*(9*a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(5*a^2 + 21*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (32*a^2*b*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a^2*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3, x, 8, (6*a*(a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*b*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a^2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3, x, 8, (2*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(a^2 + 9*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (2*b*(a^2 - 3*b^2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^3, x, 8, (2*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*b*(9*a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (16*a*b^2*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(-1/2)*(a + b*Sec[c + d*x])^3, x, 9, -((6*b*(5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/d + (8*a*b^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)) + (6*b*(5*a^2 + b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(-3/2)*(a + b*Sec[c + d*x])^3, x, 10, -((2*a*(5*a^2 + 9*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*b*(21*a^2 + 5*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (32*a*b^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*b*(21*a^2 + 5*b^2)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*a^2 + 9*b^2)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*b^2*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x]), x, 11, (2*(3*a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*b*(a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2])/(3*a^4*d) + (2*b^4*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) - (2*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)} -{Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x]), x, 10, (-2*b*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2])/(3*a^3*d) - (2*b^3*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x]), x, 9, (2*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*b*EllipticF[(c + d*x)/2, 2])/(a^2*d) + (2*b^2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d)} -{1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])), x, 5, (2*EllipticF[(c + d*x)/2, 2])/(a*d) - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*(a + b)*d)} -{1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])), x, 3, (2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d)} -{1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])), x, 7, (-2*EllipticE[(c + d*x)/2, 2])/(b*d) - (2*a*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) + (2*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])), x, 11, (2*a*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*EllipticF[(c + d*x)/2, 2])/(3*b*d) + (2*a^2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d) + (2*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) - (2*a*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2, x, 11, -((b*(4*a^2 - 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d)) + ((2*a^4 + 16*a^2*b^2 - 15*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^4*(a^2 - b^2)*d) - (b^3*(7*a^2 - 5*b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^2, x, 10, ((2*a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) - (b*(4*a^2 - 3*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + (b^2*(5*a^2 - 3*b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + (b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2), x, 10, (b*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) + ((2*a^2 - b^2)*EllipticF[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) - (b*(3*a^2 - b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) - (b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2), x, 10, -(EllipticE[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d)) - (b*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) + ((a^2 + b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*(a + b)^2*d) + (a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2), x, 10, (a*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + EllipticF[(1/2)*(c + d*x), 2]/((a^2 - b^2)*d) + ((a^2 - 3*b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b*(a + b)^2*d) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2), x, 11, -(((3*a^2 - 2*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d)) - (a*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) - (a*(3*a^2 - 5*b^2)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))} - - -{Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3, x, 12, -((b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d)) + ((8*a^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^5*(a^2 - b^2)^2*d) - (b^3*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(13*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^3, x, 11, ((8*a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(8*a^4 - 11*a^2*b^2 + 5*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + (b^2*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) + (b^2*(11*a^2 - 5*b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3), x, 11, (3*b*(3*a^2 - b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4 - 5*a^2*b^2 + 3*b^4)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - (3*b*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) - (b*(7*a^2 - b^2)*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3), x, 11, -(((5*a^2 + b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d)) - (b*(7*a^2 - b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*(a + b)^3*d) + (a*Sin[c + d*x])/(2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) + (3*(a^2 + b^2)*Sin[c + d*x])/(4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3), x, 11, ((a^2 + 5*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b*(a + b)^3*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2) + (a*(a^2 - 7*b^2)*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3), x, 11, (3*a*(a^2 - 3*b^2)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) + ((a^2 - 7*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b*(a^2 - b^2)^2*d) + (3*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^2*(a + b)^3*d) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2) - (3*a^2*(a^2 - 3*b^2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]))} -{1/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3), x, 12, -(((15*a^4 - 29*a^2*b^2 + 8*b^4)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d)) - (a*(5*a^2 - 11*b^2)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) - (a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (a^2*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2) - (a^2*(5*a^2 - 11*b^2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]], x, 10, (-4*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]], x, 9, (2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*Sqrt[a + b*Sec[c + d*x]], x, 4, (2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{Sqrt[a + b*Sec[c + d*x]]/Cos[c + d*x]^(1/2), x, 8, (2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{Sqrt[a + b*Sec[c + d*x]]/Cos[c + d*x]^(3/2), x, 13, (b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2), x, 11, (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(41*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (16*b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2), x, 10, (2*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(5*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (4*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2), x, 9, (2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (8*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(3/2), x, 12, (2*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{(a + b*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(1/2), x, 13, ((2*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (3*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2), x, 14, (7*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (5*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + (5*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2), x, 12, (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(163*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2 + 75*b^2)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (38*a*b*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a^2*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2), x, 11, (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(21*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*(29*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(21*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (6*a*b*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d) + (2*a^2*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2), x, 10, (16*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 23*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (22*a*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a^2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2), x, 13, (2*a*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (14*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a^2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(5/2), x, 13, (b*(4*a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (5*a*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(1/2), x, 14, (a*(8*a^2 + 11*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (9*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + (9*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2), x, 15, (b*(59*a^2 + 16*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (5*a*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((33*a^2 + 16*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + (13*a*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Cos[c + d*x]^(3/2)) + ((33*a^2 + 16*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^(5/2)/Sqrt[a + b*Sec[c + d*x]], x, 10, (-2*b*(7*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2 + 8*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (8*b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)} -{Cos[c + d*x]^(3/2)/Sqrt[a + b*Sec[c + d*x]], x, 9, (2*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (4*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^(1/2)/Sqrt[a + b*Sec[c + d*x]], x, 8, (-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{1/(Cos[c + d*x]^(1/2)*Sqrt[a + b*Sec[c + d*x]]), x, 4, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]), x, 4, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]), x, 13, (Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{1/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]), x, 14, -(a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (3*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) - (3*a*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(3/2), x, 11, -((8*b*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(5*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) + (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(3*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*d) + (2*(a^2 - 6*b^2)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2), x, 10, (2*(a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*b*(5*a^2 - 8*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 4*b^2)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^(1/2)/(a + b*Sec[c + d*x])^(3/2), x, 9, (-4*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(3/2)), x, 9, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)), x, 6, (-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/((a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)), x, 10, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)), x, 14, (Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (3*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + ((3*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2), x, 11, (2*(a^4 + 16*a^2*b^2 - 16*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{Cos[c + d*x]^(1/2)/(a + b*Sec[c + d*x])^(5/2), x, 10, -((2*b*(9*a^2 - 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (8*b^2*(2*a^2 - b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, (2*(3*a^2 - 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (4*b*(3*a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*b*(5*a^2 - b^2)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, (-2*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*(a^2 + b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (8*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(a^2 - 5*b^2)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)), x, 14, (-2*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*a*(3*a^2 - 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*a^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m with n symbolic*) - - -{(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^3, x, 8, If[$VersionNumber>=8, -((b*(b^2*(1 - n) + 3*a^2*(2 - n))*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a*(a^2*(1 - n) - 3*b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)*(2 - n)) + (b^2*(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n)), -((b*(b^2*(1 - n) + 3*a^2*(2 - n))*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 - n)*n*Sqrt[Sin[e + f*x]^2])) - (a*(a^2*(1 - n) - 3*b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 - 2*n)*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(2 - 3*n + n^2)) + (b^2*(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 - n))]} -{(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^2, x, 7, If[$VersionNumber>=8, -((2*a*b*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - ((a^2*(1 - n) - b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n)*(1 + n)*Sqrt[Sin[e + f*x]^2]) + (b^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n)), -((2*a*b*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - ((a^2*(1 - n) - b^2*n)*Cos[e + f*x]*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (b^2*(d*Cos[e + f*x])^n*Tan[e + f*x])/(f*(1 - n))]} -{(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^1, x, 5, -((b*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*n*Sqrt[Sin[e + f*x]^2])) - (a*(d*Cos[e + f*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*(1 + n)*Sqrt[Sin[e + f*x]^2])} -{(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x])^1, x, 7, (a*AppellF1[1/2, (1/2)*(-1 - n), 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((1/2)*(-1 - n))*Sin[e + f*x])/((a^2 - b^2)*f) - (b*AppellF1[1/2, -(n/2), 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(d*Cos[e + f*x])^n*Sin[e + f*x])/((Cos[e + f*x]^2)^(n/2)*((a^2 - b^2)*f))} -{(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x])^2, x, 10, (a^2*AppellF1[1/2, (1/2)*(-3 - n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((1/2)*(-1 - n))*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (1/2)*(-1 - n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((1/2)*(-1 - n))*Sin[e + f*x])/((a^2 - b^2)^2*f) - (2*a*b*AppellF1[1/2, (1/2)*(-2 - n), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(d*Cos[e + f*x])^n*Sin[e + f*x])/((Cos[e + f*x]^2)^(n/2)*((a^2 - b^2)^2*f))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m deleted file mode 100644 index 79bb72e..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.3 (d sin)^n (a+b sec)^m.m +++ /dev/null @@ -1,525 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+b Sec[e+f x])^m when a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+a Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sec[c + d*x])*Sin[c + d*x]^9, x, 5, -((a*Cos[c + d*x])/d) + (2*a*Cos[c + d*x]^2)/d + (4*a*Cos[c + d*x]^3)/(3*d) - (3*a*Cos[c + d*x]^4)/(2*d) - (6*a*Cos[c + d*x]^5)/(5*d) + (2*a*Cos[c + d*x]^6)/(3*d) + (4*a*Cos[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]^8)/(8*d) - (a*Cos[c + d*x]^9)/(9*d) - (a*Log[Cos[c + d*x]])/d} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^7, x, 5, -((a*Cos[c + d*x])/d) + (3*a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*a*Cos[c + d*x]^4)/(4*d) - (3*a*Cos[c + d*x]^5)/(5*d) + (a*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (a*Log[Cos[c + d*x]])/d} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^5, x, 5, -((a*Cos[c + d*x])/d) + (a*Cos[c + d*x]^2)/d + (2*a*Cos[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]^4)/(4*d) - (a*Cos[c + d*x]^5)/(5*d) - (a*Log[Cos[c + d*x]])/d} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^3, x, 5, -((a*Cos[c + d*x])/d) + (a*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/(3*d) - (a*Log[Cos[c + d*x]])/d} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^1, x, 4, -((a*Cos[c + d*x])/d) - (a*Log[Cos[c + d*x]])/d} -{Csc[c + d*x]^1*(a + a*Sec[c + d*x]), x, 6, (a*Log[1 - Cos[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d} -{Csc[c + d*x]^3*(a + a*Sec[c + d*x]), x, 5, -a^2/(2*d*(a - a*Cos[c + d*x])) + (3*a*Log[1 - Cos[c + d*x]])/(4*d) - (a*Log[Cos[c + d*x]])/d + (a*Log[1 + Cos[c + d*x]])/(4*d)} -{Csc[c + d*x]^5*(a + a*Sec[c + d*x]), x, 5, -a^3/(8*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^2/(8*d*(a + a*Cos[c + d*x])) + (11*a*Log[1 - Cos[c + d*x]])/(16*d) - (a*Log[Cos[c + d*x]])/d + (5*a*Log[1 + Cos[c + d*x]])/(16*d)} -{Csc[c + d*x]^7*(a + a*Sec[c + d*x]), x, 5, -a^4/(24*d*(a - a*Cos[c + d*x])^3) - (5*a^3)/(32*d*(a - a*Cos[c + d*x])^2) - a^2/(2*d*(a - a*Cos[c + d*x])) - a^3/(32*d*(a + a*Cos[c + d*x])^2) - (3*a^2)/(16*d*(a + a*Cos[c + d*x])) + (21*a*Log[1 - Cos[c + d*x]])/(32*d) - (a*Log[Cos[c + d*x]])/d + (11*a*Log[1 + Cos[c + d*x]])/(32*d)} - -{(a + a*Sec[c + d*x])*Sin[c + d*x]^8, x, 11, (35*a*x)/128 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (35*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a*Sin[c + d*x]^3)/(3*d) - (35*a*Cos[c + d*x]*Sin[c + d*x]^3)/(192*d) - (a*Sin[c + d*x]^5)/(5*d) - (7*a*Cos[c + d*x]*Sin[c + d*x]^5)/(48*d) - (a*Sin[c + d*x]^7)/(7*d) - (a*Cos[c + d*x]*Sin[c + d*x]^7)/(8*d)} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^6, x, 10, (5*a*x)/16 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a*Sin[c + d*x]^3)/(3*d) - (5*a*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (a*Sin[c + d*x]^5)/(5*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^4, x, 9, (3*a*x)/8 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Sin[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{(a + a*Sec[c + d*x])*Sin[c + d*x]^2, x, 7, (a*x)/2 + (a*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + a*Sec[c + d*x]), x, 7, (a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Csc[c + d*x])/d} -{Csc[c + d*x]^4*(a + a*Sec[c + d*x]), x, 8, (a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^6*(a + a*Sec[c + d*x]), x, 8, (a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (2*a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d)} -{Csc[c + d*x]^8*(a + a*Sec[c + d*x]), x, 8, (a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/d - (3*a*Cot[c + d*x]^5)/(5*d) - (a*Cot[c + d*x]^7)/(7*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d)} -{Csc[c + d*x]^10*(a + a*Sec[c + d*x]), x, 8, (a*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (4*a*Cot[c + d*x]^3)/(3*d) - (6*a*Cot[c + d*x]^5)/(5*d) - (4*a*Cot[c + d*x]^7)/(7*d) - (a*Cot[c + d*x]^9)/(9*d) - (a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/(3*d) - (a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d)} - - -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^9, x, 5, (3*a^2*Cos[c + d*x])/d + (4*a^2*Cos[c + d*x]^2)/d - (2*a^2*Cos[c + d*x]^3)/(3*d) - (3*a^2*Cos[c + d*x]^4)/d - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^6)/(3*d) + (3*a^2*Cos[c + d*x]^7)/(7*d) - (a^2*Cos[c + d*x]^8)/(4*d) - (a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^7, x, 5, (2*a^2*Cos[c + d*x])/d + (3*a^2*Cos[c + d*x]^2)/d - (3*a^2*Cos[c + d*x]^4)/(2*d) - (2*a^2*Cos[c + d*x]^5)/(5*d) + (a^2*Cos[c + d*x]^6)/(3*d) + (a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^5, x, 5, (a^2*Cos[c + d*x])/d + (2*a^2*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (a^2*Cos[c + d*x]^4)/(2*d) - (a^2*Cos[c + d*x]^5)/(5*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^3, x, 5, (a^2*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^1, x, 5, -((a^2*Cos[c + d*x])/d) - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{Csc[c + d*x]^1*(a + a*Sec[c + d*x])^2, x, 5, (2*a^2*Log[1 - Cos[c + d*x]])/d - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{Csc[c + d*x]^3*(a + a*Sec[c + d*x])^2, x, 5, -(a^3/(d*(a - a*Cos[c + d*x]))) + (2*a^2*Log[1 - Cos[c + d*x]])/d - (2*a^2*Log[Cos[c + d*x]])/d + (a^2*Sec[c + d*x])/d} -{Csc[c + d*x]^5*(a + a*Sec[c + d*x])^2, x, 5, -a^4/(4*d*(a - a*Cos[c + d*x])^2) - (5*a^3)/(4*d*(a - a*Cos[c + d*x])) + (17*a^2*Log[1 - Cos[c + d*x]])/(8*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(8*d) + (a^2*Sec[c + d*x])/d} -{Csc[c + d*x]^7*(a + a*Sec[c + d*x])^2, x, 5, -a^5/(12*d*(a - a*Cos[c + d*x])^3) - (3*a^4)/(8*d*(a - a*Cos[c + d*x])^2) - (23*a^3)/(16*d*(a - a*Cos[c + d*x])) + a^3/(16*d*(a + a*Cos[c + d*x])) + (9*a^2*Log[1 - Cos[c + d*x]])/(4*d) - (2*a^2*Log[Cos[c + d*x]])/d - (a^2*Log[1 + Cos[c + d*x]])/(4*d) + (a^2*Sec[c + d*x])/d} -{Csc[c + d*x]^9*(a + a*Sec[c + d*x])^2, x, 5, -a^6/(32*d*(a - a*Cos[c + d*x])^4) - (7*a^5)/(48*d*(a - a*Cos[c + d*x])^3) - (15*a^4)/(32*d*(a - a*Cos[c + d*x])^2) - (51*a^3)/(32*d*(a - a*Cos[c + d*x])) + a^4/(64*d*(a + a*Cos[c + d*x])^2) + (9*a^3)/(64*d*(a + a*Cos[c + d*x])) + (303*a^2*Log[1 - Cos[c + d*x]])/(128*d) - (2*a^2*Log[Cos[c + d*x]])/d - (47*a^2*Log[1 + Cos[c + d*x]])/(128*d) + (a^2*Sec[c + d*x])/d} - -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^8, x, 27, -((245*a^2*x)/128) + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d + (139*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (11*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (17*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (2*a^2*Sin[c + d*x]^3)/(3*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) - (2*a^2*Sin[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^6, x, 18, (-25*a^2*x)/16 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (7*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a^2*Sin[c + d*x]^3)/(3*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) + (a^2*Tan[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^4, x, 14, (-9*a^2*x)/8 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Tan[c + d*x])/d} -{(a + a*Sec[c + d*x])^2*Sin[c + d*x]^2, x, 9, -(a^2*x)/2 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*Tan[c + d*x])/d} -{Csc[c + d*x]^2*(a + a*Sec[c + d*x])^2, x, 11, (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Cot[c + d*x])/d - (2*a^2*Csc[c + d*x])/d + (a^2*Tan[c + d*x])/d} -{Csc[c + d*x]^4*(a + a*Sec[c + d*x])^2, x, 8, (2*a^2*ArcTanh[Sin[c + d*x]])/d + (10*a^2*Tan[c + d*x])/(3*d) - (2*a^2*Tan[c + d*x])/(d*(1 - Cos[c + d*x])) - (a^4*Tan[c + d*x])/(3*d*(a - a*Cos[c + d*x])^2)} -{Csc[c + d*x]^6*(a + a*Sec[c + d*x])^2, x, 12, (2*a^2*ArcTanh[Sin[c + d*x]])/d - (4*a^2*Cot[c + d*x])/d - (5*a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) + (a^2*Tan[c + d*x])/d} -{Csc[c + d*x]^8*(a + a*Sec[c + d*x])^2, x, 12, (2*a^2*ArcTanh[Sin[c + d*x]])/d - (5*a^2*Cot[c + d*x])/d - (3*a^2*Cot[c + d*x]^3)/d - (7*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d} -{Csc[c + d*x]^10*(a + a*Sec[c + d*x])^2, x, 12, (2*a^2*ArcTanh[Sin[c + d*x]])/d - (6*a^2*Cot[c + d*x])/d - (14*a^2*Cot[c + d*x]^3)/(3*d) - (16*a^2*Cot[c + d*x]^5)/(5*d) - (9*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d) + (a^2*Tan[c + d*x])/d} - - -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^9, x, 5, (11*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (14*a^3*Cos[c + d*x]^3)/(3*d) - (7*a^3*Cos[c + d*x]^4)/(2*d) + (6*a^3*Cos[c + d*x]^5)/(5*d) + (11*a^3*Cos[c + d*x]^6)/(6*d) + (a^3*Cos[c + d*x]^7)/(7*d) - (3*a^3*Cos[c + d*x]^8)/(8*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^7, x, 5, (8*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (2*a^3*Cos[c + d*x]^3)/d - (2*a^3*Cos[c + d*x]^4)/d + (a^3*Cos[c + d*x]^6)/(2*d) + (a^3*Cos[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^5, x, 5, (5*a^3*Cos[c + d*x])/d + (5*a^3*Cos[c + d*x]^2)/(2*d) - (a^3*Cos[c + d*x]^3)/(3*d) - (3*a^3*Cos[c + d*x]^4)/(4*d) - (a^3*Cos[c + d*x]^5)/(5*d) - (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^3, x, 4, (2*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^1, x, 5, -((a^3*Cos[c + d*x])/d) - (3*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^1*(a + a*Sec[c + d*x])^3, x, 5, (4*a^3*Log[1 - Cos[c + d*x]])/d - (4*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^3*(a + a*Sec[c + d*x])^3, x, 5, (-2*a^4)/(d*(a - a*Cos[c + d*x])) + (5*a^3*Log[1 - Cos[c + d*x]])/d - (5*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^5*(a + a*Sec[c + d*x])^3, x, 5, -a^5/(2*d*(a - a*Cos[c + d*x])^2) - (3*a^4)/(d*(a - a*Cos[c + d*x])) + (6*a^3*Log[1 - Cos[c + d*x]])/d - (6*a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^7*(a + a*Sec[c + d*x])^3, x, 5, -a^6/(6*d*(a - a*Cos[c + d*x])^3) - (7*a^5)/(8*d*(a - a*Cos[c + d*x])^2) - (31*a^4)/(8*d*(a - a*Cos[c + d*x])) + (111*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (7*a^3*Log[Cos[c + d*x]])/d + (a^3*Log[1 + Cos[c + d*x]])/(16*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^9*(a + a*Sec[c + d*x])^3, x, 5, -a^7/(16*d*(a - a*Cos[c + d*x])^4) - a^6/(3*d*(a - a*Cos[c + d*x])^3) - (39*a^5)/(32*d*(a - a*Cos[c + d*x])^2) - (75*a^4)/(16*d*(a - a*Cos[c + d*x])) - a^4/(32*d*(a + a*Cos[c + d*x])) + (501*a^3*Log[1 - Cos[c + d*x]])/(64*d) - (8*a^3*Log[Cos[c + d*x]])/d + (11*a^3*Log[1 + Cos[c + d*x]])/(64*d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)} - -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^8, x, 29, -((805*a^3*x)/128) - (a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (603*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (293*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a^3*Sin[c + d*x]^3)/(3*d) - (2*a^3*Sin[c + d*x]^5)/(5*d) - (3*a^3*Sin[c + d*x]^7)/(7*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^6, x, 18, (-85*a^3*x)/16 + (a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Sin[c + d*x])/d + (43*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a^3*Sin[c + d*x]^3)/(3*d) - (3*a^3*Sin[c + d*x]^5)/(5*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^4, x, 16, (-33*a^3*x)/8 + (3*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (2*a^3*Sin[c + d*x])/d + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^3*Sin[c + d*x]^3)/d + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^3*Sin[c + d*x]^2, x, 11, (-5*a^3*x)/2 + (5*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^3*Sin[c + d*x])/d - (a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + a*Sec[c + d*x])^3, x, 9, (9*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (4*a^3*Sin[c + d*x])/(d*(1 - Cos[c + d*x])) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^4*(a + a*Sec[c + d*x])^3, x, 11, (11*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (2*a^3*Sin[c + d*x])/(3*d*(1 - Cos[c + d*x])^2) - (17*a^3*Sin[c + d*x])/(3*d*(1 - Cos[c + d*x])) + (3*a^3*Tan[c + d*x])/d + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^6*(a + a*Sec[c + d*x])^3, x, 10, (13*a^3*ArcTanh[Sin[c + d*x]])/(2*d) + (152*a^3*Tan[c + d*x])/(15*d) + (13*a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (a^6*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a - a*Cos[c + d*x])^3) - (11*a^5*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a - a*Cos[c + d*x])^2) - (76*a^6*Sec[c + d*x]*Tan[c + d*x])/(15*d*(a^3 - a^3*Cos[c + d*x]))} -{Csc[c + d*x]^8*(a + a*Sec[c + d*x])^3, x, 17, (15*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (13*a^3*Cot[c + d*x])/d - (7*a^3*Cot[c + d*x]^3)/d - (3*a^3*Cot[c + d*x]^5)/d - (4*a^3*Cot[c + d*x]^7)/(7*d) - (15*a^3*Csc[c + d*x])/(2*d) - (5*a^3*Csc[c + d*x]^3)/(2*d) - (3*a^3*Csc[c + d*x]^5)/(2*d) - (15*a^3*Csc[c + d*x]^7)/(14*d) + (a^3*Csc[c + d*x]^7*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d} -{Csc[c + d*x]^10*(a + a*Sec[c + d*x])^3, x, 17, (17*a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (16*a^3*Cot[c + d*x])/d - (34*a^3*Cot[c + d*x]^3)/(3*d) - (36*a^3*Cot[c + d*x]^5)/(5*d) - (19*a^3*Cot[c + d*x]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (17*a^3*Csc[c + d*x])/(2*d) - (17*a^3*Csc[c + d*x]^3)/(6*d) - (17*a^3*Csc[c + d*x]^5)/(10*d) - (17*a^3*Csc[c + d*x]^7)/(14*d) - (17*a^3*Csc[c + d*x]^9)/(18*d) + (a^3*Csc[c + d*x]^9*Sec[c + d*x]^2)/(2*d) + (3*a^3*Tan[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[c + d*x]^9/(a + a*Sec[c + d*x]), x, 7, Cos[c + d*x]^3/(3*a*d) - (3*Cos[c + d*x]^5)/(5*a*d) + (3*Cos[c + d*x]^7)/(7*a*d) - Cos[c + d*x]^9/(9*a*d) + Sin[c + d*x]^8/(8*a*d)} -{Sin[c + d*x]^7/(a + a*Sec[c + d*x]), x, 7, Cos[c + d*x]^3/(3*a*d) - (2*Cos[c + d*x]^5)/(5*a*d) + Cos[c + d*x]^7/(7*a*d) + Sin[c + d*x]^6/(6*a*d)} -{Sin[c + d*x]^5/(a + a*Sec[c + d*x]), x, 7, Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*a*d) + Sin[c + d*x]^4/(4*a*d)} -{Sin[c + d*x]^3/(a + a*Sec[c + d*x]), x, 6, Cos[c + d*x]^3/(3*a*d) + Sin[c + d*x]^2/(2*a*d)} -{Sin[c + d*x]^1/(a + a*Sec[c + d*x]), x, 5, -(Cos[c + d*x]/(a*d)) + Log[1 + Cos[c + d*x]]/(a*d)} -{Csc[c + d*x]^1/(a + a*Sec[c + d*x]), x, 6, -(ArcTanh[Cos[c + d*x]]/(2*a*d)) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - Csc[c + d*x]^2/(2*a*d)} -{Csc[c + d*x]^3/(a + a*Sec[c + d*x]), x, 7, -(ArcTanh[Cos[c + d*x]]/(8*a*d)) - (Cot[c + d*x]*Csc[c + d*x])/(8*a*d) + (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d) - Csc[c + d*x]^4/(4*a*d)} -{Csc[c + d*x]^5/(a + a*Sec[c + d*x]), x, 8, -(ArcTanh[Cos[c + d*x]]/(16*a*d)) - (Cot[c + d*x]*Csc[c + d*x])/(16*a*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(24*a*d) + (Cot[c + d*x]*Csc[c + d*x]^5)/(6*a*d) - Csc[c + d*x]^6/(6*a*d)} - -{Sin[c + d*x]^8/(a + a*Sec[c + d*x]), x, 9, -((5*x)/(128*a)) - (5*Cos[c + d*x]*Sin[c + d*x])/(128*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(64*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x]^3)/(48*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^5)/(8*a*d) + Sin[c + d*x]^7/(7*a*d)} -{Sin[c + d*x]^6/(a + a*Sec[c + d*x]), x, 8, -(x/(16*a)) - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a*d) + Sin[c + d*x]^5/(5*a*d)} -{Sin[c + d*x]^4/(a + a*Sec[c + d*x]), x, 7, -(x/(8*a)) - (Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) + Sin[c + d*x]^3/(3*a*d)} -{Sin[c + d*x]^2/(a + a*Sec[c + d*x]), x, 5, -(x/(2*a)) + Sin[c + d*x]/(a*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Csc[c + d*x]^2/(a + a*Sec[c + d*x]), x, 6, Cot[c + d*x]^3/(3*a*d) - Csc[c + d*x]^3/(3*a*d)} -{Csc[c + d*x]^4/(a + a*Sec[c + d*x]), x, 7, Cot[c + d*x]^3/(3*a*d) + Cot[c + d*x]^5/(5*a*d) - Csc[c + d*x]^5/(5*a*d)} -{Csc[c + d*x]^6/(a + a*Sec[c + d*x]), x, 7, Cot[c + d*x]^3/(3*a*d) + (2*Cot[c + d*x]^5)/(5*a*d) + Cot[c + d*x]^7/(7*a*d) - Csc[c + d*x]^7/(7*a*d)} -{Csc[c + d*x]^8/(a + a*Sec[c + d*x]), x, 7, Cot[c + d*x]^3/(3*a*d) + (3*Cot[c + d*x]^5)/(5*a*d) + (3*Cot[c + d*x]^7)/(7*a*d) + Cot[c + d*x]^9/(9*a*d) - Csc[c + d*x]^9/(9*a*d)} -{Csc[c + d*x]^10/(a + a*Sec[c + d*x]), x, 7, Cot[c + d*x]^3/(3*a*d) + (4*Cot[c + d*x]^5)/(5*a*d) + (6*Cot[c + d*x]^7)/(7*a*d) + (4*Cot[c + d*x]^9)/(9*a*d) + Cot[c + d*x]^11/(11*a*d) - Csc[c + d*x]^11/(11*a*d)} - - -{Sin[c + d*x]^11/(a + a*Sec[c + d*x])^2, x, 5, (4*(a - a*Cos[c + d*x])^6)/(3*a^8*d) - (4*(a - a*Cos[c + d*x])^7)/(a^9*d) + (19*(a - a*Cos[c + d*x])^8)/(4*a^10*d) - (25*(a - a*Cos[c + d*x])^9)/(9*a^11*d) + (4*(a - a*Cos[c + d*x])^10)/(5*a^12*d) - (a - a*Cos[c + d*x])^11/(11*a^13*d)} -{Sin[c + d*x]^9/(a + a*Sec[c + d*x])^2, x, 5, (4*(a - a*Cos[c + d*x])^5)/(5*a^7*d) - (2*(a - a*Cos[c + d*x])^6)/(a^8*d) + (13*(a - a*Cos[c + d*x])^7)/(7*a^9*d) - (3*(a - a*Cos[c + d*x])^8)/(4*a^10*d) + (a - a*Cos[c + d*x])^9/(9*a^11*d)} -{Sin[c + d*x]^7/(a + a*Sec[c + d*x])^2, x, 5, -(Cos[c + d*x]^3/(3*a^2*d)) + Cos[c + d*x]^4/(2*a^2*d) - Cos[c + d*x]^6/(3*a^2*d) + Cos[c + d*x]^7/(7*a^2*d)} -{Sin[c + d*x]^5/(a + a*Sec[c + d*x])^2, x, 5, -(Cos[c + d*x]^3/(3*a^2*d)) + Cos[c + d*x]^4/(2*a^2*d) - Cos[c + d*x]^5/(5*a^2*d)} -{Sin[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 5, (2*Cos[c + d*x])/(a^2*d) - Cos[c + d*x]^2/(a^2*d) + Cos[c + d*x]^3/(3*a^2*d) - (2*Log[1 + Cos[c + d*x]])/(a^2*d)} -{Sin[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 5, -(Cos[c + d*x]/(a^2*d)) + 1/(d*(a^2 + a^2*Cos[c + d*x])) + (2*Log[1 + Cos[c + d*x]])/(a^2*d)} -{Csc[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 6, -(ArcTanh[Cos[c + d*x]]/(4*a^2*d)) + 1/(4*d*(a + a*Cos[c + d*x])^2) - 3/(4*d*(a^2 + a^2*Cos[c + d*x]))} -{Csc[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 4, -((a + 2*a*Cos[c + d*x])/(6*d*(1 - Cos[c + d*x])*(a + a*Cos[c + d*x])^3))} -{Csc[c + d*x]^5/(a + a*Sec[c + d*x])^2, x, 6, ArcTanh[Cos[c + d*x]]/(64*a^2*d) - 1/(64*d*(a - a*Cos[c + d*x])^2) + a^2/(32*d*(a + a*Cos[c + d*x])^4) - a/(48*d*(a + a*Cos[c + d*x])^3) - 1/(32*d*(a + a*Cos[c + d*x])^2) - 1/(64*d*(a^2 - a^2*Cos[c + d*x])) - 1/(32*d*(a^2 + a^2*Cos[c + d*x]))} - -{Sin[c + d*x]^8/(a + a*Sec[c + d*x])^2, x, 16, (11*x)/(128*a^2) + (11*Cos[c + d*x]*Sin[c + d*x])/(128*a^2*d) - (7*Cos[c + d*x]^3*Sin[c + d*x])/(64*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(16*a^2*d) - (Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*a^2*d) - (Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*a^2*d) - (2*Sin[c + d*x]^5)/(5*a^2*d) + (2*Sin[c + d*x]^7)/(7*a^2*d)} -{Sin[c + d*x]^6/(a + a*Sec[c + d*x])^2, x, 7, (3*x)/(16*a^2) - (3*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) - (Cos[c + d*x]*Sin[c + d*x]^3)/(8*a^2*d) - ((a - a*Cos[c + d*x])^3*Sin[c + d*x]^3)/(6*a^5*d) - Sin[c + d*x]^5/(10*a^2*d)} -{Sin[c + d*x]^4/(a + a*Sec[c + d*x])^2, x, 11, (7*x)/(8*a^2) - (2*Sin[c + d*x])/(a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) + (2*Sin[c + d*x]^3)/(3*a^2*d)} -{Sin[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 9, -((5*x)/(2*a^2)) + (2*Sin[c + d*x])/(a^2*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (2*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x]))} -{Csc[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 11, -(Cot[c + d*x]^3/(3*a^2*d)) - (2*Cot[c + d*x]^5)/(5*a^2*d) - (2*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d)} -{Csc[c + d*x]^4/(a + a*Sec[c + d*x])^2, x, 13, -(Cot[c + d*x]^3/(3*a^2*d)) - (3*Cot[c + d*x]^5)/(5*a^2*d) - (2*Cot[c + d*x]^7)/(7*a^2*d) - (2*Csc[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d)} -{Csc[c + d*x]^6/(a + a*Sec[c + d*x])^2, x, 13, -(Cot[c + d*x]^3/(3*a^2*d)) - (4*Cot[c + d*x]^5)/(5*a^2*d) - (5*Cot[c + d*x]^7)/(7*a^2*d) - (2*Cot[c + d*x]^9)/(9*a^2*d) - (2*Csc[c + d*x]^7)/(7*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d)} -{Csc[c + d*x]^8/(a + a*Sec[c + d*x])^2, x, 13, -(Cot[c + d*x]^3/(3*a^2*d)) - Cot[c + d*x]^5/(a^2*d) - (9*Cot[c + d*x]^7)/(7*a^2*d) - (7*Cot[c + d*x]^9)/(9*a^2*d) - (2*Cot[c + d*x]^11)/(11*a^2*d) - (2*Csc[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x]^11)/(11*a^2*d)} - - -{Sin[c + d*x]^11/(a + a*Sec[c + d*x])^3, x, 5, (2*(a - a*Cos[c + d*x])^6)/(3*a^9*d) - (16*(a - a*Cos[c + d*x])^7)/(7*a^10*d) + (25*(a - a*Cos[c + d*x])^8)/(8*a^11*d) - (19*(a - a*Cos[c + d*x])^9)/(9*a^12*d) + (7*(a - a*Cos[c + d*x])^10)/(10*a^13*d) - (a - a*Cos[c + d*x])^11/(11*a^14*d)} -{Sin[c + d*x]^9/(a + a*Sec[c + d*x])^3, x, 5, -Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(3*a^3*d) - (2*Cos[c + d*x]^7)/(7*a^3*d) + (3*Cos[c + d*x]^8)/(8*a^3*d) - Cos[c + d*x]^9/(9*a^3*d)} -{Sin[c + d*x]^7/(a + a*Sec[c + d*x])^3, x, 5, -Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(2*a^3*d) + Cos[c + d*x]^7/(7*a^3*d)} -{Sin[c + d*x]^5/(a + a*Sec[c + d*x])^3, x, 5, (-4*Cos[c + d*x])/(a^3*d) + (2*Cos[c + d*x]^2)/(a^3*d) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (3*Cos[c + d*x]^4)/(4*a^3*d) - Cos[c + d*x]^5/(5*a^3*d) + (4*Log[1 + Cos[c + d*x]])/(a^3*d)} -{Sin[c + d*x]^3/(a + a*Sec[c + d*x])^3, x, 5, (5*Cos[c + d*x])/(a^3*d) - (3*Cos[c + d*x]^2)/(2*a^3*d) + Cos[c + d*x]^3/(3*a^3*d) - 2/(d*(a^3 + a^3*Cos[c + d*x])) - (7*Log[1 + Cos[c + d*x]])/(a^3*d)} -{Sin[c + d*x]/(a + a*Sec[c + d*x])^3, x, 5, -(Cos[c + d*x]/(a^3*d)) - 1/(2*a*d*(a + a*Cos[c + d*x])^2) + 3/(d*(a^3 + a^3*Cos[c + d*x])) + (3*Log[1 + Cos[c + d*x]])/(a^3*d)} -{Csc[c + d*x]/(a + a*Sec[c + d*x])^3, x, 6, -ArcTanh[Cos[c + d*x]]/(8*a^3*d) - 1/(6*d*(a + a*Cos[c + d*x])^3) + 5/(8*a*d*(a + a*Cos[c + d*x])^2) - 7/(8*d*(a^3 + a^3*Cos[c + d*x]))} -{Csc[c + d*x]^3/(a + a*Sec[c + d*x])^3, x, 5, ArcTanh[Cos[c + d*x]]/(32*a^3*d) - a/(16*d*(a + a*Cos[c + d*x])^4) + 1/(6*d*(a + a*Cos[c + d*x])^3) - 3/(32*a*d*(a + a*Cos[c + d*x])^2) - 1/(32*d*(a^3 - a^3*Cos[c + d*x])) - 1/(16*d*(a^3 + a^3*Cos[c + d*x]))} -{Csc[c + d*x]^5/(a + a*Sec[c + d*x])^3, x, 6, (3*ArcTanh[Cos[c + d*x]])/(128*a^3*d) - 1/(128*a*d*(a - a*Cos[c + d*x])^2) - a^2/(40*d*(a + a*Cos[c + d*x])^5) + (3*a)/(64*d*(a + a*Cos[c + d*x])^4) - 1/(64*a*d*(a + a*Cos[c + d*x])^2) - 3/(128*d*(a^3 + a^3*Cos[c + d*x]))} - -{Sin[c + d*x]^8/(a + a*Sec[c + d*x])^3, x, 19, (-29*x)/(128*a^3) - (29*Cos[c + d*x]*Sin[c + d*x])/(128*a^3*d) - (29*Cos[c + d*x]^3*Sin[c + d*x])/(192*a^3*d) + (23*Cos[c + d*x]^5*Sin[c + d*x])/(48*a^3*d) + (Cos[c + d*x]^7*Sin[c + d*x])/(8*a^3*d) + (4*Sin[c + d*x]^3)/(3*a^3*d) - (7*Sin[c + d*x]^5)/(5*a^3*d) + (3*Sin[c + d*x]^7)/(7*a^3*d)} -{Sin[c + d*x]^6/(a + a*Sec[c + d*x])^3, x, 15, -((23*x)/(16*a^3)) + (4*Sin[c + d*x])/(a^3*d) - (23*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (23*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a^3*d) - (7*Sin[c + d*x]^3)/(3*a^3*d) + (3*Sin[c + d*x]^5)/(5*a^3*d)} -{Sin[c + d*x]^4/(a + a*Sec[c + d*x])^3, x, 13, (51*x)/(8*a^3) - (7*Sin[c + d*x])/(a^3*d) + (19*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*d) - (4*Sin[c + d*x])/(a^3*d*(1 + Cos[c + d*x])) + Sin[c + d*x]^3/(a^3*d)} -{Sin[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 10, -((11*x)/(2*a^3)) + (3*Sin[c + d*x])/(a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (2*Sin[c + d*x])/(3*a^3*d*(1 + Cos[c + d*x])^2) + (19*Sin[c + d*x])/(3*a^3*d*(1 + Cos[c + d*x]))} -{Csc[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 15, (3*Cot[c + d*x]^5)/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) - Csc[c + d*x]^3/(a^3*d) + (7*Csc[c + d*x]^5)/(5*a^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)} -{Csc[c + d*x]^4/(a + a*Sec[c + d*x])^3, x, 16, (3*Cot[c + d*x]^5)/(5*a^3*d) + Cot[c + d*x]^7/(a^3*d) + (4*Cot[c + d*x]^9)/(9*a^3*d) - (3*Csc[c + d*x]^5)/(5*a^3*d) + Csc[c + d*x]^7/(a^3*d) - (4*Csc[c + d*x]^9)/(9*a^3*d)} -{Csc[c + d*x]^6/(a + a*Sec[c + d*x])^3, x, 16, (3*Cot[c + d*x]^5)/(5*a^3*d) + (10*Cot[c + d*x]^7)/(7*a^3*d) + (11*Cot[c + d*x]^9)/(9*a^3*d) + (4*Cot[c + d*x]^11)/(11*a^3*d) - (3*Csc[c + d*x]^7)/(7*a^3*d) + (7*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*d)} -{Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3, x, 16, (3*Cot[c + d*x]^5)/(5*a^3*d) + (13*Cot[c + d*x]^7)/(7*a^3*d) + (7*Cot[c + d*x]^9)/(3*a^3*d) + (15*Cot[c + d*x]^11)/(11*a^3*d) + (4*Cot[c + d*x]^13)/(13*a^3*d) - Csc[c + d*x]^9/(3*a^3*d) + (7*Csc[c + d*x]^11)/(11*a^3*d) - (4*Csc[c + d*x]^13)/(13*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^(m/2) (a+a Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2), x, 11, -((a*e^(5/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (a*e^(5/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (6*a*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*e*(e*Sin[c + d*x])^(3/2))/(3*d) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d)} -{(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2), x, 11, (a*e^(3/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (a*e^(3/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*a*e*Sqrt[e*Sin[c + d*x]])/d - (2*a*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d)} -{(a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]], x, 9, -((a*Sqrt[e]*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (a*Sqrt[e]*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])} -{(a + a*Sec[c + d*x])/Sqrt[e*Sin[c + d*x]], x, 9, (a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (2*a*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(3/2), x, 11, -((a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2))) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2)) - (2*a)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(5/2), x, 11, (a*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) + (a*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) - (2*a)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]])} - - -{(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2), x, 15, -((2*a^2*e^(5/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (2*a^2*e^(5/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d - (9*a^2*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (4*a^2*e*(e*Sin[c + d*x])^(3/2))/(3*d) - (2*a^2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d) + (a^2*e*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/d} -{(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2), x, 15, (2*a^2*e^(3/2)*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (2*a^2*e^(3/2)*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d - (a^2*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (4*a^2*e*Sqrt[e*Sin[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d) + (a^2*e*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/d} -{(a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]], x, 13, -((2*a^2*Sqrt[e]*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d) + (2*a^2*Sqrt[e]*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/d + (a^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (a^2*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/(d*e)} -{(a + a*Sec[c + d*x])^2/Sqrt[e*Sin[c + d*x]], x, 13, (2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*Sqrt[e]) + (3*a^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (a^2*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/(d*e)} -{(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(3/2), x, 16, -((2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2))) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(3/2)) - (4*a^2)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a^2*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a^2*Sec[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) + (3*a^2*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2))/(d*e^3)} -{(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(5/2), x, 16, (2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) - (4*a^2)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2*Sec[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (7*a^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) + (5*a^2*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x]), x, 8, -((4*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a*d*Sqrt[e*Sin[c + d*x]])) - (2*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a*d) + (2*e^3*Cos[c + d*x]^3*Sqrt[e*Sin[c + d*x]])/(7*a*d) + (2*e*(e*Sin[c + d*x])^(5/2))/(5*a*d)} -{(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x]), x, 7, -((4*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a*d*Sqrt[Sin[c + d*x]])) + (2*e*(e*Sin[c + d*x])^(3/2))/(3*a*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a*d)} -{(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x]), x, 7, -((4*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a*d*Sqrt[e*Sin[c + d*x]])) + (2*e*Sqrt[e*Sin[c + d*x]])/(a*d) - (2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a*d)} -{Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x]), x, 7, -((2*e)/(a*d*Sqrt[e*Sin[c + d*x]])) + (2*e*Cos[c + d*x])/(a*d*Sqrt[e*Sin[c + d*x]]) + (4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]]), x, 7, -((2*e)/(3*a*d*(e*Sin[c + d*x])^(3/2))) + (2*e*Cos[c + d*x])/(3*a*d*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a*d*Sqrt[e*Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)), x, 8, -((2*e)/(5*a*d*(e*Sin[c + d*x])^(5/2))) + (2*e*Cos[c + d*x])/(5*a*d*(e*Sin[c + d*x])^(5/2)) - (4*Cos[c + d*x])/(5*a*d*e*Sqrt[e*Sin[c + d*x]]) - (4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a*d*e^2*Sqrt[Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)), x, 8, -((2*e)/(7*a*d*(e*Sin[c + d*x])^(7/2))) + (2*e*Cos[c + d*x])/(7*a*d*(e*Sin[c + d*x])^(7/2)) - (4*Cos[c + d*x])/(21*a*d*e*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a*d*e^2*Sqrt[e*Sin[c + d*x]])} - - -{(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2, x, 14, (52*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]]) - (4*e^3*Sqrt[e*Sin[c + d*x]])/(a^2*d) + (26*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) + (2*e^3*Cos[c + d*x]^3*Sqrt[e*Sin[c + d*x]])/(7*a^2*d) + (4*e*(e*Sin[c + d*x])^(5/2))/(5*a^2*d)} -{(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2, x, 14, (4*e^3)/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (2*e^3*Cos[c + d*x])/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (2*e^3*Cos[c + d*x]^3)/(a^2*d*Sqrt[e*Sin[c + d*x]]) - (44*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]]) + (4*e*(e*Sin[c + d*x])^(3/2))/(3*a^2*d) - (12*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a^2*d)} -{(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2, x, 14, (4*e^3)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (2*e^3*Cos[c + d*x])/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (2*e^3*Cos[c + d*x]^3)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) - (4*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (4*e*Sqrt[e*Sin[c + d*x]])/(a^2*d) - (4*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a^2*d)} -{Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x])^2, x, 15, (4*e^3)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (2*e^3*Cos[c + d*x])/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (2*e^3*Cos[c + d*x]^3)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) - (4*e)/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (16*e*Cos[c + d*x])/(5*a^2*d*Sqrt[e*Sin[c + d*x]]) + (28*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]), x, 15, (4*e^3)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (2*e^3*Cos[c + d*x])/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (2*e^3*Cos[c + d*x]^3)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) - (4*e)/(3*a^2*d*(e*Sin[c + d*x])^(3/2)) + (16*e*Cos[c + d*x])/(21*a^2*d*(e*Sin[c + d*x])^(3/2)) + (20*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)), x, 17, (4*e^3)/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (2*e^3*Cos[c + d*x])/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (2*e^3*Cos[c + d*x]^3)/(9*a^2*d*(e*Sin[c + d*x])^(9/2)) - (4*e)/(5*a^2*d*(e*Sin[c + d*x])^(5/2)) + (16*e*Cos[c + d*x])/(45*a^2*d*(e*Sin[c + d*x])^(5/2)) - (4*Cos[c + d*x])/(15*a^2*d*e*Sqrt[e*Sin[c + d*x]]) - (4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(15*a^2*d*e^2*Sqrt[Sin[c + d*x]])} -{1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)), x, 17, (4*e^3)/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (2*e^3*Cos[c + d*x])/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (2*e^3*Cos[c + d*x]^3)/(11*a^2*d*(e*Sin[c + d*x])^(11/2)) - (4*e)/(7*a^2*d*(e*Sin[c + d*x])^(7/2)) + (16*e*Cos[c + d*x])/(77*a^2*d*(e*Sin[c + d*x])^(7/2)) - (4*Cos[c + d*x])/(231*a^2*d*e*(e*Sin[c + d*x])^(3/2)) + (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(231*a^2*d*e^2*Sqrt[e*Sin[c + d*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form Sin[e+f x]^m (a+a Sec[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^m (a+a Sec[e+f x])^n with m symbolic*) - - -{(a + a*Sec[c + d*x])^3*(e*Sin[c + d*x])^m, x, 9, (a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (1/(d*e*(1 + m)))*(3*a^3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))} -{(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^m, x, 7, (a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (1/(d*e*(1 + m)))*(a^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))} -{(a + a*Sec[c + d*x])^1*(e*Sin[c + d*x])^m, x, 5, (a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^1, x, 5, -((e*(e*Sin[c + d*x])^(-1 + m))/(a*d*(1 - m))) + (e*Cos[c + d*x]*Hypergeometric2F1[-(1/2), (1/2)*(-1 + m), (1 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-1 + m))/(a*d*(1 - m)*Sqrt[Cos[c + d*x]^2])} -{(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^2, x, 9, (2*e^3*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)) - (e^3*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (1/2)*(-3 + m), (1/2)*(-1 + m), Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)*Sqrt[Cos[c + d*x]^2]) - (e^3*Cos[c + d*x]*Hypergeometric2F1[-(1/2), (1/2)*(-3 + m), (1/2)*(-1 + m), Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-3 + m))/(a^2*d*(3 - m)*Sqrt[Cos[c + d*x]^2]) - (2*e*(e*Sin[c + d*x])^(-1 + m))/(a^2*d*(1 - m))} -{(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^3, x, 12, -((4*e^5*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m))) + (e^5*Cos[c + d*x]*Hypergeometric2F1[-(5/2), (1/2)*(-5 + m), (1/2)*(-3 + m), Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m)*Sqrt[Cos[c + d*x]^2]) + (3*e^5*Cos[c + d*x]*Hypergeometric2F1[-(3/2), (1/2)*(-5 + m), (1/2)*(-3 + m), Sin[c + d*x]^2]*(e*Sin[c + d*x])^(-5 + m))/(a^3*d*(5 - m)*Sqrt[Cos[c + d*x]^2]) + (7*e^3*(e*Sin[c + d*x])^(-3 + m))/(a^3*d*(3 - m)) - (3*e*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(1 - m))} - - -{(a + a*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m, x, 5, (2*a*e*AppellF1[-(1/2), (1 - m)/2, (1/2)*(-2 - m), 1/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^(-1 + m))/((1 + Cos[c + d*x])^(m/2)*d)} -{(a + a*Sec[c + d*x])^(1/2)*(e*Sin[c + d*x])^m, x, 5, -((2*e*AppellF1[1/2, (1 - m)/2, -(m/2), 3/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^(-1 + m))/((1 + Cos[c + d*x])^(m/2)*d))} -{(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^(1/2), x, 5, -((2*e*AppellF1[3/2, (1 - m)/2, (2 - m)/2, 5/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*(1 + Cos[c + d*x])^(1 - m/2)*(e*Sin[c + d*x])^(-1 + m))/(3*d*Sqrt[a + a*Sec[c + d*x]]))} -{(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2), x, 5, -((2*e*AppellF1[5/2, (1 - m)/2, (4 - m)/2, 7/2, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]^2*(1 + Cos[c + d*x])^(1 - m/2)*(e*Sin[c + d*x])^(-1 + m))/(5*a*d*Sqrt[a + a*Sec[c + d*x]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^m (a+a Sec[e+f x])^n with n symbolic*) - - -{(a + a*Sec[c + d*x])^n*(e*Sin[c + d*x])^m, x, 5, -((e*AppellF1[1 - n, (1 - m)/2, (1/2)*(1 - m - 2*n), 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^((1 - m)/2)*Cos[c + d*x]*(1 + Cos[c + d*x])^((1/2)*(1 - m - 2*n))*(a + a*Sec[c + d*x])^n*(e*Sin[c + d*x])^(-1 + m))/(d*(1 - n)))} - - -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^7, x, 4, If[$VersionNumber>=8, -(((3 - n)*(8 - n)*(16 - n)*Hypergeometric2F1[6, 4 + n, 5 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(4 + n))/(42*a^4*d*(1 - n)*(4 + n))) - (Cos[c + d*x]^7*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(1 - n)) + (Cos[c + d*x]^7*(a + a*Sec[c + d*x])^(4 + n)*(6*(8 - n) - (108 - 25*n + n^2)*Sec[c + d*x]))/(42*a^4*d*(1 - n)), -(((3 - n)*(8 - n)*(16 - n)*Hypergeometric2F1[6, 4 + n, 5 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(4 + n))/(42*a^4*d*(4 - 3*n - n^2))) - (Cos[c + d*x]^7*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(1 - n)) + (Cos[c + d*x]^7*(a + a*Sec[c + d*x])^(4 + n)*(6*(8 - n) - (108 - 25*n + n^2)*Sec[c + d*x]))/(42*a^4*d*(1 - n))]} -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^5, x, 4, ((12 - n)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3 + n))/(20*a^3*d) - (Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3 + n))/(5*a^3*d) + ((32 - 13*n + n^2)*Hypergeometric2F1[4, 3 + n, 4 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3 + n))/(20*a^3*d*(3 + n))} -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^3, x, 3, (Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(2 + n))/(3*a^2*d) - ((4 - n)*Hypergeometric2F1[3, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2 + n))/(3*a^2*d*(2 + n))} -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^1, x, 2, (Hypergeometric2F1[2, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Csc[c + d*x]^1*(a + a*Sec[c + d*x])^n, x, 2, -((Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + Sec[c + d*x])]*(a + a*Sec[c + d*x])^n)/(2*d*n))} -{Csc[c + d*x]^3*(a + a*Sec[c + d*x])^n, x, 4, -((a*(2 - n)*(a + a*Sec[c + d*x])^(-1 + n))/(4*d*(1 - n))) + (a*(a + a*Sec[c + d*x])^(-1 + n))/(2*d*(1 - Sec[c + d*x])) - ((2 + n)*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + Sec[c + d*x])]*(a + a*Sec[c + d*x])^n)/(8*d*n)} -{Csc[c + d*x]^5*(a + a*Sec[c + d*x])^n, x, 5, (a^2*(12 + 9*n + n^2)*Hypergeometric2F1[1, -2 + n, -1 + n, (1/2)*(1 + Sec[c + d*x])]*(a + a*Sec[c + d*x])^(-2 + n))/(16*d*(2 - n)) + (a^2*(3 + n)*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(-2 + n))/(4*d*(1 - n)*(1 - Sec[c + d*x])^2) - (a^2*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(-2 + n))/(d*(1 - n)*(1 - Sec[c + d*x])^2) - (a^2*(a + a*Sec[c + d*x])^(-2 + n)*(12 + 4*n - 7*n^2 - n^3 - 2*(1 - n)*(6 + n)*Sec[c + d*x]))/(8*d*(2 - 3*n + n^2)*(1 - Sec[c + d*x]))} - -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^4, x, 11, -((AppellF1[1 - n, -(1/2), 1/2 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 + Cos[c + d*x])^(1/2 - n)*(n - n*Cos[c + d*x])*Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sqrt[1 - Cos[c + d*x]])) - (Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/d + (2^(1/2 + n)*AppellF1[1/2, -4 + n, 1/2 - n, 3/2, 1 - Cos[c + d*x], (1/2)*(1 - Cos[c + d*x])]*Cos[c + d*x]^n*(1 + Cos[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^2, x, 6, -((AppellF1[1 - n, -(1/2), -(1/2) - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(1/2 - n)*Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/(d*(1 - n)))} -{Csc[c + d*x]^2*(a + a*Sec[c + d*x])^n, x, 4, -((Cot[c + d*x]*(a + a*Sec[c + d*x])^n)/d) + (2^(-(1/2) + n)*n*Hypergeometric2F1[1/2, 3/2 - n, 3/2, (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d} -{Csc[c + d*x]^4*(a + a*Sec[c + d*x])^n, x, 7, If[$VersionNumber>=8, ((2 - n + n^2)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 - 4*n^2)*(1 - Cos[c + d*x])^2) - (a^4*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])^2) - (a^3*(4 - n)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 8*n + 4*n^2)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])) + (n*(7 - 3*n - n^2)*Cos[c + d*x]*((1 + Cos[c + d*x])/(1 - Cos[c + d*x]))^(-(1/2) - n)*Hypergeometric2F1[-(1/2) - n, 1 - n, 2 - n, -((2*Cos[c + d*x])/(1 - Cos[c + d*x]))]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 - 2*n)*(1 - n)*(1 + 2*n)*(1 - Cos[c + d*x])^2), ((2 - n + n^2)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 - 4*n^2)*(1 - Cos[c + d*x])^2) - (a^4*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])^2) - (a^3*(4 - n)*Cos[c + d*x]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 8*n + 4*n^2)*(a - a*Cos[c + d*x])^2*(a + a*Cos[c + d*x])) + (n*(7 - 3*n - n^2)*Cos[c + d*x]*((1 + Cos[c + d*x])/(1 - Cos[c + d*x]))^(-(1/2) - n)*Hypergeometric2F1[-(1/2) - n, 1 - n, 2 - n, -((2*Cos[c + d*x])/(1 - Cos[c + d*x]))]*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 5*n - 10*n^2 + 20*n^3 - 8*n^4)*(1 - Cos[c + d*x])^2)]} - - -{(a + a*Sec[c + d*x])^n*Sin[c + d*x]^(3/2), x, 5, -((AppellF1[1 - n, -(1/4), -(1/4) - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*Cos[c + d*x]*(1 + Cos[c + d*x])^(-(1/4) - n)*(a + a*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]])/(d*(1 - n)*(1 - Cos[c + d*x])^(1/4)))} -{(a + a*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]], x, 5, -((AppellF1[1 - n, 1/4, 1/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(1/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(1/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sqrt[Sin[c + d*x]]))} -{(a + a*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]], x, 5, -((AppellF1[1 - n, 3/4, 3/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(3/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(3/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sin[c + d*x]^(3/2)))} -{(a + a*Sec[c + d*x])^n/Sin[c + d*x]^(3/2), x, 5, -((AppellF1[1 - n, 5/4, 5/4 - n, 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(1 - Cos[c + d*x])^(5/4)*Cos[c + d*x]*(1 + Cos[c + d*x])^(5/4 - n)*(a + a*Sec[c + d*x])^n)/(d*(1 - n)*Sin[c + d*x]^(5/2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sin[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[c + d*x]^7*(a + b*Sec[c + d*x]), x, 5, -((a*Cos[c + d*x])/d) + (3*b*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/d - (3*b*Cos[c + d*x]^4)/(4*d) - (3*a*Cos[c + d*x]^5)/(5*d) + (b*Cos[c + d*x]^6)/(6*d) + (a*Cos[c + d*x]^7)/(7*d) - (b*Log[Cos[c + d*x]])/d} -{Sin[c + d*x]^5*(a + b*Sec[c + d*x]), x, 5, -((a*Cos[c + d*x])/d) + (b*Cos[c + d*x]^2)/d + (2*a*Cos[c + d*x]^3)/(3*d) - (b*Cos[c + d*x]^4)/(4*d) - (a*Cos[c + d*x]^5)/(5*d) - (b*Log[Cos[c + d*x]])/d} -{Sin[c + d*x]^3*(a + b*Sec[c + d*x]), x, 5, -((a*Cos[c + d*x])/d) + (b*Cos[c + d*x]^2)/(2*d) + (a*Cos[c + d*x]^3)/(3*d) - (b*Log[Cos[c + d*x]])/d} -{Sin[c + d*x]^1*(a + b*Sec[c + d*x]), x, 4, -((a*Cos[c + d*x])/d) - (b*Log[Cos[c + d*x]])/d} -{Csc[c + d*x]^1*(a + b*Sec[c + d*x]), x, 5, -((a*ArcTanh[Cos[c + d*x]])/d) + (b*Log[Tan[c + d*x]])/d} -{Csc[c + d*x]^3*(a + b*Sec[c + d*x]), x, 7, -((a*ArcTanh[Cos[c + d*x]])/(2*d)) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d) + (b*Log[Tan[c + d*x]])/d} -{Csc[c + d*x]^5*(a + b*Sec[c + d*x]), x, 9, -((3*a*ArcTanh[Cos[c + d*x]])/(8*d)) - (b*Cot[c + d*x]^2)/d - (b*Cot[c + d*x]^4)/(4*d) - (3*a*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (b*Log[Tan[c + d*x]])/d} -{Csc[c + d*x]^7*(a + b*Sec[c + d*x]), x, 10, -((5*a*ArcTanh[Cos[c + d*x]])/(16*d)) - (3*b*Cot[c + d*x]^2)/(2*d) - (3*b*Cot[c + d*x]^4)/(4*d) - (b*Cot[c + d*x]^6)/(6*d) - (5*a*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (5*a*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d) + (b*Log[Tan[c + d*x]])/d} - -{Sin[c + d*x]^6*(a + b*Sec[c + d*x]), x, 10, (5*a*x)/16 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (b*Sin[c + d*x]^3)/(3*d) - (5*a*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (b*Sin[c + d*x]^5)/(5*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d)} -{Sin[c + d*x]^4*(a + b*Sec[c + d*x]), x, 9, (3*a*x)/8 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Sin[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sin[c + d*x]^2*(a + b*Sec[c + d*x]), x, 7, (a*x)/2 + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + b*Sec[c + d*x]), x, 7, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (b*Csc[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Sec[c + d*x]), x, 8, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) - (b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d)} -{Csc[c + d*x]^6*(a + b*Sec[c + d*x]), x, 8, (b*ArcTanh[Sin[c + d*x]])/d - (a*Cot[c + d*x])/d - (2*a*Cot[c + d*x]^3)/(3*d) - (a*Cot[c + d*x]^5)/(5*d) - (b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/(3*d) - (b*Csc[c + d*x]^5)/(5*d)} - - -{Sin[c + d*x]^5*(a + b*Sec[c + d*x])^2, x, 5, -(((a^2 - 2*b^2)*Cos[c + d*x])/d) + (2*a*b*Cos[c + d*x]^2)/d + ((2*a^2 - b^2)*Cos[c + d*x]^3)/(3*d) - (a*b*Cos[c + d*x]^4)/(2*d) - (a^2*Cos[c + d*x]^5)/(5*d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d} -{Sin[c + d*x]^3*(a + b*Sec[c + d*x])^2, x, 5, -(((a^2 - b^2)*Cos[c + d*x])/d) + (a*b*Cos[c + d*x]^2)/d + (a^2*Cos[c + d*x]^3)/(3*d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d} -{Sin[c + d*x]^1*(a + b*Sec[c + d*x])^2, x, 5, -((a^2*Cos[c + d*x])/d) - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Sec[c + d*x])/d} -{Csc[c + d*x]^1*(a + b*Sec[c + d*x])^2, x, 5, ((a + b)^2*Log[1 - Cos[c + d*x]])/(2*d) - (2*a*b*Log[Cos[c + d*x]])/d - ((a - b)^2*Log[1 + Cos[c + d*x]])/(2*d) + (b^2*Sec[c + d*x])/d} -{Csc[c + d*x]^3*(a + b*Sec[c + d*x])^2, x, 6, -(((2*a*b + (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*d)) + ((a + b)*(a + 3*b)*Log[1 - Cos[c + d*x]])/(4*d) - (2*a*b*Log[Cos[c + d*x]])/d - ((a - 3*b)*(a - b)*Log[1 + Cos[c + d*x]])/(4*d) + (b^2*Sec[c + d*x])/d} - -{Sin[c + d*x]^6*(a + b*Sec[c + d*x])^2, x, 12, (5/16)*(a^2 - 6*b^2)*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - ((11*a^2 - 18*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((13*a^2 - 6*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a*b*Sin[c + d*x]^3)/(3*d) - (2*a*b*Sin[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x])/d} -{Sin[c + d*x]^4*(a + b*Sec[c + d*x])^2, x, 7, (3/8)*(a^2 - 4*b^2)*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (b*(28*a^2 + b^2)*Sin[c + d*x])/(6*a*d) - ((39*a^2 + 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(24*d) - ((12*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(12*a*b*d) + ((b + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*a*d) + ((b + a*Cos[c + d*x])^3*Tan[c + d*x])/(b*d)} -{Sin[c + d*x]^2*(a + b*Sec[c + d*x])^2, x, 10, (a^2*x)/2 - b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^2*(a + b*Sec[c + d*x])^2, x, 8, (2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + b^2)*Cot[c + d*x])/d - (2*a*b*Csc[c + d*x])/d + (b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Sec[c + d*x])^2, x, 9, (2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + 2*b^2)*Cot[c + d*x])/d - ((a^2 + b^2)*Cot[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^6*(a + b*Sec[c + d*x])^2, x, 9, (2*a*b*ArcTanh[Sin[c + d*x]])/d - ((a^2 + 3*b^2)*Cot[c + d*x])/d - ((2*a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*d) - ((a^2 + b^2)*Cot[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x])/d} - - -{Sin[c + d*x]^5*(a + b*Sec[c + d*x])^3, x, 5, -((a*(a^2 - 6*b^2)*Cos[c + d*x])/d) + (b*(6*a^2 - b^2)*Cos[c + d*x]^2)/(2*d) + (a*(2*a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*d) - (3*a^2*b*Cos[c + d*x]^4)/(4*d) - (a^3*Cos[c + d*x]^5)/(5*d) - (b*(3*a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} -{Sin[c + d*x]^3*(a + b*Sec[c + d*x])^3, x, 4, -((a*(a^2 - 3*b^2)*Cos[c + d*x])/d) + (3*a^2*b*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} -{Sin[c + d*x]^1*(a + b*Sec[c + d*x])^3, x, 5, -((a^3*Cos[c + d*x])/d) - (3*a^2*b*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^1*(a + b*Sec[c + d*x])^3, x, 5, ((a + b)^3*Log[1 - Cos[c + d*x]])/(2*d) - (b*(3*a^2 + b^2)*Log[Cos[c + d*x]])/d - ((a - b)^3*Log[1 + Cos[c + d*x]])/(2*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} -{Csc[c + d*x]^3*(a + b*Sec[c + d*x])^3, x, 6, -((a^2*(b*(3 + b^2/a^2) + a*(1 + (3*b^2)/a^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*d)) + ((a + b)^2*(a + 4*b)*Log[1 - Cos[c + d*x]])/(4*d) - (b*(3*a^2 + 2*b^2)*Log[Cos[c + d*x]])/d - ((a - 4*b)*(a - b)^2*Log[1 + Cos[c + d*x]])/(4*d) + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} - -{Sin[c + d*x]^6*(a + b*Sec[c + d*x])^3, x, 21, (5*a^3*x)/16 - (45/8)*a*b^2*x + (3*a^2*b*ArcTanh[Sin[c + d*x]])/d - (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b*Sin[c + d*x])/d + (5*b^3*Sin[c + d*x])/(2*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (a^2*b*Sin[c + d*x]^3)/d + (5*b^3*Sin[c + d*x]^3)/(6*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (3*a^2*b*Sin[c + d*x]^5)/(5*d) - (a^3*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d) + (45*a*b^2*Tan[c + d*x])/(8*d) - (15*a*b^2*Sin[c + d*x]^2*Tan[c + d*x])/(8*d) - (3*a*b^2*Sin[c + d*x]^4*Tan[c + d*x])/(4*d) + (b^3*Sin[c + d*x]^3*Tan[c + d*x]^2)/(2*d)} -{Sin[c + d*x]^4*(a + b*Sec[c + d*x])^3, x, 8, (3/8)*a*(a^2 - 12*b^2)*x + (3*b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (b*(17*a^2 - b^2)*Sin[c + d*x])/(2*d) - (a*(21*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - ((6*a^2 - b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(4*b*d) - ((4*a^2 - b^2)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*b^2*d) + (a*(b + a*Cos[c + d*x])^4*Tan[c + d*x])/(b^2*d) + ((b + a*Cos[c + d*x])^4*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sin[c + d*x]^2*(a + b*Sec[c + d*x])^3, x, 8, (1/2)*a*(a^2 - 6*b^2)*x + (b*(6*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (15*a^2*b*Sin[c + d*x])/(2*d) - (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*a*(b + a*Cos[c + d*x])^2*Tan[c + d*x])/(2*d) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Csc[c + d*x]^2*(a + b*Sec[c + d*x])^3, x, 15, (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (3*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (3*a*b^2*Cot[c + d*x])/d - (3*a^2*b*Csc[c + d*x])/d - (3*b^3*Csc[c + d*x])/(2*d) + (b^3*Csc[c + d*x]*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^4*(a + b*Sec[c + d*x])^3, x, 17, (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (6*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (a*b^2*Cot[c + d*x]^3)/d - (3*a^2*b*Csc[c + d*x])/d - (5*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (5*b^3*Csc[c + d*x]^3)/(6*d) + (b^3*Csc[c + d*x]^3*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d} -{Csc[c + d*x]^6*(a + b*Sec[c + d*x])^3, x, 17, (3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (7*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (3*a*b^2*Cot[c + d*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a*b^2*Cot[c + d*x]^5)/(5*d) - (3*a^2*b*Csc[c + d*x])/d - (7*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (7*b^3*Csc[c + d*x]^3)/(6*d) - (3*a^2*b*Csc[c + d*x]^5)/(5*d) - (7*b^3*Csc[c + d*x]^5)/(10*d) + (b^3*Csc[c + d*x]^5*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[c + d*x]^7/(a + b*Sec[c + d*x]), x, 5, -(((a^2 - b^2)^3*Cos[c + d*x])/(a^7*d)) - (b*(3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^2)/(2*a^6*d) + ((3*a^4 - 3*a^2*b^2 + b^4)*Cos[c + d*x]^3)/(3*a^5*d) + (b*(3*a^2 - b^2)*Cos[c + d*x]^4)/(4*a^4*d) - ((3*a^2 - b^2)*Cos[c + d*x]^5)/(5*a^3*d) - (b*Cos[c + d*x]^6)/(6*a^2*d) + Cos[c + d*x]^7/(7*a*d) + (b*(a^2 - b^2)^3*Log[b + a*Cos[c + d*x]])/(a^8*d)} -{Sin[c + d*x]^5/(a + b*Sec[c + d*x]), x, 5, -(((a^2 - b^2)^2*Cos[c + d*x])/(a^5*d)) - (b*(2*a^2 - b^2)*Cos[c + d*x]^2)/(2*a^4*d) + ((2*a^2 - b^2)*Cos[c + d*x]^3)/(3*a^3*d) + (b*Cos[c + d*x]^4)/(4*a^2*d) - Cos[c + d*x]^5/(5*a*d) + (b*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]])/(a^6*d)} -{Sin[c + d*x]^3/(a + b*Sec[c + d*x]), x, 5, -(((a^2 - b^2)*Cos[c + d*x])/(a^3*d)) - (b*Cos[c + d*x]^2)/(2*a^2*d) + Cos[c + d*x]^3/(3*a*d) + (b*(a^2 - b^2)*Log[b + a*Cos[c + d*x]])/(a^4*d)} -{Sin[c + d*x]^1/(a + b*Sec[c + d*x]), x, 5, -(Cos[c + d*x]/(a*d)) + (b*Log[b + a*Cos[c + d*x]])/(a^2*d)} -{Csc[c + d*x]^1/(a + b*Sec[c + d*x]), x, 4, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)} -{Csc[c + d*x]^3/(a + b*Sec[c + d*x]), x, 6, ((b - a*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d) + (a*Log[1 - Cos[c + d*x]])/(4*(a + b)^2*d) - (a*Log[1 + Cos[c + d*x]])/(4*(a - b)^2*d) + (a^2*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)} -{Csc[c + d*x]^5/(a + b*Sec[c + d*x]), x, 7, ((4*a^2*b - a*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)*d) + (a*(3*a + b)*Log[1 - Cos[c + d*x]])/(16*(a + b)^3*d) - (a*(3*a - b)*Log[1 + Cos[c + d*x]])/(16*(a - b)^3*d) + (a^4*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)} - -{Sin[c + d*x]^6/(a + b*Sec[c + d*x]), x, 7, ((5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*x)/(16*a^7) - (2*(a - b)^(5/2)*b*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^7*d) + ((16*b*(a^2 - b^2)^2 - a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*Cos[c + d*x])*Sin[c + d*x])/(16*a^6*d) + ((8*b*(a^2 - b^2) - a*(5*a^2 - 6*b^2)*Cos[c + d*x])*Sin[c + d*x]^3)/(24*a^4*d) + ((6*b - 5*a*Cos[c + d*x])*Sin[c + d*x]^5)/(30*a^2*d)} -{Sin[c + d*x]^4/(a + b*Sec[c + d*x]), x, 6, ((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*d) + ((8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/(8*a^4*d) + ((4*b - 3*a*Cos[c + d*x])*Sin[c + d*x]^3)/(12*a^2*d)} -{Sin[c + d*x]^2/(a + b*Sec[c + d*x]), x, 5, ((a^2 - 2*b^2)*x)/(2*a^3) - (2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*d) + ((2*b - a*Cos[c + d*x])*Sin[c + d*x])/(2*a^2*d)} -{Csc[c + d*x]^2/(a + b*Sec[c + d*x]), x, 5, -((2*a*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d)) + ((b - a*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)*d)} -{Csc[c + d*x]^4/(a + b*Sec[c + d*x]), x, 6, -((2*a^3*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + ((3*a^2*b - a*(2*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x])/(3*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^3)/(3*(a^2 - b^2)*d)} -{Csc[c + d*x]^6/(a + b*Sec[c + d*x]), x, 7, -((2*a^5*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + ((15*a^4*b - a*(8*a^4 + 9*a^2*b^2 - 2*b^4)*Cos[c + d*x])*Csc[c + d*x])/(15*(a^2 - b^2)^3*d) + ((5*a^2*b - a*(4*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^3)/(15*(a^2 - b^2)^2*d) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^5)/(5*(a^2 - b^2)*d)} - - -{Sin[c + d*x]^7/(a + b*Sec[c + d*x])^2, x, 5, -(((a^2 - 7*b^2)*(a^2 - b^2)^2*Cos[c + d*x])/(a^8*d)) - (3*b*(a^2 - b^2)^2*Cos[c + d*x]^2)/(a^7*d) + ((3*a^4 - 9*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(3*a^6*d) + (b*(3*a^2 - 2*b^2)*Cos[c + d*x]^4)/(2*a^5*d) - (3*(a^2 - b^2)*Cos[c + d*x]^5)/(5*a^4*d) - (b*Cos[c + d*x]^6)/(3*a^3*d) + Cos[c + d*x]^7/(7*a^2*d) + (b^2*(a^2 - b^2)^3)/(a^9*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]])/(a^9*d)} -{Sin[c + d*x]^5/(a + b*Sec[c + d*x])^2, x, 5, -(((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x])/(a^6*d)) - (2*b*(a^2 - b^2)*Cos[c + d*x]^2)/(a^5*d) + ((2*a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*a^4*d) + (b*Cos[c + d*x]^4)/(2*a^3*d) - Cos[c + d*x]^5/(5*a^2*d) + (b^2*(a^2 - b^2)^2)/(a^7*d*(b + a*Cos[c + d*x])) + (2*b*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]])/(a^7*d)} -{Sin[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 5, -(((a^2 - 3*b^2)*Cos[c + d*x])/(a^4*d)) - (b*Cos[c + d*x]^2)/(a^3*d) + Cos[c + d*x]^3/(3*a^2*d) + (b^2*(a^2 - b^2))/(a^5*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 2*b^2)*Log[b + a*Cos[c + d*x]])/(a^5*d)} -{Sin[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 5, -(Cos[c + d*x]/(a^2*d)) + b^2/(a^3*d*(b + a*Cos[c + d*x])) + (2*b*Log[b + a*Cos[c + d*x]])/(a^3*d)} -{Csc[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 5, b^2/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^2*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)^2*d) + (2*a*b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^2*d)} -{Csc[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 6, (a*b^2)/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x])) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^2*d) + ((a - b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^3*d) - ((a + b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^3*d) + (2*a*b*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)} -{Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2, x, 7, (a^3*b^2)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((8*a*b*(a^2 + b^2) - (3*a^4 + 12*a^2*b^2 + b^4)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^3*d) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^2*d) + ((3*a^2 - 4*a*b - b^2)*Log[1 - Cos[c + d*x]])/(16*(a + b)^4*d) - ((3*a^2 + 4*a*b - b^2)*Log[1 + Cos[c + d*x]])/(16*(a - b)^4*d) + (2*a^3*b*(a^2 + 2*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} - -{Sin[c + d*x]^6/(a + b*Sec[c + d*x])^2, x, 10, ((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*x)/(16*a^8) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*(2*a^2 - 7*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Sin[c + d*x])/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 + 35*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^4*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*b*d*(b + a*Cos[c + d*x])) + (a*Cos[c + d*x]^4*Sin[c + d*x])/(6*b^2*d*(b + a*Cos[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(10*a^3*b^2*d*(b + a*Cos[c + d*x])) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(30*a^2*d*(b + a*Cos[c + d*x])) - (Cos[c + d*x]^6*Sin[c + d*x])/(6*a*d*(b + a*Cos[c + d*x]))} -{Sin[c + d*x]^4/(a + b*Sec[c + d*x])^2, x, 8, ((3*a^4 - 36*a^2*b^2 + 40*b^4)*x)/(8*a^6) - (2*Sqrt[a - b]*b*Sqrt[a + b]*(2*a^2 - 5*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*d) + (b*(11*a^2 - 15*b^2)*Sin[c + d*x])/(3*a^5*d) - ((13*a^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^4*d) + ((3*a^2 - 5*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^3*b*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d) - ((a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(a^2*b*d*(b + a*Cos[c + d*x]))} -{Sin[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 8, ((a^2 - 6*b^2)*x)/(2*a^4) - (2*b*(2*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + (3*b*Sin[c + d*x])/(a^3*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (Cos[c + d*x]^2*Sin[c + d*x])/(a*d*(b + a*Cos[c + d*x]))} -{Csc[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 11, -((4*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (a*b^2*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Csc[c + d*x]^4/(a + b*Sec[c + d*x])^2, x, 15, -((2*a^2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (4*a^2*b*(a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])^2) - ((a - b)*Sin[c + d*x])/(4*(a + b)^3*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])^2) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])) + ((a + b)*Sin[c + d*x])/(4*(a - b)^3*d*(1 + Cos[c + d*x])) + (a^3*b^2*Sin[c + d*x])/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))} - - -{Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3, x, 5, -(((a^6 - 18*a^4*b^2 + 45*a^2*b^4 - 28*b^6)*Cos[c + d*x])/(a^9*d)) - (3*b*(3*a^4 - 10*a^2*b^2 + 7*b^4)*Cos[c + d*x]^2)/(2*a^8*d) + ((a^4 - 6*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/(a^7*d) + (b*(9*a^2 - 10*b^2)*Cos[c + d*x]^4)/(4*a^6*d) - (3*(a^2 - 2*b^2)*Cos[c + d*x]^5)/(5*a^5*d) - (b*Cos[c + d*x]^6)/(2*a^4*d) + Cos[c + d*x]^7/(7*a^3*d) - (b^3*(a^2 - b^2)^3)/(2*a^10*d*(b + a*Cos[c + d*x])^2) + (3*b^2*(a^2 - 3*b^2)*(a^2 - b^2)^2)/(a^10*d*(b + a*Cos[c + d*x])) + (3*b*(a^2 - b^2)*(a^4 - 9*a^2*b^2 + 12*b^4)*Log[b + a*Cos[c + d*x]])/(a^10*d)} -{Sin[c + d*x]^5/(a + b*Sec[c + d*x])^3, x, 5, -(((a^4 - 12*a^2*b^2 + 15*b^4)*Cos[c + d*x])/(a^7*d)) - (b*(3*a^2 - 5*b^2)*Cos[c + d*x]^2)/(a^6*d) + (2*(a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*a^5*d) + (3*b*Cos[c + d*x]^4)/(4*a^4*d) - Cos[c + d*x]^5/(5*a^3*d) - (b^3*(a^2 - b^2)^2)/(2*a^8*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^4 - 10*a^2*b^2 + 7*b^4))/(a^8*d*(b + a*Cos[c + d*x])) + (b*(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]])/(a^8*d)} -{Sin[c + d*x]^3/(a + b*Sec[c + d*x])^3, x, 5, -(((a^2 - 6*b^2)*Cos[c + d*x])/(a^5*d)) - (3*b*Cos[c + d*x]^2)/(2*a^4*d) + Cos[c + d*x]^3/(3*a^3*d) - (b^3*(a^2 - b^2))/(2*a^6*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 - 5*b^2))/(a^6*d*(b + a*Cos[c + d*x])) + (b*(3*a^2 - 10*b^2)*Log[b + a*Cos[c + d*x]])/(a^6*d)} -{Sin[c + d*x]^1/(a + b*Sec[c + d*x])^3, x, 5, -(Cos[c + d*x]/(a^3*d)) - b^3/(2*a^4*d*(b + a*Cos[c + d*x])^2) + (3*b^2)/(a^4*d*(b + a*Cos[c + d*x])) + (3*b*Log[b + a*Cos[c + d*x]])/(a^4*d)} -{Csc[c + d*x]^1/(a + b*Sec[c + d*x])^3, x, 5, -(b^3/(2*a^2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2)) + (b^2*(3*a^2 - b^2))/(a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(2*(a + b)^3*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)^3*d) + (b*(3*a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^3*d)} -{Csc[c + d*x]^3/(a + b*Sec[c + d*x])^3, x, 5, -(b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2)) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3, x, 7, -((a^2*b^3)/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])^2)) + (3*a^2*b^2*(a^2 + b^2))/((a^2 - b^2)^4*d*(b + a*Cos[c + d*x])) + ((4*b*(3*a^4 + 8*a^2*b^2 + b^4) - 3*a*(a^4 + 10*a^2*b^2 + 5*b^4)*Cos[c + d*x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^4*d) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^3*d) + (3*a*(a - 3*b)*Log[1 - Cos[c + d*x]])/(16*(a + b)^5*d) - (3*a*(a + 3*b)*Log[1 + Cos[c + d*x]])/(16*(a - b)^5*d) + (3*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^5*d)} - -{Sin[c + d*x]^6/(a + b*Sec[c + d*x])^3, x, 11, ((5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*x)/(16*a^9) - (Sqrt[a - b]*b*Sqrt[a + b]*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^9*d) + (b*(213*a^4 - 985*a^2*b^2 + 840*b^4)*Sin[c + d*x])/(30*a^8*d) - ((43*a^4 - 244*a^2*b^2 + 224*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^7*d) + ((45*a^4 - 291*a^2*b^2 + 280*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(30*a^6*b*d) - ((24*a^4 - 169*a^2*b^2 + 168*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^5*b^2*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(4*b*d*(b + a*Cos[c + d*x])^2) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(10*b^2*d*(b + a*Cos[c + d*x])^2) + ((9*a^4 - 60*a^2*b^2 + 56*b^4)*Cos[c + d*x]^5*Sin[c + d*x])/(60*a^3*b^2*d*(b + a*Cos[c + d*x])^2) + (4*b*Cos[c + d*x]^6*Sin[c + d*x])/(15*a^2*d*(b + a*Cos[c + d*x])^2) - (Cos[c + d*x]^7*Sin[c + d*x])/(6*a*d*(b + a*Cos[c + d*x])^2) + ((15*a^4 - 110*a^2*b^2 + 112*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(20*a^4*b^2*d*(b + a*Cos[c + d*x]))} -{Sin[c + d*x]^4/(a + b*Sec[c + d*x])^3, x, 9, (3*(a^4 - 24*a^2*b^2 + 40*b^4)*x)/(8*a^7) - (3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^7*Sqrt[a - b]*Sqrt[a + b]*d) + (b*(13*a^2 - 30*b^2)*Sin[c + d*x])/(2*a^6*d) - (3*(7*a^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^5*d) + ((3*a^2 - 10*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^4*b*d) - ((4*a^2 - 15*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b*d*(b + a*Cos[c + d*x])^2) + ((2*a^2 - 7*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b^2*d*(b + a*Cos[c + d*x]))} -{Sin[c + d*x]^2/(a + b*Sec[c + d*x])^3, x, 9, ((a^2 - 12*b^2)*x)/(2*a^5) - (b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(11*a^2 - 12*b^2)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*d) - ((5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(2*a*d*(b + a*Cos[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{Csc[c + d*x]^2/(a + b*Sec[c + d*x])^3, x, 16, -((2*b^3*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(7/2)*(a + b)^(7/2)*d)) - (2*a*b*(3*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b^3*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(7/2)*(a + b)^(7/2)*d) - Sin[c + d*x]/(2*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^3*d*(1 + Cos[c + d*x])) - (b^3*Sin[c + d*x])/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (3*b^4*Sin[c + d*x])/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (b^2*(3*a^2 - b^2)*Sin[c + d*x])/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))} -{Csc[c + d*x]^4/(a + b*Sec[c + d*x])^3, x, 20, -((2*a*b^3*(3*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d)) - (a*b^3*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d) - (2*a*b*(3*a^4 + 8*a^2*b^2 + b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(9/2)*(a + b)^(9/2)*d) - Sin[c + d*x]/(12*(a + b)^3*d*(1 - Cos[c + d*x])^2) - ((a - 2*b)*Sin[c + d*x])/(4*(a + b)^4*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(12*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^3*d*(1 + Cos[c + d*x])^2) + Sin[c + d*x]/(12*(a - b)^3*d*(1 + Cos[c + d*x])) + ((a + 2*b)*Sin[c + d*x])/(4*(a - b)^4*d*(1 + Cos[c + d*x])) - (a^2*b^3*Sin[c + d*x])/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])^2) + (3*a^2*b^4*Sin[c + d*x])/(2*(a^2 - b^2)^4*d*(b + a*Cos[c + d*x])) + (a^2*b^2*(3*a^2 + b^2)*Sin[c + d*x])/((a^2 - b^2)^4*d*(b + a*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^(m/2) (a+b Sec[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x]), x, 15, -((b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d)) - (b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (2*(5*a^4 - 28*a^2*b^2 + 21*b^4)*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a^5*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(21*b*(a^2 - b^2) - a*(5*a^2 - 7*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(21*a^4*d) + (2*e*(7*b - 5*a*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*a^2*d)} -{(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x]), x, 14, (b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) - (b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) - (b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^4*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^4*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*(3*a^2 - 5*b^2)*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^3*d*Sqrt[Sin[c + d*x]]) + (2*e*(5*b - 3*a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^2*d)} -{(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x]), x, 14, -((b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*d)) - (b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*d) + (2*(a^2 - 3*b^2)*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^3*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^3*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^3*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*e*(3*b - a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^2*d)} -{(e*Sin[c + d*x])^(1/2)/(a + b*Sec[c + d*x]), x, 13, (b*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(1/4)*d) - (b^2*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^2*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a*d*Sqrt[Sin[c + d*x]])} -{1/(e*Sin[c + d*x])^(1/2)/(a + b*Sec[c + d*x]), x, 13, -((b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(3/4)*d*Sqrt[e])) - (b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]])} -{1/(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x]), x, 14, (Sqrt[a]*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(5/4)*d*e^(3/2)) - (Sqrt[a]*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(5/4)*d*e^(3/2)) + (2*(b - a*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - (b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]])} -{1/(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x]), x, 14, -((a^(3/2)*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2))) - (a^(3/2)*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) + (2*(b - a*Cos[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) + (a*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (a*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]])} -{1/(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x]), x, 15, (a^(5/2)*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(9/4)*d*e^(7/2)) - (a^(5/2)*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(9/4)*d*e^(7/2)) + (2*(b - a*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(5/2)) + (2*(5*a^2*b - a*(3*a^2 + 2*b^2)*Cos[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Sin[c + d*x]]) - (a^2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(a - Sqrt[a^2 - b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (a^2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(a + Sqrt[a^2 - b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*a*(3*a^2 + 2*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Sin[c + d*x]])} - - -{(e*Sin[c + d*x])^(9/2)/(a + b*Sec[c + d*x])^2, x, 35, -((7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(13/2)*d)) + (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(13/2)*d) + (7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(13/2)*d) - (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(13/2)*d) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (14*e^4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(15*a^2*d*Sqrt[Sin[c + d*x]]) - (7*b^2*(3*a^2 - 5*b^2)*e^4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]]) - (4*b^2*(8*a^2 - 5*b^2)*e^4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]]) - (14*e^3*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(45*a^2*d) - (7*b^2*e^3*(5*b - 3*a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^5*d) + (4*b*e^3*(5*(a^2 - b^2) + 3*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(7/2))/(7*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(7/2))/(9*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(7/2))/(a^3*d*(b + a*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x])^2, x, 35, (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(11/2)*d) + (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(11/2)*d) + (10*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c + d*x]]) - (5*b^2*(a^2 - 3*b^2)*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c + d*x]]) - (4*b^2*(4*a^2 - 3*b^2)*e^4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (10*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) - (5*b^2*e^3*(3*b - a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e^3*(3*(a^2 - b^2) + a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(5/2))/(5*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(5/2))/(a^3*d*(b + a*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 32, -((3*b^3*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d)) + (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (3*b^3*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(9/2)*(a^2 - b^2)^(1/4)*d) - (2*b*(a^2 - b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(9/2)*d) + (3*b^4*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^5*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^5*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^5*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)*e^3*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^5*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (6*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(5*a^2*d*Sqrt[Sin[c + d*x]]) - (7*b^2*e^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^4*d*Sqrt[Sin[c + d*x]]) + (4*b*e*(e*Sin[c + d*x])^(3/2))/(3*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*a^2*d) + (b^2*e*(e*Sin[c + d*x])^(3/2))/(a^3*d*(b + a*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 32, (b^3*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) + (b^3*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(7/2)*(a^2 - b^2)^(3/4)*d) - (2*b*(a^2 - b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(7/2)*d) + (2*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^2*d*Sqrt[e*Sin[c + d*x]]) - (5*b^2*e^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^4*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^4*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^4*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^4*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)*e^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^4*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (4*b*e*Sqrt[e*Sin[c + d*x]])/(a^3*d) - (2*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*a^2*d) + (b^2*e*Sqrt[e*Sin[c + d*x]])/(a^3*d*(b + a*Cos[c + d*x]))} -{(e*Sin[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 27, (b^3*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) + (2*b*Sqrt[e]*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (b^3*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(5/2)*(a^2 - b^2)^(5/4)*d) - (2*b*Sqrt[e]*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(5/2)*(a^2 - b^2)^(1/4)*d) - (2*b^2*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^3*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^3*(a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^3*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (b^4*e*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^3*(a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^2*d*Sqrt[Sin[c + d*x]]) - (b^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) + (b^2*(e*Sin[c + d*x])^(3/2))/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))} -{1/(e*Sin[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 27, -((3*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqrt[e])) - (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*Sqrt[e]) - (3*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqrt[e]) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) + (b^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*Sqrt[e*Sin[c + d*x]])/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))} -{1/(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 33, (5*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(9/4)*d*e^(3/2)) + (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (5*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(9/4)*d*e^(3/2)) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(5/4)*d*e^(3/2)) - (2*Cos[c + d*x])/(a^2*d*e*Sqrt[e*Sin[c + d*x]]) + b^2/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (4*b*(a - b*Cos[c + d*x]))/(a^2*(a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*a*b - (3*a^2 + 2*b^2)*Cos[c + d*x]))/(a^2*(a^2 - b^2)^2*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^2*d*e^2*Sqrt[Sin[c + d*x]]) - (4*b^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]]) - (b^2*(3*a^2 + 2*b^2)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2)^2*d*e^2*Sqrt[Sin[c + d*x]])} -{1/(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 33, -((7*Sqrt[a]*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)^(11/4)*d*e^(5/2))) - (2*Sqrt[a]*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (7*Sqrt[a]*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)^(11/4)*d*e^(5/2)) - (2*Sqrt[a]*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d*e^(5/2)) - (2*Cos[c + d*x])/(3*a^2*d*e*(e*Sin[c + d*x])^(3/2)) + b^2/(a*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (4*b*(a - b*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (b^2*(7*a*b - (5*a^2 + 2*b^2)*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + (2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (4*b^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*a^2 + 2*b^2)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -(* {Sin[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]], x, 11, -(((a - b)*Sqrt[a + b]*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(7680*a^5*f*Sqrt[b + a*Cos[e + f*x]])) + (1/(7680*a^5*f*Sqrt[b + a*Cos[e + f*x]]))*Sqrt[a + b]*(10080*a^5 - 3248*a^4*b - 928*a^3*b^2 + 1416*a^2*b^3 + 210*a*b^4 - 315*b^5)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]] - (Sqrt[a + b]*(320*a^6 + 240*a^4*b^2 - 100*a^2*b^4 + 21*b^6)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(512*a^6*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(3248*a^4 - 1416*a^2*b^2 + 315*b^4)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(7680*a^5*f) - ((880*a^4 - 416*a^2*b^2 + 105*b^4)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(1280*a^4*f) - (2*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(b*f) + ((1920*a^4 - 428*a^2*b^2 + 105*b^4)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(960*a^4*b*f) + (8*a*Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*b^2*f) - ((1280*a^4 - 260*a^2*b^2 + 63*b^4)*Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(480*a^3*b^2*f) + (3*b*Cos[e + f*x]^3*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(20*a^2*f) - (Cos[e + f*x]^4*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(6*a*f)} -{Sin[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]], x, 9, -(((a - b)*Sqrt[a + b]*(68*a^2 - 15*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(192*a^3*f*Sqrt[b + a*Cos[e + f*x]])) + (Sqrt[a + b]*(264*a^3 - 68*a^2*b - 10*a*b^2 + 15*b^3)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(192*a^3*f*Sqrt[b + a*Cos[e + f*x]]) - (Sqrt[a + b]*(48*a^4 + 24*a^2*b^2 - 5*b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^4*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(68*a^2 - 15*b^2)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(192*a^3*f) - (5*(4*a^2 - b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(32*a^2*f) - (5*b*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(24*a^2*f) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f)} -{Sin[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]], x, 9, -(((a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a*f)) + ((6*a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a*f) - (Sqrt[a + b]*(4*a^2 + b^2)*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f) - (b*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f) - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(2*f)} *) -{Sin[e + f*x]^0*Sqrt[a + b*Sec[e + f*x]], x, 1, -((2*Cot[e + f*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[e + f*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[e + f*x]))/(a + b*Sec[e + f*x]))]*Sqrt[(b*(1 + Sec[e + f*x]))/(a + b*Sec[e + f*x])]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f))} -{Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]], x, 2, (Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/f} -(* {Csc[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]], x, 0, 0} -{Csc[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]], x, 0, 0} *) - - -(* {Sin[e + f*x]^4*(a + b*Sec[e + f*x])^(3/2), x, 10, -(((a - b)*Sqrt[a + b]*(236*a^2 + 3*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^2*f*Sqrt[b + a*Cos[e + f*x]])) + (Sqrt[a + b]*(216*a^3 - 236*a^2*b + 2*a*b^2 - 3*b^3)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^2*f*Sqrt[b + a*Cos[e + f*x]]) - (3*Sqrt[a + b]*(16*a^4 - 24*a^2*b^2 + b^4)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(64*a^3*f*Sqrt[b + a*Cos[e + f*x]]) - (b*(236*a^2 + 3*b^2)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(64*a^2*f) - (3*(28*a^2 + b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(32*a*f) - ((16*a^2 + b^2)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(8*a*b*f) + (2*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(b*f) + (Cos[e + f*x]*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*a*f)} -{Sin[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2), x, 10, -((13*(a - b)*Sqrt[a + b]*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*f*Sqrt[b + a*Cos[e + f*x]])) + ((14*a - 13*b)*Sqrt[a + b]*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*f*Sqrt[b + a*Cos[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 3*b^2)*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))]*Sqrt[a + b*Sec[e + f*x]])/(4*a*f*Sqrt[b + a*Cos[e + f*x]]) - (13*b*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(4*f) - (5*a*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(2*f) + (2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/f} *) -{Sin[e + f*x]^0*(a + b*Sec[e + f*x])^(3/2), x, 5, -((2*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f) + (2*(2*a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f} -{Csc[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2), x, 4, -((3*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f) + (3*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (Cot[e + f*x]*(a + b*Sec[e + f*x])^(3/2))/f} -(* {Csc[e + f*x]^4*(a + b*Sec[e + f*x])^(3/2), x, 0, 0} -{Csc[e + f*x]^6*(a + b*Sec[e + f*x])^(3/2), x, 0, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -(* {Sin[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]], x, 9, ((a - b)*Sqrt[a + b]*(188*a^2 - 105*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(120*a^3 - 188*a^2*b - 70*a*b^2 + 105*b^3)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(48*a^4 - 72*a^2*b^2 + 35*b^4)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(12*a^2 - 7*b^2)*(b + a*Cos[e + f*x])*Sin[e + f*x])/(96*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (7*b*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(4*a*f*Sqrt[a + b*Sec[e + f*x]]) + (b*(188*a^2 - 105*b^2)*(b + a*Cos[e + f*x])*Tan[e + f*x])/(192*a^4*f*Sqrt[a + b*Sec[e + f*x]])} -{Sin[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]], x, 9, (3*(a - b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) - ((2*a - 3*b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 3*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - ((b + a*Cos[e + f*x])*Sin[e + f*x])/(2*a*f*Sqrt[a + b*Sec[e + f*x]]) + (3*b*(b + a*Cos[e + f*x])*Tan[e + f*x])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]])} *) -{Sin[e + f*x]^0/Sqrt[a + b*Sec[e + f*x]], x, 1, -((2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f))} -{Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]], x, 6, (Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(Sqrt[a + b]*f) - (Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(Sqrt[a + b]*f) - Cot[e + f*x]/(f*Sqrt[a + b*Sec[e + f*x]]) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])} -(* {Csc[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]], x, 0, 0} -{Csc[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]], x, 0, 0} *) - - -(* {Sin[e + f*x]^4/(a + b*Sec[e + f*x])^(3/2), x, 10, (3*(a - b)*Sqrt[a + b]*(92*a^2 - 105*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (3*Sqrt[a + b]*(56*a^3 - 92*a^2*b - 70*a*b^2 + 105*b^3)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]]) - (3*Sqrt[a + b]*(16*a^4 - 120*a^2*b^2 + 105*b^4)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(64*a^6*f*Sqrt[a + b*Sec[e + f*x]]) - (2*(a^2 - b^2)*Cos[e + f*x]^2*Sin[e + f*x])/(a^2*b*f*Sqrt[a + b*Sec[e + f*x]]) - (21*(4*a^2 - 5*b^2)*(b + a*Cos[e + f*x])*Sin[e + f*x])/(32*a^4*f*Sqrt[a + b*Sec[e + f*x]]) + ((16*a^2 - 21*b^2)*Cos[e + f*x]*(b + a*Cos[e + f*x])*Sin[e + f*x])/(8*a^3*b*f*Sqrt[a + b*Sec[e + f*x]]) + (Cos[e + f*x]^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(4*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (3*b*(92*a^2 - 105*b^2)*(b + a*Cos[e + f*x])*Tan[e + f*x])/(64*a^5*f*Sqrt[a + b*Sec[e + f*x]])} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2), x, 10, (15*(a - b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(2*a - 3*b)*Sqrt[a + b]*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]]) - (Sqrt[a + b]*(4*a^2 - 15*b^2)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(4*a^4*f*Sqrt[a + b*Sec[e + f*x]]) + (2*Cos[e + f*x]*Sin[e + f*x])/(a*f*Sqrt[a + b*Sec[e + f*x]]) - (5*(b + a*Cos[e + f*x])*Sin[e + f*x])/(2*a^2*f*Sqrt[a + b*Sec[e + f*x]]) + (15*b*(b + a*Cos[e + f*x])*Tan[e + f*x])/(4*a^3*f*Sqrt[a + b*Sec[e + f*x]])} *) -{Sin[e + f*x]^0/(a + b*Sec[e + f*x])^(3/2), x, 6, (2*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*Sqrt[a + b]*f) - (2*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*Sqrt[a + b]*f) - (2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^2*f) + (2*b^2*Tan[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2), x, 6, (4*a*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*f) - ((3*a - b)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*f) - Cot[e + f*x]/(f*(a + b*Sec[e + f*x])^(3/2)) + (b^2*Tan[e + f*x])/((a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (4*a*b^2*Tan[e + f*x])/((a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])} -(* {Csc[e + f*x]^4/(a + b*Sec[e + f*x])^(3/2), x, 0, 0} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x])^(3/2), x, 0, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^m (a+b Sec[e+f x])^n with m symbolic*) - - -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^3, x, 9, (a^3*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (3*a^2*b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (b^3*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (3*a*b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^2, x, 9, (a^2*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]*(e*Sin[c + d*x])^m)/(d*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (2*a*b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)) + (b^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))} -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^1, x, 5, (a*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[Cos[c + d*x]^2]) + (b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^1, x, 4, -((b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^2*d*(1 - m))) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])} -{(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^2, x, 6, -((2*b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(1 - m))) + (b^2*e*AppellF1[2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^3*d*(2 - m)*(b + a*Cos[c + d*x])) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a^2*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])} -{(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^3, x, 7, -((3*b*e*AppellF1[1 - m, (1 - m)/2, (1 - m)/2, 2 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(1 - m))) - (b^3*e*AppellF1[3 - m, (1 - m)/2, (1 - m)/2, 4 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(3 - m)*(b + a*Cos[c + d*x])^2) + (3*b^2*e*AppellF1[2 - m, (1 - m)/2, (1 - m)/2, 3 - m, -((a - b)/(b + a*Cos[c + d*x])), (a + b)/(b + a*Cos[c + d*x])]*(-((a*(1 - Cos[c + d*x]))/(b + a*Cos[c + d*x])))^((1 - m)/2)*((a*(1 + Cos[c + d*x]))/(b + a*Cos[c + d*x]))^((1 - m)/2)*(e*Sin[c + d*x])^(-1 + m))/(a^4*d*(2 - m)*(b + a*Cos[c + d*x])) + (Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*(e*Sin[c + d*x])^(1 + m))/(a^3*d*e*(1 + m)*Sqrt[Cos[c + d*x]^2])} - - -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m, x]} -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^(1/2), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]*(e*Sin[c + d*x])^m, x]} -{(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(1/2), x, 0, Unintegrable[(e*Sin[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]} -{(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Sin[e+f x])^m (a+b Sec[e+f x])^n with n symbolic*) - - -{(e*Sin[c + d*x])^m*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*(e*Sin[c + d*x])^m, x]} - - -{Sin[c + d*x]^5*(a + b*Sec[c + d*x])^n, x, 6, (b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^2*d*(1 + n)) - (2*b^3*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^4*d*(1 + n)) + (b^5*Hypergeometric2F1[6, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^6*d*(1 + n))} -{Sin[c + d*x]^3*(a + b*Sec[c + d*x])^n, x, 3, (b*(6*a^2 - b^2*(2 - 3*n + n^2))*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(6*a^4*d*(1 + n)) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(1 + n)*(2*a - b*(2 - n)*Sec[c + d*x]))/(6*a^2*d)} -{Sin[c + d*x]^1*(a + b*Sec[c + d*x])^n, x, 2, (b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a^2*d*(1 + n))} -{Csc[c + d*x]^1*(a + b*Sec[c + d*x])^n, x, 4, (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n))} -{Csc[c + d*x]^3*(a + b*Sec[c + d*x])^n, x, 9, (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)^2*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)^2*d*(1 + n))} - -{Sin[c + d*x]^4*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4, x]} -{Sin[c + d*x]^2*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2, x]} -{Csc[c + d*x]^2*(a + b*Sec[c + d*x])^n, x, 4, -((Cot[c + d*x]*(a + b*Sec[c + d*x])^n)/d) + (Sqrt[2]*b*n*AppellF1[1/2, 1/2, 1 - n, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^n*((a + b)*d*Sqrt[1 + Sec[c + d*x]]))} -{Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n, x, -1, -((3*AppellF1[-(1/2), 5/2, -n, 1/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*Cot[c + d*x]*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^n)/(((a + b*Sec[c + d*x])/(a + b))^n*(2*Sqrt[2]*d))) - (AppellF1[-(3/2), 5/2, -n, -(1/2), (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*Cot[c + d*x]^3*(1 + Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^n)/(((a + b*Sec[c + d*x])/(a + b))^n*(6*Sqrt[2]*d)) + (AppellF1[1/2, 3/2, -n, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^n*(Sqrt[2]*d*Sqrt[1 + Sec[c + d*x]])) + (AppellF1[1/2, 5/2, -n, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^n*(2*Sqrt[2]*d*Sqrt[1 + Sec[c + d*x]]))} - - -{Sin[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^(3/2), x]} -{Sin[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]], x]} -{1/Sin[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]], x]} -{1/Sin[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2), x]} - - -(* ::Title:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Sec[e+f x])^m when a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e Csc[e+f x])^(m/2) (a+a Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x]), x, 11, -((2*a*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d)) - (2*a*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) + (a*e^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*e^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d)} -{(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x]), x, 11, -((2*a*e*Sqrt[e*Csc[c + d*x]])/d) - (2*a*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (a*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (2*a*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/d} -{Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x]), x, 9, (a*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (a*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/d} -{(a + a*Sec[c + d*x])/Sqrt[e*Csc[c + d*x]], x, 9, -((a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(3/2), x, 11, -((2*a)/(d*e*Sqrt[e*Csc[c + d*x]])) - (2*a*Cos[c + d*x])/(3*d*e*Sqrt[e*Csc[c + d*x]]) + (a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(3*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(5/2), x, 11, -((a*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (a*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (6*a*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (2*a*Sin[c + d*x])/(3*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x])/(5*d*e^2*Sqrt[e*Csc[c + d*x]])} - - -{(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2, x, 15, -((2*a^2*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d)) - (4*a^2*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*d) - (2*a^2*e^2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]]*Sec[c + d*x])/(3*d) + (2*a^2*e^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*e^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (7*a^2*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*d) + (5*a^2*e^2*Sqrt[e*Csc[c + d*x]]*Tan[c + d*x])/(3*d)} -{(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2, x, 15, -((4*a^2*e*Sqrt[e*Csc[c + d*x]])/d) - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Sqrt[e*Csc[c + d*x]]*Sec[c + d*x])/d - (2*a^2*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (5*a^2*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/d + (3*a^2*e*Sqrt[e*Csc[c + d*x]]*Sin[c + d*x]*Tan[c + d*x])/d} -{Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2, x, 12, (2*a^2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (3*a^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/d + (a^2*Sqrt[e*Csc[c + d*x]]*Tan[c + d*x])/d} -{(a + a*Sec[c + d*x])^2/Sqrt[e*Csc[c + d*x]], x, 12, -((2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (a^2*Tan[c + d*x])/(d*Sqrt[e*Csc[c + d*x]])} -{(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2), x, 14, -((4*a^2)/(d*e*Sqrt[e*Csc[c + d*x]])) - (2*a^2*Cos[c + d*x])/(3*d*e*Sqrt[e*Csc[c + d*x]]) + (a^2*Sec[c + d*x])/(d*e*Sqrt[e*Csc[c + d*x]]) + (2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (a^2*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(3*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(5/2), x, 14, -((2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (9*a^2*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (4*a^2*Sin[c + d*x])/(3*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*a^2*Cos[c + d*x]*Sin[c + d*x])/(5*d*e^2*Sqrt[e*Csc[c + d*x]]) + (a^2*Tan[c + d*x])/(d*e^2*Sqrt[e*Csc[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x]), x, 8, -((4*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a*d)) + (2*e^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a*d) - (2*e^2*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a*d) + (4*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a*d)} -{(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x]), x, 8, -((4*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/(5*a*d)) + (2*e*Cot[c + d*x]*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(5*a*d) - (2*e*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(5*a*d) - (4*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(5*a*d)} -{Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x]), x, 7, (2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) - (2*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a*d) + (4*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(3*a*d)} -{1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])), x, 7, (2*Cot[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) - (2*Csc[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) + (4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(a*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])), x, 7, 2/(a*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x])/(3*a*d*e*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(3*a*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])), x, 7, -((4*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(5*a*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])) + (2*Sin[c + d*x])/(3*a*d*e^2*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*e^2*Sqrt[e*Csc[c + d*x]])} -{1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])), x, 8, -((2*Cos[c + d*x])/(21*a*d*e^3*Sqrt[e*Csc[c + d*x]])) + (2*Cos[c + d*x]^3)/(7*a*d*e^3*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(21*a*d*e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (2*Sin[c + d*x]^2)/(5*a*d*e^3*Sqrt[e*Csc[c + d*x]])} - - -{(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2, x, 16, -((4*e^2*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(231*a^2*d)) + (16*e^2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(77*a^2*d) - (2*e^2*Cot[c + d*x]^3*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) - (4*e^2*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (2*e^2*Cot[c + d*x]*Csc[c + d*x]^4*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) + (4*e^2*Csc[c + d*x]^5*Sqrt[e*Csc[c + d*x]])/(11*a^2*d) + (4*e^2*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(231*a^2*d)} -{(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2, x, 16, -((4*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/(15*a^2*d)) + (16*e*Cot[c + d*x]*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(45*a^2*d) - (2*e*Cot[c + d*x]^3*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) - (4*e*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(5*a^2*d) - (2*e*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) + (4*e*Csc[c + d*x]^4*Sqrt[e*Csc[c + d*x]])/(9*a^2*d) - (4*e*Sqrt[e*Csc[c + d*x]]*EllipticE[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(15*a^2*d)} -{Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x])^2, x, 14, (16*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a^2*d) - (2*Cot[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (4*Csc[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a^2*d) - (2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (20*Sqrt[e*Csc[c + d*x]]*EllipticF[(1/2)*(c - Pi/2 + d*x), 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d)} -{1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 14, (16*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (4*Csc[c + d*x])/(a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]*Csc[c + d*x]^2)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (28*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(5*a^2*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2), x, 13, 4/(a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (4*Cos[c + d*x])/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cos[c + d*x]*Cot[c + d*x]^2)/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x]^2)/(3*a^2*d*e*Sqrt[e*Csc[c + d*x]]) - (4*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(a^2*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])} -{1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2), x, 13, -((2*Cot[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]])) - (2*Cos[c + d*x]^2*Cot[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) + (4*Csc[c + d*x])/(a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) - (44*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/(5*a^2*d*e^2*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (4*Sin[c + d*x])/(3*a^2*d*e^2*Sqrt[e*Csc[c + d*x]]) - (12*Cos[c + d*x]*Sin[c + d*x])/(5*a^2*d*e^2*Sqrt[e*Csc[c + d*x]])} -{1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2), x, 13, -(4/(a^2*d*e^3*Sqrt[e*Csc[c + d*x]])) + (26*Cos[c + d*x])/(21*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]) + (2*Cos[c + d*x]^3)/(7*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]) + (52*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(21*a^2*d*e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) + (4*Sin[c + d*x]^2)/(5*a^2*d*e^3*Sqrt[e*Csc[c + d*x]])} - - -(* ::Section:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Sec[e+f x])^m*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m deleted file mode 100644 index f3a8bc0..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.1.4 (d tan)^n (a+b sec)^m.m +++ /dev/null @@ -1,590 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Sec[e+f x])^m*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+a Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^n (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tan[c + d*x]^9*(a + a*Sec[c + d*x]), x, 3, -((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d - (2*a*Sec[c + d*x]^2)/d - (4*a*Sec[c + d*x]^3)/(3*d) + (3*a*Sec[c + d*x]^4)/(2*d) + (6*a*Sec[c + d*x]^5)/(5*d) - (2*a*Sec[c + d*x]^6)/(3*d) - (4*a*Sec[c + d*x]^7)/(7*d) + (a*Sec[c + d*x]^8)/(8*d) + (a*Sec[c + d*x]^9)/(9*d)} -{Tan[c + d*x]^7*(a + a*Sec[c + d*x]), x, 3, (a*Log[Cos[c + d*x]])/d - (a*Sec[c + d*x])/d + (3*a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c + d*x]^4)/(4*d) - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^6)/(6*d) + (a*Sec[c + d*x]^7)/(7*d)} -{Tan[c + d*x]^5*(a + a*Sec[c + d*x]), x, 3, -((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d - (a*Sec[c + d*x]^2)/d - (2*a*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]^4)/(4*d) + (a*Sec[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^3*(a + a*Sec[c + d*x]), x, 3, (a*Log[Cos[c + d*x]])/d - (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^3)/(3*d)} -{Tan[c + d*x]^1*(a + a*Sec[c + d*x]), x, 3, -((a*Log[Cos[c + d*x]])/d) + (a*Sec[c + d*x])/d} -{Cot[c + d*x]^1*(a + a*Sec[c + d*x]), x, 2, (a*Log[1 - Cos[c + d*x]])/d} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x]), x, 3, -(a/(2*d*(1 - Cos[c + d*x]))) - (3*a*Log[1 - Cos[c + d*x]])/(4*d) - (a*Log[1 + Cos[c + d*x]])/(4*d)} -{Cot[c + d*x]^5*(a + a*Sec[c + d*x]), x, 3, -(a/(8*d*(1 - Cos[c + d*x])^2)) + (3*a)/(4*d*(1 - Cos[c + d*x])) + a/(8*d*(1 + Cos[c + d*x])) + (11*a*Log[1 - Cos[c + d*x]])/(16*d) + (5*a*Log[1 + Cos[c + d*x]])/(16*d)} -{Cot[c + d*x]^7*(a + a*Sec[c + d*x]), x, 3, -(a/(24*d*(1 - Cos[c + d*x])^3)) + (9*a)/(32*d*(1 - Cos[c + d*x])^2) - (15*a)/(16*d*(1 - Cos[c + d*x])) + a/(32*d*(1 + Cos[c + d*x])^2) - a/(4*d*(1 + Cos[c + d*x])) - (21*a*Log[1 - Cos[c + d*x]])/(32*d) - (11*a*Log[1 + Cos[c + d*x]])/(32*d)} - -{Tan[c + d*x]^8*(a + a*Sec[c + d*x]), x, 6, a*x + (35*a*ArcTanh[Sin[c + d*x]])/(128*d) - ((128*a + 35*a*Sec[c + d*x])*Tan[c + d*x])/(128*d) + ((64*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^3)/(192*d) - ((48*a + 35*a*Sec[c + d*x])*Tan[c + d*x]^5)/(240*d) + ((8*a + 7*a*Sec[c + d*x])*Tan[c + d*x]^7)/(56*d)} -{Tan[c + d*x]^6*(a + a*Sec[c + d*x]), x, 5, (-a)*x - (5*a*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*a*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*a*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)} -{Tan[c + d*x]^4*(a + a*Sec[c + d*x]), x, 4, a*x + (3*a*ArcTanh[Sin[c + d*x]])/(8*d) - ((8*a + 3*a*Sec[c + d*x])*Tan[c + d*x])/(8*d) + ((4*a + 3*a*Sec[c + d*x])*Tan[c + d*x]^3)/(12*d)} -{Tan[c + d*x]^2*(a + a*Sec[c + d*x]), x, 3, (-a)*x - (a*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x]), x, 2, (-a)*x - (Cot[c + d*x]*(a + a*Sec[c + d*x]))/d} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x]), x, 3, a*x - (Cot[c + d*x]^3*(a + a*Sec[c + d*x]))/(3*d) + (Cot[c + d*x]*(3*a + 2*a*Sec[c + d*x]))/(3*d)} -{Cot[c + d*x]^6*(a + a*Sec[c + d*x]), x, 4, (-a)*x - (Cot[c + d*x]^5*(a + a*Sec[c + d*x]))/(5*d) + (Cot[c + d*x]^3*(5*a + 4*a*Sec[c + d*x]))/(15*d) - (Cot[c + d*x]*(15*a + 8*a*Sec[c + d*x]))/(15*d)} -{Cot[c + d*x]^8*(a + a*Sec[c + d*x]), x, 5, a*x - (Cot[c + d*x]^7*(a + a*Sec[c + d*x]))/(7*d) + (Cot[c + d*x]^5*(7*a + 6*a*Sec[c + d*x]))/(35*d) + (Cot[c + d*x]*(35*a + 16*a*Sec[c + d*x]))/(35*d) - (Cot[c + d*x]^3*(35*a + 24*a*Sec[c + d*x]))/(105*d)} -{Cot[c + d*x]^10*(a + a*Sec[c + d*x]), x, 6, (-a)*x - (Cot[c + d*x]^9*(a + a*Sec[c + d*x]))/(9*d) + (Cot[c + d*x]^7*(9*a + 8*a*Sec[c + d*x]))/(63*d) - (Cot[c + d*x]^5*(21*a + 16*a*Sec[c + d*x]))/(105*d) + (Cot[c + d*x]^3*(105*a + 64*a*Sec[c + d*x]))/(315*d) - (Cot[c + d*x]*(315*a + 128*a*Sec[c + d*x]))/(315*d)} - - -{Tan[c + d*x]^9*(a + a*Sec[c + d*x])^2, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d - (3*a^2*Sec[c + d*x]^2)/(2*d) - (8*a^2*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^4)/(2*d) + (12*a^2*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^6)/(3*d) - (8*a^2*Sec[c + d*x]^7)/(7*d) - (3*a^2*Sec[c + d*x]^8)/(8*d) + (2*a^2*Sec[c + d*x]^9)/(9*d) + (a^2*Sec[c + d*x]^10)/(10*d)} -{Tan[c + d*x]^7*(a + a*Sec[c + d*x])^2, x, 3, (a^2*Log[Cos[c + d*x]])/d - (2*a^2*Sec[c + d*x])/d + (a^2*Sec[c + d*x]^2)/d + (2*a^2*Sec[c + d*x]^3)/d - (6*a^2*Sec[c + d*x]^5)/(5*d) - (a^2*Sec[c + d*x]^6)/(3*d) + (2*a^2*Sec[c + d*x]^7)/(7*d) + (a^2*Sec[c + d*x]^8)/(8*d)} -{Tan[c + d*x]^5*(a + a*Sec[c + d*x])^2, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d - (a^2*Sec[c + d*x]^2)/(2*d) - (4*a^2*Sec[c + d*x]^3)/(3*d) - (a^2*Sec[c + d*x]^4)/(4*d) + (2*a^2*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^6)/(6*d)} -{Tan[c + d*x]^3*(a + a*Sec[c + d*x])^2, x, 3, (a^2*Log[Cos[c + d*x]])/d - (2*a^2*Sec[c + d*x])/d + (2*a^2*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^4)/(4*d)} -{Tan[c + d*x]^1*(a + a*Sec[c + d*x])^2, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a^2*Sec[c + d*x])/d + (a^2*Sec[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^1*(a + a*Sec[c + d*x])^2, x, 3, (2*a^2*Log[1 - Cos[c + d*x]])/d - (a^2*Log[Cos[c + d*x]])/d} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x])^2, x, 3, -(a^2/(d*(1 - Cos[c + d*x]))) - (a^2*Log[1 - Cos[c + d*x]])/d} -{Cot[c + d*x]^5*(a + a*Sec[c + d*x])^2, x, 3, -(a^2/(4*d*(1 - Cos[c + d*x])^2)) + (5*a^2)/(4*d*(1 - Cos[c + d*x])) + (7*a^2*Log[1 - Cos[c + d*x]])/(8*d) + (a^2*Log[1 + Cos[c + d*x]])/(8*d)} -{Cot[c + d*x]^7*(a + a*Sec[c + d*x])^2, x, 3, -(a^2/(12*d*(1 - Cos[c + d*x])^3)) + a^2/(2*d*(1 - Cos[c + d*x])^2) - (23*a^2)/(16*d*(1 - Cos[c + d*x])) - a^2/(16*d*(1 + Cos[c + d*x])) - (13*a^2*Log[1 - Cos[c + d*x]])/(16*d) - (3*a^2*Log[1 + Cos[c + d*x]])/(16*d)} -{Cot[c + d*x]^9*(a + a*Sec[c + d*x])^2, x, 3, -(a^2/(32*d*(1 - Cos[c + d*x])^4)) + (11*a^2)/(48*d*(1 - Cos[c + d*x])^3) - (3*a^2)/(4*d*(1 - Cos[c + d*x])^2) + (51*a^2)/(32*d*(1 - Cos[c + d*x])) - a^2/(64*d*(1 + Cos[c + d*x])^2) + (9*a^2)/(64*d*(1 + Cos[c + d*x])) + (99*a^2*Log[1 - Cos[c + d*x]])/(128*d) + (29*a^2*Log[1 + Cos[c + d*x]])/(128*d)} - -{Tan[c + d*x]^6*(a + a*Sec[c + d*x])^2, x, 12, (-a^2)*x - (5*a^2*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*Tan[c + d*x])/d + (5*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a^2*Tan[c + d*x]^3)/(3*d) - (5*a^2*Sec[c + d*x]*Tan[c + d*x]^3)/(12*d) + (a^2*Tan[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]*Tan[c + d*x]^5)/(3*d) + (a^2*Tan[c + d*x]^7)/(7*d)} -{Tan[c + d*x]^4*(a + a*Sec[c + d*x])^2, x, 10, a^2*x + (3*a^2*ArcTanh[Sin[c + d*x]])/(4*d) - (a^2*Tan[c + d*x])/d - (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x]^3)/(2*d) + (a^2*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^2*(a + a*Sec[c + d*x])^2, x, 8, (-a^2)*x - (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x])^2, x, 8, (-a^2)*x - (2*a^2*Cot[c + d*x])/d - (2*a^2*Csc[c + d*x])/d} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x])^2, x, 9, a^2*x + (a^2*Cot[c + d*x])/d - (2*a^2*Cot[c + d*x]^3)/(3*d) + (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + a*Sec[c + d*x])^2, x, 11, (-a^2)*x - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x])/d + (4*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x]^5)/(5*d)} -{Cot[c + d*x]^8*(a + a*Sec[c + d*x])^2, x, 12, a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) + (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/d + (6*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d)} -{Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2, x, 13, (-a^2)*x - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) + (a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x]^3)/(3*d) - (12*a^2*Csc[c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d)} - - -{Tan[c + d*x]^9*(a + a*Sec[c + d*x])^3, x, 3, -((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/(2*d) - (11*a^3*Sec[c + d*x]^3)/(3*d) - (3*a^3*Sec[c + d*x]^4)/(2*d) + (14*a^3*Sec[c + d*x]^5)/(5*d) + (7*a^3*Sec[c + d*x]^6)/(3*d) - (6*a^3*Sec[c + d*x]^7)/(7*d) - (11*a^3*Sec[c + d*x]^8)/(8*d) - (a^3*Sec[c + d*x]^9)/(9*d) + (3*a^3*Sec[c + d*x]^10)/(10*d) + (a^3*Sec[c + d*x]^11)/(11*d)} -{Tan[c + d*x]^7*(a + a*Sec[c + d*x])^3, x, 3, (a^3*Log[Cos[c + d*x]])/d - (3*a^3*Sec[c + d*x])/d + (8*a^3*Sec[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]^4)/(2*d) - (6*a^3*Sec[c + d*x]^5)/(5*d) - (4*a^3*Sec[c + d*x]^6)/(3*d) + (3*a^3*Sec[c + d*x]^8)/(8*d) + (a^3*Sec[c + d*x]^9)/(9*d)} -{Tan[c + d*x]^5*(a + a*Sec[c + d*x])^3, x, 3, -((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d) - (5*a^3*Sec[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]^6)/(2*d) + (a^3*Sec[c + d*x]^7)/(7*d)} -{Tan[c + d*x]^3*(a + a*Sec[c + d*x])^3, x, 3, (a^3*Log[Cos[c + d*x]])/d - (3*a^3*Sec[c + d*x])/d - (a^3*Sec[c + d*x]^2)/d + (2*a^3*Sec[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]^4)/(4*d) + (a^3*Sec[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^1*(a + a*Sec[c + d*x])^3, x, 3, -((a^3*Log[Cos[c + d*x]])/d) + (3*a^3*Sec[c + d*x])/d + (3*a^3*Sec[c + d*x]^2)/(2*d) + (a^3*Sec[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^1*(a + a*Sec[c + d*x])^3, x, 3, (4*a^3*Log[1 - Cos[c + d*x]])/d - (3*a^3*Log[Cos[c + d*x]])/d + (a^3*Sec[c + d*x])/d} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x])^3, x, 3, -((2*a^3)/(d*(1 - Cos[c + d*x]))) - (a^3*Log[1 - Cos[c + d*x]])/d} -{Cot[c + d*x]^5*(a + a*Sec[c + d*x])^3, x, 3, -(a^3/(2*d*(1 - Cos[c + d*x])^2)) + (2*a^3)/(d*(1 - Cos[c + d*x])) + (a^3*Log[1 - Cos[c + d*x]])/d} -{Cot[c + d*x]^7*(a + a*Sec[c + d*x])^3, x, 3, -(a^3/(6*d*(1 - Cos[c + d*x])^3)) + (7*a^3)/(8*d*(1 - Cos[c + d*x])^2) - (17*a^3)/(8*d*(1 - Cos[c + d*x])) - (15*a^3*Log[1 - Cos[c + d*x]])/(16*d) - (a^3*Log[1 + Cos[c + d*x]])/(16*d)} -{Cot[c + d*x]^9*(a + a*Sec[c + d*x])^3, x, 3, -(a^3/(16*d*(1 - Cos[c + d*x])^4)) + (5*a^3)/(12*d*(1 - Cos[c + d*x])^3) - (39*a^3)/(32*d*(1 - Cos[c + d*x])^2) + (9*a^3)/(4*d*(1 - Cos[c + d*x])) + a^3/(32*d*(1 + Cos[c + d*x])) + (57*a^3*Log[1 - Cos[c + d*x]])/(64*d) + (7*a^3*Log[1 + Cos[c + d*x]])/(64*d)} - -{Tan[c + d*x]^6*(a + a*Sec[c + d*x])^3, x, 17, (-a^3)*x - (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) + (a^3*Tan[c + d*x])/d + (115*a^3*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) - (a^3*Tan[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]*Tan[c + d*x]^3)/(8*d) - (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*d) + (a^3*Tan[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]*Tan[c + d*x]^5)/(2*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*d) + (3*a^3*Tan[c + d*x]^7)/(7*d)} -{Tan[c + d*x]^4*(a + a*Sec[c + d*x])^3, x, 14, a^3*x + (19*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (a^3*Tan[c + d*x])/d - (17*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (a^3*Tan[c + d*x]^3)/(3*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d) + (3*a^3*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^2*(a + a*Sec[c + d*x])^3, x, 11, (-a^3)*x - (13*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*Tan[c + d*x])/d + (11*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x])^3, x, 11, (-a^3)*x + (a^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*Cot[c + d*x])/d - (4*a^3*Csc[c + d*x])/d} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x])^3, x, 11, a^3*x + (a^3*Cot[c + d*x])/d - (4*a^3*Cot[c + d*x]^3)/(3*d) + (3*a^3*Csc[c + d*x])/d - (4*a^3*Csc[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + a*Sec[c + d*x])^3, x, 14, (-a^3)*x - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (3*a^3*Csc[c + d*x])/d + (7*a^3*Csc[c + d*x]^3)/(3*d) - (4*a^3*Csc[c + d*x]^5)/(5*d)} -{Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3, x, 15, a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (4*a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10*a^3*Csc[c + d*x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*Csc[c + d*x]^7)/(7*d)} -{Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3, x, 16, (-a^3)*x - (a^3*Cot[c + d*x])/d + (a^3*Cot[c + d*x]^3)/(3*d) - (a^3*Cot[c + d*x]^5)/(5*d) + (a^3*Cot[c + d*x]^7)/(7*d) - (4*a^3*Cot[c + d*x]^9)/(9*d) - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x]^3)/(3*d) - (21*a^3*Csc[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/(9*d)} -{Cot[c + d*x]^12*(a + a*Sec[c + d*x])^3, x, 17, a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) + (3*a^3*Csc[c + d*x])/d - (16*a^3*Csc[c + d*x]^3)/(3*d) + (34*a^3*Csc[c + d*x]^5)/(5*d) - (36*a^3*Csc[c + d*x]^7)/(7*d) + (19*a^3*Csc[c + d*x]^9)/(9*d) - (4*a^3*Csc[c + d*x]^11)/(11*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tan[c + d*x]^9/(a + a*Sec[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(a*d)) - Sec[c + d*x]/(a*d) - (3*Sec[c + d*x]^2)/(2*a*d) + Sec[c + d*x]^3/(a*d) + (3*Sec[c + d*x]^4)/(4*a*d) - (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^7/(7*a*d)} -{Tan[c + d*x]^7/(a + a*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d) + Sec[c + d*x]^2/(a*d) - (2*Sec[c + d*x]^3)/(3*a*d) - Sec[c + d*x]^4/(4*a*d) + Sec[c + d*x]^5/(5*a*d)} -{Tan[c + d*x]^5/(a + a*Sec[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(a*d)) - Sec[c + d*x]/(a*d) - Sec[c + d*x]^2/(2*a*d) + Sec[c + d*x]^3/(3*a*d)} -{Tan[c + d*x]^3/(a + a*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) + Sec[c + d*x]/(a*d)} -{Tan[c + d*x]^1/(a + a*Sec[c + d*x]), x, 2, -(Log[1 + Cos[c + d*x]]/(a*d))} -{Cot[c + d*x]^1/(a + a*Sec[c + d*x]), x, 3, 1/(2*a*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(4*a*d) + (3*Log[1 + Cos[c + d*x]])/(4*a*d)} -{Cot[c + d*x]^3/(a + a*Sec[c + d*x]), x, 3, -(1/(8*a*d*(1 - Cos[c + d*x]))) + 1/(8*a*d*(1 + Cos[c + d*x])^2) - 3/(4*a*d*(1 + Cos[c + d*x])) - (5*Log[1 - Cos[c + d*x]])/(16*a*d) - (11*Log[1 + Cos[c + d*x]])/(16*a*d)} -{Cot[c + d*x]^5/(a + a*Sec[c + d*x]), x, 3, -(1/(32*a*d*(1 - Cos[c + d*x])^2)) + 1/(4*a*d*(1 - Cos[c + d*x])) + 1/(24*a*d*(1 + Cos[c + d*x])^3) - 9/(32*a*d*(1 + Cos[c + d*x])^2) + 15/(16*a*d*(1 + Cos[c + d*x])) + (11*Log[1 - Cos[c + d*x]])/(32*a*d) + (21*Log[1 + Cos[c + d*x]])/(32*a*d)} - -{Tan[c + d*x]^8/(a + a*Sec[c + d*x]), x, 6, x/a - (5*ArcTanh[Sin[c + d*x]])/(16*a*d) - ((16 - 5*Sec[c + d*x])*Tan[c + d*x])/(16*a*d) + ((8 - 5*Sec[c + d*x])*Tan[c + d*x]^3)/(24*a*d) - ((6 - 5*Sec[c + d*x])*Tan[c + d*x]^5)/(30*a*d)} -{Tan[c + d*x]^6/(a + a*Sec[c + d*x]), x, 5, -(x/a) + (3*ArcTanh[Sin[c + d*x]])/(8*a*d) + ((8 - 3*Sec[c + d*x])*Tan[c + d*x])/(8*a*d) - ((4 - 3*Sec[c + d*x])*Tan[c + d*x]^3)/(12*a*d)} -{Tan[c + d*x]^4/(a + a*Sec[c + d*x]), x, 4, x/a - ArcTanh[Sin[c + d*x]]/(2*a*d) - ((2 - Sec[c + d*x])*Tan[c + d*x])/(2*a*d)} -{Tan[c + d*x]^2/(a + a*Sec[c + d*x]), x, 3, -(x/a) + ArcTanh[Sin[c + d*x]]/(a*d)} -{Cot[c + d*x]^2/(a + a*Sec[c + d*x]), x, 4, -(x/a) - (Cot[c + d*x]*(3 - 2*Sec[c + d*x]))/(3*a*d) + (Cot[c + d*x]^3*(1 - Sec[c + d*x]))/(3*a*d)} -{Cot[c + d*x]^4/(a + a*Sec[c + d*x]), x, 5, x/a + (Cot[c + d*x]*(15 - 8*Sec[c + d*x]))/(15*a*d) - (Cot[c + d*x]^3*(5 - 4*Sec[c + d*x]))/(15*a*d) + (Cot[c + d*x]^5*(1 - Sec[c + d*x]))/(5*a*d)} -{Cot[c + d*x]^6/(a + a*Sec[c + d*x]), x, 6, -(x/a) + (Cot[c + d*x]^3*(35 - 24*Sec[c + d*x]))/(105*a*d) - (Cot[c + d*x]*(35 - 16*Sec[c + d*x]))/(35*a*d) - (Cot[c + d*x]^5*(7 - 6*Sec[c + d*x]))/(35*a*d) + (Cot[c + d*x]^7*(1 - Sec[c + d*x]))/(7*a*d)} - - -{Tan[c + d*x]^9/(a + a*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) - Sec[c + d*x]^2/(2*a^2*d) + (4*Sec[c + d*x]^3)/(3*a^2*d) - Sec[c + d*x]^4/(4*a^2*d) - (2*Sec[c + d*x]^5)/(5*a^2*d) + Sec[c + d*x]^6/(6*a^2*d)} -{Tan[c + d*x]^7/(a + a*Sec[c + d*x])^2, x, 3, Log[Cos[c + d*x]]/(a^2*d) + (2*Sec[c + d*x])/(a^2*d) - (2*Sec[c + d*x]^3)/(3*a^2*d) + Sec[c + d*x]^4/(4*a^2*d)} -{Tan[c + d*x]^5/(a + a*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^2/(2*a^2*d)} -{Tan[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) + (2*Log[1 + Cos[c + d*x]])/(a^2*d)} -{Tan[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 3, -(1/(a^2*d*(1 + Cos[c + d*x]))) - Log[1 + Cos[c + d*x]]/(a^2*d)} -{Cot[c + d*x]^1/(a + a*Sec[c + d*x])^2, x, 3, -(1/(4*a^2*d*(1 + Cos[c + d*x])^2)) + 5/(4*a^2*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(8*a^2*d) + (7*Log[1 + Cos[c + d*x]])/(8*a^2*d)} -{Cot[c + d*x]^3/(a + a*Sec[c + d*x])^2, x, 3, -(1/(16*a^2*d*(1 - Cos[c + d*x]))) - 1/(12*a^2*d*(1 + Cos[c + d*x])^3) + 1/(2*a^2*d*(1 + Cos[c + d*x])^2) - 23/(16*a^2*d*(1 + Cos[c + d*x])) - (3*Log[1 - Cos[c + d*x]])/(16*a^2*d) - (13*Log[1 + Cos[c + d*x]])/(16*a^2*d)} -{Cot[c + d*x]^5/(a + a*Sec[c + d*x])^2, x, 3, -(1/(64*a^2*d*(1 - Cos[c + d*x])^2)) + 9/(64*a^2*d*(1 - Cos[c + d*x])) - 1/(32*a^2*d*(1 + Cos[c + d*x])^4) + 11/(48*a^2*d*(1 + Cos[c + d*x])^3) - 3/(4*a^2*d*(1 + Cos[c + d*x])^2) + 51/(32*a^2*d*(1 + Cos[c + d*x])) + (29*Log[1 - Cos[c + d*x]])/(128*a^2*d) + (99*Log[1 + Cos[c + d*x]])/(128*a^2*d)} - -{Tan[c + d*x]^8/(a + a*Sec[c + d*x])^2, x, 11, x/a^2 - (3*ArcTanh[Sin[c + d*x]])/(4*a^2*d) - Tan[c + d*x]/(a^2*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(4*a^2*d) + Tan[c + d*x]^3/(3*a^2*d) - (Sec[c + d*x]*Tan[c + d*x]^3)/(2*a^2*d) + Tan[c + d*x]^5/(5*a^2*d)} -{Tan[c + d*x]^6/(a + a*Sec[c + d*x])^2, x, 9, -(x/a^2) + ArcTanh[Sin[c + d*x]]/(a^2*d) + Tan[c + d*x]/(a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(a^2*d) + Tan[c + d*x]^3/(3*a^2*d)} -{Tan[c + d*x]^4/(a + a*Sec[c + d*x])^2, x, 5, x/a^2 - (2*ArcTanh[Sin[c + d*x]])/(a^2*d) + Tan[c + d*x]/(a^2*d)} -{Tan[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 9, -(x/a^2) + (2*Tan[c + d*x])/(a*d*(a + a*Sec[c + d*x])), -(x/a^2) - (2*Cot[c + d*x])/(a^2*d) + (2*Csc[c + d*x])/(a^2*d)} -{Cot[c + d*x]^2/(a + a*Sec[c + d*x])^2, x, 12, -(x/a^2) - Cot[c + d*x]/(a^2*d) + Cot[c + d*x]^3/(3*a^2*d) - (2*Cot[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x])/(a^2*d) - (4*Csc[c + d*x]^3)/(3*a^2*d) + (2*Csc[c + d*x]^5)/(5*a^2*d)} -{Cot[c + d*x]^4/(a + a*Sec[c + d*x])^2, x, 13, x/a^2 + Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d) + Cot[c + d*x]^5/(5*a^2*d) - (2*Cot[c + d*x]^7)/(7*a^2*d) - (2*Csc[c + d*x])/(a^2*d) + (2*Csc[c + d*x]^3)/(a^2*d) - (6*Csc[c + d*x]^5)/(5*a^2*d) + (2*Csc[c + d*x]^7)/(7*a^2*d)} -{Cot[c + d*x]^6/(a + a*Sec[c + d*x])^2, x, 14, -(x/a^2) - Cot[c + d*x]/(a^2*d) + Cot[c + d*x]^3/(3*a^2*d) - Cot[c + d*x]^5/(5*a^2*d) + Cot[c + d*x]^7/(7*a^2*d) - (2*Cot[c + d*x]^9)/(9*a^2*d) + (2*Csc[c + d*x])/(a^2*d) - (8*Csc[c + d*x]^3)/(3*a^2*d) + (12*Csc[c + d*x]^5)/(5*a^2*d) - (8*Csc[c + d*x]^7)/(7*a^2*d) + (2*Csc[c + d*x]^9)/(9*a^2*d)} - - -{Tan[c + d*x]^11/(a + a*Sec[c + d*x])^3, x, 3, Log[Cos[c + d*x]]/(a^3*d) + (3*Sec[c + d*x])/(a^3*d) - Sec[c + d*x]^2/(2*a^3*d) - (5*Sec[c + d*x]^3)/(3*a^3*d) + (5*Sec[c + d*x]^4)/(4*a^3*d) + Sec[c + d*x]^5/(5*a^3*d) - Sec[c + d*x]^6/(2*a^3*d) + Sec[c + d*x]^7/(7*a^3*d)} -{Tan[c + d*x]^9/(a + a*Sec[c + d*x])^3, x, 3, -(Log[Cos[c + d*x]]/(a^3*d)) - (3*Sec[c + d*x])/(a^3*d) + Sec[c + d*x]^2/(a^3*d) + (2*Sec[c + d*x]^3)/(3*a^3*d) - (3*Sec[c + d*x]^4)/(4*a^3*d) + Sec[c + d*x]^5/(5*a^3*d)} -{Tan[c + d*x]^7/(a + a*Sec[c + d*x])^3, x, 3, Log[Cos[c + d*x]]/(a^3*d) + (3*Sec[c + d*x])/(a^3*d) - (3*Sec[c + d*x]^2)/(2*a^3*d) + Sec[c + d*x]^3/(3*a^3*d)} -{Tan[c + d*x]^5/(a + a*Sec[c + d*x])^3, x, 3, (3*Log[Cos[c + d*x]])/(a^3*d) - (4*Log[1 + Cos[c + d*x]])/(a^3*d) + Sec[c + d*x]/(a^3*d)} -{Tan[c + d*x]^3/(a + a*Sec[c + d*x])^3, x, 3, 2/(a^3*d*(1 + Cos[c + d*x])) + Log[1 + Cos[c + d*x]]/(a^3*d)} -{Tan[c + d*x]^1/(a + a*Sec[c + d*x])^3, x, 3, 1/(2*a^3*d*(1 + Cos[c + d*x])^2) - 2/(a^3*d*(1 + Cos[c + d*x])) - Log[1 + Cos[c + d*x]]/(a^3*d)} -{Cot[c + d*x]^1/(a + a*Sec[c + d*x])^3, x, 3, 1/(6*a^3*d*(1 + Cos[c + d*x])^3) - 7/(8*a^3*d*(1 + Cos[c + d*x])^2) + 17/(8*a^3*d*(1 + Cos[c + d*x])) + Log[1 - Cos[c + d*x]]/(16*a^3*d) + (15*Log[1 + Cos[c + d*x]])/(16*a^3*d)} -{Cot[c + d*x]^3/(a + a*Sec[c + d*x])^3, x, 3, -(1/(32*a^3*d*(1 - Cos[c + d*x]))) + 1/(16*a^3*d*(1 + Cos[c + d*x])^4) - 5/(12*a^3*d*(1 + Cos[c + d*x])^3) + 39/(32*a^3*d*(1 + Cos[c + d*x])^2) - 9/(4*a^3*d*(1 + Cos[c + d*x])) - (7*Log[1 - Cos[c + d*x]])/(64*a^3*d) - (57*Log[1 + Cos[c + d*x]])/(64*a^3*d)} -{Cot[c + d*x]^5/(a + a*Sec[c + d*x])^3, x, 3, -(1/(128*a^3*d*(1 - Cos[c + d*x])^2)) + 5/(64*a^3*d*(1 - Cos[c + d*x])) + 1/(40*a^3*d*(1 + Cos[c + d*x])^5) - 13/(64*a^3*d*(1 + Cos[c + d*x])^4) + 35/(48*a^3*d*(1 + Cos[c + d*x])^3) - 99/(64*a^3*d*(1 + Cos[c + d*x])^2) + 303/(128*a^3*d*(1 + Cos[c + d*x])) + (37*Log[1 - Cos[c + d*x]])/(256*a^3*d) + (219*Log[1 + Cos[c + d*x]])/(256*a^3*d)} - -{Tan[c + d*x]^12/(a + a*Sec[c + d*x])^3, x, 18, x/a^3 - (125*ArcTanh[Sin[c + d*x]])/(128*a^3*d) - Tan[c + d*x]/(a^3*d) + (115*Sec[c + d*x]*Tan[c + d*x])/(128*a^3*d) + (5*Sec[c + d*x]^3*Tan[c + d*x])/(64*a^3*d) + Tan[c + d*x]^3/(3*a^3*d) - (5*Sec[c + d*x]*Tan[c + d*x]^3)/(8*a^3*d) - (5*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*a^3*d) - Tan[c + d*x]^5/(5*a^3*d) + (Sec[c + d*x]*Tan[c + d*x]^5)/(2*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*a^3*d) - (3*Tan[c + d*x]^7)/(7*a^3*d)} -{Tan[c + d*x]^10/(a + a*Sec[c + d*x])^3, x, 15, -(x/a^3) + (19*ArcTanh[Sin[c + d*x]])/(16*a^3*d) + Tan[c + d*x]/(a^3*d) - (17*Sec[c + d*x]*Tan[c + d*x])/(16*a^3*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(8*a^3*d) - Tan[c + d*x]^3/(3*a^3*d) + (3*Sec[c + d*x]*Tan[c + d*x]^3)/(4*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*a^3*d) - (3*Tan[c + d*x]^5)/(5*a^3*d)} -{Tan[c + d*x]^8/(a + a*Sec[c + d*x])^3, x, 12, x/a^3 - (13*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - Tan[c + d*x]/(a^3*d) + (11*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d) - Tan[c + d*x]^3/(a^3*d)} -{Tan[c + d*x]^6/(a + a*Sec[c + d*x])^3, x, 6, -(x/a^3) + (7*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (5*Tan[c + d*x])/(2*a^3*d) - ((1 - Sec[c + d*x])*Tan[c + d*x])/(2*a^3*d)} -{Tan[c + d*x]^4/(a + a*Sec[c + d*x])^3, x, 12, x/a^3 + ArcTanh[Sin[c + d*x]]/(a^3*d) - (4*Tan[c + d*x])/(a^2*d*(a + a*Sec[c + d*x])), x/a^3 + ArcTanh[Sin[c + d*x]]/(a^3*d) + (4*Cot[c + d*x])/(a^3*d) - (4*Csc[c + d*x])/(a^3*d)} -{Tan[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 12, -(x/a^3) + (2*Tan[c + d*x])/(a^2*d*(a + a*Sec[c + d*x])) - Tan[c + d*x]^3/(3*d*(a + a*Sec[c + d*x])^3), -(x/a^3) - Cot[c + d*x]/(a^3*d) + (4*Cot[c + d*x]^3)/(3*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (4*Csc[c + d*x]^3)/(3*a^3*d)} -{Cot[c + d*x]^2/(a + a*Sec[c + d*x])^3, x, 16, -(x/a^3) - Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + (4*Cot[c + d*x]^7)/(7*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (10*Csc[c + d*x]^3)/(3*a^3*d) + (11*Csc[c + d*x]^5)/(5*a^3*d) - (4*Csc[c + d*x]^7)/(7*a^3*d)} -{Cot[c + d*x]^4/(a + a*Sec[c + d*x])^3, x, 17, x/a^3 + Cot[c + d*x]/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + Cot[c + d*x]^5/(5*a^3*d) - Cot[c + d*x]^7/(7*a^3*d) + (4*Cot[c + d*x]^9)/(9*a^3*d) - (3*Csc[c + d*x])/(a^3*d) + (13*Csc[c + d*x]^3)/(3*a^3*d) - (21*Csc[c + d*x]^5)/(5*a^3*d) + (15*Csc[c + d*x]^7)/(7*a^3*d) - (4*Csc[c + d*x]^9)/(9*a^3*d)} -{Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3, x, 18, -(x/a^3) - Cot[c + d*x]/(a^3*d) + Cot[c + d*x]^3/(3*a^3*d) - Cot[c + d*x]^5/(5*a^3*d) + Cot[c + d*x]^7/(7*a^3*d) - Cot[c + d*x]^9/(9*a^3*d) + (4*Cot[c + d*x]^11)/(11*a^3*d) + (3*Csc[c + d*x])/(a^3*d) - (16*Csc[c + d*x]^3)/(3*a^3*d) + (34*Csc[c + d*x]^5)/(5*a^3*d) - (36*Csc[c + d*x]^7)/(7*a^3*d) + (19*Csc[c + d*x]^9)/(9*a^3*d) - (4*Csc[c + d*x]^11)/(11*a^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^(n/2) (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2), x, 17, (a*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (6*a*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Sin[2*c + 2*d*x]]) - (6*a*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (2*e*(5*a + 3*a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2))/(15*d)} -{(a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2), x, 16, (a*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[e*Tan[c + d*x]]) + (2*e*(3*a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(3*d)} -{(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]], x, 16, -((a*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + (a*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e)} -{(a + a*Sec[c + d*x])/Sqrt[e*Tan[c + d*x]], x, 15, -((a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e])) + (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[e*Tan[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(3/2), x, 17, (a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - (2*(a + a*Sec[c + d*x]))/(d*e*Sqrt[e*Tan[c + d*x]]) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*e^2*Sqrt[Sin[2*c + 2*d*x]]) + (2*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e^3)} -{(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(5/2), x, 16, (a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (2*(a + a*Sec[c + d*x]))/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*e^2*Sqrt[e*Tan[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Tan[c + d*x])^(7/2), x, 18, -((a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2))) + (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (2*(a + a*Sec[c + d*x]))/(5*d*e*(e*Tan[c + d*x])^(5/2)) + (2*(5*a + 3*a*Sec[c + d*x]))/(5*d*e^3*Sqrt[e*Tan[c + d*x]]) + (6*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*e^4*Sqrt[Sin[2*c + 2*d*x]]) - (6*a*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d*e^5)} - - -{(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2), x, 21, (a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (12*a^2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a^2*e*(e*Tan[c + d*x])^(3/2))/(3*d) - (12*a^2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (4*a^2*e*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (2*a^2*(e*Tan[c + d*x])^(7/2))/(7*d*e)} -{(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(3/2), x, 20, (a^2*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a^2*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (2*a^2*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*Sqrt[e*Tan[c + d*x]]) + (2*a^2*e*Sqrt[e*Tan[c + d*x]])/d + (4*a^2*e*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*d) + (2*a^2*(e*Tan[c + d*x])^(5/2))/(5*d*e)} -{(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]], x, 19, -((a^2*Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d)) + (a^2*Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) + (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (a^2*Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*a^2*(e*Tan[c + d*x])^(3/2))/(3*d*e) + (4*a^2*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e)} -{(a + a*Sec[c + d*x])^2/Sqrt[e*Tan[c + d*x]], x, 18, -((a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e])) + (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) + (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[e*Tan[c + d*x]]) + (2*a^2*Sqrt[e*Tan[c + d*x]])/(d*e)} -{(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(3/2), x, 20, (a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(3/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(3/2)) - (4*a^2)/(d*e*Sqrt[e*Tan[c + d*x]]) - (4*a^2*Cos[c + d*x])/(d*e*Sqrt[e*Tan[c + d*x]]) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d*e^2*Sqrt[Sin[2*c + 2*d*x]])} -{(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(5/2), x, 20, (a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) - (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(5/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(5/2)) - (4*a^2)/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (4*a^2*Sec[c + d*x])/(3*d*e*(e*Tan[c + d*x])^(3/2)) - (2*a^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*e^2*Sqrt[e*Tan[c + d*x]])} -{(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2), x, 22, -((a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2))) + (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*e^(7/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(7/2)) - (4*a^2)/(5*d*e*(e*Tan[c + d*x])^(5/2)) - (4*a^2*Sec[c + d*x])/(5*d*e*(e*Tan[c + d*x])^(5/2)) + (2*a^2)/(d*e^3*Sqrt[e*Tan[c + d*x]]) + (12*a^2*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Tan[c + d*x]]) + (12*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*e^4*Sqrt[Sin[2*c + 2*d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x]), x, 18, (e^(11/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(11/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (5*e^6*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a*d*Sqrt[e*Tan[c + d*x]]) + (2*e^5*(21 - 5*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(21*a*d) - (2*e^3*(7 - 5*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2))/(35*a*d)} -{(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x]), x, 18, -((e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (6*e^4*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a*d*Sqrt[Sin[2*c + 2*d*x]]) - (6*e^3*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a*d) - (2*e^3*(5 - 3*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2))/(15*a*d)} -{(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x]), x, 17, -((e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^4*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*Sqrt[e*Tan[c + d*x]]) - (2*e^3*(3 - Sec[c + d*x])*Sqrt[e*Tan[c + d*x]])/(3*a*d)} -{(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x]), x, 17, (e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a*d)} -{(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x]), x, 16, (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(a*d*Sqrt[e*Tan[c + d*x]])} -{Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x]), x, 18, -((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (2*e*(1 - Sec[c + d*x]))/(a*d*Sqrt[e*Tan[c + d*x]]) - (2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a*d*e)} -{1/((a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]), x, 17, -(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + (2*e*(1 - Sec[c + d*x]))/(3*a*d*(e*Tan[c + d*x])^(3/2)) - (EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*Sqrt[e*Tan[c + d*x]])} -{1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(3/2)), x, 19, ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(3/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(3/2)) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(3/2)) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(3/2)) + (2*e*(1 - Sec[c + d*x]))/(5*a*d*(e*Tan[c + d*x])^(5/2)) - (2*(5 - 3*Sec[c + d*x]))/(5*a*d*e*Sqrt[e*Tan[c + d*x]]) + (6*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a*d*e^2*Sqrt[Sin[2*c + 2*d*x]]) - (6*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a*d*e^3)} -{1/((a + a*Sec[c + d*x])*(e*Tan[c + d*x])^(5/2)), x, 18, ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(5/2)) - ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*e^(5/2)) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(5/2)) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*e^(5/2)) + (2*e*(1 - Sec[c + d*x]))/(7*a*d*(e*Tan[c + d*x])^(7/2)) - (2*(7 - 5*Sec[c + d*x]))/(21*a*d*e*(e*Tan[c + d*x])^(3/2)) + (5*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a*d*e^2*Sqrt[e*Tan[c + d*x]])} - - -{(e*Tan[c + d*x])^(13/2)/(a + a*Sec[c + d*x])^2, x, 22, (e^(13/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(13/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(13/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(13/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (12*e^6*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a^2*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e^5*(e*Tan[c + d*x])^(3/2))/(3*a^2*d) + (12*e^5*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a^2*d) - (4*e^5*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*a^2*d) + (2*e^3*(e*Tan[c + d*x])^(7/2))/(7*a^2*d)} -{(e*Tan[c + d*x])^(11/2)/(a + a*Sec[c + d*x])^2, x, 21, (e^(11/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(11/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(11/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (2*e^6*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*Sqrt[e*Tan[c + d*x]]) + (2*e^5*Sqrt[e*Tan[c + d*x]])/(a^2*d) - (4*e^5*Sec[c + d*x]*Sqrt[e*Tan[c + d*x]])/(3*a^2*d) + (2*e^3*(e*Tan[c + d*x])^(5/2))/(5*a^2*d)} -{(e*Tan[c + d*x])^(9/2)/(a + a*Sec[c + d*x])^2, x, 20, -((e^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (e^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(9/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (4*e^4*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a^2*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e^3*(e*Tan[c + d*x])^(3/2))/(3*a^2*d) - (4*e^3*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(a^2*d)} -{(e*Tan[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2, x, 19, -((e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (e^(7/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(7/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (2*e^4*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (2*e^3*Sqrt[e*Tan[c + d*x]])/(a^2*d)} -{(e*Tan[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2, x, 21, (e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) + (e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (4*e^3*Cos[c + d*x])/(a^2*d*Sqrt[e*Tan[c + d*x]]) + (4*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(a^2*d*Sqrt[Sin[2*c + 2*d*x]])} -{(e*Tan[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2, x, 21, (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) - (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) + (4*e^3*Sec[c + d*x])/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) + (2*e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*Sqrt[e*Tan[c + d*x]])} -{Sqrt[e*Tan[c + d*x]]/(a + a*Sec[c + d*x])^2, x, 23, -((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a^2*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a^2*d) - (4*e^3)/(5*a^2*d*(e*Tan[c + d*x])^(5/2)) + (4*e^3*Sec[c + d*x])/(5*a^2*d*(e*Tan[c + d*x])^(5/2)) + (2*e)/(a^2*d*Sqrt[e*Tan[c + d*x]]) - (12*e*Cos[c + d*x])/(5*a^2*d*Sqrt[e*Tan[c + d*x]]) - (12*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*a^2*d*Sqrt[Sin[2*c + 2*d*x]])} -{1/((a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]), x, 23, -(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) - (4*e^3)/(7*a^2*d*(e*Tan[c + d*x])^(7/2)) + (4*e^3*Sec[c + d*x])/(7*a^2*d*(e*Tan[c + d*x])^(7/2)) + (2*e)/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) - (20*e*Sec[c + d*x])/(21*a^2*d*(e*Tan[c + d*x])^(3/2)) - (10*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a^2*d*Sqrt[e*Tan[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^n (a+a Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^5, x, 8, (-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d) - (6*(a + a*Sec[c + d*x])^(7/2))/(7*a^3*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^4*d)} -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^3, x, 6, (2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c + d*x])^(3/2))/(3*a*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^2*d)} -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x], x, 4, (-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sec[c + d*x]])/d} -{Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]], x, 6, (2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]], x, 8, -((2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d) + (7*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*d) + a/(4*d*Sqrt[a + a*Sec[c + d*x]]) + a/(2*d*(1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]], x, 10, (2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (107*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) + (43*a^2)/(96*d*(a + a*Sec[c + d*x])^(3/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(3/2)) - (15*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) - (21*a)/(64*d*Sqrt[a + a*Sec[c + d*x]])} - -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^6, x, 4, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(d*(a + a*Sec[c + d*x])^(7/2)) + (10*a^5*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (2*a^6*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))} -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^4, x, 4, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^3*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^4*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))} -{Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x]^2, x, 4, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2))} -{Cot[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]], x, 5, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*d) - (Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d} -{Cot[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]], x, 7, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (9*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(8*Sqrt[2]*d) + (7*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(8*d) + (Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12*a*d) - (Cos[c + d*x]*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(3/2))/(4*a*d)} -{Cot[c + d*x]^6*Sqrt[a + a*Sec[c + d*x]], x, 9, -((2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (151*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(128*Sqrt[2]*d) - (105*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(128*d) - (23*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(192*a*d) + (87*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(160*a^2*d) - (17*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(5/2))/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(5/2))/(16*a^2*d)} - - -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^5, x, 9, (-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d) - (2*(a + a*Sec[c + d*x])^(9/2))/(3*a^3*d) + (2*(a + a*Sec[c + d*x])^(11/2))/(11*a^4*d)} -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^3, x, 7, (2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a*Sqrt[a + a*Sec[c + d*x]])/d - (2*(a + a*Sec[c + d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*a*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^2*d)} -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x], x, 5, (-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a*Sqrt[a + a*Sec[c + d*x]])/d + (2*(a + a*Sec[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]*(a + a*Sec[c + d*x])^(3/2), x, 6, (2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2), x, 7, (-2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (5*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d) + (a*Sqrt[a + a*Sec[c + d*x]])/(2*d*(1 - Sec[c + d*x]))} -{Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2), x, 9, (2*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (71*a^(3/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*d) + (7*a^2)/(32*d*Sqrt[a + a*Sec[c + d*x]]) - a^2/(4*d*(1 - Sec[c + d*x])^2*Sqrt[a + a*Sec[c + d*x]]) - (13*a^2)/(16*d*(1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]])} - -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^6, x, 4, -((2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^3*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (30*a^5*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (34*a^6*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (14*a^7*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2)) + (2*a^8*Tan[c + d*x]^13)/(13*d*(a + a*Sec[c + d*x])^(13/2))} -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^4, x, 4, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (14*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (10*a^5*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^6*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2))} -{(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x]^2, x, 4, -((2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a^2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*Tan[c + d*x]^3)/(d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^4*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2), x, 3, -((2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) - (2*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2), x, 6, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*d) + (3*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(2*d) - (Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(3/2), x, 8, -((2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (11*a^(3/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*d) - (21*a*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(16*d) + (5*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(24*d) + (3*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(20*a*d) - (Cos[c + d*x]*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(5/2))/(4*a*d)} - - -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^5, x, 10, (-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d + (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^2*d) - (6*(a + a*Sec[c + d*x])^(11/2))/(11*a^3*d) + (2*(a + a*Sec[c + d*x])^(13/2))/(13*a^4*d)} -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^3, x, 8, (2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d - (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) - (2*(a + a*Sec[c + d*x])^(5/2))/(5*d) - (2*(a + a*Sec[c + d*x])^(7/2))/(7*a*d) + (2*(a + a*Sec[c + d*x])^(9/2))/(9*a^2*d)} -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x], x, 6, (-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d + (2*a*(a + a*Sec[c + d*x])^(3/2))/(3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*d)} -{Cot[c + d*x]*(a + a*Sec[c + d*x])^(5/2), x, 7, (2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (2*a^2*Sqrt[a + a*Sec[c + d*x]])/d} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2), x, 7, (-2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d + (3*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d) + (a^2*Sqrt[a + a*Sec[c + d*x]])/(d*(1 - Sec[c + d*x]))} -{Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2), x, 8, (2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (43*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*d) - (a^2*Sqrt[a + a*Sec[c + d*x]])/(4*d*(1 - Sec[c + d*x])^2) - (11*a^2*Sqrt[a + a*Sec[c + d*x]])/(16*d*(1 - Sec[c + d*x]))} - -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^6, x, 4, -((2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^5*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (62*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (98*a^7*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (62*a^8*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2)) + (18*a^9*Tan[c + d*x]^13)/(13*d*(a + a*Sec[c + d*x])^(13/2)) + (2*a^10*Tan[c + d*x]^15)/(15*d*(a + a*Sec[c + d*x])^(15/2))} -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^4, x, 4, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (6*a^5*Tan[c + d*x]^5)/(d*(a + a*Sec[c + d*x])^(5/2)) + (34*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (14*a^7*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2)) + (2*a^8*Tan[c + d*x]^11)/(11*d*(a + a*Sec[c + d*x])^(11/2))} -{(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x]^2, x, 4, -((2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (2*a^3*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (14*a^4*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^5*Tan[c + d*x]^5)/(d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^6*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2), x, 3, -((2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) - (4*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2), x, 4, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/d - (2*a*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(3*d)} -{Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2), x, 7, -((2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d) + (a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4*Sqrt[2]*d) - (7*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4*d) + (a*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(2*d) - (Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tan[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]], x, 7, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^2*d) - (6*(a + a*Sec[c + d*x])^(5/2))/(5*a^3*d) + (2*(a + a*Sec[c + d*x])^(7/2))/(7*a^4*d)} -{Tan[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]], x, 5, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*Sqrt[a + a*Sec[c + d*x]])/(a*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^2*d)} -{Tan[c + d*x]/Sqrt[a + a*Sec[c + d*x]], x, 3, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)} -{Cot[c + d*x]/Sqrt[a + a*Sec[c + d*x]], x, 7, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) - 1/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]], x, 9, -((2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) + (9*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d) - a/(12*d*(a + a*Sec[c + d*x])^(3/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) + 7/(8*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]], x, 11, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (151*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*Sqrt[a]*d) + (87*a^2)/(160*d*(a + a*Sec[c + d*x])^(5/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(5/2)) - (17*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2)) + (23*a)/(192*d*(a + a*Sec[c + d*x])^(3/2)) - 105/(128*d*Sqrt[a + a*Sec[c + d*x]])} - -{Tan[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]], x, 4, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) - (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (6*a^3*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2)) + (2*a^4*Tan[c + d*x]^9)/(9*d*(a + a*Sec[c + d*x])^(9/2))} -{Tan[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]], x, 5, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a^2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))} -{Tan[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]], x, 3, (-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (2*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]], x, 6, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (7*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4*Sqrt[2]*Sqrt[a]*d) - (Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4*a*d) - (Cos[c + d*x]*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^2*Sqrt[a + a*Sec[c + d*x]])/(4*a*d)} -{Cot[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]], x, 8, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (107*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(64*Sqrt[2]*Sqrt[a]*d) + (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(64*a*d) + (43*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(96*a^2*d) - (15*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(3/2))/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(3/2))/(16*a^2*d)} -{Cot[c + d*x]^6/Sqrt[a + a*Sec[c + d*x]], x, 10, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (835*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (189*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(512*a*d) - (323*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(768*a^2*d) + (579*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(640*a^3*d) - (101*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(5/2))/(128*a^3*d) - (23*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(5/2))/(192*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^6*(a + a*Sec[c + d*x])^(5/2))/(48*a^3*d)} - - -{Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2), x, 6, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a^2*d) - (2*(a + a*Sec[c + d*x])^(3/2))/(a^3*d) + (2*(a + a*Sec[c + d*x])^(5/2))/(5*a^4*d)} -{Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2), x, 4, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*Sqrt[a + a*Sec[c + d*x]])/(a^2*d)} -{Tan[c + d*x]/(a + a*Sec[c + d*x])^(3/2), x, 4, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + 2/(a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]/(a + a*Sec[c + d*x])^(3/2), x, 8, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - 1/(3*d*(a + a*Sec[c + d*x])^(3/2)) - 3/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2), x, 10, -((2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + (11*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*a^(3/2)*d) - (3*a)/(20*d*(a + a*Sec[c + d*x])^(5/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2)) + 5/(24*d*(a + a*Sec[c + d*x])^(3/2)) + 21/(16*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2), x, 12, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (203*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*a^(3/2)*d) + (139*a^2)/(224*d*(a + a*Sec[c + d*x])^(7/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(7/2)) - (19*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + (15*a)/(64*d*(a + a*Sec[c + d*x])^(5/2)) - 53/(384*d*(a + a*Sec[c + d*x])^(3/2)) - 309/(256*a*d*Sqrt[a + a*Sec[c + d*x]])} - -{Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2), x, 5, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2)) + (2*a*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2)) + (2*a^2*Tan[c + d*x]^7)/(7*d*(a + a*Sec[c + d*x])^(7/2))} -{Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2), x, 4, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (2*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]]) + (2*Tan[c + d*x]^3)/(3*d*(a + a*Sec[c + d*x])^(3/2))} -{Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2), x, 4, (-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + (2*Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(a^(3/2)*d)} -{Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2), x, 7, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + (71*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(32*Sqrt[2]*a^(3/2)*d) + (7*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(32*a^2*d) - (13*Cos[c + d*x]*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^2*Sqrt[a + a*Sec[c + d*x]])/(32*a^2*d) - (Cos[c + d*x]^2*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^4*Sqrt[a + a*Sec[c + d*x]])/(16*a^2*d)} -{Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2), x, 9, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - (533*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d) - (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(256*a^2*d) + (277*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(384*a^3*d) - (81*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(3/2))/(128*a^3*d) - (7*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(3/2))/(64*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^6*(a + a*Sec[c + d*x])^(3/2))/(48*a^3*d)} -{Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(3/2), x, 11, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + (16363*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(8192*Sqrt[2]*a^(3/2)*d) - (21*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(8192*a^2*d) - (8171*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12288*a^3*d) + (12267*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(10240*a^4*d) - (2045*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(5/2))/(2048*a^4*d) - (511*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(5/2))/(3072*a^4*d) - (29*Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^6*(a + a*Sec[c + d*x])^(5/2))/(768*a^4*d) - (Cos[c + d*x]^4*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^8*(a + a*Sec[c + d*x])^(5/2))/(128*a^4*d)} - - -{Tan[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2), x, 5, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - (6*Sqrt[a + a*Sec[c + d*x]])/(a^3*d) + (2*(a + a*Sec[c + d*x])^(3/2))/(3*a^4*d)} -{Tan[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2), x, 4, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - 4/(a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Tan[c + d*x]/(a + a*Sec[c + d*x])^(5/2), x, 5, (-2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) + 2/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + 2/(a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]/(a + a*Sec[c + d*x])^(5/2), x, 9, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - 1/(5*d*(a + a*Sec[c + d*x])^(5/2)) - 1/(2*a*d*(a + a*Sec[c + d*x])^(3/2)) - 7/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2), x, 11, -((2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d)) + (13*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*a^(5/2)*d) - (5*a)/(28*d*(a + a*Sec[c + d*x])^(7/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + 3/(40*d*(a + a*Sec[c + d*x])^(5/2)) + 19/(48*a*d*(a + a*Sec[c + d*x])^(3/2)) + 51/(32*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2), x, 13, (2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - (263*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(512*Sqrt[2]*a^(5/2)*d) + (199*a^2)/(288*d*(a + a*Sec[c + d*x])^(9/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(9/2)) - (21*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(9/2)) + (135*a)/(448*d*(a + a*Sec[c + d*x])^(7/2)) + 7/(640*d*(a + a*Sec[c + d*x])^(5/2)) - 83/(256*a*d*(a + a*Sec[c + d*x])^(3/2)) - 761/(512*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - -{Tan[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2), x, 4, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (2*Tan[c + d*x])/(a^2*d*Sqrt[a + a*Sec[c + d*x]]) - (2*Tan[c + d*x]^3)/(3*a*d*(a + a*Sec[c + d*x])^(3/2)) + (2*Tan[c + d*x]^5)/(5*d*(a + a*Sec[c + d*x])^(5/2))} -{Tan[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2), x, 5, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (4*Sqrt[2]*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(a^(5/2)*d) + (2*Tan[c + d*x])/(a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Tan[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2), x, 5, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (3*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[2]*a^(5/2)*d) + (Sec[(1/2)*(c + d*x)]^2*Sin[c + d*x])/(2*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cot[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2), x, 8, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (319*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(128*Sqrt[2]*a^(5/2)*d) + (63*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(128*a^3*d) - (191*Cos[c + d*x]*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^2*Sqrt[a + a*Sec[c + d*x]])/(384*a^3*d) - (19*Cos[c + d*x]^2*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^4*Sqrt[a + a*Sec[c + d*x]])/(192*a^3*d) - (Cos[c + d*x]^3*Cot[c + d*x]*Sec[(1/2)*(c + d*x)]^6*Sqrt[a + a*Sec[c + d*x]])/(48*a^3*d)} -{Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2), x, 10, (2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - (9683*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d) - (1491*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4096*a^3*d) + (5587*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(6144*a^4*d) - (1527*Cos[c + d*x]*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(3/2))/(2048*a^4*d) - (145*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(3/2))/(1024*a^4*d) - (9*Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^6*(a + a*Sec[c + d*x])^(3/2))/(256*a^4*d) - (Cos[c + d*x]^4*Cot[c + d*x]^3*Sec[(1/2)*(c + d*x)]^8*(a + a*Sec[c + d*x])^(3/2))/(128*a^4*d)} -{Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2), x, 12, -((2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + (74461*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(32768*Sqrt[2]*a^(5/2)*d) + (8925*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(32768*a^3*d) - (41693*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(49152*a^4*d) + (58077*Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(40960*a^5*d) - (9467*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^2*(a + a*Sec[c + d*x])^(5/2))/(8192*a^5*d) - (2473*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^4*(a + a*Sec[c + d*x])^(5/2))/(12288*a^5*d) - (155*Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^6*(a + a*Sec[c + d*x])^(5/2))/(3072*a^5*d) - (7*Cos[c + d*x]^4*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^8*(a + a*Sec[c + d*x])^(5/2))/(512*a^5*d) - (Cos[c + d*x]^5*Cot[c + d*x]^5*Sec[(1/2)*(c + d*x)]^10*(a + a*Sec[c + d*x])^(5/2))/(320*a^5*d)} - - -{Tan[e + f*x]^2/(a + a*Sec[e + f*x])^(9/2), x, -7, -((2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(9/2)*f)) + (91*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(32*Sqrt[2]*a^(9/2)*f) + Tan[e + f*x]/(3*a*f*(a + a*Sec[e + f*x])^(7/2)) + (11*Tan[e + f*x])/(24*a^2*f*(a + a*Sec[e + f*x])^(5/2)) + (27*Tan[e + f*x])/(32*a^3*f*(a + a*Sec[e + f*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+a Sec[e+f x])^m with m and/or n symbolic*) - - -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^n, x, 1, (2^(1 + m + n)*AppellF1[(1 + m)/2, m + n, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(1 + m + n)*(a + a*Sec[c + d*x])^n*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} - - -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^3, x, 8, (3*a^3*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (3*a^3*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^3*(Cos[c + d*x]^2)^((4 + m)/2)*Hypergeometric2F1[(1 + m)/2, (4 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]^3*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^2, x, 7, (a^2*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a^2*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (2*a^2*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^1, x, 4, (a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)) + (a*(Cos[c + d*x]^2)^((2 + m)/2)*Hypergeometric2F1[(1 + m)/2, (2 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^1, x, 5, (e*Hypergeometric2F1[1, (1/2)*(-1 + m), (1 + m)/2, -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-1 + m))/(a*d*(1 - m)) - (e*(Cos[c + d*x]^2)^(m/2)*Hypergeometric2F1[(1/2)*(-1 + m), m/2, (1 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-1 + m))/(a*d*(1 - m))} -{(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^2, x, 8, -((e^3*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m))) - (e^3*Hypergeometric2F1[1, (1/2)*(-3 + m), (1/2)*(-1 + m), -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m)) + (2*e^3*(Cos[c + d*x]^2)^((1/2)*(-2 + m))*Hypergeometric2F1[(1/2)*(-3 + m), (1/2)*(-2 + m), (1/2)*(-1 + m), Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-3 + m))/(a^2*d*(3 - m))} -{(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^3, x, 9, (3*e^5*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) + (e^5*Hypergeometric2F1[1, (1/2)*(-5 + m), (1/2)*(-3 + m), -Tan[c + d*x]^2]*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) - (3*e^5*(Cos[c + d*x]^2)^((1/2)*(-4 + m))*Hypergeometric2F1[(1/2)*(-5 + m), (1/2)*(-4 + m), (1/2)*(-3 + m), Sin[c + d*x]^2]*Sec[c + d*x]*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m)) - (e^5*(Cos[c + d*x]^2)^((1/2)*(-2 + m))*Hypergeometric2F1[(1/2)*(-5 + m), (1/2)*(-2 + m), (1/2)*(-3 + m), Sin[c + d*x]^2]*Sec[c + d*x]^3*(e*Tan[c + d*x])^(-5 + m))/(a^3*d*(5 - m))} - - -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^(3/2), x, 1, (2^(5/2 + m)*AppellF1[(1 + m)/2, 3/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(5/2 + m)*(a + a*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m*(a + a*Sec[c + d*x])^(1/2), x, 1, (2^(3/2 + m)*AppellF1[(1 + m)/2, 1/2 + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(3/2 + m)*Sqrt[a + a*Sec[c + d*x]]*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m))} -{(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^(1/2), x, 1, (2^(1/2 + m)*AppellF1[(1 + m)/2, -(1/2) + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(1/2 + m)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)*Sqrt[a + a*Sec[c + d*x]])} -{(e*Tan[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2), x, 1, (2^(-(1/2) + m)*AppellF1[(1 + m)/2, -(3/2) + m, 1, (3 + m)/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(-(1/2) + m)*(e*Tan[c + d*x])^(1 + m))/(d*e*(1 + m)*(a + a*Sec[c + d*x])^(3/2))} - - -{Tan[c + d*x]^7*(a + a*Sec[c + d*x])^n, x, 4, (7*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(4 + n)) + (Hypergeometric2F1[1, 4 + n, 5 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(4 + n))/(a^4*d*(4 + n)) - (5*(a + a*Sec[c + d*x])^(5 + n))/(a^5*d*(5 + n)) + (a + a*Sec[c + d*x])^(6 + n)/(a^6*d*(6 + n))} -{Tan[c + d*x]^5*(a + a*Sec[c + d*x])^n, x, 4, -((3*(a + a*Sec[c + d*x])^(3 + n))/(a^3*d*(3 + n))) - (Hypergeometric2F1[1, 3 + n, 4 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3 + n))/(a^3*d*(3 + n)) + (a + a*Sec[c + d*x])^(4 + n)/(a^4*d*(4 + n))} -{Tan[c + d*x]^3*(a + a*Sec[c + d*x])^n, x, 3, (a + a*Sec[c + d*x])^(2 + n)/(a^2*d*(2 + n)) + (Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2 + n))/(a^2*d*(2 + n))} -{Tan[c + d*x]^1*(a + a*Sec[c + d*x])^n, x, 2, -((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)))} -{Cot[c + d*x]^1*(a + a*Sec[c + d*x])^n, x, 4, -((Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + Sec[c + d*x])]*(a + a*Sec[c + d*x])^n)/(2*d*n)) + (Hypergeometric2F1[1, n, 1 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^n)/(d*n)} -{Cot[c + d*x]^3*(a + a*Sec[c + d*x])^n, x, 5, -((a*(4 - n)*Hypergeometric2F1[1, -1 + n, n, (1/2)*(1 + Sec[c + d*x])]*(a + a*Sec[c + d*x])^(-1 + n))/(4*d*(1 - n))) + (a*Hypergeometric2F1[1, -1 + n, n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(-1 + n))/(d*(1 - n)) + (a*(a + a*Sec[c + d*x])^(-1 + n))/(2*d*(1 - Sec[c + d*x]))} - -{Tan[c + d*x]^4*(a + a*Sec[c + d*x])^n, x, 1, (2^(5 + n)*AppellF1[5/2, 4 + n, 1, 7/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(5 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^5)/(5*d)} -{Tan[c + d*x]^2*(a + a*Sec[c + d*x])^n, x, 1, (2^(3 + n)*AppellF1[3/2, 2 + n, 1, 5/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(3 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^2*(a + a*Sec[c + d*x])^n, x, 1, -((2^(-1 + n)*AppellF1[-(1/2), -2 + n, 1, 1/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d*x]*(1/(1 + Sec[c + d*x]))^(-1 + n)*(a + a*Sec[c + d*x])^n)/d)} -{Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n, x, 1, -((2^(-3 + n)*AppellF1[-(3/2), -4 + n, 1, -(1/2), -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d*x]^3*(1/(1 + Sec[c + d*x]))^(-3 + n)*(a + a*Sec[c + d*x])^n)/(3*d))} - - -{Tan[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^n, x, 1, (1/(5*d))*(2^(7/2 + n)*AppellF1[5/4, 3/2 + n, 1, 9/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(5/2 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(5/2))} -{Tan[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^n, x, 1, (1/(3*d))*(2^(5/2 + n)*AppellF1[3/4, 1/2 + n, 1, 7/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(3/2 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^(3/2))} -{1/Tan[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^n, x, 1, (1/d)*(2^(3/2 + n)*AppellF1[1/4, -(1/2) + n, 1, 5/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(1/2 + n)*(a + a*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]])} -{1/Tan[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^n, x, 1, -((2^(1/2 + n)*AppellF1[-(1/4), -(3/2) + n, 1, 3/4, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*(1/(1 + Sec[c + d*x]))^(-(1/2) + n)*(a + a*Sec[c + d*x])^n)/(d*Sqrt[Tan[c + d*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^n (a+a Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cot[e+f x])^(n/2) (a+a Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x]), x, 17, -((2*(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])*Tan[c + d*x])/(3*d)) - (a*(e*Cot[c + d*x])^(5/2)*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]]*Tan[c + d*x]^2)/(3*d) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) + (a*(e*Cot[c + d*x])^(5/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d) - (a*(e*Cot[c + d*x])^(5/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d)} -{(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x]), x, 18, -((2*(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])*Tan[c + d*x])/d) - (2*a*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a*(e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (a*(e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (2*a*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/d} -{Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x]), x, 16, (a*Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/d - (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) + (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) - (a*Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (a*Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d)} -{(a + a*Sec[c + d*x])/Sqrt[e*Cot[c + d*x]], x, 17, (2*a*Sin[c + d*x])/(d*Sqrt[e*Cot[c + d*x]]) - (2*a*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - (a*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])} -{(a + a*Sec[c + d*x])/(e*Cot[c + d*x])^(3/2), x, 17, (2*Cot[c + d*x]*(3*a + a*Sec[c + d*x]))/(3*d*(e*Cot[c + d*x])^(3/2)) - (a*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*(e*Cot[c + d*x])^(3/2)) + (a*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (a*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))} - - -{(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2, x, 21, -((4*a^2*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x])/(3*d)) - (4*a^2*(e*Cot[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (2*a^2*(e*Cot[c + d*x])^(5/2)*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]]*Tan[c + d*x]^2)/(3*d) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) + (a^2*(e*Cot[c + d*x])^(5/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d) - (a^2*(e*Cot[c + d*x])^(5/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d)} -{(e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2, x, 21, -((4*a^2*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x])/d) - (4*a^2*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x])/d - (4*a^2*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*(e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d) + (a^2*(e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*d)} -{Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2, x, 19, (2*a^2*Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/d - (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) + (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*d) - (a^2*Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (a^2*Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*d) + (2*a^2*Sqrt[e*Cot[c + d*x]]*Tan[c + d*x])/d} -{(a + a*Sec[c + d*x])^2/Sqrt[e*Cot[c + d*x]], x, 20, (4*a^2*Sin[c + d*x])/(d*Sqrt[e*Cot[c + d*x]]) - (4*a^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - (a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + (2*a^2*Tan[c + d*x])/(3*d*Sqrt[e*Cot[c + d*x]])} -{(a + a*Sec[c + d*x])^2/(e*Cot[c + d*x])^(3/2), x, 21, (2*a^2*Cot[c + d*x])/(d*(e*Cot[c + d*x])^(3/2)) + (4*a^2*Csc[c + d*x])/(3*d*(e*Cot[c + d*x])^(3/2)) - (2*a^2*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*(e*Cot[c + d*x])^(3/2)) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (a^2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (2*a^2*Tan[c + d*x])/(5*d*(e*Cot[c + d*x])^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]), x, 20, (2*Cot[c + d*x]*(e*Cot[c + d*x])^(3/2)*(1 - Sec[c + d*x]))/(5*a*d) - (2*(e*Cot[c + d*x])^(3/2)*(5 - 3*Sec[c + d*x])*Tan[c + d*x])/(5*a*d) + (6*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(5*a*d*Sqrt[Sin[2*c + 2*d*x]]) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*a*d) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*a*d) - ((e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*a*d) + ((e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2*Sqrt[2]*a*d) - (6*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/(5*a*d)} -{Sqrt[e*Cot[c + d*x]]/(a + a*Sec[c + d*x]), x, 18, (2*Cot[c + d*x]*Sqrt[e*Cot[c + d*x]]*(1 - Sec[c + d*x]))/(3*a*d) - (Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d) - (Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*a*d) + (Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan[c + d*x]])/(2*Sqrt[2]*a*d)} -{1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])), x, 19, (2*Cot[c + d*x]*(1 - Sec[c + d*x]))/(a*d*Sqrt[e*Cot[c + d*x]]) + (2*Sin[c + d*x])/(a*d*Sqrt[e*Cot[c + d*x]]) - (2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])} -{1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])), x, 17, (Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(a*d*(e*Cot[c + d*x])^(3/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))} -{1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])), x, 18, (2*Cos[c + d*x]*Cot[c + d*x])/(a*d*(e*Cot[c + d*x])^(5/2)) - (2*Cos[c + d*x]*Cot[c + d*x]^2*EllipticE[c - Pi/4 + d*x, 2])/(a*d*(e*Cot[c + d*x])^(5/2)*Sqrt[Sin[2*c + 2*d*x]]) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))} -{1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])), x, 18, -((2*Cot[c + d*x]^3*(3 - Sec[c + d*x]))/(3*a*d*(e*Cot[c + d*x])^(7/2))) - (Cot[c + d*x]^3*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d*(e*Cot[c + d*x])^(7/2)) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))} -{1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])), x, 19, -((6*Cos[c + d*x]*Cot[c + d*x]^3)/(5*a*d*(e*Cot[c + d*x])^(9/2))) - (2*Cot[c + d*x]^3*(5 - 3*Sec[c + d*x]))/(15*a*d*(e*Cot[c + d*x])^(9/2)) + (6*Cos[c + d*x]*Cot[c + d*x]^4*EllipticE[c - Pi/4 + d*x, 2])/(5*a*d*(e*Cot[c + d*x])^(9/2)*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2))} - - -{1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 24, (2*Cot[c + d*x])/(a^2*d*Sqrt[e*Cot[c + d*x]]) - (12*Cos[c + d*x]*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) - (4*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) + (4*Cot[c + d*x]^2*Csc[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) - (12*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(5*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])} -{1/((e*Cot[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2), x, 22, -((4*Cot[c + d*x]^3)/(3*a^2*d*(e*Cot[c + d*x])^(3/2))) + (4*Cot[c + d*x]^2*Csc[c + d*x])/(3*a^2*d*(e*Cot[c + d*x])^(3/2)) + (2*Cot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*(e*Cot[c + d*x])^(3/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))} -{1/((e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2), x, 22, -((4*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(5/2))) + (4*Cos[c + d*x]*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(5/2)) + (4*Cos[c + d*x]*Cot[c + d*x]^2*EllipticE[c - Pi/4 + d*x, 2])/(a^2*d*(e*Cot[c + d*x])^(5/2)*Sqrt[Sin[2*c + 2*d*x]]) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))} -{1/((e*Cot[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2), x, 20, (2*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(7/2)) - (2*Cot[c + d*x]^3*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(a^2*d*(e*Cot[c + d*x])^(7/2)) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) - Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2)) + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(7/2)*Tan[c + d*x]^(7/2))} -{1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])^2), x, 21, (2*Cot[c + d*x]^3)/(3*a^2*d*(e*Cot[c + d*x])^(9/2)) - (4*Cos[c + d*x]*Cot[c + d*x]^3)/(a^2*d*(e*Cot[c + d*x])^(9/2)) + (4*Cos[c + d*x]*Cot[c + d*x]^4*EllipticE[c - Pi/4 + d*x, 2])/(a^2*d*(e*Cot[c + d*x])^(9/2)*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x]^(9/2))} -{1/((e*Cot[c + d*x])^(11/2)*(a + a*Sec[c + d*x])^2), x, 22, (2*Cot[c + d*x]^3)/(5*a^2*d*(e*Cot[c + d*x])^(11/2)) + (2*Cot[c + d*x]^5)/(a^2*d*(e*Cot[c + d*x])^(11/2)) - (4*Cot[c + d*x]^4*Csc[c + d*x])/(3*a^2*d*(e*Cot[c + d*x])^(11/2)) + (2*Cot[c + d*x]^5*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*a^2*d*(e*Cot[c + d*x])^(11/2)) + ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*(e*Cot[c + d*x])^(11/2)*Tan[c + d*x]^(11/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^m (a+b Sec[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^n (a+b Sec[e+f x])^m*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Sec[c + d*x])*Tan[c + d*x]^7, x, 7, (a*Log[Cos[c + d*x]])/d - (16*b*Sec[c + d*x])/(35*d) + ((35*a + 16*b*Sec[c + d*x])*Tan[c + d*x]^2)/(70*d) - ((35*a + 24*b*Sec[c + d*x])*Tan[c + d*x]^4)/(140*d) + ((7*a + 6*b*Sec[c + d*x])*Tan[c + d*x]^6)/(42*d)} -{(a + b*Sec[c + d*x])*Tan[c + d*x]^5, x, 6, -((a*Log[Cos[c + d*x]])/d) + (8*b*Sec[c + d*x])/(15*d) - ((15*a + 8*b*Sec[c + d*x])*Tan[c + d*x]^2)/(30*d) + ((5*a + 4*b*Sec[c + d*x])*Tan[c + d*x]^4)/(20*d)} -{(a + b*Sec[c + d*x])*Tan[c + d*x]^3, x, 5, (a*Log[Cos[c + d*x]])/d - (2*b*Sec[c + d*x])/(3*d) + ((3*a + 2*b*Sec[c + d*x])*Tan[c + d*x]^2)/(6*d)} -{(a + b*Sec[c + d*x])*Tan[c + d*x]^1, x, 4, -((a*Log[Cos[c + d*x]])/d) + (b*Sec[c + d*x])/d} -{Cot[c + d*x]^1*(a + b*Sec[c + d*x]), x, 5, ((a + b)*Log[1 - Cos[c + d*x]])/(2*d) + ((a - b)*Log[1 + Cos[c + d*x]])/(2*d)} -{Cot[c + d*x]^3*(a + b*Sec[c + d*x]), x, 6, -(((2*a + b)*Log[1 - Cos[c + d*x]])/(4*d)) - ((2*a - b)*Log[1 + Cos[c + d*x]])/(4*d) - (Cot[c + d*x]^2*(a + b*Sec[c + d*x]))/(2*d)} -{Cot[c + d*x]^5*(a + b*Sec[c + d*x]), x, 7, ((8*a + 3*b)*Log[1 - Cos[c + d*x]])/(16*d) + ((8*a - 3*b)*Log[1 + Cos[c + d*x]])/(16*d) - (Cot[c + d*x]^4*(a + b*Sec[c + d*x]))/(4*d) + (Cot[c + d*x]^2*(4*a + 3*b*Sec[c + d*x]))/(8*d)} -{Cot[c + d*x]^7*(a + b*Sec[c + d*x]), x, 8, -(((16*a + 5*b)*Log[1 - Cos[c + d*x]])/(32*d)) - ((16*a - 5*b)*Log[1 + Cos[c + d*x]])/(32*d) - (Cot[c + d*x]^6*(a + b*Sec[c + d*x]))/(6*d) + (Cot[c + d*x]^4*(6*a + 5*b*Sec[c + d*x]))/(24*d) - (Cot[c + d*x]^2*(8*a + 5*b*Sec[c + d*x]))/(16*d)} - -{(a + b*Sec[c + d*x])*Tan[c + d*x]^6, x, 5, (-a)*x - (5*b*ArcTanh[Sin[c + d*x]])/(16*d) + ((16*a + 5*b*Sec[c + d*x])*Tan[c + d*x])/(16*d) - ((8*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^3)/(24*d) + ((6*a + 5*b*Sec[c + d*x])*Tan[c + d*x]^5)/(30*d)} -{(a + b*Sec[c + d*x])*Tan[c + d*x]^4, x, 4, a*x + (3*b*ArcTanh[Sin[c + d*x]])/(8*d) - ((8*a + 3*b*Sec[c + d*x])*Tan[c + d*x])/(8*d) + ((4*a + 3*b*Sec[c + d*x])*Tan[c + d*x]^3)/(12*d)} -{(a + b*Sec[c + d*x])*Tan[c + d*x]^2, x, 3, (-a)*x - (b*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cot[c + d*x]^2*(a + b*Sec[c + d*x]), x, 2, (-a)*x - (Cot[c + d*x]*(a + b*Sec[c + d*x]))/d} -{Cot[c + d*x]^4*(a + b*Sec[c + d*x]), x, 3, a*x - (Cot[c + d*x]^3*(a + b*Sec[c + d*x]))/(3*d) + (Cot[c + d*x]*(3*a + 2*b*Sec[c + d*x]))/(3*d)} -{Cot[c + d*x]^6*(a + b*Sec[c + d*x]), x, 4, (-a)*x - (Cot[c + d*x]^5*(a + b*Sec[c + d*x]))/(5*d) + (Cot[c + d*x]^3*(5*a + 4*b*Sec[c + d*x]))/(15*d) - (Cot[c + d*x]*(15*a + 8*b*Sec[c + d*x]))/(15*d)} -{Cot[c + d*x]^8*(a + b*Sec[c + d*x]), x, 5, a*x - (Cot[c + d*x]^7*(a + b*Sec[c + d*x]))/(7*d) + (Cot[c + d*x]^5*(7*a + 6*b*Sec[c + d*x]))/(35*d) + (Cot[c + d*x]*(35*a + 16*b*Sec[c + d*x]))/(35*d) - (Cot[c + d*x]^3*(35*a + 24*b*Sec[c + d*x]))/(105*d)} - - -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^9, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - (2*a^2*Sec[c + d*x]^2)/d - (8*a*b*Sec[c + d*x]^3)/(3*d) + (3*a^2*Sec[c + d*x]^4)/(2*d) + (12*a*b*Sec[c + d*x]^5)/(5*d) - (2*a^2*Sec[c + d*x]^6)/(3*d) - (8*a*b*Sec[c + d*x]^7)/(7*d) + (a^2*Sec[c + d*x]^8)/(8*d) + (2*a*b*Sec[c + d*x]^9)/(9*d) + (b^2*Tan[c + d*x]^10)/(10*d), -((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - ((4*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) - (8*a*b*Sec[c + d*x]^3)/(3*d) + ((3*a^2 - 2*b^2)*Sec[c + d*x]^4)/(2*d) + (12*a*b*Sec[c + d*x]^5)/(5*d) - ((2*a^2 - 3*b^2)*Sec[c + d*x]^6)/(3*d) - (8*a*b*Sec[c + d*x]^7)/(7*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^8)/(8*d) + (2*a*b*Sec[c + d*x]^9)/(9*d) + (b^2*Sec[c + d*x]^10)/(10*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^7, x, 3, (a^2*Log[Cos[c + d*x]])/d - (2*a*b*Sec[c + d*x])/d + (3*a^2*Sec[c + d*x]^2)/(2*d) + (2*a*b*Sec[c + d*x]^3)/d - (3*a^2*Sec[c + d*x]^4)/(4*d) - (6*a*b*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^6)/(6*d) + (2*a*b*Sec[c + d*x]^7)/(7*d) + (b^2*Tan[c + d*x]^8)/(8*d), (a^2*Log[Cos[c + d*x]])/d - (2*a*b*Sec[c + d*x])/d + ((3*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (2*a*b*Sec[c + d*x]^3)/d - (3*(a^2 - b^2)*Sec[c + d*x]^4)/(4*d) - (6*a*b*Sec[c + d*x]^5)/(5*d) + ((a^2 - 3*b^2)*Sec[c + d*x]^6)/(6*d) + (2*a*b*Sec[c + d*x]^7)/(7*d) + (b^2*Sec[c + d*x]^8)/(8*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^5, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - (a^2*Sec[c + d*x]^2)/d - (4*a*b*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^4)/(4*d) + (2*a*b*Sec[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^6)/(6*d), -((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d - ((2*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) - (4*a*b*Sec[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sec[c + d*x]^4)/(4*d) + (2*a*b*Sec[c + d*x]^5)/(5*d) + (b^2*Sec[c + d*x]^6)/(6*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^3, x, 3, (a^2*Log[Cos[c + d*x]])/d - (2*a*b*Sec[c + d*x])/d + ((a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (2*a*b*Sec[c + d*x]^3)/(3*d) + (b^2*Sec[c + d*x]^4)/(4*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^1, x, 3, -((a^2*Log[Cos[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]^2)/(2*d)} -{Cot[c + d*x]^1*(a + b*Sec[c + d*x])^2, x, 3, (a^2*Log[Cos[c + d*x]])/d + ((a + b)^2*Log[1 - Sec[c + d*x]])/(2*d) + ((a - b)^2*Log[1 + Sec[c + d*x]])/(2*d)} -{Cot[c + d*x]^3*(a + b*Sec[c + d*x])^2, x, 4, -((a^2*Log[Cos[c + d*x]])/d) - (a*(a + b)*Log[1 - Sec[c + d*x]])/(2*d) - (a*(a - b)*Log[1 + Sec[c + d*x]])/(2*d) - (Cot[c + d*x]^2*(a^2 + b^2 + 2*a*b*Sec[c + d*x]))/(2*d)} -{Cot[c + d*x]^5*(a + b*Sec[c + d*x])^2, x, 5, (a^2*Log[Cos[c + d*x]])/d + (a*(4*a + 3*b)*Log[1 - Sec[c + d*x]])/(8*d) + (a*(4*a - 3*b)*Log[1 + Sec[c + d*x]])/(8*d) + (a*Cot[c + d*x]^2*(2*a + 3*b*Sec[c + d*x]))/(4*d) - (Cot[c + d*x]^4*(a^2 + b^2 + 2*a*b*Sec[c + d*x]))/(4*d)} - -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^6, x, 12, (-a^2)*x - (5*a*b*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*Tan[c + d*x])/d + (5*a*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a^2*Tan[c + d*x]^3)/(3*d) - (5*a*b*Sec[c + d*x]*Tan[c + d*x]^3)/(12*d) + (a^2*Tan[c + d*x]^5)/(5*d) + (a*b*Sec[c + d*x]*Tan[c + d*x]^5)/(3*d) + (b^2*Tan[c + d*x]^7)/(7*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^4, x, 10, a^2*x + (3*a*b*ArcTanh[Sin[c + d*x]])/(4*d) - (a^2*Tan[c + d*x])/d - (3*a*b*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (a^2*Tan[c + d*x]^3)/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x]^3)/(2*d) + (b^2*Tan[c + d*x]^5)/(5*d)} -{(a + b*Sec[c + d*x])^2*Tan[c + d*x]^2, x, 8, (-a^2)*x - (a*b*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (b^2*Tan[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^2*(a + b*Sec[c + d*x])^2, x, 8, (-a^2)*x - (a^2*Cot[c + d*x])/d - (b^2*Cot[c + d*x])/d - (2*a*b*Csc[c + d*x])/d} -{Cot[c + d*x]^4*(a + b*Sec[c + d*x])^2, x, 9, a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) - (b^2*Cot[c + d*x]^3)/(3*d) + (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/(3*d)} -{Cot[c + d*x]^6*(a + b*Sec[c + d*x])^2, x, 11, (-a^2)*x - (a^2*Cot[c + d*x])/d + (a^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) - (b^2*Cot[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x])/d + (4*a*b*Csc[c + d*x]^3)/(3*d) - (2*a*b*Csc[c + d*x]^5)/(5*d)} -{Cot[c + d*x]^8*(a + b*Sec[c + d*x])^2, x, 12, a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (b^2*Cot[c + d*x]^7)/(7*d) + (2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/d + (6*a*b*Csc[c + d*x]^5)/(5*d) - (2*a*b*Csc[c + d*x]^7)/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tan[c + d*x]^9/(a + b*Sec[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(a*d)) - ((a^2 - b^2)^4*Log[a + b*Sec[c + d*x]])/(a*b^8*d) + ((a^6 - 4*a^4*b^2 + 6*a^2*b^4 - 4*b^6)*Sec[c + d*x])/(b^7*d) - (a*(a^4 - 4*a^2*b^2 + 6*b^4)*Sec[c + d*x]^2)/(2*b^6*d) + ((a^4 - 4*a^2*b^2 + 6*b^4)*Sec[c + d*x]^3)/(3*b^5*d) - (a*(a^2 - 4*b^2)*Sec[c + d*x]^4)/(4*b^4*d) + ((a^2 - 4*b^2)*Sec[c + d*x]^5)/(5*b^3*d) - (a*Sec[c + d*x]^6)/(6*b^2*d) + Sec[c + d*x]^7/(7*b*d)} -{Tan[c + d*x]^7/(a + b*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) - ((a^2 - b^2)^3*Log[a + b*Sec[c + d*x]])/(a*b^6*d) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Sec[c + d*x])/(b^5*d) - (a*(a^2 - 3*b^2)*Sec[c + d*x]^2)/(2*b^4*d) + ((a^2 - 3*b^2)*Sec[c + d*x]^3)/(3*b^3*d) - (a*Sec[c + d*x]^4)/(4*b^2*d) + Sec[c + d*x]^5/(5*b*d)} -{Tan[c + d*x]^5/(a + b*Sec[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(a*d)) - ((a^2 - b^2)^2*Log[a + b*Sec[c + d*x]])/(a*b^4*d) + ((a^2 - 2*b^2)*Sec[c + d*x])/(b^3*d) - (a*Sec[c + d*x]^2)/(2*b^2*d) + Sec[c + d*x]^3/(3*b*d)} -{Tan[c + d*x]^3/(a + b*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) - ((a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a*b^2*d) + Sec[c + d*x]/(b*d)} -{Tan[c + d*x]^1/(a + b*Sec[c + d*x]), x, 4, -(Log[Cos[c + d*x]]/(a*d)) - Log[a + b*Sec[c + d*x]]/(a*d)} -{Cot[c + d*x]^1/(a + b*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) + Log[1 - Sec[c + d*x]]/(2*(a + b)*d) + Log[1 + Sec[c + d*x]]/(2*(a - b)*d) - (b^2*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d)} -{Cot[c + d*x]^3/(a + b*Sec[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(a*d)) - ((2*a + 3*b)*Log[1 - Sec[c + d*x]])/(4*(a + b)^2*d) - ((2*a - 3*b)*Log[1 + Sec[c + d*x]])/(4*(a - b)^2*d) - (b^4*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sec[c + d*x])) + 1/(4*(a - b)*d*(1 + Sec[c + d*x]))} -{Cot[c + d*x]^5/(a + b*Sec[c + d*x]), x, 3, Log[Cos[c + d*x]]/(a*d) + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sec[c + d*x]])/(16*(a + b)^3*d) + ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sec[c + d*x]])/(16*(a - b)^3*d) - (b^6*Log[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)^3*d) - 1/(16*(a + b)*d*(1 - Sec[c + d*x])^2) - (5*a + 7*b)/(16*(a + b)^2*d*(1 - Sec[c + d*x])) - 1/(16*(a - b)*d*(1 + Sec[c + d*x])^2) - (5*a - 7*b)/(16*(a - b)^2*d*(1 + Sec[c + d*x]))} - -{Tan[c + d*x]^6/(a + b*Sec[c + d*x]), x, 15, -(x/a) + ((8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTanh[Sin[c + d*x]])/(8*b^5*d) - (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b^5*d) - (a*(a^2 - 2*b^2)*Tan[c + d*x])/(b^4*d) + ((4*a^2 - 7*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*b^3*d) - (a*Tan[c + d*x]^3)/(3*b^2*d) + (Sec[c + d*x]*Tan[c + d*x]^3)/(4*b*d), -(x/a) + (3*ArcTanh[Sin[c + d*x]])/(8*b*d) + ((a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b^5*d) - (a*Tan[c + d*x])/(b^2*d) - (a*(a^2 - 3*b^2)*Tan[c + d*x])/(b^4*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*b*d) + ((a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*b*d) - (a*Tan[c + d*x]^3)/(3*b^2*d)} -{Tan[c + d*x]^4/(a + b*Sec[c + d*x]), x, 6, x/a + ((2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b^3*d) - (a*Tan[c + d*x])/(b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Tan[c + d*x]^2/(a + b*Sec[c + d*x]), x, 7, -(x/a) + ArcTanh[Sin[c + d*x]]/(b*d) - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*b*d)} -{Cot[c + d*x]^2/(a + b*Sec[c + d*x]), x, 9, -(x/a) - (2*b^3*ArcTanh[(Sqrt[a^2 - b^2]*Tan[(1/2)*(c + d*x)])/(a + b)])/(a*(a^2 - b^2)^(3/2)*d) - (a*Cot[c + d*x])/((a^2 - b^2)*d) + (b*Csc[c + d*x])/((a^2 - b^2)*d), -((a*x)/(a^2 - b^2)) + (b^2*x)/(a*(a^2 - b^2)) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d) - (a*Cot[c + d*x])/((a^2 - b^2)*d) + (b*Csc[c + d*x])/((a^2 - b^2)*d)} -{Cot[c + d*x]^4/(a + b*Sec[c + d*x]), x, 15, x/a - (2*b^5*ArcTanh[(Sqrt[a^2 - b^2]*Tan[(1/2)*(c + d*x)])/(a + b)])/(a*(a^2 - b^2)^(5/2)*d) + (a*(a^2 - 2*b^2)*Cot[c + d*x])/((a^2 - b^2)^2*d) - (a*Cot[c + d*x]^3)/(3*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*Csc[c + d*x])/((a^2 - b^2)^2*d) + (b*Csc[c + d*x]^3)/(3*(a^2 - b^2)*d), -((a*b^2*x)/(a^2 - b^2)^2) + (b^4*x)/(a*(a^2 - b^2)^2) + (a*x)/(a^2 - b^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) - (a*b^2*Cot[c + d*x])/((a^2 - b^2)^2*d) + (a*Cot[c + d*x])/((a^2 - b^2)*d) - (a*Cot[c + d*x]^3)/(3*(a^2 - b^2)*d) + (b^3*Csc[c + d*x])/((a^2 - b^2)^2*d) - (b*Csc[c + d*x])/((a^2 - b^2)*d) + (b*Csc[c + d*x]^3)/(3*(a^2 - b^2)*d)} - - -{Tan[c + d*x]^9/(a + b*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) + ((a^2 - b^2)^3*(7*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^8*d) - (2*a*(3*a^4 - 8*a^2*b^2 + 6*b^4)*Sec[c + d*x])/(b^7*d) + ((5*a^4 - 12*a^2*b^2 + 6*b^4)*Sec[c + d*x]^2)/(2*b^6*d) - (4*a*(a^2 - 2*b^2)*Sec[c + d*x]^3)/(3*b^5*d) + ((3*a^2 - 4*b^2)*Sec[c + d*x]^4)/(4*b^4*d) - (2*a*Sec[c + d*x]^5)/(5*b^3*d) + Sec[c + d*x]^6/(6*b^2*d) + (a^2 - b^2)^4/(a*b^8*d*(a + b*Sec[c + d*x]))} -{Tan[c + d*x]^7/(a + b*Sec[c + d*x])^2, x, 3, Log[Cos[c + d*x]]/(a^2*d) + ((a^2 - b^2)^2*(5*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^6*d) - (2*a*(2*a^2 - 3*b^2)*Sec[c + d*x])/(b^5*d) + (3*(a^2 - b^2)*Sec[c + d*x]^2)/(2*b^4*d) - (2*a*Sec[c + d*x]^3)/(3*b^3*d) + Sec[c + d*x]^4/(4*b^2*d) + (a^2 - b^2)^3/(a*b^6*d*(a + b*Sec[c + d*x]))} -{Tan[c + d*x]^5/(a + b*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) + ((a^2 - b^2)*(3*a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^4*d) - (2*a*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^2/(2*b^2*d) + (a^2 - b^2)^2/(a*b^4*d*(a + b*Sec[c + d*x]))} -{Tan[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 3, Log[Cos[c + d*x]]/(a^2*d) + ((a^2 + b^2)*Log[a + b*Sec[c + d*x]])/(a^2*b^2*d) + (a^2 - b^2)/(a*b^2*d*(a + b*Sec[c + d*x]))} -{Tan[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) - Log[a + b*Sec[c + d*x]]/(a^2*d) + 1/(a*d*(a + b*Sec[c + d*x]))} -{Cot[c + d*x]^1/(a + b*Sec[c + d*x])^2, x, 3, Log[Cos[c + d*x]]/(a^2*d) + Log[1 - Sec[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sec[c + d*x]]/(2*(a - b)^2*d) - (b^2*(3*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^2*d) + b^2/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cot[c + d*x]^3/(a + b*Sec[c + d*x])^2, x, 3, -(Log[Cos[c + d*x]]/(a^2*d)) - ((a + 2*b)*Log[1 - Sec[c + d*x]])/(2*(a + b)^3*d) - ((a - 2*b)*Log[1 + Sec[c + d*x]])/(2*(a - b)^3*d) - (b^4*(5*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + 1/(4*(a - b)^2*d*(1 + Sec[c + d*x])) + b^4/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2, x, 3, Log[Cos[c + d*x]]/(a^2*d) + ((4*a^2 + 13*a*b + 12*b^2)*Log[1 - Sec[c + d*x]])/(8*(a + b)^4*d) + ((4*a^2 - 13*a*b + 12*b^2)*Log[1 + Sec[c + d*x]])/(8*(a - b)^4*d) - (b^6*(7*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)^4*d) - 1/(16*(a + b)^2*d*(1 - Sec[c + d*x])^2) - (5*a + 9*b)/(16*(a + b)^3*d*(1 - Sec[c + d*x])) - 1/(16*(a - b)^2*d*(1 + Sec[c + d*x])^2) - (5*a - 9*b)/(16*(a - b)^3*d*(1 + Sec[c + d*x])) + b^6/(a*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - -{Tan[c + d*x]^6/(a + b*Sec[c + d*x])^2, x, 16, -(x/a^2) - (a*(4*a^2 - 5*b^2)*ArcTanh[Sin[c + d*x]])/(b^5*d) + (2*(a - b)^(3/2)*(a + b)^(3/2)*(4*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*b^5*d) + ((a^2 - b^2)^2*Sin[c + d*x])/(a*b^4*d*(b + a*Cos[c + d*x])) + ((3*a^2 - 2*b^2)*Tan[c + d*x])/(b^4*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d), -(x/a^2) - (a*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(2*a^2 - 3*b^2)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*b^3*d) + (4*(a - b)^(3/2)*(a + b)^(3/2)*(2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*b^5*d) + ((a^2 - b^2)^2*Sin[c + d*x])/(a*b^4*d*(b + a*Cos[c + d*x])) + Tan[c + d*x]/(b^2*d) + (3*(a^2 - b^2)*Tan[c + d*x])/(b^4*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d)} -{Tan[c + d*x]^4/(a + b*Sec[c + d*x])^2, x, 6, x/a^2 - (2*a*ArcTanh[Sin[c + d*x]])/(b^3*d) + (2*Sqrt[a - b]*Sqrt[a + b]*(2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*b^3*d) + ((2*a^2 - b^2)*Sin[c + d*x])/(a*b^2*d*(b + a*Cos[c + d*x])) + Tan[c + d*x]/(b*d*(b + a*Cos[c + d*x]))} -{Tan[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 6, -(x/a^2) + (2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + Tan[c + d*x]/(a*d*(a + b*Sec[c + d*x]))} -{Cot[c + d*x]^2/(a + b*Sec[c + d*x])^2, x, 11, -(x/a^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - (4*b^3*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^4*Sin[c + d*x])/(a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Cot[c + d*x]^4/(a + b*Sec[c + d*x])^2, x, 15, x/a^2 - (2*b^7*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b^5*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(7/2)*(a + b)^(7/2)*d) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])^2) - Sin[c + d*x]/(12*(a + b)^2*d*(1 - Cos[c + d*x])) + ((3*a + 5*b)*Sin[c + d*x])/(4*(a + b)^3*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])^2) - ((3*a - 5*b)*Sin[c + d*x])/(4*(a - b)^3*d*(1 + Cos[c + d*x])) + Sin[c + d*x]/(12*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^6*Sin[c + d*x])/(a*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^(n/2) (a+b Sec[e+f x])^m*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x]), x, 38, (a*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) + (a*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) + (2*Sqrt[2]*Sqrt[a - b]*Sqrt[a + b]*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*b*d*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*Sqrt[a - b]*Sqrt[a + b]*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*b*d*Sqrt[Sin[c + d*x]]) - (2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(b*d*Sqrt[Sin[2*c + 2*d*x]]) + (2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(b*d)} -{(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x]), x, 35, (a*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) - (a*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*b^2*d) + (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) - ((a^2 - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) - (a*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*b^2*d) + ((a^2 - b^2)*e^(3/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*b^2*d) - (2*Sqrt[2]*Sqrt[a^2 - b^2]*e^2*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*b*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*Sqrt[a^2 - b^2]*e^2*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*b*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (e^2*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(b*d*Sqrt[e*Tan[c + d*x]])} -{(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x]), x, 21, -((Sqrt[e]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d)) + (Sqrt[e]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*d) + (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) - (Sqrt[e]*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*d) + (2*Sqrt[2]*b*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*b*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]])} -{1/(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x]), x, 19, -(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*a*d*Sqrt[e]) - (2*Sqrt[2]*b*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*Sqrt[a^2 - b^2]*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*Sqrt[a^2 - b^2]*d*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]])} -{1/(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x]), x, 39, (a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) - (b^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) + (b^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) + (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(3/2)) - (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(3/2)) - (2*(a - b*Sec[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[a - b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d*e^2*Sqrt[Sin[c + d*x]]) - (2*Sqrt[2]*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[a - b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d*e^2*Sqrt[Sin[c + d*x]]) + (2*b*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[2*c + 2*d*x]]) - (2*b*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/((a^2 - b^2)*d*e^3)} -{1/(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x]), x, 36, (a*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) - (b^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (a*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) + (b^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) + (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) - (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (a*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*(a^2 - b^2)*d*e^(5/2)) + (b^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*a*(a^2 - b^2)*d*e^(5/2)) - (2*(a - b*Sec[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Tan[c + d*x])^(3/2)) - (2*Sqrt[2]*b^3*EllipticPi[b/(a - Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)^(3/2)*d*e^2*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[2]*b^3*EllipticPi[b/(a + Sqrt[a^2 - b^2]), ArcSin[Sqrt[-Cos[c + d*x]]/Sqrt[1 + Sin[c + d*x]]], -1]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2)^(3/2)*d*e^2*Sqrt[-Cos[c + d*x]]*Sqrt[e*Tan[c + d*x]]) + (b*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Tan[c + d*x]])} - - -(* Mathematica indicates these have a closed-form antiderivative in terms of EllipticPi. *) -(* {(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 21, 0} -{(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 20, 0} -{(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 9, 0} -{1/(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 9, 0} -{1/(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 22, 0} -{1/(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 22, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^n (a+b Sec[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Tan[c + d*x]^5*Sqrt[a + b*Sec[c + d*x]], x, 5, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d) + (2*Sqrt[a + b*Sec[c + d*x]])/d - (2*a*(a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d) + (2*(a + b*Sec[c + d*x])^(9/2))/(9*b^4*d)} -{Tan[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]], x, 5, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sec[c + d*x]])/d - (2*a*(a + b*Sec[c + d*x])^(3/2))/(3*b^2*d) + (2*(a + b*Sec[c + d*x])^(5/2))/(5*b^2*d)} -{Tan[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]], x, 4, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d) + (2*Sqrt[a + b*Sec[c + d*x]])/d} -{Cot[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]], x, 7, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/d - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/d} -{Cot[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]], x, 13, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/d) + (a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(Sqrt[a - b]*d) - (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) + (a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(Sqrt[a + b]*d) + (3*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a + b]*d) - (Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]])/(2*d)} - -{Tan[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]], x, 7, -((2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d)) - (2*Sqrt[a + b]*(a + 2*b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Tan[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]], x, 1, -((2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d))} -{Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]], x, 5, (Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]])/d + (2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[c + d*x]))/(Sqrt[a + b]*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Tan[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]], x, 5, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) - (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4*d) + (2*(3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(3/2))/(3*b^4*d) - (6*a*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d) + (2*(a + b*Sec[c + d*x])^(7/2))/(7*b^4*d)} -{Tan[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]], x, 5, (2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*Sqrt[a + b*Sec[c + d*x]])/(b^2*d) + (2*(a + b*Sec[c + d*x])^(3/2))/(3*b^2*d)} -{Tan[c + d*x]^1/Sqrt[a + b*Sec[c + d*x]], x, 3, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))} -{Cot[c + d*x]^1/Sqrt[a + b*Sec[c + d*x]], x, 7, (2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)} -{Cot[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]], x, 11, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d) + Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)*d*(1 + Sec[c + d*x]))} - -{Tan[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]], x, 11, -((2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)) - (2*(a - b)*Sqrt[a + b]*(8*a^2 - 21*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (2*Sqrt[a + b]*(-8*a^2 + 2*a*b + 21*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)])/(15*b^3*d) - (8*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d), (4*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*(a - b)*Sqrt[a + b]*(8*a^2 + 9*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (4*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*(8*a^2 - 2*a*b + 9*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) - (8*a*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{Tan[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]], x, 6, -((2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d)) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{Tan[c + d*x]^0/Sqrt[a + b*Sec[c + d*x]], x, 1, -((2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))} -{Cot[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]], x, 9, (Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) - Cot[c + d*x]/(d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{Tan[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2), x, 5, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + (2*(a^2 - b^2)^2)/(a*b^4*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^2 - 2*b^2)*Sqrt[a + b*Sec[c + d*x]])/(b^4*d) - (2*a*(a + b*Sec[c + d*x])^(3/2))/(b^4*d) + (2*(a + b*Sec[c + d*x])^(5/2))/(5*b^4*d)} -{Tan[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2), x, 5, (2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*(a^2 - b^2))/(a*b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[a + b*Sec[c + d*x]])/(b^2*d)} -{Tan[c + d*x]^1/(a + b*Sec[c + d*x])^(3/2), x, 4, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + 2/(a*d*Sqrt[a + b*Sec[c + d*x]])} -{Cot[c + d*x]^1/(a + b*Sec[c + d*x])^(3/2), x, 7, (2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/((a - b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cot[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2), x, 11, -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + ((4*a - 7*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + ((4*a + 7*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) + (2*b^4)/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)^2*d*(1 + Sec[c + d*x])), -((2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d)) + ((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(2*(a - b)^(5/2)*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) + (2*b^4)/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)^2*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)^2*d*(1 + Sec[c + d*x]))} - -{Tan[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2), x, 17, (2*(8*a^4 - 11*a^2*b^2 + 3*b^4)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*b^4*Sqrt[a + b]*d) + (2*(2*a + b)*(4*a^2 + a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*b^3*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - (4*a*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d), (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (4*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(8*a^2 - 5*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (4*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) + (2*(2*a + b)*(4*a + b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - (4*a*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a^2*Sec[c + d*x]*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d)} -{Tan[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2), x, 7, (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*Tan[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]])} -{Tan[c + d*x]^0/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cot[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2), x, 14, (2*(a^2 + b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*(a - b)*(a + b)^(3/2)*d) - ((a^2 - a*b + 2*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*(a - b)*(a + b)^(3/2)*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - Cot[c + d*x]/(d*(a + b*Sec[c + d*x])^(3/2)) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(a^2 + b^2)*Tan[c + d*x])/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]), (4*a*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) - (2*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - ((3*a - b)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) + (2*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) - Cot[c + d*x]/(d*(a + b*Sec[c + d*x])^(3/2)) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (4*a*b^2*Tan[c + d*x])/((a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*b^2*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Sec[e+f x])^m with n symbolic*) - - -{(d*Tan[e + f*x])^n*(a + b*Sec[e + f*x])^3, x, 8, (3*a*b^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (a^3*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (3*a^2*b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b^3*(Cos[e + f*x]^2)^((4 + n)/2)*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n*(a + b*Sec[e + f*x])^2, x, 7, (b^2*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (a^2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (2*a*b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n*(a + b*Sec[e + f*x])^1, x, 4, (a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b*(Cos[e + f*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n))} -{(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x])^1, x, -1, (d*AppellF1[1 - n, (1 - n)/2, (1 - n)/2, 2 - n, (a + b)/(a + b*Sec[e + f*x]), (a - b)/(a + b*Sec[e + f*x])]*(-((b*(1 - Sec[e + f*x]))/(a + b*Sec[e + f*x])))^((1 - n)/2)*((b*(1 + Sec[e + f*x]))/(a + b*Sec[e + f*x]))^((1 - n)/2)*(d*Tan[e + f*x])^(-1 + n)*(-Tan[e + f*x]^2)^((1 - n)/2 + (1/2)*(-1 + n)))/(a*f*(1 - n)) - (d*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(-1 + n)*(-Tan[e + f*x]^2)^((1 - n)/2 + (1 + n)/2))/(a*f*(1 + n))} -(* {(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x])^2, x, 0, 0} *) - - -{(e*Tan[c + d*x])^m*(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[(a + b*Sec[c + d*x])^(3/2)*(e*Tan[c + d*x])^m, x]} -{(e*Tan[c + d*x])^m*(a + b*Sec[c + d*x])^(1/2), x, 0, Unintegrable[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]} -{(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(1/2), x, 0, Unintegrable[(e*Tan[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]} -{(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x, 0, Unintegrable[(e*Tan[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tan[e+f x])^n (a+b Sec[e+f x])^m with m symbolic*) - - -{(e*Tan[c + d*x])^m*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*(e*Tan[c + d*x])^m, x]} - - -{Tan[c + d*x]^5*(a + b*Sec[c + d*x])^n, x, 5, -((a*(a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(1 + n))/(b^4*d*(1 + n))) - (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) + ((3*a^2 - 2*b^2)*(a + b*Sec[c + d*x])^(2 + n))/(b^4*d*(2 + n)) - (3*a*(a + b*Sec[c + d*x])^(3 + n))/(b^4*d*(3 + n)) + (a + b*Sec[c + d*x])^(4 + n)/(b^4*d*(4 + n))} -{Tan[c + d*x]^3*(a + b*Sec[c + d*x])^n, x, 4, -((a*(a + b*Sec[c + d*x])^(1 + n))/(b^2*d*(1 + n))) + (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) + (a + b*Sec[c + d*x])^(2 + n)/(b^2*d*(2 + n))} -{Tan[c + d*x]^1*(a + b*Sec[c + d*x])^n, x, 2, -((Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)))} -{Cot[c + d*x]^1*(a + b*Sec[c + d*x])^n, x, 8, -((Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n))) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n))} -{Cot[c + d*x]^3*(a + b*Sec[c + d*x])^n, x, 10, (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a - b)*d*(1 + n)) + (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(2*(a + b)*d*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*Sec[c + d*x])/a]*(a + b*Sec[c + d*x])^(1 + n))/(a*d*(1 + n)) - (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a - b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a - b)^2*d*(1 + n)) + (b*Hypergeometric2F1[2, 1 + n, 2 + n, (a + b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])^(1 + n))/(4*(a + b)^2*d*(1 + n))} - -{Tan[c + d*x]^4*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^4, x]} -{Tan[c + d*x]^2*(a + b*Sec[c + d*x])^n, x, 9, (Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^n*(b*d*Sqrt[1 + Sec[c + d*x]])) - (Sqrt[2]*a*AppellF1[1/2, 1/2, -n, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^n*(b*d*Sqrt[1 + Sec[c + d*x]])) - Unintegrable[(a + b*Sec[c + d*x])^n, x]} -{Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[Cot[c + d*x]^2*(a + b*Sec[c + d*x])^n, x]} -{Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[Cot[c + d*x]^4*(a + b*Sec[c + d*x])^n, x]} - - -{Tan[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Tan[c + d*x]^(3/2), x]} -{Tan[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n*Sqrt[Tan[c + d*x]], x]} -{1/Tan[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n/Sqrt[Tan[c + d*x]], x]} -{1/Tan[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^n, x, 0, Unintegrable[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^(3/2), x]} - - -(* ::Section:: *) -(*Integrands of the form (d Cot[e+f x])^m (a+b Sec[e+f x])^m*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m deleted file mode 100644 index 344d612..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.10 (c+d x)^m (a+b sec)^n.m +++ /dev/null @@ -1,114 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Sec[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sec[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + a*Sec[e + f*x])*(c + d*x)^3, x, 11, (a*(c + d*x)^4)/(4*d) - (2*I*a*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*I*a*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (3*I*a*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (6*a*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (6*a*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 - (6*I*a*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))])/f^4 + (6*I*a*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4} -{(a + a*Sec[e + f*x])*(c + d*x)^2, x, 9, (a*(c + d*x)^3)/(3*d) - (2*I*a*(c + d*x)^2*ArcTan[E^(I*(e + f*x))])/f + (2*I*a*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (2*I*a*d*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (2*a*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (2*a*d^2*PolyLog[3, I*E^(I*(e + f*x))])/f^3} -{(a + a*Sec[e + f*x])*(c + d*x)^1, x, 7, (a*(c + d*x)^2)/(2*d) - (2*I*a*(c + d*x)*ArcTan[E^(I*(e + f*x))])/f + (I*a*d*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (I*a*d*PolyLog[2, I*E^(I*(e + f*x))])/f^2} -{(a + a*Sec[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + a*Sec[e + f*x])/(c + d*x), x]} -{(a + a*Sec[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + a*Sec[e + f*x])/(c + d*x)^2, x]} - - -{(a + a*Sec[e + f*x])^2*(c + d*x)^3, x, 17, -((I*a^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (4*I*a^2*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*a^2*d*(c + d*x)^2*Log[1 + E^(2*I*(e + f*x))])/f^2 + (6*I*a^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (6*I*a^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (3*I*a^2*d^2*(c + d*x)*PolyLog[2, -E^(2*I*(e + f*x))])/f^3 - (12*a^2*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (12*a^2*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 + (3*a^2*d^3*PolyLog[3, -E^(2*I*(e + f*x))])/(2*f^4) - (12*I*a^2*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))])/f^4 + (12*I*a^2*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4 + (a^2*(c + d*x)^3*Tan[e + f*x])/f} -{(a + a*Sec[e + f*x])^2*(c + d*x)^2, x, 14, -((I*a^2*(c + d*x)^2)/f) + (a^2*(c + d*x)^3)/(3*d) - (4*I*a^2*(c + d*x)^2*ArcTan[E^(I*(e + f*x))])/f + (2*a^2*d*(c + d*x)*Log[1 + E^(2*I*(e + f*x))])/f^2 + (4*I*a^2*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (4*I*a^2*d*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (I*a^2*d^2*PolyLog[2, -E^(2*I*(e + f*x))])/f^3 - (4*a^2*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (4*a^2*d^2*PolyLog[3, I*E^(I*(e + f*x))])/f^3 + (a^2*(c + d*x)^2*Tan[e + f*x])/f} -{(a + a*Sec[e + f*x])^2*(c + d*x)^1, x, 9, (a^2*(c + d*x)^2)/(2*d) - (4*I*a^2*(c + d*x)*ArcTan[E^(I*(e + f*x))])/f + (a^2*d*Log[Cos[e + f*x]])/f^2 + (2*I*a^2*d*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (2*I*a^2*d*PolyLog[2, I*E^(I*(e + f*x))])/f^2 + (a^2*(c + d*x)*Tan[e + f*x])/f} -{(a + a*Sec[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + a*Sec[e + f*x])^2/(c + d*x), x]} -{(a + a*Sec[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + a*Sec[e + f*x])^2/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(a + a*Sec[e + f*x])*(c + d*x)^3, x, 9, (I*(c + d*x)^3)/(a*f) + (c + d*x)^4/(4*a*d) - (6*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a*f^2) + (12*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a*f^3) - (12*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a*f^4) - ((c + d*x)^3*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Sec[e + f*x])*(c + d*x)^2, x, 8, (I*(c + d*x)^2)/(a*f) + (c + d*x)^3/(3*a*d) - (4*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(a*f^2) + (4*I*d^2*PolyLog[2, -E^(I*(e + f*x))])/(a*f^3) - ((c + d*x)^2*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Sec[e + f*x])*(c + d*x)^1, x, 5, (c + d*x)^2/(2*a*d) - (2*d*Log[Cos[e/2 + (f*x)/2]])/(a*f^2) - ((c + d*x)*Tan[e/2 + (f*x)/2])/(a*f)} -{1/(a + a*Sec[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Sec[e + f*x])), x]} -{1/(a + a*Sec[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Sec[e + f*x])), x]} - - -{1/(a + a*Sec[e + f*x])^2*(c + d*x)^3, x, 19, (5*I*(c + d*x)^3)/(3*a^2*f) + (c + d*x)^4/(4*a^2*d) - (10*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a^2*f^2) + (4*d^3*Log[Cos[e/2 + (f*x)/2]])/(a^2*f^4) + (20*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a^2*f^3) - (20*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a^2*f^4) - (d*(c + d*x)^2*Sec[e/2 + (f*x)/2]^2)/(2*a^2*f^2) + (2*d^2*(c + d*x)*Tan[e/2 + (f*x)/2])/(a^2*f^3) - (5*(c + d*x)^3*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Sec[e + f*x])^2*(c + d*x)^2, x, 17, (5*I*(c + d*x)^2)/(3*a^2*f) + (c + d*x)^3/(3*a^2*d) - (20*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(3*a^2*f^2) + (20*I*d^2*PolyLog[2, -E^(I*(e + f*x))])/(3*a^2*f^3) - (d*(c + d*x)*Sec[e/2 + (f*x)/2]^2)/(3*a^2*f^2) + (2*d^2*Tan[e/2 + (f*x)/2])/(3*a^2*f^3) - (5*(c + d*x)^2*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Sec[e + f*x])^2*(c + d*x)^1, x, 9, (c + d*x)^2/(2*a^2*d) - (10*d*Log[Cos[e/2 + (f*x)/2]])/(3*a^2*f^2) - (d*Sec[e/2 + (f*x)/2]^2)/(6*a^2*f^2) - (5*(c + d*x)*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)*Sec[e/2 + (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)} -{1/(a + a*Sec[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[1/((c + d*x)*(a + a*Sec[e + f*x])^2), x]} -{1/(a + a*Sec[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Sec[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + a*Sec[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + a*Sec[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + a*Sec[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m*(a + a*Sec[e + f*x]), x]} -{(c + d*x)^m/(a + a*Sec[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m/(a + a*Sec[e + f*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Sec[e + f*x])*(c + d*x)^3, x, 11, (a*(c + d*x)^4)/(4*d) - (2*I*b*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*I*b*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (3*I*b*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (6*b*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (6*b*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 - (6*I*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))])/f^4 + (6*I*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4} -{(a + b*Sec[e + f*x])*(c + d*x)^2, x, 9, (a*(c + d*x)^3)/(3*d) - (2*I*b*(c + d*x)^2*ArcTan[E^(I*(e + f*x))])/f + (2*I*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (2*I*b*d*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (2*b*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (2*b*d^2*PolyLog[3, I*E^(I*(e + f*x))])/f^3} -{(a + b*Sec[e + f*x])*(c + d*x)^1, x, 7, (a*(c + d*x)^2)/(2*d) - (2*I*b*(c + d*x)*ArcTan[E^(I*(e + f*x))])/f + (I*b*d*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (I*b*d*PolyLog[2, I*E^(I*(e + f*x))])/f^2} -{(a + b*Sec[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + b*Sec[e + f*x])/(c + d*x), x]} -{(a + b*Sec[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + b*Sec[e + f*x])/(c + d*x)^2, x]} - - -{(a + b*Sec[e + f*x])^2*(c + d*x)^3, x, 17, -((I*b^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (4*I*a*b*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*b^2*d*(c + d*x)^2*Log[1 + E^(2*I*(e + f*x))])/f^2 + (6*I*a*b*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (6*I*a*b*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (3*I*b^2*d^2*(c + d*x)*PolyLog[2, -E^(2*I*(e + f*x))])/f^3 - (12*a*b*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (12*a*b*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 + (3*b^2*d^3*PolyLog[3, -E^(2*I*(e + f*x))])/(2*f^4) - (12*I*a*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))])/f^4 + (12*I*a*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4 + (b^2*(c + d*x)^3*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x])^2*(c + d*x)^2, x, 14, -((I*b^2*(c + d*x)^2)/f) + (a^2*(c + d*x)^3)/(3*d) - (4*I*a*b*(c + d*x)^2*ArcTan[E^(I*(e + f*x))])/f + (2*b^2*d*(c + d*x)*Log[1 + E^(2*I*(e + f*x))])/f^2 + (4*I*a*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (4*I*a*b*d*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (I*b^2*d^2*PolyLog[2, -E^(2*I*(e + f*x))])/f^3 - (4*a*b*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (4*a*b*d^2*PolyLog[3, I*E^(I*(e + f*x))])/f^3 + (b^2*(c + d*x)^2*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x])^2*(c + d*x)^1, x, 9, (a^2*(c + d*x)^2)/(2*d) - (4*I*a*b*(c + d*x)*ArcTan[E^(I*(e + f*x))])/f + (b^2*d*Log[Cos[e + f*x]])/f^2 + (2*I*a*b*d*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (2*I*a*b*d*PolyLog[2, I*E^(I*(e + f*x))])/f^2 + (b^2*(c + d*x)*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + b*Sec[e + f*x])^2/(c + d*x), x]} -{(a + b*Sec[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + b*Sec[e + f*x])^2/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Sec[e + f*x]), x, 14, (c + d*x)^4/(4*a*d) + (I*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) - (I*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) + (3*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2) - (3*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2) + (6*I*b*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^3) - (6*I*b*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^3) - (6*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^4) + (6*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^4)} -{(c + d*x)^2/(a + b*Sec[e + f*x]), x, 12, (c + d*x)^3/(3*a*d) + (I*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) - (I*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) + (2*b*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2) - (2*b*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2) + (2*I*b*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^3) - (2*I*b*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^3)} -{(c + d*x)^1/(a + b*Sec[e + f*x]), x, 10, (c + d*x)^2/(2*a*d) + (I*b*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) - (I*b*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*f) + (b*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2) - (b*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*f^2)} -{1/((c + d*x)^1*(a + b*Sec[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sec[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Sec[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sec[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Sec[e + f*x])^2, x, 36, -((I*b^2*(c + d*x)^3)/(a^2*(a^2 - b^2)*f)) + (c + d*x)^4/(4*a^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + (2*I*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) - (2*I*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) - (6*I*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - (6*I*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) + (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) - (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (6*b^2*d^3*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) + (6*b^2*d^3*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) - (6*I*b^3*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + (12*I*b*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (6*I*b^3*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - (12*I*b*d^2*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (6*b^3*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) - (12*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) - (6*b^3*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) + (12*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) + (b^2*(c + d*x)^3*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b + a*Cos[e + f*x]))} -{(c + d*x)^2/(a + b*Sec[e + f*x])^2, x, 30, -((I*b^2*(c + d*x)^2)/(a^2*(a^2 - b^2)*f)) + (c + d*x)^3/(3*a^2*d) + (2*b^2*d*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (2*b^2*d*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + (2*I*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) - (2*I*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) - (2*I*b^2*d^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - (2*I*b^2*d^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - (2*b^3*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) + (4*b*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (2*b^3*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) - (4*b*d*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) - (2*I*b^3*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + (4*I*b*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (2*I*b^3*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - (4*I*b*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (b^2*(c + d*x)^2*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b + a*Cos[e + f*x]))} -{(c + d*x)^1/(a + b*Sec[e + f*x])^2, x, 21, (c + d*x)^2/(2*a^2*d) - (I*b^3*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + (2*I*b*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) - (2*I*b*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*f) + (b^2*d*Log[b + a*Cos[e + f*x]])/(a^2*(a^2 - b^2)*f^2) - (b^3*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) + (2*b*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (b^3*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) - (2*b*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (b^2*(c + d*x)*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b + a*Cos[e + f*x]))} -{1/((c + d*x)^1*(a + b*Sec[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sec[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Sec[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sec[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sec[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + b*Sec[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + b*Sec[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + b*Sec[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m*(a + b*Sec[e + f*x]), x]} -{(c + d*x)^m/(a + b*Sec[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m/(a + b*Sec[e + f*x]), x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m deleted file mode 100644 index f4ecf9a..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.11 (e x)^m (a+b sec(c+d x^n))^p.m +++ /dev/null @@ -1,179 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Sec[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sec[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sec[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Sec[c + d*x^2]), x, 10, (a*x^6)/6 - (I*b*x^4*ArcTan[E^(I*(c + d*x^2))])/d + (I*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - (I*b*x^2*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2 - (b*PolyLog[3, (-I)*E^(I*(c + d*x^2))])/d^3 + (b*PolyLog[3, I*E^(I*(c + d*x^2))])/d^3} -{x^4*(a + b*Sec[c + d*x^2]), x, 2, (a*x^5)/5 + b*Unintegrable[x^4*Sec[c + d*x^2], x]} -{x^3*(a + b*Sec[c + d*x^2]), x, 8, (a*x^4)/4 - (I*b*x^2*ArcTan[E^(I*(c + d*x^2))])/d + ((I/2)*b*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - ((I/2)*b*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2} -{x^2*(a + b*Sec[c + d*x^2]), x, 2, (a*x^3)/3 + b*Unintegrable[x^2*Sec[c + d*x^2], x]} -{x*(a + b*Sec[c + d*x^2]), x, 4, (a*x^2)/2 + (b*ArcTanh[Sin[c + d*x^2]])/(2*d)} -{(a + b*Sec[c + d*x^2])/x, x, 2, a*Log[x] + b*Unintegrable[Sec[c + d*x^2]/x, x]} -{(a + b*Sec[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Sec[c + d*x^2]/x^2, x]} - - -{x^5*(a + b*Sec[c + d*x^2])^2, x, 15, ((-I/2)*b^2*x^4)/d + (a^2*x^6)/6 - ((2*I)*a*b*x^4*ArcTan[E^(I*(c + d*x^2))])/d + (b^2*x^2*Log[1 + E^((2*I)*(c + d*x^2))])/d^2 + ((2*I)*a*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - ((2*I)*a*b*x^2*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2 - ((I/2)*b^2*PolyLog[2, -E^((2*I)*(c + d*x^2))])/d^3 - (2*a*b*PolyLog[3, (-I)*E^(I*(c + d*x^2))])/d^3 + (2*a*b*PolyLog[3, I*E^(I*(c + d*x^2))])/d^3 + (b^2*x^4*Tan[c + d*x^2])/(2*d)} -{x^4*(a + b*Sec[c + d*x^2])^2, x, 0, Unintegrable[x^4*(a + b*Sec[c + d*x^2])^2, x]} -{x^3*(a + b*Sec[c + d*x^2])^2, x, 10, (a^2*x^4)/4 - ((2*I)*a*b*x^2*ArcTan[E^(I*(c + d*x^2))])/d + (b^2*Log[Cos[c + d*x^2]])/(2*d^2) + (I*a*b*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - (I*a*b*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2 + (b^2*x^2*Tan[c + d*x^2])/(2*d)} -{x^2*(a + b*Sec[c + d*x^2])^2, x, 0, Unintegrable[x^2*(a + b*Sec[c + d*x^2])^2, x]} -{x*(a + b*Sec[c + d*x^2])^2, x, 5, (a^2*x^2)/2 + (a*b*ArcTanh[Sin[c + d*x^2]])/d + (b^2*Tan[c + d*x^2])/(2*d)} -{(a + b*Sec[c + d*x^2])^2/x, x, 0, Unintegrable[(a + b*Sec[c + d*x^2])^2/x, x]} -{(a + b*Sec[c + d*x^2])^2/x^2, x, 0, Unintegrable[(a + b*Sec[c + d*x^2])^2/x^2, x]} - - -{x*Sec[a + b*x^2]^7, x, 5, (5*ArcTanh[Sin[a + b*x^2]])/(32*b) + (5*Sec[a + b*x^2]*Tan[a + b*x^2])/(32*b) + (5*Sec[a + b*x^2]^3*Tan[a + b*x^2])/(48*b) + (Sec[a + b*x^2]^5*Tan[a + b*x^2])/(12*b)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Sec[c + d*x^2]), x, 13, x^6/(6*a) + ((I/2)*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((I/2)*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (I*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (I*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3)} -{x^4/(a + b*Sec[c + d*x^2]), x, 0, Unintegrable[x^4/(a + b*Sec[c + d*x^2]), x]} -{x^3/(a + b*Sec[c + d*x^2]), x, 11, x^4/(4*a) + ((I/2)*b*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((I/2)*b*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(2*a*Sqrt[-a^2 + b^2]*d^2) - (b*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(2*a*Sqrt[-a^2 + b^2]*d^2)} -{x^2/(a + b*Sec[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Sec[c + d*x^2]), x]} -{x/(a + b*Sec[c + d*x^2]), x, 4, x^2/(2*a) - (b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(x*(a + b*Sec[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Sec[c + d*x^2])), x]} -{(a + b*Sec[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Sec[c + d*x^2]/x^2, x]} - - -{x^5/(a + b*Sec[c + d*x^2])^2, x, 31, ((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((I/2)*b^3*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (I*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^4*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (2*b*x^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (I*b^3*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (b^2*x^4*Sin[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x^2]))} -{x^4/(a + b*Sec[c + d*x^2])^2, x, 0, Unintegrable[x^4/(a + b*Sec[c + d*x^2])^2, x]} -{x^3/(a + b*Sec[c + d*x^2])^2, x, 22, x^4/(4*a^2) - ((I/2)*b^3*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (I*b*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (b^2*Log[b + a*Cos[c + d*x^2]])/(2*a^2*(a^2 - b^2)*d^2) - (b^3*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) + (b*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^2*x^2*Sin[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x^2]))} -{x^2/(a + b*Sec[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Sec[c + d*x^2])^2, x]} -{x/(a + b*Sec[c + d*x^2])^2, x, 6, x^2/(2*a^2) - (b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*Tan[c + d*x^2])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x^2]))} -{1/(x*(a + b*Sec[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Sec[c + d*x^2])^2), x]} -{1/(x^2*(a + b*Sec[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sec[c + d*x^2])^2), x]} -{1/(x^3*(a + b*Sec[c + d*x^2])^2), x, 0, Unintegrable[1/(x^3*(a + b*Sec[c + d*x^2])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sec[c+d x^(1/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sec[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Sec[c + d*Sqrt[x]]), x, 20, (a*x^4)/4 - ((4*I)*b*x^(7/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + ((14*I)*b*x^3*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((14*I)*b*x^3*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (84*b*x^(5/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (84*b*x^(5/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((420*I)*b*x^2*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((420*I)*b*x^2*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (1680*b*x^(3/2)*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (1680*b*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 + ((5040*I)*b*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((5040*I)*b*x*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6 - (10080*b*Sqrt[x]*PolyLog[7, (-I)*E^(I*(c + d*Sqrt[x]))])/d^7 + (10080*b*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))])/d^7 - ((10080*I)*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))])/d^8 + ((10080*I)*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))])/d^8} -{x^2*(a + b*Sec[c + d*Sqrt[x]]), x, 16, (a*x^3)/3 - ((4*I)*b*x^(5/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + ((10*I)*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((10*I)*b*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (40*b*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (40*b*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((120*I)*b*x*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((120*I)*b*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (240*b*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (240*b*Sqrt[x]*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 + ((240*I)*b*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((240*I)*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6} -{x*(a + b*Sec[c + d*Sqrt[x]]), x, 12, (a*x^2)/2 - ((4*I)*b*x^(3/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + ((6*I)*b*x*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((6*I)*b*x*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (12*b*Sqrt[x]*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (12*b*Sqrt[x]*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((12*I)*b*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((12*I)*b*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4} -{(a + b*Sec[c + d*Sqrt[x]])/x, x, 2, a*Log[x] + b*Unintegrable[Sec[c + d*Sqrt[x]]/x, x]} -{(a + b*Sec[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Sec[c + d*Sqrt[x]]/x^2, x]} - - -{x^3*(a + b*Sec[c + d*Sqrt[x]])^2, x, 30, ((-2*I)*b^2*x^(7/2))/d + (a^2*x^4)/4 - ((8*I)*a*b*x^(7/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (14*b^2*x^3*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((28*I)*a*b*x^3*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((28*I)*a*b*x^3*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((42*I)*b^2*x^(5/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 + (105*b^2*x^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((840*I)*a*b*x^2*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((840*I)*a*b*x^2*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + ((210*I)*b^2*x^(3/2)*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*x^(3/2)*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 - (315*b^2*x*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 + ((10080*I)*a*b*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((10080*I)*a*b*x*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6 - ((315*I)*b^2*Sqrt[x]*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, (-I)*E^(I*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))])/d^7 + (315*b^2*PolyLog[7, -E^((2*I)*(c + d*Sqrt[x]))])/(2*d^8) - ((20160*I)*a*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))])/d^8 + ((20160*I)*a*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))])/d^8 + (2*b^2*x^(7/2)*Tan[c + d*Sqrt[x]])/d} -{x^2*(a + b*Sec[c + d*Sqrt[x]])^2, x, 24, ((-2*I)*b^2*x^(5/2))/d + (a^2*x^3)/3 - ((8*I)*a*b*x^(5/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (10*b^2*x^2*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((20*I)*a*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*a*b*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*b^2*x^(3/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (80*a*b*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 + (30*b^2*x*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((240*I)*a*b*x*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((240*I)*a*b*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + ((30*I)*b^2*Sqrt[x]*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 - (15*b^2*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))])/d^6 + ((480*I)*a*b*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((480*I)*a*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6 + (2*b^2*x^(5/2)*Tan[c + d*Sqrt[x]])/d} -{x*(a + b*Sec[c + d*Sqrt[x]])^2, x, 18, ((-2*I)*b^2*x^(3/2))/d + (a^2*x^2)/2 - ((8*I)*a*b*x^(3/2)*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (6*b^2*x*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((12*I)*a*b*x*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((12*I)*a*b*x*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (24*a*b*Sqrt[x]*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (24*a*b*Sqrt[x]*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 + (3*b^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((24*I)*a*b*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((24*I)*a*b*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (2*b^2*x^(3/2)*Tan[c + d*Sqrt[x]])/d} -{(a + b*Sec[c + d*Sqrt[x]])^2/x, x, 0, Unintegrable[(a + b*Sec[c + d*Sqrt[x]])^2/x, x]} -{(a + b*Sec[c + d*Sqrt[x]])^2/x^2, x, 0, Unintegrable[(a + b*Sec[c + d*Sqrt[x]])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Sec[c + d*Sqrt[x]]), x, 23, x^4/(4*a) + ((2*I)*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (14*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (14*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((84*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((84*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) - ((1680*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + ((1680*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + (5040*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) - (5040*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + ((10080*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - ((10080*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8)} -{x^2/(a + b*Sec[c + d*Sqrt[x]]), x, 19, x^3/(3*a) + ((2*I)*b*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (10*b*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (10*b*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((40*I)*b*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((40*I)*b*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) - ((240*I)*b*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + ((240*I)*b*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + (240*b*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) - (240*b*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6)} -{x/(a + b*Sec[c + d*Sqrt[x]]), x, 15, x^2/(2*a) + ((2*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (6*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (6*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((12*I)*b*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((12*I)*b*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (12*b*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (12*b*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4)} -{1/(x*(a + b*Sec[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x*(a + b*Sec[c + d*Sqrt[x]])), x]} -{(a + b*Sec[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Sec[c + d*Sqrt[x]]/x^2, x]} - - -{x^3/(a + b*Sec[c + d*Sqrt[x]])^2, x, 61, ((-2*I)*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) + (14*b^2*x^3*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (14*b^2*x^3*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((84*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((84*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (14*b^3*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (14*b^3*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (420*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (420*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((84*I)*b^3*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((168*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((84*I)*b^3*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((168*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((1680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((1680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (420*b^3*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (420*b^3*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (5040*b^2*x*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) - (5040*b^2*x*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) + ((1680*I)*b^3*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((3360*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((1680*I)*b^3*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((3360*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((10080*I)*b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - ((10080*I)*b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040*b^3*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (5040*b^3*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (10080*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^2*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) - ((10080*I)*b^3*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + ((20160*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + ((10080*I)*b^3*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - ((20160*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) + (2*b^2*x^(7/2)*Sin[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*Sqrt[x]]))} -{x^2/(a + b*Sec[c + d*Sqrt[x]])^2, x, 49, ((-2*I)*b^2*x^(5/2))/(a^2*(a^2 - b^2)*d) + x^3/(3*a^2) + (10*b^2*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (10*b^2*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(5/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((40*I)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((40*I)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (10*b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (20*b*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (10*b^3*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (20*b*x^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (120*b^2*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (120*b^2*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((40*I)*b^3*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((80*I)*b*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((40*I)*b^3*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((80*I)*b*x^(3/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((240*I)*b^2*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((240*I)*b^2*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (120*b^3*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (240*b*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (120*b^3*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (240*b*x*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (240*b^2*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) - (240*b^2*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) + ((240*I)*b^3*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((480*I)*b*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((240*I)*b^3*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((480*I)*b*Sqrt[x]*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - (240*b^3*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (480*b*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (240*b^3*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (480*b*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (2*b^2*x^(5/2)*Sin[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*Sqrt[x]]))} -{x/(a + b*Sec[c + d*Sqrt[x]])^2, x, 37, ((-2*I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (6*b^3*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b^3*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (12*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((12*I)*b^3*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((24*I)*b*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((12*I)*b^3*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((24*I)*b*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (2*b^2*x^(3/2)*Sin[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*Sqrt[x]]))} -{1/(x*(a + b*Sec[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x*(a + b*Sec[c + d*Sqrt[x]])^2), x]} -{1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^(m/2) (a+b Sec[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^(3/2)*(a + b*Sec[c + d*Sqrt[x]]), x, 14, (2*a*x^(5/2))/5 - ((4*I)*b*x^2*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + ((8*I)*b*x^(3/2)*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((8*I)*b*x^(3/2)*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (24*b*x*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (24*b*x*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((48*I)*b*Sqrt[x]*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((48*I)*b*Sqrt[x]*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (48*b*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (48*b*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5} -{Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]]), x, 10, (2*a*x^(3/2))/3 - ((4*I)*b*x*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + ((4*I)*b*Sqrt[x]*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((4*I)*b*Sqrt[x]*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (4*b*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (4*b*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3} -{(a + b*Sec[c + d*Sqrt[x]])/Sqrt[x], x, 4, 2*a*Sqrt[x] + (2*b*ArcTanh[Sin[c + d*Sqrt[x]]])/d} -{(a + b*Sec[c + d*Sqrt[x]])/x^(3/2), x, 2, (-2*a)/Sqrt[x] + b*Unintegrable[Sec[c + d*Sqrt[x]]/x^(3/2), x]} -{(a + b*Sec[c + d*Sqrt[x]])/x^(5/2), x, 2, (-2*a)/(3*x^(3/2)) + b*Unintegrable[Sec[c + d*Sqrt[x]]/x^(5/2), x]} - - -{x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2, x, 21, ((-2*I)*b^2*x^2)/d + (2*a^2*x^(5/2))/5 - ((8*I)*a*b*x^2*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (8*b^2*x^(3/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((16*I)*a*b*x^(3/2)*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((16*I)*a*b*x^(3/2)*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((12*I)*b^2*x*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (48*a*b*x*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (48*a*b*x*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 + (12*b^2*Sqrt[x]*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((96*I)*a*b*Sqrt[x]*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((96*I)*a*b*Sqrt[x]*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + ((6*I)*b^2*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (96*a*b*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (96*a*b*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 + (2*b^2*x^2*Tan[c + d*Sqrt[x]])/d} -{Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2, x, 15, ((-2*I)*b^2*x)/d + (2*a^2*x^(3/2))/3 - ((8*I)*a*b*x*ArcTan[E^(I*(c + d*Sqrt[x]))])/d + (4*b^2*Sqrt[x]*Log[1 + E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((8*I)*a*b*Sqrt[x]*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((8*I)*a*b*Sqrt[x]*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - ((2*I)*b^2*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (8*a*b*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (8*a*b*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 + (2*b^2*x*Tan[c + d*Sqrt[x]])/d} -{(a + b*Sec[c + d*Sqrt[x]])^2/Sqrt[x], x, 5, 2*a^2*Sqrt[x] + (4*a*b*ArcTanh[Sin[c + d*Sqrt[x]]])/d + (2*b^2*Tan[c + d*Sqrt[x]])/d} -{(a + b*Sec[c + d*Sqrt[x]])^2/x^(3/2), x, 0, Unintegrable[(a + b*Sec[c + d*Sqrt[x]])^2/x^(3/2), x]} -{(a + b*Sec[c + d*Sqrt[x]])^2/x^(5/2), x, 0, Unintegrable[(a + b*Sec[c + d*Sqrt[x]])^2/x^(5/2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^(3/2)/(a + b*Sec[c + d*Sqrt[x]]), x, 17, (2*x^(5/2))/(5*a) + ((2*I)*b*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (8*b*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (8*b*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((24*I)*b*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((24*I)*b*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (48*b*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (48*b*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) - ((48*I)*b*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + ((48*I)*b*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5)} -{Sqrt[x]/(a + b*Sec[c + d*Sqrt[x]]), x, 13, (2*x^(3/2))/(3*a) + ((2*I)*b*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (4*b*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (4*b*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((4*I)*b*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((4*I)*b*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3)} -{1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])), x, 4, (2*Sqrt[x])/a - (4*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])), x]} -{1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])), x]} - - -{x^(3/2)/(a + b*Sec[c + d*Sqrt[x]])^2, x, 43, ((-2*I)*b^2*x^2)/(a^2*(a^2 - b^2)*d) + (2*x^(5/2))/(5*a^2) + (8*b^2*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (8*b^2*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((24*I)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((24*I)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (16*b*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (16*b*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((24*I)*b^3*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((48*I)*b*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((24*I)*b^3*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((48*I)*b*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((48*I)*b^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((48*I)*b^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (96*b*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (96*b*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + ((48*I)*b^3*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((96*I)*b*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((48*I)*b^3*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((96*I)*b*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (2*b^2*x^2*Sin[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*Sqrt[x]]))} -{Sqrt[x]/(a + b*Sec[c + d*Sqrt[x]])^2, x, 31, ((-2*I)*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (8*b*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - ((4*I)*b^3*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((8*I)*b*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((4*I)*b^3*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((8*I)*b*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (2*b^2*x*Sin[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*Sqrt[x]]))} -{1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2), x, 6, (2*Sqrt[x])/a^2 - (4*b*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (2*b^2*Tan[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*Sqrt[x]]))} -{1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]} -{1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sec[c+d x^n])^p*) - - -{(e*x)^m*(a + b*Sec[c + d*x^n])^p, x, 1, ((e*x)^m*Unintegrable[x^m*(a + b*Sec[c + d*x^n])^p, x])/x^m} - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(e*x)^(-1 + n)*(a + b*Sec[c + d*x^n]), x, 5, (a*(e*x)^n)/(e*n) + (b*(e*x)^n*ArcTanh[Sin[c + d*x^n]])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n]), x, 9, (a*(e*x)^(2*n))/(2*e*n) - ((2*I)*b*(e*x)^(2*n)*ArcTan[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (I*b*(e*x)^(2*n)*PolyLog[2, (-I)*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b*(e*x)^(2*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n]), x, 11, (a*(e*x)^(3*n))/(3*e*n) - ((2*I)*b*(e*x)^(3*n)*ArcTan[E^(I*(c + d*x^n))])/(d*e*n*x^n) + ((2*I)*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, I*E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)*(a + b*Sec[c + d*x^n])^2, x, 6, (a^2*(e*x)^n)/(e*n) + (2*a*b*(e*x)^n*ArcTanh[Sin[c + d*x^n]])/(d*e*n*x^n) + (b^2*(e*x)^n*Tan[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n])^2, x, 11, (a^2*(e*x)^(2*n))/(2*e*n) - ((4*I)*a*b*(e*x)^(2*n)*ArcTan[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[Cos[c + d*x^n]])/(d^2*e*n*x^(2*n)) + ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, (-I)*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) + (b^2*(e*x)^(2*n)*Tan[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n])^2, x, 16, (a^2*(e*x)^(3*n))/(3*e*n) - (I*b^2*(e*x)^(3*n))/(d*e*n*x^n) - ((4*I)*a*b*(e*x)^(3*n)*ArcTan[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + E^((2*I)*(c + d*x^n))])/(d^2*e*n*x^(2*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b^2*(e*x)^(3*n)*PolyLog[2, -E^((2*I)*(c + d*x^n))])/(d^3*e*n*x^(3*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[3, I*E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (b^2*(e*x)^(3*n)*Tan[c + d*x^n])/(d*e*n*x^n)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(e*x)^(-1 + n)/(a + b*Sec[c + d*x^n]), x, 5, (e*x)^n/(a*e*n) - (2*b*(e*x)^n*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^n)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*e*n*x^n)} -{(e*x)^(-1 + 2*n)/(a + b*Sec[c + d*x^n]), x, 12, (e*x)^(2*n)/(2*a*e*n) + (I*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (I*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n]), x, 14, (e*x)^(3*n)/(3*a*e*n) + (I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)/(a + b*Sec[c + d*x^n])^2, x, 7, (e*x)^n/(a^2*e*n) - (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^n)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d*e*n*x^n) + (b^2*(e*x)^n*Tan[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Sec[c + d*x^n]))} -{(e*x)^(-1 + 2*n)/(a + b*Sec[c + d*x^n])^2, x, 23, (e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[b + a*Cos[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b^2*(e*x)^(2*n)*Sin[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cos[c + d*x^n]))} -{(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2, x, 32, (e*x)^(3*n)/(3*a^2*e*n) - (I*b^2*(e*x)^(3*n))/(x^n*(a^2*(a^2 - b^2)*d*e*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2])])/(x^(2*n)*(a^2*(a^2 - b^2)*d^2*e*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + I*Sqrt[a^2 - b^2])])/(x^(2*n)*(a^2*(a^2 - b^2)*d^2*e*n)) - (I*b^3*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^n*(a^2*(-a^2 + b^2)^(3/2)*d*e*n)) + (2*I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^n*(a^2*Sqrt[-a^2 + b^2]*d*e*n)) + (I*b^3*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^n*(a^2*(-a^2 + b^2)^(3/2)*d*e*n)) - (2*I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^n*(a^2*Sqrt[-a^2 + b^2]*d*e*n)) - (2*I*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2]))])/(x^(3*n)*(a^2*(a^2 - b^2)*d^3*e*n)) - (2*I*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + I*Sqrt[a^2 - b^2]))])/(x^(3*n)*(a^2*(a^2 - b^2)*d^3*e*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(x^(2*n)*(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(x^(2*n)*(a^2*Sqrt[-a^2 + b^2]*d^2*e*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(x^(2*n)*(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(x^(2*n)*(a^2*Sqrt[-a^2 + b^2]*d^2*e*n)) - (2*I*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(x^(3*n)*(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n)) + (4*I*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(x^(3*n)*(a^2*Sqrt[-a^2 + b^2]*d^3*e*n)) + (2*I*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(x^(3*n)*(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n)) - (4*I*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(x^(3*n)*(a^2*Sqrt[-a^2 + b^2]*d^3*e*n)) + (b^2*(e*x)^(3*n)*Sin[c + d*x^n])/(x^n*(a*(a^2 - b^2)*d*e*n*(b + a*Cos[c + d*x^n])))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m deleted file mode 100644 index 4f438d0..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.m +++ /dev/null @@ -1,532 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5, x, 15, a^2*c^5*x - (19*a^2*c^5*ArcTanh[Sin[e + f*x]])/(16*f) - (a^2*c^5*Tan[e + f*x])/f + (17*a^2*c^5*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (a^2*c^5*Tan[e + f*x]^3)/(3*f) - (3*a^2*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) - (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(6*f) + (3*a^2*c^5*Tan[e + f*x]^5)/(5*f)} -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4, x, 11, a^2*c^4*x - (3*a^2*c^4*ArcTanh[Sin[e + f*x]])/(4*f) - (a^2*c^4*Tan[e + f*x])/f + (3*a^2*c^4*Sec[e + f*x]*Tan[e + f*x])/(4*f) + (a^2*c^4*Tan[e + f*x]^3)/(3*f) - (a^2*c^4*Sec[e + f*x]*Tan[e + f*x]^3)/(2*f) + (a^2*c^4*Tan[e + f*x]^5)/(5*f)} -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3, x, 5, a^2*c^3*x - (3*a^2*c^3*ArcTanh[Sin[e + f*x]])/(8*f) - (a^2*(8*c^3 - 3*c^3*Sec[e + f*x])*Tan[e + f*x])/(8*f) + (a^2*(4*c^3 - 3*c^3*Sec[e + f*x])*Tan[e + f*x]^3)/(12*f)} -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2, x, 4, a^2*c^2*x - (a^2*c^2*Tan[e + f*x])/f + (a^2*c^2*Tan[e + f*x]^3)/(3*f)} -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^1, x, 4, a^2*c*x + (a^2*c*ArcTanh[Sin[e + f*x]])/(2*f) - (c*(2*a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(2*f)} -{(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^1, x, 8, (a^2*x)/c - (a^2*ArcTanh[Sin[e + f*x]])/(c*f) - (4*a^2*Tan[e + f*x])/(c*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^2, x, 9, (a^2*x)/c^2 - (4*a^2*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x])^2) - (4*a^2*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^3, x, 12, (a^2*x)/c^3 - (4*a^2*Tan[e + f*x])/(5*c^3*f*(1 - Sec[e + f*x])^3) - (8*a^2*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x])^2) - (23*a^2*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^4, x, 15, (a^2*x)/c^4 - (4*a^2*Tan[e + f*x])/(7*c^4*f*(1 - Sec[e + f*x])^4) - (12*a^2*Tan[e + f*x])/(35*c^4*f*(1 - Sec[e + f*x])^3) - (59*a^2*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x])^2) - (164*a^2*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^5, x, 18, (a^2*x)/c^5 - (4*a^2*Tan[e + f*x])/(9*c^5*f*(1 - Sec[e + f*x])^5) - (16*a^2*Tan[e + f*x])/(63*c^5*f*(1 - Sec[e + f*x])^4) - (37*a^2*Tan[e + f*x])/(105*c^5*f*(1 - Sec[e + f*x])^3) - (179*a^2*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x])^2) - (494*a^2*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x]))} - - -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5, x, 13, a^3*c^5*x - (5*a^3*c^5*ArcTanh[Sin[e + f*x]])/(8*f) - (a^3*c^5*Tan[e + f*x])/f + (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (a^3*c^5*Tan[e + f*x]^3)/(3*f) - (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(12*f) - (a^3*c^5*Tan[e + f*x]^5)/(5*f) + (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(3*f) - (a^3*c^5*Tan[e + f*x]^7)/(7*f)} -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4, x, 6, a^3*c^4*x - (5*a^3*c^4*ArcTanh[Sin[e + f*x]])/(16*f) - (a^3*(16*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x])/(16*f) + (a^3*(8*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x]^3)/(24*f) - (a^3*(6*c^4 - 5*c^4*Sec[e + f*x])*Tan[e + f*x]^5)/(30*f)} -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3, x, 5, a^3*c^3*x - (a^3*c^3*Tan[e + f*x])/f + (a^3*c^3*Tan[e + f*x]^3)/(3*f) - (a^3*c^3*Tan[e + f*x]^5)/(5*f)} -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2, x, 5, a^3*c^2*x + (3*a^3*c^2*ArcTanh[Sin[e + f*x]])/(8*f) - (c^2*(8*a^3 + 3*a^3*Sec[e + f*x])*Tan[e + f*x])/(8*f) + (c^2*(4*a^3 + 3*a^3*Sec[e + f*x])*Tan[e + f*x]^3)/(12*f)} -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^1, x, 9, a^3*c*x + (a^3*c*ArcTanh[Sin[e + f*x]])/f - (a^3*c*Tan[e + f*x])/f - (a^3*c*Sec[e + f*x]*Tan[e + f*x])/f - (a^3*c*Tan[e + f*x]^3)/(3*f)} -{(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^1, x, 15, (a^3*x)/c - (4*a^3*ArcTanh[Sin[e + f*x]])/(c*f) + (8*a^3*Cot[e + f*x])/(c*f) + (8*a^3*Csc[e + f*x])/(c*f) - (a^3*Tan[e + f*x])/(c*f)} -{(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^2, x, 13, (a^3*x)/c^2 + (a^3*ArcTanh[Sin[e + f*x]])/(c^2*f) - (8*a^3*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x])^2) + (4*a^3*Tan[e + f*x])/(3*c^2*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^3, x, 15, (a^3*x)/c^3 - (8*a^3*Tan[e + f*x])/(5*c^3*f*(1 - Sec[e + f*x])^3) + (4*a^3*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x])^2) - (26*a^3*Tan[e + f*x])/(15*c^3*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4, x, 19, (a^3*x)/c^4 - (8*a^3*Tan[e + f*x])/(7*c^4*f*(1 - Sec[e + f*x])^4) + (4*a^3*Tan[e + f*x])/(35*c^4*f*(1 - Sec[e + f*x])^3) - (62*a^3*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x])^2) - (167*a^3*Tan[e + f*x])/(105*c^4*f*(1 - Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^5, x, 23, (a^3*x)/c^5 - (8*a^3*Tan[e + f*x])/(9*c^5*f*(1 - Sec[e + f*x])^5) + (4*a^3*Tan[e + f*x])/(63*c^5*f*(1 - Sec[e + f*x])^4) - (38*a^3*Tan[e + f*x])/(105*c^5*f*(1 - Sec[e + f*x])^3) - (181*a^3*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x])^2) - (496*a^3*Tan[e + f*x])/(315*c^5*f*(1 - Sec[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5, x, 26, (c^5*x)/a^2 - (47*c^5*ArcTanh[Sin[e + f*x]])/(2*a^2*f) + (13*c^5*Tan[e + f*x])/(2*a^2*f) + (112*c^5*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - (32*c^5*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + ((c^5 - c^5*Sec[e + f*x])*Tan[e + f*x])/(2*a^2*f), (c^5*x)/a^2 - (47*c^5*ArcTanh[Sin[e + f*x]])/(2*a^2*f) - (48*c^5*Cot[e + f*x])/(a^2*f) - (64*c^5*Cot[e + f*x]^3)/(3*a^2*f) + (33*c^5*Csc[e + f*x])/(2*a^2*f) + (131*c^5*Csc[e + f*x]^3)/(6*a^2*f) - (c^5*Csc[e + f*x]^3*Sec[e + f*x]^2)/(2*a^2*f) + (7*c^5*Tan[e + f*x])/(a^2*f)} -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4, x, 21, (c^4*x)/a^2 - (6*c^4*ArcTanh[Sin[e + f*x]])/(a^2*f) - (16*c^4*Cot[e + f*x])/(a^2*f) - (32*c^4*Cot[e + f*x]^3)/(3*a^2*f) + (32*c^4*Csc[e + f*x]^3)/(3*a^2*f) + (c^4*Tan[e + f*x])/(a^2*f)} -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3, x, 13, (c^3*x)/a^2 - (c^3*ArcTanh[Sin[e + f*x]])/(a^2*f) - (8*c^3*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) + (4*c^3*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2, x, 9, (c^2*x)/a^2 - (4*c^2*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) - (4*c^2*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^1, x, 7, (c*x)/a^2 - (2*c*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])^2) - (5*c*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^1, x, 4, x/(a^2*c) + (Cot[e + f*x]*(3 - 2*Sec[e + f*x]))/(3*a^2*c*f) - (Cot[e + f*x]^3*(1 - Sec[e + f*x]))/(3*a^2*c*f)} -{1/(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^2, x, 4, x/(a^2*c^2) + Cot[e + f*x]/(a^2*c^2*f) - Cot[e + f*x]^3/(3*a^2*c^2*f)} -{1/(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^3, x, 5, x/(a^2*c^3) + (Cot[e + f*x]^5*(1 + Sec[e + f*x]))/(5*a^2*c^3*f) - (Cot[e + f*x]^3*(5 + 4*Sec[e + f*x]))/(15*a^2*c^3*f) + (Cot[e + f*x]*(15 + 8*Sec[e + f*x]))/(15*a^2*c^3*f)} -{1/(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^4, x, 13, x/(a^2*c^4) + Cot[e + f*x]/(a^2*c^4*f) - Cot[e + f*x]^3/(3*a^2*c^4*f) + Cot[e + f*x]^5/(5*a^2*c^4*f) - (2*Cot[e + f*x]^7)/(7*a^2*c^4*f) + (2*Csc[e + f*x])/(a^2*c^4*f) - (2*Csc[e + f*x]^3)/(a^2*c^4*f) + (6*Csc[e + f*x]^5)/(5*a^2*c^4*f) - (2*Csc[e + f*x]^7)/(7*a^2*c^4*f)} -{1/(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^5, x, 17, x/(a^2*c^5) + Cot[e + f*x]/(a^2*c^5*f) - Cot[e + f*x]^3/(3*a^2*c^5*f) + Cot[e + f*x]^5/(5*a^2*c^5*f) - Cot[e + f*x]^7/(7*a^2*c^5*f) + (4*Cot[e + f*x]^9)/(9*a^2*c^5*f) + (3*Csc[e + f*x])/(a^2*c^5*f) - (13*Csc[e + f*x]^3)/(3*a^2*c^5*f) + (21*Csc[e + f*x]^5)/(5*a^2*c^5*f) - (15*Csc[e + f*x]^7)/(7*a^2*c^5*f) + (4*Csc[e + f*x]^9)/(9*a^2*c^5*f)} - - -{1/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5, x, 29, (c^5*x)/a^3 + (8*c^5*ArcTanh[Sin[e + f*x]])/(a^3*f) + (32*c^5*Cot[e + f*x])/(a^3*f) + (128*c^5*Cot[e + f*x]^3)/(3*a^3*f) + (128*c^5*Cot[e + f*x]^5)/(5*a^3*f) - (16*c^5*Csc[e + f*x])/(a^3*f) + (64*c^5*Csc[e + f*x]^3)/(3*a^3*f) - (128*c^5*Csc[e + f*x]^5)/(5*a^3*f) - (c^5*Tan[e + f*x])/(a^3*f)} -{1/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4, x, 20, (c^4*x)/a^3 + (c^4*ArcTanh[Sin[e + f*x]])/(a^3*f) - (3*c^4*Tan[e + f*x])/(a^3*f*(1 + Sec[e + f*x])^3) - (c^4*Sec[e + f*x]^2*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (14*c^4*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^2) - (23*c^4*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3, x, 15, (c^3*x)/a^3 - (8*c^3*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) + (4*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x])^2) - (26*c^3*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2, x, 12, (c^2*x)/a^3 - (4*c^2*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) - (8*c^2*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x])^2) - (23*c^2*Tan[e + f*x])/(15*a^3*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^1, x, 9, (c*x)/a^3 - (2*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^3) - (3*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x])^2) - (8*c*Tan[e + f*x])/(5*a^3*f*(1 + Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^1, x, 12, x/(a^3*c) + Cot[e + f*x]/(a^3*c*f) - Cot[e + f*x]^3/(3*a^3*c*f) + (2*Cot[e + f*x]^5)/(5*a^3*c*f) - (2*Csc[e + f*x])/(a^3*c*f) + (4*Csc[e + f*x]^3)/(3*a^3*c*f) - (2*Csc[e + f*x]^5)/(5*a^3*c*f)} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^2, x, 5, x/(a^3*c^2) + (Cot[e + f*x]*(15 - 8*Sec[e + f*x]))/(15*a^3*c^2*f) - (Cot[e + f*x]^3*(5 - 4*Sec[e + f*x]))/(15*a^3*c^2*f) + (Cot[e + f*x]^5*(1 - Sec[e + f*x]))/(5*a^3*c^2*f)} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^3, x, 5, x/(a^3*c^3) + Cot[e + f*x]/(a^3*c^3*f) - Cot[e + f*x]^3/(3*a^3*c^3*f) + Cot[e + f*x]^5/(5*a^3*c^3*f)} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4, x, 6, x/(a^3*c^4) - (Cot[e + f*x]^7*(1 + Sec[e + f*x]))/(7*a^3*c^4*f) + (Cot[e + f*x]^5*(7 + 6*Sec[e + f*x]))/(35*a^3*c^4*f) + (Cot[e + f*x]*(35 + 16*Sec[e + f*x]))/(35*a^3*c^4*f) - (Cot[e + f*x]^3*(35 + 24*Sec[e + f*x]))/(105*a^3*c^4*f)} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^5, x, 14, x/(a^3*c^5) + Cot[e + f*x]/(a^3*c^5*f) - Cot[e + f*x]^3/(3*a^3*c^5*f) + Cot[e + f*x]^5/(5*a^3*c^5*f) - Cot[e + f*x]^7/(7*a^3*c^5*f) + (2*Cot[e + f*x]^9)/(9*a^3*c^5*f) + (2*Csc[e + f*x])/(a^3*c^5*f) - (8*Csc[e + f*x]^3)/(3*a^3*c^5*f) + (12*Csc[e + f*x]^5)/(5*a^3*c^5*f) - (8*Csc[e + f*x]^7)/(7*a^3*c^5*f) + (2*Csc[e + f*x]^9)/(9*a^3*c^5*f)} -{1/(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^6, x, 18, x/(a^3*c^6) + Cot[e + f*x]/(a^3*c^6*f) - Cot[e + f*x]^3/(3*a^3*c^6*f) + Cot[e + f*x]^5/(5*a^3*c^6*f) - Cot[e + f*x]^7/(7*a^3*c^6*f) + Cot[e + f*x]^9/(9*a^3*c^6*f) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + (3*Csc[e + f*x])/(a^3*c^6*f) - (16*Csc[e + f*x]^3)/(3*a^3*c^6*f) + (34*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (36*Csc[e + f*x]^7)/(7*a^3*c^6*f) + (19*Csc[e + f*x]^9)/(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^(m/2) (c-c Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^4, x, 5, (2*Sqrt[a]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^4*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^4*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^3*c^4*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) + (2*a^4*c^4*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3, x, 5, (2*Sqrt[a]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^3*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2, x, 5, (2*Sqrt[a]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^1, x, 4, (2*Sqrt[a]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^1, x, 4, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f)} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^2, x, 5, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^2*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^2*f)} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^3, x, 6, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^3*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^2*c^3*f)} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^4, x, 7, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a*c^4*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^2*c^4*f) - (2*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a^3*c^4*f)} - - -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3, x, 6, (2*a^(3/2)*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^4*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) - (2*a^5*c^3*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))} -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2, x, 6, (2*a^(3/2)*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) + (2*a^4*c^2*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))} -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^1, x, 5, (2*a^(3/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^2*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a^3*c*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^1, x, 4, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (4*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f)} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^2, x, 5, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^2*f) - (4*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^2*f)} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^3, x, 6, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^3*f) + (4*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a*c^3*f)} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^4, x, 7, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*a*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^4*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a*c^4*f) - (4*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a^2*c^4*f)} - - -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^3, x, 5, (2*a^(5/2)*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^4*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^5*c^3*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) - (6*a^6*c^3*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2)) - (2*a^7*c^3*Tan[e + f*x]^9)/(9*f*(a + a*Sec[e + f*x])^(9/2))} -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2, x, 5, (2*a^(5/2)*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^4*c^2*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) + (6*a^5*c^2*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2)) + (2*a^6*c^2*Tan[e + f*x]^7)/(7*f*(a + a*Sec[e + f*x])^(7/2))} -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^1, x, 5, (2*a^(5/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f - (2*a^3*c*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a^4*c*Tan[e + f*x]^3)/(f*(a + a*Sec[e + f*x])^(3/2)) - (2*a^5*c*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^1, x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (8*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c*f) - (2*a^3*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^2, x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^2*f) - (8*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^2*f)} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^3, x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^3*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^3*f) + (8*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*c^3*f)} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^4, x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^4*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^4*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^4*f) - (8*Cot[e + f*x]^7*(a + a*Sec[e + f*x])^(7/2))/(7*a*c^4*f)} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^5, x, 5, (2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c^5*f) + (2*a^2*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c^5*f) - (2*a*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*c^5*f) + (2*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*c^5*f) + (8*Cot[e + f*x]^9*(a + a*Sec[e + f*x])^(9/2))/(9*a^2*c^5*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^4, x, 8, (2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (16*Sqrt[2]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (14*c^4*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a*c^4*Tan[e + f*x]^3)/(f*(a + a*Sec[e + f*x])^(3/2)) + (2*a^2*c^4*Tan[e + f*x]^5)/(5*f*(a + a*Sec[e + f*x])^(5/2))} -{1/Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^3, x, 7, (2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (8*Sqrt[2]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (6*c^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*a*c^3*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2))} -{1/Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^2, x, 6, (2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (4*Sqrt[2]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f) + (2*c^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{1/Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^1, x, 5, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (2*Sqrt[2]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f)} -{1/Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^1, x, 6, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f)} -{1/Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^2, x, 7, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c^2*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(2*Sqrt[2]*Sqrt[a]*c^2*f) + (3*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(2*a*c^2*f) - (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(3*a^2*c^2*f)} -{1/Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^3, x, 8, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c^3*f) - ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(4*Sqrt[2]*Sqrt[a]*c^3*f) + (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(4*a*c^3*f) - (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(2*a^2*c^3*f) + (Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(5*a^3*c^3*f)} - - -{1/(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^4, x, 8, (2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) + (12*Sqrt[2]*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (14*c^4*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) + (8*c^4*Tan[e + f*x]^3)/(3*f*(a + a*Sec[e + f*x])^(3/2)) - (a*c^4*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x]*Tan[e + f*x]^4)/(f*(a + a*Sec[e + f*x])^(5/2))} -{1/(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^3, x, 7, (2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) + (2*Sqrt[2]*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (4*c^3*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) + (c^3*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x]*Tan[e + f*x]^2)/(f*(a + a*Sec[e + f*x])^(3/2))} -{1/(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^2, x, 7, (2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (Sqrt[2]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (2*c^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)), (2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (Sqrt[2]*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(3/2)*f) - (c^2*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^1, x, 6, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (3*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) - (c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)), (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - (3*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) - (c*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x])/(2*a*f*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^1, x, 7, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c*f) - (7*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(4*Sqrt[2]*a^(3/2)*c*f) + (Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(4*a^2*c*f) + (Cos[e + f*x]*Cot[e + f*x]*Sec[(1/2)*(e + f*x)]^2*Sqrt[a + a*Sec[e + f*x]])/(4*a^2*c*f)} -{1/(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^2, x, 8, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c^2*f) - (9*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(8*Sqrt[2]*a^(3/2)*c^2*f) + (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(8*a^2*c^2*f) + (Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(12*a^3*c^2*f) - (Cos[e + f*x]*Cot[e + f*x]^3*Sec[(1/2)*(e + f*x)]^2*(a + a*Sec[e + f*x])^(3/2))/(4*a^3*c^2*f)} -{1/(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^3, x, 9, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*c^3*f) - (11*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(16*Sqrt[2]*a^(3/2)*c^3*f) + (21*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(16*a^2*c^3*f) - (5*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(24*a^3*c^3*f) - (3*Cot[e + f*x]^5*(a + a*Sec[e + f*x])^(5/2))/(20*a^4*c^3*f) + (Cos[e + f*x]*Cot[e + f*x]^5*Sec[(1/2)*(e + f*x)]^2*(a + a*Sec[e + f*x])^(5/2))/(4*a^4*c^3*f)} - - -{1/(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^5, x, 9, (2*c^5*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (23*Sqrt[2]*c^5*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(a^(5/2)*f) + (21*c^5*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (19*c^5*Tan[e + f*x]^3)/(6*a*f*(a + a*Sec[e + f*x])^(3/2)) + (3*c^5*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x]*Tan[e + f*x]^4)/(4*f*(a + a*Sec[e + f*x])^(5/2)) + (a*c^5*Sec[(1/2)*(e + f*x)]^4*Sin[e + f*x]^2*Tan[e + f*x]^5)/(4*f*(a + a*Sec[e + f*x])^(7/2))} -{1/(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^4, x, 8, (2*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (11*c^4*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*a^(5/2)*f) + (7*c^4*Tan[e + f*x])/(2*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c^4*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x]*Tan[e + f*x]^2)/(4*a*f*(a + a*Sec[e + f*x])^(3/2)) - (c^4*Sec[(1/2)*(e + f*x)]^4*Sin[e + f*x]^2*Tan[e + f*x]^3)/(4*f*(a + a*Sec[e + f*x])^(5/2))} -{1/(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^3, x, 7, (2*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (7*c^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(2*Sqrt[2]*a^(5/2)*f) - (c^3*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x])/(4*a^2*f*Sqrt[a + a*Sec[e + f*x]]) + (c^3*Sec[(1/2)*(e + f*x)]^4*Sin[e + f*x]^2*Tan[e + f*x])/(4*a*f*(a + a*Sec[e + f*x])^(3/2))} -{1/(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^2, x, 7, (2*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (11*c^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(4*Sqrt[2]*a^(5/2)*f) - (3*c^2*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c^2*Cos[e + f*x]*Sec[(1/2)*(e + f*x)]^4*Sin[e + f*x])/(4*a^2*f*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^1, x, 7, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (23*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(8*Sqrt[2]*a^(5/2)*f) - (c*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(5/2)) - (7*c*Tan[e + f*x])/(8*a*f*(a + a*Sec[e + f*x])^(3/2)), (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - (23*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(8*Sqrt[2]*a^(5/2)*f) - (7*c*Sec[(1/2)*(e + f*x)]^2*Sin[e + f*x])/(16*a^2*f*Sqrt[a + a*Sec[e + f*x]]) - (c*Cos[e + f*x]*Sec[(1/2)*(e + f*x)]^4*Sin[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^1, x, 8, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*c*f) - (71*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(32*Sqrt[2]*a^(5/2)*c*f) - (7*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(32*a^3*c*f) + (13*Cos[e + f*x]*Cot[e + f*x]*Sec[(1/2)*(e + f*x)]^2*Sqrt[a + a*Sec[e + f*x]])/(32*a^3*c*f) + (Cos[e + f*x]^2*Cot[e + f*x]*Sec[(1/2)*(e + f*x)]^4*Sqrt[a + a*Sec[e + f*x]])/(16*a^3*c*f)} -{1/(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^2, x, 9, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*c^2*f) - (107*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(64*Sqrt[2]*a^(5/2)*c^2*f) + (21*Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(64*a^3*c^2*f) + (43*Cot[e + f*x]^3*(a + a*Sec[e + f*x])^(3/2))/(96*a^4*c^2*f) - (15*Cos[e + f*x]*Cot[e + f*x]^3*Sec[(1/2)*(e + f*x)]^2*(a + a*Sec[e + f*x])^(3/2))/(32*a^4*c^2*f) - (Cos[e + f*x]^2*Cot[e + f*x]^3*Sec[(1/2)*(e + f*x)]^4*(a + a*Sec[e + f*x])^(3/2))/(16*a^4*c^2*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^(m/2) (c-c Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2), x, 5, (a*c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2), x, 4, (a*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2), x, 3, (a*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(1/2), x, 2, (a*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(1/2), x, 2, (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(3/2), x, 3, -((a*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2))) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(5/2), x, 4, -((a*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2))) - (a*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x])^(7/2), x, 5, -((a*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2))) - (a*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2), x, 5, (a^2*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (a^2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2), x, 3, (a^2*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^2*c^2*Tan[e + f*x]^3)/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(1/2), x, 3, (a^2*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a*c*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(1/2), x, 3, (a^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*a^2*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(3/2), x, 3, -((2*a^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2))) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(5/2), x, 4, -((a^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2))) - (a^2*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)/(c - c*Sec[e + f*x])^(7/2), x, 5, -((2*a^2*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2))) - (a^2*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^2*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^2*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2), x, 4, (a^3*c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^3*c^3*Tan[e + f*x]^3)/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^3*c^3*Tan[e + f*x]^5)/(4*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2), x, 5, (a^3*c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c^2*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]]) + (c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(1/2), x, 4, (a^3*c*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (a^2*c*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) - (a*c*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(1/2), x, 3, (a^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*a^3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a^3*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(3/2), x, 3, -((4*a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2))) + (a^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(5/2), x, 3, -((2*a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2))) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(7/2), x, 4, -((4*a^3*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2))) - (a^3*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(9/2), x, 5, -((a^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(9/2))) - (a^3*Tan[e + f*x])/(2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(11/2), x, 6, -((4*a^3*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(11/2))) - (a^3*Tan[e + f*x])/(3*c^2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^3*Tan[e + f*x])/(2*c^3*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^5*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(c - c*Sec[e + f*x])^(7/2)/Sqrt[a + a*Sec[e + f*x]], x, 3, (c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (8*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]^2*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(5/2)/Sqrt[a + a*Sec[e + f*x]], x, 3, (c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*c^3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (c^3*Sec[e + f*x]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(3/2)/Sqrt[a + a*Sec[e + f*x]], x, 3, (c^2*Log[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*c^2*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(1/2)/Sqrt[a + a*Sec[e + f*x]], x, 2, (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(1/2)), x, 2, (Log[Sin[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)), x, 3, Tan[e + f*x]/(2*c*f*(1 - Cos[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]), (Log[Cos[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*c*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)), x, 3, (Log[Cos[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (7*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(8*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(8*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(4*c^2*f*(1 - Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (3*Tan[e + f*x])/(4*c^2*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(3/2), x, 3, (c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c^4*Sec[e + f*x]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (8*c^4*Tan[e + f*x])/(a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(3/2), x, 3, -((4*c^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]])) + (c^3*Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(3/2), x, 3, -((2*c^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]])) + (c^2*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(1/2)/(a + a*Sec[e + f*x])^(3/2), x, 3, -((c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]])) + (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(1/2)), x, 3, (Log[Cos[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(4*a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(4*a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2)), x, 3, Cot[e + f*x]/(2*a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)), x, 3, (Log[Cos[e + f*x]]*Tan[e + f*x])/(a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (11*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(16*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (5*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(16*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a*c^2*f*(1 - Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a*c^2*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a*c^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^(5/2), x, 3, (c^4*Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*c^4*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (4*c^4*Tan[e + f*x])/(a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (4*c^4*Tan[e + f*x])/(a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^(5/2), x, 3, -((2*c^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])) + (c^3*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^(5/2), x, 4, -((c^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])) - (c^2*Tan[e + f*x])/(a*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c^2*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(c - c*Sec[e + f*x])^(1/2)/(a + a*Sec[e + f*x])^(5/2), x, 4, -((c*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])) - (c*Tan[e + f*x])/(a*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) + (c*Log[1 + Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(1/2)), x, 3, (Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (7*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(8*a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (3*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)), x, 3, (Log[Cos[e + f*x]]*Tan[e + f*x])/(a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (5*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (11*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(16*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1 - Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(8*a^2*c*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(2*a^2*c*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{1/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)), x, 4, Cot[e + f*x]/(2*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Cot[e + f*x]^3/(4*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n with m and/or n symbolic*) - - -{(1 + Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x, 2, (2^(1/2 + m)*AppellF1[1/2 + n, 1/2 - m, 1, 3/2 + n, (1/2)*(1 - Sec[e + f*x]), 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[1 + Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x, 3, (2^(1/2 + n)*c*AppellF1[1/2 + m, 1/2 - n, 1, 3/2 + m, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(f*(1 + 2*m))} - - -{(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^n, x, 3, (2^(1/2 + n)*c*AppellF1[7/2, 1/2 - n, 1, 9/2, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(7*f)} -{(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^n, x, 3, (2^(1/2 + n)*c*AppellF1[5/2, 1/2 - n, 1, 7/2, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(5*f)} -{(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x])^n, x, 3, (2^(1/2 + n)*c*AppellF1[3/2, 1/2 - n, 1, 5/2, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(3*f)} -{1/(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x])^n, x, 3, -((2^(1/2 + n)*c*AppellF1[-(1/2), 1/2 - n, 1, 1/2, (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])))} -{1/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^n, x, 3, -((2^(1/2 + n)*c*AppellF1[-(3/2), 1/2 - n, 1, -(1/2), (1/2)*(1 + Sec[e + f*x]), 1 + Sec[e + f*x]]*(1 - Sec[e + f*x])^(1/2 - n)*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2))} - - -{(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^n, x, 4, (6*a^3*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) - (2*a^3*(c - c*Sec[e + f*x])^(1 + n)*Tan[e + f*x])/(c*f*(3 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^n, x, 3, (2*a^2*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(1/2)*(c - c*Sec[e + f*x])^n, x, 2, (2*a*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(1/2)*(c - c*Sec[e + f*x])^n, x, 4, -((Hypergeometric2F1[1, 1/2 + n, 3/2 + n, (1/2)*(1 - Sec[e + f*x])]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])) + (2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^n, x, 5, -(((5 - 2*n)*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, (1/2)*(1 - Sec[e + f*x])]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(4*a*f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]])) + (2*Hypergeometric2F1[1, 1/2 + n, 3/2 + n, 1 - Sec[e + f*x]]*(c - c*Sec[e + f*x])^n*Tan[e + f*x])/(a*f*(1 + 2*n)*Sqrt[a + a*Sec[e + f*x]]) - ((c - c*Sec[e + f*x])^n*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sec[e + f*x]]/(c + c*Sec[e + f*x]), x, 6, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) - (Sqrt[2]*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(c*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(c + d*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x]), x, 3, -((2*c*Cot[e + f*x]*EllipticPi[c/(c + d), ArcSin[Sqrt[c + d]/Sqrt[c + d*Sec[e + f*x]]], (c - d)/(c + d)]*Sqrt[-((d*(1 - Sec[e + f*x]))/(c + d*Sec[e + f*x]))]*Sqrt[(d*(1 + Sec[e + f*x]))/(c + d*Sec[e + f*x])]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f)) - ((c - d)*EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])/(a*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])} -{(c + d*Sec[e + f*x])^(1/2)/(a + a*Sec[e + f*x]), x, 3, -((2*Cot[e + f*x]*EllipticPi[c/(c + d), ArcSin[Sqrt[c + d]/Sqrt[c + d*Sec[e + f*x]]], (c - d)/(c + d)]*Sqrt[-((d*(1 - Sec[e + f*x]))/(c + d*Sec[e + f*x]))]*Sqrt[(d*(1 + Sec[e + f*x]))/(c + d*Sec[e + f*x])]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f)) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])/(a*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])} -{1/((c + d*Sec[e + f*x])^(1/2)*(a + a*Sec[e + f*x])), x, 5, (2*Sqrt[c + d]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*(c - d)*f) - (2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(c + d)/c, ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*c*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^(m/2) (c+d Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^4, x, 5, (2*a*d*(2*c + d)*(2*c^2 + 2*c*d + d^2)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^4*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(6*c^2 + 8*c*d + 3*d^2)*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(4*c + 3*d)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^4*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*a^2*f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3, x, 5, (2*a*d*(3*c^2 + 3*c*d + d^2)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(3*c + 2*d)*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*a*f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2, x, 5, (2*a*d*(2*c + d)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(3/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(a - a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^1, x, 4, (2*Sqrt[a]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a*d*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^1, x, 5, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) - (2*Sqrt[a]*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(c*Sqrt[c + d]*f)} -{Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2, x, 7, (2*a^(3/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(3/2)*Sqrt[d]*(3*c + 2*d)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a*d*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3, x, 8, (2*a^(3/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(3/2)*Sqrt[d]*(15*c^2 + 20*c*d + 8*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c^3*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a*d*Tan[e + f*x])/(2*c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) - (a*d*(7*c + 4*d)*Tan[e + f*x])/(4*c^2*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} - - -{(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3, x, 6, (2*a^(5/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(6*c + 13*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(35*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(2*(36*c^3 + 243*c^2*d + 189*c*d^2 + 52*d^3) + d*(24*c^2 + 111*c*d + 52*d^2)*Sec[e + f*x])*Tan[e + f*x])/(105*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2, x, 5, (2*a^(5/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(2*(6*c^2 + 25*c*d + 9*d^2) + d*(4*c + 9*d)*Sec[e + f*x])*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^1, x, 5, (2*a^(3/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a^2*(3*c + 4*d)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(3*f)} -{(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^1, x, 5, (2*a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(c*f) + (2*a^(3/2)*(c - d)*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(c*Sqrt[d]*Sqrt[c + d]*f)} -{(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2, x, 7, (2*a^(5/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(5/2)*(c^2 - 3*c*d - 2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*Sqrt[d]*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - d)*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3, x, 8, (2*a^(5/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(5/2)*(3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c^3*Sqrt[d]*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^2*(c - d)*Tan[e + f*x])/(2*c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (a^2*(3*c^2 - 7*c*d - 4*d^2)*Tan[e + f*x])/(4*c^2*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} - - -{(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3, x, 5, (2*a^3*(3*c^3 + 12*c^2*d + 12*c*d^2 + 4*d^3)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*(3*c^2 + 15*c*d + 13*d^2)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (6*d^2*(c + 2*d)*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*(a - a*Sec[e + f*x])^4*Tan[e + f*x])/(9*a*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^3 + 12*c^2*d + 24*c*d^2 + 12*d^3)*(a^3 - a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2, x, 5, (2*a^3*(c + 2*d)*(3*c + 2*d)*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a*d*(2*c + 5*d)*(a - a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^2*(a - a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[a + a*Sec[e + f*x]]) - (2*(c^2 + 8*c*d + 8*d^2)*(a^3 - a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^1, x, 6, (2*a^(5/2)*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/f + (2*a^3*(35*c + 32*d)*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*(5*c + 8*d)*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*a*d*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)} -{(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^1, x, 7, (2*a^3*Tan[e + f*x])/(d*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*d^(3/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2, x, 10, (2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*d^(3/2)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^(7/2)*(c - d)*Sqrt[c + d]*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*d^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^3*(c - d)^2*Tan[e + f*x])/(c*d*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3, x, 14, (2*a^(7/2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*a^(7/2)*(c - d)^2*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*d^(3/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^(7/2)*(c - d)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*d^(3/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^(7/2)*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^3*(c - d)^2*Tan[e + f*x])/(2*c*d*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (a^3*(c - d)*Tan[e + f*x])/(c^2*d*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) - (3*a^3*(c - d)^2*Tan[e + f*x])/(4*c*d*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{1/Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3, x, 9, (2*(3*c - d)*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (2*d^3*(1 - Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2, x, 7, (2*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*(c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^1, x, 5, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*f) - (Sqrt[2]*(c - d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*f)} -{1/Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^1, x, 8, (2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f) + (2*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*c*(c - d)*Sqrt[c + d]*f)} -{1/Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2, x, 12, (2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c*(c - d)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c - d)^2*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(c*(c^2 - d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{1/Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3, x, 16, (2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*(c - d)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*d^(3/2)*(3*c^2 - 3*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(c^3*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(2*c*(c^2 - d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^2*Tan[e + f*x])/(4*c*(c - d)*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((2*c - d)*d^2*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} - - -{1/(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3, x, 10, (2*d^3*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^3*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - d)^2*(c + 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2, x, 10, -(((c - d)^2*Tan[e + f*x])/(2*a*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])) + (2*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^1, x, 6, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(3/2)*f) - ((5*c - d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(2*Sqrt[2]*a^(3/2)*f) - ((c - d)*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2))} -{1/(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^1, x, 12, -(Tan[e + f*x]/(2*a*(c - d)*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c*(c - d)^2*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2, x, 15, -(Tan[e + f*x]/(2*a*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 3*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*(3*c - d)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^2*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^3*Tan[e + f*x])/(a*c*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3, x, 19, If[$VersionNumber>=8, -(Tan[e + f*x]/(2*a*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 4*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*Sqrt[a]*c*(c - d)^2*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((3*c - d)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^2*(c - d)^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*(6*c^2 - 4*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^3*(c - d)^4*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^3*Tan[e + f*x])/(2*a*c*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) - ((3*c - d)*d^3*Tan[e + f*x])/(a*c^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) - (3*d^3*Tan[e + f*x])/(4*a*c*(c^2 - d^2)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])), -(Tan[e + f*x]/(2*a*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]])) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(Sqrt[a]*c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 4*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sqrt[a]*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*Sqrt[a]*c*(c - d)^2*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((3*c - d)*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^2*(c - d)^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*(6*c^2 - 4*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c^3*(c - d)^4*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d^3*Tan[e + f*x])/(2*a*c*(c - d)^2*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) - ((3*c - d)*d^3*Tan[e + f*x])/(a*c^2*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) - (3*d^3*Tan[e + f*x])/(4*a*c*(c^2 - d^2)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))]} - - -{1/(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3, x, 14, -(((c - d)^3*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]])) - (3*(c - d)^3*Tan[e + f*x])/(16*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*(c + 2*d)*Tan[e + f*x])/(2*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)^2*(c + 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^3 - d^3)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2, x, 14, -(((c - d)^2*Tan[e + f*x])/(4*a^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]])) - (3*(c - d)^2*Tan[e + f*x])/(16*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - ((c^2 - d^2)*Tan[e + f*x])/(2*a^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*c^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*(c - d)^2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c^2 - d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^1, x, 7, (2*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(5/2)*f) - ((43*c - 3*d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(16*Sqrt[2]*a^(5/2)*f) - ((c - d)*Tan[e + f*x])/(4*f*(a + a*Sec[e + f*x])^(5/2)) - ((11*c - 3*d)*Tan[e + f*x])/(16*a*f*(a + a*Sec[e + f*x])^(3/2))} -{1/(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^1, x, 16, -(Tan[e + f*x]/(4*a^2*(c - d)*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]])) - ((c - 2*d)*Tan[e + f*x])/(2*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 3*c*d + 3*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{1/(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2, x, 19, -(Tan[e + f*x]/(4*a^2*(c - d)^2*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]])) - ((c - 3*d)*Tan[e + f*x])/(2*a^2*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 3*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 4*c*d + 6*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c*(c - d)^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*(4*c - d)*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^2*(c - d)^4*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^4*Tan[e + f*x])/(a^2*c*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} -{1/(a + a*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3, x, 23, -(Tan[e + f*x]/(4*a^2*(c - d)^3*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]])) - ((c - 4*d)*Tan[e + f*x])/(2*a^2*(c - d)^4*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)^3*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 4*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 5*c*d + 10*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^5*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*a^(3/2)*c*(c - d)^3*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((4*c - d)*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^2*(c - d)^4*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*d^(7/2)*(10*c^2 - 5*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c^3*(c - d)^5*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^4*Tan[e + f*x])/(2*a^2*c*(c - d)^3*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^4*Tan[e + f*x])/(4*a^2*c*(c - d)^3*(c + d)^2*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((4*c - d)*d^4*Tan[e + f*x])/(a^2*c^2*(c - d)^4*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(1/2), x, 5, (2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/f} -{Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(1/2), x, 2, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[c]*f)} -{Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x])^(3/2), x, 5, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(c^(3/2)*f) - (2*a*d*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]], x, 5, (2*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f) - (Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f)} -{1/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]), x, 5, (2*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c]*f) - (Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^1, x, 4, (a*x)/c + (2*(b*c - a*d)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c*Sqrt[c - d]*Sqrt[c + d]*f)} -{(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^2, x, 5, (a*x)/c^2 + (2*(b*c^3 - 2*a*c^2*d + a*d^3)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) - (d*(b*c - a*d)*Tan[e + f*x])/(c*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))} -{(a + b*Sec[e + f*x])/(c + d*Sec[e + f*x])^3, x, 6, (a*x)/c^3 + ((b*c^3*(2*c^2 + d^2) - a*d*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)*Tan[e + f*x])/(2*c*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (d*(3*b*c^3 - 5*a*c^2*d + 2*a*d^3)*Tan[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} - - -{(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^2, x, 5, (a^2*x)/c^2 + (2*(b*c - a*d)*(2*a*c^2 - b*c*d - a*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) + ((b*c - a*d)^2*Sin[e + f*x])/(c*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))} -{(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^3, x, 6, (a^2*x)/c^3 - ((3*b^2*c^4*d - 2*a*b*c^3*(2*c^2 + d^2) + a^2*(6*c^4*d - 5*c^2*d^3 + 2*d^5))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)^2*Sin[e + f*x])/(2*c^2*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(3*a*d*(2*c^2 - d^2) - b*c*(2*c^2 + d^2))*Sin[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x]))} -{(a + b*Sec[e + f*x])^2/(c + d*Sec[e + f*x])^4, x, 7, (a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^4*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d - b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2) - ((2*a*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))} - - -{(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^3, x, 6, (a^3*x)/c^3 - ((b*c - a*d)*(2*a*b*c*d*(4*c^2 - d^2) - b^2*c^2*(c^2 + 2*d^2) - a^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) + ((b*c - a*d)^2*(b + a*Cos[e + f*x])*Sin[e + f*x])/(2*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) + ((b*c - a*d)^2*(5*a*c^2 - 3*b*c*d - 2*a*d^2)*Sin[e + f*x])/(2*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x]))} -{(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^4, x, 7, If[$VersionNumber>=8, (a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])), (a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(c^4*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*(34*c^4*d - 28*c^2*d^3 + 9*d^5))*Sin[e + f*x])/(6*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x]))]} -{(a + b*Sec[e + f*x])^3/(c + d*Sec[e + f*x])^5, x, 8, (a^3*x)/c^5 - ((15*a*b^2*c^6*d*(4*c^2 + 3*d^2) - 3*a^2*b*c^5*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^3*c^5*(4*c^4 + 27*c^2*d^2 + 4*d^4) + a^3*(40*c^8*d - 40*c^6*d^3 + 63*c^4*d^5 - 36*c^2*d^7 + 8*d^9))*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(4*c^5*Sqrt[c - d]*Sqrt[c + d]*(c^2 - d^2)^4*f) + (d^2*(b + a*Cos[e + f*x])^3*Sin[e + f*x])/(4*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^4) - (d*(8*b*c^3 - 11*a*c^2*d - b*c*d^2 + 4*a*d^3)*(b + a*Cos[e + f*x])^2*Sin[e + f*x])/(12*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^3) - ((b*c - a*d)*(2*a*b*c*d*(32*c^4 + c^2*d^2 + 2*d^4) - a^2*d^2*(58*c^4 - 35*c^2*d^2 + 12*d^4) - b^2*(12*c^6 + 25*c^4*d^2 - 2*c^2*d^4))*Sin[e + f*x])/(24*c^4*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])^2) - ((b^3*c^3*d*(68*c^4 + 39*c^2*d^2 - 2*d^4) + a^2*b*c*d*(272*c^6 + 10*c^4*d^2 + 49*c^2*d^4 - 16*d^6) - 3*a*b^2*c^2*(24*c^6 + 84*c^4*d^2 - 5*c^2*d^4 + 2*d^6) - a^3*(212*c^6*d^2 - 210*c^4*d^4 + 139*c^2*d^6 - 36*d^8))*Sin[e + f*x])/(24*c^4*(c^2 - d^2)^4*f*(d + c*Cos[e + f*x]))} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^(m/2) (c+d Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -(* {Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3, x, 0, (2*(a - b)*Sqrt[a + b]*d*(2*a*d^2 + 3*b*(15*c^2 - 5*c*d + 3*d^2))*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b^2*f) + (2*Sqrt[a + b]*d*(15*a*b*c*d - 2*a^2*d^2 + 9*b^2*(5*c^2 + d^2))*Cot[(1/2)*(e + f*x)]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))])/(15*b^2*f*Sqrt[(b*(1 + Sec[e + f*x]))/(-a + b)]) - (2*c^3*Cot[e + f*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[e + f*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[e + f*x]))/(a + b*Sec[e + f*x]))]*Sqrt[(b*(1 + Sec[e + f*x]))/(a + b*Sec[e + f*x])]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f) + (2*c*d^2*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/f - (4*a*d^3*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(15*b*f) + (2*d^3*(a + b*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*b*f)} -{Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2, x, 0, (2*Sqrt[a + b]*d*(6*b*c + a*d)*Cot[(1/2)*(e + f*x)]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))])/(3*b*f*Sqrt[(b*(1 + Sec[e + f*x]))/(-a + b)]) + (2*(a - b)*Sqrt[a + b]*(6*c - d)*d*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b*f) - (2*c^2*Cot[e + f*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[e + f*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sec[e + f*x]))/(a + b*Sec[e + f*x]))]*Sqrt[(b*(1 + Sec[e + f*x]))/(a + b*Sec[e + f*x])]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f) + (2*d^2*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(3*f)} *) -{Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^1, x, 5, -((2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f)) + (2*Sqrt[a + b]*(b*(c - d) + a*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f} -{Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^1, x, 3, -((2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(c*f)) + (2*(b*c - a*d)*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -(* {Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2, x, 0, 0} -{Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* {(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3, x, 0, 0} -{(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2, x, 0, 0} *) -{(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^1, x, 6, -((2*(a - b)*Sqrt[a + b]*(3*b*c + 4*a*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b*f)) + (2*Sqrt[a + b]*(a*b*(6*c - 4*d) - b^2*(3*c - d) + 3*a^2*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b*f) - (2*a*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*d*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(3*f)} -{(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^1, x, 5, (2*b*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(d*f) - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(c*f) - (2*(b*c - a*d)^2*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -(* {(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2, x, 0, 0} -{(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* {(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3, x, 0, 0} -{(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2, x, 0, 0} *) -{(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^1, x, 7, -((2*(a - b)*Sqrt[a + b]*(35*a*b*c + 23*a^2*d + 9*b^2*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f)) + (2*Sqrt[a + b]*(a^2*b*(45*c - 23*d) - a*b^2*(35*c - 17*d) + b^3*(5*c - 9*d) + 15*a^3*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(15*b*f) - (2*a^2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f + (2*b*(5*b*c + 8*a*d)*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(15*f) + (2*b*d*(a + b*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)} -(* {(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^1, x, 0, 0} -{(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2, x, 0, 0} -{(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -(* {1/Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3, x, 0, -((2*Sqrt[a + b]*c^3*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f*Sqrt[a + b*Sec[e + f*x]])) + (2*(a - b)*Sqrt[a + b]*d^2*(-9*b*c + 2*a*d)*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*b^3*f*Sqrt[a + b*Sec[e + f*x]]) + (2*Sqrt[a + b]*d*(2*a*d^2 + b*(9*c^2 - 9*c*d + d^2))*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))]*Sqrt[(b*(1 + Sec[e + f*x]))/(-a + b)])/(3*b^2*f*Sqrt[a + b*Sec[e + f*x]]) + (2*d^3*Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/(3*b*f)} -{1/Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2, x, 0, -((2*(a - b)*Sqrt[a + b]*d^2*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticE[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f*Sqrt[a + b*Sec[e + f*x]])) - (2*Sqrt[a + b]*c^2*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f*Sqrt[a + b*Sec[e + f*x]]) + (2*Sqrt[a + b]*(2*c - d)*d*Sqrt[Cos[e + f*x]]*Sqrt[b + a*Cos[e + f*x]]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[b + a*Cos[e + f*x]]/(Sqrt[a + b]*Sqrt[Cos[e + f*x]])], (a + b)/(a - b)]*Sqrt[-((b*(-1 + Sec[e + f*x]))/(a + b))]*Sqrt[(b*(1 + Sec[e + f*x]))/(-a + b)])/(b*f*Sqrt[a + b*Sec[e + f*x]])} *) -{1/Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^1, x, 3, (2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f)} -{1/Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^1, x, 3, -((2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*c*f)) - (2*d*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(c*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -(* {1/Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^2, x, 0, 0} -{1/Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* {1/(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^3, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^2, x, 0, 0} *) -{1/(a + b*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^1, x, 6, (2*(b*c - a*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*b*Sqrt[a + b]*f) - (2*(b*c - a*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*b*Sqrt[a + b]*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^2*f) + (2*b*(b*c - a*d)*Tan[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Sec[e + f*x]])} -(* {1/(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^1, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^2, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* {1/(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^3, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^2, x, 0, 0} *) -{1/(a + b*Sec[e + f*x])^(5/2)*(c + d*Sec[e + f*x])^1, x, 7, (2*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*(6*a^2*b*c - a*b^2*c - 3*b^3*c - 3*a^3*d + a^2*b*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a^3*f) + (2*b*(b*c - a*d)*Tan[e + f*x])/(3*a*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)) + (2*b*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Tan[e + f*x])/(3*a^2*(a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])} -(* {1/(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^1, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^2, x, 0, 0} -{1/(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^3, x, 0, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -(* {(a + b*Sec[e + f*x])^(1/2)*(c + d*Sec[e + f*x])^(3/2), x, 1, 0} *) -{(a + b*Sec[e + f*x])^(1/2)*(c + d*Sec[e + f*x])^(1/2), x, 3, -((2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*f)) + (2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[(a + b)/(c + d)]*f)} -{(a + b*Sec[e + f*x])^(1/2)/(c + d*Sec[e + f*x])^(1/2), x, 1, -((2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c*f))} -{(a + b*Sec[e + f*x])^(1/2)/(c + d*Sec[e + f*x])^(3/2), x, 5, -((2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c^2*f)) - (2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*(1 + Sec[e + f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]) - (2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)*f)} -{(a + b*Sec[e + f*x])^(1/2)/(c + d*Sec[e + f*x])^(5/2), x, 7, (2*(a - b)*Sqrt[a + b]*d*(6*b*c^3 - 7*a*c^2*d - 2*b*c*d^2 + 3*a*d^3)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b*c^2*(3*c^2 + 3*c*d - 2*d^2) - a*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])} - - -{(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(3/2), x, 6, -((2*(a - b)*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])) - (2*Sqrt[a + b]*(b*c - a*(2*c - d))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])} -{(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2), x, 7, -((2*(a - b)*Sqrt[a + b]*(3*b*c^3 - 7*a*c^2*d + b*c*d^2 + 3*a*d^3)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])) - (2*Sqrt[a + b]*(b^2*c^3*(3*c + d) - 2*a*b*c^2*(3*c^2 + 2*c*d - d^2) + a^2*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])} -{(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(7/2), x, 8, (2*(a - b)*Sqrt[a + b]*(2*a*b*c*d*(35*c^4 - 8*c^2*d^2 + 5*d^4) - a^2*d^2*(58*c^4 - 41*c^2*d^2 + 15*d^4) - b^2*(15*c^6 + 19*c^4*d^2 - 2*c^2*d^4))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*(b^2*c^3*(15*c^3 + 10*c^2*d + 9*c*d^2 - 2*d^3) - 2*a*b*c^2*(15*c^4 + 20*c^3*d - 4*c^2*d^2 - 4*c*d^3 + 5*d^4) + a^2*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(10*b*c^3 - 13*a*c^2*d - 2*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])} - - -{(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(5/2), x, 7, -((2*(a - b)*Sqrt[a + b]*(7*a*c^2 - 4*b*c*d - 3*a*d^2)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])) + (2*Sqrt[a + b]*(b^2*c^2*(c + 3*d) - a*b*c*(7*c^2 + 4*c*d - 3*d^2) + a^2*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(3*c^3*(c - d)^2*(c + d)^(3/2)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^3*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])} -{(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(7/2), x, 8, If[$VersionNumber>=8, (2*(a - b)*Sqrt[a + b]*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*c^2*d^3 + 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]]), (2*(a - b)*Sqrt[a + b]*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*c^2*d^3 + 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^4*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(b*c - a*d)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(15*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])]} -{(a + b*Sec[e + f*x])^(5/2)/(c + d*Sec[e + f*x])^(9/2), x, 9, (2*(a - b)*Sqrt[a + b]*(2*b^3*c^3*d*(133*c^4 + 62*c^2*d^2 - 3*d^4) + 2*a^2*b*c*d*(406*c^6 + 73*c^4*d^2 + 132*c^2*d^4 - 35*d^6) - a*b^2*c^2*(245*c^6 + 852*c^4*d^2 + 41*c^2*d^4 + 14*d^6) - a^3*(582*c^6*d^2 - 485*c^4*d^4 + 392*c^2*d^6 - 105*d^8))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(105*c^4*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*Sqrt[a + b]*(b^3*c^4*(35*c^4 + 231*c^3*d + 67*c^2*d^2 + 57*c*d^3 - 6*d^4) - a*b^2*c^3*(245*c^5 + 413*c^4*d + 439*c^3*d^2 + 53*c^2*d^3 - 12*c*d^4 + 14*d^5) + a^2*b*c^2*(315*c^6 + 497*c^5*d + 219*c^4*d^2 - 73*c^3*d^3 + 208*c^2*d^4 + 56*c*d^5 - 70*d^6) - a^3*d*(525*c^7 + 57*c^6*d - 699*c^5*d^2 + 214*c^4*d^3 + 672*c^3*d^4 - 280*c^2*d^5 - 210*c*d^6 + 105*d^7))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(105*c^5*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*a^2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^5*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) + (2*d^2*(b + a*Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(7*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^3*Sqrt[c + d*Sec[e + f*x]]) - (2*d*(14*b*c^3 - 19*a*c^2*d - 2*b*c*d^2 + 7*a*d^3)*(b + a*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(35*c^2*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])^2*Sqrt[c + d*Sec[e + f*x]]) - (2*(2*a*b*c*d*(91*c^4 - 2*c^2*d^2 + 7*d^4) - a^2*d^2*(162*c^4 - 101*c^2*d^2 + 35*d^4) - b^2*(35*c^6 + 67*c^4*d^2 - 6*c^2*d^4))*Sqrt[a + b*Sec[e + f*x]]*Sin[e + f*x])/(105*c^3*(c^2 - d^2)^3*f*(d + c*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(c + d*Sec[e + f*x])^(3/2)/Sqrt[a + b*Sec[e + f*x]], x, -1, -((2*c*(c + d)*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x])^(3/2)*Sqrt[((a + b)*(b*c - a*d)*(-1 + Sec[e + f*x])*(c + d*Sec[e + f*x]))/((c + d)^2*(a + b*Sec[e + f*x])^2)])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]])) + (2*d*(c + d)*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x])^(3/2)*Sqrt[-(((a + b)*((-b)*c + a*d)*(-1 + Sec[e + f*x])*(c + d*Sec[e + f*x]))/((c + d)^2*(a + b*Sec[e + f*x])^2))])/(b*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[((b*c - a*d)*(-1 + Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a*b*f*Sqrt[((a + b)*(c + d*Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x]))])} -{(c + d*Sec[e + f*x])^(1/2)/Sqrt[a + b*Sec[e + f*x]], x, 1, -((2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*f))} -{1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(1/2)), x, 3, -((2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(a*Sqrt[a + b]*c*f)) - (2*b*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(a*Sqrt[c + d]*(b*c - a*d)*f)} -{1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)), x, 6, -((2*(a - b)*Sqrt[a + b]*d^2*Cot[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f)) - (2*Sqrt[a + b]*(2*c - d)*d*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c^2*(c - d)*Sqrt[c + d]*(b*c - a*d)*f) - (2*Sqrt[a + b]*Cot[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(a*c^2*Sqrt[c + d]*f), -((2*(a - b)*Sqrt[a + b]*d^2*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*(b*c - a*d)^2*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])) - (2*Sqrt[a + b]*(2*c - d)*d*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(c^2*(c - d)*Sqrt[c + d]*(b*c - a*d)*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]) - (2*Sqrt[a + b]*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])^(3/2)*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[a + b*Sec[e + f*x]])/(a*c^2*Sqrt[c + d]*f*Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])} -(* {1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(5/2)), x, 0, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^(m/3) (c+d Sec[e+f x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(1/3), x, 1, ((d + c*Cos[e + f*x])^(1/3)*(a + b*Sec[e + f*x])^(1/3)*Unintegrable[(b + a*Cos[e + f*x])^(1/3)/(d + c*Cos[e + f*x])^(1/3), x])/((b + a*Cos[e + f*x])^(1/3)*(c + d*Sec[e + f*x])^(1/3))} -{(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(4/3), x]} -{(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(1/3)/(c + d*Sec[e + f*x])^(7/3), x]} - - -{(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(2/3), x, 1, ((d + c*Cos[e + f*x])^(2/3)*(a + b*Sec[e + f*x])^(2/3)*Unintegrable[(b + a*Cos[e + f*x])^(2/3)/(d + c*Cos[e + f*x])^(2/3), x])/((b + a*Cos[e + f*x])^(2/3)*(c + d*Sec[e + f*x])^(2/3))} -{(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(5/3), x]} -{(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(2/3)/(c + d*Sec[e + f*x])^(8/3), x]} - - -{(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(4/3), x, 1, ((d + c*Cos[e + f*x])^(4/3)*(a + b*Sec[e + f*x])^(4/3)*Unintegrable[(b + a*Cos[e + f*x])^(4/3)/(d + c*Cos[e + f*x])^(4/3), x])/((b + a*Cos[e + f*x])^(4/3)*(c + d*Sec[e + f*x])^(4/3))} -{(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(7/3), x]} -{(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x, 0, Unintegrable[(a + b*Sec[e + f*x])^(4/3)/(c + d*Sec[e + f*x])^(10/3), x]} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Title:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c (d Sec[e+f x])^p)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c (d Sec[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (c (d Sec[e+f x])^p)^n with n and p symbolic*) - - -{(a + a*Sec[e + f*x])^m*(c*(d*Sec[e + f*x])^p)^n, x, 4, -((AppellF1[n*p, 1/2, 1/2 - m, 1 + n*p, Sec[e + f*x], -Sec[e + f*x]]*(c*(d*Sec[e + f*x])^p)^n*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*p*Sqrt[1 - Sec[e + f*x]]))} - - -{(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^3, x, 8, (a^3*(7 + 4*n*p)*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*(2 + n*p)*Sqrt[Sin[e + f*x]^2]) - (a^3*(1 + 4*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(5 + 2*n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p)) + ((c*(d*Sec[e + f*x])^p)^n*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n*p))} -{(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^2, x, 7, (2*a^2*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a^2*(1 + 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a^2*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p))} -{(c*(d*Sec[e + f*x])^p)^n*(a + a*Sec[e + f*x])^1, x, 6, (a*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n*p)*Sqrt[Sin[e + f*x]^2])} -{(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x])^1, x, 7, ((c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(a + a*Sec[e + f*x])) - (Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) + ((1 - n*p)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (1/2)*(2 - n*p), (1/2)*(4 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(a*f*(2 - n*p)*Sqrt[Sin[e + f*x]^2])} -{(c*(d*Sec[e + f*x])^p)^n/(a + a*Sec[e + f*x])^2, x, 8, (2*(2 - n*p)*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - ((3 - 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - (2*(2 - n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - ((c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c (d Sec[e+f x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (c (d Sec[e+f x])^p)^n with n and p symbolic*) - - -{(a + b*Sec[e + f*x])^m*(c*(d*Sec[e + f*x])^p)^n, x, 1, ((c*(d*Sec[e + f*x])^p)^n*Unintegrable[(d*Sec[e + f*x])^(n*p)*(a + b*Sec[e + f*x])^m, x])/(d*Sec[e + f*x])^(n*p)} - - -{(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^3, x, 8, (b*(b^2*(1 + n*p) + 3*a^2*(2 + n*p))*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*(2 + n*p)*Sqrt[Sin[e + f*x]^2]) - (a*(3*b^2*n*p + a^2*(1 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (a*b^2*(5 + 2*n*p)*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p)*(2 + n*p)) + (b^2*(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])*Tan[e + f*x])/(f*(2 + n*p))} -{(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^2, x, 7, (2*a*b*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - ((b^2*n*p + a^2*(1 + n*p))*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n^2*p^2)*Sqrt[Sin[e + f*x]^2]) + (b^2*(c*(d*Sec[e + f*x])^p)^n*Tan[e + f*x])/(f*(1 + n*p))} -{(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^1, x, 6, (b*Hypergeometric2F1[1/2, -((n*p)/2), (1/2)*(2 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*n*p*Sqrt[Sin[e + f*x]^2]) - (a*Cos[e + f*x]*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Cos[e + f*x]^2]*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/(f*(1 - n*p)*Sqrt[Sin[e + f*x]^2])} -{(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x])^1, x, 7, -((b*AppellF1[1/2, (n*p)/2, 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^((n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)*f)) + (a*AppellF1[1/2, (1/2)*(-1 + n*p), 1, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n*p))*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)*f)} -{(c*(d*Sec[e + f*x])^p)^n/(a + b*Sec[e + f*x])^2, x, 10, -((2*a*b*AppellF1[1/2, (1/2)*(-2 + n*p), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*(Cos[e + f*x]^2)^((n*p)/2)*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f)) + (a^2*AppellF1[1/2, (1/2)*(-3 + n*p), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n*p))*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f) + (b^2*AppellF1[1/2, (1/2)*(-1 + n*p), 2, 3/2, Sin[e + f*x]^2, (a^2*Sin[e + f*x]^2)/(a^2 - b^2)]*Cos[e + f*x]*(Cos[e + f*x]^2)^((1/2)*(-1 + n*p))*(c*(d*Sec[e + f*x])^p)^n*Sin[e + f*x])/((a^2 - b^2)^2*f)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n.m deleted file mode 100644 index 92fe71e..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n.m +++ /dev/null @@ -1,673 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^4, x, 12, (7*a*c^4*ArcTanh[Sin[e + f*x]])/(8*f) - (a*c^4*Sec[e + f*x]*Tan[e + f*x])/(8*f) - (3*a*c^4*Sec[e + f*x]^3*Tan[e + f*x])/(4*f) + (4*a*c^4*Tan[e + f*x]^3)/(3*f) + (a*c^4*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^3, x, 9, (5*a*c^3*ArcTanh[Sin[e + f*x]])/(8*f) - (3*a*c^3*Sec[e + f*x]*Tan[e + f*x])/(8*f) - (a*c^3*Sec[e + f*x]^3*Tan[e + f*x])/(4*f) + (2*a*c^3*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^2, x, 6, (a*c^2*ArcTanh[Sin[e + f*x]])/(2*f) - (a*c^2*Sec[e + f*x]*Tan[e + f*x])/(2*f) + (a*c^2*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^1, x, 3, (a*c*ArcTanh[Sin[e + f*x]])/(2*f) - (a*c*Sec[e + f*x]*Tan[e + f*x])/(2*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])/(c - c*Sec[e + f*x])^1, x, 2, -((a*ArcTanh[Sin[e + f*x]])/(c*f)) - (2*a*Tan[e + f*x])/(f*(c - c*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])/(c - c*Sec[e + f*x])^2, x, 1, -(((a + a*Sec[e + f*x])*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^2))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])/(c - c*Sec[e + f*x])^3, x, 2, -(((a + a*Sec[e + f*x])*Tan[e + f*x])/(5*f*(c - c*Sec[e + f*x])^3)) - ((a + a*Sec[e + f*x])*Tan[e + f*x])/(15*c*f*(c - c*Sec[e + f*x])^2)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])/(c - c*Sec[e + f*x])^4, x, 3, -(((a + a*Sec[e + f*x])*Tan[e + f*x])/(7*f*(c - c*Sec[e + f*x])^4)) - (2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(35*c*f*(c - c*Sec[e + f*x])^3) - (2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(105*f*(c^2 - c^2*Sec[e + f*x])^2)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])/(c - c*Sec[e + f*x])^5, x, 4, -(((a + a*Sec[e + f*x])*Tan[e + f*x])/(9*f*(c - c*Sec[e + f*x])^5)) - ((a + a*Sec[e + f*x])*Tan[e + f*x])/(21*c*f*(c - c*Sec[e + f*x])^4) - (2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(105*c^2*f*(c - c*Sec[e + f*x])^3) - (2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(315*c*f*(c^2 - c^2*Sec[e + f*x])^2)} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5, x, 14, (9*a^2*c^5*ArcTanh[Sin[e + f*x]])/(16*f) - (3*a^2*c^5*Sec[e + f*x]*Tan[e + f*x])/(16*f) - (3*a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (a^2*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) + (a^2*c^5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(2*f) - (4*a^2*c^5*Tan[e + f*x]^5)/(5*f) - (a^2*c^5*Tan[e + f*x]^7)/(7*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4, x, 11, (7*a^2*c^4*ArcTanh[Sin[e + f*x]])/(16*f) - (5*a^2*c^4*Sec[e + f*x]*Tan[e + f*x])/(16*f) - (a^2*c^4*Sec[e + f*x]^3*Tan[e + f*x])/(8*f) + (a^2*c^4*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) + (a^2*c^4*Sec[e + f*x]^3*Tan[e + f*x]^3)/(6*f) - (2*a^2*c^4*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3, x, 7, (3*a^2*c^3*ArcTanh[Sin[e + f*x]])/(8*f) - (3*a^2*c^3*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (a^2*c^3*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) - (a^2*c^3*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2, x, 4, (3*a^2*c^2*ArcTanh[Sin[e + f*x]])/(8*f) - (3*a^2*c^2*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (a^2*c^2*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^1, x, 6, (a^2*c*ArcTanh[Sin[e + f*x]])/(2*f) - (a^2*c*Sec[e + f*x]*Tan[e + f*x])/(2*f) - (a^2*c*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^1, x, 5, -((3*a^2*ArcTanh[Sin[e + f*x]])/(c*f)) - (3*a^2*Tan[e + f*x])/(c*f) - (2*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^2, x, 3, (a^2*ArcTanh[Sin[e + f*x]])/(c^2*f) - (2*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^2) + (2*a^2*Tan[e + f*x])/(f*(c^2 - c^2*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^3, x, 1, -(((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(c - c*Sec[e + f*x])^3))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^4, x, 2, -(((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(7*f*(c - c*Sec[e + f*x])^4)) - ((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(35*c*f*(c - c*Sec[e + f*x])^3)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^5, x, 3, -(((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(9*f*(c - c*Sec[e + f*x])^5)) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(63*c*f*(c - c*Sec[e + f*x])^4) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(315*c^2*f*(c - c*Sec[e + f*x])^3)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2/(c - c*Sec[e + f*x])^6, x, 4, -(((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(11*f*(c - c*Sec[e + f*x])^6)) - ((a + a*Sec[e + f*x])^2*Tan[e + f*x])/(33*c*f*(c - c*Sec[e + f*x])^5) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(231*c^2*f*(c - c*Sec[e + f*x])^4) - (2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(1155*f*(c^2 - c^2*Sec[e + f*x])^3)} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6, x, 16, (55*a^3*c^6*ArcTanh[Sin[e + f*x]])/(128*f) - (25*a^3*c^6*Sec[e + f*x]*Tan[e + f*x])/(128*f) - (15*a^3*c^6*Sec[e + f*x]^3*Tan[e + f*x])/(64*f) + (5*a^3*c^6*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) + (5*a^3*c^6*Sec[e + f*x]^3*Tan[e + f*x]^3)/(16*f) - (a^3*c^6*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (3*a^3*c^6*Sec[e + f*x]^3*Tan[e + f*x]^5)/(8*f) + (4*a^3*c^6*Tan[e + f*x]^7)/(7*f) + (a^3*c^6*Tan[e + f*x]^9)/(9*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5, x, 13, (45*a^3*c^5*ArcTanh[Sin[e + f*x]])/(128*f) - (35*a^3*c^5*Sec[e + f*x]*Tan[e + f*x])/(128*f) - (5*a^3*c^5*Sec[e + f*x]^3*Tan[e + f*x])/(64*f) + (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) + (5*a^3*c^5*Sec[e + f*x]^3*Tan[e + f*x]^3)/(48*f) - (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (a^3*c^5*Sec[e + f*x]^3*Tan[e + f*x]^5)/(8*f) + (2*a^3*c^5*Tan[e + f*x]^7)/(7*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4, x, 8, (5*a^3*c^4*ArcTanh[Sin[e + f*x]])/(16*f) - (5*a^3*c^4*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (5*a^3*c^4*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) - (a^3*c^4*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) + (a^3*c^4*Tan[e + f*x]^7)/(7*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3, x, 5, (5*a^3*c^3*ArcTanh[Sin[e + f*x]])/(16*f) - (5*a^3*c^3*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (5*a^3*c^3*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) - (a^3*c^3*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2, x, 7, (3*a^3*c^2*ArcTanh[Sin[e + f*x]])/(8*f) - (3*a^3*c^2*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (a^3*c^2*Sec[e + f*x]*Tan[e + f*x]^3)/(4*f) + (a^3*c^2*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^1, x, 9, (5*a^3*c*ArcTanh[Sin[e + f*x]])/(8*f) - (3*a^3*c*Sec[e + f*x]*Tan[e + f*x])/(8*f) - (a^3*c*Sec[e + f*x]^3*Tan[e + f*x])/(4*f) - (2*a^3*c*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^1, x, 6, -((15*a^3*ArcTanh[Sin[e + f*x]])/(2*c*f)) - (10*a^3*Tan[e + f*x])/(c*f) - (5*a^3*Sec[e + f*x]*Tan[e + f*x])/(2*c*f) - (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^2, x, 6, (5*a^3*ArcTanh[Sin[e + f*x]])/(c^2*f) + (5*a^3*Tan[e + f*x])/(c^2*f) - (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^2) + (10*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*(c^2 - c^2*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^3, x, 4, -((a^3*ArcTanh[Sin[e + f*x]])/(c^3*f)) - (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(c - c*Sec[e + f*x])^3) + (2*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*c*f*(c - c*Sec[e + f*x])^2) - (2*a^3*Tan[e + f*x])/(f*(c^3 - c^3*Sec[e + f*x]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^4, x, 1, -(((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*(c - c*Sec[e + f*x])^4))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^5, x, 2, -(((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(9*f*(c - c*Sec[e + f*x])^5)) - ((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(63*c*f*(c - c*Sec[e + f*x])^4)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^6, x, 3, -(((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(11*f*(c - c*Sec[e + f*x])^6)) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(99*c*f*(c - c*Sec[e + f*x])^5) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(693*c^2*f*(c - c*Sec[e + f*x])^4)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3/(c - c*Sec[e + f*x])^7, x, 4, -(((a + a*Sec[e + f*x])^3*Tan[e + f*x])/(13*f*(c - c*Sec[e + f*x])^7)) - (3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(143*c*f*(c - c*Sec[e + f*x])^6) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(429*c^2*f*(c - c*Sec[e + f*x])^5) - (2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(3003*c^3*f*(c - c*Sec[e + f*x])^4)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x]), x, 10, -((35*c^4*ArcTanh[Sin[e + f*x]])/(2*a*f)) + (28*c^4*Tan[e + f*x])/(a*f) - (21*c^4*Sec[e + f*x]*Tan[e + f*x])/(2*a*f) + (2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) + (7*c^4*Tan[e + f*x]^3)/(3*a*f)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x]), x, 6, -((15*c^3*ArcTanh[Sin[e + f*x]])/(2*a*f)) + (10*c^3*Tan[e + f*x])/(a*f) - (5*c^3*Sec[e + f*x]*Tan[e + f*x])/(2*a*f) + (2*c*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x]), x, 5, -((3*c^2*ArcTanh[Sin[e + f*x]])/(a*f)) + (3*c^2*Tan[e + f*x])/(a*f) + (2*(c^2 - c^2*Sec[e + f*x])*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^1/(a + a*Sec[e + f*x]), x, 2, -((c*ArcTanh[Sin[e + f*x]])/(a*f)) + (2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])), x, 3, Csc[e + f*x]/(a*c*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^2), x, 6, -(Cot[e + f*x]^3/(3*a*c^2*f)) + Csc[e + f*x]/(a*c^2*f) - Csc[e + f*x]^3/(3*a*c^2*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^3), x, 10, (2*Cot[e + f*x]^5)/(5*a*c^3*f) + Csc[e + f*x]/(a*c^3*f) - Csc[e + f*x]^3/(a*c^3*f) + (2*Csc[e + f*x]^5)/(5*a*c^3*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^4), x, 13, -(Cot[e + f*x]^5/(5*a*c^4*f)) - (4*Cot[e + f*x]^7)/(7*a*c^4*f) + Csc[e + f*x]/(a*c^4*f) - (2*Csc[e + f*x]^3)/(a*c^4*f) + (9*Csc[e + f*x]^5)/(5*a*c^4*f) - (4*Csc[e + f*x]^7)/(7*a*c^4*f)} - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^2, x, 11, (105*c^5*ArcTanh[Sin[e + f*x]])/(2*a^2*f) - (84*c^5*Tan[e + f*x])/(a^2*f) + (63*c^5*Sec[e + f*x]*Tan[e + f*x])/(2*a^2*f) - (6*c^2*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(f*(a^2 + a^2*Sec[e + f*x])) + (2*c*(c - c*Sec[e + f*x])^4*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (7*c^5*Tan[e + f*x]^3)/(a^2*f)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^2, x, 7, (35*c^4*ArcTanh[Sin[e + f*x]])/(2*a^2*f) - (70*c^4*Tan[e + f*x])/(3*a^2*f) + (35*c^4*Sec[e + f*x]*Tan[e + f*x])/(6*a^2*f) + (2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (14*(c^2 - c^2*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^2, x, 6, (5*c^3*ArcTanh[Sin[e + f*x]])/(a^2*f) - (5*c^3*Tan[e + f*x])/(a^2*f) + (2*c*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (10*(c^3 - c^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^2, x, 3, (c^2*ArcTanh[Sin[e + f*x]])/(a^2*f) - (2*c^2*Tan[e + f*x])/(f*(a^2 + a^2*Sec[e + f*x])) + (2*(c^2 - c^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^1/(a + a*Sec[e + f*x])^2, x, 1, ((c - c*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^1), x, 6, Cot[e + f*x]^3/(3*a^2*c*f) + Csc[e + f*x]/(a^2*c*f) - Csc[e + f*x]^3/(3*a^2*c*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^2), x, 3, Csc[e + f*x]/(a^2*c^2*f) - Csc[e + f*x]^3/(3*a^2*c^2*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^3), x, 7, Cot[e + f*x]^5/(5*a^2*c^3*f) + Csc[e + f*x]/(a^2*c^3*f) - (2*Csc[e + f*x]^3)/(3*a^2*c^3*f) + Csc[e + f*x]^5/(5*a^2*c^3*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^4), x, 10, -((2*Cot[e + f*x]^7)/(7*a^2*c^4*f)) + Csc[e + f*x]/(a^2*c^4*f) - (4*Csc[e + f*x]^3)/(3*a^2*c^4*f) + Csc[e + f*x]^5/(a^2*c^4*f) - (2*Csc[e + f*x]^7)/(7*a^2*c^4*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^5), x, 13, Cot[e + f*x]^7/(7*a^2*c^5*f) + (4*Cot[e + f*x]^9)/(9*a^2*c^5*f) + Csc[e + f*x]/(a^2*c^5*f) - (7*Csc[e + f*x]^3)/(3*a^2*c^5*f) + (3*Csc[e + f*x]^5)/(a^2*c^5*f) - (13*Csc[e + f*x]^7)/(7*a^2*c^5*f) + (4*Csc[e + f*x]^9)/(9*a^2*c^5*f)} - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^6/(a + a*Sec[e + f*x])^3, x, 12, -((231*c^6*ArcTanh[Sin[e + f*x]])/(2*a^3*f)) + (924*c^6*Tan[e + f*x])/(5*a^3*f) - (693*c^6*Sec[e + f*x]*Tan[e + f*x])/(10*a^3*f) - (22*c^2*(c - c*Sec[e + f*x])^4*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + (2*c*(c - c*Sec[e + f*x])^5*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + (66*(c^2 - c^2*Sec[e + f*x])^3*Tan[e + f*x])/(5*f*(a^3 + a^3*Sec[e + f*x])) + (77*c^6*Tan[e + f*x]^3)/(5*a^3*f)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^5/(a + a*Sec[e + f*x])^3, x, 8, -((63*c^5*ArcTanh[Sin[e + f*x]])/(2*a^3*f)) + (42*c^5*Tan[e + f*x])/(a^3*f) - (21*c^5*Sec[e + f*x]*Tan[e + f*x])/(2*a^3*f) - (6*c^2*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(5*a*f*(a + a*Sec[e + f*x])^2) + (2*c*(c - c*Sec[e + f*x])^4*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + (42*c*(c^2 - c^2*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(a^3 + a^3*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^4/(a + a*Sec[e + f*x])^3, x, 7, -((7*c^4*ArcTanh[Sin[e + f*x]])/(a^3*f)) + (7*c^4*Tan[e + f*x])/(a^3*f) + (2*c*(c - c*Sec[e + f*x])^3*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - (14*(c^2 - c^2*Sec[e + f*x])^2*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + (14*(c^4 - c^4*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a^3 + a^3*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^3/(a + a*Sec[e + f*x])^3, x, 4, -((c^3*ArcTanh[Sin[e + f*x]])/(a^3*f)) + (2*c^3*Tan[e + f*x])/(f*(a^3 + a^3*Sec[e + f*x])) + (2*c*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - (2*(c^3 - c^3*Sec[e + f*x])*Tan[e + f*x])/(3*a*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^2/(a + a*Sec[e + f*x])^3, x, 1, ((c - c*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^1/(a + a*Sec[e + f*x])^3, x, 2, ((c - c*Sec[e + f*x])*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((c - c*Sec[e + f*x])*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^1), x, 10, -((2*Cot[e + f*x]^5)/(5*a^3*c*f)) + Csc[e + f*x]/(a^3*c*f) - Csc[e + f*x]^3/(a^3*c*f) + (2*Csc[e + f*x]^5)/(5*a^3*c*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^2), x, 7, -(Cot[e + f*x]^5/(5*a^3*c^2*f)) + Csc[e + f*x]/(a^3*c^2*f) - (2*Csc[e + f*x]^3)/(3*a^3*c^2*f) + Csc[e + f*x]^5/(5*a^3*c^2*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^3), x, 4, Csc[e + f*x]/(a^3*c^3*f) - (2*Csc[e + f*x]^3)/(3*a^3*c^3*f) + Csc[e + f*x]^5/(5*a^3*c^3*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^4), x, 7, -(Cot[e + f*x]^7/(7*a^3*c^4*f)) + Csc[e + f*x]/(a^3*c^4*f) - Csc[e + f*x]^3/(a^3*c^4*f) + (3*Csc[e + f*x]^5)/(5*a^3*c^4*f) - Csc[e + f*x]^7/(7*a^3*c^4*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5), x, 10, (2*Cot[e + f*x]^9)/(9*a^3*c^5*f) + Csc[e + f*x]/(a^3*c^5*f) - (5*Csc[e + f*x]^3)/(3*a^3*c^5*f) + (9*Csc[e + f*x]^5)/(5*a^3*c^5*f) - Csc[e + f*x]^7/(a^3*c^5*f) + (2*Csc[e + f*x]^9)/(9*a^3*c^5*f)} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^6), x, 13, -(Cot[e + f*x]^9/(9*a^3*c^6*f)) - (4*Cot[e + f*x]^11)/(11*a^3*c^6*f) + Csc[e + f*x]/(a^3*c^6*f) - (8*Csc[e + f*x]^3)/(3*a^3*c^6*f) + (22*Csc[e + f*x]^5)/(5*a^3*c^6*f) - (4*Csc[e + f*x]^7)/(a^3*c^6*f) + (17*Csc[e + f*x]^9)/(9*a^3*c^6*f) - (4*Csc[e + f*x]^11)/(11*a^3*c^6*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c-c Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(7/2), x, 4, -((256*c^4*(a + a*Sec[e + f*x])*Tan[e + f*x])/(315*f*Sqrt[c - c*Sec[e + f*x]])) - (64*c^3*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(105*f) - (8*c^2*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(21*f) - (2*c*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(9*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2), x, 3, -((64*c^3*(a + a*Sec[e + f*x])*Tan[e + f*x])/(105*f*Sqrt[c - c*Sec[e + f*x]])) - (16*c^2*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(35*f) - (2*c*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(7*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2), x, 2, -((8*c^2*(a + a*Sec[e + f*x])*Tan[e + f*x])/(15*f*Sqrt[c - c*Sec[e + f*x]])) - (2*c*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]], x, 1, -((2*c*(a + a*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/Sqrt[c - c*Sec[e + f*x]], x, 3, (-2*Sqrt[2]*a*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f) + (2*a*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^(3/2), x, 3, (a*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[2]*c^(3/2)*f) - (a*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c - c*Sec[e + f*x])^(5/2), x, 4, (a*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(8*Sqrt[2]*c^(5/2)*f) - (a*Tan[e + f*x])/(2*f*(c - c*Sec[e + f*x])^(5/2)) + (a*Tan[e + f*x])/(8*c*f*(c - c*Sec[e + f*x])^(3/2))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(7/2), x, 4, -((256*c^4*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(1155*f*Sqrt[c - c*Sec[e + f*x]])) - (64*c^3*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(231*f) - (8*c^2*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(33*f) - (2*c*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(11*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2), x, 3, -((64*c^3*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(315*f*Sqrt[c - c*Sec[e + f*x]])) - (16*c^2*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(63*f) - (2*c*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(9*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2), x, 2, -((8*c^2*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(35*f*Sqrt[c - c*Sec[e + f*x]])) - (2*c*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(7*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]], x, 1, -((2*c*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/Sqrt[c - c*Sec[e + f*x]], x, 4, -((4*Sqrt[2]*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f)) + (16*a^2*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]]) - (2*a^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*c*f), -((4*Sqrt[2]*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f)) + (4*a^2*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) + (2*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(3/2), x, 4, (3*Sqrt[2]*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(c^(3/2)*f) - (2*a^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2)) - (2*a^2*Tan[e + f*x])/(c*f*Sqrt[c - c*Sec[e + f*x]]), (3*Sqrt[2]*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(c^(3/2)*f) - ((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2)) - (3*a^2*Tan[e + f*x])/(c*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(5/2), x, 4, -((3*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(4*Sqrt[2]*c^(5/2)*f)) - (a^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(5/2)) + (5*a^2*Tan[e + f*x])/(4*c*f*(c - c*Sec[e + f*x])^(3/2)), -((3*a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(4*Sqrt[2]*c^(5/2)*f)) - ((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(2*f*(c - c*Sec[e + f*x])^(5/2)) + (3*a^2*Tan[e + f*x])/(4*c*f*(c - c*Sec[e + f*x])^(3/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c - c*Sec[e + f*x])^(7/2), x, 5, -((a^2*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(16*Sqrt[2]*c^(7/2)*f)) - ((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^(7/2)) + (a^2*Tan[e + f*x])/(4*c*f*(c - c*Sec[e + f*x])^(5/2)) - (a^2*Tan[e + f*x])/(16*c^2*f*(c - c*Sec[e + f*x])^(3/2))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(7/2), x, 4, -((256*c^4*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(3003*f*Sqrt[c - c*Sec[e + f*x]])) - (64*c^3*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(429*f) - (24*c^2*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(143*f) - (2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(13*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2), x, 3, -((64*c^3*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(693*f*Sqrt[c - c*Sec[e + f*x]])) - (16*c^2*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(99*f) - (2*c*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(11*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2), x, 2, -((8*c^2*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(63*f*Sqrt[c - c*Sec[e + f*x]])) - (2*c*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(9*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]], x, 1, -((2*c*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/Sqrt[c - c*Sec[e + f*x]], x, 5, (-8*Sqrt[2]*a^3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(Sqrt[c]*f) + (8*a^3*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) + (2*a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[c - c*Sec[e + f*x]]) + (4*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^(3/2), x, 5, (10*Sqrt[2]*a^3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(c^(3/2)*f) - (a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2)) - (10*a^3*Tan[e + f*x])/(c*f*Sqrt[c - c*Sec[e + f*x]]) - (5*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(3*c*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c - c*Sec[e + f*x])^(5/2), x, 5, (-15*a^3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(2*Sqrt[2]*c^(5/2)*f) - (a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(2*f*(c - c*Sec[e + f*x])^(5/2)) + (5*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(4*c*f*(c - c*Sec[e + f*x])^(3/2)) + (15*a^3*Tan[e + f*x])/(4*c^2*f*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x]), x, 4, (128*c^4*Tan[e + f*x])/(5*a*f*Sqrt[c - c*Sec[e + f*x]]) + (32*c^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(5*a*f) + (12*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*a*f) + (2*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x]), x, 3, (32*c^3*Tan[e + f*x])/(3*a*f*Sqrt[c - c*Sec[e + f*x]]) + (8*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*a*f) + (2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x]), x, 2, (4*c^2*Tan[e + f*x])/(a*f*Sqrt[c - c*Sec[e + f*x]]) + (2*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x]), x, 1, (2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]), x, 3, -(ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])]/(Sqrt[2]*a*Sqrt[c]*f)) + Tan[e + f*x]/(f*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)), x, 4, (-3*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(4*Sqrt[2]*a*c^(3/2)*f) - (3*Tan[e + f*x])/(4*a*f*(c - c*Sec[e + f*x])^(3/2)) + Tan[e + f*x]/(f*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)), x, 5, (-15*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(32*Sqrt[2]*a*c^(5/2)*f) - (5*Tan[e + f*x])/(8*a*f*(c - c*Sec[e + f*x])^(5/2)) + Tan[e + f*x]/(f*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)) - (15*Tan[e + f*x])/(32*a*c*f*(c - c*Sec[e + f*x])^(3/2))} - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^2, x, 4, -((64*c^4*Tan[e + f*x])/(3*a^2*f*Sqrt[c - c*Sec[e + f*x]])) - (16*c^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*a^2*f) - (4*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*(a^2 + a^2*Sec[e + f*x])) + (2*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^2, x, 3, (-16*c^3*Tan[e + f*x])/(3*a^2*f*Sqrt[c - c*Sec[e + f*x]]) - (8*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])) + (2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^2, x, 2, -((4*c^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])) + (2*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^2, x, 1, (2*c*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]), x, 4, -(ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])]/(2*Sqrt[2]*a^2*Sqrt[c]*f)) + Tan[e + f*x]/(3*f*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]) + Tan[e + f*x]/(2*f*(a^2 + a^2*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)), x, 5, (-5*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(8*Sqrt[2]*a^2*c^(3/2)*f) - (5*Tan[e + f*x])/(8*a^2*f*(c - c*Sec[e + f*x])^(3/2)) + Tan[e + f*x]/(3*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)) + (5*Tan[e + f*x])/(6*f*(a^2 + a^2*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)), x, 6, (-35*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(64*Sqrt[2]*a^2*c^(5/2)*f) - (35*Tan[e + f*x])/(48*a^2*f*(c - c*Sec[e + f*x])^(5/2)) + Tan[e + f*x]/(3*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)) + (7*Tan[e + f*x])/(6*f*(a^2 + a^2*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)) - (35*Tan[e + f*x])/(64*a^2*c*f*(c - c*Sec[e + f*x])^(3/2))} - - -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(7/2)/(a + a*Sec[e + f*x])^3, x, 4, (32*c^4*Tan[e + f*x])/(5*a^3*f*Sqrt[c - c*Sec[e + f*x]]) + (16*c^3*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(5*f*(a^3 + a^3*Sec[e + f*x])) - (4*c^2*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*a*f*(a + a*Sec[e + f*x])^2) + (2*c*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2)/(a + a*Sec[e + f*x])^3, x, 3, (16*c^3*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]]) - (8*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + (2*c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)} -{Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2)/(a + a*Sec[e + f*x])^3, x, 2, -((4*c^2*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]])) + (2*c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)} -{Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]]/(a + a*Sec[e + f*x])^3, x, 1, (2*c*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]), x, 5, -(ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])]/(4*Sqrt[2]*a^3*Sqrt[c]*f)) + Tan[e + f*x]/(5*f*(a + a*Sec[e + f*x])^3*Sqrt[c - c*Sec[e + f*x]]) + Tan[e + f*x]/(6*a*f*(a + a*Sec[e + f*x])^2*Sqrt[c - c*Sec[e + f*x]]) + Tan[e + f*x]/(4*f*(a^3 + a^3*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)), x, 6, (-7*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(16*Sqrt[2]*a^3*c^(3/2)*f) - (7*Tan[e + f*x])/(16*a^3*f*(c - c*Sec[e + f*x])^(3/2)) + Tan[e + f*x]/(5*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(3/2)) + (7*Tan[e + f*x])/(30*a*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(3/2)) + (7*Tan[e + f*x])/(12*f*(a^3 + a^3*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)), x, 7, (-63*ArcTan[(Sqrt[c]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sec[e + f*x]])])/(128*Sqrt[2]*a^3*c^(5/2)*f) - (21*Tan[e + f*x])/(32*a^3*f*(c - c*Sec[e + f*x])^(5/2)) + Tan[e + f*x]/(5*f*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^(5/2)) + (3*Tan[e + f*x])/(10*a*f*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x])^(5/2)) + (21*Tan[e + f*x])/(20*f*(a^3 + a^3*Sec[e + f*x])*(c - c*Sec[e + f*x])^(5/2)) - (63*Tan[e + f*x])/(128*a^3*c*f*(c - c*Sec[e + f*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^(m/2) (c-c Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2), x, 1, (a*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2), x, 1, (a*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]], x, 1, -((c*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]], x, 1, (a*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(3/2), x, 1, -((Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(2*f*(c - c*Sec[e + f*x])^(3/2)))} -{(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - c*Sec[e + f*x])^(5/2), x, 1, -((a*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2), x, 2, (a^2*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(10*f*Sqrt[a + a*Sec[e + f*x]]) + (a*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2), x, 2, (a^2*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(6*f*Sqrt[a + a*Sec[e + f*x]]) + (a*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2), x, 2, -((c^2*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]])) - (c*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]], x, 1, -((c*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/Sqrt[c - c*Sec[e + f*x]], x, 2, (2*a^2*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(3/2), x, 2, -((a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2))) - (a^2*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(5/2), x, 1, -((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(4*f*(c - c*Sec[e + f*x])^(5/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(7/2), x, 2, -(((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(6*f*(c - c*Sec[e + f*x])^(7/2))) - ((a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(24*c*f*(c - c*Sec[e + f*x])^(5/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(9/2), x, 2, -((a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(4*f*(c - c*Sec[e + f*x])^(9/2))) + (a^2*Tan[e + f*x])/(12*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(3/2))/(c - c*Sec[e + f*x])^(11/2), x, 2, -((a*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(5*f*(c - c*Sec[e + f*x])^(11/2))) + (a^2*Tan[e + f*x])/(20*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(9/2))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(7/2), x, 3, (a^3*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(15*f*Sqrt[a + a*Sec[e + f*x]]) + (2*a^2*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(15*f) + (a*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(7/2)*Tan[e + f*x])/(6*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2), x, 3, -((2*c^3*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(15*f*Sqrt[c - c*Sec[e + f*x]])) - (c^2*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(5*f) - (c*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2), x, 2, -((c^2*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(6*f*Sqrt[c - c*Sec[e + f*x]])) - (c*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]], x, 1, -((c*(a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(3*f*Sqrt[c - c*Sec[e + f*x]]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/Sqrt[c - c*Sec[e + f*x]], x, 3, (4*a^3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*a^2*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[c - c*Sec[e + f*x]]) + (a*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(3/2), x, 3, -((a*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])^(3/2))) - (4*a^3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (2*a^2*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(c*f*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(5/2), x, 3, -(a*(a + a*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*(c - c*Sec[e + f*x])^(5/2)) + (a^2*Sqrt[a + a*Sec[e + f*x]]*Tan[e + f*x])/(c*f*(c - c*Sec[e + f*x])^(3/2)) + (a^3*Log[1 - Sec[e + f*x]]*Tan[e + f*x])/(c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(7/2), x, 1, -(((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(6*f*(c - c*Sec[e + f*x])^(7/2)))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(9/2), x, 2, -(((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(8*f*(c - c*Sec[e + f*x])^(9/2))) - ((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(48*c*f*(c - c*Sec[e + f*x])^(7/2))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^(5/2))/(c - c*Sec[e + f*x])^(11/2), x, 3, -(((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(10*f*(c - c*Sec[e + f*x])^(11/2))) - ((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(40*c*f*(c - c*Sec[e + f*x])^(9/2)) - ((a + a*Sec[e + f*x])^(5/2)*Tan[e + f*x])/(240*c^2*f*(c - c*Sec[e + f*x])^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/Sqrt[a + a*Sec[e + f*x]], x, 3, (-4*c^3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (2*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) - (c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*Sqrt[a + a*Sec[e + f*x]])} -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/Sqrt[a + a*Sec[e + f*x]], x, 2, (-2*c^2*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]])} -{(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/Sqrt[a + a*Sec[e + f*x]], x, 1, -((c*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]))} -{Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]), x, 2, -((ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]))} -{Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)), x, 3, -Tan[e + f*x]/(2*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) - (ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)), x, 4, -Tan[e + f*x]/(4*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(5/2)) - Tan[e + f*x]/(4*c*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) - (ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(4*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^(3/2), x, 3, (4*c^3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (2*c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]) + (c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2))} -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^(3/2), x, 2, (c^2*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + (c*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])^(3/2))} -{(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(3/2), x, 1, (Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]), x, 3, Tan[e + f*x]/(2*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) - (ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(2*a*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(3/2)), x, 3, Csc[e + f*x]/(2*a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(2*a*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)), x, 4, (3*Csc[e + f*x])/(8*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - Tan[e + f*x]/(4*f*(a + a*Sec[e + f*x])^(3/2)*(c - c*Sec[e + f*x])^(5/2)) - (3*ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(8*a*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(5/2))/(a + a*Sec[e + f*x])^(5/2), x, 3, -((c^3*Log[1 + Sec[e + f*x]]*Tan[e + f*x])/(a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])) - (c^2*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(a*f*(a + a*Sec[e + f*x])^(3/2)) + (c*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(5/2))} -{(Sec[e + f*x]*(c - c*Sec[e + f*x])^(3/2))/(a + a*Sec[e + f*x])^(5/2), x, 1, ((c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(4*f*(a + a*Sec[e + f*x])^(5/2))} -{(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(5/2), x, 1, (c*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]), x, 4, Tan[e + f*x]/(4*f*(a + a*Sec[e + f*x])^(5/2)*Sqrt[c - c*Sec[e + f*x]]) + Tan[e + f*x]/(4*a*f*(a + a*Sec[e + f*x])^(3/2)*Sqrt[c - c*Sec[e + f*x]]) - (ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(4*a^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)), x, 4, (3*Csc[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) + Tan[e + f*x]/(4*f*(a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(3/2)) - (3*ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(8*a^2*c*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(5/2)), x, 4, (3*Csc[e + f*x])/(8*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (Cot[e + f*x]^2*Csc[e + f*x])/(4*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (3*ArcTanh[Cos[e + f*x]]*Tan[e + f*x])/(8*a^2*c^2*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n with m and/or n symbolic*) - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^n, x, 3, -((2^(1/2 + n)*c*Hypergeometric2F1[1/2 + m, 1/2 - n, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(1 - Sec[e + f*x])^(1/2 - n)*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 + n)*Tan[e + f*x])/(f*(1 + 2*m)))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^2, x, 3, (2^(1/2 + m)*a*Hypergeometric2F1[5/2, 1/2 - m, 7/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(1/2 - m)*(a + a*Sec[e + f*x])^(-1 + m)*(c - c*Sec[e + f*x])^2*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^1, x, 3, (2^(1/2 + m)*a*Hypergeometric2F1[3/2, 1/2 - m, 5/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(1/2 - m)*(a + a*Sec[e + f*x])^(-1 + m)*(c - c*Sec[e + f*x])*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^1, x, 3, -((2^(1/2 + m)*a*Hypergeometric2F1[-(1/2), 1/2 - m, 1/2, (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(1/2 - m)*(a + a*Sec[e + f*x])^(-1 + m)*Tan[e + f*x])/(f*(c - c*Sec[e + f*x])))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^2, x, 3, -((2^(1/2 + m)*a*Hypergeometric2F1[-(3/2), 1/2 - m, -(1/2), (1/2)*(1 - Sec[e + f*x])]*(1 + Sec[e + f*x])^(1/2 - m)*(a + a*Sec[e + f*x])^(-1 + m)*Tan[e + f*x])/(3*f*(c - c*Sec[e + f*x])^2))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(5/2), x, 3, -((64*c^3*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sec[e + f*x]])) - (16*c^2*(a + a*Sec[e + f*x])^m*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*(15 + 16*m + 4*m^2)) - (2*c*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(3/2)*Tan[e + f*x])/(f*(5 + 2*m))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(3/2), x, 2, -((8*c^2*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(3 + 8*m + 4*m^2)*Sqrt[c - c*Sec[e + f*x]])) - (2*c*(a + a*Sec[e + f*x])^m*Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(f*(3 + 2*m))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(1/2), x, 1, -((2*c*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[c - c*Sec[e + f*x]]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(1/2), x, 2, -((Hypergeometric2F1[1, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*(1 + 2*m)*Sqrt[c - c*Sec[e + f*x]]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(3/2), x, 2, -((Hypergeometric2F1[2, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(2*c*f*(1 + 2*m)*Sqrt[c - c*Sec[e + f*x]]))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(5/2), x, 2, -((Hypergeometric2F1[3, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(4*c^2*f*(1 + 2*m)*Sqrt[c - c*Sec[e + f*x]]))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m + 3), x, 3, -(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-3 - m)*Tan[e + f*x])/(f*(1 + 2*m))) + (2*(a + a*Sec[e + f*x])^(1 + m)*(c - c*Sec[e + f*x])^(-3 - m)*Tan[e + f*x])/(a*f*(3 + 8*m + 4*m^2)) - (2*(a + a*Sec[e + f*x])^(2 + m)*(c - c*Sec[e + f*x])^(-3 - m)*Tan[e + f*x])/(a^2*f*(1 + 2*m)*(15 + 16*m + 4*m^2))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m + 2), x, 2, -(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(f*(1 + 2*m))) + ((a + a*Sec[e + f*x])^(1 + m)*(c - c*Sec[e + f*x])^(-2 - m)*Tan[e + f*x])/(a*f*(3 + 8*m + 4*m^2))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m + 1), x, 1, -(((a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 - m)*Tan[e + f*x])/(f*(1 + 2*m)))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m + 0), x, 3, -((2^(1/2 - m)*c*Hypergeometric2F1[1/2 + m, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(1 - Sec[e + f*x])^(1/2 + m)*(a + a*Sec[e + f*x])^m*(c - c*Sec[e + f*x])^(-1 - m)*Tan[e + f*x])/(f*(1 + 2*m)))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m - 1), x, 3, -((2^(3/2 - m)*c*Hypergeometric2F1[-(1/2) + m, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(1 - Sec[e + f*x])^(-(1/2) + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/((c - c*Sec[e + f*x])^m*(f*(1 + 2*m))))} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^m/(c - c*Sec[e + f*x])^(m - 2), x, 3, -((2^(5/2 - m)*c^2*Hypergeometric2F1[-(3/2) + m, 1/2 + m, 3/2 + m, (1/2)*(1 + Sec[e + f*x])]*(1 - Sec[e + f*x])^(-(1/2) + m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/((c - c*Sec[e + f*x])^m*(f*(1 + 2*m))))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^m (c-c Sec[e+f x])*) - - -{Sec[e + f*x]^2*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]), x, 10, (a^3*c*ArcTanh[Sin[e + f*x]])/(4*f) + (a^3*c*Sec[e + f*x]*Tan[e + f*x])/(4*f) - (a^3*c*Sec[e + f*x]^3*Tan[e + f*x])/(2*f) - (2*a^3*c*Tan[e + f*x]^3)/(3*f) - (a^3*c*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]^2*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]), x, 7, (a^2*c*ArcTanh[Sin[e + f*x]])/(8*f) + (a^2*c*Sec[e + f*x]*Tan[e + f*x])/(8*f) - (a^2*c*Sec[e + f*x]^3*Tan[e + f*x])/(4*f) - (a^2*c*Tan[e + f*x]^3)/(3*f)} -{Sec[e + f*x]^2*(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x]), x, 3, -((a*c*Tan[e + f*x]^3)/(3*f))} -{Sec[e + f*x]^2/(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x]), x, 5, (2*c*ArcTanh[Sin[e + f*x]])/(a*f) - (c*Tan[e + f*x])/(a*f) - (2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]^2/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]), x, 4, -((c*ArcTanh[Sin[e + f*x]])/(a^2*f)) + (7*c*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - (2*c*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} -{Sec[e + f*x]^2/(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x]), x, 3, -((2*c*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3)) + (11*c*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) - (4*c*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x]))} - - -{(g*Sec[e + f*x])^p*(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]), x, 5, -((a^2*c*(Cos[e + f*x]^2)^((3 + p)/2)*Hypergeometric2F1[3/2, (3 + p)/2, 5/2, Sin[e + f*x]^2]*(g*Sec[e + f*x])^p*Tan[e + f*x]^3)/(3*f)) - (a^2*c*(Cos[e + f*x]^2)^((4 + p)/2)*Hypergeometric2F1[3/2, (4 + p)/2, 5/2, Sin[e + f*x]^2]*(g*Sec[e + f*x])^(1 + p)*Tan[e + f*x]^3)/(3*f*g)} -{(g*Sec[e + f*x])^p*(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x]), x, 2, -((a*c*(Cos[e + f*x]^2)^((3 + p)/2)*Hypergeometric2F1[3/2, (3 + p)/2, 5/2, Sin[e + f*x]^2]*(g*Sec[e + f*x])^p*Tan[e + f*x]^3)/(3*f))} -{(g*Sec[e + f*x])^p/(a + a*Sec[e + f*x])^1*(c - c*Sec[e + f*x]), x, 6, -((c*g*(1 - 2*p)*Hypergeometric2F1[1/2, (1 - p)/2, (3 - p)/2, Cos[e + f*x]^2]*(g*Sec[e + f*x])^(-1 + p)*Sin[e + f*x])/(a*f*(1 - p)*Sqrt[Sin[e + f*x]^2])) + (2*c*Hypergeometric2F1[1/2, -(p/2), (2 - p)/2, Cos[e + f*x]^2]*(g*Sec[e + f*x])^p*Sin[e + f*x])/(a*f*Sqrt[Sin[e + f*x]^2]) - (2*c*(g*Sec[e + f*x])^p*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{(g*Sec[e + f*x])^p/(a + a*Sec[e + f*x])^2*(c - c*Sec[e + f*x]), x, 7, -((c*g*(3 - 4*p)*Hypergeometric2F1[1/2, (1 - p)/2, (3 - p)/2, Cos[e + f*x]^2]*(g*Sec[e + f*x])^(-1 + p)*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2])) + (c*(5 - 4*p)*Hypergeometric2F1[1/2, -(p/2), (2 - p)/2, Cos[e + f*x]^2]*(g*Sec[e + f*x])^p*Sin[e + f*x])/(3*a^2*f*Sqrt[Sin[e + f*x]^2]) - (c*(5 - 4*p)*(g*Sec[e + f*x])^p*Tan[e + f*x])/(3*a^2*f*(1 + Sec[e + f*x])) - (2*c*(g*Sec[e + f*x])^p*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^(m/2) / (c-c Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(g*Sec[e + f*x])^(3/2)*Sqrt[a + a*Sec[e + f*x]]/(c - c*Sec[e + f*x]), x, 5, -((2*Sqrt[a]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(c*f)) + (2*g*Cot[e + f*x]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])/(c*f), -((2*a*g*Sqrt[g*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) + (2*a*g^(3/2)*ArcTan[(Sqrt[c]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[c - c*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])), x, 4, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f)) + (Cot[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f), -(Tan[e + f*x]/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) - (ArcTanh[Sqrt[c - c*Sec[e + f*x]]/(Sqrt[2]*Sqrt[c])]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{Sec[e + f*x]^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])), x, 8, -((2*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f)) + ArcTanh[(Sqrt[a]*Sqrt[Sec[e + f*x]]*Sin[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Csc[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f*Sqrt[Sec[e + f*x]]), -((Sec[e + f*x]^(3/2)*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) + (2*ArcTan[(Sqrt[c]*Sqrt[Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]]]*Tan[e + f*x])/(Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]]]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -{(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])), x, 4, -((g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*Sqrt[a]*c*f)) + (g*Cot[e + f*x]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f), -((g*Sqrt[g*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) + (g^(3/2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[c - c*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} -{(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])), x, 8, -((2*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*c*f)) + (g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[2]*Sqrt[a]*c*f) + (g^2*Cot[e + f*x]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])/(a*c*f), -((g^2*Sqrt[g*Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) + (2*g^(5/2)*ArcTan[(Sqrt[c]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[c - c*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]) - (g^(5/2)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[g*Sec[e + f*x]])/(Sqrt[g]*Sqrt[c - c*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[2]*Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^(m/2) (c-c Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]]), x, 3, (Log[Tan[e + f*x]]*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -{(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c - d*Sec[e + f*x]), x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c - d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[c - d]*Sqrt[d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^4, x, 8, (a*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*ArcTanh[Sin[e + f*x]])/(8*f) + (a*(12*c^4 + 95*c^3*d + 112*c^2*d^2 + 80*c*d^3 + 16*d^4)*Tan[e + f*x])/(30*f) + (a*d*(24*c^3 + 130*c^2*d + 116*c*d^2 + 45*d^3)*Sec[e + f*x]*Tan[e + f*x])/(120*f) + (a*(12*c^2 + 35*c*d + 16*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(60*f) + (a*(4*c + 5*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*f) + (a*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3, x, 7, (a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*ArcTanh[Sin[e + f*x]])/(8*f) + (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3)*Tan[e + f*x])/(6*f) + (a*d*(6*c^2 + 20*c*d + 9*d^2)*Sec[e + f*x]*Tan[e + f*x])/(24*f) + (a*(3*c + 4*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + (a*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2, x, 6, (a*(2*c^2 + 2*c*d + d^2)*ArcTanh[Sin[e + f*x]])/(2*f) + (2*a*(c^2 + 3*c*d + d^2)*Tan[e + f*x])/(3*f) + (a*d*(2*c + 3*d)*Sec[e + f*x]*Tan[e + f*x])/(6*f) + (a*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^1, x, 5, (a*(2*c + d)*ArcTanh[Sin[e + f*x]])/(2*f) + (a*(c + d)*Tan[e + f*x])/f + (a*d*Sec[e + f*x]*Tan[e + f*x])/(2*f)} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^1, x, 5, (a*ArcTanh[Sin[e + f*x]])/(d*f) - (2*a*Sqrt[c - d]*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(d*Sqrt[c + d]*f)} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^2, x, 5, (2*a*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*(c + d)^(3/2)*f) + (a*Tan[e + f*x])/((c + d)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^3, x, 6, (a*(2*c - d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(5/2)*f) + (a*Tan[e + f*x])/(2*(c + d)*f*(c + d*Sec[e + f*x])^2) + (a*(c - 2*d)*Tan[e + f*x])/(2*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x]))/(c + d*Sec[e + f*x])^4, x, 7, (a*(2*c^2 - 2*c*d + d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(7/2)*f) + (a*Tan[e + f*x])/(3*(c + d)*f*(c + d*Sec[e + f*x])^3) + (a*(2*c - 3*d)*Tan[e + f*x])/(6*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x])^2) + (a*(c - 4*d)*(2*c - d)*Tan[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sec[e + f*x]))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^4, x, 9, (a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*ArcTanh[Sin[e + f*x]])/(16*f) - (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c^2*d^3 - 288*c*d^4 - 64*d^5)*Tan[e + f*x])/(60*d*f) - (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d^4)*Sec[e + f*x]*Tan[e + f*x])/(240*f) - (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(120*d*f) - (a^2*(4*c^2 - 48*c*d - 55*d^2)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(120*d*f) - (a^2*(c - 12*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(30*d*f) + (a^2*(c + d*Sec[e + f*x])^5*Tan[e + f*x])/(6*d*f), (a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*Tan[e + f*x])/(16*f) + (a^3*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(8*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(48*f) + (d*(9*c + 2*d)*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(30*f) + (d*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(6*f) + (1/(120*f))*(d*(a + a*Sec[e + f*x])^2*(2*(52*c^3 + 56*c^2*d + 48*c*d^2 + 9*d^3) + d*(48*c^2 + 32*c*d + 19*d^2)*Sec[e + f*x])*Tan[e + f*x])} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3, x, 8, (3*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*ArcTanh[Sin[e + f*x]])/(8*f) - (a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*Tan[e + f*x])/(10*d*f) - (a^2*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*Sec[e + f*x]*Tan[e + f*x])/(40*f) - (a^2*(c^2 - 10*c*d - 12*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(20*d*f) - (a^2*(c - 10*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*d*f) + (a^2*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*d*f), (3*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*Tan[e + f*x])/(8*f) + (3*a^3*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(8*f) + (d*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(5*f) + (d*(a + a*Sec[e + f*x])^2*(2*(8*c^2 + 5*c*d + 2*d^2) + d*(7*c + 2*d)*Sec[e + f*x])*Tan[e + f*x])/(20*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2, x, 8, (a^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTanh[Sin[e + f*x]])/(8*f) - (a^2*(c^3 - 8*c^2*d - 20*c*d^2 - 8*d^3)*Tan[e + f*x])/(6*d*f) - (a^2*(2*c*(c - 8*d) - 21*d^2)*Sec[e + f*x]*Tan[e + f*x])/(24*f) - (a^2*(c - 8*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(12*d*f) + (a^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(4*d*f), (a^2*(12*c^2 + 16*c*d + 7*d^2)*Tan[e + f*x])/(8*f) + (a^3*(12*c^2 + 16*c*d + 7*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d*(5*c + 2*d)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + ((12*c^2 + 16*c*d + 7*d^2)*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(24*f) + (d*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^1, x, 6, (a^2*(3*c + 2*d)*ArcTanh[Sin[e + f*x]])/(2*f) + (2*a^2*(3*c + 2*d)*Tan[e + f*x])/(3*f) + (a^2*(3*c + 2*d)*Sec[e + f*x]*Tan[e + f*x])/(6*f) + (d*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*f)} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^1, x, 8, -((a^2*(c - 2*d)*ArcTanh[Sin[e + f*x]])/(d^2*f)) + (2*a^2*(c - d)^(3/2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(d^2*Sqrt[c + d]*f) + (a^2*Tan[e + f*x])/(d*f), (a^2*Tan[e + f*x])/(d*f) - (2*a^3*(c - 2*d)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(d^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^3*(c - d)^(3/2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(d^2*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^2, x, 8, (a^2*ArcTanh[Sin[e + f*x]])/(d^2*f) - (2*a^2*Sqrt[c - d]*(c + 2*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(d^2*(c + d)^(3/2)*f) - (a^2*(c - d)*Tan[e + f*x])/(d*(c + d)*f*(c + d*Sec[e + f*x])), (2*a^3*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(d^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^3*Sqrt[c - d]*(c + 2*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(d^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (a^2*(c - d)*Tan[e + f*x])/(d*(c + d)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^3, x, 5, (3*a^2*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(Sqrt[c - d]*(c + d)^(5/2)*f) + ((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(2*(c + d)*f*(c + d*Sec[e + f*x])^2) + (3*a^2*Tan[e + f*x])/(2*(c + d)^2*f*(c + d*Sec[e + f*x])), -((3*a^3*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[c - d]*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])) + ((a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(2*(c + d)*f*(c + d*Sec[e + f*x])^2) + (3*a^2*Tan[e + f*x])/(2*(c + d)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^4, x, 6, (a^2*(3*c - 2*d)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(7/2)*f) - (d*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) + ((3*c - 2*d)*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(6*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x])^2) + (a^2*(3*c - 2*d)*Tan[e + f*x])/(2*(c - d)*(c + d)^3*f*(c + d*Sec[e + f*x])), -((a^3*(3*c - 2*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/((c - d)^(3/2)*(c + d)^(7/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])) - (d*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) + ((3*c - 2*d)*(a^2 + a^2*Sec[e + f*x])*Tan[e + f*x])/(6*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x])^2) + (a^2*(3*c - 2*d)*Tan[e + f*x])/(2*(c - d)*(c + d)^3*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^5, x, 8, (a^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(4*(c - d)^(5/2)*(c + d)^(9/2)*f) - (a^2*(c - d)*Tan[e + f*x])/(4*d*(c + d)*f*(c + d*Sec[e + f*x])^4) + (a^2*(c + 8*d)*Tan[e + f*x])/(12*d*(c + d)^2*f*(c + d*Sec[e + f*x])^3) + (a^2*(2*c^2 + 16*c*d - 21*d^2)*Tan[e + f*x])/(24*(c - d)*d*(c + d)^3*f*(c + d*Sec[e + f*x])^2) + (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Tan[e + f*x])/(24*(c - d)^2*d*(c + d)^4*f*(c + d*Sec[e + f*x])), -((a^3*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(4*(c - d)^(5/2)*(c + d)^(9/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])) - (a^2*(c - d)*Tan[e + f*x])/(4*d*(c + d)*f*(c + d*Sec[e + f*x])^4) + (a^2*(c + 8*d)*Tan[e + f*x])/(12*d*(c + d)^2*f*(c + d*Sec[e + f*x])^3) + (a^2*(2*c^2 + 16*c*d - 21*d^2)*Tan[e + f*x])/(24*(c - d)*d*(c + d)^3*f*(c + d*Sec[e + f*x])^2) + (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Tan[e + f*x])/(24*(c - d)^2*d*(c + d)^4*f*(c + d*Sec[e + f*x]))} - - -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3, x, 9, (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*ArcTanh[Sin[e + f*x]])/(16*f) + (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Tan[e + f*x])/(16*f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(48*f) + (a*(3*c + 8*d)*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(30*f) + (a*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(6*f) + (a*(a + a*Sec[e + f*x])^2*(2*(4*c^3 + 74*c^2*d + 66*c*d^2 + 21*d^3) + d*(6*c^2 + 62*c*d + 31*d^2)*Sec[e + f*x])*Tan[e + f*x])/(120*f), (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Tan[e + f*x])/(16*f) + (a^4*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(8*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(120*f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(48*f) + (d*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(6*f) + (d*(a + a*Sec[e + f*x])^3*(70*c^2 + 54*c*d + 19*d^2 + 4*d*(8*c + 3*d)*Sec[e + f*x])*Tan[e + f*x])/(120*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2, x, 9, (a^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTanh[Sin[e + f*x]])/(8*f) + (a^3*(2*c^4 - 15*c^3*d + 72*c^2*d^2 + 180*c*d^3 + 76*d^4)*Tan[e + f*x])/(30*d^2*f) + (a^3*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Sec[e + f*x]*Tan[e + f*x])/(120*d*f) + (a^3*(2*c^2 - 15*c*d + 76*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(60*d^2*f) - (a^3*(2*c - 11*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*d^2*f) + ((a^3 + a^3*Sec[e + f*x])*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(5*d*f), (a^3*(20*c^2 + 30*c*d + 13*d^2)*Tan[e + f*x])/(8*f) + (a^4*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(4*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a*(20*c^2 + 30*c*d + 13*d^2)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(60*f) + (3*d*(2*c + d)*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(20*f) + ((20*c^2 + 30*c*d + 13*d^2)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(24*f) + (d*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1, x, 10, (5*a^3*(4*c + 3*d)*ArcTanh[Sin[e + f*x]])/(8*f) + (a^3*(4*c + 3*d)*Tan[e + f*x])/f + (3*a^3*(4*c + 3*d)*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (d*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(4*f) + (a^3*(4*c + 3*d)*Tan[e + f*x]^3)/(12*f)} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^1, x, 9, (a^3*ArcTanh[Sin[e + f*x]])/(2*d*f) + (a^3*(c^2 - 3*c*d + 3*d^2)*ArcTanh[Sin[e + f*x]])/(d^3*f) - (2*a^3*(c - d)^(5/2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(d^3*Sqrt[c + d]*f) - (a^3*(c - 3*d)*Tan[e + f*x])/(d^2*f) + (a^3*Sec[e + f*x]*Tan[e + f*x])/(2*d*f), -((a^3*(2*c - 5*d)*Tan[e + f*x])/(2*d^2*f)) + (a^4*(2*c^2 - 6*c*d + 7*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(d^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*a^4*(c - d)^(5/2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(d^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(2*d*f)} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2, x, 9, -((a^3*(2*c - 3*d)*ArcTanh[Sin[e + f*x]])/(d^3*f)) + (2*a^3*(c - d)^(3/2)*(2*c + 3*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(d^3*(c + d)^(3/2)*f) + (2*a^3*c*Tan[e + f*x])/(d^2*(c + d)*f) - ((c - d)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(d*(c + d)*f*(c + d*Sec[e + f*x])), (2*a^3*c*Tan[e + f*x])/(d^2*(c + d)*f) - (2*a^4*(2*c - 3*d)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(d^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^4*(c - d)^(3/2)*(2*c + 3*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(d^3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(d*(c + d)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^3, x, 9, (a^3*ArcTanh[Sin[e + f*x]])/(d^3*f) - (a^3*Sqrt[c - d]*(2*c^2 + 6*c*d + 7*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(d^3*(c + d)^(5/2)*f) - ((c - d)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(2*d*(c + d)*f*(c + d*Sec[e + f*x])^2) - (a^3*(c - d)*(2*c + 5*d)*Tan[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sec[e + f*x])), (2*a^4*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(d^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (a^4*Sqrt[c - d]*(2*c^2 + 6*c*d + 7*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(d^3*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(2*d*(c + d)*f*(c + d*Sec[e + f*x])^2) - (a^3*(c - d)*(2*c + 5*d)*Tan[e + f*x])/(2*d^2*(c + d)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^4, x, 6, (5*a^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*(c + d)^(7/2)*f) + (a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*(c + d)*f*(c + d*Sec[e + f*x])^3) - (5*a^3*(c - d)*Tan[e + f*x])/(6*d*(c + d)^2*f*(c + d*Sec[e + f*x])^2) + (5*a^3*(c + 4*d)*Tan[e + f*x])/(6*d*(c + d)^3*f*(c + d*Sec[e + f*x])), -((5*a^4*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(Sqrt[c - d]*(c + d)^(7/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])) + (a*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(3*(c + d)*f*(c + d*Sec[e + f*x])^3) + (5*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(6*(c + d)^2*f*(c + d*Sec[e + f*x])^2) + (5*a^3*Tan[e + f*x])/(2*(c + d)^3*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^5, x, 7, (5*a^3*(4*c - 3*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*(c - d)^(3/2)*(c + d)^(9/2)*f) - (d*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(4*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) + (a*(4*c - 3*d)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(12*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x])^3) - (5*a^3*(4*c - 3*d)*Tan[e + f*x])/(24*d*(c + d)^3*f*(c + d*Sec[e + f*x])^2) + (5*a^3*(4*c - 3*d)*(c + 4*d)*Tan[e + f*x])/(24*(c - d)*d*(c + d)^4*f*(c + d*Sec[e + f*x])), -((5*a^4*(4*c - 3*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(4*(c - d)^(3/2)*(c + d)^(9/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])) - (d*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(4*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) + (a*(4*c - 3*d)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(12*(c - d)*(c + d)^2*f*(c + d*Sec[e + f*x])^3) + (5*(4*c - 3*d)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(24*(c - d)*(c + d)^3*f*(c + d*Sec[e + f*x])^2) + (5*a^3*(4*c - 3*d)*Tan[e + f*x])/(8*(c - d)*(c + d)^4*f*(c + d*Sec[e + f*x]))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x]), x, 7, (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*ArcTanh[Sin[e + f*x]])/(2*a*f) - ((3*c - 4*d)*d*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*a*f) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - (d*(4*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3) + d*(6*c^2 - 20*c*d + 9*d^2)*Sec[e + f*x])*Tan[e + f*x])/(6*a*f), (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((3*c - 4*d)*d*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*a*f) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - (d*(4*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3) + d*(6*c^2 - 20*c*d + 9*d^2)*Sec[e + f*x])*Tan[e + f*x])/(6*a*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x]), x, 6, (3*d*(2*c^2 - 2*c*d + d^2)*ArcTanh[Sin[e + f*x]])/(2*a*f) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - (d*(4*(c^2 - 3*c*d + d^2) + (2*c - 3*d)*d*Sec[e + f*x])*Tan[e + f*x])/(2*a*f), (3*d*(2*c^2 - 2*c*d + d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) - (d*(4*(c^2 - 3*c*d + d^2) + (2*c - 3*d)*d*Sec[e + f*x])*Tan[e + f*x])/(2*a*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x]), x, 6, (d^2*Tan[e + f*x])/(a*f) + ((c - d)^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) + ((2*c - d)*d*ArcTanh[Sin[e + f*x]])/(a*f), (d^2*Tan[e + f*x])/(a*f) + ((c - d)^2*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])) + (2*(2*c - d)*d*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + a*Sec[e + f*x]), x, 3, (d*ArcTanh[Sin[e + f*x]])/(a*f) + ((c - d)*Tan[e + f*x])/(f*(a + a*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^1), x, 4, Tan[e + f*x]/((c - d)*f*(a + a*Sec[e + f*x])) - (2*d*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(a*(c - d)^(3/2)*Sqrt[c + d]*f), Tan[e + f*x]/((c - d)*f*(a + a*Sec[e + f*x])) + (2*d*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/((c - d)^(3/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2), x, 6, ((c + 2*d)*Tan[e + f*x])/((c - d)^2*(c + d)*f*(a + a*Sec[e + f*x])) - (2*d*(2*c + d)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(a*(c - d)^(5/2)*(c + d)^(3/2)*f) - (d*Tan[e + f*x])/((c^2 - d^2)*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])), ((c + 2*d)*Tan[e + f*x])/((c - d)^2*(c + d)*f*(a + a*Sec[e + f*x])) + (2*d*(2*c + d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/((c - d)^(5/2)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d*Tan[e + f*x])/((c^2 - d^2)*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^3), x, 7, (-3*d*(2*c^2 + 2*c*d + d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a*(c - d)^(7/2)*(c + d)^(5/2)*f) + (d*(2*c + 3*d)*Tan[e + f*x])/(2*a*(c - d)^2*(c + d)*f*(c + d*Sec[e + f*x])^2) + Tan[e + f*x]/((c - d)*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + (d*(2*c + d)*(c + 4*d)*Tan[e + f*x])/(2*a*(c - d)^3*(c + d)^2*f*(c + d*Sec[e + f*x])), If[$VersionNumber>=8, ((2*c + d)*(c + 4*d)*Tan[e + f*x])/(2*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])) + (3*d*(2*c^2 + 2*c*d + d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/((c - d)^(7/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) - (d*(4*c + d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])), ((2*c + d)*(c + 4*d)*Tan[e + f*x])/(2*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])) + (3*d*(2*c^2 + 2*c*d + d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/((c - d)^(7/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) - (d*(4*c + d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])*(c + d*Sec[e + f*x]))]} - - -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^2, x, 8, (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*ArcTanh[Sin[e + f*x]])/(2*a^2*f) - (d*(c^2 + 10*c*d - 12*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*a^2*f) + ((c - d)*(c + 10*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])) + ((c - d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (d*(4*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4) + d*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Sec[e + f*x])*Tan[e + f*x])/(6*a^2*f), (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d*(c^2 + 10*c*d - 12*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*a^2*f) + ((c - d)*(c + 10*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])) + ((c - d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (d*(4*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4) + d*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*Sec[e + f*x])*Tan[e + f*x])/(6*a^2*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^2, x, 7, (d^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTanh[Sin[e + f*x]])/(2*a^2*f) + ((c - d)*(c + 8*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (d*(4*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3) + d*(2*c^2 + 16*c*d - 21*d^2)*Sec[e + f*x])*Tan[e + f*x])/(6*a^2*f), (d^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(c + 8*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) - (d*(4*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3) + d*(2*c^2 + 16*c*d - 21*d^2)*Sec[e + f*x])*Tan[e + f*x])/(6*a^2*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^2, x, 6, ((3*c - 2*d)*d^2*ArcTanh[Sin[e + f*x]])/(a^2*f) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + ((c^3 + 4*c^2*d - 12*c*d^2 + 10*d^3 - (c - 4*d)*d^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])), (2*(3*c - 2*d)*d^2*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + ((c^3 + 4*c^2*d - 12*c*d^2 + 10*d^3 - (c - 4*d)*d^2*Sec[e + f*x])*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^2, x, 6, ((c - d)^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + (d^2*ArcTanh[Sin[e + f*x]])/(a^2*f) + ((c - d)*(c + 5*d)*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x])), ((c - d)^2*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + (2*d^2*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(c + 5*d)*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + a*Sec[e + f*x])^2, x, 2, ((c - d)*Tan[e + f*x])/(3*f*(a + a*Sec[e + f*x])^2) + ((c + 2*d)*Tan[e + f*x])/(3*f*(a^2 + a^2*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^1), x, 6, Tan[e + f*x]/(3*(c - d)*f*(a + a*Sec[e + f*x])^2) + (2*d^2*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(a^2*(c - d)^(5/2)*Sqrt[c + d]*f) + ((c - 4*d)*Tan[e + f*x])/(3*(c - d)^2*f*(a^2 + a^2*Sec[e + f*x])), Tan[e + f*x]/(3*(c - d)*f*(a + a*Sec[e + f*x])^2) - (2*d^2*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a*(c - d)^(5/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - 4*d)*Tan[e + f*x])/(3*(c - d)^2*f*(a^2 + a^2*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2), x, 7, (2*d^2*(3*c + 2*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^2*(c - d)^(7/2)*(c + d)^(3/2)*f) + (d*(c^2 - 6*c*d - 10*d^2)*Tan[e + f*x])/(3*a^2*(c - d)^3*(c + d)*f*(c + d*Sec[e + f*x])) + ((c - 6*d)*Tan[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*(c + d*Sec[e + f*x])) + Tan[e + f*x]/(3*(c - d)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])), ((c + 4*d)*Tan[e + f*x])/(3*(c - d)^2*(c + d)*f*(a + a*Sec[e + f*x])^2) - (2*d^2*(3*c + 2*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a*(c - d)^(7/2)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c^2 - 6*c*d - 10*d^2)*Tan[e + f*x])/(3*(c - d)^3*(c + d)*f*(a^2 + a^2*Sec[e + f*x])) - (d*Tan[e + f*x])/((c^2 - d^2)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3), x, 8, (d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^2*(c - d)^(9/2)*(c + d)^(5/2)*f) + (d*(2*c^2 - 16*c*d - 21*d^2)*Tan[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sec[e + f*x])^2) + ((c - 8*d)*Tan[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + Tan[e + f*x]/(3*(c - d)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) + (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Tan[e + f*x])/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sec[e + f*x])), If[$VersionNumber>=8, ((2*c^2 + 22*c*d + 11*d^2)*Tan[e + f*x])/(6*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^2) - (d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a*(c - d)^(9/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Tan[e + f*x])/(6*(c - d)^4*(c + d)^2*f*(a^2 + a^2*Sec[e + f*x])) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) - (d*(5*c + 2*d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])), ((2*c^2 + 22*c*d + 11*d^2)*Tan[e + f*x])/(6*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^2) - (d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a*(c - d)^(9/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Tan[e + f*x])/(6*(c - d)^4*(c + d)^2*f*(a^2 + a^2*Sec[e + f*x])) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) - (d*(5*c + 2*d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x]))]} - - -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^6)/(a + a*Sec[e + f*x])^3, x, 9, (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*ArcTanh[Sin[e + f*x]])/(2*a^3*f) - (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5)*Tan[e + f*x])/(15*a^3*f) - (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*Sec[e + f*x]*Tan[e + f*x])/(30*a^3*f) - (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*a^3*f) + ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])) + ((c - d)*(2*c + 13*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^5*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3), (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*a^3*f) + ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])) + ((c - d)*(2*c + 13*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^5*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - (d*(4*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5) + d*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*Sec[e + f*x])*Tan[e + f*x])/(30*a^3*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + a*Sec[e + f*x])^3, x, 8, (d^3*(20*c^2 - 30*c*d + 13*d^2)*ArcTanh[Sin[e + f*x]])/(2*a^3*f) - (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*Tan[e + f*x])/(15*a^3*f) - (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Sec[e + f*x]*Tan[e + f*x])/(30*a^3*f) + ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])) + ((c - d)*(2*c + 11*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3), (d^3*(20*c^2 - 30*c*d + 13*d^2)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])) + ((c - d)*(2*c + 11*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) - (d*(4*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4) + d*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*Sec[e + f*x])*Tan[e + f*x])/(30*a^3*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + a*Sec[e + f*x])^3, x, 7, ((4*c - 3*d)*d^3*ArcTanh[Sin[e + f*x]])/(a^3*f) + ((c - d)*(2*c + 9*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((2*c^4 + 8*c^3*d + 21*c^2*d^2 - 88*c*d^3 + 72*d^4 - d^2*(2*c^2 + 10*c*d - 27*d^2)*Sec[e + f*x])*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x])), (2*(4*c - 3*d)*d^3*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(2*c + 9*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((c - d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((2*c^4 + 8*c^3*d + 21*c^2*d^2 - 88*c*d^3 + 72*d^4 - d^2*(2*c^2 + 10*c*d - 27*d^2)*Sec[e + f*x])*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + a*Sec[e + f*x])^3, x, 6, (d^3*ArcTanh[Sin[e + f*x]])/(a^3*f) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((c - d)*(2*(2*c^2 + 8*c*d + 11*d^2) + (2*c^2 + 11*c*d + 29*d^2)*Sec[e + f*x])*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2), (2*d^3*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(a^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c - d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((c - d)*(2*(2*c^2 + 8*c*d + 11*d^2) + (2*c^2 + 11*c*d + 29*d^2)*Sec[e + f*x])*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + a*Sec[e + f*x])^3, x, 4, ((c - d)^2*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + (2*(c - d)*(c + 4*d)*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((2*c^2 + 6*c*d + 7*d^2)*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + a*Sec[e + f*x])^3, x, 3, ((c - d)*Tan[e + f*x])/(5*f*(a + a*Sec[e + f*x])^3) + ((2*c + 3*d)*Tan[e + f*x])/(15*a*f*(a + a*Sec[e + f*x])^2) + ((2*c + 3*d)*Tan[e + f*x])/(15*f*(a^3 + a^3*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1), x, 7, (-2*d^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^3*(c - d)^(7/2)*Sqrt[c + d]*f) + Tan[e + f*x]/(5*(c - d)*f*(a + a*Sec[e + f*x])^3) + ((2*c - 7*d)*Tan[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sec[e + f*x])^2) + ((2*c^2 - 9*c*d + 22*d^2)*Tan[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sec[e + f*x])), Tan[e + f*x]/(5*(c - d)*f*(a + a*Sec[e + f*x])^3) + ((2*c - 7*d)*Tan[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sec[e + f*x])^2) + (2*d^3*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a^2*(c - d)^(7/2)*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c^2 - 9*c*d + 22*d^2)*Tan[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2), x, 8, (-2*d^3*(4*c + 3*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^3*(c - d)^(9/2)*(c + d)^(3/2)*f) + (d*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*Tan[e + f*x])/(15*a^3*(c - d)^4*(c + d)*f*(c + d*Sec[e + f*x])) + Tan[e + f*x]/(5*(c - d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])) + ((2*c - 9*d)*Tan[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])) + ((2*c^2 - 12*c*d + 45*d^2)*Tan[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sec[e + f*x])*(c + d*Sec[e + f*x])), ((c + 6*d)*Tan[e + f*x])/(5*(c - d)^2*(c + d)*f*(a + a*Sec[e + f*x])^3) + ((2*c^2 - 10*c*d - 27*d^2)*Tan[e + f*x])/(15*a*(c - d)^3*(c + d)*f*(a + a*Sec[e + f*x])^2) + (2*d^3*(4*c + 3*d)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a^2*(c - d)^(9/2)*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*Tan[e + f*x])/(15*(c - d)^4*(c + d)*f*(a^3 + a^3*Sec[e + f*x])) - (d*Tan[e + f*x])/((c^2 - d^2)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x]))} -{Sec[e + f*x]/((a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3), x, 9, -((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^3*(c - d)^(11/2)*(c + d)^(5/2)*f)) + (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Tan[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*Sec[e + f*x])^2) + Tan[e + f*x]/(5*(c - d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) + ((2*c - 11*d)*Tan[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) + ((2*c^2 - 15*c*d + 76*d^2)*Tan[e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + (d*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Tan[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sec[e + f*x])), If[$VersionNumber>=8, ((2*c^2 + 39*c*d + 22*d^2)*Tan[e + f*x])/(10*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^3) + ((4*c^3 - 26*c^2*d - 184*c*d^2 - 109*d^3)*Tan[e + f*x])/(30*a*(c - d)^4*(c + d)^2*f*(a + a*Sec[e + f*x])^2) + (d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a^2*(c - d)^(11/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Tan[e + f*x])/(30*(c - d)^5*(c + d)^2*f*(a^3 + a^3*Sec[e + f*x])) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) - (3*d*(2*c + d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])), ((2*c^2 + 39*c*d + 22*d^2)*Tan[e + f*x])/(10*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^3) + ((4*c^3 - 26*c^2*d - 184*c*d^2 - 109*d^3)*Tan[e + f*x])/(30*a*(c - d)^4*(c + d)^2*f*(a + a*Sec[e + f*x])^2) + (d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x])/(a^2*(c - d)^(11/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Tan[e + f*x])/(30*(c - d)^5*(c + d)^2*f*(a^3 + a^3*Sec[e + f*x])) - (d*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) - (3*d*(2*c + d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x]))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]], x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[d]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]*Sqrt[c + d*Sec[e + f*x]]/Sqrt[a + a*Sec[e + f*x]], x, 5, (Sqrt[2]*Sqrt[c - d]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f) + (2*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*f)} -{Sec[e + f*x]/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]), x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 (a+a Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]), x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[c - d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[c - d]*f)) + (2*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/(Sqrt[a]*Sqrt[d]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^(m/2) / (c+d Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(Sec[e + f*x]*Sqrt[a + a*Sec[e + f*x]])/(c + d*Sec[e + f*x]), x, 2, (2*Sqrt[a]*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[d]*Sqrt[c + d]*f)} - - -{(g*Sec[e + f*x])^(3/2)*Sqrt[a + a*Sec[e + f*x]]/(c + d*Sec[e + f*x]), x, 5, (2*Sqrt[a]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(d*f) - (2*Sqrt[a]*Sqrt[c]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(d*Sqrt[c + d]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 5, (Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[d]*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)} -{Sec[e + f*x]^2/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 5, -((Sqrt[2]*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f)) + (2*c*ArcTan[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[d]*Sqrt[c + d]*f)} - - -{(g*Sec[e + f*x])^(3/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 5, -((Sqrt[2]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f)) + (2*Sqrt[c]*g^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*Sqrt[c + d]*f)} -{(g*Sec[e + f*x])^(5/2)/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 8, (2*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*d*f) + (Sqrt[2]*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[g]*Tan[e + f*x])/(Sqrt[2]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*f) - (2*c^(3/2)*g^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Tan[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*(c - d)*d*Sqrt[c + d]*f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+b Sec[e+f x])^m (c+d Sec[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^4, x, 8, ((8*a*c^4 + 16*b*c^3*d + 24*a*c^2*d^2 + 12*b*c*d^3 + 3*a*d^4)*ArcTanh[Sin[e + f*x]])/(8*f) + ((12*b*c^4 + 95*a*c^3*d + 112*b*c^2*d^2 + 80*a*c*d^3 + 16*b*d^4)*Tan[e + f*x])/(30*f) + (d*(24*b*c^3 + 130*a*c^2*d + 116*b*c*d^2 + 45*a*d^3)*Sec[e + f*x]*Tan[e + f*x])/(120*f) + ((12*b*c^2 + 35*a*c*d + 16*b*d^2)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(60*f) + ((4*b*c + 5*a*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*f) + (b*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^3, x, 7, ((8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*ArcTanh[Sin[e + f*x]])/(8*f) + ((4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4*c*d^2))*Tan[e + f*x])/(6*f) + (d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*Sec[e + f*x]*Tan[e + f*x])/(24*f) + ((3*b*c + 4*a*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + (b*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2, x, 6, ((2*b*c*d + a*(2*c^2 + d^2))*ArcTanh[Sin[e + f*x]])/(2*f) + (2*(3*a*c*d + b*(c^2 + d^2))*Tan[e + f*x])/(3*f) + (d*(2*b*c + 3*a*d)*Sec[e + f*x]*Tan[e + f*x])/(6*f) + (b*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^1, x, 5, ((2*a*c + b*d)*ArcTanh[Sin[e + f*x]])/(2*f) + ((b*c + a*d)*Tan[e + f*x])/f + (b*d*Sec[e + f*x]*Tan[e + f*x])/(2*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^1, x, 5, (b*ArcTanh[Sin[e + f*x]])/(d*f) - (2*(b*c - a*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*d*Sqrt[c + d]*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^2, x, 5, (2*(a*c - b*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*f) + ((b*c - a*d)*Tan[e + f*x])/((c^2 - d^2)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^3, x, 6, -(((3*b*c*d - a*(2*c^2 + d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*f)) + ((b*c - a*d)*Tan[e + f*x])/(2*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - ((3*a*c*d - b*(c^2 + 2*d^2))*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x]))/(c + d*Sec[e + f*x])^4, x, 7, ((2*a*c^3 - 4*b*c^2*d + 3*a*c*d^2 - b*d^3)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(7/2)*(c + d)^(7/2)*f) + ((b*c - a*d)*Tan[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) + ((2*b*c^2 - 5*a*c*d + 3*b*d^2)*Tan[e + f*x])/(6*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x])^2) + ((2*b*c^3 - 11*a*c^2*d + 13*b*c*d^2 - 4*a*d^3)*Tan[e + f*x])/(6*(c^2 - d^2)^3*f*(c + d*Sec[e + f*x]))} - - -(* {Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^4, x, 8, ((16*a*b*d*(4*c^3 + 3*c*d^2) + 2*a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) + b^2*(8*c^4 + 36*c^2*d^2 + 5*d^4))*ArcTanh[Sin[e + f*x]])/(16*f) + ((10*a^2*c*d^2*(19*c^2 + 16*d^2) + 16*a*b*d*(3*c^4 + 28*c^2*d^2 + 4*d^4) - b^2*(4*c^5 - 121*c^3*d^2 - 128*c*d^4))*Tan[e + f*x])/(60*d*f) + ((10*a^2*d^2*(26*c^2 + 9*d^2) + 16*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(8*c^4 - 178*c^2*d^2 - 75*d^4))*Sec[e + f*x]*Tan[e + f*x])/(240*f) + ((70*a^2*c*d^2 + 16*a*b*d*(3*c^2 + 4*d^2) - b^2*(4*c^3 - 53*c*d^2))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(120*d*f) + ((5*(6*a^2 + 5*b^2)*d^2 - 4*b*c*(b*c - 12*a*d))*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(120*d*f) - (b*(b*c - 12*a*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(30*d*f) + (b^2*(c + d*Sec[e + f*x])^5*Tan[e + f*x])/(6*d*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3, x, 7, ((6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*Tan[e + f*x])/(30*d*f) + ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*Sec[e + f*x]*Tan[e + f*x])/(120*f) + ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(60*d*f) - (b*(b*c - 10*a*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*d*f) + (b^2*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*d*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2, x, 6, ((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*Tan[e + f*x])/(6*b*f) + ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Sec[e + f*x]*Tan[e + f*x])/(24*f) + (d*(8*b*c - a*d)*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(12*b*f) + (d^2*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(4*b*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^1, x, 5, ((2*a^2*c + b^2*c + 2*a*b*d)*ArcTanh[Sin[e + f*x]])/(2*f) + (2*(3*a*b*c + a^2*d + b^2*d)*Tan[e + f*x])/(3*f) + (b*(3*b*c + 2*a*d)*Sec[e + f*x]*Tan[e + f*x])/(6*f) + (d*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(3*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^1, x, 6, -((b*(b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(d^2*f)) + (2*(b*c - a*d)^2*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*d^2*Sqrt[c + d]*f) + (b^2*Tan[e + f*x])/(d*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^2, x, 6, (b^2*ArcTanh[Sin[e + f*x]])/(d^2*f) - (2*(b*c - a*d)*(b*c^2 + a*c*d - 2*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*d^2*(c + d)^(3/2)*f) - ((b*c - a*d)^2*Tan[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) + ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*Tan[e + f*x])/(2*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^4, x, 7, -(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) + ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Tan[e + f*x])/(6*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x])^2) - ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Tan[e + f*x])/(6*d*(c^2 - d^2)^3*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^5, x, 8, -((10*a*b*d*(4*c^3 + 3*c*d^2) - a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*Tan[e + f*x])/(4*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) + ((b*c - a*d)*(7*a*c*d + b*(c^2 - 8*d^2))*Tan[e + f*x])/(12*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x])^3) - ((a^2*d^2*(26*c^2 + 9*d^2) - 2*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(2*c^4 - 25*c^2*d^2 - 12*d^4))*Tan[e + f*x])/(24*d*(c^2 - d^2)^3*f*(c + d*Sec[e + f*x])^2) - ((5*a^2*c*d^2*(10*c^2 + 11*d^2) - 2*a*b*d*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^2*(2*c^5 - 39*c^3*d^2 - 68*c*d^4))*Tan[e + f*x])/(24*d*(c^2 - d^2)^4*f*(c + d*Sec[e + f*x]))} *) - - -(* {Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3, x, 8, ((18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]])/(16*f) + ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*Tan[e + f*x])/(60*d^2*f) + ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*Sec[e + f*x]*Tan[e + f*x])/(240*d*f) + ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(120*d^2*f) - (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(120*d^2*f) - (b^2*(2*b*c - 13*a*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(30*d^2*f) + (b^2*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(6*d*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2, x, 7, ((24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*Tan[e + f*x])/(30*b*f) + ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*Sec[e + f*x]*Tan[e + f*x])/(120*f) + ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(60*b*f) + (d*(10*b*c - a*d)*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(20*b*f) + (d^2*(a + b*Sec[e + f*x])^4*Tan[e + f*x])/(5*b*f)} -{Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1, x, 6, ((8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*ArcTanh[Sin[e + f*x]])/(8*f) + ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*Tan[e + f*x])/(6*f) + (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*Sec[e + f*x]*Tan[e + f*x])/(24*f) + ((4*b*c + 3*a*d)*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + (d*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(4*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^1, x, 7, -(b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*ArcTanh[Sin[e + f*x]])/(2*d^3*f) - (2*(b*c - a*d)^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*d^3*Sqrt[c + d]*f) - (b^2*(2*b*c - 5*a*d)*Tan[e + f*x])/(2*d^2*f) + (b^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(2*d*f)} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2, x, 7, -((b^2*(2*b*c - 3*a*d)*ArcTanh[Sin[e + f*x]])/(d^3*f)) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*d^3*(c + d)^(3/2)*f) - (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*Tan[e + f*x])/(d^2*(c^2 - d^2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^3, x, 7, (b^3*ArcTanh[Sin[e + f*x]])/(d^3*f) - ((b*c - a*d)*(2*a*b*c*d*(c^2 - 4*d^2) + a^2*d^2*(2*c^2 + d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 6*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*d^3*(c + d)^(5/2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*Tan[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^4, x, 7, -(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) - ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Tan[e + f*x])/(6*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x])^2) + ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*Tan[e + f*x])/(6*(c - d)^3*d^2*(c + d)^3*f*(c + d*Sec[e + f*x]))} -{(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^5, x, 8, -((5*b^3*c*d*(3*c^2 + 4*d^2) + 15*a^2*b*d*(4*c^3 + 3*c*d^2) - a^3*(8*c^4 + 24*c^2*d^2 + 3*d^4) - 3*a*b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(4*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) - ((b*c - a*d)^2*(2*b*c^2 + 7*a*c*d - 9*b*d^2)*Tan[e + f*x])/(12*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x])^3) + ((b*c - a*d)*(a^2*d^2*(26*c^2 + 9*d^2) + a*b*(8*c^3*d - 78*c*d^3) + b^2*(2*c^4 - 3*c^2*d^2 + 36*d^4))*Tan[e + f*x])/(24*(c - d)^3*d^2*(c + d)^3*f*(c + d*Sec[e + f*x])^2) - ((5*a^3*c*d^3*(10*c^2 + 11*d^2) - 3*a*b^2*c*d*(2*c^4 - 39*c^2*d^2 - 68*d^4) - 3*a^2*b*d^2*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^3*(2*c^6 - 5*c^4*d^2 + 84*c^2*d^4 + 24*d^6))*Tan[e + f*x])/(24*d^2*(c^2 - d^2)^4*f*(c + d*Sec[e + f*x]))} *) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x]), x, 12, (d^3*(4*b*c - a*d)*ArcTanh[Sin[e + f*x]])/(2*b^2*f) + (d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*ArcTanh[Sin[e + f*x]])/(b^4*f) + (2*(b*c - a*d)^4*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*f) + (d^4*Tan[e + f*x])/(b*f) + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Tan[e + f*x])/(b^3*f) + (d^3*(4*b*c - a*d)*Sec[e + f*x]*Tan[e + f*x])/(2*b^2*f) + (d^4*Tan[e + f*x]^3)/(3*b*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x]), x, 10, (d^3*ArcTanh[Sin[e + f*x]])/(2*b*f) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*ArcTanh[Sin[e + f*x]])/(b^3*f) + (2*(b*c - a*d)^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*f) + (d^2*(3*b*c - a*d)*Tan[e + f*x])/(b^2*f) + (d^3*Sec[e + f*x]*Tan[e + f*x])/(2*b*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x]), x, 8, (d*(2*b*c - a*d)*ArcTanh[Sin[e + f*x]])/(b^2*f) + (2*(b*c - a*d)^2*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*f) + (d^2*Tan[e + f*x])/(b*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + b*Sec[e + f*x]), x, 5, (d*ArcTanh[Sin[e + f*x]])/(b*f) + (2*(b*c - a*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*f)} -{Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^1), x, 6, (2*b*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(b*c - a*d)*f) - (2*d*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(b*c - a*d)*f)} -{Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2), x, 7, (2*b^2*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(b*c - a*d)^2*f) - (2*d*(2*b*c^2 - a*c*d - b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^2*f) + (d^2*Sin[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))} -(* {Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^3), x, 9, (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^3*f) + (d^2*Tan[e + f*x])/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) + (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*Tan[e + f*x])/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} *) - - -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2, x, 16, (d^4*(5*b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(2*b^3*f) + (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*ArcTanh[Sin[e + f*x]])/(b^5*f) + (2*(b*c - a*d)^5*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*b^3*(a + b)^(3/2)*f) + (2*(b*c - a*d)^4*(b*c + 4*a*d)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^5*Sqrt[a + b]*f) - ((b*c - a*d)^5*Sin[e + f*x])/(b^4*(a^2 - b^2)*f*(b + a*Cos[e + f*x])) + (d^5*Tan[e + f*x])/(b^2*f) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Tan[e + f*x])/(b^4*f) + (d^4*(5*b*c - 2*a*d)*Sec[e + f*x]*Tan[e + f*x])/(2*b^3*f) + (d^5*Tan[e + f*x]^3)/(3*b^2*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x])^2, x, 14, (d^4*ArcTanh[Sin[e + f*x]])/(2*b^2*f) + (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[Sin[e + f*x]])/(b^4*f) + (2*(b*c - a*d)^4*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*b^2*(a + b)^(3/2)*f) + (2*(b*c - a*d)^3*(b*c + 3*a*d)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^4*Sqrt[a + b]*f) - ((b*c - a*d)^4*Sin[e + f*x])/(b^3*(a^2 - b^2)*f*(b + a*Cos[e + f*x])) + (2*d^3*(2*b*c - a*d)*Tan[e + f*x])/(b^3*f) + (d^4*Sec[e + f*x]*Tan[e + f*x])/(2*b^2*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x])^2, x, 12, (d^2*(3*b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(b^3*f) + (2*(b*c - a*d)^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*b*(a + b)^(3/2)*f) + (2*(b*c - a*d)^2*(b*c + 2*a*d)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^3*Sqrt[a + b]*f) - ((b*c - a*d)^3*Sin[e + f*x])/(b^2*(a^2 - b^2)*f*(b + a*Cos[e + f*x])) + (d^3*Tan[e + f*x])/(b^2*f)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x])^2, x, 10, (d^2*ArcTanh[Sin[e + f*x]])/(b^2*f) + (2*(b*c - a*d)^2*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*f) + (2*(b^2*c^2 - a^2*d^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^2*Sqrt[a + b]*f) - ((b*c - a*d)^2*Sin[e + f*x])/(b*(a^2 - b^2)*f*(b + a*Cos[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + b*Sec[e + f*x])^2, x, 5, (2*(a*c - b*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*f) - ((b*c - a*d)*Tan[e + f*x])/((a^2 - b^2)*f*(a + b*Sec[e + f*x]))} -{Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^1), x, 7, (2*b*(a*b*c - 2*a^2*d + b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(e + f*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^2*f) + (2*d^2*ArcTanh[(Sqrt[c - d]*Tan[(1/2)*(e + f*x)])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(b*c - a*d)^2*f) - (b^2*Sin[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(b + a*Cos[e + f*x]))} -(* {Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2), x, 9, (2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^3*f) - (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sec[e + f*x])) - (b^2*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x]))} -{Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3), x, 10, (2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^4*f) - (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (b^2*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + (d*(3*a^3*c*d^3 - 3*a*b^2*c*d^3 - a^2*b*d^2*(7*c^2 - 4*d^2) - b^3*(2*c^4 - 11*c^2*d^2 + 6*d^4))*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} *) - - -(* {(Sec[e + f*x]*(c + d*Sec[e + f*x])^6)/(a + b*Sec[e + f*x])^3, x, 10, (d^3*(72*a^2*b*c*d^2 - 20*a^3*d^3 - 3*a*b^2*d*(30*c^2 + d^2) + b^3*(40*c^3 + 6*c*d^2))*ArcTanh[Sin[e + f*x]])/(2*b^6*f) + ((b*c - a*d)^4*(8*a^3*b*c*d - 14*a*b^3*c*d + 20*a^4*d^2 + a^2*b^2*(2*c^2 - 47*d^2) + b^4*(c^2 + 30*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^6*(a + b)^(5/2)*f) - (d*(216*a^5*b*c*d^4 - 60*a^6*d^5 - a^4*b^2*d^3*(250*c^2 - 101*d^2) + 6*a^3*b^3*c*d^2*(13*c^2 - 63*d^2) - 6*a*b^5*c*(3*c^4 + 37*c^2*d^2 - 18*d^4) + 2*a^2*b^4*d*(9*c^4 + 232*c^2*d^2 - 14*d^4) + b^6*d*(63*c^4 - 88*c^2*d^2 - 4*d^4))*Tan[e + f*x])/(6*b^5*(a^2 - b^2)^2*f) + (d^2*(68*a^4*b*c*d^3 - 30*a^5*d^4 - 2*b^5*c*d*(15*c^2 - 7*d^2) - 12*a^3*b^2*d^2*(3*c^2 - 4*d^2) - 2*a^2*b^3*c*d*(3*c^2 + 59*d^2) + 9*a*b^4*(c^4 + 10*c^2*d^2 - d^4))*Sec[e + f*x]*Tan[e + f*x])/(6*b^4*(a^2 - b^2)^2*f) - (d*(27*a^3*b*c*d^2 - 20*a^4*d^3 + 31*a^2*b^2*d^3 + b^4*d*(27*c^2 - 2*d^2) - 9*a*b^3*c*(c^2 + 6*d^2))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(6*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 5*a^2*d - 8*b^2*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^3, x, 9, -(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*ArcTanh[Sin[e + f*x]])/(2*b^5*f) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*f) + (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Tan[e + f*x])/(2*b^4*(a^2 - b^2)^2*f) - (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*Sec[e + f*x]*Tan[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x])^3, x, 8, (d^3*(4*b*c - 3*a*d)*ArcTanh[Sin[e + f*x]])/(b^4*f) + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*f) - (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Tan[e + f*x])/(2*b^3*(a^2 - b^2)*f) - (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*Tan[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x])^3, x, 7, (d^3*ArcTanh[Sin[e + f*x]])/(b^3*f) + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*f) - ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x])^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2) - ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*Tan[e + f*x])/(2*b*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x]))} -{(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + b*Sec[e + f*x])^3, x, 6, ((2*a^2*c + b^2*c - 3*a*b*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*f) - ((b*c - a*d)*Tan[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2) - ((3*a*b*c - a^2*d - 2*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x]))} -{Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1), x, 9, -((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(b*c - a*d)^3*f) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2) - (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x]))} -{Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2), x, 10, -((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^4*f) + (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])) - (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x]))} -{Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3), x, 11, -((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^5*f) + (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) - (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) - (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+b Sec[e+f x])^(m/2) (c+d Sec[e+f x])^n*) - - -{(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + d*Sec[e + f*x]), x, 3, (2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(d*f) - (2*(b*c - a*d)*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] (a+b Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]], x, 1, (2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(d*Sqrt[(a + b)/(c + d)]*f)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]), x, 1, (2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(Sqrt[c + d]*(b*c - a*d)*f)} - - -{Sec[e + f*x]/(Sqrt[2 + 3*Sec[e + f*x]]*Sqrt[-4 + 5*Sec[e + f*x]]), x, 1, (2*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[2 + 3*Sec[e + f*x]]/(Sqrt[5]*Sqrt[-4 + 5*Sec[e + f*x]])], 45]*(4 - 5*Sec[e + f*x])*Sqrt[(1 - Sec[e + f*x])/(4 - 5*Sec[e + f*x])]*Sqrt[(1 + Sec[e + f*x])/(4 - 5*Sec[e + f*x])])/f} -{Sec[e + f*x]/(Sqrt[2 + 3*Sec[e + f*x]]*Sqrt[4 - 5*Sec[e + f*x]]), x, 1, (2*I*Cot[e + f*x]*EllipticF[I*ArcSinh[(Sqrt[5]*Sqrt[4 - 5*Sec[e + f*x]])/Sqrt[2 + 3*Sec[e + f*x]]], 1/45]*Sqrt[(1 - Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*Sqrt[(1 + Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*(2 + 3*Sec[e + f*x]))/(3*Sqrt[5]*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^2 (a+b Sec[e+f x])^(m/2) (c+d Sec[e+f x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]), x, 3, (2*Cot[e + f*x]*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[(a + b)/(c + d)]*Sqrt[c + d*Sec[e + f*x]])/Sqrt[a + b*Sec[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e + f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))]*(a + b*Sec[e + f*x]))/(b*d*Sqrt[(a + b)/(c + d)]*f) - (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(b*Sqrt[c + d]*(b*c - a*d)*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^(3/2) (a+b Sec[e+f x])^m (c+d Sec[e+f x])^(n/2)*) - - -(* {(g*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(5/2)/(a + b*Sec[e + f*x]), x, 0, 0} -{(g*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(3/2)/(a + b*Sec[e + f*x]), x, 0, 0} *) -{(g*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(1/2)/(a + b*Sec[e + f*x]), x, 7, (2*d*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[2, (1/2)*(e + f*x), (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/(b*f*Sqrt[c + d*Sec[e + f*x]]) + (2*(b*c - a*d)*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/(b*(a + b)*f*Sqrt[c + d*Sec[e + f*x]])} -{(g*Sec[e + f*x])^(3/2)/((c + d*Sec[e + f*x])^(1/2)*(a + b*Sec[e + f*x])), x, 3, (2*g*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*a)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]])} -(* {(g*Sec[e + f*x])^(3/2)/((c + d*Sec[e + f*x])^(3/2)*(a + b*Sec[e + f*x])), x, 0, 0} -{(g*Sec[e + f*x])^(3/2)/((c + d*Sec[e + f*x])^(5/2)*(a + b*Sec[e + f*x])), x, 0, 0} *) - - -{(Sqrt[g*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])/(a + b*Cos[e + f*x]), x, 8, (2*d*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[2, (1/2)*(e + f*x), (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/(a*f*Sqrt[c + d*Sec[e + f*x]]) + (2*(a*c - b*d)*Sqrt[(d + c*Cos[e + f*x])/(c + d)]*EllipticPi[(2*b)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d)]*Sqrt[g*Sec[e + f*x]])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sec[e+f x])^(m/2) / (c+c Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + c*Sec[e + f*x]), x, 1, (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[a + b*Sec[e + f*x]])/(c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])} - - -{(g*Sec[e + f*x])^(3/2)*Sqrt[a + b*Sec[e + f*x]]/(c + c*Sec[e + f*x]), x, 11, (g*(b + a*Cos[e + f*x])*EllipticE[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + ((a - b)*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) + (2*b*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*Cos[e + f*x])*Sqrt[g*Sec[e + f*x]]*Sin[e + f*x])/(f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^1/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])), x, 3, -((2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*c*f)) + (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[a + b*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])} -{Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])), x, 3, (2*a*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/((a - b)*b*c*f) - (EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (a - b)/(a + b)]*Sqrt[1/(1 + Sec[e + f*x])]*Sqrt[a + b*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])} - - -{(g*Sec[e + f*x])^(3/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])), x, 7, (g*(b + a*Cos[e + f*x])*EllipticE[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]]) + (g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) - (g*(b + a*Cos[e + f*x])*Sqrt[g*Sec[e + f*x]]*Sin[e + f*x])/((a - b)*f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])} -{(g*Sec[e + f*x])^(5/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + c*Sec[e + f*x])), x, 11, -((g^2*(b + a*Cos[e + f*x])*EllipticE[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((a - b)*c*f*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[a + b*Sec[e + f*x]])) - (g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticF[(1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) + (2*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(c*f*Sqrt[a + b*Sec[e + f*x]]) + (g^2*(b + a*Cos[e + f*x])*Sqrt[g*Sec[e + f*x]]*Sin[e + f*x])/((a - b)*f*(c + c*Cos[e + f*x])*Sqrt[a + b*Sec[e + f*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (g Sec[e+f x])^p (a+b Sec[e+f x])^(m/2) / (c+d Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]])/(c + d*Sec[e + f*x]), x, 3, (2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(d*f) - (2*(b*c - a*d)*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} - - -{(g*Sec[e + f*x])^(3/2)*Sqrt[a + b*Sec[e + f*x]]/(c + d*Sec[e + f*x]), x, 7, (2*b*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*f*Sqrt[a + b*Sec[e + f*x]]) - (2*(b*c - a*d)*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[e + f*x]^1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 1, (2*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/((c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} -{Sec[e + f*x]^2/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 3, (2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b*d*f) - (2*c*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/(a + b)]*Tan[e + f*x])/(d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])} - - -{(g*Sec[e + f*x])^(3/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 3, (2*g*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/((c + d)*f*Sqrt[a + b*Sec[e + f*x]])} -{(g*Sec[e + f*x])^(5/2)/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])), x, 7, (2*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[2, (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*f*Sqrt[a + b*Sec[e + f*x]]) - (2*c*g^2*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*EllipticPi[(2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b)]*Sqrt[g*Sec[e + f*x]])/(d*(c + d)*f*Sqrt[a + b*Sec[e + f*x]])} - - -(* ::Title:: *) -(*Integrands of the form (c Sec[e+f x])^p (d Tan[e+f x])^m (a+b Sec[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sec[e+f x])^p (d Tan[e+f x])^m (a-a Sec[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x] Tan[e+f x]^m (a-a Sec[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[e + f*x]*Tan[e + f*x]^4/(c - c*Sec[e + f*x])^7, x, 4, Cot[(1/2)*(e + f*x)]^5/(20*c^7*f) - Cot[(1/2)*(e + f*x)]^7/(14*c^7*f) + Cot[(1/2)*(e + f*x)]^9/(36*c^7*f)} -{Sec[e + f*x]*Tan[e + f*x]^4/(c - c*Sec[e + f*x])^8, x, 4, Cot[(1/2)*(e + f*x)]^5/(40*c^8*f) - (3*Cot[(1/2)*(e + f*x)]^7)/(56*c^8*f) + Cot[(1/2)*(e + f*x)]^9/(24*c^8*f) - Cot[(1/2)*(e + f*x)]^11/(88*c^8*f)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m deleted file mode 100644 index 6c3c14c..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).m +++ /dev/null @@ -1,1058 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[c+d x])^m (b Sec[c+d x])^n (A+B Sec[c+d x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^0*(b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 8, -((6*b^3*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*A*b^2*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (6*b^2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*b*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*B*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 7, -((2*A*b^2*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*A*b*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*B*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^(1/2)*(A + B*Sec[c + d*x]), x, 6, -((2*b*B*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/d + (2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(1/2), x, 5, (2*A*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(3/2), x, 6, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*A*Sin[c + d*x])/(3*b*d*Sqrt[b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(5/2), x, 7, (6*A*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*A*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(3/2)) + (2*B*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Sec[c + d*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 6, (3*A*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(4/3), 1/2, -(1/3), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(8/3)*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 6, (3*A*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 5, -((3*A*b*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*b^2*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*b*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*b^3*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])) - (3*b^2*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 6, (3*A*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(5/3), 1/2, -(2/3), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(10/3)*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 6, (3*A*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2])} -{(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 5, (3*A*b*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*b^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*b*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*b^3*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) - (3*b^2*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3), x, 6, (3*A*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3), x, 6, -((3*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])} -{(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(2/3), x, 5, -((3*A*b*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3), x, 6, -((3*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3), x, 6, (3*A*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2])} - - -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3), x, 6, -((3*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3), x, 6, -((3*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(4/3), x, 5, -((3*A*b*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3), x, 6, -((3*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3), x, 6, -((3*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^n with m symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]), x, 6, (3*A*b*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[1/2, (1/6)*(-4 - 3*m), (1/6)*(2 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-2 - 3*m), (1/6)*(4 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(2 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]), x, 6, -((3*A*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(1/3), x, 6, -((3*A*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3), x, 6, -((3*A*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(2 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3), x, 6, -((3*A*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(b*d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^n with n symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, -((A*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, (1/2)*(-m - n), (1/2)*(2 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2])} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, (A*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-2 - n), -(n/2), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, (A*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2])} -{(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 5, -((A*b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, -((A*b^2*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])) - (b*B*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, -((A*b^3*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*B*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2])} - - -{Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, (2*A*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]), x, 6, -((2*A*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])} -{((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 6, -((2*A*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])) - (2*B*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])} -{((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 6, -((2*A*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])) - (2*B*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 7, (3*a*(A + B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 4*B)*Tan[c + d*x])/(5*d) + (3*a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(A + B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*B*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 4*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 6, (a*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(A + B)*Tan[c + d*x])/d + (a*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(A + B)*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 6, (a*(A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 2*B)*Tan[c + d*x])/(3*d) + (a*(A + B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 5, (a*(2*A + B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(A + B)*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 4, a*A*x + (a*(A + B)*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 3, a*(A + B)*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 4, (a*(A + 2*B)*x)/2 + (a*(A + B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 5, (a*(A + B)*x)/2 + (a*(2*A + 3*B)*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 6, (1/8)*a*(3*A + 4*B)*x + (a*(A + B)*Sin[c + d*x])/d + (a*(3*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x]), x, 7, (3/8)*a*(A + B)*x + (a*(4*A + 5*B)*Sin[c + d*x])/(5*d) + (3*a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*B)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 7, (a^2*(7*A + 6*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(10*A + 9*B)*Tan[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*A + 6*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(5*d) + (a^2*(10*A + 9*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 7, (a^2*(8*A + 7*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*A + 7*B)*Tan[c + d*x])/(6*d) + (a^2*(8*A + 7*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A - B)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 6, (a^2*(3*A + 2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(3*A + 2*B)*Tan[c + d*x])/(3*d) + (a^2*(3*A + 2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 5, a^2*A*x + (a^2*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B)*Tan[c + d*x])/(2*d) + (B*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 4, a^2*(2*A + B)*x + (a^2*(A + 2*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A - B)*Sin[c + d*x])/d + (B*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 4, (1/2)*a^2*(3*A + 4*B)*x + (a^2*B*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A + 2*B)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 5, (1/2)*a^2*(2*A + 3*B)*x + (2*a^2*(2*A + 3*B)*Sin[c + d*x])/(3*d) + (a^2*(2*A + 3*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 6, (1/8)*a^2*(7*A + 8*B)*x + (a^2*(4*A + 5*B)*Sin[c + d*x])/(3*d) + (a^2*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*A + 4*B)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^2, x, 7, (1/8)*a^2*(6*A + 7*B)*x + (a^2*(9*A + 10*B)*Sin[c + d*x])/(5*d) + (a^2*(6*A + 7*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(6*A + 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(9*A + 10*B)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 8, (a^3*(26*A + 23*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(19*A + 17*B)*Tan[c + d*x])/(5*d) + (a^3*(26*A + 23*B)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(22*A + 21*B)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (a*B*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((3*A + 4*B)*Sec[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^3*(19*A + 17*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 11, (a^3*(15*A + 13*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(15*A + 13*B)*Tan[c + d*x])/(5*d) + (3*a^3*(15*A + 13*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*A - B)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (B*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(15*A + 13*B)*Tan[c + d*x]^3)/(60*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 10, (5*a^3*(4*A + 3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(4*A + 3*B)*Tan[c + d*x])/d + (3*a^3*(4*A + 3*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (a^3*(4*A + 3*B)*Tan[c + d*x]^3)/(12*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 6, a^3*A*x + (a^3*(7*A + 5*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A + B)*Tan[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*A + 5*B)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 5, a^3*(3*A + B)*x + (a^3*(6*A + 7*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*B*Sin[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((A + 2*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 5, (1/2)*a^3*(7*A + 6*B)*x + (a^3*(A + 3*B)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*A*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 5, (1/2)*a^3*(5*A + 7*B)*x + (a^3*B*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((5*A + 3*B)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 8, (5/8)*a^3*(3*A + 4*B)*x + (a^3*(3*A + 4*B)*Sin[c + d*x])/d + (3*a^3*(3*A + 4*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(3*A + 4*B)*Sin[c + d*x]^3)/(12*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 7, (1/8)*a^3*(13*A + 15*B)*x + (a^3*(38*A + 45*B)*Sin[c + d*x])/(15*d) + (a^3*(13*A + 15*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*(43*A + 45*B)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((7*A + 5*B)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(20*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^3, x, 8, (1/16)*a^3*(23*A + 26*B)*x + (a^3*(17*A + 19*B)*Sin[c + d*x])/(5*d) + (a^3*(23*A + 26*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(21*A + 22*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*A + 3*B)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(17*A + 19*B)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 14, (7*a^4*(8*A + 7*B)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(8*A + 7*B)*Tan[c + d*x])/(5*d) + (27*a^4*(8*A + 7*B)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(8*A + 7*B)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + ((6*A - B)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (B*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(8*A + 7*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 13, (7*a^4*(5*A + 4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (8*a^4*(5*A + 4*B)*Tan[c + d*x])/(5*d) + (27*a^4*(5*A + 4*B)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + (a^4*(5*A + 4*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (B*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + (4*a^4*(5*A + 4*B)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 7, a^4*A*x + (a^4*(48*A + 35*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*(8*A + 7*B)*Tan[c + d*x])/(8*d) + (a*B*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + ((4*A + 7*B)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + ((32*A + 35*B)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 6, a^4*(4*A + B)*x + (a^4*(13*A + 12*B)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*(A + 2*B)*Sin[c + d*x])/(2*d) + (a*B*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((A + 2*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((9*A + 11*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 6, (1/2)*a^4*(13*A + 8*B)*x + (a^4*(8*A + 13*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(A - B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((A + 6*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 6, (1/2)*a^4*(12*A + 13*B)*x + (a^4*(A + 4*B)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(2*A + B)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((2*A + B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((8*A - 3*B)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 6, (1/8)*a^4*(35*A + 48*B)*x + (a^4*B*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 8*B)*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + ((7*A + 4*B)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + ((35*A + 32*B)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 11, (7/8)*a^4*(4*A + 5*B)*x + (8*a^4*(4*A + 5*B)*Sin[c + d*x])/(5*d) + (27*a^4*(4*A + 5*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^4*(4*A + 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) - (4*a^4*(4*A + 5*B)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 8, (7/16)*a^4*(7*A + 8*B)*x + (a^4*(72*A + 83*B)*Sin[c + d*x])/(15*d) + (7*a^4*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(159*A + 176*B)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*A + 2*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + ((73*A + 72*B)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d)} -{Cos[c + d*x]^7*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^4, x, 9, (1/16)*a^4*(44*A + 49*B)*x + (a^4*(227*A + 252*B)*Sin[c + d*x])/(35*d) + (a^4*(44*A + 49*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(276*A + 301*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((10*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*d) + (7*(A + B)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(227*A + 252*B)*Sin[c + d*x]^3)/(105*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 6, (3*(A - B)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((3*A - 4*B)*Tan[c + d*x])/(a*d) + (3*(A - B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A - 4*B)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 6, -(((2*A - 3*B)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(A - B)*Tan[c + d*x])/(a*d) - ((2*A - 3*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 5, ((A - B)*ArcTanh[Sin[c + d*x]])/(a*d) + (B*Tan[c + d*x])/(a*d) - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 3, (B*ArcTanh[Sin[c + d*x]])/(a*d) + ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 2, (A*x)/a - ((A - B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 4, -(((A - B)*x)/a) + ((2*A - B)*Sin[c + d*x])/(a*d) - ((A - B)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 5, ((3*A - 2*B)*x)/(2*a) - (2*(A - B)*Sin[c + d*x])/(a*d) + ((3*A - 2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]), x, 6, -((3*(A - B)*x)/(2*a)) + ((4*A - 3*B)*Sin[c + d*x])/(a*d) - (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A - 3*B)*Sin[c + d*x]^3)/(3*a*d)} - - -{Sec[c + d*x]^5*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 7, ((7*A - 10*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(2*A - 3*B)*Tan[c + d*x])/(a^2*d) + ((7*A - 10*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((7*A - 10*B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(2*A - 3*B)*Tan[c + d*x]^3)/(3*a^2*d)} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 7, -(((4*A - 7*B)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + (2*(5*A - 8*B)*Tan[c + d*x])/(3*a^2*d) - ((4*A - 7*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((5*A - 8*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 6, ((A - 2*B)*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((A - 4*B)*Tan[c + d*x])/(3*a^2*d) - ((A - 2*B)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 4, (B*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*A - 5*B)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 2, ((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((A + 2*B)*Tan[c + d*x])/(3*d*(a^2 + a^2*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 3, (A*x)/a^2 - ((4*A - B)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 5, -(((2*A - B)*x)/a^2) + (2*(5*A - 2*B)*Sin[c + d*x])/(3*a^2*d) - ((2*A - B)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 6, ((7*A - 4*B)*x)/(2*a^2) - (2*(8*A - 5*B)*Sin[c + d*x])/(3*a^2*d) + ((7*A - 4*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*A - 5*B)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2, x, 7, -(((10*A - 7*B)*x)/(2*a^2)) + (4*(3*A - 2*B)*Sin[c + d*x])/(a^2*d) - ((10*A - 7*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*A - 7*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(3*A - 2*B)*Sin[c + d*x]^3)/(3*a^2*d)} - - -{Sec[c + d*x]^5*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 8, -(((6*A - 13*B)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (8*(9*A - 19*B)*Tan[c + d*x])/(15*a^3*d) - ((6*A - 13*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*A - 11*B)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (4*(9*A - 19*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 7, ((A - 3*B)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((7*A - 27*B)*Tan[c + d*x])/(15*a^3*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*A - 9*B)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((A - 3*B)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 5, (B*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((2*A - 7*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((4*A - 29*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 3, -(((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3)) + ((3*A - 8*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + 7*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 3, ((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 3*B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 4, (A*x)/a^3 - ((A - B)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 2*B)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (2*(11*A - B)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 6, -(((3*A - B)*x)/a^3) + (2*(36*A - 11*B)*Sin[c + d*x])/(15*a^3*d) - ((A - B)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - 4*B)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 7, ((13*A - 6*B)*x)/(2*a^3) - (8*(19*A - 9*B)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (4*(19*A - 9*B)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3, x, 8, -(((23*A - 13*B)*x)/(2*a^3)) + (4*(34*A - 19*B)*Sin[c + d*x])/(5*a^3*d) - ((23*A - 13*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A - 13*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A - 19*B)*Sin[c + d*x]^3)/(15*a^3*d)} - - -{Sec[c + d*x]^6*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 9, -(((8*A - 21*B)*ArcTanh[Sin[c + d*x]])/(2*a^4*d)) + (8*(83*A - 216*B)*Tan[c + d*x])/(105*a^4*d) - ((8*A - 21*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) + ((52*A - 129*B)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*(83*A - 216*B)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((A - 2*B)*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^5*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 8, ((A - 4*B)*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((55*A - 244*B)*Tan[c + d*x])/(105*a^4*d) + ((25*A - 88*B)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((A - 4*B)*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((5*A - 12*B)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 6, (B*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((6*A - 55*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((12*A - 215*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((3*A - 10*B)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 4, -(((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4)) + ((4*A + 3*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(4*A + 3*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((4*A + 3*B)*Tan[c + d*x])/(15*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 4, -(((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4)) + ((4*A - 11*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((8*A + 13*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((8*A + 13*B)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 4, ((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((3*A + 4*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + (2*(3*A + 4*B)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + (2*(3*A + 4*B)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 5, (A*x)/a^4 - ((55*A - 6*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (2*(80*A - 3*B)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((10*A - 3*B)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 7, -(((4*A - B)*x)/a^4) + (8*(83*A - 20*B)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 8, ((21*A - 8*B)*x)/(2*a^4) - (8*(216*A - 83*B)*Sin[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A - 52*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(216*A - 83*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4, x, 9, -(((44*A - 21*B)*x)/(2*a^4)) + (8*(227*A - 108*B)*Sin[c + d*x])/(35*a^4*d) - ((44*A - 21*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((178*A - 87*B)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((44*A - 21*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((16*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (8*(227*A - 108*B)*Sin[c + d*x]^3)/(105*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 5, (4*a*(9*A + 8*B)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*A + 8*B)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(9*A + 8*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (4*(9*A + 8*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 4, (2*a*(7*A + 6*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(7*A + 6*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A + 6*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 3, (2*a*(5*A + 7*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*A - 2*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 2, (2*a*(3*A + B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 4, (2*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*B*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 3, (Sqrt[a]*(A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 4, (Sqrt[a]*(3*A + 4*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(3*A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 5, (Sqrt[a]*(5*A + 6*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 6*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(5*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/2), x, 6, (5*Sqrt[a]*(7*A + 8*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (5*a*(7*A + 8*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (5*a*(7*A + 8*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(7*A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 5, (2*a^2*(39*A + 34*B)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(9*A + 10*B)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(39*A + 34*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*B*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(39*A + 34*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 4, (8*a^2*(21*A + 19*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(21*A + 19*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A - 2*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 3, (8*a^2*(5*A + 3*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 5, (2*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(3*A + 4*B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 4, (a^(3/2)*(3*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(A - 2*B)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 4, (a^(3/2)*(7*A + 12*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 5, (a^(3/2)*(11*A + 14*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(11*A + 14*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2), x, 6, (a^(3/2)*(75*A + 88*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(75*A + 88*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(9*A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 6, (2*a^3*(803*A + 710*B)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(209*A + 194*B)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(803*A + 710*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a^2*(11*A + 14*B)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*a*(803*A + 710*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*a*B*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 5, (64*a^3*(15*A + 13*B)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(15*A + 13*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(15*A + 13*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*A - 2*B)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*B*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 4, (64*a^3*(7*A + 5*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(7*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*A + 5*B)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 6, (2*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(35*A + 32*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(5*A + 8*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*B*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 5, (a^(5/2)*(5*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*(3*A + 14*B)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(A + 2*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 5, (a^(5/2)*(19*A + 20*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(9*A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 4*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 5, (a^(5/2)*(25*A + 38*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(3*A + 2*B)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 6, (a^(5/2)*(163*A + 200*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(163*A + 200*B)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(95*A + 104*B)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(11*A + 8*B)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2), x, 7, (a^(5/2)*(283*A + 326*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(283*A + 326*B)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(157*A + 170*B)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 6, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*A - 37*B)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(7*A - 31*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 5, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(5*A - 7*B)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*A - B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 4, -((Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*A - 2*B)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*B*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 3, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*B*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 5, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 6, -(((A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 7, ((7*A - 4*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/2), x, 8, -(((9*A - 14*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 6, ((11*A - 15*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((65*A - 93*B)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*A - 9*B)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((35*A - 39*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 5, -(((7*A - 11*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 13*B)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((3*A - 7*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 4, ((3*A - 7*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*B*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 3, ((A + 3*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 7, -(((3*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 8, ((19*A - 12*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A - 6*B)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 9, -(((47*A - 38*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d)) + ((17*A - 13*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (7*(3*A - 2*B)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A - 12*B)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 3*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 6, -(((75*A - 163*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) + ((A - B)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((9*A - 17*B)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((93*A - 197*B)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 95*B)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 5, ((19*A - 75*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((A - 9*B)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 4, ((5*A + 19*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 13*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 4, ((3*A + 5*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*A + 5*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 3*B)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 8, -(((5*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - 7*B)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A - 11*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 9, ((39*A - 20*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A - 11*B)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - (7*(9*A - 5*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A - 15*B)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a - a Sec[e+f x])^(m/2) (A+A Sec[e+f x])*) - - -{Sec[c + d*x]^0*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(1/2), x, 5, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^1*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(1/2), x, 6, (3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(1/2), x, 7, (11*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (5*A*Sin[c + d*x])/(4*d*Sqrt[a - a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(1/2), x, 8, (23*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*Sqrt[a]*d) - (2*Sqrt[2]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[a]*d) + (9*A*Sin[c + d*x])/(8*d*Sqrt[a - a*Sec[c + d*x]]) + (7*A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a - a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a - a*Sec[c + d*x]])} - - -{Sec[c + d*x]^0*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2), x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(3/2)*d) - (3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Tan[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)), (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(3/2)*d) - (3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) + (A*Csc[(1/2)*(c + d*x)]^2*Sin[c + d*x])/(2*a*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2), x, 7, (5*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(3/2)*d) - (7*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (2*A*Sin[c + d*x])/(a*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2), x, 8, (31*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(3/2)*d) - (11*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (13*A*Sin[c + d*x])/(4*a*d*Sqrt[a - a*Sec[c + d*x]]) + (3*A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2), x, 9, (85*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(3/2)*d) - (15*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a - a*Sec[c + d*x])^(3/2)) + (35*A*Sin[c + d*x])/(8*a*d*Sqrt[a - a*Sec[c + d*x]]) + (25*A*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a - a*Sec[c + d*x]]) + (4*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a - a*Sec[c + d*x]])} - - -{Sec[c + d*x]^0*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2), x, 7, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(5/2)*d) - (23*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Tan[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (7*A*Tan[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)), (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(5/2)*d) - (23*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) + (7*A*Csc[(1/2)*(c + d*x)]^2*Sin[c + d*x])/(16*a^2*d*Sqrt[a - a*Sec[c + d*x]]) - (A*Cos[c + d*x]*Csc[(1/2)*(c + d*x)]^4*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2), x, 8, (7*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(a^(5/2)*d) - (79*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (11*A*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (23*A*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2), x, 9, (59*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(4*a^(5/2)*d) - (167*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (15*A*Cos[c + d*x]*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (49*A*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (23*A*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d*Sqrt[a - a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2), x, 10, (203*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a - a*Sec[c + d*x]]])/(8*a^(5/2)*d) - (287*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sec[c + d*x]])])/(8*Sqrt[2]*a^(5/2)*d) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a - a*Sec[c + d*x])^(5/2)) - (19*A*Cos[c + d*x]^2*Sin[c + d*x])/(8*a*d*(a - a*Sec[c + d*x])^(3/2)) + (21*A*Sin[c + d*x])/(2*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (119*A*Cos[c + d*x]*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x]]) + (77*A*Cos[c + d*x]^2*Sin[c + d*x])/(24*a^2*d*Sqrt[a - a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 9, (-6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (6*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*B*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 8, (-2*a*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(A + B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 7, (-2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 6, (2*a*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 6, (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 7, (2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 8, (6*a*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 9, (-4*a^2*(4*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^2*(7*A + 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(7*A + 9*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*B*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 8, (-4*a^2*(5*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^2*(5*A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(5*A + 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*B*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 7, (-4*a^2*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(3*A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 7, (4*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 7, (4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(7*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 8, (4*a^2*(3*A + 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(9*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (4*a^2*(6*A + 7*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2), x, 9, (4*a^2*(8*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(11*A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (4*a^2*(8*A + 9*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 6*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]), x, 10, (-4*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(21*A + 17*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(13*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (4*a^3*(24*A + 23*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*A + 13*B)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]), x, 9, (-4*a^3*(9*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(9*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(42*A + 41*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*B*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*A + 11*B)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 8, (-4*a^3*(5*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(20*A + 21*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*B*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(5*A + 9*B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 8, (4*a^3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (20*a^3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - B)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 8, (4*a^3*(9*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(6*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(9*A + 5*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 8, (4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(41*A + 42*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(11*A + 7*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2), x, 9, (4*a^3*(17*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(23*A + 24*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^3*(11*A + 13*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(13*A + 9*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2), x, 10, (4*a^3*(15*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(21*A + 22*B)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^3*(15*A + 17*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 121*B)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(15*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 9, (3*(5*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - (3*(5*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) + (5*(A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((5*A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 8, (-3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((3*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((3*A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 7, ((A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 6, -(((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])), x, 6, ((3*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, (-3*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A - 3*B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, (3*(7*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - (5*(A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A - 5*B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (5*(A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])), x, 9, (-21*(A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(9*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) + ((9*A - 7*B)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (7*(A - B)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(9*A - 7*B)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(d*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))} - - -{(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 9, -(((4*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) - (5*(A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((4*A - 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) - (5*(A - 2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) + ((4*A - 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 8, ((A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((2*A - 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 7, (B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - (B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 7, -((A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((2*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 7, ((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 8, -(((7*A - 4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (5*(2*A - B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (5*(2*A - B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A - 4*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 9, (7*(8*A - 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A - 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (7*(8*A - 5*B)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A - 2*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A - 2*B)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A - B)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} - - -{(Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 10, (-7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(7*A - 17*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((13*A - 33*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 9, ((9*A - 49*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((9*A - 49*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A - 8*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A - 13*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 8, ((A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 6*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((A + 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 8, -((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A + 4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 8, -((9*A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*A + 2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3), x, 8, ((49*A - 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 9, (-7*(17*A - 7*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((33*A - 13*B)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - (7*(17*A - 7*B)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 10, (7*(33*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((21*A - 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + (7*(33*A - 17*B)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((21*A - 11*B)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - ((12*A - 7*B)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - (3*(21*A - 11*B)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 5, (Sqrt[a]*(6*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(6*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 4, (Sqrt[a]*(4*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(4*A + 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 3, (Sqrt[a]*(2*A + B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 3, (2*Sqrt[a]*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 2, (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 3, (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(4*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 4, (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(6*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(6*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 6, (a^(3/2)*(88*A + 75*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(88*A + 75*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 5, (a^(3/2)*(14*A + 11*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(14*A + 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(6*A + 7*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 4, (a^(3/2)*(12*A + 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(4*A + 5*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])/Sqrt[Sec[c + d*x]], x, 4, (a^(3/2)*(2*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(2*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(3/2), x, 4, (2*a^(3/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(4*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(5/2), x, 3, (8*a^2*(3*A + 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(7/2), x, 4, (2*a^2*(8*A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(52*A + 63*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(9/2), x, 5, (2*a^2*(10*A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(34*A + 39*B)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(34*A + 39*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, (a^(5/2)*(326*A + 283*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(326*A + 283*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(170*A + 157*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(10*A + 13*B)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (a^(5/2)*(200*A + 163*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(200*A + 163*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(104*A + 95*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 11*B)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 5, (a^(5/2)*(38*A + 25*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(54*A + 49*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sqrt[Sec[c + d*x]], x, 5, (a^(5/2)*(20*A + 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(4*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(4*A + 7*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(3/2), x, 5, (a^(5/2)*(2*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(14*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(2*A - 3*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(5/2), x, 5, (2*a^(5/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(32*A + 35*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(7/2), x, 4, (64*a^3*(5*A + 7*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(9/2), x, 5, (2*a^3*(124*A + 135*B)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(292*A + 345*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(4*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])/Sec[c + d*x]^(11/2), x, 6, (2*a^3*(194*A + 209*B)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(710*A + 803*B)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(710*A + 803*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(14*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/Sqrt[a + a*Sec[c + d*x]], x, 7, -(((4*A - 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((4*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/Sqrt[a + a*Sec[c + d*x]], x, 6, ((2*A - B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])/Sqrt[a + a*Sec[c + d*x]], x, 5, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 3, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 5, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]), x, 6, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A - 91*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 8, -(((12*A - 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d)) + ((9*A - 13*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((6*A - 7*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 7, ((2*A - 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((A - 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2), x, 3, ((3*A + B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 4, -(((7*A - 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 5, ((11*A - 7*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A - 3*B)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A - 15*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 6, -(((15*A - 11*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 5*B)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B)*Sin[c + d*x])/(30*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((147*A - 95*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 8, ((2*A - 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 15*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((11*A - 35*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*A - 11*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 4, ((5*A + 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2), x, 5, ((19*A + 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)), ((19*A + 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((9*A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 5, -(((75*A - 19*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 6, ((163*A - 75*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((95*A - 39*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A - 147*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 7, -(((283*A - 163*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A - 85*B)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A - 1495*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/3) (A+B Sec[e+f x])*) - - -{(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3), x, 9, (3*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*B*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/3), x, 8, (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(4/3), x, 9, (3*B*Tan[c + d*x])/(5*a*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-(5/6), 1/2, 1, 1/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - -{(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(4/3), x, 11, (3*a*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) - (15*(1 + Sqrt[3])*a*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*3^(1/4)*a*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3), x, 10, (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*B*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(2/3), x, 11, (3*B*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-(1/6), 1/2, 1, 5/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) + (3*(1 + Sqrt[3])*B*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) - (3*2^(1/3)*3^(1/4)*B*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) - (3^(3/4)*(1 - Sqrt[3])*B*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]) with m and/or n symbolic*) - - -{(c*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x, 7, -((B*AppellF1[n, 1/2, -(1/2) - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(c*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]])) - ((A - B)*AppellF1[n, 1/2, 1/2 - m, 1 + n, Sec[e + f*x], -Sec[e + f*x]]*(c*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(-(1/2) - m)*(a + a*Sec[e + f*x])^m*Tan[e + f*x])/(f*n*Sqrt[1 - Sec[e + f*x]])} - - -{Sec[c + d*x]^(-1 - n)*(A + B*Sec[c + d*x])*(a + a*Sec[c + d*x])^n, x, 4, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n))) + ((B + A*n + B*n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, -((2*Sec[c + d*x])/(1 - Sec[c + d*x]))]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 6, ((4*a*A + 3*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((A*b + a*B)*Tan[c + d*x])/d + ((4*a*A + 3*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((A*b + a*B)*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 6, ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*a*A + 2*b*B)*Tan[c + d*x])/(3*d) + ((A*b + a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*B*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 5, ((2*a*A + b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + ((A*b + a*B)*Tan[c + d*x])/d + (b*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 4, a*A*x + ((A*b + a*B)*ArcTanh[Sin[c + d*x]])/d + (b*B*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 3, (A*b + a*B)*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 4, (1/2)*(a*A + 2*b*B)*x + ((A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 5, (1/2)*(A*b + a*B)*x + ((2*a*A + 3*b*B)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 6, (1/8)*(3*a*A + 4*b*B)*x + ((A*b + a*B)*Sin[c + d*x])/d + ((3*a*A + 4*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*b^2*B + 5*a*(2*A*b + a*B))*Tan[c + d*x])/(5*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*(5*A*b + 6*a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (b*B*Sec[c + d*x]^3*(a + b*Sec[c + d*x])*Tan[c + d*x])/(5*d) + ((4*b^2*B + 5*a*(2*A*b + a*B))*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, ((8*a*A*b + 4*a^2*B + 3*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*Tan[c + d*x])/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A*b - a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 6, ((2*a^2*A + A*b^2 + 2*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a*A*b + a^2*B + b^2*B)*Tan[c + d*x])/(3*d) + (b*(3*A*b + 2*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (B*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 5, a^2*A*x + ((4*a*A*b + 2*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(2*A*b + 3*a*B)*Tan[c + d*x])/(2*d) + (b*B*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 5, a*(2*A*b + a*B)*x + (b*(A*b + 2*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*A*Sin[c + d*x])/d + (b^2*B*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 5, (1/2)*(a^2*A + 2*A*b^2 + 4*a*b*B)*x + (b^2*B*ArcTanh[Sin[c + d*x]])/d + (a*(2*A*b + a*B)*Sin[c + d*x])/d + (a^2*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 5, (1/2)*(2*a*A*b + a^2*B + 2*b^2*B)*x + ((2*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, (1/8)*(3*a^2*A + 4*A*b^2 + 8*a*b*B)*x + ((2*a*A*b + a^2*B + b^2*B)*Sin[c + d*x])/d + ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(2*A*b + a*B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, (1/8)*(6*a*A*b + 3*a^2*B + 4*b^2*B)*x + ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, ((12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*Tan[c + d*x])/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*A*b - a*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (B*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 7, ((8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Tan[c + d*x])/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 6, a^3*A*x + ((6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(9*a*A*b + 8*a^2*B + 2*b^2*B)*Tan[c + d*x])/(3*d) + (b^2*(3*A*b + 5*a*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*B*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 6, a^2*(3*A*b + a*B)*x + (b*(6*a*A*b + 6*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*a*A - b*B)*Sin[c + d*x])/(2*d) + (b*B*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (b^2*(A*b + 2*a*B)*Tan[c + d*x])/d, a^2*(3*A*b + a*B)*x + (b*(6*a*A*b + 6*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a^2*A - A*b^2 - 3*a*b*B)*Tan[c + d*x])/d - (b^2*(2*a*A - b*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 6, (1/2)*a*(a^2*A + 6*A*b^2 + 6*a*b*B)*x + (b^2*(A*b + 3*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a*A - 2*b*B)*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 6, (1/2)*(3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*x + (b^3*B*ArcTanh[Sin[c + d*x]])/d + (a*(2*a^2*A + 8*A*b^2 + 9*a*b*B)*Sin[c + d*x])/(3*d) + (a^2*(5*A*b + 3*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 6, (1/8)*(3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*x + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*d) + (a*(3*a^2*A + 10*A*b^2 + 12*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(3*A*b + 2*a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (1/8)*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*x + ((4*a^3*A + 14*a*A*b^2 + 15*a^2*b*B + 5*b^3*B)*Sin[c + d*x])/(5*d) + ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(7*A*b + 5*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a*(4*a^2*A + 12*A*b^2 + 15*a*b*B)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 9, ((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*ArcTanh[Sin[c + d*x]])/(16*d) + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Tan[c + d*x])/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) + ((6*A*b - a*B)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (B*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 8, ((8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*ArcTanh[Sin[c + d*x]])/(8*d) + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*Tan[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (B*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, a^4*A*x + ((32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(34*a^2*A*b + 4*A*b^3 + 19*a^3*B + 16*a*b^2*B)*Tan[c + d*x])/(6*d) + (b^2*(32*a*A*b + 26*a^2*B + 9*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b*B*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, a^3*(4*A*b + a*B)*x + (b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d - (b*(6*a^3*A - 12*a*A*b^2 - 17*a^2*b*B - 2*b^3*B)*Tan[c + d*x])/(3*d) - (b^2*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*a*A - b*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, (1/2)*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*x + (b^2*(8*a*A*b + 12*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (b*(13*a^2*A*b - 2*A*b^3 + 4*a^3*B - 8*a*b^2*B)*Tan[c + d*x])/(2*d) - (b^2*(6*a*A*b + 2*a^2*B - b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, (1/2)*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*x + (b^3*(A*b + 4*a*B)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*Sin[c + d*x])/(3*d) + (a*(2*A*b + a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^2*(8*a*A*b + 3*a^2*B - 6*b^2*B)*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, (1/8)*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*x + (b^4*B*ArcTanh[Sin[c + d*x]])/d + (a*(16*a^2*A*b + 19*A*b^3 + 4*a^3*B + 34*a*b^2*B)*Sin[c + d*x])/(6*d) + (a^2*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*(7*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 7, (1/8)*(12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*x + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*Sin[c + d*x])/(15*d) + (a*(60*a^2*A*b + 56*A*b^3 + 15*a^3*B + 110*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a^2*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (a*(8*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (a*A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 9, (1/16)*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*x + ((48*a^3*A*b + 53*a*A*b^3 + 12*a^4*B + 87*a^2*b^2*B + 15*b^4*B)*Sin[c + d*x])/(15*d) + ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (a*(3*A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (a*A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - (a*(16*a^2*A*b + 13*A*b^3 + 4*a^3*B + 27*a*b^2*B)*Sin[c + d*x]^3)/(15*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 8, ((2*a^2 + b^2)*(A*b - a*B)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*Tan[c + d*x])/(3*b^3*d) + ((A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (B*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 7, -(((2*a*A*b - 2*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^3*d)) + (2*a^2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((A*b - a*B)*Tan[c + d*x])/(b^2*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 7, ((A*b - a*B)*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*a*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (B*Tan[c + d*x])/(b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 5, (B*ArcTanh[Sin[c + d*x]])/(b*d) + (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 4, (A*x)/a - (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 5, -(((A*b - a*B)*x)/a^2) + (2*b*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 6, ((a^2*A + 2*A*b^2 - 2*a*b*B)*x)/(2*a^3) - (2*b^2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 7, -(((a^2 + 2*b^2)*(A*b - a*B)*x)/(2*a^4)) + (2*b^3*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B)*Sin[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 8, ((3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*x)/(8*a^5) - (2*b^4*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*a^2 + 3*b^2)*(A*b - a*B)*Sin[c + d*x])/(3*a^4*d) + ((3*a^2*A + 4*A*b^2 - 4*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 8, -(((4*a*A*b - 6*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^4*d)) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 7, ((A*b - 2*a*B)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (B*Tan[c + d*x])/(b^2*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 6, (B*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(A*b - a*B)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 5, (2*(a*A - b*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((A*b - a*B)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 5, (A*x)/a^2 - (2*(2*a^2*A*b - A*b^3 - a^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 6, -(((2*A*b - a*B)*x)/a^3) + (2*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2*A - 2*A*b^2 + a*b*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 7, ((a^2*A + 6*A*b^2 - 4*a*b*B)*x)/(2*a^4) - (2*b^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 8, -(((2*a^2*A*b + 8*A*b^3 - a^3*B - 6*a*b^2*B)*x)/(2*a^5)) + (2*b^3*(5*a^2*A*b - 4*A*b^3 - 4*a^3*B + 3*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((2*a^4*A + 7*a^2*A*b^2 - 12*A*b^4 - 6*a^3*b*B + 9*a*b^3*B)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) - ((2*a^2*A*b - 4*A*b^3 - a^3*B + 3*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) + ((a^2*A - 4*A*b^2 + 3*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^5*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 9, -(((6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^5*d)) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 8, ((A*b - 3*a*B)*ArcTanh[Sin[c + d*x]])/(b^4*d) - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 7, (B*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((a^2*A*b^3 + 2*A*b^5 - 2*a^5*B + 5*a^3*b^2*B - 6*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 6, -(((3*a*A*b - a^2*B - 2*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(A*b - a*B)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 6, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b - a*B)*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 6, (A*x)/a^3 - ((6*a^4*A*b - 5*a^2*A*b^3 + 2*A*b^5 - 2*a^5*B - a^3*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 7, -(((3*A*b - a*B)*x)/a^4) + (b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(6*a^2*A*b - 3*A*b^3 - 4*a^3*B + a*b^2*B)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 8, ((a^2*A + 12*A*b^2 - 6*a*b*B)*x)/(2*a^5) - (b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^5*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 9, ((A*b - 4*a*B)*ArcTanh[Sin[c + d*x]])/(b^5*d) - (a*(2*a^6*A*b - 7*a^4*A*b^3 + 8*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 28*a^5*b^2*B - 35*a^3*b^4*B + 20*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((3*a^3*A*b - 8*a*A*b^3 - 12*a^4*B + 23*a^2*b^2*B - 6*b^4*B)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 8, (B*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 34*a*b^4*B)*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 7, ((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*(A*b - a*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 7, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b - a*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 7, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b - a*B)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Tan[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Tan[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 7, (A*x)/a^4 - ((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 8, -(((4*A*b - a*B)*x)/a^5) + (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4, x, 9, ((a^2*A + 20*A*b^2 - 8*a*b*B)*x)/(2*a^6) - (b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - - -{(b*B/a + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 4, (b*B*x)/a^2 + (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*d)} -{(a*B/b + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 2, (B*x)/b} - -{(a + b*Sec[c + d*x])/(b + a*Sec[c + d*x])^2, x, 5, (a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^2*d) - (a*Tan[c + d*x])/(b*d*(b + a*Sec[c + d*x]))} -{(3 + Sec[c + d*x])/(2 - Sec[c + d*x]), x, 4, (3*x)/2 - (5*Log[Cos[(1/2)*(c + d*x)] - Sqrt[3]*Sin[(1/2)*(c + d*x)]])/(2*Sqrt[3]*d) + (5*Log[Cos[(1/2)*(c + d*x)] + Sqrt[3]*Sin[(1/2)*(c + d*x)]])/(2*Sqrt[3]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 7, (-2*(a - b)*Sqrt[a + b]*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) + 18*a*b^2*(A - 2*B) + 12*a^2*b*(2*A - B) - 16*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*A*b - 75*A*b^3 - 8*a^3*B - 13*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b + a*B)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*B*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)} -{Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 6, (2*(a - b)*Sqrt[a + b]*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) + 2*a*b*(7*A - 3*B) - 8*a^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(7*a*A*b - 4*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*A*b + a*B)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b*d) + (2*B*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 5, (-2*(a - b)*Sqrt[a + b]*(5*a*A*b - 2*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*(a - b)*Sqrt[a + b]*(5*A*b - 2*a*B - 9*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*A*b - 2*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 4, (-2*(a - b)*Sqrt[a + b]*(3*A*b + a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*(a - b)*Sqrt[a + b]*(3*A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 5, (-2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(A*b + (a - b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{Cos[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 6, (A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A + 2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 7, ((a - b)*Sqrt[a + b]*(A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2*A - A*b^2 + 4*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 8, ((a - b)*Sqrt[a + b]*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) + (Sqrt[a + b]*(2*a + b)*(8*a*A - 3*A*b + 6*a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (Sqrt[a + b]*(4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 7, (2*(a - b)*Sqrt[a + b]*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) - 3*a*b^2*(57*A - 13*B) - 6*a^2*b*(3*A - B) + 8*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*A*b - 8*a^2*B - 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*A*b - 4*a*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 6, (-2*(a - b)*Sqrt[a + b]*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) + 6*a^2*B - a*(21*A*b - 57*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(21*a*A*b - 6*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*B*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 5, (-2*(a - b)*Sqrt[a + b]*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(a - b)*Sqrt[a + b]*(15*a*A - 5*A*b - 3*a*B + 9*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(5*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 6, (-2*(a - b)*Sqrt[a + b]*(3*A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*Sqrt[a + b]*(b^2*(3*A - B) - 3*a^2*B - a*(6*A*b - 4*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 6, ((a - b)*Sqrt[a + b]*(a*A - 2*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(2*b*(A - B) + a*(A + 4*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(3*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 7, ((a - b)*Sqrt[a + b]*(5*A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(2*a*A + 5*A*b + 4*a*B + 8*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*A + 3*A*b^2 + 12*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((5*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 8, ((a - b)*Sqrt[a + b]*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*A + 14*a*A*b + 3*A*b^2 + 12*a^2*B + 30*a*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) - (Sqrt[a + b]*(12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((7*A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a*A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 8, (2*(a - b)*Sqrt[a + b]*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(a - b)*Sqrt[a + b]*(6*a*b^3*(209*A - 505*B) - 3*b^4*(539*A - 225*B) - 15*a^2*b^2*(121*A - 19*B) + 40*a^4*B - a^3*(110*A*b - 30*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*A*b - 1254*a*A*b^3 - 40*a^4*B - 285*a^2*b^2*B - 675*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*A*b - 539*A*b^3 - 40*a^3*B - 335*a*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) - (2*(22*a*A*b - 8*a^2*B - 81*b^2*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*A*b - 4*a*B)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*B*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, (-2*(a - b)*Sqrt[a + b]*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) - 6*a*b^2*(60*A - 19*B) + 15*a^2*b*(3*A - 11*B) - 10*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*B*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (-2*(a - b)*Sqrt[a + b]*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*A - 25*B) - 8*a*b*(7*A - 15*B) + 15*a^2*(7*A - B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*B*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, (-2*(a - b)*Sqrt[a + b]*(35*a*A*b + 23*a^2*B + 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*Sqrt[a + b]*(a^2*b*(45*A - 23*B) - a*b^2*(35*A - 17*B) + b^3*(5*A - 9*B) + 15*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*(5*A*b + 8*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*B*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, ((a - b)*Sqrt[a + b]*(3*a^2*A - 6*A*b^2 - 14*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(2*a*b*(9*A - 7*B) - 2*b^2*(3*A - B) + 3*a^2*(A + 6*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (a*Sqrt[a + b]*(5*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*a*A - 2*b*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, ((a - b)*Sqrt[a + b]*(9*a*A*b + 4*a^2*B - 8*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(8*b^2*(A - B) + 2*a^2*(A + 2*B) + 3*a*b*(3*A + 8*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*A + 15*A*b^2 + 20*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (a*(7*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 8, ((a - b)*Sqrt[a + b]*(16*a^2*A + 33*A*b^2 + 54*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2*A + 26*a*A*b + 33*A*b^2 + 12*a^2*B + 54*a*b*B + 48*b^2*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2*A + 33*A*b^2 + 54*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(3*A*b + 2*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 9, ((a - b)*Sqrt[a + b]*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B) + 4*a^2*b*(71*A + 52*B) + 2*a*b^2*(59*A + 132*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(11*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 5, (2*(a - b)*Sqrt[a + b]*(10*a*A*b - 8*a^2*B - 9*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (2*Sqrt[a + b]*(b^2*(5*A - 9*B) - 8*a^2*B + 2*a*b*(5*A + B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*B*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 4, (-2*(a - b)*Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) - (2*Sqrt[a + b]*(3*A*b - (2*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 3, (-2*(a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(A - B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 3, (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 6, (A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (A*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b]*(A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 7, -((a - b)*Sqrt[a + b]*(3*A*b - 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*b*d) - (Sqrt[a + b]*(3*A*b - 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(4*a^2*A + 3*A*b^2 - 4*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - ((3*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]], x, 8, ((a - b)*Sqrt[a + b]*(16*a^2*A + 15*A*b^2 - 18*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*b*d) + (Sqrt[a + b]*(16*a^2*A - 10*a*A*b + 15*A*b^2 + 12*a^2*B - 18*a*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*d) + (Sqrt[a + b]*(4*a^2*A*b + 5*A*b^3 - 8*a^3*B - 6*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^4*d) + ((16*a^2*A + 15*A*b^2 - 18*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^3*d) - ((5*A*b - 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a^2*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)} - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 5, -((2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d)) - (2*(2*a + b)*(3*A*b - (4*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*(A*b - a*B)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 4, (2*(a*A*b - 2*a^2*B + b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b - (2*a + b)*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(A*b - a*B)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 4, (-2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*(A + B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(A*b - a*B)*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*(A*b - a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*(A*b - a*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b*(A*b - a*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 7, ((a^2*A - 3*A*b^2 + 2*a*b*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((3*A*b + a*(A - 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2*A - 3*A*b^2 + 2*a*b*B)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 8, -((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*b*Sqrt[a + b]*d) - ((15*A*b^2 + a*b*(5*A - 12*B) - 2*a^2*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(4*a^2*A + 15*A*b^2 - 12*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - ((5*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2), x, 9, ((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^4*b*Sqrt[a + b]*d) + ((105*A*b^3 + 5*a*b^2*(7*A - 18*B) + 4*a^3*(4*A + 3*B) - 6*a^2*b*(A + 5*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^4*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a^2*A*b + 35*A*b^3 - 8*a^3*B - 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^5*d) + ((16*a^2*A + 35*A*b^2 - 30*a*b*B)*Sin[c + d*x])/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((7*A*b - 6*a*B)*Cos[c + d*x]*Sin[c + d*x])/(12*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Tan[c + d*x])/(24*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 6, (-2*(8*a^4*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) + (2*(9*a*b^3*(A - B) + b^4*(3*A - B) + 16*a^4*B - 2*a^2*b^2*(3*A + 8*B) - a^3*(8*A*b - 12*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(a*A*b - 2*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 5, (2*(2*a^3*A*b - 6*a*A*b^3 - 8*a^4*B + 15*a^2*b^2*B - 3*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) + (2*(2*a^2*b*(A - 3*B) - 3*b^3*(A - B) - 8*a^3*B + 3*a*b^2*(A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (2*a^2*(A*b - a*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 5, (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*B - 3*b^2*(A + B) + a*b*(A + 3*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 5, (-2*(4*a*A*b - a^2*B - 3*b^2*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(3*a*A - A*b + a*B - 3*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*(A*b - a*B)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 7, (2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*(6*a^2*A*b - a*A*b^2 - 3*A*b^3 - 3*a^3*B + a^2*b*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 8, ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) - ((15*A*b^3 + a*b^2*(5*A - 6*B) - 3*a^3*(A - 4*B) - a^2*b*(21*A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*Sqrt[a + b]*(a^2 - b^2)*d) + (Sqrt[a + b]*(5*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^2*A - 5*A*b^2 + 2*a*b*B)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (b*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2), x, 9, -((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*(a - b)*b*(a + b)^(3/2)*d) + ((105*A*b^4 + 5*a*b^3*(7*A - 12*B) + 6*a^4*(A + 2*B) - 5*a^2*b^2*(27*A + 4*B) - a^3*(27*A*b - 84*b*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(4*a^2*A + 35*A*b^2 - 20*a*b*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - ((7*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(Sec[e + f*x]*(A + A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]], x, 1, (-2*A*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f)} -{(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]], x, 1, (2*A*Sqrt[a - b]*(a + b)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a - b]], (a - b)/(a + b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(b^2*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 8, -((2*(5*a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*a*A + 3*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(A*b + a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 7, -((2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(A*b + a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 6, (2*(a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*b*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 6, (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 7, (2*(3*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x]), x, 8, (6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a*A + 7*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 9, -((2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(7*A*b + 9*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 8, -((2*(3*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(3*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(5*A*b + 7*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, (2*(a^2*A - A*b^2 - 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(3*A*b + 5*a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, -((2*(b^2*B - a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*b^2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 8, (2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 9, (2*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(7*b^2*B + 5*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(2*A*b + a*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(7*b^2*B + 5*a*(2*A*b + a*B))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 10, -((2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(27*a*A*b + 22*a^2*B + 7*b^2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*b^2*(9*A*b + 13*a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 9, -((2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(15*a*A*b + 14*a^2*B + 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b^2*(5*A*b + 9*a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(2*a^2*A - 3*A*b^2 - 9*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(a*A - b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(9*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(a*A - 5*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(11*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*a*(5*a^2*A + 18*A*b^2 + 21*a*b*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 9, (2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(13*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*a*(7*a^2*A + 22*A*b^2 + 27*a*b*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{Sec[c + d*x]^(-11/2)*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 10, (2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(15*A*b + 11*a*B)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*a*(9*a^2*A + 26*A*b^2 + 33*a*b*B)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 11, (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^2*d) + (2*a^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) - (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*b^3*d) + (2*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*d) + (2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 10, -((2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*d)) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b*d) - (2*a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) + (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 7, -((2*B*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*d)) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a + b)*d) + (2*B*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 5, (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 7, (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 9, -((2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(a^2*A + 3*A*b^2 - 3*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]), x, 10, (2*(3*a^2*A + 5*A*b^2 - 5*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(a^2 + 3*b^2)*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^4*d) + (2*b^3*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 11, -(((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) - (a*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) + ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 10, ((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d) + ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d) + ((a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^2*(a + b)^2*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 9, -(((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*(a^2 - b^2)*d)) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 9, ((A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*d) + ((2*a^2*A - A*b^2 - a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 9, ((2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) - ((4*a^2*A*b - 3*A*b^3 - 2*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) + (b*(5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2, x, 10, -(((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d)) + ((2*a^4*A + 16*a^2*A*b^2 - 15*A*b^4 - 12*a^3*b*B + 9*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) - (b^2*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 12, -(((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d)) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + ((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 11, ((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 10, ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d) + ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 10, -(((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 10, ((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4*A - 5*a^2*A*b^2 + 3*A*b^4 - 7*a^3*b*B + a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 10, ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((24*a^4*A*b - 33*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + (b*(35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3, x, 11, -(((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d)) + ((8*a^6*A + 128*a^4*A*b^2 - 223*a^2*A*b^4 + 105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b^2*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 13, ((4*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b + a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d) + (B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 12, ((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 11, (2*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 8, (2*A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 9, (-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 10, (2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 14, ((42*a*A*b + 17*a^2*B + 16*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((30*a*A*b + 3*a^2*B + 16*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d) + ((6*A*b + 7*a*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (b*B*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 13, ((8*a^2*A + 4*A*b^2 + 7*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + 5*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b + 5*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 12, ((2*a*A*b + 2*a^2*B + b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A - b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 12, (2*(a^2*A - A*b^2 + 3*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 9, (2*(a^2 - b^2)*(3*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(6*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 10, (2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(8*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2), x, 11, (2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sec[c + d*x]^(3/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 15, ((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*d*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (b*(8*A*b + 11*a*B)*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (b*B*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 14, ((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*A*b + 33*a^2*B + 16*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (b*(2*A*b + 3*a*B)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]], x, 13, ((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (b*(4*A*b + 7*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (b*B*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2), x, 13, ((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((14*a*A*b + 6*a^2*B - 3*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(2*a*A - 3*b*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2), x, 13, (2*(8*a^2*A*b - 8*A*b^3 + 5*a^3*B + 10*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(8*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2), x, 10, (2*(a^2 - b^2)*(25*a^2*A + 15*A*b^2 + 56*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(10*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2), x, 11, (2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(4*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2), x, 12, (2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3465*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 13, ((4*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*A*b - 3*a*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (B*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 12, (B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 7, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 7, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]), x, 8, (2*(a^2*A + 2*A*b^2 - 3*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]), x, 9, (-2*(7*a^2*A*b + 8*A*b^3 - 5*a^3*B - 10*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 13, (B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A*b - 3*a^2*B + b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 9, (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b - a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 8, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b - a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)), x, 8, (-2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)), x, 9, (2*(a^2*A + 8*A*b^2 - 6*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)), x, 10, (-2*(12*a^2*A*b + 48*A*b^3 - 5*a^3*B - 40*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^4*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 13, (2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 9, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2*A + A*b^2 - 4*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 9, (2*(3*a^2*A - 2*A*b^2 - a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)), x, 9, (-2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)), x, 11, (-2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/3) (A+B Sec[e+f x])*) - - -{(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^(2/3), x, 4, (Sqrt[2]*B*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(2/3), x]} -{(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^(1/3), x, 4, (Sqrt[2]*B*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(1/3), x]} -{(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3), x, 4, (Sqrt[2]*B*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(1/3), x]} -{(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3), x, 4, (Sqrt[2]*B*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(2/3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]) with m and/or n symbolic*) - - -{(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x, 0, Unintegrable[(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x]} - - -{Sec[c + d*x]^m*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^4, x, 9, If[$VersionNumber>=8, (b*(A*b^3*(8 + 6*m + m^2) + 4*a*b^2*B*(8 + 6*m + m^2) + 2*a^3*B*(19 + 8*m + m^2) + a^2*A*b*(68 + 37*m + 5*m^2))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(3 + m)*(4 + m)) + (b^2*(b^2*B*(3 + m)^2 + 2*a*A*b*(4 + m)^2 + a^2*B*(26 + 9*m + m^2))*Sec[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)*(4 + m)) + (b*(A*b*(4 + m) + a*B*(7 + m))*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(d*(4 + m)) - ((A*b^4*m*(2 + m) + 4*a*b^3*B*m*(2 + m) + 6*a^2*A*b^2*m*(3 + m) + 4*a^3*b*B*m*(3 + m) + a^4*A*(3 + 4*m + m^2))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m)*(1 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2]) + ((b^4*B*(3 + 4*m + m^2) + 4*a*A*b^3*(4 + 5*m + m^2) + 6*a^2*b^2*B*(4 + 5*m + m^2) + 4*a^3*A*b*(8 + 6*m + m^2) + a^4*B*(8 + 6*m + m^2))*Hypergeometric2F1[1/2, -(m/2), (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2]), (b*(A*b^3*(8 + 6*m + m^2) + 4*a*b^2*B*(8 + 6*m + m^2) + 2*a^3*B*(19 + 8*m + m^2) + a^2*A*b*(68 + 37*m + 5*m^2))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(4 + m)*(3 + 4*m + m^2)) + (b^2*(b^2*B*(3 + m)^2 + 2*a*A*b*(4 + m)^2 + a^2*B*(26 + 9*m + m^2))*Sec[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(4 + m)*(6 + 5*m + m^2)) + (b*(A*b*(4 + m) + a*B*(7 + m))*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(d*(3 + m)*(4 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(d*(4 + m)) - ((A*b^4*m*(2 + m) + 4*a*b^3*B*m*(2 + m) + 6*a^2*A*b^2*m*(3 + m) + 4*a^3*b*B*m*(3 + m) + a^4*A*(3 + 4*m + m^2))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(3 + m)*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((b^4*B*(3 + 4*m + m^2) + 4*a*A*b^3*(4 + 5*m + m^2) + 6*a^2*b^2*B*(4 + 5*m + m^2) + 4*a^3*A*b*(8 + 6*m + m^2) + a^4*B*(8 + 6*m + m^2))*Hypergeometric2F1[1/2, -(m/2), (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*(4 + m)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^3, x, 8, (b*(b^2*B*(2 + m) + 3*a*A*b*(3 + m) + 2*a^2*B*(4 + m))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(3 + m)) + (b^2*(A*b*(3 + m) + a*B*(5 + m))*Sec[c + d*x]^(2 + m)*Sin[c + d*x])/(d*(2 + m)*(3 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(d*(3 + m)) - ((b^3*B*m*(2 + m) + 3*a*A*b^2*m*(3 + m) + 3*a^2*b*B*m*(3 + m) + a^3*A*(3 + 4*m + m^2))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(3 + m)*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((A*b^3*(1 + m) + 3*a*b^2*B*(1 + m) + 3*a^2*A*b*(2 + m) + a^3*B*(2 + m))*Hypergeometric2F1[1/2, -(m/2), (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^m*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^2, x, 7, (b*(A*b*(2 + m) + a*B*(3 + m))*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)*(2 + m)) + (b*B*Sec[c + d*x]^(1 + m)*(a + b*Sec[c + d*x])*Sin[c + d*x])/(d*(2 + m)) - ((A*b^2*m + 2*a*b*B*m + a^2*A*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((b^2*B*(1 + m) + a*(2*A*b + a*B)*(2 + m))*Hypergeometric2F1[1/2, -(m/2), (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*(2 + m)*Sqrt[Sin[c + d*x]^2])} -{Sec[c + d*x]^m*(A + B*Sec[c + d*x])*(a + b*Sec[c + d*x])^1, x, 6, (b*B*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m)) - ((b*B*m + a*A*(1 + m))*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(1 - m^2)*Sqrt[Sin[c + d*x]^2]) + ((A*b + a*B)*Hypergeometric2F1[1/2, -(m/2), (2 - m)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*m*Sqrt[Sin[c + d*x]^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 8, (6*a*(A + B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 7, (2*a*(3*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 6, (2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 6, (2*a*(A - B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/d + (2*a*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 7, (-2*a*(A + B)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 8, (-2*a*(5*A + 3*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 9, (4*a^2*(8*A + 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^2*(5*A + 6*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (4*a^2*(8*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(11*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 8, (4*a^2*(3*A + 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(6*A + 7*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(9*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 7, (4*a^2*(4*A + 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 7, (4*a^2*A*EllipticE[(1/2)*(c + d*x), 2])/d + (4*a^2*(2*A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 7, -((4*a^2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (4*a^2*(3*A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(3*A + 5*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 8, -((4*a^2*(5*A + 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(2*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(5*A + 7*B)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 4*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 9, -((4*a^2*(4*A + 3*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(7*A + 6*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(7*A + 9*B)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 6*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(4*A + 3*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*B*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 7, (3*(7*A - 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - (5*(A - B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - (5*(A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 6, -((3*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((5*A - 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((5*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]), x, 5, ((3*A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - ((A - B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])), x, 5, -(((A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((A + B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 6, ((A - 3*B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((A - B)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - ((A - 3*B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 7, -((3*(A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) - ((3*A - 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - ((3*A - 5*B)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(A - B)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 8, (7*(8*A - 5*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*(3*A - 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*(3*A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (7*(8*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A - 2*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 7, -(((7*A - 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (5*(2*A - B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (5*(2*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2, x, 6, ((4*A - B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - ((5*A - 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((5*A - 2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 6, -((A*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((2*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 6, (B*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + ((A + 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 7, ((A - 4*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + ((2*A - 5*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((A - 4*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((2*A - 5*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2), x, 8, -(((4*A - 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) - (5*(A - 2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*(A - 2*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + ((4*A - 7*B)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((4*A - 7*B)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) + ((A - B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 8, -((7*(17*A - 7*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((33*A - 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((33*A - 13*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - (7*(17*A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3, x, 7, ((49*A - 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - ((13*A - 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3), x, 7, -(((9*A + B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((3*A + B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 7, -(((A - B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((A + B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 7, ((A + 9*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((A + 3*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) + ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((A - 6*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3), x, 8, ((9*A - 49*B)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((3*A - 13*B)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((9*A - 49*B)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) + ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + ((3*A - 8*B)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((3*A - 13*B)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 6, (32*a*(8*A + 9*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(8*A + 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(8*A + 9*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(8*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 5, (16*a*(6*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(6*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 4, (4*a*(4*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 3, (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 4, (2*Sqrt[a]*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 4, (Sqrt[a]*(2*A + B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*B*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 5, (Sqrt[a]*(4*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*A + 3*B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2), x, 6, (Sqrt[a]*(6*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*A + 5*B)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 7, (32*a^2*(168*A + 187*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(168*A + 187*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(168*A + 187*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(168*A + 187*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(12*A + 11*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 6, (16*a^2*(34*A + 39*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(34*A + 39*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(34*A + 39*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(10*A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 5, (4*a^2*(52*A + 63*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 4, (8*a^2*(3*A + 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 5, (2*a^(3/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 5, (a^(3/2)*(2*A + 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(2*A - B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 5, (a^(3/2)*(12*A + 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(4*A + 5*B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 6, (a^(3/2)*(14*A + 11*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(6*A + 7*B)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(14*A + 11*B)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2), x, 7, (a^(3/2)*(88*A + 75*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(8*A + 9*B)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(88*A + 75*B)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 7, (16*a^3*(710*A + 803*B)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(710*A + 803*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(710*A + 803*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(194*A + 209*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(14*A + 11*B)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (4*a^3*(292*A + 345*B)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(292*A + 345*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(124*A + 135*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(4*A + 3*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 5, (64*a^3*(5*A + 7*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (2*a^(5/2)*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 35*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (a^(5/2)*(2*A + 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(14*A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(2*A - 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 6, (a^(5/2)*(20*A + 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(4*A - 9*B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(4*A + 7*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 6, (a^(5/2)*(38*A + 25*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(54*A + 49*B)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(2*A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 7, (a^(5/2)*(200*A + 163*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(104*A + 95*B)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(200*A + 163*B)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(8*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2), x, 8, (a^(5/2)*(326*A + 283*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(170*A + 157*B)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(326*A + 283*B)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(10*A + 13*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (a*B*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]], x, 7, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]], x, 6, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]], x, 5, (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]], x, 4, -((Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 6, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 7, ((2*A - B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 8, -(((4*A - 7*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d)) + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((4*A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2), x, 7, -(((15*A - 11*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((147*A - 95*B)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A - 5*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2), x, 6, ((11*A - 7*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A - 15*B)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2), x, 5, -(((7*A - 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 4, ((3*A + B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 7, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 8, ((2*A - 3*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) - ((5*A - 9*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) - ((A - 3*B)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)), x, 9, -(((12*A - 19*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d)) + ((9*A - 13*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) - ((A - 2*B)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((6*A - 7*B)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2), x, 8, -(((283*A - 163*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A - 1495*B)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A - 85*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2), x, 7, ((163*A - 75*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A - 147*B)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((95*A - 39*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2), x, 6, -(((75*A - 19*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 6, ((19*A + 5*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) - ((9*A - B)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 5, ((5*A + 3*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 8, (2*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((3*A - 11*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)), x, 9, ((2*A - 5*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) - ((43*A - 115*B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) + ((A - B)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 15*B)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) - ((11*A - 35*B)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^m (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 8, (6*(A*b + a*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*a*A + 7*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*a*A + 7*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 7, (2*(3*a*A + 5*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 6, (2*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(a*A + 3*b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]), x, 6, (2*(a*A - b*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/d + (2*b*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x])/Cos[c + d*x]^(1/2), x, 7, (-2*(A*b + a*B)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*a*A + b*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x])/Cos[c + d*x]^(3/2), x, 8, (-2*(5*a*A + 3*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*a*A + 3*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 7, (2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*a^2*A + 7*b*(A*b + 2*a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(9*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 6, (2*(3*a^2*A + 5*b*(A*b + 2*a*B))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(7*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 6, (2*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a^2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]), x, 6, (2*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x])/Cos[c + d*x]^(1/2), x, 7, -((2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b^2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x])/Cos[c + d*x]^(3/2), x, 8, -((2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b^2*B*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*b*(A*b + 2*a*B)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]), x, 8, (2*(3*a^2*A + 5*A*b^2 - 5*a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d) - (2*(a^2 + 3*b^2)*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^4*d) + (2*b^3*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]), x, 7, (-2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(a^2*A + 3*A*b^2 - 3*a*b*B)*EllipticF[(c + d*x)/2, 2])/(3*a^3*d) - (2*b^2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]), x, 6, (2*A*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(a^2*d) + (2*b*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])), x, 4, (2*A*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - (2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a*(a + b)*d)} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])), x, 6, (-2*B*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) + (2*B*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])), x, 8, (-2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*b*d) - (2*a*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d) + (2*B*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) + (2*(A*b - a*B)*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])), x, 9, (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d) + (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d) + (2*a^2*(A*b - a*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*B*Sin[c + d*x])/(5*b*d*Cos[c + d*x]^(5/2)) + (2*(A*b - a*B)*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(3/2)) - (2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sin[c + d*x])/(5*b^3*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2, x, 8, -(((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d)) + ((2*a^4*A + 16*a^2*A*b^2 - 15*A*b^4 - 12*a^3*b*B + 9*a*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^4*(a^2 - b^2)*d) - (b^2*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2, x, 7, ((2*a^2*A - 3*A*b^2 + a*b*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) - ((4*a^2*A*b - 3*A*b^3 - 2*a^3*B + a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) + (b*(5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2), x, 7, ((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) + ((2*a^2*A - A*b^2 - a*b*B)*EllipticF[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) - ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*(a + b)^2*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2), x, 7, -(((A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(a*(a^2 - b^2)*d) + ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a*(a - b)*b*(a + b)^2*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2), x, 8, ((a*A*b - 3*a^2*B + 2*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d) + ((A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(b*(a^2 - b^2)*d) + ((a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^2*(a + b)^2*d) - ((a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2), x, 9, -(((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d)) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*(a^2 - b^2)*d) - (a*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3, x, 9, -(((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d)) + ((8*a^6*A + 128*a^4*A*b^2 - 223*a^2*A*b^4 + 105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B)*EllipticF[(1/2)*(c + d*x), 2])/(12*a^5*(a^2 - b^2)^2*d) - (b^2*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3, x, 8, ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - ((24*a^4*A*b - 33*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + (b*(35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3), x, 8, ((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) + ((8*a^4*A - 5*a^2*A*b^2 + 3*A*b^4 - 7*a^3*b*B + a*b^3*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*(a + b)^3*d) + (b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3), x, 8, -(((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b*(a + b)^3*d) - ((A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3), x, 8, ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b*(a^2 - b^2)^2*d) + ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^2*(a + b)^3*d) + (a*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - (a*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3), x, 9, ((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) + ((a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*EllipticF[(1/2)*(c + d*x), 2])/(4*b^2*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3), x, 10, -(((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*EllipticE[(1/2)*(c + d*x), 2])/(4*b^4*(a^2 - b^2)^2*d)) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*EllipticF[(1/2)*(c + d*x), 2])/(12*b^3*(a^2 - b^2)^2*d) - (a*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*(a - b)^2*b^4*(a + b)^3*d) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) + ((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (a*(A*b - a*B)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 11, (2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 10, (-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 9, (2*A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]), x, 12, (2*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 13, ((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 14, ((4*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((4*a*A*b - a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*A*b + a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 12, (2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(10*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 11, (2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(8*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 10, (2*(a^2 - b^2)*(3*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(6*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 13, (2*(a^2*A - A*b^2 + 3*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]), x, 13, ((2*a*A*b + 2*a^2*B + b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 14, ((8*a^2*A + 4*A*b^2 + 7*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 15, ((42*a*A*b + 17*a^2*B + 16*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + ((6*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Cos[c + d*x]^(3/2)) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 13, (2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(14*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*a*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 12, (2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(4*A*b + 3*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 11, (2*(a^2 - b^2)*(25*a^2*A + 15*A*b^2 + 56*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(10*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*a*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 14, (2*(8*a^2*A*b - 8*A*b^3 + 5*a^3*B + 10*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(8*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 14, ((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((14*a*A*b + 6*a^2*B - 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(2*a*A - 3*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]), x, 14, ((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(4*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]], x, 15, ((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(2*A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2), x, 16, ((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(192*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(64*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (b*(8*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(3/2)) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + (b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 10, (-2*(7*a^2*A*b + 8*A*b^3 - 5*a^3*B - 10*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 9, (2*(a^2*A + 2*A*b^2 - 3*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]], x, 8, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 8, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]), x, 13, (B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]), x, 14, ((4*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) + ((4*A*b - 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 11, (-2*(12*a^2*A*b + 48*A*b^3 - 5*a^3*B - 40*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 10, (2*(a^2*A + 8*A*b^2 - 6*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2), x, 9, (-2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)), x, 9, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)), x, 10, (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)), x, 14, (B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((2*a*A*b - 3*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)), x, 15, ((4*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a*A*b - 15*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 5*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 12, (-2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^5*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 11, (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2), x, 10, (-2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)), x, 10, (2*(3*a^2*A - 2*A*b^2 - a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b - a*B)*Sin[c + d*x])/(3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(3*a^2*A + A*b^2 - 4*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)), x, 14, (2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)), x, 15, -((2*a*A*b - 5*a^2*B + 3*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m deleted file mode 100644 index e356073..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).m +++ /dev/null @@ -1,137 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (A+C Sec[e+f x]^2)*) - - -{Sec[c + d*x]^6*(A + C*Sec[c + d*x]^2), x, 3, ((7*A + 6*C)*Tan[c + d*x])/(7*d) + (C*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + (2*(7*A + 6*C)*Tan[c + d*x]^3)/(21*d) + ((7*A + 6*C)*Tan[c + d*x]^5)/(35*d)} -{Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2), x, 4, ((6*A + 5*C)*ArcTanh[Sin[c + d*x]])/(16*d) + ((6*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((6*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (C*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2), x, 3, ((5*A + 4*C)*Tan[c + d*x])/(5*d) + (C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*A + 4*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2), x, 3, ((4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2), x, 3, ((3*A + 2*C)*Tan[c + d*x])/(3*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2), x, 2, ((2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2), x, 3, A*x + (C*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2), x, 2, (C*ArcTanh[Sin[c + d*x]])/d + (A*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2), x, 2, (1/2)*(A + 2*C)*x + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2), x, 3, ((A + C)*Sin[c + d*x])/d - (A*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2), x, 3, (1/8)*(3*A + 4*C)*x + ((3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2), x, 4, ((A + C)*Sin[c + d*x])/d - ((2*A + C)*Sin[c + d*x]^3)/(3*d) + (A*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2), x, 4, (1/16)*(5*A + 6*C)*x + ((5*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*A + 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} - - -{Sec[c + d*x]^m*(-C*m/(m + 1) + C*Sec[c + d*x]^2), x, 1, (C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + m))} -{Sec[c + d*x]^m*(A - A*(m + 1)/m*Sec[c + d*x]^2), x, 1, -((A*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -{(A + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(5/2), x, 4, (2*b^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*d) + (2*b*(7*A + 5*C)*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(21*d) + (2*C*(b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{(A + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(3/2), x, 4, -((2*b^2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b*(5*A + 3*C)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*C*(b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{(A + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(1/2), x, 3, (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*C*Sqrt[b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/2), x, 3, (2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*C*Tan[c + d*x])/(d*Sqrt[b*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2), x, 3, (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*A*Tan[c + d*x])/(3*d*(b*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2), x, 3, (2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*A*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/2))} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2), x, 4, (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^4*d) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b^3*d*Sqrt[b*Sec[c + d*x]]) + (2*A*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/2))} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2), x, 4, (2*(7*A + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*b^4*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*(7*A + 9*C)*Sin[c + d*x])/(45*b^3*d*(b*Sec[c + d*x])^(3/2)) + (2*A*Tan[c + d*x])/(9*d*(b*Sec[c + d*x])^(9/2))} - - -{(3 + 3*Sec[c + d*x]^2)/Sec[c + d*x]^(1/2), x, 1, (6*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (A+C Sec[e+f x]^2) when A (m+1)+C m=0*) - - -{Sec[e + f*x]^m*(m - (m + 1)*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]^(1 + m)*Sin[e + f*x])/f)} - -{Sec[e + f*x]^5*(5 - 6*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]^5*Tan[e + f*x])/f)} -{Sec[e + f*x]^4*(4 - 5*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]^4*Tan[e + f*x])/f)} -{Sec[e + f*x]^3*(3 - 4*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]^3*Tan[e + f*x])/f)} -{Sec[e + f*x]^2*(2 - 3*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]^2*Tan[e + f*x])/f)} -{Sec[e + f*x]^1*(1 - 2*Sec[e + f*x]^2), x, 1, -((Sec[e + f*x]*Tan[e + f*x])/f)} -{Sec[e + f*x]^0*(0 - 1*Sec[e + f*x]^2), x, 2, -(Tan[e + f*x]/f)} -{Cos[e + f*x]^1*(-1 - 0*Sec[e + f*x]^2), x, 1, -(Sin[e + f*x]/f)} -{Cos[e + f*x]^2*(-2 + 1*Sec[e + f*x]^2), x, 1, -((Cos[e + f*x]*Sin[e + f*x])/f)} -{Cos[e + f*x]^3*(-3 + 2*Sec[e + f*x]^2), x, 1, -((Cos[e + f*x]^2*Sin[e + f*x])/f)} -{Cos[e + f*x]^4*(-4 + 3*Sec[e + f*x]^2), x, 1, -((Cos[e + f*x]^3*Sin[e + f*x])/f)} -{Cos[e + f*x]^5*(-5 + 4*Sec[e + f*x]^2), x, 1, -((Cos[e + f*x]^4*Sin[e + f*x])/f)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*C*ArcTanh[Sin[c + d*x]])/(8*d) + (B*Tan[c + d*x])/d + (3*C*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (B*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (B*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Tan[c + d*x])/d + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (C*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (C*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (B*ArcTanh[Sin[c + d*x]])/d + (C*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, B*x + (C*ArcTanh[Sin[c + d*x]])/d} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 3, C*x + (B*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (B*x)/2 + (C*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (C*x)/2 + (B*Sin[c + d*x])/d + (C*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (B*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*B*x)/8 + (C*Sin[c + d*x])/d + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (C*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*C*x)/8 + (B*Sin[c + d*x])/d + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(3/2), x, 10, -((6*b^2*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (6*b*C*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*C*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*b*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(1/2), x, 9, -((2*b*B*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*C*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*b*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/2), x, 8, -((2*C*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(b*d) + (2*C*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2), x, 7, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(b^2*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2), x, 8, (2*C*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*B*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Sec[c + d*x]])} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2), x, 9, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^4*d) + (2*B*Sin[c + d*x])/(5*b^2*d*(b*Sec[c + d*x])^(3/2)) + (2*C*Sin[c + d*x])/(3*b^3*d*Sqrt[b*Sec[c + d*x]])} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2), x, 10, (6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (10*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^5*d) + (2*B*Sin[c + d*x])/(7*b^2*d*(b*Sec[c + d*x])^(5/2)) + (2*C*Sin[c + d*x])/(5*b^3*d*(b*Sec[c + d*x])^(3/2)) + (10*B*Sin[c + d*x])/(21*b^4*d*Sqrt[b*Sec[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*B*ArcTanh[Sin[c + d*x]])/(8*d) + ((5*A + 4*C)*Tan[c + d*x])/(5*d) + (3*B*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*A + 4*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, ((4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (B*Tan[c + d*x])/d + ((4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (B*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (B*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A + 2*C)*Tan[c + d*x])/(3*d) + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, ((2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (B*Tan[c + d*x])/d + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, A*x + (B*ArcTanh[Sin[c + d*x]])/d + (C*Tan[c + d*x])/d} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, B*x + (C*ArcTanh[Sin[c + d*x]])/d + (A*Sin[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (1/2)*(A + 2*C)*x + (B*Sin[c + d*x])/d + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (B*x)/2 + ((A + C)*Sin[c + d*x])/d + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (A*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (1/8)*(3*A + 4*C)*x + (B*Sin[c + d*x])/d + ((3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (B*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (3*B*x)/8 + ((A + C)*Sin[c + d*x])/d + (3*B*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((2*A + C)*Sin[c + d*x]^3)/(3*d) + (A*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/16)*(5*A + 6*C)*x + (B*Sin[c + d*x])/d + ((5*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*A + 6*C)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(3/2), x, 8, -((2*b^2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*b*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*b*(5*A + 3*C)*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*B*(b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*C*(b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(b*Sec[c + d*x])^(1/2), x, 7, -((2*b*B*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])) + (2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*d) + (2*B*Sqrt[b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*C*Sqrt[b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/2), x, 6, (2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(b*d) + (2*C*Tan[c + d*x])/(d*Sqrt[b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(3/2), x, 6, (2*B*EllipticE[(1/2)*(c + d*x), 2])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^2*d) + (2*A*Tan[c + d*x])/(3*d*(b*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(5/2), x, 7, (2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(3*b^3*d) + (2*B*Sin[c + d*x])/(3*b^2*d*Sqrt[b*Sec[c + d*x]]) + (2*A*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(7/2), x, 8, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[b*Sec[c + d*x]])/(21*b^4*d) + (2*B*Sin[c + d*x])/(5*b^2*d*(b*Sec[c + d*x])^(3/2)) + (2*(5*A + 7*C)*Sin[c + d*x])/(21*b^3*d*Sqrt[b*Sec[c + d*x]]) + (2*A*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/2))} - - -(* ::Title:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m deleted file mode 100644 index 00f38be..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.4.2 (a+b sec)^m (d sec)^n (A+B sec+C sec^2).m +++ /dev/null @@ -1,2325 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (b Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2)*) -(**) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[e+f x])^m (b Sec[e+f x])^n (A+C Sec[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^(m/2) (A+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (b Sec[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 4, (3*(10*A + 7*C)*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b^2*d)} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 4, (3*(7*A + 4*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b*d)} -{(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 3, -((3*b*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 4, -((3*b^2*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) + (3*b*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 4, -((3*b*(2*A + 5*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*b^2*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3))} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 4, (3*(13*A + 10*C)*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(91*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(10/3)*Tan[c + d*x])/(13*b^2*d)} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 4, (3*(10*A + 7*C)*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b*d)} -{(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 3, (3*b*(7*A + 4*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 4, -((3*b^2*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*b*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 4, -((3*b^3*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) + (3*b^2*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 4, (3*(8*A + 5*C)*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b^2*d)} -{(Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 4, -((3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d)} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/3), x, 3, -((3*b*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) + (3*C*Tan[c + d*x])/(2*d*(b*Sec[c + d*x])^(1/3))} -{(Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 4, -((3*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*b*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))} -{(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 4, -((3*b*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*b^2*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))} - - -{(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 4, -((3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)} -{(Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 4, -((3*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))} -{(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3), x, 3, -((3*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))} -{(Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 4, -((3*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*b*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))} -{(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 4, -((3*b*(7*A + 10*C)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])) + (3*A*b^2*Tan[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (b Sec[c+d x])^n with m symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (3*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 24*m + 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 - 12*m - 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(8 - 6*m - 9*m^2)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 4, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(8 + 6*m - 9*m^2)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 4, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 + 12*m - 9*m^2)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 4, If[$VersionNumber>=8, -((3*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3))) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(1 - 3*m)*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), -((3*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3))) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(7 - 24*m + 9*m^2)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+C Sec[c+d x]^2) (b Sec[c+d x])^n with n symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*(1 + m + n)*Sqrt[Sin[c + d*x]^2]), (C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m^2 - 2*m*n - n^2)*Sqrt[Sin[c + d*x]^2])]} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, ((C*(2 + n) + A*(3 + n))*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(2 + n)*Tan[c + d*x])/(b^2*d*(3 + n))} -{Sec[c + d*x]^1*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, ((C*(1 + n) + A*(2 + n))*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(1 + n)*Tan[c + d*x])/(b*d*(2 + n))} -{Sec[c + d*x]^0*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 3, If[$VersionNumber>=8, -((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n)), -((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n^2)*Sqrt[Sin[c + d*x]^2])) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))]} -{Cos[c + d*x]^1*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, (b^2*(C*(1 - n) - A*n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*n*Sqrt[Sin[c + d*x]^2]) + (b*C*(b*Sec[c + d*x])^(-1 + n)*Tan[c + d*x])/(d*n)} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, -((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(1 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n)), -((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - 4*n + n^2)*Sqrt[Sin[c + d*x]^2])) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n))]} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, -((b^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(2 - n)*(4 - n)*Sqrt[Sin[c + d*x]^2])) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n)), -((b^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(8 - 6*n + n^2)*Sqrt[Sin[c + d*x]^2])) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n))]} - - -{Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(21 + 20*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 12*n + 4*n^2)*Sqrt[Sin[c + d*x]^2])]} -{Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x, 4, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 + 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 4*n - 4*n^2)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 4, If[$VersionNumber>=8, (2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 + 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]), (2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 4*n - 4*n^2)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 4, If[$VersionNumber>=8, -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]])) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]])) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 12*n + 4*n^2)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 4, If[$VersionNumber>=8, -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2))) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (1/4)*(7 - 2*n), (1/4)*(11 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(7 - 2*n)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2))) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (1/4)*(7 - 2*n), (1/4)*(11 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(21 - 20*n + 4*n^2)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[c+d x])^m (b Sec[c+d x])^n (B Sec[c+d x]+C Sec[c+d x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^m (B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^(m/2) (B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^n with n symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (B*Hypergeometric2F1[1/2, (1/2)*(-m - n), (1/2)*(2 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2]) + (C*Hypergeometric2F1[1/2, (1/2)*(-1 - m - n), (1/2)*(1 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Sec[c+d x])^m (b Sec[c+d x])^n (A+B Sec[c+d x]+C Sec[c+d x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form Sec[c+d x]^(m/2) (A+B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^(n/3)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*(11*A + 8*C)*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(55*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(4/3), 1/2, -(1/3), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(8/3)*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*b^2*d)} -{Sec[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*(8*A + 5*C)*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b*d)} -{(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, -((3*b*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^2*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*b*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*C*Tan[c + d*x])/(2*d*(b*Sec[c + d*x])^(1/3))} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^2*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*b*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^2*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^3*B*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2])) - (3*b^2*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^3*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*(13*A + 10*C)*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(91*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(5/3), 1/2, -(2/3), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(10/3)*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(10/3)*Tan[c + d*x])/(13*b^2*d)} -{Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (3*(10*A + 7*C)*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b*d)} -{(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (3*b*(7*A + 4*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^2*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*b*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) + (3*b*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^3*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) - (3*b^2*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*b^2*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((3*b^3*B*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) - (3*b^2*(2*A + 5*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^3*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, (3*(7*A + 4*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b^2*d)} -{(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, -((3*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b*d)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(2/3), x, 6, -((3*b*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))} -{(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, -((3*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b*d)} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, (3*(7*A + 4*C)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b^2*d)} -{(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, (3*(10*A + 7*C)*Hypergeometric2F1[-(2/3), 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(7/6), 1/2, -(1/6), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b^3*d)} - - -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, -((3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)} -{(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, -((3*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3), x, 6, -((3*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))} -{(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, -((3*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, -((3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])) + (3*B*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)} -{(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, (3*(8*A + 5*C)*Hypergeometric2F1[-(1/3), 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-(5/6), 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b^3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^n with m symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (3*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[1/2, (1/6)*(-4 - 3*m), (1/6)*(2 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 24*m + 9*m^2)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[1/2, (1/6)*(-4 - 3*m), (1/6)*(2 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-2 - 3*m), (1/6)*(4 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(2 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 - 12*m - 9*m^2)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-2 - 3*m), (1/6)*(4 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(2 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(8 - 6*m - 9*m^2)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (1/6)*(-1 - 3*m), (1/6)*(5 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3), x, 7, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(8 + 6*m - 9*m^2)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(1 - 3*m), (1/6)*(7 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3), x, 7, If[$VersionNumber>=8, (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(2 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]), (3*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(5 - 3*m), (1/6)*(11 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 + 12*m - 9*m^2)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(2 - 3*m), (1/6)*(8 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(2 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])]} -{(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3), x, 7, If[$VersionNumber>=8, -((3*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3))) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(1 - 3*m)*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(b*d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]), -((3*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3))) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (1/6)*(7 - 3*m), (1/6)*(13 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(7 - 24*m + 9*m^2)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1/6)*(4 - 3*m), (1/6)*(10 - 3*m), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(b*d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]+C Sec[c+d x]^2) (b Sec[c+d x])^n with n symbolic*) - - -{Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*(1 + m + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-m - n), (1/2)*(2 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2]), (C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m^2 - 2*m*n - n^2)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-m - n), (1/2)*(2 - m - n), Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2])]} - - -{Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((C*(2 + n) + A*(3 + n))*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-2 - n), -(n/2), Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(2 + n)*Tan[c + d*x])/(b^2*d*(3 + n))} -{Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((C*(1 + n) + A*(2 + n))*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (1/2)*(-1 - n), (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(1 + n)*Tan[c + d*x])/(b*d*(2 + n))} -{(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, If[$VersionNumber>=8, -((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n)), -((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n^2)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, -(n/2), (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))]} -{Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (b^2*(C*(1 - n) - A*n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*n*Sqrt[Sin[c + d*x]^2]) - (b*B*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]) + (b*C*(b*Sec[c + d*x])^(-1 + n)*Tan[c + d*x])/(d*n)} -{Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, -((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(1 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*B*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n)), -((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - 4*n + n^2)*Sqrt[Sin[c + d*x]^2])) - (b^2*B*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n))]} -{Cos[c + d*x]^3*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, -((b^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(2 - n)*(4 - n)*Sqrt[Sin[c + d*x]^2])) - (b^3*B*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n)), -((b^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(8 - 6*n + n^2)*Sqrt[Sin[c + d*x]^2])) - (b^3*B*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n))]} - - -{Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-5 - 2*n), (1/4)*(-1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(21 + 20*n + 4*n^2)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-5 - 2*n), (1/4)*(-1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 12*n + 4*n^2)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-3 - 2*n), (1/4)*(1 - 2*n), Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, If[$VersionNumber>=8, (2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 + 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2]), (2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 4*n - 4*n^2)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (1/4)*(-1 - 2*n), (1/4)*(3 - 2*n), Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, If[$VersionNumber>=8, (2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 + 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]), (2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 4*n - 4*n^2)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(1 - 2*n), (1/4)*(5 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 7, If[$VersionNumber>=8, -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]])) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]])) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 12*n + 4*n^2)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(3 - 2*n), (1/4)*(7 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])]} -{((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 7, If[$VersionNumber>=8, -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2))) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (1/4)*(7 - 2*n), (1/4)*(11 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(7 - 2*n)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]), -((2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2))) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (1/4)*(7 - 2*n), (1/4)*(11 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(21 - 20*n + 4*n^2)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1/4)*(5 - 2*n), (1/4)*(9 - 2*n), Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (a*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 4*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (a*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 6, (a*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, a*A*x + (a*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, a*A*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (a*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, (1/2)*a*(A + 2*C)*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, (1/2)*a*(A + 2*C)*x + (a*(2*A + 3*C)*Sin[c + d*x])/(3*d) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (1/8)*a*(3*A + 4*C)*x + (a*(A + C)*Sin[c + d*x])/d + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*A*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (1/8)*a*(3*A + 4*C)*x + (a*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 8, (a^2*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^2*(4*A + 3*C)*Tan[c + d*x])/(3*d) + (a^2*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + ((10*A + 3*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(30*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(5*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(10*a*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 7, (a^2*(12*A + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(12*A + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(12*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 6, a^2*A*x + (a^2*(2*A + C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Tan[c + d*x])/d + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + (C*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 6, 2*a^2*A*x + (a^2*(2*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/d - (a^2*(2*A - 3*C)*Tan[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 5, (1/2)*a^2*(3*A + 2*C)*x + (2*a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 5, a^2*(A + 2*C)*x + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Sin[c + d*x])/d + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 6, (1/8)*a^2*(7*A + 12*C)*x + (a^2*(7*A + 12*C)*Sin[c + d*x])/(6*d) + (a^2*(7*A + 12*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 7, (1/4)*a^2*(3*A + 4*C)*x + (a^2*(18*A + 25*C)*Sin[c + d*x])/(15*d) + (a^2*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*(9*A + 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(10*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 8, (1/16)*a^2*(11*A + 14*C)*x + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a^2*(11*A + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(9*A + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (2*a^2*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 12, (a^3*(30*A + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 23*C)*Tan[c + d*x])/(10*d) + (3*a^3*(30*A + 23*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A + 7*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(120*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(10*a*d) + (a^3*(30*A + 23*C)*Tan[c + d*x]^3)/(120*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 11, (a^3*(20*A + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(20*A + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) - (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(20*A + 13*C)*Tan[c + d*x]^3)/(60*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 7, a^3*A*x + (a^3*(28*A + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(4*A + 3*C)*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (C*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(4*a*d) + ((4*A + 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(8*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 7, 3*a^3*A*x + (a^3*(6*A + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + (5*a^3*C*Tan[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*a*d) - ((6*A - 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, (1/2)*a^3*(7*A + 2*C)*x + (a^3*(2*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A - C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((A - 4*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, (1/2)*a^3*(5*A + 6*C)*x + (3*a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*A*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((5*A - 6*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, (1/8)*a^3*(15*A + 28*C)*x + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(3*A + 4*C)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(4*a*d) + ((5*A + 4*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 9, (1/8)*a^3*(13*A + 20*C)*x + (a^3*(13*A + 20*C)*Sin[c + d*x])/(5*d) + (3*a^3*(13*A + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (3*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) - (a^3*(13*A + 20*C)*Sin[c + d*x]^3)/(60*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 8, (1/16)*a^3*(23*A + 30*C)*x + (a^3*(34*A + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(23*A + 30*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*a*d) + ((31*A + 30*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(120*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 15, (a^4*(14*A + 11*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (16*a^4*(14*A + 11*C)*Tan[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(105*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(21*a*d) + (8*a^4*(14*A + 11*C)*Tan[c + d*x]^3)/(105*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 14, (7*a^4*(10*A + 7*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(10*A + 7*C)*Tan[c + d*x])/(5*d) + (27*a^4*(10*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(10*A + 7*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) - (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(10*A + 7*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 8, a^4*A*x + (a^4*(12*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*(10*A + 7*C)*Tan[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 8, 4*a^4*A*x + (a^4*(52*A + 35*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/d + (5*a^4*(4*A + 7*C)*Tan[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) - ((12*A - 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - ((12*A - 35*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 7, (1/2)*a^4*(13*A + 2*C)*x + (2*a^4*(2*A + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A - 2*C)*Sin[c + d*x])/(2*d) - (a*(3*A - 2*C)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((3*A + 22*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 7, 2*a^4*(3*A + 2*C)*x + (a^4*(2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(2*A - C)*Sin[c + d*x])/(2*d) + (2*a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((4*A - 9*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 7, (1/8)*a^4*(35*A + 52*C)*x + (4*a^4*C*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 4*C)*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((7*A + 4*C)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) - ((35*A - 12*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 7, (1/2)*a^4*(7*A + 12*C)*x + (a^4*C*ArcTanh[Sin[c + d*x]])/d + (a^4*(7*A + 10*C)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + ((7*A + 5*C)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + ((7*A + 8*C)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 12, (7/16)*a^4*(7*A + 10*C)*x + (4*a^4*(7*A + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (2*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) - (2*a^4*(7*A + 10*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^7*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^4, x, 9, (1/4)*a^4*(11*A + 14*C)*x + (a^4*(454*A + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(11*A + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^4*(247*A + 308*C)*Cos[c + d*x]^2*Sin[c + d*x])/(210*d) + (2*a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((8*A + 7*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + ((109*A + 126*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(210*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, (3*(4*A + 5*C)*ArcTanh[Sin[c + d*x]])/(8*a*d) - ((3*A + 4*C)*Tan[c + d*x])/(a*d) + (3*(4*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + ((4*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A + 4*C)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, -(((2*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((3*A + 4*C)*Tan[c + d*x])/(a*d) - ((2*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((3*A + 4*C)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, ((2*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((A + 2*C)*Tan[c + d*x])/(a*d) + ((2*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, -((C*ArcTanh[Sin[c + d*x]])/(a*d)) + (C*Tan[c + d*x])/(a*d) + ((A + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, (A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(a*d) - ((A + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, -((A*x)/a) + ((2*A + C)*Sin[c + d*x])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 5, ((3*A + 2*C)*x)/(2*a) - ((2*A + C)*Sin[c + d*x])/(a*d) + ((3*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, -(((3*A + 2*C)*x)/(2*a)) + ((4*A + 3*C)*Sin[c + d*x])/(a*d) - ((3*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, (3*(5*A + 4*C)*x)/(8*a) - ((4*A + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((5*A + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, -(((2*A + 5*C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((5*A + 12*C)*Tan[c + d*x])/(a^2*d) - ((2*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (2*(2*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((5*A + 12*C)*Tan[c + d*x]^3)/(3*a^2*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, ((2*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(A + 4*C)*Tan[c + d*x])/(3*a^2*d) + ((2*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (2*(A + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 6, -((2*C*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((A + 4*C)*Tan[c + d*x])/(3*a^2*d) + (2*C*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 4, (C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A - 5*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2), (C*ArcTanh[Sin[c + d*x]])/(a^2*d) + (2*(A - 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 3, (A*x)/a^2 - (2*(2*A - C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 5, -((2*A*x)/a^2) + ((10*A + C)*Sin[c + d*x])/(3*a^2*d) - (2*A*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 6, ((7*A + 2*C)*x)/(2*a^2) - (4*(4*A + C)*Sin[c + d*x])/(3*a^2*d) + ((7*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*(4*A + C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, -(((5*A + 2*C)*x)/a^2) + ((12*A + 5*C)*Sin[c + d*x])/(a^2*d) - ((5*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (2*(5*A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - ((12*A + 5*C)*Sin[c + d*x]^3)/(3*a^2*d)} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, ((2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A + 76*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 7, -((3*C*ArcTanh[Sin[c + d*x]])/(a^3*d)) + ((2*A + 27*C)*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (3*C*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 5, (C*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((3*A - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((6*A - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 3, -(((A + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3)) + ((A - C)*Tan[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 4, (A*x)/a^3 - ((A + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((22*A - 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 6, -((3*A*x)/a^3) + (2*(36*A + C)*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (3*A*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 7, ((13*A + 2*C)*x)/(2*a^3) - (2*(76*A + 11*C)*Sin[c + d*x])/(15*a^3*d) + ((13*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A + C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((76*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, -(((23*A + 6*C)*x)/(2*a^3)) + (4*(34*A + 9*C)*Sin[c + d*x])/(5*a^3*d) - ((23*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A + 9*C)*Sin[c + d*x]^3)/(15*a^3*d)} - - -{Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 9, ((2*A + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(5*A + 54*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (16*(5*A + 54*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 8, -((4*C*ArcTanh[Sin[c + d*x]])/(a^4*d)) + (2*(3*A + 122*C)*Tan[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*C*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (2*(A - 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 6, (C*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((8*A - 55*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((16*A - 215*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (2*(2*A - 5*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 4, ((23*A - 54*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*(2*A + 9*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(3*A - 4*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 4, -(((A + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4)) + (2*(4*A - 3*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((6*A + 13*C)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((6*A + 13*C)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 5, (A*x)/a^4 - ((55*A - 8*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (8*(20*A - C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(5*A - 2*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 7, -((4*A*x)/a^4) + (2*(332*A + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 3*C)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*A*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(6*A - C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 8, ((21*A + 2*C)*x)/(2*a^4) - (32*(54*A + 5*C)*Sin[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (16*(54*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*A*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 9, -((2*(11*A + 2*C)*x)/a^4) + (4*(454*A + 83*C)*Sin[c + d*x])/(35*a^4*d) - (2*(11*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(a^4*d) - ((178*A + 31*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(11*A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(8*A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (4*(454*A + 83*C)*Sin[c + d*x]^3)/(105*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (4*a*(99*A + 80*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(99*A + 80*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(99*A + 80*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(99*A + 80*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)} -{Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (2*a*(21*A + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 4, (2*a*(35*A + 27*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A + 18*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)} -{Sec[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 3, (2*a*(15*A + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (4*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)} -{Sec[c + d*x]^0*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (2*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*C*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d - (a*(A - 2*C)*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 4, (Sqrt[a]*(3*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*A*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(5*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (Sqrt[a]*(35*A + 48*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(35*A + 48*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(35*A + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (2*a^2*(143*A + 112*C)*Tan[c + d*x])/(165*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(33*A + 28*C)*Sec[c + d*x]^3*Tan[c + d*x])/(231*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1155*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(385*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 5, (8*a^2*(63*A + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(21*a*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 4, (8*a^2*(35*A + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) - (4*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (2*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(5*A + 4*C)*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (3*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (a^2*(3*A - 8*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(7*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(A - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(11*A + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(19*A + 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(75*A + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 112*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*A + 16*C)*Cos[c + d*x]*Sin[c + d*x])/(32*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (a^(3/2)*(133*A + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*A + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 176*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (2*a^3*(10439*A + 8368*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2717*A + 2224*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(10439*A + 8368*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(143*A + 136*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (10*a*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (64*a^3*(33*A + 25*C)*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(231*d) + (2*(99*A + 26*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*d) + (10*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*a*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 5, (64*a^3*(21*A + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(21*A + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(21*A + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) - (4*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (2*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(49*A + 32*C)*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(7*A + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (5*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d + (a^3*(15*A + 64*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(15*A - 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(19*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(27*A - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) - (a*(3*A - 4*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (5*a^(5/2)*(5*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(3*A - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (5*a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(163*A + 304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(299*A + 432*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(17*A + 16*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(283*A + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 400*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (a^(5/2)*(1015*A + 1304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1015*A + 1304*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1015*A + 1304*C)*Cos[c + d*x]*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(109*A + 136*C)*Cos[c + d*x]^2*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(23*A + 24*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A + 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(21*A + 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(21*A + 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A + 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 5, -((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A + 14*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 4, (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*C*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, -((A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, ((7*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (A*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 8, -((9*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - (A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 9, ((107*A + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A + 16*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, ((11*A + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((455*A + 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((35*A + 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((7*A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 6, -(((7*A + 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((15*A + 31*C)*Tan[c + d*x])/(5*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*A + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(10*a^2*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 5, ((3*A + 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((3*A + 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((3*A + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 4, ((A - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 5*C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, (-3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((9*A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 8, ((19*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A + 2*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A + C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 9, -((47*A + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d) + ((17*A + 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*(7*A + 4*C)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 7, -(((75*A + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A + 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A + 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((195*A + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 6, ((19*A + 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((A + 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((21*A + 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + (5*(3*A + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 5, (5*(A - 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((3*A - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 4, ((3*A + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 9*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 8, (-5*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((115*A + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A + 3*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 9, ((39*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A + 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((63*A + 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 9, -((2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 8, -((2*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 7, (2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 7, (2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 8, (2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (2*a*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 10, (-16*a^2*(3*A + 2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (16*a^2*(3*A + 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(7*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 19*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*C*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 9, (-4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(35*A + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*C*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (-16*a^2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(15*A + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (8*C*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 8, (16*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(7*A - 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 8, (4*a^2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(33*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (16*a^2*(2*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(19*A + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 10, (4*a^2*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(25*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(89*A + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^2*(7*A + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a^2*(25*A + 33*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 11, (-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(7*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(143*A + 105*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(44*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*C*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(33*A + 35*C)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(231*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 10, (-4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a^3*(21*A + 16*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*C*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 73*C)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(70*A + 53*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + (12*C*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d) + (2*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (4*a^3*(5*A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d) - (2*(5*A - 9*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 9, (4*a^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (8*a^3*(3*A - 10*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(5*a*d*Sqrt[Sec[c + d*x]]) - (2*(9*A - 5*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 9, (4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(41*A - 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (12*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 5*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (4*a^3*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(16*A + 21*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sec[c + d*x]^(5/2)) + (2*(73*A + 63*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 10, (4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a^3*(35*A + 44*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 143*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(33*a*d*Sec[c + d*x]^(7/2)) + (2*(35*A + 33*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2), x, 11, (4*a^3*(175*A + 221*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(95*A + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (40*a^3*(118*A + 143*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (4*a^3*(175*A + 221*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(95*A + 121*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + (12*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(9/2)) + (2*(145*A + 143*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 9, (-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((3*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((5*A + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 8, ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((3*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 7, -(((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])), x, 6, ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, -(((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, (3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} - - -{(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 9, ((A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*(A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 8, (-4*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((A - 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 7, -(((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 7, (4*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 8, -(((7*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (2*(5*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 9, (4*(14*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*(14*A + 5*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A + C)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} - - -{(Sec[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 10, ((9*A + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((9*A + 119*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((A + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - ((9*A + 119*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 9, ((A - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(A - 4*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - 13*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -((A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(2*A - 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -((9*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(3*A - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3), x, 8, ((49*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*(4*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 9, -((119*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + ((11*A + C)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - ((119*A + 9*C)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 10, (7*(33*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((63*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(33*A + 7*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((63*A + 13*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (2*(6*A + C)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - ((63*A + 13*C)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(8*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 4, (Sqrt[a]*(8*A + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 4, (Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*(2*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 4, (2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 3, (2*a*(7*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 4, (2*a*A*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(24*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (2*a*A*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(16*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (a^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(176*A + 133*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 67*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(112*A + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(112*A + 75*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 13*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(32*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(24*A + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(24*A + 19*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 5, (a^(3/2)*(8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(8*A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 5, (3*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(8*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 5, (2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(4*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 4, (8*a^2*(19*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (2*a^2*(52*A + 63*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(136*A + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 6, (2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(33*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1304*A + 1015*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(136*A + 109*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(400*A + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(400*A + 283*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1040*A + 787*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(304*A + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(432*A + 299*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 6, (5*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(24*A - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 31*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (5*a*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 6, (a^(5/2)*(8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(56*A - 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 21*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (a*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 6, (5*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(64*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A - 15*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 6, (2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(32*A + 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (64*a^3*(13*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (2*a*(13*A + 21*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (10*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 6, (2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (10*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2), x, 7, (2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(15015*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (10*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 8, -(((8*A + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((8*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, ((8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 6, -((C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 6, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 5, -((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]), x, 6, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])} - - -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 7, -((3*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((A + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 4, -(((7*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 5, ((11*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 6, -(((15*A + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A + 5*C)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 5*C)*Sin[c + d*x])/(10*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((49*A + 25*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 8, -((5*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((3*A + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((A - 15*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 4, ((19*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 5, -((5*(15*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 6, ((163*A + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A + C)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 7, -(((283*A + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A + 5*C)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A + 45*C)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A + 195*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/3) (A+C Sec[e+f x]^2)*) - - -{(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(2/3), x, 10, (3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d*(1 + Sec[c + d*x])) - (3^(3/4)*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3), x, 9, (3*C*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3), x, 9, -((3*(A + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^(4/3))) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(A - 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3), x, 10, -((3*(A + C)*Tan[c + d*x])/(11*d*(a + a*Sec[c + d*x])^(7/3))) - (3*(4*A - 7*C)*Tan[c + d*x])/(55*a^2*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-(5/6), 1/2, 1, 1/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a^2*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(4*A - 7*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(55*2^(1/3)*a^2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - -{(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(4/3), x, 12, (3*a*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d) - (15*(1 + Sqrt[3])*a*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*2^(1/3)*3^(1/4)*a*C*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(1/3), x, 11, (3*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*3^(1/4)*C*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3), x, 11, -((3*(A + C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3))) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(A + 2*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(A + 2*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(A + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3), x, 12, -((3*(A + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3))) - (3*(2*A - 5*C)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-(1/6), 1/2, 1, 5/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) - (3*(1 + Sqrt[3])*(2*A - 5*C)*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(2*A - 5*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(2*A - 5*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2) with m and/or n symbolic*) - - -{Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^n, x, 8, (C*Sec[c + d*x]^(1 + m)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) + (2^(3/2 + n)*C*n*AppellF1[1/2, 1 - m, -(1/2) - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n)) + (2^(1/2 + n)*(C*(m - n) + A*(1 + m + n))*AppellF1[1/2, 1 - m, 1/2 - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n))} - - -{Sec[c + d*x]^(-n - 1)*(A + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^n, x, 8, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n))) - ((C - A*n + C*n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, -((2*Sec[c + d*x])/(1 - Sec[c + d*x]))]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x])) + (2^(3/2 + n)*C*AppellF1[1/2, 1 + n, -(1/2) - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d} - - -{Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2) + (((-a)*A*n - a*C*(1 + n)*Sec[c + d*x])*(a + a*Sec[c + d*x])^n)/(Sec[c + d*x]^n*(a*(1 + n))), x, 16, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (a*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (a*(B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*B + 2*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, (a*(2*B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, a*B*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 4, a*(B + C)*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 5, (1/2)*a*(B + 2*C)*x + (a*(B + C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 6, (1/2)*a*(B + C)*x + (a*(2*B + 3*C)*Sin[c + d*x])/(3*d) + (a*(B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x]), x, 7, (1/8)*a*(3*B + 4*C)*x + (a*(B + C)*Sin[c + d*x])/d + (a*(3*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(B + C)*Sin[c + d*x]^3)/(3*d)} - - -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 8, (a^2*(7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(10*B + 9*C)*Tan[c + d*x])/(5*d) + (a^2*(7*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*B + 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (C*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(5*d) + (a^2*(10*B + 9*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 8, (a^2*(8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*B + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*B - C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 7, (a^2*(3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(3*B + 2*C)*Tan[c + d*x])/(3*d) + (a^2*(3*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 6, a^2*B*x + (a^2*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*B + 3*C)*Tan[c + d*x])/(2*d) + (C*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 5, a^2*(2*B + C)*x + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(B - C)*Sin[c + d*x])/d + (C*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 5, (1/2)*a^2*(3*B + 4*C)*x + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*B + 2*C)*Sin[c + d*x])/(2*d) + (B*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 6, (1/2)*a^2*(2*B + 3*C)*x + (2*a^2*(2*B + 3*C)*Sin[c + d*x])/(3*d) + (a^2*(2*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 7, (1/8)*a^2*(7*B + 8*C)*x + (a^2*(4*B + 5*C)*Sin[c + d*x])/(3*d) + (a^2*(7*B + 8*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^2, x, 8, (1/8)*a^2*(6*B + 7*C)*x + (a^2*(9*B + 10*C)*Sin[c + d*x])/(5*d) + (a^2*(6*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(6*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(9*B + 10*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 12, (a^3*(15*B + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(15*B + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(15*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*B - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(15*B + 13*C)*Tan[c + d*x]^3)/(60*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 11, (5*a^3*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(4*B + 3*C)*Tan[c + d*x])/d + (3*a^3*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (a^3*(4*B + 3*C)*Tan[c + d*x]^3)/(12*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 7, a^3*B*x + (a^3*(7*B + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*B + 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, a^3*(3*B + C)*x + (a^3*(6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*C*Sin[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((B + 2*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, (1/2)*a^3*(7*B + 6*C)*x + (a^3*(B + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*B*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((B - 2*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 6, (1/2)*a^3*(5*B + 7*C)*x + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((5*B + 3*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 9, (5/8)*a^3*(3*B + 4*C)*x + (a^3*(3*B + 4*C)*Sin[c + d*x])/d + (3*a^3*(3*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(3*B + 4*C)*Sin[c + d*x]^3)/(12*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 8, (1/8)*a^3*(13*B + 15*C)*x + (a^3*(38*B + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(13*B + 15*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*(43*B + 45*C)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (a*B*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((7*B + 5*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(20*d)} -{Cos[c + d*x]^7*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^3, x, 9, (1/16)*a^3*(23*B + 26*C)*x + (a^3*(17*B + 19*C)*Sin[c + d*x])/(5*d) + (a^3*(23*B + 26*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(21*B + 22*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*B + 3*C)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(17*B + 19*C)*Sin[c + d*x]^3)/(15*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, (3*(B - C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((3*B - 4*C)*Tan[c + d*x])/(a*d) + (3*(B - C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*B - 4*C)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, -(((2*B - 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + (2*(B - C)*Tan[c + d*x])/(a*d) - ((2*B - 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, ((B - C)*ArcTanh[Sin[c + d*x]])/(a*d) + (C*Tan[c + d*x])/(a*d) - ((B - C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, (C*ArcTanh[Sin[c + d*x]])/(a*d) + ((B - C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 3, (B*x)/a - ((B - C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 5, -(((B - C)*x)/a) + ((2*B - C)*Sin[c + d*x])/(a*d) - ((B - C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, ((3*B - 2*C)*x)/(2*a) - (2*(B - C)*Sin[c + d*x])/(a*d) + ((3*B - 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, -((3*(B - C)*x)/(2*a)) + ((4*B - 3*C)*Sin[c + d*x])/(a*d) - (3*(B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*B - 3*C)*Sin[c + d*x]^3)/(3*a*d)} - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 8, -(((4*B - 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + (2*(5*B - 8*C)*Tan[c + d*x])/(3*a^2*d) - ((4*B - 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((5*B - 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, ((B - 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((B - 4*C)*Tan[c + d*x])/(3*a^2*d) - ((B - 2*C)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 5, (C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*B - 5*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 3, ((B + 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 4, (B*x)/a^2 - ((4*B - C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 6, -(((2*B - C)*x)/a^2) + (2*(5*B - 2*C)*Sin[c + d*x])/(3*a^2*d) - ((2*B - C)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, ((7*B - 4*C)*x)/(2*a^2) - (2*(8*B - 5*C)*Sin[c + d*x])/(3*a^2*d) + ((7*B - 4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*B - 5*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 8, -(((10*B - 7*C)*x)/(2*a^2)) + (4*(3*B - 2*C)*Sin[c + d*x])/(a^2*d) - ((10*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*B - 7*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(3*B - 2*C)*Sin[c + d*x]^3)/(3*a^2*d)} - - -{Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 9, -(((6*B - 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d)) + (8*(9*B - 19*C)*Tan[c + d*x])/(15*a^3*d) - ((6*B - 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*B - 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (4*(9*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, ((B - 3*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((7*B - 27*C)*Tan[c + d*x])/(15*a^3*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((B - 3*C)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 6, (C*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((2*B - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((4*B - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 4, -(((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3)) + ((3*B - 8*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*B + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 4, ((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*B + 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((2*B + 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 5, (B*x)/a^3 - ((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*B - 2*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (2*(11*B - C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 7, -(((3*B - C)*x)/a^3) + (2*(36*B - 11*C)*Sin[c + d*x])/(15*a^3*d) - ((B - C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*B - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*B - C)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, ((13*B - 6*C)*x)/(2*a^3) - (8*(19*B - 9*C)*Sin[c + d*x])/(15*a^3*d) + ((13*B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (4*(19*B - 9*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (32*a*(11*B + 10*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(11*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + 10*C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^5*Tan[c + d*x])/(11*d*Sqrt[a + a*Sec[c + d*x]]) - (64*(11*B + 10*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (32*(11*B + 10*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)} -{Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (4*a*(9*B + 8*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*B + 8*C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(9*B + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (4*(9*B + 8*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*a*(7*B + 6*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(7*B + 6*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B + 6*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)} -{Sec[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (2*a*(5*B + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)} -{Sec[c + d*x]^0*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 3, (2*a*(3*B + C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*Sqrt[a]*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (Sqrt[a]*(B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*B*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(3*B + 4*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(3*B + 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (Sqrt[a]*(5*B + 6*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*B + 6*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(5*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (4*a^2*(187*B + 168*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(187*B + 168*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(11*B + 12*C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*a*(187*B + 168*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(187*B + 168*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a^2*(39*B + 34*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(9*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(39*B + 34*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(39*B + 34*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (8*a^2*(21*B + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(21*B + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (8*a^2*(5*B + 3*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a^(3/2)*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(3*B + 4*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(3*B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(B - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(7*B + 12*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*B + 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(11*B + 14*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(11*B + 14*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(3/2)*(75*B + 88*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*B + 88*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(75*B + 88*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(9*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (4*a^3*(4615*B + 4184*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(4615*B + 4184*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(299*B + 280*C)*Sec[c + d*x]^4*Tan[c + d*x])/(1287*d*Sqrt[a + a*Sec[c + d*x]]) - (8*a^2*(4615*B + 4184*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(13*B + 16*C)*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(143*d) + (4*a*(4615*B + 4184*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (2*a*C*Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(13*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a^3*(803*B + 710*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(209*B + 194*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(803*B + 710*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a^2*(11*B + 14*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*a*(803*B + 710*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*a*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (64*a^3*(15*B + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(15*B + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(15*B + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (64*a^3*(7*B + 5*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(7*B + 5*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a^(5/2)*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(35*B + 32*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(5*B + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(5*B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*(3*B + 14*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(B + 2*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(19*B + 20*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(9*B - 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(B - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(25*B + 38*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*B + 54*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(3*B + 2*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(163*B + 200*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(163*B + 200*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(95*B + 104*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(11*B + 8*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (a^(5/2)*(283*B + 326*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*B + 326*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(283*B + 326*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(157*B + 170*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*B + 10*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 8, (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(111*B - 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(3*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(93*B - 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)} -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, -((Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*B - 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(7*B - 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(5*B - 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 5, -((Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*B - 2*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 4, (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (2*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, -(((B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 8, ((7*B - 4*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((B - 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 9, -((9*B - 14*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*B - 2*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 8, -((15*B - 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((651*B - 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((63*B - 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((7*B - 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((273*B - 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)} -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, ((11*B - 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((65*B - 93*C)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((35*B - 39*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 6, -((7*B - 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((9*B - 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((3*B - 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 5, ((3*B - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*C*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 4, ((B + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, (2*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 8, -(((3*B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*B - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*B - C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 9, ((19*B - 12*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*B - 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*B - 6*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 8, ((163*B - 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((13*B - 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((985*B - 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((85*B - 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((475*B - 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)} -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 7, -((75*B - 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((9*B - 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((93*B - 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((39*B - 95*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 6, ((19*B - 75*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((B - 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 5, ((5*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 5, ((3*B + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*B + 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 8, (2*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*B - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*B - 3*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 9, -(((5*B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*B - 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*B - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*B - 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a*(4*A + 3*(B + C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 5*B + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*(B + C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 5*B + 4*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a*(4*A + 4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*(B + C))*Tan[c + d*x])/(3*d) + (a*(4*A + 4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a*(2*A + B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 3*B + 2*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, a*A*x + (a*(2*A + 2*B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, a*(A + B)*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (a*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (1/2)*a*(A + 2*(B + C))*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*(A + B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (1/2)*a*(A + B + 2*C)*x + (a*(2*A + 3*(B + C))*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/8)*a*(3*A + 4*(B + C))*x + (a*(A + B + C)*Sin[c + d*x])/d + (a*(3*A + 4*(B + C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/8)*a*(3*(A + B) + 4*C)*x + (a*(4*A + 5*(B + C))*Sin[c + d*x])/(5*d) + (a*(3*(A + B) + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*(B + C))*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (a^2*(14*A + 12*B + 11*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^2*(10*A + 9*B + 8*C)*Tan[c + d*x])/(5*d) + (a^2*(14*A + 12*B + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*(10*A + 12*B + 9*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((3*B + C)*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^2*(10*A + 9*B + 8*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (a^2*(8*A + 7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*A + 7*B + 6*C)*Tan[c + d*x])/(6*d) + (a^2*(8*A + 7*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((20*A - 5*B + 6*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(5*d) + ((5*B + 2*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*a*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^2*(12*A + 8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(12*A + 8*B + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(12*A + 8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*B - C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, a^2*A*x + (a^2*(4*A + 3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B + 2*C)*Tan[c + d*x])/(2*d) + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*B + 2*C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, a^2*(2*A + B)*x + (a^2*(2*A + 4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/d - (a^2*(2*A - 2*B - 3*C)*Tan[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (1/2)*a^2*(3*A + 4*B + 2*C)*x + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A + 2*B - 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (1/2)*a^2*(2*A + 3*B + 4*C)*x + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B + 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((2*A + 3*B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (1/8)*a^2*(7*A + 8*B + 12*C)*x + (a^2*(7*A + 8*B + 12*C)*Sin[c + d*x])/(6*d) + (a^2*(7*A + 8*B + 12*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((A + 2*B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/8)*a^2*(6*A + 7*B + 8*C)*x + (a^2*(18*A + 20*B + 25*C)*Sin[c + d*x])/(15*d) + (a^2*(6*A + 7*B + 8*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(18*A + 25*B + 20*C)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((2*A + 5*B)*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(20*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (1/16)*a^2*(11*A + 12*B + 14*C)*x + (a^2*(8*A + 9*B + 10*C)*Sin[c + d*x])/(5*d) + (a^2*(11*A + 12*B + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(9*A + 12*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((A + 3*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^2*(8*A + 9*B + 10*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (a^3*(26*A + 23*B + 21*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(133*A + 119*B + 108*C)*Tan[c + d*x])/(35*d) + (a^3*(26*A + 23*B + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(154*A + 147*B + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(280*d) + (C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(7*d) + ((7*B + 3*C)*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Sec[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^3*(133*A + 119*B + 108*C)*Tan[c + d*x]^3)/(105*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 12, (a^3*(30*A + 26*B + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(120*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + ((2*B + C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(10*a*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x]^3)/(120*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 11, (a^3*(20*A + 15*B + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(20*A + 15*B + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(20*A + 15*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*B - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(20*A + 15*B + 13*C)*Tan[c + d*x]^3)/(60*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, a^3*A*x + (a^3*(28*A + 20*B + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(4*A + 4*B + 3*C)*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + ((4*B + 3*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*a*d) + ((12*A + 20*B + 15*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, a^3*(3*A + B)*x + (a^3*(6*A + 7*B + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*a*d) - ((6*A - 3*B - 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (1/2)*a^3*(7*A + 6*B + 2*C)*x + (a^3*(2*A + 6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A - C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((A - 2*B - 4*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (1/2)*a^3*(5*A + 7*B + 6*C)*x + (a^3*(B + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((A + B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((5*A + 3*B - 6*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (1/8)*a^3*(15*A + 20*B + 28*C)*x + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(3*A + 4*(B + C))*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + ((3*A + 4*B)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(12*a*d) + ((15*A + 20*B + 12*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (1/8)*a^3*(13*A + 15*B + 20*C)*x + (a^3*(13*A + 15*B + 20*C)*Sin[c + d*x])/(5*d) + (3*a^3*(13*A + 15*B + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((3*A + 5*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) - (a^3*(13*A + 15*B + 20*C)*Sin[c + d*x]^3)/(60*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (1/16)*a^3*(23*A + 26*B + 30*C)*x + (a^3*(34*A + 38*B + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(23*A + 26*B + 30*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(73*A + 86*B + 90*C)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((A + 2*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*a*d) + ((31*A + 42*B + 30*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(120*d)} -{Cos[c + d*x]^7*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (1/16)*a^3*(21*A + 23*B + 26*C)*x + (a^3*(108*A + 119*B + 133*C)*Sin[c + d*x])/(35*d) + (a^3*(21*A + 23*B + 26*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(129*A + 147*B + 154*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((3*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(108*A + 119*B + 133*C)*Sin[c + d*x]^3)/(105*d)} - - -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, (a^4*(56*A + 49*B + 44*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(56*A + 49*B + 44*C)*Tan[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]*Tan[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]^3*Tan[c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(210*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(42*a*d) + (2*a^4*(56*A + 49*B + 44*C)*Tan[c + d*x]^3)/(105*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, (7*a^4*(10*A + 8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(10*A + 8*B + 7*C)*Tan[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + ((6*B - C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(10*A + 8*B + 7*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, a^4*A*x + (a^4*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(40*A + 35*B + 28*C)*Tan[c + d*x])/(8*d) + (a*(5*B + 4*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, a^4*(4*A + B)*x + (a^4*(52*A + 48*B + 35*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/d + (5*a^4*(4*A + 8*B + 7*C)*Tan[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) - ((12*A - 4*B - 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - ((12*A - 32*B - 35*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/2)*a^4*(13*A + 8*B + 2*C)*x + (a^4*(8*A + 13*B + 12*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(A - B - 2*C)*Sin[c + d*x])/(2*d) - (a*(3*A - 2*C)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - ((A - B - 2*C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((3*A + 18*B + 22*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/2)*a^4*(12*A + 13*B + 8*C)*x + (a^4*(2*A + 8*B + 13*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(2*A + B - C)*Sin[c + d*x])/(2*d) + (a*(4*A + 3*B)*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - ((2*A + B - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((8*A - 3*B - 18*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/8)*a^4*(35*A + 48*B + 52*C)*x + (a^4*(B + 4*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 8*B + 4*C)*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((7*A + 8*B + 4*C)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) - ((35*A + 32*B - 12*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (1/8)*a^4*(28*A + 35*B + 48*C)*x + (a^4*C*ArcTanh[Sin[c + d*x]])/d + (a^4*(28*A + 35*B + 40*C)*Sin[c + d*x])/(8*d) + (a*(4*A + 5*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + ((28*A + 35*B + 20*C)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((28*A + 35*B + 32*C)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 12, (7/16)*a^4*(7*A + 8*B + 10*C)*x + (4*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((2*A + 3*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) - (2*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x]^3)/(15*d)} -{Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (1/16)*a^4*(44*A + 49*B + 56*C)*x + (a^4*(454*A + 504*B + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Cos[c + d*x]^2*Sin[c + d*x])/(840*d) + (a*(4*A + 7*B)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((16*A + 21*B + 14*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((436*A + 511*B + 504*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(840*d)} -{Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (1/128)*a^4*(323*A + 352*B + 392*C)*x + (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x])/(35*d) + (a^4*(323*A + 352*B + 392*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2007*A + 2208*B + 2408*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*(A + 2*B)*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(14*d) + (A*Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(8*d) + ((61*A + 80*B + 56*C)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(7*A + 8*(B + C))*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x]^3)/(105*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, (3*(4*A - 4*B + 5*C)*ArcTanh[Sin[c + d*x]])/(8*a*d) - ((3*A - 4*B + 4*C)*Tan[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A - 4*B + 4*C)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, -(((2*A - 3*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d)) + ((3*A - 3*B + 4*C)*Tan[c + d*x])/(a*d) - ((2*A - 3*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((3*A - 3*B + 4*C)*Tan[c + d*x]^3)/(3*a*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, ((2*A - 2*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((A - 2*B + 2*C)*Tan[c + d*x])/(a*d) + ((2*A - 2*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, ((B - C)*ArcTanh[Sin[c + d*x]])/(a*d) + (C*Tan[c + d*x])/(a*d) + ((A - B + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, (A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 4, -(((A - B)*x)/a) + ((2*A - B + C)*Sin[c + d*x])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 5, ((3*A - 2*B + 2*C)*x)/(2*a) - ((2*A - 2*B + C)*Sin[c + d*x])/(a*d) + ((3*A - 2*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 6, -(((3*A - 3*B + 2*C)*x)/(2*a)) + ((4*A - 3*B + 3*C)*Sin[c + d*x])/(a*d) - ((3*A - 3*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A - 3*B + 3*C)*Sin[c + d*x]^3)/(3*a*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]), x, 7, (3*(5*A - 4*B + 4*C)*x)/(8*a) - ((4*A - 4*B + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A - 4*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((5*A - 4*B + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((4*A - 4*B + 3*C)*Sin[c + d*x]^3)/(3*a*d)} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, -(((4*A - 7*B + 10*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d)) + ((5*A - 8*B + 12*C)*Tan[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((5*A - 8*B + 12*C)*Tan[c + d*x]^3)/(3*a^2*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, ((2*A - 4*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(2*A - 5*B + 8*C)*Tan[c + d*x])/(3*a^2*d) + ((2*A - 4*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((2*A - 5*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 6, ((B - 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A - B + 4*C)*Tan[c + d*x])/(3*a^2*d) - ((B - 2*C)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 4, (C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A + 2*B - 5*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2), (C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*A + B - 4*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 3, (A*x)/a^2 - ((4*A - B - 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 5, -(((2*A - B)*x)/a^2) + ((10*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d) - ((2*A - B)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 6, ((7*A - 4*B + 2*C)*x)/(2*a^2) - (2*(8*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d) + ((7*A - 4*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*A - 5*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2, x, 7, -(((10*A - 7*B + 4*C)*x)/(2*a^2)) + ((12*A - 8*B + 5*C)*Sin[c + d*x])/(a^2*d) - ((10*A - 7*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*A - 7*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - ((12*A - 8*B + 5*C)*Sin[c + d*x]^3)/(3*a^2*d)} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, ((2*A - 6*B + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A - 36*B + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A - 6*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A - 6*B + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A - 36*B + 76*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 7, ((B - 3*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((2*A - 7*B + 27*C)*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A + 4*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((B - 3*C)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 5, (C*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((3*A + 2*B - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((6*A + 4*B - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 3, -(((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3)) + ((A - C)*Tan[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 3*B + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 4, (A*x)/a^3 - ((A - B + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 2*B - 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((22*A - 2*B - 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 6, -(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - 4*B - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 7, ((13*A - 6*B + 2*C)*x)/(2*a^3) - (2*(76*A - 36*B + 11*C)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 6*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A - 6*B + C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((76*A - 36*B + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3, x, 8, -(((23*A - 13*B + 6*C)*x)/(2*a^3)) + (4*(34*A - 19*B + 9*C)*Sin[c + d*x])/(5*a^3*d) - ((23*A - 13*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A - 8*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A - 13*B + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A - 19*B + 9*C)*Sin[c + d*x]^3)/(15*a^3*d)} - - -{Sec[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 9, ((2*A - 8*B + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(20*A - 83*B + 216*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A - 52*B + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(20*A - 83*B + 216*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((B - 2*C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 8, ((B - 4*C)*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((6*A - 55*B + 244*C)*Tan[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((B - 4*C)*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((2*A + 5*B - 12*C)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 6, (C*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((8*A + 6*B - 55*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((16*A + 12*B - 215*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((4*A + 3*B - 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 4, ((23*A - 2*B - 54*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((8*A + 13*B + 36*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((6*A + B - 8*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 4, -(((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4)) + ((8*A - B - 6*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((6*A + 8*B + 13*C)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((6*A + 8*B + 13*C)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 5, (A*x)/a^4 - ((55*A - 6*B - 8*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (2*(80*A - 3*B - 4*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((10*A - 3*B - 4*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 7, -(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4, x, 8, ((21*A - 8*B + 2*C)*x)/(2*a^4) - (8*(216*A - 83*B + 20*C)*Sin[c + d*x])/(105*a^4*d) + ((21*A - 8*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A - 52*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(216*A - 83*B + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (4*a*(99*A + 88*B + 80*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)} -{Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*a*(21*A + 18*B + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (2*a*(35*A + 49*B + 27*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A - 14*B + 18*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(7*d) + (2*(7*B + C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)} -{Sec[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 3, (2*a*(15*A + 5*B + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)} -{Sec[c + d*x]^0*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*(3*B + C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d - (a*(A - 2*C)*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (Sqrt[a]*(3*A + 4*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(5*A + 6*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 6*B + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (Sqrt[a]*(35*A + 40*B + 48*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(35*A + 40*B + 48*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a^2*(429*A + 374*B + 336*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (8*a^2*(63*A + 57*B + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(21*a*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (8*a^2*(35*A + 21*B + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(35*A + 21*B + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(15*A + 20*B + 12*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(3*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (a^2*(3*A - 6*B - 8*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(7*A + 12*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(A - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(11*A + 14*B + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(19*A + 30*B + 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 2*B)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(75*A + 88*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B + 112*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(39*A + 56*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 8*B)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(3/2)*(133*A + 150*B + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*A + 150*B + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} - - -{Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a^3*(10439*A + 9230*B + 8368*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2717*A + 2522*B + 2224*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (2*a*(13*B + 5*C)*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)} -{Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (64*a^3*(165*A + 143*B + 125*C)*Tan[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(165*A + 143*B + 125*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*d) + (2*(11*B + 5*C)*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*a*d)} -{Sec[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (64*a^3*(21*A + 15*B + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)} -{Sec[c + d*x]^0*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(245*A + 224*B + 160*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(35*A + 56*B + 40*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(5*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d + (a^3*(15*A + 70*B + 64*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(15*A - 10*B - 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(19*A + 20*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(27*A - 12*B - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 4*B - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) - (a*(3*A - 4*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(25*A + 38*B + 40*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(3*A + 2*B - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*(5*A + 6*B)*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(163*A + 200*B + 304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(299*A + 392*B + 432*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(17*A + 24*B + 16*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(5*A + 8*B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(283*A + 326*B + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B + 400*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(A + 2*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (a^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1015*A + 1132*B + 1304*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Cos[c + d*x]*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Cos[c + d*x]^2*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(5*A + 12*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A - 111*B + 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A - 49*B + 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A - 7*B + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 5, -((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A - 10*B + 14*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 4, (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*(3*B - 2*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 6, -(((A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 7, ((7*A - 4*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 8, -(((9*A - 14*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]], x, 9, ((107*A - 72*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A - 8*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, ((11*A - 15*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((455*A - 651*B + 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((35*A - 63*B + 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 7*B + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((245*A - 273*B + 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 6, -(((7*A - 11*B + 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((45*A - 65*B + 93*C)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 5*B + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((15*A - 35*B + 39*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 5, ((3*A - 7*B + 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((3*A - 9*B + 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((3*A - 3*B + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 4, ((A + 3*B - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*C*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]]), ((A + 3*B - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 5*C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - B - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 7, -(((3*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 8, ((19*A - 12*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A - 6*B + 2*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2), x, 9, -(((47*A - 38*B + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d)) + ((17*A - 13*B + 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((21*A - 14*B + 12*C)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A - 12*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 3*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 7, -(((75*A - 163*B + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A - 85*B + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 6, ((19*A - 75*B + 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((A - 9*B + 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((21*A - 93*B + 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((15*A - 39*B + 95*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 5, ((5*A + 19*B - 75*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((3*A + 5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 4, ((3*A + 5*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A + B - 9*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 3*B - 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 8, -(((5*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - 7*B - C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A - 11*B + 3*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2), x, 9, ((39*A - 20*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B + 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A - 11*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((63*A - 35*B + 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A - 15*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, -((2*a*(5*A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(5*A + 3*(B + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, -((2*a*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*a*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 5*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 7, (2*a*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 7, (2*a*(A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 7, (2*a*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 8, (2*a*(3*(A + B) + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*(B + C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (2*a*(7*A + 9*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*(A + B) + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*(B + C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*(A + B) + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (-4*a^2*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(12*A + 9*B + 8*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*B + 4*C)*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (-4*a^2*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(5*A + 4*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(35*A + 49*B + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*B + 4*C)*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d)} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 8, (-4*a^2*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(5*B + 4*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 8, (4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 3*B - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 8, (4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(7*A + 5*B - 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(4*A + 5*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 8, (4*a^2*(3*A + 4*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(4*A + 7*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (4*a^2*(8*A + 9*B + 12*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 6*B + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(4*A + 9*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 10, (4*a^2*(7*A + 8*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(89*A + 121*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^2*(7*A + 8*B + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(50*A + 55*B + 66*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(4*A + 11*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 11, (-4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(21*A + 17*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(11*B + 6*C)*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(99*A + 143*B + 105*C)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(693*d)} -{Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (-4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A + 21*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(42*A + 41*B + 32*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 9, (-4*a^3*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + (2*(7*B + 6*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d) + (2*(5*A + 9*B + 7*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 9, (4*a^3*(5*A - 5*B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A + 20*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d) - (2*(5*A - 5*B - 9*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 9, (4*a^3*(9*A + 5*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(6*A - 5*B - 20*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(6*A + 5*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*(9*A + 5*B - 5*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 9, (4*a^3*(7*A + 9*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(41*A + 42*B - 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(6*A + 7*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 9, (4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(2*A + 3*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sec[c + d*x]^(5/2)) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 10, (4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 121*B + 143*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(6*A + 11*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(99*a*d*Sec[c + d*x]^(7/2)) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2), x, 11, (4*a^3*(175*A + 195*B + 221*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(95*A + 105*B + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(236*A + 273*B + 286*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (4*a^3*(175*A + 195*B + 221*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(95*A + 105*B + 121*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + (2*(6*A + 13*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(9/2)) + (2*(145*A + 195*B + 143*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 9, (-3*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(5*A - 5*B + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((3*A - 5*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((5*A - 5*B + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 8, ((A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - 3*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((3*A - 3*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 7, -(((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])), x, 6, ((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, -(((3*A - 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A - 3*B + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, (3*(7*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((5*A - 5*B + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])), x, 9, -((3*(7*A - 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a*d)) + (5*(9*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*a*d) + ((9*A - 7*B + 7*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((7*A - 7*B + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(9*A - 7*B + 7*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 9, ((A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - 4*B + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((2*A - 5*B + 10*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A - 4*B + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 8, ((B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((B - 4*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((A + 2*B - 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 7, -(((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 7, ((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - 2*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 8, -(((7*A - 4*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 9, ((56*A - 35*B + 20*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A - 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((56*A - 35*B + 20*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A - 2*B + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A - 2*B + C)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)} - - -{(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 10, ((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((9*A - 49*B + 119*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((B - 2*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - ((9*A - 49*B + 119*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 9, ((A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + 9*B - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + 3*B - 13*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -((A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*A + B - 6*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - B - 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -((9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*A - B - 4*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3), x, 8, ((49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 9, -((119*A - 49*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 10, (7*(33*A - 17*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((63*A - 33*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((63*A - 33*B + 13*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - ((12*A - 7*B + 2*C)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - ((63*A - 33*B + 13*C)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 40*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(8*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(4*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 4, (Sqrt[a]*(2*B + C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*(2*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 4, (2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 3, (2*a*(7*A + 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 4, (2*a*(A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(24*A + 28*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (2*a*(A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(16*A + 18*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(3/2)*(176*A + 150*B + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(176*A + 150*B + 133*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 90*B + 67*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(10*B + 3*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(112*A + 88*B + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(112*A + 88*B + 75*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(48*A + 56*B + 39*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + 3*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (a^(3/2)*(24*A + 14*B + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(2*B + C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 5, (a^(3/2)*(8*A + 12*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(8*A - 4*B - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*B + 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 5, (a^(3/2)*(2*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(8*A + 6*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 5, (2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(3*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (2*a^2*(52*A + 72*B + 63*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(136*A + 156*B + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 6, (2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -{Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(680*A + 628*B + 545*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)} -{Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(400*A + 326*B + 283*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1040*A + 950*B + 787*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(2*B + C)*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(5/2)*(304*A + 200*B + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(8*B + 5*C)*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(1/2), x, 6, (a^(5/2)*(40*A + 38*B + 25*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(24*A - 54*B - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 42*B + 31*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(6*B + 5*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 6, (a^(5/2)*(8*A + 20*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(56*A + 12*B - 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 12*B - 21*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (a*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 6, (a^(5/2)*(2*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(64*A + 70*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A + 10*B - 15*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(A + B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 6, (2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (2*a*(13*A + 15*B + 21*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(5*A + 9*B)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 6, (2*a^3*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*a*(5*A + 11*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2), x, 7, (2*a^3*(2224*A + 2522*B + 2717*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(15015*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*a*(5*A + 13*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 8, -(((8*A - 14*B + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((8*A - 2*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + ((6*B - C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, ((8*A - 4*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((4*B - C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 6, ((2*B - C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]]), x, 6, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 4, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 5, -((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A - 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]), x, 6, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A - 91*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])} - -{(Sqrt[Sec[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 6, ((2*A*b + 2*a*B - b*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(a - b)*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (b*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 8, ((8*A - 12*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((2*A - 6*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((A - B + 2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 7, ((2*B - 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 6, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2)), x, 4, -(((7*A - 3*B - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 5, ((11*A - 7*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A - 15*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 6, -(((15*A - 11*B + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B + 15*C)*Sin[c + d*x])/(30*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((147*A - 95*B + 75*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 8, ((2*B - 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{(Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 4, ((19*A + 5*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2)), x, 5, -(((75*A - 19*B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 6, ((163*A - 75*B + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A - 147*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 7, -(((283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A - 85*B + 45*C)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B + 195*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/3) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(2/3), x, 10, (3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*(5*B + 2*C)*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(10*d*(1 + Sec[c + d*x])) - (3^(3/4)*(5*B + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3), x, 9, (3*C*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*(2*B - C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3), x, 9, -((3*(A - B + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^(4/3))) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(A - B - 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3), x, 10, -((3*(A - B + C)*Tan[c + d*x])/(11*d*(a + a*Sec[c + d*x])^(7/3))) - (3*(4*A - 4*B - 7*C)*Tan[c + d*x])/(55*a^2*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-(5/6), 1/2, 1, 1/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a^2*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(4*A - 4*B - 7*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(55*2^(1/3)*a^2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(4/3), x, 12, (3*a*(7*B + 4*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d) - (15*(1 + Sqrt[3])*a*(7*B + 4*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*3^(1/4)*a*(7*B + 4*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(14*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*(7*B + 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(28*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^(1/3), x, 11, (3*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(4*B + C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*3^(1/4)*(4*B + C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(4*B + C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3), x, 11, -((3*(A - B + C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3))) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(A - B + 2*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(A - B + 2*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(A - B + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3), x, 12, -((3*(A - B + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3))) - (3*(2*A - 2*B - 5*C)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-(1/6), 1/2, 1, 5/6, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) - (3*(1 + Sqrt[3])*(2*A - 2*B - 5*C)*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(2*A - 2*B - 5*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(2*A - 2*B - 5*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (1/4)*(2 + Sqrt[3])]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2) when m and n symbolic*) - - -{Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + a*Sec[c + d*x])^n, x, 8, (C*Sec[c + d*x]^(1 + m)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) + (2^(3/2 + n)*(C*n + B*(1 + m + n))*AppellF1[1/2, 1 - m, -(1/2) - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n)) + (2^(1/2 + n)*(C*(m - n) + A*(1 + m + n) - B*(1 + m + n))*AppellF1[1/2, 1 - m, 1/2 - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n))} - - -{Sec[c + d*x]^(-n - 1)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, If[$VersionNumber>=8, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n))) + ((A*n + B*(1 + n) - C*(1 + n))*Hypergeometric2F1[1/2 - n, -n, 1 - n, -((2*Sec[c + d*x])/(1 - Sec[c + d*x]))]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x])) + (2^(3/2 + n)*C*AppellF1[1/2, 1 + n, -(1/2) - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n))) + ((B - C + A*n + B*n - C*n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, -((2*Sec[c + d*x])/(1 - Sec[c + d*x]))]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x])) + (2^(3/2 + n)*C*AppellF1[1/2, 1 + n, -(1/2) - n, 3/2, 1 - Sec[c + d*x], (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d]} - - -{Sec[c + d*x]^(-n - 1)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2) + (((-a)*(B + A*n + B*n) - a*C*(1 + n)*Sec[c + d*x])*(a + a*Sec[c + d*x])^n)/(Sec[c + d*x]^n*(a*(1 + n))), x, 16, (A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(Sec[c + d*x]^n*(d*(1 + n)))} - - -{(a + a*Sec[c + d*x])^m*(B - C + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (Sqrt[2]*(B - C)*AppellF1[3/2 + m, 1/2, 1, 5/2 + m, (1/2)*(1 + Sec[c + d*x]), 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/(d*(3 + 2*m)*Sqrt[1 - Sec[c + d*x]]) + (2^(3/2 + m)*C*Hypergeometric2F1[1/2, -(1/2) - m, 3/2, (1/2)*(1 - Sec[c + d*x])]*(1 + Sec[c + d*x])^(-(1/2) - m)*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/d} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (d Sec[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2)*) -(**) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (a*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(5*A + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (b*(5*A + 4*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (b*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (b*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 6, (a*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, a*A*x + (b*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, A*b*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (b*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (1/2)*a*(A + 2*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d + (A*b*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (1/2)*b*(A + 2*C)*x + (a*(2*A + 3*C)*Sin[c + d*x])/(3*d) + (A*b*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (1/8)*a*(3*A + 4*C)*x + (b*(A + C)*Sin[c + d*x])/d + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (A*b*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (1/8)*b*(3*A + 4*C)*x + (a*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (b*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (a*b*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(4*d) + ((a^4*C + 2*a^2*b^2*(5*A + 3*C) + 2*b^4*(5*A + 4*C))*Tan[c + d*x])/(15*b^2*d) + (a*(20*A*b^2 + 2*a^2*C + 13*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(60*b*d) + ((a^2*C + 2*b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(30*b^2*d) - (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(10*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 7, ((4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Tan[c + d*x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a^2*A*x + (a*b*(2*A + C)*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 2*(a^2 + b^2)*C)*Tan[c + d*x])/(3*d) + (a*b*C*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, 2*a*A*b*x + ((2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (2*a*b*(A - C)*Tan[c + d*x])/d - (b^2*(2*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/2)*(2*A*b^2 + a^2*(A + 2*C))*x + (2*a*b*C*ArcTanh[Sin[c + d*x]])/d + (a*A*b*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(A - 2*C)*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a*b*(A + 2*C)*x + (b^2*C*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (a*A*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/8)*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x + (2*a*b*(2*A + 3*C)*Sin[c + d*x])/(3*d) + ((2*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*b*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (1/4)*a*b*(3*A + 4*C)*x + ((a^2 + b^2)*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a*b*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*A*b*Cos[c + d*x]^3*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - ((2*A*b^2 + a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (a*(2*a^4*C + 24*b^4*(5*A + 4*C) + a^2*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) + ((4*a^4*C + 12*a^2*b^2*(5*A + 3*C) + 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])/(240*b*d) + (a*(30*A*b^2 + 2*a^2*C + 21*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b^2*d) + ((2*a^2*C + 5*b^2*(6*A + 5*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) - (a*C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 8, (a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) - ((3*a^4*C - 4*b^4*(5*A + 4*C) - 4*a^2*b^2*(20*A + 13*C))*Tan[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) - ((3*a^2*C - 4*b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) - (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, a^3*A*x + (b*(12*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(6*A*b^2 + (a^2 + 4*b^2)*C)*Tan[c + d*x])/(2*d) + (b*(2*a^2*C + b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, 3*a^2*A*b*x + (a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d - (b*(a^2*(6*A - 8*C) - b^2*(3*A + 2*C))*Tan[c + d*x])/(3*d) - (a*b^2*(6*A - 5*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*a*(6*A*b^2 + a^2*(A + 2*C))*x + (b*(2*A*b^2 + (6*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*A*b*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (3*a*b^2*(3*A - 2*C)*Tan[c + d*x])/(2*d) - (b^3*(4*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*b*(2*A*b^2 + 3*a^2*(A + 2*C))*x + (3*a*b^2*C*ArcTanh[Sin[c + d*x]])/d + (a*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^3*(5*A - 6*C)*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x + (b^3*C*ArcTanh[Sin[c + d*x]])/d + (b*(A*b^2 + a^2*(4*A + 6*C))*Sin[c + d*x])/(2*d) + (a*(2*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*x + (a*(15*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Sin[c + d*x])/(15*d) + (3*b*(2*A*b^2 + 5*a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a*(3*A*b^2 + 2*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (3*A*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (1/16)*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*x + (b*(9*a^2*(4*A + 5*C) + b^2*(11*A + 15*C))*Sin[c + d*x])/(15*d) + (a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A*b^2 + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (A*b*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - (b*(A*b^2 + 3*a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*ArcTanh[Sin[c + d*x]])/(4*d) + ((2*a^6*C + 8*b^6*(7*A + 6*C) + a^4*b^2*(42*A + 23*C) + 8*a^2*b^4*(49*A + 39*C))*Tan[c + d*x])/(105*b^2*d) + (a*(4*a^4*C + 12*a^2*b^2*(7*A + 4*C) + b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(420*b*d) + ((2*a^4*C + 8*b^4*(7*A + 6*C) + 3*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(210*b^2*d) + (a*(42*A*b^2 + 2*a^2*C + 31*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(210*b^2*d) + ((a^2*C + 3*b^2*(7*A + 6*C))*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(105*b^2*d) - (a*C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(21*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 9, ((8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - (a*(4*a^4*C - 32*b^4*(5*A + 4*C) - a^2*b^2*(190*A + 121*C))*Tan[c + d*x])/(60*b*d) - ((8*a^4*C - 15*b^4*(6*A + 5*C) - 2*a^2*b^2*(130*A + 89*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) - ((4*a^2*C - 5*b^2*(6*A + 5*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) - (a*C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, a^4*A*x + (a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*a^4*C + 2*b^4*(5*A + 4*C) + a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/(15*d) + (a*b*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*a^2*C + b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, 4*a^3*A*b*x + ((8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/d - (a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (b^2*(a^2*(24*A - 26*C) - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*b*(12*A - 7*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/2)*a^2*(12*A*b^2 + a^2*(A + 2*C))*x + (2*a*b*(2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/d + (2*A*b*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - (b^2*(a^2*(39*A - 34*C) - 2*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (a*b^3*(9*A - 4*C)*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (b^2*(15*A - 2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, 2*a*b*(2*A*b^2 + a^2*(A + 2*C))*x + (b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*A*b^2 + a^2*(2*A + 3*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + (2*A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - (2*a*b*(b^2*(11*A - 6*C) + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) - (b^2*(3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/8)*(8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x + (4*a*b^3*C*ArcTanh[Sin[c + d*x]])/d + (a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Sin[c + d*x])/(12*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/2)*a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x + (b^4*C*ArcTanh[Sin[c + d*x]])/d + ((6*A*b^4 + 2*a^4*(4*A + 5*C) + a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(15*d) + (a*b*(6*A*b^2 + a^2*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*A*b^2 + a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (A*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/16)*(8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x + (4*a*b*(5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((12*A*b^2 + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(120*d) + (2*A*b*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^7*(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (1/4)*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*x + ((12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + (a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Cos[c + d*x]^3*Sin[c + d*x])/(210*d) + ((2*A*b^2 + a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*A*b*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) - ((4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*x]^3)/(105*d)} - - -{(a^2 - b^2*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 8, a^5*x + (b*(24*a^4 - 8*a^2*b^2 - 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b^2*(5*a^2 - 4*b^2)*Tan[c + d*x])/(2*d) + (b^3*(2*a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a*b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) - (b^2*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{(a^2 - b^2*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 7, a^4*x + (a*b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/d + (b^2*(a^2 - 2*b^2)*Tan[c + d*x])/(3*d) - (a*b^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{(a^2 - b^2*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^1, x, 6, a^3*x + (b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (a*b^2*Tan[c + d*x])/(2*d) - (b^2*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, -((a*(2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d)) + (2*a^2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2*C + b^2*(3*A + 2*C))*Tan[c + d*x])/(3*b^3*d) - (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 7, ((2*a^2*C + b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*C*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, -((a*C*ArcTanh[Sin[c + d*x]])/(b^2*d)) + (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, (A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(b*d) - (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 5, -((A*b*x)/a^2) + (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, ((2*A*b^2 + a^2*(A + 2*C))*x)/(2*a^3) - (2*b*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - (A*b*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 7, -((b*(2*A*b^2 + a^2*(A + 2*C))*x)/(2*a^4)) + (2*b^2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^3*d) - (A*b*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, ((8*A*b^4 + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/(8*a^5) - (2*b^3*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (A*b*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} - - -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, ((2*A*b^2 + (6*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, -((2*a*C*ArcTanh[Sin[c + d*x]])/(b^3*d)) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 6, (C*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 5, (A*x)/a^2 - (2*b*(2*a^2*A - A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 6, -((2*A*b*x)/a^3) + (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b^2 - a^2*(A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, ((6*A*b^2 + a^2*(A + 2*C))*x)/(2*a^4) - (2*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(3*A*b^2 - a^2*(2*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - a^2*(A - 2*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, -((b*(4*A*b^2 + a^2*(A + 2*C))*x)/a^5) + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((12*A*b^4 - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) + (b*(2*A*b^2 - a^2*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((4*A*b^2 - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 9, ((2*A*b^2 + (12*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (a*(a^2*b^2*(2*A - 21*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*b^2*(A - 10*C) - b^4*(4*A - C) + 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, -((3*a*C*ArcTanh[Sin[c + d*x]])/(b^4*d)) + ((2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, (C*ArcTanh[Sin[c + d*x]])/(b^3*d) - (a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, ((a^2*(2*A + C) + b^2*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, (A*x)/a^3 + (b*(5*a^2*A*b^2 - 2*A*b^4 - 3*a^4*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((2*A*b^4 - a^4*C - a^2*b^2*(5*A + 2*C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, -((3*A*b*x)/a^4) - ((15*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^2*A*b^2 - 6*A*b^4 - a^4*(2*A - 3*C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, ((12*A*b^2 + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 9, -((4*a*C*ArcTanh[Sin[c + d*x]])/(b^5*d)) - ((2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (12*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 8, (C*ArcTanh[Sin[c + d*x]])/(b^4*d) + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*b^6*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, -((b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*b^2*(2*A - 5*C) + 2*a^4*C + b^4*(13*A + 18*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, (a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, (A*x)/a^4 - (b*(7*a^2*A*b^4 - 2*A*b^6 - a^4*b^2*(8*A - C) + 4*a^6*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 8, -((4*A*b*x)/a^5) - ((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 9, ((20*A*b^2 + a^2*(A + 2*C))*x)/(2*a^6) + ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(12*A - C) - 8*a^8*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((20*A*b^6 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - - -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^1, x, 3, a*x - (b*ArcTanh[Sin[c + d*x]])/d} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 5, x - (4*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, x/a - (2*b*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d) + (2*b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, x/a^2 - (2*b*(4*a^4 - 2*a^2*b^2 + b^4)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(4*a^2 - b^2)*Tan[c + d*x])/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 7, (2*(a - b)*Sqrt[a + b]*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) + (2*(a - b)*Sqrt[a + b]*(16*a^3*C + 12*a^2*b*C + 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*a*(21*A*b^2 + 8*a^2*C + 13*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) - (2*(6*a^2*C - 7*b^2*(9*A + 7*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*a*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)} -{Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (-2*a*(a - b)*Sqrt[a + b]*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(a - b)*Sqrt[a + b]*(35*A*b^2 + (8*a^2 + 6*a*b + 25*b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(8*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (2*(a - b)*Sqrt[a + b]*(2*a^2*C - 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(15*A*b + 2*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) - (4*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (-2*a*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b]*(3*A*b - (a - b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, ((a - b)*Sqrt[a + b]*(A - 2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A*b + 2*(a - b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 7, (A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b^2 - 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 8, -((a - b)*Sqrt[a + b]*(3*A*b^2 - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) + (Sqrt[a + b]*(2*a*A*b - 3*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (b*Sqrt[a + b]*(A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) - ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + (A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 9, ((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^3*d) - (Sqrt[a + b]*(10*a*A*b^2 - 15*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(7*A + 12*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^3*d) + (Sqrt[a + b]*(5*A*b^4 + 8*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^4*d) + (b*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d) - ((5*A*b^2 - 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a^2*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + (A*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 8, (4*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1155*b^5*d) + (2*(a - b)*Sqrt[a + b]*(16*a^4*C + 12*a^3*b*C + 6*a^2*b^2*(11*A + 8*C) - 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1155*b^4*d) + (2*(8*a^4*C + 25*b^4*(11*A + 9*C) + a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1155*b^3*d) + (4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1155*b^2*d) + (2*(a^2*C + 3*b^2*(11*A + 9*C))*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(231*b*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(33*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^3*C + 6*a^2*b*C - 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (-4*a*(a - b)*Sqrt[a + b]*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(105*a*A*b - 35*A*b^2 + 6*a^2*C + 57*a*b*C - 25*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) - (2*(6*a^2*C - 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(a^2*C + b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^2*d) - (2*Sqrt[a + b]*(a^2*C - 2*a*b*(5*A + 2*C) + b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (a*(a - b)*Sqrt[a + b]*(3*A - 8*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(a*b*(3*A - 8*C) + 6*a^2*C + 2*b^2*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (3*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*A - 2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*(5*A - 8*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a*A + 5*A*b + 16*a*C - 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (3*A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(3*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*A + 14*a*A*b + 3*A*b^2 + 24*a^2*C + 48*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) + (b*Sqrt[a + b]*(A*b^2 - 12*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + (A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 9, -((a - b)*Sqrt[a + b]*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (Sqrt[a + b]*(2*a*A*b^2 - 3*A*b^3 + 8*a^3*(3*A + 4*C) + a^2*(52*A*b + 80*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) - (Sqrt[a + b]*(3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(64*a^2*d) + ((A*b^2 + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*a*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 9, (2*(a - b)*Sqrt[a + b]*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^5*d) + (2*(a - b)*Sqrt[a + b]*(240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) + 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) - 6*a*b^4*(2717*A + 2174*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^4*d) + (2*a*(120*a^4*C + 5*a^2*b^2*(143*A + 79*C) + b^4*(23309*A + 18973*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^3*d) - (2*(90*a^4*C - 539*b^4*(13*A + 11*C) - 15*a^2*b^2*(715*A + 543*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9009*b*d) + (2*(15*a^2*C + 11*b^2*(13*A + 11*C))*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (10*a*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (-2*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^4*d) - (2*(a - b)*Sqrt[a + b]*(8*a^4*C + 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) - 6*a*b^3*(132*A + 101*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^3*d) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (2*(a - b)*Sqrt[a + b]*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*C + 21*b^3*(9*A + 7*C) + 15*a^2*b*(21*A + 11*C) - 6*a*b^2*(28*A + 19*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (-2*a*(a - b)*Sqrt[a + b]*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b^2*d) - (2*Sqrt[a + b]*(3*a^3*C - 9*a^2*b*(7*A + 3*C) - b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(3*a^2*C + b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(a^2*(15*A - 46*C) - 6*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (Sqrt[a + b]*(a^2*b*(15*A - 46*C) + 30*a^3*C - 6*b^3*(5*A + 3*C) + 2*a*b^2*(45*A + 17*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (5*a*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d - (a*b*(15*A - 16*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (a*(a - b)*Sqrt[a + b]*(27*A - 56*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) + (Sqrt[a + b]*(a*b*(27*A - 56*C) + 8*b^2*(3*A + C) + 6*a^2*(A + 12*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (5*A*b*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b^2*(21*A - 8*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2*A + 26*a*A*b + 33*A*b^2 + 24*a^2*C + 144*a*b*C - 48*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (5*b*Sqrt[a + b]*(A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (5*A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 9, ((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(15*A*b^3 + 24*a^3*(3*A + 4*C) + 4*a^2*b*(71*A + 108*C) + 2*a*b^2*(59*A + 192*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(5*A*b^4 - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((5*A*b^2 + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} - - -{(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(4*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) + (2*Sqrt[a + b]*(10*a^3 - 4*a^2*b - 4*a*b^2 + 3*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) - (2*a^3*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) - (2*b^2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sqrt[a + b*Sec[c + d*x]]*(a^2 - b^2*Sec[c + d*x]^2), x, 7, (2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) + (2*Sqrt[a + b]*(3*a^2 + a*b - b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a^2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 6, (4*a*(a - b)*Sqrt[a + b]*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + b]*(48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) + 2*a*b^2*(35*A + 22*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(24*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) - (12*a*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)} -{(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*(8*a^2*C + 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(8*a^2*C - 2*a*b*C + 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (8*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{(Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 4, (4*a*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(3*A*b + (2*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{(Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 6, (A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 7, (-3*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (A*(2*a - 3*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - (3*A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)} -{(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 8, ((a - b)*Sqrt[a + b]*(15*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*b*d) - (Sqrt[a + b]*(10*a*A*b - 15*A*b^2 - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*d) + (b*Sqrt[a + b]*(5*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^4*d) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^3*d) - (5*A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a^2*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)} - - -{(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 6, (-2*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^5*Sqrt[a + b]*d) - (2*(16*a^3*C + 12*a^2*b*C + 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^4*Sqrt[a + b]*d) - (2*(A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a*(5*A*b^2 + 8*a^2*C - 3*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 + 6*a^2*C - b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 5, (2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(3*A*b^2 + (8*a^2 + 6*a*b + b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*a*(A*b^2 + a^2*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)} -{(Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 4, (-2*(A*b^2 + 2*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b - (2*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*(A*b^2 + a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*Sqrt[a + b]*d) - (2*(A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 7, -(((3*A*b^2 - a^2*(A - 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d)) + ((a*A*b + 3*A*b^2 + 2*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + (3*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(3*A*b^2 - a^2*(A - 2*C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 8, ((15*A*b^2 - a^2*(7*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - ((5*a*A*b + 15*A*b^2 - 2*a^2*(A - 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(15*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - (5*A*b*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(15*A*b^2 - a^2*(7*A - 8*C))*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 6, (4*a*(a^2*b^2*(A - 14*C) - b^4*(3*A - 4*C) + 8*a^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^5*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(2*a^2*b^2*(A - 8*C) + 3*a*b^3*(A - 3*C) + 16*a^4*C + 12*a^3*b*C - b^4*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (4*a*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 5, (2*(3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(3*b^3*(A - C) + 8*a^3*C + 6*a^2*b*C - a*b^2*(A + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sec[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 5, (-4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C + 3*a*b*(A + C) - b^2*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2), x, 7, (-2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(6*a^2*A*b - a*A*b^2 - 3*A*b^3 - a^3*C + 3*a^2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 8, -((26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + ((21*a^2*A*b^2 - 5*a*A*b^3 - 15*A*b^4 + a^3*b*(3*A - 2*C) + 6*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + (5*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(5*A*b^2 - a^2*(3*A - 2*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 9, -((105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(85*A - 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + ((35*a*A*b^3 + 105*A*b^4 + 6*a^4*(A - 8*C) - 3*a^2*b^2*(45*A - 8*C) - a^3*(27*A*b - 8*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(35*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - (7*A*b*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) + (b^2*(35*A*b^2 - a^2*(27*A - 8*C))*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(85*A - 12*C))*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2), x, 8, (-2*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*b^2*Sqrt[a + b]*(a^2 - b^2)^2*d) + (2*(36*a^2*A*b^3 - 5*a*A*b^4 - 15*A*b^5 + 3*a^5*C + a^3*b^2*(13*A + 5*C) - 3*a^4*b*(15*A + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*b*Sqrt[a + b]*(a^2 - b^2)^2*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) - (2*(5*A*b^4 - 3*a^4*C - a^2*b^2*(13*A + 5*C))*Tan[c + d*x])/(15*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Tan[c + d*x])/(15*a^3*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(a^2 - b^2*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]], x, 7, (2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2), x, 4, (-2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2), x, 7, (4*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (4*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (4*b^2*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2), x, 8, (2*(11*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*(9*a^2 - 2*a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (4*b^2*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(11*a^2 - 3*b^2)*Tan[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])), x, 8, (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 11, -((2*A*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]])) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^(n/3) (A+C Sec[e+f x]^2)*) - - -{(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(2/3), x]} -{(A + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(1/3), x]} -{(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(1/3), x]} -{(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(2/3), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 8, (3*(b*B + a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((5*a*B + 4*b*C)*Tan[c + d*x])/(5*d) + (3*(b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*a*B + 4*b*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, ((4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((b*B + a*C)*Tan[c + d*x])/d + ((4*a*B + 3*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((b*B + a*C)*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*a*B + 2*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, ((2*a*B + b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, a*B*x + ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (b*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 4, (b*B + a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (1/2)*(a*B + 2*b*C)*x + ((b*B + a*C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 6, (1/2)*(b*B + a*C)*x + ((2*a*B + 3*b*C)*Sin[c + d*x])/(3*d) + ((b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (1/8)*(3*a*B + 4*b*C)*x + ((b*B + a*C)*Sin[c + d*x])/d + ((3*a*B + 4*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((b*B + a*C)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 8, (3/8)*(b*B + a*C)*x + ((4*a*B + 5*b*C)*Sin[c + d*x])/(5*d) + (3*(b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a*B + 5*b*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, ((4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*b^2*C + 5*a*(2*b*B + a*C))*Tan[c + d*x])/(5*d) + ((4*a^2*B + 3*b^2*B + 6*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*(5*b*B + 6*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (b*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])*Tan[c + d*x])/(5*d) + ((4*b^2*C + 5*a*(2*b*B + a*C))*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, ((8*a*b*B + 4*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Tan[c + d*x])/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, ((2*a^2*B + b^2*B + 2*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a*b*B + a^2*C + b^2*C)*Tan[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a^2*B*x + ((4*a*b*B + 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(2*b*B + 3*a*C)*Tan[c + d*x])/(2*d) + (b*C*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a*(2*b*B + a*C)*x + (b*(b*B + 2*a*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*B*Sin[c + d*x])/d + (b^2*C*Tan[c + d*x])/d} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/2)*(a^2*B + 2*b^2*B + 4*a*b*C)*x + (b^2*C*ArcTanh[Sin[c + d*x]])/d + (a*(2*b*B + a*C)*Sin[c + d*x])/d + (a^2*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/2)*(2*a*b*B + a^2*C + 2*b^2*C)*x + ((2*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(3*d) + (a*(2*b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (1/8)*(3*a^2*B + 4*b^2*B + 8*a*b*C)*x + ((2*a*b*B + a^2*C + b^2*C)*Sin[c + d*x])/d + ((3*a^2*B + 4*b^2*B + 8*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(2*b*B + a*C)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (1/8)*(6*a*b*B + 3*a^2*C + 4*b^2*C)*x + ((4*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d) + ((6*a*b*B + 3*a^2*C + 4*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, ((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*ArcTanh[Sin[c + d*x]])/(16*d) + ((15*a^2*b*B + 4*b^3*B + 5*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(5*d) + ((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (b*(18*a*b*B + 14*a^2*C + 5*b^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b^2*(3*b*B + 4*a*C)*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (b*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((15*a^2*b*B + 4*b^3*B + 5*a^3*C + 12*a*b^2*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, ((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Tan[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, ((8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b*(20*a*b*B + 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, a^3*B*x + ((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(9*a*b*B + 8*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b^2*(3*b*B + 5*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, a^2*(3*b*B + a*C)*x + (b*(6*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*B*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a^2*B - b^2*B - 3*a*b*C)*Tan[c + d*x])/d - (b^2*(2*a*B - b*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*a*(a^2*B + 6*b^2*B + 6*a*b*C)*x + (b^2*(b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*b*B + a*C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a*B - 2*b*C)*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*(3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*x + (b^3*C*ArcTanh[Sin[c + d*x]])/d + (a*(2*a^2*B + 8*b^2*B + 9*a*b*C)*Sin[c + d*x])/(3*d) + (a^2*(5*b*B + 3*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a*B*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*(3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*x + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*Sin[c + d*x])/(3*d) + (a*(3*a^2*B + 10*b^2*B + 12*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(3*b*B + 2*a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (1/8)*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*x + ((4*a^3*B + 14*a*b^2*B + 15*a^2*b*C + 5*b^3*C)*Sin[c + d*x])/(5*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(7*b*B + 5*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*B*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a*(4*a^2*B + 12*b^2*B + 15*a*b*C)*Sin[c + d*x]^3)/(15*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 9, ((2*a^2 + b^2)*(b*B - a*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, -(((2*a*b*B - 2*a^2*C - b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^3*d)) + (2*a^2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, ((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*a*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, (C*ArcTanh[Sin[c + d*x]])/(b*d) + (2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 5, (B*x)/a - (2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, -(((b*B - a*C)*x)/a^2) + (2*b*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 7, ((a^2*B + 2*b^2*B - 2*a*b*C)*x)/(2*a^3) - (2*b^2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((b*B - a*C)*Sin[c + d*x])/(a^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, -(((a^2 + 2*b^2)*(b*B - a*C)*x)/(2*a^4)) + (2*b^3*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2*B + 3*b^2*B - 3*a*b*C)*Sin[c + d*x])/(3*a^3*d) - ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)} - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 9, -(((4*a*b*B - 6*a^2*C - b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d)) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, ((b*B - 2*a*C)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) - (a^2*(b*B - a*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, (C*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(b*B - a*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 5, (2*(a*B - b*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((b*B - a*C)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 6, (B*x)/a^2 - (2*(2*a^2*b*B - b^3*B - a^3*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, -(((2*b*B - a*C)*x)/a^3) + (2*b*(3*a^2*b*B - 2*b^3*B - 2*a^3*C + a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2*B - 2*b^2*B + a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, ((a^2*B + 6*b^2*B - 4*a*b*C)*x)/(2*a^4) - (2*b^2*(4*a^2*b*B - 3*b^3*B - 3*a^3*C + 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*a^2*b*B - 3*b^3*B - a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*B - 3*b^2*B + 2*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 9, ((b*B - 3*a*C)*ArcTanh[Sin[c + d*x]])/(b^4*d) - (a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C - 12*a*b^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*b*B - 3*a^2*C + 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, (C*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 6*a*b^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(b*B - a*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, -(((3*a*b*B - a^2*C - 2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(b*B - a*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, ((2*a^2*B + b^2*B - 3*a*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((b*B - a*C)*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*a*b*B - a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, (B*x)/a^3 - ((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, -(((3*b*B - a*C)*x)/a^4) + (b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4*B - 11*a^2*b^2*B + 6*b^4*B + 5*a^3*b*C - 2*a*b^3*C)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) + 18*a*b^2*(B - 2*C) + 12*a^2*b*(2*B - C) - 16*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*b*B - 75*b^3*B - 8*a^3*C - 13*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*b*B - 6*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + a*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)} -{Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(a - b)*Sqrt[a + b]*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - 19*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) + 2*a*b*(7*B - 3*C) - 8*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(7*a*b*B - 4*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*b*B + a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (-2*(a - b)*Sqrt[a + b]*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*(a - b)*Sqrt[a + b]*(5*b*B - 2*a*C - 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (-2*(a - b)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*(a - b)*Sqrt[a + b]*(3*B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(b*(B - C) + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(B + 2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(b*B + 2*a*(B + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2*B - b^2*B + 4*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (B*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, -((2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 108*a^3*b^2*C + 2088*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d)) - (2*(a - b)*Sqrt[a + b]*(3*a*b^3*(143*B - 471*C) - 3*b^4*(539*B - 225*C) + 6*a^2*b^2*(11*B - 24*C) + 4*a^3*b*(22*B - 9*C) - 48*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 144*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 204*a*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^3*d) - (2*(44*a*b*B - 24*a^2*C - 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 33*a^2*b^2*C - 147*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) - 3*a*b^2*(57*B - 13*C) - 6*a^2*b*(3*B - C) + 8*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*b*B - 8*a^2*C - 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) - (2*(a - b)*Sqrt[a + b]*(a*b*(21*B - 57*C) - b^2*(63*B - 25*C) - 6*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (-2*(a - b)*Sqrt[a + b]*(20*a*b*B + 3*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(a - b)*Sqrt[a + b]*(15*a*B - 5*b*B - 3*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(3*b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b]*(a*b*(6*B - 4*C) - b^2*(3*B - C) + 3*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*a*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*(a*B - 2*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(2*b*(B - C) + a*(B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(3*b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(5*b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(2*a*B + 5*b*B + 4*a*C + 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*B + 3*b^2*B + 12*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((5*b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, ((a - b)*Sqrt[a + b]*(16*a^2*B + 3*b^2*B + 30*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*B + 14*a*b*B + 3*b^2*B + 12*a^2*C + 30*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) - (Sqrt[a + b]*(12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((7*b*B + 6*a*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a*B*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} - - -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(a - b)*Sqrt[a + b]*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 255*a^3*b^2*C - 3705*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(a - b)*Sqrt[a + b]*(6*a*b^3*(209*B - 505*C) - 3*b^4*(539*B - 225*C) - a^3*b*(110*B - 30*C) - 15*a^2*b^2*(121*B - 19*C) + 40*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 335*a*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) - (2*(22*a*b*B - 8*a^2*C - 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) - 6*a*b^2*(60*B - 19*C) + 15*a^2*b*(3*B - 11*C) - 10*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) - 8*a*b*(7*B - 15*C) + 15*a^2*(7*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(35*a*b*B + 23*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*Sqrt[a + b]*(a^2*b*(45*B - 23*C) - a*b^2*(35*B - 17*C) + b^3*(5*B - 9*C) + 15*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a^2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*(5*b*B + 8*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(3*a^2*B - 6*b^2*B - 14*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(2*a*b*(9*B - 7*C) - 2*b^2*(3*B - C) + 3*a^2*(B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (a*Sqrt[a + b]*(5*b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*a*B - 2*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(9*a*b*B + 4*a^2*C - 8*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(8*b^2*(B - C) + 2*a^2*(B + 2*C) + 3*a*b*(3*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*B + 15*b^2*B + 20*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (a*(7*b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, ((a - b)*Sqrt[a + b]*(16*a^2*B + 33*b^2*B + 54*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(4*a^2*(4*B + 3*C) + 3*b^2*(11*B + 16*C) + a*b*(26*B + 54*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(3*b*B + 2*a*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, ((a - b)*Sqrt[a + b]*(284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*b^3*B + 8*a^3*(9*B + 16*C) + 4*a^2*b*(71*B + 52*C) + 2*a*b^2*(59*B + 132*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2*B + 59*b^2*B + 104*a*b*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(11*b*B + 8*a*C)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 7, (-2*(a - b)*Sqrt[a + b]*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 44*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) - (2*Sqrt[a + b]*(b^3*(63*B - 25*C) - 48*a^3*C + 4*a^2*b*(14*B + 3*C) - 2*a*b^2*(7*B + 22*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(28*a*b*B - 24*a^2*C - 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)} -{(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 6, (2*(a - b)*Sqrt[a + b]*(10*a*b*B - 8*a^2*C - 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (2*Sqrt[a + b]*(b^2*(5*B - 9*C) - 8*a^2*C + 2*a*b*(5*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{(Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) - (2*Sqrt[a + b]*(3*b*B - 2*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]], x, 4, (-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)} -{(Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 4, (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 7, ((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b]*(b*B - 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} - - -{(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 7, (2*(40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^5*Sqrt[a + b]*d) + (2*(b^3*(5*B - 9*C) + 4*a^2*b*(10*B - 9*C) + 6*a*b^2*(5*B - 2*C) - 48*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*Sqrt[a + b]*d) + (2*a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*b*B - 6*a^2*C + b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 6, (-2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) - (2*(2*a + b)*(3*b*B - 4*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*(b*B - a*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)} -{(Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 5, (2*(a*b*B - 2*a^2*C + b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(b*(B - C) - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(b*B - a*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2), x, 5, (-2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*(B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(b*B - a*C)*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 7, (2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*(b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 8, ((a^2*B - 3*b^2*B + 2*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((3*b*B + a*(B - 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (B*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2*B - 3*b^2*B + 2*a*b*C)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 7, (-2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 16*a^5*C + 28*a^3*b^2*C - 8*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) - (2*(a^3*b*(8*B - 12*C) - 9*a*b^3*(B - C) - b^4*(3*B - C) - 16*a^4*C + 2*a^2*b^2*(3*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 6, (2*(2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) + (2*(2*a^2*b*(B - 3*C) - 3*b^3*(B - C) - 8*a^3*C + 3*a*b^2*(B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (2*a^2*(b*B - a*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sec[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 6, (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C - 3*b^2*(B + C) + a*b*(B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(b*B - a*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2), x, 6, (-2*(4*a*b*B - a^2*C - 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(3*a*B - b*B + a*C - 3*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*(b*B - a*C)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*a*b*B - a^2*C - 3*b^2*C)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 8, (2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*(6*a^2*b*B - a*b^2*B - 3*b^3*B - 3*a^3*C + a^2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b*(b*B - a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2), x, 7, (-2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*(a - b)^2*b^2*(a + b)^(5/2)*d) + (2*(3*a^2*(5*B + C) - 8*a*b*(B + 3*C) + b^2*(9*B + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*Sqrt[a + b]*(a^2 - b^2)^2*d) - (2*(b*B - a*C)*Tan[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) - (2*(8*a*b*B - 3*a^2*C - 5*b^2*C)*Tan[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*Tan[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])), x, 6, (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 8, (2*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/3) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, ((4*a*A + 3*b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((5*A*b + 5*a*B + 4*b*C)*Tan[c + d*x])/(5*d) + ((4*a*A + 3*b*B + 3*a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*A*b + 5*a*B + 4*b*C)*Tan[c + d*x]^3)/(15*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, ((4*A*b + 4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*a*A + 2*b*B + 2*a*C)*Tan[c + d*x])/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 6, ((b*B + a*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b + 3*a*B + 2*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, a*A*x + ((2*A*b + 2*a*B + b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (A*b + a*B)*x + ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (b*C*Tan[c + d*x])/d} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (1/2)*(2*b*B + a*(A + 2*C))*x + (b*C*ArcTanh[Sin[c + d*x]])/d + ((A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 5, (1/2)*(A*b + a*B + 2*b*C)*x + ((2*a*A + 3*b*B + 3*a*C)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (1/8)*(3*a*A + 4*b*B + 4*a*C)*x + ((A*b + a*B + b*C)*Sin[c + d*x])/d + ((3*a*A + 4*b*B + 4*a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (1/8)*(3*A*b + 3*a*B + 4*b*C)*x + ((4*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(5*d) + ((3*A*b + 3*a*B + 4*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a*A + 5*b*B + 5*a*C)*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((20*a*b*B + 5*a^2*(3*A + 2*C) + 2*b^2*(5*A + 4*C))*Tan[c + d*x])/(15*d) + ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((5*A*b^2 + 10*a*b*B + 2*a^2*C + 4*b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (b*(5*b*B + 2*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(5*d), ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) - ((5*a^3*b*B - 40*a*b^3*B - 2*a^4*C - 4*a^2*b^2*(5*A + 3*C) - 4*b^4*(5*A + 4*C))*Tan[c + d*x])/(30*b^2*d) - ((10*a^2*b*B - 45*b^3*B - 4*a^3*C - 2*a*b^2*(20*A + 13*C))*Sec[c + d*x]*Tan[c + d*x])/(120*b*d) + ((20*A*b^2 - 5*a*b*B + 2*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b^2*d) + ((5*b*B - 2*a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 7, ((8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*b*B + 4*b^3*B - a^3*C + 4*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a^2*A*x + ((2*a^2*B + b^2*B + 2*a*b*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b^2 + 6*a*b*B + 2*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, a*(2*A*b + a*B)*x + ((2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a*A - b*B - 2*a*C)*Tan[c + d*x])/d - (b^2*(2*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/2)*(2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*x + (b*(b*B + 2*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(A*b + a*B)*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(A - 2*C)*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/2)*(a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*x + (b^2*C*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (a*(2*A*b + 3*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 6, (1/8)*(8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x + ((4*a*A*b + 2*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(3*d) + ((2*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A*b + 2*a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (1/8)*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*x + ((10*a*b*B + a^2*(4*A + 5*C) + b^2*(4*A + 5*C))*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + 5*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - ((2*A*b^2 + 10*a*b*B + a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, ((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - ((6*a^4*b*B - 104*a^2*b^3*B - 32*b^5*B - 2*a^5*C - 24*a*b^4*(5*A + 4*C) - a^3*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) - ((12*a^3*b*B - 142*a*b^3*B - 4*a^4*C - 12*a^2*b^2*(5*A + 3*C) - 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])/(240*b*d) - ((6*a^2*b*B - 32*b^3*B - 2*a^3*C - 3*a*b^2*(10*A + 7*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b^2*d) + ((30*A*b^2 - 6*a*b*B + 2*a^2*C + 25*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) + ((3*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 8, ((12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 4*b^4*(5*A + 4*C) + 4*a^2*b^2*(20*A + 13*C))*Tan[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + a*b^2*(100*A + 71*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((4*b^2*(5*A + 4*C) + 3*a*(5*b*B - a*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, a^3*A*x + ((8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, a^2*(3*A*b + a*B)*x + ((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (b*(9*a*b*B - a^2*(6*A - 8*C) + b^2*(3*A + 2*C))*Tan[c + d*x])/(3*d) - (b^2*(6*a*A - 3*b*B - 5*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*x + (b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b + 2*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (b*(9*a*A*b + 4*a^2*B - 2*b^2*B - 6*a*b*C)*Tan[c + d*x])/(2*d) - (b^2*(4*A*b + 2*a*B - b*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/2)*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*x + (b^2*(b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^2*(5*A*b + 3*a*B - 6*b*C)*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*x + (b^3*C*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Sin[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 7, (1/8)*(3*a^3*B + 12*a*b^2*B + 4*b^3*(A + 2*C) + 3*a^2*b*(3*A + 4*C))*x + ((30*a^2*b*B + 15*b^3*B + 15*a*b^2*(2*A + 3*C) + 2*a^3*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((6*A*b^3 + 15*a^3*B + 50*a*b^2*B + 15*a^2*b*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a*(3*A*b^2 + 15*a*b*B + 2*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + ((3*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (1/16)*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*x + ((12*a^3*B + 42*a*b^2*B + 9*a^2*b*(4*A + 5*C) + b^3*(11*A + 15*C))*Sin[c + d*x])/(15*d) + ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - ((A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)} - - -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, ((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - ((28*a^5*b*B - 847*a^3*b^3*B - 896*a*b^5*B - 8*a^6*C - 32*b^6*(7*A + 6*C) - 4*a^4*b^2*(42*A + 23*C) - 32*a^2*b^4*(49*A + 39*C))*Tan[c + d*x])/(420*b^2*d) - ((56*a^4*b*B - 1246*a^2*b^3*B - 525*b^5*B - 16*a^5*C - 48*a^3*b^2*(7*A + 4*C) - 4*a*b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(1680*b*d) - ((28*a^3*b*B - 371*a*b^3*B - 8*a^4*C - 32*b^4*(7*A + 6*C) - 12*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(840*b^2*d) - ((28*a^2*b*B - 175*b^3*B - 8*a^3*C - 4*a*b^2*(42*A + 31*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(840*b^2*d) + ((42*A*b^2 - 7*a*b*B + 2*a^2*C + 36*b^2*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(210*b^2*d) + ((7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(42*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 9, ((32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*Tan[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, a^4*A*x + ((8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/(30*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, a^3*(4*A*b + a*B)*x + ((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/2)*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*x + (b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A*b + a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - (b*(12*a^3*B - 24*a*b^2*B + a^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/2)*a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*x + (b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((12*A*b^2 + 15*a*b*B + a^2*(4*A + 6*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*A*b + 3*a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - (b*(39*a^2*b*B - 6*b^3*B + 4*a*b^2*(11*A - 6*C) + 4*a^3*(2*A + 3*C))*Tan[c + d*x])/(6*d) - (b^2*(18*a*b*B + 3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/8)*(8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x + (b^3*(b*B + 4*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(12*A*b^3 + 8*a^3*B + 36*a*b^2*B + a^2*b*(23*A + 36*C))*Sin[c + d*x])/(12*d) + ((4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) + ((A*b + a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) - (b^2*(32*a*b*B + 2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Tan[c + d*x])/(24*d)} -{Cos[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/8)*(3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x + (b^4*C*ArcTanh[Sin[c + d*x]])/d + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(30*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((12*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((4*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 8, (1/16)*(24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x + ((8*a^4*B + 60*a^2*b^2*B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*(4*A*b^3 + 16*a^3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((12*A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(120*d) + ((2*A*b + 3*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^7*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (1/16)*(5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*x + ((336*a^3*b*B + 371*a*b^3*B + 12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + ((5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((4*A*b + 7*a*B)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) - ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*x]^3)/(105*d)} - - -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 8, a^4*(b*B - a*C)*x + (b*(32*a^3*b*B + 16*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (b^2*(34*a^2*b*B + 4*b^3*B - 15*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b^3*(32*a*b*B - 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (b^2*(4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b^2*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 7, a^3*(b*B - a*C)*x + (b*(6*a^2*b*B + b^3*B - 4*a^3*C + 2*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b^2*(9*a*b*B - a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b^3*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b^2*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^1, x, 6, a^2*(b*B - a*C)*x + (b*(4*a*b*B - 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b^2*(2*b*B + a*C)*Tan[c + d*x])/(2*d) + (b^2*C*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, ((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 7, ((b^2*(2*A + C) - 2*a*(b*B - a*C))*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, ((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, (A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d)} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 5, -(((A*b - a*B)*x)/a^2) + (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 6, ((2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*x)/(2*a^3) - (2*b*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 7, -(((2*A*b^3 - a^3*B - 2*a*b^2*B + a^2*b*(A + 2*C))*x)/(2*a^4)) + (2*b^2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)} -{Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, ((8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/(8*a^5) - (2*b^3*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((3*A*b^3 - 2*a^3*B - 3*a*b^2*B + a^2*b*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*d) + ((4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 9, ((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C))*Tan[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 2*a*b^2*(A - C) - 4*a^3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, ((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - a*b^2*(A - 2*C) - 3*a^3*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, ((b*B - 2*a*C)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 6, (C*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 5, (A*x)/a^2 - (2*(2*a^2*A*b - A*b^3 - a^3*B + a^2*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 6, -(((2*A*b - a*B)*x)/a^3) + (2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b^2 - a*b*B - a^2*(A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 7, ((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*x)/(2*a^4) - (2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((3*A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 8, -(((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*x)/(2*a^5)) + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 9, ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 21*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, ((b*B - 3*a*C)*ArcTanh[Sin[c + d*x]])/(b^4*d) + ((2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, (C*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 3*a*b^4*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, -(((3*a*b*B - a^2*(2*A + C) - b^2*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, (A*x)/a^3 + ((5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((2*A*b^4 + 3*a^3*b*B - a^4*C - a^2*b^2*(5*A + 2*C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 7, -(((3*A*b - a*B)*x)/a^4) - ((15*a^2*A*b^4 - 6*A*b^6 + 6*a^5*b*B - 5*a^3*b^3*B + 2*a*b^5*B - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 2*a*b^3*B - a^4*(2*A - 3*C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A*b^4 + 4*a^3*b*B - a*b^3*B - 2*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 8, ((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 9, ((b*B - 4*a*C)*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 8, (C*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, ((a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + a^3*b^2*(2*A - 5*C) + 2*a^5*C + a*b^4*(13*A + 18*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, -(((4*a^2*b*B + b^3*B - a^3*(2*A + C) - a*b^2*(3*A + 4*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2*b*B + 3*b^3*B + a^3*C - a*b^2*(5*A + 6*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((2*a^3*b*B + 13*a*b^3*B + a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, (A*x)/a^4 - ((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 8, -(((4*A*b - a*B)*x)/a^5) - ((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} -{Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 9, ((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/(2*a^6) + (b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - - -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^1, x, 3, (b*B - a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 5, ((b*B - a*C)*x)/a - (2*b*(b*B - 2*a*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 6, ((b*B - a*C)*x)/a^2 - (2*b*(2*a^2*b*B - b^3*B - 3*a^3*C + a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4, x, 7, ((b*B - a*C)*x)/a^3 - (b*(6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 8*a^5*C + 4*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(5*a^2*b*B - 2*b^3*B - 8*a^3*C + 2*a*b^2*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5, x, 8, ((b*B - a*C)*x)/a^4 - (b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B + 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 6*a^2*b^2*(7*A + 4*C) + 21*b^4*(9*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(12*a^2*b*(2*B - C) - 16*a^3*C - 6*a*b^2*(7*A - 3*B + 6*C) - 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*b*B - 75*b^3*B - 8*a^3*C - a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + a*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)} -{Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*(a - b)*Sqrt[a + b]*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - a*b^2*(35*A + 19*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(a - b)*Sqrt[a + b]*(35*A*b^2 - b^2*(63*B - 25*C) + 8*a^2*C - a*(14*b*B - 6*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (-2*(a - b)*Sqrt[a + b]*(3*b^2*(5*A + 3*C) + a*(5*b*B - 2*a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(15*A*b - 5*b*B + 2*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)} -{Sec[c + d*x]^0*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (-2*(a - b)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b]*(3*A*b + (a - b)*(3*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^1*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, ((a - b)*Sqrt[a + b]*(A - 2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A*b + 2*b*B + 2*a*C - 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (Sqrt[a + b]*(A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*(A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 2*B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b^2 - 4*a*b*B - 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, -((a - b)*Sqrt[a + b]*(3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) - (Sqrt[a + b]*(3*A*b^2 - 2*a*b*(A + 3*B) - 4*a^2*(4*A + 3*B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (Sqrt[a + b]*(A*b^3 + 8*a^3*B - 2*a*b^2*B + 4*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) - ((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} - - -{Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]*(4*a^3*b*(22*B - 9*C) - 48*a^4*C - 6*a^2*b^2*(33*A - 11*B + 24*C) + 3*b^4*(275*A - 539*B + 225*C) - 3*a*b^3*(627*A - 143*B + 471*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(44*a^3*b*B - 968*a*b^3*B - 24*a^4*C - 75*b^4*(11*A + 9*C) - 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(33*a^2*b*B + 539*b^3*B - 18*a^3*C + 6*a*b^2*(132*A + 101*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b*d) + (2*(11*b*B + 3*a*C)*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)} -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*b*(3*B - C) - 8*a^3*C - 3*a*b^2*(21*A - 57*B + 13*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (-2*(a - b)*Sqrt[a + b]*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*C + 3*a*b*(35*A - 7*B + 19*C) - b^2*(35*A - 63*B + 25*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*Sqrt[a + b]*(3*a^2*(5*B - C) + 2*a*b*(15*A - 10*B + 6*C) - b^2*(15*A - 5*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*(3*a*A - 6*b*B - 8*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(a*b*(3*A + 12*B - 8*C) + 6*a^2*C + 2*b^2*(3*A - 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (Sqrt[a + b]*(3*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*A - 2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, ((a - b)*Sqrt[a + b]*(5*A*b + 4*a*B - 8*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(b*(5*A + 8*B - 8*C) + 2*a*(A + 2*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(3*A*b^2 + 12*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((3*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(4*A + 3*B + 6*C) + 2*a*b*(7*A + 15*B + 24*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) + (Sqrt[a + b]*(A*b^3 - 8*a^3*B - 6*a*b^2*B - 12*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((A*b + 2*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, -((a - b)*Sqrt[a + b]*(9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^2*b*d) - (Sqrt[a + b]*(9*A*b^3 - 6*a*b^2*(A + 4*B) - 8*a^3*(9*A + 16*B + 12*C) - 4*a^2*b*(39*A + 28*B + 60*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^2*d) - (Sqrt[a + b]*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} - - -{Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(a - b)*Sqrt[a + b]*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 15*a^3*b^2*(33*A + 17*C) - 15*a*b^4*(319*A + 247*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*b*(11*B - 3*C) - 40*a^4*C - 15*a^2*b^2*(33*A - 121*B + 19*C) - 3*b^4*(275*A - 539*B + 225*C) + 6*a*b^3*(660*A - 209*B + 505*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 5*a*b^2*(99*A + 67*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 - 22*a*b*B + 8*a^2*C + 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)} -{Sec[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(161*A + 93*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*C + 15*a^2*b*(21*A - 3*B + 11*C) - 6*a*b^2*(28*A - 60*B + 19*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)} -{Sec[c + d*x]^0*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*Sqrt[a + b]*(15*a^3*(7*B - C) + b^3*(35*A - 63*B + 25*C) + a^2*b*(315*A - 161*B + 135*C) - a*b^2*(245*A - 119*B + 145*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)} -{Cos[c + d*x]^1*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, -((a - b)*Sqrt[a + b]*(70*a*b*B - a^2*(15*A - 46*C) + 6*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (Sqrt[a + b]*(a^2*b*(15*A + 90*B - 46*C) + 30*a^3*C - 2*b^3*(15*A - 5*B + 9*C) + 2*a*b^2*(45*A - 35*B + 17*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (a*Sqrt[a + b]*(5*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d - (b*(15*a*A - 10*b*B - 16*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*b*d) + (Sqrt[a + b]*(a*b*(27*A + 72*B - 56*C) + 8*b^2*(3*A - 3*B + C) + 6*a^2*(A + 2*(B + 6*C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b*(21*A*b + 12*a*B - 8*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)} -{Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, ((a - b)*Sqrt[a + b]*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(3*b^2*(11*A + 16*(B - C)) + 4*a^2*(4*A + 3*B + 6*C) + 2*a*b*(13*A + 27*B + 72*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 20*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((15*A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + ((5*A*b + 6*a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, ((a - b)*Sqrt[a + b]*(15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B + 12*C) + 4*a^2*b*(71*A + 52*B + 108*C) + 2*a*b^2*(59*A + 132*B + 192*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(5*A*b^4 - 160*a^3*b*B - 40*a*b^3*B - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((5*A*b^2 + 24*a*b*B + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + ((5*A*b + 8*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, -((a - b)*Sqrt[a + b]*(45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*b*d) - (Sqrt[a + b]*(45*A*b^4 - 30*a*b^3*(A + 5*B) - 16*a^4*(64*A + 45*B + 80*C) - 8*a^3*b*(193*A + 355*B + 260*C) - 4*a^2*b^2*(423*A + 295*B + 660*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*d) - (Sqrt[a + b]*(3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(128*a^3*d) - ((45*A*b^4 - 2840*a^3*b*B - 150*a*b^3*B - 256*a^4*(4*A + 5*C) - 12*a^2*b^2*(141*A + 220*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(1920*a^2*d) + ((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + ((A*b + 2*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 6, (-2*(a - b)*Sqrt[a + b]*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 2*a*b^2*(35*A + 22*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + b]*(48*a^3*C - 4*a^2*b*(14*B + 3*C) + 2*a*b^2*(35*A + 7*B + 22*C) + b^3*(35*A - 63*B + 25*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(15*A*b^2 - b^2*(5*B - 9*C) + 8*a^2*C - 2*a*b*(5*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)} -{(Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 4, (-2*(a - b)*Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(3*A*b - b*(3*B - C) + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]], x, 5, (-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{(Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 6, (A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)} -{(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 7, -((a - b)*Sqrt[a + b]*(3*A*b - 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*b*d) - (Sqrt[a + b]*(3*A*b - 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(3*A*b^2 - 4*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - ((3*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)} - - -{(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*(40*a^3*b*B - 25*a*b^3*B - 6*a^2*b^2*(5*A - 4*C) - 48*a^4*C + 3*b^4*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^5*Sqrt[a + b]*d) + (2*(a^2*b*(40*B - 36*C) - 48*a^3*C - 6*a*b^2*(5*A - 5*B + 2*C) - b^3*(15*A - 5*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*Sqrt[a + b]*d) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 3*a*b^2*(5*A - 3*C) - 24*a^3*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 5, (-2*(6*a^2*b*B - 3*b^3*B - a*b^2*(3*A - 5*C) - 8*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(3*A*b^2 - (2*a + b)*(b*(3*B - C) - 4*a*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)} -{(Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 4, (-2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b + b*(B - C) - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2), x, 6, (2*(A*b^2 - a*(b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*Sqrt[a + b]*d) - (2*(A*b - a*(B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 7, -(((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d)) + ((3*A*b^2 + a*b*(A - 2*B) + 2*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 8, ((15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*b*Sqrt[a + b]*d) - ((15*A*b^2 + a*b*(5*A - 12*B) - 2*a^2*(A + 2*B - 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(15*A*b^2 - 12*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - ((5*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 6, (-2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 2*a^3*b^2*(A - 14*C) + 2*a*b^4*(3*A - 4*C) - 16*a^5*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^5*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(a^3*b*(8*B - 12*C) - 2*a^2*b^2*(A - 3*B - 8*C) - 3*a*b^3*(A + 3*B - 3*C) - 16*a^4*C + b^4*(3*A - 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^4 + a*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C))*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)} -{(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 5, (2*(2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(2*a^2*b*(B - 3*C) - 3*b^3*(A + B - C) - 8*a^3*C + a*b^2*(A + 3*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sec[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 5, (-2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C + a*b*(3*A + B + 3*C) - b^2*(A + 3*(B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{Sec[c + d*x]^0*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2), x, 7, (2*(7*a^2*A*b^2 - 3*A*b^4 - 4*a^3*b*B + a^4*C + 3*a^2*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*A*b^2 + 3*A*b^3 + a^3*(3*B + C) - a^2*b*(6*A + B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*b*Sqrt[a + b]*(a^2 - b^2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Cos[c + d*x]^1*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 8, -((26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) - ((15*A*b^4 + a*b^3*(5*A - 6*B) - a^2*b^2*(21*A + 2*B) - 6*a^4*C - a^3*b*(3*A - 2*(6*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*b*Sqrt[a + b]*(a^2 - b^2)*d) + (Sqrt[a + b]*(5*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -{(a + b*Sec[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2), x, 8, (-2*(a - b)*Sqrt[a + b]*(35*a*b*B - 12*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*Sqrt[a + b]*(a*b^2*(35*B - 12*C) - b^3*(5*B - 9*C) + 30*a^3*C - 3*a^2*b*(15*B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*a^2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b^2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)} -{Sqrt[a + b*Sec[c + d*x]]*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2), x, 7, (-2*(a - b)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*Sqrt[a + b]*(b^2*(3*B - C) - a*b*(6*B - C) + 3*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]], x, 6, (-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2), x, 4, (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2), x, 7, (2*(b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*(b*B - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2), x, 8, (2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*(3*b^3*B + a*b^2*(B - 3*C) + 9*a^3*C - 2*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 10, -((2*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + (2*b*C*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 9, -((2*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*b*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 8, -((2*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(b*B + a*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(b*B + a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, -((2*(b*B - a*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d) + (2*(3*A*b + 3*a*B + b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(b*B + a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (2*(A*b + a*B - b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*b*B + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*b*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 7, (2*(3*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 8, (2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 9, (2*(7*a*A + 9*b*B + 9*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*A*b + 5*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*a*A + 9*b*B + 9*a*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(5*A*b + 5*a*B + 7*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-11/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x]), x, 10, (2*(7*A*b + 7*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (10*(9*a*A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*A*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 9*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (10*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 10, -((2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 4*a*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*C*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 9, -((2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, -((2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*A*b^2 + 10*a*b*B + 4*a^2*C + 3*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(5*b*B + 4*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (2*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(3*b*B - 2*a*(A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(A - C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(4*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 8, (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^2, x, 9, (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(4*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 10, -((2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, -((2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d)) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(7*b*B + 6*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(45*a*b*B - a^2*(10*A - 42*C) + 3*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(5*a*A - 5*b*B - 9*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) - (2*b*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(10*a^2*B - 15*b^2*B + 3*a*b*(7*A - 15*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(11*A*b + 7*a*B - 35*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*(6*A*b + 7*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 9, (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (2*(2*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{Sec[c + d*x]^(-11/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^3, x, 10, (2*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b*(7*A + 9*C))*Sin[c + d*x])/(495*d*Sec[c + d*x]^(3/2)) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(6*A*b + 11*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 11, -((2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, -((2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d)) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(1098*a^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(315*d) + (2*(9*b*B + 8*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(609*a^2*b*B + 63*b^3*B - a^3*(70*A - 366*C) + 84*a*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b*(35*a*A - 21*b*B - 39*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(105*d) - (2*b*(7*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(10*a^3*B - 60*a*b^2*B + a^2*b*(31*A - 87*C) - 3*b^3*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(5*a^2*B - 5*b^2*B + 14*a*b*(A - C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) - (2*b*(11*A*b + 5*a*B - 3*b*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (2*(8*A*b + 5*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b*(217*a^2*b*B - 105*b^3*B + 12*a*b^2*(19*A - 35*C) + 10*a^3*(5*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b + 7*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{Sec[c + d*x]^(-9/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(162*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b + 9*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{Sec[c + d*x]^(-11/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 10, (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*(8*A*b + 11*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} -{Sec[c + d*x]^(-13/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^4, x, 11, (2*(364*a^3*b*B + 468*a*b^3*B + 39*b^4*(3*A + 5*C) + 78*a^2*b^2*(7*A + 9*C) + a^4*(77*A + 91*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + a^2*(2518*A*b + 3146*b*C))*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(192*A*b^4 + 4004*a^3*b*B + 3458*a*b^3*B + 77*a^4*(11*A + 13*C) + 11*a^2*b^2*(491*A + 637*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(48*A*b^2 + 221*a*b*B + 11*a^2*(11*A + 13*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(8*A*b + 13*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 11, -((2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d)) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^2*d) - (2*a*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*b^3*d) + (2*(b*B - a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*d) + (2*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d)} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 10, -((2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*d)) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 9, -((2*C*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b*d)) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 8, (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 9, -((2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(3*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 10, (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(3*A*b^3 - a^3*B - 3*a*b^2*B + a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^4*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])} -{Sec[c + d*x]^(-7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]), x, 11, -((2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^4*d)) + (2*(21*A*b^4 - 7*a^3*b*B - 21*a*b^3*B + 7*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(21*a^5*d) - (2*b^3*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^5*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(5*a^2*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 11, -(((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) + ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 10, -(((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 9, ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) - ((A*b^2 + a*b*B - a^2*(2*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 9, -(((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + ((3*A*b^3 + 2*a^3*B - a*b^2*B - a^2*b*(4*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 10, ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) - ((15*A*b^4 + 12*a^3*b*B - 9*a*b^3*B - a^2*b^2*(16*A - 3*C) - 2*a^4*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) + (b*(5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2, x, 11, -(((35*A*b^4 + 20*a^3*b*B - 25*a*b^3*B - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d)) + ((21*A*b^5 + 2*a^5*B + 16*a^3*b^2*B - 15*a*b^4*B - a^2*b^3*(20*A - 9*C) - 4*a^4*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(3*a^5*(a^2 - b^2)*d) - (b^2*(7*A*b^4 + 7*a^3*b*B - 5*a*b^3*B - 3*a^2*b^2*(3*A - C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(a^5*(a - b)*(a + b)^2*d) - ((7*A*b^2 - 5*a*b*B - a^2*(2*A - 5*C))*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + ((7*A*b^3 + 2*a^3*B - 5*a*b^2*B - a^2*(4*A*b - 3*b*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))} - - -{Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 12, -(((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d)) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*b^3*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 11, ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 10, -(((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d)) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 10, -(((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d)) + ((3*A*b^4 - 7*a^3*b*B + a*b^3*B - a^2*b^2*(5*A - 3*C) + a^4*(8*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 10, ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B - a^2*b^3*(33*A + C) + a^4*b*(24*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))} -{Sec[c + d*x]^(-3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3, x, 11, -(((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d)) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, ((24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d) + ((6*b*B + a*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*b*d) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 13, ((8*a*A + 4*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B + a*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B + a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 12, ((2*a*B + b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A - C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 12, (-2*(A*b^2 - a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 9, (-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 10, (2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 11, (-2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b*(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*a*d*Sec[c + d*x]^(5/2)) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^3*d*Sqrt[Sec[c + d*x]])} - - -{Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, ((136*a^2*b*B + 128*b^3*B - 3*a^3*C + 12*a*b^2*(28*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*b*d) + ((8*b*B + 3*a*C)*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)} -{Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, ((42*a*b*B + 8*b^2*(3*A + 2*C) + a^2*(48*A + 17*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d) + ((2*b*B + a*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 13, ((8*a^2*B + 4*b^2*B + a*b*(8*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*a*A - 4*b*B - 5*a*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B + 3*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 13, ((6*a*b*B - b^2*(2*A - 3*C) + 2*a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b + 6*a*B - 3*b*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 13, -((2*(3*A*b^3 - 5*a^3*B + 5*a*b^2*B - 3*a^2*b*(A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a*d*Sqrt[a + b*Sec[c + d*x]])) + (2*b^2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 10, (2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 11, (2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} - - -{Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, ((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*d*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]], x, 14, ((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + ((6*b*B + 5*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x, 14, ((48*a^2*b*B + 12*b^3*B + 8*a^3*(A + 3*C) + a*b^2*(16*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(12*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(12*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(8*a*A - 12*b*B - 21*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (b*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x, 14, ((10*a^3*B + 20*a*b^2*B - b^3*(16*A - 15*C) + 4*a^2*b*(4*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*b*B + 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(16*A*b + 10*a*B - 15*b*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(A*b + a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x, 14, (-2*(15*A*b^4 - 56*a^3*b*B + 56*a*b^3*B + 10*a^2*b^2*(A - 7*C) - 5*a^4*(5*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 7*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2), x, 11, (-2*(a^2 - b^2)*(10*A*b^3 - 75*a^3*B - 45*a*b^2*B - 6*a^2*b*(19*A + 28*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 9*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2), x, 12, (2*(a^2 - b^2)*(40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3465*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 11*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 13, ((4*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B - 3*a*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B - 3*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 12, ((2*A + C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (C*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 11, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]), x, 8, (2*(2*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]), x, 9, (-2*(8*A*b^3 - 5*a^3*B - 10*a*b^2*B + a^2*b*(7*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]), x, 10, (2*(48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 5*a^4*(5*A + 7*C) + 2*a^2*b^2*(16*A + 35*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Sec[c + d*x]^(3/2)) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Sqrt[Sec[c + d*x]])} - -{(Sqrt[Sec[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 13, ((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d} - - -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 13, (C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 12, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)), x, 8, -((2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]])) - (2*(2*A*b^2 - a*b*B - a^2*(A - C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)), x, 9, (2*(8*A*b^2 - 6*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)), x, 10, -((2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^4*d*Sqrt[a + b*Sec[c + d*x]])) - (2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])} - - -{(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 14, ((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^3*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)} -{(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 13, -((2*(A*b^2 - a*(b*B - a*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 9, -((2*(2*A*b^2 + a*b*B - a^2*(3*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) - (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 + 2*a^3*b*B + 2*a*b^3*B + a^4*C - 5*a^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)), x, 9, (2*(8*A*b^3 + 3*a^3*B - 2*a*b^2*B - a^2*b*(9*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)), x, 10, -((2*(16*A*b^4 + 9*a^3*b*B - 8*a*b^3*B - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])) - (2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)), x, 11, (2*(128*A*b^5 + 5*a^5*B + 80*a^3*b^2*B - 80*a*b^4*B - 4*a^2*b^3*(29*A - 10*C) - a^4*b*(17*A + 45*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) - 4*a^2*b^4*(53*A - 10*C) + 3*a^6*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*(8*A*b^4 + 9*a^3*b*B - 5*a*b^3*B - 2*a^2*b^2*(6*A - C) - 6*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B + a^4*(3*A - 35*C) - a^2*b^2*(71*A - 15*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (2*(64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B + 2*a^4*b*(7*A - 20*C) - 2*a^2*b^3*(49*A - 10*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/3) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(2/3), x]} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Unintegrable[(a + b*Sec[c + d*x])^(1/3), x]} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(1/3), x]} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3), x, 8, (Sqrt[2]*C*AppellF1[1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Unintegrable[1/(a + b*Sec[c + d*x])^(2/3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2) with m and/or n symbolic*) - - -{(a + b*Sec[c + d*x])^m*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2), x, 5, (Sqrt[2]*b*(a + b)*C*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Sec[c + d*x]), (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^m*Tan[c + d*x])/(((a + b*Sec[c + d*x])/(a + b))^m*(d*Sqrt[1 + Sec[c + d*x]])) + (b*B - a*C)*Unintegrable[(a + b*Sec[c + d*x])^(1 + m), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+a Sec[e+f x])^m (d Cos[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(A + C*Sec[c + d*x]^2)*Cos[c + d*x]^(9/2), x, 4, (2*(7*A + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{(A + C*Sec[c + d*x]^2)*Cos[c + d*x]^(7/2), x, 4, (2*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{(A + C*Sec[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 3, (2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{(A + C*Sec[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 3, (2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(A + C*Sec[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 3, (2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 3, (2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 4, -((2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 4, (2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 8, (2*a*(7*A + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 7, (2*a*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 6, (2*a*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 6, (2*a*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2), x, 6, (2*a*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, -((2*a*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, -((2*a*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 10, (4*a^2*(7*A + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (8*a^2*(25*A + 33*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (8*a^2*(25*A + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(89*A + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 9, (16*a^2*(2*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 8, (4*a^2*(3*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(33*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 8, (16*a^2*A*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^2*(A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(7*A - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 5*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 8, (4*a^2*(A - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (8*a^2*(A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2), x, 8, -((16*a^2*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^2*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a^2*(15*A + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 9, -((4*a^2*(5*A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (8*a^2*(7*A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(35*A + 33*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 10, -((16*a^2*(3*A + 2*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^2*(7*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a^2*(21*A + 19*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (16*a^2*(3*A + 2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 11, (4*a^3*(175*A + 221*C)*EllipticE[(1/2)*(c + d*x), 2])/(195*d) + (4*a^3*(95*A + 121*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(95*A + 121*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(175*A + 221*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(118*A + 143*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (12*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(145*A + 143*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)} -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 10, (4*a^3*(5*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (8*a^3*(35*A + 44*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(35*A + 33*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 9, (4*a^3*(17*A + 27*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (8*a^3*(16*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(73*A + 63*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 9, (4*a^3*(7*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (4*a^3*(41*A - 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 7*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) + (2*(11*A - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 9, (4*a^3*(9*A - 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (8*a^3*(3*A - 10*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + (2*(3*A - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 9, (4*a^3*(5*A - 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) - (4*a^3*(5*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)) + (2*(5*A + 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2), x, 9, -((4*a^3*(5*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(35*A + 13*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (8*a^3*(70*A + 53*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (12*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 7*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 10, -((4*a^3*(27*A + 17*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (4*a^3*(21*A + 11*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(27*A + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 11, -((4*a^3*(7*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (4*a^3*(143*A + 105*C)*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (8*a^3*(44*A + 35*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)) + (4*a^3*(143*A + 105*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(7*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Cos[c + d*x]^(9/2)) + (2*(33*A + 35*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 8, -((3*(7*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d)) + (5*(9*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*a*d) + (5*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((7*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((9*A + 7*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 7, (3*(7*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 6, -(((3*A + C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((5*A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 5, ((3*A + C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - ((A - C)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])), x, 6, -(((A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d)) + ((A - C)*EllipticF[(1/2)*(c + d*x), 2])/(a*d) + ((A + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, ((A + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(a*d) + ((3*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((A + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, -((3*(5*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a*d)) - ((3*A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a*d) + ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 8, (4*(14*A + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*a^2*d) - (5*(3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - (5*(3*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*(14*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 7, -(((7*A + C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (2*(5*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 6, (4*A*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) - ((5*A - C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2), x, 6, -(((A - C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (2*(A + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 7, -((4*C*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + ((A - 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (4*C*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 8, ((A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d) + (2*(A + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*a^2*d) + (2*(A + 5*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((A + 7*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A + 7*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 9, (7*(33*A + 7*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - ((63*A + 13*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((63*A + 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(33*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(6*A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((63*A + 13*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -(((119*A + 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((11*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 7, ((49*A - C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) - ((13*A - C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3), x, 7, -(((9*A - C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((3*A + C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(3*A - 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 7, -(((A - 9*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d)) + ((A + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(2*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 8, ((A - 49*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((A - 13*C)*EllipticF[(1/2)*(c + d*x), 2])/(6*a^3*d) - ((A - 49*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((A - 13*C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3), x, 9, ((9*A + 119*C)*EllipticE[(1/2)*(c + d*x), 2])/(10*a^3*d) + ((A + 11*C)*EllipticF[(1/2)*(c + d*x), 2])/(2*a^3*d) + ((A + 11*C)*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - ((9*A + 119*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*C*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((9*A + 119*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 6, (16*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (4*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 4, (2*a*(7*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 5, (Sqrt[a]*(8*A + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*C*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (Sqrt[a]*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*C*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*A + 5*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*C*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 7, (16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(112*A + 143*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(28*A + 33*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(231*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(33*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (4*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 5, (8*a^2*(19*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (6*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (3*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(8*A - 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(8*A - 5*C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 6, (a^(3/2)*(24*A + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 19*C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (a^(3/2)*(112*A + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(16*A + 13*C)*Sin[c + d*x])/(32*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(112*A + 75*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, (a^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 8, (16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 143*C)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d) + (10*a*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)} -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (4*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(232*A + 297*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 33*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (10*a*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 6, (64*a^3*(13*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(13*A + 21*C)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (10*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 49*C)*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (5*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(64*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A - 15*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(56*A - 27*C)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (a*(4*A - 3*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2), x, 7, (5*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (5*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 7, (a^(5/2)*(304*A + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 299*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + (5*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, (a^(5/2)*(400*A + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 9, (a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(7/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 6, -((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 5, (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]), x, 7, -((C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 8, ((8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (C*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 9, -(((8*A + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) - (C*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((8*A + 7*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 7, -(((15*A + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + 25*C)*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 6, ((11*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 5, -(((7*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)), x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 8, -((3*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d)) + ((A + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A + 3*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 9, ((8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A + 2*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - ((2*A + 7*C)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 8, -(((283*A + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A + 735*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A + 195*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A + 45*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 7, ((163*A + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (5*(19*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 6, -((5*(15*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)), x, 5, ((19*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - 7*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 8, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 11*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 9, -((5*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d)) + ((3*A + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((A - 15*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + 35*C)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Cos[c + d*x]^(9/2), x, 7, (6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (10*B*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Cos[c + d*x]^(7/2), x, 6, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Cos[c + d*x]^(5/2), x, 5, (2*C*EllipticE[(1/2)*(c + d*x), 2])/d + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Cos[c + d*x]^(3/2), x, 4, (2*B*EllipticE[(1/2)*(c + d*x), 2])/d + (2*C*EllipticF[(1/2)*(c + d*x), 2])/d} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Cos[c + d*x]^(1/2), x, 5, -((2*C*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 6, -((2*B*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 7, -((6*C*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*B*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*C*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (6*B*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*A + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]], x, 6, (-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 7, (-2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2), x, 8, (-6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*a*(7*A + 9*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*a*(5*(A + B) + 7*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(5*(A + B) + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*(B + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(A + B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a*(3*(A + B) + 5*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(5*A + 7*(B + C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a*(3*A + 5*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*a*(A + B + 3*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a*(A + B - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(A + 3*(B + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a*(A - B - C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*a*(3*A + 3*B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]], x, 7, -((2*a*(5*A + 5*B + 3*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(3*A + B + C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 5*B + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 8, -((2*a*(5*A + 3*(B + C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*a*(7*A + 7*B + 5*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(7*A + 7*B + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*(B + C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (4*a^2*(7*A + 8*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^2*(50*A + 55*B + 66*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(7*A + 8*B + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(89*A + 121*B + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(4*A + 11*B)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (4*a^2*(8*A + 9*B + 12*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(4*A + 9*B)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (4*a^2*(3*A + 4*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(4*A + 7*B)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A + 5*B - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 5*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(A - 3*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (-4*a^2*(5*B + 4*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 9, (-4*a^2*(5*A + 4*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 4*B + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))} -{(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2), x, 10, (-4*a^2*(12*A + 9*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 6*B + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(12*A + 9*B + 8*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(9*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (4*a^3*(15*A + 17*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(6*A + 11*B)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(105*A + 143*B + 99*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (4*a^3*(17*A + 21*B + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(73*A + 99*B + 63*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (4*a^3*(7*A + 9*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(41*A + 42*B - 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 7*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) + (2*(11*A + 7*B - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (4*a^3*(9*A + 5*B - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^3*(6*A - 5*B - 20*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + (2*(3*A - 15*B - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (4*a^3*(5*A - 5*B - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(5*A + 20*B + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(15*A + 35*B + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (-4*a^3*(5*A + 9*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 9*B + 7*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 10, (-4*a^3*(27*A + 21*B + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(27*A + 21*B + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(63*A + 99*B + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))} - - -{Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 11, (8*a^4*(185*A + 208*B + 247*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (8*a^4*(100*A + 113*B + 132*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^4*(100*A + 113*B + 132*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^4*(5255*A + 6019*B + 6721*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(8*A + 13*B)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(13*d) + (2*(13*A + 17*B + 11*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (4*(1355*A + 1612*B + 1573*C)*Cos[c + d*x]^(3/2)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(9009*d)} -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (8*a^4*(16*A + 19*B + 24*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(113*A + 132*B + 187*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^4*(667*A + 803*B + 913*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1155*d) + (2*a*(8*A + 11*B)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d) + (2*(43*A + 55*B + 33*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (4*(769*A + 946*B + 891*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3465*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (8*a^4*(19*A + 24*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(12*A + 17*B + 28*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(73*A + 83*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(A - 9*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(5*A + 3*B - 21*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (4*(86*A + 81*B - 126*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(315*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (8*a^4*(8*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(17*A + 28*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(83*A + 7*B - 175*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(3*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A - 7*B - 21*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (4*(27*A - 42*B - 175*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(105*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (56*a^4*(A - C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(4*A + 5*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^4*(A - 25*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(5*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 15*B + 19*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (4*(6*A + 25*B + 34*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (-8*a^4*(7*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(35*A + 28*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^4*(175*A + 287*B + 253*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(35*A + 77*B + 73*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*(175*A + 238*B + 197*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (-8*a^4*(21*A + 24*B + 19*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(28*A + 17*B + 12*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(287*A + 253*B + 193*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*a*(9*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(63*A + 117*B + 97*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (4*(21*A + 24*B + 19*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(45*d*Cos[c + d*x]^(3/2))} -{(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(1/2), x, 11, (-8*a^4*(24*A + 19*B + 16*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(187*A + 132*B + 113*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^4*(913*A + 803*B + 667*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)) + (8*a^4*(24*A + 19*B + 16*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(11*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (2*(33*A + 55*B + 43*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (4*(891*A + 946*B + 769*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 8, (-3*(7*A - 7*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(9*A - 7*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(9*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((7*A - 7*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((9*A - 7*B + 7*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 7, (3*(7*A - 5*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A - 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 6, -(((3*A - 3*B + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((5*A - 3*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]), x, 5, ((3*A - B + C)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((A - B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])), x, 6, -(((A - B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A + B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A - B + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])), x, 7, ((A - 3*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((3*A - 3*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A - 3*B + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((A - 3*B + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])), x, 8, (-3*(5*A - 5*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((3*A - 5*B + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 9, (-7*(11*A - 8*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) + (5*(30*A - 21*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*a^2*d) + (5*(30*A - 21*B + 14*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a^2*d) - (7*(11*A - 8*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((30*A - 21*B + 14*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a^2*d) - ((11*A - 8*B + 5*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 8, ((56*A - 35*B + 20*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(3*A - 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(3*A - 2*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((56*A - 35*B + 20*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A - 2*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 7, -(((7*A - 4*B + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((10*A - 5*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2, x, 6, ((4*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - 2*B - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^2), x, 6, -(((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A + B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2), x, 7, ((B - 4*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A + 2*B - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((B - 4*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2), x, 8, ((A - 4*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A - 5*B + 10*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B + 10*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((A - 4*B + 7*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - 4*B + 7*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2), x, 9, -((20*A - 35*B + 56*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(5/2)) - (5*(A - 2*B + 3*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(5*a^2*d*Sqrt[Cos[c + d*x]]) - ((A - 2*B + 3*C)*Sin[c + d*x])/(a^2*d*Cos[c + d*x]^(5/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2)} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 9, (7*(33*A - 17*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((63*A - 33*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((63*A - 33*B + 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((12*A - 7*B + 2*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((63*A - 33*B + 13*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 8, -((119*A - 49*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3, x, 7, ((49*A - 9*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^3), x, 7, -((9*A + B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A + B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3), x, 7, -((A - B - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3), x, 8, ((A + 9*B - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + 9*B - 49*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((A + 3*B - 13*C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3), x, 9, ((9*A - 49*B + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B + 33*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((3*A - 13*B + 33*C)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) + ((B - 2*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4, x, 9, -((176*A - 57*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((339*A - 108*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((339*A - 108*B + 17*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d) - ((43*A - 15*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])^2) - ((176*A - 57*B + 8*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((13*A - 6*B - C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4, x, 8, ((57*A - 8*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) - ((108*A - 17*B - 4*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((141*A - 29*B - 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((108*A - 17*B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((11*A - 4*B - 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^4), x, 8, -((8*A + B)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((17*A + 4*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((83*A + B - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) + ((8*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((9*A - 2*B - 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4), x, 8, -((A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 3*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((41*A + 15*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4), x, 8, ((B + 8*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((3*A + 4*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((15*A - B - 83*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((B + 8*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A + 2*B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4), x, 9, ((A + 8*B - 57*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 17*B - 108*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((A + 8*B - 57*C)*Sin[c + d*x])/(10*a^4*d*Sqrt[Cos[c + d*x]]) + ((13*A + 29*B - 141*C)*Sin[c + d*x])/(210*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2) + ((4*A + 17*B - 108*C)*Sin[c + d*x])/(42*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+a Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (16*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (4*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 4, (2*a*(7*A + 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (Sqrt[a]*(2*B + C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(1/2), x, 5, (Sqrt[a]*(8*A + 4*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*(4*B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 6, (Sqrt[a]*(8*A + 6*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*(6*B + C)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*A + 6*B + 5*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))} -{(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 7, (Sqrt[a]*(48*A + 40*B + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*(8*B + C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (16*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(84*A + 110*B + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (4*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 72*B + 63*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 3*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 5, (8*a^2*(19*A + 21*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(3*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(2*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(8*A + 6*B - 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (a^(3/2)*(8*A + 12*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(8*A - 4*B - 5*C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(1/2), x, 6, (a^(3/2)*(24*A + 14*B + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(2*B + C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 7, (a^(3/2)*(112*A + 88*B + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(48*A + 56*B + 39*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(112*A + 88*B + 75*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 8, (a^(3/2)*(176*A + 150*B + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(10*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (16*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2224*A + 2522*B + 2717*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(5*A + 13*B)*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)} -{Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (4*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(1160*A + 1364*B + 1485*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(5*A + 11*B)*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (64*a^3*(13*A + 15*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(13*A + 15*B + 21*C)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(5*A + 9*B)*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(2*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(64*A + 70*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A + 10*B - 15*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(8*A + 20*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(56*A + 12*B - 27*C)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 12*B - 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (a*(4*A - 3*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (a^(5/2)*(40*A + 38*B + 25*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(24*A - 54*B - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 42*B + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (a*(6*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(1/2), x, 7, (a^(5/2)*(304*A + 200*B + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + (a*(8*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, (a^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*(2*B + C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2), x, 9, (a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(680*A + 628*B + 545*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d*Cos[c + d*x]^(7/2)) + (a*(12*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 6, -((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 5, (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*Sqrt[a + a*Sec[c + d*x]]), x, 7, ((2*B - C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]), x, 8, ((8*A - 4*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((4*B - C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]), x, 9, -(((8*A - 14*B + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + ((6*B - C)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((8*A - 2*B + 7*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - -{(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]], x, 7, (2*b*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(a - b)*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 7, -(((15*A - 11*B + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((147*A - 95*B + 75*C)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A - 5*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 6, ((11*A - 7*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A - 15*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2), x, 5, -(((7*A - 3*B - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d)) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(3/2)), x, 7, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)), x, 8, ((2*B - 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 3*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)), x, 9, ((8*A - 12*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 2*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - ((2*A - 6*B + 7*C)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 8, -(((283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A - 1495*B + 735*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B + 195*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A - 85*B + 45*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 7, ((163*A - 75*B + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A - 147*B + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((95*A - 39*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])} -{(Cos[c + d*x]^(1/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2), x, 6, -(((75*A - 19*B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d)) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(1/2)*(a + a*Sec[c + d*x])^(5/2)), x, 5, ((19*A + 5*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)), x, 8, (2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)), x, 9, ((2*B - 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (a+b Sec[e+f x])^m (d Cos[e+f x])^n (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m (A+C Sec[e+f x]^2)*) - - -(* ::Section:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m (B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^m (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(7*a*A + 9*b*B + 9*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(5*A*b + 5*a*B + 7*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*A*b + 5*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*a*A + 9*b*B + 9*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(A*b + a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(A*b + a*B + 3*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, (2*(A*b + a*B - b*C)*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(3*b*B + a*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^(1/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 6, -((2*(b*B - a*(A - C))*EllipticE[(1/2)*(c + d*x), 2])/d) + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(1/2), x, 7, -((2*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(b*B + a*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 5*a*B + 3*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 8, -((2*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 5*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(5*a*A + 3*b*B + 3*a*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(4*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(4*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(2*A*b + a*B - 6*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(A - 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, (2*(a^2*B - b^2*B + 2*a*b*(A - C))*EllipticE[(1/2)*(c + d*x), 2])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(3*b*B + 4*a*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 7, -((2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(5*b*B + 4*a*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 + 10*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 8, -((2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} - - -{Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(6*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(21*a*b*B + 6*b^2*(3*A - 7*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(11*A*b + 7*a*B - 35*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(A - 7*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(a^2*B - 6*b^2*B + 3*a*b*(A - 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(3*a*A - 15*b*B - 35*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(b*B + 2*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, (2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*b*(15*A*b^2 + 35*a*b*B + 24*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(5*a*A - 5*b*B - 9*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(5*b*B + 6*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 8, -((2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d)) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (2*(7*b*B + 6*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 9, -((2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(3*b*B + 2*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -{Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(1/2)*(c + d*x), 2])/(231*d) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(8*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(15*a^3*B + 117*a*b^2*B + 2*b^3*(31*A - 63*C) + 12*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a^2*(162*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*a*(5*A*b + 3*a*B - 21*b*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*a*(A - 9*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*a*(28*a^2*b*B - 42*b^3*B + 3*a*b^2*(13*A - 49*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(54*a*A*b + 21*a^2*B - 105*b^2*B - 350*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(a*A - 7*b*B - 21*a*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(3*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*d) + (2*a*(5*a^3*B - 105*a*b^2*B + 4*a^2*b*(5*A - 33*C) - 6*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a^2*(50*a*b*B - a^2*(3*A - 59*C) + 3*b^2*(5*A + 3*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(5*A*b^2 + 15*a*b*B + 16*a^2*C + 3*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(5*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, (2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(413*a^2*b*B + 63*b^3*B + 192*a^3*C + 2*a*b^2*(175*A + 101*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*a^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(35*A*b^2 + 77*a*b*B + 48*a^2*C + 25*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(7*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 9, -((2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(1/2)*(c + d*x), 2])/(15*d)) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(1/2)*(c + d*x), 2])/(21*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(1098*a^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(9*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x, 8, (2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(5*a^3*d) - (2*(3*A*b^3 - a^3*B - 3*a*b^2*B + a^2*b*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^4*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x, 7, -((2*(A*b - a*B)*EllipticE[(1/2)*(c + d*x), 2])/(a^2*d)) + (2*(3*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^3*d) - (2*b*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a + b)*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x, 6, (2*A*EllipticE[(1/2)*(c + d*x), 2])/(a*d) - (2*(A*b - a*B)*EllipticF[(1/2)*(c + d*x), 2])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a + b)*d)} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])), x, 7, -((2*C*EllipticE[(1/2)*(c + d*x), 2])/(b*d)) + (2*A*EllipticF[(1/2)*(c + d*x), 2])/(a*d) - (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a*b*(a + b)*d) + (2*C*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])), x, 8, -((2*(b*B - a*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*d)) + (2*C*EllipticF[(1/2)*(c + d*x), 2])/(3*b*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(b^2*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])), x, 9, -((2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(5*b^3*d)) + (2*(b*B - a*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*d) - (2*a*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Cos[c + d*x]^(5/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sin[c + d*x])/(5*b^3*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2, x, 8, ((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) - ((15*A*b^4 + 12*a^3*b*B - 9*a*b^3*B - a^2*b^2*(16*A - 3*C) - 2*a^4*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(3*a^4*(a^2 - b^2)*d) + (b*(5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^4*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2, x, 7, -(((3*A*b^2 - a*b*B - a^2*(2*A - C))*EllipticE[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d)) + ((3*A*b^3 + 2*a^3*B - a*b^2*B - a^2*b*(4*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(a^3*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^3*(a - b)*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2), x, 7, ((A*b^2 - a*(b*B - a*C))*EllipticE[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d) - ((A*b^2 + a*b*B - a^2*(2*A + C))*EllipticF[(1/2)*(c + d*x), 2])/(a^2*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2), x, 8, -(((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^2*(a^2 - b^2)*d)) - ((A*b^2 - a*(b*B - a*C))*EllipticF[(1/2)*(c + d*x), 2])/(a*b*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(a*b^2*(a + b)*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2), x, 9, -(((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*EllipticE[(1/2)*(c + d*x), 2])/(b^3*(a^2 - b^2)*d)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*EllipticF[(1/2)*(c + d*x), 2])/(3*b^2*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/((a - b)*b^3*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))} - - -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3, x, 9, -(((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - a^2*b^3*(65*A - 3*C) + a^4*(24*A*b - 9*b*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d)) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3, x, 8, ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - ((15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B - a^2*b^3*(33*A + C) + a^4*b*(24*A + 7*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a^4*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((5*A*b^4 + 7*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(11*A + 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3), x, 8, -(((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d)) + ((3*A*b^4 - 7*a^3*b*B + a*b^3*B - a^2*b^2*(5*A - 3*C) + a^4*(8*A + 3*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a^3*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3), x, 8, -(((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d)) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3), x, 9, ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(1/2)*(c + d*x), 2])/(4*b^3*(a^2 - b^2)^2*d) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*EllipticF[(1/2)*(c + d*x), 2])/(4*a*b^2*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*a)/(a + b), (1/2)*(c + d*x), 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2) + ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cos[e+f x])^(n/2) (a+b Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[c + d*x]^(9/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 12, (-2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b*(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*a*d) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 11, (2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(105*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 10, (-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 13, (-2*(A*b^2 - a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 13, ((2*a*B + b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 14, ((8*a*A + 4*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B + a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]])} -{(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 15, ((24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(24*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(8*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + ((6*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*b*d*Cos[c + d*x]^(3/2)) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]])} - - -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 12, (2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(315*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 11, (2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, (-2*(3*A*b^3 - 5*a^3*B + 5*a*b^2*B - 3*a^2*b*(A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, ((6*a*b*B - b^2*(2*A - 3*C) + 2*a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b + 6*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(2*A - 3*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 14, ((8*a^2*B + 4*b^2*B + a*b*(8*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*a*A - 4*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((4*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 15, ((42*a*b*B + 8*b^2*(3*A + 2*C) + a^2*(48*A + 17*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(8*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((2*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 16, ((136*a^2*b*B + 128*b^3*B - 3*a^3*C + 12*a*b^2*(28*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(192*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(64*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((8*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*b*d*Cos[c + d*x]^(3/2)) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))} - - -{Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 13, (2*(a^2 - b^2)*(40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)} -{Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 12, (-2*(a^2 - b^2)*(10*A*b^3 - 75*a^3*B - 45*a*b^2*B - 6*a^2*b*(19*A + 28*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(5*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)} -{Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, (-2*(15*A*b^4 - 56*a^3*b*B + 56*a*b^3*B + 10*a^2*b^2*(A - 7*C) - 5*a^4*(5*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(5*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)} -{Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, ((10*a^3*B + 20*a*b^2*B - b^3*(16*A - 15*C) + 4*a^2*b*(4*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*b*B + 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(16*A*b + 10*a*B - 15*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)} -{Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, ((48*a^2*b*B + 12*b^3*B + 8*a^3*(A + 3*C) + a*b^2*(16*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(12*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(8*a*A - 12*b*B - 21*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - 3*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)} -{Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x, 15, ((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + ((6*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]], x, 16, ((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(192*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(64*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((8*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))} -{((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2), x, 17, ((1330*a^3*b*B + 3560*a*b^3*B - 15*a^4*C + 256*b^4*(5*A + 4*C) + 4*a^2*b^2*(1180*A + 809*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(1920*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(128*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(1920*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + ((590*a^2*b*B + 360*b^3*B + 15*a^3*C + 4*a*b^2*(260*A + 193*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*b*d*Cos[c + d*x]^(3/2)) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B + a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 11, (2*(48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 5*a^4*(5*A + 7*C) + 2*a^2*b^2*(16*A + 35*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(105*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^3*d) - (2*(6*A*b - 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a^2*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*a*d)} -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 10, (-2*(8*A*b^3 - 5*a^3*B - 10*a*b^2*B + a^2*b*(7*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 9, (2*(2*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 12, (-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x, 13, ((2*A + C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]), x, 14, ((4*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) + ((4*b*B - 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])} - -{(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]], x, 13, (2*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 11, -((2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) - (2*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 10, (2*(8*A*b^2 - 6*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x, 9, -((2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) - (2*(2*A*b^2 - a*b*B - a^2*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)), x, 13, (2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)), x, 14, (C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])} - - -{(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 12, (2*(128*A*b^5 + 5*a^5*B + 80*a^3*b^2*B - 80*a*b^4*B - 4*a^2*b^3*(29*A - 10*C) - a^4*b*(17*A + 45*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(15*a^5*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) - 4*a^2*b^4*(53*A - 10*C) + 3*a^6*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(8*A*b^4 + 9*a^3*b*B - 5*a*b^3*B - 2*a^2*b^2*(6*A - C) - 6*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B + 2*a^4*b*(7*A - 20*C) - 2*a^2*b^3*(49*A - 10*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d) + (2*(48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B + a^4*(3*A - 35*C) - a^2*b^2*(71*A - 15*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d)} -{(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 11, -((2*(16*A*b^4 + 9*a^3*b*B - 8*a*b^3*B - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) - (2*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 2*a^2*b^3*(14*A - C) + a^4*(8*A*b - 6*b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)} -{(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x, 10, (2*(8*A*b^3 + 3*a^3*B - 2*a*b^2*B - a^2*b*(9*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)), x, 10, -((2*(2*A*b^2 + a*b*B - a^2*(3*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) - (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 + 2*a^3*b*B + 2*a*b^3*B + a^4*C - 5*a^2*b^2*(A + C))*Sin[c + d*x])/(3*a*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)), x, 14, -((2*(A*b^2 - a*(b*B - a*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])} -{(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)), x, 15, ((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*a)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*a)/(a + b)])/(b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(1/2)*(c + d*x), (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m deleted file mode 100644 index 3d4432e..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.5 Secant/4.5.7 (d trig)^m (a+b (c sec)^n)^p.m +++ /dev/null @@ -1,760 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^7, x, 3, -(((a - 3*b)*Cos[e + f*x])/f) + ((a - b)*Cos[e + f*x]^3)/f - ((3*a - b)*Cos[e + f*x]^5)/(5*f) + (a*Cos[e + f*x]^7)/(7*f) + (b*Sec[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^5, x, 3, -(((a - 2*b)*Cos[e + f*x])/f) + ((2*a - b)*Cos[e + f*x]^3)/(3*f) - (a*Cos[e + f*x]^5)/(5*f) + (b*Sec[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^3, x, 3, -(((a - b)*Cos[e + f*x])/f) + (a*Cos[e + f*x]^3)/(3*f) + (b*Sec[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^1, x, 3, -((a*Cos[e + f*x])/f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^1*(a + b*Sec[e + f*x]^2), x, 3, -(((a + b)*ArcTanh[Cos[e + f*x]])/f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2), x, 4, -((a + 3*b)*ArcTanh[Cos[e + f*x]])/(2*f) - ((a + b)*Cot[e + f*x]*Csc[e + f*x])/(2*f) + (b*Sec[e + f*x])/f} -{Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2), x, 5, (-3*(a + 5*b)*ArcTanh[Cos[e + f*x]])/(8*f) - ((3*a + 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - ((a + b)*Cot[e + f*x]*Csc[e + f*x]^3)/(4*f) + (b*Sec[e + f*x])/f} - -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^6, x, 6, (5*(a - 6*b)*x)/16 - ((11*a - 18*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - (a*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^4, x, 5, (3*(a - 4*b)*x)/8 - ((5*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x])/(4*f) + (b*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^2, x, 4, ((a - 2*b)*x)/2 - (a*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b*Tan[e + f*x])/f} -{(a + b*Sec[e + f*x]^2)*Sin[e + f*x]^0, x, 3, a*x + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2), x, 3, -(((a + b)*Cot[e + f*x])/f) + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2), x, 3, -(((a + 2*b)*Cot[e + f*x])/f) - ((a + b)*Cot[e + f*x]^3)/(3*f) + (b*Tan[e + f*x])/f} -{Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2), x, 3, -(((a + 3*b)*Cot[e + f*x])/f) - ((2*a + 3*b)*Cot[e + f*x]^3)/(3*f) - ((a + b)*Cot[e + f*x]^5)/(5*f) + (b*Tan[e + f*x])/f} - - -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^5, x, 3, -(((a^2 - 4*a*b + b^2)*Cos[e + f*x])/f) + (2*a*(a - b)*Cos[e + f*x]^3)/(3*f) - (a^2*Cos[e + f*x]^5)/(5*f) + (2*(a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^3, x, 3, -((a*(a - 2*b)*Cos[e + f*x])/f) + (a^2*Cos[e + f*x]^3)/(3*f) + ((2*a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^1, x, 3, -((a^2*Cos[e + f*x])/f) + (2*a*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^1*(a + b*Sec[e + f*x]^2)^2, x, 4, -(((a + b)^2*ArcTanh[Cos[e + f*x]])/f) + (b*(2*a + b)*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2, x, 5, -((a + b)*(a + 5*b)*ArcTanh[Cos[e + f*x]])/(2*f) - ((3*a^2 + 6*a*b + 5*b^2)*Cot[e + f*x]*Csc[e + f*x])/(6*f) + (b*(6*a + 5*b)*Sec[e + f*x])/(3*f) + (b^2*Csc[e + f*x]^2*Sec[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2, x, 6, -((3*a^2 + 30*a*b + 35*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - ((3*a + 7*b)^2*Cot[e + f*x]*Csc[e + f*x])/(24*f) - ((3*a^2 + 6*a*b + 7*b^2)*Cot[e + f*x]*Csc[e + f*x]^3)/(12*f) + (b*(6*a + 7*b)*Sec[e + f*x])/(3*f) + (b^2*Csc[e + f*x]^4*Sec[e + f*x]^3)/(3*f)} - -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^6, x, 7, (5*(a^2 - 12*a*b + 8*b^2)*x)/16 - ((3*a^2 - 36*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a*(a - 12*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - ((a^2 - 12*a*b + 12*b^2)*Tan[e + f*x])/(6*f) + (a^2*Sin[e + f*x]^6*Tan[e + f*x])/(6*f) + (b^2*Tan[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^4, x, 6, ((3*a^2 - 24*a*b + 8*b^2)*x)/8 - (a*(a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - ((a^2 - 8*a*b + 4*b^2)*Tan[e + f*x])/(4*f) + (a^2*Sin[e + f*x]^4*Tan[e + f*x])/(4*f) + (b^2*Tan[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^2, x, 5, (a*(a - 4*b)*x)/2 - (a*(a - 4*b)*Tan[e + f*x])/(2*f) + (a^2*Sin[e + f*x]^2*Tan[e + f*x])/(2*f) + (b^2*Tan[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)^2*Sin[e + f*x]^0, x, 4, a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2, x, 3, -(((a + b)^2*Cot[e + f*x])/f) + (2*b*(a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2, x, 3, -(((a + b)*(a + 3*b)*Cot[e + f*x])/f) - ((a + b)^2*Cot[e + f*x]^3)/(3*f) + (b*(2*a + 3*b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2, x, 3, -(((a^2 + 6*a*b + 6*b^2)*Cot[e + f*x])/f) - (2*(a + b)*(a + 2*b)*Cot[e + f*x]^3)/(3*f) - ((a + b)^2*Cot[e + f*x]^5)/(5*f) + (2*b*(a + 2*b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 4, (Sqrt[b]*(a + b)^2*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(7/2)*f) - ((a + b)^2*Cos[e + f*x])/(a^3*f) + ((2*a + b)*Cos[e + f*x]^3)/(3*a^2*f) - Cos[e + f*x]^5/(5*a*f)} -{Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 4, (Sqrt[b]*(a + b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(5/2)*f) - ((a + b)*Cos[e + f*x])/(a^2*f) + Cos[e + f*x]^3/(3*a*f)} -{Sin[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 3, (Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(a^(3/2)*f) - Cos[e + f*x]/(a*f)} -{Csc[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 4, (Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*f) - ArcTanh[Cos[e + f*x]]/((a + b)*f)} -{Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 5, (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/((a + b)^2*f) - ((a - b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^2*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f)} -{Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 6, (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/((a + b)^3*f) - ((3*a^2 - 6*a*b - b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^3*f) - ((3*a - b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*(a + b)*f)} - -{Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 7, ((5*a^3 + 30*a^2*b + 40*a*b^2 + 16*b^3)*x)/(16*a^4) - (Sqrt[b]*(a + b)^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^4*f) - ((11*a^2 + 18*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f) + ((3*a + 2*b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f)} -{Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 6, ((3*a^2 + 12*a*b + 8*b^2)*x)/(8*a^3) - (Sqrt[b]*(a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*f) - ((5*a + 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 5, ((a + 2*b)*x)/(2*a^2) - (Sqrt[b]*Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^2*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f)} -{Sin[e + f*x]^0/(a + b*Sec[e + f*x]^2), x, 3, x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(3/2)*f)) - Cot[e + f*x]/((a + b)*f)} -{Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 4, -((a*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(5/2)*f)) - (a*Cot[e + f*x])/((a + b)^2*f) - Cot[e + f*x]^3/(3*(a + b)*f)} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 4, -((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/((a + b)^(7/2)*f)) - (a^2*Cot[e + f*x])/((a + b)^3*f) - ((2*a + b)*Cot[e + f*x]^3)/(3*(a + b)^2*f) - Cot[e + f*x]^5/(5*(a + b)*f)} - - -{Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 6, (Sqrt[b]*(a + b)*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(9/2)*f) - ((a + b)*(3*a + 7*b)*Cos[e + f*x])/(2*a^4*f) + ((a + b)*(3*a + 7*b)*Cos[e + f*x]^3)/(6*a^3*b*f) - Cos[e + f*x]^5/(5*a^2*f) - ((a + b)^2*Cos[e + f*x]^5)/(2*a^2*b*f*(b + a*Cos[e + f*x]^2))} -{Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 5, (Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(7/2)*f) - ((a + 2*b)*Cos[e + f*x])/(a^3*f) + Cos[e + f*x]^3/(3*a^2*f) - (b*(a + b)*Cos[e + f*x])/(2*a^3*f*(b + a*Cos[e + f*x]^2))} -{Sin[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 4, (3*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(5/2)*f) - (3*Cos[e + f*x])/(2*a^2*f) + Cos[e + f*x]^3/(2*a*f*(b + a*Cos[e + f*x]^2))} -{Csc[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 5, (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*f) - ArcTanh[Cos[e + f*x]]/((a + b)^2*f) - (b*Cos[e + f*x])/(2*a*(a + b)*f*(b + a*Cos[e + f*x]^2))} -{Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 6, ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*Sqrt[a]*(a + b)^3*f) - ((a - 3*b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^3*f) + ((a - b)*Cos[e + f*x])/(2*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*(b + a*Cos[e + f*x]^2))} -{Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 7, (3*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(2*(a + b)^4*f) - (3*(a^2 - 6*a*b + b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^4*f) + (3*a*(a - 3*b)*Cos[e + f*x])/(8*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - ((a - 5*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - (Cot[e + f*x]*Csc[e + f*x]^3)/(4*(a + b)*f*(b + a*Cos[e + f*x]^2))} - -{Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 8, ((5*a^3 + 60*a^2*b + 120*a*b^2 + 64*b^3)*x)/(16*a^5) - (Sqrt[b]*(a + b)^(3/2)*(3*a + 8*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^5*f) - ((33*a^2 + 82*a*b + 48*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)) + ((9*a + 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)) - (b*(19*a^2 + 52*a*b + 32*b^2)*Tan[e + f*x])/(16*a^4*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 7, (3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^4*f) - ((5*a + 6*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)) - (3*b*(3*a + 4*b)*Tan[e + f*x])/(8*a^3*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 6, ((a + 4*b)*x)/(2*a^3) - (Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^3*Sqrt[a + b]*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)) - (b*Tan[e + f*x])/(a^2*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^0/(a + b*Sec[e + f*x]^2)^2, x, 5, x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 4, (-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) - (3*Cot[e + f*x])/(2*(a + b)^2*f) + Cot[e + f*x]/(2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 5, -((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) - ((a - b)*Cot[e + f*x])/((a + b)^3*f) - Cot[e + f*x]^3/(3*(a + b)^2*f) - (a*b*Tan[e + f*x])/(2*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 6, -(a*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*(a + b)^(9/2)*f) - ((5*a^2 - 10*a*b - b^2)*Cot[e + f*x])/(5*(a + b)^4*f) - ((10*a + 3*b)*Cot[e + f*x]^3)/(15*(a + b)^3*f) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)) - (b*(5*a^2 + 2*b^2)*Tan[e + f*x])/(10*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))} - - -{Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 6, (Sqrt[b]*(15*a^2 + 70*a*b + 63*b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(11/2)*f) - ((3*a^2 + 14*a*b + 13*b^2)*Cos[e + f*x])/(2*a^5*f) + ((a + 3*b)*(3*a + 5*b)*Cos[e + f*x]^3)/(12*a^4*b*f) - Cos[e + f*x]^5/(5*a^3*f) - ((a + b)^2*Cos[e + f*x]^7)/(4*a^2*b*f*(b + a*Cos[e + f*x]^2)^2) - (b*(a + b)*(3*a + 11*b)*Cos[e + f*x])/(8*a^5*f*(b + a*Cos[e + f*x]^2))} -{Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 6, (5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(9/2)*f) - ((a + 3*b)*Cos[e + f*x])/(a^4*f) + Cos[e + f*x]^3/(3*a^3*f) + (b^2*(a + b)*Cos[e + f*x])/(4*a^4*f*(b + a*Cos[e + f*x]^2)^2) - (b*(9*a + 13*b)*Cos[e + f*x])/(8*a^4*f*(b + a*Cos[e + f*x]^2))} -{Sin[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 5, (15*Sqrt[b]*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(7/2)*f) - (15*Cos[e + f*x])/(8*a^3*f) + Cos[e + f*x]^5/(4*a*f*(b + a*Cos[e + f*x]^2)^2) + (5*Cos[e + f*x]^3)/(8*a^2*f*(b + a*Cos[e + f*x]^2))} -{Csc[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 6, (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3*f) - ArcTanh[Cos[e + f*x]]/((a + b)^3*f) - (b*Cos[e + f*x]^3)/(4*a*(a + b)*f*(b + a*Cos[e + f*x]^2)^2) - (b*(7*a + 3*b)*Cos[e + f*x])/(8*a^2*(a + b)^2*f*(b + a*Cos[e + f*x]^2))} -{Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 7, (Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*f) - ((a - 5*b)*ArcTanh[Cos[e + f*x]])/(2*(a + b)^4*f) - ((2*a - b)*b*Cos[e + f*x])/(4*a*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) + ((4*a^2 - 9*a*b - b^2)*Cos[e + f*x])/(8*a*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - (Cos[e + f*x]*Cot[e + f*x]^2)/(2*(a + b)*f*(b + a*Cos[e + f*x]^2)^2)} -{Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 8, (3*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[a]*Cos[e + f*x])/Sqrt[b]])/(8*Sqrt[a]*(a + b)^5*f) - (3*(a^2 - 10*a*b + 5*b^2)*ArcTanh[Cos[e + f*x]])/(8*(a + b)^5*f) + ((a^2 - 9*a*b + 2*b^2)*Cos[e + f*x])/(8*(a + b)^3*f*(b + a*Cos[e + f*x]^2)^2) + (3*(a^2 - 6*a*b + b^2)*Cos[e + f*x])/(8*(a + b)^4*f*(b + a*Cos[e + f*x]^2)) - ((a - 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*(b + a*Cos[e + f*x]^2)^2)} - -{Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 9, (5*(a + 2*b)*(a^2 + 16*a*b + 16*b^2)*x)/(16*a^6) - (5*Sqrt[b]*Sqrt[a + b]*(a + 4*b)*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^6*f) - ((33*a^2 + 110*a*b + 80*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) + ((9*a + 10*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (5*b*(9*a^2 + 32*a*b + 24*b^2)*Tan[e + f*x])/(48*a^4*f*(a + b + b*Tan[e + f*x]^2)^2) - (5*b*(5*a^2 + 20*a*b + 16*b^2)*Tan[e + f*x])/(16*a^5*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 8, (3*(a^2 + 12*a*b + 16*b^2)*x)/(8*a^5) - (3*Sqrt[b]*(5*a^2 + 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^5*Sqrt[a + b]*f) - ((5*a + 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 12*b)*Tan[e + f*x])/(8*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*b*(a + 2*b)*Tan[e + f*x])/(2*a^4*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 7, ((a + 6*b)*x)/(2*a^4) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a + 12*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Sin[e + f*x]^0/(a + b*Sec[e + f*x]^2)^3, x, 6, x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 5, (-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) - (15*Cot[e + f*x])/(8*(a + b)^3*f) + Cot[e + f*x]/(4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (5*Cot[e + f*x])/(8*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 6, (-5*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - ((a - 2*b)*Cot[e + f*x])/((a + b)^4*f) - Cot[e + f*x]^3/(3*(a + b)^3*f) - (a*b*Tan[e + f*x])/(4*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^2) - ((7*a - 4*b)*b*Tan[e + f*x])/(8*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 7, -(Sqrt[b]*(15*a^2 - 40*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(11/2)*f) - ((5*a^2 - 20*a*b + 2*b^2)*Cot[e + f*x])/(5*(a + b)^5*f) - ((10*a + b)*Cot[e + f*x]^3)/(15*(a + b)^4*f) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(5*a^2 + 4*b^2)*Tan[e + f*x])/(20*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(35*a^2 - 40*a*b + 24*b^2)*Tan[e + f*x])/(40*(a + b)^5*f*(a + b + b*Tan[e + f*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^5, x, 6, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f + (2*(5*a + b)*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(15*a^2*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2))/(5*a*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^3, x, 5, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f + (Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^1, x, 4, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f} -{Csc[e + f*x]^1*Sqrt[a + b*Sec[e + f*x]^2], x, 6, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - (Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f} -{Csc[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2], x, 7, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - ((a + 2*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*Sqrt[a + b]*f) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x, 8, (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(3/2)*f) - ((3*a + 4*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)} - -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^6, x, 9, ((5*a^3 - 15*a^2*b - 5*a*b^2 - b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(5/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a - b)*(5*a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a^2*f) - ((5*a - b)*Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a*f) - (Cos[e + f*x]*Sin[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^4, x, 8, ((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((3*a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a*f) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^2, x, 7, ((a - b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[a]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Sin[e + f*x]^0, x, 6, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f} -{Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2], x, 4, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f} -{Csc[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2], x, 5, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f)} -{Csc[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2], x, 6, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (2*(5*a + 4*b)*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(15*(a + b)^2*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(3/2))/(5*(a + b)*f)} - - -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^5, x, 7, ((3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 4*b)*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*a*f) - ((3*a - 4*b)*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f) + (2*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(5/2))/(3*a*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(5/2))/(5*a*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^3, x, 6, ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 2*b)*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*a*f) - ((3*a - 2*b)*Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(3*a*f) + (Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(5/2))/(3*a*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^1, x, 5, (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + (3*b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f) - (Cos[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/f} -{Csc[e + f*x]^1*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - ((a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/f + (b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x, 8, (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - (Sqrt[a + b]*(a + 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/f - (Cot[e + f*x]*Csc[e + f*x]*(a + b*Sec[e + f*x]^2)^(3/2))/(2*f)} -{Csc[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x, 9, (3*Sqrt[b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*f) - (3*(a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*Sqrt[a + b]*f) + (3*(a + 4*b)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - (3*(a + 2*b)*Csc[e + f*x]^2*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - (Cot[e + f*x]*Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(4*f)} - -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^6, x, 10, ((5*a^3 - 45*a^2*b + 15*a*b^2 + b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(3/2)*f) + ((3*a - 5*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) - ((5*a^2 - 26*a*b + b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a*f) + ((5*a^2 - 40*a*b + 3*b^2)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a*f) + ((5*a - 3*b)*Sin[e + f*x]^4*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*f) - (Cos[e + f*x]*Sin[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(3/2))/(6*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^4, x, 9, (3*(a^2 - 6*a*b + b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) - (3*(a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (3*(a - b)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) - (Cos[e + f*x]*Sin[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^2, x, 8, (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f - (Cos[e + f*x]*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(2*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Sin[e + f*x]^0, x, 7, (a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2), x, 5, (3*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (3*b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f) - (Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/f} -{Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2), x, 6, (Sqrt[b]*(3*a + 5*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 5*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*(a + b)*f) - ((3*a + 5*b)*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f) - (Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(5/2))/(3*(a + b)*f)} -{Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (Sqrt[b]*(3*a + 7*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 7*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*(a + b)*f) - ((3*a + 7*b)*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(3*(a + b)*f) - (2*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(5/2))/(3*(a + b)*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(5/2))/(5*(a + b)*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sin[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 4, -((15*a^2 + 20*a*b + 8*b^2)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(15*a^3*f) + (2*(5*a + 2*b)*Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(15*a^2*f) - (Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2])/(5*a*f)} -{Sin[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 3, -((3*a + 2*b)*Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(3*a^2*f) + (Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2])/(3*a*f)} -{Sin[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 2, -((Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(a*f))} -{Csc[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 3, -(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/(Sqrt[a + b]*f))} -{Csc[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 5, -((a*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(3/2)*f)) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(2*(a + b)*f)} -{Csc[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 6, -((3*a^2*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(5/2)*f)) - ((5*a + 2*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)^2*f) - (Cot[e + f*x]^3*Csc[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2])/(4*(a + b)*f)} - -{Sin[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 7, (5*(a + b)^3*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(7/2)*f) - ((33*a^2 + 40*a*b + 15*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^3*f) + ((9*a + 5*b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*a*f)} -{Sin[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 6, (3*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(5/2)*f) - ((5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*a*f)} -{Sin[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 5, ((a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*f)} -{Sin[e + f*x]^0/Sqrt[a + b*Sec[e + f*x]^2], x, 3, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)} -{Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 2, -((Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/((a + b)*f))} -{Csc[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 3, -((3*a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)^2*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f)} -{Csc[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 4, -((15*a^2 + 10*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^3*f) - (2*(5*a + 3*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*(a + b)*f)} - - -{Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, -((15*a^2 + 40*a*b + 24*b^2)*Cos[e + f*x])/(15*a^3*f*Sqrt[a + b*Sec[e + f*x]^2]) + (2*(5*a + 3*b)*Cos[e + f*x]^3)/(15*a^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cos[e + f*x]^5/(5*a*f*Sqrt[a + b*Sec[e + f*x]^2]) - (2*b*(15*a^2 + 40*a*b + 24*b^2)*Sec[e + f*x])/(15*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, -(((3*a + 4*b)*Cos[e + f*x])/(3*a^2*f*Sqrt[a + b*Sec[e + f*x]^2])) + Cos[e + f*x]^3/(3*a*f*Sqrt[a + b*Sec[e + f*x]^2]) - (2*b*(3*a + 4*b)*Sec[e + f*x])/(3*a^3*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 3, -(Cos[e + f*x]/(a*f*Sqrt[a + b*Sec[e + f*x]^2])) - (2*b*Sec[e + f*x])/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/((a + b)^(3/2)*f)) - (b*Sec[e + f*x])/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 6, -(((a - 2*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(5/2)*f)) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) - (3*b*Sec[e + f*x])/(2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 7, -((3*a*(a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(7/2)*f)) - (5*a*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2]) - ((13*a - 2*b)*b*Sec[e + f*x])/(8*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])} - -{Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 8, (5*(a + b)^2*(a + 7*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(9/2)*f) - ((a + b)*(33*a + 35*b)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((9*a + 7*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (b*(81*a^2 + 190*a*b + 105*b^2)*Tan[e + f*x])/(48*a^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (3*(a + b)*(a + 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(7/2)*f) - (5*(a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (b*(13*a + 15*b)*Tan[e + f*x])/(8*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 6, ((a + 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(5/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (3*b*Tan[e + f*x])/(2*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 3, -(Cot[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f*x])/((a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, -((3*a - b)*Cot[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*(3*a - b)*b*Tan[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, -((15*a^2 - 10*a*b - b^2)*Cot[e + f*x])/(15*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*(5*a + 2*b)*Cot[e + f*x]^3)/(15*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - Cot[e + f*x]^5/(5*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - (2*b*(15*a^2 - 10*a*b - b^2)*Tan[e + f*x])/(15*(a + b)^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])} - - -{Sin[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, -((5*a^2 + 20*a*b + 16*b^2)*Cos[e + f*x])/(5*a^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (2*(5*a + 4*b)*Cos[e + f*x]^3)/(15*a^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cos[e + f*x]^5/(5*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 + 20*a*b + 16*b^2)*Sec[e + f*x])/(15*a^4*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 + 20*a*b + 16*b^2)*Sec[e + f*x])/(15*a^5*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 5, -(((a + 2*b)*Cos[e + f*x])/(a^2*f*(a + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(a + 2*b)*Sec[e + f*x])/(3*a^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + 2*b)*Sec[e + f*x])/(3*a^4*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Sin[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 4, -(Cos[e + f*x]/(a*f*(a + b*Sec[e + f*x]^2)^(3/2))) - (4*b*Sec[e + f*x])/(3*a^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (8*b*Sec[e + f*x])/(3*a^3*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]]/((a + b)^(5/2)*f)) - (b*Sec[e + f*x])/(3*a*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(5*a + 2*b)*Sec[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 7, -(((a - 4*b)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(2*(a + b)^(7/2)*f)) - (Cot[e + f*x]*Csc[e + f*x])/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (5*b*Sec[e + f*x])/(6*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((13*a - 2*b)*b*Sec[e + f*x])/(6*a*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Csc[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 8, -(((3*a^2 - 24*a*b + 8*b^2)*ArcTanh[(Sqrt[a + b]*Sec[e + f*x])/Sqrt[a + b*Sec[e + f*x]^2]])/(8*(a + b)^(9/2)*f)) - ((5*a - 2*b)*Cot[e + f*x]*Csc[e + f*x])/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - ((23*a - 12*b)*b*Sec[e + f*x])/(24*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (5*(11*a - 10*b)*b*Sec[e + f*x])/(24*(a + b)^4*f*Sqrt[a + b*Sec[e + f*x]^2])} - -{Sin[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 9, (5*(a + b)*(a^2 + 14*a*b + 21*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(11/2)*f) - ((a + b)*(11*a + 21*b)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (3*(a + b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (7*b*(a + b)*(7*a + 15*b)*Tan[e + f*x])/(48*a^4*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(113*a^2 + 420*a*b + 315*b^2)*Tan[e + f*x])/(48*a^5*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 8, ((3*a^2 + 30*a*b + 35*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(9/2)*f) - ((5*a + 7*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(23*a + 35*b)*Tan[e + f*x])/(24*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (5*b*(11*a + 21*b)*Tan[e + f*x])/(24*a^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 7, ((a + 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(7/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (5*b*Tan[e + f*x])/(6*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(13*a + 15*b)*Tan[e + f*x])/(6*a^3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sin[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 4, -(Cot[e + f*x]/((a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2))) - (4*b*Tan[e + f*x])/(3*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*b*Tan[e + f*x])/(3*(a + b)^3*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 5, -(((a - b)*Cot[e + f*x])/((a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2))) - Cot[e + f*x]^3/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (4*(a - b)*b*Tan[e + f*x])/(3*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*(a - b)*b*Tan[e + f*x])/(3*(a + b)^4*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Csc[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, -((5*a^2 - 10*a*b + b^2)*Cot[e + f*x])/(5*(a + b)^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (2*(5*a + b)*Cot[e + f*x]^3)/(15*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^5/(5*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 - 10*a*b + b^2)*Tan[e + f*x])/(15*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 - 10*a*b + b^2)*Tan[e + f*x])/(15*(a + b)^5*f*Sqrt[a + b + b*Tan[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic*) - - -{(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m, x, -1, (AppellF1[(1 + m)/2, 1/2 + p, -p, (3 + m)/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^(1/2 + p)*(a + b*Sec[e + f*x]^2)^p*(d*Sin[e + f*x])^m*Tan[e + f*x])/(((a + b - a*Sin[e + f*x]^2)/(a + b))^p*(f*(1 + m)))} - - -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^5, x, 5, ((10*a + b*(3 - 2*p))*Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(1 + p))/(15*a^2*f) - (Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(1 + p))/(5*a*f) - ((15*a^2 + 10*a*b*(1 - 2*p) + b^2*(3 - 8*p + 4*p^2))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(15*a^2*f*(1 + (b*Sec[e + f*x]^2)/a)^p)} -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^3, x, 4, (Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(1 + p))/(3*a*f) - ((3*a + b - 2*b*p)*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(3*a*f*(1 + (b*Sec[e + f*x]^2)/a)^p)} -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^1, x, 3, -((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/a)]*(a + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/a)^p))} -{Csc[e + f*x]^1*(a + b*Sec[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 1, -p, 3/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Sec[e + f*x]*(a + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/a)^p))} -{Csc[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 2, -p, 5/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p)/(3*f*(1 + (b*Sec[e + f*x]^2)/a)^p)} - -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^4, x, 3, (AppellF1[5/2, 3, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^2, x, 3, (AppellF1[3/2, 2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{(a + b*Sec[e + f*x]^2)^p*Sin[e + f*x]^0, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Csc[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p, x, 3, -((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))} -{Csc[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p, x, 4, -(Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(3*(a + b)*f) - ((3*a + 2*b*(1 + p))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(3*(a + b)*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Csc[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p, x, 5, -((10*a + b*(7 + 2*p))*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(15*(a + b)^2*f) - (Cot[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(5*(a + b)*f) - ((15*a^2 + 20*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/(a + b))]*(a + b + b*Tan[e + f*x]^2)^p)/(15*(a + b)^2*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} - - -(* ::Section:: *) -(*Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p when a+b=0*) - - -{(a - a*Sec[c + d*x]^2)^4, x, 6, a^4*x - (a^4*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/(3*d) - (a^4*Tan[c + d*x]^5)/(5*d) + (a^4*Tan[c + d*x]^7)/(7*d)} -{(a - a*Sec[c + d*x]^2)^3, x, 5, a^3*x - (a^3*Tan[c + d*x])/d + (a^3*Tan[c + d*x]^3)/(3*d) - (a^3*Tan[c + d*x]^5)/(5*d)} -{(a - a*Sec[c + d*x]^2)^2, x, 4, a^2*x - (a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)} -{(a - a*Sec[c + d*x]^2)^1, x, 3, a*x - (a*Tan[c + d*x])/d} -{1/(a - a*Sec[c + d*x]^2)^1, x, 3, x/a + Cot[c + d*x]/(a*d)} -{1/(a - a*Sec[c + d*x]^2)^2, x, 4, x/a^2 + Cot[c + d*x]/(a^2*d) - Cot[c + d*x]^3/(3*a^2*d)} -{1/(a - a*Sec[c + d*x]^2)^3, x, 5, x/a^3 + Cot[c + d*x]/(a^3*d) - Cot[c + d*x]^3/(3*a^3*d) + Cot[c + d*x]^5/(5*a^3*d)} -{1/(a - a*Sec[c + d*x]^2)^4, x, 6, x/a^4 + Cot[c + d*x]/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + Cot[c + d*x]^5/(5*a^4*d) - Cot[c + d*x]^7/(7*a^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2), x, 4, ((6*a + 5*b)*ArcTanh[Sin[e + f*x]])/(16*f) + ((6*a + 5*b)*Sec[e + f*x]*Tan[e + f*x])/(16*f) + ((6*a + 5*b)*Sec[e + f*x]^3*Tan[e + f*x])/(24*f) + (b*Sec[e + f*x]^5*Tan[e + f*x])/(6*f)} -{Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2), x, 3, ((4*a + 3*b)*ArcTanh[Sin[e + f*x]])/(8*f) + ((4*a + 3*b)*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (b*Sec[e + f*x]^3*Tan[e + f*x])/(4*f)} -{Sec[e + f*x]^1*(a + b*Sec[e + f*x]^2), x, 2, ((2*a + b)*ArcTanh[Sin[e + f*x]])/(2*f) + (b*Sec[e + f*x]*Tan[e + f*x])/(2*f)} -{Cos[e + f*x]^1*(a + b*Sec[e + f*x]^2), x, 2, (b*ArcTanh[Sin[e + f*x]])/f + (a*Sin[e + f*x])/f} -{Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2), x, 3, ((a + b)*Sin[e + f*x])/f - (a*Sin[e + f*x]^3)/(3*f)} -{Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2), x, 4, ((a + b)*Sin[e + f*x])/f - ((2*a + b)*Sin[e + f*x]^3)/(3*f) + (a*Sin[e + f*x]^5)/(5*f)} - -{Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2), x, 3, ((7*a + 6*b)*Tan[e + f*x])/(7*f) + (b*Sec[e + f*x]^6*Tan[e + f*x])/(7*f) + (2*(7*a + 6*b)*Tan[e + f*x]^3)/(21*f) + ((7*a + 6*b)*Tan[e + f*x]^5)/(35*f)} -{Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2), x, 3, ((5*a + 4*b)*Tan[e + f*x])/(5*f) + (b*Sec[e + f*x]^4*Tan[e + f*x])/(5*f) + ((5*a + 4*b)*Tan[e + f*x]^3)/(15*f)} -{Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2), x, 3, ((3*a + 2*b)*Tan[e + f*x])/(3*f) + (b*Sec[e + f*x]^2*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]^0*(a + b*Sec[e + f*x]^2), x, 3, a*x + (b*Tan[e + f*x])/f} -{Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2), x, 2, ((a + 2*b)*x)/2 + (a*Cos[e + f*x]*Sin[e + f*x])/(2*f)} -{Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2), x, 3, ((3*a + 4*b)*x)/8 + ((3*a + 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x])/(4*f)} -{Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2), x, 4, ((5*a + 6*b)*x)/16 + ((5*a + 6*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((5*a + 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a*Cos[e + f*x]^5*Sin[e + f*x])/(6*f)} - - -{Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2, x, 6, ((48*a^2 + 80*a*b + 35*b^2)*ArcTanh[Sin[e + f*x]])/(128*f) + ((48*a^2 + 80*a*b + 35*b^2)*Sec[e + f*x]*Tan[e + f*x])/(128*f) + ((48*a^2 + 80*a*b + 35*b^2)*Sec[e + f*x]^3*Tan[e + f*x])/(192*f) + (b*(10*a + 7*b)*Sec[e + f*x]^5*Tan[e + f*x])/(48*f) + (b*Sec[e + f*x]^7*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(8*f)} -{Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2, x, 5, ((8*a^2 + 12*a*b + 5*b^2)*ArcTanh[Sin[e + f*x]])/(16*f) + ((8*a^2 + 12*a*b + 5*b^2)*Sec[e + f*x]*Tan[e + f*x])/(16*f) + (b*(8*a + 5*b)*Sec[e + f*x]^3*Tan[e + f*x])/(24*f) + (b*Sec[e + f*x]^5*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(6*f)} -{Sec[e + f*x]^1*(a + b*Sec[e + f*x]^2)^2, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[Sin[e + f*x]])/(8*f) + (3*b*(2*a + b)*Sec[e + f*x]*Tan[e + f*x])/(8*f) + (b*Sec[e + f*x]^3*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(4*f)} -{Cos[e + f*x]^1*(a + b*Sec[e + f*x]^2)^2, x, 5, (b*(4*a + b)*ArcTanh[Sin[e + f*x]])/(2*f) + (a^2*Sin[e + f*x])/f + (b^2*Sec[e + f*x]*Tan[e + f*x])/(2*f)} -{Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2, x, 4, (b^2*ArcTanh[Sin[e + f*x]])/f + (a*(a + 2*b)*Sin[e + f*x])/f - (a^2*Sin[e + f*x]^3)/(3*f)} -{Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2, x, 3, ((a + b)^2*Sin[e + f*x])/f - (2*a*(a + b)*Sin[e + f*x]^3)/(3*f) + (a^2*Sin[e + f*x]^5)/(5*f)} - -{Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2, x, 3, ((a + b)^2*Tan[e + f*x])/f + (2*(a + b)*(a + 2*b)*Tan[e + f*x]^3)/(3*f) + ((a^2 + 6*a*b + 6*b^2)*Tan[e + f*x]^5)/(5*f) + (2*b*(a + 2*b)*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)} -{Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2, x, 3, ((a + b)^2*Tan[e + f*x])/f + ((a + b)*(a + 3*b)*Tan[e + f*x]^3)/(3*f) + (b*(2*a + 3*b)*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)} -{Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2, x, 3, ((a + b)^2*Tan[e + f*x])/f + (2*b*(a + b)*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)} -{Sec[e + f*x]^0*(a + b*Sec[e + f*x]^2)^2, x, 4, a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2, x, 5, (a*(a + 4*b)*x)/2 + (a^2*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b^2*Tan[e + f*x])/f} -{Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2, x, 4, ((3*a^2 + 8*a*b + 8*b^2)*x)/8 + (3*a*(a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + (a*Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2))/(4*f)} -{Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2, x, 5, ((5*a^2 + 12*a*b + 8*b^2)*x)/16 + ((5*a^2 + 12*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + (a*(5*a + 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) + (a*Cos[e + f*x]^5*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2))/(6*f)} - - -{(a + b*Sec[c + d*x]^2)^3, x, 4, a^3*x + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b^2*(3*a + 2*b)*Tan[c + d*x]^3)/(3*d) + (b^3*Tan[c + d*x]^5)/(5*d)} -{(a + b*Sec[c + d*x]^2)^4, x, 4, a^4*x + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b^2*(6*a^2 + 8*a*b + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b^3*(4*a + 3*b)*Tan[c + d*x]^5)/(5*d) + (b^4*Tan[c + d*x]^7)/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 5, -((2*a - b)*ArcTanh[Sin[e + f*x]])/(2*b^2*f) + (a^(3/2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(b^2*Sqrt[a + b]*f) + (Sec[e + f*x]*Tan[e + f*x])/(2*b*f)} -{Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 4, ArcTanh[Sin[e + f*x]]/(b*f) - (Sqrt[a]*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(b*Sqrt[a + b]*f)} -{Sec[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 2, ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*f)} -{Cos[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 3, -((b*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(3/2)*Sqrt[a + b]*f)) + Sin[e + f*x]/(a*f)} -{Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]*f) + ((a - b)*Sin[e + f*x])/(a^2*f) - Sin[e + f*x]^3/(3*a*f)} -{Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 4, -((b^3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(a^(7/2)*Sqrt[a + b]*f)) + ((a^2 - a*b + b^2)*Sin[e + f*x])/(a^3*f) - ((2*a - b)*Sin[e + f*x]^3)/(3*a^2*f) + Sin[e + f*x]^5/(5*a*f)} - -{Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 4, (a^2*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]*f) - ((a - b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)} -{Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 3, -((a*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b]*f)) + Tan[e + f*x]/(b*f)} -{Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 2, ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*f)} -{Sec[e + f*x]^0/(a + b*Sec[e + f*x]^2), x, 3, x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)} -{Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 5, ((a - 2*b)*x)/(2*a^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f)} -{Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 6, ((3*a^2 - 4*a*b + 8*b^2)*x)/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*f) + ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f)} -{Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 7, ((5*a^3 - 6*a^2*b + 8*a*b^2 - 16*b^3)*x)/(16*a^4) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a^4*Sqrt[a + b]*f) + ((5*a^2 - 6*a*b + 8*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f) + ((5*a - 6*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f)} - - -{Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 5, ArcTanh[Sin[e + f*x]]/(b^2*f) - (Sqrt[a]*(2*a + 3*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*b^2*(a + b)^(3/2)*f) - (a*Sin[e + f*x])/(2*b*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} -{Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 3, ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]]/(2*Sqrt[a]*(a + b)^(3/2)*f) + Sin[e + f*x]/(2*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} -{Sec[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 3, ((2*a + b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)*f) - (b*Sin[e + f*x])/(2*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 5, -(b*(4*a + 3*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(5/2)*(a + b)^(3/2)*f) + Sin[e + f*x]/(a^2*f) + (b^2*Sin[e + f*x])/(2*a^2*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 5, (b^2*(6*a + 5*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(7/2)*(a + b)^(3/2)*f) + ((a - 2*b)*Sin[e + f*x])/(a^3*f) - Sin[e + f*x]^3/(3*a^2*f) - (b^3*Sin[e + f*x])/(2*a^3*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 5, -(b^3*(8*a + 7*b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(2*a^(9/2)*(a + b)^(3/2)*f) + ((a^2 - 2*a*b + 3*b^2)*Sin[e + f*x])/(a^4*f) - (2*(a - b)*Sin[e + f*x]^3)/(3*a^3*f) + Sin[e + f*x]^5/(5*a^2*f) + (b^4*Sin[e + f*x])/(2*a^4*(a + b)*f*(a + b - a*Sin[e + f*x]^2))} - -{Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 5, -(a*(3*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*b^(5/2)*(a + b)^(3/2)*f) + Tan[e + f*x]/(b^2*f) + (a^2*Tan[e + f*x])/(2*b^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 3, ((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)*f) - (a*Tan[e + f*x])/(2*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]]/(2*Sqrt[b]*(a + b)^(3/2)*f) + Tan[e + f*x]/(2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^0/(a + b*Sec[e + f*x]^2)^2, x, 5, x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 6, ((a - 4*b)*x)/(2*a^3) + (b^(3/2)*(5*a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^3*(a + b)^(3/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)) + (b*(a + 2*b)*Tan[e + f*x])/(2*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 7, ((3*a^2 - 8*a*b + 24*b^2)*x)/(8*a^4) - (b^(5/2)*(7*a + 6*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^4*(a + b)^(3/2)*f) + (3*(a - 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)) + ((a - 3*b)*b*(3*a + 4*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 8, ((5*a^3 - 12*a^2*b + 24*a*b^2 - 64*b^3)*x)/(16*a^5) + (b^(7/2)*(9*a + 8*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^5*(a + b)^(3/2)*f) + ((15*a^2 - 26*a*b + 48*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)) + ((5*a - 8*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)) + (b*(5*a^3 - 7*a^2*b + 12*a*b^2 + 32*b^3)*Tan[e + f*x])/(16*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} - - -{Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 4, (3*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*Sqrt[a]*(a + b)^(5/2)*f) + Sin[e + f*x]/(4*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (3*Sin[e + f*x])/(8*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} -{Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 4, ((4*a + b)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*f) - (b*Sin[e + f*x])/(4*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + ((4*a + b)*Sin[e + f*x])/(8*a*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} -{Sec[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(5/2)*(a + b)^(5/2)*f) - (b*Cos[e + f*x]^2*Sin[e + f*x])/(4*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) - (3*b*(2*a + b)*Sin[e + f*x])/(8*a^2*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 6, -((3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)*f)) + Sin[e + f*x]/(a^3*f) - (b^3*Sin[e + f*x])/(4*a^3*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (3*b^2*(4*a + 3*b)*Sin[e + f*x])/(8*a^3*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 6, (b^2*(48*a^2 + 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(9/2)*(a + b)^(5/2)*f) + ((a - 3*b)*Sin[e + f*x])/(a^4*f) - Sin[e + f*x]^3/(3*a^3*f) + (b^4*Sin[e + f*x])/(4*a^4*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) - (b^3*(16*a + 13*b)*Sin[e + f*x])/(8*a^4*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 6, -(b^3*(80*a^2 + 140*a*b + 63*b^2)*ArcTanh[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]])/(8*a^(11/2)*(a + b)^(5/2)*f) + ((a^2 - 3*a*b + 6*b^2)*Sin[e + f*x])/(a^5*f) - ((2*a - 3*b)*Sin[e + f*x]^3)/(3*a^4*f) + Sin[e + f*x]^5/(5*a^3*f) - (b^5*Sin[e + f*x])/(4*a^5*(a + b)*f*(a + b - a*Sin[e + f*x]^2)^2) + (b^4*(20*a + 17*b)*Sin[e + f*x])/(8*a^5*(a + b)^2*f*(a + b - a*Sin[e + f*x]^2))} - -{Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 4, ((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*b^(5/2)*(a + b)^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(4*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*a*(a + 2*b)*Tan[e + f*x])/(8*b^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 4, ((a + 4*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(5/2)*f) - (a*Tan[e + f*x])/(4*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + ((a + 4*b)*Tan[e + f*x])/(8*b*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*Sqrt[b]*(a + b)^(5/2)*f) + Tan[e + f*x]/(4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*Tan[e + f*x])/(8*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Sec[e + f*x]^0/(a + b*Sec[e + f*x]^2)^3, x, 6, x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 7, ((a - 6*b)*x)/(2*a^4) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(5/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(2*a + 3*b)*Tan[e + f*x])/(4*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(4*a + 3*b)*(a + 4*b)*Tan[e + f*x])/(8*a^3*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 8, (3*(a^2 - 4*a*b + 16*b^2)*x)/(8*a^5) - (3*b^(5/2)*(21*a^2 + 36*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^5*(a + b)^(5/2)*f) + ((3*a - 8*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(3*a^2 - 7*a*b - 12*b^2)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (3*b*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Tan[e + f*x])/(8*a^4*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 9, ((5*a^3 - 18*a^2*b + 48*a*b^2 - 160*b^3)*x)/(16*a^6) + (b^(7/2)*(99*a^2 + 176*a*b + 80*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^6*(a + b)^(5/2)*f) + ((15*a^2 - 34*a*b + 80*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*(a + b + b*Tan[e + f*x]^2)^2) + (5*(a - 2*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(15*a^3 - 29*a^2*b + 64*a*b^2 + 120*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) + (b*(5*a^4 - 8*a^3*b + 17*a^2*b^2 + 116*a*b^3 + 80*b^4)*Tan[e + f*x])/(16*a^5*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} - - -{1/(a + b*Sec[c + d*x]^2)^4, x, 7, x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b]])/(16*a^4*(a + b)^(7/2)*d) - (b*Tan[c + d*x])/(6*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tan[c + d*x])/(24*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tan[c + d*x])/(16*a^3*(a + b)^3*d*(a + b + b*Tan[c + d*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^(p/2) when a+b=0*) - - -{(a - a*Sec[c + d*x]^2)^(7/2), x, 6, -((a^3*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[(-a)*Tan[c + d*x]^2])/d) - (a^3*Tan[c + d*x]*Sqrt[(-a)*Tan[c + d*x]^2])/(2*d) + (a^3*Tan[c + d*x]^3*Sqrt[(-a)*Tan[c + d*x]^2])/(4*d) - (a^3*Tan[c + d*x]^5*Sqrt[(-a)*Tan[c + d*x]^2])/(6*d)} -{(a - a*Sec[c + d*x]^2)^(5/2), x, 5, -((a^2*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[(-a)*Tan[c + d*x]^2])/d) - (a^2*Tan[c + d*x]*Sqrt[(-a)*Tan[c + d*x]^2])/(2*d) + (a^2*Tan[c + d*x]^3*Sqrt[(-a)*Tan[c + d*x]^2])/(4*d)} -{(a - a*Sec[c + d*x]^2)^(3/2), x, 4, -((a*Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[(-a)*Tan[c + d*x]^2])/d) - (a*Tan[c + d*x]*Sqrt[(-a)*Tan[c + d*x]^2])/(2*d)} -{(a - a*Sec[c + d*x]^2)^(1/2), x, 3, -((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[(-a)*Tan[c + d*x]^2])/d)} -{1/(a - a*Sec[c + d*x]^2)^(1/2), x, 3, (Log[Sin[c + d*x]]*Tan[c + d*x])/(d*Sqrt[(-a)*Tan[c + d*x]^2])} -{1/(a - a*Sec[c + d*x]^2)^(3/2), x, 4, Cot[c + d*x]/(2*a*d*Sqrt[(-a)*Tan[c + d*x]^2]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[(-a)*Tan[c + d*x]^2])} -{1/(a - a*Sec[c + d*x]^2)^(5/2), x, 5, Cot[c + d*x]/(2*a^2*d*Sqrt[(-a)*Tan[c + d*x]^2]) - Cot[c + d*x]^3/(4*a^2*d*Sqrt[(-a)*Tan[c + d*x]^2]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a^2*d*Sqrt[(-a)*Tan[c + d*x]^2])} -{1/(a - a*Sec[c + d*x]^2)^(7/2), x, 6, Cot[c + d*x]/(2*a^3*d*Sqrt[(-a)*Tan[c + d*x]^2]) - Cot[c + d*x]^3/(4*a^3*d*Sqrt[(-a)*Tan[c + d*x]^2]) + Cot[c + d*x]^5/(6*a^3*d*Sqrt[(-a)*Tan[c + d*x]^2]) + (Log[Sin[c + d*x]]*Tan[c + d*x])/(a^3*d*Sqrt[(-a)*Tan[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sec[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x, 11, -(((2*a^2 - 3*a*b - 8*b^2)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*b^2*f)) + ((2*a^2 - 3*a*b - 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*b^2*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 8*b)*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*b*f*(a + b - a*Sin[e + f*x]^2)) + ((a + 4*b)*Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(15*b*f) + (Sec[e + f*x]^3*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2], x, 10, ((a + 2*b)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*b*f) - ((a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*b*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (2*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*(a + b - a*Sin[e + f*x]^2)) + (Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(3*f)} -{Sec[e + f*x]^1*Sqrt[a + b*Sec[e + f*x]^2], x, 10, (Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/f - (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^1*Sqrt[a + b*Sec[e + f*x]^2], x, 5, (Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])} -{Cos[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2], x, 9, (Cos[e + f*x]^2*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*f) + ((2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*a*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x, 10, (2*(2*a - b)*Cos[e + f*x]^2*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*a*f) + (Cos[e + f*x]^2*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(5*a*f) + ((8*a^2 + 3*a*b - 2*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*a^2*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (2*(2*a - b)*b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^2*f*(a + b - a*Sin[e + f*x]^2))} - -{Sec[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2], x, 6, ((a + b)*(a^2 - 2*a*b + 5*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) + ((a^2 - 2*a*b + 5*b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b^2*f) - ((3*a - 5*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(6*b*f)} -{Sec[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2], x, 5, -((a - 3*b)*(a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) - ((a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*b*f)} -{Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2], x, 4, ((a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Sec[e + f*x]^0*Sqrt[a + b*Sec[e + f*x]^2], x, 6, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f} -{Cos[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2], x, 4, ((a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[a]*f) + (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Cos[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2], x, 5, ((3*a - b)*(a + b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(3/2)*f) + ((3*a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a*f) + (Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*a*f)} -{Cos[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2], x, 7, ((a + b)*(5*a^2 - 2*a*b + b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(5/2)*f) + ((3*a - b)*(5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^2*f) + ((5*a + b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a*f) + (Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)} - - -{Sec[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x, 12, -((2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(35*b^2*f)) + (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(35*b^2*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a + b)*(a^2 - 16*a*b - 16*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(35*b*f*(a + b - a*Sin[e + f*x]^2)) + ((a^2 + 11*a*b + 8*b^2)*Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(35*b*f) + (2*(4*a + 3*b)*Sec[e + f*x]^3*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(35*f) + (b*Sec[e + f*x]^5*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(7*f)} -{Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x, 11, ((3*a^2 + 13*a*b + 8*b^2)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*b*f) - ((3*a^2 + 13*a*b + 8*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*b*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*(9*a + 8*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*f*(a + b - a*Sin[e + f*x]^2)) + (2*(3*a + 2*b)*Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(15*f) + (b*Sec[e + f*x]^3*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(5*f)} -{Sec[e + f*x]^1*(a + b*Sec[e + f*x]^2)^(3/2), x, 10, (2*(2*a + b)*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*f) - (2*(2*a + b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((a + b)*(3*a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*(a + b - a*Sin[e + f*x]^2)) + (b*Sec[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Tan[e + f*x])/(3*f)} -{Cos[e + f*x]^1*(a + b*Sec[e + f*x]^2)^(3/2), x, 9, (b*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/f + ((a - b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x, 9, (a*Cos[e + f*x]^2*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*f) + (2*(a + 2*b)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(3*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*f*(a + b - a*Sin[e + f*x]^2))} -{Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x, 10, -((2*(a - 3*(a + b))*Cos[e + f*x]^2*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*f)) + (a*Cos[e + f*x]^4*Sin[e + f*x]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(5*f) + ((8*a^2 + 13*a*b + 3*b^2)*Sqrt[Cos[e + f*x]^2]*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])/(15*a*f*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b)*(4*a + 3*b)*Sqrt[Cos[e + f*x]^2]*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a*f*(a + b - a*Sin[e + f*x]^2))} - -{Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, ((a + b)^2*(3*a^2 - 10*a*b + 35*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) + ((a + b)*(3*a^2 - 10*a*b + 35*b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 10*a*b + 35*b^2)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(192*b^2*f) - ((3*a - 7*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(48*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(8*b*f)} -{Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2), x, 6, -((a - 5*b)*(a + b)^2*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) - ((a - 5*b)*(a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b*f) - ((a - 5*b)*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*b*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(6*b*f)} -{Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2), x, 5, (3*(a + b)^2*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*(a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)} -{Sec[e + f*x]^0*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (Sqrt[a]*(a + 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (a*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2), x, 5, (3*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(4*f)} -{Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2), x, 6, ((5*a - b)*(a + b)^2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(3/2)*f) + ((5*a - b)*(a + b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*a*f) + ((5*a - b)*Cos[e + f*x]^3*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(3/2))/(24*a*f) + (Cos[e + f*x]^5*Sin[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(5/2))/(6*a*f)} - - -{(a + b*Sec[c + d*x]^2)^(5/2), x, 8, (a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]])/d + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]])/(8*d) + (b*(7*a + 3*b)*Tan[c + d*x]*Sqrt[a + b + b*Tan[c + d*x]^2])/(8*d) + (b*Tan[c + d*x]*(a + b + b*Tan[c + d*x]^2)^(3/2))/(4*d)} - - -{(1 + Sec[x]^2)^(3/2), x, 6, 2*ArcSinh[Tan[x]/Sqrt[2]] + ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]] + (1/2)*Tan[x]*Sqrt[2 + Tan[x]^2]} -{Sqrt[1 + Sec[x]^2], x, 5, ArcSinh[Tan[x]/Sqrt[2]] + ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 10, (2*(a - b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*b^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 2*b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) - (2*(a - b)*Sec[e + f*x]*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(3*b^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (Sec[e + f*x]^3*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(3*b*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Sec[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 7, -((Sqrt[a]*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]], (a + b)/a]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + (Sec[e + f*x]*(a + b - a*Sin[e + f*x]^2)*Tan[e + f*x])/(b*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Sec[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 5, (EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 5, (Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]], (a + b)/a]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(Sqrt[a]*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 9, (Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(3*a*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (2*(a - b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 2*b)*b*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 10, (4*(a - b)*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(15*a^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (Cos[e + f*x]^2*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(5*a*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((8*a^2 - 7*a*b + 8*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(15*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(4*a^2 - 3*a*b + 8*b^2)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} - -{Sec[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 5, ((3*a^2 - 2*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - (3*(a - b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b^2*f) + (Sec[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*b*f)} -{Sec[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 4, -((a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b*f)} -{Sec[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 3, ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)} -{Sec[e + f*x]^0/Sqrt[a + b*Sec[e + f*x]^2], x, 3, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)} -{Cos[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 4, ((a - b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(3/2)*f) + (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*f)} -{Cos[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 6, ((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(5/2)*f) + (3*(a - b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*a^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*a*f)} -{Cos[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 7, ((a - b)*(5*a^2 + 2*a*b + 5*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(7/2)*f) + ((15*a^2 - 14*a*b + 15*b^2)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*a^3*f) + (5*(a - b)*Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*a^2*f) + (Cos[e + f*x]^5*Sin[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*a*f)} - - -{Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 10, (a*(2*a + b)*Sin[e + f*x])/(b^2*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) - ((2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(b^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(b*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (Sec[e + f*x]*Tan[e + f*x])/(b*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 7, -((a*Sin[e + f*x])/(b*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + (EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(b*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])} -{Sec[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 9, Sin[e + f*x]/((a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) - (EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(a*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 9, -((b*Sin[e + f*x])/(a*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + ((a + 2*b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(a^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (2*b*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 10, -((b*Cos[e + f*x]^2*Sin[e + f*x])/(a*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + ((a + 4*b)*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(3*a^2*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((2*a^2 - 3*a*b - 8*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a^3*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - ((a - 8*b)*b*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^3*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 11, -((b*Cos[e + f*x]^4*Sin[e + f*x])/(a*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + ((4*a^2 - 5*a*b - 24*b^2)*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(15*a^3*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((a + 6*b)*Cos[e + f*x]^2*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(5*a^2*(a + b)*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((8*a^3 - 9*a^2*b + 16*a*b^2 + 48*b^3)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(15*a^4*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (4*b*(a^2 - 2*a*b + 12*b^2)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^4*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} - -{Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, -((3*a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(b*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b^2*(a + b)*f)} -{Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - (a*Tan[e + f*x])/(b*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 2, Tan[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sec[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 6, ((a - 3*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(5/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (b*(a + 3*b)*Tan[e + f*x])/(2*a^2*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (3*(a^2 - 2*a*b + 5*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(7/2)*f) + ((3*a - 5*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((a - 3*b)*b*(3*a + 5*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 8, ((5*a^3 - 9*a^2*b + 15*a*b^2 - 35*b^3)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(9/2)*f) + ((15*a^2 - 22*a*b + 35*b^2)*Cos[e + f*x]*Sin[e + f*x])/(48*a^3*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((5*a - 7*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + (b*(15*a^3 - 17*a^2*b + 25*a*b^2 + 105*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} - - -{Sec[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, -((2*a*(a + 2*b)*Sin[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) - (a*Sin[e + f*x])/(3*b*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (2*(a + 2*b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*b^2*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*b*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Sec[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, -(((a - b)*Sin[e + f*x])/(3*b*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) + Sin[e + f*x]/(3*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((a - b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a*b*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + (EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Sec[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, (2*(2*a + b)*Sin[e + f*x])/(3*a*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) - (b*Sin[e + f*x])/(3*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) - (2*(2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a^2*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) + ((3*a + 2*b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^2*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, -((2*b*(3*a + 2*b)*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) - (b*Cos[e + f*x]^2*Sin[e + f*x])/(3*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((3*a^2 + 13*a*b + 8*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a^3*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(9*a + 8*b)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^3*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 11, -((2*b*(4*a + 3*b)*Cos[e + f*x]^2*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) - (b*Cos[e + f*x]^4*Sin[e + f*x])/(3*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((a^2 + 11*a*b + 8*b^2)*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(3*a^3*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (2*(a + 2*b)*(a^2 - 4*a*b - 4*b^2)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(3*a^4*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a^2 - 16*a*b - 16*b^2)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(3*a^4*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} -{Cos[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 12, -((2*b*(5*a + 4*b)*Cos[e + f*x]^4*Sin[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])) - (b*Cos[e + f*x]^6*Sin[e + f*x])/(3*a*(a + b)*f*(a + b - a*Sin[e + f*x]^2)*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + (2*(2*a^3 - 3*a^2*b - 42*a*b^2 - 32*b^3)*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(15*a^4*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((3*a^2 + 61*a*b + 48*b^2)*Cos[e + f*x]^2*Sin[e + f*x]*(a + b - a*Sin[e + f*x]^2))/(15*a^3*(a + b)^2*f*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]) + ((8*a^4 - 11*a^3*b + 27*a^2*b^2 + 184*a*b^3 + 128*b^4)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*(a + b - a*Sin[e + f*x]^2))/(15*a^5*(a + b)^2*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(4*a^3 - 9*a^2*b + 120*a*b^2 + 128*b^3)*EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(15*a^5*(a + b)*f*Sqrt[Cos[e + f*x]^2]*Sqrt[Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2)])} - -{Sec[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 5, ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - (a*Sec[e + f*x]^2*Tan[e + f*x])/(3*b*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (a*(3*a + 5*b)*Tan[e + f*x])/(3*b^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sec[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 3, (Sec[e + f*x]^2*Tan[e + f*x])/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (2*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 3, Tan[e + f*x]/(3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (2*Tan[e + f*x])/(3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Sec[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 7, ((a - 5*b)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*a^(7/2)*f) + (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(3*a + 5*b)*Tan[e + f*x])/(6*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(3*a^2 + 22*a*b + 15*b^2)*Tan[e + f*x])/(6*a^3*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 8, ((3*a^2 - 10*a*b + 35*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*a^(9/2)*f) + ((3*a - 7*b)*Cos[e + f*x]*Sin[e + f*x])/(8*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(9*a^2 - 18*a*b - 35*b^2)*Tan[e + f*x])/(24*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(9*a^3 - 15*a^2*b - 145*a*b^2 - 105*b^3)*Tan[e + f*x])/(24*a^4*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cos[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 9, (5*(a - 3*b)*(a^2 + 7*b^2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*a^(11/2)*f) + ((5*a^2 - 10*a*b + 21*b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*a^3*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((5*a - 9*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*a^2*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^5*Sin[e + f*x])/(6*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^3 - 25*a^2*b + 49*a*b^2 + 105*b^3)*Tan[e + f*x])/(48*a^4*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + (b*(15*a^4 - 20*a^3*b + 38*a^2*b^2 + 420*a*b^3 + 315*b^4)*Tan[e + f*x])/(48*a^5*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} - - -{1/(a + b*Sec[c + d*x]^2)^(7/2), x, 7, ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + b + b*Tan[c + d*x]^2]]/(a^(7/2)*d) - (b*Tan[c + d*x])/(5*a*(a + b)*d*(a + b + b*Tan[c + d*x]^2)^(5/2)) - (b*(9*a + 5*b)*Tan[c + d*x])/(15*a^2*(a + b)^2*d*(a + b + b*Tan[c + d*x]^2)^(3/2)) - (b*(33*a^2 + 40*a*b + 15*b^2)*Tan[c + d*x])/(15*a^3*(a + b)^3*d*Sqrt[a + b + b*Tan[c + d*x]^2])} - - -{1/Sqrt[1 + Sec[x]^2], x, 3, ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic*) - - -{(d*Sec[e + f*x])^m*(a + b*Sec[e + f*x]^2)^p, x, -1, (AppellF1[m/2, 1/2, -p, (2 + m)/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/a)]*Cos[e + f*x]*(d*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2]*(a + b*Sec[e + f*x]^2)^p)/((1 + (b*Sec[e + f*x]^2)/a)^p*(f*m*Sin[e + f*x]))} - - -{Sec[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p, x, 5, (AppellF1[1/2, 2 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*Sin[e + f*x]*(Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2))^p)/((1 - (a*Sin[e + f*x]^2)/(a + b))^p*f)} -{Sec[e + f*x]^1*(a + b*Sec[e + f*x]^2)^p, x, 5, (AppellF1[1/2, 1 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*Sin[e + f*x]*(Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2))^p)/((1 - (a*Sin[e + f*x]^2)/(a + b))^p*f)} -{Cos[e + f*x]^1*(a + b*Sec[e + f*x]^2)^p, x, 5, (AppellF1[1/2, p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*Sin[e + f*x]*(Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2))^p)/((1 - (a*Sin[e + f*x]^2)/(a + b))^p*f)} -{Cos[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p, x, 5, (AppellF1[1/2, -1 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*Sin[e + f*x]*(Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2))^p)/((1 - (a*Sin[e + f*x]^2)/(a + b))^p*f)} -{Cos[e + f*x]^5*(a + b*Sec[e + f*x]^2)^p, x, 5, (AppellF1[1/2, -2 + p, -p, 3/2, Sin[e + f*x]^2, (a*Sin[e + f*x]^2)/(a + b)]*(Cos[e + f*x]^2)^p*Sin[e + f*x]*(Sec[e + f*x]^2*(a + b - a*Sin[e + f*x]^2))^p)/((1 - (a*Sin[e + f*x]^2)/(a + b))^p*f)} - -{Sec[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p, x, 5, If[$VersionNumber>=8, -(((3*a - 2*b*(2 + p))*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p), -(((3*a - 2*b*(2 + p))*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b^2*f*(15 + 16*p + 4*p^2))) + (Sec[e + f*x]^2*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 - 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/((1 + (b*Tan[e + f*x]^2)/(a + b))^p*(b^2*f*(15 + 16*p + 4*p^2)))]} -{Sec[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p, x, 4, (Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p)) - ((a - 2*b*(1 + p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(b*f*(3 + 2*p)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Sec[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p, x, 3, (Hypergeometric2F1[1/2, -p, 3/2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Sec[e + f*x]^0*(a + b*Sec[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Cos[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Cos[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 3, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Cos[e + f*x]^6*(a + b*Sec[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 4, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^5, x, 4, -((a*Log[Cos[e + f*x]])/f) - ((2*a - b)*Sec[e + f*x]^2)/(2*f) + ((a - 2*b)*Sec[e + f*x]^4)/(4*f) + (b*Sec[e + f*x]^6)/(6*f)} -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^3, x, 4, (a*Log[Cos[e + f*x]])/f + ((a - b)*Sec[e + f*x]^2)/(2*f) + (b*Sec[e + f*x]^4)/(4*f)} -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^1, x, 3, -((a*Log[Cos[e + f*x]])/f) + (b*Sec[e + f*x]^2)/(2*f)} -{Cot[e + f*x]^1*(a + b*Sec[e + f*x]^2), x, 4, -((b*Log[Cos[e + f*x]])/f) + ((a + b)*Log[Sin[e + f*x]])/f} -{Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2), x, 4, -((a + b)*Csc[e + f*x]^2)/(2*f) - (a*Log[Sin[e + f*x]])/f} -{Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2), x, 4, ((2*a + b)*Csc[e + f*x]^2)/(2*f) - ((a + b)*Csc[e + f*x]^4)/(4*f) + (a*Log[Sin[e + f*x]])/f} - -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^6, x, 4, -(a*x) + (a*Tan[e + f*x])/f - (a*Tan[e + f*x]^3)/(3*f) + (a*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)} -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^4, x, 4, a*x - (a*Tan[e + f*x])/f + (a*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)} -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^2, x, 4, -(a*x) + (a*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)} -{(a + b*Sec[e + f*x]^2)*Tan[e + f*x]^0, x, 3, a*x + (b*Tan[e + f*x])/f} -{Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2), x, 4, -(a*x) - ((a + b)*Cot[e + f*x])/f} -{Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2), x, 4, a*x + (a*Cot[e + f*x])/f - ((a + b)*Cot[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2), x, 4, -(a*x) - (a*Cot[e + f*x])/f + (a*Cot[e + f*x]^3)/(3*f) - ((a + b)*Cot[e + f*x]^5)/(5*f)} - - -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^5, x, 4, -((a^2*Log[Cos[e + f*x]])/f) - (a*(a - b)*Sec[e + f*x]^2)/f + ((a^2 - 4*a*b + b^2)*Sec[e + f*x]^4)/(4*f) + ((a - b)*b*Sec[e + f*x]^6)/(3*f) + (b^2*Sec[e + f*x]^8)/(8*f)} -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^3, x, 4, (a^2*Log[Cos[e + f*x]])/f + (a*(a - 2*b)*Sec[e + f*x]^2)/(2*f) + ((2*a - b)*b*Sec[e + f*x]^4)/(4*f) + (b^2*Sec[e + f*x]^6)/(6*f)} -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^1, x, 4, -((a^2*Log[Cos[e + f*x]])/f) + (a*b*Sec[e + f*x]^2)/f + (b^2*Sec[e + f*x]^4)/(4*f)} -{Cot[e + f*x]^1*(a + b*Sec[e + f*x]^2)^2, x, 4, -((b*(2*a + b)*Log[Cos[e + f*x]])/f) + ((a + b)^2*Log[Sin[e + f*x]])/f + (b^2*Sec[e + f*x]^2)/(2*f)} -{Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2, x, 4, -((a + b)^2*Csc[e + f*x]^2)/(2*f) - (b^2*Log[Cos[e + f*x]])/f - ((a^2 - b^2)*Log[Sin[e + f*x]])/f} -{Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^2, x, 4, (a*(a + b)*Csc[e + f*x]^2)/f - ((a + b)^2*Csc[e + f*x]^4)/(4*f) + (a^2*Log[Sin[e + f*x]])/f} - -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^6, x, 4, -(a^2*x) + (a^2*Tan[e + f*x])/f - (a^2*Tan[e + f*x]^3)/(3*f) + (a^2*Tan[e + f*x]^5)/(5*f) + (b*(2*a + b)*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)} -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^4, x, 4, a^2*x - (a^2*Tan[e + f*x])/f + (a^2*Tan[e + f*x]^3)/(3*f) + (b*(2*a + b)*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)} -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^2, x, 4, -(a^2*x) + (a^2*Tan[e + f*x])/f + (b*(2*a + b)*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)} -{(a + b*Sec[e + f*x]^2)^2*Tan[e + f*x]^0, x, 4, a^2*x + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^2, x, 4, -(a^2*x) - ((a + b)^2*Cot[e + f*x])/f + (b^2*Tan[e + f*x])/f} -{Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^2, x, 4, a^2*x + ((a^2 - b^2)*Cot[e + f*x])/f - ((a + b)^2*Cot[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^2, x, 4, -(a^2*x) - (a^2*Cot[e + f*x])/f + ((a^2 - b^2)*Cot[e + f*x]^3)/(3*f) - ((a + b)^2*Cot[e + f*x]^5)/(5*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 4, ((a + 2*b)*Log[Cos[e + f*x]])/(b^2*f) - ((a + b)^2*Log[b + a*Cos[e + f*x]^2])/(2*a*b^2*f) + Sec[e + f*x]^2/(2*b*f)} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 4, -(Log[Cos[e + f*x]]/(b*f)) + ((a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a*b*f)} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 2, -Log[b + a*Cos[e + f*x]^2]/(2*a*f)} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^2), x, 4, (b*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)*f) + Log[Sin[e + f*x]]/((a + b)*f)} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2), x, 4, -Csc[e + f*x]^2/(2*(a + b)*f) - (b^2*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)^2*f) - ((a + 2*b)*Log[Sin[e + f*x]])/((a + b)^2*f)} -{Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2), x, 4, ((2*a + 3*b)*Csc[e + f*x]^2)/(2*(a + b)^2*f) - Csc[e + f*x]^4/(4*(a + b)*f) + (b^3*Log[b + a*Cos[e + f*x]^2])/(2*a*(a + b)^3*f) + ((a^2 + 3*a*b + 3*b^2)*Log[Sin[e + f*x]])/((a + b)^3*f)} - -{Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 7, -(x/a) + ((a + b)^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*b^(5/2)*f) - ((a + 2*b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)} -{Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 6, x/a - ((a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*b^(3/2)*f) + Tan[e + f*x]/(b*f)} -{Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 5, -(x/a) + (Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*Sqrt[b]*f)} -{Tan[e + f*x]^0/(a + b*Sec[e + f*x]^2), x, 3, x/a + (Sqrt[b]*ArcTan[(Sqrt[a + b]*Cot[e + f*x])/Sqrt[b]])/(a*Sqrt[a + b]*f)} -{Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2), x, 6, -(x/a) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(3/2)*f) - Cot[e + f*x]/((a + b)*f)} -{Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2), x, 7, x/a - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(5/2)*f) + ((a + 2*b)*Cot[e + f*x])/((a + b)^2*f) - Cot[e + f*x]^3/(3*(a + b)*f)} -{Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2), x, 8, -(x/a) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a*(a + b)^(7/2)*f) - ((a^2 + 3*a*b + 3*b^2)*Cot[e + f*x])/((a + b)^3*f) + ((a + 2*b)*Cot[e + f*x]^3)/(3*(a + b)^2*f) - Cot[e + f*x]^5/(5*(a + b)*f)} - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 4, -(a + b)^2/(2*a^2*b*f*(b + a*Cos[e + f*x]^2)) - Log[Cos[e + f*x]]/(b^2*f) - ((a^(-2) - b^(-2))*Log[b + a*Cos[e + f*x]^2])/(2*f)} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 4, (a + b)/(2*a^2*f*(b + a*Cos[e + f*x]^2)) + Log[b + a*Cos[e + f*x]^2]/(2*a^2*f)} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 4, -b/(2*a^2*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^2*f)} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^2)^2, x, 4, b^2/(2*a^2*(a + b)*f*(b + a*Cos[e + f*x]^2)) + (b*(2*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^2*f) + Log[Sin[e + f*x]]/((a + b)^2*f)} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^2, x, 4, -b^3/(2*a^2*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) - Csc[e + f*x]^2/(2*(a + b)^2*f) - (b^2*(3*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^3*f) - ((a + 3*b)*Log[Sin[e + f*x]])/((a + b)^3*f)} -{Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^2, x, 4, b^4/(2*a^2*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) + ((a + 2*b)*Csc[e + f*x]^2)/((a + b)^3*f) - Csc[e + f*x]^4/(4*(a + b)^2*f) + (b^3*(4*a + b)*Log[b + a*Cos[e + f*x]^2])/(2*a^2*(a + b)^4*f) + ((a^2 + 4*a*b + 6*b^2)*Log[Sin[e + f*x]])/((a + b)^4*f)} - -{Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 7, -(x/a^2) - ((3*a - 2*b)*(a + b)^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*b^(5/2)*f) + ((3*a + b)*Tan[e + f*x])/(2*a*b^2*f) - ((a + b)*Tan[e + f*x]^3)/(2*a*b*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 6, x/a^2 + ((a - 2*b)*Sqrt[a + b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*b^(3/2)*f) - ((a + b)*Tan[e + f*x])/(2*a*b*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 6, -(x/a^2) + ((a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*Sqrt[b]*Sqrt[a + b]*f) + Tan[e + f*x]/(2*a*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^0/(a + b*Sec[e + f*x]^2)^2, x, 5, x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*f) - (b*Tan[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^2, x, 7, -(x/a^2) + (b^(3/2)*(5*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(5/2)*f) - ((2*a - b)*Cot[e + f*x])/(2*a*(a + b)^2*f) - (b*Cot[e + f*x])/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^2, x, 8, x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*f) + ((2*a^2 + 6*a*b - b^2)*Cot[e + f*x])/(2*a*(a + b)^3*f) - ((2*a - 3*b)*Cot[e + f*x]^3)/(6*a*(a + b)^2*f) - (b*Cot[e + f*x]^3)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^2, x, 9, -(x/a^2) + (b^(7/2)*(9*a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(9/2)*f) - ((2*a^3 + 8*a^2*b + 12*a*b^2 - b^3)*Cot[e + f*x])/(2*a*(a + b)^4*f) + ((2*a^2 + 6*a*b - 3*b^2)*Cot[e + f*x]^3)/(6*a*(a + b)^3*f) - ((2*a - 5*b)*Cot[e + f*x]^5)/(10*a*(a + b)^2*f) - (b*Cot[e + f*x]^5)/(2*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 4, (a + b)^2/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) - (a + b)/(a^3*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 4, -(b*(a + b))/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) + (a + 2*b)/(2*a^3*f*(b + a*Cos[e + f*x]^2)) + Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 4, b^2/(4*a^3*f*(b + a*Cos[e + f*x]^2)^2) - b/(a^3*f*(b + a*Cos[e + f*x]^2)) - Log[b + a*Cos[e + f*x]^2]/(2*a^3*f)} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^2)^3, x, 4, -b^3/(4*a^3*(a + b)*f*(b + a*Cos[e + f*x]^2)^2) + (b^2*(3*a + 2*b))/(2*a^3*(a + b)^2*f*(b + a*Cos[e + f*x]^2)) + (b*(3*a^2 + 3*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^3*f) + Log[Sin[e + f*x]]/((a + b)^3*f)} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^3, x, 4, b^4/(4*a^3*(a + b)^2*f*(b + a*Cos[e + f*x]^2)^2) - (b^3*(2*a + b))/(a^3*(a + b)^3*f*(b + a*Cos[e + f*x]^2)) - Csc[e + f*x]^2/(2*(a + b)^3*f) - (b^2*(6*a^2 + 4*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^4*f) - ((a + 4*b)*Log[Sin[e + f*x]])/((a + b)^4*f)} -{Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^3, x, 4, -b^5/(4*a^3*(a + b)^3*f*(b + a*Cos[e + f*x]^2)^2) + (b^4*(5*a + 2*b))/(2*a^3*(a + b)^4*f*(b + a*Cos[e + f*x]^2)) + ((2*a + 5*b)*Csc[e + f*x]^2)/(2*(a + b)^4*f) - Csc[e + f*x]^4/(4*(a + b)^3*f) + (b^3*(10*a^2 + 5*a*b + b^2)*Log[b + a*Cos[e + f*x]^2])/(2*a^3*(a + b)^5*f) + ((a^2 + 5*a*b + 10*b^2)*Log[Sin[e + f*x]])/((a + b)^5*f)} - -{Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 7, -(x/a^3) + (Sqrt[a + b]*(3*a^2 - 4*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(4*a*b*f*(a + b + b*Tan[e + f*x]^2)^2) - ((3*a - 4*b)*(a + b)*Tan[e + f*x])/(8*a^2*b^2*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 7, x/a^3 + ((a^2 - 4*a*b - 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*b^(3/2)*Sqrt[a + b]*f) - ((a + b)*Tan[e + f*x])/(4*a*b*f*(a + b + b*Tan[e + f*x]^2)^2) + ((a - 4*b)*Tan[e + f*x])/(8*a^2*b*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 7, -(x/a^3) + ((3*a^2 + 12*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*Sqrt[b]*(a + b)^(3/2)*f) + Tan[e + f*x]/(4*a*f*(a + b + b*Tan[e + f*x]^2)^2) + ((3*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)*f*(a + b + b*Tan[e + f*x]^2))} -{Tan[e + f*x]^0/(a + b*Sec[e + f*x]^2)^3, x, 6, x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*f) - (b*Tan[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(7*a + 4*b)*Tan[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3, x, 8, -(x/a^3) + (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(7/2)*f) - ((8*a^2 - 11*a*b - 4*b^2)*Cot[e + f*x])/(8*a^2*(a + b)^3*f) - (b*Cot[e + f*x])/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(9*a + 4*b)*Cot[e + f*x])/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3, x, 9, x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*f) + ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Cot[e + f*x])/(8*a^2*(a + b)^4*f) - ((8*a^2 - 39*a*b - 12*b^2)*Cot[e + f*x]^3)/(24*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^3)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a + 4*b)*Cot[e + f*x]^3)/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} -{Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^3, x, 10, -(x/a^3) + (b^(7/2)*(99*a^2 + 44*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(11/2)*f) - ((8*a^4 + 40*a^3*b + 80*a^2*b^2 - 19*a*b^3 - 4*b^4)*Cot[e + f*x])/(8*a^2*(a + b)^5*f) + ((8*a^3 + 32*a^2*b - 51*a*b^2 - 12*b^3)*Cot[e + f*x]^3)/(24*a^2*(a + b)^4*f) - ((8*a^2 - 75*a*b - 20*b^2)*Cot[e + f*x]^5)/(40*a^2*(a + b)^3*f) - (b*Cot[e + f*x]^5)/(4*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(13*a + 4*b)*Cot[e + f*x]^5)/(8*a^2*(a + b)^2*f*(a + b + b*Tan[e + f*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^5, x, 7, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sec[e + f*x]^2]/f - ((a + 2*b)*(a + b*Sec[e + f*x]^2)^(3/2))/(3*b^2*f) + (a + b*Sec[e + f*x]^2)^(5/2)/(5*b^2*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^3, x, 6, (Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - Sqrt[a + b*Sec[e + f*x]^2]/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*b*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^1, x, 5, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sec[e + f*x]^2]/f} -{Cot[e + f*x]^1*Sqrt[a + b*Sec[e + f*x]^2], x, 7, (Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/f} -{Cot[e + f*x]^3*Sqrt[a + b*Sec[e + f*x]^2], x, 8, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + ((2*a + b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*Sqrt[a + b]*f) - (Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^5*Sqrt[a + b*Sec[e + f*x]^2], x, 9, (Sqrt[a]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((8*a^2 + 12*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(3/2)*f) + ((4*a + 3*b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)*f) - (Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)} - -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^6, x, 10, -((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((a^3 + 5*a^2*b + 15*a*b^2 - 5*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) - ((a - b)*(a + 5*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b^2*f) + ((a - 5*b)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*b*f) + (Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^4, x, 9, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a^2 + 6*a*b - 3*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 3*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^2, x, 8, -((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((a - b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x]^0, x, 6, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f} -{Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2], x, 6, -((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f} -{Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2], x, 7, (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + ((3*a + 2*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f)} -{Cot[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2], x, 8, -((Sqrt[a]*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - ((15*a^2 + 25*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - ((b - 5*(a + b))*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*f)} - - -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^5, x, 8, -((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sec[e + f*x]^2])/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*f) - ((a + 2*b)*(a + b*Sec[e + f*x]^2)^(5/2))/(5*b^2*f) + (a + b*Sec[e + f*x]^2)^(7/2)/(7*b^2*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^3, x, 7, (a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - (a*Sqrt[a + b*Sec[e + f*x]^2])/f - (a + b*Sec[e + f*x]^2)^(3/2)/(3*f) + (a + b*Sec[e + f*x]^2)^(5/2)/(5*b*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^1, x, 6, -((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sec[e + f*x]^2])/f + (a + b*Sec[e + f*x]^2)^(3/2)/(3*f)} -{Cot[e + f*x]^1*(a + b*Sec[e + f*x]^2)^(3/2), x, 8, (a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/f + (b*Sqrt[a + b*Sec[e + f*x]^2])/f} -{Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2), x, 8, -((a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f) + ((2*a - b)*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*f) - ((a + b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^5*(a + b*Sec[e + f*x]^2)^(3/2), x, 9, (a^(3/2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/f - ((8*a^2 + 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*Sqrt[a + b]*f) + ((4*a - b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*f) - ((a + b)*Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*f)} - -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^6, x, 11, -((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) - ((3*a^3 + 17*a^2*b - 55*a*b^2 - 5*b^3)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 50*a*b - 5*b^2)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(192*b*f) + ((9*a + b)*Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(48*f) + (b*Tan[e + f*x]^7*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^4, x, 10, (a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a - b)*(a^2 + 10*a*b + b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) + ((a^2 - 8*a*b - b^2)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(16*b*f) + ((7*a + b)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(24*f) + (b*Tan[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(6*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^2, x, 9, -((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((3*a^2 - 6*a*b - b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + ((5*a + b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f) + (b*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*f)} -{(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^0, x, 7, (a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*f)} -{Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(3/2), x, 8, -((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f - ((a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/f} -{Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^(3/2), x, 7, (a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f + ((3*a - b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f) - ((a + b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*f)} -{Cot[e + f*x]^6*(a + b*Sec[e + f*x]^2)^(3/2), x, 8, -((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) - ((15*a^2 + 10*a*b - 2*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)*f) + ((5*a - b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*f) - ((a + b)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 6, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) - ((a + 2*b)*Sqrt[a + b*Sec[e + f*x]^2])/(b^2*f) + (a + b*Sec[e + f*x]^2)^(3/2)/(3*b^2*f)} -{Tan[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 5, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) + Sqrt[a + b*Sec[e + f*x]^2]/(b*f)} -{Tan[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 4, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))} -{Cot[e + f*x]^1/Sqrt[a + b*Sec[e + f*x]^2], x, 7, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/(Sqrt[a + b]*f)} -{Cot[e + f*x]^3/Sqrt[a + b*Sec[e + f*x]^2], x, 8, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(3/2)*f) - (Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(2*(a + b)*f)} -{Cot[e + f*x]^5/Sqrt[a + b*Sec[e + f*x]^2], x, 9, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f) - ((8*a^2 + 20*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(5/2)*f) + ((4*a + 7*b)*Cot[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]^2])/(8*(a + b)^2*f) - (Cot[e + f*x]^4*Sqrt[a + b*Sec[e + f*x]^2])/(4*(a + b)*f)} - -{Tan[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 9, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) + ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - ((3*a + 7*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*b^2*f) + (Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(4*b*f)} -{Tan[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 8, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f) - ((a + 3*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*b*f)} -{Tan[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 7, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)} -{Tan[e + f*x]^0/Sqrt[a + b*Sec[e + f*x]^2], x, 3, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)} -{Cot[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]^2], x, 6, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) - (Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/((a + b)*f)} -{Cot[e + f*x]^4/Sqrt[a + b*Sec[e + f*x]^2], x, 7, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f) + ((3*a + 5*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)^2*f) - (Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*(a + b)*f)} -{Cot[e + f*x]^6/Sqrt[a + b*Sec[e + f*x]^2], x, 8, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(Sqrt[a]*f)) - ((15*a^2 + 40*a*b + 33*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^3*f) + ((5*a + 9*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*(a + b)^2*f) - (Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*(a + b)*f)} - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 6, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + (a + b)^2/(a*b^2*f*Sqrt[a + b*Sec[e + f*x]^2]) + Sqrt[a + b*Sec[e + f*x]^2]/(b^2*f)} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - (a + b)/(a*b*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(3/2), x, 8, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(3/2)*f) - b/(a*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(3/2), x, 9, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(5/2)*f) - ((a - 2*b)*b)/(2*a*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cot[e + f*x]^2/(2*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(3/2), x, 10, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f) - ((8*a^2 + 28*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(7/2)*f) + (b*(4*a^2 + 11*a*b - 8*b^2))/(8*a*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2]) + ((4*a + 9*b)*Cot[e + f*x]^2)/(8*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2]) - Cot[e + f*x]^4/(4*(a + b)*f*Sqrt[a + b*Sec[e + f*x]^2])} - -{Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 9, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - ((3*a + 5*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(a*b*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a + 2*b)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(2*a*b^2*f)} -{Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 8, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - ((a + b)*Tan[e + f*x])/(a*b*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 5, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) + Tan[e + f*x]/(a*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Tan[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2), x, 7, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - (b*Cot[e + f*x])/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((a - b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(a*(a + b)^2*f)} -{Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(3/2), x, 8, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f) - (b*Cot[e + f*x]^3)/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((3*a - b)*(a + 3*b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a*(a + b)^3*f) - ((a - 3*b)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a*(a + b)^2*f)} -{Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(3/2), x, 9, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(3/2)*f)) - (b*Cot[e + f*x]^5)/(a*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((15*a^3 + 55*a^2*b + 73*a*b^2 - 15*b^3)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a*(a + b)^4*f) + ((5*a^2 + 14*a*b - 15*b^2)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a*(a + b)^3*f) - ((a - 5*b)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*a*(a + b)^2*f)} - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + (a + b)^2/(3*a*b^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (a^(-2) - b^(-2))/(f*Sqrt[a + b*Sec[e + f*x]^2])} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - (a + b)/(3*a*b*f*(a + b*Sec[e + f*x]^2)^(3/2)) - 1/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + 1/(3*a*f*(a + b*Sec[e + f*x]^2)^(3/2)) + 1/(a^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^2)^(5/2), x, 9, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]]/((a + b)^(5/2)*f) - b/(3*a*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(2*a + b))/(a^2*(a + b)^2*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, -(ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + ((2*a + 7*b)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(2*(a + b)^(7/2)*f) - ((3*a - 2*b)*b)/(6*a*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cot[e + f*x]^2/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) - (b*(a^2 - 6*a*b - 2*b^2))/(2*a^2*(a + b)^3*f*Sqrt[a + b*Sec[e + f*x]^2])} -{Cot[e + f*x]^5/(a + b*Sec[e + f*x]^2)^(5/2), x, 11, ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f) - ((8*a^2 + 36*a*b + 63*b^2)*ArcTanh[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) + (b*(12*a^2 + 39*a*b - 8*b^2))/(24*a*(a + b)^3*f*(a + b*Sec[e + f*x]^2)^(3/2)) + ((4*a + 11*b)*Cot[e + f*x]^2)/(8*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^(3/2)) - Cot[e + f*x]^4/(4*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2)) + (b*(4*a^3 + 15*a^2*b - 32*a*b^2 - 8*b^3))/(8*a^2*(a + b)^4*f*Sqrt[a + b*Sec[e + f*x]^2])} - -{Tan[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 9, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - ((a + b)*Tan[e + f*x]^3)/(3*a*b*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((a^(-2) - b^(-2))*Tan[e + f*x])/(f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 7, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - ((a + b)*Tan[e + f*x])/(3*a*b*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((a - 3*b)*Tan[e + f*x])/(3*a^2*b*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 7, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) + Tan[e + f*x]/(3*a*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) + ((2*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Tan[e + f*x]^0/(a + b*Sec[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(5*a + 3*b)*Tan[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2])} -{Cot[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(5/2), x, 8, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cot[e + f*x])/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(7*a + 3*b)*Cot[e + f*x])/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((a - 3*b)*(3*a + b)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^3*f)} -{Cot[e + f*x]^4/(a + b*Sec[e + f*x]^2)^(5/2), x, 9, ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(3*a + b)*Cot[e + f*x]^3)/(a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) + ((a - b)*(3*a^2 + 14*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^4*f) - ((a^2 - 10*a*b - 3*b^2)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(3*a^2*(a + b)^3*f)} -{Cot[e + f*x]^6/(a + b*Sec[e + f*x]^2)^(5/2), x, 10, -(ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cot[e + f*x]^5)/(3*a*(a + b)*f*(a + b + b*Tan[e + f*x]^2)^(3/2)) - (b*(11*a + 3*b)*Cot[e + f*x]^5)/(3*a^2*(a + b)^2*f*Sqrt[a + b + b*Tan[e + f*x]^2]) - ((15*a^4 + 70*a^3*b + 128*a^2*b^2 - 70*a*b^3 - 15*b^4)*Cot[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a^2*(a + b)^5*f) + ((5*a^3 + 19*a^2*b - 65*a*b^2 - 15*b^3)*Cot[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(15*a^2*(a + b)^4*f) - ((a^2 - 20*a*b - 5*b^2)*Cot[e + f*x]^5*Sqrt[a + b + b*Tan[e + f*x]^2])/(5*a^2*(a + b)^3*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic*) - - -{(a + b*Sec[e + f*x]^2)^p*(d*Tan[e + f*x])^m, x, 4, (AppellF1[(1 + m)/2, 1, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*(d*Tan[e + f*x])^(1 + m)*(a + b + b*Tan[e + f*x]^2)^p)/(d*f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} - - -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^5, x, 5, -((a + 2*b)*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*b^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p)) + (a + b*Sec[e + f*x]^2)^(2 + p)/(2*b^2*f*(2 + p))} -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^3, x, 4, (a + b*Sec[e + f*x]^2)^(1 + p)/(2*b*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))} -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^1, x, 3, -(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))} -{Cot[e + f*x]^1*(a + b*Sec[e + f*x]^2)^p, x, 5, -(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sec[e + f*x]^2)/(a + b)]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))} -{Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^p, x, 6, -((Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)*f)) + ((a + b - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sec[e + f*x]^2)/(a + b)]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*(a + b)^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sec[e + f*x]^2)/a]*(a + b*Sec[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))} - -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^4, x, 4, (AppellF1[5/2, 1, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^5*(a + b + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^2, x, 4, (AppellF1[3/2, 1, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{(a + b*Sec[e + f*x]^2)^p*Tan[e + f*x]^0, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Tan[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} -{Cot[e + f*x]^2*(a + b*Sec[e + f*x]^2)^p, x, 4, -((AppellF1[-1/2, 1, -p, 1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cot[e + f*x]*(a + b + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p))} -{Cot[e + f*x]^4*(a + b*Sec[e + f*x]^2)^p, x, 4, -(AppellF1[-3/2, 1, -p, -1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/(a + b))]*Cot[e + f*x]^3*(a + b + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/(a + b))^p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^3)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tan[e + f*x]^5*(a + b*Sec[e + f*x]^3), x, 3, -((a*Log[Cos[e + f*x]])/f) - (a*Sec[e + f*x]^2)/f + (b*Sec[e + f*x]^3)/(3*f) + (a*Sec[e + f*x]^4)/(4*f) - (2*b*Sec[e + f*x]^5)/(5*f) + (b*Sec[e + f*x]^7)/(7*f)} -{Tan[e + f*x]^3*(a + b*Sec[e + f*x]^3), x, 3, (a*Log[Cos[e + f*x]])/f + (a*Sec[e + f*x]^2)/(2*f) - (b*Sec[e + f*x]^3)/(3*f) + (b*Sec[e + f*x]^5)/(5*f)} -{Tan[e + f*x]^1*(a + b*Sec[e + f*x]^3), x, 3, -((a*Log[Cos[e + f*x]])/f) + (b*Sec[e + f*x]^3)/(3*f)} -{Cot[e + f*x]^1*(a + b*Sec[e + f*x]^3), x, 3, ((a + b)*Log[1 - Cos[e + f*x]])/(2*f) + ((a - b)*Log[1 + Cos[e + f*x]])/(2*f) + (b*Sec[e + f*x])/f} -{Cot[e + f*x]^3*(a + b*Sec[e + f*x]^3), x, 5, -(((a + b*Cos[e + f*x])*Csc[e + f*x]^2)/(2*f)) - ((2*a - b)*Log[1 - Cos[e + f*x]])/(4*f) - ((2*a + b)*Log[1 + Cos[e + f*x]])/(4*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tan[e + f*x]^5/(a + b*Sec[e + f*x]^3), x, 11, -(((a^(2/3) + 2*b^(2/3))*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*b^(4/3)*f)) - ((a^(2/3) - 2*b^(2/3))*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*b^(4/3)*f) + ((a^(2/3) - 2*b^(2/3))*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*b^(4/3)*f) - Log[b + a*Cos[e + f*x]^3]/(3*a*f) + Sec[e + f*x]/(b*f)} -{Tan[e + f*x]^3/(a + b*Sec[e + f*x]^3), x, 9, ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*a^(1/3)*b^(2/3)*f) - Log[b^(1/3) + a^(1/3)*Cos[e + f*x]]/(3*a^(1/3)*b^(2/3)*f) + Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2]/(6*a^(1/3)*b^(2/3)*f) + Log[b + a*Cos[e + f*x]^3]/(3*a*f)} -{Tan[e + f*x]^1/(a + b*Sec[e + f*x]^3), x, 2, -(Log[b + a*Cos[e + f*x]^3]/(3*a*f))} -{Cot[e + f*x]^1/(a + b*Sec[e + f*x]^3), x, 11, -((b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*(a^(4/3) + a^(2/3)*b^(2/3) + b^(4/3))*f)) + Log[1 - Cos[e + f*x]]/(2*(a + b)*f) + Log[1 + Cos[e + f*x]]/(2*(a - b)*f) - ((a^(2/3) + b^(2/3))*b^(2/3)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)*f) + ((a^(2/3) + b^(2/3))*b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*(a^2 - b^2)*f) - (b^2*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)*f)} -{Cot[e + f*x]^3/(a + b*Sec[e + f*x]^3), x, 11, (b^(4/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(b^(1/3) - 2*a^(1/3)*Cos[e + f*x])/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2*f) - 1/(4*(a + b)*f*(1 - Cos[e + f*x])) - 1/(4*(a - b)*f*(1 + Cos[e + f*x])) - ((2*a + 5*b)*Log[1 - Cos[e + f*x]])/(4*(a + b)^2*f) - ((2*a - 5*b)*Log[1 + Cos[e + f*x]])/(4*(a - b)^2*f) - (b^(4/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(1/3) + a^(1/3)*Cos[e + f*x]])/(3*a^(1/3)*(a^2 - b^2)^2*f) + (b^(4/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[b^(2/3) - a^(1/3)*b^(1/3)*Cos[e + f*x] + a^(2/3)*Cos[e + f*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2*f) - (b^2*(2*a^2 + b^2)*Log[b + a*Cos[e + f*x]^3])/(3*a*(a^2 - b^2)^2*f)} - - -(* ::Section:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^n)^p when p symbolic*) - - -{(d*Tan[e + f*x])^m*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p*(d*Tan[e + f*x])^m, x]} - - -{Tan[e + f*x]^5*(a + b*(c*Sec[e + f*x])^n)^p, x, 15, -((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p))) - (Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p)/((1 + (b*(c*Sec[e + f*x])^n)/a)^p*f) + (Hypergeometric2F1[4/n, -p, (4 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^4*(a + b*(c*Sec[e + f*x])^n)^p)/((1 + (b*(c*Sec[e + f*x])^n)/a)^p*(4*f))} -{Tan[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p, x, 11, (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p)) + (Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*Sec[e + f*x])^n)/a)]*Sec[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p)/((1 + (b*(c*Sec[e + f*x])^n)/a)^p*(2*f))} -{Tan[e + f*x]^1*(a + b*(c*Sec[e + f*x])^n)^p, x, 5, -((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*(c*Sec[e + f*x])^n)/a]*(a + b*(c*Sec[e + f*x])^n)^(1 + p))/(a*f*n*(1 + p)))} -{Cot[e + f*x]^1*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[Cot[e + f*x]*(a + b*(c*Sec[e + f*x])^n)^p, x]} -{Cot[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[Cot[e + f*x]^3*(a + b*(c*Sec[e + f*x])^n)^p, x]} - -{Tan[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p*Tan[e + f*x]^2, x]} -{Tan[e + f*x]^0*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[(a + b*(c*Sec[e + f*x])^n)^p, x]} -{Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x, 0, Unintegrable[Cot[e + f*x]^2*(a + b*(c*Sec[e + f*x])^n)^p, x]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.0 (a csc)^m (b trg)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.0 (a csc)^m (b trg)^n.m deleted file mode 100644 index 39f2571..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.0 (a csc)^m (b trg)^n.m +++ /dev/null @@ -1,150 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (b Csc[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[c+d x]^n*) - - -{Csc[a + b*x]^1, x, 1, -(ArcTanh[Cos[a + b*x]]/b)} -{Csc[a + b*x]^2, x, 2, -(Cot[a + b*x]/b)} -{Csc[a + b*x]^3, x, 2, -(ArcTanh[Cos[a + b*x]]/(2*b)) - (Cot[a + b*x]*Csc[a + b*x])/(2*b)} -{Csc[a + b*x]^4, x, 2, -(Cot[a + b*x]/b) - Cot[a + b*x]^3/(3*b)} -{Csc[a + b*x]^5, x, 3, -((3*ArcTanh[Cos[a + b*x]])/(8*b)) - (3*Cot[a + b*x]*Csc[a + b*x])/(8*b) - (Cot[a + b*x]*Csc[a + b*x]^3)/(4*b)} -{Csc[a + b*x]^6, x, 2, -(Cot[a + b*x]/b) - (2*Cot[a + b*x]^3)/(3*b) - Cot[a + b*x]^5/(5*b)} -{Csc[a + b*x]^7, x, 4, -((5*ArcTanh[Cos[a + b*x]])/(16*b)) - (5*Cot[a + b*x]*Csc[a + b*x])/(16*b) - (5*Cot[a + b*x]*Csc[a + b*x]^3)/(24*b) - (Cot[a + b*x]*Csc[a + b*x]^5)/(6*b)} -{Csc[a + b*x]^8, x, 2, -(Cot[a + b*x]/b) - Cot[a + b*x]^3/b - (3*Cot[a + b*x]^5)/(5*b) - Cot[a + b*x]^7/(7*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Csc[c+d x])^(n/2)*) - - -{Csc[a + b*x]^(7/2), x, 4, -((6*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(5*b)) - (2*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(5*b) - (6*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(5*b)} -{Csc[a + b*x]^(5/2), x, 3, -((2*Cos[a + b*x]*Csc[a + b*x]^(3/2))/(3*b)) + (2*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b)} -{Csc[a + b*x]^(3/2), x, 3, -((2*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/b) - (2*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} -{Csc[a + b*x]^(1/2), x, 2, (2*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} -{1/Csc[a + b*x]^(1/2), x, 2, (2*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} -{1/Csc[a + b*x]^(3/2), x, 3, -((2*Cos[a + b*x])/(3*b*Sqrt[Csc[a + b*x]])) + (2*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b)} -{1/Csc[a + b*x]^(5/2), x, 3, -((2*Cos[a + b*x])/(5*b*Csc[a + b*x]^(3/2))) + (6*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(5*b)} -{1/Csc[a + b*x]^(7/2), x, 4, -((2*Cos[a + b*x])/(7*b*Csc[a + b*x]^(5/2))) - (10*Cos[a + b*x])/(21*b*Sqrt[Csc[a + b*x]]) + (10*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(21*b)} - - -{(c*Csc[a + b*x])^(7/2), x, 4, -((6*c^3*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/(5*b)) - (2*c*Cos[a + b*x]*(c*Csc[a + b*x])^(5/2))/(5*b) - (6*c^4*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(5*b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])} -{(c*Csc[a + b*x])^(5/2), x, 3, -((2*c*Cos[a + b*x]*(c*Csc[a + b*x])^(3/2))/(3*b)) + (2*c^2*Sqrt[c*Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b)} -{(c*Csc[a + b*x])^(3/2), x, 3, -((2*c*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/b) - (2*c^2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])} -{(c*Csc[a + b*x])^(1/2), x, 2, (2*Sqrt[c*Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b} -{1/(c*Csc[a + b*x])^(1/2), x, 2, (2*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])} -{1/(c*Csc[a + b*x])^(3/2), x, 3, -((2*Cos[a + b*x])/(3*b*c*Sqrt[c*Csc[a + b*x]])) + (2*Sqrt[c*Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b*c^2)} -{1/(c*Csc[a + b*x])^(5/2), x, 3, -((2*Cos[a + b*x])/(5*b*c*(c*Csc[a + b*x])^(3/2))) + (6*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(5*b*c^2*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])} -{1/(c*Csc[a + b*x])^(7/2), x, 4, -((2*Cos[a + b*x])/(7*b*c*(c*Csc[a + b*x])^(5/2))) - (10*Cos[a + b*x])/(21*b*c^3*Sqrt[c*Csc[a + b*x]]) + (10*Sqrt[c*Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(21*b*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Csc[c+d x])^(n/3)*) - - -{Csc[a + b*x]^(4/3), x, 2, -((3*Cos[a + b*x]*Csc[a + b*x]^(1/3)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Sin[a + b*x]^2])/(b*Sqrt[Cos[a + b*x]^2]))} -{Csc[a + b*x]^(2/3), x, 2, (3*Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2])/(b*Sqrt[Cos[a + b*x]^2]*Csc[a + b*x]^(1/3))} -{Csc[a + b*x]^(1/3), x, 2, (3*Cos[a + b*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[a + b*x]^2])/(2*b*Sqrt[Cos[a + b*x]^2]*Csc[a + b*x]^(2/3))} -{1/Csc[a + b*x]^(1/3), x, 2, (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2])/(4*b*Sqrt[Cos[a + b*x]^2]*Csc[a + b*x]^(4/3))} -{1/Csc[a + b*x]^(2/3), x, 2, (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[a + b*x]^2])/(5*b*Sqrt[Cos[a + b*x]^2]*Csc[a + b*x]^(5/3))} -{1/Csc[a + b*x]^(4/3), x, 2, (3*Cos[a + b*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2])/(7*b*Sqrt[Cos[a + b*x]^2]*Csc[a + b*x]^(7/3))} - - -{(c*Csc[a + b*x])^(4/3), x, 2, -((3*c*Cos[a + b*x]*(c*Csc[a + b*x])^(1/3)*Hypergeometric2F1[-(1/6), 1/2, 5/6, Sin[a + b*x]^2])/(b*Sqrt[Cos[a + b*x]^2]))} -{(c*Csc[a + b*x])^(2/3), x, 2, (3*c*Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2])/(b*Sqrt[Cos[a + b*x]^2]*(c*Csc[a + b*x])^(1/3))} -{(c*Csc[a + b*x])^(1/3), x, 2, (3*c*Cos[a + b*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[a + b*x]^2])/(2*b*Sqrt[Cos[a + b*x]^2]*(c*Csc[a + b*x])^(2/3))} -{1/(c*Csc[a + b*x])^(1/3), x, 2, (3*c*Cos[a + b*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2])/(4*b*Sqrt[Cos[a + b*x]^2]*(c*Csc[a + b*x])^(4/3))} -{1/(c*Csc[a + b*x])^(2/3), x, 2, (3*c*Cos[a + b*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[a + b*x]^2])/(5*b*Sqrt[Cos[a + b*x]^2]*(c*Csc[a + b*x])^(5/3))} -{1/(c*Csc[a + b*x])^(4/3), x, 2, (3*c*Cos[a + b*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2])/(7*b*Sqrt[Cos[a + b*x]^2]*(c*Csc[a + b*x])^(7/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Csc[c+d x])^n with n symbolic*) - - -{Csc[a + b*x]^n, x, 2, (Cos[a + b*x]*Csc[a + b*x]^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[a + b*x]^2])/(b*(1 - n)*Sqrt[Cos[a + b*x]^2])} - - -{(c*Csc[a + b*x])^n, x, 2, (c*Cos[a + b*x]*(c*Csc[a + b*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Sin[a + b*x]^2])/(b*(1 - n)*Sqrt[Cos[a + b*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Csc[c+d x]^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Csc[c+d x]^p)^(n/2) with p positive integer*) - - -{(Csc[x]^2)^(7/2), x, 5, (-(5/16))*ArcSinh[Cot[x]] - (5/16)*Cot[x]*Sqrt[Csc[x]^2] - (5/24)*Cot[x]*(Csc[x]^2)^(3/2) - (1/6)*Cot[x]*(Csc[x]^2)^(5/2)} -{(Csc[x]^2)^(5/2), x, 4, (-(3/8))*ArcSinh[Cot[x]] - (3/8)*Cot[x]*Sqrt[Csc[x]^2] - (1/4)*Cot[x]*(Csc[x]^2)^(3/2)} -{(Csc[x]^2)^(3/2), x, 3, (-(1/2))*ArcSinh[Cot[x]] - (1/2)*Cot[x]*Sqrt[Csc[x]^2]} -{(Csc[x]^2)^(1/2), x, 2, -ArcSinh[Cot[x]]} -{1/(Csc[x]^2)^(1/2), x, 2, -(Cot[x]/Sqrt[Csc[x]^2])} -{1/(Csc[x]^2)^(3/2), x, 3, -(Cot[x]/(3*(Csc[x]^2)^(3/2))) - (2*Cot[x])/(3*Sqrt[Csc[x]^2])} -{1/(Csc[x]^2)^(5/2), x, 4, -(Cot[x]/(5*(Csc[x]^2)^(5/2))) - (4*Cot[x])/(15*(Csc[x]^2)^(3/2)) - (8*Cot[x])/(15*Sqrt[Csc[x]^2])} -{1/(Csc[x]^2)^(7/2), x, 5, -(Cot[x]/(7*(Csc[x]^2)^(7/2))) - (6*Cot[x])/(35*(Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*(Csc[x]^2)^(3/2)) - (16*Cot[x])/(35*Sqrt[Csc[x]^2])} - - -{(a*Csc[x]^2)^(7/2), x, 6, (-(5/16))*a^(7/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]] - (5/16)*a^3*Cot[x]*Sqrt[a*Csc[x]^2] - (5/24)*a^2*Cot[x]*(a*Csc[x]^2)^(3/2) - (1/6)*a*Cot[x]*(a*Csc[x]^2)^(5/2)} -{(a*Csc[x]^2)^(5/2), x, 5, (-(3/8))*a^(5/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]] - (3/8)*a^2*Cot[x]*Sqrt[a*Csc[x]^2] - (1/4)*a*Cot[x]*(a*Csc[x]^2)^(3/2)} -{(a*Csc[x]^2)^(3/2), x, 4, (-(1/2))*a^(3/2)*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]] - (1/2)*a*Cot[x]*Sqrt[a*Csc[x]^2]} -{(a*Csc[x]^2)^(1/2), x, 3, (-Sqrt[a])*ArcTanh[(Sqrt[a]*Cot[x])/Sqrt[a*Csc[x]^2]]} -{1/(a*Csc[x]^2)^(1/2), x, 2, -(Cot[x]/Sqrt[a*Csc[x]^2])} -{1/(a*Csc[x]^2)^(3/2), x, 3, -(Cot[x]/(3*(a*Csc[x]^2)^(3/2))) - (2*Cot[x])/(3*a*Sqrt[a*Csc[x]^2])} -{1/(a*Csc[x]^2)^(5/2), x, 4, -(Cot[x]/(5*(a*Csc[x]^2)^(5/2))) - (4*Cot[x])/(15*a*(a*Csc[x]^2)^(3/2)) - (8*Cot[x])/(15*a^2*Sqrt[a*Csc[x]^2])} -{1/(a*Csc[x]^2)^(7/2), x, 5, -(Cot[x]/(7*(a*Csc[x]^2)^(7/2))) - (6*Cot[x])/(35*a*(a*Csc[x]^2)^(5/2)) - (8*Cot[x])/(35*a^2*(a*Csc[x]^2)^(3/2)) - (16*Cot[x])/(35*a^3*Sqrt[a*Csc[x]^2])} - - -{(a*Csc[x]^3)^(5/2), x, 7, (-(154/585))*a^2*Cot[x]*Sqrt[a*Csc[x]^3] - (22/117)*a^2*Cot[x]*Csc[x]^2*Sqrt[a*Csc[x]^3] - (2/13)*a^2*Cot[x]*Csc[x]^4*Sqrt[a*Csc[x]^3] - (154/195)*a^2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + (154/195)*a^2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2)} -{(a*Csc[x]^3)^(3/2), x, 5, (-(10/21))*a*Cos[x]*Sqrt[a*Csc[x]^3] - (2/7)*a*Cot[x]*Csc[x]*Sqrt[a*Csc[x]^3] - (10/21)*a*Sqrt[a*Csc[x]^3]*EllipticF[Pi/4 - x/2, 2]*Sin[x]^(3/2)} -{(a*Csc[x]^3)^(1/2), x, 4, -2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[Pi/4 - x/2, 2]*Sin[x]^(3/2)} -{1/(a*Csc[x]^3)^(1/2), x, 4, -((2*Cot[x])/(3*Sqrt[a*Csc[x]^3])) - (2*EllipticF[Pi/4 - x/2, 2])/(3*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2))} -{1/(a*Csc[x]^3)^(3/2), x, 5, -((14*Cos[x])/(45*a*Sqrt[a*Csc[x]^3])) - (14*EllipticE[Pi/4 - x/2, 2])/(15*a*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2)) - (2*Cos[x]*Sin[x]^2)/(9*a*Sqrt[a*Csc[x]^3])} -{1/(a*Csc[x]^3)^(5/2), x, 7, -((26*Cot[x])/(77*a^2*Sqrt[a*Csc[x]^3])) - (26*EllipticF[Pi/4 - x/2, 2])/(77*a^2*Sqrt[a*Csc[x]^3]*Sin[x]^(3/2)) - (78*Cos[x]*Sin[x])/(385*a^2*Sqrt[a*Csc[x]^3]) - (26*Cos[x]*Sin[x]^3)/(165*a^2*Sqrt[a*Csc[x]^3]) - (2*Cos[x]*Sin[x]^5)/(15*a^2*Sqrt[a*Csc[x]^3])} - - -{(a*Csc[x]^4)^(7/2), x, 3, -2*a^3*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - 3*a^3*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - (20/7)*a^3*Cos[x]^2*Cot[x]^5*Sqrt[a*Csc[x]^4] - (5/3)*a^3*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4] - (6/11)*a^3*Cos[x]^2*Cot[x]^9*Sqrt[a*Csc[x]^4] - (1/13)*a^3*Cos[x]^2*Cot[x]^11*Sqrt[a*Csc[x]^4] - a^3*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]} -{(a*Csc[x]^4)^(5/2), x, 3, (-(4/3))*a^2*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - (6/5)*a^2*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - (4/7)*a^2*Cos[x]^2*Cot[x]^5*Sqrt[a*Csc[x]^4] - (1/9)*a^2*Cos[x]^2*Cot[x]^7*Sqrt[a*Csc[x]^4] - a^2*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]} -{(a*Csc[x]^4)^(3/2), x, 3, (-(2/3))*a*Cos[x]^2*Cot[x]*Sqrt[a*Csc[x]^4] - (1/5)*a*Cos[x]^2*Cot[x]^3*Sqrt[a*Csc[x]^4] - a*Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x]} -{(a*Csc[x]^4)^(1/2), x, 3, (-Cos[x])*Sqrt[a*Csc[x]^4]*Sin[x]} -{1/(a*Csc[x]^4)^(1/2), x, 3, -(Cot[x]/(2*Sqrt[a*Csc[x]^4])) + (x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])} -{1/(a*Csc[x]^4)^(3/2), x, 5, -((5*Cot[x])/(16*a*Sqrt[a*Csc[x]^4])) + (5*x*Csc[x]^2)/(16*a*Sqrt[a*Csc[x]^4]) - (5*Cos[x]*Sin[x])/(24*a*Sqrt[a*Csc[x]^4]) - (Cos[x]*Sin[x]^3)/(6*a*Sqrt[a*Csc[x]^4])} -{1/(a*Csc[x]^4)^(5/2), x, 7, -((63*Cot[x])/(256*a^2*Sqrt[a*Csc[x]^4])) + (63*x*Csc[x]^2)/(256*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x])/(128*a^2*Sqrt[a*Csc[x]^4]) - (21*Cos[x]*Sin[x]^3)/(160*a^2*Sqrt[a*Csc[x]^4]) - (9*Cos[x]*Sin[x]^5)/(80*a^2*Sqrt[a*Csc[x]^4]) - (Cos[x]*Sin[x]^7)/(10*a^2*Sqrt[a*Csc[x]^4])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ((b Csc[c+d x])^p)^n with n symbolic*) - - -{((b*Csc[c + d*x])^p)^n, x, 3, (Cos[c + d*x]*((b*Csc[c + d*x])^p)^n*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Sin[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n*p)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a (b Csc[c+d x])^p)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a (b Csc[c+d x])^p)^n with n symbolic*) - - -{(a*(b*Csc[c + d*x])^p)^n, x, 3, (Cos[c + d*x]*(a*(b*Csc[c + d*x])^p)^n*Hypergeometric2F1[1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), Sin[c + d*x]^2]*Sin[c + d*x])/(d*(1 - n*p)*Sqrt[Cos[c + d*x]^2])} - - -(* ::Title:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Trg[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Csc[e+f x])^m (b Csc[e+f x])^n*) - - -{(a*Csc[e + f*x])^m*(b*Csc[e + f*x])^n, x, 3, (a*Cos[e + f*x]*(a*Csc[e + f*x])^(-1 + m)*(b*Csc[e + f*x])^n*Hypergeometric2F1[1/2, (1/2)*(1 - m - n), (1/2)*(3 - m - n), Sin[e + f*x]^2])/(f*(1 - m - n)*Sqrt[Cos[e + f*x]^2])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.2 (d csc)^n (a+b csc)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.2 (d csc)^n (a+b csc)^m.m deleted file mode 100644 index 872ee14..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.2 (d csc)^n (a+b csc)^m.m +++ /dev/null @@ -1,170 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Csc[e+f x])^m (d Csc[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[e+f x]^n (a+a Csc[e+f x])^m*) - - -{Csc[x]^5/(a + a*Csc[x]), x, 6, (3*ArcTanh[Cos[x]])/(2*a) - (4*Cot[x])/a - (4*Cot[x]^3)/(3*a) + (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^3)/(a + a*Csc[x])} -{Csc[x]^4/(a + a*Csc[x]), x, 6, -((3*ArcTanh[Cos[x]])/(2*a)) + (2*Cot[x])/a - (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^2)/(a + a*Csc[x])} -{Csc[x]^3/(a + a*Csc[x]), x, 4, ArcTanh[Cos[x]]/a - Cot[x]/a - Cot[x]/(a + a*Csc[x])} -{Csc[x]^2/(a + a*Csc[x]), x, 3, -(ArcTanh[Cos[x]]/a) + Cot[x]/(a + a*Csc[x])} -{Csc[x]^1/(a + a*Csc[x]), x, 1, -(Cot[x]/(a + a*Csc[x]))} -{Csc[c + d*x]^0/(a + a*Csc[c + d*x]), x, 2, x/a + Cot[c + d*x]/(d*(a + a*Csc[c + d*x]))} -{Sin[x]^1/(a + a*Csc[x]), x, 4, -(x/a) - (2*Cos[x])/a + Cos[x]/(a + a*Csc[x])} -{Sin[x]^2/(a + a*Csc[x]), x, 5, (3*x)/(2*a) + (2*Cos[x])/a - (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x])/(a + a*Csc[x])} -{Sin[x]^3/(a + a*Csc[x]), x, 6, -((3*x)/(2*a)) - (4*Cos[x])/a + (4*Cos[x]^3)/(3*a) + (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^2)/(a + a*Csc[x])} -{Sin[x]^4/(a + a*Csc[x]), x, 7, (15*x)/(8*a) + (4*Cos[x])/a - (4*Cos[x]^3)/(3*a) - (15*Cos[x]*Sin[x])/(8*a) - (5*Cos[x]*Sin[x]^3)/(4*a) + (Cos[x]*Sin[x]^3)/(a + a*Csc[x])} - - -{Csc[c + d*x]^0/(a + a*Csc[c + d*x])^2, x, 3, x/a^2 + (4*Cot[c + d*x])/(3*a^2*d*(1 + Csc[c + d*x])) + Cot[c + d*x]/(3*d*(a + a*Csc[c + d*x])^2)} - - -{Csc[c + d*x]^0/(a + a*Csc[c + d*x])^3, x, 4, x/a^3 + Cot[c + d*x]/(5*d*(a + a*Csc[c + d*x])^3) + (7*Cot[c + d*x])/(15*a*d*(a + a*Csc[c + d*x])^2) + (22*Cot[c + d*x])/(15*d*(a^3 + a^3*Csc[c + d*x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[e+f x]^n (a+a Csc[e+f x])^(m/2)*) - - -{(a + a*Csc[x])^(5/2), x, 5, -2*a^(5/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (14*a^3*Cot[x])/(3*Sqrt[a + a*Csc[x]]) - (2/3)*a^2*Cot[x]*Sqrt[a + a*Csc[x]]} -{(a + a*Csc[x])^(3/2), x, 4, -2*a^(3/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (2*a^2*Cot[x])/Sqrt[a + a*Csc[x]]} -{(a + a*Csc[x])^(1/2), x, 2, -2*Sqrt[a]*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]]} -{1/(a + a*Csc[x])^(1/2), x, 5, -((2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/Sqrt[a]) + (Sqrt[2]*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/Sqrt[a]} -{1/(a + a*Csc[x])^(3/2), x, 6, -((2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(3/2)) + (5*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/(2*Sqrt[2]*a^(3/2)) + Cot[x]/(2*(a + a*Csc[x])^(3/2))} -{1/(a + a*Csc[x])^(5/2), x, 7, -((2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(5/2)) + (43*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/(16*Sqrt[2]*a^(5/2)) + Cot[x]/(4*(a + a*Csc[x])^(5/2)) + (11*Cot[x])/(16*a*(a + a*Csc[x])^(3/2))} - - -(* ::Subsection:: *) -(*Integrands of the form (d Csc[e+f x])^(n/2) (a+a Csc[e+f x])^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^(n/2) (a+a Csc[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[Csc[e + f*x]]*Sqrt[a + a*Csc[e + f*x]], x, 2, -((2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a + a*Csc[e + f*x]]])/f)} -{Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]], x, 2, -((2*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a - a*Csc[e + f*x]]])/f)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^(n/3) (a+a Csc[e+f x])^(m/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[a + a*Csc[c + d*x]]*Csc[c + d*x]^(4/3), x, 4, -((6*a*Cos[c + d*x]*Csc[c + d*x]^(4/3))/(5*d*Sqrt[a + a*Csc[c + d*x]])) - (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(5*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} -{Sqrt[a + a*Csc[c + d*x]]*Csc[c + d*x]^(1/3), x, 3, -((2*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]))} -{Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(2/3), x, 4, -((3*a*Cos[c + d*x]*Csc[c + d*x]^(1/3))/(2*d*Sqrt[a + a*Csc[c + d*x]])) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} - -{Sqrt[a + a*Csc[c + d*x]]*Csc[c + d*x]^(5/3), x, 6, (24*a*Cot[c + d*x])/(7*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (6*a*Cos[c + d*x]*Csc[c + d*x]^(5/3))/(7*d*Sqrt[a + a*Csc[c + d*x]]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (8*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} -{Sqrt[a + a*Csc[c + d*x]]*Csc[c + d*x]^(2/3), x, 5, (6*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (2*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} -{Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(1/3), x, 6, -((3*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]])) - (3*a*Cos[c + d*x]*Csc[c + d*x]^(2/3))/(d*Sqrt[a + a*Csc[c + d*x]]) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) - (Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} -{Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(4/3), x, 7, -((15*a*Cot[c + d*x])/(8*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]])) - (3*a*Cos[c + d*x])/(4*d*Csc[c + d*x]^(1/3)*Sqrt[a + a*Csc[c + d*x]]) - (15*a*Cos[c + d*x]*Csc[c + d*x]^(2/3))/(8*d*Sqrt[a + a*Csc[c + d*x]]) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(16*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) - (5*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(4*Sqrt[2]*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+a Csc[e+f x])^m with n symbolic*) - - -{Csc[c + d*x]^n*Sqrt[a + a*Csc[c + d*x]], x, 2, -((2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a + a*Csc[c + d*x]]))} -{Csc[c + d*x]^n*Sqrt[a - a*Csc[c + d*x]], x, 3, -((2*a*Cos[c + d*x]*Csc[c + d*x]^(1 + n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Csc[c + d*x]])/((-Csc[c + d*x])^n*(d*Sqrt[a - a*Csc[c + d*x]])))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+a Csc[e+f x])^m with m symbolic*) - - -{Csc[e + f*x]^3*(a + a*Csc[e + f*x])^m, x, 5, If[$VersionNumber>=8, (Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (Cot[e + f*x]*(a + a*Csc[e + f*x])^(1 + m))/(a*f*(2 + m)) - (2^(1/2 + m)*(1 + m + m^2)*Cot[e + f*x]*(1 + Csc[e + f*x])^(-(1/2) - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Csc[e + f*x])])/(f*(1 + m)*(2 + m)), (Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (Cot[e + f*x]*(a + a*Csc[e + f*x])^(1 + m))/(a*f*(2 + m)) - (2^(1/2 + m)*(1 + m + m^2)*Cot[e + f*x]*(1 + Csc[e + f*x])^(-(1/2) - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Csc[e + f*x])])/(f*(2 + 3*m + m^2))]} -{Csc[e + f*x]^2*(a + a*Csc[e + f*x])^m, x, 4, -((Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*m*Cot[e + f*x]*(1 + Csc[e + f*x])^(-(1/2) - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Csc[e + f*x])])/(f*(1 + m))} -{Csc[e + f*x]^1*(a + a*Csc[e + f*x])^m, x, 3, -((2^(1/2 + m)*Cot[e + f*x]*(1 + Csc[e + f*x])^(-(1/2) - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1/2)*(1 - Csc[e + f*x])])/f)} -{Csc[e + f*x]^0*(a + a*Csc[e + f*x])^m, x, 3, -((Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1/2)*(1 + Csc[e + f*x]), 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))} -{Sin[e + f*x]^1*(a + a*Csc[e + f*x])^m, x, 3, (Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1/2)*(1 + Csc[e + f*x]), 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]])} -{Sin[e + f*x]^2*(a + a*Csc[e + f*x])^m, x, 3, -((Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1/2)*(1 + Csc[e + f*x]), 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))} - - -(* ::Section:: *) -(*Integrands of the form (a+a Csc[e+f x])^m (d Sin[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[e+f x]^m (a+b Csc[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Csc[c + d*x])^4, x, 6, a^4*x - (2*a*b*(2*a^2 + b^2)*ArcTanh[Cos[c + d*x]])/d - (b^2*(17*a^2 + 2*b^2)*Cot[c + d*x])/(3*d) - (4*a*b^3*Cot[c + d*x]*Csc[c + d*x])/(3*d) - (b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^2)/(3*d)} -{(a + b*Csc[c + d*x])^3, x, 5, a^3*x - (b*(6*a^2 + b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*b^2*Cot[c + d*x])/(2*d) - (b^2*Cot[c + d*x]*(a + b*Csc[c + d*x]))/(2*d)} -{(a + b*Csc[c + d*x])^2, x, 4, a^2*x - (2*a*b*ArcTanh[Cos[c + d*x]])/d - (b^2*Cot[c + d*x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Csc[x]^5/(a + b*Csc[x]), x, 9, (a*(2*a^2 + b^2)*ArcTanh[Cos[x]])/(2*b^4) - (2*a^4*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]) - ((3*a^2 + 2*b^2)*Cot[x])/(3*b^3) + (a*Cot[x]*Csc[x])/(2*b^2) - (Cot[x]*Csc[x]^2)/(3*b)} -{Csc[x]^4/(a + b*Csc[x]), x, 8, -(((2*a^2 + b^2)*ArcTanh[Cos[x]])/(2*b^3)) + (2*a^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) + (a*Cot[x])/b^2 - (Cot[x]*Csc[x])/(2*b)} -{Csc[x]^3/(a + b*Csc[x]), x, 7, (a*ArcTanh[Cos[x]])/b^2 - (2*a^2*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cot[x]/b} -{Csc[x]^2/(a + b*Csc[x]), x, 6, -(ArcTanh[Cos[x]]/b) + (2*a*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])} -{Csc[x]^1/(a + b*Csc[x]), x, 4, -((2*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])} -{Csc[c + d*x]^0/(a + b*Csc[c + d*x]), x, 4, x/a + (2*b*ArcTanh[(a + b*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d)} -{Sin[x]^1/(a + b*Csc[x]), x, 6, -((b*x)/a^2) - (2*b^2*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) - Cos[x]/a} -{Sin[x]^2/(a + b*Csc[x]), x, 7, ((a^2 + 2*b^2)*x)/(2*a^3) + (2*b^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) + (b*Cos[x])/a^2 - (Cos[x]*Sin[x])/(2*a)} -{Sin[x]^3/(a + b*Csc[x]), x, 8, -((b*(a^2 + 2*b^2)*x)/(2*a^4)) - (2*b^4*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) - ((2*a^2 + 3*b^2)*Cos[x])/(3*a^3) + (b*Cos[x]*Sin[x])/(2*a^2) - (Cos[x]*Sin[x]^2)/(3*a)} -{Sin[x]^4/(a + b*Csc[x]), x, 9, ((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) + (2*b^5*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]) + (b*(2*a^2 + 3*b^2)*Cos[x])/(3*a^4) - ((3*a^2 + 4*b^2)*Cos[x]*Sin[x])/(8*a^3) + (b*Cos[x]*Sin[x]^2)/(3*a^2) - (Cos[x]*Sin[x]^3)/(4*a)} - - -{1/(a + b*Csc[c + d*x])^2, x, 6, x/a^2 + (2*b*(2*a^2 - b^2)*ArcTanh[(a + b*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d) - (b^2*Cot[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*x]))} -{1/(a + b*Csc[c + d*x])^3, x, 7, x/a^3 + (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(a + b*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d) - (b^2*Cot[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^2) - (b^2*(5*a^2 - 2*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x]))} -{1/(a + b*Csc[c + d*x])^4, x, 8, x/a^4 + (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(a + b*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(7/2)*d) - (b^2*Cot[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^3) - (b^2*(8*a^2 - 3*b^2)*Cot[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x])^2) - (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Cot[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Csc[c + d*x]))} - - -{1/(3 + 5*Csc[c + d*x]), x, 2, -(x/12) - (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(6*d)} -{1/(5 + 3*Csc[c + d*x]), x, 5, x/5 + (3*Log[3*Cos[(1/2)*(c + d*x)] + Sin[(1/2)*(c + d*x)]])/(20*d) - (3*Log[Cos[(1/2)*(c + d*x)] + 3*Sin[(1/2)*(c + d*x)]])/(20*d)} - - -(* ::Subsection:: *) -(*Integrands of the form Csc[e+f x]^m (a+b Csc[e+f x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n with m symbolic*) - - -{Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m, x, 8, -((Cot[e + f*x]*(a + b*Csc[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Csc[e + f*x]), (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(((a + b*Csc[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Csc[e + f*x]])) - (Sqrt[2]*(a^2 + b^2*(1 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Csc[e + f*x]), (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(((a + b*Csc[e + f*x])/(a + b))^m*(b^2*f*(2 + m)*Sqrt[1 + Csc[e + f*x]]))} -{Csc[e + f*x]^2*(a + b*Csc[e + f*x])^m, x, 7, -((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - Csc[e + f*x]), (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(((a + b*Csc[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Csc[e + f*x]]))) + (Sqrt[2]*a*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Csc[e + f*x]), (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(((a + b*Csc[e + f*x])/(a + b))^m*(b*f*Sqrt[1 + Csc[e + f*x]]))} -{Csc[e + f*x]^1*(a + b*Csc[e + f*x])^m, x, 3, -((Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Csc[e + f*x]), (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(((a + b*Csc[e + f*x])/(a + b))^m*(f*Sqrt[1 + Csc[e + f*x]])))} -{Csc[e + f*x]^0*(a + b*Csc[e + f*x])^m, x, 0, Unintegrable[(a + b*Csc[e + f*x])^m, x]} -{Sin[e + f*x]^1*(a + b*Csc[e + f*x])^m, x, 0, Unintegrable[(a + b*Csc[e + f*x])^m*Sin[e + f*x], x]} -{Sin[e + f*x]^2*(a + b*Csc[e + f*x])^m, x, 0, Unintegrable[(a + b*Csc[e + f*x])^m*Sin[e + f*x]^2, x]} - - -(* ::Section:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Sin[e+f x])^n*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.3 (d cos)^n (a+b csc)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.3 (d cos)^n (a+b csc)^m.m deleted file mode 100644 index 6cf7d00..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.3 (d cos)^n (a+b csc)^m.m +++ /dev/null @@ -1,52 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e Cos[c+d x])^m (a+a Csc[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e Cos[c+d x])^m (a+a Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a+a Csc[c+d x])^n*) - - -{Cos[x]^4/(a + a*Csc[x]), x, 7, -(x/(8*a)) - Cos[x]^3/(3*a) - (Cos[x]*Sin[x])/(8*a) + (Cos[x]^3*Sin[x])/(4*a)} -{Cos[x]^3/(a + a*Csc[x]), x, 6, Sin[x]^2/(2*a) - Sin[x]^3/(3*a)} -{Cos[x]^2/(a + a*Csc[x]), x, 5, -(x/(2*a)) - Cos[x]/a + (Cos[x]*Sin[x])/(2*a)} -{Cos[x]^1/(a + a*Csc[x]), x, 5, -(Log[1 + Sin[x]]/a) + Sin[x]/a} -{Sec[x]^1/(a + a*Csc[x]), x, 6, ArcTanh[Sin[x]]/(2*a) + Sec[x]^2/(2*a) - (Sec[x]*Tan[x])/(2*a)} -{Sec[x]^2/(a + a*Csc[x]), x, 6, Sec[x]^3/(3*a) - Tan[x]^3/(3*a)} -{Sec[x]^3/(a + a*Csc[x]), x, 7, ArcTanh[Sin[x]]/(8*a) + Sec[x]^4/(4*a) + (Sec[x]*Tan[x])/(8*a) - (Sec[x]^3*Tan[x])/(4*a)} -{Sec[x]^4/(a + a*Csc[x]), x, 7, Sec[x]^5/(5*a) - Tan[x]^3/(3*a) - Tan[x]^5/(5*a)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[c+d x]^m (a+a Csc[c+d x])^(n/2)*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (e Cos[c+d x])^m (a+b Csc[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e Cos[c+d x])^m (a+b Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a+b Csc[c+d x])^n*) - - -{Cos[x]^4/(a + b*Csc[x]), x, 7, ((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) + (2*b*(a^2 - b^2)^(3/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/a^5 - (Cos[x]^3*(4*b - 3*a*Sin[x]))/(12*a^2) - (Cos[x]*(8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Sin[x]))/(8*a^4)} -{Cos[x]^3/(a + b*Csc[x]), x, 5, -((b*(a^2 - b^2)*Log[b + a*Sin[x]])/a^4) + ((a^2 - b^2)*Sin[x])/a^3 + (b*Sin[x]^2)/(2*a^2) - Sin[x]^3/(3*a)} -{Cos[x]^2/(a + b*Csc[x]), x, 6, ((a^2 - 2*b^2)*x)/(2*a^3) + (2*b*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/a^3 - (Cos[x]*(2*b - a*Sin[x]))/(2*a^2)} -{Cos[x]^1/(a + b*Csc[x]), x, 5, -((b*Log[b + a*Sin[x]])/a^2) + Sin[x]/a} -{Sec[x]^1/(a + b*Csc[x]), x, 4, -(Log[1 - Sin[x]]/(2*(a + b))) + Log[1 + Sin[x]]/(2*(a - b)) - (b*Log[b + a*Sin[x]])/(a^2 - b^2)} -{Sec[x]^2/(a + b*Csc[x]), x, 6, (2*a*b*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (Sec[x]*(b - a*Sin[x]))/(a^2 - b^2)} -{Sec[x]^3/(a + b*Csc[x]), x, 6, -((a*Log[1 - Sin[x]])/(4*(a + b)^2)) + (a*Log[1 + Sin[x]])/(4*(a - b)^2) - (a^2*b*Log[b + a*Sin[x]])/(a^2 - b^2)^2 - (Sec[x]^2*(b - a*Sin[x]))/(2*(a^2 - b^2))} -{Sec[x]^4/(a + b*Csc[x]), x, 7, (2*a^3*b*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Sec[x]^3*(b - a*Sin[x]))/(3*(a^2 - b^2)) - (Sec[x]*(3*a^2*b - a*(2*a^2 + b^2)*Sin[x]))/(3*(a^2 - b^2)^2)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[c+d x]^m (a+b Csc[c+d x])^(n/2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.4 (d cot)^n (a+b csc)^m.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.4 (d cot)^n (a+b csc)^m.m deleted file mode 100644 index f435825..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.1.4 (d cot)^n (a+b csc)^m.m +++ /dev/null @@ -1,60 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e Tan[c+d x])^m (a+a Csc[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e Tan[c+d x])^m (a+a Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (a+a Csc[c+d x])^n*) - - -{Tan[x]^4/(a + a*Csc[x]), x, 5, x/a - ((15 - 8*Csc[x])*Tan[x])/(15*a) + ((5 - 4*Csc[x])*Tan[x]^3)/(15*a) - ((1 - Csc[x])*Tan[x]^5)/(5*a)} -{Tan[x]^3/(a + a*Csc[x]), x, 3, (5*Log[1 - Sin[x]])/(16*a) + (11*Log[1 + Sin[x]])/(16*a) + 1/(8*a*(1 - Sin[x])) - 1/(8*a*(1 + Sin[x])^2) + 3/(4*a*(1 + Sin[x]))} -{Tan[x]^2/(a + a*Csc[x]), x, 4, -(x/a) + ((3 - 2*Csc[x])*Tan[x])/(3*a) - ((1 - Csc[x])*Tan[x]^3)/(3*a)} -{Tan[x]^1/(a + a*Csc[x]), x, 3, -(Log[1 - Sin[x]]/(4*a)) - (3*Log[1 + Sin[x]])/(4*a) - 1/(2*a*(1 + Sin[x]))} -{Cot[x]^1/(a + a*Csc[x]), x, 2, Log[1 + Sin[x]]/a} -{Cot[x]^2/(a + a*Csc[x]), x, 3, -(x/a) - ArcTanh[Cos[x]]/a} -{Cot[x]^3/(a + a*Csc[x]), x, 3, -(Csc[x]/a) - Log[Sin[x]]/a} -{Cot[x]^4/(a + a*Csc[x]), x, 4, x/a + ArcTanh[Cos[x]]/(2*a) + (Cot[x]*(2 - Csc[x]))/(2*a)} -{Cot[x]^5/(a + a*Csc[x]), x, 3, Csc[x]/a + Csc[x]^2/(2*a) - Csc[x]^3/(3*a) + Log[Sin[x]]/a} -{Cot[x]^6/(a + a*Csc[x]), x, 5, -(x/a) - (3*ArcTanh[Cos[x]])/(8*a) + (Cot[x]^3*(4 - 3*Csc[x]))/(12*a) - (Cot[x]*(8 - 3*Csc[x]))/(8*a)} -{Cot[x]^7/(a + a*Csc[x]), x, 3, -(Csc[x]/a) - Csc[x]^2/a + (2*Csc[x]^3)/(3*a) + Csc[x]^4/(4*a) - Csc[x]^5/(5*a) - Log[Sin[x]]/a} - - -(* ::Subsection:: *) -(*Integrands of the form Tan[c+d x]^m (a+a Csc[c+d x])^(n/2)*) - - -(* ::Title:: *) -(*Integrands of the form (e Tan[c+d x])^m (a+b Csc[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e Tan[c+d x])^m (a+b Csc[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[c+d x]^m (a+b Csc[c+d x])^n*) - - -{Tan[x]^5/(a + b*Csc[x]), x, 3, 1/(16*(a + b)*(1 - Csc[x])^2) + (5*a + 7*b)/(16*(a + b)^2*(1 - Csc[x])) + 1/(16*(a - b)*(1 + Csc[x])^2) + (5*a - 7*b)/(16*(a - b)^2*(1 + Csc[x])) - ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Csc[x]])/(16*(a + b)^3) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Csc[x]])/(16*(a - b)^3) + (b^6*Log[a + b*Csc[x]])/(a*(a^2 - b^2)^3) - Log[Sin[x]]/a} -{Tan[x]^3/(a + b*Csc[x]), x, 3, -(1/(4*(a + b)*(1 - Csc[x]))) - 1/(4*(a - b)*(1 + Csc[x])) + ((2*a + 3*b)*Log[1 - Csc[x]])/(4*(a + b)^2) + ((2*a - 3*b)*Log[1 + Csc[x]])/(4*(a - b)^2) + (b^4*Log[a + b*Csc[x]])/(a*(a^2 - b^2)^2) + Log[Sin[x]]/a} -{Tan[x]^1/(a + b*Csc[x]), x, 3, -(Log[1 - Csc[x]]/(2*(a + b))) - Log[1 + Csc[x]]/(2*(a - b)) + (b^2*Log[a + b*Csc[x]])/(a*(a^2 - b^2)) - Log[Sin[x]]/a} -{Cot[x]^1/(a + b*Csc[x]), x, 4, Log[a + b*Csc[x]]/a + Log[Sin[x]]/a} -{Cot[x]^3/(a + b*Csc[x]), x, 3, -(Csc[x]/b) - ((1 - a^2/b^2)*Log[a + b*Csc[x]])/a - Log[Sin[x]]/a} -{Cot[x]^5/(a + b*Csc[x]), x, 3, -(((a^2 - 2*b^2)*Csc[x])/b^3) + (a*Csc[x]^2)/(2*b^2) - Csc[x]^3/(3*b) + ((a^2 - b^2)^2*Log[a + b*Csc[x]])/(a*b^4) + Log[Sin[x]]/a} -{Cot[x]^7/(a + b*Csc[x]), x, 3, -(((a^4 - 3*a^2*b^2 + 3*b^4)*Csc[x])/b^5) + (a*(a^2 - 3*b^2)*Csc[x]^2)/(2*b^4) - ((a^2 - 3*b^2)*Csc[x]^3)/(3*b^3) + (a*Csc[x]^4)/(4*b^2) - Csc[x]^5/(5*b) + ((a^2 - b^2)^3*Log[a + b*Csc[x]])/(a*b^6) - Log[Sin[x]]/a} - -{Tan[x]^4/(a + b*Csc[x]), x, 16, x/a + (2*b^5*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) - (b^3*Sec[x])/(a^2 - b^2)^2 + (b*Sec[x])/(a^2 - b^2) - (b*Sec[x]^3)/(3*(a^2 - b^2)) + (a*b^2*Tan[x])/(a^2 - b^2)^2 - (a*Tan[x])/(a^2 - b^2) + (a*Tan[x]^3)/(3*(a^2 - b^2)), -((a*b^2*x)/(a^2 - b^2)^2) + (b^4*x)/(a*(a^2 - b^2)^2) + (a*x)/(a^2 - b^2) + (2*b^5*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) - (b^3*Sec[x])/(a^2 - b^2)^2 + (b*Sec[x])/(a^2 - b^2) - (b*Sec[x]^3)/(3*(a^2 - b^2)) + (a*b^2*Tan[x])/(a^2 - b^2)^2 - (a*Tan[x])/(a^2 - b^2) + (a*Tan[x]^3)/(3*(a^2 - b^2))} -{Tan[x]^2/(a + b*Csc[x]), x, 10, -(x/a) + (2*b^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(3/2)) - (b*Sec[x])/(a^2 - b^2) + (a*Tan[x])/(a^2 - b^2), -((a*x)/(a^2 - b^2)) + (b^2*x)/(a*(a^2 - b^2)) + (2*b^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(3/2)) - (b*Sec[x])/(a^2 - b^2) + (a*Tan[x])/(a^2 - b^2)} -{Cot[x]^2/(a + b*Csc[x]), x, 8, -(x/a) - ArcTanh[Cos[x]]/b + (2*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b)} -{Cot[x]^4/(a + b*Csc[x]), x, 7, x/a - ((2*a^2 - 3*b^2)*ArcTanh[Cos[x]])/(2*b^3) + (2*(a^2 - b^2)^(3/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b^3) + (a*Cot[x])/b^2 - (Cot[x]*Csc[x])/(2*b)} -{Cot[x]^6/(a + b*Csc[x]), x, 16, -(x/a) - (3*ArcTanh[Cos[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTanh[Cos[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTanh[Cos[x]])/b^5 + (2*(a^2 - b^2)^(5/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*b^5) + (a*Cot[x])/b^2 + (a*(a^2 - 3*b^2)*Cot[x])/b^4 + (a*Cot[x]^3)/(3*b^2) - (3*Cot[x]*Csc[x])/(8*b) - ((a^2 - 3*b^2)*Cot[x]*Csc[x])/(2*b^3) - (Cot[x]*Csc[x]^3)/(4*b)} - - -(* ::Subsection:: *) -(*Integrands of the form Tan[c+d x]^m (a+b Csc[c+d x])^(n/2)*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.11 (e x)^m (a+b csc(c+d x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.11 (e x)^m (a+b csc(c+d x^n))^p.m deleted file mode 100644 index 59c657c..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.11 (e x)^m (a+b csc(c+d x^n))^p.m +++ /dev/null @@ -1,182 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Csc[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csc[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Csc[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Csc[c + d*x^2]), x, 10, (a*x^6)/6 - (b*x^4*ArcTanh[E^(I*(c + d*x^2))])/d + (I*b*x^2*PolyLog[2, -E^(I*(c + d*x^2))])/d^2 - (I*b*x^2*PolyLog[2, E^(I*(c + d*x^2))])/d^2 - (b*PolyLog[3, -E^(I*(c + d*x^2))])/d^3 + (b*PolyLog[3, E^(I*(c + d*x^2))])/d^3} -{x^4*(a + b*Csc[c + d*x^2]), x, 2, (a*x^5)/5 + b*Unintegrable[x^4*Csc[c + d*x^2], x]} -{x^3*(a + b*Csc[c + d*x^2]), x, 8, (a*x^4)/4 - (b*x^2*ArcTanh[E^(I*(c + d*x^2))])/d + ((I/2)*b*PolyLog[2, -E^(I*(c + d*x^2))])/d^2 - ((I/2)*b*PolyLog[2, E^(I*(c + d*x^2))])/d^2} -{x^2*(a + b*Csc[c + d*x^2]), x, 2, (a*x^3)/3 + b*Unintegrable[x^2*Csc[c + d*x^2], x]} -{x*(a + b*Csc[c + d*x^2]), x, 4, (a*x^2)/2 - (b*ArcTanh[Cos[c + d*x^2]])/(2*d)} -{(a + b*Csc[c + d*x^2])/x, x, 2, a*Log[x] + b*Unintegrable[Csc[c + d*x^2]/x, x]} -{(a + b*Csc[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Csc[c + d*x^2]/x^2, x]} - - -{x^5*(a + b*Csc[c + d*x^2])^2, x, 15, ((-I/2)*b^2*x^4)/d + (a^2*x^6)/6 - (2*a*b*x^4*ArcTanh[E^(I*(c + d*x^2))])/d - (b^2*x^4*Cot[c + d*x^2])/(2*d) + (b^2*x^2*Log[1 - E^((2*I)*(c + d*x^2))])/d^2 + ((2*I)*a*b*x^2*PolyLog[2, -E^(I*(c + d*x^2))])/d^2 - ((2*I)*a*b*x^2*PolyLog[2, E^(I*(c + d*x^2))])/d^2 - ((I/2)*b^2*PolyLog[2, E^((2*I)*(c + d*x^2))])/d^3 - (2*a*b*PolyLog[3, -E^(I*(c + d*x^2))])/d^3 + (2*a*b*PolyLog[3, E^(I*(c + d*x^2))])/d^3} -{x^4*(a + b*Csc[c + d*x^2])^2, x, 0, Unintegrable[x^4*(a + b*Csc[c + d*x^2])^2, x]} -{x^3*(a + b*Csc[c + d*x^2])^2, x, 10, (a^2*x^4)/4 - (2*a*b*x^2*ArcTanh[E^(I*(c + d*x^2))])/d - (b^2*x^2*Cot[c + d*x^2])/(2*d) + (b^2*Log[Sin[c + d*x^2]])/(2*d^2) + (I*a*b*PolyLog[2, -E^(I*(c + d*x^2))])/d^2 - (I*a*b*PolyLog[2, E^(I*(c + d*x^2))])/d^2} -{x^2*(a + b*Csc[c + d*x^2])^2, x, 0, Unintegrable[x^2*(a + b*Csc[c + d*x^2])^2, x]} -{x*(a + b*Csc[c + d*x^2])^2, x, 5, (a^2*x^2)/2 - (a*b*ArcTanh[Cos[c + d*x^2]])/d - (b^2*Cot[c + d*x^2])/(2*d)} -{(a + b*Csc[c + d*x^2])^2/x, x, 0, Unintegrable[(a + b*Csc[c + d*x^2])^2/x, x]} -{(a + b*Csc[c + d*x^2])^2/x^2, x, 0, Unintegrable[(a + b*Csc[c + d*x^2])^2/x^2, x]} - - -{x*Csc[a + b*x^2]^7, x, 5, (-5*ArcTanh[Cos[a + b*x^2]])/(32*b) - (5*Cot[a + b*x^2]*Csc[a + b*x^2])/(32*b) - (5*Cot[a + b*x^2]*Csc[a + b*x^2]^3)/(48*b) - (Cot[a + b*x^2]*Csc[a + b*x^2]^5)/(12*b)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Csc[c + d*x^2]), x, 13, x^6/(6*a) + ((I/2)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((I/2)*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + (I*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (I*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3)} -{x^4/(a + b*Csc[c + d*x^2]), x, 0, Unintegrable[x^4/(a + b*Csc[c + d*x^2]), x]} -{x^3/(a + b*Csc[c + d*x^2]), x, 11, x^4/(4*a) + ((I/2)*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((I/2)*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d^2) - (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d^2)} -{x^2/(a + b*Csc[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Csc[c + d*x^2]), x]} -{x/(a + b*Csc[c + d*x^2]), x, 5, x^2/(2*a) + (b*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d)} -{1/(x*(a + b*Csc[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Csc[c + d*x^2])), x]} -{(a + b*Csc[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Csc[c + d*x^2]/x^2, x]} - - -{x^5/(a + b*Csc[c + d*x^2])^2, x, 31, ((-I/2)*b^2*x^4)/(a^2*(a^2 - b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^2*x^2*Log[1 + (a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((I/2)*b^3*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (I*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^4*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (I*b^2*PolyLog[2, -((a*E^(I*(c + d*x^2)))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (2*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (I*b^3*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((2*I)*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (I*b^3*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((2*I)*b*PolyLog[3, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (b^2*x^4*Cos[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2]))} -{x^4/(a + b*Csc[c + d*x^2])^2, x, 0, Unintegrable[x^4/(a + b*Csc[c + d*x^2])^2, x]} -{x^3/(a + b*Csc[c + d*x^2])^2, x, 22, x^4/(4*a^2) - ((I/2)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (I*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((I/2)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (I*b*x^2*Log[1 - (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + (b^2*Log[b + a*Sin[c + d*x^2]])/(2*a^2*(a^2 - b^2)*d^2) - (b^3*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) + (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^3*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, (I*a*E^(I*(c + d*x^2)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^2*x^2*Cos[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2]))} -{x^2/(a + b*Csc[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Csc[c + d*x^2])^2, x]} -{x/(a + b*Csc[c + d*x^2])^2, x, 7, x^2/(2*a^2) + (b*(2*a^2 - b^2)*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d) - (b^2*Cot[c + d*x^2])/(2*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x^2]))} -{1/(x*(a + b*Csc[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Csc[c + d*x^2])^2), x]} -{1/(x^2*(a + b*Csc[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Csc[c + d*x^2])^2), x]} -{1/(x^3*(a + b*Csc[c + d*x^2])^2), x, 0, Unintegrable[1/(x^3*(a + b*Csc[c + d*x^2])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csc[c+d x^(1/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Csc[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Csc[c + d*Sqrt[x]]), x, 20, (a*x^4)/4 - (4*b*x^(7/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((14*I)*b*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((14*I)*b*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (84*b*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (84*b*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((420*I)*b*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((420*I)*b*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (1680*b*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (1680*b*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 + ((5040*I)*b*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))])/d^6 - ((5040*I)*b*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6 - (10080*b*Sqrt[x]*PolyLog[7, -E^(I*(c + d*Sqrt[x]))])/d^7 + (10080*b*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))])/d^7 - ((10080*I)*b*PolyLog[8, -E^(I*(c + d*Sqrt[x]))])/d^8 + ((10080*I)*b*PolyLog[8, E^(I*(c + d*Sqrt[x]))])/d^8} -{x^2*(a + b*Csc[c + d*Sqrt[x]]), x, 16, (a*x^3)/3 - (4*b*x^(5/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((10*I)*b*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((10*I)*b*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (40*b*x^(3/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (40*b*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((120*I)*b*x*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((120*I)*b*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (240*b*Sqrt[x]*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (240*b*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 + ((240*I)*b*PolyLog[6, -E^(I*(c + d*Sqrt[x]))])/d^6 - ((240*I)*b*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6} -{x*(a + b*Csc[c + d*Sqrt[x]]), x, 12, (a*x^2)/2 - (4*b*x^(3/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((6*I)*b*x*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((6*I)*b*x*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (12*b*Sqrt[x]*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (12*b*Sqrt[x]*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((12*I)*b*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((12*I)*b*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4} -{(a + b*Csc[c + d*Sqrt[x]])/x, x, 2, a*Log[x] + b*Unintegrable[Csc[c + d*Sqrt[x]]/x, x]} -{(a + b*Csc[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Csc[c + d*Sqrt[x]]/x^2, x]} - - -{x^3*(a + b*Csc[c + d*Sqrt[x]])^2, x, 30, ((-2*I)*b^2*x^(7/2))/d + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^(7/2)*Cot[c + d*Sqrt[x]])/d + (14*b^2*x^3*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((28*I)*a*b*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((28*I)*a*b*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((42*I)*b^2*x^(5/2)*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 + (105*b^2*x^2*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((840*I)*a*b*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((840*I)*a*b*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + ((210*I)*b^2*x^(3/2)*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 - (315*b^2*x*PolyLog[5, E^((2*I)*(c + d*Sqrt[x]))])/d^6 + ((10080*I)*a*b*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))])/d^6 - ((10080*I)*a*b*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6 - ((315*I)*b^2*Sqrt[x]*PolyLog[6, E^((2*I)*(c + d*Sqrt[x]))])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, -E^(I*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))])/d^7 + (315*b^2*PolyLog[7, E^((2*I)*(c + d*Sqrt[x]))])/(2*d^8) - ((20160*I)*a*b*PolyLog[8, -E^(I*(c + d*Sqrt[x]))])/d^8 + ((20160*I)*a*b*PolyLog[8, E^(I*(c + d*Sqrt[x]))])/d^8} -{x^2*(a + b*Csc[c + d*Sqrt[x]])^2, x, 24, ((-2*I)*b^2*x^(5/2))/d + (a^2*x^3)/3 - (8*a*b*x^(5/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^(5/2)*Cot[c + d*Sqrt[x]])/d + (10*b^2*x^2*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((20*I)*a*b*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*a*b*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((20*I)*b^2*x^(3/2)*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (80*a*b*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 + (30*b^2*x*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((240*I)*a*b*x*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((240*I)*a*b*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + ((30*I)*b^2*Sqrt[x]*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5 - (15*b^2*PolyLog[5, E^((2*I)*(c + d*Sqrt[x]))])/d^6 + ((480*I)*a*b*PolyLog[6, -E^(I*(c + d*Sqrt[x]))])/d^6 - ((480*I)*a*b*PolyLog[6, E^(I*(c + d*Sqrt[x]))])/d^6} -{x*(a + b*Csc[c + d*Sqrt[x]])^2, x, 18, ((-2*I)*b^2*x^(3/2))/d + (a^2*x^2)/2 - (8*a*b*x^(3/2)*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^(3/2)*Cot[c + d*Sqrt[x]])/d + (6*b^2*x*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((12*I)*a*b*x*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((12*I)*a*b*x*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((6*I)*b^2*Sqrt[x]*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (24*a*b*Sqrt[x]*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (24*a*b*Sqrt[x]*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 + (3*b^2*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((24*I)*a*b*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((24*I)*a*b*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4} -{(a + b*Csc[c + d*Sqrt[x]])^2/x, x, 0, Unintegrable[(a + b*Csc[c + d*Sqrt[x]])^2/x, x]} -{(a + b*Csc[c + d*Sqrt[x]])^2/x^2, x, 0, Unintegrable[(a + b*Csc[c + d*Sqrt[x]])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Csc[c + d*Sqrt[x]]), x, 23, x^4/(4*a) + ((2*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (14*b*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (14*b*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((84*I)*b*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((84*I)*b*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((1680*I)*b*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((1680*I)*b*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + (5040*b*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) - (5040*b*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) + ((10080*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^7) - ((10080*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^8)} -{x^2/(a + b*Csc[c + d*Sqrt[x]]), x, 19, x^3/(3*a) + ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (10*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (10*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((40*I)*b*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((40*I)*b*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) - (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6)} -{x/(a + b*Csc[c + d*Sqrt[x]]), x, 15, x^2/(2*a) + ((2*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (6*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (6*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((12*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((12*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (12*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (12*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4)} -{1/(x*(a + b*Csc[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x*(a + b*Csc[c + d*Sqrt[x]])), x]} -{(a + b*Csc[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Csc[c + d*Sqrt[x]]/x^2, x]} - - -{x^3/(a + b*Csc[c + d*Sqrt[x]])^2, x, 61, ((-2*I)*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) + (14*b^2*x^3*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (14*b^2*x^3*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(7/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((84*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((84*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (14*b^3*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (14*b^3*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (420*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (420*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((84*I)*b^3*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((168*I)*b*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((84*I)*b^3*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((168*I)*b*x^(5/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((1680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((1680*I)*b^2*x^(3/2)*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (420*b^3*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (420*b^3*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (5040*b^2*x*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) - (5040*b^2*x*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) + ((1680*I)*b^3*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((3360*I)*b*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((1680*I)*b^3*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((3360*I)*b*x^(3/2)*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((10080*I)*b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - ((10080*I)*b^2*Sqrt[x]*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040*b^3*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (5040*b^3*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (10080*b*x*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^2*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^8) - ((10080*I)*b^3*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + ((20160*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^7) + ((10080*I)*b^3*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - ((20160*I)*b*Sqrt[x]*PolyLog[7, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (2*b^2*x^(7/2)*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))} -{x^2/(a + b*Csc[c + d*Sqrt[x]])^2, x, 49, ((-2*I)*b^2*x^(5/2))/(a^2*(a^2 - b^2)*d) + x^3/(3*a^2) + (10*b^2*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (10*b^2*x^2*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((40*I)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((40*I)*b^2*x^(3/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (10*b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (20*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (10*b^3*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (20*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (120*b^2*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (120*b^2*x*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((40*I)*b^3*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((80*I)*b*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((40*I)*b^3*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((80*I)*b*x^(3/2)*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((240*I)*b^2*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((240*I)*b^2*Sqrt[x]*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (120*b^3*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (240*b*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (120*b^3*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (240*b*x*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (240*b^2*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) - (240*b^2*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^6) + ((240*I)*b^3*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((480*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((240*I)*b^3*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((480*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - (240*b^3*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (480*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (240*b^3*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (480*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^6) - (2*b^2*x^(5/2)*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))} -{x/(a + b*Csc[c + d*Sqrt[x]])^2, x, 37, ((-2*I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (6*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((12*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (6*b^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b^3*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (12*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((24*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((12*I)*b^3*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((24*I)*b*Sqrt[x]*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (2*b^2*x^(3/2)*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))} -{1/(x*(a + b*Csc[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x*(a + b*Csc[c + d*Sqrt[x]])^2), x]} -{1/(x^2*(a + b*Csc[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^2*(a + b*Csc[c + d*Sqrt[x]])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^(m/2) (a+b Csc[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^(3/2)*(a + b*Csc[c + d*Sqrt[x]]), x, 14, (2*a*x^(5/2))/5 - (4*b*x^2*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((8*I)*b*x^(3/2)*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((8*I)*b*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (24*b*x*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (24*b*x*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 - ((48*I)*b*Sqrt[x]*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((48*I)*b*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + (48*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (48*b*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5} -{Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]]), x, 10, (2*a*x^(3/2))/3 - (4*b*x*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((4*I)*b*Sqrt[x]*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((4*I)*b*Sqrt[x]*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (4*b*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (4*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3} -{(a + b*Csc[c + d*Sqrt[x]])/Sqrt[x], x, 4, 2*a*Sqrt[x] - (2*b*ArcTanh[Cos[c + d*Sqrt[x]]])/d} -{(a + b*Csc[c + d*Sqrt[x]])/x^(3/2), x, 2, (-2*a)/Sqrt[x] + b*Unintegrable[Csc[c + d*Sqrt[x]]/x^(3/2), x]} -{(a + b*Csc[c + d*Sqrt[x]])/x^(5/2), x, 2, (-2*a)/(3*x^(3/2)) + b*Unintegrable[Csc[c + d*Sqrt[x]]/x^(5/2), x]} - - -{x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2, x, 21, ((-2*I)*b^2*x^2)/d + (2*a^2*x^(5/2))/5 - (8*a*b*x^2*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x^2*Cot[c + d*Sqrt[x]])/d + (8*b^2*x^(3/2)*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((16*I)*a*b*x^(3/2)*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((16*I)*a*b*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((12*I)*b^2*x*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (48*a*b*x*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (48*a*b*x*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3 + (12*b^2*Sqrt[x]*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))])/d^4 - ((96*I)*a*b*Sqrt[x]*PolyLog[4, -E^(I*(c + d*Sqrt[x]))])/d^4 + ((96*I)*a*b*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))])/d^4 + ((6*I)*b^2*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))])/d^5 + (96*a*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))])/d^5 - (96*a*b*PolyLog[5, E^(I*(c + d*Sqrt[x]))])/d^5} -{Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2, x, 15, ((-2*I)*b^2*x)/d + (2*a^2*x^(3/2))/3 - (8*a*b*x*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d - (2*b^2*x*Cot[c + d*Sqrt[x]])/d + (4*b^2*Sqrt[x]*Log[1 - E^((2*I)*(c + d*Sqrt[x]))])/d^2 + ((8*I)*a*b*Sqrt[x]*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((8*I)*a*b*Sqrt[x]*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - ((2*I)*b^2*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))])/d^3 - (8*a*b*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (8*a*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3} -{(a + b*Csc[c + d*Sqrt[x]])^2/Sqrt[x], x, 5, 2*a^2*Sqrt[x] - (4*a*b*ArcTanh[Cos[c + d*Sqrt[x]]])/d - (2*b^2*Cot[c + d*Sqrt[x]])/d} -{(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x, 0, Unintegrable[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x]} -{(a + b*Csc[c + d*Sqrt[x]])^2/x^(5/2), x, 0, Unintegrable[(a + b*Csc[c + d*Sqrt[x]])^2/x^(5/2), x]} - - -{Csc[Sqrt[x]]^3/Sqrt[x], x, 3, -ArcTanh[Cos[Sqrt[x]]] - Cot[Sqrt[x]]*Csc[Sqrt[x]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^(3/2)/(a + b*Csc[c + d*Sqrt[x]]), x, 17, (2*x^(5/2))/(5*a) + ((2*I)*b*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (8*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (8*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((24*I)*b*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((24*I)*b*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (48*b*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (48*b*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((48*I)*b*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((48*I)*b*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5)} -{Sqrt[x]/(a + b*Csc[c + d*Sqrt[x]]), x, 13, (2*x^(3/2))/(3*a) + ((2*I)*b*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - ((2*I)*b*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (4*b*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (4*b*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((4*I)*b*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((4*I)*b*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3)} -{1/(Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])), x, 5, (2*Sqrt[x])/a + (4*b*ArcTanh[(a + b*Tan[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d)} -{1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])), x]} -{1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])), x]} - - -{x^(3/2)/(a + b*Csc[c + d*Sqrt[x]])^2, x, 43, ((-2*I)*b^2*x^2)/(a^2*(a^2 - b^2)*d) + (2*x^(5/2))/(5*a^2) + (8*b^2*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (8*b^2*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x^2*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((24*I)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((24*I)*b^2*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (8*b^3*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (16*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (8*b^3*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (16*b*x^(3/2)*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) - ((24*I)*b^3*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((48*I)*b*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((24*I)*b^3*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((48*I)*b*x*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((48*I)*b^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + ((48*I)*b^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^5) + (48*b^3*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (96*b*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (48*b^3*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (96*b*Sqrt[x]*PolyLog[4, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^4) + ((48*I)*b^3*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - ((96*I)*b*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - ((48*I)*b^3*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + ((96*I)*b*PolyLog[5, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^5) - (2*b^2*x^2*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))} -{Sqrt[x]/(a + b*Csc[c + d*Sqrt[x]])^2, x, 31, ((-2*I)*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) - ((2*I)*b^3*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((4*I)*b*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) + ((2*I)*b^3*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((4*I)*b*x*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((4*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (8*b*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (4*b^3*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) - ((4*I)*b^3*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + ((8*I)*b*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((4*I)*b^3*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - ((8*I)*b*PolyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (2*b^2*x*Cos[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*Sqrt[x]]))} -{1/(Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2), x, 7, (2*Sqrt[x])/a^2 + (4*b*(2*a^2 - b^2)*ArcTanh[(a + b*Tan[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d) - (2*b^2*Cot[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*Sqrt[x]]))} -{1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]} -{1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csc[c+d x^n])^p*) - - -{(e*x)^m*(a + b*Csc[c + d*x^n])^p, x, 1, ((e*x)^m*Unintegrable[x^m*(a + b*Csc[c + d*x^n])^p, x])/x^m} - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(e*x)^(-1 + n)*(a + b*Csc[c + d*x^n]), x, 5, (a*(e*x)^n)/(e*n) - (b*(e*x)^n*ArcTanh[Cos[c + d*x^n]])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n]), x, 9, (a*(e*x)^(2*n))/(2*e*n) - (2*b*(e*x)^(2*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (I*b*(e*x)^(2*n)*PolyLog[2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b*(e*x)^(2*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]), x, 11, (a*(e*x)^(3*n))/(3*e*n) - (2*b*(e*x)^(3*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) + ((2*I)*b*(e*x)^(3*n)*PolyLog[2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, -E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)*(a + b*Csc[c + d*x^n])^2, x, 6, (a^2*(e*x)^n)/(e*n) - (2*a*b*(e*x)^n*ArcTanh[Cos[c + d*x^n]])/(d*e*n*x^n) - (b^2*(e*x)^n*Cot[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n])^2, x, 11, (a^2*(e*x)^(2*n))/(2*e*n) - (4*a*b*(e*x)^(2*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Cot[c + d*x^n])/(d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[Sin[c + d*x^n]])/(d^2*e*n*x^(2*n)) + ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n])^2, x, 16, (a^2*(e*x)^(3*n))/(3*e*n) - (I*b^2*(e*x)^(3*n))/(d*e*n*x^n) - (4*a*b*(e*x)^(3*n)*ArcTanh[E^(I*(c + d*x^n))])/(d*e*n*x^n) - (b^2*(e*x)^(3*n)*Cot[c + d*x^n])/(d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 - E^((2*I)*(c + d*x^n))])/(d^2*e*n*x^(2*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, -E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b^2*(e*x)^(3*n)*PolyLog[2, E^((2*I)*(c + d*x^n))])/(d^3*e*n*x^(3*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[3, -E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[3, E^(I*(c + d*x^n))])/(d^3*e*n*x^(3*n))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n]), x, 6, (e*x)^n/(a*e*n) + (2*b*(e*x)^n*ArcTanh[(a + b*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d*e*n*x^n)} -{(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n]), x, 12, (e*x)^(2*n)/(2*a*e*n) + (I*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (I*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]), x, 14, (e*x)^(3*n)/(3*a*e*n) + (I*b*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (I*b*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (2*b*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n])^2, x, 8, (e*x)^n/(a^2*e*n) + (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTanh[(a + b*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d*e*n*x^n) - (b^2*(e*x)^n*Cot[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Csc[c + d*x^n]))} -{(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n])^2, x, 23, (e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(2*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[b + a*Sin[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^2*(e*x)^(2*n)*Cos[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Sin[c + d*x^n]))} -{(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2, x, 32, (e*x)^(3*n)/(3*a^2*e*n) - (I*b^2*(e*x)^(3*n))/(x^n*(a^2*(a^2 - b^2)*d*e*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(I*b - Sqrt[a^2 - b^2])])/(x^(2*n)*(a^2*(a^2 - b^2)*d^2*e*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(I*b + Sqrt[a^2 - b^2])])/(x^(2*n)*(a^2*(a^2 - b^2)*d^2*e*n)) - (I*b^3*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^n*(a^2*(-a^2 + b^2)^(3/2)*d*e*n)) + (2*I*b*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^n*(a^2*Sqrt[-a^2 + b^2]*d*e*n)) + (I*b^3*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^n*(a^2*(-a^2 + b^2)^(3/2)*d*e*n)) - (2*I*b*(e*x)^(3*n)*Log[1 - (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^n*(a^2*Sqrt[-a^2 + b^2]*d*e*n)) - (2*I*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(I*b - Sqrt[a^2 - b^2]))])/(x^(3*n)*(a^2*(a^2 - b^2)*d^3*e*n)) - (2*I*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(I*b + Sqrt[a^2 - b^2]))])/(x^(3*n)*(a^2*(a^2 - b^2)*d^3*e*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^(2*n)*(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^(2*n)*(a^2*Sqrt[-a^2 + b^2]*d^2*e*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^(2*n)*(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^(2*n)*(a^2*Sqrt[-a^2 + b^2]*d^2*e*n)) - (2*I*b^3*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^(3*n)*(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n)) + (4*I*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(x^(3*n)*(a^2*Sqrt[-a^2 + b^2]*d^3*e*n)) + (2*I*b^3*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^(3*n)*(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n)) - (4*I*b*(e*x)^(3*n)*PolyLog[3, (I*a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(x^(3*n)*(a^2*Sqrt[-a^2 + b^2]*d^3*e*n)) - (b^2*(e*x)^(3*n)*Cos[c + d*x^n])/(x^n*(a*(a^2 - b^2)*d*e*n*(b + a*Sin[c + d*x^n])))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.3.1 (a+b csc)^m (d csc)^n (A+B csc).m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.3.1 (a+b csc)^m (d csc)^n (A+B csc).m deleted file mode 100644 index 882a016..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.3.1 (a+b csc)^m (d csc)^n (A+B csc).m +++ /dev/null @@ -1,76 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n (A+B Csc[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+a Csc[e+f x])^m (d Csc[e+f x])^n (A+B Csc[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[c+d x]^m (A+B Csc[c+d x]) (a+a Csc[c+d x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[c+d x]^m (A+B Csc[c+d x]) (a+a Csc[c+d x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[c+d x]^m (A+A Csc[c+d x]) (a+a Csc[c+d x])^n*) - - -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^3, x, 7, -((7*a*A*ArcTanh[Cos[c + d*x]])/(8*d)) - (2*a*A*Cot[c + d*x])/d - (2*a*A*Cot[c + d*x]^3)/(3*d) - (7*a*A*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^2, x, 7, -((a*A*ArcTanh[Cos[c + d*x]])/d) - (5*a*A*Cot[c + d*x])/(3*d) - (a*A*Cot[c + d*x]*Csc[c + d*x])/d - (a*A*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d)} -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^1, x, 6, -((3*a*A*ArcTanh[Cos[c + d*x]])/(2*d)) - (2*a*A*Cot[c + d*x])/d - (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Sin[c + d*x]^1, x, 5, 2*a*A*x - (a*A*ArcTanh[Cos[c + d*x]])/d - (a*A*Cos[c + d*x])/d} -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Sin[c + d*x]^2, x, 5, (3*a*A*x)/2 - (2*a*A*Cos[c + d*x])/d - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(A + A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Sin[c + d*x]^3, x, 7, a*A*x - (2*a*A*Cos[c + d*x])/d + (a*A*Cos[c + d*x]^3)/(3*d) - (a*A*Cos[c + d*x]*Sin[c + d*x])/d} - - -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^3, x, 4, -((a*A*ArcTanh[Cos[c + d*x]])/(8*d)) - (a*A*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^2, x, 3, (a*A*Cot[c + d*x]^3)/(3*d)} -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^1, x, 3, -((a*A*ArcTanh[Cos[c + d*x]])/(2*d)) + (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Sin[c + d*x]^1, x, 4, (a*A*ArcTanh[Cos[c + d*x]])/d - (a*A*Cos[c + d*x])/d} -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Sin[c + d*x]^2, x, 3, (-(1/2))*a*A*x - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(A + A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Sin[c + d*x]^3, x, 3, (a*A*Cos[c + d*x]^3)/(3*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[c+d x]^m (A-A Csc[c+d x]) (a+a Csc[c+d x])^n*) - - -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^3, x, 4, -((a*A*ArcTanh[Cos[c + d*x]])/(8*d)) - (a*A*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^2, x, 3, (a*A*Cot[c + d*x]^3)/(3*d)} -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1*Csc[c + d*x]^1, x, 3, -((a*A*ArcTanh[Cos[c + d*x]])/(2*d)) + (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1/Csc[c + d*x]^1, x, 4, (a*A*ArcTanh[Cos[c + d*x]])/d - (a*A*Cos[c + d*x])/d} -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1/Csc[c + d*x]^2, x, 3, (-(1/2))*a*A*x - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(A - A*Csc[c + d*x])*(a + a*Csc[c + d*x])^1/Csc[c + d*x]^3, x, 3, (a*A*Cos[c + d*x]^3)/(3*d)} - - -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^3, x, 7, -((7*a*A*ArcTanh[Cos[c + d*x]])/(8*d)) + (2*a*A*Cot[c + d*x])/d + (2*a*A*Cot[c + d*x]^3)/(3*d) - (7*a*A*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a*A*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d)} -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^2, x, 7, (a*A*ArcTanh[Cos[c + d*x]])/d - (5*a*A*Cot[c + d*x])/(3*d) + (a*A*Cot[c + d*x]*Csc[c + d*x])/d - (a*A*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d)} -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1*Csc[c + d*x]^1, x, 6, -((3*a*A*ArcTanh[Cos[c + d*x]])/(2*d)) + (2*a*A*Cot[c + d*x])/d - (a*A*Cot[c + d*x]*Csc[c + d*x])/(2*d)} -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1/Csc[c + d*x]^1, x, 5, -2*a*A*x - (a*A*ArcTanh[Cos[c + d*x]])/d - (a*A*Cos[c + d*x])/d} -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1/Csc[c + d*x]^2, x, 5, (3*a*A*x)/2 + (2*a*A*Cos[c + d*x])/d - (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{(A - A*Csc[c + d*x])*(a - a*Csc[c + d*x])^1/Csc[c + d*x]^3, x, 7, (-a)*A*x - (2*a*A*Cos[c + d*x])/d + (a*A*Cos[c + d*x]^3)/(3*d) + (a*A*Cos[c + d*x]*Sin[c + d*x])/d} - - -(* ::Section:: *) -(*Integrands of the form (a+a Csc[e+f x])^m (d Sin[e+f x])^n (A+B Csc[e+f x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n (A+B Csc[e+f x])*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[c+d x]^m (A+B Csc[c+d x]) (a+b Csc[c+d x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Csc[c+d x]^m (A+B Csc[c+d x]) (a+b Csc[c+d x])^(n/2)*) - - -(* ::Section:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Sin[e+f x])^n (A+B Csc[e+f x])*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.4.2 (a+b csc)^m (d csc)^n (A+B csc+C csc^2).m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.4.2 (a+b csc)^m (d csc)^n (A+B csc+C csc^2).m deleted file mode 100644 index 60daf05..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.4.2 (a+b csc)^m (d csc)^n (A+B csc+C csc^2).m +++ /dev/null @@ -1,23 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Csc[e+f x])^m (d Csc[e+f x])^n (A+B Csc[e+f x]+C Csc[e+f x]^2)*) - - -(* ::Section:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Csc[e+f x])^m (A+C Csc[e+f x]^2)*) - - -(* ::Section:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Csc[e+f x])^m (B Csc[e+f x]+C Csc[e+f x]^2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^n (a+b Csc[e+f x])^m (A+B Csc[e+f x]+C Csc[e+f x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Csc[e+f x])^(n/2) (a+b Csc[e+f x])^m (A+B Csc[e+f x]+C Csc[e+f x]^2)*) - - -{((a + b*Csc[x])*(A + B*Csc[x] + C*Csc[x]^2))/Csc[x]^(1/2), x, 7, -2*(b*B + a*C)*Cos[x]*Sqrt[Csc[x]] - (2/3)*b*C*Cos[x]*Csc[x]^(3/2) + 2*(b*B - a*(A - C))*Sqrt[Csc[x]]*EllipticE[Pi/4 - x/2, 2]*Sqrt[Sin[x]] - (2/3)*(3*A*b + 3*a*B + b*C)*Sqrt[Csc[x]]*EllipticF[Pi/4 - x/2, 2]*Sqrt[Sin[x]]} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.7 (d trig)^m (a+b (c csc)^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.7 (d trig)^m (a+b (c csc)^n)^p.m deleted file mode 100644 index e3f4f02..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.6 Cosecant/4.6.7 (d trig)^m (a+b (c csc)^n)^p.m +++ /dev/null @@ -1,67 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (a+b Csc[c+d x]^m)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b Csc[c+d x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Csc[c+d x]^2)^n*) - - -(* ::Subsubsection::Closed:: *) -(*n*) - - -{(a + b*Csc[c + d*x]^2)^4, x, 4, a^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Cot[c + d*x])/d - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Cot[c + d*x]^3)/(3*d) - (b^3*(4*a + 3*b)*Cot[c + d*x]^5)/(5*d) - (b^4*Cot[c + d*x]^7)/(7*d)} -{(a + b*Csc[c + d*x]^2)^3, x, 4, a^3*x - (b*(3*a^2 + 3*a*b + b^2)*Cot[c + d*x])/d - (b^2*(3*a + 2*b)*Cot[c + d*x]^3)/(3*d) - (b^3*Cot[c + d*x]^5)/(5*d)} -{(a + b*Csc[c + d*x]^2)^2, x, 4, a^2*x - (b*(2*a + b)*Cot[c + d*x])/d - (b^2*Cot[c + d*x]^3)/(3*d)} -{(a + b*Csc[c + d*x]^2)^1, x, 3, a*x - (b*Cot[c + d*x])/d} -{1/(a + b*Csc[c + d*x]^2)^1, x, 3, x/a - (Sqrt[b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]])/(a*Sqrt[a + b]*d)} -{1/(a + b*Csc[c + d*x]^2)^2, x, 5, x/a^2 + (Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*d) + (b*Cot[c + d*x])/(2*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2))} -{1/(a + b*Csc[c + d*x]^2)^3, x, 6, x/a^3 + (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*d) + (b*Cot[c + d*x])/(4*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2)^2) + (b*(7*a + 4*b)*Cot[c + d*x])/(8*a^2*(a + b)^2*d*(a + b + b*Cot[c + d*x]^2))} -{1/(a + b*Csc[c + d*x]^2)^4, x, 7, x/a^4 + (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b]])/(16*a^4*(a + b)^(7/2)*d) + (b*Cot[c + d*x])/(6*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2)^3) + (b*(11*a + 6*b)*Cot[c + d*x])/(24*a^2*(a + b)^2*d*(a + b + b*Cot[c + d*x]^2)^2) + (b*(19*a^2 + 22*a*b + 8*b^2)*Cot[c + d*x])/(16*a^3*(a + b)^3*d*(a + b + b*Cot[c + d*x]^2))} - - -(* ::Subsubsection::Closed:: *) -(*n/2*) - - -{(a + b*Csc[c + d*x]^2)^(5/2), x, 8, -((a^(5/2)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/(8*d) - (b*(7*a + 3*b)*Cot[c + d*x]*Sqrt[a + b + b*Cot[c + d*x]^2])/(8*d) - (b*Cot[c + d*x]*(a + b + b*Cot[c + d*x]^2)^(3/2))/(4*d)} -{(a + b*Csc[c + d*x]^2)^(3/2), x, 7, -((a^(3/2)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/(2*d) - (b*Cot[c + d*x]*Sqrt[a + b + b*Cot[c + d*x]^2])/(2*d)} -{(a + b*Csc[c + d*x]^2)^(1/2), x, 6, -((Sqrt[a]*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]])/d} -{1/(a + b*Csc[c + d*x]^2)^(1/2), x, 3, -(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b*Csc[c + d*x]^2]]/(Sqrt[a]*d)), -(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(Sqrt[a]*d))} -{1/(a + b*Csc[c + d*x]^2)^(3/2), x, 4, -(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a + b)*d*Sqrt[a + b + b*Cot[c + d*x]^2])} -{1/(a + b*Csc[c + d*x]^2)^(5/2), x, 6, -(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(5/2)*d)) + (b*Cot[c + d*x])/(3*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2)^(3/2)) + (b*(5*a + 3*b)*Cot[c + d*x])/(3*a^2*(a + b)^2*d*Sqrt[a + b + b*Cot[c + d*x]^2])} -{1/(a + b*Csc[c + d*x]^2)^(7/2), x, 7, -(ArcTan[(Sqrt[a]*Cot[c + d*x])/Sqrt[a + b + b*Cot[c + d*x]^2]]/(a^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a + b)*d*(a + b + b*Cot[c + d*x]^2)^(5/2)) + (b*(9*a + 5*b)*Cot[c + d*x])/(15*a^2*(a + b)^2*d*(a + b + b*Cot[c + d*x]^2)^(3/2)) + (b*(33*a^2 + 40*a*b + 15*b^2)*Cot[c + d*x])/(15*a^3*(a + b)^3*d*Sqrt[a + b + b*Cot[c + d*x]^2])} - - -{(1 + Csc[x]^2)^(3/2), x, 6, -2*ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]] - (1/2)*Cot[x]*Sqrt[2 + Cot[x]^2]} -{Sqrt[1 + Csc[x]^2], x, 5, -ArcSinh[Cot[x]/Sqrt[2]] - ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]]} -{1/Sqrt[1 + Csc[x]^2], x, 3, -ArcTan[Cot[x]/Sqrt[2 + Cot[x]^2]]} - - -{(1 - Csc[x]^2)^(3/2), x, 4, (1/2)*Cot[x]*Sqrt[-Cot[x]^2] + Sqrt[-Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{Sqrt[1 - Csc[x]^2], x, 3, Sqrt[-Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{1/Sqrt[1 - Csc[x]^2], x, 3, -((Cot[x]*Log[Cos[x]])/Sqrt[-Cot[x]^2])} - - -{(-1 + Csc[x]^2)^(3/2), x, 4, (-(1/2))*(Cot[x]^2)^(3/2)*Tan[x] - Sqrt[Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{Sqrt[-1 + Csc[x]^2], x, 3, Sqrt[Cot[x]^2]*Log[Sin[x]]*Tan[x]} -{1/Sqrt[-1 + Csc[x]^2], x, 3, -((Cot[x]*Log[Cos[x]])/Sqrt[Cot[x]^2])} - - -{(-1 - Csc[x]^2)^(3/2), x, 7, -2*ArcTan[Cot[x]/Sqrt[-2 - Cot[x]^2]] - ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]] + (1/2)*Cot[x]*Sqrt[-2 - Cot[x]^2]} -{Sqrt[-1 - Csc[x]^2], x, 6, ArcTan[Cot[x]/Sqrt[-2 - Cot[x]^2]] + ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]]} -{1/Sqrt[-1 - Csc[x]^2], x, 3, -ArcTanh[Cot[x]/Sqrt[-2 - Cot[x]^2]]} - - -(* ::Subsection:: *) -(*Integrands of the form (a+b Csc[c+d x]^3)^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (a+b Csc[c+d x]^4)^n*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.1 (c trig)^m (d trig)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.1 (c trig)^m (d trig)^n.m deleted file mode 100644 index 5923a02..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.1 (c trig)^m (d trig)^n.m +++ /dev/null @@ -1,489 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[a+b x]^m Trig[2 a+2 b x]^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Sin[2 a+2 b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Sin[2 a+2 b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]*Sin[2*a + 2*b*x]^7, x, 4, (128*Sin[a + b*x]^9)/(9*b) - (384*Sin[a + b*x]^11)/(11*b) + (384*Sin[a + b*x]^13)/(13*b) - (128*Sin[a + b*x]^15)/(15*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^6, x, 4, -((64*Cos[a + b*x]^7)/(7*b)) + (64*Cos[a + b*x]^9)/(3*b) - (192*Cos[a + b*x]^11)/(11*b) + (64*Cos[a + b*x]^13)/(13*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^5, x, 4, (32*Sin[a + b*x]^7)/(7*b) - (64*Sin[a + b*x]^9)/(9*b) + (32*Sin[a + b*x]^11)/(11*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^4, x, 4, -((16*Cos[a + b*x]^5)/(5*b)) + (32*Cos[a + b*x]^7)/(7*b) - (16*Cos[a + b*x]^9)/(9*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^3, x, 4, (8*Sin[a + b*x]^5)/(5*b) - (8*Sin[a + b*x]^7)/(7*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^2, x, 4, -((4*Cos[a + b*x]^3)/(3*b)) + (4*Cos[a + b*x]^5)/(5*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^1, x, 1, Sin[a + b*x]/(2*b) - Sin[3*a + 3*b*x]/(6*b)} -{Sin[a + b*x]*Csc[2*a + 2*b*x]^1, x, 2, ArcTanh[Sin[a + b*x]]/(2*b)} -{Sin[a + b*x]*Csc[2*a + 2*b*x]^2, x, 4, -(ArcTanh[Cos[a + b*x]]/(4*b)) + Sec[a + b*x]/(4*b)} -{Sin[a + b*x]*Csc[2*a + 2*b*x]^3, x, 5, (3*ArcTanh[Sin[a + b*x]])/(16*b) - (3*Csc[a + b*x])/(16*b) + (Csc[a + b*x]*Sec[a + b*x]^2)/(16*b)} -{Sin[a + b*x]*Csc[2*a + 2*b*x]^4, x, 6, -((5*ArcTanh[Cos[a + b*x]])/(32*b)) + (5*Sec[a + b*x])/(32*b) + (5*Sec[a + b*x]^3)/(96*b) - (Csc[a + b*x]^2*Sec[a + b*x]^3)/(32*b)} -{Sin[a + b*x]*Csc[2*a + 2*b*x]^5, x, 7, (35*ArcTanh[Sin[a + b*x]])/(256*b) - (35*Csc[a + b*x])/(256*b) - (35*Csc[a + b*x]^3)/(768*b) + (7*Csc[a + b*x]^3*Sec[a + b*x]^2)/(256*b) + (Csc[a + b*x]^3*Sec[a + b*x]^4)/(128*b)} - - -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^5, x, 5, (4*Sin[a + b*x]^8)/b - (32*Sin[a + b*x]^10)/(5*b) + (8*Sin[a + b*x]^12)/(3*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^4, x, 6, (3*x)/16 - (3*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(32*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^3)/(16*b) - Sin[2*a + 2*b*x]^5/(20*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^3, x, 4, (4*Sin[a + b*x]^6)/(3*b) - Sin[a + b*x]^8/b} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^2, x, 5, x/4 - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(8*b) - Sin[2*a + 2*b*x]^3/(12*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^1, x, 3, Sin[a + b*x]^4/(2*b)} -{Sin[a + b*x]^2*Csc[2*a + 2*b*x]^1, x, 2, -(Log[Cos[a + b*x]]/(2*b))} -{Sin[a + b*x]^2*Csc[2*a + 2*b*x]^2, x, 3, Tan[a + b*x]/(4*b)} -{Sin[a + b*x]^2*Csc[2*a + 2*b*x]^3, x, 4, Log[Tan[a + b*x]]/(8*b) + Tan[a + b*x]^2/(16*b)} -{Sin[a + b*x]^2*Csc[2*a + 2*b*x]^4, x, 4, -(Cot[a + b*x]/(16*b)) + Tan[a + b*x]/(8*b) + Tan[a + b*x]^3/(48*b)} -{Sin[a + b*x]^2*Csc[2*a + 2*b*x]^5, x, 5, -(Cot[a + b*x]^2/(64*b)) + (3*Log[Tan[a + b*x]])/(32*b) + (3*Tan[a + b*x]^2)/(64*b) + Tan[a + b*x]^4/(128*b)} - - -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^5, x, 4, (32*Sin[a + b*x]^9)/(9*b) - (64*Sin[a + b*x]^11)/(11*b) + (32*Sin[a + b*x]^13)/(13*b)} -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^4, x, 4, -((16*Cos[a + b*x]^5)/(5*b)) + (48*Cos[a + b*x]^7)/(7*b) - (16*Cos[a + b*x]^9)/(3*b) + (16*Cos[a + b*x]^11)/(11*b)} -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^3, x, 4, (8*Sin[a + b*x]^7)/(7*b) - (8*Sin[a + b*x]^9)/(9*b)} -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2, x, 4, -((4*Cos[a + b*x]^3)/(3*b)) + (8*Cos[a + b*x]^5)/(5*b) - (4*Cos[a + b*x]^7)/(7*b)} -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^1, x, 3, (2*Sin[a + b*x]^5)/(5*b)} -{Sin[a + b*x]^3*Csc[2*a + 2*b*x]^1, x, 4, ArcTanh[Sin[a + b*x]]/(2*b) - Sin[a + b*x]/(2*b)} -{Sin[a + b*x]^3*Csc[2*a + 2*b*x]^2, x, 3, Sec[a + b*x]/(4*b)} -{Sin[a + b*x]^3*Csc[2*a + 2*b*x]^3, x, 3, ArcTanh[Sin[a + b*x]]/(16*b) + (Sec[a + b*x]*Tan[a + b*x])/(16*b)} -{Sin[a + b*x]^3*Csc[2*a + 2*b*x]^4, x, 5, -(ArcTanh[Cos[a + b*x]]/(16*b)) + Sec[a + b*x]/(16*b) + Sec[a + b*x]^3/(48*b)} -{Sin[a + b*x]^3*Csc[2*a + 2*b*x]^5, x, 6, (15*ArcTanh[Sin[a + b*x]])/(256*b) - (15*Csc[a + b*x])/(256*b) + (5*Csc[a + b*x]*Sec[a + b*x]^2)/(256*b) + (Csc[a + b*x]*Sec[a + b*x]^4)/(128*b)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Csc[a + b*x]*Sin[2*a + 2*b*x]^8, x, 4, -((256*Cos[a + b*x]^9)/(9*b)) + (768*Cos[a + b*x]^11)/(11*b) - (768*Cos[a + b*x]^13)/(13*b) + (256*Cos[a + b*x]^15)/(15*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^7, x, 4, (128*Sin[a + b*x]^7)/(7*b) - (128*Sin[a + b*x]^9)/(3*b) + (384*Sin[a + b*x]^11)/(11*b) - (128*Sin[a + b*x]^13)/(13*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^6, x, 4, -((64*Cos[a + b*x]^7)/(7*b)) + (128*Cos[a + b*x]^9)/(9*b) - (64*Cos[a + b*x]^11)/(11*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^5, x, 4, (32*Sin[a + b*x]^5)/(5*b) - (64*Sin[a + b*x]^7)/(7*b) + (32*Sin[a + b*x]^9)/(9*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^4, x, 4, -((16*Cos[a + b*x]^5)/(5*b)) + (16*Cos[a + b*x]^7)/(7*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^3, x, 4, (8*Sin[a + b*x]^3)/(3*b) - (8*Sin[a + b*x]^5)/(5*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^2, x, 3, -((4*Cos[a + b*x]^3)/(3*b))} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^1, x, 2, (2*Sin[a + b*x])/b} -{Csc[a + b*x]*Csc[2*a + 2*b*x]^1, x, 4, ArcTanh[Sin[a + b*x]]/(2*b) - Csc[a + b*x]/(2*b)} -{Csc[a + b*x]*Csc[2*a + 2*b*x]^2, x, 5, -((3*ArcTanh[Cos[a + b*x]])/(8*b)) + (3*Sec[a + b*x])/(8*b) - (Csc[a + b*x]^2*Sec[a + b*x])/(8*b)} -{Csc[a + b*x]*Csc[2*a + 2*b*x]^3, x, 6, (5*ArcTanh[Sin[a + b*x]])/(16*b) - (5*Csc[a + b*x])/(16*b) - (5*Csc[a + b*x]^3)/(48*b) + (Csc[a + b*x]^3*Sec[a + b*x]^2)/(16*b)} -{Csc[a + b*x]*Csc[2*a + 2*b*x]^4, x, 7, -((35*ArcTanh[Cos[a + b*x]])/(128*b)) + (35*Sec[a + b*x])/(128*b) + (35*Sec[a + b*x]^3)/(384*b) - (7*Csc[a + b*x]^2*Sec[a + b*x]^3)/(128*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(64*b)} - - -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^8, x, 9, (5*x)/8 + (5*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(12*b) + (Cos[a + b*x]^5*Sin[a + b*x])/(3*b) + (2*Cos[a + b*x]^7*Sin[a + b*x])/(7*b) - (16*Cos[a + b*x]^9*Sin[a + b*x])/(7*b) - (160*Cos[a + b*x]^9*Sin[a + b*x]^3)/(21*b) - (128*Cos[a + b*x]^9*Sin[a + b*x]^5)/(7*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^7, x, 5, -((16*Cos[a + b*x]^8)/b) + (128*Cos[a + b*x]^10)/(5*b) - (32*Cos[a + b*x]^12)/(3*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^6, x, 7, (3*x)/4 + (3*Cos[a + b*x]*Sin[a + b*x])/(4*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(2*b) + (2*Cos[a + b*x]^5*Sin[a + b*x])/(5*b) - (12*Cos[a + b*x]^7*Sin[a + b*x])/(5*b) - (32*Cos[a + b*x]^7*Sin[a + b*x]^3)/(5*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^5, x, 4, -((16*Cos[a + b*x]^6)/(3*b)) + (4*Cos[a + b*x]^8)/b} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^4, x, 5, x + (Cos[a + b*x]*Sin[a + b*x])/b + (2*Cos[a + b*x]^3*Sin[a + b*x])/(3*b) - (8*Cos[a + b*x]^5*Sin[a + b*x])/(3*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^3, x, 3, -((2*Cos[a + b*x]^4)/b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^2, x, 3, 2*x + (2*Cos[a + b*x]*Sin[a + b*x])/b} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^1, x, 2, (2*Log[Sin[a + b*x]])/b} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^1, x, 4, -(Cot[a + b*x]^2/(4*b)) + Log[Tan[a + b*x]]/(2*b)} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^2, x, 4, -(Cot[a + b*x]/(2*b)) - Cot[a + b*x]^3/(12*b) + Tan[a + b*x]/(4*b)} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^3, x, 5, -((3*Cot[a + b*x]^2)/(16*b)) - Cot[a + b*x]^4/(32*b) + (3*Log[Tan[a + b*x]])/(8*b) + Tan[a + b*x]^2/(16*b)} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^4, x, 4, -((3*Cot[a + b*x])/(8*b)) - Cot[a + b*x]^3/(12*b) - Cot[a + b*x]^5/(80*b) + Tan[a + b*x]/(4*b) + Tan[a + b*x]^3/(48*b)} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^5, x, 5, -((5*Cot[a + b*x]^2)/(32*b)) - (5*Cot[a + b*x]^4)/(128*b) - Cot[a + b*x]^6/(192*b) + (5*Log[Tan[a + b*x]])/(16*b) + (5*Tan[a + b*x]^2)/(64*b) + Tan[a + b*x]^4/(128*b)} -{Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6, x, 4, -((5*Cot[a + b*x])/(16*b)) - (5*Cot[a + b*x]^3)/(64*b) - (3*Cot[a + b*x]^5)/(160*b) - Cot[a + b*x]^7/(448*b) + (15*Tan[a + b*x])/(64*b) + Tan[a + b*x]^3/(32*b) + Tan[a + b*x]^5/(320*b)} - - -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10, x, 4, -((1024*Cos[a + b*x]^11)/(11*b)) + (3072*Cos[a + b*x]^13)/(13*b) - (1024*Cos[a + b*x]^15)/(5*b) + (1024*Cos[a + b*x]^17)/(17*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^9, x, 4, (512*Sin[a + b*x]^7)/(7*b) - (2048*Sin[a + b*x]^9)/(9*b) + (3072*Sin[a + b*x]^11)/(11*b) - (2048*Sin[a + b*x]^13)/(13*b) + (512*Sin[a + b*x]^15)/(15*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^8, x, 4, -((256*Cos[a + b*x]^9)/(9*b)) + (512*Cos[a + b*x]^11)/(11*b) - (256*Cos[a + b*x]^13)/(13*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^7, x, 4, (128*Sin[a + b*x]^5)/(5*b) - (384*Sin[a + b*x]^7)/(7*b) + (128*Sin[a + b*x]^9)/(3*b) - (128*Sin[a + b*x]^11)/(11*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^6, x, 4, -((64*Cos[a + b*x]^7)/(7*b)) + (64*Cos[a + b*x]^9)/(9*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^5, x, 4, (32*Sin[a + b*x]^3)/(3*b) - (64*Sin[a + b*x]^5)/(5*b) + (32*Sin[a + b*x]^7)/(7*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^4, x, 3, -((16*Cos[a + b*x]^5)/(5*b))} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^3, x, 3, (8*Sin[a + b*x])/b - (8*Sin[a + b*x]^3)/(3*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^2, x, 4, -((4*ArcTanh[Cos[a + b*x]])/b) + (4*Cos[a + b*x])/b} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^1, x, 3, -((2*Csc[a + b*x])/b)} -{Csc[a + b*x]^3*Csc[2*a + 2*b*x]^1, x, 5, ArcTanh[Sin[a + b*x]]/(2*b) - Csc[a + b*x]/(2*b) - Csc[a + b*x]^3/(6*b)} -{Csc[a + b*x]^3*Csc[2*a + 2*b*x]^2, x, 6, -((15*ArcTanh[Cos[a + b*x]])/(32*b)) + (15*Sec[a + b*x])/(32*b) - (5*Csc[a + b*x]^2*Sec[a + b*x])/(32*b) - (Csc[a + b*x]^4*Sec[a + b*x])/(16*b)} -{Csc[a + b*x]^3*Csc[2*a + 2*b*x]^3, x, 6, (7*ArcTanh[Sin[a + b*x]])/(16*b) - (7*Csc[a + b*x])/(16*b) - (7*Csc[a + b*x]^3)/(48*b) - (7*Csc[a + b*x]^5)/(80*b) + (Csc[a + b*x]^5*Sec[a + b*x]^2)/(16*b)} -{Csc[a + b*x]^3*Csc[2*a + 2*b*x]^4, x, 8, -((105*ArcTanh[Cos[a + b*x]])/(256*b)) + (105*Sec[a + b*x])/(256*b) + (35*Sec[a + b*x]^3)/(256*b) - (21*Csc[a + b*x]^2*Sec[a + b*x]^3)/(256*b) - (3*Csc[a + b*x]^4*Sec[a + b*x]^3)/(128*b) - (Csc[a + b*x]^6*Sec[a + b*x]^3)/(96*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Sin[2 a+2 b x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2), x, 4, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(32*b)) + (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(32*b) - (5*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(16*b) + (5*Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(24*b) - (Cos[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(6*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2), x, 3, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(16*b)) - (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(16*b) + (3*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(8*b) - (Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(4*b)} -{Sin[a + b*x]*Sin[2*a + 2*b*x]^(1/2), x, 2, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(4*b)) + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(4*b) - (Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b)} -{Sin[a + b*x]/Sin[2*a + 2*b*x]^(1/2), x, 1, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(2*b)) - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(2*b)} -{Sin[a + b*x]/Sin[2*a + 2*b*x]^(3/2), x, 1, Sin[a + b*x]/(b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]/Sin[2*a + 2*b*x]^(5/2), x, 2, Sin[a + b*x]/(3*b*Sin[2*a + 2*b*x]^(3/2)) - (2*Cos[a + b*x])/(3*b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]/Sin[2*a + 2*b*x]^(7/2), x, 3, Sin[a + b*x]/(5*b*Sin[2*a + 2*b*x]^(5/2)) - (4*Cos[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(3/2)) + (8*Sin[a + b*x])/(15*b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]/Sin[2*a + 2*b*x]^(9/2), x, 4, Sin[a + b*x]/(7*b*Sin[2*a + 2*b*x]^(7/2)) - (6*Cos[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(5/2)) + (8*Sin[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(3/2)) - (16*Cos[a + b*x])/(35*b*Sqrt[Sin[2*a + 2*b*x]])} - - -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2), x, 4, (5*EllipticF[a - Pi/4 + b*x, 2])/(42*b) - (5*Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(42*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(5/2))/(14*b) - Sin[2*a + 2*b*x]^(9/2)/(18*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2), x, 3, (3*EllipticE[a - Pi/4 + b*x, 2])/(10*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(10*b) - Sin[2*a + 2*b*x]^(7/2)/(14*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2), x, 3, EllipticF[a - Pi/4 + b*x, 2]/(6*b) - (Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(6*b) - Sin[2*a + 2*b*x]^(5/2)/(10*b)} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(1/2), x, 2, EllipticE[a - Pi/4 + b*x, 2]/(2*b) - Sin[2*a + 2*b*x]^(3/2)/(6*b)} -{Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(1/2), x, 2, EllipticF[a - Pi/4 + b*x, 2]/(2*b) - Sqrt[Sin[2*a + 2*b*x]]/(2*b)} -{Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2), x, 2, -(EllipticE[a - Pi/4 + b*x, 2]/(2*b)) + Sin[a + b*x]^2/(b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2), x, 2, EllipticF[a - Pi/4 + b*x, 2]/(6*b) + Sin[a + b*x]^2/(3*b*Sin[2*a + 2*b*x]^(3/2))} -{Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2), x, 3, -((3*EllipticE[a - Pi/4 + b*x, 2])/(10*b)) + Sin[a + b*x]^2/(5*b*Sin[2*a + 2*b*x]^(5/2)) - (3*Cos[2*a + 2*b*x])/(10*b*Sqrt[Sin[2*a + 2*b*x]])} - - -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2), x, 4, -((7*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(64*b)) - (7*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(64*b) + (7*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(32*b) - (7*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(48*b) - (Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(12*b)} -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^(1/2), x, 3, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(32*b)) + (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(32*b) - (5*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(16*b) - (Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(8*b)} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(1/2), x, 2, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(8*b)) - (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(8*b) - (Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(4*b)} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2), x, 3, ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(4*b) - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(4*b) + Sin[a + b*x]/(b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2), x, 1, Sin[a + b*x]^3/(3*b*Sin[2*a + 2*b*x]^(3/2))} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2), x, 2, Sin[a + b*x]^3/(5*b*Sin[2*a + 2*b*x]^(5/2)) + Sin[a + b*x]/(5*b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(9/2), x, 3, Sin[a + b*x]^3/(7*b*Sin[2*a + 2*b*x]^(7/2)) + (2*Sin[a + b*x])/(21*b*Sin[2*a + 2*b*x]^(3/2)) - (4*Cos[a + b*x])/(21*b*Sqrt[Sin[2*a + 2*b*x]])} -{Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(11/2), x, 4, Sin[a + b*x]^3/(9*b*Sin[2*a + 2*b*x]^(9/2)) + Sin[a + b*x]/(15*b*Sin[2*a + 2*b*x]^(5/2)) - (4*Cos[a + b*x])/(45*b*Sin[2*a + 2*b*x]^(3/2)) + (8*Sin[a + b*x])/(45*b*Sqrt[Sin[2*a + 2*b*x]])} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{Csc[a + b*x]*Sin[2*a + 2*b*x]^(7/2), x, 5, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(16*b)) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(16*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(8*b) - (5*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(12*b) + (Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(3*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^(5/2), x, 4, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(8*b)) + (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(8*b) - (3*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(4*b) + (Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(2*b)} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^(3/2), x, 3, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(2*b)) - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(2*b) + (Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/b} -{Csc[a + b*x]*Sin[2*a + 2*b*x]^(1/2), x, 2, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/b) + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/b} -{Csc[a + b*x]/Sin[2*a + 2*b*x]^(1/2), x, 1, -((Csc[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/b)} -{Csc[a + b*x]/Sin[2*a + 2*b*x]^(3/2), x, 3, -((2*Cos[a + b*x])/(3*b*Sin[2*a + 2*b*x]^(3/2))) + (4*Sin[a + b*x])/(3*b*Sqrt[Sin[2*a + 2*b*x]])} -{Csc[a + b*x]/Sin[2*a + 2*b*x]^(5/2), x, 4, -((2*Cos[a + b*x])/(5*b*Sin[2*a + 2*b*x]^(5/2))) + (8*Sin[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(3/2)) - (16*Cos[a + b*x])/(15*b*Sqrt[Sin[2*a + 2*b*x]])} -{Csc[a + b*x]/Sin[2*a + 2*b*x]^(7/2), x, 5, -((2*Cos[a + b*x])/(7*b*Sin[2*a + 2*b*x]^(7/2))) + (12*Sin[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(5/2)) - (16*Cos[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(3/2)) + (32*Sin[a + b*x])/(35*b*Sqrt[Sin[2*a + 2*b*x]])} - - -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(9/2), x, 4, (6*EllipticE[a - Pi/4 + b*x, 2])/(5*b) - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(5*b) - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(7/2))/(7*b) + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(11/2))/(7*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2), x, 4, (2*EllipticF[a - Pi/4 + b*x, 2])/(3*b) - (2*Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(3*b) - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(5/2))/(5*b) + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(9/2))/(5*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2), x, 3, (2*EllipticE[a - Pi/4 + b*x, 2])/b - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(3*b) + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2))/(3*b)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2), x, 3, (2*EllipticF[a - Pi/4 + b*x, 2])/b - (2*Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/b + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2))/b} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(1/2), x, 2, -((2*EllipticE[a - Pi/4 + b*x, 2])/b) - (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2))/b} -{Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(1/2), x, 2, (2*EllipticF[a - Pi/4 + b*x, 2])/(3*b) - (Csc[a + b*x]^2*Sqrt[Sin[2*a + 2*b*x]])/(3*b)} -{Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2), x, 3, -((6*EllipticE[a - Pi/4 + b*x, 2])/(5*b)) - (6*Cos[2*a + 2*b*x])/(5*b*Sqrt[Sin[2*a + 2*b*x]]) - Csc[a + b*x]^2/(5*b*Sqrt[Sin[2*a + 2*b*x]])} -{Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2), x, 3, (10*EllipticF[a - Pi/4 + b*x, 2])/(21*b) - (10*Cos[2*a + 2*b*x])/(21*b*Sin[2*a + 2*b*x]^(3/2)) - Csc[a + b*x]^2/(7*b*Sin[2*a + 2*b*x]^(3/2))} -{Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2), x, 4, -((14*EllipticE[a - Pi/4 + b*x, 2])/(15*b)) - (14*Cos[2*a + 2*b*x])/(45*b*Sin[2*a + 2*b*x]^(5/2)) - Csc[a + b*x]^2/(9*b*Sin[2*a + 2*b*x]^(5/2)) - (14*Cos[2*a + 2*b*x])/(15*b*Sqrt[Sin[2*a + 2*b*x]])} -{Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(9/2), x, 4, (30*EllipticF[a - Pi/4 + b*x, 2])/(77*b) - (18*Cos[2*a + 2*b*x])/(77*b*Sin[2*a + 2*b*x]^(7/2)) - Csc[a + b*x]^2/(11*b*Sin[2*a + 2*b*x]^(7/2)) - (30*Cos[2*a + 2*b*x])/(77*b*Sin[2*a + 2*b*x]^(3/2))} - - -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(9/2), x, 7, -((7*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(8*b)) + (7*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(8*b) - (7*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(4*b) + (7*Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(6*b) - (14*Cos[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(15*b) + (4*Sin[a + b*x]*Sin[2*a + 2*b*x]^(7/2))/(5*b) + (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(11/2))/(5*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(7/2), x, 6, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(4*b)) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(4*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b) - (5*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(3*b) + (4*Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(3*b) + (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(9/2))/(3*b)} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(5/2), x, 5, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/b) + (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/b - (6*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/b + (4*Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/b + (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(7/2))/b} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2), x, 4, (2*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/b + (2*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/b - (4*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/b - (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(5/2))/b} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(1/2), x, 1, -((Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2))/(3*b))} -{Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(1/2), x, 2, -((4*Csc[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(5*b)) - (Csc[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]])/(5*b)} -{Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2), x, 4, -((16*Cos[a + b*x])/(21*b*Sin[2*a + 2*b*x]^(3/2))) - Csc[a + b*x]^3/(7*b*Sqrt[Sin[2*a + 2*b*x]]) + (32*Sin[a + b*x])/(21*b*Sqrt[Sin[2*a + 2*b*x]])} -{Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2), x, 5, -((8*Cos[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(5/2))) - Csc[a + b*x]^3/(9*b*Sin[2*a + 2*b*x]^(3/2)) + (32*Sin[a + b*x])/(45*b*Sin[2*a + 2*b*x]^(3/2)) - (64*Cos[a + b*x])/(45*b*Sqrt[Sin[2*a + 2*b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Sin[2 a+2 b x]^n with m symbolic*) - - -{Sin[a + b*x]^3*Sin[2*a + 2*b*x]^m, x, 2, ((Cos[a + b*x]^2)^((1 - m)/2)*Hypergeometric2F1[(1 - m)/2, (4 + m)/2, (6 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]^3*Sin[2*a + 2*b*x]^m*Tan[a + b*x])/(b*(4 + m))} -{Sin[a + b*x]^2*Sin[2*a + 2*b*x]^m, x, 2, ((Cos[a + b*x]^2)^((1 - m)/2)*Hypergeometric2F1[(1 - m)/2, (3 + m)/2, (5 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]^2*Sin[2*a + 2*b*x]^m*Tan[a + b*x])/(b*(3 + m))} -{Sin[a + b*x]^1*Sin[2*a + 2*b*x]^m, x, 2, ((Cos[a + b*x]^2)^((1 - m)/2)*Hypergeometric2F1[(1 - m)/2, (2 + m)/2, (4 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]*Sin[2*a + 2*b*x]^m*Tan[a + b*x])/(b*(2 + m))} -{Csc[a + b*x]^1*Sin[2*a + 2*b*x]^m, x, 2, ((Cos[a + b*x]^2)^((1 - m)/2)*Hypergeometric2F1[(1 - m)/2, m/2, (2 + m)/2, Sin[a + b*x]^2]*Sec[a + b*x]*Sin[2*a + 2*b*x]^m)/(b*m)} -{Csc[a + b*x]^2*Sin[2*a + 2*b*x]^m, x, 2, -(((Cos[a + b*x]^2)^((1 - m)/2)*Csc[a + b*x]*Hypergeometric2F1[(1 - m)/2, (1/2)*(-1 + m), (1 + m)/2, Sin[a + b*x]^2]*Sec[a + b*x]*Sin[2*a + 2*b*x]^m)/(b*(1 - m)))} -{Csc[a + b*x]^3*Sin[2*a + 2*b*x]^m, x, 2, -(((Cos[a + b*x]^2)^((1 - m)/2)*Csc[a + b*x]^2*Hypergeometric2F1[(1 - m)/2, (1/2)*(-2 + m), m/2, Sin[a + b*x]^2]*Sec[a + b*x]*Sin[2*a + 2*b*x]^m)/(b*(2 - m)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[2 a+2 b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[2 a+2 b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]*Sin[2*a + 2*b*x]^7, x, 4, -((128*Cos[a + b*x]^9)/(9*b)) + (384*Cos[a + b*x]^11)/(11*b) - (384*Cos[a + b*x]^13)/(13*b) + (128*Cos[a + b*x]^15)/(15*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^6, x, 4, (64*Sin[a + b*x]^7)/(7*b) - (64*Sin[a + b*x]^9)/(3*b) + (192*Sin[a + b*x]^11)/(11*b) - (64*Sin[a + b*x]^13)/(13*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^5, x, 4, -((32*Cos[a + b*x]^7)/(7*b)) + (64*Cos[a + b*x]^9)/(9*b) - (32*Cos[a + b*x]^11)/(11*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^4, x, 4, (16*Sin[a + b*x]^5)/(5*b) - (32*Sin[a + b*x]^7)/(7*b) + (16*Sin[a + b*x]^9)/(9*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^3, x, 4, -((8*Cos[a + b*x]^5)/(5*b)) + (8*Cos[a + b*x]^7)/(7*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^2, x, 4, (4*Sin[a + b*x]^3)/(3*b) - (4*Sin[a + b*x]^5)/(5*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^1, x, 1, -(Cos[a + b*x]/(2*b)) - Cos[3*a + 3*b*x]/(6*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^1, x, 2, -(ArcTanh[Cos[a + b*x]]/(2*b))} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^2, x, 4, ArcTanh[Sin[a + b*x]]/(4*b) - Csc[a + b*x]/(4*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^3, x, 5, -((3*ArcTanh[Cos[a + b*x]])/(16*b)) + (3*Sec[a + b*x])/(16*b) - (Csc[a + b*x]^2*Sec[a + b*x])/(16*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^4, x, 6, (5*ArcTanh[Sin[a + b*x]])/(32*b) - (5*Csc[a + b*x])/(32*b) - (5*Csc[a + b*x]^3)/(96*b) + (Csc[a + b*x]^3*Sec[a + b*x]^2)/(32*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^5, x, 7, -((35*ArcTanh[Cos[a + b*x]])/(256*b)) + (35*Sec[a + b*x])/(256*b) + (35*Sec[a + b*x]^3)/(768*b) - (7*Csc[a + b*x]^2*Sec[a + b*x]^3)/(256*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(128*b)} - - -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^5, x, 5, -((4*Cos[a + b*x]^8)/b) + (32*Cos[a + b*x]^10)/(5*b) - (8*Cos[a + b*x]^12)/(3*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^4, x, 6, (3*x)/16 - (3*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(32*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^3)/(16*b) + Sin[2*a + 2*b*x]^5/(20*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^3, x, 4, -((4*Cos[a + b*x]^6)/(3*b)) + Cos[a + b*x]^8/b} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^2, x, 5, x/4 - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(8*b) + Sin[2*a + 2*b*x]^3/(12*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^1, x, 3, -(Cos[a + b*x]^4/(2*b))} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^1, x, 2, Log[Sin[a + b*x]]/(2*b)} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^2, x, 3, -(Cot[a + b*x]/(4*b))} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^3, x, 4, -(Cot[a + b*x]^2/(16*b)) + Log[Tan[a + b*x]]/(8*b)} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^4, x, 4, -(Cot[a + b*x]/(8*b)) - Cot[a + b*x]^3/(48*b) + Tan[a + b*x]/(16*b)} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^5, x, 5, -((3*Cot[a + b*x]^2)/(64*b)) - Cot[a + b*x]^4/(128*b) + (3*Log[Tan[a + b*x]])/(32*b) + Tan[a + b*x]^2/(64*b)} - - -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5, x, 4, -((32*Cos[a + b*x]^9)/(9*b)) + (64*Cos[a + b*x]^11)/(11*b) - (32*Cos[a + b*x]^13)/(13*b)} -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^4, x, 4, (16*Sin[a + b*x]^5)/(5*b) - (48*Sin[a + b*x]^7)/(7*b) + (16*Sin[a + b*x]^9)/(3*b) - (16*Sin[a + b*x]^11)/(11*b)} -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^3, x, 4, -((8*Cos[a + b*x]^7)/(7*b)) + (8*Cos[a + b*x]^9)/(9*b)} -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^2, x, 4, (4*Sin[a + b*x]^3)/(3*b) - (8*Sin[a + b*x]^5)/(5*b) + (4*Sin[a + b*x]^7)/(7*b)} -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^1, x, 3, -((2*Cos[a + b*x]^5)/(5*b))} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^1, x, 4, -(ArcTanh[Cos[a + b*x]]/(2*b)) + Cos[a + b*x]/(2*b)} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^2, x, 3, -(Csc[a + b*x]/(4*b))} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^3, x, 3, -(ArcTanh[Cos[a + b*x]]/(16*b)) - (Cot[a + b*x]*Csc[a + b*x])/(16*b)} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^4, x, 5, ArcTanh[Sin[a + b*x]]/(16*b) - Csc[a + b*x]/(16*b) - Csc[a + b*x]^3/(48*b)} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^5, x, 6, -((15*ArcTanh[Cos[a + b*x]])/(256*b)) + (15*Sec[a + b*x])/(256*b) - (5*Csc[a + b*x]^2*Sec[a + b*x])/(256*b) - (Csc[a + b*x]^4*Sec[a + b*x])/(128*b)} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[2 a+2 b x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]*Sin[2*a + 2*b*x]^(5/2), x, 4, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(32*b)) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(32*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(16*b) - (5*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(24*b) + (Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(6*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2), x, 3, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(16*b)) + (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(16*b) - (3*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(8*b) + (Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(4*b)} -{Cos[a + b*x]*Sin[2*a + 2*b*x]^(1/2), x, 2, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(4*b)) - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(4*b) + (Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^(1/2), x, 1, -(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(2*b)) + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(2*b)} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^(3/2), x, 1, -(Cos[a + b*x]/(b*Sqrt[Sin[2*a + 2*b*x]]))} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^(5/2), x, 2, -(Cos[a + b*x]/(3*b*Sin[2*a + 2*b*x]^(3/2))) + (2*Sin[a + b*x])/(3*b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^(7/2), x, 3, -(Cos[a + b*x]/(5*b*Sin[2*a + 2*b*x]^(5/2))) + (4*Sin[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(3/2)) - (8*Cos[a + b*x])/(15*b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]/Sin[2*a + 2*b*x]^(9/2), x, 4, -(Cos[a + b*x]/(7*b*Sin[2*a + 2*b*x]^(7/2))) + (6*Sin[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(5/2)) - (8*Cos[a + b*x])/(35*b*Sin[2*a + 2*b*x]^(3/2)) + (16*Sin[a + b*x])/(35*b*Sqrt[Sin[2*a + 2*b*x]])} - - -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2), x, 4, (5*EllipticF[a - Pi/4 + b*x, 2])/(42*b) - (5*Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(42*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(5/2))/(14*b) + Sin[2*a + 2*b*x]^(9/2)/(18*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2), x, 3, (3*EllipticE[a - Pi/4 + b*x, 2])/(10*b) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(10*b) + Sin[2*a + 2*b*x]^(7/2)/(14*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2), x, 3, EllipticF[a - Pi/4 + b*x, 2]/(6*b) - (Cos[2*a + 2*b*x]*Sqrt[Sin[2*a + 2*b*x]])/(6*b) + Sin[2*a + 2*b*x]^(5/2)/(10*b)} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(1/2), x, 2, EllipticE[a - Pi/4 + b*x, 2]/(2*b) + Sin[2*a + 2*b*x]^(3/2)/(6*b)} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(1/2), x, 2, EllipticF[a - Pi/4 + b*x, 2]/(2*b) + Sqrt[Sin[2*a + 2*b*x]]/(2*b)} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2), x, 2, -(EllipticE[a - Pi/4 + b*x, 2]/(2*b)) - Cos[a + b*x]^2/(b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2), x, 2, EllipticF[a - Pi/4 + b*x, 2]/(6*b) - Cos[a + b*x]^2/(3*b*Sin[2*a + 2*b*x]^(3/2))} -{Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2), x, 3, -((3*EllipticE[a - Pi/4 + b*x, 2])/(10*b)) - Cos[a + b*x]^2/(5*b*Sin[2*a + 2*b*x]^(5/2)) - (3*Cos[2*a + 2*b*x])/(10*b*Sqrt[Sin[2*a + 2*b*x]])} - - -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2), x, 4, -((7*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(64*b)) + (7*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(64*b) - (7*Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(32*b) + (7*Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(48*b) + (Cos[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(12*b)} -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^(1/2), x, 3, -((5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(32*b)) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(32*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(16*b) + (Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(8*b)} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(1/2), x, 2, -((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(8*b)) + (3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])/(8*b) + (Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(4*b)} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2), x, 3, ArcSin[Cos[a + b*x] - Sin[a + b*x]]/(4*b) + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]]/(4*b) - Cos[a + b*x]/(b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2), x, 1, -(Cos[a + b*x]^3/(3*b*Sin[2*a + 2*b*x]^(3/2)))} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2), x, 2, -(Cos[a + b*x]^3/(5*b*Sin[2*a + 2*b*x]^(5/2))) - Cos[a + b*x]/(5*b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(9/2), x, 3, -(Cos[a + b*x]^3/(7*b*Sin[2*a + 2*b*x]^(7/2))) - (2*Cos[a + b*x])/(21*b*Sin[2*a + 2*b*x]^(3/2)) + (4*Sin[a + b*x])/(21*b*Sqrt[Sin[2*a + 2*b*x]])} -{Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(11/2), x, 4, -(Cos[a + b*x]^3/(9*b*Sin[2*a + 2*b*x]^(9/2))) - Cos[a + b*x]/(15*b*Sin[2*a + 2*b*x]^(5/2)) + (4*Sin[a + b*x])/(45*b*Sin[2*a + 2*b*x]^(3/2)) - (8*Cos[a + b*x])/(45*b*Sqrt[Sin[2*a + 2*b*x]])} - - -(* 2*Cos[x]/Sqrt[Sin[2*x]] == Csc[x]*Sqrt[Sin[2*x]] *) -{Cos[x]/Sqrt[Sin[2*x]], x, 1, (-(1/2))*ArcSin[Cos[x] - Sin[x]] + (1/2)*Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]} -{Csc[x]*Sqrt[Sin[2*x]], x, 2, -ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[2 a+2 b x]^n with m symbolic*) - - -{Cos[a + b*x]^3*Sin[2*a + 2*b*x]^m, x, 2, -((1/(b*(4 + m)))*(Cos[a + b*x]^3*Cot[a + b*x]*Hypergeometric2F1[(1 - m)/2, (4 + m)/2, (6 + m)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^((1 - m)/2)*Sin[2*a + 2*b*x]^m))} -{Cos[a + b*x]^2*Sin[2*a + 2*b*x]^m, x, 2, -((1/(b*(3 + m)))*(Cos[a + b*x]^2*Cot[a + b*x]*Hypergeometric2F1[(1 - m)/2, (3 + m)/2, (5 + m)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^((1 - m)/2)*Sin[2*a + 2*b*x]^m))} -{Cos[a + b*x]^1*Sin[2*a + 2*b*x]^m, x, 2, -((1/(b*(2 + m)))*(Cos[a + b*x]*Cot[a + b*x]*Hypergeometric2F1[(1 - m)/2, (2 + m)/2, (4 + m)/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^((1 - m)/2)*Sin[2*a + 2*b*x]^m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[a+b x]^n Sin[2 a+2 b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Sin[a+b x]^n Sin[2 a+2 b x]^p*) - - -{Cos[a + b*x]^2*Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2, x, 4, -((4*Cos[a + b*x]^5)/(5*b)) + (8*Cos[a + b*x]^7)/(7*b) - (4*Cos[a + b*x]^9)/(9*b)} - - -(* ::Subsection:: *) -(*Integrands of the form Cos[a+b x]^m Sin[a+b x]^n Sin[2 a+2 b x]^(p/2)*) - - -(* ::Title:: *) -(*Integrands of the form Trig[a+b x]^m Trig[c+d x]^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Trig[c+d x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Sin[c+d x]^n*) - - -{Sin[a + b*x]*Sin[c + d*x]^n, x, 10, -((2^(-1 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, (b - d*n)/(2*d), (1/2)*(2 + b/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(b - d*n))) - (2^(-1 - n)*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((b + d*n)/(2*d)), 1 - (b + d*n)/(2*d), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(b + d*n))} - -{Sin[a + b*x]*Sin[c + d*x]^3, x, 6, -(Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d))) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) - (3*Sin[a + c + (b + d)*x])/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} -{Sin[a + b*x]*Sin[c + d*x]^2, x, 5, -(Cos[a + b*x]/(2*b)) + Cos[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cos[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Sin[a + b*x]*Sin[c + d*x]^1, x, 4, Sin[a - c + (b - d)*x]/(2*(b - d)) - Sin[a + c + (b + d)*x]/(2*(b + d))} -{Sin[a + b*x]*Csc[c + b*x]^1, x, 3, x*Cos[a - c] + (Log[Sin[c + b*x]]*Sin[a - c])/b} -{Sin[a + b*x]*Csc[c + b*x]^2, x, 4, -((ArcTanh[Cos[c + b*x]]*Cos[a - c])/b) - (Csc[c + b*x]*Sin[a - c])/b} -{Sin[a + b*x]*Csc[c + b*x]^3, x, 5, -((Cos[a - c]*Cot[c + b*x])/b) - (Csc[c + b*x]^2*Sin[a - c])/(2*b)} -{Sin[a + b*x]*Csc[c + b*x]^4, x, 5, -((ArcTanh[Cos[c + b*x]]*Cos[a - c])/(2*b)) - (Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(2*b) - (Csc[c + b*x]^3*Sin[a - c])/(3*b)} -{Sin[a + b*x]*Csc[c + b*x]^5, x, 5, -((Cos[a - c]*Cot[c + b*x])/b) - (Cos[a - c]*Cot[c + b*x]^3)/(3*b) - (Csc[c + b*x]^4*Sin[a - c])/(4*b)} -{Sin[a + b*x]*Csc[c + b*x]^6, x, 6, -((3*ArcTanh[Cos[c + b*x]]*Cos[a - c])/(8*b)) - (3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(8*b) - (Cos[a - c]*Cot[c + b*x]*Csc[c + b*x]^3)/(4*b) - (Csc[c + b*x]^5*Sin[a - c])/(5*b)} - - -{Sin[a + b*x]^2*Sin[c + d*x]^n, x, 15, -((I*2^(-2 - n)*E^((-I)*(2*a + c*n) - I*(2*b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*(-((2*b)/d) - n), -n, (1/2)*(2 - (2*b)/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(2*b + d*n))) + (I*2^(-2 - n)*E^(I*(2*a - c*n) + I*(2*b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*((2*b)/d - n), -n, (1/2)*(2 + (2*b)/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(2*b - d*n)) + (I*2^(-1 - n)*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -(n/2), 1 - n/2, E^(2*I*(c + d*x))])/((1 - E^(2*I*(c + d*x)))^n*(d*n))} - -{Sin[a + b*x]^2*Sin[c + d*x]^1, x, 5, -(Cos[2*a - c + (2*b - d)*x]/(4*(2*b - d))) - Cos[c + d*x]/(2*d) + Cos[2*a + c + (2*b + d)*x]/(4*(2*b + d))} -{Sin[a + b*x]^2*Sin[c + d*x]^2, x, 6, x/4 - Sin[2*a + 2*b*x]/(8*b) + Sin[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) - Sin[2*c + 2*d*x]/(8*d) + Sin[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} -{Sin[a + b*x]^2*Sin[c + d*x]^3, x, 8, Cos[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) - (3*Cos[2*a - c + (2*b - d)*x])/(16*(2*b - d)) - (3*Cos[c + d*x])/(8*d) + Cos[3*c + 3*d*x]/(24*d) + (3*Cos[2*a + c + (2*b + d)*x])/(16*(2*b + d)) - Cos[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} - - -{Sin[a + b*x]^3*Sin[c + d*x]^n, x, 18, (2^(-3 - n)*E^(I*(3*a - c*n) + I*(3*b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*((3*b)/d - n), -n, (1/2)*(2 + (3*b)/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(3*b - d*n)) - (3*2^(-3 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, (b - d*n)/(2*d), (1/2)*(2 + b/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(b - d*n)) - (3*2^(-3 - n)*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((b + d*n)/(2*d)), 1 - (b + d*n)/(2*d), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(b + d*n)) + (2^(-3 - n)*E^((-I)*(3*a + c*n) - I*(3*b + d*n)*x + I*n*(c + d*x))*(I/E^(I*(c + d*x)) - I*E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((3*b + d*n)/(2*d)), (1/2)*(2 - (3*b)/d - n), E^(2*I*(c + d*x))])/((1 - E^(2*I*c + 2*I*d*x))^n*(3*b + d*n))} - -{Sin[a + b*x]^3*Sin[c + d*x]^1, x, 6, (3*Sin[a - c + (b - d)*x])/(8*(b - d)) - Sin[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Sin[a + c + (b + d)*x])/(8*(b + d)) + Sin[3*a + c + (3*b + d)*x]/(8*(3*b + d))} -{Sin[a + b*x]^3*Sin[c + d*x]^2, x, 8, -((3*Cos[a + b*x])/(8*b)) + Cos[3*a + 3*b*x]/(24*b) + (3*Cos[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) - Cos[3*a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) + (3*Cos[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) - Cos[3*a + 2*c + (3*b + 2*d)*x]/(16*(3*b + 2*d))} -{Sin[a + b*x]^3*Sin[c + d*x]^3, x, 10, -((3*Sin[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d))) + (9*Sin[a - c + (b - d)*x])/(32*(b - d)) + Sin[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) - (3*Sin[3*a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Sin[a + c + (b + d)*x])/(32*(b + d)) - Sin[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Sin[3*a + c + (3*b + d)*x])/(32*(3*b + d)) + (3*Sin[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Cos[c+d x]^n*) - - -{Sin[a + b*x]*Cos[c + d*x]^n, x, 8, -((2^(-1 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, (b - d*n)/(2*d), (1/2)*(2 + b/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(b - d*n))) - (2^(-1 - n)*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((b + d*n)/(2*d)), 1 - (b + d*n)/(2*d), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(b + d*n))} - -{Sin[a + b*x]*Cos[c + d*x]^3, x, 6, -(Cos[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d))) - (3*Cos[a - c + (b - d)*x])/(8*(b - d)) - (3*Cos[a + c + (b + d)*x])/(8*(b + d)) - Cos[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} -{Sin[a + b*x]*Cos[c + d*x]^2, x, 5, -(Cos[a + b*x]/(2*b)) - Cos[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) - Cos[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Sin[a + b*x]*Cos[c + d*x]^1, x, 4, -(Cos[a - c + (b - d)*x]/(2*(b - d))) - Cos[a + c + (b + d)*x]/(2*(b + d))} -{Sin[a + b*x]*Sec[c + b*x]^1, x, 3, -((Cos[a - c]*Log[Cos[c + b*x]])/b) + x*Sin[a - c]} -{Sin[a + b*x]*Sec[c + b*x]^2, x, 4, (Cos[a - c]*Sec[c + b*x])/b + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/b} -{Sin[a + b*x]*Sec[c + b*x]^3, x, 5, (Cos[a - c]*Sec[c + b*x]^2)/(2*b) + (Sin[a - c]*Tan[c + b*x])/b} -{Sin[a + b*x]*Sec[c + b*x]^4, x, 5, (Cos[a - c]*Sec[c + b*x]^3)/(3*b) + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/(2*b) + (Sec[c + b*x]*Sin[a - c]*Tan[c + b*x])/(2*b)} -{Sin[a + b*x]*Sec[c + b*x]^5, x, 5, (Cos[a - c]*Sec[c + b*x]^4)/(4*b) + (Sin[a - c]*Tan[c + b*x])/b + (Sin[a - c]*Tan[c + b*x]^3)/(3*b)} -{Sin[a + b*x]*Sec[c + b*x]^6, x, 6, (Cos[a - c]*Sec[c + b*x]^5)/(5*b) + (3*ArcTanh[Sin[c + b*x]]*Sin[a - c])/(8*b) + (3*Sec[c + b*x]*Sin[a - c]*Tan[c + b*x])/(8*b) + (Sec[c + b*x]^3*Sin[a - c]*Tan[c + b*x])/(4*b)} - - -{Sin[a + b*x]^2*Cos[c + d*x]^n, x, 11, -((I*2^(-2 - n)*E^((-I)*(2*a + c*n) - I*(2*b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*(-((2*b)/d) - n), -n, (1/2)*(2 - (2*b)/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(2*b + d*n))) + (I*2^(-2 - n)*E^(I*(2*a - c*n) + I*(2*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*((2*b)/d - n), -n, (1/2)*(2 + (2*b)/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(2*b - d*n)) + (I*2^(-1 - n)*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -(n/2), 1 - n/2, -E^(2*I*(c + d*x))])/((1 + E^(2*I*(c + d*x)))^n*(d*n))} - -{Sin[a + b*x]^2*Cos[c + d*x]^1, x, 5, -(Sin[2*a - c + (2*b - d)*x]/(4*(2*b - d))) + Sin[c + d*x]/(2*d) - Sin[2*a + c + (2*b + d)*x]/(4*(2*b + d))} -{Sin[a + b*x]^2*Cos[c + d*x]^2, x, 6, x/4 - Sin[2*a + 2*b*x]/(8*b) - Sin[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) + Sin[2*c + 2*d*x]/(8*d) - Sin[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} -{Sin[a + b*x]^2*Cos[c + d*x]^3, x, 8, -(Sin[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d))) - (3*Sin[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sin[c + d*x])/(8*d) + Sin[3*c + 3*d*x]/(24*d) - (3*Sin[2*a + c + (2*b + d)*x])/(16*(2*b + d)) - Sin[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} - - -{Sin[a + b*x]^3*Cos[c + d*x]^n, x, 14, (2^(-3 - n)*E^(I*(3*a - c*n) + I*(3*b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[(1/2)*((3*b)/d - n), -n, (1/2)*(2 + (3*b)/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(3*b - d*n)) - (3*2^(-3 - n)*E^(I*(a - c*n) + I*(b - d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, (b - d*n)/(2*d), (1/2)*(2 + b/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(b - d*n)) - (3*2^(-3 - n)*E^((-I)*(a + c*n) - I*(b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((b + d*n)/(2*d)), 1 - (b + d*n)/(2*d), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(b + d*n)) + (2^(-3 - n)*E^((-I)*(3*a + c*n) - I*(3*b + d*n)*x + I*n*(c + d*x))*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^n*Hypergeometric2F1[-n, -((3*b + d*n)/(2*d)), (1/2)*(2 - (3*b)/d - n), -E^(2*I*(c + d*x))])/((1 + E^(2*I*c + 2*I*d*x))^n*(3*b + d*n))} - -{Sin[a + b*x]^3*Cos[c + d*x]^1, x, 6, -((3*Cos[a - c + (b - d)*x])/(8*(b - d))) + Cos[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cos[a + c + (b + d)*x])/(8*(b + d)) + Cos[3*a + c + (3*b + d)*x]/(8*(3*b + d))} -{Sin[a + b*x]^3*Cos[c + d*x]^2, x, 8, -((3*Cos[a + b*x])/(8*b)) + Cos[3*a + 3*b*x]/(24*b) - (3*Cos[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cos[3*a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cos[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cos[3*a + 2*c + (3*b + 2*d)*x]/(16*(3*b + 2*d))} -{Sin[a + b*x]^3*Cos[c + d*x]^3, x, 10, -((3*Cos[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d))) - (9*Cos[a - c + (b - d)*x])/(32*(b - d)) + Cos[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Cos[3*a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cos[a + c + (b + d)*x])/(32*(b + d)) + Cos[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Cos[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Cos[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -{Cos[a + b*x]/Sin[c + b*x]^1, x, 3, (Cos[a - c]*Log[Sin[c + b*x]])/b - x*Sin[a - c]} -{Cos[a + b*x]/Sin[c + b*x]^2, x, 4, -((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b} -{Cos[a + b*x]/Sin[c + b*x]^3, x, 5, -((Cos[a - c]*Csc[c + b*x]^2)/(2*b)) + (Cot[c + b*x]*Sin[a - c])/b} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[a+b x]^m Tan[c+d x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sin[a + b*x]*Tan[c + b*x]^3, x, 9, -((3*ArcTanh[Sin[c + b*x]]*Cos[a - c])/(2*b)) + (Sec[c + b*x]*Sin[a - c])/b + Sin[a + b*x]/b + (Cos[a - c]*Sec[c + b*x]*Tan[c + b*x])/(2*b)} -{Sin[a + b*x]*Tan[c + b*x]^2, x, 6, Cos[a + b*x]/b + (Cos[a - c]*Sec[c + b*x])/b + (ArcTanh[Sin[c + b*x]]*Sin[a - c])/b} -{Sin[a + b*x]*Tan[c + b*x]^1, x, 3, (ArcTanh[Sin[c + b*x]]*Cos[a - c])/b - Sin[a + b*x]/b} -{Sin[a + b*x]*Cot[c + b*x]^1, x, 3, -((ArcTanh[Cos[c + b*x]]*Sin[a - c])/b) + Sin[a + b*x]/b} -{Sin[a + b*x]*Cot[c + b*x]^2, x, 6, -((ArcTanh[Cos[c + b*x]]*Cos[a - c])/b) + Cos[a + b*x]/b - (Csc[c + b*x]*Sin[a - c])/b} -{Sin[a + b*x]*Cot[c + b*x]^3, x, 9, -((Cos[a - c]*Csc[c + b*x])/b) + (3*ArcTanh[Cos[c + b*x]]*Sin[a - c])/(2*b) - (Cot[c + b*x]*Csc[c + b*x]*Sin[a - c])/(2*b) - Sin[a + b*x]/b} - - -{Sin[a + b*x]*Tan[c + d*x], x, 6, I/(E^(I*(a + b*x))*(2*b)) + (I*E^(I*(a + b*x)))/(2*b) - (I*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), -E^(2*I*(c + d*x))])/(E^(I*(a + b*x))*b) - (I*E^(I*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^(2*I*(c + d*x))])/b} -{Sin[a + b*x]*Cot[c + d*x], x, 6, -(I/(E^(I*(a + b*x))*(2*b))) - (I*E^(I*(a + b*x)))/(2*b) + (I*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), E^(2*I*(c + d*x))])/(E^(I*(a + b*x))*b) + (I*E^(I*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*I*(c + d*x))])/b} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Trig[c+d x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Cos[c+d x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]*Cos[c + d*x]^3, x, 6, Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) + (3*Sin[a + c + (b + d)*x])/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} -{Cos[a + b*x]*Cos[c + d*x]^2, x, 5, Sin[a + b*x]/(2*b) + Sin[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Sin[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Cos[a + b*x]*Cos[c + d*x]^1, x, 4, Sin[a - c + (b - d)*x]/(2*(b - d)) + Sin[a + c + (b + d)*x]/(2*(b + d))} -{Cos[a + b*x]*Sec[c + b*x]^1, x, 3, x*Cos[a - c] + (Log[Cos[c + b*x]]*Sin[a - c])/b} -{Cos[a + b*x]*Sec[c + b*x]^2, x, 4, (ArcTanh[Sin[c + b*x]]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b} -{Cos[a + b*x]*Sec[c + b*x]^3, x, 5, -((Sec[c + b*x]^2*Sin[a - c])/(2*b)) + (Cos[a - c]*Tan[c + b*x])/b} - - -{Cos[a + b*x]^2*Cos[c + d*x]^3, x, 8, Sin[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sin[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sin[c + d*x])/(8*d) + Sin[3*c + 3*d*x]/(24*d) + (3*Sin[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sin[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} -{Cos[a + b*x]^2*Cos[c + d*x]^2, x, 6, x/4 + Sin[2*a + 2*b*x]/(8*b) + Sin[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) + Sin[2*c + 2*d*x]/(8*d) + Sin[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} - - -{Cos[a + b*x]^3*Cos[c + d*x]^3, x, 10, (3*Sin[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) + (9*Sin[a - c + (b - d)*x])/(32*(b - d)) + Sin[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Sin[3*a - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sin[a + c + (b + d)*x])/(32*(b + d)) + Sin[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Sin[3*a + c + (3*b + d)*x])/(32*(3*b + d)) + (3*Sin[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[a+b x]^m Tan[c+d x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Cos[a + b*x]*Tan[c + b*x]^3, x, 9, Cos[a + b*x]/b + (Cos[a - c]*Sec[c + b*x])/b + (3*ArcTanh[Sin[c + b*x]]*Sin[a - c])/(2*b) - (Sec[c + b*x]*Sin[a - c]*Tan[c + b*x])/(2*b)} -{Cos[a + b*x]*Tan[c + b*x]^2, x, 6, (ArcTanh[Sin[c + b*x]]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b - Sin[a + b*x]/b} -{Cos[a + b*x]*Tan[c + b*x]^1, x, 3, -(Cos[a + b*x]/b) - (ArcTanh[Sin[c + b*x]]*Sin[a - c])/b} -{Cos[a + b*x]*Cot[c + b*x]^1, x, 3, -((ArcTanh[Cos[c + b*x]]*Cos[a - c])/b) + Cos[a + b*x]/b} -{Cos[a + b*x]*Cot[c + b*x]^2, x, 6, -((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b - Sin[a + b*x]/b} -{Cos[a + b*x]*Cot[c + b*x]^3, x, 9, (3*ArcTanh[Cos[c + b*x]]*Cos[a - c])/(2*b) - Cos[a + b*x]/b - (Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(2*b) + (Csc[c + b*x]*Sin[a - c])/b} - - -{Cos[a + b*x]*Tan[c + d*x], x, 6, 1/(E^(I*(a + b*x))*(2*b)) - E^(I*(a + b*x))/(2*b) - Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), -E^(2*I*(c + d*x))]/(E^(I*(a + b*x))*b) + (E^(I*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^(2*I*(c + d*x))])/b} -{Cos[a + b*x]*Cot[c + d*x], x, 6, -(1/(E^(I*(a + b*x))*(2*b))) + E^(I*(a + b*x))/(2*b) + Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), E^(2*I*(c + d*x))]/(E^(I*(a + b*x))*b) - (E^(I*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*I*(c + d*x))])/b} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection:: *) -(*Integrands of the form Cos[a+b x]^m Cot[c+d x]^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tan[a+b x]^m Trig[c+d x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tan[a+b x]^m Tan[c+d x]^n*) - - -(* ::Subsubsection:: *) -(*m>0*) - - -(* ::Subsubsection:: *) -(*m<0*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.2 trig^m (a trig+b trig)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.2 trig^m (a trig+b trig)^n.m deleted file mode 100644 index be61c35..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.2 trig^m (a trig+b trig)^n.m +++ /dev/null @@ -1,490 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Trig[c+d x]^m (a Trig[c+d x]+b Trig[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sin[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sin[x]^3*(a*Cos[x] + b*Sin[x]), x, 7, (3*b*x)/8 - (3/8)*b*Cos[x]*Sin[x] - (1/4)*b*Cos[x]*Sin[x]^3 + (1/4)*a*Sin[x]^4} -{Sin[x]^2*(a*Cos[x] + b*Sin[x]), x, 6, (-b)*Cos[x] + (1/3)*b*Cos[x]^3 + (1/3)*a*Sin[x]^3} -{Sin[x]^1*(a*Cos[x] + b*Sin[x]), x, 6, (b*x)/2 - (1/2)*b*Cos[x]*Sin[x] + (1/2)*a*Sin[x]^2} -{Sin[x]^0*(a*Cos[x] + b*Sin[x]), x, 3, (-b)*Cos[x] + a*Sin[x]} -{Csc[x]^1*(a*Cos[x] + b*Sin[x]), x, 3, b*x + a*Log[Sin[x]]} -{Csc[x]^2*(a*Cos[x] + b*Sin[x]), x, 5, (-b)*ArcTanh[Cos[x]] - a*Csc[x]} -{Csc[x]^3*(a*Cos[x] + b*Sin[x]), x, 6, (-b)*Cot[x] - (1/2)*a*Csc[x]^2} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]^3/(a*Cos[x] + b*Sin[x]), x, 5, (a^2*b*x)/(a^2 + b^2)^2 + (b*x)/(2*(a^2 + b^2)) - (a^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Cos[x]*Sin[x])/(2*(a^2 + b^2)) - (a*Sin[x]^2)/(2*(a^2 + b^2))} -{Sin[x]^2/(a*Cos[x] + b*Sin[x]), x, 4, -((a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Cos[x])/(a^2 + b^2) - (a*Sin[x])/(a^2 + b^2)} -{Sin[x]^1/(a*Cos[x] + b*Sin[x]), x, 2, (b*x)/(a^2 + b^2) - (a*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)} -{Sin[x]^0/(a*Cos[x] + b*Sin[x]), x, 2, -(ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])} -{Csc[x]^1/(a*Cos[x] + b*Sin[x]), x, 3, Log[Sin[x]]/a - Log[a*Cos[x] + b*Sin[x]]/a} -{Csc[x]^2/(a*Cos[x] + b*Sin[x]), x, 4, (b*ArcTanh[Cos[x]])/a^2 - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^2 - Csc[x]/a} -{Csc[x]^3/(a*Cos[x] + b*Sin[x]), x, 6, (b*Cot[x])/a^2 - Csc[x]^2/(2*a) + ((a^2 + b^2)*Log[Sin[x]])/a^3 - ((a^2 + b^2)*Log[a*Cos[x] + b*Sin[x]])/a^3} - - -{Sin[x]^3/(a*Cos[x] + b*Sin[x])^2, x, -19, (6*a^2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (3*a*(a^2 - b^2) + a*(a^2 + b^2)*Cos[2*x] - b*(a^2 + b^2)*Sin[2*x])/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))} -{Sin[x]^2/(a*Cos[x] + b*Sin[x])^2, x, 4, -(((a^2 - b^2)*x)/(a^2 + b^2)^2) + a/((a^2 + b^2)*(b + a*Cot[x])) - (2*a*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2} -{Sin[x]^1/(a*Cos[x] + b*Sin[x])^2, x, 3, -((b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + a/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))} -{Sin[x]^0/(a*Cos[x] + b*Sin[x])^2, x, 1, Sin[x]/(a*(a*Cos[x] + b*Sin[x]))} -{Csc[x]^1/(a*Cos[x] + b*Sin[x])^2, x, 4, -(ArcTanh[Cos[x]]/a^2) + (b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + 1/(a*(a*Cos[x] + b*Sin[x]))} -{Csc[x]^2/(a*Cos[x] + b*Sin[x])^2, x, 3, -(Cot[x]/a^2) - (2*b*Log[Tan[x]])/a^3 + (2*b*Log[a + b*Tan[x]])/a^3 - (1/b + b/a^2)/(a + b*Tan[x])} -{Csc[x]^3/(a*Cos[x] + b*Sin[x])^2, x, 11, -(ArcTanh[Cos[x]]/(2*a^2)) - (2*b^2*ArcTanh[Cos[x]])/a^4 - ((a^2 + b^2)*ArcTanh[Cos[x]])/a^4 + (3*b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^4 + (2*b*Csc[x])/a^3 - (Cot[x]*Csc[x])/(2*a^2) + (a^2 + b^2)/(a^3*(a*Cos[x] + b*Sin[x]))} - - -{Sin[x]^3/(a*Cos[x] + b*Sin[x])^3, x, 5, -((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + a/(2*(a^2 + b^2)*(b + a*Cot[x])^2) + (2*a*b)/((a^2 + b^2)^2*(b + a*Cot[x])) + (a*(a^2 - 3*b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3} -{Sin[x]^2/(a*Cos[x] + b*Sin[x])^3, x, -13, -(((a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (a*(3*a*b*Cos[x] + (a^2 + 4*b^2)*Sin[x]))/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])^2)} -{Sin[x]^1/(a*Cos[x] + b*Sin[x])^3, x, 2, 1/(2*a*(b + a*Cot[x])^2), Tan[x]^2/(2*a*(a + b*Tan[x])^2)} -{Sin[x]^0/(a*Cos[x] + b*Sin[x])^3, x, 3, -(ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2))) - (b*Cos[x] - a*Sin[x])/(2*(a^2 + b^2)*(a*Cos[x] + b*Sin[x])^2)} -{Csc[x]^1/(a*Cos[x] + b*Sin[x])^3, x, 3, Log[Tan[x]]/a^3 - Log[a + b*Tan[x]]/a^3 + (1/a + a/b^2)/(2*(a + b*Tan[x])^2) + (1/a^2 - 1/b^2)/(a + b*Tan[x])} -{Csc[x]^2/(a*Cos[x] + b*Sin[x])^3, x, 12, (3*b*ArcTanh[Cos[x]])/a^4 - ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*a^2*Sqrt[a^2 + b^2]) - (2*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^4 - Csc[x]/a^3 - (b*Cos[x] - a*Sin[x])/(2*a^2*(a*Cos[x] + b*Sin[x])^2) - (2*b)/(a^3*(a*Cos[x] + b*Sin[x]))} -{Csc[x]^3/(a*Cos[x] + b*Sin[x])^3, x, 3, (3*b*Cot[x])/a^4 - Cot[x]^2/(2*a^3) + (2*(a^2 + 3*b^2)*Log[Tan[x]])/a^5 - (2*(a^2 + 3*b^2)*Log[a + b*Tan[x]])/a^5 + (a^2 + b^2)^2/(2*a^3*b^2*(a + b*Tan[x])^2) - ((a^2 - 3*b^2)*(a^2 + b^2))/(a^4*b^2*(a + b*Tan[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n when a^2+b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n/Sin[c + d*x]^n, x, 1, -((I*Hypergeometric2F1[1, n, 1 + n, (-(1/2))*I*(I + Cot[c + d*x])]*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n)/(Sin[c + d*x]^n*(2*d*n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 8, (5*a*x)/16 - (b*Cos[c + d*x]^6)/(6*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, -((b*Cos[c + d*x]^5)/(5*d)) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 7, (3*a*x)/8 - (b*Cos[c + d*x]^4)/(4*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, -((b*Cos[c + d*x]^3)/(3*d)) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, (a*x)/2 + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^0*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 3, -((b*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d} -{Sec[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 3, a*x - (b*Log[Cos[c + d*x]])/d} -{Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 5, (a*ArcTanh[Sin[c + d*x]])/d + (b*Sec[c + d*x])/d} -{Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, (b*Sec[c + d*x]^2)/(2*d) + (a*Tan[c + d*x])/d} -{Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, (a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, (b*Sec[c + d*x]^4)/(4*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 7, (3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Sec[c + d*x]^5)/(5*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, (b*Sec[c + d*x]^6)/(6*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)} - - -{Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 9, -((2*a*b*Cos[c + d*x]^7)/(7*d)) + (a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/d + (b^2*Sin[c + d*x]^3)/(3*d) + (3*a^2*Sin[c + d*x]^5)/(5*d) - (2*b^2*Sin[c + d*x]^5)/(5*d) - (a^2*Sin[c + d*x]^7)/(7*d) + (b^2*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 12, (5*a^2*x)/16 + (b^2*x)/16 - (a*b*Cos[c + d*x]^6)/(3*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)} -{Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 9, -((2*a*b*Cos[c + d*x]^5)/(5*d)) + (a^2*Sin[c + d*x])/d - (2*a^2*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^5)/(5*d) - (b^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 10, (3*a^2*x)/8 + (b^2*x)/8 - (a*b*Cos[c + d*x]^4)/(2*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)} -{Cos[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 8, -((2*a*b*Cos[c + d*x]^3)/(3*d)) + (a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^3)/(3*d)} -{Cos[c + d*x]^0*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 2, (1/2)*(a^2 + b^2)*x - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(2*d)} -{Sec[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 7, (b^2*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Cos[c + d*x])/d + (a^2*Sin[c + d*x])/d - (b^2*Sin[c + d*x])/d} -{Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, (a^2 - b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 7, (a^2*ArcTanh[Sin[c + d*x]])/d - (b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 2, ((b + a*Cot[c + d*x])^3*Tan[c + d*x]^3)/(3*b*d)} -{Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 9, (a^2*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, (a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + ((a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*Tan[c + d*x]^4)/(2*d) + (b^2*Tan[c + d*x]^5)/(5*d)} -{Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 11, (3*a^2*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (2*a*b*Sec[c + d*x]^5)/(5*d) + (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (b^2*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)} -{Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, (a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + ((2*a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*Tan[c + d*x]^4)/d + ((a^2 + 2*b^2)*Tan[c + d*x]^5)/(5*d) + (a*b*Tan[c + d*x]^6)/(3*d) + (b^2*Tan[c + d*x]^7)/(7*d)} - - -{Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 17, (35*a^3*x)/128 + (15/128)*a*b^2*x - (b^3*Cos[c + d*x]^6)/(6*d) - (3*a^2*b*Cos[c + d*x]^8)/(8*d) + (b^3*Cos[c + d*x]^8)/(8*d) + (35*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)} -{Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 12, -((b^3*Cos[c + d*x]^5)/(5*d)) - (3*a^2*b*Cos[c + d*x]^7)/(7*d) + (b^3*Cos[c + d*x]^7)/(7*d) + (a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^3)/d + (a*b^2*Sin[c + d*x]^3)/d + (3*a^3*Sin[c + d*x]^5)/(5*d) - (6*a*b^2*Sin[c + d*x]^5)/(5*d) - (a^3*Sin[c + d*x]^7)/(7*d) + (3*a*b^2*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 15, (5*a^3*x)/16 + (3/16)*a*b^2*x - (a^2*b*Cos[c + d*x]^6)/(2*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(2*d) + (b^3*Sin[c + d*x]^4)/(4*d) - (b^3*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 12, -((b^3*Cos[c + d*x]^3)/(3*d)) - (3*a^2*b*Cos[c + d*x]^5)/(5*d) + (b^3*Cos[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x])/d - (2*a^3*Sin[c + d*x]^3)/(3*d) + (a*b^2*Sin[c + d*x]^3)/d + (a^3*Sin[c + d*x]^5)/(5*d) - (3*a*b^2*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 4, (3/8)*a*(a^2 + b^2)*x + (3*a*(b + a*Cot[c + d*x])*(a - b*Cot[c + d*x])*Sin[c + d*x]^2)/(8*d) + ((b + a*Cot[c + d*x])^3*Sin[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^0*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 2, -(((a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (b*Cos[c + d*x] - a*Sin[c + d*x])^3/(3*d)} -{Sec[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 7, (1/2)*a*(a^2 + 3*b^2)*x - (b^3*Log[Sin[c + d*x]])/d + (b^3*Log[Tan[c + d*x]])/d + ((b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 10, (3*a*b^2*ArcTanh[Sin[c + d*x]])/d - (3*a^2*b*Cos[c + d*x])/d + (b^3*Cos[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x])/d} -{Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 4, a*(a^2 - 3*b^2)*x - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (2*a*b^2*Tan[c + d*x])/d + (b*(a + b*Tan[c + d*x])^2)/(2*d)} -{Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 9, (a^3*ArcTanh[Sin[c + d*x]])/d - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a^2*b*Sec[c + d*x])/d - (b^3*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^3)/(3*d) + (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 2, ((b + a*Cot[c + d*x])^4*Tan[c + d*x]^4)/(4*b*d)} -{Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 12, (a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*b*Sec[c + d*x]^3)/d - (b^3*Sec[c + d*x]^3)/(3*d) + (b^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} -{Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, (a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b*(3*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) + (b^3*Tan[c + d*x]^6)/(6*d)} -{Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 14, (3*a^3*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (3*a^2*b*Sec[c + d*x]^5)/(5*d) - (b^3*Sec[c + d*x]^5)/(5*d) + (b^3*Sec[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, (a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(2*a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b*(6*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (a*(a^2 + 6*b^2)*Tan[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Tan[c + d*x]^6)/(6*d) + (3*a*b^2*Tan[c + d*x]^7)/(7*d) + (b^3*Tan[c + d*x]^8)/(8*d)} -{Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 16, (5*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a*b^2*ArcTanh[Sin[c + d*x]])/(128*d) + (3*a^2*b*Sec[c + d*x]^7)/(7*d) - (b^3*Sec[c + d*x]^7)/(7*d) + (b^3*Sec[c + d*x]^9)/(9*d) + (5*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (5*a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (3*a*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(8*d)} -{Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, (a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(a^2 + b^2)*Tan[c + d*x]^3)/d + (b*(9*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (3*a*(a^2 + 3*b^2)*Tan[c + d*x]^5)/(5*d) + (b*(3*a^2 + b^2)*Tan[c + d*x]^6)/(2*d) + (a*(a^2 + 9*b^2)*Tan[c + d*x]^7)/(7*d) + (3*b*(a^2 + b^2)*Tan[c + d*x]^8)/(8*d) + (a*b^2*Tan[c + d*x]^9)/(3*d) + (b^3*Tan[c + d*x]^10)/(10*d)} - - -{Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 15, -((4*a*b^3*Cos[c + d*x]^7)/(7*d)) - (4*a^3*b*Cos[c + d*x]^9)/(9*d) + (4*a*b^3*Cos[c + d*x]^9)/(9*d) + (a^4*Sin[c + d*x])/d - (4*a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d + (6*a^4*Sin[c + d*x]^5)/(5*d) - (18*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (4*a^4*Sin[c + d*x]^7)/(7*d) + (18*a^2*b^2*Sin[c + d*x]^7)/(7*d) - (2*b^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^9)/(9*d) - (2*a^2*b^2*Sin[c + d*x]^9)/(3*d) + (b^4*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 22, (35*a^4*x)/128 + (15/64)*a^2*b^2*x + (3*b^4*x)/128 - (2*a*b^3*Cos[c + d*x]^6)/(3*d) - (a^3*b*Cos[c + d*x]^8)/(2*d) + (a*b^3*Cos[c + d*x]^8)/(2*d) + (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*b^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (3*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) + (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (b^4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^4*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a^2*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)} -{Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 15, -((4*a*b^3*Cos[c + d*x]^5)/(5*d)) - (4*a^3*b*Cos[c + d*x]^7)/(7*d) + (4*a*b^3*Cos[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/d + (2*a^2*b^2*Sin[c + d*x]^3)/d + (3*a^4*Sin[c + d*x]^5)/(5*d) - (12*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/(7*d) + (6*a^2*b^2*Sin[c + d*x]^7)/(7*d) - (b^4*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 19, (5*a^4*x)/16 + (3/8)*a^2*b^2*x + (b^4*x)/16 - (2*a^3*b*Cos[c + d*x]^6)/(3*d) + (5*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a^2*b^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/d - (b^4*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d) + (a*b^3*Sin[c + d*x]^4)/d - (2*a*b^3*Sin[c + d*x]^6)/(3*d)} -{Cos[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 14, -((4*a*b^3*Cos[c + d*x]^3)/(3*d)) - (4*a^3*b*Cos[c + d*x]^5)/(5*d) + (4*a*b^3*Cos[c + d*x]^5)/(5*d) + (a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d + (a^4*Sin[c + d*x]^5)/(5*d) - (6*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d)} -{Cos[c + d*x]^0*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (3/8)*(a^2 + b^2)^2*x - (3*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(8*d) - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(4*d)} -{Sec[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 14, (b^4*ArcTanh[Sin[c + d*x]])/d - (4*a*b^3*Cos[c + d*x])/d - (4*a^3*b*Cos[c + d*x]^3)/(3*d) + (4*a*b^3*Cos[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x])/d - (b^4*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d - (b^4*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 7, (1/2)*(a^4 + 6*a^2*b^2 - 3*b^4)*x - (4*a*b^3*Log[Sin[c + d*x]])/d + (4*a*b^3*Log[Tan[c + d*x]])/d + ((4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d) + (b^4*Tan[c + d*x])/d} -{Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 14, (6*a^2*b^2*ArcTanh[Sin[c + d*x]])/d - (3*b^4*ArcTanh[Sin[c + d*x]])/(2*d) - (4*a^3*b*Cos[c + d*x])/d + (4*a*b^3*Cos[c + d*x])/d + (4*a*b^3*Sec[c + d*x])/d + (a^4*Sin[c + d*x])/d - (6*a^2*b^2*Sin[c + d*x])/d + (3*b^4*Sin[c + d*x])/(2*d) + (b^4*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 5, (a^4 - 6*a^2*b^2 + b^4)*x - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(3*a^2 - b^2)*Tan[c + d*x])/d + (a*b*(a + b*Tan[c + d*x])^2)/d + (b*(a + b*Tan[c + d*x])^3)/(3*d)} -{Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 12, (a^4*ArcTanh[Sin[c + d*x]])/d - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/d + (3*b^4*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*b*Sec[c + d*x])/d - (4*a*b^3*Sec[c + d*x])/d + (4*a*b^3*Sec[c + d*x]^3)/(3*d) + (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/d - (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d)} -{Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 2, ((b + a*Cot[c + d*x])^5*Tan[c + d*x]^5)/(5*b*d)} -{Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 16, (a^4*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(4*d) + (b^4*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^3*b*Sec[c + d*x]^3)/(3*d) - (4*a*b^3*Sec[c + d*x]^3)/(3*d) + (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (b^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (3*a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) - (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d)} -{Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (a^2*(a^2 + 6*b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*(a^2 + b^2)*Tan[c + d*x]^4)/d + (b^2*(6*a^2 + b^2)*Tan[c + d*x]^5)/(5*d) + (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d)} -{Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 19, (3*a^4*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (4*a^3*b*Sec[c + d*x]^5)/(5*d) - (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (3*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x])/d - (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)} -{Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (2*a^2*(a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*(2*a^2 + b^2)*Tan[c + d*x]^4)/d + ((a^4 + 12*a^2*b^2 + b^4)*Tan[c + d*x]^5)/(5*d) + (2*a*b*(a^2 + 2*b^2)*Tan[c + d*x]^6)/(3*d) + (2*b^2*(3*a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (a*b^3*Tan[c + d*x]^8)/(2*d) + (b^4*Tan[c + d*x]^9)/(9*d)} -{Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 22, (5*a^4*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a^2*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(256*d) + (4*a^3*b*Sec[c + d*x]^7)/(7*d) - (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (4*a*b^3*Sec[c + d*x]^9)/(9*d) + (5*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(256*d) + (5*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (5*a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(32*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(160*d) + (3*a^2*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(4*d) - (3*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(80*d) + (b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*d)} -{Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (a^2*(a^2 + 2*b^2)*Tan[c + d*x]^3)/d + (a*b*(3*a^2 + b^2)*Tan[c + d*x]^4)/d + ((3*a^4 + 18*a^2*b^2 + b^4)*Tan[c + d*x]^5)/(5*d) + (2*a*b*(a^2 + b^2)*Tan[c + d*x]^6)/d + ((a^4 + 18*a^2*b^2 + 3*b^4)*Tan[c + d*x]^7)/(7*d) + (a*b*(a^2 + 3*b^2)*Tan[c + d*x]^8)/(2*d) + (b^2*(2*a^2 + b^2)*Tan[c + d*x]^9)/(3*d) + (2*a*b^3*Tan[c + d*x]^10)/(5*d) + (b^4*Tan[c + d*x]^11)/(11*d)} - - -{Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 29, (63*a^5*x)/256 + (35/128)*a^3*b^2*x + (15/256)*a*b^4*x - (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) - (a^4*b*Cos[c + d*x]^10)/(2*d) + (a^2*b^3*Cos[c + d*x]^10)/d + (63*a^5*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (35*a^3*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (21*a^5*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (35*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (21*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) + (7*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^4*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) + (9*a^5*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (a^3*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a*b^4*Cos[c + d*x]^7*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^9*Sin[c + d*x])/(10*d) - (a^3*b^2*Cos[c + d*x]^9*Sin[c + d*x])/d - (a*b^4*Cos[c + d*x]^7*Sin[c + d*x]^3)/(2*d) + (b^5*Sin[c + d*x]^6)/(6*d) - (b^5*Sin[c + d*x]^8)/(4*d) + (b^5*Sin[c + d*x]^10)/(10*d)} -{Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 18, -((b^5*Cos[c + d*x]^5)/(5*d)) - (10*a^2*b^3*Cos[c + d*x]^7)/(7*d) + (2*b^5*Cos[c + d*x]^7)/(7*d) - (5*a^4*b*Cos[c + d*x]^9)/(9*d) + (10*a^2*b^3*Cos[c + d*x]^9)/(9*d) - (b^5*Cos[c + d*x]^9)/(9*d) + (a^5*Sin[c + d*x])/d - (4*a^5*Sin[c + d*x]^3)/(3*d) + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) + (6*a^5*Sin[c + d*x]^5)/(5*d) - (6*a^3*b^2*Sin[c + d*x]^5)/d + (a*b^4*Sin[c + d*x]^5)/d - (4*a^5*Sin[c + d*x]^7)/(7*d) + (30*a^3*b^2*Sin[c + d*x]^7)/(7*d) - (10*a*b^4*Sin[c + d*x]^7)/(7*d) + (a^5*Sin[c + d*x]^9)/(9*d) - (10*a^3*b^2*Sin[c + d*x]^9)/(9*d) + (5*a*b^4*Sin[c + d*x]^9)/(9*d)} -{Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 25, (35*a^5*x)/128 + (25/64)*a^3*b^2*x + (15/128)*a*b^4*x - (5*a^2*b^3*Cos[c + d*x]^6)/(3*d) - (5*a^4*b*Cos[c + d*x]^8)/(8*d) + (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) + (35*a^5*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (25*a^3*b^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^5*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (5*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (5*a^3*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d) + (b^5*Sin[c + d*x]^6)/(6*d) - (b^5*Sin[c + d*x]^8)/(8*d)} -{Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 18, -((b^5*Cos[c + d*x]^3)/(3*d)) - (2*a^2*b^3*Cos[c + d*x]^5)/d + (2*b^5*Cos[c + d*x]^5)/(5*d) - (5*a^4*b*Cos[c + d*x]^7)/(7*d) + (10*a^2*b^3*Cos[c + d*x]^7)/(7*d) - (b^5*Cos[c + d*x]^7)/(7*d) + (a^5*Sin[c + d*x])/d - (a^5*Sin[c + d*x]^3)/d + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) + (3*a^5*Sin[c + d*x]^5)/(5*d) - (4*a^3*b^2*Sin[c + d*x]^5)/d + (a*b^4*Sin[c + d*x]^5)/d - (a^5*Sin[c + d*x]^7)/(7*d) + (10*a^3*b^2*Sin[c + d*x]^7)/(7*d) - (5*a*b^4*Sin[c + d*x]^7)/(7*d)} -{Cos[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 5, (5/16)*a*(a^2 + b^2)^2*x + (5*a*(a^2 + b^2)*(b + a*Cot[c + d*x])*(a - b*Cot[c + d*x])*Sin[c + d*x]^2)/(16*d) + (5*a*(b + a*Cot[c + d*x])^3*(a - b*Cot[c + d*x])*Sin[c + d*x]^4)/(24*d) + ((b + a*Cot[c + d*x])^5*Sin[c + d*x]^6)/(6*d)} -{Cos[c + d*x]^0*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 3, -(((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (2*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])^3)/(3*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])^5/(5*d)} -{Sec[c + d*x]^1*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 8, (1/8)*a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x - (b^5*Log[Sin[c + d*x]])/d + (b^5*Log[Tan[c + d*x]])/d + ((4*b*(5*a^4 - b^4) + 5*a*(a^2 - 3*b^2)*(a^2 + b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(8*d) - ((b*(5*a^4 - 10*a^2*b^2 + b^4) + a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cot[c + d*x])*Sin[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 17, (5*a*b^4*ArcTanh[Sin[c + d*x]])/d - (10*a^2*b^3*Cos[c + d*x])/d + (2*b^5*Cos[c + d*x])/d - (5*a^4*b*Cos[c + d*x]^3)/(3*d) + (10*a^2*b^3*Cos[c + d*x]^3)/(3*d) - (b^5*Cos[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x])/d + (a^5*Sin[c + d*x])/d - (5*a*b^4*Sin[c + d*x])/d - (a^5*Sin[c + d*x]^3)/(3*d) + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) - (5*a*b^4*Sin[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 7, (1/2)*a*(a^4 + 10*a^2*b^2 - 15*b^4)*x - (2*b^3*(5*a^2 - b^2)*Log[Sin[c + d*x]])/d + (2*b^3*(5*a^2 - b^2)*Log[Tan[c + d*x]])/d + ((b*(5*a^4 - 10*a^2*b^2 + b^4) + a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d) + (5*a*b^4*Tan[c + d*x])/d + (b^5*Tan[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 17, (10*a^3*b^2*ArcTanh[Sin[c + d*x]])/d - (15*a*b^4*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*b*Cos[c + d*x])/d + (10*a^2*b^3*Cos[c + d*x])/d - (b^5*Cos[c + d*x])/d + (10*a^2*b^3*Sec[c + d*x])/d - (2*b^5*Sec[c + d*x])/d + (b^5*Sec[c + d*x]^3)/(3*d) + (a^5*Sin[c + d*x])/d - (10*a^3*b^2*Sin[c + d*x])/d + (15*a*b^4*Sin[c + d*x])/(2*d) + (5*a*b^4*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 6, a*(a^4 - 10*a^2*b^2 + 5*b^4)*x - (b*(5*a^4 - 10*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (4*a*b^2*(a^2 - b^2)*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) + (2*a*b*(a + b*Tan[c + d*x])^3)/(3*d) + (b*(a + b*Tan[c + d*x])^4)/(4*d)} -{Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 15, (a^5*ArcTanh[Sin[c + d*x]])/d - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/d + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*b*Sec[c + d*x])/d - (10*a^2*b^3*Sec[c + d*x])/d + (b^5*Sec[c + d*x])/d + (10*a^2*b^3*Sec[c + d*x]^3)/(3*d) - (2*b^5*Sec[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x]^5)/(5*d) + (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/d - (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (5*a*b^4*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d)} -{Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 2, ((b + a*Cot[c + d*x])^6*Tan[c + d*x]^6)/(6*b*d)} -{Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 19, (a^5*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(4*d) + (5*a*b^4*ArcTanh[Sin[c + d*x]])/(16*d) + (5*a^4*b*Sec[c + d*x]^3)/(3*d) - (10*a^2*b^3*Sec[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x]^3)/(3*d) + (2*a^2*b^3*Sec[c + d*x]^5)/d - (2*b^5*Sec[c + d*x]^5)/(5*d) + (b^5*Sec[c + d*x]^7)/(7*d) + (a^5*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (5*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) - (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d)} -{Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 3, (a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (a^3*(a^2 + 10*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*(a^2 + 2*b^2)*Tan[c + d*x]^4)/(4*d) + (a*b^2*(2*a^2 + b^2)*Tan[c + d*x]^5)/d + (b^3*(10*a^2 + b^2)*Tan[c + d*x]^6)/(6*d) + (5*a*b^4*Tan[c + d*x]^7)/(7*d) + (b^5*Tan[c + d*x]^8)/(8*d)} -{Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 22, (3*a^5*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (a^4*b*Sec[c + d*x]^5)/d - (2*a^2*b^3*Sec[c + d*x]^5)/d + (b^5*Sec[c + d*x]^5)/(5*d) + (10*a^2*b^3*Sec[c + d*x]^7)/(7*d) - (2*b^5*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^9)/(9*d) + (3*a^5*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^5*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(12*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (5*a^3*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) - (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)} -{Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 3, (a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (2*a^3*(a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*(a^2 + b^2)*Tan[c + d*x]^4)/(2*d) + (a*(a^4 + 20*a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/(5*d) + (b*(5*a^4 + 20*a^2*b^2 + b^4)*Tan[c + d*x]^6)/(6*d) + (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (b^3*(5*a^2 + b^2)*Tan[c + d*x]^8)/(4*d) + (5*a*b^4*Tan[c + d*x]^9)/(9*d) + (b^5*Tan[c + d*x]^10)/(10*d)} -{Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^5, x, 25, (5*a^5*ArcTanh[Sin[c + d*x]])/(16*d) - (25*a^3*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(256*d) + (5*a^4*b*Sec[c + d*x]^7)/(7*d) - (10*a^2*b^3*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^7)/(7*d) + (10*a^2*b^3*Sec[c + d*x]^9)/(9*d) - (2*b^5*Sec[c + d*x]^9)/(9*d) + (b^5*Sec[c + d*x]^11)/(11*d) + (5*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (25*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(256*d) + (5*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (25*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(96*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (5*a^3*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(24*d) + (a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/(32*d) + (5*a^3*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(4*d) - (3*a*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(16*d) + (a*b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 9, (a*b^4*x)/(a^2 + b^2)^3 + (a*b^2*x)/(2*(a^2 + b^2)^2) + (3*a*x)/(8*(a^2 + b^2)) + (b^3*Cos[c + d*x]^2)/(2*(a^2 + b^2)^2*d) + (b*Cos[c + d*x]^4)/(4*(a^2 + b^2)*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)^2*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*(a^2 + b^2)*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*(a^2 + b^2)*d)} -{Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 7, -((b^4*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) + (b^3*Cos[c + d*x])/((a^2 + b^2)^2*d) + (b*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a*b^2*Sin[c + d*x])/((a^2 + b^2)^2*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d) - (a*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d)} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 5, (a*b^2*x)/(a^2 + b^2)^2 + (a*x)/(2*(a^2 + b^2)) + (b*Cos[c + d*x]^2)/(2*(a^2 + b^2)*d) + (b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)*d)} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 4, -((b^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) + (b*Cos[c + d*x])/((a^2 + b^2)*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d)} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 2, (a*x)/(a^2 + b^2) + (b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 2, -(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(Sqrt[a^2 + b^2]*d))} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 3, -(Log[Cos[c + d*x]]/(b*d)) + Log[a*Cos[c + d*x] + b*Sin[c + d*x]]/(b*d)} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 4, -((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 6, -(((a^2 + b^2)*Log[Cos[c + d*x]])/(b^3*d)) + ((a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b^3*d) + Sec[c + d*x]^2/(2*b*d) - (a*Tan[c + d*x])/(b^2*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 7, -((a*ArcTanh[Sin[c + d*x]])/(2*b^2*d)) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((a^2 + b^2)^(3/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) + ((a^2 + b^2)*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^3/(3*b*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d)} -{Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 9, -(((a^2 + b^2)^2*Log[Cos[c + d*x]])/(b^5*d)) + ((a^2 + b^2)^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b^5*d) + ((a^2 + b^2)*Sec[c + d*x]^2)/(2*b^3*d) + Sec[c + d*x]^4/(4*b*d) - (a*Tan[c + d*x])/(b^2*d) - (a*(a^2 + b^2)*Tan[c + d*x])/(b^4*d) - (a*Tan[c + d*x]^3)/(3*b^2*d)} -{Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]), x, 11, -((3*a*ArcTanh[Sin[c + d*x]])/(8*b^2*d)) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (a*(a^2 + b^2)^2*ArcTanh[Sin[c + d*x]])/(b^6*d) - ((a^2 + b^2)^(5/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) + ((a^2 + b^2)^2*Sec[c + d*x])/(b^5*d) + ((a^2 + b^2)*Sec[c + d*x]^3)/(3*b^3*d) + Sec[c + d*x]^5/(5*b*d) - (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*b^2*d) - (a*(a^2 + b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d) - (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*b^2*d)} - - -{Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 7, ((a^4 + 6*a^2*b^2 - 3*b^4)*x)/(2*(a^2 + b^2)^3) + b^4/(a*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + (4*a*b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - ((2*a*b - (a^2 - b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*(a^2 + b^2)^2*d)} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, -11, -((3*a*b^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) + (2*a*b*Cos[c + d*x])/((a^2 + b^2)^2*d) + ((a^2 - b^2)*Sin[c + d*x])/((a^2 + b^2)^2*d) - b^3/((a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 4, ((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - b/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, -((a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) - b/((a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 1, Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 4, ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]*d) - 1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, (1/a + a/b^2)/(d*(b + a*Cot[c + d*x])) - (2*a*Log[b + a*Cot[c + d*x]])/(b^3*d) - (2*a*Log[Tan[c + d*x]])/(b^3*d) + Tan[c + d*x]/(b^2*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 11, (2*a^2*ArcTanh[Sin[c + d*x]])/(b^4*d) + ArcTanh[Sin[c + d*x]]/(2*b^2*d) + ((a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^4*d) + (3*a*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) - (2*a*Sec[c + d*x])/(b^3*d) - (a^2 + b^2)/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2, x, 3, (a^2 + b^2)^2/(a*b^4*d*(b + a*Cot[c + d*x])) - (4*a*(a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^5*d) - (4*a*(a^2 + b^2)*Log[Tan[c + d*x]])/(b^5*d) + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(b^4*d) - (a*Tan[c + d*x]^2)/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d)} - - -{Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, -15, -((3*b^2*(4*a^2 - b^2)*ArcTanh[(b - a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d)) + (b*(3*a^2 - b^2)*Cos[c + d*x])/((a^2 + b^2)^3*d) + (a*(a^2 - 3*b^2)*Sin[c + d*x])/((a^2 + b^2)^3*d) + (b^4*Sin[c + d*x])/(2*a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (b^3*(8*a^2 + b^2))/(2*a*(a^2 + b^2)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 5, (a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - b/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, -6, (2*(2*a^2 - b^2)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(2*d*(a^2 + b^2)^(5/2)) - (b*((4*a^2 + b^2)*Cos[c + d*x] + 3*a*b*Sin[c + d*x]))/(2*d*(a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 2, -(1/(2*b*d*(a + b*Tan[c + d*x])^2)), -(Cot[c + d*x]^2/(2*b*d*(b + a*Cot[c + d*x])^2))} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, -(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2)*d)) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, -((1/b + b/a^2)/(2*d*(b + a*Cot[c + d*x])^2)) + (1/a^2 - 1/b^2)/(d*(b + a*Cot[c + d*x])) + Log[b + a*Cot[c + d*x]]/(b^3*d) + Log[Tan[c + d*x]]/(b^3*d)} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 12, -((3*a*ArcTanh[Sin[c + d*x]])/(b^4*d)) - (2*a^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]*d) - ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*b^2*Sqrt[a^2 + b^2]*d) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) + Sec[c + d*x]/(b^3*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*b^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (2*a)/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, -((a^2 + b^2)^2/(2*a^2*b^3*d*(b + a*Cot[c + d*x])^2)) - ((3*a^2 - b^2)*(a^2 + b^2))/(a^2*b^4*d*(b + a*Cot[c + d*x])) + (2*(3*a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^5*d) + (2*(3*a^2 + b^2)*Log[Tan[c + d*x]])/(b^5*d) - (3*a*Tan[c + d*x])/(b^4*d) + Tan[c + d*x]^2/(2*b^3*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 31, -((4*a^3*ArcTanh[Sin[c + d*x]])/(b^6*d)) - (3*a*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (6*a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^6*d) - (8*a^2*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^4*d) - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) + (4*a^2*Sec[c + d*x])/(b^5*d) + (2*(a^2 + b^2)*Sec[c + d*x])/(b^5*d) + Sec[c + d*x]^3/(3*b^3*d) - ((a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (4*a*(a^2 + b^2))/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (3*a*Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d)} -{Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x])^3, x, 3, -((a^2 + b^2)^3/(2*a^2*b^5*d*(b + a*Cot[c + d*x])^2)) - ((5*a^2 - b^2)*(a^2 + b^2)^2)/(a^2*b^6*d*(b + a*Cot[c + d*x])) + (3*(a^2 + b^2)*(5*a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^7*d) + (3*(a^2 + b^2)*(5*a^2 + b^2)*Log[Tan[c + d*x]])/(b^7*d) - (a*(10*a^2 + 9*b^2)*Tan[c + d*x])/(b^6*d) + (3*(2*a^2 + b^2)*Tan[c + d*x]^2)/(2*b^5*d) - (a*Tan[c + d*x]^3)/(b^4*d) + Tan[c + d*x]^4/(4*b^3*d)} - - -{Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 6, ((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, -7, (a*(2*a^2 - 3*b^2)*ArcTanh[(-b + a*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d) + (-3*(3*a^4*b - a^2*b^3 + b^5)*Cos[2*(c + d*x)] + (1/2)*b*(-9*a^2 + b^2)*(2*(a^2 + b^2) + 3*a*b*Sin[2*(c + d*x)]))/(6*(a^2 + b^2)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 2, -(Cot[c + d*x]^3/(3*b*d*(b + a*Cot[c + d*x])^3))} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 5, -((a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*(a^2 + b^2)^(5/2)*d)) - b/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 2, -((b*Cos[c + d*x] - a*Sin[c + d*x])/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)) + (2*Sin[c + d*x])/(3*a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 8, ArcTanh[Sin[c + d*x]]/(b^4*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^2*(a^2 + b^2)^(3/2)*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]*d) - 1/(3*b*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - 1/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (a^2 + b^2)^2/(3*a^3*b^2*d*(b + a*Cot[c + d*x])^3) + (a/b^3 - b/a^3)/(d*(b + a*Cot[c + d*x])^2) + (1/a^3 + (3*a)/b^4)/(d*(b + a*Cot[c + d*x])) - (4*a*Log[b + a*Cot[c + d*x]])/(b^5*d) - (4*a*Log[Tan[c + d*x]])/(b^5*d) + Tan[c + d*x]/(b^4*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 32, (8*a^2*ArcTanh[Sin[c + d*x]])/(b^6*d) + ArcTanh[Sin[c + d*x]]/(2*b^4*d) + (2*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^6*d) + (4*a^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*Sqrt[a^2 + b^2]*d) + (3*a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^4*Sqrt[a^2 + b^2]*d) + (6*a*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) - (4*a*Sec[c + d*x])/(b^5*d) - (a^2 + b^2)/(3*b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (4*a^2)/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (2*(a^2 + b^2))/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4, x, 3, (a^2 + b^2)^3/(3*a^3*b^4*d*(b + a*Cot[c + d*x])^3) + (2*a^6 + 3*a^4*b^2 - b^6)/(a^3*b^5*d*(b + a*Cot[c + d*x])^2) + (10*a^6 + 9*a^4*b^2 + b^6)/(a^3*b^6*d*(b + a*Cot[c + d*x])) - (4*a*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]])/(b^7*d) - (4*a*(5*a^2 + 3*b^2)*Log[Tan[c + d*x]])/(b^7*d) + ((10*a^2 + 3*b^2)*Tan[c + d*x])/(b^6*d) - (2*a*Tan[c + d*x]^2)/(b^5*d) + Tan[c + d*x]^3/(3*b^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a Cos[c+d x]+b Sin[c+d x])^n when a^2+b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 9, (5*x)/(16*a) + (I*Cos[c + d*x]^6)/(6*a*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)} -{Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, (I*Cos[c + d*x]^5)/(5*a*d) + Sin[c + d*x]/(a*d) - (2*Sin[c + d*x]^3)/(3*a*d) + Sin[c + d*x]^5/(5*a*d)} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 8, (3*x)/(8*a) + (I*Cos[c + d*x]^4)/(4*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, (I*Cos[c + d*x]^3)/(3*a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^3/(3*a*d)} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 2, x/(2*a) + (I*Cos[c + d*x])/(2*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 1, I/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 4, x/a + (I*Log[Cos[c + d*x]])/(a*d)} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 6, ArcTanh[Sin[c + d*x]]/(a*d) - (I*Sec[c + d*x])/(a*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, -((I*Sec[c + d*x]^2)/(2*a*d)) + Tan[c + d*x]/(a*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, ArcTanh[Sin[c + d*x]]/(2*a*d) - (I*Sec[c + d*x]^3)/(3*a*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)} -{Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, -((I*Sec[c + d*x]^4)/(4*a*d)) + Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d)} -{Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 8, (3*ArcTanh[Sin[c + d*x]])/(8*a*d) - (I*Sec[c + d*x]^5)/(5*a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)} -{Sec[c + d*x]^7/(a*Cos[c + d*x] + I*a*Sin[c + d*x]), x, 7, -((I*Sec[c + d*x]^6)/(6*a*d)) + Tan[c + d*x]/(a*d) + (2*Tan[c + d*x]^3)/(3*a*d) + Tan[c + d*x]^5/(5*a*d)} - - -{Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 10, (2*I*Cos[c + d*x]^7)/(7*a^2*d) + Sin[c + d*x]/(a^2*d) - (4*Sin[c + d*x]^3)/(3*a^2*d) + Sin[c + d*x]^5/(a^2*d) - (2*Sin[c + d*x]^7)/(7*a^2*d)} -{Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 5, x/(4*a^2) - 1/(16*a^2*d*(I - Cot[c + d*x])) - 1/(12*a^2*d*(I + Cot[c + d*x])^3) - (3*I)/(8*a^2*d*(I + Cot[c + d*x])^2) + 11/(16*a^2*d*(I + Cot[c + d*x]))} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 10, (2*I*Cos[c + d*x]^5)/(5*a^2*d) + Sin[c + d*x]/(a^2*d) - Sin[c + d*x]^3/(a^2*d) + (2*Sin[c + d*x]^5)/(5*a^2*d)} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 3, x/(4*a^2) + (I*Cos[c + d*x]^2)/(4*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2) + (I*Cos[c + d*x])/(4*d*(a^2*Cos[c + d*x] + I*a^2*Sin[c + d*x]))} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 9, (2*I*Cos[c + d*x]^3)/(3*a^2*d) + Sin[c + d*x]/(a^2*d) - (2*Sin[c + d*x]^3)/(3*a^2*d)} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 1, I/(2*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 8, -(ArcTanh[Sin[c + d*x]]/(a^2*d)) + (2*I*Cos[c + d*x])/(a^2*d) + (2*Sin[c + d*x])/(a^2*d)} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 4, (2*x)/a^2 + (2*I*Log[Sin[c + d*x]])/(a^2*d) - (2*I*Log[Tan[c + d*x]])/(a^2*d) - Tan[c + d*x]/(a^2*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 8, (3*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*I*Sec[c + d*x])/(a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 3, -((I*(I - Cot[c + d*x])^3*Tan[c + d*x]^3)/(3*a^2*d))} -{Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 10, (5*ArcTanh[Sin[c + d*x]])/(8*a^2*d) - (2*I*Sec[c + d*x]^3)/(3*a^2*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(8*a^2*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(4*a^2*d)} -{Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 4, Tan[c + d*x]/(a^2*d) - (I*Tan[c + d*x]^2)/(a^2*d) - (I*Tan[c + d*x]^4)/(2*a^2*d) - Tan[c + d*x]^5/(5*a^2*d)} - - -{Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 5, (5*x)/(32*a^3) - 1/(32*a^3*d*(I - Cot[c + d*x])) + I/(16*a^3*d*(I + Cot[c + d*x])^4) - 1/(3*a^3*d*(I + Cot[c + d*x])^3) - (23*I)/(32*a^3*d*(I + Cot[c + d*x])^2) + 13/(16*a^3*d*(I + Cot[c + d*x]))} -{Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 13, -((I*Cos[c + d*x]^5)/(5*a^3*d)) + (4*I*Cos[c + d*x]^7)/(7*a^3*d) + Sin[c + d*x]/(a^3*d) - (2*Sin[c + d*x]^3)/(a^3*d) + (9*Sin[c + d*x]^5)/(5*a^3*d) - (4*Sin[c + d*x]^7)/(7*a^3*d)} -{Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 4, x/(8*a^3) + (I*Cos[c + d*x]^3)/(6*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3) + (I*Cos[c + d*x]^2)/(8*a*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2) + (I*Cos[c + d*x])/(8*d*(a^3*Cos[c + d*x] + I*a^3*Sin[c + d*x]))} -{Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 13, -((I*Cos[c + d*x]^3)/(3*a^3*d)) + (4*I*Cos[c + d*x]^5)/(5*a^3*d) + Sin[c + d*x]/(a^3*d) - (5*Sin[c + d*x]^3)/(3*a^3*d) + (4*Sin[c + d*x]^5)/(5*a^3*d)} -{Cos[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 2, (I*Cot[c + d*x]^2)/(2*a^3*d*(I + Cot[c + d*x])^2)} -{Cos[c + d*x]^0/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 1, I/(3*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)} -{Sec[c + d*x]^1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 4, -(x/a^3) + 2/(a^3*d*(I + Cot[c + d*x])) - (I*Log[Sin[c + d*x]])/(a^3*d) + (I*Log[Tan[c + d*x]])/(a^3*d)} -{Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 11, -((3*ArcTanh[Sin[c + d*x]])/(a^3*d)) + (4*I*Cos[c + d*x])/(a^3*d) + (I*Sec[c + d*x])/(a^3*d) + (4*Sin[c + d*x])/(a^3*d)} -{Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 4, (4*x)/a^3 + (4*I*Log[Sin[c + d*x]])/(a^3*d) - (4*I*Log[Tan[c + d*x]])/(a^3*d) - (3*Tan[c + d*x])/(a^3*d) + (I*Tan[c + d*x]^2)/(2*a^3*d)} -{Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 10, (5*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (4*I*Sec[c + d*x])/(a^3*d) + (I*Sec[c + d*x]^3)/(3*a^3*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d)} -{Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 3, (I*(I - Cot[c + d*x])^4*Tan[c + d*x]^4)/(4*a^3*d)} -{Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 13, (7*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - (4*I*Sec[c + d*x]^3)/(3*a^3*d) + (I*Sec[c + d*x]^5)/(5*a^3*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - (3*Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d)} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n/Cos[c + d*x]^n, x, 1, -((I*Hypergeometric2F1[1, n, 1 + n, (1/2)*(1 + I*Tan[c + d*x])]*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n)/(Cos[c + d*x]^n*(2*d*n)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[c+d x]^m (a Sec[c+d x]+b Tan[c+d x])^n*) - - -{1/(Sec[x] + Tan[x]), x, 3, Log[1 + Sin[x]]} -{Sin[x]/(Sec[x] + Tan[x]), x, 4, -Log[1 + Sin[x]] + Sin[x]} -{Cos[x]/(Sec[x] + Tan[x]), x, 3, x + Cos[x]} -{Tan[x]/(Sec[x] + Tan[x]), x, 3, x + Cos[x]/(1 + Sin[x])} -{Cot[x]/(Sec[x] + Tan[x]), x, 4, -x - ArcTanh[Cos[x]]} -{Sec[x]/(Sec[x] + Tan[x]), x, 2, -(Cos[x]/(1 + Sin[x]))} -{Csc[x]/(Sec[x] + Tan[x]), x, 5, Log[Sin[x]] - Log[1 + Sin[x]]} - - -{1/(Sec[x] - Tan[x]), x, 3, -Log[1 - Sin[x]]} -{Sin[x]/(Sec[x] - Tan[x]), x, 4, -Log[1 - Sin[x]] - Sin[x]} -{Cos[x]/(Sec[x] - Tan[x]), x, 3, x - Cos[x]} -{Tan[x]/(Sec[x] - Tan[x]), x, 3, -x + Cos[x]/(1 - Sin[x])} -{Cot[x]/(Sec[x] - Tan[x]), x, 4, x - ArcTanh[Cos[x]]} -{Sec[x]/(Sec[x] - Tan[x]), x, 2, Cos[x]/(1 - Sin[x])} -{Csc[x]/(Sec[x] - Tan[x]), x, 5, -Log[1 - Sin[x]] + Log[Sin[x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[c+d x]^m (a Csc[c+d x]+b Cot[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Csc[c + d*x]*(Cot[c + d*x] + Csc[c + d*x]), x, 4, -(Cot[c + d*x]/d) - Csc[c + d*x]/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sin[x]/(Csc[x] + Cot[x]), x, 3, x - Sin[x]} -{Cos[x]/(Csc[x] + Cot[x]), x, 4, -Cos[x] + Log[1 + Cos[x]]} -{Tan[x]/(Csc[x] + Cot[x]), x, 4, -x+ArcTanh[Sin[x]]} -{Cot[x]/(Csc[x] + Cot[x]), x, 3, x - Sin[x]/(1 + Cos[x])} -{Sec[x]/(Csc[x] + Cot[x]), x, 5, -Log[Cos[x]] + Log[1 + Cos[x]]} -{Csc[x]/(Csc[x] + Cot[x]), x, 2, Sin[x]/(1 + Cos[x])} - - -{Sin[x]/(Csc[x] - Cot[x]), x, 3, x + Sin[x]} -{Cos[x]/(Csc[x] - Cot[x]), x, 4, Cos[x] + Log[1 - Cos[x]]} -{Tan[x]/(Csc[x] - Cot[x]), x, 4, x + ArcTanh[Sin[x]]} -{Cot[x]/(Csc[x] - Cot[x]), x, 3, -x - Sin[x]/(1 - Cos[x])} -{Sec[x]/(Csc[x] - Cot[x]), x, 5, Log[1 - Cos[x]] - Log[Cos[x]]} -{Csc[x]/(Csc[x] - Cot[x]), x, 2, -(Sin[x]/(1 - Cos[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[c+d x]^m (a Csc[c+d x]+b Sin[c+d x])^n*) - - -{1/(Csc[c + d*x] + Sin[c + d*x]), x, 3, -(ArcTanh[Cos[c + d*x]/Sqrt[2]]/(Sqrt[2]*d))} -{Sin[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 4, x - x/Sqrt[2] - ArcTan[(Cos[c + d*x]*Sin[c + d*x])/(1 + Sqrt[2] + Sin[c + d*x]^2)]/(Sqrt[2]*d)} -{Cos[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 2, Log[1 + Sin[c + d*x]^2]/(2*d)} -{Tan[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 4, -(ArcTan[Sin[c + d*x]]/(2*d)) + ArcTanh[Sin[c + d*x]]/(2*d)} -{Cot[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 2, ArcTan[Sin[c + d*x]]/d} -{Sec[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 3, ArcTanh[Sin[c + d*x]^2]/(2*d)} -{Csc[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]), x, 2, x/Sqrt[2] + ArcTan[(Cos[c + d*x]*Sin[c + d*x])/(1 + Sqrt[2] + Sin[c + d*x]^2)]/(Sqrt[2]*d)} - - -{1/(Csc[c + d*x] - Sin[c + d*x]), x, 3, Sec[c + d*x]/d} -{Sin[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 3, -x + Tan[c + d*x]/d} -{Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 2, -(Log[Cos[c + d*x]]/d)} -{Tan[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 3, -(ArcTanh[Sin[c + d*x]]/(2*d)) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Cot[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 2, ArcTanh[Sin[c + d*x]]/d} -{Sec[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 2, Sec[c + d*x]^2/(2*d)} -{Csc[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]), x, 2, Tan[c + d*x]/d} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[c+d x]^m (a Sin[c+d x]+b Tan[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a Sin[c+d x]+b Tan[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 6, -((b*Cos[c + d*x]^3)/(3*d)) - (a*Cos[c + d*x]^4)/(4*d)} -{Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 6, -((a*Cos[c + d*x]^3)/(3*d)) + (b*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, -((b + a*Cos[c + d*x])^2/(2*a*d)), -((b*Cos[c + d*x])/d) + (a*Sin[c + d*x]^2)/(2*d)} -{Cos[c + d*x]^0*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 3, -((a*Cos[c + d*x])/d) - (b*Log[Cos[c + d*x]])/d} -{Sec[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, -((a*Log[Cos[c + d*x]])/d) + (b*Sec[c + d*x])/d} -{Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 6, (a*Sec[c + d*x])/d + (b*Sec[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 6, (a*Sec[c + d*x]^2)/(2*d) + (b*Sec[c + d*x]^3)/(3*d)} - - -{Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 6, (a*b*x)/4 - (a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((4*a^2 + b^2)*Sin[c + d*x]^3)/(30*d) + (b*(b + a*Cos[c + d*x])*Sin[c + d*x]^3)/(10*d) + ((b + a*Cos[c + d*x])^2*Sin[c + d*x]^3)/(5*d)} -{Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 5, (1/8)*(a^2 + 4*b^2)*x - ((a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Sin[c + d*x]^3)/(12*d) + (a*(b + a*Cos[c + d*x])*Sin[c + d*x]^3)/(4*d)} -{Cos[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 7, a*b*x + (b^2*ArcTanh[Sin[c + d*x]])/d + ((a^2 - 2*b^2)*Sin[c + d*x])/(3*d) - (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) - ((b + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)} -{Cos[c + d*x]^0*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 10, (a^2*x)/2 - b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d} -{Sec[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 7, -2*a*b*x + ((2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*Sin[c + d*x])/(2*d) + (a*b*Tan[c + d*x])/d + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)} -{Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 7, (-a^2)*x - (a*b*ArcTanh[Sin[c + d*x]])/d + ((2*a^2 - b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)} -{Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 9, -(((4*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(8*d)) - (2*a*b*Tan[c + d*x])/(3*d) + ((2*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)} - - -{Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 4, -(((a^2 - b^2)*(b + a*Cos[c + d*x])^4)/(4*a^3*d)) - (2*b*(b + a*Cos[c + d*x])^5)/(5*a^3*d) + (b + a*Cos[c + d*x])^6/(6*a^3*d)} -{Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, -((3*a*b^2*Cos[c + d*x])/d) - (b*(3*a^2 - b^2)*Cos[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Cos[c + d*x]^4)/(4*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (b^3*Log[Cos[c + d*x]])/d} -{Cos[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, -((b*(3*a^2 - b^2)*Cos[c + d*x])/d) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^2)/(2*d) + (a^2*b*Cos[c + d*x]^3)/d + (a^3*Cos[c + d*x]^4)/(4*d) - (3*a*b^2*Log[Cos[c + d*x]])/d + (b^3*Sec[c + d*x])/d} -{Cos[c + d*x]^0*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 4, -((a*(a^2 - 3*b^2)*Cos[c + d*x])/d) + (3*a^2*b*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)} -{Sec[c + d*x]^1*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, (3*a^2*b*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Log[Cos[c + d*x]])/d + (b*(3*a^2 - b^2)*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x]^2)/(2*d) + (b^3*Sec[c + d*x]^3)/(3*d)} -{Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, (a^3*Cos[c + d*x])/d + (3*a^2*b*Log[Cos[c + d*x]])/d + (a*(a^2 - 3*b^2)*Sec[c + d*x])/d + (b*(3*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (a*b^2*Sec[c + d*x]^3)/d + (b^3*Sec[c + d*x]^4)/(4*d)} -{Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, (a^3*Log[Cos[c + d*x]])/d - (3*a^2*b*Sec[c + d*x])/d + (a*(a^2 - 3*b^2)*Sec[c + d*x]^2)/(2*d) + (b*(3*a^2 - b^2)*Sec[c + d*x]^3)/(3*d) + (3*a*b^2*Sec[c + d*x]^4)/(4*d) + (b^3*Sec[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, -((b*Cos[c + d*x])/(a^2*d)) + Cos[c + d*x]^2/(2*a*d) + Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (b^4*Log[b + a*Cos[c + d*x]])/(a^3*(a^2 - b^2)*d)} -{Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, Cos[c + d*x]/(a*d) + Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b^3*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)*d)} -{Cos[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (b^2*Log[b + a*Cos[c + d*x]])/(a*(a^2 - b^2)*d)} -{Cos[c + d*x]^0/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 4, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)} -{Sec[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 7, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (a*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)} -{Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[Cos[c + d*x]]/(b*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (a^2*Log[b + a*Cos[c + d*x]])/(b*(a^2 - b^2)*d)} -{Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]), x, 5, Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + (a*Log[Cos[c + d*x]])/(b^2*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (a^3*Log[b + a*Cos[c + d*x]])/(b^2*(a^2 - b^2)*d) + Sec[c + d*x]/(b*d)} - - -{Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 12, (2*b*x)/a^3 + (2*b^6*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (2*b^4*(5*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(a^2*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (b^5*Sin[c + d*x])/(a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 11, -(x/a^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - (4*b^3*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^4*Sin[c + d*x])/(a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Cos[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 11, (2*b^4*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) + (2*b^2*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (b^3*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Cos[c + d*x]^0/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 11, -((4*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (a*b^2*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} -{Sec[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 6, (2*a*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Csc[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x])) - ((a^2 + 2*b^2 - 3*a*b*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)^2*d)} -{Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 6, -((6*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*Csc[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x])) + ((3*a*b - (2*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)^2*d)} -{Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 12, ArcTanh[Sin[c + d*x]]/(b^2*d) + (2*a^3*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (2*a^3*(a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^2*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (a^4*Sin[c + d*x])/(b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))} - - -{Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 6, b^6/(2*a^3*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (2*b^5*(3*a^2 - b^2))/(a^3*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - ((2*a + 5*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((2*a - 5*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (b^4*(15*a^4 - 4*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/(a^3*(a^2 - b^2)^4*d)} -{Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 6, -(b^5/(2*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2)) + (b^4*(5*a^2 - b^2))/(a^2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - ((a + 4*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((a - 4*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (2*b^3*(5*a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Cos[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 6, b^4/(2*a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (4*a*b^3)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - (3*b*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + (3*b*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (6*a*b^2*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Cos[c + d*x]^0/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, -(b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2)) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Sec[c + d*x]^1/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 6, (a*b^2)/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (2*a*b*(a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((2*a - b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((2*a + b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (a*(a^4 + 8*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 6, -((3*a^2*b)/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2)) + (3*a^2*(a^2 + 3*b^2))/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (3*a*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - (3*a*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (6*a^2*b*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} -{Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 5, (a*(2*a^2 + b^2))/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (a*b*(11*a^2 + b^2))/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a - b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((4*a + b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((4*a - b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (2*a^3*(a^2 + 5*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[c+d x]^m (a Sin[c+d x]+b Tan[c+d x])^n with m symbolic*) - - -{Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^3, x, 4, (b^3*Cos[c + d*x]^(-2 + m))/(d*(2 - m)) + (3*a*b^2*Cos[c + d*x]^(-1 + m))/(d*(1 - m)) - (b*(3*a^2 - b^2)*Cos[c + d*x]^m)/(d*m) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^(1 + m))/(d*(1 + m)) + (3*a^2*b*Cos[c + d*x]^(2 + m))/(d*(2 + m)) + (a^3*Cos[c + d*x]^(3 + m))/(d*(3 + m))} -{Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^2, x, 8, ((a^2 - 2*b^2)*Cos[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*m*(2 + m)) - (2*a*b*Cos[c + d*x]^m*Sin[c + d*x])/(d*(2 + 3*m + m^2)) - (Cos[c + d*x]^(-1 + m)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*(2 + m)) - ((a^2*(1 - m) - b^2*(2 + m))*Cos[c + d*x]^(-1 + m)*Hypergeometric2F1[1/2, (1/2)*(-1 + m), (1 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - m)*m*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Cos[c + d*x]^m*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*m*(1 + m)*Sqrt[Sin[c + d*x]^2])} -{Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^1, x, 6, -((b*Cos[c + d*x]^m)/(d*m)) - (a*Cos[c + d*x]^(1 + m))/(d*(1 + m))} -{Cos[c + d*x]^m/(a*Sin[c + d*x] + b*Tan[c + d*x])^1, x, 7, (Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -Cos[c + d*x]])/(2*(a - b)*d*(2 + m)) - (Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, Cos[c + d*x]])/(2*(a + b)*d*(2 + m)) - (a^2*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((a*Cos[c + d*x])/b)])/(b*(a^2 - b^2)*d*(2 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cos[c+d x]^m Sin[c+d x]^n (a Cos[c+d x]+b Sin[c+d x])^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[x]*Sin[x]^1/(a*Cos[x] + b*Sin[x]), x, 5, (a*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) + (b*Sin[x])/(a^2 + b^2)} -{Cos[x]*Sin[x]^2/(a*Cos[x] + b*Sin[x]), x, 7, -((a*b^2*x)/(a^2 + b^2)^2) + (a*x)/(2*(a^2 + b^2)) + (a^2*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (a*Cos[x]*Sin[x])/(2*(a^2 + b^2)) + (b*Sin[x]^2)/(2*(a^2 + b^2))} -{Cos[x]*Sin[x]^3/(a*Cos[x] + b*Sin[x]), x, 9, (a^3*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x])/(a^2 + b^2) + (a*Cos[x]^3)/(3*(a^2 + b^2)) + (a^2*b*Sin[x])/(a^2 + b^2)^2 + (b*Sin[x]^3)/(3*(a^2 + b^2))} - -{Cos[x]^2*Sin[x]^1/(a*Cos[x] + b*Sin[x]), x, 7, -((a^2*b*x)/(a^2 + b^2)^2) + (b*x)/(2*(a^2 + b^2)) - (a*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 + (b*Cos[x]*Sin[x])/(2*(a^2 + b^2)) + (a*Sin[x]^2)/(2*(a^2 + b^2))} -{Cos[x]^2*Sin[x]^2/(a*Cos[x] + b*Sin[x]), x, 10, -((a^2*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (a^2*b*Cos[x])/(a^2 + b^2)^2 - (b*Cos[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Sin[x])/(a^2 + b^2)^2 + (a*Sin[x]^3)/(3*(a^2 + b^2))} -{Cos[x]^2*Sin[x]^3/(a*Cos[x] + b*Sin[x]), x, 13, (a^2*b^3*x)/(a^2 + b^2)^3 - (a^2*b*x)/(2*(a^2 + b^2)^2) + (b*x)/(8*(a^2 + b^2)) - (a^3*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (a^2*b*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (b*Cos[x]*Sin[x])/(8*(a^2 + b^2)) - (b*Cos[x]^3*Sin[x])/(4*(a^2 + b^2)) - (a*b^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (a*Sin[x]^4)/(4*(a^2 + b^2))} - -{Cos[x]^3*Sin[x]^1/(a*Cos[x] + b*Sin[x]), x, 9, (a*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x]^3)/(3*(a^2 + b^2)) - (a^2*b*Sin[x])/(a^2 + b^2)^2 + (b*Sin[x])/(a^2 + b^2) - (b*Sin[x]^3)/(3*(a^2 + b^2))} -{Cos[x]^3*Sin[x]^2/(a*Cos[x] + b*Sin[x]), x, 13, (a^3*b^2*x)/(a^2 + b^2)^3 - (a*b^2*x)/(2*(a^2 + b^2)^2) + (a*x)/(8*(a^2 + b^2)) - (b*Cos[x]^4)/(4*(a^2 + b^2)) + (a^2*b^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*Cos[x]*Sin[x])/(8*(a^2 + b^2)) - (a*Cos[x]^3*Sin[x])/(4*(a^2 + b^2)) - (a^2*b*Sin[x]^2)/(2*(a^2 + b^2)^2)} -{Cos[x]^3*Sin[x]^3/(a*Cos[x] + b*Sin[x]), x, 17, (a^3*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (a^3*b^2*Cos[x])/(a^2 + b^2)^3 + (a*b^2*Cos[x]^3)/(3*(a^2 + b^2)^2) - (a*Cos[x]^3)/(3*(a^2 + b^2)) + (a*Cos[x]^5)/(5*(a^2 + b^2)) + (a^2*b^3*Sin[x])/(a^2 + b^2)^3 - (a^2*b*Sin[x]^3)/(3*(a^2 + b^2)^2) + (b*Sin[x]^3)/(3*(a^2 + b^2)) - (b*Sin[x]^5)/(5*(a^2 + b^2))} - - -{Cos[x]*Sin[x]^1/(a*Cos[x] + b*Sin[x])^2, x, 6, (2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Sin[x])/((a^2 + b^2)*(a*Cos[x] + b*Sin[x])), (2*a*b*x)/(a^2 + b^2)^2 - (a^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 + (b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Sin[x])/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))} -{Cos[x]*Sin[x]^2/(a*Cos[x] + b*Sin[x])^2, x, 13, -((a*(a^2 - 2*b^2)*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - (2*a*b*Cos[x])/(a^2 + b^2)^2 - ((a^2 - b^2)*Sin[x])/(a^2 + b^2)^2 - (a^2*b)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])), -((a^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (2*a*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (2*a*b*Cos[x])/(a^2 + b^2)^2 - (a^2*Sin[x])/(a^2 + b^2)^2 + (b^2*Sin[x])/(a^2 + b^2)^2 - (a^2*b)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))} -{Cos[x]*Sin[x]^3/(a*Cos[x] + b*Sin[x])^2, x, 17, (b*(3*a^3 - a*b^2)*x)/(a^2 + b^2)^3 - (a^2*(a^2 - 3*b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 - ((a^2 - b^2)*Sin[x]^2)/(2*(a^2 + b^2)^2) - (a^2*b*Sin[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])), (a^3*b*x)/(a^2 + b^2)^3 - (a*b^3*x)/(a^2 + b^2)^3 + (a*b*(a^2 - b^2)*x)/(a^2 + b^2)^3 + (a*b*x)/(a^2 + b^2)^2 - (a^2*b)/((a^2 + b^2)^2*(b + a*Cot[x])) - (a^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (3*a^2*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 - (a^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (b^2*Sin[x]^2)/(2*(a^2 + b^2)^2)} - -{Cos[x]^2*Sin[x]^1/(a*Cos[x] + b*Sin[x])^2, x, 13, -((b*(-2*a^2 + b^2)*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((a^2 - b^2)*Cos[x])/(a^2 + b^2)^2 + (2*a*b*Sin[x])/(a^2 + b^2)^2 + (a*b^2)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])), (2*a^2*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^2*Cos[x])/(a^2 + b^2)^2 + (b^2*Cos[x])/(a^2 + b^2)^2 + (2*a*b*Sin[x])/(a^2 + b^2)^2 + (a*b^2)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))} -{Cos[x]^2*Sin[x]^2/(a*Cos[x] + b*Sin[x])^2, x, 21, ((a^4 - 6*a^2*b^2 + b^4)*x)/(2*(a^2 + b^2)^3) + (2*a*b*(a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + ((-a^2 + b^2)*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*b*Sin[x]^2)/(a^2 + b^2)^2 + (a*b^2*Sin[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])), -((4*a^2*b^2*x)/(a^2 + b^2)^3) + (a^2*x)/(2*(a^2 + b^2)^2) + (b^2*x)/(2*(a^2 + b^2)^2) + (2*a^3*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (2*a*b^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (b^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*b*Sin[x]^2)/(a^2 + b^2)^2 + (a*b^2*Sin[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))} -{Cos[x]^2*Sin[x]^3/(a*Cos[x] + b*Sin[x])^2, x, 33, (a^2*b*(2*a^2 - 3*b^2)*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (a^2*(a^2 - 3*b^2)*Cos[x])/(a^2 + b^2)^3 + ((a^2 - b^2)*Cos[x]^3)/(3*(a^2 + b^2)^2) + (2*a*b*(a^2 - b^2)*Sin[x])/(a^2 + b^2)^3 + (2*a*b*Sin[x]^3)/(3*(a^2 + b^2)^2) + (a^3*b^2)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x])), (2*a^4*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (3*a^2*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (4*a^2*b^2*Cos[x])/(a^2 + b^2)^3 - (a^2*Cos[x])/(a^2 + b^2)^2 + (a^2*Cos[x]^3)/(3*(a^2 + b^2)^2) - (b^2*Cos[x]^3)/(3*(a^2 + b^2)^2) + (2*a^3*b*Sin[x])/(a^2 + b^2)^3 - (2*a*b^3*Sin[x])/(a^2 + b^2)^3 + (2*a*b*Sin[x]^3)/(3*(a^2 + b^2)^2) + (a^3*b^2)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))} - -{Cos[x]^3*Sin[x]^1/(a*Cos[x] + b*Sin[x])^2, x, 17, -((a*b*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (b^2*(3*a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 + ((a^2 - b^2)*Sin[x]^2)/(2*(a^2 + b^2)^2) + (a*b^2*Cos[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])), -((a^3*b*x)/(a^2 + b^2)^3) + (a*b^3*x)/(a^2 + b^2)^3 - (a*b*(a^2 - b^2)*x)/(a^2 + b^2)^3 + (a*b*x)/(a^2 + b^2)^2 + (b^2*Cos[x]^2)/(2*(a^2 + b^2)^2) - (3*a^2*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (b^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 + (a^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (a*b^2)/((a^2 + b^2)^2*(a + b*Tan[x]))} -{Cos[x]^3*Sin[x]^2/(a*Cos[x] + b*Sin[x])^2, x, 33, -((a*b^2*(3*a^2 - 2*b^2)*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (2*a*b*(a^2 - b^2)*Cos[x])/(a^2 + b^2)^3 - (2*a*b*Cos[x]^3)/(3*(a^2 + b^2)^2) - (b^2*(3*a^2 - b^2)*Sin[x])/(a^2 + b^2)^3 + ((a^2 - b^2)*Sin[x]^3)/(3*(a^2 + b^2)^2) - (a^2*b^3)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x])), -((3*a^3*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (2*a*b^4*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (2*a^3*b*Cos[x])/(a^2 + b^2)^3 - (2*a*b^3*Cos[x])/(a^2 + b^2)^3 - (2*a*b*Cos[x]^3)/(3*(a^2 + b^2)^2) - (4*a^2*b^2*Sin[x])/(a^2 + b^2)^3 + (b^2*Sin[x])/(a^2 + b^2)^2 + (a^2*Sin[x]^3)/(3*(a^2 + b^2)^2) - (b^2*Sin[x]^3)/(3*(a^2 + b^2)^2) - (a^2*b^3)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))} -{Cos[x]^3*Sin[x]^3/(a*Cos[x] + b*Sin[x])^2, x, 48, -((3*a*b*(a^4 - 6*a^2*b^2 + b^4)*x)/(4*(a^2 + b^2)^4)) - (b^2*Cos[x]^4)/(4*(a^2 + b^2)^2) - (3*a^2*b^2*(a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (a*b*(5*a^2 - 3*b^2)*Cos[x]*Sin[x])/(4*(a^2 + b^2)^3) - (a*b*Cos[x]^3*Sin[x])/(2*(a^2 + b^2)^2) - (2*a^2*b^2*Sin[x]^2)/(a^2 + b^2)^3 + (a^2*Sin[x]^4)/(4*(a^2 + b^2)^2) - (a^2*b^3*Sin[x])/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x])), (6*a^3*b^3*x)/(a^2 + b^2)^4 - (a^3*b*x)/(a^2 + b^2)^3 - (a*b^3*x)/(a^2 + b^2)^3 + (a*b*x)/(4*(a^2 + b^2)^2) - (b^2*Cos[x]^4)/(4*(a^2 + b^2)^2) - (3*a^4*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (3*a^2*b^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (a^3*b*Cos[x]*Sin[x])/(a^2 + b^2)^3 - (a*b^3*Cos[x]*Sin[x])/(a^2 + b^2)^3 + (a*b*Cos[x]*Sin[x])/(4*(a^2 + b^2)^2) - (a*b*Cos[x]^3*Sin[x])/(2*(a^2 + b^2)^2) - (2*a^2*b^2*Sin[x]^2)/(a^2 + b^2)^3 + (a^2*Sin[x]^4)/(4*(a^2 + b^2)^2) - (a^2*b^3*Sin[x])/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))} - - -{Tan[x]/(a*Sin[x] + b*Cos[x]), x, 5, ArcTanh[Sin[x]]/a + (b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])} - - -{Cot[x]/(a*Sin[x] + b*Cos[x]), x, 5, -(ArcTanh[Cos[x]]/b) + (a*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.3 (c+d x)^m trig^n trig^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.3 (c+d x)^m trig^n trig^p.m deleted file mode 100644 index 54b6ff1..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.3 (c+d x)^m trig^n trig^p.m +++ /dev/null @@ -1,753 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^n Sin[a+b x]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^1 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x] Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^m, x, 5, -((2^(-3 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b)) - (2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b)} - -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^4, x, 5, (3*c*d^3*x)/(2*b^3) + (3*d^4*x^2)/(4*b^3) - (c + d*x)^4/(4*b) - (3*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b^4) + (d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/b^2 + (3*d^4*Sin[a + b*x]^2)/(4*b^5) - (3*d^2*(c + d*x)^2*Sin[a + b*x]^2)/(2*b^3) + ((c + d*x)^4*Sin[a + b*x]^2)/(2*b)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^3, x, 5, (3*d^3*x)/(8*b^3) - (c + d*x)^3/(4*b) - (3*d^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) - (3*d^2*(c + d*x)*Sin[a + b*x]^2)/(4*b^3) + ((c + d*x)^3*Sin[a + b*x]^2)/(2*b)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^2, x, 3, -((c*d*x)/(2*b)) - (d^2*x^2)/(4*b) + (d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) - (d^2*Sin[a + b*x]^2)/(4*b^3) + ((c + d*x)^2*Sin[a + b*x]^2)/(2*b)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^1, x, 3, -((d*x)/(4*b)) + (d*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) + ((c + d*x)*Sin[a + b*x]^2)/(2*b)} -{Cos[a + b*x]*Sin[a + b*x]/(c + d*x)^1, x, 5, (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d) + (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Cos[a + b*x]*Sin[a + b*x]/(c + d*x)^2, x, 6, (b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^2 - Sin[2*a + 2*b*x]/(2*d*(c + d*x)) - (b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Cos[a + b*x]*Sin[a + b*x]/(c + d*x)^3, x, 7, -((b*Cos[2*a + 2*b*x])/(2*d^2*(c + d*x))) - (b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^3 - Sin[2*a + 2*b*x]/(4*d*(c + d*x)^2) - (b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^3} -{Cos[a + b*x]*Sin[a + b*x]/(c + d*x)^4, x, 8, -((b*Cos[2*a + 2*b*x])/(6*d^2*(c + d*x)^2)) - (2*b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(3*d^4) - Sin[2*a + 2*b*x]/(6*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(3*d^3*(c + d*x)) + (2*b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)} - -{Sin[x]*Cos[x]/x^1, x, 3, (1/2)*SinIntegral[2*x]} -{Sin[x]*Cos[x]/x^2, x, 4, CosIntegral[2*x] - Sin[2*x]/(2*x)} -{Sin[x]*Cos[x]/x^3, x, 5, -(Cos[2*x]/(2*x)) - Sin[2*x]/(4*x^2) - SinIntegral[2*x]} - - -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^m, x, 8, -((I*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b))) + (I*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b)) + (I*3^(-1 - m)*E^(3*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((3*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(8*b)) - (I*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, (3*I*b*(c + d*x))/d])/(E^(3*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(8*b))} - -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^4, x, 9, -((160*d^3*(c + d*x)*Cos[a + b*x])/(27*b^4)) + (8*d*(c + d*x)^3*Cos[a + b*x])/(9*b^2) + (160*d^4*Sin[a + b*x])/(27*b^5) - (8*d^2*(c + d*x)^2*Sin[a + b*x])/(3*b^3) - (8*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(27*b^4) + (4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^2) + (8*d^4*Sin[a + b*x]^3)/(81*b^5) - (4*d^2*(c + d*x)^2*Sin[a + b*x]^3)/(9*b^3) + ((c + d*x)^4*Sin[a + b*x]^3)/(3*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^3, x, 7, -((14*d^3*Cos[a + b*x])/(9*b^4)) + (2*d*(c + d*x)^2*Cos[a + b*x])/(3*b^2) + (2*d^3*Cos[a + b*x]^3)/(27*b^4) - (4*d^2*(c + d*x)*Sin[a + b*x])/(3*b^3) + (d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b^2) - (2*d^2*(c + d*x)*Sin[a + b*x]^3)/(9*b^3) + ((c + d*x)^3*Sin[a + b*x]^3)/(3*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^2, x, 4, (4*d*(c + d*x)*Cos[a + b*x])/(9*b^2) - (4*d^2*Sin[a + b*x])/(9*b^3) + (2*d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^2) - (2*d^2*Sin[a + b*x]^3)/(27*b^3) + ((c + d*x)^2*Sin[a + b*x]^3)/(3*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^1, x, 3, (d*Cos[a + b*x])/(3*b^2) - (d*Cos[a + b*x]^3)/(9*b^2) + ((c + d*x)*Sin[a + b*x]^3)/(3*b)} -{Cos[a + b*x]*Sin[a + b*x]^2/(c + d*x)^1, x, 8, (Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(4*d) - (Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(4*d) - (Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) + (Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{Cos[a + b*x]*Sin[a + b*x]^2/(c + d*x)^2, x, 10, -(Cos[a + b*x]/(4*d*(c + d*x))) + Cos[3*a + 3*b*x]/(4*d*(c + d*x)) + (3*b*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d^2) - (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d^2) - (b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{Cos[a + b*x]*Sin[a + b*x]^2/(c + d*x)^3, x, 12, -(Cos[a + b*x]/(8*d*(c + d*x)^2)) + Cos[3*a + 3*b*x]/(8*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^3) + (b*Sin[a + b*x])/(8*d^2*(c + d*x)) - (3*b*Sin[3*a + 3*b*x])/(8*d^2*(c + d*x)) + (b^2*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^3) - (9*b^2*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} -{Cos[a + b*x]*Sin[a + b*x]^2/(c + d*x)^4, x, 14, -(Cos[a + b*x]/(12*d*(c + d*x)^3)) + (b^2*Cos[a + b*x])/(24*d^3*(c + d*x)) + Cos[3*a + 3*b*x]/(12*d*(c + d*x)^3) - (3*b^2*Cos[3*a + 3*b*x])/(8*d^3*(c + d*x)) - (9*b^3*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(8*d^4) + (b^3*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(24*d^4) + (b*Sin[a + b*x])/(24*d^2*(c + d*x)^2) - (b*Sin[3*a + 3*b*x])/(8*d^2*(c + d*x)^2) + (b^3*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(24*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^4)} - - -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^m, x, 8, -((2^(-4 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b)) - (2^(-4 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b) + (E^(4*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((4*I*b*(c + d*x))/d)])/(2^(2*(3 + m))*(-((I*b*(c + d*x))/d))^m*b) + ((c + d*x)^m*Gamma[1 + m, (4*I*b*(c + d*x))/d])/(2^(2*(3 + m))*E^(4*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b)} - -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^4, x, 9, (45*c*d^3*x)/(64*b^3) + (45*d^4*x^2)/(128*b^3) - (3*(c + d*x)^4)/(32*b) - (45*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(64*b^4) + (3*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^2) + (45*d^4*Sin[a + b*x]^2)/(128*b^5) - (9*d^2*(c + d*x)^2*Sin[a + b*x]^2)/(16*b^3) - (3*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^3)/(32*b^4) + (d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^3)/(4*b^2) + (3*d^4*Sin[a + b*x]^4)/(128*b^5) - (3*d^2*(c + d*x)^2*Sin[a + b*x]^4)/(16*b^3) + ((c + d*x)^4*Sin[a + b*x]^4)/(4*b)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^3, x, 9, (45*d^3*x)/(256*b^3) - (3*(c + d*x)^3)/(32*b) - (45*d^3*Cos[a + b*x]*Sin[a + b*x])/(256*b^4) + (9*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(32*b^2) - (9*d^2*(c + d*x)*Sin[a + b*x]^2)/(32*b^3) - (3*d^3*Cos[a + b*x]*Sin[a + b*x]^3)/(128*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^3)/(16*b^2) - (3*d^2*(c + d*x)*Sin[a + b*x]^4)/(32*b^3) + ((c + d*x)^3*Sin[a + b*x]^4)/(4*b)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^2, x, 4, -((3*c*d*x)/(16*b)) - (3*d^2*x^2)/(32*b) + (3*d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(16*b^2) - (3*d^2*Sin[a + b*x]^2)/(32*b^3) + (d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^3)/(8*b^2) - (d^2*Sin[a + b*x]^4)/(32*b^3) + ((c + d*x)^2*Sin[a + b*x]^4)/(4*b)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^1, x, 4, -((3*d*x)/(32*b)) + (3*d*Cos[a + b*x]*Sin[a + b*x])/(32*b^2) + (d*Cos[a + b*x]*Sin[a + b*x]^3)/(16*b^2) + ((c + d*x)*Sin[a + b*x]^4)/(4*b)} -{Cos[a + b*x]*Sin[a + b*x]^3/(c + d*x)^1, x, 8, -((CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(8*d)) + (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(4*d) + (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(4*d) - (Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(8*d)} -{Cos[a + b*x]*Sin[a + b*x]^3/(c + d*x)^2, x, 10, (b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(2*d^2) - (b*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(2*d^2) - Sin[2*a + 2*b*x]/(4*d*(c + d*x)) + Sin[4*a + 4*b*x]/(8*d*(c + d*x)) - (b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d^2) + (b*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(2*d^2)} -{Cos[a + b*x]*Sin[a + b*x]^3/(c + d*x)^3, x, 12, -((b*Cos[2*a + 2*b*x])/(4*d^2*(c + d*x))) + (b*Cos[4*a + 4*b*x])/(4*d^2*(c + d*x)) + (b^2*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/d^3 - (b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d^3) - Sin[2*a + 2*b*x]/(8*d*(c + d*x)^2) + Sin[4*a + 4*b*x]/(16*d*(c + d*x)^2) - (b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d^3) + (b^2*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/d^3} -{Cos[a + b*x]*Sin[a + b*x]^3/(c + d*x)^4, x, 14, -((b*Cos[2*a + 2*b*x])/(12*d^2*(c + d*x)^2)) + (b*Cos[4*a + 4*b*x])/(12*d^2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(3*d^4) + (4*b^3*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(3*d^4) - Sin[2*a + 2*b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(6*d^3*(c + d*x)) + Sin[4*a + 4*b*x]/(24*d*(c + d*x)^3) - (b^2*Sin[4*a + 4*b*x])/(3*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4) - (4*b^3*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(3*d^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cos[a + b*x]*Csc[a + b*x]*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Cot[a + b*x], x]} - -{Cos[a + b*x]*Csc[a + b*x]*(c + d*x)^4, x, 7, -((I*(c + d*x)^5)/(5*d)) + ((c + d*x)^4*Log[1 - E^(2*I*(a + b*x))])/b - (2*I*d*(c + d*x)^3*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + (3*d^2*(c + d*x)^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3 + (3*I*d^3*(c + d*x)*PolyLog[4, E^(2*I*(a + b*x))])/b^4 - (3*d^4*PolyLog[5, E^(2*I*(a + b*x))])/(2*b^5)} -{Cos[a + b*x]*Csc[a + b*x]*(c + d*x)^3, x, 6, -((I*(c + d*x)^4)/(4*d)) + ((c + d*x)^3*Log[1 - E^(2*I*(a + b*x))])/b - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{Cos[a + b*x]*Csc[a + b*x]*(c + d*x)^2, x, 5, -((I*(c + d*x)^3)/(3*d)) + ((c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b - (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3)} -{Cos[a + b*x]*Csc[a + b*x]*(c + d*x)^1, x, 4, -((I*(c + d*x)^2)/(2*d)) + ((c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Cos[a + b*x]*Csc[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Cot[a + b*x]/(c + d*x), x]} -{Cos[a + b*x]*Csc[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Cot[a + b*x]/(c + d*x)^2, x]} - - -{Cos[a + b*x]*Csc[a + b*x]^2*(c + d*x)^m, x, 0, CannotIntegrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x], x]} - -{Cos[a + b*x]*Csc[a + b*x]^2*(c + d*x)^4, x, 10, -((8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2) - ((c + d*x)^4*Csc[a + b*x])/b + (12*I*d^2*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (12*I*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - (24*I*d^4*PolyLog[4, -E^(I*(a + b*x))])/b^5 + (24*I*d^4*PolyLog[4, E^(I*(a + b*x))])/b^5} -{Cos[a + b*x]*Csc[a + b*x]^2*(c + d*x)^3, x, 8, -((6*d*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^2) - ((c + d*x)^3*Csc[a + b*x])/b + (6*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (6*I*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (6*d^3*PolyLog[3, E^(I*(a + b*x))])/b^4} -{Cos[a + b*x]*Csc[a + b*x]^2*(c + d*x)^2, x, 6, -((4*d*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^2) - ((c + d*x)^2*Csc[a + b*x])/b + (2*I*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (2*I*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3} -{Cos[a + b*x]*Csc[a + b*x]^2*(c + d*x)^1, x, 2, -((d*ArcTanh[Cos[a + b*x]])/b^2) - ((c + d*x)*Csc[a + b*x])/b} -{Cos[a + b*x]*Csc[a + b*x]^2/(c + d*x)^1, x, 0, CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x), x]} -{Cos[a + b*x]*Csc[a + b*x]^2/(c + d*x)^2, x, 0, CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x)^2, x]} - - -{Cos[a + b*x]*Csc[a + b*x]^3*(c + d*x)^m, x, 0, CannotIntegrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x]^2, x]} - -{Cos[a + b*x]*Csc[a + b*x]^3*(c + d*x)^4, x, 7, -((2*I*d*(c + d*x)^3)/b^2) - (2*d*(c + d*x)^3*Cot[a + b*x])/b^2 - ((c + d*x)^4*Csc[a + b*x]^2)/(2*b) + (6*d^2*(c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b^3 - (6*I*d^3*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^4 + (3*d^4*PolyLog[3, E^(2*I*(a + b*x))])/b^5} -{Cos[a + b*x]*Csc[a + b*x]^3*(c + d*x)^3, x, 6, -((3*I*d*(c + d*x)^2)/(2*b^2)) - (3*d*(c + d*x)^2*Cot[a + b*x])/(2*b^2) - ((c + d*x)^3*Csc[a + b*x]^2)/(2*b) + (3*d^2*(c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^4)} -{Cos[a + b*x]*Csc[a + b*x]^3*(c + d*x)^2, x, 3, -((d*(c + d*x)*Cot[a + b*x])/b^2) - ((c + d*x)^2*Csc[a + b*x]^2)/(2*b) + (d^2*Log[Sin[a + b*x]])/b^3} -{Cos[a + b*x]*Csc[a + b*x]^3*(c + d*x)^1, x, 3, -((d*Cot[a + b*x])/(2*b^2)) - ((c + d*x)*Csc[a + b*x]^2)/(2*b)} -{Cos[a + b*x]*Csc[a + b*x]^3/(c + d*x)^1, x, 0, CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x), x]} -{Cos[a + b*x]*Csc[a + b*x]^3/(c + d*x)^2, x, 0, CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x]^2)/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Cos[a+b x] Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(5/2), x, 10, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(4*b) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(16*b^2)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(3/2), x, 9, -(((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(4*b)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(32*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(32*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(16*b^2)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(1/2), x, 8, -((Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(4*b)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(8*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(8*b^(3/2))} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(1/2), x, 8, -((Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(4*b)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(8*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(8*b^(3/2))} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(3/2), x, 9, -(((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(4*b)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(32*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(32*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(16*b^2)} -{Cos[a + b*x]*Sin[a + b*x]*(c + d*x)^(5/2), x, 10, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(4*b) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(16*b^2)} - - -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(5/2), x, 18, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(144*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(3/2), x, 16, (3*d*Sqrt[c + d*x]*Cos[a + b*x])/(8*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(24*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/(4*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(1/2), x, 14, -((Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2)) + (Sqrt[c + d*x]*Sin[a + b*x])/(4*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(1/2), x, 14, -((Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2)) + (Sqrt[c + d*x]*Sin[a + b*x])/(4*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(3/2), x, 16, (3*d*Sqrt[c + d*x]*Cos[a + b*x])/(8*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(24*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/(4*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(12*b)} -{Cos[a + b*x]*Sin[a + b*x]^2*(c + d*x)^(5/2), x, 18, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(144*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)} - - -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(5/2), x, 18, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) - (15*d^2*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(2048*b^3) + ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) - (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*b^2)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(3/2), x, 16, -(((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(8*b)) + ((c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(32*b) + (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(64*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(64*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(32*b^2) - (3*d*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(256*b^2)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(1/2), x, 14, -((Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(8*b)) + (Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(32*b) - (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(16*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(16*b^(3/2))} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(1/2), x, 14, -((Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(8*b)) + (Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(32*b) - (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(16*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(16*b^(3/2))} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(3/2), x, 16, -(((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(8*b)) + ((c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(32*b) + (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(64*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(64*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(32*b^2) - (3*d*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(256*b^2)} -{Cos[a + b*x]*Sin[a + b*x]^3*(c + d*x)^(5/2), x, 18, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) - (15*d^2*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(2048*b^3) + ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) - (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*b^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x], x, 8, -(E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(8*b*(((-I)*b*(c + d*x))/d)^m) - ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(8*b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) - (3^(-1 - m)*E^((3*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/(8*b*(((-I)*b*(c + d*x))/d)^m) - (3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(8*b*E^((3*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x], x, 9, (-160*d^4*Cos[a + b*x])/(27*b^5) + (8*d^2*(c + d*x)^2*Cos[a + b*x])/(3*b^3) - (8*d^4*Cos[a + b*x]^3)/(81*b^5) + (4*d^2*(c + d*x)^2*Cos[a + b*x]^3)/(9*b^3) - ((c + d*x)^4*Cos[a + b*x]^3)/(3*b) - (160*d^3*(c + d*x)*Sin[a + b*x])/(27*b^4) + (8*d*(c + d*x)^3*Sin[a + b*x])/(9*b^2) - (8*d^3*(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(27*b^4) + (4*d*(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x])/(9*b^2)} -{(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x], x, 7, (4*d^2*(c + d*x)*Cos[a + b*x])/(3*b^3) + (2*d^2*(c + d*x)*Cos[a + b*x]^3)/(9*b^3) - ((c + d*x)^3*Cos[a + b*x]^3)/(3*b) - (14*d^3*Sin[a + b*x])/(9*b^4) + (2*d*(c + d*x)^2*Sin[a + b*x])/(3*b^2) + (d*(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(3*b^2) + (2*d^3*Sin[a + b*x]^3)/(27*b^4)} -{(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x], x, 4, (4*d^2*Cos[a + b*x])/(9*b^3) + (2*d^2*Cos[a + b*x]^3)/(27*b^3) - ((c + d*x)^2*Cos[a + b*x]^3)/(3*b) + (4*d*(c + d*x)*Sin[a + b*x])/(9*b^2) + (2*d*(c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(9*b^2)} -{(c + d*x)^1*Cos[a + b*x]^2*Sin[a + b*x], x, 3, -((c + d*x)*Cos[a + b*x]^3)/(3*b) + (d*Sin[a + b*x])/(3*b^2) - (d*Sin[a + b*x]^3)/(9*b^2)} -{(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^1, x, 8, (CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^2, x, 10, (b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(4*d^2) - Sin[a + b*x]/(4*d*(c + d*x)) - Sin[3*a + 3*b*x]/(4*d*(c + d*x)) - (b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d^2) - (3*b*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^3, x, 12, -(b*Cos[a + b*x])/(8*d^2*(c + d*x)) - (3*b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)) - (9*b^2*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(8*d^3) - (b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d^3) - Sin[a + b*x]/(8*d*(c + d*x)^2) - Sin[3*a + 3*b*x]/(8*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^3) - (9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} -{(Cos[a + b*x]^2*Sin[a + b*x])/(c + d*x)^4, x, 14, -(b*Cos[a + b*x])/(24*d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)^2) - (b^3*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(24*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^4) - Sin[a + b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[a + b*x])/(24*d^3*(c + d*x)) - Sin[3*a + 3*b*x]/(12*d*(c + d*x)^3) + (3*b^2*Sin[3*a + 3*b*x])/(8*d^3*(c + d*x)) + (b^3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(24*d^4) + (9*b^3*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^4)} - - -{(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 5, (c + d*x)^(1 + m)/(8*d*(1 + m)) + (I*E^((4*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*(((-I)*b*(c + d*x))/d)^m) - (I*(c + d*x)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*E^((4*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 7, (c + d*x)^5/(40*d) + (3*d^3*(c + d*x)*Cos[4*a + 4*b*x])/(256*b^4) - (d*(c + d*x)^3*Cos[4*a + 4*b*x])/(32*b^2) - (3*d^4*Sin[4*a + 4*b*x])/(1024*b^5) + (3*d^2*(c + d*x)^2*Sin[4*a + 4*b*x])/(128*b^3) - ((c + d*x)^4*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 6, (c + d*x)^4/(32*d) + (3*d^3*Cos[4*a + 4*b*x])/(1024*b^4) - (3*d*(c + d*x)^2*Cos[4*a + 4*b*x])/(128*b^2) + (3*d^2*(c + d*x)*Sin[4*a + 4*b*x])/(256*b^3) - ((c + d*x)^3*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 5, (c + d*x)^3/(24*d) - (d*(c + d*x)*Cos[4*a + 4*b*x])/(64*b^2) + (d^2*Sin[4*a + 4*b*x])/(256*b^3) - ((c + d*x)^2*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^1*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 4, (c + d*x)^2/(16*d) - (d*Cos[4*a + 4*b*x])/(128*b^2) - ((c + d*x)*Sin[4*a + 4*b*x])/(32*b)} -{(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^1, x, 5, -(Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(8*d) + Log[c + d*x]/(8*d) + (Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(8*d)} -{(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^2, x, 6, -1/(8*d*(c + d*x)) + Cos[4*a + 4*b*x]/(8*d*(c + d*x)) + (b*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(2*d^2) + (b*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(2*d^2)} -{(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^3, x, 7, -1/(16*d*(c + d*x)^2) + Cos[4*a + 4*b*x]/(16*d*(c + d*x)^2) + (b^2*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/d^3 - (b*Sin[4*a + 4*b*x])/(4*d^2*(c + d*x)) - (b^2*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/d^3} -{(Cos[a + b*x]^2*Sin[a + b*x]^2)/(c + d*x)^4, x, 8, -1/(24*d*(c + d*x)^3) + Cos[4*a + 4*b*x]/(24*d*(c + d*x)^3) - (b^2*Cos[4*a + 4*b*x])/(3*d^3*(c + d*x)) - (4*b^3*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(3*d^4) - (b*Sin[4*a + 4*b*x])/(12*d^2*(c + d*x)^2) - (4*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(3*d^4)} - - -{(c + d*x)^m*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 11, -(E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(16*b*(((-I)*b*(c + d*x))/d)^m) - ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(16*b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) - (3^(-1 - m)*E^((3*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/(32*b*(((-I)*b*(c + d*x))/d)^m) - (3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(32*b*E^((3*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + (5^(-1 - m)*E^((5*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d])/(32*b*(((-I)*b*(c + d*x))/d)^m) + (5^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d])/(32*b*E^((5*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 17, (-3*d^4*Cos[a + b*x])/b^5 + (3*d^2*(c + d*x)^2*Cos[a + b*x])/(2*b^3) - ((c + d*x)^4*Cos[a + b*x])/(8*b) - (d^4*Cos[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Cos[3*a + 3*b*x])/(36*b^3) - ((c + d*x)^4*Cos[3*a + 3*b*x])/(48*b) + (3*d^4*Cos[5*a + 5*b*x])/(6250*b^5) - (3*d^2*(c + d*x)^2*Cos[5*a + 5*b*x])/(500*b^3) + ((c + d*x)^4*Cos[5*a + 5*b*x])/(80*b) - (3*d^3*(c + d*x)*Sin[a + b*x])/b^4 + (d*(c + d*x)^3*Sin[a + b*x])/(2*b^2) - (d^3*(c + d*x)*Sin[3*a + 3*b*x])/(54*b^4) + (d*(c + d*x)^3*Sin[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Sin[5*a + 5*b*x])/(1250*b^4) - (d*(c + d*x)^3*Sin[5*a + 5*b*x])/(100*b^2)} -{(c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 14, (3*d^2*(c + d*x)*Cos[a + b*x])/(4*b^3) - ((c + d*x)^3*Cos[a + b*x])/(8*b) + (d^2*(c + d*x)*Cos[3*a + 3*b*x])/(72*b^3) - ((c + d*x)^3*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*(c + d*x)*Cos[5*a + 5*b*x])/(1000*b^3) + ((c + d*x)^3*Cos[5*a + 5*b*x])/(80*b) - (3*d^3*Sin[a + b*x])/(4*b^4) + (3*d*(c + d*x)^2*Sin[a + b*x])/(8*b^2) - (d^3*Sin[3*a + 3*b*x])/(216*b^4) + (d*(c + d*x)^2*Sin[3*a + 3*b*x])/(48*b^2) + (3*d^3*Sin[5*a + 5*b*x])/(5000*b^4) - (3*d*(c + d*x)^2*Sin[5*a + 5*b*x])/(400*b^2)} -{(c + d*x)^2*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 11, (d^2*Cos[a + b*x])/(4*b^3) - ((c + d*x)^2*Cos[a + b*x])/(8*b) + (d^2*Cos[3*a + 3*b*x])/(216*b^3) - ((c + d*x)^2*Cos[3*a + 3*b*x])/(48*b) - (d^2*Cos[5*a + 5*b*x])/(1000*b^3) + ((c + d*x)^2*Cos[5*a + 5*b*x])/(80*b) + (d*(c + d*x)*Sin[a + b*x])/(4*b^2) + (d*(c + d*x)*Sin[3*a + 3*b*x])/(72*b^2) - (d*(c + d*x)*Sin[5*a + 5*b*x])/(200*b^2)} -{(c + d*x)^1*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 8, -((c + d*x)*Cos[a + b*x])/(8*b) - ((c + d*x)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)*Cos[5*a + 5*b*x])/(80*b) + (d*Sin[a + b*x])/(8*b^2) + (d*Sin[3*a + 3*b*x])/(144*b^2) - (d*Sin[5*a + 5*b*x])/(400*b^2)} -{(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^1, x, 11, -(CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(16*d) + (CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(16*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d) - (Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*d)} -{(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^2, x, 14, (b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(8*d^2) + (3*b*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(16*d^2) - (5*b*Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*c)/d + 5*b*x])/(16*d^2) - Sin[a + b*x]/(8*d*(c + d*x)) - Sin[3*a + 3*b*x]/(16*d*(c + d*x)) + Sin[5*a + 5*b*x]/(16*d*(c + d*x)) - (b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^2) - (3*b*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d^2) + (5*b*Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*d^2)} -{(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^3, x, 17, -(b*Cos[a + b*x])/(16*d^2*(c + d*x)) - (3*b*Cos[3*a + 3*b*x])/(32*d^2*(c + d*x)) + (5*b*Cos[5*a + 5*b*x])/(32*d^2*(c + d*x)) + (25*b^2*CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(32*d^3) - (9*b^2*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(32*d^3) - (b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(16*d^3) - Sin[a + b*x]/(16*d*(c + d*x)^2) - Sin[3*a + 3*b*x]/(32*d*(c + d*x)^2) + Sin[5*a + 5*b*x]/(32*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(16*d^3) - (9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(32*d^3) + (25*b^2*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(32*d^3)} -{(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^4, x, 20, -(b*Cos[a + b*x])/(48*d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(32*d^2*(c + d*x)^2) + (5*b*Cos[5*a + 5*b*x])/(96*d^2*(c + d*x)^2) - (b^3*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(48*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(32*d^4) + (125*b^3*Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*c)/d + 5*b*x])/(96*d^4) - Sin[a + b*x]/(24*d*(c + d*x)^3) + (b^2*Sin[a + b*x])/(48*d^3*(c + d*x)) - Sin[3*a + 3*b*x]/(48*d*(c + d*x)^3) + (3*b^2*Sin[3*a + 3*b*x])/(32*d^3*(c + d*x)) + Sin[5*a + 5*b*x]/(48*d*(c + d*x)^3) - (25*b^2*Sin[5*a + 5*b*x])/(96*d^3*(c + d*x)) + (b^3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(48*d^4) + (9*b^3*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(32*d^4) - (125*b^3*Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(96*d^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x], x, 4, (E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b)) + ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b)) + Unintegrable[(c + d*x)^m*Csc[a + b*x], x]} - -{(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x], x, 17, -((2*(c + d*x)^4*ArcTanh[E^(I*(a + b*x))])/b) + (24*d^4*Cos[a + b*x])/b^5 - (12*d^2*(c + d*x)^2*Cos[a + b*x])/b^3 + ((c + d*x)^4*Cos[a + b*x])/b + (4*I*d*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (4*I*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))])/b^2 - (12*d^2*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (12*d^2*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))])/b^3 - (24*I*d^3*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (24*I*d^3*(c + d*x)*PolyLog[4, E^(I*(a + b*x))])/b^4 + (24*d^4*PolyLog[5, -E^(I*(a + b*x))])/b^5 - (24*d^4*PolyLog[5, E^(I*(a + b*x))])/b^5 + (24*d^3*(c + d*x)*Sin[a + b*x])/b^4 - (4*d*(c + d*x)^3*Sin[a + b*x])/b^2} -{(c + d*x)^3*Cos[a + b*x]*Cot[a + b*x], x, 14, -((2*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b) - (6*d^2*(c + d*x)*Cos[a + b*x])/b^3 + ((c + d*x)^3*Cos[a + b*x])/b + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (6*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + (6*d^3*Sin[a + b*x])/b^4 - (3*d*(c + d*x)^2*Sin[a + b*x])/b^2} -{(c + d*x)^2*Cos[a + b*x]*Cot[a + b*x], x, 11, -((2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b) - (2*d^2*Cos[a + b*x])/b^3 + ((c + d*x)^2*Cos[a + b*x])/b + (2*I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (2*d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, E^(I*(a + b*x))])/b^3 - (2*d*(c + d*x)*Sin[a + b*x])/b^2} -{(c + d*x)^1*Cos[a + b*x]*Cot[a + b*x], x, 8, -((2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b) + ((c + d*x)*Cos[a + b*x])/b + (I*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, E^(I*(a + b*x))])/b^2 - (d*Sin[a + b*x])/b^2} -{(Cos[a + b*x]*Cot[a + b*x])/(c + d*x)^1, x, 4, Unintegrable[Csc[a + b*x]/(c + d*x), x] - (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{(Cos[a + b*x]*Cot[a + b*x])/(c + d*x)^2, x, 5, -((b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2) + Unintegrable[Csc[a + b*x]/(c + d*x)^2, x] + Sin[a + b*x]/(d*(c + d*x)) + (b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} - - -{(c + d*x)^m*Cot[a + b*x]^2, x, 0, Unintegrable[(c + d*x)^m*Cot[a + b*x]^2, x]} - -{(c + d*x)^4*Cot[a + b*x]^2, x, 8, -((I*(c + d*x)^4)/b) - (c + d*x)^5/(5*d) - ((c + d*x)^4*Cot[a + b*x])/b + (4*d*(c + d*x)^3*Log[1 - E^(2*I*(a + b*x))])/b^2 - (6*I*d^2*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/b^3 + (6*d^3*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/b^4 + (3*I*d^4*PolyLog[4, E^(2*I*(a + b*x))])/b^5} -{(c + d*x)^3*Cot[a + b*x]^2, x, 7, -((I*(c + d*x)^3)/b) - (c + d*x)^4/(4*d) - ((c + d*x)^3*Cot[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b^2 - (3*I*d^2*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^4)} -{(c + d*x)^2*Cot[a + b*x]^2, x, 6, -((I*(c + d*x)^2)/b) - (c + d*x)^3/(3*d) - ((c + d*x)^2*Cot[a + b*x])/b + (2*d*(c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b^2 - (I*d^2*PolyLog[2, E^(2*I*(a + b*x))])/b^3} -{(c + d*x)^1*Cot[a + b*x]^2, x, 3, -(c*x) - (d*x^2)/2 - ((c + d*x)*Cot[a + b*x])/b + (d*Log[Sin[a + b*x]])/b^2} -{Cot[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Cot[a + b*x]^2/(c + d*x), x]} -{Cot[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Cot[a + b*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^m*Cot[a + b*x]^2*Csc[a + b*x], x, 1, -Unintegrable[(c + d*x)^m*Csc[a + b*x], x] + Unintegrable[(c + d*x)^m*Csc[a + b*x]^3, x]} - -{(c + d*x)^4*Cot[a + b*x]^2*Csc[a + b*x], x, 31, -((12*d^2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^3) + ((c + d*x)^4*ArcTanh[E^(I*(a + b*x))])/b - (2*d*(c + d*x)^3*Csc[a + b*x])/b^2 - ((c + d*x)^4*Cot[a + b*x]*Csc[a + b*x])/(2*b) + (12*I*d^3*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^4 - (2*I*d*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (12*I*d^3*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^4 + (2*I*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))])/b^2 - (12*d^4*PolyLog[3, -E^(I*(a + b*x))])/b^5 + (6*d^2*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (12*d^4*PolyLog[3, E^(I*(a + b*x))])/b^5 - (6*d^2*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))])/b^3 + (12*I*d^3*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))])/b^4 - (12*I*d^3*(c + d*x)*PolyLog[4, E^(I*(a + b*x))])/b^4 - (12*d^4*PolyLog[5, -E^(I*(a + b*x))])/b^5 + (12*d^4*PolyLog[5, E^(I*(a + b*x))])/b^5} -{(c + d*x)^3*Cot[a + b*x]^2*Csc[a + b*x], x, 25, -((6*d^2*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^3) + ((c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (3*d*(c + d*x)^2*Csc[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]*Csc[a + b*x])/(2*b) + (3*I*d^3*PolyLog[2, -E^(I*(a + b*x))])/b^4 - (3*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, E^(I*(a + b*x))])/b^4 + (3*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 - (3*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 + (3*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 - (3*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4} -{(c + d*x)^2*Cot[a + b*x]^2*Csc[a + b*x], x, 17, ((c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b - (d^2*ArcTanh[Cos[a + b*x]])/b^3 - (d*(c + d*x)*Csc[a + b*x])/b^2 - ((c + d*x)^2*Cot[a + b*x]*Csc[a + b*x])/(2*b) - (I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 + (I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 + (d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 - (d^2*PolyLog[3, E^(I*(a + b*x))])/b^3} -{(c + d*x)^1*Cot[a + b*x]^2*Csc[a + b*x], x, 12, ((c + d*x)*ArcTanh[E^(I*(a + b*x))])/b - (d*Csc[a + b*x])/(2*b^2) - ((c + d*x)*Cot[a + b*x]*Csc[a + b*x])/(2*b) - (I*d*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) + (I*d*PolyLog[2, E^(I*(a + b*x))])/(2*b^2)} -{(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x)^1, x, 1, -Unintegrable[Csc[a + b*x]/(c + d*x), x] + Unintegrable[Csc[a + b*x]^3/(c + d*x), x]} -{(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x)^2, x, 1, -Unintegrable[Csc[a + b*x]/(c + d*x)^2, x] + Unintegrable[Csc[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Cos[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 18, (15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(144*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(12*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(8*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(72*b^2)} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 16, -((c + d*x)^(3/2)*Cos[a + b*x])/(4*b) - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(12*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(8*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(24*b^2)} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 14, -(Sqrt[c + d*x]*Cos[a + b*x])/(4*b) - (Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(12*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2))} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 14, -(Sqrt[c + d*x]*Cos[a + b*x])/(4*b) - (Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(12*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(12*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(4*b^(3/2))} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 16, -((c + d*x)^(3/2)*Cos[a + b*x])/(4*b) - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(12*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(24*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(24*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(8*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(24*b^2)} -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x], x, 18, (15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(144*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(12*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(8*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(72*b^2)} - - -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 10, (c + d*x)^(7/2)/(28*d) - (5*d*(c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(256*b^2) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^2*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 9, (c + d*x)^(5/2)/(20*d) - (3*d*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(256*b^2) + (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - ((c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 8, (c + d*x)^(3/2)/(12*d) + (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 8, (c + d*x)^(3/2)/(12*d) + (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 9, (c + d*x)^(5/2)/(20*d) - (3*d*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(256*b^2) + (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - ((c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(32*b)} -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^2, x, 10, (c + d*x)^(7/2)/(28*d) - (5*d*(c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(256*b^2) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^2*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Sin[4*a + 4*b*x])/(32*b)} - - -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 26, (15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(32*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(1600*b^3) + ((c + d*x)^(5/2)*Cos[5*a + 5*b*x])/(80*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(16*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(160*b^2)} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 23, -((c + d*x)^(3/2)*Cos[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(80*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(800*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(96*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(800*b^2)} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 20, -(Sqrt[c + d*x]*Cos[a + b*x])/(8*b) - (Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(48*b) + (Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(80*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2))} -{(c + d*x)^(1/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 20, -(Sqrt[c + d*x]*Cos[a + b*x])/(8*b) - (Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(48*b) + (Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(80*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2))} -{(c + d*x)^(3/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 23, -((c + d*x)^(3/2)*Cos[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(48*b) + ((c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(80*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(800*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(96*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(800*b^2)} -{(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3, x, 26, (15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(32*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(1600*b^3) + ((c + d*x)^(5/2)*Cos[5*a + 5*b*x])/(80*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(16*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(160*b^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x], x, 8, -((2^(-4 - m)*E^((2*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m)) - (2^(-4 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) - (E^((4*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*(((-I)*b*(c + d*x))/d)^m) - ((c + d*x)^m*Gamma[1 + m, ((4*I)*b*(c + d*x))/d])/(2^(2*(3 + m))*b*E^((4*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x], x, 9, (-45*c*d^3*x)/(64*b^3) - (45*d^4*x^2)/(128*b^3) + (3*(c + d*x)^4)/(32*b) - (45*d^4*Cos[a + b*x]^2)/(128*b^5) + (9*d^2*(c + d*x)^2*Cos[a + b*x]^2)/(16*b^3) - (3*d^4*Cos[a + b*x]^4)/(128*b^5) + (3*d^2*(c + d*x)^2*Cos[a + b*x]^4)/(16*b^3) - ((c + d*x)^4*Cos[a + b*x]^4)/(4*b) - (45*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(64*b^4) + (3*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^2) - (3*d^3*(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x])/(32*b^4) + (d*(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b^2)} -{(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x], x, 9, (-45*d^3*x)/(256*b^3) + (3*(c + d*x)^3)/(32*b) + (9*d^2*(c + d*x)*Cos[a + b*x]^2)/(32*b^3) + (3*d^2*(c + d*x)*Cos[a + b*x]^4)/(32*b^3) - ((c + d*x)^3*Cos[a + b*x]^4)/(4*b) - (45*d^3*Cos[a + b*x]*Sin[a + b*x])/(256*b^4) + (9*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(32*b^2) - (3*d^3*Cos[a + b*x]^3*Sin[a + b*x])/(128*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x])/(16*b^2)} -{(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x], x, 4, (3*c*d*x)/(16*b) + (3*d^2*x^2)/(32*b) + (3*d^2*Cos[a + b*x]^2)/(32*b^3) + (d^2*Cos[a + b*x]^4)/(32*b^3) - ((c + d*x)^2*Cos[a + b*x]^4)/(4*b) + (3*d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(16*b^2) + (d*(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x])/(8*b^2)} -{(c + d*x)^1*Cos[a + b*x]^3*Sin[a + b*x], x, 4, (3*d*x)/(32*b) - ((c + d*x)*Cos[a + b*x]^4)/(4*b) + (3*d*Cos[a + b*x]*Sin[a + b*x])/(32*b^2) + (d*Cos[a + b*x]^3*Sin[a + b*x])/(16*b^2)} -{(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^1, x, 8, (CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/(8*d) + (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(4*d) + (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(4*d) + (Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(8*d)} -{(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^2, x, 10, (b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(2*d^2) + (b*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(2*d^2) - Sin[2*a + 2*b*x]/(4*d*(c + d*x)) - Sin[4*a + 4*b*x]/(8*d*(c + d*x)) - (b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d^2) - (b*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(2*d^2)} -{(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^3, x, 12, -(b*Cos[2*a + 2*b*x])/(4*d^2*(c + d*x)) - (b*Cos[4*a + 4*b*x])/(4*d^2*(c + d*x)) - (b^2*CosIntegral[(4*b*c)/d + 4*b*x]*Sin[4*a - (4*b*c)/d])/d^3 - (b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d^3) - Sin[2*a + 2*b*x]/(8*d*(c + d*x)^2) - Sin[4*a + 4*b*x]/(16*d*(c + d*x)^2) - (b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d^3) - (b^2*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/d^3} -{(Cos[a + b*x]^3*Sin[a + b*x])/(c + d*x)^4, x, 14, -(b*Cos[2*a + 2*b*x])/(12*d^2*(c + d*x)^2) - (b*Cos[4*a + 4*b*x])/(12*d^2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(3*d^4) - (4*b^3*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(3*d^4) - Sin[2*a + 2*b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(6*d^3*(c + d*x)) - Sin[4*a + 4*b*x]/(24*d*(c + d*x)^3) + (b^2*Sin[4*a + 4*b*x])/(3*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4) + (4*b^3*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(3*d^4)} - - -{(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 11, ((-I/16)*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) + ((I/16)*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + ((I/32)*3^(-1 - m)*E^((3*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) - ((I/32)*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(b*E^((3*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + ((I/32)*5^(-1 - m)*E^((5*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) - ((I/32)*5^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d])/(b*E^((5*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 17, (-3*d^3*(c + d*x)*Cos[a + b*x])/b^4 + (d*(c + d*x)^3*Cos[a + b*x])/(2*b^2) + (d^3*(c + d*x)*Cos[3*a + 3*b*x])/(54*b^4) - (d*(c + d*x)^3*Cos[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Cos[5*a + 5*b*x])/(1250*b^4) - (d*(c + d*x)^3*Cos[5*a + 5*b*x])/(100*b^2) + (3*d^4*Sin[a + b*x])/b^5 - (3*d^2*(c + d*x)^2*Sin[a + b*x])/(2*b^3) + ((c + d*x)^4*Sin[a + b*x])/(8*b) - (d^4*Sin[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Sin[3*a + 3*b*x])/(36*b^3) - ((c + d*x)^4*Sin[3*a + 3*b*x])/(48*b) - (3*d^4*Sin[5*a + 5*b*x])/(6250*b^5) + (3*d^2*(c + d*x)^2*Sin[5*a + 5*b*x])/(500*b^3) - ((c + d*x)^4*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 14, (-3*d^3*Cos[a + b*x])/(4*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x])/(8*b^2) + (d^3*Cos[3*a + 3*b*x])/(216*b^4) - (d*(c + d*x)^2*Cos[3*a + 3*b*x])/(48*b^2) + (3*d^3*Cos[5*a + 5*b*x])/(5000*b^4) - (3*d*(c + d*x)^2*Cos[5*a + 5*b*x])/(400*b^2) - (3*d^2*(c + d*x)*Sin[a + b*x])/(4*b^3) + ((c + d*x)^3*Sin[a + b*x])/(8*b) + (d^2*(c + d*x)*Sin[3*a + 3*b*x])/(72*b^3) - ((c + d*x)^3*Sin[3*a + 3*b*x])/(48*b) + (3*d^2*(c + d*x)*Sin[5*a + 5*b*x])/(1000*b^3) - ((c + d*x)^3*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 11, (d*(c + d*x)*Cos[a + b*x])/(4*b^2) - (d*(c + d*x)*Cos[3*a + 3*b*x])/(72*b^2) - (d*(c + d*x)*Cos[5*a + 5*b*x])/(200*b^2) - (d^2*Sin[a + b*x])/(4*b^3) + ((c + d*x)^2*Sin[a + b*x])/(8*b) + (d^2*Sin[3*a + 3*b*x])/(216*b^3) - ((c + d*x)^2*Sin[3*a + 3*b*x])/(48*b) + (d^2*Sin[5*a + 5*b*x])/(1000*b^3) - ((c + d*x)^2*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^1*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 8, (d*Cos[a + b*x])/(8*b^2) - (d*Cos[3*a + 3*b*x])/(144*b^2) - (d*Cos[5*a + 5*b*x])/(400*b^2) + ((c + d*x)*Sin[a + b*x])/(8*b) - ((c + d*x)*Sin[3*a + 3*b*x])/(48*b) - ((c + d*x)*Sin[5*a + 5*b*x])/(80*b)} -{(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^1, x, 11, (Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(8*d) - (Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(16*d) - (Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*c)/d + 5*b*x])/(16*d) - (Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d) + (Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d) + (Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*d)} -{(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^2, x, 14, -Cos[a + b*x]/(8*d*(c + d*x)) + Cos[3*a + 3*b*x]/(16*d*(c + d*x)) + Cos[5*a + 5*b*x]/(16*d*(c + d*x)) + (5*b*CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(16*d^2) + (3*b*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(16*d^2) - (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d^2) - (b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^2) + (3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d^2) + (5*b*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*d^2)} -{(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^3, x, 17, -Cos[a + b*x]/(16*d*(c + d*x)^2) + Cos[3*a + 3*b*x]/(32*d*(c + d*x)^2) + Cos[5*a + 5*b*x]/(32*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(16*d^3) + (9*b^2*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(32*d^3) + (25*b^2*Cos[5*a - (5*b*c)/d]*CosIntegral[(5*b*c)/d + 5*b*x])/(32*d^3) + (b*Sin[a + b*x])/(16*d^2*(c + d*x)) - (3*b*Sin[3*a + 3*b*x])/(32*d^2*(c + d*x)) - (5*b*Sin[5*a + 5*b*x])/(32*d^2*(c + d*x)) + (b^2*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(16*d^3) - (9*b^2*Sin[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(32*d^3) - (25*b^2*Sin[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(32*d^3)} -{(Cos[a + b*x]^3*Sin[a + b*x]^2)/(c + d*x)^4, x, 20, -Cos[a + b*x]/(24*d*(c + d*x)^3) + (b^2*Cos[a + b*x])/(48*d^3*(c + d*x)) + Cos[3*a + 3*b*x]/(48*d*(c + d*x)^3) - (3*b^2*Cos[3*a + 3*b*x])/(32*d^3*(c + d*x)) + Cos[5*a + 5*b*x]/(48*d*(c + d*x)^3) - (25*b^2*Cos[5*a + 5*b*x])/(96*d^3*(c + d*x)) - (125*b^3*CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/(96*d^4) - (9*b^3*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(32*d^4) + (b^3*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(48*d^4) + (b*Sin[a + b*x])/(48*d^2*(c + d*x)^2) - (b*Sin[3*a + 3*b*x])/(32*d^2*(c + d*x)^2) - (5*b*Sin[5*a + 5*b*x])/(96*d^2*(c + d*x)^2) + (b^3*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(48*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(32*d^4) - (125*b^3*Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(96*d^4)} - - -{(c + d*x)^m*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 8, (-3*2^(-7 - m)*E^((2*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-2*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) - (3*2^(-7 - m)*(c + d*x)^m*Gamma[1 + m, ((2*I)*b*(c + d*x))/d])/(b*E^((2*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + (2^(-7 - m)*3^(-1 - m)*E^((6*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-6*I)*b*(c + d*x))/d])/(b*(((-I)*b*(c + d*x))/d)^m) + (2^(-7 - m)*3^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((6*I)*b*(c + d*x))/d])/(b*E^((6*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m)} - -{(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 12, (-9*d^4*Cos[2*a + 2*b*x])/(128*b^5) + (9*d^2*(c + d*x)^2*Cos[2*a + 2*b*x])/(64*b^3) - (3*(c + d*x)^4*Cos[2*a + 2*b*x])/(64*b) + (d^4*Cos[6*a + 6*b*x])/(10368*b^5) - (d^2*(c + d*x)^2*Cos[6*a + 6*b*x])/(576*b^3) + ((c + d*x)^4*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*(c + d*x)*Sin[2*a + 2*b*x])/(64*b^4) + (3*d*(c + d*x)^3*Sin[2*a + 2*b*x])/(32*b^2) + (d^3*(c + d*x)*Sin[6*a + 6*b*x])/(1728*b^4) - (d*(c + d*x)^3*Sin[6*a + 6*b*x])/(288*b^2)} -{(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 10, (9*d^2*(c + d*x)*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^3*Cos[2*a + 2*b*x])/(64*b) - (d^2*(c + d*x)*Cos[6*a + 6*b*x])/(1152*b^3) + ((c + d*x)^3*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*Sin[2*a + 2*b*x])/(256*b^4) + (9*d*(c + d*x)^2*Sin[2*a + 2*b*x])/(128*b^2) + (d^3*Sin[6*a + 6*b*x])/(6912*b^4) - (d*(c + d*x)^2*Sin[6*a + 6*b*x])/(384*b^2)} -{(c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 8, (3*d^2*Cos[2*a + 2*b*x])/(128*b^3) - (3*(c + d*x)^2*Cos[2*a + 2*b*x])/(64*b) - (d^2*Cos[6*a + 6*b*x])/(3456*b^3) + ((c + d*x)^2*Cos[6*a + 6*b*x])/(192*b) + (3*d*(c + d*x)*Sin[2*a + 2*b*x])/(64*b^2) - (d*(c + d*x)*Sin[6*a + 6*b*x])/(576*b^2)} -{(c + d*x)^1*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 6, (-3*(c + d*x)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)*Cos[6*a + 6*b*x])/(192*b) + (3*d*Sin[2*a + 2*b*x])/(128*b^2) - (d*Sin[6*a + 6*b*x])/(1152*b^2)} -{(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^1, x, 8, -(CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d])/(32*d) + (3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(32*d) + (3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(32*d) - (Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(32*d)} -{(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^2, x, 10, (3*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(16*d^2) - (3*b*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*c)/d + 6*b*x])/(16*d^2) - (3*Sin[2*a + 2*b*x])/(32*d*(c + d*x)) + Sin[6*a + 6*b*x]/(32*d*(c + d*x)) - (3*b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(16*d^2) + (3*b*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(16*d^2)} -{(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^3, x, 12, (-3*b*Cos[2*a + 2*b*x])/(32*d^2*(c + d*x)) + (3*b*Cos[6*a + 6*b*x])/(32*d^2*(c + d*x)) + (9*b^2*CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d])/(16*d^3) - (3*b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(16*d^3) - (3*Sin[2*a + 2*b*x])/(64*d*(c + d*x)^2) + Sin[6*a + 6*b*x]/(64*d*(c + d*x)^2) - (3*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(16*d^3) + (9*b^2*Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(16*d^3)} -{(Cos[a + b*x]^3*Sin[a + b*x]^3)/(c + d*x)^4, x, 14, -(b*Cos[2*a + 2*b*x])/(32*d^2*(c + d*x)^2) + (b*Cos[6*a + 6*b*x])/(32*d^2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(8*d^4) + (9*b^3*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*c)/d + 6*b*x])/(8*d^4) - Sin[2*a + 2*b*x]/(32*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(16*d^3*(c + d*x)) + Sin[6*a + 6*b*x]/(96*d*(c + d*x)^3) - (3*b^2*Sin[6*a + 6*b*x])/(16*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(8*d^4) - (9*b^3*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(8*d^4)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(c + d*x)^m*Cos[a + b*x]^2*Cot[a + b*x], x, 6, (2^(-3 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b) + (2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b) + Unintegrable[(c + d*x)^m*Cot[a + b*x], x]} - -{(c + d*x)^4*Cos[a + b*x]^2*Cot[a + b*x], x, 13, -((3*c*d^3*x)/(2*b^3)) - (3*d^4*x^2)/(4*b^3) + (c + d*x)^4/(4*b) - (I*(c + d*x)^5)/(5*d) + ((c + d*x)^4*Log[1 - E^(2*I*(a + b*x))])/b - (2*I*d*(c + d*x)^3*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + (3*d^2*(c + d*x)^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3 + (3*I*d^3*(c + d*x)*PolyLog[4, E^(2*I*(a + b*x))])/b^4 - (3*d^4*PolyLog[5, E^(2*I*(a + b*x))])/(2*b^5) + (3*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b^4) - (d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/b^2 - (3*d^4*Sin[a + b*x]^2)/(4*b^5) + (3*d^2*(c + d*x)^2*Sin[a + b*x]^2)/(2*b^3) - ((c + d*x)^4*Sin[a + b*x]^2)/(2*b)} -{(c + d*x)^3*Cos[a + b*x]^2*Cot[a + b*x], x, 12, -((3*d^3*x)/(8*b^3)) + (c + d*x)^3/(4*b) - (I*(c + d*x)^4)/(4*d) + ((c + d*x)^3*Log[1 - E^(2*I*(a + b*x))])/b - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4) + (3*d^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^4) - (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) + (3*d^2*(c + d*x)*Sin[a + b*x]^2)/(4*b^3) - ((c + d*x)^3*Sin[a + b*x]^2)/(2*b)} -{(c + d*x)^2*Cos[a + b*x]^2*Cot[a + b*x], x, 9, (c*d*x)/(2*b) + (d^2*x^2)/(4*b) - (I*(c + d*x)^3)/(3*d) + ((c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b - (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (d^2*Sin[a + b*x]^2)/(4*b^3) - ((c + d*x)^2*Sin[a + b*x]^2)/(2*b)} -{(c + d*x)^1*Cos[a + b*x]^2*Cot[a + b*x], x, 8, (d*x)/(4*b) - (I*(c + d*x)^2)/(2*d) + ((c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (d*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) - ((c + d*x)*Sin[a + b*x]^2)/(2*b)} -{(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x)^1, x, 6, Unintegrable[Cot[a + b*x]/(c + d*x), x] - (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d) - (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{(Cos[a + b*x]^2*Cot[a + b*x])/(c + d*x)^2, x, 7, -((b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^2) + Unintegrable[Cot[a + b*x]/(c + d*x)^2, x] + Sin[2*a + 2*b*x]/(2*d*(c + d*x)) + (b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} - - -{(c + d*x)^m*Cos[a + b*x]*Cot[a + b*x]^2, x, 4, CannotIntegrate[(c + d*x)^m*Cot[a + b*x]*Csc[a + b*x], x] + (I*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b)) - (I*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b))} - -{(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x]^2, x, 16, -((8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2) + (24*d^3*(c + d*x)*Cos[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cos[a + b*x])/b^2 - ((c + d*x)^4*Csc[a + b*x])/b + (12*I*d^2*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (12*I*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - (24*I*d^4*PolyLog[4, -E^(I*(a + b*x))])/b^5 + (24*I*d^4*PolyLog[4, E^(I*(a + b*x))])/b^5 - (24*d^4*Sin[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Sin[a + b*x])/b^3 - ((c + d*x)^4*Sin[a + b*x])/b} -{(c + d*x)^3*Cos[a + b*x]*Cot[a + b*x]^2, x, 13, -((6*d*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^2) + (6*d^3*Cos[a + b*x])/b^4 - (3*d*(c + d*x)^2*Cos[a + b*x])/b^2 - ((c + d*x)^3*Csc[a + b*x])/b + (6*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (6*I*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (6*d^3*PolyLog[3, E^(I*(a + b*x))])/b^4 + (6*d^2*(c + d*x)*Sin[a + b*x])/b^3 - ((c + d*x)^3*Sin[a + b*x])/b} -{(c + d*x)^2*Cos[a + b*x]*Cot[a + b*x]^2, x, 10, -((4*d*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^2) - (2*d*(c + d*x)*Cos[a + b*x])/b^2 - ((c + d*x)^2*Csc[a + b*x])/b + (2*I*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - (2*I*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3 + (2*d^2*Sin[a + b*x])/b^3 - ((c + d*x)^2*Sin[a + b*x])/b} -{(c + d*x)^1*Cos[a + b*x]*Cot[a + b*x]^2, x, 5, -((d*ArcTanh[Cos[a + b*x]])/b^2) - (d*Cos[a + b*x])/b^2 - ((c + d*x)*Csc[a + b*x])/b - ((c + d*x)*Sin[a + b*x])/b} -{(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^1, x, 4, -((Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d) + CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x), x] + (Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{(Cos[a + b*x]*Cot[a + b*x]^2)/(c + d*x)^2, x, 5, Cos[a + b*x]/(d*(c + d*x)) + CannotIntegrate[(Cot[a + b*x]*Csc[a + b*x])/(c + d*x)^2, x] + (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^2 + (b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} - - -{(c + d*x)^m*Cot[a + b*x]^3, x, 0, Unintegrable[(c + d*x)^m*Cot[a + b*x]^3, x]} - -{(c + d*x)^4*Cot[a + b*x]^3, x, 15, -((2*I*d*(c + d*x)^3)/b^2) - (c + d*x)^4/(2*b) + (I*(c + d*x)^5)/(5*d) - (2*d*(c + d*x)^3*Cot[a + b*x])/b^2 - ((c + d*x)^4*Cot[a + b*x]^2)/(2*b) + (6*d^2*(c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b^3 - ((c + d*x)^4*Log[1 - E^(2*I*(a + b*x))])/b - (6*I*d^3*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^4 + (2*I*d*(c + d*x)^3*PolyLog[2, E^(2*I*(a + b*x))])/b^2 + (3*d^4*PolyLog[3, E^(2*I*(a + b*x))])/b^5 - (3*d^2*(c + d*x)^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*(c + d*x)*PolyLog[4, E^(2*I*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, E^(2*I*(a + b*x))])/(2*b^5)} -{(c + d*x)^3*Cot[a + b*x]^3, x, 13, -((3*I*d*(c + d*x)^2)/(2*b^2)) - (c + d*x)^3/(2*b) + (I*(c + d*x)^4)/(4*d) - (3*d*(c + d*x)^2*Cot[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]^2)/(2*b) + (3*d^2*(c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b^3 - ((c + d*x)^3*Log[1 - E^(2*I*(a + b*x))])/b - (3*I*d^3*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^4) + (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{(c + d*x)^2*Cot[a + b*x]^3, x, 9, -((c*d*x)/b) - (d^2*x^2)/(2*b) + (I*(c + d*x)^3)/(3*d) - (d*(c + d*x)*Cot[a + b*x])/b^2 - ((c + d*x)^2*Cot[a + b*x]^2)/(2*b) - ((c + d*x)^2*Log[1 - E^(2*I*(a + b*x))])/b + (d^2*Log[Sin[a + b*x]])/b^3 + (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3)} -{(c + d*x)^1*Cot[a + b*x]^3, x, 7, -((d*x)/(2*b)) + (I*(c + d*x)^2)/(2*d) - (d*Cot[a + b*x])/(2*b^2) - ((c + d*x)*Cot[a + b*x]^2)/(2*b) - ((c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b + (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Cot[a + b*x]^3/(c + d*x)^1, x, 0, Unintegrable[Cot[a + b*x]^3/(c + d*x), x]} -{Cot[a + b*x]^3/(c + d*x)^2, x, 0, Unintegrable[Cot[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Cos[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 18, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) + (15*d^2*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) + (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*b^2)} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 16, -((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(8*b) - ((c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(32*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(64*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(64*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(32*b^2) + (3*d*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(256*b^2)} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 14, -(Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(8*b) - (Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(32*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(16*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(16*b^(3/2))} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 14, -(Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(8*b) - (Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(32*b) + (Sqrt[d]*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(3/2)) + (Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(16*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(64*b^(3/2)) - (Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(16*b^(3/2))} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 16, -((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(8*b) - ((c + d*x)^(3/2)*Cos[4*a + 4*b*x])/(32*b) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(64*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(512*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(64*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(32*b^2) + (3*d*Sqrt[c + d*x]*Sin[4*a + 4*b*x])/(256*b^2)} -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x], x, 18, (15*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(128*b^3) - ((c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(8*b) + (15*d^2*Sqrt[c + d*x]*Cos[4*a + 4*b*x])/(2048*b^3) - ((c + d*x)^(5/2)*Cos[4*a + 4*b*x])/(32*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[4*a - (4*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(4096*b^(7/2)) - (15*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(256*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(2*Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[4*a - (4*b*c)/d])/(4096*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(256*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(32*b^2) + (5*d*(c + d*x)^(3/2)*Sin[4*a + 4*b*x])/(256*b^2)} - - -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 26, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(16*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(160*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(32*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(48*b) + (3*d^2*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(1600*b^3) - ((c + d*x)^(5/2)*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 23, (3*d*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(800*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(800*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(48*b) - ((c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 20, -(Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2)) + (Sqrt[c + d*x]*Sin[a + b*x])/(8*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(48*b) - (Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 20, -(Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2)) + (Sqrt[c + d*x]*Sin[a + b*x])/(8*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(48*b) - (Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 23, (3*d*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(800*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(5/2)) + (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(800*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(48*b) - ((c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(80*b)} -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^2, x, 26, (5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(16*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(160*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(32*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(48*b) + (3*d^2*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(1600*b^3) - ((c + d*x)^(5/2)*Sin[5*a + 5*b*x])/(80*b)} - - -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 18, (45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(18432*b^(7/2)) - (45*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(18432*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) + (15*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*b^2)} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 16, (-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(512*b^(5/2)) + (d^(3/2)*Sqrt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(512*b^(5/2)) + (9*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^2)} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 14, (-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(3/2))} -{(c + d*x)^(1/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 14, (-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(3/2))} -{(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 16, (-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(512*b^(5/2)) + (d^(3/2)*Sqrt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(512*b^(5/2)) + (9*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^2)} -{(c + d*x)^(5/2)*Cos[a + b*x]^3*Sin[a + b*x]^3, x, 18, (45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(18432*b^(7/2)) - (45*d^(5/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(18432*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) + (15*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*b^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^4 Sin[a+b x]^p*) - - -{x^3*Cos[x]^2*Cot[x]^2, x, 12, (3*x^2)/8 - I*x^3 - (3*x^4)/8 + (3*Cos[x]^2)/8 - (3*x^2*Cos[x]^2)/4 - x^3*Cot[x] + 3*x^2*Log[1 - E^((2*I)*x)] - (3*I)*x*PolyLog[2, E^((2*I)*x)] + (3*PolyLog[3, E^((2*I)*x)])/2 + (3*x*Cos[x]*Sin[x])/4 - (x^3*Cos[x]*Sin[x])/2} -{x^2*Cos[x]^2*Cot[x]^2, x, 11, x/4 - I*x^2 - x^3/2 - (x*Cos[x]^2)/2 - x^2*Cot[x] + 2*x*Log[1 - E^((2*I)*x)] - I*PolyLog[2, E^((2*I)*x)] + (Cos[x]*Sin[x])/4 - (x^2*Cos[x]*Sin[x])/2} -{x^1*Cos[x]^2*Cot[x]^2, x, 6, (-3*x^2)/4 - Cos[x]^2/4 - x*Cot[x] + Log[Sin[x]] - (x*Cos[x]*Sin[x])/2} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^5 Sin[a+b x]^p*) - - -{x^3*Cos[x]^2*Cot[x]^3, x, 26, (3*x)/8 - ((3*I)/2)*x^2 - (3*x^3)/4 + (I/2)*x^4 - (3*x^2*Cot[x])/2 - (x^3*Cot[x]^2)/2 + 3*x*Log[1 - E^((2*I)*x)] - 2*x^3*Log[1 - E^((2*I)*x)] - ((3*I)/2)*PolyLog[2, E^((2*I)*x)] + (3*I)*x^2*PolyLog[2, E^((2*I)*x)] - 3*x*PolyLog[3, E^((2*I)*x)] - ((3*I)/2)*PolyLog[4, E^((2*I)*x)] - (3*Cos[x]*Sin[x])/8 + (3*x^2*Cos[x]*Sin[x])/4 - (3*x*Sin[x]^2)/4 + (x^3*Sin[x]^2)/2} -{x^2*Cos[x]^2*Cot[x]^3, x, 19, (-3*x^2)/4 + ((2*I)/3)*x^3 - x*Cot[x] - (x^2*Cot[x]^2)/2 - 2*x^2*Log[1 - E^((2*I)*x)] + Log[Sin[x]] + (2*I)*x*PolyLog[2, E^((2*I)*x)] - PolyLog[3, E^((2*I)*x)] + (x*Cos[x]*Sin[x])/2 - Sin[x]^2/4 + (x^2*Sin[x]^2)/2} -{x^1*Cos[x]^2*Cot[x]^3, x, 16, (-3*x)/4 + I*x^2 - Cot[x]/2 - (x*Cot[x]^2)/2 - 2*x*Log[1 - E^((2*I)*x)] + I*PolyLog[2, E^((2*I)*x)] + (Cos[x]*Sin[x])/4 + (x*Sin[x]^2)/2} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^1 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x] Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sec[a + b*x]*Sin[a + b*x]*(c + d*x)^m, x, 0, Unintegrable[(c + d*x)^m*Tan[a + b*x], x]} - -{Sec[a + b*x]*Sin[a + b*x]*(c + d*x)^4, x, 7, (I*(c + d*x)^5)/(5*d) - ((c + d*x)^4*Log[1 + E^(2*I*(a + b*x))])/b + (2*I*d*(c + d*x)^3*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*(c + d*x)*PolyLog[4, -E^(2*I*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^(2*I*(a + b*x))])/(2*b^5)} -{Sec[a + b*x]*Sin[a + b*x]*(c + d*x)^3, x, 6, (I*(c + d*x)^4)/(4*d) - ((c + d*x)^3*Log[1 + E^(2*I*(a + b*x))])/b + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4)} -{Sec[a + b*x]*Sin[a + b*x]*(c + d*x)^2, x, 5, (I*(c + d*x)^3)/(3*d) - ((c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b + (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3)} -{Sec[a + b*x]*Sin[a + b*x]*(c + d*x)^1, x, 4, (I*(c + d*x)^2)/(2*d) - ((c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2)} -{Sec[a + b*x]*Sin[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Tan[a + b*x]/(c + d*x), x]} -{Sec[a + b*x]*Sin[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Tan[a + b*x]/(c + d*x)^2, x]} - - -{Sec[a + b*x]*Sin[a + b*x]^2*(c + d*x)^m, x, 4, (I*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b)) - (I*(c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b)) + Unintegrable[(c + d*x)^m*Sec[a + b*x], x]} - -{Sec[a + b*x]*Sin[a + b*x]^2*(c + d*x)^3, x, 14, -((2*I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b) + (6*d^3*Cos[a + b*x])/b^4 - (3*d*(c + d*x)^2*Cos[a + b*x])/b^2 + (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (6*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4 + (6*d^2*(c + d*x)*Sin[a + b*x])/b^3 - ((c + d*x)^3*Sin[a + b*x])/b} -{Sec[a + b*x]*Sin[a + b*x]^2*(c + d*x)^2, x, 11, -((2*I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b) - (2*d*(c + d*x)*Cos[a + b*x])/b^2 + (2*I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + (2*d^2*Sin[a + b*x])/b^3 - ((c + d*x)^2*Sin[a + b*x])/b} -{Sec[a + b*x]*Sin[a + b*x]^2*(c + d*x)^1, x, 8, -((2*I*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b) - (d*Cos[a + b*x])/b^2 + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - ((c + d*x)*Sin[a + b*x])/b} -{Sec[a + b*x]*Sin[a + b*x]^2/(c + d*x)^1, x, 4, -((Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d) + Unintegrable[Sec[a + b*x]/(c + d*x), x] + (Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{Sec[a + b*x]*Sin[a + b*x]^2/(c + d*x)^2, x, 5, Cos[a + b*x]/(d*(c + d*x)) + Unintegrable[Sec[a + b*x]/(c + d*x)^2, x] + (b*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^2 + (b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} - - -{Sec[a + b*x]*Sin[a + b*x]^3*(c + d*x)^m, x, 6, (2^(-3 - m)*E^(2*I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((2*I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*b) + (2^(-3 - m)*(c + d*x)^m*Gamma[1 + m, (2*I*b*(c + d*x))/d])/(E^(2*I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*b) + Unintegrable[(c + d*x)^m*Tan[a + b*x], x]} - -{Sec[a + b*x]*Sin[a + b*x]^3*(c + d*x)^3, x, 12, -((3*d^3*x)/(8*b^3)) + (c + d*x)^3/(4*b) + (I*(c + d*x)^4)/(4*d) - ((c + d*x)^3*Log[1 + E^(2*I*(a + b*x))])/b + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) + (3*d^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^4) - (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) + (3*d^2*(c + d*x)*Sin[a + b*x]^2)/(4*b^3) - ((c + d*x)^3*Sin[a + b*x]^2)/(2*b)} -{Sec[a + b*x]*Sin[a + b*x]^3*(c + d*x)^2, x, 9, (c*d*x)/(2*b) + (d^2*x^2)/(4*b) + (I*(c + d*x)^3)/(3*d) - ((c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b + (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) - (d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (d^2*Sin[a + b*x]^2)/(4*b^3) - ((c + d*x)^2*Sin[a + b*x]^2)/(2*b)} -{Sec[a + b*x]*Sin[a + b*x]^3*(c + d*x)^1, x, 8, (d*x)/(4*b) + (I*(c + d*x)^2)/(2*d) - ((c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (d*Cos[a + b*x]*Sin[a + b*x])/(4*b^2) - ((c + d*x)*Sin[a + b*x]^2)/(2*b)} -{Sec[a + b*x]*Sin[a + b*x]^3/(c + d*x)^1, x, 6, Unintegrable[Tan[a + b*x]/(c + d*x), x] - (CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d) - (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Sec[a + b*x]*Sin[a + b*x]^3/(c + d*x)^2, x, 7, -((b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^2) + Unintegrable[Tan[a + b*x]/(c + d*x)^2, x] + Sin[2*a + 2*b*x]/(2*d*(c + d*x)) + (b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[a + b*x]*Csc[a + b*x]*(c + d*x)^m, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x], x]} - -{Sec[a + b*x]*Csc[a + b*x]*(c + d*x)^4, x, 12, -((2*(c + d*x)^4*ArcTanh[E^(2*I*(a + b*x))])/b) + (2*I*d*(c + d*x)^3*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)^3*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 + (3*d^2*(c + d*x)^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*(c + d*x)*PolyLog[4, -E^(2*I*(a + b*x))])/b^4 + (3*I*d^3*(c + d*x)*PolyLog[4, E^(2*I*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^(2*I*(a + b*x))])/(2*b^5) - (3*d^4*PolyLog[5, E^(2*I*(a + b*x))])/(2*b^5)} -{Sec[a + b*x]*Csc[a + b*x]*(c + d*x)^3, x, 10, -((2*(c + d*x)^3*ArcTanh[E^(2*I*(a + b*x))])/b) + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{Sec[a + b*x]*Csc[a + b*x]*(c + d*x)^2, x, 8, -((2*(c + d*x)^2*ArcTanh[E^(2*I*(a + b*x))])/b) + (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3)} -{Sec[a + b*x]*Csc[a + b*x]*(c + d*x)^1, x, 6, -((2*(c + d*x)*ArcTanh[E^(2*I*(a + b*x))])/b) + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Sec[a + b*x]*Csc[a + b*x]/(c + d*x)^1, x, 1, 2*Unintegrable[Csc[2*a + 2*b*x]/(c + d*x), x]} -{Sec[a + b*x]*Csc[a + b*x]/(c + d*x)^2, x, 1, 2*Unintegrable[Csc[2*a + 2*b*x]/(c + d*x)^2, x]} - - -{Sec[a + b*x]*Csc[a + b*x]^2*(c + d*x)^m, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x], x]} - -{Sec[a + b*x]*Csc[a + b*x]^2*(c + d*x)^3, x, 23, -((2*I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b) - (6*d*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^2 - ((c + d*x)^3*Csc[a + b*x])/b + (6*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^3 + (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (6*I*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, -E^(I*(a + b*x))])/b^4 - (6*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (6*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, E^(I*(a + b*x))])/b^4 - (6*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4} -{Sec[a + b*x]*Csc[a + b*x]^2*(c + d*x)^2, x, 19, -((2*I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b) - (4*d*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^2 - ((c + d*x)^2*Csc[a + b*x])/b + (2*I*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 + (2*I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (2*I*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (2*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3} -{Sec[a + b*x]*Csc[a + b*x]^2*(c + d*x)^1, x, 10, -((2*I*d*x*ArcTan[E^(I*(a + b*x))])/b) - (d*ArcTanh[Cos[a + b*x]])/b^2 - (d*x*ArcTanh[Sin[a + b*x]])/b + ((c + d*x)*ArcTanh[Sin[a + b*x]])/b - ((c + d*x)*Csc[a + b*x])/b + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2} -{Sec[a + b*x]*Csc[a + b*x]^2/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x), x]} -{Sec[a + b*x]*Csc[a + b*x]^2/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]^2*Sec[a + b*x])/(c + d*x)^2, x]} - - -{Sec[a + b*x]*Csc[a + b*x]^3*(c + d*x)^m, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x], x]} - -{Sec[a + b*x]*Csc[a + b*x]^3*(c + d*x)^3, x, 22, -((3*I*d*(c + d*x)^2)/(2*b^2)) - (c + d*x)^3/(2*b) - (2*(c + d*x)^3*ArcTanh[E^(2*I*(a + b*x))])/b - (3*d*(c + d*x)^2*Cot[a + b*x])/(2*b^2) - ((c + d*x)^3*Cot[a + b*x]^2)/(2*b) + (3*d^2*(c + d*x)*Log[1 - E^(2*I*(a + b*x))])/b^3 + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^4) - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4)} -{Sec[a + b*x]*Csc[a + b*x]^3*(c + d*x)^2, x, 17, -((c*d*x)/b) - (d^2*x^2)/(2*b) - (2*(c + d*x)^2*ArcTanh[E^(2*I*(a + b*x))])/b - (d*(c + d*x)*Cot[a + b*x])/b^2 - ((c + d*x)^2*Cot[a + b*x]^2)/(2*b) + (d^2*Log[Sin[a + b*x]])/b^3 + (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3)} -{Sec[a + b*x]*Csc[a + b*x]^3*(c + d*x)^1, x, 11, -((d*x)/(2*b)) - (2*d*x*ArcTanh[E^(2*I*(a + b*x))])/b - (d*Cot[a + b*x])/(2*b^2) - ((c + d*x)*Cot[a + b*x]^2)/(2*b) - (d*x*Log[Tan[a + b*x]])/b + ((c + d*x)*Log[Tan[a + b*x]])/b + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2)} -{Sec[a + b*x]*Csc[a + b*x]^3/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x), x]} -{Sec[a + b*x]*Csc[a + b*x]^3/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x])/(c + d*x)^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Sec[a+b x] Sin[a+b x]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x], x, 0, CannotIntegrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x], x]} - -{(c + d*x)^4*Sec[a + b*x]*Tan[a + b*x], x, 10, ((8*I)*d*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b^2 - ((12*I)*d^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((12*I)*d^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + (24*d^3*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (24*d^3*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + ((24*I)*d^4*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^5 - ((24*I)*d^4*PolyLog[4, I*E^(I*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x])/b} -{(c + d*x)^3*Sec[a + b*x]*Tan[a + b*x], x, 8, ((6*I)*d*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b^2 - ((6*I)*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((6*I)*d^2*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + ((c + d*x)^3*Sec[a + b*x])/b} -{(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x], x, 6, ((4*I)*d*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2 - ((2*I)*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((2*I)*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + ((c + d*x)^2*Sec[a + b*x])/b} -{(c + d*x)^1*Sec[a + b*x]*Tan[a + b*x], x, 2, -((d*ArcTanh[Sin[a + b*x]])/b^2) + ((c + d*x)*Sec[a + b*x])/b} -{(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^1, x, 0, CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x]} -{(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x, 0, CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x]} - - -{(c + d*x)^m*Tan[a + b*x]^2, x, 0, Unintegrable[(c + d*x)^m*Tan[a + b*x]^2, x]} - -{(c + d*x)^3*Tan[a + b*x]^2, x, 7, -((I*(c + d*x)^3)/b) - (c + d*x)^4/(4*d) + (3*d*(c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b^2 - (3*I*d^2*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^4) + ((c + d*x)^3*Tan[a + b*x])/b} -{(c + d*x)^2*Tan[a + b*x]^2, x, 6, -((I*(c + d*x)^2)/b) - (c + d*x)^3/(3*d) + (2*d*(c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b^2 - (I*d^2*PolyLog[2, -E^(2*I*(a + b*x))])/b^3 + ((c + d*x)^2*Tan[a + b*x])/b} -{(c + d*x)^1*Tan[a + b*x]^2, x, 3, -(c*x) - (d*x^2)/2 + (d*Log[Cos[a + b*x]])/b^2 + ((c + d*x)*Tan[a + b*x])/b} -{Tan[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Tan[a + b*x]^2/(c + d*x), x]} -{Tan[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Tan[a + b*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^m*Sin[a + b*x]*Tan[a + b*x]^2, x, 4, (E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, -((I*b*(c + d*x))/d)])/((-((I*b*(c + d*x))/d))^m*(2*b)) + ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m*(2*b)) + CannotIntegrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x], x]} - -{(c + d*x)^3*Sin[a + b*x]*Tan[a + b*x]^2, x, 13, (6*I*d*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*Cos[a + b*x])/b^3 + ((c + d*x)^3*Cos[a + b*x])/b - (6*I*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (6*I*d^2*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + ((c + d*x)^3*Sec[a + b*x])/b + (6*d^3*Sin[a + b*x])/b^4 - (3*d*(c + d*x)^2*Sin[a + b*x])/b^2} -{(c + d*x)^2*Sin[a + b*x]*Tan[a + b*x]^2, x, 10, (4*I*d*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2 - (2*d^2*Cos[a + b*x])/b^3 + ((c + d*x)^2*Cos[a + b*x])/b - (2*I*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (2*I*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + ((c + d*x)^2*Sec[a + b*x])/b - (2*d*(c + d*x)*Sin[a + b*x])/b^2} -{(c + d*x)^1*Sin[a + b*x]*Tan[a + b*x]^2, x, 5, -((d*ArcTanh[Sin[a + b*x]])/b^2) + ((c + d*x)*Cos[a + b*x])/b + ((c + d*x)*Sec[a + b*x])/b - (d*Sin[a + b*x])/b^2} -{(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^1, x, 4, CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x] - (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{(Sin[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2, x, 5, -((b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2) + CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x] + Sin[a + b*x]/(d*(c + d*x)) + (b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^2, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^2, x]} - -{(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^2, x, 27, (8*I*d*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b^2 - (2*(c + d*x)^4*ArcTanh[E^(I*(a + b*x))])/b + (4*I*d*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (12*I*d^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (12*I*d^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (4*I*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))])/b^2 - (12*d^2*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (24*d^3*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (24*d^3*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (12*d^2*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))])/b^3 - (24*I*d^3*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (24*I*d^4*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^5 - (24*I*d^4*PolyLog[4, I*E^(I*(a + b*x))])/b^5 + (24*I*d^3*(c + d*x)*PolyLog[4, E^(I*(a + b*x))])/b^4 + (24*d^4*PolyLog[5, -E^(I*(a + b*x))])/b^5 - (24*d^4*PolyLog[5, E^(I*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x])/b} -{(c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^2, x, 23, (6*I*d*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b^2 - (2*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (6*I*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (6*I*d^2*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (3*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (6*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (6*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (6*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + ((c + d*x)^3*Sec[a + b*x])/b} -{(c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^2, x, 19, (4*I*d*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2 - (2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b + (2*I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (2*I*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (2*I*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (2*I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (2*d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (2*d^2*PolyLog[3, E^(I*(a + b*x))])/b^3 + ((c + d*x)^2*Sec[a + b*x])/b} -{(c + d*x)^1*Csc[a + b*x]*Sec[a + b*x]^2, x, 10, (-2*d*x*ArcTanh[E^(I*(a + b*x))])/b - (c*ArcTanh[Cos[a + b*x]])/b - (d*ArcTanh[Sin[a + b*x]])/b^2 + (I*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, E^(I*(a + b*x))])/b^2 + (c*Sec[a + b*x])/b + (d*x*Sec[a + b*x])/b, -((2*d*x*ArcTanh[E^(I*(a + b*x))])/b) + (d*x*ArcTanh[Cos[a + b*x]])/b - ((c + d*x)*ArcTanh[Cos[a + b*x]])/b - (d*ArcTanh[Sin[a + b*x]])/b^2 + (I*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (I*d*PolyLog[2, E^(I*(a + b*x))])/b^2 + ((c + d*x)*Sec[a + b*x])/b} -{(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x), x]} -{(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]*Sec[a + b*x]^2)/(c + d*x)^2, x]} - - -{(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^2, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^2, x]} - -{(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^2, x, 7, -((2*I*(c + d*x)^3)/b) - (2*(c + d*x)^3*Cot[2*a + 2*b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(4*I*(a + b*x))])/b^2 - (3*I*d^2*(c + d*x)*PolyLog[2, E^(4*I*(a + b*x))])/(2*b^3) + (3*d^3*PolyLog[3, E^(4*I*(a + b*x))])/(8*b^4)} -{(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x]^2, x, 6, -((2*I*(c + d*x)^2)/b) - (2*(c + d*x)^2*Cot[2*a + 2*b*x])/b + (2*d*(c + d*x)*Log[1 - E^(4*I*(a + b*x))])/b^2 - (I*d^2*PolyLog[2, E^(4*I*(a + b*x))])/(2*b^3)} -{(c + d*x)^1*Csc[a + b*x]^2*Sec[a + b*x]^2, x, 3, -((2*(c + d*x)*Cot[2*a + 2*b*x])/b) + (d*Log[Sin[2*a + 2*b*x]])/b^2} -{(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x)^1, x, 1, 4*Unintegrable[Csc[2*a + 2*b*x]^2/(c + d*x), x]} -{(Csc[a + b*x]^2*Sec[a + b*x]^2)/(c + d*x)^2, x, 1, 4*Unintegrable[Csc[2*a + 2*b*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]} - -{(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 64, (12*I*c*d^2*x*ArcTan[E^(I*(a + b*x))])/b^2 + (6*I*d^3*x^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*d^3*x*ArcTanh[E^(I*(a + b*x))])/b^3 - (3*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b - (3*c*d^2*ArcTanh[Cos[a + b*x]])/b^3 - (3*c^2*d*ArcTanh[Sin[a + b*x]])/b^2 - (3*c^2*d*Csc[a + b*x])/(2*b^2) - (3*c*d^2*x*Csc[a + b*x])/b^2 - (3*d^3*x^2*Csc[a + b*x])/(2*b^2) + (3*I*d^3*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (9*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (6*I*c*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - (6*I*d^3*x*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (6*I*c*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + (6*I*d^3*x*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (3*I*d^3*PolyLog[2, E^(I*(a + b*x))])/b^4 - (9*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/(2*b^2) - (9*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (9*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (9*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (9*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + (3*(c + d*x)^3*Sec[a + b*x])/(2*b) - ((c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{(c + d*x)^2*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 36, (4*I*d^2*x*ArcTan[E^(I*(a + b*x))])/b^2 - (3*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b - (d^2*ArcTanh[Cos[a + b*x]])/b^3 - (2*c*d*ArcTanh[Sin[a + b*x]])/b^2 - (c*d*Csc[a + b*x])/b^2 - (d^2*x*Csc[a + b*x])/b^2 + (3*I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (2*I*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + (2*I*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (3*I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (3*d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (3*d^2*PolyLog[3, E^(I*(a + b*x))])/b^3 + (3*(c + d*x)^2*Sec[a + b*x])/(2*b) - ((c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{(c + d*x)^1*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 13, -((3*d*x*ArcTanh[E^(I*(a + b*x))])/b) - (3*c*ArcTanh[Cos[a + b*x]])/(2*b) - (d*ArcTanh[Sin[a + b*x]])/b^2 - (d*Csc[a + b*x])/(2*b^2) + (3*I*d*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (3*I*d*PolyLog[2, E^(I*(a + b*x))])/(2*b^2) + (3*(c + d*x)*Sec[a + b*x])/(2*b) - ((c + d*x)*Csc[a + b*x]^2*Sec[a + b*x])/(2*b), -((3*d*x*ArcTanh[E^(I*(a + b*x))])/b) + (3*d*x*ArcTanh[Cos[a + b*x]])/(2*b) - (3*(c + d*x)*ArcTanh[Cos[a + b*x]])/(2*b) - (d*ArcTanh[Sin[a + b*x]])/b^2 - (d*Csc[a + b*x])/(2*b^2) + (3*I*d*PolyLog[2, -E^(I*(a + b*x))])/(2*b^2) - (3*I*d*PolyLog[2, E^(I*(a + b*x))])/(2*b^2) + (3*(c + d*x)*Sec[a + b*x])/(2*b) - ((c + d*x)*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x), x]} -{(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/(c + d*x)^2, x]} - - -{x^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 0, CannotIntegrate[x^m*Csc[a + b*x]^3*Sec[a + b*x]^2, x]} - -{x^3*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 40, ((6*I)*x^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*x*ArcTanh[E^(I*(a + b*x))])/b^3 - (3*x^3*ArcTanh[E^(I*(a + b*x))])/b - (3*x^2*Csc[a + b*x])/(2*b^2) + ((3*I)*PolyLog[2, -E^(I*(a + b*x))])/b^4 + (((9*I)/2)*x^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((6*I)*x*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((6*I)*x*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - ((3*I)*PolyLog[2, E^(I*(a + b*x))])/b^4 - (((9*I)/2)*x^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (9*x*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + (9*x*PolyLog[3, E^(I*(a + b*x))])/b^3 - ((9*I)*PolyLog[4, -E^(I*(a + b*x))])/b^4 + ((9*I)*PolyLog[4, E^(I*(a + b*x))])/b^4 + (3*x^3*Sec[a + b*x])/(2*b) - (x^3*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{x^2*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 29, ((4*I)*x*ArcTan[E^(I*(a + b*x))])/b^2 - (3*x^2*ArcTanh[E^(I*(a + b*x))])/b - ArcTanh[Cos[a + b*x]]/b^3 - (x*Csc[a + b*x])/b^2 + ((3*I)*x*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((2*I)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((2*I)*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - ((3*I)*x*PolyLog[2, E^(I*(a + b*x))])/b^2 - (3*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (3*PolyLog[3, E^(I*(a + b*x))])/b^3 + (3*x^2*Sec[a + b*x])/(2*b) - (x^2*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{x*Csc[a + b*x]^3*Sec[a + b*x]^2, x, 13, (-3*x*ArcTanh[E^(I*(a + b*x))])/b - ArcTanh[Sin[a + b*x]]/b^2 - Csc[a + b*x]/(2*b^2) + (((3*I)/2)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (((3*I)/2)*PolyLog[2, E^(I*(a + b*x))])/b^2 + (3*x*Sec[a + b*x])/(2*b) - (x*Csc[a + b*x]^2*Sec[a + b*x])/(2*b)} -{(Csc[a + b*x]^3*Sec[a + b*x]^2)/x, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x, x]} -{(Csc[a + b*x]^3*Sec[a + b*x]^2)/x^2, x, 0, CannotIntegrate[(Csc[a + b*x]^3*Sec[a + b*x]^2)/x^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Sec[a+b x]^2 Sin[a+b x]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + d*x)^m*Sec[a + b*x]^2*Tan[a + b*x], x, 0, CannotIntegrate[(c + d*x)^m*Sec[a + b*x]^2*Tan[a + b*x], x]} - -{(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x], x, 7, (2*I*d*(c + d*x)^3)/b^2 - (6*d^2*(c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b^3 + (6*I*d^3*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^4 - (3*d^4*PolyLog[3, -E^(2*I*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2} -{(c + d*x)^3*Sec[a + b*x]^2*Tan[a + b*x], x, 6, (3*I*d*(c + d*x)^2)/(2*b^2) - (3*d^2*(c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b^3 + (3*I*d^3*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^4) + ((c + d*x)^3*Sec[a + b*x]^2)/(2*b) - (3*d*(c + d*x)^2*Tan[a + b*x])/(2*b^2)} -{(c + d*x)^2*Sec[a + b*x]^2*Tan[a + b*x], x, 3, -((d^2*Log[Cos[a + b*x]])/b^3) + ((c + d*x)^2*Sec[a + b*x]^2)/(2*b) - (d*(c + d*x)*Tan[a + b*x])/b^2} -{(c + d*x)^1*Sec[a + b*x]^2*Tan[a + b*x], x, 3, ((c + d*x)*Sec[a + b*x]^2)/(2*b) - (d*Tan[a + b*x])/(2*b^2)} -{(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x)^1, x, 0, CannotIntegrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x), x]} -{(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2, x, 0, CannotIntegrate[(Sec[a + b*x]^2*Tan[a + b*x])/(c + d*x)^2, x]} - - -{(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^2, x, 1, -Unintegrable[(c + d*x)^m*Sec[a + b*x], x] + Unintegrable[(c + d*x)^m*Sec[a + b*x]^3, x]} - -{(c + d*x)^3*Sec[a + b*x]*Tan[a + b*x]^2, x, 25, -((6*I*d^2*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^3) + (I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b + (3*I*d^3*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^4 - (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, I*E^(I*(a + b*x))])/b^4 + (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 - (3*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + (3*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 - (3*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4 - (3*d*(c + d*x)^2*Sec[a + b*x])/(2*b^2) + ((c + d*x)^3*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{(c + d*x)^2*Sec[a + b*x]*Tan[a + b*x]^2, x, 17, (I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b + (d^2*ArcTanh[Sin[a + b*x]])/b^3 - (I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 + (I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 + (d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 - (d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (d*(c + d*x)*Sec[a + b*x])/b^2 + ((c + d*x)^2*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{(c + d*x)^1*Sec[a + b*x]*Tan[a + b*x]^2, x, 12, (I*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b - (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) + (I*d*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)*Sec[a + b*x]*Tan[a + b*x])/(2*b)} -{(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^1, x, 1, -Unintegrable[Sec[a + b*x]/(c + d*x), x] + Unintegrable[Sec[a + b*x]^3/(c + d*x), x]} -{(Sec[a + b*x]*Tan[a + b*x]^2)/(c + d*x)^2, x, 1, -Unintegrable[Sec[a + b*x]/(c + d*x)^2, x] + Unintegrable[Sec[a + b*x]^3/(c + d*x)^2, x]} - - -{(c + d*x)^m*Tan[a + b*x]^3, x, 0, Unintegrable[(c + d*x)^m*Tan[a + b*x]^3, x]} - -{(c + d*x)^3*Tan[a + b*x]^3, x, 13, (3*I*d*(c + d*x)^2)/(2*b^2) + (c + d*x)^3/(2*b) - (I*(c + d*x)^4)/(4*d) - (3*d^2*(c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b^3 + ((c + d*x)^3*Log[1 + E^(2*I*(a + b*x))])/b + (3*I*d^3*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^4) - (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) - (3*d*(c + d*x)^2*Tan[a + b*x])/(2*b^2) + ((c + d*x)^3*Tan[a + b*x]^2)/(2*b)} -{(c + d*x)^2*Tan[a + b*x]^3, x, 9, (c*d*x)/b + (d^2*x^2)/(2*b) - (I*(c + d*x)^3)/(3*d) + ((c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b - (d^2*Log[Cos[a + b*x]])/b^3 - (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 + (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) - (d*(c + d*x)*Tan[a + b*x])/b^2 + ((c + d*x)^2*Tan[a + b*x]^2)/(2*b)} -{(c + d*x)^1*Tan[a + b*x]^3, x, 7, (d*x)/(2*b) - (I*(c + d*x)^2)/(2*d) + ((c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b - (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (d*Tan[a + b*x])/(2*b^2) + ((c + d*x)*Tan[a + b*x]^2)/(2*b)} -{Tan[a + b*x]^3/(c + d*x)^1, x, 0, Unintegrable[Tan[a + b*x]^3/(c + d*x), x]} -{Tan[a + b*x]^3/(c + d*x)^2, x, 0, Unintegrable[Tan[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^3, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]*Sec[a + b*x]^3, x]} - -{(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3, x, 25, (2*I*d*(c + d*x)^3)/b^2 + (c + d*x)^4/(2*b) - (2*(c + d*x)^4*ArcTanh[E^(2*I*(a + b*x))])/b - (6*d^2*(c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b^3 + (6*I*d^3*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^4 + (2*I*d*(c + d*x)^3*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)^3*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (3*d^4*PolyLog[3, -E^(2*I*(a + b*x))])/b^5 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 + (3*d^2*(c + d*x)^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*(c + d*x)*PolyLog[4, -E^(2*I*(a + b*x))])/b^4 + (3*I*d^3*(c + d*x)*PolyLog[4, E^(2*I*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^(2*I*(a + b*x))])/(2*b^5) - (3*d^4*PolyLog[5, E^(2*I*(a + b*x))])/(2*b^5) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2 + ((c + d*x)^4*Tan[a + b*x]^2)/(2*b)} -{(c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^3, x, 22, (3*I*d*(c + d*x)^2)/(2*b^2) + (c + d*x)^3/(2*b) - (2*(c + d*x)^3*ArcTanh[E^(2*I*(a + b*x))])/b - (3*d^2*(c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b^3 + (3*I*d^3*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^4) + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(4*b^4) - (3*d*(c + d*x)^2*Tan[a + b*x])/(2*b^2) + ((c + d*x)^3*Tan[a + b*x]^2)/(2*b)} -{(c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^3, x, 17, (c*d*x)/b + (d^2*x^2)/(2*b) - (2*(c + d*x)^2*ArcTanh[E^(2*I*(a + b*x))])/b - (d^2*Log[Cos[a + b*x]])/b^3 + (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/(2*b^3) - (d*(c + d*x)*Tan[a + b*x])/b^2 + ((c + d*x)^2*Tan[a + b*x]^2)/(2*b)} -{(c + d*x)^1*Csc[a + b*x]*Sec[a + b*x]^3, x, 11, (d*x)/(2*b) - (2*d*x*ArcTanh[E^((2*I)*a + (2*I)*b*x)])/b + (c*Log[Tan[a + b*x]])/b + ((I/2)*d*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - ((I/2)*d*PolyLog[2, E^((2*I)*(a + b*x))])/b^2 - (d*Tan[a + b*x])/(2*b^2) + (c*Tan[a + b*x]^2)/(2*b) + (d*x*Tan[a + b*x]^2)/(2*b), (d*x)/(2*b) - (2*d*x*ArcTanh[E^(2*I*(a + b*x))])/b - (d*x*Log[Tan[a + b*x]])/b + ((c + d*x)*Log[Tan[a + b*x]])/b + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^2) - (d*Tan[a + b*x])/(2*b^2) + ((c + d*x)*Tan[a + b*x]^2)/(2*b)} -{(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x), x]} -{(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]*Sec[a + b*x]^3)/(c + d*x)^2, x]} - - -{(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^3, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^2*Sec[a + b*x]^3, x]} - -{(c + d*x)^3*Csc[a + b*x]^2*Sec[a + b*x]^3, x, 44, -((6*I*d^2*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^3) - (3*I*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b - (6*d*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^2 - (3*(c + d*x)^3*Csc[a + b*x])/(2*b) + (6*I*d^2*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^3 + (3*I*d^3*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^4 + (9*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (3*I*d^3*PolyLog[2, I*E^(I*(a + b*x))])/b^4 - (9*I*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (6*I*d^2*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, -E^(I*(a + b*x))])/b^4 - (9*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (9*d^2*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^3 + (6*d^3*PolyLog[3, E^(I*(a + b*x))])/b^4 - (9*I*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^4 + (9*I*d^3*PolyLog[4, I*E^(I*(a + b*x))])/b^4 - (3*d*(c + d*x)^2*Sec[a + b*x])/(2*b^2) + ((c + d*x)^3*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b)} -{(c + d*x)^2*Csc[a + b*x]^2*Sec[a + b*x]^3, x, 31, -((3*I*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b) + (2*d^2*x*ArcTanh[E^(I*(a + b*x))])/b^2 - (6*d*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b^2 - (d^2*x*ArcTanh[Cos[a + b*x]])/b^2 + (d*(c + d*x)*ArcTanh[Cos[a + b*x]])/b^2 + (d^2*ArcTanh[Sin[a + b*x]])/b^3 - (3*(c + d*x)^2*Csc[a + b*x])/(2*b) + (2*I*d^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 + (3*I*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 - (3*I*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (2*I*d^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (3*d^2*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^3 + (3*d^2*PolyLog[3, I*E^(I*(a + b*x))])/b^3 - (d*(c + d*x)*Sec[a + b*x])/b^2 + ((c + d*x)^2*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b)} -{(c + d*x)^1*Csc[a + b*x]^2*Sec[a + b*x]^3, x, 13, -((3*I*d*x*ArcTan[E^(I*(a + b*x))])/b) - (d*ArcTanh[Cos[a + b*x]])/b^2 + (3*c*ArcTanh[Sin[a + b*x]])/(2*b) - (3*(c + d*x)*Csc[a + b*x])/(2*b) + (3*I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (3*I*d*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b), -((3*I*d*x*ArcTan[E^(I*(a + b*x))])/b) - (d*ArcTanh[Cos[a + b*x]])/b^2 - (3*d*x*ArcTanh[Sin[a + b*x]])/(2*b) + (3*(c + d*x)*ArcTanh[Sin[a + b*x]])/(2*b) - (3*(c + d*x)*Csc[a + b*x])/(2*b) + (3*I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/(2*b^2) - (3*I*d*PolyLog[2, I*E^(I*(a + b*x))])/(2*b^2) - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)*Csc[a + b*x]*Sec[a + b*x]^2)/(2*b)} -{(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x)^1, x, 0, CannotIntegrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x), x]} -{(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x)^2, x, 0, CannotIntegrate[(Csc[a + b*x]^2*Sec[a + b*x]^3)/(c + d*x)^2, x]} - - -{(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^3, x, 0, CannotIntegrate[(c + d*x)^m*Csc[a + b*x]^3*Sec[a + b*x]^3, x]} - -{(c + d*x)^3*Csc[a + b*x]^3*Sec[a + b*x]^3, x, 16, -((6*d^2*(c + d*x)*ArcTanh[E^(2*I*(a + b*x))])/b^3) - (4*(c + d*x)^3*ArcTanh[E^(2*I*(a + b*x))])/b - (3*d*(c + d*x)^2*Csc[2*a + 2*b*x])/b^2 - (2*(c + d*x)^3*Cot[2*a + 2*b*x]*Csc[2*a + 2*b*x])/b + (3*I*d^3*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^4) + (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (3*I*d^3*PolyLog[2, E^(2*I*(a + b*x))])/(2*b^4) - (3*I*d*(c + d*x)^2*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, E^(2*I*(a + b*x))])/b^3 - (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(2*b^4) + (3*I*d^3*PolyLog[4, E^(2*I*(a + b*x))])/(2*b^4)} -{(c + d*x)^2*Csc[a + b*x]^3*Sec[a + b*x]^3, x, 10, -((4*(c + d*x)^2*ArcTanh[E^(2*I*(a + b*x))])/b) - (d^2*ArcTanh[Cos[2*a + 2*b*x]])/b^3 - (2*d*(c + d*x)*Csc[2*a + 2*b*x])/b^2 - (2*(c + d*x)^2*Cot[2*a + 2*b*x]*Csc[2*a + 2*b*x])/b + (2*I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (2*I*d*(c + d*x)*PolyLog[2, E^(2*I*(a + b*x))])/b^2 - (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 + (d^2*PolyLog[3, E^(2*I*(a + b*x))])/b^3} -{(c + d*x)^1*Csc[a + b*x]^3*Sec[a + b*x]^3, x, 7, -((4*(c + d*x)*ArcTanh[E^(2*I*(a + b*x))])/b) - (d*Csc[2*a + 2*b*x])/b^2 - (2*(c + d*x)*Cot[2*a + 2*b*x]*Csc[2*a + 2*b*x])/b + (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 - (I*d*PolyLog[2, E^(2*I*(a + b*x))])/b^2} -{(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x)^1, x, 1, 8*Unintegrable[Csc[2*a + 2*b*x]^3/(c + d*x), x]} -{(Csc[a + b*x]^3*Sec[a + b*x]^3)/(c + d*x)^2, x, 1, 8*Unintegrable[Csc[2*a + 2*b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Sec[a+b x]^3 Sin[a+b x]^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^(n/2) Sin[a+b x]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^(n/2) Sin[a+b x]^1*) - - -{x*Cos[a + b*x]^(5/2)*Sin[a + b*x], x, 4, (-2*x*Cos[a + b*x]^(7/2))/(7*b) + (20*EllipticF[(a + b*x)/2, 2])/(147*b^2) + (20*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(147*b^2) + (4*Cos[a + b*x]^(5/2)*Sin[a + b*x])/(49*b^2)} -{x*Cos[a + b*x]^(3/2)*Sin[a + b*x], x, 3, (-2*x*Cos[a + b*x]^(5/2))/(5*b) + (12*EllipticE[(a + b*x)/2, 2])/(25*b^2) + (4*Cos[a + b*x]^(3/2)*Sin[a + b*x])/(25*b^2)} -{x*Sqrt[Cos[a + b*x]]*Sin[a + b*x], x, 3, (-2*x*Cos[a + b*x]^(3/2))/(3*b) + (4*EllipticF[(a + b*x)/2, 2])/(9*b^2) + (4*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(9*b^2)} -{(x*Sin[a + b*x])/Sqrt[Cos[a + b*x]], x, 2, (-2*x*Sqrt[Cos[a + b*x]])/b + (4*EllipticE[(a + b*x)/2, 2])/b^2} -{(x*Sin[a + b*x])/Cos[a + b*x]^(3/2), x, 2, (2*x)/(b*Sqrt[Cos[a + b*x]]) - (4*EllipticF[(a + b*x)/2, 2])/b^2} -{(x*Sin[a + b*x])/Cos[a + b*x]^(5/2), x, 3, (2*x)/(3*b*Cos[a + b*x]^(3/2)) + (4*EllipticE[(a + b*x)/2, 2])/(3*b^2) - (4*Sin[a + b*x])/(3*b^2*Sqrt[Cos[a + b*x]])} -{(x*Sin[a + b*x])/Cos[a + b*x]^(7/2), x, 3, (2*x)/(5*b*Cos[a + b*x]^(5/2)) - (4*EllipticF[(a + b*x)/2, 2])/(15*b^2) - (4*Sin[a + b*x])/(15*b^2*Cos[a + b*x]^(3/2))} -{(x*Sin[a + b*x])/Cos[a + b*x]^(9/2), x, 4, (2*x)/(7*b*Cos[a + b*x]^(7/2)) + (12*EllipticE[(a + b*x)/2, 2])/(35*b^2) - (4*Sin[a + b*x])/(35*b^2*Cos[a + b*x]^(5/2)) - (12*Sin[a + b*x])/(35*b^2*Sqrt[Cos[a + b*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^(n/2) Sin[a+b x]^1*) - - -{x*Sec[a + b*x]^(9/2)*Sin[a + b*x], x, 5, (12*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(35*b^2) + (2*x*Sec[a + b*x]^(7/2))/(7*b) - (12*Sqrt[Sec[a + b*x]]*Sin[a + b*x])/(35*b^2) - (4*Sec[a + b*x]^(5/2)*Sin[a + b*x])/(35*b^2)} -{x*Sec[a + b*x]^(7/2)*Sin[a + b*x], x, 4, (-4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(15*b^2) + (2*x*Sec[a + b*x]^(5/2))/(5*b) - (4*Sec[a + b*x]^(3/2)*Sin[a + b*x])/(15*b^2)} -{x*Sec[a + b*x]^(5/2)*Sin[a + b*x], x, 4, (4*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(3*b^2) + (2*x*Sec[a + b*x]^(3/2))/(3*b) - (4*Sqrt[Sec[a + b*x]]*Sin[a + b*x])/(3*b^2)} -{x*Sec[a + b*x]^(3/2)*Sin[a + b*x], x, 3, (2*x*Sqrt[Sec[a + b*x]])/b - (4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/b^2} -{x*Sqrt[Sec[a + b*x]]*Sin[a + b*x], x, 3, (-2*x)/(b*Sqrt[Sec[a + b*x]]) + (4*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/b^2} -{(x*Sin[a + b*x])/Sqrt[Sec[a + b*x]], x, 4, (-2*x)/(3*b*Sec[a + b*x]^(3/2)) + (4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(9*b^2) + (4*Sin[a + b*x])/(9*b^2*Sqrt[Sec[a + b*x]])} -{(x*Sin[a + b*x])/Sec[a + b*x]^(3/2), x, 4, (-2*x)/(5*b*Sec[a + b*x]^(5/2)) + (12*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(25*b^2) + (4*Sin[a + b*x])/(25*b^2*Sec[a + b*x]^(3/2))} -{(x*Sin[a + b*x])/Sec[a + b*x]^(5/2), x, 5, (-2*x)/(7*b*Sec[a + b*x]^(7/2)) + (20*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(147*b^2) + (4*Sin[a + b*x])/(49*b^2*Sec[a + b*x]^(5/2)) + (20*Sin[a + b*x])/(147*b^2*Sqrt[Sec[a + b*x]])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^n Sin[a+b x]^(p/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^1 Sin[a+b x]^(p/2)*) - - -{x*Cos[a + b*x]*Sin[a + b*x]^(5/2), x, 4, -((20*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(147*b^2)) + (20*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(147*b^2) + (4*Cos[a + b*x]*Sin[a + b*x]^(5/2))/(49*b^2) + (2*x*Sin[a + b*x]^(7/2))/(7*b)} -{x*Cos[a + b*x]*Sin[a + b*x]^(3/2), x, 3, -((12*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(25*b^2)) + (4*Cos[a + b*x]*Sin[a + b*x]^(3/2))/(25*b^2) + (2*x*Sin[a + b*x]^(5/2))/(5*b)} -{x*Cos[a + b*x]*Sqrt[Sin[a + b*x]], x, 3, -((4*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(9*b^2)) + (4*Cos[a + b*x]*Sqrt[Sin[a + b*x]])/(9*b^2) + (2*x*Sin[a + b*x]^(3/2))/(3*b)} -{(x*Cos[a + b*x])/Sqrt[Sin[a + b*x]], x, 2, -((4*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/b^2) + (2*x*Sqrt[Sin[a + b*x]])/b} -{(x*Cos[a + b*x])/Sin[a + b*x]^(3/2), x, 2, (4*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/b^2 - (2*x)/(b*Sqrt[Sin[a + b*x]])} -{(x*Cos[a + b*x])/Sin[a + b*x]^(5/2), x, 3, -((4*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(3*b^2)) - (2*x)/(3*b*Sin[a + b*x]^(3/2)) - (4*Cos[a + b*x])/(3*b^2*Sqrt[Sin[a + b*x]])} -{(x*Cos[a + b*x])/Sin[a + b*x]^(7/2), x, 3, (4*EllipticF[(1/2)*(a - Pi/2 + b*x), 2])/(15*b^2) - (2*x)/(5*b*Sin[a + b*x]^(5/2)) - (4*Cos[a + b*x])/(15*b^2*Sin[a + b*x]^(3/2))} -{(x*Cos[a + b*x])/Sin[a + b*x]^(9/2), x, 4, -((12*EllipticE[(1/2)*(a - Pi/2 + b*x), 2])/(35*b^2)) - (2*x)/(7*b*Sin[a + b*x]^(7/2)) - (4*Cos[a + b*x])/(35*b^2*Sin[a + b*x]^(5/2)) - (12*Cos[a + b*x])/(35*b^2*Sqrt[Sin[a + b*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Cos[a+b x]^1 Csc[a+b x]^(p/2)*) - - -{x*Cos[a + b*x]*Csc[a + b*x]^(9/2), x, 5, -((12*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(35*b^2)) - (4*Cos[a + b*x]*Csc[a + b*x]^(5/2))/(35*b^2) - (2*x*Csc[a + b*x]^(7/2))/(7*b) - (12*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(35*b^2)} -{x*Cos[a + b*x]*Csc[a + b*x]^(7/2), x, 4, -((4*Cos[a + b*x]*Csc[a + b*x]^(3/2))/(15*b^2)) - (2*x*Csc[a + b*x]^(5/2))/(5*b) + (4*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(15*b^2)} -{x*Cos[a + b*x]*Csc[a + b*x]^(5/2), x, 4, -((4*Cos[a + b*x]*Sqrt[Csc[a + b*x]])/(3*b^2)) - (2*x*Csc[a + b*x]^(3/2))/(3*b) - (4*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(3*b^2)} -{x*Cos[a + b*x]*Csc[a + b*x]^(3/2), x, 3, -((2*x*Sqrt[Csc[a + b*x]])/b) + (4*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b^2} -{x*Cos[a + b*x]*Sqrt[Csc[a + b*x]], x, 3, (2*x)/(b*Sqrt[Csc[a + b*x]]) - (4*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/b^2} -{(x*Cos[a + b*x])/Sqrt[Csc[a + b*x]], x, 4, (2*x)/(3*b*Csc[a + b*x]^(3/2)) + (4*Cos[a + b*x])/(9*b^2*Sqrt[Csc[a + b*x]]) - (4*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(9*b^2)} -{(x*Cos[a + b*x])/Csc[a + b*x]^(3/2), x, 4, (2*x)/(5*b*Csc[a + b*x]^(5/2)) + (4*Cos[a + b*x])/(25*b^2*Csc[a + b*x]^(3/2)) - (12*Sqrt[Csc[a + b*x]]*EllipticE[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(25*b^2)} -{(x*Cos[a + b*x])/Csc[a + b*x]^(5/2), x, 5, (2*x)/(7*b*Csc[a + b*x]^(7/2)) + (4*Cos[a + b*x])/(49*b^2*Csc[a + b*x]^(5/2)) + (20*Cos[a + b*x])/(147*b^2*Sqrt[Csc[a + b*x]]) - (20*Sqrt[Csc[a + b*x]]*EllipticF[(1/2)*(a - Pi/2 + b*x), 2]*Sqrt[Sin[a + b*x]])/(147*b^2)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (c+d x)^m Trig[a+b x]^n Trig[n (a+b x)]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Csc[a+b x]^n Sin[n (a+b x)]^p*) - - -{x*Csc[x]*Sin[3*x], x, 6, x^2/2 + (3*Cos[x]^2)/4 + 2*x*Cos[x]*Sin[x] - Sin[x]^2/4} - - -{(c + d*x)^4*Csc[x]*Sin[3*x], x, 14, (3*d^4*x)/2 - d*(c + d*x)^3 + (c + d*x)^5/(5*d) - (9/2)*d^3*(c + d*x)*Cos[x]^2 + 3*d*(c + d*x)^3*Cos[x]^2 + 3*d^4*Cos[x]*Sin[x] - 6*d^2*(c + d*x)^2*Cos[x]*Sin[x] + 2*(c + d*x)^4*Cos[x]*Sin[x] + (3/2)*d^3*(c + d*x)*Sin[x]^2 - d*(c + d*x)^3*Sin[x]^2} -{(c + d*x)^3*Csc[x]*Sin[3*x], x, 10, (-(3/2))*c*d^2*x - (3*d^3*x^2)/4 + (c + d*x)^4/(4*d) - (9/8)*d^3*Cos[x]^2 + (9/4)*d*(c + d*x)^2*Cos[x]^2 - 3*d^2*(c + d*x)*Cos[x]*Sin[x] + 2*(c + d*x)^3*Cos[x]*Sin[x] + (3/8)*d^3*Sin[x]^2 - (3/4)*d*(c + d*x)^2*Sin[x]^2} -{(c + d*x)^2*Csc[x]*Sin[3*x], x, 10, -((d^2*x)/2) + (c + d*x)^3/(3*d) + (3/2)*d*(c + d*x)*Cos[x]^2 - d^2*Cos[x]*Sin[x] + 2*(c + d*x)^2*Cos[x]*Sin[x] - (1/2)*d*(c + d*x)*Sin[x]^2} -{(c + d*x)^1*Csc[x]*Sin[3*x], x, 6, c*x + (d*x^2)/2 + (3/4)*d*Cos[x]^2 + 2*(c + d*x)*Cos[x]*Sin[x] - (1/4)*d*Sin[x]^2} -{Csc[x]*Sin[3*x]/(c + d*x)^1, x, 12, (2*Cos[(2*c)/d]*CosIntegral[(2*c)/d + 2*x])/d + Log[c + d*x]/d + (2*Sin[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d} -{Csc[x]*Sin[3*x]/(c + d*x)^2, x, 12, -((3*Cos[x]^2)/(d*(c + d*x))) + (4*CosIntegral[(2*c)/d + 2*x]*Sin[(2*c)/d])/d^2 + Sin[x]^2/(d*(c + d*x)) - (4*Cos[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d^2} -{Csc[x]*Sin[3*x]/(c + d*x)^3, x, 16, -((3*Cos[x]^2)/(2*d*(c + d*x)^2)) - (4*Cos[(2*c)/d]*CosIntegral[(2*c)/d + 2*x])/d^3 + (4*Cos[x]*Sin[x])/(d^2*(c + d*x)) + Sin[x]^2/(2*d*(c + d*x)^2) - (4*Sin[(2*c)/d]*SinIntegral[(2*c)/d + 2*x])/d^3} - - -{(c + d*x)^4*Csc[a + b*x]*Sin[3*a + 3*b*x], x, 14, (3*d^4*x)/(2*b^4) - (d*(c + d*x)^3)/b^2 + (c + d*x)^5/(5*d) - (9*d^3*(c + d*x)*Cos[a + b*x]^2)/(2*b^4) + (3*d*(c + d*x)^3*Cos[a + b*x]^2)/b^2 + (3*d^4*Cos[a + b*x]*Sin[a + b*x])/b^5 - (6*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/b^3 + (2*(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x])/b + (3*d^3*(c + d*x)*Sin[a + b*x]^2)/(2*b^4) - (d*(c + d*x)^3*Sin[a + b*x]^2)/b^2} -{(c + d*x)^3*Csc[a + b*x]*Sin[3*a + 3*b*x], x, 10, -((3*c*d^2*x)/(2*b^2)) - (3*d^3*x^2)/(4*b^2) + (c + d*x)^4/(4*d) - (9*d^3*Cos[a + b*x]^2)/(8*b^4) + (9*d*(c + d*x)^2*Cos[a + b*x]^2)/(4*b^2) - (3*d^2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^3 + (2*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/b + (3*d^3*Sin[a + b*x]^2)/(8*b^4) - (3*d*(c + d*x)^2*Sin[a + b*x]^2)/(4*b^2)} -{(c + d*x)^2*Csc[a + b*x]*Sin[3*a + 3*b*x], x, 10, -((d^2*x)/(2*b^2)) + (c + d*x)^3/(3*d) + (3*d*(c + d*x)*Cos[a + b*x]^2)/(2*b^2) - (d^2*Cos[a + b*x]*Sin[a + b*x])/b^3 + (2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/b - (d*(c + d*x)*Sin[a + b*x]^2)/(2*b^2)} -{(c + d*x)^1*Csc[a + b*x]*Sin[3*a + 3*b*x], x, 6, c*x + (d*x^2)/2 + (3*d*Cos[a + b*x]^2)/(4*b^2) + (2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b - (d*Sin[a + b*x]^2)/(4*b^2)} -{Csc[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^1, x, 12, (2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d + Log[c + d*x]/d - (2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d} -{Csc[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^2, x, 12, -((3*Cos[a + b*x]^2)/(d*(c + d*x))) - (4*b*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^2 + Sin[a + b*x]^2/(d*(c + d*x)) - (4*b*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Csc[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^3, x, 16, -((3*Cos[a + b*x]^2)/(2*d*(c + d*x)^2)) - (4*b^2*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^3 + (4*b*Cos[a + b*x]*Sin[a + b*x])/(d^2*(c + d*x)) + Sin[a + b*x]^2/(2*d*(c + d*x)^2) + (4*b^2*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^3} -{Csc[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^4, x, 16, -((2*b^2)/(3*d^3*(c + d*x))) - Cos[a + b*x]^2/(d*(c + d*x)^3) + (2*b^2*Cos[a + b*x]^2)/(d^3*(c + d*x)) + (8*b^3*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(3*d^4) + (4*b*Cos[a + b*x]*Sin[a + b*x])/(3*d^2*(c + d*x)^2) + Sin[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*Sin[a + b*x]^2)/(3*d^3*(c + d*x)) + (8*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)} - - -{(c + d*x)^3*Csc[a + b*x]^2*Sin[3*a + 3*b*x], x, 20, -((6*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b) - (24*d^2*(c + d*x)*Cos[a + b*x])/b^3 + (4*(c + d*x)^3*Cos[a + b*x])/b + (9*I*d*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (9*I*d*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^2 - (18*d^2*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (18*d^2*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^3 - (18*I*d^3*PolyLog[4, -E^(I*(a + b*x))])/b^4 + (18*I*d^3*PolyLog[4, E^(I*(a + b*x))])/b^4 + (24*d^3*Sin[a + b*x])/b^4 - (12*d*(c + d*x)^2*Sin[a + b*x])/b^2} -{(c + d*x)^2*Csc[a + b*x]^2*Sin[3*a + 3*b*x], x, 16, -((6*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b) - (8*d^2*Cos[a + b*x])/b^3 + (4*(c + d*x)^2*Cos[a + b*x])/b + (6*I*d*(c + d*x)*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (6*I*d*(c + d*x)*PolyLog[2, E^(I*(a + b*x))])/b^2 - (6*d^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (6*d^2*PolyLog[3, E^(I*(a + b*x))])/b^3 - (8*d*(c + d*x)*Sin[a + b*x])/b^2} -{(c + d*x)^1*Csc[a + b*x]^2*Sin[3*a + 3*b*x], x, 12, -((6*(c + d*x)*ArcTanh[E^(I*(a + b*x))])/b) + (4*(c + d*x)*Cos[a + b*x])/b + (3*I*d*PolyLog[2, -E^(I*(a + b*x))])/b^2 - (3*I*d*PolyLog[2, E^(I*(a + b*x))])/b^2 - (4*d*Sin[a + b*x])/b^2} -{Csc[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^1, x, 9, 3*Unintegrable[Csc[a + b*x]/(c + d*x), x] - (4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d - (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{Csc[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^2, x, 11, -((4*b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2) + 3*Unintegrable[Csc[a + b*x]/(c + d*x)^2, x] + (4*Sin[a + b*x])/(d*(c + d*x)) + (4*b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} -{Csc[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^3, x, 13, (2*b*Cos[a + b*x])/(d^2*(c + d*x)) + 3*Unintegrable[Csc[a + b*x]/(c + d*x)^3, x] + (2*b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^3 + (2*Sin[a + b*x])/(d*(c + d*x)^2) + (2*b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^3} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^n Sin[n (a+b x)]^p*) - - -{(c + d*x)^4*Sec[a + b*x]*Sin[3*a + 3*b*x], x, 20, (6*c*d^3*x)/b^3 + (3*d^4*x^2)/b^3 - (c + d*x)^4/b - (I*(c + d*x)^5)/(5*d) + ((c + d*x)^4*Log[1 + E^(2*I*(a + b*x))])/b - (2*I*d*(c + d*x)^3*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 + (3*d^2*(c + d*x)^2*PolyLog[3, -E^(2*I*(a + b*x))])/b^3 + (3*I*d^3*(c + d*x)*PolyLog[4, -E^(2*I*(a + b*x))])/b^4 - (3*d^4*PolyLog[5, -E^(2*I*(a + b*x))])/(2*b^5) - (6*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^4 + (4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/b^2 + (3*d^4*Sin[a + b*x]^2)/b^5 - (6*d^2*(c + d*x)^2*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^4*Sin[a + b*x]^2)/b} -{(c + d*x)^3*Sec[a + b*x]*Sin[3*a + 3*b*x], x, 19, (3*d^3*x)/(2*b^3) - (c + d*x)^3/b - (I*(c + d*x)^4)/(4*d) + ((c + d*x)^3*Log[1 + E^(2*I*(a + b*x))])/b - (3*I*d*(c + d*x)^2*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) + (3*d^2*(c + d*x)*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (3*I*d^3*PolyLog[4, -E^(2*I*(a + b*x))])/(4*b^4) - (3*d^3*Cos[a + b*x]*Sin[a + b*x])/(2*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/b^2 - (3*d^2*(c + d*x)*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^3*Sin[a + b*x]^2)/b} -{(c + d*x)^2*Sec[a + b*x]*Sin[3*a + 3*b*x], x, 14, -((2*c*d*x)/b) - (d^2*x^2)/b - (I*(c + d*x)^3)/(3*d) + ((c + d*x)^2*Log[1 + E^(2*I*(a + b*x))])/b - (I*d*(c + d*x)*PolyLog[2, -E^(2*I*(a + b*x))])/b^2 + (d^2*PolyLog[3, -E^(2*I*(a + b*x))])/(2*b^3) + (2*d*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^2 - (d^2*Sin[a + b*x]^2)/b^3 + (2*(c + d*x)^2*Sin[a + b*x]^2)/b} -{(c + d*x)^1*Sec[a + b*x]*Sin[3*a + 3*b*x], x, 13, -((d*x)/b) - (I*(c + d*x)^2)/(2*d) + ((c + d*x)*Log[1 + E^(2*I*(a + b*x))])/b - (I*d*PolyLog[2, -E^(2*I*(a + b*x))])/(2*b^2) + (d*Cos[a + b*x]*Sin[a + b*x])/b^2 + (2*(c + d*x)*Sin[a + b*x]^2)/b} -{Sec[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^1, x, 13, -Unintegrable[Tan[a + b*x]/(c + d*x), x] + (2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d + (2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d} -{Sec[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^2, x, 15, (4*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/d^2 - Unintegrable[Tan[a + b*x]/(c + d*x)^2, x] - (2*Sin[2*a + 2*b*x])/(d*(c + d*x)) - (4*b*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Sec[a + b*x]*Sin[3*a + 3*b*x]/(c + d*x)^3, x, 17, -((2*b*Cos[2*a + 2*b*x])/(d^2*(c + d*x))) - Unintegrable[Tan[a + b*x]/(c + d*x)^3, x] - (4*b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d^3 - Sin[2*a + 2*b*x]/(d*(c + d*x)^2) - (4*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d^3} - - -{(c + d*x)^3*Sec[a + b*x]^2*Sin[3*a + 3*b*x], x, 19, -((6*I*d*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b^2) + (24*d^2*(c + d*x)*Cos[a + b*x])/b^3 - (4*(c + d*x)^3*Cos[a + b*x])/b + (6*I*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - (6*I*d^2*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 + (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 - ((c + d*x)^3*Sec[a + b*x])/b - (24*d^3*Sin[a + b*x])/b^4 + (12*d*(c + d*x)^2*Sin[a + b*x])/b^2} -{(c + d*x)^2*Sec[a + b*x]^2*Sin[3*a + 3*b*x], x, 15, -((4*I*d*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b^2) + (8*d^2*Cos[a + b*x])/b^3 - (4*(c + d*x)^2*Cos[a + b*x])/b + (2*I*d^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 - (2*I*d^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 - ((c + d*x)^2*Sec[a + b*x])/b + (8*d*(c + d*x)*Sin[a + b*x])/b^2} -{(c + d*x)^1*Sec[a + b*x]^2*Sin[3*a + 3*b*x], x, 9, (d*ArcTanh[Sin[a + b*x]])/b^2 - (4*(c + d*x)*Cos[a + b*x])/b - ((c + d*x)*Sec[a + b*x])/b + (4*d*Sin[a + b*x])/b^2} -{Sec[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^1, x, 9, -CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x), x] + (4*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d + (4*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d} -{Sec[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^2, x, 11, (4*b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2 - CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^2, x] - (4*Sin[a + b*x])/(d*(c + d*x)) - (4*b*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^2} -{Sec[a + b*x]^2*Sin[3*a + 3*b*x]/(c + d*x)^3, x, 13, -((2*b*Cos[a + b*x])/(d^2*(c + d*x))) - CannotIntegrate[(Sec[a + b*x]*Tan[a + b*x])/(c + d*x)^3, x] - (2*b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d^3 - (2*Sin[a + b*x])/(d*(c + d*x)^2) - (2*b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d^3} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Sec[a+b x]^n Cos[n (a+b x)]^p*) - - -{x*Cos[2*x]*Sec[x], x, 12, 2*I*x*ArcTan[E^(I*x)] + 2*Cos[x] - I*PolyLog[2, (-I)*E^(I*x)] + I*PolyLog[2, I*E^(I*x)] + 2*x*Sin[x]} -{x*Cos[2*x]*Sec[x]^2, x, 5, x^2 - Log[Cos[x]] - x*Tan[x]} -{x*Cos[2*x]*Sec[x]^3, x, 19, -3*I*x*ArcTan[E^(I*x)] + (3/2)*I*PolyLog[2, (-I)*E^(I*x)] - (3/2)*I*PolyLog[2, I*E^(I*x)] + Sec[x]/2 - (1/2)*x*Sec[x]*Tan[x]} - - -(* ::Section:: *) -(*Integrands of the form (c+d x)^m Csc[a+b x]^n Cos[n (a+b x)]^p*) diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.4 x^m (a+b trig^n)^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.4 x^m (a+b trig^n)^p.m deleted file mode 100644 index 377cf05..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.4 x^m (a+b trig^n)^p.m +++ /dev/null @@ -1,39 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form x^m (a+b trig(c+d x)^n)^p*) - - -(* ::Section:: *) -(*Integrands of the form x^m (a+b Sin[c+d x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sin[c+d x]^2)^p*) - - -{x/(a + b*Sin[x]^2), x, 9, -((I*x*Log[1 - (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x*Log[1 - (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - PolyLog[2, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])]/(4*Sqrt[a]*Sqrt[a + b]) + PolyLog[2, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])]/(4*Sqrt[a]*Sqrt[a + b])} -{x^2/(a + b*Sin[x]^2), x, 11, -((I*x^2*Log[1 - (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x^2*Log[1 - (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (I*PolyLog[3, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b]) + (I*PolyLog[3, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b])} -{x^3/(a + b*Sin[x]^2), x, 13, -((I*x^3*Log[1 - (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x^3*Log[1 - (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (3*x^2*PolyLog[2, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b]) + (3*x^2*PolyLog[2, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b]) - (3*I*x*PolyLog[3, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b]) + (3*I*x*PolyLog[3, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(4*Sqrt[a]*Sqrt[a + b]) + (3*PolyLog[4, (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(8*Sqrt[a]*Sqrt[a + b]) - (3*PolyLog[4, (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(8*Sqrt[a]*Sqrt[a + b])} - - -{x/(a + b*Sin[c + d*x]^2)^2, x, 12, -((I*(2*a + b)*x*Log[1 - (b*E^(2*I*(c + d*x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(4*a^(3/2)*(a + b)^(3/2)*d)) + (I*(2*a + b)*x*Log[1 - (b*E^(2*I*(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(4*a^(3/2)*(a + b)^(3/2)*d) - Log[2*a + b - b*Cos[2*c + 2*d*x]]/(4*a*(a + b)*d^2) - ((2*a + b)*PolyLog[2, (b*E^(2*I*(c + d*x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(8*a^(3/2)*(a + b)^(3/2)*d^2) + ((2*a + b)*PolyLog[2, (b*E^(2*I*(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(8*a^(3/2)*(a + b)^(3/2)*d^2) + (b*x*Sin[2*c + 2*d*x])/(2*a*(a + b)*d*(2*a + b - b*Cos[2*c + 2*d*x]))} - - -{x*Sqrt[Sin[x]^2], x, 3, Sqrt[Sin[x]^2] - x*Cot[x]*Sqrt[Sin[x]^2]} - - -(* ::Section:: *) -(*Integrands of the form x^m (a+b Cos[c+d x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Cos[c+d x]^2)^p*) - - -{x/(a + b*Cos[x]^2), x, 9, -((I*x*Log[1 + (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x*Log[1 + (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - PolyLog[2, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b]) + PolyLog[2, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])} -{x^2/(a + b*Cos[x]^2), x, 11, -((I*x^2*Log[1 + (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x^2*Log[1 + (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) - (I*PolyLog[3, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (I*PolyLog[3, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b])} -{x^3/(a + b*Cos[x]^2), x, 13, -((I*x^3*Log[1 + (b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b])) + (I*x^3*Log[1 + (b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (3*x^2*PolyLog[2, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*x^2*PolyLog[2, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) - (3*I*x*PolyLog[3, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*I*x*PolyLog[3, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*PolyLog[4, -((b*E^(2*I*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b]) - (3*PolyLog[4, -((b*E^(2*I*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b])} - - -{x/(a + b*Cos[c + d*x]^2)^2, x, 12, -((I*(2*a + b)*x*Log[1 + (b*E^(2*I*(c + d*x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(4*a^(3/2)*(a + b)^(3/2)*d)) + (I*(2*a + b)*x*Log[1 + (b*E^(2*I*(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(4*a^(3/2)*(a + b)^(3/2)*d) - Log[2*a + b + b*Cos[2*c + 2*d*x]]/(4*a*(a + b)*d^2) - ((2*a + b)*PolyLog[2, -((b*E^(2*I*(c + d*x)))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(8*a^(3/2)*(a + b)^(3/2)*d^2) + ((2*a + b)*PolyLog[2, -((b*E^(2*I*(c + d*x)))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*a^(3/2)*(a + b)^(3/2)*d^2) - (b*x*Sin[2*c + 2*d*x])/(2*a*(a + b)*d*(2*a + b + b*Cos[2*c + 2*d*x]))} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.5 x^m trig(a+b log(c x^n))^p.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.5 x^m trig(a+b log(c x^n))^p.m deleted file mode 100644 index cb180a0..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.5 x^m trig(a+b log(c x^n))^p.m +++ /dev/null @@ -1,741 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Problems for integrands of the form (e x)^m Trig[d (a+b Log[c x^n])]^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p*) - - -{x^2*Sin[a + b*Log[c*x^n]], x, 1, -((b*n*x^3*Cos[a + b*Log[c*x^n]])/(9 + b^2*n^2)) + (3*x^3*Sin[a + b*Log[c*x^n]])/(9 + b^2*n^2)} -{x^1*Sin[a + b*Log[c*x^n]], x, 1, -((b*n*x^2*Cos[a + b*Log[c*x^n]])/(4 + b^2*n^2)) + (2*x^2*Sin[a + b*Log[c*x^n]])/(4 + b^2*n^2)} -{x^0*Sin[a + b*Log[c*x^n]], x, 1, -((b*n*x*Cos[a + b*Log[c*x^n]])/(1 + b^2*n^2)) + (x*Sin[a + b*Log[c*x^n]])/(1 + b^2*n^2)} -{Sin[a + b*Log[c*x^n]]/x^1, x, 2, -(Cos[a + b*Log[c*x^n]]/(b*n))} -{Sin[a + b*Log[c*x^n]]/x^2, x, 1, -((b*n*Cos[a + b*Log[c*x^n]])/((1 + b^2*n^2)*x)) - Sin[a + b*Log[c*x^n]]/((1 + b^2*n^2)*x)} -{Sin[a + b*Log[c*x^n]]/x^3, x, 1, -((b*n*Cos[a + b*Log[c*x^n]])/((4 + b^2*n^2)*x^2)) - (2*Sin[a + b*Log[c*x^n]])/((4 + b^2*n^2)*x^2)} - - -{x^2*Sin[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x^3)/(3*(9 + 4*b^2*n^2)) - (2*b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(9 + 4*b^2*n^2) + (3*x^3*Sin[a + b*Log[c*x^n]]^2)/(9 + 4*b^2*n^2)} -{x^1*Sin[a + b*Log[c*x^n]]^2, x, 2, (b^2*n^2*x^2)/(4*(1 + b^2*n^2)) - (b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*(1 + b^2*n^2)) + (x^2*Sin[a + b*Log[c*x^n]]^2)/(2*(1 + b^2*n^2))} -{x^0*Sin[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x)/(1 + 4*b^2*n^2) - (2*b*n*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 + 4*b^2*n^2) + (x*Sin[a + b*Log[c*x^n]]^2)/(1 + 4*b^2*n^2)} -{Sin[a + b*Log[c*x^n]]^2/x^1, x, 3, Log[x]/2 - (Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*b*n)} -{Sin[a + b*Log[c*x^n]]^2/x^2, x, 2, -((2*b^2*n^2)/((1 + 4*b^2*n^2)*x)) - (2*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1 + 4*b^2*n^2)*x) - Sin[a + b*Log[c*x^n]]^2/((1 + 4*b^2*n^2)*x)} -{Sin[a + b*Log[c*x^n]]^2/x^3, x, 2, -((b^2*n^2)/(4*(1 + b^2*n^2)*x^2)) - (b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*(1 + b^2*n^2)*x^2) - Sin[a + b*Log[c*x^n]]^2/(2*(1 + b^2*n^2)*x^2)} - - -{x^2*Sin[a + b*Log[c*x^n]]^3, x, 2, -((2*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]])/(3*(9 + 10*b^2*n^2 + b^4*n^4))) + (2*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]])/(9 + 10*b^2*n^2 + b^4*n^4) - (b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(3*(1 + b^2*n^2)) + (x^3*Sin[a + b*Log[c*x^n]]^3)/(3*(1 + b^2*n^2))} -{x^1*Sin[a + b*Log[c*x^n]]^3, x, 2, -((6*b^3*n^3*x^2*Cos[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4)) + (12*b^2*n^2*x^2*Sin[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(4 + 9*b^2*n^2) + (2*x^2*Sin[a + b*Log[c*x^n]]^3)/(4 + 9*b^2*n^2)} -{x^0*Sin[a + b*Log[c*x^n]]^3, x, 2, -((6*b^3*n^3*x*Cos[a + b*Log[c*x^n]])/(1 + 10*b^2*n^2 + 9*b^4*n^4)) + (6*b^2*n^2*x*Sin[a + b*Log[c*x^n]])/(1 + 10*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(1 + 9*b^2*n^2) + (x*Sin[a + b*Log[c*x^n]]^3)/(1 + 9*b^2*n^2)} -{Sin[a + b*Log[c*x^n]]^3/x^1, x, 3, -(Cos[a + b*Log[c*x^n]]/(b*n)) + Cos[a + b*Log[c*x^n]]^3/(3*b*n)} -{Sin[a + b*Log[c*x^n]]^3/x^2, x, 2, -((6*b^3*n^3*Cos[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x)) - (6*b^2*n^2*Sin[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x) - (3*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/((1 + 9*b^2*n^2)*x) - Sin[a + b*Log[c*x^n]]^3/((1 + 9*b^2*n^2)*x)} -{Sin[a + b*Log[c*x^n]]^3/x^3, x, 2, -((6*b^3*n^3*Cos[a + b*Log[c*x^n]])/((16 + 40*b^2*n^2 + 9*b^4*n^4)*x^2)) - (12*b^2*n^2*Sin[a + b*Log[c*x^n]])/((16 + 40*b^2*n^2 + 9*b^4*n^4)*x^2) - (3*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/((4 + 9*b^2*n^2)*x^2) - (2*Sin[a + b*Log[c*x^n]]^3)/((4 + 9*b^2*n^2)*x^2)} - - -{x^2*Sin[a + b*Log[c*x^n]]^4, x, 3, (8*b^4*n^4*x^3)/(81 + 180*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(81 + 180*b^2*n^2 + 64*b^4*n^4) + (36*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]]^2)/(81 + 180*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(9 + 16*b^2*n^2) + (3*x^3*Sin[a + b*Log[c*x^n]]^4)/(9 + 16*b^2*n^2)} -{x^1*Sin[a + b*Log[c*x^n]]^4, x, 3, (3*b^4*n^4*x^2)/(4*(1 + 5*b^2*n^2 + 4*b^4*n^4)) - (3*b^3*n^3*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)) + (3*b^2*n^2*x^2*Sin[a + b*Log[c*x^n]]^2)/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)) - (b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(1 + 4*b^2*n^2) + (x^2*Sin[a + b*Log[c*x^n]]^4)/(2*(1 + 4*b^2*n^2))} -{x^0*Sin[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x)/(1 + 20*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Sin[a + b*Log[c*x^n]]^2)/(1 + 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(1 + 16*b^2*n^2) + (x*Sin[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2)} -{Sin[a + b*Log[c*x^n]]^4/x^1, x, 4, 3*Log[x]/8 - (3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(8*b*n) - (Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(4*b*n)} -{Sin[a + b*Log[c*x^n]]^4/x^2, x, 3, -((24*b^4*n^4)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x)) - (24*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (12*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (4*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 16*b^2*n^2)*x) - Sin[a + b*Log[c*x^n]]^4/((1 + 16*b^2*n^2)*x)} -{Sin[a + b*Log[c*x^n]]^4/x^3, x, 3, -((3*b^4*n^4)/(4*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2)) - (3*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (3*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 4*b^2*n^2)*x^2) - Sin[a + b*Log[c*x^n]]^4/(2*(1 + 4*b^2*n^2)*x^2)} - - -{Sin[Log[a + b*x]], x, 2, -(((a + b*x)*Cos[Log[a + b*x]])/(2*b)) + ((a + b*x)*Sin[Log[a + b*x]])/(2*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when b^2 n^2 p^2+(m+1)^2=0*) - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, -((E^((a*(1 + m))/(Sqrt[-((1 + m)^2/n^2)]*n))*x^(1 + m)*(c*x^n)^((1 + m)/n))/(4*Sqrt[-((1 + m)^2/n^2)]*n)) + (E^((a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*(1 + m)*x^(1 + m)*Log[x])/((c*x^n)^((1 + m)/n)*(2*Sqrt[-((1 + m)^2/n^2)]*n))} - -{x^2*Sin[a + Sqrt[-(2+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, ((1/12)*Sqrt[-(1/n^2)]*n*x^3*(c*x^n)^(3/n))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/2)*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^3*Log[x])/(c*x^n)^(3/n)} -{x^1*Sin[a + Sqrt[-(1+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, ((1/8)*Sqrt[-(1/n^2)]*n*x^2*(c*x^n)^(2/n))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/2)*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^2*Log[x])/(c*x^n)^(2/n)} -{x^0*Sin[a + Sqrt[-(0+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, ((1/4)*Sqrt[-(1/n^2)]*n*x*(c*x^n)^(1/n))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/2)*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x*Log[x])/(c*x^n)^n^(-1)} -{Sin[a + Sqrt[-(-1+1)^2/(1^2*n^2)]*Log[c*x^n]]^1/x^1, x, 2, Log[x]*Sin[a]} -{Sin[a + Sqrt[-(-2+1)^2/(1^2*n^2)]*Log[c*x^n]]^1/x^2, x, 3, (E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^n^(-1)*(4*x)) + (Sqrt[-(1/n^2)]*n*(c*x^n)^(1/n)*Log[x])/(E^(a*Sqrt[-(1/n^2)]*n)*(2*x))} -{Sin[a + Sqrt[-(-3+1)^2/(1^2*n^2)]*Log[c*x^n]]^1/x^3, x, 3, (E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^(2/n)*(8*x^2)) + (Sqrt[-(1/n^2)]*n*(c*x^n)^(2/n)*Log[x])/(E^(a*Sqrt[-(1/n^2)]*n)*(2*x^2))} - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x^(1 + m)/(2*(1 + m)) - (x^(1 + m)*(c*x^n)^((1 + m)/n))/(E^((2*a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*(8*(1 + m))) - ((1/4)*E^((2*a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*x^(1 + m)*Log[x])/(c*x^n)^((1 + m)/n)} - -{x^2*Sin[a + Sqrt[-(2+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x^3/6 - ((1/24)*x^3*(c*x^n)^(3/n))/E^(2*a*Sqrt[-(1/n^2)]*n) - ((1/4)*E^(2*a*Sqrt[-(1/n^2)]*n)*x^3*Log[x])/(c*x^n)^(3/n)} -{x^1*Sin[a + Sqrt[-(1+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x^2/4 - ((1/16)*x^2*(c*x^n)^(2/n))/E^(2*a*Sqrt[-(1/n^2)]*n) - ((1/4)*E^(2*a*Sqrt[-(1/n^2)]*n)*x^2*Log[x])/(c*x^n)^(2/n)} -{x^0*Sin[a + Sqrt[-(0+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x/2 - ((1/8)*x*(c*x^n)^(1/n))/E^(2*a*Sqrt[-(1/n^2)]*n) - ((1/4)*E^(2*a*Sqrt[-(1/n^2)]*n)*x*Log[x])/(c*x^n)^n^(-1)} -{Sin[a + Sqrt[-(-1+1)^2/(2^2*n^2)]*Log[c*x^n]]^2/x^1, x, 2, Log[x]*Sin[a]^2} -{Sin[a + Sqrt[-(-2+1)^2/(2^2*n^2)]*Log[c*x^n]]^2/x^2, x, 3, -(1/(2*x)) + E^(2*a*Sqrt[-(1/n^2)]*n)/((c*x^n)^n^(-1)*(8*x)) - ((c*x^n)^(1/n)*Log[x])/(E^(2*a*Sqrt[-(1/n^2)]*n)*(4*x))} -{Sin[a + Sqrt[-(-3+1)^2/(2^2*n^2)]*Log[c*x^n]]^2/x^3, x, 3, -(1/(4*x^2)) + E^(2*a*Sqrt[-(1/n^2)]*n)/((c*x^n)^(2/n)*(16*x^2)) - ((c*x^n)^(2/n)*Log[x])/(E^(2*a*Sqrt[-(1/n^2)]*n)*(4*x^2))} - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(2^2*n^2)]*Log[c*x^n]]^3, x, 2, -((4*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]])/(5*(1 + m)^2)) + (8*x^(1 + m)*Sin[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]])/(5*(1 + m)) + (6*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]]*Sin[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]]^2)/(5*(1 + m)^2) - (4*x^(1 + m)*Sin[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]]^3)/(5*(1 + m))} - -{x^2*Sin[a + Sqrt[-(2+1)^2/(3^2*n^2)]*Log[c*x^n]]^3, x, 3, ((-(3/16))*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^3)/(c*x^n)^n^(-1) + ((3/32)*Sqrt[-(1/n^2)]*n*x^3*(c*x^n)^(1/n))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/48)*Sqrt[-(1/n^2)]*n*x^3*(c*x^n)^(3/n))/E^(3*a*Sqrt[-(1/n^2)]*n) + ((1/8)*E^(3*a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^3*Log[x])/(c*x^n)^(3/n)} -{x^1*Sin[a + Sqrt[-(1+1)^2/(3^2*n^2)]*Log[c*x^n]]^3, x, 3, ((-(9/32))*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^2)/(c*x^n)^(2/(3*n)) + ((9/64)*Sqrt[-(1/n^2)]*n*x^2*(c*x^n)^(2/(3*n)))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/32)*Sqrt[-(1/n^2)]*n*x^2*(c*x^n)^(2/n))/E^(3*a*Sqrt[-(1/n^2)]*n) + ((1/8)*E^(3*a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x^2*Log[x])/(c*x^n)^(2/n)} -{x^0*Sin[a + Sqrt[-(0+1)^2/(3^2*n^2)]*Log[c*x^n]]^3, x, 3, ((-(9/16))*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x)/(c*x^n)^(1/(3*n)) + ((9/32)*Sqrt[-(1/n^2)]*n*x*(c*x^n)^(1/(3*n)))/E^(a*Sqrt[-(1/n^2)]*n) - ((1/16)*Sqrt[-(1/n^2)]*n*x*(c*x^n)^(1/n))/E^(3*a*Sqrt[-(1/n^2)]*n) + ((1/8)*E^(3*a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n*x*Log[x])/(c*x^n)^n^(-1)} -{Sin[a + Sqrt[-(-1+1)^2/(3^2*n^2)]*Log[c*x^n]]^3/x^1, x, 2, Log[x]*Sin[a]^3} -{Sin[a + Sqrt[-(-2+1)^2/(3^2*n^2)]*Log[c*x^n]]^3/x^2, x, 3, -((E^(3*a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^n^(-1)*(16*x))) + (9*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^(1/(3*n))*(32*x)) - (9*Sqrt[-(1/n^2)]*n*(c*x^n)^(1/(3*n)))/(E^(a*Sqrt[-(1/n^2)]*n)*(16*x)) - (Sqrt[-(1/n^2)]*n*(c*x^n)^(1/n)*Log[x])/(E^(3*a*Sqrt[-(1/n^2)]*n)*(8*x))} -{Sin[a + Sqrt[-(-3+1)^2/(3^2*n^2)]*Log[c*x^n]]^3/x^3, x, 3, -((E^(3*a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^(2/n)*(32*x^2))) + (9*E^(a*Sqrt[-(1/n^2)]*n)*Sqrt[-(1/n^2)]*n)/((c*x^n)^(2/(3*n))*(64*x^2)) - (9*Sqrt[-(1/n^2)]*n*(c*x^n)^(2/(3*n)))/(E^(a*Sqrt[-(1/n^2)]*n)*(32*x^2)) - (Sqrt[-(1/n^2)]*n*(c*x^n)^(2/n)*Log[x])/(E^(3*a*Sqrt[-(1/n^2)]*n)*(8*x^2))} - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(1^2*2^2)]*Log[c*x^2]]^1, x, 3, -((E^((a*(1 + m))/Sqrt[-(1 + m)^2])*x^(1 + m)*(c*x^2)^((1 + m)/2))/(4*Sqrt[-(1 + m)^2])) + (E^((a*Sqrt[-(1 + m)^2])/(1 + m))*(1 + m)*x^(1 + m)*(c*x^2)^((1/2)*(-1 - m))*Log[x])/(2*Sqrt[-(1 + m)^2])} - -{x^0*Sin[a + Sqrt[-(0+1)^2/(1^2*2^2)]*Log[c*x^2]]^1, x, 3, (I*c*x^3)/(E^(I*a)*(4*Sqrt[c*x^2])) - (I*E^(I*a)*x*Log[x])/(2*Sqrt[c*x^2])} - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(2^2*2^2)]*Log[c*x^2]]^2, x, 3, x^(1 + m)/(2*(1 + m)) - (E^((2*a*(1 + m))/Sqrt[-(1 + m)^2])*x^(1 + m)*(c*x^2)^((1 + m)/2))/(8*(1 + m)) - ((1/4)*x^(1 + m)*(c*x^2)^((1/2)*(-1 - m))*Log[x])/E^((2*a*(1 + m))/Sqrt[-(1 + m)^2])} - -{x^0*Sin[a + Sqrt[-(0+1)^2/(2^2*2^2)]*Log[c*x^2]]^2, x, 3, x/2 - (c*x^3)/(E^(2*I*a)*(8*Sqrt[c*x^2])) - (E^(2*I*a)*x*Log[x])/(4*Sqrt[c*x^2])} - - -{x^m*Sin[a + Sqrt[-(m+1)^2/(3^2*2^2)]*Log[c*x^2]]^3, x, 3, (9*E^((a*Sqrt[-(1 + m)^2])/(1 + m))*x^(1 + m)*(c*x^2)^((1/6)*(-1 - m)))/(16*Sqrt[-(1 + m)^2]) - (9*E^((a*(1 + m))/Sqrt[-(1 + m)^2])*x^(1 + m)*(c*x^2)^((1 + m)/6))/(32*Sqrt[-(1 + m)^2]) + (E^((3*a*(1 + m))/Sqrt[-(1 + m)^2])*x^(1 + m)*(c*x^2)^((1 + m)/2))/(16*Sqrt[-(1 + m)^2]) - ((1 + m)*x^(1 + m)*(c*x^2)^((1/2)*(-1 - m))*Log[x])/(E^((3*a*(1 + m))/Sqrt[-(1 + m)^2])*(8*Sqrt[-(1 + m)^2]))} - -{x^0*Sin[a + Sqrt[-(0+1)^2/(3^2*2^2)]*Log[c*x^2]]^3, x, 3, -((I*c*x^3)/(E^(3*I*a)*(16*Sqrt[c*x^2]))) - (9*I*E^(I*a)*x)/(16*(c*x^2)^(1/6)) + ((9/32)*I*x*(c*x^2)^(1/6))/E^(I*a) + (I*E^(3*I*a)*x*Log[x])/(8*Sqrt[c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^1*Sqrt[Sin[a + b*Log[c*x^n]]], x, 3, (2*x^2*Hypergeometric2F1[-(1/2), (1/4)*(-1 - (4*I)/(b*n)), (1/4)*(3 - (4*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sin[a + b*Log[c*x^n]]])/((4 - I*b*n)*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)])} -{x^0*Sqrt[Sin[a + b*Log[c*x^n]]], x, 3, (2*x*Hypergeometric2F1[-(1/2), -((2*I + b*n)/(4*b*n)), (1/4)*(3 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sin[a + b*Log[c*x^n]]])/((2 - I*b*n)*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)])} -{Sqrt[Sin[a + b*Log[c*x^n]]]/x^1, x, 2, (2*EllipticE[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2])/(b*n)} -{Sqrt[Sin[a + b*Log[c*x^n]]]/x^2, x, 3, -((2*Hypergeometric2F1[-(1/2), (1/4)*(-1 + (2*I)/(b*n)), (1/4)*(3 + (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sin[a + b*Log[c*x^n]]])/((2 + I*b*n)*x*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]))} -{Sqrt[Sin[a + b*Log[c*x^n]]]/x^3, x, 3, -((2*Hypergeometric2F1[-(1/2), (1/4)*(-1 + (4*I)/(b*n)), (1/4)*(3 + (4*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sin[a + b*Log[c*x^n]]])/((4 + I*b*n)*x^2*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]))} - - -{x^1*Sin[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^2*Hypergeometric2F1[-(3/2), (1/4)*(-3 - (4*I)/(b*n)), (1/4)*(1 - (4*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^(3/2))/((4 - 3*I*b*n)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2))} -{x^0*Sin[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*Hypergeometric2F1[-(3/2), (1/4)*(-3 - (2*I)/(b*n)), (1/4)*(1 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^(3/2))/((2 - 3*I*b*n)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2))} -{Sin[a + b*Log[c*x^n]]^(3/2)/x^1, x, 3, (2*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2])/(3*b*n) - (2*Cos[a + b*Log[c*x^n]]*Sqrt[Sin[a + b*Log[c*x^n]]])/(3*b*n)} -{Sin[a + b*Log[c*x^n]]^(3/2)/x^2, x, 3, -((2*Hypergeometric2F1[-(3/2), (1/4)*(-3 + (2*I)/(b*n)), (1/4)*(1 + (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^(3/2))/((2 + 3*I*b*n)*x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)))} -{Sin[a + b*Log[c*x^n]]^(3/2)/x^3, x, 3, -((2*Hypergeometric2F1[-(3/2), (1/4)*(-3 + (4*I)/(b*n)), (1/4)*(1 + (4*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^(3/2))/((4 + 3*I*b*n)*x^2*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^0/Sqrt[Sin[a + b*Log[c*x^n]]], x, 3, (2*x*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]*Hypergeometric2F1[1/2, (1/4)*(1 - (2*I)/(b*n)), (1/4)*(5 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 + I*b*n)*Sqrt[Sin[a + b*Log[c*x^n]]])} -{1/(x^1*Sqrt[Sin[a + b*Log[c*x^n]]]), x, 2, (2*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2])/(b*n)} - - -{x^0/Sin[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Hypergeometric2F1[3/2, (1/4)*(3 - (2*I)/(b*n)), (1/4)*(7 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 + 3*I*b*n)*Sin[a + b*Log[c*x^n]]^(3/2))} -{1/(x^1*Sin[a + b*Log[c*x^n]]^(3/2)), x, 3, -((2*EllipticE[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2])/(b*n)) - (2*Cos[a + b*Log[c*x^n]])/(b*n*Sqrt[Sin[a + b*Log[c*x^n]]])} - - -{x^0/Sin[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Hypergeometric2F1[5/2, (1/4)*(5 - (2*I)/(b*n)), (1/4)*(9 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 + 5*I*b*n)*Sin[a + b*Log[c*x^n]]^(5/2))} -{1/(x^1*Sin[a + b*Log[c*x^n]]^(5/2)), x, 3, (2*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2])/(3*b*n) - (2*Cos[a + b*Log[c*x^n]])/(3*b*n*Sin[a + b*Log[c*x^n]]^(3/2))} - - -{1/Sin[a - 2*I*Log[c*x]]^(3/2), x, 3, (1 - c^4*E^(2*I*a)*x^4)/(E^(2*I*a)*(2*c^4*x^3*Sin[a - 2*I*Log[c*x]]^(3/2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when m symbolic*) - - -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^4, x, 3, (24*b^4*d^4*n^4*(e*x)^(1 + m))/(e*(1 + m)*((1 + m)^2 + 4*b^2*d^2*n^2)*((1 + m)^2 + 16*b^2*d^2*n^2)) - (24*b^3*d^3*n^3*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + 4*b^2*d^2*n^2)*((1 + m)^2 + 16*b^2*d^2*n^2)) + (12*b^2*d^2*(1 + m)*n^2*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^2)/(e*((1 + m)^2 + 4*b^2*d^2*n^2)*((1 + m)^2 + 16*b^2*d^2*n^2)) - (4*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^3)/(e*((1 + m)^2 + 16*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^4)/(e*((1 + m)^2 + 16*b^2*d^2*n^2))} -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^3, x, 2, -((6*b^3*d^3*n^3*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2 + 9*b^2*d^2*n^2))) + (6*b^2*d^2*(1 + m)*n^2*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2 + 9*b^2*d^2*n^2)) - (3*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^2)/(e*((1 + m)^2 + 9*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^3)/(e*((1 + m)^2 + 9*b^2*d^2*n^2))} -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^2, x, 2, (2*b^2*d^2*n^2*(e*x)^(1 + m))/(e*(1 + m)*((1 + m)^2 + 4*b^2*d^2*n^2)) - (2*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + 4*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^2)/(e*((1 + m)^2 + 4*b^2*d^2*n^2))} -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^1, x, 1, -((b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))} - - -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(3/2), x, 3, (2*(e*x)^(1 + m)*Hypergeometric2F1[-(3/2), -((2*I + 2*I*m + 3*b*d*n)/(4*b*d*n)), -((2*I + 2*I*m - b*d*n)/(4*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)]*Sin[d*(a + b*Log[c*x^n])]^(3/2))/(e*(2 + 2*m - 3*I*b*d*n)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2))} -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(1/2), x, 3, (2*(e*x)^(1 + m)*Hypergeometric2F1[-(1/2), -((2*I + 2*I*m + b*d*n)/(4*b*d*n)), -((2*I + 2*I*m - 3*b*d*n)/(4*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)]*Sqrt[Sin[d*(a + b*Log[c*x^n])]])/(e*(2 + 2*m - I*b*d*n)*Sqrt[1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])} -{(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(1/2), x, 3, (2*(e*x)^(1 + m)*Sqrt[1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)]*Hypergeometric2F1[1/2, -((2*I + 2*I*m - b*d*n)/(4*b*d*n)), -((2*I + 2*I*m - 5*b*d*n)/(4*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(2 + 2*m + I*b*d*n)*Sqrt[Sin[d*(a + b*Log[c*x^n])]])} -{(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(3/2), x, 3, (2*(e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^(3/2)*Hypergeometric2F1[3/2, -((2*I + 2*I*m - 3*b*d*n)/(4*b*d*n)), -((2*I + 2*I*m - 7*b*d*n)/(4*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(2 + 2*m + 3*I*b*d*n)*Sin[d*(a + b*Log[c*x^n])]^(3/2))} -{(e*x)^m/Sin[d*(a + b*Log[c*x^n])]^(5/2), x, 3, (2*(e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^(5/2)*Hypergeometric2F1[5/2, -((2*I + 2*I*m - 5*b*d*n)/(4*b*d*n)), -((2*I + 2*I*m - 9*b*d*n)/(4*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(2 + 2*m + 5*I*b*d*n)*Sin[d*(a + b*Log[c*x^n])]^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sin[d (a+b Log[c x^n])]^p when p symbolic*) - - -{(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^p, x, 3, ((e*x)^(1 + m)*Hypergeometric2F1[-p, -((I + I*m + b*d*n*p)/(2*b*d*n)), (1/2)*(2 - (I*(1 + m))/(b*d*n) - p), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)]*Sin[d*(a + b*Log[c*x^n])]^p)/((1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(e*(1 + m - I*b*d*n*p)))} - - -{x^2*Sin[a + b*Log[c*x^n]]^p, x, 3, (x^3*Hypergeometric2F1[-p, -((3*I + b*n*p)/(2*b*n)), (1/2)*(2 - (3*I)/(b*n) - p), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^p)/((1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*(3 - I*b*n*p))} -{x^1*Sin[a + b*Log[c*x^n]]^p, x, 3, (x^2*Hypergeometric2F1[(1/2)*(-((2*I)/(b*n)) - p), -p, (1/2)*(2 - (2*I)/(b*n) - p), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^p)/((1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*(2 - I*b*n*p))} -{x^0*Sin[a + b*Log[c*x^n]]^p, x, 3, (x*Hypergeometric2F1[-p, -((I + b*n*p)/(2*b*n)), (1/2)*(2 - I/(b*n) - p), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^p)/((1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*(1 - I*b*n*p))} -{Sin[a + b*Log[c*x^n]]^p/x^1, x, 2, (Cos[a + b*Log[c*x^n]]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Sin[a + b*Log[c*x^n]]^2]*Sin[a + b*Log[c*x^n]]^(1 + p))/(b*n*(1 + p)*Sqrt[Cos[a + b*Log[c*x^n]]^2])} -{Sin[a + b*Log[c*x^n]]^p/x^2, x, 3, -((Hypergeometric2F1[(1/2)*(I/(b*n) - p), -p, (1/2)*(2 + I/(b*n) - p), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^p)/((1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*((1 + I*b*n*p)*x)))} -{Sin[a + b*Log[c*x^n]]^p/x^3, x, 3, -((Hypergeometric2F1[(1/2)*((2*I)/(b*n) - p), -p, (1/2)*(2 + (2*I)/(b*n) - p), E^(2*I*a)*(c*x^n)^(2*I*b)]*Sin[a + b*Log[c*x^n]]^p)/((1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*((2 + I*b*n*p)*x^2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Cos[a + b*Log[c*x^n]], x, 1, (3*x^3*Cos[a + b*Log[c*x^n]])/(9 + b^2*n^2) + (b*n*x^3*Sin[a + b*Log[c*x^n]])/(9 + b^2*n^2)} -{x^1*Cos[a + b*Log[c*x^n]], x, 1, (2*x^2*Cos[a + b*Log[c*x^n]])/(4 + b^2*n^2) + (b*n*x^2*Sin[a + b*Log[c*x^n]])/(4 + b^2*n^2)} -{x^0*Cos[a + b*Log[c*x^n]], x, 1, (x*Cos[a + b*Log[c*x^n]])/(1 + b^2*n^2) + (b*n*x*Sin[a + b*Log[c*x^n]])/(1 + b^2*n^2)} -{Cos[a + b*Log[c*x^n]]/x^1, x, 2, Sin[a + b*Log[c*x^n]]/(b*n)} -{Cos[a + b*Log[c*x^n]]/x^2, x, 1, -(Cos[a + b*Log[c*x^n]]/((1 + b^2*n^2)*x)) + (b*n*Sin[a + b*Log[c*x^n]])/((1 + b^2*n^2)*x)} - - -{x^2*Cos[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x^3)/(3*(9 + 4*b^2*n^2)) + (3*x^3*Cos[a + b*Log[c*x^n]]^2)/(9 + 4*b^2*n^2) + (2*b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(9 + 4*b^2*n^2)} -{x^1*Cos[a + b*Log[c*x^n]]^2, x, 2, (b^2*n^2*x^2)/(4*(1 + b^2*n^2)) + (x^2*Cos[a + b*Log[c*x^n]]^2)/(2*(1 + b^2*n^2)) + (b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*(1 + b^2*n^2))} -{x^0*Cos[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x)/(1 + 4*b^2*n^2) + (x*Cos[a + b*Log[c*x^n]]^2)/(1 + 4*b^2*n^2) + (2*b*n*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 + 4*b^2*n^2)} -{Cos[a + b*Log[c*x^n]]^2/x^1, x, 3, Log[x]/2 + (Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*b*n)} -{Cos[a + b*Log[c*x^n]]^2/x^2, x, 2, -((2*b^2*n^2)/((1 + 4*b^2*n^2)*x)) - Cos[a + b*Log[c*x^n]]^2/((1 + 4*b^2*n^2)*x) + (2*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1 + 4*b^2*n^2)*x)} - - -{x^2*Cos[a + b*Log[c*x^n]]^3, x, 2, (2*b^2*n^2*x^3*Cos[a + b*Log[c*x^n]])/(9 + 10*b^2*n^2 + b^4*n^4) + (x^3*Cos[a + b*Log[c*x^n]]^3)/(3*(1 + b^2*n^2)) + (2*b^3*n^3*x^3*Sin[a + b*Log[c*x^n]])/(3*(9 + 10*b^2*n^2 + b^4*n^4)) + (b*n*x^3*Cos[a + b*Log[c*x^n]]^2*Sin[a + b*Log[c*x^n]])/(3*(1 + b^2*n^2))} -{x^1*Cos[a + b*Log[c*x^n]]^3, x, 2, (12*b^2*n^2*x^2*Cos[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4) + (2*x^2*Cos[a + b*Log[c*x^n]]^3)/(4 + 9*b^2*n^2) + (6*b^3*n^3*x^2*Sin[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4) + (3*b*n*x^2*Cos[a + b*Log[c*x^n]]^2*Sin[a + b*Log[c*x^n]])/(4 + 9*b^2*n^2)} -{x^0*Cos[a + b*Log[c*x^n]]^3, x, 2, (6*b^2*n^2*x*Cos[a + b*Log[c*x^n]])/(1 + 10*b^2*n^2 + 9*b^4*n^4) + (x*Cos[a + b*Log[c*x^n]]^3)/(1 + 9*b^2*n^2) + (6*b^3*n^3*x*Sin[a + b*Log[c*x^n]])/(1 + 10*b^2*n^2 + 9*b^4*n^4) + (3*b*n*x*Cos[a + b*Log[c*x^n]]^2*Sin[a + b*Log[c*x^n]])/(1 + 9*b^2*n^2)} -{Cos[a + b*Log[c*x^n]]^3/x^1, x, 3, Sin[a + b*Log[c*x^n]]/(b*n) - Sin[a + b*Log[c*x^n]]^3/(3*b*n)} -{Cos[a + b*Log[c*x^n]]^3/x^2, x, 2, -((6*b^2*n^2*Cos[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x)) - Cos[a + b*Log[c*x^n]]^3/((1 + 9*b^2*n^2)*x) + (6*b^3*n^3*Sin[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x) + (3*b*n*Cos[a + b*Log[c*x^n]]^2*Sin[a + b*Log[c*x^n]])/((1 + 9*b^2*n^2)*x)} - - -{x^0*Cos[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x)/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Cos[a + b*Log[c*x^n]]^2)/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (x*Cos[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2) + (24*b^3*n^3*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (4*b*n*x*Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])/(1 + 16*b^2*n^2)} -{Cos[a + b*Log[c*x^n]]^4/x^1, x, 4, 3*Log[x]/8 + (3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(8*b*n) + (Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])/(4*b*n)} - - -{Cos[Log[6 + 3*x]], x, 2, (1/2)*(2 + x)*Cos[Log[3*(2 + x)]] + (1/2)*(2 + x)*Sin[Log[3*(2 + x)]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when b^2 n^2 p^2+(m+1)^2=0*) - - -{x^m*Cos[a + Sqrt[-(m+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, (E^((a*(1 + m))/(Sqrt[-((1 + m)^2/n^2)]*n))*x^(1 + m)*(c*x^n)^((1 + m)/n))/(4*(1 + m)) + ((1/2)*E^((a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*x^(1 + m)*Log[x])/(c*x^n)^((1 + m)/n)} - -{x^0*Cos[a + Sqrt[-(0+1)^2/(1^2*n^2)]*Log[c*x^n]]^1, x, 3, ((1/4)*x*(c*x^n)^(1/n))/E^(a*Sqrt[-(1/n^2)]*n) + ((1/2)*E^(a*Sqrt[-(1/n^2)]*n)*x*Log[x])/(c*x^n)^n^(-1)} - - -{x^m*Cos[a + Sqrt[-(m+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x^(1 + m)/(2*(1 + m)) + (x^(1 + m)*(c*x^n)^((1 + m)/n))/(E^((2*a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*(8*(1 + m))) + ((1/4)*E^((2*a*Sqrt[-((1 + m)^2/n^2)]*n)/(1 + m))*x^(1 + m)*Log[x])/(c*x^n)^((1 + m)/n)} - -{x^0*Cos[a + Sqrt[-(0+1)^2/(2^2*n^2)]*Log[c*x^n]]^2, x, 3, x/2 + ((1/8)*x*(c*x^n)^(1/n))/E^(2*a*Sqrt[-(1/n^2)]*n) + ((1/4)*E^(2*a*Sqrt[-(1/n^2)]*n)*x*Log[x])/(c*x^n)^n^(-1)} - - -{x^m*Cos[a + Sqrt[-(m+1)^2/(2^2*n^2)]*Log[c*x^n]]^3, x, 2, (8*x^(1 + m)*Cos[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]])/(5*(1 + m)) - (4*x^(1 + m)*Cos[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]]^3)/(5*(1 + m)) + (4*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Sin[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]])/(5*(1 + m)^2) - (6*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]]^2*Sin[a + (1/2)*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n]])/(5*(1 + m)^2)} - -{x^0*Cos[a + Sqrt[-(0+1)^2/(3^2*n^2)]*Log[c*x^n]]^3, x, 3, ((9/16)*E^(a*Sqrt[-(1/n^2)]*n)*x)/(c*x^n)^(1/(3*n)) + ((9/32)*x*(c*x^n)^(1/(3*n)))/E^(a*Sqrt[-(1/n^2)]*n) + ((1/16)*x*(c*x^n)^(1/n))/E^(3*a*Sqrt[-(1/n^2)]*n) + ((1/8)*E^(3*a*Sqrt[-(1/n^2)]*n)*x*Log[x])/(c*x^n)^n^(-1)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^0*Sqrt[Cos[a + b*Log[c*x^n]]], x, 3, (2*x*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[-(1/2), -((2*I + b*n)/(4*b*n)), (1/4)*(3 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - I*b*n)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)])} -{Sqrt[Cos[a + b*Log[c*x^n]]]/x^1, x, 2, (2*EllipticE[(a + b*Log[c*x^n])/2, 2])/(b*n)} - - -{x^0*Cos[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*Cos[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[-(3/2), (1/4)*(-3 - (2*I)/(b*n)), (1/4)*(1 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - 3*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2))} -{Cos[a + b*Log[c*x^n]]^(3/2)/x^1, x, 3, (2*EllipticF[(1/2)*(a + b*Log[c*x^n]), 2])/(3*b*n) + (2*Sqrt[Cos[a + b*Log[c*x^n]]]*Sin[a + b*Log[c*x^n]])/(3*b*n)} - - -{x^0*Cos[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*Cos[a + b*Log[c*x^n]]^(5/2)*Hypergeometric2F1[-(5/2), (1/4)*(-5 - (2*I)/(b*n)), -((2*I + b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - 5*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2))} -{Cos[a + b*Log[c*x^n]]^(5/2)/x^1, x, 3, (6*EllipticE[(1/2)*(a + b*Log[c*x^n]), 2])/(5*b*n) + (2*Cos[a + b*Log[c*x^n]]^(3/2)*Sin[a + b*Log[c*x^n]])/(5*b*n)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^0/Sqrt[Cos[a + b*Log[c*x^n]]], x, 3, (2*x*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Hypergeometric2F1[1/2, (1/4)*(1 - (2*I)/(b*n)), (1/4)*(5 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + I*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])} -{1/(x^1*Sqrt[Cos[a + b*Log[c*x^n]]]), x, 2, (2*EllipticF[(a + b*Log[c*x^n])/2, 2])/(b*n)} - - -{x^0/Cos[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Hypergeometric2F1[3/2, (1/4)*(3 - (2*I)/(b*n)), (1/4)*(7 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 3*I*b*n)*Cos[a + b*Log[c*x^n]]^(3/2))} -{1/(x*Cos[a + b*Log[c*x^n]]^(3/2)), x, 3, -((2*EllipticE[(1/2)*(a + b*Log[c*x^n]), 2])/(b*n)) + (2*Sin[a + b*Log[c*x^n]])/(b*n*Sqrt[Cos[a + b*Log[c*x^n]]])} - - -{x^0/Cos[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Hypergeometric2F1[5/2, (1/4)*(5 - (2*I)/(b*n)), (1/4)*(9 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 5*I*b*n)*Cos[a + b*Log[c*x^n]]^(5/2))} -{1/(x*Cos[a + b*Log[c*x^n]]^(5/2)), x, 3, (2*EllipticF[(1/2)*(a + b*Log[c*x^n]), 2])/(3*b*n) + (2*Sin[a + b*Log[c*x^n]])/(3*b*n*Cos[a + b*Log[c*x^n]]^(3/2))} - - -{1/Cos[a - 2*I*Log[c*x]]^(3/2), x, 3, -((1 + c^4*E^(2*I*a)*x^4)/(E^(2*I*a)*(2*c^4*x^3*Cos[a - 2*I*Log[c*x]]^(3/2))))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when m symbolic*) - - -{x^m*Cos[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x^(1 + m))/((1 + m)*((1 + m)^2 + 4*b^2*n^2)*((1 + m)^2 + 16*b^2*n^2)) + (12*b^2*(1 + m)*n^2*x^(1 + m)*Cos[a + b*Log[c*x^n]]^2)/(((1 + m)^2 + 4*b^2*n^2)*((1 + m)^2 + 16*b^2*n^2)) + ((1 + m)*x^(1 + m)*Cos[a + b*Log[c*x^n]]^4)/((1 + m)^2 + 16*b^2*n^2) + (24*b^3*n^3*x^(1 + m)*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(((1 + m)^2 + 4*b^2*n^2)*((1 + m)^2 + 16*b^2*n^2)) + (4*b*n*x^(1 + m)*Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])/((1 + m)^2 + 16*b^2*n^2)} -{x^m*Cos[a + b*Log[c*x^n]]^3, x, 2, (6*b^2*(1 + m)*n^2*x^(1 + m)*Cos[a + b*Log[c*x^n]])/(((1 + m)^2 + b^2*n^2)*((1 + m)^2 + 9*b^2*n^2)) + ((1 + m)*x^(1 + m)*Cos[a + b*Log[c*x^n]]^3)/((1 + m)^2 + 9*b^2*n^2) + (6*b^3*n^3*x^(1 + m)*Sin[a + b*Log[c*x^n]])/(((1 + m)^2 + b^2*n^2)*((1 + m)^2 + 9*b^2*n^2)) + (3*b*n*x^(1 + m)*Cos[a + b*Log[c*x^n]]^2*Sin[a + b*Log[c*x^n]])/((1 + m)^2 + 9*b^2*n^2)} -{x^m*Cos[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x^(1 + m))/((1 + m)*((1 + m)^2 + 4*b^2*n^2)) + ((1 + m)*x^(1 + m)*Cos[a + b*Log[c*x^n]]^2)/((1 + m)^2 + 4*b^2*n^2) + (2*b*n*x^(1 + m)*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1 + m)^2 + 4*b^2*n^2)} -{x^m*Cos[a + b*Log[c*x^n]], x, 1, ((1 + m)*x^(1 + m)*Cos[a + b*Log[c*x^n]])/((1 + m)^2 + b^2*n^2) + (b*n*x^(1 + m)*Sin[a + b*Log[c*x^n]])/((1 + m)^2 + b^2*n^2)} - - -{x^m*Cos[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*Cos[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[-(3/2), -((2*I + 2*I*m + 3*b*n)/(4*b*n)), -((2*I + 2*I*m - b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m - 3*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2))} -{x^m*Cos[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Sqrt[Cos[a + b*Log[c*x^n]]]*Hypergeometric2F1[-(1/2), -((2*I + 2*I*m + b*n)/(4*b*n)), -((2*I + 2*I*m - 3*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m - I*b*n)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)])} -{x^m/Cos[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Hypergeometric2F1[1/2, -((2*I + 2*I*m - b*n)/(4*b*n)), -((2*I + 2*I*m - 5*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m + I*b*n)*Sqrt[Cos[a + b*Log[c*x^n]]])} -{x^m/Cos[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Hypergeometric2F1[3/2, -((2*I + 2*I*m - 3*b*n)/(4*b*n)), -((2*I + 2*I*m - 7*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m + 3*I*b*n)*Cos[a + b*Log[c*x^n]]^(3/2))} -{x^m/Cos[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x^(1 + m)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Hypergeometric2F1[5/2, -((2*I + 2*I*m - 5*b*n)/(4*b*n)), -((2*I + 2*I*m - 9*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m + 5*I*b*n)*Cos[a + b*Log[c*x^n]]^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cos[d (a+b Log[c x^n])]^p when p symbolic*) - - -{(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p, x, 3, ((e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]^p*Hypergeometric2F1[-p, -((I + I*m + b*d*n*p)/(2*b*d*n)), (1/2)*(2 - (I*(1 + m))/(b*d*n) - p), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/((1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(e*(1 + m - I*b*d*n*p)))} - - -{x^1*Cos[a + b*Log[c*x^n]]^p, x, 3, (x^2*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[(1/2)*(-((2*I)/(b*n)) - p), -p, (1/2)*(2 - (2*I)/(b*n) - p), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 + E^(2*I*a)*(c*x^n)^(2*I*b))^p*(2 - I*b*n*p))} -{x^0*Cos[a + b*Log[c*x^n]]^p, x, 3, (x*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[-p, -((I + b*n*p)/(2*b*n)), (1/2)*(2 - I/(b*n) - p), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 + E^(2*I*a)*(c*x^n)^(2*I*b))^p*(1 - I*b*n*p))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Tan[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tan[a+b Log[x]]^p*) - - -{x^3*Tan[a + I*Log[x]], x, 5, (-I)*E^((2*I)*a)*x^2 + (I/4)*x^4 + I*E^((4*I)*a)*Log[E^((2*I)*a) + x^2]} -{x^2*Tan[a + I*Log[x]], x, 5, (-2*I)*E^((2*I)*a)*x + (I/3)*x^3 + (2*I)*E^((3*I)*a)*ArcTan[x/E^(I*a)]} -{x^1*Tan[a + I*Log[x]], x, 5, (I/2)*x^2 - I*E^((2*I)*a)*Log[E^((2*I)*a) + x^2]} -{x^0*Tan[a + I*Log[x]], x, 4, I*x - (2*I)*E^(I*a)*ArcTan[x/E^(I*a)]} -{Tan[a + I*Log[x]]/x^1, x, 2, I*Log[Cos[a + I*Log[x]]]} -{Tan[a + I*Log[x]]/x^2, x, 4, I/x + ((2*I)*ArcTan[x/E^(I*a)])/E^(I*a)} -{Tan[a + I*Log[x]]/x^3, x, 4, I/(2*x^2) - (I*Log[1 + E^(2*I*a)/x^2])/E^(2*I*a)} -{Tan[a + I*Log[x]]/x^4, x, 5, I/(3*x^3) - (2*I)/(E^(2*I*a)*x) - (2*I*ArcTan[x/E^(I*a)])/E^(3*I*a)} - - -{x^3*Tan[a + I*Log[x]]^2, x, 5, 2*E^((2*I)*a)*x^2 - x^4/4 - (2*E^((6*I)*a))/(E^((2*I)*a) + x^2) - 4*E^((4*I)*a)*Log[E^((2*I)*a) + x^2]} -{x^2*Tan[a + I*Log[x]]^2, x, 6, 6*E^(2*I*a)*x - x^3/3 - (2*E^(2*I*a)*x^3)/(E^(2*I*a) + x^2) - 6*E^(3*I*a)*ArcTan[x/E^(I*a)]} -{x^1*Tan[a + I*Log[x]]^2, x, 5, -x^2/2 + (2*E^((4*I)*a))/(E^((2*I)*a) + x^2) + 2*E^((2*I)*a)*Log[E^((2*I)*a) + x^2]} -{x^0*Tan[a + I*Log[x]]^2, x, 6, -x - (2*E^(2*I*a)*x)/(E^(2*I*a) + x^2) + 2*E^(I*a)*ArcTan[x/E^(I*a)]} -{Tan[a + I*Log[x]]^2/x^1, x, 3, -Log[x] - I*Tan[a + I*Log[x]]} -{Tan[a + I*Log[x]]^2/x^2, x, 5, E^(2*I*a)/(x*(E^(2*I*a) + x^2)) + (3*x)/(E^(2*I*a) + x^2) + (2*ArcTan[x/E^(I*a)])/E^(I*a)} -{Tan[a + I*Log[x]]^2/x^3, x, 4, -2/(E^((2*I)*a)*(1 + E^((2*I)*a)/x^2)) + 1/(2*x^2) - (2*Log[1 + E^((2*I)*a)/x^2])/E^((2*I)*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tan[a+b Log[x]]^p with m symbolic*) - - -{(e*x)^m*Tan[a + I*Log[x]]^1, x, 4, -((I*(e*x)^(1 + m))/(e*(1 + m))) + (2*I*(e*x)^(1 + m)*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, -(E^(2*I*a)/x^2)])/(e*(1 + m))} -{(e*x)^m*Tan[a + I*Log[x]]^2, x, 5, -((x*(e*x)^m)/(1 + m)) + (2*x*(e*x)^m)/(1 + E^(2*I*a)/x^2) - 2*x*(e*x)^m*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, -(E^(2*I*a)/x^2)]} -{(e*x)^m*Tan[a + I*Log[x]]^3, x, 6, -((I*(1 - m)*m*x*(e*x)^m)/(2*(1 + m))) + (I*(1 - E^(2*I*a)/x^2)^2*x*(e*x)^m)/(2*(1 + E^(2*I*a)/x^2)^2) + (I*(E^(2*I*a)*(3 + m) + (E^(4*I*a)*(1 - m))/x^2)*x*(e*x)^m)/(E^(2*I*a)*(2*(1 + E^(2*I*a)/x^2))) - (I*(3 + 2*m + m^2)*x*(e*x)^m*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, -(E^(2*I*a)/x^2)])/(1 + m)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tan[a+b Log[x]]^p with p symbolic*) -(**) - - -{Tan[a + b*Log[x]]^p, x, 4, (x*((I*(1 - E^(2*I*a)*x^(2*I*b)))/(1 + E^(2*I*a)*x^(2*I*b)))^p*(1 + E^(2*I*a)*x^(2*I*b))^p*AppellF1[-(I/(2*b)), -p, p, 1 - I/(2*b), E^(2*I*a)*x^(2*I*b), (-E^(2*I*a))*x^(2*I*b)])/(1 - E^(2*I*a)*x^(2*I*b))^p} -{(e*x)^m*Tan[a + b*Log[x]]^p, x, 4, ((e*x)^(1 + m)*((I*(1 - E^(2*I*a)*x^(2*I*b)))/(1 + E^(2*I*a)*x^(2*I*b)))^p*(1 + E^(2*I*a)*x^(2*I*b))^p*AppellF1[-((I*(1 + m))/(2*b)), -p, p, 1 - (I*(1 + m))/(2*b), E^(2*I*a)*x^(2*I*b), (-E^(2*I*a))*x^(2*I*b)])/((1 - E^(2*I*a)*x^(2*I*b))^p*(e*(1 + m)))} - - -{Tan[a + 1*Log[x]]^p, x, 4, (((I*(1 - E^(2*I*a)*x^(2*I)))/(1 + E^(2*I*a)*x^(2*I)))^p*(1 + E^(2*I*a)*x^(2*I))^p*x*AppellF1[-(I/2), -p, p, 1 - I/2, E^(2*I*a)*x^(2*I), (-E^(2*I*a))*x^(2*I)])/(1 - E^(2*I*a)*x^(2*I))^p} -{Tan[a + 2*Log[x]]^p, x, 4, (((I*(1 - E^(2*I*a)*x^(4*I)))/(1 + E^(2*I*a)*x^(4*I)))^p*(1 + E^(2*I*a)*x^(4*I))^p*x*AppellF1[-(I/4), -p, p, 1 - I/4, E^(2*I*a)*x^(4*I), (-E^(2*I*a))*x^(4*I)])/(1 - E^(2*I*a)*x^(4*I))^p} -{Tan[a + 3*Log[x]]^p, x, 4, (((I*(1 - E^(2*I*a)*x^(6*I)))/(1 + E^(2*I*a)*x^(6*I)))^p*(1 + E^(2*I*a)*x^(6*I))^p*x*AppellF1[-(I/6), -p, p, 1 - I/6, E^(2*I*a)*x^(6*I), (-E^(2*I*a))*x^(6*I)])/(1 - E^(2*I*a)*x^(6*I))^p} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tan[a+b Log[c x^n]]^p*) - - -{x^3*Tan[d*(a + b*Log[c*x^n])], x, 4, (-I/4)*x^4 + (I/2)*x^4*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]} -{x^2*Tan[d*(a + b*Log[c*x^n])], x, 4, (-I/3)*x^3 + ((2*I)/3)*x^3*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]} -{x^1*Tan[d*(a + b*Log[c*x^n])], x, 4, (-I/2)*x^2 + I*x^2*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]} -{x^0*Tan[d*(a + b*Log[c*x^n])], x, 4, (-I)*x + (2*I)*x*Hypergeometric2F1[1, (-I/2)/(b*d*n), 1 - (I/2)/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))]} -{Tan[d*(a + b*Log[c*x^n])]/x^1, x, 2, -(Log[Cos[a*d + b*d*Log[c*x^n]]]/(b*d*n))} -{Tan[d*(a + b*Log[c*x^n])]/x^2, x, 4, I/x - (2*I*Hypergeometric2F1[1, I/(2*b*d*n), 1 + I/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/x} -{Tan[d*(a + b*Log[c*x^n])]/x^3, x, 4, (I/2)/x^2 - (I*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])/x^2} - - -{x^3*Tan[d*(a + b*Log[c*x^n])]^2, x, 5, ((4*I - b*d*n)*x^4)/(4*b*d*n) + (I*x^4*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^4*Hypergeometric2F1[1, -((2*I)/(b*d*n)), 1 - (2*I)/(b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^2*Tan[d*(a + b*Log[c*x^n])]^2, x, 5, ((3*I - b*d*n)*x^3)/(3*b*d*n) + (I*x^3*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^3*Hypergeometric2F1[1, -((3*I)/(2*b*d*n)), 1 - (3*I)/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^1*Tan[d*(a + b*Log[c*x^n])]^2, x, 5, ((2*I - b*d*n)*x^2)/(2*b*d*n) + (I*x^2*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^2*Hypergeometric2F1[1, -(I/(b*d*n)), 1 - I/(b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^0*Tan[d*(a + b*Log[c*x^n])]^2, x, 5, ((I - b*d*n)*x)/(b*d*n) + (I*x*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x*Hypergeometric2F1[1, -(I/(2*b*d*n)), 1 - I/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{Tan[d*(a + b*Log[c*x^n])]^2/x^1, x, 3, -Log[x] + Tan[a*d + b*d*Log[c*x^n]]/(b*d*n)} -{Tan[d*(a + b*Log[c*x^n])]^2/x^2, x, 5, (1 + I/(b*d*n))/x + (I*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*x*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*Hypergeometric2F1[1, I/(2*b*d*n), 1 + I/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n*x)} -{Tan[d*(a + b*Log[c*x^n])]^2/x^3, x, 5, (1 + (2*I)/(b*d*n))/(2*x^2) + (I*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*x^2*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*n*x^2)} - - -{Tan[a + b*Log[c*x^n]]^3/x, x, 3, Log[Cos[a + b*Log[c*x^n]]]/(b*n) + Tan[a + b*Log[c*x^n]]^2/(2*b*n)} -{Tan[a + b*Log[c*x^n]]^4/x, x, 4, Log[x] - Tan[a + b*Log[c*x^n]]/(b*n) + Tan[a + b*Log[c*x^n]]^3/(3*b*n)} -{Tan[a + b*Log[c*x^n]]^5/x, x, 4, -(Log[Cos[a + b*Log[c*x^n]]]/(b*n)) - Tan[a + b*Log[c*x^n]]^2/(2*b*n) + Tan[a + b*Log[c*x^n]]^4/(4*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tan[a+b Log[c x^n]]^p with m symbolic*) - - -{(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^1, x, 4, -((I*(e*x)^(1 + m))/(e*(1 + m))) + (2*I*(e*x)^(1 + m)*Hypergeometric2F1[1, -((I*(1 + m))/(2*b*d*n)), 1 - (I*(1 + m))/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(e*(1 + m))} -{(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^2, x, 5, ((I*(1 + m) - b*d*n)*(e*x)^(1 + m))/(b*d*e*(1 + m)*n) + (I*(e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*e*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*(e*x)^(1 + m)*Hypergeometric2F1[1, -((I*(1 + m))/(2*b*d*n)), 1 - (I*(1 + m))/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b*d*e*n)} -{(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^3, x, 6, -(((I*(1 + m) - b*d*n)*(1 + m + 2*I*b*d*n)*(e*x)^(1 + m))/(2*b^2*d^2*e*(1 + m)*n^2)) - ((e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^2)/(2*b*d*e*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^2) - (I*(e*x)^(1 + m)*((E^(2*I*a*d)*(1 + m - 2*I*b*d*n))/n - (E^(4*I*a*d)*(1 + m + 2*I*b*d*n)*(c*x^n)^(2*I*b*d))/n))/(E^(2*I*a*d)*(2*b^2*d^2*e*n*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))) + (I*(1 + 2*m + m^2 - 2*b^2*d^2*n^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, -((I*(1 + m))/(2*b*d*n)), 1 - (I*(1 + m))/(2*b*d*n), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(b^2*d^2*e*(1 + m)*n^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tan[a+b Log[c x^n]]^p with p symbolic*) - - -{Tan[d*(a + b*Log[c*x^n])]^p, x, 5, (x*((I*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*AppellF1[-(I/(2*b*d*n)), -p, p, 1 - I/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p} -{(e*x)^m*Tan[d*(a + b*Log[c*x^n])]^p, x, 5, ((e*x)^(1 + m)*((I*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))^p*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*AppellF1[-((I*(1 + m))/(2*b*d*n)), -p, p, 1 - (I*(1 + m))/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/((1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(e*(1 + m)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tan[a+b Log[c x^n]]^(p/2)*) - - -{Tan[a + b*Log[c*x^n]]^(5/2)/x, x, 13, ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + (2*Tan[a + b*Log[c*x^n]]^(3/2))/(3*b*n)} -{Tan[a + b*Log[c*x^n]]^(3/2)/x, x, 13, ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + (2*Sqrt[Tan[a + b*Log[c*x^n]]])/(b*n)} -{Tan[a + b*Log[c*x^n]]^(1/2)/x, x, 12, -(ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{1/(x*Tan[a + b*Log[c*x^n]]^(1/2)), x, 12, -(ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{1/(x*Tan[a + b*Log[c*x^n]]^(3/2)), x, 13, ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - 2/(b*n*Sqrt[Tan[a + b*Log[c*x^n]]])} -{1/(x*Tan[a + b*Log[c*x^n]]^(5/2)), x, 13, ArcTan[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + Log[1 - Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Tan[a + b*Log[c*x^n]]] + Tan[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - 2/(3*b*n*Tan[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cot[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cot[a+b Log[x]]^p*) - - -{x^3*Cot[a + I*Log[x]], x, 5, (-I)*E^((2*I)*a)*x^2 - (I/4)*x^4 - I*E^((4*I)*a)*Log[E^((2*I)*a) - x^2]} -{x^2*Cot[a + I*Log[x]], x, 5, (-2*I)*E^((2*I)*a)*x - (I/3)*x^3 + (2*I)*E^((3*I)*a)*ArcTanh[x/E^(I*a)]} -{x^1*Cot[a + I*Log[x]], x, 5, (-I/2)*x^2 - I*E^((2*I)*a)*Log[E^((2*I)*a) - x^2]} -{x^0*Cot[a + I*Log[x]], x, 4, (-I)*x + (2*I)*E^(I*a)*ArcTanh[x/E^(I*a)]} -{Cot[a + I*Log[x]]/x^1, x, 2, (-I)*Log[Sin[a + I*Log[x]]]} -{Cot[a + I*Log[x]]/x^2, x, 4, (-I)/x + ((2*I)*ArcTanh[x/E^(I*a)])/E^(I*a)} -{Cot[a + I*Log[x]]/x^3, x, 4, -(I/(2*x^2)) - (I*Log[1 - E^(2*I*a)/x^2])/E^(2*I*a)} -{Cot[a + I*Log[x]]/x^4, x, 5, -(I/(3*x^3)) - (2*I)/(E^(2*I*a)*x) + (2*I*ArcTanh[x/E^(I*a)])/E^(3*I*a)} - - -{x^3*Cot[a + I*Log[x]]^2, x, 5, -2*E^((2*I)*a)*x^2 - x^4/4 - (2*E^((6*I)*a))/(E^((2*I)*a) - x^2) - 4*E^((4*I)*a)*Log[E^((2*I)*a) - x^2]} -{x^2*Cot[a + I*Log[x]]^2, x, 6, -6*E^(2*I*a)*x - x^3/3 - (2*E^(2*I*a)*x^3)/(E^(2*I*a) - x^2) + 6*E^(3*I*a)*ArcTanh[x/E^(I*a)]} -{x^1*Cot[a + I*Log[x]]^2, x, 5, -x^2/2 - (2*E^((4*I)*a))/(E^((2*I)*a) - x^2) - 2*E^((2*I)*a)*Log[E^((2*I)*a) - x^2]} -{x^0*Cot[a + I*Log[x]]^2, x, 6, -x - (2*E^(2*I*a)*x)/(E^(2*I*a) - x^2) + 2*E^(I*a)*ArcTanh[x/E^(I*a)]} -{Cot[a + I*Log[x]]^2/x^1, x, 3, I*Cot[a + I*Log[x]] - Log[x]} -{Cot[a + I*Log[x]]^2/x^2, x, 5, E^(2*I*a)/(x*(E^(2*I*a) - x^2)) - (3*x)/(E^(2*I*a) - x^2) - (2*ArcTanh[x/E^(I*a)])/E^(I*a)} -{Cot[a + I*Log[x]]^2/x^3, x, 4, 2/(E^((2*I)*a)*(1 - E^((2*I)*a)/x^2)) + 1/(2*x^2) + (2*Log[1 - E^((2*I)*a)/x^2])/E^((2*I)*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cot[a+b Log[x]]^p with m symbolic*) - - -{(e*x)^m*Cot[a + I*Log[x]]^1, x, 4, (I*(e*x)^(1 + m))/(e*(1 + m)) - (2*I*(e*x)^(1 + m)*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, E^(2*I*a)/x^2])/(e*(1 + m))} -{(e*x)^m*Cot[a + I*Log[x]]^2, x, 5, -((x*(e*x)^m)/(1 + m)) + (2*x*(e*x)^m)/(1 - E^(2*I*a)/x^2) - 2*x*(e*x)^m*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, E^(2*I*a)/x^2]} -{(e*x)^m*Cot[a + I*Log[x]]^3, x, 6, (I*(1 - m)*m*x*(e*x)^m)/(2*(1 + m)) - (I*(1 + E^(2*I*a)/x^2)^2*x*(e*x)^m)/(2*(1 - E^(2*I*a)/x^2)^2) - (I*(3 + m - (E^(2*I*a)*(1 - m))/x^2)*x*(e*x)^m)/(2*(1 - E^(2*I*a)/x^2)) + (I*(3 + 2*m + m^2)*x*(e*x)^m*Hypergeometric2F1[1, (1/2)*(-1 - m), (1 - m)/2, E^(2*I*a)/x^2])/(1 + m)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cot[a+b Log[x]]^p with p symbolic*) -(**) - - -{Cot[a + b*Log[x]]^p, x, 4, (x*(1 - E^(2*I*a)*x^(2*I*b))^p*(-((I*(1 + E^(2*I*a)*x^(2*I*b)))/(1 - E^(2*I*a)*x^(2*I*b))))^p*AppellF1[-(I/(2*b)), p, -p, 1 - I/(2*b), E^(2*I*a)*x^(2*I*b), (-E^(2*I*a))*x^(2*I*b)])/(1 + E^(2*I*a)*x^(2*I*b))^p} -{(e*x)^m*Cot[a + b*Log[x]]^p, x, 4, ((e*x)^(1 + m)*(1 - E^(2*I*a)*x^(2*I*b))^p*(-((I*(1 + E^(2*I*a)*x^(2*I*b)))/(1 - E^(2*I*a)*x^(2*I*b))))^p*AppellF1[-((I*(1 + m))/(2*b)), p, -p, 1 - (I*(1 + m))/(2*b), E^(2*I*a)*x^(2*I*b), (-E^(2*I*a))*x^(2*I*b)])/((1 + E^(2*I*a)*x^(2*I*b))^p*(e*(1 + m)))} - - -{Cot[a + 1*Log[x]]^p, x, 4, ((1 - E^(2*I*a)*x^(2*I))^p*(-((I*(1 + E^(2*I*a)*x^(2*I)))/(1 - E^(2*I*a)*x^(2*I))))^p*x*AppellF1[-(I/2), p, -p, 1 - I/2, E^(2*I*a)*x^(2*I), (-E^(2*I*a))*x^(2*I)])/(1 + E^(2*I*a)*x^(2*I))^p} -{Cot[a + 2*Log[x]]^p, x, 4, ((1 - E^(2*I*a)*x^(4*I))^p*(-((I*(1 + E^(2*I*a)*x^(4*I)))/(1 - E^(2*I*a)*x^(4*I))))^p*x*AppellF1[-(I/4), p, -p, 1 - I/4, E^(2*I*a)*x^(4*I), (-E^(2*I*a))*x^(4*I)])/(1 + E^(2*I*a)*x^(4*I))^p} -{Cot[a + 3*Log[x]]^p, x, 4, ((1 - E^(2*I*a)*x^(6*I))^p*(-((I*(1 + E^(2*I*a)*x^(6*I)))/(1 - E^(2*I*a)*x^(6*I))))^p*x*AppellF1[-(I/6), p, -p, 1 - I/6, E^(2*I*a)*x^(6*I), (-E^(2*I*a))*x^(6*I)])/(1 + E^(2*I*a)*x^(6*I))^p} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cot[a+b Log[c x^n]]^p*) - - -{x^3*Cot[d*(a + b*Log[c*x^n])], x, 4, (I/4)*x^4 - (I/2)*x^4*Hypergeometric2F1[1, (-2*I)/(b*d*n), 1 - (2*I)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]} -{x^2*Cot[d*(a + b*Log[c*x^n])], x, 4, (I/3)*x^3 - ((2*I)/3)*x^3*Hypergeometric2F1[1, ((-3*I)/2)/(b*d*n), 1 - ((3*I)/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]} -{x^1*Cot[d*(a + b*Log[c*x^n])], x, 4, (I/2)*x^2 - I*x^2*Hypergeometric2F1[1, (-I)/(b*d*n), 1 - I/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]} -{x^0*Cot[d*(a + b*Log[c*x^n])], x, 4, I*x - (2*I)*x*Hypergeometric2F1[1, (-I/2)/(b*d*n), 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)]} -{Cot[d*(a + b*Log[c*x^n])]/x^1, x, 2, Log[Sin[a*d + b*d*Log[c*x^n]]]/(b*d*n)} -{Cot[d*(a + b*Log[c*x^n])]/x^2, x, 4, -(I/x) + (2*I*Hypergeometric2F1[1, I/(2*b*d*n), 1 + I/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/x} -{Cot[d*(a + b*Log[c*x^n])]/x^3, x, 4, (-I/2)/x^2 + (I*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/x^2} - - -{x^3*Cot[d*(a + b*Log[c*x^n])]^2, x, 5, ((4*I - b*d*n)*x^4)/(4*b*d*n) + (I*x^4*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^4*Hypergeometric2F1[1, -((2*I)/(b*d*n)), 1 - (2*I)/(b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^2*Cot[d*(a + b*Log[c*x^n])]^2, x, 5, ((3*I - b*d*n)*x^3)/(3*b*d*n) + (I*x^3*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^3*Hypergeometric2F1[1, -((3*I)/(2*b*d*n)), 1 - (3*I)/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^1*Cot[d*(a + b*Log[c*x^n])]^2, x, 5, ((2*I - b*d*n)*x^2)/(2*b*d*n) + (I*x^2*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x^2*Hypergeometric2F1[1, -(I/(b*d*n)), 1 - I/(b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{x^0*Cot[d*(a + b*Log[c*x^n])]^2, x, 5, ((I - b*d*n)*x)/(b*d*n) + (I*x*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*x*Hypergeometric2F1[1, -(I/(2*b*d*n)), 1 - I/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n)} -{Cot[d*(a + b*Log[c*x^n])]^2/x^1, x, 3, -(Cot[a*d + b*d*Log[c*x^n]]/(b*d*n)) - Log[x]} -{Cot[d*(a + b*Log[c*x^n])]^2/x^2, x, 5, (1 + I/(b*d*n))/x + (I*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*x*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*Hypergeometric2F1[1, I/(2*b*d*n), 1 + I/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n*x)} -{Cot[d*(a + b*Log[c*x^n])]^2/x^3, x, 5, (1 + (2*I)/(b*d*n))/(2*x^2) + (I*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*n*x^2*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*Hypergeometric2F1[1, I/(b*d*n), 1 + I/(b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*n*x^2)} - - -{Cot[a + b*Log[c*x^n]]^3/x, x, 3, -(Cot[a + b*Log[c*x^n]]^2/(2*b*n)) - Log[Sin[a + b*Log[c*x^n]]]/(b*n)} -{Cot[a + b*Log[c*x^n]]^4/x, x, 4, Cot[a + b*Log[c*x^n]]/(b*n) - Cot[a + b*Log[c*x^n]]^3/(3*b*n) + Log[x]} -{Cot[a + b*Log[c*x^n]]^5/x, x, 4, Cot[a + b*Log[c*x^n]]^2/(2*b*n) - Cot[a + b*Log[c*x^n]]^4/(4*b*n) + Log[Sin[a + b*Log[c*x^n]]]/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cot[a+b Log[c x^n]]^p with m symbolic*) - - -{(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^1, x, 4, (I*(e*x)^(1 + m))/(e*(1 + m)) - ((2*I)*(e*x)^(1 + m)*Hypergeometric2F1[1, ((-I/2)*(1 + m))/(b*d*n), 1 - ((I/2)*(1 + m))/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)])/(e*(1 + m))} -{(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^2, x, 5, ((I*(1 + m) - b*d*n)*(e*x)^(1 + m))/(b*d*e*(1 + m)*n) + (I*(e*x)^(1 + m)*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(b*d*e*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))) - (2*I*(e*x)^(1 + m)*Hypergeometric2F1[1, -((I*(1 + m))/(2*b*d*n)), 1 - (I*(1 + m))/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b*d*e*n)} -{(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^3, x, 6, ((I*(1 + m) - b*d*n)*(1 + m + 2*I*b*d*n)*(e*x)^(1 + m))/(2*b^2*d^2*e*(1 + m)*n^2) + ((e*x)^(1 + m)*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^2)/(2*b*d*e*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^2) + (I*(e*x)^(1 + m)*((E^(2*I*a*d)*(1 + m - 2*I*b*d*n))/n + (E^(4*I*a*d)*(1 + m + 2*I*b*d*n)*(c*x^n)^(2*I*b*d))/n))/(E^(2*I*a*d)*(2*b^2*d^2*e*n*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))) - (I*(1 + 2*m + m^2 - 2*b^2*d^2*n^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, -((I*(1 + m))/(2*b*d*n)), 1 - (I*(1 + m))/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(b^2*d^2*e*(1 + m)*n^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Cot[a+b Log[c x^n]]^p with p symbolic*) - - -{Cot[d*(a + b*Log[c*x^n])]^p, x, 5, (x*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-((I*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))))^p*AppellF1[-(I/(2*b*d*n)), p, -p, 1 - I/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p} -{(e*x)^m*Cot[d*(a + b*Log[c*x^n])]^p, x, 5, ((e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(-((I*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d)))/(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))))^p*AppellF1[-((I*(1 + m))/(2*b*d*n)), p, -p, 1 - (I*(1 + m))/(2*b*d*n), E^(2*I*a*d)*(c*x^n)^(2*I*b*d), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)])/((1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*(e*(1 + m)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cot[a+b Log[c x^n]]^(p/2)*) - - -{Cot[a + b*Log[c*x^n]]^(5/2)/x, x, 13, -(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - (2*Cot[a + b*Log[c*x^n]]^(3/2))/(3*b*n) + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{Cot[a + b*Log[c*x^n]]^(3/2)/x, x, 13, -(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - (2*Sqrt[Cot[a + b*Log[c*x^n]]])/(b*n) - Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{Cot[a + b*Log[c*x^n]]^(1/2)/x, x, 12, ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{1/(x*Cot[a + b*Log[c*x^n]]^(1/2)), x, 12, ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) - ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{1/(x*Cot[a + b*Log[c*x^n]]^(3/2)), x, 13, -(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + 2/(b*n*Sqrt[Cot[a + b*Log[c*x^n]]]) + Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) - Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} -{1/(x*Cot[a + b*Log[c*x^n]]^(5/2)), x, 13, -(ArcTan[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n)) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]]]/(Sqrt[2]*b*n) + 2/(3*b*n*Cot[a + b*Log[c*x^n]]^(3/2)) - Log[1 - Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n) + Log[1 + Sqrt[2]*Sqrt[Cot[a + b*Log[c*x^n]]] + Cot[a + b*Log[c*x^n]]]/(2*Sqrt[2]*b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p*) - - -{x^2*Sec[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x^3*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - (3*I)/(b*n)), (3/2)*(1 - I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(3 + I*b*n)} -{x^1*Sec[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - (2*I)/(b*n)), (1/2)*(3 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(2 + I*b*n)} -{x^0*Sec[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - I/(b*n)), (1/2)*(3 - I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + I*b*n)} -{Sec[a + b*Log[c*x^n]]/x^1, x, 2, ArcTanh[Sin[a + b*Log[c*x^n]]]/(b*n)} -{Sec[a + b*Log[c*x^n]]/x^2, x, 3, -((2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 + I/(b*n)), (1/2)*(3 + I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - I*b*n)*x))} -{Sec[a + b*Log[c*x^n]]/x^3, x, 3, -((2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 + (2*I)/(b*n)), (1/2)*(3 + (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - I*b*n)*x^2))} - - -{x^2*Sec[a + b*Log[c*x^n]]^2, x, 3, (4*E^(2*I*a)*x^3*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, (1/2)*(2 - (3*I)/(b*n)), (1/2)*(4 - (3*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(3 + 2*I*b*n)} -{x^1*Sec[a + b*Log[c*x^n]]^2, x, 3, (2*E^(2*I*a)*x^2*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, 1 - I/(b*n), 2 - I/(b*n), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + I*b*n)} -{x^0*Sec[a + b*Log[c*x^n]]^2, x, 3, (4*E^(2*I*a)*x*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, (1/2)*(2 - I/(b*n)), (1/2)*(4 - I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + 2*I*b*n)} -{Sec[a + b*Log[c*x^n]]^2/x^1, x, 3, Tan[a + b*Log[c*x^n]]/(b*n)} -{Sec[a + b*Log[c*x^n]]^2/x^2, x, 3, -((4*E^(2*I*a)*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, (1/2)*(2 + I/(b*n)), (1/2)*(4 + I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - 2*I*b*n)*x))} -{Sec[a + b*Log[c*x^n]]^2/x^3, x, 3, -((2*E^(2*I*a)*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, 1 + I/(b*n), 2 + I/(b*n), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - I*b*n)*x^2))} - - -{x^1*Sec[a + b*Log[c*x^n]]^3, x, 3, (8*E^(3*I*a)*x^2*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, (1/2)*(3 - (2*I)/(b*n)), (1/2)*(5 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(2 + 3*I*b*n)} -{x^0*Sec[a + b*Log[c*x^n]]^3, x, 3, (8*E^(3*I*a)*x*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, (1/2)*(3 - I/(b*n)), (1/2)*(5 - I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + 3*I*b*n)} -{Sec[a + b*Log[c*x^n]]^3/x^1, x, 3, ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]])/(2*b*n)} -{Sec[a + b*Log[c*x^n]]^3/x^2, x, 3, -((8*E^(3*I*a)*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, (1/2)*(3 + I/(b*n)), (1/2)*(5 + I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - 3*I*b*n)*x))} -{Sec[a + b*Log[c*x^n]]^3/x^3, x, 3, -((8*E^(3*I*a)*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, (1/2)*(3 + (2*I)/(b*n)), (1/2)*(5 + (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - 3*I*b*n)*x^2))} - - -{x^1*Sec[a + b*Log[c*x^n]]^4, x, 3, (8*E^(4*I*a)*x^2*(c*x^n)^(4*I*b)*Hypergeometric2F1[4, 2 - I/(b*n), 3 - I/(b*n), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + 2*I*b*n)} -{x^0*Sec[a + b*Log[c*x^n]]^4, x, 3, (16*E^(4*I*a)*x*(c*x^n)^(4*I*b)*Hypergeometric2F1[4, (1/2)*(4 - I/(b*n)), (1/2)*(6 - I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + 4*I*b*n)} -{Sec[a + b*Log[c*x^n]]^4/x^1, x, 3, Tan[a + b*Log[c*x^n]]/(b*n) + Tan[a + b*Log[c*x^n]]^3/(3*b*n)} -{Sec[a + b*Log[c*x^n]]^4/x^2, x, 3, -((16*E^(4*I*a)*(c*x^n)^(4*I*b)*Hypergeometric2F1[4, (1/2)*(4 + I/(b*n)), (1/2)*(6 + I/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - 4*I*b*n)*x))} -{Sec[a + b*Log[c*x^n]]^4/x^3, x, 3, -((8*E^(4*I*a)*(c*x^n)^(4*I*b)*Hypergeometric2F1[4, 2 + I/(b*n), 3 + I/(b*n), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((1 - 2*I*b*n)*x^2))} - - -{2*b^2*n^2*Sec[a + b*Log[c*x^n]]^3 - (1 + b^2*n^2)*Sec[a + b*Log[c*x^n]], x, -7, (-x)*Sec[a + b*Log[c*x^n]] + b*n*x*Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when b^2 n^2 (p-2)^2+(m+1)^2=0*) - - -{x^m*Sec[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]]^3, x, -3, (x^(1 + m)*Sec[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]])/(2*(1 + m)) + (x^(1 + m)*Sec[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]]*Tan[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]])/(2*Sqrt[-(1 + m)^2])} - - -{x^1*Sec[a + 2*Log[c*x^I]]^3, x, 3, (E^(I*a)*(c*x^I)^(2*I)*x^2)/(1 + E^(2*I*a)*(c*x^I)^(4*I))^2} -{x^0*Sec[a + 2*Log[c*x^(I/2)]]^3, x, 3, (1/2)*x*Sec[a + 2*Log[c*x^(I/2)]] - (1/2)*I*x*Sec[a + 2*Log[c*x^(I/2)]]*Tan[a + 2*Log[c*x^(I/2)]], (2*E^(I*a)*(c*x^(I/2))^(2*I)*x)/(1 + E^(2*I*a)*(c*x^(I/2))^(4*I))^2} - - -{Sec[a + 2*Log[c/x^(I/2)]]^3, x, 3, (2*E^(3*I*a)*(c/x^(I/2))^(6*I)*x)/(1 + E^(2*I*a)*(c/x^(I/2))^(4*I))^2} - - -{Sec[a + I/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, ((2 - p)*x*(1 + E^(2*I*a)*(c*x^n)^(2/(n*(2 - p))))*Sec[a - (I*Log[c*x^n])/(n*(2 - p))]^p)/(E^(2*I*a)*(c*x^n)^(2/(n*(2 - p)))*(2*(1 - p)))} -{Sec[a - I/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, ((2 - p)*x*(1 + E^(2*I*a)/(c*x^n)^(2/(n*(2 - p))))*Sec[a + (I*Log[c*x^n])/(n*(2 - p))]^p)/(2*(1 - p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^0*Sqrt[Sec[a + b*Log[c*x^n]]], x, 3, (2*x*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Hypergeometric2F1[1/2, (1/4)*(1 - (2*I)/(b*n)), (1/4)*(5 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sqrt[Sec[a + b*Log[c*x^n]]])/(2 + I*b*n)} -{Sqrt[Sec[a + b*Log[c*x^n]]]/x^1, x, 3, (2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(b*n)} - - -{x^0*Sec[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Hypergeometric2F1[3/2, (1/4)*(3 - (2*I)/(b*n)), (1/4)*(7 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^(3/2))/(2 + 3*I*b*n)} -{Sec[a + b*Log[c*x^n]]^(3/2)/x^1, x, 4, -((2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(1/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(b*n)) + (2*Sqrt[Sec[a + b*Log[c*x^n]]]*Sin[a + b*Log[c*x^n]])/(b*n)} - - -{x^0*Sec[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Hypergeometric2F1[5/2, (1/4)*(5 - (2*I)/(b*n)), (1/4)*(9 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^(5/2))/(2 + 5*I*b*n)} -{Sec[a + b*Log[c*x^n]]^(5/2)/x^1, x, 4, (2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(3*b*n) + (2*Sec[a + b*Log[c*x^n]]^(3/2)*Sin[a + b*Log[c*x^n]])/(3*b*n)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^0/Sqrt[Sec[a + b*Log[c*x^n]]], x, 3, (2*x*Hypergeometric2F1[-(1/2), -((2*I + b*n)/(4*b*n)), (1/4)*(3 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - I*b*n)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sec[a + b*Log[c*x^n]]])} -{1/(x*Sqrt[Sec[a + b*Log[c*x^n]]]), x, 3, (2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(b*n)} - - -{x^0/Sec[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*Hypergeometric2F1[-(3/2), (1/4)*(-3 - (2*I)/(b*n)), (1/4)*(1 - (2*I)/(b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - 3*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Sec[a + b*Log[c*x^n]]^(3/2))} -{1/(x*Sec[a + b*Log[c*x^n]]^(3/2)), x, 4, (2*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(3*b*n) + (2*Sin[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Sec[a + b*Log[c*x^n]]])} - - -{x^0/Sec[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*Hypergeometric2F1[-(5/2), (1/4)*(-5 - (2*I)/(b*n)), -((2*I + b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 - 5*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Sec[a + b*Log[c*x^n]]^(5/2))} -{1/(x*Sec[a + b*Log[c*x^n]]^(5/2)), x, 4, (6*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(1/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(5*b*n) + (2*Sin[a + b*Log[c*x^n]])/(5*b*n*Sec[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when m symbolic*) - - -{x^m*Sec[a + b*Log[c*x^n]]^3, x, 3, (8*E^(3*I*a)*x^(1 + m)*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, -((I*(1 + m) - 3*b*n)/(2*b*n)), -((I*(1 + m) - 5*b*n)/(2*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + m + 3*I*b*n)} -{x^m*Sec[a + b*Log[c*x^n]]^2, x, 3, (4*E^(2*I*a)*x^(1 + m)*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, -((I*(1 + m) - 2*b*n)/(2*b*n)), -((I*(1 + m) - 4*b*n)/(2*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + m + 2*I*b*n)} -{x^m*Sec[a + b*Log[c*x^n]]^1, x, 3, (2*E^(I*a)*x^(1 + m)*(c*x^n)^(I*b)*Hypergeometric2F1[1, -((I + I*m - b*n)/(2*b*n)), -((I*(1 + m) - 3*b*n)/(2*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/(1 + m + I*b*n)} - - -{x^m*Sec[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x^(1 + m)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Hypergeometric2F1[5/2, -((2*I + 2*I*m - 5*b*n)/(4*b*n)), -((2*I + 2*I*m - 9*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^(5/2))/(2 + 2*m + 5*I*b*n)} -{x^m*Sec[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Hypergeometric2F1[3/2, -((2*I + 2*I*m - 3*b*n)/(4*b*n)), -((2*I + 2*I*m - 7*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^(3/2))/(2 + 2*m + 3*I*b*n)} -{x^m*Sec[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Hypergeometric2F1[1/2, -((2*I + 2*I*m - b*n)/(4*b*n)), -((2*I + 2*I*m - 5*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sqrt[Sec[a + b*Log[c*x^n]]])/(2 + 2*m + I*b*n)} -{x^m/Sec[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Hypergeometric2F1[-(1/2), -((2*I + 2*I*m + b*n)/(4*b*n)), -((2*I + 2*I*m - 3*b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m - I*b*n)*Sqrt[1 + E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Sec[a + b*Log[c*x^n]]])} -{x^m/Sec[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*Hypergeometric2F1[-(3/2), -((2*I + 2*I*m + 3*b*n)/(4*b*n)), -((2*I + 2*I*m - b*n)/(4*b*n)), (-E^(2*I*a))*(c*x^n)^(2*I*b)])/((2 + 2*m - 3*I*b*n)*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Sec[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Sec[d (a+b Log[c x^n])]^p when p symbolic*) - - -{(e*x)^m*Sec[d*(a + b*Log[c*x^n])]^p, x, 3, ((e*x)^(1 + m)*(1 + E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*Hypergeometric2F1[p, -((I + I*m - b*d*n*p)/(2*b*d*n)), (1/2)*(2 - (I*(1 + m))/(b*d*n) + p), (-E^(2*I*a*d))*(c*x^n)^(2*I*b*d)]*Sec[d*(a + b*Log[c*x^n])]^p)/(e*(1 + m + I*b*d*n*p))} - - -{x^1*Sec[a + b*Log[c*x^n]]^p, x, 3, (x^2*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^p*Hypergeometric2F1[p, (1/2)*(-((2*I)/(b*n)) + p), (1/2)*(2 - (2*I)/(b*n) + p), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^p)/(2 + I*b*n*p)} -{x^0*Sec[a + b*Log[c*x^n]]^p, x, 3, (x*(1 + E^(2*I*a)*(c*x^n)^(2*I*b))^p*Hypergeometric2F1[p, -((I - b*n*p)/(2*b*n)), (1/2)*(2 - I/(b*n) + p), (-E^(2*I*a))*(c*x^n)^(2*I*b)]*Sec[a + b*Log[c*x^n]]^p)/(1 + I*b*n*p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p*) - - -{x^2*Csc[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x^3*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - (3*I)/(b*n)), (3/2)*(1 - I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(3*I - b*n)} -{x^1*Csc[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - (2*I)/(b*n)), (1/2)*(3 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2*I - b*n)} -{x^0*Csc[a + b*Log[c*x^n]], x, 3, (2*E^(I*a)*x*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 - I/(b*n)), (1/2)*(3 - I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(I - b*n)} -{Csc[a + b*Log[c*x^n]]/x^1, x, 2, -(ArcTanh[Cos[a + b*Log[c*x^n]]]/(b*n))} -{Csc[a + b*Log[c*x^n]]/x^2, x, 3, -((2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 + I/(b*n)), (1/2)*(3 + I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((I + b*n)*x))} -{Csc[a + b*Log[c*x^n]]/x^3, x, 3, -((2*E^(I*a)*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1/2)*(1 + (2*I)/(b*n)), (1/2)*(3 + (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2*I + b*n)*x^2))} - - -{x^0*Csc[a + b*Log[c*x^n]]^2, x, 3, -((4*E^(2*I*a)*x*(c*x^n)^(2*I*b)*Hypergeometric2F1[2, (1/2)*(2 - I/(b*n)), (1/2)*(4 - I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(1 + 2*I*b*n))} -{Csc[a + b*Log[c*x^n]]^2/x^1, x, 3, -(Cot[a + b*Log[c*x^n]]/(b*n))} - - -{x^0*Csc[a + b*Log[c*x^n]]^3, x, 3, -((8*E^(3*I*a)*x*(c*x^n)^(3*I*b)*Hypergeometric2F1[3, (1/2)*(3 - I/(b*n)), (1/2)*(5 - I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(I - 3*b*n))} -{Csc[a + b*Log[c*x^n]]^3/x^1, x, 3, -(ArcTanh[Cos[a + b*Log[c*x^n]]]/(2*b*n)) - (Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]])/(2*b*n)} - - -{x^0*Csc[a + b*Log[c*x^n]]^4, x, 3, (16*E^(4*I*a)*x*(c*x^n)^(4*I*b)*Hypergeometric2F1[4, (1/2)*(4 - I/(b*n)), (1/2)*(6 - I/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(1 + 4*I*b*n)} -{Csc[a + b*Log[c*x^n]]^4/x^1, x, 3, -(Cot[a + b*Log[c*x^n]]/(b*n)) - Cot[a + b*Log[c*x^n]]^3/(3*b*n)} - - -{2*b^2*n^2*Csc[a + b*Log[c*x^n]]^3 - (1 + b^2*n^2)*Csc[a + b*Log[c*x^n]], x, -7, (-x)*Csc[a + b*Log[c*x^n]] - b*n*x*Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when b^2 n^2 (p-2)^2+(m+1)^2=0*) - - -{x^m*Csc[a + 2*Log[c*x^(Sqrt[-(1 + m)^2]/2)]]^3, x, -3, (x^(1 + m)*Csc[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]])/(2*(1 + m)) - (x^(1 + m)*Cot[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]]*Csc[a + 2*Log[c*x^((1/2)*Sqrt[-(1 + m)^2])]])/(2*Sqrt[-(1 + m)^2])} - - -{x^1*Csc[a + 2*Log[c*x^I]]^3, x, 3, -((I*E^(I*a)*(c*x^I)^(2*I)*x^2)/(1 - E^(2*I*a)*(c*x^I)^(4*I))^2)} -{Csc[a + 2*Log[c*x^(I/2)]]^3, x, 3, (1/2)*x*Csc[a + 2*Log[c*x^(I/2)]] + (1/2)*I*x*Cot[a + 2*Log[c*x^(I/2)]]*Csc[a + 2*Log[c*x^(I/2)]], -((2*I*E^(I*a)*(c*x^(I/2))^(2*I)*x)/(1 - E^(2*I*a)*(c*x^(I/2))^(4*I))^2)} - - -{Csc[a + 2*Log[c/x^(I/2)]]^3, x, 3, (2*I*E^(3*I*a)*(c/x^(I/2))^(6*I)*x)/(1 - E^(2*I*a)*(c/x^(I/2))^(4*I))^2} - - -{Csc[a + I/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, -(((2 - p)*x*(1 - E^(2*I*a)*(c*x^n)^(2/(n*(2 - p))))*Csc[a - (I*Log[c*x^n])/(n*(2 - p))]^p)/(E^(2*I*a)*(c*x^n)^(2/(n*(2 - p)))*(2*(1 - p))))} -{Csc[a - I/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, ((2 - p)*x*(1 - E^(2*I*a)/(c*x^n)^(2/(n*(2 - p))))*Csc[a + (I*Log[c*x^n])/(n*(2 - p))]^p)/(2*(1 - p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^0*Sqrt[Csc[a + b*Log[c*x^n]]], x, 3, (2*x*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Csc[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, (1/4)*(1 - (2*I)/(b*n)), (1/4)*(5 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + I*b*n)} -{Sqrt[Csc[a + b*Log[c*x^n]]]/x, x, 3, (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n)} - - -{x^0*Csc[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[3/2, (1/4)*(3 - (2*I)/(b*n)), (1/4)*(7 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + 3*I*b*n)} -{Csc[a + b*Log[c*x^n]]^(3/2)/x, x, 4, -((2*Cos[a + b*Log[c*x^n]]*Sqrt[Csc[a + b*Log[c*x^n]]])/(b*n)) - (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n)} - - -{x^0*Csc[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Csc[a + b*Log[c*x^n]]^(5/2)*Hypergeometric2F1[5/2, (1/4)*(5 - (2*I)/(b*n)), (1/4)*(9 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + 5*I*b*n)} -{Csc[a + b*Log[c*x^n]]^(5/2)/x, x, 4, -((2*Cos[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]]^(3/2))/(3*b*n)) + (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(3*b*n)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^0/Sqrt[Csc[a + b*Log[c*x^n]]], x, 3, (2*x*Hypergeometric2F1[-(1/2), -((2*I + b*n)/(4*b*n)), (1/4)*(3 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 - I*b*n)*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Csc[a + b*Log[c*x^n]]])} -{1/(x*Sqrt[Csc[a + b*Log[c*x^n]]]), x, 3, (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(b*n)} - - -{x^0/Csc[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x*Hypergeometric2F1[-(3/2), (1/4)*(-3 - (2*I)/(b*n)), (1/4)*(1 - (2*I)/(b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 - 3*I*b*n)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2))} -{1/(x*Csc[a + b*Log[c*x^n]]^(3/2)), x, 4, -((2*Cos[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Csc[a + b*Log[c*x^n]]])) + (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(3*b*n)} - - -{x^0/Csc[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x*Hypergeometric2F1[-(5/2), (1/4)*(-5 - (2*I)/(b*n)), -((2*I + b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 - 5*I*b*n)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Csc[a + b*Log[c*x^n]]^(5/2))} -{1/(x*Csc[a + b*Log[c*x^n]]^(5/2)), x, 4, -((2*Cos[a + b*Log[c*x^n]])/(5*b*n*Csc[a + b*Log[c*x^n]]^(3/2))) + (6*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(1/2)*(a - Pi/2 + b*Log[c*x^n]), 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(5*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when m symbolic*) - - -{(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^3, x, 3, -((8*E^(3*I*a*d)*(e*x)^(1 + m)*(c*x^n)^(3*I*b*d)*Hypergeometric2F1[3, -((I*(1 + m) - 3*b*d*n)/(2*b*d*n)), -((I*(1 + m) - 5*b*d*n)/(2*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(I*(1 + m) - 3*b*d*n)))} -{(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^2, x, 3, -((4*E^(2*I*a*d)*(e*x)^(1 + m)*(c*x^n)^(2*I*b*d)*Hypergeometric2F1[2, -((I*(1 + m) - 2*b*d*n)/(2*b*d*n)), -((I*(1 + m) - 4*b*d*n)/(2*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(1 + m + 2*I*b*d*n)))} -{(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^1, x, 3, (2*E^(I*a*d)*(e*x)^(1 + m)*(c*x^n)^(I*b*d)*Hypergeometric2F1[1, -((I + I*m - b*d*n)/(2*b*d*n)), -((I*(1 + m) - 3*b*d*n)/(2*b*d*n)), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(I*(1 + m) - b*d*n))} - - -{x^m*Csc[a + b*Log[c*x^n]]^(5/2), x, 3, (2*x^(1 + m)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(5/2)*Csc[a + b*Log[c*x^n]]^(5/2)*Hypergeometric2F1[5/2, -((2*I + 2*I*m - 5*b*n)/(4*b*n)), -((2*I + 2*I*m - 9*b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + 2*m + 5*I*b*n)} -{x^m*Csc[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[3/2, -((2*I + 2*I*m - 3*b*n)/(4*b*n)), -((2*I + 2*I*m - 7*b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + 2*m + 3*I*b*n)} -{x^m*Csc[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Csc[a + b*Log[c*x^n]]]*Hypergeometric2F1[1/2, -((2*I + 2*I*m - b*n)/(4*b*n)), -((2*I + 2*I*m - 5*b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + 2*m + I*b*n)} -{x^m/Csc[a + b*Log[c*x^n]]^(1/2), x, 3, (2*x^(1 + m)*Hypergeometric2F1[-(1/2), -((2*I + 2*I*m + b*n)/(4*b*n)), -((2*I + 2*I*m - 3*b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 + 2*m - I*b*n)*Sqrt[1 - E^(2*I*a)*(c*x^n)^(2*I*b)]*Sqrt[Csc[a + b*Log[c*x^n]]])} -{x^m/Csc[a + b*Log[c*x^n]]^(3/2), x, 3, (2*x^(1 + m)*Hypergeometric2F1[-(3/2), -((2*I + 2*I*m + 3*b*n)/(4*b*n)), -((2*I + 2*I*m - b*n)/(4*b*n)), E^(2*I*a)*(c*x^n)^(2*I*b)])/((2 + 2*m - 3*I*b*n)*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Csc[d (a+b Log[c x^n])]^p when p symbolic*) - - -{(e*x)^m*Csc[d*(a + b*Log[c*x^n])]^p, x, 3, ((e*x)^(1 + m)*(1 - E^(2*I*a*d)*(c*x^n)^(2*I*b*d))^p*Csc[d*(a + b*Log[c*x^n])]^p*Hypergeometric2F1[p, -((I + I*m - b*d*n*p)/(2*b*d*n)), (1/2)*(2 - (I*(1 + m))/(b*d*n) + p), E^(2*I*a*d)*(c*x^n)^(2*I*b*d)])/(e*(1 + m + I*b*d*n*p))} - - -{x^1*Csc[a + b*Log[c*x^n]]^p, x, 3, (x^2*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*Csc[a + b*Log[c*x^n]]^p*Hypergeometric2F1[p, (1/2)*(-((2*I)/(b*n)) + p), (1/2)*(2 - (2*I)/(b*n) + p), E^(2*I*a)*(c*x^n)^(2*I*b)])/(2 + I*b*n*p)} -{x^0*Csc[a + b*Log[c*x^n]]^p, x, 3, (x*(1 - E^(2*I*a)*(c*x^n)^(2*I*b))^p*Csc[a + b*Log[c*x^n]]^p*Hypergeometric2F1[p, -((I - b*n*p)/(2*b*n)), (1/2)*(2 - I/(b*n) + p), E^(2*I*a)*(c*x^n)^(2*I*b)])/(1 + I*b*n*p)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.6 f^(a+b x+c x^2) trig(d+e x+f x^2)^n.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.6 f^(a+b x+c x^2) trig(d+e x+f x^2)^n.m deleted file mode 100644 index 9a326b6..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.6 f^(a+b x+c x^2) trig(d+e x+f x^2)^n.m +++ /dev/null @@ -1,282 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands involving products of exponentials and trig functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Trig[d+e x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Sin[d+e x]^n*) - - -{F^(c*(a + b*x))*Sin[d + e*x]^n, x, 2, -((F^(c*(a + b*x))*Hypergeometric2F1[-n, -((e*n + I*b*c*Log[F])/(2*e)), (1/2)*(2 - n - (I*b*c*Log[F])/e), E^(2*I*(d + e*x))]*Sin[d + e*x]^n)/((1 - E^(2*I*(d + e*x)))^n*(I*e*n - b*c*Log[F])))} - - -{F^(c*(a + b*x))*Sin[d + e*x]^3, x, 2, -((6*e^3*F^(c*(a + b*x))*Cos[d + e*x])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4)) + (6*b*c*e^2*F^(c*(a + b*x))*Log[F]*Sin[d + e*x])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) - (3*e*F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x]^2)/(9*e^2 + b^2*c^2*Log[F]^2) + (b*c*F^(c*(a + b*x))*Log[F]*Sin[d + e*x]^3)/(9*e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sin[d + e*x]^2, x, 2, (2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) - (2*e*F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2) + (b*c*F^(c*(a + b*x))*Log[F]*Sin[d + e*x]^2)/(4*e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sin[d + e*x]^1, x, 1, -((e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)) + (b*c*F^(c*(a + b*x))*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Csc[d + e*x]^1, x, 1, -((2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), (1/2)*(3 - (I*b*c*Log[F])/e), E^(2*I*(d + e*x))])/(e - I*b*c*Log[F]))} -{F^(c*(a + b*x))*Csc[d + e*x]^2, x, 1, -((4*E^(2*I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/(2*e), 2 - (I*b*c*Log[F])/(2*e), E^(2*I*(d + e*x))])/(2*I*e + b*c*Log[F]))} -{F^(c*(a + b*x))*Csc[d + e*x]^3, x, 2, -((F^(c*(a + b*x))*Cot[d + e*x]*Csc[d + e*x])/(2*e)) - (b*c*F^(c*(a + b*x))*Csc[d + e*x]*Log[F])/(2*e^2) - (E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), (1/2)*(3 - (I*b*c*Log[F])/e), E^(2*I*(d + e*x))]*(e + I*b*c*Log[F]))/e^2} -{F^(c*(a + b*x))*Csc[d + e*x]^4, x, 2, -((F^(c*(a + b*x))*Cot[d + e*x]*Csc[d + e*x]^2)/(3*e)) - (b*c*F^(c*(a + b*x))*Csc[d + e*x]^2*Log[F])/(6*e^2) + (2*E^(2*I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/(2*e), 2 - (I*b*c*Log[F])/(2*e), E^(2*I*(d + e*x))]*(2*I*e - b*c*Log[F]))/(3*e^2)} - - -{E^x*Sin[x]^4, x, 3, (24*E^x)/85 - (24/85)*E^x*Cos[x]*Sin[x] + (12/85)*E^x*Sin[x]^2 - (4/17)*E^x*Cos[x]*Sin[x]^3 + (1/17)*E^x*Sin[x]^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Cos[d+e x]^n*) - - -{F^(c*(a + b*x))*Cos[d + e*x]^n, x, 2, -((F^(c*(a + b*x))*Cos[d + e*x]^n*Hypergeometric2F1[-n, -((e*n + I*b*c*Log[F])/(2*e)), (1/2)*(2 - n - (I*b*c*Log[F])/e), -E^(2*I*(d + e*x))])/((1 + E^(2*I*(d + e*x)))^n*(I*e*n - b*c*Log[F])))} - - -{F^(c*(a + b*x))*Cos[d + e*x]^3, x, 2, (b*c*F^(c*(a + b*x))*Cos[d + e*x]^3*Log[F])/(9*e^2 + b^2*c^2*Log[F]^2) + (6*b*c*e^2*F^(c*(a + b*x))*Cos[d + e*x]*Log[F])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*Cos[d + e*x]^2*Sin[d + e*x])/(9*e^2 + b^2*c^2*Log[F]^2) + (6*e^3*F^(c*(a + b*x))*Sin[d + e*x])/(9*e^4 + 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4)} -{F^(c*(a + b*x))*Cos[d + e*x]^2, x, 2, (2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) + (b*c*F^(c*(a + b*x))*Cos[d + e*x]^2*Log[F])/(4*e^2 + b^2*c^2*Log[F]^2) + (2*e*F^(c*(a + b*x))*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Cos[d + e*x]^1, x, 1, (b*c*F^(c*(a + b*x))*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F]^2) + (e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sec[d + e*x]^1, x, 1, (2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), (1/2)*(3 - (I*b*c*Log[F])/e), -E^(2*I*(d + e*x))])/(I*e + b*c*Log[F])} -{F^(c*(a + b*x))*Sec[d + e*x]^2, x, 1, (4*E^(2*I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/(2*e), 2 - (I*b*c*Log[F])/(2*e), -E^(2*I*(d + e*x))])/(2*I*e + b*c*Log[F])} -{F^(c*(a + b*x))*Sec[d + e*x]^3, x, 2, -((E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[1, (e - I*b*c*Log[F])/(2*e), (1/2)*(3 - (I*b*c*Log[F])/e), -E^(2*I*(d + e*x))]*(I*e - b*c*Log[F]))/e^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sec[d + e*x])/(2*e^2) + (F^(c*(a + b*x))*Sec[d + e*x]*Tan[d + e*x])/(2*e)} -{F^(c*(a + b*x))*Sec[d + e*x]^4, x, 2, -((2*E^(2*I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/(2*e), 2 - (I*b*c*Log[F])/(2*e), -E^(2*I*(d + e*x))]*(2*I*e - b*c*Log[F]))/(3*e^2)) - (b*c*F^(c*(a + b*x))*Log[F]*Sec[d + e*x]^2)/(6*e^2) + (F^(c*(a + b*x))*Sec[d + e*x]^2*Tan[d + e*x])/(3*e)} - - -{E^x*Cos[x]^4, x, 3, (24*E^x)/85 + (12/85)*E^x*Cos[x]^2 + (1/17)*E^x*Cos[x]^4 + (24/85)*E^x*Cos[x]*Sin[x] + (4/17)*E^x*Cos[x]^3*Sin[x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Tan[d+e x]^n*) - - -{E^(c*(a + b*x))*Tan[d + e*x]^3, x, 6, (I*E^(c*(a + b*x)))/(b*c) - (6*I*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c) + (12*I*E^(c*(a + b*x))*Hypergeometric2F1[2, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c) - (8*I*E^(c*(a + b*x))*Hypergeometric2F1[3, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Tan[d + e*x]^2, x, 5, -(E^(c*(a + b*x))/(b*c)) + (4*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[2, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Tan[d + e*x]^1, x, 4, -((I*E^(c*(a + b*x)))/(b*c)) + (2*I*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), -E^(2*I*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Cot[d + e*x]^1, x, 4, (I*E^(c*(a + b*x)))/(b*c) - (2*I*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Cot[d + e*x]^2, x, 5, -(E^(c*(a + b*x))/(b*c)) + (4*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[2, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Cot[d + e*x]^3, x, 6, -((I*E^(c*(a + b*x)))/(b*c)) + (6*I*E^(c*(a + b*x))*Hypergeometric2F1[1, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c) - (12*I*E^(c*(a + b*x))*Hypergeometric2F1[2, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c) + (8*I*E^(c*(a + b*x))*Hypergeometric2F1[3, -((I*b*c)/(2*e)), 1 - (I*b*c)/(2*e), E^(2*I*(d + e*x))])/(b*c)} - - -{F^(a + b*x)*Tan[Pi/4 - (c + d*x)/2], x, 5, (I*F^(a + b*x))/(b*Log[F]) - (2*I*F^(a + b*x)*Hypergeometric2F1[1, -((I*b*Log[F])/d), 1 - (I*b*Log[F])/d, I*E^(I*(c + d*x))])/(b*Log[F])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Sec[d+e x]^n*) - - -{F^(c*(a + b*x))*Sec[d + e*x]^n, x, 2, ((1 + E^(2*I*(d + e*x)))^n*F^(a*c + b*c*x)*Hypergeometric2F1[n, (e*n - I*b*c*Log[F])/(2*e), (1/2)*(2 + n - (I*b*c*Log[F])/e), -E^(2*I*(d + e*x))]*Sec[d + e*x]^n)/(I*e*n + b*c*Log[F])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Csc[d+e x]^n*) - - -{F^(c*(a + b*x))*Csc[d + e*x]^n, x, 2, -(((1 - E^(-2*I*(d + e*x)))^n*F^(a*c + b*c*x)*Csc[d + e*x]^n*Hypergeometric2F1[n, (e*n + I*b*c*Log[F])/(2*e), (1/2)*(2 + n + (I*b*c*Log[F])/e), E^(-2*I*(d + e*x))])/(I*e*n - b*c*Log[F]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m F^(c (a+b x)) Trig[d+e x]^n*) - - -(* {(f*x)^m*F^(c*(a + b*x))*Sin[d + e*x]^3, x, 10, -((3*F^(a*c)*(f*x)^m*Gamma[1 + m, x*(I*e - b*c*Log[F])])/(E^(I*d)*(x*(I*e - b*c*Log[F]))^m*(8*(e + I*b*c*Log[F])))) + (F^(a*c)*(f*x)^m*Gamma[1 + m, x*(3*I*e - b*c*Log[F])])/(E^(3*I*d)*(x*(3*I*e - b*c*Log[F]))^m*(8*(3*e + I*b*c*Log[F]))) - (3*E^(I*d)*F^(a*c)*(f*x)^m*Gamma[1 + m, (-x)*(I*e + b*c*Log[F])])/(((-x)*(I*e + b*c*Log[F]))^m*(8*(e - I*b*c*Log[F]))) + (E^(3*I*d)*F^(a*c)*(f*x)^m*Gamma[1 + m, (-x)*(3*I*e + b*c*Log[F])])/(((-x)*(3*I*e + b*c*Log[F]))^m*(8*(3*e - I*b*c*Log[F])))} -{(f*x)^m*F^(c*(a + b*x))*Sin[d + e*x]^2, x, 7, (F^(a*c)*(f*x)^m*Gamma[1 + m, (-b)*c*x*Log[F]])/(((-b)*c*x*Log[F])^m*(2*b*c*Log[F])) + (F^(a*c)*(f*x)^m*Gamma[1 + m, x*(2*I*e - b*c*Log[F])])/(E^(2*I*d)*(x*(2*I*e - b*c*Log[F]))^m*(4*(2*I*e - b*c*Log[F]))) - (E^(2*I*d)*F^(a*c)*(f*x)^m*Gamma[1 + m, (-x)*(2*I*e + b*c*Log[F])])/(((-x)*(2*I*e + b*c*Log[F]))^m*(4*(2*I*e + b*c*Log[F])))} *) -{(f*x)^m*F^(c*(a + b*x))*Sin[d + e*x]^1, x, -1, -((F^(a*c)*(f*x)^m*Gamma[1 + m, x*(I*e - b*c*Log[F])])/(E^(I*d)*(x*(I*e - b*c*Log[F]))^m*(2*(e + I*b*c*Log[F])))) - (E^(I*d)*F^(a*c)*(f*x)^m*Gamma[1 + m, (-x)*(I*e + b*c*Log[F])])/(((-x)*(I*e + b*c*Log[F]))^m*(2*(e - I*b*c*Log[F])))} -{(f*x)^m*F^(c*(a + b*x))/Sin[d + e*x]^1, x, 1, CannotIntegrate[F^(a*c + b*c*x)*(f*x)^m*Csc[d + e*x], x]} -{(f*x)^m*F^(c*(a + b*x))/Sin[d + e*x]^2, x, 1, CannotIntegrate[F^(a*c + b*c*x)*(f*x)^m*Csc[d + e*x]^2, x]} - - -{f*F^(c*(a + b*x))*(f*x)^(-2 + m)*(e*x*Cos[d + e*x] + (-1 + m + b*c*x*Log[F])*Sin[d + e*x]), x, 11, F^(a*c + b*c*x)*(f*x)^(-1 + m)*Sin[d + e*x]} -{f*F^(c*(a + b*x))*(f*x)^m*(e*x*Cos[d + e*x] + (1 + m + b*c*x*Log[F])*Sin[d + e*x]), x, -6, f*F^(c*(a + b*x))*x*(f*x)^m*Sin[d + e*x]} -{(F^(c*(a + b*x))*(f*x)^m*(e*x*Cos[d + e*x] + (m + b*c*x*Log[F])*Sin[d + e*x]))/x, x, 7, F^(a*c + b*c*x)*(f*x)^m*Sin[d + e*x]} - - -(* {F^(c*(a + b*x))*x*(e*x*Cos[d + e*x] + (2 + b*c*x*Log[F])*Sin[d + e*x]), x, 0, F^(c*(a + b*x))*x^2*Sin[d + e*x]} *) -{F^(c*(a + b*x))*(e*x*Cos[d + e*x] + (1 + b*c*x*Log[F])*Sin[d + e*x]), x, 14, F^(c*(a + b*x))*x*Sin[d + e*x], (e^3*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)^2 + (b^2*c^2*e*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F]^2)/(e^2 + b^2*c^2*Log[F]^2)^2 - (e*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - (b*c*e^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)^2 - (b^3*c^3*F^(a*c + b*c*x)*Log[F]^3*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)^2 + (e^2*F^(a*c + b*c*x)*x*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b*c*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b^2*c^2*F^(a*c + b*c*x)*x*Log[F]^2*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*(e*Cos[d + e*x] + b*c*Log[F]*Sin[d + e*x]), x, 1, F^(c*(a + b*x))*Sin[d + e*x]} -{(F^(c*(a + b*x))*(e*x*Cos[d + e*x] + (-1 + b*c*x*Log[F])*Sin[d + e*x]))/x^2, x, 6, (F^(a*c + b*c*x)*Sin[d + e*x])/x} -{(F^(c*(a + b*x))*(e*x*Cos[d + e*x] + (-2 + b*c*x*Log[F])*Sin[d + e*x]))/x^3, x, 10, (F^(a*c + b*c*x)*Sin[d + e*x])/x^2} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(a+b x) Trig[c+d x]^n Trig[c+d x]^p*) - - -{E^(a + b*x)*Cos[c + d*x]^1*Sin[c + d*x]^1, x, 3, -((d*E^(a + b*x)*Cos[2*c + 2*d*x])/(b^2 + 4*d^2)) + (b*E^(a + b*x)*Sin[2*c + 2*d*x])/(2*(b^2 + 4*d^2))} -{E^(a + b*x)*Cos[c + d*x]^1*Sin[c + d*x]^2, x, 4, (b*E^(a + b*x)*Cos[c + d*x])/(4*(b^2 + d^2)) - (b*E^(a + b*x)*Cos[3*c + 3*d*x])/(4*(b^2 + 9*d^2)) + (d*E^(a + b*x)*Sin[c + d*x])/(4*(b^2 + d^2)) - (3*d*E^(a + b*x)*Sin[3*c + 3*d*x])/(4*(b^2 + 9*d^2))} -{E^(a + b*x)*Cos[c + d*x]^1*Sin[c + d*x]^3, x, 4, -((d*E^(a + b*x)*Cos[2*c + 2*d*x])/(2*(b^2 + 4*d^2))) + (d*E^(a + b*x)*Cos[4*c + 4*d*x])/(2*(b^2 + 16*d^2)) + (b*E^(a + b*x)*Sin[2*c + 2*d*x])/(4*(b^2 + 4*d^2)) - (b*E^(a + b*x)*Sin[4*c + 4*d*x])/(8*(b^2 + 16*d^2))} - -{E^(a + b*x)*Cos[c + d*x]^2*Sin[c + d*x]^1, x, 4, -((d*E^(a + b*x)*Cos[c + d*x])/(4*(b^2 + d^2))) - (3*d*E^(a + b*x)*Cos[3*c + 3*d*x])/(4*(b^2 + 9*d^2)) + (b*E^(a + b*x)*Sin[c + d*x])/(4*(b^2 + d^2)) + (b*E^(a + b*x)*Sin[3*c + 3*d*x])/(4*(b^2 + 9*d^2))} -{E^(a + b*x)*Cos[c + d*x]^2*Sin[c + d*x]^2, x, 4, E^(a + b*x)/(8*b) - (b*E^(a + b*x)*Cos[4*c + 4*d*x])/(8*(b^2 + 16*d^2)) - (d*E^(a + b*x)*Sin[4*c + 4*d*x])/(2*(b^2 + 16*d^2))} -{E^(a + b*x)*Cos[c + d*x]^2*Sin[c + d*x]^3, x, 5, -((d*E^(a + b*x)*Cos[c + d*x])/(8*(b^2 + d^2))) - (3*d*E^(a + b*x)*Cos[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) + (5*d*E^(a + b*x)*Cos[5*c + 5*d*x])/(16*(b^2 + 25*d^2)) + (b*E^(a + b*x)*Sin[c + d*x])/(8*(b^2 + d^2)) + (b*E^(a + b*x)*Sin[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) - (b*E^(a + b*x)*Sin[5*c + 5*d*x])/(16*(b^2 + 25*d^2))} - -{E^(a + b*x)*Cos[c + d*x]^3*Sin[c + d*x]^1, x, 4, -((d*E^(a + b*x)*Cos[2*c + 2*d*x])/(2*(b^2 + 4*d^2))) - (d*E^(a + b*x)*Cos[4*c + 4*d*x])/(2*(b^2 + 16*d^2)) + (b*E^(a + b*x)*Sin[2*c + 2*d*x])/(4*(b^2 + 4*d^2)) + (b*E^(a + b*x)*Sin[4*c + 4*d*x])/(8*(b^2 + 16*d^2))} -{E^(a + b*x)*Cos[c + d*x]^3*Sin[c + d*x]^2, x, 5, (b*E^(a + b*x)*Cos[c + d*x])/(8*(b^2 + d^2)) - (b*E^(a + b*x)*Cos[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) - (b*E^(a + b*x)*Cos[5*c + 5*d*x])/(16*(b^2 + 25*d^2)) + (d*E^(a + b*x)*Sin[c + d*x])/(8*(b^2 + d^2)) - (3*d*E^(a + b*x)*Sin[3*c + 3*d*x])/(16*(b^2 + 9*d^2)) - (5*d*E^(a + b*x)*Sin[5*c + 5*d*x])/(16*(b^2 + 25*d^2))} -{E^(a + b*x)*Cos[c + d*x]^3*Sin[c + d*x]^3, x, 4, -((3*d*E^(a + b*x)*Cos[2*c + 2*d*x])/(16*(b^2 + 4*d^2))) + (3*d*E^(a + b*x)*Cos[6*c + 6*d*x])/(16*(b^2 + 36*d^2)) + (3*b*E^(a + b*x)*Sin[2*c + 2*d*x])/(32*(b^2 + 4*d^2)) - (b*E^(a + b*x)*Sin[6*c + 6*d*x])/(32*(b^2 + 36*d^2))} - - -{E^x*x*Sin[x], x, 4, (1/2)*E^x*Cos[x] - (1/2)*E^x*x*Cos[x] + (1/2)*E^x*x*Sin[x]} -{E^x*x^2*Sin[x], x, 11, (-(1/2))*E^x*Cos[x] + E^x*x*Cos[x] - (1/2)*E^x*x^2*Cos[x] - (1/2)*E^x*Sin[x] + (1/2)*E^x*x^2*Sin[x]} - -{E^x*x*Cos[x], x, 4, (1/2)*E^x*x*Cos[x] - (1/2)*E^x*Sin[x] + (1/2)*E^x*x*Sin[x]} -{E^x*x^2*Cos[x], x, 11, (-(1/2))*E^x*Cos[x] + (1/2)*E^x*x^2*Cos[x] + (1/2)*E^x*Sin[x] - E^x*x*Sin[x] + (1/2)*E^x*x^2*Sin[x]} - - -{E^(3*x)*(-5*Cos[4*x] + 2*Sin[4*x]), x, 4, (-(23/25))*E^(3*x)*Cos[4*x] - (14/25)*E^(3*x)*Sin[4*x]} - -{Sin[x]/E^x + E^x*Sin[x], x, 3, ((-(1/2))*Cos[x])/E^x - (1/2)*E^x*Cos[x] - ((1/2)*Sin[x])/E^x + (1/2)*E^x*Sin[x]} - - -{F^(a + b*x)*(Cos[c + d*x]/(e + e*Sin[c + d*x])), x, 5, (I*F^(a + b*x))/(b*e*Log[F]) - (2*I*F^(a + b*x)*Hypergeometric2F1[1, -((I*b*Log[F])/d), 1 - (I*b*Log[F])/d, I*E^(I*(c + d*x))])/(b*e*Log[F])} -{F^(a + b*x)*(Cos[c + d*x]/(e - e*Sin[c + d*x])), x, 5, -((I*F^(a + b*x))/(b*e*Log[F])) + (2*I*F^(a + b*x)*Hypergeometric2F1[1, -((I*b*Log[F])/d), 1 - (I*b*Log[F])/d, (-I)*E^(I*(c + d*x))])/(b*e*Log[F])} -{F^(a + b*x)*(Sin[c + d*x]/(e + e*Cos[c + d*x])), x, 5, -((I*F^(a + b*x))/(b*e*Log[F])) + (2*I*F^(a + b*x)*Hypergeometric2F1[1, -((I*b*Log[F])/d), 1 - (I*b*Log[F])/d, -E^(I*(c + d*x))])/(b*e*Log[F])} -{F^(a + b*x)*(Sin[c + d*x]/(e - e*Cos[c + d*x])), x, 5, (I*F^(a + b*x))/(b*e*Log[F]) - (2*I*F^(a + b*x)*Hypergeometric2F1[1, -((I*b*Log[F])/d), 1 - (I*b*Log[F])/d, E^(I*(c + d*x))])/(b*e*Log[F])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m F^(a+b x+c x^2) Trig[d+e x+f x^2]^m*) - - -{E^(x^2)*Sin[b*x], x, 6, (1/4)*I*E^(b^2/4)*Sqrt[Pi]*Erfi[(1/2)*((-I)*b + 2*x)] - (1/4)*I*E^(b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(I*b + 2*x)]} -{E^(x^2)*Cos[b*x], x, 6, (1/4)*E^(b^2/4)*Sqrt[Pi]*Erfi[(1/2)*((-I)*b + 2*x)] + (1/4)*E^(b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(I*b + 2*x)]} -{E^(x^2)*Sin[a + b*x], x, 6, (1/4)*I*E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[(1/2)*((-I)*b + 2*x)] - (1/4)*I*E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(I*b + 2*x)]} -{E^(x^2)*Cos[a + b*x], x, 6, (1/4)*E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[(1/2)*((-I)*b + 2*x)] + (1/4)*E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(I*b + 2*x)]} - -{E^(2*x^2)*x*Cos[2*x^2], x, 2, (1/8)*E^(2*x^2)*Cos[2*x^2] + (1/8)*E^(2*x^2)*Sin[2*x^2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form F^(a+b x) Trig[F^(a+b x)]*) - - -{E^x*Sin[E^x], x, 2, -Cos[E^x]} - -{E^x*Csc[E^x]*Sec[E^x], x, 3, Log[Tan[E^x]]} -{E^x*Cos[E^x], x, 2, Sin[E^x]} -{E^(2*x)*Cos[E^(2*x)], x, 2, Sin[E^(2*x)]/2} -{Cos[E^(-2*x)]/E^(2*x), x, 2, -Sin[E^(-2*x)]/2} -{E^(2*x)*Cos[E^x], x, 3, Cos[E^x] + E^x*Sin[E^x]} -{E^(-1 + 3*x)*Cos[E^(-1 + 3*x)]*Sin[1 + E^(-1 + 3*x)], x, 4, -Cos[1 + 2*E^(-1 + 3*x)]/12 + (E^(-1 + 3*x)*Sin[1])/6} - -{E^x*Tan[E^x], x, 2, -Log[Cos[E^x]]} - -{E^x*Sec[E^x], x, 2, ArcTanh[Sin[E^x]]} -{E^x*Sec[E^x]*Tan[E^x], x, 3, Sec[E^x]} - -{E^x*Csc[E^x]^2, x, 3, -Cot[E^x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x+c x^2) Sin[d+e x+f x^2]^m*) - - -{E^x*Sin[a + b*x], x, 1, -((b*E^x*Cos[a + b*x])/(1 + b^2)) + (E^x*Sin[a + b*x])/(1 + b^2)} -{E^x*Sin[a + c*x^2], x, 6, ((-1)^(3/4)*E^((I/4)*(4*a + c^(-1)))*Sqrt[Pi]*Erf[((-1)^(1/4)*(1 + (2*I)*c*x))/(2*Sqrt[c])])/(4*Sqrt[c]) + ((-1)^(3/4)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(1 - (2*I)*c*x))/(2*Sqrt[c])])/(4*Sqrt[c]*E^((I/4)*(4*a + c^(-1))))} -{E^x*Sin[a + b*x + c*x^2], x, 6, ((-1)^(3/4)*E^((1/4)*I*(4*a + (1 + I*b)^2/c))*Sqrt[Pi]*Erf[((-1)^(1/4)*(1 + I*b + 2*I*c*x))/(2*Sqrt[c])])/(4*Sqrt[c]) + ((-1)^(3/4)*E^((-I)*a + (I*(I + b)^2)/(4*c))*Sqrt[Pi]*Erfi[((-1)^(1/4)*(1 - I*b - 2*I*c*x))/(2*Sqrt[c])])/(4*Sqrt[c])} - -{E^x^2*Sin[a + b*x], x, 6, (I/4)*E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[((-I)*b + 2*x)/2] - (I/4)*E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(I*b + 2*x)/2]} -{E^x^2*Sin[a + c*x^2], x, 4, ((I/4)*Sqrt[Pi]*Erfi[Sqrt[1 - I*c]*x])/(Sqrt[1 - I*c]*E^(I*a)) - ((I/4)*E^(I*a)*Sqrt[Pi]*Erfi[Sqrt[1 + I*c]*x])/Sqrt[1 + I*c]} -{E^x^2*Sin[a + b*x + c*x^2], x, 6, -((I*Sqrt[Pi]*Erfi[(I*b - 2*(1 - I*c)*x)/(2*Sqrt[1 - I*c])])/(E^(I*(a - b^2/(4*I + 4*c)))*(4*Sqrt[1 - I*c]))) - (I*E^(I*a + b^2/(4*(1 + I*c)))*Sqrt[Pi]*Erfi[(I*b + 2*(1 + I*c)*x)/(2*Sqrt[1 + I*c])])/(4*Sqrt[1 + I*c])} - - -{f^(a + b*x)*Sin[d + f*x^2], x, 8, ((-1)^(3/4)*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*((2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/4 - ((-1)^(3/4)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*((2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/(4*E^((I/4)*(4*d + (b^2*Log[f]^2)/f)))} -{f^(a + b*x)*Sin[d + f*x^2]^2, x, 9, (1/16 + I/16)*E^((2*I)*d + ((I/8)*b^2*Log[f]^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erf[((1/4 + I/4)*((4*I)*f*x + b*Log[f]))/Sqrt[f]] + ((1/16 + I/16)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((1/4 + I/4)*((4*I)*f*x - b*Log[f]))/Sqrt[f]])/E^((I/8)*(16*d + (b^2*Log[f]^2)/f)) + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Sin[d + f*x^2]^3, x, 14, (3*(-1)^(3/4)*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*((2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/16 + (1/16 - I/16)*E^((3*I)*d + ((I/12)*b^2*Log[f]^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erf[((1/2 + I/2)*((6*I)*f*x + b*Log[f]))/(Sqrt[6]*Sqrt[f])] - (3*(-1)^(3/4)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*((2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/(16*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))) - ((1/16 - I/16)*f^(-1/2 + a)*Sqrt[Pi/6]*Erfi[((1/2 + I/2)*((6*I)*f*x - b*Log[f]))/(Sqrt[6]*Sqrt[f])])/E^((I/12)*(36*d + (b^2*Log[f]^2)/f))} - -{f^(a + b*x)*Sin[d + e*x + f*x^2], x, 8, ((-1)^(3/4)*E^((I/4)*(4*d + (I*e + b*Log[f])^2/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*(I*e + (2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/4 - ((-1)^(3/4)*E^((-I)*d + ((I/4)*(e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(I*e + (2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/4} -{f^(a + b*x)*Sin[d + e*x + f*x^2]^2, x, 9, (1/16 + I/16)*E^((2*I)*d + ((I/8)*((2*I)*e + b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erf[((1/4 + I/4)*((2*I)*e + (4*I)*f*x + b*Log[f]))/Sqrt[f]] + (1/16 + I/16)*E^((-2*I)*d + ((I/8)*(2*e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((1/4 + I/4)*((2*I)*e + (4*I)*f*x - b*Log[f]))/Sqrt[f]] + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Sin[d + e*x + f*x^2]^3, x, 14, (3*(-1)^(3/4)*E^((I/4)*(4*d + (I*e + b*Log[f])^2/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*(I*e + (2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/16 + (1/16 - I/16)*E^((3*I)*d + ((I/12)*((3*I)*e + b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erf[((1/2 + I/2)*((3*I)*e + (6*I)*f*x + b*Log[f]))/(Sqrt[6]*Sqrt[f])] - (3*(-1)^(3/4)*E^((-I)*d + ((I/4)*(e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(I*e + (2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/16 - (1/16 - I/16)*E^((-3*I)*d + ((I/12)*(3*e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erfi[((1/2 + I/2)*((3*I)*e + (6*I)*f*x - b*Log[f]))/(Sqrt[6]*Sqrt[f])]} - - -{f^(a + c*x^2)*Sin[d + e*x], x, 8, If[$VersionNumber>=8, ((-I/4)*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) - ((I/4)*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]), (I*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) - (I*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Sin[d + e*x]^2, x, 9, If[$VersionNumber>=8, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^((-2*I)*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) - (E^((2*I)*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*I*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) - (E^(2*I*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Sin[d + e*x]^3, x, 14, If[$VersionNumber>=8, (((-3*I)/16)*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) + ((I/16)*E^((-3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) - (((3*I)/16)*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) + ((I/16)*E^((3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]), (3*I*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (I*E^(-3*I*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*I*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (I*E^(3*I*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + c*x^2)*Sin[d + f*x^2], x, 6, ((I/4)*f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(E^(I*d)*Sqrt[I*f - c*Log[f]]) - ((I/4)*E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[f]]])/Sqrt[I*f + c*Log[f]]} -{f^(a + c*x^2)*Sin[d + f*x^2]^2, x, 7, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (f^a*Sqrt[Pi]*Erf[x*Sqrt[(2*I)*f - c*Log[f]]])/(8*E^((2*I)*d)*Sqrt[(2*I)*f - c*Log[f]]) - (E^((2*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(2*I)*f + c*Log[f]]])/(8*Sqrt[(2*I)*f + c*Log[f]])} -{f^(a + c*x^2)*Sin[d + f*x^2]^3, x, 10, (((3*I)/16)*f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(E^(I*d)*Sqrt[I*f - c*Log[f]]) - ((I/16)*f^a*Sqrt[Pi]*Erf[x*Sqrt[(3*I)*f - c*Log[f]]])/(E^((3*I)*d)*Sqrt[(3*I)*f - c*Log[f]]) - (((3*I)/16)*E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[f]]])/Sqrt[I*f + c*Log[f]] + ((I/16)*E^((3*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(3*I)*f + c*Log[f]]])/Sqrt[(3*I)*f + c*Log[f]]} - -{f^(a + c*x^2)*Sin[d + e*x + f*x^2], x, 8, (I*E^((-I)*d - e^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]]) - (I*E^(I*d + e^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + c*x^2)*Sin[d + e*x + f*x^2]^2, x, 9, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*I*d - e^2/(2*I*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + x*(2*I*f - c*Log[f]))/Sqrt[2*I*f - c*Log[f]]])/(8*Sqrt[2*I*f - c*Log[f]]) - (E^(2*I*d + e^2/(2*I*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + x*(2*I*f + c*Log[f]))/Sqrt[2*I*f + c*Log[f]]])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + c*x^2)*Sin[d + e*x + f*x^2]^3, x, 14, (3*I*E^((-I)*d - e^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) - (I*E^(-3*I*d - (9*e^2)/(4*(3*I*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(3*I*e + 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) - (3*I*E^(I*d + e^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (I*E^(3*I*d + (9*e^2)/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*I*e + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - - -{f^(a + b*x + c*x^2)*Sin[d + e*x], x, 8, If[$VersionNumber>=8, ((-I/4)*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) - ((I/4)*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]), (I*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) - (I*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Sin[d + e*x]^2, x, 10, If[$VersionNumber>=8, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^((-2*I)*d + (2*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((2*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) - (E^((2*I)*d - ((2*I)*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((2*I)*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*I*d + (2*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((2*I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) - (E^(2*I*d - (2*I*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Sin[d + e*x]^3, x, 14, If[$VersionNumber>=8, (((-3*I)/16)*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) + ((I/16)*E^((-3*I)*d + (3*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) - (((3*I)/16)*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]) + ((I/16)*E^((3*I)*d - ((3*I)*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(Sqrt[c]*Sqrt[Log[f]]), (3*I*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (I*E^(-3*I*d + (3*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*I*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (I*E^(3*I*d - (3*I*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + b*x + c*x^2)*Sin[d + f*x^2], x, 8, -((I*E^((-I)*d + (b^2*Log[f]^2)/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]])) - (I*E^(I*d - (b^2*Log[f]^2)/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sin[d + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*I*d + (b^2*Log[f]^2)/(8*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*I*f - c*Log[f]))/(2*Sqrt[2*I*f - c*Log[f]])])/(8*Sqrt[2*I*f - c*Log[f]]) - (E^(2*I*d - (b^2*Log[f]^2)/(8*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(2*I*f + c*Log[f]))/(2*Sqrt[2*I*f + c*Log[f]])])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sin[d + f*x^2]^3, x, 14, -((3*I*E^((-I)*d + (b^2*Log[f]^2)/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]])) + (I*E^(-3*I*d + (b^2*Log[f]^2)/(12*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) - (3*I*E^(I*d - (b^2*Log[f]^2)/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (I*E^(3*I*d - (b^2*Log[f]^2)/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Sin[d + e*x + f*x^2], x, 8, (I*E^((-I)*d - (e + I*b*Log[f])^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]]) - (I*E^(I*d + (e - I*b*Log[f])^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sin[d + e*x + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*I*d - (2*e + I*b*Log[f])^2/(8*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(2*I*e - b*Log[f] + 2*x*(2*I*f - c*Log[f]))/(2*Sqrt[2*I*f - c*Log[f]])])/(8*Sqrt[2*I*f - c*Log[f]]) - (E^(2*I*d + (2*e - I*b*Log[f])^2/(8*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*I*e + b*Log[f] + 2*x*(2*I*f + c*Log[f]))/(2*Sqrt[2*I*f + c*Log[f]])])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sin[d + e*x + f*x^2]^3, x, 14, (3*I*E^((-I)*d - (e + I*b*Log[f])^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) - (I*E^(-3*I*d - (3*e + I*b*Log[f])^2/(4*(3*I*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(3*I*e - b*Log[f] + 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) - (3*I*E^(I*d + (e - I*b*Log[f])^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (I*E^(3*I*d - (3*I*e + b*Log[f])^2/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*I*e + b*Log[f] + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Sin[a + b*x + e*x^2], x, 8, If[$VersionNumber>=8, (I*Sqrt[Pi]*Erf[(b*(I - Log[f]) + 2*x*(I*e - c*Log[f]))/(2*Sqrt[I*e - c*Log[f]])])/(E^((I - Log[f])*(a - (b^2*(I - Log[f]))/(4*I*e - 4*c*Log[f])))*(4*Sqrt[I*e - c*Log[f]])) - (I*E^((I + Log[f])*(a - (b^2*(I + Log[f]))/(4*I*e + 4*c*Log[f])))*Sqrt[Pi]*Erfi[(b*(I + Log[f]) + 2*x*(I*e + c*Log[f]))/(2*Sqrt[I*e + c*Log[f]])])/(4*Sqrt[I*e + c*Log[f]]), -((I*Sqrt[Pi]*Erf[-((b*(I - Log[f]) + 2*x*(I*e - c*Log[f]))/(2*Sqrt[I*e - c*Log[f]]))])/(E^((I - Log[f])*(a - (b^2*(I - Log[f]))/(4*I*e - 4*c*Log[f])))*(4*Sqrt[I*e - c*Log[f]]))) - (I*E^((I + Log[f])*(a - (b^2*(I + Log[f]))/(4*I*e + 4*c*Log[f])))*Sqrt[Pi]*Erfi[(b*(I + Log[f]) + 2*x*(I*e + c*Log[f]))/(2*Sqrt[I*e + c*Log[f]])])/(4*Sqrt[I*e + c*Log[f]])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x+c x^2) Cos[d+e x+f x^2]^m*) - - -{E^x*Cos[a + b*x], x, 1, (E^x*Cos[a + b*x])/(1 + b^2) + (b*E^x*Sin[a + b*x])/(1 + b^2)} -{E^x*Cos[a + c*x^2], x, 6, -((-1)^(1/4)*E^((I/4)*(4*a + c^(-1)))*Sqrt[Pi]*Erf[((-1)^(1/4)*(1 + (2*I)*c*x))/(2*Sqrt[c])])/(4*Sqrt[c]) + ((-1)^(1/4)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(1 - (2*I)*c*x))/(2*Sqrt[c])])/(4*Sqrt[c]*E^((I/4)*(4*a + c^(-1))))} -{E^x*Cos[a + b*x + c*x^2], x, 6, -(((-1)^(1/4)*E^((1/4)*I*(4*a + (1 + I*b)^2/c))*Sqrt[Pi]*Erf[((-1)^(1/4)*(1 + I*b + 2*I*c*x))/(2*Sqrt[c])])/(4*Sqrt[c])) + ((-1)^(1/4)*E^((-I)*a + (I*(I + b)^2)/(4*c))*Sqrt[Pi]*Erfi[((-1)^(1/4)*(1 - I*b - 2*I*c*x))/(2*Sqrt[c])])/(4*Sqrt[c])} - -{E^x^2*Cos[a + b*x], x, 6, (E^((-I)*a + b^2/4)*Sqrt[Pi]*Erfi[((-I)*b + 2*x)/2])/4 + (E^(I*a + b^2/4)*Sqrt[Pi]*Erfi[(I*b + 2*x)/2])/4} -{E^x^2*Cos[a + c*x^2], x, 4, (Sqrt[Pi]*Erfi[Sqrt[1 - I*c]*x])/(4*Sqrt[1 - I*c]*E^(I*a)) + (E^(I*a)*Sqrt[Pi]*Erfi[Sqrt[1 + I*c]*x])/(4*Sqrt[1 + I*c])} -{E^x^2*Cos[a + b*x + c*x^2], x, 6, -((Sqrt[Pi]*Erfi[(I*b - 2*(1 - I*c)*x)/(2*Sqrt[1 - I*c])])/(E^(I*(a - b^2/(4*I + 4*c)))*(4*Sqrt[1 - I*c]))) + (E^(I*a + b^2/(4*(1 + I*c)))*Sqrt[Pi]*Erfi[(I*b + 2*(1 + I*c)*x)/(2*Sqrt[1 + I*c])])/(4*Sqrt[1 + I*c])} - - -{f^(a + b*x)*Cos[d + f*x^2], x, 8, -((-1)^(1/4)*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*((2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/4 - ((-1)^(1/4)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*((2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/(4*E^((I/4)*(4*d + (b^2*Log[f]^2)/f)))} -{f^(a + b*x)*Cos[d + f*x^2]^2, x, 9, (-1/16 - I/16)*E^((2*I)*d + ((I/8)*b^2*Log[f]^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erf[((1/4 + I/4)*((4*I)*f*x + b*Log[f]))/Sqrt[f]] - ((1/16 + I/16)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((1/4 + I/4)*((4*I)*f*x - b*Log[f]))/Sqrt[f]])/E^((I/8)*(16*d + (b^2*Log[f]^2)/f)) + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Cos[d + f*x^2]^3, x, 14, (-3*(-1)^(1/4)*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*((2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/16 - (1/16 + I/16)*E^((3*I)*d + ((I/12)*b^2*Log[f]^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erf[((1/2 + I/2)*((6*I)*f*x + b*Log[f]))/(Sqrt[6]*Sqrt[f])] - (3*(-1)^(1/4)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*((2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/(16*E^((I/4)*(4*d + (b^2*Log[f]^2)/f))) - ((1/16 + I/16)*f^(-1/2 + a)*Sqrt[Pi/6]*Erfi[((1/2 + I/2)*((6*I)*f*x - b*Log[f]))/(Sqrt[6]*Sqrt[f])])/E^((I/12)*(36*d + (b^2*Log[f]^2)/f))} - -{f^(a + b*x)*Cos[d + e*x + f*x^2], x, 8, -((-1)^(1/4)*E^((I/4)*(4*d + (I*e + b*Log[f])^2/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*(I*e + (2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/4 - ((-1)^(1/4)*E^((-I)*d + ((I/4)*(e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(I*e + (2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/4} -{f^(a + b*x)*Cos[d + e*x + f*x^2]^2, x, 9, (-1/16 - I/16)*E^((2*I)*d + ((I/8)*((2*I)*e + b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erf[((1/4 + I/4)*((2*I)*e + (4*I)*f*x + b*Log[f]))/Sqrt[f]] - (1/16 + I/16)*E^((-2*I)*d + ((I/8)*(2*e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((1/4 + I/4)*((2*I)*e + (4*I)*f*x - b*Log[f]))/Sqrt[f]] + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Cos[d + e*x + f*x^2]^3, x, 14, (-3*(-1)^(1/4)*E^((I/4)*(4*d + (I*e + b*Log[f])^2/f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[((-1)^(1/4)*(I*e + (2*I)*f*x + b*Log[f]))/(2*Sqrt[f])])/16 - (1/16 + I/16)*E^((3*I)*d + ((I/12)*((3*I)*e + b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erf[((1/2 + I/2)*((3*I)*e + (6*I)*f*x + b*Log[f]))/(Sqrt[6]*Sqrt[f])] - (3*(-1)^(1/4)*E^((-I)*d + ((I/4)*(e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi]*Erfi[((-1)^(1/4)*(I*e + (2*I)*f*x - b*Log[f]))/(2*Sqrt[f])])/16 - (1/16 + I/16)*E^((-3*I)*d + ((I/12)*(3*e + I*b*Log[f])^2)/f)*f^(-1/2 + a)*Sqrt[Pi/6]*Erfi[((1/2 + I/2)*((3*I)*e + (6*I)*f*x - b*Log[f]))/(Sqrt[6]*Sqrt[f])]} - - -{f^(a + c*x^2)*Cos[d + e*x], x, 8, If[$VersionNumber>=8, -(E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), (E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Cos[d + e*x]^2, x, 9, If[$VersionNumber>=8, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^((-2*I)*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^((2*I)*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*I*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*I*d + e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Cos[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^((-3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^((3*I)*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^((-I)*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*I*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*I*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*I*d + (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*I*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + c*x^2)*Cos[d + f*x^2], x, 6, (f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(4*E^(I*d)*Sqrt[I*f - c*Log[f]]) + (E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[f]]])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + c*x^2)*Cos[d + f*x^2]^2, x, 7, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[(2*I)*f - c*Log[f]]])/(8*E^((2*I)*d)*Sqrt[(2*I)*f - c*Log[f]]) + (E^((2*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(2*I)*f + c*Log[f]]])/(8*Sqrt[(2*I)*f + c*Log[f]])} -{f^(a + c*x^2)*Cos[d + f*x^2]^3, x, 10, (3*f^a*Sqrt[Pi]*Erf[x*Sqrt[I*f - c*Log[f]]])/(16*E^(I*d)*Sqrt[I*f - c*Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[(3*I)*f - c*Log[f]]])/(16*E^((3*I)*d)*Sqrt[(3*I)*f - c*Log[f]]) + (3*E^(I*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[I*f + c*Log[f]]])/(16*Sqrt[I*f + c*Log[f]]) + (E^((3*I)*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[(3*I)*f + c*Log[f]]])/(16*Sqrt[(3*I)*f + c*Log[f]])} - -{f^(a + c*x^2)*Cos[d + e*x + f*x^2], x, 8, (E^((-I)*d - e^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]]) + (E^(I*d + e^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + c*x^2)*Cos[d + e*x + f*x^2]^2, x, 9, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*I*d - e^2/(2*I*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + x*(2*I*f - c*Log[f]))/Sqrt[2*I*f - c*Log[f]]])/(8*Sqrt[2*I*f - c*Log[f]]) + (E^(2*I*d + e^2/(2*I*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + x*(2*I*f + c*Log[f]))/Sqrt[2*I*f + c*Log[f]]])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + c*x^2)*Cos[d + e*x + f*x^2]^3, x, 14, (3*E^((-I)*d - e^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) + (E^(-3*I*d - (9*e^2)/(4*(3*I*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(3*I*e + 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) + (3*E^(I*d + e^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (E^(3*I*d + (9*e^2)/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*I*e + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - - -{f^(a + b*x + c*x^2)*Cos[d + e*x], x, 8, If[$VersionNumber>=8, -(E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), (E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Cos[d + e*x]^2, x, 10, If[$VersionNumber>=8, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^((-2*I)*d + (2*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((2*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^((2*I)*d - ((2*I)*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((2*I)*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*I*d + (2*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((2*I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*I*d - (2*I*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Cos[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^((-3*I)*d + (3*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^((3*I)*d - ((3*I)*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[((3*I)*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^((-I)*d + (e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*I*d + (3*e + I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*I*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(I*d + (e - I*b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*I*d - (3*I*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*I*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + b*x + c*x^2)*Cos[d + f*x^2], x, 8, -((E^((-I)*d + (b^2*Log[f]^2)/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]])) + (E^(I*d - (b^2*Log[f]^2)/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cos[d + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*I*d + (b^2*Log[f]^2)/(8*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*I*f - c*Log[f]))/(2*Sqrt[2*I*f - c*Log[f]])])/(8*Sqrt[2*I*f - c*Log[f]]) + (E^(2*I*d - (b^2*Log[f]^2)/(8*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(2*I*f + c*Log[f]))/(2*Sqrt[2*I*f + c*Log[f]])])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cos[d + f*x^2]^3, x, 14, -((3*E^((-I)*d + (b^2*Log[f]^2)/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]])) - (E^(-3*I*d + (b^2*Log[f]^2)/(12*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) + (3*E^(I*d - (b^2*Log[f]^2)/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (E^(3*I*d - (b^2*Log[f]^2)/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Cos[d + e*x + f*x^2], x, 8, (E^((-I)*d - (e + I*b*Log[f])^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(4*Sqrt[I*f - c*Log[f]]) + (E^(I*d + (e - I*b*Log[f])^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(4*Sqrt[I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cos[d + e*x + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*I*d - (2*e + I*b*Log[f])^2/(8*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(2*I*e - b*Log[f] + 2*x*(2*I*f - c*Log[f]))/(2*Sqrt[2*I*f - c*Log[f]])])/(8*Sqrt[2*I*f - c*Log[f]]) + (E^(2*I*d + (2*e - I*b*Log[f])^2/(8*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*I*e + b*Log[f] + 2*x*(2*I*f + c*Log[f]))/(2*Sqrt[2*I*f + c*Log[f]])])/(8*Sqrt[2*I*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cos[d + e*x + f*x^2]^3, x, 14, (3*E^((-I)*d - (e + I*b*Log[f])^2/(4*I*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(I*e - b*Log[f] + 2*x*(I*f - c*Log[f]))/(2*Sqrt[I*f - c*Log[f]])])/(16*Sqrt[I*f - c*Log[f]]) + (E^(-3*I*d - (3*e + I*b*Log[f])^2/(4*(3*I*f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(3*I*e - b*Log[f] + 2*x*(3*I*f - c*Log[f]))/(2*Sqrt[3*I*f - c*Log[f]])])/(16*Sqrt[3*I*f - c*Log[f]]) + (3*E^(I*d + (e - I*b*Log[f])^2/(4*I*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(I*e + b*Log[f] + 2*x*(I*f + c*Log[f]))/(2*Sqrt[I*f + c*Log[f]])])/(16*Sqrt[I*f + c*Log[f]]) + (E^(3*I*d - (3*I*e + b*Log[f])^2/(4*(3*I*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*I*e + b*Log[f] + 2*x*(3*I*f + c*Log[f]))/(2*Sqrt[3*I*f + c*Log[f]])])/(16*Sqrt[3*I*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Cos[a + b*x + e*x^2], x, 8, If[$VersionNumber>=8, (Sqrt[Pi]*Erf[(b*(I - Log[f]) + 2*x*(I*e - c*Log[f]))/(2*Sqrt[I*e - c*Log[f]])])/(E^((I - Log[f])*(a - (b^2*(I - Log[f]))/(4*I*e - 4*c*Log[f])))*(4*Sqrt[I*e - c*Log[f]])) + (E^((I + Log[f])*(a - (b^2*(I + Log[f]))/(4*I*e + 4*c*Log[f])))*Sqrt[Pi]*Erfi[(b*(I + Log[f]) + 2*x*(I*e + c*Log[f]))/(2*Sqrt[I*e + c*Log[f]])])/(4*Sqrt[I*e + c*Log[f]]), -((Sqrt[Pi]*Erf[-((b*(I - Log[f]) + 2*x*(I*e - c*Log[f]))/(2*Sqrt[I*e - c*Log[f]]))])/(E^((I - Log[f])*(a - (b^2*(I - Log[f]))/(4*I*e - 4*c*Log[f])))*(4*Sqrt[I*e - c*Log[f]]))) + (E^((I + Log[f])*(a - (b^2*(I + Log[f]))/(4*I*e + 4*c*Log[f])))*Sqrt[Pi]*Erfi[(b*(I + Log[f]) + 2*x*(I*e + c*Log[f]))/(2*Sqrt[I*e + c*Log[f]])])/(4*Sqrt[I*e + c*Log[f]])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form F^(c (a+b x)) (f + g Trig[d+e x])^n*) - - -{F^(c*(a + b*x))*(f + f*Sin[d + e*x])^2, x, 8, (f^2*F^(a*c + b*c*x))/(b*c*Log[F]) - (2*e*f^2*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) + (2*b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) - (2*e*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2) + (b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x]^2)/(4*e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*(f + f*Sin[d + e*x])^1, x, 6, (f*F^(a*c + b*c*x))/(b*c*Log[F]) - (e*f*F^(a*c + b*c*x)*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (b*c*f*F^(a*c + b*c*x)*Log[F]*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))/(f + f*Sin[d + e*x])^1, x, 2, -((2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/e, 2 - (I*b*c*Log[F])/e, I*E^(I*(d + e*x))])/(f*(e - I*b*c*Log[F])))} -{F^(c*(a + b*x))/(f + f*Sin[d + e*x])^2, x, 3, -((F^(c*(a + b*x))*Cot[d/2 + Pi/4 + (e*x)/2]*Csc[d/2 + Pi/4 + (e*x)/2]^2)/(6*e*f^2)) - (b*c*F^(c*(a + b*x))*Csc[d/2 + Pi/4 + (e*x)/2]^2*Log[F])/(6*e^2*f^2) - (2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/e, 2 - (I*b*c*Log[F])/e, I*E^(I*(d + e*x))]*(e + I*b*c*Log[F]))/(3*e^2*f^2)} - - -{F^(c*(a + b*x))*(f + f*Cos[d + e*x])^2, x, 8, (f^2*F^(a*c + b*c*x))/(b*c*Log[F]) + (2*b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F]^2) + (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 + b^2*c^2*Log[F]^2)) + (b*c*f^2*F^(a*c + b*c*x)*Cos[d + e*x]^2*Log[F])/(4*e^2 + b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Cos[d + e*x]*Sin[d + e*x])/(4*e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*(f + f*Cos[d + e*x])^1, x, 6, (f*F^(a*c + b*c*x))/(b*c*Log[F]) + (b*c*f*F^(a*c + b*c*x)*Cos[d + e*x]*Log[F])/(e^2 + b^2*c^2*Log[F]^2) + (e*f*F^(a*c + b*c*x)*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))/(f + f*Cos[d + e*x])^1, x, 2, (2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/e, 2 - (I*b*c*Log[F])/e, -E^(I*(d + e*x))])/(f*(I*e + b*c*Log[F]))} -{F^(c*(a + b*x))/(f + f*Cos[d + e*x])^2, x, 3, -((2*E^(I*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 - (I*b*c*Log[F])/e, 2 - (I*b*c*Log[F])/e, -E^(I*(d + e*x))]*(I*e - b*c*Log[F]))/(3*e^2*f^2)) - (b*c*F^(c*(a + b*x))*Log[F]*Sec[d/2 + (e*x)/2]^2)/(6*e^2*f^2) + (F^(c*(a + b*x))*Sec[d/2 + (e*x)/2]^2*Tan[d/2 + (e*x)/2])/(6*e*f^2)} diff --git a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.7 Trig functions.m b/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.7 Trig functions.m deleted file mode 100644 index 795fec1..0000000 --- a/test/methods/rule_based/test_files/4 Trig functions/4.7 Miscellaneous/4.7.7 Trig functions.m +++ /dev/null @@ -1,1822 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Miscellaneous Integration Problems Involving Trig Functions*) - - -(* ::Section::Closed:: *) -(*Rectification problems*) - - -(* Following integrands are equal. *) -{2/(3 - Cos[4 + 6*x]), x, 2, x/Sqrt[2] + ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] - Cos[4 + 6*x])]/(3*Sqrt[2])} -{2*Csc[4 + 6*x]/(3*Csc[4 + 6*x] - Cot[4 + 6*x]), x, 3, x/Sqrt[2] + ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] - Cos[4 + 6*x])]/(3*Sqrt[2])} -{1/(1 + Sin[2 + 3*x]^2), x, 2, x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])} -{1/(2 - Cos[2 + 3*x]^2), x, 2, x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])} -{1/(2*Sin[2 + 3*x]^2 + Cos[2 + 3*x]^2), x, 2, x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])} -{Sec[2 + 3*x]^2/(1 + 2*Tan[2 + 3*x]^2), x, 2, x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])} -{Csc[2 + 3*x]^2/(2 + Cot[2 + 3*x]^2), x, 2, x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])} - - -(* Following integrands are equal. *) -{2/(1 - 3*Cos[4 + 6*x]), x, 3, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{2*Csc[4 + 6*x]/(Csc[4 + 6*x] - 3*Cot[4 + 6*x]), x, 4, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(-1 + 3*Sin[2 + 3*x]^2), x, 2, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(2 - 3*Cos[2 + 3*x]^2), x, 2, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(2*Sin[2 + 3*x]^2 - Cos[2 + 3*x]^2), x, 2, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{Sec[2 + 3*x]^2/(-1 + 2*Tan[2 + 3*x]^2), x, 2, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} -{Csc[2 + 3*x]^2/(2 - Cot[2 + 3*x]^2), x, 2, Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])} - - -(* Following integrands are equal. *) -{2/(3 + Cos[4 + 6*x]), x, 2, x/Sqrt[2] - ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] + Cos[4 + 6*x])]/(3*Sqrt[2])} -{2*Csc[4 + 6*x]/(3*Csc[4 + 6*x] + Cot[4 + 6*x]), x, 3, x/Sqrt[2] - ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] + Cos[4 + 6*x])]/(3*Sqrt[2])} -{1/(2 - Sin[2 + 3*x]^2), x, 2, x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])} -{1/(1 + Cos[2 + 3*x]^2), x, 2, x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])} -{1/(2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2), x, 2, x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])} -{Sec[2 + 3*x]^2/(2 + Tan[2 + 3*x]^2), x, 2, x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])} -{Csc[2 + 3*x]^2/(1 + 2*Cot[2 + 3*x]^2), x, 2, x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])} - - -(* Following integrands are equal. *) -{-2/(1 + 3*Cos[4 + 6*x]), x, 3, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{-2*Csc[4 + 6*x]/(Csc[4 + 6*x] + 3*Cot[4 + 6*x]), x, 4, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(-2 + 3*Sin[2 + 3*x]^2), x, 2, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(1 - 3*Cos[2 + 3*x]^2), x, 2, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{1/(-2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2), x, 2, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{Sec[2 + 3*x]^2/(-2 + Tan[2 + 3*x]^2), x, 2, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} -{Csc[2 + 3*x]^2/(1 - 2*Cot[2 + 3*x]^2), x, 2, Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])} - - -(* ::Section::Closed:: *) -(*Integrands involving sines*) - - -{(x + Sin[x])^2, x, 6, x/2 + x^3/3 - 2*x*Cos[x] + 2*Sin[x] - (1/2)*Cos[x]*Sin[x]} -{(x + Sin[x])^3, x, 9, (3*x^2)/4 + x^4/4 + 5*Cos[x] - 3*x^2*Cos[x] + Cos[x]^3/3 + 6*x*Sin[x] - (3/2)*x*Cos[x]*Sin[x] + (3*Sin[x]^2)/4} - - -{Sin[a + b*x]/(c + d*x^2), x, 8, -((CosIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sin[a - (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d])) + (CosIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x]*Sin[a + (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a + (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a - (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])} -{Sin[a + b*x]/(c + d*x + e*x^2), x, 8, (CosIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sin[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] - (CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] + (Cos[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e]} - - -{Sin[Sqrt[-7 + x]]/Sqrt[-7 + x], x, 3, -2*Cos[Sqrt[-7 + x]]} - - -{Sin[x]*Sqrt[b - a/x^2]/Sqrt[a - b*x^2], x, 3, (Sqrt[b - a/x^2]*x*SinIntegral[x])/Sqrt[a - b*x^2]} - - -{1/(x*(1 + Sin[Log[x]])), x, 2, -(Cos[Log[x]]/(1 + Sin[Log[x]]))} - - -(* ::Subsection::Closed:: *) -(*Sin[(a+b x)/(c+d x)]^n*) - - -{Sin[(a + b*x)/(c + d*x)]^1, x, 5, ((b*c - a*d)*Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2 + ((c + d*x)*Sin[(a + b*x)/(c + d*x)])/d + ((b*c - a*d)*Sin[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2} -{Sin[(a + b*x)/(c + d*x)]^2, x, 6, ((b*c - a*d)*CosIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sin[(2*b)/d])/d^2 + ((c + d*x)*Sin[(a + b*x)/(c + d*x)]^2)/d - ((b*c - a*d)*Cos[(2*b)/d]*SinIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2} -{Sin[(a + b*x)/(c + d*x)]^3, x, 9, (3*(b*c - a*d)*Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Cos[(3*b)/d]*CosIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sin[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*Sin[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sin[(3*b)/d]*SinIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1-a^2 x^2)^m Sin[Sqrt[1-a x]/Sqrt[1+a x]]^n*) - - -{Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2), x, 5, -((3*SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/(4*a)) + SinIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)} -{Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x, 4, CosIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(2*a) - Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)} -{Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1/(1 - a^2*x^2), x, 2, -(SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)} -{1/(Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1*(1 - a^2*x^2)), x, 1, Unintegrable[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/((1 - a*x)*(1 + a*x)), x]} -{1/(Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2*(1 - a^2*x^2)), x, 1, Unintegrable[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/((1 - a*x)*(1 + a*x)), x]} - - -(* ::Section::Closed:: *) -(*Integrands involving cosines*) - - -{(x + Cos[x])^2, x, 6, x/2 + x^3/3 + 2*Cos[x] + 2*x*Sin[x] + (1/2)*Cos[x]*Sin[x]} -{(x + Cos[x])^3, x, 9, (3*x^2)/4 + x^4/4 + 6*x*Cos[x] + (3*Cos[x]^2)/4 - 5*Sin[x] + 3*x^2*Sin[x] + (3/2)*x*Cos[x]*Sin[x] - Sin[x]^3/3} - - -{Cos[a + b*x]/(c + d*x^2), x, 8, (Cos[a + (b*Sqrt[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a - (b*Sqrt[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a + (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a - (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])} -{Cos[a + b*x]/(c + d*x + e*x^2), x, 8, (Cos[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Sin[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] + (Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e]} - - -{(x*Cos[Sqrt[1 + x^2]])/Sqrt[1 + x^2], x, 4, Sin[Sqrt[1 + x^2]]} -{(x*Cos[Sqrt[3]*Sqrt[2 + x^2]])/Sqrt[2 + x^2], x, 4, Sin[Sqrt[3]*Sqrt[2 + x^2]]/Sqrt[3]} -{((-1 + 2*x)*Cos[Sqrt[6 + 3*(-1 + 2*x)^2]])/Sqrt[6 + 3*(-1 + 2*x)^2], x, 5, (1/6)*Sin[Sqrt[3]*Sqrt[2 + (-1 + 2*x)^2]]} - - -(* ::Subsection::Closed:: *) -(*Cos[(a+b x)/(c+d x)]^n*) - - -{Cos[(a + b*x)/(c + d*x)], x, 5, ((c + d*x)*Cos[(a + b*x)/(c + d*x)])/d - ((b*c - a*d)*CosIntegral[(b*c - a*d)/(d*(c + d*x))]*Sin[b/d])/d^2 + ((b*c - a*d)*Cos[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2} -{Cos[(a + b*x)/(c + d*x)]^2, x, 6, ((c + d*x)*Cos[(a + b*x)/(c + d*x)]^2)/d - ((b*c - a*d)*CosIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sin[(2*b)/d])/d^2 + ((b*c - a*d)*Cos[(2*b)/d]*SinIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1-a^2 x^2)^m Cos[Sqrt[1-a x]/Sqrt[1+a x]]^n*) - - -{Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2), x, 5, -((3*CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/(4*a)) - CosIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)} -{Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x, 4, -(CosIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(2*a)) - Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)} -{Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1/(1 - a^2*x^2), x, 2, -(CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)} -{1/(Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1*(1 - a^2*x^2)), x, 1, Unintegrable[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/((1 - a*x)*(1 + a*x)), x]} -{1/(Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2*(1 - a^2*x^2)), x, 1, Unintegrable[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/((1 - a*x)*(1 + a*x)), x]} - - -(* ::Section::Closed:: *) -(*Integrands involving tangents*) - - -{Tan[Sqrt[x]]/Sqrt[x], x, 2, -2*Log[Cos[Sqrt[x]]]} -{Tan[Sqrt[x]]^2/Sqrt[x], x, 3, -2*Sqrt[x] + 2*Tan[Sqrt[x]]} - - -{Sqrt[x]*Tan[Sqrt[x]], x, 6, (2/3)*I*x^(3/2) - 2*x*Log[1 + E^(2*I*Sqrt[x])] + 2*I*Sqrt[x]*PolyLog[2, -E^(2*I*Sqrt[x])] - PolyLog[3, -E^(2*I*Sqrt[x])]} - - -{x*Tan[a + b*x + c*x^2] + (b*Tan[a + b*x + c*x^2])/(2*c), x, 2, -(Log[Cos[a + b*x + c*x^2]]/(2*c))} - - -(* ::Section::Closed:: *) -(*Integrands involving cotangents*) - - -{Cot[Sqrt[x]]^2/Sqrt[x], x, 3, -2*Sqrt[x] - 2*Cot[Sqrt[x]]} - - -(* ::Section::Closed:: *) -(*Integrands involving secants*) - - -{Sqrt[a + b*Sec[c + d*x]]/(1 + Cos[c + d*x]), x, 2, (EllipticE[ArcSin[Tan[c + d*x]/(1 + Sec[c + d*x])], (a - b)/(a + b)]*Sqrt[1/(1 + Sec[c + d*x])]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])} - - -{Sec[a + b*x]*Sec[2*a + 2*b*x], x, 4, -(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b} -{Sec[a + b*x]*Sec[2*(a + b*x)], x, 4, -(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[a+b x]^n Trig[c+d x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Trig[m x]^p Trig[n x]^q*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Trig[m x] Sin[n x]*) - - -{Sin[2*x]*Sin[x],x, 1, Sin[x]/2 - (1/6)*Sin[3*x]} -{Sin[3*x]*Sin[x],x, 1, (1/4)*Sin[2*x] - (1/8)*Sin[4*x]} -{Sin[4*x]*Sin[x],x, 1, (1/6)*Sin[3*x] - (1/10)*Sin[5*x]} -{Sin[m*x]*Sin[x],x, 4, Sin[(1 - m)*x]/(2*(1 - m)) - Sin[(1 + m)*x]/(2*(1 + m))} - - -{Cos[2*x]*Sin[x],x, 1, Cos[x]/2 - (1/6)*Cos[3*x]} -{Cos[3*x]*Sin[x],x, 1, (1/4)*Cos[2*x] - (1/8)*Cos[4*x]} -{Cos[4*x]*Sin[x],x, 1, (1/6)*Cos[3*x] - (1/10)*Cos[5*x]} -{Cos[m*x]*Sin[x],x, 4, -(Cos[(1 - m)*x]/(2*(1 - m))) - Cos[(1 + m)*x]/(2*(1 + m))} - - -{Tan[2*x]*Sin[x], x, 4, ArcTanh[Sqrt[2]*Sin[x]]/Sqrt[2] - Sin[x]} -{Tan[3*x]*Sin[x], x, 9, (-(1/6))*Log[1 - 2*Sin[x]] - (1/6)*Log[1 - Sin[x]] + (1/6)*Log[1 + Sin[x]] + (1/6)*Log[1 + 2*Sin[x]] - Sin[x]} -{Tan[4*x]*Sin[x], x, 5, (1/4)*Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]] + (1/4)*Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]] - Sin[x]} -{Tan[5*x]*Sin[x], x, 10, (1/5)*ArcTanh[Sin[x]] - (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Sin[x]] - (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[x]] + (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Sin[x]] + (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Sin[x]] - Sin[x]} -{Tan[6*x]*Sin[x], x, 10, ArcTanh[Sqrt[2]*Sin[x]]/(3*Sqrt[2]) + (1/6)*Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]] + (1/6)*Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]] - Sin[x]} -{Tan[n*x]*Sin[x], x, 6, ((1/2)*I)/E^(I*x) + (1/2)*I*E^(I*x) - (I*Hypergeometric2F1[1, -(1/(2*n)), 1 - 1/(2*n), -E^(2*I*n*x)])/E^(I*x) - I*E^(I*x)*Hypergeometric2F1[1, 1/(2*n), (1/2)*(2 + 1/n), -E^(2*I*n*x)]} - - -{Cot[2*x]*Sin[x], x, 3, (-(1/2))*ArcTanh[Sin[x]] + Sin[x]} -{Cot[3*x]*Sin[x], x, 3, -(ArcTanh[(2*Sin[x])/Sqrt[3]]/Sqrt[3]) + Sin[x]} -{Cot[4*x]*Sin[x], x, 6, (-(1/4))*ArcTanh[Sin[x]] - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + Sin[x]} -{Cot[5*x]*Sin[x], x, 6, (-(1/5))*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]] - (1/5)*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTanh[Sqrt[(2/5)*(5 + Sqrt[5])]*Sin[x]] + Sin[x]} -{Cot[6*x]*Sin[x], x, 7, (-(1/6))*ArcTanh[Sin[x]] - (1/6)*ArcTanh[2*Sin[x]] - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3]) + Sin[x]} - - -{Sec[2*x]*Sin[x], x, 2, ArcTanh[Sqrt[2]*Cos[x]]/Sqrt[2]} -{Sec[3*x]*Sin[x], x, 5, (1/3)*Log[Cos[x]] - (1/6)*Log[3 - 4*Cos[x]^2]} -{Sec[4*x]*Sin[x], x, 4, -(ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])])) + ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])} -{Sec[5*x]*Sin[x], x, 7, (-(1/5))*Log[Cos[x]] + (1/20)*(1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Cos[x]^2] + (1/20)*(1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Cos[x]^2]} -{Sec[6*x]*Sin[x], x, 7, -(ArcTanh[Sqrt[2]*Cos[x]]/(3*Sqrt[2])) + ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])} - - -{Csc[2*x]*Sin[x], x, 2, (1/2)*ArcTanh[Sin[x]]} -{Csc[3*x]*Sin[x], x, 2, -(Log[Sqrt[3]*Cos[x] - Sin[x]]/(2*Sqrt[3])) + Log[Sqrt[3]*Cos[x] + Sin[x]]/(2*Sqrt[3])} -{Csc[4*x]*Sin[x], x, 4, (-(1/4))*ArcTanh[Sin[x]] + ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2])} -{Csc[5*x]*Sin[x], x, 4, (-(1/10))*Sqrt[(1/2)*(5 - Sqrt[5])]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] - Sin[x]] + (1/10)*Sqrt[(1/2)*(5 + Sqrt[5])]*Log[Sqrt[5 + 2*Sqrt[5]]*Cos[x] - Sin[x]] + (1/10)*Sqrt[(1/2)*(5 - Sqrt[5])]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] + Sin[x]] - (1/10)*Sqrt[(1/2)*(5 + Sqrt[5])]*Log[Sqrt[5 + 2*Sqrt[5]]*Cos[x] + Sin[x]]} -{Csc[6*x]*Sin[x], x, 7, (1/6)*ArcTanh[Sin[x]] + (1/6)*ArcTanh[2*Sin[x]] - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3])} - -{Csc[x]*Sin[3*x], x, 3, x + 2*Cos[x]*Sin[x]} -{Csc[3*x]*Sin[6*x], x, 2, (2*Sin[3*x])/3} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Trig[m x] Cos[n x]*) - - -{Sin[2*x]*Cos[x], x, 1, -(Cos[x]/2) - (1/6)*Cos[3*x]} -{Sin[3*x]*Cos[x], x, 1, (-(1/4))*Cos[2*x] - (1/8)*Cos[4*x]} -{Sin[4*x]*Cos[x], x, 1, (-(1/6))*Cos[3*x] - (1/10)*Cos[5*x]} -{Sin[m*x]*Cos[x], x, 4, Cos[(1 - m)*x]/(2*(1 - m)) - Cos[(1 + m)*x]/(2*(1 + m))} - - -{Cos[2*x]*Cos[x], x, 1, Sin[x]/2 + (1/6)*Sin[3*x]} -{Cos[3*x]*Cos[x], x, 1, (1/4)*Sin[2*x] + (1/8)*Sin[4*x]} -{Cos[4*x]*Cos[x], x, 1, (1/6)*Sin[3*x] + (1/10)*Sin[5*x]} -{Cos[m*x]*Cos[x], x, 4, Sin[(1 - m)*x]/(2*(1 - m)) + Sin[(1 + m)*x]/(2*(1 + m))} - - -{Tan[2*x]*Cos[x], x, 4, ArcTanh[Sqrt[2]*Cos[x]]/Sqrt[2] - Cos[x]} -{Tan[3*x]*Cos[x], x, 3, ArcTanh[(2*Cos[x])/Sqrt[3]]/Sqrt[3] - Cos[x]} -{Tan[4*x]*Cos[x], x, 6, (1/4)*Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[2]]] + (1/4)*Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[2]]] - Cos[x]} -{Tan[5*x]*Cos[x], x, 6, (1/5)*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cos[x]] + (1/5)*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTanh[Sqrt[(2/5)*(5 + Sqrt[5])]*Cos[x]] - Cos[x]} -{Tan[6*x]*Cos[x], x, 10, ArcTanh[Sqrt[2]*Cos[x]]/(3*Sqrt[2]) + (1/6)*Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[3]]] + (1/6)*Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[3]]] - Cos[x]} - - -{Cot[2*x]*Cos[x], x, 4, (-(1/2))*ArcTanh[Cos[x]] + Cos[x]} -{Cot[3*x]*Cos[x], x, 9, Cos[x] + (1/6)*Log[1 - 2*Cos[x]] + (1/6)*Log[1 - Cos[x]] - (1/6)*Log[1 + Cos[x]] - (1/6)*Log[1 + 2*Cos[x]]} -{Cot[4*x]*Cos[x], x, 6, (-(1/4))*ArcTanh[Cos[x]] - ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2]) + Cos[x]} -{Cot[5*x]*Cos[x], x, 10, (-(1/5))*ArcTanh[Cos[x]] + Cos[x] + (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]] + (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Cos[x]] - (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]] - (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[x]]} -{Cot[6*x]*Cos[x], x, 7, (-(1/6))*ArcTanh[Cos[x]] - (1/6)*ArcTanh[2*Cos[x]] - ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3]) + Cos[x]} -{Cot[n*x]*Cos[x], x, 6, -(1/2)/E^(I*x) + E^(I*x)/2 + Hypergeometric2F1[1, -(1/(2*n)), 1 - 1/(2*n), E^(2*I*n*x)]/E^(I*x) - E^(I*x)*Hypergeometric2F1[1, 1/(2*n), (1/2)*(2 + 1/n), E^(2*I*n*x)]} - - -{Sec[2*x]*Cos[x], x, 2, ArcTanh[Sqrt[2]*Sin[x]]/Sqrt[2]} -{Sec[3*x]*Cos[x], x, 2, -(Log[Cos[x] - Sqrt[3]*Sin[x]]/(2*Sqrt[3])) + Log[Cos[x] + Sqrt[3]*Sin[x]]/(2*Sqrt[3])} -{Sec[4*x]*Cos[x], x, 4, ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])} -{Sec[5*x]*Cos[x], x, 4, (1/10)*Sqrt[(1/2)*(5 - Sqrt[5])]*Log[Cos[x] - Sqrt[5 - 2*Sqrt[5]]*Sin[x]] - (1/10)*Sqrt[(1/2)*(5 - Sqrt[5])]*Log[Cos[x] + Sqrt[5 - 2*Sqrt[5]]*Sin[x]] - (1/10)*Sqrt[(1/2)*(5 + Sqrt[5])]*Log[Cos[x] - Sqrt[5 + 2*Sqrt[5]]*Sin[x]] + (1/10)*Sqrt[(1/2)*(5 + Sqrt[5])]*Log[Cos[x] + Sqrt[5 + 2*Sqrt[5]]*Sin[x]]} -{Sec[6*x]*Cos[x], x, 7, -(ArcTanh[Sqrt[2]*Sin[x]]/(3*Sqrt[2])) + ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])} - -{Sec[x]*Cos[2*x], x, 3, -ArcTanh[Sin[x]] + 2*Sin[x]} -{Sec[2*x]*Cos[4*x], x, 3, -ArcTanh[Sin[2*x]]/2 + Sin[2*x]} - - -{Csc[2*x]*Cos[x], x, 2, (-(1/2))*ArcTanh[Cos[x]]} -{Csc[3*x]*Cos[x], x, 5, (1/3)*Log[Sin[x]] - (1/6)*Log[3 - 4*Sin[x]^2]} -{Csc[4*x]*Cos[x], x, 4, (-(1/4))*ArcTanh[Cos[x]] + ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2])} -{Csc[5*x]*Cos[x], x, 7, (1/5)*Log[Sin[x]] - (1/20)*(1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Sin[x]^2] - (1/20)*(1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Sin[x]^2]} -{Csc[6*x]*Cos[x], x, 7, (-(1/6))*ArcTanh[Cos[x]] - (1/6)*ArcTanh[2*Cos[x]] + ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3])} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Trig[m x]^p Trig[n x]^q*) - - -{Cos[6*x]^3*Sin[x], x, 6, (3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 - Cos[19*x]/152} -{Cos[6*x]^3*Sin[9*x], x, 6, (-(1/8))*Cos[3*x] + (1/72)*Cos[9*x] - (1/40)*Cos[15*x] - (1/216)*Cos[27*x]} - -{Cos[2*x]*Sin[6*x]^2, x, 5, (1/4)*Sin[2*x] - (1/40)*Sin[10*x] - (1/56)*Sin[14*x]} - -{Cos[x]*Sin[6*x]^2, x, 5, Sin[x]/2 - (1/44)*Sin[11*x] - (1/52)*Sin[13*x]} -{Cos[x]*Sin[6*x]^3, x, 6, (-3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 + Cos[19*x]/152} -{Cos[7*x]*Sin[6*x]^3, x, 6, (3*Cos[x])/8 + Cos[11*x]/88 - (3*Cos[13*x])/104 + Cos[25*x]/200} -{Cos[3*x]^2*Sin[2*x]^3, x, 7, (-(3/16))*Cos[2*x] + (3/64)*Cos[4*x] + (1/48)*Cos[6*x] - (3/128)*Cos[8*x] + (1/192)*Cos[12*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Trig[a+b x] Trig[c+d x] when b^2-d^2=0*) - - -{Sin[a + b*x]*Sin[c + b*x], x, 3, (1/2)*x*Cos[a - c] - Sin[a + c + 2*b*x]/(4*b)} -{Sin[a + b*x]*Sin[c - b*x], x, 3, (-(1/2))*x*Cos[a + c] + Sin[a - c + 2*b*x]/(4*b)} - - -{Cos[a + b*x]*Cos[c + b*x], x, 3, (1/2)*x*Cos[a - c] + Sin[a + c + 2*b*x]/(4*b)} -{Cos[a + b*x]*Cos[c - b*x], x, 3, (1/2)*x*Cos[a + c] + Sin[a - c + 2*b*x]/(4*b)} - - -{Tan[a + b*x]*Tan[c + b*x], x, 4, -x - (Cot[a - c]*Log[Cos[a + b*x]])/b + (Cot[a - c]*Log[Cos[c + b*x]])/b} -{Tan[a + b*x]*Tan[c - b*x], x, 4, x - (Cot[a + c]*Log[Cos[c - b*x]])/b + (Cot[a + c]*Log[Cos[a + b*x]])/b} - - -{Cot[a + b*x]*Cot[c + b*x], x, 4, -x - (Cot[a - c]*Log[Sin[a + b*x]])/b + (Cot[a - c]*Log[Sin[c + b*x]])/b} -{Cot[a + b*x]*Cot[c - b*x], x, 4, x - (Cot[a + c]*Log[Sin[c - b*x]])/b + (Cot[a + c]*Log[Sin[a + b*x]])/b} - - -{Sec[a + b*x]*Sec[c + b*x], x, 3, -((Csc[a - c]*Log[Cos[a + b*x]])/b) + (Csc[a - c]*Log[Cos[c + b*x]])/b} -{Sec[a + b*x]*Sec[c - b*x], x, 3, (Csc[a + c]*Log[Cos[c - b*x]])/b - (Csc[a + c]*Log[Cos[a + b*x]])/b} - - -{Csc[a + b*x]*Csc[c + b*x], x, 3, -((Csc[a - c]*Log[Sin[a + b*x]])/b) + (Csc[a - c]*Log[Sin[c + b*x]])/b} -{Csc[a + b*x]*Csc[c - b*x], x, 3, -((Csc[a + c]*Log[Sin[c - b*x]])/b) + (Csc[a + c]*Log[Sin[a + b*x]])/b} - - -(* ::Section::Closed:: *) -(*Integrands of the form (Trig[a+b x] Trig[a+b x])^m*) - - -{(Sin[x]*Tan[x])^(1/2), x, 2, -2*Cot[x]*Sqrt[Sin[x]*Tan[x]]} -{(Sin[x]*Tan[x])^(3/2), x, 3, (8/3)*Csc[x]*Sqrt[Sin[x]*Tan[x]] - (2/3)*Sin[x]*Sqrt[Sin[x]*Tan[x]]} -{(Sin[x]*Tan[x])^(5/2), x, 4, (64/15)*Cot[x]*Sqrt[Sin[x]*Tan[x]] + (16/15)*Tan[x]*Sqrt[Sin[x]*Tan[x]] - (2/5)*Sin[x]^2*Tan[x]*Sqrt[Sin[x]*Tan[x]]} - - -{(Cos[x]*Cot[x])^(1/2), x, 2, 2*Sqrt[Cos[x]*Cot[x]]*Tan[x]} -{(Cos[x]*Cot[x])^(3/2), x, 3, (2/3)*Cos[x]*Sqrt[Cos[x]*Cot[x]] - (8/3)*Sqrt[Cos[x]*Cot[x]]*Sec[x]} -{(Cos[x]*Cot[x])^(5/2), x, 4, (-(16/15))*Cot[x]*Sqrt[Cos[x]*Cot[x]] + (2/5)*Cos[x]^2*Cot[x]*Sqrt[Cos[x]*Cot[x]] - (64/15)*Sqrt[Cos[x]*Cot[x]]*Tan[x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^q Trig[x]^m (a+b Trig[x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^q Trig[x]^m (a+b Trig[x]^n)^p*) - - -{x*Cos[x]/(a + b*Sin[x])^2, x, 4, (2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]) - x/(b*(a + b*Sin[x]))} -{x*Cos[x]/(a + b*Sin[x])^3, x, 6, (a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)) - x/(2*b*(a + b*Sin[x])^2) + Cos[x]/(2*(a^2 - b^2)*(a + b*Sin[x]))} - - -{x*Sin[x]/(a + b*Cos[x])^2, x, 3, -((2*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])) + x/(b*(a + b*Cos[x]))} -{x*Sin[x]/(a + b*Cos[x])^3, x, 5, -((a*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*b*(a + b)^(3/2))) + x/(2*b*(a + b*Cos[x])^2) + Sin[x]/(2*(a^2 - b^2)*(a + b*Cos[x]))} - - -{x*Sec[x]^2/(a + b*Tan[x])^2, x, 3, (a*x)/(b*(a^2 + b^2)) + Log[a*Cos[x] + b*Sin[x]]/(a^2 + b^2) - x/(b*(a + b*Tan[x]))} -{x*Csc[x]^2/(a + b*Cot[x])^2, x, 3, -((a*x)/(b*(a^2 + b^2))) + x/(b*(a + b*Cot[x])) + Log[b*Cos[x] + a*Sin[x]]/(a^2 + b^2)} - - -{Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)} -{x*Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2), x, 9, -((I*x*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2])/(2*Sqrt[a]*Sqrt[b]*d)) + (I*x*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2])/(2*Sqrt[a]*Sqrt[b]*d) - PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2)]/(4*Sqrt[a]*Sqrt[b]*d^2) + PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2)]/(4*Sqrt[a]*Sqrt[b]*d^2)} -{x^2*Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2), x, 11, -((I*x^2*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2])/(2*Sqrt[a]*Sqrt[b]*d)) + (I*x^2*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2])/(2*Sqrt[a]*Sqrt[b]*d) - (x*PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2)])/(2*Sqrt[a]*Sqrt[b]*d^2) + (x*PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2)])/(2*Sqrt[a]*Sqrt[b]*d^2) + (I*PolyLog[3, -(((Sqrt[a] - Sqrt[b])*E^(2*I*(c + d*x)))/(Sqrt[a] + Sqrt[b]))])/(4*Sqrt[a]*Sqrt[b]*d^3) - (I*PolyLog[3, -(((Sqrt[a] + Sqrt[b])*E^(2*I*(c + d*x)))/(Sqrt[a] - Sqrt[b]))])/(4*Sqrt[a]*Sqrt[b]*d^3)} - - -{Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2 + c*Sec[c + d*x]^2), x, 2, ArcTan[(Sqrt[b + c]*Tan[c + d*x])/Sqrt[a + c]]/(Sqrt[a + c]*Sqrt[b + c]*d)} -{x*Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2 + c*Sec[c + d*x]^2), x, 9, -((I*x*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(2*Sqrt[a + c]*Sqrt[b + c]*d)) + (I*x*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(2*Sqrt[a + c]*Sqrt[b + c]*d) - PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2) + PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2)} -{x^2*Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2 + c*Sec[c + d*x]^2), x, 11, -((I*x^2*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(2*Sqrt[a + c]*Sqrt[b + c]*d)) + (I*x^2*Log[1 + ((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(2*Sqrt[a + c]*Sqrt[b + c]*d) - (x*PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) + (x*PolyLog[2, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) - (I*PolyLog[3, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))])/(4*Sqrt[a + c]*Sqrt[b + c]*d^3) + (I*PolyLog[3, -(((a - b)*E^(2*I*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))])/(4*Sqrt[a + c]*Sqrt[b + c]*d^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^p (a+b Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]], x, 5, -((6*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4) + (3*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (6*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 + (x^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f} -{x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]], x, 4, (2*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 + (x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f} -{x^1*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]], x, 3, (Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 + (x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f} -{Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]/x^1, x, 4, Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x]} -{Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]/x^2, x, 5, -((Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x) - f*CosIntegral[f*x]*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - f*Cos[e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x]} -{Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]/x^3, x, 6, -((Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2)) - (1/2)*f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] + (1/2)*f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] + (f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)} - - -{x^3*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2), x, 11, -((6*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4) + (3*c*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 + (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^3) - (3*c*x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) - (3*c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^4) + (3*c*x^2*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f^2) + (x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (6*c*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (3*c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)} -{x^2*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2), x, 8, (2*c*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (3*c*x^2*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) + (c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*f^2) + (x^2*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (2*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)} -{x^1*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2), x, 3, (c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) + (c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f^2) + (x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f)} -{Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2)/x^1, x, 11, c*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] + (1/2)*c*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - c*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] + (1/2)*c*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x]} -{Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2)/x^2, x, 13, -((c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x) + c*f*Cos[2*e]*CosIntegral[2*f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - c*f*CosIntegral[f*x]*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])/(2*x) - c*f*Cos[e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] - c*f*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x]} -{Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2)/x^3, x, 15, -((c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2)) - (c*f*Cos[2*e + 2*f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x) - (1/2)*c*f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - c*f^2*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])/(4*x^2) + (1/2)*c*f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] - c*f^2*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x] + (c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(g + h*x)^3*Sqrt[a - a*Sin[e + f*x]]/Sqrt[c + c*Sin[e + f*x]], x, 20, -((I*a*(g + h*x)^4*Cos[e + f*x])/(4*h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (2*I*a*(g + h*x)^3*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)^3*Cos[e + f*x]*Log[1 + E^(2*I*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*I*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (3*I*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (3*I*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, -E^(2*I*(e + f*x))])/(2*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (6*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, (-I)*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, I*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, -E^(2*I*(e + f*x))])/(2*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (6*I*a*h^3*Cos[e + f*x]*PolyLog[4, (-I)*E^(I*(e + f*x))])/(f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*I*a*h^3*Cos[e + f*x]*PolyLog[4, I*E^(I*(e + f*x))])/(f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*I*a*h^3*Cos[e + f*x]*PolyLog[4, -E^(2*I*(e + f*x))])/(4*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} -{(g + h*x)^2*Sqrt[a - a*Sin[e + f*x]]/Sqrt[c + c*Sin[e + f*x]], x, 17, -((I*a*(g + h*x)^3*Cos[e + f*x])/(3*h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (2*I*a*(g + h*x)^2*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)^2*Cos[e + f*x]*Log[1 + E^(2*I*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*I*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (2*I*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (I*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, -E^(2*I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (2*a*h^2*Cos[e + f*x]*PolyLog[3, (-I)*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*h^2*Cos[e + f*x]*PolyLog[3, I*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*h^2*Cos[e + f*x]*PolyLog[3, -E^(2*I*(e + f*x))])/(2*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} -{(g + h*x)^1*Sqrt[a - a*Sin[e + f*x]]/Sqrt[c + c*Sin[e + f*x]], x, 14, -((I*a*(g + h*x)^2*Cos[e + f*x])/(2*h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (2*I*a*(g + h*x)*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)*Cos[e + f*x]*Log[1 + E^(2*I*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (I*a*h*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (I*a*h*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (I*a*h*Cos[e + f*x]*PolyLog[2, -E^(2*I*(e + f*x))])/(2*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} -{Sqrt[a - a*Sin[e + f*x]]/((g + h*x)^1*Sqrt[c + c*Sin[e + f*x]]), x, 5, (a*Cos[e + f*x]*Unintegrable[Sec[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*Cos[e + f*x]*Unintegrable[Tan[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} - - -{x^3*Sqrt[a - a*Sin[e + f*x]]/(c + c*Sin[e + f*x])^(3/2), x, 51, -((3*a*x^2)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (3*I*a*x^2*Cos[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (12*I*a*x*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*a*x*Cos[e + f*x]*Log[1 + E^(2*I*(e + f*x))])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*I*a*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (6*I*a*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (3*I*a*Cos[e + f*x]*PolyLog[2, -E^(2*I*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^3*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*a*x^2*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^3*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} -{x^2*Sqrt[a - a*Sin[e + f*x]]/(c + c*Sin[e + f*x])^(3/2), x, 34, -((2*a*x)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) + (2*a*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^2*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*x*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^2*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} -{x^1*Sqrt[a - a*Sin[e + f*x]]/(c + c*Sin[e + f*x])^(3/2), x, 26, -(a/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (a*x*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])} - - -{(z^2*Sqrt[1 + Cos[z]])/Sqrt[1 - Cos[z]], z, 15, -((I*z^3*Sin[z])/(3*Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]])) - (2*z^2*ArcTanh[E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (z^2*Log[1 - E^(2*I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (2*I*z*PolyLog[2, -E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (2*I*z*PolyLog[2, E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (I*z*PolyLog[2, E^(2*I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (2*PolyLog[3, -E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (2*PolyLog[3, E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (PolyLog[3, E^(2*I*z)]*Sin[z])/(2*Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Trig[x]^m)^p / (a+b Trig[x]^n)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Trig[x]) (a+b Trig[x])^n*) - - -{(A + B*Sec[x])*(a + a*Cos[x])^1, x, 5, a*(A + B)*x + a*B*ArcTanh[Sin[x]] + a*A*Sin[x]} -{(A + B*Sec[x])*(a + a*Cos[x])^2, x, 6, (1/2)*a^2*(3*A + 4*B)*x + a^2*B*ArcTanh[Sin[x]] + (1/2)*a^2*(3*A + 2*B)*Sin[x] + (1/2)*A*(a^2 + a^2*Cos[x])*Sin[x]} -{(A + B*Sec[x])*(a + a*Cos[x])^3, x, 7, (1/2)*a^3*(5*A + 7*B)*x + a^3*B*ArcTanh[Sin[x]] + (5/2)*a^3*(A + B)*Sin[x] + (1/3)*a*A*(a + a*Cos[x])^2*Sin[x] + (1/6)*(5*A + 3*B)*(a^3 + a^3*Cos[x])*Sin[x]} -{(A + B*Sec[x])*(a + a*Cos[x])^4, x, 8, (1/8)*a^4*(35*A + 48*B)*x + a^4*B*ArcTanh[Sin[x]] + (5/8)*a^4*(7*A + 8*B)*Sin[x] + (1/4)*a*A*(a + a*Cos[x])^3*Sin[x] + (1/12)*(7*A + 4*B)*(a^2 + a^2*Cos[x])^2*Sin[x] + (1/24)*(35*A + 32*B)*(a^4 + a^4*Cos[x])*Sin[x]} - - -{(A + B*Sec[x])/(a + a*Cos[x])^1, x, 4, (B*ArcTanh[Sin[x]])/a + ((A - B)*Sin[x])/(a + a*Cos[x])} -{(A + B*Sec[x])/(a + a*Cos[x])^2, x, 5, (B*ArcTanh[Sin[x]])/a^2 + ((A - 4*B)*Sin[x])/(3*a^2*(1 + Cos[x])) + ((A - B)*Sin[x])/(3*(a + a*Cos[x])^2)} -{(A + B*Sec[x])/(a + a*Cos[x])^3, x, 6, (B*ArcTanh[Sin[x]])/a^3 + ((A - B)*Sin[x])/(5*(a + a*Cos[x])^3) + ((2*A - 7*B)*Sin[x])/(15*a*(a + a*Cos[x])^2) + (2*(A - 11*B)*Sin[x])/(15*(a^3 + a^3*Cos[x]))} -{(A + B*Sec[x])/(a + a*Cos[x])^4, x, 7, (B*ArcTanh[Sin[x]])/a^4 + ((6*A - 55*B)*Sin[x])/(105*a^4*(1 + Cos[x])^2) + (2*(3*A - 80*B)*Sin[x])/(105*a^4*(1 + Cos[x])) + ((A - B)*Sin[x])/(7*(a + a*Cos[x])^4) + ((3*A - 10*B)*Sin[x])/(35*a*(a + a*Cos[x])^3)} - - -{(A + B*Sec[x])*(a + a*Cos[x])^(5/2), x, 6, 2*a^(5/2)*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a^3*(32*A + 35*B)*Sin[x])/(15*Sqrt[a + a*Cos[x]]) + (2/15)*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[x]]*Sin[x] + (2/5)*a*A*(a + a*Cos[x])^(3/2)*Sin[x]} -{(A + B*Sec[x])*(a + a*Cos[x])^(3/2), x, 5, 2*a^(3/2)*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a^2*(4*A + 3*B)*Sin[x])/(3*Sqrt[a + a*Cos[x]]) + (2/3)*a*A*Sqrt[a + a*Cos[x]]*Sin[x]} -{(A + B*Sec[x])*(a + a*Cos[x])^(1/2), x, 4, 2*Sqrt[a]*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a*A*Sin[x])/Sqrt[a + a*Cos[x]]} -{(A + B*Sec[x])/(a + a*Cos[x])^(1/2), x, 6, (2*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/Sqrt[a] + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/Sqrt[a]} -{(A + B*Sec[x])/(a + a*Cos[x])^(3/2), x, 7, (2*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/a^(3/2) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/(2*Sqrt[2]*a^(3/2)) + ((A - B)*Sin[x])/(2*(a + a*Cos[x])^(3/2))} -{(A + B*Sec[x])/(a + a*Cos[x])^(5/2), x, 8, (2*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/a^(5/2) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/(16*Sqrt[2]*a^(5/2)) + ((A - B)*Sin[x])/(4*(a + a*Cos[x])^(5/2)) + ((3*A - 11*B)*Sin[x])/(16*a*(a + a*Cos[x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (A+B Trig[x]) (a+b Trig[x])^n*) - - -{x*((b + a*Sin[x])/(a + b*Sin[x])^2), x, 3, Log[a + b*Sin[x]]/b - (x*Cos[x])/(a + b*Sin[x])} -{x*((b + a*Cos[x])/(a + b*Cos[x])^2), x, 3, Log[a + b*Cos[x]]/b + (x*Sin[x])/(a + b*Cos[x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Trig[x]^m)^p / (a+b Trig[x]^n)*) -(**) - - -{(1 + Sin[x]^2)/(1 - Sin[x]^2), x, 4, -x + 2*Tan[x]} -{(1 - Sin[x]^2)/(1 + Sin[x]^2), x, 3, -x + Sqrt[2]*x + Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]} - - -{(1 + Cos[x]^2)/(1 - Cos[x]^2), x, 4, -x - 2*Cot[x]} -{(1 - Cos[x]^2)/(1 + Cos[x]^2), x, 3, -x + Sqrt[2]*x - Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]} - - -{(-1 + c^2/d^2 + Sin[x]^2)/(c + d*Cos[x]), x, 4, (c*x)/d^2 - Sin[x]/d} -{(a + b*Sin[x]^2)/(c + d*Cos[x]), x, 8, (b*c*x)/d^2 + (2*a*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]) - (2*b*Sqrt[c - d]*Sqrt[c + d]*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/d^2 - (b*Sin[x])/d} - -{(a + b*Sin[x]^2)/(c + c*Cos[x]^2), x, 5, -((b*x)/c) + ((a + 2*b)*x)/(Sqrt[2]*c) - ((a + 2*b)*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(Sqrt[2]*c)} -{(a + b*Sin[x]^2)/(c - c*Cos[x]^2), x, 3, (b*x)/c - (a*Cot[x])/c} -{(a + b*Sin[x]^2)/(c + d*Cos[x]^2), x, 4, -((b*x)/d) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c]*Tan[x])/Sqrt[c + d]])/(Sqrt[c]*d*Sqrt[c + d])} - - -{(-1 + c^2/d^2 + Cos[x]^2)/(c + d*Sin[x]), x, 4, (c*x)/d^2 + Cos[x]/d} -{(a + b*Cos[x]^2)/(c + d*Sin[x]), x, 10, (b*c*x)/d^2 + (2*a*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - (2*b*Sqrt[c^2 - d^2]*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/d^2 + (b*Cos[x])/d} - -{(a + b*Cos[x]^2)/(c + c*Sin[x]^2), x, 4, -((b*x)/c) + ((a + 2*b)*x)/(Sqrt[2]*c) + ((a + 2*b)*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)])/(Sqrt[2]*c)} -{(a + b*Cos[x]^2)/(c - c*Sin[x]^2), x, 3, (b*x)/c + (a*Tan[x])/c} -{(a + b*Cos[x]^2)/(c + d*Sin[x]^2), x, 4, -((b*x)/d) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c + d]*Tan[x])/Sqrt[c]])/(Sqrt[c]*d*Sqrt[c + d])} - - -{(a + b*Sec[x]^2)/(c + d*Cos[x]), x, 6, (2*(a*c^2 + b*d^2)*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/(c^2*Sqrt[c - d]*Sqrt[c + d]) - (b*d*ArcTanh[Sin[x]])/c^2 + (b*Tan[x])/c} -{(a + b*Csc[x]^2)/(c + d*Sin[x]), x, 7, (2*(a*c^2 + b*d^2)*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/(c^2*Sqrt[c^2 - d^2]) + (b*d*ArcTanh[Cos[x]])/c^2 - (b*Cot[x])/c} - - -(* {Sqrt[1 + Sin[x]]/(1 - Tan[x]^2), x, 0, 0} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a Trig[c+d x] + b Trig[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x] + b Sin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x] + b Sin[c+d x])^n*) - - -{(a*Cos[c+d*x] + b*Sin[c+d*x])^n, x, 2, -((Cos[c + d*x - ArcTan[a, b]]^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x - ArcTan[a, b]]^2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^n*Sin[c + d*x - ArcTan[a, b]])/(((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n*(d*(1 + n)*Sqrt[Sin[c + d*x - ArcTan[a, b]]^2])))} -{(2*Cos[c+d*x] + 3*Sin[c+d*x])^n, x, 2, -((13^(n/2)*Cos[c + d*x - ArcTan[3/2]]^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(d*(1 + n)*Sqrt[Sin[c + d*x - ArcTan[3/2]]^2]))} - - -{(a*Cos[c+d*x] + b*Sin[c+d*x])^7, x, 3, -(((a^2 + b^2)^3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + ((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x])^3)/d - (3*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])^5)/(5*d) + (b*Cos[c + d*x] - a*Sin[c + d*x])^7/(7*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^6, x, 4, (5/16)*(a^2 + b^2)^3*x - (5*(a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(16*d) - (5*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(24*d) - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)/(6*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^5, x, 3, -(((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (2*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])^3)/(3*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])^5/(5*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^4, x, 3, (3/8)*(a^2 + b^2)^2*x - (3*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(8*d) - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(4*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^3, x, 2, -(((a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (b*Cos[c + d*x] - a*Sin[c + d*x])^3/(3*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^2, x, 2, (1/2)*(a^2 + b^2)*x - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(2*d)} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^1, x, 3, -((b*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^1, x, 2, -(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(Sqrt[a^2 + b^2]*d))} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^2, x, 1, Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^3, x, 3, -(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2)*d)) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^4, x, 2, -((b*Cos[c + d*x] - a*Sin[c + d*x])/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)) + (2*Sin[c + d*x])/(3*a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^5, x, 4, -((3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(8*(a^2 + b^2)^(5/2)*d)) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(4*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(8*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^6, x, 3, -((b*Cos[c + d*x] - a*Sin[c + d*x])/(5*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)) - (4*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(15*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (8*Sin[c + d*x])/(15*a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x] + b Sin[c+d x])^(n/2)*) - - -{(a*Cos[c+d*x] + b*Sin[c+d*x])^(7/2), x, 4, -((10*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(21*d)) - (2*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2))/(7*d) + (10*(a^2 + b^2)^2*EllipticF[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(21*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^(5/2), x, 3, -((2*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2))/(5*d)) + (6*(a^2 + b^2)*EllipticE[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(5*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^(3/2), x, 3, -((2*(b*Cos[c + d*x] - a*Sin[c + d*x])*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(3*d)) + (2*(a^2 + b^2)*EllipticF[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(3*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])} -{(a*Cos[c+d*x] + b*Sin[c+d*x])^(1/2), x, 2, (2*EllipticE[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^(1/2), x, 2, (2*EllipticF[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^(3/2), x, 3, -((2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/((a^2 + b^2)*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])) - (2*EllipticE[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^(5/2), x, 3, -((2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2))) + (2*EllipticF[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(3*(a^2 + b^2)*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])} -{1/(a*Cos[c+d*x] + b*Sin[c+d*x])^(7/2), x, 4, -((2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(5*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2))) - (6*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(5*(a^2 + b^2)^2*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]]) - (6*EllipticE[(1/2)*(c + d*x - ArcTan[a, b]), 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(5*(a^2 + b^2)^2*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])} - - -{(2*Cos[c+d*x] + 3*Sin[c+d*x])^(7/2), x, 4, (130*13^(3/4)*EllipticF[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(21*d) - (130*(3*Cos[c + d*x] - 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])/(21*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2))/(7*d)} -{(2*Cos[c+d*x] + 3*Sin[c+d*x])^(5/2), x, 3, (78*13^(1/4)*EllipticE[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(5*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2))/(5*d)} -{(2*Cos[c+d*x] + 3*Sin[c+d*x])^(3/2), x, 3, (2*13^(3/4)*EllipticF[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(3*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])/(3*d)} -{(2*Cos[c+d*x] + 3*Sin[c+d*x])^(1/2), x, 2, (2*13^(1/4)*EllipticE[(1/2)*(c + d*x - ArcTan[3/2]), 2])/d} -{1/(2*Cos[c+d*x] + 3*Sin[c+d*x])^(1/2), x, 2, (2*EllipticF[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(13^(1/4)*d)} -{1/(2*Cos[c+d*x] + 3*Sin[c+d*x])^(3/2), x, 3, -((2*EllipticE[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(13^(3/4)*d)) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(13*d*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])} -{1/(2*Cos[c+d*x] + 3*Sin[c+d*x])^(5/2), x, 3, (2*EllipticF[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(39*13^(1/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(39*d*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2))} -{1/(2*Cos[c+d*x] + 3*Sin[c+d*x])^(7/2), x, 4, -((6*EllipticE[(1/2)*(c + d*x - ArcTan[3/2]), 2])/(65*13^(3/4)*d)) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(65*d*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2)) - (6*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(845*d*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x] + i a Sin[c+d x])^n*) - - -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n, x, 1, -((I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n)/(d*n))} - -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4, x, 1, -((I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)/(4*d))} -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 1, -((I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)/(3*d))} -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 1, -((I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)/(2*d))} -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^1, x, 3, -((I*a*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^1, x, 1, I/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2, x, 1, I/(2*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3, x, 1, I/(3*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4, x, 1, I/(4*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x] + i a Sin[c+d x])^(n/2)*) - - -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2), x, 1, -((2*I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2))/(5*d))} -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2), x, 1, -((2*I*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2))/(3*d))} -{(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(1/2), x, 1, -((2*I*Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]])/d)} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(1/2), x, 1, (2*I)/(d*Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]])} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2), x, 1, (2*I)/(3*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2))} -{1/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2), x, 1, (2*I)/(5*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[c+d x] + b Tan[c+d x])^n*) - - -{(a*Sec[x] + b*Tan[x])^5, x, 8, (-(1/16))*(a + b)^3*(3*a^2 - 9*a*b + 8*b^2)*Log[1 - Sin[x]] + (1/16)*(a - b)^3*(3*a^2 + 9*a*b + 8*b^2)*Log[1 + Sin[x]] - (1/8)*a*(7 - (3*a^2)/b^2)*b^4*Sin[x] + (1/4)*Sec[x]^4*(b + a*Sin[x])*(a + b*Sin[x])^4 + (1/8)*Sec[x]^2*(a + b*Sin[x])^2*(2*b*(a^2 - 2*b^2) + a*(3*a^2 - 5*b^2)*Sin[x])} -{(a*Sec[x] + b*Tan[x])^4, x, 4, b^4*x + (4/3)*a*b*(a^2 - 2*b^2)*Cos[x] + (1/3)*b^2*(2*a^2 - 3*b^2)*Cos[x]*Sin[x] + (1/3)*Sec[x]^3*(b + a*Sin[x])*(a + b*Sin[x])^3 - (1/3)*Sec[x]*(a + b*Sin[x])^2*(a*b - (2*a^2 - 3*b^2)*Sin[x])} -{(a*Sec[x] + b*Tan[x])^3, x, 7, (-(1/4))*(a - 2*b)*(a + b)^2*Log[1 - Sin[x]] + (1/4)*(a - b)^2*(a + 2*b)*Log[1 + Sin[x]] + (1/2)*a*b^2*Sin[x] + (1/2)*Sec[x]^2*(b + a*Sin[x])*(a + b*Sin[x])^2} -{(a*Sec[x] + b*Tan[x])^2, x, 4, (-b^2)*x + a*b*Cos[x] + Sec[x]*(b + a*Sin[x])*(a + b*Sin[x])} -{(a*Sec[x] + b*Tan[x])^1, x, 3, a*ArcTanh[Sin[x]] - b*Log[Cos[x]]} -{1/(a*Sec[x] + b*Tan[x])^1, x, 3, Log[a + b*Sin[x]]/b} -{1/(a*Sec[x] + b*Tan[x])^2, x, 6, -(x/b^2) + (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cos[x]/(b*(a + b*Sin[x]))} -{1/(a*Sec[x] + b*Tan[x])^3, x, 4, -(Log[a + b*Sin[x]]/b^3) + (a^2 - b^2)/(2*b^3*(a + b*Sin[x])^2) - (2*a)/(b^3*(a + b*Sin[x]))} -{1/(a*Sec[x] + b*Tan[x])^4, x, 8, x/b^4 - (a*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)) - Cos[x]^3/(3*b*(a + b*Sin[x])^3) + (a*Cos[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (Cos[x]*(2*(a^2 - b^2) + a*b*Sin[x]))/(2*b^3*(a^2 - b^2)*(a + b*Sin[x]))} -{1/(a*Sec[x] + b*Tan[x])^5, x, 4, Log[a + b*Sin[x]]/b^5 - (a^2 - b^2)^2/(4*b^5*(a + b*Sin[x])^4) + (4*a*(a^2 - b^2))/(3*b^5*(a + b*Sin[x])^3) - (3*a^2 - b^2)/(b^5*(a + b*Sin[x])^2) + (4*a)/(b^5*(a + b*Sin[x]))} - - -{(Sec[x] + Tan[x])^5, x, 4, -Log[1 - Sin[x]] + 2/(1 - Sin[x])^2 - 4/(1 - Sin[x])} -{(Sec[x] + Tan[x])^4, x, 5, x + (2*Cos[x]^3)/(3*(1 - Sin[x])^3) - (2*Cos[x])/(1 - Sin[x])} -{(Sec[x] + Tan[x])^3, x, 4, Log[1 - Sin[x]] + 2/(1 - Sin[x])} -{(Sec[x] + Tan[x])^2, x, 4, -x + (2*Cos[x])/(1 - Sin[x])} -{(Sec[x] + Tan[x])^1, x, 3, -2*Log[Cos[(1/4)*(Pi + 2*x)]], ArcTanh[Sin[x]] - Log[Cos[x]]} -{1/(Sec[x] + Tan[x])^1, x, 3, Log[1 + Sin[x]]} -{1/(Sec[x] + Tan[x])^2, x, 3, -x - (2*Cos[x])/(1 + Sin[x])} -{1/(Sec[x] + Tan[x])^3, x, 4, -Log[1 + Sin[x]] - 2/(1 + Sin[x])} -{1/(Sec[x] + Tan[x])^4, x, 4, x - (2*Cos[x]^3)/(3*(1 + Sin[x])^3) + (2*Cos[x])/(1 + Sin[x])} -{1/(Sec[x] + Tan[x])^5, x, 4, Log[1 + Sin[x]] - 2/(1 + Sin[x])^2 + 4/(1 + Sin[x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cot[c+d x] + b Csc[c+d x])^n*) - - -{(a*Cot[x] + b*Csc[x])^5, x, 8, (1/8)*a^2*b*(7*a^2 - 3*b^2)*Cos[x] + (1/8)*(b + a*Cos[x])^2*(2*a*(2*a^2 - b^2) + b*(5*a^2 - 3*b^2)*Cos[x])*Csc[x]^2 - (1/4)*(b + a*Cos[x])^4*(a + b*Cos[x])*Csc[x]^4 + (1/16)*(a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[1 - Cos[x]] + (1/16)*(a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[1 + Cos[x]]} -{(a*Cot[x] + b*Csc[x])^4, x, 4, a^4*x + (1/3)*(b + a*Cos[x])^2*(a*b + (3*a^2 - 2*b^2)*Cos[x])*Csc[x] - (1/3)*(b + a*Cos[x])^3*(a + b*Cos[x])*Csc[x]^3 + (4/3)*a*b*(2*a^2 - b^2)*Sin[x] + (1/3)*a^2*(3*a^2 - 2*b^2)*Cos[x]*Sin[x]} -{(a*Cot[x] + b*Csc[x])^3, x, 7, (-(1/2))*a^2*b*Cos[x] - (1/2)*(b + a*Cos[x])^2*(a + b*Cos[x])*Csc[x]^2 - (1/4)*(2*a - b)*(a + b)^2*Log[1 - Cos[x]] - (1/4)*(a - b)^2*(2*a + b)*Log[1 + Cos[x]]} -{(a*Cot[x] + b*Csc[x])^2, x, 4, (-a^2)*x - (b + a*Cos[x])*(a + b*Cos[x])*Csc[x] - a*b*Sin[x]} -{(a*Cot[x] + b*Csc[x])^1, x, 3, (-b)*ArcTanh[Cos[x]] + a*Log[Sin[x]]} -{1/(a*Cot[x] + b*Csc[x])^1, x, 3, -(Log[b + a*Cos[x]]/a)} -{1/(a*Cot[x] + b*Csc[x])^2, x, 5, -(x/a^2) + (2*b*ArcTanh[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) + Sin[x]/(a*(b + a*Cos[x]))} -{1/(a*Cot[x] + b*Csc[x])^3, x, 4, (a^2 - b^2)/(2*a^3*(b + a*Cos[x])^2) + (2*b)/(a^3*(b + a*Cos[x])) + Log[b + a*Cos[x]]/a^3} -{1/(a*Cot[x] + b*Csc[x])^4, x, 7, x/a^4 - (b*(3*a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)) - ((2*(a^2 - b^2) - a*b*Cos[x])*Sin[x])/(2*a^3*(a^2 - b^2)*(b + a*Cos[x])) + Sin[x]^3/(3*a*(b + a*Cos[x])^3) + (b*Sin[x]^3)/(2*a*(a^2 - b^2)*(b + a*Cos[x])^2)} -{1/(a*Cot[x] + b*Csc[x])^5, x, 4, (a^2 - b^2)^2/(4*a^5*(b + a*Cos[x])^4) + (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cos[x])^3) - (a^2 - 3*b^2)/(a^5*(b + a*Cos[x])^2) - (4*b)/(a^5*(b + a*Cos[x])) - Log[b + a*Cos[x]]/a^5} - - -{(Csc[x] + Cot[x])^5, x, 4, -(2/(1 - Cos[x])^2) + 4/(1 - Cos[x]) + Log[1 - Cos[x]]} -{(Csc[x] + Cot[x])^4, x, 5, x + (2*Sin[x])/(1 - Cos[x]) - (2*Sin[x]^3)/(3*(1 - Cos[x])^3)} -{(Csc[x] + Cot[x])^3, x, 4, -(2/(1 - Cos[x])) - Log[1 - Cos[x]]} -{(Csc[x] + Cot[x])^2, x, 4, -x - (2*Sin[x])/(1 - Cos[x])} -{(Csc[x] + Cot[x])^1, x, 3, -ArcTanh[Cos[x]] + Log[Sin[x]]} -{1/(Csc[x] + Cot[x])^1, x, 3, -Log[1 + Cos[x]]} -{1/(Csc[x] + Cot[x])^2, x, 3, -x + (2*Sin[x])/(1 + Cos[x])} -{1/(Csc[x] + Cot[x])^3, x, 4, 2/(1 + Cos[x]) + Log[1 + Cos[x]]} -{1/(Csc[x] + Cot[x])^4, x, 4, x - (2*Sin[x])/(1 + Cos[x]) + (2*Sin[x]^3)/(3*(1 + Cos[x])^3)} -{1/(Csc[x] + Cot[x])^5, x, 4, 2/(1 + Cos[x])^2 - 4/(1 + Cos[x]) - Log[1 + Cos[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Csc[c+d x] + b Sin[c+d x])^n*) - - -(* Note that Csc[x]-Sin[x] == Cos[x]*Cot[x] *) -{(Csc[x] - Sin[x])^4, x, 6, (35*x)/8 + (35*Cot[x])/8 - (35*Cot[x]^3)/24 + (7/8)*Cos[x]^2*Cot[x]^3 + (1/4)*Cos[x]^4*Cot[x]^3} -{(Csc[x] - Sin[x])^3, x, 6, (5/2)*ArcTanh[Cos[x]] - (5*Cos[x])/2 - (5*Cos[x]^3)/6 - (1/2)*Cos[x]^3*Cot[x]^2} -{(Csc[x] - Sin[x])^2, x, 4, -((3*x)/2) - (3*Cot[x])/2 + (1/2)*Cos[x]^2*Cot[x]} -{(Csc[x] - Sin[x]), x, 3, -ArcTanh[Cos[x]] + Cos[x]} -{1/(Csc[x] - Sin[x])^1, x, 3, Sec[x]} -{1/(Csc[x] - Sin[x])^2, x, 2, Tan[x]^3/3} -{1/(Csc[x] - Sin[x])^3, x, 4, (-(1/3))*Sec[x]^3 + Sec[x]^5/5} -{1/(Csc[x] - Sin[x])^4, x, 2, Tan[x]^5/5 + Tan[x]^7/7} -{1/(Csc[x] - Sin[x])^5, x, 4, Sec[x]^5/5 - (2*Sec[x]^7)/7 + Sec[x]^9/9} -{1/(Csc[x] - Sin[x])^6, x, 3, Tan[x]^7/7 + (2*Tan[x]^9)/9 + Tan[x]^11/11} -{1/(Csc[x] - Sin[x])^7, x, 4, (-(1/7))*Sec[x]^7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13} - - -{(Csc[x] - Sin[x])^(7/2), x, 6, (8/7)*Cos[x]*Cot[x]^2*Sqrt[Cos[x]*Cot[x]] + (2/7)*Cos[x]^3*Cot[x]^2*Sqrt[Cos[x]*Cot[x]] - (64/35)*Cot[x]*Sqrt[Cos[x]*Cot[x]]*Csc[x] + (256/35)*Sqrt[Cos[x]*Cot[x]]*Sec[x]} -{(Csc[x] - Sin[x])^(5/2), x, 5, (-(16/15))*Cot[x]*Sqrt[Cos[x]*Cot[x]] + (2/5)*Cos[x]^2*Cot[x]*Sqrt[Cos[x]*Cot[x]] - (64/15)*Sqrt[Cos[x]*Cot[x]]*Tan[x]} -{(Csc[x] - Sin[x])^(3/2), x, 4, (2/3)*Cos[x]*Sqrt[Cos[x]*Cot[x]] - (8/3)*Sqrt[Cos[x]*Cot[x]]*Sec[x]} -{(Csc[x] - Sin[x])^(1/2), x, 3, 2*Sqrt[Cos[x]*Cot[x]]*Tan[x]} -{1/(Csc[x] - Sin[x])^(1/2), x, 8, (ArcTan[Sqrt[-Sin[x]]]*Cos[x])/(Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]]) - (ArcTanh[Sqrt[-Sin[x]]]*Cos[x])/(Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]])} -{1/(Csc[x] - Sin[x])^(3/2), x, 9, Sec[x]/(2*Sqrt[Cos[x]*Cot[x]]) + (ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]]) + (ArcTanh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]])} -{1/(Csc[x] - Sin[x])^(5/2), x, 10, -((3*ArcTan[Sqrt[-Sin[x]]]*Cos[x])/(32*Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]])) + (3*ArcTanh[Sqrt[-Sin[x]]]*Cos[x])/(32*Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]]) - (3*Tan[x])/(16*Sqrt[Cos[x]*Cot[x]]) + (Sec[x]^2*Tan[x])/(4*Sqrt[Cos[x]*Cot[x]])} -{1/(Csc[x] - Sin[x])^(7/2), x, 11, (5*Sec[x])/(192*Sqrt[Cos[x]*Cot[x]]) - (5*Sec[x]^3)/(48*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTanh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*Cot[x]]) + (Sec[x]^3*Tan[x]^2)/(6*Sqrt[Cos[x]*Cot[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[c+d x] + b Cos[c+d x])^n*) - - -(* Note that Sec[x]-Cos[x] == Sin[x]*Tan[x] *) -{(Sec[x] - Cos[x])^4, x, 6, (35*x)/8 - (35*Tan[x])/8 + (35*Tan[x]^3)/24 - (7/8)*Sin[x]^2*Tan[x]^3 - (1/4)*Sin[x]^4*Tan[x]^3} -{(Sec[x] - Cos[x])^3, x, 6, (-(5/2))*ArcTanh[Sin[x]] + (5*Sin[x])/2 + (5*Sin[x]^3)/6 + (1/2)*Sin[x]^3*Tan[x]^2} -{(Sec[x] - Cos[x])^2, x, 4, -((3*x)/2) + (3*Tan[x])/2 - (1/2)*Sin[x]^2*Tan[x]} -{(Sec[x] - Cos[x])^1, x, 3, ArcTanh[Sin[x]] - Sin[x]} -{1/(Sec[x] - Cos[x])^1, x, 3, -Csc[x]} -{1/(Sec[x] - Cos[x])^2, x, 2, (-(1/3))*Cot[x]^3} -{1/(Sec[x] - Cos[x])^3, x, 4, Csc[x]^3/3 - Csc[x]^5/5} -{1/(Sec[x] - Cos[x])^4, x, 2, (-(1/5))*Cot[x]^5 - Cot[x]^7/7} -{1/(Sec[x] - Cos[x])^5, x, 4, (-(1/5))*Csc[x]^5 + (2*Csc[x]^7)/7 - Csc[x]^9/9} -{1/(Sec[x] - Cos[x])^6, x, 3, (-(1/7))*Cot[x]^7 - (2*Cot[x]^9)/9 - Cot[x]^11/11} -{1/(Sec[x] - Cos[x])^7, x, 4, Csc[x]^7/7 - Csc[x]^9/3 + (3*Csc[x]^11)/11 - Csc[x]^13/13} - - -{(Sec[x] - Cos[x])^(7/2), x, 6, (-(256/35))*Csc[x]*Sqrt[Sin[x]*Tan[x]] + (64/35)*Sec[x]*Tan[x]*Sqrt[Sin[x]*Tan[x]] - (8/7)*Sin[x]*Tan[x]^2*Sqrt[Sin[x]*Tan[x]] - (2/7)*Sin[x]^3*Tan[x]^2*Sqrt[Sin[x]*Tan[x]]} -{(Sec[x] - Cos[x])^(5/2), x, 5, (64/15)*Cot[x]*Sqrt[Sin[x]*Tan[x]] + (16/15)*Tan[x]*Sqrt[Sin[x]*Tan[x]] - (2/5)*Sin[x]^2*Tan[x]*Sqrt[Sin[x]*Tan[x]]} -{(Sec[x] - Cos[x])^(3/2), x, 4, (8/3)*Csc[x]*Sqrt[Sin[x]*Tan[x]] - (2/3)*Sin[x]*Sqrt[Sin[x]*Tan[x]]} -{(Sec[x] - Cos[x])^(1/2), x, 3, -2*Cot[x]*Sqrt[Sin[x]*Tan[x]]} -{1/(Sec[x] - Cos[x])^(1/2), x, 8, (ArcTan[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])} -{1/(Sec[x] - Cos[x])^(3/2), x, 9, -(Csc[x]/(2*Sqrt[Sin[x]*Tan[x]])) + (ArcTan[Sqrt[Cos[x]]]*Sin[x])/(4*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) + (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(4*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])} -{1/(Sec[x] - Cos[x])^(5/2), x, 10, (3*Cot[x])/(16*Sqrt[Sin[x]*Tan[x]]) - (Cot[x]*Csc[x]^2)/(4*Sqrt[Sin[x]*Tan[x]]) - (3*ArcTan[Sqrt[Cos[x]]]*Sin[x])/(32*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) + (3*ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(32*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])} -{1/(Sec[x] - Cos[x])^(7/2), x, 11, -((5*Csc[x])/(192*Sqrt[Sin[x]*Tan[x]])) + (5*Csc[x]^3)/(48*Sqrt[Sin[x]*Tan[x]]) - (Cot[x]^2*Csc[x]^3)/(6*Sqrt[Sin[x]*Tan[x]]) - (5*ArcTan[Sqrt[Cos[x]]]*Sin[x])/(128*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (5*ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(128*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sin[c+d x] + b Tan[c+d x])^n*) - - -{(Sin[x] + Tan[x])^4, x, 18, -((61*x)/8) - 2*ArcTanh[Sin[x]] + (19/8)*Cos[x]*Sin[x] + (1/4)*Cos[x]^3*Sin[x] - (4*Sin[x]^3)/3 + 5*Tan[x] + 2*Sec[x]*Tan[x] + Tan[x]^3/3} -{(Sin[x] + Tan[x])^3, x, 4, 2*Cos[x] + (3*Cos[x]^2)/2 + Cos[x]^3/3 - 2*Log[Cos[x]] + 3*Sec[x] + Sec[x]^2/2} -{(Sin[x] + Tan[x])^2, x, 9, -(x/2) + 2*ArcTanh[Sin[x]] - 2*Sin[x] - (1/2)*Cos[x]*Sin[x] + Tan[x]} -{(Sin[x] + Tan[x])^1, x, 3, -Cos[x] - Log[Cos[x]]} -{1/(Sin[x] + Tan[x])^1, x, 6, (-(1/2))*ArcTanh[Cos[x]] + (1/2)*Cot[x]*Csc[x] - Csc[x]^2/2} -{1/(Sin[x] + Tan[x])^2, x, 11, (-(1/3))*Cot[x]^3 - (2*Cot[x]^5)/5 - (2*Csc[x]^3)/3 + (2*Csc[x]^5)/5} -{1/(Sin[x] + Tan[x])^3, x, 5, (1/32)*ArcTanh[Cos[x]] - 1/(32*(1 - Cos[x])) - 1/(16*(1 + Cos[x])^4) + 1/(6*(1 + Cos[x])^3) - 3/(32*(1 + Cos[x])^2) - 1/(16*(1 + Cos[x]))} -{1/(Sin[x] + Tan[x])^4, x, 18, (-(1/5))*Cot[x]^5 - (9*Cot[x]^7)/7 - (16*Cot[x]^9)/9 - (8*Cot[x]^11)/11 - (4*Csc[x]^5)/5 + (16*Csc[x]^7)/7 - (20*Csc[x]^9)/9 + (8*Csc[x]^11)/11} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Trig[x]) (a Cos[x] + b Sin[x])^n*) - - -{(A + C*Sin[x])/(b*Cos[x] + c*Sin[x]), x, 3, (c*C*x)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] - (b*C*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2, x, 3, -((c*C*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) + (b*C - A*c*Cos[x] + A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))} -{(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3, x, 4, -((A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2))) + (b*C - A*c*Cos[x] + A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (c^2*C*Cos[x] - b*c*C*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))} - - -{(A + B*Cos[x])/(b*Cos[x] + c*Sin[x]), x, 3, (b*B*x)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + (B*c*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^2, x, 3, -((b*B*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) - (B*c + A*c*Cos[x] - A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))} -{(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^3, x, 4, -((A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2))) - (B*c + A*c*Cos[x] - A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (b*B*c*Cos[x] - b^2*B*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Trig[d+e x]^m (a + b Trig[d+e x] + c Trig[d+e x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2-c^2=0*) - - -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4, x, 6, (35*(b^2 + c^2)^2*x)/8 - (35*c*(b^2 + c^2)^(3/2)*Cos[d + e*x])/(8*e) + (35*b*(b^2 + c^2)^(3/2)*Sin[d + e*x])/(8*e) - (35*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(24*e) - (7*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e)} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3, x, 5, (5*(b^2 + c^2)^(3/2)*x)/2 - (5*c*(b^2 + c^2)*Cos[d + e*x])/(2*e) + (5*b*(b^2 + c^2)*Sin[d + e*x])/(2*e) - (5*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(6*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2, x, 4, (3*(b^2 + c^2)*x)/2 - (3*c*Sqrt[b^2 + c^2]*Cos[d + e*x])/(2*e) + (3*b*Sqrt[b^2 + c^2]*Sin[d + e*x])/(2*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(2*e)} -{Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x], x, 3, Sqrt[b^2 + c^2]*x - (c*Cos[d + e*x])/e + (b*Sin[d + e*x])/e} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1), x, 1, -((c - Sqrt[b^2 + c^2]*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])))} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-2), x, 2, -(c*Cos[d + e*x] - b*Sin[d + e*x])/(3*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c - Sqrt[b^2 + c^2]*Sin[d + e*x])/(3*c*Sqrt[b^2 + c^2]*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3), x, 3, -(c*Cos[d + e*x] - b*Sin[d + e*x])/(5*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(15*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(15*c*(b^2 + c^2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4), x, 4, -(c*Cos[d + e*x] - b*Sin[d + e*x])/(7*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)^(3/2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(35*c*(b^2 + c^2)^(3/2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))} - - -(* ::Subsubsection::Closed:: *) -(*a-b=0*) - - -{(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3, x, 5, 4*a*(5*a^2 + 3*c^2)*x - (4*c*(15*a^2 + 4*c^2)*Cos[d + e*x])/(3*e) + (4*a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(3*e) - (20*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*e) - (8*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)} -{(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2, x, 4, 2*(3*a^2 + c^2)*x - (6*a*c*Cos[d + e*x])/e + (6*a^2*Sin[d + e*x])/e - (2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))/e} -{(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^1, x, 3, 2*a*x - (2*c*Cos[d + e*x])/e + (2*a*Sin[d + e*x])/e} -{1/(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^1, x, 2, Log[a + c*Tan[(1/2)*(d + e*x)]]/(2*c*e)} -{1/(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2, x, 4, -((a*Log[a + c*Tan[(1/2)*(d + e*x)]])/(4*c^3*e)) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(4*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3, x, 4, ((3*a^2 + c^2)*Log[a + c*Tan[(1/2)*(d + e*x)]])/(16*c^5*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(16*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(16*c^4*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^4, x, 5, -((a*(5*a^2 + 3*c^2)*Log[a + c*Tan[(1/2)*(d + e*x)]])/(32*c^7*e)) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(48*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(96*c^4*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] - a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))} - - -{1/(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^1, x, 2, Log[1 + Tan[(1/2)*(d + e*x)]]/(2*a*e)} -{1/(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^2, x, 4, -(Log[1 + Tan[(1/2)*(d + e*x)]]/(4*a^2*e)) - (a*Cos[d + e*x] - a*Sin[d + e*x])/(4*e*(a^3 + a^3*Cos[d + e*x] + a^3*Sin[d + e*x]))} -{1/(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^3, x, 4, Log[1 + Tan[(1/2)*(d + e*x)]]/(4*a^3*e) - (a*Cos[d + e*x] - a*Sin[d + e*x])/(16*e*(a^2 + a^2*Cos[d + e*x] + a^2*Sin[d + e*x])^2) + (3*(Cos[d + e*x] - Sin[d + e*x]))/(16*e*(a^3 + a^3*Cos[d + e*x] + a^3*Sin[d + e*x]))} -{1/(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^4, x, 5, -(Log[1 + Tan[(1/2)*(d + e*x)]]/(4*a^4*e)) - (Cos[d + e*x] - Sin[d + e*x])/(48*a*e*(a + a*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(Cos[d + e*x] - Sin[d + e*x]))/(96*e*(a^2 + a^2*Cos[d + e*x] + a^2*Sin[d + e*x])^2) - (19*(a*Cos[d + e*x] - a*Sin[d + e*x]))/(96*e*(a^5 + a^5*Cos[d + e*x] + a^5*Sin[d + e*x]))} - - -(* ::Subsubsection::Closed:: *) -(*a+b=0*) - - -{(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3, x, 5, 4*a*(5*a^2 + 3*c^2)*x - (4*c*(15*a^2 + 4*c^2)*Cos[d + e*x])/(3*e) - (4*a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(3*e) - (20*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*e) - (8*(c*Cos[d + e*x] + a*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)} -{(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2, x, 4, 2*(3*a^2 + c^2)*x - (6*a*c*Cos[d + e*x])/e - (6*a^2*Sin[d + e*x])/e - (2*(c*Cos[d + e*x] + a*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))/e} -{(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^1, x, 3, 2*a*x - (2*c*Cos[d + e*x])/e - (2*a*Sin[d + e*x])/e} -{1/(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^1, x, 2, -(Log[a + c*Cot[(1/2)*(d + e*x)]]/(2*c*e))} -{1/(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2, x, 4, (a*Log[a + c*Cot[(1/2)*(d + e*x)]])/(4*c^3*e) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(4*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3, x, 4, -(((3*a^2 + c^2)*Log[a + c*Cot[(1/2)*(d + e*x)]])/(16*c^5*e)) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(16*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x]))/(16*c^4*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^4, x, 5, (a*(5*a^2 + 3*c^2)*Log[a + c*Cot[(1/2)*(d + e*x)]])/(32*c^7*e) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(48*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x]))/(96*c^4*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] + a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))} - - -(* ::Subsubsection::Closed:: *) -(*a-c=0*) - - -{(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^3, x, 5, 4*a*(5*a^2 + 3*b^2)*x - (4*a*(15*a^2 + 4*b^2)*Cos[d + e*x])/(3*e) + (4*b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(3*e) - (8*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e) - (20*(a + b*Cos[d + e*x] + a*Sin[d + e*x])*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(3*e)} -{(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^2, x, 4, 2*(3*a^2 + b^2)*x - (6*a^2*Cos[d + e*x])/e + (6*a*b*Sin[d + e*x])/e - (2*(a + b*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - b*Sin[d + e*x]))/e} -{(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^1, x, 3, 2*a*x - (2*a*Cos[d + e*x])/e + (2*b*Sin[d + e*x])/e} -{1/(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^1, x, 2, -(Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]]/(2*b*e))} -{1/(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^2, x, 4, (a*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(4*b^3*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(4*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))} -{1/(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^3, x, 4, -(((3*a^2 + b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(16*b^5*e)) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(16*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2) + (3*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(16*b^4*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))} -{1/(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^4, x, 5, (a*(5*a^2 + 3*b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(48*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2) - (a*(15*a^2 + 4*b^2)*Cos[d + e*x] - b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))} - - -(* ::Subsubsection::Closed:: *) -(*a+c=0*) - - -{(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^3, x, 5, 4*a*(5*a^2 + 3*b^2)*x + (4*a*(15*a^2 + 4*b^2)*Cos[d + e*x])/(3*e) + (4*b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(3*e) + (8*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2*(a*Cos[d + e*x] + b*Sin[d + e*x]))/(3*e) + (20*(a + b*Cos[d + e*x] - a*Sin[d + e*x])*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(3*e)} -{(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^2, x, 4, 2*(3*a^2 + b^2)*x + (6*a^2*Cos[d + e*x])/e + (6*a*b*Sin[d + e*x])/e + (2*(a + b*Cos[d + e*x] - a*Sin[d + e*x])*(a*Cos[d + e*x] + b*Sin[d + e*x]))/e} -{(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^1, x, 3, 2*a*x + (2*a*Cos[d + e*x])/e + (2*b*Sin[d + e*x])/e} -{1/(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^1, x, 2, Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]]/(2*b*e)} -{1/(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^2, x, 4, -((a*Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(4*b^3*e)) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(4*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))} -{1/(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^3, x, 4, ((3*a^2 + b^2)*Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(16*b^5*e) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(16*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2) - (3*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(16*b^4*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))} -{1/(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^4, x, 5, -((a*(5*a^2 + 3*b^2)*Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e)) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(48*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^3) - (5*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2) + (a*(15*a^2 + 4*b^2)*Cos[d + e*x] + b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))} - - -(* ::Subsubsection::Closed:: *) -(*a,b,c*) - - -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^4, x, 6, ((8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*x)/8 - (5*a*c*(10*a^2 + 11*(b^2 + c^2))*Cos[d + e*x])/(24*e) + (5*a*b*(10*a^2 + 11*(b^2 + c^2))*Sin[d + e*x])/(24*e) - (7*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e) - ((a + b*Cos[d + e*x] + c*Sin[d + e*x])*(c*(26*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(26*a^2 + 9*(b^2 + c^2))*Sin[d + e*x]))/(24*e)} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3, x, 5, (a*(2*a^2 + 3*(b^2 + c^2))*x)/2 - (c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x])/(6*e) + (b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*e) - (5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))/(6*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2, x, 4, ((2*a^2 + b^2 + c^2)*x)/2 - (3*a*c*Cos[d + e*x])/(2*e) + (3*a*b*Sin[d + e*x])/(2*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))/(2*e)} -{a + b*Cos[d + e*x] + c*Sin[d + e*x], x, 3, a*x - (c*Cos[d + e*x])/e + (b*Sin[d + e*x])/e} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^1, x, 3, (2*ArcTan[(c + (a - b)*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*e)} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2, x, 5, (2*a*ArcTan[(c + (a - b)*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(3/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/((a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3, x, 5, ((2*a^2 + b^2 + c^2)*ArcTan[(c + (a - b)*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(5/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^4, x, 6, (a*(2*a^2 + 3*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(7/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(6*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x] - b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*(a^2 - b^2 - c^2)^3*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} - - -(* {1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^1, x, 1, (2*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*e)} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2, x, 3, (2*a*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(3/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/((a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3, x, 3, ((2*a^2 + b^2 + c^2)*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(5/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} -{1/(a + b*Cos[d + e*x] + c*Sin[d + e*x])^4, x, 4, (a*(2*a^2 + 3*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(7/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(6*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x] - b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*(a^2 - b^2 - c^2)^3*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Cos[d+e x] + c Sin[d+e x])^(n/2)*) - - -{(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2), x, 7, (796*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(15*e) + (64*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e) - (32*(5*Cos[d + e*x] - 3*Sin[d + e*x])*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/(15*e) - (2*(5*Cos[d + e*x] - 3*Sin[d + e*x])*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2))/(5*e)} -{(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2), x, 6, (16*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(3*e) + (20*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e) - (2*(5*Cos[d + e*x] - 3*Sin[d + e*x])*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/(3*e)} -{Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]], x, 2, (2*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/e} -{1/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]], x, 2, (2*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e)} -{(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-3/2), x, 3, -(Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(15*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(15*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])} -{(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-5/2), x, 7, (4*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(675*e) + EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15]/(45*Sqrt[2 + Sqrt[34]]*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(45*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2)) + (4*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(675*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])} -{(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-7/2), x, 8, (-199*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(101250*e) - (8*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(3375*Sqrt[2 + Sqrt[34]]*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(75*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2)) + (8*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(3375*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2)) - (199*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(101250*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])} - - -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2), x, 7, (-16*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(5*e) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*(a^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2), x, 6, (-2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e) + (8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 2, (2*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])} -{1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 2, (2*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2), x, 3, (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/((a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/((a^2 - b^2 - c^2)*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2), x, 7, (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (8*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (2*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2), x, 8, (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(5*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + (16*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*(a^2 - b^2 - c^2)^2*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*(c*(23*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(23*a^2 + 9*(b^2 + c^2))*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])} - - -{(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2), x, 3, (-320*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (16*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(5*e)} -{(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2), x, 2, (-40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e)} -{Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]], x, 1, (-2*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])} -{1/Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]], x, 3, (Sqrt[2/5]*ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])])/e} -{(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2), x, 4, ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])]/(10*Sqrt[10]*e) - (3*Cos[d + e*x] - 4*Sin[d + e*x])/(10*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))} -{(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2), x, 5, (3*ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])])/(400*Sqrt[10]*e) - (3*Cos[d + e*x] - 4*Sin[d + e*x])/(20*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)) - (3*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(400*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))} - - -{(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2), x, 4, (6400*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(7*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (320*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(7*e) + (24*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(7*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2))/(7*e)} -{(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2), x, 3, (-320*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) + (16*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(5*e)} -{(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2), x, 2, (40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e)} -{Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]], x, 1, (-2*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])} -{1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]], x, 3, -((Sqrt[2/5]*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/e)} -{(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2), x, 4, ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])]/(10*Sqrt[10]*e) + (3*Cos[d + e*x] - 4*Sin[d + e*x])/(10*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))} -{(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2), x, 5, -((3*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/(400*Sqrt[10]*e)) + (3*Cos[d + e*x] - 4*Sin[d + e*x])/(20*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)) - (3*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(400*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))} - - -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(7/2), x, 4, (-256*(b^2 + c^2)^(3/2)*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]]) - (64*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(35*e) - (24*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(35*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2))/(7*e)} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2), x, 3, (-64*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(15*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]]) - (16*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(5*e)} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2), x, 2, (-8*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]]) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e)} -{Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 1, (-2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(e*Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{1/Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 3, (Sqrt[2]*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e)} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2), x, 4, ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])]/(2*Sqrt[2]*(b^2 + c^2)^(3/4)*e) - (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))} -{(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2), x, 5, (3*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/(16*Sqrt[2]*(b^2 + c^2)^(5/4)*e) - (c*Cos[d + e*x] - b*Sin[d + e*x])/(4*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(16*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))} - - -{(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2), x, 3, (-64*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(15*e*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (16*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(5*e)} -{(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2), x, 2, (8*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]]) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e)} -{Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 1, (-2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(e*Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]])} -{1/Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]], x, 3, -((Sqrt[2]*ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e))} -{(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2), x, 4, ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])]/(2*Sqrt[2]*(b^2 + c^2)^(3/4)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*Sqrt[b^2 + c^2]*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))} -{(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2), x, 5, -((3*ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/(16*Sqrt[2]*(b^2 + c^2)^(5/4)*e)) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(4*Sqrt[b^2 + c^2]*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(16*(b^2 + c^2)*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[d+e x]^m (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -{Sin[x]/(a + b*Cos[x] + c*Sin[x]), x, 4, (c*x)/(b^2 + c^2) - (2*a*c*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) - (b*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{Sin[x]/(1 + Cos[x] + Sin[x]), x, 3, x/2 - Log[Cos[x/2] + Sin[x/2]], x/2 - (1/2)*Log[1 + Cos[x] + Sin[x]] - (1/2)*Log[1 + Tan[x/2]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[d+e x]^m (a + b Tan[d+e x] + c Sec[d+e x])^n*) - - -{Sec[x]^0/(a + b*Tan[x] + c*Sec[x]), x, 5, (a*x)/(a^2 + b^2) + (2*a*c*ArcTanh[(b - (a - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((a^2 + b^2)*Sqrt[a^2 + b^2 - c^2]) + (b*Log[c + a*Cos[x] + b*Sin[x]])/(a^2 + b^2)} -{Sec[x]^1/(a + b*Tan[x] + c*Sec[x]), x, 4, -((2*ArcTanh[(b - (a - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2])} -{Sec[x]^2/(a + b*Tan[x] + c*Sec[x]), x, 10, -((2*a*c*ArcTanh[(b - (a - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((b^2 - c^2)*Sqrt[a^2 + b^2 - c^2])) - Log[1 - Tan[x/2]]/(b + c) - Log[1 + Tan[x/2]]/(b - c) + (b*Log[a + c + 2*b*Tan[x/2] - (a - c)*Tan[x/2]^2])/(b^2 - c^2)} - - -{(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)/Sec[d + e*x]^(3/2), x, 7, -((2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))) + (8*b*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)} -{(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(1/2)/Sec[d + e*x]^(1/2), x, 3, (2*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(e*Sqrt[Sec[d + e*x]]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])} -{Sec[d + e*x]^(1/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(1/2), x, 3, (2*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[Sec[d + e*x]]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])} -{Sec[d + e*x]^(3/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2), x, 4, -((2*Sec[d + e*x]^(3/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))) - (2*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))} -{Sec[d + e*x]^(5/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2), x, 8, -((2*Sec[d + e*x]^(5/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))) + (8*Sec[d + e*x]^(5/2)*(b*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*b*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))} - - -{Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2), x, 7, -((2*Cos[d + e*x]^(3/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))) + (8*b*Cos[d + e*x]^(3/2)*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x])*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*Cos[d + e*x]^(3/2)*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)} -{Cos[d + e*x]^(1/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(1/2), x, 3, (2*Sqrt[Cos[d + e*x]]*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])} -{1/(Cos[d + e*x]^(1/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(1/2)), x, 3, (2*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])} -{1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)), x, 4, -((2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))) - (2*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*Cos[d + e*x]^(3/2)*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))} -{1/(Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)), x, 8, -((2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))) + (8*(b*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*(a^2 - b^2 + c^2)^2*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*b*EllipticE[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*Cos[d + e*x]^(5/2)*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*EllipticF[(1/2)*(d + e*x - ArcTan[a, c]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csc[d+e x]^m (a + b Cot[d+e x] + c Csc[d+e x])^n*) - - -{Csc[x]^0/(a + b*Cot[x] + c*Csc[x]), x, 5, (a*x)/(a^2 + b^2) + (2*a*c*ArcTanh[(a - (b - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((a^2 + b^2)*Sqrt[a^2 + b^2 - c^2]) - (b*Log[c + b*Cos[x] + a*Sin[x]])/(a^2 + b^2)} -{Csc[x]^1/(a + b*Cot[x] + c*Csc[x]), x, 4, -((2*ArcTanh[(a - (b - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2])} -{Csc[x]^2/(a + b*Cot[x] + c*Csc[x]), x, 9, -((2*a*c*ArcTanh[(a - (b - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((b^2 - c^2)*Sqrt[a^2 + b^2 - c^2])) + Log[Tan[x/2]]/(b + c) - (b*Log[b + c + 2*a*Tan[x/2] - (b - c)*Tan[x/2]^2])/(b^2 - c^2)} - -{Csc[x]^1/(2 + 2*Cot[x] + 3*Csc[x]), x, 4, x + 2*ArcTan[(Cos[x] - Sin[x])/(2 + Cos[x] + Sin[x])]} - - -{(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(3/2)/Csc[d + e*x]^(3/2), x, 7, (8*b*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])])/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))} -{(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(1/2)/Csc[d + e*x]^(1/2), x, 3, (2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[Csc[d + e*x]]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])} -{Csc[d + e*x]^(1/2)/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(1/2), x, 3, (2*Sqrt[Csc[d + e*x]]*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])} -{Csc[d + e*x]^(3/2)/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(3/2), x, 4, -((2*Csc[d + e*x]^(3/2)*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])) - (2*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2))} -{Csc[d + e*x]^(5/2)/(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(5/2), x, 8, (8*b*Csc[d + e*x]^(5/2)*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*Csc[d + e*x]^(5/2)*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) - (2*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (8*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2))} - - -{Sin[d + e*x]^(3/2)*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(3/2), x, 7, (8*b*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sin[d + e*x]^(3/2))/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sin[d + e*x]^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))} -{Sin[d + e*x]^(1/2)*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(1/2), x, 3, (2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[Sin[d + e*x]])/(e*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])} -{1/(Sin[d + e*x]^(1/2)*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(1/2)), x, 3, (2*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]])} -{1/(Sin[d + e*x]^(3/2)*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(3/2)), x, 4, -((2*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))} -{1/(Sin[d + e*x]^(5/2)*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^(5/2)), x, 8, (8*b*EllipticE[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*EllipticF[(1/2)*(d + e*x - ArcTan[c, a]), (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)) + (8*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a Trig[c+d x]^2 + b Trig[c+d x]^2)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cos[c+d x]^2 + b Sin[c+d x]^2)^n*) - - -{1/(Cos[x]^2 + Sin[x]^2), x, 2, x} -{1/(Cos[x]^2 + Sin[x]^2)^2, x, 2, x} -{1/(Cos[x]^2 + Sin[x]^2)^3, x, 2, x} - -{1/(Cos[x]^2 - Sin[x]^2), x, 2, (1/2)*ArcTanh[2*Cos[x]*Sin[x]]} -{1/(Cos[x]^2 - Sin[x]^2)^2, x, 2, Tan[x]/(1 - Tan[x]^2)} -{1/(Cos[x]^2 - Sin[x]^2)^3, x, 4, (1/4)*ArcTanh[2*Cos[x]*Sin[x]] + (Sec[x]^2*Tan[x])/(2*(1 - Tan[x]^2)^2)} - - -{1/(Cos[x]^2 + a^2*Sin[x]^2), x, 2, ArcTan[a*Tan[x]]/a} -{1/(b^2*Cos[x]^2 + Sin[x]^2), x, 2, ArcTan[Tan[x]/b]/b} -{1/(b^2*Cos[x]^2 + a^2*Sin[x]^2), x, 2, ArcTan[(a*Tan[x])/b]/(a*b)} -{1/(4*Cos[1 + 2*x]^2 + 3*Sin[1 + 2*x]^2), x, 2, x/(2*Sqrt[3]) - ArcTan[(Cos[1 + 2*x]*Sin[1 + 2*x])/(3 + 2*Sqrt[3] + Cos[1 + 2*x]^2)]/(4*Sqrt[3])} - - -{Sin[x]^2/(a*Cos[x]^2 + b*Sin[x]^2), x, 4, -(x/(a - b)) + (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/((a - b)*Sqrt[b])} -{Cos[x]^2/(a*Cos[x]^2 + b*Sin[x]^2), x, 4, x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a - b))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Sec[c+d x]^2 + b Tan[c+d x]^2)^n*) - - -{1/(Sec[x]^2 + Tan[x]^2)^1, x, 4, -x + Sqrt[2]*x + Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]} -{1/(Sec[x]^2 + Tan[x]^2)^2, x, 6, x - x/Sqrt[2] - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/Sqrt[2] + Tan[x]/(1 + 2*Tan[x]^2)} -{1/(Sec[x]^2 + Tan[x]^2)^3, x, 6, -x + (7*x)/(4*Sqrt[2]) + (7*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)])/(4*Sqrt[2]) + Tan[x]/(2*(1 + 2*Tan[x]^2)^2) - Tan[x]/(4*(1 + 2*Tan[x]^2))} - -{1/(Sec[x]^2 - Tan[x]^2)^1, x, 2, x} -{1/(Sec[x]^2 - Tan[x]^2)^2, x, 2, x} -{1/(Sec[x]^2 - Tan[x]^2)^3, x, 2, x} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Cot[c+d x]^2 + b Csc[c+d x]^2)^n*) - - -{1/(Cot[x]^2 + Csc[x]^2)^1, x, 4, -x + Sqrt[2]*x - Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]} -{1/(Cot[x]^2 + Csc[x]^2)^2, x, 6, x - x/Sqrt[2] + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/Sqrt[2] - Tan[x]/(2 + Tan[x]^2)} -{1/(Cot[x]^2 + Csc[x]^2)^3, x, 6, -x + (7*x)/(4*Sqrt[2]) - (7*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(4*Sqrt[2]) - Tan[x]^3/(2*(2 + Tan[x]^2)^2) + Tan[x]/(4*(2 + Tan[x]^2))} - -{1/(Cot[x]^2 - Csc[x]^2)^1, x, 2, -x} -{1/(Cot[x]^2 - Csc[x]^2)^2, x, 2, x} -{1/(Cot[x]^2 - Csc[x]^2)^3, x, 2, -x} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a + b Trig[d+e x]^2 + c Trig[d+e x]^2)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a + b Cos[d+e x]^2 + c Sin[d+e x]^2)^n*) - - -{1/(a + b*Cos[x]^2 + c*Sin[x]^2), x, 2, ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]/(Sqrt[a + b]*Sqrt[a + c])} -{x/(a + b*Cos[x]^2 + c*Sin[x]^2), x, 9, -((I*x*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c])) + (I*x*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c]) - PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c]) + PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])} -{x^2/(a + b*Cos[x]^2 + c*Sin[x]^2), x, 11, -((I*x^2*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c])) + (I*x^2*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c]) - (x*PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sqrt[a + c]) + (x*PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sqrt[a + c]) - (I*PolyLog[3, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c]) + (I*PolyLog[3, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c])} -(* {x^3/(a + b*Cos[x]^2 + c*Sin[x]^2), x, 13, -((I*x^3*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c])) + (I*x^3*Log[1 + ((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(2*Sqrt[a + b]*Sqrt[a + c]) - (3*x^2*PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c]) + (3*x^2*PolyLog[2, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c]) - (3*I*x*PolyLog[3, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c]) + (3*I*x*PolyLog[3, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(4*Sqrt[a + b]*Sqrt[a + c]) + (3*PolyLog[4, -(((b - c)*E^(2*I*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(8*Sqrt[a + b]*Sqrt[a + c]) - (3*PolyLog[4, -(((b - c)*E^(2*I*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(8*Sqrt[a + b]*Sqrt[a + c])} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e Sin[d+e x])^m (a + b Sin[d+e x] + c Sin[d+e x]^2)^n*) - - -(* {(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^3, x, 8, (a*(5*a^6 + 120*a^4*b^2 + 240*a^2*b^4 + 64*b^6)*x)/16 - (b*(512*a^6 + 2789*a^4*b^2 + 1664*a^2*b^4 + 40*b^6)*Cos[d + e*x])/(140*e) - (a*(175*a^6 + 2502*a^4*b^2 + 2248*a^2*b^4 + 80*b^6)*Cos[d + e*x]*Sin[d + e*x])/(560*e) - (b*(337*a^4 + 624*a^2*b^2 + 40*b^4)*Cos[d + e*x]*(b + a*Sin[d + e*x])^2)/(280*e) - ((175*a^4 + 992*a^2*b^2 + 120*b^4)*Cos[d + e*x]*(b + a*Sin[d + e*x])^3)/(840*e) - (b*(113*a^2 + 30*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^4)/(210*e) - ((7*a^2 + 6*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^5)/(42*e) - (b*Cos[d + e*x]*(b + a*Sin[d + e*x])^6)/(7*e)} *) -{(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^2, x, 5, (3*a*(a^4 + 12*a^2*b^2 + 8*b^4)*x)/8 - (b*(32*a^4 + 69*a^2*b^2 + 4*b^4)*Cos[d + e*x])/(10*e) - (a*(15*a^4 + 82*a^2*b^2 + 8*b^4)*Cos[d + e*x]*Sin[d + e*x])/(40*e) - (b*(17*a^2 + 4*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^2)/(20*e) - ((5*a^2 + 4*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^3)/(20*e) - (b*Cos[d + e*x]*(b + a*Sin[d + e*x])^4)/(5*e)} -{(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2), x, 2, (1/2)*a*(a^2 + 4*b^2)*x + ((a^4 - 8*a^2*b^2 - 3*b^4)*Cos[d + e*x])/(3*b*e) + (a*(a^2 - 6*b^2)*Cos[d + e*x]*Sin[d + e*x])/(6*e) - (a^2*Cos[d + e*x]*(a + b*Sin[d + e*x])^2)/(3*b*e)} -{(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2), x, 3, -(Cos[d + e*x]/(e*(b + a*Sin[d + e*x])))} -{(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^2, x, 9, (2*a*b*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*e) - Cos[d + e*x]/(3*e*(b + a*Sin[d + e*x])^3) + (b*Cos[d + e*x])/(3*(a^2 - b^2)*e*(b + a*Sin[d + e*x])^2) - ((2*a^2 + b^2)*Cos[d + e*x])/(3*(a^2 - b^2)^2*e*(b + a*Sin[d + e*x]))} -(* {(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^3, x, 9, (a*b*(3*a^2 + 4*b^2)*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*e) - Cos[d + e*x]/(5*e*(b + a*Sin[d + e*x])^5) + (b*Cos[d + e*x])/(5*(a^2 - b^2)*e*(b + a*Sin[d + e*x])^4) - ((4*a^2 + 3*b^2)*Cos[d + e*x])/(15*(a^2 - b^2)^2*e*(b + a*Sin[d + e*x])^3) + (b*(29*a^2 + 6*b^2)*Cos[d + e*x])/(30*(a^2 - b^2)^3*e*(b + a*Sin[d + e*x])^2) - ((16*a^4 + 83*a^2*b^2 + 6*b^4)*Cos[d + e*x])/(30*(a^2 - b^2)^4*e*(b + a*Sin[d + e*x]))} *) - - -{(d + e*Sin[x])/(a + b*Sin[x] + c*Sin[x]^2), x, 7, (Sqrt[2]*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + (Sqrt[2]*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]} - - -(* {(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2), x, 7, -(b*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(6*e) - ((32*a^6 + 544*a^4*b^2 + 559*a^2*b^4 + 20*b^6)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(60*e*(b + a*Sin[d + e*x])^5) - ((32*a^4 + 179*a^2*b^2 + 20*b^4)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(120*e*(b + a*Sin[d + e*x])^3) - (b*(79*a^2 + 20*b^2)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(120*e*(b + a*Sin[d + e*x])^2) - ((6*a^2 + 5*b^2)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(30*e*(b + a*Sin[d + e*x])) + (7*a^6*b*(5*a^4 + 20*a^2*b^2 + 8*b^4)*x*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(16*(a*b + a^2*Sin[d + e*x])^5) - (a^6*b*(397*a^4 + 718*a^2*b^2 + 40*b^4)*Cos[d + e*x]*Sin[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))/(240*e*(a*b + a^2*Sin[d + e*x])^5)} *) -{(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2), x, 4, -(b*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(4*e) - ((4*a^4 + 28*a^2*b^2 + 3*b^4)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(6*e*(b + a*Sin[d + e*x])^3) - ((4*a^2 + 3*b^2)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(12*e*(b + a*Sin[d + e*x])) + (5*a^4*b*(3*a^2 + 4*b^2)*x*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(8*(a*b + a^2*Sin[d + e*x])^3) - (a^4*b*(29*a^2 + 6*b^2)*Cos[d + e*x]*Sin[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(24*e*(a*b + a^2*Sin[d + e*x])^3)} -{(a + b*Sin[d + e*x])*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2], x, 2, -(((a^2 + b^2)*Cos[d + e*x]*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(e*(b + a*Sin[d + e*x]))) + (3*a^2*b*x*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(2*(a*b + a^2*Sin[d + e*x])) - (a^2*b*Cos[d + e*x]*Sin[d + e*x]*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(2*e*(a*b + a^2*Sin[d + e*x]))} -{(a + b*Sin[d + e*x])/Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2], x, 5, (b*x*(b + a*Sin[d + e*x]))/(a*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2]) - (2*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Sin[d + e*x]))/(a*e*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])} -{(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2), x, 8, -(Cos[d + e*x]*(b + a*Sin[d + e*x]))/(2*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2)) - (ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]]*(a*b + a^2*Sin[d + e*x])^3)/(a^2*(a^2 - b^2)^(3/2)*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2)) + (b*Cos[d + e*x]*(a*b + a^2*Sin[d + e*x])^3)/(2*(a^2 - b^2)*e*(a^3*b + a^4*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))} -(* {(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2), x, 8, -(Cos[d + e*x]*(b + a*Sin[d + e*x]))/(4*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2)) + (b*Cos[d + e*x]*(b + a*Sin[d + e*x])^2)/(4*(a^2 - b^2)*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2)) + (b*(13*a^2 + 2*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^4)/(8*(a^2 - b^2)^3*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2)) - (3*(a^2 + 4*b^2)*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]]*(a*b + a^2*Sin[d + e*x])^5)/(4*a^4*(a^2 - b^2)^(7/2)*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2)) - ((3*a^2 + 2*b^2)*Cos[d + e*x]*(a*b + a^2*Sin[d + e*x])^5)/(8*a*(a^2 - b^2)^2*e*(a^2*b + a^3*Sin[d + e*x])^2*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(5/2))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e Cos[d+e x])^m (a + b Cos[d+e x] + c Cos[d+e x]^2)^n*) - - -{(a + b*Cos[x])/(b^2 + 2*a*b*Cos[x] + a^2*Cos[x]^2), x, 3, Sin[x]/(b + a*Cos[x])} -{(d + e*Cos[x])/(a + b*Cos[x] + c*Cos[x]^2), x, 5, (2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e Tan[d+e x])^m (a + b Tan[d+e x] + c Tan[d+e x]^2)^n*) - - -(* {(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^3, x, 10, -(a*(a^2 + b^2)*(a^4 - 10*a^2*b^2 + 5*b^4)*x) - (b*(a^2 + b^2)*(5*a^4 - 10*a^2*b^2 + b^4)*Log[Cos[d + e*x]])/e + ((a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*(b + a*Tan[d + e*x]))/e - (b*(3*a^2 - b^2)*(a^2 + b^2)*(b + a*Tan[d + e*x])^2)/(2*e) - ((a^4 - b^4)*(b + a*Tan[d + e*x])^3)/(3*e) + (b*(a^2 + b^2)*(b + a*Tan[d + e*x])^4)/(4*e) + ((a^2 + b^2)*(b + a*Tan[d + e*x])^5)/(5*e) + (b*(b + a*Tan[d + e*x])^6)/(6*e)} *) -{(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^2, x, 7, a*(a^2 - 3*b^2)*(a^2 + b^2)*x + (b*(3*a^2 - b^2)*(a^2 + b^2)*Log[Cos[d + e*x]])/e - (a*(a^4 - b^4)*Tan[d + e*x])/e + (b*(a^2 + b^2)*(b + a*Tan[d + e*x])^2)/(2*e) + ((a^2 + b^2)*(b + a*Tan[d + e*x])^3)/(3*e) + (b*(b + a*Tan[d + e*x])^4)/(4*e)} -{(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2), x, 3, (-a)*(a^2 + b^2)*x - (b*(a^2 + b^2)*Log[Cos[d + e*x]])/e + (2*a*b^2*Tan[d + e*x])/e + (a^2*(a + b*Tan[d + e*x])^2)/(2*b*e)} -{(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2), x, 4, -((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^2) + (b*(3*a^2 - b^2)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]])/((a^2 + b^2)^2*e) - (a^2 - b^2)/((a^2 + b^2)*e*(b + a*Tan[d + e*x]))} -{(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^2, x, 6, (a*(a^4 - 10*a^2*b^2 + 5*b^4)*x)/(a^2 + b^2)^4 - (b*(5*a^4 - 10*a^2*b^2 + b^4)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]])/((a^2 + b^2)^4*e) - (a^2 - b^2)/(3*(a^2 + b^2)*e*(b + a*Tan[d + e*x])^3) - (b*(3*a^2 - b^2))/(2*(a^2 + b^2)^2*e*(b + a*Tan[d + e*x])^2) + (a^4 - 6*a^2*b^2 + b^4)/((a^2 + b^2)^3*e*(b + a*Tan[d + e*x]))} -(* {(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^3, x, 10, -((a*(a^6 - 21*a^4*b^2 + 35*a^2*b^4 - 7*b^6)*x)/(a^2 + b^2)^6) + (b*(7*a^6 - 35*a^4*b^2 + 21*a^2*b^4 - b^6)*Log[Cos[d + e*x]])/((a^2 + b^2)^6*e) + (b*(7*a^6 - 35*a^4*b^2 + 21*a^2*b^4 - b^6)*Log[b + a*Tan[d + e*x]])/((a^2 + b^2)^6*e) - (a^2 - b^2)/(5*(a^2 + b^2)*e*(b + a*Tan[d + e*x])^5) - (b*(3*a^2 - b^2))/(4*(a^2 + b^2)^2*e*(b + a*Tan[d + e*x])^4) + (a^4 - 6*a^2*b^2 + b^4)/(3*(a^2 + b^2)^3*e*(b + a*Tan[d + e*x])^3) + (b*(5*a^4 - 10*a^2*b^2 + b^4))/(2*(a^2 + b^2)^4*e*(b + a*Tan[d + e*x])^2) - (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/((a^2 + b^2)^5*e*(b + a*Tan[d + e*x]))} *) - - -(* {(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2), x, 9, (b*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(5*e) - ((a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Log[Cos[d + e*x]]*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(e*(b + a*Tan[d + e*x])^5) + (b*(a^2 + b^2)*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(3*e*(b + a*Tan[d + e*x])^2) + ((a^2 + b^2)*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(4*e*(b + a*Tan[d + e*x])) + (4*a^6*b*(a^4 - b^4)*x*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(a*b + a^2*Tan[d + e*x])^5 - (a*(a^4 - b^4)*(a^2*b + a^3*Tan[d + e*x])^2*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(2*e*(a*b + a^2*Tan[d + e*x])^5) - (b*(3*a^2 - b^2)*(a^2 + b^2)*(a^5*b + a^6*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))/(e*(a*b + a^2*Tan[d + e*x])^5)} *) -{(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2), x, 6, (b*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(3*e) + ((a^4 - b^4)*Log[Cos[d + e*x]]*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(e*(b + a*Tan[d + e*x])^3) + ((a^2 + b^2)*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(2*e*(b + a*Tan[d + e*x])) - (2*a^4*b*(a^2 + b^2)*x*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(a*b + a^2*Tan[d + e*x])^3 + (a^4*b*(a^2 + b^2)*Tan[d + e*x]*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(e*(a*b + a^2*Tan[d + e*x])^3)} -{(a + b*Tan[d + e*x])*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2], x, 3, -(((a^2 + b^2)*Log[Cos[d + e*x]]*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])/(e*(b + a*Tan[d + e*x]))) + (a^2*b*Tan[d + e*x]*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])/(e*(a*b + a^2*Tan[d + e*x]))} -{(a + b*Tan[d + e*x])/Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2], x, 3, ((a^2 - b^2)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]]*(b + a*Tan[d + e*x]))/((a^2 + b^2)*e*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2]) + (2*b*x*(a*b + a^2*Tan[d + e*x]))/((a^2 + b^2)*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])} -{(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2), x, 5, -(((a^2 - b^2)*(b + a*Tan[d + e*x]))/(2*(a^2 + b^2)*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))) - ((a^4 - 6*a^2*b^2 + b^4)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]]*(b + a*Tan[d + e*x])^3)/((a^2 + b^2)^3*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)) - (4*b*(a^2 - b^2)*x*(a*b + a^2*Tan[d + e*x])^3)/(a^2*(a^2 + b^2)^3*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)) - (b*(3*a^2 - b^2)*(a*b + a^2*Tan[d + e*x])^3)/((a^2 + b^2)^2*e*(a^3*b + a^4*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))} -(* {(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2), x, 9, -((a^2 - b^2)*(b + a*Tan[d + e*x]))/(4*(a^2 + b^2)*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) - (b*(3*a^2 - b^2)*(b + a*Tan[d + e*x])^2)/(3*(a^2 + b^2)^2*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) + ((a^4 - 6*a^2*b^2 + b^4)*(b + a*Tan[d + e*x])^3)/(2*(a^2 + b^2)^3*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) + ((a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Log[Cos[d + e*x]]*(b + a*Tan[d + e*x])^5)/((a^2 + b^2)^5*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) + ((a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Log[b + a*Tan[d + e*x]]*(b + a*Tan[d + e*x])^5)/((a^2 + b^2)^5*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) + (2*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*x*(a*b + a^2*Tan[d + e*x])^5)/(a^4*(a^2 + b^2)^5*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2)) + (b*(5*a^4 - 10*a^2*b^2 + b^4)*(a*b + a^2*Tan[d + e*x])^5)/((a^2 + b^2)^4*e*(a^5*b + a^6*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(5/2))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e Sec[d+e x])^m (a + b Sec[d+e x] + c Sec[d+e x]^2)^n*) - - -(* {(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^3, x, 11, a*b^6*x + (a^2*b*(487*a^4 + 1620*a^2*b^2 + 348*b^4)*ArcTanh[Sin[d + e*x]])/(240*e) + (b*(64*a^6 + 1065*a^4*b^2 + 1446*a^2*b^4 + 120*b^6)*ArcTanh[Sin[d + e*x]])/(120*e) + (a*(32*a^6 + 776*a^4*b^2 + 1473*a^2*b^4 + 234*b^6)*Tan[d + e*x])/(60*e) + (a^2*b*(487*a^4 + 1620*a^2*b^2 + 348*b^4)*Sec[d + e*x]*Tan[d + e*x])/(240*e) + ((32*a^4 + 321*a^2*b^2 + 114*b^4)*(a*b + a^2*Sec[d + e*x])^2*Tan[d + e*x])/(120*a*e) + (b*(109*a^2 + 74*b^2)*(a*b + a^2*Sec[d + e*x])^3*Tan[d + e*x])/(120*a^2*e) + ((6*a^2 + 11*b^2)*(a*b + a^2*Sec[d + e*x])^4*Tan[d + e*x])/(30*a^3*e) + (b*(a*b + a^2*Sec[d + e*x])^5*Tan[d + e*x])/(6*a^4*e)} *) -{(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2, x, 8, a*b^4*x + (b*(19*a^4 + 56*a^2*b^2 + 8*b^4)*ArcTanh[Sin[d + e*x]])/(8*e) + (a*(4*a^4 + 50*a^2*b^2 + 19*b^4)*Tan[d + e*x])/(6*e) + (a^2*b*(41*a^2 + 26*b^2)*Sec[d + e*x]*Tan[d + e*x])/(24*e) + ((4*a^2 + 7*b^2)*(a*b + a^2*Sec[d + e*x])^2*Tan[d + e*x])/(12*a*e) + (b*(a*b + a^2*Sec[d + e*x])^3*Tan[d + e*x])/(4*a^2*e)} -{(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^1, x, 5, a*b^2*x + (b*(5*a^2 + 2*b^2)*ArcTanh[Sin[d + e*x]])/(2*e) + (a*(a^2 + 2*b^2)*Tan[d + e*x])/e + (a^2*b*Sec[d + e*x]*Tan[d + e*x])/(2*e)} -{(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^1, x, 6, (a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/(b^2*e) - (a^2*Tan[d + e*x])/(b*e*(a*b + a^2*Sec[d + e*x]))} -{(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2, x, 8, (a*x)/b^4 - ((a^2 - 2*b^2)*(2*a^4 - a^2*b^2 + b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*e) - (a*(3*a^2 - 5*b^2)*Tan[d + e*x])/(6*b^2*(a^2 - b^2)*e*(b + a*Sec[d + e*x])^2) - (a*(6*a^4 - 11*a^2*b^2 + 11*b^4)*Tan[d + e*x])/(6*b^3*(a^2 - b^2)^2*e*(b + a*Sec[d + e*x])) - (a^4*Tan[d + e*x])/(3*b*e*(a*b + a^2*Sec[d + e*x])^3)} -(* {(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^3, x, 8, (a*x)/b^6 - ((8*a^10 - 36*a^8*b^2 + 63*a^6*b^4 - 55*a^4*b^6 - 8*b^10)*ArcTan[(Sqrt[a^2 - b^2]*Tan[(1/2)*(d + e*x)])/(a + b)])/(4*b^6*(a^2 - b^2)^(9/2)*e) - (a^6*Tan[d + e*x])/(5*b*e*(a*b + a^2*Sec[d + e*x])^5) - (a^5*(5*a^2 - 9*b^2)*Tan[d + e*x])/(20*b^2*(a^2 - b^2)*e*(a*b + a^2*Sec[d + e*x])^4) - (a^4*(20*a^4 - 39*a^2*b^2 + 47*b^4)*Tan[d + e*x])/(60*b^3*(a^2 - b^2)^2*e*(a*b + a^2*Sec[d + e*x])^3) - (a^3*(60*a^6 - 175*a^4*b^2 + 129*a^2*b^4 - 154*b^6)*Tan[d + e*x])/(120*b^4*(a^2 - b^2)^3*e*(a*b + a^2*Sec[d + e*x])^2) - (a^6*(120*a^8 - 460*a^6*b^2 + 649*a^4*b^4 - 163*a^2*b^6 + 274*b^8)*Tan[d + e*x])/(120*b^5*(a^2 - b^2)^4*e*(a^5*b + a^6*Sec[d + e*x]))} *) - - -(* {(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2), x, 10, (b^2*(187*a^4 + 523*a^2*b^2 + 60*b^4)*ArcTanh[Sin[d + e*x]]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2))/(60*e*(b + a*Sec[d + e*x])^5) + (a^6*b^5*x*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2))/(a*b + a^2*Sec[d + e*x])^5 + (a^7*(45*a^4 + 451*a^2*b^2 + 154*b^4)*ArcTanh[Sin[d + e*x]]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2))/(120*e*(a*b + a^2*Sec[d + e*x])^5) + (a^6*b*(116*a^4 + 457*a^2*b^2 + 107*b^4)*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)*Tan[d + e*x])/(30*e*(a*b + a^2*Sec[d + e*x])^5) + (a^7*(45*a^4 + 451*a^2*b^2 + 154*b^4)*Sec[d + e*x]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)*Tan[d + e*x])/(120*e*(a*b + a^2*Sec[d + e*x])^5) + (a^2*b*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)*Tan[d + e*x])/(5*e*(a*b + a^2*Sec[d + e*x])) + ((5*a^2 + 9*b^2)*(a^2*b + a^3*Sec[d + e*x])^3*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)*Tan[d + e*x])/(20*e*(a*b + a^2*Sec[d + e*x])^5) + (b*(71*a^2 + 47*b^2)*(a^3*b + a^4*Sec[d + e*x])^2*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)*Tan[d + e*x])/(60*e*(a*b + a^2*Sec[d + e*x])^5)} *) -{(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2), x, 7, ((a^4 + 9*a^2*b^2 + 2*b^4)*ArcTanh[Sin[d + e*x]]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))/(2*e*(b + a*Sec[d + e*x])^3) + (a^4*b^3*x*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))/(a*b + a^2*Sec[d + e*x])^3 + (a^4*b*(11*a^2 + 8*b^2)*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(3*e*(a*b + a^2*Sec[d + e*x])^3) + (a^5*(3*a^2 + 5*b^2)*Sec[d + e*x]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(6*e*(a*b + a^2*Sec[d + e*x])^3) + (b*(a^2*b + a^3*Sec[d + e*x])^2*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(3*e*(a*b + a^2*Sec[d + e*x])^3)} -{(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(1/2), x, 5, ((a^2 + b^2)*ArcTanh[Sin[d + e*x]]*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])/(e*(b + a*Sec[d + e*x])) + (a^2*b*x*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])/(a*b + a^2*Sec[d + e*x]) + (a^2*b*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2]*Tan[d + e*x])/(e*(a*b + a^2*Sec[d + e*x]))} -{(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(1/2), x, 5, -((2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]]*(b + a*Sec[d + e*x]))/(b*e*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])) + (x*(a*b + a^2*Sec[d + e*x]))/(b*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])} -{(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2), x, 7, -(((2*a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(1/2)*(d + e*x)])/Sqrt[a + b]]*(b + a*Sec[d + e*x])^3)/((a - b)^(3/2)*b^3*(a + b)^(3/2)*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))) + (x*(a*b + a^2*Sec[d + e*x])^3)/(a^2*b^3*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((a*b + a^2*Sec[d + e*x])*Tan[d + e*x])/(2*b*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((2*a^2 - 3*b^2)*(a*b + a^2*Sec[d + e*x])^3*Tan[d + e*x])/(2*b^2*(a^2 - b^2)*e*(a^2*b + a^3*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))} -(* {(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2), x, 7, -(((8*a^8 - 28*a^6*b^2 + 35*a^4*b^4 - 8*a^2*b^6 + 8*b^8)*ArcTan[(Sqrt[a^2 - b^2]*Tan[(1/2)*(d + e*x)])/(a + b)]*(b + a*Sec[d + e*x])^5)/(4*b^5*(a^2 - b^2)^(7/2)*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2))) + (x*(a*b + a^2*Sec[d + e*x])^5)/(a^4*b^5*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)) - ((a*b + a^2*Sec[d + e*x])*Tan[d + e*x])/(4*b*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)) - ((4*a^2 - 7*b^2)*(a*b + a^2*Sec[d + e*x])^2*Tan[d + e*x])/(12*a*b^2*(a^2 - b^2)*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)) - ((12*a^4 - 23*a^2*b^2 + 26*b^4)*(a*b + a^2*Sec[d + e*x])^5*Tan[d + e*x])/(24*b^3*(a^2 - b^2)^2*e*(a^2*b + a^3*Sec[d + e*x])^2*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2)) - ((24*a^6 - 68*a^4*b^2 + 49*a^2*b^4 - 50*b^6)*(a*b + a^2*Sec[d + e*x])^5*Tan[d + e*x])/(24*b^4*(a^2 - b^2)^3*e*(a^4*b + a^5*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(5/2))} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form (A + B Trig[x] + C Trig[x]) (b Trig[x] + c Trig[x])^n*) - - -{(Cos[x] - I*Sin[x])/(Cos[x] + I*Sin[x]), x, 1, (1/2)*I*(Cos[x] - I*Sin[x])^2} -{(Cos[x] + I*Sin[x])/(Cos[x] - I*Sin[x]), x, 1, -(I/(2*(Cos[x] - I*Sin[x])^2))} -{(Cos[x] - Sin[x])/(Cos[x] + Sin[x]), x, 1, Log[Cos[x] + Sin[x]]} - - -{(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]), x, 1, ((b*B + c*C)*x)/(b^2 + c^2) + ((B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2, x, 3, -(((b*B + c*C)*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) - (B*c - b*C)/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))} -{(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3, x, 3, -((B*c - b*C)/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2)) + ((b*B + c*C)*Sin[x])/(b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x]))} - - -{(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]), x, 3, ((b*B + c*C)*x)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + ((B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2, x, 3, -(((b*B + c*C)*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) - (B*c - b*C + A*c*Cos[x] - A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))} -{(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3, x, 4, -((A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2))) - (B*c - b*C + A*c*Cos[x] - A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (c*(b*B + c*C)*Cos[x] - b*(b*B + c*C)*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A + B Cos[d+e x] + C Sin[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Cos[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -{(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^1, x, 4, (b*B*x)/(b^2 + c^2) - (2*(a*b*B - A*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + (B*c*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^2, x, 4, (2*(a*A - b*B)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) + (B*c + A*c*Cos[x] - (A*b - a*B)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))} -{(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^3, x, 5, ((2*a^2*A - 3*a*b*B + A*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) + (B*c + A*c*Cos[x] - (A*b - a*B)*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*B*c + (3*a*A - 2*b*B)*c*Cos[x] - (3*a*A*b - a^2*B - 2*b^2*B)*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))} - - -{(A + B*Cos[x])/(a + b*Cos[x] + I*b*Sin[x]), x, 1, ((2*a*A - b*B)*x)/(2*a^2) + (I*B*Cos[x])/(2*a) + (I*(2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*a^2*b) + (B*Sin[x])/(2*a)} -{(A + B*Cos[x])/(a + b*Cos[x] - I*b*Sin[x]), x, 1, ((2*a*A - b*B)*x)/(2*a^2) - (I*B*Cos[x])/(2*a) - (I*(2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cos[x] - I*b*Sin[x]])/(2*a^2*b) + (B*Sin[x])/(2*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + C Sin[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -{(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^1, x, 4, (c*C*x)/(b^2 + c^2) + (2*(A*(b^2 + c^2) - a*c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) - (b*C*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2, x, 4, (2*(a*A - c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) - (b*C - (A*c - a*C)*Cos[x] + A*b*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))} -{(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3, x, 5, ((2*a^2*A + A*(b^2 + c^2) - 3*a*c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) - (b*C - (A*c - a*C)*Cos[x] + A*b*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) - (a*b*C - (3*a*A*c - a^2*C - 2*c^2*C)*Cos[x] + b*(3*a*A - 2*c*C)*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))} - - -{(A + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]), x, 1, ((2*a*A - I*b*C)*x)/(2*a^2) - (C*Cos[x])/(2*a) + ((2*I*a*A*b - a^2*C + b^2*C)*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*a^2*b) + (I*C*Sin[x])/(2*a)} -{(A + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]), x, 1, ((2*a*A + I*b*C)*x)/(2*a^2) - (C*Cos[x])/(2*a) - ((2*I*a*A*b + a^2*C - b^2*C)*Log[a + b*Cos[x] - I*b*Sin[x]])/(2*a^2*b) - (I*C*Sin[x])/(2*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (B Cos[d+e x] + C Sin[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -{(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]), x, 4, ((b*B + c*C)*x)/(b^2 + c^2) - (2*a*(b*B + c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + ((B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2, x, 4, -((2*(b*B + c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2)) + (B*c - b*C - a*C*Cos[x] + a*B*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))} -{(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3, x, 5, -((3*a*(b*B + c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2)) + (B*c - b*C - a*C*Cos[x] + a*B*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*(B*c - b*C) - (2*b*B*c + (a^2 + 2*c^2)*C)*Cos[x] + (a^2*B + 2*b*(b*B + c*C))*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))} - - -{(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]), x, 1, -((b*(B + I*C)*x)/(2*a^2)) - ((I*b^2*(B + I*C) + a^2*(I*B + C))*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*a^2*b) + ((I*B - C)*(Cos[x] - I*Sin[x]))/(2*a)} -{(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]), x, 1, -((b*(B - I*C)*x)/(2*a^2)) + ((I*a^2*(B + I*C) + b^2*(I*B + C))*Log[a + b*Cos[x] - I*b*Sin[x]])/(2*a^2*b) - ((I*B + C)*(Cos[x] + I*Sin[x]))/(2*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Cos[d+e x] + C Sin[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^n*) - - -{(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]), x, 4, ((b*B + c*C)*x)/(b^2 + c^2) + (2*(A*(b^2 + c^2) - a*(b*B + c*C))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + ((B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)} -{(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2, x, 4, (2*(a*A - b*B - c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) + (B*c - b*C + (A*c - a*C)*Cos[x] - (A*b - a*B)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))} -{(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3, x, 5, ((2*a^2*A + A*(b^2 + c^2) - 3*a*(b*B + c*C))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) + (B*c - b*C + (A*c - a*C)*Cos[x] - (A*b - a*B)*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*(B*c - b*C) + (3*a*A*c - a^2*C - 2*c*(b*B + c*C))*Cos[x] - (3*a*A*b - a^2*B - 2*b*(b*B + c*C))*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))} - - -{(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]), x, 1, ((2*a*A - b*(B + I*C))*x)/(2*a^2) + (I*(2*a*A*b - a^2*(B - I*C) - b^2*(B + I*C))*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*a^2*b) + ((I*B - C)*(Cos[x] - I*Sin[x]))/(2*a)} -{(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]), x, 1, ((2*a*A - b*B + I*b*C)*x)/(2*a^2) - (I*(2*a*A*b - b^2*(B - I*C) - a^2*(B + I*C))*Log[a + b*Cos[x] - I*b*Sin[x]])/(2*a^2*b) - ((I*B + C)*(Cos[x] + I*Sin[x]))/(2*a)} - - -{(b^2 + c^2 + a*b*Cos[x] + a*c*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2, x, 1, -((c*Cos[x] - b*Sin[x])/(a + b*Cos[x] + c*Sin[x])), -((c*(a^2 - b^2 - c^2)*Cos[x] - b*(a^2 - b^2 - c^2)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Cos[d+e x] + C Sin[d+e x]) (a + b Cos[d+e x] + c Sin[d+e x])^(n/2)*) - - -{(d + b*e*Cos[x] + c*e*Sin[x])*(a + b*Cos[x] + c*Sin[x])^(5/2), x, 8, (2*(161*a^2*d + 63*(b^2 + c^2)*d + 15*a^3*e + 145*a*(b^2 + c^2)*e)*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(105*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(105*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2/7)*(a + b*Cos[x] + c*Sin[x])^(5/2)*(c*e*Cos[x] - b*e*Sin[x]) - (2/35)*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*(7*d + 5*a*e)*Cos[x] - b*(7*d + 5*a*e)*Sin[x]) - (2/105)*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Cos[x] - b*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Sin[x])} -{(d + b*e*Cos[x] + c*e*Sin[x])*(a + b*Cos[x] + c*Sin[x])^(3/2), x, 7, (2*(20*a*d + 3*a^2*e + 9*(b^2 + c^2)*e)*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(15*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*(5*d + 3*a*e)*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(15*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2/5)*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*e*Cos[x] - b*e*Sin[x]) - (2/15)*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*(5*d + 3*a*e)*Cos[x] - b*(5*d + 3*a*e)*Sin[x])} -{(d + b*e*Cos[x] + c*e*Sin[x])*(a + b*Cos[x] + c*Sin[x])^(1/2), x, 6, (2*(3*d + a*e)*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*e*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(3*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2/3)*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*e*Cos[x] - b*e*Sin[x])} -{(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(1/2), x, 5, (2*e*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])] + (2*(d - a*e)*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]]} -{(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(3/2), x, 6, (2*(d - a*e)*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/((a^2 - b^2 - c^2)*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) + (2*e*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]] + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]])} -{(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(5/2), x, 7, (2*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*EllipticE[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*(a^2 - b^2 - c^2)^2*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(d - a*e)*EllipticF[(1/2)*(x - ArcTan[b, c]), (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]]) + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^(3/2)) + (2*(c*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Cos[x] - b*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)^2*Sqrt[a + b*Cos[x] + c*Sin[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Cos[d+e x] + C Sin[d+e x]) (a + c Sin[d+e x])^n*) - - -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x]), x, 7, (C*x)/c + (2*(A*c - a*C)*ArcTan[(c + a*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - c^2]])/(c*Sqrt[a^2 - c^2]*e) + (B*Log[a + c*Sin[d + e*x]])/(c*e)} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^2, x, 8, (2*(a*A - c*C)*ArcTan[(c + a*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - c^2]])/((a^2 - c^2)^(3/2)*e) - B/(c*e*(a + c*Sin[d + e*x])) + ((A*c - a*C)*Cos[d + e*x])/((a^2 - c^2)*e*(a + c*Sin[d + e*x]))} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^3, x, 9, ((2*a^2*A + A*c^2 - 3*a*c*C)*ArcTan[(c + a*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - c^2]])/((a^2 - c^2)^(5/2)*e) - B/(2*c*e*(a + c*Sin[d + e*x])^2) + ((A*c - a*C)*Cos[d + e*x])/(2*(a^2 - c^2)*e*(a + c*Sin[d + e*x])^2) + ((3*a*A*c - a^2*C - 2*c^2*C)*Cos[d + e*x])/(2*(a^2 - c^2)^2*e*(a + c*Sin[d + e*x]))} -{(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^4, x, 10, ((2*a^3*A + 3*a*A*c^2 - 4*a^2*c*C - c^3*C)*ArcTan[(c + a*Tan[(1/2)*(d + e*x)])/Sqrt[a^2 - c^2]])/((a^2 - c^2)^(7/2)*e) - B/(3*c*e*(a + c*Sin[d + e*x])^3) + ((A*c - a*C)*Cos[d + e*x])/(3*(a^2 - c^2)*e*(a + c*Sin[d + e*x])^3) + ((5*a*A*c - 2*a^2*C - 3*c^2*C)*Cos[d + e*x])/(6*(a^2 - c^2)^2*e*(a + c*Sin[d + e*x])^2) + ((11*a^2*A*c + 4*A*c^3 - 2*a^3*C - 13*a*c^2*C)*Cos[d + e*x])/(6*(a^2 - c^2)^3*e*(a + c*Sin[d + e*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a + b Trig[c+d x] Trig[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Trig[c+d x] Trig[c+d x])^n*) - - -{(a + b*Cos[c + d*x]*Sin[c + d*x])^m, x, 4, -((AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - Sin[2*c + 2*d*x]), (b*(1 - Sin[2*c + 2*d*x]))/(2*a + b)]*Cos[2*c + 2*d*x]*(a + (1/2)*b*Sin[2*c + 2*d*x])^m)/(((2*a + b*Sin[2*c + 2*d*x])/(2*a + b))^m*(Sqrt[2]*d*Sqrt[1 + Sin[2*c + 2*d*x]])))} - -{(a + b*Cos[c + d*x]*Sin[c + d*x])^3, x, 3, (1/8)*a*(8*a^2 + 3*b^2)*x - (b*(16*a^2 + b^2)*Cos[2*c + 2*d*x])/(24*d) - (5*a*b^2*Cos[2*c + 2*d*x]*Sin[2*c + 2*d*x])/(48*d) - (b*Cos[2*c + 2*d*x]*(2*a + b*Sin[2*c + 2*d*x])^2)/(48*d)} -{(a + b*Cos[c + d*x]*Sin[c + d*x])^2, x, 2, (1/8)*(8*a^2 + b^2)*x - (a*b*Cos[2*c + 2*d*x])/(2*d) - (b^2*Cos[2*c + 2*d*x]*Sin[2*c + 2*d*x])/(16*d)} -{(a + b*Cos[c + d*x]*Sin[c + d*x])^1, x, 3, a*x + (b*Sin[c + d*x]^2)/(2*d)} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^1, x, 4, (2*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/(Sqrt[4*a^2 - b^2]*d)} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^2, x, 6, (8*a*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/((4*a^2 - b^2)^(3/2)*d) + (2*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x]))} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^3, x, 7, (4*(8*a^2 + b^2)*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/((4*a^2 - b^2)^(5/2)*d) + (2*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x])^2) + (12*a*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)^2*d*(2*a + b*Sin[2*c + 2*d*x]))} - - -{(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2), x, 8, -((2*Sqrt[2]*a*b*Cos[2*c + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(15*d)) - (b*Cos[2*c + 2*d*x]*(2*a + b*Sin[2*c + 2*d*x])^(3/2))/(20*Sqrt[2]*d) + ((92*a^2 + 9*b^2)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(60*Sqrt[2]*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - (2*Sqrt[2]*a*(4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(15*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])} -{(a + b*Cos[c + d*x]*Sin[c + d*x])^(3/2), x, 7, -((b*Cos[2*c + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(6*Sqrt[2]*d)) + (2*Sqrt[2]*a*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(3*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - ((4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(6*Sqrt[2]*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])} -{(a + b*Cos[c + d*x]*Sin[c + d*x])^(1/2), x, 3, (EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(Sqrt[2]*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^(1/2), x, 3, (Sqrt[2]*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^(3/2), x, 5, (2*Sqrt[2]*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]]) + (2*Sqrt[2]*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/((4*a^2 - b^2)*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])} -{1/(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2), x, 8, (4*Sqrt[2]*b*Cos[2*c + 2*d*x])/(3*(4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x])^(3/2)) + (32*Sqrt[2]*a*b*Cos[2*c + 2*d*x])/(3*(4*a^2 - b^2)^2*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]]) + (32*Sqrt[2]*a*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(3*(4*a^2 - b^2)^2*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - (4*Sqrt[2]*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(3*(4*a^2 - b^2)*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a + b Trig[c+d x] Trig[c+d x])^n*) - - -{x^3/(a + b*Sin[x]*Cos[x]), x, 13, -((I*x^3*Log[1 - (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2]) + (I*x^3*Log[1 - (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - (3*x^2*PolyLog[2, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) + (3*x^2*PolyLog[2, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) - (3*I*x*PolyLog[3, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) + (3*I*x*PolyLog[3, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) + (3*PolyLog[4, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/(4*Sqrt[4*a^2 - b^2]) - (3*PolyLog[4, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/(4*Sqrt[4*a^2 - b^2])} -{x^2/(a + b*Sin[x]*Cos[x]), x, 11, -((I*x^2*Log[1 - (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2]) + (I*x^2*Log[1 - (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - (x*PolyLog[2, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (x*PolyLog[2, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - (I*PolyLog[3, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) + (I*PolyLog[3, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2])} -{x^1/(a + b*Sin[x]*Cos[x]), x, 9, -((I*x*Log[1 - (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2]) + (I*x*Log[1 - (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - PolyLog[2, (I*b*E^(2*I*x))/(2*a - Sqrt[4*a^2 - b^2])]/(2*Sqrt[4*a^2 - b^2]) + PolyLog[2, (I*b*E^(2*I*x))/(2*a + Sqrt[4*a^2 - b^2])]/(2*Sqrt[4*a^2 - b^2])} -{1/(x^1*(a + b*Sin[x]*Cos[x])), x, 1, Unintegrable[1/(x*(a + (1/2)*b*Sin[2*x])), x]} - - -{((b*x)^(2 - n)*Sin[a*x]^n)/(a*c*x*Cos[a*x] - c*Sin[a*x])^2, x, 1, (b*(b*x)^(1 - n)*Sin[a*x]^(-1 + n))/(a^2*(a*c^2*x*Cos[a*x] - c^2*Sin[a*x])) + (b^2*(1 - n)*Unintegrable[Sin[a*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)} -{((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2, x, 1, -((b*(b*x)^(1 - n)*Cos[a*x]^(-1 + n))/(a^2*(c^2*Cos[a*x] + a*c^2*x*Sin[a*x]))) + (b^2*(1 - n)*Unintegrable[Cos[a*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b x)^m Trig[a x]^n (c Trig[a x]+d x Trig[a x])^p*) - - -{Sin[a*x]^6/(x^4*(a*x*Cos[a*x] - Sin[a*x])^2), x, 15, a^2/x + (a*Cos[a*x]*Sin[a*x])/x^2 + Sin[a*x]^2/x^3 - (10*a^2*Sin[a*x]^2)/x + (Cos[a*x]*Sin[a*x]^3)/(a*x^4) - (8*a*Cos[a*x]*Sin[a*x]^3)/(3*x^2) + Sin[a*x]^4/(a^2*x^5) - (4*Sin[a*x]^4)/(3*x^3) + (32*a^2*Sin[a*x]^4)/(3*x) + Sin[a*x]^5/(a^2*x^5*(a*x*Cos[a*x] - Sin[a*x])) - (2/3)*a^3*SinIntegral[2*a*x] + (16/3)*a^3*SinIntegral[4*a*x]} -{Sin[a*x]^5/(x^3*(a*x*Cos[a*x] - Sin[a*x])^2), x, 11, (a*Cos[a*x])/x + Sin[a*x]/x^2 + (Cos[a*x]*Sin[a*x]^2)/(a*x^3) - (9*a*Cos[a*x]*Sin[a*x]^2)/(2*x) + Sin[a*x]^3/(a^2*x^4) - (3*Sin[a*x]^3)/(2*x^2) + Sin[a*x]^4/(a^2*x^4*(a*x*Cos[a*x] - Sin[a*x])) - (1/8)*a^2*SinIntegral[a*x] + (27/8)*a^2*SinIntegral[3*a*x]} -{Sin[a*x]^4/(x^2*(a*x*Cos[a*x] - Sin[a*x])^2), x, 6, 1/x + (Cos[a*x]*Sin[a*x])/(a*x^2) + Sin[a*x]^2/(a^2*x^3) - (2*Sin[a*x]^2)/x + Sin[a*x]^3/(a^2*x^3*(a*x*Cos[a*x] - Sin[a*x])) + 2*a*SinIntegral[2*a*x]} -{Sin[a*x]^3/(x^1*(a*x*Cos[a*x] - Sin[a*x])^2), x, 4, Cos[a*x]/(a*x) + Sin[a*x]/(a^2*x^2) + Sin[a*x]^2/(a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])) + SinIntegral[a*x]} -{Sin[a*x]^2/(x^0*(a*x*Cos[a*x] - Sin[a*x])^2), x, 1, 1/(a^2*x) + Sin[a*x]/(a^2*x*(a*x*Cos[a*x] - Sin[a*x]))} -{x^1*Sin[a*x]^1/(a*x*Cos[a*x] - Sin[a*x])^2, x, 1, 1/(a^2*(a*x*Cos[a*x] - Sin[a*x]))} -{x^2*Sin[a*x]^0/(a*x*Cos[a*x] - Sin[a*x])^2, x, 3, -(Cot[a*x]/a^3) + (x*Csc[a*x])/(a^2*(a*x*Cos[a*x] - Sin[a*x]))} -{x^3*Csc[a*x]^1/(a*x*Cos[a*x] - Sin[a*x])^2, x, 7, -((2*x*ArcTanh[E^(I*a*x)])/a^3) - Csc[a*x]/a^4 - (x*Cot[a*x]*Csc[a*x])/a^3 + (I*PolyLog[2, -E^(I*a*x)])/a^4 - (I*PolyLog[2, E^(I*a*x)])/a^4 + (x^2*Csc[a*x]^2)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))} -{x^4*Csc[a*x]^2/(a*x*Cos[a*x] - Sin[a*x])^2, x, 9, -((2*I*x^2)/a^3) - Cot[a*x]/a^5 - (2*x^2*Cot[a*x])/a^3 - (x*Csc[a*x]^2)/a^4 - (x^2*Cot[a*x]*Csc[a*x]^2)/a^3 + (4*x*Log[1 - E^(2*I*a*x)])/a^4 - (2*I*PolyLog[2, E^(2*I*a*x)])/a^5 + (x^3*Csc[a*x]^3)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))} - - -{Cos[a*x]^6/(x^4*(Cos[a*x] + a*x*Sin[a*x])^2), x, 15, a^2/x + Cos[a*x]^2/x^3 - (10*a^2*Cos[a*x]^2)/x + Cos[a*x]^4/(a^2*x^5) - (4*Cos[a*x]^4)/(3*x^3) + (32*a^2*Cos[a*x]^4)/(3*x) - (a*Cos[a*x]*Sin[a*x])/x^2 - (Cos[a*x]^3*Sin[a*x])/(a*x^4) + (8*a*Cos[a*x]^3*Sin[a*x])/(3*x^2) - Cos[a*x]^5/(a^2*x^5*(Cos[a*x] + a*x*Sin[a*x])) + (2/3)*a^3*SinIntegral[2*a*x] + (16/3)*a^3*SinIntegral[4*a*x]} -{Cos[a*x]^5/(x^3*(Cos[a*x] + a*x*Sin[a*x])^2), x, 11, Cos[a*x]/x^2 + Cos[a*x]^3/(a^2*x^4) - (3*Cos[a*x]^3)/(2*x^2) - (1/8)*a^2*CosIntegral[a*x] - (27/8)*a^2*CosIntegral[3*a*x] - (a*Sin[a*x])/x - (Cos[a*x]^2*Sin[a*x])/(a*x^3) + (9*a*Cos[a*x]^2*Sin[a*x])/(2*x) - Cos[a*x]^4/(a^2*x^4*(Cos[a*x] + a*x*Sin[a*x]))} -{Cos[a*x]^4/(x^2*(Cos[a*x] + a*x*Sin[a*x])^2), x, 6, 1/x + Cos[a*x]^2/(a^2*x^3) - (2*Cos[a*x]^2)/x - (Cos[a*x]*Sin[a*x])/(a*x^2) - Cos[a*x]^3/(a^2*x^3*(Cos[a*x] + a*x*Sin[a*x])) - 2*a*SinIntegral[2*a*x]} -{Cos[a*x]^3/(x^1*(Cos[a*x] + a*x*Sin[a*x])^2), x, 4, Cos[a*x]/(a^2*x^2) + CosIntegral[a*x] - Sin[a*x]/(a*x) - Cos[a*x]^2/(a^2*x^2*(Cos[a*x] + a*x*Sin[a*x]))} -{Cos[a*x]^2/(x^0*(Cos[a*x] + a*x*Sin[a*x])^2), x, 1, 1/(a^2*x) - Cos[a*x]/(a^2*x*(Cos[a*x] + a*x*Sin[a*x]))} -{x^1*Cos[a*x]^1/(Cos[a*x] + a*x*Sin[a*x])^2, x, 1, -(1/(a^2*(Cos[a*x] + a*x*Sin[a*x])))} -{x^2*Cos[a*x]^0/(Cos[a*x] + a*x*Sin[a*x])^2, x, 3, -((x*Sec[a*x])/(a^2*(Cos[a*x] + a*x*Sin[a*x]))) + Tan[a*x]/a^3} -{x^3*Sec[a*x]^1/(Cos[a*x] + a*x*Sin[a*x])^2, x, 7, -((2*I*x*ArcTan[E^(I*a*x)])/a^3) + (I*PolyLog[2, (-I)*E^(I*a*x)])/a^4 - (I*PolyLog[2, I*E^(I*a*x)])/a^4 - Sec[a*x]/a^4 - (x^2*Sec[a*x]^2)/(a^2*(Cos[a*x] + a*x*Sin[a*x])) + (x*Sec[a*x]*Tan[a*x])/a^3} -{x^4*Sec[a*x]^2/(Cos[a*x] + a*x*Sin[a*x])^2, x, 9, -((2*I*x^2)/a^3) + (4*x*Log[1 + E^(2*I*a*x)])/a^4 - (2*I*PolyLog[2, -E^(2*I*a*x)])/a^5 - (x*Sec[a*x]^2)/a^4 - (x^3*Sec[a*x]^3)/(a^2*(Cos[a*x] + a*x*Sin[a*x])) + Tan[a*x]/a^5 + (2*x^2*Tan[a*x])/a^3 + (x^2*Sec[a*x]^2*Tan[a*x])/a^3} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (c Tan[a+b x] Tan[2 (a+b x)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sec[2 (a+b x)]^m (c Tan[a+b x] Tan[2 (a+b x)])^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sec[2*(a + b*x)]^4*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 5, (-2*c*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sec[2*a + 2*b*x]^3*Tan[2*a + 2*b*x])/(7*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (4*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(35*b) - (6*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(35*b*c)} -{Sec[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 4, (7*c*Tan[2*a + 2*b*x])/(15*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (2*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(15*b) + ((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b*c)} -{Sec[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 3, -(c*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b)} -{Sec[2*(a + b*x)]^1*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 2, (c*Tan[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Sec[2*(a + b*x)]^0*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 3, -((Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/b)} -{Cos[2*(a + b*x)]^1*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 4, (Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b) - (c*Sin[2*a + 2*b*x])/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 5, (-3*Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b) + (3*c*Sin[2*a + 2*b*x])/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 6, (5*Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) - (5*c*Sin[2*a + 2*b*x])/(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (5*c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} - - -{Sec[2*(a + b*x)]^4*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 7, (34*c^2*Tan[2*a + 2*b*x])/(45*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (17*c^2*Sec[2*a + 2*b*x]^3*Tan[2*a + 2*b*x])/(63*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Sec[2*a + 2*b*x]^4*Tan[2*a + 2*b*x])/(9*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (68*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(315*b) + (34*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(105*b)} -{Sec[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 5, (-76*c^2*Tan[2*a + 2*b*x])/(105*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (19*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(105*b) + (2*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(35*b) + ((-c + c*Sec[2*a + 2*b*x])^(5/2)*Tan[2*a + 2*b*x])/(7*b*c)} -{Sec[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 4, (4*c^2*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(5*b) + ((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b)} -{Sec[2*(a + b*x)]^1*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 3, (-4*c^2*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b)} -{Sec[2*(a + b*x)]^0*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 5, (c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/b + (c^2*Tan[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^1*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 6, (-3*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b) + (c^2*Sin[2*a + 2*b*x])/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 6, (7*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b) - (7*c^2*Sin[2*a + 2*b*x])/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 7, (-11*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) + (11*c^2*Sin[2*a + 2*b*x])/(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (11*c^2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sec[2*(a + b*x)]^4/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 6, -(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + (14*Tan[2*a + 2*b*x])/(15*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sec[2*a + 2*b*x]^2*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(15*b*c)} -{Sec[2*(a + b*x)]^3/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 5, -(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + (2*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b*c)} -{Sec[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 4, -(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + Tan[2*a + 2*b*x]/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Sec[2*(a + b*x)]^1/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 3, -(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]))} -{Sec[2*(a + b*x)]^0/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 6, ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])} -{Cos[2*(a + b*x)]^1/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 7, ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(2*b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]) + Sin[2*a + 2*b*x]/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]], x, 8, (7*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]) + Sin[2*a + 2*b*x]/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])} - - -{Sec[2*(a + b*x)]^4/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 6, (-11*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - (Sec[2*a + 2*b*x]^2*Tan[2*a + 2*b*x])/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) + (13*Tan[2*a + 2*b*x])/(6*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (7*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(12*b*c^2)} -{Sec[2*(a + b*x)]^3/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 5, (-7*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) + Tan[2*a + 2*b*x]/(b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Sec[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 4, (-3*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))} -{Sec[2*(a + b*x)]^1/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 4, ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))} -{Sec[2*(a + b*x)]^0/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 7, -(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(b*c^(3/2))) + (5*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))} -{Cos[2*(a + b*x)]^1/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 8, (-3*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b*c^(3/2)) + (9*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Sin[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) - (3*Sin[2*a + 2*b*x])/(4*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])} -{Cos[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2), x, 9, (-19*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b*c^(3/2)) + (13*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) - (7*Sin[2*a + 2*b*x])/(8*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(2*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form u Sin[2 x]^(p/2)*) - - -{Cot[x]*Csc[x]/Sqrt[Sin[2*x]], x, 3, -((2*Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]]))} - - -{(Csc[x]^2*Sec[x])/(Sqrt[Sin[2*x]]*(-2 + Tan[x])), x, 6, Cos[x]/(2*Sqrt[Sin[2*x]]) + (Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]]) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x])/(2*Sqrt[2]*Sqrt[Sin[2*x]]*Sqrt[Tan[x]])} - - -{(Cos[x]^2*Sin[x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)), x, 6, (Cos[x]^4*Sin[x])/(3*Sin[2*x]^(5/2)) + (Cos[x]^3*Sin[x]^2)/(2*Sin[2*x]^(5/2)) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x]^5)/(2*Sqrt[2]*Sin[2*x]^(5/2)*Tan[x]^(5/2))} - - -{(Cos[x]^3*Cos[2*x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)), x, 6, Cos[x]^5/(5*Sin[2*x]^(5/2)) + (Cos[x]^4*Sin[x])/(6*Sin[2*x]^(5/2)) - (3*Cos[x]^3*Sin[x]^2)/(4*Sin[2*x]^(5/2)) + (3*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x]^5)/(4*Sqrt[2]*Sin[2*x]^(5/2)*Tan[x]^(5/2))} - - -(* ::Section::Closed:: *) -(*Products of functions of a trig function and its derivative*) - - -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])*(a*Sin[c + d*x] + b*Sec[c + d*x])^n, x, 1, (b*Sec[c + d*x] + a*Sin[c + d*x])^(1 + n)/(d*(1 + n))} - -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])*(a*Sin[c + d*x] + b*Sec[c + d*x])^3, x, 1, (b*Sec[c + d*x] + a*Sin[c + d*x])^4/(4*d)} -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])*(a*Sin[c + d*x] + b*Sec[c + d*x])^2, x, 1, (b*Sec[c + d*x] + a*Sin[c + d*x])^3/(3*d)} -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])*(a*Sin[c + d*x] + b*Sec[c + d*x])^1, x, 1, (b*Sec[c + d*x] + a*Sin[c + d*x])^2/(2*d)} -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])/(a*Sin[c + d*x] + b*Sec[c + d*x])^1, x, 1, Log[b*Sec[c + d*x] + a*Sin[c + d*x]]/d} -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])/(a*Sin[c + d*x] + b*Sec[c + d*x])^2, x, 1, -(1/(d*(b*Sec[c + d*x] + a*Sin[c + d*x])))} -{(a Cos[c+d x]+b Sec[c+d x] Tan[c+d x])/(a*Sin[c + d*x] + b*Sec[c + d*x])^3, x, 1, -(1/(2*d*(b*Sec[c + d*x] + a*Sin[c + d*x])^2))} - - -{Sin[a + b*x]*F[c, d, Cos[a + b*x], r, s], x, 1, CannotIntegrate[F[c, d, Cos[a + b*x], r, s]*Sin[a + b*x], x]} -{Cos[a + b*x]*F[c, d, Sin[a + b*x], r, s], x, 1, CannotIntegrate[Cos[a + b*x]*F[c, d, Sin[a + b*x], r, s], x]} -{Sec[a + b*x]^2*F[c, d, Tan[a + b*x], r, s], x, 1, CannotIntegrate[F[c, d, Tan[a + b*x], r, s]*Sec[a + b*x]^2, x]} -{Csc[a + b*x]^2*F[c, d, Cot[a + b*x], r, s], x, 1, CannotIntegrate[Csc[a + b*x]^2*F[c, d, Cot[a + b*x], r, s], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Cos[a+b x]] Sin[a+b x]^n when n odd*) - - -{Sin[x]/(a + b*Cos[x]), x, 2, -(Log[a + b*Cos[x]]/b)} -{Sin[x]*(a + b*Cos[x])^n, x, 2, -((a + b*Cos[x])^(1 + n)/(b*(1 + n)))} -{Sin[x]/Sqrt[1 + Cos[x]^2], x, 2, -ArcSinh[Cos[x]]} -{Sin[x]*Cos[Cos[x]], x, 2, -Sin[Cos[x]]} -{Sin[x]*Cos[x]*Cos[Cos[x]]*Sin[Cos[x]], x, 4, Cos[x]/4 - (1/4)*Cos[Cos[x]]*Sin[Cos[x]] - (1/2)*Cos[x]*Sin[Cos[x]]^2} -{Sin[x]*Cos[Cos[x]]*Sin[6*Cos[x]]^2, x, 6, (-(1/2))*Sin[Cos[x]] + (1/44)*Sin[11*Cos[x]] + (1/52)*Sin[13*Cos[x]]} -{Sin[x]*Cos[x]^3*(a + b*Cos[x]^2)^3, x, 4, (a*(a + b*Cos[x]^2)^4)/(8*b^2) - (a + b*Cos[x]^2)^5/(10*b^2)} -{Sin[3*x]*Sin[Cos[3*x]], x, 2, Cos[Cos[3*x]]/3} -{Sin[1 + 3*x]*Cos[1 + 3*x]*E^Cos[1 + 3*x], x, 3, (1/3)*E^Cos[1 + 3*x] - (1/3)*E^Cos[1 + 3*x]*Cos[1 + 3*x]} -{Sin[x]*Cos[x]^2/Sqrt[1 - Cos[x]^6], x, 3, (-(1/3))*ArcSin[Cos[x]^3]} - - -{Sin[x]^5/Sqrt[1 - 5*Cos[x]], x, 3, (1152*Sqrt[1 - 5*Cos[x]])/3125 + (64*(1 - 5*Cos[x])^(3/2))/3125 - (88*(1 - 5*Cos[x])^(5/2))/15625 - (8*(1 - 5*Cos[x])^(7/2))/21875 + (2*(1 - 5*Cos[x])^(9/2))/28125} - - -{E^(n*Cos[a+b*x])*Sin[a+b*x], x, 2, -(E^(n*Cos[a + b*x])/(b*n))} -{E^(n*Cos[a*c+b*c*x])*Sin[c*(a+b*x)], x, 2, -(E^(n*Cos[c*(a + b*x)])/(b*c*n))} -{E^(n*Cos[c*(a+b*x)])*Sin[a*c+b*c*x], x, 2, -(E^(n*Cos[a*c + b*c*x])/(b*c*n))} - - -{E^(n*Cos[a+b*x])*Tan[a+b*x], x, 2, -(ExpIntegralEi[n*Cos[a + b*x]]/b)} -{E^(n*Cos[a*c+b*c*x])*Tan[c*(a+b*x)], x, 2, -(ExpIntegralEi[n*Cos[c*(a + b*x)]]/(b*c))} -{E^(n*Cos[c*(a+b*x)])*Tan[a*c+b*c*x], x, 2, -(ExpIntegralEi[n*Cos[a*c + b*c*x]]/(b*c))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Sin[a+b x]] Cos[a+b x]^n when n odd*) - - -{Cos[x]/(a + b*Sin[x]), x, 2, Log[a + b*Sin[x]]/b} -{Cos[x]*(a + b*Sin[x])^n, x, 2, (a + b*Sin[x])^(1 + n)/(b*(1 + n))} -{Cos[x]/Sqrt[1 + Sin[x]^2], x, 2, ArcSinh[Sin[x]]} -{Cos[x]/Sqrt[4 - Sin[x]^2], x, 2, ArcSin[Sin[x]/2]} -{Cos[3*x]/Sqrt[4 - Sin[3*x]^2], x, 2, ArcSin[Sin[3*x]/2]/3} -{Cos[x]*Sqrt[1 + Csc[x]], x, 4, ArcTanh[Sqrt[1 + Csc[x]]] + Sqrt[1 + Csc[x]]*Sin[x]} -{Cos[x]*Sqrt[4 - Sin[x]^2], x, 3, 2*ArcSin[Sin[x]/2] + (Sin[x]*Sqrt[4 - Sin[x]^2])/2} -{Cos[x]*Sin[x]*Sqrt[1 + Sin[x]^2], x, 2, (1/3)*(1 + Sin[x]^2)^(3/2)} -{Cos[x]/Sqrt[2*Sin[x] + Sin[x]^2], x, 3, 2*ArcTanh[Sin[x]/Sqrt[2*Sin[x] + Sin[x]^2]]} -{Cos[x]*Cos[Sin[x]], x, 2, Sin[Sin[x]]} -{Cos[x]*Cos[Sin[x]]*Cos[Sin[Sin[x]]], x, 3, Sin[Sin[Sin[x]]]} -{Cos[x]*Sec[Sin[x]], x, 2, ArcTanh[Sin[Sin[x]]]} -{Cos[x]*Sin[x]^3*(a + b*Sin[x]^2)^3, x, 4, -((a*(a + b*Sin[x]^2)^4)/(8*b^2)) + (a + b*Sin[x]^2)^5/(10*b^2)} -{Cos[x]*Sin[x]*E^Sin[x], x, 3, -E^Sin[x] + E^Sin[x]*Sin[x]} -{Cos[x]^3/Sqrt[Sin[x]^3], x, 4, -((2*Sin[x])/Sqrt[Sin[x]^3]) - (2/3)*Sqrt[Sin[x]^3]} - - -{E^Sqrt[Sin[x]]*Cos[x]/Sqrt[Sin[x]], x, 2, 2*E^Sqrt[Sin[x]]} -{E^(4 + Sin[x])*Cos[x], x, 2, E^(4 + Sin[x])} - - -{E^(Cos[x]*Sin[x])*Cos[2*x], x, 2, E^((1/2)*Sin[2*x])} -{E^(Cos[x/2]*Sin[x/2])*Cos[x], x, 2, 2*E^(Sin[x]/2)} - - -{E^(n*Sin[a+b*x])*Cos[a+b*x], x, 2, E^(n*Sin[a + b*x])/(b*n)} -{E^(n*Sin[a*c+b*c*x])*Cos[c*(a+b*x)], x, 2, E^(n*Sin[c*(a + b*x)])/(b*c*n)} -{E^(n*Sin[c*(a+b*x)])*Cos[a*c+b*c*x], x, 2, E^(n*Sin[a*c + b*c*x])/(b*c*n)} - - -{E^(n*Sin[a+b*x])*Cot[a+b*x], x, 2, ExpIntegralEi[n*Sin[a + b*x]]/b} -{E^(n*Sin[a*c+b*c*x])*Cot[c*(a+b*x)], x, 2, ExpIntegralEi[n*Sin[c*(a + b*x)]]/(b*c)} -{E^(n*Sin[c*(a+b*x)])*Cot[a*c+b*c*x], x, 2, ExpIntegralEi[n*Sin[a*c + b*c*x]]/(b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Tan[a+b x]] Sec[a+b x]^n when n even*) - - -{Sec[x]^2/(a + b*Tan[x]), x, 2, Log[a + b*Tan[x]]/b} -{Sec[x]^2/(1 - Tan[x]^2), x, 2, (1/2)*ArcTanh[2*Cos[x]*Sin[x]]} -{Sec[x]^2/(9 + Tan[x]^2), x, 2, x/3 - (1/3)*ArcTan[(2*Cos[x]*Sin[x])/(1 + 2*Cos[x]^2)]} -{Sec[x]^2*(a + b*Tan[x])^n, x, 2, (a + b*Tan[x])^(1 + n)/(b*(1 + n))} -{Sec[x]^2*(1 + 1/(1 + Tan[x]^2)), x, 3, x + Tan[x]} -{Sec[x]^2*(2 + Tan[x]^2)/(1 + Tan[x]^2), x, 4, x + Tan[x]} -{Sec[x]^2/(2 + 2*Tan[x] + Tan[x]^2), x, 3, x - ArcTan[(1 - 2*Cos[x]^2 + Cos[x]*Sin[x])/(2 + Cos[x]^2 + 2*Cos[x]*Sin[x])]} -{Sec[x]^2/(Tan[x]^2 + Tan[x]^3), x, 3, -Cot[x] + Log[1 + Cot[x]], -Cot[x] - Log[Tan[x]] + Log[1 + Tan[x]]} -{Sec[x]^2/(-Tan[x]^2 + Tan[x]^3), x, 3, Cot[x] + Log[1 - Cot[x]], Cot[x] + Log[1 - Tan[x]] - Log[Tan[x]]} -{Sec[x]^2/(3 - 4*Tan[x]^3), x, 7, x/(3*2^(2/3)*3^(1/6)) - ArcTan[(6^(2/3) - 2*6^(2/3)*Cos[x]^2 + 2*(3 - 2*6^(1/3))*Cos[x]*Sin[x])/(3*2^(2/3)*3^(1/6) + 4*6^(1/3) + (6 - 4*6^(1/3))*Cos[x]^2 + 2*6^(2/3)*Cos[x]*Sin[x])]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^(2/3)*Tan[x]]/(3*6^(2/3)) + Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tan[x] + 2*2^(1/3)*Tan[x]^2]/(6*6^(2/3))} -{Sec[x]^2/(11 - 5*Tan[x] + 5*Tan[x]^2), x, 3, (2*x)/Sqrt[195] - (2*ArcTan[(-5 + 10*Cos[x]^2 + 12*Cos[x]*Sin[x])/(10 + Sqrt[195] + 12*Cos[x]^2 - 10*Cos[x]*Sin[x])])/Sqrt[195]} -{Sec[x]^2*(a + b*Tan[x])/(c + d*Tan[x]), x, 3, -(((b*c - a*d)*Log[c + d*Tan[x]])/d^2) + (b*Tan[x])/d} -{Sec[x]^2*(a + b*Tan[x])^2/(c + d*Tan[x]), x, 3, ((b*c - a*d)^2*Log[c + d*Tan[x]])/d^3 - (b*(b*c - a*d)*Tan[x])/d^2 + (a + b*Tan[x])^2/(2*d)} -{Sec[x]^2*(a + b*Tan[x])^3/(c + d*Tan[x]), x, 3, -(((b*c - a*d)^3*Log[c + d*Tan[x]])/d^4) + (b*(b*c - a*d)^2*Tan[x])/d^3 - ((b*c - a*d)*(a + b*Tan[x])^2)/(2*d^2) + (a + b*Tan[x])^3/(3*d)} -{Sec[x]^2*Tan[x]^2/(2 + Tan[x]^3)^2, x, 2, -1/(3*(2 + Tan[x]^3))} -{Sec[x]^2*Tan[x]^6*(1 + Tan[x]^2)^3, x, 4, Tan[x]^7/7 + Tan[x]^9/3 + (3*Tan[x]^11)/11 + Tan[x]^13/13} -{Sec[x]^2*(2 + Tan[x]^2)/(1 + Tan[x]^3), x, 5, (2*x)/Sqrt[3] + (2*ArcTan[(1 - 2*Cos[x]^2)/(2 + Sqrt[3] - 2*Cos[x]*Sin[x])])/Sqrt[3] + Log[1 + Tan[x]]} -{Sec[x]^2*(1 + Cos[x]^2), x, 2, x + Tan[x]} -{Sec[x]^2/(1 + Sec[x]^2 - 3*Tan[x]), x, 4, -Log[Cos[x] - Sin[x]] + Log[2*Cos[x] - Sin[x]]} -{Sec[x]^2/Sqrt[4 - Sec[x]^2], x, 2, ArcSin[Tan[x]/Sqrt[3]]} -{Sec[x]^2/Sqrt[1 - 4*Tan[x]^2], x, 2, ArcSin[2*Tan[x]]/2} -{Sec[x]^2/Sqrt[-4 + Tan[x]^2], x, 3, ArcTanh[Tan[x]/Sqrt[-4 + Tan[x]^2]]} -{Sec[x]^2*Sqrt[1 - Cot[x]^2], x, 3, ArcSin[Cot[x]] + Sqrt[1 - Cot[x]^2]*Tan[x]} -{Sec[x]^2*Sqrt[1 - Tan[x]^2], x, 3, (1/2)*ArcSin[Tan[x]] + (1/2)*Tan[x]*Sqrt[1 - Tan[x]^2]} -{Sec[x]^2*E^Tan[x], x, 2, E^Tan[x]} - - -{Sec[x]^4*(-1 + Sec[x]^2)^2*Tan[x], x, 4, Tan[x]^6/6 + Tan[x]^8/8} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Cot[a+b x]] Csc[a+b x]^n when n even*) - - -{Csc[x]^2/(a + b*Cot[x]), x, 2, -(Log[a + b*Cot[x]]/b)} -{Csc[x]^2*(a + b*Cot[x])^n, x, 2, -((a + b*Cot[x])^(1 + n)/(b*(1 + n)))} -{Csc[x]^2*(1 + Sin[x]^2), x, 2, x - Cot[x]} -{Csc[x]^2*(1 + 1/(1 + Cot[x]^2)), x, 4, x - Cot[x]} -{Csc[x]^2*(a + b*Cot[x])/(c + d*Cot[x]), x, 3, -((b*Cot[x])/d) + ((b*c - a*d)*Log[c + d*Cot[x]])/d^2} -{Csc[x]^2*(a + b*Cot[x])^2/(c + d*Cot[x]), x, 3, (b*(b*c - a*d)*Cot[x])/d^2 - (a + b*Cot[x])^2/(2*d) - ((b*c - a*d)^2*Log[c + d*Cot[x]])/d^3} -{Csc[x]^2*(a + b*Cot[x])^3/(c + d*Cot[x]), x, 3, -((b*(b*c - a*d)^2*Cot[x])/d^3) + ((b*c - a*d)*(a + b*Cot[x])^2)/(2*d^2) - (a + b*Cot[x])^3/(3*d) + ((b*c - a*d)^3*Log[c + d*Cot[x]])/d^4} -{Csc[x]^2/E^Cot[x], x, 2, E^(-Cot[x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Sec[a+b x]] Sec[a+b x] Tan[a+b x]*) - - -{Sec[x]*Tan[x]/(a + b*Sec[x]), x, 4, Log[a + b*Sec[x]]/b, -(Log[Cos[x]]/b) + Log[b + a*Cos[x]]/b} -{Sec[x]*Tan[x]/(1 + Sec[x]^2), x, 2, -ArcTan[Cos[x]]} -{Sec[x]*Tan[x]/(9 + 4*Sec[x]^2), x, 2, (-(1/6))*ArcTan[(3*Cos[x])/2]} -{Sec[x]*Tan[x]/(Sec[x] + Sec[x]^2), x, 2, -Log[1 + Cos[x]]} -{Sec[x]*Tan[x]/Sqrt[4 + Sec[x]^2], x, 3, ArcCsch[2*Cos[x]]} -{Sec[x]*Tan[x]/Sqrt[1 + Cos[x]^2], x, 2, Sqrt[1 + Cos[x]^2]*Sec[x]} -{Sec[x]*Tan[x]*E^Sec[x], x, 2, E^Sec[x]} -{Sec[x]*Tan[x]*2^Sec[x], x, 2, 2^Sec[x]/Log[2]} - -{Sec[2*x]*Tan[2*x]/(1 + Sec[2*x])^(3/2), x, 2, -(1/Sqrt[1 + Sec[2*x]])} -{Sec[3*x]*Tan[3*x]*Sqrt[1 + 5*Cos[3*x]^2], x, 3, (-(1/3))*Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]] + (1/3)*Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x]} -{Sec[3*x]*Tan[3*x]/Sqrt[1 + 5*Cos[3*x]^2], x, 2, (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Csc[a+b x]] Csc[a+b x] Cot[a+b x]*) - - -{(Csc[x]*Cot[x])/(a + b*Csc[x]), x, 4, -(Log[a + b*Csc[x]]/b), Log[Sin[x]]/b - Log[b + a*Sin[x]]/b} -{5^Csc[3*x]*Cot[3*x]*Csc[3*x], x, 2, -5^Csc[3*x]/(3*Log[5])} -{(Cot[x]*Csc[x])/(1 + Csc[x]^2), x, 2, ArcTan[Sin[x]]} -{(Cot[6*x]*Csc[6*x])/(5 - 11*Csc[6*x]^2)^2, x, 3, -(ArcTanh[Sqrt[5/11]*Sin[6*x]]/(60*Sqrt[55])) + Sin[6*x]/(60*(11 - 5*Sin[6*x]^2))} -{(Cot[x]*Csc[x])/Sqrt[1 + Sin[x]^2], x, 2, -(Csc[x]*Sqrt[1 + Sin[x]^2])} -{(Cot[5*x]*Csc[5*x]^3)/Sqrt[1 + Sin[5*x]^2], x, 3, (2/15)*Csc[5*x]*Sqrt[1 + Sin[5*x]^2] - (1/15)*Csc[5*x]^3*Sqrt[1 + Sin[5*x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Sin[(a+b x)/2]] Sin[a+b x]*) - - -{E^(n*Sin[a+b*x])*Sin[2*a+2*b*x], x, 4, -((2*E^(n*Sin[a + b*x]))/(b*n^2)) + (2*E^(n*Sin[a + b*x])*Sin[a + b*x])/(b*n)} -{E^(n*Sin[a+b*x])*Sin[2*(a+b*x)], x, 4, -((2*E^(n*Sin[a + b*x]))/(b*n^2)) + (2*E^(n*Sin[a + b*x])*Sin[a + b*x])/(b*n)} -{E^(n*Sin[a/2+b/2*x])*Sin[a+b*x], x, 4, -((4*E^(n*Sin[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Sin[a/2 + (b*x)/2])*Sin[a/2 + (b*x)/2])/(b*n)} -{E^(n*Sin[(a+b*x)/2])*Sin[a+b*x], x, 4, -((4*E^(n*Sin[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Sin[a/2 + (b*x)/2])*Sin[a/2 + (b*x)/2])/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Cos[(a+b x)/2]] Sin[a+b x]*) - - -{E^(n*Cos[a+b*x])*Sin[2*a+2*b*x], x, 4, (2*E^(n*Cos[a + b*x]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)} -{E^(n*Cos[a+b*x])*Sin[2*(a+b*x)], x, 4, (2*E^(n*Cos[a + b*x]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)} -{E^(n*Cos[a/2+b/2*x])*Sin[a+b*x], x, 4, (4*E^(n*Cos[a/2 + (b*x)/2]))/(b*n^2) - (4*E^(n*Cos[a/2 + (b*x)/2])*Cos[a/2 + (b*x)/2])/(b*n)} -{E^(n*Cos[(a+b*x)/2])*Sin[a+b*x], x, 4, (4*E^(n*Cos[a/2 + (b*x)/2]))/(b*n^2) - (4*E^(n*Cos[a/2 + (b*x)/2])*Cos[a/2 + (b*x)/2])/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Tan[a+b x]] when n even*) - - -{Csc[x]*Log[Tan[x]]*Sec[x], x, 1, Log[Tan[x]]^2/2} -{Csc[2*x]*Log[Tan[x]], x, 1, Log[Tan[x]]^2/4} - - -{E^(Cos[x]^2 + Sin[x]^2), x, 3, E*x} - - -(* ::Section::Closed:: *) -(*Problems from Calculus textbooks*) - - -(* ::Subsection::Closed:: *) -(*Anton Calculus, 4th Edition*) - - -{x*Sec[x]^2, x, 2, Log[Cos[x]] + x*Tan[x]} -{x*Cos[x^2]^4, x, 4, (3*x^2)/16 + (3/16)*Cos[x^2]*Sin[x^2] + (1/8)*Cos[x^2]^3*Sin[x^2]} - -{Sqrt[Cos[x]]*Sin[x], x, 2, (-2*Cos[x]^(3/2))/3} -{Tan[E^(-2*x)]/E^(2*x), x, 2, Log[Cos[E^(-2*x)]]/2} -{(Sec[x]*Sin[2*x])/(1 + Cos[x]), x, 3, -2*Log[1 + Cos[x]]} -{x*Sec[3*x]^2, x, 2, (1/9)*Log[Cos[3*x]] + (1/3)*x*Tan[3*x]} -{Cos[2*Pi*x]/E^(2*Pi*x), x, 1, -(Cos[2*Pi*x]/(E^(2*Pi*x)*(4*Pi))) + Sin[2*Pi*x]/(E^(2*Pi*x)*(4*Pi))} -{Cos[x]^12*Sin[x]^10 - Cos[x]^10*Sin[x]^12, x, -25, (Cos[x]^11*Sin[x]^11)/11} - - -(* ::Subsection::Closed:: *) -(*Ayres Calculus, 1964 edition*) - - -{x*Cot[x^2], x, 2, Log[Sin[x^2]]/2} -{x*Sec[x^2]^2, x, 3, Tan[x^2]/2} -{Sin[8*x]/(9 + Sin[4*x]^4), x, 4, ArcTan[Sin[4*x]^2/3]/12} -{Cos[2*x]/(8 + Sin[2*x]^2), x, 2, ArcTan[Sin[2*x]/(2*Sqrt[2])]/(4*Sqrt[2])} -{x*(Cos[x^2]^3 - Sin[x^2]^3), x, 8, Cos[x^2]/2 - (1/6)*Cos[x^2]^3 + Sin[x^2]/2 - (1/6)*Sin[x^2]^3} -{Cos[x]*Sin[x]/(1 - Cos[x]), x, 3, Cos[x] + Log[1 - Cos[x]]} - - -(* ::Subsection::Closed:: *) -(*Edwards and Penney Calculus*) - - -{x*Cos[x^2], x, 2, Sin[x^2]/2} -{x^2*Cos[4*x^3], x, 2, Sin[4*x^3]/12} -{x^3*Cos[x^4], x, 2, Sin[x^4]/4} -{x*Sin[x^2/2], x, 2, -Cos[x^2/2]} -{x*Sec[x^2]*Tan[x^2], x, 3, Sec[x^2]/2} -{Tan[1/x]^2/x^2, x, 3, x^(-1) - Tan[x^(-1)]} -{x*Tan[1 + x^2], x, 2, -Log[Cos[1 + x^2]]/2} -{Sin[Pi*(1 + 2*x)], x, 1, Cos[2*Pi*x]/(2*Pi)} - -{(Cot[x] + Csc[x]^2)/(1 - Cos[x]^2), x, 3, -Cot[x] - Cot[x]^2/2 - Cot[x]^3/3} - - -(* ::Subsection::Closed:: *) -(*Grossman Calculus*) - - -{x^2*Cos[4*x^3]*Cos[5*x^3], x, 6, Sin[x^3]/6 + (1/54)*Sin[9*x^3]} -{x^14*Sin[x^3], x, 6, -8*Cos[x^3] + 4*x^6*Cos[x^3] - (1/3)*x^12*Cos[x^3] - 8*x^3*Sin[x^3] + (4/3)*x^9*Sin[x^3]} -{(x^2*Sin[2*x^3])/E^(3*x^3), x, 2, ((-(2/39))*Cos[2*x^3])/E^(3*x^3) - ((1/13)*Sin[2*x^3])/E^(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Hughes, Hallet, Gleason, et al Calculus, 2nd Edition*) - - -{2*x*Cos[x^2], x, 3, Sin[x^2]} -{3*x^2*Cos[7 + x^3], x, 3, Sin[7 + x^3]} -{(1 + x^2)^(-1) + Sin[x], x, 3, ArcTan[x] - Cos[x]} -{x*Sin[1 + x^2], x, 2, -Cos[1 + x^2]/2} -{x*Cos[1 + x^2], x, 2, Sin[1 + x^2]/2} -{1 + x^2*Cos[x^3], x, 3, x + Sin[x^3]/3} -{x^2*Sin[1 + x^3], x, 2, -Cos[1 + x^3]/3} -{12*x^2*Cos[x^3], x, 3, 4*Sin[x^3]} -{(1 + x)*Sin[1 + x], x, 2, -((1 + x)*Cos[1 + x]) + Sin[1 + x]} -{x^5*Cos[x^3], x, 3, Cos[x^3]/3 + (1/3)*x^3*Sin[x^3]} -{Cos[x]/E^(3*x), x, 1, ((-(3/10))*Cos[x])/E^(3*x) + ((1/10)*Sin[x])/E^(3*x)} -{x^3*Sin[x^2], x, 3, (-(1/2))*x^2*Cos[x^2] + Sin[x^2]/2} -{x^3*Cos[x^2], x, 3, Cos[x^2]/2 + (1/2)*x^2*Sin[x^2]} -{Cos[x]*Cos[2*Sin[x]], x, 2, (1/2)*Sin[2*Sin[x]]} -{(Cos[x]*Sin[x])/(1 + Cos[x]^2), x, 2, (-(1/2))*Log[1 + Cos[x]^2]} -{(1 + Cos[x])*(x + Sin[x])^3, x, 1, (x + Sin[x])^4/4} - - -(* ::Subsection::Closed:: *) -(*Spivak Calculus*) - - -{(1 + Cos[x])*Csc[x]^2, x, 3, -Cot[x] - Csc[x]} -{Sin[x]*Tan[x]^2, x, 3, Cos[x] + Sec[x]} -{E^Sin[x]*Sec[x]^2*(x*Cos[x]^3 - Sin[x]), x, If[$VersionNumber<9, -3, -2], E^Sin[x]*(-1 + x*Cos[x])*Sec[x]} - - -(* ::Subsection::Closed:: *) -(*Stewart Calculus*) - - -{x*Csc[x]^2, x, 2, -(x*Cot[x]) + Log[Sin[x]]} -{Cos[x]*Sin[Pi/6 + x], x, 3, x/4 - (1/4)*Cos[Pi/6 + 2*x]} -{x*Sin[x^2]^3, x, 3, (-(1/2))*Cos[x^2] + (1/6)*Cos[x^2]^3} -{Sin[x]^2*Tan[x], x, 3, Cos[x]^2/2 - Log[Cos[x]]} -{Cos[x]^2*Cot[x]^3, x, 4, (-(1/2))*Csc[x]^2 - 2*Log[Sin[x]] + Sin[x]^2/2} -{Sec[x]*(1 - Sin[x]), x, 2, Log[1 + Sin[x]]} -{(1 + Cos[x])*Csc[x], x, 2, Log[1 - Cos[x]]} -{Cos[x]^2*(1 - Tan[x]^2), x, 2, Cos[x]*Sin[x]} -{Csc[2*x]*(Cos[x] + Sin[x]), x, 6, (-(1/2))*ArcTanh[Cos[x]] + (1/2)*ArcTanh[Sin[x]]} -{(Cos[x]*(-3 + 2*Sin[x]))/(2 - 3*Sin[x] + Sin[x]^2), x, 2, Log[2 - 3*Sin[x] + Sin[x]^2]} -{(Cos[x]^2*Sin[x])/(5 + Cos[x]^2), x, 3, Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]] - Cos[x]} -{Cos[x]/(Sin[x] + Sin[x]^2), x, 2, Log[Sin[x]] - Log[1 + Sin[x]]} -{Cos[x]/(Sin[x] + Sin[x]^Sqrt[2]), x, 5, Log[Sin[x]] - (1 + Sqrt[2])*Log[1 + Sin[x]^(-1 + Sqrt[2])]} -{1/(2*Sin[x] + Sin[2*x]), x, 4, (1/4)*Log[Tan[x/2]] + (1/8)*Tan[x/2]^2} -{(-3 + 4*x + x^2)*Sin[2*x], x, 8, (7/4)*Cos[2*x] - 2*x*Cos[2*x] - (1/2)*x^2*Cos[2*x] + Sin[2*x] + (1/2)*x*Sin[2*x]} -{Cos[4*x]/E^(3*x), x, 1, ((-(3/25))*Cos[4*x])/E^(3*x) + ((4/25)*Sin[4*x])/E^(3*x)} -{(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]], x, 3, -2*Sqrt[1 + Sin[x]] + (2/3)*(1 + Sin[x])^(3/2)} -{x + 60*Cos[x]^5*Sin[x]^4, x, 4, x^2/2 + 12*Sin[x]^5 - (120*Sin[x]^7)/7 + (20*Sin[x]^9)/3} - - -(* ::Subsection::Closed:: *) -(*Thomas Calculus, 8th Edition*) - - -{Cos[x]*(Sec[x] + Tan[x]), x, 3, x - Cos[x]} -{Cos[x]*(Sec[x]^3 + Tan[x]), x, 5, -Cos[x] + Tan[x]} -{(-(Cot[x]*Csc[x]) + Csc[x]^2)/2, x, 6, -(Cot[x]/2) + Csc[x]/2} -{-Csc[x]^2 + Sin[2*x], x, 4, -Cos[2*x]/2 + Cot[x]} -{2*Cot[2*x] - 3*Sin[3*x], x, 3, Cos[3*x] + Log[Sin[2*x]]} -{x*Sin[2*x^2], x, 2, -Cos[2*x^2]/4} -{-(Cos[1 - x]*Sin[1 - x]*Sqrt[1 + Sin[1 - x]^2]), x, 2, (1/3)*(1 + Sin[1 - x]^2)^(3/2)} -{(Cos[1/x]*Sin[1/x])/x^2, x, 1, (-(1/2))*Sin[1/x]^2} -{Cos[(1 + 3*x)/2]*Sin[(1 + 3*x)/2]^3, x, 2, (1/6)*Sin[1/2 + (3*x)/2]^4} -{4*x*Tan[x^2], x, 3, -2*Log[Cos[x^2]]} -{x*Sec[5 - x^2], x, 2, -ArcTanh[Sin[5 - x^2]]/2} -{Csc[x^(-1)]/x^2, x, 2, ArcTanh[Cos[1/x]]} -{(Csc[x] - Sec[x])*(Cos[x] + Sin[x]), x, 4, Log[Cos[x]] + Log[Sin[x]], 2*Log[Cos[x]] + Log[Tan[x]]} -{-Cos[3*x]*Sin[2*x] + Cos[2*x]*Sin[3*x], x, 3, -Cos[x]} -{4*x*Sec[2*x]^2, x, 3, Log[Cos[2*x]] + 2*x*Tan[2*x]} -{4*Sin[x]^2*Tan[x]^2, x, 5, -6*x + 6*Tan[x] - 2*Sin[x]^2*Tan[x]} -{Cos[x]^4*Cot[x]^2, x, 5, -((15*x)/8) - (15*Cot[x])/8 + (5/8)*Cos[x]^2*Cot[x] + (1/4)*Cos[x]^4*Cot[x]} -{16*Cos[x]^2*Sin[x]^2, x, 4, 2*x + 2*Cos[x]*Sin[x] - 4*Cos[x]^3*Sin[x]} -{8*Cos[x]^2*Sin[x]^4, x, 5, x/2 + (1/2)*Cos[x]*Sin[x] - Cos[x]^3*Sin[x] - (4/3)*Cos[x]^3*Sin[x]^3} -{35*Cos[x]^3*Sin[x]^4, x, 4, 7*Sin[x]^5 - 5*Sin[x]^7} -{4*Cos[x]^4*Sin[x]^4, x, 6, (3*x)/32 + (3/32)*Cos[x]*Sin[x] + (1/16)*Cos[x]^3*Sin[x] - (1/4)*Cos[x]^5*Sin[x] - (1/2)*Cos[x]^5*Sin[x]^3} -{Cos[x]/(-Sin[x] + Sin[x]^3), x, 5, Log[Cos[x]] - Log[Sin[x]]} - - -(* ::Section::Closed:: *) -(*Problems from integration competitions*) - - -(* ::Subsection::Closed:: *) -(*MIT Integration Competition*) - - -{-1 + 2*Cos[x]^2 + Cos[x]*Sin[x], x, 5, Cos[x]*Sin[x] + Sin[x]^2/2} - - -(* ::Subsection::Closed:: *) -(*North Texas University Integration Competition*) - - -{Cos[x]^2 + Sin[x]^2, x, 5, x} -{-Cos[x]^2 + Sin[x]^2, x, 5, -(Cos[x]*Sin[x])} -{2^Sin[x]*Cos[x], x, 2, 2^Sin[x]/Log[2]} - - -(* ::Subsection::Closed:: *) -(*University of Wisconsin Integration Competition*) - - -{Tan[x]^3 + Tan[x]^5, x, 6, Tan[x]^4/4} -{x*Sec[x]*(2 + x*Tan[x]), x, 13, x^2*Sec[x]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving trig functions*) - - -{(Cot[Sqrt[x]]*Csc[Sqrt[x]])/Sqrt[x], x, 3, -2*Csc[Sqrt[x]]} -{(Cos[Sqrt[x]]*Sin[Sqrt[x]])/Sqrt[x], x, 1, Sin[Sqrt[x]]^2} -{(Sec[Sqrt[x]]*Tan[Sqrt[x]])/Sqrt[x], x, 3, 2*Sec[Sqrt[x]]} - - -{Sin[x]^2/(a + b*Sin[2*x]), x, 9, ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) - Log[a + b*Sin[2*x]]/(4*b), ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) - Log[Cos[x]]/(2*b) - Log[a + 2*b*Tan[x] + a*Tan[x]^2]/(4*b)} -{Cos[x]^2/(a + b*Sin[2*x]), x, 8, ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) + Log[a + b*Sin[2*x]]/(4*b), ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) + Log[Cos[x]]/(2*b) + Log[a + 2*b*Tan[x] + a*Tan[x]^2]/(4*b)} - -{Sin[x]^2/(a + b*Cos[2*x]), x, 4, -(x/(2*b)) + (Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x])/Sqrt[a + b]])/(2*Sqrt[a - b]*b)} -{Cos[x]^2/(a + b*Cos[2*x]), x, 4, x/(2*b) - (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[x])/Sqrt[a + b]])/(2*b*Sqrt[a + b])} - - -{Tan[c + d*x]/Sqrt[a*Sin[c + d*x]^2], x, 3, ArcTanh[Sqrt[a*Sin[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d)} -{Cot[c + d*x]/Sqrt[a*Cos[c + d*x]^2], x, 3, -(ArcTanh[Sqrt[a*Cos[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d))} - - -{(x*Cos[x^2])/Sqrt[Sin[x^2]], x, 1, Sqrt[Sin[x^2]]} - - -{Cos[x]/Sqrt[1 - Cos[2*x]], x, 4, (Log[Sin[x]]*Sin[x])/(Sqrt[2]*Sqrt[Sin[x]^2])} - - -{Cos[Log[x]]^2*Sin[Log[x]]^2/x, x, 4, Log[x]/8 + (1/8)*Cos[Log[x]]*Sin[Log[x]] - (1/4)*Cos[Log[x]]^3*Sin[Log[x]]} - - -{Sin[x]^3/(Cos[x]^3 + Sin[x]^3), x, 7, x/2 - (1/6)*Log[Cos[x] + Sin[x]] + (1/3)*Log[2 - Sin[2*x]], x/2 + (1/2)*Log[Cos[x]] - (1/6)*Log[1 + Tan[x]] + (1/3)*Log[1 - Tan[x] + Tan[x]^2]} -{Cos[x]^3/(Cos[x]^3 + Sin[x]^3), x, 7, x/2 + (1/6)*Log[Cos[x] + Sin[x]] - (1/3)*Log[2 - Sin[2*x]], x/2 - (1/2)*Log[Cos[x]] + (1/6)*Log[1 + Tan[x]] - (1/3)*Log[1 - Tan[x] + Tan[x]^2]} - - -{Sec[x]/(-5 + Cos[x]^2 + 4*Sin[x]), x, 4, (-(4/9))*Log[2 - Sin[x]] + (1/2)*Log[1 - Sin[x]] - (1/18)*Log[1 + Sin[x]] + 1/(3*(2 - Sin[x]))} - - -(* Nonidempotent expansion results in infinite recursion: *) -(* {(x*Cos[x] - Sin[x])/(x - Sin[x])^2, x, -7, x/(x - Sin[x])} *) -(* {x/(x - Cos[x])^2, x, 1, Unintegrable[x/(x - Cos[x])^2, x]} *) -(* {Cos[x]/(x - Cos[x])^2, x, 1, Unintegrable[Cos[x]/(x - Cos[x])^2, x]} *) -(* {(Cos[x] + x*Sin[x])/(x - Cos[x])^2, x, 0, -x/(x - Cos[x])} *) - - -{1/(Cos[x]^(3/2)*Sqrt[3*Cos[x] + Sin[x]]), x, -5, (2*Sqrt[3*Cos[x] + Sin[x]])/Sqrt[Cos[x]]} -{(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2), x, -1, -Log[Sin[x]] + 2*Log[-Sqrt[Cos[x]] + Sqrt[Cos[x] + Sin[x]]] + (2*Sqrt[Cos[x] + Sin[x]])/Sqrt[Cos[x]]} -{(Cos[x] + Sin[x])/Sqrt[1 + Sin[2*x]], x, -17, (x*Sqrt[1 + Sin[2*x]])/(Cos[x] + Sin[x])} -{Sec[x]*Sqrt[Sec[x] + Tan[x]], x, 4, 2*Sqrt[Sec[x]*(1 + Sin[x])]} - -{Sec[x]*Sqrt[4 + 3*Sec[x]]*Tan[x], x, 2, (2*(4 + 3*Sec[x])^(3/2))/9} -{Sec[x]*Sqrt[1 + Sec[x]]*Tan[x]^3, x, 6, (-(4/5))*(1 + Sec[x])^(5/2) + (2/7)*(1 + Sec[x])^(7/2)} -{Csc[x]*Sqrt[1 + Csc[x]]*Cot[x]^3, x, 6, (4/5)*(1 + Csc[x])^(5/2) - (2/7)*(1 + Csc[x])^(7/2)} - -{Sqrt[Csc[x]]*(x*Cos[x] - 4*Sec[x]*Tan[x]), x, 8, (2*x)/Sqrt[Csc[x]] - (4*Sec[x])/Csc[x]^(3/2)} - - -{Cot[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3, x, 10, (-(35/16))*Sqrt[Cot[x]^2] + (35/48)*Cos[x]^2*Sqrt[Cot[x]^2] + (7/24)*Cos[x]^4*Sqrt[Cot[x]^2] + (1/6)*Cos[x]^6*Sqrt[Cot[x]^2] - (35/16)*x*Sqrt[Cot[x]^2]*Tan[x], (35/16)*ArcTan[Sqrt[-1 + Csc[x]^2]] - (35/16)*Sqrt[-1 + Csc[x]^2] + (35/48)*(-1 + Csc[x]^2)^(3/2)*Sin[x]^2 + (7/24)*(-1 + Csc[x]^2)^(5/2)*Sin[x]^4 + (1/6)*(-1 + Csc[x]^2)^(7/2)*Sin[x]^6} -{Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3, x, 7, Sqrt[Cot[x]^2]*Sin[x] + (1/3)*Cos[x]^2*Sqrt[Cot[x]^2]*Sin[x] + (1/5)*Cos[x]^4*Sqrt[Cot[x]^2]*Sin[x] + (1/7)*Cos[x]^6*Sqrt[Cot[x]^2]*Sin[x] - ArcTanh[Cos[x]]*Sqrt[Cot[x]^2]*Tan[x]} - - -{(x^1*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2], x, 6, -((2*x*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]) + (I*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (I*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]} -{(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2], x, 8, -((2*x^2*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]) + (2*I*x*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (2*I*x*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (2*PolyLog[3, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (2*PolyLog[3, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]} -{(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2], x, 10, -((2*x^3*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]) + (3*I*x^2*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (3*I*x^2*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (6*x*PolyLog[3, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (6*x*PolyLog[3, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (6*I*PolyLog[4, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (6*I*PolyLog[4, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]} - - -{(x^1*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4], x, 5, -((I*x^2*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4])) + (x*Log[1 - E^(2*I*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - (I*PolyLog[2, E^(2*I*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4])} -{(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4], x, 6, -((I*x^3*Sec[x]^2)/(3*Sqrt[a*Sec[x]^4])) + (x^2*Log[1 - E^(2*I*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - (I*x*PolyLog[2, E^(2*I*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (PolyLog[3, E^(2*I*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4])} -{(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4], x, 7, -((I*x^4*Sec[x]^2)/(4*Sqrt[a*Sec[x]^4])) + (x^3*Log[1 - E^(2*I*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - (3*I*x^2*PolyLog[2, E^(2*I*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4]) + (3*x*PolyLog[3, E^(2*I*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4]) + (3*I*PolyLog[4, E^(2*I*x)]*Sec[x]^2)/(4*Sqrt[a*Sec[x]^4])} - - -{(x^1*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^2], x, 10, x*Sqrt[a*Sec[x]^2] - 2*x*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] + I*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - I*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2]} -{(x^2*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^2], x, 17, x^2*Sqrt[a*Sec[x]^2] + 4*I*x*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^2*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] + 2*I*x*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - 2*I*Cos[x]*PolyLog[2, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] + 2*I*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - 2*I*x*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2] - 2*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 2*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]^2]} -{(x^3*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^2], x, 21, x^3*Sqrt[a*Sec[x]^2] + 6*I*x^2*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^3*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] + 3*I*x^2*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*I*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*I*x*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - 3*I*x^2*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*x*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*Cos[x]*PolyLog[3, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*Cos[x]*PolyLog[3, I*E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*x*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*I*Cos[x]*PolyLog[4, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*I*Cos[x]*PolyLog[4, E^(I*x)]*Sqrt[a*Sec[x]^2]} - - -{(x^1*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^4], x, 12, (1/2)*x*Cos[x]^2*Sqrt[a*Sec[x]^4] - 2*x*ArcTanh[E^(2*I*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] + (1/2)*I*Cos[x]^2*PolyLog[2, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (1/2)*I*Cos[x]^2*PolyLog[2, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (1/2)*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (1/2)*x*Sqrt[a*Sec[x]^4]*Sin[x]^2} -{(x^2*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^4], x, 16, (1/2)*x^2*Cos[x]^2*Sqrt[a*Sec[x]^4] - 2*x^2*ArcTanh[E^(2*I*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] - Cos[x]^2*Log[Cos[x]]*Sqrt[a*Sec[x]^4] + I*x*Cos[x]^2*PolyLog[2, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] - I*x*Cos[x]^2*PolyLog[2, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (1/2)*Cos[x]^2*PolyLog[3, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] + (1/2)*Cos[x]^2*PolyLog[3, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - x*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (1/2)*x^2*Sqrt[a*Sec[x]^4]*Sin[x]^2} -{(x^3*Csc[x]*Sec[x])*Sqrt[a*Sec[x]^4], x, 21, (3/2)*I*x^2*Cos[x]^2*Sqrt[a*Sec[x]^4] + (1/2)*x^3*Cos[x]^2*Sqrt[a*Sec[x]^4] - 2*x^3*ArcTanh[E^(2*I*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] - 3*x*Cos[x]^2*Log[1 + E^(2*I*x)]*Sqrt[a*Sec[x]^4] + (3/2)*I*Cos[x]^2*PolyLog[2, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] + (3/2)*I*x^2*Cos[x]^2*PolyLog[2, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (3/2)*I*x^2*Cos[x]^2*PolyLog[2, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (3/2)*x*Cos[x]^2*PolyLog[3, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] + (3/2)*x*Cos[x]^2*PolyLog[3, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (3/4)*I*Cos[x]^2*PolyLog[4, -E^(2*I*x)]*Sqrt[a*Sec[x]^4] + (3/4)*I*Cos[x]^2*PolyLog[4, E^(2*I*x)]*Sqrt[a*Sec[x]^4] - (3/2)*x^2*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (1/2)*x^3*Sqrt[a*Sec[x]^4]*Sin[x]^2} - - -{Sin[x]*Sin[2*x]*Sin[3*x], x, 5, (-(1/8))*Cos[2*x] - (1/16)*Cos[4*x] + (1/24)*Cos[6*x]} -{Cos[x]*Cos[2*x]*Cos[3*x], x, 5, x/4 + (1/8)*Sin[2*x] + (1/16)*Sin[4*x] + (1/24)*Sin[6*x]} -{Cos[x]*Sin[2*x]*Sin[3*x], x, 5, x/4 + (1/8)*Sin[2*x] - (1/16)*Sin[4*x] - (1/24)*Sin[6*x]} -{Cos[2*x]*Cos[3*x]*Sin[x], x, 5, (-(1/8))*Cos[2*x] + (1/16)*Cos[4*x] - (1/24)*Cos[6*x]} - - -{x*Sin[x^2], x, 2, -Cos[x^2]/2} -{(-Cos[x] + Sin[x])*(Cos[x] + Sin[x])^5, x, 1, -(Cos[x] + Sin[x])^6/6} -{2*x*Sec[x]^2*Tan[x], x, 4, x*Sec[x]^2 - Tan[x]} -{(1 + Cos[x]^2)/(1 + Cos[2*x]), x, 3, x/2 + Tan[x]/2} - - -{Sin[x]/(Cos[x]^3 - Cos[x]^5), x, 4, Log[Tan[x]] + Tan[x]^2/2, -Log[Cos[x]] + Log[Sin[x]] + Sec[x]^2/2} -{Sec[x]*(5 - 11*Sec[x]^5)^2*Tan[x], x, 3, 25*Sec[x] - (55*Sec[x]^6)/3 + 11*Sec[x]^11} -{Sin[5*x]^3*Tan[5*x]^3, x, 5, (-(1/2))*ArcTanh[Sin[5*x]] + (1/2)*Sin[5*x] + (1/6)*Sin[5*x]^3 + (1/10)*Sin[5*x]^3*Tan[5*x]^2} -{Sin[5*x]^3*Tan[5*x]^4, x, 3, (-(3/5))*Cos[5*x] + (1/15)*Cos[5*x]^3 - (3/5)*Sec[5*x] + (1/15)*Sec[5*x]^3} -{Sin[6*x]^5*Tan[6*x]^3, x, 5, (-(7/12))*ArcTanh[Sin[6*x]] + (7/12)*Sin[6*x] + (7/36)*Sin[6*x]^3 + (7/60)*Sin[6*x]^5 + (1/12)*Sin[6*x]^5*Tan[6*x]^2} -{(-1 + Sec[2*x]^2)^3*Sin[2*x], x, 4, (1/2)*Cos[2*x] + (3/2)*Sec[2*x] - (1/2)*Sec[2*x]^3 + (1/10)*Sec[2*x]^5} -{Sin[x]*Tan[x]^5, x, 5, (15/8)*ArcTanh[Sin[x]] - (15*Sin[x])/8 - (5/8)*Sin[x]*Tan[x]^2 + (1/4)*Sin[x]*Tan[x]^4} -{Cos[2*x]^5*Cot[2*x]^4, x, 3, 2*Csc[2*x] - (1/6)*Csc[2*x]^3 + 3*Sin[2*x] - (2/3)*Sin[2*x]^3 + (1/10)*Sin[2*x]^5} - -{Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5, x, 5, (-(28/3))*Csc[3*x] + (8/9)*Csc[3*x]^3 - (1/15)*Csc[3*x]^5 - (56/3)*Sin[3*x] + (70/9)*Sin[3*x]^3 - (56/15)*Sin[3*x]^5 + (4/3)*Sin[3*x]^7 - (8/27)*Sin[3*x]^9 + (1/33)*Sin[3*x]^11} -{Cot[2*x]*(-1 + Csc[2*x]^2)^2*(1 - Sin[2*x]^2)^2, x, 5, Csc[2*x]^2 - (1/8)*Csc[2*x]^4 + 3*Log[Sin[2*x]] - Sin[2*x]^2 + (1/8)*Sin[2*x]^4} -{Cos[2*x]*(-1 + Csc[2*x]^2)^4*(1 - Sin[2*x]^2)^2, x, 5, 10*Csc[2*x] - (5/2)*Csc[2*x]^3 + (3/5)*Csc[2*x]^5 - (1/14)*Csc[2*x]^7 + (15/2)*Sin[2*x] - Sin[2*x]^3 + (1/10)*Sin[2*x]^5} -{Cot[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^2, x, 5, (-(5/3))*Csc[3*x]^2 + (5/12)*Csc[3*x]^4 - (1/18)*Csc[3*x]^6 - (10/3)*Log[Sin[3*x]] + (5/6)*Sin[3*x]^2 - (1/12)*Sin[3*x]^4} -{(1 + Cot[9*x]^2)^2*(1 + Tan[9*x]^2)^3, x, 5, (-(4/9))*Cot[9*x] - (1/27)*Cot[9*x]^3 + (2/3)*Tan[9*x] + (4/27)*Tan[9*x]^3 + (1/45)*Tan[9*x]^5} -{(Cos[x]*(9 - 7*Sin[x]^3)^2)/(1 - Sin[x]^2), x, 7, -2*Log[1 - Sin[x]] + 128*Log[1 + Sin[x]] - 49*Sin[x] + 63*Sin[x]^2 - (49*Sin[x]^3)/3 - (49*Sin[x]^5)/5} - -{Cos[2*x]^4*Cot[2*x]^5, x, 4, Csc[2*x]^2 - (1/8)*Csc[2*x]^4 + 3*Log[Sin[2*x]] - Sin[2*x]^2 + (1/8)*Sin[2*x]^4} -{(Sec[x]*Tan[x]^2)/(4 + 3*Sec[x]), x, 7, (-(4/9))*ArcTanh[Sin[x]] - (1/9)*Sqrt[7]*Log[Sqrt[7]*Cos[x/2] - Sin[x/2]] + (1/9)*Sqrt[7]*Log[Sqrt[7]*Cos[x/2] + Sin[x/2]] + Tan[x]/3} -{x*Sec[1 + x]*Tan[1 + x], x, 2, -ArcTanh[Sin[1 + x]] + x*Sec[1 + x]} -{Sin[2*x]/Sqrt[9 - Sin[x]^2], x, 3, -2*Sqrt[9 - Sin[x]^2]} -{Sin[2*x]/Sqrt[9 - Cos[x]^4], x, 5, -ArcSin[Cos[x]^2/3]} -{Cos[x^(-1)]/x^5, x, 5, 6*Cos[1/x] - (3*Cos[1/x])/x^2 - Sin[1/x]/x^3 + (6*Sin[1/x])/x} -{Cos[1 + x]^3*Sin[1 + x]^3, x, 3, (1/4)*Sin[1 + x]^4 - (1/6)*Sin[1 + x]^6} -{(1 + 2*x)^3*Sin[1 + 2*x]^2, x, 4, -((3*x)/4) - (3*x^2)/4 + (1/16)*(1 + 2*x)^4 + (3/8)*(1 + 2*x)*Cos[1 + 2*x]*Sin[1 + 2*x] - (1/4)*(1 + 2*x)^3*Cos[1 + 2*x]*Sin[1 + 2*x] - (3/16)*Sin[1 + 2*x]^2 + (3/8)*(1 + 2*x)^2*Sin[1 + 2*x]^2} -{(-1 + Sec[x])/(1 - Tan[x]), x, 6, -(x/2) + ArcTanh[(Cos[x]*(1 + Tan[x]))/Sqrt[2]]/Sqrt[2] + (1/2)*Log[Cos[x] - Sin[x]]} -{x^2*Cos[3*x]*Cos[5*x], x, 8, (1/4)*x*Cos[2*x] + (1/64)*x*Cos[8*x] - (1/8)*Sin[2*x] + (1/4)*x^2*Sin[2*x] - (1/512)*Sin[8*x] + (1/16)*x^2*Sin[8*x]} - - -(* Unfortunately the simpler antiderivative Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Cos[x]]*Sqrt[Sin[x]])/(Cos[x] - Sin[x])] is unnecessarily discontinuous. *) -{(Cos[x] + Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]]), x, -22, (-Sqrt[2])*ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]} - - -{Sec[x]^2*(1 + Sin[x]), x, 3, Sec[x] + Tan[x]} - -{10*x^9*Cos[x^5*Log[x]] - x^10*(x^4 + 5*x^4*Log[x])*Sin[x^5*Log[x]], x, -4, x^10*Cos[x^5*Log[x]]} -{Cos[x/2]^2*Tan[Pi/4 + x/2], x, -1, x/2 - Cos[x]/2 - Log[Cos[Pi/4 + x/2]]} - -{(2 + 3*x)^2*Sin[x]^3, x, 6, 14*Cos[x] - (2/3)*(2 + 3*x)^2*Cos[x] - (2*Cos[x]^3)/3 + 4*(2 + 3*x)*Sin[x] - (1/3)*(2 + 3*x)^2*Cos[x]*Sin[x]^2 + (2/3)*(2 + 3*x)*Sin[x]^3} -{Sec[x]^(1 + m)*Sin[x], x, 2, Sec[x]^m/m} -{Cos[a + b*x]^n*Sin[a + b*x]^(-2 - n), x, 1, -((Cos[a + b*x]^(1 + n)*Sin[a + b*x]^(-1 - n))/(b*(1 + n)))} -{1/(Sec[x] + Sin[x]*Tan[x]), x, 3, ArcTan[Sin[x]]} -{(a + b*x + c*x^2)*Sin[x], x, 8, (-a)*Cos[x] + 2*c*Cos[x] - b*x*Cos[x] - c*x^2*Cos[x] + b*Sin[x] + 2*c*x*Sin[x]} -{Sin[x^5]/x, x, 1, SinIntegral[x^5]/5} -{Sin[2^x]/(1 + 2^x), x, 7, (CosIntegral[1 + 2^x]*Sin[1])/Log[2] + SinIntegral[2^x]/Log[2] - (Cos[1]*SinIntegral[1 + 2^x])/Log[2]} - -{x*Cos[2*x^2]*Sin[2*x^2]^(3/4), x, 1, Sin[2*x^2]^(7/4)/7} -{x*Sec[x^2]^2*Tan[x^2]^2, x, 1, Tan[x^2]^3/6} -{x^2*Cos[a + b*x^3]^7*Sin[a + b*x^3], x, 1, -Cos[a + b*x^3]^8/(24*b)} -{x^5*Cos[a + b*x^3]^7*Sin[a + b*x^3], x, 7, (35*x^3)/(3072*b) - (x^3*Cos[a + b*x^3]^8)/(24*b) + (35*Cos[a + b*x^3]*Sin[a + b*x^3])/(3072*b^2) + (35*Cos[a + b*x^3]^3*Sin[a + b*x^3])/(4608*b^2) + (7*Cos[a + b*x^3]^5*Sin[a + b*x^3])/(1152*b^2) + (Cos[a + b*x^3]^7*Sin[a + b*x^3])/(192*b^2)} -{x^5*Sec[a + b*x^3]^7*Tan[a + b*x^3], x, 6, -((5*ArcTanh[Sin[a + b*x^3]])/(336*b^2)) + (x^3*Sec[a + b*x^3]^7)/(21*b) - (5*Sec[a + b*x^3]*Tan[a + b*x^3])/(336*b^2) - (5*Sec[a + b*x^3]^3*Tan[a + b*x^3])/(504*b^2) - (Sec[a + b*x^3]^5*Tan[a + b*x^3])/(126*b^2)} - -{Sec[x^(-1)]^2/x^2, x, 3, -Tan[x^(-1)]} -{3*x^2*Cos[x^3], x, 3, Sin[x^3]} - -{(1 + 2*x)*Sec[1 + 2*x]^2, x, 2, (1/2)*Log[Cos[1 + 2*x]] + (1/2)*(1 + 2*x)*Tan[1 + 2*x]} - - -(* Problems requiring simplification of irreducible integrands *) -{(x^2*Cos[a + b*x])/Sqrt[3*Sin[a + b*x] + x^3] + x^4/(Sqrt[x^3 + 3*Sin[a + b*x]]*b) + (4*x*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b), x, -1, (2*x^2*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b)} -{x^2*(Cos[a + b*x]/Sqrt[3*Sin[a + b*x] + x^3]), x, 0, CannotIntegrate[(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]], x]} - - -{(Cos[x] + Sin[x])/(E^(-x) + Sin[x]), x, -5, Log[1 + E^x*Sin[x]]} - - -{Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3)^1, x, 7, (3*b*x)/8 - (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3)^2, x, 9, (5*a*b*x)/8 - ((a^2 + b^2)*Cos[c + d*x])/d + ((2*a^2 + 3*b^2)*Cos[c + d*x]^3)/(3*d) - ((a^2 + 3*b^2)*Cos[c + d*x]^5)/(5*d) + (b^2*Cos[c + d*x]^7)/(7*d) - (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (5*a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^5)/(3*d)} - - -{Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3)^1, x, 7, (1/8)*(4*a + 3*c)*x - (b*Cos[c + d*x])/d + (b*Cos[c + d*x]^3)/(3*d) - ((4*a + 3*c)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (c*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)} -{Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3)^2, x, 16, (3*a*b*x)/4 + (5*b*c*x)/8 - (a^2*Cos[c + d*x])/d - (c^2*Cos[c + d*x])/d - ((b^2 + 2*a*c)*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (c^2*Cos[c + d*x]^3)/d + (2*(b^2 + 2*a*c)*Cos[c + d*x]^3)/(3*d) - (3*c^2*Cos[c + d*x]^5)/(5*d) - ((b^2 + 2*a*c)*Cos[c + d*x]^5)/(5*d) + (c^2*Cos[c + d*x]^7)/(7*d) - (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (5*b*c*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(2*d) - (5*b*c*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) - (b*c*Cos[c + d*x]*Sin[c + d*x]^5)/(3*d)} - - -{Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x])^1, x, 7, (c*x)/2 - (a*Cos[c + d*x])/d + (2*b*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/d - (c*Cos[c + d*x]*Sin[c + d*x])/(2*d)} -{Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x])^2, x, 11, b^2*x + a*c*x - (a^2*Cos[c + d*x])/d - (c^2*Cos[c + d*x])/d + (c^2*Cos[c + d*x]^3)/(3*d) + (4*a*b*EllipticE[(1/2)*(c - Pi/2 + d*x), 2])/d + (4*b*c*EllipticF[(1/2)*(c - Pi/2 + d*x), 2])/(3*d) - (4*b*c*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(3*d) - (a*c*Cos[c + d*x]*Sin[c + d*x])/d} - - -{f^(a + b*x)*(Cos[c + d*x] + I*Sin[c + d*x])^n, x, 4, ((E^(I*(c + d*x)))^n*f^(a + b*x))/(I*d*n + b*Log[f])} -{f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n, x, 4, -(((E^((-I)*(c + d*x)))^n*f^(a + b*x))/(I*d*n - b*Log[f]))} - - -{(Cos[a + b*x]^5 - Sin[a + b*x]^5)/(Cos[a + b*x]^5 + Sin[a + b*x]^5), x, 7, Log[Cos[a + b*x]]/b + Log[1 + Tan[a + b*x]]/(5*b) - (4*Log[2 - (1 - Sqrt[5])*Tan[a + b*x] + 2*Tan[a + b*x]^2])/(5*(1 - Sqrt[5])*b) - (4*Log[2 - (1 + Sqrt[5])*Tan[a + b*x] + 2*Tan[a + b*x]^2])/(5*(1 + Sqrt[5])*b)} -{(Cos[a + b*x]^4 - Sin[a + b*x]^4)/(Cos[a + b*x]^4 + Sin[a + b*x]^4), x, 4, -(Log[1 - Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b)) + Log[1 + Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b)} -{(Cos[a + b*x]^3 - Sin[a + b*x]^3)/(Cos[a + b*x]^3 + Sin[a + b*x]^3), x, 5, -(Log[Cos[a + b*x]]/b) + Log[1 + Tan[a + b*x]]/(3*b) - (2*Log[1 - Tan[a + b*x] + Tan[a + b*x]^2])/(3*b)} -{(Cos[a + b*x]^2 - Sin[a + b*x]^2)/(Cos[a + b*x]^2 + Sin[a + b*x]^2), x, 6, (Cos[a + b*x]*Sin[a + b*x])/b} -{(Cos[a + b*x]^1 - Sin[a + b*x]^1)/(Cos[a + b*x]^1 + Sin[a + b*x]^1), x, 1, Log[Cos[a + b*x] + Sin[a + b*x]]/b} -{(Sec[a + b*x]^1 - Csc[a + b*x]^1)/(Sec[a + b*x]^1 + Csc[a + b*x]^1), x, 4, -(Log[Cos[a + b*x] + Sin[a + b*x]]/b)} -{(Sec[a + b*x]^2 - Csc[a + b*x]^2)/(Sec[a + b*x]^2 + Csc[a + b*x]^2), x, 2, -((Cos[a + b*x]*Sin[a + b*x])/b)} -{(Sec[a + b*x]^3 - Csc[a + b*x]^3)/(Sec[a + b*x]^3 + Csc[a + b*x]^3), x, 5, Log[Cos[a + b*x]]/b - Log[1 + Tan[a + b*x]]/(3*b) + (2*Log[1 - Tan[a + b*x] + Tan[a + b*x]^2])/(3*b)} -{(Sec[a + b*x]^4 - Csc[a + b*x]^4)/(Sec[a + b*x]^4 + Csc[a + b*x]^4), x, 4, Log[1 - Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b) - Log[1 + Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m deleted file mode 100644 index 8f2183a..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.2 (d x)^m (a+b arcsin(c x))^n.m +++ /dev/null @@ -1,401 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcSin[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b x)^m ArcSin[a x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSin[a x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcSin[a*x], x, 4, Sqrt[1 - a^2*x^2]/(5*a^5) - (2*(1 - a^2*x^2)^(3/2))/(15*a^5) + (1 - a^2*x^2)^(5/2)/(25*a^5) + (1/5)*x^5*ArcSin[a*x]} -{x^3*ArcSin[a*x], x, 4, (3*x*Sqrt[1 - a^2*x^2])/(32*a^3) + (x^3*Sqrt[1 - a^2*x^2])/(16*a) - (3*ArcSin[a*x])/(32*a^4) + (1/4)*x^4*ArcSin[a*x]} -{x^2*ArcSin[a*x], x, 4, Sqrt[1 - a^2*x^2]/(3*a^3) - (1 - a^2*x^2)^(3/2)/(9*a^3) + (1/3)*x^3*ArcSin[a*x]} -{x^1*ArcSin[a*x], x, 3, (x*Sqrt[1 - a^2*x^2])/(4*a) - ArcSin[a*x]/(4*a^2) + (1/2)*x^2*ArcSin[a*x]} -{x^0*ArcSin[a*x], x, 2, Sqrt[1 - a^2*x^2]/a + x*ArcSin[a*x]} -{ArcSin[a*x]/x^1, x, 5, (-(1/2))*I*ArcSin[a*x]^2 + ArcSin[a*x]*Log[1 - E^(2*I*ArcSin[a*x])] - (1/2)*I*PolyLog[2, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]/x^2, x, 4, -(ArcSin[a*x]/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{ArcSin[a*x]/x^3, x, 2, -((a*Sqrt[1 - a^2*x^2])/(2*x)) - ArcSin[a*x]/(2*x^2)} -{ArcSin[a*x]/x^4, x, 5, -((a*Sqrt[1 - a^2*x^2])/(6*x^2)) - ArcSin[a*x]/(3*x^3) - (1/6)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{ArcSin[a*x]/x^5, x, 3, -((a*Sqrt[1 - a^2*x^2])/(12*x^3)) - (a^3*Sqrt[1 - a^2*x^2])/(6*x) - ArcSin[a*x]/(4*x^4)} -{ArcSin[a*x]/x^6, x, 6, -((a*Sqrt[1 - a^2*x^2])/(20*x^4)) - (3*a^3*Sqrt[1 - a^2*x^2])/(40*x^2) - ArcSin[a*x]/(5*x^5) - (3/40)*a^5*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{x^4*ArcSin[a*x]^2, x, 7, -((16*x)/(75*a^4)) - (8*x^3)/(225*a^2) - (2*x^5)/125 + (16*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(75*a^5) + (8*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(75*a^3) + (2*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(25*a) + (1/5)*x^5*ArcSin[a*x]^2} -{x^3*ArcSin[a*x]^2, x, 6, -((3*x^2)/(32*a^2)) - x^4/32 + (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(16*a^3) + (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(8*a) - (3*ArcSin[a*x]^2)/(32*a^4) + (1/4)*x^4*ArcSin[a*x]^2} -{x^2*ArcSin[a*x]^2, x, 5, -((4*x)/(9*a^2)) - (2*x^3)/27 + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^3) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a) + (1/3)*x^3*ArcSin[a*x]^2} -{x^1*ArcSin[a*x]^2, x, 4, -(x^2/4) + (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a) - ArcSin[a*x]^2/(4*a^2) + (1/2)*x^2*ArcSin[a*x]^2} -{x^0*ArcSin[a*x]^2, x, 3, -2*x + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a + x*ArcSin[a*x]^2} -{ArcSin[a*x]^2/x^1, x, 6, (-(1/3))*I*ArcSin[a*x]^3 + ArcSin[a*x]^2*Log[1 - E^(2*I*ArcSin[a*x])] - I*ArcSin[a*x]*PolyLog[2, E^(2*I*ArcSin[a*x])] + (1/2)*PolyLog[3, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^2/x^2, x, 7, -(ArcSin[a*x]^2/x) - 4*a*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + 2*I*a*PolyLog[2, -E^(I*ArcSin[a*x])] - 2*I*a*PolyLog[2, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^2/x^3, x, 3, -((a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) - ArcSin[a*x]^2/(2*x^2) + a^2*Log[x]} -{ArcSin[a*x]^2/x^4, x, 9, -(a^2/(3*x)) - (a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*x^2) - ArcSin[a*x]^2/(3*x^3) - (2/3)*a^3*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + (1/3)*I*a^3*PolyLog[2, -E^(I*ArcSin[a*x])] - (1/3)*I*a^3*PolyLog[2, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^2/x^5, x, 5, -(a^2/(12*x^2)) - (a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(6*x^3) - (a^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*x) - ArcSin[a*x]^2/(4*x^4) + (1/3)*a^4*Log[x]} - - -{x^4*ArcSin[a*x]^3, x, 14, -((298*Sqrt[1 - a^2*x^2])/(375*a^5)) + (76*(1 - a^2*x^2)^(3/2))/(1125*a^5) - (6*(1 - a^2*x^2)^(5/2))/(625*a^5) - (16*x*ArcSin[a*x])/(25*a^4) - (8*x^3*ArcSin[a*x])/(75*a^2) - (6/125)*x^5*ArcSin[a*x] + (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(25*a^5) + (4*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(25*a^3) + (3*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(25*a) + (1/5)*x^5*ArcSin[a*x]^3} -{x^3*ArcSin[a*x]^3, x, 11, -((45*x*Sqrt[1 - a^2*x^2])/(256*a^3)) - (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) + (45*ArcSin[a*x])/(256*a^4) - (9*x^2*ArcSin[a*x])/(32*a^2) - (3/32)*x^4*ArcSin[a*x] + (9*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(32*a^3) + (3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(16*a) - (3*ArcSin[a*x]^3)/(32*a^4) + (1/4)*x^4*ArcSin[a*x]^3} -{x^2*ArcSin[a*x]^3, x, 9, -((14*Sqrt[1 - a^2*x^2])/(9*a^3)) + (2*(1 - a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSin[a*x])/(3*a^2) - (2/9)*x^3*ArcSin[a*x] + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a) + (1/3)*x^3*ArcSin[a*x]^3} -{x^1*ArcSin[a*x]^3, x, 6, -((3*x*Sqrt[1 - a^2*x^2])/(8*a)) + (3*ArcSin[a*x])/(8*a^2) - (3/4)*x^2*ArcSin[a*x] + (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(4*a) - ArcSin[a*x]^3/(4*a^2) + (1/2)*x^2*ArcSin[a*x]^3} -{x^0*ArcSin[a*x]^3, x, 4, -((6*Sqrt[1 - a^2*x^2])/a) - 6*x*ArcSin[a*x] + (3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + x*ArcSin[a*x]^3} -{ArcSin[a*x]^3/x^1, x, 7, (-(1/4))*I*ArcSin[a*x]^4 + ArcSin[a*x]^3*Log[1 - E^(2*I*ArcSin[a*x])] - (3/2)*I*ArcSin[a*x]^2*PolyLog[2, E^(2*I*ArcSin[a*x])] + (3/2)*ArcSin[a*x]*PolyLog[3, E^(2*I*ArcSin[a*x])] + (3/4)*I*PolyLog[4, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^3/x^2, x, 9, -(ArcSin[a*x]^3/x) - 6*a*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] + 6*I*a*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - 6*I*a*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - 6*a*PolyLog[3, -E^(I*ArcSin[a*x])] + 6*a*PolyLog[3, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^3/x^3, x, 7, (-(3/2))*I*a^2*ArcSin[a*x]^2 - (3*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*x) - ArcSin[a*x]^3/(2*x^2) + 3*a^2*ArcSin[a*x]*Log[1 - E^(2*I*ArcSin[a*x])] - (3/2)*I*a^2*PolyLog[2, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^3/x^4, x, 14, -((a^2*ArcSin[a*x])/x) - (a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*x^2) - ArcSin[a*x]^3/(3*x^3) - a^3*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] - a^3*ArcTanh[Sqrt[1 - a^2*x^2]] + I*a^3*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - I*a^3*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - a^3*PolyLog[3, -E^(I*ArcSin[a*x])] + a^3*PolyLog[3, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^3/x^5, x, 10, -((a^3*Sqrt[1 - a^2*x^2])/(4*x)) - (a^2*ArcSin[a*x])/(4*x^2) - (1/2)*I*a^4*ArcSin[a*x]^2 - (a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(4*x^3) - (a^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*x) - ArcSin[a*x]^3/(4*x^4) + a^4*ArcSin[a*x]*Log[1 - E^(2*I*ArcSin[a*x])] - (1/2)*I*a^4*PolyLog[2, E^(2*I*ArcSin[a*x])]} - - -{x^5*ArcSin[a*x]^4, x, 23, (245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(576*a^5) - (65*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(864*a^3) - (x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(54*a) + (245*ArcSin[a*x]^2)/(1152*a^6) - (5*x^2*ArcSin[a*x]^2)/(16*a^4) - (5*x^4*ArcSin[a*x]^2)/(48*a^2) - (1/18)*x^6*ArcSin[a*x]^2 + (5*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(36*a^3) + (x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(9*a) - (5*ArcSin[a*x]^4)/(96*a^6) + (1/6)*x^6*ArcSin[a*x]^4} -{x^4*ArcSin[a*x]^4, x, 19, (16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(5625*a^5) - (1088*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(5625*a^3) - (24*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(625*a) - (32*x*ArcSin[a*x]^2)/(25*a^4) - (16*x^3*ArcSin[a*x]^2)/(75*a^2) - (12/125)*x^5*ArcSin[a*x]^2 + (32*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(75*a^5) + (16*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(75*a^3) + (4*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(25*a) + (1/5)*x^5*ArcSin[a*x]^4} -{x^3*ArcSin[a*x]^4, x, 14, (45*x^2)/(128*a^2) + (3*x^4)/128 - (45*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(64*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(32*a) + (45*ArcSin[a*x]^2)/(128*a^4) - (9*x^2*ArcSin[a*x]^2)/(16*a^2) - (3/16)*x^4*ArcSin[a*x]^2 + (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(8*a^3) + (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(4*a) - (3*ArcSin[a*x]^4)/(32*a^4) + (1/4)*x^4*ArcSin[a*x]^4} -{x^2*ArcSin[a*x]^4, x, 11, (160*x)/(27*a^2) + (8*x^3)/81 - (160*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a^3) - (8*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a) - (8*x*ArcSin[a*x]^2)/(3*a^2) - (4/9)*x^3*ArcSin[a*x]^2 + (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(9*a^3) + (4*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(9*a) + (1/3)*x^3*ArcSin[a*x]^4} -{x^1*ArcSin[a*x]^4, x, 7, (3*x^2)/4 - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a) + (3*ArcSin[a*x]^2)/(4*a^2) - (3/2)*x^2*ArcSin[a*x]^2 + (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a - ArcSin[a*x]^4/(4*a^2) + (1/2)*x^2*ArcSin[a*x]^4} -{x^0*ArcSin[a*x]^4, x, 5, 24*x - (24*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a - 12*x*ArcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a + x*ArcSin[a*x]^4} -{ArcSin[a*x]^4/x^1, x, 8, (-(1/5))*I*ArcSin[a*x]^5 + ArcSin[a*x]^4*Log[1 - E^(2*I*ArcSin[a*x])] - 2*I*ArcSin[a*x]^3*PolyLog[2, E^(2*I*ArcSin[a*x])] + 3*ArcSin[a*x]^2*PolyLog[3, E^(2*I*ArcSin[a*x])] + 3*I*ArcSin[a*x]*PolyLog[4, E^(2*I*ArcSin[a*x])] - (3/2)*PolyLog[5, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^4/x^2, x, 11, -(ArcSin[a*x]^4/x) - 8*a*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 12*I*a*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - 12*I*a*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 24*a*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 24*a*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - 24*I*a*PolyLog[4, -E^(I*ArcSin[a*x])] + 24*I*a*PolyLog[4, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^4/x^3, x, 8, -2*I*a^2*ArcSin[a*x]^3 - (2*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x - ArcSin[a*x]^4/(2*x^2) + 6*a^2*ArcSin[a*x]^2*Log[1 - E^(2*I*ArcSin[a*x])] - 6*I*a^2*ArcSin[a*x]*PolyLog[2, E^(2*I*ArcSin[a*x])] + 3*a^2*PolyLog[3, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^4/x^4, x, 19, -((2*a^2*ArcSin[a*x]^2)/x) - (2*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*x^2) - ArcSin[a*x]^4/(3*x^3) - 8*a^3*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] - (4/3)*a^3*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 4*I*a^3*PolyLog[2, -E^(I*ArcSin[a*x])] + 2*I*a^3*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - 4*I*a^3*PolyLog[2, E^(I*ArcSin[a*x])] - 2*I*a^3*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 4*a^3*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 4*a^3*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - 4*I*a^3*PolyLog[4, -E^(I*ArcSin[a*x])] + 4*I*a^3*PolyLog[4, E^(I*ArcSin[a*x])]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^6/ArcSin[a*x], x, 7, (5*CosIntegral[ArcSin[a*x]])/(64*a^7) - (9*CosIntegral[3*ArcSin[a*x]])/(64*a^7) + (5*CosIntegral[5*ArcSin[a*x]])/(64*a^7) - CosIntegral[7*ArcSin[a*x]]/(64*a^7)} -{x^5/ArcSin[a*x], x, 6, (5*SinIntegral[2*ArcSin[a*x]])/(32*a^6) - SinIntegral[4*ArcSin[a*x]]/(8*a^6) + SinIntegral[6*ArcSin[a*x]]/(32*a^6)} -{x^4/ArcSin[a*x], x, 6, CosIntegral[ArcSin[a*x]]/(8*a^5) - (3*CosIntegral[3*ArcSin[a*x]])/(16*a^5) + CosIntegral[5*ArcSin[a*x]]/(16*a^5)} -{x^3/ArcSin[a*x], x, 5, SinIntegral[2*ArcSin[a*x]]/(4*a^4) - SinIntegral[4*ArcSin[a*x]]/(8*a^4)} -{x^2/ArcSin[a*x], x, 5, CosIntegral[ArcSin[a*x]]/(4*a^3) - CosIntegral[3*ArcSin[a*x]]/(4*a^3)} -{x^1/ArcSin[a*x], x, 4, SinIntegral[2*ArcSin[a*x]]/(2*a^2)} -{x^0/ArcSin[a*x], x, 2, CosIntegral[ArcSin[a*x]]/a} -{1/(x^1*ArcSin[a*x]), x, 0, Unintegrable[1/(x*ArcSin[a*x]), x]} -{1/(x^2*ArcSin[a*x]), x, 0, Unintegrable[1/(x^2*ArcSin[a*x]), x]} - - -{x^6/ArcSin[a*x]^2, x, 6, -((x^6*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - (5*SinIntegral[ArcSin[a*x]])/(64*a^7) + (27*SinIntegral[3*ArcSin[a*x]])/(64*a^7) - (25*SinIntegral[5*ArcSin[a*x]])/(64*a^7) + (7*SinIntegral[7*ArcSin[a*x]])/(64*a^7)} -{x^5/ArcSin[a*x]^2, x, 5, -((x^5*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + (5*CosIntegral[2*ArcSin[a*x]])/(16*a^6) - CosIntegral[4*ArcSin[a*x]]/(2*a^6) + (3*CosIntegral[6*ArcSin[a*x]])/(16*a^6)} -{x^4/ArcSin[a*x]^2, x, 5, -((x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/(8*a^5) + (9*SinIntegral[3*ArcSin[a*x]])/(16*a^5) - (5*SinIntegral[5*ArcSin[a*x]])/(16*a^5)} -{x^3/ArcSin[a*x]^2, x, 4, -((x^3*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + CosIntegral[2*ArcSin[a*x]]/(2*a^4) - CosIntegral[4*ArcSin[a*x]]/(2*a^4)} -{x^2/ArcSin[a*x]^2, x, 4, -((x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/(4*a^3) + (3*SinIntegral[3*ArcSin[a*x]])/(4*a^3)} -{x^1/ArcSin[a*x]^2, x, 2, -((x*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + CosIntegral[2*ArcSin[a*x]]/a^2} -{x^0/ArcSin[a*x]^2, x, 3, -(Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/a} -{1/(x^1*ArcSin[a*x]^2), x, 0, Unintegrable[1/(x*ArcSin[a*x]^2), x]} -{1/(x^2*ArcSin[a*x]^2), x, 0, Unintegrable[1/(x^2*ArcSin[a*x]^2), x]} - - -{x^4/ArcSin[a*x]^3, x, 14, -((x^4*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]^2)) - (2*x^3)/(a^2*ArcSin[a*x]) + (5*x^5)/(2*ArcSin[a*x]) - CosIntegral[ArcSin[a*x]]/(16*a^5) + (27*CosIntegral[3*ArcSin[a*x]])/(32*a^5) - (25*CosIntegral[5*ArcSin[a*x]])/(32*a^5)} -{x^3/ArcSin[a*x]^3, x, 12, -((x^3*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]^2)) - (3*x^2)/(2*a^2*ArcSin[a*x]) + (2*x^4)/ArcSin[a*x] - SinIntegral[2*ArcSin[a*x]]/(2*a^4) + SinIntegral[4*ArcSin[a*x]]/a^4} -{x^2/ArcSin[a*x]^3, x, 10, -((x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]^2)) - x/(a^2*ArcSin[a*x]) + (3*x^3)/(2*ArcSin[a*x]) - CosIntegral[ArcSin[a*x]]/(8*a^3) + (9*CosIntegral[3*ArcSin[a*x]])/(8*a^3)} -{x^1/ArcSin[a*x]^3, x, 7, -((x*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]^2)) - 1/(2*a^2*ArcSin[a*x]) + x^2/ArcSin[a*x] - SinIntegral[2*ArcSin[a*x]]/a^2} -{x^0/ArcSin[a*x]^3, x, 4, -(Sqrt[1 - a^2*x^2]/(2*a*ArcSin[a*x]^2)) + x/(2*ArcSin[a*x]) - CosIntegral[ArcSin[a*x]]/(2*a)} -{1/(x^1*ArcSin[a*x]^3), x, 0, Unintegrable[1/(x*ArcSin[a*x]^3), x]} -{1/(x^2*ArcSin[a*x]^3), x, 0, Unintegrable[1/(x^2*ArcSin[a*x]^3), x]} - - -{x^4/ArcSin[a*x]^4, x, 12, -((x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3)) - (2*x^3)/(3*a^2*ArcSin[a*x]^2) + (5*x^5)/(6*ArcSin[a*x]^2) - (2*x^2*Sqrt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(48*a^5) - (27*SinIntegral[3*ArcSin[a*x]])/(32*a^5) + (125*SinIntegral[5*ArcSin[a*x]])/(96*a^5)} -{x^3/ArcSin[a*x]^4, x, 9, -((x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3)) - x^2/(2*a^2*ArcSin[a*x]^2) + (2*x^4)/(3*ArcSin[a*x]^2) - (x*Sqrt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (8*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]) - CosIntegral[2*ArcSin[a*x]]/(3*a^4) + (4*CosIntegral[4*ArcSin[a*x]])/(3*a^4)} -{x^2/ArcSin[a*x]^4, x, 10, -((x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3)) - x/(3*a^2*ArcSin[a*x]^2) + x^3/(2*ArcSin[a*x]^2) - Sqrt[1 - a^2*x^2]/(3*a^3*ArcSin[a*x]) + (3*x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(24*a^3) - (9*SinIntegral[3*ArcSin[a*x]])/(8*a^3)} -{x^1/ArcSin[a*x]^4, x, 5, -((x*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^3)) - 1/(6*a^2*ArcSin[a*x]^2) + x^2/(3*ArcSin[a*x]^2) + (2*x*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]) - (2*CosIntegral[2*ArcSin[a*x]])/(3*a^2)} -{x^0/ArcSin[a*x]^4, x, 5, -(Sqrt[1 - a^2*x^2]/(3*a*ArcSin[a*x]^3)) + x/(6*ArcSin[a*x]^2) + Sqrt[1 - a^2*x^2]/(6*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(6*a)} -{1/(x^1*ArcSin[a*x]^4), x, 0, Unintegrable[1/(x*ArcSin[a*x]^4), x]} -{1/(x^2*ArcSin[a*x]^4), x, 0, Unintegrable[1/(x^2*ArcSin[a*x]^4), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSin[a x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Sqrt[ArcSin[a*x]], x, 10, (1/5)*x^5*Sqrt[ArcSin[a*x]] - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5) + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(80*a^5)} -{x^3*Sqrt[ArcSin[a*x]], x, 8, -((3*Sqrt[ArcSin[a*x]])/(32*a^4)) + (1/4)*x^4*Sqrt[ArcSin[a*x]] - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(64*a^4) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(16*a^4)} -{x^2*Sqrt[ArcSin[a*x]], x, 8, (1/3)*x^3*Sqrt[ArcSin[a*x]] - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^3) + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(12*a^3)} -{x^1*Sqrt[ArcSin[a*x]], x, 6, -(Sqrt[ArcSin[a*x]]/(4*a^2)) + (1/2)*x^2*Sqrt[ArcSin[a*x]] + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a^2)} -{x^0*Sqrt[ArcSin[a*x]], x, 4, x*Sqrt[ArcSin[a*x]] - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a} -{Sqrt[ArcSin[a*x]]/x^1, x, 0, Unintegrable[Sqrt[ArcSin[a*x]]/x, x]} - - -{x^4*ArcSin[a*x]^(3/2), x, 23, (4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^3) + (3*x^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(50*a) + (1/5)*x^5*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(32*a^5) - (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5), (4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^3) + (3*x^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(50*a) + (1/5)*x^5*ArcSin[a*x]^(3/2) - (11*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(400*a^5) - (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(25*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(50*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5) - (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5)} -{x^3*ArcSin[a*x]^(3/2), x, 16, (9*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(64*a^3) + (3*x^3*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(32*a) - (3*ArcSin[a*x]^(3/2))/(32*a^4) + (1/4)*x^4*ArcSin[a*x]^(3/2) + (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(512*a^4) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(64*a^4)} -{x^2*ArcSin[a*x]^(3/2), x, 13, (Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(3*a^3) + (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(6*a) + (1/3)*x^3*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(24*a^3)} -{x^1*ArcSin[a*x]^(3/2), x, 8, (3*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(8*a) - ArcSin[a*x]^(3/2)/(4*a^2) + (1/2)*x^2*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32*a^2)} -{x^0*ArcSin[a*x]^(3/2), x, 5, (3*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(2*a) + x*ArcSin[a*x]^(3/2) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a)} -{ArcSin[a*x]^(3/2)/x^1, x, 0, Unintegrable[ArcSin[a*x]^(3/2)/x, x]} - - -{x^4*ArcSin[a*x]^(5/2), x, 26, -((2*x*Sqrt[ArcSin[a*x]])/(5*a^4)) - (x^3*Sqrt[ArcSin[a*x]])/(15*a^2) - (3/100)*x^5*Sqrt[ArcSin[a*x]] + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^3) + (x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(10*a) + (1/5)*x^5*ArcSin[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(32*a^5) - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(192*a^5) + (3*Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(1600*a^5), -((2*x*Sqrt[ArcSin[a*x]])/(5*a^4)) - (x^3*Sqrt[ArcSin[a*x]])/(15*a^2) - (3/100)*x^5*Sqrt[ArcSin[a*x]] + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(15*a^3) + (x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(10*a) + (1/5)*x^5*ArcSin[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(32*a^5) - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(60*a^5) - (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(1600*a^5)} -{x^3*ArcSin[a*x]^(5/2), x, 18, (225*Sqrt[ArcSin[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcSin[a*x]])/(256*a^2) - (15/256)*x^4*Sqrt[ArcSin[a*x]] + (15*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(64*a^3) + (5*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(32*a) - (3*ArcSin[a*x]^(5/2))/(32*a^4) + (1/4)*x^4*ArcSin[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4096*a^4) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(256*a^4)} -{x^2*ArcSin[a*x]^(5/2), x, 15, -((5*x*Sqrt[ArcSin[a*x]])/(6*a^2)) - (5/36)*x^3*Sqrt[ArcSin[a*x]] + (5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(9*a^3) + (5*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(18*a) + (1/3)*x^3*ArcSin[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^3) - (5*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(144*a^3)} -{x^1*ArcSin[a*x]^(5/2), x, 9, (15*Sqrt[ArcSin[a*x]])/(64*a^2) - (15/32)*x^2*Sqrt[ArcSin[a*x]] + (5*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(8*a) - ArcSin[a*x]^(5/2)/(4*a^2) + (1/2)*x^2*ArcSin[a*x]^(5/2) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(128*a^2)} -{x^0*ArcSin[a*x]^(5/2), x, 6, (-(15/4))*x*Sqrt[ArcSin[a*x]] + (5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(2*a) + x*ArcSin[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a)} -{ArcSin[a*x]^(5/2)/x^1, x, 0, Unintegrable[ArcSin[a*x]^(5/2)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/Sqrt[ArcSin[a*x]], x, 9, (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5) + (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^5)} -{x^3/Sqrt[ArcSin[a*x]], x, 7, -((Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^4)) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(4*a^4)} -{x^2/Sqrt[ArcSin[a*x]], x, 7, (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^3) - (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^3)} -{x^1/Sqrt[ArcSin[a*x]], x, 5, (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(2*a^2)} -{x^0/Sqrt[ArcSin[a*x]], x, 3, (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a} -{1/(x^1*Sqrt[ArcSin[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[ArcSin[a*x]]), x]} -{1/(x^2*Sqrt[ArcSin[a*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[ArcSin[a*x]]), x]} - - -{x^6/ArcSin[a*x]^(3/2), x, 10, -((2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (5*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7) + (9*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7) - (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7) + (Sqrt[(7*Pi)/2]*FresnelS[Sqrt[14/Pi]*Sqrt[ArcSin[a*x]]])/(16*a^7)} -{x^5/ArcSin[a*x]^(3/2), x, 8, -((2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^6 + (Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a^6) + (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a^6)} -{x^4/ArcSin[a*x]^(3/2), x, 8, -((2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5) - (Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(4*a^5)} -{x^3/ArcSin[a*x]^(3/2), x, 6, -((2*x^3*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^4 + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^4} -{x^2/ArcSin[a*x]^(3/2), x, 6, -((2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^3 + (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^3} -{x^1/ArcSin[a*x]^(3/2), x, 3, -((2*x*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) + (2*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/a^2} -{x^0/ArcSin[a*x]^(3/2), x, 4, -((2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcSin[a*x]])) - (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a} -{1/(x^1*ArcSin[a*x]^(3/2)), x, 0, Unintegrable[1/(x*ArcSin[a*x]^(3/2)), x]} - - -{x^4/ArcSin[a*x]^(5/2), x, 19, -((2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) - (16*x^3)/(3*a^2*Sqrt[ArcSin[a*x]]) + (20*x^5)/(3*Sqrt[ArcSin[a*x]]) - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5) - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(6*a^5), -((2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) - (16*x^3)/(3*a^2*Sqrt[ArcSin[a*x]]) + (20*x^5)/(3*Sqrt[ArcSin[a*x]]) - (25*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5) + (4*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a^5 + (25*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(2*a^5) - (4*Sqrt[(2*Pi)/3]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^5 - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(6*a^5)} -{x^3/ArcSin[a*x]^(5/2), x, 15, -((2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) - (4*x^2)/(a^2*Sqrt[ArcSin[a*x]]) + (16*x^4)/(3*Sqrt[ArcSin[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^4) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a^4)} -{x^2/ArcSin[a*x]^(5/2), x, 13, -((2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) - (8*x)/(3*a^2*Sqrt[ArcSin[a*x]]) + (4*x^3)/Sqrt[ArcSin[a*x]] - (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^3} -{x^1/ArcSin[a*x]^(5/2), x, 8, -((2*x*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) - 4/(3*a^2*Sqrt[ArcSin[a*x]]) + (8*x^2)/(3*Sqrt[ArcSin[a*x]]) - (8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a^2)} -{x^0/ArcSin[a*x]^(5/2), x, 5, -((2*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]^(3/2))) + (4*x)/(3*Sqrt[ArcSin[a*x]]) - (4*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a)} -{1/(x^1*ArcSin[a*x]^(5/2)), x, 0, Unintegrable[1/(x*ArcSin[a*x]^(5/2)), x]} - - -{x^4/ArcSin[a*x]^(7/2), x, 17, -((2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2))) - (16*x^3)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^5)/(3*ArcSin[a*x]^(3/2)) - (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcSin[a*x]]) + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^5) - (5*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/a^5 + (8*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(3*a^5)} -{x^3/ArcSin[a*x]^(7/2), x, 12, -((2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2))) - (4*x^2)/(5*a^2*ArcSin[a*x]^(3/2)) + (16*x^4)/(15*ArcSin[a*x]^(3/2)) - (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcSin[a*x]]) + (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSin[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^4) - (16*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(15*a^4)} -{x^2/ArcSin[a*x]^(7/2), x, 13, -((2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2))) - (8*x)/(15*a^2*ArcSin[a*x]^(3/2)) + (4*x^3)/(5*ArcSin[a*x]^(3/2)) - (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcSin[a*x]]) + (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcSin[a*x]]) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a^3) - (6*Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(5*a^3)} -{x^1/ArcSin[a*x]^(7/2), x, 6, -((2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2))) - 4/(15*a^2*ArcSin[a*x]^(3/2)) + (8*x^2)/(15*ArcSin[a*x]^(3/2)) + (32*x*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSin[a*x]]) - (32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(15*a^2)} -{x^0/ArcSin[a*x]^(7/2), x, 6, -((2*Sqrt[1 - a^2*x^2])/(5*a*ArcSin[a*x]^(5/2))) + (4*x)/(15*ArcSin[a*x]^(3/2)) + (8*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcSin[a*x]]) + (8*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(15*a)} -{1/(x^1*ArcSin[a*x]^(7/2)), x, 0, Unintegrable[1/(x*ArcSin[a*x]^(7/2)), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (b x)^(m/2) ArcSin[a x]^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (b x)^(m/2) ArcSin[a x]^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcSin[a x]^n with m symbolic*) - - -{(b*x)^m*ArcSin[a*x]^4, x, 1, ((b*x)^(1 + m)*ArcSin[a*x]^4)/(b*(1 + m)) - (4*a*Unintegrable[((b*x)^(1 + m)*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2], x])/(b*(1 + m))} -{(b*x)^m*ArcSin[a*x]^3, x, 1, ((b*x)^(1 + m)*ArcSin[a*x]^3)/(b*(1 + m)) - (3*a*Unintegrable[((b*x)^(1 + m)*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2], x])/(b*(1 + m))} -{(b*x)^m*ArcSin[a*x]^2, x, 2, If[$VersionNumber>=8, ((b*x)^(1 + m)*ArcSin[a*x]^2)/(b*(1 + m)) - (2*a*(b*x)^(2 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m)) + (2*a^2*(b*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(b^3*(1 + m)*(2 + m)*(3 + m)), ((b*x)^(1 + m)*ArcSin[a*x]^2)/(b*(1 + m)) - (2*a*(b*x)^(2 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m)) + (2*a^2*(b*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(b^3*(3 + m)*(2 + 3*m + m^2))]} -{(b*x)^m*ArcSin[a*x]^1, x, 2, ((b*x)^(1 + m)*ArcSin[a*x])/(b*(1 + m)) - (a*(b*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m))} -{(b*x)^m/ArcSin[a*x]^1, x, 0, Unintegrable[(b*x)^m/ArcSin[a*x], x]} -{(b*x)^m/ArcSin[a*x]^2, x, 0, Unintegrable[(b*x)^m/ArcSin[a*x]^2, x]} - - -{(b*x)^m*ArcSin[a*x]^(3/2), x, 0, Unintegrable[(b*x)^m*ArcSin[a*x]^(3/2), x]} -{(b*x)^m*ArcSin[a*x]^(1/2), x, 0, Unintegrable[(b*x)^m*Sqrt[ArcSin[a*x]], x]} -{(b*x)^m/ArcSin[a*x]^(1/2), x, 0, Unintegrable[(b*x)^m/Sqrt[ArcSin[a*x]], x]} -{(b*x)^m/ArcSin[a*x]^(3/2), x, 0, Unintegrable[(b*x)^m/ArcSin[a*x]^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcSin[a x]^n with n symbolic*) - - -{(b*x)^m*ArcSin[a*x]^n, x, 0, Unintegrable[(b*x)^m*ArcSin[a*x]^n, x]} - - -{x^3*ArcSin[a*x]^n, x, 9, -((2^(-4 - n)*ArcSin[a*x]^n*Gamma[1 + n, -2*I*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*a^4)) - (2^(-4 - n)*ArcSin[a*x]^n*Gamma[1 + n, 2*I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*a^4) + (ArcSin[a*x]^n*Gamma[1 + n, -4*I*ArcSin[a*x]])/(2^(2*(3 + n))*((-I)*ArcSin[a*x])^n*a^4) + (ArcSin[a*x]^n*Gamma[1 + n, 4*I*ArcSin[a*x]])/(2^(2*(3 + n))*(I*ArcSin[a*x])^n*a^4)} -{x^2*ArcSin[a*x]^n, x, 9, -((I*ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(8*a^3))) + (I*ArcSin[a*x]^n*Gamma[1 + n, I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(8*a^3)) + (I*3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, -3*I*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(8*a^3)) - (I*3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, 3*I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(8*a^3))} -{x^1*ArcSin[a*x]^n, x, 6, -((2^(-3 - n)*ArcSin[a*x]^n*Gamma[1 + n, -2*I*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*a^2)) - (2^(-3 - n)*ArcSin[a*x]^n*Gamma[1 + n, 2*I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*a^2)} -{x^0*ArcSin[a*x]^n, x, 4, -((I*ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(2*a))) + (I*ArcSin[a*x]^n*Gamma[1 + n, I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(2*a))} -{ArcSin[a*x]^n/x^1, x, 0, Unintegrable[ArcSin[a*x]^n/x, x]} -{ArcSin[a*x]^n/x^2, x, 0, Unintegrable[ArcSin[a*x]^n/x^2, x]} - - -{(b*x)^(3/2)*ArcSin[a*x]^n, x, 0, Unintegrable[(b*x)^(3/2)*ArcSin[a*x]^n, x]} -{(b*x)^(1/2)*ArcSin[a*x]^n, x, 0, Unintegrable[Sqrt[b*x]*ArcSin[a*x]^n, x]} -{ArcSin[a*x]^n/(b*x)^(1/2), x, 0, Unintegrable[ArcSin[a*x]^n/Sqrt[b*x], x]} -{ArcSin[a*x]^n/(b*x)^(3/2), x, 0, Unintegrable[ArcSin[a*x]^n/(b*x)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*(a + b*ArcSin[c*x]), x, 4, (3*b*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*x^3*Sqrt[1 - c^2*x^2])/(16*c) - (3*b*ArcSin[c*x])/(32*c^4) + (1/4)*x^4*(a + b*ArcSin[c*x])} -{x^2*(a + b*ArcSin[c*x]), x, 4, (b*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*(1 - c^2*x^2)^(3/2))/(9*c^3) + (1/3)*x^3*(a + b*ArcSin[c*x])} -{x^1*(a + b*ArcSin[c*x]), x, 3, (b*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*ArcSin[c*x])/(4*c^2) + (1/2)*x^2*(a + b*ArcSin[c*x])} -{x^0*(a + b*ArcSin[c*x]), x, 3, a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]} -{(a + b*ArcSin[c*x])/x^1, x, 5, -((I*(a + b*ArcSin[c*x])^2)/(2*b)) + (a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] - (1/2)*I*b*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(a + b*ArcSin[c*x])/x^2, x, 4, -((a + b*ArcSin[c*x])/x) - b*c*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(a + b*ArcSin[c*x])/x^3, x, 2, -((b*c*Sqrt[1 - c^2*x^2])/(2*x)) - (a + b*ArcSin[c*x])/(2*x^2)} -{(a + b*ArcSin[c*x])/x^4, x, 5, -((b*c*Sqrt[1 - c^2*x^2])/(6*x^2)) - (a + b*ArcSin[c*x])/(3*x^3) - (1/6)*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{x^2*(a + b*ArcSin[c*x])^2, x, 5, -((4*b^2*x)/(9*c^2)) - (2*b^2*x^3)/27 + (4*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (2*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (1/3)*x^3*(a + b*ArcSin[c*x])^2} -{x^1*(a + b*ArcSin[c*x])^2, x, 4, (-(1/4))*b^2*x^2 + (b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) - (a + b*ArcSin[c*x])^2/(4*c^2) + (1/2)*x^2*(a + b*ArcSin[c*x])^2} -{x^0*(a + b*ArcSin[c*x])^2, x, 3, -2*b^2*x + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + x*(a + b*ArcSin[c*x])^2} -{(a + b*ArcSin[c*x])^2/x^1, x, 6, -((I*(a + b*ArcSin[c*x])^3)/(3*b)) + (a + b*ArcSin[c*x])^2*Log[1 - E^(2*I*ArcSin[c*x])] - I*b*(a + b*ArcSin[c*x])*PolyLog[2, E^(2*I*ArcSin[c*x])] + (1/2)*b^2*PolyLog[3, E^(2*I*ArcSin[c*x])]} -{(a + b*ArcSin[c*x])^2/x^2, x, 7, -((a + b*ArcSin[c*x])^2/x) - 4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + 2*I*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])] - 2*I*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])]} - - -{x^2*(a + b*ArcSin[c*x])^3, x, 10, -((4*a*b^2*x)/(3*c^2)) - (14*b^3*Sqrt[1 - c^2*x^2])/(9*c^3) + (2*b^3*(1 - c^2*x^2)^(3/2))/(27*c^3) - (4*b^3*x*ArcSin[c*x])/(3*c^2) - (2/9)*b^2*x^3*(a + b*ArcSin[c*x]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^3) + (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c) + (1/3)*x^3*(a + b*ArcSin[c*x])^3} -{x^1*(a + b*ArcSin[c*x])^3, x, 6, -((3*b^3*x*Sqrt[1 - c^2*x^2])/(8*c)) + (3*b^3*ArcSin[c*x])/(8*c^2) - (3/4)*b^2*x^2*(a + b*ArcSin[c*x]) + (3*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*c) - (a + b*ArcSin[c*x])^3/(4*c^2) + (1/2)*x^2*(a + b*ArcSin[c*x])^3} -{x^0*(a + b*ArcSin[c*x])^3, x, 5, -6*a*b^2*x - (6*b^3*Sqrt[1 - c^2*x^2])/c - 6*b^3*x*ArcSin[c*x] + (3*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c + x*(a + b*ArcSin[c*x])^3} -{(a + b*ArcSin[c*x])^3/x^1, x, 7, -((I*(a + b*ArcSin[c*x])^4)/(4*b)) + (a + b*ArcSin[c*x])^3*Log[1 - E^(2*I*ArcSin[c*x])] - (3/2)*I*b*(a + b*ArcSin[c*x])^2*PolyLog[2, E^(2*I*ArcSin[c*x])] + (3/2)*b^2*(a + b*ArcSin[c*x])*PolyLog[3, E^(2*I*ArcSin[c*x])] + (3/4)*I*b^3*PolyLog[4, E^(2*I*ArcSin[c*x])]} -{(a + b*ArcSin[c*x])^3/x^2, x, 9, -((a + b*ArcSin[c*x])^3/x) - 6*b*c*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])] + 6*I*b^2*c*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - 6*I*b^2*c*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])] - 6*b^3*c*PolyLog[3, -E^(I*ArcSin[c*x])] + 6*b^3*c*PolyLog[3, E^(I*ArcSin[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcSin[c*x]), x, 9, (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^3) - (Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^3) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^3) - (Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^3)} -{x^1/(a + b*ArcSin[c*x]), x, 6, -((CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(2*b*c^2)) + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c^2)} -{x^0/(a + b*ArcSin[c*x]), x, 4, (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c)} -{1/(x^1*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcSin[c*x])), x]} -{1/(x^2*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x^2*(a + b*ArcSin[c*x])), x]} - - -{x^2/(a + b*ArcSin[c*x])^2, x, 8, -((x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b^2*c^3) - (3*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b^2*c^3) - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^3) + (3*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3)} -{x^1/(a + b*ArcSin[c*x])^2, x, 4, -((x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) + (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2) + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2)} -{x^0/(a + b*ArcSin[c*x])^2, x, 5, -(Sqrt[1 - c^2*x^2]/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c)} -{1/(x^1*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x*(a + b*ArcSin[c*x])^2), x]} -{1/(x^2*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x^2*(a + b*ArcSin[c*x])^2), x]} - - -{x^2/(a + b*ArcSin[c*x])^3, x, 16, -((x^2*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcSin[c*x])^2)) - x/(b^2*c^2*(a + b*ArcSin[c*x])) + (3*x^3)/(2*b^2*(a + b*ArcSin[c*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^3*c^3) + (9*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(8*b^3*c^3) - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^3*c^3) + (9*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(8*b^3*c^3)} -{x^1/(a + b*ArcSin[c*x])^3, x, 9, -((x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcSin[c*x])^2)) - 1/(2*b^2*c^2*(a + b*ArcSin[c*x])) + x^2/(b^2*(a + b*ArcSin[c*x])) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^3*c^2) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^3*c^2)} -{x^0/(a + b*ArcSin[c*x])^3, x, 6, -(Sqrt[1 - c^2*x^2]/(2*b*c*(a + b*ArcSin[c*x])^2)) + x/(2*b^2*(a + b*ArcSin[c*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(2*b^3*c) - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(2*b^3*c)} -{1/(x^1*(a + b*ArcSin[c*x])^3), x, 0, Unintegrable[1/(x*(a + b*ArcSin[c*x])^3), x]} -{1/(x^2*(a + b*ArcSin[c*x])^3), x, 0, Unintegrable[1/(x^2*(a + b*ArcSin[c*x])^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*(a + b*ArcSin[c*x])^(1/2), x, 14, (1/3)*x^3*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*c^3) + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(12*c^3) + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)} -{x^1*(a + b*ArcSin[c*x])^(1/2), x, 9, -(Sqrt[a + b*ArcSin[c*x]]/(4*c^2)) + (1/2)*x^2*Sqrt[a + b*ArcSin[c*x]] + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*c^2) + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*c^2)} -{x^0*(a + b*ArcSin[c*x])^(1/2), x, 7, x*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c} -{(a + b*ArcSin[c*x])^(1/2)/x^1, x, 0, Unintegrable[Sqrt[a + b*ArcSin[c*x]]/x, x]} -{(a + b*ArcSin[c*x])^(1/2)/x^2, x, 0, Unintegrable[Sqrt[a + b*ArcSin[c*x]]/x^2, x]} - - -{x^2*(a + b*ArcSin[c*x])^(3/2), x, 22, (b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c^3) + (b*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c) + (1/3)*x^3*(a + b*ArcSin[c*x])^(3/2) - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)} -{x^1*(a + b*ArcSin[c*x])^(3/2), x, 11, (3*b*x*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(8*c) - (a + b*ArcSin[c*x])^(3/2)/(4*c^2) + (1/2)*x^2*(a + b*ArcSin[c*x])^(3/2) - (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*c^2) + (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*c^2)} -{x^0*(a + b*ArcSin[c*x])^(3/2), x, 8, (3*b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + x*(a + b*ArcSin[c*x])^(3/2) - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c)} -{(a + b*ArcSin[c*x])^(3/2)/x^1, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(3/2)/x, x]} -{(a + b*ArcSin[c*x])^(3/2)/x^2, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(3/2)/x^2, x]} - - -{x^2*(a + b*ArcSin[c*x])^(5/2), x, 24, -((5*b^2*x*Sqrt[a + b*ArcSin[c*x]])/(6*c^2)) - (5/36)*b^2*x^3*Sqrt[a + b*ArcSin[c*x]] + (5*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(9*c^3) + (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(18*c) + (1/3)*x^3*(a + b*ArcSin[c*x])^(5/2) + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*c^3) - (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(144*c^3) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)} -{x^1*(a + b*ArcSin[c*x])^(5/2), x, 12, (15*b^2*Sqrt[a + b*ArcSin[c*x]])/(64*c^2) - (15/32)*b^2*x^2*Sqrt[a + b*ArcSin[c*x]] + (5*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(8*c) - (a + b*ArcSin[c*x])^(5/2)/(4*c^2) + (1/2)*x^2*(a + b*ArcSin[c*x])^(5/2) - (15*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*c^2) - (15*b^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*c^2)} -{x^0*(a + b*ArcSin[c*x])^(5/2), x, 9, (-(15/4))*b^2*x*Sqrt[a + b*ArcSin[c*x]] + (5*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(3/2))/(2*c) + x*(a + b*ArcSin[c*x])^(5/2) + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*c) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c)} -{(a + b*ArcSin[c*x])^(5/2)/x^1, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(5/2)/x, x]} -{(a + b*ArcSin[c*x])^(5/2)/x^2, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(5/2)/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcSin[c*x])^(1/2), x, 13, (Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) - (Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) + (Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*c^3) - (Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*c^3)} -{x^1/(a + b*ArcSin[c*x])^(1/2), x, 8, (Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(2*Sqrt[b]*c^2) - (Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(2*Sqrt[b]*c^2)} -{x^0/(a + b*ArcSin[c*x])^(1/2), x, 6, (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c) + (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c)} -{1/(x^1*(a + b*ArcSin[c*x])^(1/2)), x, 0, Unintegrable[1/(x*Sqrt[a + b*ArcSin[c*x]]), x]} -{1/(x^2*(a + b*ArcSin[c*x])^(1/2)), x, 0, Unintegrable[1/(x^2*Sqrt[a + b*ArcSin[c*x]]), x]} - - -{x^2/(a + b*ArcSin[c*x])^(3/2), x, 12, -((2*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) - (Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3)} -{x^1/(a + b*ArcSin[c*x])^(3/2), x, 6, -((2*x*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]])) + (2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*c^2) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*c^2)} -{x^0/(a + b*ArcSin[c*x])^(3/2), x, 7, -((2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c)} -{1/(x^1*(a + b*ArcSin[c*x])^(3/2)), x, 0, Unintegrable[1/(x*(a + b*ArcSin[c*x])^(3/2)), x]} -{1/(x^2*(a + b*ArcSin[c*x])^(3/2)), x, 0, Unintegrable[1/(x^2*(a + b*ArcSin[c*x])^(3/2)), x]} - - -{x^2/(a + b*ArcSin[c*x])^(5/2), x, 22, -((2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2))) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcSin[c*x]]) + (4*x^3)/(b^2*Sqrt[a + b*ArcSin[c*x]]) - (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(5/2)*c^3) - (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*c^3)} -{x^1/(a + b*ArcSin[c*x])^(5/2), x, 11, -((2*x*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2))) - 4/(3*b^2*c^2*Sqrt[a + b*ArcSin[c*x]]) + (8*x^2)/(3*b^2*Sqrt[a + b*ArcSin[c*x]]) - (8*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*c^2) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*c^2)} -{x^0/(a + b*ArcSin[c*x])^(5/2), x, 8, -((2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcSin[c*x])^(3/2))) + (4*x)/(3*b^2*Sqrt[a + b*ArcSin[c*x]]) - (4*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(3*b^(5/2)*c) - (4*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c)} -{1/(x^1*(a + b*ArcSin[c*x])^(5/2)), x, 0, Unintegrable[1/(x*(a + b*ArcSin[c*x])^(5/2)), x]} -{1/(x^2*(a + b*ArcSin[c*x])^(5/2)), x, 0, Unintegrable[1/(x^2*(a + b*ArcSin[c*x])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcSin[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d*x)^(5/2)*(a + b*ArcSin[c*x]), x, 5, (20*b*d^2*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(147*c^3) + (4*b*(d*x)^(5/2)*Sqrt[1 - c^2*x^2])/(49*c) + (2*(d*x)^(7/2)*(a + b*ArcSin[c*x]))/(7*d) - (20*b*d^(5/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(147*c^(7/2))} -{(d*x)^(3/2)*(a + b*ArcSin[c*x]), x, 7, (4*b*(d*x)^(3/2)*Sqrt[1 - c^2*x^2])/(25*c) + (2*(d*x)^(5/2)*(a + b*ArcSin[c*x]))/(5*d) - (12*b*d^(3/2)*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(25*c^(5/2)) + (12*b*d^(3/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(25*c^(5/2))} -{(d*x)^(1/2)*(a + b*ArcSin[c*x]), x, 4, (4*b*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(9*c) + (2*(d*x)^(3/2)*(a + b*ArcSin[c*x]))/(3*d) - (4*b*Sqrt[d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(9*c^(3/2))} -{(a + b*ArcSin[c*x])/(d*x)^(1/2), x, 6, (2*Sqrt[d*x]*(a + b*ArcSin[c*x]))/d - (4*b*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(Sqrt[c]*Sqrt[d]) + (4*b*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(Sqrt[c]*Sqrt[d])} -{(a + b*ArcSin[c*x])/(d*x)^(3/2), x, 3, -((2*(a + b*ArcSin[c*x]))/(d*Sqrt[d*x])) + (4*b*Sqrt[c]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/d^(3/2)} -{(a + b*ArcSin[c*x])/(d*x)^(5/2), x, 7, -((4*b*c*Sqrt[1 - c^2*x^2])/(3*d^2*Sqrt[d*x])) - (2*(a + b*ArcSin[c*x]))/(3*d*(d*x)^(3/2)) - (4*b*c^(3/2)*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(3*d^(5/2)) + (4*b*c^(3/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(3*d^(5/2))} - - -{(d*x)^(5/2)*(a + b*ArcSin[c*x])^2, x, 2, (2*(d*x)^(7/2)*(a + b*ArcSin[c*x])^2)/(7*d) - (8*b*c*(d*x)^(9/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 9/4, 13/4, c^2*x^2])/(63*d^2) + (16*b^2*c^2*(d*x)^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, c^2*x^2])/(693*d^3)} -{(d*x)^(3/2)*(a + b*ArcSin[c*x])^2, x, 2, (2*(d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*d) - (8*b*c*(d*x)^(7/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 7/4, 11/4, c^2*x^2])/(35*d^2) + (16*b^2*c^2*(d*x)^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, c^2*x^2])/(315*d^3)} -{(d*x)^(1/2)*(a + b*ArcSin[c*x])^2, x, 2, (2*(d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*d) - (8*b*c*(d*x)^(5/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(15*d^2) + (16*b^2*c^2*(d*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(105*d^3)} -{(a + b*ArcSin[c*x])^2/(d*x)^(1/2), x, 2, (2*Sqrt[d*x]*(a + b*ArcSin[c*x])^2)/d - (8*b*c*(d*x)^(3/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2])/(3*d^2) + (16*b^2*c^2*(d*x)^(5/2)*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, c^2*x^2])/(15*d^3)} -{(a + b*ArcSin[c*x])^2/(d*x)^(3/2), x, 2, -((2*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d*x])) + (8*b*c*Sqrt[d*x]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2])/d^2 - (16*b^2*c^2*(d*x)^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(3*d^3)} -{(a + b*ArcSin[c*x])^2/(d*x)^(5/2), x, 2, -((2*(a + b*ArcSin[c*x])^2)/(3*d*(d*x)^(3/2))) - (8*b*c*(a + b*ArcSin[c*x])*Hypergeometric2F1[-(1/4), 1/2, 3/4, c^2*x^2])/(3*d^2*Sqrt[d*x]) + (16*b^2*c^2*Sqrt[d*x]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, c^2*x^2])/(3*d^3)} - - -{(d*x)^(3/2)*(a + b*ArcSin[c*x])^3, x, 1, (2*(d*x)^(5/2)*(a + b*ArcSin[c*x])^3)/(5*d) - (6*b*c*Unintegrable[((d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2], x])/(5*d)} -{(d*x)^(1/2)*(a + b*ArcSin[c*x])^3, x, 1, (2*(d*x)^(3/2)*(a + b*ArcSin[c*x])^3)/(3*d) - (2*b*c*Unintegrable[((d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2], x])/d} -{(a + b*ArcSin[c*x])^3/(d*x)^(1/2), x, 1, (2*Sqrt[d*x]*(a + b*ArcSin[c*x])^3)/d - (6*b*c*Unintegrable[(Sqrt[d*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2], x])/d} -{(a + b*ArcSin[c*x])^3/(d*x)^(3/2), x, 1, -((2*(a + b*ArcSin[c*x])^3)/(d*Sqrt[d*x])) + (6*b*c*Unintegrable[(a + b*ArcSin[c*x])^2/(Sqrt[d*x]*Sqrt[1 - c^2*x^2]), x])/d} -{(a + b*ArcSin[c*x])^3/(d*x)^(5/2), x, 1, -((2*(a + b*ArcSin[c*x])^3)/(3*d*(d*x)^(3/2))) + (2*b*c*Unintegrable[(a + b*ArcSin[c*x])^2/((d*x)^(3/2)*Sqrt[1 - c^2*x^2]), x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*x)^(3/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[(d*x)^(3/2)/(a + b*ArcSin[c*x]), x]} -{(d*x)^(1/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[Sqrt[d*x]/(a + b*ArcSin[c*x]), x]} -{1/((d*x)^(1/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(Sqrt[d*x]*(a + b*ArcSin[c*x])), x]} -{1/((d*x)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d*x)^(3/2)*(a + b*ArcSin[c*x])), x]} - - -{(d*x)^(3/2)/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[(d*x)^(3/2)/(a + b*ArcSin[c*x])^2, x]} -{(d*x)^(1/2)/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[Sqrt[d*x]/(a + b*ArcSin[c*x])^2, x]} -{1/((d*x)^(1/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(Sqrt[d*x]*(a + b*ArcSin[c*x])^2), x]} -{1/((d*x)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d*x)^(3/2)*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^m (a+b ArcSin[c x])^n with m symbolic*) - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^m (a+b ArcSin[c x])^n with n symbolic*) diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m deleted file mode 100644 index a60ce8e..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.4 (f x)^m (d+e x^2)^p (a+b arcsin(c x))^n.m +++ /dev/null @@ -1,1290 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]), x, 5, (2*b*d*Sqrt[1 - c^2*x^2])/(35*c^5) + (b*d*(1 - c^2*x^2)^(3/2))/(105*c^5) - (8*b*d*(1 - c^2*x^2)^(5/2))/(175*c^5) + (b*d*(1 - c^2*x^2)^(7/2))/(49*c^5) + (1/5)*d*x^5*(a + b*ArcSin[c*x]) - (1/7)*c^2*d*x^7*(a + b*ArcSin[c*x])} -{x^3*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]), x, 6, (b*d*x*Sqrt[1 - c^2*x^2])/(24*c^3) + (b*d*x^3*Sqrt[1 - c^2*x^2])/(36*c) - (1/36)*b*c*d*x^5*Sqrt[1 - c^2*x^2] - (b*d*ArcSin[c*x])/(24*c^4) + (1/4)*d*x^4*(a + b*ArcSin[c*x]) - (1/6)*c^2*d*x^6*(a + b*ArcSin[c*x])} -{x^2*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]), x, 5, (2*b*d*Sqrt[1 - c^2*x^2])/(15*c^3) + (b*d*(1 - c^2*x^2)^(3/2))/(45*c^3) - (b*d*(1 - c^2*x^2)^(5/2))/(25*c^3) + (1/3)*d*x^3*(a + b*ArcSin[c*x]) - (1/5)*c^2*d*x^5*(a + b*ArcSin[c*x])} -{x^1*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]), x, 4, (3*b*d*x*Sqrt[1 - c^2*x^2])/(32*c) + (b*d*x*(1 - c^2*x^2)^(3/2))/(16*c) + (3*b*d*ArcSin[c*x])/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(4*c^2)} -{x^0*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]), x, 5, (2*b*d*Sqrt[1 - c^2*x^2])/(3*c) + (b*d*(1 - c^2*x^2)^(3/2))/(9*c) + d*x*(a + b*ArcSin[c*x]) - (1/3)*c^2*d*x^3*(a + b*ArcSin[c*x])} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^1, x, 8, (-(1/4))*b*c*d*x*Sqrt[1 - c^2*x^2] - (1/4)*b*d*ArcSin[c*x] + (1/2)*d*(1 - c^2*x^2)*(a + b*ArcSin[c*x]) - (I*d*(a + b*ArcSin[c*x])^2)/(2*b) + d*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] - (1/2)*I*b*d*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^2, x, 6, (-b)*c*d*Sqrt[1 - c^2*x^2] - (d*(a + b*ArcSin[c*x]))/x - c^2*d*x*(a + b*ArcSin[c*x]) - b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^3, x, 8, -((b*c*d*Sqrt[1 - c^2*x^2])/(2*x)) - (1/2)*b*c^2*d*ArcSin[c*x] - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*x^2) + (I*c^2*d*(a + b*ArcSin[c*x])^2)/(2*b) - c^2*d*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] + (1/2)*I*b*c^2*d*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^4, x, 6, -((b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2)) - (d*(a + b*ArcSin[c*x]))/(3*x^3) + (c^2*d*(a + b*ArcSin[c*x]))/x + (5/6)*b*c^3*d*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]), x, 6, (8*b*d^2*Sqrt[1 - c^2*x^2])/(315*c^5) + (4*b*d^2*(1 - c^2*x^2)^(3/2))/(945*c^5) + (b*d^2*(1 - c^2*x^2)^(5/2))/(525*c^5) - (10*b*d^2*(1 - c^2*x^2)^(7/2))/(441*c^5) + (b*d^2*(1 - c^2*x^2)^(9/2))/(81*c^5) + (1/5)*d^2*x^5*(a + b*ArcSin[c*x]) - (2/7)*c^2*d^2*x^7*(a + b*ArcSin[c*x]) + (1/9)*c^4*d^2*x^9*(a + b*ArcSin[c*x])} -{x^3*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]), x, 7, (73*b*d^2*x*Sqrt[1 - c^2*x^2])/(3072*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2])/(4608*c) - (43*b*c*d^2*x^5*Sqrt[1 - c^2*x^2])/1152 + (1/64)*b*c^3*d^2*x^7*Sqrt[1 - c^2*x^2] - (73*b*d^2*ArcSin[c*x])/(3072*c^4) + (1/4)*d^2*x^4*(a + b*ArcSin[c*x]) - (1/3)*c^2*d^2*x^6*(a + b*ArcSin[c*x]) + (1/8)*c^4*d^2*x^8*(a + b*ArcSin[c*x])} -{x^2*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (8*b*d^2*Sqrt[1 - c^2*x^2])/(105*c^3) + (4*b*d^2*(1 - c^2*x^2)^(3/2))/(315*c^3) + (b*d^2*(1 - c^2*x^2)^(5/2))/(175*c^3) - (b*d^2*(1 - c^2*x^2)^(7/2))/(49*c^3) + (1/3)*d^2*x^3*(a + b*ArcSin[c*x]) - (2/5)*c^2*d^2*x^5*(a + b*ArcSin[c*x]) + (1/7)*c^4*d^2*x^7*(a + b*ArcSin[c*x])} -{x^1*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (5*b*d^2*x*Sqrt[1 - c^2*x^2])/(96*c) + (5*b*d^2*x*(1 - c^2*x^2)^(3/2))/(144*c) + (b*d^2*x*(1 - c^2*x^2)^(5/2))/(36*c) + (5*b*d^2*ArcSin[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(6*c^2)} -{x^0*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (8*b*d^2*Sqrt[1 - c^2*x^2])/(15*c) + (4*b*d^2*(1 - c^2*x^2)^(3/2))/(45*c) + (b*d^2*(1 - c^2*x^2)^(5/2))/(25*c) + d^2*x*(a + b*ArcSin[c*x]) - (2/3)*c^2*d^2*x^3*(a + b*ArcSin[c*x]) + (1/5)*c^4*d^2*x^5*(a + b*ArcSin[c*x])} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^1, x, 12, (-(11/32))*b*c*d^2*x*Sqrt[1 - c^2*x^2] - (1/16)*b*c*d^2*x*(1 - c^2*x^2)^(3/2) - (11/32)*b*d^2*ArcSin[c*x] + (1/2)*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]) + (1/4)*d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]) - (I*d^2*(a + b*ArcSin[c*x])^2)/(2*b) + d^2*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] - (1/2)*I*b*d^2*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^2, x, 7, (-(5/3))*b*c*d^2*Sqrt[1 - c^2*x^2] - (1/9)*b*c*d^2*(1 - c^2*x^2)^(3/2) - (d^2*(a + b*ArcSin[c*x]))/x - 2*c^2*d^2*x*(a + b*ArcSin[c*x]) + (1/3)*c^4*d^2*x^3*(a + b*ArcSin[c*x]) - b*c*d^2*ArcTanh[Sqrt[1 - c^2*x^2]]} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^3, x, 12, (-(1/4))*b*c^3*d^2*x*Sqrt[1 - c^2*x^2] - (b*c*d^2*(1 - c^2*x^2)^(3/2))/(2*x) - (1/4)*b*c^2*d^2*ArcSin[c*x] - c^2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(2*x^2) + (I*c^2*d^2*(a + b*ArcSin[c*x])^2)/b - 2*c^2*d^2*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] + I*b*c^2*d^2*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^4, x, 7, b*c^3*d^2*Sqrt[1 - c^2*x^2] - (b*c*d^2*Sqrt[1 - c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSin[c*x]))/(3*x^3) + (2*c^2*d^2*(a + b*ArcSin[c*x]))/x + c^4*d^2*x*(a + b*ArcSin[c*x]) + (11/6)*b*c^3*d^2*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (16*b*d^3*Sqrt[1 - c^2*x^2])/(1155*c^5) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(3465*c^5) + (2*b*d^3*(1 - c^2*x^2)^(5/2))/(1925*c^5) + (b*d^3*(1 - c^2*x^2)^(7/2))/(1617*c^5) - (4*b*d^3*(1 - c^2*x^2)^(9/2))/(297*c^5) + (b*d^3*(1 - c^2*x^2)^(11/2))/(121*c^5) + (1/5)*d^3*x^5*(a + b*ArcSin[c*x]) - (3/7)*c^2*d^3*x^7*(a + b*ArcSin[c*x]) + (1/3)*c^4*d^3*x^9*(a + b*ArcSin[c*x]) - (1/11)*c^6*d^3*x^11*(a + b*ArcSin[c*x])} -{x^3*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]), x, 8, (49*b*d^3*x*Sqrt[1 - c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 - c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 - c^2*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 - c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 - c^2*x^2)^(9/2))/(100*c^3) + (49*b*d^3*ArcSin[c*x])/(5120*c^4) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x]))/(8*c^4) + (d^3*(1 - c^2*x^2)^5*(a + b*ArcSin[c*x]))/(10*c^4)} -{x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (16*b*d^3*Sqrt[1 - c^2*x^2])/(315*c^3) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(945*c^3) + (2*b*d^3*(1 - c^2*x^2)^(5/2))/(525*c^3) + (b*d^3*(1 - c^2*x^2)^(7/2))/(441*c^3) - (b*d^3*(1 - c^2*x^2)^(9/2))/(81*c^3) + (1/3)*d^3*x^3*(a + b*ArcSin[c*x]) - (3/5)*c^2*d^3*x^5*(a + b*ArcSin[c*x]) + (3/7)*c^4*d^3*x^7*(a + b*ArcSin[c*x]) - (1/9)*c^6*d^3*x^9*(a + b*ArcSin[c*x])} -{x^1*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]), x, 6, (35*b*d^3*x*Sqrt[1 - c^2*x^2])/(1024*c) + (35*b*d^3*x*(1 - c^2*x^2)^(3/2))/(1536*c) + (7*b*d^3*x*(1 - c^2*x^2)^(5/2))/(384*c) + (b*d^3*x*(1 - c^2*x^2)^(7/2))/(64*c) + (35*b*d^3*ArcSin[c*x])/(1024*c^2) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x]))/(8*c^2)} -{x^0*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (16*b*d^3*Sqrt[1 - c^2*x^2])/(35*c) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(105*c) + (6*b*d^3*(1 - c^2*x^2)^(5/2))/(175*c) + (b*d^3*(1 - c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSin[c*x]) - c^2*d^3*x^3*(a + b*ArcSin[c*x]) + (3/5)*c^4*d^3*x^5*(a + b*ArcSin[c*x]) - (1/7)*c^6*d^3*x^7*(a + b*ArcSin[c*x])} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^1, x, 17, (-(19/48))*b*c*d^3*x*Sqrt[1 - c^2*x^2] - (7/72)*b*c*d^3*x*(1 - c^2*x^2)^(3/2) - (1/36)*b*c*d^3*x*(1 - c^2*x^2)^(5/2) - (19/48)*b*d^3*ArcSin[c*x] + (1/2)*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]) + (1/4)*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]) + (1/6)*d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]) - (I*d^3*(a + b*ArcSin[c*x])^2)/(2*b) + d^3*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] - (1/2)*I*b*d^3*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^2, x, 7, (-(11/5))*b*c*d^3*Sqrt[1 - c^2*x^2] - (1/5)*b*c*d^3*(1 - c^2*x^2)^(3/2) - (1/25)*b*c*d^3*(1 - c^2*x^2)^(5/2) - (d^3*(a + b*ArcSin[c*x]))/x - 3*c^2*d^3*x*(a + b*ArcSin[c*x]) + c^4*d^3*x^3*(a + b*ArcSin[c*x]) - (1/5)*c^6*d^3*x^5*(a + b*ArcSin[c*x]) - b*c*d^3*ArcTanh[Sqrt[1 - c^2*x^2]]} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^3, x, 17, (3/32)*b*c^3*d^3*x*Sqrt[1 - c^2*x^2] - (7/16)*b*c^3*d^3*x*(1 - c^2*x^2)^(3/2) - (b*c*d^3*(1 - c^2*x^2)^(5/2))/(2*x) + (3/32)*b*c^2*d^3*ArcSin[c*x] - (3/2)*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]) - (3/4)*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]) - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(2*x^2) + (3*I*c^2*d^3*(a + b*ArcSin[c*x])^2)/(2*b) - 3*c^2*d^3*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])] + (3/2)*I*b*c^2*d^3*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^4, x, 8, (8/3)*b*c^3*d^3*Sqrt[1 - c^2*x^2] - (b*c*d^3*Sqrt[1 - c^2*x^2])/(6*x^2) + (1/9)*b*c^3*d^3*(1 - c^2*x^2)^(3/2) - (d^3*(a + b*ArcSin[c*x]))/(3*x^3) + (3*c^2*d^3*(a + b*ArcSin[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcSin[c*x]) - (1/3)*c^6*d^3*x^3*(a + b*ArcSin[c*x]) + (17/6)*b*c^3*d^3*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2), x, 12, -((4*b*Sqrt[1 - c^2*x^2])/(3*c^5*d)) + (b*(1 - c^2*x^2)^(3/2))/(9*c^5*d) - (x*(a + b*ArcSin[c*x]))/(c^4*d) - (x^3*(a + b*ArcSin[c*x]))/(3*c^2*d) - (2*I*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d) + (I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d) - (I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d)} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2), x, 8, -(b*x*Sqrt[1 - c^2*x^2])/(4*c^3*d) + (b*ArcSin[c*x])/(4*c^4*d) - (x^2*(a + b*ArcSin[c*x]))/(2*c^2*d) + ((I/2)*(a + b*ArcSin[c*x])^2)/(b*c^4*d) - ((a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d) + ((I/2)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d)} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2), x, 8, -((b*Sqrt[1 - c^2*x^2])/(c^3*d)) - (x*(a + b*ArcSin[c*x]))/(c^2*d) - ((2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d) + (I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d) - (I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d)} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2), x, 5, ((I/2)*(a + b*ArcSin[c*x])^2)/(b*c^2*d) - ((a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^2*d) + ((I/2)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^2*d)} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2), x, 6, ((-2*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*d) + (I*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d) - (I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d)} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)), x, 7, -((2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d) + (I*b*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(2*d) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d)} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)), x, 10, -((a + b*ArcSin[c*x])/(d*x)) - ((2*I)*c*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d + (I*b*c*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - (I*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/d} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)), x, 9, -((b*c*Sqrt[1 - c^2*x^2])/(2*d*x)) - (a + b*ArcSin[c*x])/(2*d*x^2) - (2*c^2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d + (I*b*c^2*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(2*d) - (I*b*c^2*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d)} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)), x, 15, -(b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) - (c^2*(a + b*ArcSin[c*x]))/(d*x) - ((2*I)*c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d - (7*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (I*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - (I*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d} - - -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 12, -b/(2*c^5*d^2*Sqrt[1 - c^2*x^2]) + (b*Sqrt[1 - c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcSin[c*x]))/(2*c^4*d^2) + (x^3*(a + b*ArcSin[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) + ((3*I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d^2) + (((3*I)/2)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^2)} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 8, -(b*x)/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (b*ArcSin[c*x])/(2*c^4*d^2) + (x^2*(a + b*ArcSin[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - ((I/2)*(a + b*ArcSin[c*x])^2)/(b*c^4*d^2) + ((a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d^2) - ((I/2)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d^2)} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 8, -b/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) + (I*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d^2) - ((I/2)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^2) + ((I/2)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d^2)} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 2, -(b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(2*c^2*d^2*(1 - c^2*x^2))} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 8, -b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (I*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*d^2) + ((I/2)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d^2) - ((I/2)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d^2)} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)^2), x, 9, -((b*c*x)/(2*d^2*Sqrt[1 - c^2*x^2])) + (a + b*ArcSin[c*x])/(2*d^2*(1 - c^2*x^2)) - (2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d^2 + (I*b*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(2*d^2) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^2)} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^2), x, 13, -((b*c)/(2*d^2*Sqrt[1 - c^2*x^2])) - (a + b*ArcSin[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (3*I*c*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d^2 - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (3*I*b*c*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(2*d^2) - (3*I*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*d^2)} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^2), x, 12, -((b*c)/(2*d^2*x*Sqrt[1 - c^2*x^2])) + (c^2*(a + b*ArcSin[c*x]))/(d^2*(1 - c^2*x^2)) - (a + b*ArcSin[c*x])/(2*d^2*x^2*(1 - c^2*x^2)) - (4*c^2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d^2 + (I*b*c^2*PolyLog[2, -E^(2*I*ArcSin[c*x])])/d^2 - (I*b*c^2*PolyLog[2, E^(2*I*ArcSin[c*x])])/d^2} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^2), x, 19, -((b*c^3)/(3*d^2*Sqrt[1 - c^2*x^2])) - (b*c)/(6*d^2*x^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcSin[c*x]))/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (5*I*c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/d^2 - (13*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d^2) + (5*I*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(2*d^2) - (5*I*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*d^2)} - - -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 12, -b/(12*c^5*d^3*(1 - c^2*x^2)^(3/2)) + (5*b)/(8*c^5*d^3*Sqrt[1 - c^2*x^2]) + (x^3*(a + b*ArcSin[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a + b*ArcSin[c*x]))/(8*c^4*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d^3) + (((3*I)/8)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d^3) - (((3*I)/8)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^3)} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 4, -(b*x^3)/(12*c*d^3*(1 - c^2*x^2)^(3/2)) + (b*x)/(4*c^3*d^3*Sqrt[1 - c^2*x^2]) - (b*ArcSin[c*x])/(4*c^4*d^3) + (x^4*(a + b*ArcSin[c*x]))/(4*d^3*(1 - c^2*x^2)^2)} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 10, -b/(12*c^3*d^3*(1 - c^2*x^2)^(3/2)) + b/(8*c^3*d^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (x*(a + b*ArcSin[c*x]))/(8*c^2*d^3*(1 - c^2*x^2)) + ((I/4)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d^3) - ((I/8)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^3) + ((I/8)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d^3)} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 3, -(b*x)/(12*c*d^3*(1 - c^2*x^2)^(3/2)) - (b*x)/(6*c*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(4*c^2*d^3*(1 - c^2*x^2)^2)} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 10, -b/(12*c*d^3*(1 - c^2*x^2)^(3/2)) - (3*b)/(8*c*d^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(4*d^3*(1 - c^2*x^2)^2) + (3*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*d^3) + (((3*I)/8)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d^3) - (((3*I)/8)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d^3)} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)^3), x, 12, -((b*c*x)/(12*d^3*(1 - c^2*x^2)^(3/2))) - (2*b*c*x)/(3*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcSin[c*x])/(2*d^3*(1 - c^2*x^2)) - (2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d^3 + (I*b*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(2*d^3) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^3)} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^3), x, 16, -((b*c)/(12*d^3*(1 - c^2*x^2)^(3/2))) - (7*b*c)/(8*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])/(d^3*x*(1 - c^2*x^2)^2) + (5*c^2*x*(a + b*ArcSin[c*x]))/(4*d^3*(1 - c^2*x^2)^2) + (15*c^2*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (15*I*c*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(4*d^3) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^3 + (15*I*b*c*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(8*d^3) - (15*I*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/(8*d^3)} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^3), x, 16, -((b*c)/(2*d^3*x*(1 - c^2*x^2)^(3/2))) + (5*b*c^3*x)/(12*d^3*(1 - c^2*x^2)^(3/2)) - (2*b*c^3*x)/(3*d^3*Sqrt[1 - c^2*x^2]) + (3*c^2*(a + b*ArcSin[c*x]))/(4*d^3*(1 - c^2*x^2)^2) - (a + b*ArcSin[c*x])/(2*d^3*x^2*(1 - c^2*x^2)^2) + (3*c^2*(a + b*ArcSin[c*x]))/(2*d^3*(1 - c^2*x^2)) - (6*c^2*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/d^3 + (3*I*b*c^2*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(2*d^3) - (3*I*b*c^2*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^3)} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3), x, 23, (b*c^3)/(12*d^3*(1 - c^2*x^2)^(3/2)) - (b*c)/(6*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (29*b*c^3)/(24*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a + b*ArcSin[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (35*I*c^3*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(4*d^3) - (19*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d^3) + (35*I*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(8*d^3) - (35*I*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/(8*d^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 7, (b*x^2*Sqrt[d - c^2*d*x^2])/(32*c^3*Sqrt[1 - c^2*x^2]) + (b*x^4*Sqrt[d - c^2*d*x^2])/(96*c*Sqrt[1 - c^2*x^2]) - (b*c*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c^4) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(24*c^2) + (1/6)*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c^5*Sqrt[1 - c^2*x^2])} -{x^2*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 5, (b*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) - (b*c*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (1/4)*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{x^0*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 3, -((b*c*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2])) + (1/2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^2, x, 3, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x) - (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^4, x, 3, -((b*c*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[1 - c^2*x^2])) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*d*x^3) - (b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^6, x, 4, -((b*c*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[1 - c^2*x^2])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5*d*x^5) - (2*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(15*d*x^3) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^8, x, 4, -((b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[1 - c^2*x^2]) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(7*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) - (8*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[1 - c^2*x^2])} - -{x^5*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 3, (8*b*x*Sqrt[d - c^2*d*x^2])/(105*c^5*Sqrt[1 - c^2*x^2]) + (4*b*x^3*Sqrt[d - c^2*d*x^2])/(315*c^3*Sqrt[1 - c^2*x^2]) + (b*x^5*Sqrt[d - c^2*d*x^2])/(175*c*Sqrt[1 - c^2*x^2]) - (b*c*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^6*d) + (2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^6*d^2) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d^3)} -{x^3*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 3, (2*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (b*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[1 - c^2*x^2]) - (b*c*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^4*d) + ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^4*d^2)} -{x^1*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 2, (b*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^2*d)} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^1, x, 8, -((b*c*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2]) + Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (I*b*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^3, x, 8, -((b*c*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*x^2) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2]) + (I*b*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2)/x^5, x, 10, -((b*c*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[1 - c^2*x^2])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[1 - c^2*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*x^4) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*x^2) + (c^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(4*Sqrt[1 - c^2*x^2]) - (I*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2]) + (I*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2])} - - -{x^4*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 10, (3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(64*c^2) + (1/16)*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/8)*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^5*Sqrt[1 - c^2*x^2])} -{x^2*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 8, (b*d*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c^2) + (1/8)*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/6)*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])} -{x^0*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 6, -((5*b*c*d*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2])) + (b*c^3*d*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (3/8)*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/4)*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^2, x, 6, (b*c^3*d*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) - (3/2)*c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x - (3*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*Sqrt[1 - c^2*x^2]) + (b*c*d*Sqrt[d - c^2*d*x^2]*Log[x])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^4, x, 6, -((b*c*d*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[1 - c^2*x^2])) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^3) + (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*Sqrt[1 - c^2*x^2]) - (4*b*c^3*d*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^6, x, 4, -((b*c*d*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[1 - c^2*x^2])) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(5*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*d*x^5) + (b*c^5*d*Sqrt[d - c^2*d*x^2]*Log[x])/(5*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^8, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2])) + (2*b*c^3*d*Sqrt[d - c^2*d*x^2])/(35*x^4*Sqrt[1 - c^2*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(70*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(7*d*x^7) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*d*x^5) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Log[x])/(35*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^10, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[1 - c^2*x^2])) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[1 - c^2*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[1 - c^2*x^2]) - (2*b*c^7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(9*d*x^9) - (4*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(315*d*x^5) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^12, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[1 - c^2*x^2])) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(66*x^8*Sqrt[1 - c^2*x^2]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(1386*x^6*Sqrt[1 - c^2*x^2]) - (b*c^7*d*Sqrt[d - c^2*d*x^2])/(770*x^4*Sqrt[1 - c^2*x^2]) - (4*b*c^9*d*Sqrt[d - c^2*d*x^2])/(1155*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(11*d*x^11) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(231*d*x^7) - (16*c^6*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(1155*d*x^5) + (16*b*c^11*d*Sqrt[d - c^2*d*x^2]*Log[x])/(1155*Sqrt[1 - c^2*x^2])} - -{x^7*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 4, (16*b*d*x*Sqrt[d - c^2*d*x^2])/(1155*c^7*Sqrt[1 - c^2*x^2]) + (8*b*d*x^3*Sqrt[d - c^2*d*x^2])/(3465*c^5*Sqrt[1 - c^2*x^2]) + (2*b*d*x^5*Sqrt[d - c^2*d*x^2])/(1925*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^7*Sqrt[d - c^2*d*x^2])/(1617*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*x^9*Sqrt[d - c^2*d*x^2])/(297*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^8*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(3*c^8*d^3) + ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^8*d^4)} -{x^5*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 4, (8*b*d*x*Sqrt[d - c^2*d*x^2])/(315*c^5*Sqrt[1 - c^2*x^2]) + (4*b*d*x^3*Sqrt[d - c^2*d*x^2])/(945*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^5*Sqrt[d - c^2*d*x^2])/(525*c*Sqrt[1 - c^2*x^2]) - (10*b*c*d*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^6*d) + (2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6*d^3)} -{x^3*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 4, (2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4*d^2)} -{x^1*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 3, (b*d*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*c^2*d)} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^1, x, 10, -((4*b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2])) + (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/3)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (I*b*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^3, x, 11, -((b*c*d*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2])) + (b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (3/2)*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2]) + (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2)/x^5, x, 11, -((b*c*d*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[1 - c^2*x^2])) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[1 - c^2*x^2]) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*x^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(4*x^4) - (3*c^4*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(4*Sqrt[1 - c^2*x^2]) + (3*I*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2]) - (3*I*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2])} - - -{x^4*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 14, (3*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(512*c^3*Sqrt[1 - c^2*x^2]) + (b*d^2*x^4*Sqrt[d - c^2*d*x^2])/(512*c*Sqrt[1 - c^2*x^2]) - (31*b*c*d^2*x^6*Sqrt[d - c^2*d*x^2])/(960*Sqrt[1 - c^2*x^2]) + (21*b*c^3*d^2*x^8*Sqrt[d - c^2*d*x^2])/(640*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^10*Sqrt[d - c^2*d*x^2])/(100*Sqrt[1 - c^2*x^2]) - (3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(256*c^4) - (d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (1/32)*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/16)*d*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (1/10)*x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]) + (3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(512*b*c^5*Sqrt[1 - c^2*x^2])} -{x^2*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 12, (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (5/64)*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/48)*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (1/8)*x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]) + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])} -{x^0*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 8, -((25*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2])) + (5*b*c^3*d^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (b*d^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/24)*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (1/6)*x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]) + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^2, x, 10, (9*b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (15/8)*c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (5/4)*c^2*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x - (15*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*Sqrt[1 - c^2*x^2]) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^4, x, 10, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[1 - c^2*x^2])) - (b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (5/2)*c^4*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(3*x^3) + (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*Sqrt[1 - c^2*x^2]) - (7*b*c^3*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^6, x, 10, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[1 - c^2*x^2])) + (11*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[1 - c^2*x^2]) - (c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x + (c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^3) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(5*x^5) - (c^5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*Sqrt[1 - c^2*x^2]) + (23*b*c^5*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^8, x, 4, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[1 - c^2*x^2])) + (3*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(28*x^4*Sqrt[1 - c^2*x^2]) - (3*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(14*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*d*x^7) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(7*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^10, x, 6, -((b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[1 - c^2*x^2])) + (b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(42*x^4*Sqrt[1 - c^2*x^2]) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(21*x^2*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(1 - c^2*x^2)^(7/2)*Sqrt[d - c^2*d*x^2])/(72*x^8) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(9*d*x^9) - (2*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(63*d*x^7) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^12, x, 5, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[1 - c^2*x^2])) + (23*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(792*x^8*Sqrt[1 - c^2*x^2]) - (113*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(4158*x^6*Sqrt[1 - c^2*x^2]) + (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(924*x^4*Sqrt[1 - c^2*x^2]) + (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2])/(693*x^2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(11*d*x^11) - (4*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(99*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(693*d*x^7) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(693*Sqrt[1 - c^2*x^2])} - -{x^5*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 4, (8*b*d^2*x*Sqrt[d - c^2*d*x^2])/(693*c^5*Sqrt[1 - c^2*x^2]) + (4*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(2079*c^3*Sqrt[1 - c^2*x^2]) + (b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(1155*c*Sqrt[1 - c^2*x^2]) - (113*b*c*d^2*x^7*Sqrt[d - c^2*d*x^2])/(4851*Sqrt[1 - c^2*x^2]) + (23*b*c^3*d^2*x^9*Sqrt[d - c^2*d*x^2])/(891*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6*d) + (2*(d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^6*d^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(11*c^6*d^3)} -{x^3*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 4, (2*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c^2*x^2]) - (b*c*d^2*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2]) + (19*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^4*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(9*c^4*d^2)} -{x^1*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 3, (b*d^2*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (b*c*d^2*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^2*d)} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^1, x, 13, -((23*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2])) + (11*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])/(45*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/3)*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (1/5)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^3, x, 13, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[1 - c^2*x^2])) + (7*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - (5/2)*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (5/6)*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2]) + (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2)/x^5, x, 14, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[1 - c^2*x^2])) + (9*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (15/8)*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(8*x^2) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(4*x^4) - (15*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(4*Sqrt[1 - c^2*x^2]) + (15*I*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2]) - (15*I*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(8*Sqrt[1 - c^2*x^2])} - - -{Sqrt[1 - x^2]*ArcSin[x], x, 3, -x^2/4 + (1/2)*x*Sqrt[1 - x^2]*ArcSin[x] + ArcSin[x]^2/4} - - -{Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcSin[c*x]), x, 3, (-(1/4))*b*c*Sqrt[Pi]*x^2 + (1/2)*x*Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcSin[c*x]) + (Sqrt[Pi]*(a + b*ArcSin[c*x])^2)/(4*b*c)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 5, (3*x^2)/(16*a^3) + x^4/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4*a^2) + (3*ArcSin[a*x]^2)/(16*a^5)} -{x^3*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 4, (2*x)/(3*a^3) + x^3/(9*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^2)} -{x^2*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 3, x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3)} -{x^1*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 2, x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2} -{x^0*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 1, ArcSin[a*x]^2/(2*a)} -{ArcSin[a*x]/(x^1*Sqrt[1 - a^2*x^2]), x, 6, -2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + I*PolyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]/(x^2*Sqrt[1 - a^2*x^2]), x, 2, -((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]} -{ArcSin[a*x]/(x^3*Sqrt[1 - a^2*x^2]), x, 8, -(a/(2*x)) - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*x^2) - a^2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + (1/2)*I*a^2*PolyLog[2, -E^(I*ArcSin[a*x])] - (1/2)*I*a^2*PolyLog[2, E^(I*ArcSin[a*x])]} - - -{x^5*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 6, (8*b*x*Sqrt[1 - c^2*x^2])/(15*c^5*Sqrt[d - c^2*d*x^2]) + (4*b*x^3*Sqrt[1 - c^2*x^2])/(45*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^5*Sqrt[1 - c^2*x^2])/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*c^4*d) - (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^2*d)} -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 5, (3*b*x^2*Sqrt[1 - c^2*x^2])/(16*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^4*Sqrt[1 - c^2*x^2])/(16*c*Sqrt[d - c^2*d*x^2]) - (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^4*d) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*c^2*d) + (3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c^5*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 4, (2*b*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[1 - c^2*x^2])/(9*c*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2*d)} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 3, (b*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 2, (b*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c^2*d)} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 1, (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)^(1/2)), x, 6, -((2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^(1/2)), x, 2, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(d*x)) + (b*c*Sqrt[1 - c^2*x^2]*Log[x])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(1/2)), x, 8, -((b*c*Sqrt[1 - c^2*x^2])/(2*x*Sqrt[d - c^2*d*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*d*x^2) - (c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] + (I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*Sqrt[d - c^2*d*x^2]) - (I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(1/2)), x, 4, -((b*c*Sqrt[1 - c^2*x^2])/(6*x^2*Sqrt[d - c^2*d*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*d*x) + (2*b*c^3*Sqrt[1 - c^2*x^2]*Log[x])/(3*Sqrt[d - c^2*d*x^2])} - - -{x^5*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 5, -((5*b*x*Sqrt[d - c^2*d*x^2])/(3*c^5*d^2*Sqrt[1 - c^2*x^2])) - (b*x^3*Sqrt[d - c^2*d*x^2])/(9*c^3*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c^6*d^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^6*d^3) - (b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(c^6*d^2*Sqrt[1 - c^2*x^2])} -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 7, -((b*x^2*Sqrt[1 - c^2*x^2])/(4*c^3*d*Sqrt[d - c^2*d*x^2])) + (x^3*(a + b*ArcSin[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) + (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^4*d^2) - (3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^5*d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(2*c^5*d*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 4, -((b*x*Sqrt[d - c^2*d*x^2])/(c^3*d^2*Sqrt[1 - c^2*x^2])) + (a + b*ArcSin[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c^4*d^2) - (b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(c^4*d^2*Sqrt[1 - c^2*x^2])} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 3, (x*(a + b*ArcSin[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(2*c^3*d*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 2, (a + b*ArcSin[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 2, (x*(a + b*ArcSin[c*x]))/(d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)^(3/2)), x, 8, (a + b*ArcSin[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^(3/2)), x, 5, -((a + b*ArcSin[c*x])/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcSin[c*x]))/(d*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^2*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(2*d^2*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)), x, 11, -((b*c*Sqrt[1 - c^2*x^2])/(2*d*x*Sqrt[d - c^2*d*x^2])) + (3*c^2*(a + b*ArcSin[c*x]))/(2*d*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])/(2*d*x^2*Sqrt[d - c^2*d*x^2]) - (3*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*d*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)), x, 5, -((b*c*Sqrt[d - c^2*d*x^2])/(6*d^2*x^2*Sqrt[1 - c^2*x^2])) - (a + b*ArcSin[c*x])/(3*d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcSin[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSin[c*x]))/(3*d*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d^2*Sqrt[1 - c^2*x^2]) + (b*c^3*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(2*d^2*Sqrt[1 - c^2*x^2])} - - -{x^6*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 11, -(b/(6*c^7*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (b*x^2*Sqrt[1 - c^2*x^2])/(4*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (x^5*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (5*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^6*d^3) + (5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^7*d^2*Sqrt[d - c^2*d*x^2]) - (7*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(6*c^7*d^2*Sqrt[d - c^2*d*x^2])} -{x^5*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 5, -((b*x*Sqrt[d - c^2*d*x^2])/(6*c^5*d^3*(1 - c^2*x^2)^(3/2))) + (b*x*Sqrt[d - c^2*d*x^2])/(c^5*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])/(3*c^6*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcSin[c*x]))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c^6*d^3) + (11*b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(6*c^6*d^3*Sqrt[1 - c^2*x^2])} -{x^4*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 7, -(b/(6*c^5*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (x^3*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSin[c*x]))/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 4, -((b*x*Sqrt[d - c^2*d*x^2])/(6*c^3*d^3*(1 - c^2*x^2)^(3/2))) + (a + b*ArcSin[c*x])/(3*c^4*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (5*b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(6*c^4*d^3*Sqrt[1 - c^2*x^2])} -{x^2*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 4, -(b/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (x^3*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) - (b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 3, -((b*x)/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (a + b*ArcSin[c*x])/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*c^2*d^2*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 4, -(b/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (x*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^1*(d - c^2*d*x^2)^(5/2)), x, 11, -((b*c*x)/(6*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (a + b*ArcSin[c*x])/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcSin[c*x])/(d^2*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (7*b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)), x, 5, -((b*c*Sqrt[d - c^2*d*x^2])/(6*d^3*(1 - c^2*x^2)^(3/2))) - (a + b*ArcSin[c*x])/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^3*Sqrt[1 - c^2*x^2]) + (5*b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(6*d^3*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)), x, 15, (b*c)/(4*d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (5*b*c^3*x)/(12*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (3*b*c*Sqrt[1 - c^2*x^2])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x]))/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin[c*x]))/(2*d^2*Sqrt[d - c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (13*b*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])])/(2*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])])/(2*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)), x, 5, -((b*c^3*Sqrt[d - c^2*d*x^2])/(6*d^3*(1 - c^2*x^2)^(3/2))) - (b*c*Sqrt[d - c^2*d*x^2])/(6*d^3*x^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*ArcSin[c*x]))/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (8*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d^3*Sqrt[1 - c^2*x^2]) + (4*b*c^3*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(3*d^3*Sqrt[1 - c^2*x^2])} - - -{ArcSin[a*x]/(c - a^2*c*x^2)^(7/2), x, 6, -(1/(20*a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2])) - 2/(15*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcSin[a*x])/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x])/(15*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])/(15*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(m/2) (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{((f*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2], x, 1, (2*(f*x)^(5/2)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*f) - (4*b*c*(f*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*f^2)} - - -{((f*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2], x, 1, (2*(f*x)^(5/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*f*Sqrt[d - c^2*d*x^2]) - (4*b*c*(f*x)^(7/2)*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*f^2*Sqrt[d - c^2*d*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x) and m symbolic*) - - -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^3, x, 6, If[$VersionNumber>=8, -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)^2*(7 + m)^2)) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 - c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*Sqrt[1 - c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (3*c^2*d^3*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (c^6*d^3*x^(7 + m)*(a + b*ArcSin[c*x]))/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2), -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((7 + m)^2*(15 + 8*m + m^2)^2)) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 - c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*Sqrt[1 - c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (3*c^2*d^3*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (c^6*d^3*x^(7 + m)*(a + b*ArcSin[c*x]))/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*(105 + 71*m + 15*m^2 + m^3)^2)]} -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^2, x, 5, If[$VersionNumber>=8, -((b*c*d^2*(38 + 13*m + m^2)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)^2)) + (b*c^3*d^2*x^(4 + m)*Sqrt[1 - c^2*x^2])/(5 + m)^2 + (d^2*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (2*c^2*d^2*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) + (c^4*d^2*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (b*c*d^2*(149 + 100*m + 15*m^2)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2), -((b*c*d^2*(38 + 13*m + m^2)*x^(2 + m)*Sqrt[1 - c^2*x^2])/((3 + m)^2*(5 + m)^2)) + (b*c^3*d^2*x^(4 + m)*Sqrt[1 - c^2*x^2])/(5 + m)^2 + (d^2*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (2*c^2*d^2*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) + (c^4*d^2*x^(5 + m)*(a + b*ArcSin[c*x]))/(5 + m) - (b*c*d^2*(149 + 100*m + 15*m^2)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*(15 + 8*m + m^2)^2)]} -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^1, x, 4, If[$VersionNumber>=8, -((b*c*d*x^(2 + m)*Sqrt[1 - c^2*x^2])/(3 + m)^2) + (d*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (c^2*d*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) - (b*c*d*(7 + 3*m)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*(3 + m)^2), -((b*c*d*x^(2 + m)*Sqrt[1 - c^2*x^2])/(3 + m)^2) + (d*x^(1 + m)*(a + b*ArcSin[c*x]))/(1 + m) - (c^2*d*x^(3 + m)*(a + b*ArcSin[c*x]))/(3 + m) - (b*c*d*(7 + 3*m)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((3 + m)^2*(2 + 3*m + m^2))]} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^1, x, 0, Unintegrable[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x]} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2, x, 2, (x^(1 + m)*(a + b*ArcSin[c*x]))/(2*d^2*(1 - c^2*x^2)) - (b*c*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2*d^2*(2 + m)) + ((1 - m)*Unintegrable[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x])/(2*d)} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3, x, 4, (x^(1 + m)*(a + b*ArcSin[c*x]))/(4*d^3*(1 - c^2*x^2)^2) + ((3 - m)*x^(1 + m)*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (b*c*(3 - m)*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(8*d^3*(2 + m)) - (b*c*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*(2 + m)) + ((1 - m)*(3 - m)*Unintegrable[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x])/(8*d^2)} - - -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(5/2), x, 9, If[$VersionNumber>=8, -((15*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2])) - (5*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) - (b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((12 + 8*m + m^2)*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*(6 + m)*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^(6 + m)*Sqrt[d - c^2*d*x^2])/((6 + m)^2*Sqrt[1 - c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((4 + m)*(6 + m)) + (x^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(6 + m) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c^2*x^2]) - (15*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]), -((15*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2])) - (5*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) - (b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((12 + 8*m + m^2)*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*(6 + m)*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^(6 + m)*Sqrt[d - c^2*d*x^2])/((6 + m)^2*Sqrt[1 - c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((4 + m)*(6 + m)) + (x^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(6 + m) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c^2*x^2]) - (15*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((2 + m)^2*(6 + m)*(4 + 5*m + m^2)*Sqrt[1 - c^2*x^2])]} -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(3/2), x, 6, If[$VersionNumber>=8, -((3*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((2 + m)^2*(4 + m)*Sqrt[1 - c^2*x^2])) - (b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*Sqrt[1 - c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8 + 6*m + m^2) + (x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(4 + m) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c^2*x^2]) - (3*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2*(4 + m)*Sqrt[1 - c^2*x^2]), -((3*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((2 + m)^2*(4 + m)*Sqrt[1 - c^2*x^2])) - (b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^(4 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*Sqrt[1 - c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8 + 6*m + m^2) + (x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(4 + m) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c^2*x^2]) - (3*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((2 + m)^2*(4 + 5*m + m^2)*Sqrt[1 - c^2*x^2])]} -{x^m*(a + b*ArcSin[c*x])*(d - c^2*d*x^2)^(1/2), x, 3, -((b*c*x^(2 + m)*Sqrt[d - c^2*d*x^2])/((2 + m)^2*Sqrt[1 - c^2*x^2])) + (x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2 + m) + (x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[1 - c^2*x^2]) - (b*c*x^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2*Sqrt[1 - c^2*x^2])} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(1/2), x, 1, (x^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[d - c^2*d*x^2])} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2), x, 3, (x^(1 + m)*(a + b*ArcSin[c*x]))/(d*Sqrt[d - c^2*d*x^2]) - (m*x^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(d*(1 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d*(2 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*m*x^(2 + m)*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(d*(2 + 3*m + m^2)*Sqrt[d - c^2*d*x^2])} -{x^m*(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2), x, 5, (x^(1 + m)*(a + b*ArcSin[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + ((2 - m)*x^(1 + m)*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) - ((2 - m)*m*x^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(3*d^2*(1 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*(2 - m)*x^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d^2*(2 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d^2*(2 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(2 - m)*m*x^(2 + m)*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(3*d^2*(2 + 3*m + m^2)*Sqrt[d - c^2*d*x^2])} - - -{x^m*ArcSin[a*x]/Sqrt[1 - a^2*x^2], x, 1, (x^(1 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) - (a*x^(2 + m)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/(2 + 3*m + m^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x)^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x, 11, (-304*b^2*d*x)/(3675*c^4) - (152*b^2*d*x^3)/(11025*c^2) - (38*b^2*d*x^5)/6125 + (2*b^2*c^2*d*x^7)/343 + (32*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c^5) + (16*b*d*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(525*c^3) + (4*b*d*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(21*c^5) - (4*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(35*c^5) + (2*b*d*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(49*c^5) + (2*d*x^5*(a + b*ArcSin[c*x])^2)/35 + (d*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7} -{x^3*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x, 14, -(b^2*d*x^2)/(24*c^2) - (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(12*c^3) + (b*d*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/18 - (d*(a + b*ArcSin[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSin[c*x])^2)/12 + (d*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/6} -{x^2*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x, 9, (-52*b^2*d*x)/(225*c^2) - (26*b^2*d*x^3)/675 + (2*b^2*c^2*d*x^5)/125 + (8*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(45*c^3) + (4*b*d*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(45*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(15*c^3) - (2*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(25*c^3) + (2*d*x^3*(a + b*ArcSin[c*x])^2)/15 + (d*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/5} -{x^1*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x, 7, (-5*b^2*d*x^2)/32 + (b^2*c^2*d*x^4)/32 + (3*b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c) + (b*d*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(8*c) + (3*d*(a + b*ArcSin[c*x])^2)/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(4*c^2)} -{x^0*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x, 6, (-14*b^2*d*x)/9 + (2*b^2*c^2*d*x^3)/27 + (4*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(9*c) + (2*d*x*(a + b*ArcSin[c*x])^2)/3 + (d*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/3} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^1, x, 10, (b^2*c^2*d*x^2)/4 - (b*c*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 - (d*(a + b*ArcSin[c*x])^2)/4 + (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 - ((I/3)*d*(a + b*ArcSin[c*x])^3)/b + d*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*d*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^2, x, 12, 2*b^2*c^2*d*x - 2*b*c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 2*c^2*d*x*(a + b*ArcSin[c*x])^2 - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/x - 4*b*c*d*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*b^2*c*d*PolyLog[2, E^(I*ArcSin[c*x])]} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^3, x, 10, -((b*c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/x) - (c^2*d*(a + b*ArcSin[c*x])^2)/2 - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*x^2) + ((I/3)*c^2*d*(a + b*ArcSin[c*x])^3)/b - c^2*d*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] + b^2*c^2*d*Log[x] + I*b*c^2*d*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (b^2*c^2*d*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2} -{((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^4, x, 16, -(b^2*c^2*d)/(3*x) - (b*c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (2*c^2*d*(a + b*ArcSin[c*x])^2)/(3*x) - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*x^3) + (10*b*c^3*d*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((5*I)/3)*b^2*c^3*d*PolyLog[2, -E^(I*ArcSin[c*x])] + ((5*I)/3)*b^2*c^3*d*PolyLog[2, E^(I*ArcSin[c*x])]} - - -{x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x, 16, (-4208*b^2*d^2*x)/(99225*c^4) - (2104*b^2*d^2*x^3)/(297675*c^2) - (526*b^2*d^2*x^5)/165375 + (212*b^2*c^2*d^2*x^7)/27783 - (2*b^2*c^4*d^2*x^9)/729 + (128*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(4725*c^5) + (64*b*d^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(4725*c^3) + (16*b*d^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(1575*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(189*c^5) - (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(315*c^5) - (20*b*d^2*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(441*c^5) + (2*b*d^2*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(81*c^5) + (8*d^2*x^5*(a + b*ArcSin[c*x])^2)/315 + (4*d^2*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/63 + (d^2*x^5*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/9} -{x^3*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x, 25, (-73*b^2*d^2*x^2)/(3072*c^2) - (73*b^2*d^2*x^4)/9216 + (43*b^2*c^2*d^2*x^6)/3456 - (b^2*c^4*d^2*x^8)/256 + (73*b*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(1536*c^3) + (73*b*d^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2304*c) - (25*b*c*d^2*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/576 - (b*c*d^2*x^5*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/32 - (73*d^2*(a + b*ArcSin[c*x])^2)/(3072*c^4) + (d^2*x^4*(a + b*ArcSin[c*x])^2)/24 + (d^2*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/12 + (d^2*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/8} -{x^2*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x, 14, (-1636*b^2*d^2*x)/(11025*c^2) - (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 - (2*b^2*c^4*d^2*x^7)/343 + (32*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(315*c^3) + (16*b*d^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(315*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(105*c^3) + (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(175*c^3) - (2*b*d^2*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(49*c^3) + (8*d^2*x^3*(a + b*ArcSin[c*x])^2)/105 + (4*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (d^2*x^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/7} -{x^1*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x, 9, (-25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 - c^2*x^2)^3)/(108*c^2) + (5*b*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (5*b*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(72*c) + (b*d^2*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(18*c) + (5*d^2*(a + b*ArcSin[c*x])^2)/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(6*c^2)} -{x^0*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x, 10, (-298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 - (2*b^2*c^4*d^2*x^5)/125 + (16*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(15*c) + (8*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(45*c) + (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSin[c*x])^2)/15 + (4*d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/15 + (d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^1, x, 17, (13*b^2*c^2*d^2*x^2)/32 - (b^2*c^4*d^2*x^4)/32 - (11*b*c*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 - (b*c*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/8 - (11*d^2*(a + b*ArcSin[c*x])^2)/32 + (d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 + (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 - ((I/3)*d^2*(a + b*ArcSin[c*x])^3)/b + d^2*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*d^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^2, x, 17, (32*b^2*c^2*d^2*x)/9 - (2*b^2*c^4*d^2*x^3)/27 - (10*b*c*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/3 - (2*b*c*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (8*c^2*d^2*x*(a + b*ArcSin[c*x])^2)/3 - (4*c^2*d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/3 - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - 4*b*c*d^2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^2*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*b^2*c*d^2*PolyLog[2, E^(I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^3, x, 17, -(b^2*c^4*d^2*x^2)/4 - (b*c^3*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/2 - (b*c*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x - (c^2*d^2*(a + b*ArcSin[c*x])^2)/4 - c^2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2 - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(2*x^2) + (((2*I)/3)*c^2*d^2*(a + b*ArcSin[c*x])^3)/b - 2*c^2*d^2*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] + b^2*c^2*d^2*Log[x] + (2*I)*b*c^2*d^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - b^2*c^2*d^2*PolyLog[3, E^((2*I)*ArcSin[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^4, x, 24, -(b^2*c^2*d^2)/(3*x) - 2*b^2*c^4*d^2*x + (5*b*c^3*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/3 - (b*c*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*x^2) + (8*c^4*d^2*x*(a + b*ArcSin[c*x])^2)/3 + (4*c^2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*x) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*x^3) + (22*b*c^3*d^2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((11*I)/3)*b^2*c^3*d^2*PolyLog[2, -E^(I*ArcSin[c*x])] + ((11*I)/3)*b^2*c^3*d^2*PolyLog[2, E^(I*ArcSin[c*x])]} - - -{x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x, 21, (-100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) - (12622*b^2*d^3*x^5)/6670125 + (9410*b^2*c^2*d^3*x^7)/1120581 - (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 + (256*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(17325*c^3) + (32*b*d^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(5775*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(693*c^5) - (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(1155*c^5) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(1617*c^5) - (8*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(297*c^5) + (2*b*d^3*(1 - c^2*x^2)^(11/2)*(a + b*ArcSin[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSin[c*x])^2)/1155 + (8*d^3*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/231 + (2*d^3*x^5*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/33 + (d^3*x^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/11} -{x^3*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x, 40, (-79*b^2*d^3*x^2)/(5120*c^2) - (79*b^2*d^3*x^4)/15360 + (401*b^2*c^2*d^3*x^6)/28800 - (57*b^2*c^4*d^3*x^8)/6400 + (b^2*c^6*d^3*x^10)/500 + (79*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2560*c^3) + (79*b*d^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3840*c) - (31*b*c*d^3*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/960 - (b*c*d^3*x^5*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/32 - (b*c*d^3*x^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/50 - (79*d^3*(a + b*ArcSin[c*x])^2)/(5120*c^4) + (d^3*x^4*(a + b*ArcSin[c*x])^2)/40 + (d^3*x^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/20 + (3*d^3*x^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/40 + (d^3*x^4*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/10} -{x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x, 19, (-10516*b^2*d^3*x)/(99225*c^2) - (5258*b^2*d^3*x^3)/297675 + (4198*b^2*c^2*d^3*x^5)/165375 - (374*b^2*c^4*d^3*x^7)/27783 + (2*b^2*c^6*d^3*x^9)/729 + (64*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c^3) + (32*b*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(945*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(315*c^3) + (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(525*c^3) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(441*c^3) - (2*b*d^3*(1 - c^2*x^2)^(9/2)*(a + b*ArcSin[c*x]))/(81*c^3) + (16*d^3*x^3*(a + b*ArcSin[c*x])^2)/315 + (8*d^3*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/105 + (2*d^3*x^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/21 + (d^3*x^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/9} -{x^1*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x, 11, (-175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 - c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 - c^2*x^2)^4)/(256*c^2) + (35*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(512*c) + (35*b*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(768*c) + (7*b*d^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(192*c) + (b*d^3*x*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(32*c) + (35*d^3*(a + b*ArcSin[c*x])^2)/(1024*c^2) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x])^2)/(8*c^2)} -{x^0*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x, 14, (-4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 - (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 + (32*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(35*c) + (16*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(105*c) + (12*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(175*c) + (2*b*d^3*(1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSin[c*x])^2)/35 + (8*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/35 + (6*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/35 + (d^3*x*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/7} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^1, x, 26, (71*b^2*c^2*d^3*x^2)/144 - (7*b^2*c^4*d^3*x^4)/144 - (b^2*d^3*(1 - c^2*x^2)^3)/108 - (19*b*c*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/24 - (7*b*c*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/36 - (b*c*d^3*x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/18 - (19*d^3*(a + b*ArcSin[c*x])^2)/48 + (d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 + (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/6 - ((I/3)*d^3*(a + b*ArcSin[c*x])^3)/b + d^3*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] - I*b*d^3*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] + (b^2*d^3*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^2, x, 24, (122*b^2*c^2*d^3*x)/25 - (14*b^2*c^4*d^3*x^3)/75 + (2*b^2*c^6*d^3*x^5)/125 - (22*b*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/5 - (2*b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/25 - (16*c^2*d^3*x*(a + b*ArcSin[c*x])^2)/5 - (8*c^2*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/5 - (6*c^2*d^3*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/5 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/x - 4*b*c*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + (2*I)*b^2*c*d^3*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*b^2*c*d^3*PolyLog[2, E^(I*ArcSin[c*x])]} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^3, x, 28, (-21*b^2*c^4*d^3*x^2)/32 + (b^2*c^6*d^3*x^4)/32 + (3*b*c^3*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/16 - (7*b*c^3*d^3*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/8 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x + (3*c^2*d^3*(a + b*ArcSin[c*x])^2)/32 - (3*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/2 - (3*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/4 - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(2*x^2) + (I*c^2*d^3*(a + b*ArcSin[c*x])^3)/b - 3*c^2*d^3*(a + b*ArcSin[c*x])^2*Log[1 - E^((2*I)*ArcSin[c*x])] + b^2*c^2*d^3*Log[x] + (3*I)*b*c^2*d^3*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])] - (3*b^2*c^2*d^3*PolyLog[3, E^((2*I)*ArcSin[c*x])])/2} -{((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^4, x, 31, -(b^2*c^2*d^3)/(3*x) - (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 + 5*b*c^3*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (b*c^3*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/9 - (b*c*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSin[c*x])^2)/3 + (8*c^4*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/3 + (2*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/x - (d^3*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(3*x^3) + (34*b*c^3*d^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/3 - ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, -E^(I*ArcSin[c*x])] + ((17*I)/3)*b^2*c^3*d^3*PolyLog[2, E^(I*ArcSin[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2), x, 16, (22*b^2*x)/(9*c^4*d) + (2*b^2*x^3)/(27*c^2*d) - (22*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^5*d) - (2*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3*d) - (x*(a + b*ArcSin[c*x])^2)/(c^4*d) - (x^3*(a + b*ArcSin[c*x])^2)/(3*c^2*d) - ((2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d) + ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d) - ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d) - (2*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c^5*d) + (2*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c^5*d)} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2), x, 10, (b^2*x^2)/(4*c^2*d) - (b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^3*d) + (a + b*ArcSin[c*x])^2/(4*c^4*d) - (x^2*(a + b*ArcSin[c*x])^2)/(2*c^2*d) + ((I/3)*(a + b*ArcSin[c*x])^3)/(b*c^4*d) - ((a + b*ArcSin[c*x])^2*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d) - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*c^4*d)} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2), x, 11, (2*b^2*x)/(c^2*d) - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c^3*d) - (x*(a + b*ArcSin[c*x])^2)/(c^2*d) - ((2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d) + ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d) - ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d) - (2*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c^3*d) + (2*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c^3*d)} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2), x, 6, ((I/3)*(a + b*ArcSin[c*x])^3)/(b*c^2*d) - ((a + b*ArcSin[c*x])^2*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^2*d) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^2*d) - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*c^2*d)} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2), x, 8, ((-2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c*d) + ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d) - ((2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d) - (2*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c*d) + (2*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c*d)} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)), x, 9, (-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d) + (b^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/(2*d)} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)), x, 15, -((a + b*ArcSin[c*x])^2/(d*x)) - ((2*I)*c*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d - (4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/d + ((2*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])])/d + ((2*I)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - ((2*I)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d - ((2*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])])/d - (2*b^2*c*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/d + (2*b^2*c*PolyLog[3, I*E^(I*ArcSin[c*x])])/d} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)), x, 12, -((b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(d*x)) - (a + b*ArcSin[c*x])^2/(2*d*x^2) - (2*c^2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d + (b^2*c^2*Log[x])/d + (I*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d - (I*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d - (b^2*c^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d) + (b^2*c^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/(2*d)} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)), x, 24, -(b^2*c^2)/(3*d*x) - (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*d*x^2) - (a + b*ArcSin[c*x])^2/(3*d*x^3) - (c^2*(a + b*ArcSin[c*x])^2)/(d*x) - ((2*I)*c^3*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d - (14*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d) + (((7*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d + ((2*I)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d - ((2*I)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d - (((7*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d - (2*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/d + (2*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/d} - - -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 15, (-2*b^2*x)/(c^4*d^2) - (b*(a + b*ArcSin[c*x]))/(c^5*d^2*Sqrt[1 - c^2*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c^5*d^2) + (3*x*(a + b*ArcSin[c*x])^2)/(2*c^4*d^2) + (x^3*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) + ((3*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d^2) + (b^2*ArcTanh[c*x])/(c^5*d^2) - ((3*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d^2) + ((3*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^2) + (3*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c^5*d^2) - (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c^5*d^2)} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 10, -((b*x*(a + b*ArcSin[c*x]))/(c^3*d^2*Sqrt[1 - c^2*x^2])) + (a + b*ArcSin[c*x])^2/(2*c^4*d^2) + (x^2*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) - ((I/3)*(a + b*ArcSin[c*x])^3)/(b*c^4*d^2) + ((a + b*ArcSin[c*x])^2*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^4*d^2) - (b^2*Log[1 - c^2*x^2])/(2*c^4*d^2) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c^4*d^2) + (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*c^4*d^2)} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 11, -((b*(a + b*ArcSin[c*x]))/(c^3*d^2*Sqrt[1 - c^2*x^2])) + (x*(a + b*ArcSin[c*x])^2)/(2*c^2*d^2*(1 - c^2*x^2)) + (I*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d^2) + (b^2*ArcTanh[c*x])/(c^3*d^2) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^2) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d^2) + (b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^2) - (b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c^3*d^2)} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 3, -((b*x*(a + b*ArcSin[c*x]))/(c*d^2*Sqrt[1 - c^2*x^2])) + (a + b*ArcSin[c*x])^2/(2*c^2*d^2*(1 - c^2*x^2)) - (b^2*Log[1 - c^2*x^2])/(2*c^2*d^2)} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 11, -((b*(a + b*ArcSin[c*x]))/(c*d^2*Sqrt[1 - c^2*x^2])) + (x*(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) - (I*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c*d^2) + (b^2*ArcTanh[c*x])/(c*d^2) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d^2) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d^2) - (b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(c*d^2) + (b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c*d^2)} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)^2), x, 12, -((b*c*x*(a + b*ArcSin[c*x]))/(d^2*Sqrt[1 - c^2*x^2])) + (a + b*ArcSin[c*x])^2/(2*d^2*(1 - c^2*x^2)) - (2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d^2 - (b^2*Log[1 - c^2*x^2])/(2*d^2) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d^2 - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^2 - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d^2) + (b^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/(2*d^2)} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^2), x, 20, -((b*c*(a + b*ArcSin[c*x]))/(d^2*Sqrt[1 - c^2*x^2])) - (a + b*ArcSin[c*x])^2/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) - ((3*I)*c*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^2 - (4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/d^2 + (b^2*c*ArcTanh[c*x])/d^2 + ((2*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])])/d^2 + ((3*I)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^2 - ((3*I)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2 - ((2*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])])/d^2 - (3*b^2*c*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/d^2 + (3*b^2*c*PolyLog[3, I*E^(I*ArcSin[c*x])])/d^2} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^2), x, 17, -((b*c*(a + b*ArcSin[c*x]))/(d^2*x*Sqrt[1 - c^2*x^2])) + (c^2*(a + b*ArcSin[c*x])^2)/(d^2*(1 - c^2*x^2)) - (a + b*ArcSin[c*x])^2/(2*d^2*x^2*(1 - c^2*x^2)) - (4*c^2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d^2 + (b^2*c^2*Log[x])/d^2 - (b^2*c^2*Log[1 - c^2*x^2])/(2*d^2) + ((2*I)*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d^2 - ((2*I)*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^2 - (b^2*c^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/d^2 + (b^2*c^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/d^2} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^2), x, 32, -(b^2*c^2)/(3*d^2*x) - (2*b*c^3*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(3*d^2*x^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcSin[c*x])^2)/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) - ((5*I)*c^3*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^2 - (26*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d^2) + (b^2*c^3*ArcTanh[c*x])/d^2 + (((13*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d^2 + ((5*I)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^2 - ((5*I)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2 - (((13*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d^2 - (5*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/d^2 + (5*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/d^2} - - -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 16, (b^2*x)/(12*c^4*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c^5*d^3*(1 - c^2*x^2)^(3/2)) + (5*b*(a + b*ArcSin[c*x]))/(4*c^5*d^3*Sqrt[1 - c^2*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a + b*ArcSin[c*x])^2)/(8*c^4*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^5*d^3) - (7*b^2*ArcTanh[c*x])/(6*c^5*d^3) + (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^5*d^3) - (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d^3) - (3*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(4*c^5*d^3) + (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^3)} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 8, b^2/(12*c^4*d^3*(1 - c^2*x^2)) - (b*x^3*(a + b*ArcSin[c*x]))/(6*c*d^3*(1 - c^2*x^2)^(3/2)) + (b*x*(a + b*ArcSin[c*x]))/(2*c^3*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(4*c^4*d^3) + (x^4*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) + (b^2*Log[1 - c^2*x^2])/(3*c^4*d^3)} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 15, (b^2*x)/(12*c^2*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c^3*d^3*(1 - c^2*x^2)^(3/2)) + (b*(a + b*ArcSin[c*x]))/(4*c^3*d^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x])^2)/(4*c^2*d^3*(1 - c^2*x^2)^2) - (x*(a + b*ArcSin[c*x])^2)/(8*c^2*d^3*(1 - c^2*x^2)) + ((I/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c^3*d^3) - (b^2*ArcTanh[c*x])/(6*c^3*d^3) - ((I/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*d^3) + ((I/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d^3) + (b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(4*c^3*d^3) - (b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^3*d^3)} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 5, b^2/(12*c^2*d^3*(1 - c^2*x^2)) - (b*x*(a + b*ArcSin[c*x]))/(6*c*d^3*(1 - c^2*x^2)^(3/2)) - (b*x*(a + b*ArcSin[c*x]))/(3*c*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])^2/(4*c^2*d^3*(1 - c^2*x^2)^2) - (b^2*Log[1 - c^2*x^2])/(6*c^2*d^3)} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 15, (b^2*x)/(12*d^3*(1 - c^2*x^2)) - (b*(a + b*ArcSin[c*x]))/(6*c*d^3*(1 - c^2*x^2)^(3/2)) - (3*b*(a + b*ArcSin[c*x]))/(4*c*d^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) + (3*x*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/(c*d^3) + (5*b^2*ArcTanh[c*x])/(6*c*d^3) + (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d^3) - (((3*I)/4)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d^3) - (3*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(4*c*d^3) + (3*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c*d^3)} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)^3), x, 17, b^2/(12*d^3*(1 - c^2*x^2)) - (b*c*x*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (4*b*c*x*(a + b*ArcSin[c*x]))/(3*d^3*Sqrt[1 - c^2*x^2]) + (a + b*ArcSin[c*x])^2/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcSin[c*x])^2/(2*d^3*(1 - c^2*x^2)) - (2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d^3 - (2*b^2*Log[1 - c^2*x^2])/(3*d^3) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d^3 - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^3 - (b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d^3) + (b^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/(2*d^3)} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^3), x, 27, (b^2*c^2*x)/(12*d^3*(1 - c^2*x^2)) - (b*c*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (7*b*c*(a + b*ArcSin[c*x]))/(4*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(d^3*x*(1 - c^2*x^2)^2) + (5*c^2*x*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) + (15*c^2*x*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) - (((15*I)/4)*c*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^3 - (4*b*c*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/d^3 + (11*b^2*c*ArcTanh[c*x])/(6*d^3) + ((2*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])])/d^3 + (((15*I)/4)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((15*I)/4)*b*c*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3 - ((2*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])])/d^3 - (15*b^2*c*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(4*d^3) + (15*b^2*c*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*d^3)} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^3), x, 23, (b^2*c^2)/(12*d^3*(1 - c^2*x^2)) - (b*c*(a + b*ArcSin[c*x]))/(d^3*x*(1 - c^2*x^2)^(3/2)) + (5*b*c^3*x*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (4*b*c^3*x*(a + b*ArcSin[c*x]))/(3*d^3*Sqrt[1 - c^2*x^2]) + (3*c^2*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) - (a + b*ArcSin[c*x])^2/(2*d^3*x^2*(1 - c^2*x^2)^2) + (3*c^2*(a + b*ArcSin[c*x])^2)/(2*d^3*(1 - c^2*x^2)) - (6*c^2*(a + b*ArcSin[c*x])^2*ArcTanh[E^((2*I)*ArcSin[c*x])])/d^3 + (b^2*c^2*Log[x])/d^3 - (7*b^2*c^2*Log[1 - c^2*x^2])/(6*d^3) + ((3*I)*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/d^3 - ((3*I)*b*c^2*(a + b*ArcSin[c*x])*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^3 - (3*b^2*c^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(2*d^3) + (3*b^2*c^2*PolyLog[3, E^((2*I)*ArcSin[c*x])])/(2*d^3)} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^3), x, 43, -(b^2*c^2)/(2*d^3*x) + (b^2*c^2)/(6*d^3*x*(1 - c^2*x^2)) - (b^2*c^4*x)/(12*d^3*(1 - c^2*x^2)) + (b*c^3*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2)) - (b*c*(a + b*ArcSin[c*x]))/(3*d^3*x^2*(1 - c^2*x^2)^(3/2)) - (29*b*c^3*(a + b*ArcSin[c*x]))/(12*d^3*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^2/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a + b*ArcSin[c*x])^2)/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(12*d^3*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) - (((35*I)/4)*c^3*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])])/d^3 - (38*b*c^3*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])])/(3*d^3) + (17*b^2*c^3*ArcTanh[c*x])/(6*d^3) + (((19*I)/3)*b^2*c^3*PolyLog[2, -E^(I*ArcSin[c*x])])/d^3 + (((35*I)/4)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/d^3 - (((35*I)/4)*b*c^3*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3 - (((19*I)/3)*b^2*c^3*PolyLog[2, E^(I*ArcSin[c*x])])/d^3 - (35*b^2*c^3*PolyLog[3, (-I)*E^(I*ArcSin[c*x])])/(4*d^3) + (35*b^2*c^3*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*d^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2), x, 14, (52*b^2*Sqrt[d - c^2*d*x^2])/(225*c^4) + (4*a*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (26*b^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(675*c^4) - (2*b^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^4) + (4*b^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(15*c^3*Sqrt[1 - c^2*x^2]) + (2*b*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*c*Sqrt[1 - c^2*x^2]) - (2*b*c*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^4) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^2) + (1/5)*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2} -{x^2*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2), x, 10, (b^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) - (1/32)*b^2*x^3*Sqrt[d - c^2*d*x^2] - (b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (b*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (b*c*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (1/4)*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(24*b*c^3*Sqrt[1 - c^2*x^2])} -{x^1*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2), x, 5, (4*b^2*Sqrt[d - c^2*d*x^2])/(9*c^2) + (2*b^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*c^2) + (2*b*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (2*b*c*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*c^2*d)} -{x^0*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2), x, 5, (-(1/4))*b^2*x*Sqrt[d - c^2*d*x^2] + (b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b*c*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (1/2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2)/x^1, x, 12, -2*b^2*Sqrt[d - c^2*d*x^2] - (2*a*b*c*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (2*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*I*b*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*I*b*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2)/x^2, x, 7, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x) - (I*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*Sqrt[1 - c^2*x^2]) + (2*b*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b^2*c*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2)/x^3, x, 13, -((b*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[1 - c^2*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*x^2) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*Sqrt[d - c^2*d*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2] - (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2)/x^4, x, 9, -((b^2*c^2*Sqrt[d - c^2*d*x^2])/(3*x)) - (b^2*c^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (I*c^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*d*x^3) - (2*b*c^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + (I*b^2*c^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2])} - - -{x^3*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2), x, 20, (304*b^2*d*Sqrt[d - c^2*d*x^2])/(3675*c^4) + (4*a*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (152*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11025*c^4) + (38*b^2*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(6125*c^4) - (2*b^2*d*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^4) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(35*c^3*Sqrt[1 - c^2*x^2]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(105*c*Sqrt[1 - c^2*x^2]) - (16*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(175*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^4) - (d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^2) + (3/35)*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/7)*x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2} -{x^2*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2), x, 17, -((7*b^2*d*x*Sqrt[d - c^2*d*x^2])/(1152*c^2)) - (43*b^2*d*x^3*Sqrt[d - c^2*d*x^2])/1728 + (1/108)*b^2*c^2*d*x^5*Sqrt[d - c^2*d*x^2] + (7*b^2*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2]) + (b*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*Sqrt[1 - c^2*x^2]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (1/8)*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/6)*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c^3*Sqrt[1 - c^2*x^2])} -{x^1*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2), x, 6, (16*b^2*d*Sqrt[d - c^2*d*x^2])/(75*c^2) + (8*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) + (2*b^2*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(5*c^2*d)} -{x^0*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2), x, 10, (-(17/64))*b^2*d*x*Sqrt[d - c^2*d*x^2] + (1/32)*b^2*c^2*d*x^3*Sqrt[d - c^2*d*x^2] + (17*b^2*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (3/8)*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/4)*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2]), (-(15/64))*b^2*d*x*Sqrt[d - c^2*d*x^2] - (1/32)*b^2*d*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] + (9*b^2*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (b*d*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (3/8)*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/4)*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2)/x^1, x, 17, (-(22/9))*b^2*d*Sqrt[d - c^2*d*x^2] - (2*a*b*c*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (2/27)*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] - (2*b^2*c*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*b*c*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/3)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*I*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*I*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2)/x^2, x, 14, (1/4)*b^2*c^2*d*x*Sqrt[d - c^2*d*x^2] - (5*b^2*c*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + b*c*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (3/2)*c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (I*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x - (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*Sqrt[1 - c^2*x^2]) + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b^2*c*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2)/x^3, x, 18, 2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (3*b^2*c^3*d*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[1 - c^2*x^2]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] - (3/2)*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2] - (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (3*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2)/x^4, x, 16, -((b^2*c^2*d*Sqrt[d - c^2*d*x^2])/(3*x)) - (b^2*c^3*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*Sqrt[1 - c^2*x^2]) - (b*c*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x + (4*I*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*x^3) + (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*Sqrt[1 - c^2*x^2]) - (8*b*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + (4*I*b^2*c^3*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2])} - - -{x^3*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2), x, 27, (160*b^2*d^2*Sqrt[d - c^2*d*x^2])/(3969*c^4) + (4*a*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (80*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11907*c^4) + (4*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1323*c^4) + (50*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(27783*c^4) - (2*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(729*c^4) + (4*b^2*d^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(63*c^3*Sqrt[1 - c^2*x^2]) + (2*b*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(189*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(21*Sqrt[1 - c^2*x^2]) + (38*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(441*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(81*Sqrt[1 - c^2*x^2]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^4) - (d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^2) + (1/21)*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/63)*d*x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (1/9)*x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2} -{x^2*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2), x, 25, -((359*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2)) - (1079*b^2*d^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (1/256)*b^2*c^4*d^2*x^7*Sqrt[d - c^2*d*x^2] + (359*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2]) + (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) + (5/64)*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/48)*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (1/8)*x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(384*b*c^3*Sqrt[1 - c^2*x^2])} -{x^1*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2), x, 6, (32*b^2*d^2*Sqrt[d - c^2*d*x^2])/(245*c^2) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(735*c^2) + (12*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) + (2*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (2*b*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcSin[c*x])^2)/(7*c^2*d)} -{x^0*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2), x, 16, -((245*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/1152) - (65*b^2*d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 - (1/108)*b^2*d^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2] + (115*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (5*b*d^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5/16)*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/24)*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (1/6)*x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2 + (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2)/x^1, x, 23, (-(598/225))*b^2*d^2*Sqrt[d - c^2*d*x^2] - (2*a*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] - (74/675)*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] - (2/125)*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2] - (2*b^2*c*d^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (16*b*c*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (22*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/3)*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (1/5)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2 - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*I*b*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*I*b*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2)/x^2, x, 23, (31/64)*b^2*c^2*d^2*x*Sqrt[d - c^2*d*x^2] + (1/32)*b^2*c^2*d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] - (89*b^2*c*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*Sqrt[1 - c^2*x^2]) + (15*b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + b*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (1/8)*b*c*d^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (15/8)*c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (I*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (5/4)*c^2*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x - (5*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*Sqrt[1 - c^2*x^2]) + (2*b*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b^2*c*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2)/x^3, x, 25, (40/9)*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2] + (5*a*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (2/27)*b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] + (5*b^2*c^3*d^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (b*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (5/2)*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (5/6)*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (b^2*c^2*d^2*Sqrt[d - c^2*d*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2] - (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (5*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (5*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2)/x^4, x, 27, (-(7/12))*b^2*c^4*d^2*x*Sqrt[d - c^2*d*x^2] - (b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(3*x) + (23*b^2*c^3*d^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12*Sqrt[1 - c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (7/3)*b*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (b*c*d^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + (5/2)*c^4*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (7*I*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*Sqrt[1 - c^2*x^2]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(3*x^3) + (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*Sqrt[1 - c^2*x^2]) - (14*b*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + (7*I*b^2*c^3*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 14, (16*a*b*x*Sqrt[1 - c^2*x^2])/(15*c^5*Sqrt[d - c^2*d*x^2]) + (298*b^2*(1 - c^2*x^2))/(225*c^6*Sqrt[d - c^2*d*x^2]) - (76*b^2*(1 - c^2*x^2)^2)/(675*c^6*Sqrt[d - c^2*d*x^2]) + (2*b^2*(1 - c^2*x^2)^3)/(125*c^6*Sqrt[d - c^2*d*x^2]) + (16*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) + (8*b*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(45*c^3*Sqrt[d - c^2*d*x^2]) + (2*b*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^4*d) - (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2*d)} -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 10, (15*b^2*x*(1 - c^2*x^2))/(64*c^4*Sqrt[d - c^2*d*x^2]) + (b^2*x^3*(1 - c^2*x^2))/(32*c^2*Sqrt[d - c^2*d*x^2]) - (15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(64*c^5*Sqrt[d - c^2*d*x^2]) + (3*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[d - c^2*d*x^2]) - (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^4*d) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c^5*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 9, (4*a*b*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (14*b^2*(1 - c^2*x^2))/(9*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*(1 - c^2*x^2)^2)/(27*c^4*Sqrt[d - c^2*d*x^2]) + (4*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (2*b*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^2*d)} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 5, (b^2*x*Sqrt[d - c^2*d*x^2])/(4*c^2*d) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]), (b^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 4, (2*a*b*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) + (2*b^2*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d)} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 1, (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)^(1/2)), x, 8, -((2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(1/2)), x, 6, -((I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(d*x) + (2*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(1/2)), x, 13, -((b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(x*Sqrt[d - c^2*d*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*d*x^2) - (c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (b^2*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[d - c^2*d*x^2] + (I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] - (b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2] + (b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(1/2)), x, 9, -((b^2*c^2*(1 - c^2*x^2))/(3*x*Sqrt[d - c^2*d*x^2])) - (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*x^2*Sqrt[d - c^2*d*x^2]) - (2*I*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*d*x) + (4*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/(3*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*Sqrt[d - c^2*d*x^2])} - - -{x^5*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 22, -((16*a*b*x*Sqrt[1 - c^2*x^2])/(3*c^5*d*Sqrt[d - c^2*d*x^2])) - (32*b^2*(1 - c^2*x^2))/(9*c^6*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*(1 - c^2*x^2)^2)/(27*c^6*d*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) + (2*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^4*d^2) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^6*d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^6*d*Sqrt[d - c^2*d*x^2])} -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 14, -((b^2*x*(1 - c^2*x^2))/(4*c^4*d*Sqrt[d - c^2*d*x^2])) + (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^5*d*Sqrt[d - c^2*d*x^2]) - (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^5*d*Sqrt[d - c^2*d*x^2]) + (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*c^4*d^2) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c^5*d*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 13, -((4*a*b*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2])) - (2*b^2*(1 - c^2*x^2))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (4*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (2*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c^3*d*Sqrt[d - c^2*d*x^2]) + (x^2*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2])} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 7, (x*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 7, (a + b*ArcSin[c*x])^2/(c^2*d*Sqrt[d - c^2*d*x^2]) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 6, (x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c*d*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)^(3/2)), x, 15, (a + b*ArcSin[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(3/2)), x, 14, -((a + b*ArcSin[c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (2*I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)), x, 26, -((b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(d*x*Sqrt[d - c^2*d*x^2])) + (3*c^2*(a + b*ArcSin[c*x])^2)/(2*d*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + (4*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (3*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(d*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (3*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) + (3*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)), x, 24, -((b^2*c^2*(1 - c^2*x^2))/(3*d*x*Sqrt[d - c^2*d*x^2])) - (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*d*x^2*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcSin[c*x])^2)/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*Sqrt[d - c^2*d*x^2]) - (8*I*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*d*Sqrt[d - c^2*d*x^2]) - (20*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) + (16*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (5*I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*d*Sqrt[d - c^2*d*x^2])} - - -{x^5*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 26, b^2/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (16*a*b*x*Sqrt[1 - c^2*x^2])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*(1 - c^2*x^2))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) + (16*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (b*x^3*(a + b*ArcSin[c*x]))/(3*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (11*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcSin[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcSin[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^6*d^3) - (22*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (11*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])} -{x^4*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 16, (b^2*x)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (b*x^2*(a + b*ArcSin[c*x]))/(3*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (4*I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (8*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (4*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 16, b^2/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*x*(a + b*ArcSin[c*x]))/(3*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x^2*(a + b*ArcSin[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcSin[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (10*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2])} -{x^2*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 9, (b^2*x)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*x^2*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 9, b^2/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b*x*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (a + b*ArcSin[c*x])^2/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 9, (b^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c*d^2*Sqrt[d - c^2*d*x^2]) + (4*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^1*(d - c^2*d*x^2)^(5/2)), x, 24, b^2/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (a + b*ArcSin[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcSin[c*x])^2/(d^2*Sqrt[d - c^2*d*x^2]) + (14*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (7*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (7*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)), x, 19, (b^2*c^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (16*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)), x, 38, (b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(d^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (2*b*c^3*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcSin[c*x])^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcSin[c*x])^2/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcSin[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (26*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (5*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (13*I*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (13*I*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*b^2*c^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)), x, 32, -((b^2*c^2)/(3*d^2*x*Sqrt[d - c^2*d*x^2])) + (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*x]))/(3*d^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*ArcSin[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (16*I*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*I*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2])} - - -{x^4*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 10, (15*x*Sqrt[1 - a^2*x^2])/(64*a^4) + (x^3*Sqrt[1 - a^2*x^2])/(32*a^2) - (15*ArcSin[a*x])/(64*a^5) + (3*x^2*ArcSin[a*x])/(8*a^3) + (x^4*ArcSin[a*x])/(8*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(4*a^2) + ArcSin[a*x]^3/(8*a^5)} -{x^3*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 8, (14*Sqrt[1 - a^2*x^2])/(9*a^4) - (2*(1 - a^2*x^2)^(3/2))/(27*a^4) + (4*x*ArcSin[a*x])/(3*a^3) + (2*x^3*ArcSin[a*x])/(9*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a^2)} -{x^2*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 5, (x*Sqrt[1 - a^2*x^2])/(4*a^2) - ArcSin[a*x]/(4*a^3) + (x^2*ArcSin[a*x])/(2*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*a^2) + ArcSin[a*x]^3/(6*a^3)} -{x^1*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 3, (2*Sqrt[1 - a^2*x^2])/a^2 + (2*x*ArcSin[a*x])/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2} -{x^0*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 1, ArcSin[a*x]^3/(3*a)} -{ArcSin[a*x]^2/(x^1*Sqrt[1 - a^2*x^2]), x, 8, -2*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] + 2*I*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - 2*I*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - 2*PolyLog[3, -E^(I*ArcSin[a*x])] + 2*PolyLog[3, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]), x, 6, (-I)*a*ArcSin[a*x]^2 - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/x + 2*a*ArcSin[a*x]*Log[1 - E^(2*I*ArcSin[a*x])] - I*a*PolyLog[2, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]), x, 13, -((a*ArcSin[a*x])/x) - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*x^2) - a^2*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] + I*a^2*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - I*a^2*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - a^2*PolyLog[3, -E^(I*ArcSin[a*x])] + a^2*PolyLog[3, E^(I*ArcSin[a*x])]} - - -{ArcSin[a*x]^2/(c - a^2*c*x^2)^(1/2), x, 1, (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^2/(c - a^2*c*x^2)^(3/2), x, 6, (x*ArcSin[a*x]^2)/(c*Sqrt[c - a^2*c*x^2]) - (I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(a*c*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^(2*I*ArcSin[a*x])])/(a*c*Sqrt[c - a^2*c*x^2]) - (I*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(a*c*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2), x, 9, x/(3*c^2*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]/(3*a*c^2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^2)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcSin[a*x]^2)/(3*c^2*Sqrt[c - a^2*c*x^2]) - (2*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(3*a*c^2*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^(2*I*ArcSin[a*x])])/(3*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*I*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(3*a*c^2*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^2/(c - a^2*c*x^2)^(7/2), x, 13, x/(3*c^3*Sqrt[c - a^2*c*x^2]) + x/(30*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]/(10*a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]) - (4*ArcSin[a*x])/(15*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^2)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^2)/(15*c^3*Sqrt[c - a^2*c*x^2]) - (8*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) + (16*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^(2*I*ArcSin[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*I*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x)^2 and m symbolic*) - - -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^3, x, 23, If[$VersionNumber>=8, (2*b^2*c^2*d^3*x^(3 + m))/((3 + m)*(7 + m)^2) + (30*b^2*c^2*d^3*x^(3 + m))/((3 + m)^2*(5 + m)*(7 + m)^2) + (36*b^2*c^2*d^3*x^(3 + m))/((3 + m)^2*(5 + m)^2*(7 + m)) + (12*b^2*c^2*d^3*x^(3 + m))/((3 + m)*(5 + m)^2*(7 + m)) + (48*b^2*c^2*d^3*x^(3 + m))/((3 + m)^3*(5 + m)*(7 + m)) + (10*b^2*c^2*d^3*x^(3 + m))/((7 + m)^2*(15 + 8*m + m^2)) - (10*b^2*c^4*d^3*x^(5 + m))/((5 + m)^2*(7 + m)^2) - (4*b^2*c^4*d^3*x^(5 + m))/((5 + m)*(7 + m)^2) - (12*b^2*c^4*d^3*x^(5 + m))/((5 + m)^3*(7 + m)) + (2*b^2*c^6*d^3*x^(7 + m))/(7 + m)^3 - (36*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2*(7 + m)) - (48*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)^2*(5 + m)*(7 + m)) - (30*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((7 + m)^2*(15 + 8*m + m^2)) - (10*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)*(7 + m)^2) - (12*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)^2*(7 + m)) - (2*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(7 + m)^2 + (48*d^3*x^(1 + m)*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)*(3 + 4*m + m^2)) + (24*d^3*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((7 + m)*(15 + 8*m + m^2)) + (6*d^3*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)) + (d^3*x^(1 + m)*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(7 + m) - (48*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)*(7 + m)) - (30*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)^2*(6 + 5*m + m^2)) - (36*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(7 + m)*(6 + 5*m + m^2)) - (96*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)*(6 + 11*m + 6*m^2 + m^3)) + (30*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)*(7 + m)^2) + (36*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)) + (48*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3*(5 + m)*(7 + m)) + (96*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(5 + m)*(7 + m)*(2 + 3*m + m^2)), (2*b^2*c^2*d^3*x^(3 + m))/((3 + m)*(7 + m)^2) + (30*b^2*c^2*d^3*x^(3 + m))/((3 + m)^2*(5 + m)*(7 + m)^2) + (12*b^2*c^2*d^3*x^(3 + m))/((3 + m)*(5 + m)^2*(7 + m)) + (48*b^2*c^2*d^3*x^(3 + m))/((3 + m)^3*(5 + m)*(7 + m)) + (36*b^2*c^2*d^3*x^(3 + m))/((7 + m)*(15 + 8*m + m^2)^2) + (10*b^2*c^2*d^3*x^(3 + m))/((7 + m)^2*(15 + 8*m + m^2)) - (10*b^2*c^4*d^3*x^(5 + m))/((5 + m)^2*(7 + m)^2) - (4*b^2*c^4*d^3*x^(5 + m))/((5 + m)*(7 + m)^2) - (12*b^2*c^4*d^3*x^(5 + m))/((5 + m)^3*(7 + m)) + (2*b^2*c^6*d^3*x^(7 + m))/(7 + m)^3 - (36*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2*(7 + m)) - (48*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)^2*(5 + m)*(7 + m)) - (30*b*c*d^3*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((7 + m)^2*(15 + 8*m + m^2)) - (10*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)*(7 + m)^2) - (12*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/((5 + m)^2*(7 + m)) - (2*b*c*d^3*x^(2 + m)*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(7 + m)^2 + (48*d^3*x^(1 + m)*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)*(3 + 4*m + m^2)) + (24*d^3*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((7 + m)*(15 + 8*m + m^2)) + (6*d^3*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/((5 + m)*(7 + m)) + (d^3*x^(1 + m)*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(7 + m) - (30*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)^2*(6 + 5*m + m^2)) - (36*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(7 + m)*(6 + 5*m + m^2)) - (48*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((3 + m)^2*(7 + m)*(10 + 7*m + m^2)) - (96*b*c*d^3*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(7 + m)*(6 + 11*m + 6*m^2 + m^3)) + (96*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(5 + m)*(7 + m)*(2 + 3*m + m^2)) + (30*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(7 + m)^2*(10 + 7*m + m^2)) + (48*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^3*(7 + m)*(10 + 7*m + m^2)) + (36*b^2*c^2*d^3*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((15 + 8*m + m^2)^2*(14 + 9*m + m^2))]} -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^2, x, 13, If[$VersionNumber>=8, (6*b^2*c^2*d^2*x^(3 + m))/((3 + m)^2*(5 + m)^2) + (2*b^2*c^2*d^2*x^(3 + m))/((3 + m)*(5 + m)^2) + (8*b^2*c^2*d^2*x^(3 + m))/((3 + m)^3*(5 + m)) - (2*b^2*c^4*d^2*x^(5 + m))/(5 + m)^3 - (6*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2) - (8*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)^2*(5 + m)) - (2*b*c*d^2*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5 + m)^2 + (8*d^2*x^(1 + m)*(a + b*ArcSin[c*x])^2)/((5 + m)*(3 + 4*m + m^2)) + (4*d^2*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(15 + 8*m + m^2) + (d^2*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5 + m) - (8*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)) - (6*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(6 + 5*m + m^2)) - (16*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(6 + 11*m + 6*m^2 + m^3)) + (6*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^2*(5 + m)^2) + (8*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3*(5 + m)) + (16*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(5 + m)*(2 + 3*m + m^2)), (6*b^2*c^2*d^2*x^(3 + m))/((3 + m)^2*(5 + m)^2) + (2*b^2*c^2*d^2*x^(3 + m))/((3 + m)*(5 + m)^2) + (8*b^2*c^2*d^2*x^(3 + m))/((3 + m)^3*(5 + m)) - (2*b^2*c^4*d^2*x^(5 + m))/(5 + m)^3 - (6*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)*(5 + m)^2) - (8*b*c*d^2*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((3 + m)^2*(5 + m)) - (2*b*c*d^2*x^(2 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5 + m)^2 + (8*d^2*x^(1 + m)*(a + b*ArcSin[c*x])^2)/((5 + m)*(3 + 4*m + m^2)) + (4*d^2*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(15 + 8*m + m^2) + (d^2*x^(1 + m)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5 + m) - (6*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)^2*(6 + 5*m + m^2)) - (8*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((3 + m)^2*(10 + 7*m + m^2)) - (16*b*c*d^2*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((5 + m)*(6 + 11*m + 6*m^2 + m^3)) + (8*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3*(5 + m)) + (16*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(5 + m)*(2 + 3*m + m^2)) + (6*b^2*c^2*d^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(15 + 8*m + m^2)^2)]} -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^1, x, 6, (2*b^2*c^2*d*x^(3 + m))/(3 + m)^3 - (2*b*c*d*x^(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3 + m)^2 + (2*d*x^(1 + m)*(a + b*ArcSin[c*x])^2)/(3 + 4*m + m^2) + (d*x^(1 + m)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3 + m) - (2*b*c*d*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + m)*(3 + m)^2) - (4*b*c*d*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(6 + 11*m + 6*m^2 + m^3) + (2*b^2*c^2*d*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((2 + m)*(3 + m)^3) + (4*b^2*c^2*d*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/((3 + m)^2*(2 + 3*m + m^2))} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^1, x, 0, Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2), x]} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2, x, 4, -((b*c*x^(2 + m)*(a + b*ArcSin[c*x]))/(d^2*Sqrt[1 - c^2*x^2])) + (x^(1 + m)*(a + b*ArcSin[c*x])^2)/(2*d^2*(1 - c^2*x^2)) + (b*c*(1 + m)*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(2 + m)) + (b^2*c^2*x^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, c^2*x^2])/(d^2*(3 + m)) - (b^2*c^2*(1 + m)*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^2*(6 + 5*m + m^2)) + ((1 - m)*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2), x])/(2*d)} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3, x, 10, -((b*c*x^(2 + m)*(a + b*ArcSin[c*x]))/(6*d^3*(1 - c^2*x^2)^(3/2))) - (b*c*(1 - m)*x^(2 + m)*(a + b*ArcSin[c*x]))/(6*d^3*Sqrt[1 - c^2*x^2]) - (b*c*(3 - m)*x^(2 + m)*(a + b*ArcSin[c*x]))/(4*d^3*Sqrt[1 - c^2*x^2]) + (x^(1 + m)*(a + b*ArcSin[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) + ((3 - m)*x^(1 + m)*(a + b*ArcSin[c*x])^2)/(8*d^3*(1 - c^2*x^2)) + (b*c*(1 - m)*(1 + m)*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(6*d^3*(2 + m)) + (b*c*(3 - m)*(1 + m)*x^(2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*(2 + m)) + (b^2*c^2*(1 - m)*x^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, c^2*x^2])/(6*d^3*(3 + m)) + (b^2*c^2*(3 - m)*x^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, c^2*x^2])/(4*d^3*(3 + m)) + (b^2*c^2*x^(3 + m)*Hypergeometric2F1[2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(6*d^3*(3 + m)) - (b^2*c^2*(1 - m)*(1 + m)*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(6*d^3*(6 + 5*m + m^2)) - (b^2*c^2*(3 - m)*(1 + m)*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(4*d^3*(6 + 5*m + m^2)) + ((1 - m)*(3 - m)*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2), x])/(8*d^2)} - - -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(5/2), x, 12, If[$VersionNumber>=8, (10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*(6 + m)^3) - (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d - c^2*d*x^2])/(6 + m)^3 - (30*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (10*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((12 + 8*m + m^2)*Sqrt[1 - c^2*x^2]) + (10*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)^2*Sqrt[1 - c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/((4 + m)*(6 + m)) + (x^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(6 + m) + (30*b^2*c^2*d^2*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(3 + m)*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) + (10*b^2*c^2*d^2*(10 + 3*m)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)*(3 + m)*(4 + m)^3*(6 + m)*Sqrt[1 - c^2*x^2]) + (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)*(3 + m)*(4 + m)^2*(6 + m)^3*Sqrt[1 - c^2*x^2]) + (15*d^3*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2)), (10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m)*Sqrt[d - c^2*d*x^2])/((4 + m)^2*(6 + m)^3) - (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d - c^2*d*x^2])/(6 + m)^3 - (30*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (10*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((12 + 8*m + m^2)*Sqrt[1 - c^2*x^2]) + (10*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d^2*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)*(6 + m)*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((6 + m)^2*Sqrt[1 - c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/((4 + m)*(6 + m)) + (x^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(6 + m) + (10*b^2*c^2*d^2*(10 + 3*m)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((4 + m)^3*(6 + m)*(6 + 5*m + m^2)*Sqrt[1 - c^2*x^2]) + (30*b^2*c^2*d^2*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(6 + m)*(12 + 7*m + m^2)*Sqrt[1 - c^2*x^2]) + (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((4 + m)^2*(6 + m)^3*(6 + 5*m + m^2)*Sqrt[1 - c^2*x^2]) + (15*d^3*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2))]} -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(3/2), x, 7, If[$VersionNumber>=8, (2*b^2*c^2*d*x^(3 + m)*Sqrt[d - c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 - c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)^2*Sqrt[1 - c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(4 + m) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(3 + m)*(4 + m)*Sqrt[1 - c^2*x^2]) + (2*b^2*c^2*d*(10 + 3*m)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)*(3 + m)*(4 + m)^3*Sqrt[1 - c^2*x^2]) + (3*d^2*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(8 + 6*m + m^2), (2*b^2*c^2*d*x^(3 + m)*Sqrt[d - c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 - c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((8 + 6*m + m^2)*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((4 + m)^2*Sqrt[1 - c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(4 + m) + (2*b^2*c^2*d*(10 + 3*m)*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((4 + m)^3*(6 + 5*m + m^2)*Sqrt[1 - c^2*x^2]) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(12 + 7*m + m^2)*Sqrt[1 - c^2*x^2]) + (3*d^2*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(8 + 6*m + m^2)]} -{x^m*(a + b*ArcSin[c*x])^2*(d - c^2*d*x^2)^(1/2), x, 3, -((2*b*c*x^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/((2 + m)^2*Sqrt[1 - c^2*x^2])) + (x^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2 + m) + (2*b^2*c^2*x^(3 + m)*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((2 + m)^2*(3 + m)*Sqrt[1 - c^2*x^2]) + (d*Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(2 + m)} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 0, Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x]} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2), x]} -{x^m*(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 0, Unintegrable[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]} - - -{x^m*ArcSin[a*x]^2/Sqrt[1 - a^2*x^2], x, 0, Unintegrable[(x^m*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x)^3*) - - -{ArcSin[a*x]^3*(c - a^2*c*x^2)^3, x, 24, -((413312*c^3*Sqrt[1 - a^2*x^2])/(128625*a)) - (30256*c^3*(1 - a^2*x^2)^(3/2))/(385875*a) - (2664*c^3*(1 - a^2*x^2)^(5/2))/(214375*a) - (6*c^3*(1 - a^2*x^2)^(7/2))/(2401*a) - (4322*c^3*x*ArcSin[a*x])/1225 + (1514*a^2*c^3*x^3*ArcSin[a*x])/3675 - (702*a^4*c^3*x^5*ArcSin[a*x])/6125 + (6/343)*a^6*c^3*x^7*ArcSin[a*x] + (48*c^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(35*a) + (8*c^3*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(35*a) + (18*c^3*(1 - a^2*x^2)^(5/2)*ArcSin[a*x]^2)/(175*a) + (3*c^3*(1 - a^2*x^2)^(7/2)*ArcSin[a*x]^2)/(49*a) + (16/35)*c^3*x*ArcSin[a*x]^3 + (8/35)*c^3*x*(1 - a^2*x^2)*ArcSin[a*x]^3 + (6/35)*c^3*x*(1 - a^2*x^2)^2*ArcSin[a*x]^3 + (1/7)*c^3*x*(1 - a^2*x^2)^3*ArcSin[a*x]^3} -{ArcSin[a*x]^3*(c - a^2*c*x^2)^2, x, 17, -((4144*c^2*Sqrt[1 - a^2*x^2])/(1125*a)) - (272*c^2*(1 - a^2*x^2)^(3/2))/(3375*a) - (6*c^2*(1 - a^2*x^2)^(5/2))/(625*a) - (298/75)*c^2*x*ArcSin[a*x] + (76/225)*a^2*c^2*x^3*ArcSin[a*x] - (6/125)*a^4*c^2*x^5*ArcSin[a*x] + (8*c^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(5*a) + (4*c^2*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(15*a) + (3*c^2*(1 - a^2*x^2)^(5/2)*ArcSin[a*x]^2)/(25*a) + (8/15)*c^2*x*ArcSin[a*x]^3 + (4/15)*c^2*x*(1 - a^2*x^2)*ArcSin[a*x]^3 + (1/5)*c^2*x*(1 - a^2*x^2)^2*ArcSin[a*x]^3} -{ArcSin[a*x]^3*(c - a^2*c*x^2)^1, x, 10, -((40*c*Sqrt[1 - a^2*x^2])/(9*a)) - (2*c*(1 - a^2*x^2)^(3/2))/(27*a) - (14/3)*c*x*ArcSin[a*x] + (2/9)*a^2*c*x^3*ArcSin[a*x] + (2*c*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + (c*(1 - a^2*x^2)^(3/2)*ArcSin[a*x]^2)/(3*a) + (2/3)*c*x*ArcSin[a*x]^3 + (1/3)*c*x*(1 - a^2*x^2)*ArcSin[a*x]^3} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^1, x, 10, -((2*I*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])])/(a*c)) + (3*I*ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(a*c) - (3*I*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])])/(a*c) - (6*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])])/(a*c) + (6*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])])/(a*c) - (6*I*PolyLog[4, (-I)*E^(I*ArcSin[a*x])])/(a*c) + (6*I*PolyLog[4, I*E^(I*ArcSin[a*x])])/(a*c)} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^2, x, 18, -((3*ArcSin[a*x]^2)/(2*a*c^2*Sqrt[1 - a^2*x^2])) + (x*ArcSin[a*x]^3)/(2*c^2*(1 - a^2*x^2)) - (6*I*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])])/(a*c^2) - (I*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])])/(a*c^2) + (3*I*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(a*c^2) + (3*I*ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(2*a*c^2) - (3*I*PolyLog[2, I*E^(I*ArcSin[a*x])])/(a*c^2) - (3*I*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])])/(2*a*c^2) - (3*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])])/(a*c^2) + (3*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])])/(a*c^2) - (3*I*PolyLog[4, (-I)*E^(I*ArcSin[a*x])])/(a*c^2) + (3*I*PolyLog[4, I*E^(I*ArcSin[a*x])])/(a*c^2)} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^3, x, 28, -(1/(4*a*c^3*Sqrt[1 - a^2*x^2])) + (x*ArcSin[a*x])/(4*c^3*(1 - a^2*x^2)) - ArcSin[a*x]^2/(4*a*c^3*(1 - a^2*x^2)^(3/2)) - (9*ArcSin[a*x]^2)/(8*a*c^3*Sqrt[1 - a^2*x^2]) + (x*ArcSin[a*x]^3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcSin[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - (5*I*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])])/(a*c^3) - (3*I*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])])/(4*a*c^3) + (5*I*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(2*a*c^3) + (9*I*ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(8*a*c^3) - (5*I*PolyLog[2, I*E^(I*ArcSin[a*x])])/(2*a*c^3) - (9*I*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])])/(8*a*c^3) - (9*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])])/(4*a*c^3) + (9*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])])/(4*a*c^3) - (9*I*PolyLog[4, (-I)*E^(I*ArcSin[a*x])])/(4*a*c^3) + (9*I*PolyLog[4, I*E^(I*ArcSin[a*x])])/(4*a*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^3*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcSin[a*x]^3*(c - a^2*c*x^2)^(5/2), x, 24, (865*a*c^2*x^2*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[1 - a^2*x^2]) - (65*a^3*c^2*x^4*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[1 - a^2*x^2]) - (c^2*(1 - a^2*x^2)^(5/2)*Sqrt[c - a^2*c*x^2])/(216*a) - (245/384)*c^2*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] - (65/576)*c^2*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] - (1/36)*c^2*x*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] + (115*c^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(768*a*Sqrt[1 - a^2*x^2]) - (15*a*c^2*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(32*Sqrt[1 - a^2*x^2]) + (5*c^2*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(32*a) + (c^2*(1 - a^2*x^2)^(5/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(12*a) + (5/16)*c^2*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3 + (5/24)*c*x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3 + (1/6)*x*(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^3 + (5*c^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(64*a*Sqrt[1 - a^2*x^2])} -{ArcSin[a*x]^3*(c - a^2*c*x^2)^(3/2), x, 14, (51*a*c*x^2*Sqrt[c - a^2*c*x^2])/(128*Sqrt[1 - a^2*x^2]) - (3*a^3*c*x^4*Sqrt[c - a^2*c*x^2])/(128*Sqrt[1 - a^2*x^2]) - (45/64)*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] - (3/32)*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] + (27*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(128*a*Sqrt[1 - a^2*x^2]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(16*Sqrt[1 - a^2*x^2]) + (3*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(16*a) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3 + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3 + (3*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(32*a*Sqrt[1 - a^2*x^2])} -{ArcSin[a*x]^3*(c - a^2*c*x^2)^(1/2), x, 6, (3*a*x^2*Sqrt[c - a^2*c*x^2])/(8*Sqrt[1 - a^2*x^2]) - (3/4)*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x] + (3*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(8*a*Sqrt[1 - a^2*x^2]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^2)/(4*Sqrt[1 - a^2*x^2]) + (1/2)*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3 + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^4)/(8*a*Sqrt[1 - a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{ArcSin[a*x]^3/(c - a^2*c*x^2)^(1/2), x, 1, (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^4)/(4*a*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^(3/2), x, 7, (x*ArcSin[a*x]^3)/(c*Sqrt[c - a^2*c*x^2]) - (I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(a*c*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2*Log[1 + E^(2*I*ArcSin[a*x])])/(a*c*Sqrt[c - a^2*c*x^2]) - (3*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(a*c*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^(2*I*ArcSin[a*x])])/(2*a*c*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^(5/2), x, 11, (x*ArcSin[a*x])/(c^2*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]^2/(2*a*c^2*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcSin[a*x]^3)/(3*c^2*Sqrt[c - a^2*c*x^2]) - (2*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a*c^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2*Log[1 + E^(2*I*ArcSin[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])/(2*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*PolyLog[3, -E^(2*I*ArcSin[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2])} -{ArcSin[a*x]^3/(c - a^2*c*x^2)^(7/2), x, 17, -(1/(20*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2])) + (x*ArcSin[a*x])/(c^3*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/(10*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - (3*ArcSin[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]) - (2*ArcSin[a*x]^2)/(5*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^3)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^3)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^3)/(15*c^3*Sqrt[c - a^2*c*x^2]) - (8*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2*Log[1 + E^(2*I*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*I*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^(2*I*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^(2*I*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2])} - - -{x^m*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 0, Unintegrable[(x^m*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2], x]} - -{x^4*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 13, -((45*x^2)/(128*a^3)) - (3*x^4)/(128*a) + (45*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(64*a^4) + (3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(32*a^2) - (45*ArcSin[a*x]^2)/(128*a^5) + (9*x^2*ArcSin[a*x]^2)/(16*a^3) + (3*x^4*ArcSin[a*x]^2)/(16*a) - (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(4*a^2) + (3*ArcSin[a*x]^4)/(32*a^5)} -{x^3*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 10, -((40*x)/(9*a^3)) - (2*x^3)/(27*a) + (40*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^4) + (2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(9*a^2) + (2*x*ArcSin[a*x]^2)/a^3 + (x^3*ArcSin[a*x]^2)/(3*a) - (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a^2)} -{x^2*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 6, -((3*x^2)/(8*a)) + (3*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4*a^2) - (3*ArcSin[a*x]^2)/(8*a^3) + (3*x^2*ArcSin[a*x]^2)/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(2*a^2) + ArcSin[a*x]^4/(8*a^3)} -{x^1*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 4, -((6*x)/a) + (6*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2 + (3*x*ArcSin[a*x]^2)/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2} -{x^0*ArcSin[a*x]^3/Sqrt[1 - a^2*x^2], x, 1, ArcSin[a*x]^4/(4*a)} -{ArcSin[a*x]^3/(x^1*Sqrt[1 - a^2*x^2]), x, 10, -2*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 3*I*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - 3*I*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 6*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 6*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - 6*I*PolyLog[4, -E^(I*ArcSin[a*x])] + 6*I*PolyLog[4, E^(I*ArcSin[a*x])]} -{ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]), x, 7, (-I)*a*ArcSin[a*x]^3 - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x + 3*a*ArcSin[a*x]^2*Log[1 - E^(2*I*ArcSin[a*x])] - 3*I*a*ArcSin[a*x]*PolyLog[2, E^(2*I*ArcSin[a*x])] + (3/2)*a*PolyLog[3, E^(2*I*ArcSin[a*x])]} -{ArcSin[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]), x, 18, -((3*a*ArcSin[a*x]^2)/(2*x)) - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(2*x^2) - 6*a^2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] - a^2*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + 3*I*a^2*PolyLog[2, -E^(I*ArcSin[a*x])] + (3/2)*I*a^2*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - 3*I*a^2*PolyLog[2, E^(I*ArcSin[a*x])] - (3/2)*I*a^2*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 3*a^2*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 3*a^2*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - 3*I*a^2*PolyLog[4, -E^(I*ArcSin[a*x])] + 3*I*a^2*PolyLog[4, E^(I*ArcSin[a*x])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p / (a+b ArcSin[c x)*) - - -{1/ArcSin[a*x]*(c - a^2*c*x^2)^3, x, 7, (35*c^3*CosIntegral[ArcSin[a*x]])/(64*a) + (21*c^3*CosIntegral[3*ArcSin[a*x]])/(64*a) + (7*c^3*CosIntegral[5*ArcSin[a*x]])/(64*a) + (c^3*CosIntegral[7*ArcSin[a*x]])/(64*a)} -{1/ArcSin[a*x]*(c - a^2*c*x^2)^2, x, 6, (5*c^2*CosIntegral[ArcSin[a*x]])/(8*a) + (5*c^2*CosIntegral[3*ArcSin[a*x]])/(16*a) + (c^2*CosIntegral[5*ArcSin[a*x]])/(16*a)} -{1/ArcSin[a*x]*(c - a^2*c*x^2)^1, x, 5, (3*c*CosIntegral[ArcSin[a*x]])/(4*a) + (c*CosIntegral[3*ArcSin[a*x]])/(4*a)} -{1/ArcSin[a*x]/(c - a^2*c*x^2)^1, x, 0, Unintegrable[1/((c - a^2*c*x^2)*ArcSin[a*x]), x]} -{1/ArcSin[a*x]/(c - a^2*c*x^2)^2, x, 0, Unintegrable[1/((c - a^2*c*x^2)^2*ArcSin[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) / (a+b ArcSin[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]), x, 12, -((Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^5)) - (Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (Cos[(6*a)/b]*CosIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^5) + Log[a + b*ArcSin[c*x]]/(16*b*c^5) - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^5) - (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (Sin[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^5)} -{x^3*Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]), x, 12, -((CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b*c^4)) - (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b*c^4) + (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b*c^4) + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^4) + (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^4) - (Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^4)} -{x^2*Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]), x, 6, -((Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c^3)) + Log[a + b*ArcSin[c*x]]/(8*b*c^3) - (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c^3)} -{x^1*Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]), x, 9, -((CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b*c^2)) - (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b*c^2) + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^2) + (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^2)} -{x^0*Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]), x, 6, (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c) + Log[a + b*ArcSin[c*x]]/(2*b*c) + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c)} -{Sqrt[1 - c^2*x^2]/(x^1*(a + b*ArcSin[c*x])), x, 6, (CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/b - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/b + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])), x, 3, -((c*Log[a + b*ArcSin[c*x]])/b) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])), x]} - - -{x^3*(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]), x, 15, -((3*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(64*b*c^4)) - (3*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(64*b*c^4) + (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(64*b*c^4) + (CosIntegral[(7*(a + b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/(64*b*c^4) + (3*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b*c^4) + (3*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b*c^4) - (Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b*c^4) - (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b*c^4)} -{x^2*(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]), x, 12, (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c^3) - (Cos[(6*a)/b]*CosIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) + Log[a + b*ArcSin[c*x]]/(16*b*c^3) + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c^3) - (Sin[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^3)} -{x^1*(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]), x, 12, -((CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b*c^2)) - (3*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b*c^2) - (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b*c^2) + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^2) + (3*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^2) + (Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^2)} -{x^0*(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]), x, 9, (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c) + (Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c) + (3*Log[a + b*ArcSin[c*x]])/(8*b*c) + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c) + (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c)} -{(1 - c^2*x^2)^(3/2)/(x^1*(a + b*ArcSin[c*x])), x, 15, (5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b) + (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b) - (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b) - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])), x, 9, -((c*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b)) - (3*c*Log[a + b*ArcSin[c*x]])/(2*b) - (c*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])), x]} - - -{x^3*(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]), x, 15, -((3*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(128*b*c^4)) - (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(32*b*c^4) + (3*CosIntegral[(7*(a + b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/(256*b*c^4) + (CosIntegral[(9*(a + b*ArcSin[c*x]))/b]*Sin[(9*a)/b])/(256*b*c^4) + (3*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(128*b*c^4) + (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(32*b*c^4) - (3*Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(256*b*c^4) - (Cos[(9*a)/b]*SinIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b*c^4)} -{x^2*(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]), x, 15, (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Cos[(6*a)/b]*CosIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Cos[(8*a)/b]*CosIntegral[(8*(a + b*ArcSin[c*x]))/b])/(128*b*c^3) + (5*Log[a + b*ArcSin[c*x]])/(128*b*c^3) + (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Sin[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c^3) - (Sin[(8*a)/b]*SinIntegral[(8*(a + b*ArcSin[c*x]))/b])/(128*b*c^3)} -{x^1*(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]), x, 15, -((5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(64*b*c^2)) - (9*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(64*b*c^2) - (5*CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(64*b*c^2) - (CosIntegral[(7*(a + b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/(64*b*c^2) + (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b*c^2) + (9*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b*c^2) + (5*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b*c^2) + (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b*c^2)} -{x^0*(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]), x, 12, (15*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c) + (3*Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c) + (Cos[(6*a)/b]*CosIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c) + (5*Log[a + b*ArcSin[c*x]])/(16*b*c) + (15*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(32*b*c) + (3*Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(16*b*c) + (Sin[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(32*b*c)} -{(1 - c^2*x^2)^(5/2)/(x^1*(a + b*ArcSin[c*x])), x, 27, (11*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b) + (7*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b) + (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b) - (11*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b) - (7*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b) - (Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])), x, 18, -((c*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/b) - (c*Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b) - (15*c*Log[a + b*ArcSin[c*x]])/(8*b) - (c*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/b - (c*Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 5, -(CosIntegral[2*ArcSin[a*x]]/(2*a^5)) + CosIntegral[4*ArcSin[a*x]]/(8*a^5) + (3*Log[ArcSin[a*x]])/(8*a^5)} -{x^3/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 5, (3*SinIntegral[ArcSin[a*x]])/(4*a^4) - SinIntegral[3*ArcSin[a*x]]/(4*a^4)} -{x^2/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 4, -(CosIntegral[2*ArcSin[a*x]]/(2*a^3)) + Log[ArcSin[a*x]]/(2*a^3)} -{x^2/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 4, -(CosIntegral[2*ArcSin[a*x]]/(2*a^3)) + Log[ArcSin[a*x]]/(2*a^3)} -{x^1/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 2, SinIntegral[ArcSin[a*x]]/a^2} -{x^0/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 1, Log[ArcSin[a*x]]/a} -{1/(x^1*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 0, Unintegrable[1/(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]} -{1/(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 0, Unintegrable[1/(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]} - - -{x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 12, -((5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b*c^6)) + (5*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b*c^6) - (CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b*c^6) + (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^6) - (5*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^6) + (Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^6)} -{x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 9, -((Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c^5)) + (Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c^5) + (3*Log[a + b*ArcSin[c*x]])/(8*b*c^5) - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c^5) + (Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b*c^5)} -{x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 9, -((3*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b*c^4)) + (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b*c^4) + (3*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^4) - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^4)} -{x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 6, -((Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c^3)) + Log[a + b*ArcSin[c*x]]/(2*b*c^3) - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(2*b*c^3)} -{x^1/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 4, -((CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b*c^2)) + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c^2)} -{x^0/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 1, Log[a + b*ArcSin[c*x]]/(b*c)} -{1/(x^1*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} - - -{x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{x^1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{x^0/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{1/(x^1*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} - - -{x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} -{x^1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} -{x^0/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} -{1/(x^1*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} -{1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) / (a+b ArcSin[c x]) with m symbolic*) - - -{x^m*(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]), x]} -{x^m*(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]), x]} -{x^m*(1 - c^2*x^2)^(1/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]), x]} -{x^m/((1 - c^2*x^2)^(1/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} -{x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} - - -{x^m/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x, 0, Unintegrable[x^m/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x)^2*) - - -{1/ArcSin[a*x]^2*(c - a^2*c*x^2)^3, x, 8, -((c^3*(1 - a^2*x^2)^(7/2))/(a*ArcSin[a*x])) - (35*c^3*SinIntegral[ArcSin[a*x]])/(64*a) - (63*c^3*SinIntegral[3*ArcSin[a*x]])/(64*a) - (35*c^3*SinIntegral[5*ArcSin[a*x]])/(64*a) - (7*c^3*SinIntegral[7*ArcSin[a*x]])/(64*a)} -{1/ArcSin[a*x]^2*(c - a^2*c*x^2)^2, x, 7, -((c^2*(1 - a^2*x^2)^(5/2))/(a*ArcSin[a*x])) - (5*c^2*SinIntegral[ArcSin[a*x]])/(8*a) - (15*c^2*SinIntegral[3*ArcSin[a*x]])/(16*a) - (5*c^2*SinIntegral[5*ArcSin[a*x]])/(16*a)} -{1/ArcSin[a*x]^2*(c - a^2*c*x^2)^1, x, 6, -((c*(1 - a^2*x^2)^(3/2))/(a*ArcSin[a*x])) - (3*c*SinIntegral[ArcSin[a*x]])/(4*a) - (3*c*SinIntegral[3*ArcSin[a*x]])/(4*a)} -{1/ArcSin[a*x]^2/(c - a^2*c*x^2)^1, x, 1, -(1/(a*c*Sqrt[1 - a^2*x^2]*ArcSin[a*x])) + (a*Unintegrable[x/((1 - a^2*x^2)^(3/2)*ArcSin[a*x]), x])/c} -{1/ArcSin[a*x]^2/(c - a^2*c*x^2)^2, x, 1, -(1/(a*c^2*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])) + (3*a*Unintegrable[x/((1 - a^2*x^2)^(5/2)*ArcSin[a*x]), x])/c^2} - - -{1/((1 - x^2)*ArcSin[x]^2) - x/((1 - x^2)^(3/2)*ArcSin[x]), x, 2, -(1/(Sqrt[1 - x^2]*ArcSin[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) / (a+b ArcSin[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x]} - -{(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x, 22, -((x^3*(1 - c^2*x^2))/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^4) + (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^4) - (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^4) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^4) + (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^4) - (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^4)} -{(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x, 16, -((x^2*(1 - c^2*x^2))/(b*c*(a + b*ArcSin[c*x]))) - (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(2*b^2*c^3) + (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(2*b^2*c^3)} -{(x*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x, 14, -((x*(1 - c^2*x^2))/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^2) + (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^2) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^2) + (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^2)} -{Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x])^2, x, 7, -((1 - c^2*x^2)/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^2*c) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c)} -{Sqrt[1 - c^2*x^2]/(x*(a + b*ArcSin[c*x])^2), x, 5, -((1 - c^2*x^2)/(b*c*x*(a + b*ArcSin[c*x]))) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/b^2 - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/b^2 - Unintegrable[1/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)} -{Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])^2), x, 1, -((1 - c^2*x^2)/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Unintegrable[1/(x^3*(a + b*ArcSin[c*x])), x])/(b*c)} -{Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])^2), x]} -{Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x]} - - -{(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x]} - -{(x^3*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x, 28, -((x^3*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(64*b^2*c^4) + (9*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4) - (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4) - (7*Cos[(7*a)/b]*CosIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4) + (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b^2*c^4) + (9*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4) - (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4) - (7*Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^4)} -{(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x, 19, -((x^2*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(16*b^2*c^3) - (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(4*b^2*c^3) - (3*CosIntegral[(6*(a + b*ArcSin[c*x]))/b]*Sin[(6*a)/b])/(16*b^2*c^3) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3) + (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3) + (3*Cos[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3)} -{(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x, 22, -((x*(1 - c^2*x^2)^2)/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^2) + (9*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2) + (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^2) + (9*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2) + (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^2)} -{(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x])^2, x, 10, -((1 - c^2*x^2)^2/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^2*c) + (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(2*b^2*c) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c) - (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(2*b^2*c)} -{(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])^2), x, 10, -((1 - c^2*x^2)^2/(b*c*x*(a + b*ArcSin[c*x]))) - (9*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2) - (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - (9*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2) - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - Unintegrable[(1 - c^2*x^2)/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)} -{(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])^2), x, 1, -((1 - c^2*x^2)^2/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Unintegrable[(1 - c^2*x^2)/(x^3*(a + b*ArcSin[c*x])), x])/(b*c) - (2*c*Unintegrable[(1 - c^2*x^2)/(x*(a + b*ArcSin[c*x])), x])/b} -{(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])^2), x]} -{(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])^2), x, 1, -((1 - c^2*x^2)^2/(b*c*x^4*(a + b*ArcSin[c*x]))) - (4*Unintegrable[(1 - c^2*x^2)/(x^5*(a + b*ArcSin[c*x])), x])/(b*c)} - - -{(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x]} - -{(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x, 34, -((x^3*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*c^4) + (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(32*b^2*c^4) - (21*Cos[(7*a)/b]*CosIntegral[(7*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Cos[(9*a)/b]*CosIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) + (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(128*b^2*c^4) + (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(32*b^2*c^4) - (21*Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4) - (9*Sin[(9*a)/b]*SinIntegral[(9*(a + b*ArcSin[c*x]))/b])/(256*b^2*c^4)} -{(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x, 28, -((x^2*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(16*b^2*c^3) - (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(8*b^2*c^3) - (3*CosIntegral[(6*(a + b*ArcSin[c*x]))/b]*Sin[(6*a)/b])/(16*b^2*c^3) - (CosIntegral[(8*(a + b*ArcSin[c*x]))/b]*Sin[(8*a)/b])/(16*b^2*c^3) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3) + (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b^2*c^3) + (3*Cos[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3) + (Cos[(8*a)/b]*SinIntegral[(8*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^3)} -{(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x, 28, -((x*(1 - c^2*x^2)^3)/(b*c*(a + b*ArcSin[c*x]))) + (5*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(64*b^2*c^2) + (27*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2) + (25*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2) + (7*Cos[(7*a)/b]*CosIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2) + (5*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b^2*c^2) + (27*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2) + (25*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2) + (7*Sin[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b^2*c^2)} -{(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x])^2, x, 13, -((1 - c^2*x^2)^3/(b*c*(a + b*ArcSin[c*x]))) + (15*CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(16*b^2*c) + (3*CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(4*b^2*c) + (3*CosIntegral[(6*(a + b*ArcSin[c*x]))/b]*Sin[(6*a)/b])/(16*b^2*c) - (15*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(16*b^2*c) - (3*Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(4*b^2*c) - (3*Cos[(6*a)/b]*SinIntegral[(6*(a + b*ArcSin[c*x]))/b])/(16*b^2*c)} -{(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])^2), x, 13, -((1 - c^2*x^2)^3/(b*c*x*(a + b*ArcSin[c*x]))) - (25*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2) - (25*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2) - (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2) - (25*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2) - (25*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2) - (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2) - Unintegrable[(1 - c^2*x^2)^2/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)} -{(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])^2), x, 1, -((1 - c^2*x^2)^3/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Unintegrable[(1 - c^2*x^2)^2/(x^3*(a + b*ArcSin[c*x])), x])/(b*c) - (4*c*Unintegrable[(1 - c^2*x^2)^2/(x*(a + b*ArcSin[c*x])), x])/b} -{(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2), x]} -{(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 1, -(x^m/(b*c*(a + b*ArcSin[c*x]))) + (m*Unintegrable[x^(-1 + m)/(a + b*ArcSin[c*x]), x])/(b*c)} - -{x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 13, -(x^5/(b*c*(a + b*ArcSin[c*x]))) + (5*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^6) - (15*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^6) + (5*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^6) + (5*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^6) - (15*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^6) + (5*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^6)} -{x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 10, -(x^4/(b*c*(a + b*ArcSin[c*x]))) - (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^2*c^5) + (CosIntegral[(4*(a + b*ArcSin[c*x]))/b]*Sin[(4*a)/b])/(2*b^2*c^5) + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^5) - (Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(2*b^2*c^5)} -{x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 10, -(x^3/(b*c*(a + b*ArcSin[c*x]))) + (3*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^4) - (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^4) + (3*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^4) - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^4)} -{x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 7, -(x^2/(b*c*(a + b*ArcSin[c*x]))) - (CosIntegral[(2*(a + b*ArcSin[c*x]))/b]*Sin[(2*a)/b])/(b^2*c^3) + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^3)} -{x^1/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 5, -(x/(b*c*(a + b*ArcSin[c*x]))) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c^2) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c^2)} -{x^0/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 1, -(1/(b*c*(a + b*ArcSin[c*x])))} -{1/(x^1*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 1, -(1/(b*c*x*(a + b*ArcSin[c*x]))) - Unintegrable[1/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)} -{1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x, 1, -(1/(b*c*x^2*(a + b*ArcSin[c*x]))) - (2*Unintegrable[1/(x^3*(a + b*ArcSin[c*x])), x])/(b*c)} - - -{x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} - -{x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} -{x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 1, -(x^2/(b*c*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))) + (2*Unintegrable[x/((1 - c^2*x^2)^2*(a + b*ArcSin[c*x])), x])/(b*c)} -{x^1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} -{x^0/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 1, -(1/(b*c*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))) + (2*c*Unintegrable[x/((1 - c^2*x^2)^2*(a + b*ArcSin[c*x])), x])/b} -{1/(x^1*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} -{1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} - - -{x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} - -{x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} -{x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} -{x^1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} -{x^0/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 1, -(1/(b*c*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))) + (4*c*Unintegrable[x/((1 - c^2*x^2)^3*(a + b*ArcSin[c*x])), x])/b} -{1/(x^1*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} -{1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x])^3*) - - -(* ::Subsection:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) / (a+b ArcSin[c x])^3*) - - -{1/(ArcSin[a*x]^3*Sqrt[1 - a^2*x^2]), x, 1, -(1/(2*a*ArcSin[a*x]^2))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcSin[c x)^(3/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2), x, 27, -((2*d*x^3*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (d*Sqrt[3*Pi]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*b^(3/2)*c^4) + (3*d*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*b^(3/2)*c^4) + (3*d*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*b^(3/2)*c^4) - (d*Sqrt[3*Pi]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(6*a)/b])/(8*b^(3/2)*c^4)} -{x^2*(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2), x, 32, -((2*d*x^2*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (5*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*b^(3/2)*c^3) + (d*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (5*d*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[(2*Pi)/3]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (d*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (5*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)*c^3) - (d*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) + (5*d*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*c^3) - (d*Sqrt[(2*Pi)/3]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3) - (d*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*c^3)} -{x^1*(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2), x, 17, -((2*d*x*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) + (d*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (d*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*c^2) + (d*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*c^2) + (d*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/(b^(3/2)*c^2)} -{x^0*(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2), x, 14, -((2*d*(1 - c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (3*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) - (d*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (3*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c) + (d*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c)} -{(d - c^2*d*x^2)/(x^1*(a + b*ArcSin[c*x])^(3/2)), x, 12, -((2*d*(1 - c^2*x^2)^(3/2))/(b*c*x*Sqrt[a + b*ArcSin[c*x]])) - (2*d*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/b^(3/2) - (2*d*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/b^(3/2) - (2*d*Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]]), x])/(b*c)} - - -{x^3*(d - c^2*d*x^2)^2/(a + b*ArcSin[c*x])^(3/2), x, 32, -((2*d^2*x^3*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) + (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*b^(3/2)*c^4) - (d^2*Sqrt[3*Pi]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (3*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(16*b^(3/2)*c^4) - (d^2*Sqrt[Pi]*Cos[(8*a)/b]*FresnelC[(4*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(16*b^(3/2)*c^4) + (3*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(16*b^(3/2)*c^4) + (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/(8*b^(3/2)*c^4) - (d^2*Sqrt[3*Pi]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(6*a)/b])/(16*b^(3/2)*c^4) - (d^2*Sqrt[Pi]*FresnelS[(4*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(8*a)/b])/(16*b^(3/2)*c^4)} -{x^2*(d - c^2*d*x^2)^2/(a + b*ArcSin[c*x])^(3/2), x, 42, -((2*d^2*x^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (5*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (3*d^2*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d^2*Sqrt[(7*Pi)/2]*Cos[(7*a)/b]*FresnelS[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (5*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(16*b^(3/2)*c^3) - (3*d^2*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(16*b^(3/2)*c^3) - (d^2*Sqrt[(7*Pi)/2]*FresnelC[(Sqrt[14/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(7*a)/b])/(16*b^(3/2)*c^3)} -{x^1*(d - c^2*d*x^2)^2/(a + b*ArcSin[c*x])^(3/2), x, 32, -((2*d^2*x*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) + (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*Cos[(6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*b^(3/2)*c^2) + (5*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*b^(3/2)*c^2) + (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/(b^(3/2)*c^2) + (d^2*Sqrt[3*Pi]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(6*a)/b])/(8*b^(3/2)*c^2)} -{x^0*(d - c^2*d*x^2)^2/(a + b*ArcSin[c*x])^(3/2), x, 19, -((2*d^2*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (5*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*b^(3/2)*c) - (5*d^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*b^(3/2)*c) - (d^2*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*b^(3/2)*c) + (5*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)*c) + (5*d^2*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*c) + (d^2*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*c)} -{(d - c^2*d*x^2)^2/(x^1*(a + b*ArcSin[c*x])^(3/2)), x, 25, -((2*d^2*(1 - c^2*x^2)^(5/2))/(b*c*x*Sqrt[a + b*ArcSin[c*x]])) - (d^2*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/b^(3/2) - (3*d^2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/b^(3/2) - (3*d^2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/b^(3/2) - (d^2*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/b^(3/2) - (2*d^2*Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]]), x])/(b*c)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -(* An excellent example of the importance of reducing even unintegrable expressions to simplest form so cancellations can occur. *) -{-((3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]])) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2, x, 3, -((3*x*Sqrt[ArcSin[x]])/(4*Sqrt[1 - x^2])) + ArcSin[x]^(3/2)/(2*(1 - x^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) ArcSin[c x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]], x, 15, (3/8)*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]] + (1/4)*x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]] + (c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(4*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(64*a*Sqrt[1 - a^2*x^2]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]], x, 7, (1/2)*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]] + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(3*a*Sqrt[1 - a^2*x^2]) - (Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])} -{Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(1/2), x, 1, (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(3*a*(c - a^2*c*x^2)^(1/2))} -{Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(3/2), x, 1, (x*Sqrt[ArcSin[a*x]])/(c*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcSin[a*x]]), x])/(2*c*Sqrt[c - a^2*c*x^2])} -{Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2), x, 2, (x*Sqrt[ArcSin[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSin[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(6*c^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcSin[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2), x, 17, (27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(256*a*Sqrt[1 - a^2*x^2]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(32*Sqrt[1 - a^2*x^2]) + (3*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(32*a) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2) + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2) + (3*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2))/(20*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(512*a*Sqrt[1 - a^2*x^2]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)*ArcSin[a*x]^(3/2), x, 8, (3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(16*a*Sqrt[1 - a^2*x^2]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])/(8*Sqrt[1 - a^2*x^2]) + (1/2)*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2) + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2))/(5*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32*a*Sqrt[1 - a^2*x^2])} -{ArcSin[a*x]^(3/2)/(c - a^2*c*x^2)^(1/2), x, 1, (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(5/2))/(5*a*(c - a^2*c*x^2)^(1/2))} -{ArcSin[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2), x, 1, (x*ArcSin[a*x]^(3/2))/(c*Sqrt[c - a^2*c*x^2]) - (3*a*Sqrt[1 - a^2*x^2]*Unintegrable[(x*Sqrt[ArcSin[a*x]])/(1 - a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2), x, 27, (-(225/512))*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]] - (15/256)*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]] + (45*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(256*a*Sqrt[1 - a^2*x^2]) - (15*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(32*Sqrt[1 - a^2*x^2]) + (5*c*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(32*a) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2) + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2) + (3*c*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(7/2))/(28*a*Sqrt[1 - a^2*x^2]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(4096*a*Sqrt[1 - a^2*x^2]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(128*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)*ArcSin[a*x]^(5/2), x, 10, (-(15/32))*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]] + (5*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(16*a*Sqrt[1 - a^2*x^2]) - (5*a*x^2*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))/(8*Sqrt[1 - a^2*x^2]) + (1/2)*x*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2) + (Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[1 - a^2*x^2]) + (15*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(128*a*Sqrt[1 - a^2*x^2])} -{ArcSin[a*x]^(5/2)/(c - a^2*c*x^2)^(1/2), x, 1, (2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(7/2))/(7*a*(c - a^2*c*x^2)^(1/2))} -{ArcSin[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2), x, 1, (x*ArcSin[a*x]^(5/2))/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Unintegrable[(x*ArcSin[a*x]^(3/2))/(1 - a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])} - - -{ArcSin[x/a]^(1/2)*(a^2 - x^2)^(3/2), x, 15, (3/8)*a^2*x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]] + (1/4)*x*(a^2 - x^2)^(3/2)*Sqrt[ArcSin[x/a]] + (a^3*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2))/(4*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[x/a]]])/(64*Sqrt[1 - x^2/a^2]) - (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(8*Sqrt[1 - x^2/a^2])} -{ArcSin[x/a]^(1/2)*(a^2 - x^2)^(1/2), x, 7, (1/2)*x*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]] + (a*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2))/(3*Sqrt[1 - x^2/a^2]) - (a*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelS[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(8*Sqrt[1 - x^2/a^2])} -{ArcSin[x/a]^(1/2)/(a^2 - x^2)^(1/2), x, 1, (2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])} -{ArcSin[x/a]^(1/2)/(a^2 - x^2)^(3/2), x, 1, (x*Sqrt[ArcSin[x/a]])/(a^2*Sqrt[a^2 - x^2]) - (Sqrt[1 - x^2/a^2]*Unintegrable[x/((1 - x^2/a^2)*Sqrt[ArcSin[x/a]]), x])/(2*a^3*Sqrt[a^2 - x^2])} -{ArcSin[x/a]^(1/2)/(a^2 - x^2)^(5/2), x, 2, (x*Sqrt[ArcSin[x/a]])/(3*a^2*(a^2 - x^2)^(3/2)) + (2*x*Sqrt[ArcSin[x/a]])/(3*a^4*Sqrt[a^2 - x^2]) - (Sqrt[1 - x^2/a^2]*Unintegrable[x/((1 - x^2/a^2)^2*Sqrt[ArcSin[x/a]]), x])/(6*a^5*Sqrt[a^2 - x^2]) - (Sqrt[1 - x^2/a^2]*Unintegrable[x/((1 - x^2/a^2)*Sqrt[ArcSin[x/a]]), x])/(3*a^5*Sqrt[a^2 - x^2])} - - -{ArcSin[x/a]^(3/2)*(a^2 - x^2)^(3/2), x, 17, (27*a^3*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/(256*Sqrt[1 - x^2/a^2]) - (9*a*x^2*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/(32*Sqrt[1 - x^2/a^2]) + (3*(a^2 - x^2)^(5/2)*Sqrt[ArcSin[x/a]])/(32*a*Sqrt[1 - x^2/a^2]) + (3/8)*a^2*x*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2) + (1/4)*x*(a^2 - x^2)^(3/2)*ArcSin[x/a]^(3/2) + (3*a^3*Sqrt[a^2 - x^2]*ArcSin[x/a]^(5/2))/(20*Sqrt[1 - x^2/a^2]) - (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[x/a]]])/(512*Sqrt[1 - x^2/a^2]) - (3*a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelC[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(32*Sqrt[1 - x^2/a^2])} -{ArcSin[x/a]^(3/2)*(a^2 - x^2)^(1/2), x, 8, (3*a*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/(16*Sqrt[1 - x^2/a^2]) - (3*x^2*Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]])/(8*a*Sqrt[1 - x^2/a^2]) + (1/2)*x*Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2) + (a*Sqrt[a^2 - x^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[1 - x^2/a^2]) - (3*a*Sqrt[Pi]*Sqrt[a^2 - x^2]*FresnelC[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]])/(32*Sqrt[1 - x^2/a^2])} -{ArcSin[x/a]^(3/2)/(a^2 - x^2)^(1/2), x, 1, (2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])} -{ArcSin[x/a]^(3/2)/(a^2 - x^2)^(3/2), x, 1, (x*ArcSin[x/a]^(3/2))/(a^2*Sqrt[a^2 - x^2]) - (3*Sqrt[1 - x^2/a^2]*Unintegrable[(x*Sqrt[ArcSin[x/a]])/(1 - x^2/a^2), x])/(2*a^3*Sqrt[a^2 - x^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x/(Sqrt[1 - x^2]*Sqrt[ArcSin[x]]), x, 3, Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[x]]]} - - -{(c - a^2*c*x^2)^(5/2)/Sqrt[ArcSin[a*x]], x, 9, (5*c^2*(c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]])/(8*a*Sqrt[1 - a^2*x^2]) + (3*c^2*Sqrt[Pi/2]*(c - a^2*c*x^2)^(1/2)*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(16*a*Sqrt[1 - a^2*x^2]) + (c^2*Sqrt[Pi/3]*(c - a^2*c*x^2)^(1/2)*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(32*a*Sqrt[1 - a^2*x^2]) + (15*c^2*Sqrt[Pi]*(c - a^2*c*x^2)^(1/2)*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(32*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(3/2)/Sqrt[ArcSin[a*x]], x, 7, (3*c*(c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]])/(4*a*Sqrt[1 - a^2*x^2]) + (c*Sqrt[Pi/2]*(c - a^2*c*x^2)^(1/2)*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(8*a*Sqrt[1 - a^2*x^2]) + (c*Sqrt[Pi]*(c - a^2*c*x^2)^(1/2)*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(2*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)/Sqrt[ArcSin[a*x]], x, 5, ((c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]])/(a*Sqrt[1 - a^2*x^2]) + (Sqrt[Pi]*(c - a^2*c*x^2)^(1/2)*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(2*a*Sqrt[1 - a^2*x^2])} -{1/((c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]]), x, 1, (2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(a*(c - a^2*c*x^2)^(1/2))} -{1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]]), x, 0, Unintegrable[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]]), x]} -{1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcSin[a*x]]), x, 0, Unintegrable[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcSin[a*x]]), x]} - - -{(c - a^2*c*x^2)^(5/2)/ArcSin[a*x]^(3/2), x, 10, -((2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcSin[a*x]])) - (3*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(2*a*Sqrt[1 - a^2*x^2]) - (c^2*Sqrt[3*Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[3/Pi]*Sqrt[ArcSin[a*x]]])/(8*a*Sqrt[1 - a^2*x^2]) - (15*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(8*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(3/2), x, 8, -((2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSin[a*x]])) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(a*Sqrt[1 - a^2*x^2]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)/ArcSin[a*x]^(3/2), x, 6, (-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(1/2))/(a*Sqrt[ArcSin[a*x]]) - (2*Sqrt[Pi]*(c - a^2*c*x^2)^(1/2)*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(a*Sqrt[1 - a^2*x^2])} -{1/((c - a^2*c*x^2)^(1/2)*ArcSin[a*x]^(3/2)), x, 1, (-2*Sqrt[1 - a^2*x^2])/(a*(c - a^2*c*x^2)^(1/2)*Sqrt[ArcSin[a*x]])} -{1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)), x, 1, -((2*Sqrt[1 - a^2*x^2])/(a*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]])) + (4*a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^2*Sqrt[ArcSin[a*x]]), x])/(c*Sqrt[c - a^2*c*x^2])} -{1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(3/2)), x, 1, -((2*Sqrt[1 - a^2*x^2])/(a*(c - a^2*c*x^2)^(5/2)*Sqrt[ArcSin[a*x]])) + (8*a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^3*Sqrt[ArcSin[a*x]]), x])/(c^2*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(5/2), x, 12, -((2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcSin[a*x]^(3/2))) + (16*c*x*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcSin[a*x]]) - (4*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(3*a*Sqrt[1 - a^2*x^2]) - (8*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a*Sqrt[1 - a^2*x^2])} -{(c - a^2*c*x^2)^(1/2)/ArcSin[a*x]^(5/2), x, 4, (-2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^(1/2))/(3*a*ArcSin[a*x]^(3/2)) + (8*x*(c - a^2*c*x^2)^(1/2))/(3*Sqrt[ArcSin[a*x]]) - (8*Sqrt[Pi]*(c - a^2*c*x^2)^(1/2)*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]])/(3*a*Sqrt[1 - a^2*x^2])} -{1/((c - a^2*c*x^2)^(1/2)*ArcSin[a*x]^(5/2)), x, 1, (-2*Sqrt[1 - a^2*x^2])/(3*a*(c - a^2*c*x^2)^(1/2)*ArcSin[a*x]^(3/2))} -{1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2)), x, 1, -((2*Sqrt[1 - a^2*x^2])/(3*a*(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2))) + (4*a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^2*ArcSin[a*x]^(3/2)), x])/(3*c*Sqrt[c - a^2*c*x^2])} -{1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(5/2)), x, 1, -((2*Sqrt[1 - a^2*x^2])/(3*a*(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(3/2))) + (8*a*Sqrt[1 - a^2*x^2]*Unintegrable[x/((1 - a^2*x^2)^3*ArcSin[a*x]^(3/2)), x])/(3*c^2*Sqrt[c - a^2*c*x^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^n with n symbolic*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*(d - c^2*d*x^2)^(1/2)*(a + b*ArcSin[c*x])^n, x, 6, (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*c^3*(1 + n)*Sqrt[1 - c^2*x^2]) + (I*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(2^(2*(3 + n))*E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*E^((4*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2]))} -{x^1*(d - c^2*d*x^2)^(1/2)*(a + b*ArcSin[c*x])^n, x, 9, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(8*c^2*Sqrt[1 - c^2*x^2]))) - (E^((I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(8*c^2*Sqrt[1 - c^2*x^2])) - (3^(-1 - n)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(8*c^2*Sqrt[1 - c^2*x^2])) - (3^(-1 - n)*E^((3*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(8*c^2*Sqrt[1 - c^2*x^2]))} -{x^0*(d - c^2*d*x^2)^(1/2)*(a + b*ArcSin[c*x])^n, x, 6, (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(2*b*c*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-3 - n)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (I*2^(-3 - n)*E^((2*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2]))} -{(d - c^2*d*x^2)^(1/2)*(a + b*ArcSin[c*x])^n/x^1, x, 6, (d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(2*Sqrt[d - c^2*d*x^2])) + (d*E^((I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(2*Sqrt[d - c^2*d*x^2])) + d*Unintegrable[(a + b*ArcSin[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(d - c^2*d*x^2)^(1/2)*(a + b*ArcSin[c*x])^n/x^2, x, 3, -((c*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(b*(1 + n)*Sqrt[d - c^2*d*x^2])) + d*Unintegrable[(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -{x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n, x, 12, (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(16*b*c^3*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-7 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - n)*d*E^((2*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - 2*n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*2^(-7 - 2*n)*d*E^((4*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - n)*3^(-1 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((6*I*(a + b*ArcSin[c*x]))/b)])/(E^((6*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*2^(-7 - n)*3^(-1 - n)*d*E^((6*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (6*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2]))} -{x^1*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n, x, 12, -((d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(16*c^2*Sqrt[1 - c^2*x^2]))) - (d*E^((I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(16*c^2*Sqrt[1 - c^2*x^2])) - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(3^n*E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(32*c^2*Sqrt[1 - c^2*x^2])) - (d*E^((3*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/(3^n*((I*(a + b*ArcSin[c*x]))/b)^n*(32*c^2*Sqrt[1 - c^2*x^2])) - (5^(-1 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((5*I*(a + b*ArcSin[c*x]))/b)])/(E^((5*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(32*c^2*Sqrt[1 - c^2*x^2])) - (5^(-1 - n)*d*E^((5*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (5*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(32*c^2*Sqrt[1 - c^2*x^2]))} -{x^0*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n, x, 9, (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-3 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (I*2^(-3 - n)*d*E^((2*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2])) - (I*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(2^(2*(3 + n))*E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (I*d*E^((4*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2]))} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n/x^1, x, 15, (1/(8*Sqrt[d - c^2*d*x^2]))*((5*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) + (5*d^2*E^((I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(8*Sqrt[d - c^2*d*x^2])) + (1/(8*Sqrt[d - c^2*d*x^2]))*((3^(-1 - n)*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) + (1/(8*Sqrt[d - c^2*d*x^2]))*((3^(-1 - n)*d^2*E^((3*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) + d^2*Unintegrable[(a + b*ArcSin[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n/x^2, x, 9, -((3*c*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(2*b*(1 + n)*Sqrt[d - c^2*d*x^2])) + (1/Sqrt[d - c^2*d*x^2])*((I*2^(-3 - n)*c*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) - (1/Sqrt[d - c^2*d*x^2])*((I*2^(-3 - n)*c*d^2*E^((2*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) + d^2*Unintegrable[(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -{x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n, x, 15, (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - n)*d^2*E^((2*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(2^(2*(4 + n))*E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*d^2*E^((4*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(2^(2*(4 + n))*((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((6*I*(a + b*ArcSin[c*x]))/b)])/(E^((6*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*2^(-7 - n)*3^(-1 - n)*d^2*E^((6*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (6*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2])) + (I*2^(-11 - 3*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((8*I*(a + b*ArcSin[c*x]))/b)])/(E^((8*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c^3*Sqrt[1 - c^2*x^2])) - (I*2^(-11 - 3*n)*d^2*E^((8*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (8*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c^3*Sqrt[1 - c^2*x^2]))} -{x^1*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n, x, 15, -((5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(128*c^2*Sqrt[1 - c^2*x^2]))) - (5*d^2*E^((I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (3^(1 - n)*d^2*E^((3*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((5*I*(a + b*ArcSin[c*x]))/b)])/(5^n*E^((5*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (d^2*E^((5*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (5*I*(a + b*ArcSin[c*x]))/b])/(5^n*((I*(a + b*ArcSin[c*x]))/b)^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((7*I*(a + b*ArcSin[c*x]))/b)])/(E^((7*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(128*c^2*Sqrt[1 - c^2*x^2])) - (7^(-1 - n)*d^2*E^((7*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (7*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(128*c^2*Sqrt[1 - c^2*x^2]))} -{x^0*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n, x, 12, (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(16*b*c*(1 + n)*Sqrt[1 - c^2*x^2]) - (15*I*2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (15*I*2^(-7 - n)*d^2*E^((2*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2])) - (3*I*2^(-7 - 2*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (3*I*2^(-7 - 2*n)*d^2*E^((4*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2])) - (I*2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((6*I*(a + b*ArcSin[c*x]))/b)])/(E^((6*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n*(c*Sqrt[1 - c^2*x^2])) + (I*2^(-7 - n)*3^(-1 - n)*d^2*E^((6*I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (6*I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(c*Sqrt[1 - c^2*x^2]))} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n/x^1, x, 27, (1/(16*Sqrt[d - c^2*d*x^2]))*((11*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) + (11*d^3*E^((I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(((I*(a + b*ArcSin[c*x]))/b)^n*(16*Sqrt[d - c^2*d*x^2])) - (1/(32*Sqrt[d - c^2*d*x^2]))*((5*3^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) + (1/(8*Sqrt[d - c^2*d*x^2]))*((d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c*x]))/b)])/(3^n*E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) - (1/(32*Sqrt[d - c^2*d*x^2]))*((5*3^(-1 - n)*d^3*E^((3*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) + (1/(8*Sqrt[d - c^2*d*x^2]))*((d^3*E^((3*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c*x]))/b])/(3^n*((I*(a + b*ArcSin[c*x]))/b)^n)) + (1/(32*Sqrt[d - c^2*d*x^2]))*((5^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((5*I*(a + b*ArcSin[c*x]))/b)])/(E^((5*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) + (1/(32*Sqrt[d - c^2*d*x^2]))*((5^(-1 - n)*d^3*E^((5*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (5*I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) + d^3*Unintegrable[(a + b*ArcSin[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n/x^2, x, 18, -((15*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2])) + (1/Sqrt[d - c^2*d*x^2])*((I*2^(-2 - n)*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) - (1/Sqrt[d - c^2*d*x^2])*((I*2^(-2 - n)*c*d^3*E^((2*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) + (1/Sqrt[d - c^2*d*x^2])*((I*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, -((4*I*(a + b*ArcSin[c*x]))/b)])/(2^(2*(3 + n))*E^((4*I*a)/b)*(-((I*(a + b*ArcSin[c*x]))/b))^n)) - (1/Sqrt[d - c^2*d*x^2])*((I*c*d^3*E^((4*I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, (4*I*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*((I*(a + b*ArcSin[c*x]))/b)^n)) + d^3*Unintegrable[(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m*ArcSin[a*x]^n/Sqrt[1 - a^2*x^2], x, 0, Unintegrable[(x^m*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2], x]} - -{x^3*ArcSin[a*x]^n/Sqrt[1 - a^2*x^2], x, 9, -((3*ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(8*a^4))) - (3*ArcSin[a*x]^n*Gamma[1 + n, I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(8*a^4)) + (3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, -3*I*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(8*a^4)) + (3^(-1 - n)*ArcSin[a*x]^n*Gamma[1 + n, 3*I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(8*a^4))} -{x^2*ArcSin[a*x]^n/Sqrt[1 - a^2*x^2], x, 6, ArcSin[a*x]^(1 + n)/(2*a^3*(1 + n)) + (I*2^(-3 - n)*ArcSin[a*x]^n*Gamma[1 + n, -2*I*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*a^3) - (I*2^(-3 - n)*ArcSin[a*x]^n*Gamma[1 + n, 2*I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*a^3)} -{x^1*ArcSin[a*x]^n/Sqrt[1 - a^2*x^2], x, 4, -((ArcSin[a*x]^n*Gamma[1 + n, (-I)*ArcSin[a*x]])/(((-I)*ArcSin[a*x])^n*(2*a^2))) - (ArcSin[a*x]^n*Gamma[1 + n, I*ArcSin[a*x]])/((I*ArcSin[a*x])^n*(2*a^2))} -{x^0*ArcSin[a*x]^n/Sqrt[1 - a^2*x^2], x, 1, ArcSin[a*x]^(1 + n)/(a*(1 + n))} -{ArcSin[a*x]^n/(x^1*Sqrt[1 - a^2*x^2]), x, 0, Unintegrable[ArcSin[a*x]^n/(x*Sqrt[1 - a^2*x^2]), x]} -{ArcSin[a*x]^n/(x^2*Sqrt[1 - a^2*x^2]), x, 0, Unintegrable[ArcSin[a*x]^n/(x^2*Sqrt[1 - a^2*x^2]), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^p (f+g x)^q (a+b ArcSin[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSin[c x])^1 with e f+d g=0 and c^2 d^2-e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(d + c*d*x)^(5/2)*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]), x, 13, (2*b*d^2*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*d^2*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(16*Sqrt[1 - c^2*x^2]) - (2*b*c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(16*Sqrt[1 - c^2*x^2]) + (3*d^2*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/8 + (c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/4 - (2*d^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c) + (5*d^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2])} -{(d + c*d*x)^(3/2)*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]), x, 8, (b*d*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(3*Sqrt[1 - c^2*x^2]) - (b*c*d*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(4*Sqrt[1 - c^2*x^2]) - (b*c^2*d*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(9*Sqrt[1 - c^2*x^2]) + (d*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/2 - (d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c) + (d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{(d + c*d*x)^(1/2)*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]), x, 4, -(b*c*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(4*Sqrt[1 - c^2*x^2]) + (x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/2 + (Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])/(d + c*d*x)^(1/2), x, 6, -((b*f*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x])) + (f*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])/(d + c*d*x)^(3/2), x, 8, (-2*f^2*(1 - c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (f^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*b*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (2*b*f^2*(1 - c^2*x^2)^(3/2)*Log[1 + c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])/(d + c*d*x)^(5/2), x, 6, (-2*b*f^3*(1 - c^2*x^2)^(5/2))/(3*c*(1 + c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (f^3*(1 - c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*f^3*(1 - c^2*x^2)^(5/2)*Log[1 + c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -{(d + c*d*x)^(5/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]), x, 12, (b*d*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(5*(1 - c^2*x^2)^(3/2)) - (5*b*c*d*x^2*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) - (2*b*c^2*d*x^3*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(15*(1 - c^2*x^2)^(3/2)) + (b*c^3*d*x^4*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) + (b*c^4*d*x^5*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(25*(1 - c^2*x^2)^(3/2)) + (d*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*d*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)) - (d*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(5*c) + (3*d*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(16*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]), x, 7, (-5*b*c*x^2*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) + (b*c^3*x^4*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) + (x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)) + (3*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(16*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(1/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]), x, 8, -(b*f*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(3*Sqrt[1 - c^2*x^2]) - (b*c*f*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(4*Sqrt[1 - c^2*x^2]) + (b*c^2*f*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(9*Sqrt[1 - c^2*x^2]) + (f*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/2 + (f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c) + (f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(1/2), x, 9, (-2*b*f^2*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (b*c*f^2*x^2*Sqrt[1 - c^2*x^2])/(4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (2*f^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (f^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (3*f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(3/2), x, 10, (b*f^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (4*f^3*(1 - c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (f^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (3*f^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*b*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (4*b*f^3*(1 - c^2*x^2)^(3/2)*Log[1 + c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(5/2), x, 9, (-4*b*f^4*(1 - c^2*x^2)^(5/2))/(3*c*(1 + c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*f^4*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]^2)/(2*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (2*f^4*(1 - c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*f^4*(1 - c*x)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (f^4*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (8*b*f^4*(1 - c^2*x^2)^(5/2)*Log[1 + c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -{(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]), x, 9, (-25*b*c*x^2*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))/(96*(1 - c^2*x^2)^(5/2)) + (5*b*c^3*x^4*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))/(96*(1 - c^2*x^2)^(5/2)) + (b*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*Sqrt[1 - c^2*x^2])/(36*c) + (x*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/6 + (5*x*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/(16*(1 - c^2*x^2)^2) + (5*x*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/(24*(1 - c^2*x^2)) + (5*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(32*b*c*(1 - c^2*x^2)^(5/2))} -{(d + c*d*x)^(3/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]), x, 12, -(b*f*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(5*(1 - c^2*x^2)^(3/2)) - (5*b*c*f*x^2*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) + (2*b*c^2*f*x^3*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(15*(1 - c^2*x^2)^(3/2)) + (b*c^3*f*x^4*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(16*(1 - c^2*x^2)^(3/2)) - (b*c^4*f*x^5*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))/(25*(1 - c^2*x^2)^(3/2)) + (f*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*f*x*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)) + (f*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(5*c) + (3*f*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(16*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(1/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]), x, 13, (-2*b*f^2*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*f^2*x^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(16*Sqrt[1 - c^2*x^2]) + (2*b*c^2*f^2*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*f^2*x^4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(16*Sqrt[1 - c^2*x^2]) + (3*f^2*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/8 + (c^2*f^2*x^3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/4 + (2*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c) + (5*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2])} -{(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(1/2), x, 13, (-11*b*f^3*x*Sqrt[1 - c^2*x^2])/(3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (3*b*c*f^3*x^2*Sqrt[1 - c^2*x^2])/(4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (b*c^2*f^3*x^3*Sqrt[1 - c^2*x^2])/(9*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (11*f^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (3*f^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (c*f^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (5*f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(3/2), x, 7, (3*b*f^4*x*(1 - c^2*x^2)^(3/2))/(2*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (b*c*f^4*x^2*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (5*b*f^4*(1 - c*x)^2*(1 - c^2*x^2)^(3/2))/(4*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (15*b*f^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2)/(4*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (2*f^4*(1 - c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (15*f^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (5*f^4*(1 - c*x)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (15*f^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (8*b*f^4*(1 - c^2*x^2)^(3/2)*Log[1 + c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x])/(d + c*d*x)^(5/2), x, 10, -((b*f^5*x*(1 - c^2*x^2)^(5/2))/((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))) - (8*b*f^5*(1 - c^2*x^2)^(5/2))/(3*c*(1 + c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (5*b*f^5*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]^2)/(2*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (2*f^5*(1 - c*x)^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (10*f^5*(1 - c*x)^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (5*f^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (5*f^5*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (28*b*f^5*(1 - c^2*x^2)^(5/2)*Log[1 + c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x], x, 13, (11*b*d^3*x*Sqrt[1 - c^2*x^2])/(3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (3*b*c*d^3*x^2*Sqrt[1 - c^2*x^2])/(4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (b*c^2*d^3*x^3*Sqrt[1 - c^2*x^2])/(9*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (11*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (3*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (c*d^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (5*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x], x, 9, (2*b*d^2*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (b*c*d^2*x^2*Sqrt[1 - c^2*x^2])/(4*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (3*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x], x, 6, (b*d*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) + (d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(1/2)*Sqrt[f - c*f*x]), x, 2, (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*Sqrt[f - c*f*x]), x, 5, -((f*(1 - c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))) + (b*f*(1 - c^2*x^2)^(3/2)*Log[1 + c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*Sqrt[f - c*f*x]), x, 8, -(b*f^2*(1 - c^2*x^2)^(5/2))/(3*c*(1 + c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (2*f^2*(1 - c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (f^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*f^2*(1 - c^2*x^2)^(5/2)*ArcTanh[c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*f^2*(1 - c^2*x^2)^(5/2)*Log[1 - c^2*x^2])/(6*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2), x, 7, (-3*b*d^4*x*(1 - c^2*x^2)^(3/2))/(2*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (b*c*d^4*x^2*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (5*b*d^4*(1 + c*x)^2*(1 - c^2*x^2)^(3/2))/(4*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (15*b*d^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2)/(4*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (2*d^4*(1 + c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (15*d^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (5*d^4*(1 + c*x)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (15*d^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (8*b*d^4*(1 - c^2*x^2)^(3/2)*Log[1 - c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2), x, 10, -((b*d^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))) + (4*d^3*(1 + c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (3*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*b*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (4*b*d^3*(1 - c^2*x^2)^(3/2)*Log[1 - c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2), x, 8, (2*d^2*(1 + c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) - (d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(2*b*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (2*b*d^2*(1 - c^2*x^2)^(3/2)*Log[1 - c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(1/2)*(f - c*f*x)^(3/2)), x, 5, (d*(1 + c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (b*d*(1 - c^2*x^2)^(3/2)*Log[1 - c*x])/(c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)), x, 3, (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)) + (b*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2])/(2*c*(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*(f - c*f*x)^(3/2)), x, 8, -(b*f*(1 - c^2*x^2)^(5/2))/(6*c*(1 + c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (f*(1 - c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*f*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*f*(1 - c^2*x^2)^(5/2)*ArcTanh[c*x])/(6*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*f*(1 - c^2*x^2)^(5/2)*Log[1 - c^2*x^2])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2), x, 10, (b*d^5*x*(1 - c^2*x^2)^(5/2))/((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (8*b*d^5*(1 - c^2*x^2)^(5/2))/(3*c*(1 - c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (5*b*d^5*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]^2)/(2*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*d^5*(1 + c*x)^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (10*d^5*(1 + c*x)^2*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (5*d^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (5*d^5*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (28*b*d^5*(1 - c^2*x^2)^(5/2)*Log[1 - c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2), x, 9, (-4*b*d^4*(1 - c^2*x^2)^(5/2))/(3*c*(1 - c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*d^4*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]^2)/(2*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*d^4*(1 + c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (2*d^4*(1 + c*x)*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (d^4*(1 - c^2*x^2)^(5/2)*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (8*b*d^4*(1 - c^2*x^2)^(5/2)*Log[1 - c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2), x, 6, (-2*b*d^3*(1 - c^2*x^2)^(5/2))/(3*c*(1 - c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (d^3*(1 + c*x)^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*d^3*(1 - c^2*x^2)^(5/2)*Log[1 - c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(1/2)*(f - c*f*x)^(5/2)), x, 8, -(b*d^2*(1 - c^2*x^2)^(5/2))/(3*c*(1 - c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*d^2*(1 + c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*d^2*(1 - c^2*x^2)^(5/2)*ArcTanh[c*x])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*d^2*(1 - c^2*x^2)^(5/2)*Log[1 - c^2*x^2])/(6*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*(f - c*f*x)^(5/2)), x, 8, -(b*d*(1 - c^2*x^2)^(5/2))/(6*c*(1 - c*x)*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (d*(1 + c*x)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*d*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) - (b*d*(1 - c^2*x^2)^(5/2)*ArcTanh[c*x])/(6*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*d*(1 - c^2*x^2)^(5/2)*Log[1 - c^2*x^2])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} -{(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)), x, 5, -(b*(1 - c^2*x^2)^(3/2))/(6*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)) + (b*(1 - c^2*x^2)^(5/2)*Log[1 - c^2*x^2])/(3*c*(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSin[c x])^2 with e f+d g=0 and c^2 d^2-e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(d + c*d*x)^(5/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2, x, 23, (8*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (15*b^2*d^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/64 - (b^2*c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/32 + (4*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (15*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (4*b*d^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*d^2*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (4*b*c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (3*d^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/8 + (c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/4 - (2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (5*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(24*b*c*Sqrt[1 - c^2*x^2])} -{(d + c*d*x)^(3/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2, x, 13, (4*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (b^2*d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 + (2*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) + (2*b*d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (b*c*d*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (2*b*c^2*d*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/2 - (d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{(d + c*d*x)^(1/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2, x, 6, -(b^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b*c*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/2 + (Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(1/2), x, 8, (-2*a*b*e*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b^2*e*(1 - c^2*x^2))/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b^2*e*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (e*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2), x, 19, (-2*e^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*e^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*e^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2), x, 20, ((I/3)*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b^2*e^3*(1 - c^2*x^2)^(5/2)*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*e^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b^2*e^3*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -{(d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2, x, 19, (8*b^2*d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(225*c) - (b^2*d*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/32 + (16*b^2*d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(75*c*(1 - c^2*x^2)) - (15*b^2*d*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(64*(1 - c^2*x^2)) + (2*b^2*d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(1 - c^2*x^2))/(125*c) + (9*b^2*d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*ArcSin[c*x])/(64*c*(1 - c^2*x^2)^(3/2)) + (2*b*d*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(5*(1 - c^2*x^2)^(3/2)) - (3*b*c*d*x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)^(3/2)) - (4*b*c^2*d*x^3*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(15*(1 - c^2*x^2)^(3/2)) + (2*b*c^4*d*x^5*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(25*(1 - c^2*x^2)^(3/2)) + (b*d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (d*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*d*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(8*(1 - c^2*x^2)) - (d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(5*c) + (d*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^3)/(8*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2, x, 11, -(b^2*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/32 - (15*b^2*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(64*(1 - c^2*x^2)) + (9*b^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*ArcSin[c*x])/(64*c*(1 - c^2*x^2)^(3/2)) - (3*b*c*x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)^(3/2)) + (b*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(8*(1 - c^2*x^2)) + ((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^3)/(8*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(1/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2, x, 13, (-4*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (b^2*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 - (2*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (2*b*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (b*c*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (2*b*c^2*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/2 + (e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(1/2), x, 11, (-4*b^2*e^2*(1 - c^2*x^2))/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*e^2*x*(1 - c^2*x^2))/(4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (b^2*e^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (4*b*e^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b*c*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*e^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (e^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2), x, 23, (2*a*b*e^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b^2*e^3*(1 - c^2*x^2)^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b^2*e^3*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (4*e^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*e^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*e^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (e^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (e^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((16*I)*b*e^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*b*e^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((8*I)*b^2*e^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b^2*e^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*e^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2), x, 21, (((8*I)/3)*e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (8*b^2*e^4*(1 - c^2*x^2)^(5/2)*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (8*e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (32*b*e^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((32*I)/3)*b^2*e^4*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -{(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2, x, 17, -(b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))/108 - (245*b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))/(1152*(1 - c^2*x^2)^2) - (65*b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))/(1728*(1 - c^2*x^2)) + (115*b^2*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*ArcSin[c*x])/(1152*c*(1 - c^2*x^2)^(5/2)) - (5*b*c*x^2*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x]))/(16*(1 - c^2*x^2)^(5/2)) + (5*b*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x]))/(48*c*Sqrt[1 - c^2*x^2]) + (b*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/6 + (5*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(16*(1 - c^2*x^2)^2) + (5*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(24*(1 - c^2*x^2)) + (5*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^3)/(48*b*c*(1 - c^2*x^2)^(5/2))} -{(d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2, x, 19, (-8*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(225*c) - (b^2*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/32 - (16*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(75*c*(1 - c^2*x^2)) - (15*b^2*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(64*(1 - c^2*x^2)) - (2*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(1 - c^2*x^2))/(125*c) + (9*b^2*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*ArcSin[c*x])/(64*c*(1 - c^2*x^2)^(3/2)) - (2*b*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(5*(1 - c^2*x^2)^(3/2)) - (3*b*c*e*x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)^(3/2)) + (4*b*c^2*e*x^3*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(15*(1 - c^2*x^2)^(3/2)) - (2*b*c^4*e*x^5*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(25*(1 - c^2*x^2)^(3/2)) + (b*e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*e*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(8*(1 - c^2*x^2)) + (e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(5*c) + (e*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^3)/(8*b*c*(1 - c^2*x^2)^(3/2))} -{(d + c*d*x)^(1/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2, x, 23, (-8*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (15*b^2*e^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/64 - (b^2*c^2*e^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/32 - (4*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (15*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) - (4*b*e^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*e^2*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (4*b*c^2*e^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*e^2*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (3*e^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/8 + (c^2*e^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/4 + (2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (5*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(24*b*c*Sqrt[1 - c^2*x^2])} -{((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(1/2), x, 17, (-68*b^2*e^3*(1 - c^2*x^2))/(9*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b^2*e^3*x*(1 - c^2*x^2))/(4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*e^3*(1 - c^2*x^2)^2)/(27*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*b^2*e^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (22*b*e^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b*c*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b*c^2*e^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (11*e^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*e^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (c*e^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (5*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2), x, 28, (8*a*b*e^4*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*b^2*e^4*(1 - c^2*x^2)^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b^2*e^4*x*(1 - c^2*x^2)^2)/(4*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (b^2*e^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(4*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*b^2*e^4*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b*c*e^4*x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (8*e^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*e^4*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*e^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (4*e^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (e^4*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (5*e^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(2*b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((32*I)*b*e^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (16*b*e^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((16*I)*b^2*e^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((16*I)*b^2*e^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b^2*e^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2), x, 25, (-2*a*b*e^5*x*(1 - c^2*x^2)^(5/2))/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b^2*e^5*(1 - c^2*x^2)^3)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b^2*e^5*x*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((28*I)/3)*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (5*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (16*b^2*e^5*(1 - c^2*x^2)^(5/2)*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (28*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (8*b*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (112*b*e^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((112*I)/3)*b^2*e^5*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x], x, 17, (68*b^2*d^3*(1 - c^2*x^2))/(9*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b^2*d^3*x*(1 - c^2*x^2))/(4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b^2*d^3*(1 - c^2*x^2)^2)/(27*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*b^2*d^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (22*b*d^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b*c*d^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b*c^2*d^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (11*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (c*d^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (5*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x], x, 11, (4*b^2*d^2*(1 - c^2*x^2))/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*d^2*x*(1 - c^2*x^2))/(4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (b^2*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (4*b*d^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b*c*d^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x], x, 8, (2*a*b*d*x*Sqrt[1 - c^2*x^2])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*d*(1 - c^2*x^2))/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*d*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(1/2)*Sqrt[e - c*e*x]), x, 2, (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*Sqrt[e - c*e*x]), x, 16, -((e*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))) + (e*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*e*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b*e*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b*e*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((2*I)*b^2*e*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*e*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*b^2*e*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*Sqrt[e - c*e*x]), x, 30, (-2*b^2*e^2*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*e^2*x*(1 - c^2*x^2)^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b^2*e^2*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*e^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b*e^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*c*e^2*x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*e^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (c^2*e^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*e^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*e^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((4*I)/3)*b*e^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b*e^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((2*I)/3)*b^2*e^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b^2*e^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*b^2*e^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2), x, 28, (-8*a*b*d^4*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (8*b^2*d^4*(1 - c^2*x^2)^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b^2*d^4*x*(1 - c^2*x^2)^2)/(4*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (b^2*d^4*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(4*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (8*b^2*d^4*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (b*c*d^4*x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*d^4*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*d^4*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^4*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (d^4*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (5*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(2*b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((32*I)*b*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (16*b*d^4*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((16*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((16*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b^2*d^4*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2), x, 23, (-2*a*b*d^3*x*(1 - c^2*x^2)^(3/2))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*(1 - c^2*x^2)^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (2*b^2*d^3*x*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*d^3*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (d^3*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((16*I)*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (8*b*d^3*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((8*I)*b^2*d^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((8*I)*b^2*d^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*d^3*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2), x, 19, (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((8*I)*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (4*b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((4*I)*b^2*d^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((4*I)*b^2*d^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*d^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(1/2)*(e - c*e*x)^(3/2)), x, 16, (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (d*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((4*I)*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - ((2*I)*b^2*d*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + ((2*I)*b^2*d*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*b^2*d*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)), x, 7, (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)), x, 21, -(b^2*e*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (b^2*e*x*(1 - c^2*x^2)^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*e*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (b*e*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (e*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (e*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*e*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*e*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b*e*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (4*b*e*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + ((I/3)*b^2*e*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*b^2*e*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b^2*e*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -{((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2), x, 25, (2*a*b*d^5*x*(1 - c^2*x^2)^(5/2))/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^5*(1 - c^2*x^2)^3)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^5*x*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((28*I)/3)*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (d^5*(1 - c^2*x^2)^3*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (5*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (112*b*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((112*I)/3)*b^2*d^5*(1 - c^2*x^2)^(5/2)*PolyLog[2, I/E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (8*b*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (16*b^2*d^5*(1 - c^2*x^2)^(5/2)*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (28*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (4*d^5*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} -{((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2), x, 21, (((-8*I)/3)*d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^3)/(3*b*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (32*b*d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((32*I)/3)*b^2*d^4*(1 - c^2*x^2)^(5/2)*PolyLog[2, I/E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (8*b^2*d^4*(1 - c^2*x^2)^(5/2)*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (8*d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d^4*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} -{((d + c*d*x)^(1/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2), x, 20, ((-I/3)*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (4*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((4*I)/3)*b^2*d^3*(1 - c^2*x^2)^(5/2)*PolyLog[2, I/E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (4*b^2*d^3*(1 - c^2*x^2)^(5/2)*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d^3*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(1/2)*(e - c*e*x)^(5/2)), x, 30, (2*b^2*d^2*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b^2*d^2*x*(1 - c^2*x^2)^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b^2*d^2*(1 - c^2*x^2)^(5/2)*ArcSin[c*x])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*d^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (2*b*d^2*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*c*d^2*x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (c^2*d^2*x^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d^2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((4*I)/3)*b*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*b*d^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((2*I)/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*b^2*d^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)), x, 21, (b^2*d*(1 - c^2*x^2)^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (b^2*d*x*(1 - c^2*x^2)^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*d*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*d*x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (d*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*d*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (((2*I)/3)*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (4*b*d*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - ((I/3)*b^2*d*(1 - c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + ((I/3)*b^2*d*(1 - c^2*x^2)^(5/2)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b^2*d*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} -{(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)), x, 10, (b^2*x*(1 - c^2*x^2)^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (b*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (2*x*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(3*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) + (4*b*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(3*c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (((2*I)/3)*b^2*(1 - c^2*x^2)^(5/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^(p/2) (f+g x)^(p/2) (a+b ArcSin[c x])^2 where e f+d g=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2), x, 11, (b^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(64*c^2) - (1/32)*b^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] - (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (b*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (b*c*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (1/4)*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 + (Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(24*b*c^3*Sqrt[1 - c^2*x^2])} -{x^1*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2), x, 6, (4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c^2) + (2*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c^2) + (2*b*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (2*b*c*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c^2)} -{x^0*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2), x, 6, (-(1/4))*b^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b*c*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (1/2)*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 + (Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{(a + b*ArcSin[c*x])^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)/x^1, x, 13, -2*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] - (2*a*b*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/Sqrt[1 - c^2*x^2] - (2*b^2*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 - (2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*I*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*I*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)/x^2, x, 8, -((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/x) - (I*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(3*b*Sqrt[1 - c^2*x^2]) + (2*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b^2*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} - - -{x^2*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2), x, 18, -((7*b^2*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(1152*c^2)) - (43*b^2*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/1728 + (1/108)*b^2*c^2*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] + (7*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2]) + (b*d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(16*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*e*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) + (b*c^3*d*e*x^6*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(18*Sqrt[1 - c^2*x^2]) - (d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (1/8)*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 + (1/6)*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2 + (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(48*b*c^3*Sqrt[1 - c^2*x^2])} -{x^1*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2), x, 7, (16*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(75*c^2) + (8*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(225*c^2) + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2)/(125*c^2) + (2*b*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^5*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)^2*(a + b*ArcSin[c*x])^2)/(5*c^2)} -{x^0*(a + b*ArcSin[c*x])^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2), x, 11, (-(1/32))*b^2*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2) - (15*b^2*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))/(64*(1 - c^2*x^2)) + (9*b^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*ArcSin[c*x])/(64*c*(1 - c^2*x^2)^(3/2)) - (3*b*c*x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x]))/(8*(1 - c^2*x^2)^(3/2)) + (b*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (1/4)*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2 + (3*x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(8*(1 - c^2*x^2)) + ((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^3)/(8*b*c*(1 - c^2*x^2)^(3/2))} -{(a + b*ArcSin[c*x])^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)/x^1, x, 18, (1/9)*-22*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] - (2*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/Sqrt[1 - c^2*x^2] - (2/27)*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2) - (2*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*e*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 + (1/3)*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2 - (2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*I*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*I*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, -E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] + (2*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[3, E^(I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} -{(a + b*ArcSin[c*x])^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)/x^2, x, 15, (1/4)*b^2*c^2*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] - (5*b^2*c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + b*c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (3/2)*c^2*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2 - (I*c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/x - (c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(2*b*Sqrt[1 - c^2*x^2]) + (2*b*c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2] - (I*b^2*c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^(2*I*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^2*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)), x, 6, (b^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*c^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{x^1*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)), x, 5, (2*a*b*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*(1 - c^2*x^2))/(c^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - ((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{x^0*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)), x, 2, (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(a + b*ArcSin[c*x])^2/(x^1*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)), x, 9, -((2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(a + b*ArcSin[c*x])^2/(x^2*(e - c*e*x)^(1/2)*(d + c*d*x)^(1/2)), x, 7, -((I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])) - ((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} - - -{x^2*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)), x, 8, (x*(a + b*ArcSin[c*x])^2)/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c^3*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{x^1*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)), x, 8, (a + b*ArcSin[c*x])^2/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{x^0*(a + b*ArcSin[c*x])^2/((e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)), x, 7, (x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) + (2*b*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)) - (I*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))} -{(a + b*ArcSin[c*x])^2/(x^1*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)), x, 16, (a + b*ArcSin[c*x])^2/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (4*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*I*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (2*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} -{(a + b*ArcSin[c*x])^2/(x^2*(e - c*e*x)^(3/2)*(d + c*d*x)^(3/2)), x, 15, -((a + b*ArcSin[c*x])^2/(d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])) + (2*c^2*x*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (2*I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E^(2*I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (4*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcSin[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcSin[c*x]), x, 5, (b*(7*c^2*d + 5*e)*Sqrt[1 - c^2*x^2])/(35*c^7) - (b*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(3/2))/(105*c^7) + (b*(7*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e*(1 - c^2*x^2)^(7/2))/(49*c^7) + (1/5)*d*x^5*(a + b*ArcSin[c*x]) + (1/7)*e*x^7*(a + b*ArcSin[c*x])} -{x^3*(d + e*x^2)*(a + b*ArcSin[c*x]), x, 6, (b*(9*c^2*d + 5*e)*x*Sqrt[1 - c^2*x^2])/(96*c^5) + (b*(9*c^2*d + 5*e)*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*x^5*Sqrt[1 - c^2*x^2])/(36*c) - (b*(9*c^2*d + 5*e)*ArcSin[c*x])/(96*c^6) + (1/4)*d*x^4*(a + b*ArcSin[c*x]) + (1/6)*e*x^6*(a + b*ArcSin[c*x])} -{x^2*(d + e*x^2)*(a + b*ArcSin[c*x]), x, 5, (b*(5*c^2*d + 3*e)*Sqrt[1 - c^2*x^2])/(15*c^5) - (b*(5*c^2*d + 6*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) + (b*e*(1 - c^2*x^2)^(5/2))/(25*c^5) + (1/3)*d*x^3*(a + b*ArcSin[c*x]) + (1/5)*e*x^5*(a + b*ArcSin[c*x])} -{x^1*(d + e*x^2)*(a + b*ArcSin[c*x]), x, 4, (3*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(16*c) - (b*(8*c^4*d^2 + 8*c^2*d*e + 3*e^2)*ArcSin[c*x])/(32*c^4*e) + ((d + e*x^2)^2*(a + b*ArcSin[c*x]))/(4*e)} -{x^0*(d + e*x^2)*(a + b*ArcSin[c*x]), x, 4, (b*(3*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*e*(1 - c^2*x^2)^(3/2))/(9*c^3) + d*x*(a + b*ArcSin[c*x]) + (1/3)*e*x^3*(a + b*ArcSin[c*x])} -{(d + e*x^2)*(a + b*ArcSin[c*x])/x^1, x, 12, (b*e*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e*ArcSin[c*x])/(4*c^2) - (1/2)*I*b*d*ArcSin[c*x]^2 + (1/2)*e*x^2*(a + b*ArcSin[c*x]) + b*d*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - b*d*ArcSin[c*x]*Log[x] + d*(a + b*ArcSin[c*x])*Log[x] - (1/2)*I*b*d*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)*(a + b*ArcSin[c*x])/x^2, x, 5, (b*e*Sqrt[1 - c^2*x^2])/c - (d*(a + b*ArcSin[c*x]))/x + e*x*(a + b*ArcSin[c*x]) - b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(d + e*x^2)*(a + b*ArcSin[c*x])/x^3, x, 10, -((b*c*d*Sqrt[1 - c^2*x^2])/(2*x)) - (1/2)*I*b*e*ArcSin[c*x]^2 - (d*(a + b*ArcSin[c*x]))/(2*x^2) + b*e*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - b*e*ArcSin[c*x]*Log[x] + e*(a + b*ArcSin[c*x])*Log[x] - (1/2)*I*b*e*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)*(a + b*ArcSin[c*x])/x^4, x, 6, -((b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2)) - (d*(a + b*ArcSin[c*x]))/(3*x^3) - (e*(a + b*ArcSin[c*x]))/x - (1/6)*b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{x^4*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 6, (b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*Sqrt[1 - c^2*x^2])/(315*c^9) - (2*b*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (b*(21*c^4*d^2 + 90*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (2*b*e*(9*c^2*d + 14*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^2*(1 - c^2*x^2)^(9/2))/(81*c^9) + (1/5)*d^2*x^5*(a + b*ArcSin[c*x]) + (2/7)*d*e*x^7*(a + b*ArcSin[c*x]) + (1/9)*e^2*x^9*(a + b*ArcSin[c*x])} -{x^3*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 7, (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x^3*Sqrt[1 - c^2*x^2])/(4608*c^5) + (b*e*(64*c^2*d + 21*e)*x^5*Sqrt[1 - c^2*x^2])/(1152*c^3) + (b*e^2*x^7*Sqrt[1 - c^2*x^2])/(64*c) - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*ArcSin[c*x])/(3072*c^8) + (1/4)*d^2*x^4*(a + b*ArcSin[c*x]) + (1/3)*d*e*x^6*(a + b*ArcSin[c*x]) + (1/8)*e^2*x^8*(a + b*ArcSin[c*x])} -{x^2*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt[1 - c^2*x^2])/(105*c^7) - (b*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1 - c^2*x^2)^(3/2))/(315*c^7) + (b*e*(14*c^2*d + 15*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^2*(1 - c^2*x^2)^(7/2))/(49*c^7) + (1/3)*d^2*x^3*(a + b*ArcSin[c*x]) + (2/5)*d*e*x^5*(a + b*ArcSin[c*x]) + (1/7)*e^2*x^7*(a + b*ArcSin[c*x])} -{x^1*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (b*(44*c^4*d^2 + 44*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2*x^2])/(288*c^5) + (5*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(144*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/(36*c) - (b*(2*c^2*d + e)*(8*c^4*d^2 + 8*c^2*d*e + 5*e^2)*ArcSin[c*x])/(96*c^6*e) + ((d + e*x^2)^3*(a + b*ArcSin[c*x]))/(6*e)} -{x^0*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[1 - c^2*x^2])/(15*c^5) - (2*b*e*(5*c^2*d + 3*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) + (b*e^2*(1 - c^2*x^2)^(5/2))/(25*c^5) + d^2*x*(a + b*ArcSin[c*x]) + (2/3)*d*e*x^3*(a + b*ArcSin[c*x]) + (1/5)*e^2*x^5*(a + b*ArcSin[c*x])} -{(d + e*x^2)^2*(a + b*ArcSin[c*x])/x^1, x, 14, (b*d*e*x*Sqrt[1 - c^2*x^2])/(2*c) + (3*b*e^2*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*e^2*x^3*Sqrt[1 - c^2*x^2])/(16*c) - (b*d*e*ArcSin[c*x])/(2*c^2) - (3*b*e^2*ArcSin[c*x])/(32*c^4) - (1/2)*I*b*d^2*ArcSin[c*x]^2 + d*e*x^2*(a + b*ArcSin[c*x]) + (1/4)*e^2*x^4*(a + b*ArcSin[c*x]) + b*d^2*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - b*d^2*ArcSin[c*x]*Log[x] + d^2*(a + b*ArcSin[c*x])*Log[x] - (1/2)*I*b*d^2*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)^2*(a + b*ArcSin[c*x])/x^2, x, 6, (b*e*(6*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*e^2*(1 - c^2*x^2)^(3/2))/(9*c^3) - (d^2*(a + b*ArcSin[c*x]))/x + 2*d*e*x*(a + b*ArcSin[c*x]) + (1/3)*e^2*x^3*(a + b*ArcSin[c*x]) - b*c*d^2*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(d + e*x^2)^2*(a + b*ArcSin[c*x])/x^3, x, 13, -((b*c*d^2*Sqrt[1 - c^2*x^2])/(2*x)) + (b*e^2*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e^2*ArcSin[c*x])/(4*c^2) - I*b*d*e*ArcSin[c*x]^2 - (d^2*(a + b*ArcSin[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcSin[c*x]) + 2*b*d*e*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - 2*b*d*e*ArcSin[c*x]*Log[x] + 2*d*e*(a + b*ArcSin[c*x])*Log[x] - I*b*d*e*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)^2*(a + b*ArcSin[c*x])/x^4, x, 6, (b*e^2*Sqrt[1 - c^2*x^2])/c - (b*c*d^2*Sqrt[1 - c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSin[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcSin[c*x]))/x + e^2*x*(a + b*ArcSin[c*x]) - (1/6)*b*c*d*(c^2*d + 12*e)*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{x^4*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*Sqrt[1 - c^2*x^2])/(1155*c^11) - (b*(462*c^6*d^3 + 1485*c^4*d^2*e + 1540*c^2*d*e^2 + 525*e^3)*(1 - c^2*x^2)^(3/2))/(3465*c^11) + (b*(77*c^6*d^3 + 495*c^4*d^2*e + 770*c^2*d*e^2 + 350*e^3)*(1 - c^2*x^2)^(5/2))/(1925*c^11) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^2*x^2)^(7/2))/(1617*c^11) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^(9/2))/(297*c^11) - (b*e^3*(1 - c^2*x^2)^(11/2))/(121*c^11) + (1/5)*d^3*x^5*(a + b*ArcSin[c*x]) + (3/7)*d^2*e*x^7*(a + b*ArcSin[c*x]) + (1/3)*d*e^2*x^9*(a + b*ArcSin[c*x]) + (1/11)*e^3*x^11*(a + b*ArcSin[c*x])} -{x^3*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 8, -((b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*Sqrt[1 - c^2*x^2])/(76800*c^9*e)) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(38400*c^7*e) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/(9600*c^5*e) + (b*(11*c^2*d + 18*e)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(1600*c^3*e) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^4)/(100*c*e) + (b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*ArcSin[c*x])/(5120*c^10*e^2) - (d*(d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcSin[c*x]))/(10*e^2)} -{x^2*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*Sqrt[1 - c^2*x^2])/(315*c^9) - (b*(105*c^6*d^3 + 378*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^3*(1 - c^2*x^2)^(9/2))/(81*c^9) + (1/3)*d^3*x^3*(a + b*ArcSin[c*x]) + (3/5)*d^2*e*x^5*(a + b*ArcSin[c*x]) + (3/7)*d*e^2*x^7*(a + b*ArcSin[c*x]) + (1/9)*e^3*x^9*(a + b*ArcSin[c*x])} -{x^1*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 6, (5*b*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e + 21*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(104*c^4*d^2 + 104*c^2*d*e + 35*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(1536*c^5) + (7*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^2)/(384*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(64*c) - (b*(128*c^8*d^4 + 256*c^6*d^3*e + 288*c^4*d^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*ArcSin[c*x])/(1024*c^8*e) + ((d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e)} -{x^0*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 5, (b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*Sqrt[1 - c^2*x^2])/(35*c^7) - (b*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^(3/2))/(105*c^7) + (3*b*e^2*(7*c^2*d + 5*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^3*(1 - c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSin[c*x]) + d^2*e*x^3*(a + b*ArcSin[c*x]) + (3/5)*d*e^2*x^5*(a + b*ArcSin[c*x]) + (1/7)*e^3*x^7*(a + b*ArcSin[c*x])} -{(d + e*x^2)^3*(a + b*ArcSin[c*x])/x^1, x, 19, (3*b*d^2*e*x*Sqrt[1 - c^2*x^2])/(4*c) + (9*b*d*e^2*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (5*b*e^3*x*Sqrt[1 - c^2*x^2])/(96*c^5) + (3*b*d*e^2*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (5*b*e^3*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e^3*x^5*Sqrt[1 - c^2*x^2])/(36*c) - (3*b*d^2*e*ArcSin[c*x])/(4*c^2) - (9*b*d*e^2*ArcSin[c*x])/(32*c^4) - (5*b*e^3*ArcSin[c*x])/(96*c^6) - (1/2)*I*b*d^3*ArcSin[c*x]^2 + (3/2)*d^2*e*x^2*(a + b*ArcSin[c*x]) + (3/4)*d*e^2*x^4*(a + b*ArcSin[c*x]) + (1/6)*e^3*x^6*(a + b*ArcSin[c*x]) + b*d^3*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - b*d^3*ArcSin[c*x]*Log[x] + d^3*(a + b*ArcSin[c*x])*Log[x] - (1/2)*I*b*d^3*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)^3*(a + b*ArcSin[c*x])/x^2, x, 6, (b*e*(15*c^4*d^2 + 5*c^2*d*e + e^2)*Sqrt[1 - c^2*x^2])/(5*c^5) - (b*e^2*(5*c^2*d + 2*e)*(1 - c^2*x^2)^(3/2))/(15*c^5) + (b*e^3*(1 - c^2*x^2)^(5/2))/(25*c^5) - (d^3*(a + b*ArcSin[c*x]))/x + 3*d^2*e*x*(a + b*ArcSin[c*x]) + d*e^2*x^3*(a + b*ArcSin[c*x]) + (1/5)*e^3*x^5*(a + b*ArcSin[c*x]) - b*c*d^3*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(d + e*x^2)^3*(a + b*ArcSin[c*x])/x^3, x, 15, -((b*c*d^3*Sqrt[1 - c^2*x^2])/(2*x)) + (3*b*e^2*(8*c^2*d + e)*x*Sqrt[1 - c^2*x^2])/(32*c^3) + (b*e^3*x^3*Sqrt[1 - c^2*x^2])/(16*c) - (3*b*e^2*(8*c^2*d + e)*ArcSin[c*x])/(32*c^4) - (3/2)*I*b*d^2*e*ArcSin[c*x]^2 - (d^3*(a + b*ArcSin[c*x]))/(2*x^2) + (3/2)*d*e^2*x^2*(a + b*ArcSin[c*x]) + (1/4)*e^3*x^4*(a + b*ArcSin[c*x]) + 3*b*d^2*e*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] - 3*b*d^2*e*ArcSin[c*x]*Log[x] + 3*d^2*e*(a + b*ArcSin[c*x])*Log[x] - (3/2)*I*b*d^2*e*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)^3*(a + b*ArcSin[c*x])/x^4, x, 8, (b*e^2*(9*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) - (b*c*d^3*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*e^3*(1 - c^2*x^2)^(3/2))/(9*c^3) - (d^3*(a + b*ArcSin[c*x]))/(3*x^3) - (3*d^2*e*(a + b*ArcSin[c*x]))/x + 3*d*e^2*x*(a + b*ArcSin[c*x]) + (1/3)*e^3*x^3*(a + b*ArcSin[c*x]) - (1/6)*b*c*d^2*(c^2*d + 18*e)*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{(d + e*x^2)^4*(a + b*ArcSin[c*x]), x, 5, (b*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 - c^2*x^2])/(315*c^9) - (4*b*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (2*b*e^2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (4*b*e^3*(9*c^2*d + 7*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^4*(1 - c^2*x^2)^(9/2))/(81*c^9) + d^4*x*(a + b*ArcSin[c*x]) + (4/3)*d^3*e*x^3*(a + b*ArcSin[c*x]) + (6/5)*d^2*e^2*x^5*(a + b*ArcSin[c*x]) + (4/7)*d*e^3*x^7*(a + b*ArcSin[c*x]) + (1/9)*e^4*x^9*(a + b*ArcSin[c*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2), x, 27, -((a*d*x)/e^2) - (b*d*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*Sqrt[1 - c^2*x^2])/(3*c^3*e) - (b*(1 - c^2*x^2)^(3/2))/(9*c^3*e) - (b*d*x*ArcSin[c*x])/e^2 + (x^3*(a + b*ArcSin[c*x]))/(3*e) + ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) + ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2)) + (I*b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e^(5/2)) - (I*b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(5/2)) + (I*b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e^(5/2)) - (I*b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(5/2))} -{(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2), x, 23, (b*x*Sqrt[1 - c^2*x^2])/(4*c*e) - (b*ArcSin[c*x])/(4*c^2*e) + (x^2*(a + b*ArcSin[c*x]))/(2*e) + (I*d*(a + b*ArcSin[c*x])^2)/(2*b*e^2) - (d*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) - (d*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) - (d*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) - (d*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) + (I*b*d*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) + (I*b*d*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2)} -{(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2), x, 23, (a*x)/e + (b*Sqrt[1 - c^2*x^2])/(c*e) + (b*x*ArcSin[c*x])/e + (Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(3/2)) + (I*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (I*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^(3/2))} -{(x^1*(a + b*ArcSin[c*x]))/(d + e*x^2), x, 18, -((I*(a + b*ArcSin[c*x])^2)/(2*b*e)) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e)} -{(a + b*ArcSin[c*x])/(d + e*x^2), x, 18, ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcSin[c*x])/(x^1*(d + e*x^2)), x, 25, -(((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d)) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d) + ((a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*d) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*d) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d)} -{(a + b*ArcSin[c*x])/(x^2*(d + e*x^2)), x, 24, -((a + b*ArcSin[c*x])/(d*x)) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d + (Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2))} -{(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)), x, 27, -((b*c*Sqrt[1 - c^2*x^2])/(2*d*x)) - (a + b*ArcSin[c*x])/(2*d*x^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) - (e*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d^2 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*d^2) - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*d^2) - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) + (I*b*e*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^2)} -{(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)), x, 29, -((b*c*Sqrt[1 - c^2*x^2])/(6*d*x^2)) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e*(a + b*ArcSin[c*x]))/(d^2*x) - (b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (I*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*(-d)^(5/2)) - (I*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (I*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*(-d)^(5/2)) - (I*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2))} - - -{(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x, 23, (d*(a + b*ArcSin[c*x]))/(2*e^2*(d + e*x^2)) - (I*(a + b*ArcSin[c*x])^2)/(2*b*e^2) - (b*c*Sqrt[d]*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*e^2*Sqrt[c^2*d + e]) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e^2) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^2) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e^2) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^2)} -{(x^1*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x, 3, -(a + b*ArcSin[c*x])/(2*e*(d + e*x^2)) + (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*Sqrt[d]*e*Sqrt[c^2*d + e])} -{(a + b*ArcSin[c*x])/(x^1*(d + e*x^2)^2), x, 28, (a + b*ArcSin[c*x])/(2*d*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) + ((a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d^2 + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*d^2) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^2) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*d^2) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^2) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^2)} -{(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^2), x, 30, -((b*c*Sqrt[1 - c^2*x^2])/(2*d^2*x)) - (a + b*ArcSin[c*x])/(2*d^2*x^2) - (e*(a + b*ArcSin[c*x]))/(2*d^2*(d + e*x^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]) + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 - (2*e*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (I*b*e*PolyLog[2, E^(2*I*ArcSin[c*x])])/d^3} - -{(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x, 49, (a*x)/e^2 + (b*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*x*ArcSin[c*x])/e^2 - (d*(a + b*ArcSin[c*x]))/(4*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b*ArcSin[c*x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e]) + (b*c*d*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e]) + (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*I*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(4*e^(5/2)) - (3*I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*I*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(4*e^(5/2)) - (3*I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2))} -{(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x, 46, (a + b*ArcSin[c*x])/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(3/2)*Sqrt[c^2*d + e]) - (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2))} -{(x^0*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x, 26, -((a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x))) + (a + b*ArcSin[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*Sqrt[c^2*d + e]) + (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d*Sqrt[e]*Sqrt[c^2*d + e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcSin[c*x])/(x^2*(d + e*x^2)^2), x, 50, -((a + b*ArcSin[c*x])/(d^2*x)) + (Sqrt[e]*(a + b*ArcSin[c*x]))/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcSin[c*x]))/(4*d^2*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d^2*Sqrt[c^2*d + e]) - (b*c*Sqrt[e]*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*d^2*Sqrt[c^2*d + e]) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 - (3*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) + (3*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) + (3*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2))} - - -{(x^5*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 27, (b*c*d*x*Sqrt[1 - c^2*x^2])/(8*e^2*(c^2*d + e)*(d + e*x^2)) - (d^2*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcSin[c*x]))/(e^3*(d + e*x^2)) - (I*(a + b*ArcSin[c*x])^2)/(2*b*e^3) - (b*c*Sqrt[d]*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(e^3*Sqrt[c^2*d + e]) + (b*c*Sqrt[d]*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^3) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*e^3) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*e^3) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*e^3) - (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*e^3)} -{(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 7, -(b*c*x*Sqrt[1 - c^2*x^2])/(8*e*(c^2*d + e)*(d + e*x^2)) - (b*ArcSin[c*x])/(4*d*e^2) + (x^4*(a + b*ArcSin[c*x]))/(4*d*(d + e*x^2)^2) + (b*c*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*Sqrt[d]*e^2*(c^2*d + e)^(3/2))} -{(x^1*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 4, (b*c*x*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSin[c*x])/(4*e*(d + e*x^2)^2) + (b*c*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(3/2)*e*(c^2*d + e)^(3/2))} -{(a + b*ArcSin[c*x])/(x^1*(d + e*x^2)^3), x, 32, -((b*c*e*x*Sqrt[1 - c^2*x^2])/(8*d^2*(c^2*d + e)*(d + e*x^2))) + (a + b*ArcSin[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcSin[c*x])/(2*d^2*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]) - (b*c*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(5/2)*(c^2*d + e)^(3/2)) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^3) + ((a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d^3 + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*d^3) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^3) + (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*d^3) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^3) - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^3)} -{(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^3), x, 34, -((b*c*Sqrt[1 - c^2*x^2])/(2*d^3*x)) + (b*c*e^2*x*Sqrt[1 - c^2*x^2])/(8*d^3*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSin[c*x])/(2*d^3*x^2) - (e*(a + b*ArcSin[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcSin[c*x]))/(d^3*(d + e*x^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]) + (b*c*e*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)) + (3*e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) - (3*e*(a + b*ArcSin[c*x])*Log[1 - E^(2*I*ArcSin[c*x])])/d^4 - (3*I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(2*d^4) - (3*I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^4) - (3*I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(2*d^4) - (3*I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) + (3*I*b*e*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*d^4)} - -{(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 80, (b*c*Sqrt[-d]*Sqrt[1 - c^2*x^2])/(16*e^2*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[-d]*Sqrt[1 - c^2*x^2])/(16*e^2*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (Sqrt[-d]*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (5*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[-d]*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (5*(a + b*ArcSin[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*d*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*(c^2*d + e)^(3/2)) - (5*b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*Sqrt[c^2*d + e]) + (b*c^3*d*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*(c^2*d + e)^(3/2)) - (5*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(5/2)*Sqrt[c^2*d + e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) - (3*I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) - (3*I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2))} -{(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 62, (b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2) - (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)^2) + (a + b*ArcSin[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - (b*c^3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e)^(3/2)) + (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - (I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) + (I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2))} -{(x^0*(a + b*ArcSin[c*x]))/(d + e*x^2)^3, x, 34, (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^2) - (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (b*c^3*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) - (3*I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*I*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) - (3*I*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e])} -(* {(a + b*ArcSin[c*x])/(x^2*(d + e*x^2)^3), x, 84, (b*c*e*Sqrt[1 - c^2*x^2])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*e*Sqrt[1 - c^2*x^2])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcSin[c*x])/(d^3*x) - (Sqrt[e]*(a + b*ArcSin[c*x]))/(16*(-d)^(5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (7*Sqrt[e]*(a + b*ArcSin[c*x]))/(16*d^3*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[e]*(a + b*ArcSin[c*x]))/(16*(-d)^(5/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (7*Sqrt[e]*(a + b*ArcSin[c*x]))/(16*d^3*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*(c^2*d + e)^(3/2)) - (7*b*c*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^3*Sqrt[c^2*d + e]) - (b*c^3*Sqrt[e]*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*(c^2*d + e)^(3/2)) - (7*b*c*Sqrt[e]*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^3*Sqrt[c^2*d + e]) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^3 + (15*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) - (15*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) - (15*Sqrt[e]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcSin[c x])*) - - -{(d + e*x^2)^(1/2)*(a + b*ArcSin[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x]), x]} -{(a + b*ArcSin[c*x])/(d + e*x^2)^(1/2), x, 0, Unintegrable[(a + b*ArcSin[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcSin[c*x])/(d + e*x^2)^(3/2), x, 6, (x*(a + b*ArcSin[c*x]))/(d*Sqrt[d + e*x^2]) + (b*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(d*Sqrt[e])} -{(a + b*ArcSin[c*x])/(d + e*x^2)^(5/2), x, 7, (b*c*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2]) + (x*(a + b*ArcSin[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (2*b*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(3*d^2*Sqrt[e])} -{(a + b*ArcSin[c*x])/(d + e*x^2)^(7/2), x, 8, (b*c*Sqrt[1 - c^2*x^2])/(15*d*(c^2*d + e)*(d + e*x^2)^(3/2)) + (2*b*c*(3*c^2*d + 2*e)*Sqrt[1 - c^2*x^2])/(15*d^2*(c^2*d + e)^2*Sqrt[d + e*x^2]) + (x*(a + b*ArcSin[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcSin[c*x]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcSin[c*x]))/(15*d^3*Sqrt[d + e*x^2]) + (8*b*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(15*d^3*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x, 6, If[$VersionNumber>=8, (b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2) + (b*e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^4*(5 + m)^2*(7 + m)^2) + (b*e^3*(f*x)^(6 + m)*Sqrt[1 - c^2*x^2])/(c*f^6*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSin[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(3 + m)*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^5*f^2*(2 + m)*(3 + m)*(5 + m)*(7 + m)), (b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2) + (b*e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^4*(5 + m)^2*(7 + m)^2) + (b*e^3*(f*x)^(6 + m)*Sqrt[1 - c^2*x^2])/(c*f^6*(7 + m)^2) + (d^3*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSin[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(3 + m)*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^5*f^2*(2 + m)*(3 + m)*(5 + m)*(7 + m))]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcSin[c*x]), x, 5, (b*e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c^3*f^2*(3 + m)^2*(5 + m)^2) + (b*e^2*(f*x)^(4 + m)*Sqrt[1 - c^2*x^2])/(c*f^4*(5 + m)^2) + (d^2*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSin[c*x]))/(f^5*(5 + m)) - (b*((c^4*d^2*(3 + m)*(5 + m))/(1 + m) + (e*(2 + m)*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2)))/((3 + m)*(5 + m)))*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^3*f^2*(2 + m)*(3 + m)*(5 + m))} -{(f*x)^m*(d + e*x^2)^1*(a + b*ArcSin[c*x]), x, 4, (b*e*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2])/(c*f^2*(3 + m)^2) + (d*(f*x)^(1 + m)*(a + b*ArcSin[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcSin[c*x]))/(f^3*(3 + m)) - (b*(e*(1 + m)*(2 + m) + c^2*d*(3 + m)^2)*(f*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c*f^2*(1 + m)*(2 + m)*(3 + m)^2)} -{(f*x)^m*(a + b*ArcSin[c*x])/(d + e*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2), x]} -{(f*x)^m*(a + b*ArcSin[c*x])/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^2*) - - -{(d + e*x^2)^3*(a + b*ArcSin[c*x])^2, x, 26, -2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) - (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) - (2/9)*b^2*d^2*e*x^3 - (8*b^2*d*e^2*x^3)/(75*c^2) - (16*b^2*e^3*x^3)/(735*c^4) - (6/125)*b^2*d*e^2*x^5 - (12*b^2*e^3*x^5)/(1225*c^2) - (2/343)*b^2*e^3*x^7 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d^2*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3) + (16*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^5) + (32*b*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^7) + (2*b*d^2*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (8*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c^3) + (16*b*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^5) + (6*b*d*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(245*c^3) + (2*b*e^3*x^6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(49*c) + d^3*x*(a + b*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3/5)*d*e^2*x^5*(a + b*ArcSin[c*x])^2 + (1/7)*e^3*x^7*(a + b*ArcSin[c*x])^2} -{(d + e*x^2)^2*(a + b*ArcSin[c*x])^2, x, 17, -2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) - (16*b^2*e^2*x)/(75*c^4) - (4/27)*b^2*d*e*x^3 - (8*b^2*e^2*x^3)/(225*c^2) - (2/125)*b^2*e^2*x^5 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (8*b*d*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (16*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*d*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*e^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + d^2*x*(a + b*ArcSin[c*x])^2 + (2/3)*d*e*x^3*(a + b*ArcSin[c*x])^2 + (1/5)*e^2*x^5*(a + b*ArcSin[c*x])^2} -{(d + e*x^2)^1*(a + b*ArcSin[c*x])^2, x, 10, -2*b^2*d*x - (4*b^2*e*x)/(9*c^2) - (2/27)*b^2*e*x^3 + (2*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (2*b*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + d*x*(a + b*ArcSin[c*x])^2 + (1/3)*e*x^3*(a + b*ArcSin[c*x])^2} -{(d + e*x^2)^0*(a + b*ArcSin[c*x])^2, x, 3, -2*b^2*x + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + x*(a + b*ArcSin[c*x])^2} -{(a + b*ArcSin[c*x])^2/(d + e*x^2)^1, x, 22, ((a + b*ArcSin[c*x])^2*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x])^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSin[c*x])^2*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x])^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^(p/2) (a+b ArcSin[c x])^2*) - - -{(d + e*x^2)^(1/2)*(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2, x]} -{(a + b*ArcSin[c*x])^2/(d + e*x^2)^(1/2), x, 0, Unintegrable[(a + b*ArcSin[c*x])^2/Sqrt[d + e*x^2], x]} -{(a + b*ArcSin[c*x])^2/(d + e*x^2)^(3/2), x, 0, Unintegrable[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(3/2), x]} -{(a + b*ArcSin[c*x])^2/(d + e*x^2)^(5/2), x, 0, Unintegrable[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(5/2), x]} - - -(* ::Section:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^3*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcSin[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcSin[c x])*) - - -{(d + e*x^2)^2/(a + b*ArcSin[c*x]), x, 27, (d^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (d*e*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(2*b*c^3) + (e^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^5) - (d*e*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (e^2*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (d^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (d*e*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(2*b*c^3) + (e^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b*c^5) - (d*e*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b*c^3) - (3*e^2*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b*c^5) + (e^2*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b*c^5)} -{(d + e*x^2)^1/(a + b*ArcSin[c*x]), x, 15, (d*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (e*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^3) - (e*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^3) + (d*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (e*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b*c^3) - (e*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b*c^3)} -{(d + e*x^2)^0/(a + b*ArcSin[c*x]), x, 4, (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c)} -{1/((d + e*x^2)^1*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSin[c*x])), x]} -{1/((d + e*x^2)^2*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^(p/2) / (a+b ArcSin[c x])*) - - -{(d + e*x^2)^(1/2)/(a + b*ArcSin[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x]), x]} -{1/((d + e*x^2)^(1/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcSin[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcSin[c x])^2*) - - -{(d + e*x^2)^2/(a + b*ArcSin[c*x])^2, x, 26, -((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) - (e^2*x^4*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (d^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) + (d*e*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(2*b^2*c^3) + (e^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(8*b^2*c^5) - (3*d*e*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(2*b^2*c^3) - (9*e^2*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(16*b^2*c^5) + (5*e^2*CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(16*b^2*c^5) - (d^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (d*e*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(2*b^2*c^3) - (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^5) - (5*e^2*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c*x]))/b])/(16*b^2*c^5)} -{(d + e*x^2)^1/(a + b*ArcSin[c*x])^2, x, 15, -((d*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (e*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (d*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) + (e*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b^2*c^3) - (3*e*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b^2*c^3) - (d*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (e*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^3) + (3*e*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x^2)^0/(a + b*ArcSin[c*x])^2, x, 5, -(Sqrt[1 - c^2*x^2]/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c)} -{1/((d + e*x^2)^1*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSin[c*x])^2), x]} -{1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^(p/2) / (a+b ArcSin[c x])^2*) - - -{(d + e*x^2)^(1/2)/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x])^2, x]} -{1/((d + e*x^2)^(1/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x^2)^2*(a + b*ArcSin[c*x])^(1/2), x, 42, d^2*x*Sqrt[a + b*ArcSin[c*x]] + (2/3)*d*e*x^3*Sqrt[a + b*ArcSin[c*x]] + (1/5)*e^2*x^5*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*d^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*d*e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c^3) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^5) + (Sqrt[b]*d*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(6*c^3) + (Sqrt[b]*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(16*c^5) - (Sqrt[b]*e^2*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(80*c^5) + (Sqrt[b]*d^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c + (Sqrt[b]*d*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c^3) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^5) - (Sqrt[b]*d*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(6*c^3) - (Sqrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(16*c^5) + (Sqrt[b]*e^2*Sqrt[Pi/10]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(80*c^5)} -{(d + e*x^2)^1*(a + b*ArcSin[c*x])^(1/2), x, 23, d*x*Sqrt[a + b*ArcSin[c*x]] + (1/3)*e*x^3*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c - (Sqrt[b]*e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*c^3) + (Sqrt[b]*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(12*c^3) + (Sqrt[b]*d*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c + (Sqrt[b]*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[b]*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)} -{(d + e*x^2)^0*(a + b*ArcSin[c*x])^(1/2), x, 7, x*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/c + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c} -{(a + b*ArcSin[c*x])^(1/2)/(d + e*x^2)^1, x, 0, Unintegrable[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2), x]} -{(a + b*ArcSin[c*x])^(1/2)/(d + e*x^2)^2, x, 0, Unintegrable[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]} - - -{(d + e*x^2)^1*(a + b*ArcSin[c*x])^(3/2), x, 32, (3*b*d*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + (b*e*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(3*c^3) + (b*e*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(6*c) + d*x*(a + b*ArcSin[c*x])^(3/2) + (1/3)*e*x^3*(a + b*ArcSin[c*x])^(3/2) - (3*b^(3/2)*d*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(3/2)*d*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c) - (3*b^(3/2)*e*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) + (b^(3/2)*e*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)} -{(d + e*x^2)^0*(a + b*ArcSin[c*x])^(3/2), x, 8, (3*b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]])/(2*c) + x*(a + b*ArcSin[c*x])^(3/2) - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*c) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c)} -{(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^1, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2), x]} -{(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2, x, 0, Unintegrable[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x^2)^2/(a + b*ArcSin[c*x])^(1/2), x, 39, (d*e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c^3) + (e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^5) + (d^2*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c) - (d*e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c^3) - (e^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^5) + (e^2*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^5) + (d*e*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c^3) + (e^2*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(4*Sqrt[b]*c^5) + (d^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c) - (d*e*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(Sqrt[b]*c^3) - (e^2*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(8*Sqrt[b]*c^5) + (e^2*Sqrt[Pi/10]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(5*a)/b])/(8*Sqrt[b]*c^5)} -{(d + e*x^2)^1/(a + b*ArcSin[c*x])^(1/2), x, 21, (e*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) + (d*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c) - (e*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) + (e*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*c^3) + (d*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c) - (e*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*c^3)} -{(d + e*x^2)^0/(a + b*ArcSin[c*x])^(1/2), x, 6, (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(Sqrt[b]*c) + (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c)} -{1/((d + e*x^2)^1*(a + b*ArcSin[c*x])^(1/2)), x, 0, Unintegrable[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]), x]} -{1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(1/2)), x, 0, Unintegrable[1/((d + e*x^2)^2*Sqrt[a + b*ArcSin[c*x]]), x]} - - -{(d + e*x^2)^1/(a + b*ArcSin[c*x])^(3/2), x, 21, -((2*d*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (2*e*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) - (e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (2*d*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (e*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) + (e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) + (2*d*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c) - (e*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3)} -{(d + e*x^2)^0/(a + b*ArcSin[c*x])^(3/2), x, 7, -((2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]])) - (2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(b^(3/2)*c) + (2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c)} -{1/((d + e*x^2)^1*(a + b*ArcSin[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSin[c*x])^(3/2)), x]} -{1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)), x]} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.5 Inverse sine functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.5 Inverse sine functions.m deleted file mode 100644 index 90f2bf6..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.1 Inverse sine/5.1.5 Inverse sine functions.m +++ /dev/null @@ -1,927 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcSin[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSin[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcSin[c*x]), x, 5, (7*b*d*(d + e*x)^2*Sqrt[1 - c^2*x^2])/(48*c) + (b*(d + e*x)^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x)*Sqrt[1 - c^2*x^2])/(96*c^3) - (b*(8*c^4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcSin[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcSin[c*x]))/(4*e)} -{(d + e*x)^2*(a + b*ArcSin[c*x]), x, 4, (b*(d + e*x)^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*(4*(4*c^2*d^2 + e^2) + 5*c^2*d*e*x)*Sqrt[1 - c^2*x^2])/(18*c^3) - (b*d*(2*d^2 + (3*e^2)/c^2)*ArcSin[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcSin[c*x]))/(3*e)} -{(d + e*x)^1*(a + b*ArcSin[c*x]), x, 4, (3*b*d*Sqrt[1 - c^2*x^2])/(4*c) + (b*(d + e*x)*Sqrt[1 - c^2*x^2])/(4*c) - (b*(2*d^2 + e^2/c^2)*ArcSin[c*x])/(4*e) + ((d + e*x)^2*(a + b*ArcSin[c*x]))/(2*e)} -{(d + e*x)^0*(a + b*ArcSin[c*x]), x, 3, a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]} -{(a + b*ArcSin[c*x])/(d + e*x)^1, x, 8, -((I*(a + b*ArcSin[c*x])^2)/(2*b*e)) + ((a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + ((a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e} -{(a + b*ArcSin[c*x])/(d + e*x)^2, x, 3, -((a + b*ArcSin[c*x])/(e*(d + e*x))) + (b*c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e*Sqrt[c^2*d^2 - e^2])} -{(a + b*ArcSin[c*x])/(d + e*x)^3, x, 4, (b*c*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSin[c*x])/(2*e*(d + e*x)^2) + (b*c^3*d*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e*(c^2*d^2 - e^2)^(3/2))} -{(a + b*ArcSin[c*x])/(d + e*x)^4, x, 5, (b*c*Sqrt[1 - c^2*x^2])/(6*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c^3*d*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - (a + b*ArcSin[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(6*e*(c^2*d^2 - e^2)^(5/2))} - - -{(d + e*x)^3*(a + b*ArcSin[c*x])^2, x, 18, -2*b^2*d^3*x - (4*b^2*d*e^2*x)/(3*c^2) - (3/4)*b^2*d^2*e*x^2 - (3*b^2*e^3*x^2)/(32*c^2) - (2/9)*b^2*d*e^2*x^3 - (1/32)*b^2*e^3*x^4 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3) + (3*b*d^2*e*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (2*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (b*e^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (d^4*(a + b*ArcSin[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcSin[c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcSin[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSin[c*x])^2)/(4*e)} -{(d + e*x)^2*(a + b*ArcSin[c*x])^2, x, 13, -2*b^2*d^2*x - (4*b^2*e^2*x)/(9*c^2) - (1/2)*b^2*d*e*x^2 - (2/27)*b^2*e^2*x^3 + (2*b*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*e^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (b*d*e*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (2*b*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) - (d^3*(a + b*ArcSin[c*x])^2)/(3*e) - (d*e*(a + b*ArcSin[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e)} -{(d + e*x)^1*(a + b*ArcSin[c*x])^2, x, 9, -2*b^2*d*x - (1/4)*b^2*e*x^2 + (2*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (b*e*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) - (d^2*(a + b*ArcSin[c*x])^2)/(2*e) - (e*(a + b*ArcSin[c*x])^2)/(4*c^2) + ((d + e*x)^2*(a + b*ArcSin[c*x])^2)/(2*e)} -{(d + e*x)^0*(a + b*ArcSin[c*x])^2, x, 3, -2*b^2*x + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + x*(a + b*ArcSin[c*x])^2} -{(a + b*ArcSin[c*x])^2/(d + e*x)^1, x, 10, -((I*(a + b*ArcSin[c*x])^3)/(3*b*e)) + ((a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + ((a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (2*I*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e - (2*I*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e + (2*b^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (2*b^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e} -{(a + b*ArcSin[c*x])^2/(d + e*x)^2, x, 10, -((a + b*ArcSin[c*x])^2/(e*(d + e*x))) - (2*I*b*c*(a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) + (2*I*b*c*(a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) + (2*b^2*c*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2])} -{(a + b*ArcSin[c*x])^2/(d + e*x)^3, x, 13, (b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSin[c*x])^2/(2*e*(d + e*x)^2) - (I*b*c^3*d*(a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) + (I*b*c^3*d*(a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) - (b^2*c^2*Log[d + e*x])/(e*(c^2*d^2 - e^2)) - (b^2*c^3*d*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) + (b^2*c^3*d*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x)^3/(a + b*ArcSin[c*x]), x, 27, (d^3*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(b*c) + (3*d*e^2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(4*b*c^3) - (3*d*e^2*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b*c^3) - (3*d^2*e*CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(2*b*c^2) - (e^3*CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(4*b*c^4) + (e^3*CosIntegral[(4*a)/b + 4*ArcSin[c*x]]*Sin[(4*a)/b])/(8*b*c^4) + (d^3*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b*c) + (3*d*e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b*c^3) + (3*d^2*e*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(2*b*c^2) + (e^3*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(4*b*c^4) - (3*d*e^2*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b*c^3) - (e^3*Cos[(4*a)/b]*SinIntegral[(4*a)/b + 4*ArcSin[c*x]])/(8*b*c^4)} -{(d + e*x)^2/(a + b*ArcSin[c*x]), x, 17, (d^2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(b*c) + (e^2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(4*b*c^3) - (e^2*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b*c^3) - (d*e*CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(b*c^2) + (d^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b*c) + (e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(4*b*c^3) + (d*e*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(b*c^2) - (e^2*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(4*b*c^3)} -{(d + e*x)^1/(a + b*ArcSin[c*x]), x, 11, (d*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]])/(b*c) - (e*CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b])/(2*b*c^2) + (d*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b*c) + (e*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/(2*b*c^2)} -{(d + e*x)^0/(a + b*ArcSin[c*x]), x, 4, (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(b*c) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b*c)} -{1/((d + e*x)^1*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcSin[c*x])), x]} -{1/((d + e*x)^2*(a + b*ArcSin[c*x])), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcSin[c*x])), x]} - - -{(d + e*x)^2/(a + b*ArcSin[c*x])^2, x, 19, -((d^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (2*d*e*x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) - (e^2*x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (2*d*e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2) + (d^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) + (e^2*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b^2*c^3) - (3*e^2*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/(4*b^2*c^3) - (d^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) - (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2*c^3) + (2*d*e*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2) + (3*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x)^1/(a + b*ArcSin[c*x])^2, x, 11, -((d*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x]))) - (e*x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcSin[c*x])) + (e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2) + (d*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) - (d*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c) + (e*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])/(b^2*c^2)} -{(d + e*x)^0/(a + b*ArcSin[c*x])^2, x, 5, -(Sqrt[1 - c^2*x^2]/(b*c*(a + b*ArcSin[c*x]))) + (CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(b^2*c) - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(b^2*c)} -{1/((d + e*x)^1*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcSin[c*x])^2), x]} -{1/((d + e*x)^2*(a + b*ArcSin[c*x])^2), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcSin[c*x])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSin[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSin[c x])^n with m symbolic*) - - -{(d + e*x)^m*(a + b*ArcSin[c*x])^2, x, 1, ((d + e*x)^(1 + m)*(a + b*ArcSin[c*x])^2)/(e*(1 + m)) - (2*b*c*Unintegrable[((d + e*x)^(1 + m)*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2], x])/(e*(1 + m))} -{(d + e*x)^m*(a + b*ArcSin[c*x])^1, x, 3, -((b*c*(d + e*x)^(2 + m)*Sqrt[1 - (c*(d + e*x))/(c*d - e)]*Sqrt[1 - (c*(d + e*x))/(c*d + e)]*AppellF1[2 + m, 1/2, 1/2, 3 + m, (c*(d + e*x))/(c*d - e), (c*(d + e*x))/(c*d + e)])/(e^2*(1 + m)*(2 + m)*Sqrt[1 - c^2*x^2])) + ((d + e*x)^(1 + m)*(a + b*ArcSin[c*x]))/(e*(1 + m))} -{(d + e*x)^m/(a + b*ArcSin[c*x])^1, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcSin[c*x]), x]} -{(d + e*x)^m/(a + b*ArcSin[c*x])^2, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcSin[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x^2)^(p/2) (a+b ArcSin[c x])^n where c^2 d+e=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 16, (b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[1 - c^2*x^2]) + (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) - (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) + (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*c*g^3*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (1/2)*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (3/4)*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/c^2 - (g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^4) + (g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^4) + (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) + (3*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 13, (2*b*f*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) - (2*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (1/2)*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c^2) + (1/4)*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) + (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 8, (b*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) - (b*c*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) + (1/2)*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 22, (a*Sqrt[d - c^2*d*x^2])/g - (b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[1 - c^2*x^2]) + (b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g + (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*g*Sqrt[1 - c^2*x^2]) - ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*(f + g*x)*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*(f + g*x)) - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (I*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (I*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 35, -((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g*(f + g*x)) - (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) - (b*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) + ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*(f + g*x)^2*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*(f + g*x)^2) + (a*c^2*f*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 - c^2*x^2]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])} - - -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 24, (3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) + (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) - (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[1 - c^2*x^2]) + (b*d*g^3*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(12*Sqrt[1 - c^2*x^2]) + (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (3/8)*d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c^2) + (3/8)*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/4)*d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/2)*d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (3*d*f^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^2) - (d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^4) + (d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^4) + (3*d*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) + (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 20, (2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) + (3/8)*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c^2) + (1/8)*d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/4)*d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/6)*d*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^2) + (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) + (d*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 12, (b*d*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (2*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (3/8)*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/4)*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c^2) + (3*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 29, -((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) - (b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*g*Sqrt[1 - c^2*x^2]) + (b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*g*Sqrt[1 - c^2*x^2]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^3 + (c^2*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^2) + (d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g) + (c*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*g^3*Sqrt[1 - c^2*x^2]) - (d*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)) + (a*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (I*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (I*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 71, (2*a*c^2*d*f*Sqrt[d - c^2*d*x^2])/g^3 + (a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/(g^3*(f + g*x)) - (2*b*c^3*d*f*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[1 - c^2*x^2]) + (2*b*c^2*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^3 + (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^3*(f + g*x)) + (a*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^4*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*g^4*Sqrt[1 - c^2*x^2]) - (c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^2) - (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) + (c^3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^3*Sqrt[1 - c^2*x^2]) - (d*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (c*d*f*(1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^2*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)^2) + (c*d*f*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^2*(f + g*x)) - (3*a*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (b*c*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^4*Sqrt[1 - c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2])} *) - - -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 30, (3*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) - (25*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (15*b*d^2*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) + (b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*f^3*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(256*Sqrt[1 - c^2*x^2]) + (9*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (19*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) + (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (15/64)*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/24)*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/16)*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/6)*d^2*f^3*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (3/8)*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (3*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^2) - (d^2*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^4) + (d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*c^4) + (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2]) + (15*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 26, (2*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (25*b*c*d^2*f^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*f^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d^2*f*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) + (b*d^2*f^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c^2) + (5/64)*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/24)*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/48)*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/6)*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/8)*d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^2) + (5*d^2*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 14, (b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2]) - (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) + (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (5/24)*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (1/6)*d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) - (d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 37, (a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 + (2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])/(45*g*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*g^2) - (d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g^3) + (d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*g) - (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) + (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*g^5*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 78, -((4*a*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^5) - (a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/(g^5*(f + g*x)) - (2*b*c^3*d^2*f*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*(3*c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) + (2*b*c^5*d^2*f*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (4*b*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^5 - (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^5*(f + g*x)) - (a*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^6*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*g^6*Sqrt[1 - c^2*x^2]) - (c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2) + (c^2*d^2*(3*c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^4) + (c^4*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*g^2) + (2*c^2*d^2*f*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g^3) + (c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d^2*(3*c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) - (2*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^5*Sqrt[1 - c^2*x^2]) + (d^2*(c*f - g)*(c*f + g)*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (2*c*d^2*f*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)^2) - (2*c*d^2*f*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^4*(f + g*x)) + (5*a*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^6*Sqrt[1 - c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])} *) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 13, (3*b*f^2*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) + (2*b*g^3*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (3*b*f*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) + (b*g^3*x^3*Sqrt[1 - c^2*x^2])/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c^4*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) + (3*f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 9, (2*b*f*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) + (b*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 6, (b*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (g*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) + (f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 10, -((I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 13, (g*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2]) - (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Log[f + g*x])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) - (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])} - - -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 11, -((b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2])) + ((g*(3*c^2*f^2 + g^2) + c^2*f*(c^2*f^2 + 3*g^2)*x)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) + (g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(2*c^4*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 8, ((2*f*g + (c^2*f^2 + g^2)*x)*(a + b*ArcSin[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^2*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(2*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(2*c^3*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 6, ((g + c^2*f*x)*(a + b*ArcSin[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(2*c^2*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(2*c^2*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 20, -((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2])) + (I*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (I*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(d*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + (1/2)*ArcSin[c*x]]])/(d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f + g)*Sqrt[d - c^2*d*x^2])} -(* {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 47, -((a*g^3*(1 - c^2*x^2))/(d*(c*f - g)^2*(c*f + g)^2*(f + g*x)*Sqrt[d - c^2*d*x^2])) - (b*g^3*(1 - c^2*x^2)*ArcSin[c*x])/(d*(c*f - g)^2*(c*f + g)^2*(f + g*x)*Sqrt[d - c^2*d*x^2]) - (6*a*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcTan[(g + c*f*Tan[(1/2)*ArcSin[c*x]])/Sqrt[c^2*f^2 - g^2]])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*c*g^2*Sqrt[1 - c^2*x^2]*Log[f + g*x])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(d*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(d*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (3*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (3*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(2*d*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f + g)^2*Sqrt[d - c^2*d*x^2])} *) - - -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^4, x, 13, -((b*(f + g*x)^2*(c^2*f^2 + g^2 + 2*c^2*f*g*x))/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) - (b*f*g^3*x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*c^5*d^2*Sqrt[d - c^2*d*x^2]) + ((f + g*x)*(g*(c^2*f^2 - 3*g^2) + 2*c^2*f*(c^2*f^2 - 2*g^2)*x)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(f + g*x)^3*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f*g*(2*c^2*f^2 - 5*g^2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]), -((b*(f + g*x)^2*(c^2*f^2 + g^2 + 2*c^2*f*g*x))/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) - (2*b*f*g^3*x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f*g*(2*c^2*f^2 - 5*g^2)*x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*f*g*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*c^5*d^2*Sqrt[d - c^2*d*x^2]) + ((f + g*x)*(g*(c^2*f^2 - 3*g^2) + 2*c^2*f*(c^2*f^2 - 2*g^2)*x)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(f + g*x)^3*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f*g*(2*c^2*f^2 - 5*g^2)*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(a + b*ArcSin[c*x]))/(c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(2*c*f - 3*g)*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (b*g*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*g*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*(2*c*f + 3*g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^5*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^3, x, 10, -((b*(f + g*x)*(c^2*f^2 + g^2 + 2*c^2*f*g*x))/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (2*(c*f - g)*(c*f + g)*(g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(f + g*x)^2*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*g*(c*f + g)^2*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*g*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^2, x, 10, -((b*x*(2*f*g + (c^2*f^2 + g^2)*x))/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (2*f*(g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (x*(f + g*x)^2*(a + b*ArcSin[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (b*(2*c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)*(2*c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2]), -((b*(f + g*x)^2)/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (2*f*(g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (x*(f + g*x)^2*(a + b*ArcSin[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (b*f*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b*g*(c*f + g)*Sqrt[1 - c^2*x^2]*Log[1 - c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*f*(c*f - g)*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)*g*Sqrt[1 - c^2*x^2]*Log[1 + c*x])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])*(f + g*x)^1, x, 6, -((b*(f + g*x))/(6*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])) + (2*f*x*(a + b*ArcSin[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + ((g + c^2*f*x)*(a + b*ArcSin[c*x]))/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b*g*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b*f*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^1, x, 30, If[$VersionNumber>=8, -(((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2])) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]]*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]), -(((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2])) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + (1/2)*ArcSin[c*x]]*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2])]} -(* {1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 57, (a*g^5*(1 - c^2*x^2))/(d^2*(c*f - g)^3*(c*f + g)^3*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (b*g^5*(1 - c^2*x^2)*ArcSin[c*x])/(d^2*(c*f - g)^3*(c*f + g)^3*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (10*a*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcTan[(g + c*f*Tan[(1/2)*ArcSin[c*x]])/Sqrt[c^2*f^2 - g^2]])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*c*g^4*Sqrt[1 - c^2*x^2]*Log[f + g*x])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) + (b*c*(c*f - 3*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f - g)^3*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*(c*f + 3*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) - (5*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (5*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (c*(c*f - 3*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^3*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (c*(c*f + 3*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2])} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 37, (4*b^2*f^2*g*Sqrt[d - c^2*d*x^2])/(3*c^2) + (52*b^2*g^3*Sqrt[d - c^2*d*x^2])/(225*c^4) - (1/4)*b^2*f^3*x*Sqrt[d - c^2*d*x^2] + (3*b^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) - (3/32)*b^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2] + (4*a*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (2*b^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(9*c^2) + (26*b^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(675*c^4) - (2*b^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^4) + (b^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (3*b^2*f*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*g^3*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(15*c^3*Sqrt[1 - c^2*x^2]) + (2*b*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[1 - c^2*x^2]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (2*b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) + (2*b*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*c*Sqrt[1 - c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (2*b*c*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^4) + (1/2)*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) - (g^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*c^2) + (3/4)*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/5)*g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/c^2 + (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 23, (8*b^2*f*g*Sqrt[d - c^2*d*x^2])/(9*c^2) - (1/4)*b^2*f^2*x*Sqrt[d - c^2*d*x^2] + (b^2*g^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) - (1/32)*b^2*g^2*x^3*Sqrt[d - c^2*d*x^2] + (4*b^2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*c^2) + (b^2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) - (b^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c^3*Sqrt[1 - c^2*x^2]) + (4*b*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c*Sqrt[1 - c^2*x^2]) - (4*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (1/2)*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*c^2) + (1/4)*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^2) + (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2]) + (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(24*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 13, (4*b^2*g*Sqrt[d - c^2*d*x^2])/(9*c^2) - (1/4)*b^2*f*x*Sqrt[d - c^2*d*x^2] + (2*b^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*c^2) + (b^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*c*Sqrt[1 - c^2*x^2]) + (2*b*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (2*b*c*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (1/2)*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^2) + (f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 38, (a^2*Sqrt[d - c^2*d*x^2])/g - (2*b^2*Sqrt[d - c^2*d*x^2])/g - (2*a*b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[1 - c^2*x^2]) + (2*a*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g - (2*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g*Sqrt[1 - c^2*x^2]) + (b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g + (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g*Sqrt[1 - c^2*x^2]) - ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)*Sqrt[1 - c^2*x^2]) + (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)) - (a^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*I*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*I*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2])} -(* {Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} *) - - -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 56, (16*b^2*d*f^2*g*Sqrt[d - c^2*d*x^2])/(25*c^2) + (304*b^2*d*g^3*Sqrt[d - c^2*d*x^2])/(3675*c^4) - (15/64)*b^2*d*f^3*x*Sqrt[d - c^2*d*x^2] - (7*b^2*d*f*g^2*x*Sqrt[d - c^2*d*x^2])/(384*c^2) - (43/576)*b^2*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2] + (1/36)*b^2*c^2*d*f*g^2*x^5*Sqrt[d - c^2*d*x^2] + (4*a*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (8*b^2*d*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(75*c^2) + (152*b^2*d*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11025*c^4) - (1/32)*b^2*d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] + (6*b^2*d*f^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (38*b^2*d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(6125*c^4) - (2*b^2*d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^4) + (9*b^2*d*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (7*b^2*d*f*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(384*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*d*g^3*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(35*c^3*Sqrt[1 - c^2*x^2]) + (6*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*Sqrt[1 - c^2*x^2]) + (2*b*d*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(105*c*Sqrt[1 - c^2*x^2]) - (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) - (16*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(175*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*g^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(6*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) + (b*d*f^3*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (2*d*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^4) + (3/8)*d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*c^2) - (d*g^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(35*c^2) + (3/8)*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (3/35)*d*g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/4)*d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/2)*d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/7)*d*g^3*x^4*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (3*d*f^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2]) + (d*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 36, (32*b^2*d*f*g*Sqrt[d - c^2*d*x^2])/(75*c^2) - (15/64)*b^2*d*f^2*x*Sqrt[d - c^2*d*x^2] - (7*b^2*d*g^2*x*Sqrt[d - c^2*d*x^2])/(1152*c^2) - (43*b^2*d*g^2*x^3*Sqrt[d - c^2*d*x^2])/1728 + (1/108)*b^2*c^2*d*g^2*x^5*Sqrt[d - c^2*d*x^2] + (16*b^2*d*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) - (1/32)*b^2*d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] + (4*b^2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (9*b^2*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (7*b^2*d*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c^3*Sqrt[1 - c^2*x^2]) + (4*b*d*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*c*Sqrt[1 - c^2*x^2]) - (8*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) - (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*Sqrt[1 - c^2*x^2]) + (b*d*f^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (3/8)*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*c^2) + (1/8)*d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/4)*d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/6)*d*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2]) + (d*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 19, (16*b^2*d*g*Sqrt[d - c^2*d*x^2])/(75*c^2) - (15/64)*b^2*d*f*x*Sqrt[d - c^2*d*x^2] + (8*b^2*d*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(225*c^2) - (1/32)*b^2*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2] + (2*b^2*d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*c^2) + (9*b^2*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (2*b*d*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*c*Sqrt[1 - c^2*x^2]) - (3*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (4*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(15*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*Sqrt[1 - c^2*x^2]) + (b*d*f*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (3/8)*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/4)*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*c^2) + (d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(8*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 50, -((4*b^2*d*Sqrt[d - c^2*d*x^2])/(9*g)) - (a^2*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3 + (2*b^2*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3 - (b^2*c^2*d*f*x*Sqrt[d - c^2*d*x^2])/(4*g^2) + (2*a*b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) - (2*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*g) - (2*a*b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^3 + (b^2*c*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*g^2*Sqrt[1 - c^2*x^2]) + (2*b^2*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^3*Sqrt[1 - c^2*x^2]) - (b^2*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g^3 - (2*b*c*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^2*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*g*Sqrt[1 - c^2*x^2]) + (c^2*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*g^2) + (d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*g) + (c*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g^3*Sqrt[1 - c^2*x^2]) - (d*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^2*(f + g*x)) + (a^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (2*I*a*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (I*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (2*I*a*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (I*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (2*a*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (2*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (2*a*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (2*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (2*I*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (2*I*b^2*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} *) - - -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 77, (96*b^2*d^2*f^2*g*Sqrt[d - c^2*d*x^2])/(245*c^2) + (160*b^2*d^2*g^3*Sqrt[d - c^2*d*x^2])/(3969*c^4) - (245*b^2*d^2*f^3*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2])/(12288*c^2) - (1079*b^2*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2])/18432 + (209*b^2*c^2*d^2*f*g^2*x^5*Sqrt[d - c^2*d*x^2])/4608 - (3/256)*b^2*c^4*d^2*f*g^2*x^7*Sqrt[d - c^2*d*x^2] + (4*a*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (16*b^2*d^2*f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2) + (80*b^2*d^2*g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(11907*c^4) - (65*b^2*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (36*b^2*d^2*f^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) + (4*b^2*d^2*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1323*c^4) - (1/108)*b^2*d^2*f^3*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2] + (6*b^2*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (50*b^2*d^2*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(27783*c^4) - (2*b^2*d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(729*c^4) + (115*b^2*d^2*f^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(12288*c^3*Sqrt[1 - c^2*x^2]) + (4*b^2*d^2*g^3*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(63*c^3*Sqrt[1 - c^2*x^2]) + (6*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (15*b*d^2*f*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (6*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(189*c*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*Sqrt[1 - c^2*x^2]) + (18*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(21*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*Sqrt[1 - c^2*x^2]) - (6*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) + (38*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(441*Sqrt[1 - c^2*x^2]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(81*Sqrt[1 - c^2*x^2]) + (5*b*d^2*f^3*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (2*d^2*g^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^4) + (5/16)*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) - (d^2*g^3*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(63*c^2) + (15/64)*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/21)*d^2*g^3*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/24)*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/16)*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/63)*d^2*g^3*x^4*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/6)*d^2*f^3*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (3/8)*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/9)*d^2*g^3*x^4*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (3*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(128*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 50, (64*b^2*d^2*f*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f^2*x*Sqrt[d - c^2*d*x^2])/1152 - (359*b^2*d^2*g^2*x*Sqrt[d - c^2*d*x^2])/(36864*c^2) - (1079*b^2*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*g^2*x^5*Sqrt[d - c^2*d*x^2])/13824 - (1/256)*b^2*c^4*d^2*g^2*x^7*Sqrt[d - c^2*d*x^2] + (32*b^2*d^2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (24*b^2*d^2*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (1/108)*b^2*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2] + (4*b^2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) + (359*b^2*d^2*g^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(36864*c^3*Sqrt[1 - c^2*x^2]) + (4*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d^2*f^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(128*c*Sqrt[1 - c^2*x^2]) - (4*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(384*Sqrt[1 - c^2*x^2]) + (12*b*c^3*d^2*f*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(144*Sqrt[1 - c^2*x^2]) - (4*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(32*Sqrt[1 - c^2*x^2]) + (5*b*d^2*f^2*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5/16)*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(128*c^2) + (5/64)*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/24)*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/48)*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/6)*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/8)*d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2]) + (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(384*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 25, (32*b^2*d^2*g*Sqrt[d - c^2*d*x^2])/(245*c^2) - (245*b^2*d^2*f*x*Sqrt[d - c^2*d*x^2])/1152 + (16*b^2*d^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(735*c^2) - (65*b^2*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/1728 + (12*b^2*d^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(1225*c^2) - (1/108)*b^2*d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2] + (2*b^2*d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(343*c^2) + (115*b^2*d^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(1152*c*Sqrt[1 - c^2*x^2]) + (2*b*d^2*g*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*c*Sqrt[1 - c^2*x^2]) - (5*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(16*Sqrt[1 - c^2*x^2]) - (2*b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(7*Sqrt[1 - c^2*x^2]) + (6*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(35*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(49*Sqrt[1 - c^2*x^2]) + (5*b*d^2*f*(1 - c^2*x^2)^(3/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(48*c) + (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(18*c) + (5/16)*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (5/24)*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 + (1/6)*d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(7*c^2) + (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(48*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 74, (52*b^2*d^2*Sqrt[d - c^2*d*x^2])/(225*g) + (4*b^2*d^2*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2])/(9*g^3) + (a^2*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (2*b^2*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (b^2*c^2*d^2*f*x*Sqrt[d - c^2*d*x^2])/(64*g^2) + (b^2*c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(4*g^4) + (b^2*c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2])/(32*g^2) + (4*a*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2]) - (2*a*b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (26*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(675*g) + (2*b^2*d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(27*g^3) - (2*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(125*g) + (2*a*b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^5 + (b^2*c*d^2*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(64*g^2*Sqrt[1 - c^2*x^2]) - (b^2*c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(4*g^4*Sqrt[1 - c^2*x^2]) + (4*b^2*c*d^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(15*g*Sqrt[1 - c^2*x^2]) - (2*b^2*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^5*Sqrt[1 - c^2*x^2]) + (b^2*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g^5 + (2*b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g^3*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^4*Sqrt[1 - c^2*x^2]) + (2*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(45*g*Sqrt[1 - c^2*x^2]) - (2*b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(9*g^3*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(25*g*Sqrt[1 - c^2*x^2]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*g) + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*g^4) - (c^2*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(15*g) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*g^2) + (c^4*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(5*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(3*g^3) - (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(24*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*g^4*Sqrt[1 - c^2*x^2]) + (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g^5*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)) - (a^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*I*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*I*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*a*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (2*I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (2*I*b^2*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 17, (6*b^2*f^2*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (14*b^2*g^3*(1 - c^2*x^2))/(9*c^4*Sqrt[d - c^2*d*x^2]) + (3*b^2*f*g^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*(1 - c^2*x^2)^2)/(27*c^4*Sqrt[d - c^2*d*x^2]) - (3*b^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (6*b*f^2*g*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2]) + (4*b*g^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^3*Sqrt[d - c^2*d*x^2]) + (3*b*f*g^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) + (2*b*g^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c^4*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2]) + (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(2*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 11, (4*b^2*f*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (b^2*g^2*x*(1 - c^2*x^2))/(4*c^2*Sqrt[d - c^2*d*x^2]) - (b^2*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) + (4*b*f*g*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2]) + (b*g^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 8, (2*b^2*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (2*b*g*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2]) - (g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) + (f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2]), (2*a*b*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) + (2*b^2*g*(1 - c^2*x^2))/(c^2*Sqrt[d - c^2*d*x^2]) + (2*b^2*g*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*Sqrt[d - c^2*d*x^2]) - (g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[d - c^2*d*x^2]) + (f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 12, -((I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 20, (I*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) + (g*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2]) - (2*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) - (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (2*b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) + (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) - (2*b*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*c*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) + (2*b*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])} - - -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 23, -((2*a*b*g^3*x*Sqrt[1 - c^2*x^2])/(c^3*d*Sqrt[d - c^2*d*x^2])) - (2*b^2*g^3*(1 - c^2*x^2))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*g^3*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g*(3*c^2*f^2 + g^2)*(a + b*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) + (f*(f^2 + (3*g^2)/c^2)*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (I*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) + (g^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(c^4*d*Sqrt[d - c^2*d*x^2]) - (f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(b*c^3*d*Sqrt[d - c^2*d*x^2]) + (4*I*b*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*b*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g*(3*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*f*(c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 19, (2*f*g*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + ((c^2*f^2 + g^2)*x*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*(c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (8*I*b*f*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*b*(c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (4*I*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (4*I*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 16, (g*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (f*x*(a + b*ArcSin[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (I*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c*d*Sqrt[d - c^2*d*x^2]) + (4*I*b*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*b*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(c*d*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(c*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 28, -((I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2])) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*d*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(d*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (I*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (I*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^(I*ArcSin[c*x])])/(d*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f + g)*Sqrt[d - c^2*d*x^2])} -(* {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 7, 0} *) - - -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^3, x, 37, -((I*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2])) + (I*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + (1/2)*ArcSin[c*x]])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi/4 + (1/2)*ArcSin[c*x]]*Csc[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*c^4*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^2, x, 30, (2*b^2*f*g)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*f^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*g^2*x)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f^2*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (2*b*f*g*x*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (b*g^2*x^2*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (2*f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*f*g*(a + b*ArcSin[c*x])^2)/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f^2*x*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (g^2*x^3*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (2*I*f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c*d^2*Sqrt[d - c^2*d*x^2]) + (I*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (4*I*b*f*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (4*b*f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*f*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*f^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2*(f + g*x)^1, x, 21, (b^2*g)/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*f*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*f*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (b*g*x*(a + b*ArcSin[c*x]))/(3*c*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (2*f*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (g*(a + b*ArcSin[c*x])^2)/(3*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (f*x*(a + b*ArcSin[c*x])^2)/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (2*I*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(3*c*d^2*Sqrt[d - c^2*d*x^2]) + (2*I*b*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (4*b*f*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*g*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*g*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(2*I*ArcSin[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -(* {1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 48, If[$VersionNumber>=8, (I*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 - I*ArcSin[c*x])])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 - I*ArcSin[c*x])])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 + I*ArcSin[c*x])])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 + I*ArcSin[c*x])])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 - I*ArcSin[c*x])])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 - I*ArcSin[c*x])])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 + I*ArcSin[c*x])])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 + I*ArcSin[c*x])])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - ((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]), (I*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - ((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2])]} *) -(* {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 7, 0} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d+e x^2)^(p/2) (a+b ArcSin[c x])^n where c^2 d+e=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^n*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[((a + b*ArcSin[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]} - - -{Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^3/Sqrt[1 - c^2*x^2], x, 15, (I*m*(a + b*ArcSin[c*x])^5)/(20*b^2*c) - (m*(a + b*ArcSin[c*x])^4*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(4*b*c) - (m*(a + b*ArcSin[c*x])^4*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(4*b*c) + ((a + b*ArcSin[c*x])^4*Log[h*(f + g*x)^m])/(4*b*c) + (I*m*(a + b*ArcSin[c*x])^3*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*(a + b*ArcSin[c*x])^3*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c - (3*b*m*(a + b*ArcSin[c*x])^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (3*b*m*(a + b*ArcSin[c*x])^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c - (6*I*b^2*m*(a + b*ArcSin[c*x])*PolyLog[4, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (6*I*b^2*m*(a + b*ArcSin[c*x])*PolyLog[4, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c + (6*b^3*m*PolyLog[5, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (6*b^3*m*PolyLog[5, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^2/Sqrt[1 - c^2*x^2], x, 13, (I*m*(a + b*ArcSin[c*x])^4)/(12*b^2*c) - (m*(a + b*ArcSin[c*x])^3*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(3*b*c) - (m*(a + b*ArcSin[c*x])^3*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(3*b*c) + ((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c) + (I*m*(a + b*ArcSin[c*x])^2*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*(a + b*ArcSin[c*x])^2*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c - (2*b*m*(a + b*ArcSin[c*x])*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (2*b*m*(a + b*ArcSin[c*x])*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c - (2*I*b^2*m*PolyLog[4, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (2*I*b^2*m*PolyLog[4, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^1/Sqrt[1 - c^2*x^2], x, 11, (I*m*(a + b*ArcSin[c*x])^3)/(6*b^2*c) - (m*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(2*b*c) - (m*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(2*b*c) + ((a + b*ArcSin[c*x])^2*Log[h*(f + g*x)^m])/(2*b*c) + (I*m*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c - (b*m*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (b*m*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcSin[c*x])^0/Sqrt[1 - c^2*x^2], x, 9, (I*m*ArcSin[c*x]^2)/(2*c) - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c + (ArcSin[c*x]*Log[h*(f + g*x)^m])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]/(a + b*ArcSin[c*x])^1/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Px (d+e x)^m (a+b ArcSin[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Px (d+e x)^m (a+b ArcSin[c x])*) - - -{(f + g*x)*(a + b*ArcSin[c*x])*(d + e*x)^3, x, 6, (b*e*(4*e^2*g + 25*c^2*d*(e*f + d*g))*x^2*Sqrt[1 - c^2*x^2])/(75*c^3) + (b*e^2*(e*f + 3*d*g)*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*e^3*g*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(75*c^4*d^3*f + 8*e^3*g + 50*c^2*d*e*(e*f + d*g)) + 75*c^2*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*x)*Sqrt[1 - c^2*x^2])/(2400*c^5) - (b*(8*c^2*d^2*(3*e*f + d*g) + 3*e^2*(e*f + 3*d*g))*ArcSin[c*x])/(32*c^4) + d^3*f*x*(a + b*ArcSin[c*x]) + (1/2)*d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + d*e*(e*f + d*g)*x^3*(a + b*ArcSin[c*x]) + (1/4)*e^2*(e*f + 3*d*g)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e^3*g*x^5*(a + b*ArcSin[c*x])} -{(f + g*x)*(a + b*ArcSin[c*x])*(d + e*x)^2, x, 6, (b*e*(e*f + 2*d*g)*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*e^2*g*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(32*(9*c^2*d^2*f + 2*e*(e*f + 2*d*g)) + 9*(3*e^2*g + 8*c^2*d*(2*e*f + d*g))*x)*Sqrt[1 - c^2*x^2])/(288*c^3) - (b*(3*e^2*g + 8*c^2*d*(2*e*f + d*g))*ArcSin[c*x])/(32*c^4) + d^2*f*x*(a + b*ArcSin[c*x]) + (1/2)*d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*e*(e*f + 2*d*g)*x^3*(a + b*ArcSin[c*x]) + (1/4)*e^2*g*x^4*(a + b*ArcSin[c*x])} -{(f + g*x)*(a + b*ArcSin[c*x])*(d + e*x)^1, x, 5, (b*e*g*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*(4*(9*c^2*d*f + 2*e*g) + 9*c^2*(e*f + d*g)*x)*Sqrt[1 - c^2*x^2])/(36*c^3) - (b*(e*f + d*g)*ArcSin[c*x])/(4*c^2) + d*f*x*(a + b*ArcSin[c*x]) + (1/2)*(e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*e*g*x^3*(a + b*ArcSin[c*x])} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^1, x, 14, (b*g*Sqrt[1 - c^2*x^2])/(c*e) - (I*b*(e*f - d*g)*ArcSin[c*x]^2)/(2*e^2) + (g*x*(a + b*ArcSin[c*x]))/e + (b*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 + (b*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2 - (b*(e*f - d*g)*ArcSin[c*x]*Log[d + e*x])/e^2 + ((e*f - d*g)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^2 - (I*b*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 - (I*b*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^2, x, 15, -((I*b*g*ArcSin[c*x]^2)/(2*e^2)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(e^2*(d + e*x)) + (b*c*(e*f - d*g)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (b*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 + (b*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2 - (b*g*ArcSin[c*x]*Log[d + e*x])/e^2 + (g*(a + b*ArcSin[c*x])*Log[d + e*x])/e^2 - (I*b*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^2 - (I*b*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^2} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^3, x, 7, (b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(2*e*(c^2*d^2 - e^2)*(d + e*x)) + (b*g^2*ArcSin[c*x])/(2*e^2*(e*f - d*g)) - ((f + g*x)^2*(a + b*ArcSin[c*x]))/(2*(e*f - d*g)*(d + e*x)^2) - (b*c*(2*e^2*g - c^2*d*(e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^2*(c^2*d^2 - e^2)^(3/2))} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^4, x, 6, (b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(6*e*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*(c^2*d*f - e*g)*Sqrt[1 - c^2*x^2])/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(3*e^2*(d + e*x)^3) - (g*(a + b*ArcSin[c*x]))/(2*e^2*(d + e*x)^2) + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(6*e^2*(c^2*d^2 - e^2)^(5/2))} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^5, x, 7, (b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(12*e*(c^2*d^2 - e^2)*(d + e*x)^3) - (b*c*(4*e^2*g - c^2*d*(5*e*f - d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^2*(d + e*x)^2) + (b*c^3*(4*e^2*(e*f - 4*d*g) + c^2*d^2*(11*e*f + d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^3*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(4*e^2*(d + e*x)^4) - (g*(a + b*ArcSin[c*x]))/(3*e^2*(d + e*x)^3) - (b*c^3*(4*e^4*g - c^2*d*e^2*(9*e*f - 13*d*g) - 2*c^4*d^3*(3*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(24*e^2*(c^2*d^2 - e^2)^(7/2))} -{(f + g*x)*(a + b*ArcSin[c*x])/(d + e*x)^6, x, 8, (b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(20*e*(c^2*d^2 - e^2)*(d + e*x)^4) - (b*c*(5*e^2*g - c^2*d*(7*e*f - 2*d*g))*Sqrt[1 - c^2*x^2])/(60*e*(c^2*d^2 - e^2)^2*(d + e*x)^3) + (b*c^3*(e^2*(9*e*f - 34*d*g) + c^2*d^2*(26*e*f - d*g))*Sqrt[1 - c^2*x^2])/(120*e*(c^2*d^2 - e^2)^3*(d + e*x)^2) - (b*c^3*(4*e^4*g - c^2*d*e^2*(11*e*f - 18*d*g) - c^4*d^3*(10*e*f + d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^4*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(5*e^2*(d + e*x)^5) - (g*(a + b*ArcSin[c*x]))/(4*e^2*(d + e*x)^4) + (b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(40*e^2*(c^2*d^2 - e^2)^(9/2))} - - -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])*(d + e*x)^3, x, 8, (b*(12*e^2*(e*g + 3*d*h) + 25*c^2*d*(3*e^2*f + 3*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*e*(5*e^2*h + 9*c^2*(e^2*f + 3*d*e*g + 3*d^2*h))*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e^2*(e*g + 3*d*h)*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*e^3*h*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*(32*(225*c^4*d^3*f + 24*e^2*(e*g + 3*d*h) + 50*c^2*d*(3*e^2*f + 3*d*e*g + d^2*h)) + 75*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*ArcSin[c*x])/(96*c^6) + d^3*f*x*(a + b*ArcSin[c*x]) + (1/2)*d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*e*(e^2*f + 3*d*e*g + 3*d^2*h)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e^2*(e*g + 3*d*h)*x^5*(a + b*ArcSin[c*x]) + (1/6)*e^3*h*x^6*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])*(d + e*x)^2, x, 7, (b*(12*e^2*h + 25*c^2*(e^2*f + 2*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*e*(e*g + 2*d*h)*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*e^2*h*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(225*c^4*d^2*f + 24*e^2*h + 50*c^2*(e^2*f + 2*d*e*g + d^2*h)) + 225*c^2*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5) - (b*(8*c^2*d*(2*e*f + d*g) + 3*e*(e*g + 2*d*h))*ArcSin[c*x])/(32*c^4) + d^2*f*x*(a + b*ArcSin[c*x]) + (1/2)*d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*(e^2*f + 2*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*e*(e*g + 2*d*h)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e^2*h*x^5*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])*(d + e*x)^1, x, 6, (b*(e*g + d*h)*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*e*h*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(32*(9*c^2*d*f + 2*e*g + 2*d*h) + 9*(8*c^2*(e*f + d*g) + 3*e*h)*x)*Sqrt[1 - c^2*x^2])/(288*c^3) - (b*(8*c^2*(e*f + d*g) + 3*e*h)*ArcSin[c*x])/(32*c^4) + d*f*x*(a + b*ArcSin[c*x]) + (1/2)*(e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*(e*g + d*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*e*h*x^4*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^1, x, 15, (b*(4*(e*g - d*h) + e*h*x)*Sqrt[1 - c^2*x^2])/(4*c*e^2) - (b*h*ArcSin[c*x])/(4*c^2*e) - (I*b*(e^2*f - d*e*g + d^2*h)*ArcSin[c*x]^2)/(2*e^3) + ((e*g - d*h)*x*(a + b*ArcSin[c*x]))/e^2 + (h*x^2*(a + b*ArcSin[c*x]))/(2*e) + (b*(e^2*f - d*e*g + d^2*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (b*(e^2*f - d*e*g + d^2*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 - (b*(e^2*f - d*e*g + d^2*h)*ArcSin[c*x]*Log[d + e*x])/e^3 + ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^3 - (I*b*(e^2*f - d*e*g + d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (I*b*(e^2*f - d*e*g + d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^2, x, 16, (b*h*Sqrt[1 - c^2*x^2])/(c*e^2) - (I*b*(e*g - 2*d*h)*ArcSin[c*x]^2)/(2*e^3) + (h*x*(a + b*ArcSin[c*x]))/e^2 - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)) + (b*c*(e^2*f - d*e*g + d^2*h)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (b*(e*g - 2*d*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (b*(e*g - 2*d*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 - (b*(e*g - 2*d*h)*ArcSin[c*x]*Log[d + e*x])/e^3 + ((e*g - 2*d*h)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^3 - (I*b*(e*g - 2*d*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (I*b*(e*g - 2*d*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^3, x, 16, (b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(2*e^2*(c^2*d^2 - e^2)*(d + e*x)) - (I*b*h*ArcSin[c*x]^2)/(2*e^3) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - ((e*g - 2*d*h)*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)) - (b*c*(2*e^2*(e*g - 2*d*h) - c^2*d*(e^2*f + d*e*g - 3*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^3*(c^2*d^2 - e^2)^(3/2)) + (b*h*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (b*h*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 - (b*h*ArcSin[c*x]*Log[d + e*x])/e^3 + (h*(a + b*ArcSin[c*x])*Log[d + e*x])/e^3 - (I*b*h*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (I*b*h*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^4, x, 6, (b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(6*e^2*(c^2*d^2 - e^2)*(d + e*x)^2) - (b*c*(e^2*(e*g - 2*d*h) - c^2*(d*e^2*f - d^3*h))*Sqrt[1 - c^2*x^2])/(2*e^2*(c^2*d^2 - e^2)^2*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - ((e*g - 2*d*h)*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - (h*(a + b*ArcSin[c*x]))/(e^3*(d + e*x)) + (b*c*(6*e^4*h + c^2*e^2*(e^2*f - 4*d*e*g - 5*d^2*h) + c^4*d^2*(2*e^2*f + d*e*g + 2*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(6*e^3*(c^2*d^2 - e^2)^(5/2))} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^5, x, 7, (b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^3) - (b*c*(4*e^2*(e*g - 2*d*h) - c^2*d*(5*e^2*f - d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c^2*d^2 - e^2)^2*(d + e*x)^2) + (b*c*(12*e^4*h + c^4*d^2*(11*e^2*f + d*e*g - d^2*h) + 4*c^2*e^2*(e^2*f - 4*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c^2*d^2 - e^2)^3*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x)^4) - ((e*g - 2*d*h)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - (h*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - (b*c^3*(4*e^4*(e*g - 5*d*h) - c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) - 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(24*e^3*(c^2*d^2 - e^2)^(7/2))} -{(f + g*x + h*x^2)*(a + b*ArcSin[c*x])/(d + e*x)^6, x, 8, (b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(20*e^2*(c^2*d^2 - e^2)*(d + e*x)^4) - (b*c*(5*e^2*(e*g - 2*d*h) - c^2*d*(7*e^2*f - 2*d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(60*e^2*(c^2*d^2 - e^2)^2*(d + e*x)^3) + (b*c*(20*e^4*h + c^4*d^2*(26*e^2*f - d*e*g - 4*d^2*h) + c^2*e^2*(9*e^2*f - 34*d*e*g + 19*d^2*h))*Sqrt[1 - c^2*x^2])/(120*e^2*(c^2*d^2 - e^2)^3*(d + e*x)^2) + (b*c^3*(c^4*d^3*(10*e*f + d*g) - 4*e^3*(e*g - 5*d*h) + c^2*d*e*(11*e^2*f - 18*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^4*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(5*e^3*(d + e*x)^5) - ((e*g - 2*d*h)*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x)^4) - (h*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) + (b*c^3*(20*e^6*h + 3*c^4*d^2*e^2*(24*e^2*f - 19*d*e*g - 6*d^2*h) + 2*c^6*d^4*(12*e^2*f + 3*d*e*g + 2*d^2*h) + 9*c^2*e^4*(e^2*f - 6*d*e*g + 11*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(120*e^3*(c^2*d^2 - e^2)^(9/2))} - - -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])*(d + e*x)^3, x, 9, (b*(1225*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 360*e^3*i + 588*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i))*x^2*Sqrt[1 - c^2*x^2])/(11025*c^5) + (b*(5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*(30*e^2*i + 49*c^2*(e^2*g + 3*d*e*h + 3*d^2*i))*x^4*Sqrt[1 - c^2*x^2])/(1225*c^3) + (b*e^2*(e*h + 3*d*i)*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*e^3*i*x^6*Sqrt[1 - c^2*x^2])/(49*c) + (b*(32*(11025*c^6*d^3*f + 2450*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 720*e^3*i + 1176*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i)) + 3675*c^2*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x)*Sqrt[1 - c^2*x^2])/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*ArcSin[c*x])/(96*c^6) + d^3*f*x*(a + b*ArcSin[c*x]) + (1/2)*d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5*(a + b*ArcSin[c*x]) + (1/6)*e^2*(e*h + 3*d*i)*x^6*(a + b*ArcSin[c*x]) + (1/7)*e^3*i*x^7*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])*(d + e*x)^2, x, 8, (b*(25*c^2*(e^2*f + 2*d*e*g + d^2*h) + 12*e*(e*h + 2*d*i))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*(5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*(e*h + 2*d*i)*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*e^2*i*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*(32*(225*c^4*d^2*f + 50*c^2*(e^2*f + 2*d*e*g + d^2*h) + 24*e*(e*h + 2*d*i)) + 75*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5) - (b*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*ArcSin[c*x])/(96*c^6) + d^2*f*x*(a + b*ArcSin[c*x]) + (1/2)*d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*(e^2*f + 2*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*(e^2*g + 2*d*e*h + d^2*i)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e*(e*h + 2*d*i)*x^5*(a + b*ArcSin[c*x]) + (1/6)*e^2*i*x^6*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])*(d + e*x)^1, x, 7, (b*(25*c^2*(e*g + d*h) + 12*e*i)*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*(e*h + d*i)*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*e*i*x^4*Sqrt[1 - c^2*x^2])/(25*c) + (b*(32*(225*c^4*d*f + 50*c^2*(e*g + d*h) + 24*e*i) + 225*c^2*(8*c^2*(e*f + d*g) + 3*(e*h + d*i))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5) - (b*(8*c^2*(e*f + d*g) + 3*(e*h + d*i))*ArcSin[c*x])/(32*c^4) + d*f*x*(a + b*ArcSin[c*x]) + (1/2)*(e*f + d*g)*x^2*(a + b*ArcSin[c*x]) + (1/3)*(e*g + d*h)*x^3*(a + b*ArcSin[c*x]) + (1/4)*(e*h + d*i)*x^4*(a + b*ArcSin[c*x]) + (1/5)*e*i*x^5*(a + b*ArcSin[c*x])} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])/(d + e*x)^1, x, 16, (b*i*x^2*Sqrt[1 - c^2*x^2])/(9*c*e) + (b*(4*(2*e^2*i + 9*c^2*(e^2*g - d*e*h + d^2*i)) + 9*c^2*e*(e*h - d*i)*x)*Sqrt[1 - c^2*x^2])/(36*c^3*e^3) - (b*(e*h - d*i)*ArcSin[c*x])/(4*c^2*e^2) - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]^2)/(2*e^4) + ((e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]))/e^3 + ((e*h - d*i)*x^2*(a + b*ArcSin[c*x]))/(2*e^2) + (i*x^3*(a + b*ArcSin[c*x]))/(3*e) + (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 + (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])/(d + e*x)^2, x, 18, (b*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/(c*e^3) + (b*i*x*Sqrt[1 - c^2*x^2])/(4*c*e^2) - (b*i*ArcSin[c*x])/(4*c^2*e^2) - (I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]^2)/(2*e^4) + ((e*h - 2*d*i)*x*(a + b*ArcSin[c*x]))/e^3 + (i*x^2*(a + b*ArcSin[c*x]))/(2*e^2) - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x)) + (b*c*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^4*Sqrt[c^2*d^2 - e^2]) + (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 + (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])/(d + e*x)^3, x, 30, (b*i*Sqrt[1 - c^2*x^2])/(c*e^3) + (5*b*c*d^3*i*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) - (b*c*d^2*(3*e*h + 4*d*i)*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*d*(e^2*g + 4*d*e*h - 4*d^2*i)*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) + (b*c*(e^3*f - 2*d*e^2*g + 2*d^3*i)*Sqrt[1 - c^2*x^2])/(2*e^3*(c^2*d^2 - e^2)*(d + e*x)) - (I*b*(e*h - 3*d*i)*ArcSin[c*x]^2)/(2*e^4) + (i*x*(a + b*ArcSin[c*x]))/e^3 - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x)) + (5*b*c^3*d^4*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*d^2*(3*c^2*d*h + 4*e*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^3*(c^2*d^2 - e^2)^(3/2)) + (b*c*d*(4*e^2*(e*h - 2*d*i) + c^2*(d*e^2*g + 4*d^3*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) - (b*c*(2*e^4*g - 6*d^2*e^2*i - c^2*(d*e^3*f - 4*d^4*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*e^4*(c^2*d^2 - e^2)^(3/2)) + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 + (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*(e*h - 3*d*i)*ArcSin[c*x]*Log[d + e*x])/e^4 + ((e*h - 3*d*i)*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*(e*h - 3*d*i)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4} -{(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x])/(d + e*x)^4, x, 29, (b*c*(2*e^2*f - 3*d*e*g + 6*d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^2) - (11*b*c*d^3*i*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d^2*(2*e*h + 27*d*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*d*(e^2*g - 6*d*e*h - 18*d^2*i)*Sqrt[1 - c^2*x^2])/(12*e^3*(c^2*d^2 - e^2)*(d + e*x)^2) - (b*c*(2*e^2*(e*g - 4*d*h) - c^2*d*(2*e^2*f - d*e*g - 2*d^2*h))*Sqrt[1 - c^2*x^2])/(4*e^2*(c^2*d^2 - e^2)^2*(d + e*x)) - (11*b*c^3*d^4*i*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) + (b*c*d^2*(18*e^2*i + c^2*d*(2*e*h + 9*d*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - (b*c*d*(4*e^2*(e*h + 6*d*i) - c^2*d*(e^2*g - 2*d*e*h + 6*d^2*i))*Sqrt[1 - c^2*x^2])/(4*e^3*(c^2*d^2 - e^2)^2*(d + e*x)) - (I*b*i*ArcSin[c*x]^2)/(2*e^4) - ((e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(3*e^4*(d + e*x)^3) - ((e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x]))/(2*e^4*(d + e*x)^2) - ((e*h - 3*d*i)*(a + b*ArcSin[c*x]))/(e^4*(d + e*x)) + (b*c*(4*c^4*d^2*f + 12*e^2*h + c^2*(2*e^2*f - 9*d*e*g + 6*d^2*h))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e*(c^2*d^2 - e^2)^(5/2)) - (11*b*c^3*d^3*(2*c^2*d^2 + e^2)*i*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^4*(c^2*d^2 - e^2)^(5/2)) + (b*c^3*d^2*(4*c^2*d^2*h + e*(2*e*h + 81*d*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^3*(c^2*d^2 - e^2)^(5/2)) + (b*c*d*(2*c^4*d^2*g - 36*e^2*i + c^2*(e^2*g - 18*d*e*h - 18*d^2*i))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(12*e^2*(c^2*d^2 - e^2)^(5/2)) + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 + (b*i*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4 - (b*i*ArcSin[c*x]*Log[d + e*x])/e^4 + (i*(a + b*ArcSin[c*x])*Log[d + e*x])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^4 - (I*b*i*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Px (d+e x)^m (a+b ArcSin[c x])^2*) - - -{(f + g*x)^1*(a + b*ArcSin[c*x])^2/(d + e*x)^3, x, 33, (a*b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(e*(c^2*d^2 - e^2)*(d + e*x)) + (a*b*g^2*ArcSin[c*x])/(e^2*(e*f - d*g)) + (b^2*c*(e*f - d*g)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(e*(c^2*d^2 - e^2)*(d + e*x)) + (b^2*g^2*ArcSin[c*x]^2)/(2*e^2*(e*f - d*g)) - ((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(2*(e*f - d*g)*(d + e*x)^2) - (a*b*c*(2*e^2*g - c^2*d*(e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^2*(c^2*d^2 - e^2)^(3/2)) - (2*I*b^2*c*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (I*b^2*c^3*d*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*(c^2*d^2 - e^2)^(3/2)) + (2*I*b^2*c*g*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (I*b^2*c^3*d*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*(c^2*d^2 - e^2)^(3/2)) - (b^2*c^2*(e*f - d*g)*Log[d + e*x])/(e^2*(c^2*d^2 - e^2)) - (2*b^2*c*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (b^2*c^3*d*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*(c^2*d^2 - e^2)^(3/2)) + (2*b^2*c*g*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (b^2*c^3*d*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*(c^2*d^2 - e^2)^(3/2))} -{(f + g*x)^2*(a + b*ArcSin[c*x])^2/(d + e*x)^3, x, 55, -((a^2*(e*f - d*g)^2)/(2*e^3*(d + e*x)^2)) - (2*a^2*g*(e*f - d*g))/(e^3*(d + e*x)) + (a*b*c*(e*f - d*g)^2*Sqrt[1 - c^2*x^2])/(e^2*(c^2*d^2 - e^2)*(d + e*x)) - (a*b*(e*f - d*g)^2*ArcSin[c*x])/(e^3*(d + e*x)^2) - (4*a*b*g*(e*f - d*g)*ArcSin[c*x])/(e^3*(d + e*x)) + (b^2*c*(e*f - d*g)^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(e^2*(c^2*d^2 - e^2)*(d + e*x)) - (I*a*b*g^2*ArcSin[c*x]^2)/e^3 - (b^2*(e*f - d*g)^2*ArcSin[c*x]^2)/(2*e^3*(d + e*x)^2) - (2*b^2*g*(e*f - d*g)*ArcSin[c*x]^2)/(e^3*(d + e*x)) - (I*b^2*g^2*ArcSin[c*x]^3)/(3*e^3) - (a*b*c*(e*f - d*g)*(4*e^2*g - c^2*d*(e*f + 3*d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (4*I*b^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (2*a*b*g^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (4*I*b^2*c*g*(e*f - d*g)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (I*b^2*c^3*d*(e*f - d*g)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) + (b^2*g^2*ArcSin[c*x]^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (a^2*g^2*Log[d + e*x])/e^3 - (b^2*c^2*(e*f - d*g)^2*Log[d + e*x])/(e^3*(c^2*d^2 - e^2)) - (2*I*a*b*g^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (4*b^2*c*g*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (b^2*c^3*d*(e*f - d*g)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - (2*I*b^2*g^2*ArcSin[c*x]*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 - (2*I*a*b*g^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (4*b^2*c*g*(e*f - d*g)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (b^2*c^3*d*(e*f - d*g)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*(c^2*d^2 - e^2)^(3/2)) - (2*I*b^2*g^2*ArcSin[c*x]*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e^3 + (2*b^2*g^2*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e^3} - - -{(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2*(g + h*x)^3, x, 35, -2*b^2*d*g^3*x - (16*b^2*h^2*(3*f*g + e*h)*x)/(75*c^4) - (4*b^2*g*(f*g^2 + 3*h*(e*g + d*h))*x)/(9*c^2) - (5*b^2*f*h^3*x^2)/(96*c^4) - (1/4)*b^2*g^2*(e*g + 3*d*h)*x^2 - (3*b^2*h*(3*f*g^2 + h*(3*e*g + d*h))*x^2)/(32*c^2) - (8*b^2*h^2*(3*f*g + e*h)*x^3)/(225*c^2) - (2/27)*b^2*g*(f*g^2 + 3*h*(e*g + d*h))*x^3 - (5*b^2*f*h^3*x^4)/(288*c^2) - (1/32)*b^2*h*(3*f*g^2 + h*(3*e*g + d*h))*x^4 - (2/125)*b^2*h^2*(3*f*g + e*h)*x^5 - (1/108)*b^2*f*h^3*x^6 + (2*b*d*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (16*b*h^2*(3*f*g + e*h)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*g*(f*g^2 + 3*h*(e*g + d*h))*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (5*b*f*h^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(48*c^5) + (b*g^2*(e*g + 3*d*h)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*h*(3*f*g^2 + h*(3*e*g + d*h))*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (8*b*h^2*(3*f*g + e*h)*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*g*(f*g^2 + 3*h*(e*g + d*h))*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (5*b*f*h^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(72*c^3) + (b*h*(3*f*g^2 + h*(3*e*g + d*h))*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (2*b*h^2*(3*f*g + e*h)*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) + (b*f*h^3*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (5*f*h^3*(a + b*ArcSin[c*x])^2)/(96*c^6) - (g^2*(e*g + 3*d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2) - (3*h*(3*f*g^2 + h*(3*e*g + d*h))*(a + b*ArcSin[c*x])^2)/(32*c^4) + d*g^3*x*(a + b*ArcSin[c*x])^2 + (1/2)*g^2*(e*g + 3*d*h)*x^2*(a + b*ArcSin[c*x])^2 + (1/3)*g*(f*g^2 + 3*h*(e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2 + (1/4)*h*(3*f*g^2 + h*(3*e*g + d*h))*x^4*(a + b*ArcSin[c*x])^2 + (1/5)*h^2*(3*f*g + e*h)*x^5*(a + b*ArcSin[c*x])^2 + (1/6)*f*h^3*x^6*(a + b*ArcSin[c*x])^2} -{(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2*(g + h*x)^2, x, 27, -2*b^2*d*g^2*x - (16*b^2*f*h^2*x)/(75*c^4) - (4*b^2*(f*g^2 + h*(2*e*g + d*h))*x)/(9*c^2) - (1/4)*b^2*g*(e*g + 2*d*h)*x^2 - (3*b^2*h*(2*f*g + e*h)*x^2)/(32*c^2) - (8*b^2*f*h^2*x^3)/(225*c^2) - (2/27)*b^2*(f*g^2 + h*(2*e*g + d*h))*x^3 - (1/32)*b^2*h*(2*f*g + e*h)*x^4 - (2/125)*b^2*f*h^2*x^5 + (2*b*d*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (16*b*f*h^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^5) + (4*b*(f*g^2 + h*(2*e*g + d*h))*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (b*g*(e*g + 2*d*h)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*h*(2*f*g + e*h)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (8*b*f*h^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(75*c^3) + (2*b*(f*g^2 + h*(2*e*g + d*h))*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (b*h*(2*f*g + e*h)*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) + (2*b*f*h^2*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(25*c) - (g*(e*g + 2*d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2) - (3*h*(2*f*g + e*h)*(a + b*ArcSin[c*x])^2)/(32*c^4) + d*g^2*x*(a + b*ArcSin[c*x])^2 + (1/2)*g*(e*g + 2*d*h)*x^2*(a + b*ArcSin[c*x])^2 + (1/3)*(f*g^2 + h*(2*e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2 + (1/4)*h*(2*f*g + e*h)*x^4*(a + b*ArcSin[c*x])^2 + (1/5)*f*h^2*x^5*(a + b*ArcSin[c*x])^2} -{(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2*(g + h*x)^1, x, 20, -2*b^2*d*g*x - (4*b^2*(f*g + e*h)*x)/(9*c^2) - (3*b^2*f*h*x^2)/(32*c^2) - (1/4)*b^2*(e*g + d*h)*x^2 - (2/27)*b^2*(f*g + e*h)*x^3 - (1/32)*b^2*f*h*x^4 + (2*b*d*g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*(f*g + e*h)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (3*b*f*h*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(16*c^3) + (b*(e*g + d*h)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (2*b*(f*g + e*h)*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + (b*f*h*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (3*f*h*(a + b*ArcSin[c*x])^2)/(32*c^4) - ((e*g + d*h)*(a + b*ArcSin[c*x])^2)/(4*c^2) + d*g*x*(a + b*ArcSin[c*x])^2 + (1/2)*(e*g + d*h)*x^2*(a + b*ArcSin[c*x])^2 + (1/3)*(f*g + e*h)*x^3*(a + b*ArcSin[c*x])^2 + (1/4)*f*h*x^4*(a + b*ArcSin[c*x])^2} -{(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2/(g + h*x)^1, x, 38, -((a^2*(f*g - e*h)*x)/h^2) + (2*b^2*(f*g - e*h)*x)/h^2 + (a^2*f*x^2)/(2*h) - (b^2*f*x^2)/(4*h) - (a*b*(4*(f*g - e*h) - f*h*x)*Sqrt[1 - c^2*x^2])/(2*c*h^2) - (a*b*f*ArcSin[c*x])/(2*c^2*h) - (2*a*b*(f*g - e*h)*x*ArcSin[c*x])/h^2 + (a*b*f*x^2*ArcSin[c*x])/h - (2*b^2*(f*g - e*h)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*h^2) + (b^2*f*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(2*c*h) - (b^2*f*ArcSin[c*x]^2)/(4*c^2*h) - (I*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^2)/h^3 - (b^2*(f*g - e*h)*x*ArcSin[c*x]^2)/h^2 + (b^2*f*x^2*ArcSin[c*x]^2)/(2*h) - (I*b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^3)/(3*h^3) + (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + (b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (a^2*(f*g^2 - e*g*h + d*h^2)*Log[g + h*x])/h^3 - (2*I*a*b*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*I*b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*I*a*b*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 - (2*I*b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (2*b^2*(f*g^2 - e*g*h + d*h^2)*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + (2*b^2*(f*g^2 - e*g*h + d*h^2)*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3} -{(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2/(g + h*x)^2, x, 45, (a^2*f*x)/h^2 - (2*b^2*f*x)/h^2 - (a^2*(f*g^2 - e*g*h + d*h^2))/(h^3*(g + h*x)) + (2*a*b*f*Sqrt[1 - c^2*x^2])/(c*h^2) + (2*a*b*f*x*ArcSin[c*x])/h^2 - (2*a*b*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x])/(h^3*(g + h*x)) + (2*b^2*f*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*h^2) + (I*a*b*(2*f*g - e*h)*ArcSin[c*x]^2)/h^3 + (b^2*f*x*ArcSin[c*x]^2)/h^2 - (b^2*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]^2)/(h^3*(g + h*x)) + (I*b^2*(2*f*g - e*h)*ArcSin[c*x]^3)/(3*h^3) + (2*a*b*c*(f*g^2 - e*g*h + d*h^2)*ArcTan[(h + c^2*g*x)/(Sqrt[c^2*g^2 - h^2]*Sqrt[1 - c^2*x^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (2*a*b*(2*f*g - e*h)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*I*b^2*c*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (b^2*(2*f*g - e*h)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*a*b*(2*f*g - e*h)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (2*I*b^2*c*(f*g^2 - e*g*h + d*h^2)*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) - (b^2*(2*f*g - e*h)*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 - (a^2*(2*f*g - e*h)*Log[g + h*x])/h^3 + (2*I*a*b*(2*f*g - e*h)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*c*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) + (2*I*b^2*(2*f*g - e*h)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 + (2*I*a*b*(2*f*g - e*h)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 + (2*b^2*c*(f*g^2 - e*g*h + d*h^2)*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/(h^3*Sqrt[c^2*g^2 - h^2]) + (2*I*b^2*(2*f*g - e*h)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*(2*f*g - e*h)*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])])/h^3 - (2*b^2*(2*f*g - e*h)*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])])/h^3} - - -{((e*f + 2*d*h*x + e*h*x^2)^1*(a + b*ArcSin[c*x])^2)/(d + e*x)^2, x, 20, -((2*b^2*h*x)/e) + (2*a*b*h*Sqrt[1 - c^2*x^2])/(c*e) + (2*b^2*h*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*e) + (h*x*(a + b*ArcSin[c*x])^2)/e - ((f - (d^2*h)/e^2)*(a + b*ArcSin[c*x])^2)/(d + e*x) + (2*a*b*c*(e^2*f - d^2*h)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (2*I*b^2*c*(e^2*f - d^2*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (2*I*b^2*c*(e^2*f - d^2*h)*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*(e^2*f - d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2]) + (2*b^2*c*(e^2*f - d^2*h)*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^2*Sqrt[c^2*d^2 - e^2])} -{((e*f + 2*d*h*x + e*h*x^2)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^2, x, 32, -((4*b^2*h^2*x)/(9*c^2)) - (2*b^2*h*(2*e^2*f - d^2*h)*x)/e^2 - (b^2*d*h^2*x^2)/(2*e) - (2/27)*b^2*h^2*x^3 + (a*b*h*(4*e^2*h + c^2*(36*e^2*f - 25*d^2*h))*Sqrt[1 - c^2*x^2])/(9*c^3*e^2) + (5*a*b*d*h^2*(d + e*x)*Sqrt[1 - c^2*x^2])/(9*c*e^2) + (2*a*b*h^2*(d + e*x)^2*Sqrt[1 - c^2*x^2])/(9*c*e^2) - (a*b*d*(2*c^2*d^2 + 3*e^2)*h^2*ArcSin[c*x])/(3*c^2*e^3) + (4*b^2*h^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(9*c^3) + (2*b^2*h*(2*e^2*f - d^2*h)*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*e^2) + (b^2*d*h^2*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*e) + (2*b^2*h^2*x^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(9*c) - (b^2*d^3*h^2*ArcSin[c*x]^2)/(3*e^3) - (b^2*d*h^2*ArcSin[c*x]^2)/(2*c^2*e) + (2*h*(e^2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) + (h^2*(d + e*x)^3*(a + b*ArcSin[c*x])^2)/(3*e^3) + (2*a*b*c*(e^2*f - d^2*h)^2*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (2*I*b^2*c*(e^2*f - d^2*h)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (2*I*b^2*c*(e^2*f - d^2*h)^2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2]) + (2*b^2*c*(e^2*f - d^2*h)^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/(e^3*Sqrt[c^2*d^2 - e^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcSin[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcSin[a + b*x], x, 6, -((7*a*x^2*Sqrt[1 - (a + b*x)^2])/(48*b^2)) + (x^3*Sqrt[1 - (a + b*x)^2])/(16*b) - ((4*a*(16 + 19*a^2) - (9 + 26*a^2)*(a + b*x))*Sqrt[1 - (a + b*x)^2])/(96*b^4) - ((3 + 24*a^2 + 8*a^4)*ArcSin[a + b*x])/(32*b^4) + (1/4)*x^4*ArcSin[a + b*x]} -{x^2*ArcSin[a + b*x], x, 5, (x^2*Sqrt[1 - (a + b*x)^2])/(9*b) + ((4 + 11*a^2 - 5*a*b*x)*Sqrt[1 - (a + b*x)^2])/(18*b^3) + (a*(3 + 2*a^2)*ArcSin[a + b*x])/(6*b^3) + (1/3)*x^3*ArcSin[a + b*x]} -{x^1*ArcSin[a + b*x], x, 5, -((3*a*Sqrt[1 - (a + b*x)^2])/(4*b^2)) + (x*Sqrt[1 - (a + b*x)^2])/(4*b) - ((1 + 2*a^2)*ArcSin[a + b*x])/(4*b^2) + (1/2)*x^2*ArcSin[a + b*x]} -{x^0*ArcSin[a + b*x], x, 3, Sqrt[1 - (a + b*x)^2]/b + ((a + b*x)*ArcSin[a + b*x])/b} -{ArcSin[a + b*x]/x^1, x, 9, (-(1/2))*I*ArcSin[a + b*x]^2 + ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] - I*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] - I*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]} -{ArcSin[a + b*x]/x^2, x, 4, -(ArcSin[a + b*x]/x) - (b*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/Sqrt[1 - a^2]} -{ArcSin[a + b*x]/x^3, x, 5, -((b*Sqrt[1 - (a + b*x)^2])/(2*(1 - a^2)*x)) - ArcSin[a + b*x]/(2*x^2) - (a*b^2*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(2*(1 - a^2)^(3/2))} -{ArcSin[a + b*x]/x^4, x, 6, -((b*Sqrt[1 - (a + b*x)^2])/(6*(1 - a^2)*x^2)) - (a*b^2*Sqrt[1 - (a + b*x)^2])/(2*(1 - a^2)^2*x) - ArcSin[a + b*x]/(3*x^3) - ((1 + 2*a^2)*b^3*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(6*(1 - a^2)^(5/2))} -{ArcSin[a + b*x]/x^5, x, 7, -((b*Sqrt[1 - (a + b*x)^2])/(12*(1 - a^2)*x^3)) - (5*a*b^2*Sqrt[1 - (a + b*x)^2])/(24*(1 - a^2)^2*x^2) - ((4 + 11*a^2)*b^3*Sqrt[1 - (a + b*x)^2])/(24*(1 - a^2)^3*x) - ArcSin[a + b*x]/(4*x^4) - (a*(3 + 2*a^2)*b^4*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(8*(1 - a^2)^(7/2))} - - -{x^3*ArcSin[a + b*x]^2, x, 19, (4*a*x)/(3*b^3) + (2*a^3*x)/b^3 - (3*(a + b*x)^2)/(32*b^4) - (3*a^2*(a + b*x)^2)/(4*b^4) + (2*a*(a + b*x)^3)/(9*b^4) - (a + b*x)^4/(32*b^4) - (4*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4) - (2*a^3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b^4 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(16*b^4) + (3*a^2*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(2*b^4) - (2*a*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4) + ((a + b*x)^3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(8*b^4) - (3*ArcSin[a + b*x]^2)/(32*b^4) - (3*a^2*ArcSin[a + b*x]^2)/(4*b^4) - (a^4*ArcSin[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcSin[a + b*x]^2} -{x^2*ArcSin[a + b*x]^2, x, 14, -((4*x)/(9*b^2)) - (2*a^2*x)/b^2 + (a*(a + b*x)^2)/(2*b^3) - (2*(a + b*x)^3)/(27*b^3) + (4*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(9*b^3) + (2*a^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b^3 - (a*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b^3 + (2*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(9*b^3) + (a*ArcSin[a + b*x]^2)/(2*b^3) + (a^3*ArcSin[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcSin[a + b*x]^2} -{x^1*ArcSin[a + b*x]^2, x, 10, (2*a*x)/b - (a + b*x)^2/(4*b^2) - (2*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b^2 + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(2*b^2) - ArcSin[a + b*x]^2/(4*b^2) - (a^2*ArcSin[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcSin[a + b*x]^2} -{x^0*ArcSin[a + b*x]^2, x, 4, -2*x + (2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/b + ((a + b*x)*ArcSin[a + b*x]^2)/b} -{ArcSin[a + b*x]^2/x^1, x, 11, (-(1/3))*I*ArcSin[a + b*x]^3 + ArcSin[a + b*x]^2*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + ArcSin[a + b*x]^2*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] - 2*I*ArcSin[a + b*x]*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] - 2*I*ArcSin[a + b*x]*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] + 2*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + 2*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]} -{ArcSin[a + b*x]^2/x^2, x, 11, -(ArcSin[a + b*x]^2/x) - (2*b*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/Sqrt[1 - a^2] + (2*b*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/Sqrt[1 - a^2] + (2*I*b*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/Sqrt[1 - a^2] - (2*I*b*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/Sqrt[1 - a^2], -(ArcSin[a + b*x]^2/x) + (2*I*b*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/Sqrt[-1 + a^2] - (2*I*b*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/Sqrt[-1 + a^2] + (2*b*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2] - (2*b*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2]} -{ArcSin[a + b*x]^2/x^3, x, 14, -((b*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/((1 - a^2)*x)) - ArcSin[a + b*x]^2/(2*x^2) - (I*a*b^2*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(-1 + a^2)^(3/2) + (I*a*b^2*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(-1 + a^2)^(3/2) + (b^2*Log[x])/(1 - a^2) - (a*b^2*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) + (a*b^2*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2)} -(* {ArcSin[a + b*x]^2/x^4, x, 40, -(b^2/(3*(1 - a^2)*x)) - (b*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*(1 - a^2)*x^2) - (a*b^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/((1 - a^2)^2*x) - ArcSin[a + b*x]^2/(3*x^3) + (I*a^2*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(-1 + a^2)^(5/2) - (I*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(3*(-1 + a^2)^(3/2)) - (I*a^2*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(-1 + a^2)^(5/2) + (I*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(3*(-1 + a^2)^(3/2)) + (a*b^3*Log[x])/(1 - a^2)^2 + (a^2*b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(5/2) - (b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(3*(-1 + a^2)^(3/2)) - (a^2*b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(5/2) + (b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(3*(-1 + a^2)^(3/2))} *) - - -{x^2*ArcSin[a + b*x]^3, x, 18, -((14*Sqrt[1 - (a + b*x)^2])/(9*b^3)) - (6*a^2*Sqrt[1 - (a + b*x)^2])/b^3 + (3*a*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(4*b^3) + (2*(1 - (a + b*x)^2)^(3/2))/(27*b^3) - (3*a*ArcSin[a + b*x])/(4*b^3) - (4*(a + b*x)*ArcSin[a + b*x])/(3*b^3) - (6*a^2*(a + b*x)*ArcSin[a + b*x])/b^3 + (3*a*(a + b*x)^2*ArcSin[a + b*x])/(2*b^3) - (2*(a + b*x)^3*ArcSin[a + b*x])/(9*b^3) + (2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(3*b^3) + (3*a^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/b^3 - (3*a*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b^3) + ((a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(3*b^3) + (a*ArcSin[a + b*x]^3)/(2*b^3) + (a^3*ArcSin[a + b*x]^3)/(3*b^3) + (1/3)*x^3*ArcSin[a + b*x]^3} -{x^1*ArcSin[a + b*x]^3, x, 12, (6*a*Sqrt[1 - (a + b*x)^2])/b^2 - (3*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(8*b^2) + (3*ArcSin[a + b*x])/(8*b^2) + (6*a*(a + b*x)*ArcSin[a + b*x])/b^2 - (3*(a + b*x)^2*ArcSin[a + b*x])/(4*b^2) - (3*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/b^2 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(4*b^2) - ArcSin[a + b*x]^3/(4*b^2) - (a^2*ArcSin[a + b*x]^3)/(2*b^2) + (1/2)*x^2*ArcSin[a + b*x]^3} -{x^0*ArcSin[a + b*x]^3, x, 5, -((6*Sqrt[1 - (a + b*x)^2])/b) - (6*(a + b*x)*ArcSin[a + b*x])/b + (3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/b + ((a + b*x)*ArcSin[a + b*x]^3)/b} -{ArcSin[a + b*x]^3/x^1, x, 13, (-(1/4))*I*ArcSin[a + b*x]^4 + ArcSin[a + b*x]^3*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + ArcSin[a + b*x]^3*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] - 3*I*ArcSin[a + b*x]^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] - 3*I*ArcSin[a + b*x]^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] + 6*ArcSin[a + b*x]*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + 6*ArcSin[a + b*x]*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])] + 6*I*PolyLog[4, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + 6*I*PolyLog[4, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]} -{ArcSin[a + b*x]^3/x^2, x, 13, -(ArcSin[a + b*x]^3/x) + (3*I*b*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/Sqrt[-1 + a^2] - (3*I*b*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/Sqrt[-1 + a^2] + (6*b*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2] - (6*b*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2] + (6*I*b*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2] - (6*I*b*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/Sqrt[-1 + a^2]} -(* {ArcSin[a + b*x]^3/x^3, x, 21, -((3*I*b^2*ArcSin[a + b*x]^2)/(2*(1 - a^2))) - (3*b*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*(1 - a^2)*x) - ArcSin[a + b*x]^3/(2*x^2) + (3*b^2*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/(1 - a^2) + (3*b^2*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/(1 - a^2) - (3*I*a*b^2*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(2*(-1 + a^2)^(3/2)) + (3*I*a*b^2*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(2*(-1 + a^2)^(3/2)) - (3*I*b^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/(1 - a^2) - (3*I*b^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/(1 - a^2) - (3*a*b^2*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) + (3*a*b^2*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) - (3*I*a*b^2*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) + (3*I*a*b^2*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2)} *) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/ArcSin[a + b*x], x, 14, CosIntegral[ArcSin[a + b*x]]/(4*b^3) + (a^2*CosIntegral[ArcSin[a + b*x]])/b^3 - CosIntegral[3*ArcSin[a + b*x]]/(4*b^3) - (a*SinIntegral[2*ArcSin[a + b*x]])/b^3} -{x^1/ArcSin[a + b*x], x, 10, -((a*CosIntegral[ArcSin[a + b*x]])/b^2) + SinIntegral[2*ArcSin[a + b*x]]/(2*b^2)} -{x^0/ArcSin[a + b*x], x, 3, CosIntegral[ArcSin[a + b*x]]/b} -{1/(x^1*ArcSin[a + b*x]), x, 1, Unintegrable[1/(x*ArcSin[a + b*x]), x]} - - -{x^2/ArcSin[a + b*x]^2, x, 12, -((x^2*Sqrt[1 - (a + b*x)^2])/(b*ArcSin[a + b*x])) - (2*a*CosIntegral[2*ArcSin[a + b*x]])/b^3 - ((1 + 4*a^2)*SinIntegral[ArcSin[a + b*x]])/(4*b^3) + (3*SinIntegral[3*ArcSin[a + b*x]])/(4*b^3), -((a^2*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a + b*x])) + (2*a*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a + b*x]) - ((a + b*x)^2*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a + b*x]) - (2*a*CosIntegral[2*ArcSin[a + b*x]])/b^3 - SinIntegral[ArcSin[a + b*x]]/(4*b^3) - (a^2*SinIntegral[ArcSin[a + b*x]])/b^3 + (3*SinIntegral[3*ArcSin[a + b*x]])/(4*b^3)} -{x^1/ArcSin[a + b*x]^2, x, 8, -((x*Sqrt[1 - (a + b*x)^2])/(b*ArcSin[a + b*x])) + CosIntegral[2*ArcSin[a + b*x]]/b^2 + (a*SinIntegral[ArcSin[a + b*x]])/b^2, (a*Sqrt[1 - (a + b*x)^2])/(b^2*ArcSin[a + b*x]) - ((a + b*x)*Sqrt[1 - (a + b*x)^2])/(b^2*ArcSin[a + b*x]) + CosIntegral[2*ArcSin[a + b*x]]/b^2 + (a*SinIntegral[ArcSin[a + b*x]])/b^2} -{x^0/ArcSin[a + b*x]^2, x, 4, -(Sqrt[1 - (a + b*x)^2]/(b*ArcSin[a + b*x])) - SinIntegral[ArcSin[a + b*x]]/b} -{1/(x^1*ArcSin[a + b*x]^2), x, 1, Unintegrable[1/(x*ArcSin[a + b*x]^2), x]} - - -{x^2/ArcSin[a + b*x]^3, x, 24, -((x^2*Sqrt[1 - (a + b*x)^2])/(2*b*ArcSin[a + b*x]^2)) + (a^2*(a + b*x))/(2*b^3*ArcSin[a + b*x]) - (2*a*(a + b*x)^2)/(b^3*ArcSin[a + b*x]) + (9*a + b*x)/(8*b^3*ArcSin[a + b*x]) - ((1 + 4*a^2)*CosIntegral[ArcSin[a + b*x]])/(8*b^3) + (9*CosIntegral[3*ArcSin[a + b*x]])/(8*b^3) - (3*Sin[3*ArcSin[a + b*x]])/(8*b^3*ArcSin[a + b*x]) + (2*a*SinIntegral[2*ArcSin[a + b*x]])/b^3, -((a^2*Sqrt[1 - (a + b*x)^2])/(2*b^3*ArcSin[a + b*x]^2)) + (a*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(b^3*ArcSin[a + b*x]^2) - ((a + b*x)^2*Sqrt[1 - (a + b*x)^2])/(2*b^3*ArcSin[a + b*x]^2) + a/(b^3*ArcSin[a + b*x]) - (a + b*x)/(b^3*ArcSin[a + b*x]) + (a^2*(a + b*x))/(2*b^3*ArcSin[a + b*x]) - (2*a*(a + b*x)^2)/(b^3*ArcSin[a + b*x]) + (3*(a + b*x)^3)/(2*b^3*ArcSin[a + b*x]) - CosIntegral[ArcSin[a + b*x]]/(8*b^3) - (a^2*CosIntegral[ArcSin[a + b*x]])/(2*b^3) + (9*CosIntegral[3*ArcSin[a + b*x]])/(8*b^3) + (2*a*SinIntegral[2*ArcSin[a + b*x]])/b^3} -{x^1/ArcSin[a + b*x]^3, x, 14, -((x*Sqrt[1 - (a + b*x)^2])/(2*b*ArcSin[a + b*x]^2)) - (a*(a + b*x))/(2*b^2*ArcSin[a + b*x]) - (1 - 2*(a + b*x)^2)/(2*b^2*ArcSin[a + b*x]) + (a*CosIntegral[ArcSin[a + b*x]])/(2*b^2) - SinIntegral[2*ArcSin[a + b*x]]/b^2, (a*Sqrt[1 - (a + b*x)^2])/(2*b^2*ArcSin[a + b*x]^2) - ((a + b*x)*Sqrt[1 - (a + b*x)^2])/(2*b^2*ArcSin[a + b*x]^2) - 1/(2*b^2*ArcSin[a + b*x]) - (a*(a + b*x))/(2*b^2*ArcSin[a + b*x]) + (a + b*x)^2/(b^2*ArcSin[a + b*x]) + (a*CosIntegral[ArcSin[a + b*x]])/(2*b^2) - SinIntegral[2*ArcSin[a + b*x]]/b^2} -{x^0/ArcSin[a + b*x]^3, x, 5, -(Sqrt[1 - (a + b*x)^2]/(2*b*ArcSin[a + b*x]^2)) + (a + b*x)/(2*b*ArcSin[a + b*x]) - CosIntegral[ArcSin[a + b*x]]/(2*b)} -{1/(x^1*ArcSin[a + b*x]^3), x, 1, Unintegrable[1/(x*ArcSin[a + b*x]^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*(a + b*ArcSin[c + d*x])^(1/2), x, 23, (c^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d^3 + ((c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(3*d^3) + (c*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(2*d^3) - (Sqrt[b]*c*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(4*d^3) - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d^3) - (Sqrt[b]*c^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d^3 + (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(12*d^3) + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d^3) + (Sqrt[b]*c^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d^3 - (Sqrt[b]*c*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(4*d^3) - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*d^3)} -{x^1*(a + b*ArcSin[c + d*x])^(1/2), x, 14, -((c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d^2) - (Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(4*d^2) + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d^2) + (Sqrt[b]*c*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d^2 - (Sqrt[b]*c*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d^2 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d^2)} -{x^0*(a + b*ArcSin[c + d*x])^(1/2), x, 8, ((c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d} - - -{x^1*(a + b*ArcSin[c + d*x])^(3/2), x, 16, -((3*b*c*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d^2)) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/d^2 - ((a + b*ArcSin[c + d*x])^(3/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d^2) - (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d^2) + (3*b^(3/2)*c*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d^2) + (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*d^2) + (3*b*Sqrt[a + b*ArcSin[c + d*x]]*Sin[2*ArcSin[c + d*x]])/(16*d^2)} -{x^0*(a + b*ArcSin[c + d*x])^(3/2), x, 9, (3*b*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/d - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d)} - - -{x^1*(a + b*ArcSin[c + d*x])^(5/2), x, 18, (15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(4*d^2) - (5*b*c*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSin[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]])/(64*d^2) - ((a + b*ArcSin[c + d*x])^(5/2)*Cos[2*ArcSin[c + d*x]])/(4*d^2) - (15*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*d^2) - (15*b^(5/2)*c*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d^2) + (15*b^(5/2)*c*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d^2) - (15*b^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*d^2) + (5*b*(a + b*ArcSin[c + d*x])^(3/2)*Sin[2*ArcSin[c + d*x]])/(16*d^2)} -{x^0*(a + b*ArcSin[c + d*x])^(5/2), x, 10, -((15*b^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(4*d)) + (5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(5/2))/d + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d)} - - -{x^0*(a + b*ArcSin[c + d*x])^(7/2), x, 11, -((105*b^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d)) - (35*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (7*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(7/2))/d + (105*b^(7/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*d) + (105*b^(7/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcSin[c + d*x])^(1/2), x, 20, (Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d^3) + (c^2*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d^3) - (Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d^3) - (c*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(Sqrt[b]*d^3) + (Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*d^3) + (c^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d^3) + (c*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(Sqrt[b]*d^3) - (Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*d^3)} -{x^1/(a + b*ArcSin[c + d*x])^(1/2), x, 12, -((c*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d^2)) + (Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(2*Sqrt[b]*d^2) - (c*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d^2) - (Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(2*Sqrt[b]*d^2)} -{x^0/(a + b*ArcSin[c + d*x])^(1/2), x, 7, (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d) + (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d)} - - -{x^1/(a + b*ArcSin[c + d*x])^(3/2), x, 16, (2*c*Sqrt[1 - (c + d*x)^2])/(b*d^2*Sqrt[a + b*ArcSin[c + d*x]]) - (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*d^2) + (2*c*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d^2) - (2*c*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*d^2) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*d^2)} -{x^0/(a + b*ArcSin[c + d*x])^(3/2), x, 8, -((2*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]])) - (2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*d)} - - -{x^1/(a + b*ArcSin[c + d*x])^(5/2), x, 22, (2*c*Sqrt[1 - (c + d*x)^2])/(3*b*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) - (4*c*(c + d*x))/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (4*c*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (8*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d^2) + (4*c*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d^2) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d^2)} -{x^0/(a + b*ArcSin[c + d*x])^(5/2), x, 9, -((2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2))) + (4*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (4*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) - (4*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d)} - - -{x^1/(a + b*ArcSin[c + d*x])^(7/2), x, 21, (2*c*Sqrt[1 - (c + d*x)^2])/(5*b*d^2*(a + b*ArcSin[c + d*x])^(5/2)) - (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(5*b*d^2*(a + b*ArcSin[c + d*x])^(5/2)) - 4/(15*b^2*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - (4*c*(c + d*x))/(15*b^2*d^2*(a + b*ArcSin[c + d*x])^(3/2)) + (8*(c + d*x)^2)/(15*b^2*d^2*(a + b*ArcSin[c + d*x])^(3/2)) - (8*c*Sqrt[1 - (c + d*x)^2])/(15*b^3*d^2*Sqrt[a + b*ArcSin[c + d*x]]) + (32*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(15*b^3*d^2*Sqrt[a + b*ArcSin[c + d*x]]) - (32*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(15*b^(7/2)*d^2) - (8*c*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d^2) + (8*c*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7/2)*d^2) - (32*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(15*b^(7/2)*d^2)} -{x^0/(a + b*ArcSin[c + d*x])^(7/2), x, 10, -((2*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2))) + (4*(c + d*x))/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (8*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (8*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (8*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7/2)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c+d x])^n with n symbolic*) - - -{x^m*(a + b*ArcSin[c + d*x])^n, x, 1, Unintegrable[x^m*(a + b*ArcSin[c + d*x])^n, x]} - - -{x^2*(a + b*ArcSin[c + d*x])^n, x, 22, -((I*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c + d*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*(8*d^3))) - (I*c^2*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c + d*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*(2*d^3)) + (I*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*(8*d^3)) + (I*c^2*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*(2*d^3)) + (2^(-2 - n)*c*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c + d*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*d^3) + (2^(-2 - n)*c*E^((2*I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*d^3) + (I*3^(-1 - n)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((3*I*(a + b*ArcSin[c + d*x]))/b)])/(E^((3*I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*(8*d^3)) - (I*3^(-1 - n)*E^((3*I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (3*I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*(8*d^3))} -{x^1*(a + b*ArcSin[c + d*x])^n, x, 14, (I*c*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c + d*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*(2*d^2)) - (I*c*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*(2*d^2)) - (2^(-3 - n)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((2*I*(a + b*ArcSin[c + d*x]))/b)])/(E^((2*I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*d^2) - (2^(-3 - n)*E^((2*I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (2*I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*d^2)} -{x^0*(a + b*ArcSin[c + d*x])^n, x, 5, -((I*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, -((I*(a + b*ArcSin[c + d*x]))/b)])/(E^((I*a)/b)*(-((I*(a + b*ArcSin[c + d*x]))/b))^n*(2*d))) + (I*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(((I*(a + b*ArcSin[c + d*x]))/b)^n*(2*d))} -{(a + b*ArcSin[c + d*x])^n/x^1, x, 1, Unintegrable[(a + b*ArcSin[c + d*x])^n/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSin[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^4, x, 6, (b*e^4*Sqrt[1 - (c + d*x)^2])/(5*d) - (2*b*e^4*(1 - (c + d*x)^2)^(3/2))/(15*d) + (b*e^4*(1 - (c + d*x)^2)^(5/2))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x]))/(5*d)} -{(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^3, x, 6, (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(32*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(16*d) - (3*b*e^3*ArcSin[c + d*x])/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x]))/(4*d)} -{(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^2, x, 6, (b*e^2*Sqrt[1 - (c + d*x)^2])/(3*d) - (b*e^2*(1 - (c + d*x)^2)^(3/2))/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(3*d)} -{(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^1, x, 5, (b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(4*d) - (b*e*ArcSin[c + d*x])/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x]))/(2*d)} -{(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^0, x, 4, a*x + (b*Sqrt[1 - (c + d*x)^2])/d + (b*(c + d*x)*ArcSin[c + d*x])/d} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^1, x, 7, -((I*(a + b*ArcSin[c + d*x])^2)/(2*b*d*e)) + ((a + b*ArcSin[c + d*x])*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e) - (I*b*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(2*d*e)} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^2, x, 6, -((a + b*ArcSin[c + d*x])/(d*e^2*(c + d*x))) - (b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(d*e^2)} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^3, x, 4, -((b*Sqrt[1 - (c + d*x)^2])/(2*d*e^3*(c + d*x))) - (a + b*ArcSin[c + d*x])/(2*d*e^3*(c + d*x)^2)} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^4, x, 7, -((b*Sqrt[1 - (c + d*x)^2])/(6*d*e^4*(c + d*x)^2)) - (a + b*ArcSin[c + d*x])/(3*d*e^4*(c + d*x)^3) - (b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(6*d*e^4)} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^5, x, 5, -((b*Sqrt[1 - (c + d*x)^2])/(12*d*e^5*(c + d*x)^3)) - (b*Sqrt[1 - (c + d*x)^2])/(6*d*e^5*(c + d*x)) - (a + b*ArcSin[c + d*x])/(4*d*e^5*(c + d*x)^4)} -{(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^6, x, 8, -((b*Sqrt[1 - (c + d*x)^2])/(20*d*e^6*(c + d*x)^4)) - (3*b*Sqrt[1 - (c + d*x)^2])/(40*d*e^6*(c + d*x)^2) - (a + b*ArcSin[c + d*x])/(5*d*e^6*(c + d*x)^5) - (3*b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(40*d*e^6)} - - -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^4, x, 9, (-(16/75))*b^2*e^4*x - (8*b^2*e^4*(c + d*x)^3)/(225*d) - (2*b^2*e^4*(c + d*x)^5)/(125*d) + (16*b*e^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(75*d) + (8*b*e^4*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(75*d) + (2*b*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x])^2)/(5*d)} -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^3, x, 8, -((3*b^2*e^3*(c + d*x)^2)/(32*d)) - (b^2*e^3*(c + d*x)^4)/(32*d) + (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(16*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(8*d) - (3*e^3*(a + b*ArcSin[c + d*x])^2)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/(4*d)} -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^2, x, 7, (-(4/9))*b^2*e^2*x - (2*b^2*e^2*(c + d*x)^3)/(27*d) + (4*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(9*d) + (2*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^2)/(3*d)} -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^1, x, 6, -((b^2*e*(c + d*x)^2)/(4*d)) + (b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(2*d) - (e*(a + b*ArcSin[c + d*x])^2)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(2*d)} -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^0, x, 4, -2*b^2*x + (2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^2)/d} -{(a + b*ArcSin[c + d*x])^2/(c*e + e*d*x)^1, x, 8, -((I*(a + b*ArcSin[c + d*x])^3)/(3*b*d*e)) + ((a + b*ArcSin[c + d*x])^2*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e) - (I*b*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(d*e) + (b^2*PolyLog[3, E^(2*I*ArcSin[c + d*x])])/(2*d*e)} -{(a + b*ArcSin[c + d*x])^2/(c*e + e*d*x)^2, x, 9, -((a + b*ArcSin[c + d*x])^2/(d*e^2*(c + d*x))) - (4*b*(a + b*ArcSin[c + d*x])*ArcTanh[E^(I*ArcSin[c + d*x])])/(d*e^2) + (2*I*b^2*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^2) - (2*I*b^2*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^2)} -{(a + b*ArcSin[c + d*x])^2/(c*e + e*d*x)^3, x, 5, -((b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(d*e^3*(c + d*x))) - (a + b*ArcSin[c + d*x])^2/(2*d*e^3*(c + d*x)^2) + (b^2*Log[c + d*x])/(d*e^3)} -{(a + b*ArcSin[c + d*x])^2/(c*e + e*d*x)^4, x, 11, -(b^2/(3*d*e^4*(c + d*x))) - (b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(3*d*e^4*(c + d*x)^2) - (a + b*ArcSin[c + d*x])^2/(3*d*e^4*(c + d*x)^3) - (2*b*(a + b*ArcSin[c + d*x])*ArcTanh[E^(I*ArcSin[c + d*x])])/(3*d*e^4) + (I*b^2*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(3*d*e^4) - (I*b^2*PolyLog[2, E^(I*ArcSin[c + d*x])])/(3*d*e^4)} - - -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^4, x, 17, (-(16/25))*a*b^2*e^4*x - (298*b^3*e^4*Sqrt[1 - (c + d*x)^2])/(375*d) + (76*b^3*e^4*(1 - (c + d*x)^2)^(3/2))/(1125*d) - (6*b^3*e^4*(1 - (c + d*x)^2)^(5/2))/(625*d) - (16*b^3*e^4*(c + d*x)*ArcSin[c + d*x])/(25*d) - (8*b^2*e^4*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(75*d) - (6*b^2*e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x]))/(125*d) + (8*b*e^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) + (4*b*e^4*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) + (3*b*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSin[c + d*x])^3)/(5*d)} -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^3, x, 13, -((45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(256*d)) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(128*d) + (45*b^3*e^3*ArcSin[c + d*x])/(256*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x]))/(32*d) - (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x]))/(32*d) + (9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(32*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcSin[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^3)/(4*d)} -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^2, x, 12, (-(4/3))*a*b^2*e^2*x - (14*b^3*e^2*Sqrt[1 - (c + d*x)^2])/(9*d) + (2*b^3*e^2*(1 - (c + d*x)^2)^(3/2))/(27*d) - (4*b^3*e^2*(c + d*x)*ArcSin[c + d*x])/(3*d) - (2*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/(9*d) + (2*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(3*d) + (b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^3)/(3*d)} -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^1, x, 8, -((3*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(8*d)) + (3*b^3*e*ArcSin[c + d*x])/(8*d) - (3*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x]))/(4*d) + (3*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(4*d) - (e*(a + b*ArcSin[c + d*x])^3)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^3)/(2*d)} -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^0, x, 6, -6*a*b^2*x - (6*b^3*Sqrt[1 - (c + d*x)^2])/d - (6*b^3*(c + d*x)*ArcSin[c + d*x])/d + (3*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^3)/d} -{(a + b*ArcSin[c + d*x])^3/(c*e + e*d*x)^1, x, 9, -((I*(a + b*ArcSin[c + d*x])^4)/(4*b*d*e)) + ((a + b*ArcSin[c + d*x])^3*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e) - (3*I*b*(a + b*ArcSin[c + d*x])^2*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(2*d*e) + (3*b^2*(a + b*ArcSin[c + d*x])*PolyLog[3, E^(2*I*ArcSin[c + d*x])])/(2*d*e) + (3*I*b^3*PolyLog[4, E^(2*I*ArcSin[c + d*x])])/(4*d*e)} -{(a + b*ArcSin[c + d*x])^3/(c*e + e*d*x)^2, x, 11, -((a + b*ArcSin[c + d*x])^3/(d*e^2*(c + d*x))) - (6*b*(a + b*ArcSin[c + d*x])^2*ArcTanh[E^(I*ArcSin[c + d*x])])/(d*e^2) + (6*I*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^2) - (6*I*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^2) - (6*b^3*PolyLog[3, -E^(I*ArcSin[c + d*x])])/(d*e^2) + (6*b^3*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^2)} -{(a + b*ArcSin[c + d*x])^3/(c*e + e*d*x)^3, x, 9, -((3*I*b*(a + b*ArcSin[c + d*x])^2)/(2*d*e^3)) - (3*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcSin[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcSin[c + d*x])*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e^3) - (3*I*b^3*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(2*d*e^3)} -{(a + b*ArcSin[c + d*x])^3/(c*e + e*d*x)^4, x, 16, -((b^2*(a + b*ArcSin[c + d*x]))/(d*e^4*(c + d*x))) - (b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/(2*d*e^4*(c + d*x)^2) - (a + b*ArcSin[c + d*x])^3/(3*d*e^4*(c + d*x)^3) - (b*(a + b*ArcSin[c + d*x])^2*ArcTanh[E^(I*ArcSin[c + d*x])])/(d*e^4) - (b^3*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(d*e^4) + (I*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^4) - (I*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^4) - (b^3*PolyLog[3, -E^(I*ArcSin[c + d*x])])/(d*e^4) + (b^3*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^4)} - - -{(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^3, x, 16, (45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(64*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(32*d) + (45*b^2*e^3*(a + b*ArcSin[c + d*x])^2)/(128*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(16*d) - (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2)/(16*d) + (3*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(8*d) + (b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(4*d) - (3*e^3*(a + b*ArcSin[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^4)/(4*d)} -{(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^2, x, 13, (160/27)*b^4*e^2*x + (8*b^4*e^2*(c + d*x)^3)/(81*d) - (160*b^3*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(27*d) - (8*b^3*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(27*d) - (8*b^2*e^2*(c + d*x)*(a + b*ArcSin[c + d*x])^2)/(3*d) - (4*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^2)/(9*d) + (8*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(9*d) + (4*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/(3*d)} -{(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^1, x, 9, (3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/(2*d) + (3*b^2*e*(a + b*ArcSin[c + d*x])^2)/(4*d) - (3*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)/(2*d) + (b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d - (e*(a + b*ArcSin[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^4)/(2*d)} -{(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^0, x, 6, 24*b^4*x - (24*b^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/d - (12*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])^2)/d + (4*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^4)/d} -{(a + b*ArcSin[c + d*x])^4/(c*e + e*d*x)^1, x, 10, -((I*(a + b*ArcSin[c + d*x])^5)/(5*b*d*e)) + ((a + b*ArcSin[c + d*x])^4*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e) - (2*I*b*(a + b*ArcSin[c + d*x])^3*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(d*e) + (3*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[3, E^(2*I*ArcSin[c + d*x])])/(d*e) + (3*I*b^3*(a + b*ArcSin[c + d*x])*PolyLog[4, E^(2*I*ArcSin[c + d*x])])/(d*e) - (3*b^4*PolyLog[5, E^(2*I*ArcSin[c + d*x])])/(2*d*e)} -{(a + b*ArcSin[c + d*x])^4/(c*e + e*d*x)^2, x, 13, -((a + b*ArcSin[c + d*x])^4/(d*e^2*(c + d*x))) - (8*b*(a + b*ArcSin[c + d*x])^3*ArcTanh[E^(I*ArcSin[c + d*x])])/(d*e^2) + (12*I*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^2) - (12*I*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^2) - (24*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, -E^(I*ArcSin[c + d*x])])/(d*e^2) + (24*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^2) - (24*I*b^4*PolyLog[4, -E^(I*ArcSin[c + d*x])])/(d*e^2) + (24*I*b^4*PolyLog[4, E^(I*ArcSin[c + d*x])])/(d*e^2)} -{(a + b*ArcSin[c + d*x])^4/(c*e + e*d*x)^3, x, 10, -((2*I*b*(a + b*ArcSin[c + d*x])^3)/(d*e^3)) - (2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(d*e^3*(c + d*x)) - (a + b*ArcSin[c + d*x])^4/(2*d*e^3*(c + d*x)^2) + (6*b^2*(a + b*ArcSin[c + d*x])^2*Log[1 - E^(2*I*ArcSin[c + d*x])])/(d*e^3) - (6*I*b^3*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(2*I*ArcSin[c + d*x])])/(d*e^3) + (3*b^4*PolyLog[3, E^(2*I*ArcSin[c + d*x])])/(d*e^3)} -{(a + b*ArcSin[c + d*x])^4/(c*e + e*d*x)^4, x, 21, -((2*b^2*(a + b*ArcSin[c + d*x])^2)/(d*e^4*(c + d*x))) - (2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3)/(3*d*e^4*(c + d*x)^2) - (a + b*ArcSin[c + d*x])^4/(3*d*e^4*(c + d*x)^3) - (8*b^3*(a + b*ArcSin[c + d*x])*ArcTanh[E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*b*(a + b*ArcSin[c + d*x])^3*ArcTanh[E^(I*ArcSin[c + d*x])])/(3*d*e^4) + (4*I*b^4*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^4) + (2*I*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[2, -E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*I*b^4*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^4) - (2*I*b^2*(a + b*ArcSin[c + d*x])^2*PolyLog[2, E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, -E^(I*ArcSin[c + d*x])])/(d*e^4) + (4*b^3*(a + b*ArcSin[c + d*x])*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^4) - (4*I*b^4*PolyLog[4, -E^(I*ArcSin[c + d*x])])/(d*e^4) + (4*I*b^4*PolyLog[4, E^(I*ArcSin[c + d*x])])/(d*e^4)} - - -{(a + b*ArcSin[c + d*x])^5, x, 8, 120*a*b^4*x + (120*b^5*Sqrt[1 - (c + d*x)^2])/d + (120*b^5*(c + d*x)*ArcSin[c + d*x])/d - (60*b^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2)/d - (20*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])^3)/d + (5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^4)/d + ((c + d*x)*(a + b*ArcSin[c + d*x])^5)/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^4, x, 14, (e^4*Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b*d) - (3*e^4*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(16*b*d) + (e^4*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b*d) + (e^4*Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b*d) - (3*e^4*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(16*b*d) + (e^4*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b*d)} -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^3, x, 11, -((e^3*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b])/(4*b*d)) + (e^3*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b]*Sin[(4*a)/b])/(8*b*d) + (e^3*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(4*b*d) - (e^3*Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(8*b*d)} -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^2, x, 11, (e^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(4*b*d) - (e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(4*b*d) + (e^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(4*b*d) - (e^2*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(4*b*d)} -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^1, x, 8, -((e*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b])/(2*b*d)) + (e*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(2*b*d)} -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^0, x, 5, (Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(b*d) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(b*d)} -{1/(a + b*ArcSin[c + d*x])/(c*e + e*d*x)^1, x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])), x]/e} - - -{1/(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^4, x, 13, -((e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x]))) + (e^4*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(8*b^2*d) - (9*e^4*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(16*b^2*d) + (5*e^4*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b]*Sin[(5*a)/b])/(16*b^2*d) - (e^4*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b^2*d) + (9*e^4*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(16*b^2*d) - (5*e^4*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(16*b^2*d)} -{1/(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^3, x, 10, -((e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x]))) + (e^3*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(2*b^2*d) - (e^3*Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(2*b^2*d) + (e^3*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(2*b^2*d) - (e^3*Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(2*b^2*d)} -{1/(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^2, x, 10, -((e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x]))) + (e^2*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(4*b^2*d) - (3*e^2*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(4*b^2*d) - (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(4*b^2*d)} -{1/(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^1, x, 6, -((e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*d*(a + b*ArcSin[c + d*x]))) + (e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(b^2*d) + (e*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(b^2*d)} -{1/(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^0, x, 6, -(Sqrt[1 - (c + d*x)^2]/(b*d*(a + b*ArcSin[c + d*x]))) + (CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(b^2*d) - (Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(b^2*d)} -{1/(a + b*ArcSin[c + d*x])^2/(c*e + e*d*x)^1, x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^2), x]/e} - - -{1/(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^4, x, 26, -((e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(2*b*d*(a + b*ArcSin[c + d*x])^2)) - (2*e^4*(c + d*x)^3)/(b^2*d*(a + b*ArcSin[c + d*x])) + (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcSin[c + d*x])) - (e^4*Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(16*b^3*d) + (27*e^4*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(32*b^3*d) - (25*e^4*Cos[(5*a)/b]*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(32*b^3*d) - (e^4*Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(16*b^3*d) + (27*e^4*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(32*b^3*d) - (25*e^4*Sin[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(32*b^3*d)} -{1/(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^3, x, 20, -((e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(2*b*d*(a + b*ArcSin[c + d*x])^2)) - (3*e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcSin[c + d*x])) + (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcSin[c + d*x])) + (e^3*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b])/(2*b^3*d) - (e^3*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b]*Sin[(4*a)/b])/(b^3*d) - (e^3*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(2*b^3*d) + (e^3*Cos[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(b^3*d)} -{1/(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^2, x, 18, -((e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(2*b*d*(a + b*ArcSin[c + d*x])^2)) - (e^2*(c + d*x))/(b^2*d*(a + b*ArcSin[c + d*x])) + (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSin[c + d*x])) - (e^2*Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(8*b^3*d) - (e^2*Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(8*b^3*d) + (9*e^2*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(8*b^3*d)} -{1/(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^1, x, 11, -((e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(2*b*d*(a + b*ArcSin[c + d*x])^2)) - e/(2*b^2*d*(a + b*ArcSin[c + d*x])) + (e*(c + d*x)^2)/(b^2*d*(a + b*ArcSin[c + d*x])) + (e*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b])/(b^3*d) - (e*Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(b^3*d)} -{1/(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^0, x, 7, -(Sqrt[1 - (c + d*x)^2]/(2*b*d*(a + b*ArcSin[c + d*x])^2)) + (c + d*x)/(2*b^2*d*(a + b*ArcSin[c + d*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(2*b^3*d) - (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(2*b^3*d)} -{1/(a + b*ArcSin[c + d*x])^3/(c*e + e*d*x)^1, x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^3), x]/e} - - -{1/(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^4, x, 24, -((e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3)) - (2*e^4*(c + d*x)^3)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) - (2*e^4*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(b^3*d*(a + b*ArcSin[c + d*x])) + (25*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(6*b^3*d*(a + b*ArcSin[c + d*x])) - (e^4*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(48*b^4*d) + (27*e^4*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(32*b^4*d) - (125*e^4*CosIntegral[(5*(a + b*ArcSin[c + d*x]))/b]*Sin[(5*a)/b])/(96*b^4*d) + (e^4*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(48*b^4*d) - (27*e^4*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(32*b^4*d) + (125*e^4*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSin[c + d*x]))/b])/(96*b^4*d)} -{1/(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^3, x, 17, -((e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3)) - (e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcSin[c + d*x])^2) + (2*e^3*(c + d*x)^4)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) - (e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b^3*d*(a + b*ArcSin[c + d*x])) + (8*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a + b*ArcSin[c + d*x])) - (e^3*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) + (4*e^3*Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) - (e^3*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) + (4*e^3*Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d)} -{1/(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^2, x, 18, -((e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3)) - (e^2*(c + d*x))/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSin[c + d*x])^2) - (e^2*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a + b*ArcSin[c + d*x])) + (3*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(2*b^3*d*(a + b*ArcSin[c + d*x])) - (e^2*CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(24*b^4*d) + (9*e^2*CosIntegral[(3*(a + b*ArcSin[c + d*x]))/b]*Sin[(3*a)/b])/(8*b^4*d) + (e^2*Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^4*d) - (9*e^2*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c + d*x]))/b])/(8*b^4*d)} -{1/(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^1, x, 9, -((e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^3)) - e/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) + (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSin[c + d*x])^2) + (2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b^3*d*(a + b*ArcSin[c + d*x])) - (2*e*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d) - (2*e*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c + d*x]))/b])/(3*b^4*d)} -{1/(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^0, x, 8, -(Sqrt[1 - (c + d*x)^2]/(3*b*d*(a + b*ArcSin[c + d*x])^3)) + (c + d*x)/(6*b^2*d*(a + b*ArcSin[c + d*x])^2) + Sqrt[1 - (c + d*x)^2]/(6*b^3*d*(a + b*ArcSin[c + d*x])) - (CosIntegral[(a + b*ArcSin[c + d*x])/b]*Sin[a/b])/(6*b^4*d) + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(6*b^4*d)} -{1/(a + b*ArcSin[c + d*x])^4/(c*e + e*d*x)^1, x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^4), x]/e} - - -{1/(a + b*ArcSin[c + d*x])^5, x, 9, -(Sqrt[1 - (c + d*x)^2]/(4*b*d*(a + b*ArcSin[c + d*x])^4)) + (c + d*x)/(12*b^2*d*(a + b*ArcSin[c + d*x])^3) + Sqrt[1 - (c + d*x)^2]/(24*b^3*d*(a + b*ArcSin[c + d*x])^2) - (c + d*x)/(24*b^4*d*(a + b*ArcSin[c + d*x])) + (Cos[a/b]*CosIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^5*d) + (Sin[a/b]*SinIntegral[(a + b*ArcSin[c + d*x])/b])/(24*b^5*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSin[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^3*Sqrt[a + b*ArcSin[c + d*x]], x, 16, (-3*e^3*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) + (e^3*(c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) - (Sqrt[b]*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(64*d) + (Sqrt[b]*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(16*d) + (Sqrt[b]*e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(16*d) - (Sqrt[b]*e^3*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(64*d)} -{(c*e + d*e*x)^2*Sqrt[a + b*ArcSin[c + d*x]], x, 16, (e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(3*d) - (Sqrt[b]*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d) + (Sqrt[b]*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(12*d) + (Sqrt[b]*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d) - (Sqrt[b]*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*d)} -{(c*e + d*e*x)^1*Sqrt[a + b*ArcSin[c + d*x]], x, 11, -(e*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + (Sqrt[b]*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d) + (Sqrt[b]*e*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d)} -{(c*e + d*e*x)^0*Sqrt[a + b*ArcSin[c + d*x]], x, 8, ((c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/d - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/d + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/d} -{Sqrt[a + b*ArcSin[c + d*x]]/(c*e + d*e*x)^1, x, 2, Unintegrable[Sqrt[a + b*ArcSin[c + d*x]]/(c + d*x), x]/e} - - -{(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(3/2), x, 27, (9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(64*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (3*b^(3/2)*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(512*d) - (3*b^(3/2)*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(64*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(64*d) - (3*b^(3/2)*e^3*Sqrt[Pi/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(512*d)} -{(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(3/2), x, 24, (b*e^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(3*d) + (b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(3/2))/(3*d) - (3*b^(3/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*d) + (b^(3/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(24*d) - (3*b^(3/2)*e^2*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*d) + (b^(3/2)*e^2*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*d)} -{(c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(3/2), x, 13, (3*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (e*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d) + (3*b^(3/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*d)} -{(c*e + d*e*x)^0*(a + b*ArcSin[c + d*x])^(3/2), x, 9, (3*b*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/d - (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d)} -{(a + b*ArcSin[c + d*x])^(3/2)/(c*e + d*e*x)^1, x, 2, Unintegrable[(a + b*ArcSin[c + d*x])^(3/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2), x, 29, (225*b^2*e^3*Sqrt[a + b*ArcSin[c + d*x]])/(2048*d) - (45*b^2*e^3*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(256*d) - (15*b^2*e^3*(c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/(256*d) + (15*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(64*d) + (5*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) - (3*e^3*(a + b*ArcSin[c + d*x])^(5/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^(5/2))/(4*d) + (15*b^(5/2)*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4096*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(256*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(256*d) + (15*b^(5/2)*e^3*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(4096*d)} -{(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(5/2), x, 26, (-5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(6*d) - (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/(36*d) + (5*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(9*d) + (5*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(5/2))/(3*d) + (15*b^(5/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(16*d) - (5*b^(5/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(144*d) - (15*b^(5/2)*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(16*d) + (5*b^(5/2)*e^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*d)} -{(c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(5/2), x, 14, (15*b^2*e*Sqrt[a + b*ArcSin[c + d*x]])/(64*d) - (15*b^2*e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) + (5*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(8*d) - (e*(a + b*ArcSin[c + d*x])^(5/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) - (15*b^(5/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*d) - (15*b^(5/2)*e*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*d)} -{(c*e + d*e*x)^0*(a + b*ArcSin[c + d*x])^(5/2), x, 10, (-15*b^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(4*d) + (5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(5/2))/d + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*d) - (15*b^(5/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*d)} -{(a + b*ArcSin[c + d*x])^(5/2)/(c*e + d*e*x)^1, x, 2, Unintegrable[(a + b*ArcSin[c + d*x])^(5/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(7/2), x, 35, (-175*b^3*e^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(54*d) - (35*b^3*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(216*d) - (35*b^2*e^2*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) - (35*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(3/2))/(108*d) + (7*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(9*d) + (7*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(7/2))/(3*d) + (105*b^(7/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(32*d) - (35*b^(7/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(864*d) + (105*b^(7/2)*e^2*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(32*d) - (35*b^(7/2)*e^2*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(864*d)} -{(c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(7/2), x, 16, (-105*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(128*d) + (35*b^2*e*(a + b*ArcSin[c + d*x])^(3/2))/(64*d) - (35*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) + (7*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(8*d) - (e*(a + b*ArcSin[c + d*x])^(7/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(7/2))/(2*d) + (105*b^(7/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(512*d) - (105*b^(7/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(512*d)} -{(c*e + d*e*x)^0*(a + b*ArcSin[c + d*x])^(7/2), x, 11, (-105*b^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (35*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (7*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(7/2))/d + (105*b^(7/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*d) + (105*b^(7/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*d)} -{(a + b*ArcSin[c + d*x])^(7/2)/(c*e + d*e*x)^1, x, 2, Unintegrable[(a + b*ArcSin[c + d*x])^(7/2)/(c + d*x), x]/e} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^4/Sqrt[a + b*ArcSin[c + d*x]], x, 20, (e^4*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d) - (e^4*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^4*Sqrt[Pi/10]*Cos[(5*a)/b]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^4*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(4*Sqrt[b]*d) - (e^4*Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(8*Sqrt[b]*d) + (e^4*Sqrt[Pi/10]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(5*a)/b])/(8*Sqrt[b]*d)} -{(c*e + d*e*x)^3/Sqrt[a + b*ArcSin[c + d*x]], x, 15, -(e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(4*Sqrt[b]*d) - (e^3*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(4*Sqrt[b]*d) + (e^3*Sqrt[Pi/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(8*Sqrt[b]*d)} -{(c*e + d*e*x)^2/Sqrt[a + b*ArcSin[c + d*x]], x, 15, (e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d) - (e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d) + (e^2*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*d) - (e^2*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*d)} -{(c*e + d*e*x)^1/Sqrt[a + b*ArcSin[c + d*x]], x, 10, (e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(2*Sqrt[b]*d) - (e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(2*Sqrt[b]*d)} -{(c*e + d*e*x)^0/Sqrt[a + b*ArcSin[c + d*x]], x, 7, (Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d) + (Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d)} -{1/((c*e + d*e*x)^1*Sqrt[a + b*ArcSin[c + d*x]]), x, 2, Unintegrable[1/((c + d*x)*Sqrt[a + b*ArcSin[c + d*x]]), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^(3/2), x, 19, (-2*e^4*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^4*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d) + (3*e^4*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) - (e^4*Sqrt[(5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) + (e^4*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*b^(3/2)*d) - (3*e^4*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(4*b^(3/2)*d) + (e^4*Sqrt[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(5*a)/b])/(4*b^(3/2)*d)} -{(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(3/2), x, 14, (-2*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*d) + (e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*d) - (e^3*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(b^(3/2)*d)} -{(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(3/2), x, 14, (-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (e^2*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*d) - (e^2*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*d)} -{(c*e + d*e*x)^1/(a + b*ArcSin[c + d*x])^(3/2), x, 8, (-2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) + (2*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*d) + (2*e*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*d)} -{(c*e + d*e*x)^0/(a + b*ArcSin[c + d*x])^(3/2), x, 8, (-2*Sqrt[1 - (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSin[c + d*x]]) - (2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*d)} -{1/((c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(3/2)), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^(3/2)), x]/e} - - -{(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(5/2), x, 26, (-2*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (4*e^3*(c + d*x)^2)/(b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) + (16*e^3*(c + d*x)^4)/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) + (4*e^3*Sqrt[2*Pi]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) - (4*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d) + (4*e^3*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d) - (4*e^3*Sqrt[2*Pi]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(3*b^(5/2)*d)} -{(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(5/2), x, 24, (-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (8*e^2*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) + (4*e^2*(c + d*x)^3)/(b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (e^2*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(b^(5/2)*d) - (e^2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d) + (e^2*Sqrt[6*Pi]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*d)} -{(c*e + d*e*x)^1/(a + b*ArcSin[c + d*x])^(5/2), x, 13, (-2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (4*e)/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) + (8*e*(c + d*x)^2)/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (8*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d) + (8*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*d)} -{(c*e + d*e*x)^0/(a + b*ArcSin[c + d*x])^(5/2), x, 9, (-2*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) + (4*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (4*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) - (4*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*d)} -{1/((c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(5/2)), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^(5/2)), x]/e} - - -{(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2), x, 23, (-2*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) - (4*e^3*(c + d*x)^2)/(5*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (16*e^3*(c + d*x)^4)/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) - (16*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (128*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (32*e^3*Sqrt[2*Pi]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (16*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(15*b^(7/2)*d) - (16*e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(15*b^(7/2)*d) + (32*e^3*Sqrt[2*Pi]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(15*b^(7/2)*d)} -{(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(7/2), x, 24, (-2*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) - (8*e^2*(c + d*x))/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) - (16*e^2*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (24*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (2*e^2*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (6*e^2*Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d) - (2*e^2*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7/2)*d) + (6*e^2*Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(5*b^(7/2)*d)} -{(c*e + d*e*x)^1/(a + b*ArcSin[c + d*x])^(7/2), x, 11, (-2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) - (4*e)/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (8*e*(c + d*x)^2)/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (32*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) - (32*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(15*b^(7/2)*d) - (32*e*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(15*b^(7/2)*d)} -{(c*e + d*e*x)^0/(a + b*ArcSin[c + d*x])^(7/2), x, 10, (-2*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) + (4*(c + d*x))/(15*b^2*d*(a + b*ArcSin[c + d*x])^(3/2)) + (8*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (8*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (8*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7/2)*d)} -{1/((c*e + d*e*x)^1*(a + b*ArcSin[c + d*x])^(7/2)), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSin[c + d*x])^(7/2)), x]/e} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^(m/2) (a+b ArcSin[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]), x, 7, (28*b*e^2*(e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2])/(405*d) + (4*b*(e*(c + d*x))^(7/2)*Sqrt[1 - (c + d*x)^2])/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x]))/(9*d*e) + (28*b*e^3*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(135*d*Sqrt[c + d*x])} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSin[c + d*x]), x, 6, (20*b*e^2*Sqrt[e*(c + d*x)]*Sqrt[1 - (c + d*x)^2])/(147*d) + (4*b*(e*(c + d*x))^(5/2)*Sqrt[1 - (c + d*x)^2])/(49*d) + (2*(e*(c + d*x))^(7/2)*(a + b*ArcSin[c + d*x]))/(7*d*e) - (20*b*e^(5/2)*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(147*d)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSin[c + d*x]), x, 6, (4*b*(e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2])/(25*d) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcSin[c + d*x]))/(5*d*e) + (12*b*e*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(25*d*Sqrt[c + d*x])} -{(c*e + d*e*x)^(1/2)*(a + b*ArcSin[c + d*x]), x, 5, (4*b*Sqrt[e*(c + d*x)]*Sqrt[1 - (c + d*x)^2])/(9*d) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x]))/(3*d*e) - (4*b*Sqrt[e]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(9*d)} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(1/2), x, 5, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x]))/(d*e) + (4*b*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(d*e*Sqrt[c + d*x])} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(3/2), x, 4, -((2*(a + b*ArcSin[c + d*x]))/(d*e*Sqrt[e*(c + d*x)])) + (4*b*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(d*e^(3/2))} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(5/2), x, 6, -((4*b*Sqrt[1 - (c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)])) - (2*(a + b*ArcSin[c + d*x]))/(3*d*e*(e*(c + d*x))^(3/2)) + (4*b*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(3*d*e^3*Sqrt[c + d*x])} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(7/2), x, 5, -((4*b*Sqrt[1 - (c + d*x)^2])/(15*d*e^2*(e*(c + d*x))^(3/2))) - (2*(a + b*ArcSin[c + d*x]))/(5*d*e*(e*(c + d*x))^(5/2)) + (4*b*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(15*d*e^(7/2))} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(9/2), x, 7, -((4*b*Sqrt[1 - (c + d*x)^2])/(35*d*e^2*(e*(c + d*x))^(5/2))) - (12*b*Sqrt[1 - (c + d*x)^2])/(35*d*e^4*Sqrt[e*(c + d*x)]) - (2*(a + b*ArcSin[c + d*x]))/(7*d*e*(e*(c + d*x))^(7/2)) + (12*b*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 - c - d*x]/Sqrt[2]], 2])/(35*d*e^5*Sqrt[c + d*x])} -{(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2), x, 6, -((4*b*Sqrt[1 - (c + d*x)^2])/(63*d*e^2*(e*(c + d*x))^(7/2))) - (20*b*Sqrt[1 - (c + d*x)^2])/(189*d*e^4*(e*(c + d*x))^(3/2)) - (2*(a + b*ArcSin[c + d*x]))/(9*d*e*(e*(c + d*x))^(9/2)) + (20*b*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(189*d*e^(11/2))} - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x])^2)/(9*d*e) - (8*b*(e*(c + d*x))^(11/2)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 11/4, 15/4, (c + d*x)^2])/(99*d*e^2) + (16*b^2*(e*(c + d*x))^(13/2)*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4, 17/4}, (c + d*x)^2])/(1287*d*e^3)} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSin[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(7/2)*(a + b*ArcSin[c + d*x])^2)/(7*d*e) - (8*b*(e*(c + d*x))^(9/2)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 9/4, 13/4, (c + d*x)^2])/(63*d*e^2) + (16*b^2*(e*(c + d*x))^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, (c + d*x)^2])/(693*d*e^3)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSin[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(5/2)*(a + b*ArcSin[c + d*x])^2)/(5*d*e) - (8*b*(e*(c + d*x))^(7/2)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 7/4, 11/4, (c + d*x)^2])/(35*d*e^2) + (16*b^2*(e*(c + d*x))^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, (c + d*x)^2])/(315*d*e^3)} -{(c*e + d*e*x)^(1/2)*(a + b*ArcSin[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^2)/(3*d*e) - (8*b*(e*(c + d*x))^(5/2)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2])/(15*d*e^2) + (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2])/(105*d*e^3)} -{(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(1/2), x, 3, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^2)/(d*e) - (8*b*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(3*d*e^2) + (16*b^2*(e*(c + d*x))^(5/2)*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, (c + d*x)^2])/(15*d*e^3)} -{(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(3/2), x, 3, -((2*(a + b*ArcSin[c + d*x])^2)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2])/(d*e^2) - (16*b^2*(e*(c + d*x))^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, (c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(5/2), x, 3, -((2*(a + b*ArcSin[c + d*x])^2)/(3*d*e*(e*(c + d*x))^(3/2))) - (8*b*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[-(1/4), 1/2, 3/4, (c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)]) + (16*b^2*Sqrt[e*(c + d*x)]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, (c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(7/2), x, 3, -((2*(a + b*ArcSin[c + d*x])^2)/(5*d*e*(e*(c + d*x))^(5/2))) - (8*b*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[-(3/4), 1/2, 1/4, (c + d*x)^2])/(15*d*e^2*(e*(c + d*x))^(3/2)) - (16*b^2*HypergeometricPFQ[{-(1/4), -(1/4), 1}, {1/4, 3/4}, (c + d*x)^2])/(15*d*e^3*Sqrt[e*(c + d*x)])} -{(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(9/2), x, 3, -((2*(a + b*ArcSin[c + d*x])^2)/(7*d*e*(e*(c + d*x))^(7/2))) - (8*b*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[-(5/4), 1/2, -(1/4), (c + d*x)^2])/(35*d*e^2*(e*(c + d*x))^(5/2)) - (16*b^2*HypergeometricPFQ[{-(3/4), -(3/4), 1}, {-(1/4), 1/4}, (c + d*x)^2])/(105*d*e^3*(e*(c + d*x))^(3/2))} - - -{(c*e + d*e*x)^(1/2)*(a + b*ArcSin[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^3)/(3*d*e) - (2*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^2)/Sqrt[1 - (c + d*x)^2], x])/e} -{(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(1/2), x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^3)/(d*e) - (6*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^2)/Sqrt[1 - (c + d*x)^2], x])/e} -{(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcSin[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)])) + (6*b*Unintegrable[(a + b*ArcSin[c + d*x])^2/(Sqrt[e*(c + d*x)]*Sqrt[1 - (c + d*x)^2]), x])/e} -{(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcSin[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2))) + (2*b*Unintegrable[(a + b*ArcSin[c + d*x])^2/((e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2]), x])/e} - - -{(c*e + d*e*x)^(1/2)*(a + b*ArcSin[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^4)/(3*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcSin[c + d*x])^3)/Sqrt[1 - (c + d*x)^2], x])/(3*e)} -{(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(1/2), x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^4)/(d*e) - (8*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^3)/Sqrt[1 - (c + d*x)^2], x])/e} -{(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcSin[c + d*x])^4)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Unintegrable[(a + b*ArcSin[c + d*x])^3/(Sqrt[e*(c + d*x)]*Sqrt[1 - (c + d*x)^2]), x])/e} -{(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcSin[c + d*x])^4)/(3*d*e*(e*(c + d*x))^(3/2))) + (8*b*Unintegrable[(a + b*ArcSin[c + d*x])^3/((e*(c + d*x))^(3/2)*Sqrt[1 - (c + d*x)^2]), x])/(3*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSin[c+d x])^n with m symbolic*) - - -{(a + b*ArcSin[c + d*x])^4*(c*e + e*d*x)^m, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^4)/(d*e*(1 + m)) - (4*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^3)/Sqrt[1 - (c + d*x)^2], x])/(e*(1 + m))} -{(a + b*ArcSin[c + d*x])^3*(c*e + e*d*x)^m, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^3)/(d*e*(1 + m)) - (3*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^2)/Sqrt[1 - (c + d*x)^2], x])/(e*(1 + m))} -{(a + b*ArcSin[c + d*x])^2*(c*e + e*d*x)^m, x, 3, If[$VersionNumber>=8, ((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)) + (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(d*e^3*(1 + m)*(2 + m)*(3 + m)), ((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)) + (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(d*e^3*(3 + m)*(2 + 3*m + m^2))]} -{(a + b*ArcSin[c + d*x])^1*(c*e + e*d*x)^m, x, 3, ((e*(c + d*x))^(1 + m)*(a + b*ArcSin[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m))} -{1/(a + b*ArcSin[c + d*x])*(c*e + e*d*x)^m, x, 1, Unintegrable[(e*(c + d*x))^m/(a + b*ArcSin[c + d*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1-(a+b x)^2)^(m/2) ArcSin[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3, x, 7, (3*(a + b*x)^2)/(8*b) - (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(4*b) + (3*ArcSin[a + b*x]^2)/(8*b) - (3*(a + b*x)^2*ArcSin[a + b*x]^2)/(4*b) + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^3)/(2*b) + ArcSin[a + b*x]^4/(8*b)} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2, x, 6, -(((a + b*x)*Sqrt[1 - (a + b*x)^2])/(4*b)) + ArcSin[a + b*x]/(4*b) - ((a + b*x)^2*ArcSin[a + b*x])/(2*b) + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b) + ArcSin[a + b*x]^3/(6*b)} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x], x, 4, -(a + b*x)^2/(4*b) + ((a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(2*b) + ArcSin[a + b*x]^2/(4*b)} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x], x, 5, CosIntegral[2*ArcSin[a + b*x]]/(2*b) + Log[ArcSin[a + b*x]]/(2*b)} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^2, x, 6, -((1 - (a + b*x)^2)/(b*ArcSin[a + b*x])) - SinIntegral[2*ArcSin[a + b*x]]/b} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^3, x, 4, -(1 - (a + b*x)^2)/(2*b*ArcSin[a + b*x]^2) + ((a + b*x)*Sqrt[1 - (a + b*x)^2])/(b*ArcSin[a + b*x]) - CosIntegral[2*ArcSin[a + b*x]]/b} -{Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^4, x, 9, -((1 - (a + b*x)^2)/(3*b*ArcSin[a + b*x]^3)) + ((a + b*x)*Sqrt[1 - (a + b*x)^2])/(3*b*ArcSin[a + b*x]^2) + 1/(3*b*ArcSin[a + b*x]) - (2*(a + b*x)^2)/(3*b*ArcSin[a + b*x]) + (2*SinIntegral[2*ArcSin[a + b*x]])/(3*b)} - - -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3, x, 15, (51*(a + b*x)^2)/(128*b) - (3*(a + b*x)^4)/(128*b) - (45*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(64*b) - (3*(a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(32*b) + (27*ArcSin[a + b*x]^2)/(128*b) - (9*(a + b*x)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(1 - (a + b*x)^2)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^3)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^3)/(4*b) + (3*ArcSin[a + b*x]^4)/(32*b)} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2, x, 11, -((15*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(64*b)) - ((a + b*x)*(1 - (a + b*x)^2)^(3/2))/(32*b) + (9*ArcSin[a + b*x])/(64*b) - (3*(a + b*x)^2*ArcSin[a + b*x])/(8*b) + ((1 - (a + b*x)^2)^2*ArcSin[a + b*x])/(8*b) + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^2)/(4*b) + ArcSin[a + b*x]^3/(8*b)} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x], x, 7, -((5*(a + b*x)^2)/(16*b)) + (a + b*x)^4/(16*b) + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(4*b) + (3*ArcSin[a + b*x]^2)/(16*b)} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x], x, 6, CosIntegral[2*ArcSin[a + b*x]]/(2*b) + CosIntegral[4*ArcSin[a + b*x]]/(8*b) + (3*Log[ArcSin[a + b*x]])/(8*b)} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^2, x, 7, -((1 - (a + b*x)^2)^2/(b*ArcSin[a + b*x])) - SinIntegral[2*ArcSin[a + b*x]]/b - SinIntegral[4*ArcSin[a + b*x]]/(2*b)} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^3, x, 11, -((1 - (a + b*x)^2)^2/(2*b*ArcSin[a + b*x]^2)) + (2*(a + b*x)*(1 - (a + b*x)^2)^(3/2))/(b*ArcSin[a + b*x]) - CosIntegral[2*ArcSin[a + b*x]]/b - CosIntegral[4*ArcSin[a + b*x]]/b} -{(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^4, x, 18, -((1 - (a + b*x)^2)^2/(3*b*ArcSin[a + b*x]^3)) + (2*(a + b*x)*(1 - (a + b*x)^2)^(3/2))/(3*b*ArcSin[a + b*x]^2) + (2*(1 - (a + b*x)^2))/(3*b*ArcSin[a + b*x]) - (8*(a + b*x)^2*(1 - (a + b*x)^2))/(3*b*ArcSin[a + b*x]) + (2*SinIntegral[2*ArcSin[a + b*x]])/(3*b) + (4*SinIntegral[4*ArcSin[a + b*x]])/(3*b)} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{ArcSin[a + b*x]^n/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2], x, 2, ArcSin[a + b*x]^(n + 1)/(b*(n + 1))} - - -{ArcSin[a + b*x]^2/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2], x, 2, ArcSin[a + b*x]^3/(3*b)} -{ArcSin[a + b*x]^1/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2], x, 2, ArcSin[a + b*x]^2/(2*b)} -{1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^1), x, 2, Log[ArcSin[a + b*x]]/b} -{1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2), x, 2, -(1/(b*ArcSin[a + b*x]))} -{1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3), x, 2, -1/(2*b*ArcSin[a + b*x]^2)} - - -{ArcSin[a + b*x]^3/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2), x, 8, ((-I)*ArcSin[a + b*x]^3)/b + ((a + b*x)*ArcSin[a + b*x]^3)/(b*Sqrt[1 - (a + b*x)^2]) + (3*ArcSin[a + b*x]^2*Log[1 + E^((2*I)*ArcSin[a + b*x])])/b - ((3*I)*ArcSin[a + b*x]*PolyLog[2, -E^((2*I)*ArcSin[a + b*x])])/b + (3*PolyLog[3, -E^((2*I)*ArcSin[a + b*x])])/(2*b)} -{ArcSin[a + b*x]^2/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2), x, 7, ((-I)*ArcSin[a + b*x]^2)/b + ((a + b*x)*ArcSin[a + b*x]^2)/(b*Sqrt[1 - (a + b*x)^2]) + (2*ArcSin[a + b*x]*Log[1 + E^((2*I)*ArcSin[a + b*x])])/b - (I*PolyLog[2, -E^((2*I)*ArcSin[a + b*x])])/b} -{ArcSin[a + b*x]^1/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2), x, 3, ((a + b*x)*ArcSin[a + b*x])/(b*Sqrt[1 - (a + b*x)^2]) + Log[1 - (a + b*x)^2]/(2*b)} -{1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^1), x, 1, Unintegrable[1/((1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]), x]} -{1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2), x, 2, -(1/(b*(1 - (a + b*x)^2)*ArcSin[a + b*x])) + 2*Unintegrable[(a + b*x)/((1 - (a + b*x)^2)^2*ArcSin[a + b*x]), x]} - - -{ArcSin[a + b*x]/Sqrt[c - c*(a + b*x)^2], x, 2, (Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b*Sqrt[c - c*(a + b*x)^2])} -{ArcSin[a + b*x]/Sqrt[(1 - a^2)*c - 2*a*b*c*x - b^2*c*x^2], x, 2, (Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b*Sqrt[c - c*(a + b*x)^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcSin[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c+d x^n])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c x^n])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^9*(a + b*ArcSin[c*x^2]), x, 5, (b*Sqrt[1 - c^2*x^4])/(10*c^5) - (b*(1 - c^2*x^4)^(3/2))/(15*c^5) + (b*(1 - c^2*x^4)^(5/2))/(50*c^5) + (1/10)*x^10*(a + b*ArcSin[c*x^2])} -{x^7*(a + b*ArcSin[c*x^2]), x, 6, (3*b*x^2*Sqrt[1 - c^2*x^4])/(64*c^3) + (b*x^6*Sqrt[1 - c^2*x^4])/(32*c) - (3*b*ArcSin[c*x^2])/(64*c^4) + (1/8)*x^8*(a + b*ArcSin[c*x^2])} -{x^5*(a + b*ArcSin[c*x^2]), x, 5, (b*Sqrt[1 - c^2*x^4])/(6*c^3) - (b*(1 - c^2*x^4)^(3/2))/(18*c^3) + (1/6)*x^6*(a + b*ArcSin[c*x^2])} -{x^3*(a + b*ArcSin[c*x^2]), x, 5, (b*x^2*Sqrt[1 - c^2*x^4])/(8*c) - (b*ArcSin[c*x^2])/(8*c^2) + (1/4)*x^4*(a + b*ArcSin[c*x^2])} -{x^1*(a + b*ArcSin[c*x^2]), x, 4, (a*x^2)/2 + (b*Sqrt[1 - c^2*x^4])/(2*c) + (1/2)*b*x^2*ArcSin[c*x^2]} -{(a + b*ArcSin[c*x^2])/x^1, x, 7, (-(1/4))*I*b*ArcSin[c*x^2]^2 + (1/2)*b*ArcSin[c*x^2]*Log[1 - E^(2*I*ArcSin[c*x^2])] + a*Log[x] - (1/4)*I*b*PolyLog[2, E^(2*I*ArcSin[c*x^2])]} -{(a + b*ArcSin[c*x^2])/x^3, x, 5, -((a + b*ArcSin[c*x^2])/(2*x^2)) - (1/2)*b*c*ArcTanh[Sqrt[1 - c^2*x^4]]} -{(a + b*ArcSin[c*x^2])/x^5, x, 3, -((b*c*Sqrt[1 - c^2*x^4])/(4*x^2)) - (a + b*ArcSin[c*x^2])/(4*x^4)} -{(a + b*ArcSin[c*x^2])/x^7, x, 6, -((b*c*Sqrt[1 - c^2*x^4])/(12*x^4)) - (a + b*ArcSin[c*x^2])/(6*x^6) - (1/12)*b*c^3*ArcTanh[Sqrt[1 - c^2*x^4]]} -{(a + b*ArcSin[c*x^2])/x^9, x, 4, -((b*c*Sqrt[1 - c^2*x^4])/(24*x^6)) - (b*c^3*Sqrt[1 - c^2*x^4])/(12*x^2) - (a + b*ArcSin[c*x^2])/(8*x^8)} -{(a + b*ArcSin[c*x^2])/x^11, x, 7, -((b*c*Sqrt[1 - c^2*x^4])/(40*x^8)) - (3*b*c^3*Sqrt[1 - c^2*x^4])/(80*x^4) - (a + b*ArcSin[c*x^2])/(10*x^10) - (3/80)*b*c^5*ArcTanh[Sqrt[1 - c^2*x^4]]} -{(a + b*ArcSin[c*x^2])/x^13, x, 5, -((b*c*Sqrt[1 - c^2*x^4])/(60*x^10)) - (b*c^3*Sqrt[1 - c^2*x^4])/(45*x^6) - (2*b*c^5*Sqrt[1 - c^2*x^4])/(45*x^2) - (a + b*ArcSin[c*x^2])/(12*x^12)} - -{x^6*(a + b*ArcSin[c*x^2]), x, 5, (10*b*x*Sqrt[1 - c^2*x^4])/(147*c^3) + (2*b*x^5*Sqrt[1 - c^2*x^4])/(49*c) + (1/7)*x^7*(a + b*ArcSin[c*x^2]) - (10*b*EllipticF[ArcSin[Sqrt[c]*x], -1])/(147*c^(7/2))} -{x^4*(a + b*ArcSin[c*x^2]), x, 7, (2*b*x^3*Sqrt[1 - c^2*x^4])/(25*c) + (1/5)*x^5*(a + b*ArcSin[c*x^2]) - (6*b*EllipticE[ArcSin[Sqrt[c]*x], -1])/(25*c^(5/2)) + (6*b*EllipticF[ArcSin[Sqrt[c]*x], -1])/(25*c^(5/2))} -{x^2*(a + b*ArcSin[c*x^2]), x, 4, (2*b*x*Sqrt[1 - c^2*x^4])/(9*c) + (1/3)*x^3*(a + b*ArcSin[c*x^2]) - (2*b*EllipticF[ArcSin[Sqrt[c]*x], -1])/(9*c^(3/2))} -{x^0*(a + b*ArcSin[c*x^2]), x, 7, a*x + b*x*ArcSin[c*x^2] - (2*b*EllipticE[ArcSin[Sqrt[c]*x], -1])/Sqrt[c] + (2*b*EllipticF[ArcSin[Sqrt[c]*x], -1])/Sqrt[c]} -{(a + b*ArcSin[c*x^2])/x^2, x, 3, -((a + b*ArcSin[c*x^2])/x) + 2*b*Sqrt[c]*EllipticF[ArcSin[Sqrt[c]*x], -1]} -{(a + b*ArcSin[c*x^2])/x^4, x, 7, -((2*b*c*Sqrt[1 - c^2*x^4])/(3*x)) - (a + b*ArcSin[c*x^2])/(3*x^3) - (2/3)*b*c^(3/2)*EllipticE[ArcSin[Sqrt[c]*x], -1] + (2/3)*b*c^(3/2)*EllipticF[ArcSin[Sqrt[c]*x], -1]} -{(a + b*ArcSin[c*x^2])/x^6, x, 4, -((2*b*c*Sqrt[1 - c^2*x^4])/(15*x^3)) - (a + b*ArcSin[c*x^2])/(5*x^5) + (2/15)*b*c^(5/2)*EllipticF[ArcSin[Sqrt[c]*x], -1]} -{(a + b*ArcSin[c*x^2])/x^8, x, 8, -((2*b*c*Sqrt[1 - c^2*x^4])/(35*x^5)) - (6*b*c^3*Sqrt[1 - c^2*x^4])/(35*x) - (a + b*ArcSin[c*x^2])/(7*x^7) - (6/35)*b*c^(7/2)*EllipticE[ArcSin[Sqrt[c]*x], -1] + (6/35)*b*c^(7/2)*EllipticF[ArcSin[Sqrt[c]*x], -1]} - - -{ArcSin[a*x^5]/x, x, 5, (-(1/10))*I*ArcSin[a*x^5]^2 + (1/5)*ArcSin[a*x^5]*Log[1 - E^(2*I*ArcSin[a*x^5])] - (1/10)*I*PolyLog[2, E^(2*I*ArcSin[a*x^5])]} - - -{x^2*ArcSin[Sqrt[x]], x, 8, (5/48)*Sqrt[1 - x]*Sqrt[x] + (5/72)*Sqrt[1 - x]*x^(3/2) + (1/18)*Sqrt[1 - x]*x^(5/2) + (5/96)*ArcSin[1 - 2*x] + (1/3)*x^3*ArcSin[Sqrt[x]]} -{x^1*ArcSin[Sqrt[x]], x, 7, (3/16)*Sqrt[1 - x]*Sqrt[x] + (1/8)*Sqrt[1 - x]*x^(3/2) + (3/32)*ArcSin[1 - 2*x] + (1/2)*x^2*ArcSin[Sqrt[x]]} -{x^0*ArcSin[Sqrt[x]], x, 6, (1/2)*Sqrt[1 - x]*Sqrt[x] + (1/4)*ArcSin[1 - 2*x] + x*ArcSin[Sqrt[x]]} -{ArcSin[Sqrt[x]]/x^1, x, 5, (-I)*ArcSin[Sqrt[x]]^2 + 2*ArcSin[Sqrt[x]]*Log[1 - E^(2*I*ArcSin[Sqrt[x]])] - I*PolyLog[2, E^(2*I*ArcSin[Sqrt[x]])]} -{ArcSin[Sqrt[x]]/x^2, x, 3, -(Sqrt[1 - x]/Sqrt[x]) - ArcSin[Sqrt[x]]/x} -{ArcSin[Sqrt[x]]/x^3, x, 4, -(Sqrt[1 - x]/(6*x^(3/2))) - Sqrt[1 - x]/(3*Sqrt[x]) - ArcSin[Sqrt[x]]/(2*x^2)} -{ArcSin[Sqrt[x]]/x^4, x, 5, -(Sqrt[1 - x]/(15*x^(5/2))) - (4*Sqrt[1 - x])/(45*x^(3/2)) - (8*Sqrt[1 - x])/(45*Sqrt[x]) - ArcSin[Sqrt[x]]/(3*x^3)} -{ArcSin[Sqrt[x]]/x^5, x, 6, -(Sqrt[1 - x]/(28*x^(7/2))) - (3*Sqrt[1 - x])/(70*x^(5/2)) - (2*Sqrt[1 - x])/(35*x^(3/2)) - (4*Sqrt[1 - x])/(35*Sqrt[x]) - ArcSin[Sqrt[x]]/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4*(a + b*ArcSin[c/x]), x, 7, (3/40)*b*c^3*Sqrt[1 - c^2/x^2]*x^2 + (1/20)*b*c*Sqrt[1 - c^2/x^2]*x^4 + (1/5)*x^5*(a + b*ArcSin[c/x]) + (3/40)*b*c^5*ArcTanh[Sqrt[1 - c^2/x^2]]} -{x^3*(a + b*ArcSin[c/x]), x, 4, (1/6)*b*c^3*Sqrt[1 - c^2/x^2]*x + (1/12)*b*c*Sqrt[1 - c^2/x^2]*x^3 + (1/4)*x^4*(a + b*ArcSin[c/x])} -{x^2*(a + b*ArcSin[c/x]), x, 6, (1/6)*b*c*Sqrt[1 - c^2/x^2]*x^2 + (1/3)*x^3*(a + b*ArcSin[c/x]) + (1/6)*b*c^3*ArcTanh[Sqrt[1 - c^2/x^2]]} -{x^1*(a + b*ArcSin[c/x]), x, 3, (1/2)*b*c*Sqrt[1 - c^2/x^2]*x + (1/2)*x^2*(a + b*ArcSin[c/x])} -{x^0*(a + b*ArcSin[c/x]), x, 6, a*x + b*x*ArcCsc[x/c] + b*c*ArcTanh[Sqrt[1 - c^2/x^2]]} -{(a + b*ArcSin[c/x])/x^1, x, 7, (1/2)*I*b*ArcSin[c/x]^2 - b*ArcSin[c/x]*Log[1 - E^(2*I*ArcSin[c/x])] + a*Log[x] + (1/2)*I*b*PolyLog[2, E^(2*I*ArcSin[c/x])]} -{(a + b*ArcSin[c/x])/x^2, x, 4, -((b*Sqrt[1 - c^2/x^2])/c) - a/x - (b*ArcCsc[x/c])/x} -{(a + b*ArcSin[c/x])/x^3, x, 5, -((b*Sqrt[1 - c^2/x^2])/(4*c*x)) + (b*ArcCsc[x/c])/(4*c^2) - (a + b*ArcSin[c/x])/(2*x^2)} -{(a + b*ArcSin[c/x])/x^4, x, 5, -((b*Sqrt[1 - c^2/x^2])/(3*c^3)) + (b*(1 - c^2/x^2)^(3/2))/(9*c^3) - (a + b*ArcSin[c/x])/(3*x^3)} -{(a + b*ArcSin[c/x])/x^5, x, 6, -((b*Sqrt[1 - c^2/x^2])/(16*c*x^3)) - (3*b*Sqrt[1 - c^2/x^2])/(32*c^3*x) + (3*b*ArcCsc[x/c])/(32*c^4) - (a + b*ArcSin[c/x])/(4*x^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c x^n]) with n symbolic*) - - -{x^m*(a + b*ArcSin[c*x^n]), x, 3, (x^(1 + m)*(a + b*ArcSin[c*x^n]))/(1 + m) - (b*c*n*x^(1 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n)])/((1 + m)*(1 + m + n))} - - -{x^2*(a + b*ArcSin[c*x^n]), x, 3, (1/3)*x^3*(a + b*ArcSin[c*x^n]) - (b*c*n*x^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/(2*n), (3*(1 + n))/(2*n), c^2*x^(2*n)])/(3*(3 + n))} -{x^1*(a + b*ArcSin[c*x^n]), x, 3, (1/2)*x^2*(a + b*ArcSin[c*x^n]) - (b*c*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), (1/2)*(3 + 2/n), c^2*x^(2*n)])/(2*(2 + n))} -{x^0*(a + b*ArcSin[c*x^n]), x, 4, a*x + b*x*ArcSin[c*x^n] - (b*c*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (1/2)*(3 + 1/n), c^2*x^(2*n)])/(1 + n)} -{(a + b*ArcSin[c*x^n])/x^1, x, 7, -((I*b*ArcSin[c*x^n]^2)/(2*n)) + (b*ArcSin[c*x^n]*Log[1 - E^(2*I*ArcSin[c*x^n])])/n + a*Log[x] - (I*b*PolyLog[2, E^(2*I*ArcSin[c*x^n])])/(2*n)} -{(a + b*ArcSin[c*x^n])/x^2, x, 3, -((a + b*ArcSin[c*x^n])/x) - (b*c*n*x^(-1 + n)*Hypergeometric2F1[1/2, -((1 - n)/(2*n)), (1/2)*(3 - 1/n), c^2*x^(2*n)])/(1 - n)} -{(a + b*ArcSin[c*x^n])/x^3, x, 3, -((a + b*ArcSin[c*x^n])/(2*x^2)) - (b*c*n*x^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(1 - 2/n), (1/2)*(3 - 2/n), c^2*x^(2*n)])/(2*(2 - n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSin[c+d x^n])*) - - -{x^5*(a + b*ArcSin[c + d*x^2]), x, 7, (b*x^4*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(18*d) + (b*(4 + 11*c^2 - 5*c*d*x^2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(36*d^3) + (b*c*(3 + 2*c^2)*ArcSin[c + d*x^2])/(12*d^3) + (1/6)*x^6*(a + b*ArcSin[c + d*x^2])} -{x^3*(a + b*ArcSin[c + d*x^2]), x, 7, -((3*b*c*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(8*d^2)) + (b*x^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(8*d) - (b*(1 + 2*c^2)*ArcSin[c + d*x^2])/(8*d^2) + (1/4)*x^4*(a + b*ArcSin[c + d*x^2])} -{x^1*(a + b*ArcSin[c + d*x^2]), x, 5, (a*x^2)/2 + (b*Sqrt[1 - (c + d*x^2)^2])/(2*d) + (b*(c + d*x^2)*ArcSin[c + d*x^2])/(2*d)} -{(a + b*ArcSin[c + d*x^2])/x^1, x, 12, (-(1/4))*I*b*ArcSin[c + d*x^2]^2 + (1/2)*b*ArcSin[c + d*x^2]*Log[1 - E^(I*ArcSin[c + d*x^2])/(I*c - Sqrt[1 - c^2])] + (1/2)*b*ArcSin[c + d*x^2]*Log[1 - E^(I*ArcSin[c + d*x^2])/(I*c + Sqrt[1 - c^2])] + a*Log[x] - (1/2)*I*b*PolyLog[2, E^(I*ArcSin[c + d*x^2])/(I*c - Sqrt[1 - c^2])] - (1/2)*I*b*PolyLog[2, E^(I*ArcSin[c + d*x^2])/(I*c + Sqrt[1 - c^2])]} -{(a + b*ArcSin[c + d*x^2])/x^3, x, 5, -((a + b*ArcSin[c + d*x^2])/(2*x^2)) - (b*d*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(2*Sqrt[1 - c^2])} -{(a + b*ArcSin[c + d*x^2])/x^5, x, 6, -((b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(1 - c^2)*x^2)) - (a + b*ArcSin[c + d*x^2])/(4*x^4) - (b*c*d^2*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(4*(1 - c^2)^(3/2))} -{(a + b*ArcSin[c + d*x^2])/x^7, x, 7, -((b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(12*(1 - c^2)*x^4)) - (b*c*d^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(1 - c^2)^2*x^2) - (a + b*ArcSin[c + d*x^2])/(6*x^6) - (b*(1 + 2*c^2)*d^3*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(12*(1 - c^2)^(5/2))} - -{x^4*(a + b*ArcSin[c + d*x^2]), x, 8, -((16*b*c*x*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(75*d^2)) + (2*b*x^3*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(25*d) + (1/5)*x^5*(a + b*ArcSin[c + d*x^2]) - (2*b*Sqrt[1 - c]*(1 + c)*(9 + 23*c^2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(75*d^(5/2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) + (2*b*Sqrt[1 - c]*(1 + c)*(9 + 8*c + 15*c^2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(75*d^(5/2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])} -{x^2*(a + b*ArcSin[c + d*x^2]), x, 7, (2*b*x*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(9*d) + (1/3)*x^3*(a + b*ArcSin[c + d*x^2]) + (8*b*Sqrt[1 - c]*c*(1 + c)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(9*d^(3/2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) - (2*b*Sqrt[1 - c]*(1 + c)*(1 + 3*c)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(9*d^(3/2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])} -{x^0*(a + b*ArcSin[c + d*x^2]), x, 7, a*x + b*x*ArcSin[c + d*x^2] - (2*b*Sqrt[1 - c]*(1 + c)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(Sqrt[d]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) + (2*b*Sqrt[1 - c]*(1 + c)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(Sqrt[d]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])} -{(a + b*ArcSin[c + d*x^2])/x^2, x, 4, -((a + b*ArcSin[c + d*x^2])/x) + (2*b*Sqrt[1 - c]*Sqrt[d]*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]} -{(a + b*ArcSin[c + d*x^2])/x^4, x, 8, -((2*b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(3*(1 - c^2)*x)) - (a + b*ArcSin[c + d*x^2])/(3*x^3) - (2*b*d^(3/2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(3*Sqrt[1 - c]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) + (2*b*d^(3/2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(3*Sqrt[1 - c]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])} -{(a + b*ArcSin[c + d*x^2])/x^6, x, 8, -((2*b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(15*(1 - c^2)*x^3)) - (8*b*c*d^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(15*(1 - c^2)^2*x) - (a + b*ArcSin[c + d*x^2])/(5*x^5) - (8*b*c*d^(5/2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(15*Sqrt[1 - c]*(1 - c^2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) + (2*b*(1 + 3*c)*d^(5/2)*Sqrt[1 - (d*x^2)/(1 - c)]*Sqrt[1 + (d*x^2)/(1 + c)]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[1 - c]], -((1 - c)/(1 + c))])/(15*Sqrt[1 - c]*(1 - c^2)*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])} - - -{x^3*ArcSin[a + b*x^4], x, 4, Sqrt[1 - (a + b*x^4)^2]/(4*b) + ((a + b*x^4)*ArcSin[a + b*x^4])/(4*b)} - - -{x^(n-1)*ArcSin[a + b*x^n], x, 4, Sqrt[1 - (a + b*x^n)^2]/(b*n) + ((a + b*x^n)*ArcSin[a + b*x^n])/(b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b ArcSin[c+d x^2])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcSin[c+d x^2])^n when c^2=1*) - - -{(a + b*ArcSin[1 + d*x^2])^4, x, 3, 384*b^4*x - (192*b^3*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2]))/(d*x) - 48*b^2*x*(a + b*ArcSin[1 + d*x^2])^2 + (8*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2])^3)/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^4} -{(a + b*ArcSin[1 + d*x^2])^3, x, 5, -24*a*b^2*x - (48*b^3*Sqrt[-2*d*x^2 - d^2*x^4])/(d*x) - 24*b^3*x*ArcSin[1 + d*x^2] + (6*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2])^2)/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^3} -{(a + b*ArcSin[1 + d*x^2])^2, x, 2, -8*b^2*x + (4*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2]))/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^2} -{(a + b*ArcSin[1 + d*x^2])^1, x, 4, a*x + (2*b*Sqrt[-2*d*x^2 - d^2*x^4])/(d*x) + b*x*ArcSin[1 + d*x^2]} -{1/(a + b*ArcSin[1 + d*x^2])^1, x, 1, -((x*CosIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(2*b*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))) - (x*(Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)])/(2*b*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{1/(a + b*ArcSin[1 + d*x^2])^2, x, 1, -(Sqrt[-2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcSin[1 + d*x^2]))) - (x*CosIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(4*b^2*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) + (x*(Cos[a/(2*b)] - Sin[a/(2*b)])*SinIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)])/(4*b^2*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{1/(a + b*ArcSin[1 + d*x^2])^3, x, 2, -(Sqrt[-2*d*x^2 - d^2*x^4]/(4*b*d*x*(a + b*ArcSin[1 + d*x^2])^2)) + x/(8*b^2*(a + b*ArcSin[1 + d*x^2])) + (x*CosIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(16*b^3*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) + (x*(Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a + b*ArcSin[1 + d*x^2])/(2*b)])/(16*b^3*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} - - -{(a + b*ArcSin[-1 + d*x^2])^4, x, 3, 384*b^4*x - (192*b^3*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2]))/(d*x) - 48*b^2*x*(a - b*ArcSin[1 - d*x^2])^2 + (8*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2])^3)/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^4} -{(a + b*ArcSin[-1 + d*x^2])^3, x, 5, -24*a*b^2*x - (48*b^3*Sqrt[2*d*x^2 - d^2*x^4])/(d*x) + 24*b^3*x*ArcSin[1 - d*x^2] + (6*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2])^2)/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^3} -{(a + b*ArcSin[-1 + d*x^2])^2, x, 2, -8*b^2*x + (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^2} -{(a + b*ArcSin[-1 + d*x^2])^1, x, 4, a*x + (2*b*Sqrt[2*d*x^2 - d^2*x^4])/(d*x) - b*x*ArcSin[1 - d*x^2]} -{1/(a + b*ArcSin[-1 + d*x^2])^1, x, 1, (x*CosIntegral[-((a - b*ArcSin[1 - d*x^2])/(2*b))]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(2*b*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) - (x*(Cos[a/(2*b)] - Sin[a/(2*b)])*SinIntegral[a/(2*b) - (1/2)*ArcSin[1 - d*x^2]])/(2*b*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{1/(a + b*ArcSin[-1 + d*x^2])^2, x, 1, -(Sqrt[2*d*x^2 - d^2*x^4]/(2*b*d*x*(a - b*ArcSin[1 - d*x^2]))) - (x*CosIntegral[-((a - b*ArcSin[1 - d*x^2])/(2*b))]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(4*b^2*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) - (x*(Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[a/(2*b) - (1/2)*ArcSin[1 - d*x^2]])/(4*b^2*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{1/(a + b*ArcSin[-1 + d*x^2])^3, x, 2, -(Sqrt[2*d*x^2 - d^2*x^4]/(4*b*d*x*(a - b*ArcSin[1 - d*x^2])^2)) + x/(8*b^2*(a - b*ArcSin[1 - d*x^2])) - (x*CosIntegral[-((a - b*ArcSin[1 - d*x^2])/(2*b))]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(16*b^3*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) + (x*(Cos[a/(2*b)] - Sin[a/(2*b)])*SinIntegral[a/(2*b) - (1/2)*ArcSin[1 - d*x^2]])/(16*b^3*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} - - -{ArcSin[1 + x^2]^2, x, 2, -8*x + (4*Sqrt[-2*x^2 - x^4]*ArcSin[1 + x^2])/x + x*ArcSin[1 + x^2]^2} -{ArcSin[1 - x^2]^2, x, 2, -8*x - (4*Sqrt[2*x^2 - x^4]*ArcSin[1 - x^2])/x + x*ArcSin[1 - x^2]^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcSin[c+d x^2])^(n/2) when c^2=1*) - - -{(a + b*ArcSin[1 + d*x^2])^(5/2), x, 2, -15*b^2*x*Sqrt[a + b*ArcSin[1 + d*x^2]] + (5*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2])^(3/2))/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^(5/2) - (15*Sqrt[Pi]*x*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/((1/b)^(5/2)*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) + (15*Sqrt[Pi]*x*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/((1/b)^(5/2)*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{(a + b*ArcSin[1 + d*x^2])^(3/2), x, 2, (3*b*Sqrt[-2*d*x^2 - d^2*x^4]*Sqrt[a + b*ArcSin[1 + d*x^2]])/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^(3/2) + (3*b^(3/2)*Sqrt[Pi]*x*FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]) + (3*b^(3/2)*Sqrt[Pi]*x*FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])} -{(a + b*ArcSin[1 + d*x^2])^(1/2), x, 1, x*Sqrt[a + b*ArcSin[1 + d*x^2]] + (Sqrt[Pi]*x*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[1/b]*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) - (Sqrt[Pi]*x*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[1/b]*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{1/(a + b*ArcSin[1 + d*x^2])^(1/2), x, 1, -((Sqrt[Pi]*x*FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[b]*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))) - (Sqrt[Pi]*x*FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[b]*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{1/(a + b*ArcSin[1 + d*x^2])^(3/2), x, 1, -(Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSin[1 + d*x^2]])) + ((1/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]) - ((1/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])} -{1/(a + b*ArcSin[1 + d*x^2])^(5/2), x, 2, -(Sqrt[-2*d*x^2 - d^2*x^4]/(3*b*d*x*(a + b*ArcSin[1 + d*x^2])^(3/2))) + x/(3*b^2*Sqrt[a + b*ArcSin[1 + d*x^2]]) + (Sqrt[Pi]*x*FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(3*b^(5/2)*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) + (Sqrt[Pi]*x*FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(3*b^(5/2)*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} -{1/(a + b*ArcSin[1 + d*x^2])^(7/2), x, 2, -(Sqrt[-2*d*x^2 - d^2*x^4]/(5*b*d*x*(a + b*ArcSin[1 + d*x^2])^(5/2))) + x/(15*b^2*(a + b*ArcSin[1 + d*x^2])^(3/2)) + Sqrt[-2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a + b*ArcSin[1 + d*x^2]]) - ((1/b)^(7/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(15*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]])) + ((1/b)^(7/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(15*(Cos[(1/2)*ArcSin[1 + d*x^2]] - Sin[(1/2)*ArcSin[1 + d*x^2]]))} - - -{(a + b*ArcSin[-1 + d*x^2])^(5/2), x, 2, -15*b^2*x*Sqrt[a - b*ArcSin[1 - d*x^2]] + (5*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2])^(3/2))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^(5/2) + (15*Sqrt[Pi]*x*FresnelC[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/((-(1/b))^(5/2)*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) - (15*Sqrt[Pi]*x*FresnelS[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/((-(1/b))^(5/2)*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{(a + b*ArcSin[-1 + d*x^2])^(3/2), x, 2, (3*b*Sqrt[2*d*x^2 - d^2*x^4]*Sqrt[a - b*ArcSin[1 - d*x^2]])/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^(3/2) + (3*(-b)^(3/2)*Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]) + (3*(-b)^(3/2)*Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])} -{(a + b*ArcSin[-1 + d*x^2])^(1/2), x, 1, x*Sqrt[a - b*ArcSin[1 - d*x^2]] - (Sqrt[Pi]*x*FresnelC[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[-(1/b)]*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) + (Sqrt[Pi]*x*FresnelS[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-(1/b)]*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{1/(a + b*ArcSin[-1 + d*x^2])^(1/2), x, 1, -((Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))) - (Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{1/(a + b*ArcSin[-1 + d*x^2])^(3/2), x, 1, -(Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a - b*ArcSin[1 - d*x^2]])) - ((-(1/b))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]) + ((-(1/b))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])} -{1/(a + b*ArcSin[-1 + d*x^2])^(5/2), x, 2, -(Sqrt[2*d*x^2 - d^2*x^4]/(3*b*d*x*(a - b*ArcSin[1 - d*x^2])^(3/2))) + x/(3*b^2*Sqrt[a - b*ArcSin[1 - d*x^2]]) + (Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(3*(-b)^(5/2)*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) + (Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(3*(-b)^(5/2)*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} -{1/(a + b*ArcSin[-1 + d*x^2])^(7/2), x, 2, -(Sqrt[2*d*x^2 - d^2*x^4]/(5*b*d*x*(a - b*ArcSin[1 - d*x^2])^(5/2))) + x/(15*b^2*(a - b*ArcSin[1 - d*x^2])^(3/2)) + Sqrt[2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a - b*ArcSin[1 - d*x^2]]) + ((-(1/b))^(7/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(15*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]])) - ((-(1/b))^(7/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-(1/b)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(15*(Cos[(1/2)*ArcSin[1 - d*x^2]] - Sin[(1/2)*ArcSin[1 - d*x^2]]))} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcSin[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 8, (I*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4)/(4*b*c) - ((a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 - E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (3*I*b*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) - (3*b^2*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) - (3*I*b^3*PolyLog[4, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)} -{(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, (I*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3)/(3*b*c) - ((a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Log[1 - E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (I*b*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c - (b^2*PolyLog[3, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 6, (I*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2)/(2*b*c) - ((a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*Log[1 - E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (I*b*PolyLog[2, E^(2*I*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{1/((a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse sines of exponentials*) - - -{E^x*ArcSin[E^x], x, 3, Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]} - - -{ArcSin[c*E^(a + b*x)], x, 6, -((I*ArcSin[c*E^(a + b*x)]^2)/(2*b)) + (ArcSin[c*E^(a + b*x)]*Log[1 - E^(2*I*ArcSin[c*E^(a + b*x)])])/b - (I*PolyLog[2, E^(2*I*ArcSin[c*E^(a + b*x)])])/(2*b)} - - -(* ::Title::Closed:: *) -(*Integrands involving exponentials of inverse sines*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(ArcSin[a x]^n)*) - - -{E^ArcSin[a*x]*x^3, x, 6, -((E^ArcSin[a*x]*Cos[2*ArcSin[a*x]])/(10*a^4)) + (E^ArcSin[a*x]*Cos[4*ArcSin[a*x]])/(34*a^4) + (E^ArcSin[a*x]*Sin[2*ArcSin[a*x]])/(20*a^4) - (E^ArcSin[a*x]*Sin[4*ArcSin[a*x]])/(136*a^4)} -{E^ArcSin[a*x]*x^2, x, 6, (E^ArcSin[a*x]*x)/(8*a^2) + (E^ArcSin[a*x]*Sqrt[1 - a^2*x^2])/(8*a^3) - (E^ArcSin[a*x]*Cos[3*ArcSin[a*x]])/(40*a^3) - (3*E^ArcSin[a*x]*Sin[3*ArcSin[a*x]])/(40*a^3)} -{E^ArcSin[a*x]*x^1, x, 5, -((E^ArcSin[a*x]*Cos[2*ArcSin[a*x]])/(5*a^2)) + (E^ArcSin[a*x]*Sin[2*ArcSin[a*x]])/(10*a^2)} -{E^ArcSin[a*x]*x^0, x, 2, (1/2)*E^ArcSin[a*x]*x + (E^ArcSin[a*x]*Sqrt[1 - a^2*x^2])/(2*a)} -{E^ArcSin[a*x]/x^1, x, 6, I*E^ArcSin[a*x] - 2*I*E^ArcSin[a*x]*Hypergeometric2F1[-(I/2), 1, 1 - I/2, E^(2*I*ArcSin[a*x])]} -{E^ArcSin[a*x]/x^2, x, 6, (1 - I)*a*E^((1 + I)*ArcSin[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, E^(2*I*ArcSin[a*x])] - (2 - 2*I)*a*E^((1 + I)*ArcSin[a*x])*Hypergeometric2F1[1/2 - I/2, 2, 3/2 - I/2, E^(2*I*ArcSin[a*x])]} - - -{E^(ArcSin[a*x]^2)*x^3, x, 12, (I*E*Sqrt[Pi]*Erfi[-I + ArcSin[a*x]])/(16*a^4) - (I*E*Sqrt[Pi]*Erfi[I + ArcSin[a*x]])/(16*a^4) - (I*E^4*Sqrt[Pi]*Erfi[-2*I + ArcSin[a*x]])/(32*a^4) + (I*E^4*Sqrt[Pi]*Erfi[2*I + ArcSin[a*x]])/(32*a^4)} -{E^(ArcSin[a*x]^2)*x^2, x, 12, (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a*x])])/(16*a^3) + (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a*x])])/(16*a^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(-3*I + 2*ArcSin[a*x])])/(16*a^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(3*I + 2*ArcSin[a*x])])/(16*a^3)} -{E^(ArcSin[a*x]^2)*x^1, x, 8, (I*E*Sqrt[Pi]*Erfi[-I + ArcSin[a*x]])/(8*a^2) - (I*E*Sqrt[Pi]*Erfi[I + ArcSin[a*x]])/(8*a^2)} -{E^(ArcSin[a*x]^2)*x^0, x, 7, (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a*x])])/(4*a) + (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a*x])])/(4*a)} -{E^(ArcSin[a*x]^2)/x^1, x, 2, a*CannotIntegrate[E^ArcSin[a*x]^2/(a*x), x]} -{E^(ArcSin[a*x]^2)/x^2, x, 2, a^2*CannotIntegrate[E^ArcSin[a*x]^2/(a^2*x^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(ArcSin[a+b x]^n)*) - - -{E^ArcSin[a + b*x]*x^3, x, 17, -((3*a*E^ArcSin[a + b*x]*(a + b*x))/(8*b^4)) - (a^3*E^ArcSin[a + b*x]*(a + b*x))/(2*b^4) - (3*a*E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(8*b^4) - (a^3*E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(2*b^4) - (E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/(10*b^4) - (3*a^2*E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/(5*b^4) + (3*a*E^ArcSin[a + b*x]*Cos[3*ArcSin[a + b*x]])/(40*b^4) + (E^ArcSin[a + b*x]*Cos[4*ArcSin[a + b*x]])/(34*b^4) + (E^ArcSin[a + b*x]*Sin[2*ArcSin[a + b*x]])/(20*b^4) + (3*a^2*E^ArcSin[a + b*x]*Sin[2*ArcSin[a + b*x]])/(10*b^4) + (9*a*E^ArcSin[a + b*x]*Sin[3*ArcSin[a + b*x]])/(40*b^4) - (E^ArcSin[a + b*x]*Sin[4*ArcSin[a + b*x]])/(136*b^4)} -{E^ArcSin[a + b*x]*x^2, x, 13, (E^ArcSin[a + b*x]*(a + b*x))/(8*b^3) + (a^2*E^ArcSin[a + b*x]*(a + b*x))/(2*b^3) + (E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(8*b^3) + (a^2*E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(2*b^3) + (2*a*E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/(5*b^3) - (E^ArcSin[a + b*x]*Cos[3*ArcSin[a + b*x]])/(40*b^3) - (a*E^ArcSin[a + b*x]*Sin[2*ArcSin[a + b*x]])/(5*b^3) - (3*E^ArcSin[a + b*x]*Sin[3*ArcSin[a + b*x]])/(40*b^3)} -{E^ArcSin[a + b*x]*x^1, x, 9, -((a*E^ArcSin[a + b*x]*(a + b*x))/(2*b^2)) - (a*E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(2*b^2) - (E^ArcSin[a + b*x]*Cos[2*ArcSin[a + b*x]])/(5*b^2) + (E^ArcSin[a + b*x]*Sin[2*ArcSin[a + b*x]])/(10*b^2)} -{E^ArcSin[a + b*x]*x^0, x, 2, (E^ArcSin[a + b*x]*(a + b*x))/(2*b) + (E^ArcSin[a + b*x]*Sqrt[1 - (a + b*x)^2])/(2*b)} -{E^ArcSin[a + b*x]/x^1, x, 3, b*CannotIntegrate[E^ArcSin[a + b*x]/(b*x), x]} -{E^ArcSin[a + b*x]/x^2, x, 3, b^2*CannotIntegrate[E^ArcSin[a + b*x]/(b^2*x^2), x]} - - -{E^(ArcSin[a + b*x]^2)*x^3, x, 37, (I*E*Sqrt[Pi]*Erfi[-I + ArcSin[a + b*x]])/(16*b^4) + (3*I*a^2*E*Sqrt[Pi]*Erfi[-I + ArcSin[a + b*x]])/(8*b^4) - (I*E*Sqrt[Pi]*Erfi[I + ArcSin[a + b*x]])/(16*b^4) - (3*I*a^2*E*Sqrt[Pi]*Erfi[I + ArcSin[a + b*x]])/(8*b^4) - (I*E^4*Sqrt[Pi]*Erfi[-2*I + ArcSin[a + b*x]])/(32*b^4) + (I*E^4*Sqrt[Pi]*Erfi[2*I + ArcSin[a + b*x]])/(32*b^4) - (3*a*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(16*b^4) - (a^3*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(4*b^4) - (3*a*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(16*b^4) - (a^3*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(4*b^4) + (3*a*E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(-3*I + 2*ArcSin[a + b*x])])/(16*b^4) + (3*a*E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(3*I + 2*ArcSin[a + b*x])])/(16*b^4)} -{E^(ArcSin[a + b*x]^2)*x^2, x, 27, -((I*a*E*Sqrt[Pi]*Erfi[-I + ArcSin[a + b*x]])/(4*b^3)) + (I*a*E*Sqrt[Pi]*Erfi[I + ArcSin[a + b*x]])/(4*b^3) + (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(16*b^3) + (a^2*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(4*b^3) + (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(16*b^3) + (a^2*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(4*b^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(-3*I + 2*ArcSin[a + b*x])])/(16*b^3) - (E^(9/4)*Sqrt[Pi]*Erfi[(1/2)*(3*I + 2*ArcSin[a + b*x])])/(16*b^3)} -{E^(ArcSin[a + b*x]^2)*x^1, x, 17, (I*E*Sqrt[Pi]*Erfi[-I + ArcSin[a + b*x]])/(8*b^2) - (I*E*Sqrt[Pi]*Erfi[I + ArcSin[a + b*x]])/(8*b^2) - (a*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(4*b^2) - (a*E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(4*b^2)} -{E^(ArcSin[a + b*x]^2)*x^0, x, 7, (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(-I + 2*ArcSin[a + b*x])])/(4*b) + (E^(1/4)*Sqrt[Pi]*Erfi[(1/2)*(I + 2*ArcSin[a + b*x])])/(4*b)} -{E^(ArcSin[a + b*x]^2)/x^1, x, 3, b*CannotIntegrate[E^ArcSin[a + b*x]^2/(b*x), x]} -{E^(ArcSin[a + b*x]^2)/x^2, x, 3, b^2*CannotIntegrate[E^ArcSin[a + b*x]^2/(b^2*x^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1-a^2 x^2)^(m/2) E^(ArcSin[a x]^n)*) - - -{E^ArcSin[a*x]*(1 - a^2*x^2)^(5/2), x, 7, (144*E^ArcSin[a*x])/(629*a) + (144/629)*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2] + (72*E^ArcSin[a*x]*(1 - a^2*x^2))/(629*a) + (120/629)*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2) + (30*E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(629*a) + (6/37)*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(5/2) + (E^ArcSin[a*x]*(1 - a^2*x^2)^3)/(37*a)} -{E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2), x, 6, (24*E^ArcSin[a*x])/(85*a) + (24/85)*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2] + (12*E^ArcSin[a*x]*(1 - a^2*x^2))/(85*a) + (4/17)*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2) + (E^ArcSin[a*x]*(1 - a^2*x^2)^2)/(17*a)} -{E^ArcSin[a*x]*(1 - a^2*x^2)^(1/2), x, 5, (2*E^ArcSin[a*x])/(5*a) + (2/5)*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^2] + (E^ArcSin[a*x]*(1 - a^2*x^2))/(5*a)} -{E^ArcSin[a*x]/(1 - a^2*x^2)^(1/2), x, 4, E^ArcSin[a*x]/a} -{E^ArcSin[a*x]/(1 - a^2*x^2)^(3/2), x, 4, ((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^(2*I*ArcSin[a*x])])/a} -{E^ArcSin[a*x]/(1 - a^2*x^2)^(5/2), x, 5, (E^ArcSin[a*x]*x)/(3*(1 - a^2*x^2)^(3/2)) - E^ArcSin[a*x]/(6*a*(1 - a^2*x^2)) + ((2/3 - (4*I)/3)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^(2*I*ArcSin[a*x])])/a} - - -(* ::Title::Closed:: *) -(*Miscellaneous integrands involving inverse sines*) - - -{ArcSin[c/(a + b*x)], x, 6, ((a + b*x)*ArcCsc[a/c + (b*x)/c])/b + (c*ArcTanh[Sqrt[1 - c^2/(a + b*x)^2]])/b} - - -{x/ArcSin[Sin[x]], x, -1, ArcSin[Sin[x]] + Log[ArcSin[Sin[x]]]*(-ArcSin[Sin[x]] + x*Sqrt[Cos[x]^2]*Sec[x])} - - -{ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2], x, 2, (Sqrt[(-b)*x^2]*ArcSin[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)} -{1/(ArcSin[Sqrt[1 + b*x^2]]*Sqrt[1 + b*x^2]), x, 2, (Sqrt[(-b)*x^2]*Log[ArcSin[Sqrt[1 + b*x^2]]])/(b*x)} - - -(* Following integrands are equal. *) -{x/(1 - x^2) + 1/(Sqrt[1 - x^2]*ArcSin[x]), x, 3, (-(1/2))*Log[1 - x^2] + Log[ArcSin[x]]} -{(Sqrt[1 - x^2] + x*ArcSin[x])/(ArcSin[x] - x^2*ArcSin[x]), x, -1, (-(1/2))*Log[1 - x^2] + Log[ArcSin[x]]} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.2 (d x)^m (a+b arccos(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.2 (d x)^m (a+b arccos(c x))^n.m deleted file mode 100644 index f688ccf..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.2 (d x)^m (a+b arccos(c x))^n.m +++ /dev/null @@ -1,401 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcCos[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b x)^m ArcCos[a x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCos[a x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcCos[a*x], x, 4, -Sqrt[1 - a^2*x^2]/(5*a^5) + (2*(1 - a^2*x^2)^(3/2))/(15*a^5) - (1 - a^2*x^2)^(5/2)/(25*a^5) + (x^5*ArcCos[a*x])/5} -{x^3*ArcCos[a*x], x, 4, (-3*x*Sqrt[1 - a^2*x^2])/(32*a^3) - (x^3*Sqrt[1 - a^2*x^2])/(16*a) + (x^4*ArcCos[a*x])/4 + (3*ArcSin[a*x])/(32*a^4)} -{x^2*ArcCos[a*x], x, 4, -Sqrt[1 - a^2*x^2]/(3*a^3) + (1 - a^2*x^2)^(3/2)/(9*a^3) + (x^3*ArcCos[a*x])/3} -{x^1*ArcCos[a*x], x, 3, -(x*Sqrt[1 - a^2*x^2])/(4*a) + (x^2*ArcCos[a*x])/2 + ArcSin[a*x]/(4*a^2)} -{ArcCos[a*x], x, 2, -(Sqrt[1 - a^2*x^2]/a) + x*ArcCos[a*x]} -{ArcCos[a*x]/x^1, x, 5, (-I/2)*ArcCos[a*x]^2 + ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2)*PolyLog[2, -E^((2*I)*ArcCos[a*x])]} -{ArcCos[a*x]/x^2, x, 4, -(ArcCos[a*x]/x) + a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{ArcCos[a*x]/x^3, x, 2, (a*Sqrt[1 - a^2*x^2])/(2*x) - ArcCos[a*x]/(2*x^2)} -{ArcCos[a*x]/x^4, x, 5, (a*Sqrt[1 - a^2*x^2])/(6*x^2) - ArcCos[a*x]/(3*x^3) + (a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/6} -{ArcCos[a*x]/x^5, x, 3, (a*Sqrt[1 - a^2*x^2])/(12*x^3) + (a^3*Sqrt[1 - a^2*x^2])/(6*x) - ArcCos[a*x]/(4*x^4)} -{ArcCos[a*x]/x^6, x, 6, (a*Sqrt[1 - a^2*x^2])/(20*x^4) + (3*a^3*Sqrt[1 - a^2*x^2])/(40*x^2) - ArcCos[a*x]/(5*x^5) + (3*a^5*ArcTanh[Sqrt[1 - a^2*x^2]])/40} - - -{x^4*ArcCos[a*x]^2, x, 7, (-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (16*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(75*a^5) - (8*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(75*a^3) - (2*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(25*a) + (x^5*ArcCos[a*x]^2)/5} -{x^3*ArcCos[a*x]^2, x, 6, (-3*x^2)/(32*a^2) - x^4/32 - (3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(16*a^3) - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(8*a) - (3*ArcCos[a*x]^2)/(32*a^4) + (x^4*ArcCos[a*x]^2)/4} -{x^2*ArcCos[a*x]^2, x, 5, (-4*x)/(9*a^2) - (2*x^3)/27 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(9*a^3) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(9*a) + (x^3*ArcCos[a*x]^2)/3} -{x*ArcCos[a*x]^2, x, 4, -x^2/4 - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a) - ArcCos[a*x]^2/(4*a^2) + (x^2*ArcCos[a*x]^2)/2} -{ArcCos[a*x]^2, x, 3, -2*x - (2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a + x*ArcCos[a*x]^2} -{ArcCos[a*x]^2/x, x, 6, (-I/3)*ArcCos[a*x]^3 + ArcCos[a*x]^2*Log[1 + E^((2*I)*ArcCos[a*x])] - I*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + PolyLog[3, -E^((2*I)*ArcCos[a*x])]/2} -{ArcCos[a*x]^2/x^2, x, 7, -(ArcCos[a*x]^2/x) - (4*I)*a*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + (2*I)*a*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (2*I)*a*PolyLog[2, I*E^(I*ArcCos[a*x])]} -{ArcCos[a*x]^2/x^3, x, 3, (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/x - ArcCos[a*x]^2/(2*x^2) + a^2*Log[x]} -{ArcCos[a*x]^2/x^4, x, 9, -a^2/(3*x) + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(3*x^2) - ArcCos[a*x]^2/(3*x^3) - ((2*I)/3)*a^3*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] + (I/3)*a^3*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (I/3)*a^3*PolyLog[2, I*E^(I*ArcCos[a*x])]} -{ArcCos[a*x]^2/x^5, x, 5, -a^2/(12*x^2) + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(6*x^3) + (a^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(3*x) - ArcCos[a*x]^2/(4*x^4) + (a^4*Log[x])/3} - - -{x^4*ArcCos[a*x]^3, x, 14, (298*Sqrt[1 - a^2*x^2])/(375*a^5) - (76*(1 - a^2*x^2)^(3/2))/(1125*a^5) + (6*(1 - a^2*x^2)^(5/2))/(625*a^5) - (16*x*ArcCos[a*x])/(25*a^4) - (8*x^3*ArcCos[a*x])/(75*a^2) - (6*x^5*ArcCos[a*x])/125 - (8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(25*a) + (x^5*ArcCos[a*x]^3)/5} -{x^3*ArcCos[a*x]^3, x, 11, (45*x*Sqrt[1 - a^2*x^2])/(256*a^3) + (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) - (9*x^2*ArcCos[a*x])/(32*a^2) - (3*x^4*ArcCos[a*x])/32 - (9*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(16*a) - (3*ArcCos[a*x]^3)/(32*a^4) + (x^4*ArcCos[a*x]^3)/4 - (45*ArcSin[a*x])/(256*a^4)} -{x^2*ArcCos[a*x]^3, x, 9, (14*Sqrt[1 - a^2*x^2])/(9*a^3) - (2*(1 - a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcCos[a*x])/(3*a^2) - (2*x^3*ArcCos[a*x])/9 - (2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(3*a^3) - (x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(3*a) + (x^3*ArcCos[a*x]^3)/3} -{x*ArcCos[a*x]^3, x, 6, (3*x*Sqrt[1 - a^2*x^2])/(8*a) - (3*x^2*ArcCos[a*x])/4 - (3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(4*a) - ArcCos[a*x]^3/(4*a^2) + (x^2*ArcCos[a*x]^3)/2 - (3*ArcSin[a*x])/(8*a^2)} -{ArcCos[a*x]^3, x, 4, (6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcCos[a*x] - (3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a + x*ArcCos[a*x]^3} -{ArcCos[a*x]^3/x, x, 7, (-I/4)*ArcCos[a*x]^4 + ArcCos[a*x]^3*Log[1 + E^((2*I)*ArcCos[a*x])] - ((3*I)/2)*ArcCos[a*x]^2*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + (3*ArcCos[a*x]*PolyLog[3, -E^((2*I)*ArcCos[a*x])])/2 + ((3*I)/4)*PolyLog[4, -E^((2*I)*ArcCos[a*x])]} -{ArcCos[a*x]^3/x^2, x, 9, -(ArcCos[a*x]^3/x) - (6*I)*a*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + (6*I)*a*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (6*I)*a*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - 6*a*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + 6*a*PolyLog[3, I*E^(I*ArcCos[a*x])]} -{ArcCos[a*x]^3/x^3, x, 7, ((-3*I)/2)*a^2*ArcCos[a*x]^2 + (3*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(2*x) - ArcCos[a*x]^3/(2*x^2) + 3*a^2*ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - ((3*I)/2)*a^2*PolyLog[2, -E^((2*I)*ArcCos[a*x])]} -{ArcCos[a*x]^3/x^4, x, 14, -((a^2*ArcCos[a*x])/x) + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(2*x^2) - ArcCos[a*x]^3/(3*x^3) - I*a^3*ArcCos[a*x]^2*ArcTan[E^(I*ArcCos[a*x])] + a^3*ArcTanh[Sqrt[1 - a^2*x^2]] + I*a^3*ArcCos[a*x]*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - I*a^3*ArcCos[a*x]*PolyLog[2, I*E^(I*ArcCos[a*x])] - a^3*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + a^3*PolyLog[3, I*E^(I*ArcCos[a*x])]} -{ArcCos[a*x]^3/x^5, x, 10, (a^3*Sqrt[1 - a^2*x^2])/(4*x) - (a^2*ArcCos[a*x])/(4*x^2) - (I/2)*a^4*ArcCos[a*x]^2 + (a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(4*x^3) + (a^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(2*x) - ArcCos[a*x]^3/(4*x^4) + a^4*ArcCos[a*x]*Log[1 + E^((2*I)*ArcCos[a*x])] - (I/2)*a^4*PolyLog[2, -E^((2*I)*ArcCos[a*x])]} - - -{x^5*ArcCos[a*x]^4, x, 23, (245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 + (245*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(576*a^5) + (65*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(864*a^3) + (x^5*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(54*a) + (245*ArcCos[a*x]^2)/(1152*a^6) - (5*x^2*ArcCos[a*x]^2)/(16*a^4) - (5*x^4*ArcCos[a*x]^2)/(48*a^2) - (x^6*ArcCos[a*x]^2)/18 - (5*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(24*a^5) - (5*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(36*a^3) - (x^5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9*a) - (5*ArcCos[a*x]^4)/(96*a^6) + (x^6*ArcCos[a*x]^4)/6} -{x^4*ArcCos[a*x]^4, x, 19, (16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 + (16576*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(5625*a^5) + (1088*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(5625*a^3) + (24*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(625*a) - (32*x*ArcCos[a*x]^2)/(25*a^4) - (16*x^3*ArcCos[a*x]^2)/(75*a^2) - (12*x^5*ArcCos[a*x]^2)/125 - (32*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(75*a^5) - (16*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(25*a) + (x^5*ArcCos[a*x]^4)/5} -{x^3*ArcCos[a*x]^4, x, 14, (45*x^2)/(128*a^2) + (3*x^4)/128 + (45*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(64*a^3) + (3*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(32*a) + (45*ArcCos[a*x]^2)/(128*a^4) - (9*x^2*ArcCos[a*x]^2)/(16*a^2) - (3*x^4*ArcCos[a*x]^2)/16 - (3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(8*a^3) - (x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(4*a) - (3*ArcCos[a*x]^4)/(32*a^4) + (x^4*ArcCos[a*x]^4)/4} -{x^2*ArcCos[a*x]^4, x, 11, (160*x)/(27*a^2) + (8*x^3)/81 + (160*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (8*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a) - (8*x*ArcCos[a*x]^2)/(3*a^2) - (4*x^3*ArcCos[a*x]^2)/9 - (8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9*a) + (x^3*ArcCos[a*x]^4)/3} -{x*ArcCos[a*x]^4, x, 7, (3*x^2)/4 + (3*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(2*a) + (3*ArcCos[a*x]^2)/(4*a^2) - (3*x^2*ArcCos[a*x]^2)/2 - (x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a - ArcCos[a*x]^4/(4*a^2) + (x^2*ArcCos[a*x]^4)/2} -{ArcCos[a*x]^4, x, 5, 24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*ArcCos[a*x]^4} -{ArcCos[a*x]^4/x, x, 8, (-I/5)*ArcCos[a*x]^5 + ArcCos[a*x]^4*Log[1 + E^((2*I)*ArcCos[a*x])] - (2*I)*ArcCos[a*x]^3*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + 3*ArcCos[a*x]^2*PolyLog[3, -E^((2*I)*ArcCos[a*x])] + (3*I)*ArcCos[a*x]*PolyLog[4, -E^((2*I)*ArcCos[a*x])] - (3*PolyLog[5, -E^((2*I)*ArcCos[a*x])])/2} -{ArcCos[a*x]^4/x^2, x, 11, -(ArcCos[a*x]^4/x) - (8*I)*a*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (12*I)*a*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (12*I)*a*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 24*a*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + 24*a*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (24*I)*a*PolyLog[4, (-I)*E^(I*ArcCos[a*x])] + (24*I)*a*PolyLog[4, I*E^(I*ArcCos[a*x])]} -{ArcCos[a*x]^4/x^3, x, 8, (-2*I)*a^2*ArcCos[a*x]^3 + (2*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/x - ArcCos[a*x]^4/(2*x^2) + 6*a^2*ArcCos[a*x]^2*Log[1 + E^((2*I)*ArcCos[a*x])] - (6*I)*a^2*ArcCos[a*x]*PolyLog[2, -E^((2*I)*ArcCos[a*x])] + 3*a^2*PolyLog[3, -E^((2*I)*ArcCos[a*x])]} -{ArcCos[a*x]^4/x^4, x, 19, (-2*a^2*ArcCos[a*x]^2)/x + (2*a*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(3*x^2) - ArcCos[a*x]^4/(3*x^3) - (8*I)*a^3*ArcCos[a*x]*ArcTan[E^(I*ArcCos[a*x])] - ((4*I)/3)*a^3*ArcCos[a*x]^3*ArcTan[E^(I*ArcCos[a*x])] + (4*I)*a^3*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog[2, I*E^(I*ArcCos[a*x])] - (2*I)*a^3*ArcCos[a*x]^2*PolyLog[2, I*E^(I*ArcCos[a*x])] - 4*a^3*ArcCos[a*x]*PolyLog[3, (-I)*E^(I*ArcCos[a*x])] + 4*a^3*ArcCos[a*x]*PolyLog[3, I*E^(I*ArcCos[a*x])] - (4*I)*a^3*PolyLog[4, (-I)*E^(I*ArcCos[a*x])] + (4*I)*a^3*PolyLog[4, I*E^(I*ArcCos[a*x])]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^6/ArcCos[a*x], x, 7, (-5*SinIntegral[ArcCos[a*x]])/(64*a^7) - (9*SinIntegral[3*ArcCos[a*x]])/(64*a^7) - (5*SinIntegral[5*ArcCos[a*x]])/(64*a^7) - SinIntegral[7*ArcCos[a*x]]/(64*a^7)} -{x^5/ArcCos[a*x], x, 6, (-5*SinIntegral[2*ArcCos[a*x]])/(32*a^6) - SinIntegral[4*ArcCos[a*x]]/(8*a^6) - SinIntegral[6*ArcCos[a*x]]/(32*a^6)} -{x^4/ArcCos[a*x], x, 6, -SinIntegral[ArcCos[a*x]]/(8*a^5) - (3*SinIntegral[3*ArcCos[a*x]])/(16*a^5) - SinIntegral[5*ArcCos[a*x]]/(16*a^5)} -{x^3/ArcCos[a*x], x, 5, -SinIntegral[2*ArcCos[a*x]]/(4*a^4) - SinIntegral[4*ArcCos[a*x]]/(8*a^4)} -{x^2/ArcCos[a*x], x, 5, -SinIntegral[ArcCos[a*x]]/(4*a^3) - SinIntegral[3*ArcCos[a*x]]/(4*a^3)} -{x/ArcCos[a*x], x, 4, -SinIntegral[2*ArcCos[a*x]]/(2*a^2)} -{ArcCos[a*x]^(-1), x, 2, -(SinIntegral[ArcCos[a*x]]/a)} -{1/(x*ArcCos[a*x]), x, 0, Unintegrable[1/(x*ArcCos[a*x]), x]} -{1/(x^2*ArcCos[a*x]), x, 0, Unintegrable[1/(x^2*ArcCos[a*x]), x]} - - -{x^6/ArcCos[a*x]^2, x, 6, (x^6*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - (5*CosIntegral[ArcCos[a*x]])/(64*a^7) - (27*CosIntegral[3*ArcCos[a*x]])/(64*a^7) - (25*CosIntegral[5*ArcCos[a*x]])/(64*a^7) - (7*CosIntegral[7*ArcCos[a*x]])/(64*a^7)} -{x^5/ArcCos[a*x]^2, x, 5, (x^5*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - (5*CosIntegral[2*ArcCos[a*x]])/(16*a^6) - CosIntegral[4*ArcCos[a*x]]/(2*a^6) - (3*CosIntegral[6*ArcCos[a*x]])/(16*a^6)} -{x^4/ArcCos[a*x]^2, x, 5, (x^4*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/(8*a^5) - (9*CosIntegral[3*ArcCos[a*x]])/(16*a^5) - (5*CosIntegral[5*ArcCos[a*x]])/(16*a^5)} -{x^3/ArcCos[a*x]^2, x, 4, (x^3*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[2*ArcCos[a*x]]/(2*a^4) - CosIntegral[4*ArcCos[a*x]]/(2*a^4)} -{x^2/ArcCos[a*x]^2, x, 4, (x^2*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/(4*a^3) - (3*CosIntegral[3*ArcCos[a*x]])/(4*a^3)} -{x/ArcCos[a*x]^2, x, 2, (x*Sqrt[1 - a^2*x^2])/(a*ArcCos[a*x]) - CosIntegral[2*ArcCos[a*x]]/a^2} -{ArcCos[a*x]^(-2), x, 3, Sqrt[1 - a^2*x^2]/(a*ArcCos[a*x]) - CosIntegral[ArcCos[a*x]]/a} -{1/(x*ArcCos[a*x]^2), x, 0, Unintegrable[1/(x*ArcCos[a*x]^2), x]} -{1/(x^2*ArcCos[a*x]^2), x, 0, Unintegrable[1/(x^2*ArcCos[a*x]^2), x]} - - -{x^4/ArcCos[a*x]^3, x, 14, (x^4*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - (2*x^3)/(a^2*ArcCos[a*x]) + (5*x^5)/(2*ArcCos[a*x]) + SinIntegral[ArcCos[a*x]]/(16*a^5) + (27*SinIntegral[3*ArcCos[a*x]])/(32*a^5) + (25*SinIntegral[5*ArcCos[a*x]])/(32*a^5)} -{x^3/ArcCos[a*x]^3, x, 12, (x^3*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - (3*x^2)/(2*a^2*ArcCos[a*x]) + (2*x^4)/ArcCos[a*x] + SinIntegral[2*ArcCos[a*x]]/(2*a^4) + SinIntegral[4*ArcCos[a*x]]/a^4} -{x^2/ArcCos[a*x]^3, x, 10, (x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - x/(a^2*ArcCos[a*x]) + (3*x^3)/(2*ArcCos[a*x]) + SinIntegral[ArcCos[a*x]]/(8*a^3) + (9*SinIntegral[3*ArcCos[a*x]])/(8*a^3)} -{x/ArcCos[a*x]^3, x, 7, (x*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]^2) - 1/(2*a^2*ArcCos[a*x]) + x^2/ArcCos[a*x] + SinIntegral[2*ArcCos[a*x]]/a^2} -{ArcCos[a*x]^(-3), x, 4, Sqrt[1 - a^2*x^2]/(2*a*ArcCos[a*x]^2) + x/(2*ArcCos[a*x]) + SinIntegral[ArcCos[a*x]]/(2*a)} -{1/(x*ArcCos[a*x]^3), x, 0, Unintegrable[1/(x*ArcCos[a*x]^3), x]} -{1/(x^2*ArcCos[a*x]^3), x, 0, Unintegrable[1/(x^2*ArcCos[a*x]^3), x]} - - -{x^4/ArcCos[a*x]^4, x, 12, (x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - (2*x^3)/(3*a^2*ArcCos[a*x]^2) + (5*x^5)/(6*ArcCos[a*x]^2) + (2*x^2*Sqrt[1 - a^2*x^2])/(a^3*ArcCos[a*x]) - (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcCos[a*x]) + CosIntegral[ArcCos[a*x]]/(48*a^5) + (27*CosIntegral[3*ArcCos[a*x]])/(32*a^5) + (125*CosIntegral[5*ArcCos[a*x]])/(96*a^5)} -{x^3/ArcCos[a*x]^4, x, 9, (x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - x^2/(2*a^2*ArcCos[a*x]^2) + (2*x^4)/(3*ArcCos[a*x]^2) + (x*Sqrt[1 - a^2*x^2])/(a^3*ArcCos[a*x]) - (8*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]) + CosIntegral[2*ArcCos[a*x]]/(3*a^4) + (4*CosIntegral[4*ArcCos[a*x]])/(3*a^4)} -{x^2/ArcCos[a*x]^4, x, 10, (x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - x/(3*a^2*ArcCos[a*x]^2) + x^3/(2*ArcCos[a*x]^2) + Sqrt[1 - a^2*x^2]/(3*a^3*ArcCos[a*x]) - (3*x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcCos[a*x]) + CosIntegral[ArcCos[a*x]]/(24*a^3) + (9*CosIntegral[3*ArcCos[a*x]])/(8*a^3)} -{x/ArcCos[a*x]^4, x, 5, (x*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^3) - 1/(6*a^2*ArcCos[a*x]^2) + x^2/(3*ArcCos[a*x]^2) - (2*x*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]) + (2*CosIntegral[2*ArcCos[a*x]])/(3*a^2)} -{ArcCos[a*x]^(-4), x, 5, Sqrt[1 - a^2*x^2]/(3*a*ArcCos[a*x]^3) + x/(6*ArcCos[a*x]^2) - Sqrt[1 - a^2*x^2]/(6*a*ArcCos[a*x]) + CosIntegral[ArcCos[a*x]]/(6*a)} -{1/(x*ArcCos[a*x]^4), x, 0, Unintegrable[1/(x*ArcCos[a*x]^4), x]} -{1/(x^2*ArcCos[a*x]^4), x, 0, Unintegrable[1/(x^2*ArcCos[a*x]^4), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCos[a x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Sqrt[ArcCos[a*x]], x, 10, (x^5*Sqrt[ArcCos[a*x]])/5 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5) - (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^5) - (Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(80*a^5)} -{x^3*Sqrt[ArcCos[a*x]], x, 8, (-3*Sqrt[ArcCos[a*x]])/(32*a^4) + (x^4*Sqrt[ArcCos[a*x]])/4 - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(64*a^4) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(16*a^4)} -{x^2*Sqrt[ArcCos[a*x]], x, 8, (x^3*Sqrt[ArcCos[a*x]])/3 - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^3) - (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(12*a^3)} -{x*Sqrt[ArcCos[a*x]], x, 6, -Sqrt[ArcCos[a*x]]/(4*a^2) + (x^2*Sqrt[ArcCos[a*x]])/2 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(8*a^2)} -{Sqrt[ArcCos[a*x]], x, 4, x*Sqrt[ArcCos[a*x]] - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a} -{Sqrt[ArcCos[a*x]]/x, x, 0, Unintegrable[Sqrt[ArcCos[a*x]]/x, x]} - - -{x^4*ArcCos[a*x]^(3/2), x, 23, -((4*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(25*a^5)) - (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(25*a^3) - (3*x^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(50*a) + (1/5)*x^5*ArcCos[a*x]^(3/2) + (11*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(400*a^5) + (2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(25*a^5) + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(50*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(800*a^5) + (3*Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(800*a^5)} -{x^3*ArcCos[a*x]^(3/2), x, 16, (-9*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(64*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(32*a) - (3*ArcCos[a*x]^(3/2))/(32*a^4) + (x^4*ArcCos[a*x]^(3/2))/4 + (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(512*a^4) + (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(64*a^4)} -{x^2*ArcCos[a*x]^(3/2), x, 13, -(Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(3*a^3) - (x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(6*a) + (x^3*ArcCos[a*x]^(3/2))/3 + (3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^3) + (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(24*a^3)} -{x*ArcCos[a*x]^(3/2), x, 8, (-3*x*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(8*a) - ArcCos[a*x]^(3/2)/(4*a^2) + (x^2*ArcCos[a*x]^(3/2))/2 + (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(32*a^2)} -{ArcCos[a*x]^(3/2), x, 5, (-3*Sqrt[1 - a^2*x^2]*Sqrt[ArcCos[a*x]])/(2*a) + x*ArcCos[a*x]^(3/2) + (3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(2*a)} -{ArcCos[a*x]^(3/2)/x, x, 0, Unintegrable[ArcCos[a*x]^(3/2)/x, x]} - - -{x^4*ArcCos[a*x]^(5/2), x, 26, -((2*x*Sqrt[ArcCos[a*x]])/(5*a^4)) - (x^3*Sqrt[ArcCos[a*x]])/(15*a^2) - (3/100)*x^5*Sqrt[ArcCos[a*x]] - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^5) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(10*a) + (1/5)*x^5*ArcCos[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(32*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(60*a^5) + (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(320*a^5) + (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(1600*a^5)} -{x^3*ArcCos[a*x]^(5/2), x, 18, (225*Sqrt[ArcCos[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcCos[a*x]])/(256*a^2) - (15*x^4*Sqrt[ArcCos[a*x]])/256 - (15*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(64*a^3) - (5*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(32*a) - (3*ArcCos[a*x]^(5/2))/(32*a^4) + (x^4*ArcCos[a*x]^(5/2))/4 + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4096*a^4) + (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(256*a^4)} -{x^2*ArcCos[a*x]^(5/2), x, 15, (-5*x*Sqrt[ArcCos[a*x]])/(6*a^2) - (5*x^3*Sqrt[ArcCos[a*x]])/36 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(9*a^3) - (5*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(18*a) + (x^3*ArcCos[a*x]^(5/2))/3 + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^3) + (5*Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(144*a^3)} -{x*ArcCos[a*x]^(5/2), x, 9, (15*Sqrt[ArcCos[a*x]])/(64*a^2) - (15*x^2*Sqrt[ArcCos[a*x]])/32 - (5*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(8*a) - ArcCos[a*x]^(5/2)/(4*a^2) + (x^2*ArcCos[a*x]^(5/2))/2 + (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(128*a^2)} -{ArcCos[a*x]^(5/2), x, 6, (-15*x*Sqrt[ArcCos[a*x]])/4 - (5*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^(3/2))/(2*a) + x*ArcCos[a*x]^(5/2) + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a)} -{ArcCos[a*x]^(5/2)/x, x, 0, Unintegrable[ArcCos[a*x]^(5/2)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/Sqrt[ArcCos[a*x]], x, 9, -(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^5) - (Sqrt[(3*Pi)/2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5) - (Sqrt[Pi/10]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^5)} -{x^3/Sqrt[ArcCos[a*x]], x, 7, -(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^4) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(4*a^4)} -{x^2/Sqrt[ArcCos[a*x]], x, 7, -(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^3) - (Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^3)} -{x/Sqrt[ArcCos[a*x]], x, 5, -(Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(2*a^2)} -{1/Sqrt[ArcCos[a*x]], x, 3, -((Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a)} -{1/(x*Sqrt[ArcCos[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[ArcCos[a*x]]), x]} -{1/(x^2*Sqrt[ArcCos[a*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[ArcCos[a*x]]), x]} - - -{x^6/ArcCos[a*x]^(3/2), x, 10, (2*x^6*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (5*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (9*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7) - (Sqrt[(7*Pi)/2]*FresnelC[Sqrt[14/Pi]*Sqrt[ArcCos[a*x]]])/(16*a^7)} -{x^5/ArcCos[a*x]^(3/2), x, 8, (2*x^5*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^6 - (Sqrt[3*Pi]*FresnelC[2*Sqrt[3/Pi]*Sqrt[ArcCos[a*x]]])/(8*a^6) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(8*a^6)} -{x^4/ArcCos[a*x]^(3/2), x, 8, (2*x^4*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^5) - (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^5) - (Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(4*a^5)} -{x^3/ArcCos[a*x]^(3/2), x, 6, (2*x^3*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^4 - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/a^4} -{x^2/ArcCos[a*x]^(3/2), x, 6, (2*x^2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^3 - (Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^3} -{x/ArcCos[a*x]^(3/2), x, 3, (2*x*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (2*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/a^2} -{ArcCos[a*x]^(-3/2), x, 4, (2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a} -{1/(x*ArcCos[a*x]^(3/2)), x, 0, Unintegrable[1/(x*ArcCos[a*x]^(3/2)), x]} - - -{x^4/ArcCos[a*x]^(5/2), x, 19, (2*x^4*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (16*x^3)/(3*a^2*Sqrt[ArcCos[a*x]]) + (20*x^5)/(3*Sqrt[ArcCos[a*x]]) + (25*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5) - (4*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a^5 + (25*Sqrt[Pi/6]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(2*a^5) - (4*Sqrt[(2*Pi)/3]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^5 + (5*Sqrt[(5*Pi)/2]*FresnelS[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(6*a^5)} -{x^3/ArcCos[a*x]^(5/2), x, 15, (2*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (4*x^2)/(a^2*Sqrt[ArcCos[a*x]]) + (16*x^4)/(3*Sqrt[ArcCos[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^4) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(3*a^4)} -{x^2/ArcCos[a*x]^(5/2), x, 13, (2*x^2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcCos[a*x]]) + (4*x^3)/Sqrt[ArcCos[a*x]] + (Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^3) + (Sqrt[6*Pi]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^3} -{x/ArcCos[a*x]^(5/2), x, 8, (2*x*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) - 4/(3*a^2*Sqrt[ArcCos[a*x]]) + (8*x^2)/(3*Sqrt[ArcCos[a*x]]) + (8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(3*a^2)} -{ArcCos[a*x]^(-5/2), x, 5, (2*Sqrt[1 - a^2*x^2])/(3*a*ArcCos[a*x]^(3/2)) + (4*x)/(3*Sqrt[ArcCos[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(3*a)} -{1/(x*ArcCos[a*x]^(5/2)), x, 0, Unintegrable[1/(x*ArcCos[a*x]^(5/2)), x]} - - -{x^4/ArcCos[a*x]^(7/2), x, 17, (2*x^4*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (16*x^3)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^5)/(3*ArcCos[a*x]^(3/2)) + (32*x^2*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (40*x^4*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[ArcCos[a*x]]) + (Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^5) + (5*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/a^5 - (8*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(5*a^5) + (5*Sqrt[(5*Pi)/2]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcCos[a*x]]])/(3*a^5)} -{x^3/ArcCos[a*x]^(7/2), x, 12, (2*x^3*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcCos[a*x]^(3/2)) + (16*x^4)/(15*ArcCos[a*x]^(3/2)) + (16*x*Sqrt[1 - a^2*x^2])/(5*a^3*Sqrt[ArcCos[a*x]]) - (128*x^3*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos[a*x]]) + (32*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(15*a^4)} -{x^2/ArcCos[a*x]^(7/2), x, 13, (2*x^2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - (8*x)/(15*a^2*ArcCos[a*x]^(3/2)) + (4*x^3)/(5*ArcCos[a*x]^(3/2)) + (16*Sqrt[1 - a^2*x^2])/(15*a^3*Sqrt[ArcCos[a*x]]) - (24*x^2*Sqrt[1 - a^2*x^2])/(5*a*Sqrt[ArcCos[a*x]]) + (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a^3) + (6*Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcCos[a*x]]])/(5*a^3)} -{x/ArcCos[a*x]^(7/2), x, 6, (2*x*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) - 4/(15*a^2*ArcCos[a*x]^(3/2)) + (8*x^2)/(15*ArcCos[a*x]^(3/2)) - (32*x*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos[a*x]]) + (32*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcCos[a*x]])/Sqrt[Pi]])/(15*a^2)} -{ArcCos[a*x]^(-7/2), x, 6, (2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) + (4*x)/(15*ArcCos[a*x]^(3/2)) - (8*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[ArcCos[a*x]]) + (8*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/(15*a)} -{1/(x*ArcCos[a*x]^(7/2)), x, 0, Unintegrable[1/(x*ArcCos[a*x]^(7/2)), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (b x)^(m/2) ArcCos[a x]^n*) - - -(* ::Subsection:: *) -(*Integrands of the form (b x)^(m/2) ArcCos[a x]^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcCos[a x]^n with m symbolic*) - - -{(b*x)^m*ArcCos[a*x]^4, x, 1, ((b*x)^(1 + m)*ArcCos[a*x]^4)/(b*(1 + m)) + (4*a*Unintegrable[((b*x)^(1 + m)*ArcCos[a*x]^3)/Sqrt[1 - a^2*x^2], x])/(b*(1 + m))} -{(b*x)^m*ArcCos[a*x]^3, x, 1, ((b*x)^(1 + m)*ArcCos[a*x]^3)/(b*(1 + m)) + (3*a*Unintegrable[((b*x)^(1 + m)*ArcCos[a*x]^2)/Sqrt[1 - a^2*x^2], x])/(b*(1 + m))} -{(b*x)^m*ArcCos[a*x]^2, x, 2, If[$VersionNumber>=8, ((b*x)^(1 + m)*ArcCos[a*x]^2)/(b*(1 + m)) + (2*a*(b*x)^(2 + m)*ArcCos[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m)) + (2*a^2*(b*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(b^3*(1 + m)*(2 + m)*(3 + m)), ((b*x)^(1 + m)*ArcCos[a*x]^2)/(b*(1 + m)) + (2*a*(b*x)^(2 + m)*ArcCos[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m)) + (2*a^2*(b*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(b^3*(3 + m)*(2 + 3*m + m^2))]} -{(b*x)^m*ArcCos[a*x], x, 2, ((b*x)^(1 + m)*ArcCos[a*x])/(b*(1 + m)) + (a*(b*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(b^2*(1 + m)*(2 + m))} -{(b*x)^m/ArcCos[a*x], x, 0, Unintegrable[(b*x)^m/ArcCos[a*x], x]} -{(b*x)^m/ArcCos[a*x]^2, x, 0, Unintegrable[(b*x)^m/ArcCos[a*x]^2, x]} - - -{(b*x)^m*ArcCos[a*x]^(3/2), x, 0, Unintegrable[(b*x)^m*ArcCos[a*x]^(3/2), x]} -{(b*x)^m*Sqrt[ArcCos[a*x]], x, 0, Unintegrable[(b*x)^m*Sqrt[ArcCos[a*x]], x]} -{(b*x)^m/Sqrt[ArcCos[a*x]], x, 0, Unintegrable[(b*x)^m/Sqrt[ArcCos[a*x]], x]} -{(b*x)^m/ArcCos[a*x]^(3/2), x, 0, Unintegrable[(b*x)^m/ArcCos[a*x]^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcCos[a x]^n with n symbolic*) - - -{(b*x)^m*ArcCos[a*x]^n, x, 0, Unintegrable[(b*x)^m*ArcCos[a*x]^n, x]} - - -{x^3*ArcCos[a*x]^n, x, 9, (2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]])/(a^4*((-I)*ArcCos[a*x])^n) + (2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (2*I)*ArcCos[a*x]])/(a^4*(I*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]])/(2^(2*(3 + n))*a^4*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, (4*I)*ArcCos[a*x]])/(2^(2*(3 + n))*a^4*(I*ArcCos[a*x])^n)} -{x^2*ArcCos[a*x]^n, x, 9, (ArcCos[a*x]^n*Gamma[1 + n, (-I)*ArcCos[a*x]])/(8*a^3*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, I*ArcCos[a*x]])/(8*a^3*(I*ArcCos[a*x])^n) + (3^(-1 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-3*I)*ArcCos[a*x]])/(8*a^3*((-I)*ArcCos[a*x])^n) + (3^(-1 - n)*ArcCos[a*x]^n*Gamma[1 + n, (3*I)*ArcCos[a*x]])/(8*a^3*(I*ArcCos[a*x])^n)} -{x*ArcCos[a*x]^n, x, 6, (2^(-3 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]])/(a^2*((-I)*ArcCos[a*x])^n) + (2^(-3 - n)*ArcCos[a*x]^n*Gamma[1 + n, (2*I)*ArcCos[a*x]])/(a^2*(I*ArcCos[a*x])^n)} -{ArcCos[a*x]^n, x, 4, (ArcCos[a*x]^n*Gamma[1 + n, (-I)*ArcCos[a*x]])/(2*a*((-I)*ArcCos[a*x])^n) + (ArcCos[a*x]^n*Gamma[1 + n, I*ArcCos[a*x]])/(2*a*(I*ArcCos[a*x])^n)} -{ArcCos[a*x]^n/x, x, 0, Unintegrable[ArcCos[a*x]^n/x, x]} -{ArcCos[a*x]^n/x^2, x, 0, Unintegrable[ArcCos[a*x]^n/x^2, x]} - - -{(b*x)^(3/2)*ArcCos[a*x]^n, x, 0, Unintegrable[(b*x)^(3/2)*ArcCos[a*x]^n, x]} -{Sqrt[b*x]*ArcCos[a*x]^n, x, 0, Unintegrable[Sqrt[b*x]*ArcCos[a*x]^n, x]} -{ArcCos[a*x]^n/Sqrt[b*x], x, 0, Unintegrable[ArcCos[a*x]^n/Sqrt[b*x], x]} -{ArcCos[a*x]^n/(b*x)^(3/2), x, 0, Unintegrable[ArcCos[a*x]^n/(b*x)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCos[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCos[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*(a + b*ArcCos[c*x]), x, 4, (-3*b*x*Sqrt[1 - c^2*x^2])/(32*c^3) - (b*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (x^4*(a + b*ArcCos[c*x]))/4 + (3*b*ArcSin[c*x])/(32*c^4)} -{x^2*(a + b*ArcCos[c*x]), x, 4, -(b*Sqrt[1 - c^2*x^2])/(3*c^3) + (b*(1 - c^2*x^2)^(3/2))/(9*c^3) + (x^3*(a + b*ArcCos[c*x]))/3} -{x*(a + b*ArcCos[c*x]), x, 3, -(b*x*Sqrt[1 - c^2*x^2])/(4*c) + (x^2*(a + b*ArcCos[c*x]))/2 + (b*ArcSin[c*x])/(4*c^2)} -{a + b*ArcCos[c*x], x, 3, a*x - (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcCos[c*x]} -{(a + b*ArcCos[c*x])/x, x, 5, ((-I/2)*(a + b*ArcCos[c*x])^2)/b + (a + b*ArcCos[c*x])*Log[1 + E^((2*I)*ArcCos[c*x])] - (I/2)*b*PolyLog[2, -E^((2*I)*ArcCos[c*x])]} -{(a + b*ArcCos[c*x])/x^2, x, 4, -((a + b*ArcCos[c*x])/x) + b*c*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(a + b*ArcCos[c*x])/x^3, x, 2, (b*c*Sqrt[1 - c^2*x^2])/(2*x) - (a + b*ArcCos[c*x])/(2*x^2)} -{(a + b*ArcCos[c*x])/x^4, x, 5, (b*c*Sqrt[1 - c^2*x^2])/(6*x^2) - (a + b*ArcCos[c*x])/(3*x^3) + (b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/6} - - -{x^2*(a + b*ArcCos[c*x])^2, x, 5, (-4*b^2*x)/(9*c^2) - (2*b^2*x^3)/27 - (4*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(9*c^3) - (2*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(9*c) + (x^3*(a + b*ArcCos[c*x])^2)/3} -{x*(a + b*ArcCos[c*x])^2, x, 4, -(b^2*x^2)/4 - (b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*c) - (a + b*ArcCos[c*x])^2/(4*c^2) + (x^2*(a + b*ArcCos[c*x])^2)/2} -{(a + b*ArcCos[c*x])^2, x, 3, -2*b^2*x - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/c + x*(a + b*ArcCos[c*x])^2} -{(a + b*ArcCos[c*x])^2/x, x, 6, ((-I/3)*(a + b*ArcCos[c*x])^3)/b + (a + b*ArcCos[c*x])^2*Log[1 + E^((2*I)*ArcCos[c*x])] - I*b*(a + b*ArcCos[c*x])*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + (b^2*PolyLog[3, -E^((2*I)*ArcCos[c*x])])/2} -{(a + b*ArcCos[c*x])^2/x^2, x, 7, -((a + b*ArcCos[c*x])^2/x) - (4*I)*b*c*(a + b*ArcCos[c*x])*ArcTan[E^(I*ArcCos[c*x])] + (2*I)*b^2*c*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (2*I)*b^2*c*PolyLog[2, I*E^(I*ArcCos[c*x])]} - - -{x^2*(a + b*ArcCos[c*x])^3, x, 10, (-4*a*b^2*x)/(3*c^2) + (14*b^3*Sqrt[1 - c^2*x^2])/(9*c^3) - (2*b^3*(1 - c^2*x^2)^(3/2))/(27*c^3) - (4*b^3*x*ArcCos[c*x])/(3*c^2) - (2*b^2*x^3*(a + b*ArcCos[c*x]))/9 - (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(3*c^3) - (b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(3*c) + (x^3*(a + b*ArcCos[c*x])^3)/3} -{x*(a + b*ArcCos[c*x])^3, x, 6, (3*b^3*x*Sqrt[1 - c^2*x^2])/(8*c) - (3*b^2*x^2*(a + b*ArcCos[c*x]))/4 - (3*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(4*c) - (a + b*ArcCos[c*x])^3/(4*c^2) + (x^2*(a + b*ArcCos[c*x])^3)/2 - (3*b^3*ArcSin[c*x])/(8*c^2)} -{(a + b*ArcCos[c*x])^3, x, 5, -6*a*b^2*x + (6*b^3*Sqrt[1 - c^2*x^2])/c - 6*b^3*x*ArcCos[c*x] - (3*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/c + x*(a + b*ArcCos[c*x])^3} -{(a + b*ArcCos[c*x])^3/x, x, 7, ((-I/4)*(a + b*ArcCos[c*x])^4)/b + (a + b*ArcCos[c*x])^3*Log[1 + E^((2*I)*ArcCos[c*x])] - ((3*I)/2)*b*(a + b*ArcCos[c*x])^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + (3*b^2*(a + b*ArcCos[c*x])*PolyLog[3, -E^((2*I)*ArcCos[c*x])])/2 + ((3*I)/4)*b^3*PolyLog[4, -E^((2*I)*ArcCos[c*x])]} -{(a + b*ArcCos[c*x])^3/x^2, x, 9, -((a + b*ArcCos[c*x])^3/x) - (6*I)*b*c*(a + b*ArcCos[c*x])^2*ArcTan[E^(I*ArcCos[c*x])] + (6*I)*b^2*c*(a + b*ArcCos[c*x])*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - (6*I)*b^2*c*(a + b*ArcCos[c*x])*PolyLog[2, I*E^(I*ArcCos[c*x])] - 6*b^3*c*PolyLog[3, (-I)*E^(I*ArcCos[c*x])] + 6*b^3*c*PolyLog[3, I*E^(I*ArcCos[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcCos[c*x]), x, 9, (CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(4*b*c^3) + (CosIntegral[(3*(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/(4*b*c^3) - (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(4*b*c^3) - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(4*b*c^3)} -{x^1/(a + b*ArcCos[c*x]), x, 6, (CosIntegral[(2*(a + b*ArcCos[c*x]))/b]*Sin[(2*a)/b])/(2*b*c^2) - (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcCos[c*x]))/b])/(2*b*c^2)} -{x^0/(a + b*ArcCos[c*x]), x, 4, (CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(b*c) - (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b*c)} -{1/(x^1*(a + b*ArcCos[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcCos[c*x])), x]} -{1/(x^2*(a + b*ArcCos[c*x])), x, 0, Unintegrable[1/(x^2*(a + b*ArcCos[c*x])), x]} - - -{x^2/(a + b*ArcCos[c*x])^2, x, 8, (x^2*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCos[c*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcCos[c*x])/b])/(4*b^2*c^3) - (3*Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcCos[c*x]))/b])/(4*b^2*c^3) - (Sin[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(4*b^2*c^3) - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(4*b^2*c^3)} -{x^1/(a + b*ArcCos[c*x])^2, x, 4, (x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCos[c*x])) - (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcCos[c*x]))/b])/(b^2*c^2) - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcCos[c*x]))/b])/(b^2*c^2)} -{x^0/(a + b*ArcCos[c*x])^2, x, 5, Sqrt[1 - c^2*x^2]/(b*c*(a + b*ArcCos[c*x])) - (Cos[a/b]*CosIntegral[(a + b*ArcCos[c*x])/b])/(b^2*c) - (Sin[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b^2*c)} -{1/(x^1*(a + b*ArcCos[c*x])^2), x, 0, Unintegrable[1/(x*(a + b*ArcCos[c*x])^2), x]} -{1/(x^2*(a + b*ArcCos[c*x])^2), x, 0, Unintegrable[1/(x^2*(a + b*ArcCos[c*x])^2), x]} - - -{x^2/(a + b*ArcCos[c*x])^3, x, 16, (x^2*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - x/(b^2*c^2*(a + b*ArcCos[c*x])) + (3*x^3)/(2*b^2*(a + b*ArcCos[c*x])) - (CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(8*b^3*c^3) - (9*CosIntegral[(3*(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/(8*b^3*c^3) + (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(8*b^3*c^3) + (9*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b])/(8*b^3*c^3)} -{x^1/(a + b*ArcCos[c*x])^3, x, 9, (x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - 1/(2*b^2*c^2*(a + b*ArcCos[c*x])) + x^2/(b^2*(a + b*ArcCos[c*x])) - (CosIntegral[(2*(a + b*ArcCos[c*x]))/b]*Sin[(2*a)/b])/(b^3*c^2) + (Cos[(2*a)/b]*SinIntegral[(2*(a + b*ArcCos[c*x]))/b])/(b^3*c^2)} -{x^0/(a + b*ArcCos[c*x])^3, x, 6, Sqrt[1 - c^2*x^2]/(2*b*c*(a + b*ArcCos[c*x])^2) + x/(2*b^2*(a + b*ArcCos[c*x])) - (CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(2*b^3*c) + (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(2*b^3*c)} -{1/(x^1*(a + b*ArcCos[c*x])^3), x, 0, Unintegrable[1/(x*(a + b*ArcCos[c*x])^3), x]} -{1/(x^2*(a + b*ArcCos[c*x])^3), x, 0, Unintegrable[1/(x^2*(a + b*ArcCos[c*x])^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCos[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*Sqrt[a + b*ArcCos[c*x]], x, 14, (x^3*Sqrt[a + b*ArcCos[c*x]])/3 - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(4*c^3) - (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(12*c^3) - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c^3) - (Sqrt[b]*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)} -{x*Sqrt[a + b*ArcCos[c*x]], x, 9, -Sqrt[a + b*ArcCos[c*x]]/(4*c^2) + (x^2*Sqrt[a + b*ArcCos[c*x]])/2 - (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*c^2) - (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*c^2)} -{Sqrt[a + b*ArcCos[c*x]], x, 7, x*Sqrt[a + b*ArcCos[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/c - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/c} -{Sqrt[a + b*ArcCos[c*x]]/x, x, 0, Unintegrable[Sqrt[a + b*ArcCos[c*x]]/x, x]} -{Sqrt[a + b*ArcCos[c*x]]/x^2, x, 0, Unintegrable[Sqrt[a + b*ArcCos[c*x]]/x^2, x]} - - -{x^2*(a + b*ArcCos[c*x])^(3/2), x, 22, -(b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(3*c^3) - (b*x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(6*c) + (x^3*(a + b*ArcCos[c*x])^(3/2))/3 + (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(8*c^3) + (b^(3/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(24*c^3) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(8*c^3) - (b^(3/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(24*c^3)} -{x*(a + b*ArcCos[c*x])^(3/2), x, 11, (-3*b*x*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(8*c) - (a + b*ArcCos[c*x])^(3/2)/(4*c^2) + (x^2*(a + b*ArcCos[c*x])^(3/2))/2 + (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*c^2) - (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*c^2)} -{(a + b*ArcCos[c*x])^(3/2), x, 8, (-3*b*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(2*c) + x*(a + b*ArcCos[c*x])^(3/2) + (3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(2*c) - (3*b^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2*c)} -{(a + b*ArcCos[c*x])^(3/2)/x, x, 0, Unintegrable[(a + b*ArcCos[c*x])^(3/2)/x, x]} -{(a + b*ArcCos[c*x])^(3/2)/x^2, x, 0, Unintegrable[(a + b*ArcCos[c*x])^(3/2)/x^2, x]} - - -{x^2*(a + b*ArcCos[c*x])^(5/2), x, 24, (-5*b^2*x*Sqrt[a + b*ArcCos[c*x]])/(6*c^2) - (5*b^2*x^3*Sqrt[a + b*ArcCos[c*x]])/36 - (5*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(18*c) + (x^3*(a + b*ArcCos[c*x])^(5/2))/3 + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(16*c^3) + (5*b^(5/2)*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(144*c^3) + (15*b^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(16*c^3) + (5*b^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*c^3)} -{x*(a + b*ArcCos[c*x])^(5/2), x, 12, (15*b^2*Sqrt[a + b*ArcCos[c*x]])/(64*c^2) - (15*b^2*x^2*Sqrt[a + b*ArcCos[c*x]])/32 - (5*b*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(8*c) - (a + b*ArcCos[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcCos[c*x])^(5/2))/2 + (15*b^(5/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*c^2) + (15*b^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*c^2)} -{(a + b*ArcCos[c*x])^(5/2), x, 9, (-15*b^2*x*Sqrt[a + b*ArcCos[c*x]])/4 - (5*b*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^(3/2))/(2*c) + x*(a + b*ArcCos[c*x])^(5/2) + (15*b^(5/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(4*c) + (15*b^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(4*c)} -{(a + b*ArcCos[c*x])^(5/2)/x, x, 0, Unintegrable[(a + b*ArcCos[c*x])^(5/2)/x, x]} -{(a + b*ArcCos[c*x])^(5/2)/x^2, x, 0, Unintegrable[(a + b*ArcCos[c*x])^(5/2)/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/Sqrt[a + b*ArcCos[c*x]], x, 13, -(Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) - (Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(2*Sqrt[b]*c^3) + (Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(2*Sqrt[b]*c^3) + (Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(2*Sqrt[b]*c^3)} -{x/Sqrt[a + b*ArcCos[c*x]], x, 8, -(Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(2*Sqrt[b]*c^2) + (Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(2*Sqrt[b]*c^2)} -{1/Sqrt[a + b*ArcCos[c*x]], x, 6, -((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(Sqrt[b]*c)) + (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*c)} -{1/(x*Sqrt[a + b*ArcCos[c*x]]), x, 0, Unintegrable[1/(x*Sqrt[a + b*ArcCos[c*x]]), x]} -{1/(x^2*Sqrt[a + b*ArcCos[c*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[a + b*ArcCos[c*x]]), x]} - - -{x^2/(a + b*ArcCos[c*x])^(3/2), x, 12, (2*x^2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) - (Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(3/2)*c^3) - (Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c^3) - (Sqrt[(3*Pi)/2]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(3/2)*c^3)} -{x/(a + b*ArcCos[c*x])^(3/2), x, 6, (2*x*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) - (2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*c^2) - (2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(b^(3/2)*c^2)} -{(a + b*ArcCos[c*x])^(-3/2), x, 7, (2*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcCos[c*x]]) - (2*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(3/2)*c) - (2*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(b^(3/2)*c)} -{1/(x*(a + b*ArcCos[c*x])^(3/2)), x, 0, Unintegrable[1/(x*(a + b*ArcCos[c*x])^(3/2)), x]} -{1/(x^2*(a + b*ArcCos[c*x])^(3/2)), x, 0, Unintegrable[1/(x^2*(a + b*ArcCos[c*x])^(3/2)), x]} - - -{x^2/(a + b*ArcCos[c*x])^(5/2), x, 22, (2*x^2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCos[c*x]]) + (4*x^3)/(b^2*Sqrt[a + b*ArcCos[c*x]]) + (Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(3*b^(5/2)*c^3) + (Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(b^(5/2)*c^3) - (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c^3) - (Sqrt[6*Pi]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(b^(5/2)*c^3)} -{x/(a + b*ArcCos[c*x])^(5/2), x, 11, (2*x*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) - 4/(3*b^2*c^2*Sqrt[a + b*ArcCos[c*x]]) + (8*x^2)/(3*b^2*Sqrt[a + b*ArcCos[c*x]]) + (8*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*c^2) - (8*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*c^2)} -{(a + b*ArcCos[c*x])^(-5/2), x, 8, (2*Sqrt[1 - c^2*x^2])/(3*b*c*(a + b*ArcCos[c*x])^(3/2)) + (4*x)/(3*b^2*Sqrt[a + b*ArcCos[c*x]]) + (4*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]])/(3*b^(5/2)*c) - (4*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*Sin[a/b])/(3*b^(5/2)*c)} -{1/(x*(a + b*ArcCos[c*x])^(5/2)), x, 0, Unintegrable[1/(x*(a + b*ArcCos[c*x])^(5/2)), x]} -{1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x, 0, Unintegrable[1/(x^2*(a + b*ArcCos[c*x])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcCos[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d*x)^(5/2)*(a + b*ArcCos[c*x]), x, 5, (-20*b*d^2*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(147*c^3) - (4*b*(d*x)^(5/2)*Sqrt[1 - c^2*x^2])/(49*c) + (2*(d*x)^(7/2)*(a + b*ArcCos[c*x]))/(7*d) + (20*b*d^(5/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(147*c^(7/2))} -{(d*x)^(3/2)*(a + b*ArcCos[c*x]), x, 7, -((4*b*(d*x)^(3/2)*Sqrt[1 - c^2*x^2])/(25*c)) + (2*(d*x)^(5/2)*(a + b*ArcCos[c*x]))/(5*d) + (12*b*d^(3/2)*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(25*c^(5/2)) - (12*b*d^(3/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(25*c^(5/2))} -{Sqrt[d*x]*(a + b*ArcCos[c*x]), x, 4, (-4*b*Sqrt[d*x]*Sqrt[1 - c^2*x^2])/(9*c) + (2*(d*x)^(3/2)*(a + b*ArcCos[c*x]))/(3*d) + (4*b*Sqrt[d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(9*c^(3/2))} -{(a + b*ArcCos[c*x])/Sqrt[d*x], x, 6, (2*Sqrt[d*x]*(a + b*ArcCos[c*x]))/d + (4*b*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(Sqrt[c]*Sqrt[d]) - (4*b*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(Sqrt[c]*Sqrt[d])} -{(a + b*ArcCos[c*x])/(d*x)^(3/2), x, 3, (-2*(a + b*ArcCos[c*x]))/(d*Sqrt[d*x]) - (4*b*Sqrt[c]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/d^(3/2)} -{(a + b*ArcCos[c*x])/(d*x)^(5/2), x, 7, (4*b*c*Sqrt[1 - c^2*x^2])/(3*d^2*Sqrt[d*x]) - (2*(a + b*ArcCos[c*x]))/(3*d*(d*x)^(3/2)) + (4*b*c^(3/2)*EllipticE[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(3*d^(5/2)) - (4*b*c^(3/2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/(3*d^(5/2))} - - -{(d*x)^(5/2)*(a + b*ArcCos[c*x])^2, x, 2, (2*(d*x)^(7/2)*(a + b*ArcCos[c*x])^2)/(7*d) + (8*b*c*(d*x)^(9/2)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, 9/4, 13/4, c^2*x^2])/(63*d^2) + (16*b^2*c^2*(d*x)^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, c^2*x^2])/(693*d^3)} -{(d*x)^(3/2)*(a + b*ArcCos[c*x])^2, x, 2, (2*(d*x)^(5/2)*(a + b*ArcCos[c*x])^2)/(5*d) + (8*b*c*(d*x)^(7/2)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, 7/4, 11/4, c^2*x^2])/(35*d^2) + (16*b^2*c^2*(d*x)^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, c^2*x^2])/(315*d^3)} -{Sqrt[d*x]*(a + b*ArcCos[c*x])^2, x, 2, (2*(d*x)^(3/2)*(a + b*ArcCos[c*x])^2)/(3*d) + (8*b*c*(d*x)^(5/2)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(15*d^2) + (16*b^2*c^2*(d*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(105*d^3)} -{(a + b*ArcCos[c*x])^2/Sqrt[d*x], x, 2, (2*Sqrt[d*x]*(a + b*ArcCos[c*x])^2)/d + (8*b*c*(d*x)^(3/2)*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^2])/(3*d^2) + (16*b^2*c^2*(d*x)^(5/2)*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, c^2*x^2])/(15*d^3)} -{(a + b*ArcCos[c*x])^2/(d*x)^(3/2), x, 2, (-2*(a + b*ArcCos[c*x])^2)/(d*Sqrt[d*x]) - (8*b*c*Sqrt[d*x]*(a + b*ArcCos[c*x])*Hypergeometric2F1[1/4, 1/2, 5/4, c^2*x^2])/d^2 - (16*b^2*c^2*(d*x)^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, c^2*x^2])/(3*d^3)} -{(a + b*ArcCos[c*x])^2/(d*x)^(5/2), x, 2, (-2*(a + b*ArcCos[c*x])^2)/(3*d*(d*x)^(3/2)) + (8*b*c*(a + b*ArcCos[c*x])*Hypergeometric2F1[-1/4, 1/2, 3/4, c^2*x^2])/(3*d^2*Sqrt[d*x]) + (16*b^2*c^2*Sqrt[d*x]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, c^2*x^2])/(3*d^3)} - - -{(d*x)^(3/2)*(a + b*ArcCos[c*x])^3, x, 1, (2*(d*x)^(5/2)*(a + b*ArcCos[c*x])^3)/(5*d) + (6*b*c*Unintegrable[((d*x)^(5/2)*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c^2*x^2], x])/(5*d)} -{Sqrt[d*x]*(a + b*ArcCos[c*x])^3, x, 1, (2*(d*x)^(3/2)*(a + b*ArcCos[c*x])^3)/(3*d) + (2*b*c*Unintegrable[((d*x)^(3/2)*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c^2*x^2], x])/d} -{(a + b*ArcCos[c*x])^3/Sqrt[d*x], x, 1, (2*Sqrt[d*x]*(a + b*ArcCos[c*x])^3)/d + (6*b*c*Unintegrable[(Sqrt[d*x]*(a + b*ArcCos[c*x])^2)/Sqrt[1 - c^2*x^2], x])/d} -{(a + b*ArcCos[c*x])^3/(d*x)^(3/2), x, 1, (-2*(a + b*ArcCos[c*x])^3)/(d*Sqrt[d*x]) - (6*b*c*Unintegrable[(a + b*ArcCos[c*x])^2/(Sqrt[d*x]*Sqrt[1 - c^2*x^2]), x])/d} -{(a + b*ArcCos[c*x])^3/(d*x)^(5/2), x, 1, (-2*(a + b*ArcCos[c*x])^3)/(3*d*(d*x)^(3/2)) - (2*b*c*Unintegrable[(a + b*ArcCos[c*x])^2/((d*x)^(3/2)*Sqrt[1 - c^2*x^2]), x])/d} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d*x)^(3/2)/(a + b*ArcCos[c*x]), x, 0, Unintegrable[(d*x)^(3/2)/(a + b*ArcCos[c*x]), x]} -{Sqrt[d*x]/(a + b*ArcCos[c*x]), x, 0, Unintegrable[Sqrt[d*x]/(a + b*ArcCos[c*x]), x]} -{1/(Sqrt[d*x]*(a + b*ArcCos[c*x])), x, 0, Unintegrable[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])), x]} -{1/((d*x)^(3/2)*(a + b*ArcCos[c*x])), x, 0, Unintegrable[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])), x]} - - -{(d*x)^(3/2)/(a + b*ArcCos[c*x])^2, x, 0, Unintegrable[(d*x)^(3/2)/(a + b*ArcCos[c*x])^2, x]} -{Sqrt[d*x]/(a + b*ArcCos[c*x])^2, x, 0, Unintegrable[Sqrt[d*x]/(a + b*ArcCos[c*x])^2, x]} -{1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2), x, 0, Unintegrable[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2), x]} -{1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x, 0, Unintegrable[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcCos[c x])^(n/2)*) - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^m (a+b ArcCos[c x])^n with m symbolic*) - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^m (a+b ArcCos[c x])^n with n symbolic*) diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.4 (f x)^m (d+e x^2)^p (a+b arccos(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.4 (f x)^m (d+e x^2)^p (a+b arccos(c x))^n.m deleted file mode 100644 index 1fa25ad..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.4 (f x)^m (d+e x^2)^p (a+b arccos(c x))^n.m +++ /dev/null @@ -1,99 +0,0 @@ -(* ::Package:: *) - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCos[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCos[c x])^1*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcCos[c*x])/(d - c^2*d*x^2), x, 8, (b*x*Sqrt[1 - c^2*x^2])/(4*c^3*d) - (x^2*(a + b*ArcCos[c*x]))/(2*c^2*d) + (I*(a + b*ArcCos[c*x])^2)/(2*b*c^4*d) - (b*ArcSin[c*x])/(4*c^4*d) - ((a + b*ArcCos[c*x])*Log[1 - E^(2*I*ArcCos[c*x])])/(c^4*d) + (I*b*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*c^4*d)} -{x^2*(a + b*ArcCos[c*x])/(d - c^2*d*x^2), x, 8, (b*Sqrt[1 - c^2*x^2])/(c^3*d) - (x*(a + b*ArcCos[c*x]))/(c^2*d) + (2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^3*d) - (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^3*d) + (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d)} -{x^1*(a + b*ArcCos[c*x])/(d - c^2*d*x^2), x, 5, (I*(a + b*ArcCos[c*x])^2)/(2*b*c^2*d) - ((a + b*ArcCos[c*x])*Log[1 - E^(2*I*ArcCos[c*x])])/(c^2*d) + (I*b*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*c^2*d)} -{x^0*(a + b*ArcCos[c*x])/(d - c^2*d*x^2), x, 6, (2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c*d) - (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c*d) + (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c*d)} -{(a + b*ArcCos[c*x])/(x^1*(d - c^2*d*x^2)), x, 7, (2*(a + b*ArcCos[c*x])*ArcTanh[E^(2*I*ArcCos[c*x])])/d - (I*b*PolyLog[2, -E^(2*I*ArcCos[c*x])])/(2*d) + (I*b*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*d)} -{(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)), x, 10, -((a + b*ArcCos[c*x])/(d*x)) + (2*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d + (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d - (I*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d + (I*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d} -{(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)), x, 9, (b*c*Sqrt[1 - c^2*x^2])/(2*d*x) - (a + b*ArcCos[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcCos[c*x])*ArcTanh[E^(2*I*ArcCos[c*x])])/d - (I*b*c^2*PolyLog[2, -E^(2*I*ArcCos[c*x])])/(2*d) + (I*b*c^2*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*d)} - - -{x^4*(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2, x, 12, b/(2*c^5*d^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[1 - c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcCos[c*x]))/(2*c^4*d^2) + (x^3*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^5*d^2) + (3*I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(2*c^5*d^2) - (3*I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c^5*d^2)} -{x^3*(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2, x, 8, (b*x)/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (x^2*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (I*(a + b*ArcCos[c*x])^2)/(2*b*c^4*d^2) - (b*ArcSin[c*x])/(2*c^4*d^2) + ((a + b*ArcCos[c*x])*Log[1 - E^(2*I*ArcCos[c*x])])/(c^4*d^2) - (I*b*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*c^4*d^2)} -{x^2*(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2, x, 8, b/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - ((a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^3*d^2) + (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(2*c^3*d^2) - (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c^3*d^2)} -{x^1*(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2, x, 2, (b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*c^2*d^2*(1 - c^2*x^2))} -{x^0*(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2, x, 8, b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + ((a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c*d^2) - (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(2*c*d^2) + (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c*d^2)} -{(a + b*ArcCos[c*x])/(x^1*(d - c^2*d*x^2)^2), x, 9, (b*c*x)/(2*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*d^2*(1 - c^2*x^2)) + (2*(a + b*ArcCos[c*x])*ArcTanh[E^(2*I*ArcCos[c*x])])/d^2 - (I*b*PolyLog[2, -E^(2*I*ArcCos[c*x])])/(2*d^2) + (I*b*PolyLog[2, E^(2*I*ArcCos[c*x])])/(2*d^2)} -{(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)^2), x, 13, (b*c)/(2*d^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + (3*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d^2 + (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 - (3*I*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/(2*d^2) + (3*I*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/(2*d^2)} -{(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)^2), x, 12, (b*c)/(2*d^2*x*Sqrt[1 - c^2*x^2]) + (c^2*(a + b*ArcCos[c*x]))/(d^2*(1 - c^2*x^2)) - (a + b*ArcCos[c*x])/(2*d^2*x^2*(1 - c^2*x^2)) + (4*c^2*(a + b*ArcCos[c*x])*ArcTanh[E^(2*I*ArcCos[c*x])])/d^2 - (I*b*c^2*PolyLog[2, -E^(2*I*ArcCos[c*x])])/d^2 + (I*b*c^2*PolyLog[2, E^(2*I*ArcCos[c*x])])/d^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcCos[c x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(d + e*x^2)*(a + b*ArcCos[c*x])*x^3, x, 6, -((b*(9*c^2*d + 5*e)*x*Sqrt[1 - c^2*x^2])/(96*c^5)) - (b*(9*c^2*d + 5*e)*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) - (b*e*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (1/4)*d*x^4*(a + b*ArcCos[c*x]) + (1/6)*e*x^6*(a + b*ArcCos[c*x]) + (b*(9*c^2*d + 5*e)*ArcSin[c*x])/(96*c^6)} -{(d + e*x^2)*(a + b*ArcCos[c*x])*x^2, x, 5, -((b*(5*c^2*d + 3*e)*Sqrt[1 - c^2*x^2])/(15*c^5)) + (b*(5*c^2*d + 6*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) - (b*e*(1 - c^2*x^2)^(5/2))/(25*c^5) + (1/3)*d*x^3*(a + b*ArcCos[c*x]) + (1/5)*e*x^5*(a + b*ArcCos[c*x])} -{(d + e*x^2)*(a + b*ArcCos[c*x])*x^1, x, 4, -((3*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2])/(32*c^3)) - (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(16*c) + ((d + e*x^2)^2*(a + b*ArcCos[c*x]))/(4*e) + (b*(8*c^4*d^2 + 8*c^2*d*e + 3*e^2)*ArcSin[c*x])/(32*c^4*e)} -{(d + e*x^2)*(a + b*ArcCos[c*x])*x^0, x, 4, -((b*(3*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3)) + (b*e*(1 - c^2*x^2)^(3/2))/(9*c^3) + d*x*(a + b*ArcCos[c*x]) + (1/3)*e*x^3*(a + b*ArcCos[c*x])} -{(d + e*x^2)*(a + b*ArcCos[c*x])/x^1, x, 12, -((b*e*x*Sqrt[1 - c^2*x^2])/(4*c)) + (1/2)*e*x^2*(a + b*ArcCos[c*x]) + (b*e*ArcSin[c*x])/(4*c^2) + (1/2)*I*b*d*ArcSin[c*x]^2 - b*d*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] + d*(a + b*ArcCos[c*x])*Log[x] + b*d*ArcSin[c*x]*Log[x] + (1/2)*I*b*d*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)*(a + b*ArcCos[c*x])/x^2, x, 5, -((b*e*Sqrt[1 - c^2*x^2])/c) - (d*(a + b*ArcCos[c*x]))/x + e*x*(a + b*ArcCos[c*x]) + b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]} -{(d + e*x^2)*(a + b*ArcCos[c*x])/x^3, x, 10, (b*c*d*Sqrt[1 - c^2*x^2])/(2*x) - (d*(a + b*ArcCos[c*x]))/(2*x^2) + (1/2)*I*b*e*ArcSin[c*x]^2 - b*e*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])] + e*(a + b*ArcCos[c*x])*Log[x] + b*e*ArcSin[c*x]*Log[x] + (1/2)*I*b*e*PolyLog[2, E^(2*I*ArcSin[c*x])]} -{(d + e*x^2)*(a + b*ArcCos[c*x])/x^4, x, 6, (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (d*(a + b*ArcCos[c*x]))/(3*x^3) - (e*(a + b*ArcCos[c*x]))/x + (1/6)*b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[1 - c^2*x^2]]} - - -{ArcCos[a*x]*(c + d*x^2)^2, x, 5, -(((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Sqrt[1 - a^2*x^2])/(15*a^5)) + (2*d*(5*a^2*c + 3*d)*(1 - a^2*x^2)^(3/2))/(45*a^5) - (d^2*(1 - a^2*x^2)^(5/2))/(25*a^5) + c^2*x*ArcCos[a*x] + (2/3)*c*d*x^3*ArcCos[a*x] + (1/5)*d^2*x^5*ArcCos[a*x]} - - -{ArcCos[a*x]*(c + d*x^2)^3, x, 5, -(((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Sqrt[1 - a^2*x^2])/(35*a^7)) + (d*(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^(3/2))/(105*a^7) - (3*d^2*(7*a^2*c + 5*d)*(1 - a^2*x^2)^(5/2))/(175*a^7) + (d^3*(1 - a^2*x^2)^(7/2))/(49*a^7) + c^3*x*ArcCos[a*x] + c^2*d*x^3*ArcCos[a*x] + (3/5)*c*d^2*x^5*ArcCos[a*x] + (1/7)*d^3*x^7*ArcCos[a*x]} - - -{ArcCos[a*x]*(c + d*x^2)^4, x, 5, -(((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Sqrt[1 - a^2*x^2])/(315*a^9)) + (4*d*(105*a^6*c^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(1 - a^2*x^2)^(3/2))/(945*a^9) - (2*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^(5/2))/(525*a^9) + (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^(7/2))/(441*a^9) - (d^4*(1 - a^2*x^2)^(9/2))/(81*a^9) + c^4*x*ArcCos[a*x] + (4/3)*c^3*d*x^3*ArcCos[a*x] + (6/5)*c^2*d^2*x^5*ArcCos[a*x] + (4/7)*c*d^3*x^7*ArcCos[a*x] + (1/9)*d^4*x^9*ArcCos[a*x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{ArcCos[a*x]/(c + d*x^2)^1, x, 18, (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (I*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(2*Sqrt[-c]*Sqrt[d]) - (I*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (I*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(2*Sqrt[-c]*Sqrt[d]) - (I*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d])} - - -{ArcCos[a*x]/(c + d*x^2)^2, x, 26, -(ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x))) + ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*ArcTanh[(Sqrt[d] - a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (a*ArcTanh[(Sqrt[d] + a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (I*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(4*(-c)^(3/2)*Sqrt[d]) + (I*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (I*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(4*(-c)^(3/2)*Sqrt[d]) + (I*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^(p/2) (a+b ArcCos[c x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcCos[a*x]*(c + d*x^2)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x^2]*ArcCos[a*x], x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{ArcCos[a*x]/(c + d*x^2)^(1/2), x, 0, Unintegrable[ArcCos[a*x]/Sqrt[c + d*x^2], x]} - - -{ArcCos[a*x]/(c + d*x^2)^(3/2), x, 6, (x*ArcCos[a*x])/(c*Sqrt[c + d*x^2]) - ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])]/(c*Sqrt[d])} - - -{ArcCos[a*x]/(c + d*x^2)^(5/2), x, 7, -((a*Sqrt[1 - a^2*x^2])/(3*c*(a^2*c + d)*Sqrt[c + d*x^2])) + (x*ArcCos[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCos[a*x])/(3*c^2*Sqrt[c + d*x^2]) - (2*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])])/(3*c^2*Sqrt[d])} - - -{ArcCos[a*x]/(c + d*x^2)^(7/2), x, 8, -((a*Sqrt[1 - a^2*x^2])/(15*c*(a^2*c + d)*(c + d*x^2)^(3/2))) - (2*a*(3*a^2*c + 2*d)*Sqrt[1 - a^2*x^2])/(15*c^2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcCos[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCos[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCos[a*x])/(15*c^3*Sqrt[c + d*x^2]) - (8*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])])/(15*c^3*Sqrt[d])} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.5 Inverse cosine functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.5 Inverse cosine functions.m deleted file mode 100644 index 0ad3a21..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.2 Inverse cosine/5.2.5 Inverse cosine functions.m +++ /dev/null @@ -1,308 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcCos[c x])^n*) - - -(* ::Section:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCos[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x^2)^p (a+b ArcCos[c x])^n where c^2 d+e=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcCos[c x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^3, x, 16, -((b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[1 - c^2*x^2])) - (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[1 - c^2*x^2]) + (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) - (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) + (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[1 - c^2*x^2]) - (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[1 - c^2*x^2]) + (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (b*c*g^3*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (1/2)*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*c^2) + (3/4)*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (f^2*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/c^2 - (g^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*c^4) + (g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^4) - (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) - (3*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^2, x, 13, -((2*b*f*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2])) + (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) - (b*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[1 - c^2*x^2]) + (2*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) + (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (1/2)*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*c^2) + (1/4)*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (2*f*g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) - (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2]) - (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c^3*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^1, x, 8, -((b*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2])) + (b*c*f*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - c^2*x^2]) + (b*c*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) + (1/2)*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (g*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*c^2) - (f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])/(f + g*x)^1, x, 22, (a*Sqrt[d - c^2*d*x^2])/g + (b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[1 - c^2*x^2]) + (b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g - (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*g*Sqrt[1 - c^2*x^2]) + ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(f + g*x)*Sqrt[1 - c^2*x^2]) - (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(f + g*x)) - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (I*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (I*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[1 - c^2*x^2]) + (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[1 - c^2*x^2])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])/(f + g*x)^2, x, 35, -((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g*(f + g*x)) + (b*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) - ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(f + g*x)^2) - (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) + (a*c^2*f*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 - c^2*x^2]) + (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])} - - -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])*(f + g*x)^3, x, 24, -((3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2])) - (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[1 - c^2*x^2]) + (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[1 - c^2*x^2]) - (b*d*g^3*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(12*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (3/8)*d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16*c^2) + (3/8)*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/4)*d*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/2)*d*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (3*d*f^2*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (d*g^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^4) + (d*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) - (3*d*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])*(f + g*x)^2, x, 20, -((2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2])) + (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[1 - c^2*x^2]) + (4*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[1 - c^2*x^2]) + (3/8)*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(16*c^2) + (1/8)*d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/4)*d*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/6)*d*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (2*d*f*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2]) - (d*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])*(f + g*x)^1, x, 12, -((b*d*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[1 - c^2*x^2])) + (5*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) + (2*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[1 - c^2*x^2]) - (b*c^3*d*f*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[1 - c^2*x^2]) - (b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[1 - c^2*x^2]) + (3/8)*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/4)*d*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (d*g*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*c^2) - (3*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])/(f + g*x)^1, x, 29, -((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) + (b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*g*Sqrt[1 - c^2*x^2]) - (b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*g*Sqrt[1 - c^2*x^2]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^3 + (c^2*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + (d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (c*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*g^3*Sqrt[1 - c^2*x^2]) + (d*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^2*(f + g*x)) + (a*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (I*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (I*b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[1 - c^2*x^2]) - (b*d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x])/(f + g*x)^2, x, 71, (2*a*c^2*d*f*Sqrt[d - c^2*d*x^2])/g^3 + (a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/(g^3*(f + g*x)) + (2*b*c^3*d*f*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) - (b*c^3*d*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[1 - c^2*x^2]) + (2*b*c^2*d*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^3 + (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g^3*(f + g*x)) - (b*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^4*Sqrt[1 - c^2*x^2]) - (c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) - (c^3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^3*Sqrt[1 - c^2*x^2]) + (d*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^2*(f + g*x)^2*Sqrt[1 - c^2*x^2]) + (c*d*f*(1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^2*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^2*(f + g*x)^2) - (c*d*f*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^2*(f + g*x)) + (a*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^4*Sqrt[1 - c^2*x^2]) - (3*a*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (b*c*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^4*Sqrt[1 - c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[1 - c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[1 - c^2*x^2])} *) - - -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])*(f + g*x)^3, x, 30, -((3*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2])) - (2*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[1 - c^2*x^2]) + (25*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) + (3*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(256*Sqrt[1 - c^2*x^2]) - (9*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[1 - c^2*x^2]) - (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[1 - c^2*x^2]) + (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[1 - c^2*x^2]) - (b*d^2*f^3*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(128*c^2) + (15/64)*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (5/24)*d^2*f^3*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (5/16)*d^2*f*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/6)*d^2*f^3*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (3/8)*d^2*f*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (3*d^2*f^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (d^2*g^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^4) + (d^2*g^3*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(9*c^4) - (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2]) - (15*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])*(f + g*x)^2, x, 26, -((2*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2])) + (25*b*c*d^2*f^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[1 - c^2*x^2]) + (2*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[1 - c^2*x^2]) - (6*b*c^3*d^2*f*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) - (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[1 - c^2*x^2]) + (2*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[1 - c^2*x^2]) - (b*d^2*f^2*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(128*c^2) + (5/64)*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (5/24)*d^2*f^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (5/48)*d^2*g^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/6)*d^2*f^2*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/8)*d^2*g^2*x^3*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (2*d^2*f*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (5*d^2*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2]) - (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(256*b*c^3*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])*(f + g*x)^1, x, 14, -((b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[1 - c^2*x^2])) + (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) + (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[1 - c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[1 - c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[1 - c^2*x^2]) - (b*d^2*f*(1 - c^2*x^2)^(5/2)*Sqrt[d - c^2*d*x^2])/(36*c) + (5/16)*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (5/24)*d^2*f*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) + (1/6)*d^2*f*x*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]) - (d^2*g*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(7*c^2) - (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(32*b*c*Sqrt[1 - c^2*x^2])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])/(f + g*x)^1, x, 37, (a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/g^5 - (2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])/(45*g*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(4*g^2) - (d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g^3) + (d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(5*g) + (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*g^5*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2]) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2])} -(* {(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x])/(f + g*x)^2, x, 78, -((4*a*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^5) - (a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/(g^5*(f + g*x)) + (2*b*c^3*d^2*f*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) - (4*b*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*(3*c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) - (2*b*c^5*d^2*f*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 - c^2*x^2]) + (b*c^5*d^2*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (4*b*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/g^5 - (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g^5*(f + g*x)) + (b*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^6*Sqrt[1 - c^2*x^2]) - (c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(8*g^2) + (c^2*d^2*(3*c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(2*g^4) + (c^4*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(4*g^2) + (2*c^2*d^2*f*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(3*g^3) - (c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) - (c*d^2*(3*c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) + (2*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^5*Sqrt[1 - c^2*x^2]) - (d^2*(c*f - g)*(c*f + g)*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)^2*Sqrt[1 - c^2*x^2]) + (2*c*d^2*f*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)^2) + (2*c*d^2*f*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(b*g^4*(f + g*x)) - (a*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^6*Sqrt[1 - c^2*x^2]) + (5*a*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (b*c*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^6*Sqrt[1 - c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[1 - c^2*x^2])} *) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^3, x, 13, -((3*b*f^2*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2])) - (2*b*g^3*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) - (3*b*f*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) - (b*g^3*x^3*Sqrt[1 - c^2*x^2])/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(3*c^4*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*x*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(3*c^2*Sqrt[d - c^2*d*x^2]) - (f^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^2, x, 9, -((2*b*f*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2])) - (b*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])*(f + g*x)^1, x, 6, -((b*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2])) - (g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (f*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])/(f + g*x)^1, x, 10, (I*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])/(f + g*x)^2, x, 13, (g*(1 - c^2*x^2)*(a + b*ArcCos[c*x]))/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (I*c^2*f*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[f + g*x])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) + (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*c^2*f*Sqrt[1 - c^2*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])} - - -(* ::Subsection:: *) -(*Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcCos[c x])^2*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d+e x^2)^p (a+b ArcCos[c x])^n where c^2 d+e=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d-c^2 d x^2)^(p/2) (a+b ArcCos[c x])^n*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^n/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[((a + b*ArcCos[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]} - -(* {Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^3/Sqrt[1 - c^2*x^2], x, 18, -((I*m*(a + b*ArcCos[c*x])^5)/(20*b^2*c)) + (m*(a + b*ArcCos[c*x])^4*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(4*b*c) + (m*(a + b*ArcCos[c*x])^4*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(4*b*c) - ((a + b*ArcCos[c*x])^4*Log[h*(f + g*x)^m])/(4*b*c) - (I*m*(a + b*ArcCos[c*x])^3*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c - (I*m*(a + b*ArcCos[c*x])^3*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c + (3*b*m*(a + b*ArcCos[c*x])^2*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c + (3*b*m*(a + b*ArcCos[c*x])^2*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c + (6*I*b^2*m*(a + b*ArcCos[c*x])*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c + (6*I*b^2*m*(a + b*ArcCos[c*x])*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c - (6*b^3*m*PolyLog[5, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c - (6*b^3*m*PolyLog[5, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c} *) -{Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^2/Sqrt[1 - c^2*x^2], x, 13, -((I*m*(a + b*ArcCos[c*x])^4)/(12*b^2*c)) + (m*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(3*b*c) + (m*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(3*b*c) - ((a + b*ArcCos[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c) - (I*m*(a + b*ArcCos[c*x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c - (I*m*(a + b*ArcCos[c*x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c + (2*b*m*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c + (2*b*m*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c + (2*I*b^2*m*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c + (2*I*b^2*m*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^1/Sqrt[1 - c^2*x^2], x, 11, -((I*m*(a + b*ArcCos[c*x])^3)/(6*b^2*c)) + (m*(a + b*ArcCos[c*x])^2*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(2*b*c) + (m*(a + b*ArcCos[c*x])^2*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(2*b*c) - ((a + b*ArcCos[c*x])^2*Log[h*(f + g*x)^m])/(2*b*c) - (I*m*(a + b*ArcCos[c*x])*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c - (I*m*(a + b*ArcCos[c*x])*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c + (b*m*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/c + (b*m*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcCos[c*x])^0/Sqrt[1 - c^2*x^2], x, 9, (I*m*ArcSin[c*x]^2)/(2*c) - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c + (ArcSin[c*x]*Log[h*(f + g*x)^m])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]/(a + b*ArcCos[c*x])^1/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])), x]} - - -(* ::Title:: *) -(*Integrands Involving Inverse Cosines*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a+b ArcCos[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCos[a+b x]^n*) - - -{x^3*ArcCos[a + b*x], x, 6, (7*a*x^2*Sqrt[1 - (a + b*x)^2])/(48*b^2) - (x^3*Sqrt[1 - (a + b*x)^2])/(16*b) + ((4*a*(16 + 19*a^2) - (9 + 26*a^2)*(a + b*x))*Sqrt[1 - (a + b*x)^2])/(96*b^4) + (1/4)*x^4*ArcCos[a + b*x] + ((3 + 24*a^2 + 8*a^4)*ArcSin[a + b*x])/(32*b^4)} -{x^2*ArcCos[a + b*x], x, 5, -((x^2*Sqrt[1 - (a + b*x)^2])/(9*b)) - ((4 + 11*a^2 - 5*a*b*x)*Sqrt[1 - (a + b*x)^2])/(18*b^3) + (1/3)*x^3*ArcCos[a + b*x] - (a*(3 + 2*a^2)*ArcSin[a + b*x])/(6*b^3)} -{x^1*ArcCos[a + b*x], x, 5, (3*a*Sqrt[1 - (a + b*x)^2])/(4*b^2) - (x*Sqrt[1 - (a + b*x)^2])/(4*b) + (1/2)*x^2*ArcCos[a + b*x] + ((1 + 2*a^2)*ArcSin[a + b*x])/(4*b^2)} -{x^0*ArcCos[a + b*x], x, 3, -(Sqrt[1 - (a + b*x)^2]/b) + ((a + b*x)*ArcCos[a + b*x])/b} -{ArcCos[a + b*x]/x^1, x, 9, (-(1/2))*I*ArcCos[a + b*x]^2 + ArcCos[a + b*x]*Log[1 - E^(I*ArcCos[a + b*x])/(a - I*Sqrt[1 - a^2])] + ArcCos[a + b*x]*Log[1 - E^(I*ArcCos[a + b*x])/(a + I*Sqrt[1 - a^2])] - I*PolyLog[2, E^(I*ArcCos[a + b*x])/(a - I*Sqrt[1 - a^2])] - I*PolyLog[2, E^(I*ArcCos[a + b*x])/(a + I*Sqrt[1 - a^2])]} -{ArcCos[a + b*x]/x^2, x, 4, -(ArcCos[a + b*x]/x) + (b*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/Sqrt[1 - a^2]} -{ArcCos[a + b*x]/x^3, x, 5, (b*Sqrt[1 - (a + b*x)^2])/(2*(1 - a^2)*x) - ArcCos[a + b*x]/(2*x^2) + (a*b^2*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(2*(1 - a^2)^(3/2))} -{ArcCos[a + b*x]/x^4, x, 6, (b*Sqrt[1 - (a + b*x)^2])/(6*(1 - a^2)*x^2) + (a*b^2*Sqrt[1 - (a + b*x)^2])/(2*(1 - a^2)^2*x) - ArcCos[a + b*x]/(3*x^3) + ((1 + 2*a^2)*b^3*ArcTanh[(1 - a*(a + b*x))/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/(6*(1 - a^2)^(5/2))} - - -{ArcCos[a + b*x]^3, x, 5, (6*Sqrt[1 - (a + b*x)^2])/b - (6*(a + b*x)*ArcCos[a + b*x])/b - (3*Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x]^2)/b + ((a + b*x)*ArcCos[a + b*x]^3)/b} -{ArcCos[a + b*x]^2, x, 4, -2*x - (2*Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x])/b + ((a + b*x)*ArcCos[a + b*x]^2)/b} -{1/ArcCos[a + b*x], x, 3, -(SinIntegral[ArcCos[a + b*x]]/b)} -{1/ArcCos[a + b*x]^2, x, 4, Sqrt[1 - (a + b*x)^2]/(b*ArcCos[a + b*x]) - CosIntegral[ArcCos[a + b*x]]/b} -{1/ArcCos[a + b*x]^3, x, 5, Sqrt[1 - (a + b*x)^2]/(2*b*ArcCos[a + b*x]^2) + (a + b*x)/(2*b*ArcCos[a + b*x]) + SinIntegral[ArcCos[a + b*x]]/(2*b)} - - -{ArcCos[a + b*x]^(5/2), x, 7, -((15*(a + b*x)*Sqrt[ArcCos[a + b*x]])/(4*b)) - (5*Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x]^(3/2))/(2*b) + ((a + b*x)*ArcCos[a + b*x]^(5/2))/b + (15*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/(4*b)} -{ArcCos[a + b*x]^(3/2), x, 6, -((3*Sqrt[1 - (a + b*x)^2]*Sqrt[ArcCos[a + b*x]])/(2*b)) + ((a + b*x)*ArcCos[a + b*x]^(3/2))/b + (3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/(2*b)} -{ArcCos[a + b*x]^(1/2), x, 5, ((a + b*x)*Sqrt[ArcCos[a + b*x]])/b - (Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/b} -{1/ArcCos[a + b*x]^(1/2), x, 4, -((Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/b)} -{1/ArcCos[a + b*x]^(3/2), x, 5, (2*Sqrt[1 - (a + b*x)^2])/(b*Sqrt[ArcCos[a + b*x]]) - (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/b} -{1/ArcCos[a + b*x]^(5/2), x, 6, (2*Sqrt[1 - (a + b*x)^2])/(3*b*ArcCos[a + b*x]^(3/2)) + (4*(a + b*x))/(3*b*Sqrt[ArcCos[a + b*x]]) + (4*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcCos[a + b*x]]])/(3*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCos[c+d x])^n*) - - -{1/Sqrt[a + b*ArcCos[c + d*x]], x, 7, -((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d)) + (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d)} -{1/Sqrt[a - b*ArcCos[c + d*x]], x, 7, -((Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a - b*ArcCos[c + d*x]])/Sqrt[b]])/(Sqrt[b]*d)) + (Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a - b*ArcCos[c + d*x]])/Sqrt[b]]*Sin[a/b])/(Sqrt[b]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ArcCos[a+b x]^n*) - - -{ArcCos[a + b*x]/((a*d)/b + d*x), x, 7, -((I*ArcCos[a + b*x]^2)/(2*d)) + (ArcCos[a + b*x]*Log[1 + E^(2*I*ArcCos[a + b*x])])/d - (I*PolyLog[2, -E^(2*I*ArcCos[a + b*x])])/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1-(a+b x)^2)^(m/2) ArcCos[a+b x]^n*) - - -{Sqrt[1 - x^2]*ArcCos[x], x, 3, x^2/4 + (1/2)*x*Sqrt[1 - x^2]*ArcCos[x] - ArcCos[x]^2/4} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCos[a+b x^n]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCos[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcCos[a*x^2], x, 5, -((x^2*Sqrt[1 - a^2*x^4])/(8*a)) + (1/4)*x^4*ArcCos[a*x^2] + ArcSin[a*x^2]/(8*a^2)} -{x^2*ArcCos[a*x^2], x, 4, -((2*x*Sqrt[1 - a^2*x^4])/(9*a)) + (1/3)*x^3*ArcCos[a*x^2] + (2*EllipticF[ArcSin[Sqrt[a]*x], -1])/(9*a^(3/2))} -{x^1*ArcCos[a*x^2], x, 3, -(Sqrt[1 - a^2*x^4]/(2*a)) + (1/2)*x^2*ArcCos[a*x^2]} -{x^0*ArcCos[a*x^2], x, 6, x*ArcCos[a*x^2] + (2*EllipticE[ArcSin[Sqrt[a]*x], -1])/Sqrt[a] - (2*EllipticF[ArcSin[Sqrt[a]*x], -1])/Sqrt[a]} -{ArcCos[a*x^2]/x^1, x, 5, (-(1/4))*I*ArcCos[a*x^2]^2 + (1/2)*ArcCos[a*x^2]*Log[1 + E^(2*I*ArcCos[a*x^2])] - (1/4)*I*PolyLog[2, -E^(2*I*ArcCos[a*x^2])]} -{ArcCos[a*x^2]/x^2, x, 3, -(ArcCos[a*x^2]/x) - 2*Sqrt[a]*EllipticF[ArcSin[Sqrt[a]*x], -1]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2*ArcCos[a/x], x, 6, (-(1/6))*a*Sqrt[1 - a^2/x^2]*x^2 + (1/3)*x^3*ArcSec[x/a] - (1/6)*a^3*ArcTanh[Sqrt[1 - a^2/x^2]]} -{x^1*ArcCos[a/x], x, 3, (-(1/2))*a*Sqrt[1 - a^2/x^2]*x + (1/2)*x^2*ArcSec[x/a]} -{x^0*ArcCos[a/x], x, 5, x*ArcSec[x/a] - a*ArcTanh[Sqrt[1 - a^2/x^2]]} -{ArcCos[a/x]/x^1, x, 5, (1/2)*I*ArcCos[a/x]^2 - ArcCos[a/x]*Log[1 + E^(2*I*ArcCos[a/x])] + (1/2)*I*PolyLog[2, -E^(2*I*ArcCos[a/x])]} -{ArcCos[a/x]/x^2, x, 3, Sqrt[1 - a^2/x^2]/a - ArcSec[x/a]/x} -{ArcCos[a/x]/x^3, x, 5, Sqrt[1 - a^2/x^2]/(4*a*x) - ArcCsc[x/a]/(4*a^2) - ArcSec[x/a]/(2*x^2)} -{ArcCos[a/x]/x^4, x, 5, Sqrt[1 - a^2/x^2]/(3*a^3) - (1 - a^2/x^2)^(3/2)/(9*a^3) - ArcSec[x/a]/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*n/2>0*) - - -{x^2*ArcCos[Sqrt[x]], x, 8, (-(5/48))*Sqrt[1 - x]*Sqrt[x] - (5/72)*Sqrt[1 - x]*x^(3/2) - (1/18)*Sqrt[1 - x]*x^(5/2) + (1/3)*x^3*ArcCos[Sqrt[x]] - (5/96)*ArcSin[1 - 2*x]} -{x^1*ArcCos[Sqrt[x]], x, 7, (-(3/16))*Sqrt[1 - x]*Sqrt[x] - (1/8)*Sqrt[1 - x]*x^(3/2) + (1/2)*x^2*ArcCos[Sqrt[x]] - (3/32)*ArcSin[1 - 2*x]} -{x^0*ArcCos[Sqrt[x]], x, 6, (-(1/2))*Sqrt[1 - x]*Sqrt[x] + x*ArcCos[Sqrt[x]] - (1/4)*ArcSin[1 - 2*x]} -{ArcCos[Sqrt[x]]/x^1, x, 5, (-I)*ArcCos[Sqrt[x]]^2 + 2*ArcCos[Sqrt[x]]*Log[1 + E^(2*I*ArcCos[Sqrt[x]])] - I*PolyLog[2, -E^(2*I*ArcCos[Sqrt[x]])]} -{ArcCos[Sqrt[x]]/x^2, x, 3, Sqrt[1 - x]/Sqrt[x] - ArcCos[Sqrt[x]]/x} -{ArcCos[Sqrt[x]]/x^3, x, 4, Sqrt[1 - x]/(6*x^(3/2)) + Sqrt[1 - x]/(3*Sqrt[x]) - ArcCos[Sqrt[x]]/(2*x^2)} -{ArcCos[Sqrt[x]]/x^4, x, 5, Sqrt[1 - x]/(15*x^(5/2)) + (4*Sqrt[1 - x])/(45*x^(3/2)) + (8*Sqrt[1 - x])/(45*Sqrt[x]) - ArcCos[Sqrt[x]]/(3*x^3)} -{ArcCos[Sqrt[x]]/x^5, x, 6, Sqrt[1 - x]/(28*x^(7/2)) + (3*Sqrt[1 - x])/(70*x^(5/2)) + (2*Sqrt[1 - x])/(35*x^(3/2)) + (4*Sqrt[1 - x])/(35*Sqrt[x]) - ArcCos[Sqrt[x]]/(4*x^4)} - - -{ArcCos[Sqrt[x]]/Sqrt[x], x, 3, -2*Sqrt[1 - x] + 2*Sqrt[x]*ArcCos[Sqrt[x]]} - - -(* ::Subsubsection:: *) -(*n/2<0*) - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{ArcCos[a*x^n]/x, x, 5, -((I*ArcCos[a*x^n]^2)/(2*n)) + (ArcCos[a*x^n]*Log[1 + E^(2*I*ArcCos[a*x^n])])/n - (I*PolyLog[2, -E^(2*I*ArcCos[a*x^n])])/(2*n)} -{ArcCos[a*x^5]/x, x, 5, (-(1/10))*I*ArcCos[a*x^5]^2 + (1/5)*ArcCos[a*x^5]*Log[1 + E^(2*I*ArcCos[a*x^5])] - (1/10)*I*PolyLog[2, -E^(2*I*ArcCos[a*x^5])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form u ArcCos[a+b x^n]*) - - -{x^3*ArcCos[a + b*x^4], x, 4, -(Sqrt[1 - (a + b*x^4)^2]/(4*b)) + ((a + b*x^4)*ArcCos[a + b*x^4])/(4*b)} - - -{x^(n-1)*ArcCos[a + b*x^n], x, 4, -(Sqrt[1 - (a + b*x^n)^2]/(b*n)) + ((a + b*x^n)*ArcCos[a + b*x^n])/(b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b ArcCos[c+d x^2])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcCos[c+d x^2])^n when c^2=1*) - - -{(a + b*ArcCos[1 + d*x^2])^4, x, 3, 384*b^4*x + (192*b^3*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2]))/(d*x) - 48*b^2*x*(a + b*ArcCos[1 + d*x^2])^2 - (8*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2])^3)/(d*x) + x*(a + b*ArcCos[1 + d*x^2])^4} -{(a + b*ArcCos[1 + d*x^2])^3, x, 5, -24*a*b^2*x + (48*b^3*Sqrt[-2*d*x^2 - d^2*x^4])/(d*x) - 24*b^3*x*ArcCos[1 + d*x^2] - (6*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2])^2)/(d*x) + x*(a + b*ArcCos[1 + d*x^2])^3} -{(a + b*ArcCos[1 + d*x^2])^2, x, 2, -8*b^2*x - (4*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2]))/(d*x) + x*(a + b*ArcCos[1 + d*x^2])^2} -{(a + b*ArcCos[1 + d*x^2])^1, x, 4, a*x - (2*b*Sqrt[-2*d*x^2 - d^2*x^4])/(d*x) + b*x*ArcCos[1 + d*x^2]} -{1/(a + b*ArcCos[1 + d*x^2])^1, x, 1, (x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[(-d)*x^2]) + (x*Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[(-d)*x^2])} -{1/(a + b*ArcCos[1 + d*x^2])^2, x, 1, Sqrt[-2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[1 + d*x^2])) + (x*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]*Sin[a/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2]) - (x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[(-d)*x^2])} -{1/(a + b*ArcCos[1 + d*x^2])^3, x, 2, Sqrt[-2*d*x^2 - d^2*x^4]/(4*b*d*x*(a + b*ArcCos[1 + d*x^2])^2) + x/(8*b^2*(a + b*ArcCos[1 + d*x^2])) - (x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[(-d)*x^2]) - (x*Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[(-d)*x^2])} - - -{(a + b*ArcCos[-1 + d*x^2])^4, x, 3, 384*b^4*x + (192*b^3*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2]))/(d*x) - 48*b^2*x*(a + b*ArcCos[-1 + d*x^2])^2 - (8*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2])^3)/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^4} -{(a + b*ArcCos[-1 + d*x^2])^3, x, 5, -24*a*b^2*x + (48*b^3*Sqrt[2*d*x^2 - d^2*x^4])/(d*x) - 24*b^3*x*ArcCos[-1 + d*x^2] - (6*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2])^2)/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^3} -{(a + b*ArcCos[-1 + d*x^2])^2, x, 2, -8*b^2*x - (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2]))/(d*x) + x*(a + b*ArcCos[-1 + d*x^2])^2} -{(a + b*ArcCos[-1 + d*x^2])^1, x, 4, a*x - (2*b*Sqrt[2*d*x^2 - d^2*x^4])/(d*x) + b*x*ArcCos[-1 + d*x^2]} -{1/(a + b*ArcCos[-1 + d*x^2])^1, x, 1, (x*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]*Sin[a/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]) - (x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])} -{1/(a + b*ArcCos[-1 + d*x^2])^2, x, 1, Sqrt[2*d*x^2 - d^2*x^4]/(2*b*d*x*(a + b*ArcCos[-1 + d*x^2])) - (x*Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) - (x*Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2])} -{1/(a + b*ArcCos[-1 + d*x^2])^3, x, 2, Sqrt[2*d*x^2 - d^2*x^4]/(4*b*d*x*(a + b*ArcCos[-1 + d*x^2])^2) + x/(8*b^2*(a + b*ArcCos[-1 + d*x^2])) - (x*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]*Sin[a/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2]) + (x*Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcCos[c+d x^2])^(n/2) when c^2=1*) - - -{(a + b*ArcCos[1 + d*x^2])^(5/2), x, 2, -((5*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcCos[1 + d*x^2])^(3/2))/(d*x)) + x*(a + b*ArcCos[1 + d*x^2])^(5/2) - (30*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/((1/b)^(5/2)*d*x) + (30*Sqrt[Pi]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/((1/b)^(5/2)*d*x) + (30*b^2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[(1/2)*ArcCos[1 + d*x^2]]^2)/(d*x)} -{(a + b*ArcCos[1 + d*x^2])^(3/2), x, 2, -((3*b*Sqrt[-2*d*x^2 - d^2*x^4]*Sqrt[a + b*ArcCos[1 + d*x^2]])/(d*x)) + x*(a + b*ArcCos[1 + d*x^2])^(3/2) + (6*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/((1/b)^(3/2)*d*x) + (6*Sqrt[Pi]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/((1/b)^(3/2)*d*x)} -{(a + b*ArcCos[1 + d*x^2])^(1/2), x, 1, (2*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(Sqrt[1/b]*d*x) - (2*Sqrt[Pi]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(Sqrt[1/b]*d*x) - (2*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[(1/2)*ArcCos[1 + d*x^2]]^2)/(d*x)} -{1/(a + b*ArcCos[1 + d*x^2])^(1/2), x, 1, -((2*Sqrt[1/b]*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(d*x)) - (2*Sqrt[1/b]*Sqrt[Pi]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(d*x)} -{1/(a + b*ArcCos[1 + d*x^2])^(3/2), x, 1, Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[1 + d*x^2]]) + (2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(d*x) - (2*(1/b)^(3/2)*Sqrt[Pi]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(d*x)} -{1/(a + b*ArcCos[1 + d*x^2])^(5/2), x, 2, Sqrt[-2*d*x^2 - d^2*x^4]/(3*b*d*x*(a + b*ArcCos[1 + d*x^2])^(3/2)) + x/(3*b^2*Sqrt[a + b*ArcCos[1 + d*x^2]]) + (2*(1/b)^(5/2)*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(3*d*x) + (2*(1/b)^(5/2)*Sqrt[Pi]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(3*d*x)} -{1/(a + b*ArcCos[1 + d*x^2])^(7/2), x, 2, Sqrt[-2*d*x^2 - d^2*x^4]/(5*b*d*x*(a + b*ArcCos[1 + d*x^2])^(5/2)) + x/(15*b^2*(a + b*ArcCos[1 + d*x^2])^(3/2)) - Sqrt[-2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a + b*ArcCos[1 + d*x^2]]) - (2*(1/b)^(7/2)*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(15*d*x) + (2*(1/b)^(7/2)*Sqrt[Pi]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[(1/2)*ArcCos[1 + d*x^2]])/(15*d*x)} - - -{(a + b*ArcCos[-1 + d*x^2])^(5/2), x, 2, -((5*b*Sqrt[2*d*x^2 - d^2*x^4]*(a + b*ArcCos[-1 + d*x^2])^(3/2))/(d*x)) + x*(a + b*ArcCos[-1 + d*x^2])^(5/2) - (30*b^2*Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[(1/2)*ArcCos[-1 + d*x^2]]^2)/(d*x) + (30*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/((1/b)^(5/2)*d*x) + (30*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/((1/b)^(5/2)*d*x)} -{(a + b*ArcCos[-1 + d*x^2])^(3/2), x, 2, -((3*b*Sqrt[2*d*x^2 - d^2*x^4]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/(d*x)) + x*(a + b*ArcCos[-1 + d*x^2])^(3/2) + (6*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/((1/b)^(3/2)*d*x) - (6*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/((1/b)^(3/2)*d*x)} -{(a + b*ArcCos[-1 + d*x^2])^(1/2), x, 1, (2*Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[(1/2)*ArcCos[-1 + d*x^2]]^2)/(d*x) - (2*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(Sqrt[1/b]*d*x) - (2*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(Sqrt[1/b]*d*x)} -{1/(a + b*ArcCos[-1 + d*x^2])^(1/2), x, 1, -((2*Sqrt[1/b]*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(d*x)) + (2*Sqrt[1/b]*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(d*x)} -{1/(a + b*ArcCos[-1 + d*x^2])^(3/2), x, 1, Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) - (2*(1/b)^(3/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(d*x) - (2*(1/b)^(3/2)*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(d*x)} -{1/(a + b*ArcCos[-1 + d*x^2])^(5/2), x, 2, Sqrt[2*d*x^2 - d^2*x^4]/(3*b*d*x*(a + b*ArcCos[-1 + d*x^2])^(3/2)) + x/(3*b^2*Sqrt[a + b*ArcCos[-1 + d*x^2]]) + (2*(1/b)^(5/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(3*d*x) - (2*(1/b)^(5/2)*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(3*d*x)} -{1/(a + b*ArcCos[-1 + d*x^2])^(7/2), x, 2, Sqrt[2*d*x^2 - d^2*x^4]/(5*b*d*x*(a + b*ArcCos[-1 + d*x^2])^(5/2)) + x/(15*b^2*(a + b*ArcCos[-1 + d*x^2])^(3/2)) - Sqrt[2*d*x^2 - d^2*x^4]/(15*b^3*d*x*Sqrt[a + b*ArcCos[-1 + d*x^2]]) + (2*(1/b)^(7/2)*Sqrt[Pi]*Cos[a/(2*b)]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelC[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(15*d*x) + (2*(1/b)^(7/2)*Sqrt[Pi]*Cos[(1/2)*ArcCos[-1 + d*x^2]]*FresnelS[(Sqrt[1/b]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(15*d*x)} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b ArcCos[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 8, (I*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4)/(4*b*c) - ((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 + E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (3*I*b*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) - (3*b^2*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) - (3*I*b^3*PolyLog[4, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)} -{(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, (I*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3)/(3*b*c) - ((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Log[1 + E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (I*b*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c - (b^2*PolyLog[3, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 6, (I*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2)/(2*b*c) - ((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*Log[1 + E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (I*b*PolyLog[2, -E^(2*I*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{1/((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse cosines of exponentials*) - - -{ArcCos[c*E^(a + b*x)], x, 6, -((I*ArcCos[c*E^(a + b*x)]^2)/(2*b)) + (ArcCos[c*E^(a + b*x)]*Log[1 + E^(2*I*ArcCos[c*E^(a + b*x)])])/b - (I*PolyLog[2, -E^(2*I*ArcCos[c*E^(a + b*x)])])/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse cosines*) - - -{x^3*E^ArcCos[a*x], x, 6, (E^ArcCos[a*x]*Cos[2*ArcCos[a*x]])/(10*a^4) + (E^ArcCos[a*x]*Cos[4*ArcCos[a*x]])/(34*a^4) - (E^ArcCos[a*x]*Sin[2*ArcCos[a*x]])/(20*a^4) - (E^ArcCos[a*x]*Sin[4*ArcCos[a*x]])/(136*a^4)} -{x^2*E^ArcCos[a*x], x, 6, (E^ArcCos[a*x]*x)/(8*a^2) - (E^ArcCos[a*x]*Sqrt[1 - a^2*x^2])/(8*a^3) + (3*E^ArcCos[a*x]*Cos[3*ArcCos[a*x]])/(40*a^3) - (E^ArcCos[a*x]*Sin[3*ArcCos[a*x]])/(40*a^3)} -{x^1*E^ArcCos[a*x], x, 5, (E^ArcCos[a*x]*Cos[2*ArcCos[a*x]])/(5*a^2) - (E^ArcCos[a*x]*Sin[2*ArcCos[a*x]])/(10*a^2)} -{x^0*E^ArcCos[a*x], x, 2, (1/2)*E^ArcCos[a*x]*x - (E^ArcCos[a*x]*Sqrt[1 - a^2*x^2])/(2*a)} -{E^ArcCos[a*x]/x^1, x, 6, I*E^ArcCos[a*x] - 2*I*E^ArcCos[a*x]*Hypergeometric2F1[-(I/2), 1, 1 - I/2, -E^(2*I*ArcCos[a*x])]} -{E^ArcCos[a*x]/x^2, x, 6, (1 + I)*a*E^((1 + I)*ArcCos[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^(2*I*ArcCos[a*x])] - (2 + 2*I)*a*E^((1 + I)*ArcCos[a*x])*Hypergeometric2F1[1/2 - I/2, 2, 3/2 - I/2, -E^(2*I*ArcCos[a*x])]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse cosines*) - - -{ArcCos[c/(a + b*x)], x, 6, ((a + b*x)*ArcSec[a/c + (b*x)/c])/b - (c*ArcTanh[Sqrt[1 - c^2/(a + b*x)^2]])/b} - - -{x/(Sqrt[1 - x^2]*Sqrt[ArcCos[x]]), x, 3, -Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[x]]]} -{x/(Sqrt[1 - x^2]*ArcCos[x]), x, 2, -CosIntegral[ArcCos[x]]} - - -{ArcCos[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2], x, 2, -((Sqrt[(-b)*x^2]*ArcCos[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x))} -{1/(ArcCos[Sqrt[1 + b*x^2]]*Sqrt[1 + b*x^2]), x, 2, -((Sqrt[(-b)*x^2]*Log[ArcCos[Sqrt[1 + b*x^2]]])/(b*x))} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m deleted file mode 100644 index 5b594e7..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.2 (d x)^m (a+b arctan(c x^n))^p.m +++ /dev/null @@ -1,346 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*ArcTan[c*x]), x, 4, -(b*x)/(6*c^5) + (b*x^3)/(18*c^3) - (b*x^5)/(30*c) + (b*ArcTan[c*x])/(6*c^6) + (x^6*(a + b*ArcTan[c*x]))/6} -{x^4*(a + b*ArcTan[c*x]), x, 4, (b*x^2)/(10*c^3) - (b*x^4)/(20*c) + (x^5*(a + b*ArcTan[c*x]))/5 - (b*Log[1 + c^2*x^2])/(10*c^5)} -{x^3*(a + b*ArcTan[c*x]), x, 4, (b*x)/(4*c^3) - (b*x^3)/(12*c) - (b*ArcTan[c*x])/(4*c^4) + (x^4*(a + b*ArcTan[c*x]))/4} -{x^2*(a + b*ArcTan[c*x]), x, 4, -(b*x^2)/(6*c) + (x^3*(a + b*ArcTan[c*x]))/3 + (b*Log[1 + c^2*x^2])/(6*c^3)} -{x^1*(a + b*ArcTan[c*x]), x, 3, -(b*x)/(2*c) + (b*ArcTan[c*x])/(2*c^2) + (x^2*(a + b*ArcTan[c*x]))/2} -{x^0*(a + b*ArcTan[c*x]), x, 3, a*x + b*x*ArcTan[c*x] - (b*Log[1 + c^2*x^2])/(2*c)} -{(a + b*ArcTan[c*x])/x^1, x, 3, a*Log[x] + (I/2)*b*PolyLog[2, (-I)*c*x] - (I/2)*b*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])/x^2, x, 5, -((a + b*ArcTan[c*x])/x) + b*c*Log[x] - (b*c*Log[1 + c^2*x^2])/2} -{(a + b*ArcTan[c*x])/x^3, x, 3, -(b*c)/(2*x) - (b*c^2*ArcTan[c*x])/2 - (a + b*ArcTan[c*x])/(2*x^2)} -{(a + b*ArcTan[c*x])/x^4, x, 4, -(b*c)/(6*x^2) - (a + b*ArcTan[c*x])/(3*x^3) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^2])/6} -{(a + b*ArcTan[c*x])/x^5, x, 4, -(b*c)/(12*x^3) + (b*c^3)/(4*x) + (b*c^4*ArcTan[c*x])/4 - (a + b*ArcTan[c*x])/(4*x^4)} -{(a + b*ArcTan[c*x])/x^6, x, 4, -(b*c)/(20*x^4) + (b*c^3)/(10*x^2) - (a + b*ArcTan[c*x])/(5*x^5) + (b*c^5*Log[x])/5 - (b*c^5*Log[1 + c^2*x^2])/10} - - -{x^5*(a + b*ArcTan[c*x])^2, x, 16, -(a*b*x)/(3*c^5) - (4*b^2*x^2)/(45*c^4) + (b^2*x^4)/(60*c^2) - (b^2*x*ArcTan[c*x])/(3*c^5) + (b*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*x^5*(a + b*ArcTan[c*x]))/(15*c) + (a + b*ArcTan[c*x])^2/(6*c^6) + (x^6*(a + b*ArcTan[c*x])^2)/6 + (23*b^2*Log[1 + c^2*x^2])/(90*c^6)} -{x^4*(a + b*ArcTan[c*x])^2, x, 14, (-3*b^2*x)/(10*c^4) + (b^2*x^3)/(30*c^2) + (3*b^2*ArcTan[c*x])/(10*c^5) + (b*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*x^4*(a + b*ArcTan[c*x]))/(10*c) + ((I/5)*(a + b*ArcTan[c*x])^2)/c^5 + (x^5*(a + b*ArcTan[c*x])^2)/5 + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) + ((I/5)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5} -{x^3*(a + b*ArcTan[c*x])^2, x, 11, (a*b*x)/(2*c^3) + (b^2*x^2)/(12*c^2) + (b^2*x*ArcTan[c*x])/(2*c^3) - (b*x^3*(a + b*ArcTan[c*x]))/(6*c) - (a + b*ArcTan[c*x])^2/(4*c^4) + (x^4*(a + b*ArcTan[c*x])^2)/4 - (b^2*Log[1 + c^2*x^2])/(3*c^4)} -{x^2*(a + b*ArcTan[c*x])^2, x, 9, (b^2*x)/(3*c^2) - (b^2*ArcTan[c*x])/(3*c^3) - (b*x^2*(a + b*ArcTan[c*x]))/(3*c) - ((I/3)*(a + b*ArcTan[c*x])^2)/c^3 + (x^3*(a + b*ArcTan[c*x])^2)/3 - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) - ((I/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{x^1*(a + b*ArcTan[c*x])^2, x, 6, -((a*b*x)/c) - (b^2*x*ArcTan[c*x])/c + (a + b*ArcTan[c*x])^2/(2*c^2) + (x^2*(a + b*ArcTan[c*x])^2)/2 + (b^2*Log[1 + c^2*x^2])/(2*c^2)} -{x^0*(a + b*ArcTan[c*x])^2, x, 5, (I*(a + b*ArcTan[c*x])^2)/c + x*(a + b*ArcTan[c*x])^2 + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c} -{(a + b*ArcTan[c*x])^2/x^1, x, 6, 2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{(a + b*ArcTan[c*x])^2/x^2, x, 4, (-I)*c*(a + b*ArcTan[c*x])^2 - (a + b*ArcTan[c*x])^2/x + 2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)]} -{(a + b*ArcTan[c*x])^2/x^3, x, 8, -((b*c*(a + b*ArcTan[c*x]))/x) - (c^2*(a + b*ArcTan[c*x])^2)/2 - (a + b*ArcTan[c*x])^2/(2*x^2) + b^2*c^2*Log[x] - (b^2*c^2*Log[1 + c^2*x^2])/2} -{(a + b*ArcTan[c*x])^2/x^4, x, 8, -(b^2*c^2)/(3*x) - (b^2*c^3*ArcTan[c*x])/3 - (b*c*(a + b*ArcTan[c*x]))/(3*x^2) + (I/3)*c^3*(a + b*ArcTan[c*x])^2 - (a + b*ArcTan[c*x])^2/(3*x^3) - (2*b*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + (I/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 - I*c*x)]} -{(a + b*ArcTan[c*x])^2/x^5, x, 13, -(b^2*c^2)/(12*x^2) - (b*c*(a + b*ArcTan[c*x]))/(6*x^3) + (b*c^3*(a + b*ArcTan[c*x]))/(2*x) + (c^4*(a + b*ArcTan[c*x])^2)/4 - (a + b*ArcTan[c*x])^2/(4*x^4) - (2*b^2*c^4*Log[x])/3 + (b^2*c^4*Log[1 + c^2*x^2])/3} - - -{x^5*(a + b*ArcTan[c*x])^3, x, 33, (19*b^3*x)/(60*c^5) - (b^3*x^3)/(60*c^3) - (19*b^3*ArcTan[c*x])/(60*c^6) - (4*b^2*x^2*(a + b*ArcTan[c*x]))/(15*c^4) + (b^2*x^4*(a + b*ArcTan[c*x]))/(20*c^2) - (((23*I)/30)*b*(a + b*ArcTan[c*x])^2)/c^6 - (b*x*(a + b*ArcTan[c*x])^2)/(2*c^5) + (b*x^3*(a + b*ArcTan[c*x])^2)/(6*c^3) - (b*x^5*(a + b*ArcTan[c*x])^2)/(10*c) + (a + b*ArcTan[c*x])^3/(6*c^6) + (x^6*(a + b*ArcTan[c*x])^3)/6 - (23*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^6) - (((23*I)/30)*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^6} -{x^4*(a + b*ArcTan[c*x])^3, x, 24, (-9*a*b^2*x)/(10*c^4) - (b^3*x^2)/(20*c^3) - (9*b^3*x*ArcTan[c*x])/(10*c^4) + (b^2*x^3*(a + b*ArcTan[c*x]))/(10*c^2) + (9*b*(a + b*ArcTan[c*x])^2)/(20*c^5) + (3*b*x^2*(a + b*ArcTan[c*x])^2)/(10*c^3) - (3*b*x^4*(a + b*ArcTan[c*x])^2)/(20*c) + ((I/5)*(a + b*ArcTan[c*x])^3)/c^5 + (x^5*(a + b*ArcTan[c*x])^3)/5 + (3*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(5*c^5) + (b^3*Log[1 + c^2*x^2])/(2*c^5) + (((3*I)/5)*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5 + (3*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(10*c^5)} -{x^3*(a + b*ArcTan[c*x])^3, x, 18, -(b^3*x)/(4*c^3) + (b^3*ArcTan[c*x])/(4*c^4) + (b^2*x^2*(a + b*ArcTan[c*x]))/(4*c^2) + (I*b*(a + b*ArcTan[c*x])^2)/c^4 + (3*b*x*(a + b*ArcTan[c*x])^2)/(4*c^3) - (b*x^3*(a + b*ArcTan[c*x])^2)/(4*c) - (a + b*ArcTan[c*x])^3/(4*c^4) + (x^4*(a + b*ArcTan[c*x])^3)/4 + (2*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 + (I*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4} -{x^2*(a + b*ArcTan[c*x])^3, x, 12, (a*b^2*x)/c^2 + (b^3*x*ArcTan[c*x])/c^2 - (b*(a + b*ArcTan[c*x])^2)/(2*c^3) - (b*x^2*(a + b*ArcTan[c*x])^2)/(2*c) - ((I/3)*(a + b*ArcTan[c*x])^3)/c^3 + (x^3*(a + b*ArcTan[c*x])^3)/3 - (b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (b^3*Log[1 + c^2*x^2])/(2*c^3) - (I*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 - (b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)} -{x^1*(a + b*ArcTan[c*x])^3, x, 8, (((-3*I)/2)*b*(a + b*ArcTan[c*x])^2)/c^2 - (3*b*x*(a + b*ArcTan[c*x])^2)/(2*c) + (a + b*ArcTan[c*x])^3/(2*c^2) + (x^2*(a + b*ArcTan[c*x])^3)/2 - (3*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 - (((3*I)/2)*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^2} -{x^0*(a + b*ArcTan[c*x])^3, x, 5, (I*(a + b*ArcTan[c*x])^3)/c + x*(a + b*ArcTan[c*x])^3 + (3*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c + ((3*I)*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c + (3*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c)} -{(a + b*ArcTan[c*x])^3/x^1, x, 8, 2*(a + b*ArcTan[c*x])^3*ArcTanh[1 - 2/(1 + I*c*x)] - ((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)] + ((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)] - (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/2 + ((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)] - ((3*I)/4)*b^3*PolyLog[4, -1 + 2/(1 + I*c*x)]} -{(a + b*ArcTan[c*x])^3/x^2, x, 5, (-I)*c*(a + b*ArcTan[c*x])^3 - (a + b*ArcTan[c*x])^3/x + 3*b*c*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)] - (3*I)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)] + (3*b^3*c*PolyLog[3, -1 + 2/(1 - I*c*x)])/2} -{(a + b*ArcTan[c*x])^3/x^3, x, 7, ((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2 - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*x) - (c^2*(a + b*ArcTan[c*x])^3)/2 - (a + b*ArcTan[c*x])^3/(2*x^2) + 3*b^2*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - ((3*I)/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)]} -{(a + b*ArcTan[c*x])^3/x^4, x, 14, -((b^2*c^2*(a + b*ArcTan[c*x]))/x) - (b*c^3*(a + b*ArcTan[c*x])^2)/2 - (b*c*(a + b*ArcTan[c*x])^2)/(2*x^2) + (I/3)*c^3*(a + b*ArcTan[c*x])^3 - (a + b*ArcTan[c*x])^3/(3*x^3) + b^3*c^3*Log[x] - (b^3*c^3*Log[1 + c^2*x^2])/2 - b*c^3*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)] + I*b^2*c^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)] - (b^3*c^3*PolyLog[3, -1 + 2/(1 - I*c*x)])/2} -{(a + b*ArcTan[c*x])^3/x^5, x, 16, -(b^3*c^3)/(4*x) - (b^3*c^4*ArcTan[c*x])/4 - (b^2*c^2*(a + b*ArcTan[c*x]))/(4*x^2) + I*b*c^4*(a + b*ArcTan[c*x])^2 - (b*c*(a + b*ArcTan[c*x])^2)/(4*x^3) + (3*b*c^3*(a + b*ArcTan[c*x])^2)/(4*x) + (c^4*(a + b*ArcTan[c*x])^3)/4 - (a + b*ArcTan[c*x])^3/(4*x^4) - 2*b^2*c^4*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + I*b^3*c^4*PolyLog[2, -1 + 2/(1 - I*c*x)]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^1/ArcTan[a*x], x, 0, Unintegrable[x/ArcTan[a*x], x]} -{x^0/ArcTan[a*x], x, 0, Unintegrable[1/ArcTan[a*x], x]} -{1/(x^1*ArcTan[a*x]), x, 0, Unintegrable[1/(x*ArcTan[a*x]), x]} - - -{x^1/ArcTan[a*x]^2, x, 0, Unintegrable[x/ArcTan[a*x]^2, x]} -{x^0/ArcTan[a*x]^2, x, 0, Unintegrable[1/ArcTan[a*x]^2, x]} -{1/(x^1*ArcTan[a*x]^2), x, 0, Unintegrable[1/(x*ArcTan[a*x]^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c x])^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x*Sqrt[ArcTan[a*x]], x]} -{Sqrt[ArcTan[a*x]], x, 0, Unintegrable[Sqrt[ArcTan[a*x]], x]} -{Sqrt[ArcTan[a*x]]/x, x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/x, x]} - - -{x*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x*ArcTan[a*x]^(3/2), x]} -{ArcTan[a*x]^(3/2), x, 0, Unintegrable[ArcTan[a*x]^(3/2), x]} -{ArcTan[a*x]^(3/2)/x, x, 0, Unintegrable[ArcTan[a*x]^(3/2)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x/Sqrt[ArcTan[a*x]], x]} -{1/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[1/Sqrt[ArcTan[a*x]], x]} -{1/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[ArcTan[a*x]]), x]} - - -{x/ArcTan[a*x]^(3/2), x, 0, Unintegrable[x/ArcTan[a*x]^(3/2), x]} -{1/ArcTan[a*x]^(3/2), x, 0, Unintegrable[ArcTan[a*x]^(-3/2), x]} -{1/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[1/(x*ArcTan[a*x]^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTan[c x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[x]*ArcTan[x], x, 12, -((4*Sqrt[x])/3) - (1/3)*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + (1/3)*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + (2/3)*x^(3/2)*ArcTan[x] - Log[1 - Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2]) + Log[1 + Sqrt[2]*Sqrt[x] + x]/(3*Sqrt[2])} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTan[c x])^(p/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x])^p with m symbolic*) - - -{(d*x)^m*(a + b*ArcTan[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x])^3, x]} -{(d*x)^m*(a + b*ArcTan[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x])^2, x]} -{(d*x)^m*(a + b*ArcTan[c*x])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTan[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(d^2*(1 + m)*(2 + m))} -{(d*x)^m/(a + b*ArcTan[c*x])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTan[c*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x])^p with p symbolic*) - - -{(a + b*ArcTan[c*x])^p, x, 0, Unintegrable[(a + b*ArcTan[c*x])^p, x]} - - -{(d*x)^m*(a + b*ArcTan[c*x])^p, x, 0, Unintegrable[(d*x)^m*(a + b* ArcTan[c*x])^p, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c x^2])^p*) - - -{x^7*(a + b*ArcTan[c*x^2]), x, 5, (b*x^2)/(8*c^3) - (b*x^6)/(24*c) - (b*ArcTan[c*x^2])/(8*c^4) + (1/8)*x^8*(a + b*ArcTan[c*x^2])} -{x^5*(a + b*ArcTan[c*x^2]), x, 4, -((b*x^4)/(12*c)) + (1/6)*x^6*(a + b*ArcTan[c*x^2]) + (b*Log[1 + c^2*x^4])/(12*c^3)} -{x^3*(a + b*ArcTan[c*x^2]), x, 4, -((b*x^2)/(4*c)) + (b*ArcTan[c*x^2])/(4*c^2) + (1/4)*x^4*(a + b*ArcTan[c*x^2])} -{x^1*(a + b*ArcTan[c*x^2]), x, 2, (1/2)*x^2*(a + b*ArcTan[c*x^2]) - (b*Log[1 + c^2*x^4])/(4*c)} -{(a + b*ArcTan[c*x^2])/x^1, x, 4, a*Log[x] + (1/4)*I*b*PolyLog[2, (-I)*c*x^2] - (1/4)*I*b*PolyLog[2, I*c*x^2]} -{(a + b*ArcTan[c*x^2])/x^3, x, 5, -((a + b*ArcTan[c*x^2])/(2*x^2)) + b*c*Log[x] - (1/4)*b*c*Log[1 + c^2*x^4]} -{(a + b*ArcTan[c*x^2])/x^5, x, 4, -((b*c)/(4*x^2)) - (1/4)*b*c^2*ArcTan[c*x^2] - (a + b*ArcTan[c*x^2])/(4*x^4)} -{(a + b*ArcTan[c*x^2])/x^7, x, 4, -((b*c)/(12*x^4)) - (a + b*ArcTan[c*x^2])/(6*x^6) - (1/3)*b*c^3*Log[x] + (1/12)*b*c^3*Log[1 + c^2*x^4]} - -{x^4*(a + b*ArcTan[c*x^2]), x, 11, -((2*b*x^3)/(15*c)) + (1/5)*x^5*(a + b*ArcTan[c*x^2]) - (b*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(5*Sqrt[2]*c^(5/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(5*Sqrt[2]*c^(5/2)) + (b*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]*c^(5/2)) - (b*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]*c^(5/2))} -{x^2*(a + b*ArcTan[c*x^2]), x, 11, -((2*b*x)/(3*c)) + (1/3)*x^3*(a + b*ArcTan[c*x^2]) - (b*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]*c^(3/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]*c^(3/2)) - (b*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2)) + (b*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2))} -{x^0*(a + b*ArcTan[c*x^2]), x, 11, a*x + b*x*ArcTan[c*x^2] + (b*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) + (b*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c])} -{(a + b*ArcTan[c*x^2])/x^2, x, 10, -((a + b*ArcTan[c*x^2])/x) - (b*Sqrt[c]*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/Sqrt[2] + (b*Sqrt[c]*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/Sqrt[2] - (b*Sqrt[c]*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]) + (b*Sqrt[c]*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2])} -{(a + b*ArcTan[c*x^2])/x^4, x, 11, -((2*b*c)/(3*x)) - (a + b*ArcTan[c*x^2])/(3*x^3) + (b*c^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^(3/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]) + (b*c^(3/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])} -{(a + b*ArcTan[c*x^2])/x^6, x, 11, -((2*b*c)/(15*x^3)) - (a + b*ArcTan[c*x^2])/(5*x^5) + (b*c^(5/2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(5*Sqrt[2]) - (b*c^(5/2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(5*Sqrt[2]) + (b*c^(5/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2]) - (b*c^(5/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(10*Sqrt[2])} - - -{x^7*(a + b*ArcTan[c*x^2])^2, x, 12, (a*b*x^2)/(4*c^3) + (b^2*x^4)/(24*c^2) + (b^2*x^2*ArcTan[c*x^2])/(4*c^3) - (b*x^6*(a + b*ArcTan[c*x^2]))/(12*c) - (a + b*ArcTan[c*x^2])^2/(8*c^4) + (1/8)*x^8*(a + b*ArcTan[c*x^2])^2 - (b^2*Log[1 + c^2*x^4])/(6*c^4)} -{x^5*(a + b*ArcTan[c*x^2])^2, x, 10, (b^2*x^2)/(6*c^2) - (b^2*ArcTan[c*x^2])/(6*c^3) - (b*x^4*(a + b*ArcTan[c*x^2]))/(6*c) - (I*(a + b*ArcTan[c*x^2])^2)/(6*c^3) + (1/6)*x^6*(a + b*ArcTan[c*x^2])^2 - (b*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/(3*c^3) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/(6*c^3)} -{x^3*(a + b*ArcTan[c*x^2])^2, x, 7, -((a*b*x^2)/(2*c)) - (b^2*x^2*ArcTan[c*x^2])/(2*c) + (a + b*ArcTan[c*x^2])^2/(4*c^2) + (1/4)*x^4*(a + b*ArcTan[c*x^2])^2 + (b^2*Log[1 + c^2*x^4])/(4*c^2)} -{x^1*(a + b*ArcTan[c*x^2])^2, x, 6, (I*(a + b*ArcTan[c*x^2])^2)/(2*c) + (1/2)*x^2*(a + b*ArcTan[c*x^2])^2 + (b*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/c + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/(2*c)} -{(a + b*ArcTan[c*x^2])^2/x^1, x, 7, (a + b*ArcTan[c*x^2])^2*ArcTanh[1 - 2/(1 + I*c*x^2)] - (1/2)*I*b*(a + b*ArcTan[c*x^2])*PolyLog[2, 1 - 2/(1 + I*c*x^2)] + (1/2)*I*b*(a + b*ArcTan[c*x^2])*PolyLog[2, -1 + 2/(1 + I*c*x^2)] - (1/4)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x^2)] + (1/4)*b^2*PolyLog[3, -1 + 2/(1 + I*c*x^2)]} -{(a + b*ArcTan[c*x^2])^2/x^3, x, 5, (-(1/2))*I*c*(a + b*ArcTan[c*x^2])^2 - (a + b*ArcTan[c*x^2])^2/(2*x^2) + b*c*(a + b*ArcTan[c*x^2])*Log[2 - 2/(1 - I*c*x^2)] - (1/2)*I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x^2)]} -{(a + b*ArcTan[c*x^2])^2/x^5, x, 9, -((b*c*(a + b*ArcTan[c*x^2]))/(2*x^2)) - (1/4)*c^2*(a + b*ArcTan[c*x^2])^2 - (a + b*ArcTan[c*x^2])^2/(4*x^4) + b^2*c^2*Log[x] - (1/4)*b^2*c^2*Log[1 + c^2*x^4]} - -{x^2*(a + b*ArcTan[c*x^2])^2, x, 86, -((4*a*b*x)/(3*c)) + (2/9)*I*a*b*x^3 + (4*(-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/(3*c^(3/2)) - (2*(-1)^(1/4)*a*b*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) - (4*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/(3*c^(3/2)) - (2*(-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + (2*(-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + (2*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) - (2*(-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - (2*I*b^2*x*Log[1 - I*c*x^2])/(3*c) - (1/9)*b^2*x^3*Log[1 - I*c*x^2] - ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/(3*c^(3/2)) - (1/9)*I*b*x^3*(2*a + I*b*Log[1 - I*c*x^2]) - ((-1)^(1/4)*b*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]))/(3*c^(3/2)) + (1/12)*x^3*(2*a + I*b*Log[1 - I*c*x^2])^2 + (2*I*b^2*x*Log[1 + I*c*x^2])/(3*c) - (1/3)*I*a*b*x^3*Log[1 + I*c*x^2] + ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/(3*c^(3/2)) + (1/6)*b^2*x^3*Log[1 - I*c*x^2]*Log[1 + I*c*x^2] - (1/12)*b^2*x^3*Log[1 + I*c*x^2]^2 + ((-1)^(1/4)*b^2*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(1/4)*b^2*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(6*c^(3/2)) + ((-1)^(3/4)*b^2*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(6*c^(3/2)) - ((-1)^(3/4)*b^2*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(6*c^(3/2)) - ((-1)^(1/4)*b^2*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(6*c^(3/2))} -{x^0*(a + b*ArcTan[c*x^2])^2, x, 69, a^2*x - (2*(-1)^(3/4)*a*b*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] + ((-1)^(3/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + (2*(-1)^(3/4)*a*b*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] - ((-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + (2*(-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + (2*(-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/Sqrt[c] + ((-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + I*a*b*x*Log[1 - I*c*x^2] + ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - ((-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - (1/4)*b^2*x*Log[1 - I*c*x^2]^2 - I*a*b*x*Log[1 + I*c*x^2] - ((-1)^(1/4)*b^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + ((-1)^(1/4)*b^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + (1/2)*b^2*x*Log[1 - I*c*x^2]*Log[1 + I*c*x^2] - (1/4)*b^2*x*Log[1 + I*c*x^2]^2 + ((-1)^(3/4)*b^2*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(3/4)*b^2*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) + ((-1)^(1/4)*b^2*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(1/4)*b^2*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(1/4)*b^2*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(3/4)*b^2*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c])} -{(a + b*ArcTan[c*x^2])^2/x^2, x, 47, (-1)^(1/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2 - 2*(-1)^(1/4)*a*b*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2 - 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)] + 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] + 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)] - 2*(-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))] + (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2] - (-1)^(1/4)*b*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]) - (2*a + I*b*Log[1 - I*c*x^2])^2/(4*x) + (I*a*b*Log[1 + I*c*x^2])/x + (-1)^(3/4)*b^2*Sqrt[c]*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] + (-1)^(3/4)*b^2*Sqrt[c]*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(2*x) + (b^2*Log[1 + I*c*x^2]^2)/(4*x) + (-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)] + (-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (1/2)*(-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)] + (-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/2)*(-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/2)*(-1)^(3/4)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/2)*(-1)^(1/4)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)]} -{(a + b*ArcTan[c*x^2])^2/x^4, x, 64, -((2*a*b*c)/(3*x)) - (4/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x] + (1/3)*(-1)^(3/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2 + (2/3)*(-1)^(3/4)*a*b*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x] - (4/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x] - (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2 + (2/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)] - (2/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (2/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)] - (2/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (I*b^2*c*Log[1 - I*c*x^2])/(3*x) - (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2] - (b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(3*x) - (1/3)*(-1)^(3/4)*b*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]) - (2*a + I*b*Log[1 - I*c*x^2])^2/(12*x^3) + (I*a*b*Log[1 + I*c*x^2])/(3*x^3) + (2*I*b^2*c*Log[1 + I*c*x^2])/(3*x) - (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(6*x^3) + (b^2*Log[1 + I*c*x^2]^2)/(12*x^3) + (1/3)*(-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)] + (1/3)*(-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (1/6)*(-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)] + (1/3)*(-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/6)*(-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/6)*(-1)^(1/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/6)*(-1)^(3/4)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)]} -{(a + b*ArcTan[c*x^2])^2/x^6, x, 77, -((2*a*b*c)/(15*x^3)) + (2*I*a*b*c^2)/(5*x) - (8*b^2*c^2)/(15*x) - (4/15)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x] - (1/5)*(-1)^(1/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2 + (2/5)*(-1)^(1/4)*a*b*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x] + (4/15)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x] + (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2 + (2/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)] - (2/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (2/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)] + (2/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)] - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))] - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (I*b^2*c*Log[1 - I*c*x^2])/(15*x^3) - (b^2*c^2*Log[1 - I*c*x^2])/(5*x) + (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2] - (b*c*(2*a + I*b*Log[1 - I*c*x^2]))/(15*x^3) - (I*b*c^2*(2*a + I*b*Log[1 - I*c*x^2]))/(5*x) + (1/5)*(-1)^(1/4)*b*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]) - (2*a + I*b*Log[1 - I*c*x^2])^2/(20*x^5) + (I*a*b*Log[1 + I*c*x^2])/(5*x^5) + (2*I*b^2*c*Log[1 + I*c*x^2])/(15*x^3) - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2] - (b^2*Log[1 - I*c*x^2]*Log[1 + I*c*x^2])/(10*x^5) + (b^2*Log[1 + I*c*x^2]^2)/(20*x^5) - (1/5)*(-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)] - (1/5)*(-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)] + (1/10)*(-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)] - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)] - (1/5)*(-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/10)*(-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/10)*(-1)^(3/4)*b^2*c^(5/2)*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)] + (1/10)*(-1)^(1/4)*b^2*c^(5/2)*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)]} - - -{x^3*(a + b*ArcTan[c*x^2])^3, x, 9, -((3*I*b*(a + b*ArcTan[c*x^2])^2)/(4*c^2)) - (3*b*x^2*(a + b*ArcTan[c*x^2])^2)/(4*c) + (a + b*ArcTan[c*x^2])^3/(4*c^2) + (1/4)*x^4*(a + b*ArcTan[c*x^2])^3 - (3*b^2*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/(2*c^2) - (3*I*b^3*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/(4*c^2)} -{x^1*(a + b*ArcTan[c*x^2])^3, x, 6, (I*(a + b*ArcTan[c*x^2])^3)/(2*c) + (1/2)*x^2*(a + b*ArcTan[c*x^2])^3 + (3*b*(a + b*ArcTan[c*x^2])^2*Log[2/(1 + I*c*x^2)])/(2*c) + (3*I*b^2*(a + b*ArcTan[c*x^2])*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/(2*c) + (3*b^3*PolyLog[3, 1 - 2/(1 + I*c*x^2)])/(4*c)} -{(a + b*ArcTan[c*x^2])^3/x^1, x, 9, (a + b*ArcTan[c*x^2])^3*ArcTanh[1 - 2/(1 + I*c*x^2)] - (3/4)*I*b*(a + b*ArcTan[c*x^2])^2*PolyLog[2, 1 - 2/(1 + I*c*x^2)] + (3/4)*I*b*(a + b*ArcTan[c*x^2])^2*PolyLog[2, -1 + 2/(1 + I*c*x^2)] - (3/4)*b^2*(a + b*ArcTan[c*x^2])*PolyLog[3, 1 - 2/(1 + I*c*x^2)] + (3/4)*b^2*(a + b*ArcTan[c*x^2])*PolyLog[3, -1 + 2/(1 + I*c*x^2)] + (3/8)*I*b^3*PolyLog[4, 1 - 2/(1 + I*c*x^2)] - (3/8)*I*b^3*PolyLog[4, -1 + 2/(1 + I*c*x^2)]} -{(a + b*ArcTan[c*x^2])^3/x^3, x, 6, (-(1/2))*I*c*(a + b*ArcTan[c*x^2])^3 - (a + b*ArcTan[c*x^2])^3/(2*x^2) + (3/2)*b*c*(a + b*ArcTan[c*x^2])^2*Log[2 - 2/(1 - I*c*x^2)] - (3/2)*I*b^2*c*(a + b*ArcTan[c*x^2])*PolyLog[2, -1 + 2/(1 - I*c*x^2)] + (3/4)*b^3*c*PolyLog[3, -1 + 2/(1 - I*c*x^2)]} -{(a + b*ArcTan[c*x^2])^3/x^5, x, 8, (-(3/4))*I*b*c^2*(a + b*ArcTan[c*x^2])^2 - (3*b*c*(a + b*ArcTan[c*x^2])^2)/(4*x^2) - (1/4)*c^2*(a + b*ArcTan[c*x^2])^3 - (a + b*ArcTan[c*x^2])^3/(4*x^4) + (3/2)*b^2*c^2*(a + b*ArcTan[c*x^2])*Log[2 - 2/(1 - I*c*x^2)] - (3/4)*I*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x^2)]} - -(* {x^2*(a + b*ArcTan[c*x^2])^3, x, 86, 0} -{x^0*(a + b*ArcTan[c*x^2])^3, x, 69, 0} -{(a + b*ArcTan[c*x^2])^3/x^2, x, 47, 0} -{(a + b*ArcTan[c*x^2])^3/x^4, x, 64, 0} -{(a + b*ArcTan[c*x^2])^3/x^6, x, 77, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^2])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTan[c*x^2])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^2])^3, x]} -{(d*x)^m*(a + b*ArcTan[c*x^2])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^2])^2, x]} -{(d*x)^m*(a + b*ArcTan[c*x^2])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTan[c*x^2]))/(d*(1 + m)) - (2*b*c*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, (-c^2)*x^4])/(d^3*(1 + m)*(3 + m))} -{(d*x)^m/(a + b*ArcTan[c*x^2])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTan[c*x^2]), x]} -{(d*x)^m/(a + b*ArcTan[c*x^2])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTan[c*x^2])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c x^3])^p*) - - -{x^11*(a + b*ArcTan[c*x^3]), x, 5, (b*x^3)/(12*c^3) - (b*x^9)/(36*c) - (b*ArcTan[c*x^3])/(12*c^4) + (1/12)*x^12*(a + b*ArcTan[c*x^3])} -{x^8*(a + b*ArcTan[c*x^3]), x, 4, -((b*x^6)/(18*c)) + (1/9)*x^9*(a + b*ArcTan[c*x^3]) + (b*Log[1 + c^2*x^6])/(18*c^3)} -{x^5*(a + b*ArcTan[c*x^3]), x, 4, -((b*x^3)/(6*c)) + (b*ArcTan[c*x^3])/(6*c^2) + (1/6)*x^6*(a + b*ArcTan[c*x^3])} -{x^2*(a + b*ArcTan[c*x^3]), x, 2, (1/3)*x^3*(a + b*ArcTan[c*x^3]) - (b*Log[1 + c^2*x^6])/(6*c)} -{(a + b*ArcTan[c*x^3])/x^1, x, 4, a*Log[x] + (1/6)*I*b*PolyLog[2, (-I)*c*x^3] - (1/6)*I*b*PolyLog[2, I*c*x^3]} -{(a + b*ArcTan[c*x^3])/x^4, x, 5, -((a + b*ArcTan[c*x^3])/(3*x^3)) + b*c*Log[x] - (1/6)*b*c*Log[1 + c^2*x^6]} -{(a + b*ArcTan[c*x^3])/x^7, x, 4, -((b*c)/(6*x^3)) - (1/6)*b*c^2*ArcTan[c*x^3] - (a + b*ArcTan[c*x^3])/(6*x^6)} -{(a + b*ArcTan[c*x^3])/x^10, x, 4, -((b*c)/(18*x^6)) - (a + b*ArcTan[c*x^3])/(9*x^9) - (1/3)*b*c^3*Log[x] + (1/18)*b*c^3*Log[1 + c^2*x^6]} - -{x^3*(a + b*ArcTan[c*x^3]), x, 12, -((3*b*x)/(4*c)) + (b*ArcTan[c^(1/3)*x])/(4*c^(4/3)) + (1/4)*x^4*(a + b*ArcTan[c*x^3]) - (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(8*c^(4/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(8*c^(4/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3))} -{x^0*(a + b*ArcTan[c*x^3]), x, 9, a*x + b*x*ArcTan[c*x^3] + (Sqrt[3]*b*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + (b*Log[1 + c^(2/3)*x^2])/(2*c^(1/3)) - (b*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))} -{(a + b*ArcTan[c*x^3])/x^3, x, 11, (1/2)*b*c^(2/3)*ArcTan[c^(1/3)*x] - (a + b*ArcTan[c*x^3])/(2*x^2) - (1/4)*b*c^(2/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + (1/4)*b*c^(2/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - (1/8)*Sqrt[3]*b*c^(2/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + (1/8)*Sqrt[3]*b*c^(2/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]} -{(a + b*ArcTan[c*x^3])/x^6, x, 9, -((3*b*c)/(10*x^2)) - (a + b*ArcTan[c*x^3])/(5*x^5) + (1/10)*Sqrt[3]*b*c^(5/3)*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]] + (1/10)*b*c^(5/3)*Log[1 + c^(2/3)*x^2] - (1/20)*b*c^(5/3)*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4]} - -{x^7*(a + b*ArcTan[c*x^3]), x, 12, -((3*b*x^5)/(40*c)) + (b*ArcTan[c^(1/3)*x])/(8*c^(8/3)) + (1/8)*x^8*(a + b*ArcTan[c*x^3]) - (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(16*c^(8/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(16*c^(8/3)) + (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3)) - (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))} -{x^4*(a + b*ArcTan[c*x^3]), x, 9, -((3*b*x^2)/(10*c)) + (1/5)*x^5*(a + b*ArcTan[c*x^3]) - (Sqrt[3]*b*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(10*c^(5/3)) + (b*Log[1 + c^(2/3)*x^2])/(10*c^(5/3)) - (b*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(20*c^(5/3))} -{x^1*(a + b*ArcTan[c*x^3]), x, 11, -((b*ArcTan[c^(1/3)*x])/(2*c^(2/3))) + (1/2)*x^2*(a + b*ArcTan[c*x^3]) + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))} -{(a + b*ArcTan[c*x^3])/x^2, x, 8, -((a + b*ArcTan[c*x^3])/x) - (1/2)*Sqrt[3]*b*c^(1/3)*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]] + (1/2)*b*c^(1/3)*Log[1 + c^(2/3)*x^2] - (1/4)*b*c^(1/3)*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4]} -{(a + b*ArcTan[c*x^3])/x^5, x, 12, -((3*b*c)/(4*x)) - (1/4)*b*c^(4/3)*ArcTan[c^(1/3)*x] - (a + b*ArcTan[c*x^3])/(4*x^4) + (1/8)*b*c^(4/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] - (1/8)*b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - (1/16)*Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + (1/16)*Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]} - - -{x^11*(a + b*ArcTan[c*x^3])^2, x, 12, (a*b*x^3)/(6*c^3) + (b^2*x^6)/(36*c^2) + (b^2*x^3*ArcTan[c*x^3])/(6*c^3) - (b*x^9*(a + b*ArcTan[c*x^3]))/(18*c) - (a + b*ArcTan[c*x^3])^2/(12*c^4) + (1/12)*x^12*(a + b*ArcTan[c*x^3])^2 - (b^2*Log[1 + c^2*x^6])/(9*c^4)} -{x^8*(a + b*ArcTan[c*x^3])^2, x, 10, (b^2*x^3)/(9*c^2) - (b^2*ArcTan[c*x^3])/(9*c^3) - (b*x^6*(a + b*ArcTan[c*x^3]))/(9*c) - (I*(a + b*ArcTan[c*x^3])^2)/(9*c^3) + (1/9)*x^9*(a + b*ArcTan[c*x^3])^2 - (2*b*(a + b*ArcTan[c*x^3])*Log[2/(1 + I*c*x^3)])/(9*c^3) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x^3)])/(9*c^3)} -{x^5*(a + b*ArcTan[c*x^3])^2, x, 7, -((a*b*x^3)/(3*c)) - (b^2*x^3*ArcTan[c*x^3])/(3*c) + (a + b*ArcTan[c*x^3])^2/(6*c^2) + (1/6)*x^6*(a + b*ArcTan[c*x^3])^2 + (b^2*Log[1 + c^2*x^6])/(6*c^2)} -{x^2*(a + b*ArcTan[c*x^3])^2, x, 6, (I*(a + b*ArcTan[c*x^3])^2)/(3*c) + (1/3)*x^3*(a + b*ArcTan[c*x^3])^2 + (2*b*(a + b*ArcTan[c*x^3])*Log[2/(1 + I*c*x^3)])/(3*c) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x^3)])/(3*c)} -{(a + b*ArcTan[c*x^3])^2/x^1, x, 7, (2/3)*(a + b*ArcTan[c*x^3])^2*ArcTanh[1 - 2/(1 + I*c*x^3)] - (1/3)*I*b*(a + b*ArcTan[c*x^3])*PolyLog[2, 1 - 2/(1 + I*c*x^3)] + (1/3)*I*b*(a + b*ArcTan[c*x^3])*PolyLog[2, -1 + 2/(1 + I*c*x^3)] - (1/6)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x^3)] + (1/6)*b^2*PolyLog[3, -1 + 2/(1 + I*c*x^3)]} -{(a + b*ArcTan[c*x^3])^2/x^4, x, 5, (-(1/3))*I*c*(a + b*ArcTan[c*x^3])^2 - (a + b*ArcTan[c*x^3])^2/(3*x^3) + (2/3)*b*c*(a + b*ArcTan[c*x^3])*Log[2 - 2/(1 - I*c*x^3)] - (1/3)*I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x^3)]} -{(a + b*ArcTan[c*x^3])^2/x^7, x, 9, -((b*c*(a + b*ArcTan[c*x^3]))/(3*x^3)) - (1/6)*c^2*(a + b*ArcTan[c*x^3])^2 - (a + b*ArcTan[c*x^3])^2/(6*x^6) + b^2*c^2*Log[x] - (1/6)*b^2*c^2*Log[1 + c^2*x^6]} -{(a + b*ArcTan[c*x^3])^2/x^10, x, 9, -((b^2*c^2)/(9*x^3)) - (1/9)*b^2*c^3*ArcTan[c*x^3] - (b*c*(a + b*ArcTan[c*x^3]))/(9*x^6) + (1/9)*I*c^3*(a + b*ArcTan[c*x^3])^2 - (a + b*ArcTan[c*x^3])^2/(9*x^9) - (2/9)*b*c^3*(a + b*ArcTan[c*x^3])*Log[2 - 2/(1 - I*c*x^3)] + (1/9)*I*b^2*c^3*PolyLog[2, -1 + 2/(1 - I*c*x^3)]} - -(* {x^3*(a + b*ArcTan[c*x^3])^2, x, 44, 0} -{x^0*(a + b*ArcTan[c*x^3])^2, x, 69, 0} -{(a + b*ArcTan[c*x^3])^2/x^3, x, 24, 0} -{(a + b*ArcTan[c*x^3])^2/x^6, x, 77, 0} - -{x^1*(a + b*ArcTan[c*x^3])^2, x, 28, 0} -{(a + b*ArcTan[c*x^3])^2/x^2, x, 47, 0} -{(a + b*ArcTan[c*x^3])^2/x^5, x, 46, 0} *) - - -{x^8*(a + b*ArcTan[c*x^3])^3, x, 13, (a*b^2*x^3)/(3*c^2) + (b^3*x^3*ArcTan[c*x^3])/(3*c^2) - (b*(a + b*ArcTan[c*x^3])^2)/(6*c^3) - (b*x^6*(a + b*ArcTan[c*x^3])^2)/(6*c) - (I*(a + b*ArcTan[c*x^3])^3)/(9*c^3) + (1/9)*x^9*(a + b*ArcTan[c*x^3])^3 - (b*(a + b*ArcTan[c*x^3])^2*Log[2/(1 + I*c*x^3)])/(3*c^3) - (b^3*Log[1 + c^2*x^6])/(6*c^3) - (I*b^2*(a + b*ArcTan[c*x^3])*PolyLog[2, 1 - 2/(1 + I*c*x^3)])/(3*c^3) - (b^3*PolyLog[3, 1 - 2/(1 + I*c*x^3)])/(6*c^3)} -{x^5*(a + b*ArcTan[c*x^3])^3, x, 9, -((I*b*(a + b*ArcTan[c*x^3])^2)/(2*c^2)) - (b*x^3*(a + b*ArcTan[c*x^3])^2)/(2*c) + (a + b*ArcTan[c*x^3])^3/(6*c^2) + (1/6)*x^6*(a + b*ArcTan[c*x^3])^3 - (b^2*(a + b*ArcTan[c*x^3])*Log[2/(1 + I*c*x^3)])/c^2 - (I*b^3*PolyLog[2, 1 - 2/(1 + I*c*x^3)])/(2*c^2)} -{x^2*(a + b*ArcTan[c*x^3])^3, x, 6, (I*(a + b*ArcTan[c*x^3])^3)/(3*c) + (1/3)*x^3*(a + b*ArcTan[c*x^3])^3 + (b*(a + b*ArcTan[c*x^3])^2*Log[2/(1 + I*c*x^3)])/c + (I*b^2*(a + b*ArcTan[c*x^3])*PolyLog[2, 1 - 2/(1 + I*c*x^3)])/c + (b^3*PolyLog[3, 1 - 2/(1 + I*c*x^3)])/(2*c)} -{(a + b*ArcTan[c*x^3])^3/x^1, x, 9, (2/3)*(a + b*ArcTan[c*x^3])^3*ArcTanh[1 - 2/(1 + I*c*x^3)] - (1/2)*I*b*(a + b*ArcTan[c*x^3])^2*PolyLog[2, 1 - 2/(1 + I*c*x^3)] + (1/2)*I*b*(a + b*ArcTan[c*x^3])^2*PolyLog[2, -1 + 2/(1 + I*c*x^3)] - (1/2)*b^2*(a + b*ArcTan[c*x^3])*PolyLog[3, 1 - 2/(1 + I*c*x^3)] + (1/2)*b^2*(a + b*ArcTan[c*x^3])*PolyLog[3, -1 + 2/(1 + I*c*x^3)] + (1/4)*I*b^3*PolyLog[4, 1 - 2/(1 + I*c*x^3)] - (1/4)*I*b^3*PolyLog[4, -1 + 2/(1 + I*c*x^3)]} -{(a + b*ArcTan[c*x^3])^3/x^4, x, 6, (-(1/3))*I*c*(a + b*ArcTan[c*x^3])^3 - (a + b*ArcTan[c*x^3])^3/(3*x^3) + b*c*(a + b*ArcTan[c*x^3])^2*Log[2 - 2/(1 - I*c*x^3)] - I*b^2*c*(a + b*ArcTan[c*x^3])*PolyLog[2, -1 + 2/(1 - I*c*x^3)] + (1/2)*b^3*c*PolyLog[3, -1 + 2/(1 - I*c*x^3)]} -{(a + b*ArcTan[c*x^3])^3/x^7, x, 8, (-(1/2))*I*b*c^2*(a + b*ArcTan[c*x^3])^2 - (b*c*(a + b*ArcTan[c*x^3])^2)/(2*x^3) - (1/6)*c^2*(a + b*ArcTan[c*x^3])^3 - (a + b*ArcTan[c*x^3])^3/(6*x^6) + b^2*c^2*(a + b*ArcTan[c*x^3])*Log[2 - 2/(1 - I*c*x^3)] - (1/2)*I*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x^3)]} - -(* {x^3*(a + b*ArcTan[c*x^3])^3, x, 44, 0} -{x^0*(a + b*ArcTan[c*x^3])^3, x, 69, 0} -{(a + b*ArcTan[c*x^3])^3/x^3, x, 24, 0} -{(a + b*ArcTan[c*x^3])^3/x^6, x, 77, 0} - -{x^1*(a + b*ArcTan[c*x^3])^3, x, 28, 0} -{(a + b*ArcTan[c*x^3])^3/x^2, x, 47, 0} -{(a + b*ArcTan[c*x^3])^3/x^5, x, 46, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^3])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTan[c*x^3])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^3])^3, x]} -{(d*x)^m*(a + b*ArcTan[c*x^3])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^3])^2, x]} -{(d*x)^m*(a + b*ArcTan[c*x^3])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTan[c*x^3]))/(d*(1 + m)) - (3*b*c*(d*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/6, (10 + m)/6, (-c^2)*x^6])/(d^4*(1 + m)*(4 + m))} -{(d*x)^m/(a + b*ArcTan[c*x^3])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTan[c*x^3]), x]} -{(d*x)^m/(a + b*ArcTan[c*x^3])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTan[c*x^3])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c/x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c/x])^p*) - - -{x^3*(a + b*ArcTan[c/x]), x, 5, (-(1/4))*b*c^3*x + (1/12)*b*c*x^3 + (1/4)*x^4*(a + b*ArcTan[c/x]) + (1/4)*b*c^4*ArcTan[x/c]} -{x^2*(a + b*ArcTan[c/x]), x, 5, (1/6)*b*c*x^2 + (1/3)*x^3*(a + b*ArcTan[c/x]) - (1/6)*b*c^3*Log[c^2 + x^2]} -{x^1*(a + b*ArcTan[c/x]), x, 4, (b*c*x)/2 + (1/2)*x^2*(a + b*ArcTan[c/x]) - (1/2)*b*c^2*ArcTan[x/c]} -{x^0*(a + b*ArcTan[c/x]), x, 4, a*x + b*x*ArcTan[c/x] + (1/2)*b*c*Log[c^2 + x^2]} -{(a + b*ArcTan[c/x])/x^1, x, 4, a*Log[x] - (1/2)*I*b*PolyLog[2, -((I*c)/x)] + (1/2)*I*b*PolyLog[2, (I*c)/x]} -{(a + b*ArcTan[c/x])/x^2, x, 2, -((a + b*ArcTan[c/x])/x) + (b*Log[1 + c^2/x^2])/(2*c)} -{(a + b*ArcTan[c/x])/x^3, x, 4, b/(2*c*x) - (a + b*ArcTan[c/x])/(2*x^2) + (b*ArcTan[x/c])/(2*c^2)} -{(a + b*ArcTan[c/x])/x^4, x, 5, b/(6*c*x^2) - (a + b*ArcTan[c/x])/(3*x^3) + (b*Log[x])/(3*c^3) - (b*Log[c^2 + x^2])/(6*c^3)} - - -{x^3*(a + b*ArcTan[c/x])^2, x, 14, (1/12)*b^2*c^2*x^2 - (1/2)*b*c^3*x*(a + b*ArcCot[x/c]) + (1/6)*b*c*x^3*(a + b*ArcCot[x/c]) - (1/4)*c^4*(a + b*ArcCot[x/c])^2 + (1/4)*x^4*(a + b*ArcCot[x/c])^2 - (1/3)*b^2*c^4*Log[1 + c^2/x^2] - (2/3)*b^2*c^4*Log[x]} -{x^2*(a + b*ArcTan[c/x])^2, x, 9, (1/3)*b^2*c^2*x + (1/3)*b^2*c^3*ArcCot[x/c] + (1/3)*b*c*x^2*(a + b*ArcCot[x/c]) - (1/3)*I*c^3*(a + b*ArcCot[x/c])^2 + (1/3)*x^3*(a + b*ArcCot[x/c])^2 + (2/3)*b*c^3*(a + b*ArcCot[x/c])*Log[2 - 2/(1 - (I*c)/x)] - (1/3)*I*b^2*c^3*PolyLog[2, -1 + 2/(1 - (I*c)/x)]} -{x^1*(a + b*ArcTan[c/x])^2, x, 9, b*c*x*(a + b*ArcCot[x/c]) + (1/2)*c^2*(a + b*ArcCot[x/c])^2 + (1/2)*x^2*(a + b*ArcCot[x/c])^2 + (1/2)*b^2*c^2*Log[1 + c^2/x^2] + b^2*c^2*Log[x]} -{x^0*(a + b*ArcTan[c/x])^2, x, 6, I*c*(a + b*ArcCot[x/c])^2 + x*(a + b*ArcCot[x/c])^2 - 2*b*c*(a + b*ArcCot[x/c])*Log[(2*c)/(c + I*x)] + I*b^2*c*PolyLog[2, 1 - (2*c)/(c + I*x)]} -{(a + b*ArcTan[c/x])^2/x^1, x, 7, -2*(a + b*ArcCot[x/c])^2*ArcTanh[1 - 2/(1 + (I*c)/x)] + I*b*(a + b*ArcCot[x/c])*PolyLog[2, 1 - 2/(1 + (I*c)/x)] - I*b*(a + b*ArcCot[x/c])*PolyLog[2, -1 + 2/(1 + (I*c)/x)] + (1/2)*b^2*PolyLog[3, 1 - 2/(1 + (I*c)/x)] - (1/2)*b^2*PolyLog[3, -1 + 2/(1 + (I*c)/x)]} -{(a + b*ArcTan[c/x])^2/x^2, x, 6, -((I*(a + b*ArcCot[x/c])^2)/c) - (a + b*ArcCot[x/c])^2/x - (2*b*(a + b*ArcCot[x/c])*Log[2/(1 + (I*c)/x)])/c - (I*b^2*PolyLog[2, 1 - 2/(1 + (I*c)/x)])/c} -{(a + b*ArcTan[c/x])^2/x^3, x, 7, (a*b)/(c*x) + (b^2*ArcCot[x/c])/(c*x) - (a + b*ArcCot[x/c])^2/(2*c^2) - (a + b*ArcCot[x/c])^2/(2*x^2) - (b^2*Log[1 + c^2/x^2])/(2*c^2)} - - -{x^3*(a + b*ArcTan[c/x])^3, x, 17, (1/4)*b^3*c^3*x + (1/4)*b^3*c^4*ArcCot[x/c] + (1/4)*b^2*c^2*x^2*(a + b*ArcCot[x/c]) - I*b*c^4*(a + b*ArcCot[x/c])^2 - (3/4)*b*c^3*x*(a + b*ArcCot[x/c])^2 + (1/4)*b*c*x^3*(a + b*ArcCot[x/c])^2 - (1/4)*c^4*(a + b*ArcCot[x/c])^3 + (1/4)*x^4*(a + b*ArcCot[x/c])^3 + 2*b^2*c^4*(a + b*ArcCot[x/c])*Log[2 - 2/(1 - (I*c)/x)] - I*b^3*c^4*PolyLog[2, -1 + 2/(1 - (I*c)/x)]} -{x^2*(a + b*ArcTan[c/x])^3, x, 15, b^2*c^2*x*(a + b*ArcCot[x/c]) + (1/2)*b*c^3*(a + b*ArcCot[x/c])^2 + (1/2)*b*c*x^2*(a + b*ArcCot[x/c])^2 - (1/3)*I*c^3*(a + b*ArcCot[x/c])^3 + (1/3)*x^3*(a + b*ArcCot[x/c])^3 + b*c^3*(a + b*ArcCot[x/c])^2*Log[2 - 2/(1 - (I*c)/x)] + (1/2)*b^3*c^3*Log[1 + c^2/x^2] + b^3*c^3*Log[x] - I*b^2*c^3*(a + b*ArcCot[x/c])*PolyLog[2, -1 + 2/(1 - (I*c)/x)] + (1/2)*b^3*c^3*PolyLog[3, -1 + 2/(1 - (I*c)/x)]} -{x^1*(a + b*ArcTan[c/x])^3, x, 8, (3/2)*I*b*c^2*(a + b*ArcCot[x/c])^2 + (3/2)*b*c*x*(a + b*ArcCot[x/c])^2 + (1/2)*c^2*(a + b*ArcCot[x/c])^3 + (1/2)*x^2*(a + b*ArcCot[x/c])^3 - 3*b^2*c^2*(a + b*ArcCot[x/c])*Log[2 - 2/(1 - (I*c)/x)] + (3/2)*I*b^3*c^2*PolyLog[2, -1 + 2/(1 - (I*c)/x)]} -{x^0*(a + b*ArcTan[c/x])^3, x, 6, I*c*(a + b*ArcCot[x/c])^3 + x*(a + b*ArcCot[x/c])^3 - 3*b*c*(a + b*ArcCot[x/c])^2*Log[(2*c)/(c + I*x)] + 3*I*b^2*c*(a + b*ArcCot[x/c])*PolyLog[2, 1 - (2*c)/(c + I*x)] - (3/2)*b^3*c*PolyLog[3, 1 - (2*c)/(c + I*x)]} -{(a + b*ArcTan[c/x])^3/x^1, x, 9, -2*(a + b*ArcCot[x/c])^3*ArcTanh[1 - 2/(1 + (I*c)/x)] + (3/2)*I*b*(a + b*ArcCot[x/c])^2*PolyLog[2, 1 - 2/(1 + (I*c)/x)] - (3/2)*I*b*(a + b*ArcCot[x/c])^2*PolyLog[2, -1 + 2/(1 + (I*c)/x)] + (3/2)*b^2*(a + b*ArcCot[x/c])*PolyLog[3, 1 - 2/(1 + (I*c)/x)] - (3/2)*b^2*(a + b*ArcCot[x/c])*PolyLog[3, -1 + 2/(1 + (I*c)/x)] - (3/4)*I*b^3*PolyLog[4, 1 - 2/(1 + (I*c)/x)] + (3/4)*I*b^3*PolyLog[4, -1 + 2/(1 + (I*c)/x)]} -{(a + b*ArcTan[c/x])^3/x^2, x, 6, -((I*(a + b*ArcCot[x/c])^3)/c) - (a + b*ArcCot[x/c])^3/x - (3*b*(a + b*ArcCot[x/c])^2*Log[2/(1 + (I*c)/x)])/c - (3*I*b^2*(a + b*ArcCot[x/c])*PolyLog[2, 1 - 2/(1 + (I*c)/x)])/c - (3*b^3*PolyLog[3, 1 - 2/(1 + (I*c)/x)])/(2*c)} -{(a + b*ArcTan[c/x])^3/x^3, x, 9, (3*I*b*(a + b*ArcCot[x/c])^2)/(2*c^2) + (3*b*(a + b*ArcCot[x/c])^2)/(2*c*x) - (a + b*ArcCot[x/c])^3/(2*c^2) - (a + b*ArcCot[x/c])^3/(2*x^2) + (3*b^2*(a + b*ArcCot[x/c])*Log[2/(1 + (I*c)/x)])/c^2 + (3*I*b^3*PolyLog[2, 1 - 2/(1 + (I*c)/x)])/(2*c^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^(n/2)])^p*) - - -{x^2*ArcTan[Sqrt[x]], x, 6, -(Sqrt[x]/3) + x^(3/2)/9 - x^(5/2)/15 + ArcTan[Sqrt[x]]/3 + (1/3)*x^3*ArcTan[Sqrt[x]]} -{x^1*ArcTan[Sqrt[x]], x, 5, Sqrt[x]/2 - x^(3/2)/6 - ArcTan[Sqrt[x]]/2 + (1/2)*x^2*ArcTan[Sqrt[x]]} -{x^0*ArcTan[Sqrt[x]], x, 4, -Sqrt[x] + ArcTan[Sqrt[x]] + x*ArcTan[Sqrt[x]]} -{ArcTan[Sqrt[x]]/x^1, x, 4, I*PolyLog[2, (-I)*Sqrt[x]] - I*PolyLog[2, I*Sqrt[x]]} -{ArcTan[Sqrt[x]]/x^2, x, 4, -(1/Sqrt[x]) - ArcTan[Sqrt[x]] - ArcTan[Sqrt[x]]/x} -{ArcTan[Sqrt[x]]/x^3, x, 5, -(1/(6*x^(3/2))) + 1/(2*Sqrt[x]) + ArcTan[Sqrt[x]]/2 - ArcTan[Sqrt[x]]/(2*x^2)} - - -{x^(3/2)*ArcTan[Sqrt[x]], x, 3, x/5 - x^2/10 + (2/5)*x^(5/2)*ArcTan[Sqrt[x]] - (1/5)*Log[1 + x]} -{x^(1/2)*ArcTan[Sqrt[x]], x, 3, -(x/3) + (2/3)*x^(3/2)*ArcTan[Sqrt[x]] + (1/3)*Log[1 + x]} -{ArcTan[Sqrt[x]]/x^(1/2), x, 2, 2*Sqrt[x]*ArcTan[Sqrt[x]] - Log[1 + x]} -{ArcTan[Sqrt[x]]/x^(3/2), x, 4, -((2*ArcTan[Sqrt[x]])/Sqrt[x]) + Log[x] - Log[1 + x]} -{ArcTan[Sqrt[x]]/x^(5/2), x, 3, -(1/(3*x)) - (2*ArcTan[Sqrt[x]])/(3*x^(3/2)) - Log[x]/3 + (1/3)*Log[1 + x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTan[c x^n])^p*) - - -{ArcTan[a*x^5]/x, x, 4, (1/10)*I*PolyLog[2, (-I)*a*x^5] - (1/10)*I*PolyLog[2, I*a*x^5]} - - -{ArcTan[a*x^n]/x, x, 4, (I*PolyLog[2, (-I)*a*x^n])/(2*n) - (I*PolyLog[2, I*a*x^n])/(2*n)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m deleted file mode 100644 index d7e45e1..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.3 (d+e x)^m (a+b arctan(c x^n))^p.m +++ /dev/null @@ -1,68 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^1])^p*) - - -{(d + e*x)^4*(a + b*ArcTan[c*x]), x, 6, -((b*d*e*(2*c^2*d^2 - e^2)*x)/c^3) - (b*e^2*(10*c^2*d^2 - e^2)*x^2)/(10*c^3) - (b*d*e^3*x^3)/(3*c) - (b*e^4*x^4)/(20*c) - (b*d*(c^4*d^4 - 10*c^2*d^2*e^2 + 5*e^4)*ArcTan[c*x])/(5*c^4*e) + ((d + e*x)^5*(a + b*ArcTan[c*x]))/(5*e) - (b*(5*c^4*d^4 - 10*c^2*d^2*e^2 + e^4)*Log[1 + c^2*x^2])/(10*c^5)} -{(d + e*x)^3*(a + b*ArcTan[c*x]), x, 6, -(b*e*(6*c^2*d^2 - e^2)*x)/(4*c^3) - (b*d*e^2*x^2)/(2*c) - (b*e^3*x^3)/(12*c) - (b*(c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*ArcTan[c*x])/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x]))/(4*e) - (b*d*(c*d - e)*(c*d + e)*Log[1 + c^2*x^2])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcTan[c*x]), x, 6, -((b*d*e*x)/c) - (b*e^2*x^2)/(6*c) - (b*d*(d^2 - (3*e^2)/c^2)*ArcTan[c*x])/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x]))/(3*e) - (b*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*c^3)} -{(d + e*x)^1*(a + b*ArcTan[c*x]), x, 6, -(b*e*x)/(2*c) - (b*(d^2 - e^2/c^2)*ArcTan[c*x])/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x]))/(2*e) - (b*d*Log[1 + c^2*x^2])/(2*c)} -{(a + b*ArcTan[c*x])/(d + e*x)^1, x, 4, -(((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e) - (I*b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)} -{(a + b*ArcTan[c*x])/(d + e*x)^2, x, 6, (b*c^2*d*ArcTan[c*x])/(e*(c^2*d^2 + e^2)) - (a + b*ArcTan[c*x])/(e*(d + e*x)) + (b*c*Log[d + e*x])/(c^2*d^2 + e^2) - (b*c*Log[1 + c^2*x^2])/(2*(c^2*d^2 + e^2))} -{(a + b*ArcTan[c*x])/(d + e*x)^3, x, 7, -(b*c)/(2*(c^2*d^2 + e^2)*(d + e*x)) + (b*c^2*(c*d - e)*(c*d + e)*ArcTan[c*x])/(2*e*(c^2*d^2 + e^2)^2) - (a + b*ArcTan[c*x])/(2*e*(d + e*x)^2) + (b*c^3*d*Log[d + e*x])/(c^2*d^2 + e^2)^2 - (b*c^3*d*Log[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)^2)} -{(a + b*ArcTan[c*x])/(d + e*x)^4, x, 7, -(b*c)/(6*(c^2*d^2 + e^2)*(d + e*x)^2) - (2*b*c^3*d)/(3*(c^2*d^2 + e^2)^2*(d + e*x)) + (b*c^4*d*(c^2*d^2 - 3*e^2)*ArcTan[c*x])/(3*e*(c^2*d^2 + e^2)^3) - (a + b*ArcTan[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(3*c^2*d^2 - e^2)*Log[d + e*x])/(3*(c^2*d^2 + e^2)^3) - (b*c^3*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)^3)} - - -{(d + e*x)^3*(a + b*ArcTan[c*x])^2, x, 19, (b^2*d*e^2*x)/c^2 - (a*b*e*(6*c^2*d^2 - e^2)*x)/(2*c^3) + (b^2*e^3*x^2)/(12*c^2) - (b^2*d*e^2*ArcTan[c*x])/c^3 - (b^2*e*(6*c^2*d^2 - e^2)*x*ArcTan[c*x])/(2*c^3) - (b*d*e^2*x^2*(a + b*ArcTan[c*x]))/c - (b*e^3*x^3*(a + b*ArcTan[c*x]))/(6*c) + (I*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^2)/c^3 - ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTan[c*x])^2)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x])^2)/(4*e) + (2*b*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^3 - (b^2*e^3*Log[1 + c^2*x^2])/(12*c^4) + (b^2*e*(6*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(4*c^4) + (I*b^2*d*(c*d - e)*(c*d + e)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{(d + e*x)^2*(a + b*ArcTan[c*x])^2, x, 15, (-2*a*b*d*e*x)/c + (b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTan[c*x])/(3*c^3) - (2*b^2*d*e*x*ArcTan[c*x])/c - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + ((I/3)*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^3 - (d*(d^2 - (3*e^2)/c^2)*(a + b*ArcTan[c*x])^2)/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x])^2)/(3*e) + (2*b*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (b^2*d*e*Log[1 + c^2*x^2])/c^2 + ((I/3)*b^2*(3*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{(d + e*x)^1*(a + b*ArcTan[c*x])^2, x, 12, -((a*b*e*x)/c) - (b^2*e*x*ArcTan[c*x])/c + (I*d*(a + b*ArcTan[c*x])^2)/c - ((d^2 - e^2/c^2)*(a + b*ArcTan[c*x])^2)/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x])^2)/(2*e) + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (b^2*e*Log[1 + c^2*x^2])/(2*c^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c} -{(a + b*ArcTan[c*x])^2/(d + e*x)^1, x, 1, -(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)} -{(a + b*ArcTan[c*x])^2/(d + e*x)^2, x, 13, (I*c*(a + b*ArcTan[c*x])^2)/(c^2*d^2 + e^2) + (c^2*d*(a + b*ArcTan[c*x])^2)/(e*(c^2*d^2 + e^2)) - (a + b*ArcTan[c*x])^2/(e*(d + e*x)) - (2*b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (2*b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2) + (2*b*c*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) + (I*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (I*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2) - (I*b^2*c*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)} -{(a + b*ArcTan[c*x])^2/(d + e*x)^3, x, 19, (b^2*c^3*d*ArcTan[c*x])/(c^2*d^2 + e^2)^2 - (b*c*(a + b*ArcTan[c*x]))/((c^2*d^2 + e^2)*(d + e*x)) + (I*c^3*d*(a + b*ArcTan[c*x])^2)/(c^2*d^2 + e^2)^2 + (c^2*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^2)/(2*e*(c^2*d^2 + e^2)^2) - (a + b*ArcTan[c*x])^2/(2*e*(d + e*x)^2) - (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (b^2*c^2*e*Log[d + e*x])/(c^2*d^2 + e^2)^2 + (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 - (b^2*c^2*e*Log[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)^2) + (I*b^2*c^3*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (I*b^2*c^3*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 - (I*b^2*c^3*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2} - - -{(d + e*x)^3*(a + b*ArcTan[c*x])^3, x, 29, (3*a*b^2*d*e^2*x)/c^2 - (b^3*e^3*x)/(4*c^3) + (b^3*e^3*ArcTan[c*x])/(4*c^4) + (3*b^3*d*e^2*x*ArcTan[c*x])/c^2 + (b^2*e^3*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (3*b*d*e^2*(a + b*ArcTan[c*x])^2)/(2*c^3) + ((I/4)*b*e^3*(a + b*ArcTan[c*x])^2)/c^4 - (((3*I)/4)*b*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2)/c^4 - (3*b*e*(6*c^2*d^2 - e^2)*x*(a + b*ArcTan[c*x])^2)/(4*c^3) - (3*b*d*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) - (b*e^3*x^3*(a + b*ArcTan[c*x])^2)/(4*c) + (I*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/c^3 - ((c^4*d^4 - 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTan[c*x])^3)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTan[c*x])^3)/(4*e) + (b^2*e^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(2*c^4) - (3*b^2*e*(6*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(2*c^4) + (3*b*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (3*b^3*d*e^2*Log[1 + c^2*x^2])/(2*c^3) + ((I/4)*b^3*e^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4 - (((3*I)/4)*b^3*e*(6*c^2*d^2 - e^2)*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^4 + ((3*I)*b^2*d*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (3*b^3*d*(c*d - e)*(c*d + e)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcTan[c*x])^3, x, 20, (a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTan[c*x])/c^2 - ((3*I)*b*d*e*(a + b*ArcTan[c*x])^2)/c^2 - (b*e^2*(a + b*ArcTan[c*x])^2)/(2*c^3) - (3*b*d*e*x*(a + b*ArcTan[c*x])^2)/c - (b*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) + ((I/3)*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^3)/c^3 - (d*(d^2 - (3*e^2)/c^2)*(a + b*ArcTan[c*x])^3)/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (b^3*e^2*Log[1 + c^2*x^2])/(2*c^3) - ((3*I)*b^3*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^2 + (I*b^2*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (b^3*(3*c^2*d^2 - e^2)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)} -{(d + e*x)^1*(a + b*ArcTan[c*x])^3, x, 14, (((-3*I)/2)*b*e*(a + b*ArcTan[c*x])^2)/c^2 - (3*b*e*x*(a + b*ArcTan[c*x])^2)/(2*c) + (I*d*(a + b*ArcTan[c*x])^3)/c - ((d^2 - e^2/c^2)*(a + b*ArcTan[c*x])^3)/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x])^3)/(2*e) - (3*b^2*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (3*b*d*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c - (((3*I)/2)*b^3*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^2 + ((3*I)*b^2*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c + (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c)} -{(a + b*ArcTan[c*x])^3/(d + e*x)^1, x, 1, -(((a + b*ArcTan[c*x])^3*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^3*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e) - (3*I*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e) - (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e) - (3*I*b^3*PolyLog[4, 1 - 2/(1 - I*c*x)])/(4*e) + (3*I*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(4*e)} -{(a + b*ArcTan[c*x])^3/(d + e*x)^2, x, 10, (I*c*(a + b*ArcTan[c*x])^3)/(c^2*d^2 + e^2) + (c^2*d*(a + b*ArcTan[c*x])^3)/(e*(c^2*d^2 + e^2)) - (a + b*ArcTan[c*x])^3/(e*(d + e*x)) - (3*b*c*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (3*b*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2) + (3*b*c*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) + (3*I*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2) + (3*I*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2) - (3*I*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2) - (3*b^3*c*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*(c^2*d^2 + e^2)) + (3*b^3*c*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*(c^2*d^2 + e^2)) + (3*b^3*c*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*(c^2*d^2 + e^2))} -{(a + b*ArcTan[c*x])^3/(d + e*x)^3, x, 23, (3*b*c^3*d*(a + b*ArcTan[c*x])^2)/(2*(c^2*d^2 + e^2)^2) + (3*I*b*c^2*e*(a + b*ArcTan[c*x])^2)/(2*(c^2*d^2 + e^2)^2) - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*(c^2*d^2 + e^2)*(d + e*x)) + (I*c^3*d*(a + b*ArcTan[c*x])^3)/(c^2*d^2 + e^2)^2 + (c^2*(c*d - e)*(c*d + e)*(a + b*ArcTan[c*x])^3)/(2*e*(c^2*d^2 + e^2)^2) - (a + b*ArcTan[c*x])^3/(2*e*(d + e*x)^2) - (3*b^2*c^2*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 - (3*b*c^3*d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (3*b^2*c^2*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (3*b*c^3*d*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 + (3*b^2*c^2*e*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 + (3*b*c^3*d*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 + (3*I*b^3*c^2*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*(c^2*d^2 + e^2)^2) + (3*I*b^2*c^3*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*d^2 + e^2)^2 + (3*I*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*(c^2*d^2 + e^2)^2) + (3*I*b^2*c^3*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2 + e^2)^2 - (3*I*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*(c^2*d^2 + e^2)^2) - (3*I*b^2*c^3*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(c^2*d^2 + e^2)^2 - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*(c^2*d^2 + e^2)^2) + (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*(c^2*d^2 + e^2)^2) + (3*b^3*c^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*(c^2*d^2 + e^2)^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^2])^p*) - - -{(d + e*x)^2*(a + b*ArcTan[c*x^2]), x, 17, -((2*b*e^2*x)/(3*c)) - (b*d^3*ArcTan[c*x^2])/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x^2]))/(3*e) + (b*(3*c*d^2 - e^2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]*c^(3/2)) - (b*(3*c*d^2 - e^2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]*c^(3/2)) - (b*(3*c*d^2 + e^2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2)) + (b*(3*c*d^2 + e^2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2]*c^(3/2)) - (b*d*e*Log[1 + c^2*x^4])/(2*c)} -{(d + e*x)^1*(a + b*ArcTan[c*x^2]), x, 16, -((b*d^2*ArcTan[c*x^2])/(2*e)) + ((d + e*x)^2*(a + b*ArcTan[c*x^2]))/(2*e) + (b*d*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*d*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*Sqrt[c]) - (b*d*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) + (b*d*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*Sqrt[c]) - (b*e*Log[1 + c^2*x^4])/(4*c)} -{(a + b*ArcTan[c*x^2])/(d + e*x)^1, x, 19, ((a + b*ArcTan[c*x^2])*Log[d + e*x])/e + (b*c*Log[(e*(1 - (-c^2)^(1/4)*x))/((-c^2)^(1/4)*d + e)]*Log[d + e*x])/(2*Sqrt[-c^2]*e) + (b*c*Log[-((e*(1 + (-c^2)^(1/4)*x))/((-c^2)^(1/4)*d - e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*Log[(e*(1 - Sqrt[-Sqrt[-c^2]]*x))/(Sqrt[-Sqrt[-c^2]]*d + e)]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*Log[-((e*(1 + Sqrt[-Sqrt[-c^2]]*x))/(Sqrt[-Sqrt[-c^2]]*d - e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/4)*(d + e*x))/((-c^2)^(1/4)*d - e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, (Sqrt[-Sqrt[-c^2]]*(d + e*x))/(Sqrt[-Sqrt[-c^2]]*d - e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/4)*(d + e*x))/((-c^2)^(1/4)*d + e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, (Sqrt[-Sqrt[-c^2]]*(d + e*x))/(Sqrt[-Sqrt[-c^2]]*d + e)])/(2*Sqrt[-c^2]*e)} -{(a + b*ArcTan[c*x^2])/(d + e*x)^2, x, 18, (b*c^2*d^3*ArcTan[c*x^2])/(e*(c^2*d^4 + e^4)) - (a + b*ArcTan[c*x^2])/(e*(d + e*x)) + (b*Sqrt[c]*(c*d^2 - e^2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*(c^2*d^4 + e^4)) - (b*Sqrt[c]*(c*d^2 - e^2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(Sqrt[2]*(c^2*d^4 + e^4)) - (2*b*c*d*e*Log[d + e*x])/(c^2*d^4 + e^4) - (b*Sqrt[c]*(c*d^2 + e^2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*(c^2*d^4 + e^4)) + (b*Sqrt[c]*(c*d^2 + e^2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(2*Sqrt[2]*(c^2*d^4 + e^4)) + (b*c*d*e*Log[1 + c^2*x^4])/(2*(c^2*d^4 + e^4))} - - -(* {(d + e*x)^2*(a + b*ArcTan[c*x^2])^2, x, 163, a^2*d^2*x - (4*a*b*e^2*x)/(3*c) + (2/9)*I*a*b*e^2*x^3 - (2*(-1)^(3/4)*a*b*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] + (4*(-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + ((-1)^(1/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/(3*c^(3/2)) + (I*d*e*(a + b*ArcTan[c*x^2])^2)/c + d*e*x^2*(a + b*ArcTan[c*x^2])^2 + (2*(-1)^(3/4)*a*b*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] - (2*(-1)^(1/4)*a*b*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) - (4*(-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/(3*c^(3/2)) - ((-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] - ((-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/(3*c^(3/2)) + (2*(-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - (2*(-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + (2*(-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + (2*(-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + (2*(-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) - (2*(-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + I*a*b*d^2*x*Log[1 - I*c*x^2] - (2*I*b^2*e^2*x*Log[1 - I*c*x^2])/(3*c) - (1/9)*b^2*e^2*x^3*Log[1 - I*c*x^2] + ((-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - ((-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - ((-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/(3*c^(3/2)) - (1/4)*b^2*d^2*x*Log[1 - I*c*x^2]^2 - (1/9)*I*b*e^2*x^3*(2*a + I*b*Log[1 - I*c*x^2]) - ((-1)^(1/4)*b*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*(2*a + I*b*Log[1 - I*c*x^2]))/(3*c^(3/2)) + (1/12)*e^2*x^3*(2*a + I*b*Log[1 - I*c*x^2])^2 + (2*b*d*e*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/c - I*a*b*d^2*x*Log[1 + I*c*x^2] + (2*I*b^2*e^2*x*Log[1 + I*c*x^2])/(3*c) - (1/3)*I*a*b*e^2*x^3*Log[1 + I*c*x^2] - ((-1)^(1/4)*b^2*d^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/(3*c^(3/2)) + (1/2)*b^2*d^2*x*Log[1 - I*c*x^2]*Log[1 + I*c*x^2] + (1/6)*b^2*e^2*x^3*Log[1 - I*c*x^2]*Log[1 + I*c*x^2] - (1/4)*b^2*d^2*x*Log[1 + I*c*x^2]^2 - (1/12)*b^2*e^2*x^3*Log[1 + I*c*x^2]^2 + ((-1)^(3/4)*b^2*d^2*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*e^2*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(3/4)*b^2*d^2*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*e^2*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(3/4)*b^2*d^2*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(1/4)*b^2*e^2*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(6*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) + ((-1)^(1/4)*b^2*d^2*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*e^2*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(3*c^(3/2)) - ((-1)^(1/4)*b^2*d^2*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(3/4)*b^2*e^2*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(6*c^(3/2)) - ((-1)^(1/4)*b^2*d^2*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(3/4)*b^2*e^2*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(6*c^(3/2)) - ((-1)^(3/4)*b^2*d^2*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(1/4)*b^2*e^2*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(6*c^(3/2)) + (I*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/c} *) -{(d + e*x)^1*(a + b*ArcTan[c*x^2])^2, x, 77, a^2*d*x - (2*(-1)^(3/4)*a*b*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] + ((-1)^(3/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + (I*e*(a + b*ArcTan[c*x^2])^2)/(2*c) + (1/2)*e*x^2*(a + b*ArcTan[c*x^2])^2 + (2*(-1)^(3/4)*a*b*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x])/Sqrt[c] - ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]^2)/Sqrt[c] + (2*(-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[(Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + (2*(-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - (2*(-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[-((Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x))])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + I*a*b*d*x*Log[1 - I*c*x^2] + ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 - I*c*x^2])/Sqrt[c] - (1/4)*b^2*d*x*Log[1 - I*c*x^2]^2 + (b*e*(a + b*ArcTan[c*x^2])*Log[2/(1 + I*c*x^2)])/c - I*a*b*d*x*Log[1 + I*c*x^2] - ((-1)^(1/4)*b^2*d*ArcTan[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + ((-1)^(1/4)*b^2*d*ArcTanh[(-1)^(3/4)*Sqrt[c]*x]*Log[1 + I*c*x^2])/Sqrt[c] + (1/2)*b^2*d*x*Log[1 - I*c*x^2]*Log[1 + I*c*x^2] - (1/4)*b^2*d*x*Log[1 + I*c*x^2]^2 + ((-1)^(3/4)*b^2*d*PolyLog[2, 1 - 2/(1 - (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(3/4)*b^2*d*PolyLog[2, 1 - 2/(1 + (-1)^(1/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(3/4)*b^2*d*PolyLog[2, 1 - (Sqrt[2]*((-1)^(1/4) + Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) + ((-1)^(1/4)*b^2*d*PolyLog[2, 1 - 2/(1 - (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] + ((-1)^(1/4)*b^2*d*PolyLog[2, 1 - 2/(1 + (-1)^(3/4)*Sqrt[c]*x)])/Sqrt[c] - ((-1)^(1/4)*b^2*d*PolyLog[2, 1 + (Sqrt[2]*((-1)^(3/4) + Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(1/4)*b^2*d*PolyLog[2, 1 - ((1 + I)*(1 + (-1)^(1/4)*Sqrt[c]*x))/(1 + (-1)^(3/4)*Sqrt[c]*x)])/(2*Sqrt[c]) - ((-1)^(3/4)*b^2*d*PolyLog[2, 1 - ((1 - I)*(1 + (-1)^(3/4)*Sqrt[c]*x))/(1 + (-1)^(1/4)*Sqrt[c]*x)])/(2*Sqrt[c]) + (I*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x^2)])/(2*c)} -{(a + b*ArcTan[c*x^2])^2/(d + e*x)^1, x, 0, Unintegrable[(a + b*ArcTan[c*x^2])^2/(d + e*x), x]} -{(a + b*ArcTan[c*x^2])^2/(d + e*x)^2, x, 0, Unintegrable[(a + b*ArcTan[c*x^2])^2/(d + e*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTan[c x^3])^p*) - - -{(d + e*x)^2*(a + b*ArcTan[c*x^3]), x, 24, -((b*d*e*ArcTan[c^(1/3)*x])/c^(2/3)) - (b*d^3*ArcTan[c*x^3])/(3*e) + ((d + e*x)^3*(a + b*ArcTan[c*x^3]))/(3*e) + (b*d*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(2*c^(2/3)) - (b*d*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(2*c^(2/3)) + (Sqrt[3]*b*d^2*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + (b*d^2*Log[1 + c^(2/3)*x^2])/(2*c^(1/3)) - (Sqrt[3]*b*d*e*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*c^(2/3)) + (Sqrt[3]*b*d*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*c^(2/3)) - (b*d^2*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3)) - (b*e^2*Log[1 + c^2*x^6])/(6*c)} -{(d + e*x)^1*(a + b*ArcTan[c*x^3]), x, 22, -((b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3))) - (b*d^2*ArcTan[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x^3]))/(2*e) + (b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) + (Sqrt[3]*b*d*ArcTan[(1 - 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + (b*d*Log[1 + c^(2/3)*x^2])/(2*c^(1/3)) - (Sqrt[3]*b*e*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*Log[1 - c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))} -{(a + b*ArcTan[c*x^3])/(d + e*x)^1, x, 25, ((a + b*ArcTan[c*x^3])*Log[d + e*x])/e + (b*c*Log[(e*(1 - (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + e)]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*Log[-((e*(1 + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e) + (b*c*Log[-((e*((-1)^(1/3) + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*Log[-((e*((-1)^(2/3) + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - (-1)^(2/3)*e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e) + (b*c*Log[((-1)^(2/3)*e*(1 + (-1)^(1/3)*(-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + (-1)^(2/3)*e)]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*Log[((-1)^(1/3)*e*(1 + (-1)^(2/3)*(-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + (-1)^(1/3)*e)]*Log[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + (-1)^(1/3)*e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - (-1)^(2/3)*e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + (-1)^(2/3)*e)])/(2*Sqrt[-c^2]*e)} -{(a + b*ArcTan[c*x^3])/(d + e*x)^2, x, 34, -((b*c^(2/3)*d*e^3*ArcTan[c^(1/3)*x])/(c^2*d^6 + e^6)) + (b*c^2*d^5*ArcTan[c*x^3])/(e*(c^2*d^6 + e^6)) - (a + b*ArcTan[c*x^3])/(e*(d + e*x)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 + e^3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(2*(c^2*d^6 + e^6)) + (b*c^(2/3)*d*(Sqrt[3]*c*d^3 - e^3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(2*(c^2*d^6 + e^6)) + (Sqrt[3]*b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*ArcTan[(1 + (2*c^(2/3)*x)/(-c^2)^(1/6))/Sqrt[3]])/(2*(-c^2)^(2/3)*(c^2*d^6 + e^6)) - (Sqrt[3]*b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*ArcTan[(c^(4/3) + 2*(-c^2)^(5/6)*x)/(Sqrt[3]*c^(4/3))])/(2*(-c^2)^(2/3)*(c^2*d^6 + e^6)) + (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/6) - c^(2/3)*x])/(2*(-c^2)^(2/3)*(c^2*d^6 + e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/6) + c^(2/3)*x])/(2*(-c^2)^(2/3)*(c^2*d^6 + e^6)) + (3*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 + e^6) + (b*c^(5/3)*d^4*Log[1 + c^(2/3)*x^2])/(2*(c^2*d^6 + e^6)) - (b*c^(2/3)*d*(c*d^3 - Sqrt[3]*e^3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6)) - (b*c^(2/3)*d*(c*d^3 + Sqrt[3]*e^3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(4*(c^2*d^6 + e^6)) + (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 - e^3)*Log[(-c^2)^(1/3) - c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c^2)^(2/3)*(c^2*d^6 + e^6)) - (b*c^(5/3)*e*(Sqrt[-c^2]*d^3 + e^3)*Log[(-c^2)^(1/3) + c^(2/3)*(-c^2)^(1/6)*x + c^(4/3)*x^2])/(4*(-c^2)^(2/3)*(c^2*d^6 + e^6)) - (b*c*d^2*e^2*Log[1 + c^2*x^6])/(2*(c^2*d^6 + e^6))} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m deleted file mode 100644 index 29a2ad8..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.4 u (a+b arctan(c x))^p.m +++ /dev/null @@ -1,2298 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^q (a+b ArcTan[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^p with c^2 d^2+e^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^1 with c^2 d^2+e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x]), x, 7, (b*d*x)/(4*c^3) + ((I/10)*b*d*x^2)/c^2 - (b*d*x^3)/(12*c) - (I/20)*b*d*x^4 - (b*d*ArcTan[c*x])/(4*c^4) + (d*x^4*(a + b*ArcTan[c*x]))/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x]) - ((I/10)*b*d*Log[1 + c^2*x^2])/c^4} -{x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x]), x, 7, ((I/4)*b*d*x)/c^2 - (b*d*x^2)/(6*c) - (I/12)*b*d*x^3 - ((I/4)*b*d*ArcTan[c*x])/c^3 + (d*x^3*(a + b*ArcTan[c*x]))/3 + (I/4)*c*d*x^4*(a + b*ArcTan[c*x]) + (b*d*Log[1 + c^2*x^2])/(6*c^3)} -{x*(d + I*c*d*x)*(a + b*ArcTan[c*x]), x, 7, -(b*d*x)/(2*c) - (I/6)*b*d*x^2 + (b*d*ArcTan[c*x])/(2*c^2) + (d*x^2*(a + b*ArcTan[c*x]))/2 + (I/3)*c*d*x^3*(a + b*ArcTan[c*x]) + ((I/6)*b*d*Log[1 + c^2*x^2])/c^2} -{(d + I*c*d*x)*(a + b*ArcTan[c*x]), x, 4, (-I/2)*b*d*x - ((I/2)*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x]))/c - (b*d*Log[I + c*x])/c} -{((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x, x, 8, I*a*c*d*x + I*b*c*d*x*ArcTan[c*x] + a*d*Log[x] - (I/2)*b*d*Log[1 + c^2*x^2] + (I/2)*b*d*PolyLog[2, (-I)*c*x] - (I/2)*b*d*PolyLog[2, I*c*x]} -{((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^2, x, 10, -((d*(a + b*ArcTan[c*x]))/x) + I*a*c*d*Log[x] + b*c*d*Log[x] - (b*c*d*Log[1 + c^2*x^2])/2 - (b*c*d*PolyLog[2, (-I)*c*x])/2 + (b*c*d*PolyLog[2, I*c*x])/2} -{((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^3, x, 4, -(b*c*d)/(2*x) - (d*(1 + I*c*x)^2*(a + b*ArcTan[c*x]))/(2*x^2) + I*b*c^2*d*Log[x] - I*b*c^2*d*Log[I + c*x]} -{((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^4, x, 4, -(b*c*d)/(6*x^2) - ((I/2)*b*c^2*d)/x - (d*(a + b*ArcTan[c*x]))/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[c*x]))/x^2 - (b*c^3*d*Log[x])/3 - (b*c^3*d*Log[I - c*x])/12 + (5*b*c^3*d*Log[I + c*x])/12} -{((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^5, x, 4, -(b*c*d)/(12*x^3) - ((I/6)*b*c^2*d)/x^2 + (b*c^3*d)/(4*x) - (d*(a + b*ArcTan[c*x]))/(4*x^4) - ((I/3)*c*d*(a + b*ArcTan[c*x]))/x^3 - (I/3)*b*c^4*d*Log[x] + (I/24)*b*c^4*d*Log[I - c*x] + ((7*I)/24)*b*c^4*d*Log[I + c*x]} - - -{x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]), x, 7, (5*b*d^2*x)/(12*c^3) + ((I/5)*b*d^2*x^2)/c^2 - (5*b*d^2*x^3)/(36*c) - (I/10)*b*d^2*x^4 + (b*c*d^2*x^5)/30 - (5*b*d^2*ArcTan[c*x])/(12*c^4) + (d^2*x^4*(a + b*ArcTan[c*x]))/4 + ((2*I)/5)*c*d^2*x^5*(a + b*ArcTan[c*x]) - (c^2*d^2*x^6*(a + b*ArcTan[c*x]))/6 - ((I/5)*b*d^2*Log[1 + c^2*x^2])/c^4} -{x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]), x, 7, ((I/2)*b*d^2*x)/c^2 - (4*b*d^2*x^2)/(15*c) - (I/6)*b*d^2*x^3 + (b*c*d^2*x^4)/20 - ((I/2)*b*d^2*ArcTan[c*x])/c^3 + (d^2*x^3*(a + b*ArcTan[c*x]))/3 + (I/2)*c*d^2*x^4*(a + b*ArcTan[c*x]) - (c^2*d^2*x^5*(a + b*ArcTan[c*x]))/5 + (4*b*d^2*Log[1 + c^2*x^2])/(15*c^3)} -{x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]), x, 7, (-3*b*d^2*x)/(4*c) - (I/3)*b*d^2*x^2 + (b*c*d^2*x^3)/12 + (3*b*d^2*ArcTan[c*x])/(4*c^2) + (d^2*x^2*(a + b*ArcTan[c*x]))/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTan[c*x]) - (c^2*d^2*x^4*(a + b*ArcTan[c*x]))/4 + ((I/3)*b*d^2*Log[1 + c^2*x^2])/c^2} -{(d + I*c*d*x)^2*(a + b*ArcTan[c*x]), x, 4, (-(2/3))*I*b*d^2*x - (b*d^2*(1 + I*c*x)^2)/(6*c) - (I*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/(3*c) - (4*b*d^2*Log[1 - I*c*x])/(3*c)} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x, x, 11, (2*I)*a*c*d^2*x + (b*c*d^2*x)/2 - (b*d^2*ArcTan[c*x])/2 + (2*I)*b*c*d^2*x*ArcTan[c*x] - (c^2*d^2*x^2*(a + b*ArcTan[c*x]))/2 + a*d^2*Log[x] - I*b*d^2*Log[1 + c^2*x^2] + (I/2)*b*d^2*PolyLog[2, (-I)*c*x] - (I/2)*b*d^2*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^2, x, 13, (-a)*c^2*d^2*x - b*c^2*d^2*x*ArcTan[c*x] - (d^2*(a + b*ArcTan[c*x]))/x + 2*I*a*c*d^2*Log[x] + b*c*d^2*Log[x] - b*c*d^2*PolyLog[2, (-I)*c*x] + b*c*d^2*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^3, x, 13, -(b*c*d^2)/(2*x) - (b*c^2*d^2*ArcTan[c*x])/2 - (d^2*(a + b*ArcTan[c*x]))/(2*x^2) - ((2*I)*c*d^2*(a + b*ArcTan[c*x]))/x - a*c^2*d^2*Log[x] + (2*I)*b*c^2*d^2*Log[x] - I*b*c^2*d^2*Log[1 + c^2*x^2] - (I/2)*b*c^2*d^2*PolyLog[2, (-I)*c*x] + (I/2)*b*c^2*d^2*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^4, x, 4, -(b*c*d^2)/(6*x^2) - (I*b*c^2*d^2)/x - (d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/(3*x^3) - (4*b*c^3*d^2*Log[x])/3 + (4*b*c^3*d^2*Log[I + c*x])/3} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^5, x, 4, -(b*c*d^2)/(12*x^3) - ((I/3)*b*c^2*d^2)/x^2 + (3*b*c^3*d^2)/(4*x) - (d^2*(a + b*ArcTan[c*x]))/(4*x^4) - (((2*I)/3)*c*d^2*(a + b*ArcTan[c*x]))/x^3 + (c^2*d^2*(a + b*ArcTan[c*x]))/(2*x^2) - ((2*I)/3)*b*c^4*d^2*Log[x] - (I/24)*b*c^4*d^2*Log[I - c*x] + ((17*I)/24)*b*c^4*d^2*Log[I + c*x]} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^6, x, 4, -(b*c*d^2)/(20*x^4) - ((I/6)*b*c^2*d^2)/x^3 + (4*b*c^3*d^2)/(15*x^2) + ((I/2)*b*c^4*d^2)/x - (d^2*(a + b*ArcTan[c*x]))/(5*x^5) - ((I/2)*c*d^2*(a + b*ArcTan[c*x]))/x^4 + (c^2*d^2*(a + b*ArcTan[c*x]))/(3*x^3) + (8*b*c^5*d^2*Log[x])/15 - (b*c^5*d^2*Log[I - c*x])/60 - (31*b*c^5*d^2*Log[I + c*x])/60} - - -{x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]), x, 7, (3*b*d^3*x)/(4*c^3) + (((13*I)/35)*b*d^3*x^2)/c^2 - (b*d^3*x^3)/(4*c) - ((13*I)/70)*b*d^3*x^4 + (b*c*d^3*x^5)/10 + (I/42)*b*c^2*d^3*x^6 - (3*b*d^3*ArcTan[c*x])/(4*c^4) + (d^3*x^4*(a + b*ArcTan[c*x]))/4 + ((3*I)/5)*c*d^3*x^5*(a + b*ArcTan[c*x]) - (c^2*d^3*x^6*(a + b*ArcTan[c*x]))/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x]) - (((13*I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4} -{x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]), x, 7, (((11*I)/12)*b*d^3*x)/c^2 - (7*b*d^3*x^2)/(15*c) - ((11*I)/36)*b*d^3*x^3 + (3*b*c*d^3*x^4)/20 + (I/30)*b*c^2*d^3*x^5 - (((11*I)/12)*b*d^3*ArcTan[c*x])/c^3 + (d^3*x^3*(a + b*ArcTan[c*x]))/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x]) - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x]))/5 - (I/6)*c^3*d^3*x^6*(a + b*ArcTan[c*x]) + (7*b*d^3*Log[1 + c^2*x^2])/(15*c^3)} -{x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]), x, 4, (-3*b*d^3*x)/(5*c) - (((3*I)/20)*b*d^3*(I - c*x)^2)/c^2 - (b*d^3*(I - c*x)^3)/(20*c^2) + ((I/20)*b*d^3*(I - c*x)^4)/c^2 + (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(4*c^2) - (d^3*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*c^2) + (((6*I)/5)*b*d^3*Log[I + c*x])/c^2} -{(d + I*c*d*x)^3*(a + b*ArcTan[c*x]), x, 4, (-I)*b*d^3*x - (b*d^3*(1 + I*c*x)^2)/(4*c) - (b*d^3*(1 + I*c*x)^3)/(12*c) - (I*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(4*c) - (2*b*d^3*Log[1 - I*c*x])/c} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x, x, 15, (3*I)*a*c*d^3*x + (3*b*c*d^3*x)/2 + (I/6)*b*c^2*d^3*x^2 - (3*b*d^3*ArcTan[c*x])/2 + (3*I)*b*c*d^3*x*ArcTan[c*x] - (3*c^2*d^3*x^2*(a + b*ArcTan[c*x]))/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x]) + a*d^3*Log[x] - ((5*I)/3)*b*d^3*Log[1 + c^2*x^2] + (I/2)*b*d^3*PolyLog[2, (-I)*c*x] - (I/2)*b*d^3*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^2, x, 16, -3*a*c^2*d^3*x + (I/2)*b*c^2*d^3*x - (I/2)*b*c*d^3*ArcTan[c*x] - 3*b*c^2*d^3*x*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/x - (I/2)*c^3*d^3*x^2*(a + b*ArcTan[c*x]) + (3*I)*a*c*d^3*Log[x] + b*c*d^3*Log[x] + b*c*d^3*Log[1 + c^2*x^2] - (3*b*c*d^3*PolyLog[2, (-I)*c*x])/2 + (3*b*c*d^3*PolyLog[2, I*c*x])/2} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^3, x, 16, -(b*c*d^3)/(2*x) - I*a*c^3*d^3*x - (b*c^2*d^3*ArcTan[c*x])/2 - I*b*c^3*d^3*x*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/(2*x^2) - ((3*I)*c*d^3*(a + b*ArcTan[c*x]))/x - 3*a*c^2*d^3*Log[x] + (3*I)*b*c^2*d^3*Log[x] - I*b*c^2*d^3*Log[1 + c^2*x^2] - ((3*I)/2)*b*c^2*d^3*PolyLog[2, (-I)*c*x] + ((3*I)/2)*b*c^2*d^3*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^4, x, 17, -(b*c*d^3)/(6*x^2) - (((3*I)/2)*b*c^2*d^3)/x - ((3*I)/2)*b*c^3*d^3*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/(3*x^3) - (((3*I)/2)*c*d^3*(a + b*ArcTan[c*x]))/x^2 + (3*c^2*d^3*(a + b*ArcTan[c*x]))/x - I*a*c^3*d^3*Log[x] - (10*b*c^3*d^3*Log[x])/3 + (5*b*c^3*d^3*Log[1 + c^2*x^2])/3 + (b*c^3*d^3*PolyLog[2, (-I)*c*x])/2 - (b*c^3*d^3*PolyLog[2, I*c*x])/2} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^5, x, 4, -(b*c*d^3)/(12*x^3) - ((I/2)*b*c^2*d^3)/x^2 + (7*b*c^3*d^3)/(4*x) - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(4*x^4) - (2*I)*b*c^4*d^3*Log[x] + (2*I)*b*c^4*d^3*Log[I + c*x]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^6, x, 4, -(b*c*d^3)/(20*x^4) - ((I/4)*b*c^2*d^3)/x^3 + (3*b*c^3*d^3)/(5*x^2) + (((5*I)/4)*b*c^4*d^3)/x - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(5*x^5) + ((I/20)*c*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/x^4 + (6*b*c^5*d^3*Log[x])/5 - (6*b*c^5*d^3*Log[I + c*x])/5} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^7, x, 4, -(b*c*d^3)/(30*x^5) - (((3*I)/20)*b*c^2*d^3)/x^4 + (11*b*c^3*d^3)/(36*x^3) + (((7*I)/15)*b*c^4*d^3)/x^2 - (11*b*c^5*d^3)/(12*x) - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x]))/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x]))/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x]))/x^3 + ((14*I)/15)*b*c^6*d^3*Log[x] - (I/120)*b*c^6*d^3*Log[I - c*x] - ((37*I)/40)*b*c^6*d^3*Log[I + c*x]} - - -{x^3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]), x, 7, (11*b*d^4*x)/(8*c^3) + (((24*I)/35)*b*d^4*x^2)/c^2 - (11*b*d^4*x^3)/(24*c) - ((12*I)/35)*b*d^4*x^4 + (9*b*c*d^4*x^5)/40 + ((2*I)/21)*b*c^2*d^4*x^6 - (b*c^3*d^4*x^7)/56 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (d^4*x^4*(a + b*ArcTan[c*x]))/4 + ((4*I)/5)*c*d^4*x^5*(a + b*ArcTan[c*x]) - c^2*d^4*x^6*(a + b*ArcTan[c*x]) - ((4*I)/7)*c^3*d^4*x^7*(a + b*ArcTan[c*x]) + (c^4*d^4*x^8*(a + b*ArcTan[c*x]))/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4} -{x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]), x, 4, (((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) - ((5*I)/9)*b*d^4*x^3 + (47*b*c*d^4*x^4)/140 + ((2*I)/15)*b*c^2*d^4*x^5 - (b*c^3*d^4*x^6)/42 + ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c^3 - ((I/3)*d^4*(1 + I*c*x)^6*(a + b*ArcTan[c*x]))/c^3 + ((I/7)*d^4*(1 + I*c*x)^7*(a + b*ArcTan[c*x]))/c^3 + (176*b*d^4*Log[I + c*x])/(105*c^3)} -{x*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]), x, 4, (-16*b*d^4*x)/(15*c) - (((4*I)/15)*b*d^4*(I - c*x)^2)/c^2 - (4*b*d^4*(I - c*x)^3)/(45*c^2) + ((I/30)*b*d^4*(I - c*x)^4)/c^2 + (b*d^4*(I - c*x)^5)/(30*c^2) + (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*c^2) - (d^4*(1 + I*c*x)^6*(a + b*ArcTan[c*x]))/(6*c^2) + (((32*I)/15)*b*d^4*Log[I + c*x])/c^2} -{(d + I*c*d*x)^4*(a + b*ArcTan[c*x]), x, 4, (-(8/5))*I*b*d^4*x - (2*b*d^4*(1 + I*c*x)^2)/(5*c) - (2*b*d^4*(1 + I*c*x)^3)/(15*c) - (b*d^4*(1 + I*c*x)^4)/(20*c) - (I*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*c) - (16*b*d^4*Log[1 - I*c*x])/(5*c)} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x, x, 19, (4*I)*a*c*d^4*x + (13*b*c*d^4*x)/4 + ((2*I)/3)*b*c^2*d^4*x^2 - (b*c^3*d^4*x^3)/12 - (13*b*d^4*ArcTan[c*x])/4 + (4*I)*b*c*d^4*x*ArcTan[c*x] - 3*c^2*d^4*x^2*(a + b*ArcTan[c*x]) - ((4*I)/3)*c^3*d^4*x^3*(a + b*ArcTan[c*x]) + (c^4*d^4*x^4*(a + b*ArcTan[c*x]))/4 + a*d^4*Log[x] - ((8*I)/3)*b*d^4*Log[1 + c^2*x^2] + (I/2)*b*d^4*PolyLog[2, (-I)*c*x] - (I/2)*b*d^4*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^2, x, 20, -6*a*c^2*d^4*x + (2*I)*b*c^2*d^4*x - (b*c^3*d^4*x^2)/6 - (2*I)*b*c*d^4*ArcTan[c*x] - 6*b*c^2*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/x - (2*I)*c^3*d^4*x^2*(a + b*ArcTan[c*x]) + (c^4*d^4*x^3*(a + b*ArcTan[c*x]))/3 + (4*I)*a*c*d^4*Log[x] + b*c*d^4*Log[x] + (8*b*c*d^4*Log[1 + c^2*x^2])/3 - 2*b*c*d^4*PolyLog[2, (-I)*c*x] + 2*b*c*d^4*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^3, x, 19, -(b*c*d^4)/(2*x) - (4*I)*a*c^3*d^4*x - (b*c^3*d^4*x)/2 - (4*I)*b*c^3*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/(2*x^2) - ((4*I)*c*d^4*(a + b*ArcTan[c*x]))/x + (c^4*d^4*x^2*(a + b*ArcTan[c*x]))/2 - 6*a*c^2*d^4*Log[x] + (4*I)*b*c^2*d^4*Log[x] - (3*I)*b*c^2*d^4*PolyLog[2, (-I)*c*x] + (3*I)*b*c^2*d^4*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^4, x, 20, -(b*c*d^4)/(6*x^2) - ((2*I)*b*c^2*d^4)/x + a*c^4*d^4*x - (2*I)*b*c^3*d^4*ArcTan[c*x] + b*c^4*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/(3*x^3) - ((2*I)*c*d^4*(a + b*ArcTan[c*x]))/x^2 + (6*c^2*d^4*(a + b*ArcTan[c*x]))/x - (4*I)*a*c^3*d^4*Log[x] - (19*b*c^3*d^4*Log[x])/3 + (8*b*c^3*d^4*Log[1 + c^2*x^2])/3 + 2*b*c^3*d^4*PolyLog[2, (-I)*c*x] - 2*b*c^3*d^4*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^5, x, 21, -(b*c*d^4)/(12*x^3) - (((2*I)/3)*b*c^2*d^4)/x^2 + (13*b*c^3*d^4)/(4*x) + (13*b*c^4*d^4*ArcTan[c*x])/4 - (d^4*(a + b*ArcTan[c*x]))/(4*x^4) - (((4*I)/3)*c*d^4*(a + b*ArcTan[c*x]))/x^3 + (3*c^2*d^4*(a + b*ArcTan[c*x]))/x^2 + ((4*I)*c^3*d^4*(a + b*ArcTan[c*x]))/x + a*c^4*d^4*Log[x] - ((16*I)/3)*b*c^4*d^4*Log[x] + ((8*I)/3)*b*c^4*d^4*Log[1 + c^2*x^2] + (I/2)*b*c^4*d^4*PolyLog[2, (-I)*c*x] - (I/2)*b*c^4*d^4*PolyLog[2, I*c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6, x, 4, -(b*c*d^4)/(20*x^4) - ((I/3)*b*c^2*d^4)/x^3 + (11*b*c^3*d^4)/(10*x^2) + ((3*I)*b*c^4*d^4)/x - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*x^5) + (16*b*c^5*d^4*Log[x])/5 - (16*b*c^5*d^4*Log[I + c*x])/5} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^7, x, 4, -(b*c*d^4)/(30*x^5) - ((I/5)*b*c^2*d^4)/x^4 + (5*b*c^3*d^4)/(9*x^3) + (((16*I)/15)*b*c^4*d^4)/x^2 - (13*b*c^5*d^4)/(6*x) - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(6*x^6) + ((I/30)*c*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/x^5 + ((32*I)/15)*b*c^6*d^4*Log[x] - ((32*I)/15)*b*c^6*d^4*Log[I + c*x]} -{((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^8, x, 4, -(b*c*d^4)/(42*x^6) - (((2*I)/15)*b*c^2*d^4)/x^5 + (47*b*c^3*d^4)/(140*x^4) + (((5*I)/9)*b*c^4*d^4)/x^3 - (88*b*c^5*d^4)/(105*x^2) - (((5*I)/3)*b*c^6*d^4)/x - (d^4*(a + b*ArcTan[c*x]))/(7*x^7) - (((2*I)/3)*c*d^4*(a + b*ArcTan[c*x]))/x^6 + (6*c^2*d^4*(a + b*ArcTan[c*x]))/(5*x^5) + (I*c^3*d^4*(a + b*ArcTan[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTan[c*x]))/(3*x^3) - (176*b*c^7*d^4*Log[x])/105 + (b*c^7*d^4*Log[I - c*x])/210 + (117*b*c^7*d^4*Log[I + c*x])/70} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x), x, 16, (I*a*x)/(c^3*d) - (b*x)/(2*c^3*d) + ((I/6)*b*x^2)/(c^2*d) + (b*ArcTan[c*x])/(2*c^4*d) + (I*b*x*ArcTan[c*x])/(c^3*d) + (x^2*(a + b*ArcTan[c*x]))/(2*c^2*d) - ((I/3)*x^3*(a + b*ArcTan[c*x]))/(c*d) + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d) - (((2*I)/3)*b*Log[1 + c^2*x^2])/(c^4*d) + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d)} -{(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x), x, 11, (a*x)/(c^2*d) + ((I/2)*b*x)/(c^2*d) - ((I/2)*b*ArcTan[c*x])/(c^3*d) + (b*x*ArcTan[c*x])/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x]))/(c*d) - (I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) - (b*Log[1 + c^2*x^2])/(2*c^3*d) + (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d)} -{(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x), x, 7, ((-I)*a*x)/(c*d) - (I*b*x*ArcTan[c*x])/(c*d) - ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d) + ((I/2)*b*Log[1 + c^2*x^2])/(c^2*d) - ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d)} -{(a + b*ArcTan[c*x])/(d + I*c*d*x), x, 3, (I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*d) - (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d)} -{(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)), x, 2, ((a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d + ((I/2)*b*PolyLog[2, -1 + 2/(1 + I*c*x)])/d} -{(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)), x, 8, -((a + b*ArcTan[c*x])/(d*x)) + (b*c*Log[x])/d - (b*c*Log[1 + c^2*x^2])/(2*d) - (I*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d + (b*c*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d)} -{(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)), x, 12, -(b*c)/(2*d*x) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x]))/(d*x) - (I*b*c^2*Log[x])/d + ((I/2)*b*c^2*Log[1 + c^2*x^2])/d - (c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d - ((I/2)*b*c^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d} -{(a + b*ArcTan[c*x])/(x^4*(d + I*c*d*x)), x, 17, -(b*c)/(6*d*x^2) + ((I/2)*b*c^2)/(d*x) + ((I/2)*b*c^3*ArcTan[c*x])/d - (a + b*ArcTan[c*x])/(3*d*x^3) + ((I/2)*c*(a + b*ArcTan[c*x]))/(d*x^2) + (c^2*(a + b*ArcTan[c*x]))/(d*x) - (4*b*c^3*Log[x])/(3*d) + (2*b*c^3*Log[1 + c^2*x^2])/(3*d) + (I*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d - (b*c^3*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d)} - - -{(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2, x, 16, ((-2*I)*a*x)/(c^3*d^2) + (b*x)/(2*c^3*d^2) + b/(2*c^4*d^2*(I - c*x)) - (b*ArcTan[c*x])/(c^4*d^2) - ((2*I)*b*x*ArcTan[c*x])/(c^3*d^2) - (x^2*(a + b*ArcTan[c*x]))/(2*c^2*d^2) + (I*(a + b*ArcTan[c*x]))/(c^4*d^2*(I - c*x)) - (3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^2) + (I*b*Log[1 + c^2*x^2])/(c^4*d^2) - (((3*I)/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2)} -{(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2, x, 13, -((a*x)/(c^2*d^2)) - ((I/2)*b)/(c^3*d^2*(I - c*x)) + ((I/2)*b*ArcTan[c*x])/(c^3*d^2) - (b*x*ArcTan[c*x])/(c^2*d^2) + (a + b*ArcTan[c*x])/(c^3*d^2*(I - c*x)) + ((2*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^2) + (b*Log[1 + c^2*x^2])/(2*c^3*d^2) - (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2)} -{(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2, x, 10, -b/(2*c^2*d^2*(I - c*x)) + (b*ArcTan[c*x])/(2*c^2*d^2) - (I*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2) + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2)} -{(a + b*ArcTan[c*x])/(d + I*c*d*x)^2, x, 5, ((I/2)*b)/(c*d^2*(I - c*x)) - ((I/2)*b*ArcTan[c*x])/(c*d^2) + (I*(a + b*ArcTan[c*x]))/(c*d^2*(1 + I*c*x))} -{(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^2), x, 13, b/(2*d^2*(I - c*x)) - (b*ArcTan[c*x])/(2*d^2) + (I*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) + (a*Log[x])/d^2 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^2 - ((I/2)*b*PolyLog[2, I*c*x])/d^2 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2} -{(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^2), x, 18, ((-I/2)*b*c)/(d^2*(I - c*x)) + ((I/2)*b*c*ArcTan[c*x])/d^2 - (a + b*ArcTan[c*x])/(d^2*x) + (c*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - ((2*I)*a*c*Log[x])/d^2 + (b*c*Log[x])/d^2 - ((2*I)*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 - (b*c*Log[1 + c^2*x^2])/(2*d^2) + (b*c*PolyLog[2, (-I)*c*x])/d^2 - (b*c*PolyLog[2, I*c*x])/d^2 + (b*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2} -{(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^2), x, 21, -(b*c)/(2*d^2*x) - (b*c^2)/(2*d^2*(I - c*x)) - (a + b*ArcTan[c*x])/(2*d^2*x^2) + ((2*I)*c*(a + b*ArcTan[c*x]))/(d^2*x) - (I*c^2*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (3*a*c^2*Log[x])/d^2 - ((2*I)*b*c^2*Log[x])/d^2 - (3*c^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 + (I*b*c^2*Log[1 + c^2*x^2])/d^2 - (((3*I)/2)*b*c^2*PolyLog[2, (-I)*c*x])/d^2 + (((3*I)/2)*b*c^2*PolyLog[2, I*c*x])/d^2 - (((3*I)/2)*b*c^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2} - - -{(x^4*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3, x, 21, (-3*a*x)/(c^4*d^3) - ((I/2)*b*x)/(c^4*d^3) - b/(8*c^5*d^3*(I - c*x)^2) - (((15*I)/8)*b)/(c^5*d^3*(I - c*x)) + (((19*I)/8)*b*ArcTan[c*x])/(c^5*d^3) - (3*b*x*ArcTan[c*x])/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x]))/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) + ((6*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^5*d^3) + (3*b*Log[1 + c^2*x^2])/(2*c^5*d^3) - (3*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3)} -{(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3, x, 18, (I*a*x)/(c^3*d^3) + ((I/8)*b)/(c^4*d^3*(I - c*x)^2) - (11*b)/(8*c^4*d^3*(I - c*x)) + (11*b*ArcTan[c*x])/(8*c^4*d^3) + (I*b*x*ArcTan[c*x])/(c^3*d^3) - (a + b*ArcTan[c*x])/(2*c^4*d^3*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x]))/(c^4*d^3*(I - c*x)) + (3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^3) - ((I/2)*b*Log[1 + c^2*x^2])/(c^4*d^3) + (((3*I)/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3)} -{(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3, x, 15, b/(8*c^3*d^3*(I - c*x)^2) + (((7*I)/8)*b)/(c^3*d^3*(I - c*x)) - (((7*I)/8)*b*ArcTan[c*x])/(c^3*d^3) + ((I/2)*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)^2) - (2*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^3) + (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d^3)} -{(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3, x, 5, ((-I/8)*b)/(c^2*d^3*(I - c*x)^2) + (3*b)/(8*c^2*d^3*(I - c*x)) + (b*ArcTan[c*x])/(8*c^2*d^3) + (x^2*(a + b*ArcTan[c*x]))/(2*d^3*(1 + I*c*x)^2)} -{(a + b*ArcTan[c*x])/(d + I*c*d*x)^3, x, 5, -b/(8*c*d^3*(I - c*x)^2) + ((I/8)*b)/(c*d^3*(I - c*x)) - ((I/8)*b*ArcTan[c*x])/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x]))/(c*d^3*(1 + I*c*x)^2)} -{(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^3), x, 18, ((I/8)*b)/(d^3*(I - c*x)^2) + (5*b)/(8*d^3*(I - c*x)) - (5*b*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*PolyLog[2, I*c*x])/d^3 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3} -{(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^3), x, 23, (b*c)/(8*d^3*(I - c*x)^2) - (((9*I)/8)*b*c)/(d^3*(I - c*x)) + (((9*I)/8)*b*c*ArcTan[c*x])/d^3 - (a + b*ArcTan[c*x])/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - ((3*I)*a*c*Log[x])/d^3 + (b*c*Log[x])/d^3 - ((3*I)*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 - (b*c*Log[1 + c^2*x^2])/(2*d^3) + (3*b*c*PolyLog[2, (-I)*c*x])/(2*d^3) - (3*b*c*PolyLog[2, I*c*x])/(2*d^3) + (3*b*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*d^3)} -{(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^3), x, 26, -(b*c)/(2*d^3*x) - ((I/8)*b*c^2)/(d^3*(I - c*x)^2) - (13*b*c^2)/(8*d^3*(I - c*x)) + (9*b*c^2*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*x^2) + ((3*I)*c*(a + b*ArcTan[c*x]))/(d^3*x) + (c^2*(a + b*ArcTan[c*x]))/(2*d^3*(I - c*x)^2) - ((3*I)*c^2*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - (6*a*c^2*Log[x])/d^3 - ((3*I)*b*c^2*Log[x])/d^3 - (6*c^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 + (((3*I)/2)*b*c^2*Log[1 + c^2*x^2])/d^3 - ((3*I)*b*c^2*PolyLog[2, (-I)*c*x])/d^3 + ((3*I)*b*c^2*PolyLog[2, I*c*x])/d^3 - ((3*I)*b*c^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3} - - -{(a + b*ArcTan[c*x])/(1 + I*c*x)^4, x, 5, ((-I/18)*b)/(c*(I - c*x)^3) - b/(24*c*(I - c*x)^2) + ((I/24)*b)/(c*(I - c*x)) - ((I/24)*b*ArcTan[c*x])/c + ((I/3)*(a + b*ArcTan[c*x]))/(c*(1 + I*c*x)^3)} - - -{ArcTan[a*x]/(c*x + I*a*c*x^2), x, 3, (ArcTan[a*x]*Log[2 - 2/(1 + I*a*x)])/c + ((I/2)*PolyLog[2, -1 + 2/(1 + I*a*x)])/c} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^2 with c^2 d^2+e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2, x, 27, (a*b*d*x)/(2*c^3) - (((3*I)/10)*b^2*d*x)/c^3 + (b^2*d*x^2)/(12*c^2) + ((I/30)*b^2*d*x^3)/c + (((3*I)/10)*b^2*d*ArcTan[c*x])/c^4 + (b^2*d*x*ArcTan[c*x])/(2*c^3) + ((I/5)*b*d*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) - (I/10)*b*d*x^4*(a + b*ArcTan[c*x]) - (9*d*(a + b*ArcTan[c*x])^2)/(20*c^4) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x])^2 + (((2*I)/5)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) - (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^4)} -{x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2, x, 22, ((I/2)*a*b*d*x)/c^2 + (b^2*d*x)/(3*c^2) + ((I/12)*b^2*d*x^2)/c - (b^2*d*ArcTan[c*x])/(3*c^3) + ((I/2)*b^2*d*x*ArcTan[c*x])/c^2 - (b*d*x^2*(a + b*ArcTan[c*x]))/(3*c) - (I/6)*b*d*x^3*(a + b*ArcTan[c*x]) - (((7*I)/12)*d*(a + b*ArcTan[c*x])^2)/c^3 + (d*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/4)*c*d*x^4*(a + b*ArcTan[c*x])^2 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) - ((I/3)*b^2*d*Log[1 + c^2*x^2])/c^3 - ((I/3)*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{x*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2, x, 17, -((a*b*d*x)/c) + ((I/3)*b^2*d*x)/c - ((I/3)*b^2*d*ArcTan[c*x])/c^2 - (b^2*d*x*ArcTan[c*x])/c - (I/3)*b*d*x^2*(a + b*ArcTan[c*x]) + (5*d*(a + b*ArcTan[c*x])^2)/(6*c^2) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/3)*c*d*x^3*(a + b*ArcTan[c*x])^2 - (((2*I)/3)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) + (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)} -{(d + I*c*d*x)*(a + b*ArcTan[c*x])^2, x, 9, (-I)*a*b*d*x - I*b^2*d*x*ArcTan[c*x] - ((I/2)*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x])^2)/c + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + ((I/2)*b^2*d*Log[1 + c^2*x^2])/c - (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/c} -{((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x, x, 13, -(d*(a + b*ArcTan[c*x])^2) + I*c*d*x*(a + b*ArcTan[c*x])^2 + 2*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (2*I)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^2, x, 12, (-I)*c*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/x + (2*I)*c*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + 2*b*c*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d*PolyLog[2, -1 + 2/(1 - I*c*x)] + b*c*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - b*c*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (I/2)*b^2*c*d*PolyLog[3, 1 - 2/(1 + I*c*x)] + (I/2)*b^2*c*d*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^3, x, 14, -((b*c*d*(a + b*ArcTan[c*x]))/x) + (c^2*d*(a + b*ArcTan[c*x])^2)/2 - (d*(a + b*ArcTan[c*x])^2)/(2*x^2) - (I*c*d*(a + b*ArcTan[c*x])^2)/x + b^2*c^2*d*Log[x] - (b^2*c^2*d*Log[1 + c^2*x^2])/2 + (2*I)*b*c^2*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + b^2*c^2*d*PolyLog[2, -1 + 2/(1 - I*c*x)]} -{((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^4, x, 18, -(b^2*c^2*d)/(3*x) - (b^2*c^3*d*ArcTan[c*x])/3 - (b*c*d*(a + b*ArcTan[c*x]))/(3*x^2) - (I*b*c^2*d*(a + b*ArcTan[c*x]))/x - (I/6)*c^3*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[c*x])^2)/x^2 + I*b^2*c^3*d*Log[x] - (I/2)*b^2*c^3*d*Log[1 + c^2*x^2] - (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + (I/3)*b^2*c^3*d*PolyLog[2, -1 + 2/(1 - I*c*x)]} - - -{x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2, x, 43, (5*a*b*d^2*x)/(6*c^3) - (((3*I)/5)*b^2*d^2*x)/c^3 + (31*b^2*d^2*x^2)/(180*c^2) + ((I/15)*b^2*d^2*x^3)/c - (b^2*d^2*x^4)/60 + (((3*I)/5)*b^2*d^2*ArcTan[c*x])/c^4 + (5*b^2*d^2*x*ArcTan[c*x])/(6*c^3) + (((2*I)/5)*b*d^2*x^2*(a + b*ArcTan[c*x]))/c^2 - (5*b*d^2*x^3*(a + b*ArcTan[c*x]))/(18*c) - (I/5)*b*d^2*x^4*(a + b*ArcTan[c*x]) + (b*c*d^2*x^5*(a + b*ArcTan[c*x]))/15 - (49*d^2*(a + b*ArcTan[c*x])^2)/(60*c^4) + (d^2*x^4*(a + b*ArcTan[c*x])^2)/4 + ((2*I)/5)*c*d^2*x^5*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^6*(a + b*ArcTan[c*x])^2)/6 + (((4*I)/5)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (53*b^2*d^2*Log[1 + c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^4)} -{x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2, x, 36, (I*a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + ((I/6)*b^2*d^2*x^2)/c - (b^2*d^2*x^3)/30 - (19*b^2*d^2*ArcTan[c*x])/(30*c^3) + (I*b^2*d^2*x*ArcTan[c*x])/c^2 - (8*b*d^2*x^2*(a + b*ArcTan[c*x]))/(15*c) - (I/3)*b*d^2*x^3*(a + b*ArcTan[c*x]) + (b*c*d^2*x^4*(a + b*ArcTan[c*x]))/10 - (((31*I)/30)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/2)*c*d^2*x^4*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^5*(a + b*ArcTan[c*x])^2)/5 - (16*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((2*I)/3)*b^2*d^2*Log[1 + c^2*x^2])/c^3 - (((8*I)/15)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2, x, 28, (-3*a*b*d^2*x)/(2*c) + (((2*I)/3)*b^2*d^2*x)/c - (b^2*d^2*x^2)/12 - (((2*I)/3)*b^2*d^2*ArcTan[c*x])/c^2 - (3*b^2*d^2*x*ArcTan[c*x])/(2*c) - ((2*I)/3)*b*d^2*x^2*(a + b*ArcTan[c*x]) + (b*c*d^2*x^3*(a + b*ArcTan[c*x]))/6 + (17*d^2*(a + b*ArcTan[c*x])^2)/(12*c^2) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^4*(a + b*ArcTan[c*x])^2)/4 - (((4*I)/3)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (5*b^2*d^2*Log[1 + c^2*x^2])/(6*c^2) + (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)} -{(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2, x, 12, (-2*I)*a*b*d^2*x - (b^2*d^2*x)/3 + (b^2*d^2*ArcTan[c*x])/(3*c) - (2*I)*b^2*d^2*x*ArcTan[c*x] + (b*c*d^2*x^2*(a + b*ArcTan[c*x]))/3 - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^2)/c + (8*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(3*c) + (I*b^2*d^2*Log[1 + c^2*x^2])/c - (((4*I)/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/c} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x, x, 19, a*b*c*d^2*x + b^2*c*d^2*x*ArcTan[c*x] - (5*d^2*(a + b*ArcTan[c*x])^2)/2 + (2*I)*c*d^2*x*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (4*I)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*d^2*Log[1 + c^2*x^2])/2 - 2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^2, x, 17, (-2*I)*c*d^2*(a + b*ArcTan[c*x])^2 - (d^2*(a + b*ArcTan[c*x])^2)/x - c^2*d^2*x*(a + b*ArcTan[c*x])^2 + (4*I)*c*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] + 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] + 2*b*c*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - I*b^2*c*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)] + I*b^2*c*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^3, x, 20, -((b*c*d^2*(a + b*ArcTan[c*x]))/x) + (3*c^2*d^2*(a + b*ArcTan[c*x])^2)/2 - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) - ((2*I)*c*d^2*(a + b*ArcTan[c*x])^2)/x - 2*c^2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^2*Log[x] - (b^2*c^2*d^2*Log[1 + c^2*x^2])/2 + (4*I)*b*c^2*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + 2*b^2*c^2*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] + I*b*c^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*c^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (b^2*c^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 - (b^2*c^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^4, x, 16, -(b^2*c^2*d^2)/(3*x) - (b^2*c^3*d^2*ArcTan[c*x])/3 - (b*c*d^2*(a + b*ArcTan[c*x]))/(3*x^2) - ((2*I)*b*c^2*d^2*(a + b*ArcTan[c*x]))/x - (d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^2)/(3*x^3) - (8*a*b*c^3*d^2*Log[x])/3 + (2*I)*b^2*c^3*d^2*Log[x] - (8*b*c^3*d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/3 - I*b^2*c^3*d^2*Log[1 + c^2*x^2] - ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, (-I)*c*x] + ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, I*c*x] + ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)]} - - -{x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2, x, 62, (3*a*b*d^3*x)/(2*c^3) - (((122*I)/105)*b^2*d^3*x)/c^3 + (7*b^2*d^3*x^2)/(20*c^2) + (((44*I)/315)*b^2*d^3*x^3)/c - (b^2*d^3*x^4)/20 - (I/105)*b^2*c*d^3*x^5 + (((122*I)/105)*b^2*d^3*ArcTan[c*x])/c^4 + (3*b^2*d^3*x*ArcTan[c*x])/(2*c^3) + (((26*I)/35)*b*d^3*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d^3*x^3*(a + b*ArcTan[c*x]))/(2*c) - ((13*I)/35)*b*d^3*x^4*(a + b*ArcTan[c*x]) + (b*c*d^3*x^5*(a + b*ArcTan[c*x]))/5 + (I/21)*b*c^2*d^3*x^6*(a + b*ArcTan[c*x]) - (209*d^3*(a + b*ArcTan[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTan[c*x])^2)/4 + ((3*I)/5)*c*d^3*x^5*(a + b*ArcTan[c*x])^2 - (c^2*d^3*x^6*(a + b*ArcTan[c*x])^2)/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x])^2 + (((52*I)/35)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (11*b^2*d^3*Log[1 + c^2*x^2])/(10*c^4) - (26*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(35*c^4)} -{x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2, x, 52, (((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c*x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3*b*c*d^3*x^4*(a + b*ArcTan[c*x]))/10 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c*x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*ArcTan[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x])^2)/5 - (I/6)*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90)*b^2*d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3} -{x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2, x, 38, (-5*a*b*d^3*x)/(2*c) + (((13*I)/10)*b^2*d^3*x)/c - (b^2*d^3*x^2)/4 - (I/30)*b^2*c*d^3*x^3 - (((13*I)/10)*b^2*d^3*ArcTan[c*x])/c^2 - (5*b^2*d^3*x*ArcTan[c*x])/(2*c) - ((6*I)/5)*b*d^3*x^2*(a + b*ArcTan[c*x]) + (b*c*d^3*x^3*(a + b*ArcTan[c*x]))/2 + (I/10)*b*c^2*d^3*x^4*(a + b*ArcTan[c*x]) + (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(4*c^2) - (d^3*(1 + I*c*x)^5*(a + b*ArcTan[c*x])^2)/(5*c^2) - (((12*I)/5)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c^2 + (3*b^2*d^3*Log[1 + c^2*x^2])/(2*c^2) - (6*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/(5*c^2)} -{(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2, x, 16, ((-7*I)/2)*a*b*d^3*x - b^2*d^3*x - (I/12)*b^2*c*d^3*x^2 + (b^2*d^3*ArcTan[c*x])/c - ((7*I)/2)*b^2*d^3*x*ArcTan[c*x] + b*c*d^3*x^2*(a + b*ArcTan[c*x]) + (I/6)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x]) - ((I/4)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/c + (4*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + (((11*I)/6)*b^2*d^3*Log[1 + c^2*x^2])/c - ((2*I)*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/c} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x, x, 28, 3*a*b*c*d^3*x - (I/3)*b^2*c*d^3*x + (I/3)*b^2*d^3*ArcTan[c*x] + 3*b^2*c*d^3*x*ArcTan[c*x] + (I/3)*b*c^2*d^3*x^2*(a + b*ArcTan[c*x]) - (29*d^3*(a + b*ArcTan[c*x])^2)/6 + (3*I)*c*d^3*x*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^2*(a + b*ArcTan[c*x])^2)/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x])^2 + 2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + ((20*I)/3)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (3*b^2*d^3*Log[1 + c^2*x^2])/2 - (10*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/3 - I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^2, x, 23, I*a*b*c^2*d^3*x + I*b^2*c^2*d^3*x*ArcTan[c*x] - ((9*I)/2)*c*d^3*(a + b*ArcTan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/x - 3*c^2*d^3*x*(a + b*ArcTan[c*x])^2 - (I/2)*c^3*d^3*x^2*(a + b*ArcTan[c*x])^2 + (6*I)*c*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 6*b*c*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (I/2)*b^2*c*d^3*Log[1 + c^2*x^2] + 2*b*c*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] - (3*I)*b^2*c*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + 3*b*c*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 3*b*c*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - ((3*I)/2)*b^2*c*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)] + ((3*I)/2)*b^2*c*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^3, x, 25, -((b*c*d^3*(a + b*ArcTan[c*x]))/x) + (7*c^2*d^3*(a + b*ArcTan[c*x])^2)/2 - (d^3*(a + b*ArcTan[c*x])^2)/(2*x^2) - ((3*I)*c*d^3*(a + b*ArcTan[c*x])^2)/x - I*c^3*d^3*x*(a + b*ArcTan[c*x])^2 - 6*c^2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^3*Log[x] - (2*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*c^2*d^3*Log[1 + c^2*x^2])/2 + (6*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + 3*b^2*c^2*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] + b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + (3*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - (3*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (3*b^2*c^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 - (3*b^2*c^2*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/2} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^4, x, 28, -(b^2*c^2*d^3)/(3*x) - (b^2*c^3*d^3*ArcTan[c*x])/3 - (b*c*d^3*(a + b*ArcTan[c*x]))/(3*x^2) - ((3*I)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x + ((11*I)/6)*c^3*d^3*(a + b*ArcTan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/(3*x^3) - (((3*I)/2)*c*d^3*(a + b*ArcTan[c*x])^2)/x^2 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/x - (2*I)*c^3*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (3*I)*b^2*c^3*d^3*Log[x] - ((3*I)/2)*b^2*c^3*d^3*Log[1 + c^2*x^2] - (20*b*c^3*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + ((10*I)/3)*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] - b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (I/2)*b^2*c^3*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)] - (I/2)*b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5, x, 20, -(b^2*c^2*d^3)/(12*x^2) - (I*b^2*c^3*d^3)/x - I*b^2*c^4*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(6*x^3) - (I*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^2 + (7*b*c^3*d^3*(a + b*ArcTan[c*x]))/(2*x) - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(4*x^4) - (4*I)*a*b*c^4*d^3*Log[x] - (11*b^2*c^4*d^3*Log[x])/3 - (4*I)*b*c^4*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] + (11*b^2*c^4*d^3*Log[1 + c^2*x^2])/6 + 2*b^2*c^4*d^3*PolyLog[2, (-I)*c*x] - 2*b^2*c^4*d^3*PolyLog[2, I*c*x] - 2*b^2*c^4*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^6, x, 24, -(b^2*c^2*d^3)/(30*x^3) - ((I/4)*b^2*c^3*d^3)/x^2 + (13*b^2*c^4*d^3)/(10*x) + (13*b^2*c^5*d^3*ArcTan[c*x])/10 - (b*c*d^3*(a + b*ArcTan[c*x]))/(10*x^4) - ((I/2)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^3 + (6*b*c^3*d^3*(a + b*ArcTan[c*x]))/(5*x^2) + (((5*I)/2)*b*c^4*d^3*(a + b*ArcTan[c*x]))/x - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(5*x^5) + ((I/20)*c*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/x^4 + (12*a*b*c^5*d^3*Log[x])/5 - (3*I)*b^2*c^5*d^3*Log[x] + (12*b*c^5*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/5 + ((3*I)/2)*b^2*c^5*d^3*Log[1 + c^2*x^2] + ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, (-I)*c*x] - ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, I*c*x] - ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)]} -{((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7, x, 31, -(b^2*c^2*d^3)/(60*x^4) - ((I/10)*b^2*c^3*d^3)/x^3 + (61*b^2*c^4*d^3)/(180*x^2) + (((37*I)/30)*b^2*c^5*d^3)/x + ((37*I)/30)*b^2*c^6*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(15*x^5) - (((3*I)/10)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^4 + (11*b*c^3*d^3*(a + b*ArcTan[c*x]))/(18*x^3) + (((14*I)/15)*b*c^4*d^3*(a + b*ArcTan[c*x]))/x^2 - (11*b*c^5*d^3*(a + b*ArcTan[c*x]))/(6*x) - (d^3*(a + b*ArcTan[c*x])^2)/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x])^2)/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x])^2)/x^3 + ((28*I)/15)*a*b*c^6*d^3*Log[x] + (113*b^2*c^6*d^3*Log[x])/45 + ((37*I)/20)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] + (I/60)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (113*b^2*c^6*d^3*Log[1 + c^2*x^2])/90 - (14*b^2*c^6*d^3*PolyLog[2, (-I)*c*x])/15 + (14*b^2*c^6*d^3*PolyLog[2, I*c*x])/15 + (37*b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/40 - (b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/120} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x), x, 26, -((a*b*x)/(c^3*d)) - ((I/3)*b^2*x)/(c^3*d) + ((I/3)*b^2*ArcTan[c*x])/(c^4*d) - (b^2*x*ArcTan[c*x])/(c^3*d) + ((I/3)*b*x^2*(a + b*ArcTan[c*x]))/(c^2*d) - (5*(a + b*ArcTan[c*x])^2)/(6*c^4*d) + (I*x*(a + b*ArcTan[c*x])^2)/(c^3*d) + (x^2*(a + b*ArcTan[c*x])^2)/(2*c^2*d) - ((I/3)*x^3*(a + b*ArcTan[c*x])^2)/(c*d) + (((8*I)/3)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d) + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d) + (b^2*Log[1 + c^2*x^2])/(2*c^4*d) - (4*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^4*d) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d) + (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d)} -{(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x), x, 16, (I*a*b*x)/(c^2*d) + (I*b^2*x*ArcTan[c*x])/(c^2*d) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d) + (x*(a + b*ArcTan[c*x])^2)/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c*d) + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) - (I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d) - ((I/2)*b^2*Log[1 + c^2*x^2])/(c^3*d) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) + (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) - ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d)} -{(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x), x, 9, (a + b*ArcTan[c*x])^2/(c^2*d) - (I*x*(a + b*ArcTan[c*x])^2)/(c*d) - ((2*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d) - ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d) + (b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d)} -{(a + b*ArcTan[c*x])^2/(d + I*c*d*x), x, 3, (I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c*d) - (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*d) + ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c*d)} -{(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)), x, 3, ((a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)), x, 8, ((-I)*c*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(d*x) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (I*c*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - ((I/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d} -{(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)), x, 17, -((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (3*c^2*(a + b*ArcTan[c*x])^2)/(2*d) - (a + b*ArcTan[c*x])^2/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^2)/(d*x) + (b^2*c^2*Log[x])/d - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) - ((2*I)*b*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d - (b^2*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (I*b*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - (b^2*c^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d)} -{(a + b*ArcTan[c*x])^2/(x^4*(d + I*c*d*x)), x, 26, -(b^2*c^2)/(3*d*x) - (b^2*c^3*ArcTan[c*x])/(3*d) - (b*c*(a + b*ArcTan[c*x]))/(3*d*x^2) + (I*b*c^2*(a + b*ArcTan[c*x]))/(d*x) + (((11*I)/6)*c^3*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(3*d*x^3) + ((I/2)*c*(a + b*ArcTan[c*x])^2)/(d*x^2) + (c^2*(a + b*ArcTan[c*x])^2)/(d*x) - (I*b^2*c^3*Log[x])/d + ((I/2)*b^2*c^3*Log[1 + c^2*x^2])/d - (8*b*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/(3*d) + (I*c^3*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d + (((4*I)/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (b*c^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + ((I/2)*b^2*c^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/d} - - -{(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2, x, 33, ((2*I)*a*b*x)/(c^4*d^2) - (b^2*x)/(3*c^4*d^2) + b^2/(2*c^5*d^2*(I - c*x)) - (b^2*ArcTan[c*x])/(6*c^5*d^2) + ((2*I)*b^2*x*ArcTan[c*x])/(c^4*d^2) + (b*x^2*(a + b*ArcTan[c*x]))/(3*c^3*d^2) + (I*b*(a + b*ArcTan[c*x]))/(c^5*d^2*(I - c*x)) + (((11*I)/6)*(a + b*ArcTan[c*x])^2)/(c^5*d^2) + (3*x*(a + b*ArcTan[c*x])^2)/(c^4*d^2) - (I*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x^3*(a + b*ArcTan[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTan[c*x])^2/(c^5*d^2*(I - c*x)) + (20*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^5*d^2) - ((4*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^2) - (I*b^2*Log[1 + c^2*x^2])/(c^5*d^2) + (((10*I)/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) + (4*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) - ((2*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^2)} -{(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2, x, 24, (a*b*x)/(c^3*d^2) - ((I/2)*b^2)/(c^4*d^2*(I - c*x)) + ((I/2)*b^2*ArcTan[c*x])/(c^4*d^2) + (b^2*x*ArcTan[c*x])/(c^3*d^2) + (b*(a + b*ArcTan[c*x]))/(c^4*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(c^4*d^2) - ((2*I)*x*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x^2*(a + b*ArcTan[c*x])^2)/(2*c^2*d^2) + (I*(a + b*ArcTan[c*x])^2)/(c^4*d^2*(I - c*x)) - ((4*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^2) - (3*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d^2) - (b^2*Log[1 + c^2*x^2])/(2*c^4*d^2) + (2*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2) - ((3*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2) - (3*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d^2)} -{(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2, x, 18, -b^2/(2*c^3*d^2*(I - c*x)) + (b^2*ArcTan[c*x])/(2*c^3*d^2) - (I*b*(a + b*ArcTan[c*x]))/(c^3*d^2*(I - c*x)) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x*(a + b*ArcTan[c*x])^2)/(c^2*d^2) + (a + b*ArcTan[c*x])^2/(c^3*d^2*(I - c*x)) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^2) + ((2*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d^2) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2) - (2*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2) + (I*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d^2)} -{(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2, x, 13, ((I/2)*b^2)/(c^2*d^2*(I - c*x)) - ((I/2)*b^2*ArcTan[c*x])/(c^2*d^2) - (b*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(2*c^2*d^2) - (I*(a + b*ArcTan[c*x])^2)/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d^2)} -{(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^2, x, 8, b^2/(2*c*d^2*(I - c*x)) - (b^2*ArcTan[c*x])/(2*c*d^2) + (I*b*(a + b*ArcTan[c*x]))/(c*d^2*(I - c*x)) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c*d^2) + (I*(a + b*ArcTan[c*x])^2)/(c*d^2*(1 + I*c*x))} -{(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^2), x, 19, ((-I/2)*b^2)/(d^2*(I - c*x)) + ((I/2)*b^2*ArcTan[c*x])/d^2 + (b*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (a + b*ArcTan[c*x])^2/(2*d^2) + (I*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^2), x, 23, -(b^2*c)/(2*d^2*(I - c*x)) + (b^2*c*ArcTan[c*x])/(2*d^2) - (I*b*c*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - ((I/2)*c*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(d^2*x) + (c*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) - ((4*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 - ((2*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^2} -{(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)^2), x, 31, ((I/2)*b^2*c^2)/(d^2*(I - c*x)) - ((I/2)*b^2*c^2*ArcTan[c*x])/d^2 - (b*c*(a + b*ArcTan[c*x]))/(d^2*x) - (b*c^2*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (2*c^2*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(2*d^2*x^2) + ((2*I)*c*(a + b*ArcTan[c*x])^2)/(d^2*x) - (I*c^2*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) - (6*c^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + (b^2*c^2*Log[x])/d^2 - (3*c^2*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) - ((4*I)*b*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (2*b^2*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - ((3*I)*b*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (3*b^2*c^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2)} - - -{(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3, x, 37, ((-I)*a*b*x)/(c^4*d^3) + ((I/16)*b^2)/(c^5*d^3*(I - c*x)^2) - (29*b^2)/(16*c^5*d^3*(I - c*x)) + (29*b^2*ArcTan[c*x])/(16*c^5*d^3) - (I*b^2*x*ArcTan[c*x])/(c^4*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c^5*d^3*(I - c*x)^2) - (((15*I)/4)*b*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) - (((5*I)/8)*(a + b*ArcTan[c*x])^2)/(c^5*d^3) - (3*x*(a + b*ArcTan[c*x])^2)/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)) - (6*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((6*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((I/2)*b^2*Log[1 + c^2*x^2])/(c^5*d^3) - ((3*I)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) - (6*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) + ((3*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^3)} -{(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3, x, 31, b^2/(16*c^4*d^3*(I - c*x)^2) + (((21*I)/16)*b^2)/(c^4*d^3*(I - c*x)) - (((21*I)/16)*b^2*ArcTan[c*x])/(c^4*d^3) + ((I/4)*b*(a + b*ArcTan[c*x]))/(c^4*d^3*(I - c*x)^2) - (11*b*(a + b*ArcTan[c*x]))/(4*c^4*d^3*(I - c*x)) + (3*(a + b*ArcTan[c*x])^2)/(8*c^4*d^3) + (I*x*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - (a + b*ArcTan[c*x])^2/(2*c^4*d^3*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x])^2)/(c^4*d^3*(I - c*x)) + ((2*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^3) + (3*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d^3) - (b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3) + ((3*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3) + (3*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d^3)} -{(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3, x, 26, ((-I/16)*b^2)/(c^3*d^3*(I - c*x)^2) + (13*b^2)/(16*c^3*d^3*(I - c*x)) - (13*b^2*ArcTan[c*x])/(16*c^3*d^3) + (b*(a + b*ArcTan[c*x]))/(4*c^3*d^3*(I - c*x)^2) + (((7*I)/4)*b*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)) - (((7*I)/8)*(a + b*ArcTan[c*x])^2)/(c^3*d^3) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d^3*(I - c*x)^2) - (2*(a + b*ArcTan[c*x])^2)/(c^3*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d^3) + (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^3) - ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d^3)} -{(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3, x, 13, -b^2/(16*c^2*d^3*(I - c*x)^2) - (((5*I)/16)*b^2)/(c^2*d^3*(I - c*x)) + (((5*I)/16)*b^2*ArcTan[c*x])/(c^2*d^3) - ((I/4)*b*(a + b*ArcTan[c*x]))/(c^2*d^3*(I - c*x)^2) + (3*b*(a + b*ArcTan[c*x]))/(4*c^2*d^3*(I - c*x)) + (a + b*ArcTan[c*x])^2/(8*c^2*d^3) + (x^2*(a + b*ArcTan[c*x])^2)/(2*d^3*(1 + I*c*x)^2)} -{(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^3, x, 13, ((I/16)*b^2)/(c*d^3*(I - c*x)^2) + (3*b^2)/(16*c*d^3*(I - c*x)) - (3*b^2*ArcTan[c*x])/(16*c*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c*d^3*(I - c*x)^2) + ((I/4)*b*(a + b*ArcTan[c*x]))/(c*d^3*(I - c*x)) - ((I/8)*(a + b*ArcTan[c*x])^2)/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c*d^3*(1 + I*c*x)^2)} -{(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^3), x, 32, b^2/(16*d^3*(I - c*x)^2) - (((11*I)/16)*b^2)/(d^3*(I - c*x)) + (((11*I)/16)*b^2*ArcTan[c*x])/d^3 + ((I/4)*b*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (5*b*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)) - (5*(a + b*ArcTan[c*x])^2)/(8*d^3) - (a + b*ArcTan[c*x])^2/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^3)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^3), x, 36, ((-I/16)*b^2*c)/(d^3*(I - c*x)^2) - (19*b^2*c)/(16*d^3*(I - c*x)) + (19*b^2*c*ArcTan[c*x])/(16*d^3) + (b*c*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)^2) - (((9*I)/4)*b*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + ((I/8)*c*(a + b*ArcTan[c*x])^2)/d^3 - (a + b*ArcTan[c*x])^2/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) - ((6*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 - ((3*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^3 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^3 + (3*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 - (((3*I)/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3} - - -{(a + b*ArcTan[c*x])^2/(1 + I*c*x)^4, x, 18, -b^2/(54*c*(I - c*x)^3) + (((5*I)/144)*b^2)/(c*(I - c*x)^2) + (11*b^2)/(144*c*(I - c*x)) - (11*b^2*ArcTan[c*x])/(144*c) - ((I/9)*b*(a + b*ArcTan[c*x]))/(c*(I - c*x)^3) - (b*(a + b*ArcTan[c*x]))/(12*c*(I - c*x)^2) + ((I/12)*b*(a + b*ArcTan[c*x]))/(c*(I - c*x)) - ((I/24)*(a + b*ArcTan[c*x])^2)/c + ((I/3)*(a + b*ArcTan[c*x])^2)/(c*(1 + I*c*x)^3)} - - -{ArcTan[a*x]^2/(c*x - I*a*c*x^2), x, 4, (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^3 with c^2 d^2+e^2=0*) - - -{(a + b*ArcTan[c*x])^3*(d + I*c*d*x)^3, x, 26, -3*a*b^2*d^3*x + (1/4)*I*b^3*d^3*x - (I*b^3*d^3*ArcTan[c*x])/(4*c) - 3*b^3*d^3*x*ArcTan[c*x] - (1/4)*I*b^2*c*d^3*x^2*(a + b*ArcTan[c*x]) + (7*b*d^3*(a + b*ArcTan[c*x])^2)/c - (21/4)*I*b*d^3*x*(a + b*ArcTan[c*x])^2 + (3/2)*b*c*d^3*x^2*(a + b*ArcTan[c*x])^2 + (1/4)*I*b*c^2*d^3*x^3*(a + b*ArcTan[c*x])^2 - (I*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^3)/(4*c) + (6*b*d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - (11*I*b^2*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (3*b^3*d^3*Log[1 + c^2*x^2])/(2*c) - (6*I*b^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (11*b^3*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/c} -{(a + b*ArcTan[c*x])^3*(d + I*c*d*x)^2, x, 17, (-a)*b^2*d^2*x - b^3*d^2*x*ArcTan[c*x] + (7*b*d^2*(a + b*ArcTan[c*x])^2)/(2*c) - 3*I*b*d^2*x*(a + b*ArcTan[c*x])^2 + (1/2)*b*c*d^2*x^2*(a + b*ArcTan[c*x])^2 - (I*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^3)/(3*c) + (4*b*d^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - (6*I*b^2*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (b^3*d^2*Log[1 + c^2*x^2])/(2*c) - (4*I*b^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (3*b^3*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c + (2*b^3*d^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/c} -{(a + b*ArcTan[c*x])^3*(d + I*c*d*x)^1, x, 11, (3*b*d*(a + b*ArcTan[c*x])^2)/(2*c) - (3/2)*I*b*d*x*(a + b*ArcTan[c*x])^2 - (I*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x])^3)/(2*c) + (3*b*d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - (3*I*b^2*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (3*I*b^2*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (3*b^3*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*c)} -{(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^1, x, 4, (I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c*d) - (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d) + (3*I*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c*d) + (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c*d)} -{(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^2, x, 11, -((3*I*b^3)/(4*c*d^2*(I - c*x))) + (3*I*b^3*ArcTan[c*x])/(4*c*d^2) + (3*b^2*(a + b*ArcTan[c*x]))/(2*c*d^2*(I - c*x)) - (3*b*(a + b*ArcTan[c*x])^2)/(4*c*d^2) + (3*I*b*(a + b*ArcTan[c*x])^2)/(2*c*d^2*(I - c*x)) - (I*(a + b*ArcTan[c*x])^3)/(2*c*d^2) + (I*(a + b*ArcTan[c*x])^3)/(c*d^2*(1 + I*c*x))} -{(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^3, x, 24, (3*b^3)/(64*c*d^3*(I - c*x)^2) - (21*I*b^3)/(64*c*d^3*(I - c*x)) + (21*I*b^3*ArcTan[c*x])/(64*c*d^3) + (3*I*b^2*(a + b*ArcTan[c*x]))/(16*c*d^3*(I - c*x)^2) + (9*b^2*(a + b*ArcTan[c*x]))/(16*c*d^3*(I - c*x)) - (9*b*(a + b*ArcTan[c*x])^2)/(32*c*d^3) - (3*b*(a + b*ArcTan[c*x])^2)/(8*c*d^3*(I - c*x)^2) + (3*I*b*(a + b*ArcTan[c*x])^2)/(8*c*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x])^3)/(8*c*d^3) + (I*(a + b*ArcTan[c*x])^3)/(2*c*d^3*(1 + I*c*x)^2)} -{(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^4, x, 42, (I*b^3)/(108*c*d^4*(I - c*x)^3) + (19*b^3)/(576*c*d^4*(I - c*x)^2) - (85*I*b^3)/(576*c*d^4*(I - c*x)) + (85*I*b^3*ArcTan[c*x])/(576*c*d^4) - (b^2*(a + b*ArcTan[c*x]))/(18*c*d^4*(I - c*x)^3) + (5*I*b^2*(a + b*ArcTan[c*x]))/(48*c*d^4*(I - c*x)^2) + (11*b^2*(a + b*ArcTan[c*x]))/(48*c*d^4*(I - c*x)) - (11*b*(a + b*ArcTan[c*x])^2)/(96*c*d^4) - (I*b*(a + b*ArcTan[c*x])^2)/(6*c*d^4*(I - c*x)^3) - (b*(a + b*ArcTan[c*x])^2)/(8*c*d^4*(I - c*x)^2) + (I*b*(a + b*ArcTan[c*x])^2)/(8*c*d^4*(I - c*x)) - (I*(a + b*ArcTan[c*x])^3)/(24*c*d^4) + (I*(a + b*ArcTan[c*x])^3)/(3*c*d^4*(1 + I*c*x)^3)} - - -{x^2*(a + b*ArcTan[c*x])^3/(d + I*c*d*x), x, 19, (-3*b*(a + b*ArcTan[c*x])^2)/(2*c^3*d) + (((3*I)/2)*b*x*(a + b*ArcTan[c*x])^2)/(c^2*d) + ((I/2)*(a + b*ArcTan[c*x])^3)/(c^3*d) + (x*(a + b*ArcTan[c*x])^3)/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x])^3)/(c*d) + ((3*I)*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) + (3*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d) - (I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c^3*d) - (3*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d) + ((3*I)*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) + (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d) + (3*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3*d) - (((3*I)/2)*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d) - (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c^3*d)} -{x^1*(a + b*ArcTan[c*x])^3/(d + I*c*d*x), x, 10, (a + b*ArcTan[c*x])^3/(c^2*d) - (I*x*(a + b*ArcTan[c*x])^3)/(c*d) - ((3*I)*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d) - ((a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c^2*d) + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (((3*I)/2)*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^2*d) - (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d) + (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(c^2*d)} -{x^0*(a + b*ArcTan[c*x])^3/(d + I*c*d*x), x, 4, (I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c*d) - (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d) + (((3*I)/2)*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c*d) + (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c*d)} -{(a + b*ArcTan[c*x])^3/(x^1*(d + I*c*d*x)), x, 4, ((a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d + (((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (((3*I)/4)*b^3*PolyLog[4, -1 + 2/(1 + I*c*x)])/d} -{(a + b*ArcTan[c*x])^3/(x^2*(d + I*c*d*x)), x, 10, ((-I)*c*(a + b*ArcTan[c*x])^3)/d - (a + b*ArcTan[c*x])^3/(d*x) + (3*b*c*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (I*c*(a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - ((3*I)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (3*b*c*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d) + (3*b^3*c*PolyLog[3, -1 + 2/(1 - I*c*x)])/(2*d) - (((3*I)/2)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/d - (3*b^3*c*PolyLog[4, -1 + 2/(1 + I*c*x)])/(4*d)} -{(a + b*ArcTan[c*x])^3/(x^3*(d + I*c*d*x)), x, 18, (((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2)/d - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*d*x) - (3*c^2*(a + b*ArcTan[c*x])^3)/(2*d) - (a + b*ArcTan[c*x])^3/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^3)/(d*x) + (3*b^2*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - ((3*I)*b*c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (((3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[3, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) + (((3*I)/4)*b^3*c^2*PolyLog[4, -1 + 2/(1 + I*c*x)])/d} - - -(* ::Subsection:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^4 with c^2 d^2+e^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^-1 with c^2 d^2+e^2=0*) - - -{1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x, 0, Unintegrable[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^q (a+b ArcTan[c x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m / (d+e x) (a+b ArcTan[c x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*ArcTan[c*x])/(d + e*x), x, 16, (a*d^2*x)/e^3 + (b*d*x)/(2*c*e^2) - (b*x^2)/(6*c*e) - (b*d*ArcTan[c*x])/(2*c^2*e^2) + (b*d^2*x*ArcTan[c*x])/e^3 - (d*x^2*(a + b*ArcTan[c*x]))/(2*e^2) + (x^3*(a + b*ArcTan[c*x]))/(3*e) + (d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^4 - (d^3*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b*d^2*Log[1 + c^2*x^2])/(2*c*e^3) + (b*Log[1 + c^2*x^2])/(6*c^3*e) - (I*b*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^4) + (I*b*d^3*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^4)} -{x^2*(a + b*ArcTan[c*x])/(d + e*x), x, 12, -((a*d*x)/e^2) - (b*x)/(2*c*e) + (b*ArcTan[c*x])/(2*c^2*e) - (b*d*x*ArcTan[c*x])/e^2 + (x^2*(a + b*ArcTan[c*x]))/(2*e) - (d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^3 + (d^2*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 + (b*d*Log[1 + c^2*x^2])/(2*c*e^2) + (I*b*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^3) - (I*b*d^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^3)} -{x^1*(a + b*ArcTan[c*x])/(d + e*x), x, 9, (a*x)/e + (b*x*ArcTan[c*x])/e + (d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 - (b*Log[1 + c^2*x^2])/(2*c*e) - (I*b*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^2) + (I*b*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^2)} -{x^0*(a + b*ArcTan[c*x])/(d + e*x), x, 4, -(((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e) - (I*b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x)), x, 9, (a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d + (I*b*PolyLog[2, (-I)*c*x])/(2*d) - (I*b*PolyLog[2, I*c*x])/(2*d) - (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d) + (I*b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d)} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x)), x, 14, -((a + b*ArcTan[c*x])/(d*x)) + (b*c*Log[x])/d - (a*e*Log[x])/d^2 - (e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 - (b*c*Log[1 + c^2*x^2])/(2*d) - (I*b*e*PolyLog[2, (-I)*c*x])/(2*d^2) + (I*b*e*PolyLog[2, I*c*x])/(2*d^2) + (I*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^2) - (I*b*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^2)} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x)), x, 17, -((b*c)/(2*d*x)) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) + (e*(a + b*ArcTan[c*x]))/(d^2*x) - (b*c*e*Log[x])/d^2 + (a*e^2*Log[x])/d^3 + (e^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 - (e^2*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 + (b*c*e*Log[1 + c^2*x^2])/(2*d^2) + (I*b*e^2*PolyLog[2, (-I)*c*x])/(2*d^3) - (I*b*e^2*PolyLog[2, I*c*x])/(2*d^3) - (I*b*e^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^3) + (I*b*e^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^3)} - - -{x^3*(a + b*ArcTan[c*x])^2/(d + e*x), x, 23, (a*b*d*x)/(c*e^2) + (b^2*x)/(3*c^2*e) - (b^2*ArcTan[c*x])/(3*c^3*e) + (b^2*d*x*ArcTan[c*x])/(c*e^2) - (b*x^2*(a + b*ArcTan[c*x]))/(3*c*e) + (I*d^2*(a + b*ArcTan[c*x])^2)/(c*e^3) - (d*(a + b*ArcTan[c*x])^2)/(2*c^2*e^2) - (I*(a + b*ArcTan[c*x])^2)/(3*c^3*e) + (d^2*x*(a + b*ArcTan[c*x])^2)/e^3 - (d*x^2*(a + b*ArcTan[c*x])^2)/(2*e^2) + (x^3*(a + b*ArcTan[c*x])^2)/(3*e) + (d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^4 + (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^3) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3*e) - (d^3*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b^2*d*Log[1 + c^2*x^2])/(2*c^2*e^2) - (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^4 + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e^3) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3*e) + (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 + (b^2*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^4) - (b^2*d^3*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^4)} -{x^2*(a + b*ArcTan[c*x])^2/(d + e*x), x, 14, -((a*b*x)/(c*e)) - (b^2*x*ArcTan[c*x])/(c*e) - (I*d*(a + b*ArcTan[c*x])^2)/(c*e^2) + (a + b*ArcTan[c*x])^2/(2*c^2*e) - (d*x*(a + b*ArcTan[c*x])^2)/e^2 + (x^2*(a + b*ArcTan[c*x])^2)/(2*e) - (d^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^3 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^2) + (d^2*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 + (b^2*Log[1 + c^2*x^2])/(2*c^2*e) + (I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^3 - (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e^2) - (I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 - (b^2*d^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^3) + (b^2*d^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^3)} -{x^1*(a + b*ArcTan[c*x])^2/(d + e*x), x, 8, (I*(a + b*ArcTan[c*x])^2)/(c*e) + (x*(a + b*ArcTan[c*x])^2)/e + (d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e) - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 - (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e) + (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 + (b^2*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^2)} -{x^0*(a + b*ArcTan[c*x])^2/(d + e*x), x, 1, -(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)} -{(a + b*ArcTan[c*x])^2/(x^1*(d + e*x)), x, 9, (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + e*x)), x, 13, -((I*c*(a + b*ArcTan[c*x])^2)/d) - (a + b*ArcTan[c*x])^2/(d*x) - (2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 - (e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^2)} -{(a + b*ArcTan[c*x])^2/(x^3*(d + e*x)), x, 21, -((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d) + (I*c*e*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(2*d*x^2) + (e*(a + b*ArcTan[c*x])^2)/(d^2*x) + (2*e^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + (b^2*c^2*Log[x])/d + (e^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^3 - (e^2*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) - (2*b*c*e*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + (I*b^2*c*e*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3 + (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 + (b^2*e^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^3) - (b^2*e^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^3) + (b^2*e^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^3) - (b^2*e^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^3)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/((d + e*x)*(a + b*ArcTan[c*x])), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcTan[c*x])), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTan[c x])^p with e=c^2 d*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^1 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(c + a^2*c*x^2)*ArcTan[a*x], x, 9, (c*x)/(12*a^3) - (c*x^3)/(36*a) - (a*c*x^5)/30 - (c*ArcTan[a*x])/(12*a^4) + (c*x^4*ArcTan[a*x])/4 + (a^2*c*x^6*ArcTan[a*x])/6} -{x^2*(c + a^2*c*x^2)*ArcTan[a*x], x, 9, -(c*x^2)/(15*a) - (a*c*x^4)/20 + (c*x^3*ArcTan[a*x])/3 + (a^2*c*x^5*ArcTan[a*x])/5 + (c*Log[1 + a^2*x^2])/(15*a^3)} -{x^1*(c + a^2*c*x^2)*ArcTan[a*x], x, 2, -(c*x)/(4*a) - (a*c*x^3)/12 + (c*(1 + a^2*x^2)^2*ArcTan[a*x])/(4*a^2)} -{x^0*(c + a^2*c*x^2)*ArcTan[a*x], x, 3, (-(1/6))*a*c*x^2 + c*x*ArcTan[a*x] + (1/3)*a^2*c*x^3*ArcTan[a*x] - (c*Log[1 + a^2*x^2])/(3*a), -((c*(1 + a^2*x^2))/(6*a)) + (2/3)*c*x*ArcTan[a*x] + (1/3)*c*x*(1 + a^2*x^2)*ArcTan[a*x] - (c*Log[1 + a^2*x^2])/(3*a)} -{((c + a^2*c*x^2)*ArcTan[a*x])/x^1, x, 7, -(a*c*x)/2 + (c*ArcTan[a*x])/2 + (a^2*c*x^2*ArcTan[a*x])/2 + (I/2)*c*PolyLog[2, (-I)*a*x] - (I/2)*c*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)*ArcTan[a*x])/x^2, x, 8, -((c*ArcTan[a*x])/x) + a^2*c*x*ArcTan[a*x] + a*c*Log[x] - a*c*Log[1 + a^2*x^2]} -{((c + a^2*c*x^2)*ArcTan[a*x])/x^3, x, 7, -(a*c)/(2*x) - (a^2*c*ArcTan[a*x])/2 - (c*ArcTan[a*x])/(2*x^2) + (I/2)*a^2*c*PolyLog[2, (-I)*a*x] - (I/2)*a^2*c*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)*ArcTan[a*x])/x^4, x, 10, -((a*c)/(6*x^2)) - (c*ArcTan[a*x])/(3*x^3) - (a^2*c*ArcTan[a*x])/x + (2/3)*a^3*c*Log[x] - (1/3)*a^3*c*Log[1 + a^2*x^2]} - - -{x^3*(c + a^2*c*x^2)^2*ArcTan[a*x], x, 14, (c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan[a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8} -{x^2*(c + a^2*c*x^2)^2*ArcTan[a*x], x, 14, (-4*c^2*x^2)/(105*a) - (9*a*c^2*x^4)/140 - (a^3*c^2*x^6)/42 + (c^2*x^3*ArcTan[a*x])/3 + (2*a^2*c^2*x^5*ArcTan[a*x])/5 + (a^4*c^2*x^7*ArcTan[a*x])/7 + (4*c^2*Log[1 + a^2*x^2])/(105*a^3)} -{x*(c + a^2*c*x^2)^2*ArcTan[a*x], x, 3, -(c^2*x)/(6*a) - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x])/(6*a^2)} -{(c + a^2*c*x^2)^2*ArcTan[a*x], x, 4, (-2*c^2*(1 + a^2*x^2))/(15*a) - (c^2*(1 + a^2*x^2)^2)/(20*a) + (8*c^2*x*ArcTan[a*x])/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/5 - (4*c^2*Log[1 + a^2*x^2])/(15*a)} -{((c + a^2*c*x^2)^2*ArcTan[a*x])/x, x, 12, (-3*a*c^2*x)/4 - (a^3*c^2*x^3)/12 + (3*c^2*ArcTan[a*x])/4 + a^2*c^2*x^2*ArcTan[a*x] + (a^4*c^2*x^4*ArcTan[a*x])/4 + (I/2)*c^2*PolyLog[2, (-I)*a*x] - (I/2)*c^2*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)^2*ArcTan[a*x])/x^2, x, 13, (-(1/6))*a^3*c^2*x^2 - (c^2*ArcTan[a*x])/x + 2*a^2*c^2*x*ArcTan[a*x] + (1/3)*a^4*c^2*x^3*ArcTan[a*x] + a*c^2*Log[x] - (4/3)*a*c^2*Log[1 + a^2*x^2]} -{((c + a^2*c*x^2)^2*ArcTan[a*x])/x^3, x, 11, -(a*c^2)/(2*x) - (a^3*c^2*x)/2 - (c^2*ArcTan[a*x])/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x])/2 + I*a^2*c^2*PolyLog[2, (-I)*a*x] - I*a^2*c^2*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)^2*ArcTan[a*x])/x^4, x, 13, -((a*c^2)/(6*x^2)) - (c^2*ArcTan[a*x])/(3*x^3) - (2*a^2*c^2*ArcTan[a*x])/x + a^4*c^2*x*ArcTan[a*x] + (5/3)*a^3*c^2*Log[x] - (4/3)*a^3*c^2*Log[1 + a^2*x^2]} - - -{x^3*(c + a^2*c*x^2)^3*ArcTan[a*x], x, 18, (c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcTan[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a^6*c^3*x^10*ArcTan[a*x])/10} -{x^2*(c + a^2*c*x^2)^3*ArcTan[a*x], x, 18, (-8*c^3*x^2)/(315*a) - (89*a*c^3*x^4)/1260 - (10*a^3*c^3*x^6)/189 - (a^5*c^3*x^8)/72 + (c^3*x^3*ArcTan[a*x])/3 + (3*a^2*c^3*x^5*ArcTan[a*x])/5 + (3*a^4*c^3*x^7*ArcTan[a*x])/7 + (a^6*c^3*x^9*ArcTan[a*x])/9 + (8*c^3*Log[1 + a^2*x^2])/(315*a^3)} -{x*(c + a^2*c*x^2)^3*ArcTan[a*x], x, 3, -(c^3*x)/(8*a) - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x])/(8*a^2)} -{(c + a^2*c*x^2)^3*ArcTan[a*x], x, 5, (-4*c^3*(1 + a^2*x^2))/(35*a) - (3*c^3*(1 + a^2*x^2)^2)/(70*a) - (c^3*(1 + a^2*x^2)^3)/(42*a) + (16*c^3*x*ArcTan[a*x])/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x])/7 - (8*c^3*Log[1 + a^2*x^2])/(35*a)} -{((c + a^2*c*x^2)^3*ArcTan[a*x])/x, x, 16, (-11*a*c^3*x)/12 - (7*a^3*c^3*x^3)/36 - (a^5*c^3*x^5)/30 + (11*c^3*ArcTan[a*x])/12 + (3*a^2*c^3*x^2*ArcTan[a*x])/2 + (3*a^4*c^3*x^4*ArcTan[a*x])/4 + (a^6*c^3*x^6*ArcTan[a*x])/6 + (I/2)*c^3*PolyLog[2, (-I)*a*x] - (I/2)*c^3*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)^3*ArcTan[a*x])/x^2, x, 17, (-(2/5))*a^3*c^3*x^2 - (1/20)*a^5*c^3*x^4 - (c^3*ArcTan[a*x])/x + 3*a^2*c^3*x*ArcTan[a*x] + a^4*c^3*x^3*ArcTan[a*x] + (1/5)*a^6*c^3*x^5*ArcTan[a*x] + a*c^3*Log[x] - (8/5)*a*c^3*Log[1 + a^2*x^2]} -{((c + a^2*c*x^2)^3*ArcTan[a*x])/x^3, x, 15, -(a*c^3)/(2*x) - (5*a^3*c^3*x)/4 - (a^5*c^3*x^3)/12 + (3*a^2*c^3*ArcTan[a*x])/4 - (c^3*ArcTan[a*x])/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x])/2 + (a^6*c^3*x^4*ArcTan[a*x])/4 + ((3*I)/2)*a^2*c^3*PolyLog[2, (-I)*a*x] - ((3*I)/2)*a^2*c^3*PolyLog[2, I*a*x]} -{((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4, x, 17, -((a*c^3)/(6*x^2)) - (1/6)*a^5*c^3*x^2 - (c^3*ArcTan[a*x])/(3*x^3) - (3*a^2*c^3*ArcTan[a*x])/x + 3*a^4*c^3*x*ArcTan[a*x] + (1/3)*a^6*c^3*x^3*ArcTan[a*x] + (8/3)*a^3*c^3*Log[x] - (8/3)*a^3*c^3*Log[1 + a^2*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*ArcTan[a*x]/(c + a^2*c*x^2), x, 9, -(x^2/(6*a^3*c)) - (x*ArcTan[a*x])/(a^4*c) + (x^3*ArcTan[a*x])/(3*a^2*c) + ArcTan[a*x]^2/(2*a^5*c) + (2*Log[1 + a^2*x^2])/(3*a^5*c)} -{x^3*ArcTan[a*x]/(c + a^2*c*x^2), x, 8, -x/(2*a^3*c) + ArcTan[a*x]/(2*a^4*c) + (x^2*ArcTan[a*x])/(2*a^2*c) + ((I/2)*ArcTan[a*x]^2)/(a^4*c) + (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c) + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c)} -{x^2*ArcTan[a*x]/(c + a^2*c*x^2), x, 4, (x*ArcTan[a*x])/(a^2*c) - ArcTan[a*x]^2/(2*a^3*c) - Log[1 + a^2*x^2]/(2*a^3*c)} -{x*ArcTan[a*x]/(c + a^2*c*x^2), x, 4, ((-I/2)*ArcTan[a*x]^2)/(a^2*c) - (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^2*c) - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c)} -{ArcTan[a*x]/(c + a^2*c*x^2), x, 1, ArcTan[a*x]^2/(2*a*c)} -{ArcTan[a*x]/(x*(c + a^2*c*x^2)), x, 3, ((-I/2)*ArcTan[a*x]^2)/c + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c} -{ArcTan[a*x]/(x^2*(c + a^2*c*x^2)), x, 7, -(ArcTan[a*x]/(c*x)) - (a*ArcTan[a*x]^2)/(2*c) + (a*Log[x])/c - (a*Log[1 + a^2*x^2])/(2*c)} -{ArcTan[a*x]/(x^3*(c + a^2*c*x^2)), x, 7, -a/(2*c*x) - (a^2*ArcTan[a*x])/(2*c) - ArcTan[a*x]/(2*c*x^2) + ((I/2)*a^2*ArcTan[a*x]^2)/c - (a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c + ((I/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c} -{ArcTan[a*x]/(x^4*(c + a^2*c*x^2)), x, 12, -(a/(6*c*x^2)) - ArcTan[a*x]/(3*c*x^3) + (a^2*ArcTan[a*x])/(c*x) + (a^3*ArcTan[a*x]^2)/(2*c) - (4*a^3*Log[x])/(3*c) + (2*a^3*Log[1 + a^2*x^2])/(3*c)} - - -{x^5*ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 17, -(x/(2*a^5*c^2)) + x/(4*a^5*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(4*a^6*c^2) + (x^2*ArcTan[a*x])/(2*a^4*c^2) - ArcTan[a*x]/(2*a^6*c^2*(1 + a^2*x^2)) + (I*ArcTan[a*x]^2)/(a^6*c^2) + (2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^6*c^2) + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^6*c^2)} -{x^4*ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 7, 1/(4*a^5*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(a^4*c^2) + (x*ArcTan[a*x])/(2*a^4*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(4*a^5*c^2) - Log[1 + a^2*x^2]/(2*a^5*c^2)} -{x^3*ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 8, -x/(4*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*a^4*c^2) + ArcTan[a*x]/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/(a^4*c^2) - (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c^2) - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2)} -{x^2*ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 2, -(1/(4*a^3*c^2*(1 + a^2*x^2))) - (x*ArcTan[a*x])/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a^3*c^2)} -{x*ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 3, x/(4*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^2*c^2) - ArcTan[a*x]/(2*a^2*c^2*(1 + a^2*x^2))} -{ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 2, 1/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a*c^2)} -{ArcTan[a*x]/(x*(c + a^2*c*x^2)^2), x, 7, -(a*x)/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*c^2) + ArcTan[a*x]/(2*c^2*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/c^2 + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^2), x, 10, -(a/(4*c^2*(1 + a^2*x^2))) - ArcTan[a*x]/(c^2*x) - (a^2*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (3*a*ArcTan[a*x]^2)/(4*c^2) + (a*Log[x])/c^2 - (a*Log[1 + a^2*x^2])/(2*c^2)} -{ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^2), x, 15, -a/(2*c^2*x) + (a^3*x)/(4*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(4*c^2) - ArcTan[a*x]/(2*c^2*x^2) - (a^2*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (I*a^2*ArcTan[a*x]^2)/c^2 - (2*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 + (I*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^2), x, 23, -(a/(6*c^2*x^2)) + a^3/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^2*x^3) + (2*a^2*ArcTan[a*x])/(c^2*x) + (a^4*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^2)/(4*c^2) - (7*a^3*Log[x])/(3*c^2) + (7*a^3*Log[1 + a^2*x^2])/(6*c^2)} - - -{(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 4, x^3/(16*a*c^3*(1 + a^2*x^2)^2) + (3*x)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x])/(32*a^4*c^3) + (x^4*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2)} -{(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, -1/(16*a^3*c^3*(1 + a^2*x^2)^2) + 1/(16*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x])/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^2/(16*a^3*c^3)} -{(x*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 4, x/(16*a*c^3*(1 + a^2*x^2)^2) + (3*x)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(32*a^2*c^3) - ArcTan[a*x]/(4*a^2*c^3*(1 + a^2*x^2)^2)} -{ArcTan[a*x]/(c + a^2*c*x^2)^3, x, 3, 1/(16*a*c^3*(1 + a^2*x^2)^2) + 3/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(16*a*c^3)} -{ArcTan[a*x]/(x*(c + a^2*c*x^2)^3), x, 12, -(a*x)/(16*c^3*(1 + a^2*x^2)^2) - (11*a*x)/(32*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x])/(32*c^3) + ArcTan[a*x]/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(2*c^3*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/c^3 + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3} -{ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^3), x, 14, -(a/(16*c^3*(1 + a^2*x^2)^2)) - (7*a)/(16*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(c^3*x) - (a^2*x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) - (15*a*ArcTan[a*x]^2)/(16*c^3) + (a*Log[x])/c^3 - (a*Log[1 + a^2*x^2])/(2*c^3)} -{ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^3), x, 28, -a/(2*c^3*x) + (a^3*x)/(16*c^3*(1 + a^2*x^2)^2) + (19*a^3*x)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x])/(32*c^3) - ArcTan[a*x]/(2*c^3*x^2) - (a^2*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x])/(c^3*(1 + a^2*x^2)) + (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^3 - (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 + (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3} -{ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3), x, 38, -(a/(6*c^3*x^2)) + a^3/(16*c^3*(1 + a^2*x^2)^2) + (11*a^3)/(16*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^3*x^3) + (3*a^2*ArcTan[a*x])/(c^3*x) + (a^4*x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^2)/(16*c^3) - (10*a^3*Log[x])/(3*c^3) + (5*a^3*Log[1 + a^2*x^2])/(3*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^1 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x, 12, (x*Sqrt[c + a^2*c*x^2])/(24*a^3) - (x^3*Sqrt[c + a^2*c*x^2])/(20*a) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(15*a^2) + (x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/5 + (11*Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4)} -{x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x, 8, Sqrt[c + a^2*c*x^2]/(8*a^3) - (c + a^2*c*x^2)^(3/2)/(12*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(8*a^2) + (1/4)*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(8*a^3*Sqrt[c + a^2*c*x^2])} -{x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x, 4, -(x*Sqrt[c + a^2*c*x^2])/(6*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*a^2*c) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^2)} -{Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x, 3, -Sqrt[c + a^2*c*x^2]/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x, x, 5, Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^2, x, 7, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (I*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^3, x, 6, -(a*Sqrt[c + a^2*c*x^2])/(2*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((I/2)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((I/2)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^4, x, 5, -(a*Sqrt[c + a^2*c*x^2])/(6*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*c*x^3) - (a^3*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6} - - -{x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x, 31, (3*c*x*Sqrt[c + a^2*c*x^2])/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2])/(840*a) - (a*c*x^5*Sqrt[c + a^2*c*x^2])/42 - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/35 + (a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/7 + (17*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(560*a^4)} -{x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x, 21, (c*Sqrt[c + a^2*c*x^2])/(16*a^3) + (c + a^2*c*x^2)^(3/2)/(72*a^3) - (c + a^2*c*x^2)^(5/2)/(30*a^3*c) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(16*a^2) + (7/24)*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (1/6)*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(16*a^3*Sqrt[c + a^2*c*x^2]) + (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(16*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x, 5, (-3*c*x*Sqrt[c + a^2*c*x^2])/(40*a) - (x*(c + a^2*c*x^2)^(3/2))/(20*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(5*a^2*c) - (3*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(40*a^2)} -{(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x, 4, (-3*c*Sqrt[c + a^2*c*x^2])/(8*a) - (c + a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x, x, 10, -(a*c*x*Sqrt[c + a^2*c*x^2])/6 + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (7*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^2, x, 11, -(a*c*Sqrt[c + a^2*c*x^2])/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (((3*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^3, x, 12, -(a*c*Sqrt[c + a^2*c*x^2])/(2*x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a^2*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((3*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^4, x, 13, -(a*c*Sqrt[c + a^2*c*x^2])/(6*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (7*a^3*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6 + (I*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (I*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} - - -{x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x, 76, (47*c^2*x*Sqrt[c + a^2*c*x^2])/(2688*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2])/(12096*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2])/3024 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2])/72 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/9 + (115*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(8064*a^4)} -{x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x, 51, (5*c^2*Sqrt[c + a^2*c*x^2])/(128*a^3) + (5*c*(c + a^2*c*x^2)^(3/2))/(576*a^3) + (c + a^2*c*x^2)^(5/2)/(240*a^3) - (c + a^2*c*x^2)^(7/2)/(56*a^3*c) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(128*a^2) + (59/192)*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (17/48)*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (1/8)*a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (5*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (5*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(128*a^3*Sqrt[c + a^2*c*x^2]) + (5*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(128*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x, 6, (-5*c^2*x*Sqrt[c + a^2*c*x^2])/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2))/(168*a) - (x*(c + a^2*c*x^2)^(5/2))/(42*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x])/(7*a^2*c) - (5*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(112*a^2)} -{(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x, 5, (-5*c^2*Sqrt[c + a^2*c*x^2])/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2))/(72*a) - (c + a^2*c*x^2)^(5/2)/(30*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x, x, 16, (-29*a*c^2*x*Sqrt[c + a^2*c*x^2])/120 - (a*c*x*(c + a^2*c*x^2)^(3/2))/20 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (149*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/120 + (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^2, x, 16, (-7*a*c^2*Sqrt[c + a^2*c*x^2])/8 - (a*c*(c + a^2*c*x^2)^(3/2))/12 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (((15*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((15*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^3, x, 23, -(a*c^2*Sqrt[c + a^2*c*x^2])/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2])/6 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^4, x, 25, -(a^3*c^2*Sqrt[c + a^2*c*x^2])/2 - (a*c^2*Sqrt[c + a^2*c*x^2])/(6*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6 + (((5*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^3*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 7, -(x*Sqrt[c + a^2*c*x^2])/(6*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) + (5*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^4*Sqrt[c])} -{(x^2*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, -(Sqrt[c + a^2*c*x^2]/(2*a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(2*a^3*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(2*a^3*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 3, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c])} -{ArcTan[a*x]/Sqrt[c + a^2*c*x^2], x, 2, ((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x*Sqrt[c + a^2*c*x^2]), x, 2, (-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]/(x^2*Sqrt[c + a^2*c*x^2]), x, 4, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]} -{ArcTan[a*x]/(x^3*Sqrt[c + a^2*c*x^2]), x, 4, -(a*Sqrt[c + a^2*c*x^2])/(2*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((I/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] + ((I/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]/(x^4*Sqrt[c + a^2*c*x^2]), x, 9, -(a*Sqrt[c + a^2*c*x^2])/(6*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x) + (5*a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*Sqrt[c])} - - -{(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 6, -(x/(a^3*c*Sqrt[c + a^2*c*x^2])) + ArcTan[a*x]/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^4*c^2) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^4*c^(3/2))} -{(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, -(1/(a^3*c*Sqrt[c + a^2*c*x^2])) - (x*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 2, x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(c + a^2*c*x^2)^(3/2), x, 1, 1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x*(c + a^2*c*x^2)^(3/2)), x, 5, -((a*x)/(c*Sqrt[c + a^2*c*x^2])) + ArcTan[a*x]/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(3/2)), x, 6, -(a/(c*Sqrt[c + a^2*c*x^2])) - (a^2*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^2*x) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(3/2)} -{ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^(3/2)), x, 10, (a^3*x)/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(2*c^2*x) - (a^2*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c^2*x^2) + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^(3/2)), x, 16, a^3/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(6*c^2*x^2) + (a^4*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x) + (11*a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*c^(3/2))} - - -{(x^5*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 10, -(x^3/(9*a^3*c*(c + a^2*c*x^2)^(3/2))) - (5*x)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x])/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x])/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^6*c^3) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^6*c^(5/2))} -{(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 8, 1/(9*a^5*c*(c + a^2*c*x^2)^(3/2)) - 4/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x])/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - (2*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^5*c^2*Sqrt[c + a^2*c*x^2])} -{(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 3, x^3/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x])/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 4, -(1/(9*a^3*c*(c + a^2*c*x^2)^(3/2))) + 1/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2))} -{(x*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 3, x/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x)/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(3*a^2*c*(c + a^2*c*x^2)^(3/2))} -{ArcTan[a*x]/(c + a^2*c*x^2)^(5/2), x, 2, 1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + 2/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)), x, 9, -(a*x)/(9*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x)/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(5/2)), x, 9, -a/(9*c*(c + a^2*c*x^2)^(3/2)) - (5*a)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^3*x) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(5/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^1 with d=a^2 c and m symbolic*) - - -{x^m*(c + a^2*c*x^2)^3*ArcTan[a*x], x, 10, (c^3*x^(1 + m)*ArcTan[a*x])/(1 + m) + (3*a^2*c^3*x^(3 + m)*ArcTan[a*x])/(3 + m) + (3*a^4*c^3*x^(5 + m)*ArcTan[a*x])/(5 + m) + (a^6*c^3*x^(7 + m)*ArcTan[a*x])/(7 + m) - (a*c^3*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + 3*m + m^2) - (3*a^3*c^3*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, (-a^2)*x^2])/(12 + 7*m + m^2) - (3*a^5*c^3*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, (-a^2)*x^2])/((5 + m)*(6 + m)) - (a^7*c^3*x^(8 + m)*Hypergeometric2F1[1, (8 + m)/2, (10 + m)/2, (-a^2)*x^2])/((7 + m)*(8 + m))} -{x^m*(c + a^2*c*x^2)^2*ArcTan[a*x], x, 8, (c^2*x^(1 + m)*ArcTan[a*x])/(1 + m) + (2*a^2*c^2*x^(3 + m)*ArcTan[a*x])/(3 + m) + (a^4*c^2*x^(5 + m)*ArcTan[a*x])/(5 + m) - (a*c^2*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + 3*m + m^2) - (2*a^3*c^2*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, (-a^2)*x^2])/(12 + 7*m + m^2) - (a^5*c^2*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, (-a^2)*x^2])/((5 + m)*(6 + m))} -{x^m*(c + a^2*c*x^2)*ArcTan[a*x], x, 5, (c*x^(1 + m)*ArcTan[a*x])/(1 + m) + (a^2*c*x^(3 + m)*ArcTan[a*x])/(3 + m) - (a*c*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + 3*m + m^2) - (a^3*c*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, (-a^2)*x^2])/(12 + 7*m + m^2)} -{(x^m*ArcTan[a*x])/(c + a^2*c*x^2), x, 0, Unintegrable[(x^m*ArcTan[a*x])/(c + a^2*c*x^2), x]} -{(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^2, x]} - - -{x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x]} -{x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x]} -{x^m*(c + a^2*c*x^2)^(1/2)*ArcTan[a*x], x, 3, (x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*x^(2 + m)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[1, (3 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m)^2 + (c*Unintegrable[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x])/(2 + m), (x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*c*x^(2 + m)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/((2 + m)^2*Sqrt[c + a^2*c*x^2]) + (c*Unintegrable[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x])/(2 + m)} -{(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 0, Unintegrable[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x]} -{(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^2 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2, x, 26, -(c*x^2)/(180*a^2) + (c*x^4)/60 + (c*x*ArcTan[a*x])/(6*a^3) - (c*x^3*ArcTan[a*x])/(18*a) - (a*c*x^5*ArcTan[a*x])/15 - (c*ArcTan[a*x]^2)/(12*a^4) + (c*x^4*ArcTan[a*x]^2)/4 + (a^2*c*x^6*ArcTan[a*x]^2)/6 - (7*c*Log[1 + a^2*x^2])/(90*a^4)} -{x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2, x, 24, (c*x)/(30*a^2) + (c*x^3)/30 - (c*ArcTan[a*x])/(30*a^3) - (2*c*x^2*ArcTan[a*x])/(15*a) - (a*c*x^4*ArcTan[a*x])/10 - (((2*I)/15)*c*ArcTan[a*x]^2)/a^3 + (c*x^3*ArcTan[a*x]^2)/3 + (a^2*c*x^5*ArcTan[a*x]^2)/5 - (4*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^3) - (((2*I)/15)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3} -{x*(c + a^2*c*x^2)*ArcTan[a*x]^2, x, 4, (c*(1 + a^2*x^2))/(12*a^2) - (c*x*ArcTan[a*x])/(3*a) - (c*x*(1 + a^2*x^2)*ArcTan[a*x])/(6*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(4*a^2) + (c*Log[1 + a^2*x^2])/(6*a^2)} -{(c + a^2*c*x^2)*ArcTan[a*x]^2, x, 7, (c*x)/3 - (c*(1 + a^2*x^2)*ArcTan[a*x])/(3*a) + (((2*I)/3)*c*ArcTan[a*x]^2)/a + (2*c*x*ArcTan[a*x]^2)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/3 + (4*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(3*a) + (((2*I)/3)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a} -{((c + a^2*c*x^2)*ArcTan[a*x]^2)/x, x, 12, -(a*c*x*ArcTan[a*x]) + (c*ArcTan[a*x]^2)/2 + (a^2*c*x^2*ArcTan[a*x]^2)/2 + 2*c*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (c*Log[1 + a^2*x^2])/2 - I*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c*PolyLog[3, -1 + 2/(1 + I*a*x)])/2} -{((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^2, x, 10, -((c*ArcTan[a*x]^2)/x) + a^2*c*x*ArcTan[a*x]^2 + 2*a*c*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 2*a*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c*PolyLog[2, -1 + 2/(1 - I*a*x)] + I*a*c*PolyLog[2, 1 - 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^3, x, 15, -((a*c*ArcTan[a*x])/x) - (1/2)*a^2*c*ArcTan[a*x]^2 - (c*ArcTan[a*x]^2)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c*Log[x] - (1/2)*a^2*c*Log[1 + a^2*x^2] - I*a^2*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*a^2*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (1/2)*a^2*c*PolyLog[3, 1 - 2/(1 + I*a*x)] + (1/2)*a^2*c*PolyLog[3, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^4, x, 13, -(a^2*c)/(3*x) - (a^3*c*ArcTan[a*x])/3 - (a*c*ArcTan[a*x])/(3*x^2) - ((2*I)/3)*a^3*c*ArcTan[a*x]^2 - (c*ArcTan[a*x]^2)/(3*x^3) - (a^2*c*ArcTan[a*x]^2)/x + (4*a^3*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((2*I)/3)*a^3*c*PolyLog[2, -1 + 2/(1 - I*a*x)]} - - -{x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x, 47, (-5*c^2*x^2)/(504*a^2) + (c^2*x^4)/84 + (a^2*c^2*x^6)/168 + (c^2*x*ArcTan[a*x])/(12*a^3) - (c^2*x^3*ArcTan[a*x])/(36*a) - (a*c^2*x^5*ArcTan[a*x])/12 - (a^3*c^2*x^7*ArcTan[a*x])/28 - (c^2*ArcTan[a*x]^2)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^2)/4 + (a^2*c^2*x^6*ArcTan[a*x]^2)/3 + (a^4*c^2*x^8*ArcTan[a*x]^2)/8 - (2*c^2*Log[1 + a^2*x^2])/(63*a^4)} -{x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x, 44, -(c^2*x)/(210*a^2) + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a*x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x])/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3} -{x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x, 5, (2*c^2*(1 + a^2*x^2))/(45*a^2) + (c^2*(1 + a^2*x^2)^2)/(60*a^2) - (8*c^2*x*ArcTan[a*x])/(45*a) - (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/(45*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(15*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(6*a^2) + (4*c^2*Log[1 + a^2*x^2])/(45*a^2)} -{(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x, 9, (11*c^2*x)/30 + (a^2*c^2*x^3)/30 - (4*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x])/(10*a) + (((8*I)/15)*c^2*ArcTan[a*x]^2)/a + (8*c^2*x*ArcTan[a*x]^2)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a) + (((8*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x, x, 23, (a^2*c^2*x^2)/12 - (3*a*c^2*x*ArcTan[a*x])/2 - (a^3*c^2*x^3*ArcTan[a*x])/6 + (3*c^2*ArcTan[a*x]^2)/4 + a^2*c^2*x^2*ArcTan[a*x]^2 + (a^4*c^2*x^4*ArcTan[a*x]^2)/4 + 2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (2*c^2*Log[1 + a^2*x^2])/3 - I*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^2*PolyLog[3, -1 + 2/(1 + I*a*x)])/2} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^2, x, 20, (a^2*c^2*x)/3 - (a*c^2*ArcTan[a*x])/3 - (a^3*c^2*x^2*ArcTan[a*x])/3 + ((2*I)/3)*a*c^2*ArcTan[a*x]^2 - (c^2*ArcTan[a*x]^2)/x + 2*a^2*c^2*x*ArcTan[a*x]^2 + (a^4*c^2*x^3*ArcTan[a*x]^2)/3 + (10*a*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + 2*a*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((5*I)/3)*a*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^3, x, 21, -((a*c^2*ArcTan[a*x])/x) - a^3*c^2*x*ArcTan[a*x] - (c^2*ArcTan[a*x]^2)/(2*x^2) + (1/2)*a^4*c^2*x^2*ArcTan[a*x]^2 + 4*a^2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^2*Log[x] - 2*I*a^2*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 2*I*a^2*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - a^2*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)] + a^2*c^2*PolyLog[3, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^4, x, 19, -(a^2*c^2)/(3*x) - (a^3*c^2*ArcTan[a*x])/3 - (a*c^2*ArcTan[a*x])/(3*x^2) - ((2*I)/3)*a^3*c^2*ArcTan[a*x]^2 - (c^2*ArcTan[a*x]^2)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^2)/x + a^4*c^2*x*ArcTan[a*x]^2 + 2*a^3*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] + (10*a^3*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((5*I)/3)*a^3*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + I*a^3*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)]} - - -{x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2, x, 72, (-107*c^3*x^2)/(12600*a^2) + (53*c^3*x^4)/6300 + (71*a^2*c^3*x^6)/7560 + (a^4*c^3*x^8)/360 + (c^3*x*ArcTan[a*x])/(20*a^3) - (c^3*x^3*ArcTan[a*x])/(60*a) - (9*a*c^3*x^5*ArcTan[a*x])/100 - (11*a^3*c^3*x^7*ArcTan[a*x])/140 - (a^5*c^3*x^9*ArcTan[a*x])/45 - (c^3*ArcTan[a*x]^2)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^2)/4 + (a^2*c^3*x^6*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^2)/8 + (a^6*c^3*x^10*ArcTan[a*x]^2)/10 - (26*c^3*Log[1 + a^2*x^2])/(1575*a^4)} -{x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2, x, 68, (-47*c^3*x)/(3780*a^2) + (239*c^3*x^3)/11340 + (59*a^2*c^3*x^5)/3780 + (a^4*c^3*x^7)/252 + (47*c^3*ArcTan[a*x])/(3780*a^3) - (16*c^3*x^2*ArcTan[a*x])/(315*a) - (89*a*c^3*x^4*ArcTan[a*x])/630 - (20*a^3*c^3*x^6*ArcTan[a*x])/189 - (a^5*c^3*x^8*ArcTan[a*x])/36 - (((16*I)/315)*c^3*ArcTan[a*x]^2)/a^3 + (c^3*x^3*ArcTan[a*x]^2)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^2)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^2)/7 + (a^6*c^3*x^9*ArcTan[a*x]^2)/9 - (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(315*a^3) - (((16*I)/315)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3} -{x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2, x, 6, (c^3*(1 + a^2*x^2))/(35*a^2) + (3*c^3*(1 + a^2*x^2)^2)/(280*a^2) + (c^3*(1 + a^2*x^2)^3)/(168*a^2) - (4*c^3*x*ArcTan[a*x])/(35*a) - (2*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(70*a) - (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x])/(28*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^2)/(8*a^2) + (2*c^3*Log[1 + a^2*x^2])/(35*a^2)} -{(c + a^2*c*x^2)^3*ArcTan[a*x]^2, x, 12, (38*c^3*x)/105 + (19*a^2*c^3*x^3)/315 + (a^4*c^3*x^5)/105 - (8*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(35*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (((16*I)/35)*c^3*ArcTan[a*x]^2)/a + (16*c^3*x*ArcTan[a*x]^2)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a) + (((16*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x, x, 38, (29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7*a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTan[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a*x]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (34*c^3*Log[1 + a^2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^2, x, 34, (7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*x^3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^3, x, 31, (1/12)*a^4*c^3*x^2 - (a*c^3*ArcTan[a*x])/x - (5/2)*a^3*c^3*x*ArcTan[a*x] - (1/6)*a^5*c^3*x^3*ArcTan[a*x] + (3/4)*a^2*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/(2*x^2) + (3/2)*a^4*c^3*x^2*ArcTan[a*x]^2 + (1/4)*a^6*c^3*x^4*ArcTan[a*x]^2 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2/3)*a^2*c^3*Log[1 + a^2*x^2] - 3*I*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 3*I*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3/2)*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)] + (3/2)*a^2*c^3*PolyLog[3, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^4, x, 28, -(a^2*c^3)/(3*x) + (a^4*c^3*x)/3 - (2*a^3*c^3*ArcTan[a*x])/3 - (a*c^3*ArcTan[a*x])/(3*x^2) - (a^5*c^3*x^2*ArcTan[a*x])/3 - (c^3*ArcTan[a*x]^2)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^2)/x + 3*a^4*c^3*x*ArcTan[a*x]^2 + (a^6*c^3*x^3*ArcTan[a*x]^2)/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((8*I)/3)*a^3*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((8*I)/3)*a^3*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*ArcTan[a*x]^2/(c + a^2*c*x^2), x, 17, x/(3*a^4*c) - ArcTan[a*x]/(3*a^5*c) - (x^2*ArcTan[a*x])/(3*a^3*c) - (4*I*ArcTan[a*x]^2)/(3*a^5*c) - (x*ArcTan[a*x]^2)/(a^4*c) + (x^3*ArcTan[a*x]^2)/(3*a^2*c) + ArcTan[a*x]^3/(3*a^5*c) - (8*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(3*a^5*c) - (4*I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(3*a^5*c)} -{x^3*ArcTan[a*x]^2/(c + a^2*c*x^2), x, 10, -((x*ArcTan[a*x])/(a^3*c)) + ArcTan[a*x]^2/(2*a^4*c) + (x^2*ArcTan[a*x]^2)/(2*a^2*c) + ((I/3)*ArcTan[a*x]^3)/(a^4*c) + (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^4*c) + Log[1 + a^2*x^2]/(2*a^4*c) + (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^4*c)} -{x^2*ArcTan[a*x]^2/(c + a^2*c*x^2), x, 7, (I*ArcTan[a*x]^2)/(a^3*c) + (x*ArcTan[a*x]^2)/(a^2*c) - ArcTan[a*x]^3/(3*a^3*c) + (2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^3*c) + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^3*c)} -{x*ArcTan[a*x]^2/(c + a^2*c*x^2), x, 4, ((-I/3)*ArcTan[a*x]^3)/(a^2*c) - (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^2*c) - (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c) - PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^2*c)} -{ArcTan[a*x]^2/(c + a^2*c*x^2), x, 1, ArcTan[a*x]^3/(3*a*c)} -{ArcTan[a*x]^2/(x*(c + a^2*c*x^2)), x, 4, ((-I/3)*ArcTan[a*x]^3)/c + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)} -{ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)), x, 6, ((-I)*a*ArcTan[a*x]^2)/c - ArcTan[a*x]^2/(c*x) - (a*ArcTan[a*x]^3)/(3*c) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c} -{ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)), x, 13, -((a*ArcTan[a*x])/(c*x)) - (a^2*ArcTan[a*x]^2)/(2*c) - ArcTan[a*x]^2/(2*c*x^2) + (I*a^2*ArcTan[a*x]^3)/(3*c) + (a^2*Log[x])/c - (a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (a^2*Log[1 + a^2*x^2])/(2*c) + (I*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c)} -{ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)), x, 15, -a^2/(3*c*x) - (a^3*ArcTan[a*x])/(3*c) - (a*ArcTan[a*x])/(3*c*x^2) + (((4*I)/3)*a^3*ArcTan[a*x]^2)/c - ArcTan[a*x]^2/(3*c*x^3) + (a^2*ArcTan[a*x]^2)/(c*x) + (a^3*ArcTan[a*x]^3)/(3*c) - (8*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c) + (((4*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c} - - -{(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x, 8, -1/(4*a^4*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x])/(2*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*a^4*c^2) + ArcTan[a*x]^2/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/(a^4*c^2) - (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^4*c^2) - (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2) - PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^4*c^2)} -{(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x, 4, x/(4*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^3*c^2) - ArcTan[a*x]/(2*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a^3*c^2)} -{(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x, 3, 1/(4*a^2*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a^2*c^2) - ArcTan[a*x]^2/(2*a^2*c^2*(1 + a^2*x^2))} -{ArcTan[a*x]^2/(c + a^2*c*x^2)^2, x, 4, -x/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*a*c^2) + ArcTan[a*x]/(2*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a*c^2)} -{ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^2), x, 8, -1/(4*c^2*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*c^2) + ArcTan[a*x]^2/(2*c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^2 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^2)} -{ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^2), x, 11, (a^2*x)/(4*c^2*(1 + a^2*x^2)) + (a*ArcTan[a*x])/(4*c^2) - (a*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^2)/c^2 - ArcTan[a*x]^2/(c^2*x) - (a^2*x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) - (a*ArcTan[a*x]^3)/(2*c^2) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^2), x, 22, a^2/(4*c^2*(1 + a^2*x^2)) - (a*ArcTan[a*x])/(c^2*x) + (a^3*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x]^2)/(4*c^2) - ArcTan[a*x]^2/(2*c^2*x^2) - (a^2*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + (2*I*a^2*ArcTan[a*x]^3)/(3*c^2) + (a^2*Log[x])/c^2 - (2*a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (a^2*Log[1 + a^2*x^2])/(2*c^2) + (2*I*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^2), x, 27, -a^2/(3*c^2*x) - (a^4*x)/(4*c^2*(1 + a^2*x^2)) - (7*a^3*ArcTan[a*x])/(12*c^2) - (a*ArcTan[a*x])/(3*c^2*x^2) + (a^3*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (((7*I)/3)*a^3*ArcTan[a*x]^2)/c^2 - ArcTan[a*x]^2/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^2)/(c^2*x) + (a^4*x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^3)/(6*c^2) - (14*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^2) + (((7*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2} - - -{(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3, x, 4, -(x^4/(32*c^3*(1 + a^2*x^2)^2)) + 3/(32*a^4*c^3*(1 + a^2*x^2)) + (x^3*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(16*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(32*a^4*c^3) + (x^4*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2)} -{(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3, x, 13, x/(32*a^2*c^3*(1 + a^2*x^2)^2) - x/(64*a^2*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(64*a^3*c^3) - ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^2)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(24*a^3*c^3)} -{(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3, x, 4, 1/(32*a^2*c^3*(1 + a^2*x^2)^2) + 3/(32*a^2*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(16*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(32*a^2*c^3) - ArcTan[a*x]^2/(4*a^2*c^3*(1 + a^2*x^2)^2)} -{ArcTan[a*x]^2/(c + a^2*c*x^2)^3, x, 8, -x/(32*c^3*(1 + a^2*x^2)^2) - (15*x)/(64*c^3*(1 + a^2*x^2)) - (15*ArcTan[a*x])/(64*a*c^3) + ArcTan[a*x]/(8*a*c^3*(1 + a^2*x^2)^2) + (3*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(8*a*c^3)} -{ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^3), x, 13, -1/(32*c^3*(1 + a^2*x^2)^2) - 11/(32*c^3*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) - (11*a*x*ArcTan[a*x])/(16*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x]^2)/(32*c^3) + ArcTan[a*x]^2/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]^2/(2*c^3*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^3 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^3)} -{ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^3), x, 20, (a^2*x)/(32*c^3*(1 + a^2*x^2)^2) + (31*a^2*x)/(64*c^3*(1 + a^2*x^2)) + (31*a*ArcTan[a*x])/(64*c^3) - (a*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) - (7*a*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(c^3*x) - (a^2*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) - (5*a*ArcTan[a*x]^3)/(8*c^3) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3} -{ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^3), x, 36, a^2/(32*c^3*(1 + a^2*x^2)^2) + (19*a^2)/(32*c^3*(1 + a^2*x^2)) - (a*ArcTan[a*x])/(c^3*x) + (a^3*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (19*a^3*x*ArcTan[a*x])/(16*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^2)/(32*c^3) - ArcTan[a*x]^2/(2*c^3*x^2) - (a^2*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^2)/(c^3*(1 + a^2*x^2)) + (I*a^2*ArcTan[a*x]^3)/c^3 + (a^2*Log[x])/c^3 - (3*a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - (a^2*Log[1 + a^2*x^2])/(2*c^3) + (3*I*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (3*a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)} -{ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^3), x, 48, -a^2/(3*c^3*x) - (a^4*x)/(32*c^3*(1 + a^2*x^2)^2) - (47*a^4*x)/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x])/(192*c^3) - (a*ArcTan[a*x])/(3*c^3*x^2) + (a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (11*a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^2)/(c^3*x) + (a^4*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^3)/(24*c^3) - (20*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^3) + (((10*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^2 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x, 26, -((11*Sqrt[c + a^2*c*x^2])/(60*a^4)) + (c + a^2*c*x^2)^(3/2)/(30*a^4*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10*a) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(15*a^2) + (1/5)*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (11*I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(30*a^4*Sqrt[c + a^2*c*x^2]) + (11*I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(60*a^4*Sqrt[c + a^2*c*x^2]) - (11*I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(60*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x, 35, (x*Sqrt[c + a^2*c*x^2])/(12*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^2) + (1/4)*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(4*a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^3) - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2])} -{x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x, 4, Sqrt[c + a^2*c*x^2]/(3*a^2) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*a^2*c) + (((2*I)/3)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - ((I/3)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + ((I/3)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])} -{Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x, 12, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x, x, 13, Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + ((4*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^2, x, 13, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^3, x, 24, -((a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (I*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^4, x, 7, -(a^2*Sqrt[c + a^2*c*x^2])/(3*x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*c*x^3) - (2*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((I/3)*a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((I/3)*a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} - - -{x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x, 75, -((17*c*Sqrt[c + a^2*c*x^2])/(280*a^4)) - (17*(c + a^2*c*x^2)^(3/2))/(1260*a^4) + (c + a^2*c*x^2)^(5/2)/(105*a^4*c) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(420*a) - (1/21)*a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^2) + (8/35)*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (1/7)*a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (17*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(140*a^4*Sqrt[c + a^2*c*x^2]) + (17*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) - (17*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(280*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x, 92, (c*x*Sqrt[c + a^2*c*x^2])/(36*a^2) + (1/60)*c*x^3*Sqrt[c + a^2*c*x^2] + (31*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(360*a^3) - (19*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(180*a) - (1/15)*a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(16*a^2) + (7/24)*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (1/6)*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(8*a^3*Sqrt[c + a^2*c*x^2]) - (41*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a^3) - (I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x, 5, (3*c*Sqrt[c + a^2*c*x^2])/(20*a^2) + (c + a^2*c*x^2)^(3/2)/(30*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(5*a^2*c) + (((3*I)/10)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - (((3*I)/20)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + (((3*I)/20)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])} -{(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x, 16, (c*x*Sqrt[c + a^2*c*x^2])/12 - (3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(6*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (5*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a) + (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) + (3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x, x, 18, (c*Sqrt[c + a^2*c*x^2])/3 - (a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3 + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + (((14*I)/3)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^2, x, 26, -(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + a*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^3, x, 38, -((a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) + ((4*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^4, x, 21, -(a^2*c*Sqrt[c + a^2*c*x^2])/(3*x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (14*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -{x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2, x, 203, -((115*c^2*Sqrt[c + a^2*c*x^2])/(4032*a^4)) - (115*c*(c + a^2*c*x^2)^(3/2))/(18144*a^4) - (23*(c + a^2*c*x^2)^(5/2))/(7560*a^4) + (c + a^2*c*x^2)^(7/2)/(252*a^4*c) + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6048*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/1512 - (1/36)*a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^2) + (5/21)*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (19/63)*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (1/9)*a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (115*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(2016*a^4*Sqrt[c + a^2*c*x^2]) + (115*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(4032*a^4*Sqrt[c + a^2*c*x^2]) - (115*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(4032*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2, x, 238, (43*c^2*x*Sqrt[c + a^2*c*x^2])/(4032*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2])/1680 + (1/168)*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2] + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20160*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10080*a) - (83/840)*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (1/28)*a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(128*a^2) + (59/192)*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (17/48)*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (1/8)*a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (5*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(64*a^3*Sqrt[c + a^2*c*x^2]) - (397*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(5040*a^3) - (5*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (5*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2, x, 6, (5*c^2*Sqrt[c + a^2*c*x^2])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2))/(252*a^2) + (c + a^2*c*x^2)^(5/2)/(105*a^2) - (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(21*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^2)/(7*a^2*c) + (((5*I)/28)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - (((5*I)/56)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + (((5*I)/56)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])} -{(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2, x, 21, (17*c^2*x*Sqrt[c + a^2*c*x^2])/180 + (c*x*(c + a^2*c*x^2)^(3/2))/60 - (5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(8*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(36*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(15*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (259*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a) + (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x, x, 24, (29*c^2*Sqrt[c + a^2*c*x^2])/60 + (c*(c + a^2*c*x^2)^(3/2))/30 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/60 - (a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/10 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/5 + (((149*I)/30)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^2, x, 43, (a^2*c^2*x*Sqrt[c + a^2*c*x^2])/12 - (7*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/6 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (11*a*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (15*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (15*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^3, x, 57, (a^2*c^2*Sqrt[c + a^2*c*x^2])/3 - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x - (a^3*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + (((26*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((5*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((5*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((13*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((13*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^4, x, 48, -(a^2*c^2*Sqrt[c + a^2*c*x^2])/(3*x) - a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (26*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + a^3*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((13*I)/3)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((13*I)/3)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^3*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x, 8, Sqrt[c + a^2*c*x^2]/(3*a^4*c) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - (10*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*a^4*Sqrt[c + a^2*c*x^2]) + (5*I*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(3*a^4*Sqrt[c + a^2*c*x^2]) - (5*I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(3*a^4*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x, 13, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) + ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^3*Sqrt[c]) - (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x, 3, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^2*c) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/Sqrt[c + a^2*c*x^2], x, 9, ((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x*Sqrt[c + a^2*c*x^2]), x, 9, (-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^2/(x^2*Sqrt[c + a^2*c*x^2]), x, 3, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^2/(x^3*Sqrt[c + a^2*c*x^2]), x, 14, -((a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c] - (I*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (I*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^2/(x^4*Sqrt[c + a^2*c*x^2]), x, 8, -(a^2*Sqrt[c + a^2*c*x^2])/(3*c*x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c*x) + (10*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) - (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] + (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]} - - -{(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x, 6, -2/(a^4*c*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(a^3*c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^4*c^2) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x, 12, (2*x)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (x*ArcTan[a*x]^2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x, 2, 2/(a^2*c*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(c + a^2*c*x^2)^(3/2), x, 2, (-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(3/2)), x, 12, -2/(c*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(3/2)), x, 6, (2*a^2*x)/(c*Sqrt[c + a^2*c*x^2]) - (2*a*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c^2*x) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^(3/2)), x, 27, (2*a^2)/(c*Sqrt[c + a^2*c*x^2]) + (2*a^3*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^2*x) - (a^2*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (a^2*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(3/2) - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (3*a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^(3/2)), x, 15, (-2*a^4*x)/(c*Sqrt[c + a^2*c*x^2]) - (a^2*Sqrt[c + a^2*c*x^2])/(3*c^2*x) + (2*a^3*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^2) + (a^4*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c^2*x) + (22*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*c*Sqrt[c + a^2*c*x^2]) - (((11*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (((11*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])} - - -{(x^5*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2), x, 13, 2/(27*a^6*c*(c + a^2*c*x^2)^(3/2)) - 32/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (10*x*ArcTan[a*x])/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^2)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^6*c^3) + (4*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*I*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (2*I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2])} -{(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2), x, 17, (2*x^3)/(27*a^2*c*(c + a^2*c*x^2)^(3/2)) + (22*x)/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^2*ArcTan[a*x])/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (22*ArcTan[a*x])/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x]^2)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - (2*I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])} -{(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2), x, 6, -(2/(27*a^4*c*(c + a^2*c*x^2)^(3/2))) + 14/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*x*ArcTan[a*x])/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^2)/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2), x, 4, -((2*x^3)/(27*c*(c + a^2*c*x^2)^(3/2))) - (4*x)/(9*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^2*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2))} -{(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2), x, 3, 2/(27*a^2*c*(c + a^2*c*x^2)^(3/2)) + 4/(9*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*x*ArcTan[a*x])/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(3*a^2*c*(c + a^2*c*x^2)^(3/2))} -{ArcTan[a*x]^2/(c + a^2*c*x^2)^(5/2), x, 5, (-2*x)/(27*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(5/2)), x, 16, -2/(27*c*(c + a^2*c*x^2)^(3/2)) - 22/(9*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (22*a*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]^2/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(5/2)), x, 12, (2*a^2*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a^2*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (10*a*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c^3*x) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^2 with d=a^2 c and m symbolic*) - - -{x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x]} -{x^m*(c + a^2*c*x^2)^1*ArcTan[a*x]^2, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^2, x]} -{(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^1, x, 0, Unintegrable[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2), x]} -{(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x]} - - -{x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x]} -{x^m*(c + a^2*c*x^2)^(1/2)*ArcTan[a*x]^2, x, 0, Unintegrable[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x]} -{(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(1/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x]} -{(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^3 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3, x, 52, (c*x)/(15*a^3) - (c*x^3)/(60*a) - (c*ArcTan[a*x])/(15*a^4) - (c*x^2*ArcTan[a*x])/(60*a^2) + (c*x^4*ArcTan[a*x])/20 + (((7*I)/30)*c*ArcTan[a*x]^2)/a^4 + (c*x*ArcTan[a*x]^2)/(4*a^3) - (c*x^3*ArcTan[a*x]^2)/(12*a) - (a*c*x^5*ArcTan[a*x]^2)/10 - (c*ArcTan[a*x]^3)/(12*a^4) + (c*x^4*ArcTan[a*x]^3)/4 + (a^2*c*x^6*ArcTan[a*x]^3)/6 + (7*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^4) + (((7*I)/30)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4} -{x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3, x, 34, -(c*x^2)/(20*a) + (c*x*ArcTan[a*x])/(10*a^2) + (c*x^3*ArcTan[a*x])/10 - (c*ArcTan[a*x]^2)/(20*a^3) - (c*x^2*ArcTan[a*x]^2)/(5*a) - (3*a*c*x^4*ArcTan[a*x]^2)/20 - (((2*I)/15)*c*ArcTan[a*x]^3)/a^3 + (c*x^3*ArcTan[a*x]^3)/3 + (a^2*c*x^5*ArcTan[a*x]^3)/5 - (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^3) - (((2*I)/5)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a^3)} -{x*(c + a^2*c*x^2)*ArcTan[a*x]^3, x, 8, -(c*x)/(4*a) + (c*(1 + a^2*x^2)*ArcTan[a*x])/(4*a^2) - ((I/2)*c*ArcTan[a*x]^2)/a^2 - (c*x*ArcTan[a*x]^2)/(2*a) - (c*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(4*a^2) - (c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/a^2 - ((I/2)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2} -{(c + a^2*c*x^2)*ArcTan[a*x]^3, x, 8, c*x*ArcTan[a*x] - (c*(1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a) + (((2*I)/3)*c*ArcTan[a*x]^3)/a + (2*c*x*ArcTan[a*x]^3)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/3 + (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - (c*Log[1 + a^2*x^2])/(2*a) + ((2*I)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/a} -{((c + a^2*c*x^2)*ArcTan[a*x]^3)/x, x, 17, ((-3*I)/2)*c*ArcTan[a*x]^2 - (3*a*c*x*ArcTan[a*x]^2)/2 + (c*ArcTan[a*x]^3)/2 + (a^2*c*x^2*ArcTan[a*x]^3)/2 + 2*c*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 3*c*ArcTan[a*x]*Log[2/(1 + I*a*x)] - ((3*I)/2)*c*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2, x, 11, -((c*ArcTan[a*x]^3)/x) + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2} -{((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^3, x, 16, ((-3*I)/2)*a^2*c*ArcTan[a*x]^2 - (3*a*c*ArcTan[a*x]^2)/(2*x) - (a^2*c*ArcTan[a*x]^3)/2 - (c*ArcTan[a*x]^3)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] + 3*a^2*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*a^2*c*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*a^2*c*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*a^2*c*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^4, x, 20, -((a^2*c*ArcTan[a*x])/x) - (1/2)*a^3*c*ArcTan[a*x]^2 - (a*c*ArcTan[a*x]^2)/(2*x^2) - (2/3)*I*a^3*c*ArcTan[a*x]^3 - (c*ArcTan[a*x]^3)/(3*x^3) - (a^2*c*ArcTan[a*x]^3)/x + a^3*c*Log[x] + 2*a^3*c*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (1/2)*a^3*c*Log[1 + a^2*x^2] - 2*I*a^3*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + a^3*c*PolyLog[3, -1 + 2/(1 - I*a*x)]} - - -{x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x, 106, (c^2*x)/(21*a^3) - (c^2*x^3)/(168*a) - (a*c^2*x^5)/280 - (c^2*ArcTan[a*x])/(21*a^4) - (5*c^2*x^2*ArcTan[a*x])/(168*a^2) + (c^2*x^4*ArcTan[a*x])/28 + (a^2*c^2*x^6*ArcTan[a*x])/56 + (((2*I)/21)*c^2*ArcTan[a*x]^2)/a^4 + (c^2*x*ArcTan[a*x]^2)/(8*a^3) - (c^2*x^3*ArcTan[a*x]^2)/(24*a) - (a*c^2*x^5*ArcTan[a*x]^2)/8 - (3*a^3*c^2*x^7*ArcTan[a*x]^2)/56 - (c^2*ArcTan[a*x]^3)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^3)/4 + (a^2*c^2*x^6*ArcTan[a*x]^3)/3 + (a^4*c^2*x^8*ArcTan[a*x]^3)/8 + (4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(21*a^4) + (((2*I)/21)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4} -{x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x, 73, (-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + (17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a*x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcTan[a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a*x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a^3)} -{x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x, 10, (-11*c^2*x)/(60*a) - (a*c^2*x^3)/60 + (2*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a^2) + (c^2*(1 + a^2*x^2)^2*ArcTan[a*x])/(20*a^2) - (((4*I)/15)*c^2*ArcTan[a*x]^2)/a^2 - (4*c^2*x*ArcTan[a*x]^2)/(15*a) - (2*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(15*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(10*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/(6*a^2) - (8*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^2) - (((4*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2} -{(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x, 12, -(c^2*(1 + a^2*x^2))/(20*a) + c^2*x*ArcTan[a*x] + (c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/10 - (2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2)/(5*a) - (3*c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(20*a) + (((8*I)/15)*c^2*ArcTan[a*x]^3)/a + (8*c^2*x*ArcTan[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/5 + (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a) - (c^2*Log[1 + a^2*x^2])/(2*a) + (((8*I)/5)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a)} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x, x, 36, -(a*c^2*x)/4 + (c^2*ArcTan[a*x])/4 + (a^2*c^2*x^2*ArcTan[a*x])/4 - (2*I)*c^2*ArcTan[a*x]^2 - (9*a*c^2*x*ArcTan[a*x]^2)/4 - (a^3*c^2*x^3*ArcTan[a*x]^2)/4 + (3*c^2*ArcTan[a*x]^3)/4 + a^2*c^2*x^2*ArcTan[a*x]^3 + (a^4*c^2*x^4*ArcTan[a*x]^3)/4 + 2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] - (2*I)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^2*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^2, x, 23, a^2*c^2*x*ArcTan[a*x] - (a*c^2*ArcTan[a*x]^2)/2 - (a^3*c^2*x^2*ArcTan[a*x]^2)/2 + ((2*I)/3)*a*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/x + 2*a^2*c^2*x*ArcTan[a*x]^3 + (a^4*c^2*x^3*ArcTan[a*x]^3)/3 + 5*a*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (a*c^2*Log[1 + a^2*x^2])/2 - (3*I)*a*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (5*I)*a*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (5*a*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^3, x, 25, (-3*I)*a^2*c^2*ArcTan[a*x]^2 - (3*a*c^2*ArcTan[a*x]^2)/(2*x) - (3*a^3*c^2*x*ArcTan[a*x]^2)/2 - (c^2*ArcTan[a*x]^3)/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x]^3)/2 + 4*a^2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 3*a^2*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - 3*a^2*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)] + ((3*I)/2)*a^2*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^4, x, 26, -((a^2*c^2*ArcTan[a*x])/x) - (1/2)*a^3*c^2*ArcTan[a*x]^2 - (a*c^2*ArcTan[a*x]^2)/(2*x^2) - (2/3)*I*a^3*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^3)/x + a^4*c^2*x*ArcTan[a*x]^3 + a^3*c^2*Log[x] + 3*a^3*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 5*a^3*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (1/2)*a^3*c^2*Log[1 + a^2*x^2] - 5*I*a^3*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + 3*I*a^3*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (5/2)*a^3*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)] + (3/2)*a^3*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)]} - - -{x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3, x, 184, (389*c^3*x)/(12600*a^3) - (17*c^3*x^3)/(9450*a) - (a*c^3*x^5)/252 - (a^3*c^3*x^7)/840 - (389*c^3*ArcTan[a*x])/(12600*a^4) - (107*c^3*x^2*ArcTan[a*x])/(4200*a^2) + (53*c^3*x^4*ArcTan[a*x])/2100 + (71*a^2*c^3*x^6*ArcTan[a*x])/2520 + (a^4*c^3*x^8*ArcTan[a*x])/120 + (((26*I)/525)*c^3*ArcTan[a*x]^2)/a^4 + (3*c^3*x*ArcTan[a*x]^2)/(40*a^3) - (c^3*x^3*ArcTan[a*x]^2)/(40*a) - (27*a*c^3*x^5*ArcTan[a*x]^2)/200 - (33*a^3*c^3*x^7*ArcTan[a*x]^2)/280 - (a^5*c^3*x^9*ArcTan[a*x]^2)/30 - (c^3*ArcTan[a*x]^3)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^3)/4 + (a^2*c^3*x^6*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^3)/8 + (a^6*c^3*x^10*ArcTan[a*x]^3)/10 + (52*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(525*a^4) + (((26*I)/525)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4} -{x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3, x, 132, (-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^3*x*ArcTan[a*x])/(1260*a^2) + (239*c^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3*x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a*x]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan[a*x]^2)/420 - (10*a^3*c^3*x^6*ArcTan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^6*c^3*x^9*ArcTan[a*x]^3)/9 - (16*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(105*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(105*a^3)} -{x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3, x, 13, (-19*c^3*x)/(140*a) - (19*a*c^3*x^3)/840 - (a^3*c^3*x^5)/280 + (3*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a^2) + (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(280*a^2) + (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(56*a^2) - (((6*I)/35)*c^3*ArcTan[a*x]^2)/a^2 - (6*c^3*x*ArcTan[a*x]^2)/(35*a) - (3*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(140*a) - (3*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(56*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^3)/(8*a^2) - (12*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a^2) - (((6*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2} -{(c + a^2*c*x^2)^3*ArcTan[a*x]^3, x, 17, (-13*c^3*(1 + a^2*x^2))/(210*a) - (c^3*(1 + a^2*x^2)^2)/(140*a) + (14*c^3*x*ArcTan[a*x])/15 + (13*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/105 + (c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 - (12*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(70*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(14*a) + (((16*I)/35)*c^3*ArcTan[a*x]^3)/a + (16*c^3*x*ArcTan[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/7 + (48*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(35*a) - (7*c^3*Log[1 + a^2*x^2])/(15*a) + (((48*I)/35)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (24*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a)} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x, x, 69, (-13*a*c^3*x)/30 - (a^3*c^3*x^3)/60 + (13*c^3*ArcTan[a*x])/30 + (29*a^2*c^3*x^2*ArcTan[a*x])/60 + (a^4*c^3*x^4*ArcTan[a*x])/20 - ((34*I)/15)*c^3*ArcTan[a*x]^2 - (11*a*c^3*x*ArcTan[a*x]^2)/4 - (7*a^3*c^3*x^3*ArcTan[a*x]^2)/12 - (a^5*c^3*x^5*ArcTan[a*x]^2)/10 + (11*c^3*ArcTan[a*x]^3)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^4*ArcTan[a*x]^3)/4 + (a^6*c^3*x^6*ArcTan[a*x]^3)/6 + 2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - (68*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/15 - ((34*I)/15)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^2, x, 45, -(a^3*c^3*x^2)/20 + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/20 - (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3*ArcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (33*a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/5 + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - a*c^3*Log[1 + a^2*x^2] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^3, x, 43, -(a^3*c^3*x)/4 + (a^2*c^3*ArcTan[a*x])/4 + (a^4*c^3*x^2*ArcTan[a*x])/4 - (5*I)*a^2*c^3*ArcTan[a*x]^2 - (3*a*c^3*ArcTan[a*x]^2)/(2*x) - (15*a^3*c^3*x*ArcTan[a*x]^2)/4 - (a^5*c^3*x^3*ArcTan[a*x]^2)/4 + (3*a^2*c^3*ArcTan[a*x]^3)/4 - (c^3*ArcTan[a*x]^3)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^3)/2 + (a^6*c^3*x^4*ArcTan[a*x]^3)/4 + 6*a^2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 7*a^2*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((7*I)/2)*a^2*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((9*I)/2)*a^2*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((9*I)/2)*a^2*c^3*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((9*I)/4)*a^2*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((9*I)/4)*a^2*c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^4, x, 37, -((a^2*c^3*ArcTan[a*x])/x) + a^4*c^3*x*ArcTan[a*x] - a^3*c^3*ArcTan[a*x]^2 - (a*c^3*ArcTan[a*x]^2)/(2*x^2) - (1/2)*a^5*c^3*x^2*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^3)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^3)/x + 3*a^4*c^3*x*ArcTan[a*x]^3 + (1/3)*a^6*c^3*x^3*ArcTan[a*x]^3 + a^3*c^3*Log[x] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - a^3*c^3*Log[1 + a^2*x^2] - 8*I*a^3*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + 8*I*a^3*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 4*a^3*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)] + 4*a^3*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*ArcTan[a*x]^3/(c + a^2*c*x^2), x, 19, (x*ArcTan[a*x])/(a^4*c) - ArcTan[a*x]^2/(2*a^5*c) - (x^2*ArcTan[a*x]^2)/(2*a^3*c) - (4*I*ArcTan[a*x]^3)/(3*a^5*c) - (x*ArcTan[a*x]^3)/(a^4*c) + (x^3*ArcTan[a*x]^3)/(3*a^2*c) + ArcTan[a*x]^4/(4*a^5*c) - (4*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^5*c) - Log[1 + a^2*x^2]/(2*a^5*c) - (4*I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^5*c) - (2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(a^5*c)} -{x^3*ArcTan[a*x]^3/(c + a^2*c*x^2), x, 14, (((-3*I)/2)*ArcTan[a*x]^2)/(a^4*c) - (3*x*ArcTan[a*x]^2)/(2*a^3*c) + ArcTan[a*x]^3/(2*a^4*c) + (x^2*ArcTan[a*x]^3)/(2*a^2*c) + ((I/4)*ArcTan[a*x]^4)/(a^4*c) - (3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c) + (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^4*c) - (((3*I)/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^4*c) - (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^4*c)} -{x^2*ArcTan[a*x]^3/(c + a^2*c*x^2), x, 7, (I*ArcTan[a*x]^3)/(a^3*c) + (x*ArcTan[a*x]^3)/(a^2*c) - ArcTan[a*x]^4/(4*a^3*c) + (3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^3*c) + ((3*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^3*c) + (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^3*c)} -{x*ArcTan[a*x]^3/(c + a^2*c*x^2), x, 5, ((-I/4)*ArcTan[a*x]^4)/(a^2*c) - (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^2*c) - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c) - (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^2*c) + (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^2*c)} -{ArcTan[a*x]^3/(c + a^2*c*x^2), x, 1, ArcTan[a*x]^4/(4*a*c)} -{ArcTan[a*x]^3/(x*(c + a^2*c*x^2)), x, 5, ((-I/4)*ArcTan[a*x]^4)/c + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c} -{ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)), x, 7, ((-I)*a*ArcTan[a*x]^3)/c - ArcTan[a*x]^3/(c*x) - (a*ArcTan[a*x]^4)/(4*c) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c)} -{ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)), x, 13, (((-3*I)/2)*a^2*ArcTan[a*x]^2)/c - (3*a*ArcTan[a*x]^2)/(2*c*x) - (a^2*ArcTan[a*x]^3)/(2*c) - ArcTan[a*x]^3/(2*c*x^2) + ((I/4)*a^2*ArcTan[a*x]^4)/c + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - (a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (((3*I)/2)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (3*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c) - (((3*I)/4)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c} -{ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)), x, 22, -((a^2*ArcTan[a*x])/(c*x)) - (a^3*ArcTan[a*x]^2)/(2*c) - (a*ArcTan[a*x]^2)/(2*c*x^2) + (4*I*a^3*ArcTan[a*x]^3)/(3*c) - ArcTan[a*x]^3/(3*c*x^3) + (a^2*ArcTan[a*x]^3)/(c*x) + (a^3*ArcTan[a*x]^4)/(4*c) + (a^3*Log[x])/c - (4*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (a^3*Log[1 + a^2*x^2])/(2*c) + (4*I*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (2*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/c} - - -{(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x, 11, (3*x)/(8*a^3*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(8*a^4*c^2) - (3*ArcTan[a*x])/(4*a^4*c^2*(1 + a^2*x^2)) - (3*x*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^3/(4*a^4*c^2) + ArcTan[a*x]^3/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/(a^4*c^2) - (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^4*c^2) - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2) - (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^4*c^2) + (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^4*c^2)} -{(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x, 4, 3/(8*a^3*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(8*a^3*c^2) - (3*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a^3*c^2)} -{(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x, 5, (-3*x)/(8*a*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x])/(8*a^2*c^2) + (3*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(4*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(4*a^2*c^2) - ArcTan[a*x]^3/(2*a^2*c^2*(1 + a^2*x^2))} -{ArcTan[a*x]^3/(c + a^2*c*x^2)^2, x, 4, -3/(8*a*c^2*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(8*a*c^2) + (3*ArcTan[a*x]^2)/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a*c^2)} -{ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^2), x, 11, (3*a*x)/(8*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(8*c^2) - (3*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (3*a*x*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^3/(4*c^2) + ArcTan[a*x]^3/(2*c^2*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/c^2 + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^2), x, 12, (3*a)/(8*c^2*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) + (3*a*ArcTan[a*x]^2)/(8*c^2) - (3*a*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^2 - ArcTan[a*x]^3/(c^2*x) - (a^2*x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) - (3*a*ArcTan[a*x]^4)/(8*c^2) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)} -{ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^2), x, 25, (-3*a^3*x)/(8*c^2*(1 + a^2*x^2)) - (3*a^2*ArcTan[a*x])/(8*c^2) + (3*a^2*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^2 - (3*a*ArcTan[a*x]^2)/(2*c^2*x) + (3*a^3*x*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x]^3)/(4*c^2) - ArcTan[a*x]^3/(2*c^2*x^2) - (a^2*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + ((I/2)*a^2*ArcTan[a*x]^4)/c^2 + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - (2*a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + ((3*I)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (3*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^2} -{ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2), x, 35, -((3*a^3)/(8*c^2*(1 + a^2*x^2))) - (a^2*ArcTan[a*x])/(c^2*x) - (3*a^4*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (7*a^3*ArcTan[a*x]^2)/(8*c^2) - (a*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^3*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) + (7*I*a^3*ArcTan[a*x]^3)/(3*c^2) - ArcTan[a*x]^3/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^3)/(c^2*x) + (a^4*x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^4)/(8*c^2) + (a^3*Log[x])/c^2 - (7*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (a^3*Log[1 + a^2*x^2])/(2*c^2) + (7*I*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (7*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)} - - -{(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3, x, 9, -((3*x^3)/(128*a*c^3*(1 + a^2*x^2)^2)) - (45*x)/(256*a^3*c^3*(1 + a^2*x^2)) - (27*ArcTan[a*x])/(256*a^4*c^3) - (3*x^4*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)) + (3*x^3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^3)/(32*a^4*c^3) + (x^4*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2)} -{(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3, x, 13, 3/(128*a^3*c^3*(1 + a^2*x^2)^2) - 3/(128*a^3*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x])/(64*a^2*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(128*a^3*c^3) - (3*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)^2) + (3*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^3)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^4/(32*a^3*c^3)} -{(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3, x, 9, (-3*x)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x])/(256*a^2*c^3) + (3*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^3)/(32*a^2*c^3) - ArcTan[a*x]^3/(4*a^2*c^3*(1 + a^2*x^2)^2)} -{ArcTan[a*x]^3/(c + a^2*c*x^2)^3, x, 8, -3/(128*a*c^3*(1 + a^2*x^2)^2) - 45/(128*a*c^3*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (45*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x]^2)/(128*a*c^3) + (3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^4)/(32*a*c^3)} -{ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^3), x, 21, (3*a*x)/(128*c^3*(1 + a^2*x^2)^2) + (141*a*x)/(256*c^3*(1 + a^2*x^2)) + (141*ArcTan[a*x])/(256*c^3) - (3*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (33*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)) - (3*a*x*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) - (33*a*x*ArcTan[a*x]^2)/(32*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x]^3)/(32*c^3) + ArcTan[a*x]^3/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]^3/(2*c^3*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/c^3 + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^3 - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^3} -{ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^3), x, 21, (3*a)/(128*c^3*(1 + a^2*x^2)^2) + (93*a)/(128*c^3*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (93*a^2*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) + (93*a*ArcTan[a*x]^2)/(128*c^3) - (3*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) - (21*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^3 - ArcTan[a*x]^3/(c^3*x) - (a^2*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) - (15*a*ArcTan[a*x]^4)/(32*c^3) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)} -{ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^3), x, 47, (-3*a^3*x)/(128*c^3*(1 + a^2*x^2)^2) - (237*a^3*x)/(256*c^3*(1 + a^2*x^2)) - (237*a^2*ArcTan[a*x])/(256*c^3) + (3*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (57*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)) - (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^3 - (3*a*ArcTan[a*x]^2)/(2*c^3*x) + (3*a^3*x*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) + (57*a^3*x*ArcTan[a*x]^2)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^3)/(32*c^3) - ArcTan[a*x]^3/(2*c^3*x^2) - (a^2*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^3)/(c^3*(1 + a^2*x^2)) + (((3*I)/4)*a^2*ArcTan[a*x]^4)/c^3 + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - (3*a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^3 - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (((9*I)/2)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (9*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3) - (((9*I)/4)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^3} -{ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^3), x, 57, -((3*a^3)/(128*c^3*(1 + a^2*x^2)^2)) - (141*a^3)/(128*c^3*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(c^3*x) - (3*a^4*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (141*a^4*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x]^2)/(128*c^3) - (a*ArcTan[a*x]^2)/(2*c^3*x^2) + (3*a^3*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) + (33*a^3*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) + (10*I*a^3*ArcTan[a*x]^3)/(3*c^3) - ArcTan[a*x]^3/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^3)/(c^3*x) + (a^4*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^4)/(32*c^3) + (a^3*Log[x])/c^3 - (10*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - (a^3*Log[1 + a^2*x^2])/(2*c^3) + (10*I*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (5*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^3 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x, 71, -((x*Sqrt[c + a^2*c*x^2])/(20*a^3)) - (9*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10*a^2) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^3) - (3*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(20*a) - (11*I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(20*a^4*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(15*a^2) + (1/5)*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a^4) + (11*I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2]) - (11*I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2]) - (11*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2]) + (11*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x, 40, -(Sqrt[c + a^2*c*x^2]/(4*a^3)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(8*a^2) + (1/4)*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(4*a^3*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(2*a^3*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(2*a^3*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (3*I*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2])} -{x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x, 13, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/a^2 - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*a^2*c) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a^2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2])} -{Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x, 14, (-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x, x, 22, ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^2, x, 22, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^3, x, 27, (-3*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((3*I)/2)*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^4, x, 25, -((a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*c*x^3) - (a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (I*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -{x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x, 200, (c*x*Sqrt[c + a^2*c*x^2])/(420*a^3) - (c*x^3*Sqrt[c + a^2*c*x^2])/(140*a) - (163*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(840*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60*a^2) + (1/35)*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(280*a) - (1/14)*a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (51*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(280*a^4*Sqrt[c + a^2*c*x^2]) - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^2) + (8/35)*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (1/7)*a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (23*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4) + (51*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) - (51*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) - (51*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) + (51*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x, 108, -((c*Sqrt[c + a^2*c*x^2])/(30*a^3)) - (c + a^2*c*x^2)^(3/2)/(60*a^3) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^2) + (1/20)*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (31*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(240*a^3) - (19*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(120*a) - (1/10)*a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(16*a^2) + (7/24)*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (1/6)*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(8*a^3*Sqrt[c + a^2*c*x^2]) + (41*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(60*a^3*Sqrt[c + a^2*c*x^2]) - (3*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(16*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(16*a^3*Sqrt[c + a^2*c*x^2]) - (41*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(120*a^3*Sqrt[c + a^2*c*x^2]) + (41*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(120*a^3*Sqrt[c + a^2*c*x^2]) + (3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (3*I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x, 17, -(c*x*Sqrt[c + a^2*c*x^2])/(20*a) + (9*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a^2) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a^2) - (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(40*a) - (3*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(20*a) + (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/(5*a^2*c) - (c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a^2) - (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (9*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*a^2*Sqrt[c + a^2*c*x^2]) - (9*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*a^2*Sqrt[c + a^2*c*x^2])} -{(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x, 18, -(c*Sqrt[c + a^2*c*x^2])/(4*a) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (9*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(4*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((5*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) + (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) - (((9*I)/4)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/4)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x, x, 36, c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (7*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^2, x, 37, (-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^3, x, 50, (-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) + ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (9*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (9*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^4, x, 48, -((a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((7*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -{x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3, x, 547, (85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (1/504)*a*c^2*x^5*Sqrt[c + a^2*c*x^2] - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (1/84)*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (1/24)*a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (115*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(1344*a^4*Sqrt[c + a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^2) + (5/21)*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (19/63)*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (1/9)*a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(15120*a^4) + (115*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2]) - (115*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])} -{x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3, x, 293, (13*c^2*Sqrt[c + a^2*c*x^2])/(6720*a^3) - (3*c*(c + a^2*c*x^2)^(3/2))/(560*a^3) - (c + a^2*c*x^2)^(5/2)/(280*a^3) + (43*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^2) + (29/560)*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (1/56)*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(13440*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(6720*a) - (83/560)*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (3/56)*a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(128*a^2) + (59/192)*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (17/48)*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (1/8)*a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (5*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(64*a^3*Sqrt[c + a^2*c*x^2]) + (397*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(840*a^3*Sqrt[c + a^2*c*x^2]) - (15*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(128*a^3*Sqrt[c + a^2*c*x^2]) + (15*I*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(128*a^3*Sqrt[c + a^2*c*x^2]) - (397*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(1680*a^3*Sqrt[c + a^2*c*x^2]) + (397*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(1680*a^3*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (15*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (15*I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2])} -{x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3, x, 22, (-17*c^2*x*Sqrt[c + a^2*c*x^2])/(420*a) - (c*x*(c + a^2*c*x^2)^(3/2))/(140*a) + (15*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a^2) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(35*a^2) - (15*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(56*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(14*a) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^3)/(7*a^2*c) - (37*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^2) - (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2])} -{(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3, x, 23, (-17*c^2*Sqrt[c + a^2*c*x^2])/(60*a) - (c*(c + a^2*c*x^2)^(3/2))/(60*a) + (17*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/60 + (c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/20 - (15*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(24*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(10*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - (((259*I)/60)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((15*I)/16)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((15*I)/16)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((259*I)/120)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((259*I)/120)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) - (((15*I)/8)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((15*I)/8)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x, x, 54, -(a*c^2*x*Sqrt[c + a^2*c*x^2])/20 + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^2, x, 56, -(a*c^2*Sqrt[c + a^2*c*x^2])/4 + (a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (21*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((11*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^3, x, 87, a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (3*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a^2*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((15*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (15*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (13*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (15*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((15*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((15*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^4, x, 86, -((a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (3*a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((6*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((13*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((15*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((13*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (15*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (15*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (13*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((15*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((15*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^3*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x, 24, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^4*c) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) - (5*I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^4*Sqrt[c]) + (5*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (5*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x, 15, -((3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) - (6*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(2*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(2*a^3*Sqrt[c + a^2*c*x^2]) + (3*I*Sqrt[1 + a^2*x^2]*PolyLog[2, -((I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*I*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*I*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x, 10, ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^2*c) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/Sqrt[c + a^2*c*x^2], x, 11, ((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(x*Sqrt[c + a^2*c*x^2]), x, 11, (-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^3/(x^2*Sqrt[c + a^2*c*x^2]), x, 10, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x)) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^3/(x^3*Sqrt[c + a^2*c*x^2]), x, 15, (-3*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} -{ArcTan[a*x]^3/(x^4*Sqrt[c + a^2*c*x^2]), x, 25, -((a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*c*x) + (5*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c] - ((5*I)*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((5*I)*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (5*a^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]} - - -{(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x, 14, (6*x)/(a^3*c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (3*x*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^4*c^2) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x, 14, 6/(a^3*c*Sqrt[c + a^2*c*x^2]) + (6*x*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) - (3*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) - (x*ArcTan[a*x]^3)/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x, 3, (-6*x)/(a*c*Sqrt[c + a^2*c*x^2]) + (6*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) + (3*x*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(a^2*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(c + a^2*c*x^2)^(3/2), x, 2, -6/(a*c*Sqrt[c + a^2*c*x^2]) - (6*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) + (3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(3/2)), x, 15, (6*a*x)/(c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(3/2)), x, 13, (6*a)/(c*Sqrt[c + a^2*c*x^2]) + (6*a^2*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^2*x) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])} - - -{(x^5*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2), x, 22, (2*x^3)/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) + (94*x)/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^2*ArcTan[a*x])/(9*a^4*c*(c + a^2*c*x^2)^(3/2)) - (94*ArcTan[a*x])/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*c*x^2)^(3/2)) - (5*x*ArcTan[a*x]^2)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^3)/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^3)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^6*c^3) - (6*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (6*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2])} -{(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2), x, 22, -(2/(27*a^5*c*(c + a^2*c*x^2)^(3/2))) + 68/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (22*x*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*c*x^2)^(3/2)) - (11*ArcTan[a*x]^2)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x]^3)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - (2*I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (3*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (3*I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (6*I*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*I*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])} -{(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2), x, 7, -((2*x^3)/(27*a*c*(c + a^2*c*x^2)^(3/2))) - (40*x)/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^2*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (40*ArcTan[a*x])/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^3)/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2), x, 7, 2/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) - 14/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (4*x*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*ArcTan[a*x]^2)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2))} -{(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2), x, 6, (-2*x)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*a*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(3*a^2*c*(c + a^2*c*x^2)^(3/2))} -{ArcTan[a*x]^3/(c + a^2*c*x^2)^(5/2), x, 5, -2/(27*a*c*(c + a^2*c*x^2)^(3/2)) - 40/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (40*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*ArcTan[a*x]^2)/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(5/2)), x, 22, (2*a*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (202*a*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (22*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]^3/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(5/2)), x, 19, (2*a)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a)/(9*c^2*Sqrt[c + a^2*c*x^2]) + (2*a^2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) + (94*a^2*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) - (a*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a*ArcTan[a*x]^2)/(c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^3*x) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^3 with d=a^2 c and m symbolic*) - - -{x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x]} -{x^m*(c + a^2*c*x^2)^1*ArcTan[a*x]^3, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^3, x]} -{(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^1, x, 0, Unintegrable[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2), x]} -{(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x]} - - -{x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x]} -{x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x, 0, Unintegrable[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]} -{(x^m*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[(x^m*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x]} -{(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^1 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x*(c + a^2*c*x^2))/ArcTan[a*x], x, 0, Unintegrable[(x*(c + a^2*c*x^2))/ArcTan[a*x], x]} -{(c + a^2*c*x^2)/ArcTan[a*x], x, 0, Unintegrable[(c + a^2*c*x^2)/ArcTan[a*x], x]} -{(c + a^2*c*x^2)/(x*ArcTan[a*x]), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*ArcTan[a*x]), x]} - - -{(x*(c + a^2*c*x^2)^2)/ArcTan[a*x], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^2/ArcTan[a*x], x, 0, Unintegrable[(c + a^2*c*x^2)^2/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^2/(x*ArcTan[a*x]), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]), x]} - - -{(x*(c + a^2*c*x^2)^3)/ArcTan[a*x], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^3/ArcTan[a*x], x, 0, Unintegrable[(c + a^2*c*x^2)^3/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^3/(x*ArcTan[a*x]), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^2/((c + a^2*c*x^2)*ArcTan[a*x]), x, 0, Unintegrable[x^2/((c + a^2*c*x^2)*ArcTan[a*x]), x]} -{x/((c + a^2*c*x^2)*ArcTan[a*x]), x, 0, Unintegrable[x/((c + a^2*c*x^2)*ArcTan[a*x]), x]} -{1/((c + a^2*c*x^2)*ArcTan[a*x]), x, 1, Log[ArcTan[a*x]]/(a*c)} -{1/(x*(c + a^2*c*x^2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]), x]} -{1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]), x]} - - -{x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 0, Unintegrable[x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]} -{x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 0, Unintegrable[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]} -{x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 4, -CosIntegral[2*ArcTan[a*x]]/(2*a^3*c^2) + Log[ArcTan[a*x]]/(2*a^3*c^2)} -{x/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 4, SinIntegral[2*ArcTan[a*x]]/(2*a^2*c^2)} -{1/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 4, CosIntegral[2*ArcTan[a*x]]/(2*a*c^2) + Log[ArcTan[a*x]]/(2*a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]} -{1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]), x, 0, Unintegrable[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]} - - -{x^6/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 0, Unintegrable[x^6/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]} -{x^5/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 0, Unintegrable[x^5/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]} -{x^4/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 5, -(CosIntegral[2*ArcTan[a*x]]/(2*a^5*c^3)) + CosIntegral[4*ArcTan[a*x]]/(8*a^5*c^3) + (3*Log[ArcTan[a*x]])/(8*a^5*c^3)} -{x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 5, SinIntegral[2*ArcTan[a*x]]/(4*a^4*c^3) - SinIntegral[4*ArcTan[a*x]]/(8*a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 4, -CosIntegral[4*ArcTan[a*x]]/(8*a^3*c^3) + Log[ArcTan[a*x]]/(8*a^3*c^3)} -{x/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 5, SinIntegral[2*ArcTan[a*x]]/(4*a^2*c^3) + SinIntegral[4*ArcTan[a*x]]/(8*a^2*c^3)} -{1/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 5, CosIntegral[2*ArcTan[a*x]]/(2*a*c^3) + CosIntegral[4*ArcTan[a*x]]/(8*a*c^3) + (3*Log[ArcTan[a*x]])/(8*a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]} -{1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x, 0, Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^1 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]} -{Sqrt[c + a^2*c*x^2]/ArcTan[a*x], x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/ArcTan[a*x], x]} -{Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]), x]} - - -{(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^(3/2)/ArcTan[a*x], x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]), x]} - - -{(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^(5/2)/ArcTan[a*x], x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x], x]} -{(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]} -{1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x, 0, Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]} - - -{x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 0, Unintegrable[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]} -{x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 0, Unintegrable[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]} -{x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 3, (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 3, (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]} -{1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]} - - -{x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 0, Unintegrable[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]} -{x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 0, Unintegrable[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]} -{x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 6, (3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 6, (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 6, (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 6, (3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]} -{1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 0, Unintegrable[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^1 with d=a^2 c and m symbolic*) - - -{(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]} -{(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]} -{(x^m*(c + a^2*c*x^2))/ArcTan[a*x], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/ArcTan[a*x], x]} -{x^m/((c + a^2*c*x^2)*ArcTan[a*x]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)*ArcTan[a*x]), x]} -{x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]} -{x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]} -{(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]} -{(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]} -{x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]} -{x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]} -{x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(x*(c + a^2*c*x^2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)/ArcTan[a*x]^2, x, 0, Unintegrable[(c + a^2*c*x^2)/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)/(x*ArcTan[a*x]^2), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*ArcTan[a*x]^2), x]} - - -{(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^2/ArcTan[a*x]^2, x, 0, Unintegrable[(c + a^2*c*x^2)^2/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^2), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^2), x]} - - -{(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^3/ArcTan[a*x]^2, x, 0, Unintegrable[(c + a^2*c*x^2)^3/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^2), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^2), x]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3/((c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(x^3/(a*c*ArcTan[a*x])) + (3*Unintegrable[x^2/ArcTan[a*x], x])/(a*c)} -{x^2/((c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(x^2/(a*c*ArcTan[a*x])) + (2*Unintegrable[x/ArcTan[a*x], x])/(a*c)} -{x/((c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(x/(a*c*ArcTan[a*x])) + Unintegrable[ArcTan[a*x]^(-1), x]/(a*c)} -{1/((c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(1/(a*c*ArcTan[a*x]))} -{1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(1/(a*c*x*ArcTan[a*x])) - Unintegrable[1/(x^2*ArcTan[a*x]), x]/(a*c)} -{1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(1/(a*c*x^2*ArcTan[a*x])) - (2*Unintegrable[1/(x^3*ArcTan[a*x]), x])/(a*c)} -{1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(1/(a*c*x^3*ArcTan[a*x])) - (3*Unintegrable[1/(x^4*ArcTan[a*x]), x])/(a*c)} -{1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(1/(a*c*x^4*ArcTan[a*x])) - (4*Unintegrable[1/(x^5*ArcTan[a*x]), x])/(a*c)} - - -{x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 11, -(x/(a^3*c^2*ArcTan[a*x])) + x/(a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Unintegrable[ArcTan[a*x]^(-1), x]/(a^3*c^2)} -{x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 5, -(x^2/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]]/(a^3*c^2)} -{x/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 9, -(x/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + CosIntegral[2*ArcTan[a*x]]/(a^2*c^2)} -{1/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 5, -(1/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) - SinIntegral[2*ArcTan[a*x]]/(a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 11, -(1/(a*c^2*x*ArcTan[a*x])) + (a*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/c^2 - Unintegrable[1/(x^2*ArcTan[a*x]), x]/(a*c^2)} -{1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 7, -(1/(a*c^2*x^2*ArcTan[a*x])) + a/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a*SinIntegral[2*ArcTan[a*x]])/c^2 - (2*Unintegrable[1/(x^3*ArcTan[a*x]), x])/(a*c^2)} -{1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 13, -(1/(a*c^2*x^3*ArcTan[a*x])) + a/(c^2*x*ArcTan[a*x]) - (a^3*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a^2*CosIntegral[2*ArcTan[a*x]])/c^2 - (3*Unintegrable[1/(x^4*ArcTan[a*x]), x])/(a*c^2) + (a*Unintegrable[1/(x^2*ArcTan[a*x]), x])/c^2} -{1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 9, -(1/(a*c^2*x^4*ArcTan[a*x])) + a/(c^2*x^2*ArcTan[a*x]) - a^3/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^3*SinIntegral[2*ArcTan[a*x]])/c^2 - (4*Unintegrable[1/(x^5*ArcTan[a*x]), x])/(a*c^2) + (2*a*Unintegrable[1/(x^3*ArcTan[a*x]), x])/c^2} - - -{x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 20, x/(a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - x/(a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[2*ArcTan[a*x]]/(2*a^4*c^3) - CosIntegral[4*ArcTan[a*x]]/(2*a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 12, 1/(a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 1/(a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[4*ArcTan[a*x]]/(2*a^3*c^3)} -{x/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 10, -(x/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])) + CosIntegral[2*ArcTan[a*x]]/(2*a^2*c^3) + CosIntegral[4*ArcTan[a*x]]/(2*a^2*c^3)} -{1/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 6, -(1/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])) - SinIntegral[2*ArcTan[a*x]]/(a*c^3) - SinIntegral[4*ArcTan[a*x]]/(2*a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 22, -(1/(a*c^3*x*ArcTan[a*x])) + (a*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (a*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*CosIntegral[2*ArcTan[a*x]])/(2*c^3) - CosIntegral[4*ArcTan[a*x]]/(2*c^3) - Unintegrable[1/(x^2*ArcTan[a*x]), x]/(a*c^3)} -{1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 14, -(1/(a*c^3*x^2*ArcTan[a*x])) + a/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + a/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (2*a*SinIntegral[2*ArcTan[a*x]])/c^3 + (a*SinIntegral[4*ArcTan[a*x]])/(2*c^3) - (2*Unintegrable[1/(x^3*ArcTan[a*x]), x])/(a*c^3)} -{1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 36, -(1/(a*c^3*x^3*ArcTan[a*x])) + (2*a)/(c^3*x*ArcTan[a*x]) - (a^3*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (2*a^3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (5*a^2*CosIntegral[2*ArcTan[a*x]])/(2*c^3) + (a^2*CosIntegral[4*ArcTan[a*x]])/(2*c^3) - (3*Unintegrable[1/(x^4*ArcTan[a*x]), x])/(a*c^3) + (2*a*Unintegrable[1/(x^2*ArcTan[a*x]), x])/c^3} -{1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 24, -(1/(a*c^3*x^4*ArcTan[a*x])) + (2*a)/(c^3*x^2*ArcTan[a*x]) - a^3/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (2*a^3)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*a^3*SinIntegral[2*ArcTan[a*x]])/c^3 - (a^3*SinIntegral[4*ArcTan[a*x]])/(2*c^3) - (4*Unintegrable[1/(x^5*ArcTan[a*x]), x])/(a*c^3) + (4*a*Unintegrable[1/(x^3*ArcTan[a*x]), x])/c^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^(q/2) / (a+b ArcTan[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]} -{Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^2, x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^2, x]} -{Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^2), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^2), x]} - - -{(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^2, x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^2), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^2), x]} - - -{(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x]} -{(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^2), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^2), x]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]} -{1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x, 1, -(Sqrt[c + a^2*c*x^2]/(a*c*x*ArcTan[a*x])) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/a} - - -{x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 5, x/(a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^2*c)} -{x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 5, 1/(a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^2*c)} -{x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 4, -(x/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 4, -(1/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 6, (a*x)/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - Sqrt[c + a^2*c*x^2]/(a*c^2*x*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/(a*c)} -{1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 5, a/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (a*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c} -{1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 7, -((a^3*x)/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) + (a*Sqrt[c + a^2*c*x^2])/(c^2*x*ArcTan[a*x]) + (a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c + (a*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c} -{1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 6, -(a^3/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (a^3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c - (a^2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c} - - -{x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 13, x^3/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (7*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^4*c^2)} -{x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 18, -(1/(a^5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + 2/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^5*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^4*c^2)} -{x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 7, -(x^3/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 12, 1/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 1/(a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 13, -(x/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 7, -(1/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 20, (a*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + (a*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - Sqrt[c + a^2*c*x^2]/(a*c^3*x*ArcTan[a*x]) - (5*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/(a*c^2)} -{1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 13, a/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + a/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*a*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + (3*a*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2} -{1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 28, -((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c^2} -{1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 20, -(a^3/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (11*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 - (2*a^2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(m/2) (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2*) - - -{Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x, 0, Unintegrable[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2 with m symbolic*) - - -{(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]} -{(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]} -{(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]} -{x^m/((c + a^2*c*x^2)*ArcTan[a*x]^2), x, 1, -(x^m/(a*c*ArcTan[a*x])) + (m*Unintegrable[x^(-1 + m)/ArcTan[a*x], x])/(a*c)} -{x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]} -{x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]} -{(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]} -{(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]} -{x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]} -{x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]} -{x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^3 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x*(c + a^2*c*x^2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)/ArcTan[a*x]^3, x, 0, Unintegrable[(c + a^2*c*x^2)/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)/(x*ArcTan[a*x]^3), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*ArcTan[a*x]^3), x]} - - -{(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^2/ArcTan[a*x]^3, x, 0, Unintegrable[(c + a^2*c*x^2)^2/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^3), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^3), x]} - - -{(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^3/ArcTan[a*x]^3, x, 0, Unintegrable[(c + a^2*c*x^2)^3/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^3), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^3), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/((c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -x^3/(2*a*c*ArcTan[a*x]^2) + (3*Unintegrable[x^2/ArcTan[a*x]^2, x])/(2*a*c)} -{x^2/((c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -x^2/(2*a*c*ArcTan[a*x]^2) + Unintegrable[x/ArcTan[a*x]^2, x]/(a*c)} -{x/((c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -x/(2*a*c*ArcTan[a*x]^2) + Unintegrable[ArcTan[a*x]^(-2), x]/(2*a*c)} -{1/((c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -1/(2*a*c*ArcTan[a*x]^2)} -{1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -1/(2*a*c*x*ArcTan[a*x]^2) - Unintegrable[1/(x^2*ArcTan[a*x]^2), x]/(2*a*c)} -{1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -1/(2*a*c*x^2*ArcTan[a*x]^2) - Unintegrable[1/(x^3*ArcTan[a*x]^2), x]/(a*c)} -{1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -1/(2*a*c*x^3*ArcTan[a*x]^2) - (3*Unintegrable[1/(x^4*ArcTan[a*x]^2), x])/(2*a*c)} -{1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -1/(2*a*c*x^4*ArcTan[a*x]^2) - (2*Unintegrable[1/(x^5*ArcTan[a*x]^2), x])/(a*c)} - - -{x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 7, -x/(2*a^3*c^2*ArcTan[a*x]^2) + x/(2*a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*a^4*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Unintegrable[ArcTan[a*x]^(-2), x]/(2*a^3*c^2)} -{x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 10, -x^2/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - x/(a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[2*ArcTan[a*x]]/(a^3*c^2)} -{x/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 5, -x/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (1 - a^2*x^2)/(2*a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(a^2*c^2)} -{1/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 10, -1/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + x/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 7, -1/(2*a*c^2*x*ArcTan[a*x]^2) + (a*x)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/c^2 - Unintegrable[1/(x^2*ArcTan[a*x]^2), x]/(2*a*c^2)} -{1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 12, -1/(2*a*c^2*x^2*ArcTan[a*x]^2) + a/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (a^2*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a*CosIntegral[2*ArcTan[a*x]])/c^2 - Unintegrable[1/(x^3*ArcTan[a*x]^2), x]/(a*c^2)} -{1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 9, -1/(2*a*c^2*x^3*ArcTan[a*x]^2) + a/(2*c^2*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (a^2*(1 - a^2*x^2))/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^2*SinIntegral[2*ArcTan[a*x]])/c^2 - (3*Unintegrable[1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^2) + (a*Unintegrable[1/(x^2*ArcTan[a*x]^2), x])/(2*c^2)} -{1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 14, -1/(2*a*c^2*x^4*ArcTan[a*x]^2) + a/(2*c^2*x^2*ArcTan[a*x]^2) - a^3/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (a^4*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^3*CosIntegral[2*ArcTan[a*x]])/c^2 - (2*Unintegrable[1/(x^5*ArcTan[a*x]^2), x])/(a*c^2) + (a*Unintegrable[1/(x^3*ArcTan[a*x]^2), x])/c^2} - - -{x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 25, x/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - x/(2*a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + 2/(a^4*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (1 - a^2*x^2)/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(2*a^4*c^3) + SinIntegral[4*ArcTan[a*x]]/(a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 22, 1/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - 1/(2*a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*x)/(a^2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + x/(a^2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[4*ArcTan[a*x]]/(a^3*c^3)} -{x/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 19, -x/(2*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - 2/(a^2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + 3/(2*a^2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(2*a^2*c^3) - SinIntegral[4*ArcTan[a*x]]/(a^2*c^3)} -{1/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 11, -1/(2*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + (2*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a*c^3) - CosIntegral[4*ArcTan[a*x]]/(a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 27, -1/(2*a*c^3*x*ArcTan[a*x]^2) + (a*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + (a*x)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + 2/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (1 - a^2*x^2)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (3*SinIntegral[2*ArcTan[a*x]])/(2*c^3) + SinIntegral[4*ArcTan[a*x]]/c^3 - Unintegrable[1/(x^2*ArcTan[a*x]^2), x]/(2*a*c^3)} -{1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 24, -1/(2*a*c^3*x^2*ArcTan[a*x]^2) + a/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + a/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (a^2*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (2*a*CosIntegral[2*ArcTan[a*x]])/c^3 + (a*CosIntegral[4*ArcTan[a*x]])/c^3 - Unintegrable[1/(x^3*ArcTan[a*x]^2), x]/(a*c^3)} -{1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 37, -1/(2*a*c^3*x^3*ArcTan[a*x]^2) + a/(c^3*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - (a^3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (3*a^2)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (a^2*(1 - a^2*x^2))/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (5*a^2*SinIntegral[2*ArcTan[a*x]])/(2*c^3) - (a^2*SinIntegral[4*ArcTan[a*x]])/c^3 - (3*Unintegrable[1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^3) + (a*Unintegrable[1/(x^2*ArcTan[a*x]^2), x])/c^3} -{1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 39, -1/(2*a*c^3*x^4*ArcTan[a*x]^2) + a/(c^3*x^2*ArcTan[a*x]^2) - a^3/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - a^3/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + (2*a^4*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (2*a^4*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*a^3*CosIntegral[2*ArcTan[a*x]])/c^3 - (a^3*CosIntegral[4*ArcTan[a*x]])/c^3 - (2*Unintegrable[1/(x^5*ArcTan[a*x]^2), x])/(a*c^3) + (2*a*Unintegrable[1/(x^3*ArcTan[a*x]^2), x])/c^3} - - -{x^3/((1 + a^2*x^2)*ArcTan[a*x]^3) - (3*x^2)/(2*a*ArcTan[a*x]^2), x, 2, -(x^3/(2*a*ArcTan[a*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^3 with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]} -{Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^3, x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^3, x]} -{Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^3), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^3), x]} - - -{(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^3, x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^3), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^3), x]} - - -{(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x]} -{(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]} -{1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 1, -(Sqrt[c + a^2*c*x^2]/(2*a*c*x*ArcTan[a*x]^2)) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a)} -{1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 0, Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]} -{1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 0, Unintegrable[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]} - - -{x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 6, x/(2*a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + 1/(2*a^4*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*a^4*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^2*c)} -{x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 6, 1/(2*a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - x/(2*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*a^3*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^2*c)} -{x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 5, -(x/(2*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)) - 1/(2*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 5, -(1/(2*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)) + x/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 7, (a*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - Sqrt[c + a^2*c*x^2]/(2*a*c^2*x*ArcTan[a*x]^2) + 1/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a*c)} -{1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 6, a/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (a^2*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (a*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c} -{1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 8, -((a^3*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)) + (a*Sqrt[c + a^2*c*x^2])/(2*c^2*x*ArcTan[a*x]^2) - a^2/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (a^2*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c + (a*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/(2*c)} -{1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 7, -(a^3/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)) + (a^4*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (a^3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c - (a^2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x])/c} - - -{x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 20, x^3/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + x/(2*a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - 3/(2*a^6*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 2/(a^6*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)} -{x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 27, -(1/(2*a^5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)) + 1/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + (3*x)/(2*a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - x/(a^4*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a^5*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)} -{x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 13, -(x^3/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)) + 3/(2*a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 3/(2*a^4*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 20, 1/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) - 1/(2*a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*x)/(2*a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 20, -(x/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)) - 3/(2*a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 1/(a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 14, -(1/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)) + (3*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 28, (a*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + (a*x)/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - Sqrt[c + a^2*c*x^2]/(2*a*c^3*x*ArcTan[a*x]^2) + 3/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 1/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) - Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a*c^2)} -{1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 21, a/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + a/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*a^2*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (a^2*x)/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*a*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + (9*a*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^3 with d=a^2 c and m symbolic*) - - -{(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]} -{(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]} -{(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]} -{x^m/((c + a^2*c*x^2)*ArcTan[a*x]^3), x, 1, -x^m/(2*a*c*ArcTan[a*x]^2) + (m*Unintegrable[x^(-1 + m)/ArcTan[a*x]^2, x])/(2*a*c)} -{x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]} -{x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]} -{(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]} -{(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]} -{x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]} -{x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]} -{x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(1/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(1/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]} - -{x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x, 1, (c*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(4*a^2) - Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x]/(8*a)} -{(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]} -{((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]])/x, x, 0, Unintegrable[((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]])/x, x]} - - -{x^m*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]} - -{x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x, 1, (c^2*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(6*a^2) - Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]/(12*a)} -{(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]} -{((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]])/x, x, 0, Unintegrable[((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]])/x, x]} - - -{x^m*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]} - -{x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x, 1, (c^3*(1 + a^2*x^2)^4*Sqrt[ArcTan[a*x]])/(8*a^2) - Unintegrable[(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]/(16*a)} -{(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]} -{((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]])/x, x, 0, Unintegrable[((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]])/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]} - -{(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x, 2, -((2*x*ArcTan[a*x]^(3/2))/(3*a^3*c)) + Unintegrable[x*Sqrt[ArcTan[a*x]], x]/(a^2*c) + (2*Unintegrable[ArcTan[a*x]^(3/2), x])/(3*a^3*c)} -{(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x, 2, (-2*ArcTan[a*x]^(3/2))/(3*a^3*c) + Unintegrable[Sqrt[ArcTan[a*x]], x]/(a^2*c)} -{(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x, 1, (2*x*ArcTan[a*x]^(3/2))/(3*a*c) - (2*Unintegrable[ArcTan[a*x]^(3/2), x])/(3*a*c)} -{Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2), x, 1, (2*ArcTan[a*x]^(3/2))/(3*a*c)} -{Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)), x, 1, (((-2*I)/3)*ArcTan[a*x]^(3/2))/c + (I*Unintegrable[Sqrt[ArcTan[a*x]]/(x*(I + a*x)), x])/c} -{Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)), x, 2, (-2*a*ArcTan[a*x]^(3/2))/(3*c) + Unintegrable[Sqrt[ArcTan[a*x]]/x^2, x]/c} -{Sqrt[ArcTan[a*x]]/(x^3*(c + a^2*c*x^2)), x, 2, (((2*I)/3)*a^2*ArcTan[a*x]^(3/2))/c + Unintegrable[Sqrt[ArcTan[a*x]]/x^3, x]/c - (I*a^2*Unintegrable[Sqrt[ArcTan[a*x]]/(x*(I + a*x)), x])/c} -{Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)), x, 3, (2*a^3*ArcTan[a*x]^(3/2))/(3*c) + Unintegrable[Sqrt[ArcTan[a*x]]/x^4, x]/c - (a^2*Unintegrable[Sqrt[ArcTan[a*x]]/x^2, x])/c} - - -{(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]} - -{(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]} -{(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x, 6, -((x*Sqrt[ArcTan[a*x]])/(2*a^2*c^2*(1 + a^2*x^2))) + ArcTan[a*x]^(3/2)/(3*a^3*c^2) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^3*c^2)} -{(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x, 6, Sqrt[ArcTan[a*x]]/(4*a^2*c^2) - Sqrt[ArcTan[a*x]]/(2*a^2*c^2*(1 + a^2*x^2)) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^2*c^2)} -{Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^2, x, 6, (x*Sqrt[ArcTan[a*x]])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a*c^2) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^2)} -{Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^2), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^2), x]} - - -{(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]} - -{(x^5*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^5*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]} -{(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 9, ArcTan[a*x]^(3/2)/(4*a^5*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^5*c^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^5*c^3) - (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a^5*c^3) + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a^5*c^3)} -{(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 8, -((3*Sqrt[ArcTan[a*x]])/(32*a^4*c^3)) + (x^4*Sqrt[ArcTan[a*x]])/(4*c^3*(1 + a^2*x^2)^2) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^4*c^3)} -{(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 6, ArcTan[a*x]^(3/2)/(12*a^3*c^3) + (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^3*c^3) - (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)} -{(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x, 8, (3*Sqrt[ArcTan[a*x]])/(32*a^2*c^3) - Sqrt[ArcTan[a*x]]/(4*a^2*c^3*(1 + a^2*x^2)^2) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^2*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^2*c^3)} -{Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^3, x, 9, ArcTan[a*x]^(3/2)/(4*a*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a*c^3) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a*c^3)} -{Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(1/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]} - -{x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]} -{x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x, 1, ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(3*a^2*c) - Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]/(6*a)} -{Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]} - - -{x^m*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]} - -{x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]} -{x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x, 1, ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(5*a^2*c) - Unintegrable[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x]/(10*a)} -{(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]} - - -{x^m*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]} - -{x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]} -{x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x, 1, ((c + a^2*c*x^2)^(7/2)*Sqrt[ArcTan[a*x]])/(7*a^2*c) - Unintegrable[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]/(14*a)} -{(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]} - -{(x^3*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x, 2, -((2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^4*c)) + (x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^2*c) + Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(3*a^3) - Unintegrable[x^2/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(6*a)} -{(x^2*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x, 1, (x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(2*a^2*c) - Unintegrable[x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(4*a) - Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x]/(2*a^2)} -{(x*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x, 1, (Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(a^2*c) - Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(2*a)} -{Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x]} -{Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x]} -{Sqrt[ArcTan[a*x]]/(x^2*Sqrt[c + a^2*c*x^2]), x, 1, -((Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(c*x)) + (a*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/2} -{Sqrt[ArcTan[a*x]]/(x^3*Sqrt[c + a^2*c*x^2]), x, 1, -(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(2*c*x^2) + (a*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/4 - (a^2*Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/2} -{Sqrt[ArcTan[a*x]]/(x^4*Sqrt[c + a^2*c*x^2]), x, 2, -(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*c*x) + (a*Unintegrable[1/(x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/6 - (a^3*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/3} - - -{(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]} - -{(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]} -{(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]} -{(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x, 5, -(Sqrt[ArcTan[a*x]]/(a^2*c*Sqrt[c + a^2*c*x^2])) + (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(3/2), x, 5, (x*Sqrt[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(3/2)), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(3/2)), x]} -{Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)^(3/2)), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)^(3/2)), x]} - - -{(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]} - -{(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]} -{(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x, 10, -((3*Sqrt[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2])) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(12*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x, 9, (x^3*Sqrt[ArcTan[a*x]])/(3*c*(c + a^2*c*x^2)^(3/2)) - (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x, 9, -(Sqrt[ArcTan[a*x]]/(3*a^2*c*(c + a^2*c*x^2)^(3/2))) + (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(5/2), x, 10, (3*x*Sqrt[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(12*a*c^2*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(5/2)), x, 0, Unintegrable[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(5/2)), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(3/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(3/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]} - -{x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]} -{x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x, 1, (c*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(4*a^2) - (3*Unintegrable[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/(8*a)} -{(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x, 1, -(c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(4*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/3 + (c*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/8 + (2*c*Unintegrable[ArcTan[a*x]^(3/2), x])/3} -{((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x, x]} -{((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x^2, x]} - - -{x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]} - -{x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]} -{x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x, 1, (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(6*a^2) - Unintegrable[(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]/(4*a)} -{(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x, 2, -(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(5*a) - (3*c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(40*a) + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/5 + (c^2*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/10 + (3*c*Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/80 + (8*c^2*Unintegrable[ArcTan[a*x]^(3/2), x])/15} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x, x]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x^2, x]} - - -{x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]} - -{x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]} -{x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x, 1, (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^(3/2))/(8*a^2) - (3*Unintegrable[(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x])/(16*a)} -{(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x, 3, (-6*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(35*a) - (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(140*a) - (c^3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(28*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/7 + (3*c^3*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/35 + (9*c^2*Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/280 + (c*Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/56 + (16*c^3*Unintegrable[ArcTan[a*x]^(3/2), x])/35} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x, x]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]} - -{(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x, 2, -((2*x*ArcTan[a*x]^(5/2))/(5*a^3*c)) + Unintegrable[x*ArcTan[a*x]^(3/2), x]/(a^2*c) + (2*Unintegrable[ArcTan[a*x]^(5/2), x])/(5*a^3*c)} -{(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x, 2, (-2*ArcTan[a*x]^(5/2))/(5*a^3*c) + Unintegrable[ArcTan[a*x]^(3/2), x]/(a^2*c)} -{(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x, 1, (2*x*ArcTan[a*x]^(5/2))/(5*a*c) - (2*Unintegrable[ArcTan[a*x]^(5/2), x])/(5*a*c)} -{ArcTan[a*x]^(3/2)/(c + a^2*c*x^2), x, 1, (2*ArcTan[a*x]^(5/2))/(5*a*c)} -{ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)), x, 1, (((-2*I)/5)*ArcTan[a*x]^(5/2))/c + (I*Unintegrable[ArcTan[a*x]^(3/2)/(x*(I + a*x)), x])/c} -{ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)), x, 2, (-2*a*ArcTan[a*x]^(5/2))/(5*c) + Unintegrable[ArcTan[a*x]^(3/2)/x^2, x]/c} -{ArcTan[a*x]^(3/2)/(x^3*(c + a^2*c*x^2)), x, 2, (((2*I)/5)*a^2*ArcTan[a*x]^(5/2))/c + Unintegrable[ArcTan[a*x]^(3/2)/x^3, x]/c - (I*a^2*Unintegrable[ArcTan[a*x]^(3/2)/(x*(I + a*x)), x])/c} -{ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)), x, 3, (2*a^3*ArcTan[a*x]^(5/2))/(5*c) + Unintegrable[ArcTan[a*x]^(3/2)/x^4, x]/c - (a^2*Unintegrable[ArcTan[a*x]^(3/2)/x^2, x])/c} - - -{(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]} - -{(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]} -{(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x, 7, (3*Sqrt[ArcTan[a*x]])/(16*a^3*c^2) - (3*Sqrt[ArcTan[a*x]])/(8*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^(3/2))/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a^3*c^2) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^3*c^2)} -{(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x, 7, (3*x*Sqrt[ArcTan[a*x]])/(8*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(4*a^2*c^2) - ArcTan[a*x]^(3/2)/(2*a^2*c^2*(1 + a^2*x^2)) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^2*c^2)} -{ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^2, x, 7, -((3*Sqrt[ArcTan[a*x]])/(16*a*c^2)) + (3*Sqrt[ArcTan[a*x]])/(8*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a*c^2) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^2)} -{ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^2), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^2), x]} - - -{(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]} - -{(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]} -{(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 15, (27*Sqrt[ArcTan[a*x]])/(256*a^5*c^3) + (3*x^4*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) - (9*Sqrt[ArcTan[a*x]])/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(3/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(3/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a^5*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^5*c^3) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^5*c^3)} -{(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 10, -((3*ArcTan[a*x]^(3/2))/(32*a^4*c^3)) + (x^4*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^4*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^4*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^4*c^3) - (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256*a^4*c^3)} -{(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 7, ArcTan[a*x]^(5/2)/(20*a^3*c^3) - (3*Sqrt[ArcTan[a*x]]*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) + (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^3*c^3) - (ArcTan[a*x]^(3/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)} -{(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x, 10, (3*ArcTan[a*x]^(3/2))/(32*a^2*c^3) - ArcTan[a*x]^(3/2)/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^2*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^2*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^2*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256*a^2*c^3)} -{ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^3, x, 15, -((45*Sqrt[ArcTan[a*x]])/(256*a*c^3)) + (3*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) + (9*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(3/2))/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a*c^3) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^3)} -{ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^3), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(3/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]} - -{x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]} -{x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x, 1, ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(3*a^2*c) - Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]/(2*a)} -{Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x, 1, (-3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/2 + (3*c*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 + (c*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/2} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/x, x]} - - -{x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]} - -{x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]} -{x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x, 1, ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(5*a^2*c) - (3*Unintegrable[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/(10*a)} -{(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x, 2, (-9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(16*a) - ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(8*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/4 + (9*c^2*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/32 + (c*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/16 + (3*c^2*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/8} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/x, x]} - - -{x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]} - -{x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]} -{x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x, 1, ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(3/2))/(7*a^2*c) - (3*Unintegrable[(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x])/(14*a)} -{(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x, 3, (-15*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a) - (5*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(48*a) - ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(20*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/6 + (15*c^3*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/64 + (5*c^2*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/96 + (c*Unintegrable[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/40 + (5*c^3*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/16} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]} - -{(x^3*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x, 3, -((x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a^3*c)) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*a^2*c) + Unintegrable[x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(8*a^2) + (5*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(4*a^3)} -{(x^2*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x, 2, -((3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*a^2*c) + (3*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(8*a^2) - Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x]/(2*a^2)} -{(x*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x, 1, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(a^2*c) - (3*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(2*a)} -{ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x]} -{ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x]} -{ArcTan[a*x]^(3/2)/(x^2*Sqrt[c + a^2*c*x^2]), x, 1, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(c*x)) + (3*a*Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/2} -{ArcTan[a*x]^(3/2)/(x^3*Sqrt[c + a^2*c*x^2]), x, 2, (-3*a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*c*x^2) + (3*a^2*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (a^2*Unintegrable[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2} -{ArcTan[a*x]^(3/2)/(x^4*Sqrt[c + a^2*c*x^2]), x, 3, -(a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x) + (a^2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (5*a^3*Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/4} - - -{(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]} - -{(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]} -{(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]} -{(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x, 6, (3*x*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(3/2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2), x, 5, (3*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(3/2))/(c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(3/2)), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(3/2)), x]} -{ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(3/2)), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(3/2)), x]} - - -{(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]} - -{(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]} -{(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]} -{(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 15, (x^3*Sqrt[ArcTan[a*x]])/(6*a*c*(c + a^2*c*x^2)^(3/2)) + (x*Sqrt[ArcTan[a*x]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^(3/2))/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^(3/2))/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 11, (3*Sqrt[ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^(3/2))/(3*c*(c + a^2*c*x^2)^(3/2)) - (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(24*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x, 11, (3*x*Sqrt[ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(3/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(24*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(5/2), x, 14, Sqrt[ArcTan[a*x]]/(6*a*c*(c + a^2*c*x^2)^(3/2)) + Sqrt[ArcTan[a*x]]/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(3/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^(3/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(5/2)), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(5/2)), x]} -{ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(5/2)), x, 0, Unintegrable[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(5/2)), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(5/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(5/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]} - -{x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]} -{x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x, 2, (5*c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(32*a^2) - (5*c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(24*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/(4*a^2) - (5*c*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(64*a) - (5*c*Unintegrable[ArcTan[a*x]^(3/2), x])/(12*a)} -{(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x, 1, (-5*c*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(12*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/3 + (5*c*Unintegrable[Sqrt[ArcTan[a*x]], x])/8 + (2*c*Unintegrable[ArcTan[a*x]^(5/2), x])/3} -{((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x, x]} -{((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x^2, x]} - - -{x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]} - -{x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]} -{x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x, 3, (c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Unintegrable[ArcTan[a*x]^(3/2), x])/(9*a)} -{(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x, 2, -(c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(3*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(8*a) + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/5 + (c^2*Unintegrable[Sqrt[ArcTan[a*x]], x])/2 + (3*c*Unintegrable[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/16 + (8*c^2*Unintegrable[ArcTan[a*x]^(5/2), x])/15} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x, x]} -{((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x^2, x]} - - -{x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]} - -{x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]} -{x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x, 4, (3*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(56*a^2) + (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(448*a^2) + (5*c^3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(448*a^2) - (c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(14*a) - (3*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(56*a) - (5*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(112*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^(5/2))/(8*a^2) - (3*c^3*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(112*a) - (9*c^2*Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(896*a) - (5*c*Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/(896*a) - (c^3*Unintegrable[ArcTan[a*x]^(3/2), x])/(7*a)} -{(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x, 3, (-2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(7*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(28*a) - (5*c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(84*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/7 + (3*c^3*Unintegrable[Sqrt[ArcTan[a*x]], x])/7 + (9*c^2*Unintegrable[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/56 + (5*c*Unintegrable[(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x])/56 + (16*c^3*Unintegrable[ArcTan[a*x]^(5/2), x])/35} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x, x]} -{((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x^2, x, 0, Unintegrable[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]} - -{(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x, 2, -((2*x*ArcTan[a*x]^(7/2))/(7*a^3*c)) + Unintegrable[x*ArcTan[a*x]^(5/2), x]/(a^2*c) + (2*Unintegrable[ArcTan[a*x]^(7/2), x])/(7*a^3*c)} -{(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x, 2, (-2*ArcTan[a*x]^(7/2))/(7*a^3*c) + Unintegrable[ArcTan[a*x]^(5/2), x]/(a^2*c)} -{(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x, 1, (2*x*ArcTan[a*x]^(7/2))/(7*a*c) - (2*Unintegrable[ArcTan[a*x]^(7/2), x])/(7*a*c)} -{ArcTan[a*x]^(5/2)/(c + a^2*c*x^2), x, 1, (2*ArcTan[a*x]^(7/2))/(7*a*c)} -{ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)), x, 1, (((-2*I)/7)*ArcTan[a*x]^(7/2))/c + (I*Unintegrable[ArcTan[a*x]^(5/2)/(x*(I + a*x)), x])/c} -{ArcTan[a*x]^(5/2)/(x^2*(c + a^2*c*x^2)), x, 2, (-2*a*ArcTan[a*x]^(7/2))/(7*c) + Unintegrable[ArcTan[a*x]^(5/2)/x^2, x]/c} -{ArcTan[a*x]^(5/2)/(x^3*(c + a^2*c*x^2)), x, 2, (((2*I)/7)*a^2*ArcTan[a*x]^(7/2))/c + Unintegrable[ArcTan[a*x]^(5/2)/x^3, x]/c - (I*a^2*Unintegrable[ArcTan[a*x]^(5/2)/(x*(I + a*x)), x])/c} -{ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)), x, 3, (2*a^3*ArcTan[a*x]^(7/2))/(7*c) + Unintegrable[ArcTan[a*x]^(5/2)/x^4, x]/c - (a^2*Unintegrable[ArcTan[a*x]^(5/2)/x^2, x])/c} - - -{(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]} - -{(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x, 0, Unintegrable[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]} -{(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x, 8, (15*x*Sqrt[ArcTan[a*x]])/(32*a^2*c^2*(1 + a^2*x^2)) + (5*ArcTan[a*x]^(3/2))/(16*a^3*c^2) - (5*ArcTan[a*x]^(3/2))/(8*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^(5/2))/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(7/2)/(7*a^3*c^2) - (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^3*c^2)} -{(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x, 8, -((15*Sqrt[ArcTan[a*x]])/(64*a^2*c^2)) + (15*Sqrt[ArcTan[a*x]])/(32*a^2*c^2*(1 + a^2*x^2)) + (5*x*ArcTan[a*x]^(3/2))/(8*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(4*a^2*c^2) - ArcTan[a*x]^(5/2)/(2*a^2*c^2*(1 + a^2*x^2)) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^2*c^2)} -{ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^2, x, 8, -((15*x*Sqrt[ArcTan[a*x]])/(32*c^2*(1 + a^2*x^2))) - (5*ArcTan[a*x]^(3/2))/(16*a*c^2) + (5*ArcTan[a*x]^(3/2))/(8*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(5/2))/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(7/2)/(7*a*c^2) + (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a*c^2)} -{ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^2), x, 0, Unintegrable[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^2), x]} - - -{(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]} - -{(x^5*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 0, Unintegrable[(x^5*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]} -{(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 18, (45*x*Sqrt[ArcTan[a*x]])/(128*a^4*c^3*(1 + a^2*x^2)) + (45*ArcTan[a*x]^(3/2))/(256*a^5*c^3) + (5*x^4*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) - (15*ArcTan[a*x]^(3/2))/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(5/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(5/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(28*a^5*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^5*c^3) - (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^5*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a^5*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^5*c^3)} -{(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 16, -((135*Sqrt[ArcTan[a*x]])/(2048*a^4*c^3)) - (15*x^4*Sqrt[ArcTan[a*x]])/(256*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcTan[a*x]])/(256*a^4*c^3*(1 + a^2*x^2)) + (5*x^3*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*x]^(3/2))/(64*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^(5/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(5/2))/(4*c^3*(1 + a^2*x^2)^2) + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^4*c^3) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^4*c^3)} -{(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 8, ArcTan[a*x]^(7/2)/(28*a^3*c^3) - (5*ArcTan[a*x]^(3/2)*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) - (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^3*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^3*c^3) - (ArcTan[a*x]^(5/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)} -{(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x, 16, -((225*Sqrt[ArcTan[a*x]])/(2048*a^2*c^3)) + (15*Sqrt[ArcTan[a*x]])/(256*a^2*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcTan[a*x]])/(256*a^2*c^3*(1 + a^2*x^2)) + (5*x*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*x]^(3/2))/(64*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(32*a^2*c^3) - ArcTan[a*x]^(5/2)/(4*a^2*c^3*(1 + a^2*x^2)^2) - (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^2*c^3) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^2*c^3)} -{ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3, x, 18, -((45*x*Sqrt[ArcTan[a*x]])/(128*c^3*(1 + a^2*x^2))) - (75*ArcTan[a*x]^(3/2))/(256*a*c^3) + (5*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(5/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(5/2))/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(28*a*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a*c^3) + (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a*c^3)} -{ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^3), x, 0, Unintegrable[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(5/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]} - -{x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]} -{x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x, 2, (5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^2) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/(3*a^2*c) - (5*c*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a) - (5*c*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(12*a)} -{Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x, 1, (-5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/2 + (15*c*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/8 + (c*Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/2} -{(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/x, x]} - - -{x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]} - -{x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]} -{x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x, 3, (9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a^2) + ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(16*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(8*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/(5*a^2*c) - (9*c^2*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(64*a) - (c*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(32*a) - (3*c^2*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(16*a)} -{(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x, 2, (-15*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (5*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(24*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/4 + (45*c^2*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/32 + (5*c*Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x])/16 + (3*c^2*Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/8} -{((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x]} - - -{x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]} - -{x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x, 0, Unintegrable[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]} -{x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x, 4, (75*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(448*a^2) + (25*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(672*a^2) + ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(56*a^2) - (25*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(224*a) - (25*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(336*a) - (5*x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(84*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(5/2))/(7*a^2*c) - (75*c^3*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(896*a) - (25*c^2*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(1344*a) - (c*Unintegrable[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/(112*a) - (25*c^3*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(224*a)} -{(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x, 3, (-25*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(32*a) - (25*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(144*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/6 + (75*c^3*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/64 + (25*c^2*Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x])/96 + (c*Unintegrable[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/8 + (5*c^3*Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/16} -{((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/x, x, 0, Unintegrable[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]} - -{(x^3*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x, 4, (5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^4*c) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^2*c) - (5*Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a^3) + (25*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(12*a^3)} -{(x^2*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x, 2, -((5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*a^2*c) + (15*Unintegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(8*a^2) - Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x]/(2*a^2)} -{(x*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x, 1, (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(a^2*c) - (5*Unintegrable[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(2*a)} -{ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x, 0, Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x]} -{ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x, 0, Unintegrable[ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x]} -{ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]), x, 1, -((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(c*x)) + (5*a*Unintegrable[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2} -{ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]), x, 2, (-5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*c*x^2) + (15*a^2*Unintegrable[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/8 - (a^2*Unintegrable[ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2} -{ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]), x, 4, (-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x) + (5*a^3*Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Unintegrable[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/12} - - -{(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]} - -{(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x, 0, Unintegrable[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]} -{(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x, 6, (15*Sqrt[ArcTan[a*x]])/(4*a^2*c*Sqrt[c + a^2*c*x^2]) + (5*x*ArcTan[a*x]^(3/2))/(2*a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(5/2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^2*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2), x, 6, -((15*x*Sqrt[ArcTan[a*x]])/(4*c*Sqrt[c + a^2*c*x^2])) + (5*ArcTan[a*x]^(3/2))/(2*a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(c*Sqrt[c + a^2*c*x^2]) + (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a*c*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(3/2)), x, 0, Unintegrable[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(3/2)), x]} - - -{(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]} - -{(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x, 0, Unintegrable[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]} -{(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x, 17, (45*Sqrt[ArcTan[a*x]])/(16*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (5*x^3*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^(5/2))/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^(5/2))/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(144*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x, 16, -((5*x^3*Sqrt[ArcTan[a*x]])/(36*c*(c + a^2*c*x^2)^(3/2))) - (5*x*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (5*x^2*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^(3/2))/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x, 15, (5*Sqrt[ArcTan[a*x]])/(36*a^2*c*(c + a^2*c*x^2)^(3/2)) + (5*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (5*x*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(5/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(5/2), x, 17, -((45*x*Sqrt[ArcTan[a*x]])/(16*c^2*Sqrt[c + a^2*c*x^2])) + (5*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^(3/2))/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^(5/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) + (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(144*a*c^2*Sqrt[c + a^2*c*x^2])} -{ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(5/2)), x, 0, Unintegrable[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(5/2)), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-1/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(1/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]} - -{(x*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*Sqrt[ArcTan[a*x]]), x]} - - -{(x^m*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]} - -{(x*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^2/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*Sqrt[ArcTan[a*x]]), x]} - - -{(x^m*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]} - -{(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^3/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]} - -{x/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x, 1, (2*x*Sqrt[ArcTan[a*x]])/(a*c) - (2*Unintegrable[Sqrt[ArcTan[a*x]], x])/(a*c)} -{1/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x, 1, (2*Sqrt[ArcTan[a*x]])/(a*c)} -{1/(x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} - -{x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{x^2/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 5, Sqrt[ArcTan[a*x]]/(a^3*c^2) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^3*c^2)} -{x/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 5, (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^2*c^2)} -{1/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 5, Sqrt[ArcTan[a*x]]/(a*c^2) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} - -{x^5/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^5/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{x^4/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 7, (3*Sqrt[ArcTan[a*x]])/(4*a^5*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^5*c^3) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^5*c^3)} -{x^3/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 7, -((Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^4*c^3)) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(4*a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 5, Sqrt[ArcTan[a*x]]/(4*a^3*c^3) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^3*c^3)} -{x/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 7, (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^2*c^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(4*a^2*c^3)} -{1/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 7, (3*Sqrt[ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(1/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]} - -{(x*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]} -{Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]} -{Sqrt[c + a^2*c*x^2]/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*Sqrt[ArcTan[a*x]]), x]} - - -{(x^m*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]} - -{(x*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^(3/2)/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*Sqrt[ArcTan[a*x]]), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]} - -{(x*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]} -{(c + a^2*c*x^2)^(5/2)/(x*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]} - -{x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]} -{1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} - -{x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{x/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x, 4, (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x, 4, (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} - -{x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{x^3/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 8, (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 8, (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 8, (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 8, (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x, 0, Unintegrable[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-3/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(3/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]} - -{(x*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(c + a^2*c*x^2)/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(3/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]} - -{(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^2/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(c + a^2*c*x^2)^2/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(3/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]} - -{(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^3/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(c + a^2*c*x^2)^3/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(3/2)), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x, 1, (-2*x^m)/(a*c*Sqrt[ArcTan[a*x]]) + (2*m*Unintegrable[x^(-1 + m)/Sqrt[ArcTan[a*x]], x])/(a*c)} - -{x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x, 1, (-2*x)/(a*c*Sqrt[ArcTan[a*x]]) + (2*Unintegrable[1/Sqrt[ArcTan[a*x]], x])/(a*c)} -{1/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*Sqrt[ArcTan[a*x]])} -{1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Unintegrable[1/(x^2*Sqrt[ArcTan[a*x]]), x])/(a*c)} - - -{x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]} - -{x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 1, -((2*x^4)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) + (8*Unintegrable[x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a + 4*a*Unintegrable[x^5/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 6, -((2*x^3)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) + (6*Sqrt[ArcTan[a*x]])/(a^4*c^2) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) + 2*a*Unintegrable[x^4/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 6, -((2*x^2)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^3*c^2)} -{x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 7, -((2*x)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) + (4*Sqrt[ArcTan[a*x]])/(a^2*c^2) - (8*Sqrt[ArcTan[a*x]])/(a^2*c^2*(1 + a^2*x^2)) + (4*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(a^2*c^2*(1 + a^2*x^2)) + (2*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^2*c^2)} -{1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 6, -(2/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) - (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 6, -(2/(a*c^2*x*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])) - (6*Sqrt[ArcTan[a*x]])/c^2 - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 - (2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a} -{1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (4*Unintegrable[1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 8*a*Unintegrable[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (6*Unintegrable[1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Unintegrable[1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Unintegrable[1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Unintegrable[1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]} - -{x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 13, -((2*x^3)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 15, -((2*x^2)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])) + (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^3)} -{x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 13, -((2*x)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^2*c^3)} -{1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 8, -(2/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^3) - (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 8, -(2/(a*c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])) - (15*Sqrt[ArcTan[a*x]])/(2*c^3) - (5*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*c^3) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 - (2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a} -{1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Unintegrable[1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Unintegrable[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (6*Unintegrable[1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 14*a*Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c^3*x^4*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (8*Unintegrable[1/(x^5*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 16*a*Unintegrable[1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(3/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]} - -{(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]} -{Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(3/2), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(3/2), x]} -{Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(3/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]} - -{(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(3/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]} - -{(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(3/2), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(3/2), x]} -{(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(3/2)), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]} - -{x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]} -{1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x, 1, -((2*Sqrt[c + a^2*c*x^2])/(a*c*x*Sqrt[ArcTan[a*x]])) - (2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/a} -{1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]} - - -{x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]} - -{x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 1, (-2*x^3)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (6*Unintegrable[x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a + 4*a*Unintegrable[x^4/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 5, -((2*x^2)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + 2*a*Unintegrable[x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 5, -((2*x)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])) + (2*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 5, -(2/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])) - (2*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 5, -(2/(a*c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) - (2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a} -{1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 6*a*Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (6*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 8*a*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (8*Unintegrable[1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]} - -{x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 9, -((2*x^3)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[(3*Pi)/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 17, -((2*x^2)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 17, -((2*x)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 9, -(2/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[(3*Pi)/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 9, -(2/(a*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])) - (6*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a} -{1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (4*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Unintegrable[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (6*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x, 1, -2/(a*c*x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (8*Unintegrable[1/(x^5*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 14*a*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-5/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(5/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]} - -{(x*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(c + a^2*c*x^2)/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(5/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]} - -{(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^2/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(c + a^2*c*x^2)^2/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(5/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]} - -{(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^3/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(c + a^2*c*x^2)^3/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(5/2)), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x, 1, (-2*x^m)/(3*a*c*ArcTan[a*x]^(3/2)) + (2*m*Unintegrable[x^(-1 + m)/ArcTan[a*x]^(3/2), x])/(3*a*c)} - -{x/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x, 1, (-2*x)/(3*a*c*ArcTan[a*x]^(3/2)) + (2*Unintegrable[ArcTan[a*x]^(-3/2), x])/(3*a*c)} -{1/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x, 1, -2/(3*a*c*ArcTan[a*x]^(3/2))} -{1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x, 1, -2/(3*a*c*x*ArcTan[a*x]^(3/2)) - (2*Unintegrable[1/(x^2*ArcTan[a*x]^(3/2)), x])/(3*a*c)} - - -{x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]} - -{x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 8, -((2*x^3)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) - (4*x^2)/(a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (4*x^4)/(3*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) + (16/3)*Unintegrable[x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x] + (8/3)*a^2*Unintegrable[x^5/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 8, -((2*x^2)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) - (8*x)/(3*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*Sqrt[ArcTan[a*x]])/(3*a^3*c^2) - (32*Sqrt[ArcTan[a*x]])/(3*a^3*c^2*(1 + a^2*x^2)) + (16*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(3*a^3*c^2*(1 + a^2*x^2)) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^3*c^2)} -{x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 6, -((2*x)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) - (4*(1 - a^2*x^2))/(3*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^2*c^2)} -{1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 8, -(2/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) + (8*x)/(3*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (16*Sqrt[ArcTan[a*x]])/(3*a*c^2) + (32*Sqrt[ArcTan[a*x]])/(3*a*c^2*(1 + a^2*x^2)) - (16*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(3*a*c^2*(1 + a^2*x^2)) - (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a*c^2)} -{1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 8, -(2/(3*a*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) + 4/(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Unintegrable[1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (16/3)*Unintegrable[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 8, -(2/(3*a*c^2*x^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))) + 8/(3*a^2*c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 16/(3*c^2*x*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*a*Sqrt[ArcTan[a*x]])/c^2 + (8*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Unintegrable[1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a^2 + (56/3)*Unintegrable[1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c^2*x^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(a^2*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 20/(3*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*Unintegrable[1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a^2 + (112*Unintegrable[1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + (80*a^2*Unintegrable[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3} -{1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c^2*x^4*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c^2*x^5*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 8/(c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (80*Unintegrable[1/(x^6*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (184*Unintegrable[1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + 40*a^2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]} - - -{x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]} - -{x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 24, -((2*x^3)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) - (4*x^2)/(a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*x^4)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^4*c^3) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^4*c^3)} -{x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 27, -((2*x^2)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) - (8*x)/(3*a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (8*x^3)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c^3)} -{x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 24, -((2*x)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) - 4/(3*a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*x^2)/(c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c^3) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^2*c^3)} -{1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 14, -(2/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) + (16*x)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a*c^3) - (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a*c^3)} -{1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 10, -(2/(3*a*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) + 20/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (5*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*c^3) + (20*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*c^3) + (8*Unintegrable[1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 8*Unintegrable[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 10, -(2/(3*a*c^3*x^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))) + 8/(3*a^2*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 8/(c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (30*a*Sqrt[ArcTan[a*x]])/c^3 + (5*a*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/c^3 + (20*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 + (8*Unintegrable[1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (80/3)*Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c^3*x^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 4/(a^2*c^3*x^4*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 28/(3*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (16*Unintegrable[1/(x^5*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (152*Unintegrable[1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3 + 56*a^2*Unintegrable[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c^3*x^4*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c^3*x^5*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 32/(3*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (80*Unintegrable[1/(x^6*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 80*Unintegrable[1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x] + (224*a^2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(5/2) with d=a^2 c*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]} - -{(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]} -{Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(5/2), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(5/2), x]} -{Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(5/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]} - -{(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(5/2)), x]} - - -{(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]} - -{(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(5/2), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(5/2), x]} -{(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(5/2)), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]} - -{x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]} -{1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]} -{1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x, 1, -((2*Sqrt[c + a^2*c*x^2])/(3*a*c*x*ArcTan[a*x]^(3/2))) - (2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x])/(3*a)} -{1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]} - - -{x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]} - -{x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 7, -((2*x^3)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) - (4*x^2)/(a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (8*x^4)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (44/3)*Unintegrable[x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + 8*a^2*Unintegrable[x^5/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 7, -((2*x^2)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) - (8*x)/(3*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*x^3)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c*Sqrt[c + a^2*c*x^2]) + 4*Unintegrable[x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + (8/3)*a^2*Unintegrable[x^4/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 6, -((2*x)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) - 4/(3*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 6, -(2/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) + (4*x)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a*c*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 7, -(2/(3*a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) + 8/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*c*Sqrt[c + a^2*c*x^2]) + (8*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 4*Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 7, -(2/(3*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))) + 8/(3*a^2*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 4/(c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) + (8*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (44/3)*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 16/(3*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (16*Unintegrable[1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (92*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 16*a^2*Unintegrable[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 20/(3*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (80*Unintegrable[1/(x^6*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 52*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + (80*a^2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3} - - -{x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 0, Unintegrable[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]} - -{x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 18, -((2*x^3)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) - (4*x^2)/(a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2])} -{x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 27, -((2*x^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) - (8*x)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (4*x^3)/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2])} -{x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 27, -((2*x)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) - 4/(3*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (8*x^2)/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 18, -(2/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) + (4*x)/(c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2])} -{1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 11, -(2/(3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) + 16/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (4*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (8*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (20/3)*Unintegrable[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 11, -(2/(3*a*c*x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))) + 8/(3*a^2*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 20/(3*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (20*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (20*a*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(3*c^2*Sqrt[c + a^2*c*x^2]) + (8*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (68/3)*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c*x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 8/(c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (16*Unintegrable[1/(x^5*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + 44*Unintegrable[1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x] + 40*a^2*Unintegrable[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} -{1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x, 3, -2/(3*a*c*x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c*x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 28/(3*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (80*Unintegrable[1/(x^6*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (212*Unintegrable[1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/3 + 56*a^2*Unintegrable[1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p symbolic*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p symbolic*) - - -{x^1*ArcTan[a*x]^n/(c + a^2*c*x^2), x, 1, (x*ArcTan[a*x]^(1 + n))/(a*c*(1 + n)) - Unintegrable[ArcTan[a*x]^(1 + n), x]/(a*c*(1 + n))} -{x^0*ArcTan[a*x]^n/(c + a^2*c*x^2), x, 1, ArcTan[a*x]^(1 + n)/(a*c*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p and q symbolic*) - - -{(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x, 0, Unintegrable[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTan[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcTan[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2), x, 5, (b*(3*c^2*d - 2*e)*x)/(12*c^5) - (b*(3*c^2*d - 2*e)*x^3)/(36*c^3) - (b*e*x^5)/(30*c) - (b*(3*c^2*d - 2*e)*ArcTan[c*x])/(12*c^6) + (1/4)*d*x^4*(a + b*ArcTan[c*x]) + (1/6)*e*x^6*(a + b*ArcTan[c*x])} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2), x, 4, -((b*(5*c^2*d - 3*e)*x^2)/(30*c^3)) - (b*e*x^4)/(20*c) + (1/3)*d*x^3*(a + b*ArcTan[c*x]) + (1/5)*e*x^5*(a + b*ArcTan[c*x]) + (b*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(30*c^5)} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2), x, 4, -((b*(2*c^2*d - e)*x)/(4*c^3)) - (b*e*x^3)/(12*c) - (b*(c^2*d - e)^2*ArcTan[c*x])/(4*c^4*e) + ((d + e*x^2)^2*(a + b*ArcTan[c*x]))/(4*e)} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2), x, 5, -((b*e*x^2)/(6*c)) + d*x*(a + b*ArcTan[c*x]) + (1/3)*e*x^3*(a + b*ArcTan[c*x]) - (b*(3*c^2*d - e)*Log[1 + c^2*x^2])/(6*c^3)} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^1, x, 8, -((b*e*x)/(2*c)) + (b*e*ArcTan[c*x])/(2*c^2) + (1/2)*e*x^2*(a + b*ArcTan[c*x]) + a*d*Log[x] + (1/2)*I*b*d*PolyLog[2, (-I)*c*x] - (1/2)*I*b*d*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^2, x, 4, -((d*(a + b*ArcTan[c*x]))/x) + e*x*(a + b*ArcTan[c*x]) + b*c*d*Log[x] - (b*(c^2*d + e)*Log[1 + c^2*x^2])/(2*c)} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^3, x, 8, -((b*c*d)/(2*x)) - (1/2)*b*c^2*d*ArcTan[c*x] - (d*(a + b*ArcTan[c*x]))/(2*x^2) + a*e*Log[x] + (1/2)*I*b*e*PolyLog[2, (-I)*c*x] - (1/2)*I*b*e*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^4, x, 5, -((b*c*d)/(6*x^2)) - (d*(a + b*ArcTan[c*x]))/(3*x^3) - (e*(a + b*ArcTan[c*x]))/x - (1/3)*b*c*(c^2*d - 3*e)*Log[x] + (1/6)*b*c*(c^2*d - 3*e)*Log[1 + c^2*x^2]} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^5, x, 5, -((b*c*d)/(12*x^3)) + (b*c*(c^2*d - 2*e))/(4*x) + (1/4)*b*c^2*(c^2*d - 2*e)*ArcTan[c*x] - (d*(a + b*ArcTan[c*x]))/(4*x^4) - (e*(a + b*ArcTan[c*x]))/(2*x^2)} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^6, x, 5, -((b*c*d)/(20*x^4)) + (b*c*(3*c^2*d - 5*e))/(30*x^2) - (d*(a + b*ArcTan[c*x]))/(5*x^5) - (e*(a + b*ArcTan[c*x]))/(3*x^3) + (1/15)*b*c^3*(3*c^2*d - 5*e)*Log[x] - (1/30)*b*c^3*(3*c^2*d - 5*e)*Log[1 + c^2*x^2]} -{(a + b*ArcTan[c*x])*(d + e*x^2)/x^7, x, 6, -((b*c*d)/(30*x^5)) + (b*c*(2*c^2*d - 3*e))/(36*x^3) - (b*c^3*(2*c^2*d - 3*e))/(12*x) - (1/12)*b*c^4*(2*c^2*d - 3*e)*ArcTan[c*x] - (d*(a + b*ArcTan[c*x]))/(6*x^6) - (e*(a + b*ArcTan[c*x]))/(4*x^4)} - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2)^2, x, 4, (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x)/(24*c^7) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x^3)/(72*c^5) - (b*(8*c^2*d - 3*e)*e*x^5)/(120*c^3) - (b*e^2*x^7)/(56*c) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*ArcTan[c*x])/(24*c^8) + (1/4)*d^2*x^4*(a + b*ArcTan[c*x]) + (1/3)*d*e*x^6*(a + b*ArcTan[c*x]) + (1/8)*e^2*x^8*(a + b*ArcTan[c*x])} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2)^2, x, 5, -((b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*x^2)/(210*c^5)) - (b*(14*c^2*d - 5*e)*e*x^4)/(140*c^3) - (b*e^2*x^6)/(42*c) + (1/3)*d^2*x^3*(a + b*ArcTan[c*x]) + (2/5)*d*e*x^5*(a + b*ArcTan[c*x]) + (1/7)*e^2*x^7*(a + b*ArcTan[c*x]) + (b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*Log[1 + c^2*x^2])/(210*c^7)} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2)^2, x, 4, -((b*(3*c^4*d^2 - 3*c^2*d*e + e^2)*x)/(6*c^5)) - (b*(3*c^2*d - e)*e*x^3)/(18*c^3) - (b*e^2*x^5)/(30*c) - (b*(c^2*d - e)^3*ArcTan[c*x])/(6*c^6*e) + ((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*e)} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2)^2, x, 5, -((b*(10*c^2*d - 3*e)*e*x^2)/(30*c^3)) - (b*e^2*x^4)/(20*c) + d^2*x*(a + b*ArcTan[c*x]) + (2/3)*d*e*x^3*(a + b*ArcTan[c*x]) + (1/5)*e^2*x^5*(a + b*ArcTan[c*x]) - (b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Log[1 + c^2*x^2])/(30*c^5)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^1, x, 12, -((b*d*e*x)/c) + (b*e^2*x)/(4*c^3) - (b*e^2*x^3)/(12*c) + (b*d*e*ArcTan[c*x])/c^2 - (b*e^2*ArcTan[c*x])/(4*c^4) + d*e*x^2*(a + b*ArcTan[c*x]) + (1/4)*e^2*x^4*(a + b*ArcTan[c*x]) + a*d^2*Log[x] + (1/2)*I*b*d^2*PolyLog[2, (-I)*c*x] - (1/2)*I*b*d^2*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^2, x, 4, -((b*e^2*x^2)/(6*c)) - (d^2*(a + b*ArcTan[c*x]))/x + 2*d*e*x*(a + b*ArcTan[c*x]) + (1/3)*e^2*x^3*(a + b*ArcTan[c*x]) + b*c*d^2*Log[x] - (b*(3*c^4*d^2 + 6*c^2*d*e - e^2)*Log[1 + c^2*x^2])/(6*c^3)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^3, x, 11, -((b*c*d^2)/(2*x)) - (b*e^2*x)/(2*c) - (1/2)*b*c^2*d^2*ArcTan[c*x] + (b*e^2*ArcTan[c*x])/(2*c^2) - (d^2*(a + b*ArcTan[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcTan[c*x]) + 2*a*d*e*Log[x] + I*b*d*e*PolyLog[2, (-I)*c*x] - I*b*d*e*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^4, x, 5, -((b*c*d^2)/(6*x^2)) - (d^2*(a + b*ArcTan[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcTan[c*x]))/x + e^2*x*(a + b*ArcTan[c*x]) - (1/3)*b*c*d*(c^2*d - 6*e)*Log[x] + (b*(c^4*d^2 - 6*c^2*d*e - 3*e^2)*Log[1 + c^2*x^2])/(6*c)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^5, x, 12, -((b*c*d^2)/(12*x^3)) + (b*c^3*d^2)/(4*x) - (b*c*d*e)/x + (1/4)*b*c^4*d^2*ArcTan[c*x] - b*c^2*d*e*ArcTan[c*x] - (d^2*(a + b*ArcTan[c*x]))/(4*x^4) - (d*e*(a + b*ArcTan[c*x]))/x^2 + a*e^2*Log[x] + (1/2)*I*b*e^2*PolyLog[2, (-I)*c*x] - (1/2)*I*b*e^2*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^6, x, 5, -((b*c*d^2)/(20*x^4)) + (b*c*d*(3*c^2*d - 10*e))/(30*x^2) - (d^2*(a + b*ArcTan[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcTan[c*x]))/(3*x^3) - (e^2*(a + b*ArcTan[c*x]))/x + (1/15)*b*c*(3*c^4*d^2 - 10*c^2*d*e + 15*e^2)*Log[x] - (1/30)*b*c*(3*c^4*d^2 - 10*c^2*d*e + 15*e^2)*Log[1 + c^2*x^2]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^7, x, 5, -((b*c*d^2)/(30*x^5)) + (b*c*d*(c^2*d - 3*e))/(18*x^3) - (b*c*(c^4*d^2 - 3*c^2*d*e + 3*e^2))/(6*x) - (b*(c^2*d - e)^3*ArcTan[c*x])/(6*d) - ((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*d*x^6)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^2/x^8, x, 5, -((b*c*d^2)/(42*x^6)) + (b*c*d*(5*c^2*d - 14*e))/(140*x^4) - (b*c*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2))/(210*x^2) - (d^2*(a + b*ArcTan[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcTan[c*x]))/(5*x^5) - (e^2*(a + b*ArcTan[c*x]))/(3*x^3) - (1/105)*b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[x] + (1/210)*b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[1 + c^2*x^2]} - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2)^3, x, -8, (b*(10*c^6*d^3 - 20*c^4*d^2*e + 15*c^2*d*e^2 - 4*e^3)*x)/(40*c^9) - (b*(10*c^6*d^3 - 20*c^4*d^2*e + 15*c^2*d*e^2 - 4*e^3)*x^3)/(120*c^7) - (b*e*(20*c^4*d^2 - 15*c^2*d*e + 4*e^2)*x^5)/(200*c^5) - (b*(15*c^2*d - 4*e)*e^2*x^7)/(280*c^3) - (b*e^3*x^9)/(90*c) + (b*(c^2*d - e)^4*(c^2*d + 4*e)*ArcTan[c*x])/(40*c^10*e^2) - (d*(d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcTan[c*x]))/(10*e^2)} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2)^3, x, 5, -((b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*x^2)/(630*c^7)) - (b*e*(189*c^4*d^2 - 135*c^2*d*e + 35*e^2)*x^4)/(1260*c^5) - (b*(27*c^2*d - 7*e)*e^2*x^6)/(378*c^3) - (b*e^3*x^8)/(72*c) + (1/3)*d^3*x^3*(a + b*ArcTan[c*x]) + (3/5)*d^2*e*x^5*(a + b*ArcTan[c*x]) + (3/7)*d*e^2*x^7*(a + b*ArcTan[c*x]) + (1/9)*e^3*x^9*(a + b*ArcTan[c*x]) + (b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2])/(630*c^9)} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2)^3, x, 4, -((b*(2*c^2*d - e)*(2*c^4*d^2 - 2*c^2*d*e + e^2)*x)/(8*c^7)) - (b*e*(6*c^4*d^2 - 4*c^2*d*e + e^2)*x^3)/(24*c^5) - (b*(4*c^2*d - e)*e^2*x^5)/(40*c^3) - (b*e^3*x^7)/(56*c) - (b*(c^2*d - e)^4*ArcTan[c*x])/(8*c^8*e) + ((d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*e)} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2)^3, x, 4, -((b*e*(35*c^4*d^2 - 21*c^2*d*e + 5*e^2)*x^2)/(70*c^5)) - (b*(21*c^2*d - 5*e)*e^2*x^4)/(140*c^3) - (b*e^3*x^6)/(42*c) + d^3*x*(a + b*ArcTan[c*x]) + d^2*e*x^3*(a + b*ArcTan[c*x]) + (3/5)*d*e^2*x^5*(a + b*ArcTan[c*x]) + (1/7)*e^3*x^7*(a + b*ArcTan[c*x]) - (b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Log[1 + c^2*x^2])/(70*c^7)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^1, x, 16, -((3*b*d^2*e*x)/(2*c)) + (3*b*d*e^2*x)/(4*c^3) - (b*e^3*x)/(6*c^5) - (b*d*e^2*x^3)/(4*c) + (b*e^3*x^3)/(18*c^3) - (b*e^3*x^5)/(30*c) + (3*b*d^2*e*ArcTan[c*x])/(2*c^2) - (3*b*d*e^2*ArcTan[c*x])/(4*c^4) + (b*e^3*ArcTan[c*x])/(6*c^6) + (3/2)*d^2*e*x^2*(a + b*ArcTan[c*x]) + (3/4)*d*e^2*x^4*(a + b*ArcTan[c*x]) + (1/6)*e^3*x^6*(a + b*ArcTan[c*x]) + a*d^3*Log[x] + (1/2)*I*b*d^3*PolyLog[2, (-I)*c*x] - (1/2)*I*b*d^3*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^2, x, 4, -((b*(5*c^2*d - e)*e^2*x^2)/(10*c^3)) - (b*e^3*x^4)/(20*c) - (d^3*(a + b*ArcTan[c*x]))/x + 3*d^2*e*x*(a + b*ArcTan[c*x]) + d*e^2*x^3*(a + b*ArcTan[c*x]) + (1/5)*e^3*x^5*(a + b*ArcTan[c*x]) + b*c*d^3*Log[x] - (b*(5*c^6*d^3 + 15*c^4*d^2*e - 5*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/(10*c^5)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^3, x, 15, -((b*c*d^3)/(2*x)) - (3*b*d*e^2*x)/(2*c) + (b*e^3*x)/(4*c^3) - (b*e^3*x^3)/(12*c) - (1/2)*b*c^2*d^3*ArcTan[c*x] + (3*b*d*e^2*ArcTan[c*x])/(2*c^2) - (b*e^3*ArcTan[c*x])/(4*c^4) - (d^3*(a + b*ArcTan[c*x]))/(2*x^2) + (3/2)*d*e^2*x^2*(a + b*ArcTan[c*x]) + (1/4)*e^3*x^4*(a + b*ArcTan[c*x]) + 3*a*d^2*e*Log[x] + (3/2)*I*b*d^2*e*PolyLog[2, (-I)*c*x] - (3/2)*I*b*d^2*e*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^4, x, 5, -((b*c*d^3)/(6*x^2)) - (b*e^3*x^2)/(6*c) - (d^3*(a + b*ArcTan[c*x]))/(3*x^3) - (3*d^2*e*(a + b*ArcTan[c*x]))/x + 3*d*e^2*x*(a + b*ArcTan[c*x]) + (1/3)*e^3*x^3*(a + b*ArcTan[c*x]) - (1/3)*b*c*d^2*(c^2*d - 9*e)*Log[x] + (b*(c^2*d + e)*(c^4*d^2 - 10*c^2*d*e + e^2)*Log[1 + c^2*x^2])/(6*c^3)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^5, x, 15, -((b*c*d^3)/(12*x^3)) + (b*c^3*d^3)/(4*x) - (3*b*c*d^2*e)/(2*x) - (b*e^3*x)/(2*c) + (1/4)*b*c^4*d^3*ArcTan[c*x] - (3/2)*b*c^2*d^2*e*ArcTan[c*x] + (b*e^3*ArcTan[c*x])/(2*c^2) - (d^3*(a + b*ArcTan[c*x]))/(4*x^4) - (3*d^2*e*(a + b*ArcTan[c*x]))/(2*x^2) + (1/2)*e^3*x^2*(a + b*ArcTan[c*x]) + 3*a*d*e^2*Log[x] + (3/2)*I*b*d*e^2*PolyLog[2, (-I)*c*x] - (3/2)*I*b*d*e^2*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^6, x, 5, -((b*c*d^3)/(20*x^4)) + (b*c*d^2*(c^2*d - 5*e))/(10*x^2) - (d^3*(a + b*ArcTan[c*x]))/(5*x^5) - (d^2*e*(a + b*ArcTan[c*x]))/x^3 - (3*d*e^2*(a + b*ArcTan[c*x]))/x + e^3*x*(a + b*ArcTan[c*x]) + (1/5)*b*c*d*(c^4*d^2 - 5*c^2*d*e + 15*e^2)*Log[x] - (b*(c^6*d^3 - 5*c^4*d^2*e + 15*c^2*d*e^2 + 5*e^3)*Log[1 + c^2*x^2])/(10*c)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^7, x, 17, -((b*c*d^3)/(30*x^5)) + (b*c^3*d^3)/(18*x^3) - (b*c*d^2*e)/(4*x^3) - (b*c^5*d^3)/(6*x) + (3*b*c^3*d^2*e)/(4*x) - (3*b*c*d*e^2)/(2*x) - (1/6)*b*c^6*d^3*ArcTan[c*x] + (3/4)*b*c^4*d^2*e*ArcTan[c*x] - (3/2)*b*c^2*d*e^2*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (3*d^2*e*(a + b*ArcTan[c*x]))/(4*x^4) - (3*d*e^2*(a + b*ArcTan[c*x]))/(2*x^2) + a*e^3*Log[x] + (1/2)*I*b*e^3*PolyLog[2, (-I)*c*x] - (1/2)*I*b*e^3*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^8, x, 5, -((b*c*d^3)/(42*x^6)) + (b*c*d^2*(5*c^2*d - 21*e))/(140*x^4) - (b*c*d*(5*c^4*d^2 - 21*c^2*d*e + 35*e^2))/(70*x^2) - (d^3*(a + b*ArcTan[c*x]))/(7*x^7) - (3*d^2*e*(a + b*ArcTan[c*x]))/(5*x^5) - (d*e^2*(a + b*ArcTan[c*x]))/x^3 - (e^3*(a + b*ArcTan[c*x]))/x - (1/35)*b*c*(5*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[x] + (1/70)*b*c*(5*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^3/x^9, x, 5, -((b*c*d^3)/(56*x^7)) + (b*c*d^2*(c^2*d - 4*e))/(40*x^5) - (b*c*d*(c^4*d^2 - 4*c^2*d*e + 6*e^2))/(24*x^3) + (b*c*(c^2*d - 2*e)*(c^4*d^2 - 2*c^2*d*e + 2*e^2))/(8*x) + (b*(c^2*d - e)^4*ArcTan[c*x])/(8*d) - ((d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*d*x^8)} - - -{ArcTan[a*x]*(c + d*x^2)^4, x, 4, -((d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/(630*a^7)) - (d^2*(378*a^4*c^2 - 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) - ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) - (d^4*x^8)/(72*a) + c^4*x*ArcTan[a*x] + (4/3)*c^3*d*x^3*ArcTan[a*x] + (6/5)*c^2*d^2*x^5*ArcTan[a*x] + (4/7)*c*d^3*x^7*ArcTan[a*x] + (1/9)*d^4*x^9*ArcTan[a*x] - ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(630*a^9)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2), x, 14, -((b*x)/(2*c*e)) + (b*ArcTan[c*x])/(2*c^2*e) + (x^2*(a + b*ArcTan[c*x]))/(2*e) + (d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (d*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (I*b*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^2) + (I*b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) + (I*b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2), x, 10, -(((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e) + (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e)} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)), x, 15, (a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) + (I*b*PolyLog[2, (-I)*c*x])/(2*d) - (I*b*PolyLog[2, I*c*x])/(2*d) - (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d)} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)), x, 19, -((b*c)/(2*d*x)) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) - (a*e*Log[x])/d^2 - (e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (I*b*e*PolyLog[2, (-I)*c*x])/(2*d^2) + (I*b*e*PolyLog[2, I*c*x])/(2*d^2) + (I*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^2) - (I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) - (I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)} - -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2), x, 23, (a*x)/e + (b*x*ArcTan[c*x])/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) - (I*b*Sqrt[-d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*e^(3/2)) + (I*b*Sqrt[-d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*e^(3/2)) - (I*b*Sqrt[-d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*e^(3/2)) + (I*b*Sqrt[-d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*e^(3/2)) - (b*Log[1 + c^2*x^2])/(2*c*e) + (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*e^(3/2)) + (I*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*e^(3/2))} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2), x, 19, (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) - (I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) + (I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) - (I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) + (I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)), x, 25, -((a + b*ArcTan[c*x])/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (b*c*Log[x])/d - (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(3/2)) + (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(3/2)) - (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(3/2)) + (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(3/2)) - (b*c*Log[1 + c^2*x^2])/(2*d) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(3/2)) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(3/2))} - - -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2)^2, x, 16, -((b*c^2*d*ArcTan[c*x])/(2*(c^2*d - e)*e^2)) + (d*(a + b*ArcTan[c*x]))/(2*e^2*(d + e*x^2)) + (b*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*(c^2*d - e)*e^(3/2)) - ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^2) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2)^2, x, 4, (b*c^2*ArcTan[c*x])/(2*(c^2*d - e)*e) - (a + b*ArcTan[c*x])/(2*e*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*(c^2*d - e)*Sqrt[e])} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)^2), x, 19, -((b*c^2*ArcTan[c*x])/(2*d*(c^2*d - e))) + (a + b*ArcTan[c*x])/(2*d*(d + e*x^2)) + (b*c*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c^2*d - e)) + (a*Log[x])/d^2 + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (I*b*PolyLog[2, (-I)*c*x])/(2*d^2) - (I*b*PolyLog[2, I*c*x])/(2*d^2) - (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^2) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^2), x, 22, -((b*c)/(2*d^2*x)) - (b*c^2*ArcTan[c*x])/(2*d^2) + (b*c^2*e*ArcTan[c*x])/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])/(2*d^2*x^2) - (e*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x^2)) - (b*c*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*(c^2*d - e)) - (2*a*e*Log[x])/d^3 - (2*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (I*b*e*PolyLog[2, (-I)*c*x])/d^3 + (I*b*e*PolyLog[2, I*c*x])/d^3 + (I*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 - (I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) - (I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^3)} - -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2)^2, x, 45, -((x*(a + b*ArcTan[c*x]))/(2*e*(d + e*x^2))) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(3/2)) - ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) - (I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) - (I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) - (I*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (I*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (I*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - (I*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (b*c*Log[1 + c^2*x^2])/(4*(c^2*d - e)*e) - (b*c*Log[d + e*x^2])/(4*(c^2*d - e)*e) + (I*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d]*e^(3/2)) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*Sqrt[d]*e^(3/2))} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2)^2, x, 24, (x*(a + b*ArcTan[c*x]))/(2*d*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) + (I*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (I*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (I*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + (I*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(3/2)*Sqrt[e])} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^2), x, 50, -((a + b*ArcTan[c*x])/(d^2*x)) - (e*x*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) - (Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + (b*c*Log[x])/d^2 + (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(5/2)) - (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(5/2)) + (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(5/2)) - (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(5/2)) - (I*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(5/2)) + (I*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(5/2)) + (I*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(5/2)) - (I*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[-c^2]*d^(5/2)) - (b*c*Log[1 + c^2*x^2])/(2*d^2) + (b*c*e*Log[1 + c^2*x^2])/(4*d^2*(c^2*d - e)) - (b*c*e*Log[d + e*x^2])/(4*d^2*(c^2*d - e)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(5/2)) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(5/2)) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(5/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(5/2)) - (I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(5/2)) + (I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(5/2)) - (I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(5/2)) + (I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(8*Sqrt[-c^2]*d^(5/2))} - - -{x^5*(a + b*ArcTan[c*x])/(d + e*x^2)^3, x, 21, -((b*c*d*x)/(8*(c^2*d - e)*e^2*(d + e*x^2))) + (b*c^4*d^2*ArcTan[c*x])/(4*(c^2*d - e)^2*e^3) - (b*c^2*d*ArcTan[c*x])/((c^2*d - e)*e^3) - (d^2*(a + b*ArcTan[c*x]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcTan[c*x]))/(e^3*(d + e*x^2)) + (b*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((c^2*d - e)*e^(5/2)) - (b*c*Sqrt[d]*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*(c^2*d - e)^2*e^(5/2)) - ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^3 + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^3) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^3) + (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*e^3) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^3) - (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^3)} -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2)^3, x, 6, (b*c*x)/(8*(c^2*d - e)*e*(d + e*x^2)) - (b*ArcTan[c*x])/(4*d*(c^2*d - e)^2) + (x^4*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*(c^2*d - 3*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*(c^2*d - e)^2*e^(3/2))} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2)^3, x, 5, -((b*c*x)/(8*d*(c^2*d - e)*(d + e*x^2))) + (b*c^4*ArcTan[c*x])/(4*(c^2*d - e)^2*e) - (a + b*ArcTan[c*x])/(4*e*(d + e*x^2)^2) - (b*c*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*(c^2*d - e)^2*Sqrt[e])} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)^3), x, 24, (b*c*e*x)/(8*d^2*(c^2*d - e)*(d + e*x^2)) - (b*c^4*ArcTan[c*x])/(4*d*(c^2*d - e)^2) - (b*c^2*ArcTan[c*x])/(2*d^2*(c^2*d - e)) + (a + b*ArcTan[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcTan[c*x])/(2*d^2*(d + e*x^2)) + (b*c*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*(c^2*d - e)) + (b*c*(3*c^2*d - e)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*(c^2*d - e)^2) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) + (I*b*PolyLog[2, (-I)*c*x])/(2*d^3) - (I*b*PolyLog[2, I*c*x])/(2*d^3) - (I*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^3) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^3) + (I*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^3)} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^3), x, 27, -((b*c)/(2*d^3*x)) - (b*c*e^2*x)/(8*d^3*(c^2*d - e)*(d + e*x^2)) - (b*c^2*ArcTan[c*x])/(2*d^3) + (b*c^4*e*ArcTan[c*x])/(4*d^2*(c^2*d - e)^2) + (b*c^2*e*ArcTan[c*x])/(d^3*(c^2*d - e)) - (a + b*ArcTan[c*x])/(2*d^3*x^2) - (e*(a + b*ArcTan[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcTan[c*x]))/(d^3*(d + e*x^2)) - (b*c*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(7/2)*(c^2*d - e)) - (b*c*(3*c^2*d - e)*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)*(c^2*d - e)^2) - (3*a*e*Log[x])/d^4 - (3*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^4 + (3*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^4) + (3*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^4) - (3*I*b*e*PolyLog[2, (-I)*c*x])/(2*d^4) + (3*I*b*e*PolyLog[2, I*c*x])/(2*d^4) + (3*I*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^4) - (3*I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^4) - (3*I*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^4)} - -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2)^3, x, 49, (b*c)/(8*(c^2*d - e)*e*(d + e*x^2)) - (x*(a + b*ArcTan[c*x]))/(4*e*(d + e*x^2)^2) + (x*(a + b*ArcTan[c*x]))/(8*d*e*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + (I*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) - (I*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) - (I*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) + (I*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) + (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d*(c^2*d - e)^2*e) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)*e) - (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d*(c^2*d - e)^2*e) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)*e) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) + (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2)) - (I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(3/2)*e^(3/2))} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2)^3, x, 23, -((b*c)/(8*d*(c^2*d - e)*(d + e*x^2))) + (x*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*ArcTan[c*x]))/(8*d^2*(d + e*x^2)) + (3*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (3*I*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (3*I*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (3*I*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (3*I*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d^2*(c^2*d - e)^2) + (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d^2*(c^2*d - e)^2) + (3*I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (3*I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (3*I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (3*I*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(5/2)*Sqrt[e])} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^3), x, 73, (b*c*e)/(8*d^2*(c^2*d - e)*(d + e*x^2)) - (a + b*ArcTan[c*x])/(d^3*x) - (e*x*(a + b*ArcTan[c*x]))/(4*d^2*(d + e*x^2)^2) - (7*e*x*(a + b*ArcTan[c*x]))/(8*d^3*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(7/2) - (7*Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + (b*c*Log[x])/d^3 - (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(7/2)) + (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(7/2)) - (I*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(4*(-d)^(7/2)) + (I*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(7/2)) - (7*I*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(7/2)) + (7*I*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(7/2)) + (7*I*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(7/2)) - (7*I*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(32*Sqrt[-c^2]*d^(7/2)) - (b*c*Log[1 + c^2*x^2])/(2*d^3) + (b*c*(5*c^2*d - 3*e)*e*Log[1 + c^2*x^2])/(16*d^3*(c^2*d - e)^2) + (b*c*e*Log[1 + c^2*x^2])/(4*d^3*(c^2*d - e)) - (b*c*(5*c^2*d - 3*e)*e*Log[d + e*x^2])/(16*d^3*(c^2*d - e)^2) - (b*c*e*Log[d + e*x^2])/(4*d^3*(c^2*d - e)) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(7/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(7/2)) - (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(4*(-d)^(7/2)) + (I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*(-d)^(7/2)) - (7*I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(7/2)) + (7*I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(7/2)) - (7*I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(7/2)) + (7*I*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(32*Sqrt[-c^2]*d^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcTan[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2), x, If[$VersionNumber>=8, 9, 10], If[$VersionNumber>=8, -((b*(c^2*d - 12*e)*x*Sqrt[d + e*x^2])/(120*c^3*e)) - (b*x*(d + e*x^2)^(3/2))/(20*c*e) - (d*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + (b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(15*c^5*e^2) + (b*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(120*c^5*e^(3/2)), -((b*(c^2*d - 12*e)*x*Sqrt[d + e*x^2])/(120*c^3*e)) - (b*x*(d + e*x^2)^(3/2))/(20*c*e) - (d*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + (b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(15*c^5*e^2) + (b*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(120*c^5*e^(3/2))]} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2), x, 5, (a*d*x*Sqrt[d + e*x^2])/(8*e) + (1/4)*a*x^3*Sqrt[d + e*x^2] - (a*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*e^(3/2)) + b*Unintegrable[x^2*Sqrt[d + e*x^2]*ArcTan[c*x], x]} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2), x, 7, -((b*x*Sqrt[d + e*x^2])/(6*c)) + ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e) - (b*(c^2*d - e)^(3/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*c^3*e) - (b*(3*c^2*d - 2*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(6*c^3*Sqrt[e])} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]), x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^1, x, 5, a*Sqrt[d + e*x^2] - a*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[(Sqrt[d + e*x^2]*ArcTan[c*x])/x, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^2, x, 4, -((a*Sqrt[d + e*x^2])/x) + a*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[(Sqrt[d + e*x^2]*ArcTan[c*x])/x^2, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^3, x, 5, -((a*Sqrt[d + e*x^2])/(2*x^2)) - (a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*Sqrt[d]) + b*Unintegrable[(Sqrt[d + e*x^2]*ArcTan[c*x])/x^3, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^4, x, 9, -((b*c*Sqrt[d + e*x^2])/(6*x^2)) - ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*d*x^3) + (b*c*(2*c^2*d - 3*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*Sqrt[d]) - (b*(c^2*d - e)^(3/2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d)} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^5, x, 6, -((a*Sqrt[d + e*x^2])/(4*x^4)) - (a*e*Sqrt[d + e*x^2])/(8*d*x^2) + (a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*d^(3/2)) + b*Unintegrable[(Sqrt[d + e*x^2]*ArcTan[c*x])/x^5, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2)/x^6, x, If[$VersionNumber>=8, 10, 21], If[$VersionNumber>=8, (b*c*(12*c^2*d - e)*Sqrt[d + e*x^2])/(120*d*x^2) - (b*c*(d + e*x^2)^(3/2))/(20*d*x^4) - ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(15*d^2*x^3) - (b*c*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(120*d^(3/2)) + (b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(15*d^2), -((b*c*Sqrt[d + e*x^2])/(20*x^4)) + (b*c*(3*c^2*d - e)*Sqrt[d + e*x^2])/(30*d*x^2) - (b*c*e*Sqrt[d + e*x^2])/(40*d*x^2) - ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(15*d^2*x^3) + (b*c*(3*c^2*d - e)*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(30*d^(3/2)) + (b*c*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*d^(3/2)) - (b*c*(c^2*d - e)*(3*c^2*d + 2*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(15*d^(3/2)) + (b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(15*d^2)]} - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2), x, 10, (b*(3*c^4*d^2 + 54*c^2*d*e - 40*e^2)*x*Sqrt[d + e*x^2])/(560*c^5*e) - (b*(13*c^2*d - 30*e)*x*(d + e*x^2)^(3/2))/(840*c^3*e) - (b*x*(d + e*x^2)^(5/2))/(42*c*e) - (d*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + (b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(35*c^7*e^2) + (b*(35*c^6*d^3 + 70*c^4*d^2*e - 168*c^2*d*e^2 + 80*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(560*c^7*e^(3/2))} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2), x, 6, (a*d^2*x*Sqrt[d + e*x^2])/(16*e) + (1/8)*a*d*x^3*Sqrt[d + e*x^2] + (1/6)*a*x^3*(d + e*x^2)^(3/2) - (a*d^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(16*e^(3/2)) + b*Unintegrable[x^2*(d + e*x^2)^(3/2)*ArcTan[c*x], x]} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2), x, 8, -((b*(7*c^2*d - 4*e)*x*Sqrt[d + e*x^2])/(40*c^3)) - (b*x*(d + e*x^2)^(3/2))/(20*c) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e) - (b*(c^2*d - e)^(5/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(5*c^5*e) - (b*(15*c^4*d^2 - 20*c^2*d*e + 8*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(40*c^5*Sqrt[e])} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2), x, 0, Unintegrable[(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^1, x, 6, a*d*Sqrt[d + e*x^2] + (1/3)*a*(d + e*x^2)^(3/2) - a*d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^2, x, 5, (3/2)*a*e*x*Sqrt[d + e*x^2] - (a*(d + e*x^2)^(3/2))/x + (3/2)*a*d*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x^2, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^3, x, 6, (3/2)*a*e*Sqrt[d + e*x^2] - (a*(d + e*x^2)^(3/2))/(2*x^2) - (3/2)*a*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x^3, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^4, x, 5, -((a*e*Sqrt[d + e*x^2])/x) - (a*(d + e*x^2)^(3/2))/(3*x^3) + a*e^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x^4, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^5, x, 6, -((3*a*e*Sqrt[d + e*x^2])/(8*x^2)) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*Sqrt[d]) + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2)/x^6, x, 10, (b*c*(4*c^2*d - 7*e)*Sqrt[d + e*x^2])/(40*x^2) - (b*c*(d + e*x^2)^(3/2))/(20*x^4) - ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*d*x^5) - (b*c*(8*c^4*d^2 - 20*c^2*d*e + 15*e^2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*Sqrt[d]) + (b*(c^2*d - e)^(5/2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(5*d)} - - -{x^3*(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2), x, 11, (b*(59*c^6*d^3 + 712*c^4*d^2*e - 1104*c^2*d*e^2 + 448*e^3)*x*Sqrt[d + e*x^2])/(8064*c^7*e) - (b*(69*c^4*d^2 - 520*c^2*d*e + 336*e^2)*x*(d + e*x^2)^(3/2))/(12096*c^5*e) - (b*(33*c^2*d - 56*e)*x*(d + e*x^2)^(5/2))/(3024*c^3*e) - (b*x*(d + e*x^2)^(7/2))/(72*c*e) - (d*(d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + ((d + e*x^2)^(9/2)*(a + b*ArcTan[c*x]))/(9*e^2) + (b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(63*c^9*e^2) + (b*(315*c^8*d^4 + 840*c^6*d^3*e - 3024*c^4*d^2*e^2 + 2880*c^2*d*e^3 - 896*e^4)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8064*c^9*e^(3/2))} -{x^2*(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2), x, 7, (5*a*d^3*x*Sqrt[d + e*x^2])/(128*e) + (5/64)*a*d^2*x^3*Sqrt[d + e*x^2] + (5/48)*a*d*x^3*(d + e*x^2)^(3/2) + (1/8)*a*x^3*(d + e*x^2)^(5/2) - (5*a*d^4*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(3/2)) + b*Unintegrable[x^2*(d + e*x^2)^(5/2)*ArcTan[c*x], x]} -{x^1*(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2), x, 9, -((b*(19*c^4*d^2 - 22*c^2*d*e + 8*e^2)*x*Sqrt[d + e*x^2])/(112*c^5)) - (b*(11*c^2*d - 6*e)*x*(d + e*x^2)^(3/2))/(168*c^3) - (b*x*(d + e*x^2)^(5/2))/(42*c) + ((d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e) - (b*(c^2*d - e)^(7/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(7*c^7*e) - (b*(35*c^6*d^3 - 70*c^4*d^2*e + 56*c^2*d*e^2 - 16*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(112*c^7*Sqrt[e])} -{x^0*(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2), x, 0, Unintegrable[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2)/x^1, x, 7, a*d^2*Sqrt[d + e*x^2] + (1/3)*a*d*(d + e*x^2)^(3/2) + (1/5)*a*(d + e*x^2)^(5/2) - a*d^(5/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[((d + e*x^2)^(5/2)*ArcTan[c*x])/x, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2)/x^2, x, 6, (15/8)*a*d*e*x*Sqrt[d + e*x^2] + (5/4)*a*e*x*(d + e*x^2)^(3/2) - (a*(d + e*x^2)^(5/2))/x + (15/8)*a*d^2*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[((d + e*x^2)^(5/2)*ArcTan[c*x])/x^2, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2)/x^3, x, 7, (5/2)*a*d*e*Sqrt[d + e*x^2] + (5/6)*a*e*(d + e*x^2)^(3/2) - (a*(d + e*x^2)^(5/2))/(2*x^2) - (5/2)*a*d^(3/2)*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Unintegrable[((d + e*x^2)^(5/2)*ArcTan[c*x])/x^3, x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2)/x^4, x, 6, (5/2)*a*e^2*x*Sqrt[d + e*x^2] - (5*a*e*(d + e*x^2)^(3/2))/(3*x) - (a*(d + e*x^2)^(5/2))/(3*x^3) + (5/2)*a*d*e^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Unintegrable[((d + e*x^2)^(5/2)*ArcTan[c*x])/x^4, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2)^(1/2), x, 8, -((b*x*Sqrt[d + e*x^2])/(6*c*e)) - (d*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e^2) + (b*Sqrt[c^2*d - e]*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*c^3*e^2) + (b*(3*c^2*d + 2*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(6*c^3*e^(3/2))} -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2)^(1/2), x, 4, (a*x*Sqrt[d + e*x^2])/(2*e) - (a*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2)) + b*Unintegrable[(x^2*ArcTan[c*x])/Sqrt[d + e*x^2], x]} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2)^(1/2), x, 6, (Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e - (b*Sqrt[c^2*d - e]*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(c*e) - (b*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[e])} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2)^(1/2), x, 0, Unintegrable[(a + b*ArcTan[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)^(1/2)), x, 4, -((a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d]) + b*Unintegrable[ArcTan[c*x]/(x*Sqrt[d + e*x^2]), x]} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(1/2)), x, 7, -((Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(d*x)) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d] + (b*Sqrt[c^2*d - e]*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/d} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(1/2)), x, 5, -((a*Sqrt[d + e*x^2])/(2*d*x^2)) + (a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(3/2)) + b*Unintegrable[ArcTan[c*x]/(x^3*Sqrt[d + e*x^2]), x]} -{(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(1/2)), x, 9, -((b*c*Sqrt[d + e*x^2])/(6*d*x^2)) - (Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(3*d^2*x) + (b*c*(2*c^2*d + 3*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(3/2)) - (b*Sqrt[c^2*d - e]*(c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^2)} - - -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2), x, 7, (d*(a + b*ArcTan[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e^2 - (b*(2*c^2*d - e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[c^2*d - e]*e^2) - (b*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*e^(3/2))} -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2), x, 4, -((a*x)/(e*Sqrt[d + e*x^2])) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2) + b*Unintegrable[(x^2*ArcTan[c*x])/(d + e*x^2)^(3/2), x]} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2), x, 3, -((a + b*ArcTan[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(Sqrt[c^2*d - e]*e)} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2), x, 5, (x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]) + (b*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(d*Sqrt[c^2*d - e])} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)^(3/2)), x, 5, a/(d*Sqrt[d + e*x^2]) - (a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) + b*Unintegrable[ArcTan[c*x]/(x*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(3/2)), x, 8, -((a + b*ArcTan[c*x])/(d*x*Sqrt[d + e*x^2])) - (2*e*x*(a + b*ArcTan[c*x]))/(d^2*Sqrt[d + e*x^2]) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) + (b*(c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(d^2*Sqrt[c^2*d - e])} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(3/2)), x, 6, -((3*a*e)/(2*d^2*Sqrt[d + e*x^2])) - a/(2*d*x^2*Sqrt[d + e*x^2]) + (3*a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(5/2)) + b*Unintegrable[ArcTan[c*x]/(x^3*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(3/2)), x, 14, -((b*c*Sqrt[d + e*x^2])/(6*d^2*x^2)) - (a + b*ArcTan[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcTan[c*x]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(5/2)) + (b*c*(c^2*d + 4*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(5/2)) - (b*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*Sqrt[c^2*d - e])} - - -{x^4*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 5, -((a*x^3)/(3*e*(d + e*x^2)^(3/2))) - (a*x)/(e^2*Sqrt[d + e*x^2]) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(5/2) + b*Unintegrable[(x^4*ArcTan[c*x])/(d + e*x^2)^(5/2), x]} -{x^3*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 6, (b*c*x)/(3*(c^2*d - e)*e*Sqrt[d + e*x^2]) + (d*(a + b*ArcTan[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcTan[c*x])/(e^2*Sqrt[d + e*x^2]) + (b*c*(2*c^2*d - 3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*(c^2*d - e)^(3/2)*e^2)} -{x^2*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 5, (b*c)/(3*(c^2*d - e)*e*Sqrt[d + e*x^2]) + (x^3*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d*(c^2*d - e)^(3/2))} -{x^1*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 4, -((b*c*x)/(3*d*(c^2*d - e)*Sqrt[d + e*x^2])) - (a + b*ArcTan[c*x])/(3*e*(d + e*x^2)^(3/2)) + (b*c^3*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*(c^2*d - e)^(3/2)*e)} -{x^0*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 7, -((b*c)/(3*d*(c^2*d - e)*Sqrt[d + e*x^2])) + (x*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcTan[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*(3*c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^2*(c^2*d - e)^(3/2))} -{(a + b*ArcTan[c*x])/(x^1*(d + e*x^2)^(5/2)), x, 6, a/(3*d*(d + e*x^2)^(3/2)) + a/(d^2*Sqrt[d + e*x^2]) - (a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(5/2) + b*Unintegrable[ArcTan[c*x]/(x*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(5/2)), x, 13, (b*c)/(d^2*Sqrt[d + e*x^2]) - (8*b*e)/(3*c*d^3*Sqrt[d + e*x^2]) - (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2))/(3*c*d^3*(c^2*d - e)*Sqrt[d + e*x^2]) - (a + b*ArcTan[c*x])/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*ArcTan[c*x]))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(5/2) + (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*(c^2*d - e)^(3/2))} -{(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(5/2)), x, 7, -((5*a*e)/(6*d^2*(d + e*x^2)^(3/2))) - a/(2*d*x^2*(d + e*x^2)^(3/2)) - (5*a*e)/(2*d^3*Sqrt[d + e*x^2]) + (5*a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + b*Unintegrable[ArcTan[c*x]/(x^3*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(5/2)), x, 18, -((b*c*e)/(2*d^3*Sqrt[d + e*x^2])) + (16*b*e^2)/(3*c*d^4*Sqrt[d + e*x^2]) - (b*c*(c^2*d + 6*e))/(3*d^3*Sqrt[d + e*x^2]) + (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2))/(3*c*d^4*(c^2*d - e)*Sqrt[d + e*x^2]) - (b*c)/(6*d^2*x^2*Sqrt[d + e*x^2]) - (a + b*ArcTan[c*x])/(3*d*x^3*(d + e*x^2)^(3/2)) + (2*e*(a + b*ArcTan[c*x]))/(d^2*x*(d + e*x^2)^(3/2)) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*(d + e*x^2)^(3/2)) + (16*e^2*x*(a + b*ArcTan[c*x]))/(3*d^4*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + (b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(7/2)) - (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^4*(c^2*d - e)^(3/2))} - - -{ArcTan[a*x]/(c + d*x^2)^(7/2), x, 8, -(a/(15*c*(a^2*c - d)*(c + d*x^2)^(3/2))) - (a*(7*a^2*c - 4*d))/(15*c^2*(a^2*c - d)^2*Sqrt[c + d*x^2]) + (x*ArcTan[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcTan[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcTan[a*x])/(15*c^3*Sqrt[c + d*x^2]) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(15*c^3*(a^2*c - d)^(5/2))} - - -{ArcTan[a*x]/(c + d*x^2)^(9/2), x, 8, -(a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2))) - (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) - (a*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcTan[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcTan[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcTan[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcTan[a*x])/(35*c^4*Sqrt[c + d*x^2]) + ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x]) with m symbolic*) - - -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^3, x, 4, If[$VersionNumber>=8, -((b*e*(e^2*(15 + 8*m + m^2) - 3*c^2*d*e*(21 + 10*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2))*x^(2 + m))/(c^5*(2 + m)*(3 + m)*(5 + m)*(7 + m))) + (b*e^2*(e*(5 + m) - 3*c^2*d*(7 + m))*x^(4 + m))/(c^3*(4 + m)*(5 + m)*(7 + m)) - (b*e^3*x^(6 + m))/(c*(6 + m)*(7 + m)) + (d^3*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (3*d*e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) + (e^3*x^(7 + m)*(a + b*ArcTan[c*x]))/(7 + m) + (b*(e^3*(15 + 23*m + 9*m^2 + m^3) - 3*c^2*d*e^2*(21 + 31*m + 11*m^2 + m^3) + 3*c^4*d^2*e*(35 + 47*m + 13*m^2 + m^3) - c^6*d^3*(105 + 71*m + 15*m^2 + m^3))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(c^5*(1 + m)*(2 + m)*(3 + m)*(5 + m)*(7 + m)), -((b*e*(e^2*(15 + 8*m + m^2) - 3*c^2*d*e*(21 + 10*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2))*x^(2 + m))/(c^5*(2 + m)*(7 + m)*(15 + 8*m + m^2))) + (b*e^2*(e*(5 + m) - 3*c^2*d*(7 + m))*x^(4 + m))/(c^3*(4 + m)*(5 + m)*(7 + m)) - (b*e^3*x^(6 + m))/(c*(6 + m)*(7 + m)) + (d^3*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (3*d*e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) + (e^3*x^(7 + m)*(a + b*ArcTan[c*x]))/(7 + m) + (b*(e^3*(15 + 23*m + 9*m^2 + m^3) - 3*c^2*d*e^2*(21 + 31*m + 11*m^2 + m^3) + 3*c^4*d^2*e*(35 + 47*m + 13*m^2 + m^3) - c^6*d^3*(105 + 71*m + 15*m^2 + m^3))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(c^5*(35 + 12*m + m^2)*(6 + 11*m + 6*m^2 + m^3))]} -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^2, x, 4, If[$VersionNumber>=8, (b*e*(e*(3 + m) - 2*c^2*d*(5 + m))*x^(2 + m))/(c^3*(2 + m)*(3 + m)*(5 + m)) - (b*e^2*x^(4 + m))/(c*(4 + m)*(5 + m)) + (d^2*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (2*d*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) - (b*(e^2*(3 + 4*m + m^2) - 2*c^2*d*e*(5 + 6*m + m^2) + c^4*d^2*(15 + 8*m + m^2))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(c^3*(1 + m)*(2 + m)*(3 + m)*(5 + m)), (b*e*(e*(3 + m) - 2*c^2*d*(5 + m))*x^(2 + m))/(c^3*(5 + m)*(6 + 5*m + m^2)) - (b*e^2*x^(4 + m))/(c*(4 + m)*(5 + m)) + (d^2*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (2*d*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) - (b*(e^2*(3 + 4*m + m^2) - 2*c^2*d*e*(5 + 6*m + m^2) + c^4*d^2*(15 + 8*m + m^2))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(c^3*(2 + 3*m + m^2)*(15 + 8*m + m^2))]} -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^1, x, 3, -((b*e*x^(2 + m))/(c*(6 + 5*m + m^2))) + (d*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) - (b*((c^2*d)/(1 + m) - e/(3 + m))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(c*(2 + m))} -{x^m*(a + b*ArcTan[c*x])/(d + e*x^2)^1, x, 2, (a*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2), x]} -{x^m*(a + b*ArcTan[c*x])/(d + e*x^2)^2, x, 2, (a*x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d^2*(1 + m)) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^2, x]} - - -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^(5/2), x, 3, (a*x^(1 + m)*(d + e*x^2)^(7/2)*Hypergeometric2F1[1, (8 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[x^m*(d + e*x^2)^(5/2)*ArcTan[c*x], x], (a*d^2*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-(5/2), (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Unintegrable[x^m*(d + e*x^2)^(5/2)*ArcTan[c*x], x]} -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^(3/2), x, 3, (a*x^(1 + m)*(d + e*x^2)^(5/2)*Hypergeometric2F1[1, (6 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[x^m*(d + e*x^2)^(3/2)*ArcTan[c*x], x], (a*d*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-(3/2), (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Unintegrable[x^m*(d + e*x^2)^(3/2)*ArcTan[c*x], x]} -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^(1/2), x, 3, (a*x^(1 + m)*(d + e*x^2)^(3/2)*Hypergeometric2F1[1, (4 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[x^m*Sqrt[d + e*x^2]*ArcTan[c*x], x], (a*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-(1/2), (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Unintegrable[x^m*Sqrt[d + e*x^2]*ArcTan[c*x], x]} -{x^m*(a + b*ArcTan[c*x])/(d + e*x^2)^(1/2), x, 3, (a*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[1, (2 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[(x^m*ArcTan[c*x])/Sqrt[d + e*x^2], x], (a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[d + e*x^2]) + b*Unintegrable[(x^m*ArcTan[c*x])/Sqrt[d + e*x^2], x]} -{x^m*(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2), x, 3, (a*x^(1 + m)*Hypergeometric2F1[1, m/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)*Sqrt[d + e*x^2]) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^(3/2), x], (a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)*Sqrt[d + e*x^2]) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^(3/2), x]} -{x^m*(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2), x, 3, (a*x^(1 + m)*Hypergeometric2F1[1, (1/2)*(-2 + m), (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)*(d + e*x^2)^(3/2)) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^(5/2), x], (a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d^2*(1 + m)*Sqrt[d + e*x^2]) + b*Unintegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^(5/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x]) with p symbolic*) - - -{x^m*(a + b*ArcTan[c*x])*(d + e*x^2)^p, x, 3, (a*x^(1 + m)*(d + e*x^2)^(1 + p)*Hypergeometric2F1[1, (1/2)*(3 + m + 2*p), (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Unintegrable[x^m*(d + e*x^2)^p*ArcTan[c*x], x], (a*x^(1 + m)*(d + e*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(1 + m)) + b*Unintegrable[x^m*(d + e*x^2)^p*ArcTan[c*x], x]} - - -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 2), x, 3, -((a*x^(-1 - 2*p)*(d + e*x^2)^(1 + p)*Hypergeometric2F1[1/2, 1, (1/2)*(1 - 2*p), -((e*x^2)/d)])/(d*(1 + 2*p))) + b*Unintegrable[x^(-2 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x], -((a*x^(-1 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-1 - 2*p), -p, (1/2)*(1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(1 + 2*p))) + b*Unintegrable[x^(-2 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 3), x, 4, -((b*c*x^(-1 - 2*p)*(d + e*x^2)^p*AppellF1[(1/2)*(-1 - 2*p), 1, -1 - p, (1/2)*(1 - 2*p), (-c^2)*x^2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*(1 + 3*p + 2*p^2)))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(1 + p))*(2*d*(1 + p)))} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 4), x, 3, -((a*x^(-3 - 2*p)*(d + e*x^2)^(1 + p)*Hypergeometric2F1[-(1/2), 1, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/(d*(3 + 2*p))) + b*Unintegrable[x^(-4 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x], -((a*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-3 - 2*p), -p, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(3 + 2*p))) + b*Unintegrable[x^(-4 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 5), x, 8, If[$VersionNumber>=8, -((b*(e + c^2*d*(1 + p))*x^(-3 - 2*p)*(d + e*x^2)^p*AppellF1[(1/2)*(-3 - 2*p), 1, -1 - p, (1/2)*(-1 - 2*p), (-c^2)*x^2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c*d*(1 + p)*(2 + p)*(3 + 2*p)))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(1 + p))*(2*d^2*(1 + p)*(2 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(2 + p))*(2*d*(2 + p))) + (b*e*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-3 - 2*p), -1 - p, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c*d*(6 + 13*p + 9*p^2 + 2*p^3))), -((b*(e + c^2*d*(1 + p))*x^(-3 - 2*p)*(d + e*x^2)^p*AppellF1[(1/2)*(-3 - 2*p), 1, -1 - p, (1/2)*(-1 - 2*p), (-c^2)*x^2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c*d*(3 + 2*p)*(2 + 3*p + p^2)))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(1 + p))*(2*d^2*(1 + p)*(2 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(2 + p))*(2*d*(2 + p))) + (b*e*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-3 - 2*p), -1 - p, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c*d*(6 + 13*p + 9*p^2 + 2*p^3)))]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 6), x, 3, -((a*x^(-5 - 2*p)*(d + e*x^2)^(1 + p)*Hypergeometric2F1[-(3/2), 1, (1/2)*(-3 - 2*p), -((e*x^2)/d)])/(d*(5 + 2*p))) + b*Unintegrable[x^(-6 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x], -((a*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-5 - 2*p), -p, (1/2)*(-3 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(5 + 2*p))) + b*Unintegrable[x^(-6 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 7), x, 10, If[$VersionNumber>=8, -((b*(2*e^2 + 2*c^2*d*e*(1 + p) + c^4*d^2*(2 + 3*p + p^2))*x^(-5 - 2*p)*(d + e*x^2)^p*AppellF1[(1/2)*(-5 - 2*p), 1, -1 - p, (1/2)*(-3 - 2*p), (-c^2)*x^2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c^3*d^2*(1 + p)*(2 + p)*(3 + p)*(5 + 2*p)))) - (e^2*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(1 + p))*(d^3*(1 + p)*(2 + p)*(3 + p))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(2 + p))*(d^2*(2 + p)*(3 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(3 + p))*(2*d*(3 + p))) + (b*e*(e + c^2*d*(1 + p))*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-5 - 2*p), -1 - p, (1/2)*(-3 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(c^3*d^2*(1 + p)*(2 + p)*(3 + p)*(5 + 2*p))) - (b*e^2*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-3 - 2*p), -1 - p, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(c*d^2*(1 + p)*(2 + p)*(3 + p)*(3 + 2*p))), -((b*(2*e^2 + 2*c^2*d*e*(1 + p) + c^4*d^2*(2 + 3*p + p^2))*x^(-5 - 2*p)*(d + e*x^2)^p*AppellF1[(1/2)*(-5 - 2*p), 1, -1 - p, (1/2)*(-3 - 2*p), (-c^2)*x^2, -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(2*c^3*d^2*(3 + p)*(5 + 2*p)*(2 + 3*p + p^2)))) - (e^2*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(1 + p))*(d^3*(2 + p)*(3 + 4*p + p^2))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(2 + p))*(d^2*(2 + p)*(3 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(x^(2*(3 + p))*(2*d*(3 + p))) + (b*e*(e + c^2*d*(1 + p))*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-5 - 2*p), -1 - p, (1/2)*(-3 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(c^3*d^2*(3 + p)*(5 + 2*p)*(2 + 3*p + p^2))) - (b*e^2*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-3 - 2*p), -1 - p, (1/2)*(-1 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(c*d^2*(2 + 3*p + p^2)*(9 + 9*p + 2*p^2)))]} -{(a + b*ArcTan[c*x])*(d + e*x^2)^p/x^(2*p + 8), x, 3, -((a*x^(-7 - 2*p)*(d + e*x^2)^(1 + p)*Hypergeometric2F1[-(5/2), 1, (1/2)*(-5 - 2*p), -((e*x^2)/d)])/(d*(7 + 2*p))) + b*Unintegrable[x^(-8 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x], -((a*x^(-7 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(1/2)*(-7 - 2*p), -p, (1/2)*(-5 - 2*p), -((e*x^2)/d)])/((1 + (e*x^2)/d)^p*(7 + 2*p))) + b*Unintegrable[x^(-8 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcTan[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(d + e*x^2)*(a + b*ArcTan[c*x])^2, x, 29, (a*b*d*x)/(2*c^3) - (a*b*e*x)/(3*c^5) + (b^2*d*x^2)/(12*c^2) - (4*b^2*e*x^2)/(45*c^4) + (b^2*e*x^4)/(60*c^2) + (b^2*d*x*ArcTan[c*x])/(2*c^3) - (b^2*e*x*ArcTan[c*x])/(3*c^5) - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) + (b*e*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e*x^5*(a + b*ArcTan[c*x]))/(15*c) - (d*(a + b*ArcTan[c*x])^2)/(4*c^4) + (e*(a + b*ArcTan[c*x])^2)/(6*c^6) + (1/4)*d*x^4*(a + b*ArcTan[c*x])^2 + (1/6)*e*x^6*(a + b*ArcTan[c*x])^2 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e*Log[1 + c^2*x^2])/(90*c^6)} -{x^2*(d + e*x^2)*(a + b*ArcTan[c*x])^2, x, 25, (b^2*d*x)/(3*c^2) - (3*b^2*e*x)/(10*c^4) + (b^2*e*x^3)/(30*c^2) - (b^2*d*ArcTan[c*x])/(3*c^3) + (3*b^2*e*ArcTan[c*x])/(10*c^5) - (b*d*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e*x^4*(a + b*ArcTan[c*x]))/(10*c) - (I*d*(a + b*ArcTan[c*x])^2)/(3*c^3) + (I*e*(a + b*ArcTan[c*x])^2)/(5*c^5) + (1/3)*d*x^3*(a + b*ArcTan[c*x])^2 + (1/5)*e*x^5*(a + b*ArcTan[c*x])^2 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3) + (I*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^5)} -{x^1*(d + e*x^2)*(a + b*ArcTan[c*x])^2, x, 19, -((a*b*d*x)/c) + (a*b*e*x)/(2*c^3) + (b^2*e*x^2)/(12*c^2) - (b^2*d*x*ArcTan[c*x])/c + (b^2*e*x*ArcTan[c*x])/(2*c^3) - (b*e*x^3*(a + b*ArcTan[c*x]))/(6*c) + (d*(a + b*ArcTan[c*x])^2)/(2*c^2) - (e*(a + b*ArcTan[c*x])^2)/(4*c^4) + (1/2)*d*x^2*(a + b*ArcTan[c*x])^2 + (1/4)*e*x^4*(a + b*ArcTan[c*x])^2 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) - (b^2*e*Log[1 + c^2*x^2])/(3*c^4)} -{x^0*(d + e*x^2)*(a + b*ArcTan[c*x])^2, x, 16, (b^2*e*x)/(3*c^2) - (b^2*e*ArcTan[c*x])/(3*c^3) - (b*e*x^2*(a + b*ArcTan[c*x]))/(3*c) + (I*d*(a + b*ArcTan[c*x])^2)/c - (I*e*(a + b*ArcTan[c*x])^2)/(3*c^3) + d*x*(a + b*ArcTan[c*x])^2 + (1/3)*e*x^3*(a + b*ArcTan[c*x])^2 + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (I*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3)} -{(d + e*x^2)*(a + b*ArcTan[c*x])^2/x^1, x, 14, -((a*b*e*x)/c) - (b^2*e*x*ArcTan[c*x])/c + (e*(a + b*ArcTan[c*x])^2)/(2*c^2) + (1/2)*e*x^2*(a + b*ArcTan[c*x])^2 + 2*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (b^2*e*Log[1 + c^2*x^2])/(2*c^2) - I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (1/2)*b^2*d*PolyLog[3, 1 - 2/(1 + I*c*x)] + (1/2)*b^2*d*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{(d + e*x^2)*(a + b*ArcTan[c*x])^2/x^2, x, 11, (-I)*c*d*(a + b*ArcTan[c*x])^2 + (I*e*(a + b*ArcTan[c*x])^2)/c - (d*(a + b*ArcTan[c*x])^2)/x + e*x*(a + b*ArcTan[c*x])^2 + (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + 2*b*c*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d*PolyLog[2, -1 + 2/(1 - I*c*x)] + (I*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c} -{(d + e*x^2)*(a + b*ArcTan[c*x])^2/x^3, x, 16, -((b*c*d*(a + b*ArcTan[c*x]))/x) - (1/2)*c^2*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/(2*x^2) + 2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d*Log[x] - (1/2)*b^2*c^2*d*Log[1 + c^2*x^2] - I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (1/2)*b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)] + (1/2)*b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)]} - - -{x^3*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2, x, 50, (a*b*d^2*x)/(2*c^3) - (2*a*b*d*e*x)/(3*c^5) + (a*b*e^2*x)/(4*c^7) + (b^2*d^2*x^2)/(12*c^2) - (8*b^2*d*e*x^2)/(45*c^4) + (71*b^2*e^2*x^2)/(840*c^6) + (b^2*d*e*x^4)/(30*c^2) - (3*b^2*e^2*x^4)/(140*c^4) + (b^2*e^2*x^6)/(168*c^2) + (b^2*d^2*x*ArcTan[c*x])/(2*c^3) - (2*b^2*d*e*x*ArcTan[c*x])/(3*c^5) + (b^2*e^2*x*ArcTan[c*x])/(4*c^7) - (b*d^2*x^3*(a + b*ArcTan[c*x]))/(6*c) + (2*b*d*e*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e^2*x^3*(a + b*ArcTan[c*x]))/(12*c^5) - (2*b*d*e*x^5*(a + b*ArcTan[c*x]))/(15*c) + (b*e^2*x^5*(a + b*ArcTan[c*x]))/(20*c^3) - (b*e^2*x^7*(a + b*ArcTan[c*x]))/(28*c) - (d^2*(a + b*ArcTan[c*x])^2)/(4*c^4) + (d*e*(a + b*ArcTan[c*x])^2)/(3*c^6) - (e^2*(a + b*ArcTan[c*x])^2)/(8*c^8) + (1/4)*d^2*x^4*(a + b*ArcTan[c*x])^2 + (1/3)*d*e*x^6*(a + b*ArcTan[c*x])^2 + (1/8)*e^2*x^8*(a + b*ArcTan[c*x])^2 - (b^2*d^2*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*d*e*Log[1 + c^2*x^2])/(45*c^6) - (22*b^2*e^2*Log[1 + c^2*x^2])/(105*c^8)} -{x^2*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2, x, 44, (b^2*d^2*x)/(3*c^2) - (3*b^2*d*e*x)/(5*c^4) + (11*b^2*e^2*x)/(42*c^6) + (b^2*d*e*x^3)/(15*c^2) - (5*b^2*e^2*x^3)/(126*c^4) + (b^2*e^2*x^5)/(105*c^2) - (b^2*d^2*ArcTan[c*x])/(3*c^3) + (3*b^2*d*e*ArcTan[c*x])/(5*c^5) - (11*b^2*e^2*ArcTan[c*x])/(42*c^7) - (b*d^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(7*c^5) - (b*d*e*x^4*(a + b*ArcTan[c*x]))/(5*c) + (b*e^2*x^4*(a + b*ArcTan[c*x]))/(14*c^3) - (b*e^2*x^6*(a + b*ArcTan[c*x]))/(21*c) - (I*d^2*(a + b*ArcTan[c*x])^2)/(3*c^3) + (2*I*d*e*(a + b*ArcTan[c*x])^2)/(5*c^5) - (I*e^2*(a + b*ArcTan[c*x])^2)/(7*c^7) + (1/3)*d^2*x^3*(a + b*ArcTan[c*x])^2 + (2/5)*d*e*x^5*(a + b*ArcTan[c*x])^2 + (1/7)*e^2*x^7*(a + b*ArcTan[c*x])^2 - (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(7*c^7) - (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3) + (2*I*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^5) - (I*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(7*c^7)} -{x^1*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2, x, 35, -((a*b*d^2*x)/c) + (a*b*d*e*x)/c^3 - (a*b*e^2*x)/(3*c^5) + (b^2*d*e*x^2)/(6*c^2) - (4*b^2*e^2*x^2)/(45*c^4) + (b^2*e^2*x^4)/(60*c^2) - (b^2*d^2*x*ArcTan[c*x])/c + (b^2*d*e*x*ArcTan[c*x])/c^3 - (b^2*e^2*x*ArcTan[c*x])/(3*c^5) - (b*d*e*x^3*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e^2*x^5*(a + b*ArcTan[c*x]))/(15*c) + (d^2*(a + b*ArcTan[c*x])^2)/(2*c^2) - (d*e*(a + b*ArcTan[c*x])^2)/(2*c^4) + (e^2*(a + b*ArcTan[c*x])^2)/(6*c^6) + (1/2)*d^2*x^2*(a + b*ArcTan[c*x])^2 + (1/2)*d*e*x^4*(a + b*ArcTan[c*x])^2 + (1/6)*e^2*x^6*(a + b*ArcTan[c*x])^2 + (b^2*d^2*Log[1 + c^2*x^2])/(2*c^2) - (2*b^2*d*e*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e^2*Log[1 + c^2*x^2])/(90*c^6)} -{x^0*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2, x, 30, (2*b^2*d*e*x)/(3*c^2) - (3*b^2*e^2*x)/(10*c^4) + (b^2*e^2*x^3)/(30*c^2) - (2*b^2*d*e*ArcTan[c*x])/(3*c^3) + (3*b^2*e^2*ArcTan[c*x])/(10*c^5) - (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e^2*x^4*(a + b*ArcTan[c*x]))/(10*c) + (I*d^2*(a + b*ArcTan[c*x])^2)/c - (2*I*d*e*(a + b*ArcTan[c*x])^2)/(3*c^3) + (I*e^2*(a + b*ArcTan[c*x])^2)/(5*c^5) + d^2*x*(a + b*ArcTan[c*x])^2 + (2/3)*d*e*x^3*(a + b*ArcTan[c*x])^2 + (1/5)*e^2*x^5*(a + b*ArcTan[c*x])^2 + (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (2*I*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3) + (I*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^5)} -{(d + e*x^2)^2*(a + b*ArcTan[c*x])^2/x^1, x, 25, -((2*a*b*d*e*x)/c) + (a*b*e^2*x)/(2*c^3) + (b^2*e^2*x^2)/(12*c^2) - (2*b^2*d*e*x*ArcTan[c*x])/c + (b^2*e^2*x*ArcTan[c*x])/(2*c^3) - (b*e^2*x^3*(a + b*ArcTan[c*x]))/(6*c) + (d*e*(a + b*ArcTan[c*x])^2)/c^2 - (e^2*(a + b*ArcTan[c*x])^2)/(4*c^4) + d*e*x^2*(a + b*ArcTan[c*x])^2 + (1/4)*e^2*x^4*(a + b*ArcTan[c*x])^2 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (b^2*d*e*Log[1 + c^2*x^2])/c^2 - (b^2*e^2*Log[1 + c^2*x^2])/(3*c^4) - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (1/2)*b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)] + (1/2)*b^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)]} -{(d + e*x^2)^2*(a + b*ArcTan[c*x])^2/x^2, x, 20, (b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTan[c*x])/(3*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(3*c) - I*c*d^2*(a + b*ArcTan[c*x])^2 + (2*I*d*e*(a + b*ArcTan[c*x])^2)/c - (I*e^2*(a + b*ArcTan[c*x])^2)/(3*c^3) - (d^2*(a + b*ArcTan[c*x])^2)/x + 2*d*e*x*(a + b*ArcTan[c*x])^2 + (1/3)*e^2*x^3*(a + b*ArcTan[c*x])^2 + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] + (2*I*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (I*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^3)} -{(d + e*x^2)^2*(a + b*ArcTan[c*x])^2/x^3, x, 22, -((a*b*e^2*x)/c) - (b^2*e^2*x*ArcTan[c*x])/c - (b*c*d^2*(a + b*ArcTan[c*x]))/x - (1/2)*c^2*d^2*(a + b*ArcTan[c*x])^2 + (e^2*(a + b*ArcTan[c*x])^2)/(2*c^2) - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcTan[c*x])^2 + 4*d*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^2*Log[x] - (1/2)*b^2*c^2*d^2*Log[1 + c^2*x^2] + (b^2*e^2*Log[1 + c^2*x^2])/(2*c^2) - 2*I*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + 2*I*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - b^2*d*e*PolyLog[3, 1 - 2/(1 + I*c*x)] + b^2*d*e*PolyLog[3, -1 + 2/(1 + I*c*x)]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcTan[c*x])^2/(d + e*x^2), x, 11, -((a*b*x)/(c*e)) - (b^2*x*ArcTan[c*x])/(c*e) + (a + b*ArcTan[c*x])^2/(2*c^2*e) + (x^2*(a + b*ArcTan[c*x])^2)/(2*e) + (d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (b^2*Log[1 + c^2*x^2])/(2*c^2*e) - (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (b^2*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)} -{x^2*(a + b*ArcTan[c*x])^2/(d + e*x^2), x, 10, (I*(a + b*ArcTan[c*x])^2)/(c*e) + (x*(a + b*ArcTan[c*x])^2)/e + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e) + (Sqrt[-d]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e) - (I*b*Sqrt[-d]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) + (I*b*Sqrt[-d]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) + (b^2*Sqrt[-d]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^(3/2)) - (b^2*Sqrt[-d]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^(3/2))} -{x^1*(a + b*ArcTan[c*x])^2/(d + e*x^2), x, 4, -(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e) - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e)} -{x^0*(a + b*ArcTan[c*x])^2/(d + e*x^2), x, 4, ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcTan[c*x])^2/(x^1*(d + e*x^2)), x, 12, (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)), x, 9, -((I*c*(a + b*ArcTan[c*x])^2)/d) - (a + b*ArcTan[c*x])^2/(d*x) + (Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (I*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) + (I*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) + (b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)) - (b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2))} -{(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)), x, 21, -((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d) - (a + b*ArcTan[c*x])^2/(2*d*x^2) - (2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + (b^2*c^2*Log[x])/d - (e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)} - - -{x^3*(a + b*ArcTan[c*x])^2/(d + e*x^2)^2, x, 33, -((c^2*d*(a + b*ArcTan[c*x])^2)/(2*(c^2*d - e)*e^2)) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 - (Sqrt[e]*x)/Sqrt[-d])) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 + (Sqrt[e]*x)/Sqrt[-d])) - ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 - (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + (I*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(c^2*d - e)*e^(3/2)) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (I*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(c^2*d - e)*e^(3/2)) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)} -{x^2*(a + b*ArcTan[c*x])^2/(d + e*x^2)^2, x, 38, -((I*c*(a + b*ArcTan[c*x])^2)/(2*(c^2*d - e)*e)) + (a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/((c^2*d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/((c^2*d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) - (I*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*(c^2*d - e)*e) - (I*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*(c^2*d - e)*e) + (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(c^2*d - e)*e) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) + (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(c^2*d - e)*e) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2)) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2))} -{x^1*(a + b*ArcTan[c*x])^2/(d + e*x^2)^2, x, 27, (c^2*(a + b*ArcTan[c*x])^2)/(2*(c^2*d - e)*e) - (a + b*ArcTan[c*x])^2/(4*d*e*(1 - (Sqrt[e]*x)/Sqrt[-d])) - (a + b*ArcTan[c*x])^2/(4*d*e*(1 + (Sqrt[e]*x)/Sqrt[-d])) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*(c^2*d - e)*Sqrt[e]) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*(c^2*d - e)*Sqrt[e]) + (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*(c^2*d - e)*Sqrt[e]) - (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*(c^2*d - e)*Sqrt[e])} -{x^0*(a + b*ArcTan[c*x])^2/(d + e*x^2)^2, x, 32, (I*c*(a + b*ArcTan[c*x])^2)/(2*d*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcTan[c*x])^2/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(d*(c^2*d - e)) + (b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(d*(c^2*d - e)) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d*(c^2*d - e)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) + (I*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d*(c^2*d - e)) + (I*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*d*(c^2*d - e)) - (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d*(c^2*d - e)) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) - (I*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d*(c^2*d - e)) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(3/2)*Sqrt[e]) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcTan[c*x])^2/(x^1*(d + e*x^2)^2), x, 39, -((c^2*(a + b*ArcTan[c*x])^2)/(2*d*(c^2*d - e))) + (a + b*ArcTan[c*x])^2/(4*d^2*(1 - (Sqrt[e]*x)/Sqrt[-d])) + (a + b*ArcTan[c*x])^2/(4*d^2*(1 + (Sqrt[e]*x)/Sqrt[-d])) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 - (b*c*Sqrt[e]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (b*c*Sqrt[e]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 + (I*b^2*c*Sqrt[e]*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*(c^2*d - e)) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (I*b^2*c*Sqrt[e]*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*(c^2*d - e)) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)} -{(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)^2), x, 42, -((I*c*(a + b*ArcTan[c*x])^2)/d^2) - (I*c*e*(a + b*ArcTan[c*x])^2)/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(d^2*x) + (Sqrt[e]*(a + b*ArcTan[c*x])^2)/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcTan[c*x])^2)/(4*d^2*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(d^2*(c^2*d - e)) - (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(d^2*(c^2*d - e)) - (b*c*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2*(c^2*d - e)) - (3*Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) - (b*c*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2*(c^2*d - e)) + (3*Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*d^2*(c^2*d - e)) - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - (I*b^2*c*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*d^2*(c^2*d - e)) + (I*b^2*c*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2*(c^2*d - e)) + (3*I*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) + (I*b^2*c*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2*(c^2*d - e)) - (3*I*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) - (3*b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(5/2)) + (3*b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(5/2))} -{(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)^2), x, 47, -((b*c*(a + b*ArcTan[c*x]))/(d^2*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d^2) + (c^2*e*(a + b*ArcTan[c*x])^2)/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(2*d^2*x^2) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 - (Sqrt[e]*x)/Sqrt[-d])) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 + (Sqrt[e]*x)/Sqrt[-d])) - (4*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + (b^2*c^2*Log[x])/d^2 - (2*e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^3 - (b*c*e^(3/2)*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(5/2)*(c^2*d - e)) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 + (b*c*e^(3/2)*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(5/2)*(c^2*d - e)) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + (2*I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + (2*I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3 - (2*I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (I*b^2*c*e^(3/2)*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)*(c^2*d - e)) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 - (I*b^2*c*e^(3/2)*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)*(c^2*d - e)) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/d^3 + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/d^3 - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^3)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (h x)^m (d+e Log[f+g x^2]) (a+b ArcTan[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e Log[f+g x^2]) (a+b ArcTan[c x])*) - - -{x^4*ArcTan[x]*Log[1 + x^2], x, 24, -((77*x^2)/300) + (9*x^4)/200 - (2/5)*x*ArcTan[x] + (2/15)*x^3*ArcTan[x] - (2/25)*x^5*ArcTan[x] + ArcTan[x]^2/5 + (137/300)*Log[1 + x^2] + (1/10)*x^2*Log[1 + x^2] - (1/20)*x^4*Log[1 + x^2] + (1/5)*x^5*ArcTan[x]*Log[1 + x^2] - (1/20)*Log[1 + x^2]^2} -{x^3*ArcTan[x]*Log[1 + x^2], x, 14, -((25*x)/24) + (7*x^3)/72 + (25*ArcTan[x])/24 + (1/4)*x^2*ArcTan[x] - (1/8)*x^4*ArcTan[x] + (1/4)*x*Log[1 + x^2] - (1/12)*x^3*Log[1 + x^2] - (1/4)*ArcTan[x]*Log[1 + x^2] + (1/4)*x^4*ArcTan[x]*Log[1 + x^2]} -{x^2*ArcTan[x]*Log[1 + x^2], x, 19, (5*x^2)/18 + (2/3)*x*ArcTan[x] - (2/9)*x^3*ArcTan[x] - ArcTan[x]^2/3 - (11/18)*Log[1 + x^2] - (1/6)*x^2*Log[1 + x^2] + (1/3)*x^3*ArcTan[x]*Log[1 + x^2] + (1/12)*Log[1 + x^2]^2} -{x^1*ArcTan[x]*Log[1 + x^2], x, 7, (3*x)/2 - (3*ArcTan[x])/2 - (1/2)*x^2*ArcTan[x] - (1/2)*x*Log[1 + x^2] + (1/2)*(1 + x^2)*ArcTan[x]*Log[1 + x^2]} -{x^0*ArcTan[x]*Log[1 + x^2], x, 8, -2*x*ArcTan[x] + ArcTan[x]^2 + Log[1 + x^2] + x*ArcTan[x]*Log[1 + x^2] - (1/4)*Log[1 + x^2]^2} -{ArcTan[x]*Log[1 + x^2]/x^1, x, 12, (-(1/2))*I*Log[1 + I*x]^2*Log[(-I)*x] + (1/2)*I*Log[1 - I*x]^2*Log[I*x] + I*Log[1 - I*x]*PolyLog[2, 1 - I*x] - I*Log[1 + I*x]*PolyLog[2, 1 + I*x] - (1/2)*I*(Log[1 - I*x] + Log[1 + I*x] - Log[1 + x^2])*PolyLog[2, (-I)*x] + (1/2)*I*(Log[1 - I*x] + Log[1 + I*x] - Log[1 + x^2])*PolyLog[2, I*x] - I*PolyLog[3, 1 - I*x] + I*PolyLog[3, 1 + I*x]} -{ArcTan[x]*Log[1 + x^2]/x^2, x, 8, ArcTan[x]^2 - (ArcTan[x]*Log[1 + x^2])/x - (1/4)*Log[1 + x^2]^2 - (1/2)*PolyLog[2, -x^2]} -{ArcTan[x]*Log[1 + x^2]/x^3, x, 6, ArcTan[x] - Log[1 + x^2]/(2*x) - (1/2)*ArcTan[x]*Log[1 + x^2] - (ArcTan[x]*Log[1 + x^2])/(2*x^2) + (1/2)*I*PolyLog[2, (-I)*x] - (1/2)*I*PolyLog[2, I*x]} -{ArcTan[x]*Log[1 + x^2]/x^4, x, 18, -((2*ArcTan[x])/(3*x)) - ArcTan[x]^2/3 + Log[x] - (1/2)*Log[1 + x^2] - Log[1 + x^2]/(6*x^2) - (ArcTan[x]*Log[1 + x^2])/(3*x^3) + (1/12)*Log[1 + x^2]^2 + (1/6)*PolyLog[2, -x^2]} -{ArcTan[x]*Log[1 + x^2]/x^5, x, 12, -(5/(12*x)) - (11*ArcTan[x])/12 - ArcTan[x]/(4*x^2) - Log[1 + x^2]/(12*x^3) + Log[1 + x^2]/(4*x) + (1/4)*ArcTan[x]*Log[1 + x^2] - (ArcTan[x]*Log[1 + x^2])/(4*x^4) - (1/4)*I*PolyLog[2, (-I)*x] + (1/4)*I*PolyLog[2, I*x]} -{ArcTan[x]*Log[1 + x^2]/x^6, x, 26, -(7/(60*x^2)) - (2*ArcTan[x])/(15*x^3) + (2*ArcTan[x])/(5*x) + ArcTan[x]^2/5 - (5*Log[x])/6 + (5/12)*Log[1 + x^2] - Log[1 + x^2]/(20*x^4) + Log[1 + x^2]/(10*x^2) - (ArcTan[x]*Log[1 + x^2])/(5*x^5) - (1/20)*Log[1 + x^2]^2 - (1/10)*PolyLog[2, -x^2]} - - -{x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]), x, 26, -((2*a*e*x)/(5*c^4)) - (77*b*e*x^2)/(300*c^3) + (2*a*e*x^3)/(15*c^2) + (9*b*e*x^4)/(200*c) - (2/25)*a*e*x^5 + (2*a*e*ArcTan[c*x])/(5*c^5) - (2*b*e*x*ArcTan[c*x])/(5*c^4) + (2*b*e*x^3*ArcTan[c*x])/(15*c^2) - (2/25)*b*e*x^5*ArcTan[c*x] + (b*e*ArcTan[c*x]^2)/(5*c^5) + (137*b*e*Log[1 + c^2*x^2])/(300*c^5) + (b*e*Log[1 + c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 + c^2*x^2]))/(10*c^3) - (b*x^4*(d + e*Log[1 + c^2*x^2]))/(20*c) + (1/5)*x^5*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]) - (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(10*c^5)} -{x^3*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]), x, 14, (b*(2*d - 3*e)*x)/(8*c^3) - (2*b*e*x)/(3*c^3) - (b*(2*d - e)*x^3)/(24*c) + (b*e*x^3)/(18*c) - (b*(2*d - 3*e)*ArcTan[c*x])/(8*c^4) + (2*b*e*ArcTan[c*x])/(3*c^4) + (e*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (1/8)*e*x^4*(a + b*ArcTan[c*x]) + (b*e*x*Log[1 + c^2*x^2])/(4*c^3) - (b*e*x^3*Log[1 + c^2*x^2])/(12*c) - (e*(a + b*ArcTan[c*x])*Log[1 + c^2*x^2])/(4*c^4) + (1/4)*x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])} -{x^2*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]), x, 21, (2*a*e*x)/(3*c^2) + (5*b*e*x^2)/(18*c) - (2/9)*a*e*x^3 - (2*a*e*ArcTan[c*x])/(3*c^3) + (2*b*e*x*ArcTan[c*x])/(3*c^2) - (2/9)*b*e*x^3*ArcTan[c*x] - (b*e*ArcTan[c*x]^2)/(3*c^3) - (11*b*e*Log[1 + c^2*x^2])/(18*c^3) - (b*e*Log[1 + c^2*x^2]^2)/(12*c^3) - (b*x^2*(d + e*Log[1 + c^2*x^2]))/(6*c) + (1/3)*x^3*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]) + (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(6*c^3)} -{x^1*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]), x, 7, -((b*(d - e)*x)/(2*c)) + (b*e*x)/c + (b*(d - e)*ArcTan[c*x])/(2*c^2) - (b*e*ArcTan[c*x])/c^2 + (1/2)*d*x^2*(a + b*ArcTan[c*x]) - (1/2)*e*x^2*(a + b*ArcTan[c*x]) - (b*e*x*Log[1 + c^2*x^2])/(2*c) + (e*(1 + c^2*x^2)*(a + b*ArcTan[c*x])*Log[1 + c^2*x^2])/(2*c^2)} -{x^0*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]), x, 9, -2*a*e*x - 2*b*e*x*ArcTan[c*x] + (e*(a + b*ArcTan[c*x])^2)/(b*c) + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]) - (b*(d + e*Log[1 + c^2*x^2])^2)/(4*c*e)} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^1, x, 18, a*d*Log[x] + (1/2)*I*b*e*Log[I*c*x]*Log[1 - I*c*x]^2 - (1/2)*I*b*e*Log[(-I)*c*x]*Log[1 + I*c*x]^2 + (1/2)*I*b*d*PolyLog[2, (-I)*c*x] - (1/2)*I*b*e*(Log[1 - I*c*x] + Log[1 + I*c*x] - Log[1 + c^2*x^2])*PolyLog[2, (-I)*c*x] - (1/2)*I*b*d*PolyLog[2, I*c*x] + (1/2)*I*b*e*(Log[1 - I*c*x] + Log[1 + I*c*x] - Log[1 + c^2*x^2])*PolyLog[2, I*c*x] - (1/2)*a*e*PolyLog[2, (-c^2)*x^2] + I*b*e*Log[1 - I*c*x]*PolyLog[2, 1 - I*c*x] - I*b*e*Log[1 + I*c*x]*PolyLog[2, 1 + I*c*x] - I*b*e*PolyLog[3, 1 - I*c*x] + I*b*e*PolyLog[3, 1 + I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^2, x, 6, (c*e*(a + b*ArcTan[c*x])^2)/b - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x + (1/2)*b*c*(d + e*Log[1 + c^2*x^2])*Log[1 - 1/(1 + c^2*x^2)] - (1/2)*b*c*e*PolyLog[2, 1/(1 + c^2*x^2)]} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^3, x, 10, b*c^2*e*ArcTan[c*x] + a*c^2*e*Log[x] - (1/2)*a*c^2*e*Log[1 + c^2*x^2] - (b*c*(d + e*Log[1 + c^2*x^2]))/(2*x) - (1/2)*b*c^2*ArcTan[c*x]*(d + e*Log[1 + c^2*x^2]) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(2*x^2) + (1/2)*I*b*c^2*e*PolyLog[2, (-I)*c*x] - (1/2)*I*b*c^2*e*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^4, x, 15, -((2*c^2*e*(a + b*ArcTan[c*x]))/(3*x)) - (c^3*e*(a + b*ArcTan[c*x])^2)/(3*b) + b*c^3*e*Log[x] - (1/3)*b*c^3*e*Log[1 + c^2*x^2] - (b*c*(1 + c^2*x^2)*(d + e*Log[1 + c^2*x^2]))/(6*x^2) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(3*x^3) - (1/6)*b*c^3*(d + e*Log[1 + c^2*x^2])*Log[1 - 1/(1 + c^2*x^2)] + (1/6)*b*c^3*e*PolyLog[2, 1/(1 + c^2*x^2)]} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^5, x, 15, -((a*c^2*e)/(4*x^2)) - (5*b*c^3*e)/(12*x) - (11/12)*b*c^4*e*ArcTan[c*x] - (b*c^2*e*ArcTan[c*x])/(4*x^2) - (1/2)*a*c^4*e*Log[x] + (1/4)*a*c^4*e*Log[1 + c^2*x^2] - (b*c*(d + e*Log[1 + c^2*x^2]))/(12*x^3) + (b*c^3*(d + e*Log[1 + c^2*x^2]))/(4*x) + (1/4)*b*c^4*ArcTan[c*x]*(d + e*Log[1 + c^2*x^2]) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(4*x^4) - (1/4)*I*b*c^4*e*PolyLog[2, (-I)*c*x] + (1/4)*I*b*c^4*e*PolyLog[2, I*c*x]} -{(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2])/x^6, x, 24, -((7*b*c^3*e)/(60*x^2)) - (2*c^2*e*(a + b*ArcTan[c*x]))/(15*x^3) + (2*c^4*e*(a + b*ArcTan[c*x]))/(5*x) + (c^5*e*(a + b*ArcTan[c*x])^2)/(5*b) - (5/6)*b*c^5*e*Log[x] + (19/60)*b*c^5*e*Log[1 + c^2*x^2] - (b*c*(d + e*Log[1 + c^2*x^2]))/(20*x^4) + (b*c^3*(1 + c^2*x^2)*(d + e*Log[1 + c^2*x^2]))/(10*x^2) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(5*x^5) + (1/10)*b*c^5*(d + e*Log[1 + c^2*x^2])*Log[1 - 1/(1 + c^2*x^2)] - (1/10)*b*c^5*e*PolyLog[2, 1/(1 + c^2*x^2)]} - - -{x^1*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]), x, 21, -((b*(d - e)*x)/(2*c)) + (b*e*x)/c + (b*(d - e)*ArcTan[c*x])/(2*c^2) + (1/2)*d*x^2*(a + b*ArcTan[c*x]) - (1/2)*e*x^2*(a + b*ArcTan[c*x]) - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c*Sqrt[g]) - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[2/(1 - I*c*x)])/(c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(2*c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(2*c^2*g) - (b*e*x*Log[f + g*x^2])/(2*c) - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[f + g*x^2])/(2*c^2*g) + (e*(f + g*x^2)*(a + b*ArcTan[c*x])*Log[f + g*x^2])/(2*g) + (I*b*e*(c^2*f - g)*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*c^2*g) - (I*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(4*c^2*g) - (I*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(4*c^2*g)} -{x^0*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]), x, 28, -2*a*e*x - 2*b*e*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + (I*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/(2*Sqrt[g]) - (I*b*e*Sqrt[-f]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[g]) + (I*b*e*Sqrt[-f]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/(2*Sqrt[g]) - (I*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[g]) + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]) - (b*Log[-((g*(1 + c^2*x^2))/(c^2*f - g))]*(d + e*Log[f + g*x^2]))/(2*c) - (I*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[g]) + (I*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) + (I*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) - (I*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[g]) - (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/(2*c)} -{(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2])/x^1, x, 8, b*e*CannotIntegrate[(ArcTan[c*x]*Log[f + g*x^2])/x, x] + a*d*Log[x] + (1/2)*a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2] + (1/2)*I*b*d*PolyLog[2, (-I)*c*x] - (1/2)*I*b*d*PolyLog[2, I*c*x] + (1/2)*a*e*PolyLog[2, 1 + (g*x^2)/f]} -{(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2])/x^2, x, 28, (2*a*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - (I*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/(2*Sqrt[-f]) + (I*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[-f]) - (I*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/(2*Sqrt[-f]) + (I*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[-f]) - ((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x + (1/2)*b*c*Log[-((g*x^2)/f)]*(d + e*Log[f + g*x^2]) - (1/2)*b*c*Log[-((g*(1 + c^2*x^2))/(c^2*f - g))]*(d + e*Log[f + g*x^2]) + (I*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[-f]) - (I*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) - (I*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) + (I*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/(2*Sqrt[-f]) - (1/2)*b*c*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)] + (1/2)*b*c*e*PolyLog[2, 1 + (g*x^2)/f]} -{(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2])/x^3, x, 22, (b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[2/(1 - I*c*x)])/f + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(2*f) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(2*f) - (a*e*g*Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) - (1/2)*b*c^2*ArcTan[c*x]*(d + e*Log[f + g*x^2]) - ((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/(2*x^2) + (I*b*e*g*PolyLog[2, (-I)*c*x])/(2*f) - (I*b*e*g*PolyLog[2, I*c*x])/(2*f) + (I*b*e*(c^2*f - g)*PolyLog[2, 1 - 2/(1 - I*c*x)])/(2*f) - (I*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(4*f) - (I*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(4*f)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m deleted file mode 100644 index d0ea16d..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.5 u (a+b arctan(c+d x))^p.m +++ /dev/null @@ -1,164 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcTan[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTan[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTan[c+d x])^p when d e-c f=0*) - - -{(c*e + d*e*x)^3*(a + b*ArcTan[c + d*x]), x, 6, (b*e^3*x)/4 - (b*e^3*(c + d*x)^3)/(12*d) - (b*e^3*ArcTan[c + d*x])/(4*d) + (e^3*(c + d*x)^4*(a + b*ArcTan[c + d*x]))/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x]), x, 6, -(b*e^2*(c + d*x)^2)/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcTan[c + d*x]))/(3*d) + (b*e^2*Log[1 + (c + d*x)^2])/(6*d)} -{(c*e + d*e*x)^1*(a + b*ArcTan[c + d*x]), x, 5, -(b*e*x)/2 + (b*e*ArcTan[c + d*x])/(2*d) + (e*(c + d*x)^2*(a + b*ArcTan[c + d*x]))/(2*d)} -{(a + b*ArcTan[c + d*x])/(c*e + d*e*x)^1, x, 5, (a*Log[c + d*x])/(d*e) + ((I/2)*b*PolyLog[2, (-I)*(c + d*x)])/(d*e) - ((I/2)*b*PolyLog[2, I*(c + d*x)])/(d*e)} -{(a + b*ArcTan[c + d*x])/(c*e + d*e*x)^2, x, 7, -((a + b*ArcTan[c + d*x])/(d*e^2*(c + d*x))) + (b*Log[c + d*x])/(d*e^2) - (b*Log[1 + (c + d*x)^2])/(2*d*e^2)} -{(a + b*ArcTan[c + d*x])/(c*e + d*e*x)^3, x, 5, -b/(2*d*e^3*(c + d*x)) - (b*ArcTan[c + d*x])/(2*d*e^3) - (a + b*ArcTan[c + d*x])/(2*d*e^3*(c + d*x)^2)} - - -{(c*e + d*e*x)^3*(a + b*ArcTan[c + d*x])^2, x, 13, (a*b*e^3*x)/2 + (b^2*e^3*(c + d*x)^2)/(12*d) + (b^2*e^3*(c + d*x)*ArcTan[c + d*x])/(2*d) - (b*e^3*(c + d*x)^3*(a + b*ArcTan[c + d*x]))/(6*d) - (e^3*(a + b*ArcTan[c + d*x])^2)/(4*d) + (e^3*(c + d*x)^4*(a + b*ArcTan[c + d*x])^2)/(4*d) - (b^2*e^3*Log[1 + (c + d*x)^2])/(3*d)} -{(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^2, x, 11, (b^2*e^2*x)/3 - (b^2*e^2*ArcTan[c + d*x])/(3*d) - (b*e^2*(c + d*x)^2*(a + b*ArcTan[c + d*x]))/(3*d) - ((I/3)*e^2*(a + b*ArcTan[c + d*x])^2)/d + (e^2*(c + d*x)^3*(a + b*ArcTan[c + d*x])^2)/(3*d) - (2*b*e^2*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/(3*d) - ((I/3)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d} -{(c*e + d*e*x)^1*(a + b*ArcTan[c + d*x])^2, x, 8, -(a*b*e*x) - (b^2*e*(c + d*x)*ArcTan[c + d*x])/d + (e*(a + b*ArcTan[c + d*x])^2)/(2*d) + (e*(c + d*x)^2*(a + b*ArcTan[c + d*x])^2)/(2*d) + (b^2*e*Log[1 + (c + d*x)^2])/(2*d)} -{(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^1, x, 8, (2*(a + b*ArcTan[c + d*x])^2*ArcTanh[1 - 2/(1 + I*(c + d*x))])/(d*e) - (I*b*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d*e) + (I*b*(a + b*ArcTan[c + d*x])*PolyLog[2, -1 + 2/(1 + I*(c + d*x))])/(d*e) - (b^2*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d*e) + (b^2*PolyLog[3, -1 + 2/(1 + I*(c + d*x))])/(2*d*e)} -{(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^2, x, 6, ((-I)*(a + b*ArcTan[c + d*x])^2)/(d*e^2) - (a + b*ArcTan[c + d*x])^2/(d*e^2*(c + d*x)) + (2*b*(a + b*ArcTan[c + d*x])*Log[2 - 2/(1 - I*(c + d*x))])/(d*e^2) - (I*b^2*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^2)} -{(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^3, x, 10, -((b*(a + b*ArcTan[c + d*x]))/(d*e^3*(c + d*x))) - (a + b*ArcTan[c + d*x])^2/(2*d*e^3) - (a + b*ArcTan[c + d*x])^2/(2*d*e^3*(c + d*x)^2) + (b^2*Log[c + d*x])/(d*e^3) - (b^2*Log[1 + (c + d*x)^2])/(2*d*e^3)} -{(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^4, x, 10, -b^2/(3*d*e^4*(c + d*x)) - (b^2*ArcTan[c + d*x])/(3*d*e^4) - (b*(a + b*ArcTan[c + d*x]))/(3*d*e^4*(c + d*x)^2) + ((I/3)*(a + b*ArcTan[c + d*x])^2)/(d*e^4) - (a + b*ArcTan[c + d*x])^2/(3*d*e^4*(c + d*x)^3) - (2*b*(a + b*ArcTan[c + d*x])*Log[2 - 2/(1 - I*(c + d*x))])/(3*d*e^4) + ((I/3)*b^2*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^4)} -{(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^5, x, 15, -b^2/(12*d*e^5*(c + d*x)^2) - (b*(a + b*ArcTan[c + d*x]))/(6*d*e^5*(c + d*x)^3) + (b*(a + b*ArcTan[c + d*x]))/(2*d*e^5*(c + d*x)) + (a + b*ArcTan[c + d*x])^2/(4*d*e^5) - (a + b*ArcTan[c + d*x])^2/(4*d*e^5*(c + d*x)^4) - (2*b^2*Log[c + d*x])/(3*d*e^5) + (b^2*Log[1 + (c + d*x)^2])/(3*d*e^5)} - - -{(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^3, x, 14, a*b^2*e^2*x + (b^3*e^2*(c + d*x)*ArcTan[c + d*x])/d - (b*e^2*(a + b*ArcTan[c + d*x])^2)/(2*d) - (b*e^2*(c + d*x)^2*(a + b*ArcTan[c + d*x])^2)/(2*d) - ((I/3)*e^2*(a + b*ArcTan[c + d*x])^3)/d + (e^2*(c + d*x)^3*(a + b*ArcTan[c + d*x])^3)/(3*d) - (b*e^2*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d - (b^3*e^2*Log[1 + (c + d*x)^2])/(2*d) - (I*b^2*e^2*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (b^3*e^2*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)} -{(c*e + d*e*x)^1*(a + b*ArcTan[c + d*x])^3, x, 10, (((-3*I)/2)*b*e*(a + b*ArcTan[c + d*x])^2)/d - (3*b*e*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/(2*d) + (e*(a + b*ArcTan[c + d*x])^3)/(2*d) + (e*(c + d*x)^2*(a + b*ArcTan[c + d*x])^3)/(2*d) - (3*b^2*e*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/d - (((3*I)/2)*b^3*e*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d} -{(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^1, x, 10, (2*(a + b*ArcTan[c + d*x])^3*ArcTanh[1 - 2/(1 + I*(c + d*x))])/(d*e) - (((3*I)/2)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d*e) + (((3*I)/2)*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, -1 + 2/(1 + I*(c + d*x))])/(d*e) - (3*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d*e) + (3*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, -1 + 2/(1 + I*(c + d*x))])/(2*d*e) + (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 + I*(c + d*x))])/(d*e) - (((3*I)/4)*b^3*PolyLog[4, -1 + 2/(1 + I*(c + d*x))])/(d*e)} -{(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^2, x, 7, ((-I)*(a + b*ArcTan[c + d*x])^3)/(d*e^2) - (a + b*ArcTan[c + d*x])^3/(d*e^2*(c + d*x)) + (3*b*(a + b*ArcTan[c + d*x])^2*Log[2 - 2/(1 - I*(c + d*x))])/(d*e^2) - ((3*I)*b^2*(a + b*ArcTan[c + d*x])*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^2) + (3*b^3*PolyLog[3, -1 + 2/(1 - I*(c + d*x))])/(2*d*e^2)} -{(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^3, x, 9, (((-3*I)/2)*b*(a + b*ArcTan[c + d*x])^2)/(d*e^3) - (3*b*(a + b*ArcTan[c + d*x])^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcTan[c + d*x])^3/(2*d*e^3) - (a + b*ArcTan[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcTan[c + d*x])*Log[2 - 2/(1 - I*(c + d*x))])/(d*e^3) - (((3*I)/2)*b^3*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^3)} -{(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^4, x, 16, -((b^2*(a + b*ArcTan[c + d*x]))/(d*e^4*(c + d*x))) - (b*(a + b*ArcTan[c + d*x])^2)/(2*d*e^4) - (b*(a + b*ArcTan[c + d*x])^2)/(2*d*e^4*(c + d*x)^2) + ((I/3)*(a + b*ArcTan[c + d*x])^3)/(d*e^4) - (a + b*ArcTan[c + d*x])^3/(3*d*e^4*(c + d*x)^3) + (b^3*Log[c + d*x])/(d*e^4) - (b^3*Log[1 + (c + d*x)^2])/(2*d*e^4) - (b*(a + b*ArcTan[c + d*x])^2*Log[2 - 2/(1 - I*(c + d*x))])/(d*e^4) + (I*b^2*(a + b*ArcTan[c + d*x])*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^4) - (b^3*PolyLog[3, -1 + 2/(1 - I*(c + d*x))])/(2*d*e^4)} - - -{ArcTan[1 + x]/(2 + 2*x), x, 5, (1/4)*I*PolyLog[2, (-I)*(1 + x)] - (1/4)*I*PolyLog[2, I*(1 + x)]} - - -{ArcTan[a + b*x]/((a*d)/b + d*x), x, 5, (I*PolyLog[2, (-I)*(a + b*x)])/(2*d) - (I*PolyLog[2, I*(a + b*x)])/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTan[c+d x])^(p/2) when d e-c f=0*) - - -{(a + b*x)^2*ArcTan[a + b*x]^(1/2), x, 0, Unintegrable[(a + b*x)^2*Sqrt[ArcTan[a + b*x]], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTan[c+d x])^p*) - - -{(e + f*x)^3*(a + b*ArcTan[c + d*x]), x, 7, -((b*f*(6*d^2*e^2 - 12*c*d*e*f - (1 - 6*c^2)*f^2)*x)/(4*d^3)) - (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) - (b*f^3*(c + d*x)^3)/(12*d^4) - (b*(d^4*e^4 - 4*c*d^3*e^3*f - 6*(1 - c^2)*d^2*e^2*f^2 + 4*c*(3 - c^2)*d*e*f^3 + (1 - 6*c^2 + c^4)*f^4)*ArcTan[c + d*x])/(4*d^4*f) + ((e + f*x)^4*(a + b*ArcTan[c + d*x]))/(4*f) - (b*(d*e - c*f)*(d*e + f - c*f)*(d*e - (1 + c)*f)*Log[1 + (c + d*x)^2])/(2*d^4)} -{(e + f*x)^2*(a + b*ArcTan[c + d*x]), x, 7, -((b*f*(d*e - c*f)*x)/d^2) - (b*f^2*(c + d*x)^2)/(6*d^3) - (b*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*ArcTan[c + d*x])/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcTan[c + d*x]))/(3*f) - (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*Log[1 + (c + d*x)^2])/(6*d^3)} -{(e + f*x)^1*(a + b*ArcTan[c + d*x]), x, 7, -((b*f*x)/(2*d)) - (b*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcTan[c + d*x]))/(2*f) - (b*(d*e - c*f)*Log[1 + (c + d*x)^2])/(2*d^2)} -{(e + f*x)^0*(a + b*ArcTan[c + d*x]), x, 4, a*x + (b*(c + d*x)*ArcTan[c + d*x])/d - (b*Log[1 + (c + d*x)^2])/(2*d)} -{(a + b*ArcTan[c + d*x])/(e + f*x)^1, x, 5, -(((a + b*ArcTan[c + d*x])*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcTan[c + d*x])*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f + (I*b*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(2*f) - (I*b*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f)} -{(a + b*ArcTan[c + d*x])/(e + f*x)^2, x, 8, (b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcTan[c + d*x])/(f*(e + f*x)) + (b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (b*d*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))} -{(a + b*ArcTan[c + d*x])/(e + f*x)^3, x, 9, -((b*d)/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(e + f*x))) + (b*d^2*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2) - (a + b*ArcTan[c + d*x])/(2*f*(e + f*x)^2) + (b*d^2*(d*e - c*f)*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2 - (b*d^2*(d*e - c*f)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2)} - - -{(e + f*x)^2*(a + b*ArcTan[c + d*x])^2, x, 16, (b^2*f^2*x)/(3*d^2) - (2*a*b*f*(d*e - c*f)*x)/d^2 - (b^2*f^2*ArcTan[c + d*x])/(3*d^3) - (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcTan[c + d*x])/d^3 - (b*f^2*(c + d*x)^2*(a + b*ArcTan[c + d*x]))/(3*d^3) + (I*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])^2)/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcTan[c + d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcTan[c + d*x])^2)/(3*f) + (2*b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 + (c + d*x)^2])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(3*d^3)} -{(e + f*x)^1*(a + b*ArcTan[c + d*x])^2, x, 13, -((a*b*f*x)/d) - (b^2*f*(c + d*x)*ArcTan[c + d*x])/d^2 + (I*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcTan[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcTan[c + d*x])^2)/(2*f) + (2*b*(d*e - c*f)*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 + (b^2*f*Log[1 + (c + d*x)^2])/(2*d^2) + (I*b^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2} -{(e + f*x)^0*(a + b*ArcTan[c + d*x])^2, x, 6, (I*(a + b*ArcTan[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcTan[c + d*x])^2)/d + (2*b*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/d + (I*b^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d} -{(a + b*ArcTan[c + d*x])^2/(e + f*x)^1, x, 2, -(((a + b*ArcTan[c + d*x])^2*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcTan[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f + (I*b*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f - (I*b*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (b^2*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f)} -{(a + b*ArcTan[c + d*x])^2/(e + f*x)^2, x, 25, (2*a*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) + (I*b^2*d*ArcTan[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(d*e - c*f)*ArcTan[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcTan[c + d*x])^2/(f*(e + f*x)) + (2*a*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) - (2*b^2*d*ArcTan[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (2*b^2*d*ArcTan[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (2*b^2*d*ArcTan[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (a*b*d*Log[1 + (c + d*x)^2])/(f^2 + (d*e - c*f)^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (I*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)} - - -{(e + f*x)^2*(a + b*ArcTan[c + d*x])^3, x, 21, (a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcTan[c + d*x])/d^3 - (b*f^2*(a + b*ArcTan[c + d*x])^2)/(2*d^3) - (3*I*b*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2)/d^3 - (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/d^3 - (b*f^2*(c + d*x)^2*(a + b*ArcTan[c + d*x])^2)/(2*d^3) + (I*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])^3)/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcTan[c + d*x])^3)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcTan[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^3 + (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^3 - (b^3*f^2*Log[1 + (c + d*x)^2])/(2*d^3) - (3*I*b^3*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 + (b^3*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^3)} -{(e + f*x)^1*(a + b*ArcTan[c + d*x])^3, x, 15, -((3*I*b*f*(a + b*ArcTan[c + d*x])^2)/(2*d^2)) - (3*b*f*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/(2*d^2) + (I*(d*e - c*f)*(a + b*ArcTan[c + d*x])^3)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcTan[c + d*x])^3)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcTan[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 + (3*b*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^2 - (3*I*b^3*f*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(2*d^2) + (3*I*b^2*(d*e - c*f)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2 + (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^2)} -{(e + f*x)^0*(a + b*ArcTan[c + d*x])^3, x, 6, (I*(a + b*ArcTan[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcTan[c + d*x])^3)/d + (3*b*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d + (3*I*b^2*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d + (3*b^3*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)} -{(a + b*ArcTan[c + d*x])^3/(e + f*x)^1, x, 2, -(((a + b*ArcTan[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcTan[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f + (3*I*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(2*f) - (3*I*b*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) - (3*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*b^2*(a + b*ArcTan[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) - (3*I*b^3*PolyLog[4, 1 - 2/(1 - I*(c + d*x))])/(4*f) + (3*I*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(4*f)} -{(a + b*ArcTan[c + d*x])^3/(e + f*x)^2, x, 35, (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) + (3*I*a*b^2*d*ArcTan[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcTan[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcTan[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^3*d*(d*e - c*f)*ArcTan[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcTan[c + d*x])^3/(f*(e + f*x)) + (3*a^2*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) - (6*a*b^2*d*ArcTan[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*ArcTan[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (6*a*b^2*d*ArcTan[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*ArcTan[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (6*a*b^2*d*ArcTan[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*ArcTan[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*a^2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + (3*I*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*b^3*d*ArcTan[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*I*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*I*b^3*d*ArcTan[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*a*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*b^3*d*ArcTan[c + d*x]*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTan[c+d x])^p with m symbolic*) - - -{(e + f*x)^m*(a + b*ArcTan[c + d*x])^1, x, 6, ((e + f*x)^(1 + m)*(a + b*ArcTan[c + d*x]))/(f*(1 + m)) - (I*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + I*f - c*f)])/(2*f*(d*e + (I - c)*f)*(1 + m)*(2 + m)) + (I*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)])/(2*f*(d*e - (I + c)*f)*(1 + m)*(2 + m))} -{(e + f*x)^m*(a + b*ArcTan[c + d*x])^2, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcTan[c + d*x])^2, x]} -{(e + f*x)^m*(a + b*ArcTan[c + d*x])^3, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcTan[c + d*x])^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form AF[x] (a+b ArcTan[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcTan[a + b*x], x, 7, ((1 - 6*a^2)*x)/(4*b^3) + (a*(a + b*x)^2)/(2*b^4) - (a + b*x)^3/(12*b^4) - ((1 - 6*a^2 + a^4)*ArcTan[a + b*x])/(4*b^4) + (1/4)*x^4*ArcTan[a + b*x] - (a*(1 - a^2)*Log[1 + (a + b*x)^2])/(2*b^4)} -{x^2*ArcTan[a + b*x], x, 7, (a*x)/b^2 - (a + b*x)^2/(6*b^3) - (a*(3 - a^2)*ArcTan[a + b*x])/(3*b^3) + (1/3)*x^3*ArcTan[a + b*x] + ((1 - 3*a^2)*Log[1 + (a + b*x)^2])/(6*b^3)} -{x^1*ArcTan[a + b*x], x, 7, -(x/(2*b)) + ((1 - a^2)*ArcTan[a + b*x])/(2*b^2) + (1/2)*x^2*ArcTan[a + b*x] + (a*Log[1 + (a + b*x)^2])/(2*b^2)} -{x^0*ArcTan[a + b*x], x, 3, ((a + b*x)*ArcTan[a + b*x])/b - Log[1 + (a + b*x)^2]/(2*b)} -{ArcTan[a + b*x]/x^1, x, 5, (-ArcTan[a + b*x])*Log[2/(1 - I*(a + b*x))] + ArcTan[a + b*x]*Log[(2*b*x)/((I - a)*(1 - I*(a + b*x)))] + (1/2)*I*PolyLog[2, 1 - 2/(1 - I*(a + b*x))] - (1/2)*I*PolyLog[2, 1 - (2*b*x)/((I - a)*(1 - I*(a + b*x)))]} -{ArcTan[a + b*x]/x^2, x, 7, -((a*b*ArcTan[a + b*x])/(1 + a^2)) - ArcTan[a + b*x]/x + (b*Log[x])/(1 + a^2) - (b*Log[1 + (a + b*x)^2])/(2*(1 + a^2))} -{ArcTan[a + b*x]/x^3, x, 8, -(b/(2*(1 + a^2)*x)) - ((1 - a^2)*b^2*ArcTan[a + b*x])/(2*(1 + a^2)^2) - ArcTan[a + b*x]/(2*x^2) - (a*b^2*Log[x])/(1 + a^2)^2 + (a*b^2*Log[1 + (a + b*x)^2])/(2*(1 + a^2)^2)} -{ArcTan[a + b*x]/x^4, x, 8, -(b/(6*(1 + a^2)*x^2)) + (2*a*b^2)/(3*(1 + a^2)^2*x) + (a*(3 - a^2)*b^3*ArcTan[a + b*x])/(3*(1 + a^2)^3) - ArcTan[a + b*x]/(3*x^3) - ((1 - 3*a^2)*b^3*Log[x])/(3*(1 + a^2)^3) + ((1 - 3*a^2)*b^3*Log[1 + (a + b*x)^2])/(6*(1 + a^2)^3)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x^n) ArcTan[a+b x]*) - - -{ArcTan[a + b*x]/(c + d*x^3), x, 23, -((I*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))) + (I*Log[1 - I*a - I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/6)*Log[1 - I*a - I*b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(5/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(5/6)*Log[1 - I*a - I*b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/6)*(1 - I*a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - (I*PolyLog[2, (d^(1/3)*(I - a - b*x))/(b*c^(1/3) + (I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(5/6)*PolyLog[2, -(((-1)^(1/6)*d^(1/3)*(I - a - b*x))/(I*b*c^(1/3) - (-1)^(1/6)*(I - a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/6)*PolyLog[2, -(((-1)^(1/3)*d^(1/3)*(I - a - b*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3)) + (I*PolyLog[2, -((d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (I + a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(5/6)*PolyLog[2, -(((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (-1)^(2/3)*(I + a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3))} -{ArcTan[a + b*x]/(c + d*x^2), x, 17, -((I*Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])) + (I*Log[1 - I*a - I*b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (I*Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (I*Log[1 - I*a - I*b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (I*PolyLog[2, -((Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] - (I - a)*Sqrt[d]))])/(4*Sqrt[-c]*Sqrt[d]) + (I*PolyLog[2, (Sqrt[d]*(I - a - b*x))/(b*Sqrt[-c] + (I - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (I*PolyLog[2, -((Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] - (I + a)*Sqrt[d]))])/(4*Sqrt[-c]*Sqrt[d]) + (I*PolyLog[2, (Sqrt[d]*(I + a + b*x))/(b*Sqrt[-c] + (I + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])} -{ArcTan[a + b*x]/(c + d*x), x, 5, -((ArcTan[a + b*x]*Log[2/(1 - I*(a + b*x))])/d) + (ArcTan[a + b*x]*Log[(2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d + (I*PolyLog[2, 1 - 2/(1 - I*(a + b*x))])/(2*d) - (I*PolyLog[2, 1 - (2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/(2*d)} -{ArcTan[a + b*x]/(c + d/x), x, 15, -(((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c)) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - (I*d*Log[1 - I*a - I*b*x]*Log[-((b*(d + c*x))/((I + a)*c - b*d))])/(2*c^2) + (I*d*Log[1 + I*a + I*b*x]*Log[(b*(d + c*x))/((I - a)*c + b*d)])/(2*c^2) + (I*d*PolyLog[2, (c*(I - a - b*x))/(I*c - a*c + b*d)])/(2*c^2) - (I*d*PolyLog[2, (c*(I + a + b*x))/((I + a)*c - b*d)])/(2*c^2)} -{ArcTan[a + b*x]/(c + d/x^2), x, 25, -(((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c)) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) + (I*Sqrt[d]*Log[1 + I*a + I*b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/(I*Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (I*Sqrt[d]*Log[1 - I*a - I*b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(I*Sqrt[-c] + a*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (I*Sqrt[d]*Log[1 - I*a - I*b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((I + a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (I*Sqrt[d]*Log[1 + I*a + I*b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/(I*Sqrt[-c] - a*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/(I*Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + I*a + I*b*x))/((1 + I*a)*Sqrt[-c] - I*b*Sqrt[d])])/(4*(-c)^(3/2)) + (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/(I*Sqrt[-c] + a*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/(I*Sqrt[-c] + a*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))} -{ArcTan[a + b*x]/(c + d/x^3), x, 31, -(((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c)) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - (I*d^(1/3)*Log[1 - I*a - I*b*x]*Log[-((b*(d^(1/3) + c^(1/3)*x))/((I + a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) + (I*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/3) + c^(1/3)*x))/((I - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(1/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[-((b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) + ((-1)^(1/6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(I + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(I - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(5/6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(1/6)*(1 - I*a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I - a - b*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(1/6)*c^(1/3)*(I - a - b*x))/((-1)^(1/6)*(I - a)*c^(1/3) - I*b*d^(1/3))])/(6*c^(4/3)) + (I*d^(1/3)*PolyLog[2, (c^(1/3)*(I - a - b*x))/((I - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - (I*d^(1/3)*PolyLog[2, (c^(1/3)*(I + a + b*x))/((I + a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(I + a + b*x))/((-1)^(2/3)*(I + a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I + a + b*x))/((-1)^(1/3)*(I + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3))} - - -(* {ArcTan[a + b*x]/(a + b*x^(3/2)), x, 41, ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[-(((-1)^(3/4)*((-1)^(1/4)*Sqrt[1 + I*a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a] - (-1)^(1/12)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[-(((-1)^(3/4)*((-1)^(1/4)*Sqrt[1 + I*a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a] + (-1)^(5/12)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[((-1)^(1/4)*Sqrt[1 + I*a] + Sqrt[b]*Sqrt[x])/((-1)^(1/4)*Sqrt[1 + I*a] - a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 + I*a] - (-1)^(3/4)*Sqrt[b]*Sqrt[x])/(Sqrt[1 + I*a] + (-1)^(1/12)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 + I*a] - (-1)^(3/4)*Sqrt[b]*Sqrt[x])/(Sqrt[1 + I*a] - (-1)^(5/12)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[((-1)^(1/4)*(Sqrt[1 + I*a] + (-1)^(3/4)*Sqrt[b]*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a] + a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) + ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/(a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/(3*a^(1/3)*b^(2/3)) - ((I/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(-I)*(I + a + b*x)])/(a^(1/3)*b^(2/3)) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(-I)*(I + a + b*x)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(-I)*(I + a + b*x)])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((I/3)*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - a^(1/3)*b^(1/6)))])/(a^(1/3)*b^(2/3)) - ((I/3)*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a] - a^(1/3)*b^(1/6)))])/(a^(1/3)*b^(2/3)) + ((I/3)*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) - ((I/3)*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a] + a^(1/3)*b^(1/6))])/(a^(1/3)*b^(2/3)) + ((-1)^(5/6)*PolyLog[2, ((-1)^(3/4)*b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a] - (-1)^(5/12)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*PolyLog[2, ((-1)^(5/12)*b^(1/6)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a] + (-1)^(5/12)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*PolyLog[2, ((-1)^(1/12)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a] + (-1)^(1/12)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*PolyLog[2, -((b^(1/6)*((-1)^(1/12)*a^(1/3) + (-1)^(3/4)*b^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a] - (-1)^(1/12)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3))} *) -{ArcTan[a + b*x]/(c + d*Sqrt[x]), x, 31, (2*I*Sqrt[I + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*d) - (2*I*Sqrt[I - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*d) + (I*c*Log[(d*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 - (I*c*Log[(d*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 + (I*c*Log[-((d*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (I*c*Log[-((d*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 + (I*Sqrt[x]*Log[1 - I*a - I*b*x])/d - (I*c*Log[c + d*Sqrt[x]]*Log[1 - I*a - I*b*x])/d^2 - (I*Sqrt[x]*Log[1 + I*a + I*b*x])/d + (I*c*Log[c + d*Sqrt[x]]*Log[1 + I*a + I*b*x])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d)])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)])/d^2} -{ArcTan[a + b*x]/(c + d/Sqrt[x]), x, 37, -((2*I*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2)) + (2*I*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*c^2) - (I*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d*Sqrt[x]*Log[1 - I*a - I*b*x])/c^2 + (I*d^2*Log[d + c*Sqrt[x]]*Log[1 - I*a - I*b*x])/c^3 + (I*d*Sqrt[x]*Log[1 + I*a + I*b*x])/c^2 - ((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c) - (I*d^2*Log[d + c*Sqrt[x]]*Log[1 + I*a + I*b*x])/c^3 - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d))])/c^3 + (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d))])/c^3 - (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)])/c^3 + (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/c^3} -(* {ArcTan[a + b*x]/(a + b/x^(3/2)), x, 49, -((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[-(((-1)^(3/4)*a^(1/3)*((-1)^(1/4)*Sqrt[1 + I*a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) + (-1)^(5/12)*b^(5/6)))])/(3*a^(5/3)) - ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*((-1)^(3/4)*Sqrt[1 - I*a] + Sqrt[b]*Sqrt[x]))/((-1)^(3/4)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6))])/a^(5/3) + ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*((-1)^(1/4)*Sqrt[1 + I*a] + Sqrt[b]*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a]*a^(1/3) - b^(5/6))])/a^(5/3) + ((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[((-1)^(1/12)*a^(1/3)*(Sqrt[1 - I*a] - (-1)^(1/4)*Sqrt[b]*Sqrt[x]))/((-1)^(1/12)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[((-1)^(5/12)*a^(1/3)*(Sqrt[1 - I*a] - (-1)^(1/4)*Sqrt[b]*Sqrt[x]))/((-1)^(5/12)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[((-1)^(5/12)*a^(1/3)*(Sqrt[1 - I*a] + (-1)^(1/4)*Sqrt[b]*Sqrt[x]))/((-1)^(5/12)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[((-1)^(1/12)*a^(1/3)*(Sqrt[1 - I*a] + (-1)^(1/4)*Sqrt[b]*Sqrt[x]))/((-1)^(1/12)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[((-1)^(3/4)*a^(1/3)*(Sqrt[1 - I*a] + (-1)^(1/4)*Sqrt[b]*Sqrt[x]))/((-1)^(3/4)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/a^(5/3) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 + I*a] - (-1)^(3/4)*Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) + (-1)^(1/12)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 + I*a] - (-1)^(3/4)*Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) - (-1)^(5/12)*b^(5/6))])/(3*a^(5/3)) + ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[((-1)^(1/4)*a^(1/3)*(Sqrt[1 + I*a] + (-1)^(3/4)*Sqrt[b]*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a]*a^(1/3) + b^(5/6))])/a^(5/3) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 + I*a] + (-1)^(3/4)*Sqrt[b]*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) - (-1)^(1/12)*b^(5/6))])/(3*a^(5/3)) - ((1 - I*a - I*b*x)*Log[1 - I*a - I*b*x])/(2*a*b) + ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - I*a - I*b*x])/a^(5/3) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - I*a - I*b*x])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 - I*a - I*b*x])/(3*a^(5/3)) - ((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*a*b) - ((I/3)*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/a^(5/3) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 + I*a + I*b*x])/(3*a^(5/3)) - ((I/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/((-1)^(3/4)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6)))])/a^(5/3) + ((I/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a]*a^(1/3) - b^(5/6)))])/a^(5/3) - ((I/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/((-1)^(3/4)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/a^(5/3) + ((I/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/((-1)^(1/4)*Sqrt[1 + I*a]*a^(1/3) + b^(5/6))])/a^(5/3) + ((-1)^(5/6)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/((-1)^(1/12)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*PolyLog[2, ((-1)^(3/4)*Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) - (-1)^(5/12)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) - (-1)^(1/3)*a^(1/3)*Sqrt[x]))/((-1)^(1/12)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*PolyLog[2, ((-1)^(5/12)*Sqrt[b]*(b^(1/3) - (-1)^(1/3)*a^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) + (-1)^(5/12)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/((-1)^(5/12)*Sqrt[1 - I*a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/((-1)^(5/12)*Sqrt[1 - I*a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*PolyLog[2, ((-1)^(1/12)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) + (-1)^(1/12)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/12)*b^(1/3) + (-1)^(3/4)*a^(1/3)*Sqrt[x]))/(Sqrt[1 + I*a]*a^(1/3) - (-1)^(1/12)*b^(5/6)))])/(3*a^(5/3))} *) - - -{ArcTan[a + b*x]/(1 + x^2), x, 17, (1/4)*Log[(b*(I - x))/(a + I*(1 + b))]*Log[1 - I*a - I*b*x] - (1/4)*Log[-((b*(I + x))/(a + I*(1 - b)))]*Log[1 - I*a - I*b*x] - (1/4)*Log[(b*(I - x))/(a - I*(1 - b))]*Log[1 + I*a + I*b*x] + (1/4)*Log[-((b*(I + x))/(a - I*(1 + b)))]*Log[1 + I*a + I*b*x] - (1/4)*PolyLog[2, -((I - a - b*x)/(a - I*(1 - b)))] + (1/4)*PolyLog[2, -((I - a - b*x)/(a - I*(1 + b)))] - (1/4)*PolyLog[2, (I + a + b*x)/(I + a - I*b)] + (1/4)*PolyLog[2, (I + a + b*x)/(a + I*(1 + b))]} - - -{ArcTan[d + e*x]/(a + b*x^2), x, 17, (I*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)]*Log[1 - I*d - I*e*x])/(4*Sqrt[-a]*Sqrt[b]) - (I*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*(I + d) - Sqrt[-a]*e))]*Log[1 - I*d - I*e*x])/(4*Sqrt[-a]*Sqrt[b]) - (I*Log[-((e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*(I - d) - Sqrt[-a]*e))]*Log[1 + I*d + I*e*x])/(4*Sqrt[-a]*Sqrt[b]) + (I*Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*(I - d) + Sqrt[-a]*e)]*Log[1 + I*d + I*e*x])/(4*Sqrt[-a]*Sqrt[b]) - (I*PolyLog[2, (Sqrt[b]*(I - d - e*x))/(Sqrt[b]*(I - d) - Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[b]) + (I*PolyLog[2, (Sqrt[b]*(I - d - e*x))/(Sqrt[b]*(I - d) + Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[b]) - (I*PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) - Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[b]) + (I*PolyLog[2, (Sqrt[b]*(I + d + e*x))/(Sqrt[b]*(I + d) + Sqrt[-a]*e)])/(4*Sqrt[-a]*Sqrt[b])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B x+C x^2)^q ArcTan[a+b x]*) - - -{ArcTan[d + e*x]/(a + b*x + c*x^2), x, 12, (ArcTan[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - (ArcTan[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - (I*PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*I*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 - I*(d + e*x)))])/(2*Sqrt[b^2 - 4*a*c]) + (I*PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/(2*Sqrt[b^2 - 4*a*c])} - - -{ArcTan[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 2, -((2*I*ArcTan[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b) + (I*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/b - (I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b} -{ArcTan[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2], x, 3, -((2*I*Sqrt[1 + (a + b*x)^2]*ArcTan[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])) + (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(b*Sqrt[c + c*(a + b*x)^2]) - (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])} - - -{ArcTan[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3), x, 1, Unintegrable[ArcTan[a + b*x]/(1 + (a + b*x)^2)^(1/3), x]} -{ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3), x, 1, Unintegrable[ArcTan[a + b*x]/(c + c*(a + b*x)^2)^(1/3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (A+B x+C x^2)^q ArcTan[a+b x]^p*) - - -{(a + b*x)^2*ArcTan[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 4, -(Sqrt[1 + (a + b*x)^2]/(2*b)) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcTan[a + b*x])/(2*b) + (I*ArcTan[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b - (I*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(2*b) + (I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(2*b)} -{(a + b*x)^2*ArcTan[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2], x, 5, -(Sqrt[c + c*(a + b*x)^2]/(2*b*c)) + ((a + b*x)*Sqrt[c + c*(a + b*x)^2]*ArcTan[a + b*x])/(2*b*c) + (I*Sqrt[1 + (a + b*x)^2]*ArcTan[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) - (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(2*b*Sqrt[c + c*(a + b*x)^2]) + (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(2*b*Sqrt[c + c*(a + b*x)^2])} - - -{(a + b*x)^2*ArcTan[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3), x, 1, Unintegrable[((a + b*x)^2*ArcTan[a + b*x])/(1 + (a + b*x)^2)^(1/3), x]} -{(a + b*x)^2*ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3), x, 1, Unintegrable[((a + b*x)^2*ArcTan[a + b*x])/(c + c*(a + b*x)^2)^(1/3), x]} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m deleted file mode 100644 index aec7b7e..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.6 Exponentials of inverse tangent.m +++ /dev/null @@ -1,723 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n I ArcTan[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(I*ArcTan[a*x])*x^4, x, 6, -((4*I*x^2*Sqrt[1 + a^2*x^2])/(15*a^3)) + (x^3*Sqrt[1 + a^2*x^2])/(4*a^2) + (I*x^4*Sqrt[1 + a^2*x^2])/(5*a) + ((64*I - 45*a*x)*Sqrt[1 + a^2*x^2])/(120*a^5) + (3*ArcSinh[a*x])/(8*a^5)} -{E^(I*ArcTan[a*x])*x^3, x, 5, (x^2*Sqrt[1 + a^2*x^2])/(3*a^2) + (I*x^3*Sqrt[1 + a^2*x^2])/(4*a) - ((16 + 9*I*a*x)*Sqrt[1 + a^2*x^2])/(24*a^4) + (3*I*ArcSinh[a*x])/(8*a^4)} -{E^(I*ArcTan[a*x])*x^2, x, 7, -((I*Sqrt[1 + a^2*x^2])/a^3) + (x*Sqrt[1 + a^2*x^2])/(2*a^2) + (I*(1 + a^2*x^2)^(3/2))/(3*a^3) - ArcSinh[a*x]/(2*a^3)} -{E^(I*ArcTan[a*x])*x^1, x, 3, ((2 + I*a*x)*Sqrt[1 + a^2*x^2])/(2*a^2) - (I*ArcSinh[a*x])/(2*a^2)} -{E^(I*ArcTan[a*x])*x^0, x, 3, (I*Sqrt[1 + a^2*x^2])/a + ArcSinh[a*x]/a} -{E^(I*ArcTan[a*x])/x^1, x, 6, I*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^(I*ArcTan[a*x])/x^2, x, 5, -(Sqrt[1 + a^2*x^2]/x) - I*a*ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^(I*ArcTan[a*x])/x^3, x, 6, -(Sqrt[1 + a^2*x^2]/(2*x^2)) - (I*a*Sqrt[1 + a^2*x^2])/x + (1/2)*a^2*ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^(I*ArcTan[a*x])/x^4, x, 7, -(Sqrt[1 + a^2*x^2]/(3*x^3)) - (I*a*Sqrt[1 + a^2*x^2])/(2*x^2) + (2*a^2*Sqrt[1 + a^2*x^2])/(3*x) + (1/2)*I*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^(I*ArcTan[a*x])/x^5, x, 8, -(Sqrt[1 + a^2*x^2]/(4*x^4)) - (I*a*Sqrt[1 + a^2*x^2])/(3*x^3) + (3*a^2*Sqrt[1 + a^2*x^2])/(8*x^2) + (2*I*a^3*Sqrt[1 + a^2*x^2])/(3*x) - (3/8)*a^4*ArcTanh[Sqrt[1 + a^2*x^2]]} - - -{E^((2*I)*ArcTan[a*x])*x^3, x, 3, ((-2*I)*x)/a^3 + x^2/a^2 + (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I + a*x])/a^4} -{E^((2*I)*ArcTan[a*x])*x^2, x, 3, (2*x)/a^2 + (I*x^2)/a - x^3/3 - ((2*I)*Log[I + a*x])/a^3} -{E^((2*I)*ArcTan[a*x])*x^1, x, 3, ((2*I)*x)/a - x^2/2 + (2*Log[I + a*x])/a^2} -{E^((2*I)*ArcTan[a*x])*x^0, x, 3, -x + ((2*I)*Log[I + a*x])/a} -{E^((2*I)*ArcTan[a*x])/x^1, x, 3, Log[x] - 2*Log[I + a*x]} -{E^((2*I)*ArcTan[a*x])/x^2, x, 3, -x^(-1) + (2*I)*a*Log[x] - (2*I)*a*Log[I + a*x]} -{E^((2*I)*ArcTan[a*x])/x^3, x, 3, -1/(2*x^2) - ((2*I)*a)/x - 2*a^2*Log[x] + 2*a^2*Log[I + a*x]} -{E^((2*I)*ArcTan[a*x])/x^4, x, 3, -1/(3*x^3) - (I*a)/x^2 + (2*a^2)/x - (2*I)*a^3*Log[x] + (2*I)*a^3*Log[I + a*x]} - - -{E^((3*I)*ArcTan[a*x])*x^3, x, 14, (1 + I*a*x)^3/(a^4*Sqrt[1 + a^2*x^2]) + (27*Sqrt[1 + a^2*x^2])/(4*a^4) - (x^2*Sqrt[1 + a^2*x^2])/a^2 - (I*x^3*Sqrt[1 + a^2*x^2])/(4*a) - (9*I*(2*I - 3*a*x)*Sqrt[1 + a^2*x^2])/(8*a^4) - (51*I*ArcSinh[a*x])/(8*a^4)} -{E^((3*I)*ArcTan[a*x])*x^2, x, 10, (I*(1 + I*a*x)^3)/(a^3*Sqrt[1 + a^2*x^2]) + ((28*I - 3*a*x)*Sqrt[1 + a^2*x^2])/(6*a^3) + (I*(3 + I*a*x)^2*Sqrt[1 + a^2*x^2])/(3*a^3) + (11*ArcSinh[a*x])/(2*a^3)} -{E^((3*I)*ArcTan[a*x])*x^1, x, 9, -((9*Sqrt[1 + a^2*x^2])/(2*a^2)) - (3*(1 + a^2*x^2)^(3/2))/(2*a^2*(1 - I*a*x)) - (1 + a^2*x^2)^(5/2)/(a^2*(1 - I*a*x)^3) + (9*I*ArcSinh[a*x])/(2*a^2)} -{E^((3*I)*ArcTan[a*x])*x^0, x, 5, -((2*I*(1 + I*a*x)^2)/(a*Sqrt[1 + a^2*x^2])) - (3*I*Sqrt[1 + a^2*x^2])/a - (3*ArcSinh[a*x])/a} -{E^((3*I)*ArcTan[a*x])/x^1, x, 8, (4*I*Sqrt[1 + a^2*x^2])/(I + a*x) - I*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^((3*I)*ArcTan[a*x])/x^2, x, 8, -(Sqrt[1 + a^2*x^2]/x) - (4*a*Sqrt[1 + a^2*x^2])/(I + a*x) - 3*I*a*ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^((3*I)*ArcTan[a*x])/x^3, x, 12, -(Sqrt[1 + a^2*x^2]/(2*x^2)) - (3*I*a*Sqrt[1 + a^2*x^2])/x - (4*I*a^2*Sqrt[1 + a^2*x^2])/(I + a*x) + (9/2)*a^2*ArcTanh[Sqrt[1 + a^2*x^2]]} -{E^((3*I)*ArcTan[a*x])/x^4, x, 14, -(Sqrt[1 + a^2*x^2]/(3*x^3)) - (3*I*a*Sqrt[1 + a^2*x^2])/(2*x^2) + (14*a^2*Sqrt[1 + a^2*x^2])/(3*x) + (4*a^3*Sqrt[1 + a^2*x^2])/(I + a*x) + (11/2)*I*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]} - - -{E^((4*I)*ArcTan[a*x])*x^3, x, 3, ((12*I)*x)/a^3 - (4*x^2)/a^2 - (((4*I)/3)*x^3)/a + x^4/4 + (4*I)/(a^4*(I + a*x)) + (16*Log[I + a*x])/a^4} -{E^((4*I)*ArcTan[a*x])*x^2, x, 3, (-8*x)/a^2 - ((2*I)*x^2)/a + x^3/3 - 4/(a^3*(I + a*x)) + ((12*I)*Log[I + a*x])/a^3} -{E^((4*I)*ArcTan[a*x])*x^1, x, 3, ((-4*I)*x)/a + x^2/2 - (4*I)/(a^2*(I + a*x)) - (8*Log[I + a*x])/a^2} -{E^((4*I)*ArcTan[a*x])*x^0, x, 3, x + 4/(a*(I + a*x)) - ((4*I)*Log[I + a*x])/a} -{E^((4*I)*ArcTan[a*x])/x^1, x, 3, (4*I)/(I + a*x) + Log[x]} -{E^((4*I)*ArcTan[a*x])/x^2, x, 3, -x^(-1) - (4*a)/(I + a*x) + (4*I)*a*Log[x] - (4*I)*a*Log[I + a*x]} -{E^((4*I)*ArcTan[a*x])/x^3, x, 3, -1/(2*x^2) - ((4*I)*a)/x - ((4*I)*a^2)/(I + a*x) - 8*a^2*Log[x] + 8*a^2*Log[I + a*x]} -{E^((4*I)*ArcTan[a*x])/x^4, x, 3, -1/(3*x^3) - ((2*I)*a)/x^2 + (8*a^2)/x + (4*a^3)/(I + a*x) - (12*I)*a^3*Log[x] + (12*I)*a^3*Log[I + a*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/E^(I*ArcTan[a*x]), x, 5, (x^2*Sqrt[1 + a^2*x^2])/(3*a^2) - (I*x^3*Sqrt[1 + a^2*x^2])/(4*a) - ((16 - 9*I*a*x)*Sqrt[1 + a^2*x^2])/(24*a^4) - (3*I*ArcSinh[a*x])/(8*a^4)} -{x^2/E^(I*ArcTan[a*x]), x, 7, (I*Sqrt[1 + a^2*x^2])/a^3 + (x*Sqrt[1 + a^2*x^2])/(2*a^2) - (I*(1 + a^2*x^2)^(3/2))/(3*a^3) - ArcSinh[a*x]/(2*a^3)} -{x^1/E^(I*ArcTan[a*x]), x, 3, ((2 - I*a*x)*Sqrt[1 + a^2*x^2])/(2*a^2) + (I*ArcSinh[a*x])/(2*a^2)} -{x^0/E^(I*ArcTan[a*x]), x, 3, -((I*Sqrt[1 + a^2*x^2])/a) + ArcSinh[a*x]/a} -{1/(E^(I*ArcTan[a*x])*x^1), x, 6, (-I)*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^(I*ArcTan[a*x])*x^2), x, 5, -(Sqrt[1 + a^2*x^2]/x) + I*a*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^(I*ArcTan[a*x])*x^3), x, 6, -(Sqrt[1 + a^2*x^2]/(2*x^2)) + (I*a*Sqrt[1 + a^2*x^2])/x + (1/2)*a^2*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^(I*ArcTan[a*x])*x^4), x, 7, -(Sqrt[1 + a^2*x^2]/(3*x^3)) + (I*a*Sqrt[1 + a^2*x^2])/(2*x^2) + (2*a^2*Sqrt[1 + a^2*x^2])/(3*x) - (1/2)*I*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^(I*ArcTan[a*x])*x^5), x, 8, -(Sqrt[1 + a^2*x^2]/(4*x^4)) + (I*a*Sqrt[1 + a^2*x^2])/(3*x^3) + (3*a^2*Sqrt[1 + a^2*x^2])/(8*x^2) - (2*I*a^3*Sqrt[1 + a^2*x^2])/(3*x) - (3/8)*a^4*ArcTanh[Sqrt[1 + a^2*x^2]]} - - -{x^3/E^((2*I)*ArcTan[a*x]), x, 3, ((2*I)*x)/a^3 + x^2/a^2 - (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I - a*x])/a^4} -{x^2/E^((2*I)*ArcTan[a*x]), x, 3, (2*x)/a^2 - (I*x^2)/a - x^3/3 + ((2*I)*Log[I - a*x])/a^3} -{x^1/E^((2*I)*ArcTan[a*x]), x, 3, ((-2*I)*x)/a - x^2/2 + (2*Log[I - a*x])/a^2} -{x^0/E^((2*I)*ArcTan[a*x]), x, 3, -x - ((2*I)*Log[I - a*x])/a} -{1/(E^((2*I)*ArcTan[a*x])*x^1), x, 3, Log[x] - 2*Log[I - a*x]} -{1/(E^((2*I)*ArcTan[a*x])*x^2), x, 3, -x^(-1) - (2*I)*a*Log[x] + (2*I)*a*Log[I - a*x]} -{1/(E^((2*I)*ArcTan[a*x])*x^3), x, 3, -1/(2*x^2) + ((2*I)*a)/x - 2*a^2*Log[x] + 2*a^2*Log[I - a*x]} -{1/(E^((2*I)*ArcTan[a*x])*x^4), x, 3, -1/(3*x^3) + (I*a)/x^2 + (2*a^2)/x + (2*I)*a^3*Log[x] - (2*I)*a^3*Log[I - a*x]} - - -{x^3/E^((3*I)*ArcTan[a*x]), x, 14, (1 - I*a*x)^3/(a^4*Sqrt[1 + a^2*x^2]) + (27*Sqrt[1 + a^2*x^2])/(4*a^4) - (x^2*Sqrt[1 + a^2*x^2])/a^2 + (I*x^3*Sqrt[1 + a^2*x^2])/(4*a) - (9*I*(2*I + 3*a*x)*Sqrt[1 + a^2*x^2])/(8*a^4) + (51*I*ArcSinh[a*x])/(8*a^4)} -{x^2/E^((3*I)*ArcTan[a*x]), x, 10, -((I*(1 - I*a*x)^3)/(a^3*Sqrt[1 + a^2*x^2])) - (I*(3 - I*a*x)^2*Sqrt[1 + a^2*x^2])/(3*a^3) - ((28*I + 3*a*x)*Sqrt[1 + a^2*x^2])/(6*a^3) + (11*ArcSinh[a*x])/(2*a^3)} -{x^1/E^((3*I)*ArcTan[a*x]), x, 9, -((9*Sqrt[1 + a^2*x^2])/(2*a^2)) - (3*(1 + a^2*x^2)^(3/2))/(2*a^2*(1 + I*a*x)) - (1 + a^2*x^2)^(5/2)/(a^2*(1 + I*a*x)^3) - (9*I*ArcSinh[a*x])/(2*a^2)} -{x^0/E^((3*I)*ArcTan[a*x]), x, 5, (2*I*(1 - I*a*x)^2)/(a*Sqrt[1 + a^2*x^2]) + (3*I*Sqrt[1 + a^2*x^2])/a - (3*ArcSinh[a*x])/a} -{1/(E^((3*I)*ArcTan[a*x])*x^1), x, 8, (4*I*Sqrt[1 + a^2*x^2])/(I - a*x) + I*ArcSinh[a*x] - ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^((3*I)*ArcTan[a*x])*x^2), x, 8, -(Sqrt[1 + a^2*x^2]/x) + (4*a*Sqrt[1 + a^2*x^2])/(I - a*x) + 3*I*a*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^((3*I)*ArcTan[a*x])*x^3), x, 12, -(Sqrt[1 + a^2*x^2]/(2*x^2)) + (3*I*a*Sqrt[1 + a^2*x^2])/x - (4*I*a^2*Sqrt[1 + a^2*x^2])/(I - a*x) + (9/2)*a^2*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^((3*I)*ArcTan[a*x])*x^4), x, 14, -(Sqrt[1 + a^2*x^2]/(3*x^3)) + (3*I*a*Sqrt[1 + a^2*x^2])/(2*x^2) + (14*a^2*Sqrt[1 + a^2*x^2])/(3*x) - (4*a^3*Sqrt[1 + a^2*x^2])/(I - a*x) - (11/2)*I*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]} -{1/(E^((3*I)*ArcTan[a*x])*x^5), x, 19, -(Sqrt[1 + a^2*x^2]/(4*x^4)) + (I*a*Sqrt[1 + a^2*x^2])/x^3 + (19*a^2*Sqrt[1 + a^2*x^2])/(8*x^2) - (6*I*a^3*Sqrt[1 + a^2*x^2])/x + (4*I*a^4*Sqrt[1 + a^2*x^2])/(I - a*x) - (51/8)*a^4*ArcTanh[Sqrt[1 + a^2*x^2]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/2 I ArcTan[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^((I/2)*ArcTan[a*x])*x^2, x, 15, (((-3*I)/8)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a^3 - ((I/12)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/a^3 + (x*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(3*a^2) + (((3*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((3*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((3*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((3*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{E^((I/2)*ArcTan[a*x])*x^1, x, 14, ((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*a^2) + ((1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(2*a^2) - ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(4*Sqrt[2]*a^2) + ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(4*Sqrt[2]*a^2) + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(8*Sqrt[2]*a^2) - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(8*Sqrt[2]*a^2)} -{E^((I/2)*ArcTan[a*x]), x, 13, (I*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a - (I*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (I*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + ((I/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - ((I/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{E^((I/2)*ArcTan[a*x])/x^1, x, 17, -2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{E^((I/2)*ArcTan[a*x])/x^2, x, 6, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x) - I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^((I/2)*ArcTan[a*x])/x^3, x, 7, -((I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*x)) - ((1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(2*x^2) + (1/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (1/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^((I/2)*ArcTan[a*x])/x^4, x, 9, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(3*x^3)) - (5*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(12*x^2) + (11*a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(24*x) + (3/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (3/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^((I/2)*ArcTan[a*x])/x^5, x, 10, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*x^4)) - (7*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(24*x^3) + (29*a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(96*x^2) + (83*I*a^3*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(192*x) - (11/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (11/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^((I/2)*ArcTan[a*x])/x^6, x, 11, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(5*x^5)) - (9*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(40*x^4) + (11*a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(48*x^3) + (269*I*a^3*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(960*x^2) - (611*a^4*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(1920*x) - (31/128)*I*a^5*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (31/128)*I*a^5*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -{E^(((3*I)/2)*ArcTan[a*x])*x^3, x, 15, -((41*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(64*a^4)) + (x^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4))/(4*a^2) - ((1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4)*(11 + 4*I*a*x))/(32*a^4) + (123*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (123*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (123*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{E^(((3*I)/2)*ArcTan[a*x])*x^2, x, 15, (((-17*I)/24)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a^3 - ((I/4)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4))/a^3 + (x*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4))/(3*a^2) + (((17*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((17*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((17*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((17*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{E^(((3*I)/2)*ArcTan[a*x])*x^1, x, 14, (3*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*a^2) + ((1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4))/(2*a^2) - (9*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^(((3*I)/2)*ArcTan[a*x]), x, 13, (I*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a - ((3*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + ((3*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - (((3*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (((3*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{E^(((3*I)/2)*ArcTan[a*x])/x^1, x, 17, 2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{E^(((3*I)/2)*ArcTan[a*x])/x^2, x, 6, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/x) + 3*I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - 3*I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((3*I)/2)*ArcTan[a*x])/x^3, x, 7, -((3*I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*x)) - ((1 - I*a*x)^(1/4)*(1 + I*a*x)^(7/4))/(2*x^2) - (9/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (9/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((3*I)/2)*ArcTan[a*x])/x^4, x, 9, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(3*x^3)) - (7*I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(12*x^2) + (23*a^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(24*x) - (17/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (17/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((3*I)/2)*ArcTan[a*x])/x^5, x, 10, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*x^4)) - (3*I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(8*x^3) + (15*a^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(32*x^2) + (63*I*a^3*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(64*x) + (123/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (123/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -{E^(((5*I)/2)*ArcTan[a*x])*x^3, x, 16, (475*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(64*a^4) - (4*I*x^3*(1 + I*a*x)^(5/4))/(a*(1 - I*a*x)^(1/4)) - (17*x^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(4*a^2) - (I*(521*I - 452*a*x)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(96*a^4) - (475*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (475*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{E^(((5*I)/2)*ArcTan[a*x])*x^2, x, 16, (((55*I)/8)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a^3 + (((11*I)/4)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/a^3 + ((2*I)*(1 + I*a*x)^(9/4))/(a^3*(1 - I*a*x)^(1/4)) + ((I/3)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(9/4))/a^3 - (((55*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((55*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((55*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((55*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{E^(((5*I)/2)*ArcTan[a*x])*x^1, x, 15, (-25*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*a^2) - (5*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(5/4))/(2*a^2) - (2*(1 + I*a*x)^(9/4))/(a^2*(1 - I*a*x)^(1/4)) + (25*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (25*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (25*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2) + (25*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^(((5*I)/2)*ArcTan[a*x]), x, 14, ((-5*I)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a - ((4*I)*(1 + I*a*x)^(5/4))/(a*(1 - I*a*x)^(1/4)) + ((5*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - ((5*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - (((5*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (((5*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{E^(((5*I)/2)*ArcTan[a*x])/x^1, x, 19, (8*(1 + I*a*x)^(1/4))/(1 - I*a*x)^(1/4) - 2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{E^(((5*I)/2)*ArcTan[a*x])/x^2, x, 7, (10*I*a*(1 + I*a*x)^(1/4))/(1 - I*a*x)^(1/4) - (1 + I*a*x)^(5/4)/(x*(1 - I*a*x)^(1/4)) - 5*I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - 5*I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((5*I)/2)*ArcTan[a*x])/x^3, x, 8, -((25*a^2*(1 + I*a*x)^(1/4))/(2*(1 - I*a*x)^(1/4))) - (5*I*a*(1 + I*a*x)^(5/4))/(4*x*(1 - I*a*x)^(1/4)) - (1 + I*a*x)^(9/4)/(2*x^2*(1 - I*a*x)^(1/4)) + (25/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (25/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((5*I)/2)*ArcTan[a*x])/x^4, x, 10, -((287*I*a^3*(1 + I*a*x)^(1/4))/(24*(1 - I*a*x)^(1/4))) - (1 + I*a*x)^(1/4)/(3*x^3*(1 - I*a*x)^(1/4)) - (13*I*a*(1 + I*a*x)^(1/4))/(12*x^2*(1 - I*a*x)^(1/4)) + (61*a^2*(1 + I*a*x)^(1/4))/(24*x*(1 - I*a*x)^(1/4)) + (55/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (55/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{E^(((5*I)/2)*ArcTan[a*x])/x^5, x, 11, (2467*a^4*(1 + I*a*x)^(1/4))/(192*(1 - I*a*x)^(1/4)) - (1 + I*a*x)^(1/4)/(4*x^4*(1 - I*a*x)^(1/4)) - (17*I*a*(1 + I*a*x)^(1/4))/(24*x^3*(1 - I*a*x)^(1/4)) + (113*a^2*(1 + I*a*x)^(1/4))/(96*x^2*(1 - I*a*x)^(1/4)) + (521*I*a^3*(1 + I*a*x)^(1/4))/(192*x*(1 - I*a*x)^(1/4)) - (475/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (475/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/E^((I/2)*ArcTan[a*x]), x, 15, -((11*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(64*a^4)) + (x^2*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(4*a^2) - ((1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4)*(25 - 4*I*a*x))/(96*a^4) - (11*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (11*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (11*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4) + (11*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^((I/2)*ArcTan[a*x]), x, 15, (((3*I)/8)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a^3 + ((I/12)*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/a^3 + (x*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(3*a^2) + (((3*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((3*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((3*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((3*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{x^1/E^((I/2)*ArcTan[a*x]), x, 14, ((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*a^2) + ((1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(2*a^2) + ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(4*Sqrt[2]*a^2) - ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(4*Sqrt[2]*a^2) + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(8*Sqrt[2]*a^2) - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/(8*Sqrt[2]*a^2)} -{x^0/E^((I/2)*ArcTan[a*x]), x, 13, ((-I)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a - (I*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (I*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - ((I/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + ((I/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{1/(E^((I/2)*ArcTan[a*x])*x^1), x, 17, 2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{1/(E^((I/2)*ArcTan[a*x])*x^2), x, 6, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/x) - I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^((I/2)*ArcTan[a*x])*x^3), x, 7, (I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*x) - ((1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(2*x^2) - (1/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (1/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^((I/2)*ArcTan[a*x])*x^4), x, 9, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(3*x^3)) + (5*I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(12*x^2) + (11*a^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(24*x) + (3/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (3/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^((I/2)*ArcTan[a*x])*x^5), x, 10, -(((1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*x^4)) + (7*I*a*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(24*x^3) + (29*a^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(96*x^2) - (83*I*a^3*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(192*x) + (11/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (11/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -{x^3/E^(((3*I)/2)*ArcTan[a*x]), x, 15, -((41*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(64*a^4)) + (x^2*(1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4))/(4*a^2) - ((1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4)*(11 - 4*I*a*x))/(32*a^4) - (123*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (123*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^(((3*I)/2)*ArcTan[a*x]), x, 15, (((17*I)/24)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a^3 + ((I/4)*(1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4))/a^3 + (x*(1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4))/(3*a^2) + (((17*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((17*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((17*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((17*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{x^1/E^(((3*I)/2)*ArcTan[a*x]), x, 14, (3*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*a^2) + ((1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4))/(2*a^2) + (9*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (9*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^(((-3*I)/2)*ArcTan[a*x]), x, 13, ((-I)*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/a - ((3*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + ((3*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (((3*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - (((3*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{1/(E^(((3*I)/2)*ArcTan[a*x])*x^1), x, 17, -2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{1/(E^(((3*I)/2)*ArcTan[a*x])*x^2), x, 6, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/x) + 3*I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + 3*I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((3*I)/2)*ArcTan[a*x])*x^3), x, 7, (3*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*x) - ((1 - I*a*x)^(7/4)*(1 + I*a*x)^(1/4))/(2*x^2) + (9/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (9/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((3*I)/2)*ArcTan[a*x])*x^4), x, 9, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(3*x^3)) + (7*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(12*x^2) + (23*a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(24*x) - (17/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (17/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((3*I)/2)*ArcTan[a*x])*x^5), x, 10, -(((1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(4*x^4)) + (3*I*a*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(8*x^3) + (15*a^2*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(32*x^2) - (63*I*a^3*(1 - I*a*x)^(3/4)*(1 + I*a*x)^(1/4))/(64*x) - (123/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (123/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -{x^3/E^(((5*I)/2)*ArcTan[a*x]), x, 16, (4*I*x^3*(1 - I*a*x)^(5/4))/(a*(1 + I*a*x)^(1/4)) + (475*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(64*a^4) - (17*x^2*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(4*a^2) - (I*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4)*(521*I + 452*a*x))/(96*a^4) + (475*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (475*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (475*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^(((5*I)/2)*ArcTan[a*x]), x, 16, ((-2*I)*(1 - I*a*x)^(9/4))/(a^3*(1 + I*a*x)^(1/4)) - (((55*I)/8)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a^3 - (((11*I)/4)*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/a^3 - ((I/3)*(1 - I*a*x)^(9/4)*(1 + I*a*x)^(3/4))/a^3 - (((55*I)/8)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((55*I)/8)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) - (((55*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3) + (((55*I)/16)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a^3)} -{x^1/E^(((5*I)/2)*ArcTan[a*x]), x, 15, (-2*(1 - I*a*x)^(9/4))/(a^2*(1 + I*a*x)^(1/4)) - (25*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/(4*a^2) - (5*(1 - I*a*x)^(5/4)*(1 + I*a*x)^(3/4))/(2*a^2) - (25*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (25*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (25*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2) + (25*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^(((-5*I)/2)*ArcTan[a*x]), x, 14, ((4*I)*(1 - I*a*x)^(5/4))/(a*(1 + I*a*x)^(1/4)) + ((5*I)*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))/a + ((5*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - ((5*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) + (((5*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a) - (((5*I)/2)*Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)])/(Sqrt[2]*a)} -{1/(E^(((5*I)/2)*ArcTan[a*x])*x^1), x, 19, (8*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4) + 2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)] - 2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] - (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a*x]/Sqrt[1 + I*a*x] + (Sqrt[2]*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)]/Sqrt[2]} -{1/(E^(((5*I)/2)*ArcTan[a*x])*x^2), x, 7, -((10*I*a*(1 - I*a*x)^(1/4))/(1 + I*a*x)^(1/4)) - (1 - I*a*x)^(5/4)/(x*(1 + I*a*x)^(1/4)) - 5*I*a*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + 5*I*a*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((5*I)/2)*ArcTan[a*x])*x^3), x, 8, -((25*a^2*(1 - I*a*x)^(1/4))/(2*(1 + I*a*x)^(1/4))) + (5*I*a*(1 - I*a*x)^(5/4))/(4*x*(1 + I*a*x)^(1/4)) - (1 - I*a*x)^(9/4)/(2*x^2*(1 + I*a*x)^(1/4)) - (25/4)*a^2*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] + (25/4)*a^2*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((5*I)/2)*ArcTan[a*x])*x^4), x, 10, (287*I*a^3*(1 - I*a*x)^(1/4))/(24*(1 + I*a*x)^(1/4)) - (1 - I*a*x)^(1/4)/(3*x^3*(1 + I*a*x)^(1/4)) + (13*I*a*(1 - I*a*x)^(1/4))/(12*x^2*(1 + I*a*x)^(1/4)) + (61*a^2*(1 - I*a*x)^(1/4))/(24*x*(1 + I*a*x)^(1/4)) + (55/8)*I*a^3*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (55/8)*I*a^3*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} -{1/(E^(((5*I)/2)*ArcTan[a*x])*x^5), x, 11, (2467*a^4*(1 - I*a*x)^(1/4))/(192*(1 + I*a*x)^(1/4)) - (1 - I*a*x)^(1/4)/(4*x^4*(1 + I*a*x)^(1/4)) + (17*I*a*(1 - I*a*x)^(1/4))/(24*x^3*(1 + I*a*x)^(1/4)) + (113*a^2*(1 - I*a*x)^(1/4))/(96*x^2*(1 + I*a*x)^(1/4)) - (521*I*a^3*(1 - I*a*x)^(1/4))/(192*x*(1 + I*a*x)^(1/4)) + (475/64)*a^4*ArcTan[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)] - (475/64)*a^4*ArcTanh[(1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/3 I ArcTan[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(I*ArcTan[x]/3)*x^2, x, 16, (-(19/54))*I*(1 - I*x)^(5/6)*(1 + I*x)^(1/6) - (1/18)*I*(1 - I*x)^(5/6)*(1 + I*x)^(7/6) + (1/3)*(1 - I*x)^(5/6)*(1 + I*x)^(7/6)*x + (19/162)*I*ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] - (19/162)*I*ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] - (19/81)*I*ArcTan[(1 - I*x)^(1/6)/(1 + I*x)^(1/6)] - (19*I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)])/(108*Sqrt[3]) + (19*I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)])/(108*Sqrt[3])} -{E^(I*ArcTan[x]/3)*x^1, x, 15, (1/6)*(1 - I*x)^(5/6)*(1 + I*x)^(1/6) + (1/2)*(1 - I*x)^(5/6)*(1 + I*x)^(7/6) - (1/18)*ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (1/18)*ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (1/9)*ArcTan[(1 - I*x)^(1/6)/(1 + I*x)^(1/6)] + Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)]/(12*Sqrt[3]) - Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)]/(12*Sqrt[3])} -{E^(I*ArcTan[x]/3)*x^0, x, 14, I*(1 - I*x)^(5/6)*(1 + I*x)^(1/6) - (1/3)*I*ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (1/3)*I*ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (2/3)*I*ArcTan[(1 - I*x)^(1/6)/(1 + I*x)^(1/6)] + (I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)])/(2*Sqrt[3]) - (I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)])/(2*Sqrt[3])} -{E^(I*ArcTan[x]/3)/x^1, x, 25, ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] - ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + Sqrt[3]*ArcTan[(1 - (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]] - Sqrt[3]*ArcTan[(1 + (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]] - 2*ArcTan[(1 - I*x)^(1/6)/(1 + I*x)^(1/6)] - 2*ArcTanh[(1 + I*x)^(1/6)/(1 - I*x)^(1/6)] - (1/2)*Sqrt[3]*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (1/2)*Sqrt[3]*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I*x)^(1/6)] + (1/2)*Log[1 - (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)] - (1/2)*Log[1 + (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)]} -{E^(I*ArcTan[x]/3)/x^2, x, 13, -(((1 - I*x)^(5/6)*(1 + I*x)^(1/6))/x) + (I*ArcTan[(1 - (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]])/Sqrt[3] - (I*ArcTan[(1 + (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]])/Sqrt[3] - (2/3)*I*ArcTanh[(1 + I*x)^(1/6)/(1 - I*x)^(1/6)] + (1/6)*I*Log[1 - (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)] - (1/6)*I*Log[1 + (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)]} -{E^(I*ArcTan[x]/3)/x^3, x, 14, -(((1 - I*x)^(5/6)*(1 + I*x)^(7/6))/(2*x^2)) - (I*(1 - I*x)^(5/6)*(1 + I*x)^(1/6))/(6*x) - ArcTan[(1 - (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) + ArcTan[(1 + (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) + (1/9)*ArcTanh[(1 + I*x)^(1/6)/(1 - I*x)^(1/6)] - (1/36)*Log[1 - (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)] + (1/36)*Log[1 + (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)]} -{E^(I*ArcTan[x]/3)/x^4, x, 16, -(((1 - I*x)^(5/6)*(1 + I*x)^(1/6))/(3*x^3)) - (7*I*(1 - I*x)^(5/6)*(1 + I*x)^(1/6))/(18*x^2) + (11*(1 - I*x)^(5/6)*(1 + I*x)^(1/6))/(27*x) - (19*I*ArcTan[(1 - (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]])/(54*Sqrt[3]) + (19*I*ArcTan[(1 + (2*(1 + I*x)^(1/6))/(1 - I*x)^(1/6))/Sqrt[3]])/(54*Sqrt[3]) + (19/81)*I*ArcTanh[(1 + I*x)^(1/6)/(1 - I*x)^(1/6)] - (19/324)*I*Log[1 - (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)] + (19/324)*I*Log[1 + (1 + I*x)^(1/6)/(1 - I*x)^(1/6) + (1 + I*x)^(1/3)/(1 - I*x)^(1/3)]} - - -{E^(2*I*ArcTan[x]/3)*x^2, x, 5, (-(11/27))*I*(1 - I*x)^(2/3)*(1 + I*x)^(1/3) - (1/9)*I*(1 - I*x)^(2/3)*(1 + I*x)^(4/3) + (1/3)*(1 - I*x)^(2/3)*(1 + I*x)^(4/3)*x + (22*I*ArcTan[1/Sqrt[3] - (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))])/(27*Sqrt[3]) + (11/27)*I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3)] + (11/81)*I*Log[1 + I*x]} -{E^(2*I*ArcTan[x]/3)*x^1, x, 4, (1/3)*(1 - I*x)^(2/3)*(1 + I*x)^(1/3) + (1/2)*(1 - I*x)^(2/3)*(1 + I*x)^(4/3) - (2*ArcTan[1/Sqrt[3] - (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))])/(3*Sqrt[3]) - (1/3)*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3)] - (1/9)*Log[1 + I*x]} -{E^(2*I*ArcTan[x]/3)*x^0, x, 3, I*(1 - I*x)^(2/3)*(1 + I*x)^(1/3) - (2*I*ArcTan[1/Sqrt[3] - (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))])/Sqrt[3] - I*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3)] - (1/3)*I*Log[1 + I*x]} -{E^(2*I*ArcTan[x]/3)/x^1, x, 4, Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))] + Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))] + (3/2)*Log[1 + (1 - I*x)^(1/3)/(1 + I*x)^(1/3)] + (3/2)*Log[(1 - I*x)^(1/3) - (1 + I*x)^(1/3)] + (1/2)*Log[1 + I*x] - Log[x]/2} -{E^(2*I*ArcTan[x]/3)/x^2, x, 3, -(((1 - I*x)^(2/3)*(1 + I*x)^(1/3))/x) + (2*I*ArcTan[1/Sqrt[3] + (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))])/Sqrt[3] + I*Log[(1 - I*x)^(1/3) - (1 + I*x)^(1/3)] - (1/3)*I*Log[x]} -{E^(2*I*ArcTan[x]/3)/x^3, x, 4, -(((1 - I*x)^(2/3)*(1 + I*x)^(4/3))/(2*x^2)) - (I*(1 - I*x)^(2/3)*(1 + I*x)^(1/3))/(3*x) - (2*ArcTan[1/Sqrt[3] + (2*(1 - I*x)^(1/3))/(Sqrt[3]*(1 + I*x)^(1/3))])/(3*Sqrt[3]) - (1/3)*Log[(1 - I*x)^(1/3) - (1 + I*x)^(1/3)] + Log[x]/9} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/4 I ArcTan[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^((I/4)*ArcTan[a*x])*x^2, x, 27, -((11*I*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/(32*a^3)) - (I*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(24*a^3) + (x*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(3*a^2) + (11*I*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(128*a^3) + (11*I*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(128*a^3) - (11*I*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(128*a^3) - (11*I*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(128*a^3) - (11*I*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(256*a^3) + (11*I*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(256*a^3) - (11*I*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(256*a^3) + (11*I*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(256*a^3)} -{E^((I/4)*ArcTan[a*x])*x^1, x, 26, ((1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/(8*a^2) + ((1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(2*a^2) - (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(32*a^2) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(32*a^2) + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(32*a^2) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(32*a^2) + (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(64*a^2) - (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(64*a^2) + (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(64*a^2) - (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(64*a^2)} -{E^((I/4)*ArcTan[a*x])*x^0, x, 25, (I*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/a - (I*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(4*a) - (I*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(4*a) + (I*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(4*a) + (I*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(4*a) + (I*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(8*a) - (I*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(8*a) + (I*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(8*a) - (I*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)])/(8*a)} -{E^((I/4)*ArcTan[a*x])/x^1, x, 39, -2*ArcTan[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] + Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] - Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] - Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)] - 2*ArcTanh[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] - (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)] + (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)] - (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)] + (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - I*a*x)^(1/4)/(1 + I*a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - I*a*x)^(1/8))/(1 + I*a*x)^(1/8)] + Log[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/Sqrt[2] - Log[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)]/Sqrt[2]} -{E^((I/4)*ArcTan[a*x])/x^2, x, 16, -(((1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/x) - (1/2)*I*a*ArcTan[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] + (I*a*ArcTan[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)])/(2*Sqrt[2]) - (I*a*ArcTan[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)])/(2*Sqrt[2]) - (1/2)*I*a*ArcTanh[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] + (I*a*Log[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/(4*Sqrt[2]) - (I*a*Log[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/(4*Sqrt[2])} -{E^((I/4)*ArcTan[a*x])/x^3, x, 17, -((I*a*(1 - I*a*x)^(7/8)*(1 + I*a*x)^(1/8))/(8*x)) - ((1 - I*a*x)^(7/8)*(1 + I*a*x)^(9/8))/(2*x^2) + (1/16)*a^2*ArcTan[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] - (a^2*ArcTan[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)])/(16*Sqrt[2]) + (a^2*ArcTan[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8)])/(16*Sqrt[2]) + (1/16)*a^2*ArcTanh[(1 + I*a*x)^(1/8)/(1 - I*a*x)^(1/8)] - (a^2*Log[1 - (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/(32*Sqrt[2]) + (a^2*Log[1 + (Sqrt[2]*(1 + I*a*x)^(1/8))/(1 - I*a*x)^(1/8) + (1 + I*a*x)^(1/4)/(1 - I*a*x)^(1/4)])/(32*Sqrt[2])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n I ArcTan[a x]) with m symbolic*) - - -{E^(6*I*ArcTan[a*x])*x^m, x, 4, -((x^(1 + m)*(1 + I*a*x)^2)/((1 + m)*(1 - I*a*x)^2)) + (4*I*x^(1 + m)*(I*(1 + m)^2 + a*(3 + 3*m + m^2)*x))/((1 + m)*(1 - I*a*x)^2) + (2*(3 + 4*m + 2*m^2)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x])/(1 + m)} -{E^(4*I*ArcTan[a*x])*x^m, x, 4, x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 - I*a*x) - 4*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x]} -{E^(2*I*ArcTan[a*x])*x^m, x, 3, -(x^(1 + m)/(1 + m)) + (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x])/(1 + m)} -{x^m/E^(2*I*ArcTan[a*x]), x, 3, -(x^(1 + m)/(1 + m)) + (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x])/(1 + m)} -{x^m/E^(4*I*ArcTan[a*x]), x, 4, x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 + I*a*x) - 4*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x]} -{x^m/E^(6*I*ArcTan[a*x]), x, 4, -((x^(1 + m)*(1 - I*a*x)^2)/((1 + m)*(1 + I*a*x)^2)) + (4*I*x^(1 + m)*(I*(1 + m)^2 - a*(3 + 3*m + m^2)*x))/((1 + m)*(1 + I*a*x)^2) + (2*(3 + 4*m + 2*m^2)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x])/(1 + m)} - -{E^(3*I*ArcTan[a*x])*x^m, x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m)) - (I*a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m) + (4*I*a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m)} -{E^(1*I*ArcTan[a*x])*x^m, x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m) + (I*a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m)} -{x^m/E^(1*I*ArcTan[a*x]), x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m) - (I*a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m)} -{x^m/E^(3*I*ArcTan[a*x]), x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m)) + (I*a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, (-a^2)*x^2])/(1 + m) - (4*I*a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + m)} - - -{E^(5*I/2*ArcTan[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 5/4, -5/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{E^(3*I/2*ArcTan[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 3/4, -3/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{E^(1*I/2*ArcTan[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/4, -1/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{x^m/E^(1*I/2*ArcTan[a*x]), x, 2, (x^(1 + m)*AppellF1[1 + m, -1/4, 1/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{x^m/E^(3*I/2*ArcTan[a*x]), x, 2, (x^(1 + m)*AppellF1[1 + m, -3/4, 3/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{x^m/E^(5*I/2*ArcTan[a*x]), x, 2, (x^(1 + m)*AppellF1[1 + m, -5/4, 5/4, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} - - -{E^(2*ArcTan[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, -(I/3), I/3, 2 + m, I*x, (-I)*x])/(1 + m)} -{E^(1*ArcTan[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, -(I/6), I/6, 2 + m, I*x, (-I)*x])/(1 + m)} - - -{E^(I/4*ArcTan[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/8, -1/8, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{E^(I*n*ArcTan[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, n/2, -n/2, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} - - -{E^(I*n*ArcTan[a*x])*x^3, x, 4, (x^2*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(4*a^2) - ((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2)*(6 + n^2 + 2*I*a*n*x))/(24*a^4) - (2^(-2 + n/2)*n*(8 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - I*a*x)])/(3*a^4*(2 - n))} -{E^(I*n*ArcTan[a*x])*x^2, x, 4, -((I*n*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(6*a^3)) + (x*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(3*a^2) - (I*2^(n/2)*(2 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - I*a*x)])/(3*a^3*(2 - n))} -{E^(I*n*ArcTan[a*x])*x^1, x, 3, ((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(2*a^2) + (2^(n/2)*n*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - I*a*x)])/(a^2*(2 - n))} -{E^(I*n*ArcTan[a*x])*x^0, x, 2, (I*2^(1 + n/2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - I*a*x)])/(a*(2 - n))} -{E^(I*n*ArcTan[a*x])/x^1, x, 4, (2*(1 + I*a*x)^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (1 - I*a*x)/(1 + I*a*x)])/((1 - I*a*x)^(n/2)*n) - (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - I*a*x)])/((1 - I*a*x)^(n/2)*n)} -{E^(I*n*ArcTan[a*x])/x^2, x, 2, -((4*I*a*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - I*a*x)/(1 + I*a*x)])/(2 - n))} -{E^(I*n*ArcTan[a*x])/x^3, x, 3, -(((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(2*x^2)) + (2*a^2*n*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - I*a*x)/(1 + I*a*x)])/(2 - n)} -{E^(I*n*ArcTan[a*x])/x^4, x, 5, -(((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(3*x^3)) - (I*a*n*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(6*x^2) + (2*I*a^3*(2 + n^2)*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - I*a*x)/(1 + I*a*x)])/(3*(2 - n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a+b x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n I ArcTan[a+b x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(I*ArcTan[a + b*x])*x^4, x, 8, ((3*I + 12*a - 24*I*a^2 - 16*a^3 + 8*I*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) - ((I + 8*a)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(20*b^3) + (x^3*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(5*b^2) + (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(19*I + 114*a - 86*I*a^2 - 96*a^3 - 2*(13 - 14*I*a - 36*a^2)*b*x))/(120*b^5) + ((3 - 12*I*a - 24*a^2 + 16*I*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)} -{E^(I*ArcTan[a + b*x])*x^3, x, 7, -(((3 - 12*I*a - 12*a^2 + 8*I*a^3)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4)) + (x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(4*b^2) - (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(7 - 10*I*a - 18*a^2 + 2*(I + 6*a)*b*x))/(24*b^4) + ((3*I + 12*a - 12*I*a^2 - 8*a^3)*ArcSinh[a + b*x])/(8*b^4)} -{E^(I*ArcTan[a + b*x])*x^2, x, 7, -((I + 2*a - (2*I)*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^3) - ((I + 4*a)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(6*b^3) + (x*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(3*b^2) - ((1 - (2*I)*a - 2*a^2)*ArcSinh[a + b*x])/(2*b^3)} -{E^(I*ArcTan[a + b*x])*x^1, x, 6, ((1 - (2*I)*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^2) + (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(2*b^2) - ((I + 2*a)*ArcSinh[a + b*x])/(2*b^2)} -{E^(I*ArcTan[a + b*x])*x^0, x, 5, (I*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/b + ArcSinh[a + b*x]/b} -{E^(I*ArcTan[a + b*x])/x^1, x, 8, I*ArcSinh[a + b*x] - (2*Sqrt[I - a]*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/Sqrt[I + a]} -{E^(I*ArcTan[a + b*x])/x^2, x, 4, -((Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*x)) + (2*I*b*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(Sqrt[I - a]*(I + a)^(3/2))} -{E^(I*ArcTan[a + b*x])/x^3, x, 5, -(((1 + 2*I*a)*b*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*(I - a)*(I + a)^2*x)) - (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(2*(1 + a^2)*x^2) + ((1 + 2*I*a)*b^2*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(3/2)*(I + a)^(5/2))} -{E^(I*ArcTan[a + b*x])/x^4, x, 7, -((Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(3*(1 - I*a)*x^3)) - ((3*I - 2*a)*b*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(6*(1 - I*a)*(1 + a^2)*x^2) + ((4 + 9*I*a - 2*a^2)*b^2*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(6*(1 - I*a)*(1 + a^2)^2*x) + ((2*a - I*(1 - 2*a^2))*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(5/2)*(I + a)^(7/2))} - - -{E^((2*I)*ArcTan[a + b*x])*x^4, x, 3, -((2*(1 - I*a)^3*x)/b^4) + (I*(I + a)^2*x^2)/b^3 + (2*(1 - I*a)*x^3)/(3*b^2) + (I*x^4)/(2*b) - x^5/5 + (2*I*(I + a)^4*Log[I + a + b*x])/b^5} -{E^((2*I)*ArcTan[a + b*x])*x^3, x, 3, (2*I*(I + a)^2*x)/b^3 + ((1 - I*a)*x^2)/b^2 + (2*I*x^3)/(3*b) - x^4/4 - (2*(1 - I*a)^3*Log[I + a + b*x])/b^4} -{E^((2*I)*ArcTan[a + b*x])*x^2, x, 3, (2*(1 - I*a)*x)/b^2 + (I*x^2)/b - x^3/3 + (2*I*(I + a)^2*Log[I + a + b*x])/b^3} -{E^((2*I)*ArcTan[a + b*x])*x^1, x, 3, ((2*I)*x)/b - x^2/2 + (2*(1 - I*a)*Log[I + a + b*x])/b^2} -{E^((2*I)*ArcTan[a + b*x])*x^0, x, 3, -x + ((2*I)*Log[I + a + b*x])/b} -{E^((2*I)*ArcTan[a + b*x])/x^1, x, 3, ((I - a)*Log[x])/(I + a) - (2*Log[I + a + b*x])/(1 - I*a)} -{E^((2*I)*ArcTan[a + b*x])/x^2, x, 3, -((I - a)/((I + a)*x)) - ((2*I)*b*Log[x])/(I + a)^2 + ((2*I)*b*Log[I + a + b*x])/(I + a)^2} -{E^((2*I)*ArcTan[a + b*x])/x^3, x, 3, -((I - a)/(2*(I + a)*x^2)) + (2*I*b)/((I + a)^2*x) - (2*b^2*Log[x])/(1 - I*a)^3 + (2*b^2*Log[I + a + b*x])/(1 - I*a)^3} -{E^((2*I)*ArcTan[a + b*x])/x^4, x, 3, -((I - a)/(3*(I + a)*x^3)) + (I*b)/((I + a)^2*x^2) + (2*b^2)/((1 - I*a)^3*x) - (2*I*b^3*Log[x])/(I + a)^4 + (2*I*b^3*Log[I + a + b*x])/(I + a)^4} - - -{E^((3*I)*ArcTan[a + b*x])*x^4, x, 9, -((3*(19*I + 68*a - 88*I*a^2 - 48*a^3 + 8*I*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5)) - (2*I*x^4*(1 + I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) + (3*(17*I + 16*a)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(20*b^3) - (11*x^3*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(5*b^2) - (I*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(163 - 458*I*a - 422*a^2 + 112*I*a^3 + 2*(61*I + 118*a - 52*I*a^2)*b*x))/(40*b^5) - (3*(19 - 68*I*a - 88*a^2 + 48*I*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)} -{E^((3*I)*ArcTan[a + b*x])*x^3, x, 8, (3*(17 - 44*I*a - 36*a^2 + 8*I*a^3)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - (2*I*x^3*(1 + I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) - (9*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(4*b^2) - (I*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(29*I + 54*a - 22*I*a^2 - 2*(11 - 10*I*a)*b*x))/(8*b^4) - (3*(17*I + 44*a - 36*I*a^2 - 8*a^3)*ArcSinh[a + b*x])/(8*b^4)} -{E^((3*I)*ArcTan[a + b*x])*x^2, x, 8, ((11*I + 18*a - 6*I*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^3) + ((11*I + 18*a - 6*I*a^2)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(6*b^3) - (I*(I + a)^2*(1 + I*a + I*b*x)^(5/2))/(b^3*Sqrt[1 - I*a - I*b*x]) + (I*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(5/2))/(3*b^3) + ((11 - 18*I*a - 6*a^2)*ArcSinh[a + b*x])/(2*b^3)} -{E^((3*I)*ArcTan[a + b*x])*x^1, x, 7, -((3*(3 - 2*I*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^2)) - ((3 - 2*I*a)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(2*b^2) - ((1 - I*a)*(1 + I*a + I*b*x)^(5/2))/(b^2*Sqrt[1 - I*a - I*b*x]) + (3*(3*I + 2*a)*ArcSinh[a + b*x])/(2*b^2)} -{E^((3*I)*ArcTan[a + b*x])*x^0, x, 6, ((-3*I)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/b - ((2*I)*(1 + I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) - (3*ArcSinh[a + b*x])/b} -{E^((3*I)*ArcTan[a + b*x])/x^1, x, 8, (4*Sqrt[1 + I*a + I*b*x])/((1 - I*a)*Sqrt[1 - I*a - I*b*x]) - I*ArcSinh[a + b*x] - (2*(I - a)^(3/2)*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(I + a)^(3/2)} -{E^((3*I)*ArcTan[a + b*x])/x^2, x, 5, -((6*I*b*Sqrt[1 + I*a + I*b*x])/((I + a)^2*Sqrt[1 - I*a - I*b*x])) - (1 + I*a + I*b*x)^(3/2)/((1 - I*a)*x*Sqrt[1 - I*a - I*b*x]) + (6*I*Sqrt[I - a]*b*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(I + a)^(5/2)} -{E^((3*I)*ArcTan[a + b*x])/x^3, x, 6, (3*(3*I - 2*a)*b^2*Sqrt[1 + I*a + I*b*x])/((1 + I*a)*(I + a)^3*Sqrt[1 - I*a - I*b*x]) + ((3*I - 2*a)*b*(1 + I*a + I*b*x)^(3/2))/(2*(1 + I*a)*(I + a)^2*x*Sqrt[1 - I*a - I*b*x]) - (1 + I*a + I*b*x)^(5/2)/(2*(1 + a^2)*x^2*Sqrt[1 - I*a - I*b*x]) + (3*(3 + 2*I*a)*b^2*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(Sqrt[I - a]*(I + a)^(7/2))} -{E^((3*I)*ArcTan[a + b*x])/x^4, x, 8, ((52 + 51*I*a - 2*a^2)*b^3*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^4*Sqrt[1 - I*a - I*b*x]) - ((I - a)*Sqrt[1 + I*a + I*b*x])/(3*(I + a)*x^3*Sqrt[1 - I*a - I*b*x]) + (7*I*b*Sqrt[1 + I*a + I*b*x])/(6*(I + a)^2*x^2*Sqrt[1 - I*a - I*b*x]) + ((19 + 16*I*a)*b^2*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^3*x*Sqrt[1 - I*a - I*b*x]) - ((11*I - 18*a - 6*I*a^2)*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(3/2)*(I + a)^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/E^(I*ArcTan[a + b*x]), x, 8, -(((3*I - 12*a - 24*I*a^2 + 16*a^3 + 8*I*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5)) + ((I - 8*a)*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(20*b^3) + (x^3*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(5*b^2) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(19*I - 114*a - 86*I*a^2 + 96*a^3 + 2*(13 + 14*I*a - 36*a^2)*b*x))/(120*b^5) + ((3 + 12*I*a - 24*a^2 - 16*I*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)} -{x^3/E^(I*ArcTan[a + b*x]), x, 7, -(((3 + 12*I*a - 12*a^2 - 8*I*a^3)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4)) + (x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(4*b^2) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(7 + 10*I*a - 18*a^2 - 2*(I - 6*a)*b*x))/(24*b^4) - ((3*I - 12*a - 12*I*a^2 + 8*a^3)*ArcSinh[a + b*x])/(8*b^4)} -{x^2/E^(I*ArcTan[a + b*x]), x, 7, ((I - 2*a - (2*I)*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^3) + ((I - 4*a)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(6*b^3) + (x*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(3*b^2) - ((1 + (2*I)*a - 2*a^2)*ArcSinh[a + b*x])/(2*b^3)} -{x^1/E^(I*ArcTan[a + b*x]), x, 6, ((1 + 2*I*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^2) + ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(2*b^2) + ((I - 2*a)*ArcSinh[a + b*x])/(2*b^2)} -{1/E^(I*ArcTan[a + b*x]), x, 5, ((-I)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/b + ArcSinh[a + b*x]/b} -{1/(E^(I*ArcTan[a + b*x])*x^1), x, 8, (-I)*ArcSinh[a + b*x] - (2*Sqrt[I + a]*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/Sqrt[I - a]} -{1/(E^(I*ArcTan[a + b*x])*x^2), x, 4, -((Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/((1 + I*a)*x)) - (2*I*b*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(3/2)*Sqrt[I + a])} -{1/(E^(I*ArcTan[a + b*x])*x^3), x, 5, ((1 - 2*I*a)*b*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*(I - a)^2*(I + a)*x) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(2*(1 + a^2)*x^2) + ((1 - 2*I*a)*b^2*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(5/2)*(I + a)^(3/2))} -{1/(E^(I*ArcTan[a + b*x])*x^4), x, 7, -((Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(3*(1 + I*a)*x^3)) + ((3 - 2*I*a)*b*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(6*(I - a)^2*(I + a)*x^2) + ((4 - 9*I*a - 2*a^2)*b^2*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(6*(1 + I*a)*(1 + a^2)^2*x) + ((2*a + I*(1 - 2*a^2))*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(7/2)*(I + a)^(5/2))} - - -{x^4/E^((2*I)*ArcTan[a + b*x]), x, 3, (-2*(1 + I*a)^3*x)/b^4 - (I*(I - a)^2*x^2)/b^3 + (2*(1 + I*a)*x^3)/(3*b^2) - ((I/2)*x^4)/b - x^5/5 - ((2*I)*(I - a)^4*Log[I - a - b*x])/b^5} -{x^3/E^((2*I)*ArcTan[a + b*x]), x, 3, ((-2*I)*(I - a)^2*x)/b^3 + ((1 + I*a)*x^2)/b^2 - (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 + I*a)^3*Log[I - a - b*x])/b^4} -{x^2/E^((2*I)*ArcTan[a + b*x]), x, 3, (2*(1 + I*a)*x)/b^2 - (I*x^2)/b - x^3/3 - ((2*I)*(I - a)^2*Log[I - a - b*x])/b^3} -{x^1/E^((2*I)*ArcTan[a + b*x]), x, 3, ((-2*I)*x)/b - x^2/2 + (2*(1 + I*a)*Log[I - a - b*x])/b^2} -{1/E^((2*I)*ArcTan[a + b*x]), x, 3, -x - ((2*I)*Log[I - a - b*x])/b} -{1/(E^((2*I)*ArcTan[a + b*x])*x^1), x, 3, ((I + a)*Log[x])/(I - a) - (2*Log[I - a - b*x])/(1 + I*a)} -{1/(E^((2*I)*ArcTan[a + b*x])*x^2), x, 3, -((I + a)/((I - a)*x)) + ((2*I)*b*Log[x])/(I - a)^2 - ((2*I)*b*Log[I - a - b*x])/(I - a)^2} -{1/(E^((2*I)*ArcTan[a + b*x])*x^3), x, 3, -(I + a)/(2*(I - a)*x^2) - ((2*I)*b)/((I - a)^2*x) - (2*b^2*Log[x])/(1 + I*a)^3 + (2*b^2*Log[I - a - b*x])/(1 + I*a)^3} -{1/(E^((2*I)*ArcTan[a + b*x])*x^4), x, 3, -(I + a)/(3*(I - a)*x^3) - (I*b)/((I - a)^2*x^2) + (2*b^2)/((1 + I*a)^3*x) + ((2*I)*b^3*Log[x])/(I - a)^4 - ((2*I)*b^3*Log[I - a - b*x])/(I - a)^4} - - -{x^4/E^((3*I)*ArcTan[a + b*x]), x, 9, (2*I*x^4*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + (3*(19*I - 68*a - 88*I*a^2 + 48*a^3 + 8*I*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) - (3*(17*I - 16*a)*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(20*b^3) - (11*x^3*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(5*b^2) + (I*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(163 + 458*I*a - 422*a^2 - 112*I*a^3 - 2*(61*I - 118*a - 52*I*a^2)*b*x))/(40*b^5) - (3*(19 + 68*I*a - 88*a^2 - 48*I*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)} -{x^3/E^((3*I)*ArcTan[a + b*x]), x, 8, (2*I*x^3*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + (3*(17 + 44*I*a - 36*a^2 - 8*I*a^3)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - (9*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(4*b^2) - (I*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(29*I - 54*a - 22*I*a^2 + 2*(11 + 10*I*a)*b*x))/(8*b^4) + (3*(17*I - 44*a - 36*I*a^2 + 8*a^3)*ArcSinh[a + b*x])/(8*b^4)} -{x^2/E^((3*I)*ArcTan[a + b*x]), x, 8, (I*(I - a)^2*(1 - I*a - I*b*x)^(5/2))/(b^3*Sqrt[1 + I*a + I*b*x]) - ((11*I - 18*a - 6*I*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^3) - ((11*I - 18*a - 6*I*a^2)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(6*b^3) - (I*(1 - I*a - I*b*x)^(5/2)*Sqrt[1 + I*a + I*b*x])/(3*b^3) + ((11 + 18*I*a - 6*a^2)*ArcSinh[a + b*x])/(2*b^3)} -{x^1/E^((3*I)*ArcTan[a + b*x]), x, 7, -(((1 + I*a)*(1 - I*a - I*b*x)^(5/2))/(b^2*Sqrt[1 + I*a + I*b*x])) - (3*(3 + 2*I*a)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^2) - ((3 + 2*I*a)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(2*b^2) - (3*(3*I - 2*a)*ArcSinh[a + b*x])/(2*b^2)} -{1/E^((3*I)*ArcTan[a + b*x]), x, 6, ((2*I)*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + ((3*I)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/b - (3*ArcSinh[a + b*x])/b} -{1/(E^((3*I)*ArcTan[a + b*x])*x^1), x, 8, (4*Sqrt[1 - I*a - I*b*x])/((1 + I*a)*Sqrt[1 + I*a + I*b*x]) + I*ArcSinh[a + b*x] - (2*(I + a)^(3/2)*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(I - a)^(3/2)} -{1/(E^((3*I)*ArcTan[a + b*x])*x^2), x, 5, (6*I*b*Sqrt[1 - I*a - I*b*x])/((I - a)^2*Sqrt[1 + I*a + I*b*x]) - (1 - I*a - I*b*x)^(3/2)/((1 + I*a)*x*Sqrt[1 + I*a + I*b*x]) - (6*I*Sqrt[I + a]*b*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/(I - a)^(5/2)} -{1/(E^((3*I)*ArcTan[a + b*x])*x^3), x, 6, -((3*(3*I + 2*a)*b^2*Sqrt[1 - I*a - I*b*x])/((1 + I*a)^3*(I + a)*Sqrt[1 + I*a + I*b*x])) + ((3 - 2*I*a)*b*(1 - I*a - I*b*x)^(3/2))/(2*(I - a)^2*(I + a)*x*Sqrt[1 + I*a + I*b*x]) - (1 - I*a - I*b*x)^(5/2)/(2*(1 + a^2)*x^2*Sqrt[1 + I*a + I*b*x]) + (3*(3 - 2*I*a)*b^2*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(7/2)*Sqrt[I + a])} -{1/(E^((3*I)*ArcTan[a + b*x])*x^4), x, 8, -(((52 - 51*I*a - 2*a^2)*b^3*Sqrt[1 - I*a - I*b*x])/(6*(I - a)^4*(I + a)*Sqrt[1 + I*a + I*b*x])) - ((I + a)*Sqrt[1 - I*a - I*b*x])/(3*(I - a)*x^3*Sqrt[1 + I*a + I*b*x]) - (7*I*b*Sqrt[1 - I*a - I*b*x])/(6*(I - a)^2*x^2*Sqrt[1 + I*a + I*b*x]) + ((19 - 16*I*a)*b^2*Sqrt[1 - I*a - I*b*x])/(6*(I - a)^3*(I + a)*x*Sqrt[1 + I*a + I*b*x]) + ((11*I + 18*a - 6*I*a^2)*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a - I*b*x])])/((I - a)^(9/2)*(I + a)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/2 I ArcTan[a+b x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^((I/2)*ArcTan[a + b*x])*x^2, x, 15, -((3*I + 4*a - (8*I)*a^2)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/(8*b^3) - ((I + 8*a)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(5/4))/(12*b^3) + (x*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(5/4))/(3*b^2) + ((3*I + 4*a - (8*I)*a^2)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((3*I + 4*a - (8*I)*a^2)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((3*I + 4*a - (8*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3) + ((3*I + 4*a - (8*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3)} -{E^((I/2)*ArcTan[a + b*x])*x, x, 14, ((1 - (4*I)*a)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/(4*b^2) + ((1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(5/4))/(2*b^2) - ((1 - (4*I)*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) + ((1 - (4*I)*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) + ((1 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) - ((1 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)} -{E^((I/2)*ArcTan[a + b*x]), x, 13, (I*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/b - (I*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + (I*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + ((I/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) - ((I/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b)} -{E^((I/2)*ArcTan[a + b*x])/x, x, 15, -((2*(I - a)^(1/4)*ArcTan[((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))/((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))])/(I + a)^(1/4)) - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 + I*(a + b*x))^(1/4))/(1 - I*(a + b*x))^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 + I*(a + b*x))^(1/4))/(1 - I*(a + b*x))^(1/4)] - (2*(I - a)^(1/4)*ArcTanh[((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))/((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))])/(I + a)^(1/4) - Log[1 - (Sqrt[2]*(1 + I*(a + b*x))^(1/4))/(1 - I*(a + b*x))^(1/4) + Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]]/Sqrt[2] + Log[1 + (Sqrt[2]*(1 + I*(a + b*x))^(1/4))/(1 - I*(a + b*x))^(1/4) + Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]]/Sqrt[2]} -{E^((I/2)*ArcTan[a + b*x])/x^2, x, 6, -(((I + a + b*x)*(1 + I*(a + b*x))^(1/4))/((I + a)*x*(1 - I*(a + b*x))^(1/4))) + (I*b*ArcTan[((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))/((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))])/((I - a)^(3/4)*(I + a)^(5/4)) + (I*b*ArcTanh[((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))/((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))])/((I - a)^(3/4)*(I + a)^(5/4))} - - -{E^(((3*I)/2)*ArcTan[a + b*x])*x^2, x, 15, -((17*I + 36*a - (24*I)*a^2)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(24*b^3) - ((3*I + 8*a)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(7/4))/(12*b^3) + (x*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(7/4))/(3*b^2) + ((17*I + 36*a - (24*I)*a^2)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((17*I + 36*a - (24*I)*a^2)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) + ((17*I + 36*a - (24*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3) - ((17*I + 36*a - (24*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3)} -{E^(((3*I)/2)*ArcTan[a + b*x])*x, x, 14, ((3 - (4*I)*a)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(4*b^2) + ((1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(7/4))/(2*b^2) - (3*(3 - (4*I)*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) - (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)} -{E^(((3*I)/2)*ArcTan[a + b*x]), x, 13, (I*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/b - ((3*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + ((3*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) - (((3*I)/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + (((3*I)/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b)} -{E^(((3*I)/2)*ArcTan[a + b*x])/x, x, 18, (2*(I - a)^(3/4)*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/(I + a)^(3/4) + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)] - (2*(I - a)^(3/4)*ArcTanh[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/(I + a)^(3/4) + Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2]} -{E^(((3*I)/2)*ArcTan[a + b*x])/x^2, x, 6, -(((1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/((1 - I*a)*x)) - (3*I*b*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(7/4)) + (3*I*b*ArcTanh[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(1/4)*(I + a)^(7/4))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/E^((I/2)*ArcTan[a + b*x]), x, 15, ((3*I - 4*a - (8*I)*a^2)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(8*b^3) + ((I - 8*a)*(1 - I*a - I*b*x)^(5/4)*(1 + I*a + I*b*x)^(3/4))/(12*b^3) + (x*(1 - I*a - I*b*x)^(5/4)*(1 + I*a + I*b*x)^(3/4))/(3*b^2) + ((3*I - 4*a - (8*I)*a^2)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((3*I - 4*a - (8*I)*a^2)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) + ((3*I - 4*a - (8*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3) - ((3*I - 4*a - (8*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3)} -{x/E^((I/2)*ArcTan[a + b*x]), x, 14, ((1 + 4*I*a)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(4*b^2) + ((1 - I*a - I*b*x)^(5/4)*(1 + I*a + I*b*x)^(3/4))/(2*b^2) + ((1 + 4*I*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) - ((1 + 4*I*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) + ((1 + 4*I*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) - ((1 + 4*I*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)} -{E^((-I/2)*ArcTan[a + b*x]), x, 13, ((-I)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/b - (I*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + (I*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) - ((I/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + ((I/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b)} -{1/(E^((I/2)*ArcTan[a + b*x])*x), x, 14, -((2*(I + a)^(1/4)*ArcTan[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))])/(I - a)^(1/4)) - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)] - (2*(I + a)^(1/4)*ArcTanh[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))])/(I - a)^(1/4) - Log[1 + Sqrt[1 - I*(a + b*x)]/Sqrt[1 + I*(a + b*x)] - (Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*(a + b*x)]/Sqrt[1 + I*(a + b*x)] + (Sqrt[2]*(1 - I*(a + b*x))^(1/4))/(1 + I*(a + b*x))^(1/4)]/Sqrt[2]} -{1/(E^((I/2)*ArcTan[a + b*x])*x^2), x, 5, -(((I - a - b*x)*(1 - I*(a + b*x))^(1/4))/((I - a)*x*(1 + I*(a + b*x))^(1/4))) - (I*b*ArcTan[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))])/((I - a)^(5/4)*(I + a)^(3/4)) - (I*b*ArcTanh[((I - a)^(1/4)*(1 - I*(a + b*x))^(1/4))/((I + a)^(1/4)*(1 + I*(a + b*x))^(1/4))])/((I - a)^(5/4)*(I + a)^(3/4))} - - -{x^2/E^(((3*I)/2)*ArcTan[a + b*x]), x, 15, ((17*I - 36*a - (24*I)*a^2)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/(24*b^3) + ((3*I - 8*a)*(1 - I*a - I*b*x)^(7/4)*(1 + I*a + I*b*x)^(1/4))/(12*b^3) + (x*(1 - I*a - I*b*x)^(7/4)*(1 + I*a + I*b*x)^(1/4))/(3*b^2) + ((17*I - 36*a - (24*I)*a^2)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((17*I - 36*a - (24*I)*a^2)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^3) - ((17*I - 36*a - (24*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3) + ((17*I - 36*a - (24*I)*a^2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(16*Sqrt[2]*b^3)} -{x/E^(((3*I)/2)*ArcTan[a + b*x]), x, 14, ((3 + 4*I*a)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/(4*b^2) + ((1 - I*a - I*b*x)^(7/4)*(1 + I*a + I*b*x)^(1/4))/(2*b^2) + (3*(3 + 4*I*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) - (3*(3 + 4*I*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4*Sqrt[2]*b^2) - (3*(3 + 4*I*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) + (3*(3 + 4*I*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)} -{E^(((-3*I)/2)*ArcTan[a + b*x]), x, 13, ((-I)*(1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/b - ((3*I)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + ((3*I)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) + (((3*I)/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b) - (((3*I)/2)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(Sqrt[2]*b)} -{1/(E^(((3*I)/2)*ArcTan[a + b*x])*x), x, 18, -((2*(I + a)^(3/4)*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/(I - a)^(3/4)) - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)] - (2*(I + a)^(3/4)*ArcTanh[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/(I - a)^(3/4) + Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2]} -{1/(E^(((3*I)/2)*ArcTan[a + b*x])*x^2), x, 6, -(((1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/((1 + I*a)*x)) - (3*I*b*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(7/4)*(I + a)^(1/4)) - (3*I*b*ArcTanh[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1/4))])/((I - a)^(7/4)*(I + a)^(1/4))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a+b x]) with n symbolic*) - - -{E^(n*ArcTan[a + b*x])*x^m, x, 4, (1/(1 + m))*((x^(1 + m)*(1 - I*a - I*b*x)^((I*n)/2)*(1 - (b*x)/(I - a))^((I*n)/2)*AppellF1[1 + m, -((I*n)/2), (I*n)/2, 2 + m, -((b*x)/(I + a)), (b*x)/(I - a)])/((1 + I*a + I*b*x)^((I*n)/2)*(1 + (b*x)/(I + a))^((I*n)/2)))} - - -{E^(n*ArcTan[a + b*x])*x^3, x, 4, (x^2*(1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2))/(4*b^2) - ((1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2)*(6 - 18*a^2 - 10*a*n - n^2 + 2*b*(6*a + n)*x))/(24*b^4) + (2^(-2 - (I*n)/2)*(24*a^3 + 36*a^2*n - 12*a*(2 - n^2) - n*(8 - n^2))*(1 - I*a - I*b*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a - I*b*x)])/(3*b^4*(2*I - n))} -{E^(n*ArcTan[a + b*x])*x^2, x, 4, -(((4*a + n)*(1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2))/(6*b^3)) + (x*(1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2))/(3*b^2) + ((2 - 6*a^2 - 6*a*n - n^2)*(1 - I*a - I*b*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a - I*b*x)])/(2^((I*n)/2)*(3*b^3*(2*I - n)))} -{E^(n*ArcTan[a + b*x])*x^1, x, 3, ((1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2))/(2*b^2) + ((2*a + n)*(1 - I*a - I*b*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a - I*b*x)])/(2^((I*n)/2)*(b^2*(2*I - n)))} -{E^(n*ArcTan[a + b*x])*x^0, x, 2, -((2^(1 - (I*n)/2)*(1 - I*a - I*b*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a - I*b*x)])/(b*(2*I - n)))} -{E^(n*ArcTan[a + b*x])/x^1, x, 5, (2*I*(1 - I*a - I*b*x)^((I*n)/2)*Hypergeometric2F1[1, (I*n)/2, 1 + (I*n)/2, ((I - a)*(1 - I*a - I*b*x))/((I + a)*(1 + I*a + I*b*x))])/((1 + I*a + I*b*x)^((I*n)/2)*n) - (I*2^(1 - (I*n)/2)*(1 - I*a - I*b*x)^((I*n)/2)*Hypergeometric2F1[(I*n)/2, (I*n)/2, 1 + (I*n)/2, (1/2)*(1 - I*a - I*b*x)])/n} -{E^(n*ArcTan[a + b*x])/x^2, x, 2, -((4*b*(1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(-1 - (I*n)/2)*Hypergeometric2F1[2, 1 + (I*n)/2, 2 + (I*n)/2, ((I - a)*(1 - I*a - I*b*x))/((I + a)*(1 + I*a + I*b*x))])/((I + a)^2*(2*I - n)))} -{E^(n*ArcTan[a + b*x])/x^3, x, 3, -(((1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(1 - (I*n)/2))/(2*(1 + a^2)*x^2)) - (2*b^2*(2*a - n)*(1 - I*a - I*b*x)^(1 + (I*n)/2)*(1 + I*a + I*b*x)^(-1 - (I*n)/2)*Hypergeometric2F1[2, 1 + (I*n)/2, 2 + (I*n)/2, ((I - a)*(1 - I*a - I*b*x))/((I + a)*(1 + I*a + I*b*x))])/((I - a)*(I + a)^3*(2*I - n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p E^(n ArcTan[a x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p E^(1 ArcTan[a x])*) - - -{E^ArcTan[a*x]*(c + a^2*c*x^2)^p, x, 3, (I*2^((1 - I/2) + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I/2 - p, (1 + I/2) + p, (2 + I/2) + p, (1/2)*(1 - I*a*x)])/((1 + a^2*x^2)^p*(a*((2 + I) + 2*p)))} - -{E^ArcTan[a*x]*(c + a^2*c*x^2)^2, x, 2, ((1/37 + (6*I)/37)*2^(3 - I/2)*c^2*(1 - I*a*x)^(3 + I/2)*Hypergeometric2F1[-2 + I/2, 3 + I/2, 4 + I/2, (1/2)*(1 - I*a*x)])/a} -{E^ArcTan[a*x]*(c + a^2*c*x^2)^1, x, 2, ((1/17 + (4*I)/17)*2^(2 - I/2)*c*(1 - I*a*x)^(2 + I/2)*Hypergeometric2F1[-1 + I/2, 2 + I/2, 3 + I/2, (1/2)*(1 - I*a*x)])/a} -{E^ArcTan[a*x]*(c + a^2*c*x^2)^0, x, 2, ((1/5 + (2*I)/5)*2^(1 - I/2)*(1 - I*a*x)^(1 + I/2)*Hypergeometric2F1[I/2, 1 + I/2, 2 + I/2, (1/2)*(1 - I*a*x)])/a} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^1, x, 1, E^ArcTan[a*x]/(a*c)} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 2, (2*E^ArcTan[a*x])/(5*a*c^2) + (E^ArcTan[a*x]*(1 + 2*a*x))/(5*a*c^2*(1 + a^2*x^2))} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^3, x, 3, (24*E^ArcTan[a*x])/(85*a*c^3) + (E^ArcTan[a*x]*(1 + 4*a*x))/(17*a*c^3*(1 + a^2*x^2)^2) + (12*E^ArcTan[a*x]*(1 + 2*a*x))/(85*a*c^3*(1 + a^2*x^2))} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^4, x, 4, (144*E^ArcTan[a*x])/(629*a*c^4) + (E^ArcTan[a*x]*(1 + 6*a*x))/(37*a*c^4*(1 + a^2*x^2)^3) + (30*E^ArcTan[a*x]*(1 + 4*a*x))/(629*a*c^4*(1 + a^2*x^2)^2) + (72*E^ArcTan[a*x]*(1 + 2*a*x))/(629*a*c^4*(1 + a^2*x^2))} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^5, x, 5, (8064*E^ArcTan[a*x])/(40885*a*c^5) + (E^ArcTan[a*x]*(1 + 8*a*x))/(65*a*c^5*(1 + a^2*x^2)^4) + (56*E^ArcTan[a*x]*(1 + 6*a*x))/(2405*a*c^5*(1 + a^2*x^2)^3) + (336*E^ArcTan[a*x]*(1 + 4*a*x))/(8177*a*c^5*(1 + a^2*x^2)^2) + (4032*E^ArcTan[a*x]*(1 + 2*a*x))/(40885*a*c^5*(1 + a^2*x^2))} - - -{E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2), x, 3, ((1/13 + (5*I)/13)*2^(3/2 - I/2)*c*(1 - I*a*x)^(5/2 + I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(3/2) + I/2, 5/2 + I/2, 7/2 + I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2])} -{E^ArcTan[a*x]*(c + a^2*c*x^2)^(1/2), x, 3, ((1/5 + (3*I)/5)*2^(1/2 - I/2)*(1 - I*a*x)^(3/2 + I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(1/2) + I/2, 3/2 + I/2, 5/2 + I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2])} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^(1/2), x, 3, ((1 + I)*2^(-(1/2) - I/2)*(1 - I*a*x)^(1/2 + I/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + I/2, 1/2 + I/2, 3/2 + I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[c + a^2*c*x^2])} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^(3/2), x, 1, (E^ArcTan[a*x]*(1 + a*x))/(2*a*c*Sqrt[c + a^2*c*x^2])} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^(5/2), x, 2, (E^ArcTan[a*x]*(1 + 3*a*x))/(10*a*c*(c + a^2*c*x^2)^(3/2)) + (3*E^ArcTan[a*x]*(1 + a*x))/(10*a*c^2*Sqrt[c + a^2*c*x^2])} -{E^ArcTan[a*x]/(c + a^2*c*x^2)^(7/2), x, 3, (E^ArcTan[a*x]*(1 + 5*a*x))/(26*a*c*(c + a^2*c*x^2)^(5/2)) + (E^ArcTan[a*x]*(1 + 3*a*x))/(13*a*c^2*(c + a^2*c*x^2)^(3/2)) + (3*E^ArcTan[a*x]*(1 + a*x))/(13*a*c^3*Sqrt[c + a^2*c*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p E^(2 ArcTan[a x])*) - - -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^p, x, 3, (I*2^(-I + p)*(1 - I*a*x)^((1 + I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I - p, (1 + I) + p, (2 + I) + p, (1/2)*(1 - I*a*x)])/((1 + a^2*x^2)^p*(a*((1 + I) + p)))} - -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2, x, 2, ((1/5 + (3*I)/5)*2^(1 - I)*c^2*(1 - I*a*x)^(3 + I)*Hypergeometric2F1[-2 + I, 3 + I, 4 + I, (1/2)*(1 - I*a*x)])/a} -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^1, x, 2, ((1/5 + (2*I)/5)*2^(1 - I)*c*(1 - I*a*x)^(2 + I)*Hypergeometric2F1[-1 + I, 2 + I, 3 + I, (1/2)*(1 - I*a*x)])/a} -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^0, x, 2, ((1 + I)*2^(-1 - I)*(1 - I*a*x)^(1 + I)*Hypergeometric2F1[I, 1 + I, 2 + I, (1/2)*(1 - I*a*x)])/a} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^1, x, 1, E^(2*ArcTan[a*x])/(2*a*c)} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, E^(2*ArcTan[a*x])/(8*a*c^2) + (E^(2*ArcTan[a*x])*(1 + a*x))/(4*a*c^2*(1 + a^2*x^2))} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, (3*E^(2*ArcTan[a*x]))/(40*a*c^3) + (E^(2*ArcTan[a*x])*(1 + 2*a*x))/(10*a*c^3*(1 + a^2*x^2)^2) + (3*E^(2*ArcTan[a*x])*(1 + a*x))/(20*a*c^3*(1 + a^2*x^2))} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^4, x, 4, (9*E^(2*ArcTan[a*x]))/(160*a*c^4) + (E^(2*ArcTan[a*x])*(1 + 3*a*x))/(20*a*c^4*(1 + a^2*x^2)^3) + (3*E^(2*ArcTan[a*x])*(1 + 2*a*x))/(40*a*c^4*(1 + a^2*x^2)^2) + (9*E^(2*ArcTan[a*x])*(1 + a*x))/(80*a*c^4*(1 + a^2*x^2))} - - -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2), x, 3, ((2/29 + (5*I)/29)*2^(5/2 - I)*c*(1 - I*a*x)^(5/2 + I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(3/2) + I, 5/2 + I, 7/2 + I, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2])} -{E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(1/2), x, 3, ((2/13 + (3*I)/13)*2^(3/2 - I)*(1 - I*a*x)^(3/2 + I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(1/2) + I, 3/2 + I, 5/2 + I, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2])} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 3, ((2/5 + I/5)*2^(1/2 - I)*(1 - I*a*x)^(1/2 + I)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + I, 1/2 + I, 3/2 + I, (1/2)*(1 - I*a*x)])/(a*Sqrt[c + a^2*c*x^2])} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 1, (E^(2*ArcTan[a*x])*(2 + a*x))/(5*a*c*Sqrt[c + a^2*c*x^2])} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 2, (E^(2*ArcTan[a*x])*(2 + 3*a*x))/(13*a*c*(c + a^2*c*x^2)^(3/2)) + (6*E^(2*ArcTan[a*x])*(2 + a*x))/(65*a*c^2*Sqrt[c + a^2*c*x^2])} -{E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(7/2), x, 3, (E^(2*ArcTan[a*x])*(2 + 5*a*x))/(29*a*c*(c + a^2*c*x^2)^(5/2)) + (20*E^(2*ArcTan[a*x])*(2 + 3*a*x))/(377*a*c^2*(c + a^2*c*x^2)^(3/2)) + (24*E^(2*ArcTan[a*x])*(2 + a*x))/(377*a*c^3*Sqrt[c + a^2*c*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p / E^(1 ArcTan[a x])*) - - -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^p, x, 3, (2^((1 + I/2) + p)*(1 - I*a*x)^((1 - I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[-(I/2) - p, (1 - I/2) + p, (2 - I/2) + p, (1/2)*(1 - I*a*x)])/((1 + a^2*x^2)^p*(a*((-1 - 2*I) - 2*I*p)))} - -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^2, x, 2, -(((1/37 - (6*I)/37)*2^(3 + I/2)*c^2*(1 - I*a*x)^(3 - I/2)*Hypergeometric2F1[-2 - I/2, 3 - I/2, 4 - I/2, (1/2)*(1 - I*a*x)])/a)} -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^1, x, 2, -(((1/17 - (4*I)/17)*2^(2 + I/2)*c*(1 - I*a*x)^(2 - I/2)*Hypergeometric2F1[-1 - I/2, 2 - I/2, 3 - I/2, (1/2)*(1 - I*a*x)])/a)} -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^0, x, 2, -(((1/5 - (2*I)/5)*2^(1 + I/2)*(1 - I*a*x)^(1 - I/2)*Hypergeometric2F1[-(I/2), 1 - I/2, 2 - I/2, (1/2)*(1 - I*a*x)])/a)} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^1, x, 1, -(1/(E^ArcTan[a*x]*(a*c)))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^2, x, 2, -(2/(E^ArcTan[a*x]*(5*a*c^2))) - (1 - 2*a*x)/(E^ArcTan[a*x]*(5*a*c^2*(1 + a^2*x^2)))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^3, x, 3, -(24/(E^ArcTan[a*x]*(85*a*c^3))) - (1 - 4*a*x)/(E^ArcTan[a*x]*(17*a*c^3*(1 + a^2*x^2)^2)) - (12*(1 - 2*a*x))/(E^ArcTan[a*x]*(85*a*c^3*(1 + a^2*x^2)))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^4, x, 4, -(144/(E^ArcTan[a*x]*(629*a*c^4))) - (1 - 6*a*x)/(E^ArcTan[a*x]*(37*a*c^4*(1 + a^2*x^2)^3)) - (30*(1 - 4*a*x))/(E^ArcTan[a*x]*(629*a*c^4*(1 + a^2*x^2)^2)) - (72*(1 - 2*a*x))/(E^ArcTan[a*x]*(629*a*c^4*(1 + a^2*x^2)))} - - -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2), x, 3, -(((1/13 - (5*I)/13)*2^(3/2 + I/2)*c*(1 - I*a*x)^(5/2 - I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(3/2) - I/2, 5/2 - I/2, 7/2 - I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2]))} -{1/E^ArcTan[a*x]*(c + a^2*c*x^2)^(1/2), x, 3, -(((1/5 - (3*I)/5)*2^(1/2 + I/2)*(1 - I*a*x)^(3/2 - I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(1/2) - I/2, 3/2 - I/2, 5/2 - I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2]))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^(1/2), x, 3, -(((1 - I)*2^(-(1/2) + I/2)*(1 - I*a*x)^(1/2 - I/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 - I/2, 1/2 - I/2, 3/2 - I/2, (1/2)*(1 - I*a*x)])/(a*Sqrt[c + a^2*c*x^2]))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^(3/2), x, 1, -((1 - a*x)/(E^ArcTan[a*x]*(2*a*c*Sqrt[c + a^2*c*x^2])))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^(5/2), x, 2, -((1 - 3*a*x)/(E^ArcTan[a*x]*(10*a*c*(c + a^2*c*x^2)^(3/2)))) - (3*(1 - a*x))/(E^ArcTan[a*x]*(10*a*c^2*Sqrt[c + a^2*c*x^2]))} -{1/E^ArcTan[a*x]/(c + a^2*c*x^2)^(7/2), x, 3, -((1 - 5*a*x)/(E^ArcTan[a*x]*(26*a*c*(c + a^2*c*x^2)^(5/2)))) - (1 - 3*a*x)/(E^ArcTan[a*x]*(13*a*c^2*(c + a^2*c*x^2)^(3/2))) - (3*(1 - a*x))/(E^ArcTan[a*x]*(13*a*c^3*Sqrt[c + a^2*c*x^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p / E^(2 ArcTan[a x])*) - - -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^p, x, 3, (I*2^(I + p)*(1 - I*a*x)^((1 - I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[-I - p, (1 - I) + p, (2 - I) + p, (1/2)*(1 - I*a*x)])/((1 + a^2*x^2)^p*(a*((1 - I) + p)))} - -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2, x, 2, -(((1/5 - (3*I)/5)*2^(1 + I)*c^2*(1 - I*a*x)^(3 - I)*Hypergeometric2F1[-2 - I, 3 - I, 4 - I, (1/2)*(1 - I*a*x)])/a)} -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^1, x, 2, -(((1/5 - (2*I)/5)*2^(1 + I)*c*(1 - I*a*x)^(2 - I)*Hypergeometric2F1[-1 - I, 2 - I, 3 - I, (1/2)*(1 - I*a*x)])/a)} -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^0, x, 2, -(((1 - I)*2^(-1 + I)*(1 - I*a*x)^(1 - I)*Hypergeometric2F1[-I, 1 - I, 2 - I, (1/2)*(1 - I*a*x)])/a)} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^1, x, 1, -(1/(E^(2*ArcTan[a*x])*(2*a*c)))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, -(1/(E^(2*ArcTan[a*x])*(8*a*c^2))) - (1 - a*x)/(E^(2*ArcTan[a*x])*(4*a*c^2*(1 + a^2*x^2)))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, -(3/(E^(2*ArcTan[a*x])*(40*a*c^3))) - (1 - 2*a*x)/(E^(2*ArcTan[a*x])*(10*a*c^3*(1 + a^2*x^2)^2)) - (3*(1 - a*x))/(E^(2*ArcTan[a*x])*(20*a*c^3*(1 + a^2*x^2)))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^4, x, 4, -(9/(E^(2*ArcTan[a*x])*(160*a*c^4))) - (1 - 3*a*x)/(E^(2*ArcTan[a*x])*(20*a*c^4*(1 + a^2*x^2)^3)) - (3*(1 - 2*a*x))/(E^(2*ArcTan[a*x])*(40*a*c^4*(1 + a^2*x^2)^2)) - (9*(1 - a*x))/(E^(2*ArcTan[a*x])*(80*a*c^4*(1 + a^2*x^2)))} - - -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2), x, 3, -(((2/29 - (5*I)/29)*2^(5/2 + I)*c*(1 - I*a*x)^(5/2 - I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(3/2) - I, 5/2 - I, 7/2 - I, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2]))} -{1/E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(1/2), x, 3, -(((2/13 - (3*I)/13)*2^(3/2 + I)*(1 - I*a*x)^(3/2 - I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-(1/2) - I, 3/2 - I, 5/2 - I, (1/2)*(1 - I*a*x)])/(a*Sqrt[1 + a^2*x^2]))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 3, -(((2/5 - I/5)*2^(1/2 + I)*(1 - I*a*x)^(1/2 - I)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 - I, 1/2 - I, 3/2 - I, (1/2)*(1 - I*a*x)])/(a*Sqrt[c + a^2*c*x^2]))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 1, -((2 - a*x)/(E^(2*ArcTan[a*x])*(5*a*c*Sqrt[c + a^2*c*x^2])))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 2, -((2 - 3*a*x)/(E^(2*ArcTan[a*x])*(13*a*c*(c + a^2*c*x^2)^(3/2)))) - (6*(2 - a*x))/(E^(2*ArcTan[a*x])*(65*a*c^2*Sqrt[c + a^2*c*x^2]))} -{1/E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(7/2), x, 3, -((2 - 5*a*x)/(E^(2*ArcTan[a*x])*(29*a*c*(c + a^2*c*x^2)^(5/2)))) - (20*(2 - 3*a*x))/(E^(2*ArcTan[a*x])*(377*a*c^2*(c + a^2*c*x^2)^(3/2))) - (24*(2 - a*x))/(E^(2*ArcTan[a*x])*(377*a*c^3*Sqrt[c + a^2*c*x^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p E^(I n ArcTan[a x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(5*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 3, -((2*I)/(a*(1 - I*a*x)^2)) + (4*I)/(a*(1 - I*a*x)) + (I*Log[I + a*x])/a} -{E^(4*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 5, (2*I*Sqrt[1 + I*a*x])/(a*Sqrt[1 - I*a*x]) - (2*I*(1 + I*a*x)^(3/2))/(3*a*(1 - I*a*x)^(3/2)) + ArcSinh[a*x]/a} -{E^(3*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 3, (2*Sqrt[1 + a^2*x^2])/(a*(I + a*x)*Sqrt[1 + a^2*x^2]) - (I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(a*Sqrt[1 + a^2*x^2])} -{E^(2*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 4, -((2*I*Sqrt[1 + I*a*x])/(a*Sqrt[1 - I*a*x])) - ArcSinh[a*x]/a} -{E^(1*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 2, (I*Log[I + a*x])/a} -{E^(-1*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 2, -((I*Log[I - a*x])/a)} -{E^(-2*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 4, (2*I*Sqrt[1 - I*a*x])/(a*Sqrt[1 + I*a*x]) - ArcSinh[a*x]/a} -{E^(-3*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 3, -((2*Sqrt[1 + a^2*x^2])/(a*(I - a*x)*Sqrt[1 + a^2*x^2])) + (I*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(a*Sqrt[1 + a^2*x^2])} -{E^(-4*I*ArcTan[a*x])/Sqrt[1 + a^2*x^2], x, 5, (2*I*(1 - I*a*x)^(3/2))/(3*a*(1 + I*a*x)^(3/2)) - (2*I*Sqrt[1 - I*a*x])/(a*Sqrt[1 + I*a*x]) + ArcSinh[a*x]/a} - - -{E^(5*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, -((2*I*Sqrt[1 + a^2*x^2])/(a*(1 - I*a*x)^2*Sqrt[c + a^2*c*x^2])) + (4*I*Sqrt[1 + a^2*x^2])/(a*(1 - I*a*x)*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(a*Sqrt[c + a^2*c*x^2])} -{E^(4*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 5, -((2*I*c*(1 + I*a*x)^3)/(3*a*(c + a^2*c*x^2)^(3/2))) + (2*I*(1 + I*a*x))/(a*Sqrt[c + a^2*c*x^2]) + ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a*Sqrt[c])} -{E^(3*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, (2*Sqrt[1 + a^2*x^2])/(a*(I + a*x)*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(a*Sqrt[c + a^2*c*x^2])} -{E^(2*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, -((2*I*(1 + I*a*x))/(a*Sqrt[c + a^2*c*x^2])) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a*Sqrt[c])} -{E^(1*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 3, (I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(a*Sqrt[c + a^2*c*x^2])} -{E^(-1*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 3, -((I*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(a*Sqrt[c + a^2*c*x^2]))} -{E^(-2*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, (2*I*(1 - I*a*x))/(a*Sqrt[c + a^2*c*x^2]) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a*Sqrt[c])} -{E^(-3*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 4, -((2*Sqrt[1 + a^2*x^2])/(a*(I - a*x)*Sqrt[c + a^2*c*x^2])) + (I*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(a*Sqrt[c + a^2*c*x^2])} -{E^(-4*I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x, 5, (2*I*c*(1 - I*a*x)^3)/(3*a*(c + a^2*c*x^2)^(3/2)) - (2*I*(1 - I*a*x))/(a*Sqrt[c + a^2*c*x^2]) + ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a*Sqrt[c])} - - -{E^(5*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 3, -(2/(3*a*(I + a*x)^3)) - I/(2*a*(I + a*x)^2)} -{E^(4*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 3, -((I*(1 + I*a*x)^(3/2))/(5*a*(1 - I*a*x)^(5/2))) - (I*(1 + I*a*x)^(3/2))/(15*a*(1 - I*a*x)^(3/2))} -{E^(3*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 2, -(I/(2*a*(1 - I*a*x)^2))} -{E^(2*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 3, -((I*Sqrt[1 + I*a*x])/(3*a*(1 - I*a*x)^(3/2))) - (I*Sqrt[1 + I*a*x])/(3*a*Sqrt[1 - I*a*x])} -{E^(1*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 4, 1/(2*a*(I + a*x)) + ArcTan[a*x]/(2*a)} -{E^(-1*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 4, -(1/(2*a*(I - a*x))) + ArcTan[a*x]/(2*a)} -{E^(-2*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 3, (I*Sqrt[1 - I*a*x])/(3*a*(1 + I*a*x)^(3/2)) + (I*Sqrt[1 - I*a*x])/(3*a*Sqrt[1 + I*a*x])} -{E^(-3*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 2, I/(2*a*(1 + I*a*x)^2)} -{E^(-4*I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2), x, 3, (I*(1 - I*a*x)^(3/2))/(5*a*(1 + I*a*x)^(5/2)) + (I*(1 - I*a*x)^(3/2))/(15*a*(1 + I*a*x)^(3/2))} - - -{E^(5*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 4, -((2*Sqrt[1 + a^2*x^2])/(3*a*c*(I + a*x)^3*Sqrt[c + a^2*c*x^2])) - (I*Sqrt[1 + a^2*x^2])/(2*a*c*(I + a*x)^2*Sqrt[c + a^2*c*x^2])} -{E^(4*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, -((I*c*(1 + I*a*x)^4)/(3*a*(c + a^2*c*x^2)^(5/2))) + (I*c*(1 + I*a*x)^5)/(15*a*(c + a^2*c*x^2)^(5/2))} -{E^(3*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, -((I*Sqrt[1 + a^2*x^2])/(2*a*c*(1 - I*a*x)^2*Sqrt[c + a^2*c*x^2]))} -{E^(2*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, -((2*I*(1 + I*a*x))/(3*a*(c + a^2*c*x^2)^(3/2))) + x/(3*c*Sqrt[c + a^2*c*x^2])} -{E^(1*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 5, Sqrt[1 + a^2*x^2]/(2*a*c*(I + a*x)*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(2*a*c*Sqrt[c + a^2*c*x^2])} -{E^(-1*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 5, -(Sqrt[1 + a^2*x^2]/(2*a*c*(I - a*x)*Sqrt[c + a^2*c*x^2])) + (Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(2*a*c*Sqrt[c + a^2*c*x^2])} -{E^(-2*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, (2*I*(1 - I*a*x))/(3*a*(c + a^2*c*x^2)^(3/2)) + x/(3*c*Sqrt[c + a^2*c*x^2])} -{E^(-3*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, (I*Sqrt[1 + a^2*x^2])/(2*a*c*(1 + I*a*x)^2*Sqrt[c + a^2*c*x^2])} -{E^(-4*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, (I*c*(1 - I*a*x)^4)/(3*a*(c + a^2*c*x^2)^(5/2)) - (I*c*(1 - I*a*x)^5)/(15*a*(c + a^2*c*x^2)^(5/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c+a^2 c x^2)^p E^(n ArcTan[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x]) (c+a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^2, x, 2, -((2^(3 - (I*n)/2)*c^2*(1 - I*a*x)^(3 + (I*n)/2)*Hypergeometric2F1[-2 + (I*n)/2, 3 + (I*n)/2, 4 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a*(6*I - n)))} -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^1, x, 2, -((2^(2 - (I*n)/2)*c*(1 - I*a*x)^(2 + (I*n)/2)*Hypergeometric2F1[-1 + (I*n)/2, 2 + (I*n)/2, 3 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a*(4*I - n)))} -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^0, x, 2, -((2^(1 - (I*n)/2)*(1 - I*a*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a*(2*I - n)))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2), x, 4, (E^(n*ArcTan[a*x])*(2*I + n - I*n^2))/(2*a^4*c*n) - (E^(n*ArcTan[a*x])*n*x)/(2*a^3*c) + (E^(n*ArcTan[a*x])*x^2)/(2*a^2*c) + (I*E^(n*ArcTan[a*x])*(-2 + n^2)*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, -E^(2*I*ArcTan[a*x])])/(a^4*c*n), (x^2*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(2*a^2*c)) + (I*(1 - I*a*x)^((I*n)/2)*(2 - I*n - n^2 + I*a*n^2*x))/((1 + I*a*x)^((I*n)/2)*(2*a^4*c*n)) + (2^(-1 - (I*n)/2)*(2 - n^2)*(1 - I*a*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, 1 + (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a^4*c*(2 + I*n))} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2), x, 4, -(((1 + I*n)*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^3*c*n))) + (x*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^2*c)) + (I*2^(1 - (I*n)/2)*(1 - I*a*x)^((I*n)/2)*Hypergeometric2F1[(I*n)/2, (I*n)/2, 1 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a^3*c)} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2), x, 3, (I*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^2*c*n)) - (I*2^(1 - (I*n)/2)*(1 - I*a*x)^((I*n)/2)*Hypergeometric2F1[(I*n)/2, (I*n)/2, 1 + (I*n)/2, (1/2)*(1 - I*a*x)])/(a^2*c*n)} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2), x, 1, E^(n*ArcTan[a*x])/(a*c*n)} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)), x, 3, (I*E^(n*ArcTan[a*x]))/(c*n) - (2*I*E^(n*ArcTan[a*x])*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, E^(2*I*ArcTan[a*x])])/(c*n), (I*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(c*n)) - (2*I*(1 - I*a*x)^((I*n)/2)*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, (1 + I*a*x)/(1 - I*a*x)])/((1 + I*a*x)^((I*n)/2)*(c*n))} -{E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)), x, 5, (I*a*E^(n*ArcTan[a*x])*(I + n))/(c*n) - E^(n*ArcTan[a*x])/(c*x) - (2*I*a*E^(n*ArcTan[a*x])*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, -1 + (2*I)/(I + a*x)])/c, -((a*(1 - I*n)*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(c*n))) - (1 - I*a*x)^((I*n)/2)/((1 + I*a*x)^((I*n)/2)*(c*x)) - (2*I*a*(1 - I*a*x)^((I*n)/2)*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, (1 + I*a*x)/(1 - I*a*x)])/((1 + I*a*x)^((I*n)/2)*c)} -{E^(n*ArcTan[a*x])/(x^3*(c + a^2*c*x^2)), x, 6, (I*a^2*E^(n*ArcTan[a*x])*(-2 + I*n + n^2))/(2*c*n) - E^(n*ArcTan[a*x])/(2*c*x^2) - (a*E^(n*ArcTan[a*x])*n)/(2*c*x) - (I*a^2*E^(n*ArcTan[a*x])*(-2 + n^2)*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, E^(2*I*ArcTan[a*x])])/(c*n), -((a^2*(2*I + n - I*n^2)*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(2*c*n))) - (1 - I*a*x)^((I*n)/2)/((1 + I*a*x)^((I*n)/2)*(2*c*x^2)) - (a*n*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(2*c*x)) + (I*a^2*(2 - n^2)*(1 - I*a*x)^((I*n)/2)*Hypergeometric2F1[1, -((I*n)/2), 1 - (I*n)/2, (1 + I*a*x)/(1 - I*a*x)])/((1 + I*a*x)^((I*n)/2)*(c*n))} - - -(* {x^4*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 10, ((1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2*(2*I + n)) - (2*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2*n*(4 + n^2)) + (2*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2*(2 - I*n)*n) - (2*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^5*c^2*n)) + (2^(1 + (I*n)/2)*(1 + I*a*x)^(1 - (I*n)/2)*Hypergeometric2F1[1 - (I*n)/2, -((I*n)/2), 2 - (I*n)/2, (1/2)*(1 + I*a*x)])/(a^5*c^2*(2*I + n)), -(((I - n)*(3*I + n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2*(2*I - n))) + ((3 - I*n)*x*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^4*c^2) + (x^3*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^2*c^2) + ((1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2*(2*I - n)) + (I*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^5*c^2) + ((3*I + n)*(2 + n^2)*(1 - I*a*x)^(-1 + (I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^5*c^2*(4 + n^2))) - ((3 - I*n)*(2 + n^2)*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^5*c^2*n*(4 + n^2))) + (n*(1 - I*a*x)^(1 + (I*n)/2)*Hypergeometric2F1[1 + (I*n)/2, 1 + (I*n)/2, 2 + (I*n)/2, (1/2)*(1 - I*a*x)])/(2^((I*n)/2)*(a^5*c^2*(2 + I*n)))} -{x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 10, -(((1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^4*c^2*(2 - I*n))) + (2*I*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^4*c^2*n*(4 + n^2)) + (2*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(a^4*c^2*n*(2*I + n)) - (3*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(1 - (I*n)/2))/(a^4*c^2*(2 - I*n)) + (3*(1 - I*a*x)^(-1 + (I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^4*c^2*(2 - I*n))) - (3*(1 - I*a*x)^((I*n)/2))/((1 + I*a*x)^((I*n)/2)*(a^4*c^2*n*(2*I + n))) + (2^(2 - (I*n)/2)*(1 - I*a*x)^(-1 + (I*n)/2)*Hypergeometric2F1[-1 + (I*n)/2, -1 + (I*n)/2, (I*n)/2, (1/2)*(1 - I*a*x)])/(a^4*c^2*(2 - I*n))} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, (E^(n*ArcTan[a*x])*(2 + n^2))/(a^3*c^2*n*(4 + n^2)) - (E^(n*ArcTan[a*x])*(n + 2*a*x))/(a^3*c^2*(4 + n^2)*(1 + a^2*x^2))} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, E^(n*ArcTan[a*x])/(a^2*c^2*(4 + n^2)) - (E^(n*ArcTan[a*x])*(2 - a*n*x))/(a^2*c^2*(4 + n^2)*(1 + a^2*x^2))} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, (2*E^(n*ArcTan[a*x]))/(a*c^2*n*(4 + n^2)) + (E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^2*(4 + n^2)*(1 + a^2*x^2))} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)^2), x, 6, If[$VersionNumber>=8, ((1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*(2 - I*n)) + ((n - I*(4 + n^2))*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*n*(4 + n^2)) - ((4 - I*n)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*n*(2*I + n)) - (2*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^2*(2 + I*n)), ((1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*(2 - I*n)) + ((n - I*(4 + n^2))*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*(4*n + n^3)) - ((4 - I*n)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*n*(2*I + n)) - (2*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^2*(2 + I*n))]} -{E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)^2), x, 7, (a*(3*I + n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*(2 - I*n)) - ((1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*x) + (a*(6 - 4*I*n + n^2 - I*n^3)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*n*(4 + n^2)) - (a*(6 - 4*I*n - n^2)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^2*(2 - I*n)*n) - (2*a*n*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^2*(2 + I*n))} *) - - -(* {x^4*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 4, (24*E^(n*ArcTan[a*x]))/(a^5*c^3*n*(64 + 20*n^2 + n^4)) - (4*E^(n*ArcTan[a*x])*x^3)/(a^2*c^3*(16 + n^2)*(1 + a^2*x^2)^2) + (E^(n*ArcTan[a*x])*n*x^4)/(a*c^3*(16 + n^2)*(1 + a^2*x^2)^2) - (24*E^(n*ArcTan[a*x])*x)/(a^4*c^3*(64 + 20*n^2 + n^4)*(1 + a^2*x^2)) + (12*E^(n*ArcTan[a*x])*n*x^2)/(a^3*c^3*(64 + 20*n^2 + n^4)*(1 + a^2*x^2))} -{x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 5, -((6*E^(n*ArcTan[a*x]))/(a^4*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2)^2)) + (6*E^(n*ArcTan[a*x])*n*x)/(a^3*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2)^2) + (E^(n*ArcTan[a*x])*n*(10 + n^2)*x^3)/(a*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2)^2) + (E^(n*ArcTan[a*x])*(10 + n^2)*x^4)/(c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2)^2) - (3*E^(n*ArcTan[a*x])*x^2)/(c^3*(16 + n^2)*(a + a^3*x^2)^2), -((E^(n*ArcTan[a*x])*Cos[2*ArcTan[a*x]])/(2*a^4*c^3*(4 + n^2))) + (E^(n*ArcTan[a*x])*Cos[4*ArcTan[a*x]])/(2*a^4*c^3*(16 + n^2)) + (E^(n*ArcTan[a*x])*n*Sin[2*ArcTan[a*x]])/(4*a^4*c^3*(4 + n^2)) - (E^(n*ArcTan[a*x])*n*Sin[4*ArcTan[a*x]])/(8*a^4*c^3*(16 + n^2))} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, (2*E^(n*ArcTan[a*x]))/(a^3*c^3*n*(16 + n^2)) - (E^(n*ArcTan[a*x])*(n + 4*a*x))/(a^3*c^3*(16 + n^2)*(1 + a^2*x^2)^2) + (E^(n*ArcTan[a*x])*(n + 2*a*x))/(a^3*c^3*(16 + n^2)*(1 + a^2*x^2))} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, (6*E^(n*ArcTan[a*x]))/(a^2*c^3*(4 + n^2)*(16 + n^2)) - (E^(n*ArcTan[a*x])*(4 - a*n*x))/(a^2*c^3*(16 + n^2)*(1 + a^2*x^2)^2) + (3*E^(n*ArcTan[a*x])*n*(n + 2*a*x))/(a^2*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2))} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 3, If[$VersionNumber>=8, (24*E^(n*ArcTan[a*x]))/(a*c^3*n*(4 + n^2)*(16 + n^2)) + (E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^3*(16 + n^2)*(1 + a^2*x^2)^2) + (12*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2)), (24*E^(n*ArcTan[a*x]))/(a*c^3*n*(64 + 20*n^2 + n^4)) + (E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^3*(16 + n^2)*(1 + a^2*x^2)^2) + (12*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^3*(4 + n^2)*(16 + n^2)*(1 + a^2*x^2))]} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)^3), x, 8, If[$VersionNumber>=8, ((1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 - I*n)) + ((8*I + n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(2 - I*n)*(4*I + n)) - ((64 - 10*I*n + 4*n^2 - I*n^3)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4*I - n)*n*(2*I + n)*(4*I + n)) - ((32 - 9*I*n - n^2)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(2 - I*n)*n*(4*I + n)) + ((22*n - I*(64 + 20*n^2 + I*n^3 + n^4))*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^3*n*(64 + 20*n^2 + n^4)) - (2*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^3*(2 + I*n)), ((1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 - I*n)) + ((8 - I*n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(8 - 6*I*n - n^2)) + ((64 - 10*I*n + 4*n^2 - I*n^3)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*n*(32*I + 16*n + 2*I*n^2 + n^3)) - ((32 - 9*I*n - n^2)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*n*(6*n + I*(8 - n^2))) + ((22*n - I*(64 + 20*n^2 + I*n^3 + n^4))*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^3*(64*n + 20*n^3 + n^5)) - (2*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^3*(2 + I*n))]} -{E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)^3), x, 9, If[$VersionNumber>=8, (a*(5*I + n)*(1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 - I*n)) - ((1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*x) + (a*((-2 + 4*I) + n)*((2 + 4*I) + n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(2 - I*n)*(4*I + n)) - (a*(120 - 64*I*n - 10*n^2 - 4*I*n^3 - n^4)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 + I*n)*n*(2*I + n)*(4*I + n)) - (a*(5*I + n)*(12 - 4*I*n - n^2)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 - I*n)*n*(2*I + n)) + (a*(120 - 64*I*n + 22*n^2 - 20*I*n^3 + n^4 - I*n^5)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^3*n*(64 + 20*n^2 + n^4)) - (2*a*n*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^3*(2 + I*n)), (a*(5*I + n)*(1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(4 - I*n)) - ((1 - I*a*x)^(-2 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*x) + (a*((-2 + 4*I) + n)*((2 + 4*I) + n)*(1 - I*a*x)^(-1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*(6*n + I*(8 - n^2))) + (a*(120*I + 64*n - 10*I*n^2 + 4*n^3 - I*n^4)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*n*(32*I + 16*n + 2*I*n^2 + n^3)) - (a*(5 - I*n)*(12 - 4*I*n - n^2)*(1 - I*a*x)^((I*n)/2)*(1 + I*a*x)^(-2 - (I*n)/2))/(c^3*n*(8 - 6*I*n - n^2)) + (a*(120 - 64*I*n + 22*n^2 - 20*I*n^3 + n^4 - I*n^5)*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2))/(c^3*n*(64 + 20*n^2 + n^4)) - (2*a*n*(1 - I*a*x)^(1 + (I*n)/2)*(1 + I*a*x)^(-1 - (I*n)/2)*Hypergeometric2F1[1, 1 + (I*n)/2, 2 + (I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/(c^3*(2 + I*n))]} *) - - -{E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4, x, 4, If[$VersionNumber>=8, (720*E^(n*ArcTan[a*x]))/(a*c^4*n*(4 + n^2)*(16 + n^2)*(36 + n^2)) + (E^(n*ArcTan[a*x])*(n + 6*a*x))/(a*c^4*(36 + n^2)*(1 + a^2*x^2)^3) + (30*E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^4*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2)^2) + (360*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^4*(4 + n^2)*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2)), (720*E^(n*ArcTan[a*x]))/(a*c^4*n*(36 + n^2)*(64 + 20*n^2 + n^4)) + (E^(n*ArcTan[a*x])*(n + 6*a*x))/(a*c^4*(36 + n^2)*(1 + a^2*x^2)^3) + (30*E^(n*ArcTan[a*x])*(n + 4*a*x))/(a*c^4*(16 + n^2)*(36 + n^2)*(1 + a^2*x^2)^2) + (360*E^(n*ArcTan[a*x])*(n + 2*a*x))/(a*c^4*(36 + n^2)*(64 + 20*n^2 + n^4)*(1 + a^2*x^2))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x]) (c+a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2), x, 3, -((2^(5/2 - (I*n)/2)*c*(1 - I*a*x)^((1/2)*(5 + I*n))*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(1/2)*(-3 + I*n), (1/2)*(5 + I*n), (1/2)*(7 + I*n), (1/2)*(1 - I*a*x)])/(a*(5*I - n)*Sqrt[1 + a^2*x^2]))} -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^(1/2), x, 3, -((2^(3/2 - (I*n)/2)*(1 - I*a*x)^((1/2)*(3 + I*n))*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1/2)*(1 - I*a*x)])/(a*(3*I - n)*Sqrt[1 + a^2*x^2]))} -{E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 3, -((2^(1/2 - (I*n)/2)*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a*(I - n)*Sqrt[c + a^2*c*x^2]))} - - -{x^2*(E^(n*ArcTan[a*x])*(c + a*a*c*x^2)^(3/2)), x, 5, -((c*n*(1 - I*a*x)^((1/2)*(5 + I*n))*(1 + I*a*x)^((1/2)*(5 - I*n))*Sqrt[c + a^2*c*x^2])/(30*a^3*Sqrt[1 + a^2*x^2])) + (c*x*(1 - I*a*x)^((1/2)*(5 + I*n))*(1 + I*a*x)^((1/2)*(5 - I*n))*Sqrt[c + a^2*c*x^2])/(6*a^2*Sqrt[1 + a^2*x^2]) + (2^(3/2 - (I*n)/2)*c*(5 - n^2)*(1 - I*a*x)^((1/2)*(5 + I*n))*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(1/2)*(-3 + I*n), (1/2)*(5 + I*n), (1/2)*(7 + I*n), (1/2)*(1 - I*a*x)])/(15*a^3*(5*I - n)*Sqrt[1 + a^2*x^2])} -{x^2*(E^(n*ArcTan[a*x])*(c + a*a*c*x^2)^(1/2)), x, 5, -((n*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(3 - I*n))*Sqrt[c + a^2*c*x^2])/(12*a^3*Sqrt[1 + a^2*x^2])) + (x*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(3 - I*n))*Sqrt[c + a^2*c*x^2])/(4*a^2*Sqrt[1 + a^2*x^2]) + (2^(-(1/2) - (I*n)/2)*(3 - n^2)*(1 - I*a*x)^((1/2)*(3 + I*n))*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1/2)*(1 - I*a*x)])/(3*a^3*(3*I - n)*Sqrt[1 + a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 5, (x^2*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(3*a^2*Sqrt[c + a^2*c*x^2]) - ((1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*(4 - I*n - n^2 + a*(1 + I*n)*n*x)*Sqrt[1 + a^2*x^2])/(6*a^4*(1 + I*n)*Sqrt[c + a^2*c*x^2]) + (2^(-(1/2) - (I*n)/2)*n*(5 - n^2)*(1 - I*a*x)^((1/2)*(3 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1/2)*(1 - I*a*x)])/(3*a^4*(4*n - I*(3 - n^2))*Sqrt[c + a^2*c*x^2])} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 5, If[$VersionNumber>=8, -(((1 + I*n)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*a^3*(I + n)*Sqrt[c + a^2*c*x^2])) + (x*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*a^2*Sqrt[c + a^2*c*x^2]) - (I*2^(1/2 - (I*n)/2)*(1 - n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a^3*(1 + n^2)*Sqrt[c + a^2*c*x^2]), -(((1 + I*n)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*a^3*(I + n)*Sqrt[c + a^2*c*x^2])) + (x*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*a^2*Sqrt[c + a^2*c*x^2]) - (I*2^(1/2 - (I*n)/2)*(1 - n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a^3*(1 + n^2)*Sqrt[c + a^2*c*x^2])]} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 4, If[$VersionNumber>=8, ((1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^2*(1 - I*n)*Sqrt[c + a^2*c*x^2]) - (I*2^(3/2 - (I*n)/2)*n*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a^2*(1 + n^2)*Sqrt[c + a^2*c*x^2]), ((1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^2*(1 - I*n)*Sqrt[c + a^2*c*x^2]) - (I*2^(3/2 - (I*n)/2)*n*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a^2*(1 + n^2)*Sqrt[c + a^2*c*x^2])]} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 3, -((2^(1/2 - (I*n)/2)*(1 - I*a*x)^((1/2)*(1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(1 + I*n), (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1/2)*(1 - I*a*x)])/(a*(I - n)*Sqrt[c + a^2*c*x^2]))} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)^(1/2)), x, 3, -((2*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1 - I*a*x)/(1 + I*a*x)])/((1 + I*n)*Sqrt[c + a^2*c*x^2]))} -{E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)^(1/2)), x, 4, -(((1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(x*Sqrt[c + a^2*c*x^2])) - (2*a*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1 - I*a*x)/(1 + I*a*x)])/((1 + I*n)*Sqrt[c + a^2*c*x^2])} -{E^(n*ArcTan[a*x])/(x^3*(c + a^2*c*x^2)^(1/2)), x, 6, -(((1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*x^2*Sqrt[c + a^2*c*x^2])) - (a*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(2*x*Sqrt[c + a^2*c*x^2]) + (a^2*(1 - n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(1 + I*n), (1/2)*(3 + I*n), (1 - I*a*x)/(1 + I*a*x)])/((1 + I*n)*Sqrt[c + a^2*c*x^2])} - - -(* {x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 5, (x^2*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^2*c*Sqrt[c + a^2*c*x^2]) + ((1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*(2 - 2*I*n - n^2 - a*(3 - 2*I*n)*n*x)*Sqrt[1 + a^2*x^2])/(a^4*c*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (2^(-(1/2) - (I*n)/2)*n*(1 - I*a*x)^((1/2)*(3 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(3 + I*n), (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1/2)*(1 - I*a*x)])/(a^4*c*(3*I - n)*Sqrt[c + a^2*c*x^2])} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 2, -((E^(n*ArcTan[a*x])*(n + a*x))/(a^3*c*(1 + n^2)*Sqrt[c + a^2*c*x^2])) + (2*E^(n*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[1, (1/2)*(1 - I*n), (1/2)*(3 - I*n), -E^(2*I*ArcTan[a*x])])/(a^3*c^2*(I + n)*(1 - I*a*x))} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 1, -((E^(n*ArcTan[a*x])*(1 - a*n*x))/(a^2*c*(1 + n^2)*Sqrt[c + a^2*c*x^2]))} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 1, (E^(n*ArcTan[a*x])*(n + a*x))/(a*c*(1 + n^2)*Sqrt[c + a^2*c*x^2])} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)^(3/2)), x, 6, ((1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(c*(1 - I*n)*Sqrt[c + a^2*c*x^2]) - ((2 - I*n)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(c*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (2*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1 - I*a*x)/(1 + I*a*x)])/(c*(3 + I*n)*Sqrt[c + a^2*c*x^2])} -{E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)^(3/2)), x, 7, (a*(2*I + n)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(c*(1 - I*n)*Sqrt[c + a^2*c*x^2]) - ((1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(c*x*Sqrt[c + a^2*c*x^2]) - (a*(2*I + 2*n - I*n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(c*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (2*a*n*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1 - I*a*x)/(1 + I*a*x)])/(c*(3 + I*n)*Sqrt[c + a^2*c*x^2])} *) - - -(* {x^5*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 18, If[$VersionNumber>=8, ((4*I + n)*x^3*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^3*c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) + (x^4*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) - (n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) - (3*(2*I - n)*(4*I + n)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(9 + n^2)*Sqrt[c + a^2*c*x^2]) + (3*(4*I + n)*x*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) - (2*I*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (2*n*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(3*I - n + 3*I*n^2 - n^3)*Sqrt[c + a^2*c*x^2]) + (3*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) + (3*(4*I + n)*(1 - 2*I*n + n^2)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(3*I - n)*(I + n)*(3*I + n)*Sqrt[c + a^2*c*x^2]) + (3*I*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (3*(4*I + n)*(I + 2*n + I*n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (3*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) + (2^(3/2 - (I*n)/2)*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(1 - I*a*x)])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]), ((4*I + n)*x^3*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^3*c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) + (x^4*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) - (n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) - (3*(2*I - n)*(4*I + n)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(9 + n^2)*Sqrt[c + a^2*c*x^2]) + (3*(4*I + n)*x*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) - (2*I*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (2*n*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(3*I - n + 3*I*n^2 - n^3)*Sqrt[c + a^2*c*x^2]) + (3*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) - (3*(4*I + n)*(1 - 2*I*n + n^2)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(9*I + 9*n + I*n^2 + n^3)*Sqrt[c + a^2*c*x^2]) + (3*I*n*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (3*(4*I + n)*(I + 2*n + I*n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (3*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2]) + (2^(3/2 - (I*n)/2)*n*(1 - I*a*x)^((1/2)*(-1 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 + I*n), (1/2)*(-1 + I*n), (1/2)*(1 + I*n), (1/2)*(1 - I*a*x)])/(a^6*c^2*(I + n)*Sqrt[c + a^2*c*x^2])]} -{x^4*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 15, ((1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*Sqrt[c + a^2*c*x^2]) + (3*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(4*n + I*(3 - n^2))*Sqrt[c + a^2*c*x^2]) - (6*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*(1 + n^2)*Sqrt[c + a^2*c*x^2]) - (6*I*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (4*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*Sqrt[c + a^2*c*x^2]) - (8*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(4*n + I*(3 - n^2))*Sqrt[c + a^2*c*x^2]) + (8*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-1 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*(1 + n^2)*Sqrt[c + a^2*c*x^2]) + (6*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*Sqrt[c + a^2*c*x^2]) + (6*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(1 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(4*n + I*(3 - n^2))*Sqrt[c + a^2*c*x^2]) - (4*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(3 - I*n))*Sqrt[1 + a^2*x^2])/(a^5*c^2*(3*I + n)*Sqrt[c + a^2*c*x^2]) + (2^(5/2 - (I*n)/2)*(1 - I*a*x)^((1/2)*(-3 + I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[(1/2)*(-3 + I*n), (1/2)*(-3 + I*n), (1/2)*(-1 + I*n), (1/2)*(1 - I*a*x)])/(a^5*c^2*(3*I + n)*Sqrt[c + a^2*c*x^2])} -{x^3*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 4, -((6*E^(n*ArcTan[a*x]))/(a^4*c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2])) + (6*E^(n*ArcTan[a*x])*n*x)/(a^3*c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (3*E^(n*ArcTan[a*x])*x^2)/(a^2*c^2*(9 + n^2)*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + (E^(n*ArcTan[a*x])*n*x^3)/(a*c^2*(9 + n^2)*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2])} -{x^2*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 2, -((E^(n*ArcTan[a*x])*(n + 3*a*x))/(a^3*c*(9 + n^2)*(c + a^2*c*x^2)^(3/2))) + (E^(n*ArcTan[a*x])*(3 + n^2)*(n + a*x))/(a^3*c^2*(1 + n^2)*(9 + n^2)*Sqrt[c + a^2*c*x^2])} -{x^1*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 2, -((E^(n*ArcTan[a*x])*(3 - a*n*x))/(a^2*c*(9 + n^2)*(c + a^2*c*x^2)^(3/2))) + (2*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^2*(1 + n^2)*(9 + n^2)*Sqrt[c + a^2*c*x^2])} -{x^0*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 2, (E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c*(9 + n^2)*(c + a^2*c*x^2)^(3/2)) + (6*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^2*(1 + n^2)*(9 + n^2)*Sqrt[c + a^2*c*x^2])} -{E^(n*ArcTan[a*x])/(x^1*(c + a^2*c*x^2)^(5/2)), x, 8, If[$VersionNumber>=8, ((1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) + ((6*I + n)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(1 - I*n)*(3*I + n)*Sqrt[c + a^2*c*x^2]) - ((15*I + 6*n - I*n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3*I + n)*(1 + n^2)*Sqrt[c + a^2*c*x^2]) + ((18 - 7*I*n + 2*n^2 - I*n^3)*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (2*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1 - I*a*x)/(1 + I*a*x)])/(c^2*(3 + I*n)*Sqrt[c + a^2*c*x^2]), ((1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) + ((6 - I*n)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3 - 4*I*n - n^2)*Sqrt[c + a^2*c*x^2]) - ((15*I + 6*n - I*n^2)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3*I + n)*(1 + n^2)*Sqrt[c + a^2*c*x^2]) + ((18 - 7*I*n + 2*n^2 - I*n^3)*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (2*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1 - I*a*x)/(1 + I*a*x)])/(c^2*(3 + I*n)*Sqrt[c + a^2*c*x^2])]} *) -(* {E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)^(5/2)), x, 9, (a*(4*I + n)*(1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3 - I*n)*Sqrt[c + a^2*c*x^2]) - ((1 - I*a*x)^((1/2)*(-3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*x*Sqrt[c + a^2*c*x^2]) - (a*(12 - 6*I*n - n^2)*(1 - I*a*x)^((1/2)*(-1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(1 - I*n)*(3*I + n)*Sqrt[c + a^2*c*x^2]) + (a*(24 - 15*I*n - 6*n^2 + I*n^3)*(1 - I*a*x)^((1/2)*(1 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(3*I + n)*(1 + n^2)*Sqrt[c + a^2*c*x^2]) + (a*(24*I + 18*n - 7*I*n^2 + 2*n^3 - I*n^4)*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2])/(c^2*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2]) - (2*a*n*(1 - I*a*x)^((1/2)*(3 + I*n))*(1 + I*a*x)^((1/2)*(-3 - I*n))*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1/2)*(3 + I*n), (1/2)*(5 + I*n), (1 - I*a*x)/(1 + I*a*x)])/(c^2*(3 + I*n)*Sqrt[c + a^2*c*x^2])} *) - - -(* {x^2*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(7/2)), x, 3, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(n + 5*a*x))/(a^3*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2))) + (E^(n*ArcTan[a*x])*(5 + n^2)*(n + 3*a*x))/(a^3*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (6*E^(n*ArcTan[a*x])*(5 + n^2)*(n + a*x))/(a^3*c^3*(1 + n^2)*(9 + n^2)*(25 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(n + 5*a*x))/(a^3*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2))) + (E^(n*ArcTan[a*x])*(5 + n^2)*(n + 3*a*x))/(a^3*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (6*E^(n*ArcTan[a*x])*(5 + n^2)*(n + a*x))/(a^3*c^3*(25 + n^2)*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} -{x^1*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(7/2)), x, 3, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(5 - a*n*x))/(a^2*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2))) + (4*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (24*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^3*(1 + n^2)*(9 + n^2)*(25 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(5 - a*n*x))/(a^2*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2))) + (4*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (24*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^3*(25 + n^2)*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} -{x^0*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(7/2)), x, 3, If[$VersionNumber>=8, (E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2)) + (20*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (120*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^3*(1 + n^2)*(9 + n^2)*(25 + n^2)*Sqrt[c + a^2*c*x^2]), (E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c*(25 + n^2)*(c + a^2*c*x^2)^(5/2)) + (20*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^2*(9 + n^2)*(25 + n^2)*(c + a^2*c*x^2)^(3/2)) + (120*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^3*(25 + n^2)*(9 + 10*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} *) - - -(* {x^2*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(9/2)), x, 4, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(n + 7*a*x))/(a^3*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2))) + (E^(n*ArcTan[a*x])*(7 + n^2)*(n + 5*a*x))/(a^3*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (20*E^(n*ArcTan[a*x])*(7 + n^2)*(n + 3*a*x))/(a^3*c^3*(9 + n^2)*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(3/2)) + (120*E^(n*ArcTan[a*x])*(7 + n^2)*(n + a*x))/(a^3*c^4*(1 + n^2)*(9 + n^2)*(25 + n^2)*(49 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(n + 7*a*x))/(a^3*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2))) + (E^(n*ArcTan[a*x])*(7 + n^2)*(n + 5*a*x))/(a^3*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (20*E^(n*ArcTan[a*x])*(7 + n^2)*(n + 3*a*x))/(a^3*c^3*(49 + n^2)*(225 + 34*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (120*E^(n*ArcTan[a*x])*(7 + n^2)*(n + a*x))/(a^3*c^4*(9 + 10*n^2 + n^4)*(1225 + 74*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} -{x^1*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(9/2)), x, 4, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(7 - a*n*x))/(a^2*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2))) + (6*E^(n*ArcTan[a*x])*n*(n + 5*a*x))/(a^2*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (120*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^3*(9 + n^2)*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(3/2)) + (720*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^4*(1 + n^2)*(9 + n^2)*(25 + n^2)*(49 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(7 - a*n*x))/(a^2*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2))) + (6*E^(n*ArcTan[a*x])*n*(n + 5*a*x))/(a^2*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (120*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^3*(49 + n^2)*(225 + 34*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (720*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^4*(9 + 10*n^2 + n^4)*(1225 + 74*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} -{x^0*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(9/2)), x, 4, If[$VersionNumber>=8, (E^(n*ArcTan[a*x])*(n + 7*a*x))/(a*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2)) + (42*E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (840*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^3*(9 + n^2)*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(3/2)) + (5040*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^4*(1 + n^2)*(9 + n^2)*(25 + n^2)*(49 + n^2)*Sqrt[c + a^2*c*x^2]), (E^(n*ArcTan[a*x])*(n + 7*a*x))/(a*c*(49 + n^2)*(c + a^2*c*x^2)^(7/2)) + (42*E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c^2*(25 + n^2)*(49 + n^2)*(c + a^2*c*x^2)^(5/2)) + (840*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^3*(49 + n^2)*(225 + 34*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (5040*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^4*(9 + 10*n^2 + n^4)*(1225 + 74*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} *) - - -(* {x^2*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(11/2)), x, 5, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(n + 9*a*x))/(a^3*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2))) + (E^(n*ArcTan[a*x])*(9 + n^2)*(n + 7*a*x))/(a^3*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (42*E^(n*ArcTan[a*x])*(9 + n^2)*(n + 5*a*x))/(a^3*c^3*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(5/2)) + (840*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a^3*c^4*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(3/2)) + (5040*E^(n*ArcTan[a*x])*(n + a*x))/(a^3*c^5*(1 + n^2)*(25 + n^2)*(49 + n^2)*(81 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(n + 9*a*x))/(a^3*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2))) + (E^(n*ArcTan[a*x])*(9 + n^2)*(n + 7*a*x))/(a^3*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (42*E^(n*ArcTan[a*x])*(9 + n^2)*(n + 5*a*x))/(a^3*c^3*(81 + n^2)*(1225 + 74*n^2 + n^4)*(c + a^2*c*x^2)^(5/2)) + (840*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a^3*c^4*(25 + n^2)*(3969 + 130*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (5040*E^(n*ArcTan[a*x])*(n + a*x))/(a^3*c^5*(1225 + 74*n^2 + n^4)*(81 + 82*n^2 + n^4)*Sqrt[c + a^2*c*x^2])]} -{x^1*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(11/2)), x, 5, If[$VersionNumber>=8, -((E^(n*ArcTan[a*x])*(9 - a*n*x))/(a^2*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2))) + (8*E^(n*ArcTan[a*x])*n*(n + 7*a*x))/(a^2*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (336*E^(n*ArcTan[a*x])*n*(n + 5*a*x))/(a^2*c^3*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(5/2)) + (6720*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^4*(9 + n^2)*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(3/2)) + (40320*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^5*(1 + n^2)*(9 + n^2)*(25 + n^2)*(49 + n^2)*(81 + n^2)*Sqrt[c + a^2*c*x^2]), -((E^(n*ArcTan[a*x])*(9 - a*n*x))/(a^2*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2))) + (8*E^(n*ArcTan[a*x])*n*(n + 7*a*x))/(a^2*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (336*E^(n*ArcTan[a*x])*n*(n + 5*a*x))/(a^2*c^3*(81 + n^2)*(1225 + 74*n^2 + n^4)*(c + a^2*c*x^2)^(5/2)) + (6720*E^(n*ArcTan[a*x])*n*(n + 3*a*x))/(a^2*c^4*(225 + 34*n^2 + n^4)*(3969 + 130*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (40320*E^(n*ArcTan[a*x])*n*(n + a*x))/(a^2*c^5*(1225 + 74*n^2 + n^4)*(729 + 819*n^2 + 91*n^4 + n^6)*Sqrt[c + a^2*c*x^2])]} -{x^0*(E^(n*ArcTan[a*x])/(c + a*a*c*x^2)^(11/2)), x, 5, If[$VersionNumber>=8, (E^(n*ArcTan[a*x])*(n + 9*a*x))/(a*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2)) + (72*E^(n*ArcTan[a*x])*(n + 7*a*x))/(a*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (3024*E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c^3*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(5/2)) + (60480*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^4*(9 + n^2)*(25 + n^2)*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(3/2)) + (362880*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^5*(1 + n^2)*(9 + n^2)*(25 + n^2)*(49 + n^2)*(81 + n^2)*Sqrt[c + a^2*c*x^2]), (E^(n*ArcTan[a*x])*(n + 9*a*x))/(a*c*(81 + n^2)*(c + a^2*c*x^2)^(9/2)) + (72*E^(n*ArcTan[a*x])*(n + 7*a*x))/(a*c^2*(49 + n^2)*(81 + n^2)*(c + a^2*c*x^2)^(7/2)) + (3024*E^(n*ArcTan[a*x])*(n + 5*a*x))/(a*c^3*(81 + n^2)*(1225 + 74*n^2 + n^4)*(c + a^2*c*x^2)^(5/2)) + (60480*E^(n*ArcTan[a*x])*(n + 3*a*x))/(a*c^4*(225 + 34*n^2 + n^4)*(3969 + 130*n^2 + n^4)*(c + a^2*c*x^2)^(3/2)) + (362880*E^(n*ArcTan[a*x])*(n + a*x))/(a*c^5*(1225 + 74*n^2 + n^4)*(729 + 819*n^2 + 91*n^4 + n^6)*Sqrt[c + a^2*c*x^2])]} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x]) (c+a^2 c x^2)^(p/3)*) - - -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^(1/3), x, 3, -((3*2^(4/3 - (I*n)/2)*(1 - I*a*x)^((1/6)*(8 + 3*I*n))*(c + a^2*c*x^2)^(1/3)*Hypergeometric2F1[(1/6)*(-2 + 3*I*n), (1/6)*(8 + 3*I*n), (1/6)*(14 + 3*I*n), (1/2)*(1 - I*a*x)])/(a*(8*I - 3*n)*(1 + a^2*x^2)^(1/3)))} -{E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/3), x, 3, -((3*2^(2/3 - (I*n)/2)*(1 - I*a*x)^((1/6)*(4 + 3*I*n))*(1 + a^2*x^2)^(1/3)*Hypergeometric2F1[(1/6)*(2 + 3*I*n), (1/6)*(4 + 3*I*n), (1/6)*(10 + 3*I*n), (1/2)*(1 - I*a*x)])/(a*(4*I - 3*n)*(c + a^2*c*x^2)^(1/3)))} -{E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(2/3), x, 3, -((3*2^(1/3 - (I*n)/2)*(1 - I*a*x)^((1/6)*(2 + 3*I*n))*(1 + a^2*x^2)^(2/3)*Hypergeometric2F1[(1/6)*(2 + 3*I*n), (1/6)*(4 + 3*I*n), (1/6)*(8 + 3*I*n), (1/2)*(1 - I*a*x)])/(a*(2*I - 3*n)*(c + a^2*c*x^2)^(2/3)))} -{E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(4/3), x, 3, (3*2^(-(1/3) - (I*n)/2)*(1 - I*a*x)^((1/6)*(-2 + 3*I*n))*(1 + a^2*x^2)^(1/3)*Hypergeometric2F1[(1/6)*(-2 + 3*I*n), (1/6)*(8 + 3*I*n), (1/6)*(4 + 3*I*n), (1/2)*(1 - I*a*x)])/(a*c*(2*I + 3*n)*(c + a^2*c*x^2)^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x]) (c+a^2 c x^2)^p with m symbolic*) - - -{x^m*E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^1, x, 2, (c*x^(1 + m)*AppellF1[1 + m, -1 - (I*n)/2, -1 + (I*n)/2, 2 + m, I*a*x, (-I)*a*x])/(1 + m)} -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^1, x, 2, (x^(1 + m)*AppellF1[1 + m, 1 - (I*n)/2, 1 + (I*n)/2, 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m))} -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^2, x, 2, (x^(1 + m)*AppellF1[1 + m, 2 - (I*n)/2, 2 + (I*n)/2, 2 + m, I*a*x, (-I)*a*x])/(c^2*(1 + m))} -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 2, (x^(1 + m)*AppellF1[1 + m, 3 - (I*n)/2, 3 + (I*n)/2, 2 + m, I*a*x, (-I)*a*x])/(c^3*(1 + m))} - - -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/2), x, 3, (x^(1 + m)*Sqrt[1 + a^2*x^2]*AppellF1[1 + m, (1/2)*(1 - I*n), (1/2)*(1 + I*n), 2 + m, I*a*x, (-I)*a*x])/((1 + m)*Sqrt[c + a^2*c*x^2])} -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 3, (x^(1 + m)*Sqrt[1 + a^2*x^2]*AppellF1[1 + m, (1/2)*(3 - I*n), (1/2)*(3 + I*n), 2 + m, I*a*x, (-I)*a*x])/(c*(1 + m)*Sqrt[c + a^2*c*x^2])} -{x^m*E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2), x, 3, (x^(1 + m)*Sqrt[1 + a^2*x^2]*AppellF1[1 + m, (1/2)*(5 - I*n), (1/2)*(5 + I*n), 2 + m, I*a*x, (-I)*a*x])/(c^2*(1 + m)*Sqrt[c + a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTan[a x]) (c+a^2 c x^2)^p with p symbolic*) - - -{E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^p, x, 3, (2^(1 - (I*n)/2 + p)*(1 - I*a*x)^(1 + (I*n)/2 + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[(I*n)/2 - p, 1 + (I*n)/2 + p, 2 + (I*n)/2 + p, (1/2)*(1 - I*a*x)])/((1 + a^2*x^2)^p*(a*(n - 2*I*(1 + p))))} - - -{(c + a^2*c*x^2)^p/E^(2*I*p*ArcTan[a*x]), x, 3, (I*(1 - I*a*x)^(1 + 2*p)*(c + a^2*c*x^2)^p)/((1 + a^2*x^2)^p*(a*(1 + 2*p)))} -{(c + a^2*c*x^2)^p/E^(-2*I*p*ArcTan[a*x]), x, 3, -((I*(1 + I*a*x)^(1 + 2*p)*(c + a^2*c*x^2)^p)/((1 + a^2*x^2)^p*(a*(1 + 2*p))))} - - -{x^2*E^(n*I*ArcTan[a*x])/(c + a^2*c*x^2)^(n^2/2 + 1), x, 1, (I*E^(I*n*ArcTan[a*x])*(1 - I*a*n*x))/((c + a^2*c*x^2)^(n^2/2)*(a^3*c*n*(1 - n^2)))} - -{x^2*E^(6*I*ArcTan[a*x])/(c + a^2*c*x^2)^19, x, 2, -((I + 6*a*x)/(210*a^3*c^19*(1 - I*a*x)^21*(1 + I*a*x)^15))} -{x^2*E^(4*I*ArcTan[a*x])/(c + a^2*c*x^2)^9, x, 2, -((I + 4*a*x)/(60*a^3*c^9*(1 - I*a*x)^10*(1 + I*a*x)^6))} -{x^2*E^(2*I*ArcTan[a*x])/(c + a^2*c*x^2)^3, x, 2, -((I + 2*a*x)/(6*a^3*c^3*(1 - I*a*x)^3*(1 + I*a*x)))} -{x^2/(E^(2*I*ArcTan[a*x])*(c + a^2*c*x^2)^3), x, 2, (I - 2*a*x)/(6*a^3*c^3*(1 - I*a*x)*(1 + I*a*x)^3)} -{x^2/(E^(4*I*ArcTan[a*x])*(c + a^2*c*x^2)^9), x, 2, (I - 4*a*x)/(60*a^3*c^9*(1 - I*a*x)^6*(1 + I*a*x)^10)} - -{x^2*E^(5*I*ArcTan[a*x])/(c + a^2*c*x^2)^(27/2), x, 3, -(((I + 5*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(1 - I*a*x)^15*(1 + I*a*x)^10*Sqrt[c + a^2*c*x^2]))} -{x^2*E^(3*I*ArcTan[a*x])/(c + a^2*c*x^2)^(11/2), x, 3, -(((I + 3*a*x)*Sqrt[1 + a^2*x^2])/(24*a^3*c^5*(1 - I*a*x)^6*(1 + I*a*x)^3*Sqrt[c + a^2*c*x^2]))} -{x^2*E^(1*I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x, 4, -(Sqrt[1 + a^2*x^2]/(2*a^3*c*(I + a*x)*Sqrt[c + a^2*c*x^2])) + (I*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(4*a^3*c*Sqrt[c + a^2*c*x^2]) + (3*I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(4*a^3*c*Sqrt[c + a^2*c*x^2])} -{x^2/(E^(1*I*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)), x, 4, Sqrt[1 + a^2*x^2]/(2*a^3*c*(I - a*x)*Sqrt[c + a^2*c*x^2]) - (3*I*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(4*a^3*c*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(4*a^3*c*Sqrt[c + a^2*c*x^2])} -{x^2/(E^(3*I*ArcTan[a*x])*(c + a^2*c*x^2)^(11/2)), x, 3, ((I - 3*a*x)*Sqrt[1 + a^2*x^2])/(24*a^3*c^5*(1 - I*a*x)^3*(1 + I*a*x)^6*Sqrt[c + a^2*c*x^2])} -{x^2/(E^(5*I*ArcTan[a*x])*(c + a^2*c*x^2)^(27/2)), x, 3, ((I - 5*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(1 - I*a*x)^10*(1 + I*a*x)^15*Sqrt[c + a^2*c*x^2])} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Inverse tangent functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Inverse tangent functions.m deleted file mode 100644 index 6b9585c..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.3 Inverse tangent/5.3.7 Inverse tangent functions.m +++ /dev/null @@ -1,335 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands involving inverse tangents of algebraic functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcTan[a+b x^n]*) - - -{x^3*ArcTan[a + b*x^4], x, 4, ((a + b*x^4)*ArcTan[a + b*x^4])/(4*b) - Log[1 + (a + b*x^4)^2]/(8*b)} - - -{x^(n-1)*ArcTan[a + b*x^n], x, 4, ((a + b*x^n)*ArcTan[a + b*x^n])/(b*n) - Log[1 + (a + b*x^n)^2]/(2*b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (a+b ArcTan[c x/Sqrt[d+e x^2]])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTan[c x/Sqrt[d+e x^2]]) when e=c^2*) - - -{x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 6, (5*d^2*x*Sqrt[d + e*x^2])/(96*(-e)^(5/2)) + (5*d*x^3*Sqrt[d + e*x^2])/(144*(-e)^(3/2)) + (x^5*Sqrt[d + e*x^2])/(36*Sqrt[-e]) + (x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/6 + (5*d^3*Sqrt[-e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(96*e^(7/2))} -{x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 5, (3*d*x*Sqrt[d + e*x^2])/(32*(-e)^(3/2)) + (x^3*Sqrt[d + e*x^2])/(16*Sqrt[-e]) + (x^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/4 - (3*d^2*Sqrt[-e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(32*e^(5/2))} -{x^1*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 4, (x*Sqrt[d + e*x^2])/(4*Sqrt[-e]) + (x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/2 + (d*Sqrt[-e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(4*e^(3/2))} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^1, x, 8, -((Sqrt[d]*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[e]*Sqrt[d + e*x^2])) + (Sqrt[d]*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(Sqrt[e]*Sqrt[d + e*x^2]) - (Sqrt[d]*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[x])/(Sqrt[e]*Sqrt[d + e*x^2]) + ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]*Log[x] + (Sqrt[d]*Sqrt[-e]*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[e]*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^3, x, 2, -(Sqrt[-e]*Sqrt[d + e*x^2])/(2*d*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(2*x^2)} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5, x, 3, -(Sqrt[-e]*Sqrt[d + e*x^2])/(12*d*x^3) - ((-e)^(3/2)*Sqrt[d + e*x^2])/(6*d^2*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(4*x^4)} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^7, x, 4, -(Sqrt[-e]*Sqrt[d + e*x^2])/(30*d*x^5) - (2*(-e)^(3/2)*Sqrt[d + e*x^2])/(45*d^2*x^3) - (4*(-e)^(5/2)*Sqrt[d + e*x^2])/(45*d^3*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(6*x^6)} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^9, x, 5, -(Sqrt[-e]*Sqrt[d + e*x^2])/(56*d*x^7) - (3*(-e)^(3/2)*Sqrt[d + e*x^2])/(140*d^2*x^5) - ((-e)^(5/2)*Sqrt[d + e*x^2])/(35*d^3*x^3) - (2*(-e)^(7/2)*Sqrt[d + e*x^2])/(35*d^4*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(8*x^8)} - -{x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 4, (d^3*Sqrt[d + e*x^2])/(7*(-e)^(7/2)) - (d^2*(d + e*x^2)^(3/2))/(7*(-e)^(7/2)) + (3*d*(d + e*x^2)^(5/2))/(35*(-e)^(7/2)) - (d + e*x^2)^(7/2)/(49*(-e)^(7/2)) + (x^7*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7} -{x^4*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 4, (d^2*Sqrt[d + e*x^2])/(5*(-e)^(5/2)) - (2*d*(d + e*x^2)^(3/2))/(15*(-e)^(5/2)) + (d + e*x^2)^(5/2)/(25*(-e)^(5/2)) + (x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/5} -{x^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 4, (d*Sqrt[d + e*x^2])/(3*(-e)^(3/2)) - (d + e*x^2)^(3/2)/(9*(-e)^(3/2)) + (x^3*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/3} -{x^0*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 2, Sqrt[d + e*x^2]/Sqrt[-e] + x*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^2, x, 4, -(ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x) - (Sqrt[-e]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d]} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^4, x, 5, -(Sqrt[-e]*Sqrt[d + e*x^2])/(6*d*x^2) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(3*x^3) - ((-e)^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(3/2))} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^6, x, 6, -(Sqrt[-e]*Sqrt[d + e*x^2])/(20*d*x^4) - (3*(-e)^(3/2)*Sqrt[d + e*x^2])/(40*d^2*x^2) - ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/(5*x^5) - (3*(-e)^(5/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*d^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(m/2) (a+b ArcTan[c x/Sqrt[d+e x^2]]) when e=c^2*) - - -{x^(9/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 6, (60*d^2*Sqrt[x]*Sqrt[d + e*x^2])/(847*(-e)^(5/2)) + (36*d*x^(5/2)*Sqrt[d + e*x^2])/(847*(-e)^(3/2)) + (4*x^(9/2)*Sqrt[d + e*x^2])/(121*Sqrt[-e]) + (2*x^(11/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/11 + (30*d^(11/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(847*e^(13/4)*Sqrt[d + e*x^2])} -{x^(5/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 5, (20*d*Sqrt[x]*Sqrt[d + e*x^2])/(147*(-e)^(3/2)) + (4*x^(5/2)*Sqrt[d + e*x^2])/(49*Sqrt[-e]) + (2*x^(7/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/7 - (10*d^(7/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(147*e^(9/4)*Sqrt[d + e*x^2])} -{x^(1/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 4, (4*Sqrt[x]*Sqrt[d + e*x^2])/(9*Sqrt[-e]) + (2*x^(3/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/3 + (2*d^(3/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(9*e^(5/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(3/2), x, 3, (-2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/Sqrt[x] + (2*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(d^(1/4)*e^(1/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(7/2), x, 4, (-4*Sqrt[-e]*Sqrt[d + e*x^2])/(15*d*x^(3/2)) - (2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(5*x^(5/2)) - (2*Sqrt[-e]*e^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(15*d^(5/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(11/2), x, 5, (-4*Sqrt[-e]*Sqrt[d + e*x^2])/(63*d*x^(7/2)) - (20*(-e)^(3/2)*Sqrt[d + e*x^2])/(189*d^2*x^(3/2)) - (2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(9*x^(9/2)) + (10*Sqrt[-e]*e^(7/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(189*d^(9/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(15/2), x, 6, (-4*Sqrt[-e]*Sqrt[d + e*x^2])/(143*d*x^(11/2)) - (36*(-e)^(3/2)*Sqrt[d + e*x^2])/(1001*d^2*x^(7/2)) - (60*(-e)^(5/2)*Sqrt[d + e*x^2])/(1001*d^3*x^(3/2)) - (2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(13*x^(13/2)) - (30*Sqrt[-e]*e^(11/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(1001*d^(13/4)*Sqrt[d + e*x^2])} - -{x^(7/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 7, (28*d*x^(3/2)*Sqrt[d + e*x^2])/(405*(-e)^(3/2)) + (4*x^(7/2)*Sqrt[d + e*x^2])/(81*Sqrt[-e]) - (28*d^2*Sqrt[-e]*Sqrt[x]*Sqrt[d + e*x^2])/(135*e^(5/2)*(Sqrt[d] + Sqrt[e]*x)) + (2*x^(9/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/9 + (28*d^(9/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(135*e^(11/4)*Sqrt[d + e*x^2]) - (14*d^(9/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(135*e^(11/4)*Sqrt[d + e*x^2])} -{x^(3/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]], x, 6, (4*x^(3/2)*Sqrt[d + e*x^2])/(25*Sqrt[-e]) + (12*d*Sqrt[-e]*Sqrt[x]*Sqrt[d + e*x^2])/(25*e^(3/2)*(Sqrt[d] + Sqrt[e]*x)) + (2*x^(5/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/5 - (12*d^(5/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(25*e^(7/4)*Sqrt[d + e*x^2]) + (6*d^(5/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(25*e^(7/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(1/2), x, 5, (-4*Sqrt[-e]*Sqrt[x]*Sqrt[d + e*x^2])/(Sqrt[e]*(Sqrt[d] + Sqrt[e]*x)) + 2*Sqrt[x]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] + (4*d^(1/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(3/4)*Sqrt[d + e*x^2]) - (2*d^(1/4)*Sqrt[-e]*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(3/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(5/2), x, 6, (-4*Sqrt[-e]*Sqrt[d + e*x^2])/(3*d*Sqrt[x]) + (4*Sqrt[-e^2]*Sqrt[x]*Sqrt[d + e*x^2])/(3*d*(Sqrt[d] + Sqrt[e]*x)) - (2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(3*x^(3/2)) - (4*Sqrt[-e]*e^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2]) + (2*Sqrt[-e]*e^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2])} -{ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^(9/2), x, 7, (-4*Sqrt[-e]*Sqrt[d + e*x^2])/(35*d*x^(5/2)) - (12*(-e)^(3/2)*Sqrt[d + e*x^2])/(35*d^2*Sqrt[x]) - (12*Sqrt[-e]*e^(3/2)*Sqrt[x]*Sqrt[d + e*x^2])/(35*d^2*(Sqrt[d] + Sqrt[e]*x)) - (2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(7*x^(7/2)) + (12*Sqrt[-e]*e^(5/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(35*d^(7/4)*Sqrt[d + e*x^2]) - (6*Sqrt[-e]*e^(5/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(35*d^(7/4)*Sqrt[d + e*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcTan[a+b x+c x^2]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[a+b x+c x^2]*) - - -{ArcTan[1 + x + x^2]/x^2, x, 8, (1/2)*ArcTan[1 + x] - ArcTan[1 + x + x^2]/x + Log[x]/2 - (1/2)*Log[1 + x^2] + (1/4)*Log[2 + 2*x + x^2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b ArcTan[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 9, -((2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcTanh[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c) + (3*I*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) - (3*I*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, -1 + 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) + (3*b^2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) - (3*b^2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -1 + 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) - (3*I*b^3*PolyLog[4, 1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(4*c) + (3*I*b^3*PolyLog[4, -1 + 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(4*c)} -{(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((2*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcTanh[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c) + (I*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c - (I*b*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -1 + 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c + (b^2*PolyLog[3, 1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) - (b^2*PolyLog[3, -1 + 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c)} -{(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 4, -((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/c) - (I*b*PolyLog[2, -((I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/(2*c) + (I*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/(2*c)} -{1/((a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse tangents of trig functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTan[Trig[a+b x]]*) - - -{x^m*ArcTan[Tan[a + b*x]], x, 2, -((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcTan[Tan[a + b*x]])/(1 + m)} - -{x^2*ArcTan[Tan[a + b*x]], x, 2, -((b*x^4)/12) + (1/3)*x^3*ArcTan[Tan[a + b*x]]} -{x^1*ArcTan[Tan[a + b*x]], x, 2, -((b*x^3)/6) + (1/2)*x^2*ArcTan[Tan[a + b*x]]} -{x^0*ArcTan[Tan[a + b*x]], x, 2, ArcTan[Tan[a + b*x]]^2/(2*b)} -{ArcTan[Tan[a + b*x]]/x^1, x, 2, b*x - (b*x - ArcTan[Tan[a + b*x]])*Log[x]} - - -{x^m*ArcTan[Cot[a + b*x]], x, 2, (b*x^(2 + m))/(2 + 3*m + m^2) + (x^(1 + m)*ArcTan[Cot[a + b*x]])/(1 + m)} - -{x^2*ArcTan[Cot[a + b*x]], x, 2, (b*x^4)/12 + (1/3)*x^3*ArcTan[Cot[a + b*x]]} -{x^1*ArcTan[Cot[a + b*x]], x, 2, (b*x^3)/6 + (1/2)*x^2*ArcTan[Cot[a + b*x]]} -{x^0*ArcTan[Cot[a + b*x]], x, 2, -(ArcTan[Cot[a + b*x]]^2/(2*b))} -{ArcTan[Cot[a + b*x]]/x^1, x, 2, (-b)*x + (b*x + ArcTan[Cot[a + b*x]])*Log[x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Trig[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Tan[a+b x]]*) - - -{ArcTan[Tan[a + b*x]], x, 2, ArcTan[Tan[a + b*x]]^2/(2*b)} - - -{x^2*ArcTan[c + d*Tan[a + b*x]], x, 11, (1/3)*x^3*ArcTan[c + d*Tan[a + b*x]] + (1/6)*I*x^3*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] - (1/6)*I*x^3*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] + (x^2*PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b) - (x^2*PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b) + (I*x*PolyLog[3, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b^2) - (I*x*PolyLog[3, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b^2) - PolyLog[4, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))]/(8*b^3) + PolyLog[4, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))]/(8*b^3)} -{x^1*ArcTan[c + d*Tan[a + b*x]], x, 9, (1/2)*x^2*ArcTan[c + d*Tan[a + b*x]] + (1/4)*I*x^2*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] - (1/4)*I*x^2*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] + (x*PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b) - (x*PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b) + (I*PolyLog[3, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(8*b^2) - (I*PolyLog[3, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(8*b^2)} -{x^0*ArcTan[c + d*Tan[a + b*x]], x, 7, x*ArcTan[c + d*Tan[a + b*x]] + (1/2)*I*x*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] - (1/2)*I*x*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] + PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))]/(4*b) - PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))]/(4*b)} -{ArcTan[c + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (1 + I*c)*Tan[a + b*x]], x, 7, -((b*x^4)/12) + (1/3)*x^3*ArcTan[c + (1 + I*c)*Tan[a + b*x]] - (1/6)*I*x^3*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - (x^2*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*x*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) + PolyLog[4, I*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcTan[c + (1 + I*c)*Tan[a + b*x]], x, 6, -((b*x^3)/6) + (1/2)*x^2*ArcTan[c + (1 + I*c)*Tan[a + b*x]] - (1/4)*I*x^2*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - (x*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (1 + I*c)*Tan[a + b*x]], x, 5, -((b*x^2)/2) + x*ArcTan[c + (1 + I*c)*Tan[a + b*x]] - (1/2)*I*x*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcTan[c + (1 + I*c)*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (1 + I*c)*Tan[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (-1 + I*c)*Tan[a + b*x]], x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTan[c - (1 - I*c)*Tan[a + b*x]] + (1/6)*I*x^3*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + (x^2*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*x*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) - PolyLog[4, (-I)*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcTan[c + (-1 + I*c)*Tan[a + b*x]], x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTan[c - (1 - I*c)*Tan[a + b*x]] + (1/4)*I*x^2*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + (x*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (-1 + I*c)*Tan[a + b*x]], x, 5, (b*x^2)/2 + x*ArcTan[c - (1 - I*c)*Tan[a + b*x]] + (1/2)*I*x*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcTan[c + (-1 + I*c)*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (-1 + I*c)*Tan[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Cot[a+b x]]*) - - -{ArcTan[Cot[a + b*x]], x, 2, -(ArcTan[Cot[a + b*x]]^2/(2*b))} - - -{x^2*ArcTan[c + d*Cot[a + b*x]], x, 11, (1/3)*x^3*ArcTan[c + d*Cot[a + b*x]] + (1/6)*I*x^3*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] - (1/6)*I*x^3*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] + (x^2*PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b) - (x^2*PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b) + (I*x*PolyLog[3, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b^2) - (I*x*PolyLog[3, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b^2) - PolyLog[4, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)]/(8*b^3) + PolyLog[4, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))]/(8*b^3)} -{x^1*ArcTan[c + d*Cot[a + b*x]], x, 9, (1/2)*x^2*ArcTan[c + d*Cot[a + b*x]] + (1/4)*I*x^2*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] - (1/4)*I*x^2*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] + (x*PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b) - (x*PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b) + (I*PolyLog[3, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(8*b^2) - (I*PolyLog[3, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(8*b^2)} -{x^0*ArcTan[c + d*Cot[a + b*x]], x, 7, x*ArcTan[c + d*Cot[a + b*x]] + (1/2)*I*x*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] - (1/2)*I*x*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] + PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)]/(4*b) - PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))]/(4*b)} -{ArcTan[c + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]], x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (1/6)*I*x^3*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + (x^2*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*x*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) - PolyLog[4, I*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcTan[c + (1 - I*c)*Cot[a + b*x]], x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (1/4)*I*x^2*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + (x*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (1 - I*c)*Cot[a + b*x]], x, 5, (b*x^2)/2 + x*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + (1/2)*I*x*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcTan[c + (1 - I*c)*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (1 - I*c)*Cot[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (-1 - I*c)*Cot[a + b*x]], x, 7, -((b*x^4)/12) + (1/3)*x^3*ArcTan[c - (1 + I*c)*Cot[a + b*x]] - (1/6)*I*x^3*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - (x^2*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*x*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) + PolyLog[4, (-I)*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcTan[c + (-1 - I*c)*Cot[a + b*x]], x, 6, -((b*x^3)/6) + (1/2)*x^2*ArcTan[c - (1 + I*c)*Cot[a + b*x]] - (1/4)*I*x^2*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - (x*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (-1 - I*c)*Cot[a + b*x]], x, 5, -((b*x^2)/2) + x*ArcTan[c - (1 + I*c)*Cot[a + b*x]] - (1/2)*I*x*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcTan[c + (-1 - I*c)*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (-1 - I*c)*Cot[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Hyper[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Sinh[a+b x]]*) - - -{ArcTan[Sinh[x]], x, 6, -2*x*ArcTan[E^x] + x*ArcTan[Sinh[x]] + I*PolyLog[2, (-I)*E^x] - I*PolyLog[2, I*E^x]} -{x*ArcTan[Sinh[x]], x, 8, (-x^2)*ArcTan[E^x] + (1/2)*x^2*ArcTan[Sinh[x]] + I*x*PolyLog[2, (-I)*E^x] - I*x*PolyLog[2, I*E^x] - I*PolyLog[3, (-I)*E^x] + I*PolyLog[3, I*E^x]} -{x^2*ArcTan[Sinh[x]], x, 10, (-(2/3))*x^3*ArcTan[E^x] + (1/3)*x^3*ArcTan[Sinh[x]] + I*x^2*PolyLog[2, (-I)*E^x] - I*x^2*PolyLog[2, I*E^x] - 2*I*x*PolyLog[3, (-I)*E^x] + 2*I*x*PolyLog[3, I*E^x] + 2*I*PolyLog[4, (-I)*E^x] - 2*I*PolyLog[4, I*E^x]} - - -(* ::Subsection:: *) -(*Integrands of the form x^m ArcTan[c+d Cosh[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Tanh[a+b x]]*) - - -{(e + f*x)^3*ArcTan[Tanh[a + b*x]], x, 12, -(((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f)) + ((e + f*x)^4*ArcTan[Tanh[a + b*x]])/(4*f) + (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (3*I*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) + (3*I*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3) - (3*I*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/(16*b^4) + (3*I*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/(16*b^4)} -{(e + f*x)^2*ArcTan[Tanh[a + b*x]], x, 10, -(((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f)) + ((e + f*x)^3*ArcTan[Tanh[a + b*x]])/(3*f) + (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (I*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(4*b^2) + (I*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3)} -{(e + f*x)^1*ArcTan[Tanh[a + b*x]], x, 8, -(((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f)) + ((e + f*x)^2*ArcTan[Tanh[a + b*x]])/(2*f) + (I*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (I*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) + (I*f*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2)} -{(e + f*x)^0*ArcTan[Tanh[a + b*x]], x, 6, (-x)*ArcTan[E^(2*a + 2*b*x)] + x*ArcTan[Tanh[a + b*x]] + (I*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[Tanh[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcTan[Tanh[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcTan[c + d*Tanh[a + b*x]], x, 11, (1/3)*x^3*ArcTan[c + d*Tanh[a + b*x]] + (1/6)*I*x^3*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/6)*I*x^3*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*x^2*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) - (I*x^2*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b) - (I*x*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b^2) + (I*x*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b^2) + (I*PolyLog[4, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(8*b^3) - (I*PolyLog[4, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(8*b^3)} -{x^1*ArcTan[c + d*Tanh[a + b*x]], x, 9, (1/2)*x^2*ArcTan[c + d*Tanh[a + b*x]] + (1/4)*I*x^2*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/4)*I*x^2*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*x*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) - (I*x*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b) - (I*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(8*b^2) + (I*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(8*b^2)} -{x^0*ArcTan[c + d*Tanh[a + b*x]], x, 7, x*ArcTan[c + d*Tanh[a + b*x]] + (1/2)*I*x*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/2)*I*x*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) - (I*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b)} -{ArcTan[c + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + d*Tanh[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (I + c)*Tanh[a + b*x]], x, 7, (-(1/12))*I*b*x^4 + (1/3)*x^3*ArcTan[c + (I + c)*Tanh[a + b*x]] + (1/6)*I*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) - (I*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(4*b^2) + (I*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcTan[c + (I + c)*Tanh[a + b*x]], x, 6, (-(1/6))*I*b*x^3 + (1/2)*x^2*ArcTan[c + (I + c)*Tanh[a + b*x]] + (1/4)*I*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (I + c)*Tanh[a + b*x]], x, 5, (-(1/2))*I*b*x^2 + x*ArcTan[c + (I + c)*Tanh[a + b*x]] + (1/2)*I*x*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[c + (I + c)*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (I + c)*Tanh[a + b*x]]/x, x]} - - -{x^2*ArcTan[c - (I - c)*Tanh[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTan[c - (I - c)*Tanh[a + b*x]] - (1/6)*I*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) + (I*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(4*b^2) - (I*PolyLog[4, I*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcTan[c - (I - c)*Tanh[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTan[c - (I - c)*Tanh[a + b*x]] - (1/4)*I*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcTan[c - (I - c)*Tanh[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTan[c - (I - c)*Tanh[a + b*x]] - (1/2)*I*x*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[c - (I - c)*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c - (I - c)*Tanh[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTan[c+d Coth[a+b x]]*) - - -{(e + f*x)^3*ArcTan[Coth[a + b*x]], x, 12, ((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f) + ((e + f*x)^4*ArcTan[Coth[a + b*x]])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (3*I*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) - (3*I*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2) - (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) + (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3) + (3*I*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/(16*b^4) - (3*I*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/(16*b^4)} -{(e + f*x)^2*ArcTan[Coth[a + b*x]], x, 10, ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) + ((e + f*x)^3*ArcTan[Coth[a + b*x]])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (I*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(4*b^2) - (I*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/(4*b^2) - (I*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) + (I*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3)} -{(e + f*x)^1*ArcTan[Coth[a + b*x]], x, 8, ((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f) + ((e + f*x)^2*ArcTan[Coth[a + b*x]])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (I*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) - (I*f*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2)} -{(e + f*x)^0*ArcTan[Coth[a + b*x]], x, 6, x*ArcTan[E^(2*a + 2*b*x)] + x*ArcTan[Coth[a + b*x]] - (I*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[Coth[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcTan[Coth[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcTan[c + d*Coth[a + b*x]], x, 11, (1/3)*x^3*ArcTan[c + d*Coth[a + b*x]] + (1/6)*I*x^3*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/6)*I*x^3*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*x^2*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) - (I*x^2*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b) - (I*x*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b^2) + (I*x*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b^2) + (I*PolyLog[4, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(8*b^3) - (I*PolyLog[4, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(8*b^3)} -{x^1*ArcTan[c + d*Coth[a + b*x]], x, 9, (1/2)*x^2*ArcTan[c + d*Coth[a + b*x]] + (1/4)*I*x^2*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/4)*I*x^2*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*x*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) - (I*x*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b) - (I*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(8*b^2) + (I*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(8*b^2)} -{x^0*ArcTan[c + d*Coth[a + b*x]], x, 7, x*ArcTan[c + d*Coth[a + b*x]] + (1/2)*I*x*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] - (1/2)*I*x*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] + (I*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) - (I*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b)} -{ArcTan[c + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + d*Coth[a + b*x]]/x, x]} - - -{x^2*ArcTan[c + (I + c)*Coth[a + b*x]], x, 7, (-(1/12))*I*b*x^4 + (1/3)*x^3*ArcTan[c + (I + c)*Coth[a + b*x]] + (1/6)*I*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) - (I*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(4*b^2) + (I*PolyLog[4, I*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcTan[c + (I + c)*Coth[a + b*x]], x, 6, (-(1/6))*I*b*x^3 + (1/2)*x^2*ArcTan[c + (I + c)*Coth[a + b*x]] + (1/4)*I*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcTan[c + (I + c)*Coth[a + b*x]], x, 5, (-(1/2))*I*b*x^2 + x*ArcTan[c + (I + c)*Coth[a + b*x]] + (1/2)*I*x*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[c + (I + c)*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c + (I + c)*Coth[a + b*x]]/x, x]} - - -{x^2*ArcTan[c - (I - c)*Coth[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTan[c - (I - c)*Coth[a + b*x]] - (1/6)*I*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) + (I*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(4*b^2) - (I*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcTan[c - (I - c)*Coth[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTan[c - (I - c)*Coth[a + b*x]] - (1/4)*I*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcTan[c - (I - c)*Coth[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTan[c - (I - c)*Coth[a + b*x]] - (1/2)*I*x*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcTan[c - (I - c)*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTan[c - (I - c)*Coth[a + b*x]]/x, x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse tangents of exponentials*) - - -{ArcTan[E^x], x, 4, (1/2)*I*PolyLog[2, (-I)*E^x] - (1/2)*I*PolyLog[2, I*E^x]} -{x*ArcTan[E^x], x, 7, (1/2)*I*x*PolyLog[2, (-I)*E^x] - (1/2)*I*x*PolyLog[2, I*E^x] - (1/2)*I*PolyLog[3, (-I)*E^x] + (1/2)*I*PolyLog[3, I*E^x]} -{x^2*ArcTan[E^x], x, 9, (1/2)*I*x^2*PolyLog[2, (-I)*E^x] - (1/2)*I*x^2*PolyLog[2, I*E^x] - I*x*PolyLog[3, (-I)*E^x] + I*x*PolyLog[3, I*E^x] + I*PolyLog[4, (-I)*E^x] - I*PolyLog[4, I*E^x]} - - -{ArcTan[E^(a + b*x)], x, 4, (I*PolyLog[2, (-I)*E^(a + b*x)])/(2*b) - (I*PolyLog[2, I*E^(a + b*x)])/(2*b)} -{x*ArcTan[E^(a + b*x)], x, 7, (I*x*PolyLog[2, (-I)*E^(a + b*x)])/(2*b) - (I*x*PolyLog[2, I*E^(a + b*x)])/(2*b) - (I*PolyLog[3, (-I)*E^(a + b*x)])/(2*b^2) + (I*PolyLog[3, I*E^(a + b*x)])/(2*b^2)} -{x^2*ArcTan[E^(a + b*x)], x, 9, (I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/(2*b) - (I*x^2*PolyLog[2, I*E^(a + b*x)])/(2*b) - (I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^2 + (I*x*PolyLog[3, I*E^(a + b*x)])/b^2 + (I*PolyLog[4, (-I)*E^(a + b*x)])/b^3 - (I*PolyLog[4, I*E^(a + b*x)])/b^3} - - -{ArcTan[a + b*f^(c + d*x)], x, 6, -((ArcTan[a + b*f^(c + d*x)]*Log[2/(1 - I*(a + b*f^(c + d*x)))])/(d*Log[f])) + (ArcTan[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(d*Log[f]) + (I*PolyLog[2, 1 - 2/(1 - I*(a + b*f^(c + d*x)))])/(2*d*Log[f]) - (I*PolyLog[2, 1 - (2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(2*d*Log[f])} -{x*ArcTan[a + b*f^(c + d*x)], x, 9, (1/2)*x^2*ArcTan[a + b*f^(c + d*x)] - (1/4)*I*x^2*Log[1 - (I*b*f^(c + d*x))/(1 - I*a)] + (1/4)*I*x^2*Log[1 + (I*b*f^(c + d*x))/(1 + I*a)] - (I*x*PolyLog[2, (I*b*f^(c + d*x))/(1 - I*a)])/(2*d*Log[f]) + (I*x*PolyLog[2, -((I*b*f^(c + d*x))/(1 + I*a))])/(2*d*Log[f]) + (I*PolyLog[3, (I*b*f^(c + d*x))/(1 - I*a)])/(2*d^2*Log[f]^2) - (I*PolyLog[3, -((I*b*f^(c + d*x))/(1 + I*a))])/(2*d^2*Log[f]^2), (1/4)*I*x^2*Log[1 - I*a - I*b*f^(c + d*x)] - (1/4)*I*x^2*Log[1 + I*a + I*b*f^(c + d*x)] + (1/4)*I*x^2*Log[1 - (b*f^(c + d*x))/(I - a)] - (1/4)*I*x^2*Log[1 + (b*f^(c + d*x))/(I + a)] + (I*x*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(2*d*Log[f]) - (I*x*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(2*d*Log[f]) - (I*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(2*d^2*Log[f]^2) + (I*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(2*d^2*Log[f]^2)} -{x^2*ArcTan[a + b*f^(c + d*x)], x, 11, (1/3)*x^3*ArcTan[a + b*f^(c + d*x)] - (1/6)*I*x^3*Log[1 - (I*b*f^(c + d*x))/(1 - I*a)] + (1/6)*I*x^3*Log[1 + (I*b*f^(c + d*x))/(1 + I*a)] - (I*x^2*PolyLog[2, (I*b*f^(c + d*x))/(1 - I*a)])/(2*d*Log[f]) + (I*x^2*PolyLog[2, -((I*b*f^(c + d*x))/(1 + I*a))])/(2*d*Log[f]) + (I*x*PolyLog[3, (I*b*f^(c + d*x))/(1 - I*a)])/(d^2*Log[f]^2) - (I*x*PolyLog[3, -((I*b*f^(c + d*x))/(1 + I*a))])/(d^2*Log[f]^2) - (I*PolyLog[4, (I*b*f^(c + d*x))/(1 - I*a)])/(d^3*Log[f]^3) + (I*PolyLog[4, -((I*b*f^(c + d*x))/(1 + I*a))])/(d^3*Log[f]^3), (1/6)*I*x^3*Log[1 - I*a - I*b*f^(c + d*x)] - (1/6)*I*x^3*Log[1 + I*a + I*b*f^(c + d*x)] + (1/6)*I*x^3*Log[1 - (b*f^(c + d*x))/(I - a)] - (1/6)*I*x^3*Log[1 + (b*f^(c + d*x))/(I + a)] + (I*x^2*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(2*d*Log[f]) - (I*x^2*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(2*d*Log[f]) - (I*x*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) + (I*x*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(d^2*Log[f]^2) + (I*PolyLog[4, (b*f^(c + d*x))/(I - a)])/(d^3*Log[f]^3) - (I*PolyLog[4, -((b*f^(c + d*x))/(I + a))])/(d^3*Log[f]^3)} - - -{ArcTan[E^x]/E^x, x, 5, x - ArcTan[E^x]/E^x - (1/2)*Log[1 + E^(2*x)]} - - -(* ::Title::Closed:: *) -(*Miscellaneous integrands involving inverse tangents*) - - -(* ::Section::Closed:: *) -(*Problems from Calculus textbooks*) - - -(* ::Subsubsection::Closed:: *) -(*Edwards and Penney Calculus*) - - -{ArcTan[x]/(-1 + x)^3, x, 5, 1/(4*(1 - x)) - ArcTan[x]/(2*(1 - x)^2) - (1/4)*Log[1 - x] + (1/8)*Log[1 + x^2]} -{ArcTan[1 + 2*x]/(4 + 3*x)^3, x, 9, -(1/(34*(4 + 3*x))) + (8/867)*ArcTan[1 + 2*x] - ArcTan[1 + 2*x]/(6*(4 + 3*x)^2) + (5/289)*Log[4 + 3*x] - (5/578)*Log[1 + 2*x + 2*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*Thomas Calculus, 8th Edition*) - - -{ArcTan[Sqrt[1 + x]], x, 4, -Sqrt[1 + x] + 2*ArcTan[Sqrt[1 + x]] + x*ArcTan[Sqrt[1 + x]]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse tangents*) - - -{1/((1 + x^2)*(2 + ArcTan[x])), x, 1, Log[2 + ArcTan[x]]} -{1/((a + a*x^2)*(b - 2*b*ArcTan[x])), x, 1, -(Log[1 - 2*ArcTan[x]]/(2*a*b))} - - -{(x + x^3 + (1 + x)^2*ArcTan[x])/((1 + x)^2*(1 + x^2)), x, 5, 1/(1 + x) + ArcTan[x]^2/2 + Log[1 + x]} - - -{x^3*ArcTan[Sqrt[x + 1] - Sqrt[x]], x, 9, -(Sqrt[x]/8) + x^(3/2)/24 - x^(5/2)/40 + x^(7/2)/56 + (Pi*x^4)/16 + ArcTan[Sqrt[x]]/8 - (1/8)*x^4*ArcTan[Sqrt[x]]} -{x^2*ArcTan[Sqrt[x + 1] - Sqrt[x]], x, 8, Sqrt[x]/6 - x^(3/2)/18 + x^(5/2)/30 + (Pi*x^3)/12 - ArcTan[Sqrt[x]]/6 - (1/6)*x^3*ArcTan[Sqrt[x]]} -{x^1*ArcTan[Sqrt[x + 1] - Sqrt[x]], x, 7, -(Sqrt[x]/4) + x^(3/2)/12 + (Pi*x^2)/8 + ArcTan[Sqrt[x]]/4 - (1/4)*x^2*ArcTan[Sqrt[x]]} -{x^0*ArcTan[Sqrt[x + 1] - Sqrt[x]], x, 6, Sqrt[x]/2 + (Pi*x)/4 - ArcTan[Sqrt[x]]/2 - (1/2)*x*ArcTan[Sqrt[x]]} -{ArcTan[Sqrt[x + 1] - Sqrt[x]]/x^1, x, 6, (1/4)*Pi*Log[x] - (1/2)*I*PolyLog[2, (-I)*Sqrt[x]] + (1/2)*I*PolyLog[2, I*Sqrt[x]]} -{ArcTan[Sqrt[x + 1] - Sqrt[x]]/x^2, x, 6, -(Pi/(4*x)) + 1/(2*Sqrt[x]) + ArcTan[Sqrt[x]]/2 + ArcTan[Sqrt[x]]/(2*x)} -{ArcTan[Sqrt[x + 1] - Sqrt[x]]/x^3, x, 7, -(Pi/(8*x^2)) + 1/(12*x^(3/2)) - 1/(4*Sqrt[x]) - ArcTan[Sqrt[x]]/4 + ArcTan[Sqrt[x]]/(4*x^2)} -{ArcTan[Sqrt[x + 1] - Sqrt[x]]/x^4, x, 8, -(Pi/(12*x^3)) + 1/(30*x^(5/2)) - 1/(18*x^(3/2)) + 1/(6*Sqrt[x]) + ArcTan[Sqrt[x]]/6 + ArcTan[Sqrt[x]]/(6*x^3)} - - -{ArcTan[c*x/Sqrt[a - c^2*x^2]]^m/Sqrt[d - c^2*d/a*x^2], x, 2, (Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^(1 + m))/(c*(1 + m)*Sqrt[d - (c^2*d*x^2)/a])} - -{ArcTan[c*x/Sqrt[a - c^2*x^2]]^2/Sqrt[d - c^2*d/a*x^2], x, 2, (Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^3)/(3*c*Sqrt[d - (c^2*d*x^2)/a])} -{ArcTan[c*x/Sqrt[a - c^2*x^2]]^1/Sqrt[d - c^2*d/a*x^2], x, 2, (Sqrt[a - c^2*x^2]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2)/(2*c*Sqrt[d - (c^2*d*x^2)/a])} -{1/ArcTan[c*x/Sqrt[a - c^2*x^2]]^1/Sqrt[d - c^2*d/a*x^2], x, 2, (Sqrt[a - c^2*x^2]*Log[ArcTan[(c*x)/Sqrt[a - c^2*x^2]]])/(c*Sqrt[d - (c^2*d*x^2)/a])} -{1/ArcTan[c*x/Sqrt[a - c^2*x^2]]^2/Sqrt[d - c^2*d/a*x^2], x, 2, -(Sqrt[a - c^2*x^2]/(c*Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]))} -{1/ArcTan[c*x/Sqrt[a - c^2*x^2]]^3/Sqrt[d - c^2*d/a*x^2], x, 2, -(Sqrt[a - c^2*x^2]/(2*c*Sqrt[d - (c^2*d*x^2)/a]*ArcTan[(c*x)/Sqrt[a - c^2*x^2]]^2))} - - -{ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^m/Sqrt[a + b*x^2], x, 2, (Sqrt[-((a*e^2)/b) - e^2*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^(1 + m))/(e*(1 + m)*Sqrt[a + b*x^2])} - -{ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^2/Sqrt[a + b*x^2], x, 2, (Sqrt[-((a*e^2)/b) - e^2*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^3)/(3*e*Sqrt[a + b*x^2])} -{ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^1/Sqrt[a + b*x^2], x, 2, (Sqrt[-((a*e^2)/b) - e^2*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2)/(2*e*Sqrt[a + b*x^2])} -{1/ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^1/Sqrt[a + b*x^2], x, 2, (Sqrt[-((a*e^2)/b) - e^2*x^2]*Log[ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]])/(e*Sqrt[a + b*x^2])} -{1/ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^2/Sqrt[a + b*x^2], x, 2, -(Sqrt[-((a*e^2)/b) - e^2*x^2]/(e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]))} -{1/ArcTan[e*x/Sqrt[-a*e^2/b - e^2*x^2]]^3/Sqrt[a + b*x^2], x, 2, -(Sqrt[-((a*e^2)/b) - e^2*x^2]/(2*e*Sqrt[a + b*x^2]*ArcTan[(e*x)/Sqrt[-((a*e^2)/b) - e^2*x^2]]^2))} - - -{ArcTan[c*(a + b*x)]*Log[d*(a + b*x)]/(a + b*x), x, 9, (I*Log[d*(a + b*x)]*PolyLog[2, (-I)*c*(a + b*x)])/(2*b) - (I*Log[d*(a + b*x)]*PolyLog[2, I*c*(a + b*x)])/(2*b) - (I*PolyLog[3, (-I)*c*(a + b*x)])/(2*b) + (I*PolyLog[3, I*c*(a + b*x)])/(2*b)} - - -{E^(c*(a + b*x))*ArcTan[Sinh[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcTan[Sinh[c*(a + b*x)]])/(b*c) - Log[1 + E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcTan[Cosh[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcTan[Cosh[c*(a + b*x)]])/(b*c) - ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) - ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcTan[Tanh[a*c + b*c*x]], x, 13, ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) - ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + (E^(a*c + b*c*x)*ArcTan[Tanh[c*(a + b*x)]])/(b*c) - Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) + Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)} -{E^(c*(a + b*x))*ArcTan[Coth[a*c + b*c*x]], x, 13, -(ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c)) + ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + (E^(a*c + b*c*x)*ArcTan[Coth[c*(a + b*x)]])/(b*c) + Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) - Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)} -{E^(c*(a + b*x))*ArcTan[Sech[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcTan[Sech[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcTan[Csch[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcTan[Csch[c*(a + b*x)]])/(b*c) + Log[1 + E^(2*c*(a + b*x))]/(b*c)} - - -{((a + b*ArcTan[c*x^n])*(d + e*Log[f*x^m]))/x, x, 13, a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) + (I*b*d*PolyLog[2, (-I)*c*x^n])/(2*n) + (I*b*e*Log[f*x^m]*PolyLog[2, (-I)*c*x^n])/(2*n) - (I*b*d*PolyLog[2, I*c*x^n])/(2*n) - (I*b*e*Log[f*x^m]*PolyLog[2, I*c*x^n])/(2*n) - (I*b*e*m*PolyLog[3, (-I)*c*x^n])/(2*n^2) + (I*b*e*m*PolyLog[3, I*c*x^n])/(2*n^2)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.1 Inverse cotangent functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.1 Inverse cotangent functions.m deleted file mode 100644 index 4af89cd..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.1 Inverse cotangent functions.m +++ /dev/null @@ -1,529 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands involving inverse cotangents*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCot[a x^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^5*ArcCot[a*x], x, 4, x/(6*a^5) - x^3/(18*a^3) + x^5/(30*a) + (1/6)*x^6*ArcCot[a*x] - ArcTan[a*x]/(6*a^6)} -{x^4*ArcCot[a*x], x, 4, -(x^2/(10*a^3)) + x^4/(20*a) + (1/5)*x^5*ArcCot[a*x] + Log[1 + a^2*x^2]/(10*a^5)} -{x^3*ArcCot[a*x], x, 4, -(x/(4*a^3)) + x^3/(12*a) + (1/4)*x^4*ArcCot[a*x] + ArcTan[a*x]/(4*a^4)} -{x^2*ArcCot[a*x], x, 4, x^2/(6*a) + (1/3)*x^3*ArcCot[a*x] - Log[1 + a^2*x^2]/(6*a^3)} -{x^1*ArcCot[a*x], x, 3, x/(2*a) + (1/2)*x^2*ArcCot[a*x] - ArcTan[a*x]/(2*a^2)} -{x^0*ArcCot[a*x], x, 2, x*ArcCot[a*x] + Log[1 + a^2*x^2]/(2*a)} -{ArcCot[a*x]/x^1, x, 3, (-(1/2))*I*PolyLog[2, -(I/(a*x))] + (1/2)*I*PolyLog[2, I/(a*x)]} -{ArcCot[a*x]/x^2, x, 5, -(ArcCot[a*x]/x) - a*Log[x] + (1/2)*a*Log[1 + a^2*x^2]} -{ArcCot[a*x]/x^3, x, 3, a/(2*x) - ArcCot[a*x]/(2*x^2) + (1/2)*a^2*ArcTan[a*x]} -{ArcCot[a*x]/x^4, x, 4, a/(6*x^2) - ArcCot[a*x]/(3*x^3) + (1/3)*a^3*Log[x] - (1/6)*a^3*Log[1 + a^2*x^2]} -{ArcCot[a*x]/x^5, x, 4, a/(12*x^3) - a^3/(4*x) - ArcCot[a*x]/(4*x^4) - (1/4)*a^4*ArcTan[a*x]} - - -{x^5*ArcCot[a*x]^2, x, 15, -((4*x^2)/(45*a^4)) + x^4/(60*a^2) + (x*ArcCot[a*x])/(3*a^5) - (x^3*ArcCot[a*x])/(9*a^3) + (x^5*ArcCot[a*x])/(15*a) + ArcCot[a*x]^2/(6*a^6) + (1/6)*x^6*ArcCot[a*x]^2 + (23*Log[1 + a^2*x^2])/(90*a^6)} -{x^4*ArcCot[a*x]^2, x, 14, -((3*x)/(10*a^4)) + x^3/(30*a^2) - (x^2*ArcCot[a*x])/(5*a^3) + (x^4*ArcCot[a*x])/(10*a) + (I*ArcCot[a*x]^2)/(5*a^5) + (1/5)*x^5*ArcCot[a*x]^2 + (3*ArcTan[a*x])/(10*a^5) - (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(5*a^5) + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(5*a^5)} -{x^3*ArcCot[a*x]^2, x, 10, x^2/(12*a^2) - (x*ArcCot[a*x])/(2*a^3) + (x^3*ArcCot[a*x])/(6*a) - ArcCot[a*x]^2/(4*a^4) + (1/4)*x^4*ArcCot[a*x]^2 - Log[1 + a^2*x^2]/(3*a^4)} -{x^2*ArcCot[a*x]^2, x, 9, x/(3*a^2) + (x^2*ArcCot[a*x])/(3*a) - (I*ArcCot[a*x]^2)/(3*a^3) + (1/3)*x^3*ArcCot[a*x]^2 - ArcTan[a*x]/(3*a^3) + (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(3*a^3) - (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(3*a^3)} -{x^1*ArcCot[a*x]^2, x, 5, (x*ArcCot[a*x])/a + ArcCot[a*x]^2/(2*a^2) + (1/2)*x^2*ArcCot[a*x]^2 + Log[1 + a^2*x^2]/(2*a^2)} -{x^0*ArcCot[a*x]^2, x, 5, (I*ArcCot[a*x]^2)/a + x*ArcCot[a*x]^2 - (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a} -{ArcCot[a*x]^2/x^1, x, 6, 2*ArcCot[a*x]^2*ArcCoth[1 - 2/(1 + I*a*x)] - I*ArcCot[a*x]*PolyLog[2, 1 - (2*I)/(I + a*x)] + I*ArcCot[a*x]*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - (1/2)*PolyLog[3, 1 - (2*I)/(I + a*x)] + (1/2)*PolyLog[3, 1 - (2*a*x)/(I + a*x)]} -{ArcCot[a*x]^2/x^2, x, 4, (-I)*a*ArcCot[a*x]^2 - ArcCot[a*x]^2/x - 2*a*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*PolyLog[2, -1 + 2/(1 - I*a*x)]} -{ArcCot[a*x]^2/x^3, x, 8, (a*ArcCot[a*x])/x - (1/2)*a^2*ArcCot[a*x]^2 - ArcCot[a*x]^2/(2*x^2) + a^2*Log[x] - (1/2)*a^2*Log[1 + a^2*x^2]} -{ArcCot[a*x]^2/x^4, x, 8, -(a^2/(3*x)) + (a*ArcCot[a*x])/(3*x^2) + (1/3)*I*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (1/3)*a^3*ArcTan[a*x] + (2/3)*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + (1/3)*I*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)]} -{ArcCot[a*x]^2/x^5, x, 13, -(a^2/(12*x^2)) + (a*ArcCot[a*x])/(6*x^3) - (a^3*ArcCot[a*x])/(2*x) + (1/4)*a^4*ArcCot[a*x]^2 - ArcCot[a*x]^2/(4*x^4) - (2/3)*a^4*Log[x] + (1/3)*a^4*Log[1 + a^2*x^2]} - - -{x^5*ArcCot[a*x]^3, x, 33, -((19*x)/(60*a^5)) + x^3/(60*a^3) - (4*x^2*ArcCot[a*x])/(15*a^4) + (x^4*ArcCot[a*x])/(20*a^2) + (23*I*ArcCot[a*x]^2)/(30*a^6) + (x*ArcCot[a*x]^2)/(2*a^5) - (x^3*ArcCot[a*x]^2)/(6*a^3) + (x^5*ArcCot[a*x]^2)/(10*a) + ArcCot[a*x]^3/(6*a^6) + (1/6)*x^6*ArcCot[a*x]^3 + (19*ArcTan[a*x])/(60*a^6) - (23*ArcCot[a*x]*Log[2/(1 + I*a*x)])/(15*a^6) + (23*I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(30*a^6)} -{x^4*ArcCot[a*x]^3, x, 22, x^2/(20*a^3) - (9*x*ArcCot[a*x])/(10*a^4) + (x^3*ArcCot[a*x])/(10*a^2) - (9*ArcCot[a*x]^2)/(20*a^5) - (3*x^2*ArcCot[a*x]^2)/(10*a^3) + (3*x^4*ArcCot[a*x]^2)/(20*a) + (I*ArcCot[a*x]^3)/(5*a^5) + (1/5)*x^5*ArcCot[a*x]^3 - (3*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^5) - Log[1 + a^2*x^2]/(2*a^5) + (3*I*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(5*a^5) - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(10*a^5)} -{x^3*ArcCot[a*x]^3, x, 18, x/(4*a^3) + (x^2*ArcCot[a*x])/(4*a^2) - (I*ArcCot[a*x]^2)/a^4 - (3*x*ArcCot[a*x]^2)/(4*a^3) + (x^3*ArcCot[a*x]^2)/(4*a) - ArcCot[a*x]^3/(4*a^4) + (1/4)*x^4*ArcCot[a*x]^3 - ArcTan[a*x]/(4*a^4) + (2*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a^4 - (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4} -{x^2*ArcCot[a*x]^3, x, 11, (x*ArcCot[a*x])/a^2 + ArcCot[a*x]^2/(2*a^3) + (x^2*ArcCot[a*x]^2)/(2*a) - (I*ArcCot[a*x]^3)/(3*a^3) + (1/3)*x^3*ArcCot[a*x]^3 + (ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a^3 + Log[1 + a^2*x^2]/(2*a^3) - (I*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 + PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^3)} -{x^1*ArcCot[a*x]^3, x, 8, (3*I*ArcCot[a*x]^2)/(2*a^2) + (3*x*ArcCot[a*x]^2)/(2*a) + ArcCot[a*x]^3/(2*a^2) + (1/2)*x^2*ArcCot[a*x]^3 - (3*ArcCot[a*x]*Log[2/(1 + I*a*x)])/a^2 + (3*I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(2*a^2)} -{x^0*ArcCot[a*x]^3, x, 5, (I*ArcCot[a*x]^3)/a + x*ArcCot[a*x]^3 - (3*ArcCot[a*x]^2*Log[2/(1 + I*a*x)])/a + (3*I*ArcCot[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a)} -{ArcCot[a*x]^3/x^1, x, 8, 2*ArcCot[a*x]^3*ArcCoth[1 - 2/(1 + I*a*x)] - (3/2)*I*ArcCot[a*x]^2*PolyLog[2, 1 - (2*I)/(I + a*x)] + (3/2)*I*ArcCot[a*x]^2*PolyLog[2, 1 - (2*a*x)/(I + a*x)] - (3/2)*ArcCot[a*x]*PolyLog[3, 1 - (2*I)/(I + a*x)] + (3/2)*ArcCot[a*x]*PolyLog[3, 1 - (2*a*x)/(I + a*x)] + (3/4)*I*PolyLog[4, 1 - (2*I)/(I + a*x)] - (3/4)*I*PolyLog[4, 1 - (2*a*x)/(I + a*x)]} -{ArcCot[a*x]^3/x^2, x, 5, (-I)*a*ArcCot[a*x]^3 - ArcCot[a*x]^3/x - 3*a*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] - 3*I*a*ArcCot[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] - (3/2)*a*PolyLog[3, -1 + 2/(1 - I*a*x)]} -{ArcCot[a*x]^3/x^3, x, 7, (3/2)*I*a^2*ArcCot[a*x]^2 + (3*a*ArcCot[a*x]^2)/(2*x) - (1/2)*a^2*ArcCot[a*x]^3 - ArcCot[a*x]^3/(2*x^2) + 3*a^2*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + (3/2)*I*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)]} -{ArcCot[a*x]^3/x^4, x, 14, -((a^2*ArcCot[a*x])/x) + (1/2)*a^3*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(2*x^2) + (1/3)*I*a^3*ArcCot[a*x]^3 - ArcCot[a*x]^3/(3*x^3) - a^3*Log[x] + (1/2)*a^3*Log[1 + a^2*x^2] + a^3*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] + I*a^3*ArcCot[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (1/2)*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)]} -{ArcCot[a*x]^3/x^5, x, 16, a^3/(4*x) - (a^2*ArcCot[a*x])/(4*x^2) - I*a^4*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(4*x^3) - (3*a^3*ArcCot[a*x]^2)/(4*x) + (1/4)*a^4*ArcCot[a*x]^3 - ArcCot[a*x]^3/(4*x^4) + (1/4)*a^4*ArcTan[a*x] - 2*a^4*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a^4*PolyLog[2, -1 + 2/(1 - I*a*x)]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a x]^n with m symbolic*) - - -{x^m*ArcCot[a*x]^3, x, 0, Unintegrable[x^m*ArcCot[a*x]^3, x]} -{x^m*ArcCot[a*x]^2, x, 0, Unintegrable[x^m*ArcCot[a*x]^2, x]} -{x^m*ArcCot[a*x], x, 2, (x^(1 + m)*ArcCot[a*x])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + 3*m + m^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a x]^n / (c+d x^2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcCot[x]/(1 + x^2), x, 9, x^2/6 - x*ArcCot[x] + (1/3)*x^3*ArcCot[x] - ArcCot[x]^2/2 - (2/3)*Log[1 + x^2]} -{x^3*ArcCot[x]/(1 + x^2), x, 8, x/2 + (1/2)*x^2*ArcCot[x] - (1/2)*I*ArcCot[x]^2 - ArcTan[x]/2 + ArcCot[x]*Log[2/(1 + I*x)] - (1/2)*I*PolyLog[2, 1 - 2/(1 + I*x)]} -{x^2*ArcCot[x]/(1 + x^2), x, 4, x*ArcCot[x] + ArcCot[x]^2/2 + (1/2)*Log[1 + x^2]} -{x^1*ArcCot[x]/(1 + x^2), x, 4, (1/2)*I*ArcCot[x]^2 - ArcCot[x]*Log[2/(1 + I*x)] + (1/2)*I*PolyLog[2, 1 - 2/(1 + I*x)]} -{x^0*ArcCot[x]/(1 + x^2), x, 1, (-(1/2))*ArcCot[x]^2} -{ArcCot[x]/(x^1*(1 + x^2)), x, 3, (1/2)*I*ArcCot[x]^2 + ArcCot[x]*Log[2 - 2/(1 - I*x)] + (1/2)*I*PolyLog[2, -1 + 2/(1 - I*x)]} -{ArcCot[x]/(x^2*(1 + x^2)), x, 7, -(ArcCot[x]/x) + ArcCot[x]^2/2 - Log[x] + (1/2)*Log[1 + x^2]} -{ArcCot[x]/(x^3*(1 + x^2)), x, 7, 1/(2*x) - ArcCot[x]/(2*x^2) - (1/2)*I*ArcCot[x]^2 + ArcTan[x]/2 - ArcCot[x]*Log[2 - 2/(1 - I*x)] - (1/2)*I*PolyLog[2, -1 + 2/(1 - I*x)]} -{ArcCot[x]/(x^4*(1 + x^2)), x, 12, 1/(6*x^2) - ArcCot[x]/(3*x^3) + ArcCot[x]/x - ArcCot[x]^2/2 + (4*Log[x])/3 - (2/3)*Log[1 + x^2]} - - -{x^2*ArcCot[c*x]/(1 + x^2), x, 28, x*ArcCot[c*x] - (1/2)*I*ArcTan[x]*Log[1 - I/(c*x)] + (1/2)*I*ArcTan[x]*Log[1 + I/(c*x)] + (1/2)*I*ArcTan[x]*Log[-((2*I*(I - c*x))/((1 - c)*(1 - I*x)))] - (1/2)*I*ArcTan[x]*Log[-((2*I*(I + c*x))/((1 + c)*(1 - I*x)))] + Log[1 + c^2*x^2]/(2*c) + (1/4)*PolyLog[2, 1 + (2*I*(I - c*x))/((1 - c)*(1 - I*x))] - (1/4)*PolyLog[2, 1 + (2*I*(I + c*x))/((1 + c)*(1 - I*x))]} -{x^1*ArcCot[c*x]/(1 + x^2), x, 10, (-ArcCot[c*x])*Log[2/(1 - I*c*x)] + (1/2)*ArcCot[c*x]*Log[(2*I*c*(I - x))/((1 - c)*(1 - I*c*x))] + (1/2)*ArcCot[c*x]*Log[-((2*I*c*(I + x))/((1 + c)*(1 - I*c*x)))] - (1/2)*I*PolyLog[2, 1 - 2/(1 - I*c*x)] + (1/4)*I*PolyLog[2, 1 - (2*I*c*(I - x))/((1 - c)*(1 - I*c*x))] + (1/4)*I*PolyLog[2, 1 + (2*I*c*(I + x))/((1 + c)*(1 - I*c*x))]} -{x^0*ArcCot[c*x]/(1 + x^2), x, 25, (1/2)*I*ArcTan[x]*Log[1 - I/(c*x)] - (1/2)*I*ArcTan[x]*Log[1 + I/(c*x)] - (1/2)*I*ArcTan[x]*Log[-((2*I*(I - c*x))/((1 - c)*(1 - I*x)))] + (1/2)*I*ArcTan[x]*Log[-((2*I*(I + c*x))/((1 + c)*(1 - I*x)))] - (1/4)*PolyLog[2, 1 + (2*I*(I - c*x))/((1 - c)*(1 - I*x))] + (1/4)*PolyLog[2, 1 + (2*I*(I + c*x))/((1 + c)*(1 - I*x))]} -{ArcCot[c*x]/(x^1*(1 + x^2)), x, 15, ArcCot[c*x]*Log[2/(1 - I*c*x)] - (1/2)*ArcCot[c*x]*Log[(2*I*c*(I - x))/((1 - c)*(1 - I*c*x))] - (1/2)*ArcCot[c*x]*Log[-((2*I*c*(I + x))/((1 + c)*(1 - I*c*x)))] - (1/2)*I*PolyLog[2, -(I/(c*x))] + (1/2)*I*PolyLog[2, I/(c*x)] + (1/2)*I*PolyLog[2, 1 - 2/(1 - I*c*x)] - (1/4)*I*PolyLog[2, 1 - (2*I*c*(I - x))/((1 - c)*(1 - I*c*x))] - (1/4)*I*PolyLog[2, 1 + (2*I*c*(I + x))/((1 + c)*(1 - I*c*x))]} -{ArcCot[c*x]/(x^2*(1 + x^2)), x, 31, -(ArcCot[c*x]/x) - (1/2)*I*ArcTan[x]*Log[1 - I/(c*x)] + (1/2)*I*ArcTan[x]*Log[1 + I/(c*x)] - c*Log[x] + (1/2)*I*ArcTan[x]*Log[-((2*I*(I - c*x))/((1 - c)*(1 - I*x)))] - (1/2)*I*ArcTan[x]*Log[-((2*I*(I + c*x))/((1 + c)*(1 - I*x)))] + (1/2)*c*Log[1 + c^2*x^2] + (1/4)*PolyLog[2, 1 + (2*I*(I - c*x))/((1 - c)*(1 - I*x))] - (1/4)*PolyLog[2, 1 + (2*I*(I + c*x))/((1 + c)*(1 - I*x))]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/((1 + x^2)*ArcCot[x]), x, 1, -Log[ArcCot[x]]} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{ArcCot[x]^n/(1 + x^2), x, 1, -(ArcCot[x]^(1 + n)/(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcCot[a x] (c+d x^2)^p*) - - -{ArcCot[a*x]*(c + d*x^2)^4, x, 4, (d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 - 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) + ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcCot[a*x] + (4/3)*c^3*d*x^3*ArcCot[a*x] + (6/5)*c^2*d^2*x^5*ArcCot[a*x] + (4/7)*c*d^3*x^7*ArcCot[a*x] + (1/9)*d^4*x^9*ArcCot[a*x] + ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(630*a^9)} -{ArcCot[a*x]*(c + d*x^2)^3, x, 4, (d*(35*a^4*c^2 - 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + ((21*a^2*c - 5*d)*d^2*x^4)/(140*a^3) + (d^3*x^6)/(42*a) + c^3*x*ArcCot[a*x] + c^2*d*x^3*ArcCot[a*x] + (3/5)*c*d^2*x^5*ArcCot[a*x] + (1/7)*d^3*x^7*ArcCot[a*x] + ((35*a^6*c^3 - 35*a^4*c^2*d + 21*a^2*c*d^2 - 5*d^3)*Log[1 + a^2*x^2])/(70*a^7)} -{ArcCot[a*x]*(c + d*x^2)^2, x, 5, ((10*a^2*c - 3*d)*d*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcCot[a*x] + (2/3)*c*d*x^3*ArcCot[a*x] + (1/5)*d^2*x^5*ArcCot[a*x] + ((15*a^4*c^2 - 10*a^2*c*d + 3*d^2)*Log[1 + a^2*x^2])/(30*a^5)} -{ArcCot[a*x]*(c + d*x^2)^1, x, 5, (d*x^2)/(6*a) + c*x*ArcCot[a*x] + (1/3)*d*x^3*ArcCot[a*x] + ((3*a^2*c - d)*Log[1 + a^2*x^2])/(6*a^3)} -{ArcCot[a*x]/(c + d*x^2)^1, x, 27, (I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - I/(a*x)])/(2*Sqrt[c]*Sqrt[d]) - (I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + I/(a*x)])/(2*Sqrt[c]*Sqrt[d]) - (I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*I*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) + (I*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[-((2*I*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x)))])/(2*Sqrt[c]*Sqrt[d]) - PolyLog[2, 1 - (2*I*Sqrt[c]*Sqrt[d]*(I - a*x))/((a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d]) + PolyLog[2, 1 + (2*I*Sqrt[c]*Sqrt[d]*(I + a*x))/((a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))]/(4*Sqrt[c]*Sqrt[d])} -{ArcCot[a*x]/(c + d*x^2)^2, x, 24, (x*ArcCot[a*x])/(2*c*(c + d*x^2)) + (ArcCot[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) - (I*a*Log[(Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (I*a*Log[-((Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (I*a*Log[-((Sqrt[d]*(1 - Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - (I*a*Log[(Sqrt[d]*(1 + Sqrt[-a^2]*x))/(I*Sqrt[-a^2]*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (a*Log[1 + a^2*x^2])/(4*c*(a^2*c - d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c - d)) - (I*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (I*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] - I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) - (I*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] - I*Sqrt[d])])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d]) + (I*a*PolyLog[2, (Sqrt[-a^2]*(Sqrt[c] + I*Sqrt[d]*x))/(Sqrt[-a^2]*Sqrt[c] + I*Sqrt[d])])/(8*Sqrt[-a^2]*c^(3/2)*Sqrt[d])} - - -{ArcCot[a*x]*(c + d*x^2)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x^2]*ArcCot[a*x], x]} -{ArcCot[a*x]/(c + d*x^2)^(1/2), x, 0, Unintegrable[ArcCot[a*x]/Sqrt[c + d*x^2], x]} -{ArcCot[a*x]/(c + d*x^2)^(3/2), x, 5, (x*ArcCot[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]]/(c*Sqrt[a^2*c - d])} -{ArcCot[a*x]/(c + d*x^2)^(5/2), x, 7, a/(3*c*(a^2*c - d)*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCot[a*x])/(3*c^2*Sqrt[c + d*x^2]) - ((3*a^2*c - 2*d)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(3*c^2*(a^2*c - d)^(3/2))} -{ArcCot[a*x]/(c + d*x^2)^(7/2), x, 8, a/(15*c*(a^2*c - d)*(c + d*x^2)^(3/2)) + (a*(7*a^2*c - 4*d))/(15*c^2*(a^2*c - d)^2*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCot[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCot[a*x])/(15*c^3*Sqrt[c + d*x^2]) - ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(15*c^3*(a^2*c - d)^(5/2))} -{ArcCot[a*x]/(c + d*x^2)^(9/2), x, 8, a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) + (a*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcCot[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcCot[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCot[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcCot[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))} - - -{ArcCot[x]*(a + a*x^2)^(1/2), x, 3, (1/2)*Sqrt[a + a*x^2] + (1/2)*x*Sqrt[a + a*x^2]*ArcCot[x] - (I*a*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] - (I*a*Sqrt[1 + x^2]*PolyLog[2, -((I*Sqrt[1 + I*x])/Sqrt[1 - I*x])])/(2*Sqrt[a + a*x^2]) + (I*a*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/(2*Sqrt[a + a*x^2])} -{ArcCot[x]/(a + a*x^2)^(1/2), x, 2, -((2*I*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2]) - (I*Sqrt[1 + x^2]*PolyLog[2, -((I*Sqrt[1 + I*x])/Sqrt[1 - I*x])])/Sqrt[a + a*x^2] + (I*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2]} -{ArcCot[x]/(a + a*x^2)^(3/2), x, 1, -(1/(a*Sqrt[a + a*x^2])) + (x*ArcCot[x])/(a*Sqrt[a + a*x^2])} -{ArcCot[x]/(a + a*x^2)^(5/2), x, 2, -(1/(9*a*(a + a*x^2)^(3/2))) - 2/(3*a^2*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(3*a*(a + a*x^2)^(3/2)) + (2*x*ArcCot[x])/(3*a^2*Sqrt[a + a*x^2])} -{ArcCot[x]/(a + a*x^2)^(7/2), x, 3, -(1/(25*a*(a + a*x^2)^(5/2))) - 4/(45*a^2*(a + a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(5*a*(a + a*x^2)^(5/2)) + (4*x*ArcCot[x])/(15*a^2*(a + a*x^2)^(3/2)) + (8*x*ArcCot[x])/(15*a^3*Sqrt[a + a*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a x] (c+d x^2)^p*) - - -{x^1*ArcCot[x]/(1 + x^2)^2, x, 3, -(x/(4*(1 + x^2))) - ArcCot[x]/(2*(1 + x^2)) - ArcTan[x]/4} - - -{x^1*ArcCot[x]/(1 + x^2)^3, x, 4, -(x/(16*(1 + x^2)^2)) - (3*x)/(32*(1 + x^2)) - ArcCot[x]/(4*(1 + x^2)^2) - (3*ArcTan[x])/32} - - -{x^0*ArcCot[x]/(1 + x^2)^2, x, 2, -(1/(4*(1 + x^2))) + (x*ArcCot[x])/(2*(1 + x^2)) - ArcCot[x]^2/4} - - -{ArcCot[x]^2/(1 + x^2)^2, x, 4, -(x/(4*(1 + x^2))) - ArcCot[x]/(2*(1 + x^2)) + (x*ArcCot[x]^2)/(2*(1 + x^2)) - ArcCot[x]^3/6 - ArcTan[x]/4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^5*ArcCot[a*x^2], x, 4, x^4/(12*a) + (1/6)*x^6*ArcCot[a*x^2] - Log[1 + a^2*x^4]/(12*a^3)} -{x^3*ArcCot[a*x^2], x, 4, x^2/(4*a) + (1/4)*x^4*ArcCot[a*x^2] - ArcTan[a*x^2]/(4*a^2)} -{x^1*ArcCot[a*x^2], x, 2, (1/2)*x^2*ArcCot[a*x^2] + Log[1 + a^2*x^4]/(4*a)} -{ArcCot[a*x^2]/x^1, x, 4, (-(1/4))*I*PolyLog[2, -(I/(a*x^2))] + (1/4)*I*PolyLog[2, I/(a*x^2)]} -{ArcCot[a*x^2]/x^3, x, 5, -(ArcCot[a*x^2]/(2*x^2)) - a*Log[x] + (1/4)*a*Log[1 + a^2*x^4]} -{ArcCot[a*x^2]/x^5, x, 4, a/(4*x^2) - ArcCot[a*x^2]/(4*x^4) + (1/4)*a^2*ArcTan[a*x^2]} - -{x^4*ArcCot[a*x^2], x, 11, (2*x^3)/(15*a) + (1/5)*x^5*ArcCot[a*x^2] + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2)) + Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2))} -{x^2*ArcCot[a*x^2], x, 11, (2*x)/(3*a) + (1/3)*x^3*ArcCot[a*x^2] + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) - ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(3*Sqrt[2]*a^(3/2)) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2)) - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(6*Sqrt[2]*a^(3/2))} -{x^0*ArcCot[a*x^2], x, 10, x*ArcCot[a*x^2] - ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(Sqrt[2]*Sqrt[a]) + ArcTan[1 + Sqrt[2]*Sqrt[a]*x]/(Sqrt[2]*Sqrt[a]) + Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(2*Sqrt[2]*Sqrt[a]) - Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2]/(2*Sqrt[2]*Sqrt[a])} -{ArcCot[a*x^2]/x^2, x, 10, -(ArcCot[a*x^2]/x) + (Sqrt[a]*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/Sqrt[2] - (Sqrt[a]*ArcTan[1 + Sqrt[2]*Sqrt[a]*x])/Sqrt[2] + (Sqrt[a]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(2*Sqrt[2]) - (Sqrt[a]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(2*Sqrt[2])} -{ArcCot[a*x^2]/x^4, x, 11, (2*a)/(3*x) - ArcCot[a*x^2]/(3*x^3) - (a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2]) - (a^(3/2)*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2])} - - -{x^2*ArcCot[Sqrt[x]], x, 6, Sqrt[x]/3 - x^(3/2)/9 + x^(5/2)/15 + (1/3)*x^3*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]/3} -{x^1*ArcCot[Sqrt[x]], x, 5, -(Sqrt[x]/2) + x^(3/2)/6 + (1/2)*x^2*ArcCot[Sqrt[x]] + ArcTan[Sqrt[x]]/2} -{x^0*ArcCot[Sqrt[x]], x, 4, Sqrt[x] + x*ArcCot[Sqrt[x]] - ArcTan[Sqrt[x]]} -{ArcCot[Sqrt[x]]/x^1, x, 4, (-I)*PolyLog[2, -(I/Sqrt[x])] + I*PolyLog[2, I/Sqrt[x]]} -{ArcCot[Sqrt[x]]/x^2, x, 4, 1/Sqrt[x] - ArcCot[Sqrt[x]]/x + ArcTan[Sqrt[x]]} -{ArcCot[Sqrt[x]]/x^3, x, 5, 1/(6*x^(3/2)) - 1/(2*Sqrt[x]) - ArcCot[Sqrt[x]]/(2*x^2) - ArcTan[Sqrt[x]]/2} - - -{x^(3/2)*ArcCot[Sqrt[x]], x, 3, -(x/5) + x^2/10 + (2/5)*x^(5/2)*ArcCot[Sqrt[x]] + (1/5)*Log[1 + x]} -{x^(1/2)*ArcCot[Sqrt[x]], x, 3, x/3 + (2/3)*x^(3/2)*ArcCot[Sqrt[x]] - (1/3)*Log[1 + x]} -{ArcCot[Sqrt[x]]/x^(1/2), x, 2, 2*Sqrt[x]*ArcCot[Sqrt[x]] + Log[1 + x]} -{ArcCot[Sqrt[x]]/x^(3/2), x, 4, -((2*ArcCot[Sqrt[x]])/Sqrt[x]) - Log[x] + Log[1 + x]} -{ArcCot[Sqrt[x]]/x^(5/2), x, 3, 1/(3*x) - (2*ArcCot[Sqrt[x]])/(3*x^(3/2)) + Log[x]/3 - (1/3)*Log[1 + x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{ArcCot[1/x], x, 3, x*ArcCot[1/x] - (1/2)*Log[1 + x^2]} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{ArcCot[a*x^n]/x, x, 4, -((I*PolyLog[2, -(I/(x^n*a))])/(2*n)) + (I*PolyLog[2, I/(x^n*a)])/(2*n)} - -{ArcCot[a*x^5]/x, x, 4, (-(1/10))*I*PolyLog[2, -(I/(a*x^5))] + (1/10)*I*PolyLog[2, I/(a*x^5)]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCot[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*ArcCot[a + b*x], x, 7, -(((1 - 6*a^2)*x)/(4*b^3)) - (a*(a + b*x)^2)/(2*b^4) + (a + b*x)^3/(12*b^4) + (1/4)*x^4*ArcCot[a + b*x] + ((1 - 6*a^2 + a^4)*ArcTan[a + b*x])/(4*b^4) + (a*(1 - a^2)*Log[1 + (a + b*x)^2])/(2*b^4)} -{x^2*ArcCot[a + b*x], x, 7, -((a*x)/b^2) + (a + b*x)^2/(6*b^3) + (1/3)*x^3*ArcCot[a + b*x] + (a*(3 - a^2)*ArcTan[a + b*x])/(3*b^3) - ((1 - 3*a^2)*Log[1 + (a + b*x)^2])/(6*b^3)} -{x^1*ArcCot[a + b*x], x, 7, x/(2*b) + (1/2)*x^2*ArcCot[a + b*x] - ((1 - a^2)*ArcTan[a + b*x])/(2*b^2) - (a*Log[1 + (a + b*x)^2])/(2*b^2)} -{x^0*ArcCot[a + b*x], x, 3, ((a + b*x)*ArcCot[a + b*x])/b + Log[1 + (a + b*x)^2]/(2*b)} -{ArcCot[a + b*x]/x^1, x, 5, (-ArcCot[a + b*x])*Log[2/(1 - I*(a + b*x))] + ArcCot[a + b*x]*Log[(2*b*x)/((I - a)*(1 - I*(a + b*x)))] - (1/2)*I*PolyLog[2, 1 - 2/(1 - I*(a + b*x))] + (1/2)*I*PolyLog[2, 1 - (2*b*x)/((I - a)*(1 - I*(a + b*x)))]} -{ArcCot[a + b*x]/x^2, x, 7, -(ArcCot[a + b*x]/x) + (a*b*ArcTan[a + b*x])/(1 + a^2) - (b*Log[x])/(1 + a^2) + (b*Log[1 + (a + b*x)^2])/(2*(1 + a^2))} -{ArcCot[a + b*x]/x^3, x, 8, b/(2*(1 + a^2)*x) - ArcCot[a + b*x]/(2*x^2) + ((1 - a^2)*b^2*ArcTan[a + b*x])/(2*(1 + a^2)^2) + (a*b^2*Log[x])/(1 + a^2)^2 - (a*b^2*Log[1 + (a + b*x)^2])/(2*(1 + a^2)^2)} -{ArcCot[a + b*x]/x^4, x, 8, b/(6*(1 + a^2)*x^2) - (2*a*b^2)/(3*(1 + a^2)^2*x) - ArcCot[a + b*x]/(3*x^3) - (a*(3 - a^2)*b^3*ArcTan[a + b*x])/(3*(1 + a^2)^3) + ((1 - 3*a^2)*b^3*Log[x])/(3*(1 + a^2)^3) - ((1 - 3*a^2)*b^3*Log[1 + (a + b*x)^2])/(6*(1 + a^2)^3)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x^n)^m ArcCot[a+b x]*) - - -(* {ArcCot[a + b*x]/(c + d*x^3), x, 35, -((I*Log[(d^(1/3)*(I - a - b*x))/(b*c^(1/3) + (I - a)*d^(1/3))]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3))) + (I*Log[-((I - a - b*x)/(a + b*x))]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + (I*Log[-((d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (I + a)*d^(1/3)))]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - (I*Log[(I + a + b*x)/(a + b*x)]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/6)*Log[-(((-1)^(1/3)*d^(1/3)*(I - a - b*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3)))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/6)*Log[-((I - a - b*x)/(a + b*x))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/6)*Log[((-1)^(1/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/6)*Log[(I + a + b*x)/(a + b*x)]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(5/6)*Log[((-1)^(2/3)*d^(1/3)*(I - a - b*x))/(b*c^(1/3) + (-1)^(2/3)*(I - a)*d^(1/3))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(5/6)*Log[-((I - a - b*x)/(a + b*x))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(5/6)*Log[-(((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (-1)^(2/3)*(I + a)*d^(1/3)))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(5/6)*Log[(I + a + b*x)/(a + b*x)]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - (I*PolyLog[2, (b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + (I*PolyLog[2, (b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/6)*PolyLog[2, (b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/6)*PolyLog[2, (b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(5/6)*PolyLog[2, (b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(I - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(5/6)*PolyLog[2, (b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/6)*(1 - I*a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))} *) -{ArcCot[a + b*x]/(c + d*x^2), x, 15, -((Log[(I + a + b*x)/(a + b*x)]*Log[-((b*(I*Sqrt[c] - Sqrt[d]*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x)))])/(4*Sqrt[c]*Sqrt[d])) + (Log[-((I - a - b*x)/(a + b*x))]*Log[(I*b*(Sqrt[c] + I*Sqrt[d]*x))/((b*Sqrt[c] - (1 + I*a)*Sqrt[d])*(a + b*x))])/(4*Sqrt[c]*Sqrt[d]) - (Log[-((I - a - b*x)/(a + b*x))]*Log[(b*(I*Sqrt[c] + Sqrt[d]*x))/((b*Sqrt[c] + (1 + I*a)*Sqrt[d])*(a + b*x))])/(4*Sqrt[c]*Sqrt[d]) + (Log[(I + a + b*x)/(a + b*x)]*Log[-((b*(I*Sqrt[c] + Sqrt[d]*x))/((b*Sqrt[c] + I*(I + a)*Sqrt[d])*(a + b*x)))])/(4*Sqrt[c]*Sqrt[d]) + PolyLog[2, -(((b*Sqrt[c] - I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] - (1 + I*a)*Sqrt[d])*(a + b*x)))]/(4*Sqrt[c]*Sqrt[d]) - PolyLog[2, -(((b*Sqrt[c] + I*a*Sqrt[d])*(I - a - b*x))/((b*Sqrt[c] + (1 + I*a)*Sqrt[d])*(a + b*x)))]/(4*Sqrt[c]*Sqrt[d]) - PolyLog[2, ((b*Sqrt[c] - I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + (1 - I*a)*Sqrt[d])*(a + b*x))]/(4*Sqrt[c]*Sqrt[d]) + PolyLog[2, ((b*Sqrt[c] + I*a*Sqrt[d])*(I + a + b*x))/((b*Sqrt[c] + I*(I + a)*Sqrt[d])*(a + b*x))]/(4*Sqrt[c]*Sqrt[d])} -{ArcCot[a + b*x]/(c + d*x^1), x, 5, -((ArcCot[a + b*x]*Log[2/(1 - I*(a + b*x))])/d) + (ArcCot[a + b*x]*Log[(2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/d - (I*PolyLog[2, 1 - 2/(1 - I*(a + b*x))])/(2*d) + (I*PolyLog[2, 1 - (2*b*(c + d*x))/((b*c + I*d - a*d)*(1 - I*(a + b*x)))])/(2*d)} -{ArcCot[a + b*x]/(c + d/x^1), x, 37, Log[I - a - b*x]/(2*b*c) + (I*(a + b*x)*Log[-((I - a - b*x)/(a + b*x))])/(2*b*c) + Log[I + a + b*x]/(2*b*c) - (I*(a + b*x)*Log[(I + a + b*x)/(a + b*x)])/(2*b*c) + (I*d*Log[(c*(I - a - b*x))/(I*c - a*c + b*d)]*Log[d + c*x])/(2*c^2) - (I*d*Log[-((I - a - b*x)/(a + b*x))]*Log[d + c*x])/(2*c^2) - (I*d*Log[(c*(I + a + b*x))/((I + a)*c - b*d)]*Log[d + c*x])/(2*c^2) + (I*d*Log[(I + a + b*x)/(a + b*x)]*Log[d + c*x])/(2*c^2) - (I*d*PolyLog[2, -((b*(d + c*x))/((I + a)*c - b*d))])/(2*c^2) + (I*d*PolyLog[2, (b*(d + c*x))/(I*c - a*c + b*d)])/(2*c^2), (I*x*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/(2*c) - (I*(I - a - b*x)*Log[-I + a + b*x])/(2*b*c) - (I*(I + a + b*x)*Log[I + a + b*x])/(2*b*c) - (I*x*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/(2*c) - (I*d*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x])*Log[d + c*x])/(2*c^2) + (I*d*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)])*Log[d + c*x])/(2*c^2) + (I*d*Log[I + a + b*x]*Log[-((b*(d + c*x))/((I + a)*c - b*d))])/(2*c^2) - (I*d*Log[-I + a + b*x]*Log[(b*(d + c*x))/((I - a)*c + b*d)])/(2*c^2) - (I*d*PolyLog[2, (c*(I - a - b*x))/((I - a)*c + b*d)])/(2*c^2) + (I*d*PolyLog[2, (c*(I + a + b*x))/((I + a)*c - b*d)])/(2*c^2)} -{ArcCot[a + b*x]/(c + d/x^2), x, 57, Log[I - a - b*x]/(2*b*c) + (I*(a + b*x)*Log[-((I - a - b*x)/(a + b*x))])/(2*b*c) - (I*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[-((I - a - b*x)/(a + b*x))])/(2*c^(3/2)) + Log[I + a + b*x]/(2*b*c) - (I*(a + b*x)*Log[(I + a + b*x)/(a + b*x)])/(2*b*c) + (I*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*Log[(I + a + b*x)/(a + b*x)])/(2*c^(3/2)) - (Sqrt[d]*Log[(Sqrt[c]*(I - a - b*x))/((I - a)*Sqrt[c] + I*b*Sqrt[d])]*Log[1 - (I*Sqrt[c]*x)/Sqrt[d]])/(4*c^(3/2)) + (Sqrt[d]*Log[(Sqrt[c]*(I + a + b*x))/((I + a)*Sqrt[c] - I*b*Sqrt[d])]*Log[1 - (I*Sqrt[c]*x)/Sqrt[d]])/(4*c^(3/2)) + (Sqrt[d]*Log[(Sqrt[c]*(I - a - b*x))/((I - a)*Sqrt[c] - I*b*Sqrt[d])]*Log[1 + (I*Sqrt[c]*x)/Sqrt[d]])/(4*c^(3/2)) - (Sqrt[d]*Log[(Sqrt[c]*(I + a + b*x))/((I + a)*Sqrt[c] + I*b*Sqrt[d])]*Log[1 + (I*Sqrt[c]*x)/Sqrt[d]])/(4*c^(3/2)) - (Sqrt[d]*PolyLog[2, (b*(Sqrt[d] - I*Sqrt[c]*x))/((1 + I*a)*Sqrt[c] + b*Sqrt[d])])/(4*c^(3/2)) + (Sqrt[d]*PolyLog[2, (b*(Sqrt[d] - I*Sqrt[c]*x))/(I*(I + a)*Sqrt[c] + b*Sqrt[d])])/(4*c^(3/2)) + (Sqrt[d]*PolyLog[2, -((b*(Sqrt[d] + I*Sqrt[c]*x))/((1 + I*a)*Sqrt[c] - b*Sqrt[d]))])/(4*c^(3/2)) - (Sqrt[d]*PolyLog[2, (b*(Sqrt[d] + I*Sqrt[c]*x))/((1 - I*a)*Sqrt[c] + b*Sqrt[d])])/(4*c^(3/2)), (I*x*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/(2*c) - (I*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-((I - a - b*x)/(a + b*x))] + Log[a + b*x] - Log[-I + a + b*x]))/(2*c^(3/2)) - (I*(I - a - b*x)*Log[-I + a + b*x])/(2*b*c) - (I*(I + a + b*x)*Log[I + a + b*x])/(2*b*c) - (I*x*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/(2*c) + (I*Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[I + a + b*x] + Log[(I + a + b*x)/(a + b*x)]))/(2*c^(3/2)) - (I*Sqrt[d]*Log[-I + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((I - a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) + (I*Sqrt[d]*Log[I + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) - (I*Sqrt[d]*Log[I + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((I + a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) + (I*Sqrt[d]*Log[-I + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) - (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) + (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I - a - b*x))/((I - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) - (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) + (I*Sqrt[d]*PolyLog[2, (Sqrt[-c]*(I + a + b*x))/((I + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))} - - -(* {ArcCot[a + b*x]/(a + b*x^(3/2)), x, 101, -((I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3))) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[I - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] - Sqrt[b]*Sqrt[x])/(Sqrt[I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[I - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[I - a] + Sqrt[b]*Sqrt[x])/(Sqrt[I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(1/3)*b^(2/3)) + (I*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*Log[(-(-1)^(2/3))*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[I - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - (I*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + (I*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[I - a] - a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - (I*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (I*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[I - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[I - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(5/6)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(5/6)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[I - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[I - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3))} *) -{ArcCot[a + b*x]/(c + d*Sqrt[x]), x, 55, -((2*I*Sqrt[I + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*d)) + (2*I*Sqrt[I - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*d) - (I*c*Log[(d*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 + (I*c*Log[(d*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)]*Log[c + d*Sqrt[x]])/d^2 - (I*c*Log[-((d*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 + (I*c*Log[-((d*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d))]*Log[c + d*Sqrt[x]])/d^2 + (I*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/d - (I*c*Log[c + d*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/d^2 - (I*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/d + (I*c*Log[c + d*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-I - a]*d)])/d^2 - (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-I - a]*d)])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[I - a]*d)])/d^2 + (I*c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[I - a]*d)])/d^2} -{ArcCot[a + b*x]/(c + d/Sqrt[x]), x, 65, (2*I*Sqrt[I + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[I + a]])/(Sqrt[b]*c^2) - (2*I*Sqrt[I - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[I - a]])/(Sqrt[b]*c^2) + (I*d^2*Log[(c*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (I*d^2*Log[(c*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (I*d^2*Log[(c*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + ((1 + I*a)*Log[I - a - b*x])/(2*b*c) - (I*d*Sqrt[x]*Log[-((I - a - b*x)/(a + b*x))])/c^2 + (I*x*Log[-((I - a - b*x)/(a + b*x))])/(2*c) + (I*d^2*Log[d + c*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/c^3 + ((1 - I*a)*Log[I + a + b*x])/(2*b*c) + (I*d*Sqrt[x]*Log[(I + a + b*x)/(a + b*x)])/c^2 - (I*x*Log[(I + a + b*x)/(a + b*x)])/(2*c) - (I*d^2*Log[d + c*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/c^3 + (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c - Sqrt[b]*d))])/c^3 - (I*d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c - Sqrt[b]*d))])/c^3 + (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-I - a]*c + Sqrt[b]*d)])/c^3 - (I*d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[I - a]*c + Sqrt[b]*d)])/c^3} -(* {ArcCot[a + b*x]/(a + b/x^(3/2)), x, 109, (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[I - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - (I*(I - a - b*x)*Log[-((I - a - b*x)/(a + b*x))])/(2*a*b) + (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((I - a - b*x)/(a + b*x))])/(3*a^(5/3)) + Log[a + b*x]/(a*b) - (I*(I + a + b*x)*Log[(I + a + b*x)/(a + b*x)])/(2*a*b) - (I*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*Log[(-(-1)^(2/3))*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(I + a + b*x)/(a + b*x)])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + (I*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) - (I*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) + (I*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - (I*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) - ((-1)^(5/6)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(5/6)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/6)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-I - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/6)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[I - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B x+C x^2)^m ArcCot[a+b x]^n*) - - -{ArcCot[d + e*x]/(a + b*x + c*x^2), x, 12, (ArcCot[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] - (ArcCot[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/Sqrt[b^2 - 4*a*c] + (I*PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*I*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 - I*(d + e*x)))])/(2*Sqrt[b^2 - 4*a*c]) - (I*PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(I - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 - I*(d + e*x)))])/(2*Sqrt[b^2 - 4*a*c])} - - -{ArcCot[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 2, -((2*I*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b) - (I*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/b + (I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/b} -{ArcCot[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2], x, 3, -((2*I*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])) - (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(b*Sqrt[c + c*(a + b*x)^2]) + (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2])} - - -{ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3), x, 1, Unintegrable[ArcCot[a + b*x]/(1 + (a + b*x)^2)^(1/3), x]} -{ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3), x, 1, Unintegrable[ArcCot[a + b*x]/(c + c*(a + b*x)^2)^(1/3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (A+B x+C x^2)^p ArcCot[a+b x]^n*) - - -{(a + b*x)^2*ArcCot[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 4, Sqrt[1 + (a + b*x)^2]/(2*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x])/(2*b) + (I*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/b + (I*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(2*b) - (I*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(2*b)} -{(a + b*x)^2*ArcCot[a + b*x]/Sqrt[(1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2], x, 5, Sqrt[c + c*(a + b*x)^2]/(2*b*c) + ((a + b*x)*Sqrt[c + c*(a + b*x)^2]*ArcCot[a + b*x])/(2*b*c) + (I*Sqrt[1 + (a + b*x)^2]*ArcCot[a + b*x]*ArcTan[Sqrt[1 + I*(a + b*x)]/Sqrt[1 - I*(a + b*x)]])/(b*Sqrt[c + c*(a + b*x)^2]) + (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, -((I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)])])/(2*b*Sqrt[c + c*(a + b*x)^2]) - (I*Sqrt[1 + (a + b*x)^2]*PolyLog[2, (I*Sqrt[1 + I*(a + b*x)])/Sqrt[1 - I*(a + b*x)]])/(2*b*Sqrt[c + c*(a + b*x)^2])} - - -{(a + b*x)^2*ArcCot[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(1/3), x, 1, Unintegrable[((a + b*x)^2*ArcCot[a + b*x])/(1 + (a + b*x)^2)^(1/3), x]} -{(a + b*x)^2*ArcCot[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3), x, 1, Unintegrable[((a + b*x)^2*ArcCot[a + b*x])/(c + c*(a + b*x)^2)^(1/3), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCot[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCot[c+d x])^p when d e-c f=0*) - - -{(a + b*x)^2*ArcCot[a + b*x], x, 5, (a + b*x)^2/(6*b) + ((a + b*x)^3*ArcCot[a + b*x])/(3*b) - Log[1 + (a + b*x)^2]/(6*b)} -{(a + b*x)^1*ArcCot[a + b*x], x, 4, x/2 + ((a + b*x)^2*ArcCot[a + b*x])/(2*b) - ArcTan[a + b*x]/(2*b)} -{ArcCot[a + b*x]/(a + b*x)^1, x, 4, -((I*PolyLog[2, -(I/(a + b*x))])/(2*b)) + (I*PolyLog[2, I/(a + b*x)])/(2*b)} -{ArcCot[a + b*x]/(a + b*x)^2, x, 6, -(ArcCot[a + b*x]/(b*(a + b*x))) - Log[a + b*x]/b + Log[1 + (a + b*x)^2]/(2*b)} - - -{ArcCot[1 + x]/(2 + 2*x), x, 5, (-(1/4))*I*PolyLog[2, -(I/(1 + x))] + (1/4)*I*PolyLog[2, I/(1 + x)]} - - -{ArcCot[a + b*x]/((a*d)/b + d*x), x, 5, -((I*PolyLog[2, -(I/(a + b*x))])/(2*d)) + (I*PolyLog[2, I/(a + b*x)])/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCot[c+d x])^(p/2) when d e-c f=0*) - - -{(a + b*x)^2*ArcCot[a + b*x]^(1/2), x, 0, Unintegrable[(a + b*x)^2*Sqrt[ArcCot[a + b*x]], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCot[c+d x])^p*) - - -{(e + f*x)^3*(a + b*ArcCot[c + d*x]), x, 7, (b*f*(6*d^2*e^2 - 12*c*d*e*f - (1 - 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcCot[c + d*x]))/(4*f) + (b*(d^4*e^4 - 4*c*d^3*e^3*f - 6*(1 - c^2)*d^2*e^2*f^2 + 4*c*(3 - c^2)*d*e*f^3 + (1 - 6*c^2 + c^4)*f^4)*ArcTan[c + d*x])/(4*d^4*f) + (b*(d*e - c*f)*(d*e + f - c*f)*(d*e - (1 + c)*f)*Log[1 + (c + d*x)^2])/(2*d^4)} -{(e + f*x)^2*(a + b*ArcCot[c + d*x]), x, 7, (b*f*(d*e - c*f)*x)/d^2 + (b*f^2*(c + d*x)^2)/(6*d^3) + ((e + f*x)^3*(a + b*ArcCot[c + d*x]))/(3*f) + (b*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*ArcTan[c + d*x])/(3*d^3*f) + (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*Log[1 + (c + d*x)^2])/(6*d^3)} -{(e + f*x)*(a + b*ArcCot[c + d*x]), x, 7, (b*f*x)/(2*d) + ((e + f*x)^2*(a + b*ArcCot[c + d*x]))/(2*f) + (b*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*d^2*f) + (b*(d*e - c*f)*Log[1 + (c + d*x)^2])/(2*d^2)} -{a + b*ArcCot[c + d*x], x, 4, a*x + (b*(c + d*x)*ArcCot[c + d*x])/d + (b*Log[1 + (c + d*x)^2])/(2*d)} -{(a + b*ArcCot[c + d*x])/(e + f*x), x, 5, -(((a + b*ArcCot[c + d*x])*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (I*b*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (I*b*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f)} -{(a + b*ArcCot[c + d*x])/(e + f*x)^2, x, 8, -((a + b*ArcCot[c + d*x])/(f*(e + f*x))) - (b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (b*d*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b*d*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))} -{(a + b*ArcCot[c + d*x])/(e + f*x)^3, x, 9, (b*d)/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(e + f*x)) - (a + b*ArcCot[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*(d*e + f - c*f)*(d*e - (1 + c)*f)*ArcTan[c + d*x])/(2*f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2 + (b*d^2*(d*e - c*f)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^2)} - - -{(e + f*x)^2*(a + b*ArcCot[c + d*x])^2, x, 16, (b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[c + d*x]))/(3*d^3) + (I*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^2)/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcCot[c + d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTan[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 + (c + d*x)^2])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(3*d^3)} -{(e + f*x)*(a + b*ArcCot[c + d*x])^2, x, 13, (a*b*f*x)/d + (b^2*f*(c + d*x)*ArcCot[c + d*x])/d^2 + (I*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcCot[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*f) - (2*b*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 + (b^2*f*Log[1 + (c + d*x)^2])/(2*d^2) + (I*b^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2} -{(a + b*ArcCot[c + d*x])^2, x, 6, (I*(a + b*ArcCot[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^2)/d - (2*b*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d + (I*b^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d} -{(a + b*ArcCot[c + d*x])^2/(e + f*x), x, 2, -(((a + b*ArcCot[c + d*x])^2*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (I*b*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/f + (I*b*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (b^2*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f)} -{(a + b*ArcCot[c + d*x])^2/(e + f*x)^2, x, 25, (I*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(d*e - c*f)*ArcCot[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*x])^2/(f*(e + f*x)) - (2*a*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (2*a*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) + (2*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (2*b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (2*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (a*b*d*Log[1 + (c + d*x)^2])/(f^2 + (d*e - c*f)^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (I*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)} - - -{(e + f*x)^2*(a + b*ArcCot[c + d*x])^3, x, 21, (a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + (3*I*b*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + (I*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^3)/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2)*f^2)*(a + b*ArcCot[c + d*x])^3)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^3 + (b^3*f^2*Log[1 + (c + d*x)^2])/(2*d^3) + (3*I*b^3*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 + (I*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^3 - (b^3*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^3)} -{(e + f*x)*(a + b*ArcCot[c + d*x])^3, x, 15, (3*I*b*f*(a + b*ArcCot[c + d*x])^2)/(2*d^2) + (3*b*f*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/(2*d^2) + (I*(d*e - c*f)*(a + b*ArcCot[c + d*x])^3)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcCot[c + d*x])^3)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCot[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 - (3*b*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^2 + (3*I*b^3*f*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(2*d^2) + (3*I*b^2*(d*e - c*f)*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^2 - (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^2)} -{(a + b*ArcCot[c + d*x])^3, x, 6, (I*(a + b*ArcCot[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcCot[c + d*x])^3)/d - (3*b*(a + b*ArcCot[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d + (3*I*b^2*(a + b*ArcCot[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (3*b^3*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d)} -{(a + b*ArcCot[c + d*x])^3/(e + f*x), x, 2, -(((a + b*ArcCot[c + d*x])^3*Log[2/(1 - I*(c + d*x))])/f) + ((a + b*ArcCot[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/f - (3*I*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*I*b*(a + b*ArcCot[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) - (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*f) + (3*b^2*(a + b*ArcCot[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*f) + (3*I*b^3*PolyLog[4, 1 - 2/(1 - I*(c + d*x))])/(4*f) - (3*I*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(4*f)} -{(a + b*ArcCot[c + d*x])^3/(e + f*x)^2, x, 35, (3*I*a*b^2*d*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a*b^2*d*(d*e - c*f)*ArcCot[c + d*x]^2)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (I*b^3*d*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^3*d*(d*e - c*f)*ArcCot[c + d*x]^3)/(f*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (a + b*ArcCot[c + d*x])^3/(f*(e + f*x)) - (3*a^2*b*d*(d*e - c*f)*ArcTan[c + d*x])/(f*(f^2 + (d*e - c*f)^2)) - (3*a^2*b*d*Log[e + f*x])/(f^2 + (d*e - c*f)^2) + (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (6*a*b^2*d*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*b^3*d*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*a^2*b*d*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) + (3*I*a*b^2*d*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*I*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (3*I*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*a*b^2*d*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*I*b^3*d*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCot[c+d x])^p with m symbolic*) - - -{(e + f*x)^m*(a + b*ArcCot[c + d*x]), x, 6, ((e + f*x)^(1 + m)*(a + b*ArcCot[c + d*x]))/(f*(1 + m)) + (I*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + I*f - c*f)])/(2*f*(d*e + (I - c)*f)*(1 + m)*(2 + m)) - (I*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)])/(2*f*(d*e - (I + c)*f)*(1 + m)*(2 + m))} -{(e + f*x)^m*(a + b*ArcCot[c + d*x])^2, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcCot[c + d*x])^2, x]} -{(e + f*x)^m*(a + b*ArcCot[c + d*x])^3, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcCot[c + d*x])^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCot[a+b x^n]*) - - -{x^3*ArcCot[a + b*x^4], x, 4, ((a + b*x^4)*ArcCot[a + b*x^4])/(4*b) + Log[1 + (a + b*x^4)^2]/(8*b)} - - -{x^(n-1)*ArcCot[a + b*x^n], x, 4, ((a + b*x^n)*ArcCot[a + b*x^n])/(b*n) + Log[1 + (a + b*x^n)^2]/(2*b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b ArcCot[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 9, -((2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcCoth[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c) + (3*I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (3*I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) + (3*b^2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (3*b^2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*I*b^3*PolyLog[4, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(4*c) + (3*I*b^3*PolyLog[4, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(4*c)} -{(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((2*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcCoth[1 - 2/(1 + (I*Sqrt[1 - c*x])/Sqrt[1 + c*x])])/c) + (I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (I*b*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/c + (b^2*PolyLog[3, 1 - (2*I)/(I + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (b^2*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(I + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c)} -{(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 4, -((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/c) + (I*b*PolyLog[2, -((I*Sqrt[1 + c*x])/Sqrt[1 - c*x])])/(2*c) - (I*b*PolyLog[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]])/(2*c)} -{1/((a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Trig[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Tan[a+b x]]*) - - -{ArcCot[Tan[a + b*x]], x, 2, -(ArcCot[Tan[a + b*x]]^2/(2*b))} - - -{x^2*ArcCot[c + d*Tan[a + b*x]], x, 11, (1/3)*x^3*ArcCot[c + d*Tan[a + b*x]] - (1/6)*I*x^3*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] + (1/6)*I*x^3*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] - (x^2*PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b) + (x^2*PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b) - (I*x*PolyLog[3, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b^2) + (I*x*PolyLog[3, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b^2) + PolyLog[4, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))]/(8*b^3) - PolyLog[4, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))]/(8*b^3)} -{x^1*ArcCot[c + d*Tan[a + b*x]], x, 9, (1/2)*x^2*ArcCot[c + d*Tan[a + b*x]] - (1/4)*I*x^2*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] + (1/4)*I*x^2*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] - (x*PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(4*b) + (x*PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(4*b) - (I*PolyLog[3, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))])/(8*b^2) + (I*PolyLog[3, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))])/(8*b^2)} -{x^0*ArcCot[c + d*Tan[a + b*x]], x, 7, x*ArcCot[c + d*Tan[a + b*x]] - (1/2)*I*x*Log[1 + ((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d)] + (1/2)*I*x*Log[1 + ((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d))] - PolyLog[2, -(((1 + I*c + d)*E^(2*I*a + 2*I*b*x))/(1 + I*c - d))]/(4*b) + PolyLog[2, -(((c + I*(1 - d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 + d)))]/(4*b)} -{ArcCot[c + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]], x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCot[c + (1 + I*c)*Tan[a + b*x]] + (1/6)*I*x^3*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + (x^2*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*x*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) - PolyLog[4, I*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcCot[c + (1 + I*c)*Tan[a + b*x]], x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCot[c + (1 + I*c)*Tan[a + b*x]] + (1/4)*I*x^2*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + (x*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcCot[c + (1 + I*c)*Tan[a + b*x]], x, 5, (b*x^2)/2 + x*ArcCot[c + (1 + I*c)*Tan[a + b*x]] + (1/2)*I*x*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] + PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + (1 + I*c)*Tan[a + b*x]]/x, x]} - - -{x^2*ArcCot[c - (1 - I*c)*Tan[a + b*x]], x, 7, -((b*x^4)/12) + (1/3)*x^3*ArcCot[c - (1 - I*c)*Tan[a + b*x]] - (1/6)*I*x^3*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - (x^2*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*x*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) + PolyLog[4, (-I)*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcCot[c - (1 - I*c)*Tan[a + b*x]], x, 6, -((b*x^3)/6) + (1/2)*x^2*ArcCot[c - (1 - I*c)*Tan[a + b*x]] - (1/4)*I*x^2*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - (x*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcCot[c - (1 - I*c)*Tan[a + b*x]], x, 5, -((b*x^2)/2) + x*ArcCot[c - (1 - I*c)*Tan[a + b*x]] - (1/2)*I*x*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] - PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c - (1 - I*c)*Tan[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Cot[a+b x]]*) - - -{ArcCot[Cot[a + b*x]], x, 2, ArcCot[Cot[a + b*x]]^2/(2*b)} - - -{x^2*ArcCot[c + d*Cot[a + b*x]], x, 11, (1/3)*x^3*ArcCot[c + d*Cot[a + b*x]] - (1/6)*I*x^3*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] + (1/6)*I*x^3*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] - (x^2*PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b) + (x^2*PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b) - (I*x*PolyLog[3, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b^2) + (I*x*PolyLog[3, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b^2) + PolyLog[4, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)]/(8*b^3) - PolyLog[4, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))]/(8*b^3)} -{x^1*ArcCot[c + d*Cot[a + b*x]], x, 9, (1/2)*x^2*ArcCot[c + d*Cot[a + b*x]] - (1/4)*I*x^2*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] + (1/4)*I*x^2*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] - (x*PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(4*b) + (x*PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(4*b) - (I*PolyLog[3, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)])/(8*b^2) + (I*PolyLog[3, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))])/(8*b^2)} -{x^0*ArcCot[c + d*Cot[a + b*x]], x, 7, x*ArcCot[c + d*Cot[a + b*x]] - (1/2)*I*x*Log[1 - ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)] + (1/2)*I*x*Log[1 - ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))] - PolyLog[2, ((1 + I*c - d)*E^(2*I*a + 2*I*b*x))/(1 + I*c + d)]/(4*b) + PolyLog[2, ((c + I*(1 + d))*E^(2*I*a + 2*I*b*x))/(c + I*(1 - d))]/(4*b)} -{ArcCot[c + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]], x, 7, -((b*x^4)/12) + (1/3)*x^3*ArcCot[c + (1 - I*c)*Cot[a + b*x]] - (1/6)*I*x^3*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - (x^2*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*x*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) + PolyLog[4, I*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcCot[c + (1 - I*c)*Cot[a + b*x]], x, 6, -((b*x^3)/6) + (1/2)*x^2*ArcCot[c + (1 - I*c)*Cot[a + b*x]] - (1/4)*I*x^2*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - (x*PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)])/(4*b) - (I*PolyLog[3, I*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcCot[c + (1 - I*c)*Cot[a + b*x]], x, 5, -((b*x^2)/2) + x*ArcCot[c + (1 - I*c)*Cot[a + b*x]] - (1/2)*I*x*Log[1 - I*c*E^(2*I*a + 2*I*b*x)] - PolyLog[2, I*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + (1 - I*c)*Cot[a + b*x]]/x, x]} - - -{x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]], x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + (1/6)*I*x^3*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + (x^2*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*x*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b^2) - PolyLog[4, (-I)*c*E^(2*I*a + 2*I*b*x)]/(8*b^3)} -{x^1*ArcCot[c - (1 + I*c)*Cot[a + b*x]], x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + (1/4)*I*x^2*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + (x*PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)])/(4*b) + (I*PolyLog[3, (-I)*c*E^(2*I*a + 2*I*b*x)])/(8*b^2)} -{x^0*ArcCot[c - (1 + I*c)*Cot[a + b*x]], x, 5, (b*x^2)/2 + x*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + (1/2)*I*x*Log[1 + I*c*E^(2*I*a + 2*I*b*x)] + PolyLog[2, (-I)*c*E^(2*I*a + 2*I*b*x)]/(4*b)} -{ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Hyper[a+b x]]*) - - -(* ::Subsection:: *) -(*Integrands of the form x^m ArcCot[c+d Sinh[a+b x]]*) - - -(* ::Subsection:: *) -(*Integrands of the form x^m ArcCot[c+d Cosh[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Tanh[a+b x]]*) - - -{(e + f*x)^3*ArcCot[Tanh[a + b*x]], x, 12, ((e + f*x)^4*ArcCot[Tanh[a + b*x]])/(4*f) + ((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (3*I*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) - (3*I*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2) - (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) + (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3) + (3*I*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/(16*b^4) - (3*I*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/(16*b^4)} -{(e + f*x)^2*ArcCot[Tanh[a + b*x]], x, 10, ((e + f*x)^3*ArcCot[Tanh[a + b*x]])/(3*f) + ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (I*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(4*b^2) - (I*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/(4*b^2) - (I*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) + (I*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3)} -{(e + f*x)^1*ArcCot[Tanh[a + b*x]], x, 8, ((e + f*x)^2*ArcCot[Tanh[a + b*x]])/(2*f) + ((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) + (I*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) - (I*f*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2)} -{(e + f*x)^0*ArcCot[Tanh[a + b*x]], x, 6, x*ArcCot[Tanh[a + b*x]] + x*ArcTan[E^(2*a + 2*b*x)] - (I*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[Tanh[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcCot[Tanh[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcCot[c + d*Tanh[a + b*x]], x, 11, (1/3)*x^3*ArcCot[c + d*Tanh[a + b*x]] - (1/6)*I*x^3*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/6)*I*x^3*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*x^2*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) + (I*x^2*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b) + (I*x*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b^2) - (I*x*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b^2) - (I*PolyLog[4, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(8*b^3) + (I*PolyLog[4, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(8*b^3)} -{x^1*ArcCot[c + d*Tanh[a + b*x]], x, 9, (1/2)*x^2*ArcCot[c + d*Tanh[a + b*x]] - (1/4)*I*x^2*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/4)*I*x^2*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*x*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) + (I*x*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b) + (I*PolyLog[3, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(8*b^2) - (I*PolyLog[3, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(8*b^2)} -{x^0*ArcCot[c + d*Tanh[a + b*x]], x, 7, x*ArcCot[c + d*Tanh[a + b*x]] - (1/2)*I*x*Log[1 + ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/2)*I*x*Log[1 + ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*PolyLog[2, -(((I - c - d)*E^(2*a + 2*b*x))/(I - c + d))])/(4*b) + (I*PolyLog[2, -(((I + c + d)*E^(2*a + 2*b*x))/(I + c - d))])/(4*b)} -{ArcCot[c + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + d*Tanh[a + b*x]]/x, x]} - - -{x^2*ArcCot[c + (I + c)*Tanh[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCot[c + (I + c)*Tanh[a + b*x]] - (1/6)*I*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) + (I*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(4*b^2) - (I*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcCot[c + (I + c)*Tanh[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCot[c + (I + c)*Tanh[a + b*x]] - (1/4)*I*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcCot[c + (I + c)*Tanh[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCot[c + (I + c)*Tanh[a + b*x]] - (1/2)*I*x*Log[1 + I*c*E^(2*a + 2*b*x)] - (I*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[c + (I + c)*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + (I + c)*Tanh[a + b*x]]/x, x]} - - -{x^2*ArcCot[c - (I - c)*Tanh[a + b*x]], x, 7, (-(1/12))*I*b*x^4 + (1/3)*x^3*ArcCot[c - (I - c)*Tanh[a + b*x]] + (1/6)*I*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) - (I*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(4*b^2) + (I*PolyLog[4, I*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcCot[c - (I - c)*Tanh[a + b*x]], x, 6, (-(1/6))*I*b*x^3 + (1/2)*x^2*ArcCot[c - (I - c)*Tanh[a + b*x]] + (1/4)*I*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcCot[c - (I - c)*Tanh[a + b*x]], x, 5, (-(1/2))*I*b*x^2 + x*ArcCot[c - (I - c)*Tanh[a + b*x]] + (1/2)*I*x*Log[1 - I*c*E^(2*a + 2*b*x)] + (I*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[c - (I - c)*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c - (I - c)*Tanh[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCot[c+d Coth[a+b x]]*) - - -{(e + f*x)^3*ArcCot[Coth[a + b*x]], x, 12, ((e + f*x)^4*ArcCot[Coth[a + b*x]])/(4*f) - ((e + f*x)^4*ArcTan[E^(2*a + 2*b*x)])/(4*f) + (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)^3*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (3*I*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) + (3*I*f*(e + f*x)^2*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3) - (3*I*f^3*PolyLog[5, (-I)*E^(2*a + 2*b*x)])/(16*b^4) + (3*I*f^3*PolyLog[5, I*E^(2*a + 2*b*x)])/(16*b^4)} -{(e + f*x)^2*ArcCot[Coth[a + b*x]], x, 10, ((e + f*x)^3*ArcCot[Coth[a + b*x]])/(3*f) - ((e + f*x)^3*ArcTan[E^(2*a + 2*b*x)])/(3*f) + (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)^2*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (I*f*(e + f*x)*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(4*b^2) + (I*f*(e + f*x)*PolyLog[3, I*E^(2*a + 2*b*x)])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*a + 2*b*x)])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*a + 2*b*x)])/(8*b^3)} -{(e + f*x)^1*ArcCot[Coth[a + b*x]], x, 8, ((e + f*x)^2*ArcCot[Coth[a + b*x]])/(2*f) - ((e + f*x)^2*ArcTan[E^(2*a + 2*b*x)])/(2*f) + (I*(e + f*x)*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*(e + f*x)*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b) - (I*f*PolyLog[3, (-I)*E^(2*a + 2*b*x)])/(8*b^2) + (I*f*PolyLog[3, I*E^(2*a + 2*b*x)])/(8*b^2)} -{(e + f*x)^0*ArcCot[Coth[a + b*x]], x, 6, x*ArcCot[Coth[a + b*x]] - x*ArcTan[E^(2*a + 2*b*x)] + (I*PolyLog[2, (-I)*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[2, I*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[Coth[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcCot[Coth[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcCot[c + d*Coth[a + b*x]], x, 11, (1/3)*x^3*ArcCot[c + d*Coth[a + b*x]] - (1/6)*I*x^3*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/6)*I*x^3*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*x^2*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) + (I*x^2*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b) + (I*x*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b^2) - (I*x*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b^2) - (I*PolyLog[4, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(8*b^3) + (I*PolyLog[4, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(8*b^3)} -{x^1*ArcCot[c + d*Coth[a + b*x]], x, 9, (1/2)*x^2*ArcCot[c + d*Coth[a + b*x]] - (1/4)*I*x^2*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/4)*I*x^2*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*x*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) + (I*x*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b) + (I*PolyLog[3, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(8*b^2) - (I*PolyLog[3, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(8*b^2)} -{x^0*ArcCot[c + d*Coth[a + b*x]], x, 7, x*ArcCot[c + d*Coth[a + b*x]] - (1/2)*I*x*Log[1 - ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)] + (1/2)*I*x*Log[1 - ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)] - (I*PolyLog[2, ((I - c - d)*E^(2*a + 2*b*x))/(I - c + d)])/(4*b) + (I*PolyLog[2, ((I + c + d)*E^(2*a + 2*b*x))/(I + c - d)])/(4*b)} -{ArcCot[c + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + d*Coth[a + b*x]]/x, x]} - - -{x^2*ArcCot[c + (I + c)*Coth[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCot[c + (I + c)*Coth[a + b*x]] - (1/6)*I*x^3*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*x^2*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) + (I*x*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(4*b^2) - (I*PolyLog[4, I*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcCot[c + (I + c)*Coth[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCot[c + (I + c)*Coth[a + b*x]] - (1/4)*I*x^2*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*x*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b) + (I*PolyLog[3, I*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcCot[c + (I + c)*Coth[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCot[c + (I + c)*Coth[a + b*x]] - (1/2)*I*x*Log[1 - I*c*E^(2*a + 2*b*x)] - (I*PolyLog[2, I*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[c + (I + c)*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c + (I + c)*Coth[a + b*x]]/x, x]} - - -{x^2*ArcCot[c - (I - c)*Coth[a + b*x]], x, 7, (-(1/12))*I*b*x^4 + (1/3)*x^3*ArcCot[c - (I - c)*Coth[a + b*x]] + (1/6)*I*x^3*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*x^2*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) - (I*x*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(4*b^2) + (I*PolyLog[4, (-I)*c*E^(2*a + 2*b*x)])/(8*b^3)} -{x^1*ArcCot[c - (I - c)*Coth[a + b*x]], x, 6, (-(1/6))*I*b*x^3 + (1/2)*x^2*ArcCot[c - (I - c)*Coth[a + b*x]] + (1/4)*I*x^2*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*x*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b) - (I*PolyLog[3, (-I)*c*E^(2*a + 2*b*x)])/(8*b^2)} -{x^0*ArcCot[c - (I - c)*Coth[a + b*x]], x, 5, (-(1/2))*I*b*x^2 + x*ArcCot[c - (I - c)*Coth[a + b*x]] + (1/2)*I*x*Log[1 + I*c*E^(2*a + 2*b*x)] + (I*PolyLog[2, (-I)*c*E^(2*a + 2*b*x)])/(4*b)} -{ArcCot[c - (I - c)*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCot[c - (I - c)*Coth[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e Log[f x^m]) (a+b ArcCot[c x^n])*) - - -{((a + b*ArcCot[c*x^n])*(d + e*Log[f*x^m]))/x, x, 13, a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) - (I*b*d*PolyLog[2, -(I/(x^n*c))])/(2*n) - (I*b*e*Log[f*x^m]*PolyLog[2, -(I/(x^n*c))])/(2*n) + (I*b*d*PolyLog[2, I/(x^n*c)])/(2*n) + (I*b*e*Log[f*x^m]*PolyLog[2, I/(x^n*c)])/(2*n) - (I*b*e*m*PolyLog[3, -(I/(x^n*c))])/(2*n^2) + (I*b*e*m*PolyLog[3, I/(x^n*c)])/(2*n^2)} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse cotangents of exponentials*) - - -{ArcCot[E^x], x, 4, (-(1/2))*I*PolyLog[2, -I/E^x] + (1/2)*I*PolyLog[2, I/E^x]} -{x*ArcCot[E^x], x, 7, (-(1/2))*I*x*PolyLog[2, -I/E^x] + (1/2)*I*x*PolyLog[2, I/E^x] - (1/2)*I*PolyLog[3, -I/E^x] + (1/2)*I*PolyLog[3, I/E^x]} -{x^2*ArcCot[E^x], x, 9, (-(1/2))*I*x^2*PolyLog[2, -I/E^x] + (1/2)*I*x^2*PolyLog[2, I/E^x] - I*x*PolyLog[3, -I/E^x] + I*x*PolyLog[3, I/E^x] - I*PolyLog[4, -I/E^x] + I*PolyLog[4, I/E^x]} - - -{ArcCot[E^(a + b*x)], x, 4, -((I*PolyLog[2, (-I)*E^(-a - b*x)])/(2*b)) + (I*PolyLog[2, I*E^(-a - b*x)])/(2*b)} -{x*ArcCot[E^(a + b*x)], x, 7, -((I*x*PolyLog[2, (-I)*E^(-a - b*x)])/(2*b)) + (I*x*PolyLog[2, I*E^(-a - b*x)])/(2*b) - (I*PolyLog[3, (-I)*E^(-a - b*x)])/(2*b^2) + (I*PolyLog[3, I*E^(-a - b*x)])/(2*b^2)} -{x^2*ArcCot[E^(a + b*x)], x, 9, -((I*x^2*PolyLog[2, (-I)*E^(-a - b*x)])/(2*b)) + (I*x^2*PolyLog[2, I*E^(-a - b*x)])/(2*b) - (I*x*PolyLog[3, (-I)*E^(-a - b*x)])/b^2 + (I*x*PolyLog[3, I*E^(-a - b*x)])/b^2 - (I*PolyLog[4, (-I)*E^(-a - b*x)])/b^3 + (I*PolyLog[4, I*E^(-a - b*x)])/b^3} - - -{ArcCot[a + b*f^(c + d*x)], x, 6, -((ArcCot[a + b*f^(c + d*x)]*Log[2/(1 - I*(a + b*f^(c + d*x)))])/(d*Log[f])) + (ArcCot[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(d*Log[f]) - (I*PolyLog[2, 1 - 2/(1 - I*(a + b*f^(c + d*x)))])/(2*d*Log[f]) + (I*PolyLog[2, 1 - (2*b*f^(c + d*x))/((I - a)*(1 - I*(a + b*f^(c + d*x))))])/(2*d*Log[f])} -{x*ArcCot[a + b*f^(c + d*x)], x, 25, (-(1/4))*I*x^2*Log[1 - (b*f^(c + d*x))/(I - a)] + (1/4)*I*x^2*Log[1 + (b*f^(c + d*x))/(I + a)] + (1/4)*I*x^2*Log[1 - I/(a + b*f^(c + d*x))] - (1/4)*I*x^2*Log[1 + I/(a + b*f^(c + d*x))] - (I*x*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(2*d*Log[f]) + (I*x*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(2*d*Log[f]) + (I*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(2*d^2*Log[f]^2) - (I*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(2*d^2*Log[f]^2)} -{x^2*ArcCot[a + b*f^(c + d*x)], x, 29, (-(1/6))*I*x^3*Log[1 - (b*f^(c + d*x))/(I - a)] + (1/6)*I*x^3*Log[1 + (b*f^(c + d*x))/(I + a)] + (1/6)*I*x^3*Log[1 - I/(a + b*f^(c + d*x))] - (1/6)*I*x^3*Log[1 + I/(a + b*f^(c + d*x))] - (I*x^2*PolyLog[2, (b*f^(c + d*x))/(I - a)])/(2*d*Log[f]) + (I*x^2*PolyLog[2, -((b*f^(c + d*x))/(I + a))])/(2*d*Log[f]) + (I*x*PolyLog[3, (b*f^(c + d*x))/(I - a)])/(d^2*Log[f]^2) - (I*x*PolyLog[3, -((b*f^(c + d*x))/(I + a))])/(d^2*Log[f]^2) - (I*PolyLog[4, (b*f^(c + d*x))/(I - a)])/(d^3*Log[f]^3) + (I*PolyLog[4, -((b*f^(c + d*x))/(I + a))])/(d^3*Log[f]^3)} - - -{ArcCot[E^x]/E^x, x, 5, -x - ArcCot[E^x]/E^x + (1/2)*Log[1 + E^(2*x)]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse cotangents*) - - -{1/((a + a*x^2)*(b - 2*b*ArcCot[x])), x, 1, Log[1 - 2*ArcCot[x]]/(2*a*b)} - - -{E^(c*(a + b*x))*ArcCot[Sinh[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcCot[Sinh[c*(a + b*x)]])/(b*c) + Log[1 + E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcCot[Cosh[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcCot[Cosh[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcCot[Tanh[a*c + b*c*x]], x, 13, (E^(a*c + b*c*x)*ArcCot[Tanh[c*(a + b*x)]])/(b*c) - ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) + Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) - Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)} -{E^(c*(a + b*x))*ArcCot[Coth[a*c + b*c*x]], x, 13, (E^(a*c + b*c*x)*ArcCot[Coth[c*(a + b*x)]])/(b*c) + ArcTan[1 - Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) - ArcTan[1 + Sqrt[2]*E^(a*c + b*c*x)]/(Sqrt[2]*b*c) - Log[1 + E^(2*c*(a + b*x)) - Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c) + Log[1 + E^(2*c*(a + b*x)) + Sqrt[2]*E^(a*c + b*c*x)]/(2*Sqrt[2]*b*c)} -{E^(c*(a + b*x))*ArcCot[Sech[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcCot[Sech[c*(a + b*x)]])/(b*c) - ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c) - ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] + E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcCot[Csch[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcCot[Csch[c*(a + b*x)]])/(b*c) - Log[1 + E^(2*c*(a + b*x))]/(b*c)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.2 Exponentials of inverse cotangent.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.2 Exponentials of inverse cotangent.m deleted file mode 100644 index 5848ed6..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.4 Inverse cotangent/5.4.2 Exponentials of inverse cotangent.m +++ /dev/null @@ -1,27 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands involving exponentials of inverse cotangent*) - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse cotangents*) - - -{E^ArcCot[x], x, 2, (4/5 + (8*I)/5)*((-I + x)/x)^(1 + I/2)*((I + x)/x)^(-1 - I/2)*Hypergeometric2F1[1 + I/2, 2, 2 + I/2, (1 - I/x)/(1 + I/x)]} - - -{E^ArcCot[x]/(a + a*x^2), x, 1, -(E^ArcCot[x]/a)} -{E^ArcCot[x]/(a + a*x^2)^2, x, 2, -((2*E^ArcCot[x])/(5*a^2)) - (E^ArcCot[x]*(1 - 2*x))/(5*a^2*(1 + x^2))} -{E^ArcCot[x]/(a + a*x^2)^3, x, 3, -((24*E^ArcCot[x])/(85*a^3)) - (E^ArcCot[x]*(1 - 4*x))/(17*a^3*(1 + x^2)^2) - (12*E^ArcCot[x]*(1 - 2*x))/(85*a^3*(1 + x^2))} - -{E^ArcCot[x]/(a + a*x^2)^(3/2), x, 1, -((E^ArcCot[x]*(1 - x))/(2*a*Sqrt[a + a*x^2]))} -{E^ArcCot[x]/(a + a*x^2)^(5/2), x, 2, -((E^ArcCot[x]*(1 - 3*x))/(10*a*(a + a*x^2)^(3/2))) - (3*E^ArcCot[x]*(1 - x))/(10*a^2*Sqrt[a + a*x^2])} -{E^ArcCot[x]/(a + a*x^2)^(7/2), x, 3, -((E^ArcCot[x]*(1 - 5*x))/(26*a*(a + a*x^2)^(5/2))) - (E^ArcCot[x]*(1 - 3*x))/(13*a^2*(a + a*x^2)^(3/2)) - (3*E^ArcCot[x]*(1 - x))/(13*a^3*Sqrt[a + a*x^2])} - - -{E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(1/3), x, 3, (3*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((1/6)*(2 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-2 + 3*I*n))*(1 + I/(a*x))^((1/6)*(4 - 3*I*n))*x*Hypergeometric2F1[-(1/3), (1/6)*(2 - 3*I*n), 2/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(1/3)} -{E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(2/3), x, 3, -((3*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((1/6)*(4 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-4 + 3*I*n))*(1 + I/(a*x))^((1/6)*(2 - 3*I*n))*x*Hypergeometric2F1[1/3, (1/6)*(4 - 3*I*n), 4/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(2/3))} -{E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(4/3), x, 4, -((3*E^(n*ArcCot[a*x])*(3*n - 2*a*x))/(a*c*(4 + 9*n^2)*(c + a^2*c*x^2)^(1/3))) - (6*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((1/6)*(2 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-2 + 3*I*n))*(1 + I/(a*x))^((1/6)*(4 - 3*I*n))*x*Hypergeometric2F1[-(1/3), (1/6)*(2 - 3*I*n), 2/3, (2*I)/((a + I/x)*x)])/(c*(4 + 9*n^2)*(c + a^2*c*x^2)^(1/3))} -{E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(5/3), x, 4, -((3*E^(n*ArcCot[a*x])*(3*n - 4*a*x))/(a*c*(16 + 9*n^2)*(c + a^2*c*x^2)^(2/3))) - (12*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((1/6)*(4 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-4 + 3*I*n))*(1 + I/(a*x))^((1/6)*(2 - 3*I*n))*x*Hypergeometric2F1[1/3, (1/6)*(4 - 3*I*n), 4/3, (2*I)/((a + I/x)*x)])/(c*(16 + 9*n^2)*(c + a^2*c*x^2)^(2/3))} -{E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(7/3), x, 5, If[$VersionNumber>=8, -((3*E^(n*ArcCot[a*x])*(3*n - 8*a*x))/(a*c*(64 + 9*n^2)*(c + a^2*c*x^2)^(4/3))) - (120*E^(n*ArcCot[a*x])*(3*n - 2*a*x))/(a*c^2*(4 + 9*n^2)*(64 + 9*n^2)*(c + a^2*c*x^2)^(1/3)) - (240*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((1/6)*(2 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-2 + 3*I*n))*(1 + I/(a*x))^((1/6)*(4 - 3*I*n))*x*Hypergeometric2F1[-(1/3), (1/6)*(2 - 3*I*n), 2/3, (2*I)/((a + I/x)*x)])/(c^2*(4 + 9*n^2)*(64 + 9*n^2)*(c + a^2*c*x^2)^(1/3)), -((3*E^(n*ArcCot[a*x])*(3*n - 8*a*x))/(a*c*(64 + 9*n^2)*(c + a^2*c*x^2)^(4/3))) - (120*E^(n*ArcCot[a*x])*(3*n - 2*a*x))/(a*c^2*(256 + 612*n^2 + 81*n^4)*(c + a^2*c*x^2)^(1/3)) - (240*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((1/6)*(2 - 3*I*n))*(1 - I/(a*x))^((1/6)*(-2 + 3*I*n))*(1 + I/(a*x))^((1/6)*(4 - 3*I*n))*x*Hypergeometric2F1[-(1/3), (1/6)*(2 - 3*I*n), 2/3, (2*I)/((a + I/x)*x)])/(c^2*(256 + 612*n^2 + 81*n^4)*(c + a^2*c*x^2)^(1/3))]} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m deleted file mode 100644 index dd17c00..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.1 u (a+b arcsec(c x))^n.m +++ /dev/null @@ -1,333 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcSec[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSec[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSec[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^6*(a + b*ArcSec[c*x]), x, 7, -((5*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(112*c^5)) - (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(168*c^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^6)/(42*c) + (1/7)*x^7*(a + b*ArcSec[c*x]) - (5*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(112*c^7)} -{x^5*(a + b*ArcSec[c*x]), x, 4, -((4*b*Sqrt[1 - 1/(c^2*x^2)]*x)/(45*c^5)) - (2*b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(45*c^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^5)/(30*c) + (1/6)*x^6*(a + b*ArcSec[c*x])} -{x^4*(a + b*ArcSec[c*x]), x, 6, -((3*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(40*c^3)) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(20*c) + (1/5)*x^5*(a + b*ArcSec[c*x]) - (3*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(40*c^5)} -{x^3*(a + b*ArcSec[c*x]), x, 3, -((b*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3)) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (1/4)*x^4*(a + b*ArcSec[c*x])} -{x^2*(a + b*ArcSec[c*x]), x, 5, -((b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(6*c)) + (1/3)*x^3*(a + b*ArcSec[c*x]) - (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(6*c^3)} -{x^1*(a + b*ArcSec[c*x]), x, 2, -((b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c)) + (1/2)*x^2*(a + b*ArcSec[c*x])} -{x^0*(a + b*ArcSec[c*x]), x, 5, a*x + b*x*ArcSec[c*x] - (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{(a + b*ArcSec[c*x])/x^1, x, 6, (I*(a + b*ArcSec[c*x])^2)/(2*b) - (a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])] + (1/2)*I*b*PolyLog[2, -E^(2*I*ArcSec[c*x])]} -{(a + b*ArcSec[c*x])/x^2, x, 2, b*c*Sqrt[1 - 1/(c^2*x^2)] - (a + b*ArcSec[c*x])/x} -{(a + b*ArcSec[c*x])/x^3, x, 4, (b*c*Sqrt[1 - 1/(c^2*x^2)])/(4*x) - (a + b*ArcSec[c*x])/(2*x^2) - (1/4)*b*c^2*ArcCsc[c*x]} -{(a + b*ArcSec[c*x])/x^4, x, 4, (1/3)*b*c^3*Sqrt[1 - 1/(c^2*x^2)] - (1/9)*b*c^3*(1 - 1/(c^2*x^2))^(3/2) - (a + b*ArcSec[c*x])/(3*x^3)} -{(a + b*ArcSec[c*x])/x^5, x, 5, (b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*x^3) + (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(32*x) - (a + b*ArcSec[c*x])/(4*x^4) - (3/32)*b*c^4*ArcCsc[c*x]} -{(a + b*ArcSec[c*x])/x^6, x, 4, (1/5)*b*c^5*Sqrt[1 - 1/(c^2*x^2)] - (2/15)*b*c^5*(1 - 1/(c^2*x^2))^(3/2) + (1/25)*b*c^5*(1 - 1/(c^2*x^2))^(5/2) - (a + b*ArcSec[c*x])/(5*x^5)} -{(a + b*ArcSec[c*x])/x^7, x, 6, (b*c*Sqrt[1 - 1/(c^2*x^2)])/(36*x^5) + (5*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 - 1/(c^2*x^2)])/(96*x) - (a + b*ArcSec[c*x])/(6*x^6) - (5/96)*b*c^6*ArcCsc[c*x]} - - -{x^3*(a + b*ArcSec[c*x])^2, x, 5, (b^2*x^2)/(12*c^2) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcSec[c*x]))/(3*c^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^3*(a + b*ArcSec[c*x]))/(6*c) + (1/4)*x^4*(a + b*ArcSec[c*x])^2 + (b^2*Log[x])/(3*c^4)} -{x^2*(a + b*ArcSec[c*x])^2, x, 8, (b^2*x)/(3*c^2) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcSec[c*x]))/(3*c) + (1/3)*x^3*(a + b*ArcSec[c*x])^2 + (2*I*b*(a + b*ArcSec[c*x])*ArcTan[E^(I*ArcSec[c*x])])/(3*c^3) - (I*b^2*PolyLog[2, (-I)*E^(I*ArcSec[c*x])])/(3*c^3) + (I*b^2*PolyLog[2, I*E^(I*ArcSec[c*x])])/(3*c^3)} -{x^1*(a + b*ArcSec[c*x])^2, x, 4, -((b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcSec[c*x]))/c) + (1/2)*x^2*(a + b*ArcSec[c*x])^2 + (b^2*Log[x])/c^2} -{x^0*(a + b*ArcSec[c*x])^2, x, 7, x*(a + b*ArcSec[c*x])^2 + (4*I*b*(a + b*ArcSec[c*x])*ArcTan[E^(I*ArcSec[c*x])])/c - (2*I*b^2*PolyLog[2, (-I)*E^(I*ArcSec[c*x])])/c + (2*I*b^2*PolyLog[2, I*E^(I*ArcSec[c*x])])/c} -{(a + b*ArcSec[c*x])^2/x^1, x, 6, (I*(a + b*ArcSec[c*x])^3)/(3*b) - (a + b*ArcSec[c*x])^2*Log[1 + E^(2*I*ArcSec[c*x])] + I*b*(a + b*ArcSec[c*x])*PolyLog[2, -E^(2*I*ArcSec[c*x])] - (1/2)*b^2*PolyLog[3, -E^(2*I*ArcSec[c*x])]} -{(a + b*ArcSec[c*x])^2/x^2, x, 4, (2*b^2)/x + 2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]) - (a + b*ArcSec[c*x])^2/x} -{(a + b*ArcSec[c*x])^2/x^3, x, 4, b^2/(4*x^2) - (1/2)*a*b*c^2*ArcSec[c*x] - (1/4)*b^2*c^2*ArcSec[c*x]^2 + (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]))/(2*x) + (1/2)*(c^2 - 1/x^2)*(a + b*ArcSec[c*x])^2} -{(a + b*ArcSec[c*x])^2/x^4, x, 5, (2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) + (4/9)*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]) + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]))/(9*x^2) - (a + b*ArcSec[c*x])^2/(3*x^3)} -{(a + b*ArcSec[c*x])^2/x^5, x, 5, b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3/16)*a*b*c^4*ArcSec[c*x] + (3/32)*b^2*c^4*ArcSec[c*x]^2 + (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]))/(8*x^3) + (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x]))/(16*x) - (a + b*ArcSec[c*x])^2/(4*x^4)} - - -{x^3*(a + b*ArcSec[c*x])^3, x, 10, -((b^3*Sqrt[1 - 1/(c^2*x^2)]*x)/(4*c^3)) + (b^2*x^2*(a + b*ArcSec[c*x]))/(4*c^2) + (I*b*(a + b*ArcSec[c*x])^2)/(2*c^4) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcSec[c*x])^2)/(2*c^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*x^3*(a + b*ArcSec[c*x])^2)/(4*c) + (1/4)*x^4*(a + b*ArcSec[c*x])^3 - (b^2*(a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/c^4 + (I*b^3*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*c^4)} -{x^2*(a + b*ArcSec[c*x])^3, x, 11, (b^2*x*(a + b*ArcSec[c*x]))/c^2 - (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcSec[c*x])^2)/(2*c) + (1/3)*x^3*(a + b*ArcSec[c*x])^3 + (I*b*(a + b*ArcSec[c*x])^2*ArcTan[E^(I*ArcSec[c*x])])/c^3 - (b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c^3 - (I*b^2*(a + b*ArcSec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])])/c^3 + (I*b^2*(a + b*ArcSec[c*x])*PolyLog[2, I*E^(I*ArcSec[c*x])])/c^3 + (b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])])/c^3 - (b^3*PolyLog[3, I*E^(I*ArcSec[c*x])])/c^3} -{x^1*(a + b*ArcSec[c*x])^3, x, 7, (3*I*b*(a + b*ArcSec[c*x])^2)/(2*c^2) - (3*b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcSec[c*x])^2)/(2*c) + (1/2)*x^2*(a + b*ArcSec[c*x])^3 - (3*b^2*(a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/c^2 + (3*I*b^3*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*c^2)} -{x^0*(a + b*ArcSec[c*x])^3, x, 9, x*(a + b*ArcSec[c*x])^3 + (6*I*b*(a + b*ArcSec[c*x])^2*ArcTan[E^(I*ArcSec[c*x])])/c - (6*I*b^2*(a + b*ArcSec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])])/c + (6*I*b^2*(a + b*ArcSec[c*x])*PolyLog[2, I*E^(I*ArcSec[c*x])])/c + (6*b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])])/c - (6*b^3*PolyLog[3, I*E^(I*ArcSec[c*x])])/c} -{(a + b*ArcSec[c*x])^3/x^1, x, 7, (I*(a + b*ArcSec[c*x])^4)/(4*b) - (a + b*ArcSec[c*x])^3*Log[1 + E^(2*I*ArcSec[c*x])] + (3/2)*I*b*(a + b*ArcSec[c*x])^2*PolyLog[2, -E^(2*I*ArcSec[c*x])] - (3/2)*b^2*(a + b*ArcSec[c*x])*PolyLog[3, -E^(2*I*ArcSec[c*x])] - (3/4)*I*b^3*PolyLog[4, -E^(2*I*ArcSec[c*x])]} -{(a + b*ArcSec[c*x])^3/x^2, x, 5, -6*b^3*c*Sqrt[1 - 1/(c^2*x^2)] + (6*b^2*(a + b*ArcSec[c*x]))/x + 3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2 - (a + b*ArcSec[c*x])^3/x} -{(a + b*ArcSec[c*x])^3/x^3, x, 6, -((3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(8*x)) + (3/8)*b^3*c^2*ArcSec[c*x] - (3/4)*b^2*(c^2 - 1/x^2)*(a + b*ArcSec[c*x]) + (3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(4*x) - (1/4)*c^2*(a + b*ArcSec[c*x])^3 + (1/2)*(c^2 - 1/x^2)*(a + b*ArcSec[c*x])^3} -{(a + b*ArcSec[c*x])^3/x^4, x, 8, (-(14/9))*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)] + (2/27)*b^3*c^3*(1 - 1/(c^2*x^2))^(3/2) + (2*b^2*(a + b*ArcSec[c*x]))/(9*x^3) + (4*b^2*c^2*(a + b*ArcSec[c*x]))/(3*x) + (2/3)*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2 + (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(3*x^2) - (a + b*ArcSec[c*x])^3/(3*x^3)} -{(a + b*ArcSec[c*x])^3/x^5, x, 10, -((3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(128*x^3)) - (45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)])/(256*x) - (45/256)*b^3*c^4*ArcSec[c*x] + (3*b^2*(a + b*ArcSec[c*x]))/(32*x^4) + (9*b^2*c^2*(a + b*ArcSec[c*x]))/(32*x^2) + (3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(16*x^3) + (9*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcSec[c*x])^2)/(32*x) + (3/32)*c^4*(a + b*ArcSec[c*x])^3 - (a + b*ArcSec[c*x])^3/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/(a + b*ArcSec[c*x]), x, 0, Unintegrable[x/(a + b*ArcSec[c*x]), x]} -{x^0/(a + b*ArcSec[c*x]), x, 0, Unintegrable[1/(a + b*ArcSec[c*x]), x]} -{1/(x^1*(a + b*ArcSec[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcSec[c*x])), x]} -{1/(x^2*(a + b*ArcSec[c*x])), x, 4, -((c*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/b) + (c*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/b} -{1/(x^3*(a + b*ArcSec[c*x])), x, 6, -((c^2*CosIntegral[(2*a)/b + 2*ArcSec[c*x]]*Sin[(2*a)/b])/(2*b)) + (c^2*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSec[c*x]])/(2*b)} -{1/(x^4*(a + b*ArcSec[c*x])), x, 9, -((c^3*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/(4*b)) - (c^3*CosIntegral[(3*a)/b + 3*ArcSec[c*x]]*Sin[(3*a)/b])/(4*b) + (c^3*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(4*b) + (c^3*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSec[c*x]])/(4*b)} - - -{x^1/(a + b*ArcSec[c*x])^2, x, 0, Unintegrable[x/(a + b*ArcSec[c*x])^2, x]} -{x^0/(a + b*ArcSec[c*x])^2, x, 0, Unintegrable[1/(a + b*ArcSec[c*x])^2, x]} -{1/(x^1*(a + b*ArcSec[c*x])^2), x, 0, Unintegrable[1/(x*(a + b*ArcSec[c*x])^2), x]} -{1/(x^2*(a + b*ArcSec[c*x])^2), x, 5, -((c*Sqrt[1 - 1/(c^2*x^2)])/(b*(a + b*ArcSec[c*x]))) + (c*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]])/b^2 + (c*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]])/b^2} -{1/(x^3*(a + b*ArcSec[c*x])^2), x, 7, (c^2*Cos[(2*a)/b]*CosIntegral[(2*a)/b + 2*ArcSec[c*x]])/b^2 - (c^2*Sin[2*ArcSec[c*x]])/(2*b*(a + b*ArcSec[c*x])) + (c^2*Sin[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSec[c*x]])/b^2} -{1/(x^4*(a + b*ArcSec[c*x])^2), x, 11, -((c^3*Sqrt[1 - 1/(c^2*x^2)])/(4*b*(a + b*ArcSec[c*x]))) + (c^3*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]])/(4*b^2) + (3*c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcSec[c*x]])/(4*b^2) - (c^3*Sin[3*ArcSec[c*x]])/(4*b*(a + b*ArcSec[c*x])) + (c^3*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(4*b^2) + (3*c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSec[c*x]])/(4*b^2)} - - -{x^1/(a + b*ArcSec[c*x])^3, x, 0, Unintegrable[x/(a + b*ArcSec[c*x])^3, x]} -{x^0/(a + b*ArcSec[c*x])^3, x, 0, Unintegrable[1/(a + b*ArcSec[c*x])^3, x]} -{1/(x^1*(a + b*ArcSec[c*x])^3), x, 0, Unintegrable[1/(x*(a + b*ArcSec[c*x])^3), x]} -{1/(x^2*(a + b*ArcSec[c*x])^3), x, 6, -((c*Sqrt[1 - 1/(c^2*x^2)])/(2*b*(a + b*ArcSec[c*x])^2)) - 1/(2*b^2*x*(a + b*ArcSec[c*x])) + (c*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/(2*b^3) - (c*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(2*b^3)} -{1/(x^3*(a + b*ArcSec[c*x])^3), x, 8, -((c^2*Cos[2*ArcSec[c*x]])/(2*b^2*(a + b*ArcSec[c*x]))) + (c^2*CosIntegral[(2*a)/b + 2*ArcSec[c*x]]*Sin[(2*a)/b])/b^3 - (c^2*Sin[2*ArcSec[c*x]])/(4*b*(a + b*ArcSec[c*x])^2) - (c^2*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSec[c*x]])/b^3} -{1/(x^4*(a + b*ArcSec[c*x])^3), x, 13, -((c^3*Sqrt[1 - 1/(c^2*x^2)])/(8*b*(a + b*ArcSec[c*x])^2)) - c^2/(8*b^2*x*(a + b*ArcSec[c*x])) - (3*c^3*Cos[3*ArcSec[c*x]])/(8*b^2*(a + b*ArcSec[c*x])) + (c^3*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b])/(8*b^3) + (9*c^3*CosIntegral[(3*a)/b + 3*ArcSec[c*x]]*Sin[(3*a)/b])/(8*b^3) - (c^3*Sin[3*ArcSec[c*x]])/(8*b*(a + b*ArcSec[c*x])^2) - (c^3*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/(8*b^3) - (9*c^3*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSec[c*x]])/(8*b^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSec[c x])^n with m symbolic*) - - -{(d*x)^m*(a + b*ArcSec[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcSec[c*x])^3, x]} -{(d*x)^m*(a + b*ArcSec[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcSec[c*x])^2, x]} -{(d*x)^m*(a + b*ArcSec[c*x])^1, x, 3, ((d*x)^(1 + m)*(a + b*ArcSec[c*x]))/(d*(1 + m)) - (b*(d*x)^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, 1/(c^2*x^2)])/(c*m*(1 + m))} -{(d*x)^m/(a + b*ArcSec[c*x])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcSec[c*x]), x]} -{(d*x)^m/(a + b*ArcSec[c*x])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcSec[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^p (a+b ArcSec[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSec[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcSec[c*x]), x, 11, -((b*e*(9*c^2*d^2 + e^2)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3)) - (b*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(2*c) - (b*e^3*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (b*d^4*ArcCsc[c*x])/(4*e) + ((d + e*x)^4*(a + b*ArcSec[c*x]))/(4*e) - (b*d*(2*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcSec[c*x]), x, 10, -((b*d*e*Sqrt[1 - 1/(c^2*x^2)]*x)/c) - (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(6*c) + (b*d^3*ArcCsc[c*x])/(3*e) + ((d + e*x)^3*(a + b*ArcSec[c*x]))/(3*e) - (b*(6*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(6*c^3)} -{(d + e*x)^1*(a + b*ArcSec[c*x]), x, 9, -((b*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c)) + (b*d^2*ArcCsc[c*x])/(2*e) + ((d + e*x)^2*(a + b*ArcSec[c*x]))/(2*e) - (b*d*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{(d + e*x)^0*(a + b*ArcSec[c*x]), x, 5, a*x + b*x*ArcSec[c*x] - (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{(a + b*ArcSec[c*x])/(d + e*x)^1, x, 4, ((a + b*ArcSec[c*x])*Log[1 + ((e - Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcSec[c*x]))/(c*d)])/e + ((a + b*ArcSec[c*x])*Log[1 + ((e + Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcSec[c*x]))/(c*d)])/e - ((a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e - (I*b*PolyLog[2, -(((e - Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcSec[c*x]))/(c*d))])/e - (I*b*PolyLog[2, -(((e + Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcSec[c*x]))/(c*d))])/e + (I*b*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e)} -{(a + b*ArcSec[c*x])/(d + e*x)^2, x, 7, -((b*ArcCsc[c*x])/(d*e)) - (a + b*ArcSec[c*x])/(e*(d + e*x)) - (b*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])])/(d*Sqrt[c^2*d^2 - e^2])} -{(a + b*ArcSec[c*x])/(d + e*x)^3, x, 8, (b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(2*d*(c^2*d^2 - e^2)*(e + d/x)) - (b*ArcCsc[c*x])/(2*d^2*e) - (a + b*ArcSec[c*x])/(2*e*(d + e*x)^2) - (b*(2*c^2*d^2 - e^2)*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])])/(2*d^2*(c^2*d^2 - e^2)^(3/2))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(m/2) (a+b ArcSec[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^(3/2)*(a + b*ArcSec[c*x]), x, 22, (4*b*e*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*(d + e*x)^(5/2)*(a + b*ArcSec[c*x]))/(5*e) + (28*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(d + e*x)^(1/2)*(a + b*ArcSec[c*x]), x, 15, (2*(d + e*x)^(3/2)*(a + b*ArcSec[c*x]))/(3*e) + (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcSec[c*x])/(d + e*x)^(1/2), x, 9, (2*Sqrt[d + e*x]*(a + b*ArcSec[c*x]))/e + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcSec[c*x])/(d + e*x)^(3/2), x, 6, -((2*(a + b*ArcSec[c*x]))/(e*Sqrt[d + e*x])) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcSec[c*x])/(d + e*x)^(5/2), x, 12, -((4*b*e*(1 - c^2*x^2))/(3*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])) - (2*(a + b*ArcSec[c*x]))/(3*e*(d + e*x)^(3/2)) + (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcSec[c*x])/(d + e*x)^(7/2), x, 19, -((4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcSec[c*x]))/(5*e*(d + e*x)^(5/2)) + (4*b*(7*c^2*d^2 - 3*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^3 - d*e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*d^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]), -((4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcSec[c*x]))/(5*e*(d + e*x)^(5/2)) + (16*b*c^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*d^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSec[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcSec[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 7, -((b*(42*c^2*d + 25*e)*x^2*Sqrt[-1 + c^2*x^2])/(560*c^5*Sqrt[c^2*x^2])) - (b*(42*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) - (b*e*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (1/5)*d*x^5*(a + b*ArcSec[c*x]) + (1/7)*e*x^7*(a + b*ArcSec[c*x]) - (b*(42*c^2*d + 25*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(560*c^6*Sqrt[c^2*x^2])} -{x^2*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 6, -((b*(20*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2])) - (b*e*x^4*Sqrt[-1 + c^2*x^2])/(20*c*Sqrt[c^2*x^2]) + (1/3)*d*x^3*(a + b*ArcSec[c*x]) + (1/5)*e*x^5*(a + b*ArcSec[c*x]) - (b*(20*c^2*d + 9*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])} -{x^0*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 5, -((b*e*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2])) + d*x*(a + b*ArcSec[c*x]) + (1/3)*e*x^3*(a + b*ArcSec[c*x]) - (b*(6*c^2*d + e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^2*Sqrt[c^2*x^2])} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^2, x, 4, (b*c*d*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2] - (d*(a + b*ArcSec[c*x]))/x + e*x*(a + b*ArcSec[c*x]) - (b*e*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[c^2*x^2]} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^4, x, 4, (b*c*(2*c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) + (b*c*d*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcSec[c*x]))/(3*x^3) - (e*(a + b*ArcSec[c*x]))/x} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^6, x, 5, (2*b*c^3*(12*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*Sqrt[c^2*x^2]) + (b*c*d*Sqrt[-1 + c^2*x^2])/(25*x^4*Sqrt[c^2*x^2]) + (b*c*(12*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcSec[c*x]))/(5*x^5) - (e*(a + b*ArcSec[c*x]))/(3*x^3)} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^8, x, 6, (8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) + (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt[c^2*x^2]) + (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) + (4*b*c^3*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcSec[c*x]))/(7*x^7) - (e*(a + b*ArcSec[c*x]))/(5*x^5)} - -{x^5*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 5, -((b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2])) - (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) - (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) - (b*e*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (1/6)*d*x^6*(a + b*ArcSec[c*x]) + (1/8)*e*x^8*(a + b*ArcSec[c*x])} -{x^3*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 5, -((b*(3*c^2*d + 2*e)*x*Sqrt[-1 + c^2*x^2])/(12*c^5*Sqrt[c^2*x^2])) - (b*(3*c^2*d + 4*e)*x*(-1 + c^2*x^2)^(3/2))/(36*c^5*Sqrt[c^2*x^2]) - (b*e*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + (1/4)*d*x^4*(a + b*ArcSec[c*x]) + (1/6)*e*x^6*(a + b*ArcSec[c*x])} -{x^1*(d + e*x^2)*(a + b*ArcSec[c*x]), x, 6, -((b*(2*c^2*d + e)*x*Sqrt[-1 + c^2*x^2])/(4*c^3*Sqrt[c^2*x^2])) - (b*e*x*(-1 + c^2*x^2)^(3/2))/(12*c^3*Sqrt[c^2*x^2]) + ((d + e*x^2)^2*(a + b*ArcSec[c*x]))/(4*e) - (b*c*d^2*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*e*Sqrt[c^2*x^2])} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^1, x, 11, -((b*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c)) - (1/2)*I*b*d*ArcCsc[c*x]^2 + (1/2)*e*x^2*(a + b*ArcSec[c*x]) + b*d*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])] - b*d*ArcCsc[c*x]*Log[1/x] - d*(a + b*ArcSec[c*x])*Log[1/x] - (1/2)*I*b*d*PolyLog[2, E^(2*I*ArcCsc[c*x])]} -{(d + e*x^2)*(a + b*ArcSec[c*x])/x^3, x, 13, (b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(4*x) - (1/4)*b*c^2*d*ArcCsc[c*x] - (1/2)*I*b*e*ArcCsc[c*x]^2 - (d*(a + b*ArcSec[c*x]))/(2*x^2) + b*e*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])] - b*e*ArcCsc[c*x]*Log[1/x] - e*(a + b*ArcSec[c*x])*Log[1/x] - (1/2)*I*b*e*PolyLog[2, E^(2*I*ArcCsc[c*x])]} - - -{x^2*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x, 7, -((b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(1680*c^5*Sqrt[c^2*x^2])) - (b*e*(84*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) - (b*e^2*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (1/3)*d^2*x^3*(a + b*ArcSec[c*x]) + (2/5)*d*e*x^5*(a + b*ArcSec[c*x]) + (1/7)*e^2*x^7*(a + b*ArcSec[c*x]) - (b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1680*c^6*Sqrt[c^2*x^2])} -{x^0*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x, 6, -((b*e*(40*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2])) - (b*e^2*x^4*Sqrt[-1 + c^2*x^2])/(20*c*Sqrt[c^2*x^2]) + d^2*x*(a + b*ArcSec[c*x]) + (2/3)*d*e*x^3*(a + b*ArcSec[c*x]) + (1/5)*e^2*x^5*(a + b*ArcSec[c*x]) - (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^2, x, 6, (b*c*d^2*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2] - (b*e^2*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcSec[c*x]))/x + 2*d*e*x*(a + b*ArcSec[c*x]) + (1/3)*e^2*x^3*(a + b*ArcSec[c*x]) - (b*e*(12*c^2*d + e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^2*Sqrt[c^2*x^2])} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^4, x, 6, (2*b*c*d*(c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) + (b*c*d^2*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcSec[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcSec[c*x]))/x + e^2*x*(a + b*ArcSec[c*x]) - (b*e^2*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[c^2*x^2]} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^6, x, 5, (b*c*(24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*Sqrt[-1 + c^2*x^2])/(225*Sqrt[c^2*x^2]) + (b*c*d^2*Sqrt[-1 + c^2*x^2])/(25*x^4*Sqrt[c^2*x^2]) + (2*b*c*d*(6*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcSec[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcSec[c*x]))/(3*x^3) - (e^2*(a + b*ArcSec[c*x]))/x} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^8, x, 6, (2*b*c^3*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 + c^2*x^2])/(11025*Sqrt[c^2*x^2]) + (b*c*d^2*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt[c^2*x^2]) + (2*b*c*d*(15*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) + (b*c*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 + c^2*x^2])/(11025*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcSec[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcSec[c*x]))/(5*x^5) - (e^2*(a + b*ArcSec[c*x]))/(3*x^3)} - -{x^3*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x, 5, -((b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2])) - (b*(6*c^4*d^2 + 16*c^2*d*e + 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) - (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) - (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (1/4)*d^2*x^4*(a + b*ArcSec[c*x]) + (1/3)*d*e*x^6*(a + b*ArcSec[c*x]) + (1/8)*e^2*x^8*(a + b*ArcSec[c*x])} -{x^1*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x, 6, -((b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*Sqrt[-1 + c^2*x^2])/(6*c^5*Sqrt[c^2*x^2])) - (b*e*(3*c^2*d + 2*e)*x*(-1 + c^2*x^2)^(3/2))/(18*c^5*Sqrt[c^2*x^2]) - (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + ((d + e*x^2)^3*(a + b*ArcSec[c*x]))/(6*e) - (b*c*d^3*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*e*Sqrt[c^2*x^2])} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^1, x, 12, -((b*e*(6*c^2*d + e)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3)) - (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) - (1/2)*I*b*d^2*ArcCsc[c*x]^2 + d*e*x^2*(a + b*ArcSec[c*x]) + (1/4)*e^2*x^4*(a + b*ArcSec[c*x]) + b*d^2*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])] - b*d^2*ArcCsc[c*x]*Log[1/x] - d^2*(a + b*ArcSec[c*x])*Log[1/x] - (1/2)*I*b*d^2*PolyLog[2, E^(2*I*ArcCsc[c*x])]} -{(d + e*x^2)^2*(a + b*ArcSec[c*x])/x^3, x, 14, (b*c*d^2*Sqrt[1 - 1/(c^2*x^2)])/(4*x) - (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) - (1/4)*b*c^2*d^2*ArcCsc[c*x] - I*b*d*e*ArcCsc[c*x]^2 - (d^2*(a + b*ArcSec[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcSec[c*x]) + 2*b*d*e*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])] - 2*b*d*e*ArcCsc[c*x]*Log[1/x] - 2*d*e*(a + b*ArcSec[c*x])*Log[1/x] - I*b*d*e*PolyLog[2, E^(2*I*ArcCsc[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -(* {x^3*(a + b*ArcSec[c*x])/(d + e*x^2), x, 47, -((b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c*e)) + (I*b*d*ArcSec[c*x]^2)/(2*e^2) + (x^2*(a + b*ArcSec[c*x]))/(2*e) - (I*d*(a + b*ArcSec[c*x])^2)/(2*b*e^2) - (b*d*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + (d*(a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e^2 - (a*d*Log[Sqrt[e] - Sqrt[-d]/x])/(2*e^2) - (a*d*Log[Sqrt[e] + Sqrt[-d]/x])/(2*e^2) + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - (I*b*d*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e^2)} *) -{x^2*(a + b*ArcSec[c*x])/(d + e*x^2), x, 25, (x*(a + b*ArcSec[c*x]))/e - (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e) + (Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) + (I*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (I*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^(3/2)) - (I*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2))} -{x^1*(a + b*ArcSec[c*x])/(d + e*x^2), x, 26, ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) - ((a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) + (I*b*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e)} -{x^0*(a + b*ArcSec[c*x])/(d + e*x^2), x, 19, ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcSec[c*x])/(x^1*(d + e*x^2)), x, 19, (I*(a + b*ArcSec[c*x])^2)/(2*b*d) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d)} -{(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)), x, 24, (b*c*Sqrt[1 - 1/(c^2*x^2)])/d - a/(d*x) - (b*ArcSec[c*x])/(d*x) + (Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) - (I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2))} - - -{x^5*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 31, -((b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c*e^2)) + (d*(a + b*ArcSec[c*x]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcSec[c*x]))/(2*e^2) + (b*d*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]) - (d*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + (2*d*(a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e^3 + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/e^3 + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/e^3 + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (I*b*d*PolyLog[2, -E^(2*I*ArcSec[c*x])])/e^3} -{x^3*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 29, -((a + b*ArcSec[c*x])/(2*e*(e + d/x^2))) - (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - ((a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e^2 - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^2) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^2) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + (I*b*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e^2)} -{x^1*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 7, -((a + b*ArcSec[c*x])/(2*e*(d + e*x^2))) + (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(2*d*e*Sqrt[c^2*x^2]) - (b*c*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(2*d*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[c^2*x^2])} -{(a + b*ArcSec[c*x])/(x^1*(d + e*x^2)^2), x, 24, -((e*(a + b*ArcSec[c*x]))/(2*d^2*(e + d/x^2))) + (I*(a + b*ArcSec[c*x])^2)/(2*b*d^2) - (b*Sqrt[e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^2) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^2) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2)} - -{x^4*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 51, -((d*(a + b*ArcSec[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] - d/x))) + (d*(a + b*ArcSec[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcSec[c*x]))/e^2 - (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e^2) - (b*Sqrt[d]*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) - (b*Sqrt[d]*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) + (3*Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*I*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*e^(5/2)) - (3*I*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*I*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*e^(5/2)) - (3*I*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2))} -{x^2*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 27, (a + b*ArcSec[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcSec[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*Sqrt[d]*e*Sqrt[c^2*d + e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*Sqrt[d]*e*Sqrt[c^2*d + e]) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2))} -{x^0*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 47, -((a + b*ArcSec[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x))) + (a + b*ArcSec[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^2), x, 50, (b*c*Sqrt[1 - 1/(c^2*x^2)])/d^2 - a/(d^2*x) - (b*ArcSec[c*x])/(d^2*x) + (e*(a + b*ArcSec[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcSec[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e]) + (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e]) - (3*Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) + (3*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) + (3*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2))} - - -{x^5*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 33, -((b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(8*e^2*(c^2*d + e)*(e + d/x^2)*x)) - (a + b*ArcSec[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcSec[c*x])/(2*e^2*(e + d/x^2)) - (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]) - (b*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(c^2*d + e)^(3/2)) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) - ((a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e^3 - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^3) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^3) - (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + (I*b*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e^3)} -{x^3*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 6, (b*c*x*Sqrt[-1 + c^2*x^2])/(8*e*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) + (x^4*(a + b*ArcSec[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*(c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d*e^(3/2)*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 8, -((b*c*x*Sqrt[-1 + c^2*x^2])/(8*d*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2))) - (a + b*ArcSec[c*x])/(4*e*(d + e*x^2)^2) + (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*d^2*e*Sqrt[c^2*x^2]) - (b*c*(3*c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt[e]*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])} -{(a + b*ArcSec[c*x])/(x^1*(d + e*x^2)^3), x, 28, (b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(8*d^2*(c^2*d + e)*(e + d/x^2)*x) + (e^2*(a + b*ArcSec[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSec[c*x]))/(d^3*(e + d/x^2)) + (I*(a + b*ArcSec[c*x])^2)/(2*b*d^3) - (b*Sqrt[e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(d^3*Sqrt[c^2*d + e]) + (b*Sqrt[e]*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^3) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) + (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^3) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3)} - -{x^4*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 35, (b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcSec[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcSec[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*ArcSec[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcSec[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) + (3*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^2*Sqrt[c^2*d + e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) + (3*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^2*Sqrt[c^2*d + e]) + (3*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) - (3*I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) - (3*I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2))} -{x^2*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 63, (b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcSec[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcSec[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcSec[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcSec[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - (I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) + (I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2))} -{x^0*(a + b*ArcSec[c*x])/(d + e*x^2)^3, x, 81, (b*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcSec[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcSec[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(a + b*ArcSec[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcSec[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d + e)^(3/2)) - (5*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) + (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d + e)^(3/2)) - (5*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) + (3*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSec[c*x])*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSec[c*x])*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) - (3*I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*I*b*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) - (3*I*b*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e])} -(* {(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^3), x, 96, (b*c*Sqrt[1 - 1/(c^2*x^2)])/d^3 + (b*c*e^(3/2)*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*e^(3/2)*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) - a/(d^3*x) + (a*e^2)/(4*d^3*(e + d/x^2)^2*x) - (9*a*e)/(8*d^3*(e + d/x^2)*x) + (15*a*Sqrt[e]*ArcCot[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + (b*e^(3/2)*ArcSec[c*x])/(16*(-d)^(5/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (9*b*e*ArcSec[c*x])/(16*d^3*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*e^(3/2)*ArcSec[c*x])/(16*(-d)^(5/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (9*b*e*ArcSec[c*x])/(16*d^3*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcSec[c*x])/(d^3*x) - (b*e^2*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*Sqrt[c^2*d + e]) - (b*e^2*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*Sqrt[c^2*d + e]) + (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) - (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) - (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcSec[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x, If[$VersionNumber>=8, 12, 13], (b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(1680*c^5*e^2*Sqrt[c^2*x^2]) + (b*(29*c^2*d - 25*e)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e^2*Sqrt[c^2*x^2]) - (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e^2*Sqrt[c^2*x^2]) + (d^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcSec[c*x]))/(7*e^3) + (8*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(105*e^3*Sqrt[c^2*x^2]) - (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^6*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x, If[$VersionNumber>=8, 11, 12], -((b*(c^2*d + 9*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e*Sqrt[c^2*x^2])) - (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e*Sqrt[c^2*x^2]) - (d*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e^2) - (2*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(15*e^2*Sqrt[c^2*x^2]) + (b*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(3/2)*Sqrt[c^2*x^2])} -{x^1*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x, 9, -((b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*Sqrt[c^2*x^2])) + ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e) + (b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e*Sqrt[c^2*x^2]) - (b*(3*c^2*d + e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*Sqrt[e]*Sqrt[c^2*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcSec[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x, x]} -{Sqrt[d + e*x^2]*(a + b*ArcSec[c*x])/x^3, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^3, x]} - -{x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x, 0, Unintegrable[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]} -{x^0*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]} -{Sqrt[d + e*x^2]*(a + b*ArcSec[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^2, x]} -{Sqrt[d + e*x^2]*(a + b*ArcSec[c*x])/x^4, x, 11, (2*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*d*x^3) - (2*b*c^2*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(2*c^2*d + 3*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcSec[c*x])/x^6, x, If[$VersionNumber>=8, 12, 32], If[$VersionNumber>=8, (b*c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(225*d^2*Sqrt[c^2*x^2]) + (b*c*(12*c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(225*d*x^2*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(25*d*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(15*d^2*x^3) - (b*c^2*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 30*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]), -((2*b*c*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(15*d^2*Sqrt[c^2*x^2])) + (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*Sqrt[c^2*x^2]) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d^2*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(25*x^4*Sqrt[c^2*x^2]) + (b*c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d*x^2*Sqrt[c^2*x^2]) + (b*c*(4*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(15*d^2*x^3) + (2*b*c^2*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(15*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(45*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) + (2*b*c^2*e*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(45*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (2*b*e^2*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(15*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])]} - - -{x^3*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x, 12, (b*(3*c^4*d^2 - 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(560*c^5*e*Sqrt[c^2*x^2]) - (b*(13*c^2*d + 25*e)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[c^2*x^2]) - (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[c^2*x^2]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcSec[c*x]))/(7*e^2) - (2*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(35*e^2*Sqrt[c^2*x^2]) + (b*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x, 10, -((b*(7*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(40*c^3*Sqrt[c^2*x^2])) - (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*Sqrt[c^2*x^2]) + ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e) + (b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(5*e*Sqrt[c^2*x^2]) - (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[c^2*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^1, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^3, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^3, x]} - -{x^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x, 0, Unintegrable[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]} -{x^0*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x, 0, Unintegrable[(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^2, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^2, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^4, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^4, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^6, x, 12, (b*c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d*Sqrt[c^2*x^2]) + (4*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*x^2*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(25*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*d*x^5) - (b*c^2*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(8*c^4*d^2 + 19*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x])/x^8, x, 13, (b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[c^2*x^2]) + (b*c*(120*c^4*d^2 + 159*c^2*d*e - 37*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3675*d*x^2*Sqrt[c^2*x^2]) + (b*c*(30*c^2*d + 11*e)*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*x^4*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^6*Sqrt[c^2*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(35*d^2*x^5) - (b*c^2*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 105*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x, 11, (b*(19*c^2*d - 9*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e^2*Sqrt[c^2*x^2]) - (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e^2*Sqrt[c^2*x^2]) + (d^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e^3) + (8*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(15*e^3*Sqrt[c^2*x^2]) - (b*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x, 10, -((b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e*Sqrt[c^2*x^2])) - (d*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^2) - (2*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^2*Sqrt[c^2*x^2]) + (b*(3*c^2*d - e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x, 9, (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e + (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e*Sqrt[c^2*x^2]) - (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[c^2*x^2])} -{(a + b*ArcSec[c*x])/(x^1*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x*Sqrt[d + e*x^2]), x]} -{(a + b*ArcSec[c*x])/(x^3*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x^3*Sqrt[d + e*x^2]), x]} - -{x^2*(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(x^2*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2], x]} -{x^0*(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcSec[c*x])/(x^2*Sqrt[d + e*x^2]), x, 11, (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d*Sqrt[c^2*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(d*x) - (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcSec[c*x])/(x^4*Sqrt[d + e*x^2]), x, 11, (b*c*(2*c^2*d - 5*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*x^2*Sqrt[c^2*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(3*d^2*x) - (b*c^2*(2*c^2*d - 5*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (2*b*(c^2*d - 3*e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcSec[c*x])/(x^6*Sqrt[d + e*x^2]), x, 32, (8*b*c*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(15*d^3*Sqrt[c^2*x^2]) - (4*b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d^3*Sqrt[c^2*x^2]) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d^3*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(25*d*x^4*Sqrt[c^2*x^2]) - (4*b*c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*x^2*Sqrt[c^2*x^2]) + (b*c*(4*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d^2*x^2*Sqrt[c^2*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(5*d*x^5) + (4*e*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(15*d^2*x^3) - (8*e^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/(15*d^3*x) - (8*b*c^2*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(15*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (4*b*c^2*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(45*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (8*b*c^2*e*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(45*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) + (8*b*e^2*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(15*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 10, -((b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e^2*Sqrt[c^2*x^2])) - (d^2*(a + b*ArcSec[c*x]))/(e^3*Sqrt[d + e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^3) - (8*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) + (b*(9*c^2*d - e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 9, (d*(a + b*ArcSec[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e^2 + (2*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^2*Sqrt[c^2*x^2]) - (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 4, -((a + b*ArcSec[c*x])/(e*Sqrt[d + e*x^2])) - (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(Sqrt[d]*e*Sqrt[c^2*x^2])} -{(a + b*ArcSec[c*x])/(x^1*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(3/2)), x]} - -{x^4*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]} -{x^2*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]} -{x^0*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 5, (x*(a + b*ArcSec[c*x]))/(d*Sqrt[d + e*x^2]) - (b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(3/2)), x, 10, (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d^2*Sqrt[c^2*x^2]) - (a + b*ArcSec[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcSec[c*x]))/(d^2*Sqrt[d + e*x^2]) - (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcSec[c*x])/(x^4*(d + e*x^2)^(3/2)), x, 25, (2*b*c*(c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^3*Sqrt[c^2*x^2]) - (4*b*c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3*d^3*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*x^2*Sqrt[c^2*x^2]) - (a + b*ArcSec[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcSec[c*x]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) - (2*b*c^2*(c^2*d - e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (4*b*c^2*e*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*(2*c^2*d - e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (4*b*c^2*e*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (8*b*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 10, -((b*c*d*x*Sqrt[-1 + c^2*x^2])/(3*e^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) - (d^2*(a + b*ArcSec[c*x]))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcSec[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e^3 + (8*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) - (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 7, (b*c*x*Sqrt[-1 + c^2*x^2])/(3*e*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (d*(a + b*ArcSec[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcSec[c*x])/(e^2*Sqrt[d + e*x^2]) - (2*b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*Sqrt[d]*e^2*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 5, -((b*c*x*Sqrt[-1 + c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) - (a + b*ArcSec[c*x])/(3*e*(d + e*x^2)^(3/2)) - (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*d^(3/2)*e*Sqrt[c^2*x^2])} -{(a + b*ArcSec[c*x])/(x^1*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(5/2)), x]} - -{x^6*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]} -{x^4*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]} -{x^2*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 10, -((b*c*x^2*Sqrt[-1 + c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) + (x^3*(a + b*ArcSec[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{x^0*(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2), x, 10, (b*c*e*x^2*Sqrt[-1 + c^2*x^2])/(3*d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (x*(a + b*ArcSec[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSec[c*x]))/(3*d^2*Sqrt[d + e*x^2]) - (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (2*b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(5/2)), x, 26, -((b*c*e*Sqrt[-1 + c^2*x^2])/(d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) - (4*b*c*e^2*x^2*Sqrt[-1 + c^2*x^2])/(3*d^3*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d^3*(c^2*d + e)*Sqrt[c^2*x^2]) - (a + b*ArcSec[c*x])/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*ArcSec[c*x]))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*ArcSec[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (4*b*c^2*e*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^3*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) + (8*b*e*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSec[c x]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcSec[c*x]), x, 6, If[$VersionNumber>=8, -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2])) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) - (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcSec[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSec[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSec[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSec[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]), -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(6 + m)*(8 + 6*m + m^2)*(105 + 71*m + 15*m^2 + m^3)*Sqrt[c^2*x^2])) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) - (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcSec[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSec[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSec[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSec[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x, 6, If[$VersionNumber>=8, -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2])) - (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[c^2*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcSec[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSec[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSec[c*x]))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]), -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f*(4 + m)*(5 + m)*(6 + 5*m + m^2)*Sqrt[c^2*x^2])) - (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[c^2*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcSec[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSec[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSec[c*x]))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])]} -{(f*x)^m*(d + e*x^2)^1*(a + b*ArcSec[c*x]), x, 5, -((b*e*x^(2 + m)*Sqrt[-1 + c^2*x^2])/(c*(6 + 5*m + m^2)*Sqrt[c^2*x^2])) + (d*x^(1 + m)*(a + b*ArcSec[c*x]))/(1 + m) + (e*x^(3 + m)*(a + b*ArcSec[c*x]))/(3 + m) + (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*x^(2 + m)*Sqrt[-1 + c^2*x^2]*Hypergeometric2F1[1, (2 + m)/2, (3 + m)/2, c^2*x^2])/(c*(1 + m)^2*(2 + m)*(3 + m)*Sqrt[c^2*x^2]), -((b*e*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c*f*(6 + 5*m + m^2)*Sqrt[c^2*x^2])) + (d*(f*x)^(1 + m)*(a + b*ArcSec[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcSec[c*x]))/(f^3*(3 + m)) - (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c*f*(1 + m)^2*(2 + m)*(3 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])} -{(f*x)^m*(a + b*ArcSec[c*x])/(d + e*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2), x]} -{(f*x)^m*(a + b*ArcSec[c*x])/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^2, x]} - - -{(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x, 0, Unintegrable[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]} -{(f*x)^m*(d + e*x^2)^(1/2)*(a + b*ArcSec[c*x]), x, 0, Unintegrable[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]} -{(f*x)^m*(a + b*ArcSec[c*x])/(d + e*x^2)^(1/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2], x]} -{(f*x)^m*(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^4)^p (a+b ArcSec[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^4)^(p/2) (a+b ArcSec[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^11*(a + b*ArcSec[c*x])/Sqrt[1 - c^4*x^4], x, 16, (4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (7*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) + (13*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(150*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/(70*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) + (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSec[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcSec[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcSec[c*x]))/(10*c^12) - (4*b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[1 - 1/(c^2*x^2)]*x)} -{x^7*(a + b*ArcSec[c*x])/Sqrt[1 - c^4*x^4], x, 13, (b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[1 - 1/(c^2*x^2)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(18*c^9*Sqrt[1 - 1/(c^2*x^2)]*x) + (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(30*c^9*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSec[c*x]))/(2*c^8) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcSec[c*x]))/(6*c^8) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(3*c^9*Sqrt[1 - 1/(c^2*x^2)]*x)} -{x^3*(a + b*ArcSec[c*x])/Sqrt[1 - c^4*x^4], x, 8, (b*x*Sqrt[1 - c^4*x^4])/(2*c^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSec[c*x]))/(2*c^4) - (b*x*ArcTan[Sqrt[1 - c^4*x^4]/Sqrt[-1 + c^2*x^2]])/(2*c^3*Sqrt[c^2*x^2]), (b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(2*c^5*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSec[c*x]))/(2*c^4) - (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(2*c^5*Sqrt[1 - 1/(c^2*x^2)]*x)} -{(a + b*ArcSec[c*x])/(x^1*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x*Sqrt[1 - c^4*x^4]), x]} -{(a + b*ArcSec[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcSec[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]} - - -(* ::Section:: *) -(*Integrands of the form u (a+b ArcSec[c x])^n*) diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Inverse secant functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Inverse secant functions.m deleted file mode 100644 index 891fdaf..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.5 Inverse secant/5.5.2 Inverse secant functions.m +++ /dev/null @@ -1,130 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Inverse Secants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcSec[a x^n]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSec[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcSec[a*x^5]/x, x, 7, (1/10)*I*ArcSec[a*x^5]^2 - (1/5)*ArcSec[a*x^5]*Log[1 + E^(2*I*ArcSec[a*x^5])] + (1/10)*I*PolyLog[2, -E^(2*I*ArcSec[a*x^5])]} - - -{x^3*ArcSec[Sqrt[x]], x, 4, (-(1/4))*Sqrt[-1 + x] - (1/4)*(-1 + x)^(3/2) - (3/20)*(-1 + x)^(5/2) - (1/28)*(-1 + x)^(7/2) + (1/4)*x^4*ArcSec[Sqrt[x]]} -{x^2*ArcSec[Sqrt[x]], x, 4, (-(1/3))*Sqrt[-1 + x] - (2/9)*(-1 + x)^(3/2) - (1/15)*(-1 + x)^(5/2) + (1/3)*x^3*ArcSec[Sqrt[x]]} -{x^1*ArcSec[Sqrt[x]], x, 4, (-(1/2))*Sqrt[-1 + x] - (1/6)*(-1 + x)^(3/2) + (1/2)*x^2*ArcSec[Sqrt[x]]} -{x^0*ArcSec[Sqrt[x]], x, 3, -Sqrt[-1 + x] + x*ArcSec[Sqrt[x]]} -{ArcSec[Sqrt[x]]/x^1, x, 7, I*ArcSec[Sqrt[x]]^2 - 2*ArcSec[Sqrt[x]]*Log[1 + E^(2*I*ArcSec[Sqrt[x]])] + I*PolyLog[2, -E^(2*I*ArcSec[Sqrt[x]])]} -{ArcSec[Sqrt[x]]/x^2, x, 5, Sqrt[-1 + x]/(2*x) - ArcSec[Sqrt[x]]/x + (1/2)*ArcTan[Sqrt[-1 + x]]} -{ArcSec[Sqrt[x]]/x^3, x, 6, Sqrt[-1 + x]/(8*x^2) + (3*Sqrt[-1 + x])/(16*x) - ArcSec[Sqrt[x]]/(2*x^2) + (3/16)*ArcTan[Sqrt[-1 + x]]} -{ArcSec[Sqrt[x]]/x^4, x, 7, Sqrt[-1 + x]/(18*x^3) + (5*Sqrt[-1 + x])/(72*x^2) + (5*Sqrt[-1 + x])/(48*x) - ArcSec[Sqrt[x]]/(3*x^3) + (5/48)*ArcTan[Sqrt[-1 + x]]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2*ArcSec[a/x], x, 5, (-(1/3))*a^3*Sqrt[1 - x^2/a^2] + (1/9)*a^3*(1 - x^2/a^2)^(3/2) + (1/3)*x^3*ArcCos[x/a]} -{x^1*ArcSec[a/x], x, 4, (-(1/4))*a*x*Sqrt[1 - x^2/a^2] + (1/2)*x^2*ArcCos[x/a] + (1/4)*a^2*ArcSin[x/a]} -{x^0*ArcSec[a/x], x, 3, (-a)*Sqrt[1 - x^2/a^2] + x*ArcCos[x/a]} -{ArcSec[a/x]/x^1, x, 6, (-(1/2))*I*ArcCos[x/a]^2 + ArcCos[x/a]*Log[1 + E^(2*I*ArcCos[x/a])] - (1/2)*I*PolyLog[2, -E^(2*I*ArcCos[x/a])]} -{ArcSec[a/x]/x^2, x, 5, -(ArcCos[x/a]/x) + ArcTanh[Sqrt[1 - x^2/a^2]]/a} -{ArcSec[a/x]/x^3, x, 3, Sqrt[1 - x^2/a^2]/(2*a*x) - ArcCos[x/a]/(2*x^2)} -{ArcSec[a/x]/x^4, x, 6, Sqrt[1 - x^2/a^2]/(6*a*x^2) - ArcCos[x/a]/(3*x^3) + ArcTanh[Sqrt[1 - x^2/a^2]]/(6*a^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSec[a x^n] when n symbolic*) - - -{ArcSec[a*x^n]/x, x, 7, (I*ArcSec[a*x^n]^2)/(2*n) - (ArcSec[a*x^n]*Log[1 + E^(2*I*ArcSec[a*x^n])])/n + (I*PolyLog[2, -E^(2*I*ArcSec[a*x^n])])/(2*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcSec[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcSec[a + b*x], x, 9, (a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(30*b^5) + (11*a*x^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(60*b^3) - (x^3*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(20*b^2) - ((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2])/(120*b^5) + (a^5*ArcSec[a + b*x])/(5*b^5) + (1/5)*x^5*ArcSec[a + b*x] - ((3 + 40*a^2 + 40*a^4)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(40*b^5)} -{x^3*ArcSec[a + b*x], x, 8, -(((2 + 17*a^2)*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(12*b^4)) - (x^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(12*b^2) + (a*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2])/(3*b^4) - (a^4*ArcSec[a + b*x])/(4*b^4) + (1/4)*x^4*ArcSec[a + b*x] + (a*(1 + 2*a^2)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(2*b^4)} -{x^2*ArcSec[a + b*x], x, 7, (5*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*b^3) - (x*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*b^2) + (a^3*ArcSec[a + b*x])/(3*b^3) + (1/3)*x^3*ArcSec[a + b*x] - ((1 + 6*a^2)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(6*b^3)} -{x^1*ArcSec[a + b*x], x, 6, -(((a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(2*b^2)) - (a^2*ArcSec[a + b*x])/(2*b^2) + (1/2)*x^2*ArcSec[a + b*x] + (a*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/b^2} -{x^0*ArcSec[a + b*x], x, 5, ((a + b*x)*ArcSec[a + b*x])/b - ArcTanh[Sqrt[1 - 1/(a + b*x)^2]]/b} -{ArcSec[a + b*x]/x^1, x, 14, ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]*Log[1 + E^(2*I*ArcSec[a + b*x])] - I*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] - I*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + (1/2)*I*PolyLog[2, -E^(2*I*ArcSec[a + b*x])]} -{ArcSec[a + b*x]/x^2, x, 5, -((b*ArcSec[a + b*x])/a) - ArcSec[a + b*x]/x + (2*b*ArcTan[(Sqrt[1 + a]*Tan[(1/2)*ArcSec[a + b*x]])/Sqrt[1 - a]])/(a*Sqrt[1 - a^2])} -{ArcSec[a + b*x]/x^3, x, 7, (b*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(2*a*(1 - a^2)*x) + (b^2*ArcSec[a + b*x])/(2*a^2) - ArcSec[a + b*x]/(2*x^2) - ((1 - 2*a^2)*b^2*ArcTan[(Sqrt[1 + a]*Tan[(1/2)*ArcSec[a + b*x]])/Sqrt[1 - a]])/(a^2*(1 - a^2)^(3/2))} -{ArcSec[a + b*x]/x^4, x, 8, (b*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*a*(1 - a^2)*x^2) - ((2 - 5*a^2)*b^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcSec[a + b*x])/(3*a^3) - ArcSec[a + b*x]/(3*x^3) + ((2 - 5*a^2 + 6*a^4)*b^3*ArcTan[(Sqrt[1 + a]*Tan[(1/2)*ArcSec[a + b*x]])/Sqrt[1 - a]])/(3*a^3*(1 - a^2)^(5/2))} - - -{x^3*ArcSec[a + b*x]^2, x, 20, -((a*x)/b^3) + (a + b*x)^2/(12*b^4) - ((a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/(3*b^4) - (3*a^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/b^4 + (a*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/b^4 - ((a + b*x)^3*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/(6*b^4) - (a^4*ArcSec[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcSec[a + b*x]^2 - (2*I*a*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/b^4 - (4*I*a^3*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/b^4 + Log[a + b*x]/(3*b^4) + (3*a^2*Log[a + b*x])/b^4 + (I*a*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^4 + (2*I*a^3*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^4 - (I*a*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^4 - (2*I*a^3*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^4} -{x^2*ArcSec[a + b*x]^2, x, 17, x/(3*b^2) + (2*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/b^3 - ((a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/(3*b^3) + (a^3*ArcSec[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcSec[a + b*x]^2 + (2*I*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/(3*b^3) + (4*I*a^2*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/b^3 - (2*a*Log[a + b*x])/b^3 - (I*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/(3*b^3) - (2*I*a^2*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^3 + (I*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/(3*b^3) + (2*I*a^2*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^3} -{x^1*ArcSec[a + b*x]^2, x, 11, -(((a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x])/b^2) - (a^2*ArcSec[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcSec[a + b*x]^2 - (4*I*a*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/b^2 + Log[a + b*x]/b^2 + (2*I*a*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^2 - (2*I*a*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^2} -{x^0*ArcSec[a + b*x]^2, x, 8, ((a + b*x)*ArcSec[a + b*x]^2)/b + (4*I*ArcSec[a + b*x]*ArcTan[E^(I*ArcSec[a + b*x])])/b - (2*I*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b + (2*I*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b} -{ArcSec[a + b*x]^2/x^1, x, 17, (-ArcSec[a + b*x]^2)*Log[1 + E^(2*I*ArcSec[a + b*x])] + I*ArcSec[a + b*x]*PolyLog[2, -E^(2*I*ArcSec[a + b*x])] - (1/2)*PolyLog[3, -E^(2*I*ArcSec[a + b*x])] + ArcSec[a + b*x]^2*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcSec[a + b*x]^2*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - 2*I*ArcSec[a + b*x]*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] - 2*I*ArcSec[a + b*x]*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + 2*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + 2*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])]} -{ArcSec[a + b*x]^2/x^2, x, 12, -((b*ArcSec[a + b*x]^2)/a) - ArcSec[a + b*x]^2/x - (2*I*b*ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (2*I*b*ArcSec[a + b*x]*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (2*b*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])} - - -{x^2*ArcSec[a + b*x]^3, x, 25, ((a + b*x)*ArcSec[a + b*x])/b^3 - (3*I*a*ArcSec[a + b*x]^2)/b^3 + (3*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x]^2)/b^3 - ((a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x]^2)/(2*b^3) + (a^3*ArcSec[a + b*x]^3)/(3*b^3) + (1/3)*x^3*ArcSec[a + b*x]^3 + (I*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])])/b^3 + (6*I*a^2*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])])/b^3 - ArcTanh[Sqrt[1 - 1/(a + b*x)^2]]/b^3 + (6*a*ArcSec[a + b*x]*Log[1 + E^(2*I*ArcSec[a + b*x])])/b^3 - (I*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^3 - (6*I*a^2*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^3 + (I*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^3 + (6*I*a^2*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^3 - (3*I*a*PolyLog[2, -E^(2*I*ArcSec[a + b*x])])/b^3 + PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])]/b^3 + (6*a^2*PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])])/b^3 - PolyLog[3, I*E^(I*ArcSec[a + b*x])]/b^3 - (6*a^2*PolyLog[3, I*E^(I*ArcSec[a + b*x])])/b^3} -{x^1*ArcSec[a + b*x]^3, x, 16, (3*I*ArcSec[a + b*x]^2)/(2*b^2) - (3*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcSec[a + b*x]^2)/(2*b^2) - (a^2*ArcSec[a + b*x]^3)/(2*b^2) + (1/2)*x^2*ArcSec[a + b*x]^3 - (6*I*a*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])])/b^2 - (3*ArcSec[a + b*x]*Log[1 + E^(2*I*ArcSec[a + b*x])])/b^2 + (6*I*a*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b^2 - (6*I*a*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b^2 + (3*I*PolyLog[2, -E^(2*I*ArcSec[a + b*x])])/(2*b^2) - (6*a*PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])])/b^2 + (6*a*PolyLog[3, I*E^(I*ArcSec[a + b*x])])/b^2} -{x^0*ArcSec[a + b*x]^3, x, 10, ((a + b*x)*ArcSec[a + b*x]^3)/b + (6*I*ArcSec[a + b*x]^2*ArcTan[E^(I*ArcSec[a + b*x])])/b - (6*I*ArcSec[a + b*x]*PolyLog[2, (-I)*E^(I*ArcSec[a + b*x])])/b + (6*I*ArcSec[a + b*x]*PolyLog[2, I*E^(I*ArcSec[a + b*x])])/b + (6*PolyLog[3, (-I)*E^(I*ArcSec[a + b*x])])/b - (6*PolyLog[3, I*E^(I*ArcSec[a + b*x])])/b} -{ArcSec[a + b*x]^3/x^1, x, 20, ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcSec[a + b*x]^3*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcSec[a + b*x]^3*Log[1 + E^(2*I*ArcSec[a + b*x])] - 3*I*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] - 3*I*ArcSec[a + b*x]^2*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] + (3/2)*I*ArcSec[a + b*x]^2*PolyLog[2, -E^(2*I*ArcSec[a + b*x])] + 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + 6*ArcSec[a + b*x]*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3/2)*ArcSec[a + b*x]*PolyLog[3, -E^(2*I*ArcSec[a + b*x])] + 6*I*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])] + 6*I*PolyLog[4, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])] - (3/4)*I*PolyLog[4, -E^(2*I*ArcSec[a + b*x])]} -{ArcSec[a + b*x]^3/x^2, x, 14, -((b*ArcSec[a + b*x]^3)/a) - ArcSec[a + b*x]^3/x - (3*I*b*ArcSec[a + b*x]^2*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (3*I*b*ArcSec[a + b*x]^2*Log[1 - (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (6*b*ArcSec[a + b*x]*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (6*b*ArcSec[a + b*x]*PolyLog[2, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (6*I*b*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (6*I*b*PolyLog[3, (a*E^(I*ArcSec[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b ArcSec[c+d x^n])*) - - -{x^1*(a + b*ArcSec[c + d*x^2]), x, 7, (a*x^2)/2 + (b*(c + d*x^2)*ArcSec[c + d*x^2])/(2*d) - (b*ArcTanh[Sqrt[1 - 1/(c + d*x^2)^2]])/(2*d)} - - -{x^2*(a + b*ArcSec[c + d*x^3]), x, 7, (a*x^3)/3 + (b*(c + d*x^3)*ArcSec[c + d*x^3])/(3*d) - (b*ArcTanh[Sqrt[1 - 1/(c + d*x^3)^2]])/(3*d)} - - -{x^3*(a + b*ArcSec[c + d*x^4]), x, 7, (a*x^4)/4 + (b*(c + d*x^4)*ArcSec[c + d*x^4])/(4*d) - (b*ArcTanh[Sqrt[1 - 1/(c + d*x^4)^2]])/(4*d)} - - -{x^(n-1)*ArcSec[a + b*x^n], x, 6, ((a + b*x^n)*ArcSec[a + b*x^n])/(b*n) - ArcTanh[Sqrt[1 - 1/(a + b*x^n)^2]]/(b*n)} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse secant functions of exponentials*) - - -{ArcSec[c*E^(a + b*x)], x, 7, (I*ArcSec[c*E^(a + b*x)]^2)/(2*b) - (ArcSec[c*E^(a + b*x)]*Log[1 + E^(2*I*ArcSec[c*E^(a + b*x)])])/b + (I*PolyLog[2, -E^(2*I*ArcSec[c*E^(a + b*x)])])/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse secant functions*) - - -{E^ArcSec[a*x]*x^2, x, 6, -(((12/5 + (4*I)/5)*E^((1 + 3*I)*ArcSec[a*x])*Hypergeometric2F1[3/2 - I/2, 3, 5/2 - I/2, -E^(2*I*ArcSec[a*x])])/a^3) + ((24/5 + (8*I)/5)*E^((1 + 3*I)*ArcSec[a*x])*Hypergeometric2F1[3/2 - I/2, 4, 5/2 - I/2, -E^(2*I*ArcSec[a*x])])/a^3} -{E^ArcSec[a*x]*x^1, x, 6, -(((8/5 + (4*I)/5)*E^((1 + 2*I)*ArcSec[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^(2*I*ArcSec[a*x])])/a^2) + ((16/5 + (8*I)/5)*E^((1 + 2*I)*ArcSec[a*x])*Hypergeometric2F1[1 - I/2, 3, 2 - I/2, -E^(2*I*ArcSec[a*x])])/a^2} -{E^ArcSec[a*x]*x^0, x, 5, -(((1 + I)*E^((1 + I)*ArcSec[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^(2*I*ArcSec[a*x])])/a) + ((2 + 2*I)*E^((1 + I)*ArcSec[a*x])*Hypergeometric2F1[1/2 - I/2, 2, 3/2 - I/2, -E^(2*I*ArcSec[a*x])])/a} -{E^ArcSec[a*x]/x^1, x, 6, (-I)*E^ArcSec[a*x] + 2*I*E^ArcSec[a*x]*Hypergeometric2F1[-(I/2), 1, 1 - I/2, -E^(2*I*ArcSec[a*x])]} -{E^ArcSec[a*x]/x^2, x, 3, (1/2)*a*E^ArcSec[a*x]*Sqrt[1 - 1/(a^2*x^2)] - E^ArcSec[a*x]/(2*x)} -{E^ArcSec[a*x]/x^3, x, 5, (-(1/5))*a^2*E^ArcSec[a*x]*Cos[2*ArcSec[a*x]] + (1/10)*a^2*E^ArcSec[a*x]*Sin[2*ArcSec[a*x]]} -{E^ArcSec[a*x]/x^4, x, 6, (1/8)*a^3*E^ArcSec[a*x]*Sqrt[1 - 1/(a^2*x^2)] - (a^2*E^ArcSec[a*x])/(8*x) - (3/40)*a^3*E^ArcSec[a*x]*Cos[3*ArcSec[a*x]] + (1/40)*a^3*E^ArcSec[a*x]*Sin[3*ArcSec[a*x]]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse secants*) - - -{ArcSec[a + b*x]/((a*d)/b + d*x), x, 8, (I*ArcSec[a + b*x]^2)/(2*d) - (ArcSec[a + b*x]*Log[1 + E^(2*I*ArcSec[a + b*x])])/d + (I*PolyLog[2, -E^(2*I*ArcSec[a + b*x])])/(2*d)} diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.1 u (a+b arccsc(c x))^n.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.1 u (a+b arccsc(c x))^n.m deleted file mode 100644 index 04ba47a..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.1 u (a+b arccsc(c x))^n.m +++ /dev/null @@ -1,341 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcCsc[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCsc[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCsc[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^6*(a + b*ArcCsc[c*x]), x, 7, (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(112*c^5) + (5*b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(168*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^6)/(42*c) + (x^7*(a + b*ArcCsc[c*x]))/7 + (5*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(112*c^7)} -{x^5*(a + b*ArcCsc[c*x]), x, 4, (4*b*Sqrt[1 - 1/(c^2*x^2)]*x)/(45*c^5) + (2*b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^5)/(30*c) + (x^6*(a + b*ArcCsc[c*x]))/6} -{x^4*(a + b*ArcCsc[c*x]), x, 6, (3*b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(40*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^4)/(20*c) + (x^5*(a + b*ArcCsc[c*x]))/5 + (3*b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(40*c^5)} -{x^3*(a + b*ArcCsc[c*x]), x, 3, (b*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (x^4*(a + b*ArcCsc[c*x]))/4} -{x^2*(a + b*ArcCsc[c*x]), x, 5, (b*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(6*c) + (x^3*(a + b*ArcCsc[c*x]))/3 + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(6*c^3)} -{x*(a + b*ArcCsc[c*x]), x, 2, (b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) + (x^2*(a + b*ArcCsc[c*x]))/2} -{a + b*ArcCsc[c*x], x, 5, a*x + b*x*ArcCsc[c*x] + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{(a + b*ArcCsc[c*x])/x, x, 6, ((I/2)*(a + b*ArcCsc[c*x])^2)/b - (a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])] + (I/2)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])]} -{(a + b*ArcCsc[c*x])/x^2, x, 2, -(b*c*Sqrt[1 - 1/(c^2*x^2)]) - (a + b*ArcCsc[c*x])/x} -{(a + b*ArcCsc[c*x])/x^3, x, 4, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(4*x) + (b*c^2*ArcCsc[c*x])/4 - (a + b*ArcCsc[c*x])/(2*x^2)} -{(a + b*ArcCsc[c*x])/x^4, x, 4, -(b*c^3*Sqrt[1 - 1/(c^2*x^2)])/3 + (b*c^3*(1 - 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsc[c*x])/(3*x^3)} -{(a + b*ArcCsc[c*x])/x^5, x, 5, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*x^3) - (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(32*x) + (3*b*c^4*ArcCsc[c*x])/32 - (a + b*ArcCsc[c*x])/(4*x^4)} -{(a + b*ArcCsc[c*x])/x^6, x, 4, -(b*c^5*Sqrt[1 - 1/(c^2*x^2)])/5 + (2*b*c^5*(1 - 1/(c^2*x^2))^(3/2))/15 - (b*c^5*(1 - 1/(c^2*x^2))^(5/2))/25 - (a + b*ArcCsc[c*x])/(5*x^5)} -{(a + b*ArcCsc[c*x])/x^7, x, 6, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(36*x^5) - (5*b*c^3*Sqrt[1 - 1/(c^2*x^2)])/(144*x^3) - (5*b*c^5*Sqrt[1 - 1/(c^2*x^2)])/(96*x) + (5*b*c^6*ArcCsc[c*x])/96 - (a + b*ArcCsc[c*x])/(6*x^6)} - - -{x^3*(a + b*ArcCsc[c*x])^2, x, 5, (b^2*x^2)/(12*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x]))/(3*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3*(a + b*ArcCsc[c*x]))/(6*c) + (x^4*(a + b*ArcCsc[c*x])^2)/4 + (b^2*Log[x])/(3*c^4)} -{x^2*(a + b*ArcCsc[c*x])^2, x, 8, (b^2*x)/(3*c^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x]))/(3*c) + (x^3*(a + b*ArcCsc[c*x])^2)/3 + (2*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/(3*c^3) - ((I/3)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + ((I/3)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c^3} -{x*(a + b*ArcCsc[c*x])^2, x, 4, (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x]))/c + (x^2*(a + b*ArcCsc[c*x])^2)/2 + (b^2*Log[x])/c^2} -{(a + b*ArcCsc[c*x])^2, x, 7, x*(a + b*ArcCsc[c*x])^2 + (4*b*(a + b*ArcCsc[c*x])*ArcTanh[E^(I*ArcCsc[c*x])])/c - ((2*I)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c + ((2*I)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c} -{(a + b*ArcCsc[c*x])^2/x, x, 6, ((I/3)*(a + b*ArcCsc[c*x])^3)/b - (a + b*ArcCsc[c*x])^2*Log[1 - E^((2*I)*ArcCsc[c*x])] + I*b*(a + b*ArcCsc[c*x])*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (b^2*PolyLog[3, E^((2*I)*ArcCsc[c*x])])/2} -{(a + b*ArcCsc[c*x])^2/x^2, x, 4, (2*b^2)/x - 2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]) - (a + b*ArcCsc[c*x])^2/x} -{(a + b*ArcCsc[c*x])^2/x^3, x, 4, b^2/(4*x^2) + (a*b*c^2*ArcCsc[c*x])/2 + (b^2*c^2*ArcCsc[c*x]^2)/4 - (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(2*x) - (a + b*ArcCsc[c*x])^2/(2*x^2)} -{(a + b*ArcCsc[c*x])^2/x^4, x, 5, (2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) - (4*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/9 - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(9*x^2) - (a + b*ArcCsc[c*x])^2/(3*x^3)} -{(a + b*ArcCsc[c*x])^2/x^5, x, 5, b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3*a*b*c^4*ArcCsc[c*x])/16 + (3*b^2*c^4*ArcCsc[c*x]^2)/32 - (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(8*x^3) - (3*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x]))/(16*x) - (a + b*ArcCsc[c*x])^2/(4*x^4)} - - -{x^3*(a + b*ArcCsc[c*x])^3, x, 10, (b^3*Sqrt[1 - 1/(c^2*x^2)]*x)/(4*c^3) + (b^2*x^2*(a + b*ArcCsc[c*x]))/(4*c^2) + ((I/2)*b*(a + b*ArcCsc[c*x])^2)/c^4 + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x])^2)/(2*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3*(a + b*ArcCsc[c*x])^2)/(4*c) + (x^4*(a + b*ArcCsc[c*x])^3)/4 - (b^2*(a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/c^4 + ((I/2)*b^3*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/c^4} -{x^2*(a + b*ArcCsc[c*x])^3, x, 11, (b^2*x*(a + b*ArcCsc[c*x]))/c^2 + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x])^2)/(2*c) + (x^3*(a + b*ArcCsc[c*x])^3)/3 + (b*(a + b*ArcCsc[c*x])^2*ArcTanh[E^(I*ArcCsc[c*x])])/c^3 + (b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c^3 - (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, E^(I*ArcCsc[c*x])])/c^3 + (b^3*PolyLog[3, -E^(I*ArcCsc[c*x])])/c^3 - (b^3*PolyLog[3, E^(I*ArcCsc[c*x])])/c^3} -{x*(a + b*ArcCsc[c*x])^3, x, 7, (((3*I)/2)*b*(a + b*ArcCsc[c*x])^2)/c^2 + (3*b*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*ArcCsc[c*x])^2)/(2*c) + (x^2*(a + b*ArcCsc[c*x])^3)/2 - (3*b^2*(a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/c^2 + (((3*I)/2)*b^3*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/c^2} -{(a + b*ArcCsc[c*x])^3, x, 9, x*(a + b*ArcCsc[c*x])^3 + (6*b*(a + b*ArcCsc[c*x])^2*ArcTanh[E^(I*ArcCsc[c*x])])/c - ((6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])])/c + ((6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, E^(I*ArcCsc[c*x])])/c + (6*b^3*PolyLog[3, -E^(I*ArcCsc[c*x])])/c - (6*b^3*PolyLog[3, E^(I*ArcCsc[c*x])])/c} -{(a + b*ArcCsc[c*x])^3/x, x, 7, ((I/4)*(a + b*ArcCsc[c*x])^4)/b - (a + b*ArcCsc[c*x])^3*Log[1 - E^((2*I)*ArcCsc[c*x])] + ((3*I)/2)*b*(a + b*ArcCsc[c*x])^2*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (3*b^2*(a + b*ArcCsc[c*x])*PolyLog[3, E^((2*I)*ArcCsc[c*x])])/2 - ((3*I)/4)*b^3*PolyLog[4, E^((2*I)*ArcCsc[c*x])]} -{(a + b*ArcCsc[c*x])^3/x^2, x, 5, 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)] + (6*b^2*(a + b*ArcCsc[c*x]))/x - 3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2 - (a + b*ArcCsc[c*x])^3/x} -{(a + b*ArcCsc[c*x])^3/x^3, x, 6, (3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(8*x) - (3*b^3*c^2*ArcCsc[c*x])/8 + (3*b^2*(a + b*ArcCsc[c*x]))/(4*x^2) - (3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(4*x) + (c^2*(a + b*ArcCsc[c*x])^3)/4 - (a + b*ArcCsc[c*x])^3/(2*x^2)} -{(a + b*ArcCsc[c*x])^3/x^4, x, 8, (14*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)])/9 - (2*b^3*c^3*(1 - 1/(c^2*x^2))^(3/2))/27 + (2*b^2*(a + b*ArcCsc[c*x]))/(9*x^3) + (4*b^2*c^2*(a + b*ArcCsc[c*x]))/(3*x) - (2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/3 - (b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(3*x^2) - (a + b*ArcCsc[c*x])^3/(3*x^3)} -{(a + b*ArcCsc[c*x])^3/x^5, x, 10, (3*b^3*c*Sqrt[1 - 1/(c^2*x^2)])/(128*x^3) + (45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)])/(256*x) - (45*b^3*c^4*ArcCsc[c*x])/256 + (3*b^2*(a + b*ArcCsc[c*x]))/(32*x^4) + (9*b^2*c^2*(a + b*ArcCsc[c*x]))/(32*x^2) - (3*b*c*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(16*x^3) - (9*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*(a + b*ArcCsc[c*x])^2)/(32*x) + (3*c^4*(a + b*ArcCsc[c*x])^3)/32 - (a + b*ArcCsc[c*x])^3/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/(a + b*ArcCsc[c*x]), x, 0, Unintegrable[x/(a + b*ArcCsc[c*x]), x]} -{x^0/(a + b*ArcCsc[c*x]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])^(-1), x]} -{1/(x^1*(a + b*ArcCsc[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcCsc[c*x])), x]} -{1/(x^2*(a + b*ArcCsc[c*x])), x, 4, -((c*Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]])/b) - (c*Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]])/b} -{1/(x^3*(a + b*ArcCsc[c*x])), x, 6, (c^2*CosIntegral[(2*a)/b + 2*ArcCsc[c*x]]*Sin[(2*a)/b])/(2*b) - (c^2*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcCsc[c*x]])/(2*b)} -{1/(x^4*(a + b*ArcCsc[c*x])), x, 9, -((c^3*Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]])/(4*b)) + (c^3*Cos[(3*a)/b]*CosIntegral[(3*a)/b + 3*ArcCsc[c*x]])/(4*b) - (c^3*Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]])/(4*b) + (c^3*Sin[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcCsc[c*x]])/(4*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCsc[c x])^n with m symbolic*) - - -{(d*x)^m*(a + b*ArcCsc[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcCsc[c*x])^3, x]} -{(d*x)^m*(a + b*ArcCsc[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcCsc[c*x])^2, x]} -{(d*x)^m*(a + b*ArcCsc[c*x]), x, 3, ((d*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(d*(1 + m)) + (b*(d*x)^m*Hypergeometric2F1[1/2, -m/2, 1 - m/2, 1/(c^2*x^2)])/(c*m*(1 + m))} -{(d*x)^m/(a + b*ArcCsc[c*x]), x, 0, Unintegrable[(d*x)^m/(a + b*ArcCsc[c*x]), x]} -{(d*x)^m/(a + b*ArcCsc[c*x])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcCsc[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^m (a+b ArcCsc[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCsc[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcCsc[c*x]), x, 11, (b*e*(9*c^2*d^2 + e^2)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(2*c) + (b*e^3*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) - (b*d^4*ArcCsc[c*x])/(4*e) + ((d + e*x)^4*(a + b*ArcCsc[c*x]))/(4*e) + (b*d*(2*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcCsc[c*x]), x, 10, (b*d*e*Sqrt[1 - 1/(c^2*x^2)]*x)/c + (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^2)/(6*c) - (b*d^3*ArcCsc[c*x])/(3*e) + ((d + e*x)^3*(a + b*ArcCsc[c*x]))/(3*e) + (b*(6*c^2*d^2 + e^2)*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(6*c^3)} -{(d + e*x)*(a + b*ArcCsc[c*x]), x, 9, (b*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) - (b*d^2*ArcCsc[c*x])/(2*e) + ((d + e*x)^2*(a + b*ArcCsc[c*x]))/(2*e) + (b*d*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{a + b*ArcCsc[c*x], x, 5, a*x + b*x*ArcCsc[c*x] + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/c} -{(a + b*ArcCsc[c*x])/(d + e*x), x, 4, ((a + b*ArcCsc[c*x])*Log[1 - (I*(e - Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e + ((a + b*ArcCsc[c*x])*Log[1 - (I*(e + Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e - ((a + b*ArcCsc[c*x])*Log[1 - E^(2*I*ArcCsc[c*x])])/e - (I*b*PolyLog[2, (I*(e - Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e - (I*b*PolyLog[2, (I*(e + Sqrt[(-c^2)*d^2 + e^2])*E^(I*ArcCsc[c*x]))/(c*d)])/e + (I*b*PolyLog[2, E^(2*I*ArcCsc[c*x])])/(2*e)} -{(a + b*ArcCsc[c*x])/(d + e*x)^2, x, 7, (b*ArcCsc[c*x])/(d*e) - (a + b*ArcCsc[c*x])/(e*(d + e*x)) + (b*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])])/(d*Sqrt[c^2*d^2 - e^2])} -{(a + b*ArcCsc[c*x])/(d + e*x)^3, x, 8, -(b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(2*d*(c^2*d^2 - e^2)*(e + d/x)) + (b*ArcCsc[c*x])/(2*d^2*e) - (a + b*ArcCsc[c*x])/(2*e*(d + e*x)^2) + (b*(2*c^2*d^2 - e^2)*ArcTanh[(c^2*d + e/x)/(c*Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])])/(2*d^2*(c^2*d^2 - e^2)^(3/2))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^(p/2) (a+b ArcCsc[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]), x, If[$VersionNumber>=8, 31, 27], (4*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(105*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) - (4*b*(d + e*x)^(3/2)*(1 - c^2*x^2))/(35*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^3) + (4*b*(5*c^2*d^2 - 9*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*d*(9*c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^4*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]), If[$VersionNumber>=8, (4*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(105*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) - (4*b*(d + e*x)^(3/2)*(1 - c^2*x^2))/(35*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^3) + (32*b*d^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(c^2*d^2 + 3*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (32*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^4*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]), -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(35*c^3*Sqrt[1 - 1/(c^2*x^2)])) - (8*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(105*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^3) + (4*b*d^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*(2*c^2*d^2 - 9*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (32*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^4*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])]} -{x^1*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]), x, If[$VersionNumber>=8, 24, 20], -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x)) - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) - (8*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*(3*c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]), -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x)) - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) - (8*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (8*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]), x, 15, (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{Sqrt[d + e*x]*(a + b*ArcCsc[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x, x]} -{Sqrt[d + e*x]*(a + b*ArcCsc[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x^2, x]} - - -{(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]), x, 22, -((4*b*e*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x)) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e) - (28*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcCsc[c*x])/Sqrt[d + e*x], x, 27, -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(35*c^3*e*Sqrt[1 - 1/(c^2*x^2)])) + (4*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(21*c^3*e^2*Sqrt[1 - 1/(c^2*x^2)]*x) - (2*d^3*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^4) - (24*b*d^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*(2*c^2*d^2 - 9*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (64*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^4*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsc[c*x])/Sqrt[d + e*x], x, 20, -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x)) + (2*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) + (4*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (32*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsc[c*x])/Sqrt[d + e*x], x, 14, -((2*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^2) + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (8*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsc[c*x])/Sqrt[d + e*x], x, 9, (2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsc[c*x])/(x^1*Sqrt[d + e*x]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x]), x]} -{(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x]), x]} - - -{x^3*(a + b*ArcCsc[c*x])/(d + e*x)^(3/2), x, 23, -((4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)) + (2*d^3*(a + b*ArcCsc[c*x]))/(e^4*Sqrt[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (32*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (8*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (64*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsc[c*x])/(d + e*x)^(3/2), x, 16, -((2*d^2*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x])) - (4*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^3 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (20*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (32*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsc[c*x])/(d + e*x)^(3/2), x, 11, (2*d*(a + b*ArcCsc[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^2 - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (8*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsc[c*x])/(d + e*x)^(3/2), x, 6, -((2*(a + b*ArcCsc[c*x]))/(e*Sqrt[d + e*x])) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsc[c*x])/(x^1*(d + e*x)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(3/2)), x]} -{(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)), x]} - - -{x^3*(a + b*ArcCsc[c*x])/(d + e*x)^(5/2), x, 31, -((4*b*d^2*(1 - c^2*x^2))/(3*c*e^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])) + (2*d^3*(a + b*ArcCsc[c*x]))/(3*e^4*(d + e*x)^(3/2)) - (6*d^2*(a + b*ArcCsc[c*x]))/(e^4*Sqrt[d + e*x]) - (6*d*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^4) + (8*b*d^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*e^3*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 - e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e^3*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (32*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsc[c*x])/(d + e*x)^(5/2), x, 25, (4*b*d*(1 - c^2*x^2))/(3*c*e*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*d^2*(a + b*ArcCsc[c*x]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^3 - (4*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*e^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsc[c*x])/(d + e*x)^(5/2), x, 19, -((4*b*(1 - c^2*x^2))/(3*c*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])) + (2*d*(a + b*ArcCsc[c*x]))/(3*e^2*(d + e*x)^(3/2)) - (2*(a + b*ArcCsc[c*x]))/(e^2*Sqrt[d + e*x]) + (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*e*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (8*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsc[c*x])/(d + e*x)^(5/2), x, 12, (4*b*e*(1 - c^2*x^2))/(3*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsc[c*x]))/(3*e*(d + e*x)^(3/2)) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsc[c*x])/(x^1*(d + e*x)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(5/2)), x]} -{(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(5/2)), x]} - - -{(a + b*ArcCsc[c*x])/(d + e*x)^(7/2), x, 19, (4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x)^(3/2)) + (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsc[c*x]))/(5*e*(d + e*x)^(5/2)) - (4*b*(7*c^2*d^2 - 3*e^2)*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^3 - d*e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*d^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]), (4*b*e*(1 - c^2*x^2))/(15*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x)^(3/2)) + (16*b*c*e*(1 - c^2*x^2))/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*e*(1 - c^2*x^2))/(5*c*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsc[c*x]))/(5*e*(d + e*x)^(5/2)) - (16*b*c^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^2 - e^2)^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*d^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*d^2*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCsc[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcCsc[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 7, (b*(42*c^2*d + 25*e)*x^2*Sqrt[-1 + c^2*x^2])/(560*c^5*Sqrt[c^2*x^2]) + (b*(42*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) + (b*e*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d*x^5*(a + b*ArcCsc[c*x]))/5 + (e*x^7*(a + b*ArcCsc[c*x]))/7 + (b*(42*c^2*d + 25*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(560*c^6*Sqrt[c^2*x^2])} -{x^2*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 6, (b*(20*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2]) + (b*e*x^4*Sqrt[-1 + c^2*x^2])/(20*c*Sqrt[c^2*x^2]) + (d*x^3*(a + b*ArcCsc[c*x]))/3 + (e*x^5*(a + b*ArcCsc[c*x]))/5 + (b*(20*c^2*d + 9*e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])} -{(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 5, (b*e*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) + d*x*(a + b*ArcCsc[c*x]) + (e*x^3*(a + b*ArcCsc[c*x]))/3 + (b*(6*c^2*d + e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^2*Sqrt[c^2*x^2])} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^2, x, 4, -((b*c*d*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/x + e*x*(a + b*ArcCsc[c*x]) + (b*e*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[c^2*x^2]} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^4, x, 4, -(b*c*(2*c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(3*x^3) - (e*(a + b*ArcCsc[c*x]))/x} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^6, x, 5, (-2*b*c^3*(12*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(25*x^4*Sqrt[c^2*x^2]) - (b*c*(12*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(5*x^5) - (e*(a + b*ArcCsc[c*x]))/(3*x^3)} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8, x, 6, (-8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) - (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt[c^2*x^2]) - (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) - (4*b*c^3*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcCsc[c*x]))/(7*x^7) - (e*(a + b*ArcCsc[c*x]))/(5*x^5)} - -{x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 5, (b*(4*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) + (b*(4*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d*x^6*(a + b*ArcCsc[c*x]))/6 + (e*x^8*(a + b*ArcCsc[c*x]))/8} -{x^3*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 5, (b*(3*c^2*d + 2*e)*x*Sqrt[-1 + c^2*x^2])/(12*c^5*Sqrt[c^2*x^2]) + (b*(3*c^2*d + 4*e)*x*(-1 + c^2*x^2)^(3/2))/(36*c^5*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + (d*x^4*(a + b*ArcCsc[c*x]))/4 + (e*x^6*(a + b*ArcCsc[c*x]))/6} -{x*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 6, (b*(2*c^2*d + e)*x*Sqrt[-1 + c^2*x^2])/(4*c^3*Sqrt[c^2*x^2]) + (b*e*x*(-1 + c^2*x^2)^(3/2))/(12*c^3*Sqrt[c^2*x^2]) + ((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/(4*e) + (b*c*d^2*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*e*Sqrt[c^2*x^2])} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x, x, 11, (b*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) + (I/2)*b*d*ArcCsc[c*x]^2 + (e*x^2*(a + b*ArcCsc[c*x]))/2 - b*d*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + b*d*ArcCsc[c*x]*Log[x^(-1)] - d*(a + b*ArcCsc[c*x])*Log[x^(-1)] + (I/2)*b*d*PolyLog[2, E^((2*I)*ArcCsc[c*x])]} -{((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^3, x, 13, -(b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(4*x) + (b*c^2*d*ArcCsc[c*x])/4 + (I/2)*b*e*ArcCsc[c*x]^2 - (d*(a + b*ArcCsc[c*x]))/(2*x^2) - b*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + b*e*ArcCsc[c*x]*Log[x^(-1)] - e*(a + b*ArcCsc[c*x])*Log[x^(-1)] + (I/2)*b*e*PolyLog[2, E^((2*I)*ArcCsc[c*x])]} - - -{x^2*(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x, 7, (b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 + c^2*x^2])/(1680*c^5*Sqrt[c^2*x^2]) + (b*e*(84*c^2*d + 25*e)*x^4*Sqrt[-1 + c^2*x^2])/(840*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^6*Sqrt[-1 + c^2*x^2])/(42*c*Sqrt[c^2*x^2]) + (d^2*x^3*(a + b*ArcCsc[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCsc[c*x]))/5 + (e^2*x^7*(a + b*ArcCsc[c*x]))/7 + (b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1680*c^6*Sqrt[c^2*x^2])} -{(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x, 6, (b*e*(40*c^2*d + 9*e)*x^2*Sqrt[-1 + c^2*x^2])/(120*c^3*Sqrt[c^2*x^2]) + (b*e^2*x^4*Sqrt[-1 + c^2*x^2])/(20*c*Sqrt[c^2*x^2]) + d^2*x*(a + b*ArcCsc[c*x]) + (2*d*e*x^3*(a + b*ArcCsc[c*x]))/3 + (e^2*x^5*(a + b*ArcCsc[c*x]))/5 + (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(120*c^4*Sqrt[c^2*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^2, x, 6, -((b*c*d^2*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*x^2]) + (b*e^2*x^2*Sqrt[-1 + c^2*x^2])/(6*c*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/x + 2*d*e*x*(a + b*ArcCsc[c*x]) + (e^2*x^3*(a + b*ArcCsc[c*x]))/3 + (b*e*(12*c^2*d + e)*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(6*c^2*Sqrt[c^2*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^4, x, 6, (-2*b*c*d*(c^2*d + 9*e)*Sqrt[-1 + c^2*x^2])/(9*Sqrt[c^2*x^2]) - (b*c*d^2*Sqrt[-1 + c^2*x^2])/(9*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcCsc[c*x]))/x + e^2*x*(a + b*ArcCsc[c*x]) + (b*e^2*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[c^2*x^2]} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^6, x, 5, -(b*c*(24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*Sqrt[-1 + c^2*x^2])/(225*Sqrt[c^2*x^2]) - (b*c*d^2*Sqrt[-1 + c^2*x^2])/(25*x^4*Sqrt[c^2*x^2]) - (2*b*c*d*(6*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2])/(225*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcCsc[c*x]))/(3*x^3) - (e^2*(a + b*ArcCsc[c*x]))/x} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^8, x, 6, (-2*b*c^3*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 + c^2*x^2])/(11025*Sqrt[c^2*x^2]) - (b*c*d^2*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt[c^2*x^2]) - (2*b*c*d*(15*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) - (b*c*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 + c^2*x^2])/(11025*x^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcCsc[c*x]))/(5*x^5) - (e^2*(a + b*ArcCsc[c*x]))/(3*x^3)} - -{x^3*(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x, 5, (b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 + c^2*x^2])/(24*c^7*Sqrt[c^2*x^2]) + (b*(6*c^4*d^2 + 16*c^2*d*e + 9*e^2)*x*(-1 + c^2*x^2)^(3/2))/(72*c^7*Sqrt[c^2*x^2]) + (b*e*(8*c^2*d + 9*e)*x*(-1 + c^2*x^2)^(5/2))/(120*c^7*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(7/2))/(56*c^7*Sqrt[c^2*x^2]) + (d^2*x^4*(a + b*ArcCsc[c*x]))/4 + (d*e*x^6*(a + b*ArcCsc[c*x]))/3 + (e^2*x^8*(a + b*ArcCsc[c*x]))/8} -{x^1*(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x, 6, (b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*x*Sqrt[-1 + c^2*x^2])/(6*c^5*Sqrt[c^2*x^2]) + (b*e*(3*c^2*d + 2*e)*x*(-1 + c^2*x^2)^(3/2))/(18*c^5*Sqrt[c^2*x^2]) + (b*e^2*x*(-1 + c^2*x^2)^(5/2))/(30*c^5*Sqrt[c^2*x^2]) + ((d + e*x^2)^3*(a + b*ArcCsc[c*x]))/(6*e) + (b*c*d^3*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*e*Sqrt[c^2*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^1, x, 12, (b*e*(6*c^2*d + e)*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (I/2)*b*d^2*ArcCsc[c*x]^2 + d*e*x^2*(a + b*ArcCsc[c*x]) + (e^2*x^4*(a + b*ArcCsc[c*x]))/4 - b*d^2*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + b*d^2*ArcCsc[c*x]*Log[x^(-1)] - d^2*(a + b*ArcCsc[c*x])*Log[x^(-1)] + (I/2)*b*d^2*PolyLog[2, E^((2*I)*ArcCsc[c*x])]} -{((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^3, x, 14, -(b*c*d^2*Sqrt[1 - 1/(c^2*x^2)])/(4*x) + (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c) + (b*c^2*d^2*ArcCsc[c*x])/4 + I*b*d*e*ArcCsc[c*x]^2 - (d^2*(a + b*ArcCsc[c*x]))/(2*x^2) + (e^2*x^2*(a + b*ArcCsc[c*x]))/2 - 2*b*d*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 2*b*d*e*ArcCsc[c*x]*Log[x^(-1)] - 2*d*e*(a + b*ArcCsc[c*x])*Log[x^(-1)] + I*b*d*e*PolyLog[2, E^((2*I)*ArcCsc[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2), x, 25, (x*(a + b*ArcCsc[c*x]))/e + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e) - (Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) - ((I/2)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(3/2) - ((I/2)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(3/2) + ((I/2)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(3/2)} -{(x*(a + b*ArcCsc[c*x]))/(d + e*x^2), x, 26, ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) - ((a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/e} -{(a + b*ArcCsc[c*x])/(d + e*x^2), x, 19, -((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e])} -{(a + b*ArcCsc[c*x])/(x*(d + e*x^2)), x, 19, ((I/2)*(a + b*ArcCsc[c*x])^2)/(b*d) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d} -{(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)), x, 24, -((b*c*Sqrt[1 - 1/(c^2*x^2)])/d) - a/(d*x) - (b*ArcCsc[c*x])/(d*x) - (Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - ((I/2)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(3/2) - ((I/2)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(-d)^(3/2)} - - -{(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 31, (b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c*e^2) + (d*(a + b*ArcCsc[c*x]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcCsc[c*x]))/(2*e^2) - (b*d*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]) - (d*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + (2*d*(a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e^3 + (I*b*d*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + (I*b*d*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (I*b*d*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/e^3} -{(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 29, -(a + b*ArcCsc[c*x])/(2*e*(e + d/x^2)) + (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - ((a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^2 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^2 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^2 + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/e^2} -{(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 7, -(a + b*ArcCsc[c*x])/(2*e*(d + e*x^2)) - (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(2*d*e*Sqrt[c^2*x^2]) + (b*c*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(2*d*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[c^2*x^2])} -{(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^2), x, 24, -(e*(a + b*ArcCsc[c*x]))/(2*d^2*(e + d/x^2)) + ((I/2)*(a + b*ArcCsc[c*x])^2)/(b*d^2) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^2} - -{(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 51, -(d*(a + b*ArcCsc[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) + (d*(a + b*ArcCsc[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcCsc[c*x]))/e^2 + (b*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])/(c*e^2) + (b*Sqrt[d]*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) + (b*Sqrt[d]*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*e^2*Sqrt[c^2*d + e]) - (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^(5/2) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^(5/2)} -{(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 27, (a + b*ArcCsc[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCsc[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*Sqrt[d]*e*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*Sqrt[d]*e*Sqrt[c^2*d + e]) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2))} -{(a + b*ArcCsc[c*x])/(d + e*x^2)^2, x, 47, -(a + b*ArcCsc[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x)) + (a + b*ArcCsc[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d + e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e]) + ((I/4)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(3/2)*Sqrt[e])} -{(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^2), x, 50, -((b*c*Sqrt[1 - 1/(c^2*x^2)])/d^2) - a/(d^2*x) - (b*ArcCsc[c*x])/(d^2*x) + (e*(a + b*ArcCsc[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcCsc[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e]) - (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d + e]) + (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (((3*I)/4)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(5/2) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(5/2) + (((3*I)/4)*b*Sqrt[e]*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(-d)^(5/2) - (((3*I)/4)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(-d)^(5/2)} - - -{(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3, x, 33, (b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(8*e^2*(c^2*d + e)*(e + d/x^2)*x) - (a + b*ArcCsc[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcCsc[c*x])/(2*e^2*(e + d/x^2)) + (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]) + (b*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(c^2*d + e)^(3/2)) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) - ((a + b*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])])/e^3 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/e^3} -{(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3, x, 6, -(b*c*x*Sqrt[-1 + c^2*x^2])/(8*e*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) + (x^4*(a + b*ArcCsc[c*x]))/(4*d*(d + e*x^2)^2) + (b*c*(c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d*e^(3/2)*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])} -{(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3, x, 8, (b*c*x*Sqrt[-1 + c^2*x^2])/(8*d*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) - (a + b*ArcCsc[c*x])/(4*e*(d + e*x^2)^2) - (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*d^2*e*Sqrt[c^2*x^2]) + (b*c*(3*c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt[e]*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])} -{(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^3), x, 28, -(b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(8*d^2*(c^2*d + e)*(e + d/x^2)*x) + (e^2*(a + b*ArcCsc[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcCsc[c*x]))/(d^3*(e + d/x^2)) + ((I/2)*(a + b*ArcCsc[c*x])^2)/(b*d^3) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(d^3*Sqrt[c^2*d + e]) - (b*Sqrt[e]*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3} - -{(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3, x, 35, -(b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*c*Sqrt[-d]*Sqrt[1 - 1/(c^2*x^2)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcCsc[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcCsc[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*ArcCsc[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcCsc[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) - (3*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^2*Sqrt[c^2*d + e]) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e*(c^2*d + e)^(3/2)) - (3*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*Sqrt[d]*e^2*Sqrt[c^2*d + e]) - (3*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(5/2)) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(5/2)) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(5/2)) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(5/2))} -{(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3, x, 63, -(b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*c*Sqrt[1 - 1/(c^2*x^2)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcCsc[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcCsc[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCsc[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcCsc[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d + e)^(3/2)) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(3/2)*e*Sqrt[c^2*d + e]) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((I/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/2)) - ((I/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/2)) + ((I/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/2)) - ((I/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(3/2)*e^(3/2))} -{(a + b*ArcCsc[c*x])/(d + e*x^2)^3, x, 81, -(b*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcCsc[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcCsc[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(a + b*ArcCsc[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcCsc[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d + e)^(3/2)) + (5*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) - (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d + e)^(3/2)) + (5*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d + e]) - (3*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCsc[c*x])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCsc[c*x])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) - (((3*I)/16)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcCsc[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/((-d)^(5/2)*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcCsc[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, If[$VersionNumber>=8, 12, 13], -((b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(1680*c^5*e^2*Sqrt[c^2*x^2])) - (b*(29*c^2*d - 25*e)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e^2*Sqrt[c^2*x^2]) + (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e^2*Sqrt[c^2*x^2]) + (d^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^3) - (8*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(105*e^3*Sqrt[c^2*x^2]) + (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^6*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, If[$VersionNumber>=8, 11, 12], (b*(c^2*d + 9*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e*Sqrt[c^2*x^2]) + (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e*Sqrt[c^2*x^2]) - (d*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) + (2*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(15*e^2*Sqrt[c^2*x^2]) - (b*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(3/2)*Sqrt[c^2*x^2])} -{x*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, 9, (b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*Sqrt[c^2*x^2]) + ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e) - (b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e*Sqrt[c^2*x^2]) + (b*(3*c^2*d + e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*Sqrt[e]*Sqrt[c^2*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x, x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x])/x^3, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3, x]} - -{x^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]} -{x^0*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^2, x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x])/x^4, x, 11, -((2*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[c^2*x^2])) - (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*d*x^3) + (2*b*c^2*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(2*c^2*d + 3*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x])/x^6, x, If[$VersionNumber>=8, 12, 32], If[$VersionNumber>=8, -((b*c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(225*d^2*Sqrt[c^2*x^2])) - (b*c*(12*c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(225*d*x^2*Sqrt[c^2*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(25*d*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(15*d^2*x^3) + (b*c^2*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 30*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]), (2*b*c*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(15*d^2*Sqrt[c^2*x^2]) - (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*Sqrt[c^2*x^2]) - (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d^2*Sqrt[c^2*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(25*x^4*Sqrt[c^2*x^2]) - (b*c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(45*d*x^2*Sqrt[c^2*x^2]) - (b*c*(4*c^2*d + e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(15*d^2*x^3) - (2*b*c^2*e^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(15*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*e*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(45*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*c^2*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*c^2*(8*c^2*d - e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) - (2*b*c^2*e*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(45*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2]) + (2*b*e^2*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(15*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])]} - - -{x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x, 12, -((b*(3*c^4*d^2 - 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(560*c^5*e*Sqrt[c^2*x^2])) + (b*(13*c^2*d + 25*e)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[c^2*x^2]) + (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[c^2*x^2]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^2) + (2*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(35*e^2*Sqrt[c^2*x^2]) - (b*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x, 10, (b*(7*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(40*c^3*Sqrt[c^2*x^2]) + (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*Sqrt[c^2*x^2]) + ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e) - (b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(5*e*Sqrt[c^2*x^2]) + (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[c^2*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^1, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^3, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^3, x]} - -{x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]} -{x^0*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^2, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^2, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^4, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^4, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^6, x, 12, -((b*c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*d*Sqrt[c^2*x^2])) - (4*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(75*x^2*Sqrt[c^2*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(25*x^4*Sqrt[c^2*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*d*x^5) + (b*c^2*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(8*c^4*d^2 + 19*c^2*d*e + 15*e^2)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x])/x^8, x, 13, -((b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[c^2*x^2])) - (b*c*(120*c^4*d^2 + 159*c^2*d*e - 37*e^2)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3675*d*x^2*Sqrt[c^2*x^2]) - (b*c*(30*c^2*d + 11*e)*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*x^4*Sqrt[c^2*x^2]) - (b*c*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^6*Sqrt[c^2*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(35*d^2*x^5) + (b*c^2*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 105*e^3)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x, 11, -((b*(19*c^2*d - 9*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e^2*Sqrt[c^2*x^2])) + (b*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e^2*Sqrt[c^2*x^2]) + (d^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) - (8*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(15*e^3*Sqrt[c^2*x^2]) + (b*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x, 10, (b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e*Sqrt[c^2*x^2]) - (d*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) + (2*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^2*Sqrt[c^2*x^2]) - (b*(3*c^2*d - e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x, 9, (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e - (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e*Sqrt[c^2*x^2]) + (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[c^2*x^2])} -{(a + b*ArcCsc[c*x])/(x^1*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x^2]), x]} -{(a + b*ArcCsc[c*x])/(x^3*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^3*Sqrt[d + e*x^2]), x]} - -{x^2*(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(x^2*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2], x]} -{x^0*(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x^2]), x, 11, -((b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d*Sqrt[c^2*x^2])) - (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/(d*x) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcCsc[c*x])/(x^4*Sqrt[d + e*x^2]), x, 11, -((b*c*(2*c^2*d - 5*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*Sqrt[c^2*x^2])) - (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*x^2*Sqrt[c^2*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/(3*d^2*x) + (b*c^2*(2*c^2*d - 5*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d - 3*e)*(c^2*d + e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 10, (b*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e^2*Sqrt[c^2*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) + (8*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) - (b*(9*c^2*d - e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 9, (d*(a + b*ArcCsc[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^2 - (2*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^2*Sqrt[c^2*x^2]) + (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(3/2)*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 4, -((a + b*ArcCsc[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(Sqrt[d]*e*Sqrt[c^2*x^2])} -{(a + b*ArcCsc[c*x])/(x^1*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(3/2)), x]} - -{x^4*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]} -{x^2*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]} -{x^0*(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2), x, 5, (x*(a + b*ArcCsc[c*x]))/(d*Sqrt[d + e*x^2]) + (b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^(3/2)), x, 10, -((b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(d^2*Sqrt[c^2*x^2])) - (a + b*ArcCsc[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcCsc[c*x]))/(d^2*Sqrt[d + e*x^2]) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + 2*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 10, (b*c*d*x*Sqrt[-1 + c^2*x^2])/(3*e^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) - (d^2*(a + b*ArcCsc[c*x]))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e^3 - (8*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*e^3*Sqrt[c^2*x^2]) + (b*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[c^2*x^2])} -{x^3*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 7, -((b*c*x*Sqrt[-1 + c^2*x^2])/(3*e*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) + (d*(a + b*ArcCsc[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcCsc[c*x])/(e^2*Sqrt[d + e*x^2]) + (2*b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*Sqrt[d]*e^2*Sqrt[c^2*x^2])} -{x^1*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 5, (b*c*x*Sqrt[-1 + c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) - (a + b*ArcCsc[c*x])/(3*e*(d + e*x^2)^(3/2)) + (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(3*d^(3/2)*e*Sqrt[c^2*x^2])} -{(a + b*ArcCsc[c*x])/(x^1*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(5/2)), x]} - -{x^6*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^6*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]} -{x^4*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]} -{x^2*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 10, (b*c*x^2*Sqrt[-1 + c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2]) + (x^3*(a + b*ArcCsc[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} -{x^0*(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2), x, 10, -((b*c*e*x^2*Sqrt[-1 + c^2*x^2])/(3*d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[d + e*x^2])) + (x*(a + b*ArcCsc[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCsc[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*c^2*x*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*(c^2*d + e)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d]) + (2*b*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCsc[c x]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsc[c*x]), x, 6, If[$VersionNumber>=8, (b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCsc[c*x]))/(f^7*(7 + m)) + (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]), (b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^5*f*(6 + m)*(8 + 6*m + m^2)*(105 + 71*m + 15*m^2 + m^3)*Sqrt[c^2*x^2]) + (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 + c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[c^2*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCsc[c*x]))/(f^7*(7 + m)) + (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x, 6, If[$VersionNumber>=8, (b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2]) + (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[c^2*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]), (b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c^3*f*(4 + m)*(5 + m)*(6 + 5*m + m^2)*Sqrt[c^2*x^2]) + (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 + c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[c^2*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCsc[c*x]))/(f^5*(5 + m)) + (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])]} -{(f*x)^m*(d + e*x^2)*(a + b*ArcCsc[c*x]), x, 5, (b*e*x*(f*x)^(1 + m)*Sqrt[-1 + c^2*x^2])/(c*f*(6 + 5*m + m^2)*Sqrt[c^2*x^2]) + (d*(f*x)^(1 + m)*(a + b*ArcCsc[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcCsc[c*x]))/(f^3*(3 + m)) + (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*x*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c*f*(1 + m)^2*(2 + m)*(3 + m)*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])} -{((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2), x]} -{((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x]} - - -{(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]} -{(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x, 0, Unintegrable[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]} -{((f*x)^m*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2], x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2], x]} -{((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^4)^p (a+b ArcCsc[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^4)^(p/2) (a+b ArcCsc[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^11*(a + b*ArcCsc[c*x])/Sqrt[1 - c^4*x^4], x, 16, -((4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[1 - 1/(c^2*x^2)]*x)) + (7*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (13*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(150*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) + (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/(70*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcCsc[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcCsc[c*x]))/(10*c^12) + (4*b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[1 - 1/(c^2*x^2)]*x)} -{x^7*(a + b*ArcCsc[c*x])/Sqrt[1 - c^4*x^4], x, 13, -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[1 - 1/(c^2*x^2)]*x)) + (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(18*c^9*Sqrt[1 - 1/(c^2*x^2)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(30*c^9*Sqrt[1 - 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/(2*c^8) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcCsc[c*x]))/(6*c^8) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(3*c^9*Sqrt[1 - 1/(c^2*x^2)]*x)} -{x^3*(a + b*ArcCsc[c*x])/Sqrt[1 - c^4*x^4], x, 8, -((b*x*Sqrt[1 - c^4*x^4])/(2*c^3*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2])) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/(2*c^4) + (b*x*ArcTan[Sqrt[1 - c^4*x^4]/Sqrt[-1 + c^2*x^2]])/(2*c^3*Sqrt[c^2*x^2]), -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(2*c^5*Sqrt[1 - 1/(c^2*x^2)]*x)) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/(2*c^4) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(2*c^5*Sqrt[1 - 1/(c^2*x^2)]*x)} -{(a + b*ArcCsc[c*x])/(x^1*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]), x]} -{(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]} - - -(* ::Section:: *) -(*Integrands of the form u (a+b ArcCsc[c x])^n*) diff --git a/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.2 Inverse cosecant functions.m b/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.2 Inverse cosecant functions.m deleted file mode 100644 index 2657bb5..0000000 --- a/test/methods/rule_based/test_files/5 Inverse trig functions/5.6 Inverse cosecant/5.6.2 Inverse cosecant functions.m +++ /dev/null @@ -1,129 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Inverse Cosecants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCsc[a x^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCsc[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcCsc[a*x^5]/x, x, 7, (1/10)*I*ArcCsc[a*x^5]^2 - (1/5)*ArcCsc[a*x^5]*Log[1 - E^(2*I*ArcCsc[a*x^5])] + (1/10)*I*PolyLog[2, E^(2*I*ArcCsc[a*x^5])]} - - -{x^3*ArcCsc[Sqrt[x]], x, 4, Sqrt[-1 + x]/4 + (1/4)*(-1 + x)^(3/2) + (3/20)*(-1 + x)^(5/2) + (1/28)*(-1 + x)^(7/2) + (1/4)*x^4*ArcCsc[Sqrt[x]]} -{x^2*ArcCsc[Sqrt[x]], x, 4, Sqrt[-1 + x]/3 + (2/9)*(-1 + x)^(3/2) + (1/15)*(-1 + x)^(5/2) + (1/3)*x^3*ArcCsc[Sqrt[x]]} -{x^1*ArcCsc[Sqrt[x]], x, 4, Sqrt[-1 + x]/2 + (1/6)*(-1 + x)^(3/2) + (1/2)*x^2*ArcCsc[Sqrt[x]]} -{x^0*ArcCsc[Sqrt[x]], x, 3, Sqrt[-1 + x] + x*ArcCsc[Sqrt[x]]} -{ArcCsc[Sqrt[x]]/x^1, x, 7, I*ArcCsc[Sqrt[x]]^2 - 2*ArcCsc[Sqrt[x]]*Log[1 - E^(2*I*ArcCsc[Sqrt[x]])] + I*PolyLog[2, E^(2*I*ArcCsc[Sqrt[x]])]} -{ArcCsc[Sqrt[x]]/x^2, x, 5, -(Sqrt[-1 + x]/(2*x)) - ArcCsc[Sqrt[x]]/x - (1/2)*ArcTan[Sqrt[-1 + x]]} -{ArcCsc[Sqrt[x]]/x^3, x, 6, -(Sqrt[-1 + x]/(8*x^2)) - (3*Sqrt[-1 + x])/(16*x) - ArcCsc[Sqrt[x]]/(2*x^2) - (3/16)*ArcTan[Sqrt[-1 + x]]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2*ArcCsc[a/x], x, 5, (1/3)*a^3*Sqrt[1 - x^2/a^2] - (1/9)*a^3*(1 - x^2/a^2)^(3/2) + (1/3)*x^3*ArcSin[x/a]} -{x^1*ArcCsc[a/x], x, 4, (1/4)*a*x*Sqrt[1 - x^2/a^2] - (1/4)*a^2*ArcSin[x/a] + (1/2)*x^2*ArcSin[x/a]} -{x^0*ArcCsc[a/x], x, 3, a*Sqrt[1 - x^2/a^2] + x*ArcSin[x/a]} -{ArcCsc[a/x]/x^1, x, 6, (-(1/2))*I*ArcSin[x/a]^2 + ArcSin[x/a]*Log[1 - E^(2*I*ArcSin[x/a])] - (1/2)*I*PolyLog[2, E^(2*I*ArcSin[x/a])]} -{ArcCsc[a/x]/x^2, x, 5, -(ArcSin[x/a]/x) - ArcTanh[Sqrt[1 - x^2/a^2]]/a} -{ArcCsc[a/x]/x^3, x, 3, -(Sqrt[1 - x^2/a^2]/(2*a*x)) - ArcSin[x/a]/(2*x^2)} -{ArcCsc[a/x]/x^4, x, 6, -(Sqrt[1 - x^2/a^2]/(6*a*x^2)) - ArcSin[x/a]/(3*x^3) - ArcTanh[Sqrt[1 - x^2/a^2]]/(6*a^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCsc[a x^n] when n symbolic*) - - -{ArcCsc[a*x^n]/x, x, 7, (I*ArcCsc[a*x^n]^2)/(2*n) - (ArcCsc[a*x^n]*Log[1 - E^(2*I*ArcCsc[a*x^n])])/n + (I*PolyLog[2, E^(2*I*ArcCsc[a*x^n])])/(2*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCsc[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCsc[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcCsc[a + b*x], x, 9, -((a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(30*b^5)) - (11*a*x^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(60*b^3) + (x^3*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(20*b^2) + ((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2])/(120*b^5) + (a^5*ArcCsc[a + b*x])/(5*b^5) + (1/5)*x^5*ArcCsc[a + b*x] + ((3 + 40*a^2 + 40*a^4)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(40*b^5)} -{x^3*ArcCsc[a + b*x], x, 8, ((2 + 17*a^2)*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(12*b^4) + (x^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(12*b^2) - (a*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2])/(3*b^4) - (a^4*ArcCsc[a + b*x])/(4*b^4) + (1/4)*x^4*ArcCsc[a + b*x] - (a*(1 + 2*a^2)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(2*b^4)} -{x^2*ArcCsc[a + b*x], x, 7, -((5*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*b^3)) + (x*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*b^2) + (a^3*ArcCsc[a + b*x])/(3*b^3) + (1/3)*x^3*ArcCsc[a + b*x] + ((1 + 6*a^2)*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/(6*b^3)} -{x^1*ArcCsc[a + b*x], x, 6, ((a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(2*b^2) - (a^2*ArcCsc[a + b*x])/(2*b^2) + (1/2)*x^2*ArcCsc[a + b*x] - (a*ArcTanh[Sqrt[1 - 1/(a + b*x)^2]])/b^2} -{x^0*ArcCsc[a + b*x], x, 5, ((a + b*x)*ArcCsc[a + b*x])/b + ArcTanh[Sqrt[1 - 1/(a + b*x)^2]]/b} -{ArcCsc[a + b*x]/x^1, x, 14, ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]*Log[1 - E^(2*I*ArcCsc[a + b*x])] - I*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] - I*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] + (1/2)*I*PolyLog[2, E^(2*I*ArcCsc[a + b*x])]} -{ArcCsc[a + b*x]/x^2, x, 6, -((b*ArcCsc[a + b*x])/a) - ArcCsc[a + b*x]/x - (2*b*ArcTan[(a - Tan[(1/2)*ArcCsc[a + b*x]])/Sqrt[1 - a^2]])/(a*Sqrt[1 - a^2])} -{ArcCsc[a + b*x]/x^3, x, 8, -((b*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(2*a*(1 - a^2)*x)) + (b^2*ArcCsc[a + b*x])/(2*a^2) - ArcCsc[a + b*x]/(2*x^2) + ((1 - 2*a^2)*b^2*ArcTan[(a - Tan[(1/2)*ArcCsc[a + b*x]])/Sqrt[1 - a^2]])/(a^2*(1 - a^2)^(3/2))} -{ArcCsc[a + b*x]/x^4, x, 9, -((b*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*a*(1 - a^2)*x^2)) + ((2 - 5*a^2)*b^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcCsc[a + b*x])/(3*a^3) - ArcCsc[a + b*x]/(3*x^3) - ((2 - 5*a^2 + 6*a^4)*b^3*ArcTan[(a - Tan[(1/2)*ArcCsc[a + b*x]])/Sqrt[1 - a^2]])/(3*a^3*(1 - a^2)^(5/2))} -{ArcCsc[a + b*x]/x^5, x, 10, -((b*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(12*a*(1 - a^2)*x^3)) + ((3 - 8*a^2)*b^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(24*a^2*(1 - a^2)^2*x^2) - ((6 - 17*a^2 + 26*a^4)*b^3*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2])/(24*a^3*(1 - a^2)^3*x) + (b^4*ArcCsc[a + b*x])/(4*a^4) - ArcCsc[a + b*x]/(4*x^4) + ((2 - 7*a^2 + 8*a^4 - 8*a^6)*b^4*ArcTan[(a - Tan[(1/2)*ArcCsc[a + b*x]])/Sqrt[1 - a^2]])/(4*a^4*(1 - a^2)^(7/2))} - - -{x^3*ArcCsc[a + b*x]^2, x, 20, -((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/(3*b^4) + (3*a^2*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/b^4 - (a*(a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/b^4 + ((a + b*x)^3*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/(6*b^4) - (a^4*ArcCsc[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcCsc[a + b*x]^2 - (2*a*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^4 - (4*a^3*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^4 + Log[a + b*x]/(3*b^4) + (3*a^2*Log[a + b*x])/b^4 + (I*a*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^4 + (2*I*a^3*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^4 - (I*a*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^4 - (2*I*a^3*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^4} -{x^2*ArcCsc[a + b*x]^2, x, 17, x/(3*b^2) - (2*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/b^3 + ((a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/(3*b^3) + (a^3*ArcCsc[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcCsc[a + b*x]^2 + (2*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/(3*b^3) + (4*a^2*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^3 - (2*a*Log[a + b*x])/b^3 - (I*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/(3*b^3) - (2*I*a^2*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^3 + (I*PolyLog[2, E^(I*ArcCsc[a + b*x])])/(3*b^3) + (2*I*a^2*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^3} -{x^1*ArcCsc[a + b*x]^2, x, 11, ((a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x])/b^2 - (a^2*ArcCsc[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcCsc[a + b*x]^2 - (4*a*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^2 + Log[a + b*x]/b^2 + (2*I*a*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^2 - (2*I*a*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^2} -{x^0*ArcCsc[a + b*x]^2, x, 8, ((a + b*x)*ArcCsc[a + b*x]^2)/b + (4*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b - (2*I*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b + (2*I*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b} -{ArcCsc[a + b*x]^2/x^1, x, 17, ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^2*Log[1 - E^(2*I*ArcCsc[a + b*x])] - 2*I*ArcCsc[a + b*x]*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] - 2*I*ArcCsc[a + b*x]*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] + I*ArcCsc[a + b*x]*PolyLog[2, E^(2*I*ArcCsc[a + b*x])] + 2*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] + 2*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] - (1/2)*PolyLog[3, E^(2*I*ArcCsc[a + b*x])]} -{ArcCsc[a + b*x]^2/x^2, x, 12, -((b*ArcCsc[a + b*x]^2)/a) - ArcCsc[a + b*x]^2/x - (2*I*b*ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (2*I*b*ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2]) + (2*b*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2])} - - -{x^2*ArcCsc[a + b*x]^3, x, 25, ((a + b*x)*ArcCsc[a + b*x])/b^3 - (3*I*a*ArcCsc[a + b*x]^2)/b^3 - (3*a*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x]^2)/b^3 + ((a + b*x)^2*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x]^2)/(2*b^3) + (a^3*ArcCsc[a + b*x]^3)/(3*b^3) + (1/3)*x^3*ArcCsc[a + b*x]^3 + (ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^3 + (6*a^2*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^3 + ArcTanh[Sqrt[1 - 1/(a + b*x)^2]]/b^3 + (6*a*ArcCsc[a + b*x]*Log[1 - E^(2*I*ArcCsc[a + b*x])])/b^3 - (I*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^3 - (6*I*a^2*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^3 + (I*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^3 + (6*I*a^2*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^3 - (3*I*a*PolyLog[2, E^(2*I*ArcCsc[a + b*x])])/b^3 + PolyLog[3, -E^(I*ArcCsc[a + b*x])]/b^3 + (6*a^2*PolyLog[3, -E^(I*ArcCsc[a + b*x])])/b^3 - PolyLog[3, E^(I*ArcCsc[a + b*x])]/b^3 - (6*a^2*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b^3} -{x^1*ArcCsc[a + b*x]^3, x, 16, (3*I*ArcCsc[a + b*x]^2)/(2*b^2) + (3*(a + b*x)*Sqrt[1 - 1/(a + b*x)^2]*ArcCsc[a + b*x]^2)/(2*b^2) - (a^2*ArcCsc[a + b*x]^3)/(2*b^2) + (1/2)*x^2*ArcCsc[a + b*x]^3 - (6*a*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^2 - (3*ArcCsc[a + b*x]*Log[1 - E^(2*I*ArcCsc[a + b*x])])/b^2 + (6*I*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^2 - (6*I*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^2 + (3*I*PolyLog[2, E^(2*I*ArcCsc[a + b*x])])/(2*b^2) - (6*a*PolyLog[3, -E^(I*ArcCsc[a + b*x])])/b^2 + (6*a*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b^2} -{x^0*ArcCsc[a + b*x]^3, x, 10, ((a + b*x)*ArcCsc[a + b*x]^3)/b + (6*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b - (6*I*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b + (6*I*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b + (6*PolyLog[3, -E^(I*ArcCsc[a + b*x])])/b - (6*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b} -{ArcCsc[a + b*x]^3/x^1, x, 20, ArcCsc[a + b*x]^3*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]^3*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^3*Log[1 - E^(2*I*ArcCsc[a + b*x])] - 3*I*ArcCsc[a + b*x]^2*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] - 3*I*ArcCsc[a + b*x]^2*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] + (3/2)*I*ArcCsc[a + b*x]^2*PolyLog[2, E^(2*I*ArcCsc[a + b*x])] + 6*ArcCsc[a + b*x]*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] + 6*ArcCsc[a + b*x]*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] - (3/2)*ArcCsc[a + b*x]*PolyLog[3, E^(2*I*ArcCsc[a + b*x])] + 6*I*PolyLog[4, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))] + 6*I*PolyLog[4, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))] - (3/4)*I*PolyLog[4, E^(2*I*ArcCsc[a + b*x])]} -{ArcCsc[a + b*x]^3/x^2, x, 14, -((b*ArcCsc[a + b*x]^3)/a) - ArcCsc[a + b*x]^3/x - (3*I*b*ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (3*I*b*ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (6*b*ArcCsc[a + b*x]*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2]) + (6*b*ArcCsc[a + b*x]*PolyLog[2, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2]) - (6*I*b*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2]) + (6*I*b*PolyLog[3, -((I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]))])/(a*Sqrt[1 - a^2])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCsc[a +b x^n]*) - - -{x^3*ArcCsc[a + b*x^4], x, 6, ((a + b*x^4)*ArcCsc[a + b*x^4])/(4*b) + ArcTanh[Sqrt[1 - 1/(a + b*x^4)^2]]/(4*b)} - - -{x^(n-1)*ArcCsc[a + b*x^n], x, 6, ((a + b*x^n)*ArcCsc[a + b*x^n])/(b*n) + ArcTanh[Sqrt[1 - 1/(a + b*x^n)^2]]/(b*n)} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse cosecant functions of exponentials*) - - -{ArcCsc[c*E^(a + b*x)], x, 7, (I*ArcCsc[c*E^(a + b*x)]^2)/(2*b) - (ArcCsc[c*E^(a + b*x)]*Log[1 - E^(2*I*ArcCsc[c*E^(a + b*x)])])/b + (I*PolyLog[2, E^(2*I*ArcCsc[c*E^(a + b*x)])])/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse cosecant functions*) - - -{E^ArcCsc[a*x]*x^2, x, 6, ((4/5 - (12*I)/5)*E^((1 + 3*I)*ArcCsc[a*x])*Hypergeometric2F1[3/2 - I/2, 3, 5/2 - I/2, E^(2*I*ArcCsc[a*x])])/a^3 - ((8/5 - (24*I)/5)*E^((1 + 3*I)*ArcCsc[a*x])*Hypergeometric2F1[3/2 - I/2, 4, 5/2 - I/2, E^(2*I*ArcCsc[a*x])])/a^3} -{E^ArcCsc[a*x]*x^1, x, 6, ((8/5 + (4*I)/5)*E^((1 + 2*I)*ArcCsc[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, E^(2*I*ArcCsc[a*x])])/a^2 - ((16/5 + (8*I)/5)*E^((1 + 2*I)*ArcCsc[a*x])*Hypergeometric2F1[1 - I/2, 3, 2 - I/2, E^(2*I*ArcCsc[a*x])])/a^2} -{E^ArcCsc[a*x]*x^0, x, 5, -(((1 - I)*E^((1 + I)*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, E^(2*I*ArcCsc[a*x])])/a) + ((2 - 2*I)*E^((1 + I)*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/2, 2, 3/2 - I/2, E^(2*I*ArcCsc[a*x])])/a} -{E^ArcCsc[a*x]/x^1, x, 6, (-I)*E^ArcCsc[a*x] + 2*I*E^ArcCsc[a*x]*Hypergeometric2F1[-(I/2), 1, 1 - I/2, E^(2*I*ArcCsc[a*x])]} -{E^ArcCsc[a*x]/x^2, x, 3, (-(1/2))*a*E^ArcCsc[a*x]*Sqrt[1 - 1/(a^2*x^2)] - E^ArcCsc[a*x]/(2*x)} -{E^ArcCsc[a*x]/x^3, x, 5, (1/5)*a^2*E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]] - (1/10)*a^2*E^ArcCsc[a*x]*Sin[2*ArcCsc[a*x]]} -{E^ArcCsc[a*x]/x^4, x, 6, (-(1/8))*a^3*E^ArcCsc[a*x]*Sqrt[1 - 1/(a^2*x^2)] - (a^2*E^ArcCsc[a*x])/(8*x) + (1/40)*a^3*E^ArcCsc[a*x]*Cos[3*ArcCsc[a*x]] + (3/40)*a^3*E^ArcCsc[a*x]*Sin[3*ArcCsc[a*x]]} -{E^ArcCsc[a*x]/x^5, x, 6, (1/10)*a^4*E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]] - (1/34)*a^4*E^ArcCsc[a*x]*Cos[4*ArcCsc[a*x]] - (1/20)*a^4*E^ArcCsc[a*x]*Sin[2*ArcCsc[a*x]] + (1/136)*a^4*E^ArcCsc[a*x]*Sin[4*ArcCsc[a*x]]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse cosecants*) - - -{ArcCsc[a + b*x]/((a*d)/b + d*x), x, 8, (I*ArcCsc[a + b*x]^2)/(2*d) - (ArcCsc[a + b*x]*Log[1 - E^(2*I*ArcCsc[a + b*x])])/d + (I*PolyLog[2, E^(2*I*ArcCsc[a + b*x])])/(2*d)} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.1 (c+d x)^m (a+b sinh)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.1 (c+d x)^m (a+b sinh)^n.m deleted file mode 100644 index ee141b7..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.1 (c+d x)^m (a+b sinh)^n.m +++ /dev/null @@ -1,1030 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m (b Sinh[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sinh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sinh[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^4*Sinh[a + b*x], x, 5, (24*d^4*Cosh[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Cosh[a + b*x])/b^3 + ((c + d*x)^4*Cosh[a + b*x])/b - (24*d^3*(c + d*x)*Sinh[a + b*x])/b^4 - (4*d*(c + d*x)^3*Sinh[a + b*x])/b^2} -{(c + d*x)^3*Sinh[a + b*x], x, 4, (6*d^2*(c + d*x)*Cosh[a + b*x])/b^3 + ((c + d*x)^3*Cosh[a + b*x])/b - (6*d^3*Sinh[a + b*x])/b^4 - (3*d*(c + d*x)^2*Sinh[a + b*x])/b^2} -{(c + d*x)^2*Sinh[a + b*x], x, 3, (2*d^2*Cosh[a + b*x])/b^3 + ((c + d*x)^2*Cosh[a + b*x])/b - (2*d*(c + d*x)*Sinh[a + b*x])/b^2} -{(c + d*x)*Sinh[a + b*x], x, 2, ((c + d*x)*Cosh[a + b*x])/b - (d*Sinh[a + b*x])/b^2} -{Sinh[a + b*x]/(c + d*x), x, 3, (CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/d + (Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d} -{Sinh[a + b*x]/(c + d*x)^2, x, 4, (b*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d^2 - Sinh[a + b*x]/(d*(c + d*x)) + (b*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d^2} -{Sinh[a + b*x]/(c + d*x)^3, x, 5, -(b*Cosh[a + b*x])/(2*d^2*(c + d*x)) + (b^2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(2*d^3) - Sinh[a + b*x]/(2*d*(c + d*x)^2) + (b^2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(2*d^3)} - - -{(c + d*x)^4*Sinh[a + b*x]^2, x, 6, (-3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) - (c + d*x)^5/(10*d) + (3*d^4*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^5) + (3*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d^3*(c + d*x)*Sinh[a + b*x]^2)/(2*b^4) - (d*(c + d*x)^3*Sinh[a + b*x]^2)/b^2} -{(c + d*x)^3*Sinh[a + b*x]^2, x, 4, (-3*c*d^2*x)/(4*b^2) - (3*d^3*x^2)/(8*b^2) - (c + d*x)^4/(8*d) + (3*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d^3*Sinh[a + b*x]^2)/(8*b^4) - (3*d*(c + d*x)^2*Sinh[a + b*x]^2)/(4*b^2)} -{(c + d*x)^2*Sinh[a + b*x]^2, x, 4, -(d^2*x)/(4*b^2) - (c + d*x)^3/(6*d) + (d^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (d*(c + d*x)*Sinh[a + b*x]^2)/(2*b^2)} -{(c + d*x)*Sinh[a + b*x]^2, x, 2, -(c*x)/2 - (d*x^2)/4 + ((c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (d*Sinh[a + b*x]^2)/(4*b^2)} -{Sinh[a + b*x]^2/(c + d*x), x, 5, (Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/(2*d) - Log[c + d*x]/(2*d) + (Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Sinh[a + b*x]^2/(c + d*x)^2, x, 5, (b*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d^2 - Sinh[a + b*x]^2/(d*(c + d*x)) + (b*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Sinh[a + b*x]^2/(c + d*x)^3, x, 7, (b^2*Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/d^3 - (b*Cosh[a + b*x]*Sinh[a + b*x])/(d^2*(c + d*x)) - Sinh[a + b*x]^2/(2*d*(c + d*x)^2) + (b^2*Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^3} -{Sinh[a + b*x]^2/(c + d*x)^4, x, 7, -b^2/(3*d^3*(c + d*x)) + (2*b^3*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/(3*d^4) - (b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*(c + d*x)^2) - Sinh[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*Sinh[a + b*x]^2)/(3*d^3*(c + d*x)) + (2*b^3*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)} - - -{(c + d*x)^4*Sinh[a + b*x]^3, x, 12, (-488*d^4*Cosh[a + b*x])/(27*b^5) - (80*d^2*(c + d*x)^2*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^4*Cosh[a + b*x])/(3*b) + (8*d^4*Cosh[a + b*x]^3)/(81*b^5) + (160*d^3*(c + d*x)*Sinh[a + b*x])/(9*b^4) + (8*d*(c + d*x)^3*Sinh[a + b*x])/(3*b^2) + (4*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (8*d^3*(c + d*x)*Sinh[a + b*x]^3)/(27*b^4) - (4*d*(c + d*x)^3*Sinh[a + b*x]^3)/(9*b^2)} -{(c + d*x)^3*Sinh[a + b*x]^3, x, 8, (-40*d^2*(c + d*x)*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^3*Cosh[a + b*x])/(3*b) + (40*d^3*Sinh[a + b*x])/(9*b^4) + (2*d*(c + d*x)^2*Sinh[a + b*x])/b^2 + (2*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^3) + ((c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (2*d^3*Sinh[a + b*x]^3)/(27*b^4) - (d*(c + d*x)^2*Sinh[a + b*x]^3)/(3*b^2)} -{(c + d*x)^2*Sinh[a + b*x]^3, x, 6, (-14*d^2*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^2*Cosh[a + b*x])/(3*b) + (2*d^2*Cosh[a + b*x]^3)/(27*b^3) + (4*d*(c + d*x)*Sinh[a + b*x])/(3*b^2) + ((c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (2*d*(c + d*x)*Sinh[a + b*x]^3)/(9*b^2)} -{(c + d*x)*Sinh[a + b*x]^3, x, 3, (-2*(c + d*x)*Cosh[a + b*x])/(3*b) + (2*d*Sinh[a + b*x])/(3*b^2) + ((c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (d*Sinh[a + b*x]^3)/(9*b^2)} -{Sinh[a + b*x]^3/(c + d*x), x, 8, (CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(4*d) - (3*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(4*d) - (3*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(4*d) + (Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{Sinh[a + b*x]^3/(c + d*x)^2, x, 8, (-3*b*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*c)/d + 3*b*x])/(4*d^2) - Sinh[a + b*x]^3/(d*(c + d*x)) - (3*b*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{Sinh[a + b*x]^3/(c + d*x)^3, x, 12, (9*b^2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(8*d^3) - (3*b^2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(8*d^3) - (3*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(2*d^2*(c + d*x)) - Sinh[a + b*x]^3/(2*d*(c + d*x)^2) - (3*b^2*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3*Csch[a + b*x], x, 9, (-2*(c + d*x)^3*ArcTanh[E^(a + b*x)])/b - (3*d*(c + d*x)^2*PolyLog[2, -E^(a + b*x)])/b^2 + (3*d*(c + d*x)^2*PolyLog[2, E^(a + b*x)])/b^2 + (6*d^2*(c + d*x)*PolyLog[3, -E^(a + b*x)])/b^3 - (6*d^2*(c + d*x)*PolyLog[3, E^(a + b*x)])/b^3 - (6*d^3*PolyLog[4, -E^(a + b*x)])/b^4 + (6*d^3*PolyLog[4, E^(a + b*x)])/b^4} -{(c + d*x)^2*Csch[a + b*x], x, 7, (-2*(c + d*x)^2*ArcTanh[E^(a + b*x)])/b - (2*d*(c + d*x)*PolyLog[2, -E^(a + b*x)])/b^2 + (2*d*(c + d*x)*PolyLog[2, E^(a + b*x)])/b^2 + (2*d^2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*d^2*PolyLog[3, E^(a + b*x)])/b^3} -{(c + d*x)*Csch[a + b*x], x, 5, (-2*(c + d*x)*ArcTanh[E^(a + b*x)])/b - (d*PolyLog[2, -E^(a + b*x)])/b^2 + (d*PolyLog[2, E^(a + b*x)])/b^2} -{Csch[a + b*x]/(c + d*x), x, 0, Unintegrable[Csch[a + b*x]/(c + d*x), x]} -{Csch[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Csch[a + b*x]/(c + d*x)^2, x]} - - -{(c + d*x)^3*Csch[a + b*x]^2, x, 6, -((c + d*x)^3/b) - ((c + d*x)^3*Coth[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(2*(a + b*x))])/b^2 + (3*d^2*(c + d*x)*PolyLog[2, E^(2*(a + b*x))])/b^3 - (3*d^3*PolyLog[3, E^(2*(a + b*x))])/(2*b^4)} -{(c + d*x)^2*Csch[a + b*x]^2, x, 5, -((c + d*x)^2/b) - ((c + d*x)^2*Coth[a + b*x])/b + (2*d*(c + d*x)*Log[1 - E^(2*(a + b*x))])/b^2 + (d^2*PolyLog[2, E^(2*(a + b*x))])/b^3} -{(c + d*x)*Csch[a + b*x]^2, x, 2, -(((c + d*x)*Coth[a + b*x])/b) + (d*Log[Sinh[a + b*x]])/b^2} -{Csch[a + b*x]^2/(c + d*x), x, 0, Unintegrable[Csch[a + b*x]^2/(c + d*x), x]} -{Csch[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Csch[a + b*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*Csch[a + b*x]^3, x, 15, (-6*d^2*(c + d*x)*ArcTanh[E^(a + b*x)])/b^3 + ((c + d*x)^3*ArcTanh[E^(a + b*x)])/b - (3*d*(c + d*x)^2*Csch[a + b*x])/(2*b^2) - ((c + d*x)^3*Coth[a + b*x]*Csch[a + b*x])/(2*b) - (3*d^3*PolyLog[2, -E^(a + b*x)])/b^4 + (3*d*(c + d*x)^2*PolyLog[2, -E^(a + b*x)])/(2*b^2) + (3*d^3*PolyLog[2, E^(a + b*x)])/b^4 - (3*d*(c + d*x)^2*PolyLog[2, E^(a + b*x)])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(a + b*x)])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, E^(a + b*x)])/b^3 + (3*d^3*PolyLog[4, -E^(a + b*x)])/b^4 - (3*d^3*PolyLog[4, E^(a + b*x)])/b^4} -{(c + d*x)^2*Csch[a + b*x]^3, x, 9, ((c + d*x)^2*ArcTanh[E^(a + b*x)])/b - (d^2*ArcTanh[Cosh[a + b*x]])/b^3 - (d*(c + d*x)*Csch[a + b*x])/b^2 - ((c + d*x)^2*Coth[a + b*x]*Csch[a + b*x])/(2*b) + (d*(c + d*x)*PolyLog[2, -E^(a + b*x)])/b^2 - (d*(c + d*x)*PolyLog[2, E^(a + b*x)])/b^2 - (d^2*PolyLog[3, -E^(a + b*x)])/b^3 + (d^2*PolyLog[3, E^(a + b*x)])/b^3} -{(c + d*x)*Csch[a + b*x]^3, x, 6, ((c + d*x)*ArcTanh[E^(a + b*x)])/b - (d*Csch[a + b*x])/(2*b^2) - ((c + d*x)*Coth[a + b*x]*Csch[a + b*x])/(2*b) + (d*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (d*PolyLog[2, E^(a + b*x)])/(2*b^2)} -{Csch[a + b*x]^3/(c + d*x), x, 0, Unintegrable[Csch[a + b*x]^3/(c + d*x), x]} -{Csch[a + b*x]^3/(c + d*x)^2, x, 0, Unintegrable[Csch[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Sinh[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^(5/2)*Sinh[a + b*x], x, 8, (15*d^2*Sqrt[c + d*x]*Cosh[a + b*x])/(4*b^3) + ((c + d*x)^(5/2)*Cosh[a + b*x])/b - (15*d^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (15*d^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d*(c + d*x)^(3/2)*Sinh[a + b*x])/(2*b^2)} -{(c + d*x)^(3/2)*Sinh[a + b*x], x, 7, ((c + d*x)^(3/2)*Cosh[a + b*x])/b - (3*d^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + (3*d^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) - (3*d*Sqrt[c + d*x]*Sinh[a + b*x])/(2*b^2)} -{Sqrt[c + d*x]*Sinh[a + b*x], x, 6, (Sqrt[c + d*x]*Cosh[a + b*x])/b - (Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) - (Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2))} -{Sinh[a + b*x]/Sqrt[c + d*x], x, 5, -(E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d])} -{Sinh[a + b*x]/(c + d*x)^(3/2), x, 6, (Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[b]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (2*Sinh[a + b*x])/(d*Sqrt[c + d*x])} -{Sinh[a + b*x]/(c + d*x)^(5/2), x, 7, (-4*b*Cosh[a + b*x])/(3*d^2*Sqrt[c + d*x]) - (2*b^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*b^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*Sinh[a + b*x])/(3*d*(c + d*x)^(3/2))} -{Sinh[a + b*x]/(c + d*x)^(7/2), x, 8, (-4*b*Cosh[a + b*x])/(15*d^2*(c + d*x)^(3/2)) + (4*b^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (4*b^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) - (2*Sinh[a + b*x])/(5*d*(c + d*x)^(5/2)) - (8*b^2*Sinh[a + b*x])/(15*d^3*Sqrt[c + d*x])} - - -{(c + d*x)^(5/2)*Sinh[a + b*x]^2, x, 10, (-5*d*(c + d*x)^(3/2))/(16*b^2) - (c + d*x)^(7/2)/(7*d) + (15*d^(5/2)*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(256*b^(7/2)) - (15*d^(5/2)*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(256*b^(7/2)) + ((c + d*x)^(5/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (5*d*(c + d*x)^(3/2)*Sinh[a + b*x]^2)/(8*b^2) + (15*d^2*Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(64*b^3)} -{(c + d*x)^(3/2)*Sinh[a + b*x]^2, x, 9, (-3*d*Sqrt[c + d*x])/(16*b^2) - (c + d*x)^(5/2)/(5*d) + (3*d^(3/2)*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + (3*d^(3/2)*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) - (3*d*Sqrt[c + d*x]*Sinh[a + b*x]^2)/(8*b^2)} -{Sqrt[c + d*x]*Sinh[a + b*x]^2, x, 8, -(c + d*x)^(3/2)/(3*d) + (Sqrt[d]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) + (Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(4*b)} -{Sinh[a + b*x]^2/Sqrt[c + d*x], x, 7, -(Sqrt[c + d*x]/d) + (E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d]) + (E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d])} -{Sinh[a + b*x]^2/(c + d*x)^(3/2), x, 7, -((Sqrt[b]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2)) + (Sqrt[b]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) - (2*Sinh[a + b*x]^2)/(d*Sqrt[c + d*x])} -{Sinh[a + b*x]^2/(c + d*x)^(5/2), x, 9, (2*b^(3/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*b^(3/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*Sqrt[c + d*x]) - (2*Sinh[a + b*x]^2)/(3*d*(c + d*x)^(3/2))} -{Sinh[a + b*x]^2/(c + d*x)^(7/2), x, 9, (-16*b^2)/(15*d^3*Sqrt[c + d*x]) - (8*b^(5/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (8*b^(5/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(15*d^2*(c + d*x)^(3/2)) - (2*Sinh[a + b*x]^2)/(5*d*(c + d*x)^(5/2)) - (32*b^2*Sinh[a + b*x]^2)/(15*d^3*Sqrt[c + d*x])} -{Sinh[a + b*x]^2/(c + d*x)^(9/2), x, 11, (-16*b^2)/(105*d^3*(c + d*x)^(3/2)) + (32*b^(7/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(105*d^(9/2)) + (32*b^(7/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(105*d^(9/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(35*d^2*(c + d*x)^(5/2)) - (128*b^3*Cosh[a + b*x]*Sinh[a + b*x])/(105*d^4*Sqrt[c + d*x]) - (2*Sinh[a + b*x]^2)/(7*d*(c + d*x)^(7/2)) - (32*b^2*Sinh[a + b*x]^2)/(105*d^3*(c + d*x)^(3/2))} - - -{(c + d*x)^(5/2)*Sinh[a + b*x]^3, x, 23, (-45*d^2*Sqrt[c + d*x]*Cosh[a + b*x])/(16*b^3) - (2*(c + d*x)^(5/2)*Cosh[a + b*x])/(3*b) + (5*d^2*Sqrt[c + d*x]*Cosh[3*a + 3*b*x])/(144*b^3) + (45*d^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) - (5*d^(5/2)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (45*d^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) - (5*d^(5/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sinh[a + b*x])/(3*b^2) + ((c + d*x)^(5/2)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (5*d*(c + d*x)^(3/2)*Sinh[a + b*x]^3)/(18*b^2)} -{(c + d*x)^(3/2)*Sinh[a + b*x]^3, x, 20, (-2*(c + d*x)^(3/2)*Cosh[a + b*x])/(3*b) + (9*d^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(5/2)) - (d^(3/2)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) - (9*d^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(5/2)) + (d^(3/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (d*Sqrt[c + d*x]*Sinh[a + b*x])/b^2 + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (d*Sqrt[c + d*x]*Sinh[a + b*x]^3)/(6*b^2)} -{Sqrt[c + d*x]*Sinh[a + b*x]^3, x, 14, (-3*Sqrt[c + d*x]*Cosh[a + b*x])/(4*b) + (Sqrt[c + d*x]*Cosh[3*a + 3*b*x])/(12*b) + (3*Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (3*Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2))} -{Sinh[a + b*x]^3/Sqrt[c + d*x], x, 12, (3*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) - (E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) - (3*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d])} -{Sinh[a + b*x]^3/(c + d*x)^(3/2), x, 12, (-3*Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (3*Sqrt[b]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (2*Sinh[a + b*x]^3)/(d*Sqrt[c + d*x])} -{Sinh[a + b*x]^3/(c + d*x)^(5/2), x, 18, (b^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) - (b^(3/2)*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) - (b^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) + (b^(3/2)*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) - (4*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(d^2*Sqrt[c + d*x]) - (2*Sinh[a + b*x]^3)/(3*d*(c + d*x)^(3/2))} -{Sinh[a + b*x]^3/(c + d*x)^(7/2), x, 19, -(b^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (3*b^(5/2)*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (b^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (3*b^(5/2)*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (16*b^2*Sinh[a + b*x])/(5*d^3*Sqrt[c + d*x]) - (4*b*Cosh[a + b*x]*Sinh[a + b*x]^2)/(5*d^2*(c + d*x)^(3/2)) - (2*Sinh[a + b*x]^3)/(5*d*(c + d*x)^(5/2)) - (24*b^2*Sinh[a + b*x]^3)/(5*d^3*Sqrt[c + d*x])} - - -{(d*x)^(3/2)*Sinh[f*x], x, 7, ((d*x)^(3/2)*Cosh[f*x])/f - (3*d^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) + (3*d^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) - (3*d*Sqrt[d*x]*Sinh[f*x])/(2*f^2)} -{Sqrt[d*x]*Sinh[f*x], x, 6, (Sqrt[d*x]*Cosh[f*x])/f - (Sqrt[d]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2)) - (Sqrt[d]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2))} -{Sinh[f*x]/Sqrt[d*x], x, 5, -(Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f])} -{Sinh[f*x]/(d*x)^(3/2), x, 6, (Sqrt[f]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[f]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Sinh[f*x])/(d*Sqrt[d*x])} -{Sinh[f*x]/(d*x)^(5/2), x, 7, (-4*f*Cosh[f*x])/(3*d^2*Sqrt[d*x]) - (2*f^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*f^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*Sinh[f*x])/(3*d*(d*x)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sqrt[c + d*x]*Csch[a + b*x], x, 0, Unintegrable[Sqrt[c + d*x]*Csch[a + b*x], x]} -{Csch[a + b*x]/Sqrt[c + d*x], x, 0, Unintegrable[Csch[a + b*x]/Sqrt[c + d*x], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sinh[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sinh[x]^(3/2)/x^3, x, 1, (-3*Cosh[x]*Sqrt[Sinh[x]])/(4*x) - Sinh[x]^(3/2)/(2*x^2) + (3*Unintegrable[1/(x*Sqrt[Sinh[x]]), x])/8 + (9*Unintegrable[Sinh[x]^(3/2)/x, x])/8} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x/Sinh[x]^(3/2) - x*Sqrt[Sinh[x]], x, 2, (-2*x*Cosh[x])/Sqrt[Sinh[x]] + 4*Sqrt[Sinh[x]]} -{x/Sinh[x]^(5/2) + x/(3*Sqrt[Sinh[x]]), x, 2, (-2*x*Cosh[x])/(3*Sinh[x]^(3/2)) - 4/(3*Sqrt[Sinh[x]])} -{x/Sinh[x]^(7/2) + (3*x*Sqrt[Sinh[x]])/5, x, 3, (-2*x*Cosh[x])/(5*Sinh[x]^(5/2)) - 4/(15*Sinh[x]^(3/2)) + (6*x*Cosh[x])/(5*Sqrt[Sinh[x]]) - (12*Sqrt[Sinh[x]])/5} -{x^2/Sinh[x]^(3/2) - x^2*Sqrt[Sinh[x]], x, 4, -((2*x^2*Cosh[x])/Sqrt[Sinh[x]]) + 8*x*Sqrt[Sinh[x]] - (16*I*EllipticE[Pi/4 - (I*x)/2, 2]*Sqrt[Sinh[x]])/Sqrt[I*Sinh[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sinh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(b*Sinh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(b*Sinh[e + f*x])^n, x]} - - -{(c + d*x)^m*Sinh[a + b*x]^3, x, 8, (3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b*(c + d*x))/d])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m) + (3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (3*b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Sinh[a + b*x]^2, x, 5, -(c + d*x)^(1 + m)/(2*d*(1 + m)) + (2^(-3 - m)*E^(2*a - (2*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-2*b*(c + d*x))/d])/(b*(-((b*(c + d*x))/d))^m) - (2^(-3 - m)*E^(-2*a + (2*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (2*b*(c + d*x))/d])/(b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Sinh[a + b*x], x, 3, (E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(2*b*(-((b*(c + d*x))/d))^m) + (E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(2*b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Csch[a + b*x], x, 0, Unintegrable[(c + d*x)^m*Csch[a + b*x], x]} -{(c + d*x)^m*Csch[a + b*x]^2, x, 0, Unintegrable[(c + d*x)^m*Csch[a + b*x]^2, x]} - - -{x^(3 + m)*Sinh[a + b*x], x, 3, -(E^a*x^m*Gamma[4 + m, -(b*x)])/(2*b^4*(-(b*x))^m) + (x^m*Gamma[4 + m, b*x])/(2*b^4*E^a*(b*x)^m)} -{x^(2 + m)*Sinh[a + b*x], x, 3, (E^a*x^m*Gamma[3 + m, -(b*x)])/(2*b^3*(-(b*x))^m) + (x^m*Gamma[3 + m, b*x])/(2*b^3*E^a*(b*x)^m)} -{x^(1 + m)*Sinh[a + b*x], x, 3, -(E^a*x^m*Gamma[2 + m, -(b*x)])/(2*b^2*(-(b*x))^m) + (x^m*Gamma[2 + m, b*x])/(2*b^2*E^a*(b*x)^m)} -{x^m*Sinh[a + b*x], x, 3, (E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) + (x^m*Gamma[1 + m, b*x])/(2*b*E^a*(b*x)^m)} -{x^(-1 + m)*Sinh[a + b*x], x, 3, -(E^a*x^m*Gamma[m, -(b*x)])/(2*(-(b*x))^m) + (x^m*Gamma[m, b*x])/(2*E^a*(b*x)^m)} -{x^(-2 + m)*Sinh[a + b*x], x, 3, (b*E^a*x^m*Gamma[-1 + m, -(b*x)])/(2*(-(b*x))^m) + (b*x^m*Gamma[-1 + m, b*x])/(2*E^a*(b*x)^m)} -{x^(-3 + m)*Sinh[a + b*x], x, 3, -(b^2*E^a*x^m*Gamma[-2 + m, -(b*x)])/(2*(-(b*x))^m) + (b^2*x^m*Gamma[-2 + m, b*x])/(2*E^a*(b*x)^m)} - - -{x^(3 + m)*Sinh[a + b*x]^2, x, 5, -x^(4 + m)/(2*(4 + m)) - (2^(-6 - m)*E^(2*a)*x^m*Gamma[4 + m, -2*b*x])/(b^4*(-(b*x))^m) - (2^(-6 - m)*x^m*Gamma[4 + m, 2*b*x])/(b^4*E^(2*a)*(b*x)^m)} -{x^(2 + m)*Sinh[a + b*x]^2, x, 5, -x^(3 + m)/(2*(3 + m)) + (2^(-5 - m)*E^(2*a)*x^m*Gamma[3 + m, -2*b*x])/(b^3*(-(b*x))^m) - (2^(-5 - m)*x^m*Gamma[3 + m, 2*b*x])/(b^3*E^(2*a)*(b*x)^m)} -{x^(1 + m)*Sinh[a + b*x]^2, x, 5, -x^(2 + m)/(2*(2 + m)) - (2^(-4 - m)*E^(2*a)*x^m*Gamma[2 + m, -2*b*x])/(b^2*(-(b*x))^m) - (2^(-4 - m)*x^m*Gamma[2 + m, 2*b*x])/(b^2*E^(2*a)*(b*x)^m)} -{x^m*Sinh[a + b*x]^2, x, 5, -x^(1 + m)/(2*(1 + m)) + (2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) - (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)*(b*x)^m)} -{x^(-1 + m)*Sinh[a + b*x]^2, x, 5, -x^m/(2*m) - (2^(-2 - m)*E^(2*a)*x^m*Gamma[m, -2*b*x])/(-(b*x))^m - (2^(-2 - m)*x^m*Gamma[m, 2*b*x])/(E^(2*a)*(b*x)^m)} -{x^(-2 + m)*Sinh[a + b*x]^2, x, 5, x^(-1 + m)/(2*(1 - m)) + (2^(-1 - m)*b*E^(2*a)*x^m*Gamma[-1 + m, -2*b*x])/(-(b*x))^m - (2^(-1 - m)*b*x^m*Gamma[-1 + m, 2*b*x])/(E^(2*a)*(b*x)^m)} -{x^(-3 + m)*Sinh[a + b*x]^2, x, 5, x^(-2 + m)/(2*(2 - m)) - (b^2*E^(2*a)*x^m*Gamma[-2 + m, -2*b*x])/(2^m*(-(b*x))^m) - (b^2*x^m*Gamma[-2 + m, 2*b*x])/(2^m*E^(2*a)*(b*x)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Csch[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Csch[e+f x])^(n/2)*) - - -{x/Csch[x]^(3/2) + (x*Sqrt[Csch[x]])/3, x, 4, -4/(9*Csch[x]^(3/2)) + (2*x*Cosh[x])/(3*Sqrt[Csch[x]])} -{x/Csch[x]^(5/2) + (3*x)/(5*Sqrt[Csch[x]]), x, 4, -4/(25*Csch[x]^(5/2)) + (2*x*Cosh[x])/(5*Csch[x]^(3/2))} -{x/Csch[x]^(7/2) - (5*x*Sqrt[Csch[x]])/21, x, 5, -4/(49*Csch[x]^(7/2)) + (2*x*Cosh[x])/(7*Csch[x]^(5/2)) + 20/(63*Csch[x]^(3/2)) - (10*x*Cosh[x])/(21*Sqrt[Csch[x]])} -{x^2/Csch[x]^(3/2) + (x^2*Sqrt[Csch[x]])/3, x, 7, -((8*x)/(9*Csch[x]^(3/2))) + (16*Cosh[x])/(27*Sqrt[Csch[x]]) + (2*x^2*Cosh[x])/(3*Sqrt[Csch[x]]) - (16/27)*I*Sqrt[Csch[x]]*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sinh[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sinh[e+f x])^n with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Sinh[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + I*a*Sinh[e + f*x]), x, 6, (a*(c + d*x)^4)/(4*d) + (6*I*a*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (I*a*(c + d*x)^3*Cosh[e + f*x])/f - (6*I*a*d^3*Sinh[e + f*x])/f^4 - (3*I*a*d*(c + d*x)^2*Sinh[e + f*x])/f^2} -{(c + d*x)^2*(a + I*a*Sinh[e + f*x]), x, 5, (a*(c + d*x)^3)/(3*d) + (2*I*a*d^2*Cosh[e + f*x])/f^3 + (I*a*(c + d*x)^2*Cosh[e + f*x])/f - (2*I*a*d*(c + d*x)*Sinh[e + f*x])/f^2} -{(c + d*x)*(a + I*a*Sinh[e + f*x]), x, 4, (a*(c + d*x)^2)/(2*d) + (I*a*(c + d*x)*Cosh[e + f*x])/f - (I*a*d*Sinh[e + f*x])/f^2} -{(a + I*a*Sinh[e + f*x])/(c + d*x), x, 5, (a*Log[c + d*x])/d + (I*a*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d + (I*a*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d} -{(a + I*a*Sinh[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) + (I*a*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (I*a*Sinh[e + f*x])/(d*(c + d*x)) + (I*a*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2} -{(a + I*a*Sinh[e + f*x])/(c + d*x)^3, x, 7, -(a/(2*d*(c + d*x)^2)) - (I*a*f*Cosh[e + f*x])/(2*d^2*(c + d*x)) + (I*a*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/(2*d^3) - (I*a*Sinh[e + f*x])/(2*d*(c + d*x)^2) + (I*a*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(c + d*x)^3*(a + I*a*Sinh[e + f*x])^2, x, 10, (3*a^2*c*d^2*x)/(4*f^2) + (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) + (12*I*a^2*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (2*I*a^2*(c + d*x)^3*Cosh[e + f*x])/f - (12*I*a^2*d^3*Sinh[e + f*x])/f^4 - (6*I*a^2*d*(c + d*x)^2*Sinh[e + f*x])/f^2 - (3*a^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) + (3*a^2*d^3*Sinh[e + f*x]^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*Sinh[e + f*x]^2)/(4*f^2)} -{(c + d*x)^2*(a + I*a*Sinh[e + f*x])^2, x, 9, (a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) + (4*I*a^2*d^2*Cosh[e + f*x])/f^3 + (2*I*a^2*(c + d*x)^2*Cosh[e + f*x])/f - (4*I*a^2*d*(c + d*x)*Sinh[e + f*x])/f^2 - (a^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) - (a^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) + (a^2*d*(c + d*x)*Sinh[e + f*x]^2)/(2*f^2)} -{(c + d*x)*(a + I*a*Sinh[e + f*x])^2, x, 6, (1/2)*a^2*c*x + (1/4)*a^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) + (2*I*a^2*(c + d*x)*Cosh[e + f*x])/f - (2*I*a^2*d*Sinh[e + f*x])/f^2 - (a^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) + (a^2*d*Sinh[e + f*x]^2)/(4*f^2)} -{(a + I*a*Sinh[e + f*x])^2/(c + d*x), x, 9, -(a^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (3*a^2*Log[c + d*x])/(2*d) + ((2*I)*a^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d + ((2*I)*a^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d - (a^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + I*a*Sinh[e + f*x])^2/(c + d*x)^2, x, 9, -((4*a^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^4)/(d*(c + d*x))) + (2*I*a^2*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (a^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d^2 + (2*I*a^2*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2 - (a^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + I*a*Sinh[e + f*x])^2/(c + d*x)^3, x, 15, -((2*a^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^4)/(d*(c + d*x)^2)) - (a^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 + (I*a^2*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - (4*a^2*f*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^3*Sinh[e/2 + (I*Pi)/4 + (f*x)/2])/(d^2*(c + d*x)) + (I*a^2*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 - (a^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + I*a*Sinh[e + f*x]), x, 7, (c + d*x)^3/(a*f) - (6*d*(c + d*x)^2*Log[1 + I*E^(e + f*x)])/(a*f^2) - (12*d^2*(c + d*x)*PolyLog[2, (-I)*E^(e + f*x)])/(a*f^3) + (12*d^3*PolyLog[3, (-I)*E^(e + f*x)])/(a*f^4) + ((c + d*x)^3*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f)} -{(c + d*x)^2/(a + I*a*Sinh[e + f*x]), x, 6, (c + d*x)^2/(a*f) - (4*d*(c + d*x)*Log[1 + I*E^(e + f*x)])/(a*f^2) - (4*d^2*PolyLog[2, (-I)*E^(e + f*x)])/(a*f^3) + ((c + d*x)^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f)} -{(c + d*x)^1/(a + I*a*Sinh[e + f*x]), x, 3, -((2*d*Log[Cosh[e/2 + (I*Pi)/4 + (f*x)/2]])/(a*f^2)) + ((c + d*x)*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f)} -{1/((c + d*x)^1*(a + I*a*Sinh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + I*a*Sinh[e + f*x])), x]} -{1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]} - - -{(c + d*x)^3/(a + I*a*Sinh[e + f*x])^2, x, 10, (c + d*x)^3/(3*a^2*f) - (2*d*(c + d*x)^2*Log[1 + I*E^(e + f*x)])/(a^2*f^2) + (4*d^3*Log[Cosh[e/2 + (I*Pi)/4 + (f*x)/2]])/(a^2*f^4) - (4*d^2*(c + d*x)*PolyLog[2, (-I)*E^(e + f*x)])/(a^2*f^3) + (4*d^3*PolyLog[3, (-I)*E^(e + f*x)])/(a^2*f^4) + (d*(c + d*x)^2*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2)/(2*a^2*f^2) - (2*d^2*(c + d*x)*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(a^2*f^3) + ((c + d*x)^3*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(6*a^2*f)} -{(c + d*x)^2/(a + I*a*Sinh[e + f*x])^2, x, 9, (c + d*x)^2/(3*a^2*f) - (4*d*(c + d*x)*Log[1 + I*E^(e + f*x)])/(3*a^2*f^2) - (4*d^2*PolyLog[2, (-I)*E^(e + f*x)])/(3*a^2*f^3) + (d*(c + d*x)*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2)/(3*a^2*f^2) - (2*d^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*a^2*f^3) + ((c + d*x)^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(6*a^2*f)} -{(c + d*x)^1/(a + I*a*Sinh[e + f*x])^2, x, 4, -((2*d*Log[Cosh[e/2 + (I*Pi)/4 + (f*x)/2]])/(3*a^2*f^2)) + (d*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2)/(6*a^2*f^2) + ((c + d*x)*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*a^2*f) + ((c + d*x)*Sech[e/2 + (I*Pi)/4 + (f*x)/2]^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(6*a^2*f)} -{1/((c + d*x)^1*(a + I*a*Sinh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + I*a*Sinh[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Sinh[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Sqrt[a + I*a*Sinh[e + f*x]], x, 6, -((384*x*Sqrt[a + I*a*Sinh[e + f*x]])/f^4) - (16*x^3*Sqrt[a + I*a*Sinh[e + f*x]])/f^2 + (768*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f^5 + (96*x^2*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f^3 + (2*x^4*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f} -{x^3*Sqrt[a + I*a*Sinh[e + f*x]], x, 5, -((96*Sqrt[a + I*a*Sinh[e + f*x]])/f^4) - (12*x^2*Sqrt[a + I*a*Sinh[e + f*x]])/f^2 + (48*x*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f^3 + (2*x^3*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f} -{x^2*Sqrt[a + I*a*Sinh[e + f*x]], x, 4, -((8*x*Sqrt[a + I*a*Sinh[e + f*x]])/f^2) + (16*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f^3 + (2*x^2*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f} -{x*Sqrt[a + I*a*Sinh[e + f*x]], x, 3, -((4*Sqrt[a + I*a*Sinh[e + f*x]])/f^2) + (2*x*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/f} -{Sqrt[a + I*a*Sinh[e + f*x]]/x, x, 4, I*CoshIntegral[(f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(2*e - I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + I*Cosh[(1/4)*(2*e - I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2]} -{Sqrt[a + I*a*Sinh[e + f*x]]/x^2, x, 5, -(Sqrt[a + I*a*Sinh[e + f*x]]/x) + (1/2)*f*CoshIntegral[(f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(2*e + I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (1/2)*f*Cosh[(1/4)*(2*e + I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2]} -{Sqrt[a + I*a*Sinh[e + f*x]]/x^3, x, 6, -(Sqrt[a + I*a*Sinh[e + f*x]]/(2*x^2)) + (1/8)*I*f^2*CoshIntegral[(f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(2*e - I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (1/8)*I*f^2*Cosh[(1/4)*(2*e - I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2] - (f*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(4*x)} - - -{x^3*(a + I*a*Sinh[e + f*x])^(3/2), x, 9, -((1280*a*Sqrt[a + I*a*Sinh[e + f*x]])/(9*f^4)) - (16*a*x^2*Sqrt[a + I*a*Sinh[e + f*x]])/f^2 - (64*a*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/(27*f^4) - (8*a*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f^2) + (32*a*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]])/(9*f^3) + (4*a*x^3*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f) + (640*a*x*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(9*f^3) + (8*a*x^3*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*f)} -{x^2*(a + I*a*Sinh[e + f*x])^(3/2), x, 7, -((32*a*x*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f^2)) - (16*a*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/(9*f^2) + (4*a*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f) + (224*a*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(9*f^3) + (8*a*x^2*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*f) + (32*a*Sinh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(27*f^3)} -{x*(a + I*a*Sinh[e + f*x])^(3/2), x, 4, -((16*a*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f^2)) - (8*a*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/(9*f^2) + (4*a*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]])/(3*f) + (8*a*x*Sqrt[a + I*a*Sinh[e + f*x]]*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(3*f)} -{(a + I*a*Sinh[e + f*x])^(3/2)/x, x, 9, (3/2)*I*a*CoshIntegral[(f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(2*e - I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (1/2)*I*a*CoshIntegral[(3*f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(6*e + I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (3/2)*I*a*Cosh[(1/4)*(2*e - I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2] + (1/2)*I*a*Cosh[(1/4)*(6*e + I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(3*f*x)/2]} -{(a + I*a*Sinh[e + f*x])^(3/2)/x^2, x, 9, -((2*a*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]^2*Sqrt[a + I*a*Sinh[e + f*x]])/x) - (3/4)*a*f*CoshIntegral[(3*f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(6*e - I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (3/4)*a*f*CoshIntegral[(f*x)/2]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sinh[(1/4)*(2*e + I*Pi)]*Sqrt[a + I*a*Sinh[e + f*x]] + (3/4)*a*f*Cosh[(1/4)*(2*e + I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(f*x)/2] - (3/4)*a*f*Cosh[(1/4)*(6*e - I*Pi)]*Sech[e/2 + (I*Pi)/4 + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*SinhIntegral[(3*f*x)/2]} - - -{x^3*(a + a*I*Sinh[c + d*x])^(5/2), x, 14, -((265216*a^2*Sqrt[a + I*a*Sinh[c + d*x]])/(1125*d^4)) - (128*a^2*x^2*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d^2) - (17408*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(3375*d^4) - (64*a^2*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2) - (384*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(625*d^4) - (48*a^2*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d^2) + (8704*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(1125*d^3) + (32*a^2*x^3*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d) + (192*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^3*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(125*d^3) + (8*a^2*x^3*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^3*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d) + (132608*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(1125*d^3) + (64*a^2*x^3*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(15*d)} -{x^2*(a + a*I*Sinh[c + d*x])^(5/2), x, 10, -((256*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2)) - (128*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(45*d^2) - (32*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d^2) + (32*a^2*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d) + (8*a^2*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^3*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d) + (9536*a^2*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(225*d^3) + (64*a^2*x^2*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(15*d) + (2432*a^2*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(675*d^3) + (64*a^2*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(125*d^3)} -{x^1*(a + a*I*Sinh[c + d*x])^(5/2), x, 5, -((128*a^2*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2)) - (64*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^2*Sqrt[a + I*a*Sinh[c + d*x]])/(45*d^2) - (16*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d^2) + (32*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d) + (8*a^2*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^3*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d) + (64*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(15*d)} -{(a + a*I*Sinh[c + d*x])^(5/2)/x^1, x, 12, (-(1/4))*I*a^2*CoshIntegral[(5*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(5*c)/2 - (I*Pi)/4]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/2)*I*a^2*CoshIntegral[(d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(2*c - I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/4)*I*a^2*CoshIntegral[(3*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(6*c + I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/2)*I*a^2*Cosh[(1/4)*(2*c - I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(d*x)/2] + (5/4)*I*a^2*Cosh[(1/4)*(6*c + I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(3*d*x)/2] - (1/4)*I*a^2*Cosh[(5*c)/2 - (I*Pi)/4]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(5*d*x)/2]} -{(a + a*I*Sinh[c + d*x])^(5/2)/x^2, x, 12, -((4*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/x) - (5/8)*a^2*d*CoshIntegral[(5*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(5*c)/2 + (I*Pi)/4]*Sqrt[a + I*a*Sinh[c + d*x]] - (15/8)*a^2*d*CoshIntegral[(3*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(6*c - I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/4)*a^2*d*CoshIntegral[(d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(2*c + I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/4)*a^2*d*Cosh[(1/4)*(2*c + I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(d*x)/2] - (15/8)*a^2*d*Cosh[(1/4)*(6*c - I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(3*d*x)/2] - (5/8)*a^2*d*Cosh[(5*c)/2 + (I*Pi)/4]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(5*d*x)/2]} -{(a + a*I*Sinh[c + d*x])^(5/2)/x^3, x, 21, -((2*a^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/x^2) - (25/32)*I*a^2*d^2*CoshIntegral[(5*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(5*c)/2 - (I*Pi)/4]*Sqrt[a + I*a*Sinh[c + d*x]] + (5/16)*I*a^2*d^2*CoshIntegral[(d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(2*c - I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] + (45/32)*I*a^2*d^2*CoshIntegral[(3*d*x)/2]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sinh[(1/4)*(6*c + I*Pi)]*Sqrt[a + I*a*Sinh[c + d*x]] - (5*a^2*d*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]^3*Sinh[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/x + (5/16)*I*a^2*d^2*Cosh[(1/4)*(2*c - I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(d*x)/2] + (45/32)*I*a^2*d^2*Cosh[(1/4)*(6*c + I*Pi)]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(3*d*x)/2] - (25/32)*I*a^2*d^2*Cosh[(5*c)/2 - (I*Pi)/4]*Sech[c/2 + (I*Pi)/4 + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]]*SinhIntegral[(5*d*x)/2]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/Sqrt[a + I*a*Sinh[e + f*x]], x, 10, (4*I*x^3*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(f*Sqrt[a + I*a*Sinh[e + f*x]]) + (12*I*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (12*I*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (48*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (48*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (96*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[4, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - (96*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[4, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^4*Sqrt[a + I*a*Sinh[e + f*x]])} -{x^2/Sqrt[a + I*a*Sinh[e + f*x]], x, 8, (4*I*x^2*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(f*Sqrt[a + I*a*Sinh[e + f*x]]) + (8*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (8*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (16*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (16*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^3*Sqrt[a + I*a*Sinh[e + f*x]])} -{x/Sqrt[a + I*a*Sinh[e + f*x]], x, 6, (4*I*x*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(f*Sqrt[a + I*a*Sinh[e + f*x]]) + (4*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (4*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(f^2*Sqrt[a + I*a*Sinh[e + f*x]])} -{1/(x*Sqrt[a + I*a*Sinh[e + f*x]]), x, 0, Unintegrable[1/(x*Sqrt[a + I*a*Sinh[e + f*x]]), x]} -{1/(x^2*Sqrt[a + I*a*Sinh[e + f*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[a + I*a*Sinh[e + f*x]]), x]} - - -{x^3/(a + I*a*Sinh[e + f*x])^(3/2), x, 16, (3*x^2)/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (24*I*x*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x^3*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) - (24*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) + (3*I*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + (24*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - (3*I*x^2*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (12*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (12*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (24*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[4, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) - (24*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[4, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^4*Sqrt[a + I*a*Sinh[e + f*x]]) + (x^3*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])} -{x^2/(a + I*a*Sinh[e + f*x])^(3/2), x, 10, (2*x)/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (4*ArcTan[Sinh[e/2 + (I*Pi)/4 + (f*x)/2]]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x^2*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) + (2*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (2*I*x*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (4*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (4*I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[3, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^3*Sqrt[a + I*a*Sinh[e + f*x]]) + (x^2*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])} -{x/(a + I*a*Sinh[e + f*x])^(3/2), x, 7, 1/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*x*ArcTanh[E^((1/4)*(2*e - I*Pi) + (f*x)/2)]*Cosh[e/2 + (I*Pi)/4 + (f*x)/2])/(a*f*Sqrt[a + I*a*Sinh[e + f*x]]) + (I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, -E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) - (I*Cosh[e/2 + (I*Pi)/4 + (f*x)/2]*PolyLog[2, E^((1/4)*(2*e - I*Pi) + (f*x)/2)])/(a*f^2*Sqrt[a + I*a*Sinh[e + f*x]]) + (x*Tanh[e/2 + (I*Pi)/4 + (f*x)/2])/(2*a*f*Sqrt[a + I*a*Sinh[e + f*x]])} -{1/(x*(a + I*a*Sinh[e + f*x])^(3/2)), x, 0, Unintegrable[1/(x*(a + I*a*Sinh[e + f*x])^(3/2)), x]} -{1/(x^2*(a + I*a*Sinh[e + f*x])^(3/2)), x, 0, Unintegrable[1/(x^2*(a + I*a*Sinh[e + f*x])^(3/2)), x]} - - -{x^3/(a + a*I*Sinh[c + d*x])^(5/2), x, 23, -(1/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]])) + (9*x^2)/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (10*I*x*ArcTanh[E^((1/4)*(2*c - I*Pi) + (d*x)/2)]*Cosh[c/2 + (I*Pi)/4 + (d*x)/2])/(a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*x^3*ArcTanh[E^((1/4)*(2*c - I*Pi) + (d*x)/2)]*Cosh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) - (10*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) + (9*I*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + (10*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) - (9*I*x^2*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (9*I*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[3, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (9*I*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[3, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (9*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[4, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) - (9*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[4, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(a^2*d^4*Sqrt[a + I*a*Sinh[c + d*x]]) + (x^2*Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2)/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (x*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x^3*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(16*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (x^3*Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]])} -{x^2/(a + a*I*Sinh[c + d*x])^(5/2), x, 13, (3*x)/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (5*ArcTan[Sinh[c/2 + (I*Pi)/4 + (d*x)/2]]*Cosh[c/2 + (I*Pi)/4 + (d*x)/2])/(3*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*x^2*ArcTanh[E^((1/4)*(2*c - I*Pi) + (d*x)/2)]*Cosh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (3*I*x*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(4*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (3*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[3, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[3, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(2*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (x*Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2)/(6*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - Tanh[c/2 + (I*Pi)/4 + (d*x)/2]/(6*a^2*d^3*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x^2*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(16*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (x^2*Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]])} -{x^1/(a + a*I*Sinh[c + d*x])^(5/2), x, 8, 3/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*x*ArcTanh[E^((1/4)*(2*c - I*Pi) + (d*x)/2)]*Cosh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, -E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) - (3*I*Cosh[c/2 + (I*Pi)/4 + (d*x)/2]*PolyLog[2, E^((1/4)*(2*c - I*Pi) + (d*x)/2)])/(8*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2/(12*a^2*d^2*Sqrt[a + I*a*Sinh[c + d*x]]) + (3*x*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(16*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (x*Sech[c/2 + (I*Pi)/4 + (d*x)/2]^2*Tanh[c/2 + (I*Pi)/4 + (d*x)/2])/(8*a^2*d*Sqrt[a + I*a*Sinh[c + d*x]])} -{1/(x^1*(a + a*I*Sinh[c + d*x])^(5/2)), x, 0, Unintegrable[1/(x*(a + I*a*Sinh[c + d*x])^(5/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Sinh[e+f x])^(n/3)*) - - -(* Used to hang Rubi *) -{(a + I*a*Sinh[e + f*x])^(1/3)/x, x, 0, Unintegrable[(a + I*a*Sinh[e + f*x])^(1/3)/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+I a Sinh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + I*a*Sinh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + I*a*Sinh[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + I*a*Sinh[e + f*x])^3, x, 12, (5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) - ((I/8)*3^(-1 - m)*a^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) - (3*2^(-3 - m)*a^3*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (((15*I)/8)*a^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + (((15*I)/8)*a^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (3*2^(-3 - m)*a^3*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - ((I/8)*3^(-1 - m)*a^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + I*a*Sinh[e + f*x])^2, x, 9, (3*a^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (2^(-3 - m)*a^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (I*a^2*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + (I*a^2*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (2^(-3 - m)*a^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + I*a*Sinh[e + f*x]), x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) + ((I/2)*a*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + ((I/2)*a*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + I*a*Sinh[e + f*x]), x, 0, Unintegrable[(c + d*x)^m/(a + I*a*Sinh[e + f*x]), x]} -{(c + d*x)^m/(a + I*a*Sinh[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + I*a*Sinh[e + f*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sinh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sinh[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Sinh[e + f*x]), x, 6, (a*(c + d*x)^4)/(4*d) + (6*b*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (b*(c + d*x)^3*Cosh[e + f*x])/f - (6*b*d^3*Sinh[e + f*x])/f^4 - (3*b*d*(c + d*x)^2*Sinh[e + f*x])/f^2} -{(c + d*x)^2*(a + b*Sinh[e + f*x]), x, 5, (a*(c + d*x)^3)/(3*d) + (2*b*d^2*Cosh[e + f*x])/f^3 + (b*(c + d*x)^2*Cosh[e + f*x])/f - (2*b*d*(c + d*x)*Sinh[e + f*x])/f^2} -{(c + d*x)*(a + b*Sinh[e + f*x]), x, 4, (a*(c + d*x)^2)/(2*d) + (b*(c + d*x)*Cosh[e + f*x])/f - (b*d*Sinh[e + f*x])/f^2} -{(a + b*Sinh[e + f*x])/(c + d*x), x, 5, (a*Log[c + d*x])/d + (b*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d + (b*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d} -{(a + b*Sinh[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) + (b*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 - (b*Sinh[e + f*x])/(d*(c + d*x)) + (b*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2} -{(a + b*Sinh[e + f*x])/(c + d*x)^3, x, 7, -a/(2*d*(c + d*x)^2) - (b*f*Cosh[e + f*x])/(2*d^2*(c + d*x)) + (b*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/(2*d^3) - (b*Sinh[e + f*x])/(2*d*(c + d*x)^2) + (b*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(c + d*x)^3*(a + b*Sinh[e + f*x])^2, x, 10, (-3*b^2*c*d^2*x)/(4*f^2) - (3*b^2*d^3*x^2)/(8*f^2) + (a^2*(c + d*x)^4)/(4*d) - (b^2*(c + d*x)^4)/(8*d) + (12*a*b*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (2*a*b*(c + d*x)^3*Cosh[e + f*x])/f - (12*a*b*d^3*Sinh[e + f*x])/f^4 - (6*a*b*d*(c + d*x)^2*Sinh[e + f*x])/f^2 + (3*b^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (b^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) - (3*b^2*d^3*Sinh[e + f*x]^2)/(8*f^4) - (3*b^2*d*(c + d*x)^2*Sinh[e + f*x]^2)/(4*f^2)} -{(c + d*x)^2*(a + b*Sinh[e + f*x])^2, x, 9, -(b^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(3*d) - (b^2*(c + d*x)^3)/(6*d) + (4*a*b*d^2*Cosh[e + f*x])/f^3 + (2*a*b*(c + d*x)^2*Cosh[e + f*x])/f - (4*a*b*d*(c + d*x)*Sinh[e + f*x])/f^2 + (b^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (b^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) - (b^2*d*(c + d*x)*Sinh[e + f*x]^2)/(2*f^2)} -{(c + d*x)*(a + b*Sinh[e + f*x])^2, x, 6, -(b^2*c*x)/2 - (b^2*d*x^2)/4 + (a^2*(c + d*x)^2)/(2*d) + (2*a*b*(c + d*x)*Cosh[e + f*x])/f - (2*a*b*d*Sinh[e + f*x])/f^2 + (b^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(2*f) - (b^2*d*Sinh[e + f*x]^2)/(4*f^2)} -{(a + b*Sinh[e + f*x])^2/(c + d*x), x, 10, (b^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (a^2*Log[c + d*x])/d - (b^2*Log[c + d*x])/(2*d) + (2*a*b*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d + (2*a*b*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d + (b^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + b*Sinh[e + f*x])^2/(c + d*x)^2, x, 11, -(a^2/(d*(c + d*x))) + (2*a*b*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d^2 - (2*a*b*Sinh[e + f*x])/(d*(c + d*x)) - (b^2*Sinh[e + f*x]^2)/(d*(c + d*x)) + (2*a*b*f*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + b*Sinh[e + f*x])^2/(c + d*x)^3, x, 14, -a^2/(2*d*(c + d*x)^2) - (a*b*f*Cosh[e + f*x])/(d^2*(c + d*x)) + (b^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 + (a*b*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - (a*b*Sinh[e + f*x])/(d*(c + d*x)^2) - (b^2*f*Cosh[e + f*x]*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2*Sinh[e + f*x]^2)/(2*d*(c + d*x)^2) + (a*b*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (b^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Sinh[e + f*x]), x, 12, ((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) - ((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) + (3*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (3*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (6*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3) + (6*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3) + (6*d^3*PolyLog[4, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^4) - (6*d^3*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^4)} -{(c + d*x)^2/(a + b*Sinh[e + f*x]), x, 10, ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) - ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) + (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (2*d^2*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3) + (2*d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3)} -{(c + d*x)^1/(a + b*Sinh[e + f*x]), x, 8, ((c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) - ((c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) + (d*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (d*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2)} -{1/((c + d*x)^1*(a + b*Sinh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sinh[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Sinh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sinh[e + f*x])), x]} - - -{(c + d*x)^2/(a + b*Sinh[e + f*x])^2, x, 18, -((c + d*x)^2/((a^2 + b^2)*f)) + (2*d*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*f^2) + (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*f) + (2*d*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*f^2) - (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*f) + (2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^2) + (2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^2) - (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^3) + (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^3) - (b*(c + d*x)^2*Cosh[e + f*x])/((a^2 + b^2)*f*(a + b*Sinh[e + f*x]))} -{(c + d*x)^1/(a + b*Sinh[e + f*x])^2, x, 11, (a*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*f) - (a*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*f) + (d*Log[a + b*Sinh[e + f*x]])/((a^2 + b^2)*f^2) + (a*d*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^2) - (a*d*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*f^2) - (b*(c + d*x)*Cosh[e + f*x])/((a^2 + b^2)*f*(a + b*Sinh[e + f*x]))} -{1/((c + d*x)^1*(a + b*Sinh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Sinh[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Sinh[e + f*x])^2), x]} - - -(* {(e + f*x)^2/(a + b*Sinh[c + d*x])^3, x, 53, -((3*a*(e + f*x)^2)/(2*(a^2 + b^2)^2*d)) - (2*f^2*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d^3) + (3*a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d^2) + (3*a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) - ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2)*d) + (3*a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d^2) - (3*a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2)*d) + (3*a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (3*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(5/2)*d^2) - (f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (3*a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (3*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(5/2)*d^2) + (f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(5/2)*d^3) + (f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (3*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(5/2)*d^3) - (f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^2) - (f*(e + f*x))/((a^2 + b^2)*d^2*(a + b*Sinh[c + d*x])) - (3*a*b*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x]))} *) -{(e + f*x)^1/(a + b*Sinh[c + d*x])^3, x, 35, (3*a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) - ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2)*d) - (3*a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2)*d) + (3*a*f*Log[a + b*Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d^2) + (3*a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(5/2)*d^2) - (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(3/2)*d^2) - (3*a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(5/2)*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(3/2)*d^2) - (b*(e + f*x)*Cosh[c + d*x])/(2*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^2) - f/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x])) - (3*a*b*(e + f*x)*Cosh[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x]))} -{1/((e + f*x)^1*(a + b*Sinh[c + d*x])^3), x, 0, Unintegrable[1/((e + f*x)*(a + b*Sinh[c + d*x])^3), x]} -{1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x, 0, Unintegrable[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Sinh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + b*Sinh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + b*Sinh[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + b*Sinh[e + f*x])^3, x, 18, (a^3*(c + d*x)^(1 + m))/(d*(1 + m)) - (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (3^(-1 - m)*b^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(8*f*(-((f*(c + d*x))/d))^m) + (3*2^(-3 - m)*a*b^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (3*a^2*b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) - (3*b^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(8*f*(-((f*(c + d*x))/d))^m) + (3*a^2*b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c + d*x))/d)^m) - (3*b^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m) - (3*2^(-3 - m)*a*b^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (3^(-1 - m)*b^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + b*Sinh[e + f*x])^2, x, 10, (a^2*(c + d*x)^(1 + m))/(d*(1 + m)) - (b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (2^(-3 - m)*b^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (a*b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + (a*b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (2^(-3 - m)*b^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + b*Sinh[e + f*x]), x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) + (b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) + (b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + b*Sinh[e + f*x]), x, 0, Unintegrable[(c + d*x)^m/(a + b*Sinh[e + f*x]), x]} -{(c + d*x)^m/(a + b*Sinh[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + b*Sinh[e + f*x])^2, x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Sinh[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Sinh[c+d x])^p with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n / (a+a Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Sinh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 9, (I*(e + f*x)^3)/(a*d) - ((I/4)*(e + f*x)^4)/(a*f) - ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + ((12*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + (I*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Sinh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 8, (I*(e + f*x)^2)/(a*d) - ((I/3)*(e + f*x)^3)/(a*f) - ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (I*(e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Sinh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 5, ((-I)*e*x)/a - ((I/2)*f*x^2)/a - ((2*I)*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) + (I*(e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Sinh[c + d*x]/(a + I*a*Sinh[c + d*x]), x, 2, ((-I)*x)/a - Cosh[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))} -{Sinh[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sinh[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 14, -((e + f*x)^3/(a*d)) + (e + f*x)^4/(4*a*f) - ((6*I)*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^3*Cosh[c + d*x])/(a*d) + (6*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (12*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*f^3*Sinh[c + d*x])/(a*d^4) + ((3*I)*f*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - ((e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 12, -((e + f*x)^2/(a*d)) + (e + f*x)^3/(3*a*f) - ((2*I)*f^2*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^2*Cosh[c + d*x])/(a*d) + (4*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) + (4*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + ((2*I)*f*(e + f*x)*Sinh[c + d*x])/(a*d^2) - ((e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Sinh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 8, (e*x)/a + (f*x^2)/(2*a) - (I*(e + f*x)*Cosh[c + d*x])/(a*d) + (2*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) + (I*f*Sinh[c + d*x])/(a*d^2) - ((e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Sinh[c + d*x]^2/(a + I*a*Sinh[c + d*x]), x, 4, x/a - (I*Cosh[c + d*x])/(a*d) - (I*Cosh[c + d*x])/(a*d*(1 + I*Sinh[c + d*x]))} -{Sinh[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sinh[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 19, (((3*I)/4)*e*f^2*x)/(a*d^2) + (((3*I)/8)*f^3*x^2)/(a*d^2) - (I*(e + f*x)^3)/(a*d) + (((3*I)/8)*(e + f*x)^4)/(a*f) + (6*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) + ((e + f*x)^3*Cosh[c + d*x])/(a*d) + ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((12*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (6*f^3*Sinh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x])/(a*d^2) - (((3*I)/4)*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^3) - ((I/2)*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(a*d) + (((3*I)/8)*f^3*Sinh[c + d*x]^2)/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Sinh[c + d*x]^2)/(a*d^2) - (I*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 17, ((I/4)*f^2*x)/(a*d^2) - (I*(e + f*x)^2)/(a*d) + ((I/2)*(e + f*x)^3)/(a*f) + (2*f^2*Cosh[c + d*x])/(a*d^3) + ((e + f*x)^2*Cosh[c + d*x])/(a*d) + ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) + ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (2*f*(e + f*x)*Sinh[c + d*x])/(a*d^2) - ((I/4)*f^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^3) - ((I/2)*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d) + ((I/2)*f*(e + f*x)*Sinh[c + d*x]^2)/(a*d^2) - (I*(e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Sinh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 11, (((3*I)/2)*e*x)/a + (((3*I)/4)*f*x^2)/a + ((e + f*x)*Cosh[c + d*x])/(a*d) + ((2*I)*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) - (f*Sinh[c + d*x])/(a*d^2) - ((I/2)*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d) + ((I/4)*f*Sinh[c + d*x]^2)/(a*d^2) - (I*(e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Sinh[c + d*x]^3/(a + I*a*Sinh[c + d*x]), x, 2, (((3*I)/2)*x)/a + (2*Cosh[c + d*x])/(a*d) - (((3*I)/2)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d) - (Cosh[c + d*x]*Sinh[c + d*x]^2)/(d*(a + I*a*Sinh[c + d*x]))} -{Sinh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sinh[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csch[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 17, ((-I)*(e + f*x)^3)/(a*d) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) + ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - ((12*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (I*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Csch[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 14, ((-I)*(e + f*x)^2)/(a*d) - (2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) + ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - (2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) + (2*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (2*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) - (I*(e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Csch[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 9, (-2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) + ((2*I)*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) - (f*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (f*PolyLog[2, E^(c + d*x)])/(a*d^2) - (I*(e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Csch[c + d*x]/(a + I*a*Sinh[c + d*x]), x, 3, -(ArcTanh[Cosh[c + d*x]]/(a*d)) + Cosh[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))} -{Csch[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Csch[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 24, (-2*(e + f*x)^3)/(a*d) + ((2*I)*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - ((e + f*x)^3*Coth[c + d*x])/(a*d) + (6*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d^2) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (12*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) + (3*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (12*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (3*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^4) + ((6*I)*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - ((6*I)*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - ((e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 20, (-2*(e + f*x)^2)/(a*d) + ((2*I)*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - ((e + f*x)^2*Coth[c + d*x])/(a*d) + (4*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) + (2*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^2) + ((2*I)*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (4*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((2*I)*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) + (f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - ((2*I)*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + ((2*I)*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) - ((e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 12, ((2*I)*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (2*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (I*f*PolyLog[2, -E^(c + d*x)])/(a*d^2) - (I*f*PolyLog[2, E^(c + d*x)])/(a*d^2) - ((e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Csch[c + d*x]^2/(a + I*a*Sinh[c + d*x]), x, 5, (I*ArcTanh[Cosh[c + d*x]])/(a*d) - (2*Coth[c + d*x])/(a*d) + Coth[c + d*x]/(d*(a + I*a*Sinh[c + d*x]))} -{Csch[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Csch[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 40, ((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) + (I*(e + f*x)^3*Coth[c + d*x])/(a*d) - (3*f*(e + f*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - ((6*I)*f*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((3*I)*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d^2) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (9*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - ((12*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) - (9*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) - ((3*I)*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + ((12*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + (9*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (((3*I)/2)*f^3*PolyLog[3, E^(2*(c + d*x))])/(a*d^4) + (9*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (9*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (I*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)^2*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 30, ((2*I)*(e + f*x)^2)/(a*d) + (3*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (I*(e + f*x)^2*Coth[c + d*x])/(a*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - ((4*I)*f*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d^2) - ((2*I)*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^2) + (3*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - ((4*I)*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (3*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (I*f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (3*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (3*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (I*(e + f*x)^2*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{((e + f*x)*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 19, (3*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) + (I*(e + f*x)*Coth[c + d*x])/(a*d) - (f*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - ((2*I)*f*Log[Cosh[c/2 + (I/4)*Pi + (d*x)/2]])/(a*d^2) - (I*f*Log[Sinh[c + d*x]])/(a*d^2) + (3*f*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (I*(e + f*x)*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(a*d)} -{Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]), x, 6, (3*ArcTanh[Cosh[c + d*x]])/(2*a*d) + ((2*I)*Coth[c + d*x])/(a*d) - (3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) + (Coth[c + d*x]*Csch[c + d*x])/(d*(a + I*a*Sinh[c + d*x]))} -{Csch[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Csch[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Sinh[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n / (a+b Sinh[c+d x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 14, (e + f*x)^4/(4*b*f) - (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) + (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) - (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4) + (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4)} -{((e + f*x)^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 12, (e + f*x)^3/(3*b*f) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) + (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) - (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3)} -{((e + f*x)*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 10, (e*x)/b + (f*x^2)/(2*b) - (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) + (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d) - (a*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2) + (a*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^2)} -{Sinh[c + d*x]/(a + b*Sinh[c + d*x]), x, 4, x/b + (2*a*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2]*d)} -{Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 19, -(a*(e + f*x)^4)/(4*b^2*f) + (6*f^2*(e + f*x)*Cosh[c + d*x])/(b*d^3) + ((e + f*x)^3*Cosh[c + d*x])/(b*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) - (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^3) + (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^3) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^4) - (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^4) - (6*f^3*Sinh[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x])/(b*d^2)} -{((e + f*x)^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 16, -(a*(e + f*x)^3)/(3*b^2*f) + (2*f^2*Cosh[c + d*x])/(b*d^3) + ((e + f*x)^2*Cosh[c + d*x])/(b*d) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) - (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^3) + (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^3) - (2*f*(e + f*x)*Sinh[c + d*x])/(b*d^2)} -{((e + f*x)*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 13, -((a*e*x)/b^2) - (a*f*x^2)/(2*b^2) + ((e + f*x)*Cosh[c + d*x])/(b*d) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) - (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*Sqrt[a^2 + b^2]*d) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*Sqrt[a^2 + b^2]*d^2) - (f*Sinh[c + d*x])/(b*d^2)} -{Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 6, -((a*x)/b^2) - (2*a^2*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]*d) + Cosh[c + d*x]/(b*d)} -{Sinh[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 24, (-3*e*f^2*x)/(4*b*d^2) - (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) - (e + f*x)^4/(8*b*f) - (6*a*f^2*(e + f*x)*Cosh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) + (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^4) + (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^4) + (6*a*f^3*Sinh[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Sinh[c + d*x])/(b^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (3*f^3*Sinh[c + d*x]^2)/(8*b*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x]^2)/(4*b*d^2)} -{((e + f*x)^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 21, -(f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) - (e + f*x)^3/(6*b*f) - (2*a*f^2*Cosh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) + (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) - (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^3) + (2*a*f*(e + f*x)*Sinh[c + d*x])/(b^2*d^2) + (f^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*(e + f*x)*Sinh[c + d*x]^2)/(2*b*d^2)} -{((e + f*x)*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 16, (a^2*e*x)/b^3 - (e*x)/(2*b) + (a^2*f*x^2)/(2*b^3) - (f*x^2)/(4*b) - (a*(e + f*x)*Cosh[c + d*x])/(b^2*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) + (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (a*f*Sinh[c + d*x])/(b^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*Sinh[c + d*x]^2)/(4*b*d^2)} -{Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 6, ((2*a^2 - b^2)*x)/(2*b^3) + (2*a^3*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]*d) - (a*Cosh[c + d*x])/(b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 22, (-2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4)} -{((e + f*x)^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 18, (-2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (2*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (2*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3)} -{((e + f*x)*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 14, (-2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (f*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (f*PolyLog[2, E^(c + d*x)])/(a*d^2) - (b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2)} -{Csch[c + d*x]/(a + b*Sinh[c + d*x]), x, 5, -(ArcTanh[Cosh[c + d*x]]/(a*d)) + (2*b*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)} -{Csch[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 29, -((e + f*x)^3/(a*d)) + (2*b*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^3*Coth[c + d*x])/(a*d) + (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (3*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (3*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (3*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^4) + (6*b*f^3*PolyLog[4, -E^(c + d*x)])/(a^2*d^4) - (6*b*f^3*PolyLog[4, E^(c + d*x)])/(a^2*d^4) + (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4)} -{((e + f*x)^2*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 24, -((e + f*x)^2/(a*d)) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^2*Coth[c + d*x])/(a*d) + (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (2*b*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3)} -{((e + f*x)*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 17, (2*b*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (b*f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2)} -{Csch[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 7, (b*ArcTanh[Cosh[c + d*x]])/(a^2*d) - (2*b^2*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]*d) - Coth[c + d*x]/(a*d)} -{Csch[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 45, (b*(e + f*x)^3)/(a^2*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + ((e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)^3*Coth[c + d*x])/(a^2*d) - (3*f*(e + f*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) - (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^3) + (3*b*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^4) + (3*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2*f^3*PolyLog[4, -E^(c + d*x)])/(a^3*d^4) - (3*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(c + d*x)])/(a^3*d^4) - (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^4)} -{((e + f*x)^2*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 34, (b*(e + f*x)^2)/(a^2*d) + ((e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (b*(e + f*x)^2*Coth[c + d*x])/(a^2*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (2*b*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) + (f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) - (b*f^2*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) - (f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) + (f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^3)} -{((e + f*x)*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 24, ((e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)*Coth[c + d*x])/(a^2*d) - (f*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (b*f*Log[Sinh[c + d*x]])/(a^2*d^2) + (f*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (b^2*f*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2)} -{Csch[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 7, ((a^2 - 2*b^2)*ArcTanh[Cosh[c + d*x]])/(2*a^3*d) + (2*b^3*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]*d) + (b*Coth[c + d*x])/(a^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)} -{Csch[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Csch[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n / (a+b Sinh[c+d x])^2*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Sinh[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Sinh[c+d x])^p with a^2+b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n / (a+a Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cosh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 6, (I*(e + f*x)^4)/(4*a*f) - (2*I*(e + f*x)^3*Log[1 + I*E^(c + d*x)])/(a*d) - (6*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (12*I*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (12*I*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4)} -{((e + f*x)^2*Cosh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 5, (I*(e + f*x)^3)/(3*a*f) - (2*I*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/(a*d) - (4*I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (4*I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3)} -{((e + f*x)*Cosh[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 4, (I*(e + f*x)^2)/(2*a*f) - (2*I*(e + f*x)*Log[1 + I*E^(c + d*x)])/(a*d) - (2*I*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2)} -{Cosh[c + d*x]/(a + I*a*Sinh[c + d*x]), x, 2, ((-I)*Log[I - Sinh[c + d*x]])/(a*d)} -{Cosh[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Cosh[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Cosh[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Cosh[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 6, (e + f*x)^4/(4*a*f) - ((6*I)*f^2*(e + f*x)*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^3*Cosh[c + d*x])/(a*d) + ((6*I)*f^3*Sinh[c + d*x])/(a*d^4) + ((3*I)*f*(e + f*x)^2*Sinh[c + d*x])/(a*d^2)} -{((e + f*x)^2*Cosh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 5, (e + f*x)^3/(3*a*f) - ((2*I)*f^2*Cosh[c + d*x])/(a*d^3) - (I*(e + f*x)^2*Cosh[c + d*x])/(a*d) + ((2*I)*f*(e + f*x)*Sinh[c + d*x])/(a*d^2)} -{((e + f*x)*Cosh[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 4, (e*x)/a + (f*x^2)/(2*a) - (I*(e + f*x)*Cosh[c + d*x])/(a*d) + (I*f*Sinh[c + d*x])/(a*d^2)} -{Cosh[c + d*x]^2/(a + I*a*Sinh[c + d*x]), x, 2, x/a - (I*Cosh[c + d*x])/(a*d)} -{Cosh[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 5, Log[e + f*x]/(a*f) - (I*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/(a*f) - (I*Cosh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f)} -{Cosh[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 6, -(1/(a*f*(e + f*x))) - (I*d*Cosh[c - (d*e)/f]*CoshIntegral[(d*e)/f + d*x])/(a*f^2) + (I*Sinh[c + d*x])/(a*f*(e + f*x)) - (I*d*Sinh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f^2)} - - -{((e + f*x)^3*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 10, (((-3*I)/8)*f^3*x)/(a*d^3) - ((I/4)*(e + f*x)^3)/(a*d) - (6*f^3*Cosh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(a*d) + (((3*I)/8)*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^4) + (((3*I)/4)*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - (((3*I)/4)*f^2*(e + f*x)*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^3*Sinh[c + d*x]^2)/(a*d)} -{((e + f*x)^2*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 7, ((-I/2)*e*f*x)/(a*d) - ((I/4)*f^2*x^2)/(a*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(a*d^2) + (2*f^2*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(a*d) + ((I/2)*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - ((I/4)*f^2*Sinh[c + d*x]^2)/(a*d^3) - ((I/2)*(e + f*x)^2*Sinh[c + d*x]^2)/(a*d)} -{((e + f*x)*Cosh[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 6, ((-I/4)*f*x)/(a*d) - (f*Cosh[c + d*x])/(a*d^2) + ((e + f*x)*Sinh[c + d*x])/(a*d) + ((I/4)*f*Cosh[c + d*x]*Sinh[c + d*x])/(a*d^2) - ((I/2)*(e + f*x)*Sinh[c + d*x]^2)/(a*d)} -{Cosh[c + d*x]^3/(a + I*a*Sinh[c + d*x]), x, 2, Sinh[c + d*x]/(a*d) - ((I/2)*Sinh[c + d*x]^2)/(a*d)} -{Cosh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 9, (Cosh[c - (d*e)/f]*CoshIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*CoshIntegral[(2*d*e)/f + 2*d*x]*Sinh[2*c - (2*d*e)/f])/(a*f) + (Sinh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f) - ((I/2)*Cosh[2*c - (2*d*e)/f]*SinhIntegral[(2*d*e)/f + 2*d*x])/(a*f)} -{Cosh[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 11, -(Cosh[c + d*x]/(a*f*(e + f*x))) - (I*d*Cosh[2*c - (2*d*e)/f]*CoshIntegral[(2*d*e)/f + 2*d*x])/(a*f^2) + (d*CoshIntegral[(d*e)/f + d*x]*Sinh[c - (d*e)/f])/(a*f^2) + ((I/2)*Sinh[2*c + 2*d*x])/(a*f*(e + f*x)) + (d*Cosh[c - (d*e)/f]*SinhIntegral[(d*e)/f + d*x])/(a*f^2) - (I*d*Sinh[2*c - (2*d*e)/f]*SinhIntegral[(2*d*e)/f + 2*d*x])/(a*f^2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Sech[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 22, (((-3*I)/2)*f*(e + f*x)^2)/(a*d^2) - (6*f^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^3) + ((e + f*x)^3*ArcTan[E^(c + d*x)])/(a*d) + ((3*I)*f^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*d^3) + ((3*I)*f^3*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^4) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - ((3*I)*f^3*PolyLog[2, I*E^(c + d*x)])/(a*d^4) + (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (((3*I)/2)*f^3*PolyLog[2, -E^(2*(c + d*x))])/(a*d^4) + ((3*I)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - ((3*I)*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - ((3*I)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) + ((3*I)*f^3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*Sech[c + d*x])/(2*a*d^2) + ((I/2)*(e + f*x)^3*Sech[c + d*x]^2)/(a*d) - (((3*I)/2)*f*(e + f*x)^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(2*a*d)} -{((e + f*x)^2*Sech[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 13, ((e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d) - (f^2*ArcTan[Sinh[c + d*x]])/(a*d^3) + (I*f^2*Log[Cosh[c + d*x]])/(a*d^3) - (I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(a*d^3) + (f*(e + f*x)*Sech[c + d*x])/(a*d^2) + ((I/2)*(e + f*x)^2*Sech[c + d*x]^2)/(a*d) - (I*f*(e + f*x)*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*a*d)} -{((e + f*x)*Sech[c + d*x])/(a + I*a*Sinh[c + d*x]), x, 10, ((e + f*x)*ArcTan[E^(c + d*x)])/(a*d) - ((I/2)*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + ((I/2)*f*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (f*Sech[c + d*x])/(2*a*d^2) + ((I/2)*(e + f*x)*Sech[c + d*x]^2)/(a*d) - ((I/2)*f*Tanh[c + d*x])/(a*d^2) + ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*a*d)} -{Sech[c + d*x]/(a + I*a*Sinh[c + d*x]), x, 4, ArcTan[Sinh[c + d*x]]/(2*a*d) + (I/2)/(d*(a + I*a*Sinh[c + d*x]))} -{Sech[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sech[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sech[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 20, (2*(e + f*x)^3)/(3*a*d) - (I*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (I*f^3*ArcTan[Sinh[c + d*x]])/(a*d^4) - (2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*d^2) + (f^3*Log[Cosh[c + d*x]])/(a*d^4) - (f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^3) - (2*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*d^3) + (f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - (f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) + (f^3*PolyLog[3, -E^(2*(c + d*x))])/(a*d^4) - (I*f^2*(e + f*x)*Sech[c + d*x])/(a*d^3) + (f*(e + f*x)^2*Sech[c + d*x]^2)/(2*a*d^2) + ((I/3)*(e + f*x)^3*Sech[c + d*x]^3)/(a*d) - (f^2*(e + f*x)*Tanh[c + d*x])/(a*d^3) + (2*(e + f*x)^3*Tanh[c + d*x])/(3*a*d) - ((I/2)*f*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*Sech[c + d*x]^2*Tanh[c + d*x])/(3*a*d)} -{((e + f*x)^2*Sech[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 16, (2*(e + f*x)^2)/(3*a*d) - (((2*I)/3)*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^2) - (4*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(3*a*d^2) - (f^2*PolyLog[2, (-I)*E^(c + d*x)])/(3*a*d^3) + (f^2*PolyLog[2, I*E^(c + d*x)])/(3*a*d^3) - (2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(3*a*d^3) - ((I/3)*f^2*Sech[c + d*x])/(a*d^3) + (f*(e + f*x)*Sech[c + d*x]^2)/(3*a*d^2) + ((I/3)*(e + f*x)^2*Sech[c + d*x]^3)/(a*d) - (f^2*Tanh[c + d*x])/(3*a*d^3) + (2*(e + f*x)^2*Tanh[c + d*x])/(3*a*d) - ((I/3)*f*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(3*a*d)} -{((e + f*x)*Sech[c + d*x]^2)/(a + I*a*Sinh[c + d*x]), x, 7, ((-I/6)*f*ArcTan[Sinh[c + d*x]])/(a*d^2) - (2*f*Log[Cosh[c + d*x]])/(3*a*d^2) + (f*Sech[c + d*x]^2)/(6*a*d^2) + ((I/3)*(e + f*x)*Sech[c + d*x]^3)/(a*d) + (2*(e + f*x)*Tanh[c + d*x])/(3*a*d) - ((I/6)*f*Sech[c + d*x]*Tanh[c + d*x])/(a*d^2) + ((e + f*x)*Sech[c + d*x]^2*Tanh[c + d*x])/(3*a*d)} -{Sech[c + d*x]^2/(a + I*a*Sinh[c + d*x]), x, 3, ((I/3)*Sech[c + d*x])/(d*(a + I*a*Sinh[c + d*x])) + (2*Tanh[c + d*x])/(3*a*d)} -{Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^2/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 32, ((-I/2)*f*(e + f*x)^2)/(a*d^2) - (5*f^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTan[E^(c + d*x)])/(4*a*d) + (I*f^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*d^3) + (((5*I)/2)*f^3*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^4) - (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (((5*I)/2)*f^3*PolyLog[2, I*E^(c + d*x)])/(a*d^4) + (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + ((I/2)*f^3*PolyLog[2, -E^(2*(c + d*x))])/(a*d^4) + (((9*I)/4)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - (((9*I)/4)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) + (((9*I)/4)*f^3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) - (f^3*Sech[c + d*x])/(4*a*d^4) + (9*f*(e + f*x)^2*Sech[c + d*x])/(8*a*d^2) - ((I/4)*f^2*(e + f*x)*Sech[c + d*x]^2)/(a*d^3) + (f*(e + f*x)^2*Sech[c + d*x]^3)/(4*a*d^2) + ((I/4)*(e + f*x)^3*Sech[c + d*x]^4)/(a*d) + ((I/4)*f^3*Tanh[c + d*x])/(a*d^4) - ((I/2)*f*(e + f*x)^2*Tanh[c + d*x])/(a*d^2) - (f^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(4*a*d^3) + (3*(e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(8*a*d) - ((I/4)*f*(e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^3*Sech[c + d*x]^3*Tanh[c + d*x])/(4*a*d)} -{((e + f*x)^2*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 17, (3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(4*a*d) - (5*f^2*ArcTan[Sinh[c + d*x]])/(6*a*d^3) + ((I/3)*f^2*Log[Cosh[c + d*x]])/(a*d^3) - (((3*I)/4)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (((3*I)/4)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (((3*I)/4)*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) - (((3*I)/4)*f^2*PolyLog[3, I*E^(c + d*x)])/(a*d^3) + (3*f*(e + f*x)*Sech[c + d*x])/(4*a*d^2) - ((I/12)*f^2*Sech[c + d*x]^2)/(a*d^3) + (f*(e + f*x)*Sech[c + d*x]^3)/(6*a*d^2) + ((I/4)*(e + f*x)^2*Sech[c + d*x]^4)/(a*d) - ((I/3)*f*(e + f*x)*Tanh[c + d*x])/(a*d^2) - (f^2*Sech[c + d*x]*Tanh[c + d*x])/(12*a*d^3) + (3*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(8*a*d) - ((I/6)*f*(e + f*x)*Sech[c + d*x]^2*Tanh[c + d*x])/(a*d^2) + ((e + f*x)^2*Sech[c + d*x]^3*Tanh[c + d*x])/(4*a*d)} -{((e + f*x)*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]), x, 11, (3*(e + f*x)*ArcTan[E^(c + d*x)])/(4*a*d) - (((3*I)/8)*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (((3*I)/8)*f*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (3*f*Sech[c + d*x])/(8*a*d^2) + (f*Sech[c + d*x]^3)/(12*a*d^2) + ((I/4)*(e + f*x)*Sech[c + d*x]^4)/(a*d) - ((I/4)*f*Tanh[c + d*x])/(a*d^2) + (3*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(8*a*d) + ((e + f*x)*Sech[c + d*x]^3*Tanh[c + d*x])/(4*a*d) + ((I/12)*f*Tanh[c + d*x]^3)/(a*d^2)} -{Sech[c + d*x]^3/(a + I*a*Sinh[c + d*x]), x, 4, (3*ArcTan[Sinh[c + d*x]])/(8*a*d) - (I/8)/(d*(a - I*a*Sinh[c + d*x])) + ((I/8)*a)/(d*(a + I*a*Sinh[c + d*x])^2) + (I/4)/(d*(a + I*a*Sinh[c + d*x]))} -{Sech[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]} -{Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Sinh[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n / (a+b Sinh[c+d x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^3*Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x, 11, -(e + f*x)^4/(4*b*f) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^3) - (6*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3) + (6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^4) + (6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^4)} -{(e + f*x)^2*Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x, 9, -(e + f*x)^3/(3*b*f) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) - (2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^3) - (2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3)} -{(e + f*x)^1*Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x, 7, -(e + f*x)^2/(2*b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)} -{(e + f*x)^0*Cosh[c + d*x]/(a + b*Sinh[c + d*x]), x, 2, Log[a + b*Sinh[c + d*x]]/(b*d)} -{Cosh[c + d*x]/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Cosh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{(e + f*x)^3*Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 18, -(a*(e + f*x)^4)/(4*b^2*f) + (6*f^2*(e + f*x)*Cosh[c + d*x])/(b*d^3) + ((e + f*x)^3*Cosh[c + d*x])/(b*d) + (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) + (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) - (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^3) + (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^3) + (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^4) - (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^4) - (6*f^3*Sinh[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Sinh[c + d*x])/(b*d^2)} -{(e + f*x)^2*Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 15, -(a*(e + f*x)^3)/(3*b^2*f) + (2*f^2*Cosh[c + d*x])/(b*d^3) + ((e + f*x)^2*Cosh[c + d*x])/(b*d) + (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) + (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) - (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^3) + (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^3) - (2*f*(e + f*x)*Sinh[c + d*x])/(b*d^2)} -{(e + f*x)^1*Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 12, -((a*e*x)/b^2) - (a*f*x^2)/(2*b^2) + ((e + f*x)*Cosh[c + d*x])/(b*d) + (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) + (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) - (f*Sinh[c + d*x])/(b*d^2)} -{(e + f*x)^0*Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 5, -((a*x)/b^2) - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b^2*d) + Cosh[c + d*x]/(b*d)} -{Cosh[c + d*x]^2/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Cosh[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{(e + f*x)^3*Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 21, (3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) - ((a^2 + b^2)*(e + f*x)^4)/(4*b^3*f) + (6*a*f^3*Cosh[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) - (6*a*f^2*(e + f*x)*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x])/(b^2*d) - (3*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (3*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^2)/(2*b*d)} -{(e + f*x)^2*Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 16, (e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) - ((a^2 + b^2)*(e + f*x)^3)/(3*b^3*f) + (2*a*f*(e + f*x)*Cosh[c + d*x])/(b^2*d^2) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a*f^2*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x])/(b^2*d) - (f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d^2) + (f^2*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^2*Sinh[c + d*x]^2)/(2*b*d)} -{(e + f*x)^1*Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 13, (f*x)/(4*b*d) - ((a^2 + b^2)*(e + f*x)^2)/(2*b^3*f) + (a*f*Cosh[c + d*x])/(b^2*d^2) + ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (a*(e + f*x)*Sinh[c + d*x])/(b^2*d) - (f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + ((e + f*x)*Sinh[c + d*x]^2)/(2*b*d)} -{(e + f*x)^0*Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 3, ((a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(b^3*d) - (a*Sinh[c + d*x])/(b^2*d) + Sinh[c + d*x]^2/(2*b*d)} -{Cosh[c + d*x]^3/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Cosh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(e + f*x)^3*Sech[c + d*x]/(a + b*Sinh[c + d*x]), x, 29, (2*a*(e + f*x)^3*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (b*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (3*I*a*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + (3*I*a*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^2) + (6*I*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - (6*I*a*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^3) - (6*I*a*f^3*PolyLog[4, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) + (6*I*a*f^3*PolyLog[4, I*E^(c + d*x)])/((a^2 + b^2)*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) - (3*b*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*(a^2 + b^2)*d^4)} -{(e + f*x)^2*Sech[c + d*x]/(a + b*Sinh[c + d*x]), x, 24, (2*a*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (b*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (2*I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (b*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^2) + (2*I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*I*a*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) + (b*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^3)} -{(e + f*x)^1*Sech[c + d*x]/(a + b*Sinh[c + d*x]), x, 19, (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (b*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + (I*a*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (b*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^2)} -{(e + f*x)^0*Sech[c + d*x]/(a + b*Sinh[c + d*x]), x, 6, (a*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d) - (b*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) + (b*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)*d)} -{Sech[c + d*x]/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{(e + f*x)^3*Sech[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 29, (a*(e + f*x)^3)/((a^2 + b^2)*d) - (6*b*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d^2) + (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (3*a*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) + (6*I*b*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - (6*I*b*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*a*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) - (6*I*b*f^3*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) + (6*I*b*f^3*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^4) - (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (3*a*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^4) + (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) + (b*(e + f*x)^3*Sech[c + d*x])/((a^2 + b^2)*d) + (a*(e + f*x)^3*Tanh[c + d*x])/((a^2 + b^2)*d)} -{(e + f*x)^2*Sech[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 24, (a*(e + f*x)^2)/((a^2 + b^2)*d) - (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d^2) + (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (2*a*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) + (2*I*b*f^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*I*b*f^2*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (a*f^2*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) - (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (b*(e + f*x)^2*Sech[c + d*x])/((a^2 + b^2)*d) + (a*(e + f*x)^2*Tanh[c + d*x])/((a^2 + b^2)*d)} -{(e + f*x)^1*Sech[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 15, -((b*f*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^2)) + (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a*f*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^2) + (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (b*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d) + (a*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d)} -{(e + f*x)^0*Sech[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 5, (-2*b^2*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) + (Sech[c + d*x]*(b + a*Sinh[c + d*x]))/((a^2 + b^2)*d)} -{Sech[c + d*x]^2/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{(e + f*x)^2*Sech[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 39, (2*a*b^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (a*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (b^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) + (b*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^3) - (2*I*a*b^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*I*a*b^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (b^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (2*I*a*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*I*a*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*a*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (b^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) + (a*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) + (b*(e + f*x)^2*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) - (b*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d^2) + (a*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d)} -{(e + f*x)^1*Sech[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 31, (2*a*b^2*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (b^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (I*a*b^2*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*(a^2 + b^2)*d^2) + (I*a*b^2*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a*f*PolyLog[2, I*E^(c + d*x)])/(2*(a^2 + b^2)*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (b^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (a*f*Sech[c + d*x])/(2*(a^2 + b^2)*d^2) + (b*(e + f*x)*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) - (b*f*Tanh[c + d*x])/(2*(a^2 + b^2)*d^2) + (a*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d)} -{(e + f*x)^0*Sech[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 7, (a*(a^2 + 3*b^2)*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d) - (b^3*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) + (b^3*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (Sech[c + d*x]^2*(b + a*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{Sech[c + d*x]^3/((e + f*x)^1*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Sech[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*m symbolic*) - - -{x^m*Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 0, Unintegrable[(x^m*Cosh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x]} -{x^m*Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 0, Unintegrable[(x^m*Cosh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x]} -{x^m*Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]), x, 0, Unintegrable[(x^m*Cosh[c + d*x])/(a + b*Sinh[c + d*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n / (a+b Sinh[c+d x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^1*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^2, x, 4, -((2*f*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2]*d^2)) - (e + f*x)/(b*d*(a + b*Sinh[c + d*x]))} -{(e + f*x)^2*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^2, x, 9, (2*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) - (2*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) + (2*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (2*f^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (e + f*x)^2/(b*d*(a + b*Sinh[c + d*x]))} -{(e + f*x)^3*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^2, x, 11, (3*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) - (3*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) + (6*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4) + (6*f^3*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4) - (e + f*x)^3/(b*d*(a + b*Sinh[c + d*x]))} - - -{((e + f*x)*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^2, x, 4, (-2*f*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2]*d^2) - (e + f*x)/(b*d*(a + b*Sinh[c + d*x]))} -{((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^2, x, 9, (2*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) - (2*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) + (2*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (2*f^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (e + f*x)^2/(b*d*(a + b*Sinh[c + d*x]))} -{((e + f*x)^3*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^2, x, 11, (3*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) - (3*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]*d^2) + (6*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^3) - (6*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4) + (6*f^3*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*Sqrt[a^2 + b^2]*d^4) - (e + f*x)^3/(b*d*(a + b*Sinh[c + d*x]))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n / (a+b Sinh[c+d x])^3*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(e + f*x)^1*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^3, x, 6, -((a*f*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2)*d^2)) - (e + f*x)/(2*b*d*(a + b*Sinh[c + d*x])^2) - (f*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} -{(e + f*x)^2*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^3, x, 12, (a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) - (a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) + (f^2*Log[a + b*Sinh[c + d*x]])/(b*(a^2 + b^2)*d^3) + (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*Sinh[c + d*x])^2) - (f*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} -{(e + f*x)^3*Cosh[c + d*x]/(a + b*Sinh[c + d*x])^3, x, 19, -((3*f*(e + f*x)^2)/(2*b*(a^2 + b^2)*d^2)) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) + (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) - (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sinh[c + d*x])^2) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} - - -{((e + f*x)*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3, x, 6, -((a*f*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2)*d^2)) - (e + f*x)/(2*b*d*(a + b*Sinh[c + d*x])^2) - (f*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} -{((e + f*x)^2*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3, x, 12, (a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) - (a*f*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d^2) + (f^2*Log[a + b*Sinh[c + d*x]])/(b*(a^2 + b^2)*d^3) + (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (a*f^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*Sinh[c + d*x])^2) - (f*(e + f*x)*Cosh[c + d*x])/((a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} -{((e + f*x)^3*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^3, x, 19, (-3*f*(e + f*x)^2)/(2*b*(a^2 + b^2)*d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) + (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) - (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sinh[c + d*x])^2) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^p (a+b Sinh[c+d x])^q*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^p / (a+b Sinh[c+d x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^1 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 16, (a*(e + f*x)^4)/(4*b^2*f) - (6*f^3*Cosh[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(b*d^2) - (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) - (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) + (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^3) + (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^3) - (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^4) - (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^4) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(b*d)} -{((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 13, (a*(e + f*x)^3)/(3*b^2*f) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) - (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^3) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^3) + (2*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)} -{((e + f*x)^1*Cosh[c + d*x]*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 10, (a*(e + f*x)^2)/(2*b^2*f) - (f*Cosh[c + d*x])/(b*d^2) - (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*d) - (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*d) - (a*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*d^2) - (a*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*d^2) + ((e + f*x)*Sinh[c + d*x])/(b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 4, -((a*Log[a + b*Sinh[c + d*x]])/(b^2*d)) + Sinh[c + d*x]/(b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 23, (3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*(e + f*x)*Cosh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x])/(b^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b*d^2) - (a*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) - (3*a*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*a*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) + (6*a*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*a*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*a*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a*f^3*Sinh[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Sinh[c + d*x])/(b^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 20, (f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) + (e + f*x)^3/(6*b*f) - (2*a*f^2*Cosh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x])/(b^2*d) - (f*(e + f*x)*Cosh[c + d*x]^2)/(2*b*d^2) - (a*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) - (2*a*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (2*a*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) + (2*a*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) + (2*a*f*(e + f*x)*Sinh[c + d*x])/(b^2*d^2) + (f^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{((e + f*x)^1*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 15, (a^2*e*x)/b^3 + (e*x)/(2*b) + (a^2*f*x^2)/(2*b^3) + (f*x^2)/(4*b) - (a*(e + f*x)*Cosh[c + d*x])/(b^2*d) - (f*Cosh[c + d*x]^2)/(4*b*d^2) - (a*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) - (a*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (a*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) + (a*f*Sinh[c + d*x])/(b^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{(Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 5, ((2*a^2 + b^2)*x)/(2*b^3) + (2*a*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(b^3*d) - (Cosh[c + d*x]*(2*a - b*Sinh[c + d*x]))/(2*b^2*d)} -{(Cosh[c + d*x]^2*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^2*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 30, (-3*a*f^3*x)/(8*b^2*d^3) - (a*(e + f*x)^3)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^4)/(4*b^4*f) - (6*a^2*f^3*Cosh[c + d*x])/(b^3*d^4) - (40*f^3*Cosh[c + d*x])/(9*b*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x])/(b^3*d^2) - (2*f*(e + f*x)^2*Cosh[c + d*x])/(b*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*b*d^4) - (f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*b*d^2) - (a*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (3*a*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (3*a*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (6*a*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) - (6*a*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^4) + (6*a^2*f^2*(e + f*x)*Sinh[c + d*x])/(b^3*d^3) + (40*f^2*(e + f*x)*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^3*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)^3*Sinh[c + d*x])/(3*b*d) + (3*a*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b*d) - (3*a*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^2*d)} -{((e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 23, -(a*e*f*x)/(2*b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Cosh[c + d*x])/(b^3*d^2) - (4*f*(e + f*x)*Cosh[c + d*x])/(3*b*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^2) - (a*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh[c + d*x])/(b^3*d^3) + (14*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)^2*Sinh[c + d*x])/(3*b*d) + (a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) + ((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b*d) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^2*d) + (2*f^2*Sinh[c + d*x]^3)/(27*b*d^3)} -{((e + f*x)^1*Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 17, -(a*f*x)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^2)/(2*b^4*f) - (a^2*f*Cosh[c + d*x])/(b^3*d^2) - (2*f*Cosh[c + d*x])/(3*b*d^2) - (f*Cosh[c + d*x]^3)/(9*b*d^2) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (a^2*(e + f*x)*Sinh[c + d*x])/(b^3*d) + (2*(e + f*x)*Sinh[c + d*x])/(3*b*d) + (a*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b*d) - (a*(e + f*x)*Sinh[c + d*x]^2)/(2*b^2*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]), x, 4, -((a*(a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(b^4*d)) + ((a^2 + b^2)*Sinh[c + d*x])/(b^3*d) - (a*Sinh[c + d*x]^2)/(2*b^2*d) + Sinh[c + d*x]^3/(3*b*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^3*Sinh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 39, (2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (a*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (a*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (3*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (3*I*a^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (3*I*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (3*I*a^2*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (3*a*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^2) + (6*I*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (6*I*a^2*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (6*I*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (6*I*a^2*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) + (6*a*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) - (3*a*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^3) - (6*I*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b*d^4) + (6*I*a^2*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^4) + (6*I*f^3*PolyLog[4, I*E^(c + d*x)])/(b*d^4) - (6*I*a^2*f^3*PolyLog[4, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^4) - (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) - (6*a*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) + (3*a*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*(a^2 + b^2)*d^4)} -{((e + f*x)^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 32, (2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (a*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (2*I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (2*I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (2*I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (2*I*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (2*a*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (a*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^2) + (2*I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (2*I*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (2*I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (2*I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) + (2*a*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^3) - (a*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^3)} -{((e + f*x)^1*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 25, (2*(e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (a*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) + (a*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (I*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (I*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^2*f*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (a*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (a*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (a*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^2)} -{Tanh[c + d*x]/(a + b*Sinh[c + d*x]), x, 6, (b*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d) + (a*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) - (a*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)*d)} -{Tanh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Tanh[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 36, (e + f*x)^3/(b*d) - (a^2*(e + f*x)^3)/(b*(a^2 + b^2)*d) + (6*a*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d^2) - (a*b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (a*b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (3*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*d^2) + (3*a^2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^2) - (6*I*a*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + (6*I*a*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (3*a*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (3*a*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (3*a^2*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^3) + (6*I*a*f^3*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) - (6*I*a*f^3*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^4) + (6*a*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (6*a*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^4) - (3*a^2*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^4) - (6*a*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) + (6*a*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (a*(e + f*x)^3*Sech[c + d*x])/((a^2 + b^2)*d) + ((e + f*x)^3*Tanh[c + d*x])/(b*d) - (a^2*(e + f*x)^3*Tanh[c + d*x])/(b*(a^2 + b^2)*d)} -{((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 30, (e + f*x)^2/(b*d) - (a^2*(e + f*x)^2)/(b*(a^2 + b^2)*d) + (4*a*f*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d^2) - (a*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (a*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*d^2) + (2*a^2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^2) - (2*I*a*f^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*I*a*f^2*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (2*a*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (2*a*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (f^2*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (a^2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^3) + (2*a*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (2*a*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (a*(e + f*x)^2*Sech[c + d*x])/((a^2 + b^2)*d) + ((e + f*x)^2*Tanh[c + d*x])/(b*d) - (a^2*(e + f*x)^2*Tanh[c + d*x])/(b*(a^2 + b^2)*d)} -{((e + f*x)^1*Sech[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 18, (a*f*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^2) - (a*b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (a*b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (f*Log[Cosh[c + d*x]])/(b*d^2) + (a^2*f*Log[Cosh[c + d*x]])/(b*(a^2 + b^2)*d^2) - (a*b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (a*b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (a*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d) + ((e + f*x)*Tanh[c + d*x])/(b*d) - (a^2*(e + f*x)*Tanh[c + d*x])/(b*(a^2 + b^2)*d)} -{(Sech[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 5, (2*a*b*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) - (Sech[c + d*x]*(a - b*Sinh[c + d*x]))/((a^2 + b^2)*d)} -{(Sech[c + d*x]*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sech[c + d*x]*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 49, ((e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (f^2*ArcTan[Sinh[c + d*x]])/(b*d^3) + (a^2*f^2*ArcTan[Sinh[c + d*x]])/(b*(a^2 + b^2)*d^3) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (a*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^3) - (I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (2*I*a^2*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (2*I*a^2*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a*b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (2*I*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (2*I*a^2*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (2*a*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (2*a*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (a*b^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) + (f*(e + f*x)*Sech[c + d*x])/(b*d^2) - (a^2*f*(e + f*x)*Sech[c + d*x])/(b*(a^2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d^2) + ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*(a^2 + b^2)*d)} -{((e + f*x^1)*Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 38, ((e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (a*b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a*b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b*d^2) + (I*a^2*b*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b*(a^2 + b^2)*d^2) + (I*f*PolyLog[2, I*E^(c + d*x)])/(2*b*d^2) - (I*a^2*b*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^2*f*PolyLog[2, I*E^(c + d*x)])/(2*b*(a^2 + b^2)*d^2) - (a*b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a*b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a*b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (f*Sech[c + d*x])/(2*b*d^2) - (a^2*f*Sech[c + d*x])/(2*b*(a^2 + b^2)*d^2) - (a*(e + f*x)*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*Tanh[c + d*x])/(2*(a^2 + b^2)*d^2) + ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*(a^2 + b^2)*d)} -{(Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 8, -(b*(a^2 - b^2)*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d) + (a*b^2*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (a*b^2*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sech[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^2 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 22, (3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) - (a^2*(e + f*x)^4)/(4*b^3*f) + (6*a*f^3*Cosh[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) - (6*a*f^2*(e + f*x)*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x])/(b^2*d) - (3*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (3*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^2)/(2*b*d)} -{((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 17, (e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) - (a^2*(e + f*x)^3)/(3*b^3*f) + (2*a*f*(e + f*x)*Cosh[c + d*x])/(b^2*d^2) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (2*a*f^2*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x])/(b^2*d) - (f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d^2) + (f^2*Sinh[c + d*x]^2)/(4*b*d^3) + ((e + f*x)^2*Sinh[c + d*x]^2)/(2*b*d)} -{((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 14, (f*x)/(4*b*d) - (a^2*(e + f*x)^2)/(2*b^3*f) + (a*f*Cosh[c + d*x])/(b^2*d^2) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*d) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (a*(e + f*x)*Sinh[c + d*x])/(b^2*d) - (f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + ((e + f*x)*Sinh[c + d*x]^2)/(2*b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 4, (a^2*Log[a + b*Sinh[c + d*x]])/(b^3*d) - (a*Sinh[c + d*x])/(b^2*d) + Sinh[c + d*x]^2/(2*b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 31, (-3*a*e*f^2*x)/(4*b^2*d^2) - (3*a*f^3*x^2)/(8*b^2*d^2) - (a^3*(e + f*x)^4)/(4*b^4*f) - (a*(e + f*x)^4)/(8*b^2*f) + (6*a^2*f^2*(e + f*x)*Cosh[c + d*x])/(b^3*d^3) + (4*f^2*(e + f*x)*Cosh[c + d*x])/(3*b*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x])/(b^3*d) + (3*a*f^3*Cosh[c + d*x]^2)/(8*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (3*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (3*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (6*a^2*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a^2*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a^2*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a^2*f^3*Sinh[c + d*x])/(b^3*d^4) - (14*f^3*Sinh[c + d*x])/(9*b*d^4) - (3*a^2*f*(e + f*x)^2*Sinh[c + d*x])/(b^3*d^2) - (2*f*(e + f*x)^2*Sinh[c + d*x])/(3*b*d^2) - (3*a*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d) - (f*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b*d^2) - (2*f^3*Sinh[c + d*x]^3)/(27*b*d^4)} -{((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 25, -(a*f^2*x)/(4*b^2*d^2) - (a^3*(e + f*x)^3)/(3*b^4*f) - (a*(e + f*x)^3)/(6*b^2*f) + (2*a^2*f^2*Cosh[c + d*x])/(b^3*d^3) + (4*f^2*Cosh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Cosh[c + d*x])/(b^3*d) + (a*f*(e + f*x)*Cosh[c + d*x]^2)/(2*b^2*d^2) + (2*f^2*Cosh[c + d*x]^3)/(27*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (2*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a^2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a^2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) - (2*a^2*f*(e + f*x)*Sinh[c + d*x])/(b^3*d^2) - (4*f*(e + f*x)*Sinh[c + d*x])/(9*b*d^2) - (a*f^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d) - (2*f*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b*d^2)} -{((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 19, -((a^3*e*x)/b^4) - (a*e*x)/(2*b^2) - (a^3*f*x^2)/(2*b^4) - (a*f*x^2)/(4*b^2) + (a^2*(e + f*x)*Cosh[c + d*x])/(b^3*d) + (a*f*Cosh[c + d*x]^2)/(4*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^3)/(3*b*d) + (a^2*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^2*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) + (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a^2*f*Sinh[c + d*x])/(b^3*d^2) - (f*Sinh[c + d*x])/(3*b*d^2) - (a*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d) - (f*Sinh[c + d*x]^3)/(9*b*d^2)} -{(Cosh[c + d*x]^2*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 8, -(a*(2*a^2 + b^2)*x)/(2*b^4) - (2*a^2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b^4*d) + ((3*a^2 + b^2)*Cosh[c + d*x])/(3*b^3*d) - (a*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b*d)} -{(Cosh[c + d*x]^2*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^2*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 40, (3*a^2*f^3*x)/(8*b^3*d^3) - (45*f^3*x)/(256*b*d^3) + (a^2*(e + f*x)^3)/(4*b^3*d) - (3*(e + f*x)^3)/(32*b*d) - (a^2*(a^2 + b^2)*(e + f*x)^4)/(4*b^5*f) + (6*a^3*f^3*Cosh[c + d*x])/(b^4*d^4) + (40*a*f^3*Cosh[c + d*x])/(9*b^2*d^4) + (3*a^3*f*(e + f*x)^2*Cosh[c + d*x])/(b^4*d^2) + (2*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + (9*f^2*(e + f*x)*Cosh[c + d*x]^2)/(32*b*d^3) + (2*a*f^3*Cosh[c + d*x]^3)/(27*b^2*d^4) + (a*f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*b^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]^4)/(32*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]^4)/(4*b*d) + (a^2*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) + (3*a^2*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (3*a^2*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) - (6*a^2*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^2*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) + (6*a^2*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^2*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^4) - (6*a^3*f^2*(e + f*x)*Sinh[c + d*x])/(b^4*d^3) - (40*a*f^2*(e + f*x)*Sinh[c + d*x])/(9*b^2*d^3) - (a^3*(e + f*x)^3*Sinh[c + d*x])/(b^4*d) - (2*a*(e + f*x)^3*Sinh[c + d*x])/(3*b^2*d) - (3*a^2*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^3*d^4) - (45*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*b*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^3*d^2) - (9*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(32*b*d^2) - (2*a*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d) - (3*f^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b*d^2) + (3*a^2*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b^3*d^3) + (a^2*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^3*d)} -{((e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 28, (a^2*e*f*x)/(2*b^3*d) - (3*e*f*x)/(16*b*d) + (a^2*f^2*x^2)/(4*b^3*d) - (3*f^2*x^2)/(32*b*d) - (a^2*(a^2 + b^2)*(e + f*x)^3)/(3*b^5*f) + (2*a^3*f*(e + f*x)*Cosh[c + d*x])/(b^4*d^2) + (4*a*f*(e + f*x)*Cosh[c + d*x])/(3*b^2*d^2) + (3*f^2*Cosh[c + d*x]^2)/(32*b*d^3) + (2*a*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b^2*d^2) + (f^2*Cosh[c + d*x]^4)/(32*b*d^3) + ((e + f*x)^2*Cosh[c + d*x]^4)/(4*b*d) + (a^2*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) + (2*a^2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (2*a^2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) - (2*a^2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (2*a^2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) - (2*a^3*f^2*Sinh[c + d*x])/(b^4*d^3) - (14*a*f^2*Sinh[c + d*x])/(9*b^2*d^3) - (a^3*(e + f*x)^2*Sinh[c + d*x])/(b^4*d) - (2*a*(e + f*x)^2*Sinh[c + d*x])/(3*b^2*d) - (a^2*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d^2) - (3*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(16*b*d^2) - (a*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d) - (f*(e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x])/(8*b*d^2) + (a^2*f^2*Sinh[c + d*x]^2)/(4*b^3*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^3*d) - (2*a*f^2*Sinh[c + d*x]^3)/(27*b^2*d^3)} -{((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 22, (a^2*f*x)/(4*b^3*d) - (3*f*x)/(32*b*d) - (a^2*(a^2 + b^2)*(e + f*x)^2)/(2*b^5*f) + (a^3*f*Cosh[c + d*x])/(b^4*d^2) + (2*a*f*Cosh[c + d*x])/(3*b^2*d^2) + (a*f*Cosh[c + d*x]^3)/(9*b^2*d^2) + ((e + f*x)*Cosh[c + d*x]^4)/(4*b*d) + (a^2*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) + (a^2*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^2*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) - (a^3*(e + f*x)*Sinh[c + d*x])/(b^4*d) - (2*a*(e + f*x)*Sinh[c + d*x])/(3*b^2*d) - (a^2*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^3*d^2) - (3*f*Cosh[c + d*x]*Sinh[c + d*x])/(32*b*d^2) - (a*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d) - (f*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b*d^2) + (a^2*(e + f*x)*Sinh[c + d*x]^2)/(2*b^3*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 4, (a^2*(a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(b^5*d) - (a*(a^2 + b^2)*Sinh[c + d*x])/(b^4*d) + ((a^2 + b^2)*Sinh[c + d*x]^2)/(2*b^3*d) - (a*Sinh[c + d*x]^3)/(3*b^2*d) + Sinh[c + d*x]^4/(4*b*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^3*Sinh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 46, -((e + f*x)^4/(4*b*f)) - (2*a*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^2*d) + (2*a^3*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + ((e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b*d) - (a^2*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d) + (3*I*a*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - (3*I*a^3*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (3*I*a*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + (3*I*a^3*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b*d^2) - (3*a^2*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^2) - (6*I*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^3) + (6*I*a^3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) + (6*I*a*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b^2*d^3) - (6*I*a^3*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^3) + (3*a^2*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^3) + (6*I*a*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^2*d^4) - (6*I*a^3*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) - (6*I*a*f^3*PolyLog[4, I*E^(c + d*x)])/(b^2*d^4) + (6*I*a^3*f^3*PolyLog[4, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b*d^4) - (3*a^2*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b*(a^2 + b^2)*d^4)} -{((e + f*x)^2*Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 38, -((e + f*x)^3/(3*b*f)) - (2*a*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*d) + (2*a^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + ((e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*d) - (a^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d) + (2*I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - (2*I*a^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (2*I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + (2*I*a^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*d^2) - (a^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^2) - (2*I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^3) + (2*I*a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) + (2*I*a*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*d^3) - (2*I*a^3*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^3) - (f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^3) + (a^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^3)} -{((e + f*x)*Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 30, -((e + f*x)^2/(2*b*f)) - (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*d) + (2*a^3*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d) + ((e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*d) - (a^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d) + (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - (I*a^3*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (I*a*f*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + (I*a^3*f*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^2) + (f*PolyLog[2, -E^(2*(c + d*x))])/(2*b*d^2) - (a^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*b*(a^2 + b^2)*d^2)} -{(Sinh[c + d*x]*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 7, -((a*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d)) + (b*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) + (a^2*Log[a + b*Sinh[c + d*x]])/(b*(a^2 + b^2)*d)} -{(Sinh[c + d*x]*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sinh[c + d*x]*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 45, -((a*(e + f*x)^3)/(b^2*d)) + (a^3*(e + f*x)^3)/(b^2*(a^2 + b^2)*d) + (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d^2) - (6*a^2*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*d^2) - (3*a^3*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^2) - (6*I*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^3) + (6*I*a^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (6*I*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^3) - (6*I*a^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*a^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (3*a*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^3) - (3*a^3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^3) + (6*I*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^4) - (6*I*a^2*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^4) - (6*I*f^3*PolyLog[3, I*E^(c + d*x)])/(b*d^4) + (6*I*a^2*f^3*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^4) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (3*a*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^4) + (3*a^3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^4) + (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (6*a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - ((e + f*x)^3*Sech[c + d*x])/(b*d) + (a^2*(e + f*x)^3*Sech[c + d*x])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^3*Tanh[c + d*x])/(b^2*d) + (a^3*(e + f*x)^3*Tanh[c + d*x])/(b^2*(a^2 + b^2)*d)} -{((e + f*x)^2*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 37, -((a*(e + f*x)^2)/(b^2*d)) + (a^3*(e + f*x)^2)/(b^2*(a^2 + b^2)*d) + (4*f*(e + f*x)*ArcTan[E^(c + d*x)])/(b*d^2) - (4*a^2*f*(e + f*x)*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (2*a*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^2*d^2) - (2*a^3*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^2) - (2*I*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^3) + (2*I*a^2*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (2*I*f^2*PolyLog[2, I*E^(c + d*x)])/(b*d^3) - (2*I*a^2*f^2*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (2*a^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (a*f^2*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^3) - (a^3*f^2*PolyLog[2, -E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^3) - (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (2*a^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - ((e + f*x)^2*Sech[c + d*x])/(b*d) + (a^2*(e + f*x)^2*Sech[c + d*x])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^2*Tanh[c + d*x])/(b^2*d) + (a^3*(e + f*x)^2*Tanh[c + d*x])/(b^2*(a^2 + b^2)*d)} -{((e + f*x)*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 21, (f*ArcTan[Sinh[c + d*x]])/(b*d^2) - (a^2*f*ArcTan[Sinh[c + d*x]])/(b*(a^2 + b^2)*d^2) + (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (a*f*Log[Cosh[c + d*x]])/(b^2*d^2) - (a^3*f*Log[Cosh[c + d*x]])/(b^2*(a^2 + b^2)*d^2) + (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - ((e + f*x)*Sech[c + d*x])/(b*d) + (a^2*(e + f*x)*Sech[c + d*x])/(b*(a^2 + b^2)*d) - (a*(e + f*x)*Tanh[c + d*x])/(b^2*d) + (a^3*(e + f*x)*Tanh[c + d*x])/(b^2*(a^2 + b^2)*d)} -{Tanh[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 8, (-2*a^2*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d) - (b*Sech[c + d*x])/((a^2 + b^2)*d) - (a*Tanh[c + d*x])/((a^2 + b^2)*d)} -{Tanh[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Tanh[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 53, -((a*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*d)) + (2*a^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a*f^2*ArcTan[Sinh[c + d*x]])/(b^2*d^3) - (a^3*f^2*ArcTan[Sinh[c + d*x]])/(b^2*(a^2 + b^2)*d^3) + (a^2*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^2*b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^2*b*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (f^2*Log[Cosh[c + d*x]])/(b*d^3) + (a^2*f^2*Log[Cosh[c + d*x]])/(b*(a^2 + b^2)*d^3) + (I*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^2) - (2*I*a^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*d^2) + (2*I*a^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) + (2*a^2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (2*a^2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^2*b*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) - (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^3) + (2*I*a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) + (I*a*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*d^3) - (2*I*a^3*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*a^3*f^2*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (2*a^2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (2*a^2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (a^2*b*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) - (a*f*(e + f*x)*Sech[c + d*x])/(b^2*d^2) + (a^3*f*(e + f*x)*Sech[c + d*x])/(b^2*(a^2 + b^2)*d^2) - ((e + f*x)^2*Sech[c + d*x]^2)/(2*b*d) + (a^2*(e + f*x)^2*Sech[c + d*x]^2)/(2*b*(a^2 + b^2)*d) + (f*(e + f*x)*Tanh[c + d*x])/(b*d^2) - (a^2*f*(e + f*x)*Tanh[c + d*x])/(b*(a^2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^2*d) + (a^3*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^2*(a^2 + b^2)*d)} -{((e + f*x)*Sech[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 42, -((a*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*d)) + (2*a^3*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d) + (a^2*b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^2*b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^2*b*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) + (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b^2*d^2) - (I*a^3*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^3*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b^2*(a^2 + b^2)*d^2) - (I*a*f*PolyLog[2, I*E^(c + d*x)])/(2*b^2*d^2) + (I*a^3*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*a^3*f*PolyLog[2, I*E^(c + d*x)])/(2*b^2*(a^2 + b^2)*d^2) + (a^2*b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a^2*b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^2*b*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) - (a*f*Sech[c + d*x])/(2*b^2*d^2) + (a^3*f*Sech[c + d*x])/(2*b^2*(a^2 + b^2)*d^2) - ((e + f*x)*Sech[c + d*x]^2)/(2*b*d) + (a^2*(e + f*x)*Sech[c + d*x]^2)/(2*b*(a^2 + b^2)*d) + (f*Tanh[c + d*x])/(2*b*d^2) - (a^2*f*Tanh[c + d*x])/(2*b*(a^2 + b^2)*d^2) - (a*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b^2*d) + (a^3*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b^2*(a^2 + b^2)*d)} -{(Sech[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 8, (a*(a^2 - b^2)*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d) - (a^2*b*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) + (a^2*b*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(b + a*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{(Sech[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sech[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^3 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 30, (-3*a*f^3*x)/(8*b^2*d^3) - (a*(e + f*x)^3)/(4*b^2*d) + (a^3*(e + f*x)^4)/(4*b^4*f) - (6*a^2*f^3*Cosh[c + d*x])/(b^3*d^4) + (14*f^3*Cosh[c + d*x])/(9*b*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x])/(b^3*d^2) + (2*f*(e + f*x)^2*Cosh[c + d*x])/(3*b*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*b*d^4) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^4) + (6*a^2*f^2*(e + f*x)*Sinh[c + d*x])/(b^3*d^3) - (4*f^2*(e + f*x)*Sinh[c + d*x])/(3*b*d^3) + (a^2*(e + f*x)^3*Sinh[c + d*x])/(b^3*d) + (3*a*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) - (3*a*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^2*d) - (f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b*d^2) + (2*f^2*(e + f*x)*Sinh[c + d*x]^3)/(9*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^3)/(3*b*d)} -{((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 22, -(a*e*f*x)/(2*b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a^3*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Cosh[c + d*x])/(b^3*d^2) + (4*f*(e + f*x)*Cosh[c + d*x])/(9*b*d^2) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh[c + d*x])/(b^3*d^3) - (4*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^2*d) - (2*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^2)/(9*b*d^2) + (2*f^2*Sinh[c + d*x]^3)/(27*b*d^3) + ((e + f*x)^2*Sinh[c + d*x]^3)/(3*b*d)} -{((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 18, -(a*f*x)/(4*b^2*d) + (a^3*(e + f*x)^2)/(2*b^4*f) - (a^2*f*Cosh[c + d*x])/(b^3*d^2) + (f*Cosh[c + d*x])/(3*b*d^2) - (f*Cosh[c + d*x]^3)/(9*b*d^2) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^2) + (a^2*(e + f*x)*Sinh[c + d*x])/(b^3*d) + (a*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^2*d^2) - (a*(e + f*x)*Sinh[c + d*x]^2)/(2*b^2*d) + ((e + f*x)*Sinh[c + d*x]^3)/(3*b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 4, -((a^3*Log[a + b*Sinh[c + d*x]])/(b^4*d)) + (a^2*Sinh[c + d*x])/(b^3*d) - (a*Sinh[c + d*x]^2)/(2*b^2*d) + Sinh[c + d*x]^3/(3*b*d)} -{(Cosh[c + d*x]*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 38, (3*a^2*e*f^2*x)/(4*b^3*d^2) + (3*a^2*f^3*x^2)/(8*b^3*d^2) + (a^4*(e + f*x)^4)/(4*b^5*f) + (a^2*(e + f*x)^4)/(8*b^3*f) - (e + f*x)^4/(32*b*f) - (6*a^3*f^2*(e + f*x)*Cosh[c + d*x])/(b^4*d^3) - (4*a*f^2*(e + f*x)*Cosh[c + d*x])/(3*b^2*d^3) - (a^3*(e + f*x)^3*Cosh[c + d*x])/(b^4*d) - (3*a^2*f^3*Cosh[c + d*x]^2)/(8*b^3*d^4) - (3*a^2*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b^3*d^2) - (2*a*f^2*(e + f*x)*Cosh[c + d*x]^3)/(9*b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]^3)/(3*b^2*d) - (3*f^3*Cosh[4*c + 4*d*x])/(1024*b*d^4) - (3*f*(e + f*x)^2*Cosh[4*c + 4*d*x])/(128*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (3*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (3*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) - (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^3*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^4) + (6*a^3*f^3*Sinh[c + d*x])/(b^4*d^4) + (14*a*f^3*Sinh[c + d*x])/(9*b^2*d^4) + (3*a^3*f*(e + f*x)^2*Sinh[c + d*x])/(b^4*d^2) + (2*a*f*(e + f*x)^2*Sinh[c + d*x])/(3*b^2*d^2) + (3*a^2*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^3*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^2*d^2) + (2*a*f^3*Sinh[c + d*x]^3)/(27*b^2*d^4) + (3*f^2*(e + f*x)*Sinh[4*c + 4*d*x])/(256*b*d^3) + ((e + f*x)^3*Sinh[4*c + 4*d*x])/(32*b*d)} -{((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 31, (a^2*f^2*x)/(4*b^3*d^2) + (a^4*(e + f*x)^3)/(3*b^5*f) + (a^2*(e + f*x)^3)/(6*b^3*f) - (e + f*x)^3/(24*b*f) - (2*a^3*f^2*Cosh[c + d*x])/(b^4*d^3) - (4*a*f^2*Cosh[c + d*x])/(9*b^2*d^3) - (a^3*(e + f*x)^2*Cosh[c + d*x])/(b^4*d) - (a^2*f*(e + f*x)*Cosh[c + d*x]^2)/(2*b^3*d^2) - (2*a*f^2*Cosh[c + d*x]^3)/(27*b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x]^3)/(3*b^2*d) - (f*(e + f*x)*Cosh[4*c + 4*d*x])/(64*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (2*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (2*a^3*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (2*a^3*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^3) - (2*a^3*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^3) + (2*a^3*f*(e + f*x)*Sinh[c + d*x])/(b^4*d^2) + (4*a*f*(e + f*x)*Sinh[c + d*x])/(9*b^2*d^2) + (a^2*f^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^3*d^3) + (a^2*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (2*a*f*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b^2*d^2) + (f^2*Sinh[4*c + 4*d*x])/(256*b*d^3) + ((e + f*x)^2*Sinh[4*c + 4*d*x])/(32*b*d)} -{((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 24, (a^4*e*x)/b^5 + (a^2*e*x)/(2*b^3) + (a^4*f*x^2)/(2*b^5) + (a^2*f*x^2)/(4*b^3) - (e + f*x)^2/(16*b*f) - (a^3*(e + f*x)*Cosh[c + d*x])/(b^4*d) - (a^2*f*Cosh[c + d*x]^2)/(4*b^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^3)/(3*b^2*d) - (f*Cosh[4*c + 4*d*x])/(128*b*d^2) - (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^5*d) + (a^3*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^5*d) - (a^3*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^5*d^2) + (a^3*f*Sinh[c + d*x])/(b^4*d^2) + (a*f*Sinh[c + d*x])/(3*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^3*d) + (a*f*Sinh[c + d*x]^3)/(9*b^2*d^2) + ((e + f*x)*Sinh[4*c + 4*d*x])/(32*b*d)} -{(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 9, ((8*a^4 + 4*a^2*b^2 - b^4)*x)/(8*b^5) + (2*a^3*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b^5*d) - (a*(3*a^2 + b^2)*Cosh[c + d*x])/(3*b^4*d) + ((4*a^2 + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^3*d) - (a*Cosh[c + d*x]*Sinh[c + d*x]^2)/(3*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*b*d)} -{(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^2*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 55, (-3*a^3*f^3*x)/(8*b^4*d^3) + (45*a*f^3*x)/(256*b^2*d^3) - (a^3*(e + f*x)^3)/(4*b^4*d) + (3*a*(e + f*x)^3)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^4)/(4*b^6*f) - (6*a^4*f^3*Cosh[c + d*x])/(b^5*d^4) - (40*a^2*f^3*Cosh[c + d*x])/(9*b^3*d^4) + (3*f^3*Cosh[c + d*x])/(4*b*d^4) - (3*a^4*f*(e + f*x)^2*Cosh[c + d*x])/(b^5*d^2) - (2*a^2*f*(e + f*x)^2*Cosh[c + d*x])/(b^3*d^2) + (3*f*(e + f*x)^2*Cosh[c + d*x])/(8*b*d^2) - (9*a*f^2*(e + f*x)*Cosh[c + d*x]^2)/(32*b^2*d^3) - (2*a^2*f^3*Cosh[c + d*x]^3)/(27*b^3*d^4) - (a^2*f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*b^3*d^2) - (3*a*f^2*(e + f*x)*Cosh[c + d*x]^4)/(32*b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x]^4)/(4*b^2*d) - (f^3*Cosh[3*c + 3*d*x])/(216*b*d^4) - (f*(e + f*x)^2*Cosh[3*c + 3*d*x])/(48*b*d^2) - (3*f^3*Cosh[5*c + 5*d*x])/(5000*b*d^4) - (3*f*(e + f*x)^2*Cosh[5*c + 5*d*x])/(400*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (3*a^3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (3*a^3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^2) + (6*a^3*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^3) + (6*a^3*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^3) - (6*a^3*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^4) - (6*a^3*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^4) + (6*a^4*f^2*(e + f*x)*Sinh[c + d*x])/(b^5*d^3) + (40*a^2*f^2*(e + f*x)*Sinh[c + d*x])/(9*b^3*d^3) - (3*f^2*(e + f*x)*Sinh[c + d*x])/(4*b*d^3) + (a^4*(e + f*x)^3*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)^3*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)^3*Sinh[c + d*x])/(8*b*d) + (3*a^3*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^4*d^4) + (45*a*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*b^2*d^4) + (3*a^3*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (9*a*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(32*b^2*d^2) + (2*a^2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*b^3*d^3) + (a^2*(e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (3*a*f^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*b^2*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b^2*d^2) - (3*a^3*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*b^4*d^3) - (a^3*(e + f*x)^3*Sinh[c + d*x]^2)/(2*b^4*d) + (f^2*(e + f*x)*Sinh[3*c + 3*d*x])/(72*b*d^3) + ((e + f*x)^3*Sinh[3*c + 3*d*x])/(48*b*d) + (3*f^2*(e + f*x)*Sinh[5*c + 5*d*x])/(1000*b*d^3) + ((e + f*x)^3*Sinh[5*c + 5*d*x])/(80*b*d)} -{((e + f*x)^2*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 40, -(a^3*e*f*x)/(2*b^4*d) + (3*a*e*f*x)/(16*b^2*d) - (a^3*f^2*x^2)/(4*b^4*d) + (3*a*f^2*x^2)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^3)/(3*b^6*f) - (2*a^4*f*(e + f*x)*Cosh[c + d*x])/(b^5*d^2) - (4*a^2*f*(e + f*x)*Cosh[c + d*x])/(3*b^3*d^2) + (f*(e + f*x)*Cosh[c + d*x])/(4*b*d^2) - (3*a*f^2*Cosh[c + d*x]^2)/(32*b^2*d^3) - (2*a^2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b^3*d^2) - (a*f^2*Cosh[c + d*x]^4)/(32*b^2*d^3) - (a*(e + f*x)^2*Cosh[c + d*x]^4)/(4*b^2*d) - (f*(e + f*x)*Cosh[3*c + 3*d*x])/(72*b*d^2) - (f*(e + f*x)*Cosh[5*c + 5*d*x])/(200*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (2*a^3*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^2) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^3) + (2*a^3*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^3) + (2*a^4*f^2*Sinh[c + d*x])/(b^5*d^3) + (14*a^2*f^2*Sinh[c + d*x])/(9*b^3*d^3) - (f^2*Sinh[c + d*x])/(4*b*d^3) + (a^4*(e + f*x)^2*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)^2*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)^2*Sinh[c + d*x])/(8*b*d) + (a^3*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^4*d^2) + (3*a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(16*b^2*d^2) + (a^2*(e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*(e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x])/(8*b^2*d^2) - (a^3*f^2*Sinh[c + d*x]^2)/(4*b^4*d^3) - (a^3*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^4*d) + (2*a^2*f^2*Sinh[c + d*x]^3)/(27*b^3*d^3) + (f^2*Sinh[3*c + 3*d*x])/(216*b*d^3) + ((e + f*x)^2*Sinh[3*c + 3*d*x])/(48*b*d) + (f^2*Sinh[5*c + 5*d*x])/(1000*b*d^3) + ((e + f*x)^2*Sinh[5*c + 5*d*x])/(80*b*d)} -{((e + f*x)*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 31, -(a^3*f*x)/(4*b^4*d) + (3*a*f*x)/(32*b^2*d) + (a^3*(a^2 + b^2)*(e + f*x)^2)/(2*b^6*f) - (a^4*f*Cosh[c + d*x])/(b^5*d^2) - (2*a^2*f*Cosh[c + d*x])/(3*b^3*d^2) + (f*Cosh[c + d*x])/(8*b*d^2) - (a^2*f*Cosh[c + d*x]^3)/(9*b^3*d^2) - (a*(e + f*x)*Cosh[c + d*x]^4)/(4*b^2*d) - (f*Cosh[3*c + 3*d*x])/(144*b*d^2) - (f*Cosh[5*c + 5*d*x])/(400*b*d^2) - (a^3*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^6*d) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^6*d^2) - (a^3*(a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^6*d^2) + (a^4*(e + f*x)*Sinh[c + d*x])/(b^5*d) + (2*a^2*(e + f*x)*Sinh[c + d*x])/(3*b^3*d) - ((e + f*x)*Sinh[c + d*x])/(8*b*d) + (a^3*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*b^4*d^2) + (3*a*f*Cosh[c + d*x]*Sinh[c + d*x])/(32*b^2*d^2) + (a^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*b^3*d) + (a*f*Cosh[c + d*x]^3*Sinh[c + d*x])/(16*b^2*d^2) - (a^3*(e + f*x)*Sinh[c + d*x]^2)/(2*b^4*d) + ((e + f*x)*Sinh[3*c + 3*d*x])/(48*b*d) + ((e + f*x)*Sinh[5*c + 5*d*x])/(80*b*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 4, -((a^3*(a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(b^6*d)) + (a^2*(a^2 + b^2)*Sinh[c + d*x])/(b^5*d) - (a*(a^2 + b^2)*Sinh[c + d*x]^2)/(2*b^4*d) + ((a^2 + b^2)*Sinh[c + d*x]^3)/(3*b^3*d) - (a*Sinh[c + d*x]^4)/(4*b^2*d) + Sinh[c + d*x]^5/(5*b*d)} -{(Cosh[c + d*x]^3*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^3*Sinh[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 61, (a*(e + f*x)^4)/(4*b^2*f) + (2*a^2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (6*f^3*Cosh[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(b*d^2) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b^2*d) + (a^3*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - (3*I*a^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) + (3*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (3*I*a^4*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + (3*I*a^2*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) - (3*I*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (3*I*a^4*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*d^2) + (3*a^3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^2) + (6*I*a^2*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) - (6*I*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (6*I*a^4*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - (6*I*a^2*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b^3*d^3) + (6*I*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (6*I*a^4*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (3*a*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (3*a^3*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) - (6*I*a^2*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^3*d^4) + (6*I*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b*d^4) + (6*I*a^4*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^4) + (6*I*a^2*f^3*PolyLog[4, I*E^(c + d*x)])/(b^3*d^4) - (6*I*f^3*PolyLog[4, I*E^(c + d*x)])/(b*d^4) - (6*I*a^4*f^3*PolyLog[4, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^4) - (3*a*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b^2*d^4) + (3*a^3*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b^2*(a^2 + b^2)*d^4) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(b*d)} -{((e + f*x)^2*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 50, (a*(e + f*x)^3)/(3*b^2*f) + (2*a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*d) + (a^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - (2*I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) + (2*I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (2*I*a^4*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + (2*I*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) - (2*I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (2*I*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (a*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^2) + (a^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^2) + (2*I*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) - (2*I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (2*I*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - (2*I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b^3*d^3) + (2*I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (2*I*a^4*f^2*PolyLog[3, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (a*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (a^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) + (2*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)} -{((e + f*x)*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 39, (a*(e + f*x)^2)/(2*b^2*f) + (2*a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (f*Cosh[c + d*x])/(b*d^2) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^2*d) + (a^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - (I*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) + (I*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (I*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + (I*a^2*f*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) - (I*f*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (I*a^4*f*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (a*f*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*d^2) + (a^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^2) + ((e + f*x)*Sinh[c + d*x])/(b*d)} -{(Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]), x, 7, -((b*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d)) - (a*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) - (a^3*Log[a + b*Sinh[c + d*x]])/(b^2*(a^2 + b^2)*d) + Sinh[c + d*x]/(b*d)} -{(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sinh[c + d*x]^2*Tanh[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 53, (a^2*(e + f*x)^3)/(b^3*d) - (e + f*x)^3/(b*d) - (a^4*(e + f*x)^3)/(b^3*(a^2 + b^2)*d) + (e + f*x)^4/(4*b*f) - (6*a*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*d^2) + (6*a^3*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (3*a^2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^3*d^2) + (3*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*d^2) + (3*a^4*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^2) + (6*I*a*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^3) - (6*I*a^3*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (6*I*a*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*d^3) + (6*I*a^3*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) - (3*a^2*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^3*d^3) + (3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (3*a^4*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^3) - (6*I*a*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*d^4) + (6*I*a^3*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) + (6*I*a*f^3*PolyLog[3, I*E^(c + d*x)])/(b^2*d^4) - (6*I*a^3*f^3*PolyLog[3, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^4) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*a^2*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^3*d^4) - (3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^4) - (3*a^4*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^3*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (a*(e + f*x)^3*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)^3*Sech[c + d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^3*Tanh[c + d*x])/(b^3*d) - ((e + f*x)^3*Tanh[c + d*x])/(b*d) - (a^4*(e + f*x)^3*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)} -{((e + f*x)^2*Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 44, (a^2*(e + f*x)^2)/(b^3*d) - (e + f*x)^2/(b*d) - (a^4*(e + f*x)^2)/(b^3*(a^2 + b^2)*d) + (e + f*x)^3/(3*b*f) - (4*a*f*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*d^2) + (4*a^3*f*(e + f*x)*ArcTan[E^(c + d*x)])/(b^2*(a^2 + b^2)*d^2) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (2*a^2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^3*d^2) + (2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b*d^2) + (2*a^4*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^2) + (2*I*a*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*d^3) - (2*I*a^3*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (2*I*a*f^2*PolyLog[2, I*E^(c + d*x)])/(b^2*d^3) + (2*I*a^3*f^2*PolyLog[2, I*E^(c + d*x)])/(b^2*(a^2 + b^2)*d^3) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) - (a^2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(b^3*d^3) + (f^2*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (a^4*f^2*PolyLog[2, -E^(2*(c + d*x))])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) - (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (a*(e + f*x)^2*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)^2*Sech[c + d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)^2*Tanh[c + d*x])/(b^3*d) - ((e + f*x)^2*Tanh[c + d*x])/(b*d) - (a^4*(e + f*x)^2*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)} -{((e + f*x)*Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 25, (e*x)/b + (f*x^2)/(2*b) - (a*f*ArcTan[Sinh[c + d*x]])/(b^2*d^2) + (a^3*f*ArcTan[Sinh[c + d*x]])/(b^2*(a^2 + b^2)*d^2) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) + (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)^(3/2)*d) - (a^2*f*Log[Cosh[c + d*x]])/(b^3*d^2) + (f*Log[Cosh[c + d*x]])/(b*d^2) + (a^4*f*Log[Cosh[c + d*x]])/(b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^2) + (a*(e + f*x)*Sech[c + d*x])/(b^2*d) - (a^3*(e + f*x)*Sech[c + d*x])/(b^2*(a^2 + b^2)*d) + (a^2*(e + f*x)*Tanh[c + d*x])/(b^3*d) - ((e + f*x)*Tanh[c + d*x])/(b*d) - (a^4*(e + f*x)*Tanh[c + d*x])/(b^3*(a^2 + b^2)*d)} -{(Sinh[c + d*x]*Tanh[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 9, (a^2*x)/(b*(a^2 + b^2)) + (b*x)/(a^2 + b^2) + (2*a^3*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2)*d) + (a*Sech[c + d*x])/((a^2 + b^2)*d) - (b*Tanh[c + d*x])/((a^2 + b^2)*d)} -{(Sinh[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Sinh[c + d*x]*Tanh[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^2*Tanh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 71, (a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*d) + ((e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)^2*d) - (a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (a^2*f^2*ArcTan[Sinh[c + d*x]])/(b^3*d^3) + (f^2*ArcTan[Sinh[c + d*x]])/(b*d^3) + (a^4*f^2*ArcTan[Sinh[c + d*x]])/(b^3*(a^2 + b^2)*d^3) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) + (a*f^2*Log[Cosh[c + d*x]])/(b^2*d^3) - (a^3*f^2*Log[Cosh[c + d*x]])/(b^2*(a^2 + b^2)*d^3) - (I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) - (I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + (2*I*a^4*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) + (I*a^4*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + (I*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) + (I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - (2*I*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) - (I*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (I*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) + (I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - (2*I*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^3) - (I*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - (I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b^3*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + (2*I*a^4*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^3) + (I*a^4*f^2*PolyLog[3, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (a^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) + (a^2*f*(e + f*x)*Sech[c + d*x])/(b^3*d^2) - (f*(e + f*x)*Sech[c + d*x])/(b*d^2) - (a^4*f*(e + f*x)*Sech[c + d*x])/(b^3*(a^2 + b^2)*d^2) + (a*(e + f*x)^2*Sech[c + d*x]^2)/(2*b^2*d) - (a^3*(e + f*x)^2*Sech[c + d*x]^2)/(2*b^2*(a^2 + b^2)*d) - (a*f*(e + f*x)*Tanh[c + d*x])/(b^2*d^2) + (a^3*f*(e + f*x)*Tanh[c + d*x])/(b^2*(a^2 + b^2)*d^2) + (a^2*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^3*d) - ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^4*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b^3*(a^2 + b^2)*d)} -{((e + f*x)*Tanh[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 55, (a^2*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*d) + ((e + f*x)*ArcTan[E^(c + d*x)])/(b*d) - (2*a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)^2*d) - (a^4*(e + f*x)*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) - (a^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (I*a^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b^3*d^2) - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b*d^2) + (I*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) + (I*a^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*b^3*(a^2 + b^2)*d^2) + (I*a^2*f*PolyLog[2, I*E^(c + d*x)])/(2*b^3*d^2) + (I*f*PolyLog[2, I*E^(c + d*x)])/(2*b*d^2) - (I*a^4*f*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)^2*d^2) - (I*a^4*f*PolyLog[2, I*E^(c + d*x)])/(2*b^3*(a^2 + b^2)*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (a^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (a^2*f*Sech[c + d*x])/(2*b^3*d^2) - (f*Sech[c + d*x])/(2*b*d^2) - (a^4*f*Sech[c + d*x])/(2*b^3*(a^2 + b^2)*d^2) + (a*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*d) - (a^3*(e + f*x)*Sech[c + d*x]^2)/(2*b^2*(a^2 + b^2)*d) - (a*f*Tanh[c + d*x])/(2*b^2*d^2) + (a^3*f*Tanh[c + d*x])/(2*b^2*(a^2 + b^2)*d^2) + (a^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b^3*d) - ((e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^4*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*b^3*(a^2 + b^2)*d)} -{Tanh[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 7, (b*(3*a^2 + b^2)*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d) + (a^3*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (a^3*Log[a + b*Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{Tanh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Tanh[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Section:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Sinh[c+d x]^p / (a+b Sinh[c+d x])^2*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^p (a+b Sinh[c+d x])^q*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^p / (a+b Sinh[c+d x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^1 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 18, -(((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*d)) - ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*d) + ((e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a*d) - (3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) - (6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^4) - (6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^4) + (3*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a*d^4)} -{((e + f*x)^2*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 15, -(((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*d)) - ((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*d) + ((e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d) - (2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^2) - (2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^2) + (2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^3) + (2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^3) - (f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3)} -{((e + f*x)*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 12, -(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*d)) - ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*d) + ((e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d) - (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*d^2) - (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*d^2) + (f*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2)} -{Coth[c + d*x]/(a + b*Sinh[c + d*x]), x, 4, Log[Sinh[c + d*x]]/(a*d) - Log[a + b*Sinh[c + d*x]]/(a*d)} -{Coth[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Coth[c + d*x]/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 33, (e + f*x)^4/(4*b*f) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b*d) + (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b*d) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^2) + (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^2) + (6*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^3) - (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^4) + (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^4)} -{((e + f*x)^2*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 27, (e + f*x)^3/(3*b*f) - (2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b*d) + (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b*d) - (2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^2) + (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^2) + (2*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (2*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^3) - (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^3)} -{((e + f*x)*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 21, (e*x)/b + (f*x^2)/(2*b) - (2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d) - (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b*d) + (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b*d) - (f*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (f*PolyLog[2, E^(c + d*x)])/(a*d^2) - (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^2) + (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^2)} -{(Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 6, x/b - ArcTanh[Cosh[c + d*x]]/(a*d) + (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*b*d)} -{(Cosh[c + d*x]*Coth[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]*Coth[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 34, -(e + f*x)^4/(4*a*f) + ((a^2 + b^2)*(e + f*x)^4)/(4*a*b^2*f) - (6*f^3*Cosh[c + d*x])/(b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(b*d^2) - ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b^2*d) - ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b^2*d) + ((e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a*d) - (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^2) - (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^3) + (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) - (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^4) - (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^4) + (3*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a*d^4) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(b*d)} -{((e + f*x)^2*Cosh[c + d*x]^2*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 26, -(e + f*x)^3/(3*a*f) + ((a^2 + b^2)*(e + f*x)^3)/(3*a*b^2*f) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2) - ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b^2*d) - ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b^2*d) + ((e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d) - (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^2) - (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^2) + (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^3) + (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^3) - (f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) + (2*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)} -{((e + f*x)*Cosh[c + d*x]^2*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 22, -(e + f*x)^2/(2*a*f) + ((a^2 + b^2)*(e + f*x)^2)/(2*a*b^2*f) - (f*Cosh[c + d*x])/(b*d^2) - ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b^2*d) - ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*b^2*d) + ((e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d) - ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b^2*d^2) - ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b^2*d^2) + (f*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + ((e + f*x)*Sinh[c + d*x])/(b*d)} -{(Cosh[c + d*x]^2*Coth[c + d*x])/(a + b*Sinh[c + d*x]), x, 4, Log[Sinh[c + d*x]]/(a*d) - ((a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(a*b^2*d) + Sinh[c + d*x]/(b*d)} -{(Cosh[c + d*x]^2*Coth[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]^2*Coth[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 40, -((2*b*(e + f*x)^3*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d)) - (2*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d) + (3*I*b*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - (3*I*b*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) - (6*I*b*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + (6*I*b*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (3*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) + (6*I*b*f^3*PolyLog[4, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) - (6*I*b*f^3*PolyLog[4, I*E^(c + d*x)])/((a^2 + b^2)*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^4) + (3*b^2*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a*(a^2 + b^2)*d^4) - (3*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a*d^4) + (3*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a*d^4)} -{((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 33, -((2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d)) - (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d) + (2*I*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*I*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) + (b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) - (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) - (2*I*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*I*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (b^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3)} -{((e + f*x)*Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 26, -((2*b*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d)) - (2*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d) + (I*b*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - (I*b*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^2) + (b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^2) - (f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) + (f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2)} -{(Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 7, -((b*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d)) - (a*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) + Log[Sinh[c + d*x]]/(a*d) - (b^2*Log[a + b*Sinh[c + d*x]])/(a*(a^2 + b^2)*d)} -{(Csch[c + d*x]*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 53, -((b*(e + f*x)^3)/((a^2 + b^2)*d)) - (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (6*b^2*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (3*b*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (6*I*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (6*I*b^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - (6*I*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^3) + (6*I*b^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) + (6*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (6*I*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + (6*I*b^2*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + (6*I*f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) - (6*I*b^2*f^3*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (3*b*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^4) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) + ((e + f*x)^3*Sech[c + d*x])/(a*d) - (b^2*(e + f*x)^3*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^3*Tanh[c + d*x])/((a^2 + b^2)*d)} -{((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 44, -((b*(e + f*x)^2)/((a^2 + b^2)*d)) - (4*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a*d^2) + (4*b^2*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (2*b*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) - (2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (2*I*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - (2*I*b^2*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - (2*I*f^2*PolyLog[2, I*E^(c + d*x)])/(a*d^3) + (2*I*b^2*f^2*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) - (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (b*f^2*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) + (2*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (2*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) + (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) + ((e + f*x)^2*Sech[c + d*x])/(a*d) - (b^2*(e + f*x)^2*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^2*Tanh[c + d*x])/((a^2 + b^2)*d)} -{((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 26, -((f*ArcTan[Sinh[c + d*x]])/(a*d^2)) + (b^2*f*ArcTan[Sinh[c + d*x]])/(a*(a^2 + b^2)*d^2) - (2*f*x*ArcTanh[E^(c + d*x)])/(a*d) + (f*x*ArcTanh[Cosh[c + d*x]])/(a*d) - ((e + f*x)*ArcTanh[Cosh[c + d*x]])/(a*d) - (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b*f*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^2) - (f*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (f*PolyLog[2, E^(c + d*x)])/(a*d^2) - (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + ((e + f*x)*Sech[c + d*x])/(a*d) - (b^2*(e + f*x)*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d)} -{(Csch[c + d*x]*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 10, -(ArcTanh[Cosh[c + d*x]]/(a*d)) + (2*b^3*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(3/2)*d) + Sech[c + d*x]/(a*d) - (b*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a*(a^2 + b^2)*d)} -{(Csch[c + d*x]*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 57, (e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) - (2*b^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) - (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) + (b^4*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d) + (f^2*Log[Cosh[c + d*x]])/(a*d^3) - (b^2*f^2*Log[Cosh[c + d*x]])/(a*(a^2 + b^2)*d^3) + (2*I*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*I*b^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) - (2*I*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*I*b^3*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^3) - (b^4*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)^2*d^3) + (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (b*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) - (b^2*(e + f*x)^2*Sech[c + d*x]^2)/(2*a*(a^2 + b^2)*d) - (f*(e + f*x)*Tanh[c + d*x])/(a*d^2) + (b^2*f*(e + f*x)*Tanh[c + d*x])/(a*(a^2 + b^2)*d^2) - (b*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d) - ((e + f*x)^2*Tanh[c + d*x]^2)/(2*a*d)} -{((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 43, (f*x)/(2*a*d) - (2*b^3*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (b*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (2*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) + (b^4*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d) - (f*x*Log[Tanh[c + d*x]])/(a*d) + ((e + f*x)*Log[Tanh[c + d*x]])/(a*d) + (I*b^3*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + (I*b*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*(a^2 + b^2)*d^2) - (I*b^3*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*b*f*PolyLog[2, I*E^(c + d*x)])/(2*(a^2 + b^2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) + (b^4*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)^2*d^2) - (f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) + (f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) - (b*f*Sech[c + d*x])/(2*(a^2 + b^2)*d^2) - (b^2*(e + f*x)*Sech[c + d*x]^2)/(2*a*(a^2 + b^2)*d) - (f*Tanh[c + d*x])/(2*a*d^2) + (b^2*f*Tanh[c + d*x])/(2*a*(a^2 + b^2)*d^2) - (b*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d) - ((e + f*x)*Tanh[c + d*x]^2)/(2*a*d)} -{(Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 9, -((b^3*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d)) - (b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) - (a*(a^2 + 2*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) + Log[Sinh[c + d*x]]/(a*d) - (b^4*Log[a + b*Sinh[c + d*x]])/(a*(a^2 + b^2)^2*d) + (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{(Csch[c + d*x]*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^2 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 27, (-6*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^3*Csch[c + d*x])/(a*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (6*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^3) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a^2*d^2) + (6*f^3*PolyLog[3, -E^(c + d*x)])/(a*d^4) - (6*f^3*PolyLog[3, E^(c + d*x)])/(a*d^4) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^4) - (3*b*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^2*d^4)} -{((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 22, (-4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (b*f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3)} -{((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 15, -((f*ArcTanh[Cosh[c + d*x]])/(a*d^2)) - ((e + f*x)*Csch[c + d*x])/(a*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) + (b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^2*d^2)} -{(Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 4, -(Csch[c + d*x]/(a*d)) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b*Log[a + b*Sinh[c + d*x]])/(a^2*d)} -{(Coth[c + d*x]*Csch[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Coth[c + d*x]*Csch[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 41, -((e + f*x)^3/(a*d)) + (2*b*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^3*Coth[c + d*x])/(a*d) + (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) + (3*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) + (3*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3) + (6*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) - (3*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^4) + (6*b*f^3*PolyLog[4, -E^(c + d*x)])/(a^2*d^4) - (6*b*f^3*PolyLog[4, E^(c + d*x)])/(a^2*d^4) + (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^4) - (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^4)} -{((e + f*x)^2*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 34, -((e + f*x)^2/(a*d)) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^2*Coth[c + d*x])/(a*d) + (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) + (2*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (2*b*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (2*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) + (f^2*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3) + (2*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3)} -{((e + f*x)*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 25, (2*b*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)*Coth[c + d*x])/(a*d) + (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) - (Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*d) + (f*Log[Sinh[c + d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (b*f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2)} -{Coth[c + d*x]^2/(a + b*Sinh[c + d*x]), x, 7, (b*ArcTanh[Cosh[c + d*x]])/(a^2*d) - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*d) - Coth[c + d*x]/(a*d)} -{Coth[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Coth[c + d*x]^2/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 48, (b*(e + f*x)^4)/(4*a^2*f) - ((a^2 + b^2)*(e + f*x)^4)/(4*a^2*b*f) - (6*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^3*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*b*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*d) - (b*(e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (6*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^3) + (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^2) + (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a^2*d^2) + (6*f^3*PolyLog[3, -E^(c + d*x)])/(a*d^4) - (6*f^3*PolyLog[3, E^(c + d*x)])/(a*d^4) - (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^3) - (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^4) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^4) - (3*b*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^2*d^4)} -{((e + f*x)^2*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 37, (b*(e + f*x)^3)/(3*a^2*f) - ((a^2 + b^2)*(e + f*x)^3)/(3*a^2*b*f) - (4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*b*d) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*d) - (b*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^2) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^2) - (b*f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^3) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^3) + (b*f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3)} -{((e + f*x)*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 28, (b*(e + f*x)^2)/(2*a^2*f) - ((a^2 + b^2)*(e + f*x)^2)/(2*a^2*b*f) - (f*ArcTanh[Cosh[c + d*x]])/(a*d^2) - ((e + f*x)*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*b*d) + ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*d) - (b*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d) + ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^2) + ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^2) - (b*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^2*d^2)} -{(Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 4, -(Csch[c + d*x]/(a*d)) - (b*Log[Sinh[c + d*x]])/(a^2*d) + ((a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(a^2*b*d)} -{(Cosh[c + d*x]*Coth[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Cosh[c + d*x]*Coth[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csch[c + d*x]^2*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 64, -((2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a*d)) + (2*b^2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (6*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d^2) + (2*b*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - ((e + f*x)^3*Csch[c + d*x])/(a*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) - (b^3*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d) - (6*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (3*I*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (3*I*b^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (3*I*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (3*I*b^2*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + (6*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^3) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^2*d^2) + (6*f^3*PolyLog[3, -E^(c + d*x)])/(a*d^4) - (6*I*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) + (6*I*b^2*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (6*I*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - (6*I*b^2*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - (6*f^3*PolyLog[3, E^(c + d*x)])/(a*d^4) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (3*b^3*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)*d^3) - (3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^2*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^2*d^3) + (6*I*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*d^4) - (6*I*b^2*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - (6*I*f^3*PolyLog[4, I*E^(c + d*x)])/(a*d^4) + (6*I*b^2*f^3*PolyLog[4, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (3*b^3*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a^2*(a^2 + b^2)*d^4) + (3*b*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a^2*d^4) - (3*b*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a^2*d^4)} -{((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 53, -((2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d)) + (2*b^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) - (b^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*I*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (2*I*b^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*I*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (2*I*b^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + (2*f^2*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) - (b^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d^2) + (b*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^2*d^2) - (b*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^2*d^2) - (2*I*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) + (2*I*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (2*I*f^2*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - (2*I*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (b^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)*d^3) - (b*f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^2*d^3) + (b*f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^2*d^3)} -{((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 37, -((2*f*x*ArcTan[E^(c + d*x)])/(a*d)) + (2*b^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) + (f*x*ArcTan[Sinh[c + d*x]])/(a*d) - ((e + f*x)*ArcTan[Sinh[c + d*x]])/(a*d) + (2*b*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - (f*ArcTanh[Cosh[c + d*x]])/(a*d^2) - ((e + f*x)*Csch[c + d*x])/(a*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) + (b^3*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) - (b^3*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d) + (I*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) - (I*b^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (I*f*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (I*b^2*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) + (b^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) - (b^3*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)*d^2) + (b*f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^2*d^2) - (b*f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^2*d^2)} -{(Csch[c + d*x]^2*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 7, -((a*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d)) - Csch[c + d*x]/(a*d) + (b*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b^3*Log[a + b*Sinh[c + d*x]])/(a^2*(a^2 + b^2)*d)} -{(Csch[c + d*x]^2*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^2*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* {((e + f*x)^3*Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 186, (-2*b^2*e^3*(a - b*E^(c + d*x)))/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (6*b^2*e^2*f*x)/(a*(a^2 + b^2)*d) - (6*b^2*e^2*f*x)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (6*b^3*e^2*E^(c + d*x)*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (6*b^2*e*f^2*x^2)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (6*b^3*e*E^(c + d*x)*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^2*f^3*x^3)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^3*E^(c + d*x)*f^3*x^3)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*(e + f*x)^3)/(a*d) - (6*b^3*e^2*f*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (12*b^3*e*f^2*x*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (6*b^3*f^3*x^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (6*b*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*d^2) + (2*b*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^2*d) - (2*(e + f*x)^3*Coth[2*c + 2*d*x])/(a*d) - (6*b^2*e*f^2*x*Log[1 + E^(-2*c - 2*d*x)])/(a*(a^2 + b^2)*d^2) - (3*b^2*f^3*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a*(a^2 + b^2)*d^2) + (b^4*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (3*f*(e + f*x)^2*Log[1 - E^(4*(c + d*x))])/(a*d^2) - (3*b^2*e^2*f*Log[1 + E^(2*c + 2*d*x)])/(a*(a^2 + b^2)*d^2) + (3*b^2*e*f^2*PolyLog[2, -E^(-2*c - 2*d*x)])/(a*(a^2 + b^2)*d^3) + (3*b^2*f^3*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a*(a^2 + b^2)*d^3) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) + ((6*I)*b^3*e*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + ((6*I)*b^3*f^3*x*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - ((6*I)*b*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^3) - ((6*I)*b^3*e*f^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - ((6*I)*b^3*f^3*x*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + ((6*I)*b*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*d^3) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (3*f^2*(e + f*x)*PolyLog[2, E^(4*(c + d*x))])/(2*a*d^3) + (3*b^2*f^3*PolyLog[3, -E^(-2*c - 2*d*x)])/(2*a*(a^2 + b^2)*d^4) - (6*b*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + ((6*I)*b*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^4) - ((6*I)*b^3*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) - ((6*I)*b*f^3*PolyLog[3, I*E^(c + d*x)])/(a^2*d^4) + ((6*I)*b^3*f^3*PolyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) + (6*b*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (6*b^4*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (6*b^4*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (3*f^3*PolyLog[3, E^(4*(c + d*x))])/(8*a*d^4) + (6*b*f^3*PolyLog[4, -E^(c + d*x)])/(a^2*d^4) - (6*b*f^3*PolyLog[4, E^(c + d*x)])/(a^2*d^4) + (6*b^4*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (6*b^4*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (b*(e + f*x)^3*Sech[c + d*x])/(a^2*d)} *) -{((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 51, -((2*(e + f*x)^2)/(a*d)) + (b^2*(e + f*x)^2)/(a*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*d^2) - (4*b^3*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d) - (2*(e + f*x)^2*Coth[2*c + 2*d*x])/(a*d) + (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (2*b^2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) + (2*f*(e + f*x)*Log[1 - E^(4*(c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (2*I*b*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^3) + (2*I*b^3*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + (2*I*b*f^2*PolyLog[2, I*E^(c + d*x)])/(a^2*d^3) - (2*I*b^3*f^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (2*b*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (b^2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^3) + (f^2*PolyLog[2, E^(4*(c + d*x))])/(2*a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Sech[c + d*x])/(a^2*d) + (b^3*(e + f*x)^2*Sech[c + d*x])/(a^2*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Tanh[c + d*x])/(a*(a^2 + b^2)*d)} -{((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 30, (b*f*ArcTan[Sinh[c + d*x]])/(a^2*d^2) - (b^3*f*ArcTan[Sinh[c + d*x]])/(a^2*(a^2 + b^2)*d^2) + (2*b*f*x*ArcTanh[E^(c + d*x)])/(a^2*d) - (b*f*x*ArcTanh[Cosh[c + d*x]])/(a^2*d) + (b*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a^2*d) - (2*(e + f*x)*Coth[2*c + 2*d*x])/(a*d) + (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^2*f*Log[Cosh[c + d*x]])/(a*(a^2 + b^2)*d^2) + (f*Log[Sinh[2*c + 2*d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (b*f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (b*(e + f*x)*Sech[c + d*x])/(a^2*d) + (b^3*(e + f*x)*Sech[c + d*x])/(a^2*(a^2 + b^2)*d) + (b^2*(e + f*x)*Tanh[c + d*x])/(a*(a^2 + b^2)*d)} -{(Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 13, (b*ArcTanh[Cosh[c + d*x]])/(a^2*d) - (2*b^4*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)*d) - Coth[c + d*x]/(a*d) - (b*Sech[c + d*x])/(a^2*d) + (b^2*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a^2*(a^2 + b^2)*d) - Tanh[c + d*x]/(a*d)} -{(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^2*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* {((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 233, (-2*b^2*e^2*(b + a*E^(c + d*x)))/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) + (2*b^2*e^2*(3*b + 2*a*E^(c + d*x)))/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (b^2*e^2*(4*b + 3*a*E^(c + d*x)))/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^3*e*f)/(a^2*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) + (2*b^2*e*E^(c + d*x)*f)/(a*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) + (4*b^3*e^2*x)/(a^2*(a^2 + b^2)) - (2*b^3*(2*a^2 + b^2)*e^2*x)/(a^2*(a^2 + b^2)^2) - (b*e*f*x)/(a^2*d) - (4*b^3*e*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) - (4*b^2*e*E^(c + d*x)*f*x)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) + (4*b^3*e*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^2*e*E^(c + d*x)*f*x)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^3*f^2*x)/(a^2*(a^2 + b^2)*d^2) + (2*b^3*f^2*x)/(a^2*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) + (2*b^2*E^(c + d*x)*f^2*x)/(a*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) - (b*f^2*x^2)/(2*a^2*d) - (2*b^3*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) - (2*b^2*E^(c + d*x)*f^2*x^2)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) + (2*b^3*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (b^2*E^(c + d*x)*f^2*x^2)/(a*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^4*e^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)^2*d) + (b^2*e^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (2*b^2*f^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (4*b^4*e*f*x*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)^2*d) + (2*b^2*e*f*x*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) + (2*b^4*f^2*x^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)^2*d) + (b^2*f^2*x^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d) + (f^2*ArcTan[Sinh[c + d*x]])/(a*d^3) + (2*f^2*x*ArcTanh[E^(c + d*x)])/(a*d^2) - (6*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - (f^2*x*ArcTanh[Cosh[c + d*x]])/(a*d^2) + (f*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a*d^2) - (3*(e + f*x)^2*Csch[c + d*x])/(2*a*d) - (4*b^3*e*f*x*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d) + (2*b^3*(2*a^2 + b^2)*e*f*x*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)^2*d) - (2*b^3*f^2*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d) + (b^3*(2*a^2 + b^2)*f^2*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)^2*d) + (b^5*(e + f*x)^2*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b])/(a^2*(a^2 + b^2)^2*d) + (b^5*(e + f*x)^2*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b])/(a^2*(a^2 + b^2)^2*d) - (2*b^3*e^2*Log[1 + E^(2*c + 2*d*x)])/(a^2*(a^2 + b^2)*d) + (b^3*(2*a^2 + b^2)*e^2*Log[1 + E^(2*c + 2*d*x)])/(a^2*(a^2 + b^2)^2*d) + (b^3*f^2*Log[1 + E^(2*c + 2*d*x)])/(a^2*(a^2 + b^2)*d^3) - (b*f^2*Log[Cosh[c + d*x]])/(a^2*d^3) + (2*b^3*e*f*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^2) - (b^3*(2*a^2 + b^2)*e*f*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)^2*d^2) + (2*b^3*f^2*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^2) - (b^3*(2*a^2 + b^2)*f^2*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)^2*d^2) - (2*b^5*f*(e + f*x)*PolyLog[2, -(((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^2*(a^2 + b^2)^2*d^2) - (2*b^5*f*(e + f*x)*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^2*(a^2 + b^2)^2*d^2) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) - ((2*I)*b^4*e*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) - (I*b^2*e*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - ((2*I)*b^4*f^2*x*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) - (I*b^2*f^2*x*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + ((3*I)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + ((2*I)*b^4*e*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) + (I*b^2*e*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) + ((2*I)*b^4*f^2*x*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) + (I*b^2*f^2*x*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - ((3*I)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (2*f^2*PolyLog[2, E^(c + d*x)])/(a*d^3) + (b*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^2*d^2) - (b*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^2*d^2) + (b^3*f^2*PolyLog[3, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^3) - (b^3*(2*a^2 + b^2)*f^2*PolyLog[3, -E^(-2*c - 2*d*x)])/(2*a^2*(a^2 + b^2)^2*d^3) - (2*b^5*f^2*PolyLog[3, -(((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^2*(a^2 + b^2)^2*d^3) - (2*b^5*f^2*PolyLog[3, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^2*(a^2 + b^2)^2*d^3) - ((3*I)*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) + ((2*I)*b^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^3) + (I*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + ((3*I)*f^2*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - ((2*I)*b^4*f^2*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^3) - (I*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - (b*f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^2*d^3) + (b*f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^2*d^3) - (f*(e + f*x)*Sech[c + d*x])/(a*d^2) + ((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^2)/(2*a*d) + (b*f*(e + f*x)*Tanh[c + d*x])/(a^2*d^2) + (b*(e + f*x)^2*Tanh[c + d*x]^2)/(2*a^2*d)} *) -{((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 57, -((b*f*x)/(2*a^2*d)) - (3*f*x*ArcTan[E^(c + d*x)])/(a*d) + (2*b^4*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)^2*d) + (b^2*(e + f*x)*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) + (3*f*x*ArcTan[Sinh[c + d*x]])/(2*a*d) - (3*(e + f*x)*ArcTan[Sinh[c + d*x]])/(2*a*d) + (2*b*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - (f*ArcTanh[Cosh[c + d*x]])/(a*d^2) - (3*(e + f*x)*Csch[c + d*x])/(2*a*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^2*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^2*d) - (b^5*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)^2*d) + (b*f*x*Log[Tanh[c + d*x]])/(a^2*d) - (b*(e + f*x)*Log[Tanh[c + d*x]])/(a^2*d) + (3*I*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*a*d^2) - (I*b^4*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) - (I*b^2*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*a*(a^2 + b^2)*d^2) - (3*I*f*PolyLog[2, I*E^(c + d*x)])/(2*a*d^2) + (I*b^4*f*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)^2*d^2) + (I*b^2*f*PolyLog[2, I*E^(c + d*x)])/(2*a*(a^2 + b^2)*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^2*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^2*d^2) - (b^5*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a^2*(a^2 + b^2)^2*d^2) + (b*f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^2*d^2) - (b*f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^2*d^2) - (f*Sech[c + d*x])/(2*a*d^2) + (b^2*f*Sech[c + d*x])/(2*a*(a^2 + b^2)*d^2) + (b^3*(e + f*x)*Sech[c + d*x]^2)/(2*a^2*(a^2 + b^2)*d) + ((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(2*a*d) + (b*f*Tanh[c + d*x])/(2*a^2*d^2) - (b^3*f*Tanh[c + d*x])/(2*a^2*(a^2 + b^2)*d^2) + (b^2*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*a*(a^2 + b^2)*d) + (b*(e + f*x)*Tanh[c + d*x]^2)/(2*a^2*d)} -{(Csch[c + d*x]^2*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 9, -(a*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) - (a*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) - Csch[c + d*x]/(a*d) + (b*(a^2 + 2*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (b*Log[Sinh[c + d*x]])/(a^2*d) + (b^5*Log[a + b*Sinh[c + d*x]])/(a^2*(a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(b + a*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{(Csch[c + d*x]^2*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^2*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^3 / (a+b Sinh[c+d x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 34, (-3*f*(e + f*x)^2)/(2*a*d^2) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (3*f*(e + f*x)^2*Coth[c + d*x])/(2*a*d^2) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d) - ((e + f*x)^3*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - (b^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^3) + (b^2*(e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (6*b*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^4) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2) - (6*b*f^3*PolyLog[3, -E^(c + d*x)])/(a^2*d^4) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (3*b^2*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^3*d^4)} -{((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 26, (4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (f*(e + f*x)*Coth[c + d*x])/(a*d^2) + (b*(e + f*x)^2*Csch[c + d*x])/(a^2*d) - ((e + f*x)^2*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (f^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^3*d^2) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (b^2*f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3)} -{((e + f*x)*Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 19, (b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - ((e + f*x)*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^3*d) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b^2*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2)} -{(Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 4, (b*Csch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (b^2*Log[Sinh[c + d*x]])/(a^3*d) - (b^2*Log[a + b*Sinh[c + d*x]])/(a^3*d)} -{(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 67, (b*(e + f*x)^3)/(a^2*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) - ((e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)^3*Coth[c + d*x])/(a^2*d) - (3*f*(e + f*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) + (3*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) + (3*b*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^4) - (3*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2*f^3*PolyLog[4, -E^(c + d*x)])/(a^3*d^4) + (3*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(c + d*x)])/(a^3*d^4) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^4)} -{((e + f*x)^2*Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 52, (b*(e + f*x)^2)/(a^2*d) - ((e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (b*(e + f*x)^2*Coth[c + d*x])/(a^2*d) - (f*(e + f*x)*Csch[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) - (2*b*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b*f^2*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) + (f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) - (f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3)} -{((e + f*x)*Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 38, -(((e + f*x)*ArcTanh[E^(c + d*x)])/(a*d)) - (2*b^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)*Coth[c + d*x])/(a^2*d) - (f*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) - (b*f*Log[Sinh[c + d*x]])/(a^2*d^2) - (f*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (b^2*f*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2)} -{(Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]), x, 8, -((a^2 + 2*b^2)*ArcTanh[Cosh[c + d*x]])/(2*a^3*d) + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*d) + (b*Coth[c + d*x])/(a^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)} -{(Coth[c + d*x]^2*Csch[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Coth[c + d*x]^2*Csch[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 62, (-3*f*(e + f*x)^2)/(2*a*d^2) + (e + f*x)^3/(2*a*d) - (e + f*x)^4/(4*a*f) - (b^2*(e + f*x)^4)/(4*a^3*f) + ((a^2 + b^2)*(e + f*x)^4)/(4*a^3*f) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (3*f*(e + f*x)^2*Coth[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d) - ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^3) + ((e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a*d) + (b^2*(e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (6*b*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2) - (6*b*f^3*PolyLog[3, -E^(c + d*x)])/(a^2*d^4) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) + (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3) - (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) - (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (3*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a*d^4) + (3*b^2*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^3*d^4)} -{((e + f*x)^2*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 47, (e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) - (e + f*x)^3/(3*a*f) - (b^2*(e + f*x)^3)/(3*a^3*f) + ((a^2 + b^2)*(e + f*x)^3)/(3*a^3*f) + (4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (f*(e + f*x)*Coth[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^2*Csch[c + d*x])/(a^2*d) - ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + ((e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a*d) + (b^2*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (f^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^3*d^2) + (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) + (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^3) - (b^2*f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3)} -{((e + f*x)*Coth[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 36, (f*x)/(2*a*d) - (e + f*x)^2/(2*a*f) - (b^2*(e + f*x)^2)/(2*a^3*f) + ((a^2 + b^2)*(e + f*x)^2)/(2*a^3*f) + (b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + ((e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d) + (b^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^3*d) - ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (f*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (b^2*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2)} -{Coth[c + d*x]^3/(a + b*Sinh[c + d*x]), x, 3, (b*Csch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + ((a^2 + b^2)*Log[Sinh[c + d*x]])/(a^3*d) - ((a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(a^3*d)} -{Coth[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[Coth[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{((e + f*x)^3*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 87, -((3*f*(e + f*x)^2)/(2*a*d^2)) + (e + f*x)^3/(2*a*d) + (2*b*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e + f*x)^3*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (2*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) - (3*f*(e + f*x)^2*Coth[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d) - (b^4*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^3) + (b^4*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (6*b*f^2*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (3*I*b*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^2) + (3*I*b^3*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (3*I*b*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - (3*I*b^3*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (6*b*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) - (3*b^4*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) + (3*b^4*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^2) + (3*f^3*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^3*d^2) - (3*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^3*d^2) - (6*b*f^3*PolyLog[3, -E^(c + d*x)])/(a^2*d^4) + (6*I*b*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - (6*I*b^3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (6*I*b*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + (6*I*b^3*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*b^4*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) + (6*b^4*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) - (3*b^4*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) + (3*b^2*f^2*(e + f*x)*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^3*d^3) - (6*I*b*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a^2*d^4) + (6*I*b^3*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) + (6*I*b*f^3*PolyLog[4, I*E^(c + d*x)])/(a^2*d^4) - (6*I*b^3*f^3*PolyLog[4, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^4) - (6*b^4*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^4) - (6*b^4*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^4) + (3*b^4*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*a^3*(a^2 + b^2)*d^4) + (3*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a*d^4) - (3*b^2*f^3*PolyLog[4, -E^(2*c + 2*d*x)])/(4*a^3*d^4) - (3*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a*d^4) + (3*b^2*f^3*PolyLog[4, E^(2*c + 2*d*x)])/(4*a^3*d^4)} -{((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 71, (e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) + (2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) - (f*(e + f*x)*Coth[c + d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^2*Csch[c + d*x])/(a^2*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (b^4*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (f^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - (2*I*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^2) + (2*I*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (2*I*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - (2*I*b^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d^2) + (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) - (b^2*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^3*d^2) - (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^3*d^2) + (2*I*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - (2*I*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (2*I*b*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + (2*I*b^3*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) - (b^4*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^3) - (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) + (b^2*f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (b^2*f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^3*d^3)} -{((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 49, (f*x)/(2*a*d) + (2*b*f*x*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) - (b*f*x*ArcTan[Sinh[c + d*x]])/(a^2*d) + (b*(e + f*x)*ArcTan[Sinh[c + d*x]])/(a^2*d) + (2*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) + (b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) - ((e + f*x)*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (b^4*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (f*x*Log[Tanh[c + d*x]])/(a*d) - ((e + f*x)*Log[Tanh[c + d*x]])/(a*d) - (I*b*f*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^2) + (I*b^3*f*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (I*b*f*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - (I*b^3*f*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) + (b^4*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^2) + (f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^3*d^2) - (f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) + (b^2*f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^3*d^2)} -{(Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]), x, 7, (b*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d) + (b*Csch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (a*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) - ((a^2 - b^2)*Log[Sinh[c + d*x]])/(a^3*d) - (b^4*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)*d)} -{(Csch[c + d*x]^3*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^3*Sech[c + d*x])/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* {((e + f*x)^3*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 251, (2*b^3*e^3*(a - b*E^(c + d*x)))/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (6*b^3*e^2*f*x)/(a^2*(a^2 + b^2)*d) + (6*b^3*e^2*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (6*b^4*e^2*E^(c + d*x)*f*x)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (6*b^3*e*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (6*b^4*e*E^(c + d*x)*f^2*x^2)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^3*f^3*x^3)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^4*E^(c + d*x)*f^3*x^3)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b*(e + f*x)^3)/(a^2*d) + (6*b^4*e^2*f*ArcTan[E^(c + d*x)])/(a^3*(a^2 + b^2)*d^2) + (12*e*f^2*x*ArcTan[E^(c + d*x)])/(a*d^2) + (12*b^4*e*f^2*x*ArcTan[E^(c + d*x)])/(a^3*(a^2 + b^2)*d^2) + (6*f^3*x^2*ArcTan[E^(c + d*x)])/(a*d^2) + (6*b^4*f^3*x^2*ArcTan[E^(c + d*x)])/(a^3*(a^2 + b^2)*d^2) - (6*b^2*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^3*d^2) + (3*e^2*f*ArcTan[Sinh[c + d*x]])/(a*d^2) - (6*f^3*x*ArcTanh[E^(c + d*x)])/(a*d^3) + (3*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) - (3*e*f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (2*b*(e + f*x)^3*Coth[2*c + 2*d*x])/(a^2*d) - (3*e^2*f*Csch[c + d*x])/(2*a*d^2) - (3*e*f^2*x*Csch[c + d*x])/(a*d^2) - (3*f^3*x^2*Csch[c + d*x])/(2*a*d^2) + (6*b^3*e*f^2*x*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^2) + (3*b^3*f^3*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^2) - (b^5*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (b^5*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(4*(c + d*x))])/(a^2*d^2) + (3*b^3*e^2*f*Log[1 + E^(2*c + 2*d*x)])/(a^2*(a^2 + b^2)*d^2) - (3*b^3*e*f^2*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^3) - (3*b^3*f^3*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^2*(a^2 + b^2)*d^3) - (3*f^3*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (9*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - ((6*I)*e*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((6*I)*b^4*e*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) - ((6*I)*f^3*x*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) - ((6*I)*b^4*f^3*x*PolyLog[2, (-I)*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^3*d^3) + ((6*I)*e*f^2*PolyLog[2, I*E^(c + d*x)])/(a*d^3) + ((6*I)*b^4*e*f^2*PolyLog[2, I*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) + ((6*I)*f^3*x*PolyLog[2, I*E^(c + d*x)])/(a*d^3) + ((6*I)*b^4*f^3*x*PolyLog[2, I*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^3*d^3) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) - (9*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b^5*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) + (3*b^5*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(4*(c + d*x))])/(2*a^2*d^3) - (3*b^3*f^3*PolyLog[3, -E^(-2*c - 2*d*x)])/(2*a^2*(a^2 + b^2)*d^4) - (9*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) + ((6*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) - ((6*I)*b^2*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a^3*d^4) + ((6*I)*b^4*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^4) - ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) + ((6*I)*b^2*f^3*PolyLog[3, I*E^(c + d*x)])/(a^3*d^4) - ((6*I)*b^4*f^3*PolyLog[3, I*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^4) + (9*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (6*b^5*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^3) - (6*b^5*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^3) + (3*b*f^3*PolyLog[3, E^(4*(c + d*x))])/(8*a^2*d^4) + (9*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2*f^3*PolyLog[4, -E^(c + d*x)])/(a^3*d^4) - (9*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(c + d*x)])/(a^3*d^4) - (6*b^5*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^4) + (6*b^5*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^4) - (3*(e + f*x)^3*Sech[c + d*x])/(2*a*d) + (b^2*(e + f*x)^3*Sech[c + d*x])/(a^3*d) - ((e + f*x)^3*Csch[c + d*x]^2*Sech[c + d*x])/(2*a*d)} *) -{((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 88, (2*b*(e + f*x)^2)/(a^2*d) - (b^3*(e + f*x)^2)/(a^2*(a^2 + b^2)*d) + (4*f^2*x*ArcTan[E^(c + d*x)])/(a*d^2) - (4*b^2*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^3*d^2) + (4*b^4*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^3*(a^2 + b^2)*d^2) + (2*e*f*ArcTan[Sinh[c + d*x]])/(a*d^2) + (3*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (2*b*(e + f*x)^2*Coth[2*c + 2*d*x])/(a^2*d) - (e*f*Csch[c + d*x])/(a*d^2) - (f^2*x*Csch[c + d*x])/(a*d^2) - (b^5*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (b^5*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (2*b^3*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d^2) - (2*b*f*(e + f*x)*Log[1 - E^(4*(c + d*x))])/(a^2*d^2) + (3*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (2*I*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^3) + (2*I*b^2*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^3*d^3) - (2*I*b^4*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) + (2*I*f^2*PolyLog[2, I*E^(c + d*x)])/(a*d^3) - (2*I*b^2*f^2*PolyLog[2, I*E^(c + d*x)])/(a^3*d^3) + (2*I*b^4*f^2*PolyLog[2, I*E^(c + d*x)])/(a^3*(a^2 + b^2)*d^3) - (3*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b^5*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) + (2*b^5*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) + (b^3*f^2*PolyLog[2, -E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d^3) - (b*f^2*PolyLog[2, E^(4*(c + d*x))])/(2*a^2*d^3) - (3*f^2*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) + (3*f^2*PolyLog[3, E^(c + d*x)])/(a*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b^5*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^3) - (2*b^5*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^3) - (3*(e + f*x)^2*Sech[c + d*x])/(2*a*d) + (b^2*(e + f*x)^2*Sech[c + d*x])/(a^3*d) - (b^4*(e + f*x)^2*Sech[c + d*x])/(a^3*(a^2 + b^2)*d) - ((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x])/(2*a*d) - (b^3*(e + f*x)^2*Tanh[c + d*x])/(a^2*(a^2 + b^2)*d)} -{((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 44, (f*ArcTan[Sinh[c + d*x]])/(a*d^2) - (b^2*f*ArcTan[Sinh[c + d*x]])/(a^3*d^2) + (b^4*f*ArcTan[Sinh[c + d*x]])/(a^3*(a^2 + b^2)*d^2) + (3*f*x*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*f*x*ArcTanh[E^(c + d*x)])/(a^3*d) - (3*f*x*ArcTanh[Cosh[c + d*x]])/(2*a*d) + (b^2*f*x*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (3*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(2*a*d) - (b^2*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (2*b*(e + f*x)*Coth[2*c + 2*d*x])/(a^2*d) - (f*Csch[c + d*x])/(2*a*d^2) - (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (b^3*f*Log[Cosh[c + d*x]])/(a^2*(a^2 + b^2)*d^2) - (b*f*Log[Sinh[2*c + 2*d*x]])/(a^2*d^2) + (3*f*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (3*f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (b^2*f*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) - (3*(e + f*x)*Sech[c + d*x])/(2*a*d) + (b^2*(e + f*x)*Sech[c + d*x])/(a^3*d) - (b^4*(e + f*x)*Sech[c + d*x])/(a^3*(a^2 + b^2)*d) - ((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x])/(2*a*d) - (b^3*(e + f*x)*Tanh[c + d*x])/(a^2*(a^2 + b^2)*d)} -{(Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]), x, 17, (3*ArcTanh[Cosh[c + d*x]])/(2*a*d) - (b^2*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (2*b^5*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^2)^(3/2)*d) + (b*Coth[c + d*x])/(a^2*d) - (3*Sech[c + d*x])/(2*a*d) + (b^2*Sech[c + d*x])/(a^3*d) - (Csch[c + d*x]^2*Sech[c + d*x])/(2*a*d) - (b^3*Sech[c + d*x]*(b + a*Sinh[c + d*x]))/(a^3*(a^2 + b^2)*d) + (b*Tanh[c + d*x])/(a^2*d)} -{(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^3*Sech[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* {((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 244, (2*b^3*e^2*(b + a*E^(c + d*x)))/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) - (2*b^3*e^2*(3*b + 2*a*E^(c + d*x)))/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (b^3*e^2*(4*b + 3*a*E^(c + d*x)))/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^4*e*f)/(a^3*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) - (2*b^3*e*E^(c + d*x)*f)/(a^2*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) - (4*b^4*e^2*x)/(a^3*(a^2 + b^2)) + (2*b^4*(2*a^2 + b^2)*e^2*x)/(a^3*(a^2 + b^2)^2) + (b^2*e*f*x)/(a^3*d) + (4*b^4*e*f*x)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) + (4*b^3*e*E^(c + d*x)*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) - (4*b^4*e*f*x)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^3*e*E^(c + d*x)*f*x)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) + (2*b^4*f^2*x)/(a^3*(a^2 + b^2)*d^2) - (2*b^4*f^2*x)/(a^3*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) - (2*b^3*E^(c + d*x)*f^2*x)/(a^2*(a^2 + b^2)*d^2*(1 + E^(2*c + 2*d*x))) + (b^2*f^2*x^2)/(2*a^3*d) + (2*b^4*f^2*x^2)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) + (2*b^3*E^(c + d*x)*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))^2) - (2*b^4*f^2*x^2)/(a^3*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (b^3*E^(c + d*x)*f^2*x^2)/(a^2*(a^2 + b^2)*d*(1 + E^(2*c + 2*d*x))) - (2*b^5*e^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d) - (b^3*e^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (2*b^3*f^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (4*b^5*e*f*x*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d) - (2*b^3*e*f*x*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) - (2*b^5*f^2*x^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d) - (b^3*f^2*x^2*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (3*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*d) - (b*f^2*ArcTan[Sinh[c + d*x]])/(a^2*d^3) - (2*b*f^2*x*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (6*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (4*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) + (b*f^2*x*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (b*f*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f^2*ArcTanh[Cosh[2*c + 2*d*x]])/(a*d^3) + (3*b*(e + f*x)^2*Csch[c + d*x])/(2*a^2*d) - (2*f*(e + f*x)*Csch[2*c + 2*d*x])/(a*d^2) - (2*(e + f*x)^2*Coth[2*c + 2*d*x]*Csch[2*c + 2*d*x])/(a*d) + (4*b^4*e*f*x*Log[1 + E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)*d) - (2*b^4*(2*a^2 + b^2)*e*f*x*Log[1 + E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)^2*d) + (2*b^4*f^2*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)*d) - (b^4*(2*a^2 + b^2)*f^2*x^2*Log[1 + E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)^2*d) - (b^6*(e + f*x)^2*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b])/(a^3*(a^2 + b^2)^2*d) - (b^6*(e + f*x)^2*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b])/(a^3*(a^2 + b^2)^2*d) + (2*b^4*e^2*Log[1 + E^(2*c + 2*d*x)])/(a^3*(a^2 + b^2)*d) - (b^4*(2*a^2 + b^2)*e^2*Log[1 + E^(2*c + 2*d*x)])/(a^3*(a^2 + b^2)^2*d) - (b^4*f^2*Log[1 + E^(2*c + 2*d*x)])/(a^3*(a^2 + b^2)*d^3) + (b^2*f^2*Log[Cosh[c + d*x]])/(a^3*d^3) - (2*b^4*e*f*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)*d^2) + (b^4*(2*a^2 + b^2)*e*f*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)^2*d^2) - (2*b^4*f^2*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)*d^2) + (b^4*(2*a^2 + b^2)*f^2*x*PolyLog[2, -E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)^2*d^2) + (2*b^6*f*(e + f*x)*PolyLog[2, -(((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^3*(a^2 + b^2)^2*d^2) + (2*b^6*f*(e + f*x)*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^3*(a^2 + b^2)^2*d^2) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) + ((2*I)*b^5*e*f*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) + (I*b^3*e*f*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + ((2*I)*b^5*f^2*x*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) + (I*b^3*f^2*x*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - ((3*I)*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*d^2) - ((2*I)*b^5*e*f*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) - (I*b^3*e*f*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) - ((2*I)*b^5*f^2*x*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) - (I*b^3*f^2*x*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + ((3*I)*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) + (2*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) - (b^2*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^3*d^2) - (2*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^3*d^2) - (b^4*f^2*PolyLog[3, -E^(-2*c - 2*d*x)])/(a^3*(a^2 + b^2)*d^3) + (b^4*(2*a^2 + b^2)*f^2*PolyLog[3, -E^(-2*c - 2*d*x)])/(2*a^3*(a^2 + b^2)^2*d^3) + (2*b^6*f^2*PolyLog[3, -(((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^3*(a^2 + b^2)^2*d^3) + (2*b^6*f^2*PolyLog[3, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)])/(a^3*(a^2 + b^2)^2*d^3) + ((3*I)*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - ((2*I)*b^5*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^3) - (I*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - ((3*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + ((2*I)*b^5*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^3) + (I*b^3*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(a*d^3) + (b^2*f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(a*d^3) - (b^2*f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a^3*d^3) + (b*f*(e + f*x)*Sech[c + d*x])/(a^2*d^2) - (b*(e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^2)/(2*a^2*d) - (b^2*f*(e + f*x)*Tanh[c + d*x])/(a^3*d^2) - (b^2*(e + f*x)^2*Tanh[c + d*x]^2)/(2*a^3*d)} *) -{((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 65, (b^2*f*x)/(2*a^3*d) + (3*b*f*x*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^5*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d) - (b^3*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d) - (3*b*f*x*ArcTan[Sinh[c + d*x]])/(2*a^2*d) + (3*b*(e + f*x)*ArcTan[Sinh[c + d*x]])/(2*a^2*d) - (2*b^2*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) + (4*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/(a*d) + (b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) + (3*b*(e + f*x)*Csch[c + d*x])/(2*a^2*d) - (f*Csch[2*c + 2*d*x])/(a*d^2) - (2*(e + f*x)*Coth[2*c + 2*d*x]*Csch[2*c + 2*d*x])/(a*d) - (b^6*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^2*d) - (b^6*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^2*d) + (b^6*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)^2*d) - (b^2*f*x*Log[Tanh[c + d*x]])/(a^3*d) + (b^2*(e + f*x)*Log[Tanh[c + d*x]])/(a^3*d) - (3*I*b*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*a^2*d^2) + (I*b^5*f*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) + (I*b^3*f*PolyLog[2, (-I)*E^(c + d*x)])/(2*a^2*(a^2 + b^2)*d^2) + (3*I*b*f*PolyLog[2, I*E^(c + d*x)])/(2*a^2*d^2) - (I*b^5*f*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)^2*d^2) - (I*b^3*f*PolyLog[2, I*E^(c + d*x)])/(2*a^2*(a^2 + b^2)*d^2) - (b^6*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^2*d^2) - (b^6*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^2*d^2) + (b^6*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)^2*d^2) + (f*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) - (b^2*f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a^3*d^2) - (f*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) + (b^2*f*PolyLog[2, E^(2*c + 2*d*x)])/(2*a^3*d^2) + (b*f*Sech[c + d*x])/(2*a^2*d^2) - (b^3*f*Sech[c + d*x])/(2*a^2*(a^2 + b^2)*d^2) - (b^4*(e + f*x)*Sech[c + d*x]^2)/(2*a^3*(a^2 + b^2)*d) - (b*(e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2)/(2*a^2*d) - (b^2*f*Tanh[c + d*x])/(2*a^3*d^2) + (b^4*f*Tanh[c + d*x])/(2*a^3*(a^2 + b^2)*d^2) - (b^3*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*a^2*(a^2 + b^2)*d) - (b^2*(e + f*x)*Tanh[c + d*x]^2)/(2*a^3*d)} -{(Csch[c + d*x]^3*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]), x, 9, (b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) + (b*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (b*Csch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (a*(2*a^2 + 3*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - ((2*a^2 - b^2)*Log[Sinh[c + d*x]])/(a^3*d) - (b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*d)} -{(Csch[c + d*x]^3*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x, 0, Unintegrable[(Csch[c + d*x]^3*Sech[c + d*x]^3)/((e + f*x)*(a + b*Sinh[c + d*x])), x]} - - -(* ::Section:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n Csch[c+d x]^p / (a+b Sinh[c+d x])^2*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.3 (e x)^m (a+b sinh(c+d x^n))^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.3 (e x)^m (a+b sinh(c+d x^n))^p.m deleted file mode 100644 index d92591e..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.3 (e x)^m (a+b sinh(c+d x^n))^p.m +++ /dev/null @@ -1,273 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sinh[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sinh[a + b*x^2], x, 3, (x^2*Cosh[a + b*x^2])/(2*b) - Sinh[a + b*x^2]/(2*b^2)} -{x^2*Sinh[a + b*x^2], x, 4, (x*Cosh[a + b*x^2])/(2*b) - (Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(8*b^(3/2))) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(8*b^(3/2))} -{x*Sinh[a + b*x^2], x, 2, Cosh[a + b*x^2]/(2*b)} -{Sinh[a + b*x^2], x, 3, -((Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(4*Sqrt[b]))) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(4*Sqrt[b])} -{Sinh[a + b*x^2]/x, x, 3, (1/2)*CoshIntegral[b*x^2]*Sinh[a] + (1/2)*Cosh[a]*SinhIntegral[b*x^2]} -{Sinh[a + b*x^2]/x^2, x, 4, ((1/2)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*x])/E^a + (1/2)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x] - Sinh[a + b*x^2]/x} -{Sinh[a + b*x^2]/x^3, x, 5, (1/2)*b*Cosh[a]*CoshIntegral[b*x^2] - Sinh[a + b*x^2]/(2*x^2) + (1/2)*b*Sinh[a]*SinhIntegral[b*x^2]} - - -{x^3*Sinh[a + b*x^2]^2, x, 3, -(x^4/8) + (x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b) - Sinh[a + b*x^2]^2/(8*b^2)} -{x^2*Sinh[a + b*x^2]^2, x, 6, -(x^3/6) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(E^(2*a)*(32*b^(3/2))) - (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x])/(32*b^(3/2)) + (x*Sinh[2*a + 2*b*x^2])/(8*b)} -{x*Sinh[a + b*x^2]^2, x, 3, -(x^2/4) + (Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b)} -{Sinh[a + b*x^2]^2, x, 5, -(x/2) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(E^(2*a)*(8*Sqrt[b])) + (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x])/(8*Sqrt[b])} -{Sinh[a + b*x^2]^2/x, x, 5, (1/4)*Cosh[2*a]*CoshIntegral[2*b*x^2] - Log[x]/2 + (1/4)*Sinh[2*a]*SinhIntegral[2*b*x^2]} -{Sinh[a + b*x^2]^2/x^2, x, 6, ((-(1/2))*Sqrt[b]*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/E^(2*a) + (1/2)*Sqrt[b]*E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x] - Sinh[a + b*x^2]^2/x} -{Sinh[a + b*x^2]^2/x^3, x, 7, 1/(4*x^2) - Cosh[2*(a + b*x^2)]/(4*x^2) + (1/2)*b*CoshIntegral[2*b*x^2]*Sinh[2*a] + (1/2)*b*Cosh[2*a]*SinhIntegral[2*b*x^2]} - - -{x^3*Sinh[a + b*x^2]^3, x, 4, -((x^2*Cosh[a + b*x^2])/(3*b)) + Sinh[a + b*x^2]/(3*b^2) + (x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2]^2)/(6*b) - Sinh[a + b*x^2]^3/(18*b^2)} -{x^2*Sinh[a + b*x^2]^3, x, 10, -((3*x*Cosh[a + b*x^2])/(8*b)) + (x*Cosh[3*a + 3*b*x^2])/(24*b) + (3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(32*b^(3/2))) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(E^(3*a)*(96*b^(3/2))) + (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(32*b^(3/2)) - (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(96*b^(3/2))} -{x*Sinh[a + b*x^2]^3, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(6*b)} -{Sinh[a + b*x^2]^3, x, 8, (3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(16*Sqrt[b])) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(E^(3*a)*(16*Sqrt[b])) - (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b])} -{Sinh[a + b*x^2]^3/x, x, 8, (-(3/8))*CoshIntegral[b*x^2]*Sinh[a] + (1/8)*CoshIntegral[3*b*x^2]*Sinh[3*a] - (3/8)*Cosh[a]*SinhIntegral[b*x^2] + (1/8)*Cosh[3*a]*SinhIntegral[3*b*x^2]} -{Sinh[a + b*x^2]^3/x^2, x, 9, ((-(3/8))*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*x])/E^a + ((1/8)*Sqrt[b]*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[b]*x])/E^(3*a) - (3/8)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x] + (1/8)*Sqrt[b]*E^(3*a)*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[b]*x] - Sinh[a + b*x^2]^3/x} -{Sinh[a + b*x^2]^3/x^3, x, 12, (-(3/8))*b*Cosh[a]*CoshIntegral[b*x^2] + (3/8)*b*Cosh[3*a]*CoshIntegral[3*b*x^2] + (3*Sinh[a + b*x^2])/(8*x^2) - Sinh[3*(a + b*x^2)]/(8*x^2) - (3/8)*b*Sinh[a]*SinhIntegral[b*x^2] + (3/8)*b*Sinh[3*a]*SinhIntegral[3*b*x^2]} - - -{x*Sinh[a + b*x^2]^7, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(2*b) - (3*Cosh[a + b*x^2]^5)/(10*b) + Cosh[a + b*x^2]^7/(14*b)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^2])^p with m symbolic*) - - -{(e*x)^m*Sinh[a + b*x^2]^p, x, 0, Unintegrable[(e*x)^m*Sinh[a + b*x^2]^p, x]} - - -{(e*x)^m*Sinh[a + b*x^2]^3, x, 8, -((3^(-(1/2) - m/2)*E^(3*a)*(e*x)^(1 + m)*((-b)*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, -3*b*x^2])/(16*e)) + (3*E^a*(e*x)^(1 + m)*((-b)*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, (-b)*x^2])/(16*e) - (3*(e*x)^(1 + m)*(b*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, b*x^2])/(E^a*(16*e)) + (3^(-(1/2) - m/2)*(e*x)^(1 + m)*(b*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, 3*b*x^2])/(E^(3*a)*(16*e))} -{(e*x)^m*Sinh[a + b*x^2]^2, x, 5, -((e*x)^(1 + m)/(2*e*(1 + m))) - (2^(-(7/2) - m/2)*E^(2*a)*(e*x)^(1 + m)*((-b)*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, -2*b*x^2])/e - (2^(-(7/2) - m/2)*(e*x)^(1 + m)*(b*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, 2*b*x^2])/(E^(2*a)*e)} -{(e*x)^m*Sinh[a + b*x^2]^1, x, 3, -((E^a*(e*x)^(1 + m)*((-b)*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, (-b)*x^2])/(4*e)) + ((e*x)^(1 + m)*(b*x^2)^((1/2)*(-1 - m))*Gamma[(1 + m)/2, b*x^2])/(E^a*(4*e))} -{(e*x)^m/Sinh[a + b*x^2]^1, x, 1, ((e*x)^m*Unintegrable[x^m*Csch[a + b*x^2], x])/x^m} - - -(* ::Section:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^3])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^4])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sinh[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sinh[a + b*x^4], x, 2, Cosh[a + b*x^4]/(4*b)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d / x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sinh[c+d / x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[a + b/x]*x^2, x, 7, (1/6)*b*x^2*Cosh[a + b/x] - (1/6)*b^3*Cosh[a]*CoshIntegral[b/x] + (1/6)*b^2*x*Sinh[a + b/x] + (1/3)*x^3*Sinh[a + b/x] - (1/6)*b^3*Sinh[a]*SinhIntegral[b/x]} -{Sinh[a + b/x]*x^1, x, 6, (1/2)*b*x*Cosh[a + b/x] - (1/2)*b^2*CoshIntegral[b/x]*Sinh[a] + (1/2)*x^2*Sinh[a + b/x] - (1/2)*b^2*Cosh[a]*SinhIntegral[b/x]} -{Sinh[a + b/x]*x^0, x, 5, (-b)*Cosh[a]*CoshIntegral[b/x] + x*Sinh[a + b/x] - b*Sinh[a]*SinhIntegral[b/x]} -{Sinh[a + b/x]/x^1, x, 3, (-CoshIntegral[b/x])*Sinh[a] - Cosh[a]*SinhIntegral[b/x]} -{Sinh[a + b/x]/x^2, x, 2, -(Cosh[a + b/x]/b)} -{Sinh[a + b/x]/x^3, x, 3, -(Cosh[a + b/x]/(b*x)) + Sinh[a + b/x]/b^2} -{Sinh[a + b/x]/x^4, x, 4, -((2*Cosh[a + b/x])/b^3) - Cosh[a + b/x]/(b*x^2) + (2*Sinh[a + b/x])/(b^2*x)} -{Sinh[a + b/x]/x^5, x, 5, -(Cosh[a + b/x]/(b*x^3)) - (6*Cosh[a + b/x])/(b^3*x) + (6*Sinh[a + b/x])/b^4 + (3*Sinh[a + b/x])/(b^2*x^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d / x])^p with m symbolic*) - - -{(e*x)^m*Sinh[a + b/x]^3, x, 9, (-(1/8))*3^(1 + m)*b*E^(3*a)*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, -((3*b)/x)] + (3/8)*b*E^a*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, -(b/x)] + ((3/8)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, b/x])/E^a - ((1/8)*3^(1 + m)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, (3*b)/x])/E^(3*a)} -{(e*x)^m*Sinh[a + b/x]^2, x, 6, -((x*(e*x)^m)/(2*(1 + m))) - 2^(-1 + m)*b*E^(2*a)*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, -((2*b)/x)] + (2^(-1 + m)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, (2*b)/x])/E^(2*a)} -{(e*x)^m*Sinh[a + b/x]^1, x, 4, (-(1/2))*b*E^a*(-(b/x))^m*(e*x)^m*Gamma[-1 - m, -(b/x)] - ((1/2)*b*(b/x)^m*(e*x)^m*Gamma[-1 - m, b/x])/E^a} -{(e*x)^m/Sinh[a + b/x]^1, x, 1, ((e*x)^m*Unintegrable[x^m*Csch[a + b/x], x])/x^m} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d / x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sinh[c+d / x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[a + b/x^2]*x^4, x, 7, (2/15)*b*x^3*Cosh[a + b/x^2] - ((2/15)*b^(5/2)*Sqrt[Pi]*Erf[Sqrt[b]/x])/E^a - (2/15)*b^(5/2)*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x] + (4/15)*b^2*x*Sinh[a + b/x^2] + (1/5)*x^5*Sinh[a + b/x^2]} -{Sinh[a + b/x^2]*x^3, x, 6, (1/4)*b*x^2*Cosh[a + b/x^2] - (1/4)*b^2*CoshIntegral[b/x^2]*Sinh[a] + (1/4)*x^4*Sinh[a + b/x^2] - (1/4)*b^2*Cosh[a]*SinhIntegral[b/x^2]} -{Sinh[a + b/x^2]*x^2, x, 6, (2/3)*b*x*Cosh[a + b/x^2] + ((1/3)*b^(3/2)*Sqrt[Pi]*Erf[Sqrt[b]/x])/E^a - (1/3)*b^(3/2)*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x] + (1/3)*x^3*Sinh[a + b/x^2]} -{Sinh[a + b/x^2]*x^1, x, 5, (-(1/2))*b*Cosh[a]*CoshIntegral[b/x^2] + (1/2)*x^2*Sinh[a + b/x^2] - (1/2)*b*Sinh[a]*SinhIntegral[b/x^2]} -{Sinh[a + b/x^2]*x^0, x, 5, ((-(1/2))*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/E^a - (1/2)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x] + x*Sinh[a + b/x^2]} -{Sinh[a + b/x^2]/x^1, x, 3, (-(1/2))*CoshIntegral[b/x^2]*Sinh[a] - (1/2)*Cosh[a]*SinhIntegral[b/x^2]} -{Sinh[a + b/x^2]/x^2, x, 4, (Sqrt[Pi]*Erf[Sqrt[b]/x])/(E^a*(4*Sqrt[b])) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(4*Sqrt[b])} -{Sinh[a + b/x^2]/x^3, x, 2, -(Cosh[a + b/x^2]/(2*b))} -{Sinh[a + b/x^2]/x^4, x, 5, -(Cosh[a + b/x^2]/(2*b*x)) + (Sqrt[Pi]*Erf[Sqrt[b]/x])/(E^a*(8*b^(3/2))) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(8*b^(3/2))} -{Sinh[a + b/x^2]/x^5, x, 3, -(Cosh[a + b/x^2]/(2*b*x^2)) + Sinh[a + b/x^2]/(2*b^2)} -{Sinh[a + b/x^2]/x^6, x, 6, -(Cosh[a + b/x^2]/(2*b*x^3)) + (3*Sqrt[Pi]*Erf[Sqrt[b]/x])/(E^a*(16*b^(5/2))) - (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(16*b^(5/2)) + (3*Sinh[a + b/x^2])/(4*b^2*x)} -{Sinh[a + b/x^2]/x^7, x, 4, -(Cosh[a + b/x^2]/b^3) - Cosh[a + b/x^2]/(2*b*x^4) + Sinh[a + b/x^2]/(b^2*x^2)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d / x])^p with m symbolic*) - - -{(e*x)^m*Sinh[a + b/x^2]^3, x, 9, (1/16)*3^((1 + m)/2)*E^(3*a)*(-(b/x^2))^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), -((3*b)/x^2)] - (3/16)*E^a*(-(b/x^2))^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), -(b/x^2)] + ((3/16)*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), b/x^2])/E^a - ((1/16)*3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), (3*b)/x^2])/E^(3*a)} -{(e*x)^m*Sinh[a + b/x^2]^2, x, 6, -((x*(e*x)^m)/(2*(1 + m))) + 2^((1/2)*(-5 + m))*E^(2*a)*(-(b/x^2))^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), -((2*b)/x^2)] + (2^((1/2)*(-5 + m))*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), (2*b)/x^2])/E^(2*a)} -{(e*x)^m*Sinh[a + b/x^2]^1, x, 4, (1/4)*E^a*(-(b/x^2))^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), -(b/x^2)] - ((1/4)*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(1/2)*(-1 - m), b/x^2])/E^a} -{(e*x)^m/Sinh[a + b/x^2]^1, x, 1, ((e*x)^m*Unintegrable[x^m*Csch[a + b/x^2], x])/x^m} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^(1/2)])^p*) - - -{Sinh[Sqrt[x]]/Sqrt[x], x, 2, 2*Cosh[Sqrt[x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^n])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Sinh[a + b*x^n], x, 3, -((E^a*x^3*Gamma[3/n, (-b)*x^n])/(((-b)*x^n)^(3/n)*(2*n))) + (x^3*Gamma[3/n, b*x^n])/(E^a*(b*x^n)^(3/n)*(2*n))} -{x^1*Sinh[a + b*x^n], x, 3, -((E^a*x^2*Gamma[2/n, (-b)*x^n])/(((-b)*x^n)^(2/n)*(2*n))) + (x^2*Gamma[2/n, b*x^n])/(E^a*(b*x^n)^(2/n)*(2*n))} -{x^0*Sinh[a + b*x^n], x, 3, -((E^a*x*Gamma[1/n, (-b)*x^n])/(((-b)*x^n)^n^(-1)*(2*n))) + (x*Gamma[1/n, b*x^n])/(E^a*(b*x^n)^n^(-1)*(2*n))} -{Sinh[a + b*x^n]/x^1, x, 3, (CoshIntegral[b*x^n]*Sinh[a])/n + (Cosh[a]*SinhIntegral[b*x^n])/n} -{Sinh[a + b*x^n]/x^2, x, 3, -((E^a*((-b)*x^n)^(1/n)*Gamma[-(1/n), (-b)*x^n])/(2*n*x)) + ((b*x^n)^(1/n)*Gamma[-(1/n), b*x^n])/(E^a*(2*n*x))} -{Sinh[a + b*x^n]/x^3, x, 3, -((E^a*((-b)*x^n)^(2/n)*Gamma[-(2/n), (-b)*x^n])/(2*n*x^2)) + ((b*x^n)^(2/n)*Gamma[-(2/n), b*x^n])/(E^a*(2*n*x^2))} - - -{x^2*Sinh[a + b*x^n]^2, x, 5, -(x^3/6) - (2^(-2 - 3/n)*E^(2*a)*x^3*Gamma[3/n, -2*b*x^n])/(((-b)*x^n)^(3/n)*n) - (2^(-2 - 3/n)*x^3*Gamma[3/n, 2*b*x^n])/(E^(2*a)*(b*x^n)^(3/n)*n)} -{x^1*Sinh[a + b*x^n]^2, x, 5, -(x^2/4) - (4^(-1 - 1/n)*E^(2*a)*x^2*Gamma[2/n, -2*b*x^n])/(((-b)*x^n)^(2/n)*n) - (4^(-1 - 1/n)*x^2*Gamma[2/n, 2*b*x^n])/(E^(2*a)*(b*x^n)^(2/n)*n)} -{x^0*Sinh[a + b*x^n]^2, x, 5, -(x/2) - (2^(-2 - 1/n)*E^(2*a)*x*Gamma[1/n, -2*b*x^n])/(((-b)*x^n)^n^(-1)*n) - (2^(-2 - 1/n)*x*Gamma[1/n, 2*b*x^n])/(E^(2*a)*(b*x^n)^n^(-1)*n)} -{Sinh[a + b*x^n]^2/x^1, x, 5, (Cosh[2*a]*CoshIntegral[2*b*x^n])/(2*n) - Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^n])/(2*n)} -{Sinh[a + b*x^n]^2/x^2, x, 5, 1/(2*x) - (2^(-2 + 1/n)*E^(2*a)*((-b)*x^n)^(1/n)*Gamma[-(1/n), -2*b*x^n])/(n*x) - (2^(-2 + 1/n)*(b*x^n)^(1/n)*Gamma[-(1/n), 2*b*x^n])/(E^(2*a)*(n*x))} - - -{x^2*Sinh[a + b*x^n]^3, x, 8, -((E^(3*a)*x^3*Gamma[3/n, -3*b*x^n])/(3^(3/n)*((-b)*x^n)^(3/n)*(8*n))) + (3*E^a*x^3*Gamma[3/n, (-b)*x^n])/(((-b)*x^n)^(3/n)*(8*n)) - (3*x^3*Gamma[3/n, b*x^n])/(E^a*(b*x^n)^(3/n)*(8*n)) + (x^3*Gamma[3/n, 3*b*x^n])/(3^(3/n)*E^(3*a)*(b*x^n)^(3/n)*(8*n))} -{x^1*Sinh[a + b*x^n]^3, x, 8, -((E^(3*a)*x^2*Gamma[2/n, -3*b*x^n])/(9^n^(-1)*((-b)*x^n)^(2/n)*(8*n))) + (3*E^a*x^2*Gamma[2/n, (-b)*x^n])/(((-b)*x^n)^(2/n)*(8*n)) - (3*x^2*Gamma[2/n, b*x^n])/(E^a*(b*x^n)^(2/n)*(8*n)) + (x^2*Gamma[2/n, 3*b*x^n])/(9^n^(-1)*E^(3*a)*(b*x^n)^(2/n)*(8*n))} -{x^0*Sinh[a + b*x^n]^3, x, 8, -((E^(3*a)*x*Gamma[1/n, -3*b*x^n])/(3^n^(-1)*((-b)*x^n)^n^(-1)*(8*n))) + (3*E^a*x*Gamma[1/n, (-b)*x^n])/(((-b)*x^n)^n^(-1)*(8*n)) - (3*x*Gamma[1/n, b*x^n])/(E^a*(b*x^n)^n^(-1)*(8*n)) + (x*Gamma[1/n, 3*b*x^n])/(3^n^(-1)*E^(3*a)*(b*x^n)^n^(-1)*(8*n))} -{Sinh[a + b*x^n]^3/x^1, x, 8, -((3*CoshIntegral[b*x^n]*Sinh[a])/(4*n)) + (CoshIntegral[3*b*x^n]*Sinh[3*a])/(4*n) - (3*Cosh[a]*SinhIntegral[b*x^n])/(4*n) + (Cosh[3*a]*SinhIntegral[3*b*x^n])/(4*n)} -{Sinh[a + b*x^n]^3/x^2, x, 8, -((3^(1/n)*E^(3*a)*((-b)*x^n)^(1/n)*Gamma[-(1/n), -3*b*x^n])/(8*n*x)) + (3*E^a*((-b)*x^n)^(1/n)*Gamma[-(1/n), (-b)*x^n])/(8*n*x) - (3*(b*x^n)^(1/n)*Gamma[-(1/n), b*x^n])/(E^a*(8*n*x)) + (3^(1/n)*(b*x^n)^(1/n)*Gamma[-(1/n), 3*b*x^n])/(E^(3*a)*(8*n*x))} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sinh[c+d x^n])^p with m symbolic*) - - -{(e*x)^m*(b*Sinh[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(b*Sinh[c + d*x^n])^p, x]} -{(e*x)^m*(a + b*Sinh[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Sinh[c + d*x^n])^p, x]} - - -{(e*x)^(n - 1)*(b*Sinh[c + d*x^n])^p, x, 3, ((e*x)^n*Cosh[c + d*x^n]*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, -Sinh[c + d*x^n]^2]*(b*Sinh[c + d*x^n])^(1 + p))/(x^n*(b*d*e*n*(1 + p)*Sqrt[Cosh[c + d*x^n]^2]))} -{(e*x)^(2*n - 1)*(b*Sinh[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(b*Sinh[c + d*x^n])^p, x])/(x^(2*n)*e)} - -{(e*x)^(n - 1)*(a + b*Sinh[c + d*x^n])^p, x, 5, (I*Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1/2)*(1 - I*Sinh[c + d*x^n]), (b*(1 - I*Sinh[c + d*x^n]))/(I*a + b)]*Cosh[c + d*x^n]*(a + b*Sinh[c + d*x^n])^p)/(x^n*((a + b*Sinh[c + d*x^n])/(a - I*b))^p*(d*e*n*Sqrt[1 + I*Sinh[c + d*x^n]]))} -{(e*x)^(2*n - 1)*(a + b*Sinh[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(a + b*Sinh[c + d*x^n])^p, x])/(x^(2*n)*e)} - - -{(e*x)^m*Sinh[a + b*x^n]^3, x, 8, -((E^(3*a)*(e*x)^(1 + m)*Gamma[(1 + m)/n, -3*b*x^n])/(3^((1 + m)/n)*((-b)*x^n)^((1 + m)/n)*(8*e*n))) + (3*E^a*(e*x)^(1 + m)*Gamma[(1 + m)/n, (-b)*x^n])/(((-b)*x^n)^((1 + m)/n)*(8*e*n)) - (3*(e*x)^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(E^a*(b*x^n)^((1 + m)/n)*(8*e*n)) + ((e*x)^(1 + m)*Gamma[(1 + m)/n, 3*b*x^n])/(3^((1 + m)/n)*E^(3*a)*(b*x^n)^((1 + m)/n)*(8*e*n))} -{(e*x)^m*Sinh[a + b*x^n]^2, x, 5, -((e*x)^(1 + m)/(2*e*(1 + m))) - (E^(2*a)*(e*x)^(1 + m)*Gamma[(1 + m)/n, -2*b*x^n])/(2^((1 + m + 2*n)/n)*((-b)*x^n)^((1 + m)/n)*(e*n)) - ((e*x)^(1 + m)*Gamma[(1 + m)/n, 2*b*x^n])/(2^((1 + m + 2*n)/n)*E^(2*a)*(b*x^n)^((1 + m)/n)*(e*n))} -{(e*x)^m*Sinh[a + b*x^n]^1, x, 3, -((E^a*(e*x)^(1 + m)*Gamma[(1 + m)/n, (-b)*x^n])/(((-b)*x^n)^((1 + m)/n)*(2*e*n))) + ((e*x)^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(E^a*(b*x^n)^((1 + m)/n)*(2*e*n))} -{(e*x)^m/Sinh[a + b*x^n]^2, x, 1, ((e*x)^m*Unintegrable[x^m*Csch[a + b*x^n]^2, x])/x^m} - - -{Sinh[a + b*x^n]^1/x^(n + 1), x, 5, (b*Cosh[a]*CoshIntegral[b*x^n])/n - Sinh[a + b*x^n]/(x^n*n) + (b*Sinh[a]*SinhIntegral[b*x^n])/n} -{Sinh[a + b*x^n]^2/x^(n + 1), x, 7, 1/(x^n*(2*n)) - Cosh[2*(a + b*x^n)]/(x^n*(2*n)) + (b*CoshIntegral[2*b*x^n]*Sinh[2*a])/n + (b*Cosh[2*a]*SinhIntegral[2*b*x^n])/n} -{Sinh[a + b*x^n]^3/x^(n + 1), x, 12, -((3*b*Cosh[a]*CoshIntegral[b*x^n])/(4*n)) + (3*b*Cosh[3*a]*CoshIntegral[3*b*x^n])/(4*n) + (3*Sinh[a + b*x^n])/(x^n*(4*n)) - Sinh[3*(a + b*x^n)]/(x^n*(4*n)) - (3*b*Sinh[a]*SinhIntegral[b*x^n])/(4*n) + (3*b*Sinh[3*a]*SinhIntegral[3*b*x^n])/(4*n)} - - -{x^(n/2 - 1)*Sinh[a + b*x^n], x, 4, -((Sqrt[Pi]*Erf[Sqrt[b]*x^(n/2)])/(E^a*(2*Sqrt[b]*n))) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)])/(2*Sqrt[b]*n)} - - -(* ::Title:: *) -(*Integrands of the form (e x)^m Sinh[a+b (c+d x)^n]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sinh[a+b (c+d x)^n]*) - - -{x^2*Sinh[(a + b*x)^2], x, 12, -((a*Cosh[(a + b*x)^2])/b^3) + ((a + b*x)*Cosh[(a + b*x)^2])/(2*b^3) - (Sqrt[Pi]*Erf[a + b*x])/(8*b^3) - (a^2*Sqrt[Pi]*Erf[a + b*x])/(4*b^3) - (Sqrt[Pi]*Erfi[a + b*x])/(8*b^3) + (a^2*Sqrt[Pi]*Erfi[a + b*x])/(4*b^3)} -{x^1*Sinh[(a + b*x)^2], x, 8, Cosh[(a + b*x)^2]/(2*b^2) + (a*Sqrt[Pi]*Erf[a + b*x])/(4*b^2) - (a*Sqrt[Pi]*Erfi[a + b*x])/(4*b^2)} -{x^0*Sinh[(a + b*x)^2], x, 4, -((Sqrt[Pi]*Erf[a + b*x])/(4*b)) + (Sqrt[Pi]*Erfi[a + b*x])/(4*b)} -{Sinh[(a + b*x)^2]/x^1, x, 1, b*CannotIntegrate[Sinh[(a + b*x)^2]/(b*x), x]} -{Sinh[(a + b*x)^2]/x^2, x, 1, Unintegrable[Sinh[(a + b*x)^2]/x^2, x], b^2*CannotIntegrate[Sinh[(a + b*x)^2]/(b^2*x^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sinh[a+b (c+d x)^(n/2)]*) - - -{Sinh[a + b*Sqrt[c + d*x]]*x^2, x, 16, (240*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b^5*d^3) - (24*c*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*(c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Cosh[a + b*Sqrt[c + d*x]])/(b*d^3) - (240*Sinh[a + b*Sqrt[c + d*x]])/(b^6*d^3) + (24*c*Sinh[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (2*c^2*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (120*(c + d*x)*Sinh[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*c*(c + d*x)*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (10*(c + d*x)^2*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d^3)} -{Sinh[a + b*Sqrt[c + d*x]]*x^1, x, 10, (12*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*c*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b*d^2) + (2*(c + d*x)^(3/2)*Cosh[a + b*Sqrt[c + d*x]])/(b*d^2) - (12*Sinh[a + b*Sqrt[c + d*x]])/(b^4*d^2) + (2*c*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d^2) - (6*(c + d*x)*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d^2)} -{Sinh[a + b*Sqrt[c + d*x]]*x^0, x, 4, (2*Sqrt[c + d*x]*Cosh[a + b*Sqrt[c + d*x]])/(b*d) - (2*Sinh[a + b*Sqrt[c + d*x]])/(b^2*d)} -{Sinh[a + b*Sqrt[c + d*x]]/x^1, x, 10, CoshIntegral[b*(Sqrt[c] + Sqrt[c + d*x])]*Sinh[a - b*Sqrt[c]] + CoshIntegral[b*(Sqrt[c] - Sqrt[c + d*x])]*Sinh[a + b*Sqrt[c]] - Cosh[a + b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] - Sqrt[c + d*x])] + Cosh[a - b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] + Sqrt[c + d*x])]} -{Sinh[a + b*Sqrt[c + d*x]]/x^2, x, 11, (b*d*Cosh[a + b*Sqrt[c]]*CoshIntegral[b*(Sqrt[c] - Sqrt[c + d*x])])/(2*Sqrt[c]) - (b*d*Cosh[a - b*Sqrt[c]]*CoshIntegral[b*(Sqrt[c] + Sqrt[c + d*x])])/(2*Sqrt[c]) - Sinh[a + b*Sqrt[c + d*x]]/x - (b*d*Sinh[a + b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] - Sqrt[c + d*x])])/(2*Sqrt[c]) - (b*d*Sinh[a - b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] + Sqrt[c + d*x])])/(2*Sqrt[c])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sinh[a+b (c+d x)^(n/3)]*) - - -{Sinh[a + b*(c + d*x)^(1/3)]*x^2, x, 23, (120960*Cosh[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*c^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^3) - (120*c*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*(c + d*x)^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*c*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^8*d^3) - (6*c^2*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (20160*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30*c*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (24*(c + d*x)^(7/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3)} -{Sinh[a + b*(c + d*x)^(1/3)]*x^1, x, 13, -((6*c*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^2)) + (360*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^2) - (3*c*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^2) + (60*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (3*(c + d*x)^(5/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^2) - (360*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^2) + (6*c*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^4*d^2) - (15*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^2)} -{Sinh[a + b*(c + d*x)^(1/3)]*x^0, x, 5, (6*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d) + (3*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d) - (6*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d)} -{Sinh[a + b*(c + d*x)^(1/3)]/x^1, x, 13, CoshIntegral[b*(c^(1/3) - (c + d*x)^(1/3))]*Sinh[a + b*c^(1/3)] + CoshIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))]*Sinh[a - (-1)^(1/3)*b*c^(1/3)] + CoshIntegral[(-b)*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))]*Sinh[a + (-1)^(2/3)*b*c^(1/3)] - Cosh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))] - Cosh[a + (-1)^(2/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))] + Cosh[a - (-1)^(1/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))]} -{Sinh[a + b*(c + d*x)^(1/3)]/x^2, x, 14, (b*d*Cosh[a + b*c^(1/3)]*CoshIntegral[b*(c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) + ((-1)^(2/3)*b*d*Cosh[a + (-1)^(2/3)*b*c^(1/3)]*CoshIntegral[(-b)*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-1)^(1/3)*b*d*Cosh[a - (-1)^(1/3)*b*c^(1/3)]*CoshIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))])/(3*c^(2/3)) - Sinh[a + b*(c + d*x)^(1/3)]/x - (b*d*Sinh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-1)^(2/3)*b*d*Sinh[a + (-1)^(2/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-1)^(1/3)*b*d*Sinh[a - (-1)^(1/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))])/(3*c^(2/3))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.4 (d+e x)^m sinh(a+b x+c x^2)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.4 (d+e x)^m sinh(a+b x+c x^2)^n.m deleted file mode 100644 index 22949ce..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.4 (d+e x)^m sinh(a+b x+c x^2)^n.m +++ /dev/null @@ -1,64 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form (d+e x)^m Sinh[a+b x+c x^2]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m Sinh[a+b x+c x^2]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*Sinh[a + b*x + c*x^2], x, 12, -((b*Cosh[a + b*x + c*x^2])/(4*c^2)) + (x*Cosh[a + b*x + c*x^2])/(2*c) - (b^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + (b^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))} -{x*Sinh[a + b*x + c*x^2], x, 6, Cosh[a + b*x + c*x^2]/(2*c) + (b*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))} -{Sinh[a + b*x + c*x^2], x, 5, -((E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])) + (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{Sinh[a + b*x + c*x^2]/x, x, 0, Unintegrable[Sinh[a + b*x + c*x^2]/x, x]} -{Sinh[a + b*x + c*x^2]/x^2 - b*Cosh[a + b*x + c*x^2]/x, x, 7, (1/2)*Sqrt[c]*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])] + (1/2)*Sqrt[c]*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])] - Sinh[a + b*x + c*x^2]/x} - -{x^2*Sinh[a + b*x - c*x^2], x, 12, -((b*Cosh[a + b*x - c*x^2])/(4*c^2)) - (x*Cosh[a + b*x - c*x^2])/(2*c) - (b^2*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + (b^2*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))} -{x*Sinh[a + b*x - c*x^2], x, 6, -(Cosh[a + b*x - c*x^2]/(2*c)) - (b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + (b*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))} -{Sinh[a + b*x - c*x^2], x, 5, -((E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])) + (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{Sinh[a + b*x - c*x^2]/x, x, 0, Unintegrable[Sinh[a + b*x - c*x^2]/x, x]} -{Sinh[a + b*x - c*x^2]/x^2 - b*Cosh[a + b*x - c*x^2]/x, x, 7, (1/2)*Sqrt[c]*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])] + (1/2)*Sqrt[c]*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])] - Sinh[a + b*x - c*x^2]/x} - -{x^2*Sinh[1/4 + x + x^2], x, 12, (-(1/4))*Cosh[1/4 + x + x^2] + (1/2)*x*Cosh[1/4 + x + x^2] + (3/16)*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] - (1/16)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)]} -{x*Sinh[1/4 + x + x^2], x, 6, (1/2)*Cosh[1/4 + x + x^2] - (1/8)*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] - (1/8)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)]} -{Sinh[1/4 + x + x^2], x, 5, (1/4)*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] + (1/4)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)]} -{Sinh[1/4 + x + x^2]/x, x, 0, Unintegrable[Sinh[1/4 + x + x^2]/x, x]} -{Sinh[1/4 + x + x^2]/x^2, x, 6, (-(1/2))*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] + (1/2)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)] + Unintegrable[Cosh[1/4 + x + x^2]/x, x] - Sinh[1/4 + x + x^2]/x} - - -{x^2*Sinh[a + b*x + c*x^2]^2, x, 14, -(x^3/6) + (b^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + (b^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (x*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{x*Sinh[a + b*x + c*x^2]^2, x, 8, -(x^2/4) - (b*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - (b*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + Sinh[2*a + 2*b*x + 2*c*x^2]/(8*c)} -{Sinh[a + b*x + c*x^2]^2, x, 7, -(x/2) + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c]) + (E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c])} -{Sinh[a + b*x + c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x + 2*c*x^2]/x, x] - Log[x]/2} - -{x^2*Sinh[a + b*x - c*x^2]^2, x, 14, -(x^3/6) - (b^2*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b^2*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x - 2*c*x^2])/(16*c^2) - (x*Sinh[2*a + 2*b*x - 2*c*x^2])/(8*c)} -{x*Sinh[a + b*x - c*x^2]^2, x, 8, -(x^2/4) - (b*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - (b*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - Sinh[2*a + 2*b*x - 2*c*x^2]/(8*c)} -{Sinh[a + b*x - c*x^2]^2, x, 7, -(x/2) - (E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c]) - (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c])} -{Sinh[a + b*x - c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x - 2*c*x^2]/x, x] - Log[x]/2} - -{x^2*Sinh[1/4 + x + x^2]^2, x, 14, -(x^3/6) + (1/16)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] - (1/16)*Sinh[1/2 + 2*x + 2*x^2] + (1/8)*x*Sinh[1/2 + 2*x + 2*x^2]} -{x*Sinh[1/4 + x + x^2]^2, x, 8, -(x^2/4) - (1/16)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] - (1/16)*Sqrt[Pi/2]*Erfi[(1 + 2*x)/Sqrt[2]] + (1/8)*Sinh[1/2 + 2*x + 2*x^2]} -{Sinh[1/4 + x + x^2]^2, x, 7, -(x/2) + (1/8)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] + (1/8)*Sqrt[Pi/2]*Erfi[(1 + 2*x)/Sqrt[2]]} -{Sinh[1/4 + x + x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[1/2 + 2*x + 2*x^2]/x, x] - Log[x]/2} - - -{(d + e*x)^2*Sinh[a + b*x + c*x^2], x, 12, (e*(2*c*d - b*e)*Cosh[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Cosh[a + b*x + c*x^2])/(2*c) - (e^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - ((2*c*d - b*e)^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (e^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2))} -{(d + e*x)*Sinh[a + b*x + c*x^2], x, 6, (e*Cosh[a + b*x + c*x^2])/(2*c) - ((2*c*d - b*e)*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))} -{Sinh[a + b*x + c*x^2]/(d + e*x), x, 0, Unintegrable[Sinh[a + b*x + c*x^2]/(d + e*x), x]} - -{(d + e*x)^2*Sinh[a + b*x + c*x^2]^2, x, 14, -((d + e*x)^3/(6*e)) + (e^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (e^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (e*(2*c*d - b*e)*Sinh[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (e*(d + e*x)*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{(d + e*x)*Sinh[a + b*x + c*x^2]^2, x, 8, -((d + e*x)^2/(4*e)) + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{Sinh[a + b*x + c*x^2]^2/(d + e*x), x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x + 2*c*x^2]/(d + e*x), x] - Log[d + e*x]/(2*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (d+e x)^m Sinh[a+b x+c x^2]^(n/2)*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.5 Hyperbolic sine functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.5 Hyperbolic sine functions.m deleted file mode 100644 index 6b45678..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.5 Hyperbolic sine functions.m +++ /dev/null @@ -1,734 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Sines*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Sinh[c+d x])^n (A+B Sinh[c+d x]+C Sinh[c+d x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[a+b x]^n*) - - -{Sinh[a + b*x], x, 1, Cosh[a + b*x]/b} -{Sinh[a + b*x]^2, x, 2, -(x/2) + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{Sinh[a + b*x]^3, x, 2, -(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)} -{Sinh[a + b*x]^4, x, 3, (3*x)/8 - (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]*Sinh[a + b*x]^3)/(4*b)} -{Sinh[a + b*x]^5, x, 2, Cosh[a + b*x]/b - (2*Cosh[a + b*x]^3)/(3*b) + Cosh[a + b*x]^5/(5*b)} -{Sinh[a + b*x]^6, x, 4, -((5*x)/16) + (5*Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (5*Cosh[a + b*x]*Sinh[a + b*x]^3)/(24*b) + (Cosh[a + b*x]*Sinh[a + b*x]^5)/(6*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[a+b x]^(n/2)*) - - -{Sinh[a + b*x]^(7/2), x, 4, -((10*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(21*b*Sqrt[Sinh[a + b*x]])) - (10*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]])/(21*b) + (2*Cosh[a + b*x]*Sinh[a + b*x]^(5/2))/(7*b)} -{Sinh[a + b*x]^(5/2), x, 3, (6*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(5*b*Sqrt[I*Sinh[a + b*x]]) + (2*Cosh[a + b*x]*Sinh[a + b*x]^(3/2))/(5*b)} -{Sinh[a + b*x]^(3/2), x, 3, (2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(3*b*Sqrt[Sinh[a + b*x]]) + (2*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]])/(3*b)} -{Sinh[a + b*x]^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[a + b*x]]))} -{1/Sinh[a + b*x]^(1/2), x, 2, -((2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]]))} -{1/Sinh[a + b*x]^(3/2), x, 3, -((2*Cosh[a + b*x])/(b*Sqrt[Sinh[a + b*x]])) - (2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[a + b*x]])} -{1/Sinh[a + b*x]^(5/2), x, 3, -((2*Cosh[a + b*x])/(3*b*Sinh[a + b*x]^(3/2))) + (2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(3*b*Sqrt[Sinh[a + b*x]])} -{1/Sinh[a + b*x]^(7/2), x, 4, -((2*Cosh[a + b*x])/(5*b*Sinh[a + b*x]^(5/2))) + (6*Cosh[a + b*x])/(5*b*Sqrt[Sinh[a + b*x]]) + (6*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(5*b*Sqrt[I*Sinh[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sinh[a+b x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n/2*) - - -{(b*Sinh[c + d*x])^(7/2), x, 4, -((10*I*b^4*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(21*d*Sqrt[b*Sinh[c + d*x]])) - (10*b^3*Cosh[c + d*x]*Sqrt[b*Sinh[c + d*x]])/(21*d) + (2*b*Cosh[c + d*x]*(b*Sinh[c + d*x])^(5/2))/(7*d)} -{(b*Sinh[c + d*x])^(5/2), x, 3, (6*I*b^2*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[b*Sinh[c + d*x]])/(5*d*Sqrt[I*Sinh[c + d*x]]) + (2*b*Cosh[c + d*x]*(b*Sinh[c + d*x])^(3/2))/(5*d)} -{(b*Sinh[c + d*x])^(3/2),x, 3, (2*I*b^2*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(3*d*Sqrt[b*Sinh[c + d*x]]) + (2*b*Cosh[c + d*x]*Sqrt[b*Sinh[c + d*x]])/(3*d)} -{(b*Sinh[c + d*x])^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[b*Sinh[c + d*x]])/(d*Sqrt[I*Sinh[c + d*x]]))} -{1/(b*Sinh[c + d*x])^(1/2), x, 2, -((2*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(d*Sqrt[b*Sinh[c + d*x]]))} -{1/(b*Sinh[c + d*x])^(3/2), x, 3, -((2*Cosh[c + d*x])/(b*d*Sqrt[b*Sinh[c + d*x]])) - (2*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[b*Sinh[c + d*x]])/(b^2*d*Sqrt[I*Sinh[c + d*x]])} -{1/(b*Sinh[c + d*x])^(5/2), x, 3, -((2*Cosh[c + d*x])/(3*b*d*(b*Sinh[c + d*x])^(3/2))) + (2*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(3*b^2*d*Sqrt[b*Sinh[c + d*x]])} -{1/(b*Sinh[c + d*x])^(7/2), x, 4, -((2*Cosh[c + d*x])/(5*b*d*(b*Sinh[c + d*x])^(5/2))) + (6*Cosh[c + d*x])/(5*b^3*d*Sqrt[b*Sinh[c + d*x]]) + (6*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[b*Sinh[c + d*x]])/(5*b^4*d*Sqrt[I*Sinh[c + d*x]])} - - -{(I*Sinh[c + d*x])^(7/2), x, 3, -((10*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(21*d)) + (10*I*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/(21*d) + (2*I*Cosh[c + d*x]*(I*Sinh[c + d*x])^(5/2))/(7*d)} -{(I*Sinh[c + d*x])^(5/2), x, 2, -((6*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(5*d)) + (2*I*Cosh[c + d*x]*(I*Sinh[c + d*x])^(3/2))/(5*d)} -{(I*Sinh[c + d*x])^(3/2),x, 2, -((2*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(3*d)) + (2*I*Cosh[c + d*x]*Sqrt[I*Sinh[c + d*x]])/(3*d)} -{(I*Sinh[c + d*x])^(1/2), x, 1, -((2*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/d)} -{1/(I*Sinh[c + d*x])^(1/2), x, 1, -((2*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2])/d)} -{1/(I*Sinh[c + d*x])^(3/2), x, 2, (2*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/d + (2*I*Cosh[c + d*x])/(d*Sqrt[I*Sinh[c + d*x]])} -{1/(I*Sinh[c + d*x])^(5/2), x, 2, -((2*I*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(3*d)) + (2*I*Cosh[c + d*x])/(3*d*(I*Sinh[c + d*x])^(3/2))} -{1/(I*Sinh[c + d*x])^(7/2), x, 3, (6*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(5*d) + (2*I*Cosh[c + d*x])/(5*d*(I*Sinh[c + d*x])^(5/2)) + (6*I*Cosh[c + d*x])/(5*d*Sqrt[I*Sinh[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*n/3*) - - -{(b*Sinh[c + d*x])^(4/3),x, 1, (3*Cosh[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(7/3))/(7*b*d*Sqrt[Cosh[c + d*x]^2])} -{(b*Sinh[c + d*x])^(2/3),x, 1, (3*Cosh[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(5/3))/(5*b*d*Sqrt[Cosh[c + d*x]^2])} -{(b*Sinh[c + d*x])^(1/3),x, 1, (3*Cosh[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(4/3))/(4*b*d*Sqrt[Cosh[c + d*x]^2])} -{1/(b*Sinh[c + d*x])^(1/3),x, 1, (3*Cosh[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(2/3))/(2*b*d*Sqrt[Cosh[c + d*x]^2])} -{1/(b*Sinh[c + d*x])^(2/3),x, 1, (3*Cosh[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(1/3))/(b*d*Sqrt[Cosh[c + d*x]^2])} -{1/(b*Sinh[c + d*x])^(4/3),x, 1, -((3*Cosh[c + d*x]*Hypergeometric2F1[-(1/6), 1/2, 5/6, -Sinh[c + d*x]^2])/(b*d*Sqrt[Cosh[c + d*x]^2]*(b*Sinh[c + d*x])^(1/3)))} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{(b*Sinh[c + d*x])^n, x, 1, (Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(1 + n))/(b*d*(1 + n)*Sqrt[Cosh[c + d*x]^2])} - - -{(I*Sinh[c + d*x])^n, x, 1, -((I*Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*(I*Sinh[c + d*x])^(1 + n))/(d*(1 + n)*Sqrt[Cosh[c + d*x]^2]))} -{(-I*Sinh[c + d*x])^n, x, 1, (I*Cosh[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, -Sinh[c + d*x]^2]*((-I)*Sinh[c + d*x])^(1 + n))/(d*(1 + n)*Sqrt[Cosh[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Sinh[c+d x])^n when a^2+b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sinh[x]^4/(I + Sinh[x]), x, 6, (3*I*x)/2 - 4*Cosh[x] + (4*Cosh[x]^3)/3 - (3/2)*I*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x]^3)/(I + Sinh[x])} -{Sinh[x]^3/(I + Sinh[x]), x, 2, -((3*x)/2) - 2*I*Cosh[x] + (3/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x]^2)/(I + Sinh[x])} -{Sinh[x]^2/(I + Sinh[x]), x, 3, (-I)*x + Cosh[x] + (I*Cosh[x])/(I + Sinh[x])} -{Sinh[x]^1/(I + Sinh[x]), x, 2, x - Cosh[x]/(I + Sinh[x])} -{Csch[x]^1/(I + Sinh[x]), x, 3, I*ArcTanh[Cosh[x]] + Cosh[x]/(I + Sinh[x])} -{Csch[x]^2/(I + Sinh[x]), x, 5, -ArcTanh[Cosh[x]] + 2*I*Coth[x] + Coth[x]/(I + Sinh[x])} -{Csch[x]^3/(I + Sinh[x]), x, 6, (-(3/2))*I*ArcTanh[Cosh[x]] - 2*Coth[x] + (3/2)*I*Coth[x]*Csch[x] + (Coth[x]*Csch[x])/(I + Sinh[x])} -{Csch[x]^4/(I + Sinh[x]), x, 6, (3/2)*ArcTanh[Cosh[x]] - 4*I*Coth[x] + (4/3)*I*Coth[x]^3 - (3/2)*Coth[x]*Csch[x] + (Coth[x]*Csch[x]^2)/(I + Sinh[x])} - - -{Sinh[x]^4/(I + Sinh[x])^2, x, 3, -((7*x)/2) - (16/3)*I*Cosh[x] + (7/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x]^3)/(3*(I + Sinh[x])^2) - (8*Cosh[x]*Sinh[x]^2)/(3*(I + Sinh[x]))} -{Sinh[x]^3/(I + Sinh[x])^2, x, 6, -2*I*x + (4*Cosh[x])/3 - (Cosh[x]*Sinh[x]^2)/(3*(I + Sinh[x])^2) + (2*I*Cosh[x])/(I + Sinh[x])} -{Sinh[x]^2/(I + Sinh[x])^2, x, 3, x + (I*Cosh[x])/(3*(I + Sinh[x])^2) - (5*Cosh[x])/(3*(I + Sinh[x]))} -{Sinh[x]^1/(I + Sinh[x])^2, x, 2, -(Cosh[x]/(3*(I + Sinh[x])^2)) - (2*I*Cosh[x])/(3*(I + Sinh[x]))} -{Csch[x]^1/(I + Sinh[x])^2, x, 4, ArcTanh[Cosh[x]] + Cosh[x]/(3*(I + Sinh[x])^2) - (4*I*Cosh[x])/(3*(I + Sinh[x]))} -{Csch[x]^2/(I + Sinh[x])^2, x, 6, 2*I*ArcTanh[Cosh[x]] + (10*Coth[x])/3 + Coth[x]/(3*(I + Sinh[x])^2) - (2*I*Coth[x])/(I + Sinh[x])} -{Csch[x]^3/(I + Sinh[x])^2, x, 7, (-(7/2))*ArcTanh[Cosh[x]] + (16/3)*I*Coth[x] + (7/2)*Coth[x]*Csch[x] + (Coth[x]*Csch[x])/(3*(I + Sinh[x])^2) - (8*I*Coth[x]*Csch[x])/(3*(I + Sinh[x]))} -{Csch[x]^4/(I + Sinh[x])^2, x, 7, -5*I*ArcTanh[Cosh[x]] - 12*Coth[x] + 4*Coth[x]^3 + 5*I*Coth[x]*Csch[x] + (Coth[x]*Csch[x]^2)/(3*(I + Sinh[x])^2) - (10*I*Coth[x]*Csch[x]^2)/(3*(I + Sinh[x]))} - - -{1/(1 + I*Sinh[c + d*x]), x, 1, (I*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x]))} -{1/(1 + I*Sinh[c + d*x])^2, x, 2, (I*Cosh[c + d*x])/(3*d*(1 + I*Sinh[c + d*x])^2) + (I*Cosh[c + d*x])/(3*d*(1 + I*Sinh[c + d*x]))} -{1/(1 + I*Sinh[c + d*x])^3, x, 3, (I*Cosh[c + d*x])/(5*d*(1 + I*Sinh[c + d*x])^3) + (2*I*Cosh[c + d*x])/(15*d*(1 + I*Sinh[c + d*x])^2) + (2*I*Cosh[c + d*x])/(15*d*(1 + I*Sinh[c + d*x]))} -{1/(1 + I*Sinh[c + d*x])^4, x, 4, (I*Cosh[c + d*x])/(7*d*(1 + I*Sinh[c + d*x])^4) + (3*I*Cosh[c + d*x])/(35*d*(1 + I*Sinh[c + d*x])^3) + (2*I*Cosh[c + d*x])/(35*d*(1 + I*Sinh[c + d*x])^2) + (2*I*Cosh[c + d*x])/(35*d*(1 + I*Sinh[c + d*x]))} - -{1/(1 - I*Sinh[c + d*x]), x, 1, -((I*Cosh[c + d*x])/(d*(1 - I*Sinh[c + d*x])))} -{1/(1 - I*Sinh[c + d*x])^2, x, 2, -((I*Cosh[c + d*x])/(3*d*(1 - I*Sinh[c + d*x])^2)) - (I*Cosh[c + d*x])/(3*d*(1 - I*Sinh[c + d*x]))} -{1/(1 - I*Sinh[c + d*x])^3, x, 3, -((I*Cosh[c + d*x])/(5*d*(1 - I*Sinh[c + d*x])^3)) - (2*I*Cosh[c + d*x])/(15*d*(1 - I*Sinh[c + d*x])^2) - (2*I*Cosh[c + d*x])/(15*d*(1 - I*Sinh[c + d*x]))} -{1/(1 - I*Sinh[c + d*x])^4, x, 4, -((I*Cosh[c + d*x])/(7*d*(1 - I*Sinh[c + d*x])^4)) - (3*I*Cosh[c + d*x])/(35*d*(1 - I*Sinh[c + d*x])^3) - (2*I*Cosh[c + d*x])/(35*d*(1 - I*Sinh[c + d*x])^2) - (2*I*Cosh[c + d*x])/(35*d*(1 - I*Sinh[c + d*x]))} - - -{Sinh[x]/Sqrt[a + a*I*Sinh[x]], x, 3, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[x]])])/Sqrt[a]) + (2*Cosh[x])/Sqrt[a + I*a*Sinh[x]]} -{Sinh[x]/Sqrt[a - a*I*Sinh[x]], x, 3, -((Sqrt[2]*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a - I*a*Sinh[x]])])/Sqrt[a]) + (2*Cosh[x])/Sqrt[a - I*a*Sinh[x]]} - - -{(a + a*I*Sinh[c + d*x])^(5/2), x, 3, (64*I*a^3*Cosh[c + d*x])/(15*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (16*I*a^2*Cosh[c + d*x]*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d) + (2*I*a*Cosh[c + d*x]*(a + I*a*Sinh[c + d*x])^(3/2))/(5*d)} -{(a + a*I*Sinh[c + d*x])^(3/2), x, 2, (8*I*a^2*Cosh[c + d*x])/(3*d*Sqrt[a + I*a*Sinh[c + d*x]]) + (2*I*a*Cosh[c + d*x]*Sqrt[a + I*a*Sinh[c + d*x]])/(3*d)} -{(a + a*I*Sinh[c + d*x])^(1/2), x, 1, (2*I*a*Cosh[c + d*x])/(d*Sqrt[a + I*a*Sinh[c + d*x]])} -{1/(a + a*I*Sinh[c + d*x])^(1/2), x, 2, (I*Sqrt[2]*ArcTanh[(Sqrt[a]*Cosh[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[c + d*x]])])/(Sqrt[a]*d)} -{1/(a + a*I*Sinh[c + d*x])^(3/2), x, 3, (I*ArcTanh[(Sqrt[a]*Cosh[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + (I*Cosh[c + d*x])/(2*d*(a + I*a*Sinh[c + d*x])^(3/2))} -{1/(a + a*I*Sinh[c + d*x])^(5/2), x, 4, (3*I*ArcTanh[(Sqrt[a]*Cosh[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + (I*Cosh[c + d*x])/(4*d*(a + I*a*Sinh[c + d*x])^(5/2)) + (3*I*Cosh[c + d*x])/(16*a*d*(a + I*a*Sinh[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Sinh[c+d x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sinh[x]^4/(a + b*Sinh[x]), x, 7, -((a*(2*a^2 - b^2)*x)/(2*b^4)) - (2*a^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]) - ((2 - (3*a^2)/b^2)*Cosh[x])/(3*b) - (a*Cosh[x]*Sinh[x])/(2*b^2) + (Cosh[x]*Sinh[x]^2)/(3*b)} -{Sinh[x]^3/(a + b*Sinh[x]), x, 6, ((2*a^2 - b^2)*x)/(2*b^3) + (2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]) - (a*Cosh[x])/b^2 + (Cosh[x]*Sinh[x])/(2*b)} -{Sinh[x]^2/(a + b*Sinh[x]), x, 6, -((a*x)/b^2) - (2*a^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) + Cosh[x]/b} -{Sinh[x]^1/(a + b*Sinh[x]), x, 4, x/b + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])} -{Csch[x]^1/(a + b*Sinh[x]), x, 5, -(ArcTanh[Cosh[x]]/a) + (2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])} -{Csch[x]^2/(a + b*Sinh[x]), x, 7, (b*ArcTanh[Cosh[x]])/a^2 - (2*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) - Coth[x]/a} -{Csch[x]^3/(a + b*Sinh[x]), x, 7, ((a^2 - 2*b^2)*ArcTanh[Cosh[x]])/(2*a^3) + (2*b^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) + (b*Coth[x])/a^2 - (Coth[x]*Csch[x])/(2*a)} -{Csch[x]^4/(a + b*Sinh[x]), x, 8, -((b*(a^2 - 2*b^2)*ArcTanh[Cosh[x]])/(2*a^4)) - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) + ((2*a^2 - 3*b^2)*Coth[x])/(3*a^3) + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch[x]^2)/(3*a)} - - -{Sinh[x]^4/(a + b*Sinh[x])^2, x, 7, ((6*a^2 - b^2)*x)/(2*b^4) + (2*a^3*(3*a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*(a^2 + b^2)^(3/2)) - (a*(3*a^2 + 2*b^2)*Cosh[x])/(b^3*(a^2 + b^2)) + ((3*a^2 + b^2)*Cosh[x]*Sinh[x])/(2*b^2*(a^2 + b^2)) - (a^2*Cosh[x]*Sinh[x]^2)/(b*(a^2 + b^2)*(a + b*Sinh[x]))} -{Sinh[x]^3/(a + b*Sinh[x])^2, x, 6, -((2*a*x)/b^3) - (2*a^2*(2*a^2 + 3*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*(a^2 + b^2)^(3/2)) + ((2*a^2 + b^2)*Cosh[x])/(b^2*(a^2 + b^2)) - (a^2*Cosh[x]*Sinh[x])/(b*(a^2 + b^2)*(a + b*Sinh[x]))} -{Sinh[x]^2/(a + b*Sinh[x])^2, x, 5, x/b^2 + (2*a*(a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(3/2)) - (a^2*Cosh[x])/(b*(a^2 + b^2)*(a + b*Sinh[x]))} -{Sinh[x]^1/(a + b*Sinh[x])^2, x, 5, -((2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + (a*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x]))} -{Csch[x]^1/(a + b*Sinh[x])^2, x, 6, -(ArcTanh[Cosh[x]]/a^2) + (2*b*(2*a^2 + b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)) + (b^2*Cosh[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))} -{Csch[x]^2/(a + b*Sinh[x])^2, x, 7, (2*b*ArcTanh[Cosh[x]])/a^3 - (2*b^2*(3*a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^2)^(3/2)) - ((a^2 + 2*b^2)*Coth[x])/(a^2*(a^2 + b^2)) + (b^2*Coth[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))} -{Csch[x]^3/(a + b*Sinh[x])^2, x, 8, ((a^2 - 6*b^2)*ArcTanh[Cosh[x]])/(2*a^4) + (2*b^3*(4*a^2 + 3*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*(a^2 + b^2)^(3/2)) + (b*(2*a^2 + 3*b^2)*Coth[x])/(a^3*(a^2 + b^2)) - ((a^2 + 3*b^2)*Coth[x]*Csch[x])/(2*a^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))} -{Csch[x]^4/(a + b*Sinh[x])^2, x, 9, -((b*(a^2 - 4*b^2)*ArcTanh[Cosh[x]])/a^5) - (2*b^4*(5*a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^5*(a^2 + b^2)^(3/2)) + ((2*a^4 - 7*a^2*b^2 - 12*b^4)*Coth[x])/(3*a^4*(a^2 + b^2)) + (b*(a^2 + 2*b^2)*Coth[x]*Csch[x])/(a^3*(a^2 + b^2)) - ((a^2 + 4*b^2)*Coth[x]*Csch[x]^2)/(3*a^2*(a^2 + b^2)) + (b^2*Coth[x]*Csch[x]^2)/(a*(a^2 + b^2)*(a + b*Sinh[x]))} - - -{1/(3 + 5*I*Sinh[c + d*x]), x, 4, (I*Log[3*Cosh[(1/2)*(c + d*x)] + I*Sinh[(1/2)*(c + d*x)]])/(4*d) - (I*Log[Cosh[(1/2)*(c + d*x)] + 3*I*Sinh[(1/2)*(c + d*x)]])/(4*d)} -{1/(3 + 5*I*Sinh[c + d*x])^2, x, 6, -((3*I*Log[3*Cosh[(1/2)*(c + d*x)] + I*Sinh[(1/2)*(c + d*x)]])/(64*d)) + (3*I*Log[Cosh[(1/2)*(c + d*x)] + 3*I*Sinh[(1/2)*(c + d*x)]])/(64*d) + (5*I*Cosh[c + d*x])/(16*d*(3 + 5*I*Sinh[c + d*x]))} -{1/(3 + 5*I*Sinh[c + d*x])^3, x, 7, (43*I*Log[3*Cosh[(1/2)*(c + d*x)] + I*Sinh[(1/2)*(c + d*x)]])/(2048*d) - (43*I*Log[Cosh[(1/2)*(c + d*x)] + 3*I*Sinh[(1/2)*(c + d*x)]])/(2048*d) + (5*I*Cosh[c + d*x])/(32*d*(3 + 5*I*Sinh[c + d*x])^2) - (45*I*Cosh[c + d*x])/(512*d*(3 + 5*I*Sinh[c + d*x]))} -{1/(3 + 5*I*Sinh[c + d*x])^4, x, 8, -((279*I*Log[3*Cosh[(1/2)*(c + d*x)] + I*Sinh[(1/2)*(c + d*x)]])/(32768*d)) + (279*I*Log[Cosh[(1/2)*(c + d*x)] + 3*I*Sinh[(1/2)*(c + d*x)]])/(32768*d) + (5*I*Cosh[c + d*x])/(48*d*(3 + 5*I*Sinh[c + d*x])^3) - (25*I*Cosh[c + d*x])/(512*d*(3 + 5*I*Sinh[c + d*x])^2) + (995*I*Cosh[c + d*x])/(24576*d*(3 + 5*I*Sinh[c + d*x]))} - -{1/(5 + 3*I*Sinh[c + d*x]), x, 1, x/4 - (I*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/(2*d)} -{1/(5 + 3*I*Sinh[c + d*x])^2, x, 3, (5*x)/64 - (5*I*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/(32*d) - (3*I*Cosh[c + d*x])/(16*d*(5 + 3*I*Sinh[c + d*x]))} -{1/(5 + 3*I*Sinh[c + d*x])^3, x, 4, (59*x)/2048 - (59*I*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/(1024*d) - (3*I*Cosh[c + d*x])/(32*d*(5 + 3*I*Sinh[c + d*x])^2) - (45*I*Cosh[c + d*x])/(512*d*(5 + 3*I*Sinh[c + d*x]))} -{1/(5 + 3*I*Sinh[c + d*x])^4, x, 5, (385*x)/32768 - (385*I*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/(16384*d) - (I*Cosh[c + d*x])/(16*d*(5 + 3*I*Sinh[c + d*x])^3) - (25*I*Cosh[c + d*x])/(512*d*(5 + 3*I*Sinh[c + d*x])^2) - (311*I*Cosh[c + d*x])/(8192*d*(5 + 3*I*Sinh[c + d*x]))} - - -{(a + b*Sinh[c + d*x])^5, x, 4, (1/8)*a*(8*a^4 - 40*a^2*b^2 + 15*b^4)*x + (b*(107*a^4 - 192*a^2*b^2 + 16*b^4)*Cosh[c + d*x])/(30*d) + (7*a*b^2*(22*a^2 - 23*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(120*d) + (b*(47*a^2 - 16*b^2)*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(60*d) + (9*a*b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^3)/(20*d) + (b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^4)/(5*d)} -{(a + b*Sinh[c + d*x])^4, x, 3, (1/8)*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 - 16*b^2)*Cosh[c + d*x])/(6*d) + (b^2*(26*a^2 - 9*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(24*d) + (7*a*b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(12*d) + (b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^3)/(4*d)} -{(a + b*Sinh[c + d*x])^3, x, 2, (1/2)*a*(2*a^2 - 3*b^2)*x + (2*b*(4*a^2 - b^2)*Cosh[c + d*x])/(3*d) + (5*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(6*d) + (b*Cosh[c + d*x]*(a + b*Sinh[c + d*x])^2)/(3*d)} -{(a + b*Sinh[c + d*x])^2, x, 1, (1/2)*(2*a^2 - b^2)*x + (2*a*b*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{(a + b*Sinh[c + d*x]), x, 2, a*x + (b*Cosh[c + d*x])/d} -{1/(a + b*Sinh[c + d*x]), x, 3, -((2*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d))} -{1/(a + b*Sinh[c + d*x])^2, x, 5, -((2*a*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) - (b*Cosh[c + d*x])/((a^2 + b^2)*d*(a + b*Sinh[c + d*x]))} -{1/(a + b*Sinh[c + d*x])^3, x, 6, -(((2*a^2 - b^2)*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) - (b*Cosh[c + d*x])/(2*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^2) - (3*a*b*Cosh[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x]))} -{1/(a + b*Sinh[c + d*x])^4, x, 7, -((a*(2*a^2 - 3*b^2)*ArcTanh[(b - a*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d)) - (b*Cosh[c + d*x])/(3*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^3) - (5*a*b*Cosh[c + d*x])/(6*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x])^2) - (b*(11*a^2 - 4*b^2)*Cosh[c + d*x])/(6*(a^2 + b^2)^3*d*(a + b*Sinh[c + d*x]))} - - -{(a + b*Sinh[x])^(5/2), x, 7, (16/15)*a*b*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2/5)*b*Cosh[x]*(a + b*Sinh[x])^(3/2) + (2*I*(23*a^2 - 9*b^2)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(15*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (16*I*a*(a^2 + b^2)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(15*Sqrt[a + b*Sinh[x]])} -{(a + b*Sinh[x])^(3/2), x, 6, (2/3)*b*Cosh[x]*Sqrt[a + b*Sinh[x]] + (8*I*a*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(3*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*Sqrt[a + b*Sinh[x]])} -{(a + b*Sinh[x])^(1/2), x, 2, (2*I*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/Sqrt[(a + b*Sinh[x])/(a - I*b)]} -{1/(a + b*Sinh[x])^(1/2), x, 2, (2*I*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]]} -{1/(a + b*Sinh[x])^(3/2), x, 4, -((2*b*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]])) + (2*I*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/((a^2 + b^2)*Sqrt[(a + b*Sinh[x])/(a - I*b)])} -{1/(a + b*Sinh[x])^(5/2), x, 7, -((2*b*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^(3/2))) - (8*a*b*Cosh[x])/(3*(a^2 + b^2)^2*Sqrt[a + b*Sinh[x]]) + (8*I*a*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(3*(a^2 + b^2)^2*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*(a^2 + b^2)*Sqrt[a + b*Sinh[x]])} - - -{Sinh[x]/Sqrt[a + b*Sinh[x]], x, 5, (2*I*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*a*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Sinh[c+d x]) (a+b Sinh[c+d x])^n when a^2+b^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(5/2), x, 4, (64*a^3*(7*I*A + 5*B)*Cosh[x])/(105*Sqrt[a + I*a*Sinh[x]]) + (16/105)*a^2*(7*I*A + 5*B)*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (2/35)*a*(7*I*A + 5*B)*Cosh[x]*(a + I*a*Sinh[x])^(3/2) + (2/7)*B*Cosh[x]*(a + I*a*Sinh[x])^(5/2)} -{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(3/2), x, 3, (8*a^2*(5*I*A + 3*B)*Cosh[x])/(15*Sqrt[a + I*a*Sinh[x]]) + (2/15)*a*(5*I*A + 3*B)*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (2/5)*B*Cosh[x]*(a + I*a*Sinh[x])^(3/2)} -{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(1/2), x, 2, (2*a*(3*I*A + B)*Cosh[x])/(3*Sqrt[a + I*a*Sinh[x]]) + (2/3)*B*Cosh[x]*Sqrt[a + I*a*Sinh[x]]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Sinh[x])/(I + Sinh[x]), x, 2, B*x - ((I*A + B)*Cosh[x])/(I + Sinh[x])} -{(A + B*Sinh[x])/(I + Sinh[x])^2, x, 2, -(((I*A + B)*Cosh[x])/(3*(I + Sinh[x])^2)) - ((A + 2*I*B)*Cosh[x])/(3*(I + Sinh[x]))} -{(A + B*Sinh[x])/(I + Sinh[x])^3, x, 3, -(((I*A + B)*Cosh[x])/(5*(I + Sinh[x])^3)) - ((2*A + 3*I*B)*Cosh[x])/(15*(I + Sinh[x])^2) + ((2*I*A - 3*B)*Cosh[x])/(15*(I + Sinh[x]))} -{(A + B*Sinh[x])/(I + Sinh[x])^4, x, 4, -(((I*A + B)*Cosh[x])/(7*(I + Sinh[x])^4)) - ((3*A + 4*I*B)*Cosh[x])/(35*(I + Sinh[x])^3) + (2*(3*I*A - 4*B)*Cosh[x])/(105*(I + Sinh[x])^2) + (2*(3*A + 4*I*B)*Cosh[x])/(105*(I + Sinh[x]))} - -{(A + B*Sinh[x])/(I - Sinh[x]), x, 2, (-B)*x + ((I*A - B)*Cosh[x])/(I - Sinh[x])} -{(A + B*Sinh[x])/(I - Sinh[x])^2, x, 2, ((I*A - B)*Cosh[x])/(3*(I - Sinh[x])^2) + ((A - 2*I*B)*Cosh[x])/(3*(I - Sinh[x]))} -{(A + B*Sinh[x])/(I - Sinh[x])^3, x, 3, ((I*A - B)*Cosh[x])/(5*(I - Sinh[x])^3) + ((2*A - 3*I*B)*Cosh[x])/(15*(I - Sinh[x])^2) - ((2*I*A + 3*B)*Cosh[x])/(15*(I - Sinh[x]))} -{(A + B*Sinh[x])/(I - Sinh[x])^4, x, 4, ((I*A - B)*Cosh[x])/(7*(I - Sinh[x])^4) + ((3*A - 4*I*B)*Cosh[x])/(35*(I - Sinh[x])^3) - (2*(3*I*A + 4*B)*Cosh[x])/(105*(I - Sinh[x])^2) - (2*(3*A - 4*I*B)*Cosh[x])/(105*(I - Sinh[x]))} - - -{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(1/2), x, 3, (Sqrt[2]*(I*A - B)*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[x]])])/Sqrt[a] + (2*B*Cosh[x])/Sqrt[a + I*a*Sinh[x]]} -{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(3/2), x, 3, ((I*A + 3*B)*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[x]])])/(2*Sqrt[2]*a^(3/2)) + ((I*A - B)*Cosh[x])/(2*(a + I*a*Sinh[x])^(3/2))} -{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(5/2), x, 4, ((3*I*A + 5*B)*ArcTanh[(Sqrt[a]*Cosh[x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[x]])])/(16*Sqrt[2]*a^(5/2)) + ((I*A - B)*Cosh[x])/(4*(a + I*a*Sinh[x])^(5/2)) + ((3*I*A + 5*B)*Cosh[x])/(16*a*(a + I*a*Sinh[x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Sinh[c+d x]) (a+b Sinh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Sinh[x])*(a + b*Sinh[x])^(5/2), x, 8, (2/105)*(56*a*A*b + 15*a^2*B - 25*b^2*B)*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2/35)*(7*A*b + 5*a*B)*Cosh[x]*(a + b*Sinh[x])^(3/2) + (2/7)*B*Cosh[x]*(a + b*Sinh[x])^(5/2) + (2*I*(161*a^2*A*b - 63*A*b^3 + 15*a^3*B - 145*a*b^2*B)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(105*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*(56*a*A*b + 15*a^2*B - 25*b^2*B)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(105*b*Sqrt[a + b*Sinh[x]])} -{(A + B*Sinh[x])*(a + b*Sinh[x])^(3/2), x, 7, (2/15)*(5*A*b + 3*a*B)*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2/5)*B*Cosh[x]*(a + b*Sinh[x])^(3/2) + (2*I*(20*a*A*b + 3*a^2*B - 9*b^2*B)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(15*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*(5*A*b + 3*a*B)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(15*b*Sqrt[a + b*Sinh[x]])} -{(A + B*Sinh[x])*(a + b*Sinh[x])^(1/2), x, 6, (2/3)*B*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2*I*(3*A*b + a*B)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(3*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*B*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*b*Sqrt[a + b*Sinh[x]])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Sinh[x])/(a + b*Sinh[x]), x, 4, (B*x)/b - (2*(A*b - a*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])} -{(A + B*Sinh[x])/(a + b*Sinh[x])^2, x, 5, -((2*(a*A + b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - ((A*b - a*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x]))} -{(A + B*Sinh[x])/(a + b*Sinh[x])^3, x, 6, -(((2*a^2*A - A*b^2 + 3*a*b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((A*b - a*B)*Cosh[x])/(2*(a^2 + b^2)*(a + b*Sinh[x])^2) - ((3*a*A*b - a^2*B + 2*b^2*B)*Cosh[x])/(2*(a^2 + b^2)^2*(a + b*Sinh[x]))} -{(A + B*Sinh[x])/(a + b*Sinh[x])^4, x, 7, -(((2*a^3*A - 3*a*A*b^2 + 4*a^2*b*B - b^3*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) - ((A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^3) - ((5*a*A*b - 2*a^2*B + 3*b^2*B)*Cosh[x])/(6*(a^2 + b^2)^2*(a + b*Sinh[x])^2) - ((11*a^2*A*b - 4*A*b^3 - 2*a^3*B + 13*a*b^2*B)*Cosh[x])/(6*(a^2 + b^2)^3*(a + b*Sinh[x]))} - - -{(b*B/a + B*Sinh[x])/(a + b*Sinh[x]), x, 4, (B*x)/b + (2*(a^2 - b^2)*B*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b*Sqrt[a^2 + b^2])} -{(a*B/b + B*Sinh[x])/(a + b*Sinh[x]), x, 2, (B*x)/b} - -{(a - b*Sinh[x])/(b + a*Sinh[x])^2, x, 2, -(Cosh[x]/(b + a*Sinh[x]))} -{(2 - Sinh[x])/(2 + Sinh[x]), x, 2, -x + (4*x)/Sqrt[5] - (8*ArcTanh[Cosh[x]/(2 + Sqrt[5] + Sinh[x])])/Sqrt[5]} - - -{(A + B*Sinh[x])/(a + b*Sinh[x])^(1/2), x, 5, (2*I*B*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*(A*b - a*B)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])} -{(A + B*Sinh[x])/(a + b*Sinh[x])^(3/2), x, 6, -((2*(A*b - a*B)*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]])) + (2*I*(A*b - a*B)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(b*(a^2 + b^2)*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*B*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])} -{(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2), x, 7, -((2*(A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^(3/2))) - (2*(4*a*A*b - a^2*B + 3*b^2*B)*Cosh[x])/(3*(a^2 + b^2)^2*Sqrt[a + b*Sinh[x]]) + (2*I*(4*a*A*b - a^2*B + 3*b^2*B)*EllipticE[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(3*b*(a^2 + b^2)^2*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(A*b - a*B)*EllipticF[Pi/4 - (I*x)/2, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*b*(a^2 + b^2)*Sqrt[a + b*Sinh[x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sinh[a+b x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sinh[a+b x]^2)^n*) - - -{(a*Sinh[x]^2)^(5/2),x, 4, (8/15)*a^2*Coth[x]*Sqrt[a*Sinh[x]^2] - (4/15)*a*Coth[x]*(a*Sinh[x]^2)^(3/2) + (1/5)*Coth[x]*(a*Sinh[x]^2)^(5/2)} -{(a*Sinh[x]^2)^(3/2),x, 3, (-(2/3))*a*Coth[x]*Sqrt[a*Sinh[x]^2] + (1/3)*Coth[x]*(a*Sinh[x]^2)^(3/2)} -{(a*Sinh[x]^2)^(1/2), x, 2, Coth[x]*Sqrt[a*Sinh[x]^2]} -{1/(a*Sinh[x]^2)^(1/2), x, 2, -((ArcTanh[Cosh[x]]*Sinh[x])/Sqrt[a*Sinh[x]^2])} -{1/(a*Sinh[x]^2)^(3/2), x, 3, -(Coth[x]/(2*a*Sqrt[a*Sinh[x]^2])) + (ArcTanh[Cosh[x]]*Sinh[x])/(2*a*Sqrt[a*Sinh[x]^2])} -{1/(a*Sinh[x]^2)^(5/2), x, 4, -(Coth[x]/(4*a*(a*Sinh[x]^2)^(3/2))) + (3*Coth[x])/(8*a^2*Sqrt[a*Sinh[x]^2]) - (3*ArcTanh[Cosh[x]]*Sinh[x])/(8*a^2*Sqrt[a*Sinh[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sinh[a+b x]^3)^n*) - - -{(a*Sinh[x]^3)^(5/2),x, 7, (-(26/77))*a^2*Coth[x]*Sqrt[a*Sinh[x]^3] + (26/77)*I*a^2*Csch[x]^2*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]*Sqrt[a*Sinh[x]^3] + (78/385)*a^2*Cosh[x]*Sinh[x]*Sqrt[a*Sinh[x]^3] - (26/165)*a^2*Cosh[x]*Sinh[x]^3*Sqrt[a*Sinh[x]^3] + (2/15)*a^2*Cosh[x]*Sinh[x]^5*Sqrt[a*Sinh[x]^3]} -{(a*Sinh[x]^3)^(3/2),x, 5, (-(14/45))*a*Cosh[x]*Sqrt[a*Sinh[x]^3] + (14*I*a*Csch[x]*EllipticE[Pi/4 - (I*x)/2, 2]*Sqrt[a*Sinh[x]^3])/(15*Sqrt[I*Sinh[x]]) + (2/9)*a*Cosh[x]*Sinh[x]^2*Sqrt[a*Sinh[x]^3]} -{(a*Sinh[x]^3)^(1/2), x, 4, (2/3)*Coth[x]*Sqrt[a*Sinh[x]^3] - (2/3)*I*Csch[x]^2*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]*Sqrt[a*Sinh[x]^3]} -{1/(a*Sinh[x]^3)^(1/2), x, 4, -((2*Cosh[x]*Sinh[x])/Sqrt[a*Sinh[x]^3]) + (2*I*EllipticE[Pi/4 - (I*x)/2, 2]*Sinh[x]^2)/(Sqrt[I*Sinh[x]]*Sqrt[a*Sinh[x]^3])} -{1/(a*Sinh[x]^3)^(3/2),x, 5, (10*Cosh[x])/(21*a*Sqrt[a*Sinh[x]^3]) - (2*Coth[x]*Csch[x])/(7*a*Sqrt[a*Sinh[x]^3]) + (10*I*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]*Sinh[x])/(21*a*Sqrt[a*Sinh[x]^3])} -{1/(a*Sinh[x]^3)^(5/2),x, 7, -((154*Coth[x])/(585*a^2*Sqrt[a*Sinh[x]^3])) + (22*Coth[x]*Csch[x]^2)/(117*a^2*Sqrt[a*Sinh[x]^3]) - (2*Coth[x]*Csch[x]^4)/(13*a^2*Sqrt[a*Sinh[x]^3]) + (154*Cosh[x]*Sinh[x])/(195*a^2*Sqrt[a*Sinh[x]^3]) - (154*I*EllipticE[Pi/4 - (I*x)/2, 2]*Sinh[x]^2)/(195*a^2*Sqrt[I*Sinh[x]]*Sqrt[a*Sinh[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sinh[a+b x]^4)^n*) - - -{(a*Sinh[x]^4)^(5/2),x, 7, (63/256)*a^2*Coth[x]*Sqrt[a*Sinh[x]^4] - (63/256)*a^2*x*Csch[x]^2*Sqrt[a*Sinh[x]^4] - (21/128)*a^2*Cosh[x]*Sinh[x]*Sqrt[a*Sinh[x]^4] + (21/160)*a^2*Cosh[x]*Sinh[x]^3*Sqrt[a*Sinh[x]^4] - (9/80)*a^2*Cosh[x]*Sinh[x]^5*Sqrt[a*Sinh[x]^4] + (1/10)*a^2*Cosh[x]*Sinh[x]^7*Sqrt[a*Sinh[x]^4]} -{(a*Sinh[x]^4)^(3/2),x, 5, (5/16)*a*Coth[x]*Sqrt[a*Sinh[x]^4] - (5/16)*a*x*Csch[x]^2*Sqrt[a*Sinh[x]^4] - (5/24)*a*Cosh[x]*Sinh[x]*Sqrt[a*Sinh[x]^4] + (1/6)*a*Cosh[x]*Sinh[x]^3*Sqrt[a*Sinh[x]^4]} -{(a*Sinh[x]^4)^(1/2), x, 3, (1/2)*Coth[x]*Sqrt[a*Sinh[x]^4] - (1/2)*x*Csch[x]^2*Sqrt[a*Sinh[x]^4]} -{1/(a*Sinh[x]^4)^(1/2), x, 3, -((Cosh[x]*Sinh[x])/Sqrt[a*Sinh[x]^4])} -{1/(a*Sinh[x]^4)^(3/2),x, 3, (2*Cosh[x]^2*Coth[x])/(3*a*Sqrt[a*Sinh[x]^4]) - (Cosh[x]^2*Coth[x]^3)/(5*a*Sqrt[a*Sinh[x]^4]) - (Cosh[x]*Sinh[x])/(a*Sqrt[a*Sinh[x]^4])} -{1/(a*Sinh[x]^4)^(5/2),x, 3, (4*Cosh[x]^2*Coth[x])/(3*a^2*Sqrt[a*Sinh[x]^4]) - (6*Cosh[x]^2*Coth[x]^3)/(5*a^2*Sqrt[a*Sinh[x]^4]) + (4*Cosh[x]^2*Coth[x]^5)/(7*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]^2*Coth[x]^7)/(9*a^2*Sqrt[a*Sinh[x]^4]) - (Cosh[x]*Sinh[x])/(a^2*Sqrt[a*Sinh[x]^4])} - - -(* ::Subsection:: *) -(*Integrands of the form (c Sinh[a+b x]^m)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Sinh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Sinh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2=0*) - - -{Cosh[x]^8/(I + Sinh[x]), x, 5, -((5*I*x)/16) + Cosh[x]^7/7 - (5/16)*I*Cosh[x]*Sinh[x] - (5/24)*I*Cosh[x]^3*Sinh[x] - (1/6)*I*Cosh[x]^5*Sinh[x]} -{Cosh[x]^7/(I + Sinh[x]), x, 3, -(I - Sinh[x])^4 - (4/5)*I*(I - Sinh[x])^5 + (1/6)*(I - Sinh[x])^6} -{Cosh[x]^6/(I + Sinh[x]), x, 4, -((3*I*x)/8) + Cosh[x]^5/5 - (3/8)*I*Cosh[x]*Sinh[x] - (1/4)*I*Cosh[x]^3*Sinh[x]} -{Cosh[x]^5/(I + Sinh[x]), x, 3, (-I)*Sinh[x] + Sinh[x]^2/2 - (1/3)*I*Sinh[x]^3 + Sinh[x]^4/4} -{Cosh[x]^4/(I + Sinh[x]), x, 3, -((I*x)/2) + Cosh[x]^3/3 - (1/2)*I*Cosh[x]*Sinh[x]} -{Cosh[x]^3/(I + Sinh[x]), x, 2, (-I)*Sinh[x] + Sinh[x]^2/2} -{Cosh[x]^2/(I + Sinh[x]), x, 2, (-I)*x + Cosh[x]} -{Cosh[x]^1/(I + Sinh[x]), x, 2, Log[I + Sinh[x]]} -{Sech[x]^1/(I + Sinh[x]), x, 4, (-(1/2))*I*ArcTan[Sinh[x]] - I/(2*(I + Sinh[x]))} -{Sech[x]^2/(I + Sinh[x]), x, 3, -((I*Sech[x])/(3*(I + Sinh[x]))) - (2/3)*I*Tanh[x]} -{Sech[x]^3/(I + Sinh[x]), x, 4, (-(3/8))*I*ArcTan[Sinh[x]] + I/(8*(I - Sinh[x])) + 1/(8*(I + Sinh[x])^2) - I/(4*(I + Sinh[x]))} -{Sech[x]^4/(I + Sinh[x]), x, 3, -((I*Sech[x]^3)/(5*(I + Sinh[x]))) - (4/5)*I*Tanh[x] + (4/15)*I*Tanh[x]^3} -{Sech[x]^5/(I + Sinh[x]), x, 4, (-(5/16))*I*ArcTan[Sinh[x]] - 1/(32*(I - Sinh[x])^2) + I/(8*(I - Sinh[x])) + I/(24*(I + Sinh[x])^3) + 3/(32*(I + Sinh[x])^2) - (3*I)/(16*(I + Sinh[x]))} - - -{Cosh[x]^6/(I + Sinh[x])^2, x, 4, -((5*x)/8) - (5/12)*I*Cosh[x]^3 - (5/8)*Cosh[x]*Sinh[x] + Cosh[x]^5/(4*(I + Sinh[x]))} -{Cosh[x]^5/(I + Sinh[x])^2, x, 2, (-(1/3))*(I - Sinh[x])^3} -{Cosh[x]^4/(I + Sinh[x])^2, x, 3, -((3*x)/2) - (3/2)*I*Cosh[x] + Cosh[x]^3/(2*(I + Sinh[x]))} -{Cosh[x]^3/(I + Sinh[x])^2, x, 3, -2*I*Log[I + Sinh[x]] + Sinh[x]} -{Cosh[x]^2/(I + Sinh[x])^2, x, 2, x - (2*Cosh[x])/(I + Sinh[x])} -{Cosh[x]^1/(I + Sinh[x])^2, x, 2, -(1/(I + Sinh[x]))} -{Sech[x]^1/(I + Sinh[x])^2, x, 4, (-(1/4))*ArcTan[Sinh[x]] - I/(4*(I + Sinh[x])^2) - 1/(4*(I + Sinh[x]))} -{Sech[x]^2/(I + Sinh[x])^2, x, 4, -((I*Sech[x])/(5*(I + Sinh[x])^2)) - Sech[x]/(5*(I + Sinh[x])) - (2*Tanh[x])/5} -{Sech[x]^3/(I + Sinh[x])^2, x, 4, (-(1/4))*ArcTan[Sinh[x]] + 1/(16*(I - Sinh[x])) + 1/(12*(I + Sinh[x])^3) - I/(8*(I + Sinh[x])^2) - 3/(16*(I + Sinh[x]))} -{Sech[x]^4/(I + Sinh[x])^2, x, 4, -((I*Sech[x]^3)/(7*(I + Sinh[x])^2)) - Sech[x]^3/(7*(I + Sinh[x])) - (4*Tanh[x])/7 + (4*Tanh[x]^3)/21} - - -{Cosh[x]^3/(1 + I*Sinh[x])^3, x, 3, I*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])} -{Cosh[x]^2/(1 + I*Sinh[x])^3, x, 1, (I*Cosh[x]^3)/(3*(1 + I*Sinh[x])^3)} -{Cosh[x]^1/(1 + I*Sinh[x])^3, x, 2, I/(2*(1 + I*Sinh[x])^2)} - - -{Cosh[x]^3/(1 - I*Sinh[x])^3, x, 3, (-I)*Log[I + Sinh[x]] - (2*I)/(1 - I*Sinh[x])} -{Cosh[x]^2/(1 - I*Sinh[x])^3, x, 1, -((I*Cosh[x]^3)/(3*(1 - I*Sinh[x])^3))} -{Cosh[x]^1/(1 - I*Sinh[x])^3, x, 2, -(I/(2*(1 - I*Sinh[x])^2))} - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2!=0*) - - -{Cosh[x]^7/(a + b*Sinh[x]), x, 3, ((a^2 + b^2)^3*Log[a + b*Sinh[x]])/b^7 - (a*(a^4 + 3*a^2*b^2 + 3*b^4)*Sinh[x])/b^6 + ((a^4 + 3*a^2*b^2 + 3*b^4)*Sinh[x]^2)/(2*b^5) - (a*(a^2 + 3*b^2)*Sinh[x]^3)/(3*b^4) + ((a^2 + 3*b^2)*Sinh[x]^4)/(4*b^3) - (a*Sinh[x]^5)/(5*b^2) + Sinh[x]^6/(6*b)} -{Cosh[x]^6/(a + b*Sinh[x]), x, 7, -((a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*x)/(8*b^6)) - (2*(a^2 + b^2)^(5/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^6 + Cosh[x]^5/(5*b) + (Cosh[x]^3*(4*(a^2 + b^2) - 3*a*b*Sinh[x]))/(12*b^3) + (Cosh[x]*(8*(a^2 + b^2)^2 - a*b*(4*a^2 + 7*b^2)*Sinh[x]))/(8*b^5)} -{Cosh[x]^5/(a + b*Sinh[x]), x, 3, ((a^2 + b^2)^2*Log[a + b*Sinh[x]])/b^5 - (a*(a^2 + 2*b^2)*Sinh[x])/b^4 + ((a^2 + 2*b^2)*Sinh[x]^2)/(2*b^3) - (a*Sinh[x]^3)/(3*b^2) + Sinh[x]^4/(4*b)} -{Cosh[x]^4/(a + b*Sinh[x]), x, 6, -((a*(2*a^2 + 3*b^2)*x)/(2*b^4)) - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^4 + Cosh[x]^3/(3*b) + (Cosh[x]*(2*(a^2 + b^2) - a*b*Sinh[x]))/(2*b^3)} -{Cosh[x]^3/(a + b*Sinh[x]), x, 3, ((a^2 + b^2)*Log[a + b*Sinh[x]])/b^3 - (a*Sinh[x])/b^2 + Sinh[x]^2/(2*b)} -{Cosh[x]^2/(a + b*Sinh[x]), x, 5, -((a*x)/b^2) - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^2 + Cosh[x]/b} -{Cosh[x]^1/(a + b*Sinh[x]), x, 2, Log[a + b*Sinh[x]]/b} -{Sech[x]^1/(a + b*Sinh[x]), x, 6, (a*ArcTan[Sinh[x]])/(a^2 + b^2) - (b*Log[Cosh[x]])/(a^2 + b^2) + (b*Log[a + b*Sinh[x]])/(a^2 + b^2)} -{Sech[x]^2/(a + b*Sinh[x]), x, 5, -((2*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + (Sech[x]*(b + a*Sinh[x]))/(a^2 + b^2)} -{Sech[x]^3/(a + b*Sinh[x]), x, 7, (a*(a^2 + 3*b^2)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (b^3*Log[Cosh[x]])/(a^2 + b^2)^2 + (b^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + (Sech[x]^2*(b + a*Sinh[x]))/(2*(a^2 + b^2))} -{Sech[x]^4/(a + b*Sinh[x]), x, 6, -((2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (Sech[x]^3*(b + a*Sinh[x]))/(3*(a^2 + b^2)) + (Sech[x]*(3*b^3 + a*(2*a^2 + 5*b^2)*Sinh[x]))/(3*(a^2 + b^2)^2)} -{Sech[x]^5/(a + b*Sinh[x]), x, 8, (a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(8*(a^2 + b^2)^3) - (b^5*Log[Cosh[x]])/(a^2 + b^2)^3 + (b^5*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (Sech[x]^4*(b + a*Sinh[x]))/(4*(a^2 + b^2)) + (Sech[x]^2*(4*b^3 + a*(3*a^2 + 7*b^2)*Sinh[x]))/(8*(a^2 + b^2)^2)} -{Sech[x]^6/(a + b*Sinh[x]), x, 7, -((2*b^6*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (Sech[x]^5*(b + a*Sinh[x]))/(5*(a^2 + b^2)) + (Sech[x]^3*(5*b^3 + a*(4*a^2 + 9*b^2)*Sinh[x]))/(15*(a^2 + b^2)^2) + (Sech[x]*(15*b^5 + a*(8*a^4 + 26*a^2*b^2 + 33*b^4)*Sinh[x]))/(15*(a^2 + b^2)^3)} - - -{Cosh[x]^4/(a + b*Sinh[x])^2, x, 6, (3*(2*a^2 + b^2)*x)/(2*b^4) + (6*a*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^4 - (3*Cosh[x]*(2*a - b*Sinh[x]))/(2*b^3) - Cosh[x]^3/(b*(a + b*Sinh[x]))} -{Cosh[x]^3/(a + b*Sinh[x])^2, x, 3, -((2*a*Log[a + b*Sinh[x]])/b^3) + Sinh[x]/b^2 - (a^2 + b^2)/(b^3*(a + b*Sinh[x]))} -{Cosh[x]^2/(a + b*Sinh[x])^2, x, 5, x/b^2 + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Cosh[x]/(b*(a + b*Sinh[x]))} -{Cosh[x]^1/(a + b*Sinh[x])^2, x, 2, -(1/(b*(a + b*Sinh[x])))} -{Sech[x]^1/(a + b*Sinh[x])^2, x, 7, ((a^2 - b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^2 - (2*a*b*Log[Cosh[x]])/(a^2 + b^2)^2 + (2*a*b*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 - b/((a^2 + b^2)*(a + b*Sinh[x]))} -{Sech[x]^2/(a + b*Sinh[x])^2, x, 6, -((6*a*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - (b*Sech[x])/((a^2 + b^2)*(a + b*Sinh[x])) + (Sech[x]*(3*a*b + (a^2 - 2*b^2)*Sinh[x]))/(a^2 + b^2)^2} -{Sech[x]^3/(a + b*Sinh[x])^2, x, 7, ((a^4 + 6*a^2*b^2 - 3*b^4)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^3) - (4*a*b^3*Log[Cosh[x]])/(a^2 + b^2)^3 + (4*a*b^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (b*(a^2 - 3*b^2))/(2*(a^2 + b^2)^2*(a + b*Sinh[x])) + (Sech[x]^2*(b + a*Sinh[x]))/(2*(a^2 + b^2)*(a + b*Sinh[x]))} -{Sech[x]^4/(a + b*Sinh[x])^2, x, 7, -((10*a*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) - (b*Sech[x]^3)/((a^2 + b^2)*(a + b*Sinh[x])) + (Sech[x]^3*(5*a*b + (a^2 - 4*b^2)*Sinh[x]))/(3*(a^2 + b^2)^2) + (Sech[x]*(15*a*b^3 + (2*a^4 + 9*a^2*b^2 - 8*b^4)*Sinh[x]))/(3*(a^2 + b^2)^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Sinh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2=0*) - - -{Tanh[x]^4/(I + Sinh[x]), x, 6, -Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5 - (1/5)*I*Tanh[x]^5} -{Tanh[x]^3/(I + Sinh[x]), x, 6, (3/8)*ArcTan[Sinh[x]] - (3/8)*Sech[x]*Tanh[x] - (1/4)*Sech[x]*Tanh[x]^3 - (1/4)*I*Tanh[x]^4} -{Tanh[x]^2/(I + Sinh[x]), x, 5, -Sech[x] + Sech[x]^3/3 - (1/3)*I*Tanh[x]^3} -{Tanh[x]^1/(I + Sinh[x]), x, 5, (1/2)*ArcTan[Sinh[x]] + (1/2)*I*Sech[x]^2 - (1/2)*Sech[x]*Tanh[x]} -{Coth[x]^1/(I + Sinh[x]), x, 4, (-I)*Log[Sinh[x]] + I*Log[I + Sinh[x]]} -{Coth[x]^2/(I + Sinh[x]), x, 4, -ArcTanh[Cosh[x]] + I*Coth[x]} -{Coth[x]^3/(I + Sinh[x]), x, 5, -Csch[x] + (1/2)*I*Csch[x]^2} -{Coth[x]^4/(I + Sinh[x]), x, 5, (-(1/2))*ArcTanh[Cosh[x]] + (1/3)*I*Coth[x]^3 - (1/2)*Coth[x]*Csch[x]} -{Coth[x]^5/(I + Sinh[x]), x, 5, (1/4)*I*Coth[x]^4 - Csch[x] - Csch[x]^3/3} -{Coth[x]^6/(I + Sinh[x]), x, 6, (-(3/8))*ArcTanh[Cosh[x]] + (1/5)*I*Coth[x]^5 - (3/8)*Coth[x]*Csch[x] - (1/4)*Coth[x]^3*Csch[x]} - - -{Tanh[x]^4/(I + Sinh[x])^2, x, 10, (2/3)*I*Sech[x]^3 - (4/5)*I*Sech[x]^5 + (2/7)*I*Sech[x]^7 - Tanh[x]^5/5 + (2*Tanh[x]^7)/7} -{Tanh[x]^3/(I + Sinh[x])^2, x, 4, (-(1/8))*I*ArcTan[Sinh[x]] - I/(16*(I - Sinh[x])) + I/(12*(I + Sinh[x])^3) - 1/(4*(I + Sinh[x])^2) - (3*I)/(16*(I + Sinh[x]))} -{Tanh[x]^2/(I + Sinh[x])^2, x, 10, (2/3)*I*Sech[x]^3 - (2/5)*I*Sech[x]^5 - Tanh[x]^3/3 + (2*Tanh[x]^5)/5} -{Tanh[x]^1/(I + Sinh[x])^2, x, 4, (-(1/4))*I*ArcTan[Sinh[x]] - 1/(4*(I + Sinh[x])^2) - I/(4*(I + Sinh[x]))} -{Coth[x]^1/(I + Sinh[x])^2, x, 3, -Log[Sinh[x]] + Log[I + Sinh[x]] - I/(I + Sinh[x])} -{Coth[x]^2/(I + Sinh[x])^2, x, If[$VersionNumber<9, 9, 7], If[$VersionNumber<9, 2*I*ArcTanh[Cosh[x]] + 3*Coth[x] - (2*I*Coth[x])/(I + Sinh[x]), 2*I*ArcTanh[Cosh[x]] + Coth[x] + (2*I*Coth[x])/(I - Csch[x])]} -{Coth[x]^3/(I + Sinh[x])^2, x, 3, 2*I*Csch[x] + Csch[x]^2/2 + 2*Log[Sinh[x]] - 2*Log[I + Sinh[x]]} -{Coth[x]^4/(I + Sinh[x])^2, x, 9, (-I)*ArcTanh[Cosh[x]] - 2*Coth[x] + Coth[x]^3/3 + I*Coth[x]*Csch[x]} -{Coth[x]^5/(I + Sinh[x])^2, x, 3, (-(1/2))*Csch[x]^2 + (2/3)*I*Csch[x]^3 + Csch[x]^4/4} -{Coth[x]^6/(I + Sinh[x])^2, x, 11, (-(1/4))*I*ArcTanh[Cosh[x]] - (2*Coth[x]^3)/3 + Coth[x]^5/5 + (1/4)*I*Coth[x]*Csch[x] + (1/2)*I*Coth[x]*Csch[x]^3} - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2!=0*) - - -{Tanh[x]^4/(a + b*Sinh[x]), x, 13, -((2*a^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - (a^2*b*Sech[x])/(a^2 + b^2)^2 - (b*Sech[x])/(a^2 + b^2) + (b*Sech[x]^3)/(3*(a^2 + b^2)) - (a^3*Tanh[x])/(a^2 + b^2)^2 - (a*Tanh[x]^3)/(3*(a^2 + b^2))} -{Tanh[x]^3/(a + b*Sinh[x]), x, 7, (b*(3*a^2 + b^2)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) + (a^3*Log[Cosh[x]])/(a^2 + b^2)^2 - (a^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + (Sech[x]^2*(a - b*Sinh[x]))/(2*(a^2 + b^2))} -{Tanh[x]^2/(a + b*Sinh[x]), x, 8, -((2*a^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Sech[x])/(a^2 + b^2) - (a*Tanh[x])/(a^2 + b^2)} -{Tanh[x]^1/(a + b*Sinh[x]), x, 6, (b*ArcTan[Sinh[x]])/(a^2 + b^2) + (a*Log[Cosh[x]])/(a^2 + b^2) - (a*Log[a + b*Sinh[x]])/(a^2 + b^2)} -{Coth[x]^1/(a + b*Sinh[x]), x, 4, Log[Sinh[x]]/a - Log[a + b*Sinh[x]]/a} -{Coth[x]^2/(a + b*Sinh[x]), x, 7, (b*ArcTanh[Cosh[x]])/a^2 - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^2 - Coth[x]/a} -{Coth[x]^3/(a + b*Sinh[x]), x, 3, (b*Csch[x])/a^2 - Csch[x]^2/(2*a) + ((a^2 + b^2)*Log[Sinh[x]])/a^3 - ((a^2 + b^2)*Log[a + b*Sinh[x]])/a^3} -{Coth[x]^4/(a + b*Sinh[x]), x, 7, (b*(3*a^2 + 2*b^2)*ArcTanh[Cosh[x]])/(2*a^4) - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^4 - ((4*a^2 + 3*b^2)*Coth[x])/(3*a^3) + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch[x]^2)/(3*a)} - - -{Tanh[x]^4/(a + b*Sinh[x])^2, x, 16, -((2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (8*a^3*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (4*a^3*b*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (a^4*b*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2 - ((2*a^4 - 3*a^2*b^2 - b^4)*Tanh[x])/(a^2 + b^2)^3 - ((a^2 - b^2)*Tanh[x]^3)/(3*(a^2 + b^2)^2)} -{Tanh[x]^3/(a + b*Sinh[x])^2, x, 7, (a*b*(3*a^2 - b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^3 + (a^2*(a^2 - 3*b^2)*Log[Cosh[x]])/(a^2 + b^2)^3 - (a^2*(a^2 - 3*b^2)*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + a^3/((a^2 + b^2)^2*(a + b*Sinh[x])) + (Sech[x]^2*(a^2 - b^2 - 2*a*b*Sinh[x]))/(2*(a^2 + b^2)^2)} -{Tanh[x]^2/(a + b*Sinh[x])^2, x, 13, -((2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (4*a*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (2*a*b*Sech[x])/(a^2 + b^2)^2 - (a^2*b*Cosh[x])/((a^2 + b^2)^2*(a + b*Sinh[x])) - ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2} -{Tanh[x]^1/(a + b*Sinh[x])^2, x, 6, (2*a*b*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + ((a^2 - b^2)*Log[Cosh[x]])/(a^2 + b^2)^2 - ((a^2 - b^2)*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + a/((a^2 + b^2)*(a + b*Sinh[x]))} -{Coth[x]^1/(a + b*Sinh[x])^2, x, 3, Log[Sinh[x]]/a^2 - Log[a + b*Sinh[x]]/a^2 + 1/(a*(a + b*Sinh[x]))} -{Coth[x]^2/(a + b*Sinh[x])^2, x, 8, (2*b*ArcTanh[Cosh[x]])/a^3 - (2*(a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - (2*Coth[x])/a^2 + Coth[x]/(a*(a + b*Sinh[x]))} -{Coth[x]^3/(a + b*Sinh[x])^2, x, 3, (2*b*Csch[x])/a^3 - Csch[x]^2/(2*a^2) + ((a^2 + 3*b^2)*Log[Sinh[x]])/a^4 - ((a^2 + 3*b^2)*Log[a + b*Sinh[x]])/a^4 + (a^2 + b^2)/(a^3*(a + b*Sinh[x]))} -{Coth[x]^4/(a + b*Sinh[x])^2, x, 8, (b*(3*a^2 + 4*b^2)*ArcTanh[Cosh[x]])/a^5 - (2*Sqrt[a^2 + b^2]*(a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^5 - ((7*a^2 + 12*b^2)*Coth[x])/(3*a^4) + ((a^2 + 2*b^2)*Coth[x]*Csch[x])/(a^3*b) - ((3 + (4*b^2)/a^2)*Coth[x]*Csch[x])/(3*b*(a + b*Sinh[x])) - (Coth[x]*Csch[x]^2)/(3*a*(a + b*Sinh[x]))} - - -{Coth[x]*Sqrt[a + b*Sinh[x]], x, 4, -2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Sinh[x]]} -{Coth[x]/Sqrt[a + b*Sinh[x]], x, 3, -((2*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]])/Sqrt[a])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Hyper[c+d x]) (a+b Sinh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Cosh[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Cosh[x])/(a + b*Sinh[x]), x, 7, -((2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]) + (B*Log[a + b*Sinh[x]])/b} - -{(A + B*Cosh[x])/(I + Sinh[x]), x, 5, B*Log[I + Sinh[x]] - (A*Cosh[x])/(1 - I*Sinh[x])} -{(A + B*Cosh[x])/(I - Sinh[x]), x, 5, (-B)*Log[I - Sinh[x]] + (A*Cosh[x])/(1 + I*Sinh[x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Tanh[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Tanh[x])/(a + b*Sinh[x]), x, 11, (b*B*ArcTan[Sinh[x]])/(a^2 + b^2) - (2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (a*B*Log[Cosh[x]])/(a^2 + b^2) - (a*B*Log[a + b*Sinh[x]])/(a^2 + b^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Coth[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Coth[x])/(a + b*Sinh[x]), x, 9, -((2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]) + (B*Log[Sinh[x]])/a - (B*Log[a + b*Sinh[x]])/a} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Sech[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Sech[x])/(a + b*Sinh[x]), x, 12, (a*B*ArcTan[Sinh[x]])/(a^2 + b^2) - (2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - (b*B*Log[Cosh[x]])/(a^2 + b^2) + (b*B*Log[a + b*Sinh[x]])/(a^2 + b^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Csch[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Csch[x])/(a + b*Sinh[x]), x, 6, -((B*ArcTanh[Cosh[x]])/a) - (2*(a*A - b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Hyper[c+d x]+C Hyper[c+d x]) (a+b Sinh[c+d x])^n*) - - -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x]), x, 7, (C*x)/c - (2*(A*c - a*C)*ArcTanh[(c - a*Tanh[(1/2)*(d + e*x)])/Sqrt[a^2 + c^2]])/(c*Sqrt[a^2 + c^2]*e) + (B*Log[a + c*Sinh[d + e*x]])/(c*e)} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^2, x, 8, -((2*(a*A + c*C)*ArcTanh[(c - a*Tanh[(1/2)*(d + e*x)])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^(3/2)*e)) - B/(c*e*(a + c*Sinh[d + e*x])) - ((A*c - a*C)*Cosh[d + e*x])/((a^2 + c^2)*e*(a + c*Sinh[d + e*x]))} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^3, x, 9, -(((2*a^2*A - A*c^2 + 3*a*c*C)*ArcTanh[(c - a*Tanh[(1/2)*(d + e*x)])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^(5/2)*e)) - B/(2*c*e*(a + c*Sinh[d + e*x])^2) - ((A*c - a*C)*Cosh[d + e*x])/(2*(a^2 + c^2)*e*(a + c*Sinh[d + e*x])^2) - ((3*a*A*c - a^2*C + 2*c^2*C)*Cosh[d + e*x])/(2*(a^2 + c^2)^2*e*(a + c*Sinh[d + e*x]))} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^4, x, 10, -(((2*a^3*A - 3*a*A*c^2 + 4*a^2*c*C - c^3*C)*ArcTanh[(c - a*Tanh[(1/2)*(d + e*x)])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^(7/2)*e)) - B/(3*c*e*(a + c*Sinh[d + e*x])^3) - ((A*c - a*C)*Cosh[d + e*x])/(3*(a^2 + c^2)*e*(a + c*Sinh[d + e*x])^3) - ((5*a*A*c - 2*a^2*C + 3*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^2*e*(a + c*Sinh[d + e*x])^2) - ((11*a^2*A*c - 4*A*c^3 - 2*a^3*C + 13*a*c^2*C)*Cosh[d + e*x])/(6*(a^2 + c^2)^3*e*(a + c*Sinh[d + e*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cosh[c+d x]^n (a+b Sinh[c+d x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Sinh[c+d x]^2)^p*) - - -{x^3/(a + b*Sinh[x]^2), x, 13, (x^3*Log[1 + (b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) - (x^3*Log[1 + (b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) + (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) - (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) - (3*x*PolyLog[3, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) + (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(4*Sqrt[a]*Sqrt[a - b]) + (3*PolyLog[4, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(8*Sqrt[a]*Sqrt[a - b]) - (3*PolyLog[4, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(8*Sqrt[a]*Sqrt[a - b])} -{x^2/(a + b*Sinh[x]^2), x, 11, (x^2*Log[1 + (b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) - (x^2*Log[1 + (b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) + (x*PolyLog[2, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))])/(2*Sqrt[a]*Sqrt[a - b]) - (x*PolyLog[2, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))])/(2*Sqrt[a]*Sqrt[a - b]) - PolyLog[3, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))]/(4*Sqrt[a]*Sqrt[a - b]) + PolyLog[3, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))]/(4*Sqrt[a]*Sqrt[a - b])} -{x^1/(a + b*Sinh[x]^2), x, 9, (x*Log[1 + (b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) - (x*Log[1 + (b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b)])/(2*Sqrt[a]*Sqrt[a - b]) + PolyLog[2, -((b*E^(2*x))/(2*a - 2*Sqrt[a]*Sqrt[a - b] - b))]/(4*Sqrt[a]*Sqrt[a - b]) - PolyLog[2, -((b*E^(2*x))/(2*a + 2*Sqrt[a]*Sqrt[a - b] - b))]/(4*Sqrt[a]*Sqrt[a - b])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[c+d x]^n (a+b Sinh[c+d x]^2)^p*) - - - {(Cosh[a + b*x]*(-2 + Sinh[a + b*x]^2))/x, x, 13, (-(9/4))*Cosh[a]*CoshIntegral[b*x] + (1/4)*Cosh[3*a]*CoshIntegral[3*b*x] - (9/4)*Sinh[a]*SinhIntegral[b*x] + (1/4)*Sinh[3*a]*SinhIntegral[3*b*x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (1-a^2 x^2)^m Sinh[Sqrt[1-a x]/Sqrt[1+a x]]^n*) - - -{Sinh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2), x, 5, (3*SinhIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/(4*a) - SinhIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)} -{Sinh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x, 4, -(CoshIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(2*a)) + Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)} -{Sinh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1/(1 - a^2*x^2), x, 2, -(SinhIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)} -{1/(Sinh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1*(1 - a^2*x^2)), x, 1, Unintegrable[Csch[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/((1 - a*x)*(1 + a*x)), x]} -{1/(Sinh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2*(1 - a^2*x^2)), x, 1, Unintegrable[Csch[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/((1 - a*x)*(1 + a*x)), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Sinh[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sinh[a+b Log[c x^n]]^p*) - - -{Sinh[a + b*Log[c*x^n]], x, 1, -((b*n*x*Cosh[a + b*Log[c*x^n]])/(1 - b^2*n^2)) + (x*Sinh[a + b*Log[c*x^n]])/(1 - b^2*n^2)} -{Sinh[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x)/(1 - 4*b^2*n^2) - (2*b*n*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1 - 4*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^2)/(1 - 4*b^2*n^2)} -{Sinh[a + b*Log[c*x^n]]^3, x, 2, -((6*b^3*n^3*x*Cosh[a + b*Log[c*x^n]])/(1 - 10*b^2*n^2 + 9*b^4*n^4)) + (6*b^2*n^2*x*Sinh[a + b*Log[c*x^n]])/(1 - 10*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^2)/(1 - 9*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^3)/(1 - 9*b^2*n^2)} -{Sinh[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1 - 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Sinh[a + b*Log[c*x^n]]^2)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/(1 - 16*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^4)/(1 - 16*b^2*n^2)} - - -{x^m*Sinh[a + b*Log[c*x^n]], x, 1, -((b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2)) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2)} -{x^m*Sinh[a + b*Log[c*x^n]]^2, x, 2, (2*b^2*n^2*x^(1 + m))/((1 + m)*((1 + m)^2 - 4*b^2*n^2)) - (2*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - 4*b^2*n^2) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]]^2)/((1 + m)^2 - 4*b^2*n^2)} -{x^m*Sinh[a + b*Log[c*x^n]]^3, x, 2, -((6*b^3*n^3*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2))) + (6*b^2*(1 + m)*n^2*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2)) - (3*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^2)/((1 + m)^2 - 9*b^2*n^2) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]]^3)/((1 + m)^2 - 9*b^2*n^2)} -{x^m*Sinh[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x^(1 + m))/((1 + m)*((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) - (24*b^3*n^3*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) + (12*b^2*(1 + m)*n^2*x^(1 + m)*Sinh[a + b*Log[c*x^n]]^2)/(((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) - (4*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/((1 + m)^2 - 16*b^2*n^2) + ((1 + m)*x^(1 + m)*Sinh[a + b*Log[c*x^n]]^4)/((1 + m)^2 - 16*b^2*n^2)} - - -{Sinh[a + b*Log[c*x^n]]/x, x, 2, Cosh[a + b*Log[c*x^n]]/(b*n)} -{Sinh[a + b*Log[c*x^n]]^2/x, x, 3, -Log[x]/2 + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b*n)} -{Sinh[a + b*Log[c*x^n]]^3/x, x, 3, -(Cosh[a + b*Log[c*x^n]]/(b*n)) + Cosh[a + b*Log[c*x^n]]^3/(3*b*n)} -{Sinh[a + b*Log[c*x^n]]^4/x, x, 4, 3*Log[x]/8 - (3*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(8*b*n) + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/(4*b*n)} -{Sinh[a + b*Log[c*x^n]]^5/x, x, 3, Cosh[a + b*Log[c*x^n]]/(b*n) - (2*Cosh[a + b*Log[c*x^n]]^3)/(3*b*n) + Cosh[a + b*Log[c*x^n]]^5/(5*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sinh[a+b Log[c x^n]]^(p/2)*) - - -{Sinh[a + b*Log[c*x^n]]^(5/2)/x, x, 4, (6*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(5*b*n*Sqrt[I*Sinh[a + b*Log[c*x^n]]]) + (2*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^(3/2))/(5*b*n)} -{Sinh[a + b*Log[c*x^n]]^(3/2)/x, x, 4, (2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(3*b*n*Sqrt[Sinh[a + b*Log[c*x^n]]]) + (2*Cosh[a + b*Log[c*x^n]]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(3*b*n)} -{Sqrt[Sinh[a + b*Log[c*x^n]]]/x, x, 3, -((2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))} -{1/(x*Sqrt[Sinh[a + b*Log[c*x^n]]]), x, 3, -((2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[Sinh[a + b*Log[c*x^n]]]))} -{1/(x*Sinh[a + b*Log[c*x^n]]^(3/2)), x, 4, -((2*Cosh[a + b*Log[c*x^n]])/(b*n*Sqrt[Sinh[a + b*Log[c*x^n]]])) - (2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[I*Sinh[a + b*Log[c*x^n]]])} -{1/(x*Sinh[a + b*Log[c*x^n]]^(5/2)), x, 4, -((2*Cosh[a + b*Log[c*x^n]])/(3*b*n*Sinh[a + b*Log[c*x^n]]^(3/2))) + (2*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(3*b*n*Sqrt[Sinh[a + b*Log[c*x^n]]])} - - -{Sinh[a + 2/n*Log[c*x^n]]^(5/2), x, 8, (-(1/4))*x*Sinh[a + (2*Log[c*x^n])/n]^(5/2) - (5*x*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(E^(2*a)*(c*x^n)^(4/n)*(4*(1 - 1/(E^(2*a)*(c*x^n)^(4/n)))^2)) + (5*x*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(12*(1 - 1/(E^(2*a)*(c*x^n)^(4/n)))) - (5*x*ArcCsc[E^a*(c*x^n)^(2/n)]*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(E^(3*a)*(c*x^n)^(6/n)*(4*(1 - 1/(E^(2*a)*(c*x^n)^(4/n)))^(5/2)))} -{Sqrt[Sinh[a + 2/n*Log[c*x^n]]], x, 6, (1/2)*x*Sqrt[Sinh[a + (2*Log[c*x^n])/n]] + (x*ArcCsc[E^a*(c*x^n)^(2/n)]*Sqrt[Sinh[a + (2*Log[c*x^n])/n]])/(E^a*(c*x^n)^(2/n)*(2*Sqrt[1 - 1/(E^(2*a)*(c*x^n)^(4/n))]))} -{1/Sinh[a + 2/n*Log[c*x^n]]^(3/2), x, 3, -((x*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))/(2*Sinh[a + (2*Log[c*x^n])/n]^(3/2)))} -{1/Sinh[a + 2/n*Log[c*x^n]]^(7/2), x, 4, -((x*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))/(6*Sinh[a + (2*Log[c*x^n])/n]^(7/2))) + (x*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))/(E^(2*a)*(c*x^n)^(4/n)*(15*Sinh[a + (2*Log[c*x^n])/n]^(7/2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[(a+b x)/(c+d x)]^n*) - - -{Sinh[a/(c + d*x)], x, 4, -((a*CoshIntegral[a/(c + d*x)])/d) + ((c + d*x)*Sinh[a/(c + d*x)])/d} -{Sinh[a/(c + d*x)]^2, x, 5, ((c + d*x)*Sinh[a/(c + d*x)]^2)/d - (a*SinhIntegral[(2*a)/(c + d*x)])/d} -{Sinh[a/(c + d*x)]^3, x, 6, (3*a*CoshIntegral[a/(c + d*x)])/(4*d) - (3*a*CoshIntegral[(3*a)/(c + d*x)])/(4*d) + ((c + d*x)*Sinh[a/(c + d*x)]^3)/d} - - -{Sinh[b*x/(c + d*x)], x, 5, (b*c*Cosh[b/d]*CoshIntegral[(b*c)/(d*(c + d*x))])/d^2 + ((c + d*x)*Sinh[(b*x)/(c + d*x)])/d - (b*c*Sinh[b/d]*SinhIntegral[(b*c)/(d*(c + d*x))])/d^2} -{Sinh[b*x/(c + d*x)]^2, x, 6, (b*c*CoshIntegral[(2*b*c)/(d*(c + d*x))]*Sinh[(2*b)/d])/d^2 + ((c + d*x)*Sinh[(b*x)/(c + d*x)]^2)/d - (b*c*Cosh[(2*b)/d]*SinhIntegral[(2*b*c)/(d*(c + d*x))])/d^2} -{Sinh[b*x/(c + d*x)]^3, x, 9, -((3*b*c*Cosh[b/d]*CoshIntegral[(b*c)/(d*(c + d*x))])/(4*d^2)) + (3*b*c*Cosh[(3*b)/d]*CoshIntegral[(3*b*c)/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(b*x)/(c + d*x)]^3)/d + (3*b*c*Sinh[b/d]*SinhIntegral[(b*c)/(d*(c + d*x))])/(4*d^2) - (3*b*c*Sinh[(3*b)/d]*SinhIntegral[(3*b*c)/(d*(c + d*x))])/(4*d^2)} - - -{Sinh[(a + b*x)/(c + d*x)], x, 5, ((b*c - a*d)*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2 + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)])/d - ((b*c - a*d)*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2} -{Sinh[(a + b*x)/(c + d*x)]^2, x, 6, ((b*c - a*d)*CoshIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sinh[(2*b)/d])/d^2 + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)]^2)/d - ((b*c - a*d)*Cosh[(2*b)/d]*SinhIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2} -{Sinh[(a + b*x)/(c + d*x)]^3, x, 9, -((3*(b*c - a*d)*Cosh[b/d]*CoshIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2)) + (3*(b*c - a*d)*Cosh[(3*b)/d]*CoshIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*Sinh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sinh[(3*b)/d]*SinhIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2)} - - -{Sinh[e + f*(a + b*x)/(c + d*x)], x, 6, ((b*c - a*d)*f*Cosh[e + (b*f)/d]*CoshIntegral[((b*c - a*d)*f)/(d*(c + d*x))])/d^2 + ((c + d*x)*Sinh[(c*e + a*f + d*e*x + b*f*x)/(c + d*x)])/d - ((b*c - a*d)*f*Sinh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))])/d^2} -{Sinh[e + f*(a + b*x)/(c + d*x)]^2, x, 7, ((b*c - a*d)*f*CoshIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))]*Sinh[2*(e + (b*f)/d)])/d^2 + ((c + d*x)*Sinh[(c*e + a*f + d*e*x + b*f*x)/(c + d*x)]^2)/d - ((b*c - a*d)*f*Cosh[2*(e + (b*f)/d)]*SinhIntegral[(2*(b*c - a*d)*f)/(d*(c + d*x))])/d^2} -{Sinh[e + f*(a + b*x)/(c + d*x)]^3, x, 10, -((3*(b*c - a*d)*f*Cosh[e + (b*f)/d]*CoshIntegral[((b*c - a*d)*f)/(d*(c + d*x))])/(4*d^2)) + (3*(b*c - a*d)*f*Cosh[3*(e + (b*f)/d)]*CoshIntegral[(3*(b*c - a*d)*f)/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sinh[(c*e + a*f + d*e*x + b*f*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*f*Sinh[e + (b*f)/d]*SinhIntegral[((b*c - a*d)*f)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*f*Sinh[3*(e + (b*f)/d)]*SinhIntegral[(3*(b*c - a*d)*f)/(d*(c + d*x))])/(4*d^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Sinh[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Sinh[a+b x]^m*) - - -{E^(a + b*x)*Sinh[a + b*x]^4, x, 4, -(E^(-3*a - 3*b*x)/(48*b)) + E^(-a - b*x)/(4*b) + (3*E^(a + b*x))/(8*b) - E^(3*a + 3*b*x)/(12*b) + E^(5*a + 5*b*x)/(80*b)} -{E^(a + b*x)*Sinh[a + b*x]^3, x, 5, E^(-2*a - 2*b*x)/(16*b) - (3*E^(2*a + 2*b*x))/(16*b) + E^(4*a + 4*b*x)/(32*b) + (3*x)/8} -{E^(a + b*x)*Sinh[a + b*x]^2, x, 4, -(E^(-a - b*x)/(4*b)) - E^(a + b*x)/(2*b) + E^(3*a + 3*b*x)/(12*b)} -{E^(a + b*x)*Sinh[a + b*x]^1, x, 4, E^(2*a + 2*b*x)/(4*b) - x/2} -{E^(a + b*x)*Csch[a + b*x]^1, x, 3, Log[1 - E^(2*a + 2*b*x)]/b} -{E^(a + b*x)*Csch[a + b*x]^2, x, 4, (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Csch[a + b*x]^3, x, 3, -((2*E^(4*a + 4*b*x))/(b*(1 - E^(2*a + 2*b*x))^2))} -{E^(a + b*x)*Csch[a + b*x]^4, x, 6, (8*E^(3*a + 3*b*x))/(3*b*(1 - E^(2*a + 2*b*x))^3) - (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))^2) + E^(a + b*x)/(b*(1 - E^(2*a + 2*b*x))) + ArcTanh[E^(a + b*x)]/b} -{E^(a + b*x)*Csch[a + b*x]^5, x, 5, -(4/(b*(1 - E^(2*a + 2*b*x))^4)) + 32/(3*b*(1 - E^(2*a + 2*b*x))^3) - 8/(b*(1 - E^(2*a + 2*b*x))^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^x Sinh[n x]^m*) - - -{E^x*Sinh[2*x]^2, x, 4, -(1/12)/E^(3*x) - E^x/2 + E^(5*x)/20} -{E^x*Sinh[2*x], x, 4, 1/(E^x*2) + E^(3*x)/6} -{E^x*Csch[2*x], x, 5, ArcTan[E^x] - ArcTanh[E^x]} -{E^x*Csch[2*x]^2, x, 6, E^x/(1 - E^(4*x)) - ArcTan[E^x]/2 - ArcTanh[E^x]/2} - - -{E^x*Sinh[3*x]^2, x, 4, -(1/20)/E^(5*x) - E^x/2 + E^(7*x)/28} -{E^x*Sinh[3*x], x, 4, 1/(E^(2*x)*4) + E^(4*x)/8} -{E^x*Csch[3*x], x, 9, ArcTan[(1 + 2*E^(2*x))/Sqrt[3]]/Sqrt[3] + (1/3)*Log[1 - E^(2*x)] - (1/6)*Log[1 + E^(2*x) + E^(4*x)]} -{E^x*Csch[3*x]^2, x, 13, (2*E^x)/(3*(1 - E^(6*x))) + ArcTan[(1 - 2*E^x)/Sqrt[3]]/(3*Sqrt[3]) - ArcTan[(1 + 2*E^x)/Sqrt[3]]/(3*Sqrt[3]) - (2*ArcTanh[E^x])/9 + (1/18)*Log[1 - E^x + E^(2*x)] - (1/18)*Log[1 + E^x + E^(2*x)]} - - -{E^x*Sinh[4*x]^2, x, 4, -(1/28)/E^(7*x) - E^x/2 + E^(9*x)/36} -{E^x*Sinh[4*x], x, 4, 1/(E^(3*x)*6) + E^(5*x)/10} -{E^x*Csch[4*x], x, 15, (-(1/2))*ArcTan[E^x] - ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTanh[E^x]/2 - Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) + Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])} -{E^x*Csch[4*x]^2, x, 16, E^x/(2*(1 - E^(8*x))) - ArcTan[E^x]/8 + ArcTan[1 - Sqrt[2]*E^x]/(8*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(8*Sqrt[2]) - ArcTanh[E^x]/8 + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Sinh[d+e x]^m*) - - -{F^(c*(a + b*x))*Sinh[d + e*x]^3, x, 2, -((6*e^3*F^(c*(a + b*x))*Cosh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4)) + (6*b*c*e^2*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x]^2)/(9*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^3)/(9*e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sinh[d + e*x]^2, x, 2, -((2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2))) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^2)/(4*e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sinh[d + e*x]^1, x, 1, (e*F^(c*(a + b*x))*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Csch[d + e*x]^1, x, 1, -((2*E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[1, (e + b*c*Log[F])/(2*e), (1/2)*(3 + (b*c*Log[F])/e), E^(2*(d + e*x))])/(e + b*c*Log[F]))} -{F^(c*(a + b*x))*Csch[d + e*x]^2, x, 1, (4*E^(2*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/(2*e), 2 + (b*c*Log[F])/(2*e), E^(2*(d + e*x))])/(2*e + b*c*Log[F])} -{F^(c*(a + b*x))*Csch[d + e*x]^3, x, 2, -((F^(c*(a + b*x))*Coth[d + e*x]*Csch[d + e*x])/(2*e)) - (b*c*F^(c*(a + b*x))*Csch[d + e*x]*Log[F])/(2*e^2) + (E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[1, (e + b*c*Log[F])/(2*e), (1/2)*(3 + (b*c*Log[F])/e), E^(2*(d + e*x))]*(e - b*c*Log[F]))/e^2} -{F^(c*(a + b*x))*Csch[d + e*x]^4, x, 2, -((F^(c*(a + b*x))*Coth[d + e*x]*Csch[d + e*x]^2)/(3*e)) - (b*c*F^(c*(a + b*x))*Csch[d + e*x]^2*Log[F])/(6*e^2) - (2*E^(2*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/(2*e), 2 + (b*c*Log[F])/(2*e), E^(2*(d + e*x))]*(2*e - b*c*Log[F]))/(3*e^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Sinh[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Sinh[a*c + b*c*x]^2)^(5/2), x, 6, (Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(128*b*c*E^(4*c*(a + b*x))) - (5*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(64*b*c*E^(2*c*(a + b*x))) + (5*E^(2*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(32*b*c) - (5*E^(4*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(128*b*c) + (E^(6*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(192*b*c) - (5*x*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/16} -{E^(c*(a + b*x))*(Sinh[a*c + b*c*x]^2)^(3/2), x, 6, (Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(16*b*c*E^(2*c*(a + b*x))) - (3*E^(2*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(16*b*c) + (E^(4*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(32*b*c) + (3*x*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/8} -{E^(c*(a + b*x))*Sqrt[Sinh[a*c + b*c*x]^2], x, 5, (E^(2*c*(a + b*x))*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/(4*b*c) - (x*Csch[a*c + b*c*x]*Sqrt[Sinh[a*c + b*c*x]^2])/2} -{E^(c*(a + b*x))/Sqrt[Sinh[a*c + b*c*x]^2], x, 4, (Log[1 - E^(2*c*(a + b*x))]*Sinh[a*c + b*c*x])/(b*c*Sqrt[Sinh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Sinh[a*c + b*c*x]^2)^(3/2), x, 4, (-2*E^(4*c*(a + b*x))*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2*Sqrt[Sinh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Sinh[a*c + b*c*x]^2)^(5/2), x, 6, (-4*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4*Sqrt[Sinh[a*c + b*c*x]^2]) + (32*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3*Sqrt[Sinh[a*c + b*c*x]^2]) - (8*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2*Sqrt[Sinh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Sinh[a*c + b*c*x]^2)^(7/2), x, 6, -((32*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^6*Sqrt[Sinh[a*c + b*c*x]^2])) + (192*Sinh[a*c + b*c*x])/(5*b*c*(1 - E^(2*c*(a + b*x)))^5*Sqrt[Sinh[a*c + b*c*x]^2]) - (48*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4*Sqrt[Sinh[a*c + b*c*x]^2]) + (64*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3*Sqrt[Sinh[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x+c x^2) Sinh[d+e x+f x^2]^m*) - - -{E^x*Sinh[a + b*x], x, 1, -((b*E^x*Cosh[a + b*x])/(1 - b^2)) + (E^x*Sinh[a + b*x])/(1 - b^2)} -{E^x*Sinh[a + c*x^2], x, 6, (E^(-a + 1/(4*c))*Sqrt[Pi]*Erf[(1 - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - 1/(4*c))*Sqrt[Pi]*Erfi[(1 + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{E^x*Sinh[a + b*x + c*x^2], x, 6, (E^(-a + (1 - b)^2/(4*c))*Sqrt[Pi]*Erf[(1 - b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - (1 + b)^2/(4*c))*Sqrt[Pi]*Erfi[(1 + b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} - -{E^(x^2)*Sinh[a + b*x], x, 6, (-(1/4))*E^(-a - b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(-b + 2*x)] + (1/4)*E^(a - b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(b + 2*x)]} -{E^(x^2)*Sinh[a + c*x^2], x, 4, -((Sqrt[Pi]*Erfi[Sqrt[1 - c]*x])/(E^a*(4*Sqrt[1 - c]))) + (E^a*Sqrt[Pi]*Erfi[Sqrt[1 + c]*x])/(4*Sqrt[1 + c])} -{E^(x^2)*Sinh[a + b*x + c*x^2], x, 6, (E^(-a - b^2/(4*(1 - c)))*Sqrt[Pi]*Erfi[(b - 2*(1 - c)*x)/(2*Sqrt[1 - c])])/(4*Sqrt[1 - c]) + (E^(a - b^2/(4*(1 + c)))*Sqrt[Pi]*Erfi[(b + 2*(1 + c)*x)/(2*Sqrt[1 + c])])/(4*Sqrt[1 + c])} - - -{f^(a + b*x)*Sinh[d + f*x^2], x, 8, -(E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/4} -{f^(a + b*x)*Sinh[d + f*x^2]^2, x, 9, (E^(-2*d + (b^2*Log[f]^2)/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erf[(4*f*x - b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + (E^(2*d - (b^2*Log[f]^2)/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erfi[(4*f*x + b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 - f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Sinh[d + f*x^2]^3, x, 14, (3*E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*d + (b^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 - (3*E^(d - (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (b^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16} - -{f^(a + b*x)*Sinh[d + e*x + f*x^2], x, 8, -(E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/4} -{f^(a + b*x)*Sinh[d + e*x + f*x^2]^2, x, 9, (E^(-2*d + (2*e - b*Log[f])^2/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erf[(2*e + 4*f*x - b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + (E^(2*d - (2*e + b*Log[f])^2/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erfi[(2*e + 4*f*x + b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 - f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Sinh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*d + (3*e - b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(3*e + 6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 - (3*E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (3*e + b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16} - - -{f^(a + c*x^2)*Sinh[d + e*x], x, 8, If[$VersionNumber>=8, (E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), -((E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]])) + (E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Sinh[d + e*x]^2, x, 9, If[$VersionNumber>=8, -(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), -((f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]])) + (E^(-2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Sinh[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^(-3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + c*x^2)*Sinh[d + f*x^2], x, 6, -(f^a*Sqrt[Pi]*Erf[x*Sqrt[f - c*Log[f]]])/(4*E^d*Sqrt[f - c*Log[f]]) + (E^d*f^a*Sqrt[Pi]*Erfi[x*Sqrt[f + c*Log[f]]])/(4*Sqrt[f + c*Log[f]])} -{f^(a + c*x^2)*Sinh[d + f*x^2]^2, x, 7, -(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[2*f - c*Log[f]]])/(8*E^(2*d)*Sqrt[2*f - c*Log[f]]) + (E^(2*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + c*x^2)*Sinh[d + f*x^2]^3, x, 10, (3*f^a*Sqrt[Pi]*Erf[x*Sqrt[f - c*Log[f]]])/(16*E^d*Sqrt[f - c*Log[f]]) - (f^a*Sqrt[Pi]*Erf[x*Sqrt[3*f - c*Log[f]]])/(16*E^(3*d)*Sqrt[3*f - c*Log[f]]) - (3*E^d*f^a*Sqrt[Pi]*Erfi[x*Sqrt[f + c*Log[f]]])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[3*f + c*Log[f]]])/(16*Sqrt[3*f + c*Log[f]])} - -{f^(a + c*x^2)*Sinh[d + e*x + f*x^2], x, 8, -(E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + c*x^2)*Sinh[d + e*x + f*x^2]^2, x, 9, -(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d + e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + x*(2*f - c*Log[f]))/Sqrt[2*f - c*Log[f]]])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - e^2/(2*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + x*(2*f + c*Log[f]))/Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + c*x^2)*Sinh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) - (E^(-3*d + (9*e^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e + 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) - (3*E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (9*e^2)/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - - -{f^(a + b*x + c*x^2)*Sinh[d + e*x], x, 8, If[$VersionNumber>=8, (E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), -((E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]])) + (E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Sinh[d + e*x]^2, x, 10, If[$VersionNumber>=8, -(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d - (2*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), -((f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]])) + (E^(-2*d - (2*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((2*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Sinh[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*d - (3*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^(-3*d - (3*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) - (3*E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + b*x + c*x^2)*Sinh[d + f*x^2], x, 8, (E^(-d + (b^2*Log[f]^2)/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - (b^2*Log[f]^2)/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sinh[d + f*x^2]^2, x, 10, -(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d + (b^2*Log[f]^2)/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log[f]])])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (b^2*Log[f]^2)/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sinh[d + f*x^2]^3, x, 14, (-3*E^(-d + (b^2*Log[f]^2)/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) + (E^(-3*d + (b^2*Log[f]^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) - (3*E^(d - (b^2*Log[f]^2)/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (b^2*Log[f]^2)/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Sinh[d + e*x + f*x^2], x, 8, -(E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Log[f] + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sinh[d + e*x + f*x^2]^2, x, 10, -(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d + (2*e - b*Log[f])^2/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(2*e - b*Log[f] + 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log[f]])])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Sinh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Log[f] + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) - (E^(-3*d + (3*e - b*Log[f])^2/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e - b*Log[f] + 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) - (3*E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving hyperbolic sines*) - - -{(x + Sinh[x])^2, x, 6, -(x/2) + x^3/3 + 2*x*Cosh[x] - 2*Sinh[x] + (1/2)*Cosh[x]*Sinh[x]} -{(x + Sinh[x])^3, x, 9, -((3*x^2)/4) + x^4/4 + 5*Cosh[x] + 3*x^2*Cosh[x] + Cosh[x]^3/3 - 6*x*Sinh[x] + (3/2)*x*Cosh[x]*Sinh[x] - (3*Sinh[x]^2)/4} - - -{Sinh[a + b*x]/(c + d*x^2), x, 8, -((CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sinh[a - (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d])) + (CoshIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x]*Sinh[a + (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a + (b*Sqrt[-c])/Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])} -{Sinh[a + b*x]/(c + d*x + e*x^2), x, 8, (CoshIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sinh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] - (CoshIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sinh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] + (Cosh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Cosh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.7 hyper^m (a+b sinh^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.7 hyper^m (a+b sinh^n)^p.m deleted file mode 100644 index b41d11b..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.1 Hyperbolic sine/6.1.7 hyper^m (a+b sinh^n)^p.m +++ /dev/null @@ -1,882 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2), x, 4, ((6*a - 5*b)*x)/16 - ((6*a - 5*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - 5*b)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2), x, 3, -(((a - b)*Cosh[c + d*x])/d) + ((a - 2*b)*Cosh[c + d*x]^3)/(3*d) + (b*Cosh[c + d*x]^5)/(5*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2), x, 3, -((4*a - 3*b)*x)/8 + ((4*a - 3*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^2), x, 2, ((a - b)*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^2), x, 3, a*x - (b*x)/2 + (b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^2), x, 2, -((a*ArcTanh[Cosh[c + d*x]])/d) + (b*Cosh[c + d*x])/d} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2), x, 2, b*x - (a*Coth[c + d*x])/d} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2), x, 2, ((a - 2*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2), x, 3, ((2*a - 3*b)*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d)} - - -{Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2, x, 6, (1/128)*(48*a^2 - 80*a*b + 35*b^2)*x - ((80*a^2 - 176*a*b + 93*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a^2 - 208*a*b + 139*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((16*a - 13*b)*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b^2*Cosh[c + d*x]^3*Sinh[c + d*x]^5)/(8*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2, x, 3, -(((a - b)^2*Cosh[c + d*x])/d) + ((a - 3*b)*(a - b)*Cosh[c + d*x]^3)/(3*d) + ((2*a - 3*b)*b*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2, x, 2, (-(1/16))*(8*a^2 - 12*a*b + 5*b^2)*x + ((8*a^2 - 20*a*b + 11*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((4*a - 3*b)*b*Cosh[c + d*x]^3*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]^3*Sinh[c + d*x]^3)/(6*d), (-(1/16))*(8*a^2 - 12*a*b + 5*b^2)*x + ((16*a^2 - 36*a*b + 15*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(48*d) + ((4*a - 5*b)*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(6*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^2, x, 3, ((a - b)^2*Cosh[c + d*x])/d + (2*(a - b)*b*Cosh[c + d*x]^3)/(3*d) + (b^2*Cosh[c + d*x]^5)/(5*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^2)^2, x, 1, (1/8)*(8*a^2 - 8*a*b + 3*b^2)*x + ((8*a - 3*b)*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^2, x, 4, -((a^2*ArcTanh[Cosh[c + d*x]])/d) + ((2*a - b)*b*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2, x, 4, (1/2)*(4*a - b)*b*x - (a^2*Coth[c + d*x])/d + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d), (1/2)*(4*a - b)*b*x - (a^2*Cosh[c + d*x]^2*Coth[c + d*x])/d + ((2*a^2 + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2, x, 5, (a*(a - 4*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + (b^2*Cosh[c + d*x])/d - (a^2*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2, x, 4, b^2*x + (a*(a - 2*b)*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d)} - - -{Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3, x, 7, (3/256)*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*x - ((576*a^3 - 1744*a^2*b + 1678*a*b^2 - 525*b^3)*Cosh[c + d*x]*Sinh[c + d*x])/(1280*d) + ((48*a^3 - 272*a^2*b + 314*a*b^2 - 105*b^3)*Cosh[c + d*x]^3*Sinh[c + d*x])/(640*d) + (3*(2*a - 3*b)*Cosh[c + d*x]^5*Sinh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^2)/(80*d) + (Cosh[c + d*x]^7*Sinh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^3)/(10*d) - (b*Cosh[c + d*x]^3*Sinh[c + d*x]^3*(a*(14*a - 9*b) - (22*a - 21*b)*(a - b)*Tanh[c + d*x]^2))/(160*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3, x, 3, -(((a - b)^3*Cosh[c + d*x])/d) + ((a - 4*b)*(a - b)^2*Cosh[c + d*x]^3)/(3*d) + (3*(a - 2*b)*(a - b)*b*Cosh[c + d*x]^5)/(5*d) + ((3*a - 4*b)*b^2*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3, x, 3, (-(1/128))*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*x + ((96*a^3 - 376*a^2*b + 360*a*b^2 - 105*b^3)*Cosh[c + d*x]*Sinh[c + d*x])/(384*d) + (b*(24*a^2 - 64*a*b + 35*b^2)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) + ((6*a - 7*b)*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(48*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^3)/(8*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^3, x, 3, ((a - b)^3*Cosh[c + d*x])/d + ((a - b)^2*b*Cosh[c + d*x]^3)/d + (3*(a - b)*b^2*Cosh[c + d*x]^5)/(5*d) + (b^3*Cosh[c + d*x]^7)/(7*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^2)^3, x, 2, (1/16)*(2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*x + (b*(64*a^2 - 54*a*b + 15*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(48*d) + (5*(2*a - b)*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(6*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^3, x, 4, -((a^3*ArcTanh[Cosh[c + d*x]])/d) + (b*(3*a^2 - 3*a*b + b^2)*Cosh[c + d*x])/d + ((3*a - 2*b)*b^2*Cosh[c + d*x]^3)/(3*d) + (b^3*Cosh[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3, x, 5, (3/8)*b*(8*a^2 - 4*a*b + b^2)*x - (a*(2*a + b)*(4*a + b)*Coth[c + d*x])/(8*d) + (b*Cosh[c + d*x]^4*Coth[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(4*d) + (b*Cosh[c + d*x]^2*Coth[c + d*x]*(a*(4*a + b) - (4*a - 3*b)*(a - b)*Tanh[c + d*x]^2))/(8*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3, x, 5, (a^2*(a - 6*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + ((3*a - b)*b^2*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3, x, 5, (1/2)*(6*a - b)*b^2*x + (a*(2*a^2 - 5*a*b - 2*b^2)*Coth[c + d*x])/(2*d) - (a^2*(2*a + 3*b)*Coth[c + d*x]^3)/(6*d) + (b*Cosh[c + d*x]^2*Coth[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^2)/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[c + d*x]^7/(a + b*Sinh[c + d*x]^2), x, 4, -((a^3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(7/2)*d)) + ((a^2 + a*b + b^2)*Cosh[c + d*x])/(b^3*d) - ((a + 2*b)*Cosh[c + d*x]^3)/(3*b^2*d) + Cosh[c + d*x]^5/(5*b*d)} -{Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^2), x, 6, ((8*a^2 + 4*a*b + 3*b^2)*x)/(8*b^3) - (a^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a - b]*b^3*d) - ((4*a + 3*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*b*d)} -{Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^2), x, 4, (a^2*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(5/2)*d) - ((a + b)*Cosh[c + d*x])/(b^2*d) + Cosh[c + d*x]^3/(3*b*d)} -{Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2), x, 5, -(((2*a + b)*x)/(2*b^2)) + (a^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a - b]*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2), x, 3, -((a*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(3/2)*d)) + Cosh[c + d*x]/(b*d)} -{Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2), x, 3, x/b - (Sqrt[a]*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a - b]*b*d)} -{Sinh[c + d*x]^1/(a + b*Sinh[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]]/(Sqrt[a - b]*Sqrt[b]*d)} -{Sinh[c + d*x]^0/(a + b*Sinh[c + d*x]^2), x, 2, ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)} -{Csch[c + d*x]^1/(a + b*Sinh[c + d*x]^2), x, 4, -((Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a*Sqrt[a - b]*d)) - ArcTanh[Cosh[c + d*x]]/(a*d)} -{Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2), x, 3, -((b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*Sqrt[a - b]*d)) - Coth[c + d*x]/(a*d)} -{Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2), x, 5, (b^(3/2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a^2*Sqrt[a - b]*d) + ((a + 2*b)*ArcTanh[Cosh[c + d*x]])/(2*a^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)} -{Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[a - b]*d) + ((a + b)*Coth[c + d*x])/(a^2*d) - Coth[c + d*x]^3/(3*a*d)} -{Csch[c + d*x]^5/(a + b*Sinh[c + d*x]^2), x, 6, -((b^(5/2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a^3*Sqrt[a - b]*d)) - ((3*a^2 + 4*a*b + 8*b^2)*ArcTanh[Cosh[c + d*x]])/(8*a^3*d) + ((3*a + 4*b)*Coth[c + d*x]*Csch[c + d*x])/(8*a^2*d) - (Coth[c + d*x]*Csch[c + d*x]^3)/(4*a*d)} -{Csch[c + d*x]^6/(a + b*Sinh[c + d*x]^2), x, 4, -((b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(a^(7/2)*Sqrt[a - b]*d)) - ((a^2 + a*b + b^2)*Coth[c + d*x])/(a^3*d) + ((2*a + b)*Coth[c + d*x]^3)/(3*a^2*d) - Coth[c + d*x]^5/(5*a*d)} - - -{Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2, x, 5, x/b^2 - (Sqrt[a]*(2*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*(a - b)^(3/2)*b^2*d) - (a*Tanh[c + d*x])/(2*(a - b)*b*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2, x, 3, ((a - 2*b)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(2*(a - b)^(3/2)*b^(3/2)*d) - (a*Cosh[c + d*x])/(2*(a - b)*b*d*(a - b + b*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2, x, 4, -(ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(2*Sqrt[a]*(a - b)^(3/2)*d)) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2))} -{Sinh[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]]/(2*(a - b)^(3/2)*Sqrt[b]*d) + Cosh[c + d*x]/(2*(a - b)*d*(a - b + b*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^0/(a + b*Sinh[c + d*x]^2)^2, x, 4, ((2*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(3/2)*d) - (b*Cosh[c + d*x]*Sinh[c + d*x])/(2*a*(a - b)*d*(a + b*Sinh[c + d*x]^2))} -{Csch[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^2, x, 5, -((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(2*a^2*(a - b)^(3/2)*d) - ArcTanh[Cosh[c + d*x]]/(a^2*d) - (b*Cosh[c + d*x])/(2*a*(a - b)*d*(a - b + b*Cosh[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2, x, 4, -(((4*a - 3*b)*b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a - b)^(3/2)*d)) - Coth[c + d*x]/(a*d*(a - (a - b)*Tanh[c + d*x]^2)) + ((2*a^2 - 4*a*b + 3*b^2)*Tanh[c + d*x])/(2*a^2*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2, x, 6, ((5*a - 4*b)*b^(3/2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(2*a^3*(a - b)^(3/2)*d) + ((a + 4*b)*ArcTanh[Cosh[c + d*x]])/(2*a^3*d) - ((a - 2*b)*b*Cosh[c + d*x])/(2*a^2*(a - b)*d*(a - b + b*Cosh[c + d*x]^2)) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a - b + b*Cosh[c + d*x]^2))} -{Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2, x, 5, ((6*a - 5*b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^(3/2)*d) + ((2*a^2 + a*b - 5*b^2)*Coth[c + d*x])/(2*a^3*(a - b)*d) - ((2*a - 5*b)*Coth[c + d*x]^3)/(6*a^2*(a - b)*d) - (b*Csch[c + d*x]^3*Sech[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))} - - -{Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3, x, 4, (3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^(5/2)*d) + Tanh[c + d*x]^3/(4*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - (3*Tanh[c + d*x])/(8*(a - b)^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3, x, 4, ((a - 4*b)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*(a - b)^(5/2)*b^(3/2)*d) - (a*Cosh[c + d*x])/(4*(a - b)*b*d*(a - b + b*Cosh[c + d*x]^2)^2) + ((a - 4*b)*Cosh[c + d*x])/(8*(a - b)^2*b*d*(a - b + b*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3, x, 5, -(((4*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^(5/2)*d)) + (Cosh[c + d*x]*Sinh[c + d*x])/(4*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) + ((2*a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^2))} -{Sinh[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*(a - b)^(5/2)*Sqrt[b]*d) + Cosh[c + d*x]/(4*(a - b)*d*(a - b + b*Cosh[c + d*x]^2)^2) + (3*Cosh[c + d*x])/(8*(a - b)^2*d*(a - b + b*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^0/(a + b*Sinh[c + d*x]^2)^3, x, 5, ((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(5/2)*d) - (b*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - (3*(2*a - b)*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Sinh[c + d*x]^2))} -{Csch[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^3, x, 6, -(Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*a^3*(a - b)^(5/2)*d) - ArcTanh[Cosh[c + d*x]]/(a^3*d) - (b*Cosh[c + d*x])/(4*a*(a - b)*d*(a - b + b*Cosh[c + d*x]^2)^2) - ((7*a - 4*b)*b*Cosh[c + d*x])/(8*a^2*(a - b)^2*d*(a - b + b*Cosh[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3, x, 5, -((3*b*(8*a^2 - 12*a*b + 5*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^(5/2)*d)) - ((4*a - 5*b)*(2*a - 3*b)*Coth[c + d*x])/(8*a^3*(a - b)^2*d) - (b*Csch[c + d*x]*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - (b*Coth[c + d*x]*(4*a - 5*b - (4*a - b)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3, x, 7, (b^(3/2)*(35*a^2 - 56*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(5/2)*d) + ((a + 6*b)*ArcTanh[Cosh[c + d*x]])/(2*a^4*d) - ((2*a - 3*b)*b*Cosh[c + d*x])/(4*a^2*(a - b)*d*(a - b + b*Cosh[c + d*x]^2)^2) - ((a - 4*b)*(4*a - 3*b)*b*Cosh[c + d*x])/(8*a^3*(a - b)^2*d*(a - b + b*Cosh[c + d*x]^2)) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a - b + b*Cosh[c + d*x]^2)^2)} -{Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3, x, 6, (b^2*(48*a^2 - 80*a*b + 35*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^(5/2)*d) + ((8*a^3 - 4*a^2*b - 45*a*b^2 + 35*b^3)*Coth[c + d*x])/(8*a^4*(a - b)^2*d) - ((8*a^2 - 52*a*b + 35*b^2)*Coth[c + d*x]^3)/(24*a^3*(a - b)^2*d) - (b*Csch[c + d*x]^3*Sech[c + d*x]^3)/(4*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - ((10*a - 7*b)*b*Csch[c + d*x]^3*Sech[c + d*x])/(8*a^2*(a - b)^2*d*(a - (a - b)*Tanh[c + d*x]^2))} - - -{1/(1 + Sinh[x]^2), x, 3, Tanh[x]} -{1/(1 + Sinh[x]^2)^2, x, 3, Tanh[x] - Tanh[x]^3/3} -{1/(1 + Sinh[x]^2)^3, x, 3, Tanh[x] - (2*Tanh[x]^3)/3 + Tanh[x]^5/5} - - -{1/(1 - Sinh[x]^2), x, 2, ArcTanh[Sqrt[2]*Tanh[x]]/Sqrt[2]} -{1/(1 - Sinh[x]^2)^2, x, 4, (3*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) + (Cosh[x]*Sinh[x])/(4*(1 - Sinh[x]^2))} -{1/(1 - Sinh[x]^2)^3, x, 5, (19*ArcTanh[Sqrt[2]*Tanh[x]])/(32*Sqrt[2]) + (Cosh[x]*Sinh[x])/(8*(1 - Sinh[x]^2)^2) + (9*Cosh[x]*Sinh[x])/(32*(1 - Sinh[x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((a - b)*(a + 3*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a + 3*b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(8*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/(4*b*f)} -{Sinh[e + f*x]^1*Sqrt[a + b*Sinh[e + f*x]^2], x, 4, ((a - b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*Sqrt[b]*f) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(2*f)} -{Csch[e + f*x]^1*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/f} -{Csch[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2], x, 4, ((a - b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*Sqrt[a]*f) - (Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(2*f)} -{Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((a - b)*(3*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*a^(3/2)*f) + ((3*a + b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e + f*x]*Csch[e + f*x]^3)/(4*a*f)} - -{Sinh[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2], x, 7, ((a - 4*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b*f) + (Cosh[e + f*x]*Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2])/(5*f) + ((2*a^2 + 3*a*b - 8*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((2*a^2 + 3*a*b - 8*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(15*b^2*f)} -{Sinh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, (Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - ((I/3)*(a - 2*b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*a*(a - b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sinh[e + f*x]^0*Sqrt[a + b*Sinh[e + f*x]^2], x, 2, ((-I)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2], x, 7, -((Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/f) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f} -{Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2], x, 7, ((2*a - b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) - (Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) + ((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((2*a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a*f)} - - -{Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -((a - b)^2*(a + 5*b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*b^(3/2)*f) - ((a - b)*(a + 5*b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(16*b*f) - ((a + 5*b)*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/(24*b*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(5/2))/(6*b*f)} -{Sinh[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (3*(a - b)^2*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*(a - b)*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(8*f) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/(4*f)} -{Csch[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, -((a^(3/2)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/f) + ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*f) + (b*Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(2*f)} -{Csch[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, (Sqrt[a]*(a - 3*b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/f - (a*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(2*f)} -{Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (-3*(a - b)^2*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*(a - b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e + f*x]*Csch[e + f*x]^3)/(4*f)} -{Csch[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, ((a - b)^2*(5*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(16*a^(3/2)*f) - ((a - b)*(5*a + b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(16*a*f) + ((5*a + b)*(a - b + b*Cosh[e + f*x]^2)^(3/2)*Coth[e + f*x]*Csch[e + f*x]^3)/(24*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(5/2)*Coth[e + f*x]*Csch[e + f*x]^5)/(6*a*f)} - -{Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2), x, 8, ((a^2 - 11*a*b + 8*b^2)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b*f) + (2*(4*a - 3*b)*Cosh[e + f*x]*Sinh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2])/(35*f) + (b*Cosh[e + f*x]*Sinh[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2])/(7*f) + (2*(a - 2*b)*(a^2 + 4*a*b - 4*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a^2 - 11*a*b + 8*b^2)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a - 2*b)*(a^2 + 4*a*b - 4*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(35*b^2*f)} -{Sinh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, ((3*a - 4*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*f) + (Cosh[e + f*x]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(5*f) - ((I/15)*(3*a^2 - 13*a*b + 8*b^2)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/15)*a*(3*a - 4*b)*(a - b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sinh[e + f*x]^0*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, (b*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*a*(a - b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -((a*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/f) - ((a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f} -{Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, (2*(a - 2*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (a*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) + (2*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 3*b)*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)} - - -{(a + b*Sinh[c + d*x]^2)^(5/2), x, 7, (4*(2*a - b)*b*Cosh[c + d*x]*Sinh[c + d*x]*Sqrt[a + b*Sinh[c + d*x]^2])/(15*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^(3/2))/(5*d) - ((I/15)*(23*a^2 - 23*a*b + 8*b^2)*EllipticE[I*c + I*d*x, b/a]*Sqrt[a + b*Sinh[c + d*x]^2])/(d*Sqrt[1 + (b*Sinh[c + d*x]^2)/a]) + (((4*I)/15)*a*(a - b)*(2*a - b)*EllipticF[I*c + I*d*x, b/a]*Sqrt[1 + (b*Sinh[c + d*x]^2)/a])/(d*Sqrt[a + b*Sinh[c + d*x]^2])} - - -{Sqrt[1 + Sinh[x]^2], x, 3, Sqrt[Cosh[x]^2]*Tanh[x]} -{Sqrt[-1 - Sinh[x]^2], x, 3, Sqrt[-Cosh[x]^2]*Tanh[x]} -{Sqrt[1 - Sinh[x]^2], x, 1, (-I)*EllipticE[I*x, -1]} -{Sqrt[-1 + Sinh[x]^2], x, 2, -((I*EllipticE[I*x, -1]*Sqrt[-1 + Sinh[x]^2])/Sqrt[1 - Sinh[x]^2])} -{Sqrt[a + b*Sinh[x]^2], x, 2, -((I*EllipticE[I*x, b/a]*Sqrt[a + b*Sinh[x]^2])/Sqrt[1 + (b*Sinh[x]^2)/a])} - - -{(1 + Sinh[x]^2)^(3/2), x, 4, (2/3)*Sqrt[Cosh[x]^2]*Tanh[x] + (1/3)*(Cosh[x]^2)^(3/2)*Tanh[x]} -{(-1 - Sinh[x]^2)^(3/2), x, 4, (-(2/3))*Sqrt[-Cosh[x]^2]*Tanh[x] + (1/3)*(-Cosh[x]^2)^(3/2)*Tanh[x]} -{(1 - Sinh[x]^2)^(3/2), x, 4, -2*I*EllipticE[I*x, -1] + (2/3)*I*EllipticF[I*x, -1] - (1/3)*Cosh[x]*Sinh[x]*Sqrt[1 - Sinh[x]^2]} -{(-1 + Sinh[x]^2)^(3/2), x, 6, (2*I*EllipticF[I*x, -1]*Sqrt[1 - Sinh[x]^2])/(3*Sqrt[-1 + Sinh[x]^2]) + (1/3)*Cosh[x]*Sinh[x]*Sqrt[-1 + Sinh[x]^2] + (2*I*EllipticE[I*x, -1]*Sqrt[-1 + Sinh[x]^2])/Sqrt[1 - Sinh[x]^2]} -{(a + b*Sinh[x]^2)^(3/2), x, 6, (1/3)*b*Cosh[x]*Sinh[x]*Sqrt[a + b*Sinh[x]^2] - (2*I*(2*a - b)*EllipticE[I*x, b/a]*Sqrt[a + b*Sinh[x]^2])/(3*Sqrt[1 + (b*Sinh[x]^2)/a]) + (I*a*(a - b)*EllipticF[I*x, b/a]*Sqrt[1 + (b*Sinh[x]^2)/a])/(3*Sqrt[a + b*Sinh[x]^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, -((a + b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*b^(3/2)*f) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/(2*b*f)} -{Sinh[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(Sqrt[b]*f)} -{Csch[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, -(ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(Sqrt[a]*f))} -{Csch[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, ((a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*a^(3/2)*f) - (Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(2*a*f)} - -{Sinh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 6, (Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b*f) + (2*(a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*b^2*f)} -{Sinh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, ((-I)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + (I*a*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sinh[e + f*x]^0/Sqrt[a + b*Sinh[e + f*x]^2], x, 2, ((-I)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Csch[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f)) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(a*f)} -{Csch[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 7, (2*(a + b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f) - (Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) + (2*(a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^2*f)} - - -{Sinh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(b^(3/2)*f) - (a*Cosh[e + f*x])/((a - b)*b*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} -{Sinh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 2, Cosh[e + f*x]/((a - b)*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} -{Csch[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(a^(3/2)*f)) - (b*Cosh[e + f*x])/(a*(a - b)*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} -{Csch[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, ((a + 3*b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*a^(5/2)*f) - ((a - 3*b)*b*Cosh[e + f*x])/(2*a^2*(a - b)*f*Sqrt[a - b + b*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x])/(2*a*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} - -{Sinh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, -((a*Cosh[e + f*x]*Sinh[e + f*x]^3)/((a - b)*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) + ((4*a - b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*b^2*f) + ((8*a^2 - 3*a*b - 2*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*b^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((4*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((8*a^2 - 3*a*b - 2*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*b^3*f)} -{Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -((a*Cosh[e + f*x]*Sinh[e + f*x])/((a - b)*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) - ((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((2*a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/((a - b)*b^2*f)} -{Sinh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, (Cosh[e + f*x]*Sinh[e + f*x])/((a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2]) + (I*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) - (I*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sinh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -((b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])) - (I*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, -((b*Coth[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])) - ((a - 2*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)*f) - ((a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(a^2*(a - b)*f)} - - -{Sinh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(b^(5/2)*f) - (a*(3*a - 5*b)*Cosh[e + f*x])/(3*(a - b)^2*b^2*f*Sqrt[a - b + b*Cosh[e + f*x]^2]) - (a*Cosh[e + f*x]*Sinh[e + f*x]^2)/(3*(a - b)*b*f*(a - b + b*Cosh[e + f*x]^2)^(3/2))} -{Sinh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2), x, 3, (-2*Cosh[e + f*x])/(3*(a - b)^2*f*Sqrt[a - b + b*Cosh[e + f*x]^2]) + (Cosh[e + f*x]*Sinh[e + f*x]^2)/(3*(a - b)*f*(a - b + b*Cosh[e + f*x]^2)^(3/2))} -{Sinh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 3, Cosh[e + f*x]/(3*(a - b)*f*(a - b + b*Cosh[e + f*x]^2)^(3/2)) + (2*Cosh[e + f*x])/(3*(a - b)^2*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} -{Csch[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, -(ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]/(a^(5/2)*f)) - (b*Cosh[e + f*x])/(3*a*(a - b)*f*(a - b + b*Cosh[e + f*x]^2)^(3/2)) - ((5*a - 3*b)*b*Cosh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a - b + b*Cosh[e + f*x]^2])} - -{Sinh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(a*Cosh[e + f*x]*Sinh[e + f*x]^3)/(3*(a - b)*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*a*(2*a - 3*b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)^2*b^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((8*a^2 - 13*a*b + 3*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*b^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*(2*a - 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((8*a^2 - 13*a*b + 3*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)^2*b^3*f)} -{Sinh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, -(a*Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*Sqrt[a]*(a - 2*b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*(a - b)^2*b^(3/2)*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - ((a - 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^2*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Sinh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, (Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((a + b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + ((I/3)*(a + b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)^2*b*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) - ((I/3)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/((a - b)*b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sinh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(2*a - b)*b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)^2*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Csch[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 8, -(b*Coth[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(3*a - 2*b)*b*Coth[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((3*a^2 - 13*a*b + 8*b^2)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f) - ((3*a^2 - 13*a*b + 8*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(3*a - 2*b)*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a^2 - 13*a*b + 8*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^3*(a - b)^2*f)} - - -{1/Sqrt[1 + Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[Cosh[x]^2]} -{1/Sqrt[1 - Sinh[x]^2], x, 1, (-I)*EllipticF[I*x, -1]} -{1/Sqrt[-1 + Sinh[x]^2], x, 2, -((I*EllipticF[I*x, -1]*Sqrt[1 - Sinh[x]^2])/Sqrt[-1 + Sinh[x]^2])} -{1/Sqrt[-1 - Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[-Cosh[x]^2]} -{1/Sqrt[a + b*Sinh[x]^2], x, 2, -((I*EllipticF[I*x, b/a]*Sqrt[1 + (b*Sinh[x]^2)/a])/Sqrt[a + b*Sinh[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Sinh[e+f x]) (a+b Sinh[e+f x]^2)^p when p symbolic*) - - -{(d*Sinh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p, x, 3, (d*AppellF1[1/2, (1 - m)/2, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*(d*Sinh[e + f*x])^(-1 + m)*(-Sinh[e + f*x]^2)^((1 - m)/2))/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p)} - - -{Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x, 5, If[$VersionNumber>=8, -(((3*a + 2*b*(2 + p))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) + ((3*a^2 + 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(1 + p)*Sinh[e + f*x]^2)/(b*f*(5 + 2*p)), -(((3*a + 2*b*(2 + p))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(1 + p))/(b^2*f*(15 + 16*p + 4*p^2))) + ((3*a^2 + 4*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/((1 + (b*Cosh[e + f*x]^2)/(a - b))^p*(b^2*f*(15 + 16*p + 4*p^2))) + (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(1 + p)*Sinh[e + f*x]^2)/(b*f*(5 + 2*p))]} -{Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x, 4, (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p)) - ((a + 2*b*(1 + p))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/(b*f*(3 + 2*p)*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p)} -{Sinh[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^p, x, 3, (Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p)} -{Csch[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 1, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p))} -{Csch[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 2, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p)} -{Csch[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x, 3, -((AppellF1[1/2, 3, -p, 3/2, Cosh[e + f*x]^2, -((b*Cosh[e + f*x]^2)/(a - b))]*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p)/(f*(1 + (b*Cosh[e + f*x]^2)/(a - b))^p))} - -{Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[5/2, 1/2, -p, 7/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*Sinh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(5*f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Sinh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[3/2, 2 + p, -p, 5/2, Tanh[e + f*x]^2, ((a - b)*Tanh[e + f*x]^2)/a]*(Sech[e + f*x]^2)^p*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x]^3)/(3*f*(1 - ((a - b)*Tanh[e + f*x]^2)/a)^p)} -{Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x, 3, -((AppellF1[-1/2, 1/2, -p, 1/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p))} -{Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x, 3, -(AppellF1[-3/2, 1/2, -p, -1/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(3*f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^3)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^3)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^3), x, 7, (3*a*x)/8 - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/d - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7*d) - (3*a*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^3), x, 8, (-5*b*x)/16 - (a*Cosh[c + d*x])/d + (a*Cosh[c + d*x]^3)/(3*d) + (5*b*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3), x, 6, -(a*x)/2 + (b*Cosh[c + d*x])/d - (2*b*Cosh[c + d*x]^3)/(3*d) + (b*Cosh[c + d*x]^5)/(5*d) + (a*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^3), x, 6, (3*b*x)/8 + (a*Cosh[c + d*x])/d - (3*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^3), x, 3, a*x - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^3), x, 5, -(b*x)/2 - (a*ArcTanh[Cosh[c + d*x]])/d + (b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3), x, 5, (b*Cosh[c + d*x])/d - (a*Coth[c + d*x])/d} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3), x, 4, b*x + (a*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3), x, 5, -((b*ArcTanh[Cosh[c + d*x]])/d) + (a*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*d)} - - -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2, x, 10, (-5*a*b*x)/8 - (a^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d) - (4*b^2*Cosh[c + d*x]^3)/(3*d) + (6*b^2*Cosh[c + d*x]^5)/(5*d) - (4*b^2*Cosh[c + d*x]^7)/(7*d) + (b^2*Cosh[c + d*x]^9)/(9*d) + (5*a*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - (5*a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(12*d) + (a*b*Cosh[c + d*x]*Sinh[c + d*x]^5)/(3*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^2, x, 11, -(a^2*x)/2 + (35*b^2*x)/128 + (2*a*b*Cosh[c + d*x])/d - (4*a*b*Cosh[c + d*x]^3)/(3*d) + (2*a*b*Cosh[c + d*x]^5)/(5*d) + (a^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(48*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^3)^2, x, 8, (3*a*b*x)/4 + (a^2*Cosh[c + d*x])/d - (b^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/d - (3*b^2*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*d) - (3*a*b*Cosh[c + d*x]*Sinh[c + d*x])/(4*d) + (a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(2*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^3)^2, x, 8, a^2*x - (5*b^2*x)/16 - (2*a*b*Cosh[c + d*x])/d + (2*a*b*Cosh[c + d*x]^3)/(3*d) + (5*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^3)^2, x, 7, -(a*b*x) - (a^2*ArcTanh[Cosh[c + d*x]])/d + (b^2*Cosh[c + d*x])/d - (2*b^2*Cosh[c + d*x]^3)/(3*d) + (b^2*Cosh[c + d*x]^5)/(5*d) + (a*b*Cosh[c + d*x]*Sinh[c + d*x])/d} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^2, x, 8, (3*b^2*x)/8 + (2*a*b*Cosh[c + d*x])/d - (a^2*Coth[c + d*x])/d - (3*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^2, x, 6, 2*a*b*x + (a^2*ArcTanh[Cosh[c + d*x]])/(2*d) - (b^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d) - (a^2*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^2, x, 7, -((b^2*x)/2) - (2*a*b*ArcTanh[Cosh[c + d*x]])/d + (a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2, x, 8, -((3*a^2*ArcTanh[Cosh[c + d*x]])/(8*d)) + (b^2*Cosh[c + d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^2, x, 6, b^2*x + (a*b*ArcTanh[Cosh[c + d*x]])/d - (a^2*Coth[c + d*x])/d + (2*a^2*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c + d*x]^5)/(5*d) - (a*b*Coth[c + d*x]*Csch[c + d*x])/d} -{Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2, x, 9, (5*a^2*ArcTanh[Cosh[c + d*x]])/(16*d) - (b^2*ArcTanh[Cosh[c + d*x]])/d + (2*a*b*Coth[c + d*x])/d - (2*a*b*Coth[c + d*x]^3)/(3*d) - (5*a^2*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)} - - -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3, x, 13, -(a^3*x)/2 + (105*a*b^2*x)/128 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d - (2*a^2*b*Cosh[c + d*x]^3)/d + (5*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b*Cosh[c + d*x]^5)/(5*d) - (2*b^3*Cosh[c + d*x]^5)/d + (10*b^3*Cosh[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^11)/(11*d) + (a^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (105*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(64*d) - (7*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(16*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^3)^3, x, 14, (9*a^2*b*x)/8 - (63*b^3*x)/256 + (a^3*Cosh[c + d*x])/d - (3*a*b^2*Cosh[c + d*x])/d + (3*a*b^2*Cosh[c + d*x]^3)/d - (9*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a*b^2*Cosh[c + d*x]^7)/(7*d) - (9*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (63*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d) - (21*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(128*d) + (21*b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(160*d) - (9*b^3*Cosh[c + d*x]*Sinh[c + d*x]^7)/(80*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^9)/(10*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^3)^3, x, 10, a^3*x - (15*a*b^2*x)/16 - (3*a^2*b*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x])/d + (a^2*b*Cosh[c + d*x]^3)/d - (4*b^3*Cosh[c + d*x]^3)/(3*d) + (6*b^3*Cosh[c + d*x]^5)/(5*d) - (4*b^3*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d) + (15*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(8*d) + (a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(2*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^3)^3, x, 12, (-3*a^2*b*x)/2 + (35*b^3*x)/128 - (a^3*ArcTanh[Cosh[c + d*x]])/d + (3*a*b^2*Cosh[c + d*x])/d - (2*a*b^2*Cosh[c + d*x]^3)/d + (3*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(48*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^3)^3, x, 10, (9*a*b^2*x)/8 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/d - (3*b^3*Cosh[c + d*x]^5)/(5*d) + (b^3*Cosh[c + d*x]^7)/(7*d) - (a^3*Coth[c + d*x])/d - (9*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^3)^3, x, 10, 3*a^2*b*x - (5*b^3*x)/16 + (a^3*ArcTanh[Cosh[c + d*x]])/(2*d) - (3*a*b^2*Cosh[c + d*x])/d + (a*b^2*Cosh[c + d*x]^3)/d - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) + (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^3)^3, x, 9, (-(3/2))*a*b^2*x - (3*a^2*b*ArcTanh[Cosh[c + d*x]])/d + (b^3*Cosh[c + d*x])/d - (2*b^3*Cosh[c + d*x]^3)/(3*d) + (b^3*Cosh[c + d*x]^5)/(5*d) + (a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^3, x, 11, (3*b^3*x)/8 - (3*a^3*ArcTanh[Cosh[c + d*x]])/(8*d) + (3*a*b^2*Cosh[c + d*x])/d - (3*a^2*b*Coth[c + d*x])/d + (3*a^3*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d) - (3*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^3)^3, x, 8, 3*a*b^2*x + (3*a^2*b*ArcTanh[Cosh[c + d*x]])/(2*d) - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x])/d + (2*a^3*Coth[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x]^5)/(5*d) - (3*a^2*b*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3, x, 11, -((b^3*x)/2) + (5*a^3*ArcTanh[Cosh[c + d*x]])/(16*d) - (3*a*b^2*ArcTanh[Cosh[c + d*x]])/d + (3*a^2*b*Coth[c + d*x])/d - (a^2*b*Coth[c + d*x]^3)/d - (5*a^3*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[c + d*x]^6/(a + b*Sinh[c + d*x]^3), x, 15, -((a*x)/b^2) - (2*(-1)^(2/3)*a^(4/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^2*d) - (2*(-1)^(2/3)*a^(4/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^2*d) - (2*a^(4/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*Sqrt[a^(2/3) + b^(2/3)]*b^2*d) - Cosh[c + d*x]/(b*d) + Cosh[c + d*x]^3/(3*b*d)} -{Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^3), x, 15, -(x/(2*b)) + (2*a*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]*b^(5/3)*d) + (2*a*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(5/3)*d) + (2*a*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*Sqrt[a^(2/3) + b^(2/3)]*b^(5/3)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{Sinh[c + d*x]^4/(a + b*Sinh[c + d*x]^3), x, 14, -((2*a^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d)) + (2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(4/3)*d) - (2*a^(2/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*Sqrt[a^(2/3) + b^(2/3)]*b^(4/3)*d) + Cosh[c + d*x]/(b*d)} -{Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^3), x, 13, x/b + (2*(-1)^(2/3)*a^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b*d) + (2*(-1)^(2/3)*a^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b*d) + (2*a^(1/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*Sqrt[a^(2/3) + b^(2/3)]*b*d)} -{Sinh[c + d*x]^2/(a + b*Sinh[c + d*x]^3), x, 11, -((2*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]*b^(2/3)*d)) - (2*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(2/3)*d) - (2*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*Sqrt[a^(2/3) + b^(2/3)]*b^(2/3)*d)} -{Sinh[c + d*x]^1/(a + b*Sinh[c + d*x]^3), x, 11, (2*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(1/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(1/3)*d) - (2*(-1)^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(1/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*b^(1/3)*d) + (2*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^(1/3)*Sqrt[a^(2/3) + b^(2/3)]*b^(1/3)*d)} -{Sinh[c + d*x]^0/(a + b*Sinh[c + d*x]^3), x, 11, -((2*(-1)^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)) - (2*(-1)^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) - (2*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + b^(2/3)]*d)} -{Csch[c + d*x]^1/(a + b*Sinh[c + d*x]^3), x, 14, (2*b^(1/3)*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]])/(3*a*Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]*d) + (2*b^(1/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) - ArcTanh[Cosh[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a*Sqrt[a^(2/3) + b^(2/3)]*d)} -{Csch[c + d*x]^2/(a + b*Sinh[c + d*x]^3), x, 15, -((2*b^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(4/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d)) + (2*(-1)^(1/3)*b^(2/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(4/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) - (2*b^(2/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^(4/3)*Sqrt[a^(2/3) + b^(2/3)]*d) - Coth[c + d*x]/(a*d)} -{Csch[c + d*x]^3/(a + b*Sinh[c + d*x]^3), x, 15, (2*(-1)^(2/3)*b*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(5/3)*Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (2*(-1)^(2/3)*b*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(5/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) + ArcTanh[Cosh[c + d*x]]/(2*a*d) + (2*b*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^(5/3)*Sqrt[a^(2/3) + b^(2/3)]*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)} -{Csch[c + d*x]^4/(a + b*Sinh[c + d*x]^3), x, 16, -((2*b^(4/3)*ArcTan[((-1)^(5/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]])/(3*a^2*Sqrt[(-(-1)^(2/3))*a^(2/3) - b^(2/3)]*d)) - (2*b^(4/3)*ArcTan[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(1/2)*(c + d*x)]))/Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^2*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3)]*d) + (b*ArcTanh[Cosh[c + d*x]])/(a^2*d) - (2*b^(4/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(1/2)*(c + d*x)])/Sqrt[a^(2/3) + b^(2/3)]])/(3*a^2*Sqrt[a^(2/3) + b^(2/3)]*d) + Coth[c + d*x]/(a*d) - Coth[c + d*x]^3/(3*a*d)} - - -{1/(1 + Sinh[x]^3), x, 12, -((2*(-1)^(1/6)*ArcTan[(I + (-1)^(1/6)*Tanh[x/2])/Sqrt[1 - (-1)^(1/3)]])/(3*Sqrt[1 - (-1)^(1/3)])) - (1/3)*Sqrt[2]*ArcTanh[(1 - Tanh[x/2])/Sqrt[2]] - (1/3)*(-1)^(1/6)*Log[1 + (-1)^(5/6) - (-1)^(1/6)*Tanh[x/2]] + (1/3)*(-1)^(1/6)*Log[1 + (-1)^(1/6) + (-1)^(1/3)*Tanh[x/2]]} -{1/(1 - Sinh[x]^3), x, 12, (2*(-1)^(5/6)*ArcTan[(I - (-1)^(5/6)*Tanh[x/2])/Sqrt[1 + (-1)^(2/3)]])/(3*Sqrt[1 + (-1)^(2/3)]) + (1/3)*Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]] - (1/3)*(-1)^(5/6)*Log[1 + (-1)^(5/6) + (-1)^(2/3)*Tanh[x/2]] + (1/3)*(-1)^(5/6)*Log[1 + (-1)^(1/6) + (-1)^(5/6)*Tanh[x/2]]} - - -(* ::Subsection:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^3)^(p/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^4)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^4)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^4), x, 6, (1/128)*(48*a + 35*b)*x - ((80*a + 93*b)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a + 163*b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) - (25*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x])/(8*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4), x, 3, -(((a + b)*Cosh[c + d*x])/d) + ((a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4), x, 5, (-(1/16))*(8*a + 5*b)*x + ((8*a + 11*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (13*b*Cosh[c + d*x]^3*Sinh[c + d*x])/(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^4), x, 2, ((a + b)*Cosh[c + d*x])/d - (2*b*Cosh[c + d*x]^3)/(3*d) + (b*Cosh[c + d*x]^5)/(5*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^4), x, 4, a*x + (3*b*x)/8 - (3*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^4), x, 4, -((a*ArcTanh[Cosh[c + d*x]])/d) - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4), x, 4, -((b*x)/2) - (a*Coth[c + d*x])/d + (b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4), x, 4, (a*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*Cosh[c + d*x])/d - (a*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4), x, 4, b*x + (a*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4), x, 4, -(((3*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(8*d)) + (3*a*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4), x, 3, -(((a + b)*Coth[c + d*x])/d) + (2*a*Coth[c + d*x]^3)/(3*d) - (a*Coth[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4), x, 5, ((5*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(16*d) - ((5*a + 8*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)} - - -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2, x, 3, -(((a + b)^2*Cosh[c + d*x])/d) + ((a + b)*(a + 5*b)*Cosh[c + d*x]^3)/(3*d) - (2*b*(3*a + 5*b)*Cosh[c + d*x]^5)/(5*d) + (2*b*(a + 5*b)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*Cosh[c + d*x]^9)/(9*d) + (b^2*Cosh[c + d*x]^11)/(11*d)} -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^2, x, 7, (-(1/256))*(128*a^2 + 160*a*b + 63*b^2)*x + ((128*a^2 + 352*a*b + 193*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) - (b*(416*a + 447*b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(384*d) + (b*(160*a + 513*b)*Cosh[c + d*x]^5*Sinh[c + d*x])/(480*d) - (41*b^2*Cosh[c + d*x]^7*Sinh[c + d*x])/(80*d) + (b^2*Cosh[c + d*x]^9*Sinh[c + d*x])/(10*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^4)^2, x, 3, ((a + b)^2*Cosh[c + d*x])/d - (4*b*(a + b)*Cosh[c + d*x]^3)/(3*d) + (2*b*(a + 3*b)*Cosh[c + d*x]^5)/(5*d) - (4*b^2*Cosh[c + d*x]^7)/(7*d) + (b^2*Cosh[c + d*x]^9)/(9*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^4)^2, x, 6, (1/128)*(128*a^2 + 96*a*b + 35*b^2)*x - (b*(160*a + 93*b)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (b*(96*a + 163*b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) - (25*b^2*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b^2*Cosh[c + d*x]^7*Sinh[c + d*x])/(8*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^4)^2, x, 4, -((a^2*ArcTanh[Cosh[c + d*x]])/d) - (b*(2*a + b)*Cosh[c + d*x])/d + (b*(2*a + 3*b)*Cosh[c + d*x]^3)/(3*d) - (3*b^2*Cosh[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]^7)/(7*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^2, x, 6, (-(1/16))*b*(16*a + 5*b)*x - (a^2*Coth[c + d*x])/d + (b*(16*a + 11*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (13*b^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(24*d) + (b^2*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^2, x, 5, (a^2*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(2*a + b)*Cosh[c + d*x])/d - (2*b^2*Cosh[c + d*x]^3)/(3*d) + (b^2*Cosh[c + d*x]^5)/(5*d) - (a^2*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4)^2, x, 6, (1/8)*b*(16*a + 3*b)*x + (a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) - (5*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^2*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)} -{Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^2, x, 6, -((a*(3*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(8*d)) - (b^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d) + (3*a^2*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^2, x, 5, -((b^2*x)/2) - (a*(a + 2*b)*Coth[c + d*x])/d + (2*a^2*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c + d*x]^5)/(5*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^2, x, 6, (a*(5*a + 16*b)*ArcTanh[Cosh[c + d*x]])/(16*d) + (b^2*Cosh[c + d*x])/d - (a*(5*a + 16*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)} - - -{Sinh[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3, x, 3, ((a + b)^3*Cosh[c + d*x])/d - (2*(a + b)^2*(a + 4*b)*Cosh[c + d*x]^3)/(3*d) + ((a + b)*(a^2 + 17*a*b + 28*b^2)*Cosh[c + d*x]^5)/(5*d) - (4*b*(3*a^2 + 15*a*b + 14*b^2)*Cosh[c + d*x]^7)/(7*d) + (b*(3*a^2 + 45*a*b + 70*b^2)*Cosh[c + d*x]^9)/(9*d) - (2*b^2*(9*a + 28*b)*Cosh[c + d*x]^11)/(11*d) + (b^2*(3*a + 28*b)*Cosh[c + d*x]^13)/(13*d) - (8*b^3*Cosh[c + d*x]^15)/(15*d) + (b^3*Cosh[c + d*x]^17)/(17*d)} -{Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3, x, 3, -(((a + b)^3*Cosh[c + d*x])/d) + ((a + b)^2*(a + 7*b)*Cosh[c + d*x]^3)/(3*d) - (3*b*(a + b)*(3*a + 7*b)*Cosh[c + d*x]^5)/(5*d) + (b*(3*a^2 + 30*a*b + 35*b^2)*Cosh[c + d*x]^7)/(7*d) - (5*b^2*(3*a + 7*b)*Cosh[c + d*x]^9)/(9*d) + (3*b^2*(a + 7*b)*Cosh[c + d*x]^11)/(11*d) - (7*b^3*Cosh[c + d*x]^13)/(13*d) + (b^3*Cosh[c + d*x]^15)/(15*d)} -{Sinh[c + d*x]^1*(a + b*Sinh[c + d*x]^4)^3, x, 3, ((a + b)^3*Cosh[c + d*x])/d - (2*b*(a + b)^2*Cosh[c + d*x]^3)/d + (3*b*(a + b)*(a + 5*b)*Cosh[c + d*x]^5)/(5*d) - (4*b^2*(3*a + 5*b)*Cosh[c + d*x]^7)/(7*d) + (b^2*(a + 5*b)*Cosh[c + d*x]^9)/(3*d) - (6*b^3*Cosh[c + d*x]^11)/(11*d) + (b^3*Cosh[c + d*x]^13)/(13*d)} -{Csch[c + d*x]^1*(a + b*Sinh[c + d*x]^4)^3, x, 4, -((a^3*ArcTanh[Cosh[c + d*x]])/d) - (b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x])/d + (b*(3*a^2 + 9*a*b + 5*b^2)*Cosh[c + d*x]^3)/(3*d) - (b^2*(9*a + 10*b)*Cosh[c + d*x]^5)/(5*d) + (b^2*(3*a + 10*b)*Cosh[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^11)/(11*d)} -{Csch[c + d*x]^3*(a + b*Sinh[c + d*x]^4)^3, x, 5, (a^3*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x])/d - (2*b^2*(3*a + 2*b)*Cosh[c + d*x]^3)/(3*d) + (3*b^2*(a + 2*b)*Cosh[c + d*x]^5)/(5*d) - (4*b^3*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d)} -{Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^4)^3, x, 6, -((3*a^2*(a + 8*b)*ArcTanh[Cosh[c + d*x]])/(8*d)) - (b^2*(3*a + b)*Cosh[c + d*x])/d + (b^2*(a + b)*Cosh[c + d*x]^3)/d - (3*b^3*Cosh[c + d*x]^5)/(5*d) + (b^3*Cosh[c + d*x]^7)/(7*d) + (3*a^3*Coth[c + d*x]*Csch[c + d*x])/(8*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)} -{Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4)^3, x, 7, (a^2*(5*a + 24*b)*ArcTanh[Cosh[c + d*x]])/(16*d) + (b^2*(3*a + b)*Cosh[c + d*x])/d - (2*b^3*Cosh[c + d*x]^3)/(3*d) + (b^3*Cosh[c + d*x]^5)/(5*d) - (a^2*(5*a + 24*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^3*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)} -{Csch[c + d*x]^9*(a + b*Sinh[c + d*x]^4)^3, x, 8, -((a*(35*a^2 + 144*a*b + 384*b^2)*ArcTanh[Cosh[c + d*x]])/(128*d)) - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/(3*d) + (a^2*(35*a + 144*b)*Coth[c + d*x]*Csch[c + d*x])/(128*d) - (a^2*(35*a + 144*b)*Coth[c + d*x]*Csch[c + d*x]^3)/(192*d) + (7*a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(48*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(8*d)} -{Csch[c + d*x]^11*(a + b*Sinh[c + d*x]^4)^3, x, 8, (3*a*(21*a^2 + 80*a*b + 128*b^2)*ArcTanh[Cosh[c + d*x]])/(256*d) + (b^3*Cosh[c + d*x])/d - (3*a*(21*a^2 + 80*a*b + 128*b^2)*Coth[c + d*x]*Csch[c + d*x])/(256*d) + (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^3)/(128*d) - (a^2*(21*a + 80*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(160*d) + (9*a^3*Coth[c + d*x]*Csch[c + d*x]^7)/(80*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(10*d)} -{Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3, x, 8, -(((231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*ArcTanh[Cosh[c + d*x]])/(1024*d)) + (3*a*(77*a^2 + 280*a*b + 384*b^2)*Coth[c + d*x]*Csch[c + d*x])/(1024*d) - (a*(77*a^2 + 280*a*b + 384*b^2)*Coth[c + d*x]*Csch[c + d*x]^3)/(512*d) + (7*a^2*(11*a + 40*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(640*d) - (3*a^2*(11*a + 40*b)*Coth[c + d*x]*Csch[c + d*x]^7)/(320*d) + (11*a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(120*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^11)/(12*d)} - -{Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3, x, 9, -(((1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*x)/2048) + ((1024*a^3 + 4224*a^2*b + 4632*a*b^2 + 1619*b^3)*Cosh[c + d*x]*Sinh[c + d*x])/(2048*d) - (b*(4992*a^2 + 10728*a*b + 5549*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(3072*d) + (b*(1920*a^2 + 12312*a*b + 10579*b^2)*Cosh[c + d*x]^5*Sinh[c + d*x])/(3840*d) - (b^2*(6888*a + 11821*b)*Cosh[c + d*x]^7*Sinh[c + d*x])/(4480*d) + (b^2*(504*a + 2593*b)*Cosh[c + d*x]^9*Sinh[c + d*x])/(1680*d) - (85*b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(168*d) + (b^3*Cosh[c + d*x]^13*Sinh[c + d*x])/(14*d)} -{Sinh[c + d*x]^0*(a + b*Sinh[c + d*x]^4)^3, x, 8, ((1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*x)/1024 - (b*(1920*a^2 + 2232*a*b + 793*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(1024*d) + (b*(1152*a^2 + 3912*a*b + 2279*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(1536*d) - (b^2*(3000*a + 3481*b)*Cosh[c + d*x]^5*Sinh[c + d*x])/(1920*d) + (3*b^2*(40*a + 139*b)*Cosh[c + d*x]^7*Sinh[c + d*x])/(320*d) - (61*b^3*Cosh[c + d*x]^9*Sinh[c + d*x])/(120*d) + (b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(12*d)} -{Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3, x, 8, (-(3/256))*b*(128*a^2 + 80*a*b + 21*b^2)*x - (a^3*Coth[c + d*x])/d + (b*(384*a^2 + 528*a*b + 193*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) - (b^2*(208*a + 149*b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*d) + (b^2*(80*a + 171*b)*Cosh[c + d*x]^5*Sinh[c + d*x])/(160*d) - (41*b^3*Cosh[c + d*x]^7*Sinh[c + d*x])/(80*d) + (b^3*Cosh[c + d*x]^9*Sinh[c + d*x])/(10*d)} -{Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^4)^3, x, 8, (1/128)*b*(384*a^2 + 144*a*b + 35*b^2)*x + (a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) - (3*b^2*(80*a + 31*b)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (b^2*(144*a + 163*b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) - (25*b^3*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b^3*Cosh[c + d*x]^7*Sinh[c + d*x])/(8*d)} -{Csch[c + d*x]^6*(a + b*Sinh[c + d*x]^4)^3, x, 7, (-(1/16))*b^2*(24*a + 5*b)*x - (a^2*(a + 3*b)*Coth[c + d*x])/d + (2*a^3*Coth[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x]^5)/(5*d) + (b^2*(24*a + 11*b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (13*b^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(24*d) + (b^3*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)} -{Csch[c + d*x]^8*(a + b*Sinh[c + d*x]^4)^3, x, 6, (3/8)*b^2*(8*a + b)*x + (a^2*(a + 3*b)*Coth[c + d*x])/d - (a^2*(a + b)*Coth[c + d*x]^3)/d + (3*a^3*Coth[c + d*x]^5)/(5*d) - (a^3*Coth[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)} -{Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3, x, 5, -((b^3*x)/2) - (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d + (2*a^2*(2*a + 3*b)*Coth[c + d*x]^3)/(3*d) - (3*a^2*(2*a + b)*Coth[c + d*x]^5)/(5*d) + (4*a^3*Coth[c + d*x]^7)/(7*d) - (a^3*Coth[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Csch[c + d*x]^12*(a + b*Sinh[c + d*x]^4)^3, x, 4, b^3*x + (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d - (a*(5*a^2 + 9*a*b + 3*b^2)*Coth[c + d*x]^3)/(3*d) + (a^2*(10*a + 9*b)*Coth[c + d*x]^5)/(5*d) - (a^2*(10*a + 3*b)*Coth[c + d*x]^7)/(7*d) + (5*a^3*Coth[c + d*x]^9)/(9*d) - (a^3*Coth[c + d*x]^11)/(11*d)} -{Csch[c + d*x]^14*(a + b*Sinh[c + d*x]^4)^3, x, 3, -(((a + b)^3*Coth[c + d*x])/d) + (2*a*(a + b)^2*Coth[c + d*x]^3)/d - (3*a*(a + b)*(5*a + b)*Coth[c + d*x]^5)/(5*d) + (4*a^2*(5*a + 3*b)*Coth[c + d*x]^7)/(7*d) - (a^2*(5*a + b)*Coth[c + d*x]^9)/(3*d) + (6*a^3*Coth[c + d*x]^11)/(11*d) - (a^3*Coth[c + d*x]^13)/(13*d)} -{Csch[c + d*x]^16*(a + b*Sinh[c + d*x]^4)^3, x, 3, ((a + b)^3*Coth[c + d*x])/d - ((a + b)^2*(7*a + b)*Coth[c + d*x]^3)/(3*d) + (3*a*(a + b)*(7*a + 3*b)*Coth[c + d*x]^5)/(5*d) - (a*(35*a^2 + 30*a*b + 3*b^2)*Coth[c + d*x]^7)/(7*d) + (5*a^2*(7*a + 3*b)*Coth[c + d*x]^9)/(9*d) - (3*a^2*(7*a + b)*Coth[c + d*x]^11)/(11*d) + (7*a^3*Coth[c + d*x]^13)/(13*d) - (a^3*Coth[c + d*x]^15)/(15*d)} -{Csch[c + d*x]^18*(a + b*Sinh[c + d*x]^4)^3, x, 3, -(((a + b)^3*Coth[c + d*x])/d) + (2*(a + b)^2*(4*a + b)*Coth[c + d*x]^3)/(3*d) - ((a + b)*(28*a^2 + 17*a*b + b^2)*Coth[c + d*x]^5)/(5*d) + (4*a*(14*a^2 + 15*a*b + 3*b^2)*Coth[c + d*x]^7)/(7*d) - (a*(70*a^2 + 45*a*b + 3*b^2)*Coth[c + d*x]^9)/(9*d) + (2*a^2*(28*a + 9*b)*Coth[c + d*x]^11)/(11*d) - (a^2*(28*a + 3*b)*Coth[c + d*x]^13)/(13*d) + (8*a^3*Coth[c + d*x]^15)/(15*d) - (a^3*Coth[c + d*x]^17)/(17*d)} -{Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3, x, 3, ((a + b)^3*Coth[c + d*x])/d - ((a + b)^2*(3*a + b)*Coth[c + d*x]^3)/d + (3*(a + b)*(12*a^2 + 9*a*b + b^2)*Coth[c + d*x]^5)/(5*d) - ((84*a^3 + 105*a^2*b + 30*a*b^2 + b^3)*Coth[c + d*x]^7)/(7*d) + (a*(42*a^2 + 35*a*b + 5*b^2)*Coth[c + d*x]^9)/(3*d) - (3*a*(42*a^2 + 21*a*b + b^2)*Coth[c + d*x]^11)/(11*d) + (21*a^2*(4*a + b)*Coth[c + d*x]^13)/(13*d) - (a^2*(12*a + b)*Coth[c + d*x]^15)/(5*d) + (9*a^3*Coth[c + d*x]^17)/(17*d) - (a^3*Coth[c + d*x]^19)/(19*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4), x, 6, -((a*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(7/4)*d)) + (a*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(7/4)*d) + Cosh[c + d*x]/(b*d) - Cosh[c + d*x]^3/(3*b*d)} -{Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4), x, 6, (Sqrt[a]*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)*d) + (Sqrt[a]*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)*d) - Cosh[c + d*x]/(b*d)} -{Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4), x, 4, -ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/4)*d) + ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/4)*d)} -{Sinh[c + d*x]^1/(a - b*Sinh[c + d*x]^4), x, 4, ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)*d) + ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)*d)} -{Csch[c + d*x]^1/(a - b*Sinh[c + d*x]^4), x, 7, -(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cosh[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + Sqrt[b]]*d)} -{Csch[c + d*x]^3/(a - b*Sinh[c + d*x]^4), x, 7, (b^(3/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcTanh[Cosh[c + d*x]]/(2*a*d) + (b^(3/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/2)*Sqrt[Sqrt[a] + Sqrt[b]]*d) + 1/(4*a*d*(1 - Cosh[c + d*x])) - 1/(4*a*d*(1 + Cosh[c + d*x]))} - -{Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4), x, 7, x/(2*b) - (a^(3/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2)*d) + (a^(3/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/2)*d) - 1/(4*b*d*(1 - Tanh[c + d*x])) + 1/(4*b*d*(1 + Tanh[c + d*x]))} -{Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4), x, 7, -(x/b) + (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b*d) + (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b*d)} -{Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4), x, 4, -ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*Sqrt[b]*d) + ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[b]]*Sqrt[b]*d)} -{Sinh[c + d*x]^0/(a - b*Sinh[c + d*x]^4), x, 4, ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d)} -{Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4), x, 6, -(Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Coth[c + d*x]/(a*d)} -{Csch[c + d*x]^4/(a - b*Sinh[c + d*x]^4), x, 6, (b*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + (b*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*a^(7/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) + Coth[c + d*x]/(a*d) - Coth[c + d*x]^3/(3*a*d)} - - -{Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^2, x, 7, -(Sqrt[a]*(5*Sqrt[a] - 6*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(9/4)*d) - (Sqrt[a]*(5*Sqrt[a] + 6*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(9/4)*d) + Cosh[c + d*x]/(b^2*d) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(4*(a - b)*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^2, x, 5, ((3*Sqrt[a] - 4*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(7/4)*d) - ((3*Sqrt[a] + 4*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^2, x, 5, -((Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(5/4)*d) - ((Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(5/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(4*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^2, x, 5, -ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/4)*d) + ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(8*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/4)*d) - (Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^1/(a - b*Sinh[c + d*x]^4)^2, x, 5, ((3*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(1/4)*d) + ((3*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(1/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Csch[c + d*x]^1/(a - b*Sinh[c + d*x]^4)^2, x, 11, -(b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cosh[c + d*x]]/(a^2*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*a*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} - -{Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^2, x, 14, x/b^2 - (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) - (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) - (a^(1/4)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tanh[c + d*x]/(4*(a - b)*b*d) + Tanh[c + d*x]^5/(4*b*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^2, x, 6, ((2*Sqrt[a] - 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) - ((2*Sqrt[a] + 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) + Tanh[c + d*x]/(4*(a - b)*b*d) + (Sech[c + d*x]^2*Tanh[c + d*x]^3)/(4*b*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4)^2, x, 7, ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d) - ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(8*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) - Tanh[c + d*x]/(4*a*(a - b)*d) + Tanh[c + d*x]^5/(4*a*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2, x, 5, -(((2*Sqrt[a] - Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] - Sqrt[b])^(3/2)*Sqrt[b]*d)) + ((2*Sqrt[a] + Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(5/4)*(Sqrt[a] + Sqrt[b])^(3/2)*Sqrt[b]*d) + (Tanh[c + d*x]*(a - (a + b)*Tanh[c + d*x]^2))/(4*a*(a - b)*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^0/(a - b*Sinh[c + d*x]^4)^2, x, 5, ((4*Sqrt[a] - 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - (b*Tanh[c + d*x]*(1 - 2*Tanh[c + d*x]^2))/(4*a*(a - b)*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2, x, 7, -(((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d)) + ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Coth[c + d*x]/(a^2*d) + (b*Tanh[c + d*x]*(a - (a + b)*Tanh[c + d*x]^2))/(4*a^2*(a - b)*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} - - -{Sinh[c + d*x]^9/(a - b*Sinh[c + d*x]^4)^3, x, 6, ((5*a - 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(9/4)*d) + ((5*a + 14*Sqrt[a]*Sqrt[b] + 12*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(9/4)*d) + (a*Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(8*(a - b)*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(9*a^2 - 11*a*b - 10*b^2 - 2*(2*a - 5*b)*b*Cosh[c + d*x]^2))/(32*(a - b)^2*b^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3, x, 6, (3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (Cosh[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cosh[c + d*x]^2))/(32*(a - b)^2*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4)^3, x, 6, -((3*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(5/4)*d) - ((3*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(5/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(8*(a - b)*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(a^2 - 11*a*b - 2*b^2 + 2*b*(2*a + b)*Cosh[c + d*x]^2))/(32*a*(a - b)^2*b*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^3/(a - b*Sinh[c + d*x]^4)^3, x, 6, -((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/4)*d) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(11*a + b - (5*a + b)*Cosh[c + d*x]^2))/(32*a*(a - b)^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Sinh[c + d*x]^1/(a - b*Sinh[c + d*x]^4)^3, x, 6, (3*(7*a - 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(1/4)*d) + (3*(7*a + 10*Sqrt[a]*Sqrt[b] + 4*b)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(1/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (Cosh[c + d*x]*((7*a - 3*b)*(a + 2*b) - 6*(2*a - b)*b*Cosh[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} -{Csch[c + d*x]^1/(a - b*Sinh[c + d*x]^4)^3, x, 16, -((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] - Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] - Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[Cosh[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*a*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (b*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(4*a^2*(a - b)*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)) - (b*Cosh[c + d*x]*(11*a + b - (5*a + b)*Cosh[c + d*x]^2))/(32*a^2*(a - b)^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))} - -{Sinh[c + d*x]^8/(a - b*Sinh[c + d*x]^4)^3, x, 9, -(((2*Sqrt[a] - 5*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d)) + ((2*Sqrt[a] + 5*Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) - ((a + 5*b)*Tanh[c + d*x])/(32*a*(a - b)^2*b*d) - Tanh[c + d*x]^3/(32*a*(a - b)*b*d) + Tanh[c + d*x]^9/(8*a*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) - (Sech[c + d*x]^2*Tanh[c + d*x]^5)/(32*a*b*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^6/(a - b*Sinh[c + d*x]^4)^3, x, 6, ((4*a - 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] - Sqrt[b])^(5/2)*b^(3/2)*d) - ((4*a + 10*Sqrt[a]*Sqrt[b] + 3*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(5/4)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/2)*d) + (Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/(8*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) + (Tanh[c + d*x]*((2*a*(a^2 - a*b - 8*b^2))/(a - b)^3 - ((2*a^2 + 15*a*b + 3*b^2)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a*b*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^4/(a - b*Sinh[c + d*x]^4)^3, x, 6, (3*(2*Sqrt[a] - Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d) - (3*(2*Sqrt[a] + Sqrt[b])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(7/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tanh[c + d*x]*(3*a + b - 4*(a + b)*Tanh[c + d*x]^2))/(8*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) - (Tanh[c + d*x]*((9*a^2 - 24*a*b - b^2)/(a - b)^3 - ((17*a + 3*b)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3, x, 6, -(((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] - Sqrt[b])^(5/2)*Sqrt[b]*d)) + ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) + (b*Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/(8*a*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) + (Tanh[c + d*x]*((2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 - (5*(2*a^2 + 3*a*b - b^2)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Sinh[c + d*x]^0/(a - b*Sinh[c + d*x]^4)^3, x, 6, ((32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + ((32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b^2*Tanh[c + d*x]*(3*a + b - 4*(a + b)*Tanh[c + d*x]^2))/(8*a*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) - (b*Tanh[c + d*x]*((17*a^2 - 40*a*b + 7*b^2)/(a - b)^3 - ((33*a - 13*b)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} -{Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^3, x, 8, -((3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d)) + (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - Coth[c + d*x]/(a^3*d) + (b^2*Tanh[c + d*x]*(a*(a + 3*b) - (a^2 + 6*a*b + b^2)*Tanh[c + d*x]^2))/(8*a^2*(a - b)^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)^2) + (b*Tanh[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 - ((18*a^2 + 15*a*b - 13*b^2)*Tanh[c + d*x]^2)/(a - b)^2))/(32*a^3*d*(a - 2*a*Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4))} - - -{1/(1 - Sinh[x]^4), x, 3, ArcTanh[Sqrt[2]*Tanh[x]]/(2*Sqrt[2]) + Tanh[x]/2} -{1/(1 + Sinh[x]^4), x, 10, -(ArcTan[(Sqrt[1 + Sqrt[2]] - 2*Tanh[x])/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]])) + ArcTan[(Sqrt[1 + Sqrt[2]] + 2*Tanh[x])/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]]) - (1/8)*Sqrt[1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*Tanh[x] + 2*Tanh[x]^2] + (1/8)*Sqrt[1 + Sqrt[2]]*Log[1 + Sqrt[2*(1 + Sqrt[2])]*Tanh[x] + Sqrt[2]*Tanh[x]^2]} - - -(* ::Subsection:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^4)^(p/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -{1/(a + b*Sinh[x]^5), x, 17, -((2*ArcTanh[(b^(1/5) - a^(1/5)*Tanh[x/2])/Sqrt[a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[a^(2/5) + b^(2/5)])) + (2*(-1)^(9/10)*ArcTanh[((-1)^(9/10)*((-1)^(1/5)*b^(1/5) + a^(1/5)*Tanh[x/2]))/Sqrt[(-(-1)^(4/5))*a^(2/5) + (-1)^(1/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[(-(-1)^(4/5))*a^(2/5) + (-1)^(1/5)*b^(2/5)]) + (2*(-1)^(1/5)*ArcTanh[(b^(1/5) + (-1)^(1/5)*a^(1/5)*Tanh[x/2])/Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/5)]])/(5*a^(4/5)*Sqrt[(-1)^(2/5)*a^(2/5) + b^(2/5)]) + (2*(-1)^(9/10)*ArcTanh[((-1)^(3/10)*(b^(1/5) + (-1)^(3/5)*a^(1/5)*Tanh[x/2]))/Sqrt[(-(-1)^(4/5))*a^(2/5) + (-1)^(3/5)*b^(2/5)]])/(5*a^(4/5)*Sqrt[(-(-1)^(4/5))*a^(2/5) + (-1)^(3/5)*b^(2/5)]) - (2*(-1)^(9/10)*ArcTanh[(I*b^(1/5) - (-1)^(9/10)*a^(1/5)*Tanh[x/2])/Sqrt[(-(-1)^(4/5))*a^(2/5) - b^(2/5)]])/(5*a^(4/5)*Sqrt[(-(-1)^(4/5))*a^(2/5) - b^(2/5)])} -{1/(a + b*Sinh[x]^6), x, 7, ArcTanh[(Sqrt[a^(1/3) - b^(1/3)]*Tanh[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tanh[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tanh[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])} -{1/(a + b*Sinh[x]^8), x, 9, -(ArcTanh[(Sqrt[(-a)^(1/4) - b^(1/4)]*Tanh[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)])) - ArcTanh[(Sqrt[(-a)^(1/4) - I*b^(1/4)]*Tanh[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcTanh[(Sqrt[(-a)^(1/4) + I*b^(1/4)]*Tanh[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTanh[(Sqrt[(-a)^(1/4) + b^(1/4)]*Tanh[x])/(-a)^(1/8)]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])} - - -{1/(1 + Sinh[x]^5), x, 17, -((2*(-1)^(3/5)*ArcTan[(1 + (-1)^(3/5)*Tanh[x/2])/Sqrt[-1 + (-1)^(1/5)]])/(5*Sqrt[-1 + (-1)^(1/5)])) + (2*(-1)^(9/10)*ArcTan[(I - (-1)^(9/10)*Tanh[x/2])/Sqrt[1 + (-1)^(4/5)]])/(5*Sqrt[1 + (-1)^(4/5)]) - (1/5)*Sqrt[2]*ArcTanh[(1 - Tanh[x/2])/Sqrt[2]] + (2*(-1)^(9/10)*ArcTanh[((-1)^(7/10)*(1 + (-1)^(1/5)*Tanh[x/2]))/Sqrt[(-(-1)^(2/5))*(1 + (-1)^(2/5))]])/(5*Sqrt[(-(-1)^(2/5))*(1 + (-1)^(2/5))]) - (2*(-1)^(4/5)*ArcTanh[(1 - (-1)^(4/5)*Tanh[x/2])/Sqrt[1 - (-1)^(3/5)]])/(5*Sqrt[1 - (-1)^(3/5)])} -{1/(1 + Sinh[x]^6), x, 8, ArcTanh[Sqrt[1 + (-1)^(1/3)]*Tanh[x]]/(3*Sqrt[1 + (-1)^(1/3)]) + ArcTanh[Sqrt[1 - (-1)^(2/3)]*Tanh[x]]/(3*Sqrt[1 - (-1)^(2/3)]) + Tanh[x]/3} -{1/(1 + Sinh[x]^8), x, 9, ArcTanh[Sqrt[1 - (-1)^(1/4)]*Tanh[x]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTanh[Sqrt[1 + (-1)^(1/4)]*Tanh[x]]/(4*Sqrt[1 + (-1)^(1/4)]) + ArcTanh[Sqrt[1 - (-1)^(3/4)]*Tanh[x]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTanh[Sqrt[1 + (-1)^(3/4)]*Tanh[x]]/(4*Sqrt[1 + (-1)^(3/4)])} - - -{1/(1 - Sinh[x]^5), x, 17, -((2*(-1)^(1/10)*ArcTan[(I + (-1)^(1/10)*Tanh[x/2])/Sqrt[1 - (-1)^(1/5)]])/(5*Sqrt[1 - (-1)^(1/5)])) - (2*ArcTanh[((-1)^(3/5) - Tanh[x/2])/Sqrt[1 - (-1)^(1/5)]])/(5*Sqrt[1 - (-1)^(1/5)]) + (1/5)*Sqrt[2]*ArcTanh[(1 + Tanh[x/2])/Sqrt[2]] + (2*ArcTanh[((-1)^(4/5) + Tanh[x/2])/Sqrt[1 - (-1)^(3/5)]])/(5*Sqrt[1 - (-1)^(3/5)]) - (2*(-1)^(1/10)*ArcTanh[((-1)^(3/10)*(1 + (-1)^(4/5)*Tanh[x/2]))/Sqrt[(-1)^(1/5) + (-1)^(3/5)]])/(5*Sqrt[(-1)^(1/5) + (-1)^(3/5)])} -{1/(1 - Sinh[x]^6), x, 7, ArcTanh[Sqrt[2]*Tanh[x]]/(3*Sqrt[2]) + ArcTanh[Sqrt[1 - (-1)^(1/3)]*Tanh[x]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTanh[Sqrt[1 + (-1)^(2/3)]*Tanh[x]]/(3*Sqrt[1 + (-1)^(2/3)])} -{1/(1 - Sinh[x]^8), x, 10, ArcTanh[Sqrt[1 - I]*Tanh[x]]/(4*Sqrt[1 - I]) + ArcTanh[Sqrt[1 + I]*Tanh[x]]/(4*Sqrt[1 + I]) + ArcTanh[Sqrt[2]*Tanh[x]]/(4*Sqrt[2]) + Tanh[x]/4} - - -(* ::Title::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^2)^p with a-b=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cosh[x]^5/(a + a*Sinh[x]^2), x, 3, Sinh[x]/a + Sinh[x]^3/(3*a)} -{Cosh[x]^4/(a + a*Sinh[x]^2), x, 3, x/(2*a) + (Cosh[x]*Sinh[x])/(2*a)} -{Cosh[x]^3/(a + a*Sinh[x]^2), x, 2, Sinh[x]/a} -{Cosh[x]^2/(a + a*Sinh[x]^2), x, 2, x/a} -{Cosh[x]^1/(a + a*Sinh[x]^2), x, 2, ArcTan[Sinh[x]]/a} -{Sech[x]^1/(a + a*Sinh[x]^2), x, 3, ArcTan[Sinh[x]]/(2*a) + (Sech[x]*Tanh[x])/(2*a)} -{Sech[x]^3/(a + a*Sinh[x]^2), x, 4, (3*ArcTan[Sinh[x]])/(8*a) + (3*Sech[x]*Tanh[x])/(8*a) + (Sech[x]^3*Tanh[x])/(4*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2), x, 5, (1/16)*(6*a - b)*x + ((6*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((6*a - b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x])/(6*d)} -{Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2), x, 3, (a*Sinh[c + d*x])/d + ((a + b)*Sinh[c + d*x]^3)/(3*d) + (b*Sinh[c + d*x]^5)/(5*d)} -{Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2), x, 4, (1/8)*(4*a - b)*x + ((4*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)} -{Cosh[c + d*x]^1*(a + b*Sinh[c + d*x]^2), x, 2, (a*Sinh[c + d*x])/d + (b*Sinh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^1*(a + b*Sinh[c + d*x]^2), x, 3, ((a - b)*ArcTan[Sinh[c + d*x]])/d + (b*Sinh[c + d*x])/d} -{Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2), x, 3, b*x + ((a - b)*Tanh[c + d*x])/d} -{Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2), x, 3, ((a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + ((a - b)*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2), x, 2, (a*Tanh[c + d*x])/d - ((a - b)*Tanh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2), x, 4, ((3*a + b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((3*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + ((a - b)*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2), x, 3, (a*Tanh[c + d*x])/d - ((2*a - b)*Tanh[c + d*x]^3)/(3*d) + ((a - b)*Tanh[c + d*x]^5)/(5*d)} - - -{Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2, x, 6, (1/128)*(48*a^2 - 16*a*b + 3*b^2)*x + ((48*a^2 - 16*a*b + 3*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a^2 - 16*a*b + 3*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((10*a - 3*b)*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2))/(8*d)} -{Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2, x, 3, (a^2*Sinh[c + d*x])/d + (a*(a + 2*b)*Sinh[c + d*x]^3)/(3*d) + (b*(2*a + b)*Sinh[c + d*x]^5)/(5*d) + (b^2*Sinh[c + d*x]^7)/(7*d)} -{Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2, x, 5, (1/16)*(8*a^2 - 4*a*b + b^2)*x + ((8*a^2 - 4*a*b + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) + ((8*a - 3*b)*b*Cosh[c + d*x]^3*Sinh[c + d*x])/(24*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2))/(6*d)} -{Cosh[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^2, x, 3, (a^2*Sinh[c + d*x])/d + (2*a*b*Sinh[c + d*x]^3)/(3*d) + (b^2*Sinh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^2, x, 4, ((a - b)^2*ArcTan[Sinh[c + d*x]])/d + ((2*a - b)*b*Sinh[c + d*x])/d + (b^2*Sinh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^2, x, 5, (1/2)*(4*a - 3*b)*b*x + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + ((a - b)^2*Tanh[c + d*x])/d} -{Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^2, x, 5, ((a - b)*(a + 3*b)*ArcTan[Sinh[c + d*x]])/(2*d) + (b^2*Sinh[c + d*x])/d + ((a - b)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2, x, 4, b^2*x + ((a^2 - b^2)*Tanh[c + d*x])/d - ((a - b)^2*Tanh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^2, x, 4, ((3*a^2 + 2*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (3*(a^2 - b^2)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + ((a - b)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2, x, 3, (a^2*Tanh[c + d*x])/d - (2*a*(a - b)*Tanh[c + d*x]^3)/(3*d) + ((a - b)^2*Tanh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^2, x, 5, ((5*a^2 + 2*a*b + b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((5*a^2 + 2*a*b + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(16*d) + ((a - b)*(5*a + 3*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + ((a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(6*d)} - - -{Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3, x, 7, (3/256)*(4*a - b)*(8*a^2 - 2*a*b + b^2)*x + (3*(4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) + ((4*a - b)*(8*a^2 - 2*a*b + b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(128*d) + (b*(44*a^2 - 28*a*b + 5*b^2)*Cosh[c + d*x]^5*Sinh[c + d*x])/(160*d) + (b*Cosh[c + d*x]^9*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(10*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a*(10*a - b) - 5*(a - b)*(2*a - b)*Tanh[c + d*x]^2))/(80*d)} -{Cosh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3, x, 3, (a^3*Sinh[c + d*x])/d + (a^2*(a + 3*b)*Sinh[c + d*x]^3)/(3*d) + (3*a*b*(a + b)*Sinh[c + d*x]^5)/(5*d) + (b^2*(3*a + b)*Sinh[c + d*x]^7)/(7*d) + (b^3*Sinh[c + d*x]^9)/(9*d)} -{Cosh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3, x, 6, (1/128)*(64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*x + ((64*a^3 - 48*a^2*b + 24*a*b^2 - 5*b^3)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (b*(88*a^2 - 68*a*b + 15*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2)^2)/(8*d) + (b*Cosh[c + d*x]^5*Sinh[c + d*x]*(a*(8*a - b) - (8*a - 5*b)*(a - b)*Tanh[c + d*x]^2))/(48*d)} -{Cosh[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^3, x, 3, (a^3*Sinh[c + d*x])/d + (a^2*b*Sinh[c + d*x]^3)/d + (3*a*b^2*Sinh[c + d*x]^5)/(5*d) + (b^3*Sinh[c + d*x]^7)/(7*d)} -{Sech[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^3, x, 4, ((a - b)^3*ArcTan[Sinh[c + d*x]])/d + (b*(3*a^2 - 3*a*b + b^2)*Sinh[c + d*x])/d + ((3*a - b)*b^2*Sinh[c + d*x]^3)/(3*d) + (b^3*Sinh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^3, x, 6, (3/8)*b*(8*a^2 - 12*a*b + 5*b^2)*x + (3*(4*a - 3*b)*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d) + ((a - b)^3*Tanh[c + d*x])/d} -{Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3, x, 5, ((a - b)^2*(a + 5*b)*ArcTan[Sinh[c + d*x]])/(2*d) + ((3*a - 2*b)*b^2*Sinh[c + d*x])/d + (b^3*Sinh[c + d*x]^3)/(3*d) + ((a - b)^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^3, x, 5, (1/2)*(6*a - 5*b)*b^2*x + (b^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + ((a - b)^2*(a + 2*b)*Tanh[c + d*x])/d - ((a - b)^3*Tanh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^3, x, 6, (3*(a - b)*(4*b^2 + (a + b)^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (b^3*Sinh[c + d*x])/d + (3*(a - b)^2*(a + 3*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + ((a - b)^3*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^3, x, 4, b^3*x + ((a^3 - b^3)*Tanh[c + d*x])/d - ((a - b)^2*(2*a + b)*Tanh[c + d*x]^3)/(3*d) + ((a - b)^3*Tanh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3, x, 5, ((a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sech[c + d*x]*Tanh[c + d*x])/(48*d) + (5*(a^2 - b^2)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + ((a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)} -{Sech[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3, x, 3, (a^3*Tanh[c + d*x])/d - (a^2*(a - b)*Tanh[c + d*x]^3)/d + (3*a*(a - b)^2*Tanh[c + d*x]^5)/(5*d) - ((a - b)^3*Tanh[c + d*x]^7)/(7*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cosh[c + d*x]^7/(a + b*Sinh[c + d*x]^2), x, 4, -(((a - b)^3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Sinh[c + d*x])/(b^3*d) - ((a - 3*b)*Sinh[c + d*x]^3)/(3*b^2*d) + Sinh[c + d*x]^5/(5*b*d)} -{Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2), x, 6, ((8*a^2 - 20*a*b + 15*b^2)*x)/(8*b^3) - ((a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^3*d) - ((4*a - 7*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*b*d)} -{Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2), x, 4, ((a - b)^2*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Sinh[c + d*x])/(b^2*d) + Sinh[c + d*x]^3/(3*b*d)} -{Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2), x, 5, -(((2*a - 3*b)*x)/(2*b^2)) + ((a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d)} -{Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2), x, 3, -(((a - b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d)) + Sinh[c + d*x]/(b*d)} -{Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2), x, 4, x/b - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)} -{Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)} -{Sech[c + d*x]^1/(a + b*Sinh[c + d*x]^2), x, 4, ArcTan[Sinh[c + d*x]]/((a - b)*d) - (Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)} -{Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2), x, 3, -((b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(3/2)*d)) + Tanh[c + d*x]/((a - b)*d)} -{Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2), x, 5, ((a - 3*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^2*d) + (b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^2*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*(a - b)*d)} -{Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)*d) + ((a - 2*b)*Tanh[c + d*x])/((a - b)^2*d) - Tanh[c + d*x]^3/(3*(a - b)*d)} -{Sech[c + d*x]^5/(a + b*Sinh[c + d*x]^2), x, 6, ((3*a^2 - 10*a*b + 15*b^2)*ArcTan[Sinh[c + d*x]])/(8*(a - b)^3*d) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^2*d) + (Sech[c + d*x]^3*Tanh[c + d*x])/(4*(a - b)*d)} -{Sech[c + d*x]^6/(a + b*Sinh[c + d*x]^2), x, 4, -((b^3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Tanh[c + d*x])/((a - b)^3*d) - ((2*a - 3*b)*Tanh[c + d*x]^3)/(3*(a - b)^2*d) + Tanh[c + d*x]^5/(5*(a - b)*d)} - - -{Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^2, x, 6, -(((4*a - 5*b)*x)/(2*b^3)) + ((a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^3*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d*(a - (a - b)*Tanh[c + d*x]^2)) + ((a - b)*(2*a - b)*Tanh[c + d*x])/(2*a*b^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^2, x, 5, -(((3*a^2 - 2*a*b - b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d)) + Sinh[c + d*x]/(b^2*d) + ((a - b)^2*Sinh[c + d*x])/(2*a*b^2*d*(a + b*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2, x, 5, x/b^2 - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) - ((a - b)*Tanh[c + d*x])/(2*a*b*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2, x, 3, ((a + b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)*d) - ((a - b)*Sinh[c + d*x])/(2*a*b*d*(a + b*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2, x, 3, ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a - b]*d) + Tanh[c + d*x]/(2*a*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Sinh[c + d*x]/(2*a*d*(a + b*Sinh[c + d*x]^2))} -{Sech[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^2, x, 5, ArcTan[Sinh[c + d*x]]/((a - b)^2*d) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) - (b*Sinh[c + d*x])/(2*a*(a - b)*d*(a + b*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^2, x, 5, -(((4*a - b)*b*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(5/2)*d)) + Tanh[c + d*x]/((a - b)^2*d) + (b^2*Tanh[c + d*x])/(2*a*(a - b)^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^2, x, 6, ((a - 5*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^3*d) + ((5*a - b)*b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^3*d) + (b*(a + b)*Sinh[c + d*x])/(2*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*x]*Tanh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2))} -{Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^2, x, 5, ((6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(7/2)*d) + ((a - 3*b)*Tanh[c + d*x])/((a - b)^3*d) - Tanh[c + d*x]^3/(3*(a - b)^2*d) - (b^3*Tanh[c + d*x])/(2*a*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2))} - - -{Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^3, x, 6, x/b^3 - (Sqrt[a - b]*(8*a^2 + 4*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^3*d) - ((a - b)*Tanh[c + d*x])/(4*a*b*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - ((a - b)*(4*a + 3*b)*Tanh[c + d*x])/(8*a^2*b^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2)^3, x, 4, ((3*a^2 + 2*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2)*d) - ((a - b)*Cosh[c + d*x]^2*Sinh[c + d*x])/(4*a*b*d*(a + b*Sinh[c + d*x]^2)^2) + (3*(1/a^2 - 1/b^2)*Sinh[c + d*x])/(8*d*(a + b*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3, x, 4, (3*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[a - b]*d) + Tanh[c + d*x]/(4*a*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*a^2*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3, x, 4, ((a + 3*b)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*d) - ((a - b)*Sinh[c + d*x])/(4*a*b*d*(a + b*Sinh[c + d*x]^2)^2) + ((a + 3*b)*Sinh[c + d*x])/(8*a^2*b*d*(a + b*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3, x, 4, ((4*a - 3*b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(3/2)*d) - (b*Tanh[c + d*x])/(4*a*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + ((4*a - 3*b)*Tanh[c + d*x])/(8*a^2*(a - b)*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*d) + Sinh[c + d*x]/(4*a*d*(a + b*Sinh[c + d*x]^2)^2) + (3*Sinh[c + d*x])/(8*a^2*d*(a + b*Sinh[c + d*x]^2))} -{Sech[c + d*x]^1/(a + b*Sinh[c + d*x]^2)^3, x, 6, ArcTan[Sinh[c + d*x]]/((a - b)^3*d) - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*d) - (b*Sinh[c + d*x])/(4*a*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - ((7*a - 3*b)*b*Sinh[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Sinh[c + d*x]^2)^3, x, 6, -((3*b*(8*a^2 - 4*a*b + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(7/2)*d)) + Tanh[c + d*x]/((a - b)^3*d) - (b^3*Tanh[c + d*x])/(4*a*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2)^2) + (3*(4*a - b)*b^2*Tanh[c + d*x])/(8*a^2*(a - b)^3*d*(a - (a - b)*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2)^3, x, 7, ((a - 7*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^4*d) + (b^(3/2)*(35*a^2 - 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^4*d) + (b*(2*a + b)*Sinh[c + d*x])/(4*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^2)^2) + ((4*a - b)*b*(a + 3*b)*Sinh[c + d*x])/(8*a^2*(a - b)^3*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*x]*Tanh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2)} -{Sech[c + d*x]^4/(a + b*Sinh[c + d*x]^2)^3, x, 6, (b^2*(48*a^2 - 16*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^(9/2)*d) + ((a - 4*b)*Tanh[c + d*x])/((a - b)^4*d) - Tanh[c + d*x]^3/(3*(a - b)^3*d) + (b^4*Tanh[c + d*x])/(4*a*(a - b)^4*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - ((16*a - 3*b)*b^3*Tanh[c + d*x])/(8*a^2*(a - b)^4*d*(a - (a - b)*Tanh[c + d*x]^2))} - - -{Cosh[x]^2/(1 - Sinh[x]^2), x, 4, -x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]} -{Cosh[x]^3/(1 - Sinh[x]^2), x, 3, 2*ArcTanh[Sinh[x]] - Sinh[x]} -{Cosh[x]^4/(1 - Sinh[x]^2), x, 5, -((5*x)/2) + 2*Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]] - (1/2)*Cosh[x]*Sinh[x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cosh[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -(a*(a - 4*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*b^(3/2)*f) - ((a - 4*b)*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(8*b*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*b*f)} -{Cosh[e + f*x]^1*Sqrt[a + b*Sinh[e + f*x]^2], x, 4, (a*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[b]*f) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*f)} -{Sech[e + f*x]^1*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/f} -{Sech[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2], x, 4, (a*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[a - b]*f) + (Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(2*f)} -{Sech[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, (a*(3*a - 4*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*(a - b)^(3/2)*f) + ((3*a - 4*b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(8*(a - b)*f) + (Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/(4*(a - b)*f)} - -{Cosh[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2], x, 7, -((2*(a - 3*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b*f)) + (Cosh[e + f*x]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(5*b*f) + ((2*a^2 - 7*a*b - 3*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 9*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((2*a^2 - 7*a*b - 3*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(15*b^2*f)} -{Cosh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, (Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - ((a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*b*f)} -{Cosh[e + f*x]^0*Sqrt[a + b*Sinh[e + f*x]^2], x, 2, ((-I)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2], x, 2, (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, ((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)} - - -{Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -(a^2*(a - 6*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(16*b^(3/2)*f) - (a*(a - 6*b)*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(16*b*f) - ((a - 6*b)*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(24*b*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(5/2))/(6*b*f)} -{Cosh[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (3*a^2*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*Sqrt[b]*f) + (3*a*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(8*f) + (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*f)} -{Sech[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, ((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/f + ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*f) + (b*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*f)} -{Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, (Sqrt[a - b]*(a + 2*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/f + ((a - b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(2*f)} -{Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (3*a^2*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(8*Sqrt[a - b]*f) + (3*a*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(8*f) + (Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/(4*f)} -{Sech[e + f*x]^7*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, (a^2*(5*a - 6*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(16*(a - b)^(3/2)*f) + (a*(5*a - 6*b)*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(16*(a - b)*f) + ((5*a - 6*b)*Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/(24*(a - b)*f) + (Sech[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(5/2)*Tanh[e + f*x])/(6*(a - b)*f)} - -{Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2), x, 8, ((a^2 + 9*a*b - 2*b^2)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b*f) + (2*(4*a - b)*Cosh[e + f*x]^3*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*f) + (b*Cosh[e + f*x]^5*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(7*f) + (2*(a + b)*(a^2 - 6*a*b + b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a^2 - 18*a*b + b^2)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(35*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a + b)*(a^2 - 6*a*b + b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(35*b^2*f)} -{Cosh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, (2*(3*a - b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*f) + (b*Cosh[e + f*x]^3*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(5*f) - ((3*a^2 + 7*a*b - 2*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((9*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(15*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a^2 + 7*a*b - 2*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(15*b*f)} -{Cosh[e + f*x]^0*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, (b*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*a*(a - b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, ((a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f + ((a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f} -{Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (2*(a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a - b)*Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cosh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, -((a - 2*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*b^(3/2)*f) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*b*f)} -{Cosh[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(Sqrt[b]*f)} -{Sech[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(Sqrt[a - b]*f)} -{Sech[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, ((a - 2*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*(a - b)^(3/2)*f) + (Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(2*(a - b)*f)} - -{Cosh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 6, (Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b*f) + (2*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*b^2*f)} -{Cosh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(b*f)} -{Cosh[e + f*x]^0/Sqrt[a + b*Sinh[e + f*x]^2], x, 2, ((-I)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 7, (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Sech[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, (2*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((a - 3*b)*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*f)} - - -{Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(b^(3/2)*f) - ((a - b)*Sinh[e + f*x])/(a*b*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Cosh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 2, Sinh[e + f*x]/(a*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, ((a - 4*b)*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*(a - b)^(5/2)*f) + (b*(a + 2*b)*Sinh[e + f*x])/(2*a*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + (Sech[e + f*x]*Tanh[e + f*x])/(2*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} - -{Cosh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, -(((a - b)*Cosh[e + f*x]^3*Sinh[e + f*x])/(a*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) + ((4*a - 3*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*b^2*f) + ((8*a^2 - 13*a*b + 3*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*b^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(2*a - 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - ((8*a^2 - 13*a*b + 3*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a*b^3*f)} -{Cosh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -(((a - b)*Cosh[e + f*x]*Sinh[e + f*x])/(a*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) - ((2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((2*a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(a*b^2*f)} -{Cosh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 2, (Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(Sqrt[a]*Sqrt[b]*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2])} -{Cosh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -((b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])) - (I*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (Sqrt[b]*(a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(Sqrt[a]*(a - b)^2*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - (2*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + Tanh[e + f*x]/((a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} - - -{Cosh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/(b^(5/2)*f) - ((a - b)*Cosh[e + f*x]^2*Sinh[e + f*x])/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((a - b)*(3*a + 2*b)*Sinh[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Cosh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2), x, 3, (Cosh[e + f*x]^2*Sinh[e + f*x])/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*Sinh[e + f*x])/(3*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Cosh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 3, Sinh[e + f*x]/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*Sinh[e + f*x])/(3*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Sinh[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} - -{Cosh[e + f*x]^6/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(((a - b)*Cosh[e + f*x]^3*Sinh[e + f*x])/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2))) - (2*(a - b)*(2*a + b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*a^2*b^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((8*a^2 - 3*a*b - 2*b^2)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*b^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((4*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*b^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((8*a^2 - 3*a*b - 2*b^2)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^2*b^3*f)} -{Cosh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, -(((a - b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2))) + (2*(a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*a^(3/2)*b^(3/2)*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*b*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Cosh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, (Cosh[e + f*x]*Sinh[e + f*x])/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((a - 2*b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*a^(3/2)*(a - b)*Sqrt[b]*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Cosh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(2*a - b)*b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)^2*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Sech[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, (b*(3*a + b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (Sqrt[b]*(3*a^2 + 7*a*b - 2*b^2)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*a^(3/2)*(a - b)^3*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - ((9*a - b)*b*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*(a - b)^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + Tanh[e + f*x]/((a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Cosh[e+f x])^m (a+b Sinh[e+f x]^2)^p when p symbolic*) - - -{(d*Cosh[e + f*x])^m*(a + b*Sinh[e + f*x]^2)^p, x, 3, (d*AppellF1[1/2, (1 - m)/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*(d*Cosh[e + f*x])^(-1 + m)*(Cosh[e + f*x]^2)^((1 - m)/2)*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} - - -{Cosh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x, 5, If[$VersionNumber>=8, -(((3*a - b*(7 + 2*p))*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(1 + p))/(b^2*f*(3 + 2*p)*(5 + 2*p))) + (Cosh[e + f*x]^2*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 - 2*a*b*(5 + 2*p) + b^2*(15 + 16*p + 4*p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(b^2*f*(3 + 2*p)*(5 + 2*p)*(1 + (b*Sinh[e + f*x]^2)/a)^p), -(((3*a - b*(7 + 2*p))*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(1 + p))/(b^2*f*(15 + 16*p + 4*p^2))) + (Cosh[e + f*x]^2*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(1 + p))/(b*f*(5 + 2*p)) + ((3*a^2 - 2*a*b*(5 + 2*p) + b^2*(15 + 16*p + 4*p^2))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/((1 + (b*Sinh[e + f*x]^2)/a)^p*(b^2*f*(15 + 16*p + 4*p^2)))]} -{Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x, 4, (Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(1 + p))/(b*f*(3 + 2*p)) - ((a - b*(3 + 2*p))*Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(b*f*(3 + 2*p)*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Cosh[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^p, x, 3, (Hypergeometric2F1[1/2, -p, 3/2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Sech[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^p)/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} - -{Cosh[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, -3/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Cosh[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, -1/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Cosh[e + f*x]^0*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 1/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 3/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} -{Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^p, x, 3, (AppellF1[1/2, 5/2, -p, 3/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*Sqrt[Cosh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^p*Tanh[e + f*x])/(f*(1 + (b*Sinh[e + f*x]^2)/a)^p)} - - -(* ::Section:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -{Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]]), x, 4, -((2*a*(a^4 + b^4)^2*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^10*d)) + (2*(a^4 + b^4)^2*Sqrt[Sinh[c + d*x]])/(b^9*d) - (a^3*(a^4 + 2*b^4)*Sinh[c + d*x])/(b^8*d) + (2*a^2*(a^4 + 2*b^4)*Sinh[c + d*x]^(3/2))/(3*b^7*d) - (a*(a^4 + 2*b^4)*Sinh[c + d*x]^2)/(2*b^6*d) + (2*(a^4 + 2*b^4)*Sinh[c + d*x]^(5/2))/(5*b^5*d) - (a^3*Sinh[c + d*x]^3)/(3*b^4*d) + (2*a^2*Sinh[c + d*x]^(7/2))/(7*b^3*d) - (a*Sinh[c + d*x]^4)/(4*b^2*d) + (2*Sinh[c + d*x]^(9/2))/(9*b*d)} -{Cosh[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]]), x, 4, -((2*a*(a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^6*d)) + (2*(a^4 + b^4)*Sqrt[Sinh[c + d*x]])/(b^5*d) - (a^3*Sinh[c + d*x])/(b^4*d) + (2*a^2*Sinh[c + d*x]^(3/2))/(3*b^3*d) - (a*Sinh[c + d*x]^2)/(2*b^2*d) + (2*Sinh[c + d*x]^(5/2))/(5*b*d)} -{Cosh[c + d*x]^1/(a + b*Sqrt[Sinh[c + d*x]]), x, 4, -((2*a*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^2*d)) + (2*Sqrt[Sinh[c + d*x]])/(b*d)} -{Sech[c + d*x]^1/(a + b*Sqrt[Sinh[c + d*x]]), x, 19, (b*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)*d) - (b*(a^2 - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)*d) + (a^3*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)*d) + (a*b^2*Log[Cosh[c + d*x]])/((a^4 + b^4)*d) - (2*a*b^2*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)*d) - (b*(a^2 + b^2)*Log[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(2*Sqrt[2]*(a^4 + b^4)*d) + (b*(a^2 + b^2)*Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(2*Sqrt[2]*(a^4 + b^4)*d)} -(* {Sech[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]]), x, 00, (a^2*b^3*(a^2 + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)^2*d) + (b*(a^2 - 3*b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(4*Sqrt[2]*(a^4 + b^4)*d) - (a^2*b^3*(a^2 + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)^2*d) - (b*(a^2 - 3*b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(4*Sqrt[2]*(a^4 + b^4)*d) + (a^3*b^4*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)^2*d) + (a^3*ArcTan[Sinh[c + d*x]])/(2*(a^4 + b^4)*d) - (a^2*b^3*(a^2 - b^2)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(Sqrt[2]*(a^4 + b^4)^2*d) + (b*(a^2 + 3*b^2)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(4*Sqrt[2]*(a^4 + b^4)*d) - (2*a*b^6*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^2*d) + (a*b^6*Log[1 + Sinh[c + d*x]^2])/(2*(a^4 + b^4)^2*d) - (a*(b^2 - a^2*Sinh[c + d*x]))/(2*(a^4 + b^4)*d*(1 + Sinh[c + d*x]^2)) + (b*Sqrt[Sinh[c + d*x]]*(b^2 - a^2*Sinh[c + d*x]))/(2*(a^4 + b^4)*d*(1 + Sinh[c + d*x]^2))} *) - - -{Cosh[c + d*x]^5/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 4, (2*(a^4 + b^4)*(9*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^10*d) + (2*a*(a^4 + b^4)^2)/(b^10*d*(a + b*Sqrt[Sinh[c + d*x]])) - (16*a^3*(a^4 + b^4)*Sqrt[Sinh[c + d*x]])/(b^9*d) + (a^2*(7*a^4 + 6*b^4)*Sinh[c + d*x])/(b^8*d) - (4*a*(3*a^4 + 2*b^4)*Sinh[c + d*x]^(3/2))/(3*b^7*d) + ((5*a^4 + 2*b^4)*Sinh[c + d*x]^2)/(2*b^6*d) - (8*a^3*Sinh[c + d*x]^(5/2))/(5*b^5*d) + (a^2*Sinh[c + d*x]^3)/(b^4*d) - (4*a*Sinh[c + d*x]^(7/2))/(7*b^3*d) + Sinh[c + d*x]^4/(4*b^2*d)} -{Cosh[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 4, (2*(5*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^6*d) + (2*a*(a^4 + b^4))/(b^6*d*(a + b*Sqrt[Sinh[c + d*x]])) - (8*a^3*Sqrt[Sinh[c + d*x]])/(b^5*d) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (4*a*Sinh[c + d*x]^(3/2))/(3*b^3*d) + Sinh[c + d*x]^2/(2*b^2*d)} -{Cosh[c + d*x]^1/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 4, (2*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^2*d) + (2*a)/(b^2*d*(a + b*Sqrt[Sinh[c + d*x]]))} -{Sech[c + d*x]^1/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 19, (Sqrt[2]*a*b*(a^4 - 2*a^2*b^2 - b^4)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^2*d) - (Sqrt[2]*a*b*(a^4 - 2*a^2*b^2 - b^4)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^2*d) + (a^2*(a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)^2*d) + (b^2*(3*a^4 - b^4)*Log[Cosh[c + d*x]])/((a^4 + b^4)^2*d) - (2*b^2*(3*a^4 - b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^2*d) - (a*b*(a^4 + 2*a^2*b^2 - b^4)*Log[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(Sqrt[2]*(a^4 + b^4)^2*d) + (a*b*(a^4 + 2*a^2*b^2 - b^4)*Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(Sqrt[2]*(a^4 + b^4)^2*d) + (2*a*b^2)/((a^4 + b^4)*d*(a + b*Sqrt[Sinh[c + d*x]]))} -(* {Sech[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 00, (a*b*(a^4 - 6*a^2*b^2 - b^4)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqrt[2]*(a^4 + b^4)^2*d) + (Sqrt[2]*a*b^3*(2*a^6 + 3*a^4*b^2 - 2*a^2*b^4 - b^6)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) - (a*b*(a^4 - 6*a^2*b^2 - b^4)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqrt[2]*(a^4 + b^4)^2*d) - (Sqrt[2]*a*b^3*(2*a^6 + 3*a^4*b^2 - 2*a^2*b^4 - b^6)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) + (a^2*b^4*(5*a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)^3*d) + (a^2*(a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]])/(2*(a^4 + b^4)^2*d) + (a*b*(a^4 + 6*a^2*b^2 - b^4)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(2*Sqrt[2]*(a^4 + b^4)^2*d) - (Sqrt[2]*a*b^3*(2*a^6 - 3*a^4*b^2 - 2*a^2*b^4 + b^6)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/((a^4 + b^4)^3*d) - (2*b^6*(7*a^4 - b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) + (b^6*(7*a^4 - b^4)*Log[1 + Sinh[c + d*x]^2])/(2*(a^4 + b^4)^3*d) + (2*a*b^6)/((a^4 + b^4)^2*d*(a + b*Sqrt[Sinh[c + d*x]])) - (b^2*(3*a^4 - b^4) - a^2*(a^4 - 3*b^4)*Sinh[c + d*x])/(2*(a^4 + b^4)^2*d*(1 + Sinh[c + d*x]^2)) + (a*b*Sqrt[Sinh[c + d*x]]*(2*a^2*b^2 - (a^4 - b^4)*Sinh[c + d*x]))/((a^4 + b^4)^2*d*(1 + Sinh[c + d*x]^2))} *) - - -{Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^n), x, 6, (Hypergeometric2F1[1, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a*d) + (2*Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a*d) + (Hypergeometric2F1[1, 5/n, (5 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^5)/(5*a*d)} -{Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^n), x, 5, (Hypergeometric2F1[1, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a*d) + (Hypergeometric2F1[1, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a*d)} -{Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]^n), x, 2, (Hypergeometric2F1[1, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a*d)} - - -{Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^n)^2, x, 6, (Hypergeometric2F1[2, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a^2*d) + (2*Hypergeometric2F1[2, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a^2*d) + (Hypergeometric2F1[2, 5/n, (5 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^5)/(5*a^2*d)} -{Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]^n)^2, x, 5, (Hypergeometric2F1[2, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a^2*d) + (Hypergeometric2F1[2, 3/n, (3 + n)/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x]^3)/(3*a^2*d)} -{Cosh[c + d*x]^1/(a + b*Sinh[c + d*x]^n)^2, x, 2, (Hypergeometric2F1[2, 1/n, 1 + 1/n, -((b*Sinh[c + d*x]^n)/a)]*Sinh[c + d*x])/(a^2*d)} - - -(* ::Title::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^p*) - - -{Coth[x]^1/(1 - Sinh[x]^2), x, 4, Log[Sinh[x]] - (1/2)*Log[1 - Sinh[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^(p/2) when a-b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^5, x, 5, -a^2/(3*f*(a*Cosh[e + f*x]^2)^(3/2)) + (2*a)/(f*Sqrt[a*Cosh[e + f*x]^2]) + Sqrt[a*Cosh[e + f*x]^2]/f} -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^3, x, 5, a/(f*Sqrt[a*Cosh[e + f*x]^2]) + Sqrt[a*Cosh[e + f*x]^2]/f} -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^1, x, 4, Sqrt[a*Cosh[e + f*x]^2]/f} -{Coth[e + f*x]^1*Sqrt[a + a*Sinh[e + f*x]^2], x, 5, -((Sqrt[a]*ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a*Cosh[e + f*x]^2]/f} -{Coth[e + f*x]^3*Sqrt[a + a*Sinh[e + f*x]^2], x, 7, (-3*Sqrt[a]*ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]])/(2*f) + (3*Sqrt[a*Cosh[e + f*x]^2])/(2*f) - ((a*Cosh[e + f*x]^2)^(3/2)*Csch[e + f*x]^2)/(2*a*f)} - -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^6, x, 7, -((15*ArcTan[Sinh[e + f*x]]*Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x])/(8*f)) + (15*Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/(8*f) - (5*Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x]^3)/(8*f) - (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x]^5)/(4*f)} -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^4, x, 6, (-3*ArcTan[Sinh[e + f*x]]*Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x])/(2*f) + (3*Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/(2*f) - (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x]^3)/(2*f)} -{Sqrt[a + a*Sinh[e + f*x]^2]*Tanh[e + f*x]^2, x, 5, -((ArcTan[Sinh[e + f*x]]*Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x])/f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/f} -{Coth[e + f*x]^2*Sqrt[a + a*Sinh[e + f*x]^2], x, 5, -((Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/f} -{Coth[e + f*x]^4*Sqrt[a + a*Sinh[e + f*x]^2], x, 5, (-2*Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e + f*x])/(3*f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/f} -{Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2], x, 5, (-3*Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^5*Sech[e + f*x])/(5*f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f*x])/f} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[e + f*x]^5/Sqrt[a + a*Sinh[e + f*x]^2], x, 5, -a^2/(5*f*(a*Cosh[e + f*x]^2)^(5/2)) + (2*a)/(3*f*(a*Cosh[e + f*x]^2)^(3/2)) - 1/(f*Sqrt[a*Cosh[e + f*x]^2])} -{Tanh[e + f*x]^3/Sqrt[a + a*Sinh[e + f*x]^2], x, 5, a/(3*f*(a*Cosh[e + f*x]^2)^(3/2)) - 1/(f*Sqrt[a*Cosh[e + f*x]^2])} -{Tanh[e + f*x]^1/Sqrt[a + a*Sinh[e + f*x]^2], x, 4, -(1/(f*Sqrt[a*Cosh[e + f*x]^2]))} -{Coth[e + f*x]^1/Sqrt[a + a*Sinh[e + f*x]^2], x, 4, -(ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))} -{Coth[e + f*x]^3/Sqrt[a + a*Sinh[e + f*x]^2], x, 6, -ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]]/(2*Sqrt[a]*f) - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^2)/(2*a*f)} - -{Tanh[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2], x, 5, (3*ArcTan[Sinh[e + f*x]]*Cosh[e + f*x])/(8*f*Sqrt[a*Cosh[e + f*x]^2]) - (3*Tanh[e + f*x])/(8*f*Sqrt[a*Cosh[e + f*x]^2]) - Tanh[e + f*x]^3/(4*f*Sqrt[a*Cosh[e + f*x]^2])} -{Tanh[e + f*x]^2/Sqrt[a + a*Sinh[e + f*x]^2], x, 4, (ArcTan[Sinh[e + f*x]]*Cosh[e + f*x])/(2*f*Sqrt[a*Cosh[e + f*x]^2]) - Tanh[e + f*x]/(2*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^2/Sqrt[a + a*Sinh[e + f*x]^2], x, 4, -(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2]))} -{Coth[e + f*x]^4/Sqrt[a + a*Sinh[e + f*x]^2], x, 4, -(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2])) - (Coth[e + f*x]*Csch[e + f*x]^2)/(3*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^6/Sqrt[a + a*Sinh[e + f*x]^2], x, 5, -(Coth[e + f*x]/(f*Sqrt[a*Cosh[e + f*x]^2])) - (2*Coth[e + f*x]*Csch[e + f*x]^2)/(3*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^4)/(5*f*Sqrt[a*Cosh[e + f*x]^2])} - - -{Tanh[e + f*x]^5/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, -a^2/(7*f*(a*Cosh[e + f*x]^2)^(7/2)) + (2*a)/(5*f*(a*Cosh[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cosh[e + f*x]^2)^(3/2))} -{Tanh[e + f*x]^3/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, a/(5*f*(a*Cosh[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cosh[e + f*x]^2)^(3/2))} -{Tanh[e + f*x]^1/(a + a*Sinh[e + f*x]^2)^(3/2), x, 4, -1/(3*f*(a*Cosh[e + f*x]^2)^(3/2))} -{Coth[e + f*x]^1/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, -(ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^3/(a + a*Sinh[e + f*x]^2)^(3/2), x, 6, ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]]/(2*a^(3/2)*f) - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^2)/(2*a^2*f)} - -{Tanh[e + f*x]^2/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, (ArcTan[Sinh[e + f*x]]*Cosh[e + f*x])/(8*a*f*Sqrt[a*Cosh[e + f*x]^2]) + Tanh[e + f*x]/(8*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Sech[e + f*x]^2*Tanh[e + f*x])/(4*a*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^2/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, -((ArcTan[Sinh[e + f*x]]*Cosh[e + f*x])/(a*f*Sqrt[a*Cosh[e + f*x]^2])) - Coth[e + f*x]/(a*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^4/(a + a*Sinh[e + f*x]^2)^(3/2), x, 4, -(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^6/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, -(Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sqrt[a*Cosh[e + f*x]^2])} -{Coth[e + f*x]^8/(a + a*Sinh[e + f*x]^2)^(3/2), x, 5, -((Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*f*Sqrt[a*Cosh[e + f*x]^2])) - (2*Coth[e + f*x]*Csch[e + f*x]^4)/(5*a*f*Sqrt[a*Cosh[e + f*x]^2]) - (Coth[e + f*x]*Csch[e + f*x]^6)/(7*a*f*Sqrt[a*Cosh[e + f*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^5, x, 6, -((8*a^2 - 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(8*(a - b)^(3/2)*f) + ((8*a^2 - 24*a*b + 15*b^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)^2*f) + ((8*a - 7*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/(8*(a - b)^2*f) - (Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*(a - b)*f)} -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^3, x, 5, -((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*Sqrt[a - b]*f) + ((2*a - 3*b)*Sqrt[a + b*Sinh[e + f*x]^2])/(2*(a - b)*f) + (Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/(2*(a - b)*f)} -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^1, x, 4, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/f) + Sqrt[a + b*Sinh[e + f*x]^2]/f} -{Coth[e + f*x]^1*Sqrt[a + b*Sinh[e + f*x]^2], x, 4, -((Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/f) + Sqrt[a + b*Sinh[e + f*x]^2]/f} -{Coth[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*Sqrt[a]*f) + ((2*a + b)*Sqrt[a + b*Sinh[e + f*x]^2])/(2*a*f) - (Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/(2*a*f)} -{Coth[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, -((8*a^2 + 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*a^(3/2)*f) + ((8*a^2 + 8*a*b - b^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^2*f) - ((8*a - b)*Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2))/(8*a^2*f) - (Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*a*f)} - -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^4, x, 7, -((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a - 8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*f) - ((3*a - 4*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*f) - (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^3)/(3*f)} -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^2, x, 6, (-2*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f} -{Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x]^0, x, 2, ((-I)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Coth[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2], x, 6, -((Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/f) - (2*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((a + b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f} -{Coth[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2], x, 7, -((3*a + b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) - (Coth[e + f*x]^3*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - ((7*a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a*f)} - - -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^5, x, 7, -((8*a^2 - 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(8*Sqrt[a - b]*f) + ((8*a^2 - 40*a*b + 35*b^2)*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)*f) + ((8*a^2 - 40*a*b + 35*b^2)*(a + b*Sinh[e + f*x]^2)^(3/2))/(24*(a - b)^2*f) + ((8*a - 9*b)*Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(8*(a - b)^2*f) - (Sech[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(5/2))/(4*(a - b)*f)} -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^3, x, 6, -((2*a - 5*b)*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*f) + ((2*a - 5*b)*Sqrt[a + b*Sinh[e + f*x]^2])/(2*f) + ((2*a - 5*b)*(a + b*Sinh[e + f*x]^2)^(3/2))/(6*(a - b)*f) + (Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(2*(a - b)*f)} -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^1, x, 5, -(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/f) + ((a - b)*Sqrt[a + b*Sinh[e + f*x]^2])/f + (a + b*Sinh[e + f*x]^2)^(3/2)/(3*f)} -{Coth[e + f*x]^1*(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, -((a^(3/2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/f) + (a*Sqrt[a + b*Sinh[e + f*x]^2])/f + (a + b*Sinh[e + f*x]^2)^(3/2)/(3*f)} -{Coth[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -(Sqrt[a]*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*f) + ((2*a + 3*b)*Sqrt[a + b*Sinh[e + f*x]^2])/(2*f) + ((2*a + 3*b)*(a + b*Sinh[e + f*x]^2)^(3/2))/(6*a*f) - (Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(2*a*f)} -{Coth[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, -((8*a^2 + 3*b*(8*a + b))*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((8*a^2 + 3*b*(8*a + b))*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a*f) + ((8*a^2 + 3*b*(8*a + b))*(a + b*Sinh[e + f*x]^2)^(3/2))/(24*a^2*f) - ((8*a + b)*Csch[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(5/2))/(8*a^2*f) - (Csch[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(5/2))/(4*a*f)} - -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^4, x, 8, -((3*a - 8*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (8*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 8*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (8*(a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f) + ((a - 2*b)*Sinh[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/f - ((a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^3)/(3*f)} -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^2, x, 7, (4*b*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - ((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a - 8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f) - ((a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x])/f} -{(a + b*Sinh[e + f*x]^2)^(3/2)*Tanh[e + f*x]^0, x, 6, (b*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*a*(a - b)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^2*(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, (4*b*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (Coth[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/f - ((7*a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)} -{Coth[e + f*x]^4*(a + b*Sinh[e + f*x]^2)^(3/2), x, 8, -(((a + b)*Cosh[e + f*x]^2*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/f) + ((3*a + 5*b)*Cosh[e + f*x]*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f) - (Coth[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2))/(3*f) - (8*(a + b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a + b)*(a + 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (8*(a + b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*f)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(8*(a - b)^(5/2)*f) + ((8*a - 5*b)*Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(8*(a - b)^2*f) - (Sech[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2])/(4*(a - b)*f)} -{Tanh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, -((2*a - b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*(a - b)^(3/2)*f) + (Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(2*(a - b)*f)} -{Tanh[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f))} -{Coth[e + f*x]^1/Sqrt[a + b*Sinh[e + f*x]^2], x, 3, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f))} -{Coth[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2], x, 4, -((2*a - b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*a^(3/2)*f) - (Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(2*a*f)} -{Coth[e + f*x]^5/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, -((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) - ((8*a - 3*b)*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(8*a^2*f) - (Csch[e + f*x]^4*Sqrt[a + b*Sinh[e + f*x]^2])/(4*a*f)} - -{Tanh[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 5, (-2*(2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sech[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*(a - b)*f)} -{Tanh[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 6, -((EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/((a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a])} -{Tanh[e + f*x]^0/Sqrt[a + b*Sinh[e + f*x]^2], x, 2, ((-I)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^2/Sqrt[a + b*Sinh[e + f*x]^2], x, 6, -((Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f)) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(a*f)} -{Coth[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2], x, 7, (-2*(2*a - b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f) - (Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*f) - (2*(2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*(2*a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^2*f)} - - -{Tanh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -((8*a^2 + 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(8*(a - b)^(7/2)*f) + (8*a^2 + 8*a*b - b^2)/(8*(a - b)^3*f*Sqrt[a + b*Sinh[e + f*x]^2]) + ((8*a - 3*b)*Sech[e + f*x]^2)/(8*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - Sech[e + f*x]^4/(4*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, -((2*a + b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*(a - b)^(5/2)*f) + (2*a + b)/(2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + Sech[e + f*x]^2/(2*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f)) + 1/((a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + 1/(a*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, -((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*a^(5/2)*f) + (2*a - 3*b)/(2*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - Csch[e + f*x]^2/(2*a*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -((8*a^2 - 24*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*a^(7/2)*f) + (8*a^2 - 24*a*b + 15*b^2)/(8*a^3*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((8*a - 5*b)*Csch[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - Csch[e + f*x]^4/(4*a*f*Sqrt[a + b*Sinh[e + f*x]^2])} - -{Tanh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2), x, 6, -(Sqrt[a]*Sqrt[b]*(7*a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*(a - b)^3*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*(a - b)^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (4*a*Tanh[e + f*x])/(3*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) + (Sech[e + f*x]^2*Tanh[e + f*x])/(3*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 5, (-2*Sqrt[a]*Sqrt[b]*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/((a - b)^2*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + ((a + b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - Tanh[e + f*x]/((a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(3/2), x, 4, -((b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])) - (I*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])} -{Coth[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(3/2), x, 7, Coth[e + f*x]/(a*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (2*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*f) - (2*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (2*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(a^2*f)} -{Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(3/2), x, 8, -(((a - b)*Coth[e + f*x]*Csch[e + f*x]^2)/(a*b*f*Sqrt[a + b*Sinh[e + f*x]^2])) - ((7*a - 8*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*f) + ((3*a - 4*b)*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*b*f) - ((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a - 8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^3*f)} - - -{Tanh[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -((8*a^2 + 24*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(8*(a - b)^(9/2)*f) + (8*a^2 + 24*a*b + 3*b^2)/(24*(a - b)^3*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((8*a - b)*Sech[e + f*x]^2)/(8*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - Sech[e + f*x]^4/(4*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (8*a^2 + 24*a*b + 3*b^2)/(8*(a - b)^4*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, -((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]])/(2*(a - b)^(7/2)*f) + (2*a + 3*b)/(6*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + Sech[e + f*x]^2/(2*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*a + 3*b)/(2*(a - b)^3*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Tanh[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f)) + 1/(3*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + 1/((a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^1/(a + b*Sinh[e + f*x]^2)^(5/2), x, 5, -(ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + 1/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + 1/(a^2*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^3/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, -((2*a - 5*b)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(2*a^(7/2)*f) + (2*a - 5*b)/(6*a^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - Csch[e + f*x]^2/(2*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (2*a - 5*b)/(2*a^3*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^5/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -((8*a^2 - 40*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a]])/(8*a^(9/2)*f) + (8*a^2 - 40*a*b + 35*b^2)/(24*a^3*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - ((8*a - 7*b)*Csch[e + f*x]^2)/(8*a^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - Csch[e + f*x]^4/(4*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (8*a^2 - 40*a*b + 35*b^2)/(8*a^4*f*Sqrt[a + b*Sinh[e + f*x]^2])} - -{Tanh[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(b*(5*a + 3*b)*Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)^3*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (8*Sqrt[a]*Sqrt[b]*(a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*(a - b)^4*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + ((3*a + b)*(a + 3*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^4*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - (2*(2*a + b)*Tanh[e + f*x])/(3*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + (Sech[e + f*x]^2*Tanh[e + f*x])/(3*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2))} -{Tanh[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 6, (-4*b*Cosh[e + f*x]*Sinh[e + f*x])/(3*(a - b)^2*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (Sqrt[b]*(7*a + b)*Cosh[e + f*x]*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b])/(3*Sqrt[a]*(a - b)^3*f*Sqrt[(a*Cosh[e + f*x]^2)/(a + b*Sinh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) + ((3*a + 5*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a*(a - b)^3*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) - Tanh[e + f*x]/((a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2))} -{Tanh[e + f*x]^0/(a + b*Sinh[e + f*x]^2)^(5/2), x, 7, -(b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a*(a - b)*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(2*a - b)*b*Cosh[e + f*x]*Sinh[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (((2*I)/3)*(2*a - b)*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a^2*(a - b)^2*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]) + ((I/3)*EllipticF[I*e + I*f*x, b/a]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2])} -{Coth[e + f*x]^2/(a + b*Sinh[e + f*x]^2)^(5/2), x, 8, Coth[e + f*x]/(3*a*f*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((3*a - 4*b)*Coth[e + f*x])/(3*a^2*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2]) - ((7*a - 8*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)*f) - ((7*a - 8*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 4*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*(a - b)*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((7*a - 8*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^3*(a - b)*f)} -{Coth[e + f*x]^4/(a + b*Sinh[e + f*x]^2)^(5/2), x, 9, -((a - b)*Coth[e + f*x]*Csch[e + f*x]^2)/(3*a*b*f*(a + b*Sinh[e + f*x]^2)^(3/2)) - (2*(a - 3*b)*Coth[e + f*x]*Csch[e + f*x]^2)/(3*a^2*b*f*Sqrt[a + b*Sinh[e + f*x]^2]) - (8*(a - 2*b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^4*f) + ((3*a - 8*b)*Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^3*b*f) - (8*(a - 2*b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^4*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - 8*b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^4*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + (8*(a - 2*b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^4*f)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d Tanh[e+f x])^m (a+b Sinh[e+f x]^2)^p when p symbolic*) - - -{(a + b*Sinh[e + f*x]^2)^p*(d*Tanh[e + f*x])^m, x, 3, (AppellF1[(1 + m)/2, (1 + m)/2, -p, (3 + m)/2, -Sinh[e + f*x]^2, -((b*Sinh[e + f*x]^2)/a)]*(Cosh[e + f*x]^2)^((1 + m)/2)*(a + b*Sinh[e + f*x]^2)^p*(d*Tanh[e + f*x])^(1 + m))/((1 + (b*Sinh[e + f*x]^2)/a)^p*(d*f*(1 + m)))} - - -{(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^3, x, 3, -((a - b*(1 + p))*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sinh[c + d*x]^2)/(a - b)]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a - b)^2*d*(1 + p)) + (Sech[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a - b)*d)} -{(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^1, x, 2, -(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sinh[c + d*x]^2)/(a - b)]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*(a - b)*d*(1 + p))} -{Coth[c + d*x]^1*(a + b*Sinh[c + d*x]^2)^p, x, 2, -(Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sinh[c + d*x]^2)/a]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*a*d*(1 + p))} -{Coth[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^p, x, 3, -(Csch[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*a*d) - ((a + b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sinh[c + d*x]^2)/a]*(a + b*Sinh[c + d*x]^2)^(1 + p))/(2*a^2*d*(1 + p))} - -{(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^4, x, 3, (AppellF1[5/2, 5/2, -p, 7/2, -Sinh[c + d*x]^2, -((b*Sinh[c + d*x]^2)/a)]*Sqrt[Cosh[c + d*x]^2]*Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x])/(5*d*(1 + (b*Sinh[c + d*x]^2)/a)^p)} -{(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x]^2, x, 3, (AppellF1[3/2, 3/2, -p, 5/2, -Sinh[c + d*x]^2, -((b*Sinh[c + d*x]^2)/a)]*Sqrt[Cosh[c + d*x]^2]*Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^p*Tanh[c + d*x])/(3*d*(1 + (b*Sinh[c + d*x]^2)/a)^p)} -{Coth[c + d*x]^2*(a + b*Sinh[c + d*x]^2)^p, x, 3, -((AppellF1[-1/2, -1/2, -p, 1/2, -Sinh[c + d*x]^2, -((b*Sinh[c + d*x]^2)/a)]*Sqrt[Cosh[c + d*x]^2]*Csch[c + d*x]*Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^p)/(d*(1 + (b*Sinh[c + d*x]^2)/a)^p))} -{Coth[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^p, x, 3, -(AppellF1[-3/2, -3/2, -p, -1/2, -Sinh[c + d*x]^2, -((b*Sinh[c + d*x]^2)/a)]*Sqrt[Cosh[c + d*x]^2]*Csch[c + d*x]^3*Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^p)/(3*d*(1 + (b*Sinh[c + d*x]^2)/a)^p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^3)^p*) - - -{Coth[x]^3/(a + b*Sinh[x]^3), x, 12, (b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sinh[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csch[x]^2/(2*a) + Log[Sinh[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sinh[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sinh[x] + b^(2/3)*Sinh[x]^2])/(6*a^(5/3)) - Log[a + b*Sinh[x]^3]/(3*a)} - - -{Coth[x]/Sqrt[a + b*Sinh[x]^3], x, 4, -((2*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/(3*Sqrt[a]))} - - -{Coth[x]*Sqrt[a + b*Sinh[x]^3], x, 5, (-(2/3))*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]] + (2/3)*Sqrt[a + b*Sinh[x]^3]} - - -(* ::Section:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^n)^p*) - - -{Coth[x]/Sqrt[a + b*Sinh[x]^n], x, 4, -((2*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]])/(Sqrt[a]*n))} - - -{Coth[x]*Sqrt[a + b*Sinh[x]^n], x, 5, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]])/n) + (2*Sqrt[a + b*Sinh[x]^n])/n} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.1 (c+d x)^m (a+b cosh)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.1 (c+d x)^m (a+b cosh)^n.m deleted file mode 100644 index 40cf0f9..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.1 (c+d x)^m (a+b cosh)^n.m +++ /dev/null @@ -1,370 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Cosh[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Cosh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Cosh[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^4*Cosh[a + b*x], x, 5, (-24*d^3*(c + d*x)*Cosh[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cosh[a + b*x])/b^2 + (24*d^4*Sinh[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Sinh[a + b*x])/b^3 + ((c + d*x)^4*Sinh[a + b*x])/b} -{(c + d*x)^3*Cosh[a + b*x], x, 4, (-6*d^3*Cosh[a + b*x])/b^4 - (3*d*(c + d*x)^2*Cosh[a + b*x])/b^2 + (6*d^2*(c + d*x)*Sinh[a + b*x])/b^3 + ((c + d*x)^3*Sinh[a + b*x])/b} -{(c + d*x)^2*Cosh[a + b*x], x, 3, (-2*d*(c + d*x)*Cosh[a + b*x])/b^2 + (2*d^2*Sinh[a + b*x])/b^3 + ((c + d*x)^2*Sinh[a + b*x])/b} -{(c + d*x)*Cosh[a + b*x], x, 2, -((d*Cosh[a + b*x])/b^2) + ((c + d*x)*Sinh[a + b*x])/b} -{Cosh[a + b*x]/(c + d*x), x, 3, (Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d + (Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d} -{Cosh[a + b*x]/(c + d*x)^2, x, 4, -(Cosh[a + b*x]/(d*(c + d*x))) + (b*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/d^2 + (b*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d^2} -{Cosh[a + b*x]/(c + d*x)^3, x, 5, -Cosh[a + b*x]/(2*d*(c + d*x)^2) + (b^2*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(2*d^3) - (b*Sinh[a + b*x])/(2*d^2*(c + d*x)) + (b^2*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(2*d^3)} - - -{(c + d*x)^4*Cosh[a + b*x]^2, x, 6, (3*d^4*x)/(4*b^4) + (d*(c + d*x)^3)/(2*b^2) + (c + d*x)^5/(10*d) - (3*d^3*(c + d*x)*Cosh[a + b*x]^2)/(2*b^4) - (d*(c + d*x)^3*Cosh[a + b*x]^2)/b^2 + (3*d^4*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^5) + (3*d^2*(c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) + ((c + d*x)^4*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{(c + d*x)^3*Cosh[a + b*x]^2, x, 4, (3*c*d^2*x)/(4*b^2) + (3*d^3*x^2)/(8*b^2) + (c + d*x)^4/(8*d) - (3*d^3*Cosh[a + b*x]^2)/(8*b^4) - (3*d*(c + d*x)^2*Cosh[a + b*x]^2)/(4*b^2) + (3*d^2*(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^3*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{(c + d*x)^2*Cosh[a + b*x]^2, x, 4, (d^2*x)/(4*b^2) + (c + d*x)^3/(6*d) - (d*(c + d*x)*Cosh[a + b*x]^2)/(2*b^2) + (d^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{(c + d*x)*Cosh[a + b*x]^2, x, 2, (c*x)/2 + (d*x^2)/4 - (d*Cosh[a + b*x]^2)/(4*b^2) + ((c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{Cosh[a + b*x]^2/(c + d*x), x, 5, (Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/(2*d) + Log[c + d*x]/(2*d) + (Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(2*d)} -{Cosh[a + b*x]^2/(c + d*x)^2, x, 5, -(Cosh[a + b*x]^2/(d*(c + d*x))) + (b*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d^2 + (b*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^2} -{Cosh[a + b*x]^2/(c + d*x)^3, x, 7, -Cosh[a + b*x]^2/(2*d*(c + d*x)^2) + (b^2*Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*c)/d + 2*b*x])/d^3 - (b*Cosh[a + b*x]*Sinh[a + b*x])/(d^2*(c + d*x)) + (b^2*Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d^3} -{Cosh[a + b*x]^2/(c + d*x)^4, x, 7, b^2/(3*d^3*(c + d*x)) - Cosh[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*Cosh[a + b*x]^2)/(3*d^3*(c + d*x)) + (2*b^3*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/(3*d^4) - (b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*(c + d*x)^2) + (2*b^3*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(3*d^4)} - - -{(c + d*x)^4*Cosh[a + b*x]^3, x, 12, (-160*d^3*(c + d*x)*Cosh[a + b*x])/(9*b^4) - (8*d*(c + d*x)^3*Cosh[a + b*x])/(3*b^2) - (8*d^3*(c + d*x)*Cosh[a + b*x]^3)/(27*b^4) - (4*d*(c + d*x)^3*Cosh[a + b*x]^3)/(9*b^2) + (488*d^4*Sinh[a + b*x])/(27*b^5) + (80*d^2*(c + d*x)^2*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^4*Sinh[a + b*x])/(3*b) + (4*d^2*(c + d*x)^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^3) + ((c + d*x)^4*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (8*d^4*Sinh[a + b*x]^3)/(81*b^5)} -{(c + d*x)^3*Cosh[a + b*x]^3, x, 8, (-40*d^3*Cosh[a + b*x])/(9*b^4) - (2*d*(c + d*x)^2*Cosh[a + b*x])/b^2 - (2*d^3*Cosh[a + b*x]^3)/(27*b^4) - (d*(c + d*x)^2*Cosh[a + b*x]^3)/(3*b^2) + (40*d^2*(c + d*x)*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^3*Sinh[a + b*x])/(3*b) + (2*d^2*(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^3) + ((c + d*x)^3*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b)} -{(c + d*x)^2*Cosh[a + b*x]^3, x, 6, (-4*d*(c + d*x)*Cosh[a + b*x])/(3*b^2) - (2*d*(c + d*x)*Cosh[a + b*x]^3)/(9*b^2) + (14*d^2*Sinh[a + b*x])/(9*b^3) + (2*(c + d*x)^2*Sinh[a + b*x])/(3*b) + ((c + d*x)^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (2*d^2*Sinh[a + b*x]^3)/(27*b^3)} -{(c + d*x)*Cosh[a + b*x]^3, x, 3, (-2*d*Cosh[a + b*x])/(3*b^2) - (d*Cosh[a + b*x]^3)/(9*b^2) + (2*(c + d*x)*Sinh[a + b*x])/(3*b) + ((c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b)} -{Cosh[a + b*x]^3/(c + d*x), x, 8, (3*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(4*d) + (Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*c)/d + 3*b*x])/(4*d) + (3*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(4*d) + (Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d)} -{Cosh[a + b*x]^3/(c + d*x)^2, x, 8, -(Cosh[a + b*x]^3/(d*(c + d*x))) + (3*b*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/(4*d^2) + (3*b*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/(4*d^2) + (3*b*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(4*d^2) + (3*b*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(4*d^2)} -{Cosh[a + b*x]^3/(c + d*x)^3, x, 12, -Cosh[a + b*x]^3/(2*d*(c + d*x)^2) + (3*b^2*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Cosh[3*a - (3*b*c)/d]*CoshIntegral[(3*b*c)/d + 3*b*x])/(8*d^3) - (3*b*Cosh[a + b*x]^2*Sinh[a + b*x])/(2*d^2*(c + d*x)) + (3*b^2*Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/(8*d^3) + (9*b^2*Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)} - - -{x^3*Cosh[a + b*x]^4, x, 8, (45*x^2)/(128*b^2) + (3*x^4)/32 - (45*Cosh[a + b*x]^2)/(128*b^4) - (9*x^2*Cosh[a + b*x]^2)/(16*b^2) - (3*Cosh[a + b*x]^4)/(128*b^4) - (3*x^2*Cosh[a + b*x]^4)/(16*b^2) + (45*x*Cosh[a + b*x]*Sinh[a + b*x])/(64*b^3) + (3*x^3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (3*x*Cosh[a + b*x]^3*Sinh[a + b*x])/(32*b^3) + (x^3*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} -{x^2*Cosh[a + b*x]^4, x, 8, (15*x)/(64*b^2) + x^3/8 - (3*x*Cosh[a + b*x]^2)/(8*b^2) - (x*Cosh[a + b*x]^4)/(8*b^2) + (15*Cosh[a + b*x]*Sinh[a + b*x])/(64*b^3) + (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(32*b^3) + (x^2*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} -{x^1*Cosh[a + b*x]^4, x, 3, (3*x^2)/16 - (3*Cosh[a + b*x]^2)/(16*b^2) - Cosh[a + b*x]^4/(16*b^2) + (3*x*Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (x*Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3*Sech[a + b*x], x, 9, (2*(c + d*x)^3*ArcTan[E^(a + b*x)])/b - ((3*I)*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + ((3*I)*d*(c + d*x)^2*PolyLog[2, I*E^(a + b*x)])/b^2 + ((6*I)*d^2*(c + d*x)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - ((6*I)*d^2*(c + d*x)*PolyLog[3, I*E^(a + b*x)])/b^3 - ((6*I)*d^3*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + ((6*I)*d^3*PolyLog[4, I*E^(a + b*x)])/b^4} -{(c + d*x)^2*Sech[a + b*x], x, 7, (2*(c + d*x)^2*ArcTan[E^(a + b*x)])/b - ((2*I)*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + ((2*I)*d*(c + d*x)*PolyLog[2, I*E^(a + b*x)])/b^2 + ((2*I)*d^2*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - ((2*I)*d^2*PolyLog[3, I*E^(a + b*x)])/b^3} -{(c + d*x)*Sech[a + b*x], x, 5, (2*(c + d*x)*ArcTan[E^(a + b*x)])/b - (I*d*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (I*d*PolyLog[2, I*E^(a + b*x)])/b^2} -{Sech[a + b*x]/(c + d*x), x, 0, Unintegrable[Sech[a + b*x]/(c + d*x), x]} -{Sech[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Sech[a + b*x]/(c + d*x)^2, x]} - - -{(c + d*x)^3*Sech[a + b*x]^2, x, 6, (c + d*x)^3/b - (3*d*(c + d*x)^2*Log[1 + E^(2*(a + b*x))])/b^2 - (3*d^2*(c + d*x)*PolyLog[2, -E^(2*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, -E^(2*(a + b*x))])/(2*b^4) + ((c + d*x)^3*Tanh[a + b*x])/b} -{(c + d*x)^2*Sech[a + b*x]^2, x, 5, (c + d*x)^2/b - (2*d*(c + d*x)*Log[1 + E^(2*(a + b*x))])/b^2 - (d^2*PolyLog[2, -E^(2*(a + b*x))])/b^3 + ((c + d*x)^2*Tanh[a + b*x])/b} -{(c + d*x)*Sech[a + b*x]^2, x, 2, -((d*Log[Cosh[a + b*x]])/b^2) + ((c + d*x)*Tanh[a + b*x])/b} -{Sech[a + b*x]^2/(c + d*x), x, 0, Unintegrable[Sech[a + b*x]^2/(c + d*x), x]} -{Sech[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Sech[a + b*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*Sech[a + b*x]^3, x, 15, (-6*d^2*(c + d*x)*ArcTan[E^(a + b*x)])/b^3 + ((c + d*x)^3*ArcTan[E^(a + b*x)])/b + ((3*I)*d^3*PolyLog[2, (-I)*E^(a + b*x)])/b^4 - (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((3*I)*d^3*PolyLog[2, I*E^(a + b*x)])/b^4 + (((3*I)/2)*d*(c + d*x)^2*PolyLog[2, I*E^(a + b*x)])/b^2 + ((3*I)*d^2*(c + d*x)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - ((3*I)*d^2*(c + d*x)*PolyLog[3, I*E^(a + b*x)])/b^3 - ((3*I)*d^3*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + ((3*I)*d^3*PolyLog[4, I*E^(a + b*x)])/b^4 + (3*d*(c + d*x)^2*Sech[a + b*x])/(2*b^2) + ((c + d*x)^3*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{(c + d*x)^2*Sech[a + b*x]^3, x, 9, ((c + d*x)^2*ArcTan[E^(a + b*x)])/b - (d^2*ArcTan[Sinh[a + b*x]])/b^3 - (I*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (I*d*(c + d*x)*PolyLog[2, I*E^(a + b*x)])/b^2 + (I*d^2*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (I*d^2*PolyLog[3, I*E^(a + b*x)])/b^3 + (d*(c + d*x)*Sech[a + b*x])/b^2 + ((c + d*x)^2*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{(c + d*x)*Sech[a + b*x]^3, x, 6, ((c + d*x)*ArcTan[E^(a + b*x)])/b - ((I/2)*d*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + ((I/2)*d*PolyLog[2, I*E^(a + b*x)])/b^2 + (d*Sech[a + b*x])/(2*b^2) + ((c + d*x)*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{Sech[a + b*x]^3/(c + d*x), x, 0, Unintegrable[Sech[a + b*x]^3/(c + d*x), x]} -{Sech[a + b*x]^3/(c + d*x)^2, x, 0, Unintegrable[Sech[a + b*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Cosh[e+f x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^(5/2)*Cosh[a + b*x], x, 8, (-5*d*(c + d*x)^(3/2)*Cosh[a + b*x])/(2*b^2) + (15*d^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (15*d^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) + (15*d^2*Sqrt[c + d*x]*Sinh[a + b*x])/(4*b^3) + ((c + d*x)^(5/2)*Sinh[a + b*x])/b} -{(c + d*x)^(3/2)*Cosh[a + b*x], x, 7, (-3*d*Sqrt[c + d*x]*Cosh[a + b*x])/(2*b^2) + (3*d^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + (3*d^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(5/2)) + ((c + d*x)^(3/2)*Sinh[a + b*x])/b} -{Sqrt[c + d*x]*Cosh[a + b*x], x, 6, (Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) - (Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*b^(3/2)) + (Sqrt[c + d*x]*Sinh[a + b*x])/b} -{Cosh[a + b*x]/Sqrt[c + d*x], x, 5, (E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) + (E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d])} -{Cosh[a + b*x]/(c + d*x)^(3/2), x, 6, (-2*Cosh[a + b*x])/(d*Sqrt[c + d*x]) - (Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[b]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2)} -{Cosh[a + b*x]/(c + d*x)^(5/2), x, 7, (-2*Cosh[a + b*x])/(3*d*(c + d*x)^(3/2)) + (2*b^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*b^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) - (4*b*Sinh[a + b*x])/(3*d^2*Sqrt[c + d*x])} -{Cosh[a + b*x]/(c + d*x)^(7/2), x, 8, (-2*Cosh[a + b*x])/(5*d*(c + d*x)^(5/2)) - (8*b^2*Cosh[a + b*x])/(15*d^3*Sqrt[c + d*x]) - (4*b^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (4*b^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) - (4*b*Sinh[a + b*x])/(15*d^2*(c + d*x)^(3/2))} - - -{(c + d*x)^(5/2)*Cosh[a + b*x]^2, x, 10, (5*d*(c + d*x)^(3/2))/(16*b^2) + (c + d*x)^(7/2)/(7*d) - (5*d*(c + d*x)^(3/2)*Cosh[a + b*x]^2)/(8*b^2) + (15*d^(5/2)*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(256*b^(7/2)) - (15*d^(5/2)*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(256*b^(7/2)) + ((c + d*x)^(5/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + (15*d^2*Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(64*b^3)} -{(c + d*x)^(3/2)*Cosh[a + b*x]^2, x, 9, (3*d*Sqrt[c + d*x])/(16*b^2) + (c + d*x)^(5/2)/(5*d) - (3*d*Sqrt[c + d*x]*Cosh[a + b*x]^2)/(8*b^2) + (3*d^(3/2)*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + (3*d^(3/2)*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(5/2)) + ((c + d*x)^(3/2)*Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{Sqrt[c + d*x]*Cosh[a + b*x]^2, x, 8, (c + d*x)^(3/2)/(3*d) + (Sqrt[d]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) + (Sqrt[c + d*x]*Sinh[2*a + 2*b*x])/(4*b)} -{Cosh[a + b*x]^2/Sqrt[c + d*x], x, 7, Sqrt[c + d*x]/d + (E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d]) + (E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*Sqrt[b]*Sqrt[d])} -{Cosh[a + b*x]^2/(c + d*x)^(3/2), x, 7, (-2*Cosh[a + b*x]^2)/(d*Sqrt[c + d*x]) - (Sqrt[b]*E^(-2*a + (2*b*c)/d)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[b]*E^(2*a - (2*b*c)/d)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/d^(3/2)} -{Cosh[a + b*x]^2/(c + d*x)^(5/2), x, 9, (-2*Cosh[a + b*x]^2)/(3*d*(c + d*x)^(3/2)) + (2*b^(3/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*b^(3/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(3*d^(5/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*Sqrt[c + d*x])} -{Cosh[a + b*x]^2/(c + d*x)^(7/2), x, 9, (16*b^2)/(15*d^3*Sqrt[c + d*x]) - (2*Cosh[a + b*x]^2)/(5*d*(c + d*x)^(5/2)) - (32*b^2*Cosh[a + b*x]^2)/(15*d^3*Sqrt[c + d*x]) - (8*b^(5/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) + (8*b^(5/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(15*d^(7/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(15*d^2*(c + d*x)^(3/2))} -{Cosh[a + b*x]^2/(c + d*x)^(9/2), x, 11, (16*b^2)/(105*d^3*(c + d*x)^(3/2)) - (2*Cosh[a + b*x]^2)/(7*d*(c + d*x)^(7/2)) - (32*b^2*Cosh[a + b*x]^2)/(105*d^3*(c + d*x)^(3/2)) + (32*b^(7/2)*E^(-2*a + (2*b*c)/d)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(105*d^(9/2)) + (32*b^(7/2)*E^(2*a - (2*b*c)/d)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(105*d^(9/2)) - (8*b*Cosh[a + b*x]*Sinh[a + b*x])/(35*d^2*(c + d*x)^(5/2)) - (128*b^3*Cosh[a + b*x]*Sinh[a + b*x])/(105*d^4*Sqrt[c + d*x])} - - -{(c + d*x)^(5/2)*Cosh[a + b*x]^3, x, 23, (-5*d*(c + d*x)^(3/2)*Cosh[a + b*x])/(3*b^2) - (5*d*(c + d*x)^(3/2)*Cosh[a + b*x]^3)/(18*b^2) + (45*d^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) + (5*d^(5/2)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (45*d^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(64*b^(7/2)) - (5*d^(5/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (45*d^2*Sqrt[c + d*x]*Sinh[a + b*x])/(16*b^3) + (2*(c + d*x)^(5/2)*Sinh[a + b*x])/(3*b) + ((c + d*x)^(5/2)*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b) + (5*d^2*Sqrt[c + d*x]*Sinh[3*a + 3*b*x])/(144*b^3)} -{(c + d*x)^(3/2)*Cosh[a + b*x]^3, x, 20, -((d*Sqrt[c + d*x]*Cosh[a + b*x])/b^2) - (d*Sqrt[c + d*x]*Cosh[a + b*x]^3)/(6*b^2) + (9*d^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(5/2)) + (d^(3/2)*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (9*d^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(5/2)) + (d^(3/2)*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (2*(c + d*x)^(3/2)*Sinh[a + b*x])/(3*b) + ((c + d*x)^(3/2)*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b)} -{Sqrt[c + d*x]*Cosh[a + b*x]^3, x, 14, (3*Sqrt[d]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) + (Sqrt[d]*E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) - (3*Sqrt[d]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(3/2)) - (Sqrt[d]*E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (3*Sqrt[c + d*x]*Sinh[a + b*x])/(4*b) + (Sqrt[c + d*x]*Sinh[3*a + 3*b*x])/(12*b)} -{Cosh[a + b*x]^3/Sqrt[c + d*x], x, 12, (3*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(-3*a + (3*b*c)/d)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (3*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d]) + (E^(3*a - (3*b*c)/d)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(8*Sqrt[b]*Sqrt[d])} -{Cosh[a + b*x]^3/(c + d*x)^(3/2), x, 12, (-2*Cosh[a + b*x]^3)/(d*Sqrt[c + d*x]) - (3*Sqrt[b]*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) - (Sqrt[b]*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (3*Sqrt[b]*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2)) + (Sqrt[b]*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(4*d^(3/2))} -{Cosh[a + b*x]^3/(c + d*x)^(5/2), x, 18, (-2*Cosh[a + b*x]^3)/(3*d*(c + d*x)^(3/2)) + (b^(3/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) + (b^(3/2)*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) + (b^(3/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) + (b^(3/2)*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(2*d^(5/2)) - (4*b*Cosh[a + b*x]^2*Sinh[a + b*x])/(d^2*Sqrt[c + d*x])} -{Cosh[a + b*x]^3/(c + d*x)^(7/2), x, 19, (16*b^2*Cosh[a + b*x])/(5*d^3*Sqrt[c + d*x]) - (2*Cosh[a + b*x]^3)/(5*d*(c + d*x)^(5/2)) - (24*b^2*Cosh[a + b*x]^3)/(5*d^3*Sqrt[c + d*x]) - (b^(5/2)*E^(-a + (b*c)/d)*Sqrt[Pi]*Erf[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (3*b^(5/2)*E^(-3*a + (3*b*c)/d)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (b^(5/2)*E^(a - (b*c)/d)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (3*b^(5/2)*E^(3*a - (3*b*c)/d)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (4*b*Cosh[a + b*x]^2*Sinh[a + b*x])/(5*d^2*(c + d*x)^(3/2))} - - -{(d*x)^(3/2)*Cosh[f*x], x, 7, (-3*d*Sqrt[d*x]*Cosh[f*x])/(2*f^2) + (3*d^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) + (3*d^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(8*f^(5/2)) + ((d*x)^(3/2)*Sinh[f*x])/f} -{Sqrt[d*x]*Cosh[f*x], x, 6, (Sqrt[d]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2)) - (Sqrt[d]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(4*f^(3/2)) + (Sqrt[d*x]*Sinh[f*x])/f} -{Cosh[f*x]/Sqrt[d*x], x, 5, (Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f]) + (Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(2*Sqrt[d]*Sqrt[f])} -{Cosh[f*x]/(d*x)^(3/2), x, 6, (-2*Cosh[f*x])/(d*Sqrt[d*x]) - (Sqrt[f]*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[f]*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/d^(3/2)} -{Cosh[f*x]/(d*x)^(5/2), x, 7, (-2*Cosh[f*x])/(3*d*(d*x)^(3/2)) + (2*f^(3/2)*Sqrt[Pi]*Erf[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (2*f^(3/2)*Sqrt[Pi]*Erfi[(Sqrt[f]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (4*f*Sinh[f*x])/(3*d^2*Sqrt[d*x])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sqrt[c + d*x]*Sech[a + b*x], x, 0, Unintegrable[Sqrt[c + d*x]*Sech[a + b*x], x]} -{Sech[a + b*x]/Sqrt[c + d*x], x, 0, Unintegrable[Sech[a + b*x]/Sqrt[c + d*x], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Cosh[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cosh[x]^(3/2)/x^3, x, 1, -Cosh[x]^(3/2)/(2*x^2) - (3*Sqrt[Cosh[x]]*Sinh[x])/(4*x) - (3*Unintegrable[1/(x*Sqrt[Cosh[x]]), x])/8 + (9*Unintegrable[Cosh[x]^(3/2)/x, x])/8} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]], x, 2, -4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]} -{x/Cosh[x]^(5/2) - x/(3*Sqrt[Cosh[x]]), x, 2, 4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))} -{x/Cosh[x]^(7/2) + (3*x*Sqrt[Cosh[x]])/5, x, 3, 4/(15*Cosh[x]^(3/2)) - (12*Sqrt[Cosh[x]])/5 + (2*x*Sinh[x])/(5*Cosh[x]^(5/2)) + (6*x*Sinh[x])/(5*Sqrt[Cosh[x]])} -{x^2/Cosh[x]^(3/2) + x^2*Sqrt[Cosh[x]], x, 3, -8*x*Sqrt[Cosh[x]] - (16*I)*EllipticE[(I/2)*x, 2] + (2*x^2*Sinh[x])/Sqrt[Cosh[x]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Cosh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(b*Cosh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(b*Cosh[e + f*x])^n, x]} - - -{(c + d*x)^m*Cosh[a + b*x]^3, x, 8, (3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b*(c + d*x))/d])/(8*b*(-((b*(c + d*x))/d))^m) + (3*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m) - (3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (3*b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Cosh[a + b*x]^2, x, 5, (c + d*x)^(1 + m)/(2*d*(1 + m)) + (2^(-3 - m)*E^(2*a - (2*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-2*b*(c + d*x))/d])/(b*(-((b*(c + d*x))/d))^m) - (2^(-3 - m)*E^(-2*a + (2*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (2*b*(c + d*x))/d])/(b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Cosh[a + b*x], x, 3, (E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(2*b*(-((b*(c + d*x))/d))^m) - (E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(2*b*((b*(c + d*x))/d)^m)} -{(c + d*x)^m*Sech[a + b*x], x, 0, Unintegrable[(c + d*x)^m*Sech[a + b*x], x]} -{(c + d*x)^m*Sech[a + b*x]^2, x, 0, Unintegrable[(c + d*x)^m*Sech[a + b*x]^2, x]} - - -{x^(3 + m)*Cosh[a + b*x], x, 3, -(E^a*x^m*Gamma[4 + m, -(b*x)])/(2*b^4*(-(b*x))^m) - (x^m*Gamma[4 + m, b*x])/(2*b^4*E^a*(b*x)^m)} -{x^(2 + m)*Cosh[a + b*x], x, 3, (E^a*x^m*Gamma[3 + m, -(b*x)])/(2*b^3*(-(b*x))^m) - (x^m*Gamma[3 + m, b*x])/(2*b^3*E^a*(b*x)^m)} -{x^(1 + m)*Cosh[a + b*x], x, 3, -(E^a*x^m*Gamma[2 + m, -(b*x)])/(2*b^2*(-(b*x))^m) - (x^m*Gamma[2 + m, b*x])/(2*b^2*E^a*(b*x)^m)} -{x^m*Cosh[a + b*x], x, 3, (E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) - (x^m*Gamma[1 + m, b*x])/(2*b*E^a*(b*x)^m)} -{x^(-1 + m)*Cosh[a + b*x], x, 3, -(E^a*x^m*Gamma[m, -(b*x)])/(2*(-(b*x))^m) - (x^m*Gamma[m, b*x])/(2*E^a*(b*x)^m)} -{x^(-2 + m)*Cosh[a + b*x], x, 3, (b*E^a*x^m*Gamma[-1 + m, -(b*x)])/(2*(-(b*x))^m) - (b*x^m*Gamma[-1 + m, b*x])/(2*E^a*(b*x)^m)} -{x^(-3 + m)*Cosh[a + b*x], x, 3, -(b^2*E^a*x^m*Gamma[-2 + m, -(b*x)])/(2*(-(b*x))^m) - (b^2*x^m*Gamma[-2 + m, b*x])/(2*E^a*(b*x)^m)} - - -{x^(3 + m)*Cosh[a + b*x]^2, x, 5, x^(4 + m)/(2*(4 + m)) - (2^(-6 - m)*E^(2*a)*x^m*Gamma[4 + m, -2*b*x])/(b^4*(-(b*x))^m) - (2^(-6 - m)*x^m*Gamma[4 + m, 2*b*x])/(b^4*E^(2*a)*(b*x)^m)} -{x^(2 + m)*Cosh[a + b*x]^2, x, 5, x^(3 + m)/(2*(3 + m)) + (2^(-5 - m)*E^(2*a)*x^m*Gamma[3 + m, -2*b*x])/(b^3*(-(b*x))^m) - (2^(-5 - m)*x^m*Gamma[3 + m, 2*b*x])/(b^3*E^(2*a)*(b*x)^m)} -{x^(1 + m)*Cosh[a + b*x]^2, x, 5, x^(2 + m)/(2*(2 + m)) - (2^(-4 - m)*E^(2*a)*x^m*Gamma[2 + m, -2*b*x])/(b^2*(-(b*x))^m) - (2^(-4 - m)*x^m*Gamma[2 + m, 2*b*x])/(b^2*E^(2*a)*(b*x)^m)} -{x^m*Cosh[a + b*x]^2, x, 5, x^(1 + m)/(2*(1 + m)) + (2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) - (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)*(b*x)^m)} -{x^(-1 + m)*Cosh[a + b*x]^2, x, 5, x^m/(2*m) - (2^(-2 - m)*E^(2*a)*x^m*Gamma[m, -2*b*x])/(-(b*x))^m - (2^(-2 - m)*x^m*Gamma[m, 2*b*x])/(E^(2*a)*(b*x)^m)} -{x^(-2 + m)*Cosh[a + b*x]^2, x, 5, -x^(-1 + m)/(2*(1 - m)) + (2^(-1 - m)*b*E^(2*a)*x^m*Gamma[-1 + m, -2*b*x])/(-(b*x))^m - (2^(-1 - m)*b*x^m*Gamma[-1 + m, 2*b*x])/(E^(2*a)*(b*x)^m)} -{x^(-3 + m)*Cosh[a + b*x]^2, x, 5, -x^(-2 + m)/(2*(2 - m)) - (b^2*E^(2*a)*x^m*Gamma[-2 + m, -2*b*x])/(2^m*(-(b*x))^m) - (b^2*x^m*Gamma[-2 + m, 2*b*x])/(2^m*E^(2*a)*(b*x)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sech[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Sech[e+f x])^(n/2)*) - - -{x/Sech[x]^(3/2) - (x*Sqrt[Sech[x]])/3, x, 4, -4/(9*Sech[x]^(3/2)) + (2*x*Sinh[x])/(3*Sqrt[Sech[x]])} -{x/Sech[x]^(5/2) - (3*x)/(5*Sqrt[Sech[x]]), x, 4, -4/(25*Sech[x]^(5/2)) + (2*x*Sinh[x])/(5*Sech[x]^(3/2))} -{x/Sech[x]^(7/2) - (5*x*Sqrt[Sech[x]])/21, x, 5, -4/(49*Sech[x]^(7/2)) - 20/(63*Sech[x]^(3/2)) + (2*x*Sinh[x])/(7*Sech[x]^(5/2)) + (10*x*Sinh[x])/(21*Sqrt[Sech[x]])} -{x^2/Sech[x]^(3/2) - (x^2*Sqrt[Sech[x]])/3, x, 7, (-8*x)/(9*Sech[x]^(3/2)) - ((16*I)/27)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x]] + (16*Sinh[x])/(27*Sqrt[Sech[x]]) + (2*x^2*Sinh[x])/(3*Sqrt[Sech[x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cosh[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cosh[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + a*Cosh[e + f*x]), x, 6, (a*(c + d*x)^4)/(4*d) - (6*a*d^3*Cosh[e + f*x])/f^4 - (3*a*d*(c + d*x)^2*Cosh[e + f*x])/f^2 + (6*a*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (a*(c + d*x)^3*Sinh[e + f*x])/f} -{(c + d*x)^2*(a + a*Cosh[e + f*x]), x, 5, (a*(c + d*x)^3)/(3*d) - (2*a*d*(c + d*x)*Cosh[e + f*x])/f^2 + (2*a*d^2*Sinh[e + f*x])/f^3 + (a*(c + d*x)^2*Sinh[e + f*x])/f} -{(c + d*x)*(a + a*Cosh[e + f*x]), x, 4, (a*(c + d*x)^2)/(2*d) - (a*d*Cosh[e + f*x])/f^2 + (a*(c + d*x)*Sinh[e + f*x])/f} -{(a + a*Cosh[e + f*x])/(c + d*x), x, 5, (a*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d + (a*Log[c + d*x])/d + (a*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d} -{(a + a*Cosh[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) - (a*Cosh[e + f*x])/(d*(c + d*x)) + (a*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (a*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2} -{(a + a*Cosh[e + f*x])/(c + d*x)^3, x, 7, -(a/(2*d*(c + d*x)^2)) - (a*Cosh[e + f*x])/(2*d*(c + d*x)^2) + (a*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/(2*d^3) - (a*f*Sinh[e + f*x])/(2*d^2*(c + d*x)) + (a*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(c + d*x)^3*(a + a*Cosh[e + f*x])^2, x, 10, (3*a^2*c*d^2*x)/(4*f^2) + (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) - (12*a^2*d^3*Cosh[e + f*x])/f^4 - (6*a^2*d*(c + d*x)^2*Cosh[e + f*x])/f^2 - (3*a^2*d^3*Cosh[e + f*x]^2)/(8*f^4) - (3*a^2*d*(c + d*x)^2*Cosh[e + f*x]^2)/(4*f^2) + (12*a^2*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^3*Sinh[e + f*x])/f + (3*a^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(c + d*x)^2*(a + a*Cosh[e + f*x])^2, x, 9, (a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) - (4*a^2*d*(c + d*x)*Cosh[e + f*x])/f^2 - (a^2*d*(c + d*x)*Cosh[e + f*x]^2)/(2*f^2) + (4*a^2*d^2*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^2*Sinh[e + f*x])/f + (a^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(c + d*x)*(a + a*Cosh[e + f*x])^2, x, 6, (1/2)*a^2*c*x + (1/4)*a^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (2*a^2*d*Cosh[e + f*x])/f^2 - (a^2*d*Cosh[e + f*x]^2)/(4*f^2) + (2*a^2*(c + d*x)*Sinh[e + f*x])/f + (a^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(a + a*Cosh[e + f*x])^2/(c + d*x), x, 9, (2*a^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d + (a^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (3*a^2*Log[c + d*x])/(2*d) + (2*a^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d + (a^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + a*Cosh[e + f*x])^2/(c + d*x)^2, x, 9, (-4*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)) + (a^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d^2 + (2*a^2*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (2*a^2*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2 + (a^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + a*Cosh[e + f*x])^2/(c + d*x)^3, x, 15, (-2*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)^2) + (a^2*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^3 + (a^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 - (4*a^2*f*Cosh[e/2 + (f*x)/2]^3*Sinh[e/2 + (f*x)/2])/(d^2*(c + d*x)) + (a^2*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (a^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + a*Cosh[e + f*x]), x, 7, (c + d*x)^3/(a*f) - (6*d*(c + d*x)^2*Log[1 + E^(e + f*x)])/(a*f^2) - (12*d^2*(c + d*x)*PolyLog[2, -E^(e + f*x)])/(a*f^3) + (12*d^3*PolyLog[3, -E^(e + f*x)])/(a*f^4) + ((c + d*x)^3*Tanh[e/2 + (f*x)/2])/(a*f)} -{(c + d*x)^2/(a + a*Cosh[e + f*x]), x, 6, (c + d*x)^2/(a*f) - (4*d*(c + d*x)*Log[1 + E^(e + f*x)])/(a*f^2) - (4*d^2*PolyLog[2, -E^(e + f*x)])/(a*f^3) + ((c + d*x)^2*Tanh[e/2 + (f*x)/2])/(a*f)} -{(c + d*x)/(a + a*Cosh[e + f*x]), x, 3, (-2*d*Log[Cosh[e/2 + (f*x)/2]])/(a*f^2) + ((c + d*x)*Tanh[e/2 + (f*x)/2])/(a*f)} -{1/((c + d*x)*(a + a*Cosh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + a*Cosh[e + f*x])), x]} -{1/((c + d*x)^2*(a + a*Cosh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Cosh[e + f*x])), x]} - - -{(c + d*x)^3/(a + a*Cosh[e + f*x])^2, x, 10, (c + d*x)^3/(3*a^2*f) - (2*d*(c + d*x)^2*Log[1 + E^(e + f*x)])/(a^2*f^2) + (4*d^3*Log[Cosh[e/2 + (f*x)/2]])/(a^2*f^4) - (4*d^2*(c + d*x)*PolyLog[2, -E^(e + f*x)])/(a^2*f^3) + (4*d^3*PolyLog[3, -E^(e + f*x)])/(a^2*f^4) + (d*(c + d*x)^2*Sech[e/2 + (f*x)/2]^2)/(2*a^2*f^2) - (2*d^2*(c + d*x)*Tanh[e/2 + (f*x)/2])/(a^2*f^3) + ((c + d*x)^3*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)} -{(c + d*x)^2/(a + a*Cosh[e + f*x])^2, x, 9, (c + d*x)^2/(3*a^2*f) - (4*d*(c + d*x)*Log[1 + E^(e + f*x)])/(3*a^2*f^2) - (4*d^2*PolyLog[2, -E^(e + f*x)])/(3*a^2*f^3) + (d*(c + d*x)*Sech[e/2 + (f*x)/2]^2)/(3*a^2*f^2) - (2*d^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f^3) + ((c + d*x)^2*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)} -{(c + d*x)/(a + a*Cosh[e + f*x])^2, x, 4, (-2*d*Log[Cosh[e/2 + (f*x)/2]])/(3*a^2*f^2) + (d*Sech[e/2 + (f*x)/2]^2)/(6*a^2*f^2) + ((c + d*x)*Tanh[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)*Sech[e/2 + (f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(6*a^2*f)} -{1/((c + d*x)*(a + a*Cosh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + a*Cosh[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + a*Cosh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + a*Cosh[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cosh[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*Sqrt[a + a*Cosh[c + d*x]], x, 5, (-96*Sqrt[a + a*Cosh[c + d*x]])/d^4 - (12*x^2*Sqrt[a + a*Cosh[c + d*x]])/d^2 + (48*x*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d^3 + (2*x^3*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d} -{x^2*Sqrt[a + a*Cosh[c + d*x]], x, 4, (-8*x*Sqrt[a + a*Cosh[c + d*x]])/d^2 + (16*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d^3 + (2*x^2*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d} -{x^1*Sqrt[a + a*Cosh[c + d*x]], x, 3, (-4*Sqrt[a + a*Cosh[c + d*x]])/d^2 + (2*x*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/d} -{Sqrt[a + a*Cosh[c + d*x]]/x^1, x, 4, Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2] + Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*Sinh[c/2]*SinhIntegral[(d*x)/2]} -{Sqrt[a + a*Cosh[c + d*x]]/x^2, x, 5, -(Sqrt[a + a*Cosh[c + d*x]]/x) + (d*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2]*Sinh[c/2])/2 + (d*Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*SinhIntegral[(d*x)/2])/2} -{Sqrt[a + a*Cosh[c + d*x]]/x^3, x, 6, -Sqrt[a + a*Cosh[c + d*x]]/(2*x^2) + (d^2*Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2])/8 + (d^2*Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*Sinh[c/2]*SinhIntegral[(d*x)/2])/8 - (d*Sqrt[a + a*Cosh[c + d*x]]*Tanh[c/2 + (d*x)/2])/(4*x)} - - -{x^3*Sqrt[a + a*Cosh[x]], x, 5, -96*Sqrt[a + a*Cosh[x]] - 12*x^2*Sqrt[a + a*Cosh[x]] + 48*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2] + 2*x^3*Sqrt[a + a*Cosh[x]]*Tanh[x/2]} -{x^2*Sqrt[a + a*Cosh[x]], x, 4, -8*x*Sqrt[a + a*Cosh[x]] + 16*Sqrt[a + a*Cosh[x]]*Tanh[x/2] + 2*x^2*Sqrt[a + a*Cosh[x]]*Tanh[x/2]} -{x^1*Sqrt[a + a*Cosh[x]], x, 3, -4*Sqrt[a + a*Cosh[x]] + 2*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2]} -{Sqrt[a + a*Cosh[x]]/x^1, x, 2, Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2]} -{Sqrt[a + a*Cosh[x]]/x^2, x, 3, -(Sqrt[a + a*Cosh[x]]/x) + (1/2)*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[x/2]} -{Sqrt[a + a*Cosh[x]]/x^3, x, 4, -(Sqrt[a + a*Cosh[x]]/(2*x^2)) + (1/8)*Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2] - (Sqrt[a + a*Cosh[x]]*Tanh[x/2])/(4*x)} - - -{x^3*(a + a*Cosh[x])^(3/2), x, 9, (-1280*a*Sqrt[a + a*Cosh[x]])/9 - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/27 - (8*a*x^2*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/3 + (32*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/9 + (4*a*x^3*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/3 + (640*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/9 + (8*a*x^3*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/3} -{x^2*(a + a*Cosh[x])^(3/2), x, 7, (-32*a*x*Sqrt[a + a*Cosh[x]])/3 - (16*a*x*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/9 + (4*a*x^2*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/3 + (224*a*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/9 + (8*a*x^2*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/3 + (32*a*Sqrt[a + a*Cosh[x]]*Sinh[x/2]^2*Tanh[x/2])/27} -{x^1*(a + a*Cosh[x])^(3/2), x, 4, (-16*a*Sqrt[a + a*Cosh[x]])/3 - (8*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/9 + (4*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/3 + (8*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/3} -{(a + a*Cosh[x])^(3/2)/x^1, x, 5, (3*a*Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2])/2 + (a*Sqrt[a + a*Cosh[x]]*CoshIntegral[(3*x)/2]*Sech[x/2])/2} -{(a + a*Cosh[x])^(3/2)/x^2, x, 5, (-2*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x + (3*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[x/2])/4 + (3*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[(3*x)/2])/4} -{(a + a*Cosh[x])^(3/2)/x^3, x, 7, -((a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x^2) + (3*a*Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2])/16 + (9*a*Sqrt[a + a*Cosh[x]]*CoshIntegral[(3*x)/2]*Sech[x/2])/16 - (3*a*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/(2*x)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/Sqrt[a + a*Cosh[c + d*x]], x, 10, (4*x^3*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((12*I)*x^2*Cosh[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((12*I)*x^2*Cosh[c/2 + (d*x)/2]*PolyLog[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((48*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^(c/2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((48*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((96*I)*Cosh[c/2 + (d*x)/2]*PolyLog[4, (-I)*E^(c/2 + (d*x)/2)])/(d^4*Sqrt[a + a*Cosh[c + d*x]]) + ((96*I)*Cosh[c/2 + (d*x)/2]*PolyLog[4, I*E^(c/2 + (d*x)/2)])/(d^4*Sqrt[a + a*Cosh[c + d*x]])} -{x^2/Sqrt[a + a*Cosh[c + d*x]], x, 8, (4*x^2*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((8*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((8*I)*x*Cosh[c/2 + (d*x)/2]*PolyLog[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, (-I)*E^(c/2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]]) - ((16*I)*Cosh[c/2 + (d*x)/2]*PolyLog[3, I*E^(c/2 + (d*x)/2)])/(d^3*Sqrt[a + a*Cosh[c + d*x]])} -{x/Sqrt[a + a*Cosh[c + d*x]], x, 6, (4*x*ArcTan[E^(c/2 + (d*x)/2)]*Cosh[c/2 + (d*x)/2])/(d*Sqrt[a + a*Cosh[c + d*x]]) - ((4*I)*Cosh[c/2 + (d*x)/2]*PolyLog[2, (-I)*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]]) + ((4*I)*Cosh[c/2 + (d*x)/2]*PolyLog[2, I*E^(c/2 + (d*x)/2)])/(d^2*Sqrt[a + a*Cosh[c + d*x]])} -{1/(x*Sqrt[a + a*Cosh[c + d*x]]), x, 0, Unintegrable[1/(x*Sqrt[a + a*Cosh[c + d*x]]), x]} -{1/(x^2*Sqrt[a + a*Cosh[c + d*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[a + a*Cosh[c + d*x]]), x]} - - -{x^3/(a + a*Cosh[x])^(3/2), x, 16, (3*x^2)/(a*Sqrt[a + a*Cosh[x]]) - (24*x*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((3*I)*x^2*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((3*I)*x^2*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((12*I)*x*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((12*I)*x*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((24*I)*Cosh[x/2]*PolyLog[4, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((24*I)*Cosh[x/2]*PolyLog[4, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^3*Tanh[x/2])/(2*a*Sqrt[a + a*Cosh[x]])} -{x^2/(a + a*Cosh[x])^(3/2), x, 10, (2*x)/(a*Sqrt[a + a*Cosh[x]]) + (x^2*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - (4*ArcTan[Sinh[x/2]]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - ((2*I)*x*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((2*I)*x*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + ((4*I)*Cosh[x/2]*PolyLog[3, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) - ((4*I)*Cosh[x/2]*PolyLog[3, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x^2*Tanh[x/2])/(2*a*Sqrt[a + a*Cosh[x]])} -{x/(a + a*Cosh[x])^(3/2), x, 7, 1/(a*Sqrt[a + a*Cosh[x]]) + (x*ArcTan[E^(x/2)]*Cosh[x/2])/(a*Sqrt[a + a*Cosh[x]]) - (I*Cosh[x/2]*PolyLog[2, (-I)*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (I*Cosh[x/2]*PolyLog[2, I*E^(x/2)])/(a*Sqrt[a + a*Cosh[x]]) + (x*Tanh[x/2])/(2*a*Sqrt[a + a*Cosh[x]])} -{1/(x*(a + a*Cosh[x])^(3/2)), x, 0, Unintegrable[1/(x*(a + a*Cosh[x])^(3/2)), x]} -{1/(x^2*(a + a*Cosh[x])^(3/2)), x, 0, Unintegrable[1/(x^2*(a + a*Cosh[x])^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cosh[e+f x])^(n/3)*) - - -(* Used to hang Rubi *) -{(a + a*Cosh[c + d*x])^(1/3)/x, x, 0, Unintegrable[(a + a*Cosh[c + d*x])^(1/3)/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Cosh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + a*Cosh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + a*Cosh[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + a*Cosh[e + f*x])^3, x, 12, (5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (3^(-1 - m)*a^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(8*f*(-((f*(c + d*x))/d))^m) + (3*2^(-3 - m)*a^3*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (15*a^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(8*f*(-((f*(c + d*x))/d))^m) - (15*a^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m) - (3*2^(-3 - m)*a^3*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (3^(-1 - m)*a^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + a*Cosh[e + f*x])^2, x, 9, (3*a^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (2^(-3 - m)*a^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (a^2*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) - (a^2*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (2^(-3 - m)*a^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + a*Cosh[e + f*x]), x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) + (a*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) - (a*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + a*Cosh[e + f*x]), x, 0, Unintegrable[(c + d*x)^m/(a + a*Cosh[e + f*x]), x]} -{(c + d*x)^m/(a + a*Cosh[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + a*Cosh[e + f*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cosh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cosh[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Cosh[e + f*x]), x, 6, (a*(c + d*x)^4)/(4*d) - (6*b*d^3*Cosh[e + f*x])/f^4 - (3*b*d*(c + d*x)^2*Cosh[e + f*x])/f^2 + (6*b*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (b*(c + d*x)^3*Sinh[e + f*x])/f} -{(c + d*x)^2*(a + b*Cosh[e + f*x]), x, 5, (a*(c + d*x)^3)/(3*d) - (2*b*d*(c + d*x)*Cosh[e + f*x])/f^2 + (2*b*d^2*Sinh[e + f*x])/f^3 + (b*(c + d*x)^2*Sinh[e + f*x])/f} -{(c + d*x)*(a + b*Cosh[e + f*x]), x, 4, (a*(c + d*x)^2)/(2*d) - (b*d*Cosh[e + f*x])/f^2 + (b*(c + d*x)*Sinh[e + f*x])/f} -{(a + b*Cosh[e + f*x])/(c + d*x), x, 5, (b*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d + (a*Log[c + d*x])/d + (b*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d} -{(a + b*Cosh[e + f*x])/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) - (b*Cosh[e + f*x])/(d*(c + d*x)) + (b*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (b*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2} -{(a + b*Cosh[e + f*x])/(c + d*x)^3, x, 7, -a/(2*d*(c + d*x)^2) - (b*Cosh[e + f*x])/(2*d*(c + d*x)^2) + (b*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/(2*d^3) - (b*f*Sinh[e + f*x])/(2*d^2*(c + d*x)) + (b*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/(2*d^3)} - - -{(c + d*x)^3*(a + b*Cosh[e + f*x])^2, x, 10, (3*b^2*c*d^2*x)/(4*f^2) + (3*b^2*d^3*x^2)/(8*f^2) + (a^2*(c + d*x)^4)/(4*d) + (b^2*(c + d*x)^4)/(8*d) - (12*a*b*d^3*Cosh[e + f*x])/f^4 - (6*a*b*d*(c + d*x)^2*Cosh[e + f*x])/f^2 - (3*b^2*d^3*Cosh[e + f*x]^2)/(8*f^4) - (3*b^2*d*(c + d*x)^2*Cosh[e + f*x]^2)/(4*f^2) + (12*a*b*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (2*a*b*(c + d*x)^3*Sinh[e + f*x])/f + (3*b^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (b^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(c + d*x)^2*(a + b*Cosh[e + f*x])^2, x, 9, (b^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(6*d) - (4*a*b*d*(c + d*x)*Cosh[e + f*x])/f^2 - (b^2*d*(c + d*x)*Cosh[e + f*x]^2)/(2*f^2) + (4*a*b*d^2*Sinh[e + f*x])/f^3 + (2*a*b*(c + d*x)^2*Sinh[e + f*x])/f + (b^2*d^2*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (b^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(c + d*x)*(a + b*Cosh[e + f*x])^2, x, 6, (b^2*c*x)/2 + (b^2*d*x^2)/4 + (a^2*(c + d*x)^2)/(2*d) - (2*a*b*d*Cosh[e + f*x])/f^2 - (b^2*d*Cosh[e + f*x]^2)/(4*f^2) + (2*a*b*(c + d*x)*Sinh[e + f*x])/f + (b^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)} -{(a + b*Cosh[e + f*x])^2/(c + d*x), x, 10, (2*a*b*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d + (b^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*d) + (a^2*Log[c + d*x])/d + (b^2*Log[c + d*x])/(2*d) + (2*a*b*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d + (b^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*d)} -{(a + b*Cosh[e + f*x])^2/(c + d*x)^2, x, 11, -(a^2/(d*(c + d*x))) - (2*a*b*Cosh[e + f*x])/(d*(c + d*x)) - (b^2*Cosh[e + f*x]^2)/(d*(c + d*x)) + (b^2*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d^2 + (2*a*b*f*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^2 + (2*a*b*f*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^2 + (b^2*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^2} -{(a + b*Cosh[e + f*x])^2/(c + d*x)^3, x, 14, -a^2/(2*d*(c + d*x)^2) - (a*b*Cosh[e + f*x])/(d*(c + d*x)^2) - (b^2*Cosh[e + f*x]^2)/(2*d*(c + d*x)^2) + (a*b*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^3 + (b^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 - (a*b*f*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2*f*Cosh[e + f*x]*Sinh[e + f*x])/(d^2*(c + d*x)) + (a*b*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (b^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^3} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Cosh[e + f*x]), x, 12, ((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - ((c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) + (3*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2) - (3*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2) - (6*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^3) + (6*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^3) + (6*d^3*PolyLog[4, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^4) - (6*d^3*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^4)} -{(c + d*x)^2/(a + b*Cosh[e + f*x]), x, 10, ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - ((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) + (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2) - (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2) - (2*d^2*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^3) + (2*d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^3)} -{(c + d*x)/(a + b*Cosh[e + f*x]), x, 8, ((c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) - ((c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*f) + (d*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2) - (d*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/(Sqrt[a^2 - b^2]*f^2)} -{1/((c + d*x)*(a + b*Cosh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Cosh[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Cosh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Cosh[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Cosh[e + f*x])^2, x, 22, -((c + d*x)^3/((a^2 - b^2)*f)) + (3*d*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (a*(c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (3*d*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) - (a*(c + d*x)^3*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (6*d^2*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^3) + (3*a*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) + (6*d^2*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^3) - (3*a*d*(c + d*x)^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) - (6*d^3*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^4) - (6*a*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^3) - (6*d^3*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^4) + (6*a*d^2*(c + d*x)*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^3) + (6*a*d^3*PolyLog[4, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^4) - (6*a*d^3*PolyLog[4, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^4) - (b*(c + d*x)^3*Sinh[e + f*x])/((a^2 - b^2)*f*(a + b*Cosh[e + f*x]))} -{(c + d*x)^2/(a + b*Cosh[e + f*x])^2, x, 18, -((c + d*x)^2/((a^2 - b^2)*f)) + (2*d*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (2*d*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) - (a*(c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) + (2*d^2*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) - (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^3) + (2*a*d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^3) - (b*(c + d*x)^2*Sinh[e + f*x])/((a^2 - b^2)*f*(a + b*Cosh[e + f*x]))} -{(c + d*x)/(a + b*Cosh[e + f*x])^2, x, 11, (a*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (a*(c + d*x)*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + (d*Log[a + b*Cosh[e + f*x]])/((a^2 - b^2)*f^2) + (a*d*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) - (a*d*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 - b^2]))])/((a^2 - b^2)^(3/2)*f^2) - (b*(c + d*x)*Sinh[e + f*x])/((a^2 - b^2)*f*(a + b*Cosh[e + f*x]))} -{1/((c + d*x)*(a + b*Cosh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Cosh[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Cosh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Cosh[e + f*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Cosh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + b*Cosh[e + f*x])^n, x, 0, Unintegrable[(c + d*x)^m*(a + b*Cosh[e + f*x])^n, x]} - - -{(c + d*x)^m*(a + b*Cosh[e + f*x])^3, x, 18, (a^3*(c + d*x)^(1 + m))/(d*(1 + m)) + (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (3^(-1 - m)*b^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(8*f*(-((f*(c + d*x))/d))^m) + (3*2^(-3 - m)*a*b^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (3*a^2*b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) + (3*b^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(8*f*(-((f*(c + d*x))/d))^m) - (3*a^2*b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c + d*x))/d)^m) - (3*b^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m) - (3*2^(-3 - m)*a*b^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (3^(-1 - m)*b^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + b*Cosh[e + f*x])^2, x, 10, (a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (2^(-3 - m)*b^2*E^(2*e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (a*b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) - (a*b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) - (2^(-3 - m)*b^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m*(a + b*Cosh[e + f*x]), x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) + (b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) - (b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + b*Cosh[e + f*x]), x, 0, Unintegrable[(c + d*x)^m/(a + b*Cosh[e + f*x]), x]} -{(c + d*x)^m/(a + b*Cosh[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m/(a + b*Cosh[e + f*x])^2, x]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.2 (e x)^m (a+b x^n)^p cosh.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.2 (e x)^m (a+b x^n)^p cosh.m deleted file mode 100644 index 034e34a..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.2 (e x)^m (a+b x^n)^p cosh.m +++ /dev/null @@ -1,181 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b x^n)^p Cosh[c+d x]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^1)^p Cosh[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x)*Cosh[c + d*x], x, 11, (-6*a*Cosh[c + d*x])/d^4 - (24*b*x*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/d^2 - (4*b*x^3*Cosh[c + d*x])/d^2 + (24*b*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (12*b*x^2*Sinh[c + d*x])/d^3 + (a*x^3*Sinh[c + d*x])/d + (b*x^4*Sinh[c + d*x])/d} -{x^2*(a + b*x)*Cosh[c + d*x], x, 9, (-6*b*Cosh[c + d*x])/d^4 - (2*a*x*Cosh[c + d*x])/d^2 - (3*b*x^2*Cosh[c + d*x])/d^2 + (2*a*Sinh[c + d*x])/d^3 + (6*b*x*Sinh[c + d*x])/d^3 + (a*x^2*Sinh[c + d*x])/d + (b*x^3*Sinh[c + d*x])/d} -{x*(a + b*x)*Cosh[c + d*x], x, 7, -((a*Cosh[c + d*x])/d^2) - (2*b*x*Cosh[c + d*x])/d^2 + (2*b*Sinh[c + d*x])/d^3 + (a*x*Sinh[c + d*x])/d + (b*x^2*Sinh[c + d*x])/d} -{(a + b*x)*Cosh[c + d*x], x, 2, -((b*Cosh[c + d*x])/d^2) + ((a + b*x)*Sinh[c + d*x])/d} -{((a + b*x)*Cosh[c + d*x])/x, x, 6, a*Cosh[c]*CoshIntegral[d*x] + (b*Sinh[c + d*x])/d + a*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x)*Cosh[c + d*x])/x^2, x, 9, -((a*Cosh[c + d*x])/x) + b*Cosh[c]*CoshIntegral[d*x] + a*d*CoshIntegral[d*x]*Sinh[c] + a*d*Cosh[c]*SinhIntegral[d*x] + b*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x)*Cosh[c + d*x])/x^3, x, 11, -(a*Cosh[c + d*x])/(2*x^2) - (b*Cosh[c + d*x])/x + (a*d^2*Cosh[c]*CoshIntegral[d*x])/2 + b*d*CoshIntegral[d*x]*Sinh[c] - (a*d*Sinh[c + d*x])/(2*x) + b*d*Cosh[c]*SinhIntegral[d*x] + (a*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x)*Cosh[c + d*x])/x^4, x, 13, -(a*Cosh[c + d*x])/(3*x^3) - (b*Cosh[c + d*x])/(2*x^2) - (a*d^2*Cosh[c + d*x])/(6*x) + (b*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) - (b*d*Sinh[c + d*x])/(2*x) + (a*d^3*Cosh[c]*SinhIntegral[d*x])/6 + (b*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x)*Cosh[c + d*x])/x^5, x, 15, -(a*Cosh[c + d*x])/(4*x^4) - (b*Cosh[c + d*x])/(3*x^3) - (a*d^2*Cosh[c + d*x])/(24*x^2) - (b*d^2*Cosh[c + d*x])/(6*x) + (a*d^4*Cosh[c]*CoshIntegral[d*x])/24 + (b*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(12*x^3) - (b*d*Sinh[c + d*x])/(6*x^2) - (a*d^3*Sinh[c + d*x])/(24*x) + (b*d^3*Cosh[c]*SinhIntegral[d*x])/6 + (a*d^4*Sinh[c]*SinhIntegral[d*x])/24} - - -{x^2*(a + b*x)^2*Cosh[c + d*x], x, 14, (-12*a*b*Cosh[c + d*x])/d^4 - (24*b^2*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (4*b^2*x^3*Cosh[c + d*x])/d^2 + (24*b^2*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (12*a*b*x*Sinh[c + d*x])/d^3 + (12*b^2*x^2*Sinh[c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (2*a*b*x^3*Sinh[c + d*x])/d + (b^2*x^4*Sinh[c + d*x])/d} -{x*(a + b*x)^2*Cosh[c + d*x], x, 11, (-6*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/d^2 - (4*a*b*x*Cosh[c + d*x])/d^2 - (3*b^2*x^2*Cosh[c + d*x])/d^2 + (4*a*b*Sinh[c + d*x])/d^3 + (6*b^2*x*Sinh[c + d*x])/d^3 + (a^2*x*Sinh[c + d*x])/d + (2*a*b*x^2*Sinh[c + d*x])/d + (b^2*x^3*Sinh[c + d*x])/d} -{(a + b*x)^2*Cosh[c + d*x], x, 3, (-2*b*(a + b*x)*Cosh[c + d*x])/d^2 + (2*b^2*Sinh[c + d*x])/d^3 + ((a + b*x)^2*Sinh[c + d*x])/d} -{((a + b*x)^2*Cosh[c + d*x])/x, x, 8, -((b^2*Cosh[c + d*x])/d^2) + a^2*Cosh[c]*CoshIntegral[d*x] + (2*a*b*Sinh[c + d*x])/d + (b^2*x*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x)^2*Cosh[c + d*x])/x^2, x, 10, -((a^2*Cosh[c + d*x])/x) + 2*a*b*Cosh[c]*CoshIntegral[d*x] + a^2*d*CoshIntegral[d*x]*Sinh[c] + (b^2*Sinh[c + d*x])/d + a^2*d*Cosh[c]*SinhIntegral[d*x] + 2*a*b*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x)^2*Cosh[c + d*x])/x^3, x, 14, -(a^2*Cosh[c + d*x])/(2*x^2) - (2*a*b*Cosh[c + d*x])/x + b^2*Cosh[c]*CoshIntegral[d*x] + (a^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(2*x) + 2*a*b*d*Cosh[c]*SinhIntegral[d*x] + b^2*Sinh[c]*SinhIntegral[d*x] + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x)^2*Cosh[c + d*x])/x^4, x, 17, -(a^2*Cosh[c + d*x])/(3*x^3) - (a*b*Cosh[c + d*x])/x^2 - (b^2*Cosh[c + d*x])/x - (a^2*d^2*Cosh[c + d*x])/(6*x) + a*b*d^2*Cosh[c]*CoshIntegral[d*x] + b^2*d*CoshIntegral[d*x]*Sinh[c] + (a^2*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a^2*d*Sinh[c + d*x])/(6*x^2) - (a*b*d*Sinh[c + d*x])/x + b^2*d*Cosh[c]*SinhIntegral[d*x] + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6 + a*b*d^2*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x)^2*Cosh[c + d*x])/x^5, x, 20, -(a^2*Cosh[c + d*x])/(4*x^4) - (2*a*b*Cosh[c + d*x])/(3*x^3) - (b^2*Cosh[c + d*x])/(2*x^2) - (a^2*d^2*Cosh[c + d*x])/(24*x^2) - (a*b*d^2*Cosh[c + d*x])/(3*x) + (b^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a^2*d^4*Cosh[c]*CoshIntegral[d*x])/24 + (a*b*d^3*CoshIntegral[d*x]*Sinh[c])/3 - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a*b*d*Sinh[c + d*x])/(3*x^2) - (b^2*d*Sinh[c + d*x])/(2*x) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (a*b*d^3*Cosh[c]*SinhIntegral[d*x])/3 + (b^2*d^2*Sinh[c]*SinhIntegral[d*x])/2 + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Cosh[c + d*x])/(a + b*x), x, 15, (-6*Cosh[c + d*x])/(b*d^4) - (a^2*Cosh[c + d*x])/(b^3*d^2) + (2*a*x*Cosh[c + d*x])/(b^2*d^2) - (3*x^2*Cosh[c + d*x])/(b*d^2) + (a^4*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^5 - (2*a*Sinh[c + d*x])/(b^2*d^3) - (a^3*Sinh[c + d*x])/(b^4*d) + (6*x*Sinh[c + d*x])/(b*d^3) + (a^2*x*Sinh[c + d*x])/(b^3*d) - (a*x^2*Sinh[c + d*x])/(b^2*d) + (x^3*Sinh[c + d*x])/(b*d) + (a^4*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5} -{(x^3*Cosh[c + d*x])/(a + b*x), x, 11, (a*Cosh[c + d*x])/(b^2*d^2) - (2*x*Cosh[c + d*x])/(b*d^2) - (a^3*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^4 + (2*Sinh[c + d*x])/(b*d^3) + (a^2*Sinh[c + d*x])/(b^3*d) - (a*x*Sinh[c + d*x])/(b^2*d) + (x^2*Sinh[c + d*x])/(b*d) - (a^3*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4} -{(x^2*Cosh[c + d*x])/(a + b*x), x, 8, -(Cosh[c + d*x]/(b*d^2)) + (a^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 - (a*Sinh[c + d*x])/(b^2*d) + (x*Sinh[c + d*x])/(b*d) + (a^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3} -{(x*Cosh[c + d*x])/(a + b*x), x, 6, -((a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2) + Sinh[c + d*x]/(b*d) - (a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^2} -{Cosh[c + d*x]/(a + b*x), x, 3, (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b} -{Cosh[c + d*x]/(x*(a + b*x)), x, 8, (Cosh[c]*CoshIntegral[d*x])/a - (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a + (Sinh[c]*SinhIntegral[d*x])/a - (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a} -{Cosh[c + d*x]/(x^2*(a + b*x)), x, 12, -(Cosh[c + d*x]/(a*x)) - (b*Cosh[c]*CoshIntegral[d*x])/a^2 + (b*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^2 + (d*CoshIntegral[d*x]*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - (b*Sinh[c]*SinhIntegral[d*x])/a^2 + (b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^2} -{Cosh[c + d*x]/(x^3*(a + b*x)), x, 17, -Cosh[c + d*x]/(2*a*x^2) + (b*Cosh[c + d*x])/(a^2*x) + (b^2*Cosh[c]*CoshIntegral[d*x])/a^3 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) - (b^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (b*d*CoshIntegral[d*x]*Sinh[c])/a^2 - (d*Sinh[c + d*x])/(2*a*x) - (b*d*Cosh[c]*SinhIntegral[d*x])/a^2 + (b^2*Sinh[c]*SinhIntegral[d*x])/a^3 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - (b^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3} - - -{(x^4*Cosh[c + d*x])/(a + b*x)^2, x, 15, (2*a*Cosh[c + d*x])/(b^3*d^2) - (2*x*Cosh[c + d*x])/(b^2*d^2) - (a^4*Cosh[c + d*x])/(b^5*(a + b*x)) - (4*a^3*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^5 + (a^4*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^6 + (2*Sinh[c + d*x])/(b^2*d^3) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (2*a*x*Sinh[c + d*x])/(b^3*d) + (x^2*Sinh[c + d*x])/(b^2*d) + (a^4*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5} -{(x^3*Cosh[c + d*x])/(a + b*x)^2, x, 12, -(Cosh[c + d*x]/(b^2*d^2)) + (a^3*Cosh[c + d*x])/(b^4*(a + b*x)) + (3*a^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^4 - (a^3*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 - (2*a*Sinh[c + d*x])/(b^3*d) + (x*Sinh[c + d*x])/(b^2*d) - (a^3*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5 + (3*a^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4} -{(x^2*Cosh[c + d*x])/(a + b*x)^2, x, 10, -((a^2*Cosh[c + d*x])/(b^3*(a + b*x))) - (2*a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 + (a^2*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 + Sinh[c + d*x]/(b^2*d) + (a^2*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4 - (2*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3} -{(x*Cosh[c + d*x])/(a + b*x)^2, x, 9, (a*Cosh[c + d*x])/(b^2*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^2 - (a*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^3 - (a*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^2} -{Cosh[c + d*x]/(a + b*x)^2, x, 4, -(Cosh[c + d*x]/(b*(a + b*x))) + (d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^2 + (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^2} -{Cosh[c + d*x]/(x*(a + b*x)^2), x, 12, Cosh[c + d*x]/(a*(a + b*x)) + (Cosh[c]*CoshIntegral[d*x])/a^2 - (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^2 - (d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/(a*b) + (Sinh[c]*SinhIntegral[d*x])/a^2 - (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(a*b) - (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^2} -{Cosh[c + d*x]/(x^2*(a + b*x)^2), x, 16, -(Cosh[c + d*x]/(a^2*x)) - (b*Cosh[c + d*x])/(a^2*(a + b*x)) - (2*b*Cosh[c]*CoshIntegral[d*x])/a^3 + (2*b*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/a^2 + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (2*b*Sinh[c]*SinhIntegral[d*x])/a^3 + (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^2 + (2*b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3} - - -{(x^3*Cosh[c + d*x])/(a + b*x)^3, x, 15, (a^3*Cosh[c + d*x])/(2*b^4*(a + b*x)^2) - (3*a^2*Cosh[c + d*x])/(b^4*(a + b*x)) - (3*a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^4 - (a^3*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^6) + (3*a^2*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 + Sinh[c + d*x]/(b^3*d) + (a^3*d*Sinh[c + d*x])/(2*b^5*(a + b*x)) + (3*a^2*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^5 - (3*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4 - (a^3*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^6)} -{(x^2*Cosh[c + d*x])/(a + b*x)^3, x, 14, -(a^2*Cosh[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Cosh[c + d*x])/(b^3*(a + b*x)) + (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^5) - (2*a*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^4 - (a^2*d*Sinh[c + d*x])/(2*b^4*(a + b*x)) - (2*a*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^4 + (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 + (a^2*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^5)} -{(x*Cosh[c + d*x])/(a + b*x)^3, x, 11, (a*Cosh[c + d*x])/(2*b^2*(a + b*x)^2) - Cosh[c + d*x]/(b^2*(a + b*x)) - (a*d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^4) + (d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^3 + (a*d*Sinh[c + d*x])/(2*b^3*(a + b*x)) + (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/b^3 - (a*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^4)} -{Cosh[c + d*x]/(a + b*x)^3, x, 5, -Cosh[c + d*x]/(2*b*(a + b*x)^2) + (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^3) - (d*Sinh[c + d*x])/(2*b^2*(a + b*x)) + (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^3)} -{Cosh[c + d*x]/(x*(a + b*x)^3), x, 17, Cosh[c + d*x]/(2*a*(a + b*x)^2) + Cosh[c + d*x]/(a^2*(a + b*x)) + (Cosh[c]*CoshIntegral[d*x])/a^3 - (Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^3 - (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*a*b^2) - (d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/(a^2*b) + (d*Sinh[c + d*x])/(2*a*b*(a + b*x)) + (Sinh[c]*SinhIntegral[d*x])/a^3 - (d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(a^2*b) - (Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3 - (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*a*b^2)} -{Cosh[c + d*x]/(x^2*(a + b*x)^3), x, 21, -(Cosh[c + d*x]/(a^3*x)) - (b*Cosh[c + d*x])/(2*a^2*(a + b*x)^2) - (2*b*Cosh[c + d*x])/(a^3*(a + b*x)) - (3*b*Cosh[c]*CoshIntegral[d*x])/a^4 + (3*b*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^4 + (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*a^2*b) + (d*CoshIntegral[d*x]*Sinh[c])/a^3 + (2*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/a^3 - (d*Sinh[c + d*x])/(2*a^2*(a + b*x)) + (d*Cosh[c]*SinhIntegral[d*x])/a^3 - (3*b*Sinh[c]*SinhIntegral[d*x])/a^4 + (2*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^3 + (3*b*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^4 + (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*a^2*b)} -{Cosh[c + d*x]/(x^3*(a + b*x)^3), x, 26, -Cosh[c + d*x]/(2*a^3*x^2) + (3*b*Cosh[c + d*x])/(a^4*x) + (b^2*Cosh[c + d*x])/(2*a^3*(a + b*x)^2) + (3*b^2*Cosh[c + d*x])/(a^4*(a + b*x)) + (6*b^2*Cosh[c]*CoshIntegral[d*x])/a^5 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^3) - (6*b^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/a^5 - (d^2*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*a^3) - (3*b*d*CoshIntegral[d*x]*Sinh[c])/a^4 - (3*b*d*CoshIntegral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/a^4 - (d*Sinh[c + d*x])/(2*a^3*x) + (b*d*Sinh[c + d*x])/(2*a^3*(a + b*x)) - (3*b*d*Cosh[c]*SinhIntegral[d*x])/a^4 + (6*b^2*Sinh[c]*SinhIntegral[d*x])/a^5 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) - (3*b*d*Cosh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^4 - (6*b^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/a^5 - (d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*a^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^2)^p Cosh[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x^2)*Cosh[c + d*x], x, 12, (-120*b*Cosh[c + d*x])/d^6 - (6*a*Cosh[c + d*x])/d^4 - (60*b*x^2*Cosh[c + d*x])/d^4 - (3*a*x^2*Cosh[c + d*x])/d^2 - (5*b*x^4*Cosh[c + d*x])/d^2 + (120*b*x*Sinh[c + d*x])/d^5 + (6*a*x*Sinh[c + d*x])/d^3 + (20*b*x^3*Sinh[c + d*x])/d^3 + (a*x^3*Sinh[c + d*x])/d + (b*x^5*Sinh[c + d*x])/d} -{x^2*(a + b*x^2)*Cosh[c + d*x], x, 10, (-24*b*x*Cosh[c + d*x])/d^4 - (2*a*x*Cosh[c + d*x])/d^2 - (4*b*x^3*Cosh[c + d*x])/d^2 + (24*b*Sinh[c + d*x])/d^5 + (2*a*Sinh[c + d*x])/d^3 + (12*b*x^2*Sinh[c + d*x])/d^3 + (a*x^2*Sinh[c + d*x])/d + (b*x^4*Sinh[c + d*x])/d} -{x*(a + b*x^2)*Cosh[c + d*x], x, 8, (-6*b*Cosh[c + d*x])/d^4 - (a*Cosh[c + d*x])/d^2 - (3*b*x^2*Cosh[c + d*x])/d^2 + (6*b*x*Sinh[c + d*x])/d^3 + (a*x*Sinh[c + d*x])/d + (b*x^3*Sinh[c + d*x])/d} -{(a + b*x^2)*Cosh[c + d*x], x, 6, (-2*b*x*Cosh[c + d*x])/d^2 + (2*b*Sinh[c + d*x])/d^3 + (a*Sinh[c + d*x])/d + (b*x^2*Sinh[c + d*x])/d} -{((a + b*x^2)*Cosh[c + d*x])/x, x, 7, -((b*Cosh[c + d*x])/d^2) + a*Cosh[c]*CoshIntegral[d*x] + (b*x*Sinh[c + d*x])/d + a*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x^2)*Cosh[c + d*x])/x^2, x, 7, -((a*Cosh[c + d*x])/x) + a*d*CoshIntegral[d*x]*Sinh[c] + (b*Sinh[c + d*x])/d + a*d*Cosh[c]*SinhIntegral[d*x]} -{((a + b*x^2)*Cosh[c + d*x])/x^3, x, 10, -(a*Cosh[c + d*x])/(2*x^2) + b*Cosh[c]*CoshIntegral[d*x] + (a*d^2*Cosh[c]*CoshIntegral[d*x])/2 - (a*d*Sinh[c + d*x])/(2*x) + b*Sinh[c]*SinhIntegral[d*x] + (a*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x^2)*Cosh[c + d*x])/x^4, x, 12, -(a*Cosh[c + d*x])/(3*x^3) - (b*Cosh[c + d*x])/x - (a*d^2*Cosh[c + d*x])/(6*x) + b*d*CoshIntegral[d*x]*Sinh[c] + (a*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) + b*d*Cosh[c]*SinhIntegral[d*x] + (a*d^3*Cosh[c]*SinhIntegral[d*x])/6} -{((a + b*x^2)*Cosh[c + d*x])/x^5, x, 14, -(a*Cosh[c + d*x])/(4*x^4) - (b*Cosh[c + d*x])/(2*x^2) - (a*d^2*Cosh[c + d*x])/(24*x^2) + (b*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (a*d^4*Cosh[c]*CoshIntegral[d*x])/24 - (a*d*Sinh[c + d*x])/(12*x^3) - (b*d*Sinh[c + d*x])/(2*x) - (a*d^3*Sinh[c + d*x])/(24*x) + (b*d^2*Sinh[c]*SinhIntegral[d*x])/2 + (a*d^4*Sinh[c]*SinhIntegral[d*x])/24} - - -{x^2*(a + b*x^2)^2*Cosh[c + d*x], x, 17, (-720*b^2*x*Cosh[c + d*x])/d^6 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2*a^2*x*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cosh[c + d*x])/d^4 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (48*a*b*Sinh[c + d*x])/d^5 + (2*a^2*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (24*a*b*x^2*Sinh[c + d*x])/d^3 + (a^2*x^2*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (2*a*b*x^4*Sinh[c + d*x])/d + (b^2*x^6*Sinh[c + d*x])/d} -{x*(a + b*x^2)^2*Cosh[c + d*x], x, 14, (-120*b^2*Cosh[c + d*x])/d^6 - (12*a*b*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (5*b^2*x^4*Cosh[c + d*x])/d^2 + (120*b^2*x*Sinh[c + d*x])/d^5 + (12*a*b*x*Sinh[c + d*x])/d^3 + (a^2*x*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (2*a*b*x^3*Sinh[c + d*x])/d + (b^2*x^5*Sinh[c + d*x])/d} -{(a + b*x^2)^2*Cosh[c + d*x], x, 11, (-24*b^2*x*Cosh[c + d*x])/d^4 - (4*a*b*x*Cosh[c + d*x])/d^2 - (4*b^2*x^3*Cosh[c + d*x])/d^2 + (24*b^2*Sinh[c + d*x])/d^5 + (4*a*b*Sinh[c + d*x])/d^3 + (a^2*Sinh[c + d*x])/d + (12*b^2*x^2*Sinh[c + d*x])/d^3 + (2*a*b*x^2*Sinh[c + d*x])/d + (b^2*x^4*Sinh[c + d*x])/d} -{((a + b*x^2)^2*Cosh[c + d*x])/x, x, 11, (-6*b^2*Cosh[c + d*x])/d^4 - (2*a*b*Cosh[c + d*x])/d^2 - (3*b^2*x^2*Cosh[c + d*x])/d^2 + a^2*Cosh[c]*CoshIntegral[d*x] + (6*b^2*x*Sinh[c + d*x])/d^3 + (2*a*b*x*Sinh[c + d*x])/d + (b^2*x^3*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x^2)^2*Cosh[c + d*x])/x^2, x, 10, -((a^2*Cosh[c + d*x])/x) - (2*b^2*x*Cosh[c + d*x])/d^2 + a^2*d*CoshIntegral[d*x]*Sinh[c] + (2*b^2*Sinh[c + d*x])/d^3 + (2*a*b*Sinh[c + d*x])/d + (b^2*x^2*Sinh[c + d*x])/d + a^2*d*Cosh[c]*SinhIntegral[d*x]} -{((a + b*x^2)^2*Cosh[c + d*x])/x^3, x, 12, -((b^2*Cosh[c + d*x])/d^2) - (a^2*Cosh[c + d*x])/(2*x^2) + 2*a*b*Cosh[c]*CoshIntegral[d*x] + (a^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 - (a^2*d*Sinh[c + d*x])/(2*x) + (b^2*x*Sinh[c + d*x])/d + 2*a*b*Sinh[c]*SinhIntegral[d*x] + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x^2)^2*Cosh[c + d*x])/x^4, x, 13, -(a^2*Cosh[c + d*x])/(3*x^3) - (2*a*b*Cosh[c + d*x])/x - (a^2*d^2*Cosh[c + d*x])/(6*x) + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] + (a^2*d^3*CoshIntegral[d*x]*Sinh[c])/6 + (b^2*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/(6*x^2) + 2*a*b*d*Cosh[c]*SinhIntegral[d*x] + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6} -{((a + b*x^2)^2*Cosh[c + d*x])/x^5, x, 17, -(a^2*Cosh[c + d*x])/(4*x^4) - (a*b*Cosh[c + d*x])/x^2 - (a^2*d^2*Cosh[c + d*x])/(24*x^2) + b^2*Cosh[c]*CoshIntegral[d*x] + a*b*d^2*Cosh[c]*CoshIntegral[d*x] + (a^2*d^4*Cosh[c]*CoshIntegral[d*x])/24 - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a*b*d*Sinh[c + d*x])/x - (a^2*d^3*Sinh[c + d*x])/(24*x) + b^2*Sinh[c]*SinhIntegral[d*x] + a*b*d^2*Sinh[c]*SinhIntegral[d*x] + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Cosh[c + d*x])/(a + b*x^2), x, 14, (-2*x*Cosh[c + d*x])/(b*d^2) + ((-a)^(3/2)*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)) - ((-a)^(3/2)*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(5/2)) + (2*Sinh[c + d*x])/(b*d^3) - (a*Sinh[c + d*x])/(b^2*d) + (x^2*Sinh[c + d*x])/(b*d) - ((-a)^(3/2)*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(5/2)) - ((-a)^(3/2)*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(5/2))} -{(x^3*Cosh[c + d*x])/(a + b*x^2), x, 12, -(Cosh[c + d*x]/(b*d^2)) - (a*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) + (x*Sinh[c + d*x])/(b*d) + (a*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2)} -{(x^2*Cosh[c + d*x])/(a + b*x^2), x, 11, (Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2)) + Sinh[c + d*x]/(b*d) - (Sqrt[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^(3/2)) - (Sqrt[-a]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^(3/2))} -{(x*Cosh[c + d*x])/(a + b*x^2), x, 8, (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b)} -{Cosh[c + d*x]/(a + b*x^2), x, 8, (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b])} -{Cosh[c + d*x]/(x*(a + b*x^2)), x, 13, (Cosh[c]*CoshIntegral[d*x])/a - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a) + (Sinh[c]*SinhIntegral[d*x])/a + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a)} -{Cosh[c + d*x]/(x^2*(a + b*x^2)), x, 14, -(Cosh[c + d*x]/(a*x)) + (Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/2)) - (Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2)) + (d*CoshIntegral[d*x]*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - (Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*(-a)^(3/2)) - (Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*(-a)^(3/2))} -{Cosh[c + d*x]/(x^3*(a + b*x^2)), x, 18, -Cosh[c + d*x]/(2*a*x^2) - (b*Cosh[c]*CoshIntegral[d*x])/a^2 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) + (b*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) - (d*Sinh[c + d*x])/(2*a*x) - (b*Sinh[c]*SinhIntegral[d*x])/a^2 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - (b*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) + (b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)} - - -{(x^4*Cosh[c + d*x])/(a + b*x^2)^2, x, 24, (x*Cosh[c + d*x])/(2*b^2) - (x^3*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (3*Sqrt[-a]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (3*Sqrt[-a]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^3) - (a*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^3) + Sinh[c + d*x]/(b^2*d) + (a*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^3) - (3*Sqrt[-a]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (a*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^3) - (3*Sqrt[-a]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2))} -{(x^3*Cosh[c + d*x])/(a + b*x^2)^2, x, 20, Cosh[c + d*x]/(2*b^2) - (x^2*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) - (Sqrt[-a]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) + (Sqrt[-a]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^(5/2)) - (Sqrt[-a]*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^(5/2)) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (Sqrt[-a]*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^(5/2)) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2)} -{(x^2*Cosh[c + d*x])/(a + b*x^2)^2, x, 17, -(x*Cosh[c + d*x])/(2*b*(a + b*x^2)) + (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*b^2) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*b^2) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*b^2) - (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*b^2) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))} -{(x*Cosh[c + d*x])/(a + b*x^2)^2, x, 9, -Cosh[c + d*x]/(2*b*(a + b*x^2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*Sqrt[-a]*b^(3/2)) - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*Sqrt[-a]*b^(3/2)) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*Sqrt[-a]*b^(3/2))} -{Cosh[c + d*x]/(a + b*x^2)^2, x, 18, -Cosh[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(4*a*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*a*b) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*a*b) + (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a*b) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a*b) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b])} -{Cosh[c + d*x]/(x*(a + b*x^2)^2), x, 22, Cosh[c + d*x]/(2*a*(a + b*x^2)) + (Cosh[c]*CoshIntegral[d*x])/a^2 - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*(-a)^(3/2)*Sqrt[b]) + (Sinh[c]*SinhIntegral[d*x])/a^2 - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(3/2)*Sqrt[b]) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^2) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(3/2)*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^2)} -{Cosh[c + d*x]/(x^2*(a + b*x^2)^2), x, 32, -(Cosh[c + d*x]/(a^2*x)) + (Sqrt[b]*Cosh[c + d*x])/(4*a^2*(Sqrt[-a] - Sqrt[b]*x)) - (Sqrt[b]*Cosh[c + d*x])/(4*a^2*(Sqrt[-a] + Sqrt[b]*x)) - (3*Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(5/2)) + (3*Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2)) + (d*CoshIntegral[d*x]*Sinh[c])/a^2 + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(4*a^2) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(4*a^2) + (d*Cosh[c]*SinhIntegral[d*x])/a^2 - (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*a^2) + (3*Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(4*(-a)^(5/2)) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*a^2) + (3*Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(4*(-a)^(5/2))} - - -{(x^3*Cosh[c + d*x])/(a + b*x^2)^3, x, 27, -(x^2*Cosh[c + d*x])/(4*b*(a + b*x^2)^2) - Cosh[c + d*x]/(4*b^2*(a + b*x^2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) + (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*Sqrt[-a]*b^(5/2)) - (d*x*Sinh[c + d*x])/(8*b^2*(a + b*x^2)) - (3*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*b^3) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2)) + (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*b^3)} -{(x^2*Cosh[c + d*x])/(a + b*x^2)^3, x, 28, -Cosh[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) - (x*Cosh[c + d*x])/(4*b*(a + b*x^2)^2) - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) + (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a*b^2) - (d*Sinh[c + d*x])/(8*b^2*(a + b*x^2)) + (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*Sqrt[-a]*b^(5/2)) - (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*Sqrt[-a]*b^(5/2))} -{(x*Cosh[c + d*x])/(a + b*x^2)^3, x, 19, -Cosh[c + d*x]/(4*b*(a + b*x^2)^2) - (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2) + (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - (d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(3/2)*b^(3/2)) - (d*Sinh[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*a*b^(3/2)*(Sqrt[-a] + Sqrt[b]*x)) + (d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) + (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a*b^2) + (d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a*b^2)} -{Cosh[c + d*x]/(a + b*x^2)^3, x, 28, -Cosh[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)^2) - (3*Cosh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) + Cosh[c + d*x]/(16*(-a)^(3/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)^2) + (3*Cosh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) + (3*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) - (3*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^2*b) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*(-a)^(3/2)*b*(Sqrt[-a] + Sqrt[b]*x)) + (3*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (3*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(3/2)*b^(3/2)) - (3*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) - (3*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(3/2)*b^(3/2))} -{Cosh[c + d*x]/(x*(a + b*x^2)^3), x, 41, Cosh[c + d*x]/(4*a*(a + b*x^2)^2) + Cosh[c + d*x]/(2*a^2*(a + b*x^2)) + (Cosh[c]*CoshIntegral[d*x])/a^3 - (Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) - (Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^3) + (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b) + (5*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) - (5*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(5/2)*Sqrt[b]) + (d*Sinh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*x)) - (d*Sinh[c + d*x])/(16*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*x)) + (Sinh[c]*SinhIntegral[d*x])/a^3 + (5*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^3) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^2*b) + (5*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^3) + (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^2*b)} -{Cosh[c + d*x]/(x^2*(a + b*x^2)^3), x, 60, -(Cosh[c + d*x]/(a^3*x)) - (Sqrt[b]*Cosh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)^2) + (7*Sqrt[b]*Cosh[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*Cosh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)^2) - (7*Sqrt[b]*Cosh[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) + (15*Sqrt[b]*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) + (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) - (15*Sqrt[b]*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (d*CoshIntegral[d*x]*Sinh[c])/a^3 + (7*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (7*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*a^3) + (d*Sinh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] - Sqrt[b]*x)) + (d*Sinh[c + d*x])/(16*(-a)^(5/2)*(Sqrt[-a] + Sqrt[b]*x)) + (d*Cosh[c]*SinhIntegral[d*x])/a^3 - (7*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) - (15*Sqrt[b]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(5/2)*Sqrt[b]) + (7*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) - (15*Sqrt[b]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(5/2)*Sqrt[b])} -{Cosh[c + d*x]/(x^3*(a + b*x^2)^3), x, 46, -Cosh[c + d*x]/(2*a^3*x^2) - (b*Cosh[c + d*x])/(4*a^2*(a + b*x^2)^2) - (b*Cosh[c + d*x])/(a^3*(a + b*x^2)) - (3*b*Cosh[c]*CoshIntegral[d*x])/a^4 + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a^3) + (3*b*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) - (d^2*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (3*b*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^4) - (d^2*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3) + (9*Sqrt[b]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (9*Sqrt[b]*d*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]])/(16*(-a)^(7/2)) - (d*Sinh[c + d*x])/(2*a^3*x) - (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt[-a] - Sqrt[b]*x)) + (Sqrt[b]*d*Sinh[c + d*x])/(16*a^3*(Sqrt[-a] + Sqrt[b]*x)) - (3*b*Sinh[c]*SinhIntegral[d*x])/a^4 + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a^3) + (9*Sqrt[b]*d*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*(-a)^(7/2)) - (3*b*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*a^4) + (d^2*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(16*a^3) + (9*Sqrt[b]*d*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*(-a)^(7/2)) + (3*b*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^4) - (d^2*Sinh[c - (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(16*a^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b x^3)^p Cosh[c+d x]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*x^3)*Cosh[c + d*x], x, 13, (-6*a*Cosh[c + d*x])/d^4 - (720*b*x*Cosh[c + d*x])/d^6 - (3*a*x^2*Cosh[c + d*x])/d^2 - (120*b*x^3*Cosh[c + d*x])/d^4 - (6*b*x^5*Cosh[c + d*x])/d^2 + (720*b*Sinh[c + d*x])/d^7 + (6*a*x*Sinh[c + d*x])/d^3 + (360*b*x^2*Sinh[c + d*x])/d^5 + (a*x^3*Sinh[c + d*x])/d + (30*b*x^4*Sinh[c + d*x])/d^3 + (b*x^6*Sinh[c + d*x])/d} -{x^2*(a + b*x^3)*Cosh[c + d*x], x, 11, (-120*b*Cosh[c + d*x])/d^6 - (2*a*x*Cosh[c + d*x])/d^2 - (60*b*x^2*Cosh[c + d*x])/d^4 - (5*b*x^4*Cosh[c + d*x])/d^2 + (2*a*Sinh[c + d*x])/d^3 + (120*b*x*Sinh[c + d*x])/d^5 + (a*x^2*Sinh[c + d*x])/d + (20*b*x^3*Sinh[c + d*x])/d^3 + (b*x^5*Sinh[c + d*x])/d} -{x^1*(a + b*x^3)*Cosh[c + d*x], x, 9, -((a*Cosh[c + d*x])/d^2) - (24*b*x*Cosh[c + d*x])/d^4 - (4*b*x^3*Cosh[c + d*x])/d^2 + (24*b*Sinh[c + d*x])/d^5 + (a*x*Sinh[c + d*x])/d + (12*b*x^2*Sinh[c + d*x])/d^3 + (b*x^4*Sinh[c + d*x])/d} -{(a + b*x^3)*Cosh[c + d*x], x, 7, (-6*b*Cosh[c + d*x])/d^4 - (3*b*x^2*Cosh[c + d*x])/d^2 + (a*Sinh[c + d*x])/d + (6*b*x*Sinh[c + d*x])/d^3 + (b*x^3*Sinh[c + d*x])/d} -{((a + b*x^3)*Cosh[c + d*x])/x^1, x, 8, (-2*b*x*Cosh[c + d*x])/d^2 + a*Cosh[c]*CoshIntegral[d*x] + (2*b*Sinh[c + d*x])/d^3 + (b*x^2*Sinh[c + d*x])/d + a*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x^3)*Cosh[c + d*x])/x^2, x, 8, -((b*Cosh[c + d*x])/d^2) - (a*Cosh[c + d*x])/x + a*d*CoshIntegral[d*x]*Sinh[c] + (b*x*Sinh[c + d*x])/d + a*d*Cosh[c]*SinhIntegral[d*x]} -{((a + b*x^3)*Cosh[c + d*x])/x^3, x, 8, -(a*Cosh[c + d*x])/(2*x^2) + (a*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (b*Sinh[c + d*x])/d - (a*d*Sinh[c + d*x])/(2*x) + (a*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x^3)*Cosh[c + d*x])/x^4, x, 11, -(a*Cosh[c + d*x])/(3*x^3) - (a*d^2*Cosh[c + d*x])/(6*x) + b*Cosh[c]*CoshIntegral[d*x] + (a*d^3*CoshIntegral[d*x]*Sinh[c])/6 - (a*d*Sinh[c + d*x])/(6*x^2) + (a*d^3*Cosh[c]*SinhIntegral[d*x])/6 + b*Sinh[c]*SinhIntegral[d*x]} - - -{x^1*(a + b*x^3)^2*Cosh[c + d*x], x, 17, (-5040*b^2*Cosh[c + d*x])/d^8 - (a^2*Cosh[c + d*x])/d^2 - (48*a*b*x*Cosh[c + d*x])/d^4 - (2520*b^2*x^2*Cosh[c + d*x])/d^6 - (8*a*b*x^3*Cosh[c + d*x])/d^2 - (210*b^2*x^4*Cosh[c + d*x])/d^4 - (7*b^2*x^6*Cosh[c + d*x])/d^2 + (48*a*b*Sinh[c + d*x])/d^5 + (5040*b^2*x*Sinh[c + d*x])/d^7 + (a^2*x*Sinh[c + d*x])/d + (24*a*b*x^2*Sinh[c + d*x])/d^3 + (840*b^2*x^3*Sinh[c + d*x])/d^5 + (2*a*b*x^4*Sinh[c + d*x])/d + (42*b^2*x^5*Sinh[c + d*x])/d^3 + (b^2*x^7*Sinh[c + d*x])/d} -{(a + b*x^3)^2*Cosh[c + d*x], x, 14, (-12*a*b*Cosh[c + d*x])/d^4 - (720*b^2*x*Cosh[c + d*x])/d^6 - (6*a*b*x^2*Cosh[c + d*x])/d^2 - (120*b^2*x^3*Cosh[c + d*x])/d^4 - (6*b^2*x^5*Cosh[c + d*x])/d^2 + (720*b^2*Sinh[c + d*x])/d^7 + (a^2*Sinh[c + d*x])/d + (12*a*b*x*Sinh[c + d*x])/d^3 + (360*b^2*x^2*Sinh[c + d*x])/d^5 + (2*a*b*x^3*Sinh[c + d*x])/d + (30*b^2*x^4*Sinh[c + d*x])/d^3 + (b^2*x^6*Sinh[c + d*x])/d} -{((a + b*x^3)^2*Cosh[c + d*x])/x^1, x, 14, (-120*b^2*Cosh[c + d*x])/d^6 - (4*a*b*x*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (5*b^2*x^4*Cosh[c + d*x])/d^2 + a^2*Cosh[c]*CoshIntegral[d*x] + (4*a*b*Sinh[c + d*x])/d^3 + (120*b^2*x*Sinh[c + d*x])/d^5 + (2*a*b*x^2*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (b^2*x^5*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x^3)^2*Cosh[c + d*x])/x^2, x, 13, (-2*a*b*Cosh[c + d*x])/d^2 - (a^2*Cosh[c + d*x])/x - (24*b^2*x*Cosh[c + d*x])/d^4 - (4*b^2*x^3*Cosh[c + d*x])/d^2 + a^2*d*CoshIntegral[d*x]*Sinh[c] + (24*b^2*Sinh[c + d*x])/d^5 + (2*a*b*x*Sinh[c + d*x])/d + (12*b^2*x^2*Sinh[c + d*x])/d^3 + (b^2*x^4*Sinh[c + d*x])/d + a^2*d*Cosh[c]*SinhIntegral[d*x]} -{((a + b*x^3)^2*Cosh[c + d*x])/x^3, x, 12, (-6*b^2*Cosh[c + d*x])/d^4 - (a^2*Cosh[c + d*x])/(2*x^2) - (3*b^2*x^2*Cosh[c + d*x])/d^2 + (a^2*d^2*Cosh[c]*CoshIntegral[d*x])/2 + (2*a*b*Sinh[c + d*x])/d - (a^2*d*Sinh[c + d*x])/(2*x) + (6*b^2*x*Sinh[c + d*x])/d^3 + (b^2*x^3*Sinh[c + d*x])/d + (a^2*d^2*Sinh[c]*SinhIntegral[d*x])/2} -{((a + b*x^3)^2*Cosh[c + d*x])/x^4, x, 14, -(a^2*Cosh[c + d*x])/(3*x^3) - (a^2*d^2*Cosh[c + d*x])/(6*x) - (2*b^2*x*Cosh[c + d*x])/d^2 + 2*a*b*Cosh[c]*CoshIntegral[d*x] + (a^2*d^3*CoshIntegral[d*x]*Sinh[c])/6 + (2*b^2*Sinh[c + d*x])/d^3 - (a^2*d*Sinh[c + d*x])/(6*x^2) + (b^2*x^2*Sinh[c + d*x])/d + (a^2*d^3*Cosh[c]*SinhIntegral[d*x])/6 + 2*a*b*Sinh[c]*SinhIntegral[d*x]} -{((a + b*x^3)^2*Cosh[c + d*x])/x^5, x, 15, -((b^2*Cosh[c + d*x])/d^2) - (a^2*Cosh[c + d*x])/(4*x^4) - (a^2*d^2*Cosh[c + d*x])/(24*x^2) - (2*a*b*Cosh[c + d*x])/x + (a^2*d^4*Cosh[c]*CoshIntegral[d*x])/24 + 2*a*b*d*CoshIntegral[d*x]*Sinh[c] - (a^2*d*Sinh[c + d*x])/(12*x^3) - (a^2*d^3*Sinh[c + d*x])/(24*x) + (b^2*x*Sinh[c + d*x])/d + 2*a*b*d*Cosh[c]*SinhIntegral[d*x] + (a^2*d^4*Sinh[c]*SinhIntegral[d*x])/24} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*Cosh[c + d*x])/(a + b*x^3), x, 15, -(Cosh[c + d*x]/(b*d^2)) + ((-1)^(2/3)*a^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b^(5/3)) + (a^(2/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(5/3)) + (x*Sinh[c + d*x])/(b*d) - ((-1)^(2/3)*a^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(5/3)) + (a^(2/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(5/3)) - ((-1)^(1/3)*a^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(5/3))} -{(x^3*Cosh[c + d*x])/(a + b*x^3), x, 14, ((-1)^(1/3)*a^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b^(4/3)) - (a^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) + Sinh[c + d*x]/(b*d) - ((-1)^(1/3)*a^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b^(4/3)) - (a^(1/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3)) - ((-1)^(2/3)*a^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b^(4/3))} -{(x^2*Cosh[c + d*x])/(a + b*x^3), x, 11, (Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*b) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b) - (Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*b) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*b) + (Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*b)} -{(x^1*Cosh[c + d*x])/(a + b*x^3), x, 11, -(((-1)^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(1/3)*b^(2/3))) + ((-1)^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(1/3)*b^(2/3)) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(1/3)*b^(2/3)) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(1/3)*b^(2/3))} -{Cosh[c + d*x]/(a + b*x^3), x, 11, -(((-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(2/3)*b^(1/3))) + ((-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(2/3)*b^(1/3)) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(2/3)*b^(1/3)) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3)) + ((-1)^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(2/3)*b^(1/3))} -{Cosh[c + d*x]/(x^1*(a + b*x^3)), x, 16, (Cosh[c]*CoshIntegral[d*x])/a - (Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a) - (Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a) + (Sinh[c]*SinhIntegral[d*x])/a + (Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a) - (Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a)} -{Cosh[c + d*x]/(x^2*(a + b*x^3)), x, 17, -(Cosh[c + d*x]/(a*x)) + ((-1)^(2/3)*b^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(4/3)) + (b^(1/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) + (d*CoshIntegral[d*x]*Sinh[c])/a + (d*Cosh[c]*SinhIntegral[d*x])/a - ((-1)^(2/3)*b^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(4/3)) + (b^(1/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(4/3))} -{Cosh[c + d*x]/(x^3*(a + b*x^3)), x, 18, -(Cosh[c + d*x]/(2*a*x^2)) + (d^2*Cosh[c]*CoshIntegral[d*x])/(2*a) + ((-1)^(1/3)*b^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^(5/3)) - (b^(2/3)*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3)) - (d*Sinh[c + d*x])/(2*a*x) + (d^2*Sinh[c]*SinhIntegral[d*x])/(2*a) - ((-1)^(1/3)*b^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(5/3)) - (b^(2/3)*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3))} - - -{(x^3*Cosh[c + d*x])/(a + b*x^3)^2, x, 23, -((x*Cosh[c + d*x])/(3*b*(a + b*x^3))) - ((-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(9*a^(2/3)*b^(4/3)) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(1/3)*b^(5/3)) - ((-1)^(2/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(1/3)*b^(5/3)) + ((-1)^(1/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(1/3)*b^(5/3)) + ((-1)^(2/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + ((-1)^(1/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3))} -{(x^2*Cosh[c + d*x])/(a + b*x^3)^2, x, 12, -(Cosh[c + d*x]/(3*b*(a + b*x^3))) + (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) + ((-1)^(1/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) + (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + ((-1)^(2/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3))} -{(x^1*Cosh[c + d*x])/(a + b*x^3)^2, x, 34, Cosh[c + d*x]/(3*a*b*x) - Cosh[c + d*x]/(3*b*x*(a + b*x^3)) - ((-1)^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) + ((-1)^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(9*a^(4/3)*b^(2/3)) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(9*a*b) - (d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a*b) - (d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a*b) + (d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a*b) + ((-1)^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) - (d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a*b) + ((-1)^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3))} -{Cosh[c + d*x]/(a + b*x^3)^2, x, 36, Cosh[c + d*x]/(3*a*b*x^2) - Cosh[c + d*x]/(3*b*x^2*(a + b*x^3)) - (2*(-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) + (2*(-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(9*a^(5/3)*b^(1/3)) + (2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) + (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) + ((-1)^(2/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) - ((-1)^(1/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(4/3)*b^(2/3)) - ((-1)^(2/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(4/3)*b^(2/3)) + (2*(-1)^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) + (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - ((-1)^(1/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(4/3)*b^(2/3)) + (2*(-1)^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3))} -{Cosh[c + d*x]/(x^1*(a + b*x^3)^2), x, 41, Cosh[c + d*x]/(3*a*b*x^3) - Cosh[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cosh[c]*CoshIntegral[d*x])/a^2 - (Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) - (Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(3*a^2) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + ((-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) - ((-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(5/3)*b^(1/3)) + (Sinh[c]*SinhIntegral[d*x])/a^2 - ((-1)^(1/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) + (Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) - ((-1)^(2/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2)} - - -{(x^5*Cosh[c + d*x])/(a + b*x^3)^3, x, 36, -((x^3*Cosh[c + d*x])/(6*b*(a + b*x^3)^2)) - Cosh[c + d*x]/(6*b^2*(a + b*x^3)) - ((-1)^(2/3)*d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(1/3)*b^(8/3)) + ((-1)^(1/3)*d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a^(1/3)*b^(8/3)) - (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(1/3)*b^(8/3)) + (2*d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(2/3)*b^(7/3)) - (2*(-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(2/3)*b^(7/3)) + (2*(-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(2/3)*b^(7/3)) - (d*x*Sinh[c + d*x])/(18*b^2*(a + b*x^3)) + (2*(-1)^(1/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(2/3)*b^(7/3)) + ((-1)^(2/3)*d^2*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(1/3)*b^(8/3)) + (2*d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(2/3)*b^(7/3)) - (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(1/3)*b^(8/3)) + (2*(-1)^(2/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(2/3)*b^(7/3)) + ((-1)^(1/3)*d^2*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(1/3)*b^(8/3))} -{(x^4*Cosh[c + d*x])/(a + b*x^3)^3, x, 47, Cosh[c + d*x]/(9*a*b^2*x) - (x^2*Cosh[c + d*x])/(6*b*(a + b*x^3)^2) - Cosh[c + d*x]/(9*b^2*x*(a + b*x^3)) - ((-1)^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(4/3)*b^(5/3)) - ((-1)^(1/3)*d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(2/3)*b^(7/3)) + ((-1)^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(27*a^(4/3)*b^(5/3)) + ((-1)^(2/3)*d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a^(2/3)*b^(7/3)) - (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(4/3)*b^(5/3)) + (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(2/3)*b^(7/3)) - (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(27*a*b^2) - (d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a*b^2) - (d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a*b^2) - (d*Sinh[c + d*x])/(18*b^2*(a + b*x^3)) + (d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a*b^2) + ((-1)^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(4/3)*b^(5/3)) + ((-1)^(1/3)*d^2*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(2/3)*b^(7/3)) - (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a*b^2) - (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(4/3)*b^(5/3)) + (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(2/3)*b^(7/3)) - (d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a*b^2) + ((-1)^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(4/3)*b^(5/3)) + ((-1)^(2/3)*d^2*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(2/3)*b^(7/3))} -{(x^3*Cosh[c + d*x])/(a + b*x^3)^3, x, 71, Cosh[c + d*x]/(18*a*b^2*x^2) - (x*Cosh[c + d*x])/(6*b*(a + b*x^3)^2) - Cosh[c + d*x]/(18*b^2*x^2*(a + b*x^3)) - ((-1)^(1/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a*b^2) + ((-1)^(2/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a*b^2) + (Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + (d*Sinh[c + d*x])/(18*a*b^2*x) - (d*Sinh[c + d*x])/(18*b^2*x*(a + b*x^3)) + ((-1)^(1/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a*b^2) + (Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + ((-1)^(2/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - (d^2*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2)} -{(x^2*Cosh[c + d*x])/(a + b*x^3)^3, x, 37, -(Cosh[c + d*x]/(6*b*(a + b*x^3)^2)) + ((-1)^(2/3)*d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(4/3)*b^(5/3)) - ((-1)^(1/3)*d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a^(4/3)*b^(5/3)) + (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(4/3)*b^(5/3)) + (d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) - ((-1)^(1/3)*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + ((-1)^(2/3)*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d*Sinh[c + d*x])/(18*a*b^2*x^2) - (d*Sinh[c + d*x])/(18*b^2*x^2*(a + b*x^3)) + ((-1)^(1/3)*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(5/3)*b^(4/3)) - ((-1)^(2/3)*d^2*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(4/3)*b^(5/3)) + (d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(4/3)*b^(5/3)) + ((-1)^(2/3)*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) - ((-1)^(1/3)*d^2*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(4/3)*b^(5/3))} -{(x^1*Cosh[c + d*x])/(a + b*x^3)^3, x, 89, -(Cosh[c + d*x]/(18*a*b^2*x^4)) + (2*Cosh[c + d*x])/(9*a^2*b*x) - Cosh[c + d*x]/(6*b*x*(a + b*x^3)^2) + Cosh[c + d*x]/(18*b^2*x^4*(a + b*x^3)) - (2*(-1)^(2/3)*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(1/3)*d^2*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(5/3)*b^(4/3)) + (2*(-1)^(1/3)*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(27*a^(7/3)*b^(2/3)) - ((-1)^(2/3)*d^2*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x])/(54*a^(5/3)*b^(4/3)) - (2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) - (d^2*Cosh[c - (a^(1/3)*d)/b^(1/3)]*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3)) - (2*d*CoshIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sinh[c - (a^(1/3)*d)/b^(1/3)])/(27*a^2*b) - (2*d*CoshIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^2*b) - (2*d*CoshIntegral[-(((-1)^(2/3)*a^(1/3)*d)/b^(1/3)) - d*x]*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(27*a^2*b) + (d*Sinh[c + d*x])/(18*a*b^2*x^3) - (d*Sinh[c + d*x])/(18*b^2*x^3*(a + b*x^3)) + (2*d*Cosh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) + (2*(-1)^(2/3)*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d^2*Sinh[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(5/3)*b^(4/3)) - (2*d*Cosh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) - (d^2*Sinh[c - (a^(1/3)*d)/b^(1/3)]*SinhIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3)) - (2*d*Cosh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) + (2*(-1)^(1/3)*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) - ((-1)^(2/3)*d^2*Sinh[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinhIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(5/3)*b^(4/3))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.3 (e x)^m (a+b cosh(c+d x^n))^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.3 (e x)^m (a+b cosh(c+d x^n))^p.m deleted file mode 100644 index 8ce5cdb..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.3 (e x)^m (a+b cosh(c+d x^n))^p.m +++ /dev/null @@ -1,165 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Cosh[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Cosh[a + b*x^2], x, 3, -(Cosh[a + b*x^2]/(2*b^2)) + (x^2*Sinh[a + b*x^2])/(2*b)} -{x^2*Cosh[a + b*x^2], x, 4, (Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(8*b^(3/2))) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(8*b^(3/2)) + (x*Sinh[a + b*x^2])/(2*b)} -{x*Cosh[a + b*x^2], x, 2, Sinh[a + b*x^2]/(2*b)} -{Cosh[a + b*x^2], x, 3, (Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(4*Sqrt[b])) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(4*Sqrt[b])} -{Cosh[a + b*x^2]/x, x, 3, (1/2)*Cosh[a]*CoshIntegral[b*x^2] + (1/2)*Sinh[a]*SinhIntegral[b*x^2]} -{Cosh[a + b*x^2]/x^2, x, 4, -(Cosh[a + b*x^2]/x) - ((1/2)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*x])/E^a + (1/2)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x]} -{Cosh[a + b*x^2]/x^3, x, 5, -(Cosh[a + b*x^2]/(2*x^2)) + (1/2)*b*CoshIntegral[b*x^2]*Sinh[a] + (1/2)*b*Cosh[a]*SinhIntegral[b*x^2]} - - -{x^3*Cosh[a + b*x^2]^2, x, 3, x^4/8 - Cosh[a + b*x^2]^2/(8*b^2) + (x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b)} -{x^2*Cosh[a + b*x^2]^2, x, 6, x^3/6 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(E^(2*a)*(32*b^(3/2))) - (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x])/(32*b^(3/2)) + (x*Sinh[2*a + 2*b*x^2])/(8*b)} -{x*Cosh[a + b*x^2]^2, x, 3, x^2/4 + (Cosh[a + b*x^2]*Sinh[a + b*x^2])/(4*b)} -{Cosh[a + b*x^2]^2, x, 5, x/2 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/(E^(2*a)*(8*Sqrt[b])) + (E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x])/(8*Sqrt[b])} -{Cosh[a + b*x^2]^2/x, x, 5, (1/4)*Cosh[2*a]*CoshIntegral[2*b*x^2] + Log[x]/2 + (1/4)*Sinh[2*a]*SinhIntegral[2*b*x^2]} -{Cosh[a + b*x^2]^2/x^2, x, 6, -(Cosh[a + b*x^2]^2/x) - ((1/2)*Sqrt[b]*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[b]*x])/E^(2*a) + (1/2)*Sqrt[b]*E^(2*a)*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[b]*x]} -{Cosh[a + b*x^2]^2/x^3, x, 7, -(1/(4*x^2)) - Cosh[2*(a + b*x^2)]/(4*x^2) + (1/2)*b*CoshIntegral[2*b*x^2]*Sinh[2*a] + (1/2)*b*Cosh[2*a]*SinhIntegral[2*b*x^2]} - - -{x^3*Cosh[a + b*x^2]^3, x, 4, -(Cosh[a + b*x^2]/(3*b^2)) - Cosh[a + b*x^2]^3/(18*b^2) + (x^2*Sinh[a + b*x^2])/(3*b) + (x^2*Cosh[a + b*x^2]^2*Sinh[a + b*x^2])/(6*b)} -{x^2*Cosh[a + b*x^2]^3, x, 10, (3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(32*b^(3/2))) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(E^(3*a)*(96*b^(3/2))) - (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(32*b^(3/2)) - (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(96*b^(3/2)) + (3*x*Sinh[a + b*x^2])/(8*b) + (x*Sinh[3*a + 3*b*x^2])/(24*b)} -{x*Cosh[a + b*x^2]^3, x, 3, Sinh[a + b*x^2]/(2*b) + Sinh[a + b*x^2]^3/(6*b)} -{Cosh[a + b*x^2]^3, x, 8, (3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(E^a*(16*Sqrt[b])) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(E^(3*a)*(16*Sqrt[b])) + (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b])} -{Cosh[a + b*x^2]^3/x, x, 8, (3/8)*Cosh[a]*CoshIntegral[b*x^2] + (1/8)*Cosh[3*a]*CoshIntegral[3*b*x^2] + (3/8)*Sinh[a]*SinhIntegral[b*x^2] + (1/8)*Sinh[3*a]*SinhIntegral[3*b*x^2]} -{Cosh[a + b*x^2]^3/x^2, x, 9, -(Cosh[a + b*x^2]^3/x) - ((3/8)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*x])/E^a - ((1/8)*Sqrt[b]*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[b]*x])/E^(3*a) + (3/8)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x] + (1/8)*Sqrt[b]*E^(3*a)*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[b]*x]} -{Cosh[a + b*x^2]^3/x^3, x, 12, -((3*Cosh[a + b*x^2])/(8*x^2)) - Cosh[3*(a + b*x^2)]/(8*x^2) + (3/8)*b*CoshIntegral[b*x^2]*Sinh[a] + (3/8)*b*CoshIntegral[3*b*x^2]*Sinh[3*a] + (3/8)*b*Cosh[a]*SinhIntegral[b*x^2] + (3/8)*b*Cosh[3*a]*SinhIntegral[3*b*x^2]} - - -{x*Cosh[a + b*x^2]^7, x, 3, Sinh[a + b*x^2]/(2*b) + Sinh[a + b*x^2]^3/(2*b) + (3*Sinh[a + b*x^2]^5)/(10*b) + Sinh[a + b*x^2]^7/(14*b)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^3])^p*) - - -{x^2*Cosh[x^3], x, 2, Sinh[x^3]/3} -{Cosh[1/x^5]/x^6, x, 2, -Sinh[x^(-5)]/5} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d / x^1])^p*) - - -{Cosh[a + b/x], x, 5, x*Cosh[a + b/x] - b*CoshIntegral[b/x]*Sinh[a] - b*Cosh[a]*SinhIntegral[b/x]} -{Cosh[a + b/x]/x, x, 3, (-Cosh[a])*CoshIntegral[b/x] - Sinh[a]*SinhIntegral[b/x]} -{Cosh[a + b/x]/x^2, x, 2, -(Sinh[a + b/x]/b)} -{Cosh[a + b/x]/x^3, x, 3, Cosh[a + b/x]/b^2 - Sinh[a + b/x]/(b*x)} -{Cosh[a + b/x]/x^4, x, 4, (2*Cosh[a + b/x])/(b^2*x) - (2*Sinh[a + b/x])/b^3 - Sinh[a + b/x]/(b*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d / x^2])^p*) - - -{Cosh[a + b/x^2], x, 5, x*Cosh[a + b/x^2] + ((1/2)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]/x])/E^a - (1/2)*Sqrt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x]} -{Cosh[a + b/x^2]/x, x, 3, (-(1/2))*Cosh[a]*CoshIntegral[b/x^2] - (1/2)*Sinh[a]*SinhIntegral[b/x^2]} -{Cosh[a + b/x^2]/x^2, x, 4, -((Sqrt[Pi]*Erf[Sqrt[b]/x])/(E^a*(4*Sqrt[b]))) - (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(4*Sqrt[b])} -{Cosh[a + b/x^2]/x^3, x, 2, -(Sinh[a + b/x^2]/(2*b))} -{Cosh[a + b/x^2]/x^4, x, 5, -((Sqrt[Pi]*Erf[Sqrt[b]/x])/(E^a*(8*b^(3/2)))) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]/x])/(8*b^(3/2)) - Sinh[a + b/x^2]/(2*b*x)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^n])^p*) - - -{Cosh[a + b*x^n], x, 3, -((E^a*x*Gamma[1/n, (-b)*x^n])/(((-b)*x^n)^n^(-1)*(2*n))) - (x*Gamma[1/n, b*x^n])/(E^a*(b*x^n)^n^(-1)*(2*n))} -{Cosh[a + b*x^n]/x, x, 3, (Cosh[a]*CoshIntegral[b*x^n])/n + (Sinh[a]*SinhIntegral[b*x^n])/n} - - -{Cosh[a + b*x^n]^2, x, 5, x/2 - (2^(-2 - 1/n)*E^(2*a)*x*Gamma[1/n, -2*b*x^n])/(((-b)*x^n)^n^(-1)*n) - (2^(-2 - 1/n)*x*Gamma[1/n, 2*b*x^n])/(E^(2*a)*(b*x^n)^n^(-1)*n)} -{Cosh[a + b*x^n]^2/x, x, 5, (Cosh[2*a]*CoshIntegral[2*b*x^n])/(2*n) + Log[x]/2 + (Sinh[2*a]*SinhIntegral[2*b*x^n])/(2*n)} - - -{Cosh[a + b*x^n]^3, x, 8, -((E^(3*a)*x*Gamma[1/n, -3*b*x^n])/(3^n^(-1)*((-b)*x^n)^n^(-1)*(8*n))) - (3*E^a*x*Gamma[1/n, (-b)*x^n])/(((-b)*x^n)^n^(-1)*(8*n)) - (3*x*Gamma[1/n, b*x^n])/(E^a*(b*x^n)^n^(-1)*(8*n)) - (x*Gamma[1/n, 3*b*x^n])/(3^n^(-1)*E^(3*a)*(b*x^n)^n^(-1)*(8*n))} -{Cosh[a + b*x^n]^3/x, x, 8, (3*Cosh[a]*CoshIntegral[b*x^n])/(4*n) + (Cosh[3*a]*CoshIntegral[3*b*x^n])/(4*n) + (3*Sinh[a]*SinhIntegral[b*x^n])/(4*n) + (Sinh[3*a]*SinhIntegral[3*b*x^n])/(4*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x^n])^p with m symbolic*) - - -{(e*x)^m*(b*Cosh[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(b*Cosh[c + d*x^n])^p, x]} -{(e*x)^m*(a + b*Cosh[c + d*x^n])^p, x, 0, Unintegrable[(e*x)^m*(a + b*Cosh[c + d*x^n])^p, x]} - - -{(e*x)^(n - 1)*(b*Cosh[c + d*x^n])^p, x, 3, -(((e*x)^n*(b*Cosh[c + d*x^n])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cosh[c + d*x^n]^2]*Sinh[c + d*x^n])/(x^n*(b*d*e*n*(1 + p)*Sqrt[-Sinh[c + d*x^n]^2])))} -{(e*x)^(2*n - 1)*(b*Cosh[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(b*Cosh[c + d*x^n])^p, x])/(x^(2*n)*e)} - -{(e*x)^(n - 1)*(a + b*Cosh[c + d*x^n])^p, x, 5, (Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1/2)*(1 - Cosh[c + d*x^n]), (b*(1 - Cosh[c + d*x^n]))/(a + b)]*(a + b*Cosh[c + d*x^n])^p*Sinh[c + d*x^n])/(x^n*((a + b*Cosh[c + d*x^n])/(a + b))^p*(d*e*n*Sqrt[1 + Cosh[c + d*x^n]]))} -{(e*x)^(2*n - 1)*(a + b*Cosh[c + d*x^n])^p, x, 1, ((e*x)^(2*n)*Unintegrable[x^(-1 + 2*n)*(a + b*Cosh[c + d*x^n])^p, x])/(x^(2*n)*e)} - - -{x^m*Cosh[a + b*x^n], x, 3, -((E^a*x^(1 + m)*Gamma[(1 + m)/n, (-b)*x^n])/(((-b)*x^n)^((1 + m)/n)*(2*n))) - (x^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(E^a*(b*x^n)^((1 + m)/n)*(2*n))} -{x^m*Cosh[a + b*x^n]^2, x, 5, x^(1 + m)/(2*(1 + m)) - (E^(2*a)*x^(1 + m)*Gamma[(1 + m)/n, -2*b*x^n])/(2^((1 + m + 2*n)/n)*((-b)*x^n)^((1 + m)/n)*n) - (x^(1 + m)*Gamma[(1 + m)/n, 2*b*x^n])/(2^((1 + m + 2*n)/n)*E^(2*a)*(b*x^n)^((1 + m)/n)*n)} -{x^m*Cosh[a + b*x^n]^3, x, 8, -((E^(3*a)*x^(1 + m)*Gamma[(1 + m)/n, -3*b*x^n])/(3^((1 + m)/n)*((-b)*x^n)^((1 + m)/n)*(8*n))) - (3*E^a*x^(1 + m)*Gamma[(1 + m)/n, (-b)*x^n])/(((-b)*x^n)^((1 + m)/n)*(8*n)) - (3*x^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(E^a*(b*x^n)^((1 + m)/n)*(8*n)) - (x^(1 + m)*Gamma[(1 + m)/n, 3*b*x^n])/(3^((1 + m)/n)*E^(3*a)*(b*x^n)^((1 + m)/n)*(8*n))} - - -{Cosh[a + b*x^n]/x^(n + 1), x, 5, -(Cosh[a + b*x^n]/(x^n*n)) + (b*CoshIntegral[b*x^n]*Sinh[a])/n + (b*Cosh[a]*SinhIntegral[b*x^n])/n} -{Cosh[a + b*x^n]^2/x^(n + 1), x, 7, -(1/(x^n*(2*n))) - Cosh[2*(a + b*x^n)]/(x^n*(2*n)) + (b*CoshIntegral[2*b*x^n]*Sinh[2*a])/n + (b*Cosh[2*a]*SinhIntegral[2*b*x^n])/n} -{Cosh[a + b*x^n]^3/x^(n + 1), x, 12, -((3*Cosh[a + b*x^n])/(x^n*(4*n))) - Cosh[3*(a + b*x^n)]/(x^n*(4*n)) + (3*b*CoshIntegral[b*x^n]*Sinh[a])/(4*n) + (3*b*CoshIntegral[3*b*x^n]*Sinh[3*a])/(4*n) + (3*b*Cosh[a]*SinhIntegral[b*x^n])/(4*n) + (3*b*Cosh[3*a]*SinhIntegral[3*b*x^n])/(4*n)} - - -{x^(n/2 - 1)*Cosh[a + b*x^n], x, 4, (Sqrt[Pi]*Erf[Sqrt[b]*x^(n/2)])/(E^a*(2*Sqrt[b]*n)) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x^(n/2)])/(2*Sqrt[b]*n)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e x)^m Cosh[a+b (c+d x)^n]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cosh[a+b (c+d x)^n]*) - - -{x^2*Cosh[(a + b*x)^2], x, 12, (Sqrt[Pi]*Erf[a + b*x])/(8*b^3) + (a^2*Sqrt[Pi]*Erf[a + b*x])/(4*b^3) - (Sqrt[Pi]*Erfi[a + b*x])/(8*b^3) + (a^2*Sqrt[Pi]*Erfi[a + b*x])/(4*b^3) - (a*Sinh[(a + b*x)^2])/b^3 + ((a + b*x)*Sinh[(a + b*x)^2])/(2*b^3)} -{x^1*Cosh[(a + b*x)^2], x, 8, -((a*Sqrt[Pi]*Erf[a + b*x])/(4*b^2)) - (a*Sqrt[Pi]*Erfi[a + b*x])/(4*b^2) + Sinh[(a + b*x)^2]/(2*b^2)} -{x^0*Cosh[(a + b*x)^2], x, 4, (Sqrt[Pi]*Erf[a + b*x])/(4*b) + (Sqrt[Pi]*Erfi[a + b*x])/(4*b)} -{Cosh[(a + b*x)^2]/x^1, x, 1, b*CannotIntegrate[Cosh[(a + b*x)^2]/(b*x), x]} -{Cosh[(a + b*x)^2]/x^2, x, 1, Unintegrable[Cosh[(a + b*x)^2]/x^2, x], b^2*CannotIntegrate[Cosh[(a + b*x)^2]/(b^2*x^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cosh[a+b (c+d x)^(n/2)]*) - - -{Cosh[a + b*Sqrt[c + d*x]]*x^2, x, 16, -((240*Cosh[a + b*Sqrt[c + d*x]])/(b^6*d^3)) + (24*c*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^3) - (2*c^2*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (120*(c + d*x)*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^3) + (12*c*(c + d*x)*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^3) - (10*(c + d*x)^2*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^3) + (240*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^5*d^3) - (24*c*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^3) + (2*c^2*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b*d^3) + (40*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^3) - (4*c*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^3) + (2*(c + d*x)^(5/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^3)} -{Cosh[a + b*Sqrt[c + d*x]]*x^1, x, 10, -((12*Cosh[a + b*Sqrt[c + d*x]])/(b^4*d^2)) + (2*c*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^2) - (6*(c + d*x)*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d^2) + (12*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b^3*d^2) - (2*c*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b*d^2) + (2*(c + d*x)^(3/2)*Sinh[a + b*Sqrt[c + d*x]])/(b*d^2)} -{Cosh[a + b*Sqrt[c + d*x]]*x^0, x, 4, -((2*Cosh[a + b*Sqrt[c + d*x]])/(b^2*d)) + (2*Sqrt[c + d*x]*Sinh[a + b*Sqrt[c + d*x]])/(b*d)} -{Cosh[a + b*Sqrt[c + d*x]]/x^1, x, 10, Cosh[a + b*Sqrt[c]]*CoshIntegral[b*(Sqrt[c] - Sqrt[c + d*x])] + Cosh[a - b*Sqrt[c]]*CoshIntegral[b*(Sqrt[c] + Sqrt[c + d*x])] - Sinh[a + b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] - Sqrt[c + d*x])] + Sinh[a - b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] + Sqrt[c + d*x])]} -{Cosh[a + b*Sqrt[c + d*x]]/x^2, x, 11, -(Cosh[a + b*Sqrt[c + d*x]]/x) - (b*d*CoshIntegral[b*(Sqrt[c] + Sqrt[c + d*x])]*Sinh[a - b*Sqrt[c]])/(2*Sqrt[c]) + (b*d*CoshIntegral[b*(Sqrt[c] - Sqrt[c + d*x])]*Sinh[a + b*Sqrt[c]])/(2*Sqrt[c]) - (b*d*Cosh[a + b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] - Sqrt[c + d*x])])/(2*Sqrt[c]) - (b*d*Cosh[a - b*Sqrt[c]]*SinhIntegral[b*(Sqrt[c] + Sqrt[c + d*x])])/(2*Sqrt[c])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cosh[a+b (c+d x)^(n/3)]*) - - -{Cosh[a + b*(c + d*x)^(1/3)]*x^2, x, 23, (720*c*Cosh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^8*d^3) - (6*c^2*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (20160*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30*c*(c + d*x)^(4/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (24*(c + d*x)^(7/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (120960*Sinh[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*c^2*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3) - (120*c*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*(c + d*x)^2*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^3)} -{Cosh[a + b*(c + d*x)^(1/3)]*x^1, x, 13, -((360*Cosh[a + b*(c + d*x)^(1/3)])/(b^6*d^2)) + (6*c*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^4*d^2) - (15*(c + d*x)^(4/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (6*c*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (360*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^5*d^2) - (3*c*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^2) + (60*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (3*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d^2)} -{Cosh[a + b*(c + d*x)^(1/3)]*x^0, x, 5, -((6*(c + d*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^2*d)) + (6*Sinh[a + b*(c + d*x)^(1/3)])/(b^3*d) + (3*(c + d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b*d)} -{Cosh[a + b*(c + d*x)^(1/3)]/x^1, x, 13, Cosh[a + b*c^(1/3)]*CoshIntegral[b*(c^(1/3) - (c + d*x)^(1/3))] + Cosh[a + (-1)^(2/3)*b*c^(1/3)]*CoshIntegral[(-b)*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))] + Cosh[a - (-1)^(1/3)*b*c^(1/3)]*CoshIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))] - Sinh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))] - Sinh[a + (-1)^(2/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))] + Sinh[a - (-1)^(1/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))]} -{Cosh[a + b*(c + d*x)^(1/3)]/x^2, x, 14, -(Cosh[a + b*(c + d*x)^(1/3)]/x) + (b*d*CoshIntegral[b*(c^(1/3) - (c + d*x)^(1/3))]*Sinh[a + b*c^(1/3)])/(3*c^(2/3)) - ((-1)^(1/3)*b*d*CoshIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))]*Sinh[a - (-1)^(1/3)*b*c^(1/3)])/(3*c^(2/3)) + ((-1)^(2/3)*b*d*CoshIntegral[(-b)*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))]*Sinh[a + (-1)^(2/3)*b*c^(1/3)])/(3*c^(2/3)) - (b*d*Cosh[a + b*c^(1/3)]*SinhIntegral[b*(c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-1)^(2/3)*b*d*Cosh[a + (-1)^(2/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(2/3)*c^(1/3) - (c + d*x)^(1/3))])/(3*c^(2/3)) - ((-1)^(1/3)*b*d*Cosh[a - (-1)^(1/3)*b*c^(1/3)]*SinhIntegral[b*((-1)^(1/3)*c^(1/3) + (c + d*x)^(1/3))])/(3*c^(2/3))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.4 (d+e x)^m cosh(a+b x+c x^2)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.4 (d+e x)^m cosh(a+b x+c x^2)^n.m deleted file mode 100644 index 2f0c2bc..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.4 (d+e x)^m cosh(a+b x+c x^2)^n.m +++ /dev/null @@ -1,64 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form (d+e x)^m Cosh[a+b x+c x^2]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m Cosh[a+b x+c x^2]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*Cosh[a + b*x + c*x^2], x, 12, (b^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) + (E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + (b^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*Sinh[a + b*x + c*x^2])/(4*c^2) + (x*Sinh[a + b*x + c*x^2])/(2*c)} -{x*Cosh[a + b*x + c*x^2], x, 6, -((b*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))) - (b*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + Sinh[a + b*x + c*x^2]/(2*c)} -{Cosh[a + b*x + c*x^2], x, 5, (E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c]) + (E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{Cosh[a + b*x + c*x^2]/x, x, 0, Unintegrable[Cosh[a + b*x + c*x^2]/x, x]} -{Cosh[a + b*x + c*x^2]/x^2 - b*Sinh[a + b*x + c*x^2]/x, x, 7, -(Cosh[a + b*x + c*x^2]/x) - (1/2)*Sqrt[c]*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])] + (1/2)*Sqrt[c]*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])]} - -{x^2*Cosh[a + b*x - c*x^2], x, 12, -((b^2*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2))) - (E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b^2*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) + (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - (b*Sinh[a + b*x - c*x^2])/(4*c^2) - (x*Sinh[a + b*x - c*x^2])/(2*c)} -{x*Cosh[a + b*x - c*x^2], x, 6, -((b*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))) - (b*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) - Sinh[a + b*x - c*x^2]/(2*c)} -{Cosh[a + b*x - c*x^2], x, 5, -((E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])) - (E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{Cosh[a + b*x - c*x^2]/x, x, 0, Unintegrable[Cosh[a + b*x - c*x^2]/x, x]} -{Cosh[a + b*x - c*x^2]/x^2 - b*Sinh[a + b*x - c*x^2]/x, x, 7, -(Cosh[a + b*x - c*x^2]/x) + (1/2)*Sqrt[c]*E^(a + b^2/(4*c))*Sqrt[Pi]*Erf[(b - 2*c*x)/(2*Sqrt[c])] - (1/2)*Sqrt[c]*E^(-a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b - 2*c*x)/(2*Sqrt[c])]} - -{x^2*Cosh[1/4 + x + x^2], x, 12, (-(3/16))*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] - (1/16)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)] - (1/4)*Sinh[1/4 + x + x^2] + (1/2)*x*Sinh[1/4 + x + x^2]} -{x*Cosh[1/4 + x + x^2], x, 6, (1/8)*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] - (1/8)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)] + (1/2)*Sinh[1/4 + x + x^2]} -{Cosh[1/4 + x + x^2], x, 5, (-(1/4))*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] + (1/4)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)]} -{Cosh[1/4 + x + x^2]/x, x, 0, Unintegrable[Cosh[1/4 + x + x^2]/x, x]} -{Cosh[1/4 + x + x^2]/x^2, x, 6, -(Cosh[1/4 + x + x^2]/x) + (1/2)*Sqrt[Pi]*Erf[(1/2)*(-1 - 2*x)] + (1/2)*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*x)] + Unintegrable[Sinh[1/4 + x + x^2]/x, x]} - - -{x^2*Cosh[a + b*x + c*x^2]^2, x, 14, x^3/6 + (b^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + (b^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (x*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{x*Cosh[a + b*x + c*x^2]^2, x, 8, x^2/4 - (b*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - (b*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + Sinh[2*a + 2*b*x + 2*c*x^2]/(8*c)} -{Cosh[a + b*x + c*x^2]^2, x, 7, x/2 + (E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c]) + (E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c])} -{Cosh[a + b*x + c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x + 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Cosh[a + b*x - c*x^2]^2, x, 14, x^3/6 - (b^2*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b^2*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x - 2*c*x^2])/(16*c^2) - (x*Sinh[2*a + 2*b*x - 2*c*x^2])/(8*c)} -{x*Cosh[a + b*x - c*x^2]^2, x, 8, x^2/4 - (b*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - (b*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) - Sinh[2*a + 2*b*x - 2*c*x^2]/(8*c)} -{Cosh[a + b*x - c*x^2]^2, x, 7, x/2 - (E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c]) - (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(8*Sqrt[c])} -{Cosh[a + b*x - c*x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x - 2*c*x^2]/x, x] + Log[x]/2} - -{x^2*Cosh[1/4 + x + x^2]^2, x, 14, x^3/6 + (1/16)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] - (1/16)*Sinh[1/2 + 2*x + 2*x^2] + (1/8)*x*Sinh[1/2 + 2*x + 2*x^2]} -{x*Cosh[1/4 + x + x^2]^2, x, 8, x^2/4 - (1/16)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] - (1/16)*Sqrt[Pi/2]*Erfi[(1 + 2*x)/Sqrt[2]] + (1/8)*Sinh[1/2 + 2*x + 2*x^2]} -{Cosh[1/4 + x + x^2]^2, x, 7, x/2 + (1/8)*Sqrt[Pi/2]*Erf[(1 + 2*x)/Sqrt[2]] + (1/8)*Sqrt[Pi/2]*Erfi[(1 + 2*x)/Sqrt[2]]} -{Cosh[1/4 + x + x^2]^2/x, x, 2, (1/2)*Unintegrable[Cosh[1/2 + 2*x + 2*x^2]/x, x] + Log[x]/2} - - -{(d + e*x)^2*Cosh[a + b*x + c*x^2], x, 12, (e^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (e^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) + (e*(2*c*d - b*e)*Sinh[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sinh[a + b*x + c*x^2])/(2*c)} -{(d + e*x)*Cosh[a + b*x + c*x^2], x, 6, ((2*c*d - b*e)*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + (e*Sinh[a + b*x + c*x^2])/(2*c)} -{Cosh[a + b*x + c*x^2]/(d + e*x), x, 0, Unintegrable[Cosh[a + b*x + c*x^2]/(d + e*x), x]} - -{(d + e*x)^2*Cosh[a + b*x + c*x^2]^2, x, 14, (d + e*x)^3/(6*e) + (e^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (e^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^2*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (e*(2*c*d - b*e)*Sinh[2*a + 2*b*x + 2*c*x^2])/(16*c^2) + (e*(d + e*x)*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{(d + e*x)*Cosh[a + b*x + c*x^2]^2, x, 8, (d + e*x)^2/(4*e) + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)} -{Cosh[a + b*x + c*x^2]^2/(d + e*x), x, 2, (1/2)*Unintegrable[Cosh[2*a + 2*b*x + 2*c*x^2]/(d + e*x), x] + Log[d + e*x]/(2*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection:: *) -(*Integrands of the form (d+e x)^m Cosh[a+b x+c x^2]^(n/2)*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.5 Hyperbolic cosine functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.5 Hyperbolic cosine functions.m deleted file mode 100644 index 1b99611..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.5 Hyperbolic cosine functions.m +++ /dev/null @@ -1,667 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Cosines*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x])^n (A+B Cosh[c+d x]+C Cosh[c+d x]^2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[a+b x]^n*) - - -{Cosh[a + b*x]^1, x, 1, Sinh[a + b*x]/b} -{Cosh[a + b*x]^2, x, 2, x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)} -{Cosh[a + b*x]^3, x, 2, Sinh[a + b*x]/b + Sinh[a + b*x]^3/(3*b)} -{Cosh[a + b*x]^4, x, 3, (3*x)/8 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} -{Cosh[a + b*x]^5, x, 2, Sinh[a + b*x]/b + (2*Sinh[a + b*x]^3)/(3*b) + Sinh[a + b*x]^5/(5*b)} -{Cosh[a + b*x]^6, x, 4, (5*x)/16 + (5*Cosh[a + b*x]*Sinh[a + b*x])/(16*b) + (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(24*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(6*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[a+b x]^(n/2)*) - - -{Cosh[a + b*x]^(7/2), x, 3, -((10*I*EllipticF[(1/2)*I*(a + b*x), 2])/(21*b)) + (10*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(21*b) + (2*Cosh[a + b*x]^(5/2)*Sinh[a + b*x])/(7*b)} -{Cosh[a + b*x]^(5/2), x, 2, -((6*I*EllipticE[(1/2)*I*(a + b*x), 2])/(5*b)) + (2*Cosh[a + b*x]^(3/2)*Sinh[a + b*x])/(5*b)} -{Cosh[a + b*x]^(3/2), x, 2, -((2*I*EllipticF[(1/2)*I*(a + b*x), 2])/(3*b)) + (2*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(3*b)} -{Cosh[a + b*x]^(1/2), x, 1, -((2*I*EllipticE[(1/2)*I*(a + b*x), 2])/b)} -{1/Cosh[a + b*x]^(1/2), x, 1, -((2*I*EllipticF[(1/2)*I*(a + b*x), 2])/b)} -{1/Cosh[a + b*x]^(3/2), x, 2, (2*I*EllipticE[(1/2)*I*(a + b*x), 2])/b + (2*Sinh[a + b*x])/(b*Sqrt[Cosh[a + b*x]])} -{1/Cosh[a + b*x]^(5/2), x, 2, -((2*I*EllipticF[(1/2)*I*(a + b*x), 2])/(3*b)) + (2*Sinh[a + b*x])/(3*b*Cosh[a + b*x]^(3/2))} -{1/Cosh[a + b*x]^(7/2), x, 3, (6*I*EllipticE[(1/2)*I*(a + b*x), 2])/(5*b) + (2*Sinh[a + b*x])/(5*b*Cosh[a + b*x]^(5/2)) + (6*Sinh[a + b*x])/(5*b*Sqrt[Cosh[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cosh[a+b x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n/2*) - - -{(a*Cosh[x])^(7/2), x, 4, -((10*I*a^4*Sqrt[Cosh[x]]*EllipticF[(I*x)/2, 2])/(21*Sqrt[a*Cosh[x]])) + (10/21)*a^3*Sqrt[a*Cosh[x]]*Sinh[x] + (2/7)*a*(a*Cosh[x])^(5/2)*Sinh[x]} -{(a*Cosh[x])^(5/2), x, 3, -((6*I*a^2*Sqrt[a*Cosh[x]]*EllipticE[(I*x)/2, 2])/(5*Sqrt[Cosh[x]])) + (2/5)*a*(a*Cosh[x])^(3/2)*Sinh[x]} -{(a*Cosh[x])^(3/2),x, 3, -((2*I*a^2*Sqrt[Cosh[x]]*EllipticF[(I*x)/2, 2])/(3*Sqrt[a*Cosh[x]])) + (2/3)*a*Sqrt[a*Cosh[x]]*Sinh[x]} -{(a*Cosh[x])^(1/2), x, 2, -((2*I*Sqrt[a*Cosh[x]]*EllipticE[(I*x)/2, 2])/Sqrt[Cosh[x]])} -{1/(a*Cosh[x])^(1/2), x, 2, -((2*I*Sqrt[Cosh[x]]*EllipticF[(I*x)/2, 2])/Sqrt[a*Cosh[x]])} -{1/(a*Cosh[x])^(3/2), x, 3, (2*I*Sqrt[a*Cosh[x]]*EllipticE[(I*x)/2, 2])/(a^2*Sqrt[Cosh[x]]) + (2*Sinh[x])/(a*Sqrt[a*Cosh[x]])} -{1/(a*Cosh[x])^(5/2), x, 3, -((2*I*Sqrt[Cosh[x]]*EllipticF[(I*x)/2, 2])/(3*a^2*Sqrt[a*Cosh[x]])) + (2*Sinh[x])/(3*a*(a*Cosh[x])^(3/2))} -{1/(a*Cosh[x])^(7/2), x, 4, (6*I*Sqrt[a*Cosh[x]]*EllipticE[(I*x)/2, 2])/(5*a^4*Sqrt[Cosh[x]]) + (2*Sinh[x])/(5*a*(a*Cosh[x])^(5/2)) + (6*Sinh[x])/(5*a^3*Sqrt[a*Cosh[x]])} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{(b*Cosh[c + d*x])^n, x, 1, -(((b*Cosh[c + d*x])^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cosh[c + d*x]^2]*Sinh[c + d*x])/(b*d*(1 + n)*Sqrt[-Sinh[c + d*x]^2]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x])^n when a^2-b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cosh[x]^4/(a + a*Cosh[x]), x, 6, -((3*x)/(2*a)) + (4*Sinh[x])/a - (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]^3*Sinh[x])/(a + a*Cosh[x]) + (4*Sinh[x]^3)/(3*a)} -{Cosh[x]^3/(a + a*Cosh[x]), x, 2, (3*x)/(2*a) - (2*Sinh[x])/a + (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]^2*Sinh[x])/(a + a*Cosh[x])} -{Cosh[x]^2/(a + a*Cosh[x]), x, 4, -(x/a) + Sinh[x]/a + Sinh[x]/(a*(1 + Cosh[x]))} -{Cosh[x]^1/(a + a*Cosh[x]), x, 2, x/a - Sinh[x]/(a + a*Cosh[x])} -{Sech[x]^1/(a + a*Cosh[x]), x, 3, ArcTan[Sinh[x]]/a - Sinh[x]/(a + a*Cosh[x])} -{Sech[x]^2/(a + a*Cosh[x]), x, 5, -(ArcTan[Sinh[x]]/a) + (2*Tanh[x])/a - Tanh[x]/(a + a*Cosh[x])} -{Sech[x]^3/(a + a*Cosh[x]), x, 6, (3*ArcTan[Sinh[x]])/(2*a) - (2*Tanh[x])/a + (3*Sech[x]*Tanh[x])/(2*a) - (Sech[x]*Tanh[x])/(a + a*Cosh[x])} -{Sech[x]^4/(a + a*Cosh[x]), x, 6, -((3*ArcTan[Sinh[x]])/(2*a)) + (4*Tanh[x])/a - (3*Sech[x]*Tanh[x])/(2*a) - (Sech[x]^2*Tanh[x])/(a + a*Cosh[x]) - (4*Tanh[x]^3)/(3*a)} - - -{1/(1 + Cosh[c + d*x]), x, 1, Sinh[c + d*x]/(d*(1 + Cosh[c + d*x]))} -{1/(1 + Cosh[c + d*x])^2, x, 2, Sinh[c + d*x]/(3*d*(1 + Cosh[c + d*x])^2) + Sinh[c + d*x]/(3*d*(1 + Cosh[c + d*x]))} -{1/(1 + Cosh[c + d*x])^3, x, 3, Sinh[c + d*x]/(5*d*(1 + Cosh[c + d*x])^3) + (2*Sinh[c + d*x])/(15*d*(1 + Cosh[c + d*x])^2) + (2*Sinh[c + d*x])/(15*d*(1 + Cosh[c + d*x]))} -{1/(1 + Cosh[c + d*x])^4, x, 4, Sinh[c + d*x]/(7*d*(1 + Cosh[c + d*x])^4) + (3*Sinh[c + d*x])/(35*d*(1 + Cosh[c + d*x])^3) + (2*Sinh[c + d*x])/(35*d*(1 + Cosh[c + d*x])^2) + (2*Sinh[c + d*x])/(35*d*(1 + Cosh[c + d*x]))} - -{1/(1 - Cosh[c + d*x]), x, 1, -(Sinh[c + d*x]/(d*(1 - Cosh[c + d*x])))} -{1/(1 - Cosh[c + d*x])^2, x, 2, -(Sinh[c + d*x]/(3*d*(1 - Cosh[c + d*x])^2)) - Sinh[c + d*x]/(3*d*(1 - Cosh[c + d*x]))} -{1/(1 - Cosh[c + d*x])^3, x, 3, -(Sinh[c + d*x]/(5*d*(1 - Cosh[c + d*x])^3)) - (2*Sinh[c + d*x])/(15*d*(1 - Cosh[c + d*x])^2) - (2*Sinh[c + d*x])/(15*d*(1 - Cosh[c + d*x]))} -{1/(1 - Cosh[c + d*x])^4, x, 4, -(Sinh[c + d*x]/(7*d*(1 - Cosh[c + d*x])^4)) - (3*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x])^3) - (2*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x])^2) - (2*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x]))} - - -{Cosh[x]/Sqrt[a + a*Cosh[x]], x, 3, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/Sqrt[a]) + (2*Sinh[x])/Sqrt[a + a*Cosh[x]]} -{Cosh[x]/Sqrt[a - a*Cosh[x]], x, 3, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/Sqrt[a]) + (2*Sinh[x])/Sqrt[a - a*Cosh[x]]} - - -{(a + a*Cosh[c + d*x])^(5/2), x, 3, (64*a^3*Sinh[c + d*x])/(15*d*Sqrt[a + a*Cosh[c + d*x]]) + (16*a^2*Sqrt[a + a*Cosh[c + d*x]]*Sinh[c + d*x])/(15*d) + (2*a*(a + a*Cosh[c + d*x])^(3/2)*Sinh[c + d*x])/(5*d)} -{(a + a*Cosh[c + d*x])^(3/2), x, 2, (8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a + a*Cosh[c + d*x]]) + (2*a*Sqrt[a + a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)} -{(a + a*Cosh[c + d*x])^(1/2), x, 1, (2*a*Sinh[c + d*x])/(d*Sqrt[a + a*Cosh[c + d*x]])} -{1/(a + a*Cosh[c + d*x])^(1/2), x, 2, (Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cosh[c + d*x]])])/(Sqrt[a]*d)} -{1/(a + a*Cosh[c + d*x])^(3/2), x, 3, ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cosh[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d) + Sinh[c + d*x]/(2*d*(a + a*Cosh[c + d*x])^(3/2))} -{1/(a + a*Cosh[c + d*x])^(5/2), x, 4, (3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Cosh[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + Sinh[c + d*x]/(4*d*(a + a*Cosh[c + d*x])^(5/2)) + (3*Sinh[c + d*x])/(16*a*d*(a + a*Cosh[c + d*x])^(3/2))} - - -{(a - a*Cosh[c + d*x])^(5/2), x, 3, -((64*a^3*Sinh[c + d*x])/(15*d*Sqrt[a - a*Cosh[c + d*x]])) - (16*a^2*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(15*d) - (2*a*(a - a*Cosh[c + d*x])^(3/2)*Sinh[c + d*x])/(5*d)} -{(a - a*Cosh[c + d*x])^(3/2), x, 2, -((8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a - a*Cosh[c + d*x]])) - (2*a*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)} -{(a - a*Cosh[c + d*x])^(1/2), x, 1, -((2*a*Sinh[c + d*x])/(d*Sqrt[a - a*Cosh[c + d*x]]))} -{1/(a - a*Cosh[c + d*x])^(1/2), x, 2, -((Sqrt[2]*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])])/(Sqrt[a]*d))} -{1/(a - a*Cosh[c + d*x])^(3/2), x, 3, -(ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])]/(2*Sqrt[2]*a^(3/2)*d)) - Sinh[c + d*x]/(2*d*(a - a*Cosh[c + d*x])^(3/2))} -{1/(a - a*Cosh[c + d*x])^(5/2), x, 4, -((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d)) - Sinh[c + d*x]/(4*d*(a - a*Cosh[c + d*x])^(5/2)) - (3*Sinh[c + d*x])/(16*a*d*(a - a*Cosh[c + d*x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cosh[x]^4/(a + b*Cosh[x]), x, 6, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]) + ((3*a^2 + 2*b^2)*Sinh[x])/(3*b^3) - (a*Cosh[x]*Sinh[x])/(2*b^2) + (Cosh[x]^2*Sinh[x])/(3*b)} -{Cosh[x]^3/(a + b*Cosh[x]), x, 5, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]) - (a*Sinh[x])/b^2 + (Cosh[x]*Sinh[x])/(2*b)} -{Cosh[x]^2/(a + b*Cosh[x]), x, 5, -((a*x)/b^2) + (2*a^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]) + Sinh[x]/b} -{Cosh[x]^1/(a + b*Cosh[x]), x, 3, x/b - (2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])} -{Sech[x]^1/(a + b*Cosh[x]), x, 4, ArcTan[Sinh[x]]/a - (2*b*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])} -{Sech[x]^2/(a + b*Cosh[x]), x, 6, -((b*ArcTan[Sinh[x]])/a^2) + (2*b^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) + Tanh[x]/a} -{Sech[x]^3/(a + b*Cosh[x]), x, 6, ((a^2 + 2*b^2)*ArcTan[Sinh[x]])/(2*a^3) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]) - (b*Tanh[x])/a^2 + (Sech[x]*Tanh[x])/(2*a)} -{Sech[x]^4/(a + b*Cosh[x]), x, 7, -((b*(a^2 + 2*b^2)*ArcTan[Sinh[x]])/(2*a^4)) + (2*b^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]) + ((2*a^2 + 3*b^2)*Tanh[x])/(3*a^3) - (b*Sech[x]*Tanh[x])/(2*a^2) + (Sech[x]^2*Tanh[x])/(3*a)} - - -{(a + b*Cosh[c + d*x])^5, x, 4, (1/8)*a*(8*a^4 + 40*a^2*b^2 + 15*b^4)*x + (b*(107*a^4 + 192*a^2*b^2 + 16*b^4)*Sinh[c + d*x])/(30*d) + (7*a*b^2*(22*a^2 + 23*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(120*d) + (b*(47*a^2 + 16*b^2)*(a + b*Cosh[c + d*x])^2*Sinh[c + d*x])/(60*d) + (9*a*b*(a + b*Cosh[c + d*x])^3*Sinh[c + d*x])/(20*d) + (b*(a + b*Cosh[c + d*x])^4*Sinh[c + d*x])/(5*d)} -{(a + b*Cosh[c + d*x])^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 + 16*b^2)*Sinh[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(24*d) + (7*a*b*(a + b*Cosh[c + d*x])^2*Sinh[c + d*x])/(12*d) + (b*(a + b*Cosh[c + d*x])^3*Sinh[c + d*x])/(4*d)} -{(a + b*Cosh[c + d*x])^3, x, 2, (1/2)*a*(2*a^2 + 3*b^2)*x + (2*b*(4*a^2 + b^2)*Sinh[c + d*x])/(3*d) + (5*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(6*d) + (b*(a + b*Cosh[c + d*x])^2*Sinh[c + d*x])/(3*d)} -{(a + b*Cosh[c + d*x])^2, x, 1, (1/2)*(2*a^2 + b^2)*x + (2*a*b*Sinh[c + d*x])/d + (b^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{(a + b*Cosh[c + d*x]), x, 2, a*x + (b*Sinh[c + d*x])/d} -{1/(a + b*Cosh[c + d*x]), x, 2, (2*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(a + b*Cosh[c + d*x])^2, x, 4, (2*a*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*Sinh[c + d*x])/((a^2 - b^2)*d*(a + b*Cosh[c + d*x]))} -{1/(a + b*Cosh[c + d*x])^3, x, 5, ((2*a^2 + b^2)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Sinh[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Cosh[c + d*x])^2) - (3*a*b*Sinh[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Cosh[c + d*x]))} -{1/(a + b*Cosh[c + d*x])^4, x, 6, (a*(2*a^2 + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*Sinh[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cosh[c + d*x])^3) - (5*a*b*Sinh[c + d*x])/(6*(a^2 - b^2)^2*d*(a + b*Cosh[c + d*x])^2) - (b*(11*a^2 + 4*b^2)*Sinh[c + d*x])/(6*(a^2 - b^2)^3*d*(a + b*Cosh[c + d*x]))} - - -{1/(3 + 5*Cosh[c + d*x]), x, 2, ArcTan[(1/2)*Tanh[(1/2)*(c + d*x)]]/(2*d)} -{1/(3 + 5*Cosh[c + d*x])^2, x, 4, -((3*ArcTan[(1/2)*Tanh[(1/2)*(c + d*x)]])/(32*d)) + (5*Sinh[c + d*x])/(16*d*(3 + 5*Cosh[c + d*x]))} -{1/(3 + 5*Cosh[c + d*x])^3, x, 5, (43*ArcTan[(1/2)*Tanh[(1/2)*(c + d*x)]])/(1024*d) + (5*Sinh[c + d*x])/(32*d*(3 + 5*Cosh[c + d*x])^2) - (45*Sinh[c + d*x])/(512*d*(3 + 5*Cosh[c + d*x]))} -{1/(3 + 5*Cosh[c + d*x])^4, x, 6, -((279*ArcTan[(1/2)*Tanh[(1/2)*(c + d*x)]])/(16384*d)) + (5*Sinh[c + d*x])/(48*d*(3 + 5*Cosh[c + d*x])^3) - (25*Sinh[c + d*x])/(512*d*(3 + 5*Cosh[c + d*x])^2) + (995*Sinh[c + d*x])/(24576*d*(3 + 5*Cosh[c + d*x]))} - -{1/(5 + 3*Cosh[c + d*x]), x, 1, x/4 - ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])]/(2*d)} -{1/(5 + 3*Cosh[c + d*x])^2, x, 3, (5*x)/64 - (5*ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])])/(32*d) - (3*Sinh[c + d*x])/(16*d*(5 + 3*Cosh[c + d*x]))} -{1/(5 + 3*Cosh[c + d*x])^3, x, 4, (59*x)/2048 - (59*ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])])/(1024*d) - (3*Sinh[c + d*x])/(32*d*(5 + 3*Cosh[c + d*x])^2) - (45*Sinh[c + d*x])/(512*d*(5 + 3*Cosh[c + d*x]))} -{1/(5 + 3*Cosh[c + d*x])^4, x, 5, (385*x)/32768 - (385*ArcTanh[Sinh[c + d*x]/(3 + Cosh[c + d*x])])/(16384*d) - Sinh[c + d*x]/(16*d*(5 + 3*Cosh[c + d*x])^3) - (25*Sinh[c + d*x])/(512*d*(5 + 3*Cosh[c + d*x])^2) - (311*Sinh[c + d*x])/(8192*d*(5 + 3*Cosh[c + d*x]))} - - -{(a + b*Cosh[x])^(5/2), x, 7, -((2*I*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(15*Sqrt[(a + b*Cosh[x])/(a + b)])) + (16*I*a*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(15*Sqrt[a + b*Cosh[x]]) + (16/15)*a*b*Sqrt[a + b*Cosh[x]]*Sinh[x] + (2/5)*b*(a + b*Cosh[x])^(3/2)*Sinh[x]} -{(a + b*Cosh[x])^(3/2), x, 6, -((8*I*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*Sqrt[a + b*Cosh[x]]) + (2/3)*b*Sqrt[a + b*Cosh[x]]*Sinh[x]} -{(a + b*Cosh[c + d*x])^(1/2), x, 2, -((2*I*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(1/2)*I*(c + d*x), (2*b)/(a + b)])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a + b)]))} -{1/(a + b*Cosh[x])^(1/2), x, 2, -((2*I*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cosh[x]])} -{1/(a + b*Cosh[x])^(3/2), x, 4, -((2*I*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)])) - (2*b*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*Cosh[x]])} -{1/(a + b*Cosh[x])^(5/2), x, 7, -((8*I*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)^2*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^2)*Sqrt[a + b*Cosh[x]]) - (2*b*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) - (8*a*b*Sinh[x])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]])} -{1/(a + b*Cosh[x])^(7/2), x, 8, -((2*I*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^3*Sqrt[(a + b*Cosh[x])/(a + b)])) + (16*I*a*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(15*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]]) - (2*b*Sinh[x])/(5*(a^2 - b^2)*(a + b*Cosh[x])^(5/2)) - (16*a*b*Sinh[x])/(15*(a^2 - b^2)^2*(a + b*Cosh[x])^(3/2)) - (2*b*(23*a^2 + 9*b^2)*Sinh[x])/(15*(a^2 - b^2)^3*Sqrt[a + b*Cosh[x]])} - - -{Cosh[x]/Sqrt[a + b*Cosh[x]], x, 5, -((2*I*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*a*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Cosh[c+d x]) (a+b Cosh[c+d x])^n when a^2-b^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cosh[x])*(a + a*Cosh[x])^(5/2), x, 4, (64*a^3*(7*A + 5*B)*Sinh[x])/(105*Sqrt[a + a*Cosh[x]]) + (16/105)*a^2*(7*A + 5*B)*Sqrt[a + a*Cosh[x]]*Sinh[x] + (2/35)*a*(7*A + 5*B)*(a + a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a + a*Cosh[x])^(5/2)*Sinh[x]} -{(A + B*Cosh[x])*(a + a*Cosh[x])^(3/2), x, 3, (8*a^2*(5*A + 3*B)*Sinh[x])/(15*Sqrt[a + a*Cosh[x]]) + (2/15)*a*(5*A + 3*B)*Sqrt[a + a*Cosh[x]]*Sinh[x] + (2/5)*B*(a + a*Cosh[x])^(3/2)*Sinh[x]} -{(A + B*Cosh[x])*(a + a*Cosh[x])^(1/2), x, 2, (2*a*(3*A + B)*Sinh[x])/(3*Sqrt[a + a*Cosh[x]]) + (2/3)*B*Sqrt[a + a*Cosh[x]]*Sinh[x]} - - -{(A + B*Cosh[x])*(a - a*Cosh[x])^(5/2), x, 4, -((64*a^3*(7*A - 5*B)*Sinh[x])/(105*Sqrt[a - a*Cosh[x]])) - (16/105)*a^2*(7*A - 5*B)*Sqrt[a - a*Cosh[x]]*Sinh[x] - (2/35)*a*(7*A - 5*B)*(a - a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a - a*Cosh[x])^(5/2)*Sinh[x]} -{(A + B*Cosh[x])*(a - a*Cosh[x])^(3/2), x, 3, -((8*a^2*(5*A - 3*B)*Sinh[x])/(15*Sqrt[a - a*Cosh[x]])) - (2/15)*a*(5*A - 3*B)*Sqrt[a - a*Cosh[x]]*Sinh[x] + (2/5)*B*(a - a*Cosh[x])^(3/2)*Sinh[x]} -{(A + B*Cosh[x])*(a - a*Cosh[x])^(1/2), x, 2, -((2*a*(3*A - B)*Sinh[x])/(3*Sqrt[a - a*Cosh[x]])) + (2/3)*B*Sqrt[a - a*Cosh[x]]*Sinh[x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cosh[x])/(1 + Cosh[x]), x, 2, B*x + ((A - B)*Sinh[x])/(1 + Cosh[x])} -{(A + B*Cosh[x])/(1 + Cosh[x])^2, x, 2, ((A - B)*Sinh[x])/(3*(1 + Cosh[x])^2) + ((A + 2*B)*Sinh[x])/(3*(1 + Cosh[x]))} -{(A + B*Cosh[x])/(1 + Cosh[x])^3, x, 3, ((A - B)*Sinh[x])/(5*(1 + Cosh[x])^3) + ((2*A + 3*B)*Sinh[x])/(15*(1 + Cosh[x])^2) + ((2*A + 3*B)*Sinh[x])/(15*(1 + Cosh[x]))} -{(A + B*Cosh[x])/(1 + Cosh[x])^4, x, 4, ((A - B)*Sinh[x])/(7*(1 + Cosh[x])^4) + ((3*A + 4*B)*Sinh[x])/(35*(1 + Cosh[x])^3) + (2*(3*A + 4*B)*Sinh[x])/(105*(1 + Cosh[x])^2) + (2*(3*A + 4*B)*Sinh[x])/(105*(1 + Cosh[x]))} - -{(A + B*Cosh[x])/(1 - Cosh[x]), x, 2, (-B)*x - ((A + B)*Sinh[x])/(1 - Cosh[x])} -{(A + B*Cosh[x])/(1 - Cosh[x])^2, x, 2, -(((A + B)*Sinh[x])/(3*(1 - Cosh[x])^2)) - ((A - 2*B)*Sinh[x])/(3*(1 - Cosh[x]))} -{(A + B*Cosh[x])/(1 - Cosh[x])^3, x, 3, -(((A + B)*Sinh[x])/(5*(1 - Cosh[x])^3)) - ((2*A - 3*B)*Sinh[x])/(15*(1 - Cosh[x])^2) - ((2*A - 3*B)*Sinh[x])/(15*(1 - Cosh[x]))} -{(A + B*Cosh[x])/(1 - Cosh[x])^4, x, 4, -(((A + B)*Sinh[x])/(7*(1 - Cosh[x])^4)) - ((3*A - 4*B)*Sinh[x])/(35*(1 - Cosh[x])^3) - (2*(3*A - 4*B)*Sinh[x])/(105*(1 - Cosh[x])^2) - (2*(3*A - 4*B)*Sinh[x])/(105*(1 - Cosh[x]))} - - -{(A + B*Cosh[x])/(a + a*Cosh[x])^(1/2), x, 3, (Sqrt[2]*(A - B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/Sqrt[a] + (2*B*Sinh[x])/Sqrt[a + a*Cosh[x]]} -{(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2), x, 3, ((A + 3*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/(2*Sqrt[2]*a^(3/2)) + ((A - B)*Sinh[x])/(2*(a + a*Cosh[x])^(3/2))} -{(A + B*Cosh[x])/(a + a*Cosh[x])^(5/2), x, 4, ((3*A + 5*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a + a*Cosh[x]])])/(16*Sqrt[2]*a^(5/2)) + ((A - B)*Sinh[x])/(4*(a + a*Cosh[x])^(5/2)) + ((3*A + 5*B)*Sinh[x])/(16*a*(a + a*Cosh[x])^(3/2))} - - -{(A + B*Cosh[x])/(a - a*Cosh[x])^(1/2), x, 3, -((Sqrt[2]*(A + B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/Sqrt[a]) + (2*B*Sinh[x])/Sqrt[a - a*Cosh[x]]} -{(A + B*Cosh[x])/(a - a*Cosh[x])^(3/2), x, 3, -(((A - 3*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/(2*Sqrt[2]*a^(3/2))) - ((A + B)*Sinh[x])/(2*(a - a*Cosh[x])^(3/2))} -{(A + B*Cosh[x])/(a - a*Cosh[x])^(5/2), x, 4, -(((3*A - 5*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/(16*Sqrt[2]*a^(5/2))) - ((A + B)*Sinh[x])/(4*(a - a*Cosh[x])^(5/2)) - ((3*A - 5*B)*Sinh[x])/(16*a*(a - a*Cosh[x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Cosh[c+d x]) (a+b Cosh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(A + B*Cosh[x])*(a + b*Cosh[x])^(5/2), x, 8, -((2*I*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(105*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(105*b*Sqrt[a + b*Cosh[x]]) + (2/105)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cosh[x]]*Sinh[x] + (2/35)*(7*A*b + 5*a*B)*(a + b*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a + b*Cosh[x])^(5/2)*Sinh[x]} -{(A + B*Cosh[x])*(a + b*Cosh[x])^(3/2), x, 7, -((2*I*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(15*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*(5*A*b + 3*a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(15*b*Sqrt[a + b*Cosh[x]]) + (2/15)*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x] + (2/5)*B*(a + b*Cosh[x])^(3/2)*Sinh[x]} -{(A + B*Cosh[x])*(a + b*Cosh[x])^(1/2), x, 6, -((2*I*(3*A*b + a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*b*Sqrt[a + b*Cosh[x]]) + (2/3)*B*Sqrt[a + b*Cosh[x]]*Sinh[x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(A + B*Cosh[x])/(a + b*Cosh[x]), x, 3, (B*x)/b + (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])} -{(A + B*Cosh[x])/(a + b*Cosh[x])^2, x, 4, (2*(a*A - b*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) - ((A*b - a*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x]))} -{(A + B*Cosh[x])/(a + b*Cosh[x])^3, x, 5, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) - ((A*b - a*B)*Sinh[x])/(2*(a^2 - b^2)*(a + b*Cosh[x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sinh[x])/(2*(a^2 - b^2)^2*(a + b*Cosh[x]))} -{(A + B*Cosh[x])/(a + b*Cosh[x])^4, x, 6, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)) - ((A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sinh[x])/(6*(a^2 - b^2)^2*(a + b*Cosh[x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[x])/(6*(a^2 - b^2)^3*(a + b*Cosh[x]))} - -{(b*B/a + B*Cosh[x])/(a + b*Cosh[x]), x, 3, (B*x)/b - (2*Sqrt[a - b]*Sqrt[a + b]*B*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b)} -{(a*B/b + B*Cosh[x])/(a + b*Cosh[x]), x, 2, (B*x)/b} - -{(a + b*Cosh[x])/(b + a*Cosh[x])^2, x, 2, Sinh[x]/(b + a*Cosh[x])} -{(3 + Cosh[x])/(2 - Cosh[x]), x, 2, -x + (5*x)/Sqrt[3] + (10*ArcTanh[Sinh[x]/(2 + Sqrt[3] - Cosh[x])])/Sqrt[3]} - - -{(A + B*Cosh[x])/(a + b*Cosh[x])^(1/2), x, 5, -((2*I*B*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)])) - (2*I*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])} -{(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2), x, 6, -((2*I*(A*b - a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)])) - (2*I*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*Cosh[x]])} -{(A + B*Cosh[x])/(a + b*Cosh[x])^(5/2), x, 7, -((2*I*(4*a*A*b - a^2*B - 3*b^2*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*Sinh[x])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Cosh[a+b x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cosh[a+b x]^2)^n*) - - -{(a*Cosh[x]^2)^(7/2),x, 5, (16/35)*a^3*Sqrt[a*Cosh[x]^2]*Tanh[x] + (8/35)*a^2*(a*Cosh[x]^2)^(3/2)*Tanh[x] + (6/35)*a*(a*Cosh[x]^2)^(5/2)*Tanh[x] + (1/7)*(a*Cosh[x]^2)^(7/2)*Tanh[x]} -{(a*Cosh[x]^2)^(5/2),x, 4, (8/15)*a^2*Sqrt[a*Cosh[x]^2]*Tanh[x] + (4/15)*a*(a*Cosh[x]^2)^(3/2)*Tanh[x] + (1/5)*(a*Cosh[x]^2)^(5/2)*Tanh[x]} -{(a*Cosh[x]^2)^(3/2),x, 3, (2/3)*a*Sqrt[a*Cosh[x]^2]*Tanh[x] + (1/3)*(a*Cosh[x]^2)^(3/2)*Tanh[x]} -{(a*Cosh[x]^2)^(1/2), x, 2, Sqrt[a*Cosh[x]^2]*Tanh[x]} -{1/(a*Cosh[x]^2)^(1/2), x, 2, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]} -{1/(a*Cosh[x]^2)^(3/2), x, 3, (ArcTan[Sinh[x]]*Cosh[x])/(2*a*Sqrt[a*Cosh[x]^2]) + Tanh[x]/(2*a*Sqrt[a*Cosh[x]^2])} -{1/(a*Cosh[x]^2)^(5/2), x, 4, (3*ArcTan[Sinh[x]]*Cosh[x])/(8*a^2*Sqrt[a*Cosh[x]^2]) + Tanh[x]/(4*a*(a*Cosh[x]^2)^(3/2)) + (3*Tanh[x])/(8*a^2*Sqrt[a*Cosh[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cosh[a+b x]^3)^n*) - - -{(a*Cosh[x]^3)^(5/2),x, 6, -((26*I*a^2*Sqrt[a*Cosh[x]^3]*EllipticF[(I*x)/2, 2])/(77*Cosh[x]^(3/2))) + (78/385)*a^2*Cosh[x]*Sqrt[a*Cosh[x]^3]*Sinh[x] + (26/165)*a^2*Cosh[x]^3*Sqrt[a*Cosh[x]^3]*Sinh[x] + (2/15)*a^2*Cosh[x]^5*Sqrt[a*Cosh[x]^3]*Sinh[x] + (26/77)*a^2*Sqrt[a*Cosh[x]^3]*Tanh[x]} -{(a*Cosh[x]^3)^(3/2),x, 4, -((14*I*a*Sqrt[a*Cosh[x]^3]*EllipticE[(I*x)/2, 2])/(15*Cosh[x]^(3/2))) + (14/45)*a*Sqrt[a*Cosh[x]^3]*Sinh[x] + (2/9)*a*Cosh[x]^2*Sqrt[a*Cosh[x]^3]*Sinh[x]} -{(a*Cosh[x]^3)^(1/2), x, 3, -((2*I*Sqrt[a*Cosh[x]^3]*EllipticF[(I*x)/2, 2])/(3*Cosh[x]^(3/2))) + (2/3)*Sqrt[a*Cosh[x]^3]*Tanh[x]} -{1/(a*Cosh[x]^3)^(1/2), x, 3, (2*I*Cosh[x]^(3/2)*EllipticE[(I*x)/2, 2])/Sqrt[a*Cosh[x]^3] + (2*Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^3]} -{1/(a*Cosh[x]^3)^(3/2),x, 4, -((10*I*Cosh[x]^(3/2)*EllipticF[(I*x)/2, 2])/(21*a*Sqrt[a*Cosh[x]^3])) + (10*Sinh[x])/(21*a*Sqrt[a*Cosh[x]^3]) + (2*Sech[x]*Tanh[x])/(7*a*Sqrt[a*Cosh[x]^3])} -{1/(a*Cosh[x]^3)^(5/2),x, 6, (154*I*Cosh[x]^(3/2)*EllipticE[(I*x)/2, 2])/(195*a^2*Sqrt[a*Cosh[x]^3]) + (154*Cosh[x]*Sinh[x])/(195*a^2*Sqrt[a*Cosh[x]^3]) + (154*Tanh[x])/(585*a^2*Sqrt[a*Cosh[x]^3]) + (22*Sech[x]^2*Tanh[x])/(117*a^2*Sqrt[a*Cosh[x]^3]) + (2*Sech[x]^4*Tanh[x])/(13*a^2*Sqrt[a*Cosh[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Cosh[a+b x]^4)^n*) - - -{(a*Cosh[x]^4)^(5/2),x, 7, (63/256)*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (21/128)*a^2*Cosh[x]*Sqrt[a*Cosh[x]^4]*Sinh[x] + (21/160)*a^2*Cosh[x]^3*Sqrt[a*Cosh[x]^4]*Sinh[x] + (9/80)*a^2*Cosh[x]^5*Sqrt[a*Cosh[x]^4]*Sinh[x] + (1/10)*a^2*Cosh[x]^7*Sqrt[a*Cosh[x]^4]*Sinh[x] + (63/256)*a^2*Sqrt[a*Cosh[x]^4]*Tanh[x]} -{(a*Cosh[x]^4)^(3/2),x, 5, (5/16)*a*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (5/24)*a*Cosh[x]*Sqrt[a*Cosh[x]^4]*Sinh[x] + (1/6)*a*Cosh[x]^3*Sqrt[a*Cosh[x]^4]*Sinh[x] + (5/16)*a*Sqrt[a*Cosh[x]^4]*Tanh[x]} -{(a*Cosh[x]^4)^(1/2), x, 3, (1/2)*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (1/2)*Sqrt[a*Cosh[x]^4]*Tanh[x]} -{1/(a*Cosh[x]^4)^(1/2), x, 3, (Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]} -{1/(a*Cosh[x]^4)^(3/2),x, 3, (Cosh[x]*Sinh[x])/(a*Sqrt[a*Cosh[x]^4]) - (2*Sinh[x]^2*Tanh[x])/(3*a*Sqrt[a*Cosh[x]^4]) + (Sinh[x]^2*Tanh[x]^3)/(5*a*Sqrt[a*Cosh[x]^4])} -{1/(a*Cosh[x]^4)^(5/2),x, 3, (Cosh[x]*Sinh[x])/(a^2*Sqrt[a*Cosh[x]^4]) - (4*Sinh[x]^2*Tanh[x])/(3*a^2*Sqrt[a*Cosh[x]^4]) + (6*Sinh[x]^2*Tanh[x]^3)/(5*a^2*Sqrt[a*Cosh[x]^4]) - (4*Sinh[x]^2*Tanh[x]^5)/(7*a^2*Sqrt[a*Cosh[x]^4]) + (Sinh[x]^2*Tanh[x]^7)/(9*a^2*Sqrt[a*Cosh[x]^4])} - - -(* ::Subsection:: *) -(*Integrands of the form (c Cosh[a+b x]^m)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Cosh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2=0*) - - -{Sinh[x]/(1 + Cosh[x])^2, x, 2, -(1/(1 + Cosh[x]))} -{Sinh[x]/(1 - Cosh[x])^2, x, 2, 1/(1 - Cosh[x])} -{Sinh[x]^2/(1 + Cosh[x])^2, x, 2, x - (2*Sinh[x])/(1 + Cosh[x])} -{Sinh[x]^2/(1 - Cosh[x])^2, x, 2, x + (2*Sinh[x])/(1 - Cosh[x])} -{Sinh[x]^3/(1 + Cosh[x])^2, x, 3, Cosh[x] - 2*Log[1 + Cosh[x]]} -{Sinh[x]^3/(1 - Cosh[x])^2, x, 3, Cosh[x] + 2*Log[1 - Cosh[x]]} - - -{Sinh[x]/(1 + Cosh[x])^3, x, 2, -(1/(2*(1 + Cosh[x])^2))} -{Sinh[x]/(1 - Cosh[x])^3, x, 2, 1/(2*(1 - Cosh[x])^2)} -{Sinh[x]^2/(1 + Cosh[x])^3, x, 1, Sinh[x]^3/(3*(1 + Cosh[x])^3)} -{Sinh[x]^2/(1 - Cosh[x])^3, x, 1, -(Sinh[x]^3/(3*(1 - Cosh[x])^3))} -{Sinh[x]^3/(1 + Cosh[x])^3, x, 3, 2/(1 + Cosh[x]) + Log[1 + Cosh[x]]} -{Sinh[x]^3/(1 - Cosh[x])^3, x, 3, -(2/(1 - Cosh[x])) - Log[1 - Cosh[x]]} - - -{Sinh[x]^8/(a + a*Cosh[x]), x, 5, (5*x)/(16*a) - (5*Cosh[x]*Sinh[x])/(16*a) + (5*Cosh[x]*Sinh[x]^3)/(24*a) - (Cosh[x]*Sinh[x]^5)/(6*a) + Sinh[x]^7/(7*a)} -{Sinh[x]^7/(a + a*Cosh[x]), x, 3, (a - a*Cosh[x])^4/a^5 - (4*(a - a*Cosh[x])^5)/(5*a^6) + (a - a*Cosh[x])^6/(6*a^7)} -{Sinh[x]^6/(a + a*Cosh[x]), x, 4, -((3*x)/(8*a)) + (3*Cosh[x]*Sinh[x])/(8*a) - (Cosh[x]*Sinh[x]^3)/(4*a) + Sinh[x]^5/(5*a)} -{Sinh[x]^5/(a + a*Cosh[x]), x, 3, -((2*(a - a*Cosh[x])^3)/(3*a^4)) + (a - a*Cosh[x])^4/(4*a^5)} -{Sinh[x]^4/(a + a*Cosh[x]), x, 3, x/(2*a) - (Cosh[x]*Sinh[x])/(2*a) + Sinh[x]^3/(3*a)} -{Sinh[x]^3/(a + a*Cosh[x]), x, 2, -(Cosh[x]/a) + Cosh[x]^2/(2*a)} -{Sinh[x]^2/(a + a*Cosh[x]), x, 2, -(x/a) + Sinh[x]/a} -{Sinh[x]^1/(a + a*Cosh[x]), x, 2, Log[1 + Cosh[x]]/a} -{Csch[x]^1/(a + a*Cosh[x]), x, 4, -(ArcTanh[Cosh[x]]/(2*a)) + 1/(2*(a + a*Cosh[x]))} -{Csch[x]^2/(a + a*Cosh[x]), x, 3, -((2*Coth[x])/(3*a)) + Csch[x]/(3*(a + a*Cosh[x]))} -{Csch[x]^3/(a + a*Cosh[x]), x, 4, (3*ArcTanh[Cosh[x]])/(8*a) + 1/(8*(a - a*Cosh[x])) - a/(8*(a + a*Cosh[x])^2) - 1/(4*(a + a*Cosh[x]))} -{Csch[x]^4/(a + a*Cosh[x]), x, 3, (4*Coth[x])/(5*a) - (4*Coth[x]^3)/(15*a) + Csch[x]^3/(5*(a + a*Cosh[x]))} -{Csch[x]^5/(a + a*Cosh[x]), x, 4, -((5*ArcTanh[Cosh[x]])/(16*a)) - a/(32*(a - a*Cosh[x])^2) - 1/(8*(a - a*Cosh[x])) + a^2/(24*(a + a*Cosh[x])^3) + (3*a)/(32*(a + a*Cosh[x])^2) + 3/(16*(a + a*Cosh[x]))} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2!=0*) - - -{Sinh[x]^7/(a + b*Cosh[x]), x, 3, -((a*(a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x])/b^6) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Cosh[x]^2)/(2*b^5) - (a*(a^2 - 3*b^2)*Cosh[x]^3)/(3*b^4) + ((a^2 - 3*b^2)*Cosh[x]^4)/(4*b^3) - (a*Cosh[x]^5)/(5*b^2) + Cosh[x]^6/(6*b) + ((a^2 - b^2)^3*Log[a + b*Cosh[x]])/b^7} -{Sinh[x]^6/(a + b*Cosh[x]), x, 6, -((a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6)) + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/b^6 + ((8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Cosh[x])*Sinh[x])/(8*b^5) + ((4*(a^2 - b^2) - 3*a*b*Cosh[x])*Sinh[x]^3)/(12*b^3) + Sinh[x]^5/(5*b)} -{Sinh[x]^5/(a + b*Cosh[x]), x, 3, -((a*(a^2 - 2*b^2)*Cosh[x])/b^4) + ((a^2 - 2*b^2)*Cosh[x]^2)/(2*b^3) - (a*Cosh[x]^3)/(3*b^2) + Cosh[x]^4/(4*b) + ((a^2 - b^2)^2*Log[a + b*Cosh[x]])/b^5} -{Sinh[x]^4/(a + b*Cosh[x]), x, 5, -((a*(2*a^2 - 3*b^2)*x)/(2*b^4)) + (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/b^4 + ((2*(a^2 - b^2) - a*b*Cosh[x])*Sinh[x])/(2*b^3) + Sinh[x]^3/(3*b)} -{Sinh[x]^3/(a + b*Cosh[x]), x, 3, -((a*Cosh[x])/b^2) + Cosh[x]^2/(2*b) + ((a^2 - b^2)*Log[a + b*Cosh[x]])/b^3} -{Sinh[x]^2/(a + b*Cosh[x]), x, 4, -((a*x)/b^2) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/b^2 + Sinh[x]/b} -{Sinh[x]^1/(a + b*Cosh[x]), x, 2, Log[a + b*Cosh[x]]/b} -{Csch[x]^1/(a + b*Cosh[x]), x, 6, Log[1 - Cosh[x]]/(2*(a + b)) - Log[1 + Cosh[x]]/(2*(a - b)) + (b*Log[a + b*Cosh[x]])/(a^2 - b^2)} -{Csch[x]^2/(a + b*Cosh[x]), x, 4, (2*b^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) + ((b - a*Cosh[x])*Csch[x])/(a^2 - b^2)} -{Csch[x]^3/(a + b*Cosh[x]), x, 4, ((b - a*Cosh[x])*Csch[x]^2)/(2*(a^2 - b^2)) - ((a + 2*b)*Log[1 - Cosh[x]])/(4*(a + b)^2) + ((a - 2*b)*Log[1 + Cosh[x]])/(4*(a - b)^2) + (b^3*Log[a + b*Cosh[x]])/(a^2 - b^2)^2} -{Csch[x]^4/(a + b*Cosh[x]), x, 5, (2*b^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) + ((3*b^3 + a*(2*a^2 - 5*b^2)*Cosh[x])*Csch[x])/(3*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^3)/(3*(a^2 - b^2))} -{Csch[x]^5/(a + b*Cosh[x]), x, 5, ((4*b^3 + a*(3*a^2 - 7*b^2)*Cosh[x])*Csch[x]^2)/(8*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^4)/(4*(a^2 - b^2)) + ((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Cosh[x]])/(16*(a + b)^3) - ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Cosh[x]])/(16*(a - b)^3) + (b^5*Log[a + b*Cosh[x]])/(a^2 - b^2)^3} -{Csch[x]^6/(a + b*Cosh[x]), x, 6, (2*b^6*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)) + ((15*b^5 - a*(8*a^4 - 26*a^2*b^2 + 33*b^4)*Cosh[x])*Csch[x])/(15*(a^2 - b^2)^3) + ((5*b^3 + a*(4*a^2 - 9*b^2)*Cosh[x])*Csch[x]^3)/(15*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^5)/(5*(a^2 - b^2))} - - -{Sinh[x]^2/(a + b*Cosh[x])^2, x, 4, x/b^2 - (2*a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]) - Sinh[x]/(b*(a + b*Cosh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x])^n*) - - -{Tanh[x]^4/(a + b*Cosh[x]), x, 6, (b*(3*a^2 - 2*b^2)*ArcTan[Sinh[x]])/(2*a^4) + (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/a^4 - ((4*a^2 - 3*b^2)*Tanh[x])/(3*a^3) - (b*Sech[x]*Tanh[x])/(2*a^2) + (Sech[x]^2*Tanh[x])/(3*a)} -{Tanh[x]^3/(a + b*Cosh[x]), x, 3, ((a^2 - b^2)*Log[Cosh[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cosh[x]])/a^3 - (b*Sech[x])/a^2 + Sech[x]^2/(2*a)} -{Tanh[x]^2/(a + b*Cosh[x]), x, 6, (b*ArcTan[Sinh[x]])/a^2 + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/a^2 - Tanh[x]/a} -{Tanh[x]^1/(a + b*Cosh[x]), x, 4, Log[Cosh[x]]/a - Log[a + b*Cosh[x]]/a} -{Coth[x]^1/(a + b*Cosh[x]), x, 3, Log[1 - Cosh[x]]/(2*(a + b)) + Log[1 + Cosh[x]]/(2*(a - b)) - (a*Log[a + b*Cosh[x]])/(a^2 - b^2)} -{Coth[x]^2/(a + b*Cosh[x]), x, 7, (2*a^2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) - (a*Coth[x])/(a^2 - b^2) + (b*Csch[x])/(a^2 - b^2)} -{Coth[x]^3/(a + b*Cosh[x]), x, 4, -(((a - b*Cosh[x])*Csch[x]^2)/(2*(a^2 - b^2))) + ((2*a + b)*Log[1 - Cosh[x]])/(4*(a + b)^2) + ((2*a - b)*Log[1 + Cosh[x]])/(4*(a - b)^2) - (a^3*Log[a + b*Cosh[x]])/(a^2 - b^2)^2} -{Coth[x]^4/(a + b*Cosh[x]), x, 12, (2*a^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) - (a^3*Coth[x])/(a^2 - b^2)^2 - (a*Coth[x]^3)/(3*(a^2 - b^2)) + (a^2*b*Csch[x])/(a^2 - b^2)^2 + (b*Csch[x])/(a^2 - b^2) + (b*Csch[x]^3)/(3*(a^2 - b^2))} - - -{Tanh[x]^6/(a + a*Cosh[x]), x, 6, (3*ArcTan[Sinh[x]])/(8*a) - (3*Sech[x]*Tanh[x])/(8*a) - (Sech[x]*Tanh[x]^3)/(4*a) - Tanh[x]^5/(5*a)} -{Tanh[x]^5/(a + a*Cosh[x]), x, 5, -(Sech[x]/a) + Sech[x]^3/(3*a) - Tanh[x]^4/(4*a)} -{Tanh[x]^4/(a + a*Cosh[x]), x, 5, ArcTan[Sinh[x]]/(2*a) - (Sech[x]*Tanh[x])/(2*a) - Tanh[x]^3/(3*a)} -{Tanh[x]^3/(a + a*Cosh[x]), x, 5, -(Sech[x]/a) + Sech[x]^2/(2*a)} -{Tanh[x]^2/(a + a*Cosh[x]), x, 4, ArcTan[Sinh[x]]/a - Tanh[x]/a} -{Tanh[x]^1/(a + a*Cosh[x]), x, 4, Log[Cosh[x]]/a - Log[1 + Cosh[x]]/a} -{Coth[x]^1/(a + a*Cosh[x]), x, 5, -(ArcTanh[Cosh[x]]/(2*a)) - (Coth[x]*Csch[x])/(2*a) + Csch[x]^2/(2*a)} -{Coth[x]^2/(a + a*Cosh[x]), x, 5, Coth[x]^3/(3*a) - Csch[x]/a - Csch[x]^3/(3*a)} -{Coth[x]^3/(a + a*Cosh[x]), x, 6, -((3*ArcTanh[Cosh[x]])/(8*a)) + Coth[x]^4/(4*a) - (3*Coth[x]*Csch[x])/(8*a) - (Coth[x]^3*Csch[x])/(4*a)} -{Coth[x]^4/(a + a*Cosh[x]), x, 6, Coth[x]^5/(5*a) - Csch[x]/a - (2*Csch[x]^3)/(3*a) - Csch[x]^5/(5*a)} - - -{Tanh[x]*Sqrt[a + b*Cosh[x]], x, 4, -2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[x]]} -{Tanh[x]/Sqrt[a + b*Cosh[x]], x, 3, -((2*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]])/Sqrt[a])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Hyper[c+d x]) (a+b Cosh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Sinh[c+d x]) (a+b Cosh[c+d x])^n*) - - -{(A + B*Sinh[x])/(a + b*Cosh[x]), x, 6, (2*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[a + b*Cosh[x]])/b} - -{(A + B*Sinh[x])/(1 + Cosh[x]), x, 5, B*Log[1 + Cosh[x]] + (A*Sinh[x])/(1 + Cosh[x])} -{(A + B*Sinh[x])/(1 - Cosh[x]), x, 5, (-B)*Log[1 - Cosh[x]] - (A*Sinh[x])/(1 - Cosh[x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A+B Hyper[c+d x]) (a+b Cosh[c+d x])^n*) - - -{(A + B*Tanh[x])/(a + b*Cosh[x]), x, 8, (2*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[Cosh[x]])/a - (B*Log[a + b*Cosh[x]])/a} -{(A + B*Coth[x])/(a + b*Cosh[x]), x, 7, (2*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cosh[x]])/(2*(a + b)) + (B*Log[1 + Cosh[x]])/(2*(a - b)) - (a*B*Log[a + b*Cosh[x]])/(a^2 - b^2)} -{(A + B*Sech[x])/(a + b*Cosh[x]), x, 5, (B*ArcTan[Sinh[x]])/a + (2*(a*A - b*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b])} -{(A + B*Csch[x])/(a + b*Cosh[x]), x, 11, (2*A*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cosh[x]])/(2*(a + b)) - (B*Log[1 + Cosh[x]])/(2*(a - b)) + (b*B*Log[a + b*Cosh[x]])/(a^2 - b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (A+B Hyper[c+d x]+C Hyper[c+d x]) (a+b Cosh[c+d x])^n*) - - -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x]), x, 6, (B*x)/b + (2*(A*b - a*B)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(d + e*x)])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*e) + (C*Log[a + b*Cosh[d + e*x]])/(b*e)} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^2, x, 7, (2*(a*A - b*B)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*e) - C/(b*e*(a + b*Cosh[d + e*x])) - ((A*b - a*B)*Sinh[d + e*x])/((a^2 - b^2)*e*(a + b*Cosh[d + e*x]))} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^3, x, 8, ((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*e) - C/(2*b*e*(a + b*Cosh[d + e*x])^2) - ((A*b - a*B)*Sinh[d + e*x])/(2*(a^2 - b^2)*e*(a + b*Cosh[d + e*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sinh[d + e*x])/(2*(a^2 - b^2)^2*e*(a + b*Cosh[d + e*x]))} -{(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^4, x, 9, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(d + e*x)])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*e) - C/(3*b*e*(a + b*Cosh[d + e*x])^3) - ((A*b - a*B)*Sinh[d + e*x])/(3*(a^2 - b^2)*e*(a + b*Cosh[d + e*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sinh[d + e*x])/(6*(a^2 - b^2)^2*e*(a + b*Cosh[d + e*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[d + e*x])/(6*(a^2 - b^2)^3*e*(a + b*Cosh[d + e*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Cosh[c+d x]^2)^p*) - - -{x/(a + b*Cosh[x]^2), x, 9, (x*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x*Log[1 + (b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b]) - PolyLog[2, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])} -{x^2/(a + b*Cosh[x]^2), x, 11, (x^2*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x^2*Log[1 + (b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(2*Sqrt[a]*Sqrt[a + b]) - PolyLog[3, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b]) + PolyLog[3, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))]/(4*Sqrt[a]*Sqrt[a + b])} -{x^3/(a + b*Cosh[x]^2), x, 13, (x^3*Log[1 + (b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - (x^3*Log[1 + (b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) - (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) - (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(4*Sqrt[a]*Sqrt[a + b]) + (3*PolyLog[4, -((b*E^(2*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b]) - (3*PolyLog[4, -((b*E^(2*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b]))])/(8*Sqrt[a]*Sqrt[a + b])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (1-a^2 x^2)^m Cosh[Sqrt[1-a x]/Sqrt[1+a x]]^n*) - - -{Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2), x, 5, -((3*CoshIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/(4*a)) - CoshIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)} -{Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x, 4, -(CoshIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(2*a)) - Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)} -{Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1/(1 - a^2*x^2), x, 2, -(CoshIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)} -{1/(Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^1*(1 - a^2*x^2)), x, 1, Unintegrable[Sech[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/((1 - a*x)*(1 + a*x)), x]} -{1/(Cosh[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2*(1 - a^2*x^2)), x, 1, Unintegrable[Sech[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/((1 - a*x)*(1 + a*x)), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Cosh[c+d x])^p*) - - -{x*Sinh[x]/(a + b*Cosh[x])^2, x, 3, (2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]) - x/(b*(a + b*Cosh[x]))} -{x*Sinh[x]/(a + b*Cosh[x])^3, x, 5, (a*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*b*(a + b)^(3/2)) - x/(2*b*(a + b*Cosh[x])^2) - Sinh[x]/(2*(a^2 - b^2)*(a + b*Cosh[x]))} - - -{Sinh[a + b*x]*(2 + Cosh[a + b*x]^2)/x, x, 13, (9/4)*CoshIntegral[b*x]*Sinh[a] + (1/4)*CoshIntegral[3*b*x]*Sinh[3*a] + (9/4)*Cosh[a]*SinhIntegral[b*x] + (1/4)*Cosh[3*a]*SinhIntegral[3*b*x]} - - -{(x^m*Sinh[c + d*x])/(a + b*Cosh[c + d*x]), x, 0, Unintegrable[(x^m*Sinh[c + d*x])/(a + b*Cosh[c + d*x]), x]} - -{(x^3*Sinh[c + d*x])/(a + b*Cosh[c + d*x]), x, 11, -x^4/(4*b) + (x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b*d) + (x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) + (3*x^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b*d^2) + (3*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b*d^2) - (6*x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b*d^3) - (6*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b*d^3) + (6*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b*d^4) + (6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b*d^4)} -{(x^2*Sinh[c + d*x])/(a + b*Cosh[c + d*x]), x, 9, -x^3/(3*b) + (x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b*d) + (x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) + (2*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b*d^2) + (2*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b*d^2) - (2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b*d^3) - (2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b*d^3)} -{(x*Sinh[c + d*x])/(a + b*Cosh[c + d*x]), x, 7, -x^2/(2*b) + (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b*d) + (x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b*d) + PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))]/(b*d^2) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))]/(b*d^2)} -{Sinh[c + d*x]/(a + b*Cosh[c + d*x]), x, 2, Log[a + b*Cosh[c + d*x]]/(b*d)} -{Sinh[c + d*x]/(x*(a + b*Cosh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]/(x*(a + b*Cosh[c + d*x])), x]} - - -{(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x, 0, Unintegrable[(x^m*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x]} - -{(x^3*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x, 18, -(a*x^4)/(4*b^2) - (6*Cosh[c + d*x])/(b*d^4) - (3*x^2*Cosh[c + d*x])/(b*d^2) + (Sqrt[a^2 - b^2]*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (Sqrt[a^2 - b^2]*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (3*Sqrt[a^2 - b^2]*x^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^2) - (3*Sqrt[a^2 - b^2]*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^2) - (6*Sqrt[a^2 - b^2]*x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^3) + (6*Sqrt[a^2 - b^2]*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^3) + (6*Sqrt[a^2 - b^2]*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^4) - (6*Sqrt[a^2 - b^2]*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^4) + (6*x*Sinh[c + d*x])/(b*d^3) + (x^3*Sinh[c + d*x])/(b*d)} -{(x^2*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x, 15, -(a*x^3)/(3*b^2) - (2*x*Cosh[c + d*x])/(b*d^2) + (Sqrt[a^2 - b^2]*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (Sqrt[a^2 - b^2]*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (2*Sqrt[a^2 - b^2]*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^2) - (2*Sqrt[a^2 - b^2]*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^2) - (2*Sqrt[a^2 - b^2]*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^3) + (2*Sqrt[a^2 - b^2]*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^3) + (2*Sinh[c + d*x])/(b*d^3) + (x^2*Sinh[c + d*x])/(b*d)} -{(x*Sinh[c + d*x]^2)/(a + b*Cosh[c + d*x]), x, 12, -(a*x^2)/(2*b^2) - Cosh[c + d*x]/(b*d^2) + (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (Sqrt[a^2 - b^2]*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^2*d) + (Sqrt[a^2 - b^2]*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^2*d^2) - (Sqrt[a^2 - b^2]*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^2*d^2) + (x*Sinh[c + d*x])/(b*d)} -{Sinh[c + d*x]^2/(a + b*Cosh[c + d*x]), x, 4, -((a*x)/b^2) + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/(b^2*d) + Sinh[c + d*x]/(b*d)} -{Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^2/(x*(a + b*Cosh[c + d*x])), x]} - - -{(x^m*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]), x, 0, Unintegrable[(x^m*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]), x]} - -{(x^3*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]), x, 21, (3*x)/(8*b*d^3) + x^3/(4*b*d) - ((a^2 - b^2)*x^4)/(4*b^3) - (6*a*x*Cosh[c + d*x])/(b^2*d^3) - (a*x^3*Cosh[c + d*x])/(b^2*d) + ((a^2 - b^2)*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^3*d) + ((a^2 - b^2)*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + (3*(a^2 - b^2)*x^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^2) + (3*(a^2 - b^2)*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^2) - (6*(a^2 - b^2)*x*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^3) - (6*(a^2 - b^2)*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^3) + (6*(a^2 - b^2)*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^4) + (6*(a^2 - b^2)*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^4) + (6*a*Sinh[c + d*x])/(b^2*d^4) + (3*a*x^2*Sinh[c + d*x])/(b^2*d^2) - (3*Cosh[c + d*x]*Sinh[c + d*x])/(8*b*d^4) - (3*x^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (3*x*Sinh[c + d*x]^2)/(4*b*d^3) + (x^3*Sinh[c + d*x]^2)/(2*b*d)} -{(x^2*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]), x, 16, x^2/(4*b*d) - ((a^2 - b^2)*x^3)/(3*b^3) - (2*a*Cosh[c + d*x])/(b^2*d^3) - (a*x^2*Cosh[c + d*x])/(b^2*d) + ((a^2 - b^2)*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^3*d) + ((a^2 - b^2)*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + (2*(a^2 - b^2)*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^2) + (2*(a^2 - b^2)*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^2) - (2*(a^2 - b^2)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^3) - (2*(a^2 - b^2)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^3) + (2*a*x*Sinh[c + d*x])/(b^2*d^2) - (x*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d^2) + Sinh[c + d*x]^2/(4*b*d^3) + (x^2*Sinh[c + d*x]^2)/(2*b*d)} -{(x*Sinh[c + d*x]^3)/(a + b*Cosh[c + d*x]), x, 13, x/(4*b*d) - ((a^2 - b^2)*x^2)/(2*b^3) - (a*x*Cosh[c + d*x])/(b^2*d) + ((a^2 - b^2)*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 - b^2])])/(b^3*d) + ((a^2 - b^2)*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 - b^2])])/(b^3*d) + ((a^2 - b^2)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 - b^2]))])/(b^3*d^2) + ((a^2 - b^2)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 - b^2]))])/(b^3*d^2) + (a*Sinh[c + d*x])/(b^2*d^2) - (Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (x*Sinh[c + d*x]^2)/(2*b*d)} -{Sinh[c + d*x]^3/(a + b*Cosh[c + d*x]), x, 3, -((a*Cosh[c + d*x])/(b^2*d)) + Cosh[c + d*x]^2/(2*b*d) + ((a^2 - b^2)*Log[a + b*Cosh[c + d*x]])/(b^3*d)} -{Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])), x, 0, Unintegrable[Sinh[c + d*x]^3/(x*(a + b*Cosh[c + d*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Cosh[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[a+b Log[c x^n]]^p*) - - -{Cosh[a + b*Log[c*x^n]], x, 1, (x*Cosh[a + b*Log[c*x^n]])/(1 - b^2*n^2) - (b*n*x*Sinh[a + b*Log[c*x^n]])/(1 - b^2*n^2)} -{Cosh[a + b*Log[c*x^n]]^2, x, 2, -((2*b^2*n^2*x)/(1 - 4*b^2*n^2)) + (x*Cosh[a + b*Log[c*x^n]]^2)/(1 - 4*b^2*n^2) - (2*b*n*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1 - 4*b^2*n^2)} -{Cosh[a + b*Log[c*x^n]]^3, x, 2, -((6*b^2*n^2*x*Cosh[a + b*Log[c*x^n]])/(1 - 10*b^2*n^2 + 9*b^4*n^4)) + (x*Cosh[a + b*Log[c*x^n]]^3)/(1 - 9*b^2*n^2) + (6*b^3*n^3*x*Sinh[a + b*Log[c*x^n]])/(1 - 10*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x*Cosh[a + b*Log[c*x^n]]^2*Sinh[a + b*Log[c*x^n]])/(1 - 9*b^2*n^2)} -{Cosh[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (12*b^2*n^2*x*Cosh[a + b*Log[c*x^n]]^2)/(1 - 20*b^2*n^2 + 64*b^4*n^4) + (x*Cosh[a + b*Log[c*x^n]]^4)/(1 - 16*b^2*n^2) + (24*b^3*n^3*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x*Cosh[a + b*Log[c*x^n]]^3*Sinh[a + b*Log[c*x^n]])/(1 - 16*b^2*n^2)} - - -{x^m*Cosh[a + b*Log[c*x^n]], x, 1, ((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2) - (b*n*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - b^2*n^2)} -{x^m*Cosh[a + b*Log[c*x^n]]^2, x, 2, -((2*b^2*n^2*x^(1 + m))/((1 + m)*((1 + m)^2 - 4*b^2*n^2))) + ((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^2)/((1 + m)^2 - 4*b^2*n^2) - (2*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - 4*b^2*n^2)} -{x^m*Cosh[a + b*Log[c*x^n]]^3, x, 2, -((6*b^2*(1 + m)*n^2*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2))) + ((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^3)/((1 + m)^2 - 9*b^2*n^2) + (6*b^3*n^3*x^(1 + m)*Sinh[a + b*Log[c*x^n]])/(((1 + m)^2 - 9*b^2*n^2)*((1 + m)^2 - b^2*n^2)) - (3*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^2*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - 9*b^2*n^2)} -{x^m*Cosh[a + b*Log[c*x^n]]^4, x, 3, (24*b^4*n^4*x^(1 + m))/((1 + m)*((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) - (12*b^2*(1 + m)*n^2*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^2)/(((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) + ((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^4)/((1 + m)^2 - 16*b^2*n^2) + (24*b^3*n^3*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(((1 + m)^2 - 16*b^2*n^2)*((1 + m)^2 - 4*b^2*n^2)) - (4*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^3*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - 16*b^2*n^2)} - - -{Cosh[a + b*Log[c*x^n]]/x, x, 2, Sinh[a + b*Log[c*x^n]]/(b*n)} -{Cosh[a + b*Log[c*x^n]]^2/x, x, 3, Log[x]/2 + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b*n)} -{Cosh[a + b*Log[c*x^n]]^3/x, x, 3, Sinh[a + b*Log[c*x^n]]/(b*n) + Sinh[a + b*Log[c*x^n]]^3/(3*b*n)} -{Cosh[a + b*Log[c*x^n]]^4/x, x, 4, 3*Log[x]/8 + (3*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(8*b*n) + (Cosh[a + b*Log[c*x^n]]^3*Sinh[a + b*Log[c*x^n]])/(4*b*n)} -{Cosh[a + b*Log[c*x^n]]^5/x, x, 3, Sinh[a + b*Log[c*x^n]]/(b*n) + (2*Sinh[a + b*Log[c*x^n]]^3)/(3*b*n) + Sinh[a + b*Log[c*x^n]]^5/(5*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[a+b Log[c x^n]]^(p/2)*) - - -{Cosh[a + b*Log[c*x^n]]^(5/2)/x, x, 3, -((6*I*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2])/(5*b*n)) + (2*Cosh[a + b*Log[c*x^n]]^(3/2)*Sinh[a + b*Log[c*x^n]])/(5*b*n)} -{Cosh[a + b*Log[c*x^n]]^(3/2)/x, x, 3, -((2*I*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2])/(3*b*n)) + (2*Sqrt[Cosh[a + b*Log[c*x^n]]]*Sinh[a + b*Log[c*x^n]])/(3*b*n)} -{Sqrt[Cosh[a + b*Log[c*x^n]]]/x, x, 2, -((2*I*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2])/(b*n))} -{1/(x*Sqrt[Cosh[a + b*Log[c*x^n]]]), x, 2, -((2*I*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2])/(b*n))} -{1/(x*Cosh[a + b*Log[c*x^n]]^(3/2)), x, 3, (2*I*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2])/(b*n) + (2*Sinh[a + b*Log[c*x^n]])/(b*n*Sqrt[Cosh[a + b*Log[c*x^n]]])} -{1/(x*Cosh[a + b*Log[c*x^n]]^(5/2)), x, 3, -((2*I*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2])/(3*b*n)) + (2*Sinh[a + b*Log[c*x^n]])/(3*b*n*Cosh[a + b*Log[c*x^n]]^(3/2))} - - -{Cosh[a + 2/n*Log[c*x^n]]^(5/2), x, 8, (-(1/4))*x*Cosh[a + (2*Log[c*x^n])/n]^(5/2) + (5*x*Cosh[a + (2*Log[c*x^n])/n]^(5/2))/(E^(2*a)*(c*x^n)^(4/n)*(4*(1 + 1/(E^(2*a)*(c*x^n)^(4/n)))^2)) + (5*x*Cosh[a + (2*Log[c*x^n])/n]^(5/2))/(12*(1 + 1/(E^(2*a)*(c*x^n)^(4/n)))) - (5*x*ArcCsch[E^a*(c*x^n)^(2/n)]*Cosh[a + (2*Log[c*x^n])/n]^(5/2))/(E^(3*a)*(c*x^n)^(6/n)*(4*(1 + 1/(E^(2*a)*(c*x^n)^(4/n)))^(5/2)))} -{Sqrt[Cosh[a + 2/n*Log[c*x^n]]], x, 6, (1/2)*x*Sqrt[Cosh[a + (2*Log[c*x^n])/n]] - (x*ArcCsch[E^a*(c*x^n)^(2/n)]*Sqrt[Cosh[a + (2*Log[c*x^n])/n]])/(E^a*(c*x^n)^(2/n)*(2*Sqrt[1 + 1/(E^(2*a)*(c*x^n)^(4/n))]))} -{1/Cosh[a + 2/n*Log[c*x^n]]^(3/2), x, 3, -((x*(1 + 1/(E^(2*a)*(c*x^n)^(4/n))))/(2*Cosh[a + (2*Log[c*x^n])/n]^(3/2)))} -{1/Cosh[a + 2/n*Log[c*x^n]]^(7/2), x, 4, -((x*(1 + 1/(E^(2*a)*(c*x^n)^(4/n))))/(6*Cosh[a + (2*Log[c*x^n])/n]^(7/2))) - (x*(1 + 1/(E^(2*a)*(c*x^n)^(4/n))))/(E^(2*a)*(c*x^n)^(4/n)*(15*Cosh[a + (2*Log[c*x^n])/n]^(7/2)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[(a+b x)/(c+d x)]^n*) - - -{Cosh[(a + b*x)/(c + d*x)], x, 5, ((c + d*x)*Cosh[(a + b*x)/(c + d*x)])/d + ((b*c - a*d)*CoshIntegral[(b*c - a*d)/(d*(c + d*x))]*Sinh[b/d])/d^2 - ((b*c - a*d)*Cosh[b/d]*SinhIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2} -{Cosh[(a + b*x)/(c + d*x)]^2, x, 6, ((c + d*x)*Cosh[(a + b*x)/(c + d*x)]^2)/d + ((b*c - a*d)*CoshIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sinh[(2*b)/d])/d^2 - ((b*c - a*d)*Cosh[(2*b)/d]*SinhIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[a+b x]^m*) - - -{E^(a + b*x)*Cosh[a + b*x]^4, x, 4, -(E^(-3*a - 3*b*x)/(48*b)) - E^(-a - b*x)/(4*b) + (3*E^(a + b*x))/(8*b) + E^(3*a + 3*b*x)/(12*b) + E^(5*a + 5*b*x)/(80*b)} -{E^(a + b*x)*Cosh[a + b*x]^3, x, 5, -(E^(-2*a - 2*b*x)/(16*b)) + (3*E^(2*a + 2*b*x))/(16*b) + E^(4*a + 4*b*x)/(32*b) + (3*x)/8} -{E^(a + b*x)*Cosh[a + b*x]^2, x, 4, -(E^(-a - b*x)/(4*b)) + E^(a + b*x)/(2*b) + E^(3*a + 3*b*x)/(12*b)} -{E^(a + b*x)*Cosh[a + b*x]^1, x, 4, E^(2*a + 2*b*x)/(4*b) + x/2} -{E^(a + b*x)*Sech[a + b*x]^1, x, 3, Log[1 + E^(2*a + 2*b*x)]/b} -{E^(a + b*x)*Sech[a + b*x]^2, x, 4, -((2*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x)))) + (2*ArcTan[E^(a + b*x)])/b} -{E^(a + b*x)*Sech[a + b*x]^3, x, 3, (2*E^(4*a + 4*b*x))/(b*(1 + E^(2*a + 2*b*x))^2)} -{E^(a + b*x)*Sech[a + b*x]^4, x, 6, -((8*E^(3*a + 3*b*x))/(3*b*(1 + E^(2*a + 2*b*x))^3)) - (2*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x))^2) + E^(a + b*x)/(b*(1 + E^(2*a + 2*b*x))) + ArcTan[E^(a + b*x)]/b} -{E^(a + b*x)*Sech[a + b*x]^5, x, 5, -(4/(b*(1 + E^(2*a + 2*b*x))^4)) + 32/(3*b*(1 + E^(2*a + 2*b*x))^3) - 8/(b*(1 + E^(2*a + 2*b*x))^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^x Cosh[n x]^m*) - - -{E^x*Cosh[2*x]^2, x, 4, -(1/12)/E^(3*x) + E^x/2 + E^(5*x)/20} -{E^x*Cosh[2*x], x, 4, -(1/(E^x*2)) + E^(3*x)/6} -{E^x*Sech[2*x], x, 11, -(ArcTan[1 - Sqrt[2]*E^x]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*E^x]/Sqrt[2] + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(2*Sqrt[2])} -{E^x*Sech[2*x]^2, x, 12, -(E^x/(1 + E^(4*x))) - ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) - Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) + Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])} - - -{E^x*Cosh[3*x]^2, x, 4, -(1/20)/E^(5*x) + E^x/2 + E^(7*x)/28} -{E^x*Cosh[3*x], x, 4, -(1/4)/E^(2*x) + E^(4*x)/8} -{E^x*Sech[3*x], x, 9, -(ArcTan[(1 - 2*E^(2*x))/Sqrt[3]]/Sqrt[3]) - (1/3)*Log[1 + E^(2*x)] + (1/6)*Log[1 - E^(2*x) + E^(4*x)]} -{E^x*Sech[3*x]^2, x, 13, -((2*E^x)/(3*(1 + E^(6*x)))) + (2*ArcTan[E^x])/9 - (1/9)*ArcTan[Sqrt[3] - 2*E^x] + (1/9)*ArcTan[Sqrt[3] + 2*E^x] - Log[1 - Sqrt[3]*E^x + E^(2*x)]/(6*Sqrt[3]) + Log[1 + Sqrt[3]*E^x + E^(2*x)]/(6*Sqrt[3])} - - -{E^x*Cosh[4*x]^2, x, 4, -(1/28)/E^(7*x) + E^x/2 + E^(9*x)/36} -{E^x*Cosh[4*x], x, 4, -(1/6)/E^(3*x) + E^(5*x)/10} -{E^x*Sech[4*x], x, 21, ArcTan[(Sqrt[2 - Sqrt[2]] - 2*E^x)/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] - 2*E^x)/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)]/(4*Sqrt[2*(2 - Sqrt[2])]) + Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)]/(4*Sqrt[2*(2 - Sqrt[2])]) + Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]/(4*Sqrt[2*(2 + Sqrt[2])]) - Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]/(4*Sqrt[2*(2 + Sqrt[2])])} -{E^x*Sech[4*x]^2, x, 22, -(E^x/(2*(1 + E^(8*x)))) - ArcTan[(Sqrt[2 - Sqrt[2]] - 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] - 2*E^x)/Sqrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) + ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) - (1/32)*Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] + (1/32)*Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] - (1/32)*Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)] + (1/32)*Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Cosh[d+e x]^m*) - - -{F^(c*(a + b*x))*Cosh[d + e*x]^3, x, 2, -((b*c*F^(c*(a + b*x))*Cosh[d + e*x]^3*Log[F])/(9*e^2 - b^2*c^2*Log[F]^2)) - (6*b*c*e^2*F^(c*(a + b*x))*Cosh[d + e*x]*Log[F])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4) + (3*e*F^(c*(a + b*x))*Cosh[d + e*x]^2*Sinh[d + e*x])/(9*e^2 - b^2*c^2*Log[F]^2) + (6*e^3*F^(c*(a + b*x))*Sinh[d + e*x])/(9*e^4 - 10*b^2*c^2*e^2*Log[F]^2 + b^4*c^4*Log[F]^4)} -{F^(c*(a + b*x))*Cosh[d + e*x]^2, x, 2, (2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c*(a + b*x))*Cosh[d + e*x]^2*Log[F])/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Cosh[d + e*x]^1, x, 1, -((b*c*F^(c*(a + b*x))*Cosh[d + e*x]*Log[F])/(e^2 - b^2*c^2*Log[F]^2)) + (e*F^(c*(a + b*x))*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*Sech[d + e*x]^1, x, 1, (2*E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[1, (e + b*c*Log[F])/(2*e), (1/2)*(3 + (b*c*Log[F])/e), -E^(2*(d + e*x))])/(e + b*c*Log[F])} -{F^(c*(a + b*x))*Sech[d + e*x]^2, x, 1, (4*E^(2*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/(2*e), 2 + (b*c*Log[F])/(2*e), -E^(2*(d + e*x))])/(2*e + b*c*Log[F])} -{F^(c*(a + b*x))*Sech[d + e*x]^3, x, 2, (E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[1, (e + b*c*Log[F])/(2*e), (1/2)*(3 + (b*c*Log[F])/e), -E^(2*(d + e*x))]*(e - b*c*Log[F]))/e^2 + (b*c*F^(c*(a + b*x))*Log[F]*Sech[d + e*x])/(2*e^2) + (F^(c*(a + b*x))*Sech[d + e*x]*Tanh[d + e*x])/(2*e)} -{F^(c*(a + b*x))*Sech[d + e*x]^4, x, 2, (2*E^(2*(d + e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/(2*e), 2 + (b*c*Log[F])/(2*e), -E^(2*(d + e*x))]*(2*e - b*c*Log[F]))/(3*e^2) + (b*c*F^(c*(a + b*x))*Log[F]*Sech[d + e*x]^2)/(6*e^2) + (F^(c*(a + b*x))*Sech[d + e*x]^2*Tanh[d + e*x])/(3*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Cosh[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Cosh[a*c + b*c*x]^2)^(5/2), x, 6, -(Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(128*b*c*E^(4*c*(a + b*x))) - (5*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(64*b*c*E^(2*c*(a + b*x))) + (5*E^(2*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(32*b*c) + (5*E^(4*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(128*b*c) + (E^(6*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(192*b*c) + (5*x*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/16} -{E^(c*(a + b*x))*(Cosh[a*c + b*c*x]^2)^(3/2), x, 6, -(Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(16*b*c*E^(2*c*(a + b*x))) + (3*E^(2*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(16*b*c) + (E^(4*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(32*b*c) + (3*x*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/8} -{E^(c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2], x, 5, (E^(2*c*(a + b*x))*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/(4*b*c) + (x*Sqrt[Cosh[a*c + b*c*x]^2]*Sech[a*c + b*c*x])/2} -{E^(c*(a + b*x))/Sqrt[Cosh[a*c + b*c*x]^2], x, 4, (Cosh[a*c + b*c*x]*Log[1 + E^(2*c*(a + b*x))])/(b*c*Sqrt[Cosh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Cosh[a*c + b*c*x]^2)^(3/2), x, 4, (2*E^(4*c*(a + b*x))*Cosh[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^2*Sqrt[Cosh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Cosh[a*c + b*c*x]^2)^(5/2), x, 6, (-4*Cosh[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^4*Sqrt[Cosh[a*c + b*c*x]^2]) + (32*Cosh[a*c + b*c*x])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3*Sqrt[Cosh[a*c + b*c*x]^2]) - (8*Cosh[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^2*Sqrt[Cosh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Cosh[a*c + b*c*x]^2)^(7/2), x, 6, (32*Cosh[a*c + b*c*x])/(3*b*c*(1 + E^(2*c*(a + b*x)))^6*Sqrt[Cosh[a*c + b*c*x]^2]) - (192*Cosh[a*c + b*c*x])/(5*b*c*(1 + E^(2*c*(a + b*x)))^5*Sqrt[Cosh[a*c + b*c*x]^2]) + (48*Cosh[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^4*Sqrt[Cosh[a*c + b*c*x]^2]) - (64*Cosh[a*c + b*c*x])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3*Sqrt[Cosh[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x+c x^2) Cosh[d+e x+f x^2]^m*) - - -{E^x*Cosh[a + b*x], x, 1, (E^x*Cosh[a + b*x])/(1 - b^2) - (b*E^x*Sinh[a + b*x])/(1 - b^2)} -{E^x*Cosh[a + c*x^2], x, 6, -((E^(-a + 1/(4*c))*Sqrt[Pi]*Erf[(1 - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])) + (E^(a - 1/(4*c))*Sqrt[Pi]*Erfi[(1 + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} -{E^x*Cosh[a + b*x + c*x^2], x, 6, -((E^(-a + (1 - b)^2/(4*c))*Sqrt[Pi]*Erf[(1 - b - 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])) + (E^(a - (1 + b)^2/(4*c))*Sqrt[Pi]*Erfi[(1 + b + 2*c*x)/(2*Sqrt[c])])/(4*Sqrt[c])} - -{E^(x^2)*Cosh[a + b*x], x, 6, (1/4)*E^(-a - b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(-b + 2*x)] + (1/4)*E^(a - b^2/4)*Sqrt[Pi]*Erfi[(1/2)*(b + 2*x)]} -{E^(x^2)*Cosh[a + c*x^2], x, 4, (Sqrt[Pi]*Erfi[Sqrt[1 - c]*x])/(E^a*(4*Sqrt[1 - c])) + (E^a*Sqrt[Pi]*Erfi[Sqrt[1 + c]*x])/(4*Sqrt[1 + c])} -{E^(x^2)*Cosh[a + b*x + c*x^2], x, 6, -((E^(-a - b^2/(4*(1 - c)))*Sqrt[Pi]*Erfi[(b - 2*(1 - c)*x)/(2*Sqrt[1 - c])])/(4*Sqrt[1 - c])) + (E^(a - b^2/(4*(1 + c)))*Sqrt[Pi]*Erfi[(b + 2*(1 + c)*x)/(2*Sqrt[1 + c])])/(4*Sqrt[1 + c])} - - -{f^(a + b*x)*Cosh[d + f*x^2], x, 8, (E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/4} -{f^(a + b*x)*Cosh[d + f*x^2]^2, x, 9, (E^(-2*d + (b^2*Log[f]^2)/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erf[(4*f*x - b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + (E^(2*d - (b^2*Log[f]^2)/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erfi[(4*f*x + b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Cosh[d + f*x^2]^3, x, 14, (3*E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/16 + (E^(-3*d + (b^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 + (3*E^(d - (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (b^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16} - -{f^(a + b*x)*Cosh[d + e*x + f*x^2], x, 8, (E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/4 + (E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/4} -{f^(a + b*x)*Cosh[d + e*x + f*x^2]^2, x, 9, (E^(-2*d + (2*e - b*Log[f])^2/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erf[(2*e + 4*f*x - b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + (E^(2*d - (2*e + b*Log[f])^2/(8*f))*f^(-1/2 + a)*Sqrt[Pi/2]*Erfi[(2*e + 4*f*x + b*Log[f])/(2*Sqrt[2]*Sqrt[f])])/8 + f^(a + b*x)/(2*b*Log[f])} -{f^(a + b*x)*Cosh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/16 + (E^(-3*d + (3*e - b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(3*e + 6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 + (3*E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (3*e + b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16} - - -{f^(a + c*x^2)*Cosh[d + e*x], x, 8, If[$VersionNumber>=8, -(E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), (E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Cosh[d + e*x]^2, x, 9, If[$VersionNumber>=8, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - e^2/(c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + c*x*Log[f])/(Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + c*x^2)*Cosh[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^(-3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^(-d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*e - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(d - e^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (9*e^2)/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + c*x^2)*Cosh[d + f*x^2], x, 6, (f^a*Sqrt[Pi]*Erf[x*Sqrt[f - c*Log[f]]])/(4*E^d*Sqrt[f - c*Log[f]]) + (E^d*f^a*Sqrt[Pi]*Erfi[x*Sqrt[f + c*Log[f]]])/(4*Sqrt[f + c*Log[f]])} -{f^(a + c*x^2)*Cosh[d + f*x^2]^2, x, 7, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[2*f - c*Log[f]]])/(8*E^(2*d)*Sqrt[2*f - c*Log[f]]) + (E^(2*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + c*x^2)*Cosh[d + f*x^2]^3, x, 10, (3*f^a*Sqrt[Pi]*Erf[x*Sqrt[f - c*Log[f]]])/(16*E^d*Sqrt[f - c*Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[3*f - c*Log[f]]])/(16*E^(3*d)*Sqrt[3*f - c*Log[f]]) + (3*E^d*f^a*Sqrt[Pi]*Erfi[x*Sqrt[f + c*Log[f]]])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[3*f + c*Log[f]]])/(16*Sqrt[3*f + c*Log[f]])} - -{f^(a + c*x^2)*Cosh[d + e*x + f*x^2], x, 8, (E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + c*x^2)*Cosh[d + e*x + f*x^2]^2, x, 9, (f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d + e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + x*(2*f - c*Log[f]))/Sqrt[2*f - c*Log[f]]])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - e^2/(2*f + c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + x*(2*f + c*Log[f]))/Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + c*x^2)*Cosh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + e^2/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) + (E^(-3*d + (9*e^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e + 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) + (3*E^(d - e^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (9*e^2)/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*e + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - - -{f^(a + b*x + c*x^2)*Cosh[d + e*x], x, 8, If[$VersionNumber>=8, -(E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]]), (E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(4*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Cosh[d + e*x]^2, x, 10, If[$VersionNumber>=8, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d - (2*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]]), (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d - (2*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((2*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(8*Sqrt[c]*Sqrt[Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(8*Sqrt[c]*Sqrt[Log[f]])]} -{f^(a + b*x + c*x^2)*Cosh[d + e*x]^3, x, 14, If[$VersionNumber>=8, (-3*E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) - (E^(-3*d - (3*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]), (3*E^(-d - (e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(-3*d - (3*e - b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[-((3*e - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]]))])/(16*Sqrt[c]*Sqrt[Log[f]]) + (3*E^(d - (e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[c]*Sqrt[Log[f]])])/(16*Sqrt[c]*Sqrt[Log[f]])]} - -{f^(a + b*x + c*x^2)*Cosh[d + f*x^2], x, 8, -(E^(-d + (b^2*Log[f]^2)/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - (b^2*Log[f]^2)/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cosh[d + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) - (E^(-2*d + (b^2*Log[f]^2)/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log[f]])])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (b^2*Log[f]^2)/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cosh[d + f*x^2]^3, x, 14, (-3*E^(-d + (b^2*Log[f]^2)/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) - (E^(-3*d + (b^2*Log[f]^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) + (3*E^(d - (b^2*Log[f]^2)/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (b^2*Log[f]^2)/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - -{f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2], x, 8, (E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Log[f] + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(4*Sqrt[f - c*Log[f]]) + (E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^2, x, 10, (f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(4*Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d + (2*e - b*Log[f])^2/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(2*e - b*Log[f] + 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log[f]])])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (2*e + b*Log[f])^2/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(2*e + b*Log[f] + 2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*f + c*Log[f]])} -{f^(a + b*x + c*x^2)*Cosh[d + e*x + f*x^2]^3, x, 14, (3*E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Log[f] + 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*Log[f]])])/(16*Sqrt[f - c*Log[f]]) + (E^(-3*d + (3*e - b*Log[f])^2/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(3*e - b*Log[f] + 2*x*(3*f - c*Log[f]))/(2*Sqrt[3*f - c*Log[f]])])/(16*Sqrt[3*f - c*Log[f]]) + (3*E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f]]) + (E^(3*d - (3*e + b*Log[f])^2/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(3*e + b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sqrt[3*f + c*Log[f]])])/(16*Sqrt[3*f + c*Log[f]])} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving hyperbolic cosines*) - - -{x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]], x, 2, -4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]} -{x/Cosh[x]^(5/2) - x/(3*Sqrt[Cosh[x]]), x, 2, 4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))} -{x/Cosh[x]^(7/2) + (3/5)*x*Sqrt[Cosh[x]], x, 3, 4/(15*Cosh[x]^(3/2)) - (12*Sqrt[Cosh[x]])/5 + (2*x*Sinh[x])/(5*Cosh[x]^(5/2)) + (6*x*Sinh[x])/(5*Sqrt[Cosh[x]])} -{x^2/Cosh[x]^(3/2) + x^2*Sqrt[Cosh[x]], x, 3, -8*x*Sqrt[Cosh[x]] - 16*I*EllipticE[(I*x)/2, 2] + (2*x^2*Sinh[x])/Sqrt[Cosh[x]]} - - -{(x + Cosh[x])^2, x, 6, x/2 + x^3/3 - 2*Cosh[x] + 2*x*Sinh[x] + (1/2)*Cosh[x]*Sinh[x]} -{(x + Cosh[x])^3, x, 9, (3*x^2)/4 + x^4/4 - 6*x*Cosh[x] - (3*Cosh[x]^2)/4 + 7*Sinh[x] + 3*x^2*Sinh[x] + (3/2)*x*Cosh[x]*Sinh[x] + Sinh[x]^3/3} - - -{Cosh[a + b*x]/(c + d*x^2), x, 8, (Cosh[a + (b*Sqrt[-c])/Sqrt[d]]*CoshIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a - (b*Sqrt[-c])/Sqrt[d]]*CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Sinh[a + (b*Sqrt[-c])/Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Sinh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])} -{Cosh[a + b*x]/(c + d*x + e*x^2), x, 8, (Cosh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*CoshIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Cosh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CoshIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] + (Sinh[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d - Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e] - (Sinh[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinhIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*e]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.7 hyper^m (a+b cosh^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.7 hyper^m (a+b cosh^n)^p.m deleted file mode 100644 index 119e8f3..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.2 Hyperbolic cosine/6.2.7 hyper^m (a+b cosh^n)^p.m +++ /dev/null @@ -1,263 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^2)^p when a+b=0*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[x]^4/(a - a*Cosh[x]^2), x, 3, x/(2*a) - (Cosh[x]*Sinh[x])/(2*a)} -{Sinh[x]^3/(a - a*Cosh[x]^2), x, 2, -(Cosh[x]/a)} -{Sinh[x]^2/(a - a*Cosh[x]^2), x, 2, -x/a} -{Csch[x]^2/(a - a*Cosh[x]^2), x, 3, -(Coth[x]/a) + Coth[x]^3/(3*a)} -{Csch[x]^4/(a - a*Cosh[x]^2), x, 3, Coth[x]/a - (2*Coth[x]^3)/(3*a) + Coth[x]^5/(5*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[x]^7/(a + b*Cosh[x]^2), x, 4, -(((a + b)^3*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) + ((a^2 + 3*a*b + 3*b^2)*Cosh[x])/b^3 - ((a + 3*b)*Cosh[x]^3)/(3*b^2) + Cosh[x]^5/(5*b)} -{Sinh[x]^5/(a + b*Cosh[x]^2), x, 4, ((a + b)^2*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - ((a + 2*b)*Cosh[x])/b^2 + Cosh[x]^3/(3*b)} -{Sinh[x]^3/(a + b*Cosh[x]^2), x, 3, -(((a + b)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Cosh[x]/b} -{Sinh[x]^1/(a + b*Cosh[x]^2), x, 2, ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{Csch[x]^1/(a + b*Cosh[x]^2), x, 4, -((Sqrt[b]*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b))) - ArcTanh[Cosh[x]]/(a + b)} -{Csch[x]^3/(a + b*Cosh[x]^2), x, 5, (b^(3/2)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^2) + ((a + 3*b)*ArcTanh[Cosh[x]])/(2*(a + b)^2) - (Coth[x]*Csch[x])/(2*(a + b))} -{Csch[x]^5/(a + b*Cosh[x]^2), x, 6, -((b^(5/2)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3)) - ((3*a^2 + 10*a*b + 15*b^2)*ArcTanh[Cosh[x]])/(8*(a + b)^3) + ((3*a + 7*b)*Coth[x]*Csch[x])/(8*(a + b)^2) - (Coth[x]*Csch[x]^3)/(4*(a + b))} - -{Sinh[x]^6/(a + b*Cosh[x]^2), x, 6, ((8*a^2 + 20*a*b + 15*b^2)*x)/(8*b^3) - ((a + b)^(5/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^3) - ((4*a + 7*b)*Cosh[x]*Sinh[x])/(8*b^2) + (Cosh[x]*Sinh[x]^3)/(4*b)} -{Sinh[x]^4/(a + b*Cosh[x]^2), x, 5, -(((2*a + 3*b)*x)/(2*b^2)) + ((a + b)^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^2) + (Cosh[x]*Sinh[x])/(2*b)} -{Sinh[x]^2/(a + b*Cosh[x]^2), x, 4, x/b - (Sqrt[a + b]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b)} -{Sinh[x]^0/(a + b*Cosh[x]^2), x, 2, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])} -{Csch[x]^4/(a + b*Cosh[x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + ((a + 2*b)*Coth[x])/(a + b)^2 - Coth[x]^3/(3*(a + b))} -{Csch[x]^6/(a + b*Cosh[x]^2), x, 4, -((b^3*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Coth[x])/(a + b)^3 + ((2*a + 3*b)*Coth[x]^3)/(3*(a + b)^2) - Coth[x]^5/(5*(a + b))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^3)^p*) - - -{Sinh[x]/(4 - 3*Cosh[x]^3), x, 7, ArcTan[(1 + 6^(1/3)*Cosh[x])/Sqrt[3]]/(2*2^(1/3)*3^(5/6)) - Log[2^(2/3) - 3^(1/3)*Cosh[x]]/(6*6^(1/3)) + Log[2*2^(1/3) + 2^(2/3)*3^(1/3)*Cosh[x] + 3^(2/3)*Cosh[x]^2]/(12*6^(1/3))} - - -(* ::Section:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cosh[x]^7/(a + b*Cosh[x]^2), x, 4, -((a^3*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b])) + ((a^2 - a*b + b^2)*Sinh[x])/b^3 - ((a - 2*b)*Sinh[x]^3)/(3*b^2) + Sinh[x]^5/(5*b)} -{Cosh[x]^6/(a + b*Cosh[x]^2), x, 6, ((8*a^2 - 4*a*b + 3*b^2)*x)/(8*b^3) - (a^(5/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b^3*Sqrt[a + b]) - ((4*a - 3*b)*Cosh[x]*Sinh[x])/(8*b^2) + (Cosh[x]^3*Sinh[x])/(4*b)} -{Cosh[x]^5/(a + b*Cosh[x]^2), x, 4, (a^2*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]) - ((a - b)*Sinh[x])/b^2 + Sinh[x]^3/(3*b)} -{Cosh[x]^4/(a + b*Cosh[x]^2), x, 5, -(((2*a - b)*x)/(2*b^2)) + (a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b^2*Sqrt[a + b]) + (Cosh[x]*Sinh[x])/(2*b)} -{Cosh[x]^3/(a + b*Cosh[x]^2), x, 3, -((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[x]/b} -{Cosh[x]^2/(a + b*Cosh[x]^2), x, 3, x/b - (Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b*Sqrt[a + b])} -{Cosh[x]^1/(a + b*Cosh[x]^2), x, 2, ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])} -{Cosh[x]^0/(a + b*Cosh[x]^2), x, 2, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])} -{Sech[x]^1/(a + b*Cosh[x]^2), x, 4, ArcTan[Sinh[x]]/a - (Sqrt[b]*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(a*Sqrt[a + b])} -{Sech[x]^2/(a + b*Cosh[x]^2), x, 3, -((b*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(a^(3/2)*Sqrt[a + b])) + Tanh[x]/a} -{Sech[x]^3/(a + b*Cosh[x]^2), x, 5, ((a - 2*b)*ArcTan[Sinh[x]])/(2*a^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]) + (Sech[x]*Tanh[x])/(2*a)} -{Sech[x]^4/(a + b*Cosh[x]^2), x, 4, (b^2*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]) + ((a - b)*Tanh[x])/a^2 - Tanh[x]^3/(3*a)} -{Sech[x]^5/(a + b*Cosh[x]^2), x, 6, ((3*a^2 - 4*a*b + 8*b^2)*ArcTan[Sinh[x]])/(8*a^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]) + ((3*a - 4*b)*Sech[x]*Tanh[x])/(8*a^2) + (Sech[x]^3*Tanh[x])/(4*a)} - - -{1/(a + b*Cosh[x]^2)^2, x, 4, ((2*a + b)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)) - (b*Cosh[x]*Sinh[x])/(2*a*(a + b)*(a + b*Cosh[x]^2))} - - -{1/(a + b*Cosh[x]^2)^3, x, 5, ((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(8*a^(5/2)*(a + b)^(5/2)) - (b*Cosh[x]*Sinh[x])/(4*a*(a + b)*(a + b*Cosh[x]^2)^2) - (3*b*(2*a + b)*Cosh[x]*Sinh[x])/(8*a^2*(a + b)^2*(a + b*Cosh[x]^2))} - - -{1/(1 + Cosh[x]^2), x, 2, ArcTanh[Tanh[x]/Sqrt[2]]/Sqrt[2]} -{1/(1 + Cosh[x]^2)^2, x, 4, (3*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - (Cosh[x]*Sinh[x])/(4*(1 + Cosh[x]^2))} -{1/(1 + Cosh[x]^2)^3, x, 5, (19*ArcTanh[Tanh[x]/Sqrt[2]])/(32*Sqrt[2]) - (Cosh[x]*Sinh[x])/(8*(1 + Cosh[x]^2)^2) - (9*Cosh[x]*Sinh[x])/(32*(1 + Cosh[x]^2))} - -{1/(1 - Cosh[x]^2), x, 3, Coth[x]} -{1/(1 - Cosh[x]^2)^2, x, 3, Coth[x] - Coth[x]^3/3} -{1/(1 - Cosh[x]^2)^3, x, 3, Coth[x] - (2*Coth[x]^3)/3 + Coth[x]^5/5} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Cosh[x]^2], x, 2, -((I*Sqrt[a + b*Cosh[x]^2]*EllipticE[Pi/2 + I*x, -(b/a)])/Sqrt[1 + (b*Cosh[x]^2)/a])} - -{Sqrt[1 + Cosh[x]^2], x, 1, (-I)*EllipticE[Pi/2 + I*x, -1]} -{Sqrt[1 - Cosh[x]^2], x, 3, Coth[x]*Sqrt[-Sinh[x]^2]} -{Sqrt[-1 + Cosh[x]^2], x, 3, Coth[x]*Sqrt[Sinh[x]^2]} -{Sqrt[-1 - Cosh[x]^2], x, 2, -((I*Sqrt[-1 - Cosh[x]^2]*EllipticE[Pi/2 + I*x, -1])/Sqrt[1 + Cosh[x]^2])} - - -{(a + b*Cosh[x]^2)^(3/2), x, 6, -((2*I*(2*a + b)*Sqrt[a + b*Cosh[x]^2]*EllipticE[Pi/2 + I*x, -(b/a)])/(3*Sqrt[1 + (b*Cosh[x]^2)/a])) + (I*a*(a + b)*Sqrt[1 + (b*Cosh[x]^2)/a]*EllipticF[Pi/2 + I*x, -(b/a)])/(3*Sqrt[a + b*Cosh[x]^2]) + (1/3)*b*Cosh[x]*Sqrt[a + b*Cosh[x]^2]*Sinh[x]} - -{(1 + Cosh[x]^2)^(3/2), x, 4, -2*I*EllipticE[Pi/2 + I*x, -1] + (2/3)*I*EllipticF[Pi/2 + I*x, -1] + (1/3)*Cosh[x]*Sqrt[1 + Cosh[x]^2]*Sinh[x]} -{(1 - Cosh[x]^2)^(3/2), x, 4, (2/3)*Coth[x]*Sqrt[-Sinh[x]^2] + (1/3)*Coth[x]*(-Sinh[x]^2)^(3/2)} -{(-1 + Cosh[x]^2)^(3/2), x, 4, (-(2/3))*Coth[x]*Sqrt[Sinh[x]^2] + (1/3)*Coth[x]*(Sinh[x]^2)^(3/2)} -{(-1 - Cosh[x]^2)^(3/2), x, 6, (2*I*Sqrt[-1 - Cosh[x]^2]*EllipticE[Pi/2 + I*x, -1])/Sqrt[1 + Cosh[x]^2] + (2*I*Sqrt[1 + Cosh[x]^2]*EllipticF[Pi/2 + I*x, -1])/(3*Sqrt[-1 - Cosh[x]^2]) - (1/3)*Cosh[x]*Sqrt[-1 - Cosh[x]^2]*Sinh[x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[a + b*Cosh[x]^2], x, 2, -((I*Sqrt[1 + (b*Cosh[x]^2)/a]*EllipticF[Pi/2 + I*x, -(b/a)])/Sqrt[a + b*Cosh[x]^2])} - -{1/Sqrt[1 + Cosh[x]^2], x, 1, (-I)*EllipticF[Pi/2 + I*x, -1]} -{1/Sqrt[1 - Cosh[x]^2], x, 3, -((ArcTanh[Cosh[x]]*Sinh[x])/Sqrt[-Sinh[x]^2])} -{1/Sqrt[-1 + Cosh[x]^2], x, 3, -((ArcTanh[Cosh[x]]*Sinh[x])/Sqrt[Sinh[x]^2])} -{1/Sqrt[-1 - Cosh[x]^2], x, 2, -((I*Sqrt[1 + Cosh[x]^2]*EllipticF[Pi/2 + I*x, -1])/Sqrt[-1 - Cosh[x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^3)^p*) - - -{1/(a + b*Cosh[x]^3), x, 8, (2*ArcTanh[(Sqrt[a^(1/3) - b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - b^(1/3)]*Sqrt[a^(1/3) + b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])} -{1/(a - b*Cosh[x]^3), x, 8, (2*ArcTanh[(Sqrt[a^(1/3) + b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - b^(1/3)]*Sqrt[a^(1/3) + b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + (2*ArcTanh[(Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]*Tanh[x/2])/Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]])/(3*a^(2/3)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])} -{1/(1 + Cosh[x]^3), x, 7, -((2*(-(1/3))^(1/4)*ArcTan[(-1)^(3/4)*3^(1/4)*Tanh[x/2]])/(3*(1 - (-1)^(1/3)))) - (2*(-(1/3))^(1/4)*ArcTanh[(-1)^(3/4)*3^(1/4)*Tanh[x/2]])/(3*(1 + (-1)^(2/3))) + Sinh[x]/(3*(1 + Cosh[x]))} -{1/(1 - Cosh[x]^3), x, 7, -((2*(-1)^(1/4)*ArcTan[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 - (-1)^(2/3)))) - (2*(-1)^(1/4)*ArcTanh[((-1)^(3/4)*Tanh[x/2])/3^(1/4)])/(3^(3/4)*(1 + (-1)^(1/3))) - Sinh[x]/(3*(1 - Cosh[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^4)^p*) - - -{1/(a + b*Cosh[x]^4), x, 10, (Sqrt[Sqrt[a] - Sqrt[a + b]]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sqrt[a + b]]])/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b]) - (Sqrt[Sqrt[a] - Sqrt[a + b]]*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sqrt[a + b]]])/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b]) - (Sqrt[Sqrt[a] + Sqrt[a + b]]*Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[a + b]]*Tanh[x] + Sqrt[a]*Tanh[x]^2])/(4*Sqrt[2]*a^(3/4)*Sqrt[a + b]) + (Sqrt[Sqrt[a] + Sqrt[a + b]]*Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt[Sqrt[a] + Sqrt[a + b]]*Tanh[x] + Sqrt[a]*Tanh[x]^2])/(4*Sqrt[2]*a^(3/4)*Sqrt[a + b]), ((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] - Sqrt[2]*(a + b)^(3/4)*Coth[x])/(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] - Sqrt[a + b])*ArcTan[(a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]] + Sqrt[2]*(a + b)^(3/4)*Coth[x])/(a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])])/(2*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) - ((Sqrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]*Coth[x] + (a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((Sqrt[a] + Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) + Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]*Coth[x] + (a + b)^(3/4)*Coth[x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])} - - -{1/(a - b*Cosh[x]^4), x, 4, ArcTanh[(a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ArcTanh[(a^(1/4)*Tanh[x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])} - - -{1/(1 + Cosh[x]^4), x, 10, -(ArcTan[(Sqrt[1 + Sqrt[2]] - 2*Coth[x])/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]])) + ArcTan[(Sqrt[1 + Sqrt[2]] + 2*Coth[x])/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[1 + Sqrt[2]]) - (1/8)*Sqrt[1 + Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*Coth[x] + 2*Coth[x]^2] + (1/8)*Sqrt[1 + Sqrt[2]]*Log[1 + Sqrt[2*(1 + Sqrt[2])]*Coth[x] + Sqrt[2]*Coth[x]^2]} - - -{1/(1 - Cosh[x]^4), x, 3, ArcTanh[Tanh[x]/Sqrt[2]]/(2*Sqrt[2]) + Coth[x]/2} - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -{1/(a + b*Cosh[x]^5), x, 12, (2*ArcTanh[(Sqrt[a^(1/5) - b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt[a^(1/5) + b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])} -{1/(a + b*Cosh[x]^6), x, 7, ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(1/3)*b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(2/3)*b^(1/3)])} -{1/(a + b*Cosh[x]^8), x, 9, -(ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - b^(1/4)])) - ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) - I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) - I*b^(1/4)]) - ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) + I*b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + I*b^(1/4)]) - ArcTanh[((-a)^(1/8)*Tanh[x])/Sqrt[(-a)^(1/4) + b^(1/4)]]/(4*(-a)^(7/8)*Sqrt[(-a)^(1/4) + b^(1/4)])} - -{1/(a - b*Cosh[x]^5), x, 12, (2*ArcTanh[(Sqrt[a^(1/5) + b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - b^(1/5)]*Sqrt[a^(1/5) + b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(1/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(1/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(2/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(2/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(3/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(3/5)*b^(1/5)]) + (2*ArcTanh[(Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)]*Tanh[x/2])/Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]])/(5*a^(4/5)*Sqrt[a^(1/5) - (-1)^(4/5)*b^(1/5)]*Sqrt[a^(1/5) + (-1)^(4/5)*b^(1/5)])} -{1/(a - b*Cosh[x]^6), x, 7, ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) - b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTanh[(a^(1/6)*Tanh[x])/Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)])} -{1/(a - b*Cosh[x]^8), x, 9, ArcTanh[(a^(1/8)*Tanh[x])/Sqrt[a^(1/4) - b^(1/4)]]/(4*a^(7/8)*Sqrt[a^(1/4) - b^(1/4)]) + ArcTanh[(a^(1/8)*Tanh[x])/Sqrt[a^(1/4) - I*b^(1/4)]]/(4*a^(7/8)*Sqrt[a^(1/4) - I*b^(1/4)]) + ArcTanh[(a^(1/8)*Tanh[x])/Sqrt[a^(1/4) + I*b^(1/4)]]/(4*a^(7/8)*Sqrt[a^(1/4) + I*b^(1/4)]) + ArcTanh[(a^(1/8)*Tanh[x])/Sqrt[a^(1/4) + b^(1/4)]]/(4*a^(7/8)*Sqrt[a^(1/4) + b^(1/4)])} - -{1/(1 + Cosh[x]^5), x, 11, -((2*ArcTan[Tanh[x/2]/Sqrt[-((1 - (-1)^(1/5))/(1 + (-1)^(1/5)))]])/(5*Sqrt[-1 + (-1)^(2/5)])) - (2*Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*ArcTan[Sqrt[-((1 + (-1)^(3/5))/(1 - (-1)^(3/5)))]*Tanh[x/2]])/(5*(1 + (-1)^(3/5))) + (2*ArcTanh[Sqrt[(1 - (-1)^(2/5))/(1 + (-1)^(2/5))]*Tanh[x/2]])/(5*Sqrt[1 - (-1)^(4/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(4/5))/(1 + (-1)^(4/5))]*Tanh[x/2]])/(5*Sqrt[1 + (-1)^(3/5)]) + Sinh[x]/(5*(1 + Cosh[x]))} -{1/(1 + Cosh[x]^6), x, 7, ArcTanh[Tanh[x]/Sqrt[2]]/(3*Sqrt[2]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/3)]]/(3*Sqrt[1 - (-1)^(1/3)]) + ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(2/3)]]/(3*Sqrt[1 + (-1)^(2/3)])} -{1/(1 + Cosh[x]^8), x, 9, ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(1/4)]]/(4*Sqrt[1 - (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(1/4)]]/(4*Sqrt[1 + (-1)^(1/4)]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(3/4)]]/(4*Sqrt[1 - (-1)^(3/4)]) + ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(3/4)]]/(4*Sqrt[1 + (-1)^(3/4)])} - -{1/(1 - Cosh[x]^5), x, 11, -((2*ArcTan[Tanh[x/2]/Sqrt[-((1 - (-1)^(2/5))/(1 + (-1)^(2/5)))]])/(5*Sqrt[-1 + (-1)^(4/5)])) + (2*ArcTan[Sqrt[-((1 + (-1)^(4/5))/(1 - (-1)^(4/5)))]*Tanh[x/2]])/(5*Sqrt[-1 - (-1)^(3/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(1/5))/(1 + (-1)^(1/5))]*Tanh[x/2]])/(5*Sqrt[1 - (-1)^(2/5)]) + (2*ArcTanh[Sqrt[(1 - (-1)^(3/5))/(1 + (-1)^(3/5))]*Tanh[x/2]])/(5*Sqrt[1 + (-1)^(1/5)]) - Sinh[x]/(5*(1 - Cosh[x]))} -{1/(1 - Cosh[x]^6), x, 8, ArcTanh[Tanh[x]/Sqrt[1 + (-1)^(1/3)]]/(3*Sqrt[1 + (-1)^(1/3)]) + ArcTanh[Tanh[x]/Sqrt[1 - (-1)^(2/3)]]/(3*Sqrt[1 - (-1)^(2/3)]) + Coth[x]/3} -{1/(1 - Cosh[x]^8), x, 10, ArcTanh[Tanh[x]/Sqrt[1 - I]]/(4*Sqrt[1 - I]) + ArcTanh[Tanh[x]/Sqrt[1 + I]]/(4*Sqrt[1 + I]) + ArcTanh[Tanh[x]/Sqrt[2]]/(4*Sqrt[2]) + Coth[x]/4} - - -(* ::Title:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^2)^p*) - - -{Tanh[x]/(1 + Cosh[x]^2), x, 4, Log[Cosh[x]] - (1/2)*Log[1 + Cosh[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tanh[x]*Sqrt[a + b*Cosh[x]^2], x, 4, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]] + Sqrt[a + b*Cosh[x]^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[x]/Sqrt[a + b*Cosh[x]^2], x, 3, -(ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]]/Sqrt[a])} -{Tanh[x]/Sqrt[1 + Cosh[x]^2], x, 3, -ArcTanh[Sqrt[1 + Cosh[x]^2]]} -{Tanh[x]/Sqrt[1 - Cosh[x]^2], x, 4, -ArcTanh[Sqrt[-Sinh[x]^2]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^3)^p*) - - -{Tanh[x]^3/(a + b*Cosh[x]^3), x, 11, -((b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Cosh[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))) + Log[Cosh[x]]/a + (b^(2/3)*Log[a^(1/3) + b^(1/3)*Cosh[x]])/(3*a^(5/3)) - (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Cosh[x] + b^(2/3)*Cosh[x]^2])/(6*a^(5/3)) - Log[a + b*Cosh[x]^3]/(3*a) + Sech[x]^2/(2*a)} - - -{Tanh[x]/Sqrt[a + b*Cosh[x]^3], x, 4, -((2*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]])/(3*Sqrt[a]))} -{Tanh[x]*Sqrt[a + b*Cosh[x]^3], x, 5, (-(2/3))*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]] + (2/3)*Sqrt[a + b*Cosh[x]^3]} - - -(* ::Section:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^4)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Cosh[c+d x]^n)^p*) - - -{Tanh[x]/Sqrt[a + b*Cosh[x]^n], x, 4, -((2*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/(Sqrt[a]*n))} - - -{Tanh[x]*Sqrt[a + b*Cosh[x]^n], x, 5, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/n) + (2*Sqrt[a + b*Cosh[x]^n])/n} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.1 (c+d x)^m (a+b tanh)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.1 (c+d x)^m (a+b tanh)^n.m deleted file mode 100644 index 48b5991..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.1 (c+d x)^m (a+b tanh)^n.m +++ /dev/null @@ -1,173 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Tanh[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tanh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Tanh[e+f x]^n*) - - -{(c + d*x)^3*Tanh[e + f*x], x, 6, -((c + d*x)^4/(4*d)) + ((c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + (3*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4)} -{(c + d*x)^2*Tanh[e + f*x], x, 5, -((c + d*x)^3/(3*d)) + ((c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*Tanh[e + f*x], x, 4, -((c + d*x)^2/(2*d)) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2)} -{Tanh[e + f*x]/(c + d*x)^1, x, 0, Unintegrable[Tanh[e + f*x]/(c + d*x), x]} -{Tanh[e + f*x]/(c + d*x)^2, x, 0, Unintegrable[Tanh[e + f*x]/(c + d*x)^2, x]} - - -{(c + d*x)^3*Tanh[e + f*x]^2, x, 7, -((c + d*x)^3/f) + (c + d*x)^4/(4*d) + (3*d*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f^2 + (3*d^2*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^3 - (3*d^3*PolyLog[3, -E^(2*(e + f*x))])/(2*f^4) - ((c + d*x)^3*Tanh[e + f*x])/f} -{(c + d*x)^2*Tanh[e + f*x]^2, x, 6, -((c + d*x)^2/f) + (c + d*x)^3/(3*d) + (2*d*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^2 + (d^2*PolyLog[2, -E^(2*(e + f*x))])/f^3 - ((c + d*x)^2*Tanh[e + f*x])/f} -{(c + d*x)^1*Tanh[e + f*x]^2, x, 3, c*x + (d*x^2)/2 + (d*Log[Cosh[e + f*x]])/f^2 - ((c + d*x)*Tanh[e + f*x])/f} -{Tanh[e + f*x]^2/(c + d*x)^1, x, 0, Unintegrable[Tanh[e + f*x]^2/(c + d*x), x]} -{Tanh[e + f*x]^2/(c + d*x)^2, x, 0, Unintegrable[Tanh[e + f*x]^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*Tanh[e + f*x]^3, x, 13, -((3*d*(c + d*x)^2)/(2*f^2)) + (c + d*x)^3/(2*f) - (c + d*x)^4/(4*d) + (3*d^2*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^3 + ((c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*d^3*PolyLog[2, -E^(2*(e + f*x))])/(2*f^4) + (3*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + (3*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4) - (3*d*(c + d*x)^2*Tanh[e + f*x])/(2*f^2) - ((c + d*x)^3*Tanh[e + f*x]^2)/(2*f)} -{(c + d*x)^2*Tanh[e + f*x]^3, x, 9, (c*d*x)/f + (d^2*x^2)/(2*f) - (c + d*x)^3/(3*d) + ((c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (d^2*Log[Cosh[e + f*x]])/f^3 + (d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (d*(c + d*x)*Tanh[e + f*x])/f^2 - ((c + d*x)^2*Tanh[e + f*x]^2)/(2*f)} -{(c + d*x)^1*Tanh[e + f*x]^3, x, 7, (d*x)/(2*f) - (c + d*x)^2/(2*d) + ((c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (d*Tanh[e + f*x])/(2*f^2) - ((c + d*x)*Tanh[e + f*x]^2)/(2*f)} -{Tanh[e + f*x]^3/(c + d*x)^1, x, 0, Unintegrable[Tanh[e + f*x]^3/(c + d*x), x]} -{Tanh[e + f*x]^3/(c + d*x)^2, x, 0, Unintegrable[Tanh[e + f*x]^3/(c + d*x)^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tanh[e+f x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{(c + d*x)*(b*Tanh[e + f*x])^(5/2), x, 44, (2*b^(5/2)*d*ArcTan[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(3*f^2) - ((-b)^(5/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f - ((-b)^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*f^2) + (2*b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(3*f^2) + (b^(5/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) - (b^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) + ((-b)^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - ((-b)^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - ((-b)^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/(2*f^2) - ((-b)^(5/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (b^(5/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*f^2) - (b^(5/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (b^(5/2)*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + (b^(5/2)*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + ((-b)^(5/2)*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - ((-b)^(5/2)*d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) - ((-b)^(5/2)*d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) + ((-b)^(5/2)*d*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - (4*b^2*d*Sqrt[b*Tanh[e + f*x]])/(3*f^2) - (2*b*(c + d*x)*(b*Tanh[e + f*x])^(3/2))/(3*f)} -{(c + d*x)*(b*Tanh[e + f*x])^(3/2), x, 43, -((2*b^(3/2)*d*ArcTan[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f^2) - ((-b)^(3/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*f^2) + (2*b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f^2 + (b^(3/2)*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) - (b^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) + ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/(2*f^2) - ((-b)^(3/2)*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*f^2) - (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + (b^(3/2)*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + ((-b)^(3/2)*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - ((-b)^(3/2)*d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) - ((-b)^(3/2)*d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) + ((-b)^(3/2)*d*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - (2*b*(c + d*x)*Sqrt[b*Tanh[e + f*x]])/f} -{(c + d*x)*(b*Tanh[e + f*x])^(1/2), x, 37, -((Sqrt[-b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*f^2) + (Sqrt[b]*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f + (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*f^2) - (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^2 + (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^2 - (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) - (Sqrt[b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*f^2) + (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*f^2) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/(2*f^2) - (Sqrt[-b]*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^2 - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*f^2) - (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*f^2) + (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + (Sqrt[b]*d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*f^2) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2) - (Sqrt[-b]*d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) - (Sqrt[-b]*d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*f^2) + (Sqrt[-b]*d*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*f^2)} -{(c + d*x)/(b*Tanh[e + f*x])^(1/2), x, 37, -(((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/(Sqrt[-b]*f)) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*Sqrt[-b]*f^2) + ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(Sqrt[b]*f) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*Sqrt[b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(Sqrt[b]*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(Sqrt[b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*Sqrt[b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*Sqrt[b]*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(Sqrt[-b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*Sqrt[-b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/(2*Sqrt[-b]*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(Sqrt[-b]*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*Sqrt[b]*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*Sqrt[b]*f^2) + (d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*Sqrt[b]*f^2) + (d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*Sqrt[b]*f^2) + (d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*Sqrt[-b]*f^2) - (d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*Sqrt[-b]*f^2) - (d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*Sqrt[-b]*f^2) + (d*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*Sqrt[-b]*f^2)} -{(c + d*x)/(b*Tanh[e + f*x])^(3/2), x, 43, (2*d*ArcTan[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) - ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/((-b)^(3/2)*f) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/(2*(-b)^(3/2)*f^2) + (2*d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) + ((c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(2*b^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*b^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(2*b^(3/2)*f^2) + (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(2*(-b)^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/(2*(-b)^(3/2)*f^2) - (d*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(2*b^(3/2)*f^2) - (d*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(2*b^(3/2)*f^2) + (d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*b^(3/2)*f^2) + (d*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(4*b^(3/2)*f^2) + (d*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*(-b)^(3/2)*f^2) - (d*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*(-b)^(3/2)*f^2) - (d*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/(4*(-b)^(3/2)*f^2) + (d*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/(2*(-b)^(3/2)*f^2) - (2*(c + d*x))/(b*f*Sqrt[b*Tanh[e + f*x]])} - - -{(c + d*x)^2*(b*Tanh[e + f*x])^(3/2), x, 38, (4*(-b)^(3/2)*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/f^2 + (2*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/f^3 + (4*b^(3/2)*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/f^2 + (2*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/f^3 - (4*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^3 + (4*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^3 - (2*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 - (2*b^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 - (4*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 + (2*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/f^3 + (2*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/f^3 + (4*(-b)^(3/2)*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 - (2*b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/f^3 - (2*b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/f^3 + (b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 + (b^(3/2)*d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/f^3 - (2*(-b)^(3/2)*d^2*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 + ((-b)^(3/2)*d^2*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/f^3 + ((-b)^(3/2)*d^2*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/f^3 - (2*(-b)^(3/2)*d^2*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/f^3 - (2*b*(c + d*x)^2*Sqrt[b*Tanh[e + f*x]])/f + b^2*Unintegrable[(c + d*x)^2/Sqrt[b*Tanh[e + f*x]], x]} -{(c + d*x)^2*(b*Tanh[e + f*x])^(1/2), x, 0, Unintegrable[(c + d*x)^2*Sqrt[b*Tanh[e + f*x]], x]} -{(c + d*x)^2/(b*Tanh[e + f*x])^(1/2), x, 0, Unintegrable[(c + d*x)^2/Sqrt[b*Tanh[e + f*x]], x]} -{(c + d*x)^2/(b*Tanh[e + f*x])^(3/2), x, 38, (4*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]])/((-b)^(3/2)*f^2) + (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]^2)/((-b)^(3/2)*f^3) + (4*d*(c + d*x)*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]])/(b^(3/2)*f^2) + (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]^2)/(b^(3/2)*f^3) - (4*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) + (4*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) - (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(b^(3/2)*f^3) - (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[b]]*Log[(2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(b^(3/2)*f^3) - (4*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) + (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[(2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/((-b)^(3/2)*f^3) + (2*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[-((2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])))])/((-b)^(3/2)*f^3) + (4*d^2*ArcTanh[Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]]*Log[2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] - Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - (2*Sqrt[b])/(Sqrt[b] + Sqrt[b*Tanh[e + f*x]])])/(b^(3/2)*f^3) + (d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(b^(3/2)*f^3) + (d^2*PolyLog[2, 1 - (2*Sqrt[b]*(Sqrt[-b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))])/(b^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - 2/(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) + (d^2*PolyLog[2, 1 - (2*(Sqrt[b] - Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] + Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/((-b)^(3/2)*f^3) + (d^2*PolyLog[2, 1 + (2*(Sqrt[b] + Sqrt[b*Tanh[e + f*x]]))/((Sqrt[-b] - Sqrt[b])*(1 - Sqrt[b*Tanh[e + f*x]]/Sqrt[-b]))])/((-b)^(3/2)*f^3) - (2*d^2*PolyLog[2, 1 - 2/(1 + Sqrt[b*Tanh[e + f*x]]/Sqrt[-b])])/((-b)^(3/2)*f^3) - (2*(c + d*x)^2)/(b*f*Sqrt[b*Tanh[e + f*x]]) + Unintegrable[(c + d*x)^2*Sqrt[b*Tanh[e + f*x]], x]/b^2} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{(b*Tanh[e + f*x])^(3/2)/(c + d*x), x, 0, Unintegrable[(b*Tanh[e + f*x])^(3/2)/(c + d*x), x]} -{(b*Tanh[e + f*x])^(1/2)/(c + d*x), x, 0, Unintegrable[Sqrt[b*Tanh[e + f*x]]/(c + d*x), x]} -{1/((c + d*x)*(b*Tanh[e + f*x])^(1/2)), x, 0, Unintegrable[1/((c + d*x)*Sqrt[b*Tanh[e + f*x]]), x]} -{1/((c + d*x)*(b*Tanh[e + f*x])^(3/2)), x, 0, Unintegrable[1/((c + d*x)*(b*Tanh[e + f*x])^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (b Tanh[e+f x])^n with m symbolic*) - - -{x^m*Tanh[a + b*x]^3, x, 0, Unintegrable[x^m*Tanh[a + b*x]^3, x]} -{x^m*Tanh[a + b*x]^2, x, 0, Unintegrable[x^m*Tanh[a + b*x]^2, x]} -{x^m*Tanh[a + b*x]^1, x, 0, Unintegrable[x^m*Tanh[a + b*x], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tanh[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Tanh[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + a*Tanh[e + f*x]), x, 5, (3*d^3*x)/(8*a*f^3) + (3*d*(c + d*x)^2)/(8*a*f^2) + (c + d*x)^3/(4*a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4*(a + a*Tanh[e + f*x])) - (3*d^2*(c + d*x))/(4*f^3*(a + a*Tanh[e + f*x])) - (3*d*(c + d*x)^2)/(4*f^2*(a + a*Tanh[e + f*x])) - (c + d*x)^3/(2*f*(a + a*Tanh[e + f*x]))} -{(c + d*x)^2/(a + a*Tanh[e + f*x]), x, 4, (d^2*x)/(4*a*f^2) + (c + d*x)^2/(4*a*f) + (c + d*x)^3/(6*a*d) - d^2/(4*f^3*(a + a*Tanh[e + f*x])) - (d*(c + d*x))/(2*f^2*(a + a*Tanh[e + f*x])) - (c + d*x)^2/(2*f*(a + a*Tanh[e + f*x]))} -{(c + d*x)^1/(a + a*Tanh[e + f*x]), x, 3, (d*x)/(4*a*f) + (c + d*x)^2/(4*a*d) - d/(4*f^2*(a + a*Tanh[e + f*x])) - (c + d*x)/(2*f*(a + a*Tanh[e + f*x]))} -{1/((c + d*x)^1*(a + a*Tanh[e + f*x])), x, 7, (Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) - (CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a*d) - (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)} -{1/((c + d*x)^2*(a + a*Tanh[e + f*x])), x, 7, -((f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^2)) + (f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a*d^2) + (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - 1/(d*(c + d*x)*(a + a*Tanh[e + f*x]))} -{1/((c + d*x)^3*(a + a*Tanh[e + f*x])), x, 8, -(f/(2*a*d^2*(c + d*x))) + (f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a*d^3) - (f^2*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) + (f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x)^2*(a + a*Tanh[e + f*x])) + f/(d^2*(c + d*x)*(a + a*Tanh[e + f*x]))} - - -{(c + d*x)^3/(a + a*Tanh[e + f*x])^2, x, 10, -((3*d^3*E^(-4*e - 4*f*x))/(512*a^2*f^4)) - (3*d^3*E^(-2*e - 2*f*x))/(16*a^2*f^4) - (3*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(128*a^2*f^3) - (3*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(8*a^2*f^3) - (3*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(64*a^2*f^2) - (3*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(8*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^3)/(16*a^2*f) - (E^(-2*e - 2*f*x)*(c + d*x)^3)/(4*a^2*f) + (c + d*x)^4/(16*a^2*d)} -{(c + d*x)^2/(a + a*Tanh[e + f*x])^2, x, 8, -((d^2*E^(-4*e - 4*f*x))/(128*a^2*f^3)) - (d^2*E^(-2*e - 2*f*x))/(8*a^2*f^3) - (d*E^(-4*e - 4*f*x)*(c + d*x))/(32*a^2*f^2) - (d*E^(-2*e - 2*f*x)*(c + d*x))/(4*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^2)/(16*a^2*f) - (E^(-2*e - 2*f*x)*(c + d*x)^2)/(4*a^2*f) + (c + d*x)^3/(12*a^2*d)} -{(c + d*x)^1/(a + a*Tanh[e + f*x])^2, x, 7, (3*d*x)/(16*a^2*f) - (d*x^2)/(8*a^2) + (x*(c + d*x))/(4*a^2) - d/(16*f^2*(a + a*Tanh[e + f*x])^2) - (c + d*x)/(4*f*(a + a*Tanh[e + f*x])^2) - (3*d)/(16*f^2*(a^2 + a^2*Tanh[e + f*x])) - (c + d*x)/(4*f*(a^2 + a^2*Tanh[e + f*x]))} -{1/((c + d*x)^1*(a + a*Tanh[e + f*x])^2), x, 21, (Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + (Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + Log[c + d*x]/(4*a^2*d) - (CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(4*a^2*d) - (CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a^2*d) - (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) - (Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + (Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d)} -{1/((c + d*x)^2*(a + a*Tanh[e + f*x])^2), x, 24, -(1/(4*a^2*d*(c + d*x))) - Cosh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cosh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) - (f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) + (f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(a^2*d^2) + (f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a^2*d^2) + Sinh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Sinh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + Sinh[4*e + 4*f*x]/(4*a^2*d*(c + d*x)) + (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2)} - - -{(c + d*x)^3/(a + a*Tanh[e + f*x])^3, x, 14, -((d^3*E^(-6*e - 6*f*x))/(1728*a^3*f^4)) - (9*d^3*E^(-4*e - 4*f*x))/(1024*a^3*f^4) - (9*d^3*E^(-2*e - 2*f*x))/(64*a^3*f^4) - (d^2*E^(-6*e - 6*f*x)*(c + d*x))/(288*a^3*f^3) - (9*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(256*a^3*f^3) - (9*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(32*a^3*f^3) - (d*E^(-6*e - 6*f*x)*(c + d*x)^2)/(96*a^3*f^2) - (9*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(128*a^3*f^2) - (9*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(32*a^3*f^2) - (E^(-6*e - 6*f*x)*(c + d*x)^3)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^3)/(32*a^3*f) - (3*E^(-2*e - 2*f*x)*(c + d*x)^3)/(16*a^3*f) + (c + d*x)^4/(32*a^3*d)} -{(c + d*x)^2/(a + a*Tanh[e + f*x])^3, x, 11, -((d^2*E^(-6*e - 6*f*x))/(864*a^3*f^3)) - (3*d^2*E^(-4*e - 4*f*x))/(256*a^3*f^3) - (3*d^2*E^(-2*e - 2*f*x))/(32*a^3*f^3) - (d*E^(-6*e - 6*f*x)*(c + d*x))/(144*a^3*f^2) - (3*d*E^(-4*e - 4*f*x)*(c + d*x))/(64*a^3*f^2) - (3*d*E^(-2*e - 2*f*x)*(c + d*x))/(16*a^3*f^2) - (E^(-6*e - 6*f*x)*(c + d*x)^2)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^2)/(32*a^3*f) - (3*E^(-2*e - 2*f*x)*(c + d*x)^2)/(16*a^3*f) + (c + d*x)^3/(24*a^3*d)} -{(c + d*x)^1/(a + a*Tanh[e + f*x])^3, x, 11, (11*d*x)/(96*a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) - d/(36*f^2*(a + a*Tanh[e + f*x])^3) - (c + d*x)/(6*f*(a + a*Tanh[e + f*x])^3) - (5*d)/(96*a*f^2*(a + a*Tanh[e + f*x])^2) - (c + d*x)/(8*a*f*(a + a*Tanh[e + f*x])^2) - (11*d)/(96*f^2*(a^3 + a^3*Tanh[e + f*x])) - (c + d*x)/(8*f*(a^3 + a^3*Tanh[e + f*x]))} -{1/((c + d*x)^1*(a + a*Tanh[e + f*x])^3), x, 53, (3*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^3*d) - (CoshIntegral[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(8*a^3*d) - (3*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(8*a^3*d) - (3*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(8*a^3*d) - (3*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (3*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (3*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - (Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + (Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)} -{1/((c + d*x)^2*(a + a*Tanh[e + f*x])^3), x, 60, -(1/(8*a^3*d*(c + d*x))) - (9*Cosh[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cosh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - Cosh[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) - (3*Cosh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (3*f*CoshIntegral[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(4*a^3*d^2) + (3*f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(2*a^3*d^2) + (3*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(4*a^3*d^2) + (15*Sinh[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Sinh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) + Sinh[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Sinh[4*e + 4*f*x])/(8*a^3*d*(c + d*x)) + (3*Sinh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) + (3*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*f*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) + (3*f*Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) - (3*f*Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Tanh[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + a*Tanh[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m*(a + a*Tanh[e + f*x])^2, x]} -{(c + d*x)^m*(a + a*Tanh[e + f*x])^1, x, 0, Unintegrable[(c+d x)^m (a+a Tanh[e+f x]),x]} -{(c + d*x)^m/(a + a*Tanh[e + f*x])^1, x, 2, (c + d*x)^(1 + m)/(2*a*d*(1 + m)) - (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a*f))} -{(c + d*x)^m/(a + a*Tanh[e + f*x])^2, x, 4, (c + d*x)^(1 + m)/(4*a^2*d*(1 + m)) - (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a^2*f)) - (4^(-2 - m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a^2*f))} -{(c + d*x)^m/(a + a*Tanh[e + f*x])^3, x, 5, (c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) - (3*2^(-4 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a^3*f)) - (3*2^(-5 - 2*m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a^3*f)) - (2^(-4 - m)*3^(-1 - m)*E^(-6*e + (6*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (6*f*(c + d*x))/d])/(((f*(c + d*x))/d)^m*(a^3*f))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tanh[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Tanh[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Tanh[e + f*x]), x, 8, (a*(c + d*x)^4)/(4*d) - (b*(c + d*x)^4)/(4*d) + (b*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*b*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (3*b*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + (3*b*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4)} -{(c + d*x)^2*(a + b*Tanh[e + f*x]), x, 7, (a*(c + d*x)^3)/(3*d) - (b*(c + d*x)^3)/(3*d) + (b*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (b*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (b*d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Tanh[e + f*x]), x, 6, (a*(c + d*x)^2)/(2*d) - (b*(c + d*x)^2)/(2*d) + (b*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (b*d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2)} -{(a + b*Tanh[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tanh[e + f*x])/(c + d*x), x]} -{(a + b*Tanh[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tanh[e + f*x])/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Tanh[e + f*x])^2, x, 15, -((b^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (a*b*(c + d*x)^4)/(2*d) + (b^2*(c + d*x)^4)/(4*d) + (3*b^2*d*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*b^2*d^2*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (3*a*b*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*PolyLog[3, -E^(2*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/f^3 + (3*a*b*d^3*PolyLog[4, -E^(2*(e + f*x))])/(2*f^4) - (b^2*(c + d*x)^3*Tanh[e + f*x])/f} -{(c + d*x)^2*(a + b*Tanh[e + f*x])^2, x, 13, -((b^2*(c + d*x)^2)/f) + (a^2*(c + d*x)^3)/(3*d) - (2*a*b*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(3*d) + (2*b^2*d*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (b^2*d^2*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (2*a*b*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (a*b*d^2*PolyLog[3, -E^(2*(e + f*x))])/f^3 - (b^2*(c + d*x)^2*Tanh[e + f*x])/f} -{(c + d*x)^1*(a + b*Tanh[e + f*x])^2, x, 9, b^2*c*x + (1/2)*b^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (a*b*(c + d*x)^2)/d + (2*a*b*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (b^2*d*Log[Cosh[e + f*x]])/f^2 + (a*b*d*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (b^2*(c + d*x)*Tanh[e + f*x])/f} -{(a + b*Tanh[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tanh[e + f*x])^2/(c + d*x), x]} -{(a + b*Tanh[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tanh[e + f*x])^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Tanh[e + f*x])^3, x, 28, -((3*b^3*d*(c + d*x)^2)/(2*f^2)) - (3*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) - (3*a^2*b*(c + d*x)^4)/(4*d) + (3*a*b^2*(c + d*x)^4)/(4*d) - (b^3*(c + d*x)^4)/(4*d) + (3*b^3*d^2*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (3*b^3*d^3*PolyLog[2, -E^(2*(e + f*x))])/(2*f^4) + (9*a*b^2*d^2*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (9*a^2*b*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (9*a*b^2*d^3*PolyLog[3, -E^(2*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (3*b^3*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + (9*a^2*b*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4) + (3*b^3*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4) - (3*b^3*d*(c + d*x)^2*Tanh[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Tanh[e + f*x])/f - (b^3*(c + d*x)^3*Tanh[e + f*x]^2)/(2*f)} -{(c + d*x)^2*(a + b*Tanh[e + f*x])^3, x, 22, (b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - (3*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) - (a^2*b*(c + d*x)^3)/d + (a*b^2*(c + d*x)^3)/d - (b^3*(c + d*x)^3)/(3*d) + (6*a*b^2*d*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (b^3*d^2*Log[Cosh[e + f*x]])/f^3 + (3*a*b^2*d^2*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (3*a^2*b*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 + (b^3*d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (b^3*d*(c + d*x)*Tanh[e + f*x])/f^2 - (3*a*b^2*(c + d*x)^2*Tanh[e + f*x])/f - (b^3*(c + d*x)^2*Tanh[e + f*x]^2)/(2*f)} -{(c + d*x)^1*(a + b*Tanh[e + f*x])^3, x, 16, 3*a*b^2*c*x + (b^3*d*x)/(2*f) + (3/2)*a*b^2*d*x^2 + (a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*(c + d*x)^2)/(2*d) - (b^3*(c + d*x)^2)/(2*d) + (3*a^2*b*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 + E^(2*(e + f*x))])/f + (3*a*b^2*d*Log[Cosh[e + f*x]])/f^2 + (3*a^2*b*d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) + (b^3*d*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (b^3*d*Tanh[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Tanh[e + f*x])/f - (b^3*(c + d*x)*Tanh[e + f*x]^2)/(2*f)} -{(a + b*Tanh[e + f*x])^3/(c + d*x)^1, x, 0, Unintegrable[(a + b*Tanh[e + f*x])^3/(c + d*x), x]} -{(a + b*Tanh[e + f*x])^3/(c + d*x)^2, x, 0, Unintegrable[(a + b*Tanh[e + f*x])^3/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Tanh[e + f*x]), x, 6, (c + d*x)^4/(4*(a + b)*d) - (b*(c + d*x)^3*Log[1 + (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (3*b*d*(c + d*x)^2*PolyLog[2, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/(2*(a^2 - b^2)*f^2) + (3*b*d^2*(c + d*x)*PolyLog[3, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/(2*(a^2 - b^2)*f^3) + (3*b*d^3*PolyLog[4, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/(4*(a^2 - b^2)*f^4)} -{(c + d*x)^2/(a + b*Tanh[e + f*x]), x, 5, (c + d*x)^3/(3*(a + b)*d) - (b*(c + d*x)^2*Log[1 + (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (b*d*(c + d*x)*PolyLog[2, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/((a^2 - b^2)*f^2) + (b*d^2*PolyLog[3, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/(2*(a^2 - b^2)*f^3)} -{(c + d*x)^1/(a + b*Tanh[e + f*x]), x, 4, (c + d*x)^2/(2*(a + b)*d) - (b*(c + d*x)*Log[1 + (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (b*d*PolyLog[2, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/(2*(a^2 - b^2)*f^2)} -{1/((c + d*x)^1*(a + b*Tanh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Tanh[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Tanh[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Tanh[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Tanh[e + f*x])^2, x, 28, -((2*b^2*(c + d*x)^3)/((a^2 - b^2)^2*f)) + (2*b^2*(c + d*x)^3)/((a - b)*(a + b)^2*(a - b + (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^4/(4*(a - b)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) - (2*b*(c + d*x)^3*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^3*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f) + (3*b^2*d^2*(c + d*x)*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3) - (3*b*d*(c + d*x)^2*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^2) + (3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^2) - (3*b^2*d^3*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/(2*(a^2 - b^2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^3) - (3*b^2*d^2*(c + d*x)*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3) - (3*b*d^3*PolyLog[4, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/(2*(a - b)^2*(a + b)*f^4) + (3*b^2*d^3*PolyLog[4, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/(2*(a^2 - b^2)^2*f^4)} -{(c + d*x)^2/(a + b*Tanh[e + f*x])^2, x, 24, -((2*b^2*(c + d*x)^2)/((a^2 - b^2)^2*f)) + (2*b^2*(c + d*x)^2)/((a - b)*(a + b)^2*(a - b + (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^3/(3*(a - b)^2*d) + (2*b^2*d*(c + d*x)*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) - (2*b*(c + d*x)^2*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^2*Log[1 + ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f) + (b^2*d^2*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3) - (2*b*d*(c + d*x)*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^2) + (2*b^2*d*(c + d*x)*PolyLog[2, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^2) + (b*d^2*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a - b)^2*(a + b)*f^3) - (b^2*d^2*PolyLog[3, -(((a + b)*E^(2*e + 2*f*x))/(a - b))])/((a^2 - b^2)^2*f^3)} -{(c + d*x)^1/(a + b*Tanh[e + f*x])^2, x, 5, -((c + d*x)^2/(2*(a^2 - b^2)*d)) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - b)*(a + b)^2*d*f^2) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 + (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)^2*f^2) + (a*b*d*PolyLog[2, -((a - b)/(E^(2*(e + f*x))*(a + b)))])/((a^2 - b^2)^2*f^2) + (b*(c + d*x))/((a^2 - b^2)*f*(a + b*Tanh[e + f*x]))} -{1/((c + d*x)^1*(a + b*Tanh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Tanh[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Tanh[e + f*x])^2), x]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.2 Hyperbolic tangent functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.2 Hyperbolic tangent functions.m deleted file mode 100644 index 38defc9..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.2 Hyperbolic tangent functions.m +++ /dev/null @@ -1,510 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands Involving Hyperbolic Tangents*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Tanh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^n*) - - -{Tanh[a + b*x]^6, x, 4, x - Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b) - Tanh[a + b*x]^5/(5*b)} -{Tanh[a + b*x]^5, x, 3, Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b) - Tanh[a + b*x]^4/(4*b)} -{Tanh[a + b*x]^4, x, 3, x - Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b)} -{Tanh[a + b*x]^3, x, 2, Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)} -{Tanh[a + b*x]^2, x, 2, x - Tanh[a + b*x]/b} -{Tanh[a + b*x]^1, x, 1, Log[Cosh[a + b*x]]/b} -{Coth[a + b*x]^1, x, 1, Log[Sinh[a + b*x]]/b} -{Coth[a + b*x]^2, x, 2, x - Coth[a + b*x]/b} -{Coth[a + b*x]^3, x, 2, -(Coth[a + b*x]^2/(2*b)) + Log[Sinh[a + b*x]]/b} -{Coth[a + b*x]^4, x, 3, x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b)} -{Coth[a + b*x]^5, x, 3, -(Coth[a + b*x]^2/(2*b)) - Coth[a + b*x]^4/(4*b) + Log[Sinh[a + b*x]]/b} -{Coth[a + b*x]^6, x, 4, x - Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b) - Coth[a + b*x]^5/(5*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x])^(n/2)*) - - -{(b*Tanh[c + d*x])^(7/2),x, 7, (b^(7/2)*ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d + (b^(7/2)*ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d - (2*b^3*Sqrt[b*Tanh[c + d*x]])/d - (2*b*(b*Tanh[c + d*x])^(5/2))/(5*d)} -{(b*Tanh[c + d*x])^(5/2),x, 6, -((b^(5/2)*ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d) + (b^(5/2)*ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d - (2*b*(b*Tanh[c + d*x])^(3/2))/(3*d)} -{(b*Tanh[c + d*x])^(3/2),x, 6, (b^(3/2)*ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d + (b^(3/2)*ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d - (2*b*Sqrt[b*Tanh[c + d*x]])/d} -{(b*Tanh[c + d*x])^(1/2), x, 5, -((Sqrt[b]*ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d) + (Sqrt[b]*ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]])/d} -{1/(b*Tanh[c + d*x])^(1/2), x, 5, ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(Sqrt[b]*d) + ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(Sqrt[b]*d)} -{1/(b*Tanh[c + d*x])^(3/2), x, 6, -(ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(3/2)*d)) + ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(3/2)*d) - 2/(b*d*Sqrt[b*Tanh[c + d*x]])} -{1/(b*Tanh[c + d*x])^(5/2), x, 6, ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(5/2)*d) + ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(5/2)*d) - 2/(3*b*d*(b*Tanh[c + d*x])^(3/2))} -{1/(b*Tanh[c + d*x])^(7/2), x, 7, -(ArcTan[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(7/2)*d)) + ArcTanh[Sqrt[b*Tanh[c + d*x]]/Sqrt[b]]/(b^(7/2)*d) - 2/(5*b*d*(b*Tanh[c + d*x])^(5/2)) - 2/(b^3*d*Sqrt[b*Tanh[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x])^(n/3)*) - - -{Tanh[8*x]^(1/3), x, 9, (-(1/16))*Sqrt[3]*ArcTan[(1 + 2*Tanh[8*x]^(2/3))/Sqrt[3]] - (1/16)*Log[1 - Tanh[8*x]^(2/3)] + (1/32)*Log[1 + Tanh[8*x]^(2/3) + Tanh[8*x]^(4/3)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x])^n with n symbolic*) - - -{Tanh[a + b*x]^n, x, 2, (Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Tanh[a + b*x]^2]*Tanh[a + b*x]^(1 + n))/(b*(1 + n))} -{(b*Tanh[c + d*x])^n,x, 2, (Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Tanh[c + d*x]^2]*(b*Tanh[c + d*x])^(1 + n))/(b*d*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Tanh[c+d x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x]^2)^n*) - - -{(a*Tanh[x]^2)^(3/2),x, 3, a*Coth[x]*Log[Cosh[x]]*Sqrt[a*Tanh[x]^2] - (1/2)*a*Tanh[x]*Sqrt[a*Tanh[x]^2]} -{Sqrt[a*Tanh[x]^2], x, 2, Coth[x]*Log[Cosh[x]]*Sqrt[a*Tanh[x]^2]} -{1/Sqrt[a*Tanh[x]^2], x, 2, (Log[Sinh[x]]*Tanh[x])/Sqrt[a*Tanh[x]^2]} - - -{(-Tanh[c + d*x]^2)^(5/2), x, 4, (Coth[c + d*x]*Log[Cosh[c + d*x]]*Sqrt[-Tanh[c + d*x]^2])/d - (Tanh[c + d*x]*Sqrt[-Tanh[c + d*x]^2])/(2*d) - (Tanh[c + d*x]^3*Sqrt[-Tanh[c + d*x]^2])/(4*d)} -{(-Tanh[c + d*x]^2)^(3/2), x, 3, -((Coth[c + d*x]*Log[Cosh[c + d*x]]*Sqrt[-Tanh[c + d*x]^2])/d) + (Tanh[c + d*x]*Sqrt[-Tanh[c + d*x]^2])/(2*d)} -{(-Tanh[c + d*x]^2)^(1/2), x, 2, (Coth[c + d*x]*Log[Cosh[c + d*x]]*Sqrt[-Tanh[c + d*x]^2])/d} -{1/(-Tanh[c + d*x]^2)^(1/2), x, 2, (Log[Sinh[c + d*x]]*Tanh[c + d*x])/(d*Sqrt[-Tanh[c + d*x]^2])} -{1/(-Tanh[c + d*x]^2)^(3/2), x, 3, Coth[c + d*x]/(2*d*Sqrt[-Tanh[c + d*x]^2]) - (Log[Sinh[c + d*x]]*Tanh[c + d*x])/(d*Sqrt[-Tanh[c + d*x]^2])} -{1/(-Tanh[c + d*x]^2)^(5/2), x, 4, -(Coth[c + d*x]/(2*d*Sqrt[-Tanh[c + d*x]^2])) - Coth[c + d*x]^3/(4*d*Sqrt[-Tanh[c + d*x]^2]) + (Log[Sinh[c + d*x]]*Tanh[c + d*x])/(d*Sqrt[-Tanh[c + d*x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x]^3)^n*) - - -{Sqrt[Tanh[x]^3], x, 7, -2*Coth[x]*Sqrt[Tanh[x]^3] + (ArcTan[Sqrt[Tanh[x]]]*Sqrt[Tanh[x]^3])/Tanh[x]^(3/2) + (ArcTanh[Sqrt[Tanh[x]]]*Sqrt[Tanh[x]^3])/Tanh[x]^(3/2)} - - -{(a*Tanh[x]^3)^(3/2),x, 8, (-(2/3))*a*Sqrt[a*Tanh[x]^3] - (a*ArcTan[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) + (a*ArcTanh[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) - (2/7)*a*Tanh[x]^2*Sqrt[a*Tanh[x]^3]} -{Sqrt[a*Tanh[x]^3], x, 7, -2*Coth[x]*Sqrt[a*Tanh[x]^3] + (ArcTan[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2) + (ArcTanh[Sqrt[Tanh[x]]]*Sqrt[a*Tanh[x]^3])/Tanh[x]^(3/2)} -{1/Sqrt[a*Tanh[x]^3], x, 7, -((2*Tanh[x])/Sqrt[a*Tanh[x]^3]) - (ArcTan[Sqrt[Tanh[x]]]*Tanh[x]^(3/2))/Sqrt[a*Tanh[x]^3] + (ArcTanh[Sqrt[Tanh[x]]]*Tanh[x]^(3/2))/Sqrt[a*Tanh[x]^3]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x]^4)^n*) - - -{(a*Tanh[x]^4)^(3/2),x, 5, (-a)*Coth[x]*Sqrt[a*Tanh[x]^4] + a*x*Coth[x]^2*Sqrt[a*Tanh[x]^4] - (1/3)*a*Tanh[x]*Sqrt[a*Tanh[x]^4] - (1/5)*a*Tanh[x]^3*Sqrt[a*Tanh[x]^4]} -{Sqrt[a*Tanh[x]^4], x, 3, (-Coth[x])*Sqrt[a*Tanh[x]^4] + x*Coth[x]^2*Sqrt[a*Tanh[x]^4]} -{1/Sqrt[a*Tanh[x]^4], x, 3, -(Tanh[x]/Sqrt[a*Tanh[x]^4]) + (x*Tanh[x]^2)/Sqrt[a*Tanh[x]^4]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Tanh[c+d x]^m)^n*) - - -{(b*Tanh[c + d*x]^m)^n, x, 3, (1/(d*(1 + m*n)))*(Hypergeometric2F1[1, (1/2)*(1 + m*n), (1/2)*(3 + m*n), Tanh[c + d*x]^2]*Tanh[c + d*x]*(b*Tanh[c + d*x]^m)^n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Tanh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Tanh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{(a + a*Tanh[c + d*x])^5, x, 5, 16*a^5*x + (16*a^5*Log[Cosh[c + d*x]])/d - (8*a^5*Tanh[c + d*x])/d - (2*a^2*(a + a*Tanh[c + d*x])^3)/(3*d) - (a*(a + a*Tanh[c + d*x])^4)/(4*d) - (2*a*(a^2 + a^2*Tanh[c + d*x])^2)/d} -{(a + a*Tanh[c + d*x])^4, x, 4, 8*a^4*x + (8*a^4*Log[Cosh[c + d*x]])/d - (4*a^4*Tanh[c + d*x])/d - (a*(a + a*Tanh[c + d*x])^3)/(3*d) - (a^2 + a^2*Tanh[c + d*x])^2/d} -{(a + a*Tanh[c + d*x])^3, x, 3, 4*a^3*x + (4*a^3*Log[Cosh[c + d*x]])/d - (2*a^3*Tanh[c + d*x])/d - (a*(a + a*Tanh[c + d*x])^2)/(2*d)} -{(a + a*Tanh[c + d*x])^2, x, 2, 2*a^2*x + (2*a^2*Log[Cosh[c + d*x]])/d - (a^2*Tanh[c + d*x])/d} -{1/(a + a*Tanh[c + d*x]), x, 2, x/(2*a) - 1/(2*d*(a + a*Tanh[c + d*x]))} -{1/(a + a*Tanh[c + d*x])^2, x, 3, x/(4*a^2) - 1/(4*d*(a + a*Tanh[c + d*x])^2) - 1/(4*d*(a^2 + a^2*Tanh[c + d*x]))} -{1/(a + a*Tanh[c + d*x])^3, x, 4, x/(8*a^3) - 1/(6*d*(a + a*Tanh[c + d*x])^3) - 1/(8*a*d*(a + a*Tanh[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Tanh[c + d*x]))} -{1/(a + a*Tanh[c + d*x])^4, x, 5, x/(16*a^4) - 1/(8*d*(a + a*Tanh[c + d*x])^4) - 1/(12*a*d*(a + a*Tanh[c + d*x])^3) - 1/(16*d*(a^2 + a^2*Tanh[c + d*x])^2) - 1/(16*d*(a^4 + a^4*Tanh[c + d*x]))} -{1/(a + a*Tanh[c + d*x])^5, x, 6, x/(32*a^5) - 1/(10*d*(a + a*Tanh[c + d*x])^5) - 1/(16*a*d*(a + a*Tanh[c + d*x])^4) - 1/(24*a^2*d*(a + a*Tanh[c + d*x])^3) - 1/(32*a*d*(a^2 + a^2*Tanh[c + d*x])^2) - 1/(32*d*(a^5 + a^5*Tanh[c + d*x]))} - - -{(1 + Tanh[x])^(7/2), x, 5, 8*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 8*Sqrt[1 + Tanh[x]] - (4/3)*(1 + Tanh[x])^(3/2) - (2/5)*(1 + Tanh[x])^(5/2)} -{(1 + Tanh[x])^(5/2), x, 4, 4*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 4*Sqrt[1 + Tanh[x]] - (2/3)*(1 + Tanh[x])^(3/2)} -{(1 + Tanh[x])^(3/2), x, 3, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]]} -{(1 + Tanh[x])^(1/2), x, 2, Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]} -{1/(1 + Tanh[x])^(1/2), x, 3, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Tanh[x]]} -{1/(1 + Tanh[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Tanh[x])^(3/2)) - 1/(2*Sqrt[1 + Tanh[x]])} -{1/(1 + Tanh[x])^(5/2), x, 5, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(4*Sqrt[2]) - 1/(5*(1 + Tanh[x])^(5/2)) - 1/(6*(1 + Tanh[x])^(3/2)) - 1/(4*Sqrt[1 + Tanh[x]])} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{(a + b*Tanh[c + d*x])^5, x, 5, a*(a^4 + 10*a^2*b^2 + 5*b^4)*x + (b*(5*a^4 + 10*a^2*b^2 + b^4)*Log[Cosh[c + d*x]])/d - (4*a*b^2*(a^2 + b^2)*Tanh[c + d*x])/d - (b*(3*a^2 + b^2)*(a + b*Tanh[c + d*x])^2)/(2*d) - (2*a*b*(a + b*Tanh[c + d*x])^3)/(3*d) - (b*(a + b*Tanh[c + d*x])^4)/(4*d)} -{(a + b*Tanh[c + d*x])^4, x, 4, (a^4 + 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 + b^2)*Log[Cosh[c + d*x]])/d - (b^2*(3*a^2 + b^2)*Tanh[c + d*x])/d - (a*b*(a + b*Tanh[c + d*x])^2)/d - (b*(a + b*Tanh[c + d*x])^3)/(3*d)} -{(a + b*Tanh[c + d*x])^3, x, 3, a*(a^2 + 3*b^2)*x + (b*(3*a^2 + b^2)*Log[Cosh[c + d*x]])/d - (2*a*b^2*Tanh[c + d*x])/d - (b*(a + b*Tanh[c + d*x])^2)/(2*d)} -{(a + b*Tanh[c + d*x])^2, x, 2, (a^2 + b^2)*x + (2*a*b*Log[Cosh[c + d*x]])/d - (b^2*Tanh[c + d*x])/d} -{1/(a + b*Tanh[c + d*x]),x, 2, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x]])/((a^2 - b^2)*d)} -{1/(a + b*Tanh[c + d*x])^2,x, 3, ((a^2 + b^2)*x)/(a^2 - b^2)^2 - (2*a*b*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x]])/((a^2 - b^2)^2*d) + b/((a^2 - b^2)*d*(a + b*Tanh[c + d*x]))} -{1/(a + b*Tanh[c + d*x])^3,x, 4, (a*(a^2 + 3*b^2)*x)/(a^2 - b^2)^3 - (b*(3*a^2 + b^2)*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x]])/((a^2 - b^2)^3*d) + b/(2*(a^2 - b^2)*d*(a + b*Tanh[c + d*x])^2) + (2*a*b)/((a^2 - b^2)^2*d*(a + b*Tanh[c + d*x]))} -{1/(a + b*Tanh[c + d*x])^4,x, 5, ((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2)^4 - (4*a*b*(a^2 + b^2)*Log[a*Cosh[c + d*x] + b*Sinh[c + d*x]])/((a^2 - b^2)^4*d) + b/(3*(a^2 - b^2)*d*(a + b*Tanh[c + d*x])^3) + (a*b)/((a^2 - b^2)^2*d*(a + b*Tanh[c + d*x])^2) + (b*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(a + b*Tanh[c + d*x]))} - -{1/(4 + 6*Tanh[c + d*x]), x, 2, -(x/5) + (3*Log[2*Cosh[c + d*x] + 3*Sinh[c + d*x]])/(10*d)} -{1/(4 - 6*Tanh[c + d*x]), x, 2, -(x/5) - (3*Log[2*Cosh[c + d*x] - 3*Sinh[c + d*x]])/(10*d)} - - -{Sqrt[a + b*Tanh[c + d*x]], x, 5, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tanh[c + d*x]]/Sqrt[a - b]])/d) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[c + d*x]]/Sqrt[a + b]])/d} -{1/Sqrt[a + b*Tanh[c + d*x]], x, 5, -(ArcTanh[Sqrt[a + b*Tanh[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d)) + ArcTanh[Sqrt[a + b*Tanh[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Tanh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Sinh[x]^4/(1 + Tanh[x]), x, 5, x/16 + 1/(32*(1 - Tanh[x])^2) - 1/(8*(1 - Tanh[x])) - 1/(24*(1 + Tanh[x])^3) + 5/(32*(1 + Tanh[x])^2) - 3/(16*(1 + Tanh[x]))} -{Sinh[x]^3/(1 + Tanh[x]), x, 9, (-(1/3))*Cosh[x]^3 + Cosh[x]^5/5 - Sinh[x]^5/5} -{Sinh[x]^2/(1 + Tanh[x]), x, 5, -(x/8) + 1/(8*(1 - Tanh[x])) - 1/(8*(1 + Tanh[x])^2) + 1/(4*(1 + Tanh[x]))} -{Sinh[x]^1/(1 + Tanh[x]), x, 8, Cosh[x]^3/3 - Sinh[x]^3/3} -{Csch[x]^1/(1 + Tanh[x]), x, 8, -ArcTanh[Cosh[x]] + Cosh[x] - Sinh[x]} -{Csch[x]^2/(1 + Tanh[x]), x, 3, -Coth[x] - Log[Tanh[x]] + Log[1 + Tanh[x]]} -{Csch[x]^3/(1 + Tanh[x]), x, 8, (-(1/2))*ArcTanh[Cosh[x]] + Csch[x] - (1/2)*Coth[x]*Csch[x]} -{Csch[x]^4/(1 + Tanh[x]), x, 4, Coth[x]^2/2 - Coth[x]^3/3} -{Csch[x]^5/(1 + Tanh[x]), x, 9, (1/8)*ArcTanh[Cosh[x]] - (1/8)*Coth[x]*Csch[x] + Csch[x]^3/3 - (1/4)*Coth[x]*Csch[x]^3} -{Csch[x]^6/(1 + Tanh[x]), x, 4, (-(1/2))*Coth[x]^2 + Coth[x]^3/3 + Coth[x]^4/4 - Coth[x]^5/5} -{Csch[x]^7/(1 + Tanh[x]), x, 10, (-(1/16))*ArcTanh[Cosh[x]] + (1/16)*Coth[x]*Csch[x] - (1/24)*Coth[x]*Csch[x]^3 + Csch[x]^5/5 - (1/6)*Coth[x]*Csch[x]^5} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Sinh[x]^4/(a + b*Tanh[x]), x, 5, -((a*(3*a + b)*Log[1 - Tanh[x]])/(16*(a + b)^3)) + (a*(3*a - b)*Log[1 + Tanh[x]])/(16*(a - b)^3) - (a^4*b*Log[a + b*Tanh[x]])/(a^2 - b^2)^3 - (Cosh[x]^4*(b - a*Tanh[x]))/(4*(a^2 - b^2)) + (Cosh[x]^2*(4*b*(2*a^2 - b^2) - a*(5*a^2 - b^2)*Tanh[x]))/(8*(a^2 - b^2)^2)} -{Sinh[x]^3/(a + b*Tanh[x]), x, 10, -((a^3*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - (a*b^2*Cosh[x])/(a^2 - b^2)^2 - (a*Cosh[x])/(a^2 - b^2) + (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a^2*b*Sinh[x])/(a^2 - b^2)^2 - (b*Sinh[x]^3)/(3*(a^2 - b^2))} -{Sinh[x]^2/(a + b*Tanh[x]), x, 4, (a*Log[1 - Tanh[x]])/(4*(a + b)^2) - (a*Log[1 + Tanh[x]])/(4*(a - b)^2) + (a^2*b*Log[a + b*Tanh[x]])/(a^2 - b^2)^2 - (Cosh[x]^2*(b - a*Tanh[x]))/(2*(a^2 - b^2))} -{Sinh[x]^1/(a + b*Tanh[x]), x, 6, (a*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2)} -{Csch[x]^1/(a + b*Tanh[x]), x, 6, -((b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])) - ArcTanh[Cosh[x]]/a} -{Csch[x]^2/(a + b*Tanh[x]), x, 3, -(Coth[x]/a) - (b*Log[Tanh[x]])/a^2 + (b*Log[a + b*Tanh[x]])/a^2} -{Csch[x]^3/(a + b*Tanh[x]), x, 15, (b*Sqrt[a^2 - b^2]*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/a^3 + ArcTanh[Cosh[x]]/(2*a) - (b^2*ArcTanh[Cosh[x]])/a^3 + (b*Csch[x])/a^2 - (Coth[x]*Csch[x])/(2*a)} -{Csch[x]^4/(a + b*Tanh[x]), x, 3, ((a^2 - b^2)*Coth[x])/a^3 + (b*Coth[x]^2)/(2*a^2) - Coth[x]^3/(3*a) + (b*(a^2 - b^2)*Log[Tanh[x]])/a^4 - (b*(a^2 - b^2)*Log[a + b*Tanh[x]])/a^4} -{Csch[x]^5/(a + b*Tanh[x]), x, 29, -((b*ArcTan[Sinh[x]])/a^2) + (b^3*ArcTan[Sinh[x]])/a^4 + (b*(a^2 - b^2)*ArcTan[Sinh[x]])/a^4 - (b*(a^2 - b^2)^(3/2)*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/a^5 - (3*ArcTanh[Cosh[x]])/(8*a) + (3*b^2*ArcTanh[Cosh[x]])/(2*a^3) - (b^4*ArcTanh[Cosh[x]])/a^5 - (b*Csch[x])/a^2 + (3*b^3*Csch[x])/(2*a^4) + (3*Coth[x]*Csch[x])/(8*a) + (b*Csch[x]^3)/(3*a^2) - (Coth[x]*Csch[x]^3)/(4*a) - (3*b^2*Sech[x])/(2*a^3) + (b^4*Sech[x])/a^5 + (b^2*(a^2 - b^2)*Sech[x])/a^5 - (b^2*Csch[x]^2*Sech[x])/(2*a^3) - (b^3*Csch[x]*Sech[x]^2)/(2*a^4) - (b^3*Sech[x]*Tanh[x])/(2*a^4)} -{Csch[x]^6/(a + b*Tanh[x]), x, 3, -(((a^2 - b^2)^2*Coth[x])/a^5) - (b*(2*a^2 - b^2)*Coth[x]^2)/(2*a^4) + ((2*a^2 - b^2)*Coth[x]^3)/(3*a^3) + (b*Coth[x]^4)/(4*a^2) - Coth[x]^5/(5*a) - (b*(a^2 - b^2)^2*Log[Tanh[x]])/a^6 + (b*(a^2 - b^2)^2*Log[a + b*Tanh[x]])/a^6} - - -(* Following hangs Mathematica 6 & 7: *) -{Csch[x]/(I + Tanh[x]), x, 6, I*ArcTanh[Cosh[x]] - (I*ArcTanh[(Cosh[x] + I*Sinh[x])/Sqrt[2]])/Sqrt[2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sech[c+d x]^m (a+b Tanh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Cosh[x]^4/(1 + Tanh[x]), x, 4, (5*x)/16 + 1/(32*(1 - Tanh[x])^2) + 1/(8*(1 - Tanh[x])) - 1/(24*(1 + Tanh[x])^3) - 3/(32*(1 + Tanh[x])^2) - 3/(16*(1 + Tanh[x]))} -{Cosh[x]^3/(1 + Tanh[x]), x, 3, (4*Sinh[x])/5 + (4*Sinh[x]^3)/15 - Cosh[x]^3/(5*(1 + Tanh[x]))} -{Cosh[x]^2/(1 + Tanh[x]), x, 4, (3*x)/8 + 1/(8*(1 - Tanh[x])) - 1/(8*(1 + Tanh[x])^2) - 1/(4*(1 + Tanh[x]))} -{Cosh[x]^1/(1 + Tanh[x]), x, 2, (2*Sinh[x])/3 - Cosh[x]/(3*(1 + Tanh[x]))} -{Sech[x]^1/(1 + Tanh[x]), x, 1, -(Sech[x]/(1 + Tanh[x]))} -{Sech[x]^2/(1 + Tanh[x]), x, 2, Log[1 + Tanh[x]]} -{Sech[x]^3/(1 + Tanh[x]), x, 2, ArcTan[Sinh[x]] + Sech[x]} -{Sech[x]^4/(1 + Tanh[x]), x, 2, Tanh[x] - Tanh[x]^2/2} -{Sech[x]^5/(1 + Tanh[x]), x, 3, (1/2)*ArcTan[Sinh[x]] + Sech[x]^3/3 + (1/2)*Sech[x]*Tanh[x]} -{Sech[x]^6/(1 + Tanh[x]), x, 3, (-(2/3))*(1 - Tanh[x])^3 + (1/4)*(1 - Tanh[x])^4} -{Sech[x]^7/(1 + Tanh[x]), x, 4, (3/8)*ArcTan[Sinh[x]] + Sech[x]^5/5 + (3/8)*Sech[x]*Tanh[x] + (1/4)*Sech[x]^3*Tanh[x]} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Sech[x]^8/(a + b*Tanh[x]), x, 3, -(((a^2 - b^2)^3*Log[a + b*Tanh[x]])/b^7) + (a*(a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[x])/b^6 - ((a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[x]^2)/(2*b^5) + (a*(a^2 - 3*b^2)*Tanh[x]^3)/(3*b^4) - ((a^2 - 3*b^2)*Tanh[x]^4)/(4*b^3) + (a*Tanh[x]^5)/(5*b^2) - Tanh[x]^6/(6*b)} -{Sech[x]^6/(a + b*Tanh[x]), x, 3, ((a^2 - b^2)^2*Log[a + b*Tanh[x]])/b^5 - (a*(a^2 - 2*b^2)*Tanh[x])/b^4 + ((a^2 - 2*b^2)*Tanh[x]^2)/(2*b^3) - (a*Tanh[x]^3)/(3*b^2) + Tanh[x]^4/(4*b)} -{Sech[x]^4/(a + b*Tanh[x]), x, 3, -(((a^2 - b^2)*Log[a + b*Tanh[x]])/b^3) + (a*Tanh[x])/b^2 - Tanh[x]^2/(2*b)} -{Sech[x]^2/(a + b*Tanh[x]), x, 2, Log[a + b*Tanh[x]]/b} -{Sech[x]^0/(a + b*Tanh[x]), x, 2, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{Cosh[x]^2/(a + b*Tanh[x]), x, 4, -(((a + 2*b)*Log[1 - Tanh[x]])/(4*(a + b)^2)) + ((a - 2*b)*Log[1 + Tanh[x]])/(4*(a - b)^2) + (b^3*Log[a + b*Tanh[x]])/(a^2 - b^2)^2 - (Cosh[x]^2*(b - a*Tanh[x]))/(2*(a^2 - b^2))} -{Cosh[x]^4/(a + b*Tanh[x]), x, 5, -(((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Tanh[x]])/(16*(a + b)^3)) + ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Tanh[x]])/(16*(a - b)^3) - (b^5*Log[a + b*Tanh[x]])/(a^2 - b^2)^3 - (Cosh[x]^4*(b - a*Tanh[x]))/(4*(a^2 - b^2)) + (Cosh[x]^2*(4*b^3 - a*(7 - (3*a^2)/b^2)*b^2*Tanh[x]))/(8*(a^2 - b^2)^2)} - -{Sech[x]^7/(a + b*Tanh[x]), x, 14, (a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(8*b^6) - ((a^2 - b^2)^(5/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^6 + ((a^2 - b^2)^2*Sech[x])/b^5 - ((a^2 - b^2)*Sech[x]^3)/(3*b^3) + Sech[x]^5/(5*b) - (a*(4*a^2 - 7*b^2)*Sech[x]*Tanh[x])/(8*b^4) + (a*Sech[x]^3*Tanh[x])/(4*b^2), (3*a*ArcTan[Sinh[x]])/(8*b^2) - (a*(a^2 - b^2)*ArcTan[Sinh[x]])/(2*b^4) + (a*(a^2 - b^2)^2*ArcTan[Sinh[x]])/b^6 - ((a^2 - b^2)^(5/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^6 + ((a^2 - b^2)^2*Sech[x])/b^5 - ((a^2 - b^2)*Sech[x]^3)/(3*b^3) + Sech[x]^5/(5*b) + (3*a*Sech[x]*Tanh[x])/(8*b^2) - (a*(a^2 - b^2)*Sech[x]*Tanh[x])/(2*b^4) + (a*Sech[x]^3*Tanh[x])/(4*b^2)} -{Sech[x]^5/(a + b*Tanh[x]), x, 9, -((a*(2*a^2 - 3*b^2)*ArcTan[Sinh[x]])/(2*b^4)) + ((a^2 - b^2)^(3/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^4 - ((a^2 - b^2)*Sech[x])/b^3 + Sech[x]^3/(3*b) + (a*Sech[x]*Tanh[x])/(2*b^2), (a*ArcTan[Sinh[x]])/(2*b^2) - (a*(a^2 - b^2)*ArcTan[Sinh[x]])/b^4 + ((a^2 - b^2)^(3/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^4 - ((a^2 - b^2)*Sech[x])/b^3 + Sech[x]^3/(3*b) + (a*Sech[x]*Tanh[x])/(2*b^2)} -{Sech[x]^3/(a + b*Tanh[x]), x, 5, (a*ArcTan[Sinh[x]])/b^2 - (Sqrt[a^2 - b^2]*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^2 + Sech[x]/b} -{Sech[x]^1/(a + b*Tanh[x]), x, 2, ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]]/Sqrt[a^2 - b^2]} -{Cosh[x]^1/(a + b*Tanh[x]), x, 5, -((b^2*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)} -{Cosh[x]^3/(a + b*Tanh[x]), x, 9, (b^4*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b^3*Cosh[x])/(a^2 - b^2)^2 - (b*Cosh[x]^3)/(3*(a^2 - b^2)) - (a*b^2*Sinh[x])/(a^2 - b^2)^2 + (a*Sinh[x])/(a^2 - b^2) + (a*Sinh[x]^3)/(3*(a^2 - b^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Tanh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Tanh[x]^5/(1 + Tanh[x]), x, 5, (5*x)/2 - 2*Log[Cosh[x]] - (5*Tanh[x])/2 + Tanh[x]^2 - (5*Tanh[x]^3)/6 + Tanh[x]^4/(2*(1 + Tanh[x]))} -{Tanh[x]^4/(1 + Tanh[x]), x, 4, -((3*x)/2) + 2*Log[Cosh[x]] + (3*Tanh[x])/2 - Tanh[x]^2 + Tanh[x]^3/(2*(1 + Tanh[x]))} -{Tanh[x]^3/(1 + Tanh[x]), x, 3, (3*x)/2 - Log[Cosh[x]] - (3*Tanh[x])/2 + Tanh[x]^2/(2*(1 + Tanh[x]))} -{Tanh[x]^2/(1 + Tanh[x]), x, 3, -(x/2) + Log[Cosh[x]] - 1/(2*(1 + Tanh[x]))} -{Tanh[x]^1/(1 + Tanh[x]), x, 2, x/2 + 1/(2*(1 + Tanh[x]))} -{Tanh[x]^0/(1 + Tanh[x]), x, 2, x/2 - 1/(2*(1 + Tanh[x]))} -{Coth[x]^1/(1 + Tanh[x]), x, 4, -(x/2) + Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))} -{Coth[x]^2/(1 + Tanh[x]), x, 4, (3*x)/2 - (3*Coth[x])/2 - Log[Sinh[x]] + Coth[x]/(2*(1 + Tanh[x]))} -{Coth[x]^3/(1 + Tanh[x]), x, 5, -((3*x)/2) + (3*Coth[x])/2 - Coth[x]^2 + 2*Log[Sinh[x]] + Coth[x]^2/(2*(1 + Tanh[x]))} -{Coth[x]^4/(1 + Tanh[x]), x, 6, (5*x)/2 - (5*Coth[x])/2 + Coth[x]^2 - (5*Coth[x]^3)/6 - 2*Log[Sinh[x]] + Coth[x]^3/(2*(1 + Tanh[x]))} - - -{Tanh[x]*(1 + Tanh[x])^(3/2), x, 4, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]] - (2/3)*(1 + Tanh[x])^(3/2)} -{Tanh[x]*Sqrt[1 + Tanh[x]], x, 3, Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]]} -{Tanh[x]/Sqrt[1 + Tanh[x]], x, 3, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] + 1/Sqrt[1 + Tanh[x]]} -{Tanh[x]/(1 + Tanh[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(2*Sqrt[2]) + 1/(3*(1 + Tanh[x])^(3/2)) - 1/(2*Sqrt[1 + Tanh[x]])} - -{Tanh[x]^2*(1 + Tanh[x])^(3/2), x, 4, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]] - (2/5)*(1 + Tanh[x])^(5/2)} -{Tanh[x]^2*Sqrt[1 + Tanh[x]], x, 3, Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - (2/3)*(1 + Tanh[x])^(3/2)} -{Tanh[x]^2/Sqrt[1 + Tanh[x]], x, 4, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Tanh[x]] - 2*Sqrt[1 + Tanh[x]]} -{Tanh[x]^2/(1 + Tanh[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Tanh[x])^(3/2)) + 3/(2*Sqrt[1 + Tanh[x]])} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Tanh[x]^5/(a + b*Tanh[x]), x, 7, -((b*x)/(a^2 - b^2)) + (a*Log[Cosh[x]])/(a^2 - b^2) + (a^5*Log[a + b*Tanh[x]])/(b^4*(a^2 - b^2)) - ((a^2 + b^2)*Tanh[x])/b^3 + (a*Tanh[x]^2)/(2*b^2) - Tanh[x]^3/(3*b)} -{Tanh[x]^4/(a + b*Tanh[x]), x, 6, (a*x)/(a^2 - b^2) - (b*Log[Cosh[x]])/(a^2 - b^2) - (a^4*Log[a + b*Tanh[x]])/(b^3*(a^2 - b^2)) + (a*Tanh[x])/b^2 - Tanh[x]^2/(2*b)} -{Tanh[x]^3/(a + b*Tanh[x]), x, 5, -((b*x)/(a^2 - b^2)) + (a*Log[Cosh[x]])/(a^2 - b^2) + (a^3*Log[a + b*Tanh[x]])/(b^2*(a^2 - b^2)) - Tanh[x]/b} -{Tanh[x]^2/(a + b*Tanh[x]), x, 4, -((a*x)/b^2) + (a^3*x)/(b^2*(a^2 - b^2)) + Log[Cosh[x]]/b - (a^2*Log[a*Cosh[x] + b*Sinh[x]])/(b*(a^2 - b^2))} -{Tanh[x]^1/(a + b*Tanh[x]), x, 2, -((b*x)/(a^2 - b^2)) + (a*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{Tanh[x]^0/(a + b*Tanh[x]), x, 2, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{Coth[x]^1/(a + b*Tanh[x]), x, 3, -((b*x)/(a^2 - b^2)) + Log[Sinh[x]]/a + (b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a*(a^2 - b^2))} -{Coth[x]^2/(a + b*Tanh[x]), x, 4, (a*x)/(a^2 - b^2) - Coth[x]/a - (b*Log[Sinh[x]])/a^2 - (b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2*(a^2 - b^2))} -{Coth[x]^3/(a + b*Tanh[x]), x, 5, -((b*x)/(a^2 - b^2)) + (b*Coth[x])/a^2 - Coth[x]^2/(2*a) + ((a^2 + b^2)*Log[Sinh[x]])/a^3 + (b^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^3*(a^2 - b^2))} -{Coth[x]^4/(a + b*Tanh[x]), x, 6, (a*x)/(a^2 - b^2) - ((a^2 + b^2)*Coth[x])/a^3 + (b*Coth[x]^2)/(2*a^2) - Coth[x]^3/(3*a) - (b*(a^2 + b^2)*Log[Sinh[x]])/a^4 - (b^5*Log[a*Cosh[x] + b*Sinh[x]])/(a^4*(a^2 - b^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Sech[c+d x]^2 (a+b Tanh[c+d x]^n)^p*) - - -{x*Sech[x]^2/(a + b*Tanh[x])^2, x, 3, (a*x)/(b*(a^2 - b^2)) - Log[a*Cosh[x] + b*Sinh[x]]/(a^2 - b^2) - x/(b*(a + b*Tanh[x]))} - - -{x*Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 9, (x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2)} -{x^2*Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 11, (x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + (x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))])/(2*Sqrt[-a]*Sqrt[b]*d^2) - (x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))])/(2*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^3) + PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Tanh[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tanh[a+b Log[x]]^p*) - - -{x^3*Tanh[a + 2*Log[x]], x, 4, x^4/4 - Log[1 + E^(2*a)*x^4]/(2*E^(2*a))} -{x^2*Tanh[a + 2*Log[x]], x, 11, x^3/3 + ArcTan[1 - Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^((3*a)/2)) - ArcTan[1 + Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^((3*a)/2)) - Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2]/(2*Sqrt[2]*E^((3*a)/2)) + Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2]/(2*Sqrt[2]*E^((3*a)/2))} -{x^1*Tanh[a + 2*Log[x]], x, 4, x^2/2 - ArcTan[E^a*x^2]/E^a} -{x^0*Tanh[a + 2*Log[x]], x, 11, x + ArcTan[1 - Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^(a/2)) - ArcTan[1 + Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^(a/2)) + Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2]/(2*Sqrt[2]*E^(a/2)) - Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2]/(2*Sqrt[2]*E^(a/2))} -{Tanh[a + 2*Log[x]]/x^1, x, 2, Log[Cosh[a + 2*Log[x]]]/2} -{Tanh[a + 2*Log[x]]/x^2, x, 11, x^(-1) - (E^(a/2)*ArcTan[1 - Sqrt[2]*E^(a/2)*x])/Sqrt[2] + (E^(a/2)*ArcTan[1 + Sqrt[2]*E^(a/2)*x])/Sqrt[2] + (E^(a/2)*Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2])/(2*Sqrt[2]) - (E^(a/2)*Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2])/(2*Sqrt[2])} -{Tanh[a + 2*Log[x]]/x^3, x, 4, 1/(2*x^2) + E^a*ArcTan[E^a*x^2]} - - -{x^3*Tanh[a + 2*Log[x]]^2, x, 4, x^4/4 - 1/(E^(2*a)*(1 + E^(2*a)*x^4)) - Log[1 + E^(2*a)*x^4]/E^(2*a)} -{x^2*Tanh[a + 2*Log[x]]^2, x, 12, x^3/3 + x^3/(1 + E^(2*a)*x^4) + (3*ArcTan[1 - Sqrt[2]*E^(a/2)*x])/(E^((3*a)/2)*(2*Sqrt[2])) - (3*ArcTan[1 + Sqrt[2]*E^(a/2)*x])/(E^((3*a)/2)*(2*Sqrt[2])) - (3*Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2])/(E^((3*a)/2)*(4*Sqrt[2])) + (3*Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2])/(E^((3*a)/2)*(4*Sqrt[2]))} -{x^1*Tanh[a + 2*Log[x]]^2, x, 5, x^2/2 + x^2/(1 + E^(2*a)*x^4) - ArcTan[E^a*x^2]/E^a} -{x^0*Tanh[a + 2*Log[x]]^2, x, 13, x + x/(1 + E^(2*a)*x^4) + ArcTan[1 - Sqrt[2]*E^(a/2)*x]/(E^(a/2)*(2*Sqrt[2])) - ArcTan[1 + Sqrt[2]*E^(a/2)*x]/(E^(a/2)*(2*Sqrt[2])) + Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2]/(E^(a/2)*(4*Sqrt[2])) - Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2]/(E^(a/2)*(4*Sqrt[2]))} -{Tanh[a + 2*Log[x]]^2/x^1, x, 3, Log[x] - Tanh[a + 2*Log[x]]/2} -{Tanh[a + 2*Log[x]]^2/x^2, x, 12, -(1/(x*(1 + E^(2*a)*x^4))) - (2*E^(2*a)*x^3)/(1 + E^(2*a)*x^4) + (E^(a/2)*ArcTan[1 - Sqrt[2]*E^(a/2)*x])/(2*Sqrt[2]) - (E^(a/2)*ArcTan[1 + Sqrt[2]*E^(a/2)*x])/(2*Sqrt[2]) - (E^(a/2)*Log[1 - Sqrt[2]*E^(a/2)*x + E^a*x^2])/(4*Sqrt[2]) + (E^(a/2)*Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2])/(4*Sqrt[2])} -{Tanh[a + 2*Log[x]]^2/x^3, x, 5, -(1/(2*x^2*(1 + E^(2*a)*x^4))) - (3*E^(2*a)*x^2)/(2*(1 + E^(2*a)*x^4)) - E^a*ArcTan[E^a*x^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tanh[a+b Log[x]]^p with m symbolic*) - - -{(e*x)^m*Tanh[a + 2*Log[x]]^1, x, 3, (e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(E^(2*a)*x^4)])/(e*(1 + m))} -{(e*x)^m*Tanh[a + 2*Log[x]]^2, x, 4, (e*x)^(1 + m)/(e*(1 + m)) + (e*x)^(1 + m)/(e*(1 + E^(2*a)*x^4)) - ((e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-E^(2*a))*x^4])/e} -{(e*x)^m*Tanh[a + 2*Log[x]]^3, x, 5, ((3 + m)*(5 + m)*(e*x)^(1 + m))/(8*e*(1 + m)) - ((e*x)^(1 + m)*(1 - E^(2*a)*x^4)^2)/(4*e*(1 + E^(2*a)*x^4)^2) - ((e*x)^(1 + m)*(E^(2*a)*(3 - m) + E^(4*a)*(5 + m)*x^4))/(E^(2*a)*(8*e*(1 + E^(2*a)*x^4))) - ((9 + 2*m + m^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-E^(2*a))*x^4])/(4*e*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tanh[a+b Log[x]]^p with p symbolic*) -(**) - - -{Tanh[a + b*Log[x]]^p, x, 3, (x*(-1 + E^(2*a)*x^(2*b))^p*AppellF1[1/(2*b), -p, p, (1/2)*(2 + 1/b), E^(2*a)*x^(2*b), (-E^(2*a))*x^(2*b)])/(1 - E^(2*a)*x^(2*b))^p} -{(e*x)^m*Tanh[a + b*Log[x]]^p, x, 3, ((e*x)^(1 + m)*(-1 + E^(2*a)*x^(2*b))^p*AppellF1[(1 + m)/(2*b), -p, p, 1 + (1 + m)/(2*b), E^(2*a)*x^(2*b), (-E^(2*a))*x^(2*b)])/((1 - E^(2*a)*x^(2*b))^p*(e*(1 + m)))} - - -{Tanh[a + 1/2*Log[x]]^p, x, 2, ((-1 + E^(2*a)*x)^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 - E^(2*a)*x)])/(2^p*E^(2*a)*(1 + p))} -{Tanh[a + 1/4*Log[x]]^p, x, 4, ((-1 + E^(2*a)*Sqrt[x])^(1 + p)*(1 + E^(2*a)*Sqrt[x])^(1 - p))/E^(4*a) - (2^(1 - p)*p*(-1 + E^(2*a)*Sqrt[x])^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 - E^(2*a)*Sqrt[x])])/(E^(4*a)*(1 + p))} -{Tanh[a + 1/6*Log[x]]^p, x, 5, (-E^(-6*a))*p*(-1 + E^(2*a)*x^(1/3))^(1 + p)*(1 + E^(2*a)*x^(1/3))^(1 - p) + ((-1 + E^(2*a)*x^(1/3))^(1 + p)*(1 + E^(2*a)*x^(1/3))^(1 - p)*x^(1/3))/E^(4*a) + ((1 + 2*p^2)*(-1 + E^(2*a)*x^(1/3))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 - E^(2*a)*x^(1/3))])/(2^p*E^(6*a)*(1 + p))} -{Tanh[a + 1/8*Log[x]]^p, x, 5, ((1/3)*(-1 + E^(2*a)*x^(1/4))^(1 + p)*(1 + E^(2*a)*x^(1/4))^(1 - p)*(E^(4*a)*(3 + 2*p^2) - 2*E^(6*a)*p*x^(1/4)))/E^(12*a) + ((-1 + E^(2*a)*x^(1/4))^(1 + p)*(1 + E^(2*a)*x^(1/4))^(1 - p)*Sqrt[x])/E^(4*a) - (2^(2 - p)*p*(2 + p^2)*(-1 + E^(2*a)*x^(1/4))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 - E^(2*a)*x^(1/4))])/(E^(8*a)*(3*(1 + p)))} - - -{Tanh[a + 1*Log[x]]^p, x, 3, (x*(-1 + E^(2*a)*x^2)^p*AppellF1[1/2, -p, p, 3/2, E^(2*a)*x^2, (-E^(2*a))*x^2])/(1 - E^(2*a)*x^2)^p} -{Tanh[a + 2*Log[x]]^p, x, 3, (x*(-1 + E^(2*a)*x^4)^p*AppellF1[1/4, -p, p, 5/4, E^(2*a)*x^4, (-E^(2*a))*x^4])/(1 - E^(2*a)*x^4)^p} -{Tanh[a + 3*Log[x]]^p, x, 3, (x*(-1 + E^(2*a)*x^6)^p*AppellF1[1/6, -p, p, 7/6, E^(2*a)*x^6, (-E^(2*a))*x^6])/(1 - E^(2*a)*x^6)^p} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tanh[a+b Log[c x^n]]^p*) - - -{x^3*Tanh[d*(a + b*Log[c*x^n])], x, 4, x^4/4 - (x^4*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/2} -{x^2*Tanh[d*(a + b*Log[c*x^n])], x, 4, x^3/3 - (2*x^3*Hypergeometric2F1[1, 3/(2*b*d*n), 1 + 3/(2*b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/3} -{x^1*Tanh[d*(a + b*Log[c*x^n])], x, 4, x^2/2 - x^2*Hypergeometric2F1[1, 1/(b*d*n), 1 + 1/(b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))]} -{x^0*Tanh[d*(a + b*Log[c*x^n])], x, 4, x - 2*x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))]} -{Tanh[d*(a + b*Log[c*x^n])]/x^1, x, 2, Log[Cosh[a*d + b*d*Log[c*x^n]]]/(b*d*n)} -{Tanh[d*(a + b*Log[c*x^n])]/x^2, x, 4, -(1/x) + (2*Hypergeometric2F1[1, -(1/(2*b*d*n)), 1 - 1/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/x} -{Tanh[d*(a + b*Log[c*x^n])]/x^3, x, 4, -1/(2*x^2) + Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))]/x^2} - - -{x^3*Tanh[d*(a + b*Log[c*x^n])]^2, x, 5, (1/4)*(1 + 4/(b*d*n))*x^4 + (x^4*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^4*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^2*Tanh[d*(a + b*Log[c*x^n])]^2, x, 5, (1/3)*(1 + 3/(b*d*n))*x^3 + (x^3*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^3*Hypergeometric2F1[1, 3/(2*b*d*n), 1 + 3/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^1*Tanh[d*(a + b*Log[c*x^n])]^2, x, 5, (1/2)*(1 + 2/(b*d*n))*x^2 + (x^2*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^2*Hypergeometric2F1[1, 1/(b*d*n), 1 + 1/(b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^0*Tanh[d*(a + b*Log[c*x^n])]^2, x, 5, (1 + 1/(b*d*n))*x + (x*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n)} -{Tanh[d*(a + b*Log[c*x^n])]^2/x^1, x, 3, Log[x] - Tanh[a*d + b*d*Log[c*x^n]]/(b*d*n)} -{Tanh[d*(a + b*Log[c*x^n])]^2/x^2, x, 5, -((1 - 1/(b*d*n))/x) + (1 - E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*Hypergeometric2F1[1, -(1/(2*b*d*n)), 1 - 1/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n*x)} -{Tanh[d*(a + b*Log[c*x^n])]^2/x^3, x, 5, (2 - b*d*n)/(2*b*d*n*x^2) + (1 - E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x^2*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*n*x^2)} - - -{Tanh[a + b*Log[c*x^n]]^3/x, x, 3, Log[Cosh[a + b*Log[c*x^n]]]/(b*n) - Tanh[a + b*Log[c*x^n]]^2/(2*b*n)} -{Tanh[a + b*Log[c*x^n]]^4/x, x, 4, Log[x] - Tanh[a + b*Log[c*x^n]]/(b*n) - Tanh[a + b*Log[c*x^n]]^3/(3*b*n)} -{Tanh[a + b*Log[c*x^n]]^5/x, x, 4, Log[Cosh[a + b*Log[c*x^n]]]/(b*n) - Tanh[a + b*Log[c*x^n]]^2/(2*b*n) - Tanh[a + b*Log[c*x^n]]^4/(4*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tanh[a+b Log[c x^n]]^p with m symbolic*) - - -{(e*x)^m*Tanh[d*(a + b*Log[c*x^n])]^1, x, 4, (e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/(e*(1 + m))} -{(e*x)^m*Tanh[d*(a + b*Log[c*x^n])]^2, x, 5, ((1 + m + b*d*n)*(e*x)^(1 + m))/(b*d*e*(1 + m)*n) + ((e*x)^(1 + m)*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*e*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b*d*e*n)} -{(e*x)^m*Tanh[d*(a + b*Log[c*x^n])]^3, x, 6, ((1 + m + b*d*n)*(1 + m + 2*b*d*n)*(e*x)^(1 + m))/(2*b^2*d^2*e*(1 + m)*n^2) - ((e*x)^(1 + m)*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^2)/(2*b*d*e*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^2) + ((e*x)^(1 + m)*((E^(2*a*d)*(1 + m - 2*b*d*n))/n - (E^(4*a*d)*(1 + m + 2*b*d*n)*(c*x^n)^(2*b*d))/n))/(E^(2*a*d)*(2*b^2*d^2*e*n*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))) - ((1 + 2*m + m^2 + 2*b^2*d^2*n^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(b^2*d^2*e*(1 + m)*n^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Tanh[a+b Log[c x^n]]^p with p symbolic*) - - -{Tanh[d*(a + b*Log[c*x^n])]^p, x, 4, (x*(-1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[1/(2*b*d*n), -p, p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p} -{(e*x)^m*Tanh[d*(a + b*Log[c*x^n])]^p, x, 4, ((e*x)^(1 + m)*(-1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[(1 + m)/(2*b*d*n), -p, p, 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/((1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*(e*(1 + m)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Tanh[a+b Log[c x^n]]^(p/2)*) - - -{Tanh[a + b*Log[c*x^n]]^(5/2)/x, x, 7, -(ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) - (2*Tanh[a + b*Log[c*x^n]]^(3/2))/(3*b*n)} -{Tanh[a + b*Log[c*x^n]]^(3/2)/x, x, 7, ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) - (2*Sqrt[Tanh[a + b*Log[c*x^n]]])/(b*n)} -{Sqrt[Tanh[a + b*Log[c*x^n]]]/x, x, 6, -(ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n)} -{1/(x*Sqrt[Tanh[a + b*Log[c*x^n]]]), x, 6, ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n)} -{1/(x*Tanh[a + b*Log[c*x^n]]^(3/2)), x, 7, -(ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) - 2/(b*n*Sqrt[Tanh[a + b*Log[c*x^n]]])} -{1/(x*Tanh[a + b*Log[c*x^n]]^(5/2)), x, 7, ArcTan[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Tanh[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Tanh[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[d+e x]^m (a+b Tanh[d+e x]^2+c Tanh[d+e x]^4)^n*) - - -{Tanh[x]^5/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 8, ((b - 2*c)*ArcTanh[(b + 2*c*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/(4*c^(3/2)) + ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c]) - Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4]/(2*c)} -{Tanh[x]^3/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 7, -(ArcTanh[(b + 2*c*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[c])) + ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c])} -{Tanh[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 4, ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c])} -{Coth[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 8, -(ArcTanh[(2*a + b*Tanh[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a])) + ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c])} -{Coth[x]^3/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 11, -(ArcTanh[(2*a + b*Tanh[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a])) + (b*ArcTanh[(2*a + b*Tanh[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/(4*a^(3/2)) + ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])]/(2*Sqrt[a + b + c]) - (Coth[x]^2*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])/(2*a)} - - -(* {Tanh[x]^5*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 0, 0} *) -(* {Tanh[x]^3*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 0, 0} *) -{Tanh[x]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 8, -(((b + 2*c)*ArcTanh[(b + 2*c*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])])/(4*Sqrt[c])) + (1/2)*Sqrt[a + b + c]*ArcTanh[(2*a + b + (b + 2*c)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4])] - (1/2)*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4]} -(* {Coth[x]*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 0, 0} *) -(* {Coth[x]^3*Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4], x, 0, 0} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Tanh[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Tanh[a+b x]^m*) - - -{E^(a + b*x)*Tanh[a + b*x]^4, x, 7, E^(a + b*x)/b + (8*E^(a + b*x))/(3*b*(1 + E^(2*a + 2*b*x))^3) - (14*E^(a + b*x))/(3*b*(1 + E^(2*a + 2*b*x))^2) + (5*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x))) - (3*ArcTan[E^(a + b*x)])/b} -{E^(a + b*x)*Tanh[a + b*x]^3, x, 7, E^(a + b*x)/b - (2*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x))^2) + (3*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x))) - (3*ArcTan[E^(a + b*x)])/b} -{E^(a + b*x)*Tanh[a + b*x]^2, x, 5, E^(a + b*x)/b + (2*E^(a + b*x))/(b*(1 + E^(2*a + 2*b*x))) - (2*ArcTan[E^(a + b*x)])/b} -{E^(a + b*x)*Tanh[a + b*x]^1, x, 3, E^(a + b*x)/b - (2*ArcTan[E^(a + b*x)])/b} -{E^(a + b*x)*Coth[a + b*x]^1, x, 3, E^(a + b*x)/b - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Coth[a + b*x]^2, x, 5, E^(a + b*x)/b + (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Coth[a + b*x]^3, x, 7, E^(a + b*x)/b - (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))^2) + (3*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (3*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Coth[a + b*x]^4, x, 7, E^(a + b*x)/b + (8*E^(a + b*x))/(3*b*(1 - E^(2*a + 2*b*x))^3) - (14*E^(a + b*x))/(3*b*(1 - E^(2*a + 2*b*x))^2) + (5*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (3*ArcTanh[E^(a + b*x)])/b} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^x Tanh[n x]^m*) - - -{E^x*Tanh[2*x]^2, x, 13, E^x + E^x/(1 + E^(4*x)) + ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])} -{E^x*Tanh[2*x], x, 11, E^x + ArcTan[1 - Sqrt[2]*E^x]/Sqrt[2] - ArcTan[1 + Sqrt[2]*E^x]/Sqrt[2] + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(2*Sqrt[2])} -{E^x*Coth[2*x], x, 5, E^x - ArcTan[E^x] - ArcTanh[E^x]} -{E^x*Coth[2*x]^2, x, 7, E^x + E^x/(1 - E^(4*x)) - ArcTan[E^x]/2 - ArcTanh[E^x]/2} - - -{E^x*Tanh[3*x]^2, x, 14, E^x + (2*E^x)/(3*(1 + E^(6*x))) - (2*ArcTan[E^x])/9 + (1/9)*ArcTan[Sqrt[3] - 2*E^x] - (1/9)*ArcTan[Sqrt[3] + 2*E^x] + Log[1 - Sqrt[3]*E^x + E^(2*x)]/(6*Sqrt[3]) - Log[1 + Sqrt[3]*E^x + E^(2*x)]/(6*Sqrt[3])} -{E^x*Tanh[3*x], x, 12, E^x - (2*ArcTan[E^x])/3 + (1/3)*ArcTan[Sqrt[3] - 2*E^x] - (1/3)*ArcTan[Sqrt[3] + 2*E^x] + Log[1 - Sqrt[3]*E^x + E^(2*x)]/(2*Sqrt[3]) - Log[1 + Sqrt[3]*E^x + E^(2*x)]/(2*Sqrt[3])} -{E^x*Coth[3*x], x, 12, E^x + ArcTan[(1 - 2*E^x)/Sqrt[3]]/Sqrt[3] - ArcTan[(1 + 2*E^x)/Sqrt[3]]/Sqrt[3] - (2*ArcTanh[E^x])/3 + (1/6)*Log[1 - E^x + E^(2*x)] - (1/6)*Log[1 + E^x + E^(2*x)]} -{E^x*Coth[3*x]^2, x, 14, E^x + (2*E^x)/(3*(1 - E^(6*x))) + ArcTan[(1 - 2*E^x)/Sqrt[3]]/(3*Sqrt[3]) - ArcTan[(1 + 2*E^x)/Sqrt[3]]/(3*Sqrt[3]) - (2*ArcTanh[E^x])/9 + (1/18)*Log[1 - E^x + E^(2*x)] - (1/18)*Log[1 + E^x + E^(2*x)]} - - -{E^x*Tanh[4*x]^2, x, 23, E^x + E^x/(2*(1 + E^(8*x))) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*E^x)/Sqrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) + (1/32)*Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] - (1/32)*Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] + (1/32)*Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)] - (1/32)*Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]} -{E^x*Tanh[4*x], x, 21, E^x + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*E^x)/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*E^x)/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])]) + (1/8)*Sqrt[2 - Sqrt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] - (1/8)*Sqrt[2 - Sqrt[2]]*Log[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)] + (1/8)*Sqrt[2 + Sqrt[2]]*Log[1 - Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)] - (1/8)*Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)]} -{E^x*Coth[4*x], x, 15, E^x - ArcTan[E^x]/2 + ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) - ArcTanh[E^x]/2 + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])} -{E^x*Coth[4*x]^2, x, 17, E^x + E^x/(2*(1 - E^(8*x))) - ArcTan[E^x]/8 + ArcTan[1 - Sqrt[2]*E^x]/(8*Sqrt[2]) - ArcTan[1 + Sqrt[2]*E^x]/(8*Sqrt[2]) - ArcTanh[E^x]/8 + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2])} - - -{E^x/(a - Tanh[2*x]), x, 5, -(E^x/(1 - a)) + ArcTan[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)]/((1 - a)*Sqrt[1 + a]*(1 - a^2)^(1/4)) + ArcTanh[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)]/((1 - a)*Sqrt[1 + a]*(1 - a^2)^(1/4))} -{E^x/(a - Tanh[2*x])^2, x, 7, E^x/(1 - a)^2 + E^x/((1 - a)^2*(1 + a)*(1 + a + (-1 + a)*E^(4*x))) - ((1 + 4*a)*ArcTan[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)])/(2*(1 - a)^2*(1 + a)^(3/2)*(1 - a^2)^(1/4)) - ((1 + 4*a)*ArcTanh[((1 - a)^(1/4)*E^x)/(1 + a)^(1/4)])/(2*(1 - a)^2*(1 + a)^(3/2)*(1 - a^2)^(1/4))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) Tanh[d+e x]^n*) - - -{E^(c*(a + b*x))*Tanh[d + e*x]^3, x, 6, E^(c*(a + b*x))/(b*c) - (6*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c) + (12*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c) - (8*E^(c*(a + b*x))*Hypergeometric2F1[3, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Tanh[d + e*x]^2, x, 5, E^(c*(a + b*x))/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c) + (4*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Tanh[d + e*x]^1, x, 4, E^(c*(a + b*x))/(b*c) - (2*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), -E^(2*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Coth[d + e*x]^1, x, 4, E^(c*(a + b*x))/(b*c) - (2*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Coth[d + e*x]^2, x, 5, E^(c*(a + b*x))/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) + (4*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c)} -{E^(c*(a + b*x))*Coth[d + e*x]^3, x, 6, E^(c*(a + b*x))/(b*c) - (6*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) + (12*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) - (8*E^(c*(a + b*x))*Hypergeometric2F1[3, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Tanh[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Tanh[a*c + b*c*x]^2)^(5/2), x, 9, (E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c) - (4*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^4) + (26*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3) - (55*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(6*b*c*(1 + E^(2*c*(a + b*x)))^2) + (25*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(4*b*c*(1 + E^(2*c*(a + b*x)))) - (15*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(4*b*c)} -{E^(c*(a + b*x))*(Tanh[a*c + b*c*x]^2)^(3/2), x, 8, (E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c) - (2*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^2) + (3*E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))) - (3*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c)} -{E^(c*(a + b*x))*Sqrt[Tanh[a*c + b*c*x]^2], x, 4, (E^(c*(a + b*x))*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c) - (2*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x]*Sqrt[Tanh[a*c + b*c*x]^2])/(b*c)} -{E^(c*(a + b*x))/Sqrt[Tanh[a*c + b*c*x]^2], x, 4, (E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2]) - (2*ArcTanh[E^(c*(a + b*x))]*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(3/2), x, 8, (E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2]) - (2*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2*Sqrt[Tanh[a*c + b*c*x]^2]) + (3*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))*Sqrt[Tanh[a*c + b*c*x]^2]) - (3*ArcTanh[E^(c*(a + b*x))]*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(5/2), x, 9, (E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2]) - (4*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4*Sqrt[Tanh[a*c + b*c*x]^2]) + (26*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3*Sqrt[Tanh[a*c + b*c*x]^2]) - (55*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(6*b*c*(1 - E^(2*c*(a + b*x)))^2*Sqrt[Tanh[a*c + b*c*x]^2]) + (25*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(4*b*c*(1 - E^(2*c*(a + b*x)))*Sqrt[Tanh[a*c + b*c*x]^2]) - (15*ArcTanh[E^(c*(a + b*x))]*Tanh[a*c + b*c*x])/(4*b*c*Sqrt[Tanh[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands that are functions of Tanh[a+b x]*) - - -{Sin[Tanh[a + b*x]]^3, x, 19, -((3*CosIntegral[1 - Tanh[a + b*x]]*Sin[1])/(8*b)) - (3*CosIntegral[1 + Tanh[a + b*x]]*Sin[1])/(8*b) + (CosIntegral[3 - 3*Tanh[a + b*x]]*Sin[3])/(8*b) + (CosIntegral[3 + 3*Tanh[a + b*x]]*Sin[3])/(8*b) - (Cos[3]*SinIntegral[3 - 3*Tanh[a + b*x]])/(8*b) + (3*Cos[1]*SinIntegral[1 - Tanh[a + b*x]])/(8*b) + (3*Cos[1]*SinIntegral[1 + Tanh[a + b*x]])/(8*b) - (Cos[3]*SinIntegral[3 + 3*Tanh[a + b*x]])/(8*b)} -{Sin[Tanh[a + b*x]]^2, x, 13, (Cos[2]*CosIntegral[2 - 2*Tanh[a + b*x]])/(4*b) - (Cos[2]*CosIntegral[2 + 2*Tanh[a + b*x]])/(4*b) - Log[1 - Tanh[a + b*x]]/(4*b) + Log[1 + Tanh[a + b*x]]/(4*b) + (Sin[2]*SinIntegral[2 - 2*Tanh[a + b*x]])/(4*b) - (Sin[2]*SinIntegral[2 + 2*Tanh[a + b*x]])/(4*b)} -{Sin[Tanh[a + b*x]]^1, x, 9, -((CosIntegral[1 - Tanh[a + b*x]]*Sin[1])/(2*b)) - (CosIntegral[1 + Tanh[a + b*x]]*Sin[1])/(2*b) + (Cos[1]*SinIntegral[1 - Tanh[a + b*x]])/(2*b) + (Cos[1]*SinIntegral[1 + Tanh[a + b*x]])/(2*b)} -{Csc[Tanh[a + b*x]]^1, x, 3, (-(1/2))*Unintegrable[(Csc[Tanh[a + b*x]]*Sech[a + b*x]^2)/(-1 + Tanh[a + b*x]), x] + (1/2)*Unintegrable[(Csc[Tanh[a + b*x]]*Sech[a + b*x]^2)/(1 + Tanh[a + b*x]), x]} - - -{Cos[Tanh[a + b*x]]^3, x, 19, -((Cos[3]*CosIntegral[3 - 3*Tanh[a + b*x]])/(8*b)) - (3*Cos[1]*CosIntegral[1 - Tanh[a + b*x]])/(8*b) + (3*Cos[1]*CosIntegral[1 + Tanh[a + b*x]])/(8*b) + (Cos[3]*CosIntegral[3 + 3*Tanh[a + b*x]])/(8*b) - (Sin[3]*SinIntegral[3 - 3*Tanh[a + b*x]])/(8*b) - (3*Sin[1]*SinIntegral[1 - Tanh[a + b*x]])/(8*b) + (3*Sin[1]*SinIntegral[1 + Tanh[a + b*x]])/(8*b) + (Sin[3]*SinIntegral[3 + 3*Tanh[a + b*x]])/(8*b)} -{Cos[Tanh[a + b*x]]^2, x, 13, -((Cos[2]*CosIntegral[2 - 2*Tanh[a + b*x]])/(4*b)) + (Cos[2]*CosIntegral[2 + 2*Tanh[a + b*x]])/(4*b) - Log[1 - Tanh[a + b*x]]/(4*b) + Log[1 + Tanh[a + b*x]]/(4*b) - (Sin[2]*SinIntegral[2 - 2*Tanh[a + b*x]])/(4*b) + (Sin[2]*SinIntegral[2 + 2*Tanh[a + b*x]])/(4*b)} -{Cos[Tanh[a + b*x]]^1, x, 9, -((Cos[1]*CosIntegral[1 - Tanh[a + b*x]])/(2*b)) + (Cos[1]*CosIntegral[1 + Tanh[a + b*x]])/(2*b) - (Sin[1]*SinIntegral[1 - Tanh[a + b*x]])/(2*b) + (Sin[1]*SinIntegral[1 + Tanh[a + b*x]])/(2*b)} -{Sec[Tanh[a + b*x]]^1, x, 3, (-(1/2))*Unintegrable[(Sec[Tanh[a + b*x]]*Sech[a + b*x]^2)/(-1 + Tanh[a + b*x]), x] + (1/2)*Unintegrable[(Sec[Tanh[a + b*x]]*Sech[a + b*x]^2)/(1 + Tanh[a + b*x]), x]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.7 (d hyper)^m (a+b (c tanh)^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.7 (d hyper)^m (a+b (c tanh)^n)^p.m deleted file mode 100644 index e18fa6a..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.3 Hyperbolic tangent/6.3.7 (d hyper)^m (a+b (c tanh)^n)^p.m +++ /dev/null @@ -1,461 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Tanh[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 5, (3/8)*(a + 5*b)*x - ((5*a + 9*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + ((a + b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d) - (b*Tanh[c + d*x])/d} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 3, -(((a + 2*b)*Cosh[c + d*x])/d) + ((a + b)*Cosh[c + d*x]^3)/(3*d) - (b*Sech[c + d*x])/d} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 4, (-(1/2))*(a + 3*b)*x + ((a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (b*Tanh[c + d*x])/d} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 3, ((a + b)*Cosh[c + d*x])/d + (b*Sech[c + d*x])/d} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 3, -((a*ArcTanh[Cosh[c + d*x]])/d) - (b*Sech[c + d*x])/d} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 3, -((a*Coth[c + d*x])/d) + (b*Tanh[c + d*x])/d} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 4, ((a - 2*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*Coth[c + d*x]*Csch[c + d*x])/(2*d) + (b*Sech[c + d*x])/d} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 3, ((a - b)*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*d) - (b*Tanh[c + d*x])/d} - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 6, (1/8)*(3*a^2 + 30*a*b + 35*b^2)*x - ((a + b)*(a + 9*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - ((a^2 + 10*a*b + 13*b^2)*Tanh[c + d*x])/(4*d) + ((a + b)^2*Sinh[c + d*x]^4*Tanh[c + d*x])/(4*d) - (b^2*Tanh[c + d*x]^3)/(3*d)} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 3, -(((a + b)*(a + 3*b)*Cosh[c + d*x])/d) + ((a + b)^2*Cosh[c + d*x]^3)/(3*d) - (b*(2*a + 3*b)*Sech[c + d*x])/d + (b^2*Sech[c + d*x]^3)/(3*d)} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 5, (-(1/2))*(a + b)*(a + 5*b)*x + ((a + b)*(a + 5*b)*Tanh[c + d*x])/(2*d) + ((a + b)^2*Sinh[c + d*x]^2*Tanh[c + d*x])/(2*d) + (b^2*Tanh[c + d*x]^3)/(3*d)} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 3, ((a + b)^2*Cosh[c + d*x])/d + (2*b*(a + b)*Sech[c + d*x])/d - (b^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 4, -((a^2*ArcTanh[Cosh[c + d*x]])/d) - (b*(2*a + b)*Sech[c + d*x])/d + (b^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 3, -((a^2*Coth[c + d*x])/d) + (2*a*b*Tanh[c + d*x])/d + (b^2*Tanh[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 5, (a*(a - 4*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*(a - 4*b)*Sech[c + d*x])/(2*d) - (a^2*Csch[c + d*x]^2*Sech[c + d*x])/(2*d) - (b^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 3, (a*(a - 2*b)*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) - ((2*a - b)*b*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x]^3)/(3*d)} - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 6, (3/8)*(a + b)*(a^2 + 14*a*b + 21*b^2)*x - (3*(a + b)*(a^2 + 14*a*b + 21*b^2)*Tanh[c + d*x])/(8*d) - (b*(6*a^2 + 35*a*b + 21*b^2)*Tanh[c + d*x]^3)/(8*d) - (3*b^2*(5*a + 21*b)*Tanh[c + d*x]^5)/(40*d) - (3*(a + 3*b)*Sinh[c + d*x]^2*Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2)/(8*d) + (Cosh[c + d*x]*Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3)/(4*d)} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 3, -(((a + b)^2*(a + 4*b)*Cosh[c + d*x])/d) + ((a + b)^3*Cosh[c + d*x]^3)/(3*d) - (3*b*(a + b)*(a + 2*b)*Sech[c + d*x])/d + (b^2*(3*a + 4*b)*Sech[c + d*x]^3)/(3*d) - (b^3*Sech[c + d*x]^5)/(5*d)} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 6, -((a + b)^2*(a + 7*b)*x)/2 + (a + b)^3/(4*d*(1 - Tanh[c + d*x])) + (3*b*(a + b)^2*Tanh[c + d*x])/d + (b^2*(3*a + 2*b)*Tanh[c + d*x]^3)/(3*d) + (b^3*Tanh[c + d*x]^5)/(5*d) - (a + b)^3/(4*d*(1 + Tanh[c + d*x])), (-(1/2))*(a + b)^2*(a + 7*b)*x + (b*(81*a^2 + 190*a*b + 105*b^2)*Tanh[c + d*x])/(30*d) + (b*(33*a + 35*b)*Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2))/(30*d) + (7*b*Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2)/(10*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3)/(2*d)} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 3, ((a + b)^3*Cosh[c + d*x])/d + (3*b*(a + b)^2*Sech[c + d*x])/d - (b^2*(a + b)*Sech[c + d*x]^3)/d + (b^3*Sech[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 4, -((a^3*ArcTanh[Cosh[c + d*x]])/d) - (b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x])/d + (b^2*(3*a + 2*b)*Sech[c + d*x]^3)/(3*d) - (b^3*Sech[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 3, -((a^3*Coth[c + d*x])/d) + (3*a^2*b*Tanh[c + d*x])/d + (a*b^2*Tanh[c + d*x]^3)/d + (b^3*Tanh[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 6, (a^2*(a - 6*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(81*a^2 - 28*a*b - 4*b^2)*Sech[c + d*x])/(30*d) + ((33*a - 2*b)*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2))/(30*d) + (7*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2)^2)/(10*d) - (Coth[c + d*x]*Csch[c + d*x]*(a + b - b*Sech[c + d*x]^2)^3)/(2*d)} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 3, (a^2*(a - 3*b)*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) - (3*a*(a - b)*b*Tanh[c + d*x])/d - ((3*a - b)*b^2*Tanh[c + d*x]^3)/(3*d) - (b^3*Tanh[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 6, ((3*a^2 - 6*a*b - b^2)*x)/(8*(a + b)^3) + (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/((a + b)^3*d) - ((5*a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a + b)*d)} -{Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 4, (a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(5/2)*d) - (a*Cosh[c + d*x])/((a + b)^2*d) + Cosh[c + d*x]^3/(3*(a + b)*d)} -{Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 5, -(((a - b)*x)/(2*(a + b)^2)) - (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/((a + b)^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d)} -{Sinh[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 3, -((Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(3/2)*d)) + Cosh[c + d*x]/((a + b)*d)} -{Csch[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 4, -(ArcTanh[Cosh[c + d*x]]/(a*d)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)} -{Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*d)) - Coth[c + d*x]/(a*d)} -{Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 5, ((a + 2*b)*ArcTanh[Cosh[c + d*x]])/(2*a^2*d) - (Sqrt[b]*Sqrt[a + b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)} -{Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 4, (Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*d) + ((a + b)*Coth[c + d*x])/(a^2*d) - Coth[c + d*x]^3/(3*a*d)} - - -{Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2, x, 7, (3*(a^2 - 6*a*b + b^2)*x)/(8*(a + b)^4) + (3*Sqrt[a]*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*(a + b)^4*d) - ((5*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)) + (3*(3*a - b)*b*Tanh[c + d*x])/(8*(a + b)^3*d*(a + b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 5, ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*d) - ((a - b)*Cosh[c + d*x])/((a + b)^3*d) + Cosh[c + d*x]^3/(3*(a + b)^2*d) + (a*b*Sech[c + d*x])/(2*(a + b)^3*d*(a + b - b*Sech[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 6, -(((a - 3*b)*x)/(2*(a + b)^3)) - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*Sqrt[a]*(a + b)^3*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2)) - (b*Tanh[c + d*x])/((a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 4, -((3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(5/2)*d)) + (3*Cosh[c + d*x])/(2*(a + b)^2*d) - Cosh[c + d*x]/(2*(a + b)*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 5, -(ArcTanh[Cosh[c + d*x]]/(a^2*d)) + (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*d) + (b*Sech[c + d*x])/(2*a*(a + b)*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 4, (-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*d) - (3*Coth[c + d*x])/(2*a^2*d) + Coth[c + d*x]/(2*a*d*(a + b*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 6, ((a + 4*b)*ArcTanh[Cosh[c + d*x]])/(2*a^3*d) - (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*a^3*Sqrt[a + b]*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a + b - b*Sech[c + d*x]^2)) - (b*Sech[c + d*x])/(a^2*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2, x, 5, (Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*d) + ((a + 2*b)*Coth[c + d*x])/(a^3*d) - Coth[c + d*x]^3/(3*a^2*d) + (b*(a + b)*Tanh[c + d*x])/(2*a^3*d*(a + b*Tanh[c + d*x]^2))} - - -{Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3, x, 8, (3*(a^2 - 10*a*b + 5*b^2)*x)/(8*(a + b)^5) + (3*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*(a + b)^5*d) - ((5*a - 3*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + ((7*a - 5*b)*b*Tanh[c + d*x])/(8*(a + b)^3*d*(a + b*Tanh[c + d*x]^2)^2) + (3*(a - b)*b*Tanh[c + d*x])/(2*(a + b)^4*d*(a + b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3, x, 6, (5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d) - ((a - 2*b)*Cosh[c + d*x])/((a + b)^4*d) + Cosh[c + d*x]^3/(3*(a + b)^3*d) + (a*b*Sech[c + d*x])/(4*(a + b)^3*d*(a + b - b*Sech[c + d*x]^2)^2) + ((7*a - 4*b)*b*Sech[c + d*x])/(8*(a + b)^4*d*(a + b - b*Sech[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 7, -(((a - 5*b)*x)/(2*(a + b)^4)) - (Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(3/2)*(a + b)^4*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - (3*b*Tanh[c + d*x])/(4*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) - ((11*a - b)*b*Tanh[c + d*x])/(8*a*(a + b)^3*d*(a + b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 5, -((15*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(7/2)*d)) + (15*Cosh[c + d*x])/(8*(a + b)^3*d) - Cosh[c + d*x]/(4*(a + b)*d*(a + b - b*Sech[c + d*x]^2)^2) - (5*Cosh[c + d*x])/(8*(a + b)^2*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 6, -(ArcTanh[Cosh[c + d*x]]/(a^3*d)) + (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*d) + (b*Sech[c + d*x])/(4*a*(a + b)*d*(a + b - b*Sech[c + d*x]^2)^2) + (b*(7*a + 4*b)*Sech[c + d*x])/(8*a^2*(a + b)^2*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 5, (-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*d) - (15*Coth[c + d*x])/(8*a^3*d) + Coth[c + d*x]/(4*a*d*(a + b*Tanh[c + d*x]^2)^2) + (5*Coth[c + d*x])/(8*a^2*d*(a + b*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3, x, 7, ((a + 6*b)*ArcTanh[Cosh[c + d*x]])/(2*a^4*d) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a + b - b*Sech[c + d*x]^2)^2) - (3*b*Sech[c + d*x])/(4*a^2*d*(a + b - b*Sech[c + d*x]^2)^2) - (b*(11*a + 12*b)*Sech[c + d*x])/(8*a^3*(a + b)*d*(a + b - b*Sech[c + d*x]^2))} -{Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3, x, 6, (5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(9/2)*d) + ((a + 3*b)*Coth[c + d*x])/(a^4*d) - Coth[c + d*x]^3/(3*a^3*d) + (b*(a + b)*Tanh[c + d*x])/(4*a^3*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(7*a + 11*b)*Tanh[c + d*x])/(8*a^4*d*(a + b*Tanh[c + d*x]^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Tanh[e+f x]^3)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3), x, 8, -((3*(a + 8*b)*Log[1 - Tanh[c + d*x]])/(16*d)) + (3*(a - 8*b)*Log[1 + Tanh[c + d*x]])/(16*d) - (3*a*Tanh[c + d*x])/(8*d) - (3*b*Tanh[c + d*x]^2)/(2*d) + (Sinh[c + d*x]^4*(b + a*Tanh[c + d*x]))/(4*d) - (Sinh[c + d*x]^2*Tanh[c + d*x]*(a + 8*b*Tanh[c + d*x]))/(8*d)} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3), x, 9, (5*b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*Cosh[c + d*x])/d + (a*Cosh[c + d*x]^3)/(3*d) - (5*b*Sinh[c + d*x])/(2*d) + (5*b*Sinh[c + d*x]^3)/(6*d) - (b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(2*d)} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3), x, 7, ((a + 4*b)*Log[1 - Tanh[c + d*x]])/(4*d) - ((a - 4*b)*Log[1 + Tanh[c + d*x]])/(4*d) + (a*Tanh[c + d*x])/(2*d) + (b*Tanh[c + d*x]^2)/(2*d) + (Sinh[c + d*x]^2*(b + a*Tanh[c + d*x]))/(2*d)} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^3), x, 7, (-3*b*ArcTan[Sinh[c + d*x]])/(2*d) + (a*Cosh[c + d*x])/d + (3*b*Sinh[c + d*x])/(2*d) - (b*Sinh[c + d*x]*Tanh[c + d*x]^2)/(2*d)} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^3), x, 5, (b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*ArcTanh[Cosh[c + d*x]])/d - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3), x, 3, -((a*Coth[c + d*x])/d) + (b*Tanh[c + d*x]^2)/(2*d)} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3), x, 6, (b*ArcTan[Sinh[c + d*x]])/(2*d) + (a*ArcTanh[Cosh[c + d*x]])/(2*d) - (a*Coth[c + d*x]*Csch[c + d*x])/(2*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3), x, 3, (a*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*d) + (b*Log[Tanh[c + d*x]])/d - (b*Tanh[c + d*x]^2)/(2*d)} - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2, x, 8, (3/8)*(a^2 + 21*b^2)*x + (6*a*b*Log[Cosh[c + d*x]])/d - (6*b^2*Tanh[c + d*x])/d - (a*b*Tanh[c + d*x]^2)/d - (b^2*Tanh[c + d*x]^3)/d - (b^2*Tanh[c + d*x]^5)/(5*d) + (Cosh[c + d*x]^3*Sinh[c + d*x]*(a^2 + b^2 + 2*a*b*Tanh[c + d*x]))/(4*d) - (Cosh[c + d*x]*Sinh[c + d*x]*(5*a^2 + 17*b^2 + 20*a*b*Tanh[c + d*x]))/(8*d), -((3*(a^2 + 16*a*b + 21*b^2)*Log[1 - Tanh[c + d*x]])/(16*d)) + (3*(a^2 - 16*a*b + 21*b^2)*Log[1 + Tanh[c + d*x]])/(16*d) - (3*(a^2 + 21*b^2)*Tanh[c + d*x])/(8*d) - (3*a*b*Tanh[c + d*x]^2)/d - (b^2*Tanh[c + d*x]^3)/d - (b^2*Tanh[c + d*x]^5)/(5*d) - (Sinh[c + d*x]^2*Tanh[c + d*x]*(a^2 + 13*b^2 + 16*a*b*Tanh[c + d*x]))/(8*d) + (Sinh[c + d*x]^4*(2*a*b + (a^2 + b^2)*Tanh[c + d*x]))/(4*d)} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2, x, 12, (5*a*b*ArcTan[Sinh[c + d*x]])/d - (a^2*Cosh[c + d*x])/d - (4*b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d) + (b^2*Cosh[c + d*x]^3)/(3*d) - (6*b^2*Sech[c + d*x])/d + (4*b^2*Sech[c + d*x]^3)/(3*d) - (b^2*Sech[c + d*x]^5)/(5*d) - (5*a*b*Sinh[c + d*x])/d + (5*a*b*Sinh[c + d*x]^3)/(3*d) - (a*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2, x, 7, (-(1/2))*(a^2 + 7*b^2)*x - (4*a*b*Log[Cosh[c + d*x]])/d + (3*b^2*Tanh[c + d*x])/d + (a*b*Tanh[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a^2 + b^2 + 2*a*b*Tanh[c + d*x]))/(2*d), ((a + b)*(a + 7*b)*Log[1 - Tanh[c + d*x]])/(4*d) - ((a - 7*b)*(a - b)*Log[1 + Tanh[c + d*x]])/(4*d) + ((a^2 + 7*b^2)*Tanh[c + d*x])/(2*d) + (a*b*Tanh[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d) + (Sinh[c + d*x]^2*(2*a*b + (a^2 + b^2)*Tanh[c + d*x]))/(2*d)} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^3)^2, x, 10, (-3*a*b*ArcTan[Sinh[c + d*x]])/d + (a^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x])/d + (3*b^2*Sech[c + d*x])/d - (b^2*Sech[c + d*x]^3)/d + (b^2*Sech[c + d*x]^5)/(5*d) + (3*a*b*Sinh[c + d*x])/d - (a*b*Sinh[c + d*x]*Tanh[c + d*x]^2)/d} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^3)^2, x, 8, (a*b*ArcTan[Sinh[c + d*x]])/d - (a^2*ArcTanh[Cosh[c + d*x]])/d - (b^2*Sech[c + d*x])/d + (2*b^2*Sech[c + d*x]^3)/(3*d) - (b^2*Sech[c + d*x]^5)/(5*d) - (a*b*Sech[c + d*x]*Tanh[c + d*x])/d} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^2, x, 3, -((a^2*Coth[c + d*x])/d) + (a*b*Tanh[c + d*x]^2)/d + (b^2*Tanh[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2, x, 9, (a*b*ArcTan[Sinh[c + d*x]])/d + (a^2*ArcTanh[Cosh[c + d*x]])/(2*d) - (a^2*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (b^2*Sech[c + d*x]^3)/(3*d) + (b^2*Sech[c + d*x]^5)/(5*d) + (a*b*Sech[c + d*x]*Tanh[c + d*x])/d} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2, x, 3, (a^2*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d) + (2*a*b*Log[Tanh[c + d*x]])/d - (a*b*Tanh[c + d*x]^2)/d + (b^2*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)} - - -{Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3, x, 8, (3/8)*a*(a^2 + 63*b^2)*x + (3*b*(3*a^2 + 5*b^2)*Log[Cosh[c + d*x]])/d - (18*a*b^2*Tanh[c + d*x])/d - (b*(3*a^2 + 10*b^2)*Tanh[c + d*x]^2)/(2*d) - (3*a*b^2*Tanh[c + d*x]^3)/d - (3*b^3*Tanh[c + d*x]^4)/(2*d) - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^6)/(2*d) - (b^3*Tanh[c + d*x]^8)/(8*d) + (Cosh[c + d*x]^3*Sinh[c + d*x]*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Tanh[c + d*x]))/(4*d) - (Cosh[c + d*x]*Sinh[c + d*x]*(a*(5*a^2 + 51*b^2) + 2*b*(15*a^2 + 11*b^2)*Tanh[c + d*x]))/(8*d), -((3*(a + b)*(a^2 + 23*a*b + 40*b^2)*Log[1 - Tanh[c + d*x]])/(16*d)) + (3*(a - b)*(a^2 - 23*a*b + 40*b^2)*Log[1 + Tanh[c + d*x]])/(16*d) - (3*a*(a^2 + 63*b^2)*Tanh[c + d*x])/(8*d) - (3*b*(3*a^2 + 5*b^2)*Tanh[c + d*x]^2)/(2*d) - (3*a*b^2*Tanh[c + d*x]^3)/d - (3*b^3*Tanh[c + d*x]^4)/(2*d) - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^6)/(2*d) - (b^3*Tanh[c + d*x]^8)/(8*d) + (Sinh[c + d*x]^4*(b*(3*a^2 + b^2) + a*(a^2 + 3*b^2)*Tanh[c + d*x]))/(4*d) - (Sinh[c + d*x]^2*Tanh[c + d*x]*(a*(a^2 + 39*b^2) + 4*b*(6*a^2 + 5*b^2)*Tanh[c + d*x]))/(8*d)} -{Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3, x, 20, (15*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (1155*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*Cosh[c + d*x])/d - (12*a*b^2*Cosh[c + d*x])/d + (a^3*Cosh[c + d*x]^3)/(3*d) + (a*b^2*Cosh[c + d*x]^3)/d - (18*a*b^2*Sech[c + d*x])/d + (4*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (15*a^2*b*Sinh[c + d*x])/(2*d) - (1155*b^3*Sinh[c + d*x])/(128*d) + (5*a^2*b*Sinh[c + d*x]^3)/(2*d) + (385*b^3*Sinh[c + d*x]^3)/(128*d) - (3*a^2*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(2*d) - (231*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(128*d) - (33*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^4)/(64*d) - (11*b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^6)/(48*d) - (b^3*Sinh[c + d*x]^3*Tanh[c + d*x]^8)/(8*d)} -{Sinh[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^3, x, 7, (-(1/2))*a*(a^2 + 21*b^2)*x - (b*(6*a^2 + 5*b^2)*Log[Cosh[c + d*x]])/d + (9*a*b^2*Tanh[c + d*x])/d + (b*(3*a^2 + 4*b^2)*Tanh[c + d*x]^2)/(2*d) + (2*a*b^2*Tanh[c + d*x]^3)/d + (3*b^3*Tanh[c + d*x]^4)/(4*d) + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(3*d) + (b^3*Tanh[c + d*x]^8)/(8*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a*(a^2 + 3*b^2) + b*(3*a^2 + b^2)*Tanh[c + d*x]))/(2*d), ((a + b)^2*(a + 10*b)*Log[1 - Tanh[c + d*x]])/(4*d) - ((a - 10*b)*(a - b)^2*Log[1 + Tanh[c + d*x]])/(4*d) + (a*(a^2 + 21*b^2)*Tanh[c + d*x])/(2*d) + (b*(3*a^2 + 4*b^2)*Tanh[c + d*x]^2)/(2*d) + (2*a*b^2*Tanh[c + d*x]^3)/d + (3*b^3*Tanh[c + d*x]^4)/(4*d) + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(3*d) + (b^3*Tanh[c + d*x]^8)/(8*d) + (Sinh[c + d*x]^2*(b*(3*a^2 + b^2) + a*(a^2 + 3*b^2)*Tanh[c + d*x]))/(2*d)} -{Sinh[c + d*x]^1*(a + b*Tanh[c + d*x]^3)^3, x, 17, (-9*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) - (315*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*Cosh[c + d*x])/d + (3*a*b^2*Cosh[c + d*x])/d + (9*a*b^2*Sech[c + d*x])/d - (3*a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (9*a^2*b*Sinh[c + d*x])/(2*d) + (315*b^3*Sinh[c + d*x])/(128*d) - (3*a^2*b*Sinh[c + d*x]*Tanh[c + d*x]^2)/(2*d) - (105*b^3*Sinh[c + d*x]*Tanh[c + d*x]^2)/(128*d) - (21*b^3*Sinh[c + d*x]*Tanh[c + d*x]^4)/(64*d) - (3*b^3*Sinh[c + d*x]*Tanh[c + d*x]^6)/(16*d) - (b^3*Sinh[c + d*x]*Tanh[c + d*x]^8)/(8*d)} -{Csch[c + d*x]^1*(a + b*Tanh[c + d*x]^3)^3, x, 13, (3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (35*b^3*ArcTan[Sinh[c + d*x]])/(128*d) - (a^3*ArcTanh[Cosh[c + d*x]])/d - (3*a*b^2*Sech[c + d*x])/d + (2*a*b^2*Sech[c + d*x]^3)/d - (3*a*b^2*Sech[c + d*x]^5)/(5*d) - (3*a^2*b*Sech[c + d*x]*Tanh[c + d*x])/(2*d) - (35*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (35*b^3*Sech[c + d*x]*Tanh[c + d*x]^3)/(192*d) - (7*b^3*Sech[c + d*x]*Tanh[c + d*x]^5)/(48*d) - (b^3*Sech[c + d*x]*Tanh[c + d*x]^7)/(8*d)} -{Csch[c + d*x]^2*(a + b*Tanh[c + d*x]^3)^3, x, 3, -((a^3*Coth[c + d*x])/d) + (3*a^2*b*Tanh[c + d*x]^2)/(2*d) + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^8)/(8*d)} -{Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3, x, 14, (3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (5*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*ArcTanh[Cosh[c + d*x]])/(2*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (3*a^2*b*Sech[c + d*x]*Tanh[c + d*x])/(2*d) + (5*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (5*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(64*d) - (5*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^3)/(48*d) - (b^3*Sech[c + d*x]^3*Tanh[c + d*x]^5)/(8*d)} -{Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3, x, 3, (a^3*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a^2*b*Log[Tanh[c + d*x]])/d - (3*a^2*b*Tanh[c + d*x]^2)/(2*d) + (a*b^2*Tanh[c + d*x]^3)/d - (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^6)/(6*d) - (b^3*Tanh[c + d*x]^8)/(8*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^3), x, 11, -((a^(2/3)*b^(1/3)*(a^2 + 3*a^(4/3)*b^(2/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*(a^(4/3) + a^(2/3)*b^(2/3) + b^(4/3))^3*d)) - (3*a*(a - 5*b)*Log[1 - Tanh[c + d*x]])/(16*(a + b)^3*d) + (3*a*(a + 5*b)*Log[1 + Tanh[c + d*x]])/(16*(a - b)^3*d) - (a^(2/3)*b^(1/3)*(a^4 + 7*a^2*b^2 + b^4 + 3*a^(2/3)*b^(4/3)*(2*a^2 + b^2))*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*(a^2 - b^2)^3*d) + (a^(2/3)*b^(1/3)*(a^4 + 7*a^2*b^2 + b^4 + 3*a^(2/3)*b^(4/3)*(2*a^2 + b^2))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*(a^2 - b^2)^3*d) - (a^2*b*(a^2 + 2*b^2)*Log[a + b*Tanh[c + d*x]^3])/((a^2 - b^2)^3*d) + 1/(16*(a + b)*d*(1 - Tanh[c + d*x])^2) - (5*a - b)/(16*(a + b)^2*d*(1 - Tanh[c + d*x])) - 1/(16*(a - b)*d*(1 + Tanh[c + d*x])^2) + (5*a + b)/(16*(a - b)^2*d*(1 + Tanh[c + d*x]))} -{Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^3), x, 0, I*Unintegrable[((-I)*Sinh[c + d*x]^3)/(a + b*Tanh[c + d*x]^3), x]} -{Sinh[c + d*x]^2/(a + b*Tanh[c + d*x]^3), x, 11, (a^(2/3)*b^(1/3)*(a^2 - 3*a^(2/3)*b^(4/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*(a^2 - b^2)^2*d) + ((a - 2*b)*Log[1 - Tanh[c + d*x]])/(4*(a + b)^2*d) - ((a + 2*b)*Log[1 + Tanh[c + d*x]])/(4*(a - b)^2*d) + (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*(a^2 - b^2)^2*d) - (a^(2/3)*b^(1/3)*(a^2 + 3*a^(2/3)*b^(4/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*(a^2 - b^2)^2*d) + (b*(2*a^2 + b^2)*Log[a + b*Tanh[c + d*x]^3])/(3*(a^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Tanh[c + d*x])) - 1/(4*(a - b)*d*(1 + Tanh[c + d*x]))} -{Sinh[c + d*x]^1/(a + b*Tanh[c + d*x]^3), x, 0, (-I)*Unintegrable[(I*Sinh[c + d*x])/(a + b*Tanh[c + d*x]^3), x]} -{Csch[c + d*x]^1/(a + b*Tanh[c + d*x]^3), x, 0, I*Unintegrable[((-I)*Csch[c + d*x])/(a + b*Tanh[c + d*x]^3), x]} -{Csch[c + d*x]^2/(a + b*Tanh[c + d*x]^3), x, 8, (b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d) - Coth[c + d*x]/(a*d) + (b^(1/3)*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*a^(4/3)*d) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*a^(4/3)*d)} -{Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3), x, 0, (-I)*Unintegrable[(I*Csch[c + d*x]^3)/(a + b*Tanh[c + d*x]^3), x]} -{Csch[c + d*x]^4/(a + b*Tanh[c + d*x]^3), x, 12, -((b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tanh[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(4/3)*d)) + Coth[c + d*x]/(a*d) - Coth[c + d*x]^3/(3*a*d) - (b*Log[Tanh[c + d*x]])/(a^2*d) - (b^(1/3)*Log[a^(1/3) + b^(1/3)*Tanh[c + d*x]])/(3*a^(4/3)*d) + (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tanh[c + d*x] + b^(2/3)*Tanh[c + d*x]^2])/(6*a^(4/3)*d) + (b*Log[a + b*Tanh[c + d*x]^3])/(3*a^2*d)} - - -(* ::Section:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Tanh[e+f x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Tanh[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 4, (1/8)*(3*a - b)*x + ((3*a - b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + ((a + b)*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)} -{Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 2, (a*Sinh[c + d*x])/d + ((a + b)*Sinh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 3, (1/2)*(a - b)*x + ((a + b)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Cosh[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 3, -((b*ArcTan[Sinh[c + d*x]])/d) + ((a + b)*Sinh[c + d*x])/d} -{Sech[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 3, ((2*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 2, (a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)} -{Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 4, ((4*a + b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((4*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) - (b*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 3, (a*Tanh[c + d*x])/d - ((a - b)*Tanh[c + d*x]^3)/(3*d) - (b*Tanh[c + d*x]^5)/(5*d)} - - -{Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 4, (1/8)*(3*a^2 - 2*a*b + 3*b^2)*x + (3*(a^2 - b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + ((a + b)*Cosh[c + d*x]^3*Sinh[c + d*x]*(a + b*Tanh[c + d*x]^2))/(4*d)} -{Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 4, (b^2*ArcTan[Sinh[c + d*x]])/d + ((a^2 - b^2)*Sinh[c + d*x])/d + ((a + b)^2*Sinh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 5, (1/2)*(a - 3*b)*(a + b)*x + ((a + b)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (b^2*Tanh[c + d*x])/d} -{Cosh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 5, -((b*(4*a + 3*b)*ArcTan[Sinh[c + d*x]])/(2*d)) + ((a + b)^2*Sinh[c + d*x])/d + (b^2*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]])/(8*d) - (3*b*(2*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) - (b*Sech[c + d*x]^3*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 3, (a^2*Tanh[c + d*x])/d + (2*a*b*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 5, ((8*a^2 + 4*a*b + b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((8*a^2 + 4*a*b + b^2)*Sech[c + d*x]*Tanh[c + d*x])/(16*d) - (b*(8*a + 3*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) - (b*Sech[c + d*x]^5*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(6*d)} -{Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 3, (a^2*Tanh[c + d*x])/d - (a*(a - 2*b)*Tanh[c + d*x]^3)/(3*d) - ((2*a - b)*b*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)} - - -{Cosh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 6, (3/8)*(a + b)*(a^2 - 2*a*b + 5*b^2)*x + (3*(a - 3*b)*(a + b)^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + ((a + b)^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d) - (b^3*Tanh[c + d*x])/d} -{Cosh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 5, (b^2*(6*a + 5*b)*ArcTan[Sinh[c + d*x]])/(2*d) + ((a - 2*b)*(a + b)^2*Sinh[c + d*x])/d + ((a + b)^3*Sinh[c + d*x]^3)/(3*d) - (b^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 5, (1/2)*(a - 5*b)*(a + b)^2*x + ((a + b)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (b^2*(3*a + 2*b)*Tanh[c + d*x])/d + (b^3*Tanh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 6, -((3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[Sinh[c + d*x]])/(8*d)) + ((a + b)^3*Sinh[c + d*x])/d + (3*b^2*(4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) - (b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 5, ((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) - (b*(44*a^2 + 44*a*b + 15*b^2)*Sech[c + d*x]*Tanh[c + d*x])/(48*d) - (5*b*(2*a + b)*Sech[c + d*x]^3*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) - (b*Sech[c + d*x]^5*(a + (a + b)*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)} -{Sech[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 3, (a^3*Tanh[c + d*x])/d + (a^2*b*Tanh[c + d*x]^3)/d + (3*a*b^2*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^7)/(7*d)} -{Sech[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 6, ((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*ArcTan[Sinh[c + d*x]])/(128*d) + ((64*a^3 + 48*a^2*b + 24*a*b^2 + 5*b^3)*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (b*(72*a^2 + 52*a*b + 15*b^2)*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d) - (b*(12*a + 5*b)*Sech[c + d*x]^5*(a + (a + b)*Sinh[c + d*x]^2)*Tanh[c + d*x])/(48*d) - (b*Sech[c + d*x]^7*(a + (a + b)*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(8*d)} -{Sech[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 3, (a^3*Tanh[c + d*x])/d - (a^2*(a - 3*b)*Tanh[c + d*x]^3)/(3*d) - (3*a*(a - b)*b*Tanh[c + d*x]^5)/(5*d) - ((3*a - b)*b^2*Tanh[c + d*x]^7)/(7*d) - (b^3*Tanh[c + d*x]^9)/(9*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Cosh[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 6, ((3*a^2 + 10*a*b + 15*b^2)*x)/(8*(a + b)^3) + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^3*d) + ((3*a + 7*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a + b)*d)} -{Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 4, (b^2*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)*d) + ((a + 2*b)*Sinh[c + d*x])/((a + b)^2*d) + Sinh[c + d*x]^3/(3*(a + b)*d)} -{Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 5, ((a + 3*b)*x)/(2*(a + b)^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d)} -{Cosh[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 3, (b*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2)*d) + Sinh[c + d*x]/((a + b)*d)} -{Sech[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 2, ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a + b]*d)} -{Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)} -{Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 4, -(ArcTan[Sinh[c + d*x]]/(b*d)) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)} -{Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 3, ((a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d) - Tanh[c + d*x]/(b*d)} -{Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2), x, 5, -(((2*a + 3*b)*ArcTan[Sinh[c + d*x]])/(2*b^2*d)) + ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^2*d) - (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d)} -{Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2), x, 4, ((a + b)^2*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a + 2*b)*Tanh[c + d*x])/(b^2*d) + Tanh[c + d*x]^3/(3*b*d)} - - -{Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 5, (b^2*(6*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(7/2)*d) + ((a + 3*b)*Sinh[c + d*x])/((a + b)^3*d) + Sinh[c + d*x]^3/(3*(a + b)^2*d) + (b^3*Sinh[c + d*x])/(2*a*(a + b)^3*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 6, ((a + 5*b)*x)/(2*(a + b)^3) + (b^(3/2)*(5*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^3*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2)) - ((a - b)*b*Tanh[c + d*x])/(2*a*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 5, (b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)*d) + Sinh[c + d*x]/((a + b)^2*d) + (b^2*Sinh[c + d*x])/(2*a*(a + b)^2*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 3, ((2*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)*d) + (b*Sinh[c + d*x])/(2*a*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Tanh[c + d*x]/(2*a*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a + b]*d) + Sinh[c + d*x]/(2*a*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2, x, 3, -((a - b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)*d) + ((a + b)*Tanh[c + d*x])/(2*a*b*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^2, x, 5, ArcTan[Sinh[c + d*x]]/(b^2*d) - ((2*a - b)*Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) + ((a + b)*Sinh[c + d*x])/(2*a*b*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^2, x, 5, -(((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d)) + Tanh[c + d*x]/(b^2*d) + ((a + b)^2*Tanh[c + d*x])/(2*a*b^2*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^7/(a + b*Tanh[c + d*x]^2)^2, x, 6, ((4*a + 5*b)*ArcTan[Sinh[c + d*x]])/(2*b^3*d) - ((4*a - b)*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^3*d) + ((a + b)*(2*a + b)*Sinh[c + d*x])/(2*a*b^2*d*(a + (a + b)*Sinh[c + d*x]^2)) - (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d*(a + (a + b)*Sinh[c + d*x]^2))} - - -{Cosh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 7, ((a + 7*b)*x)/(2*(a + b)^4) + (b^(3/2)*(35*a^2 + 14*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^4*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((2*a - b)*b*Tanh[c + d*x])/(4*a*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) - ((a - 3*b)*b*(4*a + b)*Tanh[c + d*x])/(8*a^2*(a + b)^3*d*(a + b*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 6, (3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(7/2)*d) + Sinh[c + d*x]/((a + b)^3*d) + (b^3*Sinh[c + d*x])/(4*a*(a + b)^3*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + b)*Sinh[c + d*x])/(8*a^2*(a + b)^3*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(5/2)*d) + (b*Cosh[c + d*x]^2*Sinh[c + d*x])/(4*a*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + (3*b*(2*a + b)*Sinh[c + d*x])/(8*a^2*(a + b)^2*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*d) + Tanh[c + d*x]/(4*a*d*(a + b*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*a^2*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3, x, 4, ((4*a + 3*b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(3/2)*d) + (b*Sinh[c + d*x])/(4*a*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + ((4*a + 3*b)*Sinh[c + d*x])/(8*a^2*(a + b)*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3, x, 4, -((a - 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(3/2)*d) + ((a + b)*Tanh[c + d*x])/(4*a*b*d*(a + b*Tanh[c + d*x]^2)^2) - ((a - 3*b)*Tanh[c + d*x])/(8*a^2*b*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[a + b]*d) + Sinh[c + d*x]/(4*a*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + (3*Sinh[c + d*x])/(8*a^2*d*(a + (a + b)*Sinh[c + d*x]^2))} -{Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3, x, 4, ((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2)*d) + ((a + b)*Sech[c + d*x]^2*Tanh[c + d*x])/(4*a*b*d*(a + b*Tanh[c + d*x]^2)^2) + (3*(1/a^2 - 1/b^2)*Tanh[c + d*x])/(8*d*(a + b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^7/(a + b*Tanh[c + d*x]^2)^3, x, 6, -(ArcTan[Sinh[c + d*x]]/(b^3*d)) + (Sqrt[a + b]*(8*a^2 - 4*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^3*d) + ((a + b)*Sinh[c + d*x])/(4*a*b*d*(a + (a + b)*Sinh[c + d*x]^2)^2) - ((4*a - 3*b)*(a + b)*Sinh[c + d*x])/(8*a^2*b^2*d*(a + (a + b)*Sinh[c + d*x]^2))} - - -(* ::Section:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Tanh[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Tanh[e+f x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 4, (a + b)*x - ((a + b)*Tanh[c + d*x])/d - ((a + b)*Tanh[c + d*x]^3)/(3*d) - (b*Tanh[c + d*x]^5)/(5*d)} -{Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 3, ((a + b)*Log[Cosh[c + d*x]])/d - ((a + b)*Tanh[c + d*x]^2)/(2*d) - (b*Tanh[c + d*x]^4)/(4*d)} -{Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 3, (a + b)*x - ((a + b)*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)} -{Tanh[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 2, ((a + b)*Log[Cosh[c + d*x]])/d - (b*Tanh[c + d*x]^2)/(2*d)} -{Tanh[c + d*x]^0*(a + b*Tanh[c + d*x]^2), x, 3, a*x + b*x - (b*Tanh[c + d*x])/d} -{Coth[c + d*x]^1*(a + b*Tanh[c + d*x]^2), x, 3, (b*Log[Cosh[c + d*x]])/d + (a*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2), x, 2, (a + b)*x - (a*Coth[c + d*x])/d} -{Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2), x, 3, -(a*Coth[c + d*x]^2)/(2*d) + ((a + b)*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2), x, 4, (a + b)*x - ((a + b)*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2), x, 4, -(((a + b)*Coth[c + d*x]^2)/(2*d)) - (a*Coth[c + d*x]^4)/(4*d) + ((a + b)*Log[Sinh[c + d*x]])/d} - - -{Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - ((a + b)^2*Tanh[c + d*x])/d - ((a + b)^2*Tanh[c + d*x]^3)/(3*d) - (b*(2*a + b)*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)} -{Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 4, ((a + b)^2*Log[Cosh[c + d*x]])/d - ((a + b)^2*Tanh[c + d*x]^2)/(2*d) - (b*(2*a + b)*Tanh[c + d*x]^4)/(4*d) - (b^2*Tanh[c + d*x]^6)/(6*d)} -{Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - ((a + b)^2*Tanh[c + d*x])/d - (b*(2*a + b)*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)} -{Tanh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 4, ((a + b)^2*Log[Cosh[c + d*x]])/d - (b*(a + b)*Tanh[c + d*x]^2)/(2*d) - (a + b*Tanh[c + d*x]^2)^2/(4*d)} -{Tanh[c + d*x]^0*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - (b*(2*a + b)*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^2, x, 4, ((a + b)^2*Log[Cosh[c + d*x]])/d + (a^2*Log[Tanh[c + d*x]])/d - (b^2*Tanh[c + d*x]^2)/(2*d)} -{Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - (a^2*Coth[c + d*x])/d - (b^2*Tanh[c + d*x])/d} -{Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2, x, 4, -((a^2*Coth[c + d*x]^2)/(2*d)) + ((a + b)^2*Log[Cosh[c + d*x]])/d + (a*(a + 2*b)*Log[Tanh[c + d*x]])/d} -{Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - (a*(a + 2*b)*Coth[c + d*x])/d - (a^2*Coth[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^2, x, 4, -((a*(a + 2*b)*Coth[c + d*x]^2)/(2*d)) - (a^2*Coth[c + d*x]^4)/(4*d) + ((a + b)^2*Log[Cosh[c + d*x]])/d + ((a + b)^2*Log[Tanh[c + d*x]])/d} -{Coth[c + d*x]^6*(a + b*Tanh[c + d*x]^2)^2, x, 4, (a + b)^2*x - ((a + b)^2*Coth[c + d*x])/d - (a*(a + 2*b)*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^2, x, 4, -(((a + b)^2*Coth[c + d*x]^2)/(2*d)) - (a*(a + 2*b)*Coth[c + d*x]^4)/(4*d) - (a^2*Coth[c + d*x]^6)/(6*d) + ((a + b)^2*Log[Cosh[c + d*x]])/d + ((a + b)^2*Log[Tanh[c + d*x]])/d} - - -{Tanh[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - ((a + b)^3*Tanh[c + d*x]^3)/(3*d) - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^2*(3*a + b)*Tanh[c + d*x]^7)/(7*d) - (b^3*Tanh[c + d*x]^9)/(9*d)} -{Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 4, ((a + b)^3*Log[Cosh[c + d*x]])/d - ((a + b)^3*Tanh[c + d*x]^2)/(2*d) - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^4)/(4*d) - (b^2*(3*a + b)*Tanh[c + d*x]^6)/(6*d) - (b^3*Tanh[c + d*x]^8)/(8*d)} -{Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*Tanh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)} -{Tanh[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 4, ((a + b)^3*Log[Cosh[c + d*x]])/d - (b*(a + b)^2*Tanh[c + d*x]^2)/(2*d) - ((a + b)*(a + b*Tanh[c + d*x]^2)^2)/(4*d) - (a + b*Tanh[c + d*x]^2)^3/(6*d)} -{Tanh[c + d*x]^0*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(3*a + b)*Tanh[c + d*x]^3)/(3*d) - (b^3*Tanh[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^1*(a + b*Tanh[c + d*x]^2)^3, x, 4, ((a + b)^3*Log[Cosh[c + d*x]])/d + (a^3*Log[Tanh[c + d*x]])/d - (b^2*(3*a + b)*Tanh[c + d*x]^2)/(2*d) - (b^3*Tanh[c + d*x]^4)/(4*d)} -{Coth[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - (a^3*Coth[c + d*x])/d - (b^2*(3*a + b)*Tanh[c + d*x])/d - (b^3*Tanh[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3, x, 4, -((a^3*Coth[c + d*x]^2)/(2*d)) + ((a + b)^3*Log[Cosh[c + d*x]])/d + (a^2*(a + 3*b)*Log[Tanh[c + d*x]])/d - (b^3*Tanh[c + d*x]^2)/(2*d)} -{Coth[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - (a^2*(a + 3*b)*Coth[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) - (b^3*Tanh[c + d*x])/d} -{Coth[c + d*x]^5*(a + b*Tanh[c + d*x]^2)^3, x, 4, -((a^2*(a + 3*b)*Coth[c + d*x]^2)/(2*d)) - (a^3*Coth[c + d*x]^4)/(4*d) + ((a + b)^3*Log[Cosh[c + d*x]])/d + (a*(a^2 + 3*a*b + 3*b^2)*Log[Tanh[c + d*x]])/d} -{Coth[c + d*x]^6*(a + b*Tanh[c + d*x]^2)^3, x, 4, (a + b)^3*x - (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d - (a^2*(a + 3*b)*Coth[c + d*x]^3)/(3*d) - (a^3*Coth[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^3, x, 4, -((a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x]^2)/(2*d)) - (a^2*(a + 3*b)*Coth[c + d*x]^4)/(4*d) - (a^3*Coth[c + d*x]^6)/(6*d) + ((a + b)^3*Log[Cosh[c + d*x]])/d + ((a + b)^3*Log[Tanh[c + d*x]])/d} - - -{Tanh[c + d*x]^0*(a + b*Tanh[c + d*x]^2)^4, x, 4, (a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(6*a^2 + 4*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^3*(4*a + b)*Tanh[c + d*x]^5)/(5*d) - (b^4*Tanh[c + d*x]^7)/(7*d)} - - -{Tanh[c + d*x]^0*(a + b*Tanh[c + d*x]^2)^5, x, 4, (a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Tanh[c + d*x]^7)/(7*d) - (b^5*Tanh[c + d*x]^9)/(9*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2), x, 4, Log[Cosh[c + d*x]]/((a + b)*d) + (a^2*Log[a + b*Tanh[c + d*x]^2])/(2*b^2*(a + b)*d) - Tanh[c + d*x]^2/(2*b*d)} -{Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 5, x/(a + b) + (a^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(b^(3/2)*(a + b)*d) - Tanh[c + d*x]/(b*d)} -{Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 4, Log[Cosh[c + d*x]]/((a + b)*d) - (a*Log[a + b*Tanh[c + d*x]^2])/(2*b*(a + b)*d)} -{Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 4, x/(a + b) - (Sqrt[a]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[b]*(a + b)*d)} -{Tanh[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 5, Log[Cosh[c + d*x]]/((a + b)*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)*d)} -{Tanh[c + d*x]^0/(a + b*Tanh[c + d*x]^2), x, 3, x/(a + b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)*d)} -{Coth[c + d*x]^1/(a + b*Tanh[c + d*x]^2), x, 4, Log[Cosh[c + d*x]]/((a + b)*d) + Log[Tanh[c + d*x]]/(a*d) - (b*Log[a + b*Tanh[c + d*x]^2])/(2*a*(a + b)*d)} -{Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 5, x/(a + b) - (b^(3/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*(a + b)*d) - Coth[c + d*x]/(a*d)} -{Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2), x, 4, -(Coth[c + d*x]^2/(2*a*d)) + Log[Cosh[c + d*x]]/((a + b)*d) + ((a - b)*Log[Tanh[c + d*x]])/(a^2*d) + (b^2*Log[a + b*Tanh[c + d*x]^2])/(2*a^2*(a + b)*d)} -{Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2), x, 6, x/(a + b) + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*(a + b)*d) - ((a - b)*Coth[c + d*x])/(a^2*d) - Coth[c + d*x]^3/(3*a*d)} - - -{Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^2, x, 4, Log[Cosh[c + d*x]]/((a + b)^2*d) - (a*(a + 2*b)*Log[a + b*Tanh[c + d*x]^2])/(2*b^2*(a + b)^2*d) - a^2/(2*b^2*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2, x, 5, x/(a + b)^2 - (Sqrt[a]*(a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*b^(3/2)*(a + b)^2*d) + (a*Tanh[c + d*x])/(2*b*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 4, Log[Cosh[c + d*x]]/((a + b)^2*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)^2*d) + a/(2*b*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 5, x/(a + b)^2 - ((a - b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]*(a + b)^2*d) - Tanh[c + d*x]/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 4, Log[Cosh[c + d*x]]/((a + b)^2*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)^2*d) - 1/(2*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^0/(a + b*Tanh[c + d*x]^2)^2, x, 5, x/(a + b)^2 + (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^2*d) + (b*Tanh[c + d*x])/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^2, x, 4, Log[Cosh[c + d*x]]/((a + b)^2*d) + Log[Tanh[c + d*x]]/(a^2*d) - (b*(2*a + b)*Log[a + b*Tanh[c + d*x]^2])/(2*a^2*(a + b)^2*d) + b/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^2, x, 6, x/(a + b)^2 - (b^(3/2)*(5*a + 3*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(5/2)*(a + b)^2*d) - ((2*a + 3*b)*Coth[c + d*x])/(2*a^2*(a + b)*d) + (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2, x, 4, -(Coth[c + d*x]^2/(2*a^2*d)) + Log[Cosh[c + d*x]]/((a + b)^2*d) + ((a - 2*b)*Log[Tanh[c + d*x]])/(a^3*d) + (b^2*(3*a + 2*b)*Log[a + b*Tanh[c + d*x]^2])/(2*a^3*(a + b)^2*d) - b^2/(2*a^2*(a + b)*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2, x, 7, x/(a + b)^2 + (b^(5/2)*(7*a + 5*b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(7/2)*(a + b)^2*d) - ((2*a^2 - 2*a*b - 5*b^2)*Coth[c + d*x])/(2*a^3*(a + b)*d) - ((2*a + 5*b)*Coth[c + d*x]^3)/(6*a^2*(a + b)*d) + (b*Coth[c + d*x]^3)/(2*a*(a + b)*d*(a + b*Tanh[c + d*x]^2))} - - -{Tanh[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3, x, 6, x/(a + b)^3 - (Sqrt[a]*(3*a^2 + 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*b^(5/2)*(a + b)^3*d) + (a*Tanh[c + d*x]^3)/(4*b*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (a*(3*a + 7*b)*Tanh[c + d*x])/(8*b^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^5/(a + b*Tanh[c + d*x]^2)^3, x, 4, Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)^3*d) - a^2/(4*b^2*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (a*(a + 2*b))/(2*b^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3, x, 6, x/(a + b)^3 - ((a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*Sqrt[a]*b^(3/2)*(a + b)^3*d) + (a*Tanh[c + d*x])/(4*b*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((a + 5*b)*Tanh[c + d*x])/(8*b*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3, x, 4, Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)^3*d) + a/(4*b*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - 1/(2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 6, x/(a + b)^3 - ((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(3/2)*Sqrt[b]*(a + b)^3*d) - Tanh[c + d*x]/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - ((3*a - b)*Tanh[c + d*x])/(8*a*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 4, Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[a + b*Tanh[c + d*x]^2]/(2*(a + b)^3*d) - 1/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - 1/(2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^0/(a + b*Tanh[c + d*x]^2)^3, x, 6, x/(a + b)^3 + (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^3*d) + (b*Tanh[c + d*x])/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(7*a + 3*b)*Tanh[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^1/(a + b*Tanh[c + d*x]^2)^3, x, 4, Log[Cosh[c + d*x]]/((a + b)^3*d) + Log[Tanh[c + d*x]]/(a^3*d) - (b*(3*a^2 + 3*a*b + b^2)*Log[a + b*Tanh[c + d*x]^2])/(2*a^3*(a + b)^3*d) + b/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(2*a + b))/(2*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3, x, 7, x/(a + b)^3 - (b^(3/2)*(35*a^2 + 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a + b)^3*d) - ((8*a^2 + 27*a*b + 15*b^2)*Coth[c + d*x])/(8*a^3*(a + b)^2*d) + (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(9*a + 5*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3, x, 4, -(Coth[c + d*x]^2/(2*a^3*d)) + Log[Cosh[c + d*x]]/((a + b)^3*d) + ((a - 3*b)*Log[Tanh[c + d*x]])/(a^4*d) + (b^2*(6*a^2 + 8*a*b + 3*b^2)*Log[a + b*Tanh[c + d*x]^2])/(2*a^4*(a + b)^3*d) - b^2/(4*a^2*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) - (b^2*(3*a + 2*b))/(2*a^3*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3, x, 8, x/(a + b)^3 + (b^(5/2)*(63*a^2 + 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(9/2)*(a + b)^3*d) - ((8*a^3 - 8*a^2*b - 55*a*b^2 - 35*b^3)*Coth[c + d*x])/(8*a^4*(a + b)^2*d) - ((8*a^2 + 55*a*b + 35*b^2)*Coth[c + d*x]^3)/(24*a^3*(a + b)^2*d) + (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(11*a + 7*b)*Coth[c + d*x]^3)/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))} - - -{Tanh[c + d*x]^0/(a + b*Tanh[c + d*x]^2)^4,x, 7, x/(a + b)^4 + (Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(16*a^(7/2)*(a + b)^4*d) + (b*Tanh[c + d*x])/(6*a*(a + b)*d*(a + b*Tanh[c + d*x]^2)^3) + (b*(11*a + 5*b)*Tanh[c + d*x])/(24*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2)^2) + (b*(19*a^2 + 16*a*b + 5*b^2)*Tanh[c + d*x])/(16*a^3*(a + b)^3*d*(a + b*Tanh[c + d*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^2)^(p/2) when a+b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[1 - Tanh[x]^2], x, 3, ArcSin[Tanh[x]]} -{Sqrt[-1 + Tanh[x]^2], x, 4, -ArcTanh[Tanh[x]/Sqrt[-Sech[x]^2]]} - - -{(1 - Tanh[x]^2)^(3/2), x, 4, (1/2)*ArcSin[Tanh[x]] + (1/2)*Sqrt[Sech[x]^2]*Tanh[x]} -{(-1 + Tanh[x]^2)^(3/2), x, 5, (1/2)*ArcTanh[Tanh[x]/Sqrt[-Sech[x]^2]] - (1/2)*Sqrt[-Sech[x]^2]*Tanh[x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[1 - Tanh[x]^2], x, 3, Tanh[x]/Sqrt[Sech[x]^2]} -{1/Sqrt[-1 + Tanh[x]^2], x, 3, Tanh[x]/Sqrt[-Sech[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Tanh[x]^5*Sqrt[a + b*Tanh[x]^2], x, 7, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2] + ((a - b)*(a + b*Tanh[x]^2)^(3/2))/(3*b^2) - (a + b*Tanh[x]^2)^(5/2)/(5*b^2)} -{Tanh[x]^4*Sqrt[a + b*Tanh[x]^2], x, 8, ((a^2 - 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(8*b^(3/2)) + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((a + 4*b)*Tanh[x]*Sqrt[a + b*Tanh[x]^2])/(8*b) - (1/4)*Tanh[x]^3*Sqrt[a + b*Tanh[x]^2]} -{Tanh[x]^3*Sqrt[a + b*Tanh[x]^2], x, 6, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2] - (a + b*Tanh[x]^2)^(3/2)/(3*b)} -{Tanh[x]^2*Sqrt[a + b*Tanh[x]^2], x, 7, -(((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(2*Sqrt[b])) + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - (1/2)*Tanh[x]*Sqrt[a + b*Tanh[x]^2]} -{Tanh[x]^1*Sqrt[a + b*Tanh[x]^2], x, 5, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2]} -{Tanh[x]^0*Sqrt[a + b*Tanh[x]^2], x, 6, (-Sqrt[b])*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]} -{Coth[x]^1*Sqrt[a + b*Tanh[x]^2], x, 7, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]} -{Coth[x]^2*Sqrt[a + b*Tanh[x]^2], x, 5, Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - Coth[x]*Sqrt[a + b*Tanh[x]^2]} -{Coth[x]^3*Sqrt[a + b*Tanh[x]^2], x, 8, -(((2*a + b)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]])/(2*Sqrt[a])) + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (1/2)*Coth[x]^2*Sqrt[a + b*Tanh[x]^2]} -{Coth[x]^4*Sqrt[a + b*Tanh[x]^2], x, 6, Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((3*a + b)*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(3*a) - (1/3)*Coth[x]^3*Sqrt[a + b*Tanh[x]^2]} -{Coth[x]^5*Sqrt[a + b*Tanh[x]^2], x, 9, -(((8*a^2 + 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]])/(8*a^(3/2))) + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - ((4*a + b)*Coth[x]^2*Sqrt[a + b*Tanh[x]^2])/(8*a) - (1/4)*Coth[x]^4*Sqrt[a + b*Tanh[x]^2]} - - -{Tanh[x]^3*(a + b*Tanh[x]^2)^(3/2), x, 7, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Tanh[x]^2] - (1/3)*(a + b*Tanh[x]^2)^(3/2) - (a + b*Tanh[x]^2)^(5/2)/(5*b)} -{Tanh[x]^2*(a + b*Tanh[x]^2)^(3/2), x, 8, -(((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(8*Sqrt[b])) + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - (1/8)*(5*a + 4*b)*Tanh[x]*Sqrt[a + b*Tanh[x]^2] - (1/4)*b*Tanh[x]^3*Sqrt[a + b*Tanh[x]^2]} -{Tanh[x]^1*(a + b*Tanh[x]^2)^(3/2), x, 6, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Tanh[x]^2] - (1/3)*(a + b*Tanh[x]^2)^(3/2)} -{Tanh[x]^0*(a + b*Tanh[x]^2)^(3/2), x, 7, (-(1/2))*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - (1/2)*b*Tanh[x]*Sqrt[a + b*Tanh[x]^2]} -{Coth[x]^1*(a + b*Tanh[x]^2)^(3/2), x, 8, (-a^(3/2))*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Tanh[x]^2]} -{Coth[x]^2*(a + b*Tanh[x]^2)^(3/2), x, 7, (-b^(3/2))*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - a*Coth[x]*Sqrt[a + b*Tanh[x]^2]} - - -{Sqrt[1 + Tanh[x]^2], x, 5, -ArcSinh[Tanh[x]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Tanh[x])/Sqrt[1 + Tanh[x]^2]]} -{Sqrt[-1 - Tanh[x]^2], x, 6, ArcTan[Tanh[x]/Sqrt[-1 - Tanh[x]^2]] - Sqrt[2]*ArcTan[(Sqrt[2]*Tanh[x])/Sqrt[-1 - Tanh[x]^2]]} - - -{(1 + Tanh[x]^2)^(3/2), x, 6, (-(5/2))*ArcSinh[Tanh[x]] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Tanh[x])/Sqrt[1 + Tanh[x]^2]] - (1/2)*Tanh[x]*Sqrt[1 + Tanh[x]^2]} -{(-1 - Tanh[x]^2)^(3/2), x, 7, (-(5/2))*ArcTan[Tanh[x]/Sqrt[-1 - Tanh[x]^2]] + 2*Sqrt[2]*ArcTan[(Sqrt[2]*Tanh[x])/Sqrt[-1 - Tanh[x]^2]] + (1/2)*Tanh[x]*Sqrt[-1 - Tanh[x]^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[x]^5/Sqrt[a + b*Tanh[x]^2], x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] + ((a - b)*Sqrt[a + b*Tanh[x]^2])/b^2 - (a + b*Tanh[x]^2)^(3/2)/(3*b^2)} -{Tanh[x]^4/Sqrt[a + b*Tanh[x]^2], x, 7, ((a - 2*b)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(2*b^(3/2)) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b] - (Tanh[x]*Sqrt[a + b*Tanh[x]^2])/(2*b)} -{Tanh[x]^3/Sqrt[a + b*Tanh[x]^2], x, 5, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Tanh[x]^2]/b} -{Tanh[x]^2/Sqrt[a + b*Tanh[x]^2], x, 6, -(ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[b]) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b]} -{Tanh[x]^1/Sqrt[a + b*Tanh[x]^2], x, 4, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]} -{Tanh[x]^0/Sqrt[a + b*Tanh[x]^2], x, 3, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b]} -{Coth[x]^1/Sqrt[a + b*Tanh[x]^2], x, 7, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]} -{Coth[x]^2/Sqrt[a + b*Tanh[x]^2], x, 5, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b] - (Coth[x]*Sqrt[a + b*Tanh[x]^2])/a} -{Coth[x]^3/Sqrt[a + b*Tanh[x]^2], x, 8, -(((2*a - b)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]])/(2*a^(3/2))) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - (Coth[x]^2*Sqrt[a + b*Tanh[x]^2])/(2*a)} - - -{Tanh[x]^5/(a + b*Tanh[x]^2)^(3/2), x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - a^2/(b^2*(a + b)*Sqrt[a + b*Tanh[x]^2]) - Sqrt[a + b*Tanh[x]^2]/b^2} -{Tanh[x]^4/(a + b*Tanh[x]^2)^(3/2), x, 7, -(ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/b^(3/2)) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) + (a*Tanh[x])/(b*(a + b)*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^3/(a + b*Tanh[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^2/(a + b*Tanh[x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) - Tanh[x]/((a + b)*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^1/(a + b*Tanh[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - 1/((a + b)*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^0/(a + b*Tanh[x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) + (b*Tanh[x])/(a*(a + b)*Sqrt[a + b*Tanh[x]^2])} -{Coth[x]^1/(a + b*Tanh[x]^2)^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/a^(3/2)) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + b/(a*(a + b)*Sqrt[a + b*Tanh[x]^2])} -{Coth[x]^2/(a + b*Tanh[x]^2)^(3/2), x, 6, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) + (b*Coth[x])/(a*(a + b)*Sqrt[a + b*Tanh[x]^2]) - ((a + 2*b)*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(a^2*(a + b))} - - -{Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2), x, 8, -(ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/b^(5/2)) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + (a*Tanh[x]^3)/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (a*(a + 2*b)*Tanh[x])/(b^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^5/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - a^2/(3*b^2*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (a*(a + 2*b))/(b^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^4/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + (a*Tanh[x])/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - ((a + 4*b)*Tanh[x])/(3*b*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^3/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + a/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^2/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) - Tanh[x]/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - ((2*a - b)*Tanh[x])/(3*a*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^1/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - 1/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Tanh[x]^0/(a + b*Tanh[x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (b*(5*a + 2*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Coth[x]^1/(a + b*Tanh[x]^2)^(5/2), x, 9, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/a^(5/2)) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + b/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])} -{Coth[x]^2/(a + b*Tanh[x]^2)^(5/2), x, 7, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + (b*Coth[x])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (b*(7*a + 4*b)*Coth[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2]) - ((3*a + 2*b)*(a + 4*b)*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(3*a^3*(a + b)^2)} - - -{1/Sqrt[1 + Tanh[x]^2], x, 3, ArcTanh[(Sqrt[2]*Tanh[x])/Sqrt[1 + Tanh[x]^2]]/Sqrt[2]} -{1/Sqrt[-1 - Tanh[x]^2], x, 3, ArcTan[(Sqrt[2]*Tanh[x])/Sqrt[-1 - Tanh[x]^2]]/Sqrt[2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^3)^p*) - - -{(a + b*Tanh[c + d*x]^3)^2, x, 6, (a^2 + b^2)*x + (2*a*b*Log[Cosh[c + d*x]])/d - (b^2*Tanh[c + d*x])/d - (a*b*Tanh[c + d*x]^2)/d - (b^2*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d), -(((a + b)^2*Log[1 - Tanh[c + d*x]])/(2*d)) + ((a - b)^2*Log[1 + Tanh[c + d*x]])/(2*d) - (b^2*Tanh[c + d*x])/d - (a*b*Tanh[c + d*x]^2)/d - (b^2*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)} - - -{1/(1 + Tanh[x]^3), x, 6, x/2 - (2*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(6*(1 + Tanh[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^4)^p*) - - -{Tanh[x]*(a + b*Tanh[x]^4)^(3/2), x, 9, (-(1/4))*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]] + (1/2)*(a + b)^(3/2)*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - (1/4)*(2*(a + b) + b*Tanh[x]^2)*Sqrt[a + b*Tanh[x]^4] - (1/6)*(a + b*Tanh[x]^4)^(3/2)} -{Tanh[x]*(a + b*Tanh[x]^4)^(1/2), x, 8, (-(1/2))*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[x]^2)/Sqrt[a + b*Tanh[x]^4]] + (1/2)*Sqrt[a + b]*ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])] - (1/2)*Sqrt[a + b*Tanh[x]^4]} -{Tanh[x]/(a + b*Tanh[x]^4)^(1/2), x, 4, ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*Sqrt[a + b])} -{Tanh[x]/(a + b*Tanh[x]^4)^(3/2), x, 6, ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*(a + b)^(3/2)) - (a - b*Tanh[x]^2)/(2*a*(a + b)*Sqrt[a + b*Tanh[x]^4])} -{Tanh[x]/(a + b*Tanh[x]^4)^(5/2), x, 7, ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*(a + b)^(5/2)) - (a - b*Tanh[x]^2)/(6*a*(a + b)*(a + b*Tanh[x]^4)^(3/2)) - (3*a^2 - b*(5*a + 2*b)*Tanh[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^4])} - - -(* ::Section:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Tanh[e+f x]^n)^p*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.1 (c+d x)^m (a+b coth)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.1 (c+d x)^m (a+b coth)^n.m deleted file mode 100644 index be1de75..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.1 (c+d x)^m (a+b coth)^n.m +++ /dev/null @@ -1,137 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (c+d x)^m (a+b Coth[e+f x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (b Coth[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Coth[e+f x]^n*) - - -{x^3*Coth[a + b*x], x, 6, -(x^4/4) + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4)} -{x^2*Coth[a + b*x], x, 5, -(x^3/3) + (x^2*Log[1 - E^(2*(a + b*x))])/b + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3)} -{x^1*Coth[a + b*x], x, 4, -(x^2/2) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2)} -{Coth[a + b*x]/x^1, x, 0, Unintegrable[Coth[a + b*x]/x, x]} -{Coth[a + b*x]/x^2, x, 0, Unintegrable[Coth[a + b*x]/x^2, x]} - - -{x^3*Coth[a + b*x]^2, x, 7, -(x^3/b) + x^4/4 - (x^3*Coth[a + b*x])/b + (3*x^2*Log[1 - E^(2*(a + b*x))])/b^2 + (3*x*PolyLog[2, E^(2*(a + b*x))])/b^3 - (3*PolyLog[3, E^(2*(a + b*x))])/(2*b^4)} -{x^2*Coth[a + b*x]^2, x, 6, -(x^2/b) + x^3/3 - (x^2*Coth[a + b*x])/b + (2*x*Log[1 - E^(2*(a + b*x))])/b^2 + PolyLog[2, E^(2*(a + b*x))]/b^3} -{x^1*Coth[a + b*x]^2, x, 3, x^2/2 - (x*Coth[a + b*x])/b + Log[Sinh[a + b*x]]/b^2} -{Coth[a + b*x]^2/x^1, x, 0, Unintegrable[Coth[a + b*x]^2/x, x]} -{Coth[a + b*x]^2/x^2, x, 0, Unintegrable[Coth[a + b*x]^2/x^2, x]} - - -{x^3*Coth[a + b*x]^3, x, 13, -((3*x^2)/(2*b^2)) + x^3/(2*b) - x^4/4 - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Coth[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*(a + b*x))])/b^3 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4) + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4)} -{x^2*Coth[a + b*x]^3, x, 9, x^2/(2*b) - x^3/3 - (x*Coth[a + b*x])/b^2 - (x^2*Coth[a + b*x]^2)/(2*b) + (x^2*Log[1 - E^(2*(a + b*x))])/b + Log[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3)} -{x^1*Coth[a + b*x]^3, x, 7, x/(2*b) - x^2/2 - Coth[a + b*x]/(2*b^2) - (x*Coth[a + b*x]^2)/(2*b) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2)} -{Coth[a + b*x]^3/x^1, x, 0, Unintegrable[Coth[a + b*x]^3/x, x]} -{Coth[a + b*x]^3/x^2, x, 0, Unintegrable[Coth[a + b*x]^3/x^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Coth[e+f x]^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Coth[e+f x])^n with a^2-b^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Coth[e+f x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + a*Coth[e + f*x]), x, 5, (3*d^3*x)/(8*a*f^3) + (3*d*(c + d*x)^2)/(8*a*f^2) + (c + d*x)^3/(4*a*f) + (c + d*x)^4/(8*a*d) - (3*d^3)/(8*f^4*(a + a*Coth[e + f*x])) - (3*d^2*(c + d*x))/(4*f^3*(a + a*Coth[e + f*x])) - (3*d*(c + d*x)^2)/(4*f^2*(a + a*Coth[e + f*x])) - (c + d*x)^3/(2*f*(a + a*Coth[e + f*x]))} -{(c + d*x)^2/(a + a*Coth[e + f*x]), x, 4, (d^2*x)/(4*a*f^2) + (c + d*x)^2/(4*a*f) + (c + d*x)^3/(6*a*d) - d^2/(4*f^3*(a + a*Coth[e + f*x])) - (d*(c + d*x))/(2*f^2*(a + a*Coth[e + f*x])) - (c + d*x)^2/(2*f*(a + a*Coth[e + f*x]))} -{(c + d*x)^1/(a + a*Coth[e + f*x]), x, 3, (d*x)/(4*a*f) + (c + d*x)^2/(4*a*d) - d/(4*f^2*(a + a*Coth[e + f*x])) - (c + d*x)/(2*f*(a + a*Coth[e + f*x]))} -{1/((c + d*x)^1*(a + a*Coth[e + f*x])), x, 7, -(Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) + Log[c + d*x]/(2*a*d) + (CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a*d) + (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d) - (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a*d)} -{1/((c + d*x)^2*(a + a*Coth[e + f*x])), x, 7, (f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) - 1/(d*(c + d*x)*(a + a*Coth[e + f*x])) - (f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a*d^2) - (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^2) + (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^2)} -{1/((c + d*x)^3*(a + a*Coth[e + f*x])), x, 8, -f/(2*a*d^2*(c + d*x)) - (f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - 1/(2*d*(c + d*x)^2*(a + a*Coth[e + f*x])) + f/(d^2*(c + d*x)*(a + a*Coth[e + f*x])) + (f^2*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a*d^3) + (f^2*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3) - (f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a*d^3)} - - -{(c + d*x)^3/(a + a*Coth[e + f*x])^2, x, 10, (-3*d^3*E^(-4*e - 4*f*x))/(512*a^2*f^4) + (3*d^3*E^(-2*e - 2*f*x))/(16*a^2*f^4) - (3*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(128*a^2*f^3) + (3*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(8*a^2*f^3) - (3*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(64*a^2*f^2) + (3*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(8*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^3)/(16*a^2*f) + (E^(-2*e - 2*f*x)*(c + d*x)^3)/(4*a^2*f) + (c + d*x)^4/(16*a^2*d)} -{(c + d*x)^2/(a + a*Coth[e + f*x])^2, x, 8, -(d^2*E^(-4*e - 4*f*x))/(128*a^2*f^3) + (d^2*E^(-2*e - 2*f*x))/(8*a^2*f^3) - (d*E^(-4*e - 4*f*x)*(c + d*x))/(32*a^2*f^2) + (d*E^(-2*e - 2*f*x)*(c + d*x))/(4*a^2*f^2) - (E^(-4*e - 4*f*x)*(c + d*x)^2)/(16*a^2*f) + (E^(-2*e - 2*f*x)*(c + d*x)^2)/(4*a^2*f) + (c + d*x)^3/(12*a^2*d)} -{(c + d*x)^1/(a + a*Coth[e + f*x])^2, x, 7, (3*d*x)/(16*a^2*f) - (d*x^2)/(8*a^2) + (x*(c + d*x))/(4*a^2) - d/(16*f^2*(a + a*Coth[e + f*x])^2) - (c + d*x)/(4*f*(a + a*Coth[e + f*x])^2) - (3*d)/(16*f^2*(a^2 + a^2*Coth[e + f*x])) - (c + d*x)/(4*f*(a^2 + a^2*Coth[e + f*x]))} -{1/((c + d*x)^1*(a + a*Coth[e + f*x])^2), x, 21, -(Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + (Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + Log[c + d*x]/(4*a^2*d) - (CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(4*a^2*d) + (CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(2*a^2*d) + (Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) - (Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) - (Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + (Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d)} -{1/((c + d*x)^2*(a + a*Coth[e + f*x])^2), x, 24, -(1/(4*a^2*d*(c + d*x))) + Cosh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cosh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + (f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) + (f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(a^2*d^2) - (f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(a^2*d^2) - Sinh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Sinh[2*e + 2*f*x]^2/(4*a^2*d*(c + d*x)) + Sinh[4*e + 4*f*x]/(4*a^2*d*(c + d*x)) - (f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2)} - - -{(c + d*x)^3/(a + a*Coth[e + f*x])^3, x, 14, (d^3*E^(-6*e - 6*f*x))/(1728*a^3*f^4) - (9*d^3*E^(-4*e - 4*f*x))/(1024*a^3*f^4) + (9*d^3*E^(-2*e - 2*f*x))/(64*a^3*f^4) + (d^2*E^(-6*e - 6*f*x)*(c + d*x))/(288*a^3*f^3) - (9*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(256*a^3*f^3) + (9*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(32*a^3*f^3) + (d*E^(-6*e - 6*f*x)*(c + d*x)^2)/(96*a^3*f^2) - (9*d*E^(-4*e - 4*f*x)*(c + d*x)^2)/(128*a^3*f^2) + (9*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (E^(-6*e - 6*f*x)*(c + d*x)^3)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^3)/(32*a^3*f) + (3*E^(-2*e - 2*f*x)*(c + d*x)^3)/(16*a^3*f) + (c + d*x)^4/(32*a^3*d)} -{(c + d*x)^2/(a + a*Coth[e + f*x])^3, x, 11, (d^2*E^(-6*e - 6*f*x))/(864*a^3*f^3) - (3*d^2*E^(-4*e - 4*f*x))/(256*a^3*f^3) + (3*d^2*E^(-2*e - 2*f*x))/(32*a^3*f^3) + (d*E^(-6*e - 6*f*x)*(c + d*x))/(144*a^3*f^2) - (3*d*E^(-4*e - 4*f*x)*(c + d*x))/(64*a^3*f^2) + (3*d*E^(-2*e - 2*f*x)*(c + d*x))/(16*a^3*f^2) + (E^(-6*e - 6*f*x)*(c + d*x)^2)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^2)/(32*a^3*f) + (3*E^(-2*e - 2*f*x)*(c + d*x)^2)/(16*a^3*f) + (c + d*x)^3/(24*a^3*d)} -{(c + d*x)^1/(a + a*Coth[e + f*x])^3, x, 11, (11*d*x)/(96*a^3*f) - (d*x^2)/(16*a^3) + (x*(c + d*x))/(8*a^3) - d/(36*f^2*(a + a*Coth[e + f*x])^3) - (c + d*x)/(6*f*(a + a*Coth[e + f*x])^3) - (5*d)/(96*a*f^2*(a + a*Coth[e + f*x])^2) - (c + d*x)/(8*a*f*(a + a*Coth[e + f*x])^2) - (11*d)/(96*f^2*(a^3 + a^3*Coth[e + f*x])) - (c + d*x)/(8*f*(a^3 + a^3*Coth[e + f*x]))} -{1/((c + d*x)^1*(a + a*Coth[e + f*x])^3), x, 53, (-3*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - (Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^3*d) + (CoshIntegral[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(8*a^3*d) - (3*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(8*a^3*d) + (3*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(8*a^3*d) + (3*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (3*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (3*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (3*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) - (Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)} -{1/((c + d*x)^2*(a + a*Coth[e + f*x])^3), x, 60, -(1/(8*a^3*d*(c + d*x))) + (9*Cosh[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cosh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) + Cosh[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Cosh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) + (3*f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) + (3*f*Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) - (3*f*CoshIntegral[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(4*a^3*d^2) + (3*f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*c*f)/d])/(2*a^3*d^2) - (3*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(4*a^3*d^2) - (15*Sinh[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Sinh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - Sinh[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Sinh[4*e + 4*f*x])/(8*a^3*d*(c + d*x)) - (3*Sinh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*f*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*f*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Sinh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (3*f*Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+a Coth[e+f x])^n with m symbolic*) - - -{(c + d*x)^m*(a + a*Coth[e + f*x])^2, x, 0, Unintegrable[(c + d*x)^m*(a + a*Coth[e + f*x])^2, x]} -{(c + d*x)^m*(a + a*Coth[e + f*x])^1, x, 0, Unintegrable[(c + d*x)^m*(a + a*Coth[e + f*x]), x]} -{(c + d*x)^m/(a + a*Coth[e + f*x])^1, x, 2, (c + d*x)^(1 + m)/(2*a*d*(1 + m)) + (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(a*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + a*Coth[e + f*x])^2, x, 4, (c + d*x)^(1 + m)/(4*a^2*d*(1 + m)) + (2^(-2 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(a^2*f*((f*(c + d*x))/d)^m) - (4^(-2 - m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d*x))/d])/(a^2*f*((f*(c + d*x))/d)^m)} -{(c + d*x)^m/(a + a*Coth[e + f*x])^3, x, 5, (c + d*x)^(1 + m)/(8*a^3*d*(1 + m)) + (3*2^(-4 - m)*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(a^3*f*((f*(c + d*x))/d)^m) - (3*2^(-5 - 2*m)*E^(-4*e + (4*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (4*f*(c + d*x))/d])/(a^3*f*((f*(c + d*x))/d)^m) + (2^(-4 - m)*3^(-1 - m)*E^(-6*e + (6*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (6*f*(c + d*x))/d])/(a^3*f*((f*(c + d*x))/d)^m)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Coth[e+f x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m (a+b Coth[e+f x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*(a + b*Coth[e + f*x]), x, 8, (a*(c + d*x)^4)/(4*d) - (b*(c + d*x)^4)/(4*d) + (b*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (3*b*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) - (3*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) + (3*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4)} -{(c + d*x)^2*(a + b*Coth[e + f*x]), x, 7, (a*(c + d*x)^3)/(3*d) - (b*(c + d*x)^3)/(3*d) + (b*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b*d*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^2 - (b*d^2*PolyLog[3, E^(2*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Coth[e + f*x]), x, 6, (a*(c + d*x)^2)/(2*d) - (b*(c + d*x)^2)/(2*d) + (b*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (b*d*PolyLog[2, E^(2*(e + f*x))])/(2*f^2)} -{(a + b*Coth[e + f*x])/(c + d*x)^1, x, 0, Unintegrable[(a + b*Coth[e + f*x])/(c + d*x), x]} -{(a + b*Coth[e + f*x])/(c + d*x)^2, x, 0, Unintegrable[(a + b*Coth[e + f*x])/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Coth[e + f*x])^2, x, 15, -((b^2*(c + d*x)^3)/f) + (a^2*(c + d*x)^4)/(4*d) - (a*b*(c + d*x)^4)/(2*d) + (b^2*(c + d*x)^4)/(4*d) - (b^2*(c + d*x)^3*Coth[e + f*x])/f + (3*b^2*d*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (3*b^2*d^2*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^3 + (3*a*b*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/f^2 - (3*b^2*d^3*PolyLog[3, E^(2*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/f^3 + (3*a*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(2*f^4)} -{(c + d*x)^2*(a + b*Coth[e + f*x])^2, x, 13, -((b^2*(c + d*x)^2)/f) + (a^2*(c + d*x)^3)/(3*d) - (2*a*b*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(3*d) - (b^2*(c + d*x)^2*Coth[e + f*x])/f + (2*b^2*d*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b^2*d^2*PolyLog[2, E^(2*(e + f*x))])/f^3 + (2*a*b*d*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^2 - (a*b*d^2*PolyLog[3, E^(2*(e + f*x))])/f^3} -{(c + d*x)^1*(a + b*Coth[e + f*x])^2, x, 9, b^2*c*x + (b^2*d*x^2)/2 + (a^2*(c + d*x)^2)/(2*d) - (a*b*(c + d*x)^2)/d - (b^2*(c + d*x)*Coth[e + f*x])/f + (2*a*b*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (b^2*d*Log[Sinh[e + f*x]])/f^2 + (a*b*d*PolyLog[2, E^(2*(e + f*x))])/f^2} -{(a + b*Coth[e + f*x])^2/(c + d*x)^1, x, 0, Unintegrable[(a + b*Coth[e + f*x])^2/(c + d*x), x]} -{(a + b*Coth[e + f*x])^2/(c + d*x)^2, x, 0, Unintegrable[(a + b*Coth[e + f*x])^2/(c + d*x)^2, x]} - - -{(c + d*x)^3*(a + b*Coth[e + f*x])^3, x, 28, (-3*b^3*d*(c + d*x)^2)/(2*f^2) - (3*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) - (3*a^2*b*(c + d*x)^4)/(4*d) + (3*a*b^2*(c + d*x)^4)/(4*d) - (b^3*(c + d*x)^4)/(4*d) - (3*b^3*d*(c + d*x)^2*Coth[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Coth[e + f*x])/f - (b^3*(c + d*x)^3*Coth[e + f*x]^2)/(2*f) + (3*b^3*d^2*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)^3*Log[1 - E^(2*(e + f*x))])/f + (3*b^3*d^3*PolyLog[2, E^(2*(e + f*x))])/(2*f^4) + (9*a*b^2*d^2*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^3 + (9*a^2*b*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) - (9*a*b^2*d^3*PolyLog[3, E^(2*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) - (3*b^3*d^2*(c + d*x)*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) + (9*a^2*b*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4) + (3*b^3*d^3*PolyLog[4, E^(2*(e + f*x))])/(4*f^4)} -{(c + d*x)^2*(a + b*Coth[e + f*x])^3, x, 22, (b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - (3*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) - (a^2*b*(c + d*x)^3)/d + (a*b^2*(c + d*x)^3)/d - (b^3*(c + d*x)^3)/(3*d) - (b^3*d*(c + d*x)*Coth[e + f*x])/f^2 - (3*a*b^2*(c + d*x)^2*Coth[e + f*x])/f - (b^3*(c + d*x)^2*Coth[e + f*x]^2)/(2*f) + (6*a*b^2*d*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b^3*d^2*Log[Sinh[e + f*x]])/f^3 + (3*a*b^2*d^2*PolyLog[2, E^(2*(e + f*x))])/f^3 + (3*a^2*b*d*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^2 + (b^3*d*(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, E^(2*(e + f*x))])/(2*f^3)} -{(c + d*x)^1*(a + b*Coth[e + f*x])^3, x, 16, 3*a*b^2*c*x + (b^3*d*x)/(2*f) + (3*a*b^2*d*x^2)/2 + (a^3*(c + d*x)^2)/(2*d) - (3*a^2*b*(c + d*x)^2)/(2*d) - (b^3*(c + d*x)^2)/(2*d) - (b^3*d*Coth[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)*Coth[e + f*x])/f - (b^3*(c + d*x)*Coth[e + f*x]^2)/(2*f) + (3*a^2*b*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (b^3*(c + d*x)*Log[1 - E^(2*(e + f*x))])/f + (3*a*b^2*d*Log[Sinh[e + f*x]])/f^2 + (3*a^2*b*d*PolyLog[2, E^(2*(e + f*x))])/(2*f^2) + (b^3*d*PolyLog[2, E^(2*(e + f*x))])/(2*f^2)} -{(a + b*Coth[e + f*x])^3/(c + d*x)^1, x, 0, Unintegrable[(a + b*Coth[e + f*x])^3/(c + d*x), x]} -{(a + b*Coth[e + f*x])^3/(c + d*x)^2, x, 0, Unintegrable[(a + b*Coth[e + f*x])^3/(c + d*x)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^3/(a + b*Coth[e + f*x]), x, 6, (c + d*x)^4/(4*(a + b)*d) - (b*(c + d*x)^3*Log[1 - (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (3*b*d*(c + d*x)^2*PolyLog[2, (a - b)/(E^(2*(e + f*x))*(a + b))])/(2*(a^2 - b^2)*f^2) + (3*b*d^2*(c + d*x)*PolyLog[3, (a - b)/(E^(2*(e + f*x))*(a + b))])/(2*(a^2 - b^2)*f^3) + (3*b*d^3*PolyLog[4, (a - b)/(E^(2*(e + f*x))*(a + b))])/(4*(a^2 - b^2)*f^4)} -{(c + d*x)^2/(a + b*Coth[e + f*x]), x, 5, (c + d*x)^3/(3*(a + b)*d) - (b*(c + d*x)^2*Log[1 - (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (b*d*(c + d*x)*PolyLog[2, (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f^2) + (b*d^2*PolyLog[3, (a - b)/(E^(2*(e + f*x))*(a + b))])/(2*(a^2 - b^2)*f^3)} -{(c + d*x)^1/(a + b*Coth[e + f*x]), x, 4, (c + d*x)^2/(2*(a + b)*d) - (b*(c + d*x)*Log[1 - (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)*f) + (b*d*PolyLog[2, (a - b)/(E^(2*(e + f*x))*(a + b))])/(2*(a^2 - b^2)*f^2)} -{1/((c + d*x)^1*(a + b*Coth[e + f*x])), x, 0, Unintegrable[1/((c + d*x)*(a + b*Coth[e + f*x])), x]} -{1/((c + d*x)^2*(a + b*Coth[e + f*x])), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Coth[e + f*x])), x]} - - -{(c + d*x)^3/(a + b*Coth[e + f*x])^2, x, 28, -((2*b^2*(c + d*x)^3)/((a^2 - b^2)^2*f)) + (2*b^2*(c + d*x)^3)/((a - b)*(a + b)^2*(a - b - (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^4/(4*(a - b)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) - (2*b*(c + d*x)^3*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^3*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f) + (3*b^2*d^2*(c + d*x)*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^3) - (3*b*d*(c + d*x)^2*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^2) + (3*b^2*d*(c + d*x)^2*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) - (3*b^2*d^3*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/(2*(a^2 - b^2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^3) - (3*b^2*d^2*(c + d*x)*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^3) - (3*b*d^3*PolyLog[4, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/(2*(a - b)^2*(a + b)*f^4) + (3*b^2*d^3*PolyLog[4, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/(2*(a^2 - b^2)^2*f^4)} -{(c + d*x)^2/(a + b*Coth[e + f*x])^2, x, 24, -((2*b^2*(c + d*x)^2)/((a^2 - b^2)^2*f)) + (2*b^2*(c + d*x)^2)/((a - b)*(a + b)^2*(a - b - (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^3/(3*(a - b)^2*d) + (2*b^2*d*(c + d*x)*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) - (2*b*(c + d*x)^2*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^2*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f) + (b^2*d^2*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^3) - (2*b*d*(c + d*x)*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^2) + (2*b^2*d*(c + d*x)*PolyLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) + (b*d^2*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^3) - (b^2*d^2*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^3)} -{(c + d*x)^1/(a + b*Coth[e + f*x])^2, x, 5, -((c + d*x)^2/(2*(a^2 - b^2)*d)) + (b*d - 2*a*c*f - 2*a*d*f*x)^2/(4*a*(a - b)*(a + b)^2*d*f^2) + (b*(c + d*x))/((a^2 - b^2)*f*(a + b*Coth[e + f*x])) + (b*(b*d - 2*a*c*f - 2*a*d*f*x)*Log[1 - (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)^2*f^2) + (a*b*d*PolyLog[2, (a - b)/(E^(2*(e + f*x))*(a + b))])/((a^2 - b^2)^2*f^2)} -{1/((c + d*x)^1*(a + b*Coth[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)*(a + b*Coth[e + f*x])^2), x]} -{1/((c + d*x)^2*(a + b*Coth[e + f*x])^2), x, 0, Unintegrable[1/((c + d*x)^2*(a + b*Coth[e + f*x])^2), x]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.2 Hyperbolic cotangent functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.2 Hyperbolic cotangent functions.m deleted file mode 100644 index dafc1e1..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.2 Hyperbolic cotangent functions.m +++ /dev/null @@ -1,465 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands Involving Hyperbolic Cotangents*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Coth[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x])^(n/2)*) - - -{(b*Coth[c + d*x])^(7/2),x, 7, (b^(7/2)*ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d + (b^(7/2)*ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d - (2*b^3*Sqrt[b*Coth[c + d*x]])/d - (2*b*(b*Coth[c + d*x])^(5/2))/(5*d)} -{(b*Coth[c + d*x])^(5/2),x, 6, -((b^(5/2)*ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d) + (b^(5/2)*ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d - (2*b*(b*Coth[c + d*x])^(3/2))/(3*d)} -{(b*Coth[c + d*x])^(3/2),x, 6, (b^(3/2)*ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d + (b^(3/2)*ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d - (2*b*Sqrt[b*Coth[c + d*x]])/d} -{(b*Coth[c + d*x])^(1/2), x, 5, -((Sqrt[b]*ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d) + (Sqrt[b]*ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]])/d} -{1/(b*Coth[c + d*x])^(1/2), x, 5, ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(Sqrt[b]*d) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(Sqrt[b]*d)} -{1/(b*Coth[c + d*x])^(3/2), x, 6, -(ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(3/2)*d)) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(3/2)*d) - 2/(b*d*Sqrt[b*Coth[c + d*x]])} -{1/(b*Coth[c + d*x])^(5/2), x, 6, ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(5/2)*d) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(5/2)*d) - 2/(3*b*d*(b*Coth[c + d*x])^(3/2))} -{1/(b*Coth[c + d*x])^(7/2), x, 7, -(ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d)) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d) - 2/(5*b*d*(b*Coth[c + d*x])^(5/2)) - 2/(b^3*d*Sqrt[b*Coth[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x])^(n/3)*) - - -{(b*Coth[c + d*x])^(4/3),x, 13, -((Sqrt[3]*b^(4/3)*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d)) + (Sqrt[3]*b^(4/3)*ArcTan[(1 + (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d) + (b^(4/3)*ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)])/d - (3*b*(b*Coth[c + d*x])^(1/3))/d - (b^(4/3)*Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d) + (b^(4/3)*Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d)} -{(b*Coth[c + d*x])^(2/3),x, 12, (Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d) - (Sqrt[3]*b^(2/3)*ArcTan[(1 + (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*d) + (b^(2/3)*ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)])/d - (b^(2/3)*Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d) + (b^(2/3)*Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)])/(4*d)} -{(b*Coth[c + d*x])^(1/3), x, 9, -((Sqrt[3]*b^(1/3)*ArcTan[(b^(2/3) + 2*(b*Coth[c + d*x])^(2/3))/(Sqrt[3]*b^(2/3))])/(2*d)) - (b^(1/3)*Log[b^(2/3) - (b*Coth[c + d*x])^(2/3)])/(2*d) + (b^(1/3)*Log[b^(4/3) + b^(2/3)*(b*Coth[c + d*x])^(2/3) + (b*Coth[c + d*x])^(4/3)])/(4*d)} -{1/(b*Coth[c + d*x])^(1/3), x, 9, (Sqrt[3]*ArcTan[(b^(2/3) + 2*(b*Coth[c + d*x])^(2/3))/(Sqrt[3]*b^(2/3))])/(2*b^(1/3)*d) - Log[b^(2/3) - (b*Coth[c + d*x])^(2/3)]/(2*b^(1/3)*d) + Log[b^(4/3) + b^(2/3)*(b*Coth[c + d*x])^(2/3) + (b*Coth[c + d*x])^(4/3)]/(4*b^(1/3)*d)} -{1/(b*Coth[c + d*x])^(2/3), x, 12, -((Sqrt[3]*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(2/3)*d)) + (Sqrt[3]*ArcTan[(1 + (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(2/3)*d) + ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)]/(b^(2/3)*d) - Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(2/3)*d) + Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(2/3)*d)} -{1/(b*Coth[c + d*x])^(4/3), x, 13, (Sqrt[3]*ArcTan[(1 - (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) - (Sqrt[3]*ArcTan[(1 + (2*(b*Coth[c + d*x])^(1/3))/b^(1/3))/Sqrt[3]])/(2*b^(4/3)*d) + ArcTanh[(b*Coth[c + d*x])^(1/3)/b^(1/3)]/(b^(4/3)*d) - 3/(b*d*(b*Coth[c + d*x])^(1/3)) - Log[b^(2/3) - b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(4/3)*d) + Log[b^(2/3) + b^(1/3)*(b*Coth[c + d*x])^(1/3) + (b*Coth[c + d*x])^(2/3)]/(4*b^(4/3)*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x])^n with n symbolic*) - - -{Coth[a + b*x]^n, x, 2, (Coth[a + b*x]^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Coth[a + b*x]^2])/(b*(1 + n))} -{(b*Coth[c + d*x])^n,x, 2, ((b*Coth[c + d*x])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, Coth[c + d*x]^2])/(b*d*(1 + n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (b Coth[c+d x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x]^2)^n*) - - -{(b*Coth[c + d*x]^2)^n,x, 3, (Coth[c + d*x]*(b*Coth[c + d*x]^2)^n*Hypergeometric2F1[1, (1/2)*(1 + 2*n), (1/2)*(3 + 2*n), Coth[c + d*x]^2])/(d*(1 + 2*n))} - -{(b*Coth[c + d*x]^2)^(3/2),x, 3, -((b*Coth[c + d*x]*Sqrt[b*Coth[c + d*x]^2])/(2*d)) + (b*Sqrt[b*Coth[c + d*x]^2]*Log[Sinh[c + d*x]]*Tanh[c + d*x])/d} -{(b*Coth[c + d*x]^2)^(1/2), x, 2, (Sqrt[b*Coth[c + d*x]^2]*Log[Sinh[c + d*x]]*Tanh[c + d*x])/d} -{1/(b*Coth[c + d*x]^2)^(1/2), x, 2, (Coth[c + d*x]*Log[Cosh[c + d*x]])/(d*Sqrt[b*Coth[c + d*x]^2])} -{1/(b*Coth[c + d*x]^2)^(3/2), x, 3, (Coth[c + d*x]*Log[Cosh[c + d*x]])/(b*d*Sqrt[b*Coth[c + d*x]^2]) - Tanh[c + d*x]/(2*b*d*Sqrt[b*Coth[c + d*x]^2])} - - -{(b*Coth[c + d*x]^2)^(4/3),x, 14, (Sqrt[3]*b*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(1/3))/(2*d*Coth[c + d*x]^(2/3)) - (Sqrt[3]*b*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(1/3))/(2*d*Coth[c + d*x]^(2/3)) + (b*ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^2)^(1/3))/(d*Coth[c + d*x]^(2/3)) - (3*b*Coth[c + d*x]*(b*Coth[c + d*x]^2)^(1/3))/(5*d) - (b*(b*Coth[c + d*x]^2)^(1/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(2/3)) + (b*(b*Coth[c + d*x]^2)^(1/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(2/3))} -{(b*Coth[c + d*x]^2)^(2/3),x, 14, -((Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(2/3))/(2*d*Coth[c + d*x]^(4/3))) + (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(2/3))/(2*d*Coth[c + d*x]^(4/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^2)^(2/3))/(d*Coth[c + d*x]^(4/3)) - ((b*Coth[c + d*x]^2)^(2/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) + ((b*Coth[c + d*x]^2)^(2/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) - (3*(b*Coth[c + d*x]^2)^(2/3)*Tanh[c + d*x])/d} -{(b*Coth[c + d*x]^2)^(1/3), x, 13, (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(1/3))/(2*d*Coth[c + d*x]^(2/3)) - (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^2)^(1/3))/(2*d*Coth[c + d*x]^(2/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^2)^(1/3))/(d*Coth[c + d*x]^(2/3)) - ((b*Coth[c + d*x]^2)^(1/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(2/3)) + ((b*Coth[c + d*x]^2)^(1/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(2/3))} -{1/(b*Coth[c + d*x]^2)^(1/3), x, 13, -((Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(2/3))/(2*d*(b*Coth[c + d*x]^2)^(1/3))) + (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(2/3))/(2*d*(b*Coth[c + d*x]^2)^(1/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(2/3))/(d*(b*Coth[c + d*x]^2)^(1/3)) - (Coth[c + d*x]^(2/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^2)^(1/3)) + (Coth[c + d*x]^(2/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^2)^(1/3))} -{1/(b*Coth[c + d*x]^2)^(2/3), x, 14, -((3*Coth[c + d*x])/(d*(b*Coth[c + d*x]^2)^(2/3))) + (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*d*(b*Coth[c + d*x]^2)^(2/3)) - (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*d*(b*Coth[c + d*x]^2)^(2/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(4/3))/(d*(b*Coth[c + d*x]^2)^(2/3)) - (Coth[c + d*x]^(4/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^2)^(2/3)) + (Coth[c + d*x]^(4/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^2)^(2/3))} -{1/(b*Coth[c + d*x]^2)^(4/3), x, 14, -((Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(2/3))/(2*b*d*(b*Coth[c + d*x]^2)^(1/3))) + (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(2/3))/(2*b*d*(b*Coth[c + d*x]^2)^(1/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(2/3))/(b*d*(b*Coth[c + d*x]^2)^(1/3)) - (Coth[c + d*x]^(2/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*b*d*(b*Coth[c + d*x]^2)^(1/3)) + (Coth[c + d*x]^(2/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*b*d*(b*Coth[c + d*x]^2)^(1/3)) - (3*Tanh[c + d*x])/(5*b*d*(b*Coth[c + d*x]^2)^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x]^3)^n*) - - -{(b*Coth[c + d*x]^3)^n,x, 3, (Coth[c + d*x]*(b*Coth[c + d*x]^3)^n*Hypergeometric2F1[1, (1/2)*(1 + 3*n), (3*(1 + n))/2, Coth[c + d*x]^2])/(d*(1 + 3*n))} - -{(b*Coth[c + d*x]^3)^(3/2),x, 8, -((2*b*Sqrt[b*Coth[c + d*x]^3])/(3*d)) - (b*ArcTan[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) + (b*ArcTanh[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) - (2*b*Coth[c + d*x]^2*Sqrt[b*Coth[c + d*x]^3])/(7*d)} -{(b*Coth[c + d*x]^3)^(1/2), x, 7, (ArcTan[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) + (ArcTanh[Sqrt[Coth[c + d*x]]]*Sqrt[b*Coth[c + d*x]^3])/(d*Coth[c + d*x]^(3/2)) - (2*Sqrt[b*Coth[c + d*x]^3]*Tanh[c + d*x])/d} -{1/(b*Coth[c + d*x]^3)^(1/2), x, 7, -((2*Coth[c + d*x])/(d*Sqrt[b*Coth[c + d*x]^3])) - (ArcTan[Sqrt[Coth[c + d*x]]]*Coth[c + d*x]^(3/2))/(d*Sqrt[b*Coth[c + d*x]^3]) + (ArcTanh[Sqrt[Coth[c + d*x]]]*Coth[c + d*x]^(3/2))/(d*Sqrt[b*Coth[c + d*x]^3])} -{1/(b*Coth[c + d*x]^3)^(3/2), x, 8, -(2/(3*b*d*Sqrt[b*Coth[c + d*x]^3])) + (ArcTan[Sqrt[Coth[c + d*x]]]*Coth[c + d*x]^(3/2))/(b*d*Sqrt[b*Coth[c + d*x]^3]) + (ArcTanh[Sqrt[Coth[c + d*x]]]*Coth[c + d*x]^(3/2))/(b*d*Sqrt[b*Coth[c + d*x]^3]) - (2*Tanh[c + d*x]^2)/(7*b*d*Sqrt[b*Coth[c + d*x]^3])} - - -{(b*Coth[c + d*x]^3)^(4/3),x, 4, -((b*(b*Coth[c + d*x]^3)^(1/3))/d) - (b*Coth[c + d*x]^2*(b*Coth[c + d*x]^3)^(1/3))/(3*d) + b*x*(b*Coth[c + d*x]^3)^(1/3)*Tanh[c + d*x]} -{(b*Coth[c + d*x]^3)^(2/3),x, 3, -(((b*Coth[c + d*x]^3)^(2/3)*Tanh[c + d*x])/d) + x*(b*Coth[c + d*x]^3)^(2/3)*Tanh[c + d*x]^2} -{(b*Coth[c + d*x]^3)^(1/3), x, 2, ((b*Coth[c + d*x]^3)^(1/3)*Log[Sinh[c + d*x]]*Tanh[c + d*x])/d} -{1/(b*Coth[c + d*x]^3)^(1/3), x, 2, (Coth[c + d*x]*Log[Cosh[c + d*x]])/(d*(b*Coth[c + d*x]^3)^(1/3))} -{1/(b*Coth[c + d*x]^3)^(2/3), x, 3, -(Coth[c + d*x]/(d*(b*Coth[c + d*x]^3)^(2/3))) + (x*Coth[c + d*x]^2)/(b*Coth[c + d*x]^3)^(2/3)} -{1/(b*Coth[c + d*x]^3)^(4/3), x, 4, -(1/(b*d*(b*Coth[c + d*x]^3)^(1/3))) + (x*Coth[c + d*x])/(b*(b*Coth[c + d*x]^3)^(1/3)) - Tanh[c + d*x]^2/(3*b*d*(b*Coth[c + d*x]^3)^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x]^4)^n*) - - -{(b*Coth[c + d*x]^4)^n,x, 3, (Coth[c + d*x]*(b*Coth[c + d*x]^4)^n*Hypergeometric2F1[1, (1/2)*(1 + 4*n), (1/2)*(3 + 4*n), Coth[c + d*x]^2])/(d*(1 + 4*n))} - -{(b*Coth[c + d*x]^4)^(3/2),x, 5, -((b*Coth[c + d*x]*Sqrt[b*Coth[c + d*x]^4])/(3*d)) - (b*Coth[c + d*x]^3*Sqrt[b*Coth[c + d*x]^4])/(5*d) - (b*Sqrt[b*Coth[c + d*x]^4]*Tanh[c + d*x])/d + b*x*Sqrt[b*Coth[c + d*x]^4]*Tanh[c + d*x]^2} -{(b*Coth[c + d*x]^4)^(1/2), x, 3, -((Sqrt[b*Coth[c + d*x]^4]*Tanh[c + d*x])/d) + x*Sqrt[b*Coth[c + d*x]^4]*Tanh[c + d*x]^2} -{1/(b*Coth[c + d*x]^4)^(1/2), x, 3, -(Coth[c + d*x]/(d*Sqrt[b*Coth[c + d*x]^4])) + (x*Coth[c + d*x]^2)/Sqrt[b*Coth[c + d*x]^4]} -{1/(b*Coth[c + d*x]^4)^(3/2), x, 5, -(Coth[c + d*x]/(b*d*Sqrt[b*Coth[c + d*x]^4])) + (x*Coth[c + d*x]^2)/(b*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*x]/(3*b*d*Sqrt[b*Coth[c + d*x]^4]) - Tanh[c + d*x]^3/(5*b*d*Sqrt[b*Coth[c + d*x]^4])} - - -{(b*Coth[c + d*x]^4)^(4/3),x, 16, -((Sqrt[3]*b*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(2*d*Coth[c + d*x]^(4/3))) + (Sqrt[3]*b*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(2*d*Coth[c + d*x]^(4/3)) + (b*ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^4)^(1/3))/(d*Coth[c + d*x]^(4/3)) - (3*b*Coth[c + d*x]*(b*Coth[c + d*x]^4)^(1/3))/(7*d) - (3*b*Coth[c + d*x]^3*(b*Coth[c + d*x]^4)^(1/3))/(13*d) - (b*(b*Coth[c + d*x]^4)^(1/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) + (b*(b*Coth[c + d*x]^4)^(1/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) - (3*b*(b*Coth[c + d*x]^4)^(1/3)*Tanh[c + d*x])/d} -{(b*Coth[c + d*x]^4)^(2/3),x, 14, (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(2/3))/(2*d*Coth[c + d*x]^(8/3)) - (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(2/3))/(2*d*Coth[c + d*x]^(8/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^4)^(2/3))/(d*Coth[c + d*x]^(8/3)) - ((b*Coth[c + d*x]^4)^(2/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(8/3)) + ((b*Coth[c + d*x]^4)^(2/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(8/3)) - (3*(b*Coth[c + d*x]^4)^(2/3)*Tanh[c + d*x])/(5*d)} -{(b*Coth[c + d*x]^4)^(1/3), x, 14, -((Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(2*d*Coth[c + d*x]^(4/3))) + (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*(b*Coth[c + d*x]^4)^(1/3))/(2*d*Coth[c + d*x]^(4/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*(b*Coth[c + d*x]^4)^(1/3))/(d*Coth[c + d*x]^(4/3)) - ((b*Coth[c + d*x]^4)^(1/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) + ((b*Coth[c + d*x]^4)^(1/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*Coth[c + d*x]^(4/3)) - (3*(b*Coth[c + d*x]^4)^(1/3)*Tanh[c + d*x])/d} -{1/(b*Coth[c + d*x]^4)^(1/3), x, 14, -((3*Coth[c + d*x])/(d*(b*Coth[c + d*x]^4)^(1/3))) + (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*d*(b*Coth[c + d*x]^4)^(1/3)) - (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*d*(b*Coth[c + d*x]^4)^(1/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(4/3))/(d*(b*Coth[c + d*x]^4)^(1/3)) - (Coth[c + d*x]^(4/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^4)^(1/3)) + (Coth[c + d*x]^(4/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^4)^(1/3))} -{1/(b*Coth[c + d*x]^4)^(2/3), x, 14, -((3*Coth[c + d*x])/(5*d*(b*Coth[c + d*x]^4)^(2/3))) - (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(8/3))/(2*d*(b*Coth[c + d*x]^4)^(2/3)) + (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(8/3))/(2*d*(b*Coth[c + d*x]^4)^(2/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(8/3))/(d*(b*Coth[c + d*x]^4)^(2/3)) - (Coth[c + d*x]^(8/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^4)^(2/3)) + (Coth[c + d*x]^(8/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*d*(b*Coth[c + d*x]^4)^(2/3))} -{1/(b*Coth[c + d*x]^4)^(4/3), x, 16, -((3*Coth[c + d*x])/(b*d*(b*Coth[c + d*x]^4)^(1/3))) + (Sqrt[3]*ArcTan[(1 - 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*b*d*(b*Coth[c + d*x]^4)^(1/3)) - (Sqrt[3]*ArcTan[(1 + 2*Coth[c + d*x]^(1/3))/Sqrt[3]]*Coth[c + d*x]^(4/3))/(2*b*d*(b*Coth[c + d*x]^4)^(1/3)) + (ArcTanh[Coth[c + d*x]^(1/3)]*Coth[c + d*x]^(4/3))/(b*d*(b*Coth[c + d*x]^4)^(1/3)) - (Coth[c + d*x]^(4/3)*Log[1 - Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*b*d*(b*Coth[c + d*x]^4)^(1/3)) + (Coth[c + d*x]^(4/3)*Log[1 + Coth[c + d*x]^(1/3) + Coth[c + d*x]^(2/3)])/(4*b*d*(b*Coth[c + d*x]^4)^(1/3)) - (3*Tanh[c + d*x])/(7*b*d*(b*Coth[c + d*x]^4)^(1/3)) - (3*Tanh[c + d*x]^3)/(13*b*d*(b*Coth[c + d*x]^4)^(1/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b Coth[c+d x]^m)^n*) - - -{(b*Coth[c + d*x]^m)^n,x, 3, (Coth[c + d*x]*(b*Coth[c + d*x]^m)^n*Hypergeometric2F1[1, (1/2)*(1 + m*n), (1/2)*(3 + m*n), Coth[c + d*x]^2])/(d*(1 + m*n))} - -{(b*Coth[c + d*x]^m)^(3/2),x, 3, (2*b*Coth[c + d*x]^(1 + m)*Sqrt[b*Coth[c + d*x]^m]*Hypergeometric2F1[1, (1/4)*(2 + 3*m), (3*(2 + m))/4, Coth[c + d*x]^2])/(d*(2 + 3*m))} -{(b*Coth[c + d*x]^m)^(1/2), x, 3, (2*Coth[c + d*x]*Sqrt[b*Coth[c + d*x]^m]*Hypergeometric2F1[1, (2 + m)/4, (6 + m)/4, Coth[c + d*x]^2])/(d*(2 + m))} -{1/(b*Coth[c + d*x]^m)^(1/2), x, 3, (2*Coth[c + d*x]*Hypergeometric2F1[1, (2 - m)/4, (6 - m)/4, Coth[c + d*x]^2])/(d*(2 - m)*Sqrt[b*Coth[c + d*x]^m])} -{1/(b*Coth[c + d*x]^m)^(3/2), x, 3, (2*Coth[c + d*x]^(1 - m)*Hypergeometric2F1[1, (1/4)*(2 - 3*m), (3*(2 - m))/4, Coth[c + d*x]^2])/(b*d*(2 - 3*m)*Sqrt[b*Coth[c + d*x]^m])} - - -{(b*Coth[c + d*x]^m)^(4/3),x, 3, (3*b*Coth[c + d*x]^(1 + m)*(b*Coth[c + d*x]^m)^(1/3)*Hypergeometric2F1[1, (1/6)*(3 + 4*m), (1/6)*(9 + 4*m), Coth[c + d*x]^2])/(d*(3 + 4*m))} -{(b*Coth[c + d*x]^m)^(2/3),x, 3, (3*Coth[c + d*x]*(b*Coth[c + d*x]^m)^(2/3)*Hypergeometric2F1[1, (1/6)*(3 + 2*m), (1/6)*(9 + 2*m), Coth[c + d*x]^2])/(d*(3 + 2*m))} -{(b*Coth[c + d*x]^m)^(1/3), x, 3, (3*Coth[c + d*x]*(b*Coth[c + d*x]^m)^(1/3)*Hypergeometric2F1[1, (3 + m)/6, (9 + m)/6, Coth[c + d*x]^2])/(d*(3 + m))} -{1/(b*Coth[c + d*x]^m)^(1/3), x, 3, (3*Coth[c + d*x]*Hypergeometric2F1[1, (3 - m)/6, (9 - m)/6, Coth[c + d*x]^2])/(d*(3 - m)*(b*Coth[c + d*x]^m)^(1/3))} -{1/(b*Coth[c + d*x]^m)^(2/3), x, 3, (3*Coth[c + d*x]*Hypergeometric2F1[1, (1/6)*(3 - 2*m), (1/6)*(9 - 2*m), Coth[c + d*x]^2])/(d*(3 - 2*m)*(b*Coth[c + d*x]^m)^(2/3))} -{1/(b*Coth[c + d*x]^m)^(4/3), x, 3, (3*Coth[c + d*x]^(1 - m)*Hypergeometric2F1[1, (1/6)*(3 - 4*m), (1/6)*(9 - 4*m), Coth[c + d*x]^2])/(b*d*(3 - 4*m)*(b*Coth[c + d*x]^m)^(1/3))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Coth[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b Coth[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{(1 + Coth[x])^5, x, 5, 16*x - 8*Coth[x] - 2*(1 + Coth[x])^2 - (2/3)*(1 + Coth[x])^3 - (1/4)*(1 + Coth[x])^4 + 16*Log[Sinh[x]]} -{(1 + Coth[x])^4, x, 4, 8*x - 4*Coth[x] - (1 + Coth[x])^2 - (1/3)*(1 + Coth[x])^3 + 8*Log[Sinh[x]]} -{(1 + Coth[x])^3, x, 3, 4*x - 2*Coth[x] - (1/2)*(1 + Coth[x])^2 + 4*Log[Sinh[x]]} -{(1 + Coth[x])^2, x, 2, 2*x - Coth[x] + 2*Log[Sinh[x]]} -{1/(1 + Coth[x]), x, 2, x/2 - 1/(2*(1 + Coth[x]))} -{1/(1 + Coth[x])^2, x, 3, x/4 - 1/(4*(1 + Coth[x])^2) - 1/(4*(1 + Coth[x]))} -{1/(1 + Coth[x])^3, x, 4, x/8 - 1/(6*(1 + Coth[x])^3) - 1/(8*(1 + Coth[x])^2) - 1/(8*(1 + Coth[x]))} -{1/(1 + Coth[x])^4, x, 5, x/16 - 1/(8*(1 + Coth[x])^4) - 1/(12*(1 + Coth[x])^3) - 1/(16*(1 + Coth[x])^2) - 1/(16*(1 + Coth[x]))} -{1/(1 + Coth[x])^5, x, 6, x/32 - 1/(10*(1 + Coth[x])^5) - 1/(16*(1 + Coth[x])^4) - 1/(24*(1 + Coth[x])^3) - 1/(32*(1 + Coth[x])^2) - 1/(32*(1 + Coth[x]))} - - -{(1 + Coth[x])^(7/2), x, 5, 8*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 8*Sqrt[1 + Coth[x]] - (4/3)*(1 + Coth[x])^(3/2) - (2/5)*(1 + Coth[x])^(5/2)} -{(1 + Coth[x])^(5/2), x, 4, 4*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 4*Sqrt[1 + Coth[x]] - (2/3)*(1 + Coth[x])^(3/2)} -{(1 + Coth[x])^(3/2), x, 3, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]]} -{(1 + Coth[x])^(1/2), x, 2, Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]} -{1/(1 + Coth[x])^(1/2), x, 3, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]]} -{1/(1 + Coth[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Coth[x])^(3/2)) - 1/(2*Sqrt[1 + Coth[x]])} -{1/(1 + Coth[x])^(5/2), x, 5, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(4*Sqrt[2]) - 1/(5*(1 + Coth[x])^(5/2)) - 1/(6*(1 + Coth[x])^(3/2)) - 1/(4*Sqrt[1 + Coth[x]])} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{(a + b*Coth[c + d*x])^5, x, 5, a*(a^4 + 10*a^2*b^2 + 5*b^4)*x - (4*a*b^2*(a^2 + b^2)*Coth[c + d*x])/d - (b*(3*a^2 + b^2)*(a + b*Coth[c + d*x])^2)/(2*d) - (2*a*b*(a + b*Coth[c + d*x])^3)/(3*d) - (b*(a + b*Coth[c + d*x])^4)/(4*d) + (b*(5*a^4 + 10*a^2*b^2 + b^4)*Log[Sinh[c + d*x]])/d} -{(a + b*Coth[c + d*x])^4, x, 4, (a^4 + 6*a^2*b^2 + b^4)*x - (b^2*(3*a^2 + b^2)*Coth[c + d*x])/d - (a*b*(a + b*Coth[c + d*x])^2)/d - (b*(a + b*Coth[c + d*x])^3)/(3*d) + (4*a*b*(a^2 + b^2)*Log[Sinh[c + d*x]])/d} -{(a + b*Coth[c + d*x])^3, x, 3, a*(a^2 + 3*b^2)*x - (2*a*b^2*Coth[c + d*x])/d - (b*(a + b*Coth[c + d*x])^2)/(2*d) + (b*(3*a^2 + b^2)*Log[Sinh[c + d*x]])/d} -{(a + b*Coth[c + d*x])^2, x, 2, (a^2 + b^2)*x - (b^2*Coth[c + d*x])/d + (2*a*b*Log[Sinh[c + d*x]])/d} -{1/(a + b*Coth[c + d*x]),x, 2, (a*x)/(a^2 - b^2) - (b*Log[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)*d)} -{1/(a + b*Coth[c + d*x])^2,x, 3, ((a^2 + b^2)*x)/(a^2 - b^2)^2 + b/((a^2 - b^2)*d*(a + b*Coth[c + d*x])) - (2*a*b*Log[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^2*d)} -{1/(a + b*Coth[c + d*x])^3,x, 4, (a*(a^2 + 3*b^2)*x)/(a^2 - b^2)^3 + b/(2*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^2) + (2*a*b)/((a^2 - b^2)^2*d*(a + b*Coth[c + d*x])) - (b*(3*a^2 + b^2)*Log[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^3*d)} -{1/(a + b*Coth[c + d*x])^4,x, 5, ((a^4 + 6*a^2*b^2 + b^4)*x)/(a^2 - b^2)^4 + b/(3*(a^2 - b^2)*d*(a + b*Coth[c + d*x])^3) + (a*b)/((a^2 - b^2)^2*d*(a + b*Coth[c + d*x])^2) + (b*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(a + b*Coth[c + d*x])) - (4*a*b*(a^2 + b^2)*Log[b*Cosh[c + d*x] + a*Sinh[c + d*x]])/((a^2 - b^2)^4*d)} - -{1/(4 + 6*Coth[c + d*x]), x, 2, -(x/5) + (3*Log[3*Cosh[c + d*x] + 2*Sinh[c + d*x]])/(10*d)} -{1/(4 - 6*Coth[c + d*x]), x, 2, -(x/5) - (3*Log[3*Cosh[c + d*x] - 2*Sinh[c + d*x]])/(10*d)} - - -{Sqrt[a + b*Coth[c + d*x]], x, 5, -((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a - b]])/d) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a + b]])/d} -{1/Sqrt[a + b*Coth[c + d*x]], x, 5, -(ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d)) + ArcTanh[Sqrt[a + b*Coth[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csch[c+d x]^m (a+b Coth[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Sinh[x]^4/(1 + Coth[x]), x, 4, (5*x)/16 + 1/(32*(1 - Coth[x])^2) + 1/(8*(1 - Coth[x])) - 1/(24*(1 + Coth[x])^3) - 3/(32*(1 + Coth[x])^2) - 3/(16*(1 + Coth[x]))} -{Sinh[x]^3/(1 + Coth[x]), x, 3, -((4*Cosh[x])/5) + (4*Cosh[x]^3)/15 - Sinh[x]^3/(5*(1 + Coth[x]))} -{Sinh[x]^2/(1 + Coth[x]), x, 4, -((3*x)/8) - 1/(8*(1 - Coth[x])) + 1/(8*(1 + Coth[x])^2) + 1/(4*(1 + Coth[x]))} -{Sinh[x]^1/(1 + Coth[x]), x, 2, (2*Cosh[x])/3 - Sinh[x]/(3*(1 + Coth[x]))} -{Csch[x]^1/(1 + Coth[x]), x, 1, -(Csch[x]/(1 + Coth[x]))} -{Csch[x]^2/(1 + Coth[x]), x, 2, -Log[1 + Coth[x]]} -{Csch[x]^3/(1 + Coth[x]), x, 2, ArcTanh[Cosh[x]] - Csch[x]} -{Csch[x]^4/(1 + Coth[x]), x, 2, Coth[x] - Coth[x]^2/2} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Sinh[x]^4/(a + b*Coth[x]), x, 5, -(((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Coth[x]])/(16*(a + b)^3)) + ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Coth[x]])/(16*(a - b)^3) - (b^5*Log[a + b*Coth[x]])/(a^2 - b^2)^3 - ((4*b^3 - a*(7 - (3*a^2)/b^2)*b^2*Coth[x])*Sinh[x]^2)/(8*(a^2 - b^2)^2) - ((b - a*Coth[x])*Sinh[x]^4)/(4*(a^2 - b^2))} -{Sinh[x]^3/(a + b*Coth[x]), x, 9, -((b^4*ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) + (a*b^2*Cosh[x])/(a^2 - b^2)^2 - (a*Cosh[x])/(a^2 - b^2) + (a*Cosh[x]^3)/(3*(a^2 - b^2)) - (b^3*Sinh[x])/(a^2 - b^2)^2 - (b*Sinh[x]^3)/(3*(a^2 - b^2))} -{Sinh[x]^2/(a + b*Coth[x]), x, 4, ((a + 2*b)*Log[1 - Coth[x]])/(4*(a + b)^2) - ((a - 2*b)*Log[1 + Coth[x]])/(4*(a - b)^2) - (b^3*Log[a + b*Coth[x]])/(a^2 - b^2)^2 - ((b - a*Coth[x])*Sinh[x]^2)/(2*(a^2 - b^2))} -{Sinh[x]^1/(a + b*Coth[x]), x, 5, -((b^2*ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2)} -{Csch[x]^1/(a + b*Coth[x]), x, 2, -(ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b^2]]/Sqrt[a^2 - b^2])} -{Csch[x]^2/(a + b*Coth[x]), x, 2, -(Log[a + b*Coth[x]]/b)} -{Csch[x]^3/(a + b*Coth[x]), x, 5, (a*ArcTanh[Cosh[x]])/b^2 - (Sqrt[a^2 - b^2]*ArcTanh[((b + a*Coth[x])*Sinh[x])/Sqrt[a^2 - b^2]])/b^2 - Csch[x]/b} -{Csch[x]^4/(a + b*Coth[x]), x, 3, (a*Coth[x])/b^2 - Coth[x]^2/(2*b) - ((a^2 - b^2)*Log[a + b*Coth[x]])/b^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Coth[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Cosh[x]^4/(1 + Coth[x]), x, 5, x/16 + 1/(32*(1 - Coth[x])^2) - 1/(8*(1 - Coth[x])) - 1/(24*(1 + Coth[x])^3) + 5/(32*(1 + Coth[x])^2) - 3/(16*(1 + Coth[x]))} -{Cosh[x]^3/(1 + Coth[x]), x, 9, Cosh[x]^5/5 - Sinh[x]^3/3 - Sinh[x]^5/5} -{Cosh[x]^2/(1 + Coth[x]), x, 5, x/8 - 1/(8*(1 - Coth[x])) + 1/(8*(1 + Coth[x])^2) - 1/(4*(1 + Coth[x]))} -{Cosh[x]^1/(1 + Coth[x]), x, 8, Cosh[x]^3/3 - Sinh[x]^3/3} -{Sech[x]^1/(1 + Coth[x]), x, 8, ArcTan[Sinh[x]] + Cosh[x] - Sinh[x]} -{Sech[x]^2/(1 + Coth[x]), x, 3, -Log[1 + Coth[x]] - Log[Tanh[x]] + Tanh[x]} -{Sech[x]^3/(1 + Coth[x]), x, 8, (-(1/2))*ArcTan[Sinh[x]] - Sech[x] + (1/2)*Sech[x]*Tanh[x]} -{Sech[x]^4/(1 + Coth[x]), x, 4, Tanh[x]^2/2 - Tanh[x]^3/3} - - -{Sech[x]^2*Sqrt[1 + Coth[x]], x, 4, ArcTanh[Sqrt[1 + Coth[x]]] + Sqrt[1 + Coth[x]]*Tanh[x]} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Cosh[x]^4/(a + b*Coth[x]), x, 5, -((a*(3*a + b)*Log[1 - Coth[x]])/(16*(a + b)^3)) + (a*(3*a - b)*Log[1 + Coth[x]])/(16*(a - b)^3) - (a^4*b*Log[a + b*Coth[x]])/(a^2 - b^2)^3 - ((4*b*(2*a^2 - b^2) - a*(5*a^2 - b^2)*Coth[x])*Sinh[x]^2)/(8*(a^2 - b^2)^2) - ((b - a*Coth[x])*Sinh[x]^4)/(4*(a^2 - b^2))} -{Cosh[x]^3/(a + b*Coth[x]), x, 10, (a^3*b*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (a^2*b*Cosh[x])/(a^2 - b^2)^2 - (b*Cosh[x]^3)/(3*(a^2 - b^2)) + (a*b^2*Sinh[x])/(a^2 - b^2)^2 + (a*Sinh[x])/(a^2 - b^2) + (a*Sinh[x]^3)/(3*(a^2 - b^2))} -{Cosh[x]^2/(a + b*Coth[x]), x, 4, -((a*Log[1 - Coth[x]])/(4*(a + b)^2)) + (a*Log[1 + Coth[x]])/(4*(a - b)^2) - (a^2*b*Log[a + b*Coth[x]])/(a^2 - b^2)^2 - ((b - a*Coth[x])*Sinh[x]^2)/(2*(a^2 - b^2))} -{Cosh[x]^1/(a + b*Coth[x]), x, 6, (a*b*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)} -{Sech[x]^1/(a + b*Coth[x]), x, 6, ArcTan[Sinh[x]]/a + (b*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])} -{Sech[x]^2/(a + b*Coth[x]), x, 3, -((b*Log[a + b*Coth[x]])/a^2) - (b*Log[Tanh[x]])/a^2 + Tanh[x]/a} -{Sech[x]^3/(a + b*Coth[x]), x, 9, ArcTan[Sinh[x]]/(2*a) - (b^2*ArcTan[Sinh[x]])/a^3 + (b*Sqrt[a^2 - b^2]*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/a^3 - (b*Sech[x])/a^2 + (Sech[x]*Tanh[x])/(2*a)} -{Sech[x]^4/(a + b*Coth[x]), x, 3, -((b*(a^2 - b^2)*Log[a + b*Coth[x]])/a^4) - (b*(a^2 - b^2)*Log[Tanh[x]])/a^4 + ((a^2 - b^2)*Tanh[x])/a^3 + (b*Tanh[x]^2)/(2*a^2) - Tanh[x]^3/(3*a)} - - -(* Following hangs Mathematica: *) -{Sech[x]/(I + 2*Coth[x]), x, 6, (-I)*ArcTan[Sinh[x]] - (2*ArcTanh[(Cosh[x] - 2*I*Sinh[x])/Sqrt[5]])/Sqrt[5]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Coth[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 = 0*) - - -{Tanh[x]^4/(1 + Coth[x]), x, 6, (5*x)/2 - 2*Log[Cosh[x]] - (5*Tanh[x])/2 + Tanh[x]^2 - (5*Tanh[x]^3)/6 + Tanh[x]^3/(2*(1 + Coth[x]))} -{Tanh[x]^3/(1 + Coth[x]), x, 5, -((3*x)/2) + 2*Log[Cosh[x]] + (3*Tanh[x])/2 - Tanh[x]^2 + Tanh[x]^2/(2*(1 + Coth[x]))} -{Tanh[x]^2/(1 + Coth[x]), x, 4, (3*x)/2 - Log[Cosh[x]] - (3*Tanh[x])/2 + Tanh[x]/(2*(1 + Coth[x]))} -{Tanh[x]^1/(1 + Coth[x]), x, 4, -(x/2) + 1/(2*(1 + Coth[x])) + Log[Cosh[x]]} -{Tanh[x]^0/(1 + Coth[x]), x, 2, x/2 - 1/(2*(1 + Coth[x]))} -{Coth[x]^1/(1 + Coth[x]), x, 2, x/2 + 1/(2*(1 + Coth[x]))} -{Coth[x]^2/(1 + Coth[x]), x, 3, -(x/2) - 1/(2*(1 + Coth[x])) + Log[Sinh[x]]} -{Coth[x]^3/(1 + Coth[x]), x, 3, (3*x)/2 - (3*Coth[x])/2 + Coth[x]^2/(2*(1 + Coth[x])) - Log[Sinh[x]]} -{Coth[x]^4/(1 + Coth[x]), x, 4, -((3*x)/2) + (3*Coth[x])/2 - Coth[x]^2 + Coth[x]^3/(2*(1 + Coth[x])) + 2*Log[Sinh[x]]} - - -{Coth[x]*(1 + Coth[x])^(3/2), x, 4, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]] - (2/3)*(1 + Coth[x])^(3/2)} -{Coth[x]*Sqrt[1 + Coth[x]], x, 3, Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]]} -{Coth[x]/Sqrt[1 + Coth[x]], x, 3, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] + 1/Sqrt[1 + Coth[x]]} -{Coth[x]/(1 + Coth[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) + 1/(3*(1 + Coth[x])^(3/2)) - 1/(2*Sqrt[1 + Coth[x]])} - - -{Coth[x]^2*(1 + Coth[x])^(3/2), x, 4, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]] - (2/5)*(1 + Coth[x])^(5/2)} -{Coth[x]^2*Sqrt[1 + Coth[x]], x, 3, Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2/3)*(1 + Coth[x])^(3/2)} -{Coth[x]^2/Sqrt[1 + Coth[x]], x, 4, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]] - 2*Sqrt[1 + Coth[x]]} -{Coth[x]^2/(1 + Coth[x])^(3/2), x, 4, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Coth[x])^(3/2)) + 3/(2*Sqrt[1 + Coth[x]])} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2 /= 0*) - - -{Tanh[x]^4/(a + b*Coth[x]), x, 6, (a*x)/(a^2 - b^2) - (b*(a^2 + b^2)*Log[Cosh[x]])/a^4 - (b^5*Log[b*Cosh[x] + a*Sinh[x]])/(a^4*(a^2 - b^2)) - ((a^2 + b^2)*Tanh[x])/a^3 + (b*Tanh[x]^2)/(2*a^2) - Tanh[x]^3/(3*a)} -{Tanh[x]^3/(a + b*Coth[x]), x, 5, -((b*x)/(a^2 - b^2)) + ((a^2 + b^2)*Log[Cosh[x]])/a^3 + (b^4*Log[b*Cosh[x] + a*Sinh[x]])/(a^3*(a^2 - b^2)) + (b*Tanh[x])/a^2 - Tanh[x]^2/(2*a)} -{Tanh[x]^2/(a + b*Coth[x]), x, 4, (a*x)/(a^2 - b^2) - (b*Log[Cosh[x]])/a^2 - (b^3*Log[b*Cosh[x] + a*Sinh[x]])/(a^2*(a^2 - b^2)) - Tanh[x]/a} -{Tanh[x]^1/(a + b*Coth[x]), x, 3, -((b*x)/(a^2 - b^2)) + Log[Cosh[x]]/a + (b^2*Log[b*Cosh[x] + a*Sinh[x]])/(a*(a^2 - b^2))} -{Tanh[x]^0/(a + b*Coth[x]), x, 2, (a*x)/(a^2 - b^2) - (b*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)} -{Coth[x]^1/(a + b*Coth[x]), x, 2, -((b*x)/(a^2 - b^2)) + (a*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)} -{Coth[x]^2/(a + b*Coth[x]), x, 4, -((a*x)/b^2) + (a^3*x)/(b^2*(a^2 - b^2)) + Log[Sinh[x]]/b - (a^2*Log[b*Cosh[x] + a*Sinh[x]])/(b*(a^2 - b^2))} -{Coth[x]^3/(a + b*Coth[x]), x, 5, -((b*x)/(a^2 - b^2)) - Coth[x]/b + (a^3*Log[a + b*Coth[x]])/(b^2*(a^2 - b^2)) + (a*Log[Sinh[x]])/(a^2 - b^2)} -{Coth[x]^4/(a + b*Coth[x]), x, 6, (a*x)/(a^2 - b^2) + (a*Coth[x])/b^2 - Coth[x]^2/(2*b) - (a^4*Log[a + b*Coth[x]])/(b^3*(a^2 - b^2)) - (b*Log[Sinh[x]])/(a^2 - b^2)} -{Coth[x]^5/(a + b*Coth[x]), x, 7, -((b*x)/(a^2 - b^2)) - ((a^2 + b^2)*Coth[x])/b^3 + (a*Coth[x]^2)/(2*b^2) - Coth[x]^3/(3*b) + (a^5*Log[a + b*Coth[x]])/(b^4*(a^2 - b^2)) + (a*Log[Sinh[x]])/(a^2 - b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Csch[c+d x]^2 (a+b Coth[c+d x]^n)^p*) - - -{x*Csch[x]^2/(a + b*Coth[x])^2, x, 3, -((a*x)/(b*(a^2 - b^2))) + x/(b*(a + b*Coth[x])) + Log[b*Cosh[x] + a*Sinh[x]]/(a^2 - b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Coth[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Coth[a+2 Log[x]]^p*) - - -{x^3*Coth[a + 2*Log[x]], x, 4, x^4/4 + Log[1 - E^(2*a)*x^4]/(2*E^(2*a))} -{x^2*Coth[a + 2*Log[x]], x, 5, x^3/3 + ArcTan[E^(a/2)*x]/E^((3*a)/2) - ArcTanh[E^(a/2)*x]/E^((3*a)/2)} -{x^1*Coth[a + 2*Log[x]], x, 4, x^2/2 - ArcTanh[E^a*x^2]/E^a} -{x^0*Coth[a + 2*Log[x]], x, 5, x - ArcTan[E^(a/2)*x]/E^(a/2) - ArcTanh[E^(a/2)*x]/E^(a/2)} -{Coth[a + 2*Log[x]]/x^1, x, 2, Log[Sinh[a + 2*Log[x]]]/2} -{Coth[a + 2*Log[x]]/x^2, x, 5, x^(-1) + E^(a/2)*ArcTan[E^(a/2)*x] - E^(a/2)*ArcTanh[E^(a/2)*x]} -{Coth[a + 2*Log[x]]/x^3, x, 4, 1/(2*x^2) - E^a*ArcTanh[E^a*x^2]} - - -{x^3*Coth[a + 2*Log[x]]^2, x, 4, x^4/4 + 1/(E^(2*a)*(1 - E^(2*a)*x^4)) + Log[1 - E^(2*a)*x^4]/E^(2*a)} -{x^2*Coth[a + 2*Log[x]]^2, x, 6, x^3/3 + x^3/(1 - E^(2*a)*x^4) + (3*ArcTan[E^(a/2)*x])/(2*E^((3*a)/2)) - (3*ArcTanh[E^(a/2)*x])/(2*E^((3*a)/2))} -{x^1*Coth[a + 2*Log[x]]^2, x, 5, x^2/2 + x^2/(1 - E^(2*a)*x^4) - ArcTanh[E^a*x^2]/E^a} -{x^0*Coth[a + 2*Log[x]]^2, x, 7, x + x/(1 - E^(2*a)*x^4) - ArcTan[E^(a/2)*x]/(2*E^(a/2)) - ArcTanh[E^(a/2)*x]/(2*E^(a/2))} -{Coth[a + 2*Log[x]]^2/x^1, x, 3, -Coth[a + 2*Log[x]]/2 + Log[x]} -{Coth[a + 2*Log[x]]^2/x^2, x, 6, -(1/(x*(1 - E^(2*a)*x^4))) + (2*E^(2*a)*x^3)/(1 - E^(2*a)*x^4) - (1/2)*E^(a/2)*ArcTan[E^(a/2)*x] + (1/2)*E^(a/2)*ArcTanh[E^(a/2)*x]} -{Coth[a + 2*Log[x]]^2/x^3, x, 5, -(1/(2*x^2*(1 - E^(2*a)*x^4))) + (3*E^(2*a)*x^2)/(2*(1 - E^(2*a)*x^4)) + E^a*ArcTanh[E^a*x^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Coth[a+b Log[x]]^p with m symbolic*) - - -{(e*x)^m*Coth[a + 2*Log[x]]^1, x, 3, (e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2*a)*x^4])/(e*(1 + m))} -{(e*x)^m*Coth[a + 2*Log[x]]^2, x, 4, (e*x)^(1 + m)/(e*(1 + m)) + (e*x)^(1 + m)/(e*(1 - E^(2*a)*x^4)) - ((e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2*a)*x^4])/e} -{(e*x)^m*Coth[a + 2*Log[x]]^3, x, 5, ((3 + m)*(5 + m)*(e*x)^(1 + m))/(8*e*(1 + m)) - ((e*x)^(1 + m)*(1 + E^(2*a)*x^4)^2)/(4*e*(1 - E^(2*a)*x^4)^2) - ((e*x)^(1 + m)*(E^(2*a)*(3 - m) - E^(4*a)*(5 + m)*x^4))/(E^(2*a)*(8*e*(1 - E^(2*a)*x^4))) - ((9 + 2*m + m^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2*a)*x^4])/(4*e*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Coth[a+b Log[x]]^p with p symbolic*) -(**) - - -{Coth[a + b*Log[x]]^p, x, 3, (x*(-1 - E^(2*a)*x^(2*b))^p*AppellF1[1/(2*b), p, -p, (1/2)*(2 + 1/b), E^(2*a)*x^(2*b), (-E^(2*a))*x^(2*b)])/(1 + E^(2*a)*x^(2*b))^p} -{(e*x)^m*Coth[a + b*Log[x]]^p, x, 3, ((e*x)^(1 + m)*(-1 - E^(2*a)*x^(2*b))^p*AppellF1[(1 + m)/(2*b), p, -p, 1 + (1 + m)/(2*b), E^(2*a)*x^(2*b), (-E^(2*a))*x^(2*b)])/((1 + E^(2*a)*x^(2*b))^p*(e*(1 + m)))} - - -{Coth[a + 1/2*Log[x]]^p, x, 2, -(((-1 - E^(2*a)*x)^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 + E^(2*a)*x)])/(2^p*E^(2*a)*(1 + p)))} -{Coth[a + 1/4*Log[x]]^p, x, 4, ((-1 - E^(2*a)*Sqrt[x])^(1 + p)*(1 - E^(2*a)*Sqrt[x])^(1 - p))/E^(4*a) - (2^(1 - p)*p*(-1 - E^(2*a)*Sqrt[x])^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 + E^(2*a)*Sqrt[x])])/(E^(4*a)*(1 + p))} -{Coth[a + 1/6*Log[x]]^p, x, 5, (p*(-1 - E^(2*a)*x^(1/3))^(1 + p)*(1 - E^(2*a)*x^(1/3))^(1 - p))/E^(6*a) + ((-1 - E^(2*a)*x^(1/3))^(1 + p)*(1 - E^(2*a)*x^(1/3))^(1 - p)*x^(1/3))/E^(4*a) - ((1 + 2*p^2)*(-1 - E^(2*a)*x^(1/3))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 + E^(2*a)*x^(1/3))])/(2^p*E^(6*a)*(1 + p))} -{Coth[a + 1/8*Log[x]]^p, x, 5, ((1/3)*(-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*(E^(4*a)*(3 + 2*p^2) + 2*E^(6*a)*p*x^(1/4)))/E^(12*a) + ((-1 - E^(2*a)*x^(1/4))^(1 + p)*(1 - E^(2*a)*x^(1/4))^(1 - p)*Sqrt[x])/E^(4*a) - (2^(2 - p)*p*(2 + p^2)*(-1 - E^(2*a)*x^(1/4))^(1 + p)*Hypergeometric2F1[p, 1 + p, 2 + p, (1/2)*(1 + E^(2*a)*x^(1/4))])/(E^(8*a)*(3*(1 + p)))} - - -{Coth[a + 1*Log[x]]^p, x, 3, (x*(-1 - E^(2*a)*x^2)^p*AppellF1[1/2, p, -p, 3/2, E^(2*a)*x^2, (-E^(2*a))*x^2])/(1 + E^(2*a)*x^2)^p} -{Coth[a + 2*Log[x]]^p, x, 3, (x*(-1 - E^(2*a)*x^4)^p*AppellF1[1/4, p, -p, 5/4, E^(2*a)*x^4, (-E^(2*a))*x^4])/(1 + E^(2*a)*x^4)^p} -{Coth[a + 3*Log[x]]^p, x, 3, (x*(-1 - E^(2*a)*x^6)^p*AppellF1[1/6, p, -p, 7/6, E^(2*a)*x^6, (-E^(2*a))*x^6])/(1 + E^(2*a)*x^6)^p} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Coth[a+b Log[c x^n]]^p*) - - -{x^3*Coth[d*(a + b*Log[c*x^n])], x, 4, x^4/4 - (x^4*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/2} -{x^2*Coth[d*(a + b*Log[c*x^n])], x, 4, x^3/3 - (2*x^3*Hypergeometric2F1[1, 3/(2*b*d*n), 1 + 3/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/3} -{x^1*Coth[d*(a + b*Log[c*x^n])], x, 4, x^2/2 - x^2*Hypergeometric2F1[1, 1/(b*d*n), 1 + 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)]} -{x^0*Coth[d*(a + b*Log[c*x^n])], x, 4, x - 2*x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)]} -{Coth[d*(a + b*Log[c*x^n])]/x^1, x, 2, Log[Sinh[a*d + b*d*Log[c*x^n]]]/(b*d*n)} -{Coth[d*(a + b*Log[c*x^n])]/x^2, x, 4, -(1/x) + (2*Hypergeometric2F1[1, -(1/(2*b*d*n)), 1 - 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/x} -{Coth[d*(a + b*Log[c*x^n])]/x^3, x, 4, -1/(2*x^2) + Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)]/x^2} - - -{x^3*Coth[d*(a + b*Log[c*x^n])]^2, x, 5, (1/4)*(1 + 4/(b*d*n))*x^4 + (x^4*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^4*Hypergeometric2F1[1, 2/(b*d*n), 1 + 2/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^2*Coth[d*(a + b*Log[c*x^n])]^2, x, 5, (1/3)*(1 + 3/(b*d*n))*x^3 + (x^3*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^3*Hypergeometric2F1[1, 3/(2*b*d*n), 1 + 3/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^1*Coth[d*(a + b*Log[c*x^n])]^2, x, 5, (1/2)*(1 + 2/(b*d*n))*x^2 + (x^2*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x^2*Hypergeometric2F1[1, 1/(b*d*n), 1 + 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n)} -{x^0*Coth[d*(a + b*Log[c*x^n])]^2, x, 5, (1 + 1/(b*d*n))*x + (x*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*x*Hypergeometric2F1[1, 1/(2*b*d*n), 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n)} -{Coth[d*(a + b*Log[c*x^n])]^2/x^1, x, 3, -(Coth[a*d + b*d*Log[c*x^n]]/(b*d*n)) + Log[x]} -{Coth[d*(a + b*Log[c*x^n])]^2/x^2, x, 5, -((1 - 1/(b*d*n))/x) + (1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*Hypergeometric2F1[1, -(1/(2*b*d*n)), 1 - 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n*x)} -{Coth[d*(a + b*Log[c*x^n])]^2/x^3, x, 5, (2 - b*d*n)/(2*b*d*n*x^2) + (1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(b*d*n*x^2*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*Hypergeometric2F1[1, -(1/(b*d*n)), 1 - 1/(b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*n*x^2)} - - -{Coth[a + b*Log[c*x^n]]^3/x, x, 3, -(Coth[a + b*Log[c*x^n]]^2/(2*b*n)) + Log[Sinh[a + b*Log[c*x^n]]]/(b*n)} -{Coth[a + b*Log[c*x^n]]^4/x, x, 4, -(Coth[a + b*Log[c*x^n]]/(b*n)) - Coth[a + b*Log[c*x^n]]^3/(3*b*n) + Log[x]} -{Coth[a + b*Log[c*x^n]]^5/x, x, 4, -(Coth[a + b*Log[c*x^n]]^2/(2*b*n)) - Coth[a + b*Log[c*x^n]]^4/(4*b*n) + Log[Sinh[a + b*Log[c*x^n]]]/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Coth[a+b Log[c x^n]]^p with m symbolic*) - - -{(e*x)^m*Coth[d*(a + b*Log[c*x^n])]^1, x, 4, (e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(e*(1 + m))} -{(e*x)^m*Coth[d*(a + b*Log[c*x^n])]^2, x, 5, ((1 + m + b*d*n)*(e*x)^(1 + m))/(b*d*e*(1 + m)*n) + ((e*x)^(1 + m)*(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))/(b*d*e*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b*d*e*n)} -{(e*x)^m*Coth[d*(a + b*Log[c*x^n])]^3, x, 6, ((1 + m + b*d*n)*(1 + m + 2*b*d*n)*(e*x)^(1 + m))/(2*b^2*d^2*e*(1 + m)*n^2) - ((e*x)^(1 + m)*(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^2)/(2*b*d*e*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d))^2) + ((e*x)^(1 + m)*((E^(2*a*d)*(1 + m - 2*b*d*n))/n + (E^(4*a*d)*(1 + m + 2*b*d*n)*(c*x^n)^(2*b*d))/n))/(E^(2*a*d)*(2*b^2*d^2*e*n*(1 - E^(2*a*d)*(c*x^n)^(2*b*d)))) - ((1 + 2*m + m^2 + 2*b^2*d^2*n^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(b^2*d^2*e*(1 + m)*n^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m Coth[a+b Log[c x^n]]^p with p symbolic*) - - -{Coth[d*(a + b*Log[c*x^n])]^p, x, 4, (x*(-1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[1/(2*b*d*n), p, -p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/(1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p} -{(e*x)^m*Coth[d*(a + b*Log[c*x^n])]^p, x, 4, ((e*x)^(1 + m)*(-1 - E^(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1[(1 + m)/(2*b*d*n), p, -p, 1 + (1 + m)/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), (-E^(2*a*d))*(c*x^n)^(2*b*d)])/((1 + E^(2*a*d)*(c*x^n)^(2*b*d))^p*(e*(1 + m)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Coth[a+b Log[c x^n]]^(p/2)*) - - -{Coth[a + b*Log[c*x^n]]^(5/2)/x, x, 7, -(ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - (2*Coth[a + b*Log[c*x^n]]^(3/2))/(3*b*n)} -{Coth[a + b*Log[c*x^n]]^(3/2)/x, x, 7, ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - (2*Sqrt[Coth[a + b*Log[c*x^n]]])/(b*n)} -{Sqrt[Coth[a + b*Log[c*x^n]]]/x, x, 6, -(ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n)} -{1/(x*Sqrt[Coth[a + b*Log[c*x^n]]]), x, 6, ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n)} -{1/(x*Coth[a + b*Log[c*x^n]]^(3/2)), x, 7, -(ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n)) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - 2/(b*n*Sqrt[Coth[a + b*Log[c*x^n]]])} -{1/(x*Coth[a + b*Log[c*x^n]]^(5/2)), x, 7, ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) + ArcTanh[Sqrt[Coth[a + b*Log[c*x^n]]]]/(b*n) - 2/(3*b*n*Coth[a + b*Log[c*x^n]]^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[d+e x]^m (a+b Coth[d+e x]^2+c Coth[d+e x]^4)^n*) - - -{Coth[x]^5/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 8, ((b - 2*c)*ArcTanh[(b + 2*c*Coth[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])])/(4*c^(3/2)) + ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b + c]) - Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4]/(2*c)} -{Coth[x]^3/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 7, -(ArcTanh[(b + 2*c*Coth[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[c])) + ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b + c])} -{Coth[x]/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 4, ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b + c])} -{Tanh[x]/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 8, -(ArcTanh[(2*a + b*Coth[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a])) + ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b + c])} -{Tanh[x]^3/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 11, -(ArcTanh[(2*a + b*Coth[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a])) + (b*ArcTanh[(2*a + b*Coth[x]^2)/(2*Sqrt[a]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])])/(4*a^(3/2)) + ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b + c]) - (Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4]*Tanh[x]^2)/(2*a)} - - -(* {Coth[x]^5*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 0, 0} *) -(* {Coth[x]^3*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 0, 0} *) -{Coth[x]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 8, -(((b + 2*c)*ArcTanh[(b + 2*c*Coth[x]^2)/(2*Sqrt[c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])])/(4*Sqrt[c])) + (1/2)*Sqrt[a + b + c]*ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])] - (1/2)*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4]} -(* {Tanh[x]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 0, 0} *) -(* {Tanh[x]^3*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4], x, 0, 0} *) - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Coth[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Coth[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Coth[a*c + b*c*x]^2)^(5/2), x, 9, (E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c) - (4*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4) + (26*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3) - (55*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(6*b*c*(1 - E^(2*c*(a + b*x)))^2) + (25*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(4*b*c*(1 - E^(2*c*(a + b*x)))) - (15*ArcTanh[E^(c*(a + b*x))]*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(4*b*c)} -{E^(c*(a + b*x))*(Coth[a*c + b*c*x]^2)^(3/2), x, 8, (E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c) - (2*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2) + (3*E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))) - (3*ArcTanh[E^(c*(a + b*x))]*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c)} -{E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2], x, 4, (E^(c*(a + b*x))*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c) - (2*ArcTanh[E^(c*(a + b*x))]*Sqrt[Coth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c)} -{E^(c*(a + b*x))/Sqrt[Coth[a*c + b*c*x]^2], x, 4, (E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2]) - (2*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Coth[a*c + b*c*x]^2)^(3/2), x, 8, (E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2]) - (2*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^2*Sqrt[Coth[a*c + b*c*x]^2]) + (3*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))*Sqrt[Coth[a*c + b*c*x]^2]) - (3*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Coth[a*c + b*c*x]^2)^(5/2), x, 9, (E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2]) - (4*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*(1 + E^(2*c*(a + b*x)))^4*Sqrt[Coth[a*c + b*c*x]^2]) + (26*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3*Sqrt[Coth[a*c + b*c*x]^2]) - (55*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(6*b*c*(1 + E^(2*c*(a + b*x)))^2*Sqrt[Coth[a*c + b*c*x]^2]) + (25*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(4*b*c*(1 + E^(2*c*(a + b*x)))*Sqrt[Coth[a*c + b*c*x]^2]) - (15*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x])/(4*b*c*Sqrt[Coth[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands that are function of Coth[c+d x]*) - - -{Sin[Coth[a + b*x]]^3, x, 19, -((3*CosIntegral[1 - Coth[a + b*x]]*Sin[1])/(8*b)) - (3*CosIntegral[1 + Coth[a + b*x]]*Sin[1])/(8*b) + (CosIntegral[3 - 3*Coth[a + b*x]]*Sin[3])/(8*b) + (CosIntegral[3 + 3*Coth[a + b*x]]*Sin[3])/(8*b) - (Cos[3]*SinIntegral[3 - 3*Coth[a + b*x]])/(8*b) + (3*Cos[1]*SinIntegral[1 - Coth[a + b*x]])/(8*b) + (3*Cos[1]*SinIntegral[1 + Coth[a + b*x]])/(8*b) - (Cos[3]*SinIntegral[3 + 3*Coth[a + b*x]])/(8*b)} -{Sin[Coth[a + b*x]]^2, x, 13, (Cos[2]*CosIntegral[2 - 2*Coth[a + b*x]])/(4*b) - (Cos[2]*CosIntegral[2 + 2*Coth[a + b*x]])/(4*b) - Log[1 - Coth[a + b*x]]/(4*b) + Log[1 + Coth[a + b*x]]/(4*b) + (Sin[2]*SinIntegral[2 - 2*Coth[a + b*x]])/(4*b) - (Sin[2]*SinIntegral[2 + 2*Coth[a + b*x]])/(4*b)} -{Sin[Coth[a + b*x]]^1, x, 9, -((CosIntegral[1 - Coth[a + b*x]]*Sin[1])/(2*b)) - (CosIntegral[1 + Coth[a + b*x]]*Sin[1])/(2*b) + (Cos[1]*SinIntegral[1 - Coth[a + b*x]])/(2*b) + (Cos[1]*SinIntegral[1 + Coth[a + b*x]])/(2*b)} -{Csc[Coth[a + b*x]]^1, x, 3, (1/2)*Unintegrable[(Csc[Coth[a + b*x]]*Csch[a + b*x]^2)/(-1 + Coth[a + b*x]), x] - (1/2)*Unintegrable[(Csc[Coth[a + b*x]]*Csch[a + b*x]^2)/(1 + Coth[a + b*x]), x]} - - -{Cos[Coth[a + b*x]]^3, x, 19, -((Cos[3]*CosIntegral[3 - 3*Coth[a + b*x]])/(8*b)) - (3*Cos[1]*CosIntegral[1 - Coth[a + b*x]])/(8*b) + (3*Cos[1]*CosIntegral[1 + Coth[a + b*x]])/(8*b) + (Cos[3]*CosIntegral[3 + 3*Coth[a + b*x]])/(8*b) - (Sin[3]*SinIntegral[3 - 3*Coth[a + b*x]])/(8*b) - (3*Sin[1]*SinIntegral[1 - Coth[a + b*x]])/(8*b) + (3*Sin[1]*SinIntegral[1 + Coth[a + b*x]])/(8*b) + (Sin[3]*SinIntegral[3 + 3*Coth[a + b*x]])/(8*b)} -{Cos[Coth[a + b*x]]^2, x, 13, -((Cos[2]*CosIntegral[2 - 2*Coth[a + b*x]])/(4*b)) + (Cos[2]*CosIntegral[2 + 2*Coth[a + b*x]])/(4*b) - Log[1 - Coth[a + b*x]]/(4*b) + Log[1 + Coth[a + b*x]]/(4*b) - (Sin[2]*SinIntegral[2 - 2*Coth[a + b*x]])/(4*b) + (Sin[2]*SinIntegral[2 + 2*Coth[a + b*x]])/(4*b)} -{Cos[Coth[a + b*x]]^1, x, 9, -((Cos[1]*CosIntegral[1 - Coth[a + b*x]])/(2*b)) + (Cos[1]*CosIntegral[1 + Coth[a + b*x]])/(2*b) - (Sin[1]*SinIntegral[1 - Coth[a + b*x]])/(2*b) + (Sin[1]*SinIntegral[1 + Coth[a + b*x]])/(2*b)} -{Sec[Coth[a + b*x]]^1, x, 3, (1/2)*Unintegrable[(Csch[a + b*x]^2*Sec[Coth[a + b*x]])/(-1 + Coth[a + b*x]), x] - (1/2)*Unintegrable[(Csch[a + b*x]^2*Sec[Coth[a + b*x]])/(1 + Coth[a + b*x]), x]} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.7 (d hyper)^m (a+b (c coth)^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.7 (d hyper)^m (a+b (c coth)^n)^p.m deleted file mode 100644 index 364a826..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.4 Hyperbolic cotangent/6.4.7 (d hyper)^m (a+b (c coth)^n)^p.m +++ /dev/null @@ -1,148 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Coth[e+f x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Coth[e+f x]^n)^p*) - - -(* ::Title:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Coth[c + d*x]^2)^5, x, 4, (a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Coth[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Coth[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Coth[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Coth[c + d*x]^7)/(7*d) - (b^5*Coth[c + d*x]^9)/(9*d)} -{(a + b*Coth[c + d*x]^2)^4, x, 4, (a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Coth[c + d*x])/d - (b^2*(6*a^2 + 4*a*b + b^2)*Coth[c + d*x]^3)/(3*d) - (b^3*(4*a + b)*Coth[c + d*x]^5)/(5*d) - (b^4*Coth[c + d*x]^7)/(7*d)} -{(a + b*Coth[c + d*x]^2)^3, x, 4, (a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Coth[c + d*x])/d - (b^2*(3*a + b)*Coth[c + d*x]^3)/(3*d) - (b^3*Coth[c + d*x]^5)/(5*d)} -{(a + b*Coth[c + d*x]^2)^2, x, 4, (a + b)^2*x - (b*(2*a + b)*Coth[c + d*x])/d - (b^2*Coth[c + d*x]^3)/(3*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/(a + b*Coth[c + d*x]^2),x, 3, x/(a + b) - (Sqrt[b]*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*d)} -{1/(a + b*Coth[c + d*x]^2)^2,x, 5, x/(a + b)^2 - (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*d) + (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b*Coth[c + d*x]^2))} -{1/(a + b*Coth[c + d*x]^2)^3,x, 6, x/(a + b)^3 - (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3*d) + (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b*Coth[c + d*x]^2)^2) + (b*(7*a + 3*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Coth[c + d*x]^2))} -{1/(a + b*Coth[c + d*x]^2)^4,x, 7, x/(a + b)^4 - (Sqrt[b]*(35*a^3 + 35*a^2*b + 21*a*b^2 + 5*b^3)*ArcTan[(Sqrt[a]*Tanh[c + d*x])/Sqrt[b]])/(16*a^(7/2)*(a + b)^4*d) + (b*Coth[c + d*x])/(6*a*(a + b)*d*(a + b*Coth[c + d*x]^2)^3) + (b*(11*a + 5*b)*Coth[c + d*x])/(24*a^2*(a + b)^2*d*(a + b*Coth[c + d*x]^2)^2) + (b*(19*a^2 + 16*a*b + 5*b^2)*Coth[c + d*x])/(16*a^3*(a + b)^3*d*(a + b*Coth[c + d*x]^2))} - -{1/(1 - 2*Coth[x]^2), x, 3, -x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^2)^(p/2) when a+b=0*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[1 - Coth[x]^2], x, 3, ArcSin[Coth[x]]} -{Sqrt[-1 + Coth[x]^2], x, 4, -ArcTanh[Coth[x]/Sqrt[Csch[x]^2]]} - - -{(1 - Coth[x]^2)^(3/2), x, 4, (1/2)*ArcSin[Coth[x]] + (1/2)*Coth[x]*Sqrt[-Csch[x]^2]} -{(-1 + Coth[x]^2)^(3/2), x, 5, (1/2)*ArcTanh[Coth[x]/Sqrt[Csch[x]^2]] - (1/2)*Coth[x]*Sqrt[Csch[x]^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[1 - Coth[x]^2], x, 3, Coth[x]/Sqrt[-Csch[x]^2]} -{1/Sqrt[-1 + Coth[x]^2], x, 3, Coth[x]/Sqrt[Csch[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Coth[x]^3*Sqrt[a + b*Coth[x]^2], x, 6, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(3/2)/(3*b)} -{Coth[x]^2*Sqrt[a + b*Coth[x]^2], x, 7, -(((a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/(2*Sqrt[b])) + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - (1/2)*Coth[x]*Sqrt[a + b*Coth[x]^2]} -{Coth[x]^1*Sqrt[a + b*Coth[x]^2], x, 5, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]} -{Coth[x]^0*Sqrt[a + b*Coth[x]^2], x, 6, (-Sqrt[b])*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]} -{Tanh[x]^1*Sqrt[a + b*Coth[x]^2], x, 7, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]] + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]} -{Tanh[x]^2*Sqrt[a + b*Coth[x]^2], x, 5, Sqrt[a + b]*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - Sqrt[a + b*Coth[x]^2]*Tanh[x]} - - -{Coth[x]^3*(a + b*Coth[x]^2)^(3/2), x, 7, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (1/3)*(a + b*Coth[x]^2)^(3/2) - (a + b*Coth[x]^2)^(5/2)/(5*b)} -{Coth[x]^2*(a + b*Coth[x]^2)^(3/2), x, 8, -(((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/(8*Sqrt[b])) + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - (1/8)*(5*a + 4*b)*Coth[x]*Sqrt[a + b*Coth[x]^2] - (1/4)*b*Coth[x]^3*Sqrt[a + b*Coth[x]^2]} -{Coth[x]^1*(a + b*Coth[x]^2)^(3/2), x, 6, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (1/3)*(a + b*Coth[x]^2)^(3/2)} -{Coth[x]^0*(a + b*Coth[x]^2)^(3/2), x, 7, (-(1/2))*Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - (1/2)*b*Coth[x]*Sqrt[a + b*Coth[x]^2]} -{Tanh[x]^1*(a + b*Coth[x]^2)^(3/2), x, 8, (-a^(3/2))*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]] + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Coth[x]^2]} -{Tanh[x]^2*(a + b*Coth[x]^2)^(3/2), x, 7, (-b^(3/2))*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - a*Sqrt[a + b*Coth[x]^2]*Tanh[x]} - - -{Sqrt[1 + Coth[x]^2], x, 5, -ArcSinh[Coth[x]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Coth[x])/Sqrt[1 + Coth[x]^2]]} -{Sqrt[-1 - Coth[x]^2], x, 6, ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]] - Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]]} - - -{(1 + Coth[x]^2)^(3/2), x, 6, (-(5/2))*ArcSinh[Coth[x]] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Coth[x])/Sqrt[1 + Coth[x]^2]] - (1/2)*Coth[x]*Sqrt[1 + Coth[x]^2]} -{(-1 - Coth[x]^2)^(3/2), x, 7, (-(5/2))*ArcTan[Coth[x]/Sqrt[-1 - Coth[x]^2]] + 2*Sqrt[2]*ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]] + (1/2)*Coth[x]*Sqrt[-1 - Coth[x]^2]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Coth[x]^3/Sqrt[a + b*Coth[x]^2], x, 5, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Coth[x]^2]/b} -{Coth[x]^2/Sqrt[a + b*Coth[x]^2], x, 6, -(ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[b]) + ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b]} -{Coth[x]^1/Sqrt[a + b*Coth[x]^2], x, 4, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]} -{Coth[x]^0/Sqrt[a + b*Coth[x]^2], x, 3, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b]} -{Tanh[x]^1/Sqrt[a + b*Coth[x]^2], x, 7, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]} -{Tanh[x]^2/Sqrt[a + b*Coth[x]^2], x, 5, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b] - (Sqrt[a + b*Coth[x]^2]*Tanh[x])/a} - - -{Coth[x]^3/(a + b*Coth[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Coth[x]^2])} -{Coth[x]^2/(a + b*Coth[x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) - Coth[x]/((a + b)*Sqrt[a + b*Coth[x]^2])} -{Coth[x]/(a + b*Coth[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - 1/((a + b)*Sqrt[a + b*Coth[x]^2])} -{Tanh[x]/(a + b*Coth[x]^2)^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/a^(3/2)) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + b/(a*(a + b)*Sqrt[a + b*Coth[x]^2])} -{Tanh[x]^2/(a + b*Coth[x]^2)^(3/2), x, 6, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) + (b*Tanh[x])/(a*(a + b)*Sqrt[a + b*Coth[x]^2]) - ((a + 2*b)*Sqrt[a + b*Coth[x]^2]*Tanh[x])/(a^2*(a + b))} - - -{Coth[x]^3/(a + b*Coth[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + a/(3*b*(a + b)*(a + b*Coth[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Coth[x]^2])} -{Coth[x]^2/(a + b*Coth[x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) - Coth[x]/(3*(a + b)*(a + b*Coth[x]^2)^(3/2)) - ((2*a - b)*Coth[x])/(3*a*(a + b)^2*Sqrt[a + b*Coth[x]^2])} -{Coth[x]/(a + b*Coth[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - 1/(3*(a + b)*(a + b*Coth[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Coth[x]^2])} -{Tanh[x]/(a + b*Coth[x]^2)^(5/2), x, 9, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/a^(5/2)) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + b/(3*a*(a + b)*(a + b*Coth[x]^2)^(3/2)) + (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Coth[x]^2])} -{Tanh[x]^2/(a + b*Coth[x]^2)^(5/2), x, 7, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Coth[x]^2)^(3/2)) + (b*(7*a + 4*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Coth[x]^2]) - ((3*a + 2*b)*(a + 4*b)*Sqrt[a + b*Coth[x]^2]*Tanh[x])/(3*a^3*(a + b)^2)} - - -{1/Sqrt[1 + Coth[x]^2], x, 3, ArcTanh[(Sqrt[2]*Coth[x])/Sqrt[1 + Coth[x]^2]]/Sqrt[2]} -{1/Sqrt[-1 - Coth[x]^2], x, 3, ArcTan[(Sqrt[2]*Coth[x])/Sqrt[-1 - Coth[x]^2]]/Sqrt[2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^3)^p*) - - -{1/(1 + Coth[x]^3), x, 6, x/2 - (2*ArcTan[(1 - 2*Coth[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(6*(1 + Coth[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^4)^p*) - - -{Coth[x]*(a + b*Coth[x]^4)^(1/2), x, 8, (-(1/2))*Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x]^2)/Sqrt[a + b*Coth[x]^4]] + (1/2)*Sqrt[a + b]*ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])] - (1/2)*Sqrt[a + b*Coth[x]^4]} -{Coth[x]/(a + b*Coth[x]^4)^(1/2), x, 4, ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(2*Sqrt[a + b])} -{Coth[x]/(a + b*Coth[x]^4)^(3/2), x, 6, ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(2*(a + b)^(3/2)) - (a - b*Coth[x]^2)/(2*a*(a + b)*Sqrt[a + b*Coth[x]^4])} - - -(* ::Section:: *) -(*Integrands of the form Coth[e+f x]^m (a+b Coth[e+f x]^n)^p*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.1 (c+d x)^m (a+b sech)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.1 (c+d x)^m (a+b sech)^n.m deleted file mode 100644 index 0072260..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.1 (c+d x)^m (a+b sech)^n.m +++ /dev/null @@ -1,44 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form (c+d x)^m Sech[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sech[a+b x]^n*) - - -{(c + d*x)^3*Sech[a + b*x], x, 9, (2*(c + d*x)^3*ArcTan[E^(a + b*x)])/b - (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(a + b*x)])/b^2 + (6*I*d^2*(c + d*x)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (6*I*d^2*(c + d*x)*PolyLog[3, I*E^(a + b*x)])/b^3 - (6*I*d^3*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + (6*I*d^3*PolyLog[4, I*E^(a + b*x)])/b^4} -{(c + d*x)^2*Sech[a + b*x], x, 7, (2*(c + d*x)^2*ArcTan[E^(a + b*x)])/b - (2*I*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (2*I*d*(c + d*x)*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*I*d^2*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (2*I*d^2*PolyLog[3, I*E^(a + b*x)])/b^3} -{(c + d*x)^1*Sech[a + b*x], x, 5, (2*(c + d*x)*ArcTan[E^(a + b*x)])/b - (I*d*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (I*d*PolyLog[2, I*E^(a + b*x)])/b^2} -{Sech[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Sech[a + b*x]/(c + d*x), x]} - - -{(c + d*x)^3*Sech[a + b*x]^2, x, 6, (c + d*x)^3/b - (3*d*(c + d*x)^2*Log[1 + E^(2*(a + b*x))])/b^2 - (3*d^2*(c + d*x)*PolyLog[2, -E^(2*(a + b*x))])/b^3 + (3*d^3*PolyLog[3, -E^(2*(a + b*x))])/(2*b^4) + ((c + d*x)^3*Tanh[a + b*x])/b} -{(c + d*x)^2*Sech[a + b*x]^2, x, 5, (c + d*x)^2/b - (2*d*(c + d*x)*Log[1 + E^(2*(a + b*x))])/b^2 - (d^2*PolyLog[2, -E^(2*(a + b*x))])/b^3 + ((c + d*x)^2*Tanh[a + b*x])/b} -{(c + d*x)^1*Sech[a + b*x]^2, x, 2, -((d*Log[Cosh[a + b*x]])/b^2) + ((c + d*x)*Tanh[a + b*x])/b} -{Sech[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Sech[a + b*x]^2/(c + d*x), x]} - - -{(c + d*x)^3*Sech[a + b*x]^3, x, 15, -((6*d^2*(c + d*x)*ArcTan[E^(a + b*x)])/b^3) + ((c + d*x)^3*ArcTan[E^(a + b*x)])/b + (3*I*d^3*PolyLog[2, (-I)*E^(a + b*x)])/b^4 - (3*I*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) - (3*I*d^3*PolyLog[2, I*E^(a + b*x)])/b^4 + (3*I*d*(c + d*x)^2*PolyLog[2, I*E^(a + b*x)])/(2*b^2) + (3*I*d^2*(c + d*x)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (3*I*d^2*(c + d*x)*PolyLog[3, I*E^(a + b*x)])/b^3 - (3*I*d^3*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + (3*I*d^3*PolyLog[4, I*E^(a + b*x)])/b^4 + (3*d*(c + d*x)^2*Sech[a + b*x])/(2*b^2) + ((c + d*x)^3*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{(c + d*x)^2*Sech[a + b*x]^3, x, 9, ((c + d*x)^2*ArcTan[E^(a + b*x)])/b - (d^2*ArcTan[Sinh[a + b*x]])/b^3 - (I*d*(c + d*x)*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (I*d*(c + d*x)*PolyLog[2, I*E^(a + b*x)])/b^2 + (I*d^2*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (I*d^2*PolyLog[3, I*E^(a + b*x)])/b^3 + (d*(c + d*x)*Sech[a + b*x])/b^2 + ((c + d*x)^2*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{(c + d*x)^1*Sech[a + b*x]^3, x, 6, ((c + d*x)*ArcTan[E^(a + b*x)])/b - (I*d*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) + (I*d*PolyLog[2, I*E^(a + b*x)])/(2*b^2) + (d*Sech[a + b*x])/(2*b^2) + ((c + d*x)*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{Sech[a + b*x]^3/(c + d*x)^1, x, 0, Unintegrable[Sech[a + b*x]^3/(c + d*x), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Sech[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Sech[a+b x]^(n/2)*) - - -{x/Sech[x]^(3/2) - (1/3)*x*Sqrt[Sech[x]], x, 4, -(4/(9*Sech[x]^(3/2))) + (2*x*Sinh[x])/(3*Sqrt[Sech[x]])} -{x/Sech[x]^(5/2) - (3/5)*x/Sqrt[Sech[x]], x, 4, -(4/(25*Sech[x]^(5/2))) + (2*x*Sinh[x])/(5*Sech[x]^(3/2))} -{x/Sech[x]^(7/2) - (5/21)*x*Sqrt[Sech[x]], x, 5, -(4/(49*Sech[x]^(7/2))) - 20/(63*Sech[x]^(3/2)) + (2*x*Sinh[x])/(7*Sech[x]^(5/2)) + (10*x*Sinh[x])/(21*Sqrt[Sech[x]])} -{x^2/Sech[x]^(3/2) - (1/3)*x^2*Sqrt[Sech[x]], x, 7, -((8*x)/(9*Sech[x]^(3/2))) - (16/27)*I*Sqrt[Cosh[x]]*EllipticF[(I*x)/2, 2]*Sqrt[Sech[x]] + (16*Sinh[x])/(27*Sqrt[Sech[x]]) + (2*x^2*Sinh[x])/(3*Sqrt[Sech[x]])} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Sech[a+b x]^(n/2)*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.2 (e x)^m (a+b sech(c+d x^n))^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.2 (e x)^m (a+b sech(c+d x^n))^p.m deleted file mode 100644 index 0a5e3f8..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.2 (e x)^m (a+b sech(c+d x^n))^p.m +++ /dev/null @@ -1,190 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sech[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Sech[c + d*x^2]), x, 10, (a*x^6)/6 + (b*x^4*ArcTan[E^(c + d*x^2)])/d - (I*b*x^2*PolyLog[2, (-I)*E^(c + d*x^2)])/d^2 + (I*b*x^2*PolyLog[2, I*E^(c + d*x^2)])/d^2 + (I*b*PolyLog[3, (-I)*E^(c + d*x^2)])/d^3 - (I*b*PolyLog[3, I*E^(c + d*x^2)])/d^3} -{x^4*(a + b*Sech[c + d*x^2]), x, 2, (a*x^5)/5 + b*Unintegrable[x^4*Sech[c + d*x^2], x]} -{x^3*(a + b*Sech[c + d*x^2]), x, 8, (a*x^4)/4 + (b*x^2*ArcTan[E^(c + d*x^2)])/d - ((I/2)*b*PolyLog[2, (-I)*E^(c + d*x^2)])/d^2 + ((I/2)*b*PolyLog[2, I*E^(c + d*x^2)])/d^2} -{x^2*(a + b*Sech[c + d*x^2]), x, 2, (a*x^3)/3 + b*Unintegrable[x^2*Sech[c + d*x^2], x]} -{x*(a + b*Sech[c + d*x^2]), x, 4, (a*x^2)/2 + (b*ArcTan[Sinh[c + d*x^2]])/(2*d)} -{(a + b*Sech[c + d*x^2])/x, x, 2, a*Log[x] + b*Unintegrable[Sech[c + d*x^2]/x, x]} -{(a + b*Sech[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Sech[c + d*x^2]/x^2, x]} - - -{x^5*(a + b*Sech[c + d*x^2])^2, x, 15, (b^2*x^4)/(2*d) + (a^2*x^6)/6 + (2*a*b*x^4*ArcTan[E^(c + d*x^2)])/d - (b^2*x^2*Log[1 + E^(2*(c + d*x^2))])/d^2 - (2*I*a*b*x^2*PolyLog[2, (-I)*E^(c + d*x^2)])/d^2 + (2*I*a*b*x^2*PolyLog[2, I*E^(c + d*x^2)])/d^2 - (b^2*PolyLog[2, -E^(2*(c + d*x^2))])/(2*d^3) + (2*I*a*b*PolyLog[3, (-I)*E^(c + d*x^2)])/d^3 - (2*I*a*b*PolyLog[3, I*E^(c + d*x^2)])/d^3 + (b^2*x^4*Tanh[c + d*x^2])/(2*d)} -{x^4*(a + b*Sech[c + d*x^2])^2, x, 0, Unintegrable[x^4*(a + b*Sech[c + d*x^2])^2, x]} -{x^3*(a + b*Sech[c + d*x^2])^2, x, 10, (a^2*x^4)/4 + (2*a*b*x^2*ArcTan[E^(c + d*x^2)])/d - (b^2*Log[Cosh[c + d*x^2]])/(2*d^2) - (I*a*b*PolyLog[2, (-I)*E^(c + d*x^2)])/d^2 + (I*a*b*PolyLog[2, I*E^(c + d*x^2)])/d^2 + (b^2*x^2*Tanh[c + d*x^2])/(2*d)} -{x^2*(a + b*Sech[c + d*x^2])^2, x, 0, Unintegrable[x^2*(a + b*Sech[c + d*x^2])^2, x]} -{x*(a + b*Sech[c + d*x^2])^2, x, 5, (a^2*x^2)/2 + (a*b*ArcTan[Sinh[c + d*x^2]])/d + (b^2*Tanh[c + d*x^2])/(2*d)} -{(a + b*Sech[c + d*x^2])^2/x, x, 0, Unintegrable[(a + b*Sech[c + d*x^2])^2/x, x]} -{(a + b*Sech[c + d*x^2])^2/x^2, x, 0, Unintegrable[(a + b*Sech[c + d*x^2])^2/x^2, x]} - - -{x*Sech[a + b*x^2]^7, x, 5, (5*ArcTan[Sinh[a + b*x^2]])/(32*b) + (5*Sech[a + b*x^2]*Tanh[a + b*x^2])/(32*b) + (5*Sech[a + b*x^2]^3*Tanh[a + b*x^2])/(48*b) + (Sech[a + b*x^2]^5*Tanh[a + b*x^2])/(12*b)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Sech[c + d*x^2]), x, 13, x^6/(6*a) - (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d) + (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d) - (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3)} -{x^4/(a + b*Sech[c + d*x^2]), x, 0, Unintegrable[x^4/(a + b*Sech[c + d*x^2]), x]} -{x^3/(a + b*Sech[c + d*x^2]), x, 11, x^4/(4*a) - (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d) + (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(2*a*Sqrt[-a^2 + b^2]*d) - (b*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(2*a*Sqrt[-a^2 + b^2]*d^2) + (b*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(2*a*Sqrt[-a^2 + b^2]*d^2)} -{x^2/(a + b*Sech[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Sech[c + d*x^2]), x]} -{x/(a + b*Sech[c + d*x^2]), x, 4, x^2/(2*a) - (b*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x^2)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(x*(a + b*Sech[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Sech[c + d*x^2])), x]} -{(a + b*Sech[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Sech[c + d*x^2]/x^2, x]} - - -{x^5/(a + b*Sech[c + d*x^2])^2, x, 31, (b^2*x^4)/(2*a^2*(a^2 - b^2)*d) + x^6/(6*a^2) - (b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d) - (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d) + (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (b^3*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (2*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (b^3*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (2*b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (2*b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (b^2*x^4*Sinh[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cosh[c + d*x^2]))} -{x^4/(a + b*Sech[c + d*x^2])^2, x, 0, Unintegrable[x^4/(a + b*Sech[c + d*x^2])^2, x]} -{x^3/(a + b*Sech[c + d*x^2])^2, x, 22, x^4/(4*a^2) + (b^3*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d) - (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (b^3*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(2*a^2*(-a^2 + b^2)^(3/2)*d) + (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (b^2*Log[b + a*Cosh[c + d*x^2]])/(2*a^2*(a^2 - b^2)*d^2) + (b^3*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (b^3*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(2*a^2*(-a^2 + b^2)^(3/2)*d^2) + (b*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (b^2*x^2*Sinh[c + d*x^2])/(2*a*(a^2 - b^2)*d*(b + a*Cosh[c + d*x^2]))} -{x^2/(a + b*Sech[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Sech[c + d*x^2])^2, x]} -{x/(a + b*Sech[c + d*x^2])^2, x, 6, x^2/(2*a^2) - (b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x^2)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*Tanh[c + d*x^2])/(2*a*(a^2 - b^2)*d*(a + b*Sech[c + d*x^2]))} -{1/(x*(a + b*Sech[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Sech[c + d*x^2])^2), x]} -{1/(x^2*(a + b*Sech[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sech[c + d*x^2])^2), x]} -{1/(x^3*(a + b*Sech[c + d*x^2])^2), x, 0, Unintegrable[1/(x^3*(a + b*Sech[c + d*x^2])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d / x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d / x])^p*) - - -{Sech[1/x]^2/x^2, x, 3, -Tanh[x^(-1)]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d x^(1/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Sech[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Sech[c + d*Sqrt[x]]), x, 20, (a*x^4)/4 + (4*b*x^(7/2)*ArcTan[E^(c + d*Sqrt[x])])/d - ((14*I)*b*x^3*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + ((14*I)*b*x^3*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 + ((84*I)*b*x^(5/2)*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - ((84*I)*b*x^(5/2)*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 - ((420*I)*b*x^2*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + ((420*I)*b*x^2*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 + ((1680*I)*b*x^(3/2)*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - ((1680*I)*b*x^(3/2)*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5 - ((5040*I)*b*x*PolyLog[6, (-I)*E^(c + d*Sqrt[x])])/d^6 + ((5040*I)*b*x*PolyLog[6, I*E^(c + d*Sqrt[x])])/d^6 + ((10080*I)*b*Sqrt[x]*PolyLog[7, (-I)*E^(c + d*Sqrt[x])])/d^7 - ((10080*I)*b*Sqrt[x]*PolyLog[7, I*E^(c + d*Sqrt[x])])/d^7 - ((10080*I)*b*PolyLog[8, (-I)*E^(c + d*Sqrt[x])])/d^8 + ((10080*I)*b*PolyLog[8, I*E^(c + d*Sqrt[x])])/d^8} -{x^2*(a + b*Sech[c + d*Sqrt[x]]), x, 16, (a*x^3)/3 + (4*b*x^(5/2)*ArcTan[E^(c + d*Sqrt[x])])/d - ((10*I)*b*x^2*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + ((10*I)*b*x^2*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 + ((40*I)*b*x^(3/2)*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - ((40*I)*b*x^(3/2)*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 - ((120*I)*b*x*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + ((120*I)*b*x*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 + ((240*I)*b*Sqrt[x]*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - ((240*I)*b*Sqrt[x]*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5 - ((240*I)*b*PolyLog[6, (-I)*E^(c + d*Sqrt[x])])/d^6 + ((240*I)*b*PolyLog[6, I*E^(c + d*Sqrt[x])])/d^6} -{x*(a + b*Sech[c + d*Sqrt[x]]), x, 12, (a*x^2)/2 + (4*b*x^(3/2)*ArcTan[E^(c + d*Sqrt[x])])/d - ((6*I)*b*x*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + ((6*I)*b*x*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 + ((12*I)*b*Sqrt[x]*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - ((12*I)*b*Sqrt[x]*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 - ((12*I)*b*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + ((12*I)*b*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4} -{(a + b*Sech[c + d*Sqrt[x]])/x, x, 2, a*Log[x] + b*Unintegrable[Sech[c + d*Sqrt[x]]/x, x]} -{(a + b*Sech[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Sech[c + d*Sqrt[x]]/x^2, x]} - - -{x^3*(a + b*Sech[c + d*Sqrt[x]])^2, x, 30, (2*b^2*x^(7/2))/d + (a^2*x^4)/4 + (8*a*b*x^(7/2)*ArcTan[E^(c + d*Sqrt[x])])/d - (14*b^2*x^3*Log[1 + E^(2*(c + d*Sqrt[x]))])/d^2 - (28*I*a*b*x^3*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + (28*I*a*b*x^3*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 - (42*b^2*x^(5/2)*PolyLog[2, -E^(2*(c + d*Sqrt[x]))])/d^3 + (168*I*a*b*x^(5/2)*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - (168*I*a*b*x^(5/2)*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 + (105*b^2*x^2*PolyLog[3, -E^(2*(c + d*Sqrt[x]))])/d^4 - (840*I*a*b*x^2*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + (840*I*a*b*x^2*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 - (210*b^2*x^(3/2)*PolyLog[4, -E^(2*(c + d*Sqrt[x]))])/d^5 + (3360*I*a*b*x^(3/2)*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - (3360*I*a*b*x^(3/2)*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5 + (315*b^2*x*PolyLog[5, -E^(2*(c + d*Sqrt[x]))])/d^6 - (10080*I*a*b*x*PolyLog[6, (-I)*E^(c + d*Sqrt[x])])/d^6 + (10080*I*a*b*x*PolyLog[6, I*E^(c + d*Sqrt[x])])/d^6 - (315*b^2*Sqrt[x]*PolyLog[6, -E^(2*(c + d*Sqrt[x]))])/d^7 + (20160*I*a*b*Sqrt[x]*PolyLog[7, (-I)*E^(c + d*Sqrt[x])])/d^7 - (20160*I*a*b*Sqrt[x]*PolyLog[7, I*E^(c + d*Sqrt[x])])/d^7 + (315*b^2*PolyLog[7, -E^(2*(c + d*Sqrt[x]))])/(2*d^8) - (20160*I*a*b*PolyLog[8, (-I)*E^(c + d*Sqrt[x])])/d^8 + (20160*I*a*b*PolyLog[8, I*E^(c + d*Sqrt[x])])/d^8 + (2*b^2*x^(7/2)*Tanh[c + d*Sqrt[x]])/d} -{x^2*(a + b*Sech[c + d*Sqrt[x]])^2, x, 24, (2*b^2*x^(5/2))/d + (a^2*x^3)/3 + (8*a*b*x^(5/2)*ArcTan[E^(c + d*Sqrt[x])])/d - (10*b^2*x^2*Log[1 + E^(2*(c + d*Sqrt[x]))])/d^2 - (20*I*a*b*x^2*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + (20*I*a*b*x^2*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 - (20*b^2*x^(3/2)*PolyLog[2, -E^(2*(c + d*Sqrt[x]))])/d^3 + (80*I*a*b*x^(3/2)*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - (80*I*a*b*x^(3/2)*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 + (30*b^2*x*PolyLog[3, -E^(2*(c + d*Sqrt[x]))])/d^4 - (240*I*a*b*x*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + (240*I*a*b*x*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 - (30*b^2*Sqrt[x]*PolyLog[4, -E^(2*(c + d*Sqrt[x]))])/d^5 + (480*I*a*b*Sqrt[x]*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - (480*I*a*b*Sqrt[x]*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5 + (15*b^2*PolyLog[5, -E^(2*(c + d*Sqrt[x]))])/d^6 - (480*I*a*b*PolyLog[6, (-I)*E^(c + d*Sqrt[x])])/d^6 + (480*I*a*b*PolyLog[6, I*E^(c + d*Sqrt[x])])/d^6 + (2*b^2*x^(5/2)*Tanh[c + d*Sqrt[x]])/d} -{x*(a + b*Sech[c + d*Sqrt[x]])^2, x, 18, (2*b^2*x^(3/2))/d + (a^2*x^2)/2 + (8*a*b*x^(3/2)*ArcTan[E^(c + d*Sqrt[x])])/d - (6*b^2*x*Log[1 + E^(2*(c + d*Sqrt[x]))])/d^2 - (12*I*a*b*x*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + (12*I*a*b*x*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 - (6*b^2*Sqrt[x]*PolyLog[2, -E^(2*(c + d*Sqrt[x]))])/d^3 + (24*I*a*b*Sqrt[x]*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - (24*I*a*b*Sqrt[x]*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 + (3*b^2*PolyLog[3, -E^(2*(c + d*Sqrt[x]))])/d^4 - (24*I*a*b*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + (24*I*a*b*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 + (2*b^2*x^(3/2)*Tanh[c + d*Sqrt[x]])/d} -{(a + b*Sech[c + d*Sqrt[x]])^2/x, x, 0, Unintegrable[(a + b*Sech[c + d*Sqrt[x]])^2/x, x]} -{(a + b*Sech[c + d*Sqrt[x]])^2/x^2, x, 0, Unintegrable[(a + b*Sech[c + d*Sqrt[x]])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Sech[c + d*Sqrt[x]]), x, 23, x^4/(4*a) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) - (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) - (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (10080*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^8)} -{x^2/(a + b*Sech[c + d*Sqrt[x]]), x, 19, x^3/(3*a) - (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) - (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) - (240*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) + (240*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6)} -{x/(a + b*Sech[c + d*Sqrt[x]]), x, 15, x^2/(2*a) - (2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (12*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (12*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (12*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (12*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4)} -{1/(x*(a + b*Sech[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x*(a + b*Sech[c + d*Sqrt[x]])), x]} -{(a + b*Sech[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Sech[c + d*Sqrt[x]]/x^2, x]} - - -{x^3/(a + b*Sech[c + d*Sqrt[x]])^2, x, 61, (2*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(4*a^2) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) - (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) - (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) + (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) - (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^7) + (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) - (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^7) - (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) - (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) + (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^8) + (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^7) - (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^8) + (2*b^2*x^(7/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))} -{x^2/(a + b*Sech[c + d*Sqrt[x]])^2, x, 49, (2*b^2*x^(5/2))/(a^2*(a^2 - b^2)*d) + x^3/(3*a^2) - (10*b^2*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (4*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (10*b^2*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (40*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (10*b^3*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (20*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (40*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (10*b^3*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (20*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (120*b^2*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (40*b^3*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (80*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (120*b^2*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (40*b^3*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (80*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (240*b^2*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (120*b^3*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (240*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (240*b^2*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) - (120*b^3*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (240*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (240*b^2*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) - (240*b^3*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + (480*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (240*b^2*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^6) + (240*b^3*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - (480*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (240*b^3*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) - (480*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) - (240*b^3*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^6) + (480*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^6) + (2*b^2*x^(5/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))} -{x/(a + b*Sech[c + d*Sqrt[x]])^2, x, 37, (2*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(2*a^2) - (6*b^2*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (4*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (6*b^2*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) + (2*b^2*x^(3/2)*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))} -{1/(x*(a + b*Sech[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x*(a + b*Sech[c + d*Sqrt[x]])^2), x]} -{1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^2*(a + b*Sech[c + d*Sqrt[x]])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^(m/2) (a+b Sech[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^(3/2)*(a + b*Sech[c + d*Sqrt[x]]), x, 14, (2*a*x^(5/2))/5 + (4*b*x^2*ArcTan[E^(c + d*Sqrt[x])])/d - ((8*I)*b*x^(3/2)*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + ((8*I)*b*x^(3/2)*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 + ((24*I)*b*x*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - ((24*I)*b*x*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 - ((48*I)*b*Sqrt[x]*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + ((48*I)*b*Sqrt[x]*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 + ((48*I)*b*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - ((48*I)*b*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5} -{Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]]), x, 10, (2*a*x^(3/2))/3 + (4*b*x*ArcTan[E^(c + d*Sqrt[x])])/d - ((4*I)*b*Sqrt[x]*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + ((4*I)*b*Sqrt[x]*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 + ((4*I)*b*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - ((4*I)*b*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3} -{(a + b*Sech[c + d*Sqrt[x]])/Sqrt[x], x, 4, 2*a*Sqrt[x] + (2*b*ArcTan[Sinh[c + d*Sqrt[x]]])/d} -{(a + b*Sech[c + d*Sqrt[x]])/x^(3/2), x, 2, (-2*a)/Sqrt[x] + b*Unintegrable[Sech[c + d*Sqrt[x]]/x^(3/2), x]} -{(a + b*Sech[c + d*Sqrt[x]])/x^(5/2), x, 2, (-2*a)/(3*x^(3/2)) + b*Unintegrable[Sech[c + d*Sqrt[x]]/x^(5/2), x]} - - -{x^(3/2)*(a + b*Sech[c + d*Sqrt[x]])^2, x, 21, (2*b^2*x^2)/d + (2/5)*a^2*x^(5/2) + (8*a*b*x^2*ArcTan[E^(c + d*Sqrt[x])])/d - (8*b^2*x^(3/2)*Log[1 + E^(2*(c + d*Sqrt[x]))])/d^2 - (16*I*a*b*x^(3/2)*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + (16*I*a*b*x^(3/2)*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 - (12*b^2*x*PolyLog[2, -E^(2*(c + d*Sqrt[x]))])/d^3 + (48*I*a*b*x*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - (48*I*a*b*x*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 + (12*b^2*Sqrt[x]*PolyLog[3, -E^(2*(c + d*Sqrt[x]))])/d^4 - (96*I*a*b*Sqrt[x]*PolyLog[4, (-I)*E^(c + d*Sqrt[x])])/d^4 + (96*I*a*b*Sqrt[x]*PolyLog[4, I*E^(c + d*Sqrt[x])])/d^4 - (6*b^2*PolyLog[4, -E^(2*(c + d*Sqrt[x]))])/d^5 + (96*I*a*b*PolyLog[5, (-I)*E^(c + d*Sqrt[x])])/d^5 - (96*I*a*b*PolyLog[5, I*E^(c + d*Sqrt[x])])/d^5 + (2*b^2*x^2*Tanh[c + d*Sqrt[x]])/d} -{Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])^2, x, 15, (2*b^2*x)/d + (2/3)*a^2*x^(3/2) + (8*a*b*x*ArcTan[E^(c + d*Sqrt[x])])/d - (4*b^2*Sqrt[x]*Log[1 + E^(2*(c + d*Sqrt[x]))])/d^2 - (8*I*a*b*Sqrt[x]*PolyLog[2, (-I)*E^(c + d*Sqrt[x])])/d^2 + (8*I*a*b*Sqrt[x]*PolyLog[2, I*E^(c + d*Sqrt[x])])/d^2 - (2*b^2*PolyLog[2, -E^(2*(c + d*Sqrt[x]))])/d^3 + (8*I*a*b*PolyLog[3, (-I)*E^(c + d*Sqrt[x])])/d^3 - (8*I*a*b*PolyLog[3, I*E^(c + d*Sqrt[x])])/d^3 + (2*b^2*x*Tanh[c + d*Sqrt[x]])/d} -{(a + b*Sech[c + d*Sqrt[x]])^2/Sqrt[x], x, 5, 2*a^2*Sqrt[x] + (4*a*b*ArcTan[Sinh[c + d*Sqrt[x]]])/d + (2*b^2*Tanh[c + d*Sqrt[x]])/d} -{(a + b*Sech[c + d*Sqrt[x]])^2/x^(3/2), x, 0, Unintegrable[(a + b*Sech[c + d*Sqrt[x]])^2/x^(3/2), x]} -{(a + b*Sech[c + d*Sqrt[x]])^2/x^(5/2), x, 0, Unintegrable[(a + b*Sech[c + d*Sqrt[x]])^2/x^(5/2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^(3/2)/(a + b*Sech[c + d*Sqrt[x]]), x, 17, (2*x^(5/2))/(5*a) - (2*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (8*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (8*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (24*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (24*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (48*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (48*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (48*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) - (48*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5)} -{Sqrt[x]/(a + b*Sech[c + d*Sqrt[x]]), x, 13, (2*x^(3/2))/(3*a) - (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (4*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (4*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + (4*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (4*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3)} -{1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])), x, 4, (2*Sqrt[x])/a - (4*b*ArcTan[(Sqrt[a - b]*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(x^(3/2)*(a + b*Sech[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Sech[c + d*Sqrt[x]])), x]} -{1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])), x]} - - -{x^(3/2)/(a + b*Sech[c + d*Sqrt[x]])^2, x, 43, (2*b^2*x^2)/(a^2*(a^2 - b^2)*d) + (2*x^(5/2))/(5*a^2) - (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (4*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) - (24*b^3*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (48*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^4) + (24*b^3*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (48*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) - (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) + (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) - (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^5) - (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^4) + (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^4) - (48*b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) + (96*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (48*b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^5) - (96*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^5) + (2*b^2*x^2*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))} -{Sqrt[x]/(a + b*Sech[c + d*Sqrt[x]])^2, x, 31, (2*b^2*x)/(a^2*(a^2 - b^2)*d) + (2*x^(3/2))/(3*a^2) - (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) + (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) + (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) + (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + (2*b^2*x*Sinh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(b + a*Cosh[c + d*Sqrt[x]]))} -{1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])^2), x, 6, (2*Sqrt[x])/a^2 - (4*b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (2*b^2*Tanh[c + d*Sqrt[x]])/(a*(a^2 - b^2)*d*(a + b*Sech[c + d*Sqrt[x]]))} -{1/(x^(3/2)*(a + b*Sech[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Sech[c + d*Sqrt[x]])^2), x]} -{1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Sech[c + d*Sqrt[x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Sech[c+d x^n])^p*) - - -{(e*x)^m*(a + b*Sech[c + d*x^n])^p, x, 1, ((e*x)^m*Unintegrable[x^m*(a + b*Sech[c + d*x^n])^p, x])/x^m} - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(e*x)^(-1 + n)*(a + b*Sech[c + d*x^n]), x, 5, (a*(e*x)^n)/(e*n) + (b*(e*x)^n*ArcTan[Sinh[c + d*x^n]])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Sech[c + d*x^n]), x, 9, (a*(e*x)^(2*n))/(2*e*n) + (2*b*(e*x)^(2*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x^n) - (I*b*(e*x)^(2*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (I*b*(e*x)^(2*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n]), x, 11, (a*(e*x)^(3*n))/(3*e*n) + (2*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x^n) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)*(a + b*Sech[c + d*x^n])^2, x, 6, (a^2*(e*x)^n)/(e*n) + (2*a*b*(e*x)^n*ArcTan[Sinh[c + d*x^n]])/(d*e*n*x^n) + (b^2*(e*x)^n*Tanh[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Sech[c + d*x^n])^2, x, 11, (a^2*(e*x)^(2*n))/(2*e*n) + (4*a*b*(e*x)^(2*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Log[Cosh[c + d*x^n]])/(d^2*e*n*x^(2*n)) - ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b^2*(e*x)^(2*n)*Tanh[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n])^2, x, 16, (a^2*(e*x)^(3*n))/(3*e*n) + (b^2*(e*x)^(3*n))/(x^n*(d*e*n)) + (4*a*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(x^n*(d*e*n)) - (2*b^2*(e*x)^(3*n)*Log[1 + E^(2*(c + d*x^n))])/(x^(2*n)*(d^2*e*n)) - (4*I*a*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(x^(2*n)*(d^2*e*n)) + (4*I*a*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(x^(2*n)*(d^2*e*n)) - (b^2*(e*x)^(3*n)*PolyLog[2, -E^(2*(c + d*x^n))])/(x^(3*n)*(d^3*e*n)) + (4*I*a*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(c + d*x^n)])/(x^(3*n)*(d^3*e*n)) - (4*I*a*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(x^(3*n)*(d^3*e*n)) + (b^2*(e*x)^(3*n)*Tanh[c + d*x^n])/(x^n*(d*e*n))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n]), x, 5, (e*x)^n/(a*e*n) - (2*b*(e*x)^n*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x^n)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d*e*n*x^n)} -{(e*x)^(-1 + 2*n)/(a + b*Sech[c + d*x^n]), x, 12, (e*x)^(2*n)/(2*a*e*n) - (b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)/(a + b*Sech[c + d*x^n]), x, 14, (e*x)^(3*n)/(3*a*e*n) - (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n])^2, x, 7, (e*x)^n/(a^2*e*n) - (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTan[(Sqrt[a - b]*Tanh[(c + d*x^n)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d*e*n*x^n) + (b^2*(e*x)^n*Tanh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Sech[c + d*x^n]))} -{(e*x)^(-1 + 2*n)/(a + b*Sech[c + d*x^n])^2, x, 23, (e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (b^2*(e*x)^(2*n)*Log[b + a*Cosh[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) + (b^2*(e*x)^(2*n)*Sinh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cosh[c + d*x^n]))} -{(e*x)^(-1 + 3*n)/(a + b*Sech[c + d*x^n])^2, x, 32, (e*x)^(3*n)/(3*a^2*e*n) + (b^2*(e*x)^(3*n))/(a^2*(a^2 - b^2)*d*e*n*x^n) - (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) - (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) + (b^2*(e*x)^(3*n)*Sinh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cosh[c + d*x^n]))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.3 Hyperbolic secant functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.3 Hyperbolic secant functions.m deleted file mode 100644 index 9b21a53..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.3 Hyperbolic secant functions.m +++ /dev/null @@ -1,398 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Secants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sech[a+b x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sech[a+b x]^n*) - - -{Sech[a + b*x], x, 1, ArcTan[Sinh[a + b*x]]/b} -{Sech[a + b*x]^2, x, 2, Tanh[a + b*x]/b} -{Sech[a + b*x]^3, x, 2, ArcTan[Sinh[a + b*x]]/(2*b) + (Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{Sech[a + b*x]^4, x, 2, Tanh[a + b*x]/b - Tanh[a + b*x]^3/(3*b)} -{Sech[a + b*x]^5, x, 3, (3*ArcTan[Sinh[a + b*x]])/(8*b) + (3*Sech[a + b*x]*Tanh[a + b*x])/(8*b) + (Sech[a + b*x]^3*Tanh[a + b*x])/(4*b)} -{Sech[a + b*x]^6, x, 2, Tanh[a + b*x]/b - (2*Tanh[a + b*x]^3)/(3*b) + Tanh[a + b*x]^5/(5*b)} - - -{Sech[7*x]^4, x, 2, (1/7)*Tanh[7*x] - (1/21)*Tanh[7*x]^3} - - -{Sech[Pi*x]^6, x, 2, Tanh[Pi*x]/Pi - (2*Tanh[Pi*x]^3)/(3*Pi) + Tanh[Pi*x]^5/(5*Pi)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sech[a+b x])^(n/2)*) - - -{Sech[a + b*x]^(5/2), x, 3, -((2*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(3*b)) + (2*Sech[a + b*x]^(3/2)*Sinh[a + b*x])/(3*b)} -{Sech[a + b*x]^(3/2), x, 3, (2*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b + (2*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/b} -{Sech[a + b*x]^(1/2), x, 2, -((2*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b)} -{1/Sech[a + b*x]^(1/2), x, 2, -((2*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b)} -{1/Sech[a + b*x]^(3/2), x, 3, -((2*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(3*b)) + (2*Sinh[a + b*x])/(3*b*Sqrt[Sech[a + b*x]])} -{1/Sech[a + b*x]^(5/2), x, 3, -((6*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(5*b)) + (2*Sinh[a + b*x])/(5*b*Sech[a + b*x]^(3/2))} - - -{(b*Sech[c + d*x])^(7/2), x, 4, (6*I*b^4*EllipticE[(1/2)*I*(c + d*x), 2])/(5*d*Sqrt[Cosh[c + d*x]]*Sqrt[b*Sech[c + d*x]]) + (6*b^3*Sqrt[b*Sech[c + d*x]]*Sinh[c + d*x])/(5*d) + (2*b*(b*Sech[c + d*x])^(5/2)*Sinh[c + d*x])/(5*d)} -{(b*Sech[c + d*x])^(5/2), x, 3, -((2*I*b^2*Sqrt[Cosh[c + d*x]]*EllipticF[(1/2)*I*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/(3*d)) + (2*b*(b*Sech[c + d*x])^(3/2)*Sinh[c + d*x])/(3*d)} -{(b*Sech[c + d*x])^(3/2), x, 3, (2*I*b^2*EllipticE[(1/2)*I*(c + d*x), 2])/(d*Sqrt[Cosh[c + d*x]]*Sqrt[b*Sech[c + d*x]]) + (2*b*Sqrt[b*Sech[c + d*x]]*Sinh[c + d*x])/d} -{(b*Sech[c + d*x])^(1/2), x, 2, -((2*I*Sqrt[Cosh[c + d*x]]*EllipticF[(1/2)*I*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/d)} -{1/(b*Sech[c + d*x])^(1/2), x, 2, -((2*I*EllipticE[(1/2)*I*(c + d*x), 2])/(d*Sqrt[Cosh[c + d*x]]*Sqrt[b*Sech[c + d*x]]))} -{1/(b*Sech[c + d*x])^(3/2), x, 3, -((2*I*Sqrt[Cosh[c + d*x]]*EllipticF[(1/2)*I*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/(3*b^2*d)) + (2*Sinh[c + d*x])/(3*b*d*Sqrt[b*Sech[c + d*x]])} -{1/(b*Sech[c + d*x])^(5/2), x, 3, -((6*I*EllipticE[(1/2)*I*(c + d*x), 2])/(5*b^2*d*Sqrt[Cosh[c + d*x]]*Sqrt[b*Sech[c + d*x]])) + (2*Sinh[c + d*x])/(5*b*d*(b*Sech[c + d*x])^(3/2))} -{1/(b*Sech[c + d*x])^(7/2), x, 4, -((10*I*Sqrt[Cosh[c + d*x]]*EllipticF[(1/2)*I*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/(21*b^4*d)) + (2*Sinh[c + d*x])/(7*b*d*(b*Sech[c + d*x])^(5/2)) + (10*Sinh[c + d*x])/(21*b^3*d*Sqrt[b*Sech[c + d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sech[a+b x])^n with n symbolic*) - - -{(b*Sech[c + d*x])^n, x, 2, -((b*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cosh[c + d*x]^2]*(b*Sech[c + d*x])^(-1 + n)*Sinh[c + d*x])/(d*(1 - n)*Sqrt[-Sinh[c + d*x]^2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Sech[a+b x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sech[a+b x]^2)^n*) - - -{(Sech[a + b*x]^2)^(7/2), x, 5, (5*ArcSin[Tanh[a + b*x]])/(16*b) + (5*Sqrt[Sech[a + b*x]^2]*Tanh[a + b*x])/(16*b) + (5*(Sech[a + b*x]^2)^(3/2)*Tanh[a + b*x])/(24*b) + ((Sech[a + b*x]^2)^(5/2)*Tanh[a + b*x])/(6*b)} -{(Sech[a + b*x]^2)^(5/2), x, 4, (3*ArcSin[Tanh[a + b*x]])/(8*b) + (3*Sqrt[Sech[a + b*x]^2]*Tanh[a + b*x])/(8*b) + ((Sech[a + b*x]^2)^(3/2)*Tanh[a + b*x])/(4*b)} -{(Sech[a + b*x]^2)^(3/2), x, 3, ArcSin[Tanh[a + b*x]]/(b*2) + (Sqrt[Sech[a + b*x]^2]*Tanh[a + b*x])/(2*b)} -{(Sech[a + b*x]^2)^(1/2), x, 2, ArcSin[Tanh[a + b*x]]/b} -{1/(Sech[a + b*x]^2)^(1/2), x, 2, Tanh[a + b*x]/(b*Sqrt[Sech[a + b*x]^2])} -{1/(Sech[a + b*x]^2)^(3/2), x, 3, Tanh[a + b*x]/(3*b*(Sech[a + b*x]^2)^(3/2)) + (2*Tanh[a + b*x])/(3*b*Sqrt[Sech[a + b*x]^2])} -{1/(Sech[a + b*x]^2)^(5/2), x, 4, Tanh[a + b*x]/(5*b*(Sech[a + b*x]^2)^(5/2)) + (4*Tanh[a + b*x])/(15*b*(Sech[a + b*x]^2)^(3/2)) + (8*Tanh[a + b*x])/(15*b*Sqrt[Sech[a + b*x]^2])} -{1/(Sech[a + b*x]^2)^(7/2), x, 5, Tanh[a + b*x]/(7*b*(Sech[a + b*x]^2)^(7/2)) + (6*Tanh[a + b*x])/(35*b*(Sech[a + b*x]^2)^(5/2)) + (8*Tanh[a + b*x])/(35*b*(Sech[a + b*x]^2)^(3/2)) + (16*Tanh[a + b*x])/(35*b*Sqrt[Sech[a + b*x]^2])} - - -{(a*Sech[x]^2)^(5/2), x, 5, (3/8)*a^(5/2)*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]] + (3/8)*a^2*Sqrt[a*Sech[x]^2]*Tanh[x] + (1/4)*a*(a*Sech[x]^2)^(3/2)*Tanh[x]} -{(a*Sech[x]^2)^(3/2), x, 4, (1/2)*a^(3/2)*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]] + (1/2)*a*Sqrt[a*Sech[x]^2]*Tanh[x]} -{(a*Sech[x]^2)^(1/2), x, 3, Sqrt[a]*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a*Sech[x]^2]]} -{1/(a*Sech[x]^2)^(1/2), x, 2, Tanh[x]/Sqrt[a*Sech[x]^2]} -{1/(a*Sech[x]^2)^(3/2), x, 3, Tanh[x]/(3*(a*Sech[x]^2)^(3/2)) + (2*Tanh[x])/(3*a*Sqrt[a*Sech[x]^2])} -{1/(a*Sech[x]^2)^(5/2), x, 4, Tanh[x]/(5*(a*Sech[x]^2)^(5/2)) + (4*Tanh[x])/(15*a*(a*Sech[x]^2)^(3/2)) + (8*Tanh[x])/(15*a^2*Sqrt[a*Sech[x]^2])} -{1/(a*Sech[x]^2)^(7/2), x, 5, Tanh[x]/(7*(a*Sech[x]^2)^(7/2)) + (6*Tanh[x])/(35*a*(a*Sech[x]^2)^(5/2)) + (8*Tanh[x])/(35*a^2*(a*Sech[x]^2)^(3/2)) + (16*Tanh[x])/(35*a^3*Sqrt[a*Sech[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sech[a+b x]^3)^n*) - - -{(a*Sech[x]^3)^(5/2), x, 7, (154/195)*I*a^2*Cosh[x]^(3/2)*EllipticE[(I*x)/2, 2]*Sqrt[a*Sech[x]^3] + (154/195)*a^2*Cosh[x]*Sqrt[a*Sech[x]^3]*Sinh[x] + (154/585)*a^2*Sqrt[a*Sech[x]^3]*Tanh[x] + (22/117)*a^2*Sech[x]^2*Sqrt[a*Sech[x]^3]*Tanh[x] + (2/13)*a^2*Sech[x]^4*Sqrt[a*Sech[x]^3]*Tanh[x]} -{(a*Sech[x]^3)^(3/2), x, 5, (-(10/21))*I*a*Cosh[x]^(3/2)*EllipticF[(I*x)/2, 2]*Sqrt[a*Sech[x]^3] + (10/21)*a*Sqrt[a*Sech[x]^3]*Sinh[x] + (2/7)*a*Sech[x]*Sqrt[a*Sech[x]^3]*Tanh[x]} -{(a*Sech[x]^3)^(1/2), x, 4, 2*I*Cosh[x]^(3/2)*EllipticE[(I*x)/2, 2]*Sqrt[a*Sech[x]^3] + 2*Cosh[x]*Sqrt[a*Sech[x]^3]*Sinh[x]} -{1/(a*Sech[x]^3)^(1/2), x, 4, -((2*I*EllipticF[(I*x)/2, 2])/(3*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3])) + (2*Tanh[x])/(3*Sqrt[a*Sech[x]^3])} -{1/(a*Sech[x]^3)^(3/2), x, 5, -((14*I*EllipticE[(I*x)/2, 2])/(15*a*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3])) + (14*Sinh[x])/(45*a*Sqrt[a*Sech[x]^3]) + (2*Cosh[x]^2*Sinh[x])/(9*a*Sqrt[a*Sech[x]^3])} -{1/(a*Sech[x]^3)^(5/2), x, 7, -((26*I*EllipticF[(I*x)/2, 2])/(77*a^2*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3])) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqrt[a*Sech[x]^3]) + (26*Cosh[x]^3*Sinh[x])/(165*a^2*Sqrt[a*Sech[x]^3]) + (2*Cosh[x]^5*Sinh[x])/(15*a^2*Sqrt[a*Sech[x]^3]) + (26*Tanh[x])/(77*a^2*Sqrt[a*Sech[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Sech[a+b x]^4)^n*) - - -{(a*Sech[x]^4)^(7/2), x, 3, a^3*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - 2*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x] + 3*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^3 - (20/7)*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5 + (5/3)*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^7 - (6/11)*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^9 + (1/13)*a^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^11} -{(a*Sech[x]^4)^(5/2), x, 3, a^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (4/3)*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x] + (6/5)*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^3 - (4/7)*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^5 + (1/9)*a^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^7} -{(a*Sech[x]^4)^(3/2), x, 3, a*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (2/3)*a*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x] + (1/5)*a*Sqrt[a*Sech[x]^4]*Sinh[x]^2*Tanh[x]^3} -{(a*Sech[x]^4)^(1/2), x, 3, Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x]} -{1/(a*Sech[x]^4)^(1/2), x, 3, (x*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4]) + Tanh[x]/(2*Sqrt[a*Sech[x]^4])} -{1/(a*Sech[x]^4)^(3/2), x, 5, (5*x*Sech[x]^2)/(16*a*Sqrt[a*Sech[x]^4]) + (5*Cosh[x]*Sinh[x])/(24*a*Sqrt[a*Sech[x]^4]) + (Cosh[x]^3*Sinh[x])/(6*a*Sqrt[a*Sech[x]^4]) + (5*Tanh[x])/(16*a*Sqrt[a*Sech[x]^4])} -{1/(a*Sech[x]^4)^(5/2), x, 7, (63*x*Sech[x]^2)/(256*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]*Sinh[x])/(128*a^2*Sqrt[a*Sech[x]^4]) + (21*Cosh[x]^3*Sinh[x])/(160*a^2*Sqrt[a*Sech[x]^4]) + (9*Cosh[x]^5*Sinh[x])/(80*a^2*Sqrt[a*Sech[x]^4]) + (Cosh[x]^7*Sinh[x])/(10*a^2*Sqrt[a*Sech[x]^4]) + (63*Tanh[x])/(256*a^2*Sqrt[a*Sech[x]^4])} - - -(* ::Subsection:: *) -(*Integrands of the form (c Sech[a+b x]^m)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Sech[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Sech[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2=0*) - - -{Sinh[x]^4/(a + a*Sech[x]), x, 7, -(x/(8*a)) - (Cosh[x]*Sinh[x])/(8*a) + (Cosh[x]^3*Sinh[x])/(4*a) - Sinh[x]^3/(3*a)} -{Sinh[x]^3/(a + a*Sech[x]), x, 6, Cosh[x]^3/(3*a) - Sinh[x]^2/(2*a)} -{Sinh[x]^2/(a + a*Sech[x]), x, 5, x/(2*a) - Sinh[x]/a + (Cosh[x]*Sinh[x])/(2*a)} -{Sinh[x]^1/(a + a*Sech[x]), x, 5, Cosh[x]/a - Log[1 + Cosh[x]]/a} -{Csch[x]^1/(a + a*Sech[x]), x, 6, -(ArcTanh[Cosh[x]]/(2*a)) - (Coth[x]*Csch[x])/(2*a) + Csch[x]^2/(2*a)} -{Csch[x]^2/(a + a*Sech[x]), x, 6, -(Coth[x]^3/(3*a)) + Csch[x]^3/(3*a)} -{Csch[x]^3/(a + a*Sech[x]), x, 7, ArcTanh[Cosh[x]]/(8*a) - (Coth[x]*Csch[x])/(8*a) - (Coth[x]*Csch[x]^3)/(4*a) + Csch[x]^4/(4*a)} -{Csch[x]^4/(a + a*Sech[x]), x, 7, Coth[x]^3/(3*a) - Coth[x]^5/(5*a) + Csch[x]^5/(5*a)} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2!=0*) - - -{Sinh[x]^4/(a + b*Sech[x]), x, 6, ((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/a^5 + ((8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Cosh[x])*Sinh[x])/(8*a^4) - ((4*b - 3*a*Cosh[x])*Sinh[x]^3)/(12*a^2)} -{Sinh[x]^3/(a + b*Sech[x]), x, 5, -(((a^2 - b^2)*Cosh[x])/a^3) - (b*Cosh[x]^2)/(2*a^2) + Cosh[x]^3/(3*a) + (b*(a^2 - b^2)*Log[b + a*Cosh[x]])/a^4} -{Sinh[x]^2/(a + b*Sech[x]), x, 5, -(((a^2 - 2*b^2)*x)/(2*a^3)) + (2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/a^3 - ((2*b - a*Cosh[x])*Sinh[x])/(2*a^2)} -{Sinh[x]^1/(a + b*Sech[x]), x, 5, Cosh[x]/a - (b*Log[b + a*Cosh[x]])/a^2} -{Csch[x]^1/(a + b*Sech[x]), x, 4, Log[1 - Cosh[x]]/(2*(a + b)) - Log[1 + Cosh[x]]/(2*(a - b)) + (b*Log[b + a*Cosh[x]])/(a^2 - b^2)} -{Csch[x]^2/(a + b*Sech[x]), x, 5, (2*a*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) + ((b - a*Cosh[x])*Csch[x])/(a^2 - b^2)} -{Csch[x]^3/(a + b*Sech[x]), x, 6, ((b - a*Cosh[x])*Csch[x]^2)/(2*(a^2 - b^2)) - (a*Log[1 - Cosh[x]])/(4*(a + b)^2) + (a*Log[1 + Cosh[x]])/(4*(a - b)^2) - (a^2*b*Log[b + a*Cosh[x]])/(a^2 - b^2)^2} -{Csch[x]^4/(a + b*Sech[x]), x, 6, -((2*a^3*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2))) - ((3*a^2*b - a*(2*a^2 + b^2)*Cosh[x])*Csch[x])/(3*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^3)/(3*(a^2 - b^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Sech[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2=0*) - - -{Cosh[x]^4/(a + a*Sech[x]), x, 7, (15*x)/(8*a) - (4*Sinh[x])/a + (15*Cosh[x]*Sinh[x])/(8*a) + (5*Cosh[x]^3*Sinh[x])/(4*a) - (Cosh[x]^3*Sinh[x])/(a + a*Sech[x]) - (4*Sinh[x]^3)/(3*a)} -{Cosh[x]^3/(a + a*Sech[x]), x, 6, -((3*x)/(2*a)) + (4*Sinh[x])/a - (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]^2*Sinh[x])/(a + a*Sech[x]) + (4*Sinh[x]^3)/(3*a)} -{Cosh[x]^2/(a + a*Sech[x]), x, 5, (3*x)/(2*a) - (2*Sinh[x])/a + (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]*Sinh[x])/(a + a*Sech[x])} -{Cosh[x]^1/(a + a*Sech[x]), x, 4, -(x/a) + (2*Sinh[x])/a - Sinh[x]/(a + a*Sech[x])} -{Sech[x]^1/(a + a*Sech[x]), x, 1, Tanh[x]/(a + a*Sech[x])} -{Sech[x]^2/(a + a*Sech[x]), x, 3, ArcTan[Sinh[x]]/a - Tanh[x]/(a + a*Sech[x])} -{Sech[x]^3/(a + a*Sech[x]), x, 4, -(ArcTan[Sinh[x]]/a) + Tanh[x]/a + Tanh[x]/(a + a*Sech[x])} -{Sech[x]^4/(a + a*Sech[x]), x, 6, (3*ArcTan[Sinh[x]])/(2*a) - (2*Tanh[x])/a + (3*Sech[x]*Tanh[x])/(2*a) - (Sech[x]^2*Tanh[x])/(a + a*Sech[x])} - - -{1/(a + a*Sech[c + d*x]), x, 2, x/a - Tanh[c + d*x]/(d*(a + a*Sech[c + d*x]))} - - -{1/(a - a*Sech[c + d*x]), x, 2, x/a - Tanh[c + d*x]/(d*(a - a*Sech[c + d*x]))} - - -{(a + a*Sech[c + d*x])^(5/2), x, 5, (2*a^(5/2)*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d + (14*a^3*Tanh[c + d*x])/(3*d*Sqrt[a + a*Sech[c + d*x]]) + (2*a^2*Sqrt[a + a*Sech[c + d*x]]*Tanh[c + d*x])/(3*d)} -{(a + a*Sech[c + d*x])^(3/2), x, 4, (2*a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d + (2*a^2*Tanh[c + d*x])/(d*Sqrt[a + a*Sech[c + d*x]])} -{(a + a*Sech[c + d*x])^(1/2), x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d} -{1/(a + a*Sech[c + d*x])^(1/2), x, 5, (2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(Sqrt[a]*d)} -{1/(a + a*Sech[c + d*x])^(3/2), x, 6, (2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/2))} - -{Sqrt[a - a*Sech[c + d*x]], x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a - a*Sech[c + d*x]]])/d} -{1/Sqrt[a - a*Sech[c + d*x]], x, 5, (2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a - a*Sech[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Sech[c + d*x]])])/(Sqrt[a]*d)} - - -{Sqrt[3 + 3*Sech[x]], x, 2, 2*Sqrt[3]*ArcTanh[Tanh[x]/Sqrt[1 + Sech[x]]]} -{Sqrt[3 - 3*Sech[x]], x, 2, 2*Sqrt[3]*ArcTanh[Tanh[x]/Sqrt[1 - Sech[x]]]} - - -(* ::Subsubsection::Closed:: *) -(*a^2-b^2!=0*) - - -{(a + b*Sech[c + d*x])^4, x, 6, a^4*x + (2*a*b*(2*a^2 + b^2)*ArcTan[Sinh[c + d*x]])/d + (b^2*(17*a^2 + 2*b^2)*Tanh[c + d*x])/(3*d) + (4*a*b^3*Sech[c + d*x]*Tanh[c + d*x])/(3*d) + (b^2*(a + b*Sech[c + d*x])^2*Tanh[c + d*x])/(3*d)} -{(a + b*Sech[c + d*x])^3, x, 5, a^3*x + (b*(6*a^2 + b^2)*ArcTan[Sinh[c + d*x]])/(2*d) + (5*a*b^2*Tanh[c + d*x])/(2*d) + (b^2*(a + b*Sech[c + d*x])*Tanh[c + d*x])/(2*d)} -{(a + b*Sech[c + d*x])^2, x, 4, a^2*x + (2*a*b*ArcTan[Sinh[c + d*x]])/d + (b^2*Tanh[c + d*x])/d} -{(a + b*Sech[c + d*x])^1, x, 2, a*x + (b*ArcTan[Sinh[c + d*x]])/d} -{1/(a + b*Sech[c + d*x])^1, x, 3, x/a - (2*b*ArcTan[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)} -{1/(a + b*Sech[c + d*x])^2, x, 5, x/a^2 - (2*b*(2*a^2 - b^2)*ArcTan[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sech[c + d*x]))} -{1/(a + b*Sech[c + d*x])^3, x, 6, x/a^3 - (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tanh[(1/2)*(c + d*x)])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*Tanh[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sech[c + d*x])^2) + (b^2*(5*a^2 - 2*b^2)*Tanh[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sech[c + d*x]))} - - -{1/Sqrt[a + b*Sech[c + d*x]], x, 1, (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)} - - -{Cosh[x]^4/(a + b*Sech[x]), x, 8, ((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) - (2*b^5*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]) - (b*(2*a^2 + 3*b^2)*Sinh[x])/(3*a^4) + ((3*a^2 + 4*b^2)*Cosh[x]*Sinh[x])/(8*a^3) - (b*Cosh[x]^2*Sinh[x])/(3*a^2) + (Cosh[x]^3*Sinh[x])/(4*a)} -{Cosh[x]^3/(a + b*Sech[x]), x, 7, -((b*(a^2 + 2*b^2)*x)/(2*a^4)) + (2*b^4*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]) + ((2*a^2 + 3*b^2)*Sinh[x])/(3*a^3) - (b*Cosh[x]*Sinh[x])/(2*a^2) + (Cosh[x]^2*Sinh[x])/(3*a)} -{Cosh[x]^2/(a + b*Sech[x]), x, 6, ((a^2 + 2*b^2)*x)/(2*a^3) - (2*b^3*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]) - (b*Sinh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)} -{Cosh[x]^1/(a + b*Sech[x]), x, 5, -((b*x)/a^2) + (2*b^2*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) + Sinh[x]/a} -{Sech[x]^1/(a + b*Sech[x]), x, 3, (2*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b])} -{Sech[x]^2/(a + b*Sech[x]), x, 5, ArcTan[Sinh[x]]/b - (2*a*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b])} -{Sech[x]^3/(a + b*Sech[x]), x, 6, -((a*ArcTan[Sinh[x]])/b^2) + (2*a^2*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]) + Tanh[x]/b} -{Sech[x]^4/(a + b*Sech[x]), x, 7, ((2*a^2 + b^2)*ArcTan[Sinh[x]])/(2*b^3) - (2*a^3*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]) - (a*Tanh[x])/b^2 + (Sech[x]*Tanh[x])/(2*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Sech[c+d x])^n when a^2-b^2=0*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tanh[x]^6/(a + a*Sech[x]), x, 5, x/a - (3*ArcTan[Sinh[x]])/(8*a) - ((8 - 3*Sech[x])*Tanh[x])/(8*a) - ((4 - 3*Sech[x])*Tanh[x]^3)/(12*a)} -{Tanh[x]^5/(a + a*Sech[x]), x, 3, Log[Cosh[x]]/a + Sech[x]/a + Sech[x]^2/(2*a) - Sech[x]^3/(3*a)} -{Tanh[x]^4/(a + a*Sech[x]), x, 4, x/a - ArcTan[Sinh[x]]/(2*a) - ((2 - Sech[x])*Tanh[x])/(2*a)} -{Tanh[x]^3/(a + a*Sech[x]), x, 3, Log[Cosh[x]]/a + Sech[x]/a} -{Tanh[x]^2/(a + a*Sech[x]), x, 3, x/a - ArcTan[Sinh[x]]/a} -{Tanh[x]^1/(a + a*Sech[x]), x, 2, Log[1 + Cosh[x]]/a} -{Coth[x]^1/(a + a*Sech[x]), x, 3, 1/(2*a*(1 + Cosh[x])) + Log[1 - Cosh[x]]/(4*a) + (3*Log[1 + Cosh[x]])/(4*a)} -{Coth[x]^2/(a + a*Sech[x]), x, 4, x/a - (Coth[x]*(3 - 2*Sech[x]))/(3*a) - (Coth[x]^3*(1 - Sech[x]))/(3*a)} -{Coth[x]^3/(a + a*Sech[x]), x, 3, 1/(8*a*(1 - Cosh[x])) - 1/(8*a*(1 + Cosh[x])^2) + 3/(4*a*(1 + Cosh[x])) + (5*Log[1 - Cosh[x]])/(16*a) + (11*Log[1 + Cosh[x]])/(16*a)} -{Coth[x]^4/(a + a*Sech[x]), x, 5, x/a - (Coth[x]*(15 - 8*Sech[x]))/(15*a) - (Coth[x]^3*(5 - 4*Sech[x]))/(15*a) - (Coth[x]^5*(1 - Sech[x]))/(5*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Sech[c+d x])^n*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tanh[x]^7/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a - ((a^2 - b^2)^3*Log[a + b*Sech[x]])/(a*b^6) + ((a^4 - 3*a^2*b^2 + 3*b^4)*Sech[x])/b^5 - (a*(a^2 - 3*b^2)*Sech[x]^2)/(2*b^4) + ((a^2 - 3*b^2)*Sech[x]^3)/(3*b^3) - (a*Sech[x]^4)/(4*b^2) + Sech[x]^5/(5*b)} -{Tanh[x]^6/(a + b*Sech[x]), x, 15, x/a - (3*ArcTan[Sinh[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTan[Sinh[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTan[Sinh[x]])/b^5 + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b^5) + (a*Tanh[x])/b^2 + (a*(a^2 - 3*b^2)*Tanh[x])/b^4 - (3*Sech[x]*Tanh[x])/(8*b) - ((a^2 - 3*b^2)*Sech[x]*Tanh[x])/(2*b^3) - (Sech[x]^3*Tanh[x])/(4*b) - (a*Tanh[x]^3)/(3*b^2)} -{Tanh[x]^5/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a + ((a^2 - b^2)^2*Log[a + b*Sech[x]])/(a*b^4) - ((a^2 - 2*b^2)*Sech[x])/b^3 + (a*Sech[x]^2)/(2*b^2) - Sech[x]^3/(3*b)} -{Tanh[x]^4/(a + b*Sech[x]), x, 6, x/a + ((2*a^2 - 3*b^2)*ArcTan[Sinh[x]])/(2*b^3) - (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b^3) - (a*Tanh[x])/b^2 + (Sech[x]*Tanh[x])/(2*b)} -{Tanh[x]^3/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a + ((1 - a^2/b^2)*Log[a + b*Sech[x]])/a + Sech[x]/b} -{Tanh[x]^2/(a + b*Sech[x]), x, 7, x/a - ArcTan[Sinh[x]]/b + (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b)} -{Tanh[x]^1/(a + b*Sech[x]), x, 4, Log[Cosh[x]]/a + Log[a + b*Sech[x]]/a} -{Coth[x]^1/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a + Log[1 - Sech[x]]/(2*(a + b)) + Log[1 + Sech[x]]/(2*(a - b)) - (b^2*Log[a + b*Sech[x]])/(a*(a^2 - b^2))} -{Coth[x]^2/(a + b*Sech[x]), x, 9, (a*x)/(a^2 - b^2) - (b^2*x)/(a*(a^2 - b^2)) + (2*b^3*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)) - (a*Coth[x])/(a^2 - b^2) + (b*Csch[x])/(a^2 - b^2)} -{Coth[x]^3/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a + ((2*a + 3*b)*Log[1 - Sech[x]])/(4*(a + b)^2) + ((2*a - 3*b)*Log[1 + Sech[x]])/(4*(a - b)^2) + (b^4*Log[a + b*Sech[x]])/(a*(a^2 - b^2)^2) - 1/(4*(a + b)*(1 - Sech[x])) - 1/(4*(a - b)*(1 + Sech[x]))} -{Coth[x]^4/(a + b*Sech[x]), x, 15, -((a*b^2*x)/(a^2 - b^2)^2) + (b^4*x)/(a*(a^2 - b^2)^2) + (a*x)/(a^2 - b^2) - (2*b^5*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)) + (a*b^2*Coth[x])/(a^2 - b^2)^2 - (a*Coth[x])/(a^2 - b^2) - (a*Coth[x]^3)/(3*(a^2 - b^2)) - (b^3*Csch[x])/(a^2 - b^2)^2 + (b*Csch[x])/(a^2 - b^2) + (b*Csch[x]^3)/(3*(a^2 - b^2))} -{Coth[x]^5/(a + b*Sech[x]), x, 3, Log[Cosh[x]]/a + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sech[x]])/(16*(a + b)^3) + ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sech[x]])/(16*(a - b)^3) - (b^6*Log[a + b*Sech[x]])/(a*(a^2 - b^2)^3) - 1/(16*(a + b)*(1 - Sech[x])^2) - (5*a + 7*b)/(16*(a + b)^2*(1 - Sech[x])) - 1/(16*(a - b)*(1 + Sech[x])^2) - (5*a - 7*b)/(16*(a - b)^2*(1 + Sech[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Sech[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Tanh[c + d*x]^5*Sqrt[a + b*Sech[c + d*x]], x, 5, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sech[c + d*x]])/d + (2*a*(a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d) + (6*a*(a + b*Sech[c + d*x])^(7/2))/(7*b^4*d) - (2*(a + b*Sech[c + d*x])^(9/2))/(9*b^4*d)} -{Tanh[c + d*x]^3*Sqrt[a + b*Sech[c + d*x]], x, 5, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sech[c + d*x]])/d - (2*a*(a + b*Sech[c + d*x])^(3/2))/(3*b^2*d) + (2*(a + b*Sech[c + d*x])^(5/2))/(5*b^2*d)} -{Tanh[c + d*x]^1*Sqrt[a + b*Sech[c + d*x]], x, 4, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sech[c + d*x]])/d} -{Coth[c + d*x]^1*Sqrt[a + b*Sech[c + d*x]], x, 7, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/d - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/d} -{Coth[c + d*x]^3*Sqrt[a + b*Sech[c + d*x]], x, 13, (2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (a*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(Sqrt[a - b]*d) + (3*b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) - (a*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(Sqrt[a + b]*d) - (3*b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a + b]*d) - (Coth[c + d*x]^2*Sqrt[a + b*Sech[c + d*x]])/(2*d)} - -{Tanh[c + d*x]^2*Sqrt[a + b*Sech[c + d*x]], x, 7, -((2*a*(a - b)*Sqrt[a + b]*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(3*b^2*d)) - (2*Sqrt[a + b]*(a + 2*b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x])/(3*d)} -{Tanh[c + d*x]^0*Sqrt[a + b*Sech[c + d*x]], x, 1, (2*Coth[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sech[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sech[c + d*x]))/(a + b*Sech[c + d*x]))]*Sqrt[(b*(1 + Sech[c + d*x]))/(a + b*Sech[c + d*x])]*(a + b*Sech[c + d*x]))/(Sqrt[a + b]*d)} -{Coth[c + d*x]^2*Sqrt[a + b*Sech[c + d*x]], x, 5, (Sqrt[a + b]*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/d - (Coth[c + d*x]*Sqrt[a + b*Sech[c + d*x]])/d + (2*Coth[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sech[c + d*x]]], (a - b)/(a + b)]*Sqrt[-((b*(1 - Sech[c + d*x]))/(a + b*Sech[c + d*x]))]*Sqrt[(b*(1 + Sech[c + d*x]))/(a + b*Sech[c + d*x])]*(a + b*Sech[c + d*x]))/(Sqrt[a + b]*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Tanh[c + d*x]^5/Sqrt[a + b*Sech[c + d*x]], x, 5, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) + (6*a*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d) - (2*(a + b*Sech[c + d*x])^(7/2))/(7*b^4*d)} -{Tanh[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]], x, 5, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - (2*a*Sqrt[a + b*Sech[c + d*x]])/(b^2*d) + (2*(a + b*Sech[c + d*x])^(3/2))/(3*b^2*d)} -{Tanh[c + d*x]^1/Sqrt[a + b*Sech[c + d*x]], x, 3, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d)} -{Coth[c + d*x]^1/Sqrt[a + b*Sech[c + d*x]], x, 7, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)} -{Coth[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]], x, 11, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d) + (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) - (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d) - Sqrt[a + b*Sech[c + d*x]]/(4*(a + b)*d*(1 - Sech[c + d*x])) - Sqrt[a + b*Sech[c + d*x]]/(4*(a - b)*d*(1 + Sech[c + d*x]))} - -{Tanh[c + d*x]^4/Sqrt[a + b*Sech[c + d*x]], x, 11, -((4*(a - b)*Sqrt[a + b]*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b^2*d)) + (2*(a - b)*Sqrt[a + b]*(8*a^2 + 9*b^2)*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(15*b^4*d) - (4*Sqrt[a + b]*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(8*a^2 - 2*a*b + 9*b^2)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(15*b^3*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d) - (8*a*Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x])/(15*b^2*d) + (2*Sech[c + d*x]*Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x])/(5*b*d)} -{Tanh[c + d*x]^2/Sqrt[a + b*Sech[c + d*x]], x, 6, -((2*(a - b)*Sqrt[a + b]*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b^2*d)) - (2*Sqrt[a + b]*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)} -{Tanh[c + d*x]^0/Sqrt[a + b*Sech[c + d*x]], x, 1, (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)} -{Coth[c + d*x]^2/Sqrt[a + b*Sech[c + d*x]], x, 9, (Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d) - Coth[c + d*x]/(d*Sqrt[a + b*Sech[c + d*x]]) - (b^2*Tanh[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]])} - - -{Tanh[c + d*x]^5/(a + b*Sech[c + d*x])^(3/2), x, 5, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (2*(a^2 - b^2)^2)/(a*b^4*d*Sqrt[a + b*Sech[c + d*x]]) - (2*(3*a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^4*d) + (2*a*(a + b*Sech[c + d*x])^(3/2))/(b^4*d) - (2*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d)} -{Tanh[c + d*x]^3/(a + b*Sech[c + d*x])^(3/2), x, 5, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (2*(a^2 - b^2))/(a*b^2*d*Sqrt[a + b*Sech[c + d*x]]) + (2*Sqrt[a + b*Sech[c + d*x]])/(b^2*d)} -{Tanh[c + d*x]^1/(a + b*Sech[c + d*x])^(3/2), x, 4, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - 2/(a*d*Sqrt[a + b*Sech[c + d*x]])} -{Coth[c + d*x]^1/(a + b*Sech[c + d*x])^(3/2), x, 7, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/((a - b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]])} -{Coth[c + d*x]^3/(a + b*Sech[c + d*x])^(3/2), x, 11, (2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(2*(a - b)^(5/2)*d) + (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) - (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) - ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) - (2*b^4)/(a*(a^2 - b^2)^2*d*Sqrt[a + b*Sech[c + d*x]]) - Sqrt[a + b*Sech[c + d*x]]/(4*(a + b)^2*d*(1 - Sech[c + d*x])) - Sqrt[a + b*Sech[c + d*x]]/(4*(a - b)^2*d*(1 + Sech[c + d*x]))} - -{Tanh[c + d*x]^4/(a + b*Sech[c + d*x])^(3/2), x, 17, -((2*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d)) + (4*a*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*a*(8*a^2 - 5*b^2)*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (4*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(2*a + b)*(4*a + b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a^2*d) - (4*a*Tanh[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]]) + (2*b^2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]]) - (2*a^2*Sech[c + d*x]*Tanh[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x])/(3*b^2*(a^2 - b^2)*d)} -{Tanh[c + d*x]^2/(a + b*Sech[c + d*x])^(3/2), x, 7, (2*(a - b)*Sqrt[a + b]*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*b^2*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*b*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a^2*d) - (2*Tanh[c + d*x])/(a*d*Sqrt[a + b*Sech[c + d*x]])} -{Tanh[c + d*x]^0/(a + b*Sech[c + d*x])^(3/2), x, 6, -((2*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d)) + (2*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]])} -{Coth[c + d*x]^2/(a + b*Sech[c + d*x])^(3/2), x, 14, (4*a*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) - (2*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - ((3*a - b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/((a - b)*(a + b)^(3/2)*d) + (2*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a^2*d) - Coth[c + d*x]/(d*(a + b*Sech[c + d*x])^(3/2)) - (b^2*Tanh[c + d*x])/((a^2 - b^2)*d*(a + b*Sech[c + d*x])^(3/2)) - (4*a*b^2*Tanh[c + d*x])/((a^2 - b^2)^2*d*Sqrt[a + b*Sech[c + d*x]]) + (2*b^2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]])} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Sech[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Sech[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Sech[a*c + b*c*x]^2)^(7/2), x, 6, (32*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x)))^6) - (192*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(5*b*c*(1 + E^(2*c*(a + b*x)))^5) + (48*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^4) - (64*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3)} -{E^(c*(a + b*x))*(Sech[a*c + b*c*x]^2)^(5/2), x, 6, (-4*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^4) + (32*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(3*b*c*(1 + E^(2*c*(a + b*x)))^3) - (8*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^2)} -{E^(c*(a + b*x))*(Sech[a*c + b*c*x]^2)^(3/2), x, 4, (2*E^(4*c*(a + b*x))*Cosh[a*c + b*c*x]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c*(1 + E^(2*c*(a + b*x)))^2)} -{E^(c*(a + b*x))*Sqrt[Sech[a*c + b*c*x]^2], x, 4, (Cosh[a*c + b*c*x]*Log[1 + E^(2*c*(a + b*x))]*Sqrt[Sech[a*c + b*c*x]^2])/(b*c)} -{E^(c*(a + b*x))/Sqrt[Sech[a*c + b*c*x]^2], x, 5, (E^(2*c*(a + b*x))*Sech[a*c + b*c*x])/(4*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (x*Sech[a*c + b*c*x])/(2*Sqrt[Sech[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Sech[a*c + b*c*x]^2)^(3/2), x, 6, -Sech[a*c + b*c*x]/(16*b*c*E^(2*c*(a + b*x))*Sqrt[Sech[a*c + b*c*x]^2]) + (3*E^(2*c*(a + b*x))*Sech[a*c + b*c*x])/(16*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (E^(4*c*(a + b*x))*Sech[a*c + b*c*x])/(32*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (3*x*Sech[a*c + b*c*x])/(8*Sqrt[Sech[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Sech[a*c + b*c*x]^2)^(5/2), x, 6, -Sech[a*c + b*c*x]/(128*b*c*E^(4*c*(a + b*x))*Sqrt[Sech[a*c + b*c*x]^2]) - (5*Sech[a*c + b*c*x])/(64*b*c*E^(2*c*(a + b*x))*Sqrt[Sech[a*c + b*c*x]^2]) + (5*E^(2*c*(a + b*x))*Sech[a*c + b*c*x])/(32*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (5*E^(4*c*(a + b*x))*Sech[a*c + b*c*x])/(128*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (E^(6*c*(a + b*x))*Sech[a*c + b*c*x])/(192*b*c*Sqrt[Sech[a*c + b*c*x]^2]) + (5*x*Sech[a*c + b*c*x])/(16*Sqrt[Sech[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Sech[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sech[b Log[c x^n]]^(p/2)*) - - -{x^5/Sech[2*Log[c*x]]^(1/2), x, 6, (2*x^2)/(21*c^4*Sqrt[Sech[2*Log[c*x]]]) + x^6/(7*Sqrt[Sech[2*Log[c*x]]]) + (Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(21*c^5*(c^4 + 1/x^4)*x*Sqrt[Sech[2*Log[c*x]]])} -{x^4/Sech[2*Log[c*x]]^(1/2), x, 3, ((c^4 + 1/x^4)*x^5)/(6*c^4*Sqrt[Sech[2*Log[c*x]]])} -{x^3/Sech[2*Log[c*x]]^(1/2), x, 8, 2/(5*c^4*Sqrt[Sech[2*Log[c*x]]]) - 2/(5*c^4*(c^2 + 1/x^2)*x^2*Sqrt[Sech[2*Log[c*x]]]) + x^4/(5*Sqrt[Sech[2*Log[c*x]]]) + (2*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticE[2*ArcCot[c*x], 1/2])/(5*c^3*(c^4 + 1/x^4)*x*Sqrt[Sech[2*Log[c*x]]]) - (Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(5*c^3*(c^4 + 1/x^4)*x*Sqrt[Sech[2*Log[c*x]]])} -{x^2/Sech[2*Log[c*x]]^(1/2), x, 6, x^3/(4*Sqrt[Sech[2*Log[c*x]]]) + ArcTanh[Sqrt[1 + 1/(c^4*x^4)]]/(4*c^4*Sqrt[1 + 1/(c^4*x^4)]*x*Sqrt[Sech[2*Log[c*x]]])} -{x^1/Sech[2*Log[c*x]]^(1/2), x, 5, x^2/(3*Sqrt[Sech[2*Log[c*x]]]) - (Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(3*c*(c^4 + 1/x^4)*x*Sqrt[Sech[2*Log[c*x]]])} -{x^0/Sech[2*Log[c*x]]^(1/2), x, 6, x/(2*Sqrt[Sech[2*Log[c*x]]]) - ArcCsch[c^2*x^2]/(2*c^2*Sqrt[1 + 1/(c^4*x^4)]*x*Sqrt[Sech[2*Log[c*x]]])} -{Sech[2*Log[c*x]]^(1/2)/x^1, x, 3, (-I)*Sqrt[Cosh[2*Log[c*x]]]*EllipticF[I*Log[c*x], 2]*Sqrt[Sech[2*Log[c*x]]]} -{Sech[2*Log[c*x]]^(1/2)/x^2, x, 5, (-(1/2))*c^2*Sqrt[1 + 1/(c^4*x^4)]*x*ArcCsch[c^2*x^2]*Sqrt[Sech[2*Log[c*x]]]} -{Sech[2*Log[c*x]]^(1/2)/x^3, x, 6, -(((c^4 + 1/x^4)*Sqrt[Sech[2*Log[c*x]]])/(c^2 + 1/x^2)) + c*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*x*EllipticE[2*ArcCot[c*x], 1/2]*Sqrt[Sech[2*Log[c*x]]] - (1/2)*c*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*x*EllipticF[2*ArcCot[c*x], 1/2]*Sqrt[Sech[2*Log[c*x]]]} -{Sech[2*Log[c*x]]^(1/2)/x^4, x, 3, (-(1/2))*(c^4 + 1/x^4)*x*Sqrt[Sech[2*Log[c*x]]]} -{Sech[2*Log[c*x]]^(1/2)/x^5, x, 5, (-(1/3))*(c^4 + 1/x^4)*Sqrt[Sech[2*Log[c*x]]] + (1/6)*c^3*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*x*EllipticF[2*ArcCot[c*x], 1/2]*Sqrt[Sech[2*Log[c*x]]]} - - -{x^8/Sech[2*Log[c*x]]^(3/2), x, 8, x/(32*c^4*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^5/(16*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^9/(12*Sech[2*Log[c*x]]^(3/2)) - ArcTanh[Sqrt[1 + 1/(c^4*x^4)]]/(32*c^12*(1 + 1/(c^4*x^4))^(3/2)*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^7/Sech[2*Log[c*x]]^(3/2), x, 7, 4/(77*c^4*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + (6*x^4)/(77*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^8/(11*Sech[2*Log[c*x]]^(3/2)) + (2*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(77*c^5*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^6/Sech[2*Log[c*x]]^(3/2), x, 3, ((c^4 + 1/x^4)*x^7)/(10*c^4*Sech[2*Log[c*x]]^(3/2))} -{x^5/Sech[2*Log[c*x]]^(3/2), x, 9, -(4/(15*c^4*(c^4 + 1/x^4)*(c^2 + 1/x^2)*x^4*Sech[2*Log[c*x]]^(3/2))) + 4/(15*c^4*(c^4 + 1/x^4)*x^2*Sech[2*Log[c*x]]^(3/2)) + (2*x^2)/(15*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^6/(9*Sech[2*Log[c*x]]^(3/2)) + (4*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticE[2*ArcCot[c*x], 1/2])/(15*c^3*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2)) - (2*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(15*c^3*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^4/Sech[2*Log[c*x]]^(3/2), x, 7, (3*x)/(16*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^5/(8*Sech[2*Log[c*x]]^(3/2)) + (3*ArcTanh[Sqrt[1 + 1/(c^4*x^4)]])/(16*c^8*(1 + 1/(c^4*x^4))^(3/2)*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^3/Sech[2*Log[c*x]]^(3/2), x, 6, 2/(7*(c^4 + 1/x^4)*Sech[2*Log[c*x]]^(3/2)) + x^4/(7*Sech[2*Log[c*x]]^(3/2)) - (2*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(7*c*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^2/Sech[2*Log[c*x]]^(3/2), x, 7, 1/(2*(c^4 + 1/x^4)*x*Sech[2*Log[c*x]]^(3/2)) + x^3/(6*Sech[2*Log[c*x]]^(3/2)) - ArcCsch[c^2*x^2]/(2*c^6*(1 + 1/(c^4*x^4))^(3/2)*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^1/Sech[2*Log[c*x]]^(3/2), x, 8, -(12/(5*(c^4 + 1/x^4)*(c^2 + 1/x^2)*x^4*Sech[2*Log[c*x]]^(3/2))) + 6/(5*(c^4 + 1/x^4)*x^2*Sech[2*Log[c*x]]^(3/2)) + x^2/(5*Sech[2*Log[c*x]]^(3/2)) + (12*c*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticE[2*ArcCot[c*x], 1/2])/(5*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2)) - (6*c*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*EllipticF[2*ArcCot[c*x], 1/2])/(5*(c^4 + 1/x^4)^2*x^3*Sech[2*Log[c*x]]^(3/2))} -{x^0/Sech[2*Log[c*x]]^(3/2), x, 7, -(3/(4*(c^4 + 1/x^4)*x^3*Sech[2*Log[c*x]]^(3/2))) + x/(4*Sech[2*Log[c*x]]^(3/2)) + (3*ArcTanh[Sqrt[1 + 1/(c^4*x^4)]])/(4*c^4*(1 + 1/(c^4*x^4))^(3/2)*x^3*Sech[2*Log[c*x]]^(3/2))} -{Sech[2*Log[c*x]]^(3/2)/x^1, x, 4, I*Sqrt[Cosh[2*Log[c*x]]]*EllipticE[I*Log[c*x], 2]*Sqrt[Sech[2*Log[c*x]]] + Sqrt[Sech[2*Log[c*x]]]*Sinh[2*Log[c*x]]} -{Sech[2*Log[c*x]]^(3/2)/x^2, x, 3, (1/2)*(c^4 + 1/x^4)*x^3*Sech[2*Log[c*x]]^(3/2)} -{Sech[2*Log[c*x]]^(3/2)/x^3, x, 5, (1/2)*(c^4 + 1/x^4)*x^2*Sech[2*Log[c*x]]^(3/2) - ((c^4 + 1/x^4)*Sqrt[(c^4 + 1/x^4)/(c^2 + 1/x^2)^2]*(c^2 + 1/x^2)*x^3*EllipticF[2*ArcCot[c*x], 1/2]*Sech[2*Log[c*x]]^(3/2))/(4*c)} -{Sech[2*Log[c*x]]^(3/2)/x^4, x, 6, (1/2)*(c^4 + 1/x^4)*x*Sech[2*Log[c*x]]^(3/2) - (1/2)*c^6*(1 + 1/(c^4*x^4))^(3/2)*x^3*ArcCsch[c^2*x^2]*Sech[2*Log[c*x]]^(3/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sech[a+b Log[c x^n]]^p*) - - -{Sech[a + b*Log[c*x^n]], x, 4, (2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + 1/n)/(2*b), (1/2)*(3 + 1/(b*n)), (-E^(2*a))*(c*x^n)^(2*b)])/(1 + b*n)} -{Sech[a + b*Log[c*x^n]]^2, x, 4, (4*E^(2*a)*x*(c*x^n)^(2*b)*Hypergeometric2F1[2, (1/2)*(2 + 1/(b*n)), (1/2)*(4 + 1/(b*n)), (-E^(2*a))*(c*x^n)^(2*b)])/(1 + 2*b*n)} -{Sech[a + b*Log[c*x^n]]^3, x, 4, (8*E^(3*a)*x*(c*x^n)^(3*b)*Hypergeometric2F1[3, (3*b + 1/n)/(2*b), (1/2)*(5 + 1/(b*n)), (-E^(2*a))*(c*x^n)^(2*b)])/(1 + 3*b*n)} -{Sech[a + b*Log[c*x^n]]^4, x, 4, (16*E^(4*a)*x*(c*x^n)^(4*b)*Hypergeometric2F1[4, (1/2)*(4 + 1/(b*n)), (1/2)*(6 + 1/(b*n)), (-E^(2*a))*(c*x^n)^(2*b)])/(1 + 4*b*n)} - -{2*b^2*n^2*Sech[a + b*Log[c*x^n]]^3 + (1 - b^2*n^2)*Sech[a + b*Log[c*x^n]], x, -9, x*Sech[a + b*Log[c*x^n]] + b*n*x*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]]} - - -{Sech[a + 2*Log[c*x^(1/2)]]^3, x, 3, (2*c^6)/(E^a*(c^4 + 1/(E^(2*a)*x^2))^2)} -{Sech[a + 2*Log[c/x^(1/2)]]^3, x, 4, (2*c^2)/(E^(3*a)*(E^(-2*a) + c^4/x^2)^2)} -{Sech[a + 1/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, (E^(2*a)*(2 - p)*x*(1 + (c*x^n)^(2/(n*(2 - p)))/E^(2*a))*Sech[a - Log[c*x^n]/(n*(2 - p))]^p)/((c*x^n)^(2/(n*(2 - p)))*(2*(1 - p)))} -{Sech[a - 1/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, ((2 - p)*x*(1 + 1/(E^(2*a)*(c*x^n)^(2/(n*(2 - p)))))*Sech[a + Log[c*x^n]/(n*(2 - p))]^p)/(2*(1 - p))} - - -{Sech[a + b*Log[c*x^n]]/x, x, 2, ArcTan[Sinh[a + b*Log[c*x^n]]]/(b*n)} -{Sech[a + b*Log[c*x^n]]^2/x, x, 3, Tanh[a + b*Log[c*x^n]]/(b*n)} -{Sech[a + b*Log[c*x^n]]^3/x, x, 3, ArcTan[Sinh[a + b*Log[c*x^n]]]/(2*b*n) + (Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(2*b*n)} -{Sech[a + b*Log[c*x^n]]^4/x, x, 3, Tanh[a + b*Log[c*x^n]]/(b*n) - Tanh[a + b*Log[c*x^n]]^3/(3*b*n)} -{Sech[a + b*Log[c*x^n]]^5/x, x, 4, (3*ArcTan[Sinh[a + b*Log[c*x^n]]])/(8*b*n) + (3*Sech[a + b*Log[c*x^n]]*Tanh[a + b*Log[c*x^n]])/(8*b*n) + (Sech[a + b*Log[c*x^n]]^3*Tanh[a + b*Log[c*x^n]])/(4*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sech[a+b Log[c x^n]]^(p/2)*) - - -{Sech[a + b*Log[c*x^n]]^(5/2)/x, x, 4, -((2*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(3*b*n)) + (2*Sech[a + b*Log[c*x^n]]^(3/2)*Sinh[a + b*Log[c*x^n]])/(3*b*n)} -{Sech[a + b*Log[c*x^n]]^(3/2)/x, x, 4, (2*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(b*n) + (2*Sqrt[Sech[a + b*Log[c*x^n]]]*Sinh[a + b*Log[c*x^n]])/(b*n)} -{Sqrt[Sech[a + b*Log[c*x^n]]]/x, x, 3, -((2*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(b*n))} -{1/(x*Sqrt[Sech[a + b*Log[c*x^n]]]), x, 3, -((2*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(b*n))} -{1/(x*Sech[a + b*Log[c*x^n]]^(3/2)), x, 4, -((2*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticF[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(3*b*n)) + (2*Sinh[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Sech[a + b*Log[c*x^n]]])} -{1/(x*Sech[a + b*Log[c*x^n]]^(5/2)), x, 4, -((6*I*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(1/2)*I*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(5*b*n)) + (2*Sinh[a + b*Log[c*x^n]])/(5*b*n*Sech[a + b*Log[c*x^n]]^(3/2))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.7 (d hyper)^m (a+b (c sech)^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.7 (d hyper)^m (a+b (c sech)^n)^p.m deleted file mode 100644 index e11d1ff..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.5 Hyperbolic secant/6.5.7 (d hyper)^m (a+b (c sech)^n)^p.m +++ /dev/null @@ -1,356 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form Sinh[e+f x]^m (a+b Sech[e+f x]^n)^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*Sech[c + d*x]^2)*Sinh[c + d*x]^4, x, 5, (3/8)*(a - 4*b)*x - ((5*a - 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d) + (b*Tanh[c + d*x])/d} -{(a + b*Sech[c + d*x]^2)*Sinh[c + d*x]^3, x, 3, -(((a - b)*Cosh[c + d*x])/d) + (a*Cosh[c + d*x]^3)/(3*d) + (b*Sech[c + d*x])/d} -{(a + b*Sech[c + d*x]^2)*Sinh[c + d*x]^2, x, 4, (-(1/2))*(a - 2*b)*x + (a*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (b*Tanh[c + d*x])/d} -{(a + b*Sech[c + d*x]^2)*Sinh[c + d*x], x, 3, (a*Cosh[c + d*x])/d - (b*Sech[c + d*x])/d} -{Csch[c + d*x]*(a + b*Sech[c + d*x]^2), x, 3, -(((a + b)*ArcTanh[Cosh[c + d*x]])/d) + (b*Sech[c + d*x])/d} -{Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2), x, 3, -(((a + b)*Coth[c + d*x])/d) - (b*Tanh[c + d*x])/d} -{Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2), x, 4, ((a + 3*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((a + b)*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (b*Sech[c + d*x])/d} -{Csch[c + d*x]^4*(a + b*Sech[c + d*x]^2), x, 3, ((a + 2*b)*Coth[c + d*x])/d - ((a + b)*Coth[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x])/d} - - -{(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^4, x, 6, (1/8)*(3*a^2 - 24*a*b + 8*b^2)*x - (a*(a - 8*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - ((a^2 - 8*a*b + 4*b^2)*Tanh[c + d*x])/(4*d) + (a^2*Sinh[c + d*x]^4*Tanh[c + d*x])/(4*d) - (b^2*Tanh[c + d*x]^3)/(3*d)} -{(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^3, x, 3, -((a*(a - 2*b)*Cosh[c + d*x])/d) + (a^2*Cosh[c + d*x]^3)/(3*d) + ((2*a - b)*b*Sech[c + d*x])/d + (b^2*Sech[c + d*x]^3)/(3*d)} -{(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x]^2, x, 5, (-(1/2))*a*(a - 4*b)*x + (a*(a - 4*b)*Tanh[c + d*x])/(2*d) + (a^2*Sinh[c + d*x]^2*Tanh[c + d*x])/(2*d) + (b^2*Tanh[c + d*x]^3)/(3*d)} -{(a + b*Sech[c + d*x]^2)^2*Sinh[c + d*x], x, 3, (a^2*Cosh[c + d*x])/d - (2*a*b*Sech[c + d*x])/d - (b^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^2, x, 4, -(((a + b)^2*ArcTanh[Cosh[c + d*x]])/d) + (b*(2*a + b)*Sech[c + d*x])/d + (b^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2, x, 3, -(((a + b)^2*Coth[c + d*x])/d) - (2*b*(a + b)*Tanh[c + d*x])/d + (b^2*Tanh[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2, x, 5, ((a + b)*(a + 5*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((3*a^2 + 6*a*b + 5*b^2)*Coth[c + d*x]*Csch[c + d*x])/(6*d) - (b*(6*a + 5*b)*Sech[c + d*x])/(3*d) + (b^2*Csch[c + d*x]^2*Sech[c + d*x]^3)/(3*d)} -{Csch[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2, x, 3, ((a + b)*(a + 3*b)*Coth[c + d*x])/d - ((a + b)^2*Coth[c + d*x]^3)/(3*d) + (b*(2*a + 3*b)*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x]^3)/(3*d)} - - -{(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^4, x, 6, (3/8)*a*(a^2 - 12*a*b + 8*b^2)*x - (3*a*(a^2 - 12*a*b + 8*b^2)*Tanh[c + d*x])/(8*d) + (b*(6*a^2 - 23*a*b - 8*b^2)*Tanh[c + d*x]^3)/(8*d) - (3*(5*a - 16*b)*b^2*Tanh[c + d*x]^5)/(40*d) - (3*(a - 2*b)*Sinh[c + d*x]^2*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^2)/(8*d) + (Cosh[c + d*x]*Sinh[c + d*x]^3*(a + b - b*Tanh[c + d*x]^2)^3)/(4*d)} -{(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^3, x, 3, -((a^2*(a - 3*b)*Cosh[c + d*x])/d) + (a^3*Cosh[c + d*x]^3)/(3*d) + (3*a*(a - b)*b*Sech[c + d*x])/d + ((3*a - b)*b^2*Sech[c + d*x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*d)} -{(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x]^2, x, 6, -(a^2*(a - 6*b)*x)/2 + a^3/(4*d*(1 - Tanh[c + d*x])) - (3*a^2*b*Tanh[c + d*x])/d + (b^2*(3*a + b)*Tanh[c + d*x]^3)/(3*d) - (b^3*Tanh[c + d*x]^5)/(5*d) - a^3/(4*d*(1 + Tanh[c + d*x])), (-(1/2))*a^2*(a - 6*b)*x - (b*(81*a^2 - 28*a*b - 4*b^2)*Tanh[c + d*x])/(30*d) - ((33*a - 2*b)*b*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2))/(30*d) - (7*b*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^2)/(10*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^3)/(2*d)} -{(a + b*Sech[c + d*x]^2)^3*Sinh[c + d*x], x, 3, (a^3*Cosh[c + d*x])/d - (3*a^2*b*Sech[c + d*x])/d - (a*b^2*Sech[c + d*x]^3)/d - (b^3*Sech[c + d*x]^5)/(5*d)} -{Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3, x, 4, -(((a + b)^3*ArcTanh[Cosh[c + d*x]])/d) + (b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x])/d + (b^2*(3*a + b)*Sech[c + d*x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3, x, 3, -(((a + b)^3*Coth[c + d*x])/d) - (3*b*(a + b)^2*Tanh[c + d*x])/d + (b^2*(a + b)*Tanh[c + d*x]^3)/d - (b^3*Tanh[c + d*x]^5)/(5*d)} -{Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3, x, 5, ((a + b)^2*(a + 7*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((a + b)^2*(a + 7*b)*Sech[c + d*x])/(2*d) - (b*(6*a^2 + 15*a*b + 7*b^2)*Sech[c + d*x]^3)/(6*d) - (b^2*(5*a + 7*b)*Sech[c + d*x]^5)/(10*d) - ((a + b)*(b + a*Cosh[c + d*x]^2)^2*Csch[c + d*x]^2*Sech[c + d*x]^5)/(2*d)} -{Csch[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3, x, 3, ((a + b)^2*(a + 4*b)*Coth[c + d*x])/d - ((a + b)^3*Coth[c + d*x]^3)/(3*d) + (3*b*(a + b)*(a + 2*b)*Tanh[c + d*x])/d - (b^2*(3*a + 4*b)*Tanh[c + d*x]^3)/(3*d) + (b^3*Tanh[c + d*x]^5)/(5*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 6, ((3*a^2 + 12*a*b + 8*b^2)*x)/(8*a^3) - (Sqrt[b]*(a + b)^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a^3*d) - ((5*a + 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*a*d)} -{Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 4, (Sqrt[b]*(a + b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(a^(5/2)*d) - ((a + b)*Cosh[c + d*x])/(a^2*d) + Cosh[c + d*x]^3/(3*a*d)} -{Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 5, -(((a + 2*b)*x)/(2*a^2)) + (Sqrt[b]*Sqrt[a + b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a^2*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d)} -{Sinh[c + d*x]/(a + b*Sech[c + d*x]^2), x, 3, -((Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(a^(3/2)*d)) + Cosh[c + d*x]/(a*d)} -{Csch[c + d*x]/(a + b*Sech[c + d*x]^2), x, 4, (Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b)*d) - ArcTanh[Cosh[c + d*x]]/((a + b)*d)} -{Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 3, (Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/((a + b)^(3/2)*d) - Coth[c + d*x]/((a + b)*d)} -{Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 5, -((Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/((a + b)^2*d)) + ((a - b)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*(a + b)*d)} -{Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 4, -((a*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/((a + b)^(5/2)*d)) + (a*Coth[c + d*x])/((a + b)^2*d) - Coth[c + d*x]^3/(3*(a + b)*d)} - - -{Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2, x, 7, (3*(a^2 + 8*a*b + 8*b^2)*x)/(8*a^4) - (3*Sqrt[b]*Sqrt[a + b]*(a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^4*d) - ((5*a + 6*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*x]^2)) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)) - (3*b*(3*a + 4*b)*Tanh[c + d*x])/(8*a^3*d*(a + b - b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 5, (Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(7/2)*d) - ((a + 2*b)*Cosh[c + d*x])/(a^3*d) + Cosh[c + d*x]^3/(3*a^2*d) - (b*(a + b)*Cosh[c + d*x])/(2*a^3*d*(b + a*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 6, -(((a + 4*b)*x)/(2*a^3)) + (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^3*Sqrt[a + b]*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)) + (b*Tanh[c + d*x])/(a^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]/(a + b*Sech[c + d*x]^2)^2, x, 4, (-3*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(5/2)*d) + (3*Cosh[c + d*x])/(2*a^2*d) - Cosh[c + d*x]^3/(2*a*d*(b + a*Cosh[c + d*x]^2))} -{Csch[c + d*x]/(a + b*Sech[c + d*x]^2)^2, x, 5, (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(3/2)*(a + b)^2*d) - ArcTanh[Cosh[c + d*x]]/((a + b)^2*d) - (b*Cosh[c + d*x])/(2*a*(a + b)*d*(b + a*Cosh[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 4, (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) - (3*Coth[c + d*x])/(2*(a + b)^2*d) + Coth[c + d*x]/(2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 6, -(((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*Sqrt[a]*(a + b)^3*d)) + ((a - 3*b)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^3*d) - ((a - b)*Cosh[c + d*x])/(2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)) - (Coth[c + d*x]*Csch[c + d*x])/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2))} -{Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2, x, 5, -(((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*d)) + ((a - b)*Coth[c + d*x])/((a + b)^3*d) - Coth[c + d*x]^3/(3*(a + b)^2*d) - (a*b*Tanh[c + d*x])/(2*(a + b)^3*d*(a + b - b*Tanh[c + d*x]^2))} - - -{Sinh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3, x, 8, (3*(a^2 + 12*a*b + 16*b^2)*x)/(8*a^5) - (3*Sqrt[b]*(5*a^2 + 20*a*b + 16*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^5*Sqrt[a + b]*d) - ((5*a + 8*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d*(a + b - b*Tanh[c + d*x]^2)^2) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*a*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(7*a + 12*b)*Tanh[c + d*x])/(8*a^3*d*(a + b - b*Tanh[c + d*x]^2)^2) - (3*b*(a + 2*b)*Tanh[c + d*x])/(2*a^4*d*(a + b - b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3, x, 6, (5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(9/2)*d) - ((a + 3*b)*Cosh[c + d*x])/(a^4*d) + Cosh[c + d*x]^3/(3*a^3*d) + (b^2*(a + b)*Cosh[c + d*x])/(4*a^4*d*(b + a*Cosh[c + d*x]^2)^2) - (b*(9*a + 13*b)*Cosh[c + d*x])/(8*a^4*d*(b + a*Cosh[c + d*x]^2))} -{Sinh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 7, -(((a + 6*b)*x)/(2*a^4)) + (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)^2) + (3*b*Tanh[c + d*x])/(4*a^2*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(11*a + 12*b)*Tanh[c + d*x])/(8*a^3*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Sinh[c + d*x]/(a + b*Sech[c + d*x]^2)^3, x, 5, (-15*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(7/2)*d) + (15*Cosh[c + d*x])/(8*a^3*d) - Cosh[c + d*x]^5/(4*a*d*(b + a*Cosh[c + d*x]^2)^2) - (5*Cosh[c + d*x]^3)/(8*a^2*d*(b + a*Cosh[c + d*x]^2))} -{Csch[c + d*x]/(a + b*Sech[c + d*x]^2)^3, x, 6, (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(5/2)*(a + b)^3*d) - ArcTanh[Cosh[c + d*x]]/((a + b)^3*d) - (b*Cosh[c + d*x]^3)/(4*a*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2) - (b*(7*a + 3*b)*Cosh[c + d*x])/(8*a^2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2))} -{Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 5, (15*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(7/2)*d) - (15*Coth[c + d*x])/(8*(a + b)^3*d) + Coth[c + d*x]/(4*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (5*Coth[c + d*x])/(8*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3, x, 7, -((Sqrt[b]*(15*a^2 - 10*a*b - b^2)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(3/2)*(a + b)^4*d)) + ((a - 5*b)*ArcTanh[Cosh[c + d*x]])/(2*(a + b)^4*d) + ((2*a - b)*b*Cosh[c + d*x])/(4*a*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)^2) - ((4*a^2 - 9*a*b - b^2)*Cosh[c + d*x])/(8*a*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - (Cosh[c + d*x]*Coth[c + d*x]^2)/(2*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2)} -{Csch[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3, x, 6, -((5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d)) + ((a - 2*b)*Coth[c + d*x])/((a + b)^4*d) - Coth[c + d*x]^3/(3*(a + b)^3*d) - (a*b*Tanh[c + d*x])/(4*(a + b)^3*d*(a + b - b*Tanh[c + d*x]^2)^2) - ((7*a - 4*b)*b*Tanh[c + d*x])/(8*(a + b)^4*d*(a + b - b*Tanh[c + d*x]^2))} - - -(* ::Title::Closed:: *) -(*Integrands of the form Cosh[e+f x]^m (a+b Sech[e+f x]^n)^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2), x, 3, ((3*a + 4*b)*x)/8 + ((3*a + 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d)} -{Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2), x, 3, ((a + b)*Sinh[c + d*x])/d + (a*Sinh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2), x, 2, ((a + 2*b)*x)/2 + (a*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)} -{Cosh[c + d*x]^1*(a + b*Sech[c + d*x]^2), x, 2, (b*ArcTan[Sinh[c + d*x]])/d + (a*Sinh[c + d*x])/d} -{Sech[c + d*x]^1*(a + b*Sech[c + d*x]^2), x, 2, ((2*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^2*(a + b*Sech[c + d*x]^2), x, 3, ((a + b)*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d), ((3*a + 2*b)*Tanh[c + d*x])/(3*d) + (b*Sech[c + d*x]^2*Tanh[c + d*x])/(3*d)} -{Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2), x, 3, ((4*a + 3*b)*ArcTan[Sinh[c + d*x]])/(8*d) + ((4*a + 3*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2), x, 3, ((a + b)*Tanh[c + d*x])/d - ((a + 2*b)*Tanh[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x]^5)/(5*d), ((5*a + 4*b)*Tanh[c + d*x])/(5*d) + (b*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d) - ((5*a + 4*b)*Tanh[c + d*x]^3)/(15*d)} - - -{Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2, x, 4, (1/8)*(3*a^2 + 8*a*b + 8*b^2)*x + (3*a*(a + 2*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]^3*Sinh[c + d*x]*(a + b - b*Tanh[c + d*x]^2))/(4*d)} -{Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2, x, 4, (b^2*ArcTan[Sinh[c + d*x]])/d + (a*(a + 2*b)*Sinh[c + d*x])/d + (a^2*Sinh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2, x, 5, (1/2)*a*(a + 4*b)*x + (a^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (b^2*Tanh[c + d*x])/d} -{Cosh[c + d*x]^1*(a + b*Sech[c + d*x]^2)^2, x, 5, (b*(4*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2*Sinh[c + d*x])/d + (b^2*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Sech[c + d*x]^1*(a + b*Sech[c + d*x]^2)^2, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (3*b*(2*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b*Sech[c + d*x]^3*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2, x, 3, ((a + b)^2*Tanh[c + d*x])/d - (2*b*(a + b)*Tanh[c + d*x]^3)/(3*d) + (b^2*Tanh[c + d*x]^5)/(5*d)} -{Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2, x, 5, ((8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x])/(16*d) + (b*(8*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + (b*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(6*d)} -{Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2, x, 3, ((a + b)^2*Tanh[c + d*x])/d - ((a + b)*(a + 3*b)*Tanh[c + d*x]^3)/(3*d) + (b*(2*a + 3*b)*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)} - - -{Cosh[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3, x, 6, (3/8)*a*(a^2 + 4*a*b + 8*b^2)*x + (3*a^2*(a + 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a^3*Cosh[c + d*x]^3*Sinh[c + d*x])/(4*d) + (b^3*Tanh[c + d*x])/d} -{Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3, x, 5, (b^2*(6*a + b)*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2*(a + 3*b)*Sinh[c + d*x])/d + (a^3*Sinh[c + d*x]^3)/(3*d) + (b^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} -{Cosh[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3, x, 5, (1/2)*a^2*(a + 6*b)*x + (a^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (b^2*(3*a + b)*Tanh[c + d*x])/d - (b^3*Tanh[c + d*x]^3)/(3*d)} -{Cosh[c + d*x]^1*(a + b*Sech[c + d*x]^2)^3, x, 6, (3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[Sinh[c + d*x]])/(8*d) + (a^3*Sinh[c + d*x])/d + (3*b^2*(4*a + b)*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d)} -{Sech[c + d*x]^1*(a + b*Sech[c + d*x]^2)^3, x, 5, ((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + (b*(44*a^2 + 44*a*b + 15*b^2)*Sech[c + d*x]*Tanh[c + d*x])/(48*d) + (5*b*(2*a + b)*Sech[c + d*x]^3*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + (b*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)} -{Sech[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3, x, 3, ((a + b)^3*Tanh[c + d*x])/d - (b*(a + b)^2*Tanh[c + d*x]^3)/d + (3*b^2*(a + b)*Tanh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)} -{Sech[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3, x, 6, ((64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*ArcTan[Sinh[c + d*x]])/(128*d) + ((64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*Sech[c + d*x]*Tanh[c + d*x])/(128*d) + (b*(72*a^2 + 92*a*b + 35*b^2)*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d) + (b*(12*a + 7*b)*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(48*d) + (b*Sech[c + d*x]^7*(a + b + a*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(8*d)} -{Sech[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3, x, 3, ((a + b)^3*Tanh[c + d*x])/d - ((a + b)^2*(a + 4*b)*Tanh[c + d*x]^3)/(3*d) + (3*b*(a + b)*(a + 2*b)*Tanh[c + d*x]^5)/(5*d) - (b^2*(3*a + 4*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Cosh[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 6, ((3*a^2 - 4*a*b + 8*b^2)*x)/(8*a^3) - (b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a^3*Sqrt[a + b]*d) + ((3*a - 4*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*a*d)} -{Cosh[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 4, (b^2*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]*d) + ((a - b)*Sinh[c + d*x])/(a^2*d) + Sinh[c + d*x]^3/(3*a*d)} -{Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 5, ((a - 2*b)*x)/(2*a^2) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a^2*Sqrt[a + b]*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d)} -{Cosh[c + d*x]/(a + b*Sech[c + d*x]^2), x, 3, -((b*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a^(3/2)*Sqrt[a + b]*d)) + Sinh[c + d*x]/(a*d)} -{Sech[c + d*x]/(a + b*Sech[c + d*x]^2), x, 2, ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)} -{Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 2, ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b]*d)} -{Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 4, ArcTan[Sinh[c + d*x]]/(b*d) - (Sqrt[a]*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(b*Sqrt[a + b]*d)} -{Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 3, -((a*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b]*d)) + Tanh[c + d*x]/(b*d)} -{Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2), x, 5, -(((2*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^2*d)) + (a^(3/2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(b^2*Sqrt[a + b]*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d)} -{Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2), x, 4, (a^2*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]*d) - ((a - b)*Tanh[c + d*x])/(b^2*d) - Tanh[c + d*x]^3/(3*b*d)} - - -{Cosh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 5, (b^2*(6*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(7/2)*(a + b)^(3/2)*d) + ((a - 2*b)*Sinh[c + d*x])/(a^3*d) + Sinh[c + d*x]^3/(3*a^2*d) - (b^3*Sinh[c + d*x])/(2*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))} -{Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 6, ((a - 4*b)*x)/(2*a^3) + (b^(3/2)*(5*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^3*(a + b)^(3/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)) + (b*(a + 2*b)*Tanh[c + d*x])/(2*a^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^2, x, 5, -((b*(4*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(5/2)*(a + b)^(3/2)*d)) + Sinh[c + d*x]/(a^2*d) + (b^2*Sinh[c + d*x])/(2*a^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]/(a + b*Sech[c + d*x]^2)^2, x, 3, ((2*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(3/2)*(a + b)^(3/2)*d) - (b*Sinh[c + d*x])/(2*a*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 3, ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]]/(2*Sqrt[b]*(a + b)^(3/2)*d) + Tanh[c + d*x]/(2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 3, ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(2*Sqrt[a]*(a + b)^(3/2)*d) + Sinh[c + d*x]/(2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2, x, 3, ((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)*d) - (a*Tanh[c + d*x])/(2*b*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2, x, 5, ArcTan[Sinh[c + d*x]]/(b^2*d) - (Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*b^2*(a + b)^(3/2)*d) - (a*Sinh[c + d*x])/(2*b*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2)^2, x, 5, -((a*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*b^(5/2)*(a + b)^(3/2)*d)) + Tanh[c + d*x]/(b^2*d) + (a^2*Tanh[c + d*x])/(2*b^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^2, x, 6, -(((4*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^3*d)) + (a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*b^3*(a + b)^(3/2)*d) + (a*(2*a + b)*Sinh[c + d*x])/(2*b^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)) + (Sech[c + d*x]*Tanh[c + d*x])/(2*b*d*(a + b + a*Sinh[c + d*x]^2))} - - -{Cosh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 7, ((a - 6*b)*x)/(2*a^4) + (b^(3/2)*(35*a^2 + 56*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(5/2)*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(2*a + 3*b)*Tanh[c + d*x])/(4*a^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (b*(4*a + 3*b)*(a + 4*b)*Tanh[c + d*x])/(8*a^3*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Cosh[c + d*x]^1/(a + b*Sech[c + d*x]^2)^3, x, 6, -((3*b*(4*(a + b)^2 + (2*a + b)^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(7/2)*(a + b)^(5/2)*d)) + Sinh[c + d*x]/(a^3*d) - (b^3*Sinh[c + d*x])/(4*a^3*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + 3*b)*Sinh[c + d*x])/(8*a^3*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^1/(a + b*Sech[c + d*x]^2)^3, x, 4, ((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(5/2)*(a + b)^(5/2)*d) - (b*Cosh[c + d*x]^2*Sinh[c + d*x])/(4*a*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) - (3*b*(2*a + b)*Sinh[c + d*x])/(8*a^2*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 4, (3*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*Sqrt[b]*(a + b)^(5/2)*d) + Tanh[c + d*x]/(4*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + (3*Tanh[c + d*x])/(8*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3, x, 4, ((4*a + b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*a^(3/2)*(a + b)^(5/2)*d) - (b*Sinh[c + d*x])/(4*a*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + ((4*a + b)*Sinh[c + d*x])/(8*a*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3, x, 4, ((a + 4*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*b^(3/2)*(a + b)^(5/2)*d) - (a*Tanh[c + d*x])/(4*b*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) + ((a + 4*b)*Tanh[c + d*x])/(8*b*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^5/(a + b*Sech[c + d*x]^2)^3, x, 4, (3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*Sqrt[a]*(a + b)^(5/2)*d) + Sinh[c + d*x]/(4*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) + (3*Sinh[c + d*x])/(8*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))} -{Sech[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3, x, 4, ((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*b^(5/2)*(a + b)^(5/2)*d) - (a*Sech[c + d*x]^2*Tanh[c + d*x])/(4*b*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (3*a*(a + 2*b)*Tanh[c + d*x])/(8*b^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3, x, 6, ArcTan[Sinh[c + d*x]]/(b^3*d) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(8*b^3*(a + b)^(5/2)*d) - (a*Sinh[c + d*x])/(4*b*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) - (a*(4*a + 7*b)*Sinh[c + d*x])/(8*b^2*(a + b)^2*d*(a + b + a*Sinh[c + d*x]^2))} - - -(* ::Title::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^n)^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^4, x, 4, a*x - (a*Tanh[c + d*x])/d - (a*Tanh[c + d*x]^3)/(3*d) + (b*Tanh[c + d*x]^5)/(5*d)} -{(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^3, x, 4, (a*Log[Cosh[c + d*x]])/d + ((a - b)*Sech[c + d*x]^2)/(2*d) + (b*Sech[c + d*x]^4)/(4*d)} -{(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^2, x, 4, a*x - (a*Tanh[c + d*x])/d + (b*Tanh[c + d*x]^3)/(3*d)} -{(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^1, x, 3, (a*Log[Cosh[c + d*x]])/d - (b*Sech[c + d*x]^2)/(2*d)} -{(a + b*Sech[c + d*x]^2)*Tanh[c + d*x]^0, x, 3, a*x + (b*Tanh[c + d*x])/d} -{Coth[c + d*x]^1*(a + b*Sech[c + d*x]^2), x, 4, -((b*Log[Cosh[c + d*x]])/d) + ((a + b)*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2), x, 4, a*x - ((a + b)*Coth[c + d*x])/d} -{Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2), x, 4, -(((a + b)*Csch[c + d*x]^2)/(2*d)) + (a*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2), x, 4, a*x - (a*Coth[c + d*x])/d - ((a + b)*Coth[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2), x, 4, -(((2*a + b)*Csch[c + d*x]^2)/(2*d)) - ((a + b)*Csch[c + d*x]^4)/(4*d) + (a*Log[Sinh[c + d*x]])/d} - - -{(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^4, x, 4, a^2*x - (a^2*Tanh[c + d*x])/d - (a^2*Tanh[c + d*x]^3)/(3*d) + (b*(2*a + b)*Tanh[c + d*x]^5)/(5*d) - (b^2*Tanh[c + d*x]^7)/(7*d)} -{(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^3, x, 4, (a^2*Log[Cosh[c + d*x]])/d + (a*(a - 2*b)*Sech[c + d*x]^2)/(2*d) + ((2*a - b)*b*Sech[c + d*x]^4)/(4*d) + (b^2*Sech[c + d*x]^6)/(6*d)} -{(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^2, x, 4, a^2*x - (a^2*Tanh[c + d*x])/d + (b*(2*a + b)*Tanh[c + d*x]^3)/(3*d) - (b^2*Tanh[c + d*x]^5)/(5*d)} -{(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^1, x, 4, (a^2*Log[Cosh[c + d*x]])/d - (a*b*Sech[c + d*x]^2)/d - (b^2*Sech[c + d*x]^4)/(4*d)} -{(a + b*Sech[c + d*x]^2)^2*Tanh[c + d*x]^0, x, 4, a^2*x + (b*(2*a + b)*Tanh[c + d*x])/d - (b^2*Tanh[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^1*(a + b*Sech[c + d*x]^2)^2, x, 4, -((b*(2*a + b)*Log[Cosh[c + d*x]])/d) + ((a + b)^2*Log[Sinh[c + d*x]])/d + (b^2*Sech[c + d*x]^2)/(2*d)} -{Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^2, x, 4, a^2*x - ((a + b)^2*Coth[c + d*x])/d - (b^2*Tanh[c + d*x])/d} -{Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2, x, 4, -(((a + b)^2*Csch[c + d*x]^2)/(2*d)) + (b^2*Log[Cosh[c + d*x]])/d + ((a^2 - b^2)*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^2, x, 4, a^2*x - ((a^2 - b^2)*Coth[c + d*x])/d - ((a + b)^2*Coth[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^2, x, 4, -((a*(a + b)*Csch[c + d*x]^2)/d) - ((a + b)^2*Csch[c + d*x]^4)/(4*d) + (a^2*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^2, x, 4, a^2*x - (a^2*Coth[c + d*x])/d - ((a^2 - b^2)*Coth[c + d*x]^3)/(3*d) - ((a + b)^2*Coth[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2, x, 5, -((a*(a + b)*Csch[c + d*x]^2)/d) - ((a + b)^2*Csch[c + d*x]^4)/(4*d) - ((b + a*Cosh[c + d*x]^2)^3*Csch[c + d*x]^6)/(6*(a + b)*d) + (a^2*Log[Sinh[c + d*x]])/d} - - -{(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4, x, 4, a^3*x - (a^3*Tanh[c + d*x])/d - (a^3*Tanh[c + d*x]^3)/(3*d) + (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^2*(3*a + 2*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)} -{(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^3, x, 5, (a^3*Log[Cosh[c + d*x]])/d - (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (3*a*b^2*Sech[c + d*x]^4)/(4*d) - (b^3*Sech[c + d*x]^6)/(6*d) + ((b + a*Cosh[c + d*x]^2)^4*Sech[c + d*x]^8)/(8*b*d)} -{(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^2, x, 4, a^3*x - (a^3*Tanh[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + 2*b)*Tanh[c + d*x]^5)/(5*d) + (b^3*Tanh[c + d*x]^7)/(7*d)} -{(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^1, x, 4, (a^3*Log[Cosh[c + d*x]])/d - (3*a^2*b*Sech[c + d*x]^2)/(2*d) - (3*a*b^2*Sech[c + d*x]^4)/(4*d) - (b^3*Sech[c + d*x]^6)/(6*d)} -{(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^0, x, 4, a^3*x + (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(3*a + 2*b)*Tanh[c + d*x]^3)/(3*d) + (b^3*Tanh[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^1*(a + b*Sech[c + d*x]^2)^3, x, 4, -((b*(3*a^2 + 3*a*b + b^2)*Log[Cosh[c + d*x]])/d) + ((a + b)^3*Log[Sinh[c + d*x]])/d + (b^2*(3*a + b)*Sech[c + d*x]^2)/(2*d) + (b^3*Sech[c + d*x]^4)/(4*d)} -{Coth[c + d*x]^2*(a + b*Sech[c + d*x]^2)^3, x, 4, a^3*x - ((a + b)^3*Coth[c + d*x])/d - (b^2*(3*a + 2*b)*Tanh[c + d*x])/d + (b^3*Tanh[c + d*x]^3)/(3*d)} -{Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3, x, 4, -(((a + b)^3*Csch[c + d*x]^2)/(2*d)) + (b^2*(3*a + 2*b)*Log[Cosh[c + d*x]])/d + ((a - 2*b)*(a + b)^2*Log[Sinh[c + d*x]])/d - (b^3*Sech[c + d*x]^2)/(2*d)} -{Coth[c + d*x]^4*(a + b*Sech[c + d*x]^2)^3, x, 4, a^3*x - ((a - 2*b)*(a + b)^2*Coth[c + d*x])/d - ((a + b)^3*Coth[c + d*x]^3)/(3*d) + (b^3*Tanh[c + d*x])/d} -{Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3, x, 4, -(((2*a - b)*(a + b)^2*Csch[c + d*x]^2)/(2*d)) - ((a + b)^3*Csch[c + d*x]^4)/(4*d) - (b^3*Log[Cosh[c + d*x]])/d + ((a^3 + b^3)*Log[Sinh[c + d*x]])/d} -{Coth[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3, x, 4, a^3*x - ((a^3 + b^3)*Coth[c + d*x])/d - ((a - 2*b)*(a + b)^2*Coth[c + d*x]^3)/(3*d) - ((a + b)^3*Coth[c + d*x]^5)/(5*d)} -{Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^3, x, 4, -((3*a^2*(a + b)*Csch[c + d*x]^2)/(2*d)) - (3*a*(a + b)^2*Csch[c + d*x]^4)/(4*d) - ((a + b)^3*Csch[c + d*x]^6)/(6*d) + (a^3*Log[Sinh[c + d*x]])/d} - - -{(a + b*Sech[c + d*x]^2)^4, x, 4, a^4*x + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tanh[c + d*x])/d - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Tanh[c + d*x]^3)/(3*d) + (b^3*(4*a + 3*b)*Tanh[c + d*x]^5)/(5*d) - (b^4*Tanh[c + d*x]^7)/(7*d)} - - -{(a + b*Sech[c + d*x]^2)^5, x, 4, a^5*x + (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 20*a^2*b + 15*a*b^2 + 4*b^3)*Tanh[c + d*x]^3)/(3*d) + (b^3*(10*a^2 + 15*a*b + 6*b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + 4*b)*Tanh[c + d*x]^7)/(7*d) + (b^5*Tanh[c + d*x]^9)/(9*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2), x, 4, -(((a + 2*b)*Log[Cosh[c + d*x]])/(b^2*d)) + ((a + b)^2*Log[b + a*Cosh[c + d*x]^2])/(2*a*b^2*d) - Sech[c + d*x]^2/(2*b*d)} -{Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 6, x/a - ((a + b)^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*b^(3/2)*d) + Tanh[c + d*x]/(b*d)} -{Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 4, -(Log[Cosh[c + d*x]]/(b*d)) + ((a + b)*Log[b + a*Cosh[c + d*x]^2])/(2*a*b*d)} -{Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 5, x/a - (Sqrt[a + b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]*d)} -{Tanh[c + d*x]^1/(a + b*Sech[c + d*x]^2), x, 2, Log[b + a*Cosh[c + d*x]^2]/(2*a*d)} -{Tanh[c + d*x]^0/(a + b*Sech[c + d*x]^2), x, 3, x/a - (Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)} -{Coth[c + d*x]^1/(a + b*Sech[c + d*x]^2), x, 4, (b*Log[b + a*Cosh[c + d*x]^2])/(2*a*(a + b)*d) + Log[Sinh[c + d*x]]/((a + b)*d)} -{Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2), x, 6, x/a - (b^(3/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*(a + b)^(3/2)*d) - Coth[c + d*x]/((a + b)*d)} -{Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2), x, 4, -(Csch[c + d*x]^2/(2*(a + b)*d)) + (b^2*Log[b + a*Cosh[c + d*x]^2])/(2*a*(a + b)^2*d) + ((a + 2*b)*Log[Sinh[c + d*x]])/((a + b)^2*d)} -{Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2), x, 7, x/a - (b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*(a + b)^(5/2)*d) - ((a + 2*b)*Coth[c + d*x])/((a + b)^2*d) - Coth[c + d*x]^3/(3*(a + b)*d)} - - -{Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2, x, 4, (a + b)^2/(2*a^2*b*d*(b + a*Cosh[c + d*x]^2)) + Log[Cosh[c + d*x]]/(b^2*d) + ((a^(-2) - b^(-2))*Log[b + a*Cosh[c + d*x]^2])/(2*d)} -{Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2, x, 6, x/a^2 + ((a - 2*b)*Sqrt[a + b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*b^(3/2)*d) - ((a + b)*Tanh[c + d*x])/(2*a*b*d*(a + b - b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 4, (a + b)/(2*a^2*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2*a^2*d)} -{Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 6, x/a^2 - ((a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*Sqrt[b]*Sqrt[a + b]*d) - Tanh[c + d*x]/(2*a*d*(a + b - b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^1/(a + b*Sech[c + d*x]^2)^2, x, 4, b/(2*a^2*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2*a^2*d)} -{Tanh[c + d*x]^0/(a + b*Sech[c + d*x]^2)^2, x, 5, x/a^2 - (Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(3/2)*d) - (b*Tanh[c + d*x])/(2*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^1/(a + b*Sech[c + d*x]^2)^2, x, 4, b^2/(2*a^2*(a + b)*d*(b + a*Cosh[c + d*x]^2)) + (b*(2*a + b)*Log[b + a*Cosh[c + d*x]^2])/(2*a^2*(a + b)^2*d) + Log[Sinh[c + d*x]]/((a + b)^2*d)} -{Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^2, x, 7, x/a^2 - (b^(3/2)*(5*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(5/2)*d) - ((2*a - b)*Coth[c + d*x])/(2*a*(a + b)^2*d) - (b*Coth[c + d*x])/(2*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^2, x, 4, b^3/(2*a^2*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)) - Csch[c + d*x]^2/(2*(a + b)^2*d) + (b^2*(3*a + b)*Log[b + a*Cosh[c + d*x]^2])/(2*a^2*(a + b)^3*d) + ((a + 3*b)*Log[Sinh[c + d*x]])/((a + b)^3*d)} -{Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^2, x, 8, x/a^2 - (b^(5/2)*(7*a + 2*b)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(2*a^2*(a + b)^(7/2)*d) - ((2*a^2 + 6*a*b - b^2)*Coth[c + d*x])/(2*a*(a + b)^3*d) - ((2*a - 3*b)*Coth[c + d*x]^3)/(6*a*(a + b)^2*d) - (b*Coth[c + d*x]^3)/(2*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} - - -{Tanh[c + d*x]^6/(a + b*Sech[c + d*x]^2)^3, x, 7, x/a^3 - (Sqrt[a + b]*(3*a^2 - 4*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*b^(5/2)*d) - ((a + b)*Tanh[c + d*x]^3)/(4*a*b*d*(a + b - b*Tanh[c + d*x]^2)^2) + ((3*a - 4*b)*(a + b)*Tanh[c + d*x])/(8*a^2*b^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^3, x, 4, -(a + b)^2/(4*a^3*d*(b + a*Cosh[c + d*x]^2)^2) + (a + b)/(a^3*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2*a^3*d)} -{Tanh[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3, x, 7, x/a^3 + ((a^2 - 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*b^(3/2)*Sqrt[a + b]*d) - ((a + b)*Tanh[c + d*x])/(4*a*b*d*(a + b - b*Tanh[c + d*x]^2)^2) + ((a - 4*b)*Tanh[c + d*x])/(8*a^2*b*d*(a + b - b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3, x, 4, -(b*(a + b))/(4*a^3*d*(b + a*Cosh[c + d*x]^2)^2) + (a + 2*b)/(2*a^3*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2*a^3*d)} -{Tanh[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 7, x/a^3 - ((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*Sqrt[b]*(a + b)^(3/2)*d) - Tanh[c + d*x]/(4*a*d*(a + b - b*Tanh[c + d*x]^2)^2) - ((3*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a + b)*d*(a + b - b*Tanh[c + d*x]^2))} -{Tanh[c + d*x]^1/(a + b*Sech[c + d*x]^2)^3, x, 4, -b^2/(4*a^3*d*(b + a*Cosh[c + d*x]^2)^2) + b/(a^3*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2*a^3*d)} -{Tanh[c + d*x]^0/(a + b*Sech[c + d*x]^2)^3, x, 6, x/a^3 - (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(5/2)*d) - (b*Tanh[c + d*x])/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(7*a + 4*b)*Tanh[c + d*x])/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^1/(a + b*Sech[c + d*x]^2)^3, x, 4, -(b^3/(4*a^3*(a + b)*d*(b + a*Cosh[c + d*x]^2)^2)) + (b^2*(3*a + 2*b))/(2*a^3*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)) + (b*(3*a^2 + 3*a*b + b^2)*Log[b + a*Cosh[c + d*x]^2])/(2*a^3*(a + b)^3*d) + Log[Sinh[c + d*x]]/((a + b)^3*d)} -{Coth[c + d*x]^2/(a + b*Sech[c + d*x]^2)^3, x, 8, x/a^3 - (b^(3/2)*(35*a^2 + 28*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(7/2)*d) - ((8*a^2 - 11*a*b - 4*b^2)*Coth[c + d*x])/(8*a^2*(a + b)^3*d) - (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(9*a + 4*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} -{Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3, x, 4, -(b^4/(4*a^3*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)^2)) + (b^3*(2*a + b))/(a^3*(a + b)^3*d*(b + a*Cosh[c + d*x]^2)) - Csch[c + d*x]^2/(2*(a + b)^3*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Log[b + a*Cosh[c + d*x]^2])/(2*a^3*(a + b)^4*d) + ((a + 4*b)*Log[Sinh[c + d*x]])/((a + b)^4*d)} -{Coth[c + d*x]^4/(a + b*Sech[c + d*x]^2)^3, x, 9, x/a^3 - (b^(5/2)*(63*a^2 + 36*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(8*a^3*(a + b)^(9/2)*d) - ((8*a^3 + 32*a^2*b - 15*a*b^2 - 4*b^3)*Coth[c + d*x])/(8*a^2*(a + b)^4*d) - ((8*a^2 - 39*a*b - 12*b^2)*Coth[c + d*x]^3)/(24*a^2*(a + b)^3*d) - (b*Coth[c + d*x]^3)/(4*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(11*a + 4*b)*Coth[c + d*x]^3)/(8*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2))} - - -{1/(a + b*Sech[c + d*x]^2)^4, x, 7, x/a^4 - (Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(16*a^4*(a + b)^(7/2)*d) - (b*Tanh[c + d*x])/(6*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^3) - (b*(11*a + 6*b)*Tanh[c + d*x])/(24*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2)^2) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tanh[c + d*x])/(16*a^3*(a + b)^3*d*(a + b - b*Tanh[c + d*x]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^(p/2) when a+b=0*) - - -{(1 - Sech[x]^2)^(3/2), x, 4, Coth[x]*Log[Cosh[x]]*Sqrt[Tanh[x]^2] - (1/2)*Coth[x]*(Tanh[x]^2)^(3/2)} -{Sqrt[1 - Sech[x]^2], x, 3, Coth[x]*Log[Cosh[x]]*Sqrt[Tanh[x]^2]} -{1/Sqrt[1 - Sech[x]^2], x, 3, (Log[Sinh[x]]*Tanh[x])/Sqrt[Tanh[x]^2]} - - -{(-1 + Sech[x]^2)^(3/2), x, 4, (-Coth[x])*Log[Cosh[x]]*Sqrt[-Tanh[x]^2] + (1/2)*Tanh[x]*Sqrt[-Tanh[x]^2]} -{Sqrt[-1 + Sech[x]^2], x, 3, Coth[x]*Log[Cosh[x]]*Sqrt[-Tanh[x]^2]} -{1/Sqrt[-1 + Sech[x]^2], x, 3, (Log[Sinh[x]]*Tanh[x])/Sqrt[-Tanh[x]^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^5, x, 7, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - Sqrt[a + b*Sech[x]^2] + ((a + 2*b)*(a + b*Sech[x]^2)^(3/2))/(3*b^2) - (a + b*Sech[x]^2)^(5/2)/(5*b^2)} -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^4, x, 9, -((a^2 + 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/(8*b^(3/2)) + Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] + ((a - 3*b)*Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/(8*b) - (Tanh[x]^3*Sqrt[a + b - b*Tanh[x]^2])/4} -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^3, x, 6, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - Sqrt[a + b*Sech[x]^2] + (a + b*Sech[x]^2)^(3/2)/(3*b)} -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^2, x, 8, -((a - b)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/(2*Sqrt[b]) + Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - (Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/2} -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^1, x, 5, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - Sqrt[a + b*Sech[x]^2]} -{Sqrt[a + b*Sech[x]^2]*Tanh[x]^0, x, 6, Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] + Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]} -{Coth[x]^1*Sqrt[a + b*Sech[x]^2], x, 7, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]]} -{Coth[x]^2*Sqrt[a + b*Sech[x]^2], x, 6, Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - Coth[x]*Sqrt[a + b - b*Tanh[x]^2]} -{Coth[x]^3*Sqrt[a + b*Sech[x]^2], x, 8, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - ((2*a + b)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]])/(2*Sqrt[a + b]) - (1/2)*Coth[x]^2*Sqrt[a + b*Sech[x]^2]} -{Coth[x]^4*Sqrt[a + b*Sech[x]^2], x, 7, Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - ((3*a + 2*b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(3*(a + b)) - (Coth[x]^3*Sqrt[a + b - b*Tanh[x]^2])/3} -{Coth[x]^5*Sqrt[a + b*Sech[x]^2], x, 9, Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - ((8*a^2 + 12*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]])/(8*(a + b)^(3/2)) - ((4*a + 3*b)*Coth[x]^2*Sqrt[a + b*Sech[x]^2])/(8*(a + b)) - (1/4)*Coth[x]^4*Sqrt[a + b*Sech[x]^2]} - - -{(a + b*Sech[x]^2)^(3/2)*Tanh[x]^3, x, 7, a^(3/2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - a*Sqrt[a + b*Sech[x]^2] - (a + b*Sech[x]^2)^(3/2)/3 + (a + b*Sech[x]^2)^(5/2)/(5*b)} -{(a + b*Sech[x]^2)^(3/2)*Tanh[x]^2, x, 9, -((3*a^2 - 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/(8*Sqrt[b]) + a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - ((5*a + b)*Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/8 + (b*Tanh[x]^3*Sqrt[a + b - b*Tanh[x]^2])/4} -{(a + b*Sech[x]^2)^(3/2)*Tanh[x]^1, x, 6, a^(3/2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - a*Sqrt[a + b*Sech[x]^2] - (a + b*Sech[x]^2)^(3/2)/3} -{(a + b*Sech[x]^2)^(3/2)*Tanh[x]^0, x, 7, (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/2 + a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] + (b*Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/2} -{Coth[x]^1*(a + b*Sech[x]^2)^(3/2), x, 8, a^(3/2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]] + b*Sqrt[a + b*Sech[x]^2]} -{Coth[x]^2*(a + b*Sech[x]^2)^(3/2), x, 8, -(b^(3/2)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]) + a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]] - (a + b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2]} - - -{(a + b*Sech[c + d*x]^2)^(5/2), x, 8, (Sqrt[b]*(15*a^2 + 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b - b*Tanh[c + d*x]^2]])/(8*d) + (a^(5/2)*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + b - b*Tanh[c + d*x]^2]])/d + (b*(7*a + 3*b)*Tanh[c + d*x]*Sqrt[a + b - b*Tanh[c + d*x]^2])/(8*d) + (b*Tanh[c + d*x]*(a + b - b*Tanh[c + d*x]^2)^(3/2))/(4*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Tanh[x]^5/Sqrt[a + b*Sech[x]^2], x, 6, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] + ((a + 2*b)*Sqrt[a + b*Sech[x]^2])/b^2 - (a + b*Sech[x]^2)^(3/2)/(3*b^2)} -{Tanh[x]^4/Sqrt[a + b*Sech[x]^2], x, 8, -((a + 3*b)*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]])/(2*b^(3/2)) + ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[a] + (Tanh[x]*Sqrt[a + b - b*Tanh[x]^2])/(2*b)} -{Tanh[x]^3/Sqrt[a + b*Sech[x]^2], x, 5, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] + Sqrt[a + b*Sech[x]^2]/b} -{Tanh[x]^2/Sqrt[a + b*Sech[x]^2], x, 7, -(ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[b]) + ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[a]} -{Tanh[x]^1/Sqrt[a + b*Sech[x]^2], x, 4, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a]} -{Tanh[x]^0/Sqrt[a + b*Sech[x]^2], x, 3, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[a]} -{Coth[x]^1/Sqrt[a + b*Sech[x]^2], x, 7, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]]/Sqrt[a + b]} -{Coth[x]^2/Sqrt[a + b*Sech[x]^2], x, 6, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/Sqrt[a] - (Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(a + b)} -{Coth[x]^3/Sqrt[a + b*Sech[x]^2], x, 8, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/Sqrt[a] - ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]])/(2*(a + b)^(3/2)) - (Coth[x]^2*Sqrt[a + b*Sech[x]^2])/(2*(a + b))} - - -{Tanh[x]^5/(a + b*Sech[x]^2)^(3/2), x, 6, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(3/2) - (a + b)^2/(a*b^2*Sqrt[a + b*Sech[x]^2]) - Sqrt[a + b*Sech[x]^2]/b^2} -{Tanh[x]^4/(a + b*Sech[x]^2)^(3/2), x, 8, ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/b^(3/2) + ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - ((a + b)*Tanh[x])/(a*b*Sqrt[a + b - b*Tanh[x]^2])} -{Tanh[x]^3/(a + b*Sech[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(3/2) - (a + b)/(a*b*Sqrt[a + b*Sech[x]^2])} -{Tanh[x]^2/(a + b*Sech[x]^2)^(3/2), x, 5, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - Tanh[x]/(a*Sqrt[a + b - b*Tanh[x]^2])} -{Tanh[x]^1/(a + b*Sech[x]^2)^(3/2), x, 5, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(3/2) - 1/(a*Sqrt[a + b*Sech[x]^2])} -{Tanh[x]^0/(a + b*Sech[x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - (b*Tanh[x])/(a*(a + b)*Sqrt[a + b - b*Tanh[x]^2])} -{Coth[x]^1/(a + b*Sech[x]^2)^(3/2), x, 8, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(3/2) - ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - b/(a*(a + b)*Sqrt[a + b*Sech[x]^2])} -{Coth[x]^2/(a + b*Sech[x]^2)^(3/2), x, 7, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(3/2) - (b*Coth[x])/(a*(a + b)*Sqrt[a + b - b*Tanh[x]^2]) - ((a - b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(a*(a + b)^2)} - - -{Tanh[x]^6/(a + b*Sech[x]^2)^(5/2), x, 9, -(ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/b^(5/2)) + ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - ((a + b)*Tanh[x]^3)/(3*a*b*(a + b - b*Tanh[x]^2)^(3/2)) - ((a^(-2) - b^(-2))*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]} -{Tanh[x]^5/(a + b*Sech[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - (a + b)^2/(3*a*b^2*(a + b*Sech[x]^2)^(3/2)) - (1/a^2 - 1/b^2)/Sqrt[a + b*Sech[x]^2]} -{Tanh[x]^4/(a + b*Sech[x]^2)^(5/2), x, 7, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - ((a + b)*Tanh[x])/(3*a*b*(a + b - b*Tanh[x]^2)^(3/2)) + ((a - 3*b)*Tanh[x])/(3*a^2*b*Sqrt[a + b - b*Tanh[x]^2])} -{Tanh[x]^3/(a + b*Sech[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - (a + b)/(3*a*b*(a + b*Sech[x]^2)^(3/2)) - 1/(a^2*Sqrt[a + b*Sech[x]^2])} -{Tanh[x]^2/(a + b*Sech[x]^2)^(5/2), x, 7, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - Tanh[x]/(3*a*(a + b - b*Tanh[x]^2)^(3/2)) - ((2*a + 3*b)*Tanh[x])/(3*a^2*(a + b)*Sqrt[a + b - b*Tanh[x]^2])} -{Tanh[x]^1/(a + b*Sech[x]^2)^(5/2), x, 6, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - 1/(3*a*(a + b*Sech[x]^2)^(3/2)) - 1/(a^2*Sqrt[a + b*Sech[x]^2])} -{Tanh[x]^0/(a + b*Sech[x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - (b*Tanh[x])/(3*a*(a + b)*(a + b - b*Tanh[x]^2)^(3/2)) - (b*(5*a + 3*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b - b*Tanh[x]^2])} -{Coth[x]^1/(a + b*Sech[x]^2)^(5/2), x, 9, ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - b/(3*a*(a + b)*(a + b*Sech[x]^2)^(3/2)) - (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Sech[x]^2])} -{Coth[x]^2/(a + b*Sech[x]^2)^(5/2), x, 8, ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b - b*Tanh[x]^2]]/a^(5/2) - (b*Coth[x])/(3*a*(a + b)*(a + b - b*Tanh[x]^2)^(3/2)) - (b*(7*a + 3*b)*Coth[x])/(3*a^2*(a + b)^2*Sqrt[a + b - b*Tanh[x]^2]) - ((a - 3*b)*(3*a + b)*Coth[x]*Sqrt[a + b - b*Tanh[x]^2])/(3*a^2*(a + b)^3)} - - -{1/(a + b*Sech[c + d*x]^2)^(7/2), x, 7, ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + b - b*Tanh[c + d*x]^2]]/(a^(7/2)*d) - (b*Tanh[c + d*x])/(5*a*(a + b)*d*(a + b - b*Tanh[c + d*x]^2)^(5/2)) - (b*(9*a + 5*b)*Tanh[c + d*x])/(15*a^2*(a + b)^2*d*(a + b - b*Tanh[c + d*x]^2)^(3/2)) - (b*(33*a^2 + 40*a*b + 15*b^2)*Tanh[c + d*x])/(15*a^3*(a + b)^3*d*Sqrt[a + b - b*Tanh[c + d*x]^2])} - - -(* ::Subsection:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^p when p symbolic*) - - -(* ::Section:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^3)^p*) - - -(* ::Section:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^4)^p*) - - -(* ::Section:: *) -(*Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^n)^p*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.1 (c+d x)^m (a+b csch)^n.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.1 (c+d x)^m (a+b csch)^n.m deleted file mode 100644 index 19ccadc..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.1 (c+d x)^m (a+b csch)^n.m +++ /dev/null @@ -1,95 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b Csch[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Csch[a+b x]^p*) - - -{(c + d*x)^3*Csch[a + b*x], x, 9, -((2*(c + d*x)^3*ArcTanh[E^(a + b*x)])/b) - (3*d*(c + d*x)^2*PolyLog[2, -E^(a + b*x)])/b^2 + (3*d*(c + d*x)^2*PolyLog[2, E^(a + b*x)])/b^2 + (6*d^2*(c + d*x)*PolyLog[3, -E^(a + b*x)])/b^3 - (6*d^2*(c + d*x)*PolyLog[3, E^(a + b*x)])/b^3 - (6*d^3*PolyLog[4, -E^(a + b*x)])/b^4 + (6*d^3*PolyLog[4, E^(a + b*x)])/b^4} -{(c + d*x)^2*Csch[a + b*x], x, 7, -((2*(c + d*x)^2*ArcTanh[E^(a + b*x)])/b) - (2*d*(c + d*x)*PolyLog[2, -E^(a + b*x)])/b^2 + (2*d*(c + d*x)*PolyLog[2, E^(a + b*x)])/b^2 + (2*d^2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*d^2*PolyLog[3, E^(a + b*x)])/b^3} -{(c + d*x)^1*Csch[a + b*x], x, 5, -((2*(c + d*x)*ArcTanh[E^(a + b*x)])/b) - (d*PolyLog[2, -E^(a + b*x)])/b^2 + (d*PolyLog[2, E^(a + b*x)])/b^2} -{1/x^1*Csch[a + b*x], x, 0, Unintegrable[Csch[a + b*x]/x, x]} - - -{(c + d*x)^3*Csch[a + b*x]^2, x, 6, -((c + d*x)^3/b) - ((c + d*x)^3*Coth[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(2*(a + b*x))])/b^2 + (3*d^2*(c + d*x)*PolyLog[2, E^(2*(a + b*x))])/b^3 - (3*d^3*PolyLog[3, E^(2*(a + b*x))])/(2*b^4)} -{(c + d*x)^2*Csch[a + b*x]^2, x, 5, -((c + d*x)^2/b) - ((c + d*x)^2*Coth[a + b*x])/b + (2*d*(c + d*x)*Log[1 - E^(2*(a + b*x))])/b^2 + (d^2*PolyLog[2, E^(2*(a + b*x))])/b^3} -{(c + d*x)^1*Csch[a + b*x]^2, x, 2, -(((c + d*x)*Coth[a + b*x])/b) + (d*Log[Sinh[a + b*x]])/b^2} -{1/x^1*Csch[a + b*x]^2, x, 0, Unintegrable[Csch[a + b*x]^2/x, x]} - - -{(c + d*x)^3*Csch[a + b*x]^3, x, 15, -((6*d^2*(c + d*x)*ArcTanh[E^(a + b*x)])/b^3) + ((c + d*x)^3*ArcTanh[E^(a + b*x)])/b - (3*d*(c + d*x)^2*Csch[a + b*x])/(2*b^2) - ((c + d*x)^3*Coth[a + b*x]*Csch[a + b*x])/(2*b) - (3*d^3*PolyLog[2, -E^(a + b*x)])/b^4 + (3*d*(c + d*x)^2*PolyLog[2, -E^(a + b*x)])/(2*b^2) + (3*d^3*PolyLog[2, E^(a + b*x)])/b^4 - (3*d*(c + d*x)^2*PolyLog[2, E^(a + b*x)])/(2*b^2) - (3*d^2*(c + d*x)*PolyLog[3, -E^(a + b*x)])/b^3 + (3*d^2*(c + d*x)*PolyLog[3, E^(a + b*x)])/b^3 + (3*d^3*PolyLog[4, -E^(a + b*x)])/b^4 - (3*d^3*PolyLog[4, E^(a + b*x)])/b^4} -{(c + d*x)^2*Csch[a + b*x]^3, x, 9, ((c + d*x)^2*ArcTanh[E^(a + b*x)])/b - (d^2*ArcTanh[Cosh[a + b*x]])/b^3 - (d*(c + d*x)*Csch[a + b*x])/b^2 - ((c + d*x)^2*Coth[a + b*x]*Csch[a + b*x])/(2*b) + (d*(c + d*x)*PolyLog[2, -E^(a + b*x)])/b^2 - (d*(c + d*x)*PolyLog[2, E^(a + b*x)])/b^2 - (d^2*PolyLog[3, -E^(a + b*x)])/b^3 + (d^2*PolyLog[3, E^(a + b*x)])/b^3} -{(c + d*x)^1*Csch[a + b*x]^3, x, 6, ((c + d*x)*ArcTanh[E^(a + b*x)])/b - (d*Csch[a + b*x])/(2*b^2) - ((c + d*x)*Coth[a + b*x]*Csch[a + b*x])/(2*b) + (d*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (d*PolyLog[2, E^(a + b*x)])/(2*b^2)} -{1/x^1*Csch[a + b*x]^3, x, 0, Unintegrable[Csch[a + b*x]^3/x, x]} - - -(* ::Subsection:: *) -(*Integrands of the form (c+d x)^(m/2) Csch[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Csch[a+b x]^(p/2)*) - - -{x/Csch[x]^(3/2) + x*Sqrt[Csch[x]]/3, x, 4, -(4/(9*Csch[x]^(3/2))) + (2*x*Cosh[x])/(3*Sqrt[Csch[x]])} -{x/Csch[x]^(5/2) + 3*x/(5*Sqrt[Csch[x]]), x, 4, -(4/(25*Csch[x]^(5/2))) + (2*x*Cosh[x])/(5*Csch[x]^(3/2))} -{x/Csch[x]^(7/2) - (5/21)*x*Sqrt[Csch[x]], x, 5, -(4/(49*Csch[x]^(7/2))) + (2*x*Cosh[x])/(7*Csch[x]^(5/2)) + 20/(63*Csch[x]^(3/2)) - (10*x*Cosh[x])/(21*Sqrt[Csch[x]])} -{x^2/Csch[x]^(3/2) + (1/3)*x^2*Sqrt[Csch[x]], x, 7, -((8*x)/(9*Csch[x]^(3/2))) + (16*Cosh[x])/(27*Sqrt[Csch[x]]) + (2*x^2*Cosh[x])/(3*Sqrt[Csch[x]]) - (16/27)*I*Sqrt[Csch[x]]*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Csch[c+d x])^p*) - - -(* ::Section:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Csch[c+d x])^p with a^2+b^2=0*) - - -(* ::Section:: *) -(*Integrands of the form (e+f x)^m Sinh[c+d x]^n (a+b Csch[c+d x])^p*) - - -(* ::Title::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Csch[c+d x])^p*) - - -(* ::Section:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Csch[c+d x])^p with a^2+b^2=0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n (a+b Csch[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m Cosh[c+d x]^n / (a+b Csch[c+d x])^1*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{((e + f*x)^3*Cosh[c + d*x])/(a + b*Csch[c + d*x]), x, 17, (b*(e + f*x)^4)/(4*a^2*f) - (6*f^3*Cosh[c + d*x])/(a*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) - (b*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^2*d) - (3*b*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^2) + (6*b*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^3) - (6*b*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^4) - (6*b*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^4) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(a*d)} -{((e + f*x)^2*Cosh[c + d*x])/(a + b*Csch[c + d*x]), x, 14, (b*(e + f*x)^3)/(3*a^2*f) - (2*f*(e + f*x)*Cosh[c + d*x])/(a*d^2) - (b*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^2*d) - (2*b*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (2*b*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^2) + (2*b*f^2*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^3) + (2*b*f^2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (2*f^2*Sinh[c + d*x])/(a*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(a*d)} -{((e + f*x)*Cosh[c + d*x])/(a + b*Csch[c + d*x]), x, 11, (b*(e + f*x)^2)/(2*a^2*f) - (f*Cosh[c + d*x])/(a*d^2) - (b*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^2*d) - (b*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^2*d) - (b*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^2*d^2) + ((e + f*x)*Sinh[c + d*x])/(a*d)} -{Cosh[c + d*x]/(a + b*Csch[c + d*x]), x, 5, -((b*Log[b + a*Sinh[c + d*x]])/(a^2*d)) + Sinh[c + d*x]/(a*d)} -{Cosh[c + d*x]/((e + f*x)*(a + b*Csch[c + d*x])), x, 1, Unintegrable[(Cosh[c + d*x]*Sinh[c + d*x])/((e + f*x)*(b + a*Sinh[c + d*x])), x]} - - -{((e + f*x)^3*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]), x, 24, (3*e*f^2*x)/(4*a*d^2) + (3*f^3*x^2)/(8*a*d^2) + (e + f*x)^4/(8*a*f) + (b^2*(e + f*x)^4)/(4*a^3*f) - (6*b*f^2*(e + f*x)*Cosh[c + d*x])/(a^2*d^3) - (b*(e + f*x)^3*Cosh[c + d*x])/(a^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*a*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*b*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^3) - (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (6*b*f^3*Sinh[c + d*x])/(a^2*d^4) + (3*b*f*(e + f*x)^2*Sinh[c + d*x])/(a^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d)} -{((e + f*x)^2*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]), x, 21, (f^2*x)/(4*a*d^2) + (e + f*x)^3/(6*a*f) + (b^2*(e + f*x)^3)/(3*a^3*f) - (2*b*f^2*Cosh[c + d*x])/(a^2*d^3) - (b*(e + f*x)^2*Cosh[c + d*x])/(a^2*d) - (f*(e + f*x)*Cosh[c + d*x]^2)/(2*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^3) + (2*b*f*(e + f*x)*Sinh[c + d*x])/(a^2*d^2) + (f^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*a*d^3) + ((e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d)} -{((e + f*x)*Cosh[c + d*x]^2)/(a + b*Csch[c + d*x]), x, 16, (e*x)/(2*a) + (b^2*e*x)/a^3 + (f*x^2)/(4*a) + (b^2*f*x^2)/(2*a^3) - (b*(e + f*x)*Cosh[c + d*x])/(a^2*d) - (f*Cosh[c + d*x]^2)/(4*a*d^2) - (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^3*d) - (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b*f*Sinh[c + d*x])/(a^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*a*d)} -{Cosh[c + d*x]^2/(a + b*Csch[c + d*x]), x, 6, ((a^2 + 2*b^2)*x)/(2*a^3) + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^3*d) - (Cosh[c + d*x]*(2*b - a*Sinh[c + d*x]))/(2*a^2*d)} - - -{((e + f*x)^3*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]), x, 31, (-3*b*f^3*x)/(8*a^2*d^3) - (b*(e + f*x)^3)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^4)/(4*a^4*f) - (40*f^3*Cosh[c + d*x])/(9*a*d^4) - (6*b^2*f^3*Cosh[c + d*x])/(a^3*d^4) - (2*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) - (3*b^2*f*(e + f*x)^2*Cosh[c + d*x])/(a^3*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*a*d^4) - (f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*a*d^2) - (b*(a^2 + b^2)*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^3) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^3) - (6*b*(a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^4) - (6*b*(a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^4) + (40*f^2*(e + f*x)*Sinh[c + d*x])/(9*a*d^3) + (6*b^2*f^2*(e + f*x)*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^3*Sinh[c + d*x])/(3*a*d) + (b^2*(e + f*x)^3*Sinh[c + d*x])/(a^3*d) + (3*b*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d^4) + (3*b*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*a^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*a*d^3) + ((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (3*b*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^3*Sinh[c + d*x]^2)/(2*a^2*d)} -{((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]), x, 24, -(b*e*f*x)/(2*a^2*d) - (b*f^2*x^2)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^3)/(3*a^4*f) - (4*f*(e + f*x)*Cosh[c + d*x])/(3*a*d^2) - (2*b^2*f*(e + f*x)*Cosh[c + d*x])/(a^3*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*a*d^2) - (b*(a^2 + b^2)*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (2*b*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (2*b*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (2*b*(a^2 + b^2)*f^2*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^3) + (2*b*(a^2 + b^2)*f^2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^3) + (14*f^2*Sinh[c + d*x])/(9*a*d^3) + (2*b^2*f^2*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^2*Sinh[c + d*x])/(3*a*d) + (b^2*(e + f*x)^2*Sinh[c + d*x])/(a^3*d) + (b*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*a^2*d^2) + ((e + f*x)^2*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (b*f^2*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^2*Sinh[c + d*x]^2)/(2*a^2*d) + (2*f^2*Sinh[c + d*x]^3)/(27*a*d^3)} -{((e + f*x)*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]), x, 18, -(b*f*x)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^2)/(2*a^4*f) - (2*f*Cosh[c + d*x])/(3*a*d^2) - (b^2*f*Cosh[c + d*x])/(a^3*d^2) - (f*Cosh[c + d*x]^3)/(9*a*d^2) - (b*(a^2 + b^2)*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (b*(a^2 + b^2)*f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (2*(e + f*x)*Sinh[c + d*x])/(3*a*d) + (b^2*(e + f*x)*Sinh[c + d*x])/(a^3*d) + (b*f*Cosh[c + d*x]*Sinh[c + d*x])/(4*a^2*d^2) + ((e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (b*(e + f*x)*Sinh[c + d*x]^2)/(2*a^2*d)} -{Cosh[c + d*x]^3/(a + b*Csch[c + d*x]), x, 5, -((b*(a^2 + b^2)*Log[b + a*Sinh[c + d*x]])/(a^4*d)) + ((a^2 + b^2)*Sinh[c + d*x])/(a^3*d) - (b*Sinh[c + d*x]^2)/(2*a^2*d) + Sinh[c + d*x]^3/(3*a*d)} - - -(* ::Subsubsection:: *) -(*n<0*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.2 (e x)^m (a+b csch(c+d x^n))^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.2 (e x)^m (a+b csch(c+d x^n))^p.m deleted file mode 100644 index 6dc3e08..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.2 (e x)^m (a+b csch(c+d x^n))^p.m +++ /dev/null @@ -1,179 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (e x)^m (a+b Csch[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csch[c+d x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Csch[c+d x^2])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*(a + b*Csch[c + d*x^2]), x, 10, (a*x^6)/6 - (b*x^4*ArcTanh[E^(c + d*x^2)])/d - (b*x^2*PolyLog[2, -E^(c + d*x^2)])/d^2 + (b*x^2*PolyLog[2, E^(c + d*x^2)])/d^2 + (b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (b*PolyLog[3, E^(c + d*x^2)])/d^3} -{x^4*(a + b*Csch[c + d*x^2]), x, 2, (a*x^5)/5 + b*Unintegrable[x^4*Csch[c + d*x^2], x]} -{x^3*(a + b*Csch[c + d*x^2]), x, 8, (a*x^4)/4 - (b*x^2*ArcTanh[E^(c + d*x^2)])/d - (b*PolyLog[2, -E^(c + d*x^2)])/(2*d^2) + (b*PolyLog[2, E^(c + d*x^2)])/(2*d^2)} -{x^2*(a + b*Csch[c + d*x^2]), x, 2, (a*x^3)/3 + b*Unintegrable[x^2*Csch[c + d*x^2], x]} -{x*(a + b*Csch[c + d*x^2]), x, 4, (a*x^2)/2 - (b*ArcTanh[Cosh[c + d*x^2]])/(2*d)} -{(a + b*Csch[c + d*x^2])/x, x, 2, a*Log[x] + b*Unintegrable[Csch[c + d*x^2]/x, x]} -{(a + b*Csch[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Csch[c + d*x^2]/x^2, x]} - - -{x^5*(a + b*Csch[c + d*x^2])^2, x, 15, -((b^2*x^4)/(2*d)) + (a^2*x^6)/6 - (2*a*b*x^4*ArcTanh[E^(c + d*x^2)])/d - (b^2*x^4*Coth[c + d*x^2])/(2*d) + (b^2*x^2*Log[1 - E^(2*(c + d*x^2))])/d^2 - (2*a*b*x^2*PolyLog[2, -E^(c + d*x^2)])/d^2 + (2*a*b*x^2*PolyLog[2, E^(c + d*x^2)])/d^2 + (b^2*PolyLog[2, E^(2*(c + d*x^2))])/(2*d^3) + (2*a*b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (2*a*b*PolyLog[3, E^(c + d*x^2)])/d^3} -{x^4*(a + b*Csch[c + d*x^2])^2, x, 0, Unintegrable[x^4*(a + b*Csch[c + d*x^2])^2, x]} -{x^3*(a + b*Csch[c + d*x^2])^2, x, 10, (a^2*x^4)/4 - (2*a*b*x^2*ArcTanh[E^(c + d*x^2)])/d - (b^2*x^2*Coth[c + d*x^2])/(2*d) + (b^2*Log[Sinh[c + d*x^2]])/(2*d^2) - (a*b*PolyLog[2, -E^(c + d*x^2)])/d^2 + (a*b*PolyLog[2, E^(c + d*x^2)])/d^2} -{x^2*(a + b*Csch[c + d*x^2])^2, x, 0, Unintegrable[x^2*(a + b*Csch[c + d*x^2])^2, x]} -{x*(a + b*Csch[c + d*x^2])^2, x, 5, (a^2*x^2)/2 - (a*b*ArcTanh[Cosh[c + d*x^2]])/d - (b^2*Coth[c + d*x^2])/(2*d)} -{(a + b*Csch[c + d*x^2])^2/x, x, 0, Unintegrable[(a + b*Csch[c + d*x^2])^2/x, x]} -{(a + b*Csch[c + d*x^2])^2/x^2, x, 0, Unintegrable[(a + b*Csch[c + d*x^2])^2/x^2, x]} - - -{x*Csch[a + b*x^2]^7, x, 5, (5*ArcTanh[Cosh[a + b*x^2]])/(32*b) - (5*Coth[a + b*x^2]*Csch[a + b*x^2])/(32*b) + (5*Coth[a + b*x^2]*Csch[a + b*x^2]^3)/(48*b) - (Coth[a + b*x^2]*Csch[a + b*x^2]^5)/(12*b)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(a + b*Csch[c + d*x^2]), x, 13, x^6/(6*a) - (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) + (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) - (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3)} -{x^4/(a + b*Csch[c + d*x^2]), x, 0, Unintegrable[x^4/(a + b*Csch[c + d*x^2]), x]} -{x^3/(a + b*Csch[c + d*x^2]), x, 11, x^4/(4*a) - (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) + (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a*Sqrt[a^2 + b^2]*d) - (b*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(2*a*Sqrt[a^2 + b^2]*d^2) + (b*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(2*a*Sqrt[a^2 + b^2]*d^2)} -{x^2/(a + b*Csch[c + d*x^2]), x, 0, Unintegrable[x^2/(a + b*Csch[c + d*x^2]), x]} -{x/(a + b*Csch[c + d*x^2]), x, 5, x^2/(2*a) + (b*ArcTanh[(a - b*Tanh[(c + d*x^2)/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)} -{1/(x*(a + b*Csch[c + d*x^2])), x, 0, Unintegrable[1/(x*(a + b*Csch[c + d*x^2])), x]} -{(a + b*Csch[c + d*x^2])/x^2, x, 2, -(a/x) + b*Unintegrable[Csch[c + d*x^2]/x^2, x]} - - -{x^5/(a + b*Csch[c + d*x^2])^2, x, 31, -(b^2*x^4)/(2*a^2*(a^2 + b^2)*d) + x^6/(6*a^2) + (b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a^2*(a^2 + b^2)^(3/2)*d) - (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a^2*(a^2 + b^2)^(3/2)*d) + (b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (b^3*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (2*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (b^3*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (2*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (2*b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (2*b*PolyLog[3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (b^2*x^4*Cosh[c + d*x^2])/(2*a*(a^2 + b^2)*d*(b + a*Sinh[c + d*x^2]))} -{x^4/(a + b*Csch[c + d*x^2])^2, x, 0, Unintegrable[x^4/(a + b*Csch[c + d*x^2])^2, x]} -{x^3/(a + b*Csch[c + d*x^2])^2, x, 22, x^4/(4*a^2) + (b^3*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(2*a^2*(a^2 + b^2)^(3/2)*d) - (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^3*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(2*a^2*(a^2 + b^2)^(3/2)*d) + (b*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (b^2*Log[b + a*Sinh[c + d*x^2]])/(2*a^2*(a^2 + b^2)*d^2) + (b^3*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(2*a^2*(a^2 + b^2)^(3/2)*d^2) - (b*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (b^3*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(2*a^2*(a^2 + b^2)^(3/2)*d^2) + (b*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (b^2*x^2*Cosh[c + d*x^2])/(2*a*(a^2 + b^2)*d*(b + a*Sinh[c + d*x^2]))} -{x^2/(a + b*Csch[c + d*x^2])^2, x, 0, Unintegrable[x^2/(a + b*Csch[c + d*x^2])^2, x]} -{x/(a + b*Csch[c + d*x^2])^2, x, 7, x^2/(2*a^2) + (b*(2*a^2 + b^2)*ArcTanh[(a - b*Tanh[(c + d*x^2)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^2*Coth[c + d*x^2])/(2*a*(a^2 + b^2)*d*(a + b*Csch[c + d*x^2]))} -{1/(x*(a + b*Csch[c + d*x^2])^2), x, 0, Unintegrable[1/(x*(a + b*Csch[c + d*x^2])^2), x]} -{1/(x^2*(a + b*Csch[c + d*x^2])^2), x, 0, Unintegrable[1/(x^2*(a + b*Csch[c + d*x^2])^2), x]} -{1/(x^3*(a + b*Csch[c + d*x^2])^2), x, 0, Unintegrable[1/(x^3*(a + b*Csch[c + d*x^2])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csch[c+d x^(1/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b Csch[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*Csch[c + d*Sqrt[x]]), x, 20, (a*x^4)/4 - (4*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (14*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (14*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (84*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (84*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (420*b*x^2*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (420*b*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (1680*b*x^(3/2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (1680*b*x^(3/2)*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (5040*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (5040*b*x*PolyLog[6, E^(c + d*Sqrt[x])])/d^6 + (10080*b*Sqrt[x]*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (10080*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (10080*b*PolyLog[8, -E^(c + d*Sqrt[x])])/d^8 + (10080*b*PolyLog[8, E^(c + d*Sqrt[x])])/d^8} -{x^2*(a + b*Csch[c + d*Sqrt[x]]), x, 16, (a*x^3)/3 - (4*b*x^(5/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (10*b*x^2*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (10*b*x^2*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (40*b*x^(3/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (40*b*x^(3/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (120*b*x*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (120*b*x*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (240*b*Sqrt[x]*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (240*b*Sqrt[x]*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (240*b*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (240*b*PolyLog[6, E^(c + d*Sqrt[x])])/d^6} -{x*(a + b*Csch[c + d*Sqrt[x]]), x, 12, (a*x^2)/2 - (4*b*x^(3/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (6*b*x*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (6*b*x*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (12*b*Sqrt[x]*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (12*b*Sqrt[x]*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (12*b*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (12*b*PolyLog[4, E^(c + d*Sqrt[x])])/d^4} -{(a + b*Csch[c + d*Sqrt[x]])/x, x, 2, a*Log[x] + b*Unintegrable[Csch[c + d*Sqrt[x]]/x, x]} -{(a + b*Csch[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Csch[c + d*Sqrt[x]]/x^2, x]} - - -{x^3*(a + b*Csch[c + d*Sqrt[x]])^2, x, 30, -((2*b^2*x^(7/2))/d) + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(7/2)*Coth[c + d*Sqrt[x]])/d + (14*b^2*x^3*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (28*a*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (28*a*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (42*b^2*x^(5/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (168*a*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (105*b^2*x^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (840*a*b*x^2*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (840*a*b*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (210*b^2*x^(3/2)*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*x^(3/2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (315*b^2*x*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/d^6 - (10080*a*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (10080*a*b*x*PolyLog[6, E^(c + d*Sqrt[x])])/d^6 + (315*b^2*Sqrt[x]*PolyLog[6, E^(2*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (315*b^2*PolyLog[7, E^(2*(c + d*Sqrt[x]))])/(2*d^8) - (20160*a*b*PolyLog[8, -E^(c + d*Sqrt[x])])/d^8 + (20160*a*b*PolyLog[8, E^(c + d*Sqrt[x])])/d^8} -{x^2*(a + b*Csch[c + d*Sqrt[x]])^2, x, 24, -((2*b^2*x^(5/2))/d) + (a^2*x^3)/3 - (8*a*b*x^(5/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(5/2)*Coth[c + d*Sqrt[x]])/d + (10*b^2*x^2*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (20*a*b*x^2*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (20*a*b*x^2*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (20*b^2*x^(3/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (80*a*b*x^(3/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (30*b^2*x*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (240*a*b*x*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (240*a*b*x*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (30*b^2*Sqrt[x]*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (15*b^2*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/d^6 - (480*a*b*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (480*a*b*PolyLog[6, E^(c + d*Sqrt[x])])/d^6} -{x*(a + b*Csch[c + d*Sqrt[x]])^2, x, 18, -((2*b^2*x^(3/2))/d) + (a^2*x^2)/2 - (8*a*b*x^(3/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(3/2)*Coth[c + d*Sqrt[x]])/d + (6*b^2*x*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (12*a*b*x*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (12*a*b*x*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (6*b^2*Sqrt[x]*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (24*a*b*Sqrt[x]*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (24*a*b*Sqrt[x]*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (3*b^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (24*a*b*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (24*a*b*PolyLog[4, E^(c + d*Sqrt[x])])/d^4} -{(a + b*Csch[c + d*Sqrt[x]])^2/x, x, 0, Unintegrable[(a + b*Csch[c + d*Sqrt[x]])^2/x, x]} -{(a + b*Csch[c + d*Sqrt[x]])^2/x^2, x, 0, Unintegrable[(a + b*Csch[c + d*Sqrt[x]])^2/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3/(a + b*Csch[c + d*Sqrt[x]]), x, 23, x^4/(4*a) - (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (14*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (84*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (420*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (1680*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (5040*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (10080*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^7) - (10080*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^7) - (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^8) + (10080*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^8)} -{x^2/(a + b*Csch[c + d*Sqrt[x]]), x, 19, x^3/(3*a) - (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (240*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (240*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6)} -{x/(a + b*Csch[c + d*Sqrt[x]]), x, 15, x^2/(2*a) - (2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (6*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (12*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (12*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (12*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (12*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4)} -{1/(x*(a + b*Csch[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x*(a + b*Csch[c + d*Sqrt[x]])), x]} -{(a + b*Csch[c + d*Sqrt[x]])/x^2, x, 2, -(a/x) + b*Unintegrable[Csch[c + d*Sqrt[x]]/x^2, x]} - - -{x^3/(a + b*Csch[c + d*Sqrt[x]])^2, x, 61, (-2*b^2*x^(7/2))/(a^2*(a^2 + b^2)*d) + x^4/(4*a^2) + (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (14*b^2*x^3*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (2*b^3*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (4*b*x^(7/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (84*b^2*x^(5/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (14*b^3*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (28*b*x^3*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (420*b^2*x^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) + (84*b^3*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (168*b*x^(5/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) + (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) + (1680*b^2*x^(3/2)*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) - (420*b^3*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) + (840*b*x^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^6) - (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) + (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) - (5040*b^2*x*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^6) + (1680*b^3*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) - (3360*b*x^(3/2)*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) + (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^7) + (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^6) - (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^6) + (10080*b^2*Sqrt[x]*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^7) - (5040*b^3*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^6) + (10080*b*x*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^6) - (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^8) - (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^7) + (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^7) - (10080*b^2*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^8) + (10080*b^3*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^7) - (20160*b*Sqrt[x]*PolyLog[7, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^7) + (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^8) - (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^8) - (10080*b^3*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^8) + (20160*b*PolyLog[8, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^8) - (2*b^2*x^(7/2)*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*Sqrt[x]]))} -{x^2/(a + b*Csch[c + d*Sqrt[x]])^2, x, 49, (-2*b^2*x^(5/2))/(a^2*(a^2 + b^2)*d) + x^3/(3*a^2) + (10*b^2*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (4*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (10*b^2*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (2*b^3*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (4*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (40*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (10*b^3*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (20*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (40*b^2*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (10*b^3*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (20*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (120*b^2*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (40*b^3*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (80*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (120*b^2*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) + (40*b^3*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (80*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (240*b^2*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) + (120*b^3*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (240*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) + (240*b^2*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) - (120*b^3*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) + (240*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (240*b^2*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^6) - (240*b^3*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) + (480*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) - (240*b^2*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^6) + (240*b^3*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) - (480*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) + (240*b^3*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^6) - (480*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^6) - (240*b^3*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^6) + (480*b*PolyLog[6, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^6) - (2*b^2*x^(5/2)*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*Sqrt[x]]))} -{x/(a + b*Csch[c + d*Sqrt[x]])^2, x, 37, (-2*b^2*x^(3/2))/(a^2*(a^2 + b^2)*d) + x^2/(2*a^2) + (6*b^2*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (4*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (6*b^2*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (2*b^3*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (4*b*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (12*b^2*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (6*b^3*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (12*b*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (12*b^2*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) + (12*b^3*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (24*b*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (12*b^3*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) + (24*b*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (2*b^2*x^(3/2)*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*Sqrt[x]]))} -{1/(x*(a + b*Csch[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x*(a + b*Csch[c + d*Sqrt[x]])^2), x]} -{1/(x^2*(a + b*Csch[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^2*(a + b*Csch[c + d*Sqrt[x]])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^(m/2) (a+b Csch[c+d x^(1/2)])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^(3/2)*(a + b*Csch[c + d*Sqrt[x]]), x, 14, (2*a*x^(5/2))/5 - (4*b*x^2*ArcTanh[E^(c + d*Sqrt[x])])/d - (8*b*x^(3/2)*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (8*b*x^(3/2)*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (24*b*x*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (24*b*x*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (48*b*Sqrt[x]*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (48*b*Sqrt[x]*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (48*b*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (48*b*PolyLog[5, E^(c + d*Sqrt[x])])/d^5} -{Sqrt[x]*(a + b*Csch[c + d*Sqrt[x]]), x, 10, (2*a*x^(3/2))/3 - (4*b*x*ArcTanh[E^(c + d*Sqrt[x])])/d - (4*b*Sqrt[x]*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (4*b*Sqrt[x]*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (4*b*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (4*b*PolyLog[3, E^(c + d*Sqrt[x])])/d^3} -{(a + b*Csch[c + d*Sqrt[x]])/Sqrt[x], x, 4, 2*a*Sqrt[x] - (2*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d} -{(a + b*Csch[c + d*Sqrt[x]])/x^(3/2), x, 2, (-2*a)/Sqrt[x] + b*Unintegrable[Csch[c + d*Sqrt[x]]/x^(3/2), x]} -{(a + b*Csch[c + d*Sqrt[x]])/x^(5/2), x, 2, (-2*a)/(3*x^(3/2)) + b*Unintegrable[Csch[c + d*Sqrt[x]]/x^(5/2), x]} - - -{x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])^2, x, 21, -((2*b^2*x^2)/d) + (2/5)*a^2*x^(5/2) - (8*a*b*x^2*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^2*Coth[c + d*Sqrt[x]])/d + (8*b^2*x^(3/2)*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (16*a*b*x^(3/2)*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (16*a*b*x^(3/2)*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (12*b^2*x*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (48*a*b*x*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (48*a*b*x*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (12*b^2*Sqrt[x]*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (96*a*b*Sqrt[x]*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (96*a*b*Sqrt[x]*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (6*b^2*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (96*a*b*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (96*a*b*PolyLog[5, E^(c + d*Sqrt[x])])/d^5} -{Sqrt[x]*(a + b*Csch[c + d*Sqrt[x]])^2, x, 15, -((2*b^2*x)/d) + (2/3)*a^2*x^(3/2) - (8*a*b*x*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x*Coth[c + d*Sqrt[x]])/d + (4*b^2*Sqrt[x]*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (8*a*b*Sqrt[x]*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (8*a*b*Sqrt[x]*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (2*b^2*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (8*a*b*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (8*a*b*PolyLog[3, E^(c + d*Sqrt[x])])/d^3} -{(a + b*Csch[c + d*Sqrt[x]])^2/Sqrt[x], x, 5, 2*a^2*Sqrt[x] - (4*a*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d - (2*b^2*Coth[c + d*Sqrt[x]])/d} -{(a + b*Csch[c + d*Sqrt[x]])^2/x^(3/2), x, 0, Unintegrable[(a + b*Csch[c + d*Sqrt[x]])^2/x^(3/2), x]} -{(a + b*Csch[c + d*Sqrt[x]])^2/x^(5/2), x, 0, Unintegrable[(a + b*Csch[c + d*Sqrt[x]])^2/x^(5/2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^(3/2)/(a + b*Csch[c + d*Sqrt[x]]), x, 17, (2*x^(5/2))/(5*a) - (2*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (8*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (8*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (24*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (24*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (48*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (48*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^4) + (48*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (48*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5)} -{Sqrt[x]/(a + b*Csch[c + d*Sqrt[x]]), x, 13, (2*x^(3/2))/(3*a) - (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (4*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (4*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (4*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (4*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3)} -{1/(Sqrt[x]*(a + b*Csch[c + d*Sqrt[x]])), x, 5, (2*Sqrt[x])/a + (4*b*ArcTanh[(a - b*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)} -{1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])), x]} -{1/(x^(5/2)*(a + b*Csch[c + d*Sqrt[x]])), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Csch[c + d*Sqrt[x]])), x]} - - -{x^(3/2)/(a + b*Csch[c + d*Sqrt[x]])^2, x, 43, (-2*b^2*x^2)/(a^2*(a^2 + b^2)*d) + (2*x^(5/2))/(5*a^2) + (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (4*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (8*b^2*x^(3/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (2*b^3*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (4*b*x^2*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (24*b^2*x*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (8*b^3*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (16*b*x^(3/2)*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) - (24*b^3*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (48*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (48*b^2*Sqrt[x]*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^4) + (24*b^3*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (48*b*x*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) + (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) - (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) + (48*b^2*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^5) - (48*b^3*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^4) + (96*b*Sqrt[x]*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (48*b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) + (96*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) + (48*b^3*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^5) - (96*b*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^5) - (2*b^2*x^2*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*Sqrt[x]]))} -{Sqrt[x]/(a + b*Csch[c + d*Sqrt[x]])^2, x, 31, (-2*b^2*x)/(a^2*(a^2 + b^2)*d) + (2*x^(3/2))/(3*a^2) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (4*b^2*Sqrt[x]*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) - (2*b^3*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (4*b*x*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) + (4*b^2*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (4*b^3*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (8*b*Sqrt[x]*PolyLog[2, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (4*b^3*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (8*b*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (2*b^2*x*Cosh[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*Sqrt[x]]))} -{1/(Sqrt[x]*(a + b*Csch[c + d*Sqrt[x]])^2), x, 7, (2*Sqrt[x])/a^2 + (4*b*(2*a^2 + b^2)*ArcTanh[(a - b*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)*d) - (2*b^2*Coth[c + d*Sqrt[x]])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*Sqrt[x]]))} -{1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(3/2)*(a + b*Csch[c + d*Sqrt[x]])^2), x]} -{1/(x^(5/2)*(a + b*Csch[c + d*Sqrt[x]])^2), x, 0, Unintegrable[1/(x^(5/2)*(a + b*Csch[c + d*Sqrt[x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b Csch[c+d x^n])^p*) - - -{(e*x)^m*(a + b*Csch[c + d*x^n])^p, x, 1, ((e*x)^m*Unintegrable[x^m*(a + b*Csch[c + d*x^n])^p, x])/x^m} - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(e*x)^(-1 + n)*(a + b*Csch[c + d*x^n]), x, 5, (a*(e*x)^n)/(e*n) - (b*(e*x)^n*ArcTanh[Cosh[c + d*x^n]])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n]), x, 9, (a*(e*x)^(2*n))/(2*e*n) - (2*b*(e*x)^(2*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*x^n) - (b*(e*x)^(2*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b*(e*x)^(2*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Csch[c + d*x^n]), x, 11, (a*(e*x)^(3*n))/(3*e*n) - (2*b*(e*x)^(3*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*x^n) - (2*b*(e*x)^(3*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, -E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, E^(c + d*x^n)])/(d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)*(a + b*Csch[c + d*x^n])^2, x, 6, (a^2*(e*x)^n)/(e*n) - (2*a*b*(e*x)^n*ArcTanh[Cosh[c + d*x^n]])/(d*e*n*x^n) - (b^2*(e*x)^n*Coth[c + d*x^n])/(d*e*n*x^n)} -{(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n])^2, x, 11, (a^2*(e*x)^(2*n))/(2*e*n) - (4*a*b*(e*x)^(2*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Coth[c + d*x^n])/(d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[Sinh[c + d*x^n]])/(d^2*e*n*x^(2*n)) - (2*a*b*(e*x)^(2*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (2*a*b*(e*x)^(2*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)*(a + b*Csch[c + d*x^n])^2, x, 16, (a^2*(e*x)^(3*n))/(3*e*n) - (b^2*(e*x)^(3*n))/(x^n*(d*e*n)) - (4*a*b*(e*x)^(3*n)*ArcTanh[E^(c + d*x^n)])/(x^n*(d*e*n)) - (b^2*(e*x)^(3*n)*Coth[c + d*x^n])/(x^n*(d*e*n)) + (2*b^2*(e*x)^(3*n)*Log[1 - E^(2*(c + d*x^n))])/(x^(2*n)*(d^2*e*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[2, -E^(c + d*x^n)])/(x^(2*n)*(d^2*e*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[2, E^(c + d*x^n)])/(x^(2*n)*(d^2*e*n)) + (b^2*(e*x)^(3*n)*PolyLog[2, E^(2*(c + d*x^n))])/(x^(3*n)*(d^3*e*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[3, -E^(c + d*x^n)])/(x^(3*n)*(d^3*e*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[3, E^(c + d*x^n)])/(x^(3*n)*(d^3*e*n))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(e*x)^(-1 + n)/(a + b*Csch[c + d*x^n]), x, 6, (e*x)^n/(a*e*n) + (2*b*(e*x)^n*ArcTanh[(a - b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n)} -{(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n]), x, 12, (e*x)^(2*n)/(2*a*e*n) - (b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n) - (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n))} -{(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n]), x, 14, (e*x)^(3*n)/(3*a*e*n) - (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n) + (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n) - (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n))} - - -{(e*x)^(-1 + n)/(a + b*Csch[c + d*x^n])^2, x, 8, (e*x)^n/(a^2*e*n) + (2*b*(2*a^2 + b^2)*(e*x)^n*ArcTanh[(a - b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) - (b^2*(e*x)^n*Coth[c + d*x^n])/(a*(a^2 + b^2)*d*e*n*x^n*(a + b*Csch[c + d*x^n]))} -{(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n])^2, x, 23, (e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) - (b^3*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[b + a*Sinh[c + d*x^n]])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^2*(e*x)^(2*n)*Cosh[c + d*x^n])/(a*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))} -{(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2, x, 32, (e*x)^(3*n)/(3*a^2*e*n) - (b^2*(e*x)^(3*n))/(a^2*(a^2 + b^2)*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3*e*n*x^(3*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) - (b^2*(e*x)^(3*n)*Cosh[c + d*x^n])/(a*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.3 Hyperbolic cosecant functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.3 Hyperbolic cosecant functions.m deleted file mode 100644 index 3c9b542..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.3 Hyperbolic cosecant functions.m +++ /dev/null @@ -1,333 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Cosecants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Csch[a+b x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Csch[a+b x]^n*) - - -{Csch[a + b*x], x, 1, -(ArcTanh[Cosh[a + b*x]]/b)} -{Csch[a + b*x]^2, x, 2, -Coth[a + b*x]/b} -{Csch[a + b*x]^3, x, 2, ArcTanh[Cosh[a + b*x]]/(2*b) - (Coth[a + b*x]*Csch[a + b*x])/(2*b)} -{Csch[a + b*x]^4, x, 2, Coth[a + b*x]/b - Coth[a + b*x]^3/(3*b)} -{Csch[a + b*x]^5, x, 3, -((3*ArcTanh[Cosh[a + b*x]])/(8*b)) + (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]*Csch[a + b*x]^3)/(4*b)} -{Csch[a + b*x]^6, x, 2, -(Coth[a + b*x]/b) + (2*Coth[a + b*x]^3)/(3*b) - Coth[a + b*x]^5/(5*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Csch[a+b x])^(n/2)*) - - -{Csch[a + b*x]^(5/2), x, 3, -((2*Cosh[a + b*x]*Csch[a + b*x]^(3/2))/(3*b)) + (2*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(3*b)} -{Csch[a + b*x]^(3/2), x, 3, -((2*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/b) - (2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} -{Csch[a + b*x]^(1/2), x, 2, -((2*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/b)} -{1/Csch[a + b*x]^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]]))} -{1/Csch[a + b*x]^(3/2), x, 3, (2*Cosh[a + b*x])/(3*b*Sqrt[Csch[a + b*x]]) + (2*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(3*b)} -{1/Csch[a + b*x]^(5/2), x, 3, (2*Cosh[a + b*x])/(5*b*Csch[a + b*x]^(3/2)) + (6*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(5*b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} - - -{(b*Csch[c + d*x])^(7/2), x, 4, (6*b^3*Cosh[c + d*x]*Sqrt[b*Csch[c + d*x]])/(5*d) - (2*b*Cosh[c + d*x]*(b*Csch[c + d*x])^(5/2))/(5*d) + (6*I*b^4*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(5*d*Sqrt[b*Csch[c + d*x]]*Sqrt[I*Sinh[c + d*x]])} -{(b*Csch[c + d*x])^(5/2), x, 3, -((2*b*Cosh[c + d*x]*(b*Csch[c + d*x])^(3/2))/(3*d)) + (2*I*b^2*Sqrt[b*Csch[c + d*x]]*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(3*d)} -{(b*Csch[c + d*x])^(3/2), x, 3, -((2*b*Cosh[c + d*x]*Sqrt[b*Csch[c + d*x]])/d) - (2*I*b^2*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(d*Sqrt[b*Csch[c + d*x]]*Sqrt[I*Sinh[c + d*x]])} -{(b*Csch[c + d*x])^(1/2), x, 2, -((2*I*Sqrt[b*Csch[c + d*x]]*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/d)} -{1/(b*Csch[c + d*x])^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(d*Sqrt[b*Csch[c + d*x]]*Sqrt[I*Sinh[c + d*x]]))} -{1/(b*Csch[c + d*x])^(3/2), x, 3, (2*Cosh[c + d*x])/(3*b*d*Sqrt[b*Csch[c + d*x]]) + (2*I*Sqrt[b*Csch[c + d*x]]*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(3*b^2*d)} -{1/(b*Csch[c + d*x])^(5/2), x, 3, (2*Cosh[c + d*x])/(5*b*d*(b*Csch[c + d*x])^(3/2)) + (6*I*EllipticE[(1/2)*(I*c - Pi/2 + I*d*x), 2])/(5*b^2*d*Sqrt[b*Csch[c + d*x]]*Sqrt[I*Sinh[c + d*x]])} -{1/(b*Csch[c + d*x])^(7/2), x, 4, (2*Cosh[c + d*x])/(7*b*d*(b*Csch[c + d*x])^(5/2)) - (10*Cosh[c + d*x])/(21*b^3*d*Sqrt[b*Csch[c + d*x]]) - (10*I*Sqrt[b*Csch[c + d*x]]*EllipticF[(1/2)*(I*c - Pi/2 + I*d*x), 2]*Sqrt[I*Sinh[c + d*x]])/(21*b^4*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Csch[a+b x])^n with n symbolic*) - - -{(b*Csch[c + d*x])^n, x, 2, (b*Cosh[c + d*x]*(b*Csch[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, -Sinh[c + d*x]^2])/(d*(1 - n)*Sqrt[Cosh[c + d*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c Csch[a+b x]^m)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Csch[a+b x]^2)^n*) - - -{(-Csch[x]^2)^(5/2), x, 4, (3/8)*ArcSin[Coth[x]] + (3/8)*Coth[x]*Sqrt[-Csch[x]^2] + (1/4)*Coth[x]*(-Csch[x]^2)^(3/2)} -{(-Csch[x]^2)^(3/2), x, 3, (1/2)*ArcSin[Coth[x]] + (1/2)*Coth[x]*Sqrt[-Csch[x]^2]} -{(-Csch[x]^2)^(1/2), x, 2, ArcSin[Coth[x]]} -{1/(-Csch[x]^2)^(1/2), x, 2, Coth[x]/Sqrt[-Csch[x]^2]} -{1/(-Csch[x]^2)^(3/2), x, 3, Coth[x]/(3*(-Csch[x]^2)^(3/2)) + (2*Coth[x])/(3*Sqrt[-Csch[x]^2])} -{1/(-Csch[x]^2)^(5/2), x, 4, Coth[x]/(5*(-Csch[x]^2)^(5/2)) + (4*Coth[x])/(15*(-Csch[x]^2)^(3/2)) + (8*Coth[x])/(15*Sqrt[-Csch[x]^2])} -{1/(-Csch[x]^2)^(7/2), x, 5, Coth[x]/(7*(-Csch[x]^2)^(7/2)) + (6*Coth[x])/(35*(-Csch[x]^2)^(5/2)) + (8*Coth[x])/(35*(-Csch[x]^2)^(3/2)) + (16*Coth[x])/(35*Sqrt[-Csch[x]^2])} - - -{(a*Csch[x]^2)^(5/2), x, 5, (-(3/8))*a^(5/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]] + (3/8)*a^2*Coth[x]*Sqrt[a*Csch[x]^2] - (1/4)*a*Coth[x]*(a*Csch[x]^2)^(3/2)} -{(a*Csch[x]^2)^(3/2), x, 4, (1/2)*a^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]] - (1/2)*a*Coth[x]*Sqrt[a*Csch[x]^2]} -{(a*Csch[x]^2)^(1/2), x, 3, (-Sqrt[a])*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]]} -{1/(a*Csch[x]^2)^(1/2), x, 2, Coth[x]/Sqrt[a*Csch[x]^2]} -{1/(a*Csch[x]^2)^(3/2), x, 3, Coth[x]/(3*(a*Csch[x]^2)^(3/2)) - (2*Coth[x])/(3*a*Sqrt[a*Csch[x]^2])} -{1/(a*Csch[x]^2)^(5/2), x, 4, Coth[x]/(5*(a*Csch[x]^2)^(5/2)) - (4*Coth[x])/(15*a*(a*Csch[x]^2)^(3/2)) + (8*Coth[x])/(15*a^2*Sqrt[a*Csch[x]^2])} -{1/(a*Csch[x]^2)^(7/2), x, 5, Coth[x]/(7*(a*Csch[x]^2)^(7/2)) - (6*Coth[x])/(35*a*(a*Csch[x]^2)^(5/2)) + (8*Coth[x])/(35*a^2*(a*Csch[x]^2)^(3/2)) - (16*Coth[x])/(35*a^3*Sqrt[a*Csch[x]^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Csch[a+b x]^3)^n*) - - -{(a*Csch[x]^3)^(5/2), x, 7, (-(154/585))*a^2*Coth[x]*Sqrt[a*Csch[x]^3] + (22/117)*a^2*Coth[x]*Csch[x]^2*Sqrt[a*Csch[x]^3] - (2/13)*a^2*Coth[x]*Csch[x]^4*Sqrt[a*Csch[x]^3] + (154/195)*a^2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x] - (154*I*a^2*Sqrt[a*Csch[x]^3]*EllipticE[Pi/4 - (I*x)/2, 2]*Sinh[x]^2)/(195*Sqrt[I*Sinh[x]])} -{(a*Csch[x]^3)^(3/2), x, 5, (10/21)*a*Cosh[x]*Sqrt[a*Csch[x]^3] - (2/7)*a*Coth[x]*Csch[x]*Sqrt[a*Csch[x]^3] + (10/21)*I*a*Sqrt[a*Csch[x]^3]*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]]*Sinh[x]} -{(a*Csch[x]^3)^(1/2), x, 4, -2*I*Sqrt[a*Csch[x]^3]*EllipticE[Pi/4 - (I*x)/2, 2]*(I*Sinh[x])^(3/2) - 2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x], -2*Cosh[x]*Sqrt[a*Csch[x]^3]*Sinh[x] + (2*I*Sqrt[a*Csch[x]^3]*EllipticE[Pi/4 - (I*x)/2, 2]*Sinh[x]^2)/Sqrt[I*Sinh[x]]} -{1/(a*Csch[x]^3)^(1/2), x, 4, (2*Coth[x])/(3*Sqrt[a*Csch[x]^3]) - (2*I*Csch[x]^2*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]])/(3*Sqrt[a*Csch[x]^3])} -{1/(a*Csch[x]^3)^(3/2), x, 5, -((14*Cosh[x])/(45*a*Sqrt[a*Csch[x]^3])) + (14*I*Csch[x]*EllipticE[Pi/4 - (I*x)/2, 2])/(15*a*Sqrt[a*Csch[x]^3]*Sqrt[I*Sinh[x]]) + (2*Cosh[x]*Sinh[x]^2)/(9*a*Sqrt[a*Csch[x]^3])} -{1/(a*Csch[x]^3)^(5/2), x, 7, -((26*Coth[x])/(77*a^2*Sqrt[a*Csch[x]^3])) + (26*I*Csch[x]^2*EllipticF[Pi/4 - (I*x)/2, 2]*Sqrt[I*Sinh[x]])/(77*a^2*Sqrt[a*Csch[x]^3]) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqrt[a*Csch[x]^3]) - (26*Cosh[x]*Sinh[x]^3)/(165*a^2*Sqrt[a*Csch[x]^3]) + (2*Cosh[x]*Sinh[x]^5)/(15*a^2*Sqrt[a*Csch[x]^3])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c Csch[a+b x]^4)^n*) - - -{(a*Csch[x]^4)^(7/2), x, 3, 2*a^3*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4] - 3*a^3*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4] + (20/7)*a^3*Cosh[x]^2*Coth[x]^5*Sqrt[a*Csch[x]^4] - (5/3)*a^3*Cosh[x]^2*Coth[x]^7*Sqrt[a*Csch[x]^4] + (6/11)*a^3*Cosh[x]^2*Coth[x]^9*Sqrt[a*Csch[x]^4] - (1/13)*a^3*Cosh[x]^2*Coth[x]^11*Sqrt[a*Csch[x]^4] - a^3*Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x]} -{(a*Csch[x]^4)^(5/2), x, 3, (4/3)*a^2*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4] - (6/5)*a^2*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4] + (4/7)*a^2*Cosh[x]^2*Coth[x]^5*Sqrt[a*Csch[x]^4] - (1/9)*a^2*Cosh[x]^2*Coth[x]^7*Sqrt[a*Csch[x]^4] - a^2*Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x]} -{(a*Csch[x]^4)^(3/2), x, 3, (2/3)*a*Cosh[x]^2*Coth[x]*Sqrt[a*Csch[x]^4] - (1/5)*a*Cosh[x]^2*Coth[x]^3*Sqrt[a*Csch[x]^4] - a*Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x]} -{(a*Csch[x]^4)^(1/2), x, 3, (-Cosh[x])*Sqrt[a*Csch[x]^4]*Sinh[x]} -{1/(a*Csch[x]^4)^(1/2), x, 3, Coth[x]/(2*Sqrt[a*Csch[x]^4]) - (x*Csch[x]^2)/(2*Sqrt[a*Csch[x]^4])} -{1/(a*Csch[x]^4)^(3/2), x, 5, (5*Coth[x])/(16*a*Sqrt[a*Csch[x]^4]) - (5*x*Csch[x]^2)/(16*a*Sqrt[a*Csch[x]^4]) - (5*Cosh[x]*Sinh[x])/(24*a*Sqrt[a*Csch[x]^4]) + (Cosh[x]*Sinh[x]^3)/(6*a*Sqrt[a*Csch[x]^4])} -{1/(a*Csch[x]^4)^(5/2), x, 7, (63*Coth[x])/(256*a^2*Sqrt[a*Csch[x]^4]) - (63*x*Csch[x]^2)/(256*a^2*Sqrt[a*Csch[x]^4]) - (21*Cosh[x]*Sinh[x])/(128*a^2*Sqrt[a*Csch[x]^4]) + (21*Cosh[x]*Sinh[x]^3)/(160*a^2*Sqrt[a*Csch[x]^4]) - (9*Cosh[x]*Sinh[x]^5)/(80*a^2*Sqrt[a*Csch[x]^4]) + (Cosh[x]*Sinh[x]^7)/(10*a^2*Sqrt[a*Csch[x]^4])} - - -(* ::Subsection:: *) -(*Integrands of the form (c Csch[a+b x]^m)^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Csch[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Csch[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2=0*) - - -{1/(a + I*a*Csch[a + b*x]), x, 2, x/a - Coth[a + b*x]/(b*(a + I*a*Csch[a + b*x]))} - - -{1/(a - I*a*Csch[a + b*x]), x, 2, x/a - Coth[a + b*x]/(b*(a - I*a*Csch[a + b*x]))} - - -{(a + a*I*Csch[c + d*x])^(5/2), x, 5, (2*a^(5/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d + (14*a^3*Coth[c + d*x])/(3*d*Sqrt[a + I*a*Csch[c + d*x]]) + (2*a^2*Coth[c + d*x]*Sqrt[a + I*a*Csch[c + d*x]])/(3*d)} -{(a + a*I*Csch[c + d*x])^(3/2), x, 4, (2*a^(3/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d + (2*a^2*Coth[c + d*x])/(d*Sqrt[a + I*a*Csch[c + d*x]])} -{(a + a*I*Csch[c + d*x])^(1/2), x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d} -{1/(a + a*I*Csch[c + d*x])^(1/2), x, 5, (2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(Sqrt[a]*d)} -{1/(a + a*I*Csch[c + d*x])^(3/2), x, 6, (2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Coth[c + d*x]/(2*d*(a + I*a*Csch[c + d*x])^(3/2))} - -{Sqrt[a - a*I*Csch[c + d*x]], x, 2, (2*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - I*a*Csch[c + d*x]]])/d} -{1/Sqrt[a - I*a*Csch[c + d*x]], x, 5, (2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - I*a*Csch[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a - I*a*Csch[c + d*x]])])/(Sqrt[a]*d)} - - -{Sqrt[3 + 3*I*Csch[x]], x, 2, 2*Sqrt[3]*ArcTanh[Coth[x]/Sqrt[1 + I*Csch[x]]]} -{Sqrt[3 - 3*I*Csch[x]], x, 2, 2*Sqrt[3]*ArcTanh[Coth[x]/Sqrt[1 - I*Csch[x]]]} -{Sqrt[-3 + 3*I*Csch[x]], x, 2, -2*Sqrt[3]*ArcTan[Coth[x]/Sqrt[-1 + I*Csch[x]]]} -{Sqrt[-3 - 3*I*Csch[x]], x, 2, -2*Sqrt[3]*ArcTan[Coth[x]/Sqrt[-1 - I*Csch[x]]]} - - -{Sinh[x]^4/(I + Csch[x]), x, 7, -((15*I*x)/8) - 4*Cosh[x] + (4*Cosh[x]^3)/3 + (15/8)*I*Cosh[x]*Sinh[x] - (5/4)*I*Cosh[x]*Sinh[x]^3 - (Cosh[x]*Sinh[x]^3)/(I + Csch[x])} -{Sinh[x]^3/(I + Csch[x]), x, 6, -((3*x)/2) + 4*I*Cosh[x] - (4/3)*I*Cosh[x]^3 + (3/2)*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x]^2)/(I + Csch[x])} -{Sinh[x]^2/(I + Csch[x]), x, 5, (3*I*x)/2 + 2*Cosh[x] - (3/2)*I*Cosh[x]*Sinh[x] - (Cosh[x]*Sinh[x])/(I + Csch[x])} -{Sinh[x]^1/(I + Csch[x]), x, 4, x - 2*I*Cosh[x] - Cosh[x]/(I + Csch[x])} -{Csch[x]^1/(I + Csch[x]), x, 1, (I*Coth[x])/(I + Csch[x])} -{Csch[x]^2/(I + Csch[x]), x, 3, -ArcTanh[Cosh[x]] + Coth[x]/(I + Csch[x])} -{Csch[x]^3/(I + Csch[x]), x, 4, I*ArcTanh[Cosh[x]] - Coth[x] - (I*Coth[x])/(I + Csch[x])} -{Csch[x]^4/(I + Csch[x]), x, 6, (3/2)*ArcTanh[Cosh[x]] + 2*I*Coth[x] - (3/2)*Coth[x]*Csch[x] + (Coth[x]*Csch[x]^2)/(I + Csch[x])} - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2!=0*) - - -{(a + b*Csch[c + d*x])^4, x, 6, a^4*x - (2*a*b*(2*a^2 - b^2)*ArcTanh[Cosh[c + d*x]])/d - (b^2*(17*a^2 - 2*b^2)*Coth[c + d*x])/(3*d) - (4*a*b^3*Coth[c + d*x]*Csch[c + d*x])/(3*d) - (b^2*Coth[c + d*x]*(a + b*Csch[c + d*x])^2)/(3*d)} -{(a + b*Csch[c + d*x])^3, x, 5, a^3*x - (b*(6*a^2 - b^2)*ArcTanh[Cosh[c + d*x]])/(2*d) - (5*a*b^2*Coth[c + d*x])/(2*d) - (b^2*Coth[c + d*x]*(a + b*Csch[c + d*x]))/(2*d)} -{(a + b*Csch[c + d*x])^2, x, 4, a^2*x - (2*a*b*ArcTanh[Cosh[c + d*x]])/d - (b^2*Coth[c + d*x])/d} -{(a + b*Csch[c + d*x])^1, x, 2, a*x - (b*ArcTanh[Cosh[c + d*x]])/d} -{1/(a + b*Csch[c + d*x])^1, x, 4, x/a + (2*b*ArcTanh[(a - b*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]*d)} -{1/(a + b*Csch[c + d*x])^2, x, 6, x/a^2 + (2*b*(2*a^2 + b^2)*ArcTanh[(a - b*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^2*Coth[c + d*x])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*x]))} -{1/(a + b*Csch[c + d*x])^3, x, 7, x/a^3 + (b*(6*a^4 + 5*a^2*b^2 + 2*b^4)*ArcTanh[(a - b*Tanh[(1/2)*(c + d*x)])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^2)^(5/2)*d) - (b^2*Coth[c + d*x])/(2*a*(a^2 + b^2)*d*(a + b*Csch[c + d*x])^2) - (b^2*(5*a^2 + 2*b^2)*Coth[c + d*x])/(2*a^2*(a^2 + b^2)^2*d*(a + b*Csch[c + d*x]))} - - -{Sinh[x]^3/(a + b*Csch[x]), x, 8, (b*(a^2 - 2*b^2)*x)/(2*a^4) - (2*b^4*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - ((2*a^2 - 3*b^2)*Cosh[x])/(3*a^3) - (b*Cosh[x]*Sinh[x])/(2*a^2) + (Cosh[x]*Sinh[x]^2)/(3*a)} -{Sinh[x]^2/(a + b*Csch[x]), x, 7, -(((a^2 - 2*b^2)*x)/(2*a^3)) + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - (b*Cosh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)} -{Sinh[x]^1/(a + b*Csch[x]), x, 6, -((b*x)/a^2) - (2*b^2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + Cosh[x]/a} -{Csch[x]^1/(a + b*Csch[x]), x, 4, -((2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2])} -{Csch[x]^2/(a + b*Csch[x]), x, 6, -(ArcTanh[Cosh[x]]/b) + (2*a*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])} -{Csch[x]^3/(a + b*Csch[x]), x, 7, (a*ArcTanh[Cosh[x]])/b^2 - (2*a^2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Coth[x]/b} -{Csch[x]^4/(a + b*Csch[x]), x, 8, -(((2*a^2 - b^2)*ArcTanh[Cosh[x]])/(2*b^3)) + (2*a^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]) + (a*Coth[x])/b^2 - (Coth[x]*Csch[x])/(2*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Csch[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2=0*) - - -{Cosh[x]^4/(I + Csch[x]), x, 7, (I*x)/8 + Cosh[x]^3/3 + (1/8)*I*Cosh[x]*Sinh[x] - (1/4)*I*Cosh[x]^3*Sinh[x]} -{Cosh[x]^3/(I + Csch[x]), x, 6, Sinh[x]^2/2 - (1/3)*I*Sinh[x]^3} -{Cosh[x]^2/(I + Csch[x]), x, 5, (I*x)/2 + Cosh[x] - (1/2)*I*Cosh[x]*Sinh[x]} -{Cosh[x]^1/(I + Csch[x]), x, 4, Log[I - Sinh[x]] - I*Sinh[x]} -{Sech[x]^1/(I + Csch[x]), x, 6, (-(1/2))*I*ArcTan[Sinh[x]] - Sech[x]^2/2 + (1/2)*I*Sech[x]*Tanh[x]} -{Sech[x]^2/(I + Csch[x]), x, 6, (-(1/3))*Sech[x]^3 - (1/3)*I*Tanh[x]^3} -{Sech[x]^3/(I + Csch[x]), x, 7, (-(1/8))*I*ArcTan[Sinh[x]] - Sech[x]^4/4 - (1/8)*I*Sech[x]*Tanh[x] + (1/4)*I*Sech[x]^3*Tanh[x]} -{Sech[x]^4/(I + Csch[x]), x, 7, (-(1/5))*Sech[x]^5 - (1/3)*I*Tanh[x]^3 + (1/5)*I*Tanh[x]^5} - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2!=0*) - - -{Cosh[x]^5/(a + b*Csch[x]), x, 5, -((b*(a^2 + b^2)^2*Log[b + a*Sinh[x]])/a^6) + ((a^2 + b^2)^2*Sinh[x])/a^5 - (b*(2*a^2 + b^2)*Sinh[x]^2)/(2*a^4) + ((2*a^2 + b^2)*Sinh[x]^3)/(3*a^3) - (b*Sinh[x]^4)/(4*a^2) + Sinh[x]^5/(5*a)} -{Cosh[x]^4/(a + b*Csch[x]), x, 7, ((3*a^4 + 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) + (2*b*(a^2 + b^2)^(3/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^5 - (Cosh[x]^3*(4*b - 3*a*Sinh[x]))/(12*a^2) - (Cosh[x]*(8*b*(a^2 + b^2) - a*(3*a^2 + 4*b^2)*Sinh[x]))/(8*a^4)} -{Cosh[x]^3/(a + b*Csch[x]), x, 5, -((b*(a^2 + b^2)*Log[b + a*Sinh[x]])/a^4) + ((a^2 + b^2)*Sinh[x])/a^3 - (b*Sinh[x]^2)/(2*a^2) + Sinh[x]^3/(3*a)} -{Cosh[x]^2/(a + b*Csch[x]), x, 6, ((a^2 + 2*b^2)*x)/(2*a^3) + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^3 - (Cosh[x]*(2*b - a*Sinh[x]))/(2*a^2)} -{Cosh[x]^1/(a + b*Csch[x]), x, 5, -((b*Log[b + a*Sinh[x]])/a^2) + Sinh[x]/a} -{Sech[x]^1/(a + b*Csch[x]), x, 4, Log[I - Sinh[x]]/(2*(I*a + b)) - Log[I + Sinh[x]]/(2*(I*a - b)) - (b*Log[b + a*Sinh[x]])/(a^2 + b^2)} -{Sech[x]^2/(a + b*Csch[x]), x, 6, (2*a*b*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (Sech[x]*(b - a*Sinh[x]))/(a^2 + b^2)} -{Sech[x]^3/(a + b*Csch[x]), x, 6, -((I*a*Log[I - Sinh[x]])/(4*(a - I*b)^2)) + (I*a*Log[I + Sinh[x]])/(4*(a + I*b)^2) - (a^2*b*Log[b + a*Sinh[x]])/(a^2 + b^2)^2 - (Sech[x]^2*(b - a*Sinh[x]))/(2*(a^2 + b^2))} -{Sech[x]^4/(a + b*Csch[x]), x, 7, (2*a^3*b*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (Sech[x]^3*(b - a*Sinh[x]))/(3*(a^2 + b^2)) - (Sech[x]*(3*a^2*b - a*(2*a^2 - b^2)*Sinh[x]))/(3*(a^2 + b^2)^2)} -{Sech[x]^5/(a + b*Csch[x]), x, 7, -((a*(3*I*a + b)*Log[I - Sinh[x]])/(16*(a - I*b)^3)) + (a*(3*a + I*b)*Log[I + Sinh[x]])/(16*(I*a - b)^3) - (a^4*b*Log[b + a*Sinh[x]])/(a^2 + b^2)^3 - (Sech[x]^4*(b - a*Sinh[x]))/(4*(a^2 + b^2)) - (Sech[x]^2*(4*a^2*b - a*(3*a^2 - b^2)*Sinh[x]))/(8*(a^2 + b^2)^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Csch[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2=0*) - - -{Tanh[x]^5/(I + Csch[x]), x, 3, (-(21/32))*I*Log[I - Sinh[x]] - (11/32)*I*Log[I + Sinh[x]] + I/(32*(1 - I*Sinh[x])^2) - I/(4*(1 - I*Sinh[x])) - I/(24*(1 + I*Sinh[x])^3) + (9*I)/(32*(1 + I*Sinh[x])^2) - (15*I)/(16*(1 + I*Sinh[x]))} -{Tanh[x]^4/(I + Csch[x]), x, 5, (-I)*x + (1/15)*(15*I - 8*Csch[x])*Tanh[x] + (1/15)*(5*I - 4*Csch[x])*Tanh[x]^3 + (1/5)*(I - Csch[x])*Tanh[x]^5} -{Tanh[x]^3/(I + Csch[x]), x, 3, (-(11/16))*I*Log[I - Sinh[x]] - (5/16)*I*Log[I + Sinh[x]] - I/(8*(1 - I*Sinh[x])) + I/(8*(1 + I*Sinh[x])^2) - (3*I)/(4*(1 + I*Sinh[x]))} -{Tanh[x]^2/(I + Csch[x]), x, 4, (-I)*x + (1/3)*(3*I - 2*Csch[x])*Tanh[x] + (1/3)*(I - Csch[x])*Tanh[x]^3} -{Tanh[x]^1/(I + Csch[x]), x, 3, (-(3/4))*I*Log[I - Sinh[x]] - (1/4)*I*Log[I + Sinh[x]] - I/(2*(1 + I*Sinh[x]))} -{Coth[x]^1/(I + Csch[x]), x, 2, (-I)*Log[I - Sinh[x]]} -{Coth[x]^2/(I + Csch[x]), x, 3, (-I)*x - ArcTanh[Cosh[x]]} -{Coth[x]^3/(I + Csch[x]), x, 3, -Csch[x] - I*Log[Sinh[x]]} -{Coth[x]^4/(I + Csch[x]), x, 4, (-I)*x - (1/2)*ArcTanh[Cosh[x]] + (1/2)*Coth[x]*(2*I - Csch[x])} -{Coth[x]^5/(I + Csch[x]), x, 3, -Csch[x] + (1/2)*I*Csch[x]^2 - Csch[x]^3/3 - I*Log[Sinh[x]]} -{Coth[x]^6/(I + Csch[x]), x, 5, (-I)*x - (3/8)*ArcTanh[Cosh[x]] + (1/12)*Coth[x]^3*(4*I - 3*Csch[x]) + (1/8)*Coth[x]*(8*I - 3*Csch[x])} - - -(* ::Subsubsection::Closed:: *) -(*a^2+b^2!=0*) - - -{Tanh[x]^5/(a + b*Csch[x]), x, 11, -((b^5*ArcTan[Sinh[x]])/(a^2 + b^2)^3) - (b^3*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (3*b*ArcTan[Sinh[x]])/(8*(a^2 + b^2)) + (b^6*Log[a + b*Csch[x]])/(a*(a^2 + b^2)^3) + Log[Sinh[x]]/a - (a*(a^4 + 3*a^2*b^2 + 3*b^4)*Log[Tanh[x]])/(a^2 + b^2)^3 + (3*b*Sech[x]*Tanh[x])/(8*(a^2 + b^2)) - ((a*(a^2 + 2*b^2) - b^3*Csch[x])*Tanh[x]^2)/(2*(a^2 + b^2)^2) - ((a - b*Csch[x])*Tanh[x]^4)/(4*(a^2 + b^2))} -{Tanh[x]^4/(a + b*Csch[x]), x, 16, (a*b^2*x)/(a^2 + b^2)^2 + (b^4*x)/(a*(a^2 + b^2)^2) + (a*x)/(a^2 + b^2) + (2*b^5*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(5/2)) + (b^3*Sech[x])/(a^2 + b^2)^2 + (b*Sech[x])/(a^2 + b^2) - (b*Sech[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Tanh[x])/(a^2 + b^2)^2 - (a*Tanh[x])/(a^2 + b^2) - (a*Tanh[x]^3)/(3*(a^2 + b^2))} -{Tanh[x]^3/(a + b*Csch[x]), x, 8, -((b^3*ArcTan[Sinh[x]])/(a^2 + b^2)^2) - (b*ArcTan[Sinh[x]])/(2*(a^2 + b^2)) + (b^4*Log[a + b*Csch[x]])/(a*(a^2 + b^2)^2) + Log[Sinh[x]]/a - (a*(a^2 + 2*b^2)*Log[Tanh[x]])/(a^2 + b^2)^2 - ((a - b*Csch[x])*Tanh[x]^2)/(2*(a^2 + b^2))} -{Tanh[x]^2/(a + b*Csch[x]), x, 10, (a*x)/(a^2 + b^2) + (b^2*x)/(a*(a^2 + b^2)) + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(3/2)) + (b*Sech[x])/(a^2 + b^2) - (a*Tanh[x])/(a^2 + b^2)} -{Tanh[x]^1/(a + b*Csch[x]), x, 6, -((b*ArcTan[Sinh[x]])/(a^2 + b^2)) + (b^2*Log[a + b*Csch[x]])/(a*(a^2 + b^2)) + Log[Sinh[x]]/a - (a*Log[Tanh[x]])/(a^2 + b^2)} -{Coth[x]^1/(a + b*Csch[x]), x, 4, Log[a + b*Csch[x]]/a + Log[Sinh[x]]/a} -{Coth[x]^2/(a + b*Csch[x]), x, 8, x/a - ArcTanh[Cosh[x]]/b + (2*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b)} -{Coth[x]^3/(a + b*Csch[x]), x, 3, -(Csch[x]/b) + (1/a + a/b^2)*Log[a + b*Csch[x]] + Log[Sinh[x]]/a} -{Coth[x]^4/(a + b*Csch[x]), x, 7, x/a - ((2*a^2 + 3*b^2)*ArcTanh[Cosh[x]])/(2*b^3) + (2*(a^2 + b^2)^(3/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b^3) + (a*Coth[x])/b^2 - (Coth[x]*Csch[x])/(2*b)} -{Coth[x]^5/(a + b*Csch[x]), x, 3, -(((a^2 + 2*b^2)*Csch[x])/b^3) + (a*Csch[x]^2)/(2*b^2) - Csch[x]^3/(3*b) + ((a^2 + b^2)^2*Log[a + b*Csch[x]])/(a*b^4) + Log[Sinh[x]]/a} -{Coth[x]^6/(a + b*Csch[x]), x, 16, x/a - (3*ArcTanh[Cosh[x]])/(8*b) + ((a^2 + 3*b^2)*ArcTanh[Cosh[x]])/(2*b^3) - ((a^4 + 3*a^2*b^2 + 3*b^4)*ArcTanh[Cosh[x]])/b^5 + (2*(a^2 + b^2)^(5/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b^5) - (a*Coth[x])/b^2 + (a*(a^2 + 3*b^2)*Coth[x])/b^4 + (a*Coth[x]^3)/(3*b^2) + (3*Coth[x]*Csch[x])/(8*b) - ((a^2 + 3*b^2)*Coth[x]*Csch[x])/(2*b^3) - (Coth[x]*Csch[x]^3)/(4*b)} -{Coth[x]^7/(a + b*Csch[x]), x, 3, -(((a^4 + 3*a^2*b^2 + 3*b^4)*Csch[x])/b^5) + (a*(a^2 + 3*b^2)*Csch[x]^2)/(2*b^4) - ((a^2 + 3*b^2)*Csch[x]^3)/(3*b^3) + (a*Csch[x]^4)/(4*b^2) - Csch[x]^5/(5*b) + ((a^2 + b^2)^3*Log[a + b*Csch[x]])/(a*b^6) + Log[Sinh[x]]/a} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Csch[c+d x]^m*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) (Csch[a c+b c x]^2)^(m/2)*) - - -{E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(7/2), x, 6, -((32*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^6)) + (192*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(5*b*c*(1 - E^(2*c*(a + b*x)))^5) - (48*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4) + (64*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3)} -{E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(5/2), x, 6, (-4*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4) + (32*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3) - (8*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2)} -{E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(3/2), x, 4, (-2*E^(4*c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^2)} -{E^(c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2], x, 4, (Sqrt[Csch[a*c + b*c*x]^2]*Log[1 - E^(2*c*(a + b*x))]*Sinh[a*c + b*c*x])/(b*c)} -{E^(c*(a + b*x))/Sqrt[Csch[a*c + b*c*x]^2], x, 5, (E^(2*c*(a + b*x))*Csch[a*c + b*c*x])/(4*b*c*Sqrt[Csch[a*c + b*c*x]^2]) - (x*Csch[a*c + b*c*x])/(2*Sqrt[Csch[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Csch[a*c + b*c*x]^2)^(3/2), x, 6, Csch[a*c + b*c*x]/(16*b*c*E^(2*c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2]) - (3*E^(2*c*(a + b*x))*Csch[a*c + b*c*x])/(16*b*c*Sqrt[Csch[a*c + b*c*x]^2]) + (E^(4*c*(a + b*x))*Csch[a*c + b*c*x])/(32*b*c*Sqrt[Csch[a*c + b*c*x]^2]) + (3*x*Csch[a*c + b*c*x])/(8*Sqrt[Csch[a*c + b*c*x]^2])} -{E^(c*(a + b*x))/(Csch[a*c + b*c*x]^2)^(5/2), x, 6, Csch[a*c + b*c*x]/(128*b*c*E^(4*c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2]) - (5*Csch[a*c + b*c*x])/(64*b*c*E^(2*c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2]) + (5*E^(2*c*(a + b*x))*Csch[a*c + b*c*x])/(32*b*c*Sqrt[Csch[a*c + b*c*x]^2]) - (5*E^(4*c*(a + b*x))*Csch[a*c + b*c*x])/(128*b*c*Sqrt[Csch[a*c + b*c*x]^2]) + (E^(6*c*(a + b*x))*Csch[a*c + b*c*x])/(192*b*c*Sqrt[Csch[a*c + b*c*x]^2]) - (5*x*Csch[a*c + b*c*x])/(16*Sqrt[Csch[a*c + b*c*x]^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Csch[a+b Log[c x^n]]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Csch[b Log[c x^n]]^(p/2)*) - - -{x^5/Csch[2*Log[c*x]]^(1/2), x, 6, -((2*x^2)/(21*c^4*Sqrt[Csch[2*Log[c*x]]])) + x^6/(7*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(21*c^7*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])} -{x^4/Csch[2*Log[c*x]]^(1/2), x, 3, ((c^4 - 1/x^4)*x^5)/(6*c^4*Sqrt[Csch[2*Log[c*x]]])} -{x^3/Csch[2*Log[c*x]]^(1/2), x, 9, -(2/(5*c^4*Sqrt[Csch[2*Log[c*x]]])) + x^4/(5*Sqrt[Csch[2*Log[c*x]]]) - (2*EllipticE[ArcCsc[c*x], -1])/(5*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(5*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])} -{x^2/Csch[2*Log[c*x]]^(1/2), x, 6, x^3/(4*Sqrt[Csch[2*Log[c*x]]]) - ArcTanh[Sqrt[1 - 1/(c^4*x^4)]]/(4*c^4*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])} -{x^1/Csch[2*Log[c*x]]^(1/2), x, 5, x^2/(3*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(3*c^3*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])} -{x^0/Csch[2*Log[c*x]]^(1/2), x, 6, x/(2*Sqrt[Csch[2*Log[c*x]]]) + ArcCsc[c^2*x^2]/(2*c^2*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])} -{Csch[2*Log[c*x]]^(1/2)/x^1, x, 3, I*Sqrt[Csch[2*Log[c*x]]]*EllipticF[Pi/4 - I*Log[c*x], 2]*Sqrt[I*Sinh[2*Log[c*x]]]} -{Csch[2*Log[c*x]]^(1/2)/x^2, x, 5, (-(1/2))*c^2*Sqrt[1 - 1/(c^4*x^4)]*x*ArcCsc[c^2*x^2]*Sqrt[Csch[2*Log[c*x]]]} -{Csch[2*Log[c*x]]^(1/2)/x^3, x, 7, (-c^3)*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticE[ArcCsc[c*x], -1] + c^3*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticF[ArcCsc[c*x], -1]} -{Csch[2*Log[c*x]]^(1/2)/x^4, x, 3, (1/2)*(c^4 - 1/x^4)*x*Sqrt[Csch[2*Log[c*x]]]} -{Csch[2*Log[c*x]]^(1/2)/x^5, x, 5, (1/3)*(c^4 - 1/x^4)*Sqrt[Csch[2*Log[c*x]]] - (1/3)*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticF[ArcCsc[c*x], -1]} - - -{x^8/Csch[2*Log[c*x]]^(3/2), x, 8, x/(32*c^4*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2)) - x^5/(16*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2)) + x^9/(12*Csch[2*Log[c*x]]^(3/2)) + ArcTanh[Sqrt[1 - 1/(c^4*x^4)]]/(32*c^12*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^7/Csch[2*Log[c*x]]^(3/2), x, 7, 4/(77*c^4*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2)) - (6*x^4)/(77*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2)) + x^8/(11*Csch[2*Log[c*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(77*c^11*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^6/Csch[2*Log[c*x]]^(3/2), x, 3, ((c^4 - 1/x^4)*x^7)/(10*c^4*Csch[2*Log[c*x]]^(3/2))} -{x^5/Csch[2*Log[c*x]]^(3/2), x, 10, 4/(15*c^4*(c^4 - 1/x^4)*x^2*Csch[2*Log[c*x]]^(3/2)) - (2*x^2)/(15*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2)) + x^6/(9*Csch[2*Log[c*x]]^(3/2)) + (4*EllipticE[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^4/Csch[2*Log[c*x]]^(3/2), x, 7, -((3*x)/(16*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2))) + x^5/(8*Csch[2*Log[c*x]]^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(c^4*x^4)]])/(16*c^8*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^3/Csch[2*Log[c*x]]^(3/2), x, 6, -(2/(7*(c^4 - 1/x^4)*Csch[2*Log[c*x]]^(3/2))) + x^4/(7*Csch[2*Log[c*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(7*c^7*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^2/Csch[2*Log[c*x]]^(3/2), x, 7, -(1/(2*(c^4 - 1/x^4)*x*Csch[2*Log[c*x]]^(3/2))) + x^3/(6*Csch[2*Log[c*x]]^(3/2)) - ArcCsc[c^2*x^2]/(2*c^6*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^1/Csch[2*Log[c*x]]^(3/2), x, 9, -(6/(5*(c^4 - 1/x^4)*x^2*Csch[2*Log[c*x]]^(3/2))) + x^2/(5*Csch[2*Log[c*x]]^(3/2)) - (12*EllipticE[ArcCsc[c*x], -1])/(5*c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)) + (12*EllipticF[ArcCsc[c*x], -1])/(5*c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{x^0/Csch[2*Log[c*x]]^(3/2), x, 7, 3/(4*(c^4 - 1/x^4)*x^3*Csch[2*Log[c*x]]^(3/2)) + x/(4*Csch[2*Log[c*x]]^(3/2)) - (3*ArcTanh[Sqrt[1 - 1/(c^4*x^4)]])/(4*c^4*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))} -{Csch[2*Log[c*x]]^(3/2)/x^1, x, 4, (-Cosh[2*Log[c*x]])*Sqrt[Csch[2*Log[c*x]]] + (I*EllipticE[Pi/4 - I*Log[c*x], 2])/(Sqrt[Csch[2*Log[c*x]]]*Sqrt[I*Sinh[2*Log[c*x]]])} -{Csch[2*Log[c*x]]^(3/2)/x^2, x, 3, (-(1/2))*(c^4 - 1/x^4)*x^3*Csch[2*Log[c*x]]^(3/2)} -{Csch[2*Log[c*x]]^(3/2)/x^3, x, 5, (-(1/2))*(c^4 - 1/x^4)*x^2*Csch[2*Log[c*x]]^(3/2) + (1/2)*c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)*EllipticF[ArcCsc[c*x], -1]} -{Csch[2*Log[c*x]]^(3/2)/x^4, x, 6, (-(1/2))*(c^4 - 1/x^4)*x*Csch[2*Log[c*x]]^(3/2) + (1/2)*c^6*(1 - 1/(c^4*x^4))^(3/2)*x^3*ArcCsc[c^2*x^2]*Csch[2*Log[c*x]]^(3/2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Csch[a+b Log[c x^n]]^p*) - - -{Csch[a + b*Log[c*x^n]]^1, x, 4, -((2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + 1/n)/(2*b), (1/2)*(3 + 1/(b*n)), E^(2*a)*(c*x^n)^(2*b)])/(1 + b*n))} -{Csch[a + b*Log[c*x^n]]^2, x, 4, (4*E^(2*a)*x*(c*x^n)^(2*b)*Hypergeometric2F1[2, (1/2)*(2 + 1/(b*n)), (1/2)*(4 + 1/(b*n)), E^(2*a)*(c*x^n)^(2*b)])/(1 + 2*b*n)} -{Csch[a + b*Log[c*x^n]]^3, x, 4, -((8*E^(3*a)*x*(c*x^n)^(3*b)*Hypergeometric2F1[3, (3*b + 1/n)/(2*b), (1/2)*(5 + 1/(b*n)), E^(2*a)*(c*x^n)^(2*b)])/(1 + 3*b*n))} -{Csch[a + b*Log[c*x^n]]^4, x, 4, (16*E^(4*a)*x*(c*x^n)^(4*b)*Hypergeometric2F1[4, (1/2)*(4 + 1/(b*n)), (1/2)*(6 + 1/(b*n)), E^(2*a)*(c*x^n)^(2*b)])/(1 + 4*b*n)} - -{2*b^2*n^2*Csch[a + b*Log[c*x^n]]^3 - (1 - b^2*n^2)*Csch[a + b*Log[c*x^n]], x, -9, (-x)*Csch[a + b*Log[c*x^n]] - b*n*x*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]} - - -{Csch[a + 2*Log[c*x^(1/2)]]^3, x, 3, -((2*c^6)/(E^a*(c^4 - 1/(E^(2*a)*x^2))^2))} -{Csch[a + 2*Log[c/x^(1/2)]]^3, x, 4, (2*c^2)/(E^(3*a)*(E^(-2*a) - c^4/x^2)^2)} -{Csch[a + 1/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, -((E^(2*a)*(2 - p)*x*(1 - (c*x^n)^(2/(n*(2 - p)))/E^(2*a))*Csch[a - Log[c*x^n]/(n*(2 - p))]^p)/((c*x^n)^(2/(n*(2 - p)))*(2*(1 - p))))} -{Csch[a - 1/(n*(-2 + p))*Log[c*x^n]]^p, x, 3, ((2 - p)*x*(1 - 1/(E^(2*a)*(c*x^n)^(2/(n*(2 - p)))))*Csch[a + Log[c*x^n]/(n*(2 - p))]^p)/(2*(1 - p))} - - -{Csch[a + b*Log[c*x^n]]/x, x, 2, -(ArcTanh[Cosh[a + b*Log[c*x^n]]]/(b*n))} -{Csch[a + b*Log[c*x^n]]^2/x, x, 3, -(Coth[a + b*Log[c*x^n]]/(b*n))} -{Csch[a + b*Log[c*x^n]]^3/x, x, 3, ArcTanh[Cosh[a + b*Log[c*x^n]]]/(2*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(2*b*n)} -{Csch[a + b*Log[c*x^n]]^4/x, x, 3, Coth[a + b*Log[c*x^n]]/(b*n) - Coth[a + b*Log[c*x^n]]^3/(3*b*n)} -{Csch[a + b*Log[c*x^n]]^5/x, x, 4, -((3*ArcTanh[Cosh[a + b*Log[c*x^n]]])/(8*b*n)) + (3*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/(8*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^3)/(4*b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Csch[a+b Log[c x^n]]^(p/2)*) - - -{Csch[a + b*Log[c*x^n]]^(5/2)/x, x, 4, -((2*Cosh[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^(3/2))/(3*b*n)) + (2*I*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(3*b*n)} -{Csch[a + b*Log[c*x^n]]^(3/2)/x, x, 4, -((2*Cosh[a + b*Log[c*x^n]]*Sqrt[Csch[a + b*Log[c*x^n]]])/(b*n)) - (2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2])/(b*n*Sqrt[Csch[a + b*Log[c*x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])} -{Sqrt[Csch[a + b*Log[c*x^n]]]/x, x, 3, -((2*I*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n))} -{1/(x*Sqrt[Csch[a + b*Log[c*x^n]]]), x, 3, -((2*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2])/(b*n*Sqrt[Csch[a + b*Log[c*x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))} -{1/(x*Csch[a + b*Log[c*x^n]]^(3/2)), x, 4, (2*Cosh[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Csch[a + b*Log[c*x^n]]]) + (2*I*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(3*b*n)} -{1/(x*Csch[a + b*Log[c*x^n]]^(5/2)), x, 4, (2*Cosh[a + b*Log[c*x^n]])/(5*b*n*Csch[a + b*Log[c*x^n]]^(3/2)) + (6*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*Log[c*x^n]), 2])/(5*b*n*Sqrt[Csch[a + b*Log[c*x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])} diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.7 (d hyper)^m (a+b (c csch)^n)^p.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.7 (d hyper)^m (a+b (c csch)^n)^p.m deleted file mode 100644 index 42415b7..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.6 Hyperbolic cosecant/6.6.7 (d hyper)^m (a+b (c csch)^n)^p.m +++ /dev/null @@ -1,63 +0,0 @@ -(* ::Package:: *) - -(* ::Section:: *) -(*Integrands of the form Hyper[c+d x]^m (a+b Csch[c+d x]^2)^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sinh[c+d x]^m (a+b Csch[c+d x]^2)^n*) - - -(* ::Subsubsection::Closed:: *) -(*n*) - - -{(a + b*Csch[c + d*x]^2)^4, x, 4, a^4*x - ((2*a - b)*b*(2*a^2 - 2*a*b + b^2)*Coth[c + d*x])/d - (b^2*(6*a^2 - 8*a*b + 3*b^2)*Coth[c + d*x]^3)/(3*d) - ((4*a - 3*b)*b^3*Coth[c + d*x]^5)/(5*d) - (b^4*Coth[c + d*x]^7)/(7*d)} -{(a + b*Csch[c + d*x]^2)^3, x, 4, a^3*x - (b*(3*a^2 - 3*a*b + b^2)*Coth[c + d*x])/d - ((3*a - 2*b)*b^2*Coth[c + d*x]^3)/(3*d) - (b^3*Coth[c + d*x]^5)/(5*d)} -{(a + b*Csch[c + d*x]^2)^2, x, 4, a^2*x - ((2*a - b)*b*Coth[c + d*x])/d - (b^2*Coth[c + d*x]^3)/(3*d)} -{(a + b*Csch[c + d*x]^2)^1, x, 3, a*x - (b*Coth[c + d*x])/d} -{1/(a + b*Csch[c + d*x]^2)^1, x, 3, x/a - (Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(a*Sqrt[a - b]*d)} -{1/(a + b*Csch[c + d*x]^2)^2, x, 5, x/a^2 - ((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(2*a^2*(a - b)^(3/2)*d) + (b*Coth[c + d*x])/(2*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2))} -{1/(a + b*Csch[c + d*x]^2)^3, x, 6, x/a^3 - (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(8*a^3*(a - b)^(5/2)*d) + (b*Coth[c + d*x])/(4*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^2) + ((7*a - 4*b)*b*Coth[c + d*x])/(8*a^2*(a - b)^2*d*(a - b + b*Coth[c + d*x]^2))} -{1/(a + b*Csch[c + d*x]^2)^4, x, 7, x/a^4 - (Sqrt[b]*(35*a^3 - 70*a^2*b + 56*a*b^2 - 16*b^3)*ArcTan[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[b]])/(16*a^4*(a - b)^(7/2)*d) + (b*Coth[c + d*x])/(6*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^3) + ((11*a - 6*b)*b*Coth[c + d*x])/(24*a^2*(a - b)^2*d*(a - b + b*Coth[c + d*x]^2)^2) + (b*(19*a^2 - 22*a*b + 8*b^2)*Coth[c + d*x])/(16*a^3*(a - b)^3*d*(a - b + b*Coth[c + d*x]^2))} - - -(* ::Subsubsection::Closed:: *) -(*n/2*) - - -{(a + b*Csch[c + d*x]^2)^(5/2), x, 8, (a^(5/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTanh[(Sqrt[b]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/(8*d) - ((7*a - 3*b)*b*Coth[c + d*x]*Sqrt[a - b + b*Coth[c + d*x]^2])/(8*d) - (b*Coth[c + d*x]*(a - b + b*Coth[c + d*x]^2)^(3/2))/(4*d)} -{(a + b*Csch[c + d*x]^2)^(3/2), x, 7, (a^(3/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/(2*d) - (b*Coth[c + d*x]*Sqrt[a - b + b*Coth[c + d*x]^2])/(2*d)} -{(a + b*Csch[c + d*x]^2)^(1/2), x, 6, (Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - (Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d} -{1/(a + b*Csch[c + d*x]^2)^(1/2), x, 3, ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + b*Csch[c + d*x]^2]]/(Sqrt[a]*d), ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(Sqrt[a]*d)} -{1/(a + b*Csch[c + d*x]^2)^(3/2), x, 4, ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(a^(3/2)*d) + (b*Coth[c + d*x])/(a*(a - b)*d*Sqrt[a - b + b*Coth[c + d*x]^2])} -{1/(a + b*Csch[c + d*x]^2)^(5/2), x, 6, ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(a^(5/2)*d) + (b*Coth[c + d*x])/(3*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^(3/2)) + ((5*a - 3*b)*b*Coth[c + d*x])/(3*a^2*(a - b)^2*d*Sqrt[a - b + b*Coth[c + d*x]^2])} -{1/(a + b*Csch[c + d*x]^2)^(7/2), x, 7, ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]]/(a^(7/2)*d) + (b*Coth[c + d*x])/(5*a*(a - b)*d*(a - b + b*Coth[c + d*x]^2)^(5/2)) + ((9*a - 5*b)*b*Coth[c + d*x])/(15*a^2*(a - b)^2*d*(a - b + b*Coth[c + d*x]^2)^(3/2)) + (b*(33*a^2 - 40*a*b + 15*b^2)*Coth[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a - b + b*Coth[c + d*x]^2])} - - -{(1 + Csch[x]^2)^(3/2), x, 4, (-(1/2))*(Coth[x]^2)^(3/2)*Tanh[x] + Sqrt[Coth[x]^2]*Log[Sinh[x]]*Tanh[x]} -{Sqrt[1 + Csch[x]^2], x, 3, Sqrt[Coth[x]^2]*Log[Sinh[x]]*Tanh[x]} -{1/Sqrt[1 + Csch[x]^2], x, 3, (Coth[x]*Log[Cosh[x]])/Sqrt[Coth[x]^2]} - - -{(1 - Csch[x]^2)^(3/2), x, 6, 2*ArcSin[Coth[x]/Sqrt[2]] + ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]] + (1/2)*Coth[x]*Sqrt[2 - Coth[x]^2]} -{Sqrt[1 - Csch[x]^2], x, 5, ArcSin[Coth[x]/Sqrt[2]] + ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]]} -{1/Sqrt[1 - Csch[x]^2], x, 3, ArcTanh[Coth[x]/Sqrt[2 - Coth[x]^2]]} - - -{(-1 + Csch[x]^2)^(3/2), x, 7, ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]] + 2*ArcTanh[Coth[x]/Sqrt[-2 + Coth[x]^2]] - (1/2)*Coth[x]*Sqrt[-2 + Coth[x]^2]} -{Sqrt[-1 + Csch[x]^2], x, 6, -ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]] - ArcTanh[Coth[x]/Sqrt[-2 + Coth[x]^2]]} -{1/Sqrt[-1 + Csch[x]^2], x, 3, ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]]} - - -{(-1 - Csch[x]^2)^(3/2), x, 4, (1/2)*Coth[x]*Sqrt[-Coth[x]^2] - Sqrt[-Coth[x]^2]*Log[Sinh[x]]*Tanh[x]} -{Sqrt[-1 - Csch[x]^2], x, 3, Sqrt[-Coth[x]^2]*Log[Sinh[x]]*Tanh[x]} -{1/Sqrt[-1 - Csch[x]^2], x, 3, (Coth[x]*Log[Cosh[x]])/Sqrt[-Coth[x]^2]} - - -(* ::Subsection:: *) -(*Integrands of the form Cosh[c+d x]^m (a+b Csch[c+d x]^2)^n*) - - -(* ::Subsection:: *) -(*Integrands of the form Tanh[c+d x]^m (a+b Csch[c+d x]^2)^n*) diff --git a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.7 Miscellaneous/6.7.1 Hyperbolic functions.m b/test/methods/rule_based/test_files/6 Hyperbolic functions/6.7 Miscellaneous/6.7.1 Hyperbolic functions.m deleted file mode 100644 index a863d19..0000000 --- a/test/methods/rule_based/test_files/6 Hyperbolic functions/6.7 Miscellaneous/6.7.1 Hyperbolic functions.m +++ /dev/null @@ -1,1920 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Functions*) - - -(* ::Section::Closed:: *) -(*Rectification problems*) - - -(* Following integrands are equal. *) -{2/(-1 + 3*Cosh[4 + 6*x]), x, 3, ArcTan[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2])} -{1/(2*Sinh[2 + 3*x]^2 + Cosh[2 + 3*x]^2), x, 2, ArcTan[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2])} -{Sech[2 + 3*x]^2/(1 + 2*Tanh[2 + 3*x]^2), x, 2, ArcTan[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2])} -{Csch[2 + 3*x]^2/(2 + Coth[2 + 3*x]^2), x, 2, ArcTan[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2])} - - -{Csch[2 + 3*x]^2/(2 - Coth[2 + 3*x]^2), x, 2, -(ArcTanh[Sqrt[2]*Tanh[2 + 3*x]]/(3*Sqrt[2]))} - -{Csch[2 + 3*x]^2/(1 + 2*Coth[2 + 3*x]^2), x, 2, ArcTan[Tanh[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])} - -{Csch[2 + 3*x]^2/(1 - 2*Coth[2 + 3*x]^2), x, 2, -(ArcTanh[Tanh[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Hyper[a+b x]^n Hyper[c+d x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[a+b x]^m Hyper[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Cosh[a+b x]^m Sinh[a+b x]^n*) - - -{Cosh[a + b*x]*Sinh[a + b*x], x, 2, Sinh[a + b*x]^2/(2*b)} -{Cosh[a + b*x]*Sinh[a + b*x]^n, x, 2, Sinh[a + b*x]^(1 + n)/(b*(1 + n))} -{Cosh[a + b*x]^3*Sinh[a + b*x]^n, x, 3, Sinh[a + b*x]^(1 + n)/(b*(1 + n)) + Sinh[a + b*x]^(3 + n)/(b*(3 + n))} -{Cosh[a + b*x]^5*Sinh[a + b*x]^n, x, 3, Sinh[a + b*x]^(1 + n)/(b*(1 + n)) + (2*Sinh[a + b*x]^(3 + n))/(b*(3 + n)) + Sinh[a + b*x]^(5 + n)/(b*(5 + n))} - -{Cosh[a + b*x]^m*Sinh[a + b*x], x, 2, Cosh[a + b*x]^(1 + m)/(b*(1 + m))} -{Cosh[a + b*x]^m*Sinh[a + b*x]^3, x, 3, -(Cosh[a + b*x]^(1 + m)/(b*(1 + m))) + Cosh[a + b*x]^(3 + m)/(b*(3 + m))} -{Cosh[a + b*x]^m*Sinh[a + b*x]^5, x, 3, Cosh[a + b*x]^(1 + m)/(b*(1 + m)) - (2*Cosh[a + b*x]^(3 + m))/(b*(3 + m)) + Cosh[a + b*x]^(5 + m)/(b*(5 + m))} - -{Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 3, -(x/8) - (Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} -{Cosh[a + b*x]^2*Sinh[a + b*x]^4, x, 4, x/16 + (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x]^3)/(6*b)} -{Cosh[a + b*x]^2*Sinh[a + b*x]^6, x, 5, -((5*x)/128) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x]^3)/(48*b) + (Cosh[a + b*x]^3*Sinh[a + b*x]^5)/(8*b)} - -{Cosh[a + b*x]^4*Sinh[a + b*x]^2, x, 4, -(x/16) - (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(24*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(6*b)} -{Cosh[a + b*x]^4*Sinh[a + b*x]^4, x, 5, (3*x)/128 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(8*b)} -{Cosh[a + b*x]^4*Sinh[a + b*x]^6, x, 6, -((3*x)/256) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(32*b) - (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^5)/(10*b)} - -{Cosh[a + b*x]^6*Sinh[a + b*x]^2, x, 5, -((5*x)/128) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(192*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(48*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(8*b)} -{Cosh[a + b*x]^6*Sinh[a + b*x]^4, x, 6, (3*x)/256 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(160*b) - (3*Cosh[a + b*x]^7*Sinh[a + b*x])/(80*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(10*b)} -{Cosh[a + b*x]^6*Sinh[a + b*x]^6, x, 7, -((5*x)/1024) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(1024*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(1536*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(384*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(24*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^5)/(12*b)} - - -{Csch[a + b*x]*Sech[a + b*x], x, 2, Log[Tanh[a + b*x]]/b} -{Csch[a + b*x]*Sech[a + b*x]^2, x, 3, -(ArcTanh[Cosh[a + b*x]]/b) + Sech[a + b*x]/b} -{Csch[a + b*x]*Sech[a + b*x]^3, x, 3, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)} -{Csch[a + b*x]*Sech[a + b*x]^4, x, 4, -(ArcTanh[Cosh[a + b*x]]/b) + Sech[a + b*x]/b + Sech[a + b*x]^3/(3*b)} -{Csch[a + b*x]*Sech[a + b*x]^5, x, 4, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/b + Tanh[a + b*x]^4/(4*b)} - -{Csch[a + b*x]^2*Sech[a + b*x], x, 3, -(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b} -{Csch[a + b*x]^2*Sech[a + b*x]^2, x, 3, -(Coth[a + b*x]/b) - Tanh[a + b*x]/b} -{Csch[a + b*x]^2*Sech[a + b*x]^3, x, 4, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) - (3*Csch[a + b*x])/(2*b) + (Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)} -{Csch[a + b*x]^2*Sech[a + b*x]^4, x, 3, -(Coth[a + b*x]/b) - (2*Tanh[a + b*x])/b + Tanh[a + b*x]^3/(3*b)} -{Csch[a + b*x]^2*Sech[a + b*x]^5, x, 5, -((15*ArcTan[Sinh[a + b*x]])/(8*b)) - (15*Csch[a + b*x])/(8*b) + (5*Csch[a + b*x]*Sech[a + b*x]^2)/(8*b) + (Csch[a + b*x]*Sech[a + b*x]^4)/(4*b)} - -{Csch[a + b*x]^3*Sech[a + b*x], x, 3, -(Coth[a + b*x]^2/(2*b)) - Log[Tanh[a + b*x]]/b} -{Csch[a + b*x]^3*Sech[a + b*x]^2, x, 4, (3*ArcTanh[Cosh[a + b*x]])/(2*b) - (3*Sech[a + b*x])/(2*b) - (Csch[a + b*x]^2*Sech[a + b*x])/(2*b)} -{Csch[a + b*x]^3*Sech[a + b*x]^3, x, 4, -(Coth[a + b*x]^2/(2*b)) - (2*Log[Tanh[a + b*x]])/b + Tanh[a + b*x]^2/(2*b)} -{Csch[a + b*x]^3*Sech[a + b*x]^4, x, 5, (5*ArcTanh[Cosh[a + b*x]])/(2*b) - (5*Sech[a + b*x])/(2*b) - (5*Sech[a + b*x]^3)/(6*b) - (Csch[a + b*x]^2*Sech[a + b*x]^3)/(2*b)} -{Csch[a + b*x]^3*Sech[a + b*x]^5, x, 4, -(Coth[a + b*x]^2/(2*b)) - (3*Log[Tanh[a + b*x]])/b + (3*Tanh[a + b*x]^2)/(2*b) - Tanh[a + b*x]^4/(4*b)} - -{Csch[a + b*x]^4*Sech[a + b*x], x, 4, ArcTan[Sinh[a + b*x]]/b + Csch[a + b*x]/b - Csch[a + b*x]^3/(3*b)} -{Csch[a + b*x]^4*Sech[a + b*x]^2, x, 3, (2*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*b) + Tanh[a + b*x]/b} -{Csch[a + b*x]^4*Sech[a + b*x]^3, x, 5, (5*ArcTan[Sinh[a + b*x]])/(2*b) + (5*Csch[a + b*x])/(2*b) - (5*Csch[a + b*x]^3)/(6*b) + (Csch[a + b*x]^3*Sech[a + b*x]^2)/(2*b)} -{Csch[a + b*x]^4*Sech[a + b*x]^4, x, 3, (3*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*b) + (3*Tanh[a + b*x])/b - Tanh[a + b*x]^3/(3*b)} -{Csch[a + b*x]^4*Sech[a + b*x]^5, x, 6, (35*ArcTan[Sinh[a + b*x]])/(8*b) + (35*Csch[a + b*x])/(8*b) - (35*Csch[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]^3*Sech[a + b*x]^2)/(8*b) + (Csch[a + b*x]^3*Sech[a + b*x]^4)/(4*b)} - -{Csch[a + b*x]^5*Sech[a + b*x], x, 4, Coth[a + b*x]^2/b - Coth[a + b*x]^4/(4*b) + Log[Tanh[a + b*x]]/b} -{Csch[a + b*x]^5*Sech[a + b*x]^2, x, 5, -((15*ArcTanh[Cosh[a + b*x]])/(8*b)) + (15*Sech[a + b*x])/(8*b) + (5*Csch[a + b*x]^2*Sech[a + b*x])/(8*b) - (Csch[a + b*x]^4*Sech[a + b*x])/(4*b)} -{Csch[a + b*x]^5*Sech[a + b*x]^3, x, 4, (3*Coth[a + b*x]^2)/(2*b) - Coth[a + b*x]^4/(4*b) + (3*Log[Tanh[a + b*x]])/b - Tanh[a + b*x]^2/(2*b)} -{Csch[a + b*x]^5*Sech[a + b*x]^4, x, 6, -((35*ArcTanh[Cosh[a + b*x]])/(8*b)) + (35*Sech[a + b*x])/(8*b) + (35*Sech[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]^2*Sech[a + b*x]^3)/(8*b) - (Csch[a + b*x]^4*Sech[a + b*x]^3)/(4*b)} -{Csch[a + b*x]^5*Sech[a + b*x]^5, x, 4, (2*Coth[a + b*x]^2)/b - Coth[a + b*x]^4/(4*b) + (6*Log[Tanh[a + b*x]])/b - (2*Tanh[a + b*x]^2)/b + Tanh[a + b*x]^4/(4*b)} - - -{Sinh[a + b*x]^(7/2)/Cosh[a + b*x]^(7/2), x, 6, -(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b - (2*Sqrt[Sinh[a + b*x]])/(b*Sqrt[Cosh[a + b*x]]) - (2*Sinh[a + b*x]^(5/2))/(5*b*Cosh[a + b*x]^(5/2))} -{Sinh[a + b*x]^(5/2)/Cosh[a + b*x]^(5/2), x, 5, -(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b - (2*Sinh[a + b*x]^(3/2))/(3*b*Cosh[a + b*x]^(3/2))} -{Sinh[a + b*x]^(3/2)/Cosh[a + b*x]^(3/2), x, 5, -(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b - (2*Sqrt[Sinh[a + b*x]])/(b*Sqrt[Cosh[a + b*x]])} -{Sinh[a + b*x]^(1/2)/Cosh[a + b*x]^(1/2), x, 4, -(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b} -{Cosh[a + b*x]^(1/2)/Sinh[a + b*x]^(1/2), x, 4, -(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b} -{Cosh[a + b*x]^(3/2)/Sinh[a + b*x]^(3/2), x, 5, -(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b - (2*Sqrt[Cosh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])} -{Cosh[a + b*x]^(5/2)/Sinh[a + b*x]^(5/2), x, 5, -(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b - (2*Cosh[a + b*x]^(3/2))/(3*b*Sinh[a + b*x]^(3/2))} -{Cosh[a + b*x]^(7/2)/Sinh[a + b*x]^(7/2), x, 6, -(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b - (2*Cosh[a + b*x]^(5/2))/(5*b*Sinh[a + b*x]^(5/2)) - (2*Sqrt[Cosh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])} - - -{Sinh[a + b*x]^(7/3)/Cosh[a + b*x]^(7/3), x, 9, -((Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(2/3))/Cosh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(2*b) + Log[1 + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3) + Sinh[a + b*x]^(4/3)/Cosh[a + b*x]^(4/3)]/(4*b) - (3*Sinh[a + b*x]^(4/3))/(4*b*Cosh[a + b*x]^(4/3))} -{Sinh[a + b*x]^(5/3)/Cosh[a + b*x]^(5/3), x, 9, -((Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(2*b) + Log[1 + Cosh[a + b*x]^(4/3)/Sinh[a + b*x]^(4/3) + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(4*b) - (3*Sinh[a + b*x]^(2/3))/(2*b*Cosh[a + b*x]^(2/3))} -{Sinh[a + b*x]^(4/3)/Cosh[a + b*x]^(4/3), x, 12, (Sqrt[3]*ArcTan[(1 - (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) - (Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) + ArcTanh[Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/b - Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) - Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b) + Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) + Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b) - (3*Sinh[a + b*x]^(1/3))/(b*Cosh[a + b*x]^(1/3))} -{Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3), x, 11, (Sqrt[3]*ArcTan[(1 - (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) - (Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) + ArcTanh[Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3)]/b - Log[1 - Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(4*b) + Log[1 + Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(4*b)} -{Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3), x, 8, -((Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(2/3))/Cosh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(2*b) + Log[1 + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3) + Sinh[a + b*x]^(4/3)/Cosh[a + b*x]^(4/3)]/(4*b)} -{Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3), x, 8, -((Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(2*b) + Log[1 + Cosh[a + b*x]^(4/3)/Sinh[a + b*x]^(4/3) + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(4*b)} -{Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3), x, 11, (Sqrt[3]*ArcTan[(1 - (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) - (Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) + ArcTanh[Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/b - Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) - Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b) + Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) + Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b)} -{Cosh[a + b*x]^(4/3)/Sinh[a + b*x]^(4/3), x, 12, (Sqrt[3]*ArcTan[(1 - (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) - (Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(1/3))/Cosh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) + ArcTanh[Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3)]/b - Log[1 - Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(4*b) + Log[1 + Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3) + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(4*b) - (3*Cosh[a + b*x]^(1/3))/(b*Sinh[a + b*x]^(1/3))} -{Cosh[a + b*x]^(5/3)/Sinh[a + b*x]^(5/3), x, 9, -((Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(2/3))/Cosh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(2*b) + Log[1 + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3) + Sinh[a + b*x]^(4/3)/Cosh[a + b*x]^(4/3)]/(4*b) - (3*Cosh[a + b*x]^(2/3))/(2*b*Sinh[a + b*x]^(2/3))} -{Cosh[a + b*x]^(7/3)/Sinh[a + b*x]^(7/3), x, 9, -((Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(2*b) + Log[1 + Cosh[a + b*x]^(4/3)/Sinh[a + b*x]^(4/3) + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(4*b) - (3*Cosh[a + b*x]^(4/3))/(4*b*Sinh[a + b*x]^(4/3))} - - -{Cosh[x]^(2/3)/Sinh[x]^(8/3), x, 1, -((3*Cosh[x]^(5/3))/(5*Sinh[x]^(5/3)))} -{Sinh[x]^(2/3)/Cosh[x]^(8/3), x, 1, (3*Sinh[x]^(5/3))/(5*Cosh[x]^(5/3))} - -{Cosh[x]*Csch[x]^(7/3), x, 2, (-3*Csch[x]^(4/3))/4} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Hyper[a+b x]^m Tanh[a+b x]^n*) - - -{Sinh[a + b*x]*Tanh[a + b*x], x, 3, -(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b} -{Sinh[a + b*x]*Tanh[a + b*x]^2, x, 3, Cosh[a + b*x]/b + Sech[a + b*x]/b} -{Sinh[a + b*x]*Tanh[a + b*x]^3, x, 4, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) + (3*Sinh[a + b*x])/(2*b) - (Sinh[a + b*x]*Tanh[a + b*x]^2)/(2*b)} -{Sinh[a + b*x]*Tanh[a + b*x]^4, x, 3, Cosh[a + b*x]/b + (2*Sech[a + b*x])/b - Sech[a + b*x]^3/(3*b)} - -{Sinh[a + b*x]^2*Tanh[a + b*x], x, 3, Cosh[a + b*x]^2/(2*b) - Log[Cosh[a + b*x]]/b} -{Sinh[a + b*x]^2*Tanh[a + b*x]^2, x, 4, -((3*x)/2) + (3*Tanh[a + b*x])/(2*b) + (Sinh[a + b*x]^2*Tanh[a + b*x])/(2*b)} -{Sinh[a + b*x]^2*Tanh[a + b*x]^3, x, 4, Cosh[a + b*x]^2/(2*b) - (2*Log[Cosh[a + b*x]])/b - Sech[a + b*x]^2/(2*b)} - -{Sinh[a + b*x]^3*Tanh[a + b*x], x, 4, ArcTan[Sinh[a + b*x]]/b - Sinh[a + b*x]/b + Sinh[a + b*x]^3/(3*b)} -{Sinh[a + b*x]^3*Tanh[a + b*x]^2, x, 3, -((2*Cosh[a + b*x])/b) + Cosh[a + b*x]^3/(3*b) - Sech[a + b*x]/b} -{Sinh[a + b*x]^3*Tanh[a + b*x]^3, x, 5, (5*ArcTan[Sinh[a + b*x]])/(2*b) - (5*Sinh[a + b*x])/(2*b) + (5*Sinh[a + b*x]^3)/(6*b) - (Sinh[a + b*x]^3*Tanh[a + b*x]^2)/(2*b)} - -{Sinh[a + b*x]^4*Tanh[a + b*x], x, 4, -(Cosh[a + b*x]^2/b) + Cosh[a + b*x]^4/(4*b) + Log[Cosh[a + b*x]]/b} - - -{Sech[a + b*x]*Tanh[a + b*x], x, 2, -Sech[a + b*x]/b} -{Sech[a + b*x]^2*Tanh[a + b*x], x, 2, -(Sech[a + b*x]^2/(2*b))} -{Sech[a + b*x]^n*Tanh[a + b*x], x, 2, -Sech[a + b*x]^n/(b*n)} - -{Sech[a + b*x]^2*Tanh[a + b*x]^2, x, 2, Tanh[a + b*x]^3/(3*b)} -{Sech[a + b*x]^2*Tanh[a + b*x]^3, x, 2, Tanh[a + b*x]^4/(4*b)} -{Sech[a + b*x]^2*Tanh[a + b*x]^n, x, 2, Tanh[a + b*x]^(1 + n)/(b*(1 + n))} - -{Sech[a + b*x]^1*Tanh[a + b*x]^3, x, 2, -(Sech[a + b*x]/b) + Sech[a + b*x]^3/(3*b)} -{Sech[a + b*x]^3*Tanh[a + b*x]^3, x, 3, -(Sech[a + b*x]^3/(3*b)) + Sech[a + b*x]^5/(5*b)} -{Sech[a + b*x]^n*Tanh[a + b*x]^3, x, 3, -(Sech[a + b*x]^n/(b*n)) + Sech[a + b*x]^(2 + n)/(b*(2 + n))} - -{Sech[a + b*x]^4*Tanh[a + b*x]^2, x, 3, Tanh[a + b*x]^3/(3*b) - Tanh[a + b*x]^5/(5*b)} -{Sech[a + b*x]^4*Sqrt[Tanh[a + b*x]], x, 3, (2*Tanh[a + b*x]^(3/2))/(3*b) - (2*Tanh[a + b*x]^(7/2))/(7*b)} -{Sech[a + b*x]^4*Tanh[a + b*x]^n, x, 3, Tanh[a + b*x]^(1 + n)/(b*(1 + n)) - Tanh[a + b*x]^(3 + n)/(b*(3 + n))} - -{Sech[a + b*x]*Tanh[a + b*x]^2, x, 2, ArcTan[Sinh[a + b*x]]/(2*b) - (Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{Sech[a + b*x]*Tanh[a + b*x]^4, x, 3, (3*ArcTan[Sinh[a + b*x]])/(8*b) - (3*Sech[a + b*x]*Tanh[a + b*x])/(8*b) - (Sech[a + b*x]*Tanh[a + b*x]^3)/(4*b)} - -{Sech[a + b*x]^3*Tanh[a + b*x]^2, x, 3, ArcTan[Sinh[a + b*x]]/(8*b) + (Sech[a + b*x]*Tanh[a + b*x])/(8*b) - (Sech[a + b*x]^3*Tanh[a + b*x])/(4*b)} - -{Sech[x]*Tanh[x]^5, x, 3, -Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5} -{Sech[x]^7*Tanh[x]^5, x, 3, (-(1/7))*Sech[x]^7 + (2*Sech[x]^9)/9 - Sech[x]^11/11} -{Sech[x]^3*Tanh[x]^4, x, 4, (1/16)*ArcTan[Sinh[x]] + (1/16)*Sech[x]*Tanh[x] - (1/8)*Sech[x]^3*Tanh[x] - (1/6)*Sech[x]^3*Tanh[x]^3} -{Sech[x]^5*Tanh[x]^2, x, 4, (1/16)*ArcTan[Sinh[x]] + (1/16)*Sech[x]*Tanh[x] + (1/24)*Sech[x]^3*Tanh[x] - (1/6)*Sech[x]^5*Tanh[x]} -{Sech[x]^8*Tanh[x]^6, x, 3, Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Hyper[a+b x]^m Coth[a+b x]^n*) - - -{Cosh[a + b*x]*Coth[a + b*x], x, 3, -(ArcTanh[Cosh[a + b*x]]/b) + Cosh[a + b*x]/b} -{Cosh[a + b*x]*Coth[a + b*x]^2, x, 3, -(Csch[a + b*x]/b) + Sinh[a + b*x]/b} -{Cosh[a + b*x]*Coth[a + b*x]^3, x, 4, -((3*ArcTanh[Cosh[a + b*x]])/(2*b)) + (3*Cosh[a + b*x])/(2*b) - (Cosh[a + b*x]*Coth[a + b*x]^2)/(2*b)} -{Cosh[a + b*x]*Coth[a + b*x]^4, x, 3, -((2*Csch[a + b*x])/b) - Csch[a + b*x]^3/(3*b) + Sinh[a + b*x]/b} - -{Cosh[a + b*x]^2*Coth[a + b*x], x, 3, Log[Sinh[a + b*x]]/b + Sinh[a + b*x]^2/(2*b)} -{Cosh[a + b*x]^2*Coth[a + b*x]^2, x, 4, (3*x)/2 - (3*Coth[a + b*x])/(2*b) + (Cosh[a + b*x]^2*Coth[a + b*x])/(2*b)} -{Cosh[a + b*x]^2*Coth[a + b*x]^3, x, 4, -(Csch[a + b*x]^2/(2*b)) + (2*Log[Sinh[a + b*x]])/b + Sinh[a + b*x]^2/(2*b)} - -{Cosh[a + b*x]^3*Coth[a + b*x], x, 4, -(ArcTanh[Cosh[a + b*x]]/b) + Cosh[a + b*x]/b + Cosh[a + b*x]^3/(3*b)} -{Cosh[a + b*x]^3*Coth[a + b*x]^2, x, 3, -(Csch[a + b*x]/b) + (2*Sinh[a + b*x])/b + Sinh[a + b*x]^3/(3*b)} -{Cosh[a + b*x]^3*Coth[a + b*x]^3, x, 5, -((5*ArcTanh[Cosh[a + b*x]])/(2*b)) + (5*Cosh[a + b*x])/(2*b) + (5*Cosh[a + b*x]^3)/(6*b) - (Cosh[a + b*x]^3*Coth[a + b*x]^2)/(2*b)} - -{Cosh[a + b*x]^4*Coth[a + b*x], x, 4, Log[Sinh[a + b*x]]/b + Sinh[a + b*x]^2/b + Sinh[a + b*x]^4/(4*b)} - - -{Coth[a + b*x]*Csch[a + b*x], x, 2, -(Csch[a + b*x]/b)} -{Coth[a + b*x]*Csch[a + b*x]^2, x, 2, -(Csch[a + b*x]^2/(2*b))} -{Coth[a + b*x]*Csch[a + b*x]^n, x, 2, -Csch[a + b*x]^n/(b*n)} - -{Coth[a + b*x]^2*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^3/(3*b)} -{Coth[a + b*x]^3*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^4/(4*b)} -{Coth[a + b*x]^n*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^(1 + n)/(b*(1 + n))} - -{Coth[a + b*x]^3*Csch[a + b*x], x, 2, -(Csch[a + b*x]/b) - Csch[a + b*x]^3/(3*b)} -{Coth[a + b*x]^3*Csch[a + b*x]^3, x, 3, -(Csch[a + b*x]^3/(3*b)) - Csch[a + b*x]^5/(5*b)} -{Coth[a + b*x]^3*Csch[a + b*x]^n, x, 3, -(Csch[a + b*x]^n/(b*n)) - Csch[a + b*x]^(2 + n)/(b*(2 + n))} - -{Coth[a + b*x]^2*Csch[a + b*x], x, 2, -(ArcTanh[Cosh[a + b*x]]/(2*b)) - (Coth[a + b*x]*Csch[a + b*x])/(2*b)} -{Coth[a + b*x]^2*Csch[a + b*x]^3, x, 3, ArcTanh[Cosh[a + b*x]]/(8*b) - (Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]*Csch[a + b*x]^3)/(4*b)} - -{Coth[a + b*x]^4*Csch[a + b*x], x, 3, -((3*ArcTanh[Cosh[a + b*x]])/(8*b)) - (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]^3*Csch[a + b*x])/(4*b)} - -{Coth[x]^2*Csch[x]^4, x, 3, Coth[x]^3/3 - Coth[x]^5/5} -{Coth[x]^3*Csch[x]^4, x, 3, (-(1/4))*Csch[x]^4 - Csch[x]^6/6} -{Coth[x]^n*Csch[x]^4, x, 3, Coth[x]^(1 + n)/(1 + n) - Coth[x]^(3 + n)/(3 + n)} - -{Coth[x]^4*Csch[x]^3, x, 4, (1/16)*ArcTanh[Cosh[x]] - (1/16)*Coth[x]*Csch[x] - (1/8)*Coth[x]*Csch[x]^3 - (1/6)*Coth[x]^3*Csch[x]^3} -{Coth[x]^4*Csch[x]^6, x, 3, (-(1/5))*Coth[x]^5 + (2*Coth[x]^7)/7 - Coth[x]^9/9} -{Coth[6*x]^5*Csch[6*x], x, 3, (-(1/6))*Csch[6*x] - (1/9)*Csch[6*x]^3 - (1/30)*Csch[6*x]^5} -{Coth[x]^7*Csch[x]^3, x, 3, (-(1/3))*Csch[x]^3 - (3*Csch[x]^5)/5 - (3*Csch[x]^7)/7 - Csch[x]^9/9} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[a+b x] Hyper[c+d x] when b^2-d^2=0*) - - -{Sinh[a + b*x]*Sinh[c + b*x], x, 3, (-(1/2))*x*Cosh[a - c] + Sinh[a + c + 2*b*x]/(4*b)} -{Sinh[a + b*x]*Sinh[c - b*x], x, 3, (1/2)*x*Cosh[a + c] - Sinh[a - c + 2*b*x]/(4*b)} - - -{Cosh[a + b*x]*Cosh[c + b*x], x, 3, (1/2)*x*Cosh[a - c] + Sinh[a + c + 2*b*x]/(4*b)} -{Cosh[a + b*x]*Cosh[c - b*x], x, 3, (1/2)*x*Cosh[a + c] + Sinh[a - c + 2*b*x]/(4*b)} - - -{Tanh[a + b*x]*Tanh[c + b*x], x, 4, x - (Coth[a - c]*Log[Cosh[a + b*x]])/b + (Coth[a - c]*Log[Cosh[c + b*x]])/b} -{Tanh[a + b*x]*Tanh[c - b*x], x, 4, -x - (Coth[a + c]*Log[Cosh[c - b*x]])/b + (Coth[a + c]*Log[Cosh[a + b*x]])/b} - - -{Coth[a + b*x]*Coth[c + b*x], x, 4, x - (Coth[a - c]*Log[Sinh[a + b*x]])/b + (Coth[a - c]*Log[Sinh[c + b*x]])/b} -{Coth[a + b*x]*Coth[c - b*x], x, 4, -x - (Coth[a + c]*Log[Sinh[c - b*x]])/b + (Coth[a + c]*Log[Sinh[a + b*x]])/b} - - -{Sech[a + b*x]*Sech[c + b*x], x, 3, (Csch[a - c]*Log[Cosh[a + b*x]])/b - (Csch[a - c]*Log[Cosh[c + b*x]])/b} -{Sech[a + b*x]*Sech[c - b*x], x, 3, -((Csch[a + c]*Log[Cosh[c - b*x]])/b) + (Csch[a + c]*Log[Cosh[a + b*x]])/b} - - -{Csch[a + b*x]*Csch[c + b*x], x, 3, -((Csch[a - c]*Log[Sinh[a + b*x]])/b) + (Csch[a - c]*Log[Sinh[c + b*x]])/b} -{Csch[a + b*x]*Csch[c - b*x], x, 3, -((Csch[a + c]*Log[Sinh[c - b*x]])/b) + (Csch[a + c]*Log[Sinh[a + b*x]])/b} - - -{Sinh[a + b*x]*Tanh[c + b*x]^1, x, 3, -((ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b) + Sinh[a + b*x]/b} -{Sinh[a + b*x]*Tanh[c + b*x]^2, x, 6, Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b} -{Sinh[a + b*x]*Tanh[c + b*x]^3, x, 9, -((3*ArcTan[Sinh[c + b*x]]*Cosh[a - c])/(2*b)) + (Sech[c + b*x]*Sinh[a - c])/b + Sinh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x]*Tanh[c + b*x])/(2*b)} - -{Sinh[a + b*x]*Coth[c + b*x]^1, x, 3, -((ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b) + Sinh[a + b*x]/b} -{Sinh[a + b*x]*Coth[c + b*x]^2, x, 6, -((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b - (Csch[c + b*x]*Sinh[a - c])/b} -{Sinh[a + b*x]*Coth[c + b*x]^3, x, 9, -((Cosh[a - c]*Csch[c + b*x])/b) - (3*ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/(2*b) - (Coth[c + b*x]*Csch[c + b*x]*Sinh[a - c])/(2*b) + Sinh[a + b*x]/b} - -{Sinh[a + b*x]*Sech[c + b*x]^1, x, 3, (Cosh[a - c]*Log[Cosh[c + b*x]])/b + x*Sinh[a - c]} -{Sinh[a + b*x]*Sech[c + b*x]^2, x, 4, -((Cosh[a - c]*Sech[c + b*x])/b) + (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b} -{Sinh[a + b*x]*Sech[c + b*x]^3, x, 5, -((Cosh[a - c]*Sech[c + b*x]^2)/(2*b)) + (Sinh[a - c]*Tanh[c + b*x])/b} - -{Sinh[a + b*x]*Csch[c + b*x]^1, x, 3, x*Cosh[a - c] + (Log[Sinh[c + b*x]]*Sinh[a - c])/b} -{Sinh[a + b*x]*Csch[c + b*x]^2, x, 4, -((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) - (Csch[c + b*x]*Sinh[a - c])/b} -{Sinh[a + b*x]*Csch[c + b*x]^3, x, 5, -((Cosh[a - c]*Coth[c + b*x])/b) - (Csch[c + b*x]^2*Sinh[a - c])/(2*b)} - - -{Cosh[a + b*x]*Tanh[c + b*x]^1, x, 3, Cosh[a + b*x]/b - (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b} -{Cosh[a + b*x]*Tanh[c + b*x]^2, x, 6, -((ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b) + (Sech[c + b*x]*Sinh[a - c])/b + Sinh[a + b*x]/b} -{Cosh[a + b*x]*Tanh[c + b*x]^3, x, 9, Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (3*ArcTan[Sinh[c + b*x]]*Sinh[a - c])/(2*b) + (Sech[c + b*x]*Sinh[a - c]*Tanh[c + b*x])/(2*b)} - -{Cosh[a + b*x]*Coth[c + b*x]^1, x, 3, -((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b} -{Cosh[a + b*x]*Coth[c + b*x]^2, x, 6, -((Cosh[a - c]*Csch[c + b*x])/b) - (ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b + Sinh[a + b*x]/b} -{Cosh[a + b*x]*Coth[c + b*x]^3, x, 9, -((3*ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/(2*b)) + Cosh[a + b*x]/b - (Cosh[a - c]*Coth[c + b*x]*Csch[c + b*x])/(2*b) - (Csch[c + b*x]*Sinh[a - c])/b} - -{Cosh[a + b*x]*Sech[c + b*x]^1, x, 3, x*Cosh[a - c] + (Log[Cosh[c + b*x]]*Sinh[a - c])/b} -{Cosh[a + b*x]*Sech[c + b*x]^2, x, 4, (ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b - (Sech[c + b*x]*Sinh[a - c])/b} -{Cosh[a + b*x]*Sech[c + b*x]^3, x, 5, -((Sech[c + b*x]^2*Sinh[a - c])/(2*b)) + (Cosh[a - c]*Tanh[c + b*x])/b} - -{Cosh[a + b*x]*Csch[c + b*x]^1, x, 3, (Cosh[a - c]*Log[Sinh[c + b*x]])/b + x*Sinh[a - c]} -{Cosh[a + b*x]*Csch[c + b*x]^2, x, 4, -((Cosh[a - c]*Csch[c + b*x])/b) - (ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/b} -{Cosh[a + b*x]*Csch[c + b*x]^3, x, 5, -((Cosh[a - c]*Csch[c + b*x]^2)/(2*b)) - (Coth[c + b*x]*Sinh[a - c])/b} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[a+b x]^m Hyper[c+d x]^n*) - - -{Sinh[a + b*x]*Sinh[c + d*x]^1, x, 4, -(Sinh[a - c + (b - d)*x]/(2*(b - d))) + Sinh[a + c + (b + d)*x]/(2*(b + d))} -{Sinh[a + b*x]*Sinh[c + d*x]^2, x, 5, -(Cosh[a + b*x]/(2*b)) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Sinh[a + b*x]*Sinh[c + d*x]^3, x, 6, -(Sinh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d))) + (3*Sinh[a - c + (b - d)*x])/(8*(b - d)) - (3*Sinh[a + c + (b + d)*x])/(8*(b + d)) + Sinh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} - -{Sinh[a + b*x]^2*Sinh[c + d*x]^2, x, 6, x/4 - Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) - Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} -{Sinh[a + b*x]^2*Sinh[c + d*x]^3, x, 8, -(Cosh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d))) + (3*Cosh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Cosh[c + d*x])/(8*d) - Cosh[3*c + 3*d*x]/(24*d) - (3*Cosh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Cosh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} - -{Sinh[a + b*x]^3*Sinh[c + d*x]^3, x, 10, (3*Sinh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Sinh[a - c + (b - d)*x])/(32*(b - d)) - Sinh[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Sinh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sinh[a + c + (b + d)*x])/(32*(b + d)) + Sinh[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) - (3*Sinh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Sinh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -{Cosh[a + b*x]*Cosh[c + d*x]^1, x, 4, Sinh[a - c + (b - d)*x]/(2*(b - d)) + Sinh[a + c + (b + d)*x]/(2*(b + d))} -{Cosh[a + b*x]*Cosh[c + d*x]^2, x, 5, Sinh[a + b*x]/(2*b) + Sinh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Sinh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Cosh[a + b*x]*Cosh[c + d*x]^3, x, 6, Sinh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sinh[a - c + (b - d)*x])/(8*(b - d)) + (3*Sinh[a + c + (b + d)*x])/(8*(b + d)) + Sinh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} - -{Cosh[a + b*x]^2*Cosh[c + d*x]^2, x, 6, x/4 + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) + Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} -{Cosh[a + b*x]^2*Cosh[c + d*x]^3, x, 8, Sinh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sinh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sinh[c + d*x])/(8*d) + Sinh[3*c + 3*d*x]/(24*d) + (3*Sinh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sinh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} - -{Cosh[a + b*x]^3*Cosh[c + d*x]^3, x, 10, (3*Sinh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) + (9*Sinh[a - c + (b - d)*x])/(32*(b - d)) + Sinh[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Sinh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sinh[a + c + (b + d)*x])/(32*(b + d)) + Sinh[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Sinh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) + (3*Sinh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -{Sinh[a + b*x]*Cosh[c + d*x]^1, x, 4, Cosh[a - c + (b - d)*x]/(2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*(b + d))} -{Sinh[a + b*x]*Cosh[c + d*x]^2, x, 5, Cosh[a + b*x]/(2*b) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))} -{Sinh[a + b*x]*Cosh[c + d*x]^3, x, 6, Cosh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Cosh[a - c + (b - d)*x])/(8*(b - d)) + (3*Cosh[a + c + (b + d)*x])/(8*(b + d)) + Cosh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))} - -{Sinh[a + b*x]^2*Cosh[c + d*x^1], x, 5, Sinh[2*a - c + (2*b - d)*x]/(4*(2*b - d)) - Sinh[c + d*x]/(2*d) + Sinh[2*a + c + (2*b + d)*x]/(4*(2*b + d))} -{Sinh[a + b*x]^2*Cosh[c + d*x]^2, x, 6, -(x/4) + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) - Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*(a + c) + 2*(b + d)*x]/(16*(b + d))} -{Sinh[a + b*x]^2*Cosh[c + d*x]^3, x, 8, Sinh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sinh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) - (3*Sinh[c + d*x])/(8*d) - Sinh[3*c + 3*d*x]/(24*d) + (3*Sinh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sinh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))} - -{Sinh[a + b*x]^3*Cosh[c + d*x]^1, x, 6, -((3*Cosh[a - c + (b - d)*x])/(8*(b - d))) + Cosh[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cosh[a + c + (b + d)*x])/(8*(b + d)) + Cosh[3*a + c + (3*b + d)*x]/(8*(3*b + d))} -{Sinh[a + b*x]^3*Cosh[c + d*x]^2, x, 8, -((3*Cosh[a + b*x])/(8*b)) + Cosh[3*a + 3*b*x]/(24*b) - (3*Cosh[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cosh[3*a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cosh[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cosh[3*a + 2*c + (3*b + 2*d)*x]/(16*(3*b + 2*d))} -{Sinh[a + b*x]^3*Cosh[c + d*x]^3, x, 10, -((3*Cosh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d))) - (9*Cosh[a - c + (b - d)*x])/(32*(b - d)) + Cosh[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Cosh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cosh[a + c + (b + d)*x])/(32*(b + d)) + Cosh[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Cosh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Cosh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))} - - -{Sinh[a + b*x]*Tanh[c + d*x], x, 6, E^(-a - b*x)/(2*b) + E^(a + b*x)/(2*b) - (E^(-a - b*x)*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), -E^(2*(c + d*x))])/b - (E^(a + b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^(2*(c + d*x))])/b} -{Sinh[a + b*x]*Coth[c + d*x], x, 6, E^(-a - b*x)/(2*b) + E^(a + b*x)/(2*b) - (E^(-a - b*x)*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), E^(2*(c + d*x))])/b - (E^(a + b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*(c + d*x))])/b} - - -{Cosh[a + b*x]*Coth[c + d*x], x, 6, -(E^(-a - b*x)/(2*b)) + E^(a + b*x)/(2*b) + (E^(-a - b*x)*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), E^(2*(c + d*x))])/b - (E^(a + b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*(c + d*x))])/b} -{Cosh[a + b*x]*Tanh[c + d*x], x, 6, -(E^(-a - b*x)/(2*b)) + E^(a + b*x)/(2*b) + (E^(-a - b*x)*Hypergeometric2F1[1, -(b/(2*d)), 1 - b/(2*d), -E^(2*(c + d*x))])/b - (E^(a + b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^(2*(c + d*x))])/b} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[m x] Hyper[n x]*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Hyper[m x] Sinh[n x]*) - - -{Sinh[2*x]*Sinh[x], x, 1, (2*Sinh[x]^3)/3, -(Sinh[x]/2) + (1/6)*Sinh[3*x]} -{Sinh[3*x]*Sinh[x], x, 1, (-(1/4))*Sinh[2*x] + (1/8)*Sinh[4*x]} -{Sinh[4*x]*Sinh[x], x, 1, (-(1/6))*Sinh[3*x] + (1/10)*Sinh[5*x]} -{Sinh[m*x]*Sinh[x], x, 4, -(Sinh[(1 - m)*x]/(2*(1 - m))) + Sinh[(1 + m)*x]/(2*(1 + m))} - - -{Cosh[2*x]*Sinh[x], x, 1, -(Cosh[x]/2) + (1/6)*Cosh[3*x]} -{Cosh[3*x]*Sinh[x], x, 1, (-(1/4))*Cosh[2*x] + (1/8)*Cosh[4*x]} -{Cosh[4*x]*Sinh[x], x, 1, (-(1/6))*Cosh[3*x] + (1/10)*Cosh[5*x]} -{Cosh[m*x]*Sinh[x], x, 4, Cosh[(1 - m)*x]/(2*(1 - m)) + Cosh[(1 + m)*x]/(2*(1 + m))} - - -{Tanh[2*x]*Sinh[x], x, 4, -(ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]) + Sinh[x]} -{Tanh[3*x]*Sinh[x], x, 5, (-(1/3))*ArcTan[Sinh[x]] - (1/3)*ArcTan[2*Sinh[x]] + Sinh[x]} -{Tanh[4*x]*Sinh[x], x, 6, (-(1/4))*Sqrt[2 - Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 + Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[2]]] + Sinh[x]} -{Tanh[5*x]*Sinh[x], x, 9, (-(1/5))*ArcTan[Sinh[x]] - (1/5)*Sqrt[(1/2)*(3 + Sqrt[5])]*ArcTan[2*Sqrt[2/(3 + Sqrt[5])]*Sinh[x]] - (1/5)*Sqrt[(1/2)*(3 - Sqrt[5])]*ArcTan[Sqrt[2*(3 + Sqrt[5])]*Sinh[x]] + Sinh[x]} -{Tanh[6*x]*Sinh[x], x, 10, -(ArcTan[Sqrt[2]*Sinh[x]]/(3*Sqrt[2])) - (1/6)*Sqrt[2 - Sqrt[3]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[3]]] - (1/6)*Sqrt[2 + Sqrt[3]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[3]]] + Sinh[x]} -{Tanh[n*x]*Sinh[x], x, 6, 1/(E^x*2) + E^x/2 - Hypergeometric2F1[1, -(1/(2*n)), 1 - 1/(2*n), -E^(2*n*x)]/E^x - E^x*Hypergeometric2F1[1, 1/(2*n), (1/2)*(2 + 1/n), -E^(2*n*x)]} - - -{Coth[2*x]*Sinh[x], x, 3, (-(1/2))*ArcTan[Sinh[x]] + Sinh[x]} -{Coth[3*x]*Sinh[x], x, 3, -(ArcTan[(2*Sinh[x])/Sqrt[3]]/Sqrt[3]) + Sinh[x]} -{Coth[4*x]*Sinh[x], x, 6, (-(1/4))*ArcTan[Sinh[x]] - ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2]) + Sinh[x]} -{Coth[5*x]*Sinh[x], x, 6, (-(1/5))*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTan[2*Sqrt[2/(5 + Sqrt[5])]*Sinh[x]] - (1/5)*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTan[Sqrt[(2/5)*(5 + Sqrt[5])]*Sinh[x]] + Sinh[x]} -{Coth[6*x]*Sinh[x], x, 7, (-(1/6))*ArcTan[Sinh[x]] - (1/6)*ArcTan[2*Sinh[x]] - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3]) + Sinh[x]} - - -{Sech[2*x]*Sinh[x], x, 2, -(ArcTanh[Sqrt[2]*Cosh[x]]/Sqrt[2])} -{Sech[3*x]*Sinh[x], x, 5, (-(1/3))*Log[Cosh[x]] + (1/6)*Log[3 - 4*Cosh[x]^2]} -{Sech[4*x]*Sinh[x], x, 4, ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])} -{Sech[5*x]*Sinh[x], x, 7, (1/5)*Log[Cosh[x]] - (1/20)*(1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Cosh[x]^2] - (1/20)*(1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Cosh[x]^2]} -{Sech[6*x]*Sinh[x], x, 7, ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) - ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])} - - -{Csch[2*x]*Sinh[x], x, 2, (1/2)*ArcTan[Sinh[x]]} -{Csch[3*x]*Sinh[x], x, 2, ArcTan[Tanh[x]/Sqrt[3]]/Sqrt[3]} -{Csch[4*x]*Sinh[x], x, 4, (-(1/4))*ArcTan[Sinh[x]] + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])} -{Csch[5*x]*Sinh[x], x, 4, (1/5)*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTan[Tanh[x]/Sqrt[5 - 2*Sqrt[5]]] - (1/5)*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTan[Tanh[x]/Sqrt[5 + 2*Sqrt[5]]]} -{Csch[6*x]*Sinh[x], x, 7, (1/6)*ArcTan[Sinh[x]] + (1/6)*ArcTan[2*Sinh[x]] - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3])} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Hyper[m x] Cosh[n x]*) - - -{Sinh[2*x]*Cosh[x], x, 1, (2*Cosh[x]^3)/3, Cosh[x]/2 + (1/6)*Cosh[3*x]} -{Sinh[3*x]*Cosh[x], x, 1, (1/4)*Cosh[2*x] + (1/8)*Cosh[4*x]} -{Sinh[4*x]*Cosh[x], x, 1, (1/6)*Cosh[3*x] + (1/10)*Cosh[5*x]} -{Sinh[m*x]*Cosh[x], x, 4, -(Cosh[(1 - m)*x]/(2*(1 - m))) + Cosh[(1 + m)*x]/(2*(1 + m))} - - -{Cosh[2*x]*Cosh[x], x, 1, Sinh[x]/2 + (1/6)*Sinh[3*x]} -{Cosh[3*x]*Cosh[x], x, 1, (1/4)*Sinh[2*x] + (1/8)*Sinh[4*x]} -{Cosh[4*x]*Cosh[x], x, 1, (1/6)*Sinh[3*x] + (1/10)*Sinh[5*x]} -{Cosh[m*x]*Cosh[x], x, 4, Sinh[(1 - m)*x]/(2*(1 - m)) + Sinh[(1 + m)*x]/(2*(1 + m))} - - -{Tanh[2*x]*Cosh[x], x, 4, -(ArcTanh[Sqrt[2]*Cosh[x]]/Sqrt[2]) + Cosh[x]} -{Tanh[3*x]*Cosh[x], x, 3, -(ArcTanh[(2*Cosh[x])/Sqrt[3]]/Sqrt[3]) + Cosh[x]} -{Tanh[4*x]*Cosh[x], x, 6, (-(1/4))*Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]] + Cosh[x]} -{Tanh[5*x]*Cosh[x], x, 6, (-(1/5))*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cosh[x]] - (1/5)*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTanh[Sqrt[(2/5)*(5 + Sqrt[5])]*Cosh[x]] + Cosh[x]} -{Tanh[6*x]*Cosh[x], x, 10, -(ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2])) - (1/6)*Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[3]]] - (1/6)*Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[3]]] + Cosh[x]} - - -{Coth[2*x]*Cosh[x], x, 4, (-(1/2))*ArcTanh[Cosh[x]] + Cosh[x]} -{Coth[3*x]*Cosh[x], x, 9, Cosh[x] + (1/6)*Log[1 - 2*Cosh[x]] + (1/6)*Log[1 - Cosh[x]] - (1/6)*Log[1 + Cosh[x]] - (1/6)*Log[1 + 2*Cosh[x]]} -{Coth[4*x]*Cosh[x], x, 6, (-(1/4))*ArcTanh[Cosh[x]] - ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2]) + Cosh[x]} -{Coth[5*x]*Cosh[x], x, 10, (-(1/5))*ArcTanh[Cosh[x]] + Cosh[x] + (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cosh[x]] + (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Cosh[x]] - (1/20)*(1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cosh[x]] - (1/20)*(1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cosh[x]]} -{Coth[6*x]*Cosh[x], x, 7, (-(1/6))*ArcTanh[Cosh[x]] - (1/6)*ArcTanh[2*Cosh[x]] - ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3]) + Cosh[x]} -{Coth[n*x]*Cosh[x], x, 6, -(1/(E^x*2)) + E^x/2 + Hypergeometric2F1[1, -(1/(2*n)), 1 - 1/(2*n), E^(2*n*x)]/E^x - E^x*Hypergeometric2F1[1, 1/(2*n), (1/2)*(2 + 1/n), E^(2*n*x)]} - - -{Sech[2*x]*Cosh[x], x, 2, ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]} -{Sech[3*x]*Cosh[x], x, 2, ArcTan[Sqrt[3]*Tanh[x]]/Sqrt[3]} -{Sech[4*x]*Cosh[x], x, 4, ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])} -{Sech[5*x]*Cosh[x], x, 4, (-(1/5))*Sqrt[(1/2)*(5 - Sqrt[5])]*ArcTan[Sqrt[5 - 2*Sqrt[5]]*Tanh[x]] + (1/5)*Sqrt[(1/2)*(5 + Sqrt[5])]*ArcTan[Sqrt[5 + 2*Sqrt[5]]*Tanh[x]]} -{Sech[6*x]*Cosh[x], x, 7, -(ArcTan[Sqrt[2]*Sinh[x]]/(3*Sqrt[2])) + ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])} - - -{Csch[2*x]*Cosh[x], x, 2, (-(1/2))*ArcTanh[Cosh[x]]} -{Csch[3*x]*Cosh[x], x, 5, (1/3)*Log[Sinh[x]] - (1/6)*Log[3 + 4*Sinh[x]^2]} -{Csch[4*x]*Cosh[x], x, 4, (-(1/4))*ArcTanh[Cosh[x]] + ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2])} -{Csch[5*x]*Cosh[x], x, 7, (1/5)*Log[Sinh[x]] - (1/20)*(1 + Sqrt[5])*Log[5 - Sqrt[5] + 8*Sinh[x]^2] - (1/20)*(1 - Sqrt[5])*Log[5 + Sqrt[5] + 8*Sinh[x]^2]} -{Csch[6*x]*Cosh[x], x, 7, (-(1/6))*ArcTanh[Cosh[x]] - (1/6)*ArcTanh[2*Cosh[x]] + ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m Hyper[a+b x]^n Hyper[a+b x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[a+b x]^n Sinh[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p=1*) - - -{x^m*Cosh[a + b*x]*Sinh[a + b*x], x, 5, (2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b)} - -{x^3*Cosh[a + b*x]*Sinh[a + b*x], x, 5, (3*x)/(8*b^3) + x^3/(4*b) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Sinh[a + b*x]^2)/(2*b)} -{x^2*Cosh[a + b*x]*Sinh[a + b*x], x, 3, x^2/(4*b) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) + (x^2*Sinh[a + b*x]^2)/(2*b)} -{x^1*Cosh[a + b*x]*Sinh[a + b*x], x, 3, x/(4*b) - (Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (x*Sinh[a + b*x]^2)/(2*b)} -{x^0*Cosh[a + b*x]*Sinh[a + b*x], x, 2, Sinh[a + b*x]^2/(2*b)} -{Cosh[a + b*x]*Sinh[a + b*x]/x^1, x, 5, (1/2)*CoshIntegral[2*b*x]*Sinh[2*a] + (1/2)*Cosh[2*a]*SinhIntegral[2*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]/x^2, x, 6, b*Cosh[2*a]*CoshIntegral[2*b*x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]/x^3, x, 7, -((b*Cosh[2*a + 2*b*x])/(2*x)) + b^2*CoshIntegral[2*b*x]*Sinh[2*a] - Sinh[2*a + 2*b*x]/(4*x^2) + b^2*Cosh[2*a]*SinhIntegral[2*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]/x^4, x, 8, -((b*Cosh[2*a + 2*b*x])/(6*x^2)) + (2/3)*b^3*Cosh[2*a]*CoshIntegral[2*b*x] - Sinh[2*a + 2*b*x]/(6*x^3) - (b^2*Sinh[2*a + 2*b*x])/(3*x) + (2/3)*b^3*Sinh[2*a]*SinhIntegral[2*b*x]} - - -{x^m*Cosh[a + b*x]^2*Sinh[a + b*x], x, 8, (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x])/(((-b)*x)^m*(8*b)) + (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(8*b)) + (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(8*b)) + (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(E^(3*a)*(b*x)^m*(8*b))} - -{x^3*Cosh[a + b*x]^2*Sinh[a + b*x], x, 7, (4*x*Cosh[a + b*x])/(3*b^3) + (2*x*Cosh[a + b*x]^3)/(9*b^3) + (x^3*Cosh[a + b*x]^3)/(3*b) - (14*Sinh[a + b*x])/(9*b^4) - (2*x^2*Sinh[a + b*x])/(3*b^2) - (x^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b^2) - (2*Sinh[a + b*x]^3)/(27*b^4)} -{x^2*Cosh[a + b*x]^2*Sinh[a + b*x], x, 4, (4*Cosh[a + b*x])/(9*b^3) + (2*Cosh[a + b*x]^3)/(27*b^3) + (x^2*Cosh[a + b*x]^3)/(3*b) - (4*x*Sinh[a + b*x])/(9*b^2) - (2*x*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^2)} -{x^1*Cosh[a + b*x]^2*Sinh[a + b*x], x, 3, (x*Cosh[a + b*x]^3)/(3*b) - Sinh[a + b*x]/(3*b^2) - Sinh[a + b*x]^3/(9*b^2)} -{x^0*Cosh[a + b*x]^2*Sinh[a + b*x], x, 2, Cosh[a + b*x]^3/(3*b)} -{Cosh[a + b*x]^2*Sinh[a + b*x]/x^1, x, 8, (1/4)*CoshIntegral[b*x]*Sinh[a] + (1/4)*CoshIntegral[3*b*x]*Sinh[3*a] + (1/4)*Cosh[a]*SinhIntegral[b*x] + (1/4)*Cosh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]/x^2, x, 10, (1/4)*b*Cosh[a]*CoshIntegral[b*x] + (3/4)*b*Cosh[3*a]*CoshIntegral[3*b*x] - Sinh[a + b*x]/(4*x) - Sinh[3*a + 3*b*x]/(4*x) + (1/4)*b*Sinh[a]*SinhIntegral[b*x] + (3/4)*b*Sinh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]/x^3, x, 12, -((b*Cosh[a + b*x])/(8*x)) - (3*b*Cosh[3*a + 3*b*x])/(8*x) + (1/8)*b^2*CoshIntegral[b*x]*Sinh[a] + (9/8)*b^2*CoshIntegral[3*b*x]*Sinh[3*a] - Sinh[a + b*x]/(8*x^2) - Sinh[3*a + 3*b*x]/(8*x^2) + (1/8)*b^2*Cosh[a]*SinhIntegral[b*x] + (9/8)*b^2*Cosh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]/x^4, x, 14, -((b*Cosh[a + b*x])/(24*x^2)) - (b*Cosh[3*a + 3*b*x])/(8*x^2) + (1/24)*b^3*Cosh[a]*CoshIntegral[b*x] + (9/8)*b^3*Cosh[3*a]*CoshIntegral[3*b*x] - Sinh[a + b*x]/(12*x^3) - (b^2*Sinh[a + b*x])/(24*x) - Sinh[3*a + 3*b*x]/(12*x^3) - (3*b^2*Sinh[3*a + 3*b*x])/(8*x) + (1/24)*b^3*Sinh[a]*SinhIntegral[b*x] + (9/8)*b^3*Sinh[3*a]*SinhIntegral[3*b*x]} - - -{x^m*Cosh[a + b*x]^3*Sinh[a + b*x], x, 8, (E^(4*a)*x^m*Gamma[1 + m, -4*b*x])/(2^(2*(3 + m))*((-b)*x)^m*b) + (2^(-4 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) + (2^(-4 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b) + (x^m*Gamma[1 + m, 4*b*x])/(2^(2*(3 + m))*E^(4*a)*(b*x)^m*b)} - -{x^3*Cosh[a + b*x]^3*Sinh[a + b*x], x, 9, -((45*x)/(256*b^3)) - (3*x^3)/(32*b) + (9*x*Cosh[a + b*x]^2)/(32*b^3) + (3*x*Cosh[a + b*x]^4)/(32*b^3) + (x^3*Cosh[a + b*x]^4)/(4*b) - (45*Cosh[a + b*x]*Sinh[a + b*x])/(256*b^4) - (9*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(32*b^2) - (3*Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b^4) - (3*x^2*Cosh[a + b*x]^3*Sinh[a + b*x])/(16*b^2)} -{x^2*Cosh[a + b*x]^3*Sinh[a + b*x], x, 4, -((3*x^2)/(32*b)) + (3*Cosh[a + b*x]^2)/(32*b^3) + Cosh[a + b*x]^4/(32*b^3) + (x^2*Cosh[a + b*x]^4)/(4*b) - (3*x*Cosh[a + b*x]*Sinh[a + b*x])/(16*b^2) - (x*Cosh[a + b*x]^3*Sinh[a + b*x])/(8*b^2)} -{x^1*Cosh[a + b*x]^3*Sinh[a + b*x], x, 4, -((3*x)/(32*b)) + (x*Cosh[a + b*x]^4)/(4*b) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(32*b^2) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(16*b^2)} -{x^0*Cosh[a + b*x]^3*Sinh[a + b*x], x, 2, Cosh[a + b*x]^4/(4*b)} -{Cosh[a + b*x]^3*Sinh[a + b*x]/x^1, x, 8, (1/4)*CoshIntegral[2*b*x]*Sinh[2*a] + (1/8)*CoshIntegral[4*b*x]*Sinh[4*a] + (1/4)*Cosh[2*a]*SinhIntegral[2*b*x] + (1/8)*Cosh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]/x^2, x, 10, (1/2)*b*Cosh[2*a]*CoshIntegral[2*b*x] + (1/2)*b*Cosh[4*a]*CoshIntegral[4*b*x] - Sinh[2*a + 2*b*x]/(4*x) - Sinh[4*a + 4*b*x]/(8*x) + (1/2)*b*Sinh[2*a]*SinhIntegral[2*b*x] + (1/2)*b*Sinh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]/x^3, x, 12, -((b*Cosh[2*a + 2*b*x])/(4*x)) - (b*Cosh[4*a + 4*b*x])/(4*x) + (1/2)*b^2*CoshIntegral[2*b*x]*Sinh[2*a] + b^2*CoshIntegral[4*b*x]*Sinh[4*a] - Sinh[2*a + 2*b*x]/(8*x^2) - Sinh[4*a + 4*b*x]/(16*x^2) + (1/2)*b^2*Cosh[2*a]*SinhIntegral[2*b*x] + b^2*Cosh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]/x^4, x, 14, -((b*Cosh[2*a + 2*b*x])/(12*x^2)) - (b*Cosh[4*a + 4*b*x])/(12*x^2) + (1/3)*b^3*Cosh[2*a]*CoshIntegral[2*b*x] + (4/3)*b^3*Cosh[4*a]*CoshIntegral[4*b*x] - Sinh[2*a + 2*b*x]/(12*x^3) - (b^2*Sinh[2*a + 2*b*x])/(6*x) - Sinh[4*a + 4*b*x]/(24*x^3) - (b^2*Sinh[4*a + 4*b*x])/(3*x) + (1/3)*b^3*Sinh[2*a]*SinhIntegral[2*b*x] + (4/3)*b^3*Sinh[4*a]*SinhIntegral[4*b*x]} - - -{Sinh[x]*Cosh[x]/x^1, x, 3, (1/2)*SinhIntegral[2*x]} -{Sinh[x]*Cosh[x]/x^2, x, 4, CoshIntegral[2*x] - Sinh[2*x]/(2*x)} -{Sinh[x]*Cosh[x]/x^3, x, 5, -(Cosh[2*x]/(2*x)) - Sinh[2*x]/(4*x^2) + SinhIntegral[2*x]} - - -(* ::Subsubsection::Closed:: *) -(*p=2*) - - -{x^m*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 8, (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x])/(((-b)*x)^m*(8*b)) - (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(8*b)) + (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(8*b)) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(E^(3*a)*(b*x)^m*(8*b))} - -{x^3*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 7, (14*Cosh[a + b*x])/(9*b^4) + (2*x^2*Cosh[a + b*x])/(3*b^2) - (2*Cosh[a + b*x]^3)/(27*b^4) - (4*x*Sinh[a + b*x])/(3*b^3) - (x^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b^2) + (2*x*Sinh[a + b*x]^3)/(9*b^3) + (x^3*Sinh[a + b*x]^3)/(3*b)} -{x^2*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 4, (4*x*Cosh[a + b*x])/(9*b^2) - (4*Sinh[a + b*x])/(9*b^3) - (2*x*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^2) + (2*Sinh[a + b*x]^3)/(27*b^3) + (x^2*Sinh[a + b*x]^3)/(3*b)} -{x^1*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 3, Cosh[a + b*x]/(3*b^2) - Cosh[a + b*x]^3/(9*b^2) + (x*Sinh[a + b*x]^3)/(3*b)} -{x^0*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 2, Sinh[a + b*x]^3/(3*b)} -{Cosh[a + b*x]*Sinh[a + b*x]^2/x^1, x, 8, (-(1/4))*Cosh[a]*CoshIntegral[b*x] + (1/4)*Cosh[3*a]*CoshIntegral[3*b*x] - (1/4)*Sinh[a]*SinhIntegral[b*x] + (1/4)*Sinh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^2/x^2, x, 10, Cosh[a + b*x]/(4*x) - Cosh[3*a + 3*b*x]/(4*x) - (1/4)*b*CoshIntegral[b*x]*Sinh[a] + (3/4)*b*CoshIntegral[3*b*x]*Sinh[3*a] - (1/4)*b*Cosh[a]*SinhIntegral[b*x] + (3/4)*b*Cosh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^2/x^3, x, 12, Cosh[a + b*x]/(8*x^2) - Cosh[3*a + 3*b*x]/(8*x^2) - (1/8)*b^2*Cosh[a]*CoshIntegral[b*x] + (9/8)*b^2*Cosh[3*a]*CoshIntegral[3*b*x] + (b*Sinh[a + b*x])/(8*x) - (3*b*Sinh[3*a + 3*b*x])/(8*x) - (1/8)*b^2*Sinh[a]*SinhIntegral[b*x] + (9/8)*b^2*Sinh[3*a]*SinhIntegral[3*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^2/x^4, x, 14, Cosh[a + b*x]/(12*x^3) + (b^2*Cosh[a + b*x])/(24*x) - Cosh[3*a + 3*b*x]/(12*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(8*x) - (1/24)*b^3*CoshIntegral[b*x]*Sinh[a] + (9/8)*b^3*CoshIntegral[3*b*x]*Sinh[3*a] + (b*Sinh[a + b*x])/(24*x^2) - (b*Sinh[3*a + 3*b*x])/(8*x^2) - (1/24)*b^3*Cosh[a]*SinhIntegral[b*x] + (9/8)*b^3*Cosh[3*a]*SinhIntegral[3*b*x]} - - -{x^m*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 5, -(x^(1 + m)/(8*(1 + m))) + (E^(4*a)*x^m*Gamma[1 + m, -4*b*x])/(2^(2*(3 + m))*((-b)*x)^m*b) - (x^m*Gamma[1 + m, 4*b*x])/(2^(2*(3 + m))*E^(4*a)*(b*x)^m*b)} - -{x^3*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 6, -(x^4/32) - (3*Cosh[4*a + 4*b*x])/(1024*b^4) - (3*x^2*Cosh[4*a + 4*b*x])/(128*b^2) + (3*x*Sinh[4*a + 4*b*x])/(256*b^3) + (x^3*Sinh[4*a + 4*b*x])/(32*b)} -{x^2*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 5, -(x^3/24) - (x*Cosh[4*a + 4*b*x])/(64*b^2) + Sinh[4*a + 4*b*x]/(256*b^3) + (x^2*Sinh[4*a + 4*b*x])/(32*b)} -{x^1*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 4, -(x^2/16) - Cosh[4*a + 4*b*x]/(128*b^2) + (x*Sinh[4*a + 4*b*x])/(32*b)} -{x^0*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 3, -(x/8) - (Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)} -{Cosh[a + b*x]^2*Sinh[a + b*x]^2/x^1, x, 5, (1/8)*Cosh[4*a]*CoshIntegral[4*b*x] - Log[x]/8 + (1/8)*Sinh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^2/x^2, x, 6, 1/(8*x) - Cosh[4*a + 4*b*x]/(8*x) + (1/2)*b*CoshIntegral[4*b*x]*Sinh[4*a] + (1/2)*b*Cosh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^2/x^3, x, 7, 1/(16*x^2) - Cosh[4*a + 4*b*x]/(16*x^2) + b^2*Cosh[4*a]*CoshIntegral[4*b*x] - (b*Sinh[4*a + 4*b*x])/(4*x) + b^2*Sinh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^2/x^4, x, 8, 1/(24*x^3) - Cosh[4*a + 4*b*x]/(24*x^3) - (b^2*Cosh[4*a + 4*b*x])/(3*x) + (4/3)*b^3*CoshIntegral[4*b*x]*Sinh[4*a] - (b*Sinh[4*a + 4*b*x])/(12*x^2) + (4/3)*b^3*Cosh[4*a]*SinhIntegral[4*b*x]} - - -{x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 11, (5^(-1 - m)*E^(5*a)*x^m*Gamma[1 + m, -5*b*x])/(((-b)*x)^m*(32*b)) + (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x])/(((-b)*x)^m*(32*b)) - (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(16*b)) + (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(16*b)) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(E^(3*a)*(b*x)^m*(32*b)) - (5^(-1 - m)*x^m*Gamma[1 + m, 5*b*x])/(E^(5*a)*(b*x)^m*(32*b))} - -{x^3*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 14, (3*Cosh[a + b*x])/(4*b^4) + (3*x^2*Cosh[a + b*x])/(8*b^2) - Cosh[3*a + 3*b*x]/(216*b^4) - (x^2*Cosh[3*a + 3*b*x])/(48*b^2) - (3*Cosh[5*a + 5*b*x])/(5000*b^4) - (3*x^2*Cosh[5*a + 5*b*x])/(400*b^2) - (3*x*Sinh[a + b*x])/(4*b^3) - (x^3*Sinh[a + b*x])/(8*b) + (x*Sinh[3*a + 3*b*x])/(72*b^3) + (x^3*Sinh[3*a + 3*b*x])/(48*b) + (3*x*Sinh[5*a + 5*b*x])/(1000*b^3) + (x^3*Sinh[5*a + 5*b*x])/(80*b)} -{x^2*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 11, (x*Cosh[a + b*x])/(4*b^2) - (x*Cosh[3*a + 3*b*x])/(72*b^2) - (x*Cosh[5*a + 5*b*x])/(200*b^2) - Sinh[a + b*x]/(4*b^3) - (x^2*Sinh[a + b*x])/(8*b) + Sinh[3*a + 3*b*x]/(216*b^3) + (x^2*Sinh[3*a + 3*b*x])/(48*b) + Sinh[5*a + 5*b*x]/(1000*b^3) + (x^2*Sinh[5*a + 5*b*x])/(80*b)} -{x^1*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 8, Cosh[a + b*x]/(8*b^2) - Cosh[3*a + 3*b*x]/(144*b^2) - Cosh[5*a + 5*b*x]/(400*b^2) - (x*Sinh[a + b*x])/(8*b) + (x*Sinh[3*a + 3*b*x])/(48*b) + (x*Sinh[5*a + 5*b*x])/(80*b)} -{x^0*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 3, Sinh[a + b*x]^3/(3*b) + Sinh[a + b*x]^5/(5*b)} -{Cosh[a + b*x]^3*Sinh[a + b*x]^2/x^1, x, 11, (-(1/8))*Cosh[a]*CoshIntegral[b*x] + (1/16)*Cosh[3*a]*CoshIntegral[3*b*x] + (1/16)*Cosh[5*a]*CoshIntegral[5*b*x] - (1/8)*Sinh[a]*SinhIntegral[b*x] + (1/16)*Sinh[3*a]*SinhIntegral[3*b*x] + (1/16)*Sinh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^2/x^2, x, 14, Cosh[a + b*x]/(8*x) - Cosh[3*a + 3*b*x]/(16*x) - Cosh[5*a + 5*b*x]/(16*x) - (1/8)*b*CoshIntegral[b*x]*Sinh[a] + (3/16)*b*CoshIntegral[3*b*x]*Sinh[3*a] + (5/16)*b*CoshIntegral[5*b*x]*Sinh[5*a] - (1/8)*b*Cosh[a]*SinhIntegral[b*x] + (3/16)*b*Cosh[3*a]*SinhIntegral[3*b*x] + (5/16)*b*Cosh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^2/x^3, x, 17, Cosh[a + b*x]/(16*x^2) - Cosh[3*a + 3*b*x]/(32*x^2) - Cosh[5*a + 5*b*x]/(32*x^2) - (1/16)*b^2*Cosh[a]*CoshIntegral[b*x] + (9/32)*b^2*Cosh[3*a]*CoshIntegral[3*b*x] + (25/32)*b^2*Cosh[5*a]*CoshIntegral[5*b*x] + (b*Sinh[a + b*x])/(16*x) - (3*b*Sinh[3*a + 3*b*x])/(32*x) - (5*b*Sinh[5*a + 5*b*x])/(32*x) - (1/16)*b^2*Sinh[a]*SinhIntegral[b*x] + (9/32)*b^2*Sinh[3*a]*SinhIntegral[3*b*x] + (25/32)*b^2*Sinh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^2/x^4, x, 20, Cosh[a + b*x]/(24*x^3) + (b^2*Cosh[a + b*x])/(48*x) - Cosh[3*a + 3*b*x]/(48*x^3) - (3*b^2*Cosh[3*a + 3*b*x])/(32*x) - Cosh[5*a + 5*b*x]/(48*x^3) - (25*b^2*Cosh[5*a + 5*b*x])/(96*x) - (1/48)*b^3*CoshIntegral[b*x]*Sinh[a] + (9/32)*b^3*CoshIntegral[3*b*x]*Sinh[3*a] + (125/96)*b^3*CoshIntegral[5*b*x]*Sinh[5*a] + (b*Sinh[a + b*x])/(48*x^2) - (b*Sinh[3*a + 3*b*x])/(32*x^2) - (5*b*Sinh[5*a + 5*b*x])/(96*x^2) - (1/48)*b^3*Cosh[a]*SinhIntegral[b*x] + (9/32)*b^3*Cosh[3*a]*SinhIntegral[3*b*x] + (125/96)*b^3*Cosh[5*a]*SinhIntegral[5*b*x]} - - -(* ::Subsubsection::Closed:: *) -(*p=3*) - - -{x^m*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 8, (E^(4*a)*x^m*Gamma[1 + m, -4*b*x])/(2^(2*(3 + m))*((-b)*x)^m*b) - (2^(-4 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) - (2^(-4 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b) + (x^m*Gamma[1 + m, 4*b*x])/(2^(2*(3 + m))*E^(4*a)*(b*x)^m*b)} - -{x^3*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 9, -((45*x)/(256*b^3)) - (3*x^3)/(32*b) + (45*Cosh[a + b*x]*Sinh[a + b*x])/(256*b^4) + (9*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(32*b^2) - (9*x*Sinh[a + b*x]^2)/(32*b^3) - (3*Cosh[a + b*x]*Sinh[a + b*x]^3)/(128*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x]^3)/(16*b^2) + (3*x*Sinh[a + b*x]^4)/(32*b^3) + (x^3*Sinh[a + b*x]^4)/(4*b)} -{x^2*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 4, -((3*x^2)/(32*b)) + (3*x*Cosh[a + b*x]*Sinh[a + b*x])/(16*b^2) - (3*Sinh[a + b*x]^2)/(32*b^3) - (x*Cosh[a + b*x]*Sinh[a + b*x]^3)/(8*b^2) + Sinh[a + b*x]^4/(32*b^3) + (x^2*Sinh[a + b*x]^4)/(4*b)} -{x^1*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 4, -((3*x)/(32*b)) + (3*Cosh[a + b*x]*Sinh[a + b*x])/(32*b^2) - (Cosh[a + b*x]*Sinh[a + b*x]^3)/(16*b^2) + (x*Sinh[a + b*x]^4)/(4*b)} -{x^0*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 2, Sinh[a + b*x]^4/(4*b)} -{Cosh[a + b*x]*Sinh[a + b*x]^3/x^1, x, 8, (-(1/4))*CoshIntegral[2*b*x]*Sinh[2*a] + (1/8)*CoshIntegral[4*b*x]*Sinh[4*a] - (1/4)*Cosh[2*a]*SinhIntegral[2*b*x] + (1/8)*Cosh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^3/x^2, x, 10, (-(1/2))*b*Cosh[2*a]*CoshIntegral[2*b*x] + (1/2)*b*Cosh[4*a]*CoshIntegral[4*b*x] + Sinh[2*a + 2*b*x]/(4*x) - Sinh[4*a + 4*b*x]/(8*x) - (1/2)*b*Sinh[2*a]*SinhIntegral[2*b*x] + (1/2)*b*Sinh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^3/x^3, x, 12, (b*Cosh[2*a + 2*b*x])/(4*x) - (b*Cosh[4*a + 4*b*x])/(4*x) - (1/2)*b^2*CoshIntegral[2*b*x]*Sinh[2*a] + b^2*CoshIntegral[4*b*x]*Sinh[4*a] + Sinh[2*a + 2*b*x]/(8*x^2) - Sinh[4*a + 4*b*x]/(16*x^2) - (1/2)*b^2*Cosh[2*a]*SinhIntegral[2*b*x] + b^2*Cosh[4*a]*SinhIntegral[4*b*x]} -{Cosh[a + b*x]*Sinh[a + b*x]^3/x^4, x, 14, (b*Cosh[2*a + 2*b*x])/(12*x^2) - (b*Cosh[4*a + 4*b*x])/(12*x^2) - (1/3)*b^3*Cosh[2*a]*CoshIntegral[2*b*x] + (4/3)*b^3*Cosh[4*a]*CoshIntegral[4*b*x] + Sinh[2*a + 2*b*x]/(12*x^3) + (b^2*Sinh[2*a + 2*b*x])/(6*x) - Sinh[4*a + 4*b*x]/(24*x^3) - (b^2*Sinh[4*a + 4*b*x])/(3*x) - (1/3)*b^3*Sinh[2*a]*SinhIntegral[2*b*x] + (4/3)*b^3*Sinh[4*a]*SinhIntegral[4*b*x]} - - -{x^m*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 11, (5^(-1 - m)*E^(5*a)*x^m*Gamma[1 + m, -5*b*x])/(((-b)*x)^m*(32*b)) - (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x])/(((-b)*x)^m*(32*b)) - (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(16*b)) - (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(16*b)) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(E^(3*a)*(b*x)^m*(32*b)) + (5^(-1 - m)*x^m*Gamma[1 + m, 5*b*x])/(E^(5*a)*(b*x)^m*(32*b))} - -{x^3*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 14, -((3*x*Cosh[a + b*x])/(4*b^3)) - (x^3*Cosh[a + b*x])/(8*b) - (x*Cosh[3*a + 3*b*x])/(72*b^3) - (x^3*Cosh[3*a + 3*b*x])/(48*b) + (3*x*Cosh[5*a + 5*b*x])/(1000*b^3) + (x^3*Cosh[5*a + 5*b*x])/(80*b) + (3*Sinh[a + b*x])/(4*b^4) + (3*x^2*Sinh[a + b*x])/(8*b^2) + Sinh[3*a + 3*b*x]/(216*b^4) + (x^2*Sinh[3*a + 3*b*x])/(48*b^2) - (3*Sinh[5*a + 5*b*x])/(5000*b^4) - (3*x^2*Sinh[5*a + 5*b*x])/(400*b^2)} -{x^2*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 11, -(Cosh[a + b*x]/(4*b^3)) - (x^2*Cosh[a + b*x])/(8*b) - Cosh[3*a + 3*b*x]/(216*b^3) - (x^2*Cosh[3*a + 3*b*x])/(48*b) + Cosh[5*a + 5*b*x]/(1000*b^3) + (x^2*Cosh[5*a + 5*b*x])/(80*b) + (x*Sinh[a + b*x])/(4*b^2) + (x*Sinh[3*a + 3*b*x])/(72*b^2) - (x*Sinh[5*a + 5*b*x])/(200*b^2)} -{x^1*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 8, -((x*Cosh[a + b*x])/(8*b)) - (x*Cosh[3*a + 3*b*x])/(48*b) + (x*Cosh[5*a + 5*b*x])/(80*b) + Sinh[a + b*x]/(8*b^2) + Sinh[3*a + 3*b*x]/(144*b^2) - Sinh[5*a + 5*b*x]/(400*b^2)} -{x^0*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 3, -(Cosh[a + b*x]^3/(3*b)) + Cosh[a + b*x]^5/(5*b)} -{Cosh[a + b*x]^2*Sinh[a + b*x]^3/x^1, x, 11, (-(1/8))*CoshIntegral[b*x]*Sinh[a] - (1/16)*CoshIntegral[3*b*x]*Sinh[3*a] + (1/16)*CoshIntegral[5*b*x]*Sinh[5*a] - (1/8)*Cosh[a]*SinhIntegral[b*x] - (1/16)*Cosh[3*a]*SinhIntegral[3*b*x] + (1/16)*Cosh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^3/x^2, x, 14, (-(1/8))*b*Cosh[a]*CoshIntegral[b*x] - (3/16)*b*Cosh[3*a]*CoshIntegral[3*b*x] + (5/16)*b*Cosh[5*a]*CoshIntegral[5*b*x] + Sinh[a + b*x]/(8*x) + Sinh[3*a + 3*b*x]/(16*x) - Sinh[5*a + 5*b*x]/(16*x) - (1/8)*b*Sinh[a]*SinhIntegral[b*x] - (3/16)*b*Sinh[3*a]*SinhIntegral[3*b*x] + (5/16)*b*Sinh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^3/x^3, x, 17, (b*Cosh[a + b*x])/(16*x) + (3*b*Cosh[3*a + 3*b*x])/(32*x) - (5*b*Cosh[5*a + 5*b*x])/(32*x) - (1/16)*b^2*CoshIntegral[b*x]*Sinh[a] - (9/32)*b^2*CoshIntegral[3*b*x]*Sinh[3*a] + (25/32)*b^2*CoshIntegral[5*b*x]*Sinh[5*a] + Sinh[a + b*x]/(16*x^2) + Sinh[3*a + 3*b*x]/(32*x^2) - Sinh[5*a + 5*b*x]/(32*x^2) - (1/16)*b^2*Cosh[a]*SinhIntegral[b*x] - (9/32)*b^2*Cosh[3*a]*SinhIntegral[3*b*x] + (25/32)*b^2*Cosh[5*a]*SinhIntegral[5*b*x]} -{Cosh[a + b*x]^2*Sinh[a + b*x]^3/x^4, x, 20, (b*Cosh[a + b*x])/(48*x^2) + (b*Cosh[3*a + 3*b*x])/(32*x^2) - (5*b*Cosh[5*a + 5*b*x])/(96*x^2) - (1/48)*b^3*Cosh[a]*CoshIntegral[b*x] - (9/32)*b^3*Cosh[3*a]*CoshIntegral[3*b*x] + (125/96)*b^3*Cosh[5*a]*CoshIntegral[5*b*x] + Sinh[a + b*x]/(24*x^3) + (b^2*Sinh[a + b*x])/(48*x) + Sinh[3*a + 3*b*x]/(48*x^3) + (3*b^2*Sinh[3*a + 3*b*x])/(32*x) - Sinh[5*a + 5*b*x]/(48*x^3) - (25*b^2*Sinh[5*a + 5*b*x])/(96*x) - (1/48)*b^3*Sinh[a]*SinhIntegral[b*x] - (9/32)*b^3*Sinh[3*a]*SinhIntegral[3*b*x] + (125/96)*b^3*Sinh[5*a]*SinhIntegral[5*b*x]} - - -{x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 8, (2^(-7 - m)*3^(-1 - m)*E^(6*a)*x^m*Gamma[1 + m, -6*b*x])/(((-b)*x)^m*b) - (3*2^(-7 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) - (3*2^(-7 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b) + (2^(-7 - m)*3^(-1 - m)*x^m*Gamma[1 + m, 6*b*x])/(E^(6*a)*(b*x)^m*b)} - -{x^3*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 10, -((9*x*Cosh[2*a + 2*b*x])/(128*b^3)) - (3*x^3*Cosh[2*a + 2*b*x])/(64*b) + (x*Cosh[6*a + 6*b*x])/(1152*b^3) + (x^3*Cosh[6*a + 6*b*x])/(192*b) + (9*Sinh[2*a + 2*b*x])/(256*b^4) + (9*x^2*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*a + 6*b*x]/(6912*b^4) - (x^2*Sinh[6*a + 6*b*x])/(384*b^2)} -{x^2*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 8, -((3*Cosh[2*a + 2*b*x])/(128*b^3)) - (3*x^2*Cosh[2*a + 2*b*x])/(64*b) + Cosh[6*a + 6*b*x]/(3456*b^3) + (x^2*Cosh[6*a + 6*b*x])/(192*b) + (3*x*Sinh[2*a + 2*b*x])/(64*b^2) - (x*Sinh[6*a + 6*b*x])/(576*b^2)} -{x^1*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 6, -((3*x*Cosh[2*a + 2*b*x])/(64*b)) + (x*Cosh[6*a + 6*b*x])/(192*b) + (3*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*a + 6*b*x]/(1152*b^2)} -{x^0*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 3, Sinh[a + b*x]^4/(4*b) + Sinh[a + b*x]^6/(6*b)} -{Cosh[a + b*x]^3*Sinh[a + b*x]^3/x^1, x, 8, (-(3/32))*CoshIntegral[2*b*x]*Sinh[2*a] + (1/32)*CoshIntegral[6*b*x]*Sinh[6*a] - (3/32)*Cosh[2*a]*SinhIntegral[2*b*x] + (1/32)*Cosh[6*a]*SinhIntegral[6*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^3/x^2, x, 10, (-(3/16))*b*Cosh[2*a]*CoshIntegral[2*b*x] + (3/16)*b*Cosh[6*a]*CoshIntegral[6*b*x] + (3*Sinh[2*a + 2*b*x])/(32*x) - Sinh[6*a + 6*b*x]/(32*x) - (3/16)*b*Sinh[2*a]*SinhIntegral[2*b*x] + (3/16)*b*Sinh[6*a]*SinhIntegral[6*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^3/x^3, x, 12, (3*b*Cosh[2*a + 2*b*x])/(32*x) - (3*b*Cosh[6*a + 6*b*x])/(32*x) - (3/16)*b^2*CoshIntegral[2*b*x]*Sinh[2*a] + (9/16)*b^2*CoshIntegral[6*b*x]*Sinh[6*a] + (3*Sinh[2*a + 2*b*x])/(64*x^2) - Sinh[6*a + 6*b*x]/(64*x^2) - (3/16)*b^2*Cosh[2*a]*SinhIntegral[2*b*x] + (9/16)*b^2*Cosh[6*a]*SinhIntegral[6*b*x]} -{Cosh[a + b*x]^3*Sinh[a + b*x]^3/x^4, x, 14, (b*Cosh[2*a + 2*b*x])/(32*x^2) - (b*Cosh[6*a + 6*b*x])/(32*x^2) - (1/8)*b^3*Cosh[2*a]*CoshIntegral[2*b*x] + (9/8)*b^3*Cosh[6*a]*CoshIntegral[6*b*x] + Sinh[2*a + 2*b*x]/(32*x^3) + (b^2*Sinh[2*a + 2*b*x])/(16*x) - Sinh[6*a + 6*b*x]/(96*x^3) - (3*b^2*Sinh[6*a + 6*b*x])/(16*x) - (1/8)*b^3*Sinh[2*a]*SinhIntegral[2*b*x] + (9/8)*b^3*Sinh[6*a]*SinhIntegral[6*b*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sech[a+b x]^n Sinh[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p=1*) - - -{x^m*Sech[a + b*x]*Sinh[a + b*x], x, 0, Unintegrable[x^m*Tanh[a + b*x], x]} - -{x^3*Sech[a + b*x]*Sinh[a + b*x], x, 6, -(x^4/4) + (x^3*Log[1 + E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, -E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, -E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, -E^(2*(a + b*x))])/(4*b^4)} -{x^2*Sech[a + b*x]*Sinh[a + b*x], x, 5, -(x^3/3) + (x^2*Log[1 + E^(2*(a + b*x))])/b + (x*PolyLog[2, -E^(2*(a + b*x))])/b^2 - PolyLog[3, -E^(2*(a + b*x))]/(2*b^3)} -{x^1*Sech[a + b*x]*Sinh[a + b*x], x, 4, -(x^2/2) + (x*Log[1 + E^(2*(a + b*x))])/b + PolyLog[2, -E^(2*(a + b*x))]/(2*b^2)} -{x^0*Sech[a + b*x]*Sinh[a + b*x], x, 1, Log[Cosh[a + b*x]]/b} -{Sech[a + b*x]*Sinh[a + b*x]/x^1, x, 0, Unintegrable[Tanh[a + b*x]/x, x]} -{Sech[a + b*x]*Sinh[a + b*x]/x^2, x, 0, Unintegrable[Tanh[a + b*x]/x^2, x]} - - -{x^m*Sech[a + b*x]^2*Sinh[a + b*x], x, 0, CannotIntegrate[x^m*Sech[a + b*x]*Tanh[a + b*x], x]} - -{x^3*Sech[a + b*x]^2*Sinh[a + b*x], x, 8, (6*x^2*ArcTan[E^(a + b*x)])/b^2 - (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 - (x^3*Sech[a + b*x])/b} -{x^2*Sech[a + b*x]^2*Sinh[a + b*x], x, 6, (4*x*ArcTan[E^(a + b*x)])/b^2 - (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 - (x^2*Sech[a + b*x])/b} -{x^1*Sech[a + b*x]^2*Sinh[a + b*x], x, 2, ArcTan[Sinh[a + b*x]]/b^2 - (x*Sech[a + b*x])/b} -{x^0*Sech[a + b*x]^2*Sinh[a + b*x], x, 2, -(Sech[a + b*x]/b)} -{Sech[a + b*x]^2*Sinh[a + b*x]/x^1, x, 0, CannotIntegrate[(Sech[a + b*x]*Tanh[a + b*x])/x, x]} -{Sech[a + b*x]^2*Sinh[a + b*x]/x^2, x, 0, CannotIntegrate[(Sech[a + b*x]*Tanh[a + b*x])/x^2, x]} - - -{x^m*Sech[a + b*x]^3*Sinh[a + b*x], x, 0, CannotIntegrate[x^m*Sech[a + b*x]^2*Tanh[a + b*x], x]} - -{x^3*Sech[a + b*x]^3*Sinh[a + b*x], x, 6, (3*x^2)/(2*b^2) - (3*x*Log[1 + E^(2*(a + b*x))])/b^3 - (3*PolyLog[2, -E^(2*(a + b*x))])/(2*b^4) - (x^3*Sech[a + b*x]^2)/(2*b) + (3*x^2*Tanh[a + b*x])/(2*b^2)} -{x^2*Sech[a + b*x]^3*Sinh[a + b*x], x, 3, -(Log[Cosh[a + b*x]]/b^3) - (x^2*Sech[a + b*x]^2)/(2*b) + (x*Tanh[a + b*x])/b^2} -{x^1*Sech[a + b*x]^3*Sinh[a + b*x], x, 3, -((x*Sech[a + b*x]^2)/(2*b)) + Tanh[a + b*x]/(2*b^2)} -{x^0*Sech[a + b*x]^3*Sinh[a + b*x], x, 2, -(Sech[a + b*x]^2/(2*b))} -{Sech[a + b*x]^3*Sinh[a + b*x]/x^1, x, 0, CannotIntegrate[(Sech[a + b*x]^2*Tanh[a + b*x])/x, x]} -{Sech[a + b*x]^3*Sinh[a + b*x]/x^2, x, 0, CannotIntegrate[(Sech[a + b*x]^2*Tanh[a + b*x])/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p=2*) - - -{x^m*Sech[a + b*x]*Sinh[a + b*x]^2, x, 4, (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b)) - (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(2*b)) - Unintegrable[x^m*Sech[a + b*x], x]} - -{x^3*Sech[a + b*x]*Sinh[a + b*x]^2, x, 14, -((2*x^3*ArcTan[E^(a + b*x)])/b) - (6*Cosh[a + b*x])/b^4 - (3*x^2*Cosh[a + b*x])/b^2 + (3*I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (3*I*x^2*PolyLog[2, I*E^(a + b*x)])/b^2 - (6*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 + (6*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[4, I*E^(a + b*x)])/b^4 + (6*x*Sinh[a + b*x])/b^3 + (x^3*Sinh[a + b*x])/b} -{x^2*Sech[a + b*x]*Sinh[a + b*x]^2, x, 11, -((2*x^2*ArcTan[E^(a + b*x)])/b) - (2*x*Cosh[a + b*x])/b^2 + (2*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (2*I*x*PolyLog[2, I*E^(a + b*x)])/b^2 - (2*I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[3, I*E^(a + b*x)])/b^3 + (2*Sinh[a + b*x])/b^3 + (x^2*Sinh[a + b*x])/b} -{x^1*Sech[a + b*x]*Sinh[a + b*x]^2, x, 8, -((2*x*ArcTan[E^(a + b*x)])/b) - Cosh[a + b*x]/b^2 + (I*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (I*PolyLog[2, I*E^(a + b*x)])/b^2 + (x*Sinh[a + b*x])/b} -{x^0*Sech[a + b*x]*Sinh[a + b*x]^2, x, 3, -(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b} -{Sech[a + b*x]*Sinh[a + b*x]^2/x^1, x, 4, Cosh[a]*CoshIntegral[b*x] - Unintegrable[Sech[a + b*x]/x, x] + Sinh[a]*SinhIntegral[b*x]} -{Sech[a + b*x]*Sinh[a + b*x]^2/x^2, x, 5, -(Cosh[a + b*x]/x) - Unintegrable[Sech[a + b*x]/x^2, x] + b*CoshIntegral[b*x]*Sinh[a] + b*Cosh[a]*SinhIntegral[b*x]} - - -{x^m*Sech[a + b*x]^2*Sinh[a + b*x]^2, x, 0, Unintegrable[x^m*Tanh[a + b*x]^2, x]} - -{x^3*Sech[a + b*x]^2*Sinh[a + b*x]^2, x, 7, -(x^3/b) + x^4/4 + (3*x^2*Log[1 + E^(2*(a + b*x))])/b^2 + (3*x*PolyLog[2, -E^(2*(a + b*x))])/b^3 - (3*PolyLog[3, -E^(2*(a + b*x))])/(2*b^4) - (x^3*Tanh[a + b*x])/b} -{x^2*Sech[a + b*x]^2*Sinh[a + b*x]^2, x, 6, -(x^2/b) + x^3/3 + (2*x*Log[1 + E^(2*(a + b*x))])/b^2 + PolyLog[2, -E^(2*(a + b*x))]/b^3 - (x^2*Tanh[a + b*x])/b} -{x^1*Sech[a + b*x]^2*Sinh[a + b*x]^2, x, 3, x^2/2 + Log[Cosh[a + b*x]]/b^2 - (x*Tanh[a + b*x])/b} -{x^0*Sech[a + b*x]^2*Sinh[a + b*x]^2, x, 2, x - Tanh[a + b*x]/b} -{Sech[a + b*x]^2*Sinh[a + b*x]^2/x^1, x, 0, Unintegrable[Tanh[a + b*x]^2/x, x]} -{Sech[a + b*x]^2*Sinh[a + b*x]^2/x^2, x, 0, Unintegrable[Tanh[a + b*x]^2/x^2, x]} - - -{x^m*Sech[a + b*x]^3*Sinh[a + b*x]^2, x, 1, Unintegrable[x^m*Sech[a + b*x], x] - Unintegrable[x^m*Sech[a + b*x]^3, x]} - -{x^3*Sech[a + b*x]^3*Sinh[a + b*x]^2, x, 25, (6*x*ArcTan[E^(a + b*x)])/b^3 + (x^3*ArcTan[E^(a + b*x)])/b - (3*I*PolyLog[2, (-I)*E^(a + b*x)])/b^4 - (3*I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) + (3*I*PolyLog[2, I*E^(a + b*x)])/b^4 + (3*I*x^2*PolyLog[2, I*E^(a + b*x)])/(2*b^2) + (3*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (3*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 - (3*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + (3*I*PolyLog[4, I*E^(a + b*x)])/b^4 - (3*x^2*Sech[a + b*x])/(2*b^2) - (x^3*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{x^2*Sech[a + b*x]^3*Sinh[a + b*x]^2, x, 17, (x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 - (I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + (I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - (I*PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 - (x^2*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{x^1*Sech[a + b*x]^3*Sinh[a + b*x]^2, x, 12, (x*ArcTan[E^(a + b*x)])/b - (I*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) + (I*PolyLog[2, I*E^(a + b*x)])/(2*b^2) - Sech[a + b*x]/(2*b^2) - (x*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{x^0*Sech[a + b*x]^3*Sinh[a + b*x]^2, x, 2, ArcTan[Sinh[a + b*x]]/(2*b) - (Sech[a + b*x]*Tanh[a + b*x])/(2*b)} -{Sech[a + b*x]^3*Sinh[a + b*x]^2/x^1, x, 1, Unintegrable[Sech[a + b*x]/x, x] - Unintegrable[Sech[a + b*x]^3/x, x]} -{Sech[a + b*x]^3*Sinh[a + b*x]^2/x^2, x, 1, Unintegrable[Sech[a + b*x]/x^2, x] - Unintegrable[Sech[a + b*x]^3/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p=3*) - - -{x^m*Sech[a + b*x]*Sinh[a + b*x]^3, x, 6, (2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b) - Unintegrable[x^m*Tanh[a + b*x], x]} - -{x^3*Sech[a + b*x]*Sinh[a + b*x]^3, x, 12, (3*x)/(8*b^3) + x^3/(4*b) + x^4/4 - (x^3*Log[1 + E^(2*(a + b*x))])/b - (3*x^2*PolyLog[2, -E^(2*(a + b*x))])/(2*b^2) + (3*x*PolyLog[3, -E^(2*(a + b*x))])/(2*b^3) - (3*PolyLog[4, -E^(2*(a + b*x))])/(4*b^4) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Sinh[a + b*x]^2)/(2*b)} -{x^2*Sech[a + b*x]*Sinh[a + b*x]^3, x, 9, x^2/(4*b) + x^3/3 - (x^2*Log[1 + E^(2*(a + b*x))])/b - (x*PolyLog[2, -E^(2*(a + b*x))])/b^2 + PolyLog[3, -E^(2*(a + b*x))]/(2*b^3) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) + (x^2*Sinh[a + b*x]^2)/(2*b)} -{x^1*Sech[a + b*x]*Sinh[a + b*x]^3, x, 8, x/(4*b) + x^2/2 - (x*Log[1 + E^(2*(a + b*x))])/b - PolyLog[2, -E^(2*(a + b*x))]/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (x*Sinh[a + b*x]^2)/(2*b)} -{x^0*Sech[a + b*x]*Sinh[a + b*x]^3, x, 3, Cosh[a + b*x]^2/(2*b) - Log[Cosh[a + b*x]]/b} -{Sech[a + b*x]*Sinh[a + b*x]^3/x^1, x, 6, -Unintegrable[Tanh[a + b*x]/x, x] + (1/2)*CoshIntegral[2*b*x]*Sinh[2*a] + (1/2)*Cosh[2*a]*SinhIntegral[2*b*x]} -{Sech[a + b*x]*Sinh[a + b*x]^3/x^2, x, 7, b*Cosh[2*a]*CoshIntegral[2*b*x] - Unintegrable[Tanh[a + b*x]/x^2, x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x]} - - -{x^m*Sech[a + b*x]^2*Sinh[a + b*x]^3, x, 4, (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b)) + (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(2*b)) - CannotIntegrate[x^m*Sech[a + b*x]*Tanh[a + b*x], x]} - -{x^3*Sech[a + b*x]^2*Sinh[a + b*x]^3, x, 13, -((6*x^2*ArcTan[E^(a + b*x)])/b^2) + (6*x*Cosh[a + b*x])/b^3 + (x^3*Cosh[a + b*x])/b + (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 - (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 + (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 + (x^3*Sech[a + b*x])/b - (6*Sinh[a + b*x])/b^4 - (3*x^2*Sinh[a + b*x])/b^2} -{x^2*Sech[a + b*x]^2*Sinh[a + b*x]^3, x, 10, -((4*x*ArcTan[E^(a + b*x)])/b^2) + (2*Cosh[a + b*x])/b^3 + (x^2*Cosh[a + b*x])/b + (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 + (x^2*Sech[a + b*x])/b - (2*x*Sinh[a + b*x])/b^2} -{x^1*Sech[a + b*x]^2*Sinh[a + b*x]^3, x, 5, -(ArcTan[Sinh[a + b*x]]/b^2) + (x*Cosh[a + b*x])/b + (x*Sech[a + b*x])/b - Sinh[a + b*x]/b^2} -{x^0*Sech[a + b*x]^2*Sinh[a + b*x]^3, x, 3, Cosh[a + b*x]/b + Sech[a + b*x]/b} -{Sech[a + b*x]^2*Sinh[a + b*x]^3/x^1, x, 4, -CannotIntegrate[(Sech[a + b*x]*Tanh[a + b*x])/x, x] + CoshIntegral[b*x]*Sinh[a] + Cosh[a]*SinhIntegral[b*x]} -{Sech[a + b*x]^2*Sinh[a + b*x]^3/x^2, x, 5, b*Cosh[a]*CoshIntegral[b*x] - CannotIntegrate[(Sech[a + b*x]*Tanh[a + b*x])/x^2, x] - Sinh[a + b*x]/x + b*Sinh[a]*SinhIntegral[b*x]} - - -{x^m*Sech[a + b*x]^3*Sinh[a + b*x]^3, x, 0, Unintegrable[x^m*Tanh[a + b*x]^3, x]} - -{x^3*Sech[a + b*x]^3*Sinh[a + b*x]^3, x, 13, -((3*x^2)/(2*b^2)) + x^3/(2*b) - x^4/4 + (3*x*Log[1 + E^(2*(a + b*x))])/b^3 + (x^3*Log[1 + E^(2*(a + b*x))])/b + (3*PolyLog[2, -E^(2*(a + b*x))])/(2*b^4) + (3*x^2*PolyLog[2, -E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, -E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, -E^(2*(a + b*x))])/(4*b^4) - (3*x^2*Tanh[a + b*x])/(2*b^2) - (x^3*Tanh[a + b*x]^2)/(2*b)} -{x^2*Sech[a + b*x]^3*Sinh[a + b*x]^3, x, 9, x^2/(2*b) - x^3/3 + (x^2*Log[1 + E^(2*(a + b*x))])/b + Log[Cosh[a + b*x]]/b^3 + (x*PolyLog[2, -E^(2*(a + b*x))])/b^2 - PolyLog[3, -E^(2*(a + b*x))]/(2*b^3) - (x*Tanh[a + b*x])/b^2 - (x^2*Tanh[a + b*x]^2)/(2*b)} -{x^1*Sech[a + b*x]^3*Sinh[a + b*x]^3, x, 7, x/(2*b) - x^2/2 + (x*Log[1 + E^(2*(a + b*x))])/b + PolyLog[2, -E^(2*(a + b*x))]/(2*b^2) - Tanh[a + b*x]/(2*b^2) - (x*Tanh[a + b*x]^2)/(2*b)} -{x^0*Sech[a + b*x]^3*Sinh[a + b*x]^3, x, 2, Log[Cosh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)} -{Sech[a + b*x]^3*Sinh[a + b*x]^3/x^1, x, 0, Unintegrable[Tanh[a + b*x]^3/x, x]} -{Sech[a + b*x]^3*Sinh[a + b*x]^3/x^2, x, 0, Unintegrable[Tanh[a + b*x]^3/x^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[a+b x]^n Csch[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p=1*) - - -{x^m*Cosh[a + b*x]*Csch[a + b*x], x, 0, Unintegrable[x^m*Coth[a + b*x], x]} - -{x^3*Cosh[a + b*x]*Csch[a + b*x], x, 6, -(x^4/4) + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4)} -{x^2*Cosh[a + b*x]*Csch[a + b*x], x, 5, -(x^3/3) + (x^2*Log[1 - E^(2*(a + b*x))])/b + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3)} -{x^1*Cosh[a + b*x]*Csch[a + b*x], x, 4, -(x^2/2) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2)} -{x^0*Cosh[a + b*x]*Csch[a + b*x], x, 1, Log[Sinh[a + b*x]]/b} -{Cosh[a + b*x]*Csch[a + b*x]/x^1, x, 0, Unintegrable[Coth[a + b*x]/x, x]} -{Cosh[a + b*x]*Csch[a + b*x]/x^2, x, 0, Unintegrable[Coth[a + b*x]/x^2, x]} - - -{x^m*Cosh[a + b*x]^2*Csch[a + b*x], x, 4, (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b)) + (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(2*b)) + Unintegrable[x^m*Csch[a + b*x], x]} - -{x^3*Cosh[a + b*x]^2*Csch[a + b*x], x, 14, -((2*x^3*ArcTanh[E^(a + b*x)])/b) + (6*x*Cosh[a + b*x])/b^3 + (x^3*Cosh[a + b*x])/b - (3*x^2*PolyLog[2, -E^(a + b*x)])/b^2 + (3*x^2*PolyLog[2, E^(a + b*x)])/b^2 + (6*x*PolyLog[3, -E^(a + b*x)])/b^3 - (6*x*PolyLog[3, E^(a + b*x)])/b^3 - (6*PolyLog[4, -E^(a + b*x)])/b^4 + (6*PolyLog[4, E^(a + b*x)])/b^4 - (6*Sinh[a + b*x])/b^4 - (3*x^2*Sinh[a + b*x])/b^2} -{x^2*Cosh[a + b*x]^2*Csch[a + b*x], x, 11, -((2*x^2*ArcTanh[E^(a + b*x)])/b) + (2*Cosh[a + b*x])/b^3 + (x^2*Cosh[a + b*x])/b - (2*x*PolyLog[2, -E^(a + b*x)])/b^2 + (2*x*PolyLog[2, E^(a + b*x)])/b^2 + (2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*PolyLog[3, E^(a + b*x)])/b^3 - (2*x*Sinh[a + b*x])/b^2} -{x^1*Cosh[a + b*x]^2*Csch[a + b*x], x, 8, -((2*x*ArcTanh[E^(a + b*x)])/b) + (x*Cosh[a + b*x])/b - PolyLog[2, -E^(a + b*x)]/b^2 + PolyLog[2, E^(a + b*x)]/b^2 - Sinh[a + b*x]/b^2} -{x^0*Cosh[a + b*x]^2*Csch[a + b*x], x, 3, -(ArcTanh[Cosh[a + b*x]]/b) + Cosh[a + b*x]/b} -{Cosh[a + b*x]^2*Csch[a + b*x]/x^1, x, 4, Unintegrable[Csch[a + b*x]/x, x] + CoshIntegral[b*x]*Sinh[a] + Cosh[a]*SinhIntegral[b*x]} -{Cosh[a + b*x]^2*Csch[a + b*x]/x^2, x, 5, b*Cosh[a]*CoshIntegral[b*x] + Unintegrable[Csch[a + b*x]/x^2, x] - Sinh[a + b*x]/x + b*Sinh[a]*SinhIntegral[b*x]} - - -{x^m*Cosh[a + b*x]^3*Csch[a + b*x], x, 6, (2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(((-b)*x)^m*b) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(E^(2*a)*(b*x)^m*b) + Unintegrable[x^m*Coth[a + b*x], x]} - -{x^3*Cosh[a + b*x]^3*Csch[a + b*x], x, 12, (3*x)/(8*b^3) + x^3/(4*b) - x^4/4 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Sinh[a + b*x]^2)/(2*b)} -{x^2*Cosh[a + b*x]^3*Csch[a + b*x], x, 9, x^2/(4*b) - x^3/3 + (x^2*Log[1 - E^(2*(a + b*x))])/b + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) + (x^2*Sinh[a + b*x]^2)/(2*b)} -{x^1*Cosh[a + b*x]^3*Csch[a + b*x], x, 8, x/(4*b) - x^2/2 + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (x*Sinh[a + b*x]^2)/(2*b)} -{x^0*Cosh[a + b*x]^3*Csch[a + b*x], x, 3, Log[Sinh[a + b*x]]/b + Sinh[a + b*x]^2/(2*b)} -{Cosh[a + b*x]^3*Csch[a + b*x]/x^1, x, 6, Unintegrable[Coth[a + b*x]/x, x] + (1/2)*CoshIntegral[2*b*x]*Sinh[2*a] + (1/2)*Cosh[2*a]*SinhIntegral[2*b*x]} -{Cosh[a + b*x]^3*Csch[a + b*x]/x^2, x, 7, b*Cosh[2*a]*CoshIntegral[2*b*x] + Unintegrable[Coth[a + b*x]/x^2, x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x]} - - -{x^1*Cosh[x]^2*Coth[x]^2, x, 6, (3*x^2)/4 - Cosh[x]^2/4 - x*Coth[x] + Log[Sinh[x]] + (1/2)*x*Cosh[x]*Sinh[x]} -{x^2*Cosh[x]^2*Coth[x]^2, x, 11, x/4 - x^2 + x^3/2 - (1/2)*x*Cosh[x]^2 - x^2*Coth[x] + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] + (1/4)*Cosh[x]*Sinh[x] + (1/2)*x^2*Cosh[x]*Sinh[x]} -{x^3*Cosh[x]^2*Coth[x]^2, x, 12, (3*x^2)/8 - x^3 + (3*x^4)/8 - (3*Cosh[x]^2)/8 - (3/4)*x^2*Cosh[x]^2 - x^3*Coth[x] + 3*x^2*Log[1 - E^(2*x)] + 3*x*PolyLog[2, E^(2*x)] - (3/2)*PolyLog[3, E^(2*x)] + (3/4)*x*Cosh[x]*Sinh[x] + (1/2)*x^3*Cosh[x]*Sinh[x]} - - -{x^1*Cosh[x]^2*Coth[x]^3, x, 16, (3*x)/4 - x^2 - Coth[x]/2 - (1/2)*x*Coth[x]^2 + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] - (1/4)*Cosh[x]*Sinh[x] + (1/2)*x*Sinh[x]^2} -{x^2*Cosh[x]^2*Coth[x]^3, x, 19, (3*x^2)/4 - (2*x^3)/3 - x*Coth[x] - (1/2)*x^2*Coth[x]^2 + 2*x^2*Log[1 - E^(2*x)] + Log[Sinh[x]] + 2*x*PolyLog[2, E^(2*x)] - PolyLog[3, E^(2*x)] - (1/2)*x*Cosh[x]*Sinh[x] + Sinh[x]^2/4 + (1/2)*x^2*Sinh[x]^2} -{x^3*Cosh[x]^2*Coth[x]^3, x, 26, (3*x)/8 - (3*x^2)/2 + (3*x^3)/4 - x^4/2 - (3/2)*x^2*Coth[x] - (1/2)*x^3*Coth[x]^2 + 3*x*Log[1 - E^(2*x)] + 2*x^3*Log[1 - E^(2*x)] + (3/2)*PolyLog[2, E^(2*x)] + 3*x^2*PolyLog[2, E^(2*x)] - 3*x*PolyLog[3, E^(2*x)] + (3/2)*PolyLog[4, E^(2*x)] - (3/8)*Cosh[x]*Sinh[x] - (3/4)*x^2*Cosh[x]*Sinh[x] + (3/4)*x*Sinh[x]^2 + (1/2)*x^3*Sinh[x]^2} - - -(* ::Subsubsection::Closed:: *) -(*p=2*) - - -{x^m*Cosh[a + b*x]*Csch[a + b*x]^2, x, 0, CannotIntegrate[x^m*Coth[a + b*x]*Csch[a + b*x], x]} - -{x^3*Cosh[a + b*x]*Csch[a + b*x]^2, x, 8, -((6*x^2*ArcTanh[E^(a + b*x)])/b^2) - (x^3*Csch[a + b*x])/b - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (6*PolyLog[3, E^(a + b*x)])/b^4} -{x^2*Cosh[a + b*x]*Csch[a + b*x]^2, x, 6, -((4*x*ArcTanh[E^(a + b*x)])/b^2) - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (2*PolyLog[2, E^(a + b*x)])/b^3} -{x^1*Cosh[a + b*x]*Csch[a + b*x]^2, x, 2, -(ArcTanh[Cosh[a + b*x]]/b^2) - (x*Csch[a + b*x])/b} -{x^0*Cosh[a + b*x]*Csch[a + b*x]^2, x, 2, -(Csch[a + b*x]/b)} -{Cosh[a + b*x]*Csch[a + b*x]^2/x^1, x, 0, CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x])/x, x]} -{Cosh[a + b*x]*Csch[a + b*x]^2/x^2, x, 0, CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x])/x^2, x]} - - -{x^m*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 0, Unintegrable[x^m*Coth[a + b*x]^2, x]} - -{x^3*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 7, -(x^3/b) + x^4/4 - (x^3*Coth[a + b*x])/b + (3*x^2*Log[1 - E^(2*(a + b*x))])/b^2 + (3*x*PolyLog[2, E^(2*(a + b*x))])/b^3 - (3*PolyLog[3, E^(2*(a + b*x))])/(2*b^4)} -{x^2*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 6, -(x^2/b) + x^3/3 - (x^2*Coth[a + b*x])/b + (2*x*Log[1 - E^(2*(a + b*x))])/b^2 + PolyLog[2, E^(2*(a + b*x))]/b^3} -{x^1*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 3, x^2/2 - (x*Coth[a + b*x])/b + Log[Sinh[a + b*x]]/b^2} -{x^0*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 2, x - Coth[a + b*x]/b} -{Cosh[a + b*x]^2*Csch[a + b*x]^2/x^1, x, 0, Unintegrable[Coth[a + b*x]^2/x, x]} -{Cosh[a + b*x]^2*Csch[a + b*x]^2/x^2, x, 0, Unintegrable[Coth[a + b*x]^2/x^2, x]} - - -{x^m*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 4, (E^a*x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b)) - (x^m*Gamma[1 + m, b*x])/(E^a*(b*x)^m*(2*b)) + CannotIntegrate[x^m*Coth[a + b*x]*Csch[a + b*x], x]} - -{x^3*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 13, -((6*x^2*ArcTanh[E^(a + b*x)])/b^2) - (6*Cosh[a + b*x])/b^4 - (3*x^2*Cosh[a + b*x])/b^2 - (x^3*Csch[a + b*x])/b - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (6*PolyLog[3, E^(a + b*x)])/b^4 + (6*x*Sinh[a + b*x])/b^3 + (x^3*Sinh[a + b*x])/b} -{x^2*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 10, -((4*x*ArcTanh[E^(a + b*x)])/b^2) - (2*x*Cosh[a + b*x])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (2*PolyLog[2, E^(a + b*x)])/b^3 + (2*Sinh[a + b*x])/b^3 + (x^2*Sinh[a + b*x])/b} -{x^1*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 5, -(ArcTanh[Cosh[a + b*x]]/b^2) - Cosh[a + b*x]/b^2 - (x*Csch[a + b*x])/b + (x*Sinh[a + b*x])/b} -{x^0*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 3, -(Csch[a + b*x]/b) + Sinh[a + b*x]/b} -{Cosh[a + b*x]^3*Csch[a + b*x]^2/x^1, x, 4, Cosh[a]*CoshIntegral[b*x] + CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x])/x, x] + Sinh[a]*SinhIntegral[b*x]} -{Cosh[a + b*x]^3*Csch[a + b*x]^2/x^2, x, 5, -(Cosh[a + b*x]/x) + CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x])/x^2, x] + b*CoshIntegral[b*x]*Sinh[a] + b*Cosh[a]*SinhIntegral[b*x]} - - -(* ::Subsubsection::Closed:: *) -(*p=3*) - - -{x^m*Cosh[a + b*x]*Csch[a + b*x]^3, x, 0, CannotIntegrate[x^m*Coth[a + b*x]*Csch[a + b*x]^2, x]} - -{x^3*Cosh[a + b*x]*Csch[a + b*x]^3, x, 6, -((3*x^2)/(2*b^2)) - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Csch[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*(a + b*x))])/b^3 + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4)} -{x^2*Cosh[a + b*x]*Csch[a + b*x]^3, x, 3, -((x*Coth[a + b*x])/b^2) - (x^2*Csch[a + b*x]^2)/(2*b) + Log[Sinh[a + b*x]]/b^3} -{x^1*Cosh[a + b*x]*Csch[a + b*x]^3, x, 3, -(Coth[a + b*x]/(2*b^2)) - (x*Csch[a + b*x]^2)/(2*b)} -{x^0*Cosh[a + b*x]*Csch[a + b*x]^3, x, 2, -(Csch[a + b*x]^2/(2*b))} -{Cosh[a + b*x]*Csch[a + b*x]^3/x^1, x, 0, CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x]^2)/x, x]} -{Cosh[a + b*x]*Csch[a + b*x]^3/x^2, x, 0, CannotIntegrate[(Coth[a + b*x]*Csch[a + b*x]^2)/x^2, x]} - - -{x^m*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 1, Unintegrable[x^m*Csch[a + b*x], x] + Unintegrable[x^m*Csch[a + b*x]^3, x]} - -{x^3*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 25, -((6*x*ArcTanh[E^(a + b*x)])/b^3) - (x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*Csch[a + b*x])/(2*b^2) - (x^3*Coth[a + b*x]*Csch[a + b*x])/(2*b) - (3*PolyLog[2, -E^(a + b*x)])/b^4 - (3*x^2*PolyLog[2, -E^(a + b*x)])/(2*b^2) + (3*PolyLog[2, E^(a + b*x)])/b^4 + (3*x^2*PolyLog[2, E^(a + b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(a + b*x)])/b^3 - (3*x*PolyLog[3, E^(a + b*x)])/b^3 - (3*PolyLog[4, -E^(a + b*x)])/b^4 + (3*PolyLog[4, E^(a + b*x)])/b^4} -{x^2*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 17, -((x^2*ArcTanh[E^(a + b*x)])/b) - ArcTanh[Cosh[a + b*x]]/b^3 - (x*Csch[a + b*x])/b^2 - (x^2*Coth[a + b*x]*Csch[a + b*x])/(2*b) - (x*PolyLog[2, -E^(a + b*x)])/b^2 + (x*PolyLog[2, E^(a + b*x)])/b^2 + PolyLog[3, -E^(a + b*x)]/b^3 - PolyLog[3, E^(a + b*x)]/b^3} -{x^1*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 12, -((x*ArcTanh[E^(a + b*x)])/b) - Csch[a + b*x]/(2*b^2) - (x*Coth[a + b*x]*Csch[a + b*x])/(2*b) - PolyLog[2, -E^(a + b*x)]/(2*b^2) + PolyLog[2, E^(a + b*x)]/(2*b^2)} -{x^0*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 2, -(ArcTanh[Cosh[a + b*x]]/(2*b)) - (Coth[a + b*x]*Csch[a + b*x])/(2*b)} -{Cosh[a + b*x]^2*Csch[a + b*x]^3/x^1, x, 1, Unintegrable[Csch[a + b*x]/x, x] + Unintegrable[Csch[a + b*x]^3/x, x]} -{Cosh[a + b*x]^2*Csch[a + b*x]^3/x^2, x, 1, Unintegrable[Csch[a + b*x]/x^2, x] + Unintegrable[Csch[a + b*x]^3/x^2, x]} - - -{x^m*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 0, Unintegrable[x^m*Coth[a + b*x]^3, x]} - -{x^3*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 13, -((3*x^2)/(2*b^2)) + x^3/(2*b) - x^4/4 - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Coth[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*(a + b*x))])/b^3 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4) + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4)} -{x^2*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 9, x^2/(2*b) - x^3/3 - (x*Coth[a + b*x])/b^2 - (x^2*Coth[a + b*x]^2)/(2*b) + (x^2*Log[1 - E^(2*(a + b*x))])/b + Log[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3)} -{x^1*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 7, x/(2*b) - x^2/2 - Coth[a + b*x]/(2*b^2) - (x*Coth[a + b*x]^2)/(2*b) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog[2, E^(2*(a + b*x))]/(2*b^2)} -{x^0*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 2, -(Coth[a + b*x]^2/(2*b)) + Log[Sinh[a + b*x]]/b} -{Cosh[a + b*x]^3*Csch[a + b*x]^3/x^1, x, 0, Unintegrable[Coth[a + b*x]^3/x, x]} -{Cosh[a + b*x]^3*Csch[a + b*x]^3/x^2, x, 0, Unintegrable[Coth[a + b*x]^3/x^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sech[a+b x]^n Csch[a+b x]^p*) - - -(* ::Subsubsection::Closed:: *) -(*p=1*) - - -{x^m*Sech[a + b*x]*Csch[a + b*x], x, 0, CannotIntegrate[x^m*Csch[a + b*x]*Sech[a + b*x], x]} - -{x^3*Sech[a + b*x]*Csch[a + b*x], x, 10, -((2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b) - (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) - (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4)} -{x^2*Sech[a + b*x]*Csch[a + b*x], x, 8, -((2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b) - (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 + (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 + PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) - PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3)} -{x^1*Sech[a + b*x]*Csch[a + b*x], x, 6, -((2*x*ArcTanh[E^(2*a + 2*b*x)])/b) - PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) + PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2)} -{x^0*Sech[a + b*x]*Csch[a + b*x], x, 2, Log[Tanh[a + b*x]]/b} -{Sech[a + b*x]*Csch[a + b*x]/x^1, x, 1, 2*Unintegrable[Csch[2*a + 2*b*x]/x, x]} -{Sech[a + b*x]*Csch[a + b*x]/x^2, x, 1, 2*Unintegrable[Csch[2*a + 2*b*x]/x^2, x]} - - -{x^m*Sech[a + b*x]^2*Csch[a + b*x], x, 0, CannotIntegrate[x^m*Csch[a + b*x]*Sech[a + b*x]^2, x]} - -{x^3*Sech[a + b*x]^2*Csch[a + b*x], x, 21, -((6*x^2*ArcTan[E^(a + b*x)])/b^2) - (2*x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*PolyLog[2, -E^(a + b*x)])/b^2 + (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (3*x^2*PolyLog[2, E^(a + b*x)])/b^2 + (6*x*PolyLog[3, -E^(a + b*x)])/b^3 - (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 + (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 - (6*x*PolyLog[3, E^(a + b*x)])/b^3 - (6*PolyLog[4, -E^(a + b*x)])/b^4 + (6*PolyLog[4, E^(a + b*x)])/b^4 + (x^3*Sech[a + b*x])/b} -{x^2*Sech[a + b*x]^2*Csch[a + b*x], x, 17, -((4*x*ArcTan[E^(a + b*x)])/b^2) - (2*x^2*ArcTanh[E^(a + b*x)])/b - (2*x*PolyLog[2, -E^(a + b*x)])/b^2 + (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 + (2*x*PolyLog[2, E^(a + b*x)])/b^2 + (2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*PolyLog[3, E^(a + b*x)])/b^3 + (x^2*Sech[a + b*x])/b} -{x^1*Sech[a + b*x]^2*Csch[a + b*x], x, 10, -(ArcTan[Sinh[a + b*x]]/b^2) - (2*x*ArcTanh[E^(a + b*x)])/b - PolyLog[2, -E^(a + b*x)]/b^2 + PolyLog[2, E^(a + b*x)]/b^2 + (x*Sech[a + b*x])/b} -{x^0*Sech[a + b*x]^2*Csch[a + b*x], x, 3, -(ArcTanh[Cosh[a + b*x]]/b) + Sech[a + b*x]/b} -{Sech[a + b*x]^2*Csch[a + b*x]/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]*Sech[a + b*x]^2)/x, x]} -{Sech[a + b*x]^2*Csch[a + b*x]/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]*Sech[a + b*x]^2)/x^2, x]} - - -{x^m*Sech[a + b*x]^3*Csch[a + b*x], x, 0, CannotIntegrate[x^m*Csch[a + b*x]*Sech[a + b*x]^3, x]} - -{x^3*Sech[a + b*x]^3*Csch[a + b*x], x, 20, -((3*x^2)/(2*b^2)) + x^3/(2*b) - (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b + (3*x*Log[1 + E^(2*(a + b*x))])/b^3 + (3*PolyLog[2, -E^(2*(a + b*x))])/(2*b^4) - (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) - (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4) - (3*x^2*Tanh[a + b*x])/(2*b^2) - (x^3*Tanh[a + b*x]^2)/(2*b)} -{x^2*Sech[a + b*x]^3*Csch[a + b*x], x, 15, x^2/(2*b) - (2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b + Log[Cosh[a + b*x]]/b^3 - (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 + (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 + PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) - PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3) - (x*Tanh[a + b*x])/b^2 - (x^2*Tanh[a + b*x]^2)/(2*b)} -{x^1*Sech[a + b*x]^3*Csch[a + b*x], x, 11, x/(2*b) - (2*x*ArcTanh[E^(2*a + 2*b*x)])/b - PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) + PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2) - Tanh[a + b*x]/(2*b^2) - (x*Tanh[a + b*x]^2)/(2*b)} -{x^0*Sech[a + b*x]^3*Csch[a + b*x], x, 3, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)} -{Sech[a + b*x]^3*Csch[a + b*x]/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x, x]} -{Sech[a + b*x]^3*Csch[a + b*x]/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p=2*) - - -{x^m*Sech[a + b*x]*Csch[a + b*x]^2, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^2*Sech[a + b*x], x]} - -{x^3*Sech[a + b*x]*Csch[a + b*x]^2, x, 21, -((2*x^3*ArcTan[E^(a + b*x)])/b) - (6*x^2*ArcTanh[E^(a + b*x)])/b^2 - (x^3*Csch[a + b*x])/b - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 + (3*I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (3*I*x^2*PolyLog[2, I*E^(a + b*x)])/b^2 + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (6*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 - (6*PolyLog[3, E^(a + b*x)])/b^4 + (6*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[4, I*E^(a + b*x)])/b^4} -{x^2*Sech[a + b*x]*Csch[a + b*x]^2, x, 17, -((2*x^2*ArcTan[E^(a + b*x)])/b) - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (2*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (2*I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - (2*I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[3, I*E^(a + b*x)])/b^3} -{x^1*Sech[a + b*x]*Csch[a + b*x]^2, x, 10, -((2*x*ArcTan[E^(a + b*x)])/b) - ArcTanh[Cosh[a + b*x]]/b^2 - (x*Csch[a + b*x])/b + (I*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (I*PolyLog[2, I*E^(a + b*x)])/b^2} -{x^0*Sech[a + b*x]*Csch[a + b*x]^2, x, 3, -(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b} -{Sech[a + b*x]*Csch[a + b*x]^2/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]^2*Sech[a + b*x])/x, x]} -{Sech[a + b*x]*Csch[a + b*x]^2/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]^2*Sech[a + b*x])/x^2, x]} - - -{x^m*Sech[a + b*x]^2*Csch[a + b*x]^2, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^2*Sech[a + b*x]^2, x]} - -{x^3*Sech[a + b*x]^2*Csch[a + b*x]^2, x, 7, -((2*x^3)/b) - (2*x^3*Coth[2*a + 2*b*x])/b + (3*x^2*Log[1 - E^(4*(a + b*x))])/b^2 + (3*x*PolyLog[2, E^(4*(a + b*x))])/(2*b^3) - (3*PolyLog[3, E^(4*(a + b*x))])/(8*b^4)} -{x^2*Sech[a + b*x]^2*Csch[a + b*x]^2, x, 6, -((2*x^2)/b) - (2*x^2*Coth[2*a + 2*b*x])/b + (2*x*Log[1 - E^(4*(a + b*x))])/b^2 + PolyLog[2, E^(4*(a + b*x))]/(2*b^3)} -{x^1*Sech[a + b*x]^2*Csch[a + b*x]^2, x, 3, -((2*x*Coth[2*a + 2*b*x])/b) + Log[Sinh[2*a + 2*b*x]]/b^2} -{x^0*Sech[a + b*x]^2*Csch[a + b*x]^2, x, 3, -(Coth[a + b*x]/b) - Tanh[a + b*x]/b} -{Sech[a + b*x]^2*Csch[a + b*x]^2/x^1, x, 1, 4*Unintegrable[Csch[2*a + 2*b*x]^2/x, x]} -{Sech[a + b*x]^2*Csch[a + b*x]^2/x^2, x, 1, 4*Unintegrable[Csch[2*a + 2*b*x]^2/x^2, x]} - - -{x^m*Sech[a + b*x]^3*Csch[a + b*x]^2, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^2*Sech[a + b*x]^3, x]} - -(* {x^3*Sech[a + b*x]^3*Csch[a + b*x]^2, x, 35, (6*x*ArcTan[E^(a + b*x)])/b^3 - (3*x^3*ArcTan[E^(a + b*x)])/b - (6*x^2*ArcTanh[E^(a + b*x)])/b^2 - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 - (3*I*PolyLog[2, (-I)*E^(a + b*x)])/b^4 + (9*I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) + (3*I*PolyLog[2, I*E^(a + b*x)])/b^4 - (9*I*x^2*PolyLog[2, I*E^(a + b*x)])/(2*b^2) + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (9*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (9*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 - (6*PolyLog[3, E^(a + b*x)])/b^4 + (9*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 - (9*I*PolyLog[4, I*E^(a + b*x)])/b^4 - (3*x^2*Sech[a + b*x])/(2*b^2) - (x^3*Csch[a + b*x]*Sech[a + b*x]^2)/b - (3*x^3*Sech[a + b*x]*Tanh[a + b*x])/(2*b)} *) -{x^2*Sech[a + b*x]^3*Csch[a + b*x]^2, x, 29, -((3*x^2*ArcTan[E^(a + b*x)])/b) + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (3*x^2*Csch[a + b*x])/(2*b) - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (3*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (3*I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - (3*I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (3*I*PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 + (x^2*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)} -{x^1*Sech[a + b*x]^3*Csch[a + b*x]^2, x, 13, -((3*x*ArcTan[E^(a + b*x)])/b) - ArcTanh[Cosh[a + b*x]]/b^2 - (3*x*Csch[a + b*x])/(2*b) + (3*I*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) - (3*I*PolyLog[2, I*E^(a + b*x)])/(2*b^2) - Sech[a + b*x]/(2*b^2) + (x*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)} -{x^0*Sech[a + b*x]^3*Csch[a + b*x]^2, x, 4, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) - (3*Csch[a + b*x])/(2*b) + (Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)} -{Sech[a + b*x]^3*Csch[a + b*x]^2/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x, x]} -{Sech[a + b*x]^3*Csch[a + b*x]^2/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*p=3*) - - -{x^m*Sech[a + b*x]*Csch[a + b*x]^3, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^3*Sech[a + b*x], x]} - -{x^3*Sech[a + b*x]*Csch[a + b*x]^3, x, 20, -((3*x^2)/(2*b^2)) + x^3/(2*b) + (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Coth[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*(a + b*x))])/b^3 + (3*PolyLog[2, E^(2*(a + b*x))])/(2*b^4) + (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) - (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^2) - (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) + (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) + (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) - (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4)} -{x^2*Sech[a + b*x]*Csch[a + b*x]^3, x, 15, x^2/(2*b) + (2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - (x*Coth[a + b*x])/b^2 - (x^2*Coth[a + b*x]^2)/(2*b) + Log[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 - (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) + PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3)} -{x^1*Sech[a + b*x]*Csch[a + b*x]^3, x, 11, x/(2*b) + (2*x*ArcTanh[E^(2*a + 2*b*x)])/b - Coth[a + b*x]/(2*b^2) - (x*Coth[a + b*x]^2)/(2*b) + PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) - PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2)} -{x^0*Sech[a + b*x]*Csch[a + b*x]^3, x, 3, -(Coth[a + b*x]^2/(2*b)) - Log[Tanh[a + b*x]]/b} -{Sech[a + b*x]*Csch[a + b*x]^3/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]^3*Sech[a + b*x])/x, x]} -{Sech[a + b*x]*Csch[a + b*x]^3/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]^3*Sech[a + b*x])/x^2, x]} - - -{x^m*Sech[a + b*x]^2*Csch[a + b*x]^3, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^3*Sech[a + b*x]^2, x]} - -{x^3*Sech[a + b*x]^2*Csch[a + b*x]^3, x, 40, (6*x^2*ArcTan[E^(a + b*x)])/b^2 - (6*x*ArcTanh[E^(a + b*x)])/b^3 + (3*x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*Csch[a + b*x])/(2*b^2) - (3*PolyLog[2, -E^(a + b*x)])/b^4 + (9*x^2*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (3*PolyLog[2, E^(a + b*x)])/b^4 - (9*x^2*PolyLog[2, E^(a + b*x)])/(2*b^2) - (9*x*PolyLog[3, -E^(a + b*x)])/b^3 + (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 + (9*x*PolyLog[3, E^(a + b*x)])/b^3 + (9*PolyLog[4, -E^(a + b*x)])/b^4 - (9*PolyLog[4, E^(a + b*x)])/b^4 - (3*x^3*Sech[a + b*x])/(2*b) - (x^3*Csch[a + b*x]^2*Sech[a + b*x])/(2*b)} -{x^2*Sech[a + b*x]^2*Csch[a + b*x]^3, x, 29, (4*x*ArcTan[E^(a + b*x)])/b^2 + (3*x^2*ArcTanh[E^(a + b*x)])/b - ArcTanh[Cosh[a + b*x]]/b^3 - (x*Csch[a + b*x])/b^2 + (3*x*PolyLog[2, -E^(a + b*x)])/b^2 - (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 - (3*x*PolyLog[2, E^(a + b*x)])/b^2 - (3*PolyLog[3, -E^(a + b*x)])/b^3 + (3*PolyLog[3, E^(a + b*x)])/b^3 - (3*x^2*Sech[a + b*x])/(2*b) - (x^2*Csch[a + b*x]^2*Sech[a + b*x])/(2*b)} -{x^1*Sech[a + b*x]^2*Csch[a + b*x]^3, x, 13, ArcTan[Sinh[a + b*x]]/b^2 + (3*x*ArcTanh[E^(a + b*x)])/b - Csch[a + b*x]/(2*b^2) + (3*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (3*PolyLog[2, E^(a + b*x)])/(2*b^2) - (3*x*Sech[a + b*x])/(2*b) - (x*Csch[a + b*x]^2*Sech[a + b*x])/(2*b)} -{x^0*Sech[a + b*x]^2*Csch[a + b*x]^3, x, 4, (3*ArcTanh[Cosh[a + b*x]])/(2*b) - (3*Sech[a + b*x])/(2*b) - (Csch[a + b*x]^2*Sech[a + b*x])/(2*b)} -{Sech[a + b*x]^2*Csch[a + b*x]^3/x^1, x, 0, CannotIntegrate[(Csch[a + b*x]^3*Sech[a + b*x]^2)/x, x]} -{Sech[a + b*x]^2*Csch[a + b*x]^3/x^2, x, 0, CannotIntegrate[(Csch[a + b*x]^3*Sech[a + b*x]^2)/x^2, x]} - - -{x^m*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 0, CannotIntegrate[x^m*Csch[a + b*x]^3*Sech[a + b*x]^3, x]} - -{x^3*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 16, -((6*x*ArcTanh[E^(2*a + 2*b*x)])/b^3) + (4*x^3*ArcTanh[E^(2*a + 2*b*x)])/b - (3*x^2*Csch[2*a + 2*b*x])/b^2 - (2*x^3*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b - (3*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^4) + (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 + (3*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^4) - (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/b^3 + (3*x*PolyLog[3, E^(2*a + 2*b*x)])/b^3 + (3*PolyLog[4, -E^(2*a + 2*b*x)])/(2*b^4) - (3*PolyLog[4, E^(2*a + 2*b*x)])/(2*b^4)} -{x^2*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 10, (4*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - ArcTanh[Cosh[2*a + 2*b*x]]/b^3 - (2*x*Csch[2*a + 2*b*x])/b^2 - (2*x^2*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + (2*x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 - (2*x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - PolyLog[3, -E^(2*a + 2*b*x)]/b^3 + PolyLog[3, E^(2*a + 2*b*x)]/b^3} -{x^1*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 7, (4*x*ArcTanh[E^(2*a + 2*b*x)])/b - Csch[2*a + 2*b*x]/b^2 - (2*x*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + PolyLog[2, -E^(2*a + 2*b*x)]/b^2 - PolyLog[2, E^(2*a + 2*b*x)]/b^2} -{x^0*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 4, -(Coth[a + b*x]^2/(2*b)) - (2*Log[Tanh[a + b*x]])/b + Tanh[a + b*x]^2/(2*b)} -{Sech[a + b*x]^3*Csch[a + b*x]^3/x^1, x, 1, 8*Unintegrable[Csch[2*a + 2*b*x]^3/x, x]} -{Sech[a + b*x]^3*Csch[a + b*x]^3/x^2, x, 1, 8*Unintegrable[Csch[2*a + 2*b*x]^3/x^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cosh[a+b x]^(n/2) Sinh[a+b x]^n*) - - -{x*Sinh[a + b*x]*Cosh[a + b*x]^(5/2), x, 4, (2*x*Cosh[a + b*x]^(7/2))/(7*b) + (20*I*EllipticF[(1/2)*I*(a + b*x), 2])/(147*b^2) - (20*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(147*b^2) - (4*Cosh[a + b*x]^(5/2)*Sinh[a + b*x])/(49*b^2)} -{x*Sinh[a + b*x]*Cosh[a + b*x]^(3/2), x, 3, (2*x*Cosh[a + b*x]^(5/2))/(5*b) + (12*I*EllipticE[(1/2)*I*(a + b*x), 2])/(25*b^2) - (4*Cosh[a + b*x]^(3/2)*Sinh[a + b*x])/(25*b^2)} -{x*Sinh[a + b*x]*Sqrt[Cosh[a + b*x]], x, 3, (2*x*Cosh[a + b*x]^(3/2))/(3*b) + (4*I*EllipticF[(1/2)*I*(a + b*x), 2])/(9*b^2) - (4*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(9*b^2)} -{x*Sinh[a + b*x]/Sqrt[Cosh[a + b*x]], x, 2, (2*x*Sqrt[Cosh[a + b*x]])/b + (4*I*EllipticE[(1/2)*I*(a + b*x), 2])/b^2} -{x*Sinh[a + b*x]/Cosh[a + b*x]^(3/2), x, 2, -((2*x)/(b*Sqrt[Cosh[a + b*x]])) - (4*I*EllipticF[(1/2)*I*(a + b*x), 2])/b^2} -{x*Sinh[a + b*x]/Cosh[a + b*x]^(5/2), x, 3, -((2*x)/(3*b*Cosh[a + b*x]^(3/2))) + (4*I*EllipticE[(1/2)*I*(a + b*x), 2])/(3*b^2) + (4*Sinh[a + b*x])/(3*b^2*Sqrt[Cosh[a + b*x]])} -{x*Sinh[a + b*x]/Cosh[a + b*x]^(7/2), x, 3, -((2*x)/(5*b*Cosh[a + b*x]^(5/2))) - (4*I*EllipticF[(1/2)*I*(a + b*x), 2])/(15*b^2) + (4*Sinh[a + b*x])/(15*b^2*Cosh[a + b*x]^(3/2))} -{x*Sinh[a + b*x]/Cosh[a + b*x]^(9/2), x, 4, -((2*x)/(7*b*Cosh[a + b*x]^(7/2))) + (12*I*EllipticE[(1/2)*I*(a + b*x), 2])/(35*b^2) + (4*Sinh[a + b*x])/(35*b^2*Cosh[a + b*x]^(5/2)) + (12*Sinh[a + b*x])/(35*b^2*Sqrt[Cosh[a + b*x]])} - - -{x*Sinh[a + b*x]*Sech[a + b*x]^(9/2), x, 5, (12*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(35*b^2) - (2*x*Sech[a + b*x]^(7/2))/(7*b) + (12*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/(35*b^2) + (4*Sech[a + b*x]^(5/2)*Sinh[a + b*x])/(35*b^2)} -{x*Sinh[a + b*x]*Sech[a + b*x]^(7/2), x, 4, -((4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(15*b^2)) - (2*x*Sech[a + b*x]^(5/2))/(5*b) + (4*Sech[a + b*x]^(3/2)*Sinh[a + b*x])/(15*b^2)} -{x*Sinh[a + b*x]*Sech[a + b*x]^(5/2), x, 4, (4*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(3*b^2) - (2*x*Sech[a + b*x]^(3/2))/(3*b) + (4*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/(3*b^2)} -{x*Sinh[a + b*x]*Sech[a + b*x]^(3/2), x, 3, -((2*x*Sqrt[Sech[a + b*x]])/b) - (4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b^2} -{x*Sinh[a + b*x]*Sech[a + b*x]^(1/2), x, 3, (2*x)/(b*Sqrt[Sech[a + b*x]]) + (4*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b^2} -{x*Sinh[a + b*x]/Sech[a + b*x]^(1/2), x, 4, (2*x)/(3*b*Sech[a + b*x]^(3/2)) + (4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(9*b^2) - (4*Sinh[a + b*x])/(9*b^2*Sqrt[Sech[a + b*x]])} -{x*Sinh[a + b*x]/Sech[a + b*x]^(3/2), x, 4, (2*x)/(5*b*Sech[a + b*x]^(5/2)) + (12*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(25*b^2) - (4*Sinh[a + b*x])/(25*b^2*Sech[a + b*x]^(3/2))} -{x*Sinh[a + b*x]/Sech[a + b*x]^(5/2), x, 5, (2*x)/(7*b*Sech[a + b*x]^(7/2)) + (20*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(147*b^2) - (4*Sinh[a + b*x])/(49*b^2*Sech[a + b*x]^(5/2)) - (20*Sinh[a + b*x])/(147*b^2*Sqrt[Sech[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sinh[a+b x]^(n/2) Cosh[a+b x]^n*) - - -{x*Cosh[a + b*x]*Sinh[a + b*x]^(5/2), x, 5, (20*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(147*b^2*Sqrt[Sinh[a + b*x]]) + (20*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]])/(147*b^2) - (4*Cosh[a + b*x]*Sinh[a + b*x]^(5/2))/(49*b^2) + (2*x*Sinh[a + b*x]^(7/2))/(7*b)} -{x*Cosh[a + b*x]*Sinh[a + b*x]^(3/2), x, 4, -((12*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(25*b^2*Sqrt[I*Sinh[a + b*x]])) - (4*Cosh[a + b*x]*Sinh[a + b*x]^(3/2))/(25*b^2) + (2*x*Sinh[a + b*x]^(5/2))/(5*b)} -{x*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]], x, 4, -((4*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(9*b^2*Sqrt[Sinh[a + b*x]])) - (4*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]])/(9*b^2) + (2*x*Sinh[a + b*x]^(3/2))/(3*b)} -{x*Cosh[a + b*x]/Sqrt[Sinh[a + b*x]], x, 3, (2*x*Sqrt[Sinh[a + b*x]])/b + (4*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(b^2*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Sinh[a + b*x]^(3/2), x, 3, -((2*x)/(b*Sqrt[Sinh[a + b*x]])) - (4*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(b^2*Sqrt[Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Sinh[a + b*x]^(5/2), x, 4, -((2*x)/(3*b*Sinh[a + b*x]^(3/2))) - (4*Cosh[a + b*x])/(3*b^2*Sqrt[Sinh[a + b*x]]) - (4*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(3*b^2*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Sinh[a + b*x]^(7/2), x, 4, -((2*x)/(5*b*Sinh[a + b*x]^(5/2))) - (4*Cosh[a + b*x])/(15*b^2*Sinh[a + b*x]^(3/2)) + (4*I*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(15*b^2*Sqrt[Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Sinh[a + b*x]^(9/2), x, 5, -((2*x)/(7*b*Sinh[a + b*x]^(7/2))) - (4*Cosh[a + b*x])/(35*b^2*Sinh[a + b*x]^(5/2)) + (12*Cosh[a + b*x])/(35*b^2*Sqrt[Sinh[a + b*x]]) + (12*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[Sinh[a + b*x]])/(35*b^2*Sqrt[I*Sinh[a + b*x]])} - - -{x*Cosh[a + b*x]*Csch[a + b*x]^(9/2), x, 5, (12*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(35*b^2) - (4*Cosh[a + b*x]*Csch[a + b*x]^(5/2))/(35*b^2) - (2*x*Csch[a + b*x]^(7/2))/(7*b) + (12*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(35*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]*Csch[a + b*x]^(7/2), x, 4, -((4*Cosh[a + b*x]*Csch[a + b*x]^(3/2))/(15*b^2)) - (2*x*Csch[a + b*x]^(5/2))/(5*b) + (4*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(15*b^2)} -{x*Cosh[a + b*x]*Csch[a + b*x]^(5/2), x, 4, -((4*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(3*b^2)) - (2*x*Csch[a + b*x]^(3/2))/(3*b) - (4*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(3*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]*Csch[a + b*x]^(3/2), x, 3, -((2*x*Sqrt[Csch[a + b*x]])/b) - (4*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/b^2} -{x*Cosh[a + b*x]*Csch[a + b*x]^(1/2), x, 3, (2*x)/(b*Sqrt[Csch[a + b*x]]) + (4*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Csch[a + b*x]^(1/2), x, 4, (2*x)/(3*b*Csch[a + b*x]^(3/2)) - (4*Cosh[a + b*x])/(9*b^2*Sqrt[Csch[a + b*x]]) - (4*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(9*b^2)} -{x*Cosh[a + b*x]/Csch[a + b*x]^(3/2), x, 4, (2*x)/(5*b*Csch[a + b*x]^(5/2)) - (4*Cosh[a + b*x])/(25*b^2*Csch[a + b*x]^(3/2)) - (12*I*EllipticE[(1/2)*(I*a - Pi/2 + I*b*x), 2])/(25*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])} -{x*Cosh[a + b*x]/Csch[a + b*x]^(5/2), x, 5, (2*x)/(7*b*Csch[a + b*x]^(7/2)) - (4*Cosh[a + b*x])/(49*b^2*Csch[a + b*x]^(5/2)) + (20*Cosh[a + b*x])/(147*b^2*Sqrt[Csch[a + b*x]]) + (20*I*Sqrt[Csch[a + b*x]]*EllipticF[(1/2)*(I*a - Pi/2 + I*b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(147*b^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (Hyper[a+b x] Hyper[a+b x])^m*) - - -{(Sinh[x]*Tanh[x])^(1/2), x, 3, 2*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]} -{(Sinh[x]*Tanh[x])^(3/2), x, 4, (8/3)*Csch[x]*Sqrt[Sinh[x]*Tanh[x]] + (2/3)*Sinh[x]*Sqrt[Sinh[x]*Tanh[x]]} -{(Sinh[x]*Tanh[x])^(5/2), x, 5, (-(64/15))*Coth[x]*Sqrt[Sinh[x]*Tanh[x]] + (16/15)*Tanh[x]*Sqrt[Sinh[x]*Tanh[x]] + (2/5)*Sinh[x]^2*Tanh[x]*Sqrt[Sinh[x]*Tanh[x]]} - - -{(Cosh[x]*Coth[x])^(1/2), x, 3, 2*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]} -{(Cosh[x]*Coth[x])^(3/2), x, 4, (2/3)*Cosh[x]*Sqrt[Cosh[x]*Coth[x]] - (8/3)*Sqrt[Cosh[x]*Coth[x]]*Sech[x]} -{(Cosh[x]*Coth[x])^(5/2), x, 5, (-(16/15))*Coth[x]*Sqrt[Cosh[x]*Coth[x]] + (2/5)*Cosh[x]^2*Coth[x]*Sqrt[Cosh[x]*Coth[x]] + (64/15)*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b Hyper[x])^n (c+d Hyper[x])^p*) - - -{(b + c + Cosh[x])/(a + b*Sinh[x]), x, 7, -((2*(b + c)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]) + Log[a + b*Sinh[x]]/b} -{(b + c + Cosh[x])/(a - b*Sinh[x]), x, 7, (2*(b + c)*ArcTanh[(b + a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - Log[a - b*Sinh[x]]/b} - -{(b + c + Sinh[x])/(a + b*Cosh[x]), x, 6, (2*(b + c)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + Log[a + b*Cosh[x]]/b} -{(b + c + Sinh[x])/(a - b*Cosh[x]), x, 6, (2*(b + c)*ArcTanh[(Sqrt[a + b]*Tanh[x/2])/Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[a + b]) - Log[a - b*Cosh[x]]/b} - - -{x*((b - a*Sinh[x])/(a + b*Sinh[x])^2), x, 3, Log[a + b*Sinh[x]]/b - (x*Cosh[x])/(a + b*Sinh[x])} -{x*((b + a*Cosh[x])/(a + b*Cosh[x])^2), x, 3, -(Log[a + b*Cosh[x]]/b) + (x*Sinh[x])/(a + b*Cosh[x])} - - -{(a + b*Sech[x])/(c + d*Cosh[x]), x, 5, (b*ArcTan[Sinh[x]])/c + (2*(a*c - b*d)*ArcTanh[(Sqrt[c - d]*Tanh[x/2])/Sqrt[c + d]])/(c*Sqrt[c - d]*Sqrt[c + d])} -{(a + b*Csch[x])/(c + d*Sinh[x]), x, 6, -((b*ArcTanh[Cosh[x]])/c) - (2*(a*c - b*d)*ArcTanh[(d - c*Tanh[x/2])/Sqrt[c^2 + d^2]])/(c*Sqrt[c^2 + d^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b Hyper[x])^n (c+d Hyper[x]^2)^p*) - - -{(1 + Sinh[x]^2)/(1 - Sinh[x]^2), x, 3, -x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]} -{(1 - Sinh[x]^2)/(1 + Sinh[x]^2), x, 4, -x + 2*Tanh[x]} - - -{(1 + Cosh[x]^2)/(1 - Cosh[x]^2), x, 4, -x + 2*Coth[x]} -{(1 - Cosh[x]^2)/(1 + Cosh[x]^2), x, 3, -x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]} - - -{(a + b*Sech[x]^2)/(c + d*Cosh[x]), x, 6, -((b*d*ArcTan[Sinh[x]])/c^2) + (2*(a*c^2 + b*d^2)*ArcTanh[(Sqrt[c - d]*Tanh[x/2])/Sqrt[c + d]])/(c^2*Sqrt[c - d]*Sqrt[c + d]) + (b*Tanh[x])/c} -{(a + b*Csch[x]^2)/(c + d*Sinh[x]), x, 7, (b*d*ArcTanh[Cosh[x]])/c^2 - (2*(a*c^2 + b*d^2)*ArcTanh[(d - c*Tanh[x/2])/Sqrt[c^2 + d^2]])/(c^2*Sqrt[c^2 + d^2]) - (b*Coth[x])/c} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a Hyper[c+d x] + b Hyper[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a Hyper[c+d x] + b Hyper[c+d x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Cosh[c+d x] + b Sinh[c+d x])^n*) - - -{(a*Cosh[x] + b*Sinh[x]), x, 3, b*Cosh[x] + a*Sinh[x]} -{(a*Cosh[x] + b*Sinh[x])^2, x, 2, (1/2)*(a^2 - b^2)*x + (1/2)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])} -{(a*Cosh[x] + b*Sinh[x])^3, x, 2, (a^2 - b^2)*(b*Cosh[x] + a*Sinh[x]) + (1/3)*(b*Cosh[x] + a*Sinh[x])^3} -{(a*Cosh[x] + b*Sinh[x])^4, x, 3, (3/8)*(a^2 - b^2)^2*x + (3/8)*(a^2 - b^2)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x]) + (1/4)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])^3} -{(a*Cosh[x] + b*Sinh[x])^5, x, 3, (a^2 - b^2)^2*(b*Cosh[x] + a*Sinh[x]) + (2/3)*(a^2 - b^2)*(b*Cosh[x] + a*Sinh[x])^3 + (1/5)*(b*Cosh[x] + a*Sinh[x])^5} - -{1/(a*Cosh[x] + b*Sinh[x]), x, 2, ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]]/Sqrt[a^2 - b^2]} -{1/(a*Cosh[x] + b*Sinh[x])^2, x, 1, Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))} -{1/(a*Cosh[x] + b*Sinh[x])^3, x, 3, ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]]/(2*(a^2 - b^2)^(3/2)) + (b*Cosh[x] + a*Sinh[x])/(2*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^2)} -{1/(a*Cosh[x] + b*Sinh[x])^4, x, 2, (b*Cosh[x] + a*Sinh[x])/(3*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^3) + (2*Sinh[x])/(3*a*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x]))} -{1/(a*Cosh[x] + b*Sinh[x])^5, x, 4, (3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(5/2)) + (b*Cosh[x] + a*Sinh[x])/(4*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^4) + (3*(b*Cosh[x] + a*Sinh[x]))/(8*(a^2 - b^2)^2*(a*Cosh[x] + b*Sinh[x])^2)} - -{(a*Cosh[x] + b*Sinh[x])^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])} -{(a*Cosh[x] + b*Sinh[x])^(3/2), x, 3, (2/3)*(b*Cosh[x] + a*Sinh[x])*Sqrt[a*Cosh[x] + b*Sinh[x]] - (2*I*(a^2 - b^2)*EllipticF[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(3*Sqrt[a*Cosh[x] + b*Sinh[x]])} -{(a*Cosh[x] + b*Sinh[x])^(5/2), x, 3, (2/5)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])^(3/2) - (6*I*(a^2 - b^2)*EllipticE[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/(5*Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])} - -{1/(a*Cosh[x] + b*Sinh[x])^(1/2), x, 2, -((2*I*EllipticF[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/Sqrt[a*Cosh[x] + b*Sinh[x]])} -{1/(a*Cosh[x] + b*Sinh[x])^(3/2), x, 3, (2*(b*Cosh[x] + a*Sinh[x]))/((a^2 - b^2)*Sqrt[a*Cosh[x] + b*Sinh[x]]) + (2*I*EllipticE[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/((a^2 - b^2)*Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])} -{1/(a*Cosh[x] + b*Sinh[x])^(5/2), x, 3, (2*(b*Cosh[x] + a*Sinh[x]))/(3*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^(3/2)) - (2*I*EllipticF[(1/2)*(I*x - ArcTan[a, (-I)*b]), 2]*Sqrt[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(3*(a^2 - b^2)*Sqrt[a*Cosh[x] + b*Sinh[x]])} - - -{(a*Cosh[c + d*x] + a*Sinh[c + d*x]), x, 3, (a*Cosh[c + d*x])/d + (a*Sinh[c + d*x])/d} -{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2, x, 1, (a*Cosh[c + d*x] + a*Sinh[c + d*x])^2/(2*d)} -{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3, x, 1, (a*Cosh[c + d*x] + a*Sinh[c + d*x])^3/(3*d)} -{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^n, x, 1, (a*Cosh[c + d*x] + a*Sinh[c + d*x])^n/(d*n)} - -{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x]), x, 1, -(1/(d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])))} -{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2, x, 1, -(1/(2*d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2))} -{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3, x, 1, -(1/(3*d*(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3))} - -{Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]], x, 1, (2*Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]])/d} -{1/Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]], x, 1, -(2/(d*Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]]))} - - -{(a*Cosh[c + d*x] - a*Sinh[c + d*x]), x, 3, -((a*Cosh[c + d*x])/d) + (a*Sinh[c + d*x])/d} -{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2, x, 1, -((a*Cosh[c + d*x] - a*Sinh[c + d*x])^2/(2*d))} -{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^3, x, 1, -((a*Cosh[c + d*x] - a*Sinh[c + d*x])^3/(3*d))} -{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^n, x, 1, -((a*Cosh[c + d*x] - a*Sinh[c + d*x])^n/(d*n))} - -{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x]), x, 1, 1/(d*(a*Cosh[c + d*x] - a*Sinh[c + d*x]))} -{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2, x, 1, 1/(2*d*(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2)} -{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x])^3, x, 1, 1/(3*d*(a*Cosh[c + d*x] - a*Sinh[c + d*x])^3)} - -{Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]], x, 1, -((2*Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]])/d)} -{1/Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]], x, 1, 2/(d*Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]])} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Sech[c+d x] + b Tanh[c+d x])^n*) - - -{(a*Sech[x] + b*Tanh[x])^5, x, 8, (1/8)*a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]] + b^5*Log[Cosh[x]] - (1/8)*a*b^2*(3*a^2 + 7*b^2)*Sinh[x] - (1/4)*Sech[x]^4*(b - a*Sinh[x])*(a + b*Sinh[x])^4 - (1/8)*Sech[x]^2*(a + b*Sinh[x])^2*(2*b*(a^2 + 2*b^2) - a*(3*a^2 + 5*b^2)*Sinh[x])} -{(a*Sech[x] + b*Tanh[x])^4, x, 4, b^4*x - (4/3)*a*b*(a^2 + 2*b^2)*Cosh[x] - (1/3)*b^2*(2*a^2 + 3*b^2)*Cosh[x]*Sinh[x] - (1/3)*Sech[x]^3*(b - a*Sinh[x])*(a + b*Sinh[x])^3 + (1/3)*Sech[x]*(a + b*Sinh[x])^2*(a*b + (2*a^2 + 3*b^2)*Sinh[x])} -{(a*Sech[x] + b*Tanh[x])^3, x, 7, (1/2)*a*(a^2 + 3*b^2)*ArcTan[Sinh[x]] + b^3*Log[Cosh[x]] - (1/2)*a*b^2*Sinh[x] - (1/2)*Sech[x]^2*(b - a*Sinh[x])*(a + b*Sinh[x])^2} -{(a*Sech[x] + b*Tanh[x])^2, x, 4, b^2*x - a*b*Cosh[x] - Sech[x]*(b - a*Sinh[x])*(a + b*Sinh[x])} -{(a*Sech[x] + b*Tanh[x]), x, 3, a*ArcTan[Sinh[x]] + b*Log[Cosh[x]]} -{1/(a*Sech[x] + b*Tanh[x]), x, 3, Log[a + b*Sinh[x]]/b} -{1/(a*Sech[x] + b*Tanh[x])^2, x, 6, x/b^2 + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Cosh[x]/(b*(a + b*Sinh[x]))} -{1/(a*Sech[x] + b*Tanh[x])^3, x, 4, Log[a + b*Sinh[x]]/b^3 - (a^2 + b^2)/(2*b^3*(a + b*Sinh[x])^2) + (2*a)/(b^3*(a + b*Sinh[x]))} -{1/(a*Sech[x] + b*Tanh[x])^4, x, 8, x/b^4 + (a*(2*a^2 + 3*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*(a^2 + b^2)^(3/2)) - Cosh[x]^3/(3*b*(a + b*Sinh[x])^3) + (a*Cosh[x]^3)/(2*b*(a^2 + b^2)*(a + b*Sinh[x])^2) - (Cosh[x]*(2*(a^2 + b^2) + a*b*Sinh[x]))/(2*b^3*(a^2 + b^2)*(a + b*Sinh[x]))} -{1/(a*Sech[x] + b*Tanh[x])^5, x, 4, Log[a + b*Sinh[x]]/b^5 - (a^2 + b^2)^2/(4*b^5*(a + b*Sinh[x])^4) + (4*a*(a^2 + b^2))/(3*b^5*(a + b*Sinh[x])^3) - (3*a^2 + b^2)/(b^5*(a + b*Sinh[x])^2) + (4*a)/(b^5*(a + b*Sinh[x]))} - - -{(Sech[x] + I*Tanh[x])^5, x, 4, I*Log[I + Sinh[x]] - (2*I)/(1 - I*Sinh[x])^2 + (4*I)/(1 - I*Sinh[x])} -{(Sech[x] + I*Tanh[x])^4, x, 5, x - (2*I*Cosh[x]^3)/(3*(1 - I*Sinh[x])^3) + (2*I*Cosh[x])/(1 - I*Sinh[x])} -{(Sech[x] + I*Tanh[x])^3, x, 4, (-I)*Log[I + Sinh[x]] - (2*I)/(1 - I*Sinh[x])} -{(Sech[x] + I*Tanh[x])^2, x, 4, -x - (2*I*Cosh[x])/(1 - I*Sinh[x])} -{(Sech[x] + I*Tanh[x]), x, 3, ArcTan[Sinh[x]] + I*Log[Cosh[x]]} -{1/(Sech[x] + I*Tanh[x]), x, 3, (-I)*Log[I - Sinh[x]]} -{1/(Sech[x] + I*Tanh[x])^2, x, 3, -x + (2*I*Cosh[x])/(1 + I*Sinh[x])} -{1/(Sech[x] + I*Tanh[x])^3, x, 4, I*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])} -{1/(Sech[x] + I*Tanh[x])^4, x, 4, x + (2*I*Cosh[x]^3)/(3*(1 + I*Sinh[x])^3) - (2*I*Cosh[x])/(1 + I*Sinh[x])} -{1/(Sech[x] + I*Tanh[x])^5, x, 4, (-I)*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])^2 - (4*I)/(1 + I*Sinh[x])} - - -{(Sech[x] - I*Tanh[x])^5, x, 4, (-I)*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])^2 - (4*I)/(1 + I*Sinh[x])} -{(Sech[x] - I*Tanh[x])^4, x, 5, x + (2*I*Cosh[x]^3)/(3*(1 + I*Sinh[x])^3) - (2*I*Cosh[x])/(1 + I*Sinh[x])} -{(Sech[x] - I*Tanh[x])^3, x, 4, I*Log[I - Sinh[x]] + (2*I)/(1 + I*Sinh[x])} -{(Sech[x] - I*Tanh[x])^2, x, 4, -x + (2*I*Cosh[x])/(1 + I*Sinh[x])} -{(Sech[x] - I*Tanh[x]), x, 3, ArcTan[Sinh[x]] - I*Log[Cosh[x]]} -{1/(Sech[x] - I*Tanh[x]), x, 3, I*Log[I + Sinh[x]]} -{1/(Sech[x] - I*Tanh[x])^2, x, 3, -x - (2*I*Cosh[x])/(1 - I*Sinh[x])} -{1/(Sech[x] - I*Tanh[x])^3, x, 4, (-I)*Log[I + Sinh[x]] - (2*I)/(1 - I*Sinh[x])} -{1/(Sech[x] - I*Tanh[x])^4, x, 4, x - (2*I*Cosh[x]^3)/(3*(1 - I*Sinh[x])^3) + (2*I*Cosh[x])/(1 - I*Sinh[x])} -{1/(Sech[x] - I*Tanh[x])^5, x, 4, I*Log[I + Sinh[x]] - (2*I)/(1 - I*Sinh[x])^2 + (4*I)/(1 - I*Sinh[x])} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Coth[c+d x] + b Csch[c+d x])^n*) - - -{(a*Coth[x] + b*Csch[x])^5, x, 8, (-(1/8))*b*(15*a^4 - 10*a^2*b^2 + 3*b^4)*ArcTanh[Cosh[x]] + (1/8)*a^2*b*(7*a^2 - 3*b^2)*Cosh[x] - (1/8)*(b + a*Cosh[x])^2*(2*a*(2*a^2 - b^2) + b*(5*a^2 - 3*b^2)*Cosh[x])*Csch[x]^2 - (1/4)*(b + a*Cosh[x])^4*(a + b*Cosh[x])*Csch[x]^4 + a^5*Log[Sinh[x]]} -{(a*Coth[x] + b*Csch[x])^4, x, 4, a^4*x - (1/3)*(b + a*Cosh[x])^2*(a*b + (3*a^2 - 2*b^2)*Cosh[x])*Csch[x] - (1/3)*(b + a*Cosh[x])^3*(a + b*Cosh[x])*Csch[x]^3 + (4/3)*a*b*(2*a^2 - b^2)*Sinh[x] + (1/3)*a^2*(3*a^2 - 2*b^2)*Cosh[x]*Sinh[x]} -{(a*Coth[x] + b*Csch[x])^3, x, 7, (-(1/2))*b*(3*a^2 - b^2)*ArcTanh[Cosh[x]] + (1/2)*a^2*b*Cosh[x] - (1/2)*(b + a*Cosh[x])^2*(a + b*Cosh[x])*Csch[x]^2 + a^3*Log[Sinh[x]]} -{(a*Coth[x] + b*Csch[x])^2, x, 4, a^2*x - (b + a*Cosh[x])*(a + b*Cosh[x])*Csch[x] + a*b*Sinh[x]} -{(a*Coth[x] + b*Csch[x])^1, x, 3, (-b)*ArcTanh[Cosh[x]] + a*Log[Sinh[x]]} -{1/(a*Coth[x] + b*Csch[x])^1, x, 3, Log[b + a*Cosh[x]]/a} -{1/(a*Coth[x] + b*Csch[x])^2, x, 5, x/a^2 - (2*b*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) - Sinh[x]/(a*(b + a*Cosh[x]))} -{1/(a*Coth[x] + b*Csch[x])^3, x, 4, (a^2 - b^2)/(2*a^3*(b + a*Cosh[x])^2) + (2*b)/(a^3*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^3} -{1/(a*Coth[x] + b*Csch[x])^4, x, 7, x/a^4 - (b*(3*a^2 - 2*b^2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)) - ((2*(a^2 - b^2) - a*b*Cosh[x])*Sinh[x])/(2*a^3*(a^2 - b^2)*(b + a*Cosh[x])) - Sinh[x]^3/(3*a*(b + a*Cosh[x])^3) - (b*Sinh[x]^3)/(2*a*(a^2 - b^2)*(b + a*Cosh[x])^2)} -{1/(a*Coth[x] + b*Csch[x])^5, x, 4, -((a^2 - b^2)^2/(4*a^5*(b + a*Cosh[x])^4)) - (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cosh[x])^3) + (a^2 - 3*b^2)/(a^5*(b + a*Cosh[x])^2) + (4*b)/(a^5*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^5} - - -{(Coth[x] + Csch[x])^5, x, 4, -(2/(1 - Cosh[x])^2) + 4/(1 - Cosh[x]) + Log[1 - Cosh[x]]} -{(Coth[x] + Csch[x])^4, x, 5, x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)} -{(Coth[x] + Csch[x])^3, x, 4, 2/(1 - Cosh[x]) + Log[1 - Cosh[x]]} -{(Coth[x] + Csch[x])^2, x, 4, x + (2*Sinh[x])/(1 - Cosh[x])} -{(Coth[x] + Csch[x])^1, x, 3, -ArcTanh[Cosh[x]] + Log[Sinh[x]]} -{1/(Coth[x] + Csch[x])^1, x, 3, Log[1 + Cosh[x]]} -{1/(Coth[x] + Csch[x])^2, x, 3, x - (2*Sinh[x])/(1 + Cosh[x])} -{1/(Coth[x] + Csch[x])^3, x, 4, 2/(1 + Cosh[x]) + Log[1 + Cosh[x]]} -{1/(Coth[x] + Csch[x])^4, x, 4, x - (2*Sinh[x])/(1 + Cosh[x]) - (2*Sinh[x]^3)/(3*(1 + Cosh[x])^3)} -{1/(Coth[x] + Csch[x])^5, x, 4, -(2/(1 + Cosh[x])^2) + 4/(1 + Cosh[x]) + Log[1 + Cosh[x]]} - - -{(-Coth[x] + Csch[x])^5, x, 4, 2/(1 + Cosh[x])^2 - 4/(1 + Cosh[x]) - Log[1 + Cosh[x]]} -{(-Coth[x] + Csch[x])^4, x, 5, x - (2*Sinh[x])/(1 + Cosh[x]) - (2*Sinh[x]^3)/(3*(1 + Cosh[x])^3)} -{(-Coth[x] + Csch[x])^3, x, 4, -(2/(1 + Cosh[x])) - Log[1 + Cosh[x]]} -{(-Coth[x] + Csch[x])^2, x, 4, x - (2*Sinh[x])/(1 + Cosh[x])} -{(-Coth[x] + Csch[x])^1, x, 3, -ArcTanh[Cosh[x]] - Log[Sinh[x]]} -{1/(-Coth[x] + Csch[x])^1, x, 3, -Log[1 - Cosh[x]]} -{1/(-Coth[x] + Csch[x])^2, x, 3, x + (2*Sinh[x])/(1 - Cosh[x])} -{1/(-Coth[x] + Csch[x])^3, x, 4, -(2/(1 - Cosh[x])) - Log[1 - Cosh[x]]} -{1/(-Coth[x] + Csch[x])^4, x, 4, x + (2*Sinh[x])/(1 - Cosh[x]) + (2*Sinh[x]^3)/(3*(1 - Cosh[x])^3)} -{1/(-Coth[x] + Csch[x])^5, x, 4, 2/(1 - Cosh[x])^2 - 4/(1 - Cosh[x]) - Log[1 - Cosh[x]]} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Csch[c+d x] + b Sinh[c+d x])^n*) - - -(* Note that Csch[x]+Sinh[x] == Cosh[x]*Coth[x] *) -{(Csch[x] + Sinh[x]), x, 3, -ArcTanh[Cosh[x]] + Cosh[x]} -{(Csch[x] + Sinh[x])^2, x, 4, (3*x)/2 - (3*Coth[x])/2 + (1/2)*Cosh[x]^2*Coth[x]} -{(Csch[x] + Sinh[x])^3, x, 6, (-(5/2))*ArcTanh[Cosh[x]] + (5*Cosh[x])/2 + (5*Cosh[x]^3)/6 - (1/2)*Cosh[x]^3*Coth[x]^2} - -{(Csch[x] + Sinh[x])^(1/2), x, 4, 2*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]} -{(Csch[x] + Sinh[x])^(3/2), x, 5, (2/3)*Cosh[x]*Sqrt[Cosh[x]*Coth[x]] - (8/3)*Sqrt[Cosh[x]*Coth[x]]*Sech[x]} -{(Csch[x] + Sinh[x])^(5/2), x, 6, (-(16/15))*Coth[x]*Sqrt[Cosh[x]*Coth[x]] + (2/5)*Cosh[x]^2*Coth[x]*Sqrt[Cosh[x]*Coth[x]] + (64/15)*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Sech[c+d x] + b Cosh[c+d x])^n*) - - -(* Note that Sech[x]-Cosh[x] == -Sinh[x]*Tanh[x] *) -{(Sech[x] - Cosh[x]), x, 3, ArcTan[Sinh[x]] - Sinh[x]} -{(Sech[x] - Cosh[x])^2, x, 4, -((3*x)/2) + (3*Tanh[x])/2 + (1/2)*Sinh[x]^2*Tanh[x]} -{(Sech[x] - Cosh[x])^3, x, 6, (-(5/2))*ArcTan[Sinh[x]] + (5*Sinh[x])/2 - (5*Sinh[x]^3)/6 + (1/2)*Sinh[x]^3*Tanh[x]^2} - -{(Sech[x] - Cosh[x])^(1/2), x, 3, 2*Coth[x]*Sqrt[(-Sinh[x])*Tanh[x]]} -{(Sech[x] - Cosh[x])^(3/2), x, 4, (-(8/3))*Csch[x]*Sqrt[(-Sinh[x])*Tanh[x]] - (2/3)*Sinh[x]*Sqrt[(-Sinh[x])*Tanh[x]]} -{(Sech[x] - Cosh[x])^(5/2), x, 5, (-(64/15))*Coth[x]*Sqrt[(-Sinh[x])*Tanh[x]] + (16/15)*Tanh[x]*Sqrt[(-Sinh[x])*Tanh[x]] + (2/5)*Sinh[x]^2*Tanh[x]*Sqrt[(-Sinh[x])*Tanh[x]]} - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form (a Sinh[c+d x] + b Tanh[c+d x])^n*) - - -{1/(Sinh[x] + Tanh[x]), x, 6, (-(1/2))*ArcTanh[Cosh[x]] - 1/(2*(1 + Cosh[x])), (-(1/2))*ArcTanh[Cosh[x]] - (1/2)*Coth[x]*Csch[x] + Csch[x]^2/2} -{1/(Sinh[x] - Tanh[x]), x, 6, (-(1/2))*ArcTanh[Cosh[x]] + 1/(2*(1 - Cosh[x])), (-(1/2))*ArcTanh[Cosh[x]] - (1/2)*Coth[x]*Csch[x] - Csch[x]^2/2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a Hyper[x] + b Hyper[x])*) - - -(* ::Subsubsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a Cosh[x] + b Sinh[x])*) - - -{Sinh[x]/(a*Cosh[x] + b*Sinh[x]), x, 2, -((b*x)/(a^2 - b^2)) + (a*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{Sinh[x]^2/(a*Cosh[x] + b*Sinh[x]), x, 4, -((a^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)} -{Sinh[x]^3/(a*Cosh[x] + b*Sinh[x]), x, 5, (a^2*b*x)/(a^2 - b^2)^2 + (b*x)/(2*(a^2 - b^2)) - (a^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 - (b*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)) + (a*Sinh[x]^2)/(2*(a^2 - b^2))} - - -{Cosh[x]/(a*Cosh[x] + b*Sinh[x]), x, 2, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{Cosh[x]^2/(a*Cosh[x] + b*Sinh[x]), x, 4, -((b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)} -{Cosh[x]^3/(a*Cosh[x] + b*Sinh[x]), x, 5, -((a*b^2*x)/(a^2 - b^2)^2) + (a*x)/(2*(a^2 - b^2)) - (b*Cosh[x]^2)/(2*(a^2 - b^2)) + (b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + (a*Cosh[x]*Sinh[x])/(2*(a^2 - b^2))} - - -{Tanh[x]/(a*Sinh[x] + b*Cosh[x]), x, 5, ArcTan[Sinh[x]]/a + (b*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])} - - -{Coth[x]/(a*Sinh[x] + b*Cosh[x]), x, 5, -(ArcTanh[Cosh[x]]/b) + (a*ArcTanh[(a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])} - - -(* ::Subsubsection:: *) -(*Integrands of the form Hyper[x]^m / (a Sech[x] + b Tanh[x])*) - - -(* ::Subsubsection:: *) -(*Integrands of the form Hyper[x]^m / (a Csch[x] + b Coth[x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a Hyper[x] + b Hyper[x])^2*) - - -{Sinh[x]/(a*Cosh[x] + b*Sinh[x])^2, x, 3, -((b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - a/((a^2 - b^2)*(a*Cosh[x] + b*Sinh[x]))} -{Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2, x, 4, ((a^2 + b^2)*x)/(a^2 - b^2)^2 - a/((a^2 - b^2)*(b + a*Coth[x])) - (2*a*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2} -{Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^2, x, 16, (3*a^2*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ((2*a^2 + b^2)*Cosh[x])/((-a^2)*b^2 + b^4) + (a*(a^2 + 2*b^2)*Sinh[x])/(b^3*(a^2 - b^2)) - a^3/(b^3*(a + b)^2*(1 - Tanh[x/2])) + a^3/((a - b)^2*b^3*(1 + Tanh[x/2])) + (2*a^2*(a + b*Tanh[x/2]))/((a^2 - b^2)^2*(a + 2*b*Tanh[x/2] + a*Tanh[x/2]^2)), -((3*a^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2))) + (2*a^2*b*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (2*a^2*(3*a^2 - b^2)*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(5/2)) + Cosh[x]/b^2 - (3*a^2*Cosh[x])/(b^2*(a^2 - b^2)) - (2*a*Sinh[x])/b^3 + (3*a^3*Sinh[x])/(b^3*(a^2 - b^2)) - a^3/(b^3*(a + b)^2*(1 - Tanh[x/2])) + a^3/((a - b)^2*b^3*(1 + Tanh[x/2])) + (2*a^2*(a + b*Tanh[x/2]))/((a^2 - b^2)^2*(a + 2*b*Tanh[x/2] + a*Tanh[x/2]^2))} - - -{Cosh[x]/(a*Cosh[x] + b*Sinh[x])^2, x, 3, (a*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + b/((a^2 - b^2)*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]^2/(a*Cosh[x] + b*Sinh[x])^2, x, 4, ((a^2 + b^2)*x)/(a^2 - b^2)^2 - (2*a*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + b/((a^2 - b^2)*(a + b*Tanh[x]))} -{Cosh[x]^3/(a*Cosh[x] + b*Sinh[x])^2, x, 8, -((3*a*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) + 1/((a + b)^2*(1 - Tanh[x/2])) - 1/((a - b)^2*(1 + Tanh[x/2])) - (2*b^3*(a + b*Tanh[x/2]))/(a*(a^2 - b^2)^2*(a + 2*b*Tanh[x/2] + a*Tanh[x/2]^2)), -((2*b^4*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2))) - (2*b^2*(3*a^2 - b^2)*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) + 1/((a + b)^2*(1 - Tanh[x/2])) - 1/((a - b)^2*(1 + Tanh[x/2])) - (2*b^3*(a + b*Tanh[x/2]))/(a*(a^2 - b^2)^2*(a + 2*b*Tanh[x/2] + a*Tanh[x/2]^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a Hyper[x] + b Hyper[x])^3*) - - -{Sinh[x]/(a*Cosh[x] + b*Sinh[x])^3, x, 2, Tanh[x]^2/(2*a*(a + b*Tanh[x])^2)} -(* {Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^3, x, 0, -(((a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (a*(3*a*b*Cosh[x] + (a^2 + 4*b^2)*Sinh[x]))/(2*(a^2 + b^2)^2*(a*Cosh[x] + b*Sinh[x])^2)} *) -{Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^3, x, 5, -((b*(3*a^2 + b^2)*x)/(a^2 - b^2)^3) - a/(2*(a^2 - b^2)*(b + a*Coth[x])^2) + (2*a*b)/((a^2 - b^2)^2*(b + a*Coth[x])) + (a*(a^2 + 3*b^2)*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3} - - -{Cosh[x]/(a*Cosh[x] + b*Sinh[x])^3, x, 2, -(Coth[x]^2/(2*b*(b + a*Coth[x])^2))} -(* {Cosh[x]^2/(a*Cosh[x] + b*Sinh[x])^3, x, 0, ((2*a^2 - b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (b*((4*a^2 + b^2)*Cosh[x] + 3*a*b*Sinh[x]))/(2*(a^2 + b^2)^2*(a*Cosh[x] + b*Sinh[x])^2)} *) -{Cosh[x]^3/(a*Cosh[x] + b*Sinh[x])^3, x, 5, (a*(a^2 + 3*b^2)*x)/(a^2 - b^2)^3 - (b*(3*a^2 + b^2)*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + b/(2*(a^2 - b^2)*(a + b*Tanh[x])^2) + (2*a*b)/((a^2 - b^2)^2*(a + b*Tanh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m Hyper[x]^n (a Hyper[x] + b Hyper[x])^p*) - - -{Cosh[x]*Sinh[x]/(a*Cosh[x] + b*Sinh[x]), x, 5, (a*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2)} -{Cosh[x]*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x]), x, 7, -((a*b^2*x)/(a^2 - b^2)^2) - (a*x)/(2*(a^2 - b^2)) + (a^2*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + (a*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)) - (b*Sinh[x]^2)/(2*(a^2 - b^2))} -{Cosh[x]*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x]), x, 9, -((a^3*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - (a*b^2*Cosh[x])/(a^2 - b^2)^2 - (a*Cosh[x])/(a^2 - b^2) + (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a^2*b*Sinh[x])/(a^2 - b^2)^2 - (b*Sinh[x]^3)/(3*(a^2 - b^2))} - -{Cosh[x]^2*Sinh[x]/(a*Cosh[x] + b*Sinh[x]), x, 7, (a^2*b*x)/(a^2 - b^2)^2 - (b*x)/(2*(a^2 - b^2)) - (a*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 - (b*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)) + (a*Sinh[x]^2)/(2*(a^2 - b^2))} -{Cosh[x]^2*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x]), x, 10, (a^2*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a^2*b*Cosh[x])/(a^2 - b^2)^2 - (b*Cosh[x]^3)/(3*(a^2 - b^2)) - (a*b^2*Sinh[x])/(a^2 - b^2)^2 + (a*Sinh[x]^3)/(3*(a^2 - b^2))} -{Cosh[x]^2*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x]), x, 13, -((a^2*b^3*x)/(a^2 - b^2)^3) - (a^2*b*x)/(2*(a^2 - b^2)^2) + (b*x)/(8*(a^2 - b^2)) + (a^3*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + (a^2*b*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) + (b*Cosh[x]*Sinh[x])/(8*(a^2 - b^2)) - (b*Cosh[x]^3*Sinh[x])/(4*(a^2 - b^2)) - (a*b^2*Sinh[x]^2)/(2*(a^2 - b^2)^2) + (a*Sinh[x]^4)/(4*(a^2 - b^2))} - -{Cosh[x]^3*Sinh[x]/(a*Cosh[x] + b*Sinh[x]), x, 9, -((a*b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - (a*b^2*Cosh[x])/(a^2 - b^2)^2 + (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a^2*b*Sinh[x])/(a^2 - b^2)^2 - (b*Sinh[x])/(a^2 - b^2) - (b*Sinh[x]^3)/(3*(a^2 - b^2))} -{Cosh[x]^3*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x]), x, 13, (a^3*b^2*x)/(a^2 - b^2)^3 - (a*b^2*x)/(2*(a^2 - b^2)^2) - (a*x)/(8*(a^2 - b^2)) - (b*Cosh[x]^4)/(4*(a^2 - b^2)) - (a^2*b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) - (a*Cosh[x]*Sinh[x])/(8*(a^2 - b^2)) + (a*Cosh[x]^3*Sinh[x])/(4*(a^2 - b^2)) + (a^2*b*Sinh[x]^2)/(2*(a^2 - b^2)^2)} -{Cosh[x]^3*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x]), x, 17, (a^3*b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (a^3*b^2*Cosh[x])/(a^2 - b^2)^3 - (a*b^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a*Cosh[x]^5)/(5*(a^2 - b^2)) - (a^2*b^3*Sinh[x])/(a^2 - b^2)^3 + (a^2*b*Sinh[x]^3)/(3*(a^2 - b^2)^2) - (b*Sinh[x]^3)/(3*(a^2 - b^2)) - (b*Sinh[x]^5)/(5*(a^2 - b^2))} - - -{Cosh[x]*Sinh[x]/(a*Cosh[x] + b*Sinh[x])^2, x, 6, -((2*a*b*x)/(a^2 - b^2)^2) + (a^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + (b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + (b*Sinh[x])/((a^2 - b^2)*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2, x, 13, -((a^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - (2*a*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (2*a*b*Cosh[x])/(a^2 - b^2)^2 + (a^2*Sinh[x])/(a^2 - b^2)^2 + (b^2*Sinh[x])/(a^2 - b^2)^2 - (a^2*b)/((a^2 - b^2)^2*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^2, x, 17, (a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 + (a*b*x)/(a^2 - b^2)^2 + (a*b*(a^2 + b^2)*x)/(a^2 - b^2)^3 - (a^2*b)/((a^2 - b^2)^2*(b + a*Coth[x])) - (a^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (3*a^2*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^2 + (a^2*Sinh[x]^2)/(2*(a^2 - b^2)^2) + (b^2*Sinh[x]^2)/(2*(a^2 - b^2)^2)} - -{Cosh[x]^2*Sinh[x]/(a*Cosh[x] + b*Sinh[x])^2, x, 13, (2*a^2*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a^2*Cosh[x])/(a^2 - b^2)^2 + (b^2*Cosh[x])/(a^2 - b^2)^2 - (2*a*b*Sinh[x])/(a^2 - b^2)^2 + (a*b^2)/((a^2 - b^2)^2*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]^2*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2, x, 21, -((4*a^2*b^2*x)/(a^2 - b^2)^3) - (a^2*x)/(2*(a^2 - b^2)^2) + (b^2*x)/(2*(a^2 - b^2)^2) + (2*a^3*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + (2*a*b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + (a^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) + (b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) - (a*b*Sinh[x]^2)/(a^2 - b^2)^2 + (a*b^2*Sinh[x])/((a^2 - b^2)^2*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]^2*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^2, x, 33, -((2*a^4*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2)) - (3*a^2*b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - (4*a^2*b^2*Cosh[x])/(a^2 - b^2)^3 - (a^2*Cosh[x])/(a^2 - b^2)^2 + (a^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) + (b^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) + (2*a^3*b*Sinh[x])/(a^2 - b^2)^3 + (2*a*b^3*Sinh[x])/(a^2 - b^2)^3 - (2*a*b*Sinh[x]^3)/(3*(a^2 - b^2)^2) - (a^3*b^2)/((a^2 - b^2)^3*(a*Cosh[x] + b*Sinh[x]))} - -{Cosh[x]^3*Sinh[x]/(a*Cosh[x] + b*Sinh[x])^2, x, 17, (a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 - (a*b*x)/(a^2 - b^2)^2 + (a*b*(a^2 + b^2)*x)/(a^2 - b^2)^3 + (b^2*Cosh[x]^2)/(2*(a^2 - b^2)^2) - (3*a^2*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (b^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^2 + (a^2*Sinh[x]^2)/(2*(a^2 - b^2)^2) + (a*b^2)/((a^2 - b^2)^2*(a + b*Tanh[x]))} -{Cosh[x]^3*Sinh[x]^2/(a*Cosh[x] + b*Sinh[x])^2, x, 33, (3*a^3*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a*b^4*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a^3*b*Cosh[x])/(a^2 - b^2)^3 + (2*a*b^3*Cosh[x])/(a^2 - b^2)^3 - (2*a*b*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (4*a^2*b^2*Sinh[x])/(a^2 - b^2)^3 + (b^2*Sinh[x])/(a^2 - b^2)^2 + (a^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (b^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (a^2*b^3)/((a^2 - b^2)^3*(a*Cosh[x] + b*Sinh[x]))} -{Cosh[x]^3*Sinh[x]^3/(a*Cosh[x] + b*Sinh[x])^2, x, 48, -((6*a^3*b^3*x)/(a^2 - b^2)^4) - (a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 + (a*b*x)/(4*(a^2 - b^2)^2) + (b^2*Cosh[x]^4)/(4*(a^2 - b^2)^2) + (3*a^4*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (3*a^2*b^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (a^3*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^3 + (a*b^3*Cosh[x]*Sinh[x])/(a^2 - b^2)^3 + (a*b*Cosh[x]*Sinh[x])/(4*(a^2 - b^2)^2) - (a*b*Cosh[x]^3*Sinh[x])/(2*(a^2 - b^2)^2) - (2*a^2*b^2*Sinh[x]^2)/(a^2 - b^2)^3 + (a^2*Sinh[x]^4)/(4*(a^2 - b^2)^2) + (a^2*b^3*Sinh[x])/((a^2 - b^2)^3*(a*Cosh[x] + b*Sinh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Hyper[x]) / (a Hyper[x] + b Hyper[x])*) - - -{(A + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x]), x, 3, -((c*C*x)/(b^2 - c^2)) + (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] + (b*C*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^2, x, 3, -((c*C*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(b^2 - c^2)^(3/2)) - (b*C - A*c*Cosh[x] - A*b*Sinh[x])/((b^2 - c^2)*(b*Cosh[x] + c*Sinh[x]))} -{(A + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^3, x, 4, (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(2*(b^2 - c^2)^(3/2)) - (b*C - A*c*Cosh[x] - A*b*Sinh[x])/(2*(b^2 - c^2)*(b*Cosh[x] + c*Sinh[x])^2) - (c^2*C*Cosh[x] + b*c*C*Sinh[x])/((b^2 - c^2)^2*(b*Cosh[x] + c*Sinh[x]))} - - -{(A + B*Cosh[x])/(b*Cosh[x] + c*Sinh[x]), x, 3, (b*B*x)/(b^2 - c^2) + (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] - (B*c*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + B*Cosh[x])/(b*Cosh[x] + c*Sinh[x])^2, x, 3, (b*B*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(b^2 - c^2)^(3/2) + (B*c + A*c*Cosh[x] + A*b*Sinh[x])/((b^2 - c^2)*(b*Cosh[x] + c*Sinh[x]))} -{(A + B*Cosh[x])/(b*Cosh[x] + c*Sinh[x])^3, x, 4, (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(2*(b^2 - c^2)^(3/2)) + (B*c + A*c*Cosh[x] + A*b*Sinh[x])/(2*(b^2 - c^2)*(b*Cosh[x] + c*Sinh[x])^2) + (b*B*c*Cosh[x] + b^2*B*Sinh[x])/((b^2 - c^2)^2*(b*Cosh[x] + c*Sinh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Hyper[x]+C Hyper[x]) / (a Hyper[x]+b Hyper[x])*) - - -{(Cosh[x] + Sinh[x])/(Cosh[x] - Sinh[x]), x, 1, (1/2)*(Cosh[x] + Sinh[x])^2} -{(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 1, -(1/(2*(Cosh[x] + Sinh[x])^2))} -{(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]), x, 1, (-I)*Log[Cosh[x] + I*Sinh[x]]} - - -{(B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x]), x, 1, ((b*B - c*C)*x)/(b^2 - c^2) - ((B*c - b*C)*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^2, x, 3, ((b*B - c*C)*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(b^2 - c^2)^(3/2) + (B*c - b*C)/((b^2 - c^2)*(b*Cosh[x] + c*Sinh[x]))} -{(B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^3, x, 3, (B*c - b*C)/(2*(b^2 - c^2)*(b*Cosh[x] + c*Sinh[x])^2) + ((b*B - c*C)*Sinh[x])/(b*(b^2 - c^2)*(b*Cosh[x] + c*Sinh[x]))} - - -{(A + B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x]), x, 3, ((b*B - c*C)*x)/(b^2 - c^2) + (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[b^2 - c^2] - ((B*c - b*C)*Log[b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^2, x, 3, ((b*B - c*C)*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(b^2 - c^2)^(3/2) + (B*c - b*C + A*c*Cosh[x] + A*b*Sinh[x])/((b^2 - c^2)*(b*Cosh[x] + c*Sinh[x]))} -{(A + B*Cosh[x] + C*Sinh[x])/(b*Cosh[x] + c*Sinh[x])^3, x, 4, (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(2*(b^2 - c^2)^(3/2)) + (B*c - b*C + A*c*Cosh[x] + A*b*Sinh[x])/(2*(b^2 - c^2)*(b*Cosh[x] + c*Sinh[x])^2) + (c*(b*B - c*C)*Cosh[x] + b*(b*B - c*C)*Sinh[x])/((b^2 - c^2)^2*(b*Cosh[x] + c*Sinh[x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a + b Hyper[d+e x] + c Hyper[d+e x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Cosh[d+e x] + c Sinh[d+e x])^n*) - - -{(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, (1/2)*a*(2*a^2 + 3*b^2 - 3*c^2)*x + (1/6)*c*(11*a^2 + 4*b^2 - 4*c^2)*Cosh[x] + (1/6)*b*(11*a^2 + 4*b^2 - 4*c^2)*Sinh[x] + (5/6)*(a*c*Cosh[x] + a*b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x]) + (1/3)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])^2} -{(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, (1/2)*(2*a^2 + b^2 - c^2)*x + (3/2)*a*c*Cosh[x] + (3/2)*a*b*Sinh[x] + (1/2)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])} -{(a + b*Cosh[x] + c*Sinh[x]), x, 3, a*x + c*Cosh[x] + b*Sinh[x]} -{1/(a + b*Cosh[x] + c*Sinh[x]), x, 3, -((2*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/Sqrt[a^2 - b^2 + c^2])} -{1/(a + b*Cosh[x] + c*Sinh[x])^2, x, 5, -((2*a*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) - (c*Cosh[x] + b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))} -{1/(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, -(((2*a^2 + b^2 - c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2)) - (c*Cosh[x] + b*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (3*(a*c*Cosh[x] + a*b*Sinh[x]))/(2*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))} -{1/(a + b*Cosh[x] + c*Sinh[x])^4, x, 6, -((a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(7/2)) - (c*Cosh[x] + b*Sinh[x])/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^3) - (5*(a*c*Cosh[x] + a*b*Sinh[x]))/(6*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2) - (c*(11*a^2 + 4*b^2 - 4*c^2)*Cosh[x] + b*(11*a^2 + 4*b^2 - 4*c^2)*Sinh[x])/(6*(a^2 - b^2 + c^2)^3*(a + b*Cosh[x] + c*Sinh[x]))} - -{(a + a*Cosh[x] + c*Sinh[x])^3, x, 5, (1/2)*a*(5*a^2 - 3*c^2)*x + (1/6)*c*(15*a^2 - 4*c^2)*Cosh[x] + (1/6)*a*(15*a^2 - 4*c^2)*Sinh[x] + (5/6)*(a*c*Cosh[x] + a^2*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x]) + (1/3)*(c*Cosh[x] + a*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x])^2} -{(a + a*Cosh[x] + c*Sinh[x])^2, x, 4, (1/2)*(3*a^2 - c^2)*x + (3/2)*a*c*Cosh[x] + (3/2)*a^2*Sinh[x] + (1/2)*(c*Cosh[x] + a*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x])} -{(a + a*Cosh[x] + c*Sinh[x]), x, 3, a*x + c*Cosh[x] + a*Sinh[x]} -{1/(a + a*Cosh[x] + c*Sinh[x]), x, 2, Log[a + c*Tanh[x/2]]/c} -{1/(a + a*Cosh[x] + c*Sinh[x])^2, x, 4, (a*Log[a + c*Tanh[x/2]])/c^3 - (c*Cosh[x] + a*Sinh[x])/(c^2*(a + a*Cosh[x] + c*Sinh[x]))} -{1/(a + a*Cosh[x] + c*Sinh[x])^3, x, 4, ((3*a^2 - c^2)*Log[a + c*Tanh[x/2]])/(2*c^5) - (c*Cosh[x] + a*Sinh[x])/(2*c^2*(a + a*Cosh[x] + c*Sinh[x])^2) - (3*(a*c*Cosh[x] + a^2*Sinh[x]))/(2*c^4*(a + a*Cosh[x] + c*Sinh[x]))} -{1/(a + a*Cosh[x] + c*Sinh[x])^4, x, 5, (a*(5*a^2 - 3*c^2)*Log[a + c*Tanh[x/2]])/(2*c^7) - (c*Cosh[x] + a*Sinh[x])/(3*c^2*(a + a*Cosh[x] + c*Sinh[x])^3) - (5*(a*c*Cosh[x] + a^2*Sinh[x]))/(6*c^4*(a + a*Cosh[x] + c*Sinh[x])^2) - (c*(15*a^2 - 4*c^2)*Cosh[x] + a*(15*a^2 - 4*c^2)*Sinh[x])/(6*c^6*(a + a*Cosh[x] + c*Sinh[x]))} - -{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^4, x, 6, (35/8)*(b^2 - c^2)^2*x + (35/8)*c*(b^2 - c^2)^(3/2)*Cosh[x] + (35/8)*b*(b^2 - c^2)^(3/2)*Sinh[x] + (35/24)*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]) + (7/12)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2 + (1/4)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3} -{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^3, x, 5, (5/2)*(b^2 - c^2)^(3/2)*x + (5/2)*c*(b^2 - c^2)*Cosh[x] + (5/2)*b*(b^2 - c^2)*Sinh[x] + (5/6)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]) + (1/3)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2} -{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^2, x, 4, (3/2)*(b^2 - c^2)*x + (3/2)*c*Sqrt[b^2 - c^2]*Cosh[x] + (3/2)*b*Sqrt[b^2 - c^2]*Sinh[x] + (1/2)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])} -{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x]), x, 3, Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]} -{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x]), x, 1, -((c + Sqrt[b^2 - c^2]*Sinh[x])/(c*(c*Cosh[x] + b*Sinh[x])))} -{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^2, x, 2, (c*Cosh[x] + b*Sinh[x])/(3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (c + Sqrt[b^2 - c^2]*Sinh[x])/(3*c*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))} -{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^3, x, 3, (c*Cosh[x] + b*Sinh[x])/(5*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Sinh[x]))/(15*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(15*c*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))} -{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^4, x, 4, (c*Cosh[x] + b*Sinh[x])/(7*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^4) + (3*(c*Cosh[x] + b*Sinh[x]))/(35*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Sinh[x]))/(35*(b^2 - c^2)^(3/2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(35*c*(b^2 - c^2)^(3/2)*(c*Cosh[x] + b*Sinh[x]))} - - -{(a + b*Cosh[x] + c*Sinh[x])^(5/2), x, 7, (16/15)*(a*c*Cosh[x] + a*b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])^(3/2) - (2*I*(23*a^2 + 9*b^2 - 9*c^2)*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(15*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (16*I*a*(a^2 - b^2 + c^2)*EllipticF[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/(15*Sqrt[a + b*Cosh[x] + c*Sinh[x]])} -{(a + b*Cosh[x] + c*Sinh[x])^(3/2), x, 6, (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]] - (8*I*a*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(3*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (2*I*(a^2 - b^2 + c^2)*EllipticF[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/(3*Sqrt[a + b*Cosh[x] + c*Sinh[x]])} -{(a + b*Cosh[x] + c*Sinh[x])^(1/2), x, 2, -((2*I*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])} -{1/(a + b*Cosh[x] + c*Sinh[x])^(1/2), x, 2, -((2*I*EllipticF[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/Sqrt[a + b*Cosh[x] + c*Sinh[x]])} -{1/(a + b*Cosh[x] + c*Sinh[x])^(3/2), x, 3, -((2*(c*Cosh[x] + b*Sinh[x]))/((a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])) - (2*I*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/((a^2 - b^2 + c^2)*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])} -{1/(a + b*Cosh[x] + c*Sinh[x])^(5/2), x, 7, -((2*(c*Cosh[x] + b*Sinh[x]))/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(3/2))) - (8*(a*c*Cosh[x] + a*b*Sinh[x]))/(3*(a^2 - b^2 + c^2)^2*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) - (8*I*a*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(3*(a^2 - b^2 + c^2)^2*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (2*I*EllipticF[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/(3*(a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])} -{1/(a + b*Cosh[x] + c*Sinh[x])^(7/2), x, 8, -((2*(c*Cosh[x] + b*Sinh[x]))/(5*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(5/2))) - (16*(a*c*Cosh[x] + a*b*Sinh[x]))/(15*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^(3/2)) - (2*I*(23*a^2 + 9*b^2 - 9*c^2)*EllipticE[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(15*(a^2 - b^2 + c^2)^3*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])]) + (16*I*a*EllipticF[(1/2)*(I*x - ArcTan[b, (-I)*c]), (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/(15*(a^2 - b^2 + c^2)^2*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) - (2*(c*(23*a^2 + 9*b^2 - 9*c^2)*Cosh[x] + b*(23*a^2 + 9*b^2 - 9*c^2)*Sinh[x]))/(15*(a^2 - b^2 + c^2)^3*Sqrt[a + b*Cosh[x] + c*Sinh[x]])} - - -{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 3, (64*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))/(15*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) + (16/15)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2)} -{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, (8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) + (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]} -{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*(c*Cosh[x] + b*Sinh[x]))/Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]} -{1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 3, (Sqrt[2]*ArcTan[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(b^2 - c^2)^(1/4)} -{1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 4, ArcTan[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])]/(2*Sqrt[2]*(b^2 - c^2)^(3/4)) + (c*Cosh[x] + b*Sinh[x])/(2*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))} -{1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 5, (3*ArcTan[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(16*Sqrt[2]*(b^2 - c^2)^(5/4)) + (c*Cosh[x] + b*Sinh[x])/(4*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2)) + (3*(c*Cosh[x] + b*Sinh[x]))/(16*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))} - -{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 3, (64*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))/(15*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) - (16/15)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2)} -{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, -((8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])) + (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]} -{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*(c*Cosh[x] + b*Sinh[x]))/Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]} -{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 3, -((Sqrt[2]*ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(b^2 - c^2)^(1/4))} -{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 4, ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])]/(2*Sqrt[2]*(b^2 - c^2)^(3/4)) - (c*Cosh[x] + b*Sinh[x])/(2*Sqrt[b^2 - c^2]*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))} -{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 5, -((3*ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(16*Sqrt[2]*(b^2 - c^2)^(5/4))) - (c*Cosh[x] + b*Sinh[x])/(4*Sqrt[b^2 - c^2]*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2)) + (3*(c*Cosh[x] + b*Sinh[x]))/(16*(b^2 - c^2)*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Tanh[d+e x] + c Sech[d+e x])^n*) - - -{1/(a + b*Tanh[x] + c*Sech[x]), x, 5, (a*x)/(a^2 - b^2) - (2*a*c*ArcTan[(b + (a - c)*Tanh[x/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2)*Sqrt[a^2 - b^2 - c^2]) - (b*Log[c + a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)} -{1/(a + b*Coth[x] + c*Csch[x]), x, 5, (a*x)/(a^2 - b^2) + (2*a*c*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((a^2 - b^2)*Sqrt[a^2 - b^2 + c^2]) - (b*Log[I*c + I*b*Cosh[x] + I*a*Sinh[x]])/(a^2 - b^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a + b Hyper[x] + c Hyper[x])*) - - -{Sinh[x]/(a + b*Cosh[x] + c*Sinh[x]), x, 4, -((c*x)/(b^2 - c^2)) - (2*a*c*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2]) + (b*Log[a + b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{Sinh[x]/(1 + Cosh[x] + Sinh[x]), x, 1, x/2 + Cosh[x]/2 - Sinh[x]/2} - -{Sech[x]/(a + b*Tanh[x] + c*Sech[x]), x, 4, (2*ArcTan[(b + (a - c)*Tanh[x/2])/Sqrt[a^2 - b^2 - c^2]])/Sqrt[a^2 - b^2 - c^2]} -{Sech[x]^2/(a + b*Tanh[x] + c*Sech[x]), x, 10, (2*c*ArcTan[Tanh[x/2]])/(b^2 + c^2) - (2*a*c*ArcTan[(b + (a - c)*Tanh[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) - (b*Log[1 + Tanh[x/2]^2])/(b^2 + c^2) + (b*Log[a + c + 2*b*Tanh[x/2] + (a - c)*Tanh[x/2]^2])/(b^2 + c^2)} - -{Csch[x]/(2 + 2*Coth[x] + 3*Csch[x]), x, 4, (-(2/3))*ArcTanh[(1/3)*(2 - Tanh[x/2])]} -{Csch[x]/(a + b*Coth[x] + c*Csch[x]), x, 4, -((2*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/Sqrt[a^2 - b^2 + c^2])} -{Csch[x]^2/(a + b*Coth[x] + c*Csch[x]), x, 9, -((2*a*c*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2])) + Log[Tanh[x/2]]/(b + c) - (b*Log[b + c + 2*a*Tanh[x/2] + (b - c)*Tanh[x/2]^2])/(b^2 - c^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Hyper[x]) / (a + b Hyper[x] + c Hyper[x])*) - - -{(A + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x]), x, 4, -((c*C*x)/(b^2 - c^2)) - (2*(A*(b^2 - c^2) + a*c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2]) + (b*C*Log[a + b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, -((2*(a*A + c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) + (b*C - (A*c - a*C)*Cosh[x] - A*b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))} -{(A + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, -(((2*a^2*A + A*(b^2 - c^2) + 3*a*c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2)) + (b*C - (A*c - a*C)*Cosh[x] - A*b*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) + (a*b*C - (3*a*A*c - a^2*C + 2*c^2*C)*Cosh[x] - b*(3*a*A + 2*c*C)*Sinh[x])/(2*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))} - - -{(A + B*Cosh[x])/(a + b*Cosh[x] + c*Sinh[x]), x, 4, (b*B*x)/(b^2 - c^2) + (2*(a*b*B - A*(b^2 - c^2))*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2]) - (B*c*Log[a + b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + B*Cosh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, -((2*(a*A - b*B)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) - (B*c + A*c*Cosh[x] + (A*b - a*B)*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))} -{(A + B*Cosh[x])/(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, -(((2*a^2*A - 3*a*b*B + A*(b^2 - c^2))*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2)) - (B*c + A*c*Cosh[x] + (A*b - a*B)*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (a*B*c + (3*a*A - 2*b*B)*c*Cosh[x] + (3*a*A*b - a^2*B - 2*b^2*B)*Sinh[x])/(2*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (A + B Hyper[x] + C Hyper[x]) / (a + b Hyper[x] + c Hyper[x])*) - - -{(B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x]), x, 4, ((b*B - c*C)*x)/(b^2 - c^2) + (2*a*(b*B - c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2]) - ((B*c - b*C)*Log[a + b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, (2*(b*B - c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2) - (B*c - b*C - a*C*Cosh[x] - a*B*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))} -{(B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, (3*a*(b*B - c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2) - (B*c - b*C - a*C*Cosh[x] - a*B*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (a*(B*c - b*C) - (2*b*B*c + (a^2 - 2*c^2)*C)*Cosh[x] - (a^2*B + 2*b*(b*B - c*C))*Sinh[x])/(2*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))} - - -{(A + B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x]), x, 4, ((b*B - c*C)*x)/(b^2 - c^2) - (2*(A*b^2 - a*b*B - A*c^2 + a*c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^2)*Sqrt[a^2 - b^2 + c^2]) - ((B*c - b*C)*Log[a + b*Cosh[x] + c*Sinh[x]])/(b^2 - c^2)} -{(A + B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, -((2*(a*A - b*B + c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) - (B*c - b*C + (A*c - a*C)*Cosh[x] + (A*b - a*B)*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))} -{(A + B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^3, x, 5, -(((2*a^2*A + A*b^2 - 3*a*b*B - A*c^2 + 3*a*c*C)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2)) - (B*c - b*C + (A*c - a*C)*Cosh[x] + (A*b - a*B)*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (a*(B*c - b*C) + (3*a*A*c - a^2*C - 2*c*(b*B - c*C))*Cosh[x] + (3*a*A*b - a^2*B - 2*b*(b*B - c*C))*Sinh[x])/(2*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))} - -{(b^2 - c^2 + a*b*Cosh[x] + a*c*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 1, (c*Cosh[x] + b*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])} - - -{(A + C*Sinh[x])/(a + b*Cosh[x] + b*Sinh[x]), x, 1, If[$VersionNumber>=8, ((2*a*A + b*C)*x)/(2*a^2) + (C*Cosh[x])/(2*a) - (1/2)*((2*A)/a - C/b + (b*C)/a^2)*Log[a + b*Cosh[x] + b*Sinh[x]] - (C*Sinh[x])/(2*a), ((2*a*A + b*C)*x)/(2*a^2) + (C*Cosh[x])/(2*a) - ((2*a*A*b - a^2*C + b^2*C)*Log[a + b*Cosh[x] + b*Sinh[x]])/(2*a^2*b) - (C*Sinh[x])/(2*a)]} -{(A + B*Cosh[x])/(a + b*Cosh[x] + b*Sinh[x]), x, 1, ((2*a*A - b*B)*x)/(2*a^2) - (B*Cosh[x])/(2*a) - ((2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cosh[x] + b*Sinh[x]])/(2*a^2*b) + (B*Sinh[x])/(2*a)} -{(A + B*Cosh[x]+C*Sinh[x])/(a + b*Cosh[x] + b*Sinh[x]), x, 1, ((2*a*A - b*(B - C))*x)/(2*a^2) - ((2*a*A*b - b^2*(B - C) - a^2*(B + C))*Log[a + b*Cosh[x] + b*Sinh[x]])/(2*a^2*b) - ((B - C)*(Cosh[x] - Sinh[x]))/(2*a)} - -{(A + C*Sinh[x])/(a + b*Cosh[x] - b*Sinh[x]), x, 1, ((2*a*A - b*C)*x)/(2*a^2) + (C*Cosh[x])/(2*a) + ((2*a*A*b + a^2*C - b^2*C)*Log[a + b*Cosh[x] - b*Sinh[x]])/(2*a^2*b) + (C*Sinh[x])/(2*a)} -{(A + B*Cosh[x])/(a + b*Cosh[x] - b*Sinh[x]), x, 1, ((2*a*A - b*B)*x)/(2*a^2) + (B*Cosh[x])/(2*a) + ((2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cosh[x] - b*Sinh[x]])/(2*a^2*b) + (B*Sinh[x])/(2*a)} -{(A + B*Cosh[x]+C*Sinh[x])/(a + b*Cosh[x] - b*Sinh[x]), x, 1, ((2*a*A - b*(B + C))*x)/(2*a^2) + ((2*a*A*b - a^2*(B - C) - b^2*(B + C))*Log[a + b*Cosh[x] - b*Sinh[x]])/(2*a^2*b) + ((B + C)*(Cosh[x] + Sinh[x]))/(2*a)} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a Hyper[c+d x]^2 + b Hyper[c+d x]^2)^p*) - - -{1/(Cosh[x]^2 + Sinh[x]^2), x, 2, ArcTan[Tanh[x]]} -{1/(Cosh[x]^2 + Sinh[x]^2)^2, x, 2, Tanh[x]/(1 + Tanh[x]^2)} -{1/(Cosh[x]^2 + Sinh[x]^2)^3, x, 4, (1/2)*ArcTan[Tanh[x]] + (Sech[x]^2*Tanh[x])/(2*(1 + Tanh[x]^2)^2)} - -{1/(Cosh[x]^2 - Sinh[x]^2), x, 2, x} -{1/(Cosh[x]^2 - Sinh[x]^2)^2, x, 2, x} -{1/(Cosh[x]^2 - Sinh[x]^2)^3, x, 2, x} - - -{1/(Sech[x]^2 + Tanh[x]^2), x, 2, x} -{1/(Sech[x]^2 + Tanh[x]^2)^2, x, 2, x} -{1/(Sech[x]^2 + Tanh[x]^2)^3, x, 2, x} - -{1/(Sech[x]^2 - Tanh[x]^2), x, 4, -x + Sqrt[2]*ArcTanh[Sqrt[2]*Tanh[x]]} -{1/(Sech[x]^2 - Tanh[x]^2)^2, x, 6, x - ArcTanh[Sqrt[2]*Tanh[x]]/Sqrt[2] + Tanh[x]/(1 - 2*Tanh[x]^2)} -{1/(Sech[x]^2 - Tanh[x]^2)^3, x, 6, -x + (7*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) + Tanh[x]/(2*(1 - 2*Tanh[x]^2)^2) - Tanh[x]/(4*(1 - 2*Tanh[x]^2))} - - -{1/(Coth[x]^2 + Csch[x]^2), x, 4, x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]} -{1/(Coth[x]^2 + Csch[x]^2)^2, x, 6, x - ArcTanh[Tanh[x]/Sqrt[2]]/Sqrt[2] - Tanh[x]/(2 - Tanh[x]^2)} -{1/(Coth[x]^2 + Csch[x]^2)^3, x, 6, x - (7*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - Tanh[x]^3/(2*(2 - Tanh[x]^2)^2) - Tanh[x]/(4*(2 - Tanh[x]^2))} - -{1/(Coth[x]^2 - Csch[x]^2), x, 2, x} -{1/(Coth[x]^2 - Csch[x]^2)^2, x, 2, x} -{1/(Coth[x]^2 - Csch[x]^2)^3, x, 2, x} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a + b Hyper[c+d x]^m + c Hyper[c+d x]^n)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m / (a + b Hyper[x] + c Hyper[x]^2)*) - - -{1/(a + b*Sinh[x] + c*Sinh[x]^2), x, 7, -((2*Sqrt[2]*c*ArcTan[(2*I*c - I*b*Tanh[x/2] + Sqrt[-b^2 + 4*a*c]*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-b^2 + 4*a*c]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])) + (2*Sqrt[2]*c*ArcTan[(2*I*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/(Sqrt[-b^2 + 4*a*c]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])} -{Sinh[x]/(a + b*Sinh[x] + c*Sinh[x]^2), x, 8, (Sqrt[2]*(I + b/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - I*b*Tanh[x/2] + Sqrt[-b^2 + 4*a*c]*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]] + (Sqrt[2]*(I - b/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]]} -{Sinh[x]^2/(a + b*Sinh[x] + c*Sinh[x]^2), x, 9, x/c - (Sqrt[2]*(I*b + (b^2 - 2*a*c)/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - (I*b - Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/(c*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]]) - (Sqrt[2]*(I*b - (b^2 - 2*a*c)/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/(c*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])} -{Sinh[x]^3/(a + b*Sinh[x] + c*Sinh[x]^2), x, 10, -((b*x)/c^2) + (Sqrt[2]*(b^3/Sqrt[-b^2 + 4*a*c] + I*(b^2 - a*c + (3*I*a*b*c)/Sqrt[-b^2 + 4*a*c]))*ArcTan[(2*I*c - I*b*Tanh[x/2] + Sqrt[-b^2 + 4*a*c]*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/(c^2*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]]) - (Sqrt[2]*(b^3/Sqrt[-b^2 + 4*a*c] - I*(b^2 - a*c - (3*I*a*b*c)/Sqrt[-b^2 + 4*a*c]))*ArcTan[(2*I*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/(c^2*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]]) + Cosh[x]/c} - -{(a + b*Sinh[x])/(b^2 - 2*a*b*Sinh[x] + a^2*Sinh[x]^2), x, 3, Cosh[x]/(b - a*Sinh[x])} -{(d + e*Sinh[x])/(a + b*Sinh[x] + c*Sinh[x]^2), x, 7, (Sqrt[2]*(I*e - (2*c*d - b*e)/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - I*b*Tanh[x/2] + Sqrt[-b^2 + 4*a*c]*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]] + (Sqrt[2]*(I*e + (2*c*d - b*e)/Sqrt[-b^2 + 4*a*c])*ArcTan[(2*I*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]]} - - -{1/(a + b*Cosh[x] + c*Cosh[x]^2), x, 5, (4*c*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (4*c*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Cosh[x]/(a + b*Cosh[x] + c*Cosh[x]^2), x, 6, (2*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Cosh[x]^2/(a + b*Cosh[x] + c*Cosh[x]^2), x, 7, x/c - (2*(b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (2*(b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} -{Cosh[x]^3/(a + b*Cosh[x] + c*Cosh[x]^2), x, 8, -((b*x)/c^2) + (2*(b^2 - a*c - b^3/Sqrt[b^2 - 4*a*c] + (3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(c^2*Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(b^2 - a*c + b^3/Sqrt[b^2 - 4*a*c] - (3*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(c^2*Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]) + Sinh[x]/c} - -{(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*Cosh[x]^2), x, 3, Sinh[x]/(b + a*Cosh[x])} -{(d + e*Cosh[x])/(a + b*Cosh[x] + c*Cosh[x]^2), x, 5, (2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tanh[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Hyper[x]^m (a Hyper[x]^n + b Hyper[x]^n)^p*) - - -{Sinh[x]^2/(a*Cosh[x]^2 + b*Sinh[x]^2), x, 4, x/(a + b) - (Sqrt[a]*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a]])/(Sqrt[b]*(a + b))} -{Cosh[x]^2/(a*Cosh[x]^2 + b*Sinh[x]^2), x, 4, x/(a + b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Tanh[x])/Sqrt[a]])/(Sqrt[a]*(a + b))} - -{Sinh[x]^3/(Cosh[x]^3 + Sinh[x]^3), x, 6, x/2 + (2*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) + 1/(6*(1 + Tanh[x]))} -{Cosh[x]^3/(Cosh[x]^3 + Sinh[x]^3), x, 6, x/2 - (2*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/(3*Sqrt[3]) - 1/(6*(1 + Tanh[x]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Csch[x] Sech[x] (a Sech[x]^n)^p*) - - -{(x^1*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^2], x, 6, -((2*x*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2]) - (PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (PolyLog[2, E^x]*Sech[x])/Sqrt[a*Sech[x]^2]} -{(x^2*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^2], x, 8, -((2*x^2*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2]) - (2*x*PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (2*x*PolyLog[2, E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (2*PolyLog[3, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (2*PolyLog[3, E^x]*Sech[x])/Sqrt[a*Sech[x]^2]} -{(x^3*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^2], x, 10, -((2*x^3*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2]) - (3*x^2*PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (3*x^2*PolyLog[2, E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (6*x*PolyLog[3, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (6*x*PolyLog[3, E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (6*PolyLog[4, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (6*PolyLog[4, E^x]*Sech[x])/Sqrt[a*Sech[x]^2]} - - -{(x^1*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^4], x, 5, -((x^2*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4])) + (x*Log[1 - E^(2*x)]*Sech[x]^2)/Sqrt[a*Sech[x]^4] + (PolyLog[2, E^(2*x)]*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4])} -{(x^2*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^4], x, 6, -((x^3*Sech[x]^2)/(3*Sqrt[a*Sech[x]^4])) + (x^2*Log[1 - E^(2*x)]*Sech[x]^2)/Sqrt[a*Sech[x]^4] + (x*PolyLog[2, E^(2*x)]*Sech[x]^2)/Sqrt[a*Sech[x]^4] - (PolyLog[3, E^(2*x)]*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4])} -{(x^3*Csch[x]*Sech[x])/Sqrt[a*Sech[x]^4], x, 7, -((x^4*Sech[x]^2)/(4*Sqrt[a*Sech[x]^4])) + (x^3*Log[1 - E^(2*x)]*Sech[x]^2)/Sqrt[a*Sech[x]^4] + (3*x^2*PolyLog[2, E^(2*x)]*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4]) - (3*x*PolyLog[3, E^(2*x)]*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4]) + (3*PolyLog[4, E^(2*x)]*Sech[x]^2)/(4*Sqrt[a*Sech[x]^4])} - - -{(x^1*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^2], x, 10, x*Sqrt[a*Sech[x]^2] - ArcTan[Sinh[x]]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2]} -{(x^2*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^2], x, 17, x^2*Sqrt[a*Sech[x]^2] - 4*x*ArcTan[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x^2*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x*Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + 2*I*Cosh[x]*PolyLog[2, (-I)*E^x]*Sqrt[a*Sech[x]^2] - 2*I*Cosh[x]*PolyLog[2, I*E^x]*Sqrt[a*Sech[x]^2] + 2*x*Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2] + 2*Cosh[x]*PolyLog[3, -E^x]*Sqrt[a*Sech[x]^2] - 2*Cosh[x]*PolyLog[3, E^x]*Sqrt[a*Sech[x]^2]} -{(x^3*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^2], x, 21, x^3*Sqrt[a*Sech[x]^2] - 6*x^2*ArcTan[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x^3*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 3*x^2*Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + 6*I*x*Cosh[x]*PolyLog[2, (-I)*E^x]*Sqrt[a*Sech[x]^2] - 6*I*x*Cosh[x]*PolyLog[2, I*E^x]*Sqrt[a*Sech[x]^2] + 3*x^2*Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2] + 6*x*Cosh[x]*PolyLog[3, -E^x]*Sqrt[a*Sech[x]^2] - 6*I*Cosh[x]*PolyLog[3, (-I)*E^x]*Sqrt[a*Sech[x]^2] + 6*I*Cosh[x]*PolyLog[3, I*E^x]*Sqrt[a*Sech[x]^2] - 6*x*Cosh[x]*PolyLog[3, E^x]*Sqrt[a*Sech[x]^2] - 6*Cosh[x]*PolyLog[4, -E^x]*Sqrt[a*Sech[x]^2] + 6*Cosh[x]*PolyLog[4, E^x]*Sqrt[a*Sech[x]^2]} - - -{(x^1*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^4], x, 12, (1/2)*x*Cosh[x]^2*Sqrt[a*Sech[x]^4] - 2*x*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] - (1/2)*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] + (1/2)*Cosh[x]^2*PolyLog[2, E^(2*x)]*Sqrt[a*Sech[x]^4] - (1/2)*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (1/2)*x*Sqrt[a*Sech[x]^4]*Sinh[x]^2} -{(x^2*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^4], x, 16, (1/2)*x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4] - 2*x^2*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] + Cosh[x]^2*Log[Cosh[x]]*Sqrt[a*Sech[x]^4] - x*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] + x*Cosh[x]^2*PolyLog[2, E^(2*x)]*Sqrt[a*Sech[x]^4] + (1/2)*Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4] - (1/2)*Cosh[x]^2*PolyLog[3, E^(2*x)]*Sqrt[a*Sech[x]^4] - x*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (1/2)*x^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2} -{(x^3*Csch[x]*Sech[x])*Sqrt[a*Sech[x]^4], x, 21, (-(3/2))*x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4] + (1/2)*x^3*Cosh[x]^2*Sqrt[a*Sech[x]^4] - 2*x^3*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] + 3*x*Cosh[x]^2*Log[1 + E^(2*x)]*Sqrt[a*Sech[x]^4] + (3/2)*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] - (3/2)*x^2*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] + (3/2)*x^2*Cosh[x]^2*PolyLog[2, E^(2*x)]*Sqrt[a*Sech[x]^4] + (3/2)*x*Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4] - (3/2)*x*Cosh[x]^2*PolyLog[3, E^(2*x)]*Sqrt[a*Sech[x]^4] - (3/4)*Cosh[x]^2*PolyLog[4, -E^(2*x)]*Sqrt[a*Sech[x]^4] + (3/4)*Cosh[x]^2*PolyLog[4, E^(2*x)]*Sqrt[a*Sech[x]^4] - (3/2)*x^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (1/2)*x^3*Sqrt[a*Sech[x]^4]*Sinh[x]^2} - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a + b Hyper[c+d x] Hyper[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a + b Hyper[c+d x] Hyper[c+d x])^n*) - - -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^m, x, 4, (I*AppellF1[1/2, 1/2, -m, 3/2, (1/2)*(1 - I*Sinh[2*c + 2*d*x]), (b*(1 - I*Sinh[2*c + 2*d*x]))/(2*I*a + b)]*Cosh[2*c + 2*d*x]*(a + (1/2)*b*Sinh[2*c + 2*d*x])^m)/(((2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b))^m*(Sqrt[2]*d*Sqrt[1 + I*Sinh[2*c + 2*d*x]]))} - -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^3, x, 3, (1/8)*a*(8*a^2 - 3*b^2)*x + (b*(16*a^2 - b^2)*Cosh[2*c + 2*d*x])/(24*d) + (5*a*b^2*Cosh[2*c + 2*d*x]*Sinh[2*c + 2*d*x])/(48*d) + (b*Cosh[2*c + 2*d*x]*(2*a + b*Sinh[2*c + 2*d*x])^2)/(48*d)} -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^2, x, 2, (1/8)*(8*a^2 - b^2)*x + (a*b*Cosh[2*c + 2*d*x])/(2*d) + (b^2*Cosh[2*c + 2*d*x]*Sinh[2*c + 2*d*x])/(16*d)} -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^1, x, 3, a*x + (b*Sinh[c + d*x]^2)/(2*d)} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^1, x, 4, -((2*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/(Sqrt[4*a^2 + b^2]*d))} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^2, x, 6, -((8*a*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/((4*a^2 + b^2)^(3/2)*d)) - (2*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2*d*x]))} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^3, x, 7, -((4*(8*a^2 - b^2)*ArcTanh[(b - 2*a*Tanh[c + d*x])/Sqrt[4*a^2 + b^2]])/((4*a^2 + b^2)^(5/2)*d)) - (2*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2*d*x])^2) - (12*a*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)^2*d*(2*a + b*Sinh[2*c + 2*d*x]))} - - -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(5/2), x, 8, (2*Sqrt[2]*a*b*Cosh[2*c + 2*d*x]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(15*d) + (b*Cosh[2*c + 2*d*x]*(2*a + b*Sinh[2*c + 2*d*x])^(3/2))/(20*Sqrt[2]*d) - (I*(92*a^2 - 9*b^2)*EllipticE[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(60*Sqrt[2]*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]) + (2*I*Sqrt[2]*a*(4*a^2 + b^2)*EllipticF[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(15*d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])} -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(3/2), x, 7, (b*Cosh[2*c + 2*d*x]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(6*Sqrt[2]*d) - (2*I*Sqrt[2]*a*EllipticE[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(3*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]) + (I*(4*a^2 + b^2)*EllipticF[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(6*Sqrt[2]*d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])} -{(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(1/2), x, 3, -((I*EllipticE[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(Sqrt[2]*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]))} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(1/2), x, 3, -((I*Sqrt[2]*EllipticF[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]]))} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(3/2), x, 5, -((2*Sqrt[2]*b*Cosh[2*c + 2*d*x])/((4*a^2 + b^2)*d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])) - (2*I*Sqrt[2]*EllipticE[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/((4*a^2 + b^2)*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])} -{1/(a + b*Cosh[c + d*x]*Sinh[c + d*x])^(5/2), x, 8, -((4*Sqrt[2]*b*Cosh[2*c + 2*d*x])/(3*(4*a^2 + b^2)*d*(2*a + b*Sinh[2*c + 2*d*x])^(3/2))) - (32*Sqrt[2]*a*b*Cosh[2*c + 2*d*x])/(3*(4*a^2 + b^2)^2*d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]]) - (32*I*Sqrt[2]*a*EllipticE[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])/(3*(4*a^2 + b^2)^2*d*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)]) + (4*I*Sqrt[2]*EllipticF[(1/2)*(2*I*c - Pi/2 + 2*I*d*x), (2*b)/(2*I*a + b)]*Sqrt[(2*a + b*Sinh[2*c + 2*d*x])/(2*a - I*b)])/(3*(4*a^2 + b^2)*d*Sqrt[2*a + b*Sinh[2*c + 2*d*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a + b Hyper[c+d x] Hyper[c+d x])^n*) - - -{x^3/(a + b*Sinh[x]*Cosh[x]), x, 13, (x^3*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^3*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) - (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) - (3*x*PolyLog[3, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) + (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) + (3*PolyLog[4, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(4*Sqrt[4*a^2 + b^2]) - (3*PolyLog[4, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(4*Sqrt[4*a^2 + b^2])} -{x^2/(a + b*Sinh[x]*Cosh[x]), x, 11, (x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (x*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - (x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - PolyLog[3, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2])} -{x^1/(a + b*Sinh[x]*Cosh[x]), x, 9, (x*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) - PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2])} -{1/(x^1*(a + b*Sinh[x]*Cosh[x])), x, 1, Unintegrable[1/(x*(a + (1/2)*b*Sinh[2*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form F^(c (a+b x)) Hyper[d+e x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) Sinh[d+e x]^n*) - - -{F^(c*(a + b*x))*Sinh[d + e*x]^n, x, 2, -((F^(c*(a + b*x))*Hypergeometric2F1[-n, -((e*n - b*c*Log[F])/(2*e)), (1/2)*(2 - n + (b*c*Log[F])/e), E^(2*(d + e*x))]*Sinh[d + e*x]^n)/((1 - E^(2*(d + e*x)))^n*(e*n - b*c*Log[F])))} - - -{E^(2*(a + b*x))*Sinh[a + b*x]^3, x, 4, E^(-a - b*x)/(8*b) + (3*E^(a + b*x))/(8*b) - E^(3*a + 3*b*x)/(8*b) + E^(5*a + 5*b*x)/(40*b)} -{E^(2*(a + b*x))*Sinh[a + b*x]^2, x, 5, -(E^(2*a + 2*b*x)/(4*b)) + E^(4*a + 4*b*x)/(16*b) + x/4} -{E^(2*(a + b*x))*Sinh[a + b*x]^1, x, 3, -(E^(a + b*x)/(2*b)) + E^(3*a + 3*b*x)/(6*b)} -{E^(2*(a + b*x))*Csch[a + b*x]^1, x, 4, (2*E^(a + b*x))/b - (2*ArcTanh[E^(a + b*x)])/b} -{E^(2*(a + b*x))*Csch[a + b*x]^2, x, 5, 2/(b*(1 - E^(2*a + 2*b*x))) + (2*Log[1 - E^(2*a + 2*b*x)])/b} -{E^(2*(a + b*x))*Csch[a + b*x]^3, x, 5, -((2*E^(3*a + 3*b*x))/(b*(1 - E^(2*a + 2*b*x))^2)) + (3*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (3*ArcTanh[E^(a + b*x)])/b} - - -{E^(a + b*x)*Sinh[c + d*x]^3, x, 2, -((6*d^3*E^(a + b*x)*Cosh[c + d*x])/(b^4 - 10*b^2*d^2 + 9*d^4)) + (6*b*d^2*E^(a + b*x)*Sinh[c + d*x])/(b^4 - 10*b^2*d^2 + 9*d^4) - (3*d*E^(a + b*x)*Cosh[c + d*x]*Sinh[c + d*x]^2)/(b^2 - 9*d^2) + (b*E^(a + b*x)*Sinh[c + d*x]^3)/(b^2 - 9*d^2)} -{E^(a + b*x)*Sinh[c + d*x]^2, x, 2, (2*d^2*E^(a + b*x))/(b*(b^2 - 4*d^2)) - (2*d*E^(a + b*x)*Cosh[c + d*x]*Sinh[c + d*x])/(b^2 - 4*d^2) + (b*E^(a + b*x)*Sinh[c + d*x]^2)/(b^2 - 4*d^2)} -{E^(a + b*x)*Sinh[c + d*x]^1, x, 1, -((d*E^(a + b*x)*Cosh[c + d*x])/(b^2 - d^2)) + (b*E^(a + b*x)*Sinh[c + d*x])/(b^2 - d^2)} -{E^(a + b*x)*Csch[c + d*x]^1, x, 1, -((2*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*d), (1/2)*(3 + b/d), E^(2*(c + d*x))])/(b + d))} -{E^(c + d*x)*Csch[a + b*x]^2, x, 1, (4*E^(c + d*x + 2*(a + b*x))*Hypergeometric2F1[2, 1 + d/(2*b), 2 + d/(2*b), E^(2*(a + b*x))])/(2*b + d)} -{E^(c + d*x)*Csch[a + b*x]^3, x, 2, -((d*E^(c + d*x)*Csch[a + b*x])/(2*b^2)) - (E^(c + d*x)*Coth[a + b*x]*Csch[a + b*x])/(2*b) + ((b - d)*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*b), (1/2)*(3 + d/b), E^(2*(a + b*x))])/b^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) Cosh[d+e x]^n*) - - -{F^(c*(a + b*x))*Cosh[d + e*x]^n, x, 2, -((F^(c*(a + b*x))*Cosh[d + e*x]^n*Hypergeometric2F1[-n, -((e*n - b*c*Log[F])/(2*e)), (1/2)*(2 - n + (b*c*Log[F])/e), -E^(2*(d + e*x))])/((1 + E^(2*(d + e*x)))^n*(e*n - b*c*Log[F])))} - - -{E^(a + b*x)*Cosh[c + d*x]^3, x, 2, -((6*b*d^2*E^(a + b*x)*Cosh[c + d*x])/(b^4 - 10*b^2*d^2 + 9*d^4)) + (b*E^(a + b*x)*Cosh[c + d*x]^3)/(b^2 - 9*d^2) + (6*d^3*E^(a + b*x)*Sinh[c + d*x])/(b^4 - 10*b^2*d^2 + 9*d^4) - (3*d*E^(a + b*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(b^2 - 9*d^2)} -{E^(a + b*x)*Cosh[c + d*x]^2, x, 2, -((2*d^2*E^(a + b*x))/(b*(b^2 - 4*d^2))) + (b*E^(a + b*x)*Cosh[c + d*x]^2)/(b^2 - 4*d^2) - (2*d*E^(a + b*x)*Cosh[c + d*x]*Sinh[c + d*x])/(b^2 - 4*d^2)} -{E^(a + b*x)*Cosh[c + d*x]^1, x, 1, (b*E^(a + b*x)*Cosh[c + d*x])/(b^2 - d^2) - (d*E^(a + b*x)*Sinh[c + d*x])/(b^2 - d^2)} -{E^(a + b*x)*Sech[c + d*x]^1, x, 1, (2*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*d), (1/2)*(3 + b/d), -E^(2*(c + d*x))])/(b + d)} -{E^(a + b*x)*Sech[c + d*x]^2, x, 1, (4*E^(a + b*x + 2*(c + d*x))*Hypergeometric2F1[2, 1 + b/(2*d), 2 + b/(2*d), -E^(2*(c + d*x))])/(b + 2*d)} -{E^(a + b*x)*Sech[c + d*x]^3, x, 2, -(((b - d)*E^(a + c + b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*d), (1/2)*(3 + b/d), -E^(2*(c + d*x))])/d^2) + (b*E^(a + b*x)*Sech[c + d*x])/(2*d^2) + (E^(a + b*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) Sech[d+e x]^n*) - - -{F^(c*(a + b*x))*Sech[d + e*x]^n, x, 2, ((1 + E^(2*(d + e*x)))^n*F^(a*c + b*c*x)*Hypergeometric2F1[n, (e*n + b*c*Log[F])/(2*e), 1 + (e*n + b*c*Log[F])/(2*e), -E^(2*(d + e*x))]*Sech[d + e*x]^n)/(e*n + b*c*Log[F])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c (a+b x)) Csch[d+e x]^n*) - - -{F^(c*(a + b*x))*Csch[d + e*x]^n, x, 2, -(((1 - E^(-2*(d + e*x)))^n*F^(a*c + b*c*x)*Csch[d + e*x]^n*Hypergeometric2F1[n, (e*n - b*c*Log[F])/(2*e), (1/2)*(2 + n - (b*c*Log[F])/e), E^(-2*(d + e*x))])/(e*n - b*c*Log[F]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form F^(c (a+b x)) (f + g Hyper[d+e x])^n*) - - -{F^(c*(a + b*x))*(f + I*f*Sinh[d + e*x])^2, x, 8, (f^2*F^(a*c + b*c*x))/(b*c*Log[F]) + (2*I*e*f^2*F^(a*c + b*c*x)*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) + (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (2*I*b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (2*e*f^2*F^(a*c + b*c*x)*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2) + (b*c*f^2*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x]^2)/(4*e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*(f + I*f*Sinh[d + e*x])^1, x, 6, (f*F^(a*c + b*c*x))/(b*c*Log[F]) + (I*e*f*F^(a*c + b*c*x)*Cosh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) - (I*b*c*f*F^(a*c + b*c*x)*Log[F]*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))/(f + I*f*Sinh[d + e*x])^1, x, 2, (2*E^((1/2)*(2*d + I*Pi + 2*e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/e, 2 + (b*c*Log[F])/e, -E^((1/2)*(2*d + I*Pi + 2*e*x))])/(f*(e + b*c*Log[F]))} -{F^(c*(a + b*x))/(f + I*f*Sinh[d + e*x])^2, x, 3, (2*E^((1/2)*(2*d + I*Pi + 2*e*x))*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/e, 2 + (b*c*Log[F])/e, -E^((1/2)*(2*d + I*Pi + 2*e*x))]*(e - b*c*Log[F]))/(3*e^2*f^2) + (b*c*F^(c*(a + b*x))*Log[F]*Sech[d/2 + (I*Pi)/4 + (e*x)/2]^2)/(6*e^2*f^2) + (F^(c*(a + b*x))*Sech[d/2 + (I*Pi)/4 + (e*x)/2]^2*Tanh[d/2 + (I*Pi)/4 + (e*x)/2])/(6*e*f^2)} - - -{F^(c*(a + b*x))*(f + f*Cosh[d + e*x])^2, x, 8, (f^2*F^(a*c + b*c*x))/(b*c*Log[F]) - (2*b*c*f^2*F^(a*c + b*c*x)*Cosh[d + e*x]*Log[F])/(e^2 - b^2*c^2*Log[F]^2) + (2*e^2*f^2*F^(a*c + b*c*x))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*f^2*F^(a*c + b*c*x)*Cosh[d + e*x]^2*Log[F])/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2) + (2*e*f^2*F^(a*c + b*c*x)*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))*(f + f*Cosh[d + e*x])^1, x, 6, (f*F^(a*c + b*c*x))/(b*c*Log[F]) - (b*c*f*F^(a*c + b*c*x)*Cosh[d + e*x]*Log[F])/(e^2 - b^2*c^2*Log[F]^2) + (e*f*F^(a*c + b*c*x)*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)} -{F^(c*(a + b*x))/(f + f*Cosh[d + e*x])^1, x, 2, (2*E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/e, 2 + (b*c*Log[F])/e, -E^(d + e*x)])/(f*(e + b*c*Log[F]))} -{F^(c*(a + b*x))/(f + f*Cosh[d + e*x])^2, x, 3, (2*E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[2, 1 + (b*c*Log[F])/e, 2 + (b*c*Log[F])/e, -E^(d + e*x)]*(e - b*c*Log[F]))/(3*e^2*f^2) + (b*c*F^(c*(a + b*x))*Log[F]*Sech[d/2 + (e*x)/2]^2)/(6*e^2*f^2) + (F^(c*(a + b*x))*Sech[d/2 + (e*x)/2]^2*Tanh[d/2 + (e*x)/2])/(6*e*f^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[c+d x]^m Sinh[c+d x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[a+b x]^m Sinh[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{E^(a + b*x)*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 4, E^(-3*a - 3*b*x)/(48*b) - E^(-a - b*x)/(8*b) - E^(3*a + 3*b*x)/(24*b) + E^(5*a + 5*b*x)/(80*b)} -{E^(a + b*x)*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 5, -(E^(-2*a - 2*b*x)/(16*b)) - E^(2*a + 2*b*x)/(16*b) + E^(4*a + 4*b*x)/(32*b) - x/8} -{E^(a + b*x)*Cosh[a + b*x]*Sinh[a + b*x]^1, x, 4, E^(-a - b*x)/(4*b) + E^(3*a + 3*b*x)/(12*b)} -{E^(a + b*x)*Cosh[a + b*x]*Csch[a + b*x]^1, x, 3, E^(a + b*x)/b - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Cosh[a + b*x]*Csch[a + b*x]^2, x, 5, 2/(b*(1 - E^(2*a + 2*b*x))) + Log[1 - E^(2*a + 2*b*x)]/b} -{E^(a + b*x)*Cosh[a + b*x]*Csch[a + b*x]^3, x, 5, -((2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))^2)) + (3*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - ArcTanh[E^(a + b*x)]/b} - - -{E^(a + b*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 5, E^(-4*a - 4*b*x)/(128*b) - E^(-2*a - 2*b*x)/(64*b) - E^(2*a + 2*b*x)/(32*b) - E^(4*a + 4*b*x)/(128*b) + E^(6*a + 6*b*x)/(192*b) + x/16} -{E^(a + b*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 4, -(E^(-3*a - 3*b*x)/(48*b)) - E^(a + b*x)/(8*b) + E^(5*a + 5*b*x)/(80*b)} -{E^(a + b*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^1, x, 5, E^(-2*a - 2*b*x)/(16*b) + E^(2*a + 2*b*x)/(16*b) + E^(4*a + 4*b*x)/(32*b) - x/8} -{E^(a + b*x)*Cosh[a + b*x]^2*Csch[a + b*x]^1, x, 5, E^(2*a + 2*b*x)/(4*b) - x/2 + Log[1 - E^(2*a + 2*b*x)]/b} -{E^(a + b*x)*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 5, E^(a + b*x)/b + (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 5, -(2/(b*(1 - E^(2*a + 2*b*x))^2)) + 4/(b*(1 - E^(2*a + 2*b*x))) + Log[1 - E^(2*a + 2*b*x)]/b} - - -{E^(a + b*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 4, E^(-5*a - 5*b*x)/(320*b) - (3*E^(-a - b*x))/(64*b) - E^(3*a + 3*b*x)/(64*b) + E^(7*a + 7*b*x)/(448*b)} -{E^(a + b*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 5, -(E^(-4*a - 4*b*x)/(128*b)) - E^(-2*a - 2*b*x)/(64*b) - E^(2*a + 2*b*x)/(32*b) + E^(4*a + 4*b*x)/(128*b) + E^(6*a + 6*b*x)/(192*b) - x/16} -{E^(a + b*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^1, x, 4, E^(-3*a - 3*b*x)/(48*b) + E^(-a - b*x)/(8*b) + E^(3*a + 3*b*x)/(24*b) + E^(5*a + 5*b*x)/(80*b)} -{E^(a + b*x)*Cosh[a + b*x]^3*Csch[a + b*x]^1, x, 5, E^(-a - b*x)/(4*b) + E^(a + b*x)/b + E^(3*a + 3*b*x)/(12*b) - (2*ArcTanh[E^(a + b*x)])/b} -{E^(a + b*x)*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 5, E^(2*a + 2*b*x)/(4*b) + 2/(b*(1 - E^(2*a + 2*b*x))) + x/2 + Log[1 - E^(2*a + 2*b*x)]/b} -{E^(a + b*x)*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 7, E^(a + b*x)/b - (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))^2) + (3*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (3*ArcTanh[E^(a + b*x)])/b} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[n (a+b x)]^m Sinh[n (a+b x)]^p*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{E^(2*(a + b*x))*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 5, E^(-2*a - 2*b*x)/(32*b) - E^(4*a + 4*b*x)/(32*b) + E^(6*a + 6*b*x)/(96*b) + x/8} -{E^(2*(a + b*x))*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 4, -(E^(-a - b*x)/(8*b)) - E^(a + b*x)/(8*b) - E^(3*a + 3*b*x)/(24*b) + E^(5*a + 5*b*x)/(40*b)} -{E^(2*(a + b*x))*Cosh[a + b*x]*Sinh[a + b*x]^1, x, 4, E^(4*a + 4*b*x)/(16*b) - x/4} -{E^(2*(a + b*x))*Cosh[a + b*x]*Csch[a + b*x]^1, x, 4, E^(2*a + 2*b*x)/(2*b) + Log[1 - E^(2*a + 2*b*x)]/b} -{E^(2*(a + b*x))*Cosh[a + b*x]*Csch[a + b*x]^2, x, 5, (2*E^(a + b*x))/b + (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (4*ArcTanh[E^(a + b*x)])/b} -{E^(2*(a + b*x))*Cosh[a + b*x]*Csch[a + b*x]^3, x, 5, -(2/(b*(1 - E^(2*a + 2*b*x))^2)) + 6/(b*(1 - E^(2*a + 2*b*x))) + (2*Log[1 - E^(2*a + 2*b*x)])/b} - - -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 4, E^(-3*a - 3*b*x)/(96*b) - E^(-a - b*x)/(32*b) + E^(a + b*x)/(16*b) - E^(3*a + 3*b*x)/(48*b) - E^(5*a + 5*b*x)/(160*b) + E^(7*a + 7*b*x)/(224*b)} -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 4, -(E^(-2*a - 2*b*x)/(32*b)) - E^(2*a + 2*b*x)/(16*b) + E^(6*a + 6*b*x)/(96*b)} -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Sinh[a + b*x]^1, x, 4, E^(-a - b*x)/(8*b) - E^(a + b*x)/(8*b) + E^(3*a + 3*b*x)/(24*b) + E^(5*a + 5*b*x)/(40*b)} -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Csch[a + b*x]^1, x, 5, (3*E^(a + b*x))/(2*b) + E^(3*a + 3*b*x)/(6*b) - (2*ArcTanh[E^(a + b*x)])/b} -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 4, E^(2*a + 2*b*x)/(2*b) + 2/(b*(1 - E^(2*a + 2*b*x))) + (2*Log[1 - E^(2*a + 2*b*x)])/b} -{E^(2*(a + b*x))*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 6, (2*E^(a + b*x))/b - (2*E^(3*a + 3*b*x))/(b*(1 - E^(2*a + 2*b*x))^2) + (3*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (5*ArcTanh[E^(a + b*x)])/b} - - -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 5, E^(-4*a - 4*b*x)/(256*b) - (3*E^(4*a + 4*b*x))/(256*b) + E^(8*a + 8*b*x)/(512*b) + (3*x)/64} -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 4, -(E^(-3*a - 3*b*x)/(96*b)) - E^(-a - b*x)/(32*b) - E^(a + b*x)/(16*b) - E^(3*a + 3*b*x)/(48*b) + E^(5*a + 5*b*x)/(160*b) + E^(7*a + 7*b*x)/(224*b)} -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Sinh[a + b*x]^1, x, 5, E^(-2*a - 2*b*x)/(32*b) + E^(4*a + 4*b*x)/(32*b) + E^(6*a + 6*b*x)/(96*b) - x/8} -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Csch[a + b*x]^1, x, 5, E^(2*a + 2*b*x)/(2*b) + E^(4*a + 4*b*x)/(16*b) - x/4 + Log[1 - E^(2*a + 2*b*x)]/b} -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 6, (5*E^(a + b*x))/(2*b) + E^(3*a + 3*b*x)/(6*b) + (2*E^(a + b*x))/(b*(1 - E^(2*a + 2*b*x))) - (4*ArcTanh[E^(a + b*x)])/b} -{E^(2*(a + b*x))*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 4, E^(2*a + 2*b*x)/(2*b) - 2/(b*(1 - E^(2*a + 2*b*x))^2) + 6/(b*(1 - E^(2*a + 2*b*x))) + (3*Log[1 - E^(2*a + 2*b*x)])/b} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{E^x*Sech[2*x]*Tanh[2*x], x, 12, -(E^(3*x)/(1 + E^(4*x))) - ArcTan[1 - Sqrt[2]*E^x]/(2*Sqrt[2]) + ArcTan[1 + Sqrt[2]*E^x]/(2*Sqrt[2]) + Log[1 - Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2]) - Log[1 + Sqrt[2]*E^x + E^(2*x)]/(4*Sqrt[2])} -{E^x*Sech[2*x]^2*Tanh[2*x], x, 13, -(E^(5*x)/(1 + E^(4*x))^2) - E^x/(4*(1 + E^(4*x))) - ArcTan[1 - Sqrt[2]*E^x]/(8*Sqrt[2]) + ArcTan[1 + Sqrt[2]*E^x]/(8*Sqrt[2]) - Log[1 - Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2]) + Log[1 + Sqrt[2]*E^x + E^(2*x)]/(16*Sqrt[2])} -{E^x*Sech[2*x]*Tanh[2*x]^2, x, 13, E^(3*x)/(1 + E^(4*x))^2 - (3*E^(3*x))/(4*(1 + E^(4*x))) - (5*ArcTan[1 - Sqrt[2]*E^x])/(8*Sqrt[2]) + (5*ArcTan[1 + Sqrt[2]*E^x])/(8*Sqrt[2]) + (5*Log[1 - Sqrt[2]*E^x + E^(2*x)])/(16*Sqrt[2]) - (5*Log[1 + Sqrt[2]*E^x + E^(2*x)])/(16*Sqrt[2])} -{E^x*Sech[2*x]^2*Tanh[2*x]^2, x, 14, (4*E^(5*x))/(3*(1 + E^(4*x))^3) - (5*E^(5*x))/(6*(1 + E^(4*x))^2) - (3*E^x)/(8*(1 + E^(4*x))) - (3*ArcTan[1 - Sqrt[2]*E^x])/(16*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*E^x])/(16*Sqrt[2]) - (3*Log[1 - Sqrt[2]*E^x + E^(2*x)])/(32*Sqrt[2]) + (3*Log[1 + Sqrt[2]*E^x + E^(2*x)])/(32*Sqrt[2])} - - -{E^x*Coth[2*x]*Csch[2*x], x, 6, E^(3*x)/(1 - E^(4*x)) + ArcTan[E^x]/2 - ArcTanh[E^x]/2} -{E^x*Coth[2*x]*Csch[2*x]^2, x, 7, -(E^(5*x)/(1 - E^(4*x))^2) + E^x/(4*(1 - E^(4*x))) - ArcTan[E^x]/8 - ArcTanh[E^x]/8} -{E^x*Coth[2*x]^2*Csch[2*x], x, 7, -(E^(3*x)/(1 - E^(4*x))^2) + (3*E^(3*x))/(4*(1 - E^(4*x))) + (5*ArcTan[E^x])/8 - (5*ArcTanh[E^x])/8} -{E^x*Coth[2*x]^2*Csch[2*x]^2, x, 8, (4*E^(5*x))/(3*(1 - E^(4*x))^3) - (5*E^(5*x))/(6*(1 - E^(4*x))^2) + (3*E^x)/(8*(1 - E^(4*x))) - (3*ArcTan[E^x])/16 - (3*ArcTanh[E^x])/16} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(a+b x) Cosh[c+d x]^m Sinh[c+d x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>=0*) - - -{E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^3, x, 4, -((b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(2*(4*b^2 - d^2))) + (b*E^(c + d*x)*Cosh[4*a + 4*b*x])/(2*(16*b^2 - d^2)) + (d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(4*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[4*a + 4*b*x])/(8*(16*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^2, x, 4, (d*E^(c + d*x)*Cosh[a + b*x])/(4*(b^2 - d^2)) - (d*E^(c + d*x)*Cosh[3*a + 3*b*x])/(4*(9*b^2 - d^2)) - (b*E^(c + d*x)*Sinh[a + b*x])/(4*(b^2 - d^2)) + (3*b*E^(c + d*x)*Sinh[3*a + 3*b*x])/(4*(9*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^1, x, 3, (b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(4*b^2 - d^2) - (d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(2*(4*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^0, x, 1, -((d*E^(c + d*x)*Cosh[a + b*x])/(b^2 - d^2)) + (b*E^(c + d*x)*Sinh[a + b*x])/(b^2 - d^2)} -{E^(c + d*x)*Cosh[a + b*x]*Csch[a + b*x]^1, x, 4, E^(c + d*x)/d - (2*E^(c + d*x)*Hypergeometric2F1[1, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d} -{E^(c + d*x)*Cosh[a + b*x]*Csch[a + b*x]^2, x, 4, -((2*E^(a + c + (b + d)*x)*Hypergeometric2F1[1, (b + d)/(2*b), (3*b + d)/(2*b), E^(2*(a + b*x))])/(b + d)) + (4*E^(a + c + (b + d)*x)*Hypergeometric2F1[2, (b + d)/(2*b), (3*b + d)/(2*b), E^(2*(a + b*x))])/(b + d)} -{E^(c + d*x)*Cosh[a + b*x]*Csch[a + b*x]^3, x, 4, (4*E^(2*a + c + (2*b + d)*x)*Hypergeometric2F1[2, (1/2)*(2 + d/b), (1/2)*(4 + d/b), E^(2*(a + b*x))])/(2*b + d) - (8*E^(2*a + c + (2*b + d)*x)*Hypergeometric2F1[3, (1/2)*(2 + d/b), (1/2)*(4 + d/b), E^(2*(a + b*x))])/(2*b + d)} - - -{E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^3, x, 5, -((b*E^(c + d*x)*Cosh[a + b*x])/(8*(b^2 - d^2))) - (3*b*E^(c + d*x)*Cosh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) + (5*b*E^(c + d*x)*Cosh[5*a + 5*b*x])/(16*(25*b^2 - d^2)) + (d*E^(c + d*x)*Sinh[a + b*x])/(8*(b^2 - d^2)) + (d*E^(c + d*x)*Sinh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[5*a + 5*b*x])/(16*(25*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 4, -(E^(c + d*x)/(8*d)) - (d*E^(c + d*x)*Cosh[4*a + 4*b*x])/(8*(16*b^2 - d^2)) + (b*E^(c + d*x)*Sinh[4*a + 4*b*x])/(2*(16*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^1, x, 4, (b*E^(c + d*x)*Cosh[a + b*x])/(4*(b^2 - d^2)) + (3*b*E^(c + d*x)*Cosh[3*a + 3*b*x])/(4*(9*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[a + b*x])/(4*(b^2 - d^2)) - (d*E^(c + d*x)*Sinh[3*a + 3*b*x])/(4*(9*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x]^0, x, 2, (2*b^2*E^(c + d*x))/(d*(4*b^2 - d^2)) - (d*E^(c + d*x)*Cosh[a + b*x]^2)/(4*b^2 - d^2) + (2*b*E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2 - d^2)} -{E^(c + d*x)*Cosh[a + b*x]^2*Csch[a + b*x]^1, x, 6, -((3*E^(-a + c - (b - d)*x))/(2*(b - d))) + E^(a + c + (b + d)*x)/(2*(b + d)) + (2*E^(-a + c - (b - d)*x)*Hypergeometric2F1[1, -((b - d)/(2*b)), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)} -{E^(c + d*x)*Cosh[a + b*x]^2*Csch[a + b*x]^2, x, 5, E^(c + d*x)/d - (4*E^(c + d*x)*Hypergeometric2F1[1, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d + (4*E^(c + d*x)*Hypergeometric2F1[2, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d} -{E^(c + d*x)*Cosh[a + b*x]^2*Csch[a + b*x]^3, x, 5, -((2*E^(a + c + (b + d)*x)*Hypergeometric2F1[1, (b + d)/(2*b), (3*b + d)/(2*b), E^(2*(a + b*x))])/(b + d)) + (8*E^(a + c + (b + d)*x)*Hypergeometric2F1[2, (b + d)/(2*b), (3*b + d)/(2*b), E^(2*(a + b*x))])/(b + d) - (8*E^(a + c + (b + d)*x)*Hypergeometric2F1[3, (b + d)/(2*b), (3*b + d)/(2*b), E^(2*(a + b*x))])/(b + d)} - - -{E^(c + d*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^3, x, 4, -((3*b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(16*(4*b^2 - d^2))) + (3*b*E^(c + d*x)*Cosh[6*a + 6*b*x])/(16*(36*b^2 - d^2)) + (3*d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(32*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[6*a + 6*b*x])/(32*(36*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^2, x, 5, (d*E^(c + d*x)*Cosh[a + b*x])/(8*(b^2 - d^2)) - (d*E^(c + d*x)*Cosh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) - (d*E^(c + d*x)*Cosh[5*a + 5*b*x])/(16*(25*b^2 - d^2)) - (b*E^(c + d*x)*Sinh[a + b*x])/(8*(b^2 - d^2)) + (3*b*E^(c + d*x)*Sinh[3*a + 3*b*x])/(16*(9*b^2 - d^2)) + (5*b*E^(c + d*x)*Sinh[5*a + 5*b*x])/(16*(25*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^1, x, 4, (b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(2*(4*b^2 - d^2)) + (b*E^(c + d*x)*Cosh[4*a + 4*b*x])/(2*(16*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(4*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[4*a + 4*b*x])/(8*(16*b^2 - d^2))} -{E^(c + d*x)*Cosh[a + b*x]^3*Sinh[a + b*x]^0, x, 2, -((6*b^2*d*E^(c + d*x)*Cosh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4)) - (d*E^(c + d*x)*Cosh[a + b*x]^3)/(9*b^2 - d^2) + (6*b^3*E^(c + d*x)*Sinh[a + b*x])/(9*b^4 - 10*b^2*d^2 + d^4) + (3*b*E^(c + d*x)*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^2 - d^2)} -{E^(c + d*x)*Cosh[a + b*x]^3*Csch[a + b*x]^1, x, 8, -((7*E^(-2*a + c - (2*b - d)*x))/(4*(2*b - d))) + E^(c + d*x)/d + E^(2*a + c + (2*b + d)*x)/(4*(2*b + d)) + (2*E^(-2*a + c - (2*b - d)*x)*Hypergeometric2F1[1, (1/2)*(-2 + d/b), d/(2*b), E^(2*(a + b*x))])/(2*b - d)} -{E^(c + d*x)*Cosh[a + b*x]^3*Csch[a + b*x]^2, x, 7, -((5*E^(-a + c - (b - d)*x))/(2*(b - d))) + E^(a + c + (b + d)*x)/(2*(b + d)) + (6*E^(-a + c - (b - d)*x)*Hypergeometric2F1[1, -((b - d)/(2*b)), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d) - (4*E^(-a + c - (b - d)*x)*Hypergeometric2F1[2, -((b - d)/(2*b)), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)} -{E^(c + d*x)*Cosh[a + b*x]^3*Csch[a + b*x]^3, x, 6, E^(c + d*x)/d - (6*E^(c + d*x)*Hypergeometric2F1[1, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d + (12*E^(c + d*x)*Hypergeometric2F1[2, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d - (8*E^(c + d*x)*Hypergeometric2F1[3, d/(2*b), 1 + d/(2*b), E^(2*(a + b*x))])/d} - - -{E^(a + b*x)*Sinh[c + d*x]^(3/2) - ((3*d^2)/(4*(b^2 - (9*d^2)/4)))*(E^(a + b*x)/Sqrt[Sinh[c + d*x]]), x, 10, -((6*d*E^(a + b*x)*Cosh[c + d*x]*Sqrt[Sinh[c + d*x]])/(4*b^2 - 9*d^2)) + (4*b*E^(a + b*x)*Sinh[c + d*x]^(3/2))/(4*b^2 - 9*d^2)} - - -(* ::Subsubsection:: *) -(*m<0*) - - -(* ::Section::Closed:: *) -(*Products of functions of a hyperbolic function and its derivative*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Cosh[a+b x]] Sinh[a+b x]^n when n odd*) - - -{E^(n*Cosh[a+b*x])*Sinh[a+b*x], x, 2, E^(n*Cosh[a + b*x])/(b*n)} -{E^(n*Cosh[a*c+b*c*x])*Sinh[c*(a+b*x)], x, 2, E^(n*Cosh[c*(a + b*x)])/(b*c*n)} -{E^(n*Cosh[c*(a+b*x)])*Sinh[a*c+b*c*x], x, 2, E^(n*Cosh[a*c + b*c*x])/(b*c*n)} - - -{E^(n*Cosh[a+b*x])*Tanh[a+b*x], x, 2, ExpIntegralEi[n*Cosh[a + b*x]]/b} -{E^(n*Cosh[a*c+b*c*x])*Tanh[c*(a+b*x)], x, 2, ExpIntegralEi[n*Cosh[c*(a + b*x)]]/(b*c)} -{E^(n*Cosh[c*(a+b*x)])*Tanh[a*c+b*c*x], x, 2, ExpIntegralEi[n*Cosh[a*c + b*c*x]]/(b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Sinh[a+b x]] Cosh[a+b x]^n when n odd*) - - -{E^(n*Sinh[a+b*x])*Cosh[a+b*x], x, 2, E^(n*Sinh[a + b*x])/(b*n)} -{E^(n*Sinh[a*c+b*c*x])*Cosh[c*(a+b*x)], x, 2, E^(n*Sinh[c*(a + b*x)])/(b*c*n)} -{E^(n*Sinh[c*(a+b*x)])*Cosh[a*c+b*c*x], x, 2, E^(n*Sinh[a*c + b*c*x])/(b*c*n)} - - -{E^(n*Sinh[a+b*x])*Coth[a+b*x], x, 2, ExpIntegralEi[n*Sinh[a + b*x]]/b} -{E^(n*Sinh[a*c+b*c*x])*Coth[c*(a+b*x)], x, 2, ExpIntegralEi[n*Sinh[c*(a + b*x)]]/(b*c)} -{E^(n*Sinh[c*(a+b*x)])*Coth[a*c+b*c*x], x, 2, ExpIntegralEi[n*Sinh[a*c + b*c*x]]/(b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Tanh[a+b x]] Sech[a+b x]^n when n even*) - - -{Sech[x]^2/(a + b*Tanh[x]), x, 2, Log[a + b*Tanh[x]]/b} -{Sech[x]^2/(1 + Tanh[x]^2), x, 2, ArcTan[Tanh[x]]} -{Sech[x]^2/(9 + Tanh[x]^2), x, 2, ArcTan[Tanh[x]/3]/3} -{Sech[x]^2*(a + b*Tanh[x])^n, x, 2, (a + b*Tanh[x])^(1 + n)/(b*(1 + n))} -{Sech[x]^2*(1 + 1/(1 - Tanh[x]^2)), x, 3, x + Tanh[x]} -{Sech[x]^2*(2 - Tanh[x]^2)/(1 - Tanh[x]^2), x, 4, x + Tanh[x]} -{Sech[x]^2/(2 + 2*Tanh[x] + Tanh[x]^2), x, 3, ArcTan[1 + Tanh[x]]} -{Sech[x]^2/(Tanh[x]^2 + Tanh[x]^3), x, 3, -Coth[x] - Log[Tanh[x]] + Log[1 + Tanh[x]]} -{Sech[x]^2/(-Tanh[x]^2 + Tanh[x]^3), x, 3, Coth[x] + Log[1 - Tanh[x]] - Log[Tanh[x]]} -{Sech[x]^2/(3 - 4*Tanh[x]^3), x, 7, ArcTan[(3^(1/3) + 2*2^(2/3)*Tanh[x])/3^(5/6)]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^(2/3)*Tanh[x]]/(3*6^(2/3)) + Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tanh[x] + 2*2^(1/3)*Tanh[x]^2]/(6*6^(2/3))} -{Sech[x]^2/(11 - 5*Tanh[x] + 5*Tanh[x]^2), x, 3, -((2*ArcTan[Sqrt[5/39]*(1 - 2*Tanh[x])])/Sqrt[195])} -{Sech[x]^2*(a + b*Tanh[x])/(c + d*Tanh[x]), x, 3, -(((b*c - a*d)*Log[c + d*Tanh[x]])/d^2) + (b*Tanh[x])/d} -{Sech[x]^2*(a + b*Tanh[x])^2/(c + d*Tanh[x]), x, 3, ((b*c - a*d)^2*Log[c + d*Tanh[x]])/d^3 - (b*(b*c - a*d)*Tanh[x])/d^2 + (a + b*Tanh[x])^2/(2*d)} -{Sech[x]^2*(a + b*Tanh[x])^3/(c + d*Tanh[x]), x, 3, -(((b*c - a*d)^3*Log[c + d*Tanh[x]])/d^4) + (b*(b*c - a*d)^2*Tanh[x])/d^3 - ((b*c - a*d)*(a + b*Tanh[x])^2)/(2*d^2) + (a + b*Tanh[x])^3/(3*d)} -{Sech[x]^2*Tanh[x]^2/(2 + Tanh[x]^3)^2, x, 2, -1/(3*(2 + Tanh[x]^3))} -{Sech[x]^2*Tanh[x]^6*(1 - Tanh[x]^2)^3, x, 4, Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13} -{Sech[x]^2*(2 + Tanh[x]^2)/(1 + Tanh[x]^3), x, 5, -((2*ArcTan[(1 - 2*Tanh[x])/Sqrt[3]])/Sqrt[3]) + Log[1 + Tanh[x]]} -{Sech[x]^2*(1 + Cosh[x]^2), x, 2, x + Tanh[x]} -{Sech[x]^2/(1 + Sech[x]^2 - 3*Tanh[x]), x, 3, (2*ArcTanh[(3 + 2*Tanh[x])/Sqrt[17]])/Sqrt[17]} -{Sech[x]^2/Sqrt[4 - Sech[x]^2], x, 2, ArcSinh[Tanh[x]/Sqrt[3]]} -{Sech[x]^2/Sqrt[1 - 4*Tanh[x]^2], x, 2, ArcSin[2*Tanh[x]]/2} -{Sech[x]^2/Sqrt[-4 + Tanh[x]^2], x, 3, ArcTanh[Tanh[x]/Sqrt[-4 + Tanh[x]^2]]} -{Sech[x]^2*Sqrt[1 + Coth[x]^2], x, 3, -ArcSinh[Coth[x]] + Sqrt[1 + Coth[x]^2]*Tanh[x]} -{Sech[x]^2*Sqrt[1 + Tanh[x]^2], x, 3, (1/2)*ArcSinh[Tanh[x]] + (1/2)*Tanh[x]*Sqrt[1 + Tanh[x]^2]} - - -{Sech[x]^4*(-1 + Sech[x]^2)^2*Tanh[x], x, 4, Tanh[x]^6/6 - Tanh[x]^8/8} - - -(* ::Subsection:: *) -(*Integrands of the form F[Coth[a+b x]] Csch[a+b x]^n when n even*) - - -(* ::Subsection:: *) -(*Integrands of the form F[Sech[a+b x]] Sech[a+b x] Tanh[a+b x]*) - - -(* ::Subsection:: *) -(*Integrands of the form F[Csch[a+b x]] Csch[a+b x] Coth[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Sinh[(a+b x)/2]] Sinh[a+b x]*) - - -{E^(n*Sinh[a+b*x])*Sinh[2*a+2*b*x], x, 4, -((2*E^(n*Sinh[a + b*x]))/(b*n^2)) + (2*E^(n*Sinh[a + b*x])*Sinh[a + b*x])/(b*n)} -{E^(n*Sinh[a+b*x])*Sinh[2*(a+b*x)], x, 4, -((2*E^(n*Sinh[a + b*x]))/(b*n^2)) + (2*E^(n*Sinh[a + b*x])*Sinh[a + b*x])/(b*n)} -{E^(n*Sinh[a/2+b/2*x])*Sinh[a+b*x], x, 4, -((4*E^(n*Sinh[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Sinh[a/2 + (b*x)/2])*Sinh[a/2 + (b*x)/2])/(b*n)} -{E^(n*Sinh[(a+b*x)/2])*Sinh[a+b*x], x, 4, -((4*E^(n*Sinh[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Sinh[a/2 + (b*x)/2])*Sinh[a/2 + (b*x)/2])/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Cosh[(a+b x)/2]] Sinh[a+b x]*) - - -{E^(n*Cosh[a+b*x])*Sinh[2*a+2*b*x], x, 4, -((2*E^(n*Cosh[a + b*x]))/(b*n^2)) + (2*E^(n*Cosh[a + b*x])*Cosh[a + b*x])/(b*n)} -{E^(n*Cosh[a+b*x])*Sinh[2*(a+b*x)], x, 4, -((2*E^(n*Cosh[a + b*x]))/(b*n^2)) + (2*E^(n*Cosh[a + b*x])*Cosh[a + b*x])/(b*n)} -{E^(n*Cosh[a/2+b/2*x])*Sinh[a+b*x], x, 4, -((4*E^(n*Cosh[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Cosh[a/2 + (b*x)/2])*Cosh[a/2 + (b*x)/2])/(b*n)} -{E^(n*Cosh[(a+b*x)/2])*Sinh[a+b*x], x, 4, -((4*E^(n*Cosh[a/2 + (b*x)/2]))/(b*n^2)) + (4*E^(n*Cosh[a/2 + (b*x)/2])*Cosh[a/2 + (b*x)/2])/(b*n)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form F[Tanh[a+b x]] when n even*) - - -{Csch[x]*Log[Tanh[x]]*Sech[x], x, 1, Log[Tanh[x]]^2/2} -{Csch[2*x]*Log[Tanh[x]], x, 1, Log[Tanh[x]]^2/4} - - -(* ::Subsection::Closed:: *) -(*Products of functions of a hyperbolic function and its derivative*) - - -{Cosh[a + b*x]*F[c, d, Sinh[a + b*x], r, s], x, 1, CannotIntegrate[Cosh[a + b*x]*F[c, d, Sinh[a + b*x], r, s], x]} -{Sinh[a + b*x]*F[c, d, Cosh[a + b*x], r, s], x, 1, CannotIntegrate[F[c, d, Cosh[a + b*x], r, s]*Sinh[a + b*x], x]} -{Sech[a + b*x]^2*F[c, d, Tanh[a + b*x], r, s], x, 1, CannotIntegrate[F[c, d, Tanh[a + b*x], r, s]*Sech[a + b*x]^2, x]} -{Csch[a + b*x]^2*F[c, d, Coth[a + b*x], r, s], x, 1, CannotIntegrate[Csch[a + b*x]^2*F[c, d, Coth[a + b*x], r, s], x]} - - -{Sech[x]*(5 - 11*Sech[x]^2)*Tanh[x], x, 3, -5*Sech[x] + (11*Sech[x]^3)/3} - - -{Csch[x]^2/(a + b*Coth[x]), x, 2, -(Log[a + b*Coth[x]]/b)} -{Csch[x]^2*(a + b*Coth[x])^n, x, 2, -((a + b*Coth[x])^(1 + n)/(b*(1 + n)))} - -{Csch[x]^2*(-1 + Sinh[x]^2), x, 2, x + Coth[x]} -{Csch[x]^2*(-1 - 1/(1 - Coth[x]^2)), x, 3, x + Coth[x]} -{Csch[x]^2*(a + b*Coth[x])/(c + d*Coth[x]), x, 3, -((b*Coth[x])/d) + ((b*c - a*d)*Log[c + d*Coth[x]])/d^2} -{Csch[x]^2*(a + b*Coth[x])^2/(c + d*Coth[x]), x, 3, (b*(b*c - a*d)*Coth[x])/d^2 - (a + b*Coth[x])^2/(2*d) - ((b*c - a*d)^2*Log[c + d*Coth[x]])/d^3} -{Csch[x]^2*(a + b*Coth[x])^3/(c + d*Coth[x]), x, 3, -((b*(b*c - a*d)^2*Coth[x])/d^3) + ((b*c - a*d)*(a + b*Coth[x])^2)/(2*d^2) - (a + b*Coth[x])^3/(3*d) + ((b*c - a*d)^3*Log[c + d*Coth[x]])/d^4} - - -{Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x], x, 4, -((a*(a + b*Cosh[x]^2)^4)/(8*b^2)) + (a + b*Cosh[x]^2)^5/(10*b^2)} -{Sinh[x]^3*(a + b*Sinh[x]^2)^3*Cosh[x], x, 4, -((a*(a + b*Sinh[x]^2)^4)/(8*b^2)) + (a + b*Sinh[x]^2)^5/(10*b^2)} - - -{Cosh[x]*Sinh[x]*Sqrt[a + b*Sinh[x]^2], x, 2, (a + b*Sinh[x]^2)^(3/2)/(3*b)} - - -{Csch[x]*Sqrt[1 + Log[Coth[x]]^2]*Sech[x], x, 3, (-(1/2))*ArcSinh[Log[Coth[x]]] - (1/2)*Log[Coth[x]]*Sqrt[1 + Log[Coth[x]]^2]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving hyperbolic functions*) - - -{(Coth[Sqrt[x]]*Csch[Sqrt[x]])/Sqrt[x], x, 3, -2*Csch[Sqrt[x]]} -{(Cosh[Sqrt[x]]*Sinh[Sqrt[x]])/Sqrt[x], x, 1, Sinh[Sqrt[x]]^2} -{(Sech[Sqrt[x]]*Tanh[Sqrt[x]])/Sqrt[x], x, 3, -2*Sech[Sqrt[x]]} - - -{Sinh[x]^2/(a + b*Sinh[2*x]), x, 9, ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2]) + Log[a + b*Sinh[2*x]]/(4*b), ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2]) + Log[Cosh[x]]/(2*b) + Log[a + 2*b*Tanh[x] - a*Tanh[x]^2]/(4*b)} -{Cosh[x]^2/(a + b*Sinh[2*x]), x, 8, -(ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2])) + Log[a + b*Sinh[2*x]]/(4*b), -(ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2])) + Log[Cosh[x]]/(2*b) + Log[a + 2*b*Tanh[x] - a*Tanh[x]^2]/(4*b)} - -{Sinh[x]^2/(a + b*Cosh[2*x]), x, 4, x/(2*b) - (Sqrt[a + b]*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a + b]])/(2*Sqrt[a - b]*b)} -{Cosh[x]^2/(a + b*Cosh[2*x]), x, 4, x/(2*b) - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a + b]])/(2*b*Sqrt[a + b])} - - -{Tanh[c + d*x]/Sqrt[a*Sinh[c+d*x]^2], x, 3, ArcTan[Sqrt[a*Sinh[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d)} -{Coth[c + d*x]/Sqrt[a*Cosh[c+d*x]^2], x, 3, -(ArcTanh[Sqrt[a*Cosh[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d))} - - -{x*Cosh[2*x]*Sech[x], x, 12, -2*x*ArcTan[E^x] - 2*Cosh[x] + I*PolyLog[2, (-I)*E^x] - I*PolyLog[2, I*E^x] + 2*x*Sinh[x]} -{x*Cosh[2*x]*Sech[x]^2, x, 5, x^2+Log[Cosh[x]]-x*Tanh[x]} -{x*Cosh[2*x]*Sech[x]^3, x, 19, 3*x*ArcTan[E^x] - (3/2)*I*PolyLog[2, (-I)*E^x] + (3/2)*I*PolyLog[2, I*E^x] - Sech[x]/2 - (1/2)*x*Sech[x]*Tanh[x]} - - -{Sqrt[Csch[x]]*(x*Cosh[x] - 4*Sech[x]*Tanh[x]), x, 8, (2*x)/Sqrt[Csch[x]] - (4*Sech[x])/Csch[x]^(3/2)} - - -{Sinh[x]*(Cosh[x] + Sinh[x]), x, 6, -(x/2) + (1/2)*Cosh[x]*Sinh[x] + Sinh[x]^2/2} - -{(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]), x, 5, (1/4)*Log[1 - Tanh[x/2]] + (3/4)*Log[1 + Tanh[x/2]] + 1/(2*(1 - Tanh[x/2])) - 1/(2*(1 + Tanh[x/2])^2) + 1/(1 + Tanh[x/2])} -{x^5*Cosh[a + b*x^3]^7*Sinh[a + b*x^3], x, 7, -((35*x^3)/(3072*b)) + (x^3*Cosh[a + b*x^3]^8)/(24*b) - (35*Cosh[a + b*x^3]*Sinh[a + b*x^3])/(3072*b^2) - (35*Cosh[a + b*x^3]^3*Sinh[a + b*x^3])/(4608*b^2) - (7*Cosh[a + b*x^3]^5*Sinh[a + b*x^3])/(1152*b^2) - (Cosh[a + b*x^3]^7*Sinh[a + b*x^3])/(192*b^2)} - -(* {Csch[x^5]/x, x, Unintegrable[Csch[x^5]/x, x]} *) - -{Cosh[x]^2/(1 + E^x), x, 4, -(1/8)/E^(2*x) + 1/(E^x*4) + E^x/4 + (3*x)/4 - Log[1 + E^x]} - - -{Sqrt[1 + Sech[x]]*Sech[x]*Tanh[x]^3, x, 6, (-(4/5))*(1 + Sech[x])^(5/2) + (2/7)*(1 + Sech[x])^(7/2)} -{Sqrt[1 + Csch[x]]*Csch[x]*Coth[x]^3, x, 5, (-(4/3))*(1 + Csch[x])^(3/2) + (4/5)*(1 + Csch[x])^(5/2) - (2/7)*(1 + Csch[x])^(7/2)} - - -{Cosh[x]^x*(Log[Cosh[x]] + x*Tanh[x]), x, 3, Cosh[x]^x} - - -(* Nonidempotent expansion results in infinite recursion: *) -(* {(x*Cosh[x] - Sinh[x])/(x - Sinh[x])^2, x, -7, x/(x - Sinh[x])} *) -(* {(-Cosh[x] + x*Sinh[x])/(x - Cosh[x])^2, x, 0, x/(x - Cosh[x])} *) - - -{F^(a + b*x)*(Cosh[c + d*x] + Sinh[c + d*x])^n, x, 4, ((E^(c + d*x))^n*F^(a + b*x))/(d*n + b*Log[F])} -{F^(a + b*x)*(Cosh[c + d*x] - Sinh[c + d*x])^n, x, 4, -(((E^(-c - d*x))^n*F^(a + b*x))/(d*n - b*Log[F]))} - - -(* {(Cosh[a + b*x]^5 - Sinh[a + b*x]^5)/(Cosh[a + b*x]^5 + Sinh[a + b*x]^5), x, 5, 0} *) -{(Cosh[a + b*x]^4 - Sinh[a + b*x]^4)/(Cosh[a + b*x]^4 + Sinh[a + b*x]^4), x, 6, -(ArcTan[1 - Sqrt[2]*Tanh[a + b*x]]/(Sqrt[2]*b)) + ArcTan[1 + Sqrt[2]*Tanh[a + b*x]]/(Sqrt[2]*b)} -{(Cosh[a + b*x]^3 - Sinh[a + b*x]^3)/(Cosh[a + b*x]^3 + Sinh[a + b*x]^3), x, 5, -((4*ArcTan[(1 - 2*Tanh[a + b*x])/Sqrt[3]])/(3*Sqrt[3]*b)) - 1/(3*b*(1 + Tanh[a + b*x]))} -{(Cosh[a + b*x]^2 - Sinh[a + b*x]^2)/(Cosh[a + b*x]^2 + Sinh[a + b*x]^2), x, 3, ArcTan[Tanh[a + b*x]]/b} -{(Cosh[a + b*x]^1 - Sinh[a + b*x]^1)/(Cosh[a + b*x]^1 + Sinh[a + b*x]^1), x, 1, -(1/(2*b*(Cosh[a + b*x] + Sinh[a + b*x])^2))} -{(Sech[a + b*x]^1 - Csch[a + b*x]^1)/(Sech[a + b*x]^1 + Csch[a + b*x]^1), x, 2, 1/(b*(1 + Tanh[a + b*x]))} -{(Sech[a + b*x]^2 - Csch[a + b*x]^2)/(Sech[a + b*x]^2 + Csch[a + b*x]^2), x, 2, -(ArcTan[Tanh[a + b*x]]/b)} -{(Sech[a + b*x]^3 - Csch[a + b*x]^3)/(Sech[a + b*x]^3 + Csch[a + b*x]^3), x, 5, (4*ArcTan[(1 - 2*Tanh[a + b*x])/Sqrt[3]])/(3*Sqrt[3]*b) + 1/(3*b*(1 + Tanh[a + b*x]))} -{(Sech[a + b*x]^4 - Csch[a + b*x]^4)/(Sech[a + b*x]^4 + Csch[a + b*x]^4), x, 6, ArcTan[1 - Sqrt[2]*Tanh[a + b*x]]/(Sqrt[2]*b) - ArcTan[1 + Sqrt[2]*Tanh[a + b*x]]/(Sqrt[2]*b)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m deleted file mode 100644 index 4f348bc..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.2 (d x)^m (a+b arcsinh(c x))^n.m +++ /dev/null @@ -1,264 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcSinh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b x)^m ArcSinh[a x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSinh[a x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcSinh[a*x], x, 4, -(Sqrt[1 + a^2*x^2]/(5*a^5)) + (2*(1 + a^2*x^2)^(3/2))/(15*a^5) - (1 + a^2*x^2)^(5/2)/(25*a^5) + (1/5)*x^5*ArcSinh[a*x]} -{x^3*ArcSinh[a*x], x, 4, (3*x*Sqrt[1 + a^2*x^2])/(32*a^3) - (x^3*Sqrt[1 + a^2*x^2])/(16*a) - (3*ArcSinh[a*x])/(32*a^4) + (1/4)*x^4*ArcSinh[a*x]} -{x^2*ArcSinh[a*x], x, 4, Sqrt[1 + a^2*x^2]/(3*a^3) - (1 + a^2*x^2)^(3/2)/(9*a^3) + (1/3)*x^3*ArcSinh[a*x]} -{x^1*ArcSinh[a*x], x, 3, -((x*Sqrt[1 + a^2*x^2])/(4*a)) + ArcSinh[a*x]/(4*a^2) + (1/2)*x^2*ArcSinh[a*x]} -{x^0*ArcSinh[a*x], x, 2, -(Sqrt[1 + a^2*x^2]/a) + x*ArcSinh[a*x]} -{ArcSinh[a*x]/x^1, x, 5, (-(1/2))*ArcSinh[a*x]^2 + ArcSinh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (1/2)*PolyLog[2, E^(2*ArcSinh[a*x])]} -{ArcSinh[a*x]/x^2, x, 4, -(ArcSinh[a*x]/x) - a*ArcTanh[Sqrt[1 + a^2*x^2]]} -{ArcSinh[a*x]/x^3, x, 2, -((a*Sqrt[1 + a^2*x^2])/(2*x)) - ArcSinh[a*x]/(2*x^2)} -{ArcSinh[a*x]/x^4, x, 5, -((a*Sqrt[1 + a^2*x^2])/(6*x^2)) - ArcSinh[a*x]/(3*x^3) + (1/6)*a^3*ArcTanh[Sqrt[1 + a^2*x^2]]} -{ArcSinh[a*x]/x^5, x, 3, -((a*Sqrt[1 + a^2*x^2])/(12*x^3)) + (a^3*Sqrt[1 + a^2*x^2])/(6*x) - ArcSinh[a*x]/(4*x^4)} -{ArcSinh[a*x]/x^6, x, 6, -((a*Sqrt[1 + a^2*x^2])/(20*x^4)) + (3*a^3*Sqrt[1 + a^2*x^2])/(40*x^2) - ArcSinh[a*x]/(5*x^5) - (3/40)*a^5*ArcTanh[Sqrt[1 + a^2*x^2]]} - - -{x^4*ArcSinh[a*x]^2, x, 7, (16*x)/(75*a^4) - (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(75*a^5) + (8*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(75*a^3) - (2*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(25*a) + (1/5)*x^5*ArcSinh[a*x]^2} -{x^3*ArcSinh[a*x]^2, x, 6, (-3*x^2)/(32*a^2) + x^4/32 + (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(16*a^3) - (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(8*a) - (3*ArcSinh[a*x]^2)/(32*a^4) + (x^4*ArcSinh[a*x]^2)/4} -{x^2*ArcSinh[a*x]^2, x, 5, -((4*x)/(9*a^2)) + (2*x^3)/27 + (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^3) - (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a) + (1/3)*x^3*ArcSinh[a*x]^2} -{x*ArcSinh[a*x]^2, x, 4, x^2/4 - (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a) + ArcSinh[a*x]^2/(4*a^2) + (x^2*ArcSinh[a*x]^2)/2} -{ArcSinh[a*x]^2, x, 3, 2*x - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + x*ArcSinh[a*x]^2} -{ArcSinh[a*x]^2/x, x, 6, -ArcSinh[a*x]^3/3 + ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - PolyLog[3, E^(2*ArcSinh[a*x])]/2} -{ArcSinh[a*x]^2/x^2, x, 7, -(ArcSinh[a*x]^2/x) - 4*a*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] - 2*a*PolyLog[2, -E^ArcSinh[a*x]] + 2*a*PolyLog[2, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^2/x^3, x, 3, -((a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) - ArcSinh[a*x]^2/(2*x^2) + a^2*Log[x]} -{ArcSinh[a*x]^2/x^4, x, 9, -(a^2/(3*x)) - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*x^2) - ArcSinh[a*x]^2/(3*x^3) + (2/3)*a^3*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + (1/3)*a^3*PolyLog[2, -E^ArcSinh[a*x]] - (1/3)*a^3*PolyLog[2, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^2/x^5, x, 5, -a^2/(12*x^2) - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(6*x^3) + (a^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*x) - ArcSinh[a*x]^2/(4*x^4) - (a^4*Log[x])/3} - - -{x^4*ArcSinh[a*x]^3, x, 14, -((298*Sqrt[1 + a^2*x^2])/(375*a^5)) + (76*(1 + a^2*x^2)^(3/2))/(1125*a^5) - (6*(1 + a^2*x^2)^(5/2))/(625*a^5) + (16*x*ArcSinh[a*x])/(25*a^4) - (8*x^3*ArcSinh[a*x])/(75*a^2) + (6/125)*x^5*ArcSinh[a*x] - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(25*a^5) + (4*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(25*a) + (1/5)*x^5*ArcSinh[a*x]^3} -{x^3*ArcSinh[a*x]^3, x, 11, (45*x*Sqrt[1 + a^2*x^2])/(256*a^3) - (3*x^3*Sqrt[1 + a^2*x^2])/(128*a) - (45*ArcSinh[a*x])/(256*a^4) - (9*x^2*ArcSinh[a*x])/(32*a^2) + (3*x^4*ArcSinh[a*x])/32 + (9*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(16*a) - (3*ArcSinh[a*x]^3)/(32*a^4) + (x^4*ArcSinh[a*x]^3)/4} -{x^2*ArcSinh[a*x]^3, x, 9, (14*Sqrt[1 + a^2*x^2])/(9*a^3) - (2*(1 + a^2*x^2)^(3/2))/(27*a^3) - (4*x*ArcSinh[a*x])/(3*a^2) + (2/9)*x^3*ArcSinh[a*x] + (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a) + (1/3)*x^3*ArcSinh[a*x]^3} -{x*ArcSinh[a*x]^3, x, 6, (-3*x*Sqrt[1 + a^2*x^2])/(8*a) + (3*ArcSinh[a*x])/(8*a^2) + (3*x^2*ArcSinh[a*x])/4 - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(4*a) + ArcSinh[a*x]^3/(4*a^2) + (x^2*ArcSinh[a*x]^3)/2} -{ArcSinh[a*x]^3, x, 4, (-6*Sqrt[1 + a^2*x^2])/a + 6*x*ArcSinh[a*x] - (3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a + x*ArcSinh[a*x]^3} -{ArcSinh[a*x]^3/x, x, 7, -ArcSinh[a*x]^4/4 + ArcSinh[a*x]^3*Log[1 - E^(2*ArcSinh[a*x])] + (3*ArcSinh[a*x]^2*PolyLog[2, E^(2*ArcSinh[a*x])])/2 - (3*ArcSinh[a*x]*PolyLog[3, E^(2*ArcSinh[a*x])])/2 + (3*PolyLog[4, E^(2*ArcSinh[a*x])])/4} -{ArcSinh[a*x]^3/x^2, x, 9, -(ArcSinh[a*x]^3/x) - 6*a*ArcSinh[a*x]^2*ArcTanh[E^ArcSinh[a*x]] - 6*a*ArcSinh[a*x]*PolyLog[2, -E^ArcSinh[a*x]] + 6*a*ArcSinh[a*x]*PolyLog[2, E^ArcSinh[a*x]] + 6*a*PolyLog[3, -E^ArcSinh[a*x]] - 6*a*PolyLog[3, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^3/x^3, x, 7, (-3*a^2*ArcSinh[a*x]^2)/2 - (3*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(2*x^2) + 3*a^2*ArcSinh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (3*a^2*PolyLog[2, E^(2*ArcSinh[a*x])])/2} -{ArcSinh[a*x]^3/x^4, x, 14, -((a^2*ArcSinh[a*x])/x) - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x^2) - ArcSinh[a*x]^3/(3*x^3) + a^3*ArcSinh[a*x]^2*ArcTanh[E^ArcSinh[a*x]] - a^3*ArcTanh[Sqrt[1 + a^2*x^2]] + a^3*ArcSinh[a*x]*PolyLog[2, -E^ArcSinh[a*x]] - a^3*ArcSinh[a*x]*PolyLog[2, E^ArcSinh[a*x]] - a^3*PolyLog[3, -E^ArcSinh[a*x]] + a^3*PolyLog[3, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^3/x^5, x, 10, -(a^3*Sqrt[1 + a^2*x^2])/(4*x) - (a^2*ArcSinh[a*x])/(4*x^2) + (a^4*ArcSinh[a*x]^2)/2 - (a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(4*x^3) + (a^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(4*x^4) - a^4*ArcSinh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] - (a^4*PolyLog[2, E^(2*ArcSinh[a*x])])/2} - - -{x^5*ArcSinh[a*x]^4, x, 23, (245*x^2)/(1152*a^4) - (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(576*a^5) + (65*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(864*a^3) - (x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(54*a) + (245*ArcSinh[a*x]^2)/(1152*a^6) + (5*x^2*ArcSinh[a*x]^2)/(16*a^4) - (5*x^4*ArcSinh[a*x]^2)/(48*a^2) + (x^6*ArcSinh[a*x]^2)/18 - (5*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(24*a^5) + (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(36*a^3) - (x^5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a) + (5*ArcSinh[a*x]^4)/(96*a^6) + (x^6*ArcSinh[a*x]^4)/6} -{x^4*ArcSinh[a*x]^4, x, 19, (16576*x)/(5625*a^4) - (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(5625*a^5) + (1088*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(5625*a^3) - (24*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(625*a) + (32*x*ArcSinh[a*x]^2)/(25*a^4) - (16*x^3*ArcSinh[a*x]^2)/(75*a^2) + (12/125)*x^5*ArcSinh[a*x]^2 - (32*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(75*a^5) + (16*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(25*a) + (1/5)*x^5*ArcSinh[a*x]^4} -{x^3*ArcSinh[a*x]^4, x, 14, (-45*x^2)/(128*a^2) + (3*x^4)/128 + (45*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(64*a^3) - (3*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(32*a) - (45*ArcSinh[a*x]^2)/(128*a^4) - (9*x^2*ArcSinh[a*x]^2)/(16*a^2) + (3*x^4*ArcSinh[a*x]^2)/16 + (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(8*a^3) - (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a) - (3*ArcSinh[a*x]^4)/(32*a^4) + (x^4*ArcSinh[a*x]^4)/4} -{x^2*ArcSinh[a*x]^4, x, 11, -((160*x)/(27*a^2)) + (8*x^3)/81 + (160*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(27*a^3) - (8*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(27*a) - (8*x*ArcSinh[a*x]^2)/(3*a^2) + (4/9)*x^3*ArcSinh[a*x]^2 + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(9*a) + (1/3)*x^3*ArcSinh[a*x]^4} -{x*ArcSinh[a*x]^4, x, 7, (3*x^2)/4 - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a) + (3*ArcSinh[a*x]^2)/(4*a^2) + (3*x^2*ArcSinh[a*x]^2)/2 - (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a + ArcSinh[a*x]^4/(4*a^2) + (x^2*ArcSinh[a*x]^4)/2} -{ArcSinh[a*x]^4, x, 5, 24*x - (24*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a + 12*x*ArcSinh[a*x]^2 - (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a + x*ArcSinh[a*x]^4} -{ArcSinh[a*x]^4/x, x, 8, -ArcSinh[a*x]^5/5 + ArcSinh[a*x]^4*Log[1 - E^(2*ArcSinh[a*x])] + 2*ArcSinh[a*x]^3*PolyLog[2, E^(2*ArcSinh[a*x])] - 3*ArcSinh[a*x]^2*PolyLog[3, E^(2*ArcSinh[a*x])] + 3*ArcSinh[a*x]*PolyLog[4, E^(2*ArcSinh[a*x])] - (3*PolyLog[5, E^(2*ArcSinh[a*x])])/2} -{ArcSinh[a*x]^4/x^2, x, 11, -(ArcSinh[a*x]^4/x) - 8*a*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 12*a*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]] + 12*a*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] + 24*a*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] - 24*a*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 24*a*PolyLog[4, -E^ArcSinh[a*x]] + 24*a*PolyLog[4, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^4/x^3, x, 8, -2*a^2*ArcSinh[a*x]^3 - (2*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/x - ArcSinh[a*x]^4/(2*x^2) + 6*a^2*ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + 6*a^2*ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - 3*a^2*PolyLog[3, E^(2*ArcSinh[a*x])]} -{ArcSinh[a*x]^4/x^4, x, 19, -((2*a^2*ArcSinh[a*x]^2)/x) - (2*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*x^2) - ArcSinh[a*x]^4/(3*x^3) - 8*a^3*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + (4/3)*a^3*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 4*a^3*PolyLog[2, -E^ArcSinh[a*x]] + 2*a^3*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]] + 4*a^3*PolyLog[2, E^ArcSinh[a*x]] - 2*a^3*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] - 4*a^3*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] + 4*a^3*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] + 4*a^3*PolyLog[4, -E^ArcSinh[a*x]] - 4*a^3*PolyLog[4, E^ArcSinh[a*x]]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^6/ArcSinh[a*x], x, 7, -((5*CoshIntegral[ArcSinh[a*x]])/(64*a^7)) + (9*CoshIntegral[3*ArcSinh[a*x]])/(64*a^7) - (5*CoshIntegral[5*ArcSinh[a*x]])/(64*a^7) + CoshIntegral[7*ArcSinh[a*x]]/(64*a^7)} -{x^5/ArcSinh[a*x], x, 6, (5*SinhIntegral[2*ArcSinh[a*x]])/(32*a^6) - SinhIntegral[4*ArcSinh[a*x]]/(8*a^6) + SinhIntegral[6*ArcSinh[a*x]]/(32*a^6)} -{x^4/ArcSinh[a*x], x, 6, CoshIntegral[ArcSinh[a*x]]/(8*a^5) - (3*CoshIntegral[3*ArcSinh[a*x]])/(16*a^5) + CoshIntegral[5*ArcSinh[a*x]]/(16*a^5)} -{x^3/ArcSinh[a*x], x, 5, -SinhIntegral[2*ArcSinh[a*x]]/(4*a^4) + SinhIntegral[4*ArcSinh[a*x]]/(8*a^4)} -{x^2/ArcSinh[a*x], x, 5, -CoshIntegral[ArcSinh[a*x]]/(4*a^3) + CoshIntegral[3*ArcSinh[a*x]]/(4*a^3)} -{x/ArcSinh[a*x], x, 4, SinhIntegral[2*ArcSinh[a*x]]/(2*a^2)} -{ArcSinh[a*x]^(-1), x, 2, CoshIntegral[ArcSinh[a*x]]/a} -{1/(x*ArcSinh[a*x]), x, 0, Unintegrable[1/(x*ArcSinh[a*x]), x]} -{1/(x^2*ArcSinh[a*x]), x, 0, Unintegrable[1/(x^2*ArcSinh[a*x]), x]} - - -{x^6/ArcSinh[a*x]^2, x, 6, -((x^6*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) - (5*SinhIntegral[ArcSinh[a*x]])/(64*a^7) + (27*SinhIntegral[3*ArcSinh[a*x]])/(64*a^7) - (25*SinhIntegral[5*ArcSinh[a*x]])/(64*a^7) + (7*SinhIntegral[7*ArcSinh[a*x]])/(64*a^7)} -{x^5/ArcSinh[a*x]^2, x, 5, -((x^5*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + (5*CoshIntegral[2*ArcSinh[a*x]])/(16*a^6) - CoshIntegral[4*ArcSinh[a*x]]/(2*a^6) + (3*CoshIntegral[6*ArcSinh[a*x]])/(16*a^6)} -{x^4/ArcSinh[a*x]^2, x, 5, -((x^4*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x]]/(8*a^5) - (9*SinhIntegral[3*ArcSinh[a*x]])/(16*a^5) + (5*SinhIntegral[5*ArcSinh[a*x]])/(16*a^5)} -{x^3/ArcSinh[a*x]^2, x, 4, -((x^3*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) - CoshIntegral[2*ArcSinh[a*x]]/(2*a^4) + CoshIntegral[4*ArcSinh[a*x]]/(2*a^4)} -{x^2/ArcSinh[a*x]^2, x, 4, -((x^2*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) - SinhIntegral[ArcSinh[a*x]]/(4*a^3) + (3*SinhIntegral[3*ArcSinh[a*x]])/(4*a^3)} -{x/ArcSinh[a*x]^2, x, 2, -((x*Sqrt[1 + a^2*x^2])/(a*ArcSinh[a*x])) + CoshIntegral[2*ArcSinh[a*x]]/a^2} -{ArcSinh[a*x]^(-2), x, 3, -(Sqrt[1 + a^2*x^2]/(a*ArcSinh[a*x])) + SinhIntegral[ArcSinh[a*x]]/a} -{1/(x*ArcSinh[a*x]^2), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^2), x]} -{1/(x^2*ArcSinh[a*x]^2), x, 0, Unintegrable[1/(x^2*ArcSinh[a*x]^2), x]} - - -{x^4/ArcSinh[a*x]^3, x, 14, -(x^4*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - (2*x^3)/(a^2*ArcSinh[a*x]) - (5*x^5)/(2*ArcSinh[a*x]) + CoshIntegral[ArcSinh[a*x]]/(16*a^5) - (27*CoshIntegral[3*ArcSinh[a*x]])/(32*a^5) + (25*CoshIntegral[5*ArcSinh[a*x]])/(32*a^5)} -{x^3/ArcSinh[a*x]^3, x, 12, -(x^3*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - (3*x^2)/(2*a^2*ArcSinh[a*x]) - (2*x^4)/ArcSinh[a*x] - SinhIntegral[2*ArcSinh[a*x]]/(2*a^4) + SinhIntegral[4*ArcSinh[a*x]]/a^4} -{x^2/ArcSinh[a*x]^3, x, 10, -(x^2*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - x/(a^2*ArcSinh[a*x]) - (3*x^3)/(2*ArcSinh[a*x]) - CoshIntegral[ArcSinh[a*x]]/(8*a^3) + (9*CoshIntegral[3*ArcSinh[a*x]])/(8*a^3)} -{x/ArcSinh[a*x]^3, x, 7, -(x*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]^2) - 1/(2*a^2*ArcSinh[a*x]) - x^2/ArcSinh[a*x] + SinhIntegral[2*ArcSinh[a*x]]/a^2} -{ArcSinh[a*x]^(-3), x, 4, -Sqrt[1 + a^2*x^2]/(2*a*ArcSinh[a*x]^2) - x/(2*ArcSinh[a*x]) + CoshIntegral[ArcSinh[a*x]]/(2*a)} -{1/(x*ArcSinh[a*x]^3), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^3), x]} -{1/(x^2*ArcSinh[a*x]^3), x, 0, Unintegrable[1/(x^2*ArcSinh[a*x]^3), x]} - - -{x^4/ArcSinh[a*x]^4, x, 12, -(x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - (2*x^3)/(3*a^2*ArcSinh[a*x]^2) - (5*x^5)/(6*ArcSinh[a*x]^2) - (2*x^2*Sqrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (25*x^4*Sqrt[1 + a^2*x^2])/(6*a*ArcSinh[a*x]) + SinhIntegral[ArcSinh[a*x]]/(48*a^5) - (27*SinhIntegral[3*ArcSinh[a*x]])/(32*a^5) + (125*SinhIntegral[5*ArcSinh[a*x]])/(96*a^5)} -{x^3/ArcSinh[a*x]^4, x, 9, -(x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - x^2/(2*a^2*ArcSinh[a*x]^2) - (2*x^4)/(3*ArcSinh[a*x]^2) - (x*Sqrt[1 + a^2*x^2])/(a^3*ArcSinh[a*x]) - (8*x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]) - CoshIntegral[2*ArcSinh[a*x]]/(3*a^4) + (4*CoshIntegral[4*ArcSinh[a*x]])/(3*a^4)} -{x^2/ArcSinh[a*x]^4, x, 10, -(x^2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - x/(3*a^2*ArcSinh[a*x]^2) - x^3/(2*ArcSinh[a*x]^2) - Sqrt[1 + a^2*x^2]/(3*a^3*ArcSinh[a*x]) - (3*x^2*Sqrt[1 + a^2*x^2])/(2*a*ArcSinh[a*x]) - SinhIntegral[ArcSinh[a*x]]/(24*a^3) + (9*SinhIntegral[3*ArcSinh[a*x]])/(8*a^3)} -{x/ArcSinh[a*x]^4, x, 5, -(x*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^3) - 1/(6*a^2*ArcSinh[a*x]^2) - x^2/(3*ArcSinh[a*x]^2) - (2*x*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]) + (2*CoshIntegral[2*ArcSinh[a*x]])/(3*a^2)} -{ArcSinh[a*x]^(-4), x, 5, -Sqrt[1 + a^2*x^2]/(3*a*ArcSinh[a*x]^3) - x/(6*ArcSinh[a*x]^2) - Sqrt[1 + a^2*x^2]/(6*a*ArcSinh[a*x]) + SinhIntegral[ArcSinh[a*x]]/(6*a)} -{1/(x*ArcSinh[a*x]^4), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^4), x]} -{1/(x^2*ArcSinh[a*x]^4), x, 0, Unintegrable[1/(x^2*ArcSinh[a*x]^4), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSinh[a x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Sqrt[ArcSinh[a*x]], x, 19, (x^5*Sqrt[ArcSinh[a*x]])/5 + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(32*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(64*a^5) + (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(320*a^5) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(64*a^5) - (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(320*a^5)} -{x^3*Sqrt[ArcSinh[a*x]], x, 14, (-3*Sqrt[ArcSinh[a*x]])/(32*a^4) + (x^4*Sqrt[ArcSinh[a*x]])/4 - (Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(256*a^4) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(32*a^4) - (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(256*a^4) + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(32*a^4)} -{x^2*Sqrt[ArcSinh[a*x]], x, 14, (x^3*Sqrt[ArcSinh[a*x]])/3 - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(16*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(48*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(16*a^3) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(48*a^3)} -{x*Sqrt[ArcSinh[a*x]], x, 9, Sqrt[ArcSinh[a*x]]/(4*a^2) + (x^2*Sqrt[ArcSinh[a*x]])/2 - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a^2) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a^2)} -{Sqrt[ArcSinh[a*x]], x, 7, x*Sqrt[ArcSinh[a*x]] + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(4*a) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(4*a)} -{Sqrt[ArcSinh[a*x]]/x, x, 0, Unintegrable[Sqrt[ArcSinh[a*x]]/x, x]} - - -{x^4*ArcSinh[a*x]^(3/2), x, 41, -((4*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(25*a^5)) + (2*x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(25*a^3) - (3*x^4*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(50*a) + (1/5)*x^5*ArcSinh[a*x]^(3/2) + (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(64*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(200*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(3200*a^5) + (3*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(3200*a^5) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(64*a^5) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(200*a^5) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(3200*a^5) + (3*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(3200*a^5)} -{x^3*ArcSinh[a*x]^(3/2), x, 25, (9*x*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(64*a^3) - (3*x^3*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(32*a) - (3*ArcSinh[a*x]^(3/2))/(32*a^4) + (1/4)*x^4*ArcSinh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(2048*a^4) + (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(128*a^4) + (3*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(2048*a^4) - (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(128*a^4)} -{x^2*ArcSinh[a*x]^(3/2), x, 22, (Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(6*a) + (1/3)*x^3*ArcSinh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(96*a^3) - (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(96*a^3)} -{x^1*ArcSinh[a*x]^(3/2), x, 11, -((3*x*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(8*a)) + ArcSinh[a*x]^(3/2)/(4*a^2) + (1/2)*x^2*ArcSinh[a*x]^(3/2) - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a^2) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a^2)} -{ArcSinh[a*x]^(3/2), x, 8, (-3*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(2*a) + x*ArcSinh[a*x]^(3/2) + (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(8*a) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(8*a)} -{ArcSinh[a*x]^(3/2)/x, x, 0, Unintegrable[ArcSinh[a*x]^(3/2)/x, x]} - - -{x^4*ArcSinh[a*x]^(5/2), x, 44, (2*x*Sqrt[ArcSinh[a*x]])/(5*a^4) - (x^3*Sqrt[ArcSinh[a*x]])/(15*a^2) + (3/100)*x^5*Sqrt[ArcSinh[a*x]] - (4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(15*a^5) + (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(10*a) + (1/5)*x^5*ArcSinh[a*x]^(5/2) + (15*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(128*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(240*a^5) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(1280*a^5) + (3*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(6400*a^5) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(128*a^5) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(240*a^5) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(1280*a^5) - (3*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(6400*a^5)} -{x^3*ArcSinh[a*x]^(5/2), x, 27, (-225*Sqrt[ArcSinh[a*x]])/(2048*a^4) - (45*x^2*Sqrt[ArcSinh[a*x]])/(256*a^2) + (15*x^4*Sqrt[ArcSinh[a*x]])/256 + (15*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(64*a^3) - (5*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(32*a) - (3*ArcSinh[a*x]^(5/2))/(32*a^4) + (x^4*ArcSinh[a*x]^(5/2))/4 - (15*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(16384*a^4) + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(512*a^4) - (15*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(16384*a^4) + (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(512*a^4)} -{x^2*ArcSinh[a*x]^(5/2), x, 24, (-5*x*Sqrt[ArcSinh[a*x]])/(6*a^2) + (5*x^3*Sqrt[ArcSinh[a*x]])/36 + (5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(9*a^3) - (5*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(18*a) + (x^3*ArcSinh[a*x]^(5/2))/3 - (15*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(64*a^3) + (5*Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(576*a^3) + (15*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(64*a^3) - (5*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(576*a^3)} -{x*ArcSinh[a*x]^(5/2), x, 12, (15*Sqrt[ArcSinh[a*x]])/(64*a^2) + (15*x^2*Sqrt[ArcSinh[a*x]])/32 - (5*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(8*a) + ArcSinh[a*x]^(5/2)/(4*a^2) + (x^2*ArcSinh[a*x]^(5/2))/2 - (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a^2) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a^2)} -{ArcSinh[a*x]^(5/2), x, 9, (15*x*Sqrt[ArcSinh[a*x]])/4 - (5*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(2*a) + x*ArcSinh[a*x]^(5/2) + (15*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(16*a) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(16*a)} -{ArcSinh[a*x]^(5/2)/x, x, 0, Unintegrable[ArcSinh[a*x]^(5/2)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/Sqrt[ArcSinh[a*x]], x, 18, (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(32*a^5) + (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(32*a^5)} -{x^3/Sqrt[ArcSinh[a*x]], x, 13, -(Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(32*a^4) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(8*a^4) + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(32*a^4) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(8*a^4)} -{x^2/Sqrt[ArcSinh[a*x]], x, 13, -(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(8*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(8*a^3) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(8*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(8*a^3)} -{x/Sqrt[ArcSinh[a*x]], x, 8, -(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a^2) + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a^2)} -{1/Sqrt[ArcSinh[a*x]], x, 6, (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(2*a) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(2*a)} -{1/(x*Sqrt[ArcSinh[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[ArcSinh[a*x]]), x]} -{1/(x^2*Sqrt[ArcSinh[a*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[ArcSinh[a*x]]), x]} - - -{x^4/ArcSinh[a*x]^(3/2), x, 17, -((2*x^4*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]])) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) - (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(8*a^5) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(16*a^5)} -{x^3/ArcSinh[a*x]^(3/2), x, 12, -((2*x^3*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]])) + (Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(4*a^4) - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(2*a^4) + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a^4) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(2*a^4)} -{x^2/ArcSinh[a*x]^(3/2), x, 12, -((2*x^2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]])) + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(4*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(4*a^3) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(4*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(4*a^3)} -{x/ArcSinh[a*x]^(3/2), x, 6, (-2*x*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/a^2 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/a^2} -{ArcSinh[a*x]^(-3/2), x, 7, (-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/a + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/a} -{1/(x*ArcSinh[a*x]^(3/2)), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^(3/2)), x]} - - -{x^4/ArcSinh[a*x]^(5/2), x, 34, -((2*x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2))) - (16*x^3)/(3*a^2*Sqrt[ArcSinh[a*x]]) - (20*x^5)/(3*Sqrt[ArcSinh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(12*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(8*a^5) + (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(24*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(12*a^5) - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(8*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(24*a^5)} -{x^3/ArcSinh[a*x]^(5/2), x, 24, (-2*x^3*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (4*x^2)/(a^2*Sqrt[ArcSinh[a*x]]) - (16*x^4)/(3*Sqrt[ArcSinh[a*x]]) - (2*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(3*a^4) + (Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a^4) + (2*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(3*a^4) - (Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a^4)} -{x^2/ArcSinh[a*x]^(5/2), x, 22, (-2*x^2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (8*x)/(3*a^2*Sqrt[ArcSinh[a*x]]) - (4*x^3)/Sqrt[ArcSinh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(2*a^3) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(2*a^3)} -{x/ArcSinh[a*x]^(5/2), x, 11, (-2*x*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - 4/(3*a^2*Sqrt[ArcSinh[a*x]]) - (8*x^2)/(3*Sqrt[ArcSinh[a*x]]) - (2*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a^2) + (2*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a^2)} -{1/ArcSinh[a*x]^(5/2), x, 8, (-2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcSinh[a*x]]) + (2*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(3*a)} -{1/(x*ArcSinh[a*x]^(5/2)), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^(5/2)), x]} - - -{x^4/ArcSinh[a*x]^(7/2), x, 32, -((2*x^4*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2))) - (16*x^3)/(15*a^2*ArcSinh[a*x]^(3/2)) - (4*x^5)/(3*ArcSinh[a*x]^(3/2)) - (32*x^2*Sqrt[1 + a^2*x^2])/(5*a^3*Sqrt[ArcSinh[a*x]]) - (40*x^4*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(30*a^5) + (9*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(20*a^5) - (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(12*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(30*a^5) - (9*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(20*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/(12*a^5)} -{x^3/ArcSinh[a*x]^(7/2), x, 21, (-2*x^3*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (4*x^2)/(5*a^2*ArcSinh[a*x]^(3/2)) - (16*x^4)/(15*ArcSinh[a*x]^(3/2)) - (16*x*Sqrt[1 + a^2*x^2])/(5*a^3*Sqrt[ArcSinh[a*x]]) - (128*x^3*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) + (16*Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/(15*a^4) - (4*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(15*a^4) - (4*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^4)} -{x^2/ArcSinh[a*x]^(7/2), x, 22, (-2*x^2*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (8*x)/(15*a^2*ArcSinh[a*x]^(3/2)) - (4*x^3)/(5*ArcSinh[a*x]^(3/2)) - (16*Sqrt[1 + a^2*x^2])/(15*a^3*Sqrt[ArcSinh[a*x]]) - (24*x^2*Sqrt[1 + a^2*x^2])/(5*a*Sqrt[ArcSinh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(15*a^3) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(5*a^3) - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(15*a^3) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(5*a^3)} -{x/ArcSinh[a*x]^(7/2), x, 9, (-2*x*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - 4/(15*a^2*ArcSinh[a*x]^(3/2)) - (8*x^2)/(15*ArcSinh[a*x]^(3/2)) - (32*x*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^2)} -{ArcSinh[a*x]^(-7/2), x, 9, (-2*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (4*x)/(15*ArcSinh[a*x]^(3/2)) - (8*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) - (4*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(15*a) + (4*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(15*a)} -{1/(x*ArcSinh[a*x]^(7/2)), x, 0, Unintegrable[1/(x*ArcSinh[a*x]^(7/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcSin[a x]^n with m symbolic*) - - -{x^m*ArcSinh[a*x]^4, x, 1, (x^(1 + m)*ArcSinh[a*x]^4)/(1 + m) - (4*a*Unintegrable[(x^(1 + m)*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x])/(1 + m)} -{x^m*ArcSinh[a*x]^3, x, 1, (x^(1 + m)*ArcSinh[a*x]^3)/(1 + m) - (3*a*Unintegrable[(x^(1 + m)*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x])/(1 + m)} -{x^m*ArcSinh[a*x]^2, x, 2, (x^(1 + m)*ArcSinh[a*x]^2)/(1 + m) - (2*a*x^(2 + m)*ArcSinh[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-a^2)*x^2])/(2 + 3*m + m^2) + (2*a^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (-a^2)*x^2])/(6 + 11*m + 6*m^2 + m^3)} -{x^m*ArcSinh[a*x]^1, x, 2, (x^(1 + m)*ArcSinh[a*x])/(1 + m) - (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2)} -{x^m/ArcSinh[a*x]^1, x, 0, Unintegrable[x^m/ArcSinh[a*x], x]} -{x^m/ArcSinh[a*x]^2, x, 0, Unintegrable[x^m/ArcSinh[a*x]^2, x]} - - -{x^m*ArcSinh[a*x]^(5/2), x, 0, Unintegrable[x^m*ArcSinh[a*x]^(5/2), x]} -{x^m*ArcSinh[a*x]^(3/2), x, 0, Unintegrable[x^m*ArcSinh[a*x]^(3/2), x]} -{x^m*Sqrt[ArcSinh[a*x]], x, 0, Unintegrable[x^m*Sqrt[ArcSinh[a*x]], x]} -{x^m/Sqrt[ArcSinh[a*x]], x, 0, Unintegrable[x^m/Sqrt[ArcSinh[a*x]], x]} -{x^m/ArcSinh[a*x]^(3/2), x, 0, Unintegrable[x^m/ArcSinh[a*x]^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcSin[a x]^n with n symbolic*) - - -{(b*x)^m*ArcSinh[a*x]^n, x, 0, Unintegrable[(b*x)^m*ArcSinh[a*x]^n, x]} - - -{x^4*ArcSinh[a*x]^n, x, 12, (5^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -5*ArcSinh[a*x]])/((-ArcSinh[a*x])^n*(32*a^5)) - (ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(3^n*(-ArcSinh[a*x])^n*(32*a^5)) + (ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/((-ArcSinh[a*x])^n*(16*a^5)) - Gamma[1 + n, ArcSinh[a*x]]/(16*a^5) + Gamma[1 + n, 3*ArcSinh[a*x]]/(3^n*(32*a^5)) - (5^(-1 - n)*Gamma[1 + n, 5*ArcSinh[a*x]])/(32*a^5)} -{x^3*ArcSinh[a*x]^n, x, 9, (ArcSinh[a*x]^n*Gamma[1 + n, -4*ArcSinh[a*x]])/(2^(2*(3 + n))*(-ArcSinh[a*x])^n*a^4) - (2^(-4 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/((-ArcSinh[a*x])^n*a^4) - (2^(-4 - n)*Gamma[1 + n, 2*ArcSinh[a*x]])/a^4 + Gamma[1 + n, 4*ArcSinh[a*x]]/(2^(2*(3 + n))*a^4)} -{x^2*ArcSinh[a*x]^n, x, 9, (3^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/((-ArcSinh[a*x])^n*(8*a^3)) - (ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/((-ArcSinh[a*x])^n*(8*a^3)) + Gamma[1 + n, ArcSinh[a*x]]/(8*a^3) - (3^(-1 - n)*Gamma[1 + n, 3*ArcSinh[a*x]])/(8*a^3)} -{x^1*ArcSinh[a*x]^n, x, 6, (2^(-3 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/((-ArcSinh[a*x])^n*a^2) + (2^(-3 - n)*Gamma[1 + n, 2*ArcSinh[a*x]])/a^2} -{x^0*ArcSinh[a*x]^n, x, 4, (ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(2*a*(-ArcSinh[a*x])^n) - Gamma[1 + n, ArcSinh[a*x]]/(2*a)} -{ArcSinh[a*x]^n/x^1, x, 0, Unintegrable[ArcSinh[a*x]^n/x, x]} -{ArcSinh[a*x]^n/x^2, x, 0, Unintegrable[ArcSinh[a*x]^n/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSinh[c x]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*(a + b*ArcSinh[c*x])^(1/2), x, 14, (1/3)*x^3*Sqrt[a + b*ArcSinh[c*x]] - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c^3) + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3) + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(16*c^3)) - (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(48*c^3))} -{x^1*(a + b*ArcSinh[c*x])^(1/2), x, 9, Sqrt[a + b*ArcSinh[c*x]]/(4*c^2) + (1/2)*x^2*Sqrt[a + b*ArcSinh[c*x]] - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*c^2) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(16*c^2))} -{x^0*(a + b*ArcSinh[c*x])^(1/2), x, 7, x*Sqrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(4*c))} - - -{x^2*(a + b*ArcSinh[c*x])^(3/2), x, 22, (b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(3*c^3) - (b*x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(6*c) + (1/3)*x^3*(a + b*ArcSinh[c*x])^(3/2) - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3) + (b^(3/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3) - (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(32*c^3)) + (b^(3/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(96*c^3))} -{x^1*(a + b*ArcSinh[c*x])^(3/2), x, 11, -((3*b*x*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(8*c)) + (a + b*ArcSinh[c*x])^(3/2)/(4*c^2) + (1/2)*x^2*(a + b*ArcSinh[c*x])^(3/2) - (3*b^(3/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^2) + (3*b^(3/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(64*c^2))} -{x^0*(a + b*ArcSinh[c*x])^(3/2), x, 8, -((3*b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c)) + x*(a + b*ArcSinh[c*x])^(3/2) + (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(8*c))} - - -{x^2*(a + b*ArcSinh[c*x])^(5/2), x, 24, -((5*b^2*x*Sqrt[a + b*ArcSinh[c*x]])/(6*c^2)) + (5/36)*b^2*x^3*Sqrt[a + b*ArcSinh[c*x]] + (5*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(18*c) + (1/3)*x^3*(a + b*ArcSinh[c*x])^(5/2) - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(64*c^3) + (5*b^(5/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(576*c^3) + (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(64*c^3)) - (5*b^(5/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(576*c^3))} -{x^1*(a + b*ArcSinh[c*x])^(5/2), x, 12, (15*b^2*Sqrt[a + b*ArcSinh[c*x]])/(64*c^2) + (15/32)*b^2*x^2*Sqrt[a + b*ArcSinh[c*x]] - (5*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(8*c) + (a + b*ArcSinh[c*x])^(5/2)/(4*c^2) + (1/2)*x^2*(a + b*ArcSinh[c*x])^(5/2) - (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(256*c^2))} -{x^0*(a + b*ArcSinh[c*x])^(5/2), x, 9, (15/4)*b^2*x*Sqrt[a + b*ArcSinh[c*x]] - (5*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(2*c) + x*(a + b*ArcSinh[c*x])^(5/2) + (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(16*c))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcSinh[c*x])^(1/2), x, 13, -((E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3)) + (E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3) - (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(8*Sqrt[b]*c^3)) + (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(8*Sqrt[b]*c^3))} -{x^1/(a + b*ArcSinh[c*x])^(1/2), x, 8, -((E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^2)) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(4*Sqrt[b]*c^2))} -{x^0/(a + b*ArcSinh[c*x])^(1/2), x, 6, (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*c))} - - -{x^2/(a + b*ArcSinh[c*x])^(3/2), x, 12, -((2*x^2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]])) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) - (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) - (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(4*b^(3/2)*c^3)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(4*b^(3/2)*c^3))} -{x^1/(a + b*ArcSinh[c*x])^(3/2), x, 6, -((2*x*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]])) + (E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(b^(3/2)*c^2))} -{x^0/(a + b*ArcSinh[c*x])^(3/2), x, 7, -((2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]])) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(b^(3/2)*c))} - - -{x^2/(a + b*ArcSinh[c*x])^(5/2), x, 22, -((2*x^2*Sqrt[1 + c^2*x^2])/(3*b*c*(a + b*ArcSinh[c*x])^(3/2))) - (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcSinh[c*x]]) - (4*x^3)/(b^2*Sqrt[a + b*ArcSinh[c*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(6*b^(5/2)*c^3) + (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3) - (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(6*b^(5/2)*c^3)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(2*b^(5/2)*c^3))} -{x^1/(a + b*ArcSinh[c*x])^(5/2), x, 11, -((2*x*Sqrt[1 + c^2*x^2])/(3*b*c*(a + b*ArcSinh[c*x])^(3/2))) - 4/(3*b^2*c^2*Sqrt[a + b*ArcSinh[c*x]]) - (8*x^2)/(3*b^2*Sqrt[a + b*ArcSinh[c*x]]) - (2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(3*b^(5/2)*c^2) + (2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(3*b^(5/2)*c^2))} -{x^0/(a + b*ArcSinh[c*x])^(5/2), x, 8, -((2*Sqrt[1 + c^2*x^2])/(3*b*c*(a + b*ArcSinh[c*x])^(3/2))) - (4*x)/(3*b^2*Sqrt[a + b*ArcSinh[c*x]]) + (2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(3*b^(5/2)*c) + (2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(3*b^(5/2)*c))} - - -{x^2/(a + b*ArcSinh[c*x])^(7/2), x, 22, -((2*x^2*Sqrt[1 + c^2*x^2])/(5*b*c*(a + b*ArcSinh[c*x])^(5/2))) - (8*x)/(15*b^2*c^2*(a + b*ArcSinh[c*x])^(3/2)) - (4*x^3)/(5*b^2*(a + b*ArcSinh[c*x])^(3/2)) - (16*Sqrt[1 + c^2*x^2])/(15*b^3*c^3*Sqrt[a + b*ArcSinh[c*x]]) - (24*x^2*Sqrt[1 + c^2*x^2])/(5*b^3*c*Sqrt[a + b*ArcSinh[c*x]]) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c^3) - (3*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(5*b^(7/2)*c^3) - (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(15*b^(7/2)*c^3)) + (3*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(5*b^(7/2)*c^3))} -{x^1/(a + b*ArcSinh[c*x])^(7/2), x, 9, -((2*x*Sqrt[1 + c^2*x^2])/(5*b*c*(a + b*ArcSinh[c*x])^(5/2))) - 4/(15*b^2*c^2*(a + b*ArcSinh[c*x])^(3/2)) - (8*x^2)/(15*b^2*(a + b*ArcSinh[c*x])^(3/2)) - (32*x*Sqrt[1 + c^2*x^2])/(15*b^3*c*Sqrt[a + b*ArcSinh[c*x]]) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2) + (8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(15*b^(7/2)*c^2))} -{x^0/(a + b*ArcSinh[c*x])^(7/2), x, 9, -((2*Sqrt[1 + c^2*x^2])/(5*b*c*(a + b*ArcSinh[c*x])^(5/2))) - (4*x)/(15*b^2*(a + b*ArcSinh[c*x])^(3/2)) - (8*Sqrt[1 + c^2*x^2])/(15*b^3*c*Sqrt[a + b*ArcSinh[c*x]]) - (4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c) + (4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(E^(a/b)*(15*b^(7/2)*c))} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m deleted file mode 100644 index 44e8b1d..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.4 (f x)^m (d+e x^2)^p (a+b arcsinh(c x))^n.m +++ /dev/null @@ -1,1165 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]), x, 5, (-2*b*d*Sqrt[1 + c^2*x^2])/(35*c^5) - (b*d*(1 + c^2*x^2)^(3/2))/(105*c^5) + (8*b*d*(1 + c^2*x^2)^(5/2))/(175*c^5) - (b*d*(1 + c^2*x^2)^(7/2))/(49*c^5) + (d*x^5*(a + b*ArcSinh[c*x]))/5 + (c^2*d*x^7*(a + b*ArcSinh[c*x]))/7} -{x^3*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]), x, 6, (b*d*x*Sqrt[1 + c^2*x^2])/(24*c^3) - (b*d*x^3*Sqrt[1 + c^2*x^2])/(36*c) - (b*c*d*x^5*Sqrt[1 + c^2*x^2])/36 - (b*d*ArcSinh[c*x])/(24*c^4) + (d*x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*d*x^6*(a + b*ArcSinh[c*x]))/6} -{x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]), x, 5, (2*b*d*Sqrt[1 + c^2*x^2])/(15*c^3) + (b*d*(1 + c^2*x^2)^(3/2))/(45*c^3) - (b*d*(1 + c^2*x^2)^(5/2))/(25*c^3) + (d*x^3*(a + b*ArcSinh[c*x]))/3 + (c^2*d*x^5*(a + b*ArcSinh[c*x]))/5} -{x^1*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]), x, 4, (-3*b*d*x*Sqrt[1 + c^2*x^2])/(32*c) - (b*d*x*(1 + c^2*x^2)^(3/2))/(16*c) - (3*b*d*ArcSinh[c*x])/(32*c^2) + (d*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(4*c^2)} -{x^0*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]), x, 5, (-2*b*d*Sqrt[1 + c^2*x^2])/(3*c) - (b*d*(1 + c^2*x^2)^(3/2))/(9*c) + d*x*(a + b*ArcSinh[c*x]) + (c^2*d*x^3*(a + b*ArcSinh[c*x]))/3} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^1, x, 8, (-(1/4))*b*c*d*x*Sqrt[1 + c^2*x^2] - (1/4)*b*d*ArcSinh[c*x] + (1/2)*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]) + (d*(a + b*ArcSinh[c*x])^2)/(2*b) + d*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (1/2)*b*d*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^2, x, 6, -(b*c*d*Sqrt[1 + c^2*x^2]) - (d*(a + b*ArcSinh[c*x]))/x + c^2*d*x*(a + b*ArcSinh[c*x]) - b*c*d*ArcTanh[Sqrt[1 + c^2*x^2]]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^3, x, 8, -((b*c*d*Sqrt[1 + c^2*x^2])/(2*x)) + (1/2)*b*c^2*d*ArcSinh[c*x] - (d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*x^2) + (c^2*d*(a + b*ArcSinh[c*x])^2)/(2*b) + c^2*d*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (1/2)*b*c^2*d*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^4, x, 6, -(b*c*d*Sqrt[1 + c^2*x^2])/(6*x^2) - (d*(a + b*ArcSinh[c*x]))/(3*x^3) - (c^2*d*(a + b*ArcSinh[c*x]))/x - (5*b*c^3*d*ArcTanh[Sqrt[1 + c^2*x^2]])/6} - - -{x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 6, (-8*b*d^2*Sqrt[1 + c^2*x^2])/(315*c^5) - (4*b*d^2*(1 + c^2*x^2)^(3/2))/(945*c^5) - (b*d^2*(1 + c^2*x^2)^(5/2))/(525*c^5) + (10*b*d^2*(1 + c^2*x^2)^(7/2))/(441*c^5) - (b*d^2*(1 + c^2*x^2)^(9/2))/(81*c^5) + (d^2*x^5*(a + b*ArcSinh[c*x]))/5 + (2*c^2*d^2*x^7*(a + b*ArcSinh[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcSinh[c*x]))/9} -{x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 7, (73*b*d^2*x*Sqrt[1 + c^2*x^2])/(3072*c^3) - (73*b*d^2*x^3*Sqrt[1 + c^2*x^2])/(4608*c) - (43*b*c*d^2*x^5*Sqrt[1 + c^2*x^2])/1152 - (b*c^3*d^2*x^7*Sqrt[1 + c^2*x^2])/64 - (73*b*d^2*ArcSinh[c*x])/(3072*c^4) + (d^2*x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*d^2*x^6*(a + b*ArcSinh[c*x]))/3 + (c^4*d^2*x^8*(a + b*ArcSinh[c*x]))/8} -{x^2*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 5, (8*b*d^2*Sqrt[1 + c^2*x^2])/(105*c^3) + (4*b*d^2*(1 + c^2*x^2)^(3/2))/(315*c^3) + (b*d^2*(1 + c^2*x^2)^(5/2))/(175*c^3) - (b*d^2*(1 + c^2*x^2)^(7/2))/(49*c^3) + (d^2*x^3*(a + b*ArcSinh[c*x]))/3 + (2*c^2*d^2*x^5*(a + b*ArcSinh[c*x]))/5 + (c^4*d^2*x^7*(a + b*ArcSinh[c*x]))/7} -{x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 5, (-5*b*d^2*x*Sqrt[1 + c^2*x^2])/(96*c) - (5*b*d^2*x*(1 + c^2*x^2)^(3/2))/(144*c) - (b*d^2*x*(1 + c^2*x^2)^(5/2))/(36*c) - (5*b*d^2*ArcSinh[c*x])/(96*c^2) + (d^2*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(6*c^2)} -{(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 5, (-8*b*d^2*Sqrt[1 + c^2*x^2])/(15*c) - (4*b*d^2*(1 + c^2*x^2)^(3/2))/(45*c) - (b*d^2*(1 + c^2*x^2)^(5/2))/(25*c) + d^2*x*(a + b*ArcSinh[c*x]) + (2*c^2*d^2*x^3*(a + b*ArcSinh[c*x]))/3 + (c^4*d^2*x^5*(a + b*ArcSinh[c*x]))/5} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x, x, 12, (-(11/32))*b*c*d^2*x*Sqrt[1 + c^2*x^2] - (1/16)*b*c*d^2*x*(1 + c^2*x^2)^(3/2) - (11/32)*b*d^2*ArcSinh[c*x] + (1/2)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]) + (1/4)*d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]) + (d^2*(a + b*ArcSinh[c*x])^2)/(2*b) + d^2*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (1/2)*b*d^2*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^2, x, 7, (-5*b*c*d^2*Sqrt[1 + c^2*x^2])/3 - (b*c*d^2*(1 + c^2*x^2)^(3/2))/9 - (d^2*(a + b*ArcSinh[c*x]))/x + 2*c^2*d^2*x*(a + b*ArcSinh[c*x]) + (c^4*d^2*x^3*(a + b*ArcSinh[c*x]))/3 - b*c*d^2*ArcTanh[Sqrt[1 + c^2*x^2]]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^3, x, 12, (1/4)*b*c^3*d^2*x*Sqrt[1 + c^2*x^2] - (b*c*d^2*(1 + c^2*x^2)^(3/2))/(2*x) + (1/4)*b*c^2*d^2*ArcSinh[c*x] + c^2*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]) - (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(2*x^2) + (c^2*d^2*(a + b*ArcSinh[c*x])^2)/b + 2*c^2*d^2*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - b*c^2*d^2*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^4, x, 7, -(b*c^3*d^2*Sqrt[1 + c^2*x^2]) - (b*c*d^2*Sqrt[1 + c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSinh[c*x]))/(3*x^3) - (2*c^2*d^2*(a + b*ArcSinh[c*x]))/x + c^4*d^2*x*(a + b*ArcSinh[c*x]) - (11*b*c^3*d^2*ArcTanh[Sqrt[1 + c^2*x^2]])/6} - - -{x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 5, (-16*b*d^3*Sqrt[1 + c^2*x^2])/(1155*c^5) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(3465*c^5) - (2*b*d^3*(1 + c^2*x^2)^(5/2))/(1925*c^5) - (b*d^3*(1 + c^2*x^2)^(7/2))/(1617*c^5) + (4*b*d^3*(1 + c^2*x^2)^(9/2))/(297*c^5) - (b*d^3*(1 + c^2*x^2)^(11/2))/(121*c^5) + (d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^2*d^3*x^7*(a + b*ArcSinh[c*x]))/7 + (c^4*d^3*x^9*(a + b*ArcSinh[c*x]))/3 + (c^6*d^3*x^11*(a + b*ArcSinh[c*x]))/11} -{x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 8, (49*b*d^3*x*Sqrt[1 + c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 + c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 + c^2*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 + c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 + c^2*x^2)^(9/2))/(100*c^3) + (49*b*d^3*ArcSinh[c*x])/(5120*c^4) - (d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/(8*c^4) + (d^3*(1 + c^2*x^2)^5*(a + b*ArcSinh[c*x]))/(10*c^4)} -{x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 5, (16*b*d^3*Sqrt[1 + c^2*x^2])/(315*c^3) + (8*b*d^3*(1 + c^2*x^2)^(3/2))/(945*c^3) + (2*b*d^3*(1 + c^2*x^2)^(5/2))/(525*c^3) + (b*d^3*(1 + c^2*x^2)^(7/2))/(441*c^3) - (b*d^3*(1 + c^2*x^2)^(9/2))/(81*c^3) + (d^3*x^3*(a + b*ArcSinh[c*x]))/3 + (3*c^2*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcSinh[c*x]))/7 + (c^6*d^3*x^9*(a + b*ArcSinh[c*x]))/9} -{x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 6, (-35*b*d^3*x*Sqrt[1 + c^2*x^2])/(1024*c) - (35*b*d^3*x*(1 + c^2*x^2)^(3/2))/(1536*c) - (7*b*d^3*x*(1 + c^2*x^2)^(5/2))/(384*c) - (b*d^3*x*(1 + c^2*x^2)^(7/2))/(64*c) - (35*b*d^3*ArcSinh[c*x])/(1024*c^2) + (d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x]))/(8*c^2)} -{(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 5, (-16*b*d^3*Sqrt[1 + c^2*x^2])/(35*c) - (8*b*d^3*(1 + c^2*x^2)^(3/2))/(105*c) - (6*b*d^3*(1 + c^2*x^2)^(5/2))/(175*c) - (b*d^3*(1 + c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSinh[c*x]) + c^2*d^3*x^3*(a + b*ArcSinh[c*x]) + (3*c^4*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (c^6*d^3*x^7*(a + b*ArcSinh[c*x]))/7} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x, x, 17, (-(19/48))*b*c*d^3*x*Sqrt[1 + c^2*x^2] - (7/72)*b*c*d^3*x*(1 + c^2*x^2)^(3/2) - (1/36)*b*c*d^3*x*(1 + c^2*x^2)^(5/2) - (19/48)*b*d^3*ArcSinh[c*x] + (1/2)*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]) + (1/4)*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]) + (1/6)*d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]) + (d^3*(a + b*ArcSinh[c*x])^2)/(2*b) + d^3*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (1/2)*b*d^3*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^2, x, 7, (-11*b*c*d^3*Sqrt[1 + c^2*x^2])/5 - (b*c*d^3*(1 + c^2*x^2)^(3/2))/5 - (b*c*d^3*(1 + c^2*x^2)^(5/2))/25 - (d^3*(a + b*ArcSinh[c*x]))/x + 3*c^2*d^3*x*(a + b*ArcSinh[c*x]) + c^4*d^3*x^3*(a + b*ArcSinh[c*x]) + (c^6*d^3*x^5*(a + b*ArcSinh[c*x]))/5 - b*c*d^3*ArcTanh[Sqrt[1 + c^2*x^2]]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^3, x, 17, (-(3/32))*b*c^3*d^3*x*Sqrt[1 + c^2*x^2] + (7/16)*b*c^3*d^3*x*(1 + c^2*x^2)^(3/2) - (b*c*d^3*(1 + c^2*x^2)^(5/2))/(2*x) - (3/32)*b*c^2*d^3*ArcSinh[c*x] + (3/2)*c^2*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]) + (3/4)*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]) - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(2*x^2) + (3*c^2*d^3*(a + b*ArcSinh[c*x])^2)/(2*b) + 3*c^2*d^3*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])] - (3/2)*b*c^2*d^3*PolyLog[2, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^4, x, 8, (-(8/3))*b*c^3*d^3*Sqrt[1 + c^2*x^2] - (b*c*d^3*Sqrt[1 + c^2*x^2])/(6*x^2) - (1/9)*b*c^3*d^3*(1 + c^2*x^2)^(3/2) - (d^3*(a + b*ArcSinh[c*x]))/(3*x^3) - (3*c^2*d^3*(a + b*ArcSinh[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcSinh[c*x]) + (1/3)*c^6*d^3*x^3*(a + b*ArcSinh[c*x]) - (17/6)*b*c^3*d^3*ArcTanh[Sqrt[1 + c^2*x^2]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x, 12, (4*b*Sqrt[1 + c^2*x^2])/(3*c^5*d) - (b*(1 + c^2*x^2)^(3/2))/(9*c^5*d) - (x*(a + b*ArcSinh[c*x]))/(c^4*d) + (x^3*(a + b*ArcSinh[c*x]))/(3*c^2*d) + (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^5*d) - (I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d) + (I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d)} -{(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x, 8, -(b*x*Sqrt[1 + c^2*x^2])/(4*c^3*d) + (b*ArcSinh[c*x])/(4*c^4*d) + (x^2*(a + b*ArcSinh[c*x]))/(2*c^2*d) + (a + b*ArcSinh[c*x])^2/(2*b*c^4*d) - ((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c^4*d) - (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*c^4*d)} -{(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x, 8, -((b*Sqrt[1 + c^2*x^2])/(c^3*d)) + (x*(a + b*ArcSinh[c*x]))/(c^2*d) - (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^3*d) + (I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d) - (I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d)} -{(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x, 5, -(a + b*ArcSinh[c*x])^2/(2*b*c^2*d) + ((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c^2*d) + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*c^2*d)} -{(a + b*ArcSinh[c*x])/(d + c^2*d*x^2), x, 6, (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*d) - (I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d) + (I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d)} -{(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)), x, 7, (-2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d - (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d) + (b*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d)} -{(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)), x, 10, -((a + b*ArcSinh[c*x])/(d*x)) - (2*c*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d - (b*c*ArcTanh[Sqrt[1 + c^2*x^2]])/d + (I*b*c*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d - (I*b*c*PolyLog[2, I*E^ArcSinh[c*x]])/d} -{(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)), x, 9, -(b*c*Sqrt[1 + c^2*x^2])/(2*d*x) - (a + b*ArcSinh[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d + (b*c^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d) - (b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d)} -{(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)), x, 15, -(b*c*Sqrt[1 + c^2*x^2])/(6*d*x^2) - (a + b*ArcSinh[c*x])/(3*d*x^3) + (c^2*(a + b*ArcSinh[c*x]))/(d*x) + (2*c^3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d + (7*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(6*d) - (I*b*c^3*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d + (I*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]])/d} - - -{(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2, x, 12, b/(2*c^5*d^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[1 + c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcSinh[c*x]))/(2*c^4*d^2) - (x^3*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*(1 + c^2*x^2)) - (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^5*d^2) + (((3*I)/2)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d^2)} -{(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2, x, 8, -(b*x)/(2*c^3*d^2*Sqrt[1 + c^2*x^2]) + (b*ArcSinh[c*x])/(2*c^4*d^2) - (x^2*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])^2/(2*b*c^4*d^2) + ((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c^4*d^2) + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*c^4*d^2)} -{(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2, x, 8, -b/(2*c^3*d^2*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x]))/(2*c^2*d^2*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^3*d^2) - ((I/2)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d^2) + ((I/2)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^2)} -{(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2, x, 2, (b*x)/(2*c*d^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(2*c^2*d^2*(1 + c^2*x^2))} -{(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^2, x, 8, b/(2*c*d^2*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(2*d^2*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*d^2) - ((I/2)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d^2) + ((I/2)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d^2)} -{(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^2), x, 9, -(b*c*x)/(2*d^2*Sqrt[1 + c^2*x^2]) + (a + b*ArcSinh[c*x])/(2*d^2*(1 + c^2*x^2)) - (2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d^2 - (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d^2) + (b*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d^2)} -{(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^2), x, 13, -(b*c)/(2*d^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(d^2*x*(1 + c^2*x^2)) - (3*c^2*x*(a + b*ArcSinh[c*x]))/(2*d^2*(1 + c^2*x^2)) - (3*c*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d^2 - (b*c*ArcTanh[Sqrt[1 + c^2*x^2]])/d^2 + (((3*I)/2)*b*c*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^2 - (((3*I)/2)*b*c*PolyLog[2, I*E^ArcSinh[c*x]])/d^2} -{(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^2), x, 12, -(b*c)/(2*d^2*x*Sqrt[1 + c^2*x^2]) - (c^2*(a + b*ArcSinh[c*x]))/(d^2*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])/(2*d^2*x^2*(1 + c^2*x^2)) + (4*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d^2 + (b*c^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^2 - (b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2} -{(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^2), x, 19, (b*c^3)/(3*d^2*Sqrt[1 + c^2*x^2]) - (b*c)/(6*d^2*x^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(3*d^2*x^3*(1 + c^2*x^2)) + (5*c^2*(a + b*ArcSinh[c*x]))/(3*d^2*x*(1 + c^2*x^2)) + (5*c^4*x*(a + b*ArcSinh[c*x]))/(2*d^2*(1 + c^2*x^2)) + (5*c^3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/d^2 + (13*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(6*d^2) - (5*I*b*c^3*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(2*d^2) + (5*I*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]])/(2*d^2)} - - -{(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3, x, 12, b/(12*c^5*d^3*(1 + c^2*x^2)^(3/2)) - (5*b)/(8*c^5*d^3*Sqrt[1 + c^2*x^2]) - (x^3*(a + b*ArcSinh[c*x]))/(4*c^2*d^3*(1 + c^2*x^2)^2) - (3*x*(a + b*ArcSinh[c*x]))/(8*c^4*d^3*(1 + c^2*x^2)) + (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(4*c^5*d^3) - (((3*I)/8)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^3) + (((3*I)/8)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d^3)} -{(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3, x, 4, (b*x^3)/(12*c*d^3*(1 + c^2*x^2)^(3/2)) + (b*x)/(4*c^3*d^3*Sqrt[1 + c^2*x^2]) - (b*ArcSinh[c*x])/(4*c^4*d^3) + (x^4*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2)} -{(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3, x, 10, -b/(12*c^3*d^3*(1 + c^2*x^2)^(3/2)) + b/(8*c^3*d^3*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x]))/(4*c^2*d^3*(1 + c^2*x^2)^2) + (x*(a + b*ArcSinh[c*x]))/(8*c^2*d^3*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(4*c^3*d^3) - ((I/8)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d^3) + ((I/8)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^3)} -{(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3, x, 3, (b*x)/(12*c*d^3*(1 + c^2*x^2)^(3/2)) + (b*x)/(6*c*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(4*c^2*d^3*(1 + c^2*x^2)^2)} -{(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^3, x, 10, b/(12*c*d^3*(1 + c^2*x^2)^(3/2)) + (3*b)/(8*c*d^3*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) + (3*x*(a + b*ArcSinh[c*x]))/(8*d^3*(1 + c^2*x^2)) + (3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(4*c*d^3) - (((3*I)/8)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d^3) + (((3*I)/8)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d^3)} -{(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^3), x, 12, -(b*c*x)/(12*d^3*(1 + c^2*x^2)^(3/2)) - (2*b*c*x)/(3*d^3*Sqrt[1 + c^2*x^2]) + (a + b*ArcSinh[c*x])/(4*d^3*(1 + c^2*x^2)^2) + (a + b*ArcSinh[c*x])/(2*d^3*(1 + c^2*x^2)) - (2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d^3 - (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d^3) + (b*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d^3)} -{(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^3), x, 16, -(b*c)/(12*d^3*(1 + c^2*x^2)^(3/2)) - (7*b*c)/(8*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(d^3*x*(1 + c^2*x^2)^2) - (5*c^2*x*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) - (15*c^2*x*(a + b*ArcSinh[c*x]))/(8*d^3*(1 + c^2*x^2)) - (15*c*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(4*d^3) - (b*c*ArcTanh[Sqrt[1 + c^2*x^2]])/d^3 + (((15*I)/8)*b*c*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^3 - (((15*I)/8)*b*c*PolyLog[2, I*E^ArcSinh[c*x]])/d^3} -{(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^3), x, 16, -(b*c)/(2*d^3*x*(1 + c^2*x^2)^(3/2)) - (5*b*c^3*x)/(12*d^3*(1 + c^2*x^2)^(3/2)) + (2*b*c^3*x)/(3*d^3*Sqrt[1 + c^2*x^2]) - (3*c^2*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) - (a + b*ArcSinh[c*x])/(2*d^3*x^2*(1 + c^2*x^2)^2) - (3*c^2*(a + b*ArcSinh[c*x]))/(2*d^3*(1 + c^2*x^2)) + (6*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/d^3 + (3*b*c^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(2*d^3) - (3*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d^3)} -{(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^3), x, 23, -((b*c^3)/(12*d^3*(1 + c^2*x^2)^(3/2))) - (b*c)/(6*d^3*x^2*(1 + c^2*x^2)^(3/2)) + (29*b*c^3)/(24*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(3*d^3*x^3*(1 + c^2*x^2)^2) + (7*c^2*(a + b*ArcSinh[c*x]))/(3*d^3*x*(1 + c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSinh[c*x]))/(12*d^3*(1 + c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSinh[c*x]))/(8*d^3*(1 + c^2*x^2)) + (35*c^3*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(4*d^3) + (19*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/(6*d^3) - (35*I*b*c^3*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(8*d^3) + (35*I*b*c^3*PolyLog[2, I*E^ArcSinh[c*x]])/(8*d^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (Pi+Pi c^2 x^2)^(p/2) (a+b ArcSinh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]), x, 3, (2*b*Sqrt[Pi]*x)/(15*c^3) - (b*Sqrt[Pi]*x^3)/(45*c) - (1/25)*b*c*Sqrt[Pi]*x^5 - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^4*Pi) + ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4*Pi^2)} -{x^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]), x, 5, -((b*Sqrt[Pi]*x^2)/(16*c)) - (1/16)*b*c*Sqrt[Pi]*x^4 + (Sqrt[Pi]*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (1/4)*x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) - (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(16*b*c^3)} -{x^1*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]), x, 2, -((b*Sqrt[Pi]*x)/(3*c)) - (1/9)*b*c*Sqrt[Pi]*x^3 + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*Pi)} -{x^0*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]), x, 3, (-(1/4))*b*c*Sqrt[Pi]*x^2 + (1/2)*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(4*b*c)} -{(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^1, x, 8, (-b)*c*Sqrt[Pi]*x + Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) - 2*Sqrt[Pi]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - b*Sqrt[Pi]*PolyLog[2, -E^ArcSinh[c*x]] + b*Sqrt[Pi]*PolyLog[2, E^ArcSinh[c*x]]} -{(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^2, x, 3, -((Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x) + (c*Sqrt[Pi]*(a + b*ArcSinh[c*x])^2)/(2*b) + b*c*Sqrt[Pi]*Log[x]} -{(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^3, x, 8, -((b*c*Sqrt[Pi])/(2*x)) - (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2*x^2) - c^2*Sqrt[Pi]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - (1/2)*b*c^2*Sqrt[Pi]*PolyLog[2, -E^ArcSinh[c*x]] + (1/2)*b*c^2*Sqrt[Pi]*PolyLog[2, E^ArcSinh[c*x]]} -{(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^4, x, 3, -((b*c*Sqrt[Pi])/(6*x^2)) - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*Pi*x^3) + (1/3)*b*c^3*Sqrt[Pi]*Log[x]} - - -{x^3*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 4, (2*b*Pi^(3/2)*x)/(35*c^3) - (b*Pi^(3/2)*x^3)/(105*c) - (8/175)*b*c*Pi^(3/2)*x^5 - (1/49)*b*c^3*Pi^(3/2)*x^7 - ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4*Pi) + ((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*Pi^2)} -{x^2*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 8, -((b*Pi^(3/2)*x^2)/(32*c)) - (7/96)*b*c*Pi^(3/2)*x^4 - (1/36)*b*c^3*Pi^(3/2)*x^6 + (Pi^(3/2)*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (1/8)*Pi*x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (1/6)*x^3*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - (Pi^(3/2)*(a + b*ArcSinh[c*x])^2)/(32*b*c^3)} -{x^1*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 3, -((b*Pi^(3/2)*x)/(5*c)) - (2/15)*b*c*Pi^(3/2)*x^3 - (1/25)*b*c^3*Pi^(3/2)*x^5 + ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2*Pi)} -{x^0*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 6, (-(5/16))*b*c*Pi^(3/2)*x^2 - (1/16)*b*c^3*Pi^(3/2)*x^4 + (3/8)*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (1/4)*x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + (3*Pi^(3/2)*(a + b*ArcSinh[c*x])^2)/(16*b*c)} -{((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^1, x, 10, (-(4/3))*b*c*Pi^(3/2)*x - (1/9)*b*c^3*Pi^(3/2)*x^3 + Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (1/3)*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - 2*Pi^(3/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - b*Pi^(3/2)*PolyLog[2, -E^ArcSinh[c*x]] + b*Pi^(3/2)*PolyLog[2, E^ArcSinh[c*x]]} -{((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2, x, 6, (-(1/4))*b*c^3*Pi^(3/2)*x^2 + (3/2)*c^2*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x + (3*c*Pi^(3/2)*(a + b*ArcSinh[c*x])^2)/(4*b) + b*c*Pi^(3/2)*Log[x]} -{((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3, x, 11, -((b*c*Pi^(3/2))/(2*x)) - b*c^3*Pi^(3/2)*x + (3/2)*c^2*Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - 3*c^2*Pi^(3/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - (3/2)*b*c^2*Pi^(3/2)*PolyLog[2, -E^ArcSinh[c*x]] + (3/2)*b*c^2*Pi^(3/2)*PolyLog[2, E^ArcSinh[c*x]]} -{((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4, x, 6, -((b*c*Pi^(3/2))/(6*x^2)) - (c^2*Pi*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x - ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x^3) + (c^3*Pi^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*b) + (4/3)*b*c^3*Pi^(3/2)*Log[x]} - - -{x^3*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 4, (2*b*Pi^(5/2)*x)/(63*c^3) - (b*Pi^(5/2)*x^3)/(189*c) - (1/21)*b*c*Pi^(5/2)*x^5 - (19/441)*b*c^3*Pi^(5/2)*x^7 - (1/81)*b*c^5*Pi^(5/2)*x^9 - ((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*Pi) + ((Pi + c^2*Pi*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*Pi^2)} -{x^2*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 12, -((5*b*Pi^(5/2)*x^2)/(256*c)) - (59/768)*b*c*Pi^(5/2)*x^4 - (17/288)*b*c^3*Pi^(5/2)*x^6 - (1/64)*b*c^5*Pi^(5/2)*x^8 + (5*Pi^(5/2)*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (5/64)*Pi^2*x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (5/48)*Pi*x^3*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + (1/8)*x^3*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]) - (5*Pi^(5/2)*(a + b*ArcSinh[c*x])^2)/(256*b*c^3)} -{x^1*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 3, -((b*Pi^(5/2)*x)/(7*c)) - (1/7)*b*c*Pi^(5/2)*x^3 - (3/35)*b*c^3*Pi^(5/2)*x^5 - (1/49)*b*c^5*Pi^(5/2)*x^7 + ((Pi + c^2*Pi*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^2*Pi)} -{x^0*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 8, (-(25/96))*b*c*Pi^(5/2)*x^2 - (5/96)*b*c^3*Pi^(5/2)*x^4 - (b*Pi^(5/2)*(1 + c^2*x^2)^3)/(36*c) + (5/16)*Pi^2*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (5/24)*Pi*x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + (1/6)*x*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]) + (5*Pi^(5/2)*(a + b*ArcSinh[c*x])^2)/(32*b*c)} -{((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^1, x, 13, (-(23/15))*b*c*Pi^(5/2)*x - (11/45)*b*c^3*Pi^(5/2)*x^3 - (1/25)*b*c^5*Pi^(5/2)*x^5 + Pi^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (1/3)*Pi*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + (1/5)*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]) - 2*Pi^(5/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - b*Pi^(5/2)*PolyLog[2, -E^ArcSinh[c*x]] + b*Pi^(5/2)*PolyLog[2, E^ArcSinh[c*x]]} -{((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^2, x, 10, (-(9/16))*b*c^3*Pi^(5/2)*x^2 - (1/16)*b*c^5*Pi^(5/2)*x^4 + (15/8)*c^2*Pi^2*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (5/4)*c^2*Pi*x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x + (15*c*Pi^(5/2)*(a + b*ArcSinh[c*x])^2)/(16*b) + b*c*Pi^(5/2)*Log[x]} -{((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3, x, 13, -((b*c*Pi^(5/2))/(2*x)) - (7/3)*b*c^3*Pi^(5/2)*x - (1/9)*b*c^5*Pi^(5/2)*x^3 + (5/2)*c^2*Pi^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) + (5/6)*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - 5*c^2*Pi^(5/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - (5/2)*b*c^2*Pi^(5/2)*PolyLog[2, -E^ArcSinh[c*x]] + (5/2)*b*c^2*Pi^(5/2)*PolyLog[2, E^ArcSinh[c*x]]} -{((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4, x, 10, -((b*c*Pi^(5/2))/(6*x^2)) - (1/4)*b*c^5*Pi^(5/2)*x^2 + (5/2)*c^4*Pi^2*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]) - (5*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x) - ((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(3*x^3) + (5*c^3*Pi^(5/2)*(a + b*ArcSinh[c*x])^2)/(4*b) + (7/3)*b*c^3*Pi^(5/2)*Log[x]} - - -{Sqrt[1 + x^2]*ArcSinh[x], x, 3, -x^2/4 + (x*Sqrt[1 + x^2]*ArcSinh[x])/2 + ArcSinh[x]^2/4} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^5*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2], x, 6, -((8*b*x)/(15*c^5*Sqrt[Pi])) + (4*b*x^3)/(45*c^3*Sqrt[Pi]) - (b*x^5)/(25*c*Sqrt[Pi]) + (8*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(15*c^6*Pi) - (4*x^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(15*c^4*Pi) + (x^4*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2*Pi)} -{(x^4*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2], x, 5, (3*b*x^2)/(16*c^3*Sqrt[Pi]) - (b*x^4)/(16*c*Sqrt[Pi]) - (3*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(8*c^4*Pi) + (x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2*Pi) + (3*(a + b*ArcSinh[c*x])^2)/(16*b*c^5*Sqrt[Pi])} -{(x^3*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2], x, 4, (2*b*x)/(3*c^3*Sqrt[Pi]) - (b*x^3)/(9*c*Sqrt[Pi]) - (2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(3*c^4*Pi) + (x^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2*Pi)} -{(x^2*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2], x, 3, -((b*x^2)/(4*c*Sqrt[Pi])) + (x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2*Pi) - (a + b*ArcSinh[c*x])^2/(4*b*c^3*Sqrt[Pi])} -{(x^1*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2], x, 2, -((b*x)/(c*Sqrt[Pi])) + (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^2*Pi)} -{x^0*(a + b*ArcSinh[c*x])/Sqrt[Pi + c^2*Pi*x^2], x, 1, (a + b*ArcSinh[c*x])^2/(2*b*c*Sqrt[Pi])} -{(a + b*ArcSinh[c*x])/(x^1*Sqrt[Pi + c^2*Pi*x^2]), x, 6, -((2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[Pi]) - (b*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[Pi] + (b*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[Pi]} -{(a + b*ArcSinh[c*x])/(x^2*Sqrt[Pi + c^2*Pi*x^2]), x, 2, -((Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(Pi*x)) + (b*c*Log[x])/Sqrt[Pi]} -{(a + b*ArcSinh[c*x])/(x^3*Sqrt[Pi + c^2*Pi*x^2]), x, 8, -((b*c)/(2*Sqrt[Pi]*x)) - (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2*Pi*x^2) + (c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[Pi] + (b*c^2*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[Pi]) - (b*c^2*PolyLog[2, E^ArcSinh[c*x]])/(2*Sqrt[Pi])} -{(a + b*ArcSinh[c*x])/(x^4*Sqrt[Pi + c^2*Pi*x^2]), x, 4, -((b*c)/(6*Sqrt[Pi]*x^2)) - (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(3*Pi*x^3) + (2*c^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(3*Pi*x) - (2*b*c^3*Log[x])/(3*Sqrt[Pi])} - - -{(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2), x, 5, (5*b*x)/(3*c^5*Pi^(3/2)) - (b*x^3)/(9*c^3*Pi^(3/2)) - (a + b*ArcSinh[c*x])/(c^6*Pi*Sqrt[Pi + c^2*Pi*x^2]) - (2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^6*Pi^3) + (b*ArcTan[c*x])/(c^6*Pi^(3/2))} -{(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2), x, 7, -((b*x^2)/(4*c^3*Pi^(3/2))) - (x^3*(a + b*ArcSinh[c*x]))/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2]) + (3*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2*c^4*Pi^2) - (3*(a + b*ArcSinh[c*x])^2)/(4*b*c^5*Pi^(3/2)) - (b*Log[1 + c^2*x^2])/(2*c^5*Pi^(3/2))} -{(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2), x, 4, -((b*x)/(c^3*Pi^(3/2))) + (a + b*ArcSinh[c*x])/(c^4*Pi*Sqrt[Pi + c^2*Pi*x^2]) + (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^4*Pi^2) - (b*ArcTan[c*x])/(c^4*Pi^(3/2))} -{(x^2*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2), x, 3, -((x*(a + b*ArcSinh[c*x]))/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (a + b*ArcSinh[c*x])^2/(2*b*c^3*Pi^(3/2)) + (b*Log[1 + c^2*x^2])/(2*c^3*Pi^(3/2))} -{(x^1*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2), x, 2, -((a + b*ArcSinh[c*x])/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (b*ArcTan[c*x])/(c^2*Pi^(3/2))} -{x^0*(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2), x, 2, (x*(a + b*ArcSinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*Log[1 + c^2*x^2])/(2*c*Pi^(3/2))} -{(a + b*ArcSinh[c*x])/(x^1*(Pi + c^2*Pi*x^2)^(3/2)), x, 8, (a + b*ArcSinh[c*x])/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*ArcTan[c*x])/Pi^(3/2) - (2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Pi^(3/2) - (b*PolyLog[2, -E^ArcSinh[c*x]])/Pi^(3/2) + (b*PolyLog[2, E^ArcSinh[c*x]])/Pi^(3/2)} -{(a + b*ArcSinh[c*x])/(x^2*(Pi + c^2*Pi*x^2)^(3/2)), x, 5, -((a + b*ArcSinh[c*x])/(Pi*x*Sqrt[Pi + c^2*Pi*x^2])) - (2*c^2*x*(a + b*ArcSinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*Log[x])/Pi^(3/2) + (b*c*Log[1 + c^2*x^2])/(2*Pi^(3/2))} -{(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(3/2)), x, 11, -((b*c)/(2*Pi^(3/2)*x)) - (3*c^2*(a + b*ArcSinh[c*x]))/(2*Pi*Sqrt[Pi + c^2*Pi*x^2]) - (a + b*ArcSinh[c*x])/(2*Pi*x^2*Sqrt[Pi + c^2*Pi*x^2]) + (b*c^2*ArcTan[c*x])/Pi^(3/2) + (3*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Pi^(3/2) + (3*b*c^2*PolyLog[2, -E^ArcSinh[c*x]])/(2*Pi^(3/2)) - (3*b*c^2*PolyLog[2, E^ArcSinh[c*x]])/(2*Pi^(3/2))} -{(a + b*ArcSinh[c*x])/(x^4*(Pi + c^2*Pi*x^2)^(3/2)), x, 5, -((b*c)/(6*Pi^(3/2)*x^2)) - (a + b*ArcSinh[c*x])/(3*Pi*x^3*Sqrt[Pi + c^2*Pi*x^2]) + (4*c^2*(a + b*ArcSinh[c*x]))/(3*Pi*x*Sqrt[Pi + c^2*Pi*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x]))/(3*Pi*Sqrt[Pi + c^2*Pi*x^2]) - (5*b*c^3*Log[x])/(3*Pi^(3/2)) - (b*c^3*Log[1 + c^2*x^2])/(2*Pi^(3/2))} - - -{(x^6*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 11, -((b*x^2)/(4*c^5*Pi^(5/2))) - b/(6*c^7*Pi^(5/2)*(1 + c^2*x^2)) - (x^5*(a + b*ArcSinh[c*x]))/(3*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (5*x^3*(a + b*ArcSinh[c*x]))/(3*c^4*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (5*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2*c^6*Pi^3) - (5*(a + b*ArcSinh[c*x])^2)/(4*b*c^7*Pi^(5/2)) - (7*b*Log[1 + c^2*x^2])/(6*c^7*Pi^(5/2))} -{(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 5, -((b*x)/(c^5*Pi^(5/2))) + (b*x)/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])/(3*c^6*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (2*(a + b*ArcSinh[c*x]))/(c^6*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^6*Pi^3) - (11*b*ArcTan[c*x])/(6*c^6*Pi^(5/2))} -{(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 7, b/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)) - (x^3*(a + b*ArcSinh[c*x]))/(3*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x]))/(c^4*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (a + b*ArcSinh[c*x])^2/(2*b*c^5*Pi^(5/2)) + (2*b*Log[1 + c^2*x^2])/(3*c^5*Pi^(5/2))} -{(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 4, -((b*x)/(6*c^3*Pi^(5/2)*(1 + c^2*x^2))) + (a + b*ArcSinh[c*x])/(3*c^4*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (a + b*ArcSinh[c*x])/(c^4*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (5*b*ArcTan[c*x])/(6*c^4*Pi^(5/2))} -{(x^2*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 4, -(b/(6*c^3*Pi^(5/2)*(1 + c^2*x^2))) + (x^3*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (b*Log[1 + c^2*x^2])/(6*c^3*Pi^(5/2))} -{(x^1*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2), x, 3, (b*x)/(6*c*Pi^(5/2)*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])/(3*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (b*ArcTan[c*x])/(6*c^2*Pi^(5/2))} -{x^0*(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(5/2), x, 4, b/(6*c*Pi^(5/2)*(1 + c^2*x^2)) + (x*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x]))/(3*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) - (b*Log[1 + c^2*x^2])/(3*c*Pi^(5/2))} -{(a + b*ArcSinh[c*x])/(x^1*(Pi + c^2*Pi*x^2)^(5/2)), x, 11, -((b*c*x)/(6*Pi^(5/2)*(1 + c^2*x^2))) + (a + b*ArcSinh[c*x])/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (a + b*ArcSinh[c*x])/(Pi^2*Sqrt[Pi + c^2*Pi*x^2]) - (7*b*ArcTan[c*x])/(6*Pi^(5/2)) - (2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Pi^(5/2) - (b*PolyLog[2, -E^ArcSinh[c*x]])/Pi^(5/2) + (b*PolyLog[2, E^ArcSinh[c*x]])/Pi^(5/2)} -{(a + b*ArcSinh[c*x])/(x^2*(Pi + c^2*Pi*x^2)^(5/2)), x, 5, -((b*c)/(6*Pi^(5/2)*(1 + c^2*x^2))) - (a + b*ArcSinh[c*x])/(Pi*x*(Pi + c^2*Pi*x^2)^(3/2)) - (4*c^2*x*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x]))/(3*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (b*c*Log[x])/Pi^(5/2) + (5*b*c*Log[1 + c^2*x^2])/(6*Pi^(5/2))} -{(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(5/2)), x, 15, -((3*b*c)/(4*Pi^(5/2)*x)) + (b*c)/(4*Pi^(5/2)*x*(1 + c^2*x^2)) + (5*b*c^3*x)/(12*Pi^(5/2)*(1 + c^2*x^2)) - (5*c^2*(a + b*ArcSinh[c*x]))/(6*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (a + b*ArcSinh[c*x])/(2*Pi*x^2*(Pi + c^2*Pi*x^2)^(3/2)) - (5*c^2*(a + b*ArcSinh[c*x]))/(2*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (13*b*c^2*ArcTan[c*x])/(6*Pi^(5/2)) + (5*c^2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Pi^(5/2) + (5*b*c^2*PolyLog[2, -E^ArcSinh[c*x]])/(2*Pi^(5/2)) - (5*b*c^2*PolyLog[2, E^ArcSinh[c*x]])/(2*Pi^(5/2))} -{(a + b*ArcSinh[c*x])/(x^4*(Pi + c^2*Pi*x^2)^(5/2)), x, 5, -((b*c)/(6*Pi^(5/2)*x^2)) + (b*c^3)/(6*Pi^(5/2)*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])/(3*Pi*x^3*(Pi + c^2*Pi*x^2)^(3/2)) + (2*c^2*(a + b*ArcSinh[c*x]))/(Pi*x*(Pi + c^2*Pi*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSinh[c*x]))/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSinh[c*x]))/(3*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) - (8*b*c^3*Log[x])/(3*Pi^(5/2)) - (4*b*c^3*Log[1 + c^2*x^2])/(3*Pi^(5/2))} - - -{ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2), x, 6, 1/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + 2/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x])/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x])/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x])/(15*c^3*Sqrt[c + a^2*c*x^2]) - (4*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])/(15*a*c^3*Sqrt[c + a^2*c*x^2])} - - -{(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2], x, 5, (3*x^2)/(16*a^3) - x^4/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) + (3*ArcSinh[a*x]^2)/(16*a^5)} -{(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2], x, 4, (2*x)/(3*a^3) - x^3/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^2)} -{(x^2*ArcSinh[a*x])/Sqrt[1 + a^2*x^2], x, 3, -x^2/(4*a) + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)} -{(x^1*ArcSinh[a*x])/Sqrt[1 + a^2*x^2], x, 2, -(x/a) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2} -{ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 1, ArcSinh[a*x]^2/(2*a)} -{ArcSinh[a*x]/(x^1*Sqrt[1 + a^2*x^2]), x, 6, -2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] - PolyLog[2, -E^ArcSinh[a*x]] + PolyLog[2, E^ArcSinh[a*x]]} -{ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]), x, 2, -((Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) + a*Log[x]} -{ArcSinh[a*x]/(x^3*Sqrt[1 + a^2*x^2]), x, 8, -a/(2*x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*x^2) + a^2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + (a^2*PolyLog[2, -E^ArcSinh[a*x]])/2 - (a^2*PolyLog[2, E^ArcSinh[a*x]])/2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]), x, 3, (2*b*x*Sqrt[d + c^2*d*x^2])/(15*c^3*Sqrt[1 + c^2*x^2]) - (b*x^3*Sqrt[d + c^2*d*x^2])/(45*c*Sqrt[1 + c^2*x^2]) - (b*c*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^4*d) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4*d^2)} -{x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]), x, 5, -(b*x^2*Sqrt[d + c^2*d*x^2])/(16*c*Sqrt[1 + c^2*x^2]) - (b*c*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/4 - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c^3*Sqrt[1 + c^2*x^2])} -{x^1*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]), x, 2, -(b*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2]) - (b*c*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*d)} -{x^0*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]), x, 3, -(b*c*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^1, x, 8, -((b*c*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2]) + Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^2, x, 3, -((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x) + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*Sqrt[1 + c^2*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[x])/Sqrt[1 + c^2*x^2]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^3, x, 8, -((b*c*Sqrt[d + c^2*d*x^2])/(2*x*Sqrt[1 + c^2*x^2])) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*x^2) - (c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*c^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (b*c^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2])} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4, x, 3, -((b*c*Sqrt[d + c^2*d*x^2])/(6*x^2*Sqrt[1 + c^2*x^2])) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*d*x^3) + (b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*Sqrt[1 + c^2*x^2])} - - -{x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 4, (2*b*d*x*Sqrt[d + c^2*d*x^2])/(35*c^3*Sqrt[1 + c^2*x^2]) - (b*d*x^3*Sqrt[d + c^2*d*x^2])/(105*c*Sqrt[1 + c^2*x^2]) - (8*b*c*d*x^5*Sqrt[d + c^2*d*x^2])/(175*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4*d) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*d^2)} -{x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 8, -(b*d*x^2*Sqrt[d + c^2*d*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^6*Sqrt[d + c^2*d*x^2])/(36*Sqrt[1 + c^2*x^2]) + (d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 - (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])} -{x^1*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 3, -(b*d*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2*d)} -{x^0*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 6, (-5*b*c*d*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) + (3*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^1, x, 10, -((4*b*c*d*x*Sqrt[d + c^2*d*x^2])/(3*Sqrt[1 + c^2*x^2])) - (b*c^3*d*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/3)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - (2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2, x, 6, -(b*c^3*d*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) + (3*c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x + (3*c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*Sqrt[1 + c^2*x^2]) + (b*c*d*Sqrt[d + c^2*d*x^2]*Log[x])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3, x, 11, -((b*c*d*Sqrt[d + c^2*d*x^2])/(2*x*Sqrt[1 + c^2*x^2])) - (b*c^3*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (3/2)*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - (3*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4, x, 6, -(b*c*d*Sqrt[d + c^2*d*x^2])/(6*x^2*Sqrt[1 + c^2*x^2]) - (c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x^3) + (c^3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*Sqrt[1 + c^2*x^2]) + (4*b*c^3*d*Sqrt[d + c^2*d*x^2]*Log[x])/(3*Sqrt[1 + c^2*x^2])} - - -{x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 4, (2*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c^2*x^2]) - (b*c*d^2*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^4*d) + ((d + c^2*d*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(9*c^4*d^2)} -{x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 12, (-5*b*d^2*x^2*Sqrt[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d + c^2*d*x^2])/(768*Sqrt[1 + c^2*x^2]) - (17*b*c^3*d^2*x^6*Sqrt[d + c^2*d*x^2])/(288*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/64 + (5*d*x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/48 + (x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/8 - (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt[1 + c^2*x^2])} -{x^1*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 3, -(b*d^2*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) - (b*c*d^2*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d^2*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^2*d)} -{x^0*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 8, (-25*b*c*d^2*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (b*d^2*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*d*x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/24 + (x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^1, x, 13, -((23*b*c*d^2*x*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2])) - (11*b*c^3*d^2*x^3*Sqrt[d + c^2*d*x^2])/(45*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/3)*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + (1/5)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]) - (2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^2, x, 10, (-9*b*c^3*d^2*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) + (15*c^2*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (5*c^2*d*x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x + (15*c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*Sqrt[1 + c^2*x^2]) + (b*c*d^2*Sqrt[d + c^2*d*x^2]*Log[x])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3, x, 13, -((b*c*d^2*Sqrt[d + c^2*d*x^2])/(2*x*Sqrt[1 + c^2*x^2])) - (7*b*c^3*d^2*x*Sqrt[d + c^2*d*x^2])/(3*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + (5/2)*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/6)*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - (5*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4, x, 10, -(b*c*d^2*Sqrt[d + c^2*d*x^2])/(6*x^2*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) + (5*c^4*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*Sqrt[1 + c^2*x^2]) + (7*b*c^3*d^2*Sqrt[d + c^2*d*x^2]*Log[x])/(3*Sqrt[1 + c^2*x^2])} - - -{Sqrt[1 + x^2]*ArcSinh[x], x, 3, -x^2/4 + (x*Sqrt[1 + x^2]*ArcSinh[x])/2 + ArcSinh[x]^2/4} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 6, (-8*b*x*Sqrt[1 + c^2*x^2])/(15*c^5*Sqrt[d + c^2*d*x^2]) + (4*b*x^3*Sqrt[1 + c^2*x^2])/(45*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^5*Sqrt[1 + c^2*x^2])/(25*c*Sqrt[d + c^2*d*x^2]) + (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15*c^6*d) - (4*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15*c^4*d) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2*d)} -{x^4*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 5, (3*b*x^2*Sqrt[1 + c^2*x^2])/(16*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^4*Sqrt[1 + c^2*x^2])/(16*c*Sqrt[d + c^2*d*x^2]) - (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^4*d) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2*d) + (3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c^5*Sqrt[d + c^2*d*x^2])} -{x^3*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 4, (2*b*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^3*Sqrt[1 + c^2*x^2])/(9*c*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^4*d) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2*d)} -{x^2*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 3, -(b*x^2*Sqrt[1 + c^2*x^2])/(4*c*Sqrt[d + c^2*d*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2*d) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^3*Sqrt[d + c^2*d*x^2])} -{x^1*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 2, -((b*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2])) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^2*d)} -{x^0*(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2], x, 1, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^1*Sqrt[d + c^2*d*x^2]), x, 6, -((2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2]} -{(a + b*ArcSinh[c*x])/(x^2*Sqrt[d + c^2*d*x^2]), x, 2, -((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(d*x)) + (b*c*Sqrt[1 + c^2*x^2]*Log[x])/Sqrt[d + c^2*d*x^2]} -{(a + b*ArcSinh[c*x])/(x^3*Sqrt[d + c^2*d*x^2]), x, 8, -((b*c*Sqrt[1 + c^2*x^2])/(2*x*Sqrt[d + c^2*d*x^2])) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*d*x^2) + (c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] + (b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[d + c^2*d*x^2]) - (b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^4*Sqrt[d + c^2*d*x^2]), x, 4, -(b*c*Sqrt[1 + c^2*x^2])/(6*x^2*Sqrt[d + c^2*d*x^2]) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x^3) + (2*c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x) - (2*b*c^3*Sqrt[1 + c^2*x^2]*Log[x])/(3*Sqrt[d + c^2*d*x^2])} - - -{x^5*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 5, (5*b*x*Sqrt[d + c^2*d*x^2])/(3*c^5*d^2*Sqrt[1 + c^2*x^2]) - (b*x^3*Sqrt[d + c^2*d*x^2])/(9*c^3*d^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(c^6*d*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^6*d^2) + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^6*d^3) + (b*Sqrt[d + c^2*d*x^2]*ArcTan[c*x])/(c^6*d^2*Sqrt[1 + c^2*x^2])} -{x^4*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 7, -(b*x^2*Sqrt[1 + c^2*x^2])/(4*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x]))/(c^2*d*Sqrt[d + c^2*d*x^2]) + (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^4*d^2) - (3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^5*d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c^5*d*Sqrt[d + c^2*d*x^2])} -{x^3*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 4, -((b*x*Sqrt[d + c^2*d*x^2])/(c^3*d^2*Sqrt[1 + c^2*x^2])) + (a + b*ArcSinh[c*x])/(c^4*d*Sqrt[d + c^2*d*x^2]) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^4*d^2) - (b*Sqrt[d + c^2*d*x^2]*ArcTan[c*x])/(c^4*d^2*Sqrt[1 + c^2*x^2])} -{x^2*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 3, -((x*(a + b*ArcSinh[c*x]))/(c^2*d*Sqrt[d + c^2*d*x^2])) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c^3*d*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c^3*d*Sqrt[d + c^2*d*x^2])} -{x^1*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 2, -((a + b*ArcSinh[c*x])/(c^2*d*Sqrt[d + c^2*d*x^2])) + (b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(c^2*d*Sqrt[d + c^2*d*x^2])} -{x^0*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2), x, 2, (x*(a + b*ArcSinh[c*x]))/(d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c*d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^1*(d + c^2*d*x^2)^(3/2)), x, 8, (a + b*ArcSinh[c*x])/(d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(d*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^(3/2)), x, 5, -((a + b*ArcSinh[c*x])/(d*x*Sqrt[d + c^2*d*x^2])) - (2*c^2*x*(a + b*ArcSinh[c*x]))/(d*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[x])/(d^2*Sqrt[1 + c^2*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[1 + c^2*x^2])/(2*d^2*Sqrt[1 + c^2*x^2])} -{(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^(3/2)), x, 11, -((b*c*Sqrt[1 + c^2*x^2])/(2*d*x*Sqrt[d + c^2*d*x^2])) - (3*c^2*(a + b*ArcSinh[c*x]))/(2*d*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])/(2*d*x^2*Sqrt[d + c^2*d*x^2]) + (b*c^2*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(d*Sqrt[d + c^2*d*x^2]) + (3*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (3*b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*d*Sqrt[d + c^2*d*x^2]) - (3*b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(3/2)), x, 5, -((b*c*Sqrt[d + c^2*d*x^2])/(6*d^2*x^2*Sqrt[1 + c^2*x^2])) - (a + b*ArcSinh[c*x])/(3*d*x^3*Sqrt[d + c^2*d*x^2]) + (4*c^2*(a + b*ArcSinh[c*x]))/(3*d*x*Sqrt[d + c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x]))/(3*d*Sqrt[d + c^2*d*x^2]) - (5*b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*d^2*Sqrt[1 + c^2*x^2]) - (b*c^3*Sqrt[d + c^2*d*x^2]*Log[1 + c^2*x^2])/(2*d^2*Sqrt[1 + c^2*x^2])} - - -{x^6*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 11, -b/(6*c^7*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (b*x^2*Sqrt[1 + c^2*x^2])/(4*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (x^5*(a + b*ArcSinh[c*x]))/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (5*x^3*(a + b*ArcSinh[c*x]))/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (5*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*c^6*d^3) - (5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^7*d^2*Sqrt[d + c^2*d*x^2]) - (7*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(6*c^7*d^2*Sqrt[d + c^2*d*x^2])} -{x^5*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 5, (b*x*Sqrt[d + c^2*d*x^2])/(6*c^5*d^3*(1 + c^2*x^2)^(3/2)) - (b*x*Sqrt[d + c^2*d*x^2])/(c^5*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(3*c^6*d*(d + c^2*d*x^2)^(3/2)) + (2*(a + b*ArcSinh[c*x]))/(c^6*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^6*d^3) - (11*b*Sqrt[d + c^2*d*x^2]*ArcTan[c*x])/(6*c^6*d^3*Sqrt[1 + c^2*x^2])} -{x^4*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 7, b/(6*c^5*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x]))/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x]))/(c^4*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (2*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2])} -{x^3*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 4, -((b*x*Sqrt[d + c^2*d*x^2])/(6*c^3*d^3*(1 + c^2*x^2)^(3/2))) + (a + b*ArcSinh[c*x])/(3*c^4*d*(d + c^2*d*x^2)^(3/2)) - (a + b*ArcSinh[c*x])/(c^4*d^2*Sqrt[d + c^2*d*x^2]) + (5*b*Sqrt[d + c^2*d*x^2]*ArcTan[c*x])/(6*c^4*d^3*Sqrt[1 + c^2*x^2])} -{x^2*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 4, -b/(6*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (x^3*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) - (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(6*c^3*d^2*Sqrt[d + c^2*d*x^2])} -{x^1*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 3, (b*x)/(6*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) + (b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(6*c^2*d^2*Sqrt[d + c^2*d*x^2])} -{x^0*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2), x, 4, b/(6*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (x*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(3*c*d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^1*(d + c^2*d*x^2)^(5/2)), x, 11, -((b*c*x)/(6*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])) + (a + b*ArcSinh[c*x])/(3*d*(d + c^2*d*x^2)^(3/2)) + (a + b*ArcSinh[c*x])/(d^2*Sqrt[d + c^2*d*x^2]) - (7*b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(6*d^2*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^(5/2)), x, 5, -((b*c*Sqrt[d + c^2*d*x^2])/(6*d^3*(1 + c^2*x^2)^(3/2))) - (a + b*ArcSinh[c*x])/(d*x*(d + c^2*d*x^2)^(3/2)) - (4*c^2*x*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[x])/(d^3*Sqrt[1 + c^2*x^2]) + (5*b*c*Sqrt[d + c^2*d*x^2]*Log[1 + c^2*x^2])/(6*d^3*Sqrt[1 + c^2*x^2])} -{(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^(5/2)), x, 15, (b*c)/(4*d^2*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (5*b*c^3*x)/(12*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (3*b*c*Sqrt[1 + c^2*x^2])/(4*d^2*x*Sqrt[d + c^2*d*x^2]) - (5*c^2*(a + b*ArcSinh[c*x]))/(6*d*(d + c^2*d*x^2)^(3/2)) - (a + b*ArcSinh[c*x])/(2*d*x^2*(d + c^2*d*x^2)^(3/2)) - (5*c^2*(a + b*ArcSinh[c*x]))/(2*d^2*Sqrt[d + c^2*d*x^2]) + (13*b*c^2*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(6*d^2*Sqrt[d + c^2*d*x^2]) + (5*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (5*b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*d^2*Sqrt[d + c^2*d*x^2]) - (5*b*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, E^ArcSinh[c*x]])/(2*d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(5/2)), x, 5, (b*c^3*Sqrt[d + c^2*d*x^2])/(6*d^3*(1 + c^2*x^2)^(3/2)) - (b*c*Sqrt[d + c^2*d*x^2])/(6*d^3*x^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])/(3*d*x^3*(d + c^2*d*x^2)^(3/2)) + (2*c^2*(a + b*ArcSinh[c*x]))/(d*x*(d + c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[d + c^2*d*x^2]) - (8*b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*d^3*Sqrt[1 + c^2*x^2]) - (4*b*c^3*Sqrt[d + c^2*d*x^2]*Log[1 + c^2*x^2])/(3*d^3*Sqrt[1 + c^2*x^2])} - - -{ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2), x, 6, 1/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + 2/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x])/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x])/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x])/(15*c^3*Sqrt[c + a^2*c*x^2]) - (4*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])/(15*a*c^3*Sqrt[c + a^2*c*x^2])} - - -{x^4*ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 5, (3*x^2)/(16*a^3) - x^4/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) + (3*ArcSinh[a*x]^2)/(16*a^5)} -{x^3*ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 4, (2*x)/(3*a^3) - x^3/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(3*a^2)} -{x^2*ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 3, -x^2/(4*a) + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3)} -{x^1*ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 2, -(x/a) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2} -{x^0*ArcSinh[a*x]/Sqrt[1 + a^2*x^2], x, 1, ArcSinh[a*x]^2/(2*a)} -{ArcSinh[a*x]/(x^1*Sqrt[1 + a^2*x^2]), x, 6, -2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] - PolyLog[2, -E^ArcSinh[a*x]] + PolyLog[2, E^ArcSinh[a*x]]} -{ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]), x, 2, -((Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) + a*Log[x]} -{ArcSinh[a*x]/(x^3*Sqrt[1 + a^2*x^2]), x, 8, -a/(2*x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*x^2) + a^2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + (a^2*PolyLog[2, -E^ArcSinh[a*x]])/2 - (a^2*PolyLog[2, E^ArcSinh[a*x]])/2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x]) and m symbolic*) - - -{x^m*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]), x, 6, If[$VersionNumber>=8, -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 + c^2*x^2])/((3 + m)^2*(5 + m)^2*(7 + m)^2)) - (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 + c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*Sqrt[1 + c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (3*c^2*d^3*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSinh[c*x]))/(5 + m) + (c^6*d^3*x^(7 + m)*(a + b*ArcSinh[c*x]))/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2), -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*x^(2 + m)*Sqrt[1 + c^2*x^2])/((7 + m)^2*(15 + 8*m + m^2)^2)) - (b*c^3*d^3*(9 + m)*(13 + 2*m)*x^(4 + m)*Sqrt[1 + c^2*x^2])/((5 + m)^2*(7 + m)^2) - (b*c^5*d^3*x^(6 + m)*Sqrt[1 + c^2*x^2])/(7 + m)^2 + (d^3*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (3*c^2*d^3*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) + (3*c^4*d^3*x^(5 + m)*(a + b*ArcSinh[c*x]))/(5 + m) + (c^6*d^3*x^(7 + m)*(a + b*ArcSinh[c*x]))/(7 + m) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/((2 + 3*m + m^2)*(105 + 71*m + 15*m^2 + m^3)^2)]} -{x^m*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]), x, 5, If[$VersionNumber>=8, -((b*c*d^2*(38 + 13*m + m^2)*x^(2 + m)*Sqrt[1 + c^2*x^2])/((3 + m)^2*(5 + m)^2)) - (b*c^3*d^2*x^(4 + m)*Sqrt[1 + c^2*x^2])/(5 + m)^2 + (d^2*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (2*c^2*d^2*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) + (c^4*d^2*x^(5 + m)*(a + b*ArcSinh[c*x]))/(5 + m) - (b*c*d^2*(149 + 100*m + 15*m^2)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/((1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2), -((b*c*d^2*(38 + 13*m + m^2)*x^(2 + m)*Sqrt[1 + c^2*x^2])/((3 + m)^2*(5 + m)^2)) - (b*c^3*d^2*x^(4 + m)*Sqrt[1 + c^2*x^2])/(5 + m)^2 + (d^2*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (2*c^2*d^2*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) + (c^4*d^2*x^(5 + m)*(a + b*ArcSinh[c*x]))/(5 + m) - (b*c*d^2*(149 + 100*m + 15*m^2)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/((2 + 3*m + m^2)*(15 + 8*m + m^2)^2)]} -{x^m*(d + c^2*d*x^2)^1*(a + b*ArcSinh[c*x]), x, 4, If[$VersionNumber>=8, -((b*c*d*x^(2 + m)*Sqrt[1 + c^2*x^2])/(3 + m)^2) + (d*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (c^2*d*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) - (b*c*d*(7 + 3*m)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/((1 + m)*(2 + m)*(3 + m)^2), -((b*c*d*x^(2 + m)*Sqrt[1 + c^2*x^2])/(3 + m)^2) + (d*x^(1 + m)*(a + b*ArcSinh[c*x]))/(1 + m) + (c^2*d*x^(3 + m)*(a + b*ArcSinh[c*x]))/(3 + m) - (b*c*d*(7 + 3*m)*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/((3 + m)^2*(2 + 3*m + m^2))]} -{x^m*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^1, x, 0, Unintegrable[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x]} -{x^m*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^2, x, 2, (x^(1 + m)*(a + b*ArcSinh[c*x]))/(2*d^2*(1 + c^2*x^2)) - (b*c*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(2*d^2*(2 + m)) + ((1 - m)*Unintegrable[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x])/(2*d)} -{x^m*(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^3, x, 4, (x^(1 + m)*(a + b*ArcSinh[c*x]))/(4*d^3*(1 + c^2*x^2)^2) + ((3 - m)*x^(1 + m)*(a + b*ArcSinh[c*x]))/(8*d^3*(1 + c^2*x^2)) - (b*c*(3 - m)*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(8*d^3*(2 + m)) - (b*c*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(4*d^3*(2 + m)) + ((1 - m)*(3 - m)*Unintegrable[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x])/(8*d^2)} - - -{x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]), x, 9, If[$VersionNumber>=8, (-15*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (5*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((12 + 8*m + m^2)*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*(6 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^(6 + m)*Sqrt[d + c^2*d*x^2])/((6 + m)^2*Sqrt[1 + c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((4 + m)*(6 + m)) + (x^(1 + m)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(6 + m) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/((6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 + c^2*x^2]) - (15*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((1 + m)*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]), -((15*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2])) - (5*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((12 + 8*m + m^2)*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*(6 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^(6 + m)*Sqrt[d + c^2*d*x^2])/((6 + m)^2*Sqrt[1 + c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((4 + m)*(6 + m)) + (x^(1 + m)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(6 + m) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/((6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 + c^2*x^2]) - (15*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2])/((2 + m)^2*(6 + m)*(4 + 5*m + m^2)*Sqrt[1 + c^2*x^2])]} -{x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]), x, 6, If[$VersionNumber>=8, (-3*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]) - (b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8 + 6*m + m^2) + (x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(4 + m) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/((8 + 14*m + 7*m^2 + m^3)*Sqrt[1 + c^2*x^2]) - (3*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((1 + m)*(2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]), -((3*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2])) - (b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8 + 6*m + m^2) + (x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(4 + m) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/((8 + 14*m + 7*m^2 + m^3)*Sqrt[1 + c^2*x^2]) - (3*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2])/((2 + m)^2*(4 + 5*m + m^2)*Sqrt[1 + c^2*x^2])]} -{x^m*(d + c^2*d*x^2)^(1/2)*(a + b*ArcSinh[c*x]), x, 3, -((b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2])/((2 + m)^2*Sqrt[1 + c^2*x^2])) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2 + m) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/((2 + 3*m + m^2)*Sqrt[1 + c^2*x^2]) - (b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((1 + m)*(2 + m)^2*Sqrt[1 + c^2*x^2])} -{(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(1/2), x, 1, (x^(1 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/((1 + m)*Sqrt[d + c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 + c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/((2 + 3*m + m^2)*Sqrt[d + c^2*d*x^2])} -{(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2), x, 3, (x^(1 + m)*(a + b*ArcSinh[c*x]))/(d*Sqrt[d + c^2*d*x^2]) - (m*x^(1 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(d*(1 + m)*Sqrt[d + c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(d*(2 + m)*Sqrt[d + c^2*d*x^2]) + (b*c*m*x^(2 + m)*Sqrt[1 + c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2])/(d*(2 + 3*m + m^2)*Sqrt[d + c^2*d*x^2])} -{(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2), x, 5, (x^(1 + m)*(a + b*ArcSinh[c*x]))/(3*d*(d + c^2*d*x^2)^(3/2)) + ((2 - m)*x^(1 + m)*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[d + c^2*d*x^2]) - ((2 - m)*m*x^(1 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(3*d^2*(1 + m)*Sqrt[d + c^2*d*x^2]) - (b*c*(2 - m)*x^(2 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(3*d^2*(2 + m)*Sqrt[d + c^2*d*x^2]) - (b*c*x^(2 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, (-c^2)*x^2])/(3*d^2*(2 + m)*Sqrt[d + c^2*d*x^2]) + (b*c*(2 - m)*m*x^(2 + m)*Sqrt[1 + c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2])/(3*d^2*(2 + 3*m + m^2)*Sqrt[d + c^2*d*x^2])} - - -{(x^m*ArcSinh[a*x])/Sqrt[1 + a^2*x^2], x, 1, (x^(1 + m)*ArcSinh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(a^2*x^2)])/(1 + m) - (a*x^(2 + m)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(a^2*x^2)])/(2 + 3*m + m^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2, x, 11, (304*b^2*d*x)/(3675*c^4) - (152*b^2*d*x^3)/(11025*c^2) + (38*b^2*d*x^5)/6125 + (2*b^2*c^2*d*x^7)/343 - (32*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(525*c^5) + (16*b*d*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(525*c^3) - (4*b*d*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(175*c) - (2*b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(21*c^5) + (4*b*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(35*c^5) - (2*b*d*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(49*c^5) + (2*d*x^5*(a + b*ArcSinh[c*x])^2)/35 + (d*x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/7} -{x^3*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2, x, 14, -(b^2*d*x^2)/(24*c^2) + (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(12*c^3) - (b*d*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/18 - (d*(a + b*ArcSinh[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSinh[c*x])^2)/12 + (d*x^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/6} -{x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2, x, 9, (-52*b^2*d*x)/(225*c^2) + (26*b^2*d*x^3)/675 + (2*b^2*c^2*d*x^5)/125 + (8*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(45*c^3) - (4*b*d*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(45*c) + (2*b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(15*c^3) - (2*b*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(25*c^3) + (2*d*x^3*(a + b*ArcSinh[c*x])^2)/15 + (d*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/5} -{x^1*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2, x, 7, (5*b^2*d*x^2)/32 + (b^2*c^2*d*x^4)/32 - (3*b*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(16*c) - (b*d*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(8*c) - (3*d*(a + b*ArcSinh[c*x])^2)/(32*c^2) + (d*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(4*c^2)} -{x^0*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2, x, 6, (14*b^2*d*x)/9 + (2*b^2*c^2*d*x^3)/27 - (4*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c) - (2*b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(9*c) + (2*d*x*(a + b*ArcSinh[c*x])^2)/3 + (d*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^1, x, 10, (1/4)*b^2*c^2*d*x^2 - (1/2)*b*c*d*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (1/4)*d*(a + b*ArcSinh[c*x])^2 + (1/2)*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2 + (d*(a + b*ArcSinh[c*x])^3)/(3*b) + d*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] - b*d*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (1/2)*b^2*d*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^2, x, 12, 2*b^2*c^2*d*x - 2*b*c*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) + 2*c^2*d*x*(a + b*ArcSinh[c*x])^2 - (d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/x - 4*b*c*d*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - 2*b^2*c*d*PolyLog[2, -E^ArcSinh[c*x]] + 2*b^2*c*d*PolyLog[2, E^ArcSinh[c*x]]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^3, x, 10, -((b*c*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/x) + (1/2)*c^2*d*(a + b*ArcSinh[c*x])^2 - (d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*x^2) + (c^2*d*(a + b*ArcSinh[c*x])^3)/(3*b) + c^2*d*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] + b^2*c^2*d*Log[x] - b*c^2*d*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (1/2)*b^2*c^2*d*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^4, x, 16, -(b^2*c^2*d)/(3*x) - (b*c*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) - (2*c^2*d*(a + b*ArcSinh[c*x])^2)/(3*x) - (d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*x^3) - (10*b*c^3*d*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (5*b^2*c^3*d*PolyLog[2, -E^ArcSinh[c*x]])/3 + (5*b^2*c^3*d*PolyLog[2, E^ArcSinh[c*x]])/3} - - -{x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 16, (4208*b^2*d^2*x)/(99225*c^4) - (2104*b^2*d^2*x^3)/(297675*c^2) + (526*b^2*d^2*x^5)/165375 + (212*b^2*c^2*d^2*x^7)/27783 + (2*b^2*c^4*d^2*x^9)/729 - (128*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^5) + (64*b*d^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4725*c^3) - (16*b*d^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(1575*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(189*c^5) + (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(315*c^5) + (20*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(441*c^5) - (2*b*d^2*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(81*c^5) + (8*d^2*x^5*(a + b*ArcSinh[c*x])^2)/315 + (4*d^2*x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/63 + (d^2*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/9} -{x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 25, (-73*b^2*d^2*x^2)/(3072*c^2) + (73*b^2*d^2*x^4)/9216 + (43*b^2*c^2*d^2*x^6)/3456 + (b^2*c^4*d^2*x^8)/256 + (73*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(1536*c^3) - (73*b*d^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2304*c) - (25*b*c*d^2*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/576 - (b*c*d^2*x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/32 - (73*d^2*(a + b*ArcSinh[c*x])^2)/(3072*c^4) + (d^2*x^4*(a + b*ArcSinh[c*x])^2)/24 + (d^2*x^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/12 + (d^2*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/8} -{x^2*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 14, (-1636*b^2*d^2*x)/(11025*c^2) + (818*b^2*d^2*x^3)/33075 + (136*b^2*c^2*d^2*x^5)/6125 + (2*b^2*c^4*d^2*x^7)/343 + (32*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(315*c^3) - (16*b*d^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(315*c) + (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(105*c^3) + (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(175*c^3) - (2*b*d^2*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(49*c^3) + (8*d^2*x^3*(a + b*ArcSinh[c*x])^2)/105 + (4*d^2*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/35 + (d^2*x^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/7} -{x^1*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 9, (25*b^2*d^2*x^2)/288 + (5*b^2*c^2*d^2*x^4)/288 + (b^2*d^2*(1 + c^2*x^2)^3)/(108*c^2) - (5*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(48*c) - (5*b*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(72*c) - (b*d^2*x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(18*c) - (5*d^2*(a + b*ArcSinh[c*x])^2)/(96*c^2) + (d^2*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(6*c^2)} -{x^0*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 10, (298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 + (2*b^2*c^4*d^2*x^5)/125 - (16*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(15*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(45*c) - (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSinh[c*x])^2)/15 + (4*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/15 + (d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^1, x, 17, (13/32)*b^2*c^2*d^2*x^2 + (1/32)*b^2*c^4*d^2*x^4 - (11/16)*b*c*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (1/8)*b*c*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - (11/32)*d^2*(a + b*ArcSinh[c*x])^2 + (1/2)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2 + (1/4)*d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2 + (d^2*(a + b*ArcSinh[c*x])^3)/(3*b) + d^2*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] - b*d^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (1/2)*b^2*d^2*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^2, x, 17, (32*b^2*c^2*d^2*x)/9 + (2*b^2*c^4*d^2*x^3)/27 - (10*b*c*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/3 - (2*b*c*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/9 + (8*c^2*d^2*x*(a + b*ArcSinh[c*x])^2)/3 + (4*c^2*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x - 4*b*c*d^2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - 2*b^2*c*d^2*PolyLog[2, -E^ArcSinh[c*x]] + 2*b^2*c*d^2*PolyLog[2, E^ArcSinh[c*x]]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^3, x, 17, (1/4)*b^2*c^4*d^2*x^2 + (1/2)*b*c^3*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (b*c*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x + (1/4)*c^2*d^2*(a + b*ArcSinh[c*x])^2 + c^2*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2 - (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(2*x^2) + (2*c^2*d^2*(a + b*ArcSinh[c*x])^3)/(3*b) + 2*c^2*d^2*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] + b^2*c^2*d^2*Log[x] - 2*b*c^2*d^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - b^2*c^2*d^2*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^4, x, 24, -(b^2*c^2*d^2)/(3*x) + 2*b^2*c^4*d^2*x - (5*b*c^3*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/3 - (b*c*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*x^2) + (8*c^4*d^2*x*(a + b*ArcSinh[c*x])^2)/3 - (4*c^2*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*x) - (d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*x^3) - (22*b*c^3*d^2*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (11*b^2*c^3*d^2*PolyLog[2, -E^ArcSinh[c*x]])/3 + (11*b^2*c^3*d^2*PolyLog[2, E^ArcSinh[c*x]])/3} - - -{x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 21, (100976*b^2*d^3*x)/(4002075*c^4) - (50488*b^2*d^3*x^3)/(12006225*c^2) + (12622*b^2*d^3*x^5)/6670125 + (9410*b^2*c^2*d^3*x^7)/1120581 + (182*b^2*c^4*d^3*x^9)/29403 + (2*b^2*c^6*d^3*x^11)/1331 - (256*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(17325*c^5) + (128*b*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(17325*c^3) - (32*b*d^3*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(5775*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(693*c^5) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(1155*c^5) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(1617*c^5) + (8*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(297*c^5) - (2*b*d^3*(1 + c^2*x^2)^(11/2)*(a + b*ArcSinh[c*x]))/(121*c^5) + (16*d^3*x^5*(a + b*ArcSinh[c*x])^2)/1155 + (8*d^3*x^5*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/231 + (2*d^3*x^5*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/33 + (d^3*x^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/11} -{x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 40, (-79*b^2*d^3*x^2)/(5120*c^2) + (79*b^2*d^3*x^4)/15360 + (401*b^2*c^2*d^3*x^6)/28800 + (57*b^2*c^4*d^3*x^8)/6400 + (b^2*c^6*d^3*x^10)/500 + (79*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2560*c^3) - (79*b*d^3*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3840*c) - (31*b*c*d^3*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/960 - (b*c*d^3*x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/32 - (b*c*d^3*x^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/50 - (79*d^3*(a + b*ArcSinh[c*x])^2)/(5120*c^4) + (d^3*x^4*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/20 + (3*d^3*x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/40 + (d^3*x^4*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/10} -{x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 19, (-10516*b^2*d^3*x)/(99225*c^2) + (5258*b^2*d^3*x^3)/297675 + (4198*b^2*c^2*d^3*x^5)/165375 + (374*b^2*c^4*d^3*x^7)/27783 + (2*b^2*c^6*d^3*x^9)/729 + (64*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(945*c^3) - (32*b*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(945*c) + (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(315*c^3) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(525*c^3) + (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(441*c^3) - (2*b*d^3*(1 + c^2*x^2)^(9/2)*(a + b*ArcSinh[c*x]))/(81*c^3) + (16*d^3*x^3*(a + b*ArcSinh[c*x])^2)/315 + (8*d^3*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/105 + (2*d^3*x^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/21 + (d^3*x^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/9} -{x^1*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 11, (175*b^2*d^3*x^2)/3072 + (35*b^2*c^2*d^3*x^4)/3072 + (7*b^2*d^3*(1 + c^2*x^2)^3)/(1152*c^2) + (b^2*d^3*(1 + c^2*x^2)^4)/(256*c^2) - (35*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(512*c) - (35*b*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(768*c) - (7*b*d^3*x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(192*c) - (b*d^3*x*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(32*c) - (35*d^3*(a + b*ArcSinh[c*x])^2)/(1024*c^2) + (d^3*(1 + c^2*x^2)^4*(a + b*ArcSinh[c*x])^2)/(8*c^2)} -{x^0*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 14, (4322*b^2*d^3*x)/3675 + (1514*b^2*c^2*d^3*x^3)/11025 + (234*b^2*c^4*d^3*x^5)/6125 + (2*b^2*c^6*d^3*x^7)/343 - (32*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(35*c) - (16*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(105*c) - (12*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(175*c) - (2*b*d^3*(1 + c^2*x^2)^(7/2)*(a + b*ArcSinh[c*x]))/(49*c) + (16*d^3*x*(a + b*ArcSinh[c*x])^2)/35 + (8*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/35 + (6*d^3*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/35 + (d^3*x*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/7} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^1, x, 26, (71/144)*b^2*c^2*d^3*x^2 + (7/144)*b^2*c^4*d^3*x^4 + (1/108)*b^2*d^3*(1 + c^2*x^2)^3 - (19/24)*b*c*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (7/36)*b*c*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - (1/18)*b*c*d^3*x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]) - (19/48)*d^3*(a + b*ArcSinh[c*x])^2 + (1/2)*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2 + (1/4)*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2 + (1/6)*d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2 + (d^3*(a + b*ArcSinh[c*x])^3)/(3*b) + d^3*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] - b*d^3*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (1/2)*b^2*d^3*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^2, x, 24, (122*b^2*c^2*d^3*x)/25 + (14*b^2*c^4*d^3*x^3)/75 + (2*b^2*c^6*d^3*x^5)/125 - (22*b*c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/5 - (2*b*c*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/5 - (2*b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/25 + (16*c^2*d^3*x*(a + b*ArcSinh[c*x])^2)/5 + (8*c^2*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/5 + (6*c^2*d^3*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5 - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/x - 4*b*c*d^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - 2*b^2*c*d^3*PolyLog[2, -E^ArcSinh[c*x]] + 2*b^2*c*d^3*PolyLog[2, E^ArcSinh[c*x]]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^3, x, 28, (21/32)*b^2*c^4*d^3*x^2 + (1/32)*b^2*c^6*d^3*x^4 - (3/16)*b*c^3*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) + (7/8)*b*c^3*d^3*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]) - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x - (3/32)*c^2*d^3*(a + b*ArcSinh[c*x])^2 + (3/2)*c^2*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2 + (3/4)*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2 - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(2*x^2) + (c^2*d^3*(a + b*ArcSinh[c*x])^3)/b + 3*c^2*d^3*(a + b*ArcSinh[c*x])^2*Log[1 - E^(-2*ArcSinh[c*x])] + b^2*c^2*d^3*Log[x] - 3*b*c^2*d^3*(a + b*ArcSinh[c*x])*PolyLog[2, E^(-2*ArcSinh[c*x])] - (3/2)*b^2*c^2*d^3*PolyLog[3, E^(-2*ArcSinh[c*x])]} -{((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4, x, 31, -(b^2*c^2*d^3)/(3*x) + (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 - 5*b*c^3*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) + (b*c^3*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/9 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSinh[c*x])^2)/3 + (8*c^4*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (2*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(3*x^3) - (34*b*c^3*d^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (17*b^2*c^3*d^3*PolyLog[2, -E^ArcSinh[c*x]])/3 + (17*b^2*c^3*d^3*PolyLog[2, E^ArcSinh[c*x]])/3} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2), x, 16, (-22*b^2*x)/(9*c^4*d) + (2*b^2*x^3)/(27*c^2*d) + (22*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^5*d) - (2*b*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3*d) - (x*(a + b*ArcSinh[c*x])^2)/(c^4*d) + (x^3*(a + b*ArcSinh[c*x])^2)/(3*c^2*d) + (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c^5*d) - ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d) + ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d) + ((2*I)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^5*d) - ((2*I)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^5*d)} -{x^3*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2), x, 10, (b^2*x^2)/(4*c^2*d) - (b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^3*d) + (a + b*ArcSinh[c*x])^2/(4*c^4*d) + (x^2*(a + b*ArcSinh[c*x])^2)/(2*c^2*d) + (a + b*ArcSinh[c*x])^3/(3*b*c^4*d) - ((a + b*ArcSinh[c*x])^2*Log[1 + E^(2*ArcSinh[c*x])])/(c^4*d) - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^4*d) + (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*c^4*d)} -{x^2*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2), x, 11, (2*b^2*x)/(c^2*d) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^3*d) + (x*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c^3*d) + ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d) - ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d) - ((2*I)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^3*d) + ((2*I)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^3*d)} -{x^1*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2), x, 6, -(a + b*ArcSinh[c*x])^3/(3*b*c^2*d) + ((a + b*ArcSinh[c*x])^2*Log[1 + E^(2*ArcSinh[c*x])])/(c^2*d) + (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^2*d) - (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*c^2*d)} -{x^0*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2), x, 8, (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c*d) - ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d) + ((2*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d) + ((2*I)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c*d) - ((2*I)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c*d)} -{(a + b*ArcSinh[c*x])^2/(x^1*(d + c^2*d*x^2)), x, 9, (-2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d + (b*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d + (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*d) - (b^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d)} -{(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)), x, 15, -((a + b*ArcSinh[c*x])^2/(d*x)) - (2*c*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/d - (4*b*c*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/d - (2*b^2*c*PolyLog[2, -E^ArcSinh[c*x]])/d + ((2*I)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d - ((2*I)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d + (2*b^2*c*PolyLog[2, E^ArcSinh[c*x]])/d - ((2*I)*b^2*c*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d + ((2*I)*b^2*c*PolyLog[3, I*E^ArcSinh[c*x]])/d} -{(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)), x, 12, -((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(d*x)) - (a + b*ArcSinh[c*x])^2/(2*d*x^2) + (2*c^2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d + (b^2*c^2*Log[x])/d + (b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d - (b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d - (b^2*c^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*d) + (b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d)} -{(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)), x, 24, -(b^2*c^2)/(3*d*x) - (b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x^2) - (a + b*ArcSinh[c*x])^2/(3*d*x^3) + (c^2*(a + b*ArcSinh[c*x])^2)/(d*x) + (2*c^3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/d + (14*b*c^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(3*d) + (7*b^2*c^3*PolyLog[2, -E^ArcSinh[c*x]])/(3*d) - ((2*I)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d + ((2*I)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d - (7*b^2*c^3*PolyLog[2, E^ArcSinh[c*x]])/(3*d) + ((2*I)*b^2*c^3*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d - ((2*I)*b^2*c^3*PolyLog[3, I*E^ArcSinh[c*x]])/d} - - -{x^4*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2, x, 15, (2*b^2*x)/(c^4*d^2) + (b*(a + b*ArcSinh[c*x]))/(c^5*d^2*Sqrt[1 + c^2*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^5*d^2) + (3*x*(a + b*ArcSinh[c*x])^2)/(2*c^4*d^2) - (x^3*(a + b*ArcSinh[c*x])^2)/(2*c^2*d^2*(1 + c^2*x^2)) - (3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c^5*d^2) - (b^2*ArcTan[c*x])/(c^5*d^2) + ((3*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^2) - ((3*I)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d^2) - ((3*I)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^5*d^2) + ((3*I)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^5*d^2)} -{x^3*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2, x, 10, -((b*x*(a + b*ArcSinh[c*x]))/(c^3*d^2*Sqrt[1 + c^2*x^2])) + (a + b*ArcSinh[c*x])^2/(2*c^4*d^2) - (x^2*(a + b*ArcSinh[c*x])^2)/(2*c^2*d^2*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])^3/(3*b*c^4*d^2) + ((a + b*ArcSinh[c*x])^2*Log[1 + E^(2*ArcSinh[c*x])])/(c^4*d^2) + (b^2*Log[1 + c^2*x^2])/(2*c^4*d^2) + (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^4*d^2) - (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*c^4*d^2)} -{x^2*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2, x, 11, -((b*(a + b*ArcSinh[c*x]))/(c^3*d^2*Sqrt[1 + c^2*x^2])) - (x*(a + b*ArcSinh[c*x])^2)/(2*c^2*d^2*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c^3*d^2) + (b^2*ArcTan[c*x])/(c^3*d^2) - (I*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d^2) + (I*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^2) + (I*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^3*d^2) - (I*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^3*d^2)} -{x^1*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2, x, 3, (b*x*(a + b*ArcSinh[c*x]))/(c*d^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(2*c^2*d^2*(1 + c^2*x^2)) - (b^2*Log[1 + c^2*x^2])/(2*c^2*d^2)} -{x^0*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2, x, 11, (b*(a + b*ArcSinh[c*x]))/(c*d^2*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x])^2)/(2*d^2*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(c*d^2) - (b^2*ArcTan[c*x])/(c*d^2) - (I*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d^2) + (I*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d^2) + (I*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c*d^2) - (I*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c*d^2)} -{(a + b*ArcSinh[c*x])^2/(x^1*(d + c^2*d*x^2)^2), x, 12, -((b*c*x*(a + b*ArcSinh[c*x]))/(d^2*Sqrt[1 + c^2*x^2])) + (a + b*ArcSinh[c*x])^2/(2*d^2*(1 + c^2*x^2)) - (2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d^2 + (b^2*Log[1 + c^2*x^2])/(2*d^2) - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^2 + (b*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2 + (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*d^2) - (b^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^2)} -{(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^2), x, 20, -((b*c*(a + b*ArcSinh[c*x]))/(d^2*Sqrt[1 + c^2*x^2])) - (a + b*ArcSinh[c*x])^2/(d^2*x*(1 + c^2*x^2)) - (3*c^2*x*(a + b*ArcSinh[c*x])^2)/(2*d^2*(1 + c^2*x^2)) - (3*c*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/d^2 + (b^2*c*ArcTan[c*x])/d^2 - (4*b*c*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/d^2 - (2*b^2*c*PolyLog[2, -E^ArcSinh[c*x]])/d^2 + ((3*I)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^2 - ((3*I)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d^2 + (2*b^2*c*PolyLog[2, E^ArcSinh[c*x]])/d^2 - ((3*I)*b^2*c*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d^2 + ((3*I)*b^2*c*PolyLog[3, I*E^ArcSinh[c*x]])/d^2} -{(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^2), x, 17, -((b*c*(a + b*ArcSinh[c*x]))/(d^2*x*Sqrt[1 + c^2*x^2])) - (c^2*(a + b*ArcSinh[c*x])^2)/(d^2*(1 + c^2*x^2)) - (a + b*ArcSinh[c*x])^2/(2*d^2*x^2*(1 + c^2*x^2)) + (4*c^2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d^2 + (b^2*c^2*Log[x])/d^2 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^2 - (2*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2 - (b^2*c^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/d^2 + (b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d^2} -{(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^2), x, 32, -(b^2*c^2)/(3*d^2*x) + (2*b*c^3*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[1 + c^2*x^2]) - (b*c*(a + b*ArcSinh[c*x]))/(3*d^2*x^2*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(3*d^2*x^3*(1 + c^2*x^2)) + (5*c^2*(a + b*ArcSinh[c*x])^2)/(3*d^2*x*(1 + c^2*x^2)) + (5*c^4*x*(a + b*ArcSinh[c*x])^2)/(2*d^2*(1 + c^2*x^2)) + (5*c^3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/d^2 - (b^2*c^3*ArcTan[c*x])/d^2 + (26*b*c^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(3*d^2) + (13*b^2*c^3*PolyLog[2, -E^ArcSinh[c*x]])/(3*d^2) - ((5*I)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^2 + ((5*I)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d^2 - (13*b^2*c^3*PolyLog[2, E^ArcSinh[c*x]])/(3*d^2) + ((5*I)*b^2*c^3*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d^2 - ((5*I)*b^2*c^3*PolyLog[3, I*E^ArcSinh[c*x]])/d^2} - - -{x^4*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3, x, 16, -(b^2*x)/(12*c^4*d^3*(1 + c^2*x^2)) + (b*(a + b*ArcSinh[c*x]))/(6*c^5*d^3*(1 + c^2*x^2)^(3/2)) - (5*b*(a + b*ArcSinh[c*x]))/(4*c^5*d^3*Sqrt[1 + c^2*x^2]) - (x^3*(a + b*ArcSinh[c*x])^2)/(4*c^2*d^3*(1 + c^2*x^2)^2) - (3*x*(a + b*ArcSinh[c*x])^2)/(8*c^4*d^3*(1 + c^2*x^2)) + (3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*c^5*d^3) + (7*b^2*ArcTan[c*x])/(6*c^5*d^3) - (((3*I)/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^5*d^3) + (((3*I)/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d^3) + (((3*I)/4)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^5*d^3) - (((3*I)/4)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^5*d^3)} -{x^3*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3, x, 8, -b^2/(12*c^4*d^3*(1 + c^2*x^2)) + (b*x^3*(a + b*ArcSinh[c*x]))/(6*c*d^3*(1 + c^2*x^2)^(3/2)) + (b*x*(a + b*ArcSinh[c*x]))/(2*c^3*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(4*c^4*d^3) + (x^4*(a + b*ArcSinh[c*x])^2)/(4*d^3*(1 + c^2*x^2)^2) - (b^2*Log[1 + c^2*x^2])/(3*c^4*d^3)} -{x^2*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3, x, 15, (b^2*x)/(12*c^2*d^3*(1 + c^2*x^2)) - (b*(a + b*ArcSinh[c*x]))/(6*c^3*d^3*(1 + c^2*x^2)^(3/2)) + (b*(a + b*ArcSinh[c*x]))/(4*c^3*d^3*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x])^2)/(4*c^2*d^3*(1 + c^2*x^2)^2) + (x*(a + b*ArcSinh[c*x])^2)/(8*c^2*d^3*(1 + c^2*x^2)) + ((a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*c^3*d^3) - (b^2*ArcTan[c*x])/(6*c^3*d^3) - ((I/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^3*d^3) + ((I/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^3) + ((I/4)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c^3*d^3) - ((I/4)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c^3*d^3)} -{x^1*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3, x, 5, b^2/(12*c^2*d^3*(1 + c^2*x^2)) + (b*x*(a + b*ArcSinh[c*x]))/(6*c*d^3*(1 + c^2*x^2)^(3/2)) + (b*x*(a + b*ArcSinh[c*x]))/(3*c*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(4*c^2*d^3*(1 + c^2*x^2)^2) - (b^2*Log[1 + c^2*x^2])/(6*c^2*d^3)} -{x^0*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3, x, 15, -(b^2*x)/(12*d^3*(1 + c^2*x^2)) + (b*(a + b*ArcSinh[c*x]))/(6*c*d^3*(1 + c^2*x^2)^(3/2)) + (3*b*(a + b*ArcSinh[c*x]))/(4*c*d^3*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x])^2)/(4*d^3*(1 + c^2*x^2)^2) + (3*x*(a + b*ArcSinh[c*x])^2)/(8*d^3*(1 + c^2*x^2)) + (3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*c*d^3) - (5*b^2*ArcTan[c*x])/(6*c*d^3) - (((3*I)/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*d^3) + (((3*I)/4)*b*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/(c*d^3) + (((3*I)/4)*b^2*PolyLog[3, (-I)*E^ArcSinh[c*x]])/(c*d^3) - (((3*I)/4)*b^2*PolyLog[3, I*E^ArcSinh[c*x]])/(c*d^3)} -{(a + b*ArcSinh[c*x])^2/(x^1*(d + c^2*d*x^2)^3), x, 17, -b^2/(12*d^3*(1 + c^2*x^2)) - (b*c*x*(a + b*ArcSinh[c*x]))/(6*d^3*(1 + c^2*x^2)^(3/2)) - (4*b*c*x*(a + b*ArcSinh[c*x]))/(3*d^3*Sqrt[1 + c^2*x^2]) + (a + b*ArcSinh[c*x])^2/(4*d^3*(1 + c^2*x^2)^2) + (a + b*ArcSinh[c*x])^2/(2*d^3*(1 + c^2*x^2)) - (2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d^3 + (2*b^2*Log[1 + c^2*x^2])/(3*d^3) - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^3 + (b*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^3 + (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*d^3) - (b^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^3)} -{(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^3), x, 27, (b^2*c^2*x)/(12*d^3*(1 + c^2*x^2)) - (b*c*(a + b*ArcSinh[c*x]))/(6*d^3*(1 + c^2*x^2)^(3/2)) - (7*b*c*(a + b*ArcSinh[c*x]))/(4*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(d^3*x*(1 + c^2*x^2)^2) - (5*c^2*x*(a + b*ArcSinh[c*x])^2)/(4*d^3*(1 + c^2*x^2)^2) - (15*c^2*x*(a + b*ArcSinh[c*x])^2)/(8*d^3*(1 + c^2*x^2)) - (15*c*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*d^3) + (11*b^2*c*ArcTan[c*x])/(6*d^3) - (4*b*c*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/d^3 - (2*b^2*c*PolyLog[2, -E^ArcSinh[c*x]])/d^3 + (((15*I)/4)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^3 - (((15*I)/4)*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d^3 + (2*b^2*c*PolyLog[2, E^ArcSinh[c*x]])/d^3 - (((15*I)/4)*b^2*c*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d^3 + (((15*I)/4)*b^2*c*PolyLog[3, I*E^ArcSinh[c*x]])/d^3} -{(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^3), x, 23, (b^2*c^2)/(12*d^3*(1 + c^2*x^2)) - (b*c*(a + b*ArcSinh[c*x]))/(d^3*x*(1 + c^2*x^2)^(3/2)) - (5*b*c^3*x*(a + b*ArcSinh[c*x]))/(6*d^3*(1 + c^2*x^2)^(3/2)) + (4*b*c^3*x*(a + b*ArcSinh[c*x]))/(3*d^3*Sqrt[1 + c^2*x^2]) - (3*c^2*(a + b*ArcSinh[c*x])^2)/(4*d^3*(1 + c^2*x^2)^2) - (a + b*ArcSinh[c*x])^2/(2*d^3*x^2*(1 + c^2*x^2)^2) - (3*c^2*(a + b*ArcSinh[c*x])^2)/(2*d^3*(1 + c^2*x^2)) + (6*c^2*(a + b*ArcSinh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d^3 + (b^2*c^2*Log[x])/d^3 - (7*b^2*c^2*Log[1 + c^2*x^2])/(6*d^3) + (3*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/d^3 - (3*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d^3 - (3*b^2*c^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*d^3) + (3*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^3)} -{(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^3), x, 43, -(b^2*c^2)/(2*d^3*x) + (b^2*c^2)/(6*d^3*x*(1 + c^2*x^2)) + (b^2*c^4*x)/(12*d^3*(1 + c^2*x^2)) - (b*c^3*(a + b*ArcSinh[c*x]))/(6*d^3*(1 + c^2*x^2)^(3/2)) - (b*c*(a + b*ArcSinh[c*x]))/(3*d^3*x^2*(1 + c^2*x^2)^(3/2)) + (29*b*c^3*(a + b*ArcSinh[c*x]))/(12*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(3*d^3*x^3*(1 + c^2*x^2)^2) + (7*c^2*(a + b*ArcSinh[c*x])^2)/(3*d^3*x*(1 + c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSinh[c*x])^2)/(12*d^3*(1 + c^2*x^2)^2) + (35*c^4*x*(a + b*ArcSinh[c*x])^2)/(8*d^3*(1 + c^2*x^2)) + (35*c^3*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]])/(4*d^3) - (17*b^2*c^3*ArcTan[c*x])/(6*d^3) + (38*b*c^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/(3*d^3) + (19*b^2*c^3*PolyLog[2, -E^ArcSinh[c*x]])/(3*d^3) - (((35*I)/4)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]])/d^3 + (((35*I)/4)*b*c^3*(a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]])/d^3 - (19*b^2*c^3*PolyLog[2, E^ArcSinh[c*x]])/(3*d^3) + (((35*I)/4)*b^2*c^3*PolyLog[3, (-I)*E^ArcSinh[c*x]])/d^3 - (((35*I)/4)*b^2*c^3*PolyLog[3, I*E^ArcSinh[c*x]])/d^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (Pi+Pi c^2 x^2)^(p/2) (a+b ArcSinh[c x])^2*) - - -{(Pi + Pi*c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 16, (245*b^2*Pi^(5/2)*x*Sqrt[1 + c^2*x^2])/1152 + (65*b^2*Pi^(5/2)*x*(1 + c^2*x^2)^(3/2))/1728 + (1/108)*b^2*Pi^(5/2)*x*(1 + c^2*x^2)^(5/2) - (115*b^2*Pi^(5/2)*ArcSinh[c*x])/(1152*c) - (5/16)*b*c*Pi^(5/2)*x^2*(a + b*ArcSinh[c*x]) - (5*b*Pi^(5/2)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(48*c) - (b*Pi^(5/2)*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(18*c) + (5/16)*Pi^2*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2 + (5/24)*Pi*x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2 + (1/6)*x*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2 + (5*Pi^(5/2)*(a + b*ArcSinh[c*x])^3)/(48*b*c)} -{(Pi + Pi*c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 10, (15/64)*b^2*Pi^(3/2)*x*Sqrt[1 + c^2*x^2] + (1/32)*b^2*Pi^(3/2)*x*(1 + c^2*x^2)^(3/2) - (9*b^2*Pi^(3/2)*ArcSinh[c*x])/(64*c) - (3/8)*b*c*Pi^(3/2)*x^2*(a + b*ArcSinh[c*x]) - (b*Pi^(3/2)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(8*c) + (3/8)*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2 + (1/4)*x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2 + (Pi^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c)} -{(Pi + Pi*c^2*x^2)^(1/2)*(a + b*ArcSinh[c*x])^2, x, 5, (1/4)*b^2*Sqrt[Pi]*x*Sqrt[1 + c^2*x^2] - (b^2*Sqrt[Pi]*ArcSinh[c*x])/(4*c) - (1/2)*b*c*Sqrt[Pi]*x^2*(a + b*ArcSinh[c*x]) + (1/2)*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2 + (Sqrt[Pi]*(a + b*ArcSinh[c*x])^3)/(6*b*c)} -{(a + b*ArcSinh[c*x])^2/(Pi + Pi*c^2*x^2)^(1/2), x, 1, (a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])} -{(a + b*ArcSinh[c*x])^2/(Pi + Pi*c^2*x^2)^(3/2), x, 6, (a + b*ArcSinh[c*x])^2/(c*Pi^(3/2)) + (x*(a + b*ArcSinh[c*x])^2)/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (2*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*Pi^(3/2)) - (b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*Pi^(3/2))} -{(a + b*ArcSinh[c*x])^2/(Pi + Pi*c^2*x^2)^(5/2), x, 9, -((b^2*x)/(3*Pi^(5/2)*Sqrt[1 + c^2*x^2])) + (b*(a + b*ArcSinh[c*x]))/(3*c*Pi^(5/2)*(1 + c^2*x^2)) + (2*(a + b*ArcSinh[c*x])^2)/(3*c*Pi^(5/2)) + (x*(a + b*ArcSinh[c*x])^2)/(3*Pi*(Pi + c^2*Pi*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x])^2)/(3*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) - (4*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*Pi^(5/2)) - (2*b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*Pi^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x, 14, (-52*b^2*Sqrt[d + c^2*d*x^2])/(225*c^4) + (4*a*b*x*Sqrt[d + c^2*d*x^2])/(15*c^3*Sqrt[1 + c^2*x^2]) - (26*b^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(675*c^4) + (2*b^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(125*c^4) + (4*b^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(15*c^3*Sqrt[1 + c^2*x^2]) - (2*b*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(45*c*Sqrt[1 + c^2*x^2]) - (2*b*c*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(25*Sqrt[1 + c^2*x^2]) - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^4) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^2) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/5} -{x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x, 10, (b^2*x*Sqrt[d + c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d + c^2*d*x^2])/32 - (b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(64*c^3*Sqrt[1 + c^2*x^2]) - (b*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c*Sqrt[1 + c^2*x^2]) - (b*c*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8*c^2) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(24*b*c^3*Sqrt[1 + c^2*x^2])} -{x^1*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x, 5, (4*b^2*Sqrt[d + c^2*d*x^2])/(9*c^2) + (2*b^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(27*c^2) - (2*b*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c*Sqrt[1 + c^2*x^2]) - (2*b*c*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*c^2*d)} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x, 5, (b^2*x*Sqrt[d + c^2*d*x^2])/4 - (b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(4*c*Sqrt[1 + c^2*x^2]) - (b*c*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[1 + c^2*x^2])} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^1, x, 12, 2*b^2*Sqrt[d + c^2*d*x^2] - (2*a*b*c*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] - (2*b^2*c*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^2, x, 7, -((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x) + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] + (c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*Sqrt[1 + c^2*x^2]) + (2*b*c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^3, x, 13, -((b*c*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(x*Sqrt[1 + c^2*x^2])) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*x^2) - (c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b^2*c^2*Sqrt[d + c^2*d*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/Sqrt[1 + c^2*x^2] - (b*c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b^2*c^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b^2*c^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4, x, 9, -((b^2*c^2*Sqrt[d + c^2*d*x^2])/(3*x)) + (b^2*c^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(3*Sqrt[1 + c^2*x^2]) - (b*c*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (c^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*d*x^3) + (2*b*c^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (b^2*c^3*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])} - - -{x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 20, (-304*b^2*d*Sqrt[d + c^2*d*x^2])/(3675*c^4) + (4*a*b*d*x*Sqrt[d + c^2*d*x^2])/(35*c^3*Sqrt[1 + c^2*x^2]) - (152*b^2*d*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(11025*c^4) - (38*b^2*d*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(6125*c^4) + (2*b^2*d*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(343*c^4) + (4*b^2*d*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(35*c^3*Sqrt[1 + c^2*x^2]) - (2*b*d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(105*c*Sqrt[1 + c^2*x^2]) - (16*b*c*d*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(175*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(49*Sqrt[1 + c^2*x^2]) - (2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(35*c^4) + (d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(35*c^2) + (3*d*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/35 + (x^4*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/7} -{x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 17, (-7*b^2*d*x*Sqrt[d + c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d + c^2*d*x^2])/1728 + (b^2*c^2*d*x^5*Sqrt[d + c^2*d*x^2])/108 + (7*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(1152*c^3*Sqrt[1 + c^2*x^2]) - (b*d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c*Sqrt[1 + c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(48*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(18*Sqrt[1 + c^2*x^2]) + (d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*c^2) + (d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/8 + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/6 - (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(48*b*c^3*Sqrt[1 + c^2*x^2])} -{x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 6, (16*b^2*d*Sqrt[d + c^2*d*x^2])/(75*c^2) + (8*b^2*d*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(225*c^2) + (2*b^2*d*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(125*c^2) - (2*b*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c*Sqrt[1 + c^2*x^2]) - (4*b*c*d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(25*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(5*c^2*d)} -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 10, (15/64)*b^2*d*x*Sqrt[d + c^2*d*x^2] + (1/32)*b^2*d*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2] - (9*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(64*c*Sqrt[1 + c^2*x^2]) - (3*b*c*d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) - (b*d*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (3/8)*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + (1/4)*x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2 + (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(8*b*c*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x, x, 17, (22*b^2*d*Sqrt[d + c^2*d*x^2])/9 - (2*a*b*c*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (2*b^2*d*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/27 - (2*b^2*c*d*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (2*b*c*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/3 - (2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^2, x, 14, (1/4)*b^2*c^2*d*x*Sqrt[d + c^2*d*x^2] - (5*b^2*c*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(4*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + b*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3/2)*c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + (c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x + (c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*Sqrt[1 + c^2*x^2]) + (2*b*c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^3, x, 18, 2*b^2*c^2*d*Sqrt[d + c^2*d*x^2] - (3*a*b*c^3*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] - (3*b^2*c^3*d*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (b*c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(x*Sqrt[1 + c^2*x^2]) + (b*c^3*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (3*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*x^2) - (3*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b^2*c^2*d*Sqrt[d + c^2*d*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/Sqrt[1 + c^2*x^2] - (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (3*b^2*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (3*b^2*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^4, x, 16, -((b^2*c^2*d*Sqrt[d + c^2*d*x^2])/(3*x)) + (b^2*c^3*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(3*Sqrt[1 + c^2*x^2]) - (b*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) - (c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x + (4*c^3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*x^3) + (c^3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*Sqrt[1 + c^2*x^2]) + (8*b*c^3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (4*b^2*c^3*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])} - - -{x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 27, (-160*b^2*d^2*Sqrt[d + c^2*d*x^2])/(3969*c^4) + (4*a*b*d^2*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (80*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(11907*c^4) - (4*b^2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(1323*c^4) - (50*b^2*d^2*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(27783*c^4) + (2*b^2*d^2*(1 + c^2*x^2)^4*Sqrt[d + c^2*d*x^2])/(729*c^4) + (4*b^2*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(63*c^3*Sqrt[1 + c^2*x^2]) - (2*b*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(189*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(21*Sqrt[1 + c^2*x^2]) - (38*b*c^3*d^2*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(441*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^9*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(81*Sqrt[1 + c^2*x^2]) - (2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(63*c^4) + (d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(63*c^2) + (d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/21 + (5*d*x^4*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/63 + (x^4*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/9} -{x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 25, (-359*b^2*d^2*x*Sqrt[d + c^2*d*x^2])/(36864*c^2) + (1079*b^2*d^2*x^3*Sqrt[d + c^2*d*x^2])/55296 + (209*b^2*c^2*d^2*x^5*Sqrt[d + c^2*d*x^2])/13824 + (b^2*c^4*d^2*x^7*Sqrt[d + c^2*d*x^2])/256 + (359*b^2*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(36864*c^3*Sqrt[1 + c^2*x^2]) - (5*b*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(384*Sqrt[1 + c^2*x^2]) - (17*b*c^3*d^2*x^6*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(144*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(32*Sqrt[1 + c^2*x^2]) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(128*c^2) + (5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/64 + (5*d*x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/48 + (x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/8 - (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(384*b*c^3*Sqrt[1 + c^2*x^2])} -{x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 6, (32*b^2*d^2*Sqrt[d + c^2*d*x^2])/(245*c^2) + (16*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(735*c^2) + (12*b^2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(1225*c^2) + (2*b^2*d^2*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2])/(343*c^2) - (2*b*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*Sqrt[1 + c^2*x^2]) - (6*b*c^3*d^2*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(35*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^7*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(49*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(7/2)*(a + b*ArcSinh[c*x])^2)/(7*c^2*d)} -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 16, (245*b^2*d^2*x*Sqrt[d + c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/1728 + (b^2*d^2*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/108 - (115*b^2*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(1152*c*Sqrt[1 + c^2*x^2]) - (5*b*c*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*Sqrt[1 + c^2*x^2]) - (5*b*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(48*c) - (b*d^2*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(18*c) + (5*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/16 + (5*d*x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/24 + (x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(48*b*c*Sqrt[1 + c^2*x^2])} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x, x, 23, (598*b^2*d^2*Sqrt[d + c^2*d*x^2])/225 - (2*a*b*c*d^2*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (74*b^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/675 + (2*b^2*d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/125 - (2*b^2*c*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (16*b*c*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(15*Sqrt[1 + c^2*x^2]) - (22*b*c^3*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(45*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(25*Sqrt[1 + c^2*x^2]) + d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + (d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/3 + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/5 - (2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (2*b^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (2*b^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^2, x, 23, (31/64)*b^2*c^2*d^2*x*Sqrt[d + c^2*d*x^2] + (1/32)*b^2*c^2*d^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2] - (89*b^2*c*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(64*Sqrt[1 + c^2*x^2]) - (15*b*c^3*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + b*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) - (1/8)*b*c*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (15/8)*c^2*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + (c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[1 + c^2*x^2] + (5/4)*c^2*d*x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2 - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x + (5*c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(8*b*Sqrt[1 + c^2*x^2]) + (2*b*c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2] - (b^2*c*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^3, x, 25, (40*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2])/9 - (5*a*b*c^3*d^2*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (2*b^2*c^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/27 - (5*b^2*c^3*d^2*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (b*c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(x*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + (5*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/6 - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(2*x^2) - (5*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/Sqrt[1 + c^2*x^2] - (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (5*b*c^2*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (5*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (5*b^2*c^2*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4, x, 27, (7/12)*b^2*c^4*d^2*x*Sqrt[d + c^2*d*x^2] - (b^2*c^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(3*x) - (23*b^2*c^3*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(12*Sqrt[1 + c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (7/3)*b*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) - (b*c*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (5/2)*c^4*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 + (7*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2]) - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*Sqrt[1 + c^2*x^2]) + (14*b*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (7*b^2*c^3*d^2*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x, 10, (-15*x*Sqrt[1 + a^2*x^2])/(64*a^4) + (x^3*Sqrt[1 + a^2*x^2])/(32*a^2) + (15*ArcSinh[a*x])/(64*a^5) + (3*x^2*ArcSinh[a*x])/(8*a^3) - (x^4*ArcSinh[a*x])/(8*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(4*a^2) + ArcSinh[a*x]^3/(8*a^5)} -{(x^3*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x, 8, (-14*Sqrt[1 + a^2*x^2])/(9*a^4) + (2*(1 + a^2*x^2)^(3/2))/(27*a^4) + (4*x*ArcSinh[a*x])/(3*a^3) - (2*x^3*ArcSinh[a*x])/(9*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(3*a^2)} -{(x^2*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x, 5, (x*Sqrt[1 + a^2*x^2])/(4*a^2) - ArcSinh[a*x]/(4*a^3) - (x^2*ArcSinh[a*x])/(2*a) + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*a^2) - ArcSinh[a*x]^3/(6*a^3)} -{(x*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x, 3, (2*Sqrt[1 + a^2*x^2])/a^2 - (2*x*ArcSinh[a*x])/a + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a^2} -{ArcSinh[a*x]^2/Sqrt[1 + a^2*x^2], x, 1, ArcSinh[a*x]^3/(3*a)} -{ArcSinh[a*x]^2/(x*Sqrt[1 + a^2*x^2]), x, 8, -2*ArcSinh[a*x]^2*ArcTanh[E^ArcSinh[a*x]] - 2*ArcSinh[a*x]*PolyLog[2, -E^ArcSinh[a*x]] + 2*ArcSinh[a*x]*PolyLog[2, E^ArcSinh[a*x]] + 2*PolyLog[3, -E^ArcSinh[a*x]] - 2*PolyLog[3, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^2/(x^2*Sqrt[1 + a^2*x^2]), x, 6, -(a*ArcSinh[a*x]^2) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/x + 2*a*ArcSinh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + a*PolyLog[2, E^(2*ArcSinh[a*x])]} -{ArcSinh[a*x]^2/(x^3*Sqrt[1 + a^2*x^2]), x, 13, -((a*ArcSinh[a*x])/x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x^2) + a^2*ArcSinh[a*x]^2*ArcTanh[E^ArcSinh[a*x]] - a^2*ArcTanh[Sqrt[1 + a^2*x^2]] + a^2*ArcSinh[a*x]*PolyLog[2, -E^ArcSinh[a*x]] - a^2*ArcSinh[a*x]*PolyLog[2, E^ArcSinh[a*x]] - a^2*PolyLog[3, -E^ArcSinh[a*x]] + a^2*PolyLog[3, E^ArcSinh[a*x]]} - - -{(x^5*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x, 14, -((16*a*b*x*Sqrt[1 + c^2*x^2])/(15*c^5*Sqrt[d + c^2*d*x^2])) + (298*b^2*(1 + c^2*x^2))/(225*c^6*Sqrt[d + c^2*d*x^2]) - (76*b^2*(1 + c^2*x^2)^2)/(675*c^6*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2)^3)/(125*c^6*Sqrt[d + c^2*d*x^2]) - (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(15*c^5*Sqrt[d + c^2*d*x^2]) + (8*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(45*c^3*Sqrt[d + c^2*d*x^2]) - (2*b*x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c*Sqrt[d + c^2*d*x^2]) + (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^4*d) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(5*c^2*d)} -{(x^4*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x, 10, -((15*b^2*x*(1 + c^2*x^2))/(64*c^4*Sqrt[d + c^2*d*x^2])) + (b^2*x^3*(1 + c^2*x^2))/(32*c^2*Sqrt[d + c^2*d*x^2]) + (15*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(64*c^5*Sqrt[d + c^2*d*x^2]) + (3*b*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c*Sqrt[d + c^2*d*x^2]) - (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8*c^4*d) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*c^2*d) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(8*b*c^5*Sqrt[d + c^2*d*x^2])} -{(x^3*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x, 9, (4*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (14*b^2*(1 + c^2*x^2))/(9*c^4*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2)^2)/(27*c^4*Sqrt[d + c^2*d*x^2]) + (4*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (2*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2*d)} -{(x^2*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x, 5, (b^2*x*(1 + c^2*x^2))/(4*c^2*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c*Sqrt[d + c^2*d*x^2]) + (x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*c^2*d) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*c^3*Sqrt[d + c^2*d*x^2])} -{(x*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x, 4, (-2*a*b*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2))/(c^2*Sqrt[d + c^2*d*x^2]) - (2*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*Sqrt[d + c^2*d*x^2]) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^2*d)} -{(a + b*ArcSinh[c*x])^2/Sqrt[d + c^2*d*x^2], x, 1, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x*Sqrt[d + c^2*d*x^2]), x, 8, (-2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] + (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] + (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] - (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2]} -{(a + b*ArcSinh[c*x])^2/(x^2*Sqrt[d + c^2*d*x^2]), x, 6, (c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2] - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(d*x) + (2*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/Sqrt[d + c^2*d*x^2] - (b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/Sqrt[d + c^2*d*x^2]} -{(a + b*ArcSinh[c*x])^2/(x^3*Sqrt[d + c^2*d*x^2]), x, 13, -((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(x*Sqrt[d + c^2*d*x^2])) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*d*x^2) + (c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] - (b^2*c^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/Sqrt[d + c^2*d*x^2] + (b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] - (b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] - (b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2] + (b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/Sqrt[d + c^2*d*x^2]} -{(a + b*ArcSinh[c*x])^2/(x^4*Sqrt[d + c^2*d*x^2]), x, 9, -((b^2*c^2*(1 + c^2*x^2))/(3*x*Sqrt[d + c^2*d*x^2])) - (b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2*Sqrt[d + c^2*d*x^2]) - (2*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[d + c^2*d*x^2]) - (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d*x^3) + (2*c^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d*x) - (4*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[d + c^2*d*x^2]) + (2*b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[d + c^2*d*x^2])} - - -{(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x, 22, (16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d*Sqrt[d + c^2*d*x^2]) - (32*b^2*(1 + c^2*x^2))/(9*c^6*d*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2)^2)/(27*c^6*d*Sqrt[d + c^2*d*x^2]) + (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2]) - (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2]) - (2*I*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2]) + (2*I*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2])} -{(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x, 14, (b^2*x*(1 + c^2*x^2))/(4*c^4*d*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c^5*d*Sqrt[d + c^2*d*x^2]) - (b*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(c^5*d*Sqrt[d + c^2*d*x^2]) + (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*c^4*d^2) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c^5*d*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^5*d*Sqrt[d + c^2*d*x^2])} -{(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x, 13, (-4*a*b*x*Sqrt[1 + c^2*x^2])/(c^3*d*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2))/(c^4*d*Sqrt[d + c^2*d*x^2]) - (4*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c^3*d*Sqrt[d + c^2*d*x^2]) + (2*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^3*d*Sqrt[d + c^2*d*x^2]) - (x^2*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2]) + (2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^4*d^2) - (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^4*d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^4*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^4*d*Sqrt[d + c^2*d*x^2])} -{(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x, 7, -((x*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(c^3*d*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c^3*d*Sqrt[d + c^2*d*x^2]) + (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c^3*d*Sqrt[d + c^2*d*x^2]) + (b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^3*d*Sqrt[d + c^2*d*x^2])} -{(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x, 7, -((a + b*ArcSinh[c*x])^2/(c^2*d*Sqrt[d + c^2*d*x^2])) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(3/2), x, 6, (x*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(c*d*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*d*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(3/2)), x, 15, (a + b*ArcSinh[c*x])^2/(d*Sqrt[d + c^2*d*x^2]) - (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(3/2)), x, 14, -((a + b*ArcSinh[c*x])^2/(d*x*Sqrt[d + c^2*d*x^2])) - (2*c^2*x*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) - (2*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(d*Sqrt[d + c^2*d*x^2]) - (4*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/(d*Sqrt[d + c^2*d*x^2]) + (4*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(d*Sqrt[d + c^2*d*x^2]) + (b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(d*Sqrt[d + c^2*d*x^2]) + (b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^(3/2)), x, 26, -((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(d*x*Sqrt[d + c^2*d*x^2])) - (3*c^2*(a + b*ArcSinh[c*x])^2)/(2*d*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(2*d*x^2*Sqrt[d + c^2*d*x^2]) + (4*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (3*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (b^2*c^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(d*Sqrt[d + c^2*d*x^2]) + (3*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (3*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) - (3*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2]) + (3*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/(d*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)), x, 24, -(b^2*c^2*(1 + c^2*x^2))/(3*d*x*Sqrt[d + c^2*d*x^2]) - (b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*d*x^2*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(3*d*x^3*Sqrt[d + c^2*d*x^2]) + (4*c^2*(a + b*ArcSinh[c*x])^2)/(3*d*x*Sqrt[d + c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSinh[c*x])^2)/(3*d*Sqrt[d + c^2*d*x^2]) + (8*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d*Sqrt[d + c^2*d*x^2]) + (20*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2]) - (16*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2]) - (b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(d*Sqrt[d + c^2*d*x^2]) - (5*b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(3*d*Sqrt[d + c^2*d*x^2])} - - -{(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x, 26, b^2/(3*c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (2*b^2*(1 + c^2*x^2))/(c^6*d^2*Sqrt[d + c^2*d*x^2]) - (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (b*x^3*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (11*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^6*d^3) - (22*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^6*d^2*Sqrt[d + c^2*d*x^2]) + (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d^2*Sqrt[d + c^2*d*x^2]) - (((11*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d^2*Sqrt[d + c^2*d*x^2])} -{(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x, 16, -(b^2*x)/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) - (b*x^2*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^3*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (x*(a + b*ArcSinh[c*x])^2)/(c^4*d^2*Sqrt[d + c^2*d*x^2]) - (4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (8*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2]) + (4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c^5*d^2*Sqrt[d + c^2*d*x^2])} -{(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x, 16, -b^2/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) - (b*x*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (x^2*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (2*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) + (10*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^4*d^2*Sqrt[d + c^2*d*x^2]) - (((5*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^4*d^2*Sqrt[d + c^2*d*x^2]) + (((5*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^4*d^2*Sqrt[d + c^2*d*x^2])} -{(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x, 9, (b^2*x)/(3*c^2*d^2*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^3*d^2*Sqrt[d + c^2*d*x^2]) + (b*x^2*(a + b*ArcSinh[c*x]))/(3*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (x^3*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^3*d^2*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c^3*d^2*Sqrt[d + c^2*d*x^2]) - (b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c^3*d^2*Sqrt[d + c^2*d*x^2])} -{(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x, 9, b^2/(3*c^2*d^2*Sqrt[d + c^2*d*x^2]) + (b*x*(a + b*ArcSinh[c*x]))/(3*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) + (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^2*d^2*Sqrt[d + c^2*d*x^2]) - ((I/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^2*d^2*Sqrt[d + c^2*d*x^2]) + ((I/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^2*d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(5/2), x, 9, -(b^2*x)/(3*d^2*Sqrt[d + c^2*d*x^2]) + (b*(a + b*ArcSinh[c*x]))/(3*c*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) + (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c*d^2*Sqrt[d + c^2*d*x^2]) - (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*d^2*Sqrt[d + c^2*d*x^2]) - (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(5/2)), x, 24, -b^2/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*c*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) + (a + b*ArcSinh[c*x])^2/(3*d*(d + c^2*d*x^2)^(3/2)) + (a + b*ArcSinh[c*x])^2/(d^2*Sqrt[d + c^2*d*x^2]) - (14*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*d^2*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (((7*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (((7*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)), x, 19, (b^2*c^2*x)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*c*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(d*x*(d + c^2*d*x^2)^(3/2)) - (4*c^2*x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (8*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (4*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/(d^2*Sqrt[d + c^2*d*x^2]) + (16*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2]) + (5*b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2]) + (b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^(5/2)), x, 38, (b^2*c^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*c*(a + b*ArcSinh[c*x]))/(d^2*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (2*b*c^3*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (5*c^2*(a + b*ArcSinh[c*x])^2)/(6*d*(d + c^2*d*x^2)^(3/2)) - (a + b*ArcSinh[c*x])^2/(2*d*x^2*(d + c^2*d*x^2)^(3/2)) - (5*c^2*(a + b*ArcSinh[c*x])^2)/(2*d^2*Sqrt[d + c^2*d*x^2]) + (26*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*d^2*Sqrt[d + c^2*d*x^2]) + (5*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2*ArcTanh[E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (b^2*c^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(d^2*Sqrt[d + c^2*d*x^2]) + (5*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (((13*I)/3)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (((13*I)/3)*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (5*b*c^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*PolyLog[2, E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) - (5*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2]) + (5*b^2*c^2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^ArcSinh[c*x]])/(d^2*Sqrt[d + c^2*d*x^2])} -{(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(5/2)), x, 32, -(b^2*c^2)/(3*d^2*x*Sqrt[d + c^2*d*x^2]) - (2*b^2*c^4*x)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*c*(a + b*ArcSinh[c*x]))/(3*d^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]) - (a + b*ArcSinh[c*x])^2/(3*d*x^3*(d + c^2*d*x^2)^(3/2)) + (2*c^2*(a + b*ArcSinh[c*x])^2)/(d*x*(d + c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) + (16*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) + (32*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2]) - (32*b*c^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2]) - (8*b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2]) - (8*b^2*c^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^2*d*x^2])} - - -{ArcSinh[a*x]^2/(c + a^2*c*x^2)^(7/2), x, 13, -x/(3*c^3*Sqrt[c + a^2*c*x^2]) - x/(30*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + ArcSinh[a*x]/(10*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + (4*ArcSinh[a*x])/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^2)/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^2)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x]^2)/(15*c^3*Sqrt[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[1 + E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^2 and m symbolic*) - - -{x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 12, If[$VersionNumber>=8, (10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*(6 + m)^3) + (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d + c^2*d*x^2])/(6 + m)^3 - (30*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (10*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((12 + 8*m + m^2)*Sqrt[1 + c^2*x^2]) - (10*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 + c^2*x^2]) - (4*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)^2*Sqrt[1 + c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/((4 + m)*(6 + m)) + (x^(1 + m)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(6 + m) + (30*b^2*c^2*d^2*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)^2*(3 + m)*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) + (10*b^2*c^2*d^2*(10 + 3*m)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)*(3 + m)*(4 + m)^3*(6 + m)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)*(3 + m)*(4 + m)^2*(6 + m)^3*Sqrt[1 + c^2*x^2]) + (15*d^3*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2)), (10*b^2*c^2*d^2*x^(3 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^3*(6 + m)) + (2*b^2*c^2*d^2*(52 + 15*m + m^2)*x^(3 + m)*Sqrt[d + c^2*d*x^2])/((4 + m)^2*(6 + m)^3) + (2*b^2*c^4*d^2*x^(5 + m)*Sqrt[d + c^2*d*x^2])/(6 + m)^3 - (30*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (10*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)*(8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((12 + 8*m + m^2)*Sqrt[1 + c^2*x^2]) - (10*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*(6 + m)*Sqrt[1 + c^2*x^2]) - (4*b*c^3*d^2*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)*(6 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*x^(6 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((6 + m)^2*Sqrt[1 + c^2*x^2]) + (15*d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/((6 + m)*(8 + 6*m + m^2)) + (5*d*x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/((4 + m)*(6 + m)) + (x^(1 + m)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(6 + m) + (10*b^2*c^2*d^2*(10 + 3*m)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((4 + m)^3*(6 + m)*(6 + 5*m + m^2)*Sqrt[1 + c^2*x^2]) + (30*b^2*c^2*d^2*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)^2*(6 + m)*(12 + 7*m + m^2)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((4 + m)^2*(6 + m)^3*(6 + 5*m + m^2)*Sqrt[1 + c^2*x^2]) + (15*d^3*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2))]} -{x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 7, If[$VersionNumber>=8, (2*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(4 + m) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)^2*(3 + m)*(4 + m)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d*(10 + 3*m)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)*(3 + m)*(4 + m)^3*Sqrt[1 + c^2*x^2]) + (3*d^2*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(8 + 6*m + m^2), (2*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(4 + m) + (2*b^2*c^2*d*(10 + 3*m)*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((4 + m)^3*(6 + 5*m + m^2)*Sqrt[1 + c^2*x^2]) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)^2*(12 + 7*m + m^2)*Sqrt[1 + c^2*x^2]) + (3*d^2*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(8 + 6*m + m^2)]} -{x^m*(d + c^2*d*x^2)^(1/2)*(a + b*ArcSinh[c*x])^2, x, 3, -((2*b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*Sqrt[1 + c^2*x^2])) + (x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2 + m) + (2*b^2*c^2*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, (-c^2)*x^2])/((2 + m)^2*(3 + m)*Sqrt[1 + c^2*x^2]) + (d*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(2 + m)} -{x^m*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(1/2), x, 0, Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x]} -{x^m*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(3/2), x, 0, Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x]} -{x^m*(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(5/2), x, 0, Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x]} - - -{(x^m*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x, 0, Unintegrable[(x^m*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^3*) - - -{(c + a^2*c*x^2)^3*ArcSinh[a*x]^3, x, 24, (-413312*c^3*Sqrt[1 + a^2*x^2])/(128625*a) - (30256*c^3*(1 + a^2*x^2)^(3/2))/(385875*a) - (2664*c^3*(1 + a^2*x^2)^(5/2))/(214375*a) - (6*c^3*(1 + a^2*x^2)^(7/2))/(2401*a) + (4322*c^3*x*ArcSinh[a*x])/1225 + (1514*a^2*c^3*x^3*ArcSinh[a*x])/3675 + (702*a^4*c^3*x^5*ArcSinh[a*x])/6125 + (6*a^6*c^3*x^7*ArcSinh[a*x])/343 - (48*c^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(35*a) - (8*c^3*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(35*a) - (18*c^3*(1 + a^2*x^2)^(5/2)*ArcSinh[a*x]^2)/(175*a) - (3*c^3*(1 + a^2*x^2)^(7/2)*ArcSinh[a*x]^2)/(49*a) + (16*c^3*x*ArcSinh[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcSinh[a*x]^3)/7} -{(c + a^2*c*x^2)^2*ArcSinh[a*x]^3, x, 17, (-4144*c^2*Sqrt[1 + a^2*x^2])/(1125*a) - (272*c^2*(1 + a^2*x^2)^(3/2))/(3375*a) - (6*c^2*(1 + a^2*x^2)^(5/2))/(625*a) + (298*c^2*x*ArcSinh[a*x])/75 + (76*a^2*c^2*x^3*ArcSinh[a*x])/225 + (6*a^4*c^2*x^5*ArcSinh[a*x])/125 - (8*c^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(5*a) - (4*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(15*a) - (3*c^2*(1 + a^2*x^2)^(5/2)*ArcSinh[a*x]^2)/(25*a) + (8*c^2*x*ArcSinh[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/5} -{(c + a^2*c*x^2)*ArcSinh[a*x]^3, x, 10, (-40*c*Sqrt[1 + a^2*x^2])/(9*a) - (2*c*(1 + a^2*x^2)^(3/2))/(27*a) + (14*c*x*ArcSinh[a*x])/3 + (2*a^2*c*x^3*ArcSinh[a*x])/9 - (2*c*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a - (c*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(3*a) + (2*c*x*ArcSinh[a*x]^3)/3 + (c*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/3} -{ArcSinh[a*x]^3/(c + a^2*c*x^2), x, 10, (2*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(a*c) - ((3*I)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c) + ((3*I)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c) + ((6*I)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]])/(a*c) - ((6*I)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c) - ((6*I)*PolyLog[4, (-I)*E^ArcSinh[a*x]])/(a*c) + ((6*I)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c)} -{ArcSinh[a*x]^3/(c + a^2*c*x^2)^2, x, 18, (3*ArcSinh[a*x]^2)/(2*a*c^2*Sqrt[1 + a^2*x^2]) + (x*ArcSinh[a*x]^3)/(2*c^2*(1 + a^2*x^2)) - (6*ArcSinh[a*x]*ArcTan[E^ArcSinh[a*x]])/(a*c^2) + (ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(a*c^2) + ((3*I)*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c^2) - (((3*I)/2)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c^2) - ((3*I)*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c^2) + (((3*I)/2)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c^2) + ((3*I)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]])/(a*c^2) - ((3*I)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c^2) - ((3*I)*PolyLog[4, (-I)*E^ArcSinh[a*x]])/(a*c^2) + ((3*I)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c^2)} -{ArcSinh[a*x]^3/(c + a^2*c*x^2)^3, x, 28, -1/(4*a*c^3*Sqrt[1 + a^2*x^2]) - (x*ArcSinh[a*x])/(4*c^3*(1 + a^2*x^2)) + ArcSinh[a*x]^2/(4*a*c^3*(1 + a^2*x^2)^(3/2)) + (9*ArcSinh[a*x]^2)/(8*a*c^3*Sqrt[1 + a^2*x^2]) + (x*ArcSinh[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcSinh[a*x]^3)/(8*c^3*(1 + a^2*x^2)) - (5*ArcSinh[a*x]*ArcTan[E^ArcSinh[a*x]])/(a*c^3) + (3*ArcSinh[a*x]^3*ArcTan[E^ArcSinh[a*x]])/(4*a*c^3) + (((5*I)/2)*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c^3) - (((9*I)/8)*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]])/(a*c^3) - (((5*I)/2)*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c^3) + (((9*I)/8)*ArcSinh[a*x]^2*PolyLog[2, I*E^ArcSinh[a*x]])/(a*c^3) + (((9*I)/4)*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]])/(a*c^3) - (((9*I)/4)*ArcSinh[a*x]*PolyLog[3, I*E^ArcSinh[a*x]])/(a*c^3) - (((9*I)/4)*PolyLog[4, (-I)*E^ArcSinh[a*x]])/(a*c^3) + (((9*I)/4)*PolyLog[4, I*E^ArcSinh[a*x]])/(a*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^3*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^3, x, 24, (-865*a*c^2*x^2*Sqrt[c + a^2*c*x^2])/(2304*Sqrt[1 + a^2*x^2]) - (65*a^3*c^2*x^4*Sqrt[c + a^2*c*x^2])/(2304*Sqrt[1 + a^2*x^2]) - (c^2*(1 + a^2*x^2)^(5/2)*Sqrt[c + a^2*c*x^2])/(216*a) + (245*c^2*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/384 + (65*c^2*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/576 + (c^2*x*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/36 - (115*c^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(768*a*Sqrt[1 + a^2*x^2]) - (15*a*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(32*Sqrt[1 + a^2*x^2]) - (5*c^2*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(32*a) - (c^2*(1 + a^2*x^2)^(5/2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^3)/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^3)/6 + (5*c^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(64*a*Sqrt[1 + a^2*x^2])} -{(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^3, x, 14, (-51*a*c*x^2*Sqrt[c + a^2*c*x^2])/(128*Sqrt[1 + a^2*x^2]) - (3*a^3*c*x^4*Sqrt[c + a^2*c*x^2])/(128*Sqrt[1 + a^2*x^2]) + (45*c*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/64 + (3*c*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/32 - (27*c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(128*a*Sqrt[1 + a^2*x^2]) - (9*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(16*Sqrt[1 + a^2*x^2]) - (3*c*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(16*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^3)/4 + (3*c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(32*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3, x, 6, (-3*a*x^2*Sqrt[c + a^2*c*x^2])/(8*Sqrt[1 + a^2*x^2]) + (3*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x])/4 - (3*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(8*a*Sqrt[1 + a^2*x^2]) - (3*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^2)/(4*Sqrt[1 + a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3)/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^4)/(8*a*Sqrt[1 + a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{ArcSinh[a*x]^3/Sqrt[c + a^2*c*x^2], x, 1, (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^4)/(4*a*Sqrt[c + a^2*c*x^2])} -{ArcSinh[a*x]^3/(c + a^2*c*x^2)^(3/2), x, 7, (x*ArcSinh[a*x]^3)/(c*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(a*c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(a*c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(a*c*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(2*a*c*Sqrt[c + a^2*c*x^2])} -{ArcSinh[a*x]^3/(c + a^2*c*x^2)^(5/2), x, 11, -((x*ArcSinh[a*x])/(c^2*Sqrt[c + a^2*c*x^2])) + ArcSinh[a*x]^2/(2*a*c^2*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcSinh[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])/(2*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(a*c^2*Sqrt[c + a^2*c*x^2])} -{ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2), x, 17, -1/(20*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[a*x])/(c^3*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[a*x])/(10*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + (3*ArcSinh[a*x]^2)/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + (2*ArcSinh[a*x]^2)/(5*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^3)/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^3)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x]^3)/(15*c^3*Sqrt[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])/(2*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2]) + (4*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2])} - - -{(x^m*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x, 0, Unintegrable[(x^m*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x]} - -{(x^4*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x, 13, (45*x^2)/(128*a^3) - (3*x^4)/(128*a) - (45*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(64*a^4) + (3*x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(32*a^2) + (45*ArcSinh[a*x]^2)/(128*a^5) + (9*x^2*ArcSinh[a*x]^2)/(16*a^3) - (3*x^4*ArcSinh[a*x]^2)/(16*a) - (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(8*a^4) + (x^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(4*a^2) + (3*ArcSinh[a*x]^4)/(32*a^5)} -{(x^3*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x, 10, (40*x)/(9*a^3) - (2*x^3)/(27*a) - (40*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4) + (2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^2) + (2*x*ArcSinh[a*x]^2)/a^3 - (x^3*ArcSinh[a*x]^2)/(3*a) - (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*a^4) + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(3*a^2)} -{(x^2*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x, 6, (-3*x^2)/(8*a) + (3*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(4*a^2) - (3*ArcSinh[a*x]^2)/(8*a^3) - (3*x^2*ArcSinh[a*x]^2)/(4*a) + (x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*a^2) - ArcSinh[a*x]^4/(8*a^3)} -{(x*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x, 4, (-6*x)/a + (6*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2 - (3*x*ArcSinh[a*x]^2)/a + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a^2} -{ArcSinh[a*x]^3/Sqrt[1 + a^2*x^2], x, 1, ArcSinh[a*x]^4/(4*a)} -{ArcSinh[a*x]^3/(x*Sqrt[1 + a^2*x^2]), x, 10, -2*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 3*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]] + 3*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] + 6*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] - 6*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 6*PolyLog[4, -E^ArcSinh[a*x]] + 6*PolyLog[4, E^ArcSinh[a*x]]} -{ArcSinh[a*x]^3/(x^2*Sqrt[1 + a^2*x^2]), x, 7, -(a*ArcSinh[a*x]^3) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/x + 3*a*ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + 3*a*ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - (3*a*PolyLog[3, E^(2*ArcSinh[a*x])])/2} -{ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]), x, 18, (-3*a*ArcSinh[a*x]^2)/(2*x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*x^2) - 6*a^2*ArcSinh[a*x]*ArcTanh[E^ArcSinh[a*x]] + a^2*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 3*a^2*PolyLog[2, -E^ArcSinh[a*x]] + (3*a^2*ArcSinh[a*x]^2*PolyLog[2, -E^ArcSinh[a*x]])/2 + 3*a^2*PolyLog[2, E^ArcSinh[a*x]] - (3*a^2*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]])/2 - 3*a^2*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] + 3*a^2*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] + 3*a^2*PolyLog[4, -E^ArcSinh[a*x]] - 3*a^2*PolyLog[4, E^ArcSinh[a*x]]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])*) - - -{(c + a^2*c*x^2)^3/ArcSinh[a*x], x, 7, (35*c^3*CoshIntegral[ArcSinh[a*x]])/(64*a) + (21*c^3*CoshIntegral[3*ArcSinh[a*x]])/(64*a) + (7*c^3*CoshIntegral[5*ArcSinh[a*x]])/(64*a) + (c^3*CoshIntegral[7*ArcSinh[a*x]])/(64*a)} -{(c + a^2*c*x^2)^2/ArcSinh[a*x], x, 6, (5*c^2*CoshIntegral[ArcSinh[a*x]])/(8*a) + (5*c^2*CoshIntegral[3*ArcSinh[a*x]])/(16*a) + (c^2*CoshIntegral[5*ArcSinh[a*x]])/(16*a)} -{(c + a^2*c*x^2)^1/ArcSinh[a*x], x, 5, (3*c*CoshIntegral[ArcSinh[a*x]])/(4*a) + (c*CoshIntegral[3*ArcSinh[a*x]])/(4*a)} -{1/((c + a^2*c*x^2)^1*ArcSinh[a*x]), x, 0, Unintegrable[1/((c + a^2*c*x^2)*ArcSinh[a*x]), x]} -{1/((c + a^2*c*x^2)^2*ArcSinh[a*x]), x, 0, Unintegrable[1/((c + a^2*c*x^2)^2*ArcSinh[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) / (a+b ArcSinh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^4*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x, 12, -((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^5)) - (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^5) + Log[a + b*ArcSinh[c*x]]/(16*b*c^5) + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^5) + (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^5)} -{(x^3*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x, 12, (CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b*c^4) + (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b*c^4) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b*c^4) - (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^4) - (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^4) + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^4)} -{(x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x, 6, (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c^3) - Log[a + b*ArcSinh[c*x]]/(8*b*c^3) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c^3)} -{(x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x, 9, -((CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b*c^2)) - (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b*c^2) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^2) + (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^2)} -{Sqrt[1 + c^2*x^2]/(a + b*ArcSinh[c*x]), x, 6, (Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c) + Log[a + b*ArcSinh[c*x]]/(2*b*c) - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c)} -{Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])), x, 6, -((CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/b) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/b + Unintegrable[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])), x, 3, (c*Log[a + b*ArcSinh[c*x]])/b + Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])), x]} -{Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])), x]} - - -{(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]), x, 15, (3*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(64*b*c^4) + (3*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(64*b*c^4) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(64*b*c^4) - (CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b]*Sinh[(7*a)/b])/(64*b*c^4) - (3*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(64*b*c^4) - (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b*c^4) + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^4) + (Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^4)} -{(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]), x, 12, -((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3)) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c^3) + (Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) - Log[a + b*ArcSinh[c*x]]/(16*b*c^3) + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c^3) - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3)} -{(x*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]), x, 12, -((CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b*c^2)) - (3*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b*c^2) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b*c^2) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^2) + (3*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^2) + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^2)} -{(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x]), x, 9, (Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c) + (3*Log[a + b*ArcSinh[c*x]])/(8*b*c) - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c)} -{(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])), x, 15, -((5*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b)) - (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b) + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b) + (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b) + Unintegrable[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])), x, 9, (c*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b) + (3*c*Log[a + b*ArcSinh[c*x]])/(2*b) - (c*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b) + Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])), x]} - - -{(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]), x, 15, (3*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(128*b*c^4) + (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(32*b*c^4) - (3*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b]*Sinh[(7*a)/b])/(256*b*c^4) - (CoshIntegral[(9*(a + b*ArcSinh[c*x]))/b]*Sinh[(9*a)/b])/(256*b*c^4) - (3*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(128*b*c^4) - (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(32*b*c^4) + (3*Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(256*b*c^4) + (Cosh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/(256*b*c^4)} -{(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]), x, 15, -((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3)) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) + (Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) + (Cosh[(8*a)/b]*CoshIntegral[(8*(a + b*ArcSinh[c*x]))/b])/(128*b*c^3) - (5*Log[a + b*ArcSinh[c*x]])/(128*b*c^3) + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c^3) - (Sinh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcSinh[c*x]))/b])/(128*b*c^3)} -{(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]), x, 15, -((5*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(64*b*c^2)) - (9*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(64*b*c^2) - (5*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(64*b*c^2) - (CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b]*Sinh[(7*a)/b])/(64*b*c^2) + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(64*b*c^2) + (9*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b*c^2) + (5*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^2) + (Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^2)} -{(1 + c^2*x^2)^(5/2)/(a + b*ArcSinh[c*x]), x, 12, (15*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c) + (3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c) + (Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c) + (5*Log[a + b*ArcSinh[c*x]])/(16*b*c) - (15*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(32*b*c) - (3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(16*b*c) - (Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(32*b*c)} -{(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])), x, 27, -((11*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b)) - (7*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b) + (11*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b) + (7*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b) + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b) + Unintegrable[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])), x, 18, (c*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/b + (c*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b) + (15*c*Log[a + b*ArcSinh[c*x]])/(8*b) - (c*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/b - (c*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b) + Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])), x]} -{(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 5, -CoshIntegral[2*ArcSinh[a*x]]/(2*a^5) + CoshIntegral[4*ArcSinh[a*x]]/(8*a^5) + (3*Log[ArcSinh[a*x]])/(8*a^5)} -{x^3/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 5, (-3*SinhIntegral[ArcSinh[a*x]])/(4*a^4) + SinhIntegral[3*ArcSinh[a*x]]/(4*a^4)} -{x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 4, CoshIntegral[2*ArcSinh[a*x]]/(2*a^3) - Log[ArcSinh[a*x]]/(2*a^3)} -{x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 4, CoshIntegral[2*ArcSinh[a*x]]/(2*a^3) - Log[ArcSinh[a*x]]/(2*a^3)} -{x/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 2, SinhIntegral[ArcSinh[a*x]]/a^2} -{1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 1, Log[ArcSinh[a*x]]/a} -{1/(x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 0, Unintegrable[1/(x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x]} -{1/(x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x, 0, Unintegrable[1/(x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x]} - - -{x^5/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 12, -((5*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b*c^6)) + (5*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b*c^6) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b*c^6) + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^6) - (5*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^6) + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^6)} -{x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 9, -((Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c^5)) + (Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c^5) + (3*Log[a + b*ArcSinh[c*x]])/(8*b*c^5) + (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c^5) - (Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b*c^5)} -{x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 9, (3*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b*c^4) - (CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b*c^4) - (3*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^4) + (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^4)} -{x^2/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 6, (Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c^3) - Log[a + b*ArcSinh[c*x]]/(2*b*c^3) - (Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(2*b*c^3)} -{x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 4, -((CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b*c^2)) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c^2)} -{1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 1, Log[a + b*ArcSinh[c*x]]/(b*c)} -{1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} - - -{x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} -{x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} -{1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} -{1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} -{1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) / (a+b ArcSinh[c x]) with m symbolic*) - - -{x^m*(1 + c^2*x^2)^(5/2)/(a + b*ArcSinh[c*x]), x, 0, Unintegrable[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]), x]} -{x^m*(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x]), x, 0, Unintegrable[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]), x]} -{x^m*(1 + c^2*x^2)^(1/2)/(a + b*ArcSinh[c*x]), x, 0, Unintegrable[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x]} -{x^m/((1 + c^2*x^2)^(1/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} -{x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])^2*) - - -{(c + a^2*c*x^2)^3/ArcSinh[a*x]^2, x, 8, -((c^3*(1 + a^2*x^2)^(7/2))/(a*ArcSinh[a*x])) + (35*c^3*SinhIntegral[ArcSinh[a*x]])/(64*a) + (63*c^3*SinhIntegral[3*ArcSinh[a*x]])/(64*a) + (35*c^3*SinhIntegral[5*ArcSinh[a*x]])/(64*a) + (7*c^3*SinhIntegral[7*ArcSinh[a*x]])/(64*a)} -{(c + a^2*c*x^2)^2/ArcSinh[a*x]^2, x, 7, -((c^2*(1 + a^2*x^2)^(5/2))/(a*ArcSinh[a*x])) + (5*c^2*SinhIntegral[ArcSinh[a*x]])/(8*a) + (15*c^2*SinhIntegral[3*ArcSinh[a*x]])/(16*a) + (5*c^2*SinhIntegral[5*ArcSinh[a*x]])/(16*a)} -{(c + a^2*c*x^2)/ArcSinh[a*x]^2, x, 6, -((c*(1 + a^2*x^2)^(3/2))/(a*ArcSinh[a*x])) + (3*c*SinhIntegral[ArcSinh[a*x]])/(4*a) + (3*c*SinhIntegral[3*ArcSinh[a*x]])/(4*a)} -{1/((c + a^2*c*x^2)*ArcSinh[a*x]^2), x, 1, -(1/(a*c*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])) - (a*Unintegrable[x/((1 + a^2*x^2)^(3/2)*ArcSinh[a*x]), x])/c} -{1/((c + a^2*c*x^2)^2*ArcSinh[a*x]^2), x, 1, -(1/(a*c^2*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x])) - (3*a*Unintegrable[x/((1 + a^2*x^2)^(5/2)*ArcSinh[a*x]), x])/c^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) / (a+b ArcSinh[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^3*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x, 22, -((x^3*(1 + c^2*x^2))/(b*c*(a + b*ArcSinh[c*x]))) - (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^4) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^4) + (5*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^4) + (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^4) + (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^4) - (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^4)} -{(x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x, 16, -((x^2*(1 + c^2*x^2))/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c^3) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(2*b^2*c^3)} -{(x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x, 14, -((x*(1 + c^2*x^2))/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^2) + (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^2) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^2) - (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^2)} -{Sqrt[1 + c^2*x^2]/(a + b*ArcSinh[c*x])^2, x, 7, -((1 + c^2*x^2)/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c) + (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c)} -{Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])^2), x, 5, -((1 + c^2*x^2)/(b*c*x*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/b^2 - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/b^2 - Unintegrable[1/(x^2*(a + b*ArcSinh[c*x])), x]/(b*c)} -{Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])^2), x, 1, -((1 + c^2*x^2)/(b*c*x^2*(a + b*ArcSinh[c*x]))) - (2*Unintegrable[1/(x^3*(a + b*ArcSinh[c*x])), x])/(b*c)} -{Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2), x]} -{Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x]} - - -{(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x, 28, -((x^3*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(64*b^2*c^4) - (9*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4) + (5*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4) + (7*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(64*b^2*c^4) + (9*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4) - (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4) - (7*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^4)} -{(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x, 19, -((x^2*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) + (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c^3) - (3*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3) - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3) + (3*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3)} -{(x*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x, 22, -((x*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^2) + (9*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^2) + (5*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^2) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^2) - (9*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^2) - (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^2)} -{(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x])^2, x, 10, -((1 + c^2*x^2)^2/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c) + (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(2*b^2*c)} -{(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])^2), x, 10, -((1 + c^2*x^2)^2/(b*c*x*(a + b*ArcSinh[c*x]))) + (9*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2) + (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2) - (9*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2) - (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2) - Unintegrable[(1 + c^2*x^2)/(x^2*(a + b*ArcSinh[c*x])), x]/(b*c)} -{(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])^2), x, 1, -((1 + c^2*x^2)^2/(b*c*x^2*(a + b*ArcSinh[c*x]))) - (2*Unintegrable[(1 + c^2*x^2)/(x^3*(a + b*ArcSinh[c*x])), x])/(b*c) + (2*c*Unintegrable[(1 + c^2*x^2)/(x*(a + b*ArcSinh[c*x])), x])/b} -{(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]} -{(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])^2), x, 1, -((1 + c^2*x^2)^2/(b*c*x^4*(a + b*ArcSinh[c*x]))) - (4*Unintegrable[(1 + c^2*x^2)/(x^5*(a + b*ArcSinh[c*x])), x])/(b*c)} - - -{(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x, 34, -((x^3*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(128*b^2*c^4) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) + (21*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4) + (9*Cosh[(9*a)/b]*CoshIntegral[(9*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(128*b^2*c^4) + (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) - (21*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4) - (9*Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4)} -{(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x, 28, -((x^2*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) + (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(8*b^2*c^3) - (3*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3) - (CoshIntegral[(8*(a + b*ArcSinh[c*x]))/b]*Sinh[(8*a)/b])/(16*b^2*c^3) - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(8*b^2*c^3) + (3*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3) + (Cosh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3)} -{(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x, 28, -((x*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) + (5*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(64*b^2*c^2) + (27*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2) + (25*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2) + (7*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2) - (5*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(64*b^2*c^2) - (27*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2) - (25*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2) - (7*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b^2*c^2)} -{(1 + c^2*x^2)^(5/2)/(a + b*ArcSinh[c*x])^2, x, 13, -((1 + c^2*x^2)^3/(b*c*(a + b*ArcSinh[c*x]))) - (15*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c) - (3*CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c) - (3*CoshIntegral[(6*(a + b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c) + (15*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c) + (3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c) + (3*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c)} -{(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])^2), x, 13, -((1 + c^2*x^2)^3/(b*c*x*(a + b*ArcSinh[c*x]))) + (25*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2) + (25*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2) + (5*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2) - (25*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2) - (25*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2) - (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2) - Unintegrable[(1 + c^2*x^2)^2/(x^2*(a + b*ArcSinh[c*x])), x]/(b*c)} -{(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2), x, 1, -((1 + c^2*x^2)^3/(b*c*x^2*(a + b*ArcSinh[c*x]))) - (2*Unintegrable[(1 + c^2*x^2)^2/(x^3*(a + b*ArcSinh[c*x])), x])/(b*c) + (4*c*Unintegrable[(1 + c^2*x^2)^2/(x*(a + b*ArcSinh[c*x])), x])/b} -{(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]} -{(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])^2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 13, -(x^5/(b*c*(a + b*ArcSinh[c*x]))) + (5*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^6) - (15*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^6) + (5*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^6) - (5*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^6) + (15*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^6) - (5*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^6)} -{x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 10, -(x^4/(b*c*(a + b*ArcSinh[c*x]))) + (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c^5) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c^5) - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^5) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(2*b^2*c^5)} -{x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 10, -(x^3/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^4) + (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^4) - (3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^4)} -{x^2/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 7, -(x^2/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c^3) + (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^3)} -{x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 5, -(x/(b*c*(a + b*ArcSinh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c^2) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c^2)} -{1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 1, -(1/(b*c*(a + b*ArcSinh[c*x])))} -{1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 1, -(1/(b*c*x*(a + b*ArcSinh[c*x]))) - Unintegrable[1/(x^2*(a + b*ArcSinh[c*x])), x]/(b*c)} -{1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x, 1, -(1/(b*c*x^2*(a + b*ArcSinh[c*x]))) - (2*Unintegrable[1/(x^3*(a + b*ArcSinh[c*x])), x])/(b*c)} - - -{x^3/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x^3/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} -{x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 1, -(x^2/(b*c*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))) + (2*Unintegrable[x/((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])), x])/(b*c)} -{x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} -{1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 1, -(1/(b*c*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))) - (2*c*Unintegrable[x/((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])), x])/b} -{1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} -{1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} - - -{x^3/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x^3/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} -{x^2/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x^2/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} -{x/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} -{1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 1, -(1/(b*c*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))) - (4*c*Unintegrable[x/((1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])), x])/b} -{1/(x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/(x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} -{1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) / (a+b ArcSinh[c x])^2 with m symbolic*) - - -{(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x]} -{(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x]} -{(x^m*(1 + c^2*x^2)^(1/2))/(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x]} -{x^m/((1 + c^2*x^2)^(1/2)*(a + b*ArcSinh[c*x])^2), x, 1, -(x^m/(b*c*(a + b*ArcSinh[c*x]))) + (m*Unintegrable[x^(-1 + m)/(a + b*ArcSinh[c*x]), x])/(b*c)} -{x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} -{x^m/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[x^m/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) / (a+b ArcSinh[c x])^3*) - - -{1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3), x, 1, -1/(2*a*ArcSinh[a*x]^2)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p / (a+b ArcSinh[c x])^(3/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^3*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2), x, 27, (-2*d*x^3*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (3*d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (d*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) - (3*d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4*E^((2*a)/b)) + (d*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4*E^((6*a)/b))} -{(x^2*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2), x, 32, (-2*d*x^2*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3) - (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) - (d*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) - (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3*E^(a/b)) + (d*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((3*a)/b)) + (d*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3*E^((5*a)/b))} -{(x*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2), x, 17, (-2*d*x*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (d*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2) + (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(3/2)*c^2) + (d*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2*E^((4*a)/b)) + (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(3/2)*c^2*E^((2*a)/b))} -{(d + c^2*d*x^2)/(a + b*ArcSinh[c*x])^(3/2), x, 14, (-2*d*(1 + c^2*x^2)^(3/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (3*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c) - (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c) + (3*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c*E^(a/b)) + (d*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c*E^((3*a)/b))} -{(d + c^2*d*x^2)/(x*(a + b*ArcSinh[c*x])^(3/2)), x, 12, (-2*d*(1 + c^2*x^2)^(3/2))/(b*c*x*Sqrt[a + b*ArcSinh[c*x]]) + (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/b^(3/2) + (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(b^(3/2)*E^((2*a)/b)) - (2*d*Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]]), x])/(b*c)} - - -{(x^3*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2), x, 32, (-2*d^2*x^3*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) - (3*d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) + (d^2*E^((8*a)/b)*Sqrt[Pi/2]*Erf[(2*Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) + (d^2*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) - (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4*E^((4*a)/b)) - (3*d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4*E^((2*a)/b)) + (d^2*Sqrt[Pi/2]*Erfi[(2*Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4*E^((8*a)/b)) + (d^2*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4*E^((6*a)/b))} -{(x^2*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2), x, 42, (-2*d^2*x^2*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (5*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3) - (d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (3*d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (d^2*E^((7*a)/b)*Sqrt[7*Pi]*Erf[(Sqrt[7]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (5*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3*E^(a/b)) + (d^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((3*a)/b)) + (3*d^2*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((5*a)/b)) + (d^2*Sqrt[7*Pi]*Erfi[(Sqrt[7]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((7*a)/b))} -{(x*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2), x, 32, (-2*d^2*x*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2) + (5*d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2) + (d^2*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2) + (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2*E^((4*a)/b)) + (5*d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2*E^((2*a)/b)) + (d^2*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2*E^((6*a)/b))} -{(d + c^2*d*x^2)^2/(a + b*ArcSinh[c*x])^(3/2), x, 19, (-2*d^2*(1 + c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (5*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c) - (5*d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c) - (d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c) + (5*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c*E^(a/b)) + (5*d^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c*E^((3*a)/b)) + (d^2*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c*E^((5*a)/b))} -{(d + c^2*d*x^2)^2/(x*(a + b*ArcSinh[c*x])^(3/2)), x, 25, (-2*d^2*(1 + c^2*x^2)^(5/2))/(b*c*x*Sqrt[a + b*ArcSinh[c*x]]) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)) - (d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(3/2)) + (d^2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/b^(3/2) + (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*E^((4*a)/b)) - (d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(2*b^(3/2)*E^((2*a)/b)) + (d^2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(b^(3/2)*E^((2*a)/b)) - (2*d^2*Unintegrable[1/(x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]]), x])/(b*c)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) ArcSinh[c x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]], x, 24, (3*c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/8 + (x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]])/4 + (c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(4*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]], x, 10, (x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[1 + a^2*x^2]) + (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) - (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2])} -{Sqrt[ArcSinh[a*x]]/Sqrt[c + a^2*c*x^2], x, 1, (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2), x, 1, (x*Sqrt[ArcSinh[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)*Sqrt[ArcSinh[a*x]]), x])/(2*c*Sqrt[c + a^2*c*x^2])} -{Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2), x, 2, (x*Sqrt[ArcSinh[a*x]])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[a*x]])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)^2*Sqrt[ArcSinh[a*x]]), x])/(6*c^2*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)*Sqrt[ArcSinh[a*x]]), x])/(3*c^2*Sqrt[c + a^2*c*x^2])} - - -{(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2), x, 26, (-27*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(256*a*Sqrt[1 + a^2*x^2]) - (9*a*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(32*Sqrt[1 + a^2*x^2]) - (3*c*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(32*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2))/4 + (3*c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2))/(20*a*Sqrt[1 + a^2*x^2]) + (3*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(2048*a*Sqrt[1 + a^2*x^2]) + (3*c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (3*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(2048*a*Sqrt[1 + a^2*x^2]) + (3*c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2), x, 11, (-3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(16*a*Sqrt[1 + a^2*x^2]) - (3*a*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(8*Sqrt[1 + a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2))/(5*a*Sqrt[1 + a^2*x^2]) + (3*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (3*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2])} -{ArcSinh[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x, 1, (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(5/2))/(5*a*Sqrt[c + a^2*c*x^2])} -{ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2), x, 1, (x*ArcSinh[a*x]^(3/2))/(c*Sqrt[c + a^2*c*x^2]) - (3*a*Sqrt[1 + a^2*x^2]*Unintegrable[(x*Sqrt[ArcSinh[a*x]])/(1 + a^2*x^2), x])/(2*c*Sqrt[c + a^2*c*x^2])} - - -{(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2), x, 39, (225*c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/512 + (15*c*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/256 - (45*c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(256*a*Sqrt[1 + a^2*x^2]) - (15*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(32*Sqrt[1 + a^2*x^2]) - (5*c*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(32*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2))/4 + (3*c*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(7/2))/(28*a*Sqrt[1 + a^2*x^2]) + (15*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(16384*a*Sqrt[1 + a^2*x^2]) + (15*c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) - (15*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(16384*a*Sqrt[1 + a^2*x^2]) - (15*c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2), x, 13, (15*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/32 - (5*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(16*a*Sqrt[1 + a^2*x^2]) - (5*a*x^2*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/(8*Sqrt[1 + a^2*x^2]) + (x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2))/2 + (Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[1 + a^2*x^2]) + (15*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2]) - (15*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(256*a*Sqrt[1 + a^2*x^2])} -{ArcSinh[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x, 1, (2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[c + a^2*c*x^2])} -{ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2), x, 1, (x*ArcSinh[a*x]^(5/2))/(c*Sqrt[c + a^2*c*x^2]) - (5*a*Sqrt[1 + a^2*x^2]*Unintegrable[(x*ArcSinh[a*x]^(3/2))/(1 + a^2*x^2), x])/(2*c*Sqrt[c + a^2*c*x^2])} - - -{(a^2 + x^2)^(3/2)*Sqrt[ArcSinh[x/a]], x, 24, (3*a^2*x*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/8 + (x*(a^2 + x^2)^(3/2)*Sqrt[ArcSinh[x/a]])/4 + (a^3*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/(4*Sqrt[1 + x^2/a^2]) + (a^3*Sqrt[Pi]*Sqrt[a^2 + x^2]*Erf[2*Sqrt[ArcSinh[x/a]]])/(256*Sqrt[1 + x^2/a^2]) + (a^3*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2]) - (a^3*Sqrt[Pi]*Sqrt[a^2 + x^2]*Erfi[2*Sqrt[ArcSinh[x/a]]])/(256*Sqrt[1 + x^2/a^2]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2])} -{Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]], x, 10, (x*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/(3*Sqrt[1 + x^2/a^2]) + (a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2]) - (a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(16*Sqrt[1 + x^2/a^2])} -{Sqrt[ArcSinh[x/a]]/Sqrt[a^2 + x^2], x, 1, (2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(3/2))/(3*Sqrt[a^2 + x^2])} -{Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(3/2), x, 1, (x*Sqrt[ArcSinh[x/a]])/(a^2*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^2]*Unintegrable[x/((1 + x^2/a^2)*Sqrt[ArcSinh[x/a]]), x])/(2*a^3*Sqrt[a^2 + x^2])} -{Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2), x, 2, (x*Sqrt[ArcSinh[x/a]])/(3*a^2*(a^2 + x^2)^(3/2)) + (2*x*Sqrt[ArcSinh[x/a]])/(3*a^4*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^2]*Unintegrable[x/((1 + x^2/a^2)^2*Sqrt[ArcSinh[x/a]]), x])/(6*a^5*Sqrt[a^2 + x^2]) - (Sqrt[1 + x^2/a^2]*Unintegrable[x/((1 + x^2/a^2)*Sqrt[ArcSinh[x/a]]), x])/(3*a^5*Sqrt[a^2 + x^2])} - - -{(a^2 + x^2)^(3/2)*ArcSinh[x/a]^(3/2), x, 26, (-27*a^3*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(256*Sqrt[1 + x^2/a^2]) - (9*a*x^2*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(32*Sqrt[1 + x^2/a^2]) - (3*(a^2 + x^2)^(5/2)*Sqrt[ArcSinh[x/a]])/(32*a*Sqrt[1 + x^2/a^2]) + (3*a^2*x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/8 + (x*(a^2 + x^2)^(3/2)*ArcSinh[x/a]^(3/2))/4 + (3*a^3*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(20*Sqrt[1 + x^2/a^2]) + (3*a^3*Sqrt[Pi]*Sqrt[a^2 + x^2]*Erf[2*Sqrt[ArcSinh[x/a]]])/(2048*Sqrt[1 + x^2/a^2]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2]) + (3*a^3*Sqrt[Pi]*Sqrt[a^2 + x^2]*Erfi[2*Sqrt[ArcSinh[x/a]]])/(2048*Sqrt[1 + x^2/a^2]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])} -{Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2), x, 11, (-3*a*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(16*Sqrt[1 + x^2/a^2]) - (3*x^2*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(8*a*Sqrt[1 + x^2/a^2]) + (x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[1 + x^2/a^2]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])} -{ArcSinh[x/a]^(3/2)/Sqrt[a^2 + x^2], x, 1, (2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[a^2 + x^2])} -{ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2), x, 1, (x*ArcSinh[x/a]^(3/2))/(a^2*Sqrt[a^2 + x^2]) - (3*Sqrt[1 + x^2/a^2]*Unintegrable[(x*Sqrt[ArcSinh[x/a]])/(1 + x^2/a^2), x])/(2*a^3*Sqrt[a^2 + x^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]), x, 6, -(Sqrt[Pi]*Erf[Sqrt[ArcSinh[x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[x]]])/2} - - -{(c + a^2*c*x^2)^(5/2)/Sqrt[ArcSinh[a*x]], x, 18, (5*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(8*a*Sqrt[1 + a^2*x^2]) + (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[Pi/6]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[6]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[Pi/6]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[6]*Sqrt[ArcSinh[a*x]]])/(64*a*Sqrt[1 + a^2*x^2])} -{(c + a^2*c*x^2)^(3/2)/Sqrt[ArcSinh[a*x]], x, 13, (3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(4*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(32*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(32*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]/Sqrt[ArcSinh[a*x]], x, 8, (Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])/(a*Sqrt[1 + a^2*x^2]) + (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) + (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2])} -{1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]]), x, 1, (2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(a*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]), x, 0, Unintegrable[1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]), x]} -{1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcSinh[a*x]]), x, 0, Unintegrable[1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcSinh[a*x]]), x]} - - -{(c + a^2*c*x^2)^(5/2)/ArcSinh[a*x]^(3/2), x, 19, -((2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(5/2))/(a*Sqrt[ArcSinh[a*x]])) - (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(8*a*Sqrt[1 + a^2*x^2]) - (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) - (c^2*Sqrt[(3*Pi)/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[6]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) + (3*c^2*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(8*a*Sqrt[1 + a^2*x^2]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[(3*Pi)/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[6]*Sqrt[ArcSinh[a*x]]])/(16*a*Sqrt[1 + a^2*x^2])} -{(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(3/2), x, 14, -((2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(a*Sqrt[ArcSinh[a*x]])) - (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) - (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(4*a*Sqrt[1 + a^2*x^2]) + (c*Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(3/2), x, 9, (-2*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2]) + (Sqrt[Pi/2]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(a*Sqrt[1 + a^2*x^2])} -{1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2)), x, 1, (-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])} -{1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)), x, 1, -((2*Sqrt[1 + a^2*x^2])/(a*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]])) - (4*a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)^2*Sqrt[ArcSinh[a*x]]), x])/(c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(3/2)), x, 1, -((2*Sqrt[1 + a^2*x^2])/(a*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcSinh[a*x]])) - (8*a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)^3*Sqrt[ArcSinh[a*x]]), x])/(c^2*Sqrt[c + a^2*c*x^2])} - - -{(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(5/2), x, 18, -((2*Sqrt[1 + a^2*x^2]*(c + a^2*c*x^2)^(3/2))/(3*a*ArcSinh[a*x]^(3/2))) - (16*c*x*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2])/(3*Sqrt[ArcSinh[a*x]]) + (2*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erf[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqrt[Pi]*Sqrt[c + a^2*c*x^2]*Erfi[2*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*c*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2])} -{Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(5/2), x, 7, (-2*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (8*x*Sqrt[c + a^2*c*x^2])/(3*Sqrt[ArcSinh[a*x]]) + (2*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2]) + (2*Sqrt[2*Pi]*Sqrt[c + a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(3*a*Sqrt[1 + a^2*x^2])} -{1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2)), x, 1, (-2*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))} -{1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2)), x, 1, -((2*Sqrt[1 + a^2*x^2])/(3*a*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2))) - (4*a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)^2*ArcSinh[a*x]^(3/2)), x])/(3*c*Sqrt[c + a^2*c*x^2])} -{1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(5/2)), x, 1, -((2*Sqrt[1 + a^2*x^2])/(3*a*(c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(3/2))) - (8*a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((1 + a^2*x^2)^3*ArcSinh[a*x]^(3/2)), x])/(3*c^2*Sqrt[c + a^2*c*x^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^n with n symbolic*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n, x, 6, -(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(8*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n, x, 9, (3^(-1 - n)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(8*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(8*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(8*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (3^(-1 - n)*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(8*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n, x, 6, (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(2*b*c*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-3 - n)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x, x, 6, (d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c*x])/b))^n*(2*Sqrt[d + c^2*d*x^2])) + (d*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(((a + b*ArcSinh[c*x])/b)^n*(2*Sqrt[d + c^2*d*x^2])) + d*Unintegrable[(a + b*ArcSinh[c*x])^n/(x*Sqrt[d + c^2*d*x^2]), x]} -{(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x^2, x, 3, (c*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(b*(1 + n)*Sqrt[d + c^2*d*x^2]) + d*Unintegrable[(a + b*ArcSinh[c*x])^n/(x^2*Sqrt[d + c^2*d*x^2]), x]} - - -{x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n, x, 12, -(d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(16*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-7 - n)*3^(-1 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - 2*n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-7 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*d*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - 2*n)*d*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d*E^((6*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n, x, 12, (5^(-1 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b])/(32*c^2*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(32*3^n*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(16*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d*E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(16*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (d*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(32*3^n*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (5^(-1 - n)*d*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(32*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n, x, 9, (3*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[1 + c^2*x^2]) + (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-3 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-3 - n)*d*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (d*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x, x, 15, (1/(8*Sqrt[d + c^2*d*x^2]))*((3^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcSinh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) + (5*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c*x])/b))^n*(8*Sqrt[d + c^2*d*x^2])) + (5*d^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(((a + b*ArcSinh[c*x])/b)^n*(8*Sqrt[d + c^2*d*x^2])) + (1/(8*Sqrt[d + c^2*d*x^2]))*((3^(-1 - n)*d^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n) + d^2*Unintegrable[(a + b*ArcSinh[c*x])^n/(x*Sqrt[d + c^2*d*x^2]), x]} -{((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x^2, x, 9, (3*c*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(2*b*(1 + n)*Sqrt[d + c^2*d*x^2]) + (1/Sqrt[d + c^2*d*x^2])*((2^(-3 - n)*c*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcSinh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) - (1/Sqrt[d + c^2*d*x^2])*((2^(-3 - n)*c*d^2*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n) + d^2*Unintegrable[(a + b*ArcSinh[c*x])^n/(x^2*Sqrt[d + c^2*d*x^2]), x]} - - -{x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n, x, 15, (-5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-11 - 3*n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((8*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(4 + n))*c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-7 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*d^2*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (d^2*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-11 - 3*n)*d^2*E^((8*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n, x, 15, (7^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b])/(128*c^2*E^((7*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (3^(1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(128*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (d^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(128*5^n*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n, x, 12, (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(16*b*c*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c*E^((6*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (3*2^(-7 - 2*n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(c*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (15*2^(-7 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (15*2^(-7 - n)*d^2*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (3*2^(-7 - 2*n)*d^2*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n)} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x, x, 27, (1/(32*Sqrt[d + c^2*d*x^2]))*((5^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((5*(a + b*ArcSinh[c*x]))/b)])/(E^((5*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) - (1/(32*Sqrt[d + c^2*d*x^2]))*((5*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcSinh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) + (1/(8*Sqrt[d + c^2*d*x^2]))*((d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcSinh[c*x]))/b)])/(3^n*E^((3*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) + (11*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c*x])/b))^n*(16*Sqrt[d + c^2*d*x^2])) + (11*d^3*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(((a + b*ArcSinh[c*x])/b)^n*(16*Sqrt[d + c^2*d*x^2])) - (1/(32*Sqrt[d + c^2*d*x^2]))*((5*3^(-1 - n)*d^3*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n) + (d^3*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(3^n*((a + b*ArcSinh[c*x])/b)^n*(8*Sqrt[d + c^2*d*x^2])) + (1/(32*Sqrt[d + c^2*d*x^2]))*((5^(-1 - n)*d^3*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n) + d^3*Unintegrable[(a + b*ArcSinh[c*x])^n/(x*Sqrt[d + c^2*d*x^2]), x]} -{((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2, x, 18, (15*c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d + c^2*d*x^2]) + (1/Sqrt[d + c^2*d*x^2])*((c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((4*(a + b*ArcSinh[c*x]))/b)])/(2^(2*(3 + n))*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) + (1/Sqrt[d + c^2*d*x^2])*((2^(-2 - n)*c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcSinh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n)) - (1/Sqrt[d + c^2*d*x^2])*((2^(-2 - n)*c*d^3*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/((a + b*ArcSinh[c*x])/b)^n) - (1/Sqrt[d + c^2*d*x^2])*((c*d^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*((a + b*ArcSinh[c*x])/b)^n)) + d^3*Unintegrable[(a + b*ArcSinh[c*x])^n/(x^2*Sqrt[d + c^2*d*x^2]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^m*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x, 0, Unintegrable[(x^m*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x]} - -{(x^3*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x, 9, (3^(-1 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]])/(8*a^4*(-ArcSinh[a*x])^n) - (3*ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(8*a^4*(-ArcSinh[a*x])^n) - (3*Gamma[1 + n, ArcSinh[a*x]])/(8*a^4) + (3^(-1 - n)*Gamma[1 + n, 3*ArcSinh[a*x]])/(8*a^4)} -{(x^2*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x, 6, -ArcSinh[a*x]^(1 + n)/(2*a^3*(1 + n)) + (2^(-3 - n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]])/(a^3*(-ArcSinh[a*x])^n) - (2^(-3 - n)*Gamma[1 + n, 2*ArcSinh[a*x]])/a^3} -{(x*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x, 4, (ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(2*a^2*(-ArcSinh[a*x])^n) + Gamma[1 + n, ArcSinh[a*x]]/(2*a^2)} -{ArcSinh[a*x]^n/Sqrt[1 + a^2*x^2], x, 1, ArcSinh[a*x]^(1 + n)/(a*(1 + n))} -{ArcSinh[a*x]^n/(x*Sqrt[1 + a^2*x^2]), x, 0, Unintegrable[ArcSinh[a*x]^n/(x*Sqrt[1 + a^2*x^2]), x]} -{ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]), x, 0, Unintegrable[ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^p (f+g x)^q (a+b ArcSinh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSinh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSinh[c x])^1 with e f+d g=0 and c^2 d^2+e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]), x, 13, (((-2*I)/3)*b*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] - (3*b*c*d^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(16*Sqrt[1 + c^2*x^2]) - (((2*I)/9)*b*c^2*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] + (b*c^3*d^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(16*Sqrt[1 + c^2*x^2]) + (3*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/8 - (c^2*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/4 + (((2*I)/3)*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (5*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2])} -{(d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]), x, 8, ((-I/3)*b*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] - (b*c*d*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(4*Sqrt[1 + c^2*x^2]) - ((I/9)*b*c^2*d*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] + (d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/2 + ((I/3)*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])} -{Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]), x, 4, -(b*c*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(4*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/2 + (Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x], x, 6, (I*b*f*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (I*f*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (f*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2), x, 8, ((2*I)*f^2*(1 - I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b*f^2*(1 + c^2*x^2)^(3/2)*Log[I - c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2), x, 6, (((2*I)/3)*b*f^3*(1 + c^2*x^2)^(5/2))/(c*(I - c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*f^3*(1 - I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*f^3*(1 + c^2*x^2)^(5/2)*Log[I - c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]), x, 12, ((-I/5)*b*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) - (5*b*c*d*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) - (((2*I)/15)*b*c^2*d*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) - (b*c^3*d*x^4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) - ((I/25)*b*c^4*d*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) + (d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)) + ((I/5)*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (3*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(16*b*c*(1 + c^2*x^2)^(3/2))} -{(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]), x, 7, (-5*b*c*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) - (b*c^3*x^4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) + (x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)) + (3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(16*b*c*(1 + c^2*x^2)^(3/2))} -{Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]), x, 8, ((I/3)*b*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] - (b*c*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(4*Sqrt[1 + c^2*x^2]) + ((I/9)*b*c^2*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] + (f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/2 - ((I/3)*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x], x, 9, ((2*I)*b*f^2*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*f^2*x^2*Sqrt[1 + c^2*x^2])/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((2*I)*f^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (f^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*f^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2), x, 10, ((-I)*b*f^3*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*f^3*(1 - I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (I*f^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (3*f^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b*f^3*(1 + c^2*x^2)^(3/2)*Log[I - c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2), x, 9, (((4*I)/3)*b*f^4*(1 + c^2*x^2)^(5/2))/(c*(I - c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*f^4*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]^2)/(2*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*f^4*(1 - I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((2*I)*f^4*(1 - I*c*x)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f^4*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (8*b*f^4*(1 + c^2*x^2)^(5/2)*Log[I - c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]), x, 9, (-25*b*c*x^2*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(96*(1 + c^2*x^2)^(5/2)) - (5*b*c^3*x^4*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(96*(1 + c^2*x^2)^(5/2)) - (b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*Sqrt[1 + c^2*x^2])/(36*c) + (x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(16*(1 + c^2*x^2)^2) + (5*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(24*(1 + c^2*x^2)) + (5*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(32*b*c*(1 + c^2*x^2)^(5/2))} -{(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]), x, 12, ((I/5)*b*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) - (5*b*c*f*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) + (((2*I)/15)*b*c^2*f*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) - (b*c^3*f*x^4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) + ((I/25)*b*c^4*f*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) + (f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)) - ((I/5)*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (3*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(16*b*c*(1 + c^2*x^2)^(3/2))} -{Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]), x, 13, (((2*I)/3)*b*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] - (3*b*c*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(16*Sqrt[1 + c^2*x^2]) + (((2*I)/9)*b*c^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/Sqrt[1 + c^2*x^2] + (b*c^3*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(16*Sqrt[1 + c^2*x^2]) + (3*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/8 - (c^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/4 - (((2*I)/3)*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c + (5*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2])} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x], x, 13, (((11*I)/3)*b*f^3*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b*c*f^3*x^2*Sqrt[1 + c^2*x^2])/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((I/9)*b*c^2*f^3*x^3*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (((11*I)/3)*f^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*f^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((I/3)*c*f^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (5*f^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2), x, 7, (((-3*I)/2)*b*f^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b*c*f^4*x^2*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (5*b*f^4*(1 - I*c*x)^2*(1 + c^2*x^2)^(3/2))/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (15*b*f^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2)/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((2*I)*f^4*(1 - I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (((15*I)/2)*f^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (((5*I)/2)*f^4*(1 - I*c*x)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (15*f^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(2*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b*f^4*(1 + c^2*x^2)^(3/2)*Log[I - c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2), x, 10, (I*b*f^5*x*(1 + c^2*x^2)^(5/2))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((8*I)/3)*b*f^5*(1 + c^2*x^2)^(5/2))/(c*(I - c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (5*b*f^5*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]^2)/(2*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*f^5*(1 - I*c*x)^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((10*I)/3)*f^5*(1 - I*c*x)^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((5*I)*f^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (5*f^5*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (28*b*f^5*(1 + c^2*x^2)^(5/2)*Log[I - c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x], x, 13, (((-11*I)/3)*b*d^3*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b*c*d^3*x^2*Sqrt[1 + c^2*x^2])/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((I/9)*b*c^2*d^3*x^3*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (((11*I)/3)*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((I/3)*c*d^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (5*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x], x, 9, ((-2*I)*b*d^2*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*d^2*x^2*Sqrt[1 + c^2*x^2])/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x], x, 6, ((-I)*b*d*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (I*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]), x, 2, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]), x, 5, (f*(I + c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*f*(1 + c^2*x^2)^(3/2)*Log[I - c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]), x, 8, ((I/3)*b*f^2*(1 + c^2*x^2)^(5/2))/(c*(I - c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*f^2*(1 - I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*b*f^2*(1 + c^2*x^2)^(5/2)*ArcTan[c*x])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*f^2*(1 + c^2*x^2)^(5/2)*Log[1 + c^2*x^2])/(6*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2), x, 7, (((3*I)/2)*b*d^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b*c*d^4*x^2*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (5*b*d^4*(1 + I*c*x)^2*(1 + c^2*x^2)^(3/2))/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (15*b*d^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2)/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((2*I)*d^4*(1 + I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (((15*I)/2)*d^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (((5*I)/2)*d^4*(1 + I*c*x)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (15*d^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(2*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b*d^4*(1 + c^2*x^2)^(3/2)*Log[I + c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2), x, 10, (I*b*d^3*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((4*I)*d^3*(1 + I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (I*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (3*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b*d^3*(1 + c^2*x^2)^(3/2)*Log[I + c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2), x, 8, ((-2*I)*d^2*(1 + I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b*d^2*(1 + c^2*x^2)^(3/2)*Log[I + c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)), x, 5, -((d*(I - c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))) - (b*d*(1 + c^2*x^2)^(3/2)*Log[I + c*x])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)), x, 3, (x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(2*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)), x, 8, ((I/6)*b*f*(1 + c^2*x^2)^(5/2))/(c*(I - c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f*(I + c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*f*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/6)*b*f*(1 + c^2*x^2)^(5/2)*ArcTan[c*x])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*f*(1 + c^2*x^2)^(5/2)*Log[1 + c^2*x^2])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2), x, 10, ((-I)*b*d^5*x*(1 + c^2*x^2)^(5/2))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((8*I)/3)*b*d^5*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (5*b*d^5*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]^2)/(2*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*d^5*(1 + I*c*x)^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((10*I)/3)*d^5*(1 + I*c*x)^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((5*I)*d^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (5*d^5*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (28*b*d^5*(1 + c^2*x^2)^(5/2)*Log[I + c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2), x, 9, (((4*I)/3)*b*d^4*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*d^4*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]^2)/(2*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*d^4*(1 + I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((2*I)*d^4*(1 + I*c*x)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d^4*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (8*b*d^4*(1 + c^2*x^2)^(5/2)*Log[I + c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2), x, 6, (((2*I)/3)*b*d^3*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*d^3*(1 + I*c*x)^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*d^3*(1 + c^2*x^2)^(5/2)*Log[I + c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)), x, 8, ((I/3)*b*d^2*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*d^2*(1 + I*c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*b*d^2*(1 + c^2*x^2)^(5/2)*ArcTan[c*x])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*d^2*(1 + c^2*x^2)^(5/2)*Log[1 + c^2*x^2])/(6*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)), x, 8, ((I/6)*b*d*(1 + c^2*x^2)^(5/2))/(c*(I + c*x)*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (d*(I - c*x)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*d*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/6)*b*d*(1 + c^2*x^2)^(5/2)*ArcTan[c*x])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*d*(1 + c^2*x^2)^(5/2)*Log[1 + c^2*x^2])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)), x, 5, (b*(1 + c^2*x^2)^(3/2))/(6*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*(1 + c^2*x^2)^(5/2)*Log[1 + c^2*x^2])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(p/2) (f+g x)^(q/2) (a+b ArcSinh[c x])^2 with e f+d g=0 and c^2 d^2+e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2, x, 23, (((8*I)/9)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/c + (15*b^2*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/64 - (b^2*c^2*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/32 + (((4*I)/27)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2))/c - (15*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x])/(64*c*Sqrt[1 + c^2*x^2]) - (((4*I)/3)*b*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (3*b*c*d^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) - (((4*I)/9)*b*c^2*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (b*c^3*d^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (3*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/8 - (c^2*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/4 + (((2*I)/3)*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (5*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^3)/(24*b*c*Sqrt[1 + c^2*x^2])} -{(d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2, x, 13, (((4*I)/9)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/c + (b^2*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/4 + (((2*I)/27)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2))/c - (b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x])/(4*c*Sqrt[1 + c^2*x^2]) - (((2*I)/3)*b*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (b*c*d*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) - (((2*I)/9)*b*c^2*d*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/2 + ((I/3)*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[1 + c^2*x^2])} -{Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2, x, 6, (b^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/4 - (b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x])/(4*c*Sqrt[1 + c^2*x^2]) - (b*c*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[1 + c^2*x^2])} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x], x, 8, ((2*I)*a*b*f*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((2*I)*b^2*f*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*b^2*f*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (I*f*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (f*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2), x, 19, ((2*I)*f^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*f^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*I)*b*f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b*f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b^2*f^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*b^2*f^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b^2*f^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2), x, 20, -(f^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((4*I)/3)*b^2*f^3*(1 + c^2*x^2)^(5/2)*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*f^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*b*f^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*f^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]]*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (4*b*f^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (4*b^2*f^3*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 19, (((8*I)/225)*b^2*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/c + (b^2*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/32 + (((16*I)/75)*b^2*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(c*(1 + c^2*x^2)) + (15*b^2*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(64*(1 + c^2*x^2)) + (((2*I)/125)*b^2*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2))/c - (9*b^2*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*ArcSinh[c*x])/(64*c*(1 + c^2*x^2)^(3/2)) - (((2*I)/5)*b*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (3*b*c*d*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)^(3/2)) - (((4*I)/15)*b*c^2*d*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (((2*I)/25)*b*c^4*d*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (b*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(8*(1 + c^2*x^2)) + ((I/5)*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c*(1 + c^2*x^2)^(3/2))} -{(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 11, (b^2*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/32 + (15*b^2*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(64*(1 + c^2*x^2)) - (9*b^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*ArcSinh[c*x])/(64*c*(1 + c^2*x^2)^(3/2)) - (3*b*c*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)^(3/2)) - (b*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(8*(1 + c^2*x^2)) + ((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c*(1 + c^2*x^2)^(3/2))} -{Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2, x, 13, (((-4*I)/9)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/c + (b^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/4 - (((2*I)/27)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2))/c - (b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x])/(4*c*Sqrt[1 + c^2*x^2]) + (((2*I)/3)*b*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (b*c*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (((2*I)/9)*b*c^2*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/2 - ((I/3)*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[1 + c^2*x^2])} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x], x, 11, ((-4*I)*b^2*f^2*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (b^2*f^2*x*(1 + c^2*x^2))/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b^2*f^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((4*I)*b*f^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*f^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((2*I)*f^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (f^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (f^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2), x, 23, ((-2*I)*a*b*f^3*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((2*I)*b^2*f^3*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((2*I)*b^2*f^3*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*f^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*f^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*f^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (I*f^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (f^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((16*I)*b*f^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b*f^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b^2*f^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*b^2*f^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b^2*f^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2), x, 21, (-8*f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((8*I)/3)*b^2*f^4*(1 + c^2*x^2)^(5/2)*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((8*I)/3)*f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (4*b*f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]]*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (32*b*f^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (32*b^2*f^4*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 17, (b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/108 + (245*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(1152*(1 + c^2*x^2)^2) + (65*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(1728*(1 + c^2*x^2)) - (115*b^2*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*ArcSinh[c*x])/(1152*c*(1 + c^2*x^2)^(5/2)) - (5*b*c*x^2*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(16*(1 + c^2*x^2)^(5/2)) - (5*b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(48*c*Sqrt[1 + c^2*x^2]) - (b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(18*c) + (x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 + (5*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(16*(1 + c^2*x^2)^2) + (5*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(24*(1 + c^2*x^2)) + (5*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^3)/(48*b*c*(1 + c^2*x^2)^(5/2))} -{(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 19, (((-8*I)/225)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/c + (b^2*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/32 - (((16*I)/75)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(c*(1 + c^2*x^2)) + (15*b^2*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(64*(1 + c^2*x^2)) - (((2*I)/125)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2))/c - (9*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*ArcSinh[c*x])/(64*c*(1 + c^2*x^2)^(3/2)) + (((2*I)/5)*b*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (3*b*c*f*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)^(3/2)) + (((4*I)/15)*b*c^2*f*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) + (((2*I)/25)*b*c^4*f*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (b*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(8*(1 + c^2*x^2)) - ((I/5)*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c*(1 + c^2*x^2)^(3/2))} -{Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2, x, 23, (((-8*I)/9)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/c + (15*b^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/64 - (b^2*c^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/32 - (((4*I)/27)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2))/c - (15*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x])/(64*c*Sqrt[1 + c^2*x^2]) + (((4*I)/3)*b*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (3*b*c*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (((4*I)/9)*b*c^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (b*c^3*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(8*Sqrt[1 + c^2*x^2]) + (3*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/8 - (c^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/4 - (((2*I)/3)*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c + (5*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^3)/(24*b*c*Sqrt[1 + c^2*x^2])} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x], x, 17, (((-68*I)/9)*b^2*f^3*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*b^2*f^3*x*(1 + c^2*x^2))/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (((2*I)/27)*b^2*f^3*(1 + c^2*x^2)^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b^2*f^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (((22*I)/3)*b*f^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b*c*f^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (((2*I)/9)*b*c^2*f^3*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (((11*I)/3)*f^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*f^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((I/3)*c*f^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (5*f^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2), x, 28, ((-8*I)*a*b*f^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b^2*f^4*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b^2*f^4*x*(1 + c^2*x^2)^2)/(4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b^2*f^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*I)*b^2*f^4*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*c*f^4*x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*f^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*f^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (f^4*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (5*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((32*I)*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (16*b*f^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (16*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b^2*f^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2), x, 25, ((2*I)*a*b*f^5*x*(1 + c^2*x^2)^(5/2))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((2*I)*b^2*f^5*(1 + c^2*x^2)^3)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((2*I)*b^2*f^5*x*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (28*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (I*f^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (5*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((16*I)/3)*b^2*f^5*(1 + c^2*x^2)^(5/2)*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((28*I)/3)*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (8*b*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((4*I)/3)*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]]*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (112*b*f^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (112*b^2*f^5*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x], x, 17, (((68*I)/9)*b^2*d^3*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*b^2*d^3*x*(1 + c^2*x^2))/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (((2*I)/27)*b^2*d^3*(1 + c^2*x^2)^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b^2*d^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (((22*I)/3)*b*d^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*b*c*d^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (((2*I)/9)*b*c^2*d^3*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (((11*I)/3)*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (3*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((I/3)*c*d^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (5*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x], x, 11, ((4*I)*b^2*d^2*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (b^2*d^2*x*(1 + c^2*x^2))/(4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b^2*d^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(4*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((4*I)*b*d^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*d^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(2*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x], x, 8, ((-2*I)*a*b*d*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*b^2*d*(1 + c^2*x^2))/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) - ((2*I)*b^2*d*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (I*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]), x, 2, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]), x, 16, (I*f*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (f*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (f*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((4*I)*b*f*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b*f*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b^2*f*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*b^2*f*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b^2*f*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]), x, 30, (((-2*I)/3)*b^2*f^2*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*f^2*x*(1 + c^2*x^2)^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b^2*f^2*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*f^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*b*f^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*c*f^2*x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*f^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (c^2*f^2*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*f^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((4*I)/3)*b*f^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b*f^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*f^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*b^2*f^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*f^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2), x, 28, ((8*I)*a*b*d^4*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*I)*b^2*d^4*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (b^2*d^4*x*(1 + c^2*x^2)^2)/(4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b^2*d^4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/(4*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b^2*d^4*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b*c*d^4*x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((8*I)*d^4*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*d^4*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*d^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((4*I)*d^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (d^4*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (5*d^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(2*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((32*I)*b*d^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (16*b*d^4*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (16*b^2*d^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (16*b^2*d^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b^2*d^4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2), x, 23, ((2*I)*a*b*d^3*x*(1 + c^2*x^2)^(3/2))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((2*I)*b^2*d^3*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((2*I)*b^2*d^3*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - ((4*I)*d^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (I*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((16*I)*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (8*b^2*d^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (8*b^2*d^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b^2*d^3*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2), x, 19, ((-2*I)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((8*I)*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (4*b^2*d^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (4*b^2*d^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b^2*d^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)), x, 16, ((-I)*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (d*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((4*I)*b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + (2*b^2*d*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b^2*d*(1 + c^2*x^2)^(3/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b^2*d*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)), x, 7, (x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (2*b*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)) - (b^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)), x, 21, ((-I/3)*b^2*f*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*f*x*(1 + c^2*x^2)^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*f*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*b*f*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*f*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (f*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*f*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*f*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*b*f*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (4*b*f*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*f*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b^2*f*(1 + c^2*x^2)^(5/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*f*(1 + c^2*x^2)^(5/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -{((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2), x, 25, ((-2*I)*a*b*d^5*x*(1 + c^2*x^2)^(5/2))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((2*I)*b^2*d^5*(1 + c^2*x^2)^3)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((2*I)*b^2*d^5*x*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (28*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (I*d^5*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (5*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (112*b*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (112*b^2*d^5*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (8*b*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((16*I)/3)*b^2*d^5*(1 + c^2*x^2)^(5/2)*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((28*I)/3)*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((4*I)/3)*d^5*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2), x, 21, (8*d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^3)/(3*b*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (32*b*d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (32*b^2*d^4*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (4*b*d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((8*I)/3)*b^2*d^4*(1 + c^2*x^2)^(5/2)*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((8*I)/3)*d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*d^4*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2), x, 20, (d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (4*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (4*b^2*d^3*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*b*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((4*I)/3)*b^2*d^3*(1 + c^2*x^2)^(5/2)*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*d^3*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2*Sec[Pi/4 + (I/2)*ArcSinh[c*x]]^2*Tan[Pi/4 + (I/2)*ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)), x, 30, (((2*I)/3)*b^2*d^2*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*d^2*x*(1 + c^2*x^2)^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b^2*d^2*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*b*d^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b*c*d^2*x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (((2*I)/3)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (c^2*d^2*x^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((4*I)/3)*b*d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b*d^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*b^2*d^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*d^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*d^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)), x, 21, ((I/3)*b^2*d*(1 + c^2*x^2)^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*d*x*(1 + c^2*x^2)^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*d*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + ((I/3)*b*d*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - ((I/3)*d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (d*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*d*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (((2*I)/3)*b*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (4*b*d*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b^2*d*(1 + c^2*x^2)^(5/2)*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (b^2*d*(1 + c^2*x^2)^(5/2)*PolyLog[2, I*E^ArcSinh[c*x]])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*d*(1 + c^2*x^2)^(5/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} -{(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)), x, 10, -(b^2*x*(1 + c^2*x^2)^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (b*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(3*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) + (2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (4*b*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)) - (2*b^2*(1 + c^2*x^2)^(5/2)*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*c*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x^2)^4*(a + b*ArcSinh[c*x]), x, 5, -((b*(315*c^8*d^4 - 420*c^6*d^3*e + 378*c^4*d^2*e^2 - 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 + c^2*x^2])/(315*c^9)) - (4*b*e*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*(1 + c^2*x^2)^(3/2))/(945*c^9) - (2*b*e^2*(63*c^4*d^2 - 90*c^2*d*e + 35*e^2)*(1 + c^2*x^2)^(5/2))/(525*c^9) - (4*b*(9*c^2*d - 7*e)*e^3*(1 + c^2*x^2)^(7/2))/(441*c^9) - (b*e^4*(1 + c^2*x^2)^(9/2))/(81*c^9) + d^4*x*(a + b*ArcSinh[c*x]) + (4/3)*d^3*e*x^3*(a + b*ArcSinh[c*x]) + (6/5)*d^2*e^2*x^5*(a + b*ArcSinh[c*x]) + (4/7)*d*e^3*x^7*(a + b*ArcSinh[c*x]) + (1/9)*e^4*x^9*(a + b*ArcSinh[c*x])} -{(d + e*x^2)^3*(a + b*ArcSinh[c*x]), x, 5, -((b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Sqrt[1 + c^2*x^2])/(35*c^7)) - (b*e*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*(1 + c^2*x^2)^(3/2))/(105*c^7) - (3*b*(7*c^2*d - 5*e)*e^2*(1 + c^2*x^2)^(5/2))/(175*c^7) - (b*e^3*(1 + c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSinh[c*x]) + d^2*e*x^3*(a + b*ArcSinh[c*x]) + (3/5)*d*e^2*x^5*(a + b*ArcSinh[c*x]) + (1/7)*e^3*x^7*(a + b*ArcSinh[c*x])} -{(d + e*x^2)^2*(a + b*ArcSinh[c*x]), x, 5, -(b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Sqrt[1 + c^2*x^2])/(15*c^5) - (2*b*(5*c^2*d - 3*e)*e*(1 + c^2*x^2)^(3/2))/(45*c^5) - (b*e^2*(1 + c^2*x^2)^(5/2))/(25*c^5) + d^2*x*(a + b*ArcSinh[c*x]) + (2*d*e*x^3*(a + b*ArcSinh[c*x]))/3 + (e^2*x^5*(a + b*ArcSinh[c*x]))/5} -{(d + e*x^2)*(a + b*ArcSinh[c*x]), x, 4, -(b*(3*c^2*d - e)*Sqrt[1 + c^2*x^2])/(3*c^3) - (b*e*(1 + c^2*x^2)^(3/2))/(9*c^3) + d*x*(a + b*ArcSinh[c*x]) + (e*x^3*(a + b*ArcSinh[c*x]))/3} -{a + b*ArcSinh[c*x], x, 3, a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]} -{(a + b*ArcSinh[c*x])/(d + e*x^2), x, 18, ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcSinh[c*x])/(d + e*x^2)^2, x, 26, -((a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x))) + (a + b*ArcSinh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*ArcTan[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*Sqrt[e]) - (b*c*ArcTan[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/(4*d*Sqrt[c^2*d - e]*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[c*x])*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[c*x])*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e])} - - -{(d + e*x^2)^3*(a + b*ArcSinh[c*x])^2, x, 26, 2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) + (16*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(1225*c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d^2*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^5) + (32*b*e^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c) + (8*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(49*c) + d^3*x*(a + b*ArcSinh[c*x])^2 + d^2*e*x^3*(a + b*ArcSinh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSinh[c*x])^2)/5 + (e^3*x^7*(a + b*ArcSinh[c*x])^2)/7} -{(d + e*x^2)^2*(a + b*ArcSinh[c*x])^2, x, 17, 2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) + (16*b^2*e^2*x)/(75*c^4) + (4*b^2*d*e*x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2) + (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (8*b*d*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(75*c^5) - (4*b*d*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) + (8*b*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(75*c^3) - (2*b*e^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + d^2*x*(a + b*ArcSinh[c*x])^2 + (2*d*e*x^3*(a + b*ArcSinh[c*x])^2)/3 + (e^2*x^5*(a + b*ArcSinh[c*x])^2)/5} -{(d + e*x^2)^1*(a + b*ArcSinh[c*x])^2, x, 10, 2*b^2*d*x - (4*b^2*e*x)/(9*c^2) + (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3) - (2*b*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) + d*x*(a + b*ArcSinh[c*x])^2 + (e*x^3*(a + b*ArcSinh[c*x])^2)/3} -{(d + e*x^2)^0*(a + b*ArcSinh[c*x])^2, x, 3, 2*b^2*x - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + x*(a + b*ArcSinh[c*x])^2} -{(a + b*ArcSinh[c*x])^2/(d + e*x^2), x, 22, ((a + b*ArcSinh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSinh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSinh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcSinh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d + e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d + e])])/(Sqrt[-d]*Sqrt[e])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x^2)^3/(a + b*ArcSinh[c*x]), x, 42, (d^3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (3*d^2*e*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) + (3*d*e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) - (5*e^3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(64*b*c^7) + (3*d^2*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) - (9*d*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (9*e^3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) + (3*d*e^2*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (5*e^3*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) + (e^3*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (d^3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) + (3*d^2*e*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) - (3*d*e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) + (5*e^3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(64*b*c^7) - (3*d^2*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) + (9*d*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (9*e^3*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (3*d*e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (5*e^3*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7) - (e^3*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(64*b*c^7)} -{(d + e*x^2)^2/(a + b*ArcSinh[c*x]), x, 27, (d^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (d*e*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(2*b*c^3) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) + (d*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(2*b*c^3) - (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) + (e^2*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (d^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) + (d*e*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(2*b*c^3) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b*c^5) - (d*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(2*b*c^3) + (3*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5) - (e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b*c^5)} -{(d + e*x^2)^1/(a + b*ArcSinh[c*x]), x, 15, (d*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (e*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) + (e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3) - (d*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) + (e*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b*c^3) - (e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b*c^3)} -{(d + e*x^2)^0/(a + b*ArcSinh[c*x]), x, 4, (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c)} -{1/((d + e*x^2)^1*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSinh[c*x])), x]} -{1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])), x]} - - -{(d + e*x^2)^2/(a + b*ArcSinh[c*x])^2, x, 26, -((d^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (2*d*e*x^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (e^2*x^4*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (d^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (d*e*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(2*b^2*c^3) - (e^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b^2*c^5) - (3*d*e*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(2*b^2*c^3) + (9*e^2*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c^5) - (5*e^2*CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b^2*c^5) + (d^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (d*e*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(2*b^2*c^3) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(2*b^2*c^3) - (9*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^5) + (5*e^2*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^5)} -{(d + e*x^2)^1/(a + b*ArcSinh[c*x])^2, x, 15, -((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (d*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (e*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b^2*c^3) - (3*e*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*c^3) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^3) + (3*e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x^2)^0/(a + b*ArcSinh[c*x])^2, x, 5, -(Sqrt[1 + c^2*x^2]/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c)} -{1/((d + e*x^2)^1*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^2), x]} -{1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]], x, 42, d^2*x*Sqrt[a + b*ArcSinh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcSinh[c*x]])/5 + (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3) + (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5) + (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5) + (Sqrt[b]*e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sqrt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) + (Sqrt[b]*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*a)/b))} -{(d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]], x, 23, d*x*Sqrt[a + b*ArcSinh[c*x]] + (e*x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + (Sqrt[b]*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c^3) + (Sqrt[b]*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3) - (Sqrt[b]*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) + (Sqrt[b]*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(48*c^3*E^((3*a)/b))} -{Sqrt[a + b*ArcSinh[c*x]], x, 7, x*Sqrt[a + b*ArcSinh[c*x]] + (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*c*E^(a/b))} -{Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2), x, 0, Unintegrable[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2), x]} -{Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2)^2, x, 0, Unintegrable[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2)^2, x]} - - -{(d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2), x, 32, (-3*b*d*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + (b*e*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(3*c^3) - (b*e*x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(6*c) + d*x*(a + b*ArcSinh[c*x])^(3/2) + (e*x^3*(a + b*ArcSinh[c*x])^(3/2))/3 + (3*b^(3/2)*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) - (3*b^(3/2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3) + (b^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c*E^(a/b)) - (3*b^(3/2)*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(32*c^3*E^(a/b)) + (b^(3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(96*c^3*E^((3*a)/b))} -{(a + b*ArcSinh[c*x])^(3/2), x, 8, (-3*b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + x*(a + b*ArcSinh[c*x])^(3/2) + (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c*E^(a/b))} -{(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2), x, 0, Unintegrable[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2), x]} -{(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2)^2, x, 0, Unintegrable[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x^2)^2/Sqrt[a + b*ArcSinh[c*x]], x, 39, (d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5) + (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) - (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5*E^(a/b)) + (d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) - (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((5*a)/b))} -{(d + e*x^2)/Sqrt[a + b*ArcSinh[c*x]], x, 21, (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3) + (e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) - (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))} -{1/Sqrt[a + b*ArcSinh[c*x]], x, 6, (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b))} -{1/((d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]]), x, 0, Unintegrable[1/((d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]]), x]} -{1/((d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]]), x, 0, Unintegrable[1/((d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]]), x]} - - -{(d + e*x^2)/(a + b*ArcSinh[c*x])^(3/2), x, 21, (-2*d*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (2*e*x^2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) - (e*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b)) - (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3*E^(a/b)) + (e*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3*E^((3*a)/b))} -{(a + b*ArcSinh[c*x])^(-3/2), x, 7, (-2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt[a + b*ArcSinh[c*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b))} -{1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)), x]} -{1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^(p/2) (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]), x]} -{(a + b*ArcSinh[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcSinh[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcSinh[c*x])/(d + e*x^2)^(3/2), x, 6, (x*(a + b*ArcSinh[c*x]))/(d*Sqrt[d + e*x^2]) - (b*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(d*Sqrt[e])} -{(a + b*ArcSinh[c*x])/(d + e*x^2)^(5/2), x, 7, -(b*c*Sqrt[1 + c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[d + e*x^2]) + (x*(a + b*ArcSinh[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[d + e*x^2]) - (2*b*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(3*d^2*Sqrt[e])} -{(a + b*ArcSinh[c*x])/(d + e*x^2)^(7/2), x, 8, -((b*c*Sqrt[1 + c^2*x^2])/(15*d*(c^2*d - e)*(d + e*x^2)^(3/2))) - (2*b*c*(3*c^2*d - 2*e)*Sqrt[1 + c^2*x^2])/(15*d^2*(c^2*d - e)^2*Sqrt[d + e*x^2]) + (x*(a + b*ArcSinh[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcSinh[c*x]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcSinh[c*x]))/(15*d^3*Sqrt[d + e*x^2]) - (8*b*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(15*d^3*Sqrt[e])} - - -{Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x]} -{(a + b*ArcSinh[c*x])^2/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcSinh[c*x])^2/Sqrt[d + e*x^2], x]} -{(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(3/2), x, 0, Unintegrable[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(3/2), x]} -{(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(5/2), x, 0, Unintegrable[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(5/2), x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x]), x]} -{1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])), x]} - - -{Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x])^2, x]} -{1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 Inverse hyperbolic sine functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 Inverse hyperbolic sine functions.m deleted file mode 100644 index f6699ac..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.1 Inverse hyperbolic sine/7.1.5 Inverse hyperbolic sine functions.m +++ /dev/null @@ -1,737 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcSinh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m ArcSinh[c x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcSinh[c*x]^1/(d + e*x), x, 8, -(ArcSinh[c*x]^2/(2*e)) + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))]/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))]/e} - - -{ArcSinh[c*x]^2/(d + e*x), x, 10, -(ArcSinh[c*x]^3/(3*e)) + (ArcSinh[c*x]^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (2*ArcSinh[c*x]*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (2*ArcSinh[c*x]*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e - (2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e - (2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e} - - -{ArcSinh[c*x]^3/(d + e*x), x, 12, -(ArcSinh[c*x]^4/(4*e)) + (ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (3*ArcSinh[c*x]^2*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (3*ArcSinh[c*x]^2*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e - (6*ArcSinh[c*x]*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e - (6*ArcSinh[c*x]*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e + (6*PolyLog[4, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (6*PolyLog[4, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcSinh[c*x]), x, 5, -((7*b*d*(d + e*x)^2*Sqrt[1 + c^2*x^2])/(48*c)) - (b*(d + e*x)^3*Sqrt[1 + c^2*x^2])/(16*c) - (b*(4*d*(19*c^2*d^2 - 16*e^2) + e*(26*c^2*d^2 - 9*e^2)*x)*Sqrt[1 + c^2*x^2])/(96*c^3) - (b*(8*c^4*d^4 - 24*c^2*d^2*e^2 + 3*e^4)*ArcSinh[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcSinh[c*x]))/(4*e)} -{(d + e*x)^2*(a + b*ArcSinh[c*x]), x, 4, -((b*(d + e*x)^2*Sqrt[1 + c^2*x^2])/(9*c)) - (b*(4*(4*c^2*d^2 - e^2) + 5*c^2*d*e*x)*Sqrt[1 + c^2*x^2])/(18*c^3) - (b*d*(2*d^2 - (3*e^2)/c^2)*ArcSinh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcSinh[c*x]))/(3*e)} -{(d + e*x)^1*(a + b*ArcSinh[c*x]), x, 4, -((3*b*d*Sqrt[1 + c^2*x^2])/(4*c)) - (b*(d + e*x)*Sqrt[1 + c^2*x^2])/(4*c) - (b*(2*d^2 - e^2/c^2)*ArcSinh[c*x])/(4*e) + ((d + e*x)^2*(a + b*ArcSinh[c*x]))/(2*e)} -{(d + e*x)^0*(a + b*ArcSinh[c*x]), x, 3, a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]} -{(a + b*ArcSinh[c*x])/(d + e*x)^1, x, 8, -((a + b*ArcSinh[c*x])^2/(2*b*e)) + ((a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + ((a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (b*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (b*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e} -{(a + b*ArcSinh[c*x])/(d + e*x)^2, x, 3, -((a + b*ArcSinh[c*x])/(e*(d + e*x))) - (b*c*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(e*Sqrt[c^2*d^2 + e^2])} -{(a + b*ArcSinh[c*x])/(d + e*x)^3, x, 4, -((b*c*Sqrt[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)*(d + e*x))) - (a + b*ArcSinh[c*x])/(2*e*(d + e*x)^2) - (b*c^3*d*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(2*e*(c^2*d^2 + e^2)^(3/2))} -{(a + b*ArcSinh[c*x])/(d + e*x)^4, x, 5, -((b*c*Sqrt[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)*(d + e*x)^2)) - (b*c^3*d*Sqrt[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)^2*(d + e*x)) - (a + b*ArcSinh[c*x])/(3*e*(d + e*x)^3) - (b*c^3*(2*c^2*d^2 - e^2)*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(6*e*(c^2*d^2 + e^2)^(5/2))} - - -{(d + e*x)^3*(a + b*ArcSinh[c*x])^2, x, 18, 2*b^2*d^3*x - (4*b^2*d*e^2*x)/(3*c^2) + (3/4)*b^2*d^2*e*x^2 - (3*b^2*e^3*x^2)/(32*c^2) + (2/9)*b^2*d*e^2*x^3 + (1/32)*b^2*e^3*x^4 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c) - (b*e^3*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) - (d^4*(a + b*ArcSinh[c*x])^2)/(4*e) + (3*d^2*e*(a + b*ArcSinh[c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcSinh[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSinh[c*x])^2)/(4*e)} -{(d + e*x)^2*(a + b*ArcSinh[c*x])^2, x, 13, 2*b^2*d^2*x - (4*b^2*e^2*x)/(9*c^2) + (1/2)*b^2*d*e*x^2 + (2/27)*b^2*e^2*x^3 - (2*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3) - (b*d*e*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c - (2*b*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) - (d^3*(a + b*ArcSinh[c*x])^2)/(3*e) + (d*e*(a + b*ArcSinh[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a + b*ArcSinh[c*x])^2)/(3*e)} -{(d + e*x)^1*(a + b*ArcSinh[c*x])^2, x, 9, 2*b^2*d*x + (1/4)*b^2*e*x^2 - (2*b*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c - (b*e*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c) - (d^2*(a + b*ArcSinh[c*x])^2)/(2*e) + (e*(a + b*ArcSinh[c*x])^2)/(4*c^2) + ((d + e*x)^2*(a + b*ArcSinh[c*x])^2)/(2*e)} -{(d + e*x)^0*(a + b*ArcSinh[c*x])^2, x, 3, 2*b^2*x - (2*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + x*(a + b*ArcSinh[c*x])^2} -{(a + b*ArcSinh[c*x])^2/(d + e*x)^1, x, 10, -((a + b*ArcSinh[c*x])^3/(3*b*e)) + ((a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + ((a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (2*b*(a + b*ArcSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (2*b*(a + b*ArcSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e} -{(a + b*ArcSinh[c*x])^2/(d + e*x)^2, x, 10, -((a + b*ArcSinh[c*x])^2/(e*(d + e*x))) + (2*b*c*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/(e*Sqrt[c^2*d^2 + e^2]) - (2*b*c*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/(e*Sqrt[c^2*d^2 + e^2]) + (2*b^2*c*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/(e*Sqrt[c^2*d^2 + e^2]) - (2*b^2*c*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(e*Sqrt[c^2*d^2 + e^2])} -{(a + b*ArcSinh[c*x])^2/(d + e*x)^3, x, 13, -((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/((c^2*d^2 + e^2)*(d + e*x))) - (a + b*ArcSinh[c*x])^2/(2*e*(d + e*x)^2) + (b*c^3*d*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/(e*(c^2*d^2 + e^2)^(3/2)) - (b*c^3*d*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/(e*(c^2*d^2 + e^2)^(3/2)) + (b^2*c^2*Log[d + e*x])/(e*(c^2*d^2 + e^2)) + (b^2*c^3*d*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/(e*(c^2*d^2 + e^2)^(3/2)) - (b^2*c^3*d*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(e*(c^2*d^2 + e^2)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x)^3/(a + b*ArcSinh[c*x]), x, 27, (d^3*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(b*c) - (3*d*e^2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(4*b*c^3) + (3*d*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(4*b*c^3) - (3*d^2*e*CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) + (e^3*CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b])/(4*b*c^4) - (e^3*CoshIntegral[(4*a)/b + 4*ArcSinh[c*x]]*Sinh[(4*a)/b])/(8*b*c^4) - (d^3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b*c) + (3*d*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(4*b*c^3) + (3*d^2*e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]])/(2*b*c^2) - (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]])/(4*b*c^4) - (3*d*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(4*b*c^3) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcSinh[c*x]])/(8*b*c^4)} -{(d + e*x)^2/(a + b*ArcSinh[c*x]), x, 17, (d^2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(b*c) - (e^2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(4*b*c^3) + (e^2*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(4*b*c^3) - (d*e*CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b])/(b*c^2) - (d^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b*c) + (e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(4*b*c^3) + (d*e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]])/(b*c^2) - (e^2*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(4*b*c^3)} -{(d + e*x)^1/(a + b*ArcSinh[c*x]), x, 11, (d*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(b*c) - (e*CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) - (d*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b*c) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]])/(2*b*c^2)} -{(d + e*x)^0/(a + b*ArcSinh[c*x]), x, 4, (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(b*c) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b*c)} -{1/((d + e*x)^1*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcSinh[c*x])), x]} -{1/((d + e*x)^2*(a + b*ArcSinh[c*x])), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcSinh[c*x])), x]} - - -{(d + e*x)^2/(a + b*ArcSinh[c*x])^2, x, 19, -((d^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (2*d*e*x*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) - (e^2*x^2*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) + (2*d*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2) - (d^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (e^2*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(4*b^2*c^3) - (3*e^2*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*c^3) + (d^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(4*b^2*c^3) - (2*d*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x)^1/(a + b*ArcSinh[c*x])^2, x, 11, -((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x])) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2) - (d*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2)} -{(d + e*x)^0/(a + b*ArcSinh[c*x])^2, x, 5, -(Sqrt[1 + c^2*x^2]/(b*c*(a + b*ArcSinh[c*x]))) - (CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(b^2*c) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c)} -{1/((d + e*x)^1*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcSinh[c*x])^2), x]} -{1/((d + e*x)^2*(a + b*ArcSinh[c*x])^2), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcSinh[c*x])^2), x]} - - -(* ::Subsection:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n with m symbolic*) - - -{(d + e*x)^m*(a + b*ArcSinh[c*x])^2, x, 1, ((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x])^2)/(e*(1 + m)) - (2*b*c*Unintegrable[((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2], x])/(e*(1 + m))} -{(d + e*x)^m*(a + b*ArcSinh[c*x])^1, x, 3, -((b*c*(d + e*x)^(2 + m)*Sqrt[1 - (d + e*x)/(d - e/Sqrt[-c^2])]*Sqrt[1 - (d + e*x)/(d + e/Sqrt[-c^2])]*AppellF1[2 + m, 1/2, 1/2, 3 + m, (d + e*x)/(d - e/Sqrt[-c^2]), (d + e*x)/(d + e/Sqrt[-c^2])])/(e^2*(1 + m)*(2 + m)*Sqrt[1 + c^2*x^2])) + ((d + e*x)^(1 + m)*(a + b*ArcSinh[c*x]))/(e*(1 + m))} -{(d + e*x)^m/(a + b*ArcSinh[c*x])^1, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcSinh[c*x]), x]} -{(d + e*x)^m/(a + b*ArcSinh[c*x])^2, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcSinh[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x^2)^p (a+b ArcSinh[c x])^n where e=c^2 d*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^3, x, 16, -((b*f^2*g*x*Sqrt[d + c^2*d*x^2])/(c*Sqrt[1 + c^2*x^2])) + (2*b*g^3*x*Sqrt[d + c^2*d*x^2])/(15*c^3*Sqrt[1 + c^2*x^2]) - (b*c*f^3*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) - (3*b*f*g^2*x^2*Sqrt[d + c^2*d*x^2])/(16*c*Sqrt[1 + c^2*x^2]) - (b*c*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(3*Sqrt[1 + c^2*x^2]) - (b*g^3*x^3*Sqrt[d + c^2*d*x^2])/(45*c*Sqrt[1 + c^2*x^2]) - (3*b*c*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*c*g^3*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + (1/2)*f^3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (3/4)*f*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (f^2*g*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/c^2 - (g^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^4) + (g^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^4) + (f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2]) - (3*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c^3*Sqrt[1 + c^2*x^2])} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^2, x, 13, -((2*b*f*g*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2])) - (b*c*f^2*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) - (b*g^2*x^2*Sqrt[d + c^2*d*x^2])/(16*c*Sqrt[1 + c^2*x^2]) - (2*b*c*f*g*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) - (b*c*g^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) + (1/2)*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (1/4)*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (2*f*g*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (f^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2]) - (g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c^3*Sqrt[1 + c^2*x^2])} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^1, x, 8, -((b*g*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2])) - (b*c*f*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x^2]) - (b*c*g*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + (1/2)*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (g*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2])} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])/(f + g*x)^1, x, 22, (a*Sqrt[d + c^2*d*x^2])/g - (b*c*x*Sqrt[d + c^2*d*x^2])/(g*Sqrt[1 + c^2*x^2]) + (b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g - (c*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g*Sqrt[1 + c^2*x^2]) - ((1 + (c^2*f^2)/g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g*x)*Sqrt[1 + c^2*x^2]) + (Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g*x)) - (a*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^2*Sqrt[1 + c^2*x^2]) + (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[1 + c^2*x^2]) + (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[1 + c^2*x^2]) - (b*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[1 + c^2*x^2])} -{Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 35, -((a*Sqrt[d + c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g*(f + g*x)) + (a*c^3*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^2*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]) + (b*c^3*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(2*g^2*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]) - ((g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(c^2*f^2 + g^2)*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*(f + g*x)^2) + (a*c^2*f*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) - (b*c^2*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (b*c^2*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 + c^2*x^2]) - (b*c^2*f*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (b*c^2*f*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])} - - -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])*(f + g*x)^3, x, 24, -((3*b*d*f^2*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2])) + (2*b*d*g^3*x*Sqrt[d + c^2*d*x^2])/(35*c^3*Sqrt[1 + c^2*x^2]) - (5*b*c*d*f^3*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (3*b*d*f*g^2*x^2*Sqrt[d + c^2*d*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(5*Sqrt[1 + c^2*x^2]) - (b*d*g^3*x^3*Sqrt[d + c^2*d*x^2])/(105*c*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f^3*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (7*b*c*d*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(32*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d*f^2*g*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) - (8*b*c*d*g^3*x^5*Sqrt[d + c^2*d*x^2])/(175*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f*g^2*x^6*Sqrt[d + c^2*d*x^2])/(12*Sqrt[1 + c^2*x^2]) - (b*c^3*d*g^3*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) + (3/8)*d*f^3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*d*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (3/8)*d*f*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/4)*d*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/2)*d*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*d*f^2*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) - (d*g^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^4) + (d*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4) + (3*d*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2]) - (3*d*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])*(f + g*x)^2, x, 20, -((2*b*d*f*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2])) - (5*b*c*d*f^2*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*d*g^2*x^2*Sqrt[d + c^2*d*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (4*b*c*d*f*g*x^3*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (7*b*c*d*g^2*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*f*g*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) - (b*c^3*d*g^2*x^6*Sqrt[d + c^2*d*x^2])/(36*Sqrt[1 + c^2*x^2]) + (3/8)*d*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (d*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (1/8)*d*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/4)*d*f^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/6)*d*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (2*d*f*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) + (3*d*f^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2]) - (d*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])*(f + g*x)^1, x, 12, -((b*d*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2])) - (5*b*c*d*f*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (2*b*c*d*g*x^3*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (b*c^3*d*g*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + (3/8)*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/4)*d*f*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (d*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*c^2) + (3*d*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])/(f + g*x)^1, x, 29, (a*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/g^3 - (b*c*d*x*Sqrt[d + c^2*d*x^2])/(3*g*Sqrt[1 + c^2*x^2]) - (b*c*d*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2])/(g^3*Sqrt[1 + c^2*x^2]) + (b*c^3*d*f*x^2*Sqrt[d + c^2*d*x^2])/(4*g^2*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^3*Sqrt[d + c^2*d*x^2])/(9*g*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^3 - (c^2*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + (d*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g) - (c*d*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^2*Sqrt[1 + c^2*x^2]) - (c*d*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g^3*Sqrt[1 + c^2*x^2]) - (d*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)) - (a*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2]) - (b*d*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2])} -(* {(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 71, -((2*a*c^2*d*f*Sqrt[d + c^2*d*x^2])/g^3) - (a*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/(g^3*(f + g*x)) + (2*b*c^3*d*f*x*Sqrt[d + c^2*d*x^2])/(g^3*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^2*Sqrt[d + c^2*d*x^2])/(4*g^2*Sqrt[1 + c^2*x^2]) - (2*b*c^2*d*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^3 - (b*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^3*(f + g*x)) + (a*c^3*d*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^4*Sqrt[1 + c^2*x^2]) + (b*c^3*d*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(2*g^4*Sqrt[1 + c^2*x^2]) + (c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + (c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^2*Sqrt[1 + c^2*x^2]) + (c^3*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^3*Sqrt[1 + c^2*x^2]) - (d*(g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (c*d*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^4*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)^2) - (c*d*f*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^2*(f + g*x)) + (3*a*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^4*Sqrt[1 + c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*c*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^4*Sqrt[1 + c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2])} *) - - -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])*(f + g*x)^3, x, 30, -((3*b*d^2*f^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2])) + (2*b*d^2*g^3*x*Sqrt[d + c^2*d*x^2])/(63*c^3*Sqrt[1 + c^2*x^2]) - (25*b*c*d^2*f^3*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (15*b*d^2*f*g^2*x^2*Sqrt[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (b*d^2*g^3*x^3*Sqrt[d + c^2*d*x^2])/(189*c*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^3*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d + c^2*d*x^2])/(256*Sqrt[1 + c^2*x^2]) - (9*b*c^3*d^2*f^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g^3*x^5*Sqrt[d + c^2*d*x^2])/(21*Sqrt[1 + c^2*x^2]) - (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (19*b*c^3*d^2*g^3*x^7*Sqrt[d + c^2*d*x^2])/(441*Sqrt[1 + c^2*x^2]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^3*x^9*Sqrt[d + c^2*d*x^2])/(81*Sqrt[1 + c^2*x^2]) - (b*d^2*f^3*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5/16)*d^2*f^3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (15*d^2*f*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (15/64)*d^2*f*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/24)*d^2*f^3*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/16)*d^2*f*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/6)*d^2*f^3*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3/8)*d^2*f*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (3*d^2*f^2*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) - (d^2*g^3*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^4) + (d^2*g^3*(1 + c^2*x^2)^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(9*c^4) + (5*d^2*f^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1 + c^2*x^2]) - (15*d^2*f*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])*(f + g*x)^2, x, 26, -((2*b*d^2*f*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2])) - (25*b*c*d^2*f^2*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (5*b*d^2*g^2*x^2*Sqrt[d + c^2*d*x^2])/(256*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d^2*f*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f^2*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (59*b*c*d^2*g^2*x^4*Sqrt[d + c^2*d*x^2])/(768*Sqrt[1 + c^2*x^2]) - (6*b*c^3*d^2*f*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (17*b*c^3*d^2*g^2*x^6*Sqrt[d + c^2*d*x^2])/(288*Sqrt[1 + c^2*x^2]) - (2*b*c^5*d^2*f*g*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g^2*x^8*Sqrt[d + c^2*d*x^2])/(64*Sqrt[1 + c^2*x^2]) - (b*d^2*f^2*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5/16)*d^2*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5*d^2*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(128*c^2) + (5/64)*d^2*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/24)*d^2*f^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/48)*d^2*g^2*x^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/6)*d^2*f^2*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/8)*d^2*g^2*x^3*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (2*d^2*f*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1 + c^2*x^2]) - (5*d^2*g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(256*b*c^3*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])*(f + g*x)^1, x, 14, -((b*d^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2])) - (25*b*c*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g*x^7*Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*d^2*f*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5/16)*d^2*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (5/24)*d^2*f*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (1/6)*d^2*f*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (d^2*g*(1 + c^2*x^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c*Sqrt[1 + c^2*x^2])} -{(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])/(f + g*x)^1, x, 37, (a*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2])/g^5 + (2*b*c*d^2*x*Sqrt[d + c^2*d*x^2])/(15*g*Sqrt[1 + c^2*x^2]) - (b*c*d^2*(c^2*f^2 + g^2)^2*x*Sqrt[d + c^2*d*x^2])/(g^5*Sqrt[1 + c^2*x^2]) - (b*c*d^2*(c^2*f^2 + 2*g^2)*x*Sqrt[d + c^2*d*x^2])/(3*g^3*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*f*(c^2*f^2 + 2*g^2)*x^2*Sqrt[d + c^2*d*x^2])/(4*g^4*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*x^3*Sqrt[d + c^2*d*x^2])/(45*g*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*(c^2*f^2 + 2*g^2)*x^3*Sqrt[d + c^2*d*x^2])/(9*g^3*Sqrt[1 + c^2*x^2]) + (b*c^5*d^2*f*x^4*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^5*Sqrt[d + c^2*d*x^2])/(25*g*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^5 - (c^2*d^2*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 + 2*g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*g^2) - (d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g) + (d^2*(c^2*f^2 + 2*g^2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g^3) + (d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*g) + (c*d^2*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*g^2*Sqrt[1 + c^2*x^2]) - (c*d^2*f*(c^2*f^2 + 2*g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^4*Sqrt[1 + c^2*x^2]) - (c*d^2*(c^2*f^2 + g^2)^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g^5*Sqrt[1 + c^2*x^2]) - (d^2*(c^2*f^2 + g^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d^2*(c^2*f^2 + g^2)^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) - (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2]) - (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2])} -(* {(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 78, -((4*a*c^2*d^2*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/g^5) - (a*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2])/(g^5*(f + g*x)) + (2*b*c^3*d^2*f*x*Sqrt[d + c^2*d*x^2])/(3*g^3*Sqrt[1 + c^2*x^2]) + (4*b*c^3*d^2*f*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2])/(g^5*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*x^2*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*(3*c^2*f^2 + 2*g^2)*x^2*Sqrt[d + c^2*d*x^2])/(4*g^4*Sqrt[1 + c^2*x^2]) + (2*b*c^5*d^2*f*x^3*Sqrt[d + c^2*d*x^2])/(9*g^3*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (4*b*c^2*d^2*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^5 - (b*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^5*(f + g*x)) + (a*c^3*d^2*f^2*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^6*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*f^2*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(2*g^6*Sqrt[1 + c^2*x^2]) + (c^2*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*g^2) + (c^2*d^2*(3*c^2*f^2 + 2*g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^4) + (c^4*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*g^2) - (2*c^2*d^2*f*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g^3) - (c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*g^2*Sqrt[1 + c^2*x^2]) + (c*d^2*(3*c^2*f^2 + 2*g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^4*Sqrt[1 + c^2*x^2]) + (2*c^3*d^2*f*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^5*Sqrt[1 + c^2*x^2]) - (d^2*(c^2*f^2 + g^2)*(g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (2*c*d^2*f*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^6*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d^2*(c^2*f^2 + g^2)^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)^2) - (2*c*d^2*f*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^4*(f + g*x)) + (5*a*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^6*Sqrt[1 + c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*c*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^6*Sqrt[1 + c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2])} *) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^3, x, 13, -((3*b*f^2*g*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2])) + (2*b*g^3*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (3*b*f*g^2*x^2*Sqrt[1 + c^2*x^2])/(4*c*Sqrt[d + c^2*d*x^2]) - (b*g^3*x^3*Sqrt[1 + c^2*x^2])/(9*c*Sqrt[d + c^2*d*x^2]) + (3*f^2*g*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c^2*Sqrt[d + c^2*d*x^2]) - (2*g^3*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*c^4*Sqrt[d + c^2*d*x^2]) + (3*f*g^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*c^2*Sqrt[d + c^2*d*x^2]) + (g^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*c^2*Sqrt[d + c^2*d*x^2]) + (f^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2]) - (3*f*g^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^3*Sqrt[d + c^2*d*x^2])} -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^2, x, 9, -((2*b*f*g*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2])) - (b*g^2*x^2*Sqrt[1 + c^2*x^2])/(4*c*Sqrt[d + c^2*d*x^2]) + (2*f*g*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c^2*Sqrt[d + c^2*d*x^2]) + (g^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*c^2*Sqrt[d + c^2*d*x^2]) + (f^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2]) - (g^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c^3*Sqrt[d + c^2*d*x^2])} -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^1, x, 6, -((b*g*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2])) + (g*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c^2*Sqrt[d + c^2*d*x^2]) + (f*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2])} -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*(f + g*x)^0, x, 1, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2])} -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])/(f + g*x)^1, x, 10, (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2])} -{1/Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 13, -((g*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/((c^2*f^2 + g^2)*(f + g*x)*Sqrt[d + c^2*d*x^2])) + (c^2*f*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/((c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]) - (c^2*f*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/((c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[1 + c^2*x^2]*Log[f + g*x])/((c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]) + (b*c^2*f*Sqrt[1 + c^2*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/((c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]) - (b*c^2*f*Sqrt[1 + c^2*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/((c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d+e x^2)^p (a+b ArcSinh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^n*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2], x, 0, Unintegrable[((a + b*ArcSinh[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 + c^2*x^2], x]} - -(* {Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^3/Sqrt[1 + c^2*x^2], x, 18, (m*(a + b*ArcSinh[c*x])^5)/(20*b^2*c) - (m*(a + b*ArcSinh[c*x])^4*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(4*b*c) - (m*(a + b*ArcSinh[c*x])^4*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(4*b*c) + ((a + b*ArcSinh[c*x])^4*Log[h*(f + g*x)^m])/(4*b*c) - (m*(a + b*ArcSinh[c*x])^3*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*(a + b*ArcSinh[c*x])^3*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (3*b*m*(a + b*ArcSinh[c*x])^2*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (3*b*m*(a + b*ArcSinh[c*x])^2*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c - (6*b^2*m*(a + b*ArcSinh[c*x])*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (6*b^2*m*(a + b*ArcSinh[c*x])*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (6*b^3*m*PolyLog[5, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (6*b^3*m*PolyLog[5, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c} *) -{Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^2/Sqrt[1 + c^2*x^2], x, 13, (m*(a + b*ArcSinh[c*x])^4)/(12*b^2*c) - (m*(a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(3*b*c) - (m*(a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(3*b*c) + ((a + b*ArcSinh[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c) - (m*(a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*(a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (2*b*m*(a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (2*b*m*(a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c - (2*b^2*m*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (2*b^2*m*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^1/Sqrt[1 + c^2*x^2], x, 11, (m*(a + b*ArcSinh[c*x])^3)/(6*b^2*c) - (m*(a + b*ArcSinh[c*x])^2*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(2*b*c) - (m*(a + b*ArcSinh[c*x])^2*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(2*b*c) + ((a + b*ArcSinh[c*x])^2*Log[h*(f + g*x)^m])/(2*b*c) - (m*(a + b*ArcSinh[c*x])*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*(a + b*ArcSinh[c*x])*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (b*m*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (b*m*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c} -{Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^0/Sqrt[1 + c^2*x^2], x, 9, (m*ArcSinh[c*x]^2)/(2*c) - (m*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/c - (m*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/c + (ArcSinh[c*x]*Log[h*(f + g*x)^m])/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c} -{Log[h*(f + g*x)^m]/(a + b*ArcSinh[c*x])^1/Sqrt[1 + c^2*x^2], x, 0, Unintegrable[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]} - - -(* ::Title:: *) -(*Integrands Involving Inverse Hyperbolic Sines*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u (a+b ArcSinh[c +d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSinh[c+d x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcSinh[a + b*x], x, 6, (7*a*x^2*Sqrt[1 + (a + b*x)^2])/(48*b^2) - (x^3*Sqrt[1 + (a + b*x)^2])/(16*b) - ((4*a*(16 - 19*a^2) - (9 - 26*a^2)*(a + b*x))*Sqrt[1 + (a + b*x)^2])/(96*b^4) - ((3 - 24*a^2 + 8*a^4)*ArcSinh[a + b*x])/(32*b^4) + (1/4)*x^4*ArcSinh[a + b*x]} -{x^2*ArcSinh[a + b*x], x, 5, -((x^2*Sqrt[1 + (a + b*x)^2])/(9*b)) + ((4 - 11*a^2 + 5*a*b*x)*Sqrt[1 + (a + b*x)^2])/(18*b^3) - (a*(3 - 2*a^2)*ArcSinh[a + b*x])/(6*b^3) + (1/3)*x^3*ArcSinh[a + b*x]} -{x^1*ArcSinh[a + b*x], x, 5, (3*a*Sqrt[1 + (a + b*x)^2])/(4*b^2) - (x*Sqrt[1 + (a + b*x)^2])/(4*b) + ((1 - 2*a^2)*ArcSinh[a + b*x])/(4*b^2) + (1/2)*x^2*ArcSinh[a + b*x]} -{x^0*ArcSinh[a + b*x], x, 3, -(Sqrt[1 + (a + b*x)^2]/b) + ((a + b*x)*ArcSinh[a + b*x])/b} -{ArcSinh[a + b*x]/x^1, x, 9, (-(1/2))*ArcSinh[a + b*x]^2 + ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] + PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]} -{ArcSinh[a + b*x]/x^2, x, 4, -(ArcSinh[a + b*x]/x) - (b*ArcTanh[(1 + a*(a + b*x))/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])])/Sqrt[1 + a^2]} -{ArcSinh[a + b*x]/x^3, x, 5, -((b*Sqrt[1 + (a + b*x)^2])/(2*(1 + a^2)*x)) - ArcSinh[a + b*x]/(2*x^2) + (a*b^2*ArcTanh[(1 + a*(a + b*x))/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])])/(2*(1 + a^2)^(3/2))} -{ArcSinh[a + b*x]/x^4, x, 6, -((b*Sqrt[1 + (a + b*x)^2])/(6*(1 + a^2)*x^2)) + (a*b^2*Sqrt[1 + (a + b*x)^2])/(2*(1 + a^2)^2*x) - ArcSinh[a + b*x]/(3*x^3) + ((1 - 2*a^2)*b^3*ArcTanh[(1 + a*(a + b*x))/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])])/(6*(1 + a^2)^(5/2))} -{ArcSinh[a + b*x]/x^5, x, 7, -((b*Sqrt[1 + (a + b*x)^2])/(12*(1 + a^2)*x^3)) + (5*a*b^2*Sqrt[1 + (a + b*x)^2])/(24*(1 + a^2)^2*x^2) + ((4 - 11*a^2)*b^3*Sqrt[1 + (a + b*x)^2])/(24*(1 + a^2)^3*x) - ArcSinh[a + b*x]/(4*x^4) - (a*(3 - 2*a^2)*b^4*ArcTanh[(1 + a*(a + b*x))/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])])/(8*(1 + a^2)^(7/2))} - - -{x^3*ArcSinh[a + b*x]^2, x, 19, (4*a*x)/(3*b^3) - (2*a^3*x)/b^3 - (3*(a + b*x)^2)/(32*b^4) + (3*a^2*(a + b*x)^2)/(4*b^4) - (2*a*(a + b*x)^3)/(9*b^4) + (a + b*x)^4/(32*b^4) - (4*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(3*b^4) + (2*a^3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/b^4 + (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(16*b^4) - (3*a^2*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(2*b^4) + (2*a*(a + b*x)^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(3*b^4) - ((a + b*x)^3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(8*b^4) - (3*ArcSinh[a + b*x]^2)/(32*b^4) + (3*a^2*ArcSinh[a + b*x]^2)/(4*b^4) - (a^4*ArcSinh[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcSinh[a + b*x]^2} -{x^2*ArcSinh[a + b*x]^2, x, 14, -((4*x)/(9*b^2)) + (2*a^2*x)/b^2 - (a*(a + b*x)^2)/(2*b^3) + (2*(a + b*x)^3)/(27*b^3) + (4*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(9*b^3) - (2*a^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/b^3 + (a*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/b^3 - (2*(a + b*x)^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(9*b^3) - (a*ArcSinh[a + b*x]^2)/(2*b^3) + (a^3*ArcSinh[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcSinh[a + b*x]^2} -{x^1*ArcSinh[a + b*x]^2, x, 10, -((2*a*x)/b) + (a + b*x)^2/(4*b^2) + (2*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/b^2 - ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(2*b^2) + ArcSinh[a + b*x]^2/(4*b^2) - (a^2*ArcSinh[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcSinh[a + b*x]^2} -{x^0*ArcSinh[a + b*x]^2, x, 4, 2*x - (2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/b + ((a + b*x)*ArcSinh[a + b*x]^2)/b} -{ArcSinh[a + b*x]^2/x^1, x, 11, (-(1/3))*ArcSinh[a + b*x]^3 + ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] + 2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]} -{ArcSinh[a + b*x]^2/x^2, x, 11, -(ArcSinh[a + b*x]^2/x) - (2*b*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (2*b*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2] - (2*b*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (2*b*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2]} -{ArcSinh[a + b*x]^2/x^3, x, 14, -((b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*x)) - ArcSinh[a + b*x]^2/(2*x^2) + (a*b^2*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) - (a*b^2*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(3/2) + (b^2*Log[x])/(1 + a^2) + (a*b^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) - (a*b^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(3/2)} -{ArcSinh[a + b*x]^2/x^4, x, 40, -(b^2/(3*(1 + a^2)*x)) - (b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(3*(1 + a^2)*x^2) + (a*b^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)^2*x) - ArcSinh[a + b*x]^2/(3*x^3) - (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) + (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(5/2) - (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2)) - (a*b^3*Log[x])/(1 + a^2)^2 - (a^2*b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) + (b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(5/2) - (b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2))} - - -{x^2*ArcSinh[a + b*x]^3, x, 18, (14*Sqrt[1 + (a + b*x)^2])/(9*b^3) - (6*a^2*Sqrt[1 + (a + b*x)^2])/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(4*b^3) - (2*(1 + (a + b*x)^2)^(3/2))/(27*b^3) - (3*a*ArcSinh[a + b*x])/(4*b^3) - (4*(a + b*x)*ArcSinh[a + b*x])/(3*b^3) + (6*a^2*(a + b*x)*ArcSinh[a + b*x])/b^3 - (3*a*(a + b*x)^2*ArcSinh[a + b*x])/(2*b^3) + (2*(a + b*x)^3*ArcSinh[a + b*x])/(9*b^3) + (2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (3*a^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(2*b^3) - ((a + b*x)^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (a*ArcSinh[a + b*x]^3)/(2*b^3) + (a^3*ArcSinh[a + b*x]^3)/(3*b^3) + (1/3)*x^3*ArcSinh[a + b*x]^3} -{x^1*ArcSinh[a + b*x]^3, x, 12, (6*a*Sqrt[1 + (a + b*x)^2])/b^2 - (3*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(8*b^2) + (3*ArcSinh[a + b*x])/(8*b^2) - (6*a*(a + b*x)*ArcSinh[a + b*x])/b^2 + (3*(a + b*x)^2*ArcSinh[a + b*x])/(4*b^2) + (3*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/b^2 - (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(4*b^2) + ArcSinh[a + b*x]^3/(4*b^2) - (a^2*ArcSinh[a + b*x]^3)/(2*b^2) + (1/2)*x^2*ArcSinh[a + b*x]^3} -{x^0*ArcSinh[a + b*x]^3, x, 5, (-6*Sqrt[1 + (a + b*x)^2])/b + (6*(a + b*x)*ArcSinh[a + b*x])/b - (3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/b + ((a + b*x)*ArcSinh[a + b*x]^3)/b} -{ArcSinh[a + b*x]^3/x^1, x, 13, (-(1/4))*ArcSinh[a + b*x]^4 + ArcSinh[a + b*x]^3*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + ArcSinh[a + b*x]^3*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] + 3*ArcSinh[a + b*x]^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 3*ArcSinh[a + b*x]^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - 6*ArcSinh[a + b*x]*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - 6*ArcSinh[a + b*x]*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] + 6*PolyLog[4, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*PolyLog[4, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]} -{ArcSinh[a + b*x]^3/x^2, x, 13, -(ArcSinh[a + b*x]^3/x) - (3*b*ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (3*b*ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2] - (6*b*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (6*b*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2] + (6*b*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/Sqrt[1 + a^2] - (6*b*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/Sqrt[1 + a^2]} -{ArcSinh[a + b*x]^3/x^3, x, 21, -((3*b^2*ArcSinh[a + b*x]^2)/(2*(1 + a^2))) - (3*b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(2*(1 + a^2)*x) - ArcSinh[a + b*x]^3/(2*x^2) + (3*b^2*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2) + (3*a*b^2*ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(2*(1 + a^2)^(3/2)) + (3*b^2*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2) - (3*a*b^2*ArcSinh[a + b*x]^2*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(2*(1 + a^2)^(3/2)) + (3*b^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2) + (3*a*b^2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) + (3*b^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2) - (3*a*b^2*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(3/2) - (3*a*b^2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(3/2) + (3*a*b^2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(3/2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/ArcSinh[a + b*x], x, 14, -CoshIntegral[ArcSinh[a + b*x]]/(4*b^3) + (a^2*CoshIntegral[ArcSinh[a + b*x]])/b^3 + CoshIntegral[3*ArcSinh[a + b*x]]/(4*b^3) - (a*SinhIntegral[2*ArcSinh[a + b*x]])/b^3} -{x^1/ArcSinh[a + b*x], x, 10, -((a*CoshIntegral[ArcSinh[a + b*x]])/b^2) + SinhIntegral[2*ArcSinh[a + b*x]]/(2*b^2)} -{x^0*ArcSinh[a + b*x]^(-1), x, 3, CoshIntegral[ArcSinh[a + b*x]]/b} -{1/(x^1*ArcSinh[a + b*x]), x, 1, Unintegrable[1/(x*ArcSinh[a + b*x]), x]} - - -{x^2/ArcSinh[a + b*x]^2, x, 12, -((a^2*Sqrt[1 + (a + b*x)^2])/(b^3*ArcSinh[a + b*x])) + (2*a*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(b^3*ArcSinh[a + b*x]) - ((a + b*x)^2*Sqrt[1 + (a + b*x)^2])/(b^3*ArcSinh[a + b*x]) - (2*a*CoshIntegral[2*ArcSinh[a + b*x]])/b^3 - SinhIntegral[ArcSinh[a + b*x]]/(4*b^3) + (a^2*SinhIntegral[ArcSinh[a + b*x]])/b^3 + (3*SinhIntegral[3*ArcSinh[a + b*x]])/(4*b^3)} -{x^1/ArcSinh[a + b*x]^2, x, 8, (a*Sqrt[1 + (a + b*x)^2])/(b^2*ArcSinh[a + b*x]) - ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(b^2*ArcSinh[a + b*x]) + CoshIntegral[2*ArcSinh[a + b*x]]/b^2 - (a*SinhIntegral[ArcSinh[a + b*x]])/b^2} -{x^0/ArcSinh[a + b*x]^2, x, 4, -(Sqrt[1 + (a + b*x)^2]/(b*ArcSinh[a + b*x])) + SinhIntegral[ArcSinh[a + b*x]]/b} -{1/(x^1*ArcSinh[a + b*x]^2), x, 1, Unintegrable[1/(x*ArcSinh[a + b*x]^2), x]} - - -{x^2/ArcSinh[a + b*x]^3, x, 24, -((a^2*Sqrt[1 + (a + b*x)^2])/(2*b^3*ArcSinh[a + b*x]^2)) + (a*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(b^3*ArcSinh[a + b*x]^2) - ((a + b*x)^2*Sqrt[1 + (a + b*x)^2])/(2*b^3*ArcSinh[a + b*x]^2) + a/(b^3*ArcSinh[a + b*x]) - (a + b*x)/(b^3*ArcSinh[a + b*x]) - (a^2*(a + b*x))/(2*b^3*ArcSinh[a + b*x]) + (2*a*(a + b*x)^2)/(b^3*ArcSinh[a + b*x]) - (3*(a + b*x)^3)/(2*b^3*ArcSinh[a + b*x]) - CoshIntegral[ArcSinh[a + b*x]]/(8*b^3) + (a^2*CoshIntegral[ArcSinh[a + b*x]])/(2*b^3) + (9*CoshIntegral[3*ArcSinh[a + b*x]])/(8*b^3) - (2*a*SinhIntegral[2*ArcSinh[a + b*x]])/b^3} -{x^1/ArcSinh[a + b*x]^3, x, 14, (a*Sqrt[1 + (a + b*x)^2])/(2*b^2*ArcSinh[a + b*x]^2) - ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(2*b^2*ArcSinh[a + b*x]^2) - 1/(2*b^2*ArcSinh[a + b*x]) + (a*(a + b*x))/(2*b^2*ArcSinh[a + b*x]) - (a + b*x)^2/(b^2*ArcSinh[a + b*x]) - (a*CoshIntegral[ArcSinh[a + b*x]])/(2*b^2) + SinhIntegral[2*ArcSinh[a + b*x]]/b^2} -{x^0/ArcSinh[a + b*x]^3, x, 5, -Sqrt[1 + (a + b*x)^2]/(2*b*ArcSinh[a + b*x]^2) - (a + b*x)/(2*b*ArcSinh[a + b*x]) + CoshIntegral[ArcSinh[a + b*x]]/(2*b)} -{1/(x*ArcSinh[a + b*x]^3), x, 1, Unintegrable[1/(x*ArcSinh[a + b*x]^3), x]} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{x^m*(a + b*ArcSinh[c + d*x])^n, x, 1, Unintegrable[x^m*(a + b*ArcSinh[c + d*x])^n, x]} - -{x^2*(a + b*ArcSinh[c + d*x])^n, x, 22, (3^(-1 - n)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((3*(a + b*ArcSinh[c + d*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(8*d^3)) - (2^(-2 - n)*c*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((2*(a + b*ArcSinh[c + d*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*d^3) - ((a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(8*d^3)) + (c^2*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(2*d^3)) + (E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(((a + b*ArcSinh[c + d*x])/b)^n*(8*d^3)) - (c^2*E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(((a + b*ArcSinh[c + d*x])/b)^n*(2*d^3)) - (2^(-2 - n)*c*E^((2*a)/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b])/(((a + b*ArcSinh[c + d*x])/b)^n*d^3) - (3^(-1 - n)*E^((3*a)/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c + d*x]))/b])/(((a + b*ArcSinh[c + d*x])/b)^n*(8*d^3))} -{x^1*(a + b*ArcSinh[c + d*x])^n, x, 14, (2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((2*(a + b*ArcSinh[c + d*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*d^2) - (c*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(2*d^2)) + (c*E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(((a + b*ArcSinh[c + d*x])/b)^n*(2*d^2)) + (2^(-3 - n)*E^((2*a)/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b])/(((a + b*ArcSinh[c + d*x])/b)^n*d^2)} -{x^0*(a + b*ArcSinh[c + d*x])^n, x, 5, ((a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(2*d)) - (E^(a/b)*(a + b*ArcSinh[c + d*x])^n*Gamma[1 + n, (a + b*ArcSinh[c + d*x])/b])/(((a + b*ArcSinh[c + d*x])/b)^n*(2*d))} -{(a + b*ArcSinh[c + d*x])^n/x^1, x, 1, Unintegrable[(a + b*ArcSinh[c + d*x])^n/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSinh[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*(a + b*ArcSinh[c + d*x])^(1/2), x, 23, (c^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^3 + ((c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/(3*d^3) - (c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(2*d^3) - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d^3) + (Sqrt[b]*c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^3) + (Sqrt[b]*c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*d^3) + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(48*d^3) + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(16*d^3)) - (Sqrt[b]*c^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(4*d^3)) + (Sqrt[b]*c*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(8*d^3)) - (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((3*a)/b)*(48*d^3))} -{x^1*(a + b*ArcSinh[c + d*x])^(1/2), x, 14, -((c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d^2) + (Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (Sqrt[b]*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d^2) - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d^2) + (Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(4*d^2)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(16*d^2))} -{x^0*(a + b*ArcSinh[c + d*x])^(1/2), x, 8, ((c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d + (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(4*d))} - - -{x^1*(a + b*ArcSinh[c + d*x])^(3/2), x, 16, (3*b*c*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d^2) - (c*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/d^2 + ((a + b*ArcSinh[c + d*x])^(3/2)*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (3*b^(3/2)*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d^2) - (3*b^(3/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d^2) - (3*b^(3/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(8*d^2)) + (3*b^(3/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(64*d^2)) - (3*b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]])/(16*d^2)} -{x^0*(a + b*ArcSinh[c + d*x])^(3/2), x, 9, -((3*b*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d)) + ((c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/d + (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(8*d))} - - -{x^1*(a + b*ArcSinh[c + d*x])^(5/2), x, 18, -((15*b^2*c*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d^2)) + (5*b*c*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(2*d^2) - (c*(c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2))/d^2 + (15*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]])/(64*d^2) + ((a + b*ArcSinh[c + d*x])^(5/2)*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (15*b^(5/2)*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d^2) - (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d^2) + (15*b^(5/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(16*d^2)) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(256*d^2)) - (5*b*(a + b*ArcSinh[c + d*x])^(3/2)*Sinh[2*ArcSinh[c + d*x]])/(16*d^2)} -{x^0*(a + b*ArcSinh[c + d*x])^(5/2), x, 10, (15*b^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d) - (5*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2))/d + (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(16*d))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/(a + b*ArcSinh[c + d*x])^(1/2), x, 20, -((E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*Sqrt[b]*d^3)) + (c^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d^3) + (c*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*Sqrt[b]*d^3) + (E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d^3) - (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(8*Sqrt[b]*d^3)) + (c^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*d^3)) - (c*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(2*Sqrt[b]*d^3)) + (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((3*a)/b)*(8*Sqrt[b]*d^3))} -{x^1/(a + b*ArcSinh[c + d*x])^(1/2), x, 12, -((c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d^2)) - (E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d^2) - (c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*d^2)) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(4*Sqrt[b]*d^2))} -{x^0/(a + b*ArcSinh[c + d*x])^(1/2), x, 7, (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*d))} - - -{x^1/(a + b*ArcSinh[c + d*x])^(3/2), x, 16, (2*c*Sqrt[1 + (c + d*x)^2])/(b*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) + (c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d^2) + (E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(b^(3/2)*d^2) - (c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(b^(3/2)*d^2)) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(b^(3/2)*d^2))} -{x^0/(a + b*ArcSinh[c + d*x])^(3/2), x, 8, -((2*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]])) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(b^(3/2)*d))} - - -{x^1/(a + b*ArcSinh[c + d*x])^(5/2), x, 22, (2*c*Sqrt[1 + (c + d*x)^2])/(3*b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*b*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - 4/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) + (4*c*(c + d*x))/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (8*(c + d*x)^2)/(3*b^2*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d^2) - (2*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(3*b^(5/2)*d^2)) + (2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(3*b^(5/2)*d^2))} -{x^0/(a + b*ArcSinh[c + d*x])^(5/2), x, 9, -((2*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2))) - (4*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d) + (2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(3*b^(5/2)*d))} - - -{x^1/(a + b*ArcSinh[c + d*x])^(7/2), x, 21, (2*c*Sqrt[1 + (c + d*x)^2])/(5*b*d^2*(a + b*ArcSinh[c + d*x])^(5/2)) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(5*b*d^2*(a + b*ArcSinh[c + d*x])^(5/2)) - 4/(15*b^2*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) + (4*c*(c + d*x))/(15*b^2*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) - (8*(c + d*x)^2)/(15*b^2*d^2*(a + b*ArcSinh[c + d*x])^(3/2)) + (8*c*Sqrt[1 + (c + d*x)^2])/(15*b^3*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (32*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(15*b^3*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) + (4*c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d^2) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d^2) - (4*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(15*b^(7/2)*d^2)) + (8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(15*b^(7/2)*d^2))} -{x^0/(a + b*ArcSinh[c + d*x])^(7/2), x, 10, -((2*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2))) - (4*(c + d*x))/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (8*Sqrt[1 + (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(15*b^(7/2)*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSinh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x]), x, 3, ((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(d*e^2*(1 + m)*(2 + m))} - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x]), x, 6, -((b*e^4*Sqrt[1 + (c + d*x)^2])/(5*d)) + (2*b*e^4*(1 + (c + d*x)^2)^(3/2))/(15*d) - (b*e^4*(1 + (c + d*x)^2)^(5/2))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x]))/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x]), x, 6, (3*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(32*d) - (b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(16*d) - (3*b*e^3*ArcSinh[c + d*x])/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x]))/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x]), x, 6, (b*e^2*Sqrt[1 + (c + d*x)^2])/(3*d) - (b*e^2*(1 + (c + d*x)^2)^(3/2))/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(3*d)} -{(c*e + d*e*x)*(a + b*ArcSinh[c + d*x]), x, 5, -(b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(4*d) + (b*e*ArcSinh[c + d*x])/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x]))/(2*d)} -{a + b*ArcSinh[c + d*x], x, 4, a*x - (b*Sqrt[1 + (c + d*x)^2])/d + (b*(c + d*x)*ArcSinh[c + d*x])/d} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x), x, 7, (a + b*ArcSinh[c + d*x])^2/(2*b*d*e) + ((a + b*ArcSinh[c + d*x])*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e) - (b*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(2*d*e)} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^2, x, 6, -((a + b*ArcSinh[c + d*x])/(d*e^2*(c + d*x))) - (b*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(d*e^2)} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^3, x, 4, -(b*Sqrt[1 + (c + d*x)^2])/(2*d*e^3*(c + d*x)) - (a + b*ArcSinh[c + d*x])/(2*d*e^3*(c + d*x)^2)} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^4, x, 7, -(b*Sqrt[1 + (c + d*x)^2])/(6*d*e^4*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])/(3*d*e^4*(c + d*x)^3) + (b*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(6*d*e^4)} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^5, x, 5, -(b*Sqrt[1 + (c + d*x)^2])/(12*d*e^5*(c + d*x)^3) + (b*Sqrt[1 + (c + d*x)^2])/(6*d*e^5*(c + d*x)) - (a + b*ArcSinh[c + d*x])/(4*d*e^5*(c + d*x)^4)} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^6, x, 8, -(b*Sqrt[1 + (c + d*x)^2])/(20*d*e^6*(c + d*x)^4) + (3*b*Sqrt[1 + (c + d*x)^2])/(40*d*e^6*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])/(5*d*e^6*(c + d*x)^5) - (3*b*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(40*d*e^6)} - - -{(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^2, x, 3, If[$VersionNumber>=8, ((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)) + (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c + d*x)^2])/(d*e^3*(1 + m)*(2 + m)*(3 + m)), ((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*(e*(c + d*x))^(2 + m)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)) + (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c + d*x)^2])/(d*e^3*(3 + m)*(2 + 3*m + m^2))]} - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2, x, 9, (16/75)*b^2*e^4*x - (8*b^2*e^4*(c + d*x)^3)/(225*d) + (2*b^2*e^4*(c + d*x)^5)/(125*d) - (16*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(75*d) + (8*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(75*d) - (2*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^2)/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^2, x, 8, -((3*b^2*e^3*(c + d*x)^2)/(32*d)) + (b^2*e^3*(c + d*x)^4)/(32*d) + (3*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(16*d) - (b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(8*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^2)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^2)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^2, x, 7, (-(4/9))*b^2*e^2*x + (2*b^2*e^2*(c + d*x)^3)/(27*d) + (4*b*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(9*d) - (2*b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^2)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^2, x, 6, (b^2*e*(c + d*x)^2)/(4*d) - (b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*d) + (e*(a + b*ArcSinh[c + d*x])^2)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(2*d)} -{(a + b*ArcSinh[c + d*x])^2, x, 4, 2*b^2*x - (2*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/d + ((c + d*x)*(a + b*ArcSinh[c + d*x])^2)/d} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x), x, 8, (a + b*ArcSinh[c + d*x])^3/(3*b*d*e) + ((a + b*ArcSinh[c + d*x])^2*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e) - (b*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e) - (b^2*PolyLog[3, E^(-2*ArcSinh[c + d*x])])/(2*d*e)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^2, x, 9, -((a + b*ArcSinh[c + d*x])^2/(d*e^2*(c + d*x))) - (4*b*(a + b*ArcSinh[c + d*x])*ArcTanh[E^ArcSinh[c + d*x]])/(d*e^2) - (2*b^2*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^2) + (2*b^2*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^2)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^3, x, 5, -((b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(d*e^3*(c + d*x))) - (a + b*ArcSinh[c + d*x])^2/(2*d*e^3*(c + d*x)^2) + (b^2*Log[c + d*x])/(d*e^3)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^4, x, 11, -(b^2/(3*d*e^4*(c + d*x))) - (b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(3*d*e^4*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])^2/(3*d*e^4*(c + d*x)^3) + (2*b*(a + b*ArcSinh[c + d*x])*ArcTanh[E^ArcSinh[c + d*x]])/(3*d*e^4) + (b^2*PolyLog[2, -E^ArcSinh[c + d*x]])/(3*d*e^4) - (b^2*PolyLog[2, E^ArcSinh[c + d*x]])/(3*d*e^4)} - - -{(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^3, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^3)/(d*e*(1 + m)) - (3*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/(e*(1 + m))} - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^3, x, 17, (16/25)*a*b^2*e^4*x - (298*b^3*e^4*Sqrt[1 + (c + d*x)^2])/(375*d) + (76*b^3*e^4*(1 + (c + d*x)^2)^(3/2))/(1125*d) - (6*b^3*e^4*(1 + (c + d*x)^2)^(5/2))/(625*d) + (16*b^3*e^4*(c + d*x)*ArcSinh[c + d*x])/(25*d) - (8*b^2*e^4*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(75*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x]))/(125*d) - (8*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(25*d) + (4*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(25*d) - (3*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^3)/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^3, x, 13, (45*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(256*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(128*d) - (45*b^3*e^3*ArcSinh[c + d*x])/(256*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x]))/(32*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x]))/(32*d) + (9*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(32*d) - (3*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^3)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^3, x, 12, (-(4/3))*a*b^2*e^2*x + (14*b^3*e^2*Sqrt[1 + (c + d*x)^2])/(9*d) - (2*b^3*e^2*(1 + (c + d*x)^2)^(3/2))/(27*d) - (4*b^3*e^2*(c + d*x)*ArcSinh[c + d*x])/(3*d) + (2*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(9*d) + (2*b*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(3*d) - (b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^3)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3, x, 8, (-3*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(8*d) + (3*b^3*e*ArcSinh[c + d*x])/(8*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x]))/(4*d) - (3*b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(4*d) + (e*(a + b*ArcSinh[c + d*x])^3)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^3)/(2*d)} -{(a + b*ArcSinh[c + d*x])^3, x, 6, 6*a*b^2*x - (6*b^3*Sqrt[1 + (c + d*x)^2])/d + (6*b^3*(c + d*x)*ArcSinh[c + d*x])/d - (3*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcSinh[c + d*x])^3)/d} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x), x, 9, (a + b*ArcSinh[c + d*x])^4/(4*b*d*e) + ((a + b*ArcSinh[c + d*x])^3*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e) - (3*b*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(2*d*e) - (3*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[3, E^(-2*ArcSinh[c + d*x])])/(2*d*e) - (3*b^3*PolyLog[4, E^(-2*ArcSinh[c + d*x])])/(4*d*e)} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^2, x, 11, -((a + b*ArcSinh[c + d*x])^3/(d*e^2*(c + d*x))) - (6*b*(a + b*ArcSinh[c + d*x])^2*ArcTanh[E^ArcSinh[c + d*x]])/(d*e^2) - (6*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^2) + (6*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^2) + (6*b^3*PolyLog[3, -E^ArcSinh[c + d*x]])/(d*e^2) - (6*b^3*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^2)} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^3, x, 9, (3*b*(a + b*ArcSinh[c + d*x])^2)/(2*d*e^3) - (3*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcSinh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcSinh[c + d*x])*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e^3) - (3*b^3*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(2*d*e^3)} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^4, x, 16, -((b^2*(a + b*ArcSinh[c + d*x]))/(d*e^4*(c + d*x))) - (b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(2*d*e^4*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])^3/(3*d*e^4*(c + d*x)^3) + (b*(a + b*ArcSinh[c + d*x])^2*ArcTanh[E^ArcSinh[c + d*x]])/(d*e^4) - (b^3*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(d*e^4) + (b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^4) - (b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^4) - (b^3*PolyLog[3, -E^ArcSinh[c + d*x]])/(d*e^4) + (b^3*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^4)} - - -{(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^4)/(d*e*(1 + m)) - (4*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(e*(1 + m))} - -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^4, x, 16, -((45*b^4*e^3*(c + d*x)^2)/(128*d)) + (3*b^4*e^3*(c + d*x)^4)/(128*d) + (45*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(64*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(32*d) - (45*b^2*e^3*(a + b*ArcSinh[c + d*x])^2)/(128*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(16*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^2)/(16*d) + (3*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(8*d) - (b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(4*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^4)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^4, x, 13, (-(160/27))*b^4*e^2*x + (8*b^4*e^2*(c + d*x)^3)/(81*d) + (160*b^3*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(27*d) - (8*b^3*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(27*d) - (8*b^2*e^2*(c + d*x)*(a + b*ArcSinh[c + d*x])^2)/(3*d) + (4*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^2)/(9*d) + (8*b*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(9*d) - (4*b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^4)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4, x, 9, (3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*d) + (3*b^2*e*(a + b*ArcSinh[c + d*x])^2)/(4*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/(2*d) - (b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/d + (e*(a + b*ArcSinh[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^4)/(2*d)} -{(a + b*ArcSinh[c + d*x])^4, x, 6, 24*b^4*x - (24*b^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/d + (12*b^2*(c + d*x)*(a + b*ArcSinh[c + d*x])^2)/d - (4*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcSinh[c + d*x])^4)/d} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x), x, 10, (a + b*ArcSinh[c + d*x])^5/(5*b*d*e) + ((a + b*ArcSinh[c + d*x])^4*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e) - (2*b*(a + b*ArcSinh[c + d*x])^3*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e) - (3*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[3, E^(-2*ArcSinh[c + d*x])])/(d*e) - (3*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[4, E^(-2*ArcSinh[c + d*x])])/(d*e) - (3*b^4*PolyLog[5, E^(-2*ArcSinh[c + d*x])])/(2*d*e)} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^2, x, 13, -((a + b*ArcSinh[c + d*x])^4/(d*e^2*(c + d*x))) - (8*b*(a + b*ArcSinh[c + d*x])^3*ArcTanh[E^ArcSinh[c + d*x]])/(d*e^2) - (12*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^2) + (12*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^2) + (24*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, -E^ArcSinh[c + d*x]])/(d*e^2) - (24*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^2) - (24*b^4*PolyLog[4, -E^ArcSinh[c + d*x]])/(d*e^2) + (24*b^4*PolyLog[4, E^ArcSinh[c + d*x]])/(d*e^2)} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^3, x, 10, (2*b*(a + b*ArcSinh[c + d*x])^3)/(d*e^3) - (2*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(d*e^3*(c + d*x)) - (a + b*ArcSinh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) + (6*b^2*(a + b*ArcSinh[c + d*x])^2*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e^3) - (6*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e^3) - (3*b^4*PolyLog[3, E^(-2*ArcSinh[c + d*x])])/(d*e^3)} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^4, x, 21, -((2*b^2*(a + b*ArcSinh[c + d*x])^2)/(d*e^4*(c + d*x))) - (2*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(3*d*e^4*(c + d*x)^2) - (a + b*ArcSinh[c + d*x])^4/(3*d*e^4*(c + d*x)^3) - (8*b^3*(a + b*ArcSinh[c + d*x])*ArcTanh[E^ArcSinh[c + d*x]])/(d*e^4) + (4*b*(a + b*ArcSinh[c + d*x])^3*ArcTanh[E^ArcSinh[c + d*x]])/(3*d*e^4) - (4*b^4*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^4) + (2*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, -E^ArcSinh[c + d*x]])/(d*e^4) + (4*b^4*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^4) - (2*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, E^ArcSinh[c + d*x]])/(d*e^4) - (4*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, -E^ArcSinh[c + d*x]])/(d*e^4) + (4*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[3, E^ArcSinh[c + d*x]])/(d*e^4) + (4*b^4*PolyLog[4, -E^ArcSinh[c + d*x]])/(d*e^4) - (4*b^4*PolyLog[4, E^ArcSinh[c + d*x]])/(d*e^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^m/(a + b*ArcSinh[c + d*x]), x, 1, Unintegrable[(e*(c + d*x))^m/(a + b*ArcSinh[c + d*x]), x]} - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x]), x, 14, (e^4*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(8*b*d) - (3*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(16*b*d) + (e^4*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(16*b*d) - (e^4*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(8*b*d) + (3*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(16*b*d) - (e^4*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(16*b*d)} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x]), x, 11, (e^3*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b])/(4*b*d) - (e^3*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(4*a)/b])/(8*b*d) - (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(4*b*d) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(8*b*d)} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x]), x, 11, -((e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(4*b*d)) + (e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(4*b*d) + (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(4*b*d) - (e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(4*b*d)} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x]), x, 8, -((e*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b*d)) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b*d)} -{1/(a + b*ArcSinh[c + d*x]), x, 5, (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(b*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(b*d)} -{1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^2, x, 13, -((e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) - (e^4*CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(8*b^2*d) + (9*e^4*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/(16*b^2*d) - (5*e^4*CoshIntegral[(5*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(5*a)/b])/(16*b^2*d) + (e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(8*b^2*d) - (9*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(16*b^2*d) + (5*e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(16*b^2*d)} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^2, x, 10, -((e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) - (e^3*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d) + (e^3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d) + (e^3*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d) - (e^3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(2*b^2*d)} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^2, x, 10, -((e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) + (e^2*CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(4*b^2*d) - (3*e^2*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/(4*b^2*d) - (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(4*b^2*d)} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^2, x, 6, -((e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b*d*(a + b*ArcSinh[c + d*x]))) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(b^2*d) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(b^2*d)} -{(a + b*ArcSinh[c + d*x])^(-2), x, 6, -(Sqrt[1 + (c + d*x)^2]/(b*d*(a + b*ArcSinh[c + d*x]))) - (CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(b^2*d) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(b^2*d)} -{1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^2), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^3, x, 26, -((e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2)) - (2*e^4*(c + d*x)^3)/(b^2*d*(a + b*ArcSinh[c + d*x])) - (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) + (e^4*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(16*b^3*d) - (27*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(32*b^3*d) + (25*e^4*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(32*b^3*d) - (e^4*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(16*b^3*d) + (27*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(32*b^3*d) - (25*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(32*b^3*d)} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^3, x, 20, -((e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2)) - (3*e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) - (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcSinh[c + d*x])) + (e^3*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b^3*d) - (e^3*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(4*a)/b])/(b^3*d) - (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(2*b^3*d) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(b^3*d)} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^3, x, 18, -((e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2)) - (e^2*(c + d*x))/(b^2*d*(a + b*ArcSinh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) - (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(8*b^3*d) + (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(8*b^3*d) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(8*b^3*d)} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^3, x, 11, -((e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2)) - e/(2*b^2*d*(a + b*ArcSinh[c + d*x])) - (e*(c + d*x)^2)/(b^2*d*(a + b*ArcSinh[c + d*x])) - (e*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(2*a)/b])/(b^3*d) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(b^3*d)} -{(a + b*ArcSinh[c + d*x])^(-3), x, 7, -Sqrt[1 + (c + d*x)^2]/(2*b*d*(a + b*ArcSinh[c + d*x])^2) - (c + d*x)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) + (Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(2*b^3*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(2*b^3*d)} -{1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^3), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^4, x, 24, -((e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^3)) - (2*e^4*(c + d*x)^3)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (2*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b^3*d*(a + b*ArcSinh[c + d*x])) - (25*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(6*b^3*d*(a + b*ArcSinh[c + d*x])) - (e^4*CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(48*b^4*d) + (27*e^4*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/(32*b^4*d) - (125*e^4*CoshIntegral[(5*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(5*a)/b])/(96*b^4*d) + (e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(48*b^4*d) - (27*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(32*b^4*d) + (125*e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c + d*x]))/b])/(96*b^4*d)} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^4, x, 17, -((e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^3)) - (e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (2*e^3*(c + d*x)^4)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b^3*d*(a + b*ArcSinh[c + d*x])) - (8*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(3*b^3*d*(a + b*ArcSinh[c + d*x])) - (e^3*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d) + (4*e^3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d) + (e^3*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d) - (4*e^3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d)} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^4, x, 18, -((e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^3)) - (e^2*(c + d*x))/(3*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (e^2*Sqrt[1 + (c + d*x)^2])/(3*b^3*d*(a + b*ArcSinh[c + d*x])) - (3*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(2*b^3*d*(a + b*ArcSinh[c + d*x])) + (e^2*CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(24*b^4*d) - (9*e^2*CoshIntegral[(3*(a + b*ArcSinh[c + d*x]))/b]*Sinh[(3*a)/b])/(8*b^4*d) - (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(24*b^4*d) + (9*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c + d*x]))/b])/(8*b^4*d)} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^4, x, 9, -((e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^3)) - e/(6*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^2) - (2*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*b^3*d*(a + b*ArcSinh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d) - (2*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c + d*x]))/b])/(3*b^4*d)} -{(a + b*ArcSinh[c + d*x])^(-4), x, 8, -(Sqrt[1 + (c + d*x)^2]/(3*b*d*(a + b*ArcSinh[c + d*x])^3)) - (c + d*x)/(6*b^2*d*(a + b*ArcSinh[c + d*x])^2) - Sqrt[1 + (c + d*x)^2]/(6*b^3*d*(a + b*ArcSinh[c + d*x])) - (CoshIntegral[(a + b*ArcSinh[c + d*x])/b]*Sinh[a/b])/(6*b^4*d) + (Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(6*b^4*d)} -{1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^4), x]/e} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcSinh[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^4*Sqrt[a + b*ArcSinh[c + d*x]], x, 21, (e^4*(c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/(5*d) + (Sqrt[b]*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d) - (Sqrt[b]*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d) + (Sqrt[b]*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(320*d) - (Sqrt[b]*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b)) + (Sqrt[b]*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d*E^((3*a)/b)) - (Sqrt[b]*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(320*d*E^((5*a)/b))} -{(c*e + d*e*x)^3*Sqrt[a + b*ArcSinh[c + d*x]], x, 16, (-3*e^3*Sqrt[a + b*ArcSinh[c + d*x]])/(32*d) + (e^3*(c + d*x)^4*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d) - (Sqrt[b]*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d) + (Sqrt[b]*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*d) - (Sqrt[b]*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d*E^((4*a)/b)) + (Sqrt[b]*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*Sqrt[a + b*ArcSinh[c + d*x]], x, 16, (e^2*(c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/(3*d) - (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d) + (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(48*d) + (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(48*d*E^((3*a)/b))} -{(c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]], x, 11, (e*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d) - (Sqrt[b]*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d) - (Sqrt[b]*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*d*E^((2*a)/b))} -{Sqrt[a + b*ArcSinh[c + d*x]], x, 8, ((c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/d + (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*d*E^(a/b))} -{1/(c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]], x, 2, Unintegrable[Sqrt[a + b*ArcSinh[c + d*x]]/(c + d*x), x]/e} - - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(3/2), x, 43, (-4*b*e^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(25*d) - (3*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(50*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(3/2))/(5*d) + (3*b^(3/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d) - (b^(3/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(200*d) - (3*b^(3/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d) + (3*b^(3/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d) + (3*b^(3/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d*E^(a/b)) - (b^(3/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(200*d*E^((3*a)/b)) - (3*b^(3/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d*E^((3*a)/b)) + (3*b^(3/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3200*d*E^((5*a)/b))} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(3/2), x, 27, (9*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(64*d) - (3*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^(3/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^(3/2))/(4*d) - (3*b^(3/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2048*d) + (3*b^(3/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(128*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2048*d*E^((4*a)/b)) - (3*b^(3/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(128*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(3/2), x, 24, (b*e^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(3*d) - (b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^(3/2))/(3*d) - (3*b^(3/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d) + (b^(3/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(96*d) - (3*b^(3/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b)) + (b^(3/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(96*d*E^((3*a)/b))} -{(c*e + d*e*x)^1*(a + b*ArcSinh[c + d*x])^(3/2), x, 13, (-3*b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(8*d) + (e*(a + b*ArcSinh[c + d*x])^(3/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d) + (3*b^(3/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(3/2), x, 9, (-3*b*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/d + (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(3/2), x, 2, Unintegrable[(a + b*ArcSinh[c + d*x])^(3/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(5/2), x, 46, (2*b^2*e^4*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(5*d) - (b^2*e^4*(c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/(15*d) + (3*b^2*e^4*(c + d*x)^5*Sqrt[a + b*ArcSinh[c + d*x]])/(100*d) - (4*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(15*d) + (2*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(15*d) - (b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(10*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(5/2))/(5*d) + (15*b^(5/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d) - (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d) - (b^(5/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d) + (3*b^(5/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(6400*d) - (15*b^(5/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (b^(5/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(240*d*E^((3*a)/b)) + (b^(5/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1280*d*E^((3*a)/b)) - (3*b^(5/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(6400*d*E^((5*a)/b))} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(5/2), x, 29, (-225*b^2*e^3*Sqrt[a + b*ArcSinh[c + d*x]])/(2048*d) - (45*b^2*e^3*(c + d*x)^2*Sqrt[a + b*ArcSinh[c + d*x]])/(256*d) + (15*b^2*e^3*(c + d*x)^4*Sqrt[a + b*ArcSinh[c + d*x]])/(256*d) + (15*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(64*d) - (5*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(32*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^(5/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^(5/2))/(4*d) - (15*b^(5/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16384*d) + (15*b^(5/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(512*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16384*d*E^((4*a)/b)) + (15*b^(5/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(512*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(5/2), x, 26, (-5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(6*d) + (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcSinh[c + d*x]])/(36*d) + (5*b*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(9*d) - (5*b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^(5/2))/(3*d) - (15*b^(5/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d) + (5*b^(5/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(576*d) + (15*b^(5/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(64*d*E^(a/b)) - (5*b^(5/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(576*d*E^((3*a)/b))} -{(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2), x, 14, (15*b^2*e*Sqrt[a + b*ArcSinh[c + d*x]])/(64*d) + (15*b^2*e*(c + d*x)^2*Sqrt[a + b*ArcSinh[c + d*x]])/(32*d) - (5*b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(8*d) + (e*(a + b*ArcSinh[c + d*x])^(5/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(5/2))/(2*d) - (15*b^(5/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d) - (15*b^(5/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(256*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(5/2), x, 10, (15*b^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]])/(4*d) - (5*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2))/d + (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2), x, 2, Unintegrable[(a + b*ArcSinh[c + d*x])^(5/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(7/2), x, 77, (-1813*b^3*e^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(1125*d) + (119*b^3*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(1125*d) - (21*b^3*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(1000*d) + (14*b^2*e^4*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/(15*d) - (7*b^2*e^4*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^(3/2))/(45*d) + (7*b^2*e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(3/2))/(100*d) - (28*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(75*d) + (14*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(75*d) - (7*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(50*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^(7/2))/(5*d) + (105*b^(7/2)*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(256*d) - (119*b^(7/2)*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(18000*d) - (21*b^(7/2)*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d) + (21*b^(7/2)*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d) + (105*b^(7/2)*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(256*d*E^(a/b)) - (119*b^(7/2)*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(18000*d*E^((3*a)/b)) - (21*b^(7/2)*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d*E^((3*a)/b)) + (21*b^(7/2)*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64000*d*E^((5*a)/b))} -{(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(7/2), x, 42, (1575*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(4096*d) - (105*b^3*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2048*d) - (525*b^2*e^3*(a + b*ArcSinh[c + d*x])^(3/2))/(2048*d) - (105*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(256*d) + (35*b^2*e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^(3/2))/(256*d) + (21*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(64*d) - (7*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(32*d) - (3*e^3*(a + b*ArcSinh[c + d*x])^(7/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^(7/2))/(4*d) - (105*b^(7/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(131072*d) + (105*b^(7/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2048*d) + (105*b^(7/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(131072*d*E^((4*a)/b)) - (105*b^(7/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2048*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(7/2), x, 35, (175*b^3*e^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(54*d) - (35*b^3*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(216*d) - (35*b^2*e^2*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/(18*d) + (35*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^(3/2))/(108*d) + (7*b*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(9*d) - (7*b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^(7/2))/(3*d) - (105*b^(7/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d) + (35*b^(7/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3456*d) - (105*b^(7/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (35*b^(7/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3456*d*E^((3*a)/b))} -{(c*e + d*e*x)^1*(a + b*ArcSinh[c + d*x])^(7/2), x, 16, (-105*b^3*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(128*d) + (35*b^2*e*(a + b*ArcSinh[c + d*x])^(3/2))/(64*d) + (35*b^2*e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(3/2))/(32*d) - (7*b*e*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(8*d) + (e*(a + b*ArcSinh[c + d*x])^(7/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^(7/2))/(2*d) - (105*b^(7/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1024*d) + (105*b^(7/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(1024*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(7/2), x, 11, (-105*b^3*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(8*d) + (35*b^2*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2))/(4*d) - (7*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSinh[c + d*x])^(7/2))/d + (105*b^(7/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d) + (105*b^(7/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2), x, 2, Unintegrable[(a + b*ArcSinh[c + d*x])^(7/2)/(c + d*x), x]/e} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^4/Sqrt[a + b*ArcSinh[c + d*x]], x, 20, (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*Sqrt[b]*d) - (e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) + (e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(16*Sqrt[b]*d*E^(a/b)) - (e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((3*a)/b)) + (e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/Sqrt[a + b*ArcSinh[c + d*x]], x, 15, -(e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) + (e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((4*a)/b)) - (e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/Sqrt[a + b*ArcSinh[c + d*x]], x, 15, -(e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*Sqrt[b]*d) + (e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) - (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*Sqrt[b]*d*E^(a/b)) + (e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d*E^((3*a)/b))} -{(c*e + d*e*x)^1/Sqrt[a + b*ArcSinh[c + d*x]], x, 10, -(e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d) + (e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d*E^((2*a)/b))} -{1/Sqrt[a + b*ArcSinh[c + d*x]], x, 7, (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d*E^(a/b))} -{1/(c*e + d*e*x)/Sqrt[a + b*ArcSinh[c + d*x]], x, 2, Unintegrable[1/((c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]]), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(3/2), x, 19, (-2*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*b^(3/2)*d) + (3*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d) - (e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*b^(3/2)*d*E^(a/b)) - (3*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d*E^((3*a)/b)) + (e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(3/2), x, 14, (-2*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) - (e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d) + (e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d*E^((4*a)/b)) - (e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(3/2), x, 14, (-2*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*b^(3/2)*d) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) - (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(4*b^(3/2)*d*E^(a/b)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d*E^((3*a)/b))} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(3/2), x, 8, (-2*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(b^(3/2)*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(-3/2), x, 8, (-2*Sqrt[1 + (c + d*x)^2])/(b*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d*E^(a/b))} -{1/(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(3/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^(3/2)), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(5/2), x, 36, (-2*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (16*e^4*(c + d*x)^3)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (20*e^4*(c + d*x)^5)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(12*b^(5/2)*d) - (3*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*b^(5/2)*d) + (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(12*b^(5/2)*d*E^(a/b)) - (3*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*b^(5/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(5/2), x, 26, (-2*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e^3*(c + d*x)^2)/(b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (16*e^3*(c + d*x)^4)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (e^3*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((4*a)/b)) - (e^3*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(5/2), x, 24, (-2*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (8*e^2*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (4*e^2*(c + d*x)^3)/(b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(6*b^(5/2)*d) + (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*b^(5/2)*d) - (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(6*b^(5/2)*d*E^(a/b)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(2*b^(5/2)*d*E^((3*a)/b))} -{(c*e + d*e*x)^1/(a + b*ArcSinh[c + d*x])^(5/2), x, 13, (-2*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (8*e*(c + d*x)^2)/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*e*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(-5/2), x, 9, (-2*Sqrt[1 + (c + d*x)^2])/(3*b*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d) + (2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d*E^(a/b))} -{1/(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(5/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^(5/2)), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(7/2), x, 34, (-2*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (16*e^4*(c + d*x)^3)/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e^4*(c + d*x)^5)/(3*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (32*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (40*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(3*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(30*b^(7/2)*d) + (9*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(20*b^(7/2)*d) - (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(30*b^(7/2)*d*E^(a/b)) - (9*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(20*b^(7/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(7/2), x, 23, (-2*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (4*e^3*(c + d*x)^2)/(5*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (16*e^3*(c + d*x)^4)/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (16*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (128*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (16*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (4*e^3*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (16*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((4*a)/b)) - (4*e^3*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(7/2), x, 24, (-2*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (8*e^2*(c + d*x))/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (16*e^2*Sqrt[1 + (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (24*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(5*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) - (3*e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d) - (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d*E^(a/b)) + (3*e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d*E^((3*a)/b))} -{(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(7/2), x, 11, (-2*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (4*e)/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (8*e*(c + d*x)^2)/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (32*e*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) + (8*e*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (8*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((2*a)/b))} -{(a + b*ArcSinh[c + d*x])^(-7/2), x, 10, (-2*Sqrt[1 + (c + d*x)^2])/(5*b*d*(a + b*ArcSinh[c + d*x])^(5/2)) - (4*(c + d*x))/(15*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2)) - (8*Sqrt[1 + (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSinh[c + d*x]]) - (4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d*E^(a/b))} -{1/(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(7/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcSinh[c + d*x])^(7/2)), x]/e} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^(m/2) (a+b ArcSinh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x]), x, 8, (28*b*e^2*(e*(c + d*x))^(3/2)*Sqrt[1 + (c + d*x)^2])/(405*d) - (4*b*(e*(c + d*x))^(7/2)*Sqrt[1 + (c + d*x)^2])/(81*d) - (28*b*e^3*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(135*d*(1 + c + d*x)) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x]))/(9*d*e) + (28*b*e^(7/2)*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(135*d*Sqrt[1 + (c + d*x)^2]) - (14*b*e^(7/2)*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(135*d*Sqrt[1 + (c + d*x)^2])} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x]), x, 6, (20*b*e^2*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(147*d) - (4*b*(e*(c + d*x))^(5/2)*Sqrt[1 + (c + d*x)^2])/(49*d) + (2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x]))/(7*d*e) - (10*b*e^(5/2)*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(147*d*Sqrt[1 + (c + d*x)^2])} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x]), x, 7, -((4*b*(e*(c + d*x))^(3/2)*Sqrt[1 + (c + d*x)^2])/(25*d)) + (12*b*e*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(25*d*(1 + c + d*x)) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x]))/(5*d*e) - (12*b*e^(3/2)*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(25*d*Sqrt[1 + (c + d*x)^2]) + (6*b*e^(3/2)*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(25*d*Sqrt[1 + (c + d*x)^2])} -{Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x]), x, 5, -((4*b*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(9*d)) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x]))/(3*d*e) + (2*b*Sqrt[e]*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(9*d*Sqrt[1 + (c + d*x)^2])} -{(a + b*ArcSinh[c + d*x])/Sqrt[c*e + d*e*x], x, 6, -((4*b*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(d*e*(1 + c + d*x))) + (2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x]))/(d*e) + (4*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(d*Sqrt[e]*Sqrt[1 + (c + d*x)^2]) - (2*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(d*Sqrt[e]*Sqrt[1 + (c + d*x)^2])} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(3/2), x, 4, -((2*(a + b*ArcSinh[c + d*x]))/(d*e*Sqrt[e*(c + d*x)])) + (2*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(d*e^(3/2)*Sqrt[1 + (c + d*x)^2])} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(5/2), x, 7, -((4*b*Sqrt[1 + (c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)])) + (4*b*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(3*d*e^3*(1 + c + d*x)) - (2*(a + b*ArcSinh[c + d*x]))/(3*d*e*(e*(c + d*x))^(3/2)) - (4*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(3*d*e^(5/2)*Sqrt[1 + (c + d*x)^2]) + (2*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(3*d*e^(5/2)*Sqrt[1 + (c + d*x)^2])} -{(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(7/2), x, 5, -((4*b*Sqrt[1 + (c + d*x)^2])/(15*d*e^2*(e*(c + d*x))^(3/2))) - (2*(a + b*ArcSinh[c + d*x]))/(5*d*e*(e*(c + d*x))^(5/2)) - (2*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(15*d*e^(7/2)*Sqrt[1 + (c + d*x)^2])} - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^2)/(9*d*e) - (8*b*(e*(c + d*x))^(11/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 11/4, 15/4, -(c + d*x)^2])/(99*d*e^2) + (16*b^2*(e*(c + d*x))^(13/2)*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4, 17/4}, -(c + d*x)^2])/(1287*d*e^3)} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^2)/(7*d*e) - (8*b*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 9/4, 13/4, -(c + d*x)^2])/(63*d*e^2) + (16*b^2*(e*(c + d*x))^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, -(c + d*x)^2])/(693*d*e^3)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^2)/(5*d*e) - (8*b*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 7/4, 11/4, -(c + d*x)^2])/(35*d*e^2) + (16*b^2*(e*(c + d*x))^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, -(c + d*x)^2])/(315*d*e^3)} -{Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^2)/(3*d*e) - (8*b*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, -(c + d*x)^2])/(15*d*e^2) + (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2])/(105*d*e^3)} -{(a + b*ArcSinh[c + d*x])^2/Sqrt[c*e + d*e*x], x, 3, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^2)/(d*e) - (8*b*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, -(c + d*x)^2])/(3*d*e^2) + (16*b^2*(e*(c + d*x))^(5/2)*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, -(c + d*x)^2])/(15*d*e^3)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(3/2), x, 3, -((2*(a + b*ArcSinh[c + d*x])^2)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2])/(d*e^2) - (16*b^2*(e*(c + d*x))^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, -(c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(5/2), x, 3, -((2*(a + b*ArcSinh[c + d*x])^2)/(3*d*e*(e*(c + d*x))^(3/2))) - (8*b*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[-(1/4), 1/2, 3/4, -(c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)]) + (16*b^2*Sqrt[e*(c + d*x)]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, -(c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(7/2), x, 3, -((2*(a + b*ArcSinh[c + d*x])^2)/(5*d*e*(e*(c + d*x))^(5/2))) - (8*b*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[-(3/4), 1/2, 1/4, -(c + d*x)^2])/(15*d*e^2*(e*(c + d*x))^(3/2)) - (16*b^2*HypergeometricPFQ[{-(1/4), -(1/4), 1}, {1/4, 3/4}, -(c + d*x)^2])/(15*d*e^3*Sqrt[e*(c + d*x)])} - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^3)/(9*d*e) - (2*b*Unintegrable[((e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/(3*e)} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^3)/(7*d*e) - (6*b*Unintegrable[((e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/(7*e)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^3)/(5*d*e) - (6*b*Unintegrable[((e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/(5*e)} -{Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^3)/(3*d*e) - (2*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/e} -{(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x], x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^3)/(d*e) - (6*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^2)/Sqrt[1 + (c + d*x)^2], x])/e} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)])) + (6*b*Unintegrable[(a + b*ArcSinh[c + d*x])^2/(Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2]), x])/e} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2))) + (2*b*Unintegrable[(a + b*ArcSinh[c + d*x])^2/((e*(c + d*x))^(3/2)*Sqrt[1 + (c + d*x)^2]), x])/e} -{(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^3)/(5*d*e*(e*(c + d*x))^(5/2))) + (6*b*Unintegrable[(a + b*ArcSinh[c + d*x])^2/((e*(c + d*x))^(5/2)*Sqrt[1 + (c + d*x)^2]), x])/(5*e)} - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^4)/(9*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(9/2)*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(9*e)} -{(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^4)/(7*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(7/2)*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(7*e)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^4)/(5*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(5/2)*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(5*e)} -{Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^4)/(3*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/(3*e)} -{(a + b*ArcSinh[c + d*x])^4/Sqrt[c*e + d*e*x], x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^4)/(d*e) - (8*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^3)/Sqrt[1 + (c + d*x)^2], x])/e} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^4)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Unintegrable[(a + b*ArcSinh[c + d*x])^3/(Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2]), x])/e} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^4)/(3*d*e*(e*(c + d*x))^(3/2))) + (8*b*Unintegrable[(a + b*ArcSinh[c + d*x])^3/((e*(c + d*x))^(3/2)*Sqrt[1 + (c + d*x)^2]), x])/(3*e)} -{(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2), x, 2, -((2*(a + b*ArcSinh[c + d*x])^4)/(5*d*e*(e*(c + d*x))^(5/2))) + (8*b*Unintegrable[(a + b*ArcSinh[c + d*x])^3/((e*(c + d*x))^(5/2)*Sqrt[1 + (c + d*x)^2]), x])/(5*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (1+(a+b x)^2)^(m/2) ArcSinh[a +b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*m>0*) - - -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3, x, 7, -((3*(a + b*x)^2)/(8*b)) + (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(4*b) - (3*ArcSinh[a + b*x]^2)/(8*b) - (3*(a + b*x)^2*ArcSinh[a + b*x]^2)/(4*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^3)/(2*b) + ArcSinh[a + b*x]^4/(8*b)} -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2, x, 6, ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(4*b) - ArcSinh[a + b*x]/(4*b) - ((a + b*x)^2*ArcSinh[a + b*x])/(2*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(2*b) + ArcSinh[a + b*x]^3/(6*b)} -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x], x, 4, -(a + b*x)^2/(4*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(2*b) + ArcSinh[a + b*x]^2/(4*b)} -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x], x, 5, CoshIntegral[2*ArcSinh[a + b*x]]/(2*b) + Log[ArcSinh[a + b*x]]/(2*b)} -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^2, x, 6, -((1 + (a + b*x)^2)/(b*ArcSinh[a + b*x])) + SinhIntegral[2*ArcSinh[a + b*x]]/b} -{Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^3, x, 4, -(1 + (a + b*x)^2)/(2*b*ArcSinh[a + b*x]^2) - ((a + b*x)*Sqrt[1 + (a + b*x)^2])/(b*ArcSinh[a + b*x]) + CoshIntegral[2*ArcSinh[a + b*x]]/b} - - -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^3, x, 15, -((51*(a + b*x)^2)/(128*b)) - (3*(a + b*x)^4)/(128*b) + (45*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(64*b) + (3*(a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x])/(32*b) - (27*ArcSinh[a + b*x]^2)/(128*b) - (9*(a + b*x)^2*ArcSinh[a + b*x]^2)/(16*b) - (3*(1 + (a + b*x)^2)^2*ArcSinh[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^3)/(8*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^3)/(4*b) + (3*ArcSinh[a + b*x]^4)/(32*b)} -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^2, x, 11, (15*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(64*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2))/(32*b) - (9*ArcSinh[a + b*x])/(64*b) - (3*(a + b*x)^2*ArcSinh[a + b*x])/(8*b) - ((1 + (a + b*x)^2)^2*ArcSinh[a + b*x])/(8*b) + (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(8*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^2)/(4*b) + ArcSinh[a + b*x]^3/(8*b)} -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x], x, 7, -((5*(a + b*x)^2)/(16*b)) - (a + b*x)^4/(16*b) + (3*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(8*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x])/(4*b) + (3*ArcSinh[a + b*x]^2)/(16*b)} -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x], x, 6, CoshIntegral[2*ArcSinh[a + b*x]]/(2*b) + CoshIntegral[4*ArcSinh[a + b*x]]/(8*b) + (3*Log[ArcSinh[a + b*x]])/(8*b)} -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x]^2, x, 7, -((1 + (a + b*x)^2)^2/(b*ArcSinh[a + b*x])) + SinhIntegral[2*ArcSinh[a + b*x]]/b + SinhIntegral[4*ArcSinh[a + b*x]]/(2*b)} -{(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x]^3, x, 11, -(1 + (a + b*x)^2)^2/(2*b*ArcSinh[a + b*x]^2) - (2*(a + b*x)*(1 + (a + b*x)^2)^(3/2))/(b*ArcSinh[a + b*x]) + CoshIntegral[2*ArcSinh[a + b*x]]/b + CoshIntegral[4*ArcSinh[a + b*x]]/b} - - -(* ::Subsubsection::Closed:: *) -(*m<0*) - - -{ArcSinh[a + b*x]^3/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 2, ArcSinh[a + b*x]^4/(4*b)} -{ArcSinh[a + b*x]^2/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 2, ArcSinh[a + b*x]^3/(3*b)} -{ArcSinh[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2], x, 2, ArcSinh[a + b*x]^2/(2*b)} -{1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]), x, 2, Log[ArcSinh[a + b*x]]/b} -{1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2), x, 2, -(1/(b*ArcSinh[a + b*x]))} -{1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3), x, 2, -1/(2*b*ArcSinh[a + b*x]^2)} - - -{ArcSinh[a + b*x]^3/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2), x, 8, ArcSinh[a + b*x]^3/b + ((a + b*x)*ArcSinh[a + b*x]^3)/(b*Sqrt[1 + (a + b*x)^2]) - (3*ArcSinh[a + b*x]^2*Log[1 + E^(2*ArcSinh[a + b*x])])/b - (3*ArcSinh[a + b*x]*PolyLog[2, -E^(2*ArcSinh[a + b*x])])/b + (3*PolyLog[3, -E^(2*ArcSinh[a + b*x])])/(2*b)} -{ArcSinh[a + b*x]^2/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2), x, 7, ArcSinh[a + b*x]^2/b + ((a + b*x)*ArcSinh[a + b*x]^2)/(b*Sqrt[1 + (a + b*x)^2]) - (2*ArcSinh[a + b*x]*Log[1 + E^(2*ArcSinh[a + b*x])])/b - PolyLog[2, -E^(2*ArcSinh[a + b*x])]/b} -{ArcSinh[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2), x, 3, ((a + b*x)*ArcSinh[a + b*x])/(b*Sqrt[1 + (a + b*x)^2]) - Log[1 + (a + b*x)^2]/(2*b)} -{1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]), x, 1, Unintegrable[1/((1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]), x]} -{1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^2), x, 2, -(1/(b*(1 + (a + b*x)^2)*ArcSinh[a + b*x])) - 2*Unintegrable[(a + b*x)/((1 + (a + b*x)^2)^2*ArcSinh[a + b*x]), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcSinh[a x^n]^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSinh[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcSinh[a*x^2], x, 5, -((x^2*Sqrt[1 + a^2*x^4])/(8*a)) + ArcSinh[a*x^2]/(8*a^2) + (1/4)*x^4*ArcSinh[a*x^2]} -{x^2*ArcSinh[a*x^2], x, 4, -((2*x*Sqrt[1 + a^2*x^4])/(9*a)) + (1/3)*x^3*ArcSinh[a*x^2] + ((1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/(9*a^(3/2)*Sqrt[1 + a^2*x^4])} -{x^1*ArcSinh[a*x^2], x, 3, -(Sqrt[1 + a^2*x^4]/(2*a)) + (1/2)*x^2*ArcSinh[a*x^2]} -{x^0*ArcSinh[a*x^2], x, 5, -((2*x*Sqrt[1 + a^2*x^4])/(1 + a*x^2)) + x*ArcSinh[a*x^2] + (2*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticE[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4]) - ((1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/(Sqrt[a]*Sqrt[1 + a^2*x^4])} -{ArcSinh[a*x^2]/x^1, x, 5, (-(1/4))*ArcSinh[a*x^2]^2 + (1/2)*ArcSinh[a*x^2]*Log[1 - E^(2*ArcSinh[a*x^2])] + (1/4)*PolyLog[2, E^(2*ArcSinh[a*x^2])]} -{ArcSinh[a*x^2]/x^2, x, 3, -(ArcSinh[a*x^2]/x) + (Sqrt[a]*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/Sqrt[1 + a^2*x^4]} -{ArcSinh[a*x^2]/x^3, x, 5, -(ArcSinh[a*x^2]/(2*x^2)) - (1/2)*a*ArcTanh[Sqrt[1 + a^2*x^4]]} -{ArcSinh[a*x^2]/x^4, x, 6, -((2*a*Sqrt[1 + a^2*x^4])/(3*x)) + (2*a^2*x*Sqrt[1 + a^2*x^4])/(3*(1 + a*x^2)) - ArcSinh[a*x^2]/(3*x^3) - (2*a^(3/2)*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticE[2*ArcTan[Sqrt[a]*x], 1/2])/(3*Sqrt[1 + a^2*x^4]) + (a^(3/2)*(1 + a*x^2)*Sqrt[(1 + a^2*x^4)/(1 + a*x^2)^2]*EllipticF[2*ArcTan[Sqrt[a]*x], 1/2])/(3*Sqrt[1 + a^2*x^4])} - - -{ArcSinh[a*x^5]/x, x, 5, (-(1/10))*ArcSinh[a*x^5]^2 + (1/5)*ArcSinh[a*x^5]*Log[1 - E^(2*ArcSinh[a*x^5])] + (1/10)*PolyLog[2, E^(2*ArcSinh[a*x^5])]} - - -{x^2*ArcSinh[Sqrt[x]], x, 7, (-(5/48))*Sqrt[x]*Sqrt[1 + x] + (5/72)*x^(3/2)*Sqrt[1 + x] - (1/18)*x^(5/2)*Sqrt[1 + x] + (5*ArcSinh[Sqrt[x]])/48 + (1/3)*x^3*ArcSinh[Sqrt[x]]} -{x^1*ArcSinh[Sqrt[x]], x, 6, (3/16)*Sqrt[x]*Sqrt[1 + x] - (1/8)*x^(3/2)*Sqrt[1 + x] - (3*ArcSinh[Sqrt[x]])/16 + (1/2)*x^2*ArcSinh[Sqrt[x]]} -{x^0*ArcSinh[Sqrt[x]], x, 6, (-(1/2))*Sqrt[x]*Sqrt[1 + x] + ArcSinh[Sqrt[x]]/2 + x*ArcSinh[Sqrt[x]]} -{ArcSinh[Sqrt[x]]/x^1, x, 5, -ArcSinh[Sqrt[x]]^2 + 2*ArcSinh[Sqrt[x]]*Log[1 - E^(2*ArcSinh[Sqrt[x]])] + PolyLog[2, E^(2*ArcSinh[Sqrt[x]])]} -{ArcSinh[Sqrt[x]]/x^2, x, 3, -(Sqrt[1 + x]/Sqrt[x]) - ArcSinh[Sqrt[x]]/x} -{ArcSinh[Sqrt[x]]/x^3, x, 4, -(Sqrt[1 + x]/(6*x^(3/2))) + Sqrt[1 + x]/(3*Sqrt[x]) - ArcSinh[Sqrt[x]]/(2*x^2)} -{ArcSinh[Sqrt[x]]/x^4, x, 5, -(Sqrt[1 + x]/(15*x^(5/2))) + (4*Sqrt[1 + x])/(45*x^(3/2)) - (8*Sqrt[1 + x])/(45*Sqrt[x]) - ArcSinh[Sqrt[x]]/(3*x^3)} -{ArcSinh[Sqrt[x]]/x^5, x, 6, -(Sqrt[1 + x]/(28*x^(7/2))) + (3*Sqrt[1 + x])/(70*x^(5/2)) - (2*Sqrt[1 + x])/(35*x^(3/2)) + (4*Sqrt[1 + x])/(35*Sqrt[x]) - ArcSinh[Sqrt[x]]/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2*ArcSinh[a/x], x, 6, (1/6)*a*Sqrt[1 + a^2/x^2]*x^2 + (1/3)*x^3*ArcCsch[x/a] - (1/6)*a^3*ArcTanh[Sqrt[1 + a^2/x^2]]} -{x^1*ArcSinh[a/x], x, 3, (1/2)*a*Sqrt[1 + a^2/x^2]*x + (1/2)*x^2*ArcCsch[x/a]} -{x^0*ArcSinh[a/x], x, 5, x*ArcCsch[x/a] + a*ArcTanh[Sqrt[1 + a^2/x^2]]} -{ArcSinh[a/x]/x^1, x, 5, (1/2)*ArcSinh[a/x]^2 - ArcSinh[a/x]*Log[1 - E^(2*ArcSinh[a/x])] - (1/2)*PolyLog[2, E^(2*ArcSinh[a/x])]} -{ArcSinh[a/x]/x^2, x, 3, Sqrt[1 + a^2/x^2]/a - ArcCsch[x/a]/x} -{ArcSinh[a/x]/x^3, x, 5, Sqrt[1 + a^2/x^2]/(4*a*x) - ArcCsch[x/a]/(4*a^2) - ArcCsch[x/a]/(2*x^2)} -{ArcSinh[a/x]/x^4, x, 5, -(Sqrt[1 + a^2/x^2]/(3*a^3)) + (1 + a^2/x^2)^(3/2)/(9*a^3) - ArcCsch[x/a]/(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSinh[a x^n] with n symbolic*) - - -{x^m*ArcSinh[a*x^n], x, 3, (x^(1 + m)*ArcSinh[a*x^n])/(1 + m) - (a*n*x^(1 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), (-a^2)*x^(2*n)])/((1 + m)*(1 + m + n))} - -{x^2*ArcSinh[a*x^n], x, 3, (1/3)*x^3*ArcSinh[a*x^n] - (a*n*x^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/(2*n), (3*(1 + n))/(2*n), (-a^2)*x^(2*n)])/(3*(3 + n))} -{x^1*ArcSinh[a*x^n], x, 3, (1/2)*x^2*ArcSinh[a*x^n] - (a*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), (1/2)*(3 + 2/n), (-a^2)*x^(2*n)])/(2*(2 + n))} -{x^0*ArcSinh[a*x^n], x, 3, x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (1/2)*(3 + 1/n), (-a^2)*x^(2*n)])/(1 + n)} -{ArcSinh[a*x^n]/x^1, x, 5, -(ArcSinh[a*x^n]^2/(2*n)) + (ArcSinh[a*x^n]*Log[1 - E^(2*ArcSinh[a*x^n])])/n + PolyLog[2, E^(2*ArcSinh[a*x^n])]/(2*n)} -{ArcSinh[a*x^n]/x^2, x, 3, -(ArcSinh[a*x^n]/x) - (a*n*x^(-1 + n)*Hypergeometric2F1[1/2, -((1 - n)/(2*n)), (1/2)*(3 - 1/n), (-a^2)*x^(2*n)])/(1 - n)} -{ArcSinh[a*x^n]/x^3, x, 3, -(ArcSinh[a*x^n]/(2*x^2)) - (a*n*x^(-2 + n)*Hypergeometric2F1[1/2, (1/2)*(1 - 2/n), (1/2)*(3 - 2/n), (-a^2)*x^(2*n)])/(2*(2 - n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b ArcSinh[c+d x^2])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcSinh[c+d x^2])^n when c^2=-1*) - - -{(a + b*ArcSinh[I + d*x^2])^4, x, 3, 384*b^4*x - (192*b^3*Sqrt[2*I*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2]))/(d*x) + 48*b^2*x*(a + I*b*ArcSin[1 - I*d*x^2])^2 - (8*b*Sqrt[2*I*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2])^3)/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^4} -{(a + b*ArcSinh[I + d*x^2])^3, x, 5, 24*a*b^2*x - (48*b^3*Sqrt[2*I*d*x^2 + d^2*x^4])/(d*x) + 24*I*b^3*x*ArcSin[1 - I*d*x^2] - (6*b*Sqrt[2*I*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2])^2)/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^3} -{(a + b*ArcSinh[I + d*x^2])^2, x, 2, 8*b^2*x - (4*b*Sqrt[2*I*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2]))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^2} -{(a + b*ArcSinh[I + d*x^2])^1, x, 4, a*x - (2*b*Sqrt[2*I*d*x^2 + d^2*x^4])/(d*x) + I*b*x*ArcSin[1 - I*d*x^2]} -{1/(a + b*ArcSinh[I + d*x^2])^1, x, 1, (x*CosIntegral[-((I*(a + I*b*ArcSin[1 - I*d*x^2]))/(2*b))]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(2*b*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) - (x*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)])*SinIntegral[(I*a)/(2*b) - (1/2)*ArcSin[1 - I*d*x^2]])/(2*b*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} -{1/(a + b*ArcSinh[I + d*x^2])^2, x, 1, -(Sqrt[2*I*d*x^2 + d^2*x^4]/(2*b*d*x*(a + I*b*ArcSin[1 - I*d*x^2]))) + (x*CosIntegral[-((I*(a + I*b*ArcSin[1 - I*d*x^2]))/(2*b))]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(4*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) + (x*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinIntegral[(I*a)/(2*b) - (1/2)*ArcSin[1 - I*d*x^2]])/(4*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} -{1/(a + b*ArcSinh[I + d*x^2])^3, x, 2, -(Sqrt[2*I*d*x^2 + d^2*x^4]/(4*b*d*x*(a + I*b*ArcSin[1 - I*d*x^2])^2)) - x/(8*b^2*(a + I*b*ArcSin[1 - I*d*x^2])) + (x*CosIntegral[-((I*(a + I*b*ArcSin[1 - I*d*x^2]))/(2*b))]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(16*b^3*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) - (x*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)])*SinIntegral[(I*a)/(2*b) - (1/2)*ArcSin[1 - I*d*x^2]])/(16*b^3*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} - - -{(a + b*ArcSinh[-I + d*x^2])^4, x, 3, 384*b^4*x - (192*b^3*Sqrt[-2*I*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2]))/(d*x) + 48*b^2*x*(a - I*b*ArcSin[1 + I*d*x^2])^2 - (8*b*Sqrt[-2*I*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2])^3)/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^4} -{(a + b*ArcSinh[-I + d*x^2])^3, x, 5, 24*a*b^2*x - (48*b^3*Sqrt[-2*I*d*x^2 + d^2*x^4])/(d*x) - 24*I*b^3*x*ArcSin[1 + I*d*x^2] - (6*b*Sqrt[-2*I*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2])^2)/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^3} -{(a + b*ArcSinh[-I + d*x^2])^2, x, 2, 8*b^2*x - (4*b*Sqrt[-2*I*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2]))/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^2} -{(a + b*ArcSinh[-I + d*x^2])^1, x, 4, a*x - (2*b*Sqrt[-2*I*d*x^2 + d^2*x^4])/(d*x) - I*b*x*ArcSin[1 + I*d*x^2]} -{1/(a + b*ArcSinh[-I + d*x^2])^1, x, 1, -((x*CosIntegral[(I*(a - I*b*ArcSin[1 + I*d*x^2]))/(2*b)]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(2*b*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))) + (x*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinhIntegral[(a - I*b*ArcSin[1 + I*d*x^2])/(2*b)])/(2*b*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{1/(a + b*ArcSinh[-I + d*x^2])^2, x, 1, -(Sqrt[-2*I*d*x^2 + d^2*x^4]/(2*b*d*x*(a - I*b*ArcSin[1 + I*d*x^2]))) + (x*CosIntegral[(I*(a - I*b*ArcSin[1 + I*d*x^2]))/(2*b)]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(4*b^2*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) - (x*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)])*SinhIntegral[(a - I*b*ArcSin[1 + I*d*x^2])/(2*b)])/(4*b^2*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{1/(a + b*ArcSinh[-I + d*x^2])^3, x, 2, -(Sqrt[-2*I*d*x^2 + d^2*x^4]/(4*b*d*x*(a - I*b*ArcSin[1 + I*d*x^2])^2)) - x/(8*b^2*(a - I*b*ArcSin[1 + I*d*x^2])) - (x*CosIntegral[(I*(a - I*b*ArcSin[1 + I*d*x^2]))/(2*b)]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(16*b^3*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) + (x*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinhIntegral[(a - I*b*ArcSin[1 + I*d*x^2])/(2*b)])/(16*b^3*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcSinh[c+d x^2])^(n/2) when c^2=-1*) - - -{(a + b*ArcSinh[I + d*x^2])^(5/2), x, 2, 15*b^2*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]] - (5*b*Sqrt[2*I*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2) + (15*b^2*Sqrt[Pi]*x*FresnelS[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[-(I/b)]*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) - (15*Sqrt[-(I/b)]*b^3*Sqrt[Pi]*x*FresnelC[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])} -{(a + b*ArcSinh[I + d*x^2])^(3/2), x, 2, -((3*b*Sqrt[2*I*d*x^2 + d^2*x^4]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/(d*x)) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2) + (3*Sqrt[I*b]*b*Sqrt[Pi]*x*FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]) - (3*b^2*Sqrt[Pi]*x*FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[I*b]*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} -{(a + b*ArcSinh[I + d*x^2])^(1/2), x, 1, x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]] + (Sqrt[Pi]*x*FresnelS[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[-(I/b)]*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) - (Sqrt[-(I/b)]*b*Sqrt[Pi]*x*FresnelC[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])} -{1/(a + b*ArcSinh[I + d*x^2])^(1/2), x, 1, -((Sqrt[Pi]*x*FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[I*b]*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))) - (Sqrt[Pi]*x*FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[I*b]*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} -{1/(a + b*ArcSinh[I + d*x^2])^(3/2), x, 1, -(Sqrt[2*I*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])) - ((-(I/b))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]) + ((-(I/b))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])} -{1/(a + b*ArcSinh[I + d*x^2])^(5/2), x, 2, -(Sqrt[2*I*d*x^2 + d^2*x^4]/(3*b*d*x*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2))) - x/(3*b^2*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]) - (Sqrt[Pi]*x*FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(3*Sqrt[I*b]*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) - (Sqrt[Pi]*x*FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(3*Sqrt[I*b]*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} -{1/(a + b*ArcSinh[I + d*x^2])^(7/2), x, 2, -(Sqrt[2*I*d*x^2 + d^2*x^4]/(5*b*d*x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2))) - x/(15*b^2*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2)) - Sqrt[2*I*d*x^2 + d^2*x^4]/(15*b^3*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]) - ((-(I/b))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(15*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]])) + ((-(I/b))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-(I/b)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(15*b^2*(Cos[(1/2)*ArcSin[1 - I*d*x^2]] - Sin[(1/2)*ArcSin[1 - I*d*x^2]]))} - - -{(a + b*ArcSinh[-I + d*x^2])^(5/2), x, 2, 15*b^2*x*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]] - (5*b*Sqrt[-2*I*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2))/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^(5/2) + (15*b^2*Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[I/b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) - (15*b^2*Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[I/b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{(a + b*ArcSinh[-I + d*x^2])^(3/2), x, 2, -((3*b*Sqrt[-2*I*d*x^2 + d^2*x^4]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/(d*x)) + x*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2) - (3*b^2*Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) - (3*Sqrt[(-I)*b]*b*Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])} -{(a + b*ArcSinh[-I + d*x^2])^(1/2), x, 1, x*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]] + (Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[I/b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) - (Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[I/b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{1/(a + b*ArcSinh[-I + d*x^2])^(1/2), x, 1, -((Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))) - (Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{1/(a + b*ArcSinh[-I + d*x^2])^(3/2), x, 1, -(Sqrt[-2*I*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])) + ((I/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]) - ((I/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])} -{1/(a + b*ArcSinh[-I + d*x^2])^(5/2), x, 2, -(Sqrt[-2*I*d*x^2 + d^2*x^4]/(3*b*d*x*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2))) - x/(3*b^2*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]) - (Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(3*Sqrt[(-I)*b]*b^2*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) - (Sqrt[(-I)*b]*Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(3*b^3*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} -{1/(a + b*ArcSinh[-I + d*x^2])^(7/2), x, 2, -(Sqrt[-2*I*d*x^2 + d^2*x^4]/(5*b*d*x*(a - I*b*ArcSin[1 + I*d*x^2])^(5/2))) - x/(15*b^2*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2)) - Sqrt[-2*I*d*x^2 + d^2*x^4]/(15*b^3*d*x*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]) - ((I/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(15*b^2*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]])) + (Sqrt[I/b]*Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(15*b^3*(Cos[(1/2)*ArcSin[1 + I*d*x^2]] - Sin[(1/2)*ArcSin[1 + I*d*x^2]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b ArcSinh[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 8, -((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(4*b*c)) - ((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 - E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (3*b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^3*PolyLog[4, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)} -{(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(3*b*c)) - ((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Log[1 - E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b^2*PolyLog[3, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 6, -((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(2*b*c)) - ((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*Log[1 - E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b*PolyLog[2, E^(-2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{1/((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse hyperbolic sines of exponentials*) - - -(* ::Subsection::Closed:: *) -(*x^m ArcSinh[c E^(a+b x)]*) - - -{ArcSinh[c*E^(a + b*x)], x, 6, -(ArcSinh[c*E^(a + b*x)]^2/(2*b)) + (ArcSinh[c*E^(a + b*x)]*Log[1 - E^(2*ArcSinh[c*E^(a + b*x)])])/b + PolyLog[2, E^(2*ArcSinh[c*E^(a + b*x)])]/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse hyperbolic sines*) - - -(* ::Subsection::Closed:: *) -(*x^m E^(ArcSinh[a+b x]^n)*) - - -{x^3*E^ArcSinh[a + b*x], x, 5, 1/(E^(3*ArcSinh[a + b*x])*(48*b^4)) + (3*a)/(E^(2*ArcSinh[a + b*x])*(16*b^4)) - (1 - 6*a^2)/(E^ArcSinh[a + b*x]*(8*b^4)) + (a*(3 - 4*a^2)*E^(2*ArcSinh[a + b*x]))/(16*b^4) - ((1 - 6*a^2)*E^(3*ArcSinh[a + b*x]))/(24*b^4) - (3*a*E^(4*ArcSinh[a + b*x]))/(32*b^4) + E^(5*ArcSinh[a + b*x])/(80*b^4) + (a*(3 - 4*a^2)*ArcSinh[a + b*x])/(8*b^4)} -{x^2*E^ArcSinh[a + b*x], x, 5, -(1/(E^(2*ArcSinh[a + b*x])*(16*b^3))) - a/(E^ArcSinh[a + b*x]*(2*b^3)) - ((1 - 4*a^2)*E^(2*ArcSinh[a + b*x]))/(16*b^3) - (a*E^(3*ArcSinh[a + b*x]))/(6*b^3) + E^(4*ArcSinh[a + b*x])/(32*b^3) - ((1 - 4*a^2)*ArcSinh[a + b*x])/(8*b^3)} -{x^1*E^ArcSinh[a + b*x], x, 5, 1/(E^ArcSinh[a + b*x]*(4*b^2)) - (a*E^(2*ArcSinh[a + b*x]))/(4*b^2) + E^(3*ArcSinh[a + b*x])/(12*b^2) - (a*ArcSinh[a + b*x])/(2*b^2)} -{x^0*E^ArcSinh[a + b*x], x, 5, E^(2*ArcSinh[a + b*x])/(4*b) + ArcSinh[a + b*x]/(2*b)} -{E^ArcSinh[a + b*x]/x^1, x, 9, b*x + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + a*ArcSinh[a + b*x] - Sqrt[1 + a^2]*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])] + a*Log[x]} -{E^ArcSinh[a + b*x]/x^2, x, 9, -(a/x) - Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/x + b*ArcSinh[a + b*x] - (a*b*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])])/Sqrt[1 + a^2] + b*Log[x]} -{E^ArcSinh[a + b*x]/x^3, x, 6, -(a/(2*x^2)) - b/x - ((1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(2*(1 + a^2)*x^2) - (b^2*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])])/(2*(1 + a^2)^(3/2))} -{E^ArcSinh[a + b*x]/x^4, x, 7, -(a/(3*x^3)) - b/(2*x^2) + (a*b*(1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(2*(1 + a^2)^2*x^2) - (1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(3*(1 + a^2)*x^3) + (a*b^3*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])])/(2*(1 + a^2)^(5/2))} -{E^ArcSinh[a + b*x]/x^5, x, 8, -(a/(4*x^4)) - b/(3*x^3) + ((1 - 4*a^2)*b^2*(1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(8*(1 + a^2)^3*x^2) - (1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(4*(1 + a^2)*x^4) + (5*a*b*(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(12*(1 + a^2)^2*x^3) + ((1 - 4*a^2)*b^4*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])])/(8*(1 + a^2)^(7/2))} - - -{x^3*E^(ArcSinh[a + b*x]^2), x, 37, -((Sqrt[Pi]*Erfi[-2 + ArcSinh[a + b*x]])/(32*b^4*E^4)) + (Sqrt[Pi]*Erfi[-1 + ArcSinh[a + b*x]])/(16*b^4*E) - (3*a^2*Sqrt[Pi]*Erfi[-1 + ArcSinh[a + b*x]])/(8*b^4*E) - (Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(16*b^4*E) + (3*a^2*Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(8*b^4*E) + (Sqrt[Pi]*Erfi[2 + ArcSinh[a + b*x]])/(32*b^4*E^4) - (3*a*Sqrt[Pi]*Erfi[(1/2)*(-3 + 2*ArcSinh[a + b*x])])/(16*b^4*E^(9/4)) + (3*a*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(16*b^4*E^(1/4)) - (a^3*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(4*b^4*E^(1/4)) + (3*a*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(16*b^4*E^(1/4)) - (a^3*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(4*b^4*E^(1/4)) - (3*a*Sqrt[Pi]*Erfi[(1/2)*(3 + 2*ArcSinh[a + b*x])])/(16*b^4*E^(9/4))} -{x^2*E^(ArcSinh[a + b*x]^2), x, 27, (a*Sqrt[Pi]*Erfi[-1 + ArcSinh[a + b*x]])/(4*b^3*E) - (a*Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(4*b^3*E) + (Sqrt[Pi]*Erfi[(1/2)*(-3 + 2*ArcSinh[a + b*x])])/(16*b^3*E^(9/4)) - (Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(16*b^3*E^(1/4)) + (a^2*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(4*b^3*E^(1/4)) - (Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(16*b^3*E^(1/4)) + (a^2*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(4*b^3*E^(1/4)) + (Sqrt[Pi]*Erfi[(1/2)*(3 + 2*ArcSinh[a + b*x])])/(16*b^3*E^(9/4))} -{x^1*E^(ArcSinh[a + b*x]^2), x, 17, -((Sqrt[Pi]*Erfi[-1 + ArcSinh[a + b*x]])/(8*b^2*E)) + (Sqrt[Pi]*Erfi[1 + ArcSinh[a + b*x]])/(8*b^2*E) - (a*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(4*b^2*E^(1/4)) - (a*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(4*b^2*E^(1/4))} -{x^0*E^(ArcSinh[a + b*x]^2), x, 7, (Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcSinh[a + b*x])])/(4*b*E^(1/4)) + (Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcSinh[a + b*x])])/(4*b*E^(1/4))} -{E^(ArcSinh[a + b*x]^2)/x^1, x, 0, CannotIntegrate[E^ArcSinh[a + b*x]^2/x, x]} -{E^(ArcSinh[a + b*x]^2)/x^2, x, 0, CannotIntegrate[E^ArcSinh[a + b*x]^2/x^2, x]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic sines*) - - -{ArcSinh[a + b*x]/((a*d)/b + d*x), x, 7, -(ArcSinh[a + b*x]^2/(2*d)) + (ArcSinh[a + b*x]*Log[1 - E^(2*ArcSinh[a + b*x])])/d + PolyLog[2, E^(2*ArcSinh[a + b*x])]/(2*d)} - - -{x/(Sqrt[1 + x^2]*ArcSinh[x]), x, 2, SinhIntegral[ArcSinh[x]]} - - -{x^3*ArcSinh[a + b*x^4], x, 4, -(Sqrt[1 + (a + b*x^4)^2]/(4*b)) + ((a + b*x^4)*ArcSinh[a + b*x^4])/(4*b)} - -{x^(n-1)*ArcSinh[a + b*x^n], x, 4, -(Sqrt[1 + (a + b*x^n)^2]/(b*n)) + ((a + b*x^n)*ArcSinh[a + b*x^n])/(b*n)} - - -{ArcSinh[c/(a + b*x)], x, 6, ((a + b*x)*ArcCsch[a/c + (b*x)/c])/b + (c*ArcTanh[Sqrt[1 + 1/(a/c + (b*x)/c)^2]])/b} - - -{x/ArcSinh[Sinh[x]], x, -1, ArcSinh[Sinh[x]] + Log[ArcSinh[Sinh[x]]]*(-ArcSinh[Sinh[x]] + x*Sqrt[Cosh[x]^2]*Sech[x])} - - -{ArcSinh[Sqrt[-1 + b*x^2]]^n/Sqrt[-1 + b*x^2], x, 2, (Sqrt[b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)} -{1/(ArcSinh[Sqrt[-1 + b*x^2]]*Sqrt[-1 + b*x^2]), x, 2, (Sqrt[b*x^2]*Log[ArcSinh[Sqrt[-1 + b*x^2]]])/(b*x)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/(d+e x)^p (-d+e x)^q (a+b arccosh(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/(d+e x)^p (-d+e x)^q (a+b arccosh(c x))^n.m deleted file mode 100644 index 5823ef0..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/(d+e x)^p (-d+e x)^q (a+b arccosh(c x))^n.m +++ /dev/null @@ -1,32 +0,0 @@ -(* ::Package:: *) - -{(-1 + c*x)^(5/2)*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(3/2)*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(1/2)*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 3, (-(1/4))*b*c*x^2 + (1/2)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]) - (a + b*ArcCosh[c*x])^2/(4*b*c)} -{Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])/(-1 + c*x)^(1/2), x, 0, 0} -{Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])/(-1 + c*x)^(3/2), x, 0, 0} -{Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])/(-1 + c*x)^(5/2), x, 7, (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2])/(3*c*Sqrt[-1 + c*x]*(-((1 - c*x)/(1 + c*x)))^(3/2)*(1 + c*x)^(3/2)) - ((1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c*(-1 + c*x)^(3/2)) - (2*b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(3*c*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(3*c*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2))} - - -{(-1 + c*x)^(5/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 6, (5/16)*b*c*x^2 - (1/16)*b*c^3*x^4 - (3/8)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]) + (1/4)*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + (3*(a + b*ArcCosh[c*x])^2)/(16*b*c)} -{(-1 + c*x)^(1/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 0, 0} -{(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])/(-1 + c*x)^(1/2), x, 0, 0} -{(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])/(-1 + c*x)^(3/2), x, 0, 0} -{(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])/(-1 + c*x)^(5/2), x, 0, 0} - - -{(-1 + c*x)^(5/2)/Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(3/2)/Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(1/2)/Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x, 0, (-b)*x + (b*ArcCosh[c*x]^2)/(2*c) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (ArcCosh[c*x]*(a + b*ArcCosh[c*x]))/c} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(1/2)*Sqrt[1 + c*x]), x, 1, (a + b*ArcCosh[c*x])^2/(2*b*c)} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]), x, 8, -((Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(c*Sqrt[-1 + c*x])) - (2*b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(c*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2))} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(5/2)*Sqrt[1 + c*x]), x, 11, -(b/(3*c*(1 - c*x))) + (2*b*Sqrt[1 - c^2*x^2])/(3*c*(1 - c*x)^(3/2)*Sqrt[1 + c*x]) - (Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c*(-1 + c*x)^(3/2)) + (Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c*Sqrt[-1 + c*x]) - (2*b*Log[Sqrt[-1 + c*x]])/(3*c)} - - -{(-1 + c*x)^(5/2)/(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(3/2)/(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 0, 0} -{(-1 + c*x)^(1/2)/(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 0, 0} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(1/2)*(1 + c*x)^(3/2)), x, 5, (Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x]))/(c*Sqrt[1 + c*x]) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(c*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2))} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)), x, 2, -((x*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*Log[1 - c^2*x^2])/(2*c)} -{(a + b*ArcCosh[c*x])/((-1 + c*x)^(5/2)*(1 + c*x)^(3/2)), x, 19, (b*Sqrt[1 - c^2*x^2])/(6*c*(1 - c*x)^(3/2)*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*c*(-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (2*x*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(6*c*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (2*b*Log[Sqrt[-1 + c*x]])/(3*c) - (b*Log[1 + c*x])/(3*c)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m deleted file mode 100644 index ab4e348..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.2 (d x)^m (a+b arccosh(c x))^n.m +++ /dev/null @@ -1,308 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcCosh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (b x)^m ArcCosh[a x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCosh[c x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*ArcCosh[a*x], x, 6, -((8*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^5)) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(75*a^3) - (x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(25*a) + (1/5)*x^5*ArcCosh[a*x]} -{x^3*ArcCosh[a*x], x, 5, -((3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(32*a^3)) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(16*a) - (3*ArcCosh[a*x])/(32*a^4) + (1/4)*x^4*ArcCosh[a*x]} -{x^2*ArcCosh[a*x], x, 4, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3)) - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + (1/3)*x^3*ArcCosh[a*x]} -{x^1*ArcCosh[a*x], x, 3, -((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*a)) - ArcCosh[a*x]/(4*a^2) + (1/2)*x^2*ArcCosh[a*x]} -{x^0*ArcCosh[a*x], x, 2, -((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a) + x*ArcCosh[a*x]} -{ArcCosh[a*x]/x^1, x, 5, (-(1/2))*ArcCosh[a*x]^2 + ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] + (1/2)*PolyLog[2, -E^(2*ArcCosh[a*x])]} -{ArcCosh[a*x]/x^2, x, 3, -(ArcCosh[a*x]/x) + a*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]} -{ArcCosh[a*x]/x^3, x, 2, (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*x) - ArcCosh[a*x]/(2*x^2)} -{ArcCosh[a*x]/x^4, x, 5, (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*x^2) - ArcCosh[a*x]/(3*x^3) + (1/6)*a^3*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]} -{ArcCosh[a*x]/x^5, x, 4, (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(12*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*x) - ArcCosh[a*x]/(4*x^4)} -{ArcCosh[a*x]/x^6, x, 7, (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*x^4) + (3*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(40*x^2) - ArcCosh[a*x]/(5*x^5) + (3/40)*a^5*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]]} - - -{x^4*ArcCosh[a*x]^2, x, 7, (16*x)/(75*a^4) + (8*x^3)/(225*a^2) + (2*x^5)/125 - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^5) - (8*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(75*a^3) - (2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(25*a) + (x^5*ArcCosh[a*x]^2)/5} -{x^3*ArcCosh[a*x]^2, x, 6, (3*x^2)/(32*a^2) + x^4/32 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(16*a^3) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(8*a) - (3*ArcCosh[a*x]^2)/(32*a^4) + (x^4*ArcCosh[a*x]^2)/4} -{x^2*ArcCosh[a*x]^2, x, 5, (4*x)/(9*a^2) + (2*x^3)/27 - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^3) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a) + (x^3*ArcCosh[a*x]^2)/3} -{x^1*ArcCosh[a*x]^2, x, 4, x^2/4 - (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a) - ArcCosh[a*x]^2/(4*a^2) + (x^2*ArcCosh[a*x]^2)/2} -{x^0*ArcCosh[a*x]^2, x, 3, 2*x - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a + x*ArcCosh[a*x]^2} -{ArcCosh[a*x]^2/x^1, x, 6, -ArcCosh[a*x]^3/3 + ArcCosh[a*x]^2*Log[1 + E^(2*ArcCosh[a*x])] + ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])] - PolyLog[3, -E^(2*ArcCosh[a*x])]/2} -{ArcCosh[a*x]^2/x^2, x, 7, -(ArcCosh[a*x]^2/x) + 4*a*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]] - (2*I)*a*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (2*I)*a*PolyLog[2, I*E^ArcCosh[a*x]]} -{ArcCosh[a*x]^2/x^3, x, 3, (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/x - ArcCosh[a*x]^2/(2*x^2) - a^2*Log[x]} -{ArcCosh[a*x]^2/x^4, x, 9, a^2/(3*x) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*x^2) - ArcCosh[a*x]^2/(3*x^3) + (2*a^3*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/3 - (I/3)*a^3*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (I/3)*a^3*PolyLog[2, I*E^ArcCosh[a*x]]} -{ArcCosh[a*x]^2/x^5, x, 5, a^2/(12*x^2) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(6*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*x) - ArcCosh[a*x]^2/(4*x^4) - (a^4*Log[x])/3} - - -{x^4*ArcCosh[a*x]^3, x, 16, -((4144*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^5)) - (272*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5625*a^3) - (6*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(625*a) + (16*x*ArcCosh[a*x])/(25*a^4) + (8*x^3*ArcCosh[a*x])/(75*a^2) + (6/125)*x^5*ArcCosh[a*x] - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(25*a) + (1/5)*x^5*ArcCosh[a*x]^3} -{x^3*ArcCosh[a*x]^3, x, 12, -((45*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(256*a^3)) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(128*a) - (45*ArcCosh[a*x])/(256*a^4) + (9*x^2*ArcCosh[a*x])/(32*a^2) + (3/32)*x^4*ArcCosh[a*x] - (9*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(16*a) - (3*ArcCosh[a*x]^3)/(32*a^4) + (1/4)*x^4*ArcCosh[a*x]^3} -{x^2*ArcCosh[a*x]^3, x, 9, -((40*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a^3)) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (4*x*ArcCosh[a*x])/(3*a^2) + (2/9)*x^3*ArcCosh[a*x] - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a^3) - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(3*a) + (1/3)*x^3*ArcCosh[a*x]^3} -{x^1*ArcCosh[a*x]^3, x, 6, -((3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(8*a)) - (3*ArcCosh[a*x])/(8*a^2) + (3/4)*x^2*ArcCosh[a*x] - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*a) - ArcCosh[a*x]^3/(4*a^2) + (1/2)*x^2*ArcCosh[a*x]^3} -{x^0*ArcCosh[a*x]^3, x, 4, -((6*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/a) + 6*x*ArcCosh[a*x] - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + x*ArcCosh[a*x]^3} -{ArcCosh[a*x]^3/x^1, x, 7, (-(1/4))*ArcCosh[a*x]^4 + ArcCosh[a*x]^3*Log[1 + E^(2*ArcCosh[a*x])] + (3/2)*ArcCosh[a*x]^2*PolyLog[2, -E^(2*ArcCosh[a*x])] - (3/2)*ArcCosh[a*x]*PolyLog[3, -E^(2*ArcCosh[a*x])] + (3/4)*PolyLog[4, -E^(2*ArcCosh[a*x])]} -{ArcCosh[a*x]^3/x^2, x, 9, -(ArcCosh[a*x]^3/x) + 6*a*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] - 6*I*a*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]] + 6*I*a*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + 6*I*a*PolyLog[3, (-I)*E^ArcCosh[a*x]] - 6*I*a*PolyLog[3, I*E^ArcCosh[a*x]]} -{ArcCosh[a*x]^3/x^3, x, 7, (3/2)*a^2*ArcCosh[a*x]^2 + (3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCosh[a*x]^3/(2*x^2) - 3*a^2*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (3/2)*a^2*PolyLog[2, -E^(2*ArcCosh[a*x])]} -{ArcCosh[a*x]^3/x^4, x, 13, (a^2*ArcCosh[a*x])/x + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x^2) - ArcCosh[a*x]^3/(3*x^3) + a^3*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] - a^3*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]] - I*a^3*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]] + I*a^3*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + I*a^3*PolyLog[3, (-I)*E^ArcCosh[a*x]] - I*a^3*PolyLog[3, I*E^ArcCosh[a*x]]} -{ArcCosh[a*x]^3/x^5, x, 10, -(a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(4*x) + (a^2*ArcCosh[a*x])/(4*x^2) + (a^4*ArcCosh[a*x]^2)/2 + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(4*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCosh[a*x]^3/(4*x^4) - a^4*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (a^4*PolyLog[2, -E^(2*ArcCosh[a*x])])/2} - - -{x^5*ArcCosh[a*x]^4, x, 23, (245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 - (245*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(576*a^5) - (65*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(864*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(54*a) - (245*ArcCosh[a*x]^2)/(1152*a^6) + (5*x^2*ArcCosh[a*x]^2)/(16*a^4) + (5*x^4*ArcCosh[a*x]^2)/(48*a^2) + (x^6*ArcCosh[a*x]^2)/18 - (5*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(24*a^5) - (5*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(36*a^3) - (x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) - (5*ArcCosh[a*x]^4)/(96*a^6) + (x^6*ArcCosh[a*x]^4)/6} -{x^4*ArcCosh[a*x]^4, x, 19, (16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 - (16576*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(5625*a^5) - (1088*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(5625*a^3) - (24*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(625*a) + (32*x*ArcCosh[a*x]^2)/(25*a^4) + (16*x^3*ArcCosh[a*x]^2)/(75*a^2) + (12/125)*x^5*ArcCosh[a*x]^2 - (32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(75*a^5) - (16*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(25*a) + (1/5)*x^5*ArcCosh[a*x]^4} -{x^3*ArcCosh[a*x]^4, x, 14, (45*x^2)/(128*a^2) + (3*x^4)/128 - (45*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(64*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(32*a) - (45*ArcCosh[a*x]^2)/(128*a^4) + (9*x^2*ArcCosh[a*x]^2)/(16*a^2) + (3/16)*x^4*ArcCosh[a*x]^2 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(8*a^3) - (x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(4*a) - (3*ArcCosh[a*x]^4)/(32*a^4) + (1/4)*x^4*ArcCosh[a*x]^4} -{x^2*ArcCosh[a*x]^4, x, 11, (160*x)/(27*a^2) + (8*x^3)/81 - (160*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a^3) - (8*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(27*a) + (8*x*ArcCosh[a*x]^2)/(3*a^2) + (4/9)*x^3*ArcCosh[a*x]^2 - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a^3) - (4*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(9*a) + (1/3)*x^3*ArcCosh[a*x]^4} -{x^1*ArcCosh[a*x]^4, x, 7, (3*x^2)/4 - (3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(2*a) - (3*ArcCosh[a*x]^2)/(4*a^2) + (3/2)*x^2*ArcCosh[a*x]^2 - (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/a - ArcCosh[a*x]^4/(4*a^2) + (1/2)*x^2*ArcCosh[a*x]^4} -{x^0*ArcCosh[a*x]^4, x, 5, 24*x - (24*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a + 12*x*ArcCosh[a*x]^2 - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/a + x*ArcCosh[a*x]^4} -{ArcCosh[a*x]^4/x^1, x, 8, (-(1/5))*ArcCosh[a*x]^5 + ArcCosh[a*x]^4*Log[1 + E^(2*ArcCosh[a*x])] + 2*ArcCosh[a*x]^3*PolyLog[2, -E^(2*ArcCosh[a*x])] - 3*ArcCosh[a*x]^2*PolyLog[3, -E^(2*ArcCosh[a*x])] + 3*ArcCosh[a*x]*PolyLog[4, -E^(2*ArcCosh[a*x])] - (3/2)*PolyLog[5, -E^(2*ArcCosh[a*x])]} -{ArcCosh[a*x]^4/x^2, x, 11, -(ArcCosh[a*x]^4/x) + 8*a*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] - 12*I*a*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] + 12*I*a*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]] + 24*I*a*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - 24*I*a*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]] - 24*I*a*PolyLog[4, (-I)*E^ArcCosh[a*x]] + 24*I*a*PolyLog[4, I*E^ArcCosh[a*x]]} -{ArcCosh[a*x]^4/x^3, x, 8, 2*a^2*ArcCosh[a*x]^3 + (2*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/x - ArcCosh[a*x]^4/(2*x^2) - 6*a^2*ArcCosh[a*x]^2*Log[1 + E^(2*ArcCosh[a*x])] - 6*a^2*ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])] + 3*a^2*PolyLog[3, -E^(2*ArcCosh[a*x])]} -{ArcCosh[a*x]^4/x^4, x, 19, (2*a^2*ArcCosh[a*x]^2)/x + (2*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(3*x^2) - ArcCosh[a*x]^4/(3*x^3) - 8*a^3*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]] + (4/3)*a^3*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] + 4*I*a^3*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 2*I*a^3*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 4*I*a^3*PolyLog[2, I*E^ArcCosh[a*x]] + 2*I*a^3*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]] + 4*I*a^3*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - 4*I*a^3*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]] - 4*I*a^3*PolyLog[4, (-I)*E^ArcCosh[a*x]] + 4*I*a^3*PolyLog[4, I*E^ArcCosh[a*x]]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^6/ArcCosh[a*x], x, 7, (5*SinhIntegral[ArcCosh[a*x]])/(64*a^7) + (9*SinhIntegral[3*ArcCosh[a*x]])/(64*a^7) + (5*SinhIntegral[5*ArcCosh[a*x]])/(64*a^7) + SinhIntegral[7*ArcCosh[a*x]]/(64*a^7)} -{x^5/ArcCosh[a*x], x, 6, (5*SinhIntegral[2*ArcCosh[a*x]])/(32*a^6) + SinhIntegral[4*ArcCosh[a*x]]/(8*a^6) + SinhIntegral[6*ArcCosh[a*x]]/(32*a^6)} -{x^4/ArcCosh[a*x], x, 6, SinhIntegral[ArcCosh[a*x]]/(8*a^5) + (3*SinhIntegral[3*ArcCosh[a*x]])/(16*a^5) + SinhIntegral[5*ArcCosh[a*x]]/(16*a^5)} -{x^3/ArcCosh[a*x], x, 5, SinhIntegral[2*ArcCosh[a*x]]/(4*a^4) + SinhIntegral[4*ArcCosh[a*x]]/(8*a^4)} -{x^2/ArcCosh[a*x], x, 5, SinhIntegral[ArcCosh[a*x]]/(4*a^3) + SinhIntegral[3*ArcCosh[a*x]]/(4*a^3)} -{x^1/ArcCosh[a*x], x, 4, SinhIntegral[2*ArcCosh[a*x]]/(2*a^2)} -{x^0/ArcCosh[a*x], x, 2, SinhIntegral[ArcCosh[a*x]]/a} -{1/(x^1*ArcCosh[a*x]), x, 0, Unintegrable[1/(x*ArcCosh[a*x]), x]} -{1/(x^2*ArcCosh[a*x]), x, 0, Unintegrable[1/(x^2*ArcCosh[a*x]), x]} - - -{x^4/ArcCosh[a*x]^2, x, 5, -((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(8*a^5) + (9*CoshIntegral[3*ArcCosh[a*x]])/(16*a^5) + (5*CoshIntegral[5*ArcCosh[a*x]])/(16*a^5)} -{x^3/ArcCosh[a*x]^2, x, 4, -((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[2*ArcCosh[a*x]]/(2*a^4) + CoshIntegral[4*ArcCosh[a*x]]/(2*a^4)} -{x^2/ArcCosh[a*x]^2, x, 4, -((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(4*a^3) + (3*CoshIntegral[3*ArcCosh[a*x]])/(4*a^3)} -{x^1/ArcCosh[a*x]^2, x, 2, -((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[2*ArcCosh[a*x]]/a^2} -{x^0/ArcCosh[a*x]^2, x, 3, -((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/a} -{1/(x^1*ArcCosh[a*x]^2), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^2), x]} -{1/(x^2*ArcCosh[a*x]^2), x, 0, Unintegrable[1/(x^2*ArcCosh[a*x]^2), x]} - - -{x^4/ArcCosh[a*x]^3, x, 14, -((x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2)) + (2*x^3)/(a^2*ArcCosh[a*x]) - (5*x^5)/(2*ArcCosh[a*x]) + SinhIntegral[ArcCosh[a*x]]/(16*a^5) + (27*SinhIntegral[3*ArcCosh[a*x]])/(32*a^5) + (25*SinhIntegral[5*ArcCosh[a*x]])/(32*a^5)} -{x^3/ArcCosh[a*x]^3, x, 12, -((x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2)) + (3*x^2)/(2*a^2*ArcCosh[a*x]) - (2*x^4)/ArcCosh[a*x] + SinhIntegral[2*ArcCosh[a*x]]/(2*a^4) + SinhIntegral[4*ArcCosh[a*x]]/a^4} -{x^2/ArcCosh[a*x]^3, x, 10, -((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2)) + x/(a^2*ArcCosh[a*x]) - (3*x^3)/(2*ArcCosh[a*x]) + SinhIntegral[ArcCosh[a*x]]/(8*a^3) + (9*SinhIntegral[3*ArcCosh[a*x]])/(8*a^3)} -{x^1/ArcCosh[a*x]^3, x, 7, -((x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2)) + 1/(2*a^2*ArcCosh[a*x]) - x^2/ArcCosh[a*x] + SinhIntegral[2*ArcCosh[a*x]]/a^2} -{x^0/ArcCosh[a*x]^3, x, 4, -((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2)) - x/(2*ArcCosh[a*x]) + SinhIntegral[ArcCosh[a*x]]/(2*a)} -{1/(x^1*ArcCosh[a*x]^3), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^3), x]} -{1/(x^2*ArcCosh[a*x]^3), x, 0, Unintegrable[1/(x^2*ArcCosh[a*x]^3), x]} - - -{x^4/ArcCosh[a*x]^4, x, 12, -(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + (2*x^3)/(3*a^2*ArcCosh[a*x]^2) - (5*x^5)/(6*ArcCosh[a*x]^2) + (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (25*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(48*a^5) + (27*CoshIntegral[3*ArcCosh[a*x]])/(32*a^5) + (125*CoshIntegral[5*ArcCosh[a*x]])/(96*a^5)} -{x^3/ArcCosh[a*x]^4, x, 9, -(x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x^2/(2*a^2*ArcCosh[a*x]^2) - (2*x^4)/(3*ArcCosh[a*x]^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (8*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]) + CoshIntegral[2*ArcCosh[a*x]]/(3*a^4) + (4*CoshIntegral[4*ArcCosh[a*x]])/(3*a^4)} -{x^2/ArcCosh[a*x]^4, x, 10, -(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x/(3*a^2*ArcCosh[a*x]^2) - x^3/(2*ArcCosh[a*x]^2) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^3*ArcCosh[a*x]) - (3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(24*a^3) + (9*CoshIntegral[3*ArcCosh[a*x]])/(8*a^3)} -{x^1/ArcCosh[a*x]^4, x, 5, -(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + 1/(6*a^2*ArcCosh[a*x]^2) - x^2/(3*ArcCosh[a*x]^2) - (2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]) + (2*CoshIntegral[2*ArcCosh[a*x]])/(3*a^2)} -{x^0/ArcCosh[a*x]^4, x, 5, -(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) - x/(6*ArcCosh[a*x]^2) - (Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(6*a)} -{1/(x^1*ArcCosh[a*x]^4), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^4), x]} -{1/(x^2*ArcCosh[a*x]^4), x, 0, Unintegrable[1/(x^2*ArcCosh[a*x]^4), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCosh[c x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^4*Sqrt[ArcCosh[a*x]], x, 19, (1/5)*x^5*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(32*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(64*a^5) - (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(320*a^5) - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(32*a^5) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(64*a^5) - (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(320*a^5)} -{x^3*Sqrt[ArcCosh[a*x]], x, 14, -((3*Sqrt[ArcCosh[a*x]])/(32*a^4)) + (1/4)*x^4*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(256*a^4) - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(32*a^4) - (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(256*a^4) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(32*a^4)} -{x^2*Sqrt[ArcCosh[a*x]], x, 14, (1/3)*x^3*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(16*a^3) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(48*a^3) - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(16*a^3) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(48*a^3)} -{x*Sqrt[ArcCosh[a*x]], x, 9, -(Sqrt[ArcCosh[a*x]]/(4*a^2)) + (1/2)*x^2*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a^2) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a^2)} -{Sqrt[ArcCosh[a*x]], x, 7, x*Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(4*a) - (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(4*a)} -{Sqrt[ArcCosh[a*x]]/x, x, 0, Unintegrable[Sqrt[ArcCosh[a*x]]/x, x]} - - -{x^4*ArcCosh[a*x]^(3/2), x, 41, -((4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(25*a^5)) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(25*a^3) - (3*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(50*a) + (1/5)*x^5*ArcCosh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(64*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(200*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(3200*a^5) - (3*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(3200*a^5) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(64*a^5) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(200*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(3200*a^5) + (3*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(3200*a^5)} -{x^3*ArcCosh[a*x]^(3/2), x, 25, (-9*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(64*a^3) - (3*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(32*a) - (3*ArcCosh[a*x]^(3/2))/(32*a^4) + (x^4*ArcCosh[a*x]^(3/2))/4 - (3*Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(2048*a^4) - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(128*a^4) + (3*Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(2048*a^4) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(128*a^4)} -{x^2*ArcCosh[a*x]^(3/2), x, 22, -(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(3*a^3) - (x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(6*a) + (x^3*ArcCosh[a*x]^(3/2))/3 - (3*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(32*a^3) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(96*a^3) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(96*a^3)} -{x*ArcCosh[a*x]^(3/2), x, 11, (-3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(8*a) - ArcCosh[a*x]^(3/2)/(4*a^2) + (x^2*ArcCosh[a*x]^(3/2))/2 - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a^2) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a^2)} -{ArcCosh[a*x]^(3/2), x, 8, (-3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(2*a) + x*ArcCosh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(8*a) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a)} -{ArcCosh[a*x]^(3/2)/x, x, 0, Unintegrable[ArcCosh[a*x]^(3/2)/x, x]} - - -{x^4*ArcCosh[a*x]^(5/2), x, 44, (2*x*Sqrt[ArcCosh[a*x]])/(5*a^4) + (x^3*Sqrt[ArcCosh[a*x]])/(15*a^2) + (3/100)*x^5*Sqrt[ArcCosh[a*x]] - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(15*a^5) - (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(15*a^3) - (x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(10*a) + (1/5)*x^5*ArcCosh[a*x]^(5/2) - (15*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(128*a^5) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(240*a^5) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(1280*a^5) - (3*Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(6400*a^5) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(128*a^5) - (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(240*a^5) - (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(1280*a^5) - (3*Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(6400*a^5)} -{x^3*ArcCosh[a*x]^(5/2), x, 27, -((225*Sqrt[ArcCosh[a*x]])/(2048*a^4)) + (45*x^2*Sqrt[ArcCosh[a*x]])/(256*a^2) + (15/256)*x^4*Sqrt[ArcCosh[a*x]] - (15*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(64*a^3) - (5*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(32*a) - (3*ArcCosh[a*x]^(5/2))/(32*a^4) + (1/4)*x^4*ArcCosh[a*x]^(5/2) - (15*Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(16384*a^4) - (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(512*a^4) - (15*Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(16384*a^4) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(512*a^4)} -{x^2*ArcCosh[a*x]^(5/2), x, 24, (5*x*Sqrt[ArcCosh[a*x]])/(6*a^2) + (5/36)*x^3*Sqrt[ArcCosh[a*x]] - (5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(9*a^3) - (5*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(18*a) + (1/3)*x^3*ArcCosh[a*x]^(5/2) - (15*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(64*a^3) - (5*Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(576*a^3) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(64*a^3) - (5*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(576*a^3)} -{x*ArcCosh[a*x]^(5/2), x, 12, -((15*Sqrt[ArcCosh[a*x]])/(64*a^2)) + (15/32)*x^2*Sqrt[ArcCosh[a*x]] - (5*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(8*a) - ArcCosh[a*x]^(5/2)/(4*a^2) + (1/2)*x^2*ArcCosh[a*x]^(5/2) - (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a^2) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a^2)} -{ArcCosh[a*x]^(5/2), x, 9, (15/4)*x*Sqrt[ArcCosh[a*x]] - (5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(2*a) + x*ArcCosh[a*x]^(5/2) - (15*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(16*a) - (15*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(16*a)} -{ArcCosh[a*x]^(5/2)/x, x, 0, Unintegrable[ArcCosh[a*x]^(5/2)/x, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/Sqrt[ArcCosh[a*x]], x, 18, -((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(16*a^5)) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(32*a^5) - (Sqrt[Pi/5]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(32*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(32*a^5) + (Sqrt[Pi/5]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(32*a^5)} -{x^3/Sqrt[ArcCosh[a*x]], x, 13, -((Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(32*a^4)) - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(8*a^4) + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(32*a^4) + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(8*a^4)} -{x^2/Sqrt[ArcCosh[a*x]], x, 13, -((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(8*a^3)) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(8*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(8*a^3)} -{x/Sqrt[ArcCosh[a*x]], x, 8, -(Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a^2) + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a^2)} -{1/Sqrt[ArcCosh[a*x]], x, 6, -((Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(2*a)) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(2*a)} -{1/(x*Sqrt[ArcCosh[a*x]]), x, 0, Unintegrable[1/(x*Sqrt[ArcCosh[a*x]]), x]} -{1/(x^2*Sqrt[ArcCosh[a*x]]), x, 0, Unintegrable[1/(x^2*Sqrt[ArcCosh[a*x]]), x]} - - -{x^4/ArcCosh[a*x]^(3/2), x, 17, -((2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]])) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(16*a^5) + (Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(16*a^5)} -{x^3/ArcCosh[a*x]^(3/2), x, 12, -((2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]])) + (Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(4*a^4) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(2*a^4) + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(4*a^4) + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(2*a^4)} -{x^2/ArcCosh[a*x]^(3/2), x, 12, -((2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]])) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(4*a^3) + (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(4*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(4*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(4*a^3)} -{x/ArcCosh[a*x]^(3/2), x, 6, -((2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]])) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/a^2 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/a^2} -{1/ArcCosh[a*x]^(3/2), x, 7, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]])) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/a + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/a} -{1/(x*ArcCosh[a*x]^(3/2)), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^(3/2)), x]} - - -{x^4/ArcCosh[a*x]^(5/2), x, 34, -((2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2))) + (16*x^3)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (20*x^5)/(3*Sqrt[ArcCosh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(12*a^5) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(8*a^5) - (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(24*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(12*a^5) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(8*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(24*a^5)} -{x^3/ArcCosh[a*x]^(5/2), x, 24, -((2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2))) + (4*x^2)/(a^2*Sqrt[ArcCosh[a*x]]) - (16*x^4)/(3*Sqrt[ArcCosh[a*x]]) - (2*Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(3*a^4) - (Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a^4) + (2*Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(3*a^4) + (Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a^4)} -{x^2/ArcCosh[a*x]^(5/2), x, 22, -((2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2))) + (8*x)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (4*x^3)/Sqrt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(6*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*a^3)} -{x/ArcCosh[a*x]^(5/2), x, 11, -((2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2))) + 4/(3*a^2*Sqrt[ArcCosh[a*x]]) - (8*x^2)/(3*Sqrt[ArcCosh[a*x]]) - (2*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a^2) + (2*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a^2)} -{1/ArcCosh[a*x]^(5/2), x, 8, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2))) - (4*x)/(3*Sqrt[ArcCosh[a*x]]) - (2*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(3*a)} -{1/(x*ArcCosh[a*x]^(5/2)), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^(5/2)), x]} - - -{x^4/ArcCosh[a*x]^(7/2), x, 32, -((2*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2))) + (16*x^3)/(15*a^2*ArcCosh[a*x]^(3/2)) - (4*x^5)/(3*ArcCosh[a*x]^(3/2)) + (32*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a^3*Sqrt[ArcCosh[a*x]]) - (40*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(30*a^5) + (9*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(20*a^5) + (5*Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(12*a^5) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(30*a^5) + (9*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(20*a^5) + (5*Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcCosh[a*x]]])/(12*a^5)} -{x^3/ArcCosh[a*x]^(7/2), x, 21, -((2*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2))) + (4*x^2)/(5*a^2*ArcCosh[a*x]^(3/2)) - (16*x^4)/(15*ArcCosh[a*x]^(3/2)) + (16*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a^3*Sqrt[ArcCosh[a*x]]) - (128*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (16*Sqrt[Pi]*Erf[2*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (4*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (16*Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(15*a^4) + (4*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^4)} -{x^2/ArcCosh[a*x]^(7/2), x, 22, -((2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2))) + (8*x)/(15*a^2*ArcCosh[a*x]^(3/2)) - (4*x^3)/(5*ArcCosh[a*x]^(3/2)) + (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a^3*Sqrt[ArcCosh[a*x]]) - (24*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(15*a^3) + (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(5*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(15*a^3) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(5*a^3)} -{x/ArcCosh[a*x]^(7/2), x, 9, -((2*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2))) + 4/(15*a^2*ArcCosh[a*x]^(3/2)) - (8*x^2)/(15*ArcCosh[a*x]^(3/2)) - (32*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(15*a^2)} -{1/ArcCosh[a*x]^(7/2), x, 9, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(5*a*ArcCosh[a*x]^(5/2))) - (4*x)/(15*ArcCosh[a*x]^(3/2)) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*Sqrt[ArcCosh[a*x]]) + (4*Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(15*a) + (4*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(15*a)} -{1/(x*ArcCosh[a*x]^(7/2)), x, 0, Unintegrable[1/(x*ArcCosh[a*x]^(7/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcCosh[c x]^n with m symbolic*) - - -{x^m*ArcCosh[a*x]^4, x, 1, (x^(1 + m)*ArcCosh[a*x]^4)/(1 + m) - (4*a*Unintegrable[(x^(1 + m)*ArcCosh[a*x]^3)/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]), x])/(1 + m)} -{x^m*ArcCosh[a*x]^3, x, 1, (x^(1 + m)*ArcCosh[a*x]^3)/(1 + m) - (3*a*Unintegrable[(x^(1 + m)*ArcCosh[a*x]^2)/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]), x])/(1 + m)} -{x^m*ArcCosh[a*x]^2, x, 2, (x^(1 + m)*ArcCosh[a*x]^2)/(1 + m) - (2*a*x^(2 + m)*Sqrt[1 - a*x]*ArcCosh[a*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + a*x]) - (2*a^2*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, a^2*x^2])/(6 + 11*m + 6*m^2 + m^3)} -{x^m*ArcCosh[a*x]^1, x, 4, (x^(1 + m)*ArcCosh[a*x])/(1 + m) - (a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{x^m/ArcCosh[a*x]^1, x, 0, Unintegrable[x^m/ArcCosh[a*x], x]} -{x^m/ArcCosh[a*x]^2, x, 0, Unintegrable[x^m/ArcCosh[a*x]^2, x]} -{x^m/ArcCosh[a*x]^3, x, 0, Unintegrable[x^m/ArcCosh[a*x]^3, x]} - - -{x^m*ArcCosh[a*x]^(3/2), x, 0, Unintegrable[x^m*ArcCosh[a*x]^(3/2), x]} -{x^m*Sqrt[ArcCosh[a*x]], x, 0, Unintegrable[x^m*Sqrt[ArcCosh[a*x]], x]} -{x^m/Sqrt[ArcCosh[a*x]], x, 0, Unintegrable[x^m/Sqrt[ArcCosh[a*x]], x]} -{x^m/ArcCosh[a*x]^(3/2), x, 0, Unintegrable[x^m/ArcCosh[a*x]^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (b x)^m ArcCosh[c x]^n with n symbolic*) - - -{(d*x)^m*ArcCosh[a*x]^n, x, 0, Unintegrable[(d*x)^m*ArcCosh[a*x]^n, x]} - - -{x^4*ArcCosh[a*x]^n, x, 12, (5^(-1 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -5*ArcCosh[a*x]])/(32*a^5*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1 + n, -3*ArcCosh[a*x]])/(32*3^n*a^5*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1 + n, -ArcCosh[a*x]])/(16*a^5*(-ArcCosh[a*x])^n) + Gamma[1 + n, ArcCosh[a*x]]/(16*a^5) + Gamma[1 + n, 3*ArcCosh[a*x]]/(32*3^n*a^5) + (5^(-1 - n)*Gamma[1 + n, 5*ArcCosh[a*x]])/(32*a^5)} -{x^3*ArcCosh[a*x]^n, x, 9, (ArcCosh[a*x]^n*Gamma[1 + n, -4*ArcCosh[a*x]])/(2^(2*(3 + n))*a^4*(-ArcCosh[a*x])^n) + (2^(-4 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -2*ArcCosh[a*x]])/(a^4*(-ArcCosh[a*x])^n) + (2^(-4 - n)*Gamma[1 + n, 2*ArcCosh[a*x]])/a^4 + Gamma[1 + n, 4*ArcCosh[a*x]]/(2^(2*(3 + n))*a^4)} -{x^2*ArcCosh[a*x]^n, x, 9, (3^(-1 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -3*ArcCosh[a*x]])/(8*a^3*(-ArcCosh[a*x])^n) + (ArcCosh[a*x]^n*Gamma[1 + n, -ArcCosh[a*x]])/(8*a^3*(-ArcCosh[a*x])^n) + Gamma[1 + n, ArcCosh[a*x]]/(8*a^3) + (3^(-1 - n)*Gamma[1 + n, 3*ArcCosh[a*x]])/(8*a^3)} -{x*ArcCosh[a*x]^n, x, 6, (2^(-3 - n)*ArcCosh[a*x]^n*Gamma[1 + n, -2*ArcCosh[a*x]])/(a^2*(-ArcCosh[a*x])^n) + (2^(-3 - n)*Gamma[1 + n, 2*ArcCosh[a*x]])/a^2} -{ArcCosh[a*x]^n, x, 4, (ArcCosh[a*x]^n*Gamma[1 + n, -ArcCosh[a*x]])/(2*a*(-ArcCosh[a*x])^n) + Gamma[1 + n, ArcCosh[a*x]]/(2*a)} -{ArcCosh[a*x]^n/x, x, 0, Unintegrable[ArcCosh[a*x]^n/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCosh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCosh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(a + b*ArcCosh[c*x])*x^3, x, 5, -((3*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3)) - (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (3*b*ArcCosh[c*x])/(32*c^4) + (1/4)*x^4*(a + b*ArcCosh[c*x])} -{(a + b*ArcCosh[c*x])*x^2, x, 4, -((2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3)) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + (1/3)*x^3*(a + b*ArcCosh[c*x])} -{(a + b*ArcCosh[c*x])*x^1, x, 3, -((b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c)) - (b*ArcCosh[c*x])/(4*c^2) + (1/2)*x^2*(a + b*ArcCosh[c*x])} -{(a + b*ArcCosh[c*x])*x^0, x, 3, a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCosh[c*x]} -{(a + b*ArcCosh[c*x])/x^1, x, 5, (a + b*ArcCosh[c*x])^2/(2*b) + (a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (1/2)*b*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{(a + b*ArcCosh[c*x])/x^2, x, 3, -((a + b*ArcCosh[c*x])/x) + b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} -{(a + b*ArcCosh[c*x])/x^3, x, 2, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (a + b*ArcCosh[c*x])/(2*x^2)} -{(a + b*ArcCosh[c*x])/x^4, x, 5, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) - (a + b*ArcCosh[c*x])/(3*x^3) + (1/6)*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} -{(a + b*ArcCosh[c*x])/x^5, x, 4, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(12*x^3) + (b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x) - (a + b*ArcCosh[c*x])/(4*x^4)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCosh[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*Sqrt[a + b*ArcCosh[c*x]], x, 14, (x^3*Sqrt[a + b*ArcCosh[c*x]])/3 - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c^3) - (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(48*c^3) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(48*c^3*E^((3*a)/b))} -{x*Sqrt[a + b*ArcCosh[c*x]], x, 9, -Sqrt[a + b*ArcCosh[c*x]]/(4*c^2) + (x^2*Sqrt[a + b*ArcCosh[c*x]])/2 - (Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*c^2) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*c^2*E^((2*a)/b))} -{Sqrt[a + b*ArcCosh[c*x]], x, 7, x*Sqrt[a + b*ArcCosh[c*x]] - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c*E^(a/b))} - - -{x^2*(a + b*ArcCosh[c*x])^(3/2), x, 22, -(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(3*c^3) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(6*c) + (x^3*(a + b*ArcCosh[c*x])^(3/2))/3 - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3) - (b^(3/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3*E^(a/b)) + (b^(3/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3*E^((3*a)/b))} -{x*(a + b*ArcCosh[c*x])^(3/2), x, 11, (-3*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(8*c) - (a + b*ArcCosh[c*x])^(3/2)/(4*c^2) + (x^2*(a + b*ArcCosh[c*x])^(3/2))/2 - (3*b^(3/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^2) + (3*b^(3/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^2*E^((2*a)/b))} -{(a + b*ArcCosh[c*x])^(3/2), x, 8, (-3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) + x*(a + b*ArcCosh[c*x])^(3/2) - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c*E^(a/b))} - - -{x^2*(a + b*ArcCosh[c*x])^(5/2), x, 24, (5*b^2*x*Sqrt[a + b*ArcCosh[c*x]])/(6*c^2) + (5*b^2*x^3*Sqrt[a + b*ArcCosh[c*x]])/36 - (5*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(9*c^3) - (5*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(18*c) + (x^3*(a + b*ArcCosh[c*x])^(5/2))/3 - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*c^3) - (5*b^(5/2)*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^3) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*c^3*E^(a/b)) - (5*b^(5/2)*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(576*c^3*E^((3*a)/b))} -{x*(a + b*ArcCosh[c*x])^(5/2), x, 12, (-15*b^2*Sqrt[a + b*ArcCosh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcCosh[c*x]])/32 - (5*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(8*c) - (a + b*ArcCosh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcCosh[c*x])^(5/2))/2 - (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))} -{(a + b*ArcCosh[c*x])^(5/2), x, 9, (15*b^2*x*Sqrt[a + b*ArcCosh[c*x]])/4 - (5*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(3/2))/(2*c) + x*(a + b*ArcCosh[c*x])^(5/2) - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c*E^(a/b))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^2/Sqrt[a + b*ArcCosh[c*x]], x, 13, -(E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3) - (E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))} -{x/Sqrt[a + b*ArcCosh[c*x]], x, 8, -(E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^2) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^2*E^((2*a)/b))} -{1/Sqrt[a + b*ArcCosh[c*x]], x, 6, -(E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b))} - - -{x^2/(a + b*ArcCosh[c*x])^(3/2), x, 12, (-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) + (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3*E^(a/b)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3*E^((3*a)/b))} -{x/(a + b*ArcCosh[c*x])^(3/2), x, 6, (-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(b^(3/2)*c^2) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(b^(3/2)*c^2*E^((2*a)/b))} -{(a + b*ArcCosh[c*x])^(-3/2), x, 7, (-2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b))} - - -{x^2/(a + b*ArcCosh[c*x])^(5/2), x, 22, (-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + (8*x)/(3*b^2*c^2*Sqrt[a + b*ArcCosh[c*x]]) - (4*x^3)/(b^2*Sqrt[a + b*ArcCosh[c*x]]) - (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6*b^(5/2)*c^3) - (E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(6*b^(5/2)*c^3*E^(a/b)) + (Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(5/2)*c^3*E^((3*a)/b))} -{x/(a + b*ArcCosh[c*x])^(5/2), x, 11, (-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) + 4/(3*b^2*c^2*Sqrt[a + b*ArcCosh[c*x]]) - (8*x^2)/(3*b^2*Sqrt[a + b*ArcCosh[c*x]]) - (2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(3*b^(5/2)*c^2) + (2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(3*b^(5/2)*c^2*E^((2*a)/b))} -{(a + b*ArcCosh[c*x])^(-5/2), x, 8, (-2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*b*c*(a + b*ArcCosh[c*x])^(3/2)) - (4*x)/(3*b^2*Sqrt[a + b*ArcCosh[c*x]]) - (2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(3*b^(5/2)*c) + (2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(3*b^(5/2)*c*E^(a/b))} - - -{x^2/(a + b*ArcCosh[c*x])^(7/2), x, 22, (-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + (8*x)/(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3/2)) - (4*x^3)/(5*b^2*(a + b*ArcCosh[c*x])^(3/2)) + (16*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c^3*Sqrt[a + b*ArcCosh[c*x]]) - (24*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b^3*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c^3) + (3*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(5*b^(7/2)*c^3) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c^3*E^(a/b)) + (3*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(5*b^(7/2)*c^3*E^((3*a)/b))} -{x/(a + b*ArcCosh[c*x])^(7/2), x, 9, (-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + 4/(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3/2)) - (8*x^2)/(15*b^2*(a + b*ArcCosh[c*x])^(3/2)) - (32*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c*Sqrt[a + b*ArcCosh[c*x]]) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2) + (8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2*E^((2*a)/b))} -{(a + b*ArcCosh[c*x])^(-7/2), x, 9, (-2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) - (4*x)/(15*b^2*(a + b*ArcCosh[c*x])^(3/2)) - (8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c*Sqrt[a + b*ArcCosh[c*x]]) + (4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c) + (4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c*E^(a/b))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcCosh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[f*x]*(a + b*ArcCosh[c*x])^2, x, 2, (2*(f*x)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*f) - (8*b*c*(f*x)^(5/2)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(15*f^2*Sqrt[-1 + c*x]) - (16*b^2*c^2*(f*x)^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(105*f^3)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCosh[c x])^n with m symbolic*) - - -{(d*x)^m*(a + b*ArcCosh[c*x])^2, x, 2, If[$VersionNumber>=8, ((d*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(d*(1 + m)) - (2*b*c*(d*x)^(2 + m)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]) - (2*b^2*c^2*(d*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^3*(1 + m)*(2 + m)*(3 + m)), ((d*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(d*(1 + m)) - (2*b*c*(d*x)^(2 + m)*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]) - (2*b^2*c^2*(d*x)^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, c^2*x^2])/(d^3*(3 + m)*(2 + 3*m + m^2))]} -{(d*x)^m*(a + b*ArcCosh[c*x])^1, x, 4, ((d*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d*x)^m/(a + b*ArcCosh[c*x])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcCosh[c*x]), x]} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m deleted file mode 100644 index ce6a503..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.4 (f x)^m (d+e x^2)^p (a+b arccosh(c x))^n.m +++ /dev/null @@ -1,1061 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]), x, 8, -((152*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^5)) - (76*b*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^3) - (19*b*d*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c) + (1/49)*b*c*d*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (1/5)*d*x^5*(a + b*ArcCosh[c*x]) - (1/7)*c^2*d*x^7*(a + b*ArcCosh[c*x])} -{x^3*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]), x, 7, -((b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3)) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (1/36)*b*c*d*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*d*ArcCosh[c*x])/(24*c^4) + (1/4)*d*x^4*(a + b*ArcCosh[c*x]) - (1/6)*c^2*d*x^6*(a + b*ArcCosh[c*x])} -{x^2*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]), x, 6, -((26*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(225*c^3)) - (13*b*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(225*c) + (1/25)*b*c*d*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (1/3)*d*x^3*(a + b*ArcCosh[c*x]) - (1/5)*c^2*d*x^5*(a + b*ArcCosh[c*x])} -{x^1*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]), x, 4, -((3*b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c)) + (b*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(16*c) + (3*b*d*ArcCosh[c*x])/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(4*c^2)} -{x^0*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]), x, 4, -((7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c)) + (1/9)*b*c*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + d*x*(a + b*ArcCosh[c*x]) - (1/3)*c^2*d*x^3*(a + b*ArcCosh[c*x])} -{((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^1, x, 8, (1/4)*b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (1/4)*b*d*ArcCosh[c*x] + (1/2)*d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) + (d*(a + b*ArcCosh[c*x])^2)/(2*b) + d*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (1/2)*b*d*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^2, x, 5, b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (d*(a + b*ArcCosh[c*x]))/x - c^2*d*x*(a + b*ArcCosh[c*x]) + b*c*d*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} -{((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^3, x, 9, (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (1/2)*b*c^2*d*ArcCosh[c*x] - (d*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]))/(2*x^2) - (c^2*d*(a + b*ArcCosh[c*x])^2)/(2*b) - c^2*d*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] + (1/2)*b*c^2*d*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^4, x, 5, (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) - (d*(a + b*ArcCosh[c*x]))/(3*x^3) + (c^2*d*(a + b*ArcCosh[c*x]))/x - (5/6)*b*c^3*d*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} - - -{x^4*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 7, -((8*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(315*c^5)) + (4*b*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(945*c^5) - (b*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(525*c^5) - (10*b*d^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(441*c^5) - (b*d^2*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(81*c^5) + (1/5)*d^2*x^5*(a + b*ArcCosh[c*x]) - (2/7)*c^2*d^2*x^7*(a + b*ArcCosh[c*x]) + (1/9)*c^4*d^2*x^9*(a + b*ArcCosh[c*x]), (8*b*d^2*(1 - c^2*x^2))/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(945*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(525*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*d^2*(1 - c^2*x^2)^4)/(441*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^5)/(81*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/5)*d^2*x^5*(a + b*ArcCosh[c*x]) - (2/7)*c^2*d^2*x^7*(a + b*ArcCosh[c*x]) + (1/9)*c^4*d^2*x^9*(a + b*ArcCosh[c*x])} -{x^3*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 9, -((73*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3072*c^3)) - (73*b*d^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4608*c) + (43*b*c*d^2*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/1152 - (1/64)*b*c^3*d^2*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (73*b*d^2*ArcCosh[c*x])/(3072*c^4) + (1/4)*d^2*x^4*(a + b*ArcCosh[c*x]) - (1/3)*c^2*d^2*x^6*(a + b*ArcCosh[c*x]) + (1/8)*c^4*d^2*x^8*(a + b*ArcCosh[c*x]), (73*b*d^2*x*(1 - c^2*x^2))/(3072*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (73*b*d^2*x^3*(1 - c^2*x^2))/(4608*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (43*b*c*d^2*x^5*(1 - c^2*x^2))/(1152*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^7*(1 - c^2*x^2))/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/4)*d^2*x^4*(a + b*ArcCosh[c*x]) - (1/3)*c^2*d^2*x^6*(a + b*ArcCosh[c*x]) + (1/8)*c^4*d^2*x^8*(a + b*ArcCosh[c*x]) - (73*b*d^2*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(3072*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 6, -((8*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(105*c^3)) + (4*b*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(315*c^3) - (b*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(175*c^3) - (b*d^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(49*c^3) + (1/3)*d^2*x^3*(a + b*ArcCosh[c*x]) - (2/5)*c^2*d^2*x^5*(a + b*ArcCosh[c*x]) + (1/7)*c^4*d^2*x^7*(a + b*ArcCosh[c*x]), (8*b*d^2*(1 - c^2*x^2))/(105*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(315*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(175*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(1 - c^2*x^2)^4)/(49*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/3)*d^2*x^3*(a + b*ArcCosh[c*x]) - (2/5)*c^2*d^2*x^5*(a + b*ArcCosh[c*x]) + (1/7)*c^4*d^2*x^7*(a + b*ArcCosh[c*x])} -{x^1*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 5, -((5*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c)) + (5*b*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(144*c) - (b*d^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(36*c) + (5*b*d^2*ArcCosh[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*ArcCosh[c*x]))/(6*c^2)} -{x^0*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 6, -((8*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c)) + (4*b*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(45*c) - (b*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(25*c) + d^2*x*(a + b*ArcCosh[c*x]) - (2/3)*c^2*d^2*x^3*(a + b*ArcCosh[c*x]) + (1/5)*c^4*d^2*x^5*(a + b*ArcCosh[c*x]), (8*b*d^2*(1 - c^2*x^2))/(15*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(25*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*x*(a + b*ArcCosh[c*x]) - (2/3)*c^2*d^2*x^3*(a + b*ArcCosh[c*x]) + (1/5)*c^4*d^2*x^5*(a + b*ArcCosh[c*x])} -{((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^1, x, 12, (11/32)*b*c*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (1/16)*b*c*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) - (11/32)*b*d^2*ArcCosh[c*x] + (1/2)*d^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) + (1/4)*d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]) + (d^2*(a + b*ArcCosh[c*x])^2)/(2*b) + d^2*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (1/2)*b*d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^2, x, 8, (5/3)*b*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (1/9)*b*c*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) - (d^2*(a + b*ArcCosh[c*x]))/x - 2*c^2*d^2*x*(a + b*ArcCosh[c*x]) + (1/3)*c^4*d^2*x^3*(a + b*ArcCosh[c*x]) + b*c*d^2*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]], -((5*b*c*d^2*(1 - c^2*x^2))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d^2*(1 - c^2*x^2)^2)/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/x - 2*c^2*d^2*x*(a + b*ArcCosh[c*x]) + (1/3)*c^4*d^2*x^3*(a + b*ArcCosh[c*x]) + (b*c*d^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^3, x, 13, (1/4)*b*c^3*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*c*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(2*x) - (1/4)*b*c^2*d^2*ArcCosh[c*x] - c^2*d^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) - (d^2*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(2*x^2) - (c^2*d^2*(a + b*ArcCosh[c*x])^2)/b - 2*c^2*d^2*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] + b*c^2*d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^4, x, 8, (-b)*c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (b*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) - (d^2*(a + b*ArcCosh[c*x]))/(3*x^3) + (2*c^2*d^2*(a + b*ArcCosh[c*x]))/x + c^4*d^2*x*(a + b*ArcCosh[c*x]) - (11/6)*b*c^3*d^2*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]], (b*c^3*d^2*(1 - c^2*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/(3*x^3) + (2*c^2*d^2*(a + b*ArcCosh[c*x]))/x + c^4*d^2*x*(a + b*ArcCosh[c*x]) - (11*b*c^3*d^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^4*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, -((16*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1155*c^5)) + (8*b*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(3465*c^5) - (2*b*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(1925*c^5) + (b*d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(1617*c^5) + (4*b*d^3*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(297*c^5) + (b*d^3*(-1 + c*x)^(11/2)*(1 + c*x)^(11/2))/(121*c^5) + (1/5)*d^3*x^5*(a + b*ArcCosh[c*x]) - (3/7)*c^2*d^3*x^7*(a + b*ArcCosh[c*x]) + (1/3)*c^4*d^3*x^9*(a + b*ArcCosh[c*x]) - (1/11)*c^6*d^3*x^11*(a + b*ArcCosh[c*x]), (16*b*d^3*(1 - c^2*x^2))/(1155*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(3465*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^3*(1 - c^2*x^2)^3)/(1925*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^4)/(1617*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*d^3*(1 - c^2*x^2)^5)/(297*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^6)/(121*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/5)*d^3*x^5*(a + b*ArcCosh[c*x]) - (3/7)*c^2*d^3*x^7*(a + b*ArcCosh[c*x]) + (1/3)*c^4*d^3*x^9*(a + b*ArcCosh[c*x]) - (1/11)*c^6*d^3*x^11*(a + b*ArcCosh[c*x])} -{x^3*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 11, -((49*b*d^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5120*c^3)) + (49*b*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(7680*c^3) - (49*b*d^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(9600*c^3) + (7*b*d^3*x*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(1600*c^3) + (b*d^3*x*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(100*c^3) + (49*b*d^3*ArcCosh[c*x])/(5120*c^4) - (d^3*(-1 + c*x)^4*(1 + c*x)^4*(a + b*ArcCosh[c*x]))/(8*c^4) - (d^3*(-1 + c*x)^5*(1 + c*x)^5*(a + b*ArcCosh[c*x]))/(10*c^4), (49*b*d^3*x*(1 - c^2*x^2))/(5120*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (49*b*d^3*x*(1 - c^2*x^2)^2)/(7680*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (49*b*d^3*x*(1 - c^2*x^2)^3)/(9600*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*b*d^3*x*(1 - c^2*x^2)^4)/(1600*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^3*x*(1 - c^2*x^2)^5)/(100*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcCosh[c*x]))/(8*c^4) + (d^3*(1 - c^2*x^2)^5*(a + b*ArcCosh[c*x]))/(10*c^4) + (49*b*d^3*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, -((16*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(315*c^3)) + (8*b*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(945*c^3) - (2*b*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(525*c^3) + (b*d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(441*c^3) + (b*d^3*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(81*c^3) + (1/3)*d^3*x^3*(a + b*ArcCosh[c*x]) - (3/5)*c^2*d^3*x^5*(a + b*ArcCosh[c*x]) + (3/7)*c^4*d^3*x^7*(a + b*ArcCosh[c*x]) - (1/9)*c^6*d^3*x^9*(a + b*ArcCosh[c*x]), (16*b*d^3*(1 - c^2*x^2))/(315*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(945*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^3*(1 - c^2*x^2)^3)/(525*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^4)/(441*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^3*(1 - c^2*x^2)^5)/(81*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^3*(a + b*ArcCosh[c*x]))/3 - (3*c^2*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcCosh[c*x]))/7 - (c^6*d^3*x^9*(a + b*ArcCosh[c*x]))/9} -{x^1*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, -((35*b*d^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1024*c)) + (35*b*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(1536*c) - (7*b*d^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(384*c) + (b*d^3*x*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(64*c) + (35*b*d^3*ArcCosh[c*x])/(1024*c^2) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcCosh[c*x]))/(8*c^2)} -{x^0*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, -((16*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(35*c)) + (8*b*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(105*c) - (6*b*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(175*c) + (b*d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(49*c) + d^3*x*(a + b*ArcCosh[c*x]) - c^2*d^3*x^3*(a + b*ArcCosh[c*x]) + (3/5)*c^4*d^3*x^5*(a + b*ArcCosh[c*x]) - (1/7)*c^6*d^3*x^7*(a + b*ArcCosh[c*x]), (16*b*d^3*(1 - c^2*x^2))/(35*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*b*d^3*(1 - c^2*x^2)^3)/(175*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^4)/(49*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) - c^2*d^3*x^3*(a + b*ArcCosh[c*x]) + (3*c^4*d^3*x^5*(a + b*ArcCosh[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcCosh[c*x]))/7} -{((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^1, x, 17, (19/48)*b*c*d^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (7/72)*b*c*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) + (1/36)*b*c*d^3*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2) - (19/48)*b*d^3*ArcCosh[c*x] + (1/2)*d^3*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) + (1/4)*d^3*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]) + (1/6)*d^3*(1 - c^2*x^2)^3*(a + b*ArcCosh[c*x]) + (d^3*(a + b*ArcCosh[c*x])^2)/(2*b) + d^3*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] - (1/2)*b*d^3*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^2, x, 8, (11/5)*b*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (1/5)*b*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) + (1/25)*b*c*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2) - (d^3*(a + b*ArcCosh[c*x]))/x - 3*c^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d^3*x^3*(a + b*ArcCosh[c*x]) - (1/5)*c^6*d^3*x^5*(a + b*ArcCosh[c*x]) + b*c*d^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]], -((11*b*c*d^3*(1 - c^2*x^2))/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d^3*(1 - c^2*x^2)^2)/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2)^3)/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/x - 3*c^2*d^3*x*(a + b*ArcCosh[c*x]) + c^4*d^3*x^3*(a + b*ArcCosh[c*x]) - (1/5)*c^6*d^3*x^5*(a + b*ArcCosh[c*x]) + (b*c*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^3, x, 18, (-(3/32))*b*c^3*d^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (7/16)*b*c^3*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) + (b*c*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(2*x) + (3/32)*b*c^2*d^3*ArcCosh[c*x] - (3/2)*c^2*d^3*(1 - c^2*x^2)*(a + b*ArcCosh[c*x]) - (3/4)*c^2*d^3*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]) - (d^3*(1 - c^2*x^2)^3*(a + b*ArcCosh[c*x]))/(2*x^2) - (3*c^2*d^3*(a + b*ArcCosh[c*x])^2)/(2*b) - 3*c^2*d^3*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])] + (3/2)*b*c^2*d^3*PolyLog[2, -E^(-2*ArcCosh[c*x])]} -{((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4, x, 9, (-(8/3))*b*c^3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (b*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) + (1/9)*b*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2) - (d^3*(a + b*ArcCosh[c*x]))/(3*x^3) + (3*c^2*d^3*(a + b*ArcCosh[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcCosh[c*x]) - (1/3)*c^6*d^3*x^3*(a + b*ArcCosh[c*x]) - (17/6)*b*c^3*d^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]], (8*b*c^3*d^3*(1 - c^2*x^2))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^3*(1 - c^2*x^2)^2)/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/(3*x^3) + (3*c^2*d^3*(a + b*ArcCosh[c*x]))/x + 3*c^4*d^3*x*(a + b*ArcCosh[c*x]) - (1/3)*c^6*d^3*x^3*(a + b*ArcCosh[c*x]) - (17*b*c^3*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2), x, 12, (11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^5*d) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3*d) - (x*(a + b*ArcCosh[c*x]))/(c^4*d) - (x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d) + (2*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^5*d) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(c^5*d) - (b*PolyLog[2, E^ArcCosh[c*x]])/(c^5*d)} -{x^3*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2), x, 8, (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c^3*d) + (b*ArcCosh[c*x])/(4*c^4*d) - (x^2*(a + b*ArcCosh[c*x]))/(2*c^2*d) + (a + b*ArcCosh[c*x])^2/(2*b*c^4*d) - ((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^4*d) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*c^4*d)} -{x^2*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2), x, 8, (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d) - (x*(a + b*ArcCosh[c*x]))/(c^2*d) + (2*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^3*d) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(c^3*d) - (b*PolyLog[2, E^ArcCosh[c*x]])/(c^3*d)} -{x^1*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2), x, 5, (a + b*ArcCosh[c*x])^2/(2*b*c^2*d) - ((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^2*d) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*c^2*d)} -{x^0*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2), x, 6, (2*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c*d) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(c*d) - (b*PolyLog[2, E^ArcCosh[c*x]])/(c*d)} -{(a + b*ArcCosh[c*x])/(x^1*(d - c^2*d*x^2)), x, 7, (2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d)} -{(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)), x, 9, -((a + b*ArcCosh[c*x])/(d*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d + (2*c*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d + (b*c*PolyLog[2, -E^ArcCosh[c*x]])/d - (b*c*PolyLog[2, E^ArcCosh[c*x]])/d} -{(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)), x, 9, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d + (b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d) - (b*c^2*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d)} -{(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)), x, 14, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) - (c^2*(a + b*ArcCosh[c*x]))/(d*x) + (7*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) + (2*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d + (b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/d - (b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d} - - -{x^4*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 12, -((b*x^2)/(2*c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^5*d^2) + (3*x*(a + b*ArcCosh[c*x]))/(2*c^4*d^2) + (x^3*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^5*d^2) - (3*b*PolyLog[2, -E^ArcCosh[c*x]])/(2*c^5*d^2) + (3*b*PolyLog[2, E^ArcCosh[c*x]])/(2*c^5*d^2)} -{x^3*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 10, -(b/(2*c^4*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*Sqrt[-1 + c*x])/(2*c^4*d^2*Sqrt[1 + c*x]) + (b*ArcCosh[c*x])/(2*c^4*d^2) + (x^2*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*c^4*d^2) + ((a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^4*d^2) + (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*c^4*d^2)} -{x^2*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 8, -(b/(2*c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (x*(a + b*ArcCosh[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - ((a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^3*d^2) - (b*PolyLog[2, -E^ArcCosh[c*x]])/(2*c^3*d^2) + (b*PolyLog[2, E^ArcCosh[c*x]])/(2*c^3*d^2)} -{x^1*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 2, -((b*x)/(2*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a + b*ArcCosh[c*x])/(2*c^2*d^2*(1 - c^2*x^2))} -{x^0*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 8, -(b/(2*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (x*(a + b*ArcCosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + ((a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c*d^2) + (b*PolyLog[2, -E^ArcCosh[c*x]])/(2*c*d^2) - (b*PolyLog[2, E^ArcCosh[c*x]])/(2*c*d^2)} -{(a + b*ArcCosh[c*x])/(x^1*(d - c^2*d*x^2)^2), x, 9, -((b*c*x)/(2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a + b*ArcCosh[c*x])/(2*d^2*(1 - c^2*x^2)) + (2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^2 + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^2) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d^2)} -{(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^2), x, 13, -((b*c)/(2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (a + b*ArcCosh[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcCosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^2 + (3*c*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d^2 + (3*b*c*PolyLog[2, -E^ArcCosh[c*x]])/(2*d^2) - (3*b*c*PolyLog[2, E^ArcCosh[c*x]])/(2*d^2)} -{(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^2), x, 13, -((b*c)/(2*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (c^2*(a + b*ArcCosh[c*x]))/(d^2*(1 - c^2*x^2)) - (a + b*ArcCosh[c*x])/(2*d^2*x^2*(1 - c^2*x^2)) + (4*c^2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^2 + (b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/d^2 - (b*c^2*PolyLog[2, E^(2*ArcCosh[c*x])])/d^2} -{(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^2), x, 20, -((b*c^3)/(3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c)/(6*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d^2*x^3*(1 - c^2*x^2)) - (5*c^2*(a + b*ArcCosh[c*x]))/(3*d^2*x*(1 - c^2*x^2)) + (5*c^4*x*(a + b*ArcCosh[c*x]))/(2*d^2*(1 - c^2*x^2)) + (13*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^2) + (5*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/d^2 + (5*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(2*d^2) - (5*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(2*d^2)} - - -{x^4*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 13, (b*x^3)/(12*c^2*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + b/(4*c^5*d^3*Sqrt[-1 + c*x]*(1 + c*x)^(3/2)) - (b*(-1 + c*x)^(3/2))/(12*c^5*d^3*(1 + c*x)^(3/2)) + (3*b)/(8*c^5*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x^3*(a + b*ArcCosh[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (3*x*(a + b*ArcCosh[c*x]))/(8*c^4*d^3*(1 - c^2*x^2)) + (3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*c^5*d^3) + (3*b*PolyLog[2, -E^ArcCosh[c*x]])/(8*c^5*d^3) - (3*b*PolyLog[2, E^ArcCosh[c*x]])/(8*c^5*d^3)} -{x^3*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 7, (b*x^3)/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + b/(4*c^4*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*Sqrt[-1 + c*x])/(4*c^4*d^3*Sqrt[1 + c*x]) - (b*ArcCosh[c*x])/(4*c^4*d^3) + (x^4*(a + b*ArcCosh[c*x]))/(4*d^3*(1 - c^2*x^2)^2)} -{x^2*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 10, b/(12*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + b/(8*c^3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(4*c^2*d^3*(1 - c^2*x^2)^2) - (x*(a + b*ArcCosh[c*x]))/(8*c^2*d^3*(1 - c^2*x^2)) - ((a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*c^3*d^3) - (b*PolyLog[2, -E^ArcCosh[c*x]])/(8*c^3*d^3) + (b*PolyLog[2, E^ArcCosh[c*x]])/(8*c^3*d^3)} -{x^1*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 3, (b*x)/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (b*x)/(6*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(4*c^2*d^3*(1 - c^2*x^2)^2)} -{x^0*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 10, b/(12*c*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (3*b)/(8*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/(4*d^3*(1 - c^2*x^2)^2) + (3*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*c*d^3) + (3*b*PolyLog[2, -E^ArcCosh[c*x]])/(8*c*d^3) - (3*b*PolyLog[2, E^ArcCosh[c*x]])/(8*c*d^3)} -{(a + b*ArcCosh[c*x])/(x^1*(d - c^2*d*x^2)^3), x, 12, (b*c*x)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2*b*c*x)/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (a + b*ArcCosh[c*x])/(2*d^3*(1 - c^2*x^2)) + (2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^3 + (b*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^3) - (b*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d^3)} -{(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^3), x, 17, (b*c)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (7*b*c)/(8*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(d^3*x*(1 - c^2*x^2)^2) + (5*c^2*x*(a + b*ArcCosh[c*x]))/(4*d^3*(1 - c^2*x^2)^2) + (15*c^2*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^3 + (15*c*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*d^3) + (15*b*c*PolyLog[2, -E^ArcCosh[c*x]])/(8*d^3) - (15*b*c*PolyLog[2, E^ArcCosh[c*x]])/(8*d^3)} -{(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^3), x, 17, (b*c)/(2*d^3*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (5*b*c^3*x)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (2*b*c^3*x)/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*c^2*(a + b*ArcCosh[c*x]))/(4*d^3*(1 - c^2*x^2)^2) - (a + b*ArcCosh[c*x])/(2*d^3*x^2*(1 - c^2*x^2)^2) + (3*c^2*(a + b*ArcCosh[c*x]))/(2*d^3*(1 - c^2*x^2)) + (6*c^2*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/d^3 + (3*b*c^2*PolyLog[2, -E^(2*ArcCosh[c*x])])/(2*d^3) - (3*b*c^2*PolyLog[2, E^(2*ArcCosh[c*x])])/(2*d^3)} -{(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3), x, 26, -((b*c^3)/(12*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))) + (b*c)/(6*d^3*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (29*b*c^3)/(24*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a + b*ArcCosh[c*x])/(3*d^3*x^3*(1 - c^2*x^2)^2) - (7*c^2*(a + b*ArcCosh[c*x]))/(3*d^3*x*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcCosh[c*x]))/(12*d^3*(1 - c^2*x^2)^2) + (35*c^4*x*(a + b*ArcCosh[c*x]))/(8*d^3*(1 - c^2*x^2)) + (19*b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d^3) + (35*c^3*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(4*d^3) + (35*b*c^3*PolyLog[2, -E^ArcCosh[c*x]])/(8*d^3) - (35*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(8*d^3)} - - -{ArcCosh[a*x]/(c - a^2*c*x^2), x, 6, (2*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c) + PolyLog[2, -E^ArcCosh[a*x]]/(a*c) - PolyLog[2, E^ArcCosh[a*x]]/(a*c)} -{ArcCosh[a*x]/(c - a^2*c*x^2)^2, x, 8, -(1/(2*a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (x*ArcCosh[a*x])/(2*c^2*(1 - a^2*x^2)) + (ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) + PolyLog[2, -E^ArcCosh[a*x]]/(2*a*c^2) - PolyLog[2, E^ArcCosh[a*x]]/(2*a*c^2)} -{ArcCosh[a*x]/(c - a^2*c*x^2)^3, x, 10, 1/(12*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - 3/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x])/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x])/(8*c^3*(1 - a^2*x^2)) + (3*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) + (3*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) - (3*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 7, (b*x^2*Sqrt[d - c^2*d*x^2])/(32*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^4*Sqrt[d - c^2*d*x^2])/(96*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^4) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(24*c^2) + (1/6)*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 5, (b*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c^2) + (1/4)*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^0*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 3, -((b*c*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (1/2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^2, x, 3, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x) + (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^4, x, 4, -((b*c*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*d*x^3) - (b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^6, x, 4, -((b*c*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(5*d*x^5) - (2*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(15*d*x^3) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^8, x, 4, -((b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5) - (8*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - -{x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 3, (8*b*x*Sqrt[d - c^2*d*x^2])/(105*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*x^3*Sqrt[d - c^2*d*x^2])/(315*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^5*Sqrt[d - c^2*d*x^2])/(175*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^6*d) + (2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^6*d^2) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d^3)} -{x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 3, (2*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^4*d) + ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^4*d^2)} -{x^1*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]), x, 3, (b*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^2*d)} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^1, x, 8, -((b*c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^3, x, 8, -((b*c*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*x^2) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^5, x, 10, -((b*c*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*x^4) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*x^2) + (c^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*c^4*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 11, (3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(64*c^2) + (1/16)*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/8)*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(256*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 9, (b*d*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (1/8)*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/6)*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^0*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 7, -((5*b*c*d*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3/8)*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/4)*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - (3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^2, x, 7, (b*c^3*d*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3/2)*c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x + (3*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d*Sqrt[d - c^2*d*x^2]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^4, x, 7, -((b*c*d*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^3*d*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(5*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*d*x^5) + (b*c^5*d*Sqrt[d - c^2*d*x^2]*Log[x])/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (2*b*c^3*d*Sqrt[d - c^2*d*x^2])/(35*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(70*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Log[x])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(9*d*x^9) - (4*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(63*d*x^7) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(315*d*x^5) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12, x, 5, -((b*c*d*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d*Sqrt[d - c^2*d*x^2])/(66*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(1386*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^7*d*Sqrt[d - c^2*d*x^2])/(770*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c^9*d*Sqrt[d - c^2*d*x^2])/(1155*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(11*d*x^11) - (2*c^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(33*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(231*d*x^7) - (16*c^6*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(1155*d*x^5) + (16*b*c^11*d*Sqrt[d - c^2*d*x^2]*Log[x])/(1155*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - -{x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 4, (16*b*d*x*Sqrt[d - c^2*d*x^2])/(1155*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d*x^3*Sqrt[d - c^2*d*x^2])/(3465*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*x^5*Sqrt[d - c^2*d*x^2])/(1925*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^7*Sqrt[d - c^2*d*x^2])/(1617*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c*d*x^9*Sqrt[d - c^2*d*x^2])/(297*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^8*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(3*c^8*d^3) + ((d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^8*d^4)} -{x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 4, (8*b*d*x*Sqrt[d - c^2*d*x^2])/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d*x^3*Sqrt[d - c^2*d*x^2])/(945*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^5*Sqrt[d - c^2*d*x^2])/(525*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*c*d*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^6*d) + (2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d^2) - ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(9*c^6*d^3)} -{x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 4, (2*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c*d*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^4*d) + ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d^2)} -{x^1*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 4, (b*d*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^2*d)} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^1, x, 11, -((4*b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/3)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^3, x, 12, -((b*c*d*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3/2)*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^5, x, 12, -((b*c*d*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*x^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(4*x^4) - (3*c^4*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*I*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*I*b*c^4*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 16, (3*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(512*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^4*Sqrt[d - c^2*d*x^2])/(512*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (31*b*c*d^2*x^6*Sqrt[d - c^2*d*x^2])/(960*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (21*b*c^3*d^2*x^8*Sqrt[d - c^2*d*x^2])/(640*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^10*Sqrt[d - c^2*d*x^2])/(100*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(256*c^4) - (d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (1/32)*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/16)*d*x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) + (1/10)*x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]) - (3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(512*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 14, (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (5/64)*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/48)*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) + (1/8)*x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]) - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(256*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^0*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 10, -((25*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (5*b*c^3*d^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/16)*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/24)*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) + (1/6)*x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]) - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^2, x, 12, (9*b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (15/8)*c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (5/4)*c^2*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x + (15*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^4, x, 12, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/2)*c^4*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c^3*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6, x, 12, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (11*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x + (c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*x^5) + (c^5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^5*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8, x, 5, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (3*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(28*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(14*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^10, x, 6, -((b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(189*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(42*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(21*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(9*d*x^9) - (2*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(63*d*x^7) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12, x, 5, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(110*x^10*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (23*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(792*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (113*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(4158*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^7*d^2*Sqrt[d - c^2*d*x^2])/(924*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2])/(693*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(11*d*x^11) - (4*c^2*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(99*d*x^9) - (8*c^4*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(693*d*x^7) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(693*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - -{x^7*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 4, (16*b*d^2*x*Sqrt[d - c^2*d*x^2])/(3003*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(9009*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(5005*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*d^2*x^7*Sqrt[d - c^2*d*x^2])/(21021*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (53*b*c*d^2*x^9*Sqrt[d - c^2*d*x^2])/(3861*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (27*b*c^3*d^2*x^11*Sqrt[d - c^2*d*x^2])/(1573*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^13*Sqrt[d - c^2*d*x^2])/(169*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^8*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(3*c^8*d^2) - (3*(d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^8*d^3) + ((d - c^2*d*x^2)^(13/2)*(a + b*ArcCosh[c*x]))/(13*c^8*d^4)} -{x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 4, (8*b*d^2*x*Sqrt[d - c^2*d*x^2])/(693*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(2079*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(1155*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (113*b*c*d^2*x^7*Sqrt[d - c^2*d*x^2])/(4851*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^3*d^2*x^9*Sqrt[d - c^2*d*x^2])/(891*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^6*d) + (2*(d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(9*c^6*d^2) - ((d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^6*d^3)} -{x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 4, (2*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (19*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(9*c^4*d^2)} -{x^1*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 4, (b*d^2*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^2*d)} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^1, x, 15, -((23*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (11*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])/(45*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/3)*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) + (1/5)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^3, x, 15, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (7*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5/2)*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (5/6)*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^5, x, 16, -((b*c*d^2*Sqrt[d - c^2*d*x^2])/(12*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (9*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(8*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15/8)*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(8*x^2) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(4*x^4) - (15*c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*I*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (15*I*b*c^4*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{Sqrt[1 - x^2]*ArcCosh[x], x, 3, -((Sqrt[1 - x]*x^2)/(4*Sqrt[-1 + x])) + (1/2)*x*Sqrt[1 - x^2]*ArcCosh[x] - (Sqrt[1 - x]*ArcCosh[x]^2)/(4*Sqrt[-1 + x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 6, -((8*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2])) - (4*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(45*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*c^4*d) - (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2*d)} -{x^4*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 5, -((3*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c^3*Sqrt[d - c^2*d*x^2])) - (b*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c*Sqrt[d - c^2*d*x^2]) - (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c^4*d) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*c^2*d) + (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(16*b*c^5*Sqrt[d - c^2*d*x^2])} -{x^3*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 4, -((2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^3*Sqrt[d - c^2*d*x^2])) - (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^2*d)} -{x^2*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 3, -((b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c*Sqrt[d - c^2*d*x^2])) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{x^1*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 2, -((b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^2*d)} -{x^0*(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2], x, 1, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^1*Sqrt[d - c^2*d*x^2]), x, 6, (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcCosh[c*x])/(x^2*Sqrt[d - c^2*d*x^2]), x, 2, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(d*x)) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcCosh[c*x])/(x^3*Sqrt[d - c^2*d*x^2]), x, 8, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*d*x^2) + (c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*Sqrt[d - c^2*d*x^2]) + (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^4*Sqrt[d - c^2*d*x^2]), x, 4, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*d*x) - (2*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x])/(3*Sqrt[d - c^2*d*x^2])} - - -{(x^5*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2), x, 5, -((5*b*x*Sqrt[d - c^2*d*x^2])/(3*c^5*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*x^3*Sqrt[d - c^2*d*x^2])/(9*c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^6*d^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*c^6*d^3) - (b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(c^6*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2), x, 7, (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) + (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*c^4*d^2) - (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(4*b*c^5*d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c^5*d*Sqrt[d - c^2*d*x^2])} -{(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2), x, 4, -((b*x*Sqrt[d - c^2*d*x^2])/(c^3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a + b*ArcCosh[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^4*d^2) - (b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(c^4*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2), x, 4, (x*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c^3*d*Sqrt[d - c^2*d*x^2])} -{(x^1*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2), x, 3, (a + b*ArcCosh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2), x, 2, (x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^1*(d - c^2*d*x^2)^(3/2)), x, 9, (a + b*ArcCosh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(3/2)), x, 5, -((a + b*ArcCosh[c*x])/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)), x, 13, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x*Sqrt[d - c^2*d*x^2]) + (3*c^2*(a + b*ArcCosh[c*x]))/(2*d*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + (3*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*d*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)), x, 5, -((b*c*Sqrt[d - c^2*d*x^2])/(6*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (a + b*ArcCosh[c*x])/(3*d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcCosh[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{(x^5*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2), x, 5, (b*x*Sqrt[d - c^2*d*x^2])/(c^5*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*x*Sqrt[d - c^2*d*x^2])/(6*c^5*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)) + (a + b*ArcCosh[c*x])/(3*c^6*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcCosh[c*x]))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^6*d^3) + (11*b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(6*c^6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2), x, 9, (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^5*d*(d - c^2*d*x^2)^(3/2)) + (x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcCosh[c*x]))/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]), (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^5*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcCosh[c*x]))/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])} -{(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2), x, 4, (b*x*Sqrt[d - c^2*d*x^2])/(6*c^3*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (a + b*ArcCosh[c*x])/(3*c^4*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (5*b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(6*c^4*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*x*Sqrt[d - c^2*d*x^2])/(6*c^3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2))) + (a + b*ArcCosh[c*x])/(3*c^4*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (5*b*Sqrt[d - c^2*d*x^2]*ArcTanh[c*x])/(6*c^4*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2), x, 5, (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^3*d*(d - c^2*d*x^2)^(3/2)) + (x^3*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2]), (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(6*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{(x^1*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2), x, 4, (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcCosh[c*x])/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*c^2*d^2*Sqrt[d - c^2*d*x^2]), (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*c^2*d^2*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(5/2), x, 5, (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c*d*(d - c^2*d*x^2)^(3/2)) + (x*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(3*c*d^2*Sqrt[d - c^2*d*x^2]), (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^1*(d - c^2*d*x^2)^(5/2)), x, 13, (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcCosh[c*x])/(d^2*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (7*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)), x, 5, -((b*c*Sqrt[d - c^2*d*x^2])/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2))) - (a + b*ArcCosh[c*x])/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)), x, 18, (3*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(12*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x]))/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcCosh[c*x]))/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (13*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^ArcCosh[c*x]])/(2*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)), x, 5, -((b*c*Sqrt[d - c^2*d*x^2])/(6*d^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c^3*Sqrt[d - c^2*d*x^2])/(6*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)) - (a + b*ArcCosh[c*x])/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*ArcCosh[c*x]))/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcCosh[c*x]))/(3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (8*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*c^3*Sqrt[d - c^2*d*x^2]*Log[1 - c^2*x^2])/(3*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{ArcCosh[a*x]/(c - a^2*c*x^2)^(7/2), x, 8, (Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(15*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x])/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcCosh[a*x])/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x])/(15*c^3*Sqrt[c - a^2*c*x^2]) - (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(15*a*c^3*Sqrt[c - a^2*c*x^2])} - - -{(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2], x, 5, -((3*x^2*Sqrt[-1 + a*x])/(16*a^3*Sqrt[1 - a*x])) - (x^4*Sqrt[-1 + a*x])/(16*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(4*a^2) + (3*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(16*a^5*Sqrt[1 - a*x])} -{(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2], x, 4, -((2*x*Sqrt[-1 + a*x])/(3*a^3*Sqrt[1 - a*x])) - (x^3*Sqrt[-1 + a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(3*a^2)} -{(x^2*ArcCosh[a*x])/Sqrt[1 - a^2*x^2], x, 3, -((x^2*Sqrt[-1 + a*x])/(4*a*Sqrt[1 - a*x])) - (x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(4*a^3*Sqrt[1 - a*x])} -{(x^1*ArcCosh[a*x])/Sqrt[1 - a^2*x^2], x, 2, -((x*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x])) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/a^2} -{x^0*ArcCosh[a*x]/Sqrt[1 - a^2*x^2], x, 1, (Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(2*a*Sqrt[1 - a*x])} -{ArcCosh[a*x]/(x^1*Sqrt[1 - a^2*x^2]), x, 6, (2*Sqrt[-1 + a*x]*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (I*Sqrt[-1 + a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (I*Sqrt[-1 + a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]} -{ArcCosh[a*x]/(x^2*Sqrt[1 - a^2*x^2]), x, 2, -((Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/x) - (a*Sqrt[-1 + a*x]*Log[x])/Sqrt[1 - a*x]} -{ArcCosh[a*x]/(x^3*Sqrt[1 - a^2*x^2]), x, 8, (a*Sqrt[-1 + a*x])/(2*x*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x])/(2*x^2) + (a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (I*a^2*Sqrt[-1 + a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/(2*Sqrt[1 - a*x]) + (I*a^2*Sqrt[-1 + a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/(2*Sqrt[1 - a*x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(m/2) (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{((f*x)^(3/2)*(a + b*ArcCosh[c*x]))/Sqrt[1 - c^2*x^2], x, 1, (2*(f*x)^(5/2)*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*f) + (4*b*c*(f*x)^(7/2)*Sqrt[-1 + c*x]*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*f^2*Sqrt[1 - c*x])} - - -{((f*x)^(3/2)*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2], x, 1, (2*(f*x)^(5/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2])/(5*f*Sqrt[d - c^2*d*x^2]) + (4*b*c*(f*x)^(7/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/(35*f^2*Sqrt[d - c^2*d*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x]) with m symbolic*) - - -{(f*x)^m*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]), x, 8, If[$VersionNumber>=8, -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*(f*x)^(2 + m)*(1 - c^2*x^2))/(f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*(f*x)^(4 + m)*(1 - c^2*x^2))/(f^4*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^3*(f*x)^(6 + m)*(1 - c^2*x^2))/(f^6*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (3*c^2*d^3*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*c^4*d^3*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (c^6*d^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x]))/(f^7*(7 + m)) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*c*d^3*(2271 + 1329*m + 284*m^2 + 27*m^3 + m^4)*(f*x)^(2 + m)*(1 - c^2*x^2))/(f^2*(7 + m)^2*(15 + 8*m + m^2)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d^3*(9 + m)*(13 + 2*m)*(f*x)^(4 + m)*(1 - c^2*x^2))/(f^4*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^3*(f*x)^(6 + m)*(1 - c^2*x^2))/(f^6*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (3*c^2*d^3*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*c^4*d^3*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (c^6*d^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x]))/(f^7*(7 + m)) - (3*b*c*d^3*(2161 + 1813*m + 455*m^2 + 35*m^3)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(7 + m)^2*(2 + 3*m + m^2)*(15 + 8*m + m^2)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]), x, 7, If[$VersionNumber>=8, -((b*c*d^2*(38 + 13*m + m^2)*(f*x)^(2 + m)*(1 - c^2*x^2))/(f^2*(3 + m)^2*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d^2*(f*x)^(4 + m)*(1 - c^2*x^2))/(f^4*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (2*c^2*d^2*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (c^4*d^2*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (b*c*d^2*(149 + 100*m + 15*m^2)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(1 + m)*(2 + m)*(3 + m)^2*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*c*d^2*(38 + 13*m + m^2)*(f*x)^(2 + m)*(1 - c^2*x^2))/(f^2*(3 + m)^2*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (b*c^3*d^2*(f*x)^(4 + m)*(1 - c^2*x^2))/(f^4*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (2*c^2*d^2*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (c^4*d^2*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (b*c*d^2*(149 + 100*m + 15*m^2)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(2 + 3*m + m^2)*(15 + 8*m + m^2)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d - c^2*d*x^2)^1*(a + b*ArcCosh[c*x]), x, 6, If[$VersionNumber>=8, (b*c*d*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(f^2*(3 + m)^2) + (d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (c^2*d*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*c*d*(7 + 3*m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(1 + m)*(2 + m)*(3 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), (b*c*d*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(f^2*(3 + m)^2) + (d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) - (c^2*d*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*c*d*(7 + 3*m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f^2*(3 + m)^2*(2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x]} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2, x, 4, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(2*d^2*f*(1 - c^2*x^2)) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(2*d^2*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(2*d)} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3, x, 8, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(4*d^3*f*(1 - c^2*x^2)^2) + ((3 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(8*d^3*f*(1 - c^2*x^2)) - (b*c*(3 - m)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(8*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(4*d^3*f^2*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - m)*(3 - m)*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x])/(8*d^2)} - - -{(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]), x, 11, If[$VersionNumber>=8, -((b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (15*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(4 + m)*(6 + m)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (15*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (15*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(6 + m)*(8 + 6*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (15*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(2 + m)^2*(6 + m)*(4 + 5*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]), x, 7, If[$VersionNumber>=8, -((3*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(4 + m)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (3*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((3*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2])/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (3*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(2 + m)^2*(4 + 5*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d - c^2*d*x^2)^(1/2)*(a + b*ArcCosh[c*x]), x, 3, -((b*c*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2])/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f*(2 + m)) + ((f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(1/2), x, 1, ((f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(1 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2), x, 4, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d*f*Sqrt[d - c^2*d*x^2]) - (m*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(d*f*(1 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d*f^2*(2 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(d*f^2*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(5/2), x, 7, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d*f*(d - c^2*d*x^2)^(3/2)) + ((2 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d^2*f*Sqrt[d - c^2*d*x^2]) - ((2 - m)*m*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(3*d^2*f*(1 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(2 - m)*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d^2*f^2*(2 + m)*Sqrt[d - c^2*d*x^2]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d^2*f^2*(2 + m)*Sqrt[d - c^2*d*x^2]) - (b*c*(2 - m)*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(3*d^2*f^2*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])} - - -{(f*x)^m*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]), x, 11, If[$VersionNumber>=8, -((b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d1^2*d2^2*(f*x)^(6 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d1*d2*(f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(4 + m)*(6 + m)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(6 + m)*(8 + 6*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d1^2*d2^2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d1^2*d2^2*(f*x)^(6 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d1*d2*(f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]))/(f*(6 + m)) + (15*d1^2*d2^2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(6 + m)*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (15*b*c*d1^2*d2^2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(2 + m)^2*(6 + m)*(4 + 5*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]), x, 7, If[$VersionNumber>=8, -((3*b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d1*d2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d1*d2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d1*d2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(4 + m)*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (3*b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((3*b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d1*d2*(f*x)^(4 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d1*d2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]))/(f*(4 + m)) + (3*d1*d2*(f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(8 + 14*m + 7*m^2 + m^3)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (3*b*c*d1*d2*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(2 + m)^2*(4 + 5*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d1 + c*d1*x)^(1/2)*(d2 - c*d2*x)^(1/2)*(a + b*ArcCosh[c*x]), x, 3, -((b*c*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(2 + m)) + ((f*x)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(2 + 3*m + m^2)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) - (b*c*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(f*x)^m*(a + b*ArcCosh[c*x])/((d1 + c*d1*x)^(1/2)*(d2 - c*d2*x)^(1/2)), x, 1, ((f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])} -{(f*x)^m*(a + b*ArcCosh[c*x])/((d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)), x, 4, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d1*d2*f*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - (m*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(d1*d2*f*(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d1*d2*f^2*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - (b*c*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(d1*d2*f^2*(1 + m)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])} -{(f*x)^m*(a + b*ArcCosh[c*x])/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)), x, 7, ((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d1*d2*f*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)) + ((2 - m)*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(3*d1^2*d2^2*f*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - ((2 - m)*m*(f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(3*d1^2*d2^2*f*(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(2 - m)*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d1^2*d2^2*f^2*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(3*d1^2*d2^2*f^2*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) - (b*c*(2 - m)*m*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(3*d1^2*d2^2*f^2*(1 + m)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])} - - -{(f*x)^m*ArcCosh[a*x]/Sqrt[1 - a^2*x^2], x, 1, ((f*x)^(1 + m)*ArcCosh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(f*(1 + m)) + (a*(f*x)^(2 + m)*Sqrt[-1 + a*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/(f^2*(1 + m)*(2 + m)*Sqrt[1 - a*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^2*) - - -{(c - a^2*c*x^2)^3*ArcCosh[a*x]^2, x, 17, (4322*c^3*x)/3675 - (1514*a^2*c^3*x^3)/11025 + (234*a^4*c^3*x^5)/6125 - (2/343)*a^6*c^3*x^7 - (32*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(35*a) + (16*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(105*a) - (12*c^3*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x])/(175*a) + (2*c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2)*ArcCosh[a*x])/(49*a) + (16/35)*c^3*x*ArcCosh[a*x]^2 + (8/35)*c^3*x*(1 - a^2*x^2)*ArcCosh[a*x]^2 + (6/35)*c^3*x*(1 - a^2*x^2)^2*ArcCosh[a*x]^2 + (1/7)*c^3*x*(1 - a^2*x^2)^3*ArcCosh[a*x]^2} -{(c - a^2*c*x^2)^2*ArcCosh[a*x]^2, x, 12, (298*c^2*x)/225 - (76/675)*a^2*c^2*x^3 + (2/125)*a^4*c^2*x^5 - (16*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(15*a) + (8*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(45*a) - (2*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x])/(25*a) + (8/15)*c^2*x*ArcCosh[a*x]^2 + (4/15)*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^2 + (1/5)*c^2*x*(1 - a^2*x^2)^2*ArcCosh[a*x]^2} -{(c - a^2*c*x^2)^1*ArcCosh[a*x]^2, x, 7, (14*c*x)/9 - (2/27)*a^2*c*x^3 - (4*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*a) + (2*c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(9*a) + (2/3)*c*x*ArcCosh[a*x]^2 + (1/3)*c*x*(1 - a^2*x^2)*ArcCosh[a*x]^2} -{ArcCosh[a*x]^2/(c - a^2*c*x^2)^1, x, 8, (2*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (2*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (2*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (2*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (2*PolyLog[3, E^ArcCosh[a*x]])/(a*c)} -{ArcCosh[a*x]^2/(c - a^2*c*x^2)^2, x, 12, -(ArcCosh[a*x]/(a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (x*ArcCosh[a*x]^2)/(2*c^2*(1 - a^2*x^2)) + (ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - ArcTanh[a*x]/(a*c^2) + (ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(a*c^2) - (ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(a*c^2) - PolyLog[3, -E^ArcCosh[a*x]]/(a*c^2) + PolyLog[3, E^ArcCosh[a*x]]/(a*c^2)} -{ArcCosh[a*x]^2/(c - a^2*c*x^2)^3, x, 17, -(x/(12*c^3*(1 - a^2*x^2))) + ArcCosh[a*x]/(6*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (3*ArcCosh[a*x])/(4*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^2)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^2)/(8*c^3*(1 - a^2*x^2)) + (3*ArcCosh[a*x]^2*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*ArcTanh[a*x])/(6*a*c^3) + (3*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]])/(4*a*c^3) - (3*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]])/(4*a*c^3) - (3*PolyLog[3, -E^ArcCosh[a*x]])/(4*a*c^3) + (3*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2, x, 16, (-856*b^2*Sqrt[d - c^2*d*x^2])/(3375*c^4) + (22*b^2*x^2*Sqrt[d - c^2*d*x^2])/(3375*c^2) + (2*b^2*x^4*Sqrt[d - c^2*d*x^2])/125 + (4*a*b*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^2) + (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/5} -{x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2, x, 11, -(b^2*x*Sqrt[d - c^2*d*x^2])/(64*c^2) + (b^2*x^3*Sqrt[d - c^2*d*x^2])/32 - (b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(64*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(8*c^2) + (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/4 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(24*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^1*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2, x, 6, -((14*b^2*Sqrt[d - c^2*d*x^2])/(27*c^2)) + (2/27)*b^2*x^2*Sqrt[d - c^2*d*x^2] + (2*b*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*c^2*d)} -{x^0*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2, x, 5, (b^2*x*Sqrt[d - c^2*d*x^2])/4 + (b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(6*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^1, x, 12, 2*b^2*Sqrt[d - c^2*d*x^2] - (2*a*b*c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^2, x, 7, -((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x) + (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^3, x, 12, -((b*c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*x^2) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c^2*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (I*b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b^2*c^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^4, x, 11, (b^2*c^2*Sqrt[d - c^2*d*x^2])/(3*x) - (b^2*c^3*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*d*x^3) - (2*b*c^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x, 26, -((37384*b^2*d*Sqrt[d - c^2*d*x^2])/(385875*c^4)) + (3358*b^2*d*x^2*Sqrt[d - c^2*d*x^2])/(385875*c^2) + (484*b^2*d*x^4*Sqrt[d - c^2*d*x^2])/42875 - (2/343)*b^2*c^2*d*x^6*Sqrt[d - c^2*d*x^2] + (4*a*b*d*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (16*b*c*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(35*c^4) - (d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(35*c^2) + (3/35)*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (1/7)*x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2} -{x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x, 20, (7*b^2*d*x*Sqrt[d - c^2*d*x^2])/(1152*c^2) + (43*b^2*d*x^3*Sqrt[d - c^2*d*x^2])/1728 - (1/108)*b^2*c^2*d*x^5*Sqrt[d - c^2*d*x^2] + (7*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(48*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(18*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*c^2) + (1/8)*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (1/6)*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(48*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^1*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x, 8, -((16*b^2*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(75*c^2*(1 - c*x)*(1 + c*x))) - (8*b^2*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(225*c^2*(1 - c*x)*(1 + c*x)) - (2*b^2*d*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(125*c^2*(1 - c*x)*(1 + c*x)) + (2*b*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(5*c^2*d)} -{x^0*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x, 11, (15/64)*b^2*d*x*Sqrt[d - c^2*d*x^2] + (1/32)*b^2*d*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2] + (9*b^2*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3/8)*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (1/4)*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(8*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^1, x, 18, (68/27)*b^2*d*Sqrt[d - c^2*d*x^2] - (2/27)*b^2*c^2*d*x^2*Sqrt[d - c^2*d*x^2] - (2*a*b*c*d*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b^2*c*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (1/3)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^2, x, 15, (-(1/4))*b^2*c^2*d*x*Sqrt[d - c^2*d*x^2] - (5*b^2*c*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3/2)*c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^3, x, 18, -2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b^2*c^3*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3/2)*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*I*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*I*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*I*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^4, x, 18, (b^2*c^2*d*Sqrt[d - c^2*d*x^2])/(3*x) - (b^2*c^3*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x - (4*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*x^3) - (c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c^3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b^2*c^3*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x, 35, -((37384*b^2*d^2*Sqrt[d - c^2*d*x^2])/(694575*c^4)) + (3358*b^2*d^2*x^2*Sqrt[d - c^2*d*x^2])/(694575*c^2) + (484*b^2*d^2*x^4*Sqrt[d - c^2*d*x^2])/77175 - (10*b^2*c^2*d^2*x^6*Sqrt[d - c^2*d*x^2])/3087 + (4*a*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(2835*c^4*(1 - c*x)*(1 + c*x)) + (8*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(8505*c^4*(1 - c*x)*(1 + c*x)) + (2*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(4725*c^4*(1 - c*x)*(1 + c*x)) - (20*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(3969*c^4*(1 - c*x)*(1 + c*x)) + (2*b^2*d^2*(1 - c^2*x^2)^5*Sqrt[d - c^2*d*x^2])/(729*c^4*(1 - c*x)*(1 + c*x)) + (4*b^2*d^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(189*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (38*b*c^3*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^4) - (d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(63*c^2) + (1/21)*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (5/63)*d*x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 + (1/9)*x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2} -{x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x, 31, (35*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/(9216*c^2) + (215*b^2*d^2*x^3*Sqrt[d - c^2*d*x^2])/13824 - (5/864)*b^2*c^2*d^2*x^5*Sqrt[d - c^2*d*x^2] + (73*b^2*d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(12288*c^2*(1 - c*x)*(1 + c*x)) + (73*b^2*d^2*x^3*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(18432*(1 - c*x)*(1 + c*x)) - (43*b^2*c^2*d^2*x^5*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(4608*(1 - c*x)*(1 + c*x)) + (b^2*c^4*d^2*x^7*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(256*(1 - c*x)*(1 + c*x)) + (35*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(9216*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^2*x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(384*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*x^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(144*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(128*c^2) + (5/64)*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (5/48)*d*x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 + (1/8)*x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(384*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (73*b^2*d^2*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(12288*c^3*(1 - c*x)*(1 + c*x))} -{x^1*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x, 8, -((32*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(245*c^2*(1 - c*x)*(1 + c*x))) - (16*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(735*c^2*(1 - c*x)*(1 + c*x)) - (12*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(1225*c^2*(1 - c*x)*(1 + c*x)) - (2*b^2*d^2*(1 - c^2*x^2)^4*Sqrt[d - c^2*d*x^2])/(343*c^2*(1 - c*x)*(1 + c*x)) + (2*b*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^7*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x])^2)/(7*c^2*d)} -{x^0*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x, 18, (245*b^2*d^2*x*Sqrt[d - c^2*d*x^2])/1152 + (65*b^2*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/1728 + (1/108)*b^2*d^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2] + (115*b^2*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(1152*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(48*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(18*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/16)*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (5/24)*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 + (1/6)*x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(48*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^1, x, 26, (68/27)*b^2*d^2*Sqrt[d - c^2*d*x^2] - (2/27)*b^2*c^2*d^2*x^2*Sqrt[d - c^2*d*x^2] - (2*a*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (16*b^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(75*(1 - c*x)*(1 + c*x)) + (8*b^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(225*(1 - c*x)*(1 + c*x)) + (2*b^2*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(125*(1 - c*x)*(1 + c*x)) - (2*b^2*c*d^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (16*b*c*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (22*b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(45*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (1/3)*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 + (1/5)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2 - (2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*I*b^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*I*b^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^2, x, 25, (-(31/64))*b^2*c^2*d^2*x*Sqrt[d - c^2*d*x^2] - (1/32)*b^2*c^2*d^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2] - (89*b^2*c*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (15/8)*c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 + (c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5/4)*c^2*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x + (5*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(8*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^3, x, 28, (-(170/27))*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2] + (5/27)*b^2*c^4*d^2*x^2*Sqrt[d - c^2*d*x^2] + (5*a*b*c^3*d^2*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(3*(1 - c*x)*(1 + c*x)) + (b^2*c^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(9*(1 - c*x)*(1 + c*x)) + (5*b^2*c^3*d^2*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5/2)*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 - (5/6)*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c^2*d^2*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/((1 - c*x)*(1 + c*x)) - (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*I*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*I*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*I*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^4, x, 30, (7/12)*b^2*c^4*d^2*x*Sqrt[d - c^2*d*x^2] + (b^2*c^2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2])/(3*x) + (23*b^2*c^3*d^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(12*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c^5*d^2*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c^3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/2)*c^4*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2 - (7*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(3*x) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(3*x^3) - (5*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(6*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (14*b*c^3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*b^2*c^3*d^2*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^5*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x, 16, -((16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2])) - (4144*b^2*(1 - c*x)*(1 + c*x))/(3375*c^6*Sqrt[d - c^2*d*x^2]) - (272*b^2*x^2*(1 - c*x)*(1 + c*x))/(3375*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*x^4*(1 - c*x)*(1 + c*x))/(125*c^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (8*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(45*c^3*Sqrt[d - c^2*d*x^2]) - (2*b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(15*c^4*d) - (x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(5*c^2*d)} -{(x^4*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x, 11, -((15*b^2*x*(1 - c*x)*(1 + c*x))/(64*c^4*Sqrt[d - c^2*d*x^2])) - (b^2*x^3*(1 - c*x)*(1 + c*x))/(32*c^2*Sqrt[d - c^2*d*x^2]) + (15*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(64*c^5*Sqrt[d - c^2*d*x^2]) - (3*b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*c*Sqrt[d - c^2*d*x^2]) - (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(8*c^4*d) - (x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(8*b*c^5*Sqrt[d - c^2*d*x^2])} -{(x^3*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x, 9, -((4*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^3*Sqrt[d - c^2*d*x^2])) - (40*b^2*(1 - c*x)*(1 + c*x))/(27*c^4*Sqrt[d - c^2*d*x^2]) - (2*b^2*x^2*(1 - c*x)*(1 + c*x))/(27*c^2*Sqrt[d - c^2*d*x^2]) - (4*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3*Sqrt[d - c^2*d*x^2]) - (2*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c*Sqrt[d - c^2*d*x^2]) - (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^2*d)} -{(x^2*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x, 5, -((b^2*x*(1 - c*x)*(1 + c*x))/(4*c^2*Sqrt[d - c^2*d*x^2])) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(4*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c*Sqrt[d - c^2*d*x^2]) - (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*c^2*d) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2])} -{(x^1*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x, 4, -((2*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (2*b^2*(1 - c*x)*(1 + c*x))/(c^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(c*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d)} -{x^0*(a + b*ArcCosh[c*x])^2/Sqrt[d - c^2*d*x^2], x, 1, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^1*Sqrt[d - c^2*d*x^2]), x, 8, (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + ((2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - ((2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcCosh[c*x])^2/(x^2*Sqrt[d - c^2*d*x^2]), x, 6, -((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(d*x) - (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/Sqrt[d - c^2*d*x^2] + (b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcCosh[c*x])^2/(x^3*Sqrt[d - c^2*d*x^2]), x, 12, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*d*x^2) + (c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - (b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/Sqrt[d - c^2*d*x^2] - (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + (I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] + (I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2] - (I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/Sqrt[d - c^2*d*x^2]} -{(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]), x, 9, (b^2*c^2*(1 - c*x)*(1 + c*x))/(3*x*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*x^2*Sqrt[d - c^2*d*x^2]) - (2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*d*x^3) - (2*c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*d*x) - (4*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/(3*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*Sqrt[d - c^2*d*x^2])} - - -{(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x, 23, (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) + (94*b^2*(1 - c*x)*(1 + c*x))/(27*c^6*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*x^2*(1 - c*x)*(1 + c*x))/(27*c^4*d*Sqrt[d - c^2*d*x^2]) + (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(c^5*d*Sqrt[d - c^2*d*x^2]) + (2*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2])} -{(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x, 15, (b^2*x*(1 - c*x)*(1 + c*x))/(4*c^4*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(4*c^5*d*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^3*d*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(c^5*d*Sqrt[d - c^2*d*x^2]) + (3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*c^4*d^2) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(2*b*c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c^5*d*Sqrt[d - c^2*d*x^2])} -{(x^3*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x, 14, (4*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*(1 - c*x)*(1 + c*x))/(c^4*d*Sqrt[d - c^2*d*x^2]) + (4*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(c^3*d*Sqrt[d - c^2*d*x^2]) + (x^2*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^4*d^2) + (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(c^4*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(c^4*d*Sqrt[d - c^2*d*x^2])} -{(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x, 8, (x*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2]) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])} -{(x^1*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x, 8, (a + b*ArcCosh[c*x])^2/(c^2*d*Sqrt[d - c^2*d*x^2]) + (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(c^2*d*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 6, (x*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(c*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(c*d*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(c*d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^1*(d - c^2*d*x^2)^(3/2)), x, 16, (a + b*ArcCosh[c*x])^2/(d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - ((2*I)*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^2*(d - c^2*d*x^2)^(3/2)), x, 15, -((a + b*ArcCosh[c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) + (2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)), x, 27, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(d*x*Sqrt[d - c^2*d*x^2]) + (3*c^2*(a + b*ArcCosh[c*x])^2)/(2*d*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + (3*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (4*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - (2*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (3*I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - (3*I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)), x, 26, (b^2*c^2*(1 - c*x)*(1 + c*x))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d*x^2*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(3*d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcCosh[c*x])^2)/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcCosh[c*x])^2)/(3*d*Sqrt[d - c^2*d*x^2]) + (8*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*d*Sqrt[d - c^2*d*x^2]) - (20*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) - (16*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) - (5*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(3*d*Sqrt[d - c^2*d*x^2]) - (b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2])} - - -{(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x, 28, -((b^2*x^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2])) - (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) - (7*b^2*(1 - c*x)*(1 + c*x))/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (11*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcCosh[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^6*d^3) - (22*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])} -{(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x, 20, -(b^2/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])) + (b^2*(1 - c*x))/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (x*(a + b*ArcCosh[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (8*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (4*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2])} -{(x^3*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x, 18, -(b^2/(3*c^4*d^2*Sqrt[d - c^2*d*x^2])) + (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^2*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcCosh[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (10*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (5*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^4*d^2*Sqrt[d - c^2*d*x^2])} -{(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x, 12, -(b^2/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])) + (b^2*(1 - c*x))/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])} -{(x^1*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x, 10, -(b^2/(3*c^2*d^2*Sqrt[d - c^2*d*x^2])) + (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])^2/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^2*d^2*Sqrt[d - c^2*d*x^2])} -{x^0*(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 10, -((b^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2])) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*c*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^1*(d - c^2*d*x^2)^(5/2)), x, 26, -(b^2/(3*d^2*Sqrt[d - c^2*d*x^2])) + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (a + b*ArcCosh[c*x])^2/(3*d*(d - c^2*d*x^2)^(3/2)) + (a + b*ArcCosh[c*x])^2/(d^2*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (14*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (7*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (2*I*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (7*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)), x, 21, -((b^2*c^2*x)/(3*d^2*Sqrt[d - c^2*d*x^2])) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(d*x*(d - c^2*d*x^2)^(3/2)) + (4*c^2*x*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (8*c^2*x*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (8*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)), x, 41, -((b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2])) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(d^2*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (2*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x])^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])^2/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcCosh[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (26*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (13*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (13*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (5*I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (5*I*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)), x, 36, (b^2*c^2)/(3*d^2*x*Sqrt[d - c^2*d*x^2]) - (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*x^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(3*d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*ArcCosh[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (16*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (8*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2])} - - -{ArcCosh[a*x]^2/(c - a^2*c*x^2)^(7/2), x, 15, -(x/(3*c^3*Sqrt[c - a^2*c*x^2])) - x/(30*c^3*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(10*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(15*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcCosh[a*x]^2)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x]^2)/(15*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*Log[1 - E^(2*ArcCosh[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2])} - - -{x^4*ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2], x, 11, -((15*x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(64*a^4)) - (x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(32*a^2) + (15*Sqrt[-1 + a*x]*ArcCosh[a*x])/(64*a^5*Sqrt[1 - a*x]) - (3*x^2*Sqrt[-1 + a*x]*ArcCosh[a*x])/(8*a^3*Sqrt[1 - a*x]) - (x^4*Sqrt[-1 + a*x]*ArcCosh[a*x])/(8*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(4*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/(8*a^5*Sqrt[1 - a*x])} -{x^3*ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2], x, 8, -((40*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^4)) - (2*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(27*a^2) - (4*x*Sqrt[-1 + a*x]*ArcCosh[a*x])/(3*a^3*Sqrt[1 - a*x]) - (2*x^3*Sqrt[-1 + a*x]*ArcCosh[a*x])/(9*a*Sqrt[1 - a*x]) - (2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(3*a^2)} -{x^2*ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2], x, 5, -((x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(4*a^2)) + (Sqrt[-1 + a*x]*ArcCosh[a*x])/(4*a^3*Sqrt[1 - a*x]) - (x^2*Sqrt[-1 + a*x]*ArcCosh[a*x])/(2*a*Sqrt[1 - a*x]) - (x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/(6*a^3*Sqrt[1 - a*x])} -{x^1*ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2], x, 3, -((2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/a^2) - (2*x*Sqrt[-1 + a*x]*ArcCosh[a*x])/(a*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/a^2} -{x^0*ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2], x, 1, (Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/(3*a*Sqrt[1 - a*x])} -{ArcCosh[a*x]^2/(x^1*Sqrt[1 - a^2*x^2]), x, 8, (2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (2*I*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (2*I*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (2*I*Sqrt[-1 + a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (2*I*Sqrt[-1 + a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]} -{ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]), x, 6, (a*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/Sqrt[1 - a*x] - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/x - (2*a*Sqrt[-1 + a*x]*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])])/Sqrt[1 - a*x] - (a*Sqrt[-1 + a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])])/Sqrt[1 - a*x]} -{ArcCosh[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]), x, 12, (a*Sqrt[-1 + a*x]*ArcCosh[a*x])/(x*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(2*x^2) + (a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (a^2*Sqrt[-1 + a*x]*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/Sqrt[1 - a*x] - (I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (I*a^2*Sqrt[-1 + a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (I*a^2*Sqrt[-1 + a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^2 with m symbolic*) - - -(* {(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^2, x, 1, 0} -{(f*x)^m*(d - c^2*d*x^2)^1*(a + b*ArcCosh[c*x])^2, x, 1, 0} -{(f*x)^m/(d - c^2*d*x^2)^1*(a + b*ArcCosh[c*x])^2, x, 1, 0} -{(f*x)^m/(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^2, x, 1, 0} *) - - -{(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x, 22, If[$VersionNumber>=8, -((10*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3*(6 + m))) - (2*b^2*c^2*d^2*(52 + 15*m + m^2)*(f*x)^(3 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^2*(6 + m)^3*(1 - c*x)*(1 + c*x)) + (2*b^2*c^4*d^2*(f*x)^(5 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^5*(6 + m)^3*(1 - c*x)*(1 + c*x)) - (2*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (30*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (10*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)) - (30*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(6 + m)*(1 - c*x)*(1 + c*x)) - (10*b^2*c^2*d^2*(10 + 3*m)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)*(3 + m)*(4 + m)^3*(6 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)*(3 + m)*(4 + m)^2*(6 + m)^3*(1 - c*x)*(1 + c*x)) + (15*d^3*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2)), -((10*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3*(6 + m))) - (2*b^2*c^2*d^2*(52 + 15*m + m^2)*(f*x)^(3 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^2*(6 + m)^3*(1 - c*x)*(1 + c*x)) + (2*b^2*c^4*d^2*(f*x)^(5 + m)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(f^5*(6 + m)^3*(1 - c*x)*(1 + c*x)) - (2*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (30*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*c*d^2*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(6 + m)*(8 + 6*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (10*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)^2*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*c^3*d^2*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*(f*x)^(6 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^6*(6 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*d^2*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)*(8 + 6*m + m^2)) + (5*d*(f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(f*(4 + m)*(6 + m)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(f*(6 + m)) - (30*b^2*c^2*d^2*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(6 + m)*(1 - c*x)*(1 + c*x)) - (10*b^2*c^2*d^2*(10 + 3*m)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(4 + m)^3*(6 + m)*(6 + 5*m + m^2)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d^2*(264 + 130*m + 15*m^2)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(4 + m)^2*(6 + m)^3*(6 + 5*m + m^2)*(1 - c*x)*(1 + c*x)) + (15*d^3*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/((6 + m)*(8 + 6*m + m^2))]} -{(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x, 13, If[$VersionNumber>=8, -((2*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3)) - (6*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(f*(4 + m)) - (6*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d*(10 + 3*m)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)*(3 + m)*(4 + m)^3*(1 - c*x)*(1 + c*x)) + (3*d^2*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(8 + 6*m + m^2), -((2*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[d - c^2*d*x^2])/(f^3*(4 + m)^3)) - (6*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*(f*x)^(4 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^4*(4 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*d*(f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(8 + 6*m + m^2)) + ((f*x)^(1 + m)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/(f*(4 + m)) - (6*b^2*c^2*d*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(4 + m)*(1 - c*x)*(1 + c*x)) - (2*b^2*c^2*d*(10 + 3*m)*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(4 + m)^3*(6 + 5*m + m^2)*(1 - c*x)*(1 + c*x)) + (3*d^2*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(8 + 6*m + m^2)]} -{(f*x)^m*(d - c^2*d*x^2)^(1/2)*(a + b*ArcCosh[c*x])^2, x, 5, -((2*b*c*(f*x)^(2 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x)^(1 + m)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(f*(2 + m)) - (2*b^2*c^2*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(f^3*(2 + m)^2*(3 + m)*(1 - c*x)*(1 + c*x)) + (d*Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x])/(2 + m)} -{(f*x)^m*(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(1/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(5/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]} - - -{(f*x)^m*ArcCosh[c*x]^2/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^3*) - - -{(c - a^2*c*x^2)^3*ArcCosh[a*x]^3, x, 29, -((976*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(315*a)) + (16/315)*a*c^3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (7104*c^3*(1 - a^2*x^2))/(42875*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (1184*c^3*(1 - a^2*x^2)^2)/(42875*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2664*c^3*(1 - a^2*x^2)^3)/(214375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (6*c^3*(1 - a^2*x^2)^4)/(2401*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (4322*c^3*x*ArcCosh[a*x])/1225 - (1514*a^2*c^3*x^3*ArcCosh[a*x])/3675 + (702*a^4*c^3*x^5*ArcCosh[a*x])/6125 - (6/343)*a^6*c^3*x^7*ArcCosh[a*x] - (48*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(35*a) + (8*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(35*a) - (18*c^3*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]^2)/(175*a) + (3*c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2)*ArcCosh[a*x]^2)/(49*a) + (16/35)*c^3*x*ArcCosh[a*x]^3 + (8/35)*c^3*x*(1 - a^2*x^2)*ArcCosh[a*x]^3 + (6/35)*c^3*x*(1 - a^2*x^2)^2*ArcCosh[a*x]^3 + (1/7)*c^3*x*(1 - a^2*x^2)^3*ArcCosh[a*x]^3} -{(c - a^2*c*x^2)^2*ArcCosh[a*x]^3, x, 20, -((488*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(135*a)) + (8/135)*a*c^2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (16*c^2*(1 - a^2*x^2))/(125*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (8*c^2*(1 - a^2*x^2)^2)/(375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (6*c^2*(1 - a^2*x^2)^3)/(625*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (298/75)*c^2*x*ArcCosh[a*x] - (76/225)*a^2*c^2*x^3*ArcCosh[a*x] + (6/125)*a^4*c^2*x^5*ArcCosh[a*x] - (8*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(5*a) + (4*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(15*a) - (3*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]^2)/(25*a) + (8/15)*c^2*x*ArcCosh[a*x]^3 + (4/15)*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^3 + (1/5)*c^2*x*(1 - a^2*x^2)^2*ArcCosh[a*x]^3} -{(c - a^2*c*x^2)*ArcCosh[a*x]^3, x, 11, -((122*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a)) + (2/27)*a*c*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (14/3)*c*x*ArcCosh[a*x] - (2/9)*a^2*c*x^3*ArcCosh[a*x] - (2*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + (c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(3*a) + (2/3)*c*x*ArcCosh[a*x]^3 + (1/3)*c*x*(1 - a^2*x^2)*ArcCosh[a*x]^3} -{ArcCosh[a*x]^3/(c - a^2*c*x^2), x, 10, (2*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (6*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (6*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c) + (6*PolyLog[4, -E^ArcCosh[a*x]])/(a*c) - (6*PolyLog[4, E^ArcCosh[a*x]])/(a*c)} -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^2, x, 19, -((3*ArcCosh[a*x]^2)/(2*a*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (x*ArcCosh[a*x]^3)/(2*c^2*(1 - a^2*x^2)) - (6*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) + (ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c^2) - (3*PolyLog[2, -E^ArcCosh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^2) + (3*PolyLog[2, E^ArcCosh[a*x]])/(a*c^2) - (3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^2) - (3*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(a*c^2) + (3*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c^2) + (3*PolyLog[4, -E^ArcCosh[a*x]])/(a*c^2) - (3*PolyLog[4, E^ArcCosh[a*x]])/(a*c^2)} -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^3, x, 30, 1/(4*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (x*ArcCosh[a*x])/(4*c^3*(1 - a^2*x^2)) + ArcCosh[a*x]^2/(4*a*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)) - (9*ArcCosh[a*x]^2)/(8*a*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*ArcCosh[a*x]^3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*ArcCosh[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - (5*ArcCosh[a*x]*ArcTanh[E^ArcCosh[a*x]])/(a*c^3) + (3*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(4*a*c^3) - (5*PolyLog[2, -E^ArcCosh[a*x]])/(2*a*c^3) + (9*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(8*a*c^3) + (5*PolyLog[2, E^ArcCosh[a*x]])/(2*a*c^3) - (9*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(8*a*c^3) - (9*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(4*a*c^3) + (9*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(4*a*c^3) + (9*PolyLog[4, -E^ArcCosh[a*x]])/(4*a*c^3) - (9*PolyLog[4, E^ArcCosh[a*x]])/(4*a*c^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])^3*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^3, x, 29, -((865*a*c^2*x^2*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (65*a^3*c^2*x^4*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*(1 - a^2*x^2)^3*Sqrt[c - a^2*c*x^2])/(216*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (245/384)*c^2*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x] + (65/576)*c^2*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x] + (1/36)*c^2*x*(1 - a*x)^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x] + (115*c^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(768*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*c^2*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c^2*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*(1 - a^2*x^2)^3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(12*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5/16)*c^2*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3 + (5/24)*c*x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3 + (1/6)*x*(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^3 - (5*c^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3, x, 16, -((51*a*c*x^2*Sqrt[c - a^2*c*x^2])/(128*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (3*a^3*c*x^4*Sqrt[c - a^2*c*x^2])/(128*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (45/64)*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x] + (3/32)*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x] + (27*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(128*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3 + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3, x, 6, (-3*a*x^2*Sqrt[c - a^2*c*x^2])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/4 + (3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^(1/2), x, 1, (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^4)/(4*a*Sqrt[c - a^2*c*x^2])} -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^(3/2), x, 7, (x*ArcCosh[a*x]^3)/(c*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(a*c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(a*c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(a*c*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, E^(2*ArcCosh[a*x])])/(2*a*c*Sqrt[c - a^2*c*x^2])} -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2), x, 12, -((x*ArcCosh[a*x])/(c^2*Sqrt[c - a^2*c*x^2])) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*a*c^2*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcCosh[a*x]^3)/(3*c^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(3*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2])} -{ArcCosh[a*x]^3/(c - a^2*c*x^2)^(7/2), x, 20, -((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2])) - (x*ArcCosh[a*x])/(c^3*Sqrt[c - a^2*c*x^2]) - (x*ArcCosh[a*x])/(10*c^3*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(5*a*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^3)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcCosh[a*x]^3)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcCosh[a*x]^3)/(15*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, E^(2*ArcCosh[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2])} - - -{(x^4*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2], x, 13, -((45*x^2*Sqrt[-1 + a*x])/(128*a^3*Sqrt[1 - a*x])) - (3*x^4*Sqrt[-1 + a*x])/(128*a*Sqrt[1 - a*x]) - (45*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(64*a^4) - (3*x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(32*a^2) + (45*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(128*a^5*Sqrt[1 - a*x]) - (9*x^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(16*a^3*Sqrt[1 - a*x]) - (3*x^4*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(16*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(4*a^2) + (3*Sqrt[-1 + a*x]*ArcCosh[a*x]^4)/(32*a^5*Sqrt[1 - a*x])} -{(x^3*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2], x, 10, -((40*x*Sqrt[-1 + a*x])/(9*a^3*Sqrt[1 - a*x])) - (2*x^3*Sqrt[-1 + a*x])/(27*a*Sqrt[1 - a*x]) - (40*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^4) - (2*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(9*a^2) - (2*x*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(a^3*Sqrt[1 - a*x]) - (x^3*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(3*a*Sqrt[1 - a*x]) - (2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(3*a^2)} -{(x^2*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2], x, 6, -((3*x^2*Sqrt[-1 + a*x])/(8*a*Sqrt[1 - a*x])) - (3*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(4*a^2) + (3*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(8*a^3*Sqrt[1 - a*x]) - (3*x^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(4*a*Sqrt[1 - a*x]) - (x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^4)/(8*a^3*Sqrt[1 - a*x])} -{(x^1*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2], x, 4, -((6*x*Sqrt[-1 + a*x])/(a*Sqrt[1 - a*x])) - (6*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/a^2 - (3*x*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(a*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/a^2} -{x^0*ArcCosh[a*x]^3/Sqrt[1 - a^2*x^2], x, 1, (Sqrt[-1 + a*x]*ArcCosh[a*x]^4)/(4*a*Sqrt[1 - a*x])} -{ArcCosh[a*x]^3/(x^1*Sqrt[1 - a^2*x^2]), x, 10, (2*Sqrt[-1 + a*x]*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (3*I*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (3*I*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (6*I*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (6*I*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (6*I*Sqrt[-1 + a*x]*PolyLog[4, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (6*I*Sqrt[-1 + a*x]*PolyLog[4, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]} -{ArcCosh[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]), x, 7, (a*Sqrt[-1 + a*x]*ArcCosh[a*x]^3)/Sqrt[1 - a*x] - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/x - (3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*Log[1 + E^(2*ArcCosh[a*x])])/Sqrt[1 - a*x] - (3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])])/Sqrt[1 - a*x] + (3*a*Sqrt[-1 + a*x]*PolyLog[3, -E^(2*ArcCosh[a*x])])/(2*Sqrt[1 - a*x])} -{ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]), x, 18, (3*a*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(2*x*Sqrt[1 - a*x]) - (Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(2*x^2) - (6*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (3*I*a^2*Sqrt[-1 + a*x]*PolyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (3*I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]])/(2*Sqrt[1 - a*x]) - (3*I*a^2*Sqrt[-1 + a*x]*PolyLog[2, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (3*I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]])/(2*Sqrt[1 - a*x]) + (3*I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (3*I*a^2*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - (3*I*a^2*Sqrt[-1 + a*x]*PolyLog[4, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + (3*I*a^2*Sqrt[-1 + a*x]*PolyLog[4, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^3 with m symbolic*) - - -{(f*x)^m*(a + b*ArcCosh[c*x])^3/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^3)/Sqrt[1 - c^2*x^2], x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])*) - - -{(c - a^2*c*x^2)^3/ArcCosh[a*x], x, 7, (35*c^3*SinhIntegral[ArcCosh[a*x]])/(64*a) - (21*c^3*SinhIntegral[3*ArcCosh[a*x]])/(64*a) + (7*c^3*SinhIntegral[5*ArcCosh[a*x]])/(64*a) - (c^3*SinhIntegral[7*ArcCosh[a*x]])/(64*a)} -{(c - a^2*c*x^2)^2/ArcCosh[a*x], x, 6, (5*c^2*SinhIntegral[ArcCosh[a*x]])/(8*a) - (5*c^2*SinhIntegral[3*ArcCosh[a*x]])/(16*a) + (c^2*SinhIntegral[5*ArcCosh[a*x]])/(16*a)} -{(c - a^2*c*x^2)/ArcCosh[a*x], x, 5, (3*c*SinhIntegral[ArcCosh[a*x]])/(4*a) - (c*SinhIntegral[3*ArcCosh[a*x]])/(4*a)} -{1/((c - a^2*c*x^2)*ArcCosh[a*x]), x, 0, Unintegrable[1/((c - a^2*c*x^2)*ArcCosh[a*x]), x]} -{1/((c - a^2*c*x^2)^2*ArcCosh[a*x]), x, 0, Unintegrable[1/((c - a^2*c*x^2)^2*ArcCosh[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) / (a+b ArcCosh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x, 12, -((Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^5*Sqrt[-1 + c*x])) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^5*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(16*b*c^5*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^5*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^5*Sqrt[-1 + c*x])} -{(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x, 12, -((Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^4*Sqrt[-1 + c*x])) + (Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^4*Sqrt[-1 + c*x])} -{(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x, 6, (Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(8*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c^3*Sqrt[-1 + c*x])} -{(x^1*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x, 9, -((Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^2*Sqrt[-1 + c*x])) + (Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^2*Sqrt[-1 + c*x])} -{x^0*Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x]), x, 6, (Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(2*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c*Sqrt[-1 + c*x])} -{Sqrt[1 - c^2*x^2]/(x^1*(a + b*ArcCosh[c*x])), x, 6, -((Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b*Sqrt[1 - c*x])) + (Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*Sqrt[1 - c*x]) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])), x, 3, -((c*Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(b*Sqrt[1 - c*x])) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])), x]} -{Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])), x]} - - -{(x^3*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x, 15, -((3*Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(64*b*c^4*Sqrt[-1 + c*x])) + (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(64*b*c^4*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^4*Sqrt[-1 + c*x])} -{(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x, 12, (Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(16*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x])} -{(x^1*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x, 12, -((Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^2*Sqrt[-1 + c*x])) + (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^2*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^2*Sqrt[-1 + c*x])} -{x^0*(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x]), x, 9, (Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(8*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(3/2)/(x^1*(a + b*ArcCosh[c*x])), x, 15, -((5*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*Sqrt[1 - c*x])) + (Sqrt[-1 + c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*Sqrt[1 - c*x]) + (5*Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*Sqrt[1 - c*x]) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])), x, 9, (c*Sqrt[-1 + c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*Sqrt[1 - c*x]) - (3*c*Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(2*b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*Sqrt[1 - c*x]) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])), x]} - - -{(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]), x, 15, -((3*Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(128*b*c^4*Sqrt[-1 + c*x])) + (Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(32*b*c^4*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(9*a)/b]*CoshIntegral[(9*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(128*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(32*b*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcCosh[c*x]))/b])/(256*b*c^4*Sqrt[-1 + c*x])} -{(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]), x, 15, (Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(8*a)/b]*CoshIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(128*b*c^3*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(128*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(128*b*c^3*Sqrt[-1 + c*x])} -{(x^1*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]), x, 15, -((5*Sqrt[1 - c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(64*b*c^2*Sqrt[-1 + c*x])) + (9*Sqrt[1 - c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(7*a)/b]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(64*b*c^2*Sqrt[-1 + c*x]) - (9*Sqrt[1 - c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b*c^2*Sqrt[-1 + c*x])} -{x^0*(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x]), x, 12, (15*Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(6*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(16*b*c*Sqrt[-1 + c*x]) - (15*Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(16*b*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(5/2)/(x^1*(a + b*ArcCosh[c*x])), x, 27, -((11*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x])) + (7*Sqrt[-1 + c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) + (11*Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x]) - (7*Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])), x, 18, (c*Sqrt[-1 + c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*Sqrt[1 - c*x]) - (15*c*Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(8*b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b*Sqrt[1 - c*x]) + (c*Sqrt[-1 + c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*Sqrt[1 - c*x]) + Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])), x]} -{(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 5, (Sqrt[-1 + a*x]*CoshIntegral[2*ArcCosh[a*x]])/(2*a^5*Sqrt[1 - a*x]) + (Sqrt[-1 + a*x]*CoshIntegral[4*ArcCosh[a*x]])/(8*a^5*Sqrt[1 - a*x]) + (3*Sqrt[-1 + a*x]*Log[ArcCosh[a*x]])/(8*a^5*Sqrt[1 - a*x])} -{x^3/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 5, (3*Sqrt[-1 + a*x]*CoshIntegral[ArcCosh[a*x]])/(4*a^4*Sqrt[1 - a*x]) + (Sqrt[-1 + a*x]*CoshIntegral[3*ArcCosh[a*x]])/(4*a^4*Sqrt[1 - a*x])} -{x^2/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 4, (Sqrt[-1 + a*x]*CoshIntegral[2*ArcCosh[a*x]])/(2*a^3*Sqrt[1 - a*x]) + (Sqrt[-1 + a*x]*Log[ArcCosh[a*x]])/(2*a^3*Sqrt[1 - a*x])} -{x^1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 2, (Sqrt[-1 + a*x]*CoshIntegral[ArcCosh[a*x]])/(a^2*Sqrt[1 - a*x])} -{x^0/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 1, (Sqrt[-1 + a*x]*Log[ArcCosh[a*x]])/(a*Sqrt[1 - a*x])} -{1/(x^1*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 0, Unintegrable[1/(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x]} -{1/(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x, 0, Unintegrable[1/(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x]} - - -{x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 9, (3*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^4*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^4*Sqrt[1 - c*x]) - (3*Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^4*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^4*Sqrt[1 - c*x])} -{x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 6, (Sqrt[-1 + c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c^3*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(2*b*c^3*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*c^3*Sqrt[1 - c*x])} -{x^1/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 4, (Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b*c^2*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*c^2*Sqrt[1 - c*x])} -{x^0/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 1, (Sqrt[-1 + c*x]*Log[a + b*ArcCosh[c*x]])/(b*c*Sqrt[1 - c*x])} -{1/(x^1*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} - - -{x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{x^1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{x^0/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{1/(x^1*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x]) with m symbolic*) - - -{(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x, 0, Unintegrable[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x]} -{(x^m*(1 - c^2*x^2)^(1/2))/(a + b*ArcCosh[c*x]), x, 0, Unintegrable[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x]} -{x^m/((1 - c^2*x^2)^(1/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]} -{x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^2*) - - -{(c - a^2*c*x^2)^3/ArcCosh[a*x]^2, x, 8, (c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2))/(a*ArcCosh[a*x]) + (35*c^3*CoshIntegral[ArcCosh[a*x]])/(64*a) - (63*c^3*CoshIntegral[3*ArcCosh[a*x]])/(64*a) + (35*c^3*CoshIntegral[5*ArcCosh[a*x]])/(64*a) - (7*c^3*CoshIntegral[7*ArcCosh[a*x]])/(64*a)} -{(c - a^2*c*x^2)^2/ArcCosh[a*x]^2, x, 7, -((c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/2))/(a*ArcCosh[a*x])) + (5*c^2*CoshIntegral[ArcCosh[a*x]])/(8*a) - (15*c^2*CoshIntegral[3*ArcCosh[a*x]])/(16*a) + (5*c^2*CoshIntegral[5*ArcCosh[a*x]])/(16*a)} -{(c - a^2*c*x^2)/ArcCosh[a*x]^2, x, 6, (c*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2))/(a*ArcCosh[a*x]) + (3*c*CoshIntegral[ArcCosh[a*x]])/(4*a) - (3*c*CoshIntegral[3*ArcCosh[a*x]])/(4*a)} -{1/((c - a^2*c*x^2)*ArcCosh[a*x]^2), x, 1, 1/(a*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]) + (a*Unintegrable[x/((-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]), x])/c} -{1/((c - a^2*c*x^2)^2*ArcCosh[a*x]^2), x, 1, -(1/(a*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])) - (3*a*Unintegrable[x/((-1 + a*x)^(5/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]), x])/c^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) / (a+b ArcCosh[c x])^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x])^2, x, 22, -((x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c*x]))) + (Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*c^4*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c^4*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b^2*c^4*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^4*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^4*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^4*Sqrt[-1 + c*x])} -{x^2*Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x])^2, x, 16, -((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c^3*Sqrt[-1 + c*x])} -{x^1*Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x])^2, x, 14, -((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c*x]))) + (Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b^2*c^2*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^2*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^2*Sqrt[-1 + c*x])} -{x^0*Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x])^2, x, 7, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c*Sqrt[-1 + c*x])} -{Sqrt[1 - c^2*x^2]/(x^1*(a + b*ArcCosh[c*x])^2), x, 5, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*x*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Unintegrable[1/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])} -{Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])^2), x, 1, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(b*c*x^2*(a + b*ArcCosh[c*x]))) + (2*Sqrt[1 - c*x]*Unintegrable[1/(x^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])} -{Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2), x]} -{Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2), x]} - - -{x^2*(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2, x, 21, -((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c^3*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^3*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x])} -{x^1*(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2, x, 24, -((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c*x]))) + (Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*c^2*Sqrt[-1 + c*x]) - (9*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^2*Sqrt[-1 + c*x]) + (9*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^2*Sqrt[-1 + c*x])} -{x^0*(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2, x, 11, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(3/2)/(x^1*(a + b*ArcCosh[c*x])^2), x, 12, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*x*(a + b*ArcCosh[c*x]))) - (9*Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b^2*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*Sqrt[-1 + c*x]) + (9*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])^2), x, 3, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*x^2*(a + b*ArcCosh[c*x]))) - (2*Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)/(x^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x]) - (2*c*Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)/(x*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])^2), x]} -{(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])^2), x, 2, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*x^4*(a + b*ArcCosh[c*x]))) - (4*Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)/(x^5*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])} - - -{x^2*(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2, x, 30, -((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(8*b^2*c^3*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(8*(a + b*ArcCosh[c*x]))/b]*Sinh[(8*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b^2*c^3*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x])} -{x^1*(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2, x, 30, -((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) + (5*Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (27*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (7*Sqrt[1 - c*x]*CoshIntegral[(7*(a + b*ArcCosh[c*x]))/b]*Sinh[(7*a)/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) + (27*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) - (25*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(64*b^2*c^2*Sqrt[-1 + c*x]) + (7*Sqrt[1 - c*x]*Cosh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcCosh[c*x]))/b])/(64*b^2*c^2*Sqrt[-1 + c*x])} -{x^0*(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2, x, 14, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) - (15*Sqrt[1 - c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c*Sqrt[-1 + c*x]) + (15*Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(5/2)/(x^1*(a + b*ArcCosh[c*x])^2), x, 15, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*x*(a + b*ArcCosh[c*x]))) - (25*Sqrt[1 - c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*Sqrt[-1 + c*x]) - (25*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)^2/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2), x, 3, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*x^2*(a + b*ArcCosh[c*x]))) + (2*Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)^2/(x^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x]) + (4*c*Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)^2/(x*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[-1 + c*x])} -{(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])^2), x]} -{(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 13, -((x^5*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (5*Sqrt[-1 + c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*c^6*Sqrt[1 - c*x]) - (15*Sqrt[-1 + c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c^6*Sqrt[1 - c*x]) - (5*Sqrt[-1 + c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b^2*c^6*Sqrt[1 - c*x]) + (5*Sqrt[-1 + c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^6*Sqrt[1 - c*x]) + (15*Sqrt[-1 + c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^6*Sqrt[1 - c*x]) + (5*Sqrt[-1 + c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^6*Sqrt[1 - c*x])} -{x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 10, -((x^4*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (Sqrt[-1 + c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c^5*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c^5*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^5*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c^5*Sqrt[1 - c*x])} -{x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 10, -((x^3*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (3*Sqrt[-1 + c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b^2*c^4*Sqrt[1 - c*x]) - (3*Sqrt[-1 + c*x]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*c^4*Sqrt[1 - c*x]) + (3*Sqrt[-1 + c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^4*Sqrt[1 - c*x]) + (3*Sqrt[-1 + c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^4*Sqrt[1 - c*x])} -{x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 7, -((x^2*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (Sqrt[-1 + c*x]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c^3*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^3*Sqrt[1 - c*x])} -{x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 5, -((x*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (Sqrt[-1 + c*x]*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b^2*c^2*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c^2*Sqrt[1 - c*x])} -{1/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 1, -(Sqrt[-1 + c*x]/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x])))} -{1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 1, -(Sqrt[-1 + c*x]/(b*c*x*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (Sqrt[-1 + c*x]*Unintegrable[1/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[1 - c*x])} -{1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x, 1, -(Sqrt[-1 + c*x]/(b*c*x^2*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) - (2*Sqrt[-1 + c*x]*Unintegrable[1/(x^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[1 - c*x])} - - -{x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} -{x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 2, -((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x]))) + (2*Sqrt[-1 + c*x]*Unintegrable[x/((-1 + c^2*x^2)^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[1 - c*x])} -{x^1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} -{x^0/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 2, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x]))) + (2*c*Sqrt[-1 + c*x]*Unintegrable[x/((-1 + c^2*x^2)^2*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[1 - c*x])} -{1/(x^1*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} -{1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} - - -{x^4/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 2, -((x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x]))) - (4*Sqrt[-1 + c*x]*Unintegrable[x^3/((-1 + c^2*x^2)^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[1 - c*x])} -{x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} -{x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} -{x^1/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} -{x^0/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 2, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x]))) - (4*c*Sqrt[-1 + c*x]*Unintegrable[x/((-1 + c^2*x^2)^3*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[1 - c*x])} -{1/(x^1*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} -{1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^2 with m symbolic*) - - -{(f*x)^m*(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2, x, 0, Unintegrable[((f*x)^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2, x]} -{(f*x)^m*(1 - c^2*x^2)^(1/2)/(a + b*ArcCosh[c*x])^2, x, 0, Unintegrable[((f*x)^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2, x]} -{(f*x)^m/((1 - c^2*x^2)^(1/2)*(a + b*ArcCosh[c*x])^2), x, 1, -(((f*x)^m*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) + (f*m*Sqrt[-1 + c*x]*Unintegrable[(f*x)^(-1 + m)/(a + b*ArcCosh[c*x]), x])/(b*c*Sqrt[1 - c*x])} -{(f*x)^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[(f*x)^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} -{(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^3*) - - -(* ::Subsection:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p / (a+b ArcCosh[c x])^3*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^(p/2) / (a+b ArcCosh[c x])^3*) - - -{1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3), x, 1, -(Sqrt[-1 + a*x]/(2*a*Sqrt[1 - a*x]*ArcCosh[a*x]^2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^(n/2)*) - - -(* ::Subsubsection:: *) -(*n>0*) - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(x^3*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2), x, 27, (2*d*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (3*d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) - (d*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (3*d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(16*b^(3/2)*c^4)) - (d*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((6*a)/b)*(16*b^(3/2)*c^4))} -{(x^2*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2), x, 32, (2*d*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c^3) + (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) - (d*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(E^(a/b)*(8*b^(3/2)*c^3)) + (d*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(16*b^(3/2)*c^3)) - (d*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((5*a)/b)*(16*b^(3/2)*c^3))} -{(x^1*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2), x, 17, (2*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2) + (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(3/2)*c^2) - (d*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((4*a)/b)*(4*b^(3/2)*c^2)) + (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(2*b^(3/2)*c^2))} -{x^0*(d - c^2*d*x^2)/(a + b*ArcCosh[c*x])^(3/2), x, 14, (2*d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (3*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c) - (d*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c) + (3*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c*E^(a/b)) - (d*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c*E^((3*a)/b))} -{(d - c^2*d*x^2)/(x^1*(a + b*ArcCosh[c*x])^(3/2)), x, 12, (2*d*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^(3/2) - (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*b^(3/2)) + (2*d*Unintegrable[1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]]), x])/(b*c)} - - -{(x^3*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2), x, 32, -((2*d^2*x^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]])) - (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) + (3*d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) + (d^2*E^((8*a)/b)*Sqrt[Pi/2]*Erf[(2*Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) - (d^2*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*b^(3/2)*c^4) - (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((4*a)/b)*(32*b^(3/2)*c^4)) + (3*d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(32*b^(3/2)*c^4)) + (d^2*Sqrt[Pi/2]*Erfi[(2*Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((8*a)/b)*(32*b^(3/2)*c^4)) - (d^2*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((6*a)/b)*(32*b^(3/2)*c^4))} -{(x^2*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2), x, 42, -((2*d^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]])) + (5*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3) + (d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (3*d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) + (d^2*E^((7*a)/b)*Sqrt[7*Pi]*Erf[(Sqrt[7]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) + (5*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(E^(a/b)*(64*b^(3/2)*c^3)) + (d^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((3*a)/b)*(64*b^(3/2)*c^3)) - (3*d^2*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((5*a)/b)*(64*b^(3/2)*c^3)) + (d^2*Sqrt[7*Pi]*Erfi[(Sqrt[7]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((7*a)/b)*(64*b^(3/2)*c^3))} -{(x*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2), x, 32, -((2*d^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]])) - (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2) + (5*d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2) + (d^2*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^2) - (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((4*a)/b)*(4*b^(3/2)*c^2)) + (5*d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(16*b^(3/2)*c^2)) + (d^2*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((6*a)/b)*(16*b^(3/2)*c^2))} -{(d - c^2*d*x^2)^2/(a + b*ArcCosh[c*x])^(3/2), x, 19, (-2*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (5*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c) - (5*d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c) + (d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c) + (5*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*b^(3/2)*c*E^(a/b)) - (5*d^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c*E^((3*a)/b)) + (d^2*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c*E^((5*a)/b))} -{(d - c^2*d*x^2)^2/(x*(a + b*ArcCosh[c*x])^(3/2)), x, 25, -((2*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]])) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)) - (3*d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(3/2)) + (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((4*a)/b)*(4*b^(3/2))) - (3*d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(2*b^(3/2))) + (2*d^2*Unintegrable[1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]]), x])/(b*c), -((2*d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*x*Sqrt[a + b*ArcCosh[c*x]])) + (d^2*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)) + (d^2*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(3/2)) - (d^2*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/b^(3/2) + (d^2*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((4*a)/b)*(4*b^(3/2))) + (d^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*(2*b^(3/2))) - (d^2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(E^((2*a)/b)*b^(3/2)) + (2*d^2*Unintegrable[1/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]]), x])/(b*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) ArcCosh[c x]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]], x, 25, (3/8)*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]] + (1/4)*x*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]] - (c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]], x, 10, (x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{Sqrt[ArcCosh[a*x]]/Sqrt[c - a^2*c*x^2], x, 1, (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[c - a^2*c*x^2])} -{Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(3/2), x, 1, (x*Sqrt[ArcCosh[a*x]])/(c*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(2*c*Sqrt[c - a^2*c*x^2])} -{Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2), x, 3, (x*Sqrt[ArcCosh[a*x]])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[a*x]])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((1 - a^2*x^2)*Sqrt[ArcCosh[a*x]]), x])/(3*c^2*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(6*c^2*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2), x, 27, (27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2) + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2) - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(20*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2), x, 11, (3*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{ArcCosh[a*x]^(3/2)/Sqrt[c - a^2*c*x^2], x, 1, (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[c - a^2*c*x^2])} -{ArcCosh[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2), x, 1, (x*ArcCosh[a*x]^(3/2))/(c*Sqrt[c - a^2*c*x^2]) + (3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[(x*Sqrt[ArcCosh[a*x]])/(1 - a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2), x, 41, (225/512)*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]] + (15/256)*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]] + (45*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2) + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2) - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(7/2))/(28*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2), x, 13, (15*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/32 + (5*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (5*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{ArcCosh[a*x]^(5/2)/Sqrt[c - a^2*c*x^2], x, 1, (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])} -{ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2), x, 1, (x*ArcCosh[a*x]^(5/2))/(c*Sqrt[c - a^2*c*x^2]) + (5*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[(x*ArcCosh[a*x]^(3/2))/(1 - a^2*x^2), x])/(2*c*Sqrt[c - a^2*c*x^2])} - - -{(a^2 - x^2)^(3/2)*Sqrt[ArcCosh[x/a]], x, 25, (3/8)*a^2*x*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]] + (1/4)*x*(a^2 - x^2)^(3/2)*Sqrt[ArcCosh[x/a]] - (a^3*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/(4*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erf[2*Sqrt[ArcCosh[x/a]]])/(256*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erfi[2*Sqrt[ArcCosh[x/a]]])/(256*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a])} -{Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]], x, 10, (x*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/2 - (a*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/(3*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a])} -{Sqrt[ArcCosh[x/a]]/Sqrt[a^2 - x^2], x, 1, (2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])} -{Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2), x, 1, (x*Sqrt[ArcCosh[x/a]])/(a^2*Sqrt[a^2 - x^2]) + (Sqrt[-1 + x/a]*Sqrt[1 + x/a]*Unintegrable[x/((1 - x^2/a^2)*Sqrt[ArcCosh[x/a]]), x])/(2*a^3*Sqrt[a^2 - x^2])} -{Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2), x, 3, (x*Sqrt[ArcCosh[x/a]])/(3*a^2*(a^2 - x^2)^(3/2)) + (2*x*Sqrt[ArcCosh[x/a]])/(3*a^4*Sqrt[a^2 - x^2]) + (Sqrt[-1 + x/a]*Sqrt[1 + x/a]*Unintegrable[x/((1 - x^2/a^2)*Sqrt[ArcCosh[x/a]]), x])/(3*a^5*Sqrt[a^2 - x^2]) + (Sqrt[-1 + x/a]*Sqrt[1 + x/a]*Unintegrable[x/((-1 + x^2/a^2)^2*Sqrt[ArcCosh[x/a]]), x])/(6*a^5*Sqrt[a^2 - x^2])} - - -{(a^2 - x^2)^(3/2)*ArcCosh[x/a]^(3/2), x, 27, (27*a^3*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(256*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (9*a*x^2*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(32*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*(a^2 - x^2)^(5/2)*Sqrt[ArcCosh[x/a]])/(32*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3/8)*a^2*x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2) + (1/4)*x*(a^2 - x^2)^(3/2)*ArcCosh[x/a]^(3/2) - (3*a^3*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(5/2))/(20*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erf[2*Sqrt[ArcCosh[x/a]]])/(2048*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*a^3*Sqrt[Pi]*Sqrt[a^2 - x^2]*Erfi[2*Sqrt[ArcCosh[x/a]]])/(2048*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a^3*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a])} -{Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2), x, 11, (3*a*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(16*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*x^2*Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]])/(8*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/2 - (a*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(5/2))/(5*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 - x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/(64*Sqrt[-1 + x/a]*Sqrt[1 + x/a])} -{ArcCosh[x/a]^(3/2)/Sqrt[a^2 - x^2], x, 1, (2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])} -{ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(3/2), x, 1, (x*ArcCosh[x/a]^(3/2))/(a^2*Sqrt[a^2 - x^2]) + (3*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*Unintegrable[(x*Sqrt[ArcCosh[x/a]])/(1 - x^2/a^2), x])/(2*a^3*Sqrt[a^2 - x^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x/(Sqrt[1 - x^2]*Sqrt[ArcCosh[x]]), x, 6, (Sqrt[Pi]*Sqrt[-1 + x]*Erf[Sqrt[ArcCosh[x]]])/(2*Sqrt[1 - x]) + (Sqrt[Pi]*Sqrt[-1 + x]*Erfi[Sqrt[ArcCosh[x]]])/(2*Sqrt[1 - x])} - - -{(c - a^2*c*x^2)^(5/2)/Sqrt[ArcCosh[a*x]], x, 18, (-5*c^2*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*Sqrt[Pi/6]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*Sqrt[Pi/6]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(3/2)/Sqrt[ArcCosh[a*x]], x, 13, (-3*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(1/2)/Sqrt[ArcCosh[a*x]], x, 8, -((Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])) + (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{1/((c - a^2*c*x^2)^(1/2)*Sqrt[ArcCosh[a*x]]), x, 1, (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(a*Sqrt[c - a^2*c*x^2])} -{1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]), x, 0, Unintegrable[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]), x]} -{1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]), x, 0, Unintegrable[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]), x]} - - -{(c - a^2*c*x^2)^(5/2)/ArcCosh[a*x]^(3/2), x, 20, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(5/2))/(a*Sqrt[ArcCosh[a*x]])) + (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c^2*Sqrt[(3*Pi)/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c^2*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c^2*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*Sqrt[(3*Pi)/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[6]*Sqrt[ArcCosh[a*x]]])/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(3/2), x, 15, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(a*Sqrt[ArcCosh[a*x]])) + (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(1/2)/ArcCosh[a*x]^(3/2), x, 9, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(a*Sqrt[ArcCosh[a*x]])) - (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{1/((c - a^2*c*x^2)^(1/2)*ArcCosh[a*x]^(3/2)), x, 1, (-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])} -{1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)), x, 2, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]])) + (4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(c*Sqrt[c - a^2*c*x^2])} -{1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(3/2)), x, 2, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*(c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]])) - (8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^3*Sqrt[ArcCosh[a*x]]), x])/(c^2*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(5/2), x, 19, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c - a^2*c*x^2)^(3/2))/(3*a*ArcCosh[a*x]^(3/2))) - (16*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcCosh[a*x]]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (2*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2*c*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{(c - a^2*c*x^2)^(1/2)/ArcCosh[a*x]^(5/2), x, 7, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c - a^2*c*x^2])/(3*a*ArcCosh[a*x]^(3/2))) - (8*x*Sqrt[c - a^2*c*x^2])/(3*Sqrt[ArcCosh[a*x]]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2*Sqrt[2*Pi]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{1/((c - a^2*c*x^2)^(1/2)*ArcCosh[a*x]^(5/2)), x, 1, (-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))} -{1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)), x, 2, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2))) + (4*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^2*ArcCosh[a*x]^(3/2)), x])/(3*c*Sqrt[c - a^2*c*x^2])} -{1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(5/2)), x, 2, -((2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(3/2))) - (8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Unintegrable[x/((-1 + a^2*x^2)^3*ArcCosh[a*x]^(3/2)), x])/(3*c^2*Sqrt[c - a^2*c*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^n when n symbolic*) - - -(* ::Subsection:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^n when n symbolic*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])^n when n symbolic*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n, x, 6, -(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*c^3*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^1*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n, x, 9, (3^(-1 - n)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b])/(8*c^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(8*c^2*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(8*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (3^(-1 - n)*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(8*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^0*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n, x, 6, -(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2^(-3 - n)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n/x^1, x, 6, -((d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*Sqrt[d - c^2*d*x^2]))) + (d*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*Sqrt[d - c^2*d*x^2])) + d*Unintegrable[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n/x^2, x, 3, -((c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*(1 + n)*Sqrt[d - c^2*d*x^2])) + d*Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -{x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x, 12, -(d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(16*b*c^3*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2^(-7 - n)*3^(-1 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-7 - 2*n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-7 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*d*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (2^(-7 - 2*n)*d*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (2^(-7 - n)*3^(-1 - n)*d*E^((6*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^1*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x, 12, -(5^(-1 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b])/(32*c^2*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b])/(32*3^n*c^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(16*c^2*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (d*E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(16*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (d*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(32*3^n*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (5^(-1 - n)*d*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(32*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^0*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x, 9, (-3*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-3 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-3 - n)*d*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (d*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n/x^1, x, 15, (3^(-1 - n)*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcCosh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*Sqrt[d - c^2*d*x^2])) - (5*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*Sqrt[d - c^2*d*x^2])) + (5*d^2*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*Sqrt[d - c^2*d*x^2])) - (3^(-1 - n)*d^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*Sqrt[d - c^2*d*x^2])) + d^2*Unintegrable[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n/x^2, x, 9, -((3*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*(1 + n)*Sqrt[d - c^2*d*x^2])) + (2^(-3 - n)*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcCosh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Sqrt[d - c^2*d*x^2]) - (2^(-3 - n)*c*d^2*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*Sqrt[d - c^2*d*x^2]) + d^2*Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -{x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n, x, 15, (-5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2^(-11 - 3*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((8*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((6*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (2^(-7 - n)*d^2*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (d^2*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (2^(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (2^(-11 - 3*n)*d^2*E^((8*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b])/(c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^1*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n, x, 15, (7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcCosh[c*x]))/b])/(128*c^2*E^((7*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b])/(128*c^2*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (d^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(128*5^n*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (7*(a + b*ArcCosh[c*x]))/b])/(128*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{x^0*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n, x, 12, (-5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(16*b*c*(1 + n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b])/(c*E^((6*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (3*2^(-7 - 2*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(c*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (15*2^(-7 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(c*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) - (15*2^(-7 - n)*d^2*E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (3*2^(-7 - 2*n)*d^2*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d^2*E^((6*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n/x^1, x, 27, -((5^(-1 - n)*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((5*(a + b*ArcCosh[c*x]))/b)])/(E^((5*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(32*Sqrt[d - c^2*d*x^2]))) - (5*3^(-1 - n)*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcCosh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(32*Sqrt[d - c^2*d*x^2])) + (d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcCosh[c*x]))/b)])/(3^n*E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*Sqrt[d - c^2*d*x^2])) - (11*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(16*Sqrt[d - c^2*d*x^2])) + (11*d^3*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(16*Sqrt[d - c^2*d*x^2])) + (5*3^(-1 - n)*d^3*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(32*Sqrt[d - c^2*d*x^2])) - (d^3*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(3^n*((a + b*ArcCosh[c*x])/b)^n*(8*Sqrt[d - c^2*d*x^2])) + (5^(-1 - n)*d^3*E^((5*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(32*Sqrt[d - c^2*d*x^2])) + d^3*Unintegrable[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n/x^2, x, 18, -((15*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2])) - (c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((4*(a + b*ArcCosh[c*x]))/b)])/(2^(2*(3 + n))*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Sqrt[d - c^2*d*x^2]) + (2^(-2 - n)*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcCosh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Sqrt[d - c^2*d*x^2]) - (2^(-2 - n)*c*d^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*Sqrt[d - c^2*d*x^2]) + (c*d^3*E^((4*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*((a + b*ArcCosh[c*x])/b)^n*Sqrt[d - c^2*d*x^2]) + d^3*Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 9, (3^(-1 - n)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcCosh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*c^4*Sqrt[1 - c*x])) + (3*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*c^4*Sqrt[1 - c*x])) - (3*E^(a/b)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*c^4*Sqrt[1 - c*x])) - (3^(-1 - n)*E^((3*a)/b)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*c^4*Sqrt[1 - c*x]))} -{x^2*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 6, (Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*c^3*(1 + n)*Sqrt[1 - c*x]) + (2^(-3 - n)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcCosh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(c^3*Sqrt[1 - c*x])) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(c^3*Sqrt[1 - c*x]))} -{x^1*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 4, (Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*c^2*Sqrt[1 - c*x])) - (E^(a/b)*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*c^2*Sqrt[1 - c*x]))} -{x^0*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 1, (Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[1 - c*x])} -{(a + b*ArcCosh[c*x])^n/(x^1*Sqrt[1 - c^2*x^2]), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x*Sqrt[1 - c^2*x^2]), x]} -{(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[1 - c^2*x^2]), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[1 - c^2*x^2]), x]} - - -{x^3*(a + b*ArcCosh[c*x])^n/Sqrt[d - c^2*d*x^2], x, 9, (3^(-1 - n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((3*(a + b*ArcCosh[c*x]))/b)])/(E^((3*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*c^4*Sqrt[d - c^2*d*x^2])) + (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(8*c^4*Sqrt[d - c^2*d*x^2])) - (3*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*c^4*Sqrt[d - c^2*d*x^2])) - (3^(-1 - n)*E^((3*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(8*c^4*Sqrt[d - c^2*d*x^2]))} -{x^2*(a + b*ArcCosh[c*x])^n/Sqrt[d - c^2*d*x^2], x, 6, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(2*b*c^3*(1 + n)*Sqrt[d - c^2*d*x^2]) + (2^(-3 - n)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((2*(a + b*ArcCosh[c*x]))/b)])/(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(c^3*Sqrt[d - c^2*d*x^2])) - (2^(-3 - n)*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(((a + b*ArcCosh[c*x])/b)^n*(c^3*Sqrt[d - c^2*d*x^2]))} -{x^1*(a + b*ArcCosh[c*x])^n/Sqrt[d - c^2*d*x^2], x, 4, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*c^2*Sqrt[d - c^2*d*x^2])) - (E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*c^2*Sqrt[d - c^2*d*x^2]))} -{x^0*(a + b*ArcCosh[c*x])^n/Sqrt[d - c^2*d*x^2], x, 1, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[d - c^2*d*x^2])} -{(a + b*ArcCosh[c*x])^n/(x^1*Sqrt[d - c^2*d*x^2]), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]} -{(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]} - - -{x^2*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[(x^2*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]} -{x^1*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]} -{x^0*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x]} -{(a + b*ArcCosh[c*x])^n/(x^1*(d - c^2*d*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x*(d - c^2*d*x^2)^(3/2)), x]} -{(a + b*ArcCosh[c*x])^n/(x^2*(d - c^2*d*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^n/(x^2*(d - c^2*d*x^2)^(3/2)), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d-c^2 d x^2)^p (a+b ArcCosh[c x])^n when m and n symbolic*) - - -{(f*x)^m*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 0, Unintegrable[(f*x)^m*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x]} - - -{(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^n, x, 0, Unintegrable[(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^n, x]} -{(f*x)^m*(d - c^2*d*x^2)^1*(a + b*ArcCosh[c*x])^n, x, 0, Unintegrable[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x])^n, x]} -{(f*x)^m*(d - c^2*d*x^2)^0*(a + b*ArcCosh[c*x])^n, x, 0, Unintegrable[(f*x)^m*(a + b*ArcCosh[c*x])^n, x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2), x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^2, x]} - - -{(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x, 0, Unintegrable[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x]} -{(f*x)^m*(d - c^2*d*x^2)^(1/2)*(a + b*ArcCosh[c*x])^n, x, 0, Unintegrable[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n, x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(1/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2], x]} -{(f*x)^m*(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]} - - -(* ::Title:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcCosh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcCosh[c*x]), x, 7, -((8*b*(49*c^2*d + 30*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^7)) - (4*b*(49*c^2*d + 30*e)*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^5) - (b*(49*c^2*d + 30*e)*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c^3) - (b*e*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(49*c) + (1/5)*d*x^5*(a + b*ArcCosh[c*x]) + (1/7)*e*x^7*(a + b*ArcCosh[c*x])} -{x^3*(d + e*x^2)*(a + b*ArcCosh[c*x]), x, 6, -((b*(9*c^2*d + 5*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5)) - (b*(9*c^2*d + 5*e)*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(144*c^3) - (b*e*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (b*(9*c^2*d + 5*e)*ArcCosh[c*x])/(96*c^6) + (1/4)*d*x^4*(a + b*ArcCosh[c*x]) + (1/6)*e*x^6*(a + b*ArcCosh[c*x])} -{x^2*(d + e*x^2)*(a + b*ArcCosh[c*x]), x, 5, -((2*b*(25*c^2*d + 12*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(225*c^5)) - (b*(25*c^2*d + 12*e)*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(225*c^3) - (b*e*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(25*c) + (1/3)*d*x^3*(a + b*ArcCosh[c*x]) + (1/5)*e*x^5*(a + b*ArcCosh[c*x])} -{x^1*(d + e*x^2)*(a + b*ArcCosh[c*x]), x, 4, -((b*(8*c^2*d + 3*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3)) - (b*e*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (b*(8*c^2*d + 3*e)*ArcCosh[c*x])/(32*c^4) + (1/2)*d*x^2*(a + b*ArcCosh[c*x]) + (1/4)*e*x^4*(a + b*ArcCosh[c*x])} -{x^0*(d + e*x^2)*(a + b*ArcCosh[c*x]), x, 3, -((b*(9*c^2*d + 2*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3)) - (b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) + d*x*(a + b*ArcCosh[c*x]) + (1/3)*e*x^3*(a + b*ArcCosh[c*x])} -{((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^1, x, 13, -((b*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c)) - (b*e*ArcCosh[c*x])/(4*c^2) + (1/2)*e*x^2*(a + b*ArcCosh[c*x]) - (I*b*d*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d*(a + b*ArcCosh[c*x])*Log[x] - (b*d*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^2, x, 4, -((b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c) - (d*(a + b*ArcCosh[c*x]))/x + e*x*(a + b*ArcCosh[c*x]) + b*c*d*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} -{((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^3, x, 11, (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (d*(a + b*ArcCosh[c*x]))/(2*x^2) - (I*b*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + e*(a + b*ArcCosh[c*x])*Log[x] - (b*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*e*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^4, x, 4, (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) - (d*(a + b*ArcCosh[c*x]))/(3*x^3) - (e*(a + b*ArcCosh[c*x]))/x + (1/6)*b*c*(c^2*d + 6*e)*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]} - - -{x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 7, (b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(21*c^4*d^2 + 90*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*e*(9*c^2*d + 14*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^5*(a + b*ArcCosh[c*x]))/5 + (2*d*e*x^7*(a + b*ArcCosh[c*x]))/7 + (e^2*x^9*(a + b*ArcCosh[c*x]))/9} -{x^3*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 9, (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x*(1 - c^2*x^2))/(3072*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*x^3*(1 - c^2*x^2))/(4608*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(64*c^2*d + 21*e)*x^5*(1 - c^2*x^2))/(1152*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*x^7*(1 - c^2*x^2))/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/4)*d^2*x^4*(a + b*ArcCosh[c*x]) + (1/3)*d*e*x^6*(a + b*ArcCosh[c*x]) + (1/8)*e^2*x^8*(a + b*ArcCosh[c*x]) - (b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(3072*c^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 6, (b*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2))/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(1 - c^2*x^2)^2)/(315*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(14*c^2*d + 15*e)*(1 - c^2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*x^3*(a + b*ArcCosh[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCosh[c*x]))/5 + (e^2*x^7*(a + b*ArcCosh[c*x]))/7} -{x^1*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 7, (b*(44*c^4*d^2 + 44*c^2*d*e + 15*e^2)*x*(1 - c^2*x^2))/(288*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*(2*c^2*d + e)*x*(1 - c^2*x^2)*(d + e*x^2))/(144*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^2)/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/(6*e) - (b*(2*c^2*d + e)*(8*c^4*d^2 + 8*c^2*d*e + 5*e^2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(96*c^6*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^0*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 6, (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*(1 - c^2*x^2))/(15*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*e*(5*c^2*d + 3*e)*(1 - c^2*x^2)^2)/(45*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(1 - c^2*x^2)^3)/(25*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*x*(a + b*ArcCosh[c*x]) + (2*d*e*x^3*(a + b*ArcCosh[c*x]))/3 + (e^2*x^5*(a + b*ArcCosh[c*x]))/5} -{((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^1, x, 16, -((b*e*(16*c^2*d + 3*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3)) - (b*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (b*e*(16*c^2*d + 3*e)*ArcCosh[c*x])/(32*c^4) + d*e*x^2*(a + b*ArcCosh[c*x]) + (1/4)*e^2*x^4*(a + b*ArcCosh[c*x]) - (I*b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*(a + b*ArcCosh[c*x])*Log[x] - (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c)) - (3*b*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (b*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (b*d*e*ArcCosh[c*x])/(2*c^2) - (3*b*e^2*ArcCosh[c*x])/(32*c^4) + d*e*x^2*(a + b*ArcCosh[c*x]) + (1/4)*e^2*x^4*(a + b*ArcCosh[c*x]) - (I*b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^2*(a + b*ArcCosh[c*x])*Log[x] - (b*d^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d^2*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^2, x, 7, (b*e*(6*c^2*d + e)*(1 - c^2*x^2))/(3*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(1 - c^2*x^2)^2)/(9*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/x + 2*d*e*x*(a + b*ArcCosh[c*x]) + (1/3)*e^2*x^3*(a + b*ArcCosh[c*x]) + b*c*d^2*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]], (b*e*(6*c^2*d + e)*(1 - c^2*x^2))/(3*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(1 - c^2*x^2)^2)/(9*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/x + 2*d*e*x*(a + b*ArcCosh[c*x]) + (1/3)*e^2*x^3*(a + b*ArcCosh[c*x]) + (b*c*d^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^3, x, 14, (b*c*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (b*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c) - (b*e^2*ArcCosh[c*x])/(4*c^2) - (d^2*(a + b*ArcCosh[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcCosh[c*x]) - (I*b*d*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 2*d*e*(a + b*ArcCosh[c*x])*Log[x] - (2*b*d*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d*e*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^4, x, 7, (b*e^2*(1 - c^2*x^2))/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(a + b*ArcCosh[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcCosh[c*x]))/x + e^2*x*(a + b*ArcCosh[c*x]) + (b*c*d*(c^2*d + 12*e)*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, (b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*(1 - c^2*x^2))/(1155*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(462*c^6*d^3 + 1485*c^4*d^2*e + 1540*c^2*d*e^2 + 525*e^3)*(1 - c^2*x^2)^2)/(3465*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(77*c^6*d^3 + 495*c^4*d^2*e + 770*c^2*d*e^2 + 350*e^3)*(1 - c^2*x^2)^3)/(1925*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^2*x^2)^4)/(1617*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^5)/(297*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^6)/(121*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*d^2*e*x^7*(a + b*ArcCosh[c*x]))/7 + (d*e^2*x^9*(a + b*ArcCosh[c*x]))/3 + (e^3*x^11*(a + b*ArcCosh[c*x]))/11} -{x^3*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 10, -((b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*(1 - c^2*x^2))/(76800*c^9*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*(1 - c^2*x^2)*(d + e*x^2))/(38400*c^7*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*(1 - c^2*x^2)*(d + e*x^2)^2)/(9600*c^5*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(11*c^2*d + 18*e)*x*(1 - c^2*x^2)*(d + e*x^2)^3)/(1600*c^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^4)/(100*c*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*(d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcCosh[c*x]))/(10*e^2) + (b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^10*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^2*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, (b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(105*c^6*d^3 + 378*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/3)*d^3*x^3*(a + b*ArcCosh[c*x]) + (3/5)*d^2*e*x^5*(a + b*ArcCosh[c*x]) + (3/7)*d*e^2*x^7*(a + b*ArcCosh[c*x]) + (1/9)*e^3*x^9*(a + b*ArcCosh[c*x])} -{x^1*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 8, (5*b*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e + 21*e^2)*x*(1 - c^2*x^2))/(3072*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(104*c^4*d^2 + 104*c^2*d*e + 35*e^2)*x*(1 - c^2*x^2)*(d + e*x^2))/(1536*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*b*(2*c^2*d + e)*x*(1 - c^2*x^2)*(d + e*x^2)^2)/(384*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^3)/(64*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e) - (b*(128*c^8*d^4 + 256*c^6*d^3*e + 288*c^4*d^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(1024*c^8*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{x^0*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 6, (b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*(1 - c^2*x^2))/(35*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*(1 - c^2*x^2)^2)/(105*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*e^2*(7*c^2*d + 5*e)*(1 - c^2*x^2)^3)/(175*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^4)/(49*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) + d^2*e*x^3*(a + b*ArcCosh[c*x]) + (3*d*e^2*x^5*(a + b*ArcCosh[c*x]))/5 + (e^3*x^7*(a + b*ArcCosh[c*x]))/7} -{((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^1, x, 23, -((3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c)) - (9*b*d*e^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (5*b*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5) - (3*b*d*e^2*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) - (5*b*e^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(144*c^3) - (b*e^3*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (3*b*d^2*e*ArcCosh[c*x])/(4*c^2) - (9*b*d*e^2*ArcCosh[c*x])/(32*c^4) - (5*b*e^3*ArcCosh[c*x])/(96*c^6) + (3/2)*d^2*e*x^2*(a + b*ArcCosh[c*x]) + (3/4)*d*e^2*x^4*(a + b*ArcCosh[c*x]) + (1/6)*e^3*x^6*(a + b*ArcCosh[c*x]) - (I*b*d^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*(a + b*ArcCosh[c*x])*Log[x] - (b*d^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^2, x, 7, (b*e*(15*c^4*d^2 + 5*c^2*d*e + e^2)*(1 - c^2*x^2))/(5*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^2*(5*c^2*d + 2*e)*(1 - c^2*x^2)^2)/(15*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(1 - c^2*x^2)^3)/(25*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/x + 3*d^2*e*x*(a + b*ArcCosh[c*x]) + d*e^2*x^3*(a + b*ArcCosh[c*x]) + (1/5)*e^3*x^5*(a + b*ArcCosh[c*x]) + (b*c*d^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^3, x, 18, -((b*c*d^3*(1 - c^2*x^2))/(2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (3*b*e^2*(8*c^2*d + e)*x*(1 - c^2*x^2))/(32*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*x^3*(1 - c^2*x^2))/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/(2*x^2) + (3/2)*d*e^2*x^2*(a + b*ArcCosh[c*x]) + (1/4)*e^3*x^4*(a + b*ArcCosh[c*x]) - (3*I*b*d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*e^2*(8*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(32*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(2*I*ArcSin[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 3*d^2*e*(a + b*ArcCosh[c*x])*Log[x] - (3*b*d^2*e*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*I*b*d^2*e*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(2*I*ArcSin[c*x])])/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^4, x, 9, (b*e^2*(9*c^2*d + e)*(1 - c^2*x^2))/(3*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*(1 - c^2*x^2))/(6*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(1 - c^2*x^2)^2)/(9*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^3*(a + b*ArcCosh[c*x]))/(3*x^3) - (3*d^2*e*(a + b*ArcCosh[c*x]))/x + 3*d*e^2*x*(a + b*ArcCosh[c*x]) + (1/3)*e^3*x^3*(a + b*ArcCosh[c*x]) + (b*c*d^2*(c^2*d + 18*e)*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{(d + e*x^2)^4*(a + b*ArcCosh[c*x]), x, 6, (b*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*(1 - c^2*x^2))/(315*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^2)/(945*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*e^2*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^3)/(525*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*e^3*(9*c^2*d + 7*e)*(1 - c^2*x^2)^4)/(441*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^4*(1 - c^2*x^2)^5)/(81*c^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^4*x*(a + b*ArcCosh[c*x]) + (4*d^3*e*x^3*(a + b*ArcCosh[c*x]))/3 + (6*d^2*e^2*x^5*(a + b*ArcCosh[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcCosh[c*x]))/7 + (e^4*x^9*(a + b*ArcCosh[c*x]))/9} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2), x, 27, -((a*d*x)/e^2) + (b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3*e) - (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*e) - (b*d*x*ArcCosh[c*x])/e^2 + (x^3*(a + b*ArcCosh[c*x]))/(3*e) + ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(5/2)) + ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(5/2)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(5/2)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(5/2))} -{(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2), x, 23, -((b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c*e)) - (b*ArcCosh[c*x])/(4*c^2*e) + (x^2*(a + b*ArcCosh[c*x]))/(2*e) + (d*(a + b*ArcCosh[c*x])^2)/(2*b*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2) - (d*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2) - (b*d*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e^2) - (b*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) - (b*d*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e^2) - (b*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2)} -{(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2), x, 23, (a*x)/e - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e) + (b*x*ArcCosh[c*x])/e + (Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^(3/2))} -{(x*(a + b*ArcCosh[c*x]))/(d + e*x^2), x, 18, -((a + b*ArcCosh[c*x])^2/(2*b*e)) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e)} -{(a + b*ArcCosh[c*x])/(d + e*x^2), x, 18, ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcCosh[c*x])/(x*(d + e*x^2)), x, 25, (a + b*ArcCosh[c*x])^2/(b*d) + ((a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*d) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*d) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d)} -{(a + b*ArcCosh[c*x])/(x^2*(d + e*x^2)), x, 23, -((a + b*ArcCosh[c*x])/(d*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(3/2))} -{(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)), x, 27, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2) - (e*(a + b*ArcCosh[c*x])^2)/(b*d^2) - (e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^2 + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2) + (b*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^2) + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*d^2) + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*d^2) + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2)} -{(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)), x, 28, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) + (e*(a + b*ArcCosh[c*x]))/(d^2*x) + (b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) - (b*c*e*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^2 + (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2)) - (b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*(-d)^(5/2)) + (b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2)) - (b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*(-d)^(5/2)) + (b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*(-d)^(5/2))} - - -{(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x, 24, (d*(a + b*ArcCosh[c*x]))/(2*e^2*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^2) - (b*c*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*e^2*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e^2) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^2) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e^2) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^2)} -{(x^1*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x, 4, -((a + b*ArcCosh[c*x])/(2*e*(d + e*x^2))) + (b*c*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*Sqrt[d]*e*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(x^1*(d + e*x^2)^2), x, 29, (a + b*ArcCosh[c*x])/(2*d*(d + e*x^2)) + (a + b*ArcCosh[c*x])^2/(b*d^2) - (b*c*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(3/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^2 - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^2) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*d^2) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^2) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*d^2) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^2)} -{(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^2), x, 31, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^2*x) - (a + b*ArcCosh[c*x])/(2*d^2*x^2) - (e*(a + b*ArcCosh[c*x]))/(2*d^2*(d + e*x^2)) - (2*e*(a + b*ArcCosh[c*x])^2)/(b*d^3) + (b*c*e*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^3 + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/d^3 + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/d^3 + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/d^3 + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/d^3 + (b*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/d^3 + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/d^3 + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/d^3 + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/d^3 + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/d^3} - -{(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x, 49, (a*x)/e^2 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) + (b*x*ArcCosh[c*x])/e^2 - (d*(a + b*ArcCosh[c*x]))/(4*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b*ArcCosh[c*x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) - (b*c*d*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*e^(5/2))} -{(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x, 46, (a + b*ArcCosh[c*x])/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) + (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*Sqrt[-d]*e^(3/2))} -{(a + b*ArcCosh[c*x])/(d + e*x^2)^2, x, 26, -((a + b*ArcCosh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x))) + (a + b*ArcCosh[c*x])/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) - (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcCosh[c*x])/(x^2*(d + e*x^2)^2), x, 49, -((a + b*ArcCosh[c*x])/(d^2*x)) + (Sqrt[e]*(a + b*ArcCosh[c*x]))/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcCosh[c*x]))/(4*d^2*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^2 - (b*c*Sqrt[e]*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]) + (b*c*Sqrt[e]*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]) - (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(4*(-d)^(5/2))} - - -{(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x, 29, (b*c*d*x*(1 - c^2*x^2))/(8*e^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (d^2*(a + b*ArcCosh[c*x]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcCosh[c*x]))/(e^3*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^3) - (b*c*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^3*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d]*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*e^3)} -{(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x, 9, -(b*c*x*(1 - c^2*x^2))/(8*e*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) + (x^4*(a + b*ArcCosh[c*x]))/(4*d*(d + e*x^2)^2) - (b*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*(2*c^2*d + 3*e)*Sqrt[1 - c^2*x^2]*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*Sqrt[d]*e^2*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), -((b*c*x*(1 - c^2*x^2))/(8*e*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))) + (x^4*(a + b*ArcCosh[c*x]))/(4*d*(d + e*x^2)^2) - (b*Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*(2*c^2*d + 3*e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*Sqrt[d]*e^2*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(x*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x, 5, (b*c*x*(1 - c^2*x^2))/(8*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (a + b*ArcCosh[c*x])/(4*e*(d + e*x^2)^2) + (b*c*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*d^(3/2)*e*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^3), x, 34, -((b*c*e*x*(1 - c^2*x^2))/(8*d^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))) + (a + b*ArcCosh[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcCosh[c*x])/(2*d^2*(d + e*x^2)) + (a + b*ArcCosh[c*x])^2/(b*d^3) - (b*c*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*d^(5/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^3 - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^3) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^3) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^3) - ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^3) - (b*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^3) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*d^3) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^3) - (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*d^3) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^3)} -{(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^3), x, 36, (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^3*x) + (b*c*e^2*x*(1 - c^2*x^2))/(8*d^3*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (a + b*ArcCosh[c*x])/(2*d^3*x^2) - (e*(a + b*ArcCosh[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcCosh[c*x]))/(d^3*(d + e*x^2)) - (3*e*(a + b*ArcCosh[c*x])^2)/(b*d^4) + (b*c*e*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^4 + (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^4) + (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^4) + (3*b*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^4) + (3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(2*d^4) + (3*b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*d^4) + (3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(2*d^4) + (3*b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*d^4)} - -{(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x, 80, -((b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*e^2*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x))) - (b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*e^2*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*d*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(5/2)) - (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (b*c^3*d*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(5/2)) + (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*Sqrt[-d]*e^(5/2))} -{(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x, 62, -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x))) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2) - (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)^2) + (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(3/2)) + (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) - (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(3/2)) - (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(3/2)*e^(3/2))} -{(a + b*ArcCosh[c*x])/(d + e*x^2)^3, x, 34, -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x))) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^2) - (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcCosh[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) + (3*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) - (3*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(16*(-d)^(5/2)*Sqrt[e])} -(* {(a + b*ArcCosh[c*x])/(x^2*(d + e*x^2)^3), x, 102, (b*c*e*Sqrt[1 - c^2*x^2])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*e*Sqrt[1 - c^2*x^2])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(d^3*x) - (Sqrt[e]*(a + b*ArcCosh[c*x]))/(16*(-d)^(5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (7*Sqrt[e]*(a + b*ArcCosh[c*x]))/(16*d^3*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[e]*(a + b*ArcCosh[c*x]))/(16*(-d)^(5/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (7*Sqrt[e]*(a + b*ArcCosh[c*x]))/(16*d^3*(Sqrt[-d] + Sqrt[e]*x)) - (b*c^3*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*(c^2*d + e)^(3/2)) - (7*b*c*Sqrt[e]*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^3*Sqrt[c^2*d + e]) - (b*c^3*Sqrt[e]*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*(c^2*d + e)^(3/2)) - (7*b*c*Sqrt[e]*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^3*Sqrt[c^2*d + e]) - (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^3 + (15*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) - (15*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) - (15*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^(I*ArcCosh[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(7/2))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcCosh[c x])*) - - -{Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x]), x]} -{(a + b*ArcCosh[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcCosh[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcCosh[c*x])/(d + e*x^2)^(3/2), x, 7, (x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d + e*x^2]) - (b*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(d*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(d + e*x^2)^(5/2), x, 8, -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2])) + (x*(a + b*ArcCosh[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (2*b*Sqrt[1 - c^2*x^2]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(3*d^2*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), (b*c*(1 - c^2*x^2))/(3*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]) + (x*(a + b*ArcCosh[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCosh[c*x]))/(3*d^2*Sqrt[d + e*x^2]) - (2*b*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(3*d^2*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2), x, 9, (b*c*(1 - c^2*x^2))/(15*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)^(3/2)) + (2*b*c*(3*c^2*d + 2*e)*(1 - c^2*x^2))/(15*d^2*(c^2*d + e)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]) + (x*(a + b*ArcCosh[c*x]))/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcCosh[c*x]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcCosh[c*x]))/(15*d^3*Sqrt[d + e*x^2]) - (8*b*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(15*d^3*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x]) with m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcCosh[c*x]), x, 8, If[$VersionNumber>=8, (b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4))*(f*x)^(2 + m)*(1 - c^2*x^2))/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*(1 - c^2*x^2))/(c^3*f^4*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(f*x)^(6 + m)*(1 - c^2*x^2))/(c*f^6*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(3 + m)*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^5*f^2*(2 + m)*(3 + m)*(5 + m)*(7 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), (b*e*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4))*(f*x)^(2 + m)*(1 - c^2*x^2))/(c^5*f^2*(3 + m)^2*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f*x)^(4 + m)*(1 - c^2*x^2))/(c^3*f^4*(5 + m)^2*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^3*(f*x)^(6 + m)*(1 - c^2*x^2))/(c*f^6*(7 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(3 + m)*(5 + m)*(7 + m))/(1 + m) + (e*(2 + m)*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^5*f^2*(2 + m)*(3 + m)*(5 + m)*(7 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcCosh[c*x]), x, 7, (b*e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2))*(f*x)^(2 + m)*(1 - c^2*x^2))/(c^3*f^2*(3 + m)^2*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*e^2*(f*x)^(4 + m)*(1 - c^2*x^2))/(c*f^4*(5 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCosh[c*x]))/(f^5*(5 + m)) - (b*((c^4*d^2*(3 + m)*(5 + m))/(1 + m) + (e*(2 + m)*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2)))/((3 + m)*(5 + m)))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c^3*f^2*(2 + m)*(3 + m)*(5 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(f*x)^m*(d + e*x^2)^1*(a + b*ArcCosh[c*x]), x, 5, -((b*e*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*f^2*(3 + m)^2)) + (d*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) - (b*(e*(1 + m)*(2 + m) + c^2*d*(3 + m)^2)*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(c*f^2*(1 + m)*(2 + m)*(3 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d + e*x^2)^1, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2), x]} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x]} -{(f*x)^m*(a + b*ArcCosh[c*x])/(d + e*x^2)^3, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcCosh[c x])^2*) - - -{(d + e*x^2)^3*(a + b*ArcCosh[c*x])^2, x, 26, 2*b^2*d^3*x + (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) + (32*b^2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^3)/9 + (8*b^2*d*e^2*x^3)/(75*c^2) + (16*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 + (12*b^2*e^3*x^5)/(1225*c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*d^2*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c^5) - (32*b*e^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c) - (8*b*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c) - (12*b*e^3*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(49*c) + d^3*x*(a + b*ArcCosh[c*x])^2 + d^2*e*x^3*(a + b*ArcCosh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcCosh[c*x])^2)/5 + (e^3*x^7*(a + b*ArcCosh[c*x])^2)/7} -{(d + e*x^2)^2*(a + b*ArcCosh[c*x])^2, x, 17, 2*b^2*d^2*x + (8*b^2*d*e*x)/(9*c^2) + (16*b^2*e^2*x)/(75*c^4) + (4*b^2*d*e*x^3)/27 + (8*b^2*e^2*x^3)/(225*c^2) + (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (8*b*d*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(75*c^5) - (4*b*d*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c) - (8*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(75*c^3) - (2*b*e^2*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(25*c) + d^2*x*(a + b*ArcCosh[c*x])^2 + (2*d*e*x^3*(a + b*ArcCosh[c*x])^2)/3 + (e^2*x^5*(a + b*ArcCosh[c*x])^2)/5} -{(d + e*x^2)*(a + b*ArcCosh[c*x])^2, x, 10, 2*b^2*d*x + (4*b^2*e*x)/(9*c^2) + (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (2*b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c) + d*x*(a + b*ArcCosh[c*x])^2 + (e*x^3*(a + b*ArcCosh[c*x])^2)/3} -{(a + b*ArcCosh[c*x])^2, x, 3, 2*b^2*x - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c + x*(a + b*ArcCosh[c*x])^2} -{(a + b*ArcCosh[c*x])^2/(d + e*x^2), x, 22, ((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[(-c^2)*d - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[(-c^2)*d - e])])/(Sqrt[-d]*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcCosh[c x])^2*) - - -{Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2, x]} -{(a + b*ArcCosh[c*x])^2/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcCosh[c*x])^2/Sqrt[d + e*x^2], x]} -{(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(3/2), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(3/2), x]} -{(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]} - - -(* ::Section:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x])^3*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcCosh[c x])^1*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p / (a+b ArcCosh[c x])*) - - -{(d + e*x^2)^2/(a + b*ArcCosh[c*x]), x, 27, -((d^2*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c)) - (d*e*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(2*b*c^3) - (e^2*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b*c^5) - (d*e*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(2*b*c^3) - (3*e^2*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b*c^5) - (e^2*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b*c^5) + (d^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*c) + (d*e*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(2*b*c^3) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^5) + (d*e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(2*b*c^3) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5) + (e^2*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5)} -{(d + e*x^2)^1/(a + b*ArcCosh[c*x]), x, 15, -(((4*c^2*d + e)*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b*c^3)) - (e*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(4*b*c^3) + ((4*c^2*d + e)*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^3) + (e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^3), -((d*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c)) - (e*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b*c^3) - (e*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(4*b*c^3) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*c) + (e*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^3) + (e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^3)} -{(d + e*x^2)^0/(a + b*ArcCosh[c*x]), x, 4, -((CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c)) + (Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b*c)} -{1/((d + e*x^2)^1*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcCosh[c*x])), x]} -{1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) / (a+b ArcCosh[c x])*) - - -{Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]), x]} -{1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p / (a+b ArcCosh[c x])^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p / (a+b ArcCosh[c x])^2*) - - -{(d + e*x^2)^2/(a + b*ArcCosh[c*x])^2, x, 26, -((d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (2*d*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) - (e^2*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (d*e*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(2*b^2*c^3) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^5) + (3*d*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c^3) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) + (5*e^2*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) - (d^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (d*e*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(2*b^2*c^3) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b^2*c^5) - (3*d*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(2*b^2*c^3) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5) - (5*e^2*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^5)} -{(d + e*x^2)^1/(a + b*ArcCosh[c*x])^2, x, 15, -((d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (e*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^3) + (3*e*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^3) - (d*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (e*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^3) - (3*e*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x^2)^0/(a + b*ArcCosh[c*x])^2, x, 5, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c)} -{1/((d + e*x^2)^1*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^2), x]} -{1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) / (a+b ArcCosh[c x])^2*) - - -{Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x])^2, x, 0, Unintegrable[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x])^2, x]} -{1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2), x]} -{1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]} -{1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCosh[c x])^(n/2)*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcCosh[c x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]], x, 42, d^2*x*Sqrt[a + b*ArcCosh[c*x]] + (2*d*e*x^3*Sqrt[a + b*ArcCosh[c*x]])/3 + (e^2*x^5*Sqrt[a + b*ArcCosh[c*x]])/5 - (Sqrt[b]*d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c^3) - (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^5) - (Sqrt[b]*d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(24*c^3) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^5) - (Sqrt[b]*e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(320*c^5) - (Sqrt[b]*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) - (Sqrt[b]*d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c^3*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^5*E^(a/b)) - (Sqrt[b]*d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(24*c^3*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*c^5*E^((3*a)/b)) - (Sqrt[b]*e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(320*c^5*E^((5*a)/b))} -{(d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]], x, 23, d*x*Sqrt[a + b*ArcCosh[c*x]] + (e*x^3*Sqrt[a + b*ArcCosh[c*x]])/3 - (Sqrt[b]*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c^3) - (Sqrt[b]*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(48*c^3) - (Sqrt[b]*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*c^3*E^(a/b)) - (Sqrt[b]*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(48*c^3*E^((3*a)/b))} -{Sqrt[a + b*ArcCosh[c*x]], x, 7, x*Sqrt[a + b*ArcCosh[c*x]] - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*c*E^(a/b))} -{Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2), x, 0, Unintegrable[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2), x]} -{Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2)^2, x, 0, Unintegrable[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2)^2, x]} - - -{(d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2), x, 32, (-3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(3*c^3) - (b*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(6*c) + d*x*(a + b*ArcCosh[c*x])^(3/2) + (e*x^3*(a + b*ArcCosh[c*x])^(3/2))/3 - (3*b^(3/2)*d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c) - (3*b^(3/2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3) - (b^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3) + (3*b^(3/2)*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c*E^(a/b)) + (3*b^(3/2)*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3*E^(a/b)) + (b^(3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3*E^((3*a)/b))} -{(a + b*ArcCosh[c*x])^(3/2), x, 8, (-3*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) + x*(a + b*ArcCosh[c*x])^(3/2) - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c*E^(a/b))} -{(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x, 0, Unintegrable[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]} -{(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2)^2, x, 0, Unintegrable[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2)^2, x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x^2)^2/Sqrt[a + b*ArcCosh[c*x]], x, 39, -(d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5) - (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) - (e^2*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) + (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5*E^(a/b)) + (d*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((5*a)/b))} -{(d + e*x^2)/Sqrt[a + b*ArcCosh[c*x]], x, 21, -(d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3) - (e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) + (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*Sqrt[b]*c^3*E^(a/b)) + (e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*Sqrt[b]*c^3*E^((3*a)/b))} -{1/Sqrt[a + b*ArcCosh[c*x]], x, 6, -(E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b))} -{1/((d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]]), x, 0, Unintegrable[1/((d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]]), x]} -{1/((d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]]), x, 0, Unintegrable[1/((d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]]), x]} - - -{(d + e*x^2)/(a + b*ArcCosh[c*x])^(3/2), x, 21, (-2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) + (e*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b)) + (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3*E^(a/b)) + (e*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3*E^((3*a)/b))} -{(a + b*ArcCosh[c*x])^(-3/2), x, 7, (-2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c*E^(a/b))} -{1/((d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2)), x]} -{1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)), x, 0, Unintegrable[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)), x]} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 Inverse hyperbolic cosine functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 Inverse hyperbolic cosine functions.m deleted file mode 100644 index 201b77b..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.2 Inverse hyperbolic cosine/7.2.5 Inverse hyperbolic cosine functions.m +++ /dev/null @@ -1,628 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcCosh[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCosh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m ArcCosh[c x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*ArcCosh[c*x], x, 5, -((7*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(48*c)) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(16*c) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x))/(96*c^3) - ((8*c^4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c*x])/(32*c^4*e) + ((d + e*x)^4*ArcCosh[c*x])/(4*e)} -{(d + e*x)^2*ArcCosh[c*x], x, 4, -((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c)) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5*c^2*d*e*x))/(18*c^3) - (1/6)*d*((2*d^2)/e + (3*e)/c^2)*ArcCosh[c*x] + ((d + e*x)^3*ArcCosh[c*x])/(3*e)} -{(d + e*x)^1*ArcCosh[c*x], x, 4, -((3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c)) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x))/(4*c) - (1/4)*((2*d^2)/e + e/c^2)*ArcCosh[c*x] + ((d + e*x)^2*ArcCosh[c*x])/(2*e)} -{ArcCosh[c*x]/(d + e*x)^1, x, 8, -(ArcCosh[c*x]^2/(2*e)) + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))]/e + PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))]/e} -{ArcCosh[c*x]/(d + e*x)^2, x, 3, -(ArcCosh[c*x]/(e*(d + e*x))) + (2*c*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d - e]*e*Sqrt[c*d + e])} -{ArcCosh[c*x]/(d + e*x)^3, x, 4, -((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c^2*d^2 - e^2)*(d + e*x))) - ArcCosh[c*x]/(2*e*(d + e*x)^2) + (c^3*d*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/((c*d - e)^(3/2)*e*(c*d + e)^(3/2))} -{ArcCosh[c*x]/(d + e*x)^4, x, 6, If[$VersionNumber>=8, -((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2)) - (c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c*d - e)^2*(c*d + e)^2*(d + e*x)) - ArcCosh[c*x]/(3*e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2)), -((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2)) - (c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - ArcCosh[c*x]/(3*e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))]} - - -{(d + e*x)^3*ArcCosh[c*x]^2, x, 18, 2*d^3*x + (4*d*e^2*x)/(3*c^2) + (3/4)*d^2*e*x^2 + (3*e^3*x^2)/(32*c^2) + (2/9)*d*e^2*x^3 + (e^3*x^4)/32 - (2*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (4*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^3) - (3*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c) - (3*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(16*c^3) - (2*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c) - (e^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(8*c) - (d^4*ArcCosh[c*x]^2)/(4*e) - (3*d^2*e*ArcCosh[c*x]^2)/(4*c^2) - (3*e^3*ArcCosh[c*x]^2)/(32*c^4) + ((d + e*x)^4*ArcCosh[c*x]^2)/(4*e)} -{(d + e*x)^2*ArcCosh[c*x]^2, x, 13, 2*d^2*x + (4*e^2*x)/(9*c^2) + (1/2)*d*e*x^2 + (2*e^2*x^3)/27 - (2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (4*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c^3) - (d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (2*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(9*c) - (d^3*ArcCosh[c*x]^2)/(3*e) - (d*e*ArcCosh[c*x]^2)/(2*c^2) + ((d + e*x)^3*ArcCosh[c*x]^2)/(3*e)} -{(d + e*x)^1*ArcCosh[c*x]^2, x, 9, 2*d*x + (e*x^2)/4 - (2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/c - (e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(2*c) - (d^2*ArcCosh[c*x]^2)/(2*e) - (e*ArcCosh[c*x]^2)/(4*c^2) + ((d + e*x)^2*ArcCosh[c*x]^2)/(2*e)} -{ArcCosh[c*x]^2/(d + e*x)^1, x, 10, -(ArcCosh[c*x]^3/(3*e)) + (ArcCosh[c*x]^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcCosh[c*x]^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + (2*ArcCosh[c*x]*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e + (2*ArcCosh[c*x]*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/e - (2*PolyLog[3, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e - (2*PolyLog[3, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/e} -{ArcCosh[c*x]^2/(d + e*x)^2, x, 10, -(ArcCosh[c*x]^2/(e*(d + e*x))) + (2*c*ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) - (2*c*ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) + (2*c*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/(e*Sqrt[c^2*d^2 - e^2]) - (2*c*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/(e*Sqrt[c^2*d^2 - e^2])} -{ArcCosh[c*x]^2/(d + e*x)^3, x, 13, -((c*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*ArcCosh[c*x])/((c^2*d^2 - e^2)*(d + e*x))) - ArcCosh[c*x]^2/(2*e*(d + e*x)^2) + (c^3*d*ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) - (c^3*d*ArcCosh[c*x]*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) + (c^2*Log[d + e*x])/(e*(c^2*d^2 - e^2)) + (c^3*d*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/(e*(c^2*d^2 - e^2)^(3/2)) - (c^3*d*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/(e*(c^2*d^2 - e^2)^(3/2))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCosh[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcCosh[c*x]), x, 5, -((7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(48*c)) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(16*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x))/(96*c^3) - (b*(8*c^4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcCosh[c*x]))/(4*e)} -{(d + e*x)^2*(a + b*ArcCosh[c*x]), x, 4, -((b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c)) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5*c^2*d*e*x))/(18*c^3) - (b*d*(2*d^2 + (3*e^2)/c^2)*ArcCosh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcCosh[c*x]))/(3*e)} -{(d + e*x)^1*(a + b*ArcCosh[c*x]), x, 4, -((3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c)) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x))/(4*c) - (b*(2*d^2 + e^2/c^2)*ArcCosh[c*x])/(4*e) + ((d + e*x)^2*(a + b*ArcCosh[c*x]))/(2*e)} -{(a + b*ArcCosh[c*x])/(d + e*x)^1, x, 8, -((a + b*ArcCosh[c*x])^2/(2*b*e)) + ((a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + ((a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + (b*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e + (b*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/e} -{(a + b*ArcCosh[c*x])/(d + e*x)^2, x, 3, -((a + b*ArcCosh[c*x])/(e*(d + e*x))) + (2*b*c*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d - e]*e*Sqrt[c*d + e])} -{(a + b*ArcCosh[c*x])/(d + e*x)^3, x, 4, -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c^2*d^2 - e^2)*(d + e*x))) - (a + b*ArcCosh[c*x])/(2*e*(d + e*x)^2) + (b*c^3*d*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/((c*d - e)^(3/2)*e*(c*d + e)^(3/2))} -{(a + b*ArcCosh[c*x])/(d + e*x)^4, x, 6, If[$VersionNumber>=8, -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2)) - (b*c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c*d - e)^2*(c*d + e)^2*(d + e*x)) - (a + b*ArcCosh[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2)), -((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*(c^2*d^2 - e^2)*(d + e*x)^2)) - (b*c^3*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - (a + b*ArcCosh[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(3*(c*d - e)^(5/2)*e*(c*d + e)^(5/2))]} - - -{(d + e*x)^3*(a + b*ArcCosh[c*x])^2, x, 18, 2*b^2*d^3*x + (4*b^2*d*e^2*x)/(3*c^2) + (3/4)*b^2*d^2*e*x^2 + (3*b^2*e^3*x^2)/(32*c^2) + (2/9)*b^2*d*e^2*x^3 + (1/32)*b^2*e^3*x^4 - (2*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*d*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3) - (3*b*d^2*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c) - (3*b*e^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(16*c^3) - (2*b*d*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c) - (b*e^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*c) - (d^4*(a + b*ArcCosh[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcCosh[c*x])^2)/(4*c^2) - (3*e^3*(a + b*ArcCosh[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcCosh[c*x])^2)/(4*e)} -{(d + e*x)^2*(a + b*ArcCosh[c*x])^2, x, 13, 2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + (1/2)*b^2*d*e*x^2 + (2/27)*b^2*e^2*x^3 - (2*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (b*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (2*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c) - (d^3*(a + b*ArcCosh[c*x])^2)/(3*e) - (d*e*(a + b*ArcCosh[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a + b*ArcCosh[c*x])^2)/(3*e)} -{(d + e*x)^1*(a + b*ArcCosh[c*x])^2, x, 9, 2*b^2*d*x + (1/4)*b^2*e*x^2 - (2*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (b*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c) - (d^2*(a + b*ArcCosh[c*x])^2)/(2*e) - (e*(a + b*ArcCosh[c*x])^2)/(4*c^2) + ((d + e*x)^2*(a + b*ArcCosh[c*x])^2)/(2*e)} -{(a + b*ArcCosh[c*x])^2/(d + e*x)^1, x, 10, -((a + b*ArcCosh[c*x])^3/(3*b*e)) + ((a + b*ArcCosh[c*x])^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + ((a + b*ArcCosh[c*x])^2*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/e + (2*b*(a + b*ArcCosh[c*x])*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e + (2*b*(a + b*ArcCosh[c*x])*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e - (2*b^2*PolyLog[3, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/e} -{(a + b*ArcCosh[c*x])^2/(d + e*x)^2, x, 10, -((a + b*ArcCosh[c*x])^2/(e*(d + e*x))) + (2*b*c*(a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) - (2*b*c*(a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*Sqrt[c^2*d^2 - e^2]) + (2*b^2*c*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/(e*Sqrt[c^2*d^2 - e^2]) - (2*b^2*c*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/(e*Sqrt[c^2*d^2 - e^2])} -{(a + b*ArcCosh[c*x])^2/(d + e*x)^3, x, 13, -((b*c*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*(a + b*ArcCosh[c*x]))/((c^2*d^2 - e^2)*(d + e*x))) - (a + b*ArcCosh[c*x])^2/(2*e*(d + e*x)^2) + (b*c^3*d*(a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) - (b*c^3*d*(a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2])])/(e*(c^2*d^2 - e^2)^(3/2)) + (b^2*c^2*Log[d + e*x])/(e*(c^2*d^2 - e^2)) + (b^2*c^3*d*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/(e*(c^2*d^2 - e^2)^(3/2)) - (b^2*c^3*d*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2*d^2 - e^2]))])/(e*(c^2*d^2 - e^2)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(d + e*x)^3/(a + b*ArcCosh[c*x]), x, 27, -((d^3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(b*c)) - (3*d*e^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(4*b*c^3) - (3*d^2*e*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) - (e^3*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(4*b*c^4) - (3*d*e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(4*b*c^3) - (e^3*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(8*b*c^4) + (d^3*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b*c) + (3*d*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(4*b*c^3) + (3*d^2*e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(2*b*c^2) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(4*b*c^4) + (3*d*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b*c^3) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(8*b*c^4)} -{(d + e*x)^2/(a + b*ArcCosh[c*x]), x, 17, -((d^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(b*c)) - (e^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(4*b*c^3) - (d*e*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(b*c^2) - (e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(4*b*c^3) + (d^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b*c) + (e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(4*b*c^3) + (d*e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(b*c^2) + (e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(4*b*c^3)} -{(d + e*x)^1/(a + b*ArcCosh[c*x]), x, 11, -((d*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(b*c)) - (e*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(2*b*c^2) + (d*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b*c) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(2*b*c^2)} -{1/((d + e*x)^1*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcCosh[c*x])), x]} -{1/((d + e*x)^2*(a + b*ArcCosh[c*x])), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcCosh[c*x])), x]} - - -{(d + e*x)^2/(a + b*ArcCosh[c*x])^2, x, 19, -((d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (2*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) - (e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^3) + (2*d*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^2) + (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^3) - (d^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b^2*c^3) - (2*d*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^2) - (3*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b^2*c^3)} -{(d + e*x)^1/(a + b*ArcCosh[c*x])^2, x, 11, -((d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x]))) - (e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*(a + b*ArcCosh[c*x])) + (d*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^2) - (d*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(b^2*c) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b^2*c^2)} -{1/((d + e*x)^1*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcCosh[c*x])^2), x]} -{1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2), x, 0, Unintegrable[1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCosh[c x])^n with m symbolic*) - - -{(d + e*x)^m*(a + b*ArcCosh[c*x])^3, x, 1, ((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^3)/(e*(1 + m)) - (3*b*c*Unintegrable[((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))} -{(d + e*x)^m*(a + b*ArcCosh[c*x])^2, x, 1, ((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^2)/(e*(1 + m)) - (2*b*c*Unintegrable[((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))} -{(d + e*x)^m*(a + b*ArcCosh[c*x])^1, x, 3, -((Sqrt[2]*b*(c*d + e)*Sqrt[-1 + c*x]*(d + e*x)^m*AppellF1[1/2, 1/2, -1 - m, 3/2, (1/2)*(1 - c*x), (e*(1 - c*x))/(c*d + e)])/(((c*(d + e*x))/(c*d + e))^m*(c*e*(1 + m)))) + ((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(e*(1 + m))} -{(d + e*x)^m/(a + b*ArcCosh[c*x])^1, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcCosh[c*x]), x]} -{(d + e*x)^m/(a + b*ArcCosh[c*x])^2, x, 0, Unintegrable[(d + e*x)^m/(a + b*ArcCosh[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcCosh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^p (a+b ArcCosh[c x])^n*) - - -{ArcCosh[a*x]*(c + d*x^2)^4, x, 6, ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*(1 - a^2*x^2))/(315*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (4*d*(105*a^6*c^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(1 - a^2*x^2)^2)/(945*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (2*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^3)/(525*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^4)/(441*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^4*(1 - a^2*x^2)^5)/(81*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^4*x*ArcCosh[a*x] + (4/3)*c^3*d*x^3*ArcCosh[a*x] + (6/5)*c^2*d^2*x^5*ArcCosh[a*x] + (4/7)*c*d^3*x^7*ArcCosh[a*x] + (1/9)*d^4*x^9*ArcCosh[a*x]} -{ArcCosh[a*x]*(c + d*x^2)^3, x, 6, ((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*(1 - a^2*x^2))/(35*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d*(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^2)/(105*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*d^2*(7*a^2*c + 5*d)*(1 - a^2*x^2)^3)/(175*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (d^3*(1 - a^2*x^2)^4)/(49*a^7*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^3*x*ArcCosh[a*x] + c^2*d*x^3*ArcCosh[a*x] + (3/5)*c*d^2*x^5*ArcCosh[a*x] + (1/7)*d^3*x^7*ArcCosh[a*x]} -{ArcCosh[a*x]*(c + d*x^2)^2, x, 6, ((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*(1 - a^2*x^2))/(15*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (2*d*(5*a^2*c + 3*d)*(1 - a^2*x^2)^2)/(45*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^2*(1 - a^2*x^2)^3)/(25*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^2*x*ArcCosh[a*x] + (2/3)*c*d*x^3*ArcCosh[a*x] + (1/5)*d^2*x^5*ArcCosh[a*x]} -{ArcCosh[a*x]*(c + d*x^2)^1, x, 3, -(((9*a^2*c + 2*d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3)) - (d*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + c*x*ArcCosh[a*x] + (1/3)*d*x^3*ArcCosh[a*x]} -{ArcCosh[a*x]/(c + d*x^2)^1, x, 18, (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])])/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])]/(2*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d]))]/(2*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])]/(2*Sqrt[-c]*Sqrt[d])} -{ArcCosh[a*x]/(c + d*x^2)^2, x, 26, -(ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x))) + ArcCosh[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) + (a*ArcTanh[(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[1 + a*x])/(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (a*ArcTanh[(Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[1 + a*x])/(Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[-1 + a*x])])/(2*c*Sqrt[a*Sqrt[-c] - Sqrt[d]]*Sqrt[a*Sqrt[-c] + Sqrt[d]]*Sqrt[d]) - (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCosh[a*x]*Log[1 - (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCosh[a*x]*Log[1 + (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])])/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] - Sqrt[(-a^2)*c - d])]/(4*(-c)^(3/2)*Sqrt[d]) + PolyLog[2, -((Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d]))]/(4*(-c)^(3/2)*Sqrt[d]) - PolyLog[2, (Sqrt[d]*E^ArcCosh[a*x])/(a*Sqrt[-c] + Sqrt[(-a^2)*c - d])]/(4*(-c)^(3/2)*Sqrt[d])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^(p/2) (a+b ArcCosh[c x])^n*) - - -{ArcCosh[a*x]*(c + d*x^2)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x^2]*ArcCosh[a*x], x]} -{ArcCosh[a*x]/(c + d*x^2)^(1/2), x, 0, Unintegrable[ArcCosh[a*x]/Sqrt[c + d*x^2], x]} -{ArcCosh[a*x]/(c + d*x^2)^(3/2), x, 7, (x*ArcCosh[a*x])/(c*Sqrt[c + d*x^2]) - (Sqrt[-1 + a^2*x^2]*ArcTanh[(Sqrt[d]*Sqrt[-1 + a^2*x^2])/(a*Sqrt[c + d*x^2])])/(c*Sqrt[d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{ArcCosh[a*x]/(c + d*x^2)^(5/2), x, 8, (a*(1 - a^2*x^2))/(3*c*(a^2*c + d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c + d*x^2]) + (x*ArcCosh[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCosh[a*x])/(3*c^2*Sqrt[c + d*x^2]) - (2*Sqrt[-1 + a^2*x^2]*ArcTanh[(Sqrt[d]*Sqrt[-1 + a^2*x^2])/(a*Sqrt[c + d*x^2])])/(3*c^2*Sqrt[d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{ArcCosh[a*x]/(c + d*x^2)^(7/2), x, 9, (a*(1 - a^2*x^2))/(15*c*(a^2*c + d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)^(3/2)) + (2*a*(3*a^2*c + 2*d)*(1 - a^2*x^2))/(15*c^2*(a^2*c + d)^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c + d*x^2]) + (x*ArcCosh[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCosh[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCosh[a*x])/(15*c^3*Sqrt[c + d*x^2]) - (8*Sqrt[-1 + a^2*x^2]*ArcTanh[(Sqrt[d]*Sqrt[-1 + a^2*x^2])/(a*Sqrt[c + d*x^2])])/(15*c^3*Sqrt[d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} -{ArcCosh[a*x]/(c + d*x^2)^(9/2), x, 10, (a*(1 - a^2*x^2))/(35*c*(a^2*c + d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)^(5/2)) + (2*a*(5*a^2*c + 3*d)*(1 - a^2*x^2))/(105*c^2*(a^2*c + d)^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)^(3/2)) + (4*a*(11*a^4*c^2 + 15*a^2*c*d + 6*d^2)*(1 - a^2*x^2))/(105*c^3*(a^2*c + d)^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c + d*x^2]) + (x*ArcCosh[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcCosh[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCosh[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcCosh[a*x])/(35*c^4*Sqrt[c + d*x^2]) - (16*Sqrt[-1 + a^2*x^2]*ArcTanh[(Sqrt[d]*Sqrt[-1 + a^2*x^2])/(a*Sqrt[c + d*x^2])])/(35*c^4*Sqrt[d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x^2)^p (a+b ArcCosh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^3, x, 17, (b*f^2*g*x*Sqrt[d - c^2*d*x^2])/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*g^3*x*Sqrt[d - c^2*d*x^2])/(15*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^3*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*g^3*x^3*Sqrt[d - c^2*d*x^2])/(45*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*g^3*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/2)*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (3*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c^2) + (3/4)*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (f^2*g*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/c^2 - (2*g^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*c^4) - (g^3*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2) - (f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^2, x, 14, (2*b*f*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f^2*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*g^2*x^2*Sqrt[d - c^2*d*x^2])/(16*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*f*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*g^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/2)*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*c^2) + (1/4)*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (2*f*g*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^2) - (f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^1, x, 9, (b*g*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*f*x^2*Sqrt[d - c^2*d*x^2])/(4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*g*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (1/2)*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (g*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*c^2) - (f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])/(f + g*x)^1, x, 23, -((b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g*(1 - c*x)*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g - (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - ((1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^2*(1 - c*x)*(1 + c*x)) + (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])/(f + g*x)^2, x, 38, -((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) + (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(g^2*(c^2*f^2 - g^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[-((1 - c*x)/(1 + c*x))]*Sqrt[1 + c*x]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(g*Sqrt[-1 + c*x]*(f + g*x)) + (b*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)^2) - ((1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)^2) - (2*a*c^2*f*Sqrt[d - c^2*d*x^2]*ArcTanh[(Sqrt[c*f + g]*Sqrt[1 + c*x])/(Sqrt[c*f - g]*Sqrt[-1 + c*x])])/(Sqrt[c*f - g]*g^2*Sqrt[c*f + g]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} - - -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^3, x, 27, (3*b*d*f^2*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*g^3*x*Sqrt[d - c^2*d*x^2])/(35*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f^3*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*d*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*g^3*x^3*Sqrt[d - c^2*d*x^2])/(105*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f^3*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (8*b*c*d*g^3*x^5*Sqrt[d - c^2*d*x^2])/(175*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(12*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g^3*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3/8)*d*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (3*d*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (3/8)*d*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/4)*d*f^3*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/2)*d*f*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (3*d*f^2*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2) - (2*d*g^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(35*c^4) - (d*g^3*x^2*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^2) - (3*d*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^2, x, 23, (2*b*d*f*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f^2*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*g^2*x^2*Sqrt[d - c^2*d*x^2])/(32*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c*d*f*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f^2*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*b*c*d*g^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*d*f*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g^2*x^6*Sqrt[d - c^2*d*x^2])/(36*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3/8)*d*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (d*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(16*c^2) + (1/8)*d*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/4)*d*f^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/6)*d*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (2*d*f*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2) - (3*d*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^1, x, 14, (b*d*g*x*Sqrt[d - c^2*d*x^2])/(5*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*b*c*d*f*x^2*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*g*x^3*Sqrt[d - c^2*d*x^2])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*f*x^4*Sqrt[d - c^2*d*x^2])/(16*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*g*x^5*Sqrt[d - c^2*d*x^2])/(25*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3/8)*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/4)*d*f*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (d*g*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2) - (3*d*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])/(f + g*x)^1, x, -28, -((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) + (b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^2*d*(c*f - g)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a*d*(2 + 3*c*x - 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(6*g) + (b*c*d*x*(-12 - 9*c*x + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(36*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g^3 - (a*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(2*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(2 + 3*c*x - 2*c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(6*g) - (b*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2)/(4*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d*(c*f - g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^2) - (d*(c*f - g)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d*(c*f - g)^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) + (d*(c*f - g)*(c*f + g)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (2*a*d*(c*f - g)^(3/2)*(c*f + g)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTanh[(Sqrt[c*f + g]*Sqrt[1 + c*x])/(Sqrt[c*f - g]*Sqrt[-1 + c*x])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -(* {(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])/(f + g*x)^2, x, 0, 0} *) - - -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])*(f + g*x)^3, x, 35, (3*b*d^2*f^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^2*g^3*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (25*b*c*d^2*f^3*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (15*b*d^2*f*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d^2*f^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*g^3*x^3*Sqrt[d - c^2*d*x^2])/(189*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*f^3*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^2*f*g^2*x^4*Sqrt[d - c^2*d*x^2])/(256*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (9*b*c^3*d^2*f^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*g^3*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*f*g^2*x^6*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*f^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (19*b*c^3*d^2*g^3*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*f*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*g^3*x^9*Sqrt[d - c^2*d*x^2])/(81*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*f^3*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/16)*d^2*f^3*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (15*d^2*f*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (15/64)*d^2*f*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/24)*d^2*f^3*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/16)*d^2*f*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/6)*d^2*f^3*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (3/8)*d^2*f*g^2*x^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (3*d^2*f^2*g*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^2) - (2*d^2*g^3*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(63*c^4) - (d^2*g^3*x^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*c^2) - (5*d^2*f^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (15*d^2*f*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(256*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])*(f + g*x)^2, x, 31, (2*b*d^2*f*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (25*b*c*d^2*f^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*d^2*g^2*x^2*Sqrt[d - c^2*d*x^2])/(256*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d^2*f*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*f^2*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (59*b*c*d^2*g^2*x^4*Sqrt[d - c^2*d*x^2])/(768*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*b*c^3*d^2*f*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (17*b*c^3*d^2*g^2*x^6*Sqrt[d - c^2*d*x^2])/(288*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*f*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*g^2*x^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*f^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/16)*d^2*f^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (5*d^2*g^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (5/64)*d^2*g^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/24)*d^2*f^2*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/48)*d^2*g^2*x^3*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/6)*d^2*f^2*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/8)*d^2*g^2*x^3*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (2*d^2*f*g*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^2) - (5*d^2*f^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*d^2*g^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(256*b*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])*(f + g*x)^1, x, 17, (b*d^2*g*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (25*b*c*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*g*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*g*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*g*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*f*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5/16)*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (5/24)*d^2*f*x*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + (1/6)*d^2*f*x*(1 - c*x)^2*(1 + c*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) - (d^2*g*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*c^2) - (5*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(32*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -{(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])/(f + g*x)^1, x, 39, (2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^3*Sqrt[d - c^2*d*x^2])/(45*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a*d^2*(c^2*f^2 - g^2)^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g^5*(1 - c*x)*(1 + c*x)) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4*g^2) - (2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*g) - (d^2*(c^2*f^2 - 2*g^2)*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*g^3) - (c^2*d^2*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*g) + (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (d^2*(c^2*f^2 - g^2)^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^6*(1 - c*x)*(1 + c*x)) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])} -(* {(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])/(f + g*x)^2, x, 0, 0} *) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^3, x, 13, -((3*b*f^2*g*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (2*b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^3*Sqrt[d - c^2*d*x^2]) - (3*b*f*g^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c*Sqrt[d - c^2*d*x^2]) - (b*g^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*Sqrt[d - c^2*d*x^2]) - (3*f^2*g*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (2*g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(3*c^4*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*x*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) - (g^3*x^2*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(3*c^2*Sqrt[d - c^2*d*x^2]) + (f^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) + (3*f*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^2, x, 9, -((2*b*f*g*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (b*g^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*c*Sqrt[d - c^2*d*x^2]) - (2*f*g*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*(f + g*x)^1, x, 6, -((b*g*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (g*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) + (f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])/(f + g*x)^1, x, 10, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])} -{1/Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])/(f + g*x)^2, x, 13, -((g*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/((c^2*f^2 - g^2)*(f + g*x)*Sqrt[d - c^2*d*x^2])) + (c^2*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (c^2*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[f + g*x])/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2]) + (b*c^2*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) - (b*c^2*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/((c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])} - - -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^3, x, 21, (b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*(1 - c*x)*(a + b*ArcCosh[c*x]))/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + (g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^2, x, 19, -(((c*f - g)^2*(1 - c*x)*(a + b*ArcCosh[c*x]))/(2*c^3*d*Sqrt[d - c^2*d*x^2])) + ((c*f + g)^2*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^3*d*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^3*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^3*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^2*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^3*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])*(f + g*x)^1, x, 5, ((g + c^2*f*x)*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c*x])/(c*d*Sqrt[d - c^2*d*x^2]), -(((c*f - g)*(a + b*ArcCosh[c*x]))/(c^2*d*Sqrt[d - c^2*d*x^2])) + (f*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c*d*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (b*f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c*x])/(c*d*Sqrt[d - c^2*d*x^2])} -{1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])/(f + g*x)^1, x, 27, -(((1 - c*x)*(a + b*ArcCosh[c*x]))/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2])) + ((1 + c*x)*(a + b*ArcCosh[c*x]))/(2*d*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(d*(c*f + g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*d*(c*f - g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*d*(c*f + g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2])} -(* {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])/(f + g*x)^2, x, 0, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f+g x)^m (d1+c d1 x)^(p/2) (d2-c d2 x)^(p/2) (a+b ArcCosh[c x])^n*) - - -{((f + g*x)*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2], x, 7, (f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[1 - c^2*x^2]) + (g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*c^2*Sqrt[1 - c^2*x^2])) - (E^(a/b)*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*c^2*Sqrt[1 - c^2*x^2]))} -{((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 - c*x]*Sqrt[1 + c*x]), x, 7, (f*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[1 - c*x]) + (g*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*c^2*Sqrt[1 - c*x])) - (E^(a/b)*g*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*c^2*Sqrt[1 - c*x]))} -{((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]), x, 7, (f*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n*(2*c^2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])) - (E^(a/b)*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(((a + b*ArcCosh[c*x])/b)^n*(2*c^2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d+e x^2)^p (a+b ArcCosh[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Log[h (f + g x)^m] (d-c^2 d x^2)^(p/2) (a+b ArcCosh[c x])^n*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2], x, 1, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Unintegrable[((a + b*ArcCosh[c*x])^n*Log[h*(f + g*x)^m])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/Sqrt[1 - c^2*x^2]} - -(* {Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^3/Sqrt[1 - c^2*x^2], x, 0, 0} *) -{Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^2/Sqrt[1 - c^2*x^2], x, 14, (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^4)/(12*b^2*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(3*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(3*b*c*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[4, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[4, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2])} -{Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^1/Sqrt[1 - c^2*x^2], x, 12, (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b^2*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(2*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(2*b*c*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*Log[h*(f + g*x)^m])/(2*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2])} -{Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^0/Sqrt[1 - c^2*x^2], x, 9, (I*m*ArcSin[c*x]^2)/(2*c) - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c - (m*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c + (ArcSin[c*x]*Log[h*(f + g*x)^m])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c} -{Log[h*(f + g*x)^m]/(a + b*ArcCosh[c*x])^1/Sqrt[1 - c^2*x^2], x, 1, (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Unintegrable[Log[h*(f + g*x)^m]/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])), x])/Sqrt[1 - c^2*x^2]} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcCosh[c+d x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCosh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCosh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ArcCosh[a + b*x], x, 6, (7*a*x^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(48*b^2) - (x^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(16*b) + (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4*a*(16 + 19*a^2) - (9 + 26*a^2)*(a + b*x)))/(96*b^4) - ((3 + 24*a^2 + 8*a^4)*ArcCosh[a + b*x])/(32*b^4) + (1/4)*x^4*ArcCosh[a + b*x]} -{x^2*ArcCosh[a + b*x], x, 5, -((x^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(9*b)) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(4 + 11*a^2 - 5*a*b*x))/(18*b^3) + (a*(3 + 2*a^2)*ArcCosh[a + b*x])/(6*b^3) + (1/3)*x^3*ArcCosh[a + b*x]} -{x^1*ArcCosh[a + b*x], x, 5, (3*a*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*b^2) - (x*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*b) - ((1 + 2*a^2)*ArcCosh[a + b*x])/(4*b^2) + (1/2)*x^2*ArcCosh[a + b*x]} -{x^0*ArcCosh[a + b*x], x, 3, -((Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/b) + ((a + b*x)*ArcCosh[a + b*x])/b} -{ArcCosh[a + b*x]/x^1, x, 9, (-(1/2))*ArcCosh[a + b*x]^2 + ArcCosh[a + b*x]*Log[1 - E^ArcCosh[a + b*x]/(a - Sqrt[-1 + a^2])] + ArcCosh[a + b*x]*Log[1 - E^ArcCosh[a + b*x]/(a + Sqrt[-1 + a^2])] + PolyLog[2, E^ArcCosh[a + b*x]/(a - Sqrt[-1 + a^2])] + PolyLog[2, E^ArcCosh[a + b*x]/(a + Sqrt[-1 + a^2])]} -{ArcCosh[a + b*x]/x^2, x, 4, -(ArcCosh[a + b*x]/x) - (2*b*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/Sqrt[1 - a^2]} -{ArcCosh[a + b*x]/x^3, x, 5, (b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)*x) - ArcCosh[a + b*x]/(2*x^2) - (a*b^2*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a^2)^(3/2)} -{ArcCosh[a + b*x]/x^4, x, 7, (b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(6*(1 - a^2)*x^2) + (a*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)^2*x) - ArcCosh[a + b*x]/(3*x^3) - ((1 + 2*a^2)*b^3*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(3*(1 - a^2)^(5/2))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCosh[c+d x])^(n/2)*) - - -{1/Sqrt[a + b*ArcCosh[c + d*x]], x, 7, -((E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d)) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*d))} -{1/Sqrt[a - b*ArcCosh[c + d*x]], x, 7, -((E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d)) + (Sqrt[Pi]*Erfi[Sqrt[a - b*ArcCosh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(2*Sqrt[b]*d))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcCosh[c+d x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e c+e d x)^m (a+b ArcCosh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x]), x, 8, (-8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(75*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(75*d) - (b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x]))/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x]), x, 7, -((3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(32*d)) - (b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(16*d) - (3*b*e^3*ArcCosh[c + d*x])/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x]))/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x]), x, 6, (-2*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(9*d) - (b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(3*d)} -{(c*e + d*e*x)*(a + b*ArcCosh[c + d*x]), x, 5, -((b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(4*d)) - (b*e*ArcCosh[c + d*x])/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x]))/(2*d)} -{a + b*ArcCosh[c + d*x], x, 4, a*x - (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/d + (b*(c + d*x)*ArcCosh[c + d*x])/d} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x), x, 7, (a + b*ArcCosh[c + d*x])^2/(2*b*d*e) + ((a + b*ArcCosh[c + d*x])*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e) - (b*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(2*d*e)} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^2, x, 5, -((a + b*ArcCosh[c + d*x])/(d*e^2*(c + d*x))) + (b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]])/(d*e^2)} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^3, x, 4, (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(2*d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])/(2*d*e^3*(c + d*x)^2)} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^4, x, 6, (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(6*d*e^4*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])/(3*d*e^4*(c + d*x)^3) + (b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]])/(6*d*e^4)} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^5, x, 6, (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(12*d*e^5*(c + d*x)^3) + (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(6*d*e^5*(c + d*x)) - (a + b*ArcCosh[c + d*x])/(4*d*e^5*(c + d*x)^4)} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^6, x, 8, (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(20*d*e^6*(c + d*x)^4) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(40*d*e^6*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])/(5*d*e^6*(c + d*x)^5) + (3*b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]])/(40*d*e^6)} - - -{(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^2, x, 9, (16*b^2*e^4*x)/75 + (8*b^2*e^4*(c + d*x)^3)/(225*d) + (2*b^2*e^4*(c + d*x)^5)/(125*d) - (16*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(75*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(75*d) - (2*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^2)/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^2, x, 8, (3*b^2*e^3*(c + d*x)^2)/(32*d) + (b^2*e^3*(c + d*x)^4)/(32*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(16*d) - (b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(8*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^2)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^2, x, 7, (4*b^2*e^2*x)/9 + (2*b^2*e^2*(c + d*x)^3)/(27*d) - (4*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(9*d) - (2*b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^2)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^2, x, 6, (b^2*e*(c + d*x)^2)/(4*d) - (b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(2*d) - (e*(a + b*ArcCosh[c + d*x])^2)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/(2*d)} -{(a + b*ArcCosh[c + d*x])^2, x, 4, 2*b^2*x - (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/d + ((c + d*x)*(a + b*ArcCosh[c + d*x])^2)/d} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x), x, 8, (a + b*ArcCosh[c + d*x])^3/(3*b*d*e) + ((a + b*ArcCosh[c + d*x])^2*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e) - (b*(a + b*ArcCosh[c + d*x])*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (b^2*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(2*d*e)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^2, x, 9, -((a + b*ArcCosh[c + d*x])^2/(d*e^2*(c + d*x))) + (4*b*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(d*e^2) - (2*I*b^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + (2*I*b^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^3, x, 5, (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^2/(2*d*e^3*(c + d*x)^2) - (b^2*Log[c + d*x])/(d*e^3)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^4, x, 11, b^2/(3*d*e^4*(c + d*x)) + (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(3*d*e^4*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])^2/(3*d*e^4*(c + d*x)^3) + (2*b*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(3*d*e^4) - (I*b^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(3*d*e^4) + (I*b^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(3*d*e^4)} - - -{(c*e + d*e*x)^4*(a + b*ArcCosh[c + d*x])^3, x, 19, (16*a*b^2*e^4*x)/25 - (4144*b^3*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(5625*d) - (272*b^3*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5625*d) - (6*b^3*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(625*d) + (16*b^3*e^4*(c + d*x)*ArcCosh[c + d*x])/(25*d) + (8*b^2*e^4*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(75*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x]))/(125*d) - (8*b*e^4*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(25*d) - (4*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(25*d) - (3*b*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(25*d) + (e^4*(c + d*x)^5*(a + b*ArcCosh[c + d*x])^3)/(5*d)} -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^3, x, 14, -((45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(256*d)) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(128*d) - (45*b^3*e^3*ArcCosh[c + d*x])/(256*d) + (9*b^2*e^3*(c + d*x)^2*(a + b*ArcCosh[c + d*x]))/(32*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x]))/(32*d) - (9*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(32*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^3)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^3, x, 12, (4*a*b^2*e^2*x)/3 - (40*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(27*d) - (2*b^3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(27*d) + (4*b^3*e^2*(c + d*x)*ArcCosh[c + d*x])/(3*d) + (2*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x]))/(9*d) - (2*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(3*d) - (b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^3)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^3, x, 8, -((3*b^3*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(8*d)) - (3*b^3*e*ArcCosh[c + d*x])/(8*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcCosh[c + d*x]))/(4*d) - (3*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(4*d) - (e*(a + b*ArcCosh[c + d*x])^3)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^3)/(2*d)} -{(a + b*ArcCosh[c + d*x])^3, x, 6, 6*a*b^2*x - (6*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/d + (6*b^3*(c + d*x)*ArcCosh[c + d*x])/d - (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/d + ((c + d*x)*(a + b*ArcCosh[c + d*x])^3)/d} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x), x, 9, (a + b*ArcCosh[c + d*x])^4/(4*b*d*e) + ((a + b*ArcCosh[c + d*x])^3*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(2*d*e) - (3*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(2*d*e) - (3*b^3*PolyLog[4, -E^(-2*ArcCosh[c + d*x])])/(4*d*e)} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^2, x, 11, -((a + b*ArcCosh[c + d*x])^3/(d*e^2*(c + d*x))) + (6*b*(a + b*ArcCosh[c + d*x])^2*ArcTan[E^ArcCosh[c + d*x]])/(d*e^2) - (6*I*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + (6*I*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + (6*I*b^3*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) - (6*I*b^3*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^2)} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^3, x, 9, -((3*b*(a + b*ArcCosh[c + d*x])^2)/(2*d*e^3)) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(2*d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) - (3*b^2*(a + b*ArcCosh[c + d*x])*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e^3) + (3*b^3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(2*d*e^3)} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^4, x, 15, (b^2*(a + b*ArcCosh[c + d*x]))/(d*e^4*(c + d*x)) + (b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^2)/(2*d*e^4*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])^3/(3*d*e^4*(c + d*x)^3) + (b*(a + b*ArcCosh[c + d*x])^2*ArcTan[E^ArcCosh[c + d*x]])/(d*e^4) - (b^3*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]])/(d*e^4) - (I*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) + (I*b^2*(a + b*ArcCosh[c + d*x])*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^4) + (I*b^3*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - (I*b^3*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^4)} - - -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^4, x, 16, (45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) - (45*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(64*d) - (3*b^3*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(32*d) - (45*b^2*e^3*(a + b*ArcCosh[c + d*x])^2)/(128*d) + (9*b^2*e^3*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/(16*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/(16*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(8*d) - (b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(4*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^4)/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^4, x, 13, (160*b^4*e^2*x)/27 + (8*b^4*e^2*(c + d*x)^3)/(81*d) - (160*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(27*d) - (8*b^3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(27*d) + (8*b^2*e^2*(c + d*x)*(a + b*ArcCosh[c + d*x])^2)/(3*d) + (4*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^2)/(9*d) - (8*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(9*d) - (4*b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^4)/(3*d)} -{(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^4, x, 9, (3*b^4*e*(c + d*x)^2)/(4*d) - (3*b^3*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(2*d) - (3*b^2*e*(a + b*ArcCosh[c + d*x])^2)/(4*d) + (3*b^2*e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^2)/(2*d) - (b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/d - (e*(a + b*ArcCosh[c + d*x])^4)/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^4)/(2*d)} -{(a + b*ArcCosh[c + d*x])^4, x, 6, 24*b^4*x - (24*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/d + (12*b^2*(c + d*x)*(a + b*ArcCosh[c + d*x])^2)/d - (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/d + ((c + d*x)*(a + b*ArcCosh[c + d*x])^4)/d} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x), x, 10, (a + b*ArcCosh[c + d*x])^5/(5*b*d*e) + ((a + b*ArcCosh[c + d*x])^4*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e) - (2*b*(a + b*ArcCosh[c + d*x])^3*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[4, -E^(-2*ArcCosh[c + d*x])])/(d*e) - (3*b^4*PolyLog[5, -E^(-2*ArcCosh[c + d*x])])/(2*d*e)} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^2, x, 13, -((a + b*ArcCosh[c + d*x])^4/(d*e^2*(c + d*x))) + (8*b*(a + b*ArcCosh[c + d*x])^3*ArcTan[E^ArcCosh[c + d*x]])/(d*e^2) - (12*I*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + (12*I*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^2) + (24*I*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) - (24*I*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^2) - (24*I*b^4*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])/(d*e^2) + (24*I*b^4*PolyLog[4, I*E^ArcCosh[c + d*x]])/(d*e^2)} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^3, x, 10, -((2*b*(a + b*ArcCosh[c + d*x])^3)/(d*e^3)) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) - (6*b^2*(a + b*ArcCosh[c + d*x])^2*Log[1 + E^(-2*ArcCosh[c + d*x])])/(d*e^3) + (6*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[2, -E^(-2*ArcCosh[c + d*x])])/(d*e^3) + (3*b^4*PolyLog[3, -E^(-2*ArcCosh[c + d*x])])/(d*e^3)} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^4, x, 21, (2*b^2*(a + b*ArcCosh[c + d*x])^2)/(d*e^4*(c + d*x)) + (2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/(3*d*e^4*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])^4/(3*d*e^4*(c + d*x)^3) - (8*b^3*(a + b*ArcCosh[c + d*x])*ArcTan[E^ArcCosh[c + d*x]])/(d*e^4) + (4*b*(a + b*ArcCosh[c + d*x])^3*ArcTan[E^ArcCosh[c + d*x]])/(3*d*e^4) + (4*I*b^4*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - (2*I*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - (4*I*b^4*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^4) + (2*I*b^2*(a + b*ArcCosh[c + d*x])^2*PolyLog[2, I*E^ArcCosh[c + d*x]])/(d*e^4) + (4*I*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) - (4*I*b^3*(a + b*ArcCosh[c + d*x])*PolyLog[3, I*E^ArcCosh[c + d*x]])/(d*e^4) - (4*I*b^4*PolyLog[4, (-I)*E^ArcCosh[c + d*x]])/(d*e^4) + (4*I*b^4*PolyLog[4, I*E^ArcCosh[c + d*x]])/(d*e^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x]), x, 14, -((e^4*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(8*b*d)) - (3*e^4*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/(16*b*d) - (e^4*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/(16*b*d) + (e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b*d) + (3*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(16*b*d) + (e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(16*b*d)} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x]), x, 11, -((e^3*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b])/(4*b*d)) - (e^3*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(4*a)/b])/(8*b*d) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(4*b*d) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(8*b*d)} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x]), x, 11, -((e^2*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(4*b*d)) - (e^2*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/(4*b*d) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b*d) + (e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b*d)} -{(c*e + d*e*x)/(a + b*ArcCosh[c + d*x]), x, 8, -((e*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b*d)) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(2*b*d)} -{(a + b*ArcCosh[c + d*x])^(-1), x, 5, -((CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(b*d)) + (Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(b*d)} -{1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^2, x, 13, -((e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^4*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b^2*d) + (9*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(16*b^2*d) + (5*e^4*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(16*b^2*d) - (e^4*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b^2*d) - (9*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(16*b^2*d) - (5*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(16*b^2*d)} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^2, x, 10, -((e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^3*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(2*b^2*d) + (e^3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(2*b^2*d) - (e^3*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(2*b^2*d) - (e^3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(2*b^2*d)} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^2, x, 10, -((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b^2*d) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) - (3*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b^2*d)} -{(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^2, x, 6, -((e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(b^2*d) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(b^2*d)} -{(a + b*ArcCosh[c + d*x])^(-2), x, 6, -((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(b^2*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(b^2*d)} -{1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^2), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^3, x, 26, -((e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2)) + (2*e^4*(c + d*x)^3)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^4*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(16*b^3*d) - (27*e^4*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/(32*b^3*d) - (25*e^4*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(5*a)/b])/(32*b^3*d) + (e^4*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(16*b^3*d) + (27*e^4*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(32*b^3*d) + (25*e^4*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(32*b^3*d)} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^3, x, 20, -((e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2)) + (3*e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (2*e^3*(c + d*x)^4)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (e^3*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b])/(2*b^3*d) - (e^3*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(4*a)/b])/(b^3*d) + (e^3*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(2*b^3*d) + (e^3*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(b^3*d)} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^3, x, 18, -((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2)) + (e^2*(c + d*x))/(b^2*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^2*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(8*b^3*d) - (9*e^2*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*a)/b])/(8*b^3*d) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*b^3*d)} -{(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^3, x, 11, -((e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2)) + e/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e*(c + d*x)^2)/(b^2*d*(a + b*ArcCosh[c + d*x])) - (e*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(2*a)/b])/(b^3*d) + (e*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(b^3*d)} -{(a + b*ArcCosh[c + d*x])^(-3), x, 7, -(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2) - (c + d*x)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(2*b^3*d) + (Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(2*b^3*d)} -{1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^3), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^3), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^4, x, 24, -((e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3)) + (2*e^4*(c + d*x)^3)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (5*e^4*(c + d*x)^5)/(6*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (2*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b^3*d*(a + b*ArcCosh[c + d*x])) - (25*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(6*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^4*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(48*b^4*d) + (27*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(32*b^4*d) + (125*e^4*Cosh[(5*a)/b]*CoshIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(96*b^4*d) - (e^4*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(48*b^4*d) - (27*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(32*b^4*d) - (125*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c + d*x]))/b])/(96*b^4*d)} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^4, x, 17, -((e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3)) + (e^3*(c + d*x)^2)/(2*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (2*e^3*(c + d*x)^4)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b^3*d*(a + b*ArcCosh[c + d*x])) - (8*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^3*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d) + (4*e^3*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d) - (e^3*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d) - (4*e^3*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d)} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^4, x, 18, -((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3)) + (e^2*(c + d*x))/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])^2) + (e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(2*b^3*d*(a + b*ArcCosh[c + d*x])) + (e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(24*b^4*d) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*b^4*d) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(24*b^4*d) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*b^4*d)} -{(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^4, x, 9, -((e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3)) + e/(6*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d) - (2*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*d)} -{(a + b*ArcCosh[c + d*x])^(-4), x, 8, -((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^3)) - (c + d*x)/(6*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(6*b^3*d*(a + b*ArcCosh[c + d*x])) + (Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c + d*x])/b])/(6*b^4*d) - (Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(6*b^4*d)} -{1/((c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^4), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^4), x]/e} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e c+e d x)^m (a+b ArcCosh[c+d x])^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^4*Sqrt[a + b*ArcCosh[c + d*x]], x, 21, (e^4*(c + d*x)^5*Sqrt[a + b*ArcCosh[c + d*x]])/(5*d) - (Sqrt[b]*e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d) - (Sqrt[b]*e^4*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d) - (Sqrt[b]*e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(320*d) - (Sqrt[b]*e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b)) - (Sqrt[b]*e^4*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d*E^((3*a)/b)) - (Sqrt[b]*e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(320*d*E^((5*a)/b))} -{(c*e + d*e*x)^3*Sqrt[a + b*ArcCosh[c + d*x]], x, 16, (-3*e^3*Sqrt[a + b*ArcCosh[c + d*x]])/(32*d) + (e^3*(c + d*x)^4*Sqrt[a + b*ArcCosh[c + d*x]])/(4*d) - (Sqrt[b]*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*d) - (Sqrt[b]*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*d) - (Sqrt[b]*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*d*E^((4*a)/b)) - (Sqrt[b]*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*Sqrt[a + b*ArcCosh[c + d*x]], x, 16, (e^2*(c + d*x)^3*Sqrt[a + b*ArcCosh[c + d*x]])/(3*d) - (Sqrt[b]*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d) - (Sqrt[b]*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(48*d) - (Sqrt[b]*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d*E^(a/b)) - (Sqrt[b]*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(48*d*E^((3*a)/b))} -{(c*e + d*e*x)*Sqrt[a + b*ArcCosh[c + d*x]], x, 11, -(e*Sqrt[a + b*ArcCosh[c + d*x]])/(4*d) + (e*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(2*d) - (Sqrt[b]*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*d) - (Sqrt[b]*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*d*E^((2*a)/b))} -{Sqrt[a + b*ArcCosh[c + d*x]], x, 8, ((c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]])/d - (Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(4*d) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(4*d*E^(a/b))} -{1/(c*e + d*e*x)*Sqrt[a + b*ArcCosh[c + d*x]], x, 2, Unintegrable[Sqrt[a + b*ArcCosh[c + d*x]]/(c + d*x), x]/e} - - -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(3/2), x, 27, (-9*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(64*d) - (3*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^(3/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^(3/2))/(4*d) - (3*b^(3/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2048*d) - (3*b^(3/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(128*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2048*d*E^((4*a)/b)) + (3*b^(3/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(128*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^(3/2), x, 24, -(b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(3*d) - (b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(3/2))/(3*d) - (3*b^(3/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d) - (b^(3/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(96*d) + (3*b^(3/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b)) + (b^(3/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(96*d*E^((3*a)/b))} -{(c*e + d*e*x)^1*(a + b*ArcCosh[c + d*x])^(3/2), x, 13, (-3*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(8*d) - (e*(a + b*ArcCosh[c + d*x])^(3/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d) + (3*b^(3/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d*E^((2*a)/b))} -{(a + b*ArcCosh[c + d*x])^(3/2), x, 9, (-3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcCosh[c + d*x])^(3/2))/d - (3*b^(3/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*d) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(3/2), x, 2, Unintegrable[(a + b*ArcCosh[c + d*x])^(3/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^(5/2), x, 29, (-225*b^2*e^3*Sqrt[a + b*ArcCosh[c + d*x]])/(2048*d) + (45*b^2*e^3*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(256*d) + (15*b^2*e^3*(c + d*x)^4*Sqrt[a + b*ArcCosh[c + d*x]])/(256*d) - (15*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(64*d) - (5*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(32*d) - (3*e^3*(a + b*ArcCosh[c + d*x])^(5/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^(5/2))/(4*d) - (15*b^(5/2)*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16384*d) - (15*b^(5/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(512*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16384*d*E^((4*a)/b)) - (15*b^(5/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(512*d*E^((2*a)/b))} -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^(5/2), x, 26, (5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]])/(6*d) + (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcCosh[c + d*x]])/(36*d) - (5*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(9*d) - (5*b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(5/2))/(3*d) - (15*b^(5/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(64*d) - (5*b^(5/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(576*d) - (15*b^(5/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(64*d*E^(a/b)) - (5*b^(5/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(576*d*E^((3*a)/b))} -{(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(5/2), x, 14, (-15*b^2*e*Sqrt[a + b*ArcCosh[c + d*x]])/(64*d) + (15*b^2*e*(c + d*x)^2*Sqrt[a + b*ArcCosh[c + d*x]])/(32*d) - (5*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(8*d) - (e*(a + b*ArcCosh[c + d*x])^(5/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(5/2))/(2*d) - (15*b^(5/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*d) - (15*b^(5/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(256*d*E^((2*a)/b))} -{(a + b*ArcCosh[c + d*x])^(5/2), x, 10, (15*b^2*(c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]])/(4*d) - (5*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(3/2))/(2*d) + ((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2))/d - (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d) - (15*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(5/2), x, 2, Unintegrable[(a + b*ArcCosh[c + d*x])^(5/2)/(c + d*x), x]/e} - - -{(c*e + d*e*x)^2*(a + b*ArcCosh[c + d*x])^(7/2), x, 35, (-175*b^3*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(54*d) - (35*b^3*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(216*d) + (35*b^2*e^2*(c + d*x)*(a + b*ArcCosh[c + d*x])^(3/2))/(18*d) + (35*b^2*e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(3/2))/(108*d) - (7*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(9*d) - (7*b*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcCosh[c + d*x])^(7/2))/(3*d) - (105*b^(7/2)*e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d) - (35*b^(7/2)*e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d) + (105*b^(7/2)*e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(128*d*E^(a/b)) + (35*b^(7/2)*e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3456*d*E^((3*a)/b))} -{(c*e + d*e*x)^1*(a + b*ArcCosh[c + d*x])^(7/2), x, 16, (-105*b^3*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(128*d) - (35*b^2*e*(a + b*ArcCosh[c + d*x])^(3/2))/(64*d) + (35*b^2*e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(3/2))/(32*d) - (7*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(8*d) - (e*(a + b*ArcCosh[c + d*x])^(7/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(7/2))/(2*d) - (105*b^(7/2)*e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(1024*d) + (105*b^(7/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(1024*d*E^((2*a)/b))} -{(a + b*ArcCosh[c + d*x])^(7/2), x, 11, (-105*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(8*d) + (35*b^2*(c + d*x)*(a + b*ArcCosh[c + d*x])^(3/2))/(4*d) - (7*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcCosh[c + d*x])^(7/2))/d - (105*b^(7/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d) + (105*b^(7/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(32*d*E^(a/b))} -{1/(c*e + d*e*x)*(a + b*ArcCosh[c + d*x])^(7/2), x, 2, Unintegrable[(a + b*ArcCosh[c + d*x])^(7/2)/(c + d*x), x]/e} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c*e + d*e*x)^4/Sqrt[a + b*ArcCosh[c + d*x]], x, 20, -(e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*Sqrt[b]*d) - (e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) - (e^4*E^((5*a)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(16*Sqrt[b]*d*E^(a/b)) + (e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((3*a)/b)) + (e^4*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/Sqrt[a + b*ArcCosh[c + d*x]], x, 15, -(e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d) - (e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(32*Sqrt[b]*d*E^((4*a)/b)) + (e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/Sqrt[a + b*ArcCosh[c + d*x]], x, 15, -(e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*Sqrt[b]*d) - (e^2*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*Sqrt[b]*d*E^(a/b)) + (e^2*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d*E^((3*a)/b))} -{(c*e + d*e*x)^1/Sqrt[a + b*ArcCosh[c + d*x]], x, 10, -(e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d) + (e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*Sqrt[b]*d*E^((2*a)/b))} -{1/Sqrt[a + b*ArcCosh[c + d*x]], x, 7, -(E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(2*Sqrt[b]*d*E^(a/b))} -{1/(c*e + d*e*x)/Sqrt[a + b*ArcCosh[c + d*x]], x, 2, Unintegrable[1/((c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]]), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(3/2), x, 19, -((2*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(b*d*Sqrt[a + b*ArcCosh[c + d*x]])) + (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(8*b^(3/2)*d) + (3*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d) + (e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(16*b^(3/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(8*b^(3/2)*d)) + (3*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((3*a)/b)*(16*b^(3/2)*d)) + (e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((5*a)/b)*(16*b^(3/2)*d))} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^(3/2), x, 14, -((2*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(b*d*Sqrt[a + b*ArcCosh[c + d*x]])) + (e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) + (e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2*b^(3/2)*d) + (e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((4*a)/b)*(4*b^(3/2)*d)) + (e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(2*b^(3/2)*d))} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^(3/2), x, 14, -((2*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*Sqrt[a + b*ArcCosh[c + d*x]])) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(4*b^(3/2)*d) + (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(4*b^(3/2)*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(4*b^(3/2)*d)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((3*a)/b)*(4*b^(3/2)*d))} -{(c*e + d*e*x)^1/(a + b*ArcCosh[c + d*x])^(3/2), x, 8, -((2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*d*Sqrt[a + b*ArcCosh[c + d*x]])) + (e*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(b^(3/2)*d) + (e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(E^((2*a)/b)*(b^(3/2)*d))} -{(a + b*ArcCosh[c + d*x])^(-3/2), x, 8, -((2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(b*d*Sqrt[a + b*ArcCosh[c + d*x]])) + (E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(E^(a/b)*(b^(3/2)*d))} -{1/(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(3/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^(3/2)), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(5/2), x, 36, (-2*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (16*e^4*(c + d*x)^3)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (20*e^4*(c + d*x)^5)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(12*b^(5/2)*d) - (3*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*b^(5/2)*d) - (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(12*b^(5/2)*d*E^(a/b)) + (3*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*b^(5/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^(5/2), x, 26, (-2*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (4*e^3*(c + d*x)^2)/(b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (16*e^3*(c + d*x)^4)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) - (e^3*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((4*a)/b)) + (e^3*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^(5/2), x, 24, (-2*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (8*e^2*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (4*e^2*(c + d*x)^3)/(b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(6*b^(5/2)*d) - (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2*b^(5/2)*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(6*b^(5/2)*d*E^(a/b)) + (e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2*b^(5/2)*d*E^((3*a)/b))} -{(c*e + d*e*x)^1/(a + b*ArcCosh[c + d*x])^(5/2), x, 13, (-2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (4*e)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (8*e*(c + d*x)^2)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*e*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d) + (2*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(3*b^(5/2)*d*E^((2*a)/b))} -{(a + b*ArcCosh[c + d*x])^(-5/2), x, 9, (-2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (4*(c + d*x))/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d) + (2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(3*b^(5/2)*d*E^(a/b))} -{1/(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(5/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^(5/2)), x]/e} - - -{(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(7/2), x, 34, (-2*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (16*e^4*(c + d*x)^3)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (4*e^4*(c + d*x)^5)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (32*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (40*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(30*b^(7/2)*d) + (9*e^4*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(20*b^(7/2)*d) + (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(30*b^(7/2)*d*E^(a/b)) + (9*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(20*b^(7/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(12*b^(7/2)*d*E^((5*a)/b))} -{(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^(7/2), x, 23, (-2*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (4*e^3*(c + d*x)^2)/(5*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (16*e^3*(c + d*x)^4)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (16*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (128*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (16*e^3*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (4*e^3*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (16*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((4*a)/b)) + (4*e^3*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((2*a)/b))} -{(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^(7/2), x, 24, (-2*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (8*e^2*(c + d*x))/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (16*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (24*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (3*e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d*E^(a/b)) + (3*e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d*E^((3*a)/b))} -{(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(7/2), x, 11, (-2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (4*e)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (8*e*(c + d*x)^2)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (32*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (8*e*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (8*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((2*a)/b))} -{(a + b*ArcCosh[c + d*x])^(-7/2), x, 10, (-2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) - (4*(c + d*x))/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (8*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d*E^(a/b))} -{1/(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(7/2), x, 2, Unintegrable[1/((c + d*x)*(a + b*ArcCosh[c + d*x])^(7/2)), x]/e} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e c+e d x)^(m/2) (a+b ArcCosh[c+d x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcCosh[c + d*x]), x, 8, -((28*b*e^2*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(405*d)) - (4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(7/2)*Sqrt[1 + c + d*x])/(81*d) + (2*(e*(c + d*x))^(9/2)*(a + b*ArcCosh[c + d*x]))/(9*d*e) - (28*b*e^3*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(135*d*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])} -{(c*e + d*e*x)^(5/2)*(a + b*ArcCosh[c + d*x]), x, 8, -((20*b*e^2*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(147*d)) - (4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(5/2)*Sqrt[1 + c + d*x])/(49*d) + (2*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x]))/(7*d*e) - (20*b*e^(5/2)*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(147*d*Sqrt[-1 + c + d*x])} -{(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x]), x, 6, -((4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(25*d)) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x]))/(5*d*e) - (12*b*e*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(25*d*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])} -{Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x]), x, 6, -((4*b*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(9*d)) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x]))/(3*d*e) - (4*b*Sqrt[e]*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(9*d*Sqrt[-1 + c + d*x])} -{(a + b*ArcCosh[c + d*x])/Sqrt[c*e + d*e*x], x, 4, (2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x]))/(d*e) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(d*e*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(3/2), x, 4, -((2*(a + b*ArcCosh[c + d*x]))/(d*e*Sqrt[e*(c + d*x)])) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(d*e^(3/2)*Sqrt[-1 + c + d*x])} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(5/2), x, 7, (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*d*e^2*Sqrt[e*(c + d*x)]) - (2*(a + b*ArcCosh[c + d*x]))/(3*d*e*(e*(c + d*x))^(3/2)) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(3*d*e^3*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])} -{(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(7/2), x, 7, (4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*d*e^2*(e*(c + d*x))^(3/2)) - (2*(a + b*ArcCosh[c + d*x]))/(5*d*e*(e*(c + d*x))^(5/2)) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(15*d*e^(7/2)*Sqrt[-1 + c + d*x])} - - -{(c*e + d*e*x)^(7/2)*(a + b*ArcCosh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(9/2)*(a + b*ArcCosh[c + d*x])^2)/(9*d*e) - (1/(99*d*e^2*Sqrt[-1 + c + d*x]))*(8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(11/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 11/4, 15/4, (c + d*x)^2]) - (16*b^2*(e*(c + d*x))^(13/2)*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4, 17/4}, (c + d*x)^2])/(1287*d*e^3)} -{(c*e + d*e*x)^(5/2)*(a + b*ArcCosh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x])^2)/(7*d*e) - (1/(63*d*e^2*Sqrt[-1 + c + d*x]))*(8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(9/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 9/4, 13/4, (c + d*x)^2]) - (16*b^2*(e*(c + d*x))^(11/2)*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, (c + d*x)^2])/(693*d*e^3)} -{(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^2)/(5*d*e) - (1/(35*d*e^2*Sqrt[-1 + c + d*x]))*(8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 7/4, 11/4, (c + d*x)^2]) - (16*b^2*(e*(c + d*x))^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, (c + d*x)^2])/(315*d*e^3)} -{Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x])^2, x, 3, (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^2)/(3*d*e) - (8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2])/(15*d*e^2*Sqrt[-1 + c + d*x]) - (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2])/(105*d*e^3)} -{(a + b*ArcCosh[c + d*x])^2/Sqrt[c*e + d*e*x], x, 3, (2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])^2)/(d*e) - (8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(3*d*e^2*Sqrt[-1 + c + d*x]) - (16*b^2*(e*(c + d*x))^(5/2)*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, (c + d*x)^2])/(15*d*e^3)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^(3/2), x, 3, -((2*(a + b*ArcCosh[c + d*x])^2)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2])/(d*e^2*Sqrt[-1 + c + d*x]) + (16*b^2*(e*(c + d*x))^(3/2)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, (c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^(5/2), x, 3, -((2*(a + b*ArcCosh[c + d*x])^2)/(3*d*e*(e*(c + d*x))^(3/2))) - (8*b*Sqrt[1 - c - d*x]*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[-(1/4), 1/2, 3/4, (c + d*x)^2])/(3*d*e^2*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]) - (16*b^2*Sqrt[e*(c + d*x)]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, (c + d*x)^2])/(3*d*e^3)} -{(a + b*ArcCosh[c + d*x])^2/(c*e + d*e*x)^(7/2), x, 3, -((2*(a + b*ArcCosh[c + d*x])^2)/(5*d*e*(e*(c + d*x))^(5/2))) - (8*b*Sqrt[1 - c - d*x]*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[-(3/4), 1/2, 1/4, (c + d*x)^2])/(15*d*e^2*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)) + (16*b^2*HypergeometricPFQ[{-(1/4), -(1/4), 1}, {1/4, 3/4}, (c + d*x)^2])/(15*d*e^3*Sqrt[e*(c + d*x)])} - - -{(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^3)/(5*d*e) - (6*b*Unintegrable[((e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^2)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(5*e)} -{Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x])^3, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^3)/(3*d*e) - (2*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^2)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^3/Sqrt[c*e + d*e*x], x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])^3)/(d*e) - (6*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])^2)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)])) + (6*b*Unintegrable[(a + b*ArcCosh[c + d*x])^2/(Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2))) + (2*b*Unintegrable[(a + b*ArcCosh[c + d*x])^2/(Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(7/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^3)/(5*d*e*(e*(c + d*x))^(5/2))) + (6*b*Unintegrable[(a + b*ArcCosh[c + d*x])^2/(Sqrt[-1 + c + d*x]*(e*(c + d*x))^(5/2)*Sqrt[1 + c + d*x]), x])/(5*e)} - - -{(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^4)/(5*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^3)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(5*e)} -{Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x])^4, x, 2, (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^4)/(3*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^3)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(3*e)} -{(a + b*ArcCosh[c + d*x])^4/Sqrt[c*e + d*e*x], x, 2, (2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])^4)/(d*e) - (8*b*Unintegrable[(Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x])^3)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(3/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^4)/(d*e*Sqrt[e*(c + d*x)])) + (8*b*Unintegrable[(a + b*ArcCosh[c + d*x])^3/(Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x]), x])/e} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(5/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^4)/(3*d*e*(e*(c + d*x))^(3/2))) + (8*b*Unintegrable[(a + b*ArcCosh[c + d*x])^3/(Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x]), x])/(3*e)} -{(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(7/2), x, 2, -((2*(a + b*ArcCosh[c + d*x])^4)/(5*d*e*(e*(c + d*x))^(5/2))) + (8*b*Unintegrable[(a + b*ArcCosh[c + d*x])^3/(Sqrt[-1 + c + d*x]*(e*(c + d*x))^(5/2)*Sqrt[1 + c + d*x]), x])/(5*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e c+e d x)^m (a+b ArcCosh[c+d x])^n with m symbolic*) - - -{(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x])^4, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^4)/(d*e*(1 + m)) - (4*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^3)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(e*(1 + m))} -{(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x])^3, x, 2, ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^3)/(d*e*(1 + m)) - (3*b*Unintegrable[((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(e*(1 + m))} -{(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x])^2, x, 3, If[$VersionNumber>=8, ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(2 + m)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]) - (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(d*e^3*(1 + m)*(2 + m)*(3 + m)), ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(d*e*(1 + m)) - (2*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(2 + m)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]) - (2*b^2*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(d*e^3*(3 + m)*(2 + 3*m + m^2))]} -{(c*e + d*e*x)^m*(a + b*ArcCosh[c + d*x]), x, 5, ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*(1 - (c + d*x)^2)*Hypergeometric2F1[1, (3 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), ((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x]))/(d*e*(1 + m)) - (b*(e*(c + d*x))^(2 + m)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(d*e^2*(1 + m)*(2 + m)*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])} -{(c*e + d*e*x)^m/(a + b*ArcCosh[c + d*x]), x, 1, Unintegrable[(e*(c + d*x))^m/(a + b*ArcCosh[c + d*x]), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcCosh[c+d x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (a+b ArcCosh[c+d x^n])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCosh[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcCosh[a*x^5]/x, x, 5, (-(1/10))*ArcCosh[a*x^5]^2 + (1/5)*ArcCosh[a*x^5]*Log[1 + E^(2*ArcCosh[a*x^5])] + (1/10)*PolyLog[2, -E^(2*ArcCosh[a*x^5])]} - - -{x^2*ArcCosh[Sqrt[x]], x, 7, (-(5/48))*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - (5/72)*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2) - (1/18)*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2) - (5*ArcCosh[Sqrt[x]])/48 + (1/3)*x^3*ArcCosh[Sqrt[x]]} -{x^1*ArcCosh[Sqrt[x]], x, 6, (-(3/16))*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - (1/8)*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2) - (3*ArcCosh[Sqrt[x]])/16 + (1/2)*x^2*ArcCosh[Sqrt[x]]} -{x^0*ArcCosh[Sqrt[x]], x, 5, (-(1/2))*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x] - ArcCosh[Sqrt[x]]/2 + x*ArcCosh[Sqrt[x]]} -{ArcCosh[Sqrt[x]]/x^1, x, 5, -ArcCosh[Sqrt[x]]^2 + 2*ArcCosh[Sqrt[x]]*Log[1 + E^(2*ArcCosh[Sqrt[x]])] + PolyLog[2, -E^(2*ArcCosh[Sqrt[x]])]} -{ArcCosh[Sqrt[x]]/x^2, x, 3, (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/Sqrt[x] - ArcCosh[Sqrt[x]]/x} -{ArcCosh[Sqrt[x]]/x^3, x, 4, (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(6*x^(3/2)) + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(3*Sqrt[x]) - ArcCosh[Sqrt[x]]/(2*x^2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{ArcCosh[1/x], x, 3, x*ArcSech[x] + Sqrt[1/(1 + x)]*Sqrt[1 + x]*ArcSin[x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCosh[a x^n] with n symbolic*) - - -{ArcCosh[a*x^n]/x, x, 5, -(ArcCosh[a*x^n]^2/(2*n)) + (ArcCosh[a*x^n]*Log[1 + E^(2*ArcCosh[a*x^n])])/n + PolyLog[2, -E^(2*ArcCosh[a*x^n])]/(2*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (a+b ArcCosh[c+d x^2])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcCosh[c+d x^2])^n when c^2=1*) - - -{(a + b*ArcCosh[1 + d*x^2])^4, x, 3, 384*b^4*x - (192*b^3*(2*x^2 + d*x^4)*(a + b*ArcCosh[1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + 48*b^2*x*(a + b*ArcCosh[1 + d*x^2])^2 - (8*b*(2*x^2 + d*x^4)*(a + b*ArcCosh[1 + d*x^2])^3)/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + x*(a + b*ArcCosh[1 + d*x^2])^4} -{(a + b*ArcCosh[1 + d*x^2])^3, x, 7, 24*a*b^2*x - (48*b^3*Sqrt[(d*x^2)/(2 + d*x^2)]*(2 + d*x^2))/(d*x) + 24*b^3*x*ArcCosh[1 + d*x^2] - (6*b*(2*x^2 + d*x^4)*(a + b*ArcCosh[1 + d*x^2])^2)/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + x*(a + b*ArcCosh[1 + d*x^2])^3} -{(a + b*ArcCosh[1 + d*x^2])^2, x, 2, 8*b^2*x - (4*b*(2*x^2 + d*x^4)*(a + b*ArcCosh[1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + x*(a + b*ArcCosh[1 + d*x^2])^2} -{(a + b*ArcCosh[1 + d*x^2])^1, x, 6, a*x - (2*b*Sqrt[(d*x^2)/(2 + d*x^2)]*(2 + d*x^2))/(d*x) + b*x*ArcCosh[1 + d*x^2]} -{1/(a + b*ArcCosh[1 + d*x^2])^1, x, 1, (x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2]) - (x*Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])} -{1/(a + b*ArcCosh[1 + d*x^2])^2, x, 1, -((Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(2*b*d*x*(a + b*ArcCosh[1 + d*x^2]))) - (x*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) + (x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2])} -{1/(a + b*ArcCosh[1 + d*x^2])^3, x, 2, -((2*x^2 + d*x^4)/(4*b*x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]*(a + b*ArcCosh[1 + d*x^2])^2)) - x/(8*b^2*(a + b*ArcCosh[1 + d*x^2])) + (x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2]) - (x*Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2])} - - -{(a + b*ArcCosh[-1 + d*x^2])^4, x, 3, 384*b^4*x + (192*b^3*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + 48*b^2*x*(a + b*ArcCosh[-1 + d*x^2])^2 + (8*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^3)/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^4} -{(a + b*ArcCosh[-1 + d*x^2])^3, x, 5, 24*a*b^2*x - 48*b^3*Sqrt[1 - 2/(d*x^2)]*x + 24*b^3*x*ArcCosh[-1 + d*x^2] + (6*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^2)/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^3} -{(a + b*ArcCosh[-1 + d*x^2])^2, x, 2, 8*b^2*x + (4*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^2} -{(a + b*ArcCosh[-1 + d*x^2])^1, x, 4, a*x - 2*b*Sqrt[1 - 2/(d*x^2)]*x + b*x*ArcCosh[-1 + d*x^2]} -{1/(a + b*ArcCosh[-1 + d*x^2])^1, x, 1, -((x*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])) + (x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(Sqrt[2]*b*Sqrt[d*x^2])} -{1/(a + b*ArcCosh[-1 + d*x^2])^2, x, 1, -((Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(2*b*d*x*(a + b*ArcCosh[-1 + d*x^2]))) + (x*Cosh[a/(2*b)]*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2]) - (x*Sinh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(2*Sqrt[2]*b^2*Sqrt[d*x^2])} -{1/(a + b*ArcCosh[-1 + d*x^2])^3, x, 2, (2*x^2 - d*x^4)/(4*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d*x^2])^2) - x/(8*b^2*(a + b*ArcCosh[-1 + d*x^2])) - (x*CoshIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)]*Sinh[a/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2]) + (x*Cosh[a/(2*b)]*SinhIntegral[(a + b*ArcCosh[-1 + d*x^2])/(2*b)])/(8*Sqrt[2]*b^3*Sqrt[d*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (a+b ArcCosh[c+d x^2])^(n/2) when c^2=1*) - - -{(a + b*ArcCosh[1 + d*x^2])^(5/2), x, 2, -((5*b*(2*x^2 + d*x^4)*(a + b*ArcCosh[1 + d*x^2])^(3/2))/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2])) + x*(a + b*ArcCosh[1 + d*x^2])^(5/2) - (15*b^(5/2)*Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x) + (15*b^(5/2)*Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x) + (30*b^2*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Sinh[(1/2)*ArcCosh[1 + d*x^2]]^2)/(d*x)} -{(a + b*ArcCosh[1 + d*x^2])^(3/2), x, 2, -((3*b*(2*x^2 + d*x^4)*Sqrt[a + b*ArcCosh[1 + d*x^2]])/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2])) + x*(a + b*ArcCosh[1 + d*x^2])^(3/2) + (3*b^(3/2)*Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x) + (3*b^(3/2)*Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x)} -{(a + b*ArcCosh[1 + d*x^2])^(1/2), x, 1, -((Sqrt[b]*Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x)) + (Sqrt[b]*Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(d*x) + (2*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Sinh[(1/2)*ArcCosh[1 + d*x^2]]^2)/(d*x)} -{1/(a + b*ArcCosh[1 + d*x^2])^(1/2), x, 1, (Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(Sqrt[b]*d*x) + (Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(Sqrt[b]*d*x)} -{1/(a + b*ArcCosh[1 + d*x^2])^(3/2), x, 1, -((Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]])) + (Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(b^(3/2)*d*x) - (Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(b^(3/2)*d*x)} -{1/(a + b*ArcCosh[1 + d*x^2])^(5/2), x, 2, -((2*x^2 + d*x^4)/(3*b*x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]*(a + b*ArcCosh[1 + d*x^2])^(3/2))) - x/(3*b^2*Sqrt[a + b*ArcCosh[1 + d*x^2]]) + (Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(3*b^(5/2)*d*x) + (Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(3*b^(5/2)*d*x)} -{1/(a + b*ArcCosh[1 + d*x^2])^(7/2), x, 2, -((2*x^2 + d*x^4)/(5*b*x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]*(a + b*ArcCosh[1 + d*x^2])^(5/2))) - x/(15*b^2*(a + b*ArcCosh[1 + d*x^2])^(3/2)) - (Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(15*b^3*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]]) + (Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(15*b^(7/2)*d*x) - (Sqrt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[(1/2)*ArcCosh[1 + d*x^2]])/(15*b^(7/2)*d*x)} - - -{(a + b*ArcCosh[-1 + d*x^2])^(5/2), x, 2, (5*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^(3/2))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^(5/2) + (30*b^2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2)/(d*x) - (15*b^(5/2)*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(d*x) - (15*b^(5/2)*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x)} -{(a + b*ArcCosh[-1 + d*x^2])^(3/2), x, 2, (3*b*(2*x^2 - d*x^4)*Sqrt[a + b*ArcCosh[-1 + d*x^2]])/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^(3/2) + (3*b^(3/2)*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(d*x) - (3*b^(3/2)*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x)} -{(a + b*ArcCosh[-1 + d*x^2])^(1/2), x, 1, (2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]^2)/(d*x) - (Sqrt[b]*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(d*x) - (Sqrt[b]*Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(d*x)} -{1/(a + b*ArcCosh[-1 + d*x^2])^(1/2), x, 1, (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(Sqrt[b]*d*x) - (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Sqrt[b]*d*x)} -{1/(a + b*ArcCosh[-1 + d*x^2])^(3/2), x, 1, -((Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])) + (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(b^(3/2)*d*x) + (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(b^(3/2)*d*x)} -{1/(a + b*ArcCosh[-1 + d*x^2])^(5/2), x, 2, (2*x^2 - d*x^4)/(3*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d*x^2])^(3/2)) - x/(3*b^2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(3*b^(5/2)*d*x) - (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(3*b^(5/2)*d*x)} -{1/(a + b*ArcCosh[-1 + d*x^2])^(7/2), x, 2, (2*x^2 - d*x^4)/(5*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d*x^2])^(5/2)) - x/(15*b^2*(a + b*ArcCosh[-1 + d*x^2])^(3/2)) - (Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(15*b^3*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(15*b^(7/2)*d*x) + (Sqrt[Pi/2]*Cosh[(1/2)*ArcCosh[-1 + d*x^2]]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(15*b^(7/2)*d*x)} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcCosh[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 8, -((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(4*b*c)) - ((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 + E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (3*b*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^2*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c) + (3*b^3*PolyLog[4, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)} -{(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(3*b*c)) - ((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*Log[1 + E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b^2*PolyLog[3, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 6, -((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(2*b*c)) - ((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*Log[1 + E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/c + (b*PolyLog[2, -E^(-2*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*c)} -{1/((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCosh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse hyperbolic cosines of exponentials*) - - -(* ::Subsection::Closed:: *) -(*x^m ArcCosh[c E^(a+b x)]*) - - -{ArcCosh[c*E^(a + b*x)], x, 6, -(ArcCosh[c*E^(a + b*x)]^2/(2*b)) + (ArcCosh[c*E^(a + b*x)]*Log[1 + E^(2*ArcCosh[c*E^(a + b*x)])])/b + PolyLog[2, -E^(2*ArcCosh[c*E^(a + b*x)])]/(2*b)} - - -(* ::Title::Closed:: *) -(*Integrands involving exponentials of inverse hyperbolic cosines*) - - -(* ::Subsection::Closed:: *) -(*x^m E^(ArcCosh[a+b x]^n)*) - - -{x^3*E^ArcCosh[a + b*x], x, 5, 1/(E^(3*ArcCosh[a + b*x])*(48*b^4)) - (3*a)/(E^(2*ArcCosh[a + b*x])*(16*b^4)) + (1 + 6*a^2)/(E^ArcCosh[a + b*x]*(8*b^4)) - (a*(3 + 4*a^2)*E^(2*ArcCosh[a + b*x]))/(16*b^4) + ((1 + 6*a^2)*E^(3*ArcCosh[a + b*x]))/(24*b^4) - (3*a*E^(4*ArcCosh[a + b*x]))/(32*b^4) + E^(5*ArcCosh[a + b*x])/(80*b^4) + (a*(3 + 4*a^2)*ArcCosh[a + b*x])/(8*b^4)} -{x^2*E^ArcCosh[a + b*x], x, 5, 1/(E^(2*ArcCosh[a + b*x])*(16*b^3)) - a/(E^ArcCosh[a + b*x]*(2*b^3)) + ((1 + 4*a^2)*E^(2*ArcCosh[a + b*x]))/(16*b^3) - (a*E^(3*ArcCosh[a + b*x]))/(6*b^3) + E^(4*ArcCosh[a + b*x])/(32*b^3) - ((1 + 4*a^2)*ArcCosh[a + b*x])/(8*b^3)} -{x^1*E^ArcCosh[a + b*x], x, 5, 1/(E^ArcCosh[a + b*x]*(4*b^2)) - (a*E^(2*ArcCosh[a + b*x]))/(4*b^2) + E^(3*ArcCosh[a + b*x])/(12*b^2) + (a*ArcCosh[a + b*x])/(2*b^2)} -{x^0*E^ArcCosh[a + b*x], x, 5, E^(2*ArcCosh[a + b*x])/(4*b) - ArcCosh[a + b*x]/(2*b)} -{E^ArcCosh[a + b*x]/x^1, x, 9, b*x + Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x] + 2*a*ArcSinh[Sqrt[-1 + a + b*x]/Sqrt[2]] + 2*Sqrt[1 - a^2]*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])] + a*Log[x]} -{E^ArcCosh[a + b*x]/x^2, x, 9, -(a/x) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/x + 2*b*ArcSinh[Sqrt[-1 + a + b*x]/Sqrt[2]] - (2*a*b*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/Sqrt[1 - a^2] + b*Log[x]} -{E^ArcCosh[a + b*x]/x^3, x, 7, -(a/(2*x^2)) - b/x + (b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)*x) - (Sqrt[-1 + a + b*x]*(1 + a + b*x)^(3/2))/(2*(1 + a)*x^2) - (b^2*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a^2)^(3/2)} -{E^ArcCosh[a + b*x]/x^4, x, 8, -(a/(3*x^3)) - b/(2*x^2) + (a*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)^2*x) - (a*b*Sqrt[-1 + a + b*x]*(1 + a + b*x)^(3/2))/(2*(1 - a)*(1 + a)^2*x^2) + ((-1 + a + b*x)^(3/2)*(1 + a + b*x)^(3/2))/(3*(1 - a^2)*x^3) - (a*b^3*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a^2)^(5/2)} -{E^ArcCosh[a + b*x]/x^5, x, 10, -(a/(4*x^4)) - b/(3*x^3) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*x^4) + (a*b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(12*(1 - a^2)*x^3) + ((3 + 2*a^2)*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^2*x^2) + (a*(13 + 2*a^2)*b^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^3*x) - ((1 + 4*a^2)*b^4*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(4*(1 - a^2)^(7/2))} - - -{x^3*E^(ArcCosh[a + b*x]^2), x, 37, -((Sqrt[Pi]*Erfi[-2 + ArcCosh[a + b*x]])/(32*b^4*E^4)) - (Sqrt[Pi]*Erfi[-1 + ArcCosh[a + b*x]])/(16*b^4*E) - (3*a^2*Sqrt[Pi]*Erfi[-1 + ArcCosh[a + b*x]])/(8*b^4*E) + (Sqrt[Pi]*Erfi[1 + ArcCosh[a + b*x]])/(16*b^4*E) + (3*a^2*Sqrt[Pi]*Erfi[1 + ArcCosh[a + b*x]])/(8*b^4*E) + (Sqrt[Pi]*Erfi[2 + ArcCosh[a + b*x]])/(32*b^4*E^4) + (3*a*Sqrt[Pi]*Erfi[(1/2)*(-3 + 2*ArcCosh[a + b*x])])/(16*b^4*E^(9/4)) + (3*a*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(16*b^4*E^(1/4)) + (a^3*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(4*b^4*E^(1/4)) - (3*a*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(16*b^4*E^(1/4)) - (a^3*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(4*b^4*E^(1/4)) - (3*a*Sqrt[Pi]*Erfi[(1/2)*(3 + 2*ArcCosh[a + b*x])])/(16*b^4*E^(9/4))} -{x^2*E^(ArcCosh[a + b*x]^2), x, 27, (a*Sqrt[Pi]*Erfi[-1 + ArcCosh[a + b*x]])/(4*b^3*E) - (a*Sqrt[Pi]*Erfi[1 + ArcCosh[a + b*x]])/(4*b^3*E) - (Sqrt[Pi]*Erfi[(1/2)*(-3 + 2*ArcCosh[a + b*x])])/(16*b^3*E^(9/4)) - (Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(16*b^3*E^(1/4)) - (a^2*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(4*b^3*E^(1/4)) + (Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(16*b^3*E^(1/4)) + (a^2*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(4*b^3*E^(1/4)) + (Sqrt[Pi]*Erfi[(1/2)*(3 + 2*ArcCosh[a + b*x])])/(16*b^3*E^(9/4))} -{x^1*E^(ArcCosh[a + b*x]^2), x, 17, -((Sqrt[Pi]*Erfi[-1 + ArcCosh[a + b*x]])/(8*b^2*E)) + (Sqrt[Pi]*Erfi[1 + ArcCosh[a + b*x]])/(8*b^2*E) + (a*Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(4*b^2*E^(1/4)) - (a*Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(4*b^2*E^(1/4))} -{x^0*E^(ArcCosh[a + b*x]^2), x, 7, -((Sqrt[Pi]*Erfi[(1/2)*(-1 + 2*ArcCosh[a + b*x])])/(4*b*E^(1/4))) + (Sqrt[Pi]*Erfi[(1/2)*(1 + 2*ArcCosh[a + b*x])])/(4*b*E^(1/4))} -{E^(ArcCosh[a + b*x]^2)/x^1, x, 0, CannotIntegrate[E^ArcCosh[a + b*x]^2/x, x]} -{E^(ArcCosh[a + b*x]^2)/x^2, x, 0, CannotIntegrate[E^ArcCosh[a + b*x]^2/x^2, x]} - - -(* ::Title::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic cosines*) - - -{ArcCosh[a + b*x]/((a*d)/b + d*x), x, 7, -(ArcCosh[a + b*x]^2/(2*d)) + (ArcCosh[a + b*x]*Log[1 + E^(2*ArcCosh[a + b*x])])/d + PolyLog[2, -E^(2*ArcCosh[a + b*x])]/(2*d)} - - -{x/(Sqrt[-1 + x]*Sqrt[1 + x]*ArcCosh[x]), x, 2, CoshIntegral[ArcCosh[x]]} - - -{x^3*ArcCosh[a + b*x^4], x, 4, -((Sqrt[-1 + a + b*x^4]*Sqrt[1 + a + b*x^4])/(4*b)) + ((a + b*x^4)*ArcCosh[a + b*x^4])/(4*b)} - -{x^(n-1)*ArcCosh[a + b*x^n], x, 4, -((Sqrt[-1 + a + b*x^n]*Sqrt[1 + a + b*x^n])/(b*n)) + ((a + b*x^n)*ArcCosh[a + b*x^n])/(b*n)} - - -{ArcCosh[c/(a + b*x)], x, 5, ((a + b*x)*ArcSech[a/c + (b*x)/c])/b - (2*c*ArcTan[Sqrt[((1 - a/c)*c - b*x)/(a + c + b*x)]])/b} - - -{ArcCosh[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2], x, 2, (Sqrt[-1 + Sqrt[1 + b*x^2]]*Sqrt[1 + Sqrt[1 + b*x^2]]*ArcCosh[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)} -{1/(ArcCosh[Sqrt[1 + b*x^2]]*Sqrt[1 + b*x^2]), x, 2, (Sqrt[-1 + Sqrt[1 + b*x^2]]*Sqrt[1 + Sqrt[1 + b*x^2]]*Log[ArcCosh[Sqrt[1 + b*x^2]]])/(b*x)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m deleted file mode 100644 index 1ef2e69..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.2 (d x)^m (a+b arctanh(c x^n))^p.m +++ /dev/null @@ -1,464 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x])^p*) - - -{x^5*(a + b*ArcTanh[c*x]), x, 4, (b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) - (b*ArcTanh[c*x])/(6*c^6) + (1/6)*x^6*(a + b*ArcTanh[c*x])} -{x^4*(a + b*ArcTanh[c*x]), x, 4, (b*x^2)/(10*c^3) + (b*x^4)/(20*c) + (1/5)*x^5*(a + b*ArcTanh[c*x]) + (b*Log[1 - c^2*x^2])/(10*c^5)} -{x^3*(a + b*ArcTanh[c*x]), x, 4, (b*x)/(4*c^3) + (b*x^3)/(12*c) - (b*ArcTanh[c*x])/(4*c^4) + (1/4)*x^4*(a + b*ArcTanh[c*x])} -{x^2*(a + b*ArcTanh[c*x]), x, 4, (b*x^2)/(6*c) + (1/3)*x^3*(a + b*ArcTanh[c*x]) + (b*Log[1 - c^2*x^2])/(6*c^3)} -{x^1*(a + b*ArcTanh[c*x]), x, 3, (b*x)/(2*c) - (b*ArcTanh[c*x])/(2*c^2) + (1/2)*x^2*(a + b*ArcTanh[c*x])} -{x^0*(a + b*ArcTanh[c*x]), x, 3, a*x + b*x*ArcTanh[c*x] + (b*Log[1 - c^2*x^2])/(2*c)} -{(a + b*ArcTanh[c*x])/x^1, x, 1, a*Log[x] - (1/2)*b*PolyLog[2, (-c)*x] + (1/2)*b*PolyLog[2, c*x]} -{(a + b*ArcTanh[c*x])/x^2, x, 5, -((a + b*ArcTanh[c*x])/x) + b*c*Log[x] - (1/2)*b*c*Log[1 - c^2*x^2]} -{(a + b*ArcTanh[c*x])/x^3, x, 3, -((b*c)/(2*x)) + (1/2)*b*c^2*ArcTanh[c*x] - (a + b*ArcTanh[c*x])/(2*x^2)} -{(a + b*ArcTanh[c*x])/x^4, x, 4, -((b*c)/(6*x^2)) - (a + b*ArcTanh[c*x])/(3*x^3) + (1/3)*b*c^3*Log[x] - (1/6)*b*c^3*Log[1 - c^2*x^2]} -{(a + b*ArcTanh[c*x])/x^5, x, 4, -((b*c)/(12*x^3)) - (b*c^3)/(4*x) + (1/4)*b*c^4*ArcTanh[c*x] - (a + b*ArcTanh[c*x])/(4*x^4)} -{(a + b*ArcTanh[c*x])/x^6, x, 4, -((b*c)/(20*x^4)) - (b*c^3)/(10*x^2) - (a + b*ArcTanh[c*x])/(5*x^5) + (1/5)*b*c^5*Log[x] - (1/10)*b*c^5*Log[1 - c^2*x^2]} - - -{x^5*(a + b*ArcTanh[c*x])^2, x, 16, (a*b*x)/(3*c^5) + (4*b^2*x^2)/(45*c^4) + (b^2*x^4)/(60*c^2) + (b^2*x*ArcTanh[c*x])/(3*c^5) + (b*x^3*(a + b*ArcTanh[c*x]))/(9*c^3) + (b*x^5*(a + b*ArcTanh[c*x]))/(15*c) - (a + b*ArcTanh[c*x])^2/(6*c^6) + (1/6)*x^6*(a + b*ArcTanh[c*x])^2 + (23*b^2*Log[1 - c^2*x^2])/(90*c^6)} -{x^4*(a + b*ArcTanh[c*x])^2, x, 14, (3*b^2*x)/(10*c^4) + (b^2*x^3)/(30*c^2) - (3*b^2*ArcTanh[c*x])/(10*c^5) + (b*x^2*(a + b*ArcTanh[c*x]))/(5*c^3) + (b*x^4*(a + b*ArcTanh[c*x]))/(10*c) + (a + b*ArcTanh[c*x])^2/(5*c^5) + (1/5)*x^5*(a + b*ArcTanh[c*x])^2 - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^5) - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^5)} -{x^3*(a + b*ArcTanh[c*x])^2, x, 11, (a*b*x)/(2*c^3) + (b^2*x^2)/(12*c^2) + (b^2*x*ArcTanh[c*x])/(2*c^3) + (b*x^3*(a + b*ArcTanh[c*x]))/(6*c) - (a + b*ArcTanh[c*x])^2/(4*c^4) + (1/4)*x^4*(a + b*ArcTanh[c*x])^2 + (b^2*Log[1 - c^2*x^2])/(3*c^4)} -{x^2*(a + b*ArcTanh[c*x])^2, x, 9, (b^2*x)/(3*c^2) - (b^2*ArcTanh[c*x])/(3*c^3) + (b*x^2*(a + b*ArcTanh[c*x]))/(3*c) + (a + b*ArcTanh[c*x])^2/(3*c^3) + (1/3)*x^3*(a + b*ArcTanh[c*x])^2 - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^3) - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^3)} -{x^1*(a + b*ArcTanh[c*x])^2, x, 6, (a*b*x)/c + (b^2*x*ArcTanh[c*x])/c - (a + b*ArcTanh[c*x])^2/(2*c^2) + (1/2)*x^2*(a + b*ArcTanh[c*x])^2 + (b^2*Log[1 - c^2*x^2])/(2*c^2)} -{x^0*(a + b*ArcTanh[c*x])^2, x, 5, (a + b*ArcTanh[c*x])^2/c + x*(a + b*ArcTanh[c*x])^2 - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/c} -{(a + b*ArcTanh[c*x])^2/x^1, x, 6, 2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (1/2)*b^2*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*PolyLog[3, -1 + 2/(1 - c*x)]} -{(a + b*ArcTanh[c*x])^2/x^2, x, 4, c*(a + b*ArcTanh[c*x])^2 - (a + b*ArcTanh[c*x])^2/x + 2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c*PolyLog[2, -1 + 2/(1 + c*x)]} -{(a + b*ArcTanh[c*x])^2/x^3, x, 8, -((b*c*(a + b*ArcTanh[c*x]))/x) + (1/2)*c^2*(a + b*ArcTanh[c*x])^2 - (a + b*ArcTanh[c*x])^2/(2*x^2) + b^2*c^2*Log[x] - (1/2)*b^2*c^2*Log[1 - c^2*x^2]} -{(a + b*ArcTanh[c*x])^2/x^4, x, 8, -((b^2*c^2)/(3*x)) + (1/3)*b^2*c^3*ArcTanh[c*x] - (b*c*(a + b*ArcTanh[c*x]))/(3*x^2) + (1/3)*c^3*(a + b*ArcTanh[c*x])^2 - (a + b*ArcTanh[c*x])^2/(3*x^3) + (2/3)*b*c^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (1/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 + c*x)]} -{(a + b*ArcTanh[c*x])^2/x^5, x, 13, -((b^2*c^2)/(12*x^2)) - (b*c*(a + b*ArcTanh[c*x]))/(6*x^3) - (b*c^3*(a + b*ArcTanh[c*x]))/(2*x) + (1/4)*c^4*(a + b*ArcTanh[c*x])^2 - (a + b*ArcTanh[c*x])^2/(4*x^4) + (2/3)*b^2*c^4*Log[x] - (1/3)*b^2*c^4*Log[1 - c^2*x^2]} - - -{x^5*(a + b*ArcTanh[c*x])^3, x, 33, (19*b^3*x)/(60*c^5) + (b^3*x^3)/(60*c^3) - (19*b^3*ArcTanh[c*x])/(60*c^6) + (4*b^2*x^2*(a + b*ArcTanh[c*x]))/(15*c^4) + (b^2*x^4*(a + b*ArcTanh[c*x]))/(20*c^2) + (23*b*(a + b*ArcTanh[c*x])^2)/(30*c^6) + (b*x*(a + b*ArcTanh[c*x])^2)/(2*c^5) + (b*x^3*(a + b*ArcTanh[c*x])^2)/(6*c^3) + (b*x^5*(a + b*ArcTanh[c*x])^2)/(10*c) - (a + b*ArcTanh[c*x])^3/(6*c^6) + (1/6)*x^6*(a + b*ArcTanh[c*x])^3 - (23*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(15*c^6) - (23*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/(30*c^6)} -{x^4*(a + b*ArcTanh[c*x])^3, x, 24, (9*a*b^2*x)/(10*c^4) + (b^3*x^2)/(20*c^3) + (9*b^3*x*ArcTanh[c*x])/(10*c^4) + (b^2*x^3*(a + b*ArcTanh[c*x]))/(10*c^2) - (9*b*(a + b*ArcTanh[c*x])^2)/(20*c^5) + (3*b*x^2*(a + b*ArcTanh[c*x])^2)/(10*c^3) + (3*b*x^4*(a + b*ArcTanh[c*x])^2)/(20*c) + (a + b*ArcTanh[c*x])^3/(5*c^5) + (1/5)*x^5*(a + b*ArcTanh[c*x])^3 - (3*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/(5*c^5) + (b^3*Log[1 - c^2*x^2])/(2*c^5) - (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^5) + (3*b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(10*c^5)} -{x^3*(a + b*ArcTanh[c*x])^3, x, 18, (b^3*x)/(4*c^3) - (b^3*ArcTanh[c*x])/(4*c^4) + (b^2*x^2*(a + b*ArcTanh[c*x]))/(4*c^2) + (b*(a + b*ArcTanh[c*x])^2)/c^4 + (3*b*x*(a + b*ArcTanh[c*x])^2)/(4*c^3) + (b*x^3*(a + b*ArcTanh[c*x])^2)/(4*c) - (a + b*ArcTanh[c*x])^3/(4*c^4) + (1/4)*x^4*(a + b*ArcTanh[c*x])^3 - (2*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^4 - (b^3*PolyLog[2, 1 - 2/(1 - c*x)])/c^4} -{x^2*(a + b*ArcTanh[c*x])^3, x, 12, (a*b^2*x)/c^2 + (b^3*x*ArcTanh[c*x])/c^2 - (b*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (b*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + (a + b*ArcTanh[c*x])^3/(3*c^3) + (1/3)*x^3*(a + b*ArcTanh[c*x])^3 - (b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (b^3*Log[1 - c^2*x^2])/(2*c^3) - (b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)} -{x^1*(a + b*ArcTanh[c*x])^3, x, 8, (3*b*(a + b*ArcTanh[c*x])^2)/(2*c^2) + (3*b*x*(a + b*ArcTanh[c*x])^2)/(2*c) - (a + b*ArcTanh[c*x])^3/(2*c^2) + (1/2)*x^2*(a + b*ArcTanh[c*x])^3 - (3*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^2 - (3*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c^2)} -{x^0*(a + b*ArcTanh[c*x])^3, x, 5, (a + b*ArcTanh[c*x])^3/c + x*(a + b*ArcTanh[c*x])^3 - (3*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (3*b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c)} -{(a + b*ArcTanh[c*x])^3/x^1, x, 8, 2*(a + b*ArcTanh[c*x])^3*ArcTanh[1 - 2/(1 - c*x)] - (3/2)*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 - c*x)] + (3/2)*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, -1 + 2/(1 - c*x)] + (3/2)*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 - c*x)] - (3/2)*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, -1 + 2/(1 - c*x)] - (3/4)*b^3*PolyLog[4, 1 - 2/(1 - c*x)] + (3/4)*b^3*PolyLog[4, -1 + 2/(1 - c*x)]} -{(a + b*ArcTanh[c*x])^3/x^2, x, 5, c*(a + b*ArcTanh[c*x])^3 - (a + b*ArcTanh[c*x])^3/x + 3*b*c*(a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)] - 3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)] - (3/2)*b^3*c*PolyLog[3, -1 + 2/(1 + c*x)]} -{(a + b*ArcTanh[c*x])^3/x^3, x, 7, (3/2)*b*c^2*(a + b*ArcTanh[c*x])^2 - (3*b*c*(a + b*ArcTanh[c*x])^2)/(2*x) + (1/2)*c^2*(a + b*ArcTanh[c*x])^3 - (a + b*ArcTanh[c*x])^3/(2*x^2) + 3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (3/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 + c*x)]} -{(a + b*ArcTanh[c*x])^3/x^4, x, 14, -((b^2*c^2*(a + b*ArcTanh[c*x]))/x) + (1/2)*b*c^3*(a + b*ArcTanh[c*x])^2 - (b*c*(a + b*ArcTanh[c*x])^2)/(2*x^2) + (1/3)*c^3*(a + b*ArcTanh[c*x])^3 - (a + b*ArcTanh[c*x])^3/(3*x^3) + b^3*c^3*Log[x] - (1/2)*b^3*c^3*Log[1 - c^2*x^2] + b*c^3*(a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)] - b^2*c^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)] - (1/2)*b^3*c^3*PolyLog[3, -1 + 2/(1 + c*x)]} -{(a + b*ArcTanh[c*x])^3/x^5, x, 16, -((b^3*c^3)/(4*x)) + (1/4)*b^3*c^4*ArcTanh[c*x] - (b^2*c^2*(a + b*ArcTanh[c*x]))/(4*x^2) + b*c^4*(a + b*ArcTanh[c*x])^2 - (b*c*(a + b*ArcTanh[c*x])^2)/(4*x^3) - (3*b*c^3*(a + b*ArcTanh[c*x])^2)/(4*x) + (1/4)*c^4*(a + b*ArcTanh[c*x])^3 - (a + b*ArcTanh[c*x])^3/(4*x^4) + 2*b^2*c^4*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^3*c^4*PolyLog[2, -1 + 2/(1 + c*x)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTanh[c x])^p*) - - -{(d*x)^(5/2)*(a + b*ArcTanh[c*x]), x, 7, (4*b*d^2*Sqrt[d*x])/(7*c^3) + (4*b*(d*x)^(5/2))/(35*c) - (2*b*d^(5/2)*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/2)) + (2*(d*x)^(7/2)*(a + b*ArcTanh[c*x]))/(7*d) - (2*b*d^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/2))} -{(d*x)^(3/2)*(a + b*ArcTanh[c*x]), x, 6, (4*b*(d*x)^(3/2))/(15*c) + (2*b*d^(3/2)*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/2)) + (2*(d*x)^(5/2)*(a + b*ArcTanh[c*x]))/(5*d) - (2*b*d^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/2))} -{(d*x)^(1/2)*(a + b*ArcTanh[c*x]), x, 6, (4*b*Sqrt[d*x])/(3*c) - (2*b*Sqrt[d]*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2)) + (2*(d*x)^(3/2)*(a + b*ArcTanh[c*x]))/(3*d) - (2*b*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/2))} -{(a + b*ArcTanh[c*x])/(d*x)^(1/2), x, 5, (2*b*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[c]*Sqrt[d]) + (2*Sqrt[d*x]*(a + b*ArcTanh[c*x]))/d - (2*b*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[c]*Sqrt[d])} -{(a + b*ArcTanh[c*x])/(d*x)^(3/2), x, 5, (2*b*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*(a + b*ArcTanh[c*x]))/(d*Sqrt[d*x]) + (2*b*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/d^(3/2)} -{(a + b*ArcTanh[c*x])/(d*x)^(5/2), x, 6, -((4*b*c)/(3*d^2*Sqrt[d*x])) - (2*b*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*(a + b*ArcTanh[c*x]))/(3*d*(d*x)^(3/2)) + (2*b*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2))} -{(a + b*ArcTanh[c*x])/(d*x)^(7/2), x, 6, -((4*b*c)/(15*d^2*(d*x)^(3/2))) + (2*b*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2)) - (2*(a + b*ArcTanh[c*x]))/(5*d*(d*x)^(5/2)) + (2*b*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2))} -{(a + b*ArcTanh[c*x])/(d*x)^(9/2), x, 7, -((4*b*c)/(35*d^2*(d*x)^(5/2))) - (4*b*c^3)/(7*d^4*Sqrt[d*x]) - (2*b*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) - (2*(a + b*ArcTanh[c*x]))/(7*d*(d*x)^(7/2)) + (2*b*c^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTanh[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x])^3, x]} -{(d*x)^m*(a + b*ArcTanh[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x])^2, x]} -{(d*x)^m*(a + b*ArcTanh[c*x])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTanh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m))} -{(d*x)^m/(a + b*ArcTanh[c*x])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x]), x]} -{(d*x)^m/(a + b*ArcTanh[c*x])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x])^2, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a + b ArcTanh[c x])^p with p symbolic*) - - -{(a + b*ArcTanh[c*x])^p, x, 0, Unintegrable[(a + b*ArcTanh[c*x])^p, x]} - - -{(d*x)^m*(a + b*ArcTanh[c*x])^p, x, 0, Unintegrable[(d*x)^m*(a + b* ArcTanh[c*x])^p, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x^2])^p*) - - -{x^7*(a + b*ArcTanh[c*x^2]), x, 5, (b*x^2)/(8*c^3) + (b*x^6)/(24*c) - (b*ArcTanh[c*x^2])/(8*c^4) + (1/8)*x^8*(a + b*ArcTanh[c*x^2])} -{x^5*(a + b*ArcTanh[c*x^2]), x, 4, (b*x^4)/(12*c) + (1/6)*x^6*(a + b*ArcTanh[c*x^2]) + (b*Log[1 - c^2*x^4])/(12*c^3)} -{x^3*(a + b*ArcTanh[c*x^2]), x, 4, (b*x^2)/(4*c) - (b*ArcTanh[c*x^2])/(4*c^2) + (1/4)*x^4*(a + b*ArcTanh[c*x^2])} -{x^1*(a + b*ArcTanh[c*x^2]), x, 2, (1/2)*x^2*(a + b*ArcTanh[c*x^2]) + (b*Log[1 - c^2*x^4])/(4*c)} -{(a + b*ArcTanh[c*x^2])/x^1, x, 2, a*Log[x] - (1/4)*b*PolyLog[2, (-c)*x^2] + (1/4)*b*PolyLog[2, c*x^2]} -{(a + b*ArcTanh[c*x^2])/x^3, x, 5, -((a + b*ArcTanh[c*x^2])/(2*x^2)) + b*c*Log[x] - (1/4)*b*c*Log[1 - c^2*x^4]} -{(a + b*ArcTanh[c*x^2])/x^5, x, 4, -((b*c)/(4*x^2)) + (1/4)*b*c^2*ArcTanh[c*x^2] - (a + b*ArcTanh[c*x^2])/(4*x^4)} -{(a + b*ArcTanh[c*x^2])/x^7, x, 4, -((b*c)/(12*x^4)) - (a + b*ArcTanh[c*x^2])/(6*x^6) + (1/3)*b*c^3*Log[x] - (1/12)*b*c^3*Log[1 - c^2*x^4]} - -{x^4*(a + b*ArcTanh[c*x^2]), x, 5, (2*b*x^3)/(15*c) + (b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) - (b*ArcTanh[Sqrt[c]*x])/(5*c^(5/2)) + (1/5)*x^5*(a + b*ArcTanh[c*x^2])} -{x^2*(a + b*ArcTanh[c*x^2]), x, 5, (2*b*x)/(3*c) - (b*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) - (b*ArcTanh[Sqrt[c]*x])/(3*c^(3/2)) + (1/3)*x^3*(a + b*ArcTanh[c*x^2])} -{x^0*(a + b*ArcTanh[c*x^2]), x, 5, a*x + (b*ArcTan[Sqrt[c]*x])/Sqrt[c] - (b*ArcTanh[Sqrt[c]*x])/Sqrt[c] + b*x*ArcTanh[c*x^2]} -{(a + b*ArcTanh[c*x^2])/x^2, x, 4, b*Sqrt[c]*ArcTan[Sqrt[c]*x] + b*Sqrt[c]*ArcTanh[Sqrt[c]*x] - (a + b*ArcTanh[c*x^2])/x} -{(a + b*ArcTanh[c*x^2])/x^4, x, 5, -((2*b*c)/(3*x)) - (1/3)*b*c^(3/2)*ArcTan[Sqrt[c]*x] + (1/3)*b*c^(3/2)*ArcTanh[Sqrt[c]*x] - (a + b*ArcTanh[c*x^2])/(3*x^3)} -{(a + b*ArcTanh[c*x^2])/x^6, x, 5, -((2*b*c)/(15*x^3)) + (1/5)*b*c^(5/2)*ArcTan[Sqrt[c]*x] + (1/5)*b*c^(5/2)*ArcTanh[Sqrt[c]*x] - (a + b*ArcTanh[c*x^2])/(5*x^5)} - - -{x^7*(a + b*ArcTanh[c*x^2])^2, x, 12, (a*b*x^2)/(4*c^3) + (b^2*x^4)/(24*c^2) + (b^2*x^2*ArcTanh[c*x^2])/(4*c^3) + (b*x^6*(a + b*ArcTanh[c*x^2]))/(12*c) - (a + b*ArcTanh[c*x^2])^2/(8*c^4) + (1/8)*x^8*(a + b*ArcTanh[c*x^2])^2 + (b^2*Log[1 - c^2*x^4])/(6*c^4)} -{x^5*(a + b*ArcTanh[c*x^2])^2, x, 10, (b^2*x^2)/(6*c^2) - (b^2*ArcTanh[c*x^2])/(6*c^3) + (b*x^4*(a + b*ArcTanh[c*x^2]))/(6*c) + (a + b*ArcTanh[c*x^2])^2/(6*c^3) + (1/6)*x^6*(a + b*ArcTanh[c*x^2])^2 - (b*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/(3*c^3) - (b^2*PolyLog[2, 1 - 2/(1 - c*x^2)])/(6*c^3)} -{x^3*(a + b*ArcTanh[c*x^2])^2, x, 7, (a*b*x^2)/(2*c) + (b^2*x^2*ArcTanh[c*x^2])/(2*c) - (a + b*ArcTanh[c*x^2])^2/(4*c^2) + (1/4)*x^4*(a + b*ArcTanh[c*x^2])^2 + (b^2*Log[1 - c^2*x^4])/(4*c^2)} -{x^1*(a + b*ArcTanh[c*x^2])^2, x, 6, (a + b*ArcTanh[c*x^2])^2/(2*c) + (1/2)*x^2*(a + b*ArcTanh[c*x^2])^2 - (b*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c - (b^2*PolyLog[2, 1 - 2/(1 - c*x^2)])/(2*c)} -{(a + b*ArcTanh[c*x^2])^2/x^1, x, 7, (a + b*ArcTanh[c*x^2])^2*ArcTanh[1 - 2/(1 - c*x^2)] - (1/2)*b*(a + b*ArcTanh[c*x^2])*PolyLog[2, 1 - 2/(1 - c*x^2)] + (1/2)*b*(a + b*ArcTanh[c*x^2])*PolyLog[2, -1 + 2/(1 - c*x^2)] + (1/4)*b^2*PolyLog[3, 1 - 2/(1 - c*x^2)] - (1/4)*b^2*PolyLog[3, -1 + 2/(1 - c*x^2)]} -{(a + b*ArcTanh[c*x^2])^2/x^3, x, 5, (1/2)*c*(a + b*ArcTanh[c*x^2])^2 - (a + b*ArcTanh[c*x^2])^2/(2*x^2) + b*c*(a + b*ArcTanh[c*x^2])*Log[2 - 2/(1 + c*x^2)] - (1/2)*b^2*c*PolyLog[2, -1 + 2/(1 + c*x^2)]} -{(a + b*ArcTanh[c*x^2])^2/x^5, x, 9, -((b*c*(a + b*ArcTanh[c*x^2]))/(2*x^2)) + (1/4)*c^2*(a + b*ArcTanh[c*x^2])^2 - (a + b*ArcTanh[c*x^2])^2/(4*x^4) + b^2*c^2*Log[x] - (1/4)*b^2*c^2*Log[1 - c^2*x^4]} - -{x^4*(a + b*ArcTanh[c*x^2])^2, x, 102, (8*b^2*x)/(15*c^2) + (2*a*b*x^3)/(15*c) - (2/25)*a*b*x^5 + (2*a*b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) - (4*b^2*ArcTan[Sqrt[c]*x])/(15*c^(5/2)) + (I*b^2*ArcTan[Sqrt[c]*x]^2)/(5*c^(5/2)) - (4*b^2*ArcTanh[Sqrt[c]*x])/(15*c^(5/2)) - (b^2*ArcTanh[Sqrt[c]*x]^2)/(5*c^(5/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) - (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(5*c^(5/2)) - (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) - (b^2*x^3*Log[1 - c*x^2])/(15*c) + (1/25)*b^2*x^5*Log[1 - c*x^2] - (b^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(5*c^(5/2)) + (b*x^3*(2*a - b*Log[1 - c*x^2]))/(15*c) + (1/25)*b*x^5*(2*a - b*Log[1 - c*x^2]) - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(5*c^(5/2)) + (1/20)*x^5*(2*a - b*Log[1 - c*x^2])^2 + (2*b^2*x^3*Log[1 + c*x^2])/(15*c) + (1/5)*a*b*x^5*Log[1 + c*x^2] + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/(5*c^(5/2)) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(5*c^(5/2)) - (1/10)*b^2*x^5*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/20)*b^2*x^5*Log[1 + c*x^2]^2 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) + (I*b^2*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) - (I*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(10*c^(5/2)) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/(5*c^(5/2)) - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c^(5/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c^(5/2)) - (I*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(10*c^(5/2))} -{x^2*(a + b*ArcTanh[c*x^2])^2, x, 86, (4*a*b*x)/(3*c) - (2/9)*a*b*x^3 - (2*a*b*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) + (4*b^2*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) - (I*b^2*ArcTan[Sqrt[c]*x]^2)/(3*c^(3/2)) - (4*b^2*ArcTanh[Sqrt[c]*x])/(3*c^(3/2)) - (b^2*ArcTanh[Sqrt[c]*x]^2)/(3*c^(3/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(3*c^(3/2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(3*c^(3/2)) - (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/(3*c^(3/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(3*c^(3/2)) - (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (2*b^2*x*Log[1 - c*x^2])/(3*c) + (1/9)*b^2*x^3*Log[1 - c*x^2] + (b^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(3*c^(3/2)) + (1/9)*b*x^3*(2*a - b*Log[1 - c*x^2]) - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/2)) + (1/12)*x^3*(2*a - b*Log[1 - c*x^2])^2 + (2*b^2*x*Log[1 + c*x^2])/(3*c) + (1/3)*a*b*x^3*Log[1 + c*x^2] - (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2)) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2)) - (1/6)*b^2*x^3*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/12)*b^2*x^3*Log[1 + c*x^2]^2 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(3*c^(3/2)) - (I*b^2*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) + (I*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(6*c^(3/2)) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/(3*c^(3/2)) - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(6*c^(3/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(6*c^(3/2)) + (I*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(6*c^(3/2))} -{x^0*(a + b*ArcTanh[c*x^2])^2, x, 69, a^2*x + (2*a*b*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (2*a*b*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (b^2*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - a*b*x*Log[1 - c*x^2] - (b^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*ArcTanh[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (1/4)*b^2*x*Log[1 - c*x^2]^2 + a*b*x*Log[1 + c*x^2] + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (1/2)*b^2*x*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/4)*b^2*x*Log[1 + c*x^2]^2 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/Sqrt[c] + (I*b^2*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (I*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c]) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/Sqrt[c] + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (I*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c])} -{(a + b*ArcTanh[c*x^2])^2/x^2, x, 47, 2*a*b*Sqrt[c]*ArcTan[Sqrt[c]*x] + I*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]^2 + b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]^2 - 2*b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)] - 2*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)] + b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + 2*b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)] + 2*b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))] - b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] + b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] - b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2] + b*Sqrt[c]*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]) - (2*a - b*Log[1 - c*x^2])^2/(4*x) - (a*b*Log[1 + c*x^2])/x + b^2*Sqrt[c]*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2] + b^2*Sqrt[c]*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2] + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(2*x) - (b^2*Log[1 + c*x^2]^2)/(4*x) - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)] - (1/2)*I*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)] - b^2*Sqrt[c]*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)] + (1/2)*b^2*Sqrt[c]*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))] + (1/2)*b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] - (1/2)*I*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)]} -{(a + b*ArcTanh[c*x^2])^2/x^4, x, 64, -((2*a*b*c)/(3*x)) - (2/3)*a*b*c^(3/2)*ArcTan[Sqrt[c]*x] + (4/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x] - (1/3)*I*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]^2 + (4/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x] + (1/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]^2 - (2/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)] + (2/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)] - (1/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] - (2/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)] + (2/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)] - (1/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))] - (1/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] - (1/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + (b^2*c*Log[1 - c*x^2])/(3*x) + (1/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2] - (b*c*(2*a - b*Log[1 - c*x^2]))/(3*x) + (1/3)*b*c^(3/2)*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]) - (2*a - b*Log[1 - c*x^2])^2/(12*x^3) - (a*b*Log[1 + c*x^2])/(3*x^3) - (2*b^2*c*Log[1 + c*x^2])/(3*x) - (1/3)*b^2*c^(3/2)*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2] + (1/3)*b^2*c^(3/2)*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2] + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(6*x^3) - (b^2*Log[1 + c*x^2]^2)/(12*x^3) - (1/3)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)] - (1/3)*I*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)] + (1/6)*I*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] - (1/3)*I*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)] - (1/3)*b^2*c^(3/2)*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)] + (1/6)*b^2*c^(3/2)*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))] + (1/6)*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] + (1/6)*I*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)]} -{(a + b*ArcTanh[c*x^2])^2/x^6, x, 77, -((2*a*b*c)/(15*x^3)) + (2*a*b*c^2)/(5*x) - (8*b^2*c^2)/(15*x) + (2/5)*a*b*c^(5/2)*ArcTan[Sqrt[c]*x] - (4/15)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x] + (1/5)*I*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]^2 + (4/15)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x] + (1/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]^2 - (2/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)] - (2/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)] + (1/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + (2/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)] + (2/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)] - (1/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))] - (1/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] + (1/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + (b^2*c*Log[1 - c*x^2])/(15*x^3) - (b^2*c^2*Log[1 - c*x^2])/(5*x) - (1/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2] - (b*c*(2*a - b*Log[1 - c*x^2]))/(15*x^3) - (b*c^2*(2*a - b*Log[1 - c*x^2]))/(5*x) + (1/5)*b*c^(5/2)*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]) - (2*a - b*Log[1 - c*x^2])^2/(20*x^5) - (a*b*Log[1 + c*x^2])/(5*x^5) - (2*b^2*c*Log[1 + c*x^2])/(15*x^3) + (1/5)*b^2*c^(5/2)*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2] + (1/5)*b^2*c^(5/2)*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2] + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(10*x^5) - (b^2*Log[1 + c*x^2]^2)/(20*x^5) - (1/5)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)] + (1/5)*I*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)] - (1/10)*I*b^2*c^(5/2)*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)] + (1/5)*I*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)] - (1/5)*b^2*c^(5/2)*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)] + (1/10)*b^2*c^(5/2)*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))] + (1/10)*b^2*c^(5/2)*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))] - (1/10)*I*b^2*c^(5/2)*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)]} - - -{x^3*(a + b*ArcTanh[c*x^2])^3, x, 9, (3*b*(a + b*ArcTanh[c*x^2])^2)/(4*c^2) + (3*b*x^2*(a + b*ArcTanh[c*x^2])^2)/(4*c) - (a + b*ArcTanh[c*x^2])^3/(4*c^2) + (1/4)*x^4*(a + b*ArcTanh[c*x^2])^3 - (3*b^2*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/(2*c^2) - (3*b^3*PolyLog[2, 1 - 2/(1 - c*x^2)])/(4*c^2)} -{x^1*(a + b*ArcTanh[c*x^2])^3, x, 6, (a + b*ArcTanh[c*x^2])^3/(2*c) + (1/2)*x^2*(a + b*ArcTanh[c*x^2])^3 - (3*b*(a + b*ArcTanh[c*x^2])^2*Log[2/(1 - c*x^2)])/(2*c) - (3*b^2*(a + b*ArcTanh[c*x^2])*PolyLog[2, 1 - 2/(1 - c*x^2)])/(2*c) + (3*b^3*PolyLog[3, 1 - 2/(1 - c*x^2)])/(4*c)} -{(a + b*ArcTanh[c*x^2])^3/x^1, x, 9, (a + b*ArcTanh[c*x^2])^3*ArcTanh[1 - 2/(1 - c*x^2)] - (3/4)*b*(a + b*ArcTanh[c*x^2])^2*PolyLog[2, 1 - 2/(1 - c*x^2)] + (3/4)*b*(a + b*ArcTanh[c*x^2])^2*PolyLog[2, -1 + 2/(1 - c*x^2)] + (3/4)*b^2*(a + b*ArcTanh[c*x^2])*PolyLog[3, 1 - 2/(1 - c*x^2)] - (3/4)*b^2*(a + b*ArcTanh[c*x^2])*PolyLog[3, -1 + 2/(1 - c*x^2)] - (3/8)*b^3*PolyLog[4, 1 - 2/(1 - c*x^2)] + (3/8)*b^3*PolyLog[4, -1 + 2/(1 - c*x^2)]} -{(a + b*ArcTanh[c*x^2])^3/x^3, x, 6, (1/2)*c*(a + b*ArcTanh[c*x^2])^3 - (a + b*ArcTanh[c*x^2])^3/(2*x^2) + (3/2)*b*c*(a + b*ArcTanh[c*x^2])^2*Log[2 - 2/(1 + c*x^2)] - (3/2)*b^2*c*(a + b*ArcTanh[c*x^2])*PolyLog[2, -1 + 2/(1 + c*x^2)] - (3/4)*b^3*c*PolyLog[3, -1 + 2/(1 + c*x^2)]} -{(a + b*ArcTanh[c*x^2])^3/x^5, x, 8, (3/4)*b*c^2*(a + b*ArcTanh[c*x^2])^2 - (3*b*c*(a + b*ArcTanh[c*x^2])^2)/(4*x^2) + (1/4)*c^2*(a + b*ArcTanh[c*x^2])^3 - (a + b*ArcTanh[c*x^2])^3/(4*x^4) + (3/2)*b^2*c^2*(a + b*ArcTanh[c*x^2])*Log[2 - 2/(1 + c*x^2)] - (3/4)*b^3*c^2*PolyLog[2, -1 + 2/(1 + c*x^2)]} - -(* {x^2*(a + b*ArcTan[c*x^2])^3, x, 86, 0} -{x^0*(a + b*ArcTan[c*x^2])^3, x, 69, 0} -{(a + b*ArcTan[c*x^2])^3/x^2, x, 47, 0} -{(a + b*ArcTan[c*x^2])^3/x^4, x, 64, 0} -{(a + b*ArcTan[c*x^2])^3/x^6, x, 77, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTanh[c x^2])^p*) - - -{(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]), x, 16, (8*b*d*(d*x)^(3/2))/(21*c) + (2*b*d^(5/2)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/4)) + (Sqrt[2]*b*d^(5/2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/4)) - (Sqrt[2]*b*d^(5/2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/4)) + (2*(d*x)^(7/2)*(a + b*ArcTanh[c*x^2]))/(7*d) - (2*b*d^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*c^(7/4)) - (b*d^(5/2)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(7*Sqrt[2]*c^(7/4)) + (b*d^(5/2)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(7*Sqrt[2]*c^(7/4))} -{(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]), x, 16, (8*b*d*Sqrt[d*x])/(5*c) - (2*b*d^(3/2)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/4)) + (Sqrt[2]*b*d^(3/2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/4)) - (Sqrt[2]*b*d^(3/2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/4)) + (2*(d*x)^(5/2)*(a + b*ArcTanh[c*x^2]))/(5*d) - (2*b*d^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*c^(5/4)) + (b*d^(3/2)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(5*Sqrt[2]*c^(5/4)) - (b*d^(3/2)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(5*Sqrt[2]*c^(5/4))} -{(d*x)^(1/2)*(a + b*ArcTanh[c*x^2]), x, 15, (2*b*Sqrt[d]*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) - (Sqrt[2]*b*Sqrt[d]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) + (Sqrt[2]*b*Sqrt[d]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) + (2*(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]))/(3*d) - (2*b*Sqrt[d]*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*c^(3/4)) + (b*Sqrt[d]*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*c^(3/4)) - (b*Sqrt[d]*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*c^(3/4))} -{(a + b*ArcTanh[c*x^2])/(d*x)^(1/2), x, 15, -((2*b*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d])) - (Sqrt[2]*b*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) + (Sqrt[2]*b*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) + (2*Sqrt[d*x]*(a + b*ArcTanh[c*x^2]))/d - (2*b*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(c^(1/4)*Sqrt[d]) - (b*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d]) + (b*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*c^(1/4)*Sqrt[d])} -{(a + b*ArcTanh[c*x^2])/(d*x)^(3/2), x, 15, -((2*b*c^(1/4)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2)) - (Sqrt[2]*b*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (Sqrt[2]*b*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*(a + b*ArcTanh[c*x^2]))/(d*Sqrt[d*x]) + (2*b*c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (b*c^(1/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*d^(3/2)) - (b*c^(1/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(Sqrt[2]*d^(3/2))} -{(a + b*ArcTanh[c*x^2])/(d*x)^(5/2), x, 15, (2*b*c^(3/4)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (Sqrt[2]*b*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) + (Sqrt[2]*b*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (2*(a + b*ArcTanh[c*x^2]))/(3*d*(d*x)^(3/2)) + (2*b*c^(3/4)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(3*d^(5/2)) - (b*c^(3/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*d^(5/2)) + (b*c^(3/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(3*Sqrt[2]*d^(5/2))} -{(a + b*ArcTanh[c*x^2])/(d*x)^(7/2), x, 16, -((8*b*c)/(5*d^3*Sqrt[d*x])) - (2*b*c^(5/4)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2)) + (Sqrt[2]*b*c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2)) - (Sqrt[2]*b*c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2)) - (2*(a + b*ArcTanh[c*x^2]))/(5*d*(d*x)^(5/2)) + (2*b*c^(5/4)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(5*d^(7/2)) - (b*c^(5/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(5*Sqrt[2]*d^(7/2)) + (b*c^(5/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(5*Sqrt[2]*d^(7/2))} -{(a + b*ArcTanh[c*x^2])/(d*x)^(9/2), x, 16, -((8*b*c)/(21*d^3*(d*x)^(3/2))) + (2*b*c^(7/4)*ArcTan[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) + (Sqrt[2]*b*c^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) - (Sqrt[2]*b*c^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) - (2*(a + b*ArcTanh[c*x^2]))/(7*d*(d*x)^(7/2)) + (2*b*c^(7/4)*ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]])/(7*d^(9/2)) + (b*c^(7/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x - Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(7*Sqrt[2]*d^(9/2)) - (b*c^(7/4)*Log[Sqrt[d] + Sqrt[c]*Sqrt[d]*x + Sqrt[2]*c^(1/4)*Sqrt[d*x]])/(7*Sqrt[2]*d^(9/2))} - - -{(d*x)^(1/2)*(a + b*ArcTanh[c*x^2])^2, x, 238, (-(8/9))*a*b*x*Sqrt[d*x] - (2*Sqrt[2]*a*b*Sqrt[d*x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*c^(3/4)*Sqrt[x]) + (2*Sqrt[2]*a*b*Sqrt[d*x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*c^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/(3*(-c)^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(3*c^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x])))])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (4*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (4*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (Sqrt[2]*a*b*Sqrt[d*x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*c^(3/4)*Sqrt[x]) - (Sqrt[2]*a*b*Sqrt[d*x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*c^(3/4)*Sqrt[x]) + (4/9)*b^2*x*Sqrt[d*x]*Log[1 - c*x^2] + (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) + (4/9)*b*x*Sqrt[d*x]*(2*a - b*Log[1 - c*x^2]) + (2*b*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/4)*Sqrt[x]) - (2*b*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/4)*Sqrt[x]) + (1/6)*x*Sqrt[d*x]*(2*a - b*Log[1 - c*x^2])^2 + (2/3)*a*b*x*Sqrt[d*x]*Log[1 + c*x^2] - (2*b^2*Sqrt[d*x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*(-c)^(3/4)*Sqrt[x]) - (2*b^2*Sqrt[d*x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*c^(3/4)*Sqrt[x]) - (1/3)*b^2*x*Sqrt[d*x]*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/6)*b^2*x*Sqrt[d*x]*Log[1 + c*x^2]^2 + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*(-c)^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (2*I*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) + (2*b^2*Sqrt[d*x]*PolyLog[2, 1 - 2/(1 + c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*c^(3/4)*Sqrt[x]) + (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (b^2*Sqrt[d*x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*(-c)^(3/4)*Sqrt[x]) - (I*b^2*Sqrt[d*x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*c^(3/4)*Sqrt[x])} -{(a + b*ArcTanh[c*x^2])^2/(d*x)^(1/2), x, 241, (2*a^2*x)/Sqrt[d*x] - (2*Sqrt[2]*a*b*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + (2*Sqrt[2]*a*b*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/((-c)^(1/4)*Sqrt[d*x]) - (4*a*b*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/((-c)^(1/4)*Sqrt[d*x]) - (4*a*b*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(c^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x])))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (4*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (4*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (Sqrt[2]*a*b*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(c^(1/4)*Sqrt[d*x]) + (Sqrt[2]*a*b*Sqrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(c^(1/4)*Sqrt[d*x]) - (2*a*b*x*Log[1 - c*x^2])/Sqrt[d*x] - (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(c^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(c^(1/4)*Sqrt[d*x]) + (b^2*x*Log[1 - c*x^2]^2)/(2*Sqrt[d*x]) + (2*a*b*x*Log[1 + c*x^2])/Sqrt[d*x] + (2*b^2*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/((-c)^(1/4)*Sqrt[d*x]) - (2*b^2*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(c^(1/4)*Sqrt[d*x]) - (b^2*x*Log[1 - c*x^2]*Log[1 + c*x^2])/Sqrt[d*x] + (b^2*x*Log[1 + c*x^2]^2)/(2*Sqrt[d*x]) + (2*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (I*b^2*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 + (-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (b^2*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (I*b^2*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/((-c)^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 - c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (I*b^2*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*I*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) + (2*b^2*Sqrt[x]*PolyLog[2, 1 - 2/(1 + c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (b^2*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) + (b^2*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(c^(1/4)*Sqrt[d*x]) - (I*b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) - (b^2*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/((-c)^(1/4)*Sqrt[d*x]) + (I*b^2*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(c^(1/4)*Sqrt[d*x])} -{(a + b*ArcTanh[c*x^2])^2/(d*x)^(3/2), x, 197, -((2*Sqrt[2]*a*b*c^(1/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(d*Sqrt[d*x])) + (2*Sqrt[2]*a*b*c^(1/4)*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(d*Sqrt[d*x]) + (2*I*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) + (2*I*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(d*Sqrt[d*x]) - (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (4*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (4*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (4*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (Sqrt[2]*a*b*c^(1/4)*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(d*Sqrt[d*x]) - (Sqrt[2]*a*b*c^(1/4)*Sqrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(d*Sqrt[d*x]) - (2*b*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(d*Sqrt[d*x]) + (2*b*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(d*Sqrt[d*x]) - (2*a - b*Log[1 - c*x^2])^2/(2*d*Sqrt[d*x]) - (2*a*b*Log[1 + c*x^2])/(d*Sqrt[d*x]) + (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(d*Sqrt[d*x]) + (2*b^2*c^(1/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(d*Sqrt[d*x]) + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(d*Sqrt[d*x]) - (b^2*Log[1 + c*x^2]^2)/(2*d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (2*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + (-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (2*I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (2*I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) - (2*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x]) + (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) - (I*b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (b^2*(-c)^(1/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(d*Sqrt[d*x]) + (I*b^2*c^(1/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(d*Sqrt[d*x])} -{(a + b*ArcTanh[c*x^2])^2/(d*x)^(5/2), x, 197, -((2*Sqrt[2]*a*b*c^(3/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*d^2*Sqrt[d*x])) + (2*Sqrt[2]*a*b*c^(3/4)*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]])/(3*d^2*Sqrt[d*x]) - (2*I*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) - (2*I*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]^2)/(3*d^2*Sqrt[d*x]) - (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[-((2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 - c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (4*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (4*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[2/(1 + I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (4*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[2/(1 + c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[-((2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x])))])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[(2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[(2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (Sqrt[2]*a*b*c^(3/4)*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*d^2*Sqrt[d*x]) + (Sqrt[2]*a*b*c^(3/4)*Sqrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 - c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*d^2*Sqrt[d*x]) + (2*b*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*(2*a - b*Log[1 - c*x^2]))/(3*d^2*Sqrt[d*x]) - (2*a - b*Log[1 - c*x^2])^2/(6*d^2*x*Sqrt[d*x]) - (2*a*b*Log[1 + c*x^2])/(3*d^2*x*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTan[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*ArcTanh[(-c)^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) + (2*b^2*c^(3/4)*Sqrt[x]*ArcTanh[c^(1/4)*Sqrt[x]]*Log[1 + c*x^2])/(3*d^2*Sqrt[d*x]) + (b^2*Log[1 - c*x^2]*Log[1 + c*x^2])/(3*d^2*x*Sqrt[d*x]) - (b^2*Log[1 + c*x^2]^2)/(6*d^2*x*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + (-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*(-c)^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + (-c)^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + (-c)^(1/4)*Sqrt[x]))/(1 - I*(-c)^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 - c^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (2*I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((I*Sqrt[-Sqrt[-c]] + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) - c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/((I*(-c)^(1/4) + c^(1/4))*(1 - I*c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 + I)*(1 - c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) - (2*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - 2/(1 + c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[-c]]*Sqrt[x]))/((Sqrt[-Sqrt[-c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + Sqrt[-Sqrt[c]]*Sqrt[x]))/((Sqrt[-Sqrt[c]] + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 + (2*c^(1/4)*(1 - (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) - c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*c^(1/4)*(1 + (-c)^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + c^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (I*b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + I*c^(1/4))*(1 - I*(-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) + (b^2*(-c)^(3/4)*Sqrt[x]*PolyLog[2, 1 - (2*(-c)^(1/4)*(1 + c^(1/4)*Sqrt[x]))/(((-c)^(1/4) + c^(1/4))*(1 + (-c)^(1/4)*Sqrt[x]))])/(3*d^2*Sqrt[d*x]) - (I*b^2*c^(3/4)*Sqrt[x]*PolyLog[2, 1 - ((1 - I)*(1 + c^(1/4)*Sqrt[x]))/(1 - I*c^(1/4)*Sqrt[x])])/(3*d^2*Sqrt[d*x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^2])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTanh[c*x^2])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^2])^3, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^2])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^2])^2, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^2])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTanh[c*x^2]))/(d*(1 + m)) - (2*b*c*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, c^2*x^4])/(d^3*(1 + m)*(3 + m))} -{(d*x)^m/(a + b*ArcTanh[c*x^2])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^2]), x]} -{(d*x)^m/(a + b*ArcTanh[c*x^2])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^2])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x^3])^p*) - - -{x^11*(a + b*ArcTanh[c*x^3]), x, 5, (b*x^3)/(12*c^3) + (b*x^9)/(36*c) - (b*ArcTanh[c*x^3])/(12*c^4) + (1/12)*x^12*(a + b*ArcTanh[c*x^3])} -{x^8*(a + b*ArcTanh[c*x^3]), x, 4, (b*x^6)/(18*c) + (1/9)*x^9*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^2*x^6])/(18*c^3)} -{x^5*(a + b*ArcTanh[c*x^3]), x, 4, (b*x^3)/(6*c) - (b*ArcTanh[c*x^3])/(6*c^2) + (1/6)*x^6*(a + b*ArcTanh[c*x^3])} -{x^2*(a + b*ArcTanh[c*x^3]), x, 2, (1/3)*x^3*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^2*x^6])/(6*c)} -{(a + b*ArcTanh[c*x^3])/x^1, x, 2, a*Log[x] - (1/6)*b*PolyLog[2, (-c)*x^3] + (1/6)*b*PolyLog[2, c*x^3]} -{(a + b*ArcTanh[c*x^3])/x^4, x, 5, -((a + b*ArcTanh[c*x^3])/(3*x^3)) + b*c*Log[x] - (1/6)*b*c*Log[1 - c^2*x^6]} -{(a + b*ArcTanh[c*x^3])/x^7, x, 4, -((b*c)/(6*x^3)) + (1/6)*b*c^2*ArcTanh[c*x^3] - (a + b*ArcTanh[c*x^3])/(6*x^6)} -{(a + b*ArcTanh[c*x^3])/x^10, x, 4, -((b*c)/(18*x^6)) - (a + b*ArcTanh[c*x^3])/(9*x^9) + (1/3)*b*c^3*Log[x] - (1/18)*b*c^3*Log[1 - c^2*x^6]} - -{x^3*(a + b*ArcTanh[c*x^3]), x, 12, (3*b*x)/(4*c) + (Sqrt[3]*b*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(8*c^(4/3)) - (b*ArcTanh[c^(1/3)*x])/(4*c^(4/3)) + (1/4)*x^4*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(16*c^(4/3))} -{x^0*(a + b*ArcTanh[c*x^3]), x, 9, a*x + (Sqrt[3]*b*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) + b*x*ArcTanh[c*x^3] + (b*Log[1 - c^(2/3)*x^2])/(2*c^(1/3)) - (b*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))} -{(a + b*ArcTanh[c*x^3])/x^3, x, 11, (-(1/4))*Sqrt[3]*b*c^(2/3)*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]] + (1/4)*Sqrt[3]*b*c^(2/3)*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]] + (1/2)*b*c^(2/3)*ArcTanh[c^(1/3)*x] - (a + b*ArcTanh[c*x^3])/(2*x^2) - (1/8)*b*c^(2/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2] + (1/8)*b*c^(2/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2]} -{(a + b*ArcTanh[c*x^3])/x^6, x, 9, -((3*b*c)/(10*x^2)) - (1/10)*Sqrt[3]*b*c^(5/3)*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]] - (a + b*ArcTanh[c*x^3])/(5*x^5) - (1/10)*b*c^(5/3)*Log[1 - c^(2/3)*x^2] + (1/20)*b*c^(5/3)*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4]} - -{x^7*(a + b*ArcTanh[c*x^3]), x, 12, (3*b*x^5)/(40*c) - (Sqrt[3]*b*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(16*c^(8/3)) + (Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(16*c^(8/3)) - (b*ArcTanh[c^(1/3)*x])/(8*c^(8/3)) + (1/8)*x^8*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))} -{x^4*(a + b*ArcTanh[c*x^3]), x, 9, (3*b*x^2)/(10*c) - (Sqrt[3]*b*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(10*c^(5/3)) + (1/5)*x^5*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^(2/3)*x^2])/(10*c^(5/3)) - (b*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(20*c^(5/3))} -{x^1*(a + b*ArcTanh[c*x^3]), x, 11, -((Sqrt[3]*b*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3))) + (Sqrt[3]*b*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) - (b*ArcTanh[c^(1/3)*x])/(2*c^(2/3)) + (1/2)*x^2*(a + b*ArcTanh[c*x^3]) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))} -{(a + b*ArcTanh[c*x^3])/x^2, x, 8, (1/2)*Sqrt[3]*b*c^(1/3)*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]] - (a + b*ArcTanh[c*x^3])/x - (1/2)*b*c^(1/3)*Log[1 - c^(2/3)*x^2] + (1/4)*b*c^(1/3)*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4]} -{(a + b*ArcTanh[c*x^3])/x^5, x, 12, -((3*b*c)/(4*x)) + (1/8)*Sqrt[3]*b*c^(4/3)*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]] - (1/8)*Sqrt[3]*b*c^(4/3)*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]] + (1/4)*b*c^(4/3)*ArcTanh[c^(1/3)*x] - (a + b*ArcTanh[c*x^3])/(4*x^4) - (1/16)*b*c^(4/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2] + (1/16)*b*c^(4/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2]} - - -{x^11*(a + b*ArcTanh[c*x^3])^2, x, 12, (a*b*x^3)/(6*c^3) + (b^2*x^6)/(36*c^2) + (b^2*x^3*ArcTanh[c*x^3])/(6*c^3) + (b*x^9*(a + b*ArcTanh[c*x^3]))/(18*c) - (a + b*ArcTanh[c*x^3])^2/(12*c^4) + (1/12)*x^12*(a + b*ArcTanh[c*x^3])^2 + (b^2*Log[1 - c^2*x^6])/(9*c^4)} -{x^8*(a + b*ArcTanh[c*x^3])^2, x, 10, (b^2*x^3)/(9*c^2) - (b^2*ArcTanh[c*x^3])/(9*c^3) + (b*x^6*(a + b*ArcTanh[c*x^3]))/(9*c) + (a + b*ArcTanh[c*x^3])^2/(9*c^3) + (1/9)*x^9*(a + b*ArcTanh[c*x^3])^2 - (2*b*(a + b*ArcTanh[c*x^3])*Log[2/(1 - c*x^3)])/(9*c^3) - (b^2*PolyLog[2, 1 - 2/(1 - c*x^3)])/(9*c^3)} -{x^5*(a + b*ArcTanh[c*x^3])^2, x, 7, (a*b*x^3)/(3*c) + (b^2*x^3*ArcTanh[c*x^3])/(3*c) - (a + b*ArcTanh[c*x^3])^2/(6*c^2) + (1/6)*x^6*(a + b*ArcTanh[c*x^3])^2 + (b^2*Log[1 - c^2*x^6])/(6*c^2)} -{x^2*(a + b*ArcTanh[c*x^3])^2, x, 6, (a + b*ArcTanh[c*x^3])^2/(3*c) + (1/3)*x^3*(a + b*ArcTanh[c*x^3])^2 - (2*b*(a + b*ArcTanh[c*x^3])*Log[2/(1 - c*x^3)])/(3*c) - (b^2*PolyLog[2, 1 - 2/(1 - c*x^3)])/(3*c)} -{(a + b*ArcTanh[c*x^3])^2/x^1, x, 7, (2/3)*(a + b*ArcTanh[c*x^3])^2*ArcTanh[1 - 2/(1 - c*x^3)] - (1/3)*b*(a + b*ArcTanh[c*x^3])*PolyLog[2, 1 - 2/(1 - c*x^3)] + (1/3)*b*(a + b*ArcTanh[c*x^3])*PolyLog[2, -1 + 2/(1 - c*x^3)] + (1/6)*b^2*PolyLog[3, 1 - 2/(1 - c*x^3)] - (1/6)*b^2*PolyLog[3, -1 + 2/(1 - c*x^3)]} -{(a + b*ArcTanh[c*x^3])^2/x^4, x, 5, (1/3)*c*(a + b*ArcTanh[c*x^3])^2 - (a + b*ArcTanh[c*x^3])^2/(3*x^3) + (2/3)*b*c*(a + b*ArcTanh[c*x^3])*Log[2 - 2/(1 + c*x^3)] - (1/3)*b^2*c*PolyLog[2, -1 + 2/(1 + c*x^3)]} -{(a + b*ArcTanh[c*x^3])^2/x^7, x, 9, -((b*c*(a + b*ArcTanh[c*x^3]))/(3*x^3)) + (1/6)*c^2*(a + b*ArcTanh[c*x^3])^2 - (a + b*ArcTanh[c*x^3])^2/(6*x^6) + b^2*c^2*Log[x] - (1/6)*b^2*c^2*Log[1 - c^2*x^6]} -{(a + b*ArcTanh[c*x^3])^2/x^10, x, 9, -((b^2*c^2)/(9*x^3)) + (1/9)*b^2*c^3*ArcTanh[c*x^3] - (b*c*(a + b*ArcTanh[c*x^3]))/(9*x^6) + (1/9)*c^3*(a + b*ArcTanh[c*x^3])^2 - (a + b*ArcTanh[c*x^3])^2/(9*x^9) + (2/9)*b*c^3*(a + b*ArcTanh[c*x^3])*Log[2 - 2/(1 + c*x^3)] - (1/9)*b^2*c^3*PolyLog[2, -1 + 2/(1 + c*x^3)]} - -(* {x^3*(a + b*ArcTanh[c*x^3])^2, x, 44, 0} -{x^0*(a + b*ArcTanh[c*x^3])^2, x, 69, 0} -{(a + b*ArcTanh[c*x^3])^2/x^3, x, 24, 0} -{(a + b*ArcTanh[c*x^3])^2/x^6, x, 77, 0} - -{x^1*(a + b*ArcTanh[c*x^3])^2, x, 28, 0} -{(a + b*ArcTanh[c*x^3])^2/x^2, x, 47, 0} -{(a + b*ArcTanh[c*x^3])^2/x^5, x, 46, 0} *) - - -{x^8*(a + b*ArcTanh[c*x^3])^3, x, 13, (a*b^2*x^3)/(3*c^2) + (b^3*x^3*ArcTanh[c*x^3])/(3*c^2) - (b*(a + b*ArcTanh[c*x^3])^2)/(6*c^3) + (b*x^6*(a + b*ArcTanh[c*x^3])^2)/(6*c) + (a + b*ArcTanh[c*x^3])^3/(9*c^3) + (1/9)*x^9*(a + b*ArcTanh[c*x^3])^3 - (b*(a + b*ArcTanh[c*x^3])^2*Log[2/(1 - c*x^3)])/(3*c^3) + (b^3*Log[1 - c^2*x^6])/(6*c^3) - (b^2*(a + b*ArcTanh[c*x^3])*PolyLog[2, 1 - 2/(1 - c*x^3)])/(3*c^3) + (b^3*PolyLog[3, 1 - 2/(1 - c*x^3)])/(6*c^3)} -{x^5*(a + b*ArcTanh[c*x^3])^3, x, 9, (b*(a + b*ArcTanh[c*x^3])^2)/(2*c^2) + (b*x^3*(a + b*ArcTanh[c*x^3])^2)/(2*c) - (a + b*ArcTanh[c*x^3])^3/(6*c^2) + (1/6)*x^6*(a + b*ArcTanh[c*x^3])^3 - (b^2*(a + b*ArcTanh[c*x^3])*Log[2/(1 - c*x^3)])/c^2 - (b^3*PolyLog[2, 1 - 2/(1 - c*x^3)])/(2*c^2)} -{x^2*(a + b*ArcTanh[c*x^3])^3, x, 6, (a + b*ArcTanh[c*x^3])^3/(3*c) + (1/3)*x^3*(a + b*ArcTanh[c*x^3])^3 - (b*(a + b*ArcTanh[c*x^3])^2*Log[2/(1 - c*x^3)])/c - (b^2*(a + b*ArcTanh[c*x^3])*PolyLog[2, 1 - 2/(1 - c*x^3)])/c + (b^3*PolyLog[3, 1 - 2/(1 - c*x^3)])/(2*c)} -{(a + b*ArcTanh[c*x^3])^3/x^1, x, 9, (2/3)*(a + b*ArcTanh[c*x^3])^3*ArcTanh[1 - 2/(1 - c*x^3)] - (1/2)*b*(a + b*ArcTanh[c*x^3])^2*PolyLog[2, 1 - 2/(1 - c*x^3)] + (1/2)*b*(a + b*ArcTanh[c*x^3])^2*PolyLog[2, -1 + 2/(1 - c*x^3)] + (1/2)*b^2*(a + b*ArcTanh[c*x^3])*PolyLog[3, 1 - 2/(1 - c*x^3)] - (1/2)*b^2*(a + b*ArcTanh[c*x^3])*PolyLog[3, -1 + 2/(1 - c*x^3)] - (1/4)*b^3*PolyLog[4, 1 - 2/(1 - c*x^3)] + (1/4)*b^3*PolyLog[4, -1 + 2/(1 - c*x^3)]} -{(a + b*ArcTanh[c*x^3])^3/x^4, x, 6, (1/3)*c*(a + b*ArcTanh[c*x^3])^3 - (a + b*ArcTanh[c*x^3])^3/(3*x^3) + b*c*(a + b*ArcTanh[c*x^3])^2*Log[2 - 2/(1 + c*x^3)] - b^2*c*(a + b*ArcTanh[c*x^3])*PolyLog[2, -1 + 2/(1 + c*x^3)] - (1/2)*b^3*c*PolyLog[3, -1 + 2/(1 + c*x^3)]} -{(a + b*ArcTanh[c*x^3])^3/x^7, x, 8, (1/2)*b*c^2*(a + b*ArcTanh[c*x^3])^2 - (b*c*(a + b*ArcTanh[c*x^3])^2)/(2*x^3) + (1/6)*c^2*(a + b*ArcTanh[c*x^3])^3 - (a + b*ArcTanh[c*x^3])^3/(6*x^6) + b^2*c^2*(a + b*ArcTanh[c*x^3])*Log[2 - 2/(1 + c*x^3)] - (1/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 + c*x^3)]} - -(* {x^3*(a + b*ArcTanh[c*x^3])^3, x, 44, 0} -{x^0*(a + b*ArcTanh[c*x^3])^3, x, 69, 0} -{(a + b*ArcTanh[c*x^3])^3/x^3, x, 24, 0} -{(a + b*ArcTanh[c*x^3])^3/x^6, x, 77, 0} - -{x^1*(a + b*ArcTanh[c*x^3])^3, x, 28, 0} -{(a + b*ArcTanh[c*x^3])^3/x^2, x, 47, 0} -{(a + b*ArcTanh[c*x^3])^3/x^5, x, 46, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^3])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTanh[c*x^3])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^3])^3, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^3])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^3])^2, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^3])^1, x, 2, ((d*x)^(1 + m)*(a + b*ArcTanh[c*x^3]))/(d*(1 + m)) - (3*b*c*(d*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/6, (10 + m)/6, c^2*x^6])/(d^4*(1 + m)*(4 + m))} -{(d*x)^m/(a + b*ArcTanh[c*x^3])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^3]), x]} -{(d*x)^m/(a + b*ArcTanh[c*x^3])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^3])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c/x^1])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c/x])^p*) - - -{x^3*(a + b*ArcTanh[c/x]), x, 5, (1/4)*b*c^3*x + (1/12)*b*c*x^3 + (1/4)*x^4*(a + b*ArcTanh[c/x]) - (1/4)*b*c^4*ArcTanh[x/c]} -{x^2*(a + b*ArcTanh[c/x]), x, 5, (1/6)*b*c*x^2 + (1/3)*x^3*(a + b*ArcTanh[c/x]) + (1/6)*b*c^3*Log[c^2 - x^2]} -{x^1*(a + b*ArcTanh[c/x]), x, 4, (b*c*x)/2 + (1/2)*x^2*(a + b*ArcTanh[c/x]) - (1/2)*b*c^2*ArcTanh[x/c]} -{x^0*(a + b*ArcTanh[c/x]), x, 4, a*x + b*x*ArcTanh[c/x] + (1/2)*b*c*Log[c^2 - x^2]} -{(a + b*ArcTanh[c/x])/x^1, x, 2, a*Log[x] + (1/2)*b*PolyLog[2, -(c/x)] - (1/2)*b*PolyLog[2, c/x]} -{(a + b*ArcTanh[c/x])/x^2, x, 2, -((a + b*ArcTanh[c/x])/x) - (b*Log[1 - c^2/x^2])/(2*c)} -{(a + b*ArcTanh[c/x])/x^3, x, 4, -(b/(2*c*x)) - (a + b*ArcTanh[c/x])/(2*x^2) + (b*ArcTanh[x/c])/(2*c^2)} -{(a + b*ArcTanh[c/x])/x^4, x, 5, -(b/(6*c*x^2)) - (a + b*ArcTanh[c/x])/(3*x^3) + (b*Log[x])/(3*c^3) - (b*Log[c^2 - x^2])/(6*c^3)} - - -{x^3*(a + b*ArcTanh[c/x])^2, x, 14, (1/12)*b^2*c^2*x^2 + (1/2)*b*c^3*x*(a + b*ArcCoth[x/c]) + (1/6)*b*c*x^3*(a + b*ArcCoth[x/c]) - (1/4)*c^4*(a + b*ArcCoth[x/c])^2 + (1/4)*x^4*(a + b*ArcCoth[x/c])^2 + (1/3)*b^2*c^4*Log[1 - c^2/x^2] + (2/3)*b^2*c^4*Log[x]} -{x^2*(a + b*ArcTanh[c/x])^2, x, 9, (1/3)*b^2*c^2*x - (1/3)*b^2*c^3*ArcCoth[x/c] + (1/3)*b*c*x^2*(a + b*ArcCoth[x/c]) - (1/3)*c^3*(a + b*ArcCoth[x/c])^2 + (1/3)*x^3*(a + b*ArcCoth[x/c])^2 - (2/3)*b*c^3*(a + b*ArcCoth[x/c])*Log[2 - 2/(1 + c/x)] + (1/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 + c/x)]} -{x^1*(a + b*ArcTanh[c/x])^2, x, 9, b*c*x*(a + b*ArcCoth[x/c]) - (1/2)*c^2*(a + b*ArcCoth[x/c])^2 + (1/2)*x^2*(a + b*ArcCoth[x/c])^2 + (1/2)*b^2*c^2*Log[1 - c^2/x^2] + b^2*c^2*Log[x]} -{x^0*(a + b*ArcTanh[c/x])^2, x, 6, c*(a + b*ArcCoth[x/c])^2 + x*(a + b*ArcCoth[x/c])^2 - 2*b*c*(a + b*ArcCoth[x/c])*Log[(2*c)/(c - x)] - b^2*c*PolyLog[2, -((c + x)/(c - x))]} -{(a + b*ArcTanh[c/x])^2/x^1, x, 7, -2*(a + b*ArcCoth[x/c])^2*ArcTanh[1 - 2/(1 - c/x)] + b*(a + b*ArcCoth[x/c])*PolyLog[2, 1 - 2/(1 - c/x)] - b*(a + b*ArcCoth[x/c])*PolyLog[2, -1 + 2/(1 - c/x)] - (1/2)*b^2*PolyLog[3, 1 - 2/(1 - c/x)] + (1/2)*b^2*PolyLog[3, -1 + 2/(1 - c/x)]} -{(a + b*ArcTanh[c/x])^2/x^2, x, 6, -((a + b*ArcCoth[x/c])^2/c) - (a + b*ArcCoth[x/c])^2/x + (2*b*(a + b*ArcCoth[x/c])*Log[2/(1 - c/x)])/c + (b^2*PolyLog[2, 1 - 2/(1 - c/x)])/c} -{(a + b*ArcTanh[c/x])^2/x^3, x, 7, -((a*b)/(c*x)) - (b^2*ArcCoth[x/c])/(c*x) + (a + b*ArcCoth[x/c])^2/(2*c^2) - (a + b*ArcCoth[x/c])^2/(2*x^2) - (b^2*Log[1 - c^2/x^2])/(2*c^2)} - - -{x^3*(a + b*ArcTanh[c/x])^3, x, 17, (1/4)*b^3*c^3*x - (1/4)*b^3*c^4*ArcCoth[x/c] + (1/4)*b^2*c^2*x^2*(a + b*ArcCoth[x/c]) - b*c^4*(a + b*ArcCoth[x/c])^2 + (3/4)*b*c^3*x*(a + b*ArcCoth[x/c])^2 + (1/4)*b*c*x^3*(a + b*ArcCoth[x/c])^2 - (1/4)*c^4*(a + b*ArcCoth[x/c])^3 + (1/4)*x^4*(a + b*ArcCoth[x/c])^3 - 2*b^2*c^4*(a + b*ArcCoth[x/c])*Log[2 - 2/(1 + c/x)] + b^3*c^4*PolyLog[2, -1 + 2/(1 + c/x)]} -{x^2*(a + b*ArcTanh[c/x])^3, x, 15, b^2*c^2*x*(a + b*ArcCoth[x/c]) - (1/2)*b*c^3*(a + b*ArcCoth[x/c])^2 + (1/2)*b*c*x^2*(a + b*ArcCoth[x/c])^2 - (1/3)*c^3*(a + b*ArcCoth[x/c])^3 + (1/3)*x^3*(a + b*ArcCoth[x/c])^3 - b*c^3*(a + b*ArcCoth[x/c])^2*Log[2 - 2/(1 + c/x)] + (1/2)*b^3*c^3*Log[1 - c^2/x^2] + b^3*c^3*Log[x] + b^2*c^3*(a + b*ArcCoth[x/c])*PolyLog[2, -1 + 2/(1 + c/x)] + (1/2)*b^3*c^3*PolyLog[3, -1 + 2/(1 + c/x)]} -{x^1*(a + b*ArcTanh[c/x])^3, x, 8, (-(3/2))*b*c^2*(a + b*ArcCoth[x/c])^2 + (3/2)*b*c*x*(a + b*ArcCoth[x/c])^2 - (1/2)*c^2*(a + b*ArcCoth[x/c])^3 + (1/2)*x^2*(a + b*ArcCoth[x/c])^3 - 3*b^2*c^2*(a + b*ArcCoth[x/c])*Log[2 - 2/(1 + c/x)] + (3/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 + c/x)]} -{x^0*(a + b*ArcTanh[c/x])^3, x, 6, c*(a + b*ArcCoth[x/c])^3 + x*(a + b*ArcCoth[x/c])^3 - 3*b*c*(a + b*ArcCoth[x/c])^2*Log[(2*c)/(c - x)] - 3*b^2*c*(a + b*ArcCoth[x/c])*PolyLog[2, 1 - (2*c)/(c - x)] + (3/2)*b^3*c*PolyLog[3, 1 - (2*c)/(c - x)]} -{(a + b*ArcTanh[c/x])^3/x^1, x, 9, -2*(a + b*ArcCoth[x/c])^3*ArcTanh[1 - 2/(1 - c/x)] + (3/2)*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, 1 - 2/(1 - c/x)] - (3/2)*b*(a + b*ArcCoth[x/c])^2*PolyLog[2, -1 + 2/(1 - c/x)] - (3/2)*b^2*(a + b*ArcCoth[x/c])*PolyLog[3, 1 - 2/(1 - c/x)] + (3/2)*b^2*(a + b*ArcCoth[x/c])*PolyLog[3, -1 + 2/(1 - c/x)] + (3/4)*b^3*PolyLog[4, 1 - 2/(1 - c/x)] - (3/4)*b^3*PolyLog[4, -1 + 2/(1 - c/x)]} -{(a + b*ArcTanh[c/x])^3/x^2, x, 6, -((a + b*ArcCoth[x/c])^3/c) - (a + b*ArcCoth[x/c])^3/x + (3*b*(a + b*ArcCoth[x/c])^2*Log[2/(1 - c/x)])/c + (3*b^2*(a + b*ArcCoth[x/c])*PolyLog[2, 1 - 2/(1 - c/x)])/c - (3*b^3*PolyLog[3, 1 - 2/(1 - c/x)])/(2*c)} -{(a + b*ArcTanh[c/x])^3/x^3, x, 9, -((3*b*(a + b*ArcCoth[x/c])^2)/(2*c^2)) - (3*b*(a + b*ArcCoth[x/c])^2)/(2*c*x) + (a + b*ArcCoth[x/c])^3/(2*c^2) - (a + b*ArcCoth[x/c])^3/(2*x^2) + (3*b^2*(a + b*ArcCoth[x/c])*Log[2/(1 - c/x)])/c^2 + (3*b^3*PolyLog[2, 1 - 2/(1 - c/x)])/(2*c^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c/x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c/x^2])^p*) - - -{x^7*(a + b*ArcTanh[c/x^2]), x, 6, (1/8)*b*c^3*x^2 + (1/24)*b*c*x^6 + (1/8)*x^8*(a + b*ArcTanh[c/x^2]) - (1/8)*b*c^4*ArcTanh[x^2/c]} -{x^5*(a + b*ArcTanh[c/x^2]), x, 5, (1/12)*b*c*x^4 + (1/6)*x^6*(a + b*ArcTanh[c/x^2]) + (1/12)*b*c^3*Log[c^2 - x^4]} -{x^3*(a + b*ArcTanh[c/x^2]), x, 5, (1/4)*b*c*x^2 + (1/4)*x^4*(a + b*ArcTanh[c/x^2]) - (1/4)*b*c^2*ArcTanh[x^2/c]} -{x^1*(a + b*ArcTanh[c/x^2]), x, 3, (1/2)*x^2*(a + b*ArcTanh[c/x^2]) + (1/4)*b*c*Log[c^2 - x^4]} -{(a + b*ArcTanh[c/x^2])/x^1, x, 2, a*Log[x] + (1/4)*b*PolyLog[2, -(c/x^2)] - (1/4)*b*PolyLog[2, c/x^2]} -{(a + b*ArcTanh[c/x^2])/x^3, x, 2, -((a + b*ArcTanh[c/x^2])/(2*x^2)) - (b*Log[1 - c^2/x^4])/(4*c)} -{(a + b*ArcTanh[c/x^2])/x^5, x, 5, -(b/(4*c*x^2)) - (a + b*ArcTanh[c/x^2])/(4*x^4) + (b*ArcTanh[x^2/c])/(4*c^2)} -{(a + b*ArcTanh[c/x^2])/x^7, x, 5, -(b/(12*c*x^4)) - (a + b*ArcTanh[c/x^2])/(6*x^6) + (b*Log[x])/(3*c^3) - (b*Log[c^2 - x^4])/(12*c^3)} - -{x^4*(a + b*ArcTanh[c/x^2]), x, 6, (2/15)*b*c*x^3 + (1/5)*b*c^(5/2)*ArcTan[x/Sqrt[c]] + (1/5)*x^5*(a + b*ArcTanh[c/x^2]) - (1/5)*b*c^(5/2)*ArcTanh[x/Sqrt[c]]} -{x^2*(a + b*ArcTanh[c/x^2]), x, 6, (2*b*c*x)/3 - (1/3)*b*c^(3/2)*ArcTan[x/Sqrt[c]] + (1/3)*x^3*(a + b*ArcTanh[c/x^2]) - (1/3)*b*c^(3/2)*ArcTanh[x/Sqrt[c]]} -{x^0*(a + b*ArcTanh[c/x^2]), x, 6, a*x + b*Sqrt[c]*ArcTan[x/Sqrt[c]] + b*x*ArcTanh[c/x^2] - b*Sqrt[c]*ArcTanh[x/Sqrt[c]]} -{(a + b*ArcTanh[c/x^2])/x^2, x, 5, (b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (a + b*ArcTanh[c/x^2])/x + (b*ArcTanh[x/Sqrt[c]])/Sqrt[c]} -{(a + b*ArcTanh[c/x^2])/x^4, x, 6, -((2*b)/(3*c*x)) - (b*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) - (a + b*ArcTanh[c/x^2])/(3*x^3) + (b*ArcTanh[x/Sqrt[c]])/(3*c^(3/2))} -{(a + b*ArcTanh[c/x^2])/x^6, x, 6, -((2*b)/(15*c*x^3)) + (b*ArcTan[x/Sqrt[c]])/(5*c^(5/2)) - (a + b*ArcTanh[c/x^2])/(5*x^5) + (b*ArcTanh[x/Sqrt[c]])/(5*c^(5/2))} - - -{x^3*(a + b*ArcTanh[c/x^2])^2, x, 9, (1/2)*b*c*x^2*(a + b*ArcCoth[x^2/c]) - (1/4)*c^2*(a + b*ArcCoth[x^2/c])^2 + (1/4)*x^4*(a + b*ArcCoth[x^2/c])^2 + (1/4)*b^2*c^2*Log[1 - c^2/x^4] + b^2*c^2*Log[x]} -{x^1*(a + b*ArcTanh[c/x^2])^2, x, 5, (-(1/2))*c*(a + b*ArcCoth[x^2/c])^2 + (1/2)*x^2*(a + b*ArcCoth[x^2/c])^2 - b*c*(a + b*ArcCoth[x^2/c])*Log[2 - 2/(1 + c/x^2)] + (1/2)*b^2*c*PolyLog[2, -1 + 2/(1 + c/x^2)]} -{(a + b*ArcTanh[c/x^2])^2/x^1, x, 7, (-(a + b*ArcCoth[x^2/c])^2)*ArcTanh[1 - 2/(1 - c/x^2)] + (1/2)*b*(a + b*ArcCoth[x^2/c])*PolyLog[2, 1 - 2/(1 - c/x^2)] - (1/2)*b*(a + b*ArcCoth[x^2/c])*PolyLog[2, -1 + 2/(1 - c/x^2)] - (1/4)*b^2*PolyLog[3, 1 - 2/(1 - c/x^2)] + (1/4)*b^2*PolyLog[3, -1 + 2/(1 - c/x^2)]} -{(a + b*ArcTanh[c/x^2])^2/x^3, x, 6, -((a + b*ArcCoth[x^2/c])^2/(2*c)) - (a + b*ArcCoth[x^2/c])^2/(2*x^2) + (b*(a + b*ArcCoth[x^2/c])*Log[2/(1 - c/x^2)])/c + (b^2*PolyLog[2, 1 - 2/(1 - c/x^2)])/(2*c)} -{(a + b*ArcTanh[c/x^2])^2/x^5, x, 7, -((a*b)/(2*c*x^2)) - (b^2*ArcCoth[x^2/c])/(2*c*x^2) + (a + b*ArcCoth[x^2/c])^2/(4*c^2) - (a + b*ArcCoth[x^2/c])^2/(4*x^4) - (b^2*Log[1 - c^2/x^4])/(4*c^2)} - -{x^4*(a + b*ArcTanh[c/x^2])^2, x, 98, (8/15)*b^2*c^2*x + (2/15)*a*b*c*x^3 + (2/5)*a*b*c^(5/2)*ArcTan[x/Sqrt[c]] - (4/15)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]] - (1/5)*I*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]^2 - (4/15)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]] + (1/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]^2 + (2/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (1/15)*b^2*c*x^3*Log[1 - c/x^2] - (1/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + (1/15)*b*c*x^3*(2*a - b*Log[1 - c/x^2]) - (1/5)*b*c^(5/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]) + (1/20)*x^5*(2*a - b*Log[1 - c/x^2])^2 + (2/15)*b^2*c*x^3*Log[1 + c/x^2] + (1/5)*a*b*x^5*Log[1 + c/x^2] + (1/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2] - (1/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2] - (1/10)*b^2*x^5*Log[1 - c/x^2]*Log[1 + c/x^2] + (1/20)*b^2*x^5*Log[1 + c/x^2]^2 - (2/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)] + (1/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - (2/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + (1/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + (1/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (1/5)*b^2*c^(5/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + (2/5)*b^2*c^(5/2)*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)] + (1/5)*I*b^2*c^(5/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (1/5)*I*b^2*c^(5/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] - (1/10)*I*b^2*c^(5/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + (1/5)*b^2*c^(5/2)*PolyLog[2, -(x/Sqrt[c])] - (1/5)*I*b^2*c^(5/2)*PolyLog[2, -((I*x)/Sqrt[c])] + (1/5)*I*b^2*c^(5/2)*PolyLog[2, (I*x)/Sqrt[c]] - (1/5)*b^2*c^(5/2)*PolyLog[2, x/Sqrt[c]] + (1/5)*b^2*c^(5/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (1/5)*b^2*c^(5/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)] - (1/10)*b^2*c^(5/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] - (1/10)*b^2*c^(5/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] - (1/10)*I*b^2*c^(5/2)*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]} -{x^2*(a + b*ArcTanh[c/x^2])^2, x, 80, (4/3)*a*b*c*x - (2/3)*a*b*c^(3/2)*ArcTan[x/Sqrt[c]] + (4/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]] + (1/3)*I*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]^2 - (4/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]] + (1/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]^2 - (2/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (2/3)*b^2*c*x*Log[1 - c/x^2] + (1/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] - (1/3)*b*c^(3/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]) + (1/12)*x^3*(2*a - b*Log[1 - c/x^2])^2 + (2/3)*b^2*c*x*Log[1 + c/x^2] + (1/3)*a*b*x^3*Log[1 + c/x^2] - (1/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2] - (1/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2] - (1/6)*b^2*x^3*Log[1 - c/x^2]*Log[1 + c/x^2] + (1/12)*b^2*x^3*Log[1 + c/x^2]^2 + (2/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)] - (1/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - (2/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + (1/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + (1/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] - (1/3)*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + (2/3)*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)] - (1/3)*I*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] + (1/3)*I*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] + (1/6)*I*b^2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + (1/3)*b^2*c^(3/2)*PolyLog[2, -(x/Sqrt[c])] + (1/3)*I*b^2*c^(3/2)*PolyLog[2, -((I*x)/Sqrt[c])] - (1/3)*I*b^2*c^(3/2)*PolyLog[2, (I*x)/Sqrt[c]] - (1/3)*b^2*c^(3/2)*PolyLog[2, x/Sqrt[c]] + (1/3)*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (1/3)*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)] - (1/6)*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] - (1/6)*b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (1/6)*I*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]} -{x^0*(a + b*ArcTanh[c/x^2])^2, x, 100, a^2*x + 2*a*b*Sqrt[c]*ArcTan[x/Sqrt[c]] - 2*a*b*Sqrt[c]*ArcTanh[x/Sqrt[c]] - a*b*x*Log[1 - c/x^2] - b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + (1/4)*b^2*x*Log[1 - c/x^2]^2 + a*b*x*Log[1 + c/x^2] - b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2] - (1/2)*b^2*x*Log[1 - c/x^2]*Log[1 + c/x^2] + (1/4)*b^2*x*Log[1 + c/x^2]^2 - (1/2)*b^2*Sqrt[-c]*Log[1 + c/x^2]*Log[Sqrt[-c] - x] + (1/4)*b^2*Sqrt[-c]*Log[Sqrt[-c] - x]^2 - (1/2)*b^2*Sqrt[c]*Log[1 - c/x^2]*Log[Sqrt[c] - x] + (1/4)*b^2*Sqrt[c]*Log[Sqrt[c] - x]^2 - 2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - b^2*Sqrt[-c]*Log[Sqrt[-c] - x]*Log[x/Sqrt[-c]] - b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[x/Sqrt[c]] + (1/2)*b^2*Sqrt[-c]*Log[1 + c/x^2]*Log[Sqrt[-c] + x] - (1/2)*b^2*Sqrt[-c]*Log[(Sqrt[-c] - x)/(2*Sqrt[-c])]*Log[Sqrt[-c] + x] + b^2*Sqrt[-c]*Log[-(x/Sqrt[-c])]*Log[Sqrt[-c] + x] - (1/4)*b^2*Sqrt[-c]*Log[Sqrt[-c] + x]^2 + (1/2)*b^2*Sqrt[-c]*Log[Sqrt[-c] - x]*Log[(Sqrt[-c] + x)/(2*Sqrt[-c])] - 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (1/2)*b^2*Sqrt[c]*Log[1 - c/x^2]*Log[Sqrt[c] + x] - (1/2)*b^2*Sqrt[c]*Log[(Sqrt[c] - x)/(2*Sqrt[c])]*Log[Sqrt[c] + x] + b^2*Sqrt[c]*Log[-(x/Sqrt[c])]*Log[Sqrt[c] + x] - (1/4)*b^2*Sqrt[c]*Log[Sqrt[c] + x]^2 + (1/2)*b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[(Sqrt[c] + x)/(2*Sqrt[c])] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + I*b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (1/2)*I*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + b^2*Sqrt[c]*PolyLog[2, -(x/Sqrt[c])] - I*b^2*Sqrt[c]*PolyLog[2, -((I*x)/Sqrt[c])] + I*b^2*Sqrt[c]*PolyLog[2, (I*x)/Sqrt[c]] - b^2*Sqrt[c]*PolyLog[2, x/Sqrt[c]] - (1/2)*b^2*Sqrt[c]*PolyLog[2, (Sqrt[c] + x)/(2*Sqrt[c])] + (1/2)*b^2*Sqrt[-c]*PolyLog[2, (1/2)*(1 - x/Sqrt[-c])] - b^2*Sqrt[-c]*PolyLog[2, 1 - x/Sqrt[-c]] + b^2*Sqrt[-c]*PolyLog[2, 1 + x/Sqrt[-c]] - (1/2)*b^2*Sqrt[-c]*PolyLog[2, (c - Sqrt[-c]*x)/(2*c)] - b^2*Sqrt[c]*PolyLog[2, 1 - x/Sqrt[c]] + (1/2)*b^2*Sqrt[c]*PolyLog[2, 1/2 - x/(2*Sqrt[c])] + b^2*Sqrt[c]*PolyLog[2, 1 + x/Sqrt[c]] + b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (1/2)*b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] - (1/2)*b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] - (1/2)*I*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]} -{(a + b*ArcTanh[c/x^2])^2/x^2, x, 72, (2*a*b)/x - (2*a*b*ArcCot[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcCot[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcCoth[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcTan[x/Sqrt[c]])/Sqrt[c] - (I*b^2*ArcTan[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTanh[x/Sqrt[c]])/Sqrt[c] - (b^2*ArcTanh[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/Sqrt[c] - (b^2*Log[1 - c/x^2])/x + (b^2*ArcCot[x/Sqrt[c]]*Log[1 - c/x^2])/Sqrt[c] - (b*(2*a - b*Log[1 - c/x^2]))/x + (b*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/Sqrt[c] - (2*a - b*Log[1 - c/x^2])^2/(4*x) - (a*b*Log[1 + c/x^2])/x + (b^2*ArcCoth[x/Sqrt[c]]*Log[1 + c/x^2])/Sqrt[c] + (b^2*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/Sqrt[c] + (b^2*Log[1 - c/x^2]*Log[1 + c/x^2])/(2*x) - (b^2*Log[1 + c/x^2]^2)/(4*x) + (2*b^2*ArcCot[x/Sqrt[c]]*Log[2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (b^2*ArcCot[x/Sqrt[c]]*Log[((1 + I)*(1 - Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] + (2*b^2*ArcCoth[x/Sqrt[c]]*Log[2/(1 + Sqrt[c]/x)])/Sqrt[c] - (b^2*ArcCoth[x/Sqrt[c]]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]/x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]/x)))])/Sqrt[c] - (b^2*ArcCoth[x/Sqrt[c]]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]/x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]/x))])/Sqrt[c] - (b^2*ArcCot[x/Sqrt[c]]*Log[((1 - I)*(1 + Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (2*b^2*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/Sqrt[c] - (I*b^2*PolyLog[2, 1 - 2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] + (I*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/(2*Sqrt[c]) - (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]/x)])/Sqrt[c] + (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]/x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]/x))])/(2*Sqrt[c]) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]/x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]/x))])/(2*Sqrt[c]) + (I*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/(2*Sqrt[c]) - (I*b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)])/Sqrt[c] + (b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)])/Sqrt[c]} -{(a + b*ArcTanh[c/x^2])^2/x^4, x, 105, (2*a*b)/(9*x^3) - (2*a*b)/(3*c*x) - (2*a*b*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) + (4*b^2*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) + (I*b^2*ArcTan[x/Sqrt[c]]^2)/(3*c^(3/2)) + (4*b^2*ArcTanh[x/Sqrt[c]])/(3*c^(3/2)) - (b^2*ArcTanh[x/Sqrt[c]]^2)/(3*c^(3/2)) - (2*b^2*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/(3*c^(3/2)) - (b^2*Log[1 - c/x^2])/(9*x^3) + (b^2*Log[1 - c/x^2])/(3*c*x) + (b^2*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2])/(3*c^(3/2)) - (b*(2*a - b*Log[1 - c/x^2]))/(9*x^3) - (b*(2*a - b*Log[1 - c/x^2]))/(3*c*x) + (b*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/(3*c^(3/2)) - (2*a - b*Log[1 - c/x^2])^2/(12*x^3) - (a*b*Log[1 + c/x^2])/(3*x^3) - (2*b^2*Log[1 + c/x^2])/(3*c*x) - (b^2*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/(3*c^(3/2)) + (b^2*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2])/(3*c^(3/2)) + (b^2*Log[1 - c/x^2]*Log[1 + c/x^2])/(6*x^3) - (b^2*Log[1 + c/x^2]^2)/(12*x^3) + (2*b^2*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)])/(3*c^(3/2)) - (b^2*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/(3*c^(3/2)) + (2*b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/(3*c^(3/2)) - (b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/(3*c^(3/2)) - (b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))])/(3*c^(3/2)) - (b^2*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)])/(3*c^(3/2)) - (2*b^2*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/(3*c^(3/2)) - (I*b^2*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/(3*c^(3/2)) + (I*b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)])/(3*c^(3/2)) + (I*b^2*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/(6*c^(3/2)) - (b^2*PolyLog[2, -(x/Sqrt[c])])/(3*c^(3/2)) + (I*b^2*PolyLog[2, -((I*x)/Sqrt[c])])/(3*c^(3/2)) - (I*b^2*PolyLog[2, (I*x)/Sqrt[c]])/(3*c^(3/2)) + (b^2*PolyLog[2, x/Sqrt[c]])/(3*c^(3/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)])/(3*c^(3/2)) + (b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)])/(3*c^(3/2)) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/(6*c^(3/2)) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))])/(6*c^(3/2)) + (I*b^2*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)])/(6*c^(3/2))} -{(a + b*ArcTanh[c/x^2])^2/x^6, x, 130, (2*a*b)/(25*x^5) - (2*a*b)/(15*c*x^3) + (2*a*b)/(5*c^2*x) - (8*b^2)/(15*c^2*x) + (2*a*b*ArcTan[x/Sqrt[c]])/(5*c^(5/2)) - (4*b^2*ArcTan[x/Sqrt[c]])/(15*c^(5/2)) - (I*b^2*ArcTan[x/Sqrt[c]]^2)/(5*c^(5/2)) + (4*b^2*ArcTanh[x/Sqrt[c]])/(15*c^(5/2)) - (b^2*ArcTanh[x/Sqrt[c]]^2)/(5*c^(5/2)) + (2*b^2*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/(5*c^(5/2)) - (b^2*Log[1 - c/x^2])/(25*x^5) + (b^2*Log[1 - c/x^2])/(15*c*x^3) - (b^2*Log[1 - c/x^2])/(5*c^2*x) - (b^2*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2])/(5*c^(5/2)) - (b*(2*a - b*Log[1 - c/x^2]))/(25*x^5) - (b*(2*a - b*Log[1 - c/x^2]))/(15*c*x^3) - (b*(2*a - b*Log[1 - c/x^2]))/(5*c^2*x) + (b*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/(5*c^(5/2)) - (2*a - b*Log[1 - c/x^2])^2/(20*x^5) - (a*b*Log[1 + c/x^2])/(5*x^5) - (2*b^2*Log[1 + c/x^2])/(15*c*x^3) + (b^2*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/(5*c^(5/2)) + (b^2*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2])/(5*c^(5/2)) + (b^2*Log[1 - c/x^2]*Log[1 + c/x^2])/(10*x^5) - (b^2*Log[1 + c/x^2]^2)/(20*x^5) - (2*b^2*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)])/(5*c^(5/2)) + (b^2*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/(5*c^(5/2)) + (2*b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/(5*c^(5/2)) - (b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/(5*c^(5/2)) - (b^2*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))])/(5*c^(5/2)) + (b^2*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)])/(5*c^(5/2)) - (2*b^2*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/(5*c^(5/2)) + (I*b^2*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/(5*c^(5/2)) - (I*b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)])/(5*c^(5/2)) - (I*b^2*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/(10*c^(5/2)) - (b^2*PolyLog[2, -(x/Sqrt[c])])/(5*c^(5/2)) - (I*b^2*PolyLog[2, -((I*x)/Sqrt[c])])/(5*c^(5/2)) + (I*b^2*PolyLog[2, (I*x)/Sqrt[c]])/(5*c^(5/2)) + (b^2*PolyLog[2, x/Sqrt[c]])/(5*c^(5/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)])/(5*c^(5/2)) + (b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)])/(5*c^(5/2)) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/(10*c^(5/2)) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))])/(10*c^(5/2)) - (I*b^2*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)])/(10*c^(5/2))} - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTanh[c x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^2])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTanh[c/x^2])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c/x^2])^3, x]} -{(d*x)^m*(a + b*ArcTanh[c/x^2])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c/x^2])^2, x]} -{(d*x)^m*(a + b*ArcTanh[c/x^2])^1, x, 3, ((d*x)^(1 + m)*(a + b*ArcTanh[c/x^2]))/(d*(1 + m)) - (2*b*c*d*(d*x)^(-1 + m)*Hypergeometric2F1[1, (1 - m)/4, (5 - m)/4, c^2/x^4])/(1 - m^2)} -{(d*x)^m/(a + b*ArcTanh[c/x^2])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c/x^2]), x]} -{(d*x)^m/(a + b*ArcTanh[c/x^2])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c/x^2])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^(1/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c Sqrt[x]])^p*) - - -{x^3*(a + b*ArcTanh[c*Sqrt[x]]), x, 7, (b*Sqrt[x])/(4*c^7) + (b*x^(3/2))/(12*c^5) + (b*x^(5/2))/(20*c^3) + (b*x^(7/2))/(28*c) - (b*ArcTanh[c*Sqrt[x]])/(4*c^8) + (1/4)*x^4*(a + b*ArcTanh[c*Sqrt[x]])} -{x^2*(a + b*ArcTanh[c*Sqrt[x]]), x, 6, (b*Sqrt[x])/(3*c^5) + (b*x^(3/2))/(9*c^3) + (b*x^(5/2))/(15*c) - (b*ArcTanh[c*Sqrt[x]])/(3*c^6) + (1/3)*x^3*(a + b*ArcTanh[c*Sqrt[x]])} -{x^1*(a + b*ArcTanh[c*Sqrt[x]]), x, 5, (b*Sqrt[x])/(2*c^3) + (b*x^(3/2))/(6*c) - (b*ArcTanh[c*Sqrt[x]])/(2*c^4) + (1/2)*x^2*(a + b*ArcTanh[c*Sqrt[x]])} -{x^0*(a + b*ArcTanh[c*Sqrt[x]]), x, 5, (b*Sqrt[x])/c + a*x - (b*ArcTanh[c*Sqrt[x]])/c^2 + b*x*ArcTanh[c*Sqrt[x]]} -{(a + b*ArcTanh[c*Sqrt[x]])/x^1, x, 2, a*Log[x] - b*PolyLog[2, (-c)*Sqrt[x]] + b*PolyLog[2, c*Sqrt[x]]} -{(a + b*ArcTanh[c*Sqrt[x]])/x^2, x, 4, -((b*c)/Sqrt[x]) + b*c^2*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/x} -{(a + b*ArcTanh[c*Sqrt[x]])/x^3, x, 5, -((b*c)/(6*x^(3/2))) - (b*c^3)/(2*Sqrt[x]) + (1/2)*b*c^4*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/(2*x^2)} -{(a + b*ArcTanh[c*Sqrt[x]])/x^4, x, 6, -((b*c)/(15*x^(5/2))) - (b*c^3)/(9*x^(3/2)) - (b*c^5)/(3*Sqrt[x]) + (1/3)*b*c^6*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/(3*x^3)} - - -{x^3*(a + b*ArcTanh[c*Sqrt[x]])^2, x, 22, (a*b*Sqrt[x])/(2*c^7) + (71*b^2*x)/(420*c^6) + (3*b^2*x^2)/(70*c^4) + (b^2*x^3)/(84*c^2) + (b^2*Sqrt[x]*ArcTanh[c*Sqrt[x]])/(2*c^7) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(6*c^5) + (b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(10*c^3) + (b*x^(7/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(14*c) - (a + b*ArcTanh[c*Sqrt[x]])^2/(4*c^8) + (1/4)*x^4*(a + b*ArcTanh[c*Sqrt[x]])^2 + (44*b^2*Log[1 - c^2*x])/(105*c^8)} -{x^2*(a + b*ArcTanh[c*Sqrt[x]])^2, x, 17, (2*a*b*Sqrt[x])/(3*c^5) + (8*b^2*x)/(45*c^4) + (b^2*x^2)/(30*c^2) + (2*b^2*Sqrt[x]*ArcTanh[c*Sqrt[x]])/(3*c^5) + (2*b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(9*c^3) + (2*b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(15*c) - (a + b*ArcTanh[c*Sqrt[x]])^2/(3*c^6) + (1/3)*x^3*(a + b*ArcTanh[c*Sqrt[x]])^2 + (23*b^2*Log[1 - c^2*x])/(45*c^6)} -{x^1*(a + b*ArcTanh[c*Sqrt[x]])^2, x, 12, (a*b*Sqrt[x])/c^3 + (b^2*x)/(6*c^2) + (b^2*Sqrt[x]*ArcTanh[c*Sqrt[x]])/c^3 + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]]))/(3*c) - (a + b*ArcTanh[c*Sqrt[x]])^2/(2*c^4) + (1/2)*x^2*(a + b*ArcTanh[c*Sqrt[x]])^2 + (2*b^2*Log[1 - c^2*x])/(3*c^4)} -{x^0*(a + b*ArcTanh[c*Sqrt[x]])^2, x, 7, (2*a*b*Sqrt[x])/c + (2*b^2*Sqrt[x]*ArcTanh[c*Sqrt[x]])/c - (a + b*ArcTanh[c*Sqrt[x]])^2/c^2 + x*(a + b*ArcTanh[c*Sqrt[x]])^2 + (b^2*Log[1 - c^2*x])/c^2} -{(a + b*ArcTanh[c*Sqrt[x]])^2/x^1, x, 7, 4*ArcTanh[1 - 2/(1 - c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^2 - 2*b*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])] + 2*b*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[2, -1 + 2/(1 - c*Sqrt[x])] + b^2*PolyLog[3, 1 - 2/(1 - c*Sqrt[x])] - b^2*PolyLog[3, -1 + 2/(1 - c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])^2/x^2, x, 9, -((2*b*c*(a + b*ArcTanh[c*Sqrt[x]]))/Sqrt[x]) + c^2*(a + b*ArcTanh[c*Sqrt[x]])^2 - (a + b*ArcTanh[c*Sqrt[x]])^2/x + b^2*c^2*Log[x] - b^2*c^2*Log[1 - c^2*x]} -{(a + b*ArcTanh[c*Sqrt[x]])^2/x^3, x, 14, -((b^2*c^2)/(6*x)) - (b*c*(a + b*ArcTanh[c*Sqrt[x]]))/(3*x^(3/2)) - (b*c^3*(a + b*ArcTanh[c*Sqrt[x]]))/Sqrt[x] + (1/2)*c^4*(a + b*ArcTanh[c*Sqrt[x]])^2 - (a + b*ArcTanh[c*Sqrt[x]])^2/(2*x^2) + (2/3)*b^2*c^4*Log[x] - (2/3)*b^2*c^4*Log[1 - c^2*x]} - - -{x^3*(a + b*ArcTanh[c*Sqrt[x]])^3, x, 54, (47*b^3*Sqrt[x])/(70*c^7) + (23*b^3*x^(3/2))/(420*c^5) + (b^3*x^(5/2))/(140*c^3) - (47*b^3*ArcTanh[c*Sqrt[x]])/(70*c^8) + (71*b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(140*c^6) + (9*b^2*x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(70*c^4) + (b^2*x^3*(a + b*ArcTanh[c*Sqrt[x]]))/(28*c^2) + (44*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/(35*c^8) + (3*b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^7) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^5) + (3*b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(20*c^3) + (3*b*x^(7/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(28*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(4*c^8) + (1/4)*x^4*(a + b*ArcTanh[c*Sqrt[x]])^3 - (88*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/(35*c^8) - (44*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(35*c^8)} -{x^2*(a + b*ArcTanh[c*Sqrt[x]])^3, x, 34, (19*b^3*Sqrt[x])/(30*c^5) + (b^3*x^(3/2))/(30*c^3) - (19*b^3*ArcTanh[c*Sqrt[x]])/(30*c^6) + (8*b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(15*c^4) + (b^2*x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(10*c^2) + (23*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/(15*c^6) + (b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/c^5 + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(3*c^3) + (b*x^(5/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(5*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(3*c^6) + (1/3)*x^3*(a + b*ArcTanh[c*Sqrt[x]])^3 - (46*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/(15*c^6) - (23*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(15*c^6)} -{x^1*(a + b*ArcTanh[c*Sqrt[x]])^3, x, 19, (b^3*Sqrt[x])/(2*c^3) - (b^3*ArcTanh[c*Sqrt[x]])/(2*c^4) + (b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(2*c^2) + (2*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/c^4 + (3*b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*c^3) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(2*c^4) + (1/2)*x^2*(a + b*ArcTanh[c*Sqrt[x]])^3 - (4*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^4 - (2*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^4} -{x^0*(a + b*ArcTanh[c*Sqrt[x]])^3, x, 9, (3*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/c^2 + (3*b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/c - (a + b*ArcTanh[c*Sqrt[x]])^3/c^2 + x*(a + b*ArcTanh[c*Sqrt[x]])^3 - (6*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^2 - (3*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^2} -{(a + b*ArcTanh[c*Sqrt[x]])^3/x^1, x, 9, 4*ArcTanh[1 - 2/(1 - c*Sqrt[x])]*(a + b*ArcTanh[c*Sqrt[x]])^3 - 3*b*(a + b*ArcTanh[c*Sqrt[x]])^2*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])] + 3*b*(a + b*ArcTanh[c*Sqrt[x]])^2*PolyLog[2, -1 + 2/(1 - c*Sqrt[x])] + 3*b^2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[3, 1 - 2/(1 - c*Sqrt[x])] - 3*b^2*(a + b*ArcTanh[c*Sqrt[x]])*PolyLog[3, -1 + 2/(1 - c*Sqrt[x])] - (3/2)*b^3*PolyLog[4, 1 - 2/(1 - c*Sqrt[x])] + (3/2)*b^3*PolyLog[4, -1 + 2/(1 - c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])^3/x^2, x, 8, 3*b*c^2*(a + b*ArcTanh[c*Sqrt[x]])^2 - (3*b*c*(a + b*ArcTanh[c*Sqrt[x]])^2)/Sqrt[x] + c^2*(a + b*ArcTanh[c*Sqrt[x]])^3 - (a + b*ArcTanh[c*Sqrt[x]])^3/x + 6*b^2*c^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - 3*b^3*c^2*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])^3/x^3, x, 17, -((b^3*c^3)/(2*Sqrt[x])) + (1/2)*b^3*c^4*ArcTanh[c*Sqrt[x]] - (b^2*c^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*x) + 2*b*c^4*(a + b*ArcTanh[c*Sqrt[x]])^2 - (b*c*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*x^(3/2)) - (3*b*c^3*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*Sqrt[x]) + (1/2)*c^4*(a + b*ArcTanh[c*Sqrt[x]])^3 - (a + b*ArcTanh[c*Sqrt[x]])^3/(2*x^2) + 4*b^2*c^4*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - 2*b^3*c^4*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTanh[c Sqrt[x]])^p*) - - -{x^(3/2)*ArcTanh[Sqrt[x]], x, 3, x/5 + x^2/10 + (2/5)*x^(5/2)*ArcTanh[Sqrt[x]] + (1/5)*Log[1 - x]} -{Sqrt[x]*ArcTanh[Sqrt[x]], x, 3, x/3 + (2/3)*x^(3/2)*ArcTanh[Sqrt[x]] + (1/3)*Log[1 - x]} -{ArcTanh[Sqrt[x]]/Sqrt[x], x, 2, 2*Sqrt[x]*ArcTanh[Sqrt[x]] + Log[1 - x]} -{ArcTanh[Sqrt[x]]/x^(3/2), x, 4, -((2*ArcTanh[Sqrt[x]])/Sqrt[x]) - Log[1 - x] + Log[x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^(3/2)])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x^(3/2)])^p*) - - -{x^3*(a + b*ArcTanh[c*x^(3/2)]), x, 13, (3*b*x^(5/2))/(20*c) - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(8*c^(8/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/(4*c^(8/3)) + (1/4)*x^4*(a + b*ArcTanh[c*x^(3/2)]) + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(16*c^(8/3))} -{x^2*(a + b*ArcTanh[c*x^(3/2)]), x, 5, (b*x^(3/2))/(3*c) - (b*ArcTanh[c*x^(3/2)])/(3*c^2) + (1/3)*x^3*(a + b*ArcTanh[c*x^(3/2)])} -{x^1*(a + b*ArcTanh[c*x^(3/2)]), x, 13, (3*b*Sqrt[x])/(2*c) + (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(4*c^(4/3)) - (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(4*c^(4/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/(2*c^(4/3)) + (1/2)*x^2*(a + b*ArcTanh[c*x^(3/2)]) + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(8*c^(4/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(8*c^(4/3))} -{x^0*(a + b*ArcTanh[c*x^(3/2)]), x, 13, a*x - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) - (b*ArcTanh[c^(1/3)*Sqrt[x]])/c^(2/3) + b*x*ArcTanh[c*x^(3/2)] + (b*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(2/3))} -{(a + b*ArcTanh[c*x^(3/2)])/x^1, x, 2, a*Log[x] - (1/3)*b*PolyLog[2, (-c)*x^(3/2)] + (1/3)*b*PolyLog[2, c*x^(3/2)]} -{(a + b*ArcTanh[c*x^(3/2)])/x^2, x, 12, (-(1/2))*Sqrt[3]*b*c^(2/3)*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]] + (1/2)*Sqrt[3]*b*c^(2/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]] + b*c^(2/3)*ArcTanh[c^(1/3)*Sqrt[x]] - (a + b*ArcTanh[c*x^(3/2)])/x - (1/4)*b*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x] + (1/4)*b*c^(2/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]} -{(a + b*ArcTanh[c*x^(3/2)])/x^3, x, 13, -((3*b*c)/(2*Sqrt[x])) + (1/4)*Sqrt[3]*b*c^(4/3)*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]] - (1/4)*Sqrt[3]*b*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]] + (1/2)*b*c^(4/3)*ArcTanh[c^(1/3)*Sqrt[x]] - (a + b*ArcTanh[c*x^(3/2)])/(2*x^2) - (1/8)*b*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x] + (1/8)*b*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]} -{(a + b*ArcTanh[c*x^(3/2)])/x^4, x, 5, -((b*c)/(3*x^(3/2))) + (1/3)*b*c^2*ArcTanh[c*x^(3/2)] - (a + b*ArcTanh[c*x^(3/2)])/(3*x^3)} - - -{x^2*(a + b*ArcTanh[c*x^(3/2)])^2, x, 7, (2*a*b*x^(3/2))/(3*c) + (2*b^2*x^(3/2)*ArcTanh[c*x^(3/2)])/(3*c) - (a + b*ArcTanh[c*x^(3/2)])^2/(3*c^2) + (1/3)*x^3*(a + b*ArcTanh[c*x^(3/2)])^2 + (b^2*Log[1 - c^2*x^3])/(3*c^2)} -{(a + b*ArcTanh[c*x^(3/2)])^2/x^1, x, 7, (4/3)*(a + b*ArcTanh[c*x^(3/2)])^2*ArcTanh[1 - 2/(1 - c*x^(3/2))] - (2/3)*b*(a + b*ArcTanh[c*x^(3/2)])*PolyLog[2, 1 - 2/(1 - c*x^(3/2))] + (2/3)*b*(a + b*ArcTanh[c*x^(3/2)])*PolyLog[2, -1 + 2/(1 - c*x^(3/2))] + (1/3)*b^2*PolyLog[3, 1 - 2/(1 - c*x^(3/2))] - (1/3)*b^2*PolyLog[3, -1 + 2/(1 - c*x^(3/2))]} -{(a + b*ArcTanh[c*x^(3/2)])^2/x^4, x, 9, -((2*b*c*(a + b*ArcTanh[c*x^(3/2)]))/(3*x^(3/2))) + (1/3)*c^2*(a + b*ArcTanh[c*x^(3/2)])^2 - (a + b*ArcTanh[c*x^(3/2)])^2/(3*x^3) + b^2*c^2*Log[x] - (1/3)*b^2*c^2*Log[1 - c^2*x^3]} -(* -{x^3*ArcTanh[c*x^(3/2)]^2, x, 272, (9*x)/(20*c^2) + (3*Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(20*c^(8/3)) + (3*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(20*c^(8/3)) + (3*Log[1 - c^(1/3)*Sqrt[x]])/(20*c^(8/3)) + Log[1 - c^(1/3)*Sqrt[x]]^2/(16*c^(8/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(8*c^(8/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(8*c^(8/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])])/(8*c^(8/3)) + (3*Log[1 + c^(1/3)*Sqrt[x]])/(20*c^(8/3)) - (Log[(1/2)*(1 - c^(1/3)*Sqrt[x])]*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + (Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - (Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]*Log[1 + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + Log[1 + c^(1/3)*Sqrt[x]]^2/(16*c^(8/3)) - (Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(8*c^(8/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) + (Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(8*c^(8/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(8*c^(8/3)) + ((-1)^(1/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(1/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(16*c^(8/3)) + ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) - ((-1)^(1/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(1/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(16*c^(8/3)) + ((-1)^(2/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(2/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(2/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(16*c^(8/3)) - ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) - ((-1)^(2/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(2/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(2/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - ((-1)^(2/3)*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(2/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(16*c^(8/3)) - ((-1)^(2/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))])/(8*c^(8/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(2/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - (3*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(40*c^(8/3)) - (3*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(40*c^(8/3)) - (3*x^(5/2)*Log[1 - c*x^(3/2)])/(20*c) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) + (Log[1 + c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) + ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) - ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) + ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(8*c^(8/3)) + (1/16)*x^4*Log[1 - c*x^(3/2)]^2 + (3*x^(5/2)*Log[1 + c*x^(3/2)])/(20*c) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) - (Log[1 + c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) + ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) - ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) - ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(8*c^(8/3)) - (1/8)*x^4*Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)] + (1/16)*x^4*Log[1 + c*x^(3/2)]^2 - PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])]/(8*c^(8/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(8*c^(8/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(8*c^(8/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(8*c^(8/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]])/(8*c^(8/3)) - PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])]/(8*c^(8/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(8*c^(8/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(8*c^(8/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(8*c^(8/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]])/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) + ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(8*c^(8/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(8*c^(8/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(8*c^(8/3))} -{x^0*ArcTanh[c*x^(3/2)]^2, x, 200, Log[1 - c^(1/3)*Sqrt[x]]^2/(4*c^(2/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(2*c^(2/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(2*c^(2/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])])/(2*c^(2/3)) - (Log[(1/2)*(1 - c^(1/3)*Sqrt[x])]*Log[1 + c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + (Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - (Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]*Log[1 + c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + Log[1 + c^(1/3)*Sqrt[x]]^2/(4*c^(2/3)) - (Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(2*c^(2/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) + (Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(2*c^(2/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(2*c^(2/3)) + ((-1)^(1/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(1/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(4*c^(2/3)) + ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) - ((-1)^(1/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(1/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(4*c^(2/3)) + ((-1)^(2/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(2/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(2/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(4*c^(2/3)) - ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) - ((-1)^(2/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(2/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(2/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - ((-1)^(2/3)*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(2/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(4*c^(2/3)) - ((-1)^(2/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))])/(2*c^(2/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(2/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) + (Log[1 + c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) - ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) + ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) - ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) + ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(2*c^(2/3)) + (1/4)*x*Log[1 - c*x^(3/2)]^2 + (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) - (Log[1 + c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) + ((-1)^(1/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) - ((-1)^(1/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) + ((-1)^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) - ((-1)^(2/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(2*c^(2/3)) - (1/2)*x*Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)] + (1/4)*x*Log[1 + c*x^(3/2)]^2 - PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])]/(2*c^(2/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(2*c^(2/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(2*c^(2/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(2*c^(2/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]])/(2*c^(2/3)) - PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])]/(2*c^(2/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(2*c^(2/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(2*c^(2/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(2*c^(2/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]])/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) + ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(2*c^(2/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(2*c^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(2*c^(2/3))} -{ArcTanh[c*x^(3/2)]^2/x^3, x, 196, (-(3/2))*Sqrt[3]*c^(4/3)*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]] - (3/2)*Sqrt[3]*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]] - (3/2)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]] - (1/8)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]^2 + (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))] - (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))] + (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])] - (3/2)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]] + (1/4)*c^(4/3)*Log[(1/2)*(1 - c^(1/3)*Sqrt[x])]*Log[1 + c^(1/3)*Sqrt[x]] - (1/4)*c^(4/3)*Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + c^(1/3)*Sqrt[x]] + (1/4)*c^(4/3)*Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]*Log[1 + c^(1/3)*Sqrt[x]] - (1/8)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]]^2 + (1/4)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))] - (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/4)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))] + (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/8)*(-1)^(1/3)*c^(4/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2 - (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/8)*(-1)^(1/3)*c^(4/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2 - (1/4)*(-1)^(2/3)*c^(4/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/8)*(-1)^(2/3)*c^(4/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2 + (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(2/3)*c^(4/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(2/3)*c^(4/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/8)*(-1)^(2/3)*c^(4/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2 + (1/4)*(-1)^(2/3)*c^(4/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))] - (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (3/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x] + (3/4)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x] + (3*c*Log[1 - c*x^(3/2)])/(2*Sqrt[x]) + (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - (1/4)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - Log[1 - c*x^(3/2)]^2/(8*x^2) - (3*c*Log[1 + c*x^(3/2)])/(2*Sqrt[x]) - (1/4)*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (1/4)*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] - (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 - (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (1/4)*(-1)^(1/3)*c^(4/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] - (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (1/4)*(-1)^(2/3)*c^(4/3)*Log[1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)])/(4*x^2) - Log[1 + c*x^(3/2)]^2/(8*x^2) + (1/4)*c^(4/3)*PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])] + (1/4)*c^(4/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/4)*c^(4/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/4)*c^(4/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/4)*c^(4/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]] + (1/4)*c^(4/3)*PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])] + (1/4)*c^(4/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/4)*c^(4/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/4)*c^(4/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/4)*c^(4/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] + (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])] - (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] + (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (1/4)*(-1)^(1/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])] - (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] + (1/4)*(-1)^(2/3)*c^(4/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]} - -{x^4*ArcTanh[c*x^(3/2)]^2, x, 308, (9*x^2)/(70*c^2) - (24*Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(35*c^(10/3)) - (24*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(35*c^(10/3)) + Log[-1 - c^(1/3)*Sqrt[x]]^2/(20*c^(10/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - c^(1/3)*Sqrt[x])])/(10*c^(10/3)) + (24*Log[1 - c^(1/3)*Sqrt[x]])/(35*c^(10/3)) + Log[1 - c^(1/3)*Sqrt[x]]^2/(20*c^(10/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(10*c^(10/3)) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(10*c^(10/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(10*c^(10/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])])/(10*c^(10/3)) + (24*Log[1 + c^(1/3)*Sqrt[x]])/(35*c^(10/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(10*c^(10/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(10*c^(10/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(10*c^(10/3)) - ((-1)^(2/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(2/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(20*c^(10/3)) - ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) + ((-1)^(2/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(2/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(20*c^(10/3)) + ((-1)^(1/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(1/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(1/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(20*c^(10/3)) + ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) - ((-1)^(1/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(1/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) + ((-1)^(1/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(20*c^(10/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) + ((-1)^(1/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))])/(10*c^(10/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(1/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) - (12*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(35*c^(10/3)) - (12*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(35*c^(10/3)) - (3*Sqrt[x]*Log[1 - c*x^(3/2)])/(5*c^3) - (3*x^(7/2)*Log[1 - c*x^(3/2)])/(35*c) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) - ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) + ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(10*c^(10/3)) + (1/20)*x^5*Log[1 - c*x^(3/2)]^2 + (3*Sqrt[x]*Log[1 + c*x^(3/2)])/(5*c^3) + (3*x^(7/2)*Log[1 + c*x^(3/2)])/(35*c) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) - ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) + ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) + ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(10*c^(10/3)) - (1/10)*x^5*Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)] + (1/20)*x^5*Log[1 + c*x^(3/2)]^2 - PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])]/(10*c^(10/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(10*c^(10/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(10*c^(10/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(10*c^(10/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) - c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])]/(10*c^(10/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(10*c^(10/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(10*c^(10/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(10*c^(10/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(10*c^(10/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(10*c^(10/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(10*c^(10/3))} -{x^1*ArcTanh[c*x^(3/2)]^2, x, 236, -((3*Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(4/3))) - (3*Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(4/3)) + Log[-1 - c^(1/3)*Sqrt[x]]^2/(8*c^(4/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - c^(1/3)*Sqrt[x])])/(4*c^(4/3)) + (3*Log[1 - c^(1/3)*Sqrt[x]])/(2*c^(4/3)) + Log[1 - c^(1/3)*Sqrt[x]]^2/(8*c^(4/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(4*c^(4/3)) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(4*c^(4/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(4*c^(4/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])])/(4*c^(4/3)) + (3*Log[1 + c^(1/3)*Sqrt[x]])/(2*c^(4/3)) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))])/(4*c^(4/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))])/(4*c^(4/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(4*c^(4/3)) - ((-1)^(2/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(2/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(8*c^(4/3)) - ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) + ((-1)^(2/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(2/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2)/(8*c^(4/3)) + ((-1)^(1/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(1/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(1/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(8*c^(4/3)) + ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) - ((-1)^(1/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(1/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) + ((-1)^(1/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2)/(8*c^(4/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) + ((-1)^(1/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))])/(4*c^(4/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(1/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) - (3*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(4/3)) - (3*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x])/(4*c^(4/3)) - (3*Sqrt[x]*Log[1 - c*x^(3/2)])/(2*c) + (Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) - (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) + ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) - ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) - ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) + ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)])/(4*c^(4/3)) + (1/8)*x^2*Log[1 - c*x^(3/2)]^2 + (3*Sqrt[x]*Log[1 + c*x^(3/2)])/(2*c) - (Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) + (Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) - ((-1)^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) + ((-1)^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) + ((-1)^(1/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) - ((-1)^(1/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)])/(4*c^(4/3)) - (1/4)*x^2*Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)] + (1/8)*x^2*Log[1 + c*x^(3/2)]^2 - PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])]/(4*c^(4/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(4*c^(4/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(4*c^(4/3)) + PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(4*c^(4/3)) - PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) - c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])]/(4*c^(4/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))]/(4*c^(4/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]/(4*c^(4/3)) + PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))]/(4*c^(4/3)) - PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))]/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) - ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) - ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])])/(4*c^(4/3)) - ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))])/(4*c^(4/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))])/(4*c^(4/3))} -{ArcTanh[c*x^(3/2)]^2/x^2, x, 160, (-(1/4))*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]^2 + (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - c^(1/3)*Sqrt[x])] - (1/4)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]^2 + (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))] - (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))] + (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] + (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + c^(1/3)*Sqrt[x])] + (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3)))] - (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3)))] + (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[((-1)^(2/3) + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[-(((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[(-1)^(1/3) + c^(1/3)*Sqrt[x]]*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(2/3)*c^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2 + (1/2)*(-1)^(2/3)*c^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[((-1)^(1/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(1/3))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[(-1)^(1/3) - c^(1/3)*Sqrt[x]]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[-(((-1)^(1/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(1/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[-(((-1)^(1/3)*((-1)^(1/3) + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[(1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])]*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]] - (1/4)*(-1)^(2/3)*c^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]^2 - (1/2)*(-1)^(1/3)*c^(2/3)*Log[((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(1/3)*c^(2/3)*Log[-(((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[-(-1)^(2/3) + c^(1/3)*Sqrt[x]]*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(1/3)*c^(2/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2 - (1/2)*(-1)^(1/3)*c^(2/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])] + (1/2)*(-1)^(1/3)*c^(2/3)*Log[-(((-1)^(2/3)*(1 - c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[-(-1)^(2/3) - c^(1/3)*Sqrt[x]]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[((-1)^(2/3)*(1 + c^(1/3)*Sqrt[x]))/(1 + (-1)^(2/3))]*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]] + (1/4)*(-1)^(1/3)*c^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]^2 + (1/2)*(-1)^(1/3)*c^(2/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[(1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[-(((-1)^(2/3)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x]))/(1 - (-1)^(2/3)))] + (1/2)*(-1)^(1/3)*c^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]*Log[((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] + (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] + (1/2)*(-1)^(1/3)*c^(2/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 - c*x^(3/2)] - Log[1 - c*x^(3/2)]^2/(4*x) + (1/2)*c^(2/3)*Log[-1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] - (1/2)*c^(2/3)*Log[1 - c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (1/2)*(-1)^(2/3)*c^(2/3)*Log[-1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] - (1/2)*(-1)^(2/3)*c^(2/3)*Log[1 + (-1)^(1/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] - (1/2)*(-1)^(1/3)*c^(2/3)*Log[-1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (1/2)*(-1)^(1/3)*c^(2/3)*Log[1 - (-1)^(2/3)*c^(1/3)*Sqrt[x]]*Log[1 + c*x^(3/2)] + (Log[1 - c*x^(3/2)]*Log[1 + c*x^(3/2)])/(2*x) - Log[1 + c*x^(3/2)]^2/(4*x) + (1/2)*c^(2/3)*PolyLog[2, (1/2)*(1 - c^(1/3)*Sqrt[x])] + (1/2)*c^(2/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/2)*c^(2/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*c^(2/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/2)*c^(2/3)*PolyLog[2, (1 - c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) - c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) - c^(1/3)*Sqrt[x]] + (1/2)*c^(2/3)*PolyLog[2, (1/2)*(1 + c^(1/3)*Sqrt[x])] + (1/2)*c^(2/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/2)*c^(2/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*c^(2/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/2)*c^(2/3)*PolyLog[2, (1 + c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, 1/(1 - (-1)^(1/3)) + c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, 1/(1 + (-1)^(2/3)) + c^(1/3)*Sqrt[x]] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 - (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])] + (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(1/3))] - (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(2/3)*c^(2/3)*PolyLog[2, (1 + (-1)^(1/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1/2)*(1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])] + (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1 - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, ((-1)^(1/3) - (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1/2)*(1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])] + (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 - (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, (1 + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(2/3))] - (1/2)*(-1)^(1/3)*c^(2/3)*PolyLog[2, ((-1)^(1/3) + (-1)^(2/3)*c^(1/3)*Sqrt[x])/(1 + (-1)^(1/3))]} -*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x^n])^p*) - - -{x^2*(a + b*ArcTanh[c*x^n]), x, 2, (1/3)*x^3*(a + b*ArcTanh[c*x^n]) - (b*c*n*x^(3 + n)*Hypergeometric2F1[1, (3 + n)/(2*n), (3*(1 + n))/(2*n), c^2*x^(2*n)])/(3*(3 + n))} -{x^1*(a + b*ArcTanh[c*x^n]), x, 2, (1/2)*x^2*(a + b*ArcTanh[c*x^n]) - (b*c*n*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/(2*n), (1/2)*(3 + 2/n), c^2*x^(2*n)])/(2*(2 + n))} -{x^0*(a + b*ArcTanh[c*x^n]), x, 3, a*x + b*x*ArcTanh[c*x^n] - (b*c*n*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (1/2)*(3 + 1/n), c^2*x^(2*n)])/(1 + n)} -{(a + b*ArcTanh[c*x^n])/x^1, x, 2, a*Log[x] - (b*PolyLog[2, (-c)*x^n])/(2*n) + (b*PolyLog[2, c*x^n])/(2*n)} -{(a + b*ArcTanh[c*x^n])/x^2, x, 2, -((a + b*ArcTanh[c*x^n])/x) - (b*c*n*x^(-1 + n)*Hypergeometric2F1[1, -((1 - n)/(2*n)), (1/2)*(3 - 1/n), c^2*x^(2*n)])/(1 - n)} -{(a + b*ArcTanh[c*x^n])/x^3, x, 2, -((a + b*ArcTanh[c*x^n])/(2*x^2)) - (b*c*n*x^(-2 + n)*Hypergeometric2F1[1, (1/2)*(1 - 2/n), (1/2)*(3 - 2/n), c^2*x^(2*n)])/(2*(2 - n))} -{(a + b*ArcTanh[c*x^n])/x^4, x, 2, -((a + b*ArcTanh[c*x^n])/(3*x^3)) - (b*c*n*x^(-3 + n)*Hypergeometric2F1[1, -((3 - n)/(2*n)), -((3*(1 - n))/(2*n)), c^2*x^(2*n)])/(3*(3 - n))} - - -{x^1*(a + b*ArcTanh[c*x^n])^2, x, 0, Unintegrable[x*(a + b*ArcTanh[c*x^n])^2, x]} -{x^0*(a + b*ArcTanh[c*x^n])^2, x, 0, Unintegrable[(a + b*ArcTanh[c*x^n])^2, x]} -{(a + b*ArcTanh[c*x^n])^2/x^1, x, 7, (2*(a + b*ArcTanh[c*x^n])^2*ArcTanh[1 - 2/(1 - c*x^n)])/n - (b*(a + b*ArcTanh[c*x^n])*PolyLog[2, 1 - 2/(1 - c*x^n)])/n + (b*(a + b*ArcTanh[c*x^n])*PolyLog[2, -1 + 2/(1 - c*x^n)])/n + (b^2*PolyLog[3, 1 - 2/(1 - c*x^n)])/(2*n) - (b^2*PolyLog[3, -1 + 2/(1 - c*x^n)])/(2*n)} -{(a + b*ArcTanh[c*x^n])^2/x^2, x, 0, Unintegrable[(a + b*ArcTanh[c*x^n])^2/x^2, x]} -{(a + b*ArcTanh[c*x^n])^2/x^3, x, 0, Unintegrable[(a + b*ArcTanh[c*x^n])^2/x^3, x]} - - -{ArcTanh[a*x^n]/x, x, 2, -(PolyLog[2, (-a)*x^n]/(2*n)) + PolyLog[2, a*x^n]/(2*n)} - - -{ArcTanh[a*x^5]/x, x, 2, (-(1/10))*PolyLog[2, (-a)*x^5] + (1/10)*PolyLog[2, a*x^5]} - - -{ArcTanh[1/x], x, 3, x*ArcTanh[1/x] + (1/2)*Log[1 - x^2]} - - -(* ::Subsection:: *) -(*Integrands of the form (d x)^(m/2) (a+b ArcTanh[c x^n])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcTanh[c x^n])^p when m symbolic*) - - -{(d*x)^m*(a + b*ArcTanh[c*x^n])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^n])^3, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^n])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcTanh[c*x^n])^2, x]} -{(d*x)^m*(a + b*ArcTanh[c*x^n])^1, x, 3, (x*(d*x)^m*(a + b*ArcTanh[c*x^n]))/(1 + m) - (b*c*n*x^(1 + n)*(d*x)^m*Hypergeometric2F1[1, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n)])/((1 + m)*(1 + m + n))} -{(d*x)^m/(a + b*ArcTanh[c*x^n])^1, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^n]), x]} -{(d*x)^m/(a + b*ArcTanh[c*x^n])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcTanh[c*x^n])^2, x]} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m deleted file mode 100644 index 18ba5af..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.3 (d+e x)^m (a+b arctanh(c x^n))^p.m +++ /dev/null @@ -1,102 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^n])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^1])^p*) - - -{(d + e*x)^4*(a + b*ArcTanh[c*x]), x, 6, (b*d*e*(2*c^2*d^2 + e^2)*x)/c^3 + (b*e^2*(10*c^2*d^2 + e^2)*x^2)/(10*c^3) + (b*d*e^3*x^3)/(3*c) + (b*e^4*x^4)/(20*c) + ((d + e*x)^5*(a + b*ArcTanh[c*x]))/(5*e) + (b*(c*d + e)^5*Log[1 - c*x])/(10*c^5*e) - (b*(c*d - e)^5*Log[1 + c*x])/(10*c^5*e)} -{(d + e*x)^3*(a + b*ArcTanh[c*x]), x, 6, (b*e*(6*c^2*d^2 + e^2)*x)/(4*c^3) + (b*d*e^2*x^2)/(2*c) + (b*e^3*x^3)/(12*c) + ((d + e*x)^4*(a + b*ArcTanh[c*x]))/(4*e) + (b*(c*d + e)^4*Log[1 - c*x])/(8*c^4*e) - (b*(c*d - e)^4*Log[1 + c*x])/(8*c^4*e)} -{(d + e*x)^2*(a + b*ArcTanh[c*x]), x, 6, (b*d*e*x)/c + (b*e^2*x^2)/(6*c) + ((d + e*x)^3*(a + b*ArcTanh[c*x]))/(3*e) + (b*(c*d + e)^3*Log[1 - c*x])/(6*c^3*e) - (b*(c*d - e)^3*Log[1 + c*x])/(6*c^3*e)} -{(d + e*x)^1*(a + b*ArcTanh[c*x]), x, 6, (b*e*x)/(2*c) + ((d + e*x)^2*(a + b*ArcTanh[c*x]))/(2*e) + (b*(c*d + e)^2*Log[1 - c*x])/(4*c^2*e) - (b*(c*d - e)^2*Log[1 + c*x])/(4*c^2*e)} -{(a + b*ArcTanh[c*x])/(d + e*x)^1, x, 4, -(((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e) - (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)} -{(a + b*ArcTanh[c*x])/(d + e*x)^2, x, 6, -((a + b*ArcTanh[c*x])/(e*(d + e*x))) - (b*c*Log[1 - c*x])/(2*e*(c*d + e)) + (b*c*Log[1 + c*x])/(2*(c*d - e)*e) - (b*c*Log[d + e*x])/(c^2*d^2 - e^2)} -{(a + b*ArcTanh[c*x])/(d + e*x)^3, x, 4, (b*c)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])/(2*e*(d + e*x)^2) - (b*c^2*Log[1 - c*x])/(4*e*(c*d + e)^2) + (b*c^2*Log[1 + c*x])/(4*(c*d - e)^2*e) - (b*c^3*d*Log[d + e*x])/(c^2*d^2 - e^2)^2} -{(a + b*ArcTanh[c*x])/(d + e*x)^4, x, 4, If[$VersionNumber>=8, (b*c)/(6*(c^2*d^2 - e^2)*(d + e*x)^2) + (2*b*c^3*d)/(3*(c^2*d^2 - e^2)^2*(d + e*x)) - (a + b*ArcTanh[c*x])/(3*e*(d + e*x)^3) - (b*c^3*Log[1 - c*x])/(6*e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/(6*(c*d - e)^3*e) - (b*c^3*(3*c^2*d^2 + e^2)*Log[d + e*x])/(3*(c*d - e)^3*(c*d + e)^3), (b*c)/(6*(c^2*d^2 - e^2)*(d + e*x)^2) + (2*b*c^3*d)/(3*(c^2*d^2 - e^2)^2*(d + e*x)) - (a + b*ArcTanh[c*x])/(3*e*(d + e*x)^3) - (b*c^3*Log[1 - c*x])/(6*e*(c*d + e)^3) + (b*c^3*Log[1 + c*x])/(6*(c*d - e)^3*e) - (b*c^3*(3*c^2*d^2 + e^2)*Log[d + e*x])/(3*(c^2*d^2 - e^2)^3)]} - - -{(d + e*x)^3*(a + b*ArcTanh[c*x])^2, x, 19, (b^2*d*e^2*x)/c^2 + (a*b*e*(6*c^2*d^2 + e^2)*x)/(2*c^3) + (b^2*e^3*x^2)/(12*c^2) - (b^2*d*e^2*ArcTanh[c*x])/c^3 + (b^2*e*(6*c^2*d^2 + e^2)*x*ArcTanh[c*x])/(2*c^3) + (b*d*e^2*x^2*(a + b*ArcTanh[c*x]))/c + (b*e^3*x^3*(a + b*ArcTanh[c*x]))/(6*c) + (d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/c^3 - ((c^4*d^4 + 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTanh[c*x])^2)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTanh[c*x])^2)/(4*e) - (2*b*d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^3 + (b^2*e^3*Log[1 - c^2*x^2])/(12*c^4) + (b^2*e*(6*c^2*d^2 + e^2)*Log[1 - c^2*x^2])/(4*c^4) - (b^2*d*(c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/c^3} -{(d + e*x)^2*(a + b*ArcTanh[c*x])^2, x, 15, (2*a*b*d*e*x)/c + (b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTanh[c*x])/(3*c^3) + (2*b^2*d*e*x*ArcTanh[c*x])/c + (b*e^2*x^2*(a + b*ArcTanh[c*x]))/(3*c) + ((3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/(3*c^3) - (d*(d^2 + (3*e^2)/c^2)*(a + b*ArcTanh[c*x])^2)/(3*e) + ((d + e*x)^3*(a + b*ArcTanh[c*x])^2)/(3*e) - (2*b*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^3) + (b^2*d*e*Log[1 - c^2*x^2])/c^2 - (b^2*(3*c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^3)} -{(d + e*x)^1*(a + b*ArcTanh[c*x])^2, x, 12, (a*b*e*x)/c + (b^2*e*x*ArcTanh[c*x])/c + (d*(a + b*ArcTanh[c*x])^2)/c - ((d^2 + e^2/c^2)*(a + b*ArcTanh[c*x])^2)/(2*e) + ((d + e*x)^2*(a + b*ArcTanh[c*x])^2)/(2*e) - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (b^2*e*Log[1 - c^2*x^2])/(2*c^2) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/c} -{(a + b*ArcTanh[c*x])^2/(d + e*x)^1, x, 1, -(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e) - (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)} -{(a + b*ArcTanh[c*x])^2/(d + e*x)^2, x, 12, -((a + b*ArcTanh[c*x])^2/(e*(d + e*x))) + (b*c*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(e*(c*d + e)) - (b*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)*e) + (2*b*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d^2 - e^2) - (2*b*c*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2) + (b^2*c*PolyLog[2, 1 - 2/(1 - c*x)])/(2*e*(c*d + e)) + (b^2*c*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)*e) - (b^2*c*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d^2 - e^2) + (b^2*c*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)} -{(a + b*ArcTanh[c*x])^2/(d + e*x)^3, x, 18, If[$VersionNumber>=8, (b*c*(a + b*ArcTanh[c*x]))/((c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^2/(2*e*(d + e*x)^2) + (b*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*e*(c*d + e)^2) + (b^2*c^2*Log[1 - c*x])/(2*(c*d - e)*(c*d + e)^2) - (b*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*e) + (2*b*c^3*d*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) - (b^2*c^2*Log[1 + c*x])/(2*(c*d - e)^2*(c*d + e)) + (b^2*c^2*e*Log[d + e*x])/((c*d - e)^2*(c*d + e)^2) - (2*b*c^3*d*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) + (b^2*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (b^2*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (b^2*c^3*d*PolyLog[2, 1 - 2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) + (b^2*c^3*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2), (b*c*(a + b*ArcTanh[c*x]))/((c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^2/(2*e*(d + e*x)^2) + (b*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*e*(c*d + e)^2) + (b^2*c^2*Log[1 - c*x])/(2*(c*d - e)*(c*d + e)^2) - (b*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*e) + (2*b*c^3*d*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d^2 - e^2)^2 - (b^2*c^2*Log[1 + c*x])/(2*(c*d - e)^2*(c*d + e)) + (b^2*c^2*e*Log[d + e*x])/(c^2*d^2 - e^2)^2 - (2*b*c^3*d*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)^2 + (b^2*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (b^2*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (b^2*c^3*d*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d^2 - e^2)^2 + (b^2*c^3*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)^2]} - - -{(d + e*x)^3*(a + b*ArcTanh[c*x])^3, x, 29, (3*a*b^2*d*e^2*x)/c^2 + (b^3*e^3*x)/(4*c^3) - (b^3*e^3*ArcTanh[c*x])/(4*c^4) + (3*b^3*d*e^2*x*ArcTanh[c*x])/c^2 + (b^2*e^3*x^2*(a + b*ArcTanh[c*x]))/(4*c^2) - (3*b*d*e^2*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (b*e^3*(a + b*ArcTanh[c*x])^2)/(4*c^4) + (3*b*e*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2)/(4*c^4) + (3*b*e*(6*c^2*d^2 + e^2)*x*(a + b*ArcTanh[c*x])^2)/(4*c^3) + (3*b*d*e^2*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + (b*e^3*x^3*(a + b*ArcTanh[c*x])^2)/(4*c) + (d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/c^3 - ((c^4*d^4 + 6*c^2*d^2*e^2 + e^4)*(a + b*ArcTanh[c*x])^3)/(4*c^4*e) + ((d + e*x)^4*(a + b*ArcTanh[c*x])^3)/(4*e) - (b^2*e^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*c^4) - (3*b^2*e*(6*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*c^4) - (3*b*d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (3*b^3*d*e^2*Log[1 - c^2*x^2])/(2*c^3) - (b^3*e^3*PolyLog[2, 1 - 2/(1 - c*x)])/(4*c^4) - (3*b^3*e*(6*c^2*d^2 + e^2)*PolyLog[2, 1 - 2/(1 - c*x)])/(4*c^4) - (3*b^2*d*(c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (3*b^3*d*(c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcTanh[c*x])^3, x, 20, (a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTanh[c*x])/c^2 + (3*b*d*e*(a + b*ArcTanh[c*x])^2)/c^2 - (b*e^2*(a + b*ArcTanh[c*x])^2)/(2*c^3) + (3*b*d*e*x*(a + b*ArcTanh[c*x])^2)/c + (b*e^2*x^2*(a + b*ArcTanh[c*x])^2)/(2*c) + ((3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^3)/(3*c^3) - (d*(d^2 + (3*e^2)/c^2)*(a + b*ArcTanh[c*x])^3)/(3*e) + ((d + e*x)^3*(a + b*ArcTanh[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^2 - (b*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^3 + (b^3*e^2*Log[1 - c^2*x^2])/(2*c^3) - (3*b^3*d*e*PolyLog[2, 1 - 2/(1 - c*x)])/c^2 - (b^2*(3*c^2*d^2 + e^2)*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^3 + (b^3*(3*c^2*d^2 + e^2)*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^3)} -{(d + e*x)^1*(a + b*ArcTanh[c*x])^3, x, 14, (3*b*e*(a + b*ArcTanh[c*x])^2)/(2*c^2) + (3*b*e*x*(a + b*ArcTanh[c*x])^2)/(2*c) + (d*(a + b*ArcTanh[c*x])^3)/c - ((d^2 + e^2/c^2)*(a + b*ArcTanh[c*x])^3)/(2*e) + ((d + e*x)^2*(a + b*ArcTanh[c*x])^3)/(2*e) - (3*b^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c^2 - (3*b*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - (3*b^3*e*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c^2) - (3*b^2*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (3*b^3*d*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c)} -{(a + b*ArcTanh[c*x])^3/(d + e*x)^1, x, 1, -(((a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])^3*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e) - (3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e) + (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e) - (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e) + (3*b^3*PolyLog[4, 1 - 2/(1 + c*x)])/(4*e) - (3*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(4*e)} -{(a + b*ArcTanh[c*x])^3/(d + e*x)^2, x, 9, -((a + b*ArcTanh[c*x])^3/(e*(d + e*x))) + (3*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/(2*e*(c*d + e)) - (3*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(2*(c*d - e)*e) + (3*b*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^2*d^2 - e^2) - (3*b*c*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2) + (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(2*e*(c*d + e)) + (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)*e) - (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d^2 - e^2) + (3*b^2*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2) - (3*b^3*c*PolyLog[3, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)) + (3*b^3*c*PolyLog[3, 1 - 2/(1 + c*x)])/(4*(c*d - e)*e) - (3*b^3*c*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c^2*d^2 - e^2)) + (3*b^3*c*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c^2*d^2 - e^2))} -{(a + b*ArcTanh[c*x])^3/(d + e*x)^3, x, 21, If[$VersionNumber>=8, (3*b*c*(a + b*ArcTanh[c*x])^2)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^3/(2*e*(d + e*x)^2) - (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/(4*e*(c*d + e)^2) - (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) - (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(4*(c*d - e)^2*e) + (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) + (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) - (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d - e)*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (3*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*(c*d + e)) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c*d + e)^2) + (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(8*e*(c*d + e)^2) + (3*b^3*c^2*PolyLog[3, 1 - 2/(1 + c*x)])/(8*(c*d - e)^2*e) - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^2) + (3*b^3*c^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c*d + e)^2), (3*b*c*(a + b*ArcTanh[c*x])^2)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^3/(2*e*(d + e*x)^2) - (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/(4*e*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) - (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d^2 - e^2)^2 - (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(4*(c*d - e)^2*e) + (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^2*d^2 - e^2)^2 + (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)^2 - (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)^2 - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d - e)*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*(c*d + e)) + (3*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c^2*d^2 - e^2)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d^2 - e^2)^2 - (3*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c^2*d^2 - e^2)^2) + (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(c^2*d^2 - e^2)^2 - (3*b^3*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(8*e*(c*d + e)^2) + (3*b^3*c^2*PolyLog[3, 1 - 2/(1 + c*x)])/(8*(c*d - e)^2*e) - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c^2*d^2 - e^2)^2) + (3*b^3*c^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c^2*d^2 - e^2)^2)]} - - -{(a + b*ArcTanh[c*x])/(1 + 2*c*x), x, 4, ((a - b*ArcTanh[1/2])*Log[-((1 + 2*c*x)/(2*d))])/(2*c) - (b*PolyLog[2, -1 - 2*c*x])/(4*c) + (b*PolyLog[2, (1/3)*(1 + 2*c*x)])/(4*c), -(((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*c)) + ((a + b*ArcTanh[c*x])*Log[(2*(1 + 2*c*x))/(3*(1 + c*x))])/(2*c) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(4*c) - (b*PolyLog[2, 1 - (2*(1 + 2*c*x))/(3*(1 + c*x))])/(4*c)} - - -{ArcTanh[x]/(1 - Sqrt[2]*x), x, 4, -((ArcTanh[1/Sqrt[2]]*Log[1 - Sqrt[2]*x])/Sqrt[2]) - PolyLog[2, -((Sqrt[2] - 2*x)/(2 - Sqrt[2]))]/(2*Sqrt[2]) + PolyLog[2, (Sqrt[2] - 2*x)/(2 + Sqrt[2])]/(2*Sqrt[2]), (ArcTanh[x]*Log[2/(1 + x)])/Sqrt[2] - (ArcTanh[x]*Log[-((2*(1 + Sqrt[2])*(1 - Sqrt[2]*x))/(1 + x))])/Sqrt[2] - PolyLog[2, 1 - 2/(1 + x)]/(2*Sqrt[2]) + PolyLog[2, 1 + (2*(1 + Sqrt[2])*(1 - Sqrt[2]*x))/(1 + x)]/(2*Sqrt[2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^2])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^2])^p*) - - -{(d + e*x)^3*(a + b*ArcTanh[c*x^2]), x, 13, (2*b*d*e^2*x)/c + (b*e^3*x^2)/(4*c) + (b*d*(c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/c^(3/2) - (b*d*(c*d^2 + e^2)*ArcTanh[Sqrt[c]*x])/c^(3/2) + ((d + e*x)^4*(a + b*ArcTanh[c*x^2]))/(4*e) + (b*(c^2*d^4 + 6*c*d^2*e^2 + e^4)*Log[1 - c*x^2])/(8*c^2*e) - (b*(c^2*d^4 - 6*c*d^2*e^2 + e^4)*Log[1 + c*x^2])/(8*c^2*e)} -{(d + e*x)^2*(a + b*ArcTanh[c*x^2]), x, 11, (2*b*e^2*x)/(3*c) + (b*(3*c*d^2 - e^2)*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) - (b*(3*c*d^2 + e^2)*ArcTanh[Sqrt[c]*x])/(3*c^(3/2)) + ((d + e*x)^3*(a + b*ArcTanh[c*x^2]))/(3*e) + (b*d*(c*d^2 + 3*e^2)*Log[1 - c*x^2])/(6*c*e) - (b*d*(c*d^2 - 3*e^2)*Log[1 + c*x^2])/(6*c*e)} -{(d + e*x)^1*(a + b*ArcTanh[c*x^2]), x, 10, (b*d*ArcTan[Sqrt[c]*x])/Sqrt[c] - (b*d*ArcTanh[Sqrt[c]*x])/Sqrt[c] + ((d + e*x)^2*(a + b*ArcTanh[c*x^2]))/(2*e) + (b*(c*d^2 + e^2)*Log[1 - c*x^2])/(4*c*e) - (b*(c*d^2 - e^2)*Log[1 + c*x^2])/(4*c*e)} -{(a + b*ArcTanh[c*x^2])/(d + e*x)^1, x, 19, ((a + b*ArcTanh[c*x^2])*Log[d + e*x])/e - (b*Log[(e*(1 - Sqrt[-c]*x))/(Sqrt[-c]*d + e)]*Log[d + e*x])/(2*e) - (b*Log[-((e*(1 + Sqrt[-c]*x))/(Sqrt[-c]*d - e))]*Log[d + e*x])/(2*e) + (b*Log[(e*(1 - Sqrt[c]*x))/(Sqrt[c]*d + e)]*Log[d + e*x])/(2*e) + (b*Log[-((e*(1 + Sqrt[c]*x))/(Sqrt[c]*d - e))]*Log[d + e*x])/(2*e) - (b*PolyLog[2, (Sqrt[-c]*(d + e*x))/(Sqrt[-c]*d - e)])/(2*e) + (b*PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - e)])/(2*e) - (b*PolyLog[2, (Sqrt[-c]*(d + e*x))/(Sqrt[-c]*d + e)])/(2*e) + (b*PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + e)])/(2*e)} -{(a + b*ArcTanh[c*x^2])/(d + e*x)^2, x, 9, (b*Sqrt[c]*ArcTan[Sqrt[c]*x])/(c*d^2 + e^2) - (b*Sqrt[c]*ArcTanh[Sqrt[c]*x])/(c*d^2 - e^2) - (a + b*ArcTanh[c*x^2])/(e*(d + e*x)) + (2*b*c*d*e*Log[d + e*x])/(c^2*d^4 - e^4) - (b*c*d*Log[1 - c*x^2])/(2*e*(c*d^2 - e^2)) + (b*c*d*Log[1 + c*x^2])/(2*e*(c*d^2 + e^2))} -{(a + b*ArcTanh[c*x^2])/(d + e*x)^3, x, 9, -((b*c*d*e)/((c^2*d^4 - e^4)*(d + e*x))) + (b*c^(3/2)*d*ArcTan[Sqrt[c]*x])/(c*d^2 + e^2)^2 - (b*c^(3/2)*d*ArcTanh[Sqrt[c]*x])/(c*d^2 - e^2)^2 - (a + b*ArcTanh[c*x^2])/(2*e*(d + e*x)^2) + (b*c*e*(3*c^2*d^4 + e^4)*Log[d + e*x])/(c^2*d^4 - e^4)^2 - (b*c*(c*d^2 + e^2)*Log[1 - c*x^2])/(4*e*(c*d^2 - e^2)^2) + (b*c*(c*d^2 - e^2)*Log[1 + c*x^2])/(4*e*(c*d^2 + e^2)^2)} - - -(* {(d + e*x)^2*(a + b*ArcTanh[c*x^2])^2, x, 163, a^2*d^2*x + (4*a*b*e^2*x)/(3*c) - (2/9)*a*b*e^2*x^3 + (2*a*b*d^2*ArcTan[Sqrt[c]*x])/Sqrt[c] - (2*a*b*e^2*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) + (4*b^2*e^2*ArcTan[Sqrt[c]*x])/(3*c^(3/2)) + (I*b^2*d^2*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (I*b^2*e^2*ArcTan[Sqrt[c]*x]^2)/(3*c^(3/2)) - (2*a*b*d^2*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (4*b^2*e^2*ArcTanh[Sqrt[c]*x])/(3*c^(3/2)) - (b^2*d^2*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] - (b^2*e^2*ArcTanh[Sqrt[c]*x]^2)/(3*c^(3/2)) + (d*e*(a + b*ArcTanh[c*x^2])^2)/c + d*e*x^2*(a + b*ArcTanh[c*x^2])^2 + (2*b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] + (2*b^2*e^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(3*c^(3/2)) - (2*b^2*d^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (2*b^2*e^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*d^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (b^2*e^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) + (2*b^2*d^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*e^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(3*c^(3/2)) - (2*b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] - (2*b^2*e^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/Sqrt[c] + (b^2*e^2*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/(3*c^(3/2)) + (b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*e^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(3*c^(3/2)) + (b^2*d^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (b^2*e^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (2*b*d*e*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c - a*b*d^2*x*Log[1 - c*x^2] - (2*b^2*e^2*x*Log[1 - c*x^2])/(3*c) + (1/9)*b^2*e^2*x^3*Log[1 - c*x^2] - (b^2*d^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*e^2*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(3*c^(3/2)) + (b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (1/4)*b^2*d^2*x*Log[1 - c*x^2]^2 + (1/9)*b*e^2*x^3*(2*a - b*Log[1 - c*x^2]) - (b*e^2*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(3*c^(3/2)) + (1/12)*e^2*x^3*(2*a - b*Log[1 - c*x^2])^2 + a*b*d^2*x*Log[1 + c*x^2] + (2*b^2*e^2*x*Log[1 + c*x^2])/(3*c) + (1/3)*a*b*e^2*x^3*Log[1 + c*x^2] + (b^2*d^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*e^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2)) - (b^2*d^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*e^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(3*c^(3/2)) - (1/2)*b^2*d^2*x*Log[1 - c*x^2]*Log[1 + c*x^2] - (1/6)*b^2*e^2*x^3*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/4)*b^2*d^2*x*Log[1 + c*x^2]^2 + (1/12)*b^2*e^2*x^3*Log[1 + c*x^2]^2 + (b^2*d^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/Sqrt[c] + (b^2*e^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(3*c^(3/2)) + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (I*b^2*e^2*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/(3*c^(3/2)) - (I*b^2*d^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c]) + (I*b^2*e^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(6*c^(3/2)) + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (I*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/(3*c^(3/2)) + (b^2*d^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*e^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/(3*c^(3/2)) - (b^2*d^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*e^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(6*c^(3/2)) - (b^2*d^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*e^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(6*c^(3/2)) - (I*b^2*d^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c]) + (I*b^2*e^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(6*c^(3/2)) - (b^2*d*e*PolyLog[2, 1 - 2/(1 - c*x^2)])/c} *) -{(d + e*x)^1*(a + b*ArcTanh[c*x^2])^2, x, 77, a^2*d*x + (2*a*b*d*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*d*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (2*a*b*d*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] + (e*(a + b*ArcTanh[c*x^2])^2)/(2*c) + (1/2)*e*x^2*(a + b*ArcTanh[c*x^2])^2 + (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[-((2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x)))])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (b*e*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c - a*b*d*x*Log[1 - c*x^2] - (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (1/4)*b^2*d*x*Log[1 - c*x^2]^2 + a*b*d*x*Log[1 + c*x^2] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (1/2)*b^2*d*x*Log[1 - c*x^2]*Log[1 + c*x^2] + (1/4)*b^2*d*x*Log[1 + c*x^2]^2 + (b^2*d*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (I*b^2*d*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c]) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/Sqrt[c] + (b^2*d*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] - (b^2*d*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*d*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (I*b^2*d*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/(2*Sqrt[c]) - (b^2*e*PolyLog[2, 1 - 2/(1 - c*x^2)])/(2*c)} -{(a + b*ArcTanh[c*x^2])^2/(d + e*x)^1, x, 0, Unintegrable[(a + b*ArcTanh[c*x^2])^2/(d + e*x), x]} -{(a + b*ArcTanh[c*x^2])^2/(d + e*x)^2, x, 0, Unintegrable[(a + b*ArcTanh[c*x^2])^2/(d + e*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^3])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcTanh[c x^3])^p*) - - -{(d + e*x)^2*(a + b*ArcTanh[c*x^3]), x, 24, -((Sqrt[3]*b*d*e*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(2*c^(2/3))) + (Sqrt[3]*b*d*e*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(2*c^(2/3)) + (Sqrt[3]*b*d^2*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) - (b*d*e*ArcTanh[c^(1/3)*x])/c^(2/3) + ((d + e*x)^3*(a + b*ArcTanh[c*x^3]))/(3*e) + (b*d^2*Log[1 - c^(2/3)*x^2])/(2*c^(1/3)) + (b*d*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(4*c^(2/3)) - (b*d*e*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(4*c^(2/3)) + (b*(c*d^3 + e^3)*Log[1 - c*x^3])/(6*c*e) - (b*(c*d^3 - e^3)*Log[1 + c*x^3])/(6*c*e) - (b*d^2*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))} -{(d + e*x)^1*(a + b*ArcTanh[c*x^3]), x, 22, -((Sqrt[3]*b*e*ArcTan[1/Sqrt[3] - (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3))) + (Sqrt[3]*b*e*ArcTan[1/Sqrt[3] + (2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (Sqrt[3]*b*d*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/(2*c^(1/3)) - (b*e*ArcTanh[c^(1/3)*x])/(2*c^(2/3)) - (b*d^2*ArcTanh[c*x^3])/(2*e) + ((d + e*x)^2*(a + b*ArcTanh[c*x^3]))/(2*e) + (b*d*Log[1 - c^(2/3)*x^2])/(2*c^(1/3)) + (b*e*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*e*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4])/(4*c^(1/3))} -{(a + b*ArcTanh[c*x^3])/(d + e*x)^1, x, 25, ((a + b*ArcTanh[c*x^3])*Log[d + e*x])/e + (b*Log[(e*(1 - c^(1/3)*x))/(c^(1/3)*d + e)]*Log[d + e*x])/(2*e) - (b*Log[-((e*(1 + c^(1/3)*x))/(c^(1/3)*d - e))]*Log[d + e*x])/(2*e) + (b*Log[-((e*((-1)^(1/3) + c^(1/3)*x))/(c^(1/3)*d - (-1)^(1/3)*e))]*Log[d + e*x])/(2*e) - (b*Log[-((e*((-1)^(2/3) + c^(1/3)*x))/(c^(1/3)*d - (-1)^(2/3)*e))]*Log[d + e*x])/(2*e) + (b*Log[((-1)^(2/3)*e*(1 + (-1)^(1/3)*c^(1/3)*x))/(c^(1/3)*d + (-1)^(2/3)*e)]*Log[d + e*x])/(2*e) - (b*Log[((-1)^(1/3)*e*(1 + (-1)^(2/3)*c^(1/3)*x))/(c^(1/3)*d + (-1)^(1/3)*e)]*Log[d + e*x])/(2*e) - (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - e)])/(2*e) + (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + e)])/(2*e) + (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - (-1)^(1/3)*e)])/(2*e) - (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + (-1)^(1/3)*e)])/(2*e) - (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - (-1)^(2/3)*e)])/(2*e) + (b*PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + (-1)^(2/3)*e)])/(2*e)} -{(a + b*ArcTanh[c*x^3])/(d + e*x)^2, x, 19, -((Sqrt[3]*b*c^(1/3)*ArcTan[(1 - 2*c^(1/3)*x)/Sqrt[3]])/(2*(c^(2/3)*d^2 + c^(1/3)*d*e + e^2))) - (Sqrt[3]*b*c^(1/3)*(c^(1/3)*d + e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(2*(c*d^3 + e^3)) - (a + b*ArcTanh[c*x^3])/(e*(d + e*x)) + (b*c^(1/3)*(c^(1/3)*d - e)*Log[1 - c^(1/3)*x])/(2*(c*d^3 + e^3)) + (b*c^(1/3)*(c^(1/3)*d + e)*Log[1 + c^(1/3)*x])/(2*(c*d^3 - e^3)) - (3*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 - e^6) - (b*c^(1/3)*(c^(1/3)*d + e)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(4*(c*d^3 - e^3)) - (b*c^(1/3)*(c^(1/3)*d - e)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(4*(c*d^3 + e^3)) - (b*c*d^2*Log[1 - c*x^3])/(2*e*(c*d^3 + e^3)) + (b*c*d^2*Log[1 + c*x^3])/(2*e*(c*d^3 - e^3))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^q (d+e x)^m (a+b ArcTanh[c x^(1/2)])^p with c^2 d+e=0*) - - -{x^3*(a + b*ArcTanh[c*Sqrt[x]])/(1 - c^2*x), x, 19, -((11*b*Sqrt[x])/(6*c^7)) - (5*b*x^(3/2))/(18*c^5) - (b*x^(5/2))/(15*c^3) + (11*b*ArcTanh[c*Sqrt[x]])/(6*c^8) - (x*(a + b*ArcTanh[c*Sqrt[x]]))/c^6 - (x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*c^4) - (x^3*(a + b*ArcTanh[c*Sqrt[x]]))/(3*c^2) - (a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^8) + (2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^8 + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^8} -{x^2*(a + b*ArcTanh[c*Sqrt[x]])/(1 - c^2*x), x, 14, -((3*b*Sqrt[x])/(2*c^5)) - (b*x^(3/2))/(6*c^3) + (3*b*ArcTanh[c*Sqrt[x]])/(2*c^6) - (x*(a + b*ArcTanh[c*Sqrt[x]]))/c^4 - (x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*c^2) - (a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^6) + (2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^6 + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^6} -{x^1*(a + b*ArcTanh[c*Sqrt[x]])/(1 - c^2*x), x, 9, -((b*Sqrt[x])/c^3) + (b*ArcTanh[c*Sqrt[x]])/c^4 - (x*(a + b*ArcTanh[c*Sqrt[x]]))/c^2 - (a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^4) + (2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^4 + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^4} -{x^0*(a + b*ArcTanh[c*Sqrt[x]])/(1 - c^2*x), x, 5, -((a + b*ArcTanh[c*Sqrt[x]])^2/(b*c^2)) + (2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^2 + (b*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^2} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^1*(1 - c^2*x)), x, 5, (a + b*ArcTanh[c*Sqrt[x]])^2/b + 2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - b*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^2*(1 - c^2*x)), x, 9, -((b*c)/Sqrt[x]) + b*c^2*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/x + (c^2*(a + b*ArcTanh[c*Sqrt[x]])^2)/b + 2*c^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - b*c^2*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(1 - c^2*x)), x, 14, -((b*c)/(6*x^(3/2))) - (3*b*c^3)/(2*Sqrt[x]) + (3/2)*b*c^4*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/(2*x^2) - (c^2*(a + b*ArcTanh[c*Sqrt[x]]))/x + (c^4*(a + b*ArcTanh[c*Sqrt[x]])^2)/b + 2*c^4*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - b*c^4*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^4*(1 - c^2*x)), x, 20, -((b*c)/(15*x^(5/2))) - (5*b*c^3)/(18*x^(3/2)) - (11*b*c^5)/(6*Sqrt[x]) + (11/6)*b*c^6*ArcTanh[c*Sqrt[x]] - (a + b*ArcTanh[c*Sqrt[x]])/(3*x^3) - (c^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*x^2) - (c^4*(a + b*ArcTanh[c*Sqrt[x]]))/x + (c^6*(a + b*ArcTanh[c*Sqrt[x]])^2)/b + 2*c^6*(a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - b*c^6*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^q (d+e x)^m (a+b ArcTanh[c x^(1/2)])^p*) - - -{x^2*(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x), x, 20, -((b*d*Sqrt[x])/(c*e^2)) + (b*Sqrt[x])/(2*c^3*e) + (b*x^(3/2))/(6*c*e) + (b*d*ArcTanh[c*Sqrt[x]])/(c^2*e^2) - (b*ArcTanh[c*Sqrt[x]])/(2*c^4*e) - (d*x*(a + b*ArcTanh[c*Sqrt[x]]))/e^2 + (x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(2*e) - (2*d^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/e^3 + (d^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/e^3 + (d^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/e^3 + (b*d^2*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/e^3 - (b*d^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e^3) - (b*d^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e^3)} -{x^1*(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x), x, 15, (b*Sqrt[x])/(c*e) - (b*ArcTanh[c*Sqrt[x]])/(c^2*e) + (x*(a + b*ArcTanh[c*Sqrt[x]]))/e + (2*d*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/e^2 - (d*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/e^2 - (d*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/e^2 - (b*d*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/e^2 + (b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e^2) + (b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e^2)} -{x^0*(a + b*ArcTanh[c*Sqrt[x]])/(d + e*x), x, 11, -((2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/e) + ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/e + ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/e + (b*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/e - (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e) - (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*e)} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^1*(d + e*x)), x, 15, (2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/d - ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/d - ((a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/d + (a*Log[x])/d - (b*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/d + (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d) + (b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d) - (b*PolyLog[2, (-c)*Sqrt[x]])/d + (b*PolyLog[2, c*Sqrt[x]])/d} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^2*(d + e*x)), x, 19, -((b*c)/(d*Sqrt[x])) + (b*c^2*ArcTanh[c*Sqrt[x]])/d - (a + b*ArcTanh[c*Sqrt[x]])/(d*x) - (2*e*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/d^2 + (e*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/d^2 + (e*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/d^2 - (a*e*Log[x])/d^2 + (b*e*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/d^2 - (b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^2) - (b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^2) + (b*e*PolyLog[2, (-c)*Sqrt[x]])/d^2 - (b*e*PolyLog[2, c*Sqrt[x]])/d^2} -{(a + b*ArcTanh[c*Sqrt[x]])/(x^3*(d + e*x)), x, 24, -((b*c)/(6*d*x^(3/2))) - (b*c^3)/(2*d*Sqrt[x]) + (b*c*e)/(d^2*Sqrt[x]) + (b*c^4*ArcTanh[c*Sqrt[x]])/(2*d) - (b*c^2*e*ArcTanh[c*Sqrt[x]])/d^2 - (a + b*ArcTanh[c*Sqrt[x]])/(2*d*x^2) + (e*(a + b*ArcTanh[c*Sqrt[x]]))/(d^2*x) + (2*e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 + c*Sqrt[x])])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 - (e^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/d^3 + (a*e^2*Log[x])/d^3 - (b*e^2*PolyLog[2, 1 - 2/(1 + c*Sqrt[x])])/d^3 + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] - Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) + (b*e^2*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*Sqrt[x]))/((c*Sqrt[-d] + Sqrt[e])*(1 + c*Sqrt[x]))])/(2*d^3) - (b*e^2*PolyLog[2, (-c)*Sqrt[x]])/d^3 + (b*e^2*PolyLog[2, c*Sqrt[x]])/d^3} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m deleted file mode 100644 index 8bec6f9..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.4 u (a+b arctanh(c x))^p.m +++ /dev/null @@ -1,997 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^q (a+b ArcTanh[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^p when c^2 d^2-e^2=0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^1 when c^2 d^2 - e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(d + c*d*x)*(a + b*ArcTanh[c*x]), x, 7, (b*d*x)/(4*c^3) + (b*d*x^2)/(10*c^2) + (b*d*x^3)/(12*c) + (b*d*x^4)/20 + (d*x^4*(a + b*ArcTanh[c*x]))/4 + (c*d*x^5*(a + b*ArcTanh[c*x]))/5 + (9*b*d*Log[1 - c*x])/(40*c^4) - (b*d*Log[1 + c*x])/(40*c^4)} -{x^2*(d + c*d*x)*(a + b*ArcTanh[c*x]), x, 7, (b*d*x)/(4*c^2) + (b*d*x^2)/(6*c) + (b*d*x^3)/12 + (d*x^3*(a + b*ArcTanh[c*x]))/3 + (c*d*x^4*(a + b*ArcTanh[c*x]))/4 + (7*b*d*Log[1 - c*x])/(24*c^3) + (b*d*Log[1 + c*x])/(24*c^3)} -{x^1*(d + c*d*x)*(a + b*ArcTanh[c*x]), x, 7, (b*d*x)/(2*c) + (b*d*x^2)/6 + (d*x^2*(a + b*ArcTanh[c*x]))/2 + (c*d*x^3*(a + b*ArcTanh[c*x]))/3 + (5*b*d*Log[1 - c*x])/(12*c^2) - (b*d*Log[1 + c*x])/(12*c^2)} -{x^0*(d + c*d*x)*(a + b*ArcTanh[c*x]), x, 4, (b*d*x)/2 + (d*(1 + c*x)^2*(a + b*ArcTanh[c*x]))/(2*c) + (b*d*Log[1 - c*x])/c} -{((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^1, x, 6, a*c*d*x + b*c*d*x*ArcTanh[c*x] + a*d*Log[x] + (1/2)*b*d*Log[1 - c^2*x^2] - (1/2)*b*d*PolyLog[2, (-c)*x] + (1/2)*b*d*PolyLog[2, c*x]} -{((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^2, x, 8, -((d*(a + b*ArcTanh[c*x]))/x) + a*c*d*Log[x] + b*c*d*Log[x] - (1/2)*b*c*d*Log[1 - c^2*x^2] - (1/2)*b*c*d*PolyLog[2, (-c)*x] + (1/2)*b*c*d*PolyLog[2, c*x]} -{((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^3, x, 4, -(b*c*d)/(2*x) - (d*(1 + c*x)^2*(a + b*ArcTanh[c*x]))/(2*x^2) + b*c^2*d*Log[x] - b*c^2*d*Log[1 - c*x]} -{((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^4, x, 4, -(b*c*d)/(6*x^2) - (b*c^2*d)/(2*x) - (d*(a + b*ArcTanh[c*x]))/(3*x^3) - (c*d*(a + b*ArcTanh[c*x]))/(2*x^2) + (b*c^3*d*Log[x])/3 - (5*b*c^3*d*Log[1 - c*x])/12 + (b*c^3*d*Log[1 + c*x])/12} -{((d + c*d*x)*(a + b*ArcTanh[c*x]))/x^5, x, 4, -((b*c*d)/(12*x^3)) - (b*c^2*d)/(6*x^2) - (b*c^3*d)/(4*x) - (d*(a + b*ArcTanh[c*x]))/(4*x^4) - (c*d*(a + b*ArcTanh[c*x]))/(3*x^3) + (1/3)*b*c^4*d*Log[x] - (7/24)*b*c^4*d*Log[1 - c*x] - (1/24)*b*c^4*d*Log[1 + c*x]} - - -{x^3*(d + c*d*x)^2*(a + b*ArcTanh[c*x]), x, 7, (5*b*d^2*x)/(12*c^3) + (b*d^2*x^2)/(5*c^2) + (5*b*d^2*x^3)/(36*c) + (b*d^2*x^4)/10 + (b*c*d^2*x^5)/30 + (d^2*x^4*(a + b*ArcTanh[c*x]))/4 + (2*c*d^2*x^5*(a + b*ArcTanh[c*x]))/5 + (c^2*d^2*x^6*(a + b*ArcTanh[c*x]))/6 + (49*b*d^2*Log[1 - c*x])/(120*c^4) - (b*d^2*Log[1 + c*x])/(120*c^4)} -{x^2*(d + c*d*x)^2*(a + b*ArcTanh[c*x]), x, 7, (b*d^2*x)/(2*c^2) + (4*b*d^2*x^2)/(15*c) + (b*d^2*x^3)/6 + (b*c*d^2*x^4)/20 + (d^2*x^3*(a + b*ArcTanh[c*x]))/3 + (c*d^2*x^4*(a + b*ArcTanh[c*x]))/2 + (c^2*d^2*x^5*(a + b*ArcTanh[c*x]))/5 + (31*b*d^2*Log[1 - c*x])/(60*c^3) + (b*d^2*Log[1 + c*x])/(60*c^3)} -{x^1*(d + c*d*x)^2*(a + b*ArcTanh[c*x]), x, 7, (3*b*d^2*x)/(4*c) + (b*d^2*x^2)/3 + (b*c*d^2*x^3)/12 + (d^2*x^2*(a + b*ArcTanh[c*x]))/2 + (2*c*d^2*x^3*(a + b*ArcTanh[c*x]))/3 + (c^2*d^2*x^4*(a + b*ArcTanh[c*x]))/4 + (17*b*d^2*Log[1 - c*x])/(24*c^2) - (b*d^2*Log[1 + c*x])/(24*c^2)} -{x^0*(d + c*d*x)^2*(a + b*ArcTanh[c*x]), x, 4, (2/3)*b*d^2*x + (b*d^2*(1 + c*x)^2)/(6*c) + (d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x]))/(3*c) + (4*b*d^2*Log[1 - c*x])/(3*c)} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^1, x, 9, 2*a*c*d^2*x + (1/2)*b*c*d^2*x - (1/2)*b*d^2*ArcTanh[c*x] + 2*b*c*d^2*x*ArcTanh[c*x] + (1/2)*c^2*d^2*x^2*(a + b*ArcTanh[c*x]) + a*d^2*Log[x] + b*d^2*Log[1 - c^2*x^2] - (1/2)*b*d^2*PolyLog[2, (-c)*x] + (1/2)*b*d^2*PolyLog[2, c*x]} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^2, x, 11, (d^2*(-1 + c^2*x^2)*(a + b*ArcTanh[c*x]))/x + (2*a + b)*c*d^2*Log[x] - b*c*d^2*PolyLog[2, (-c)*x] + b*c*d^2*PolyLog[2, c*x], a*c^2*d^2*x + b*c^2*d^2*x*ArcTanh[c*x] - (d^2*(a + b*ArcTanh[c*x]))/x + 2*a*c*d^2*Log[x] + b*c*d^2*Log[x] - b*c*d^2*PolyLog[2, (-c)*x] + b*c*d^2*PolyLog[2, c*x]} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^3, x, 11, -((b*c*d^2)/(2*x)) + (1/2)*b*c^2*d^2*ArcTanh[c*x] - (d^2*(a + b*ArcTanh[c*x]))/(2*x^2) - (2*c*d^2*(a + b*ArcTanh[c*x]))/x + a*c^2*d^2*Log[x] + 2*b*c^2*d^2*Log[x] - b*c^2*d^2*Log[1 - c^2*x^2] - (1/2)*b*c^2*d^2*PolyLog[2, (-c)*x] + (1/2)*b*c^2*d^2*PolyLog[2, c*x]} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^4, x, 4, -(b*c*d^2)/(6*x^2) - (b*c^2*d^2)/x - (d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x]))/(3*x^3) + (4*b*c^3*d^2*Log[x])/3 - (4*b*c^3*d^2*Log[1 - c*x])/3} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^5, x, 4, -((b*c*d^2)/(12*x^3)) - (b*c^2*d^2)/(3*x^2) - (3*b*c^3*d^2)/(4*x) - (d^2*(a + b*ArcTanh[c*x]))/(4*x^4) - (2*c*d^2*(a + b*ArcTanh[c*x]))/(3*x^3) - (c^2*d^2*(a + b*ArcTanh[c*x]))/(2*x^2) + (2/3)*b*c^4*d^2*Log[x] - (17/24)*b*c^4*d^2*Log[1 - c*x] + (1/24)*b*c^4*d^2*Log[1 + c*x]} -{((d + c*d*x)^2*(a + b*ArcTanh[c*x]))/x^6, x, 4, -((b*c*d^2)/(20*x^4)) - (b*c^2*d^2)/(6*x^3) - (4*b*c^3*d^2)/(15*x^2) - (b*c^4*d^2)/(2*x) - (d^2*(a + b*ArcTanh[c*x]))/(5*x^5) - (c*d^2*(a + b*ArcTanh[c*x]))/(2*x^4) - (c^2*d^2*(a + b*ArcTanh[c*x]))/(3*x^3) + (8/15)*b*c^5*d^2*Log[x] - (31/60)*b*c^5*d^2*Log[1 - c*x] - (1/60)*b*c^5*d^2*Log[1 + c*x]} - - -{x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x]), x, 7, (3*b*d^3*x)/(4*c^3) + (13*b*d^3*x^2)/(35*c^2) + (b*d^3*x^3)/(4*c) + (13*b*d^3*x^4)/70 + (b*c*d^3*x^5)/10 + (b*c^2*d^3*x^6)/42 + (d^3*x^4*(a + b*ArcTanh[c*x]))/4 + (3*c*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (c^2*d^3*x^6*(a + b*ArcTanh[c*x]))/2 + (c^3*d^3*x^7*(a + b*ArcTanh[c*x]))/7 + (209*b*d^3*Log[1 - c*x])/(280*c^4) - (b*d^3*Log[1 + c*x])/(280*c^4)} -{x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x]), x, 7, (11*b*d^3*x)/(12*c^2) + (7*b*d^3*x^2)/(15*c) + (11*b*d^3*x^3)/36 + (3*b*c*d^3*x^4)/20 + (b*c^2*d^3*x^5)/30 + (d^3*x^3*(a + b*ArcTanh[c*x]))/3 + (3*c*d^3*x^4*(a + b*ArcTanh[c*x]))/4 + (3*c^2*d^3*x^5*(a + b*ArcTanh[c*x]))/5 + (c^3*d^3*x^6*(a + b*ArcTanh[c*x]))/6 + (37*b*d^3*Log[1 - c*x])/(40*c^3) + (b*d^3*Log[1 + c*x])/(120*c^3)} -{x^1*(d + c*d*x)^3*(a + b*ArcTanh[c*x]), x, 4, (3*b*d^3*x)/(5*c) + (3*b*d^3*(1 + c*x)^2)/(20*c^2) + (b*d^3*(1 + c*x)^3)/(20*c^2) + (b*d^3*(1 + c*x)^4)/(20*c^2) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(4*c^2) + (d^3*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*c^2) + (6*b*d^3*Log[1 - c*x])/(5*c^2)} -{x^0*(d + c*d*x)^3*(a + b*ArcTanh[c*x]), x, 4, b*d^3*x + (b*d^3*(1 + c*x)^2)/(4*c) + (b*d^3*(1 + c*x)^3)/(12*c) + (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(4*c) + (2*b*d^3*Log[1 - c*x])/c} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^1, x, 13, 3*a*c*d^3*x + (3/2)*b*c*d^3*x + (1/6)*b*c^2*d^3*x^2 - (3/2)*b*d^3*ArcTanh[c*x] + 3*b*c*d^3*x*ArcTanh[c*x] + (3/2)*c^2*d^3*x^2*(a + b*ArcTanh[c*x]) + (1/3)*c^3*d^3*x^3*(a + b*ArcTanh[c*x]) + a*d^3*Log[x] + (5/3)*b*d^3*Log[1 - c^2*x^2] - (1/2)*b*d^3*PolyLog[2, (-c)*x] + (1/2)*b*d^3*PolyLog[2, c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^2, x, 14, 3*a*c^2*d^3*x + (1/2)*b*c^2*d^3*x - (1/2)*b*c*d^3*ArcTanh[c*x] + 3*b*c^2*d^3*x*ArcTanh[c*x] - (d^3*(a + b*ArcTanh[c*x]))/x + (1/2)*c^3*d^3*x^2*(a + b*ArcTanh[c*x]) + 3*a*c*d^3*Log[x] + b*c*d^3*Log[x] + b*c*d^3*Log[1 - c^2*x^2] - (3/2)*b*c*d^3*PolyLog[2, (-c)*x] + (3/2)*b*c*d^3*PolyLog[2, c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^3, x, 14, -((b*c*d^3)/(2*x)) + a*c^3*d^3*x + (1/2)*b*c^2*d^3*ArcTanh[c*x] + b*c^3*d^3*x*ArcTanh[c*x] - (d^3*(a + b*ArcTanh[c*x]))/(2*x^2) - (3*c*d^3*(a + b*ArcTanh[c*x]))/x + 3*a*c^2*d^3*Log[x] + 3*b*c^2*d^3*Log[x] - b*c^2*d^3*Log[1 - c^2*x^2] - (3/2)*b*c^2*d^3*PolyLog[2, (-c)*x] + (3/2)*b*c^2*d^3*PolyLog[2, c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^4, x, 15, -((b*c*d^3)/(6*x^2)) - (3*b*c^2*d^3)/(2*x) + (3/2)*b*c^3*d^3*ArcTanh[c*x] - (d^3*(a + b*ArcTanh[c*x]))/(3*x^3) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x]))/x + a*c^3*d^3*Log[x] + (10/3)*b*c^3*d^3*Log[x] - (5/3)*b*c^3*d^3*Log[1 - c^2*x^2] - (1/2)*b*c^3*d^3*PolyLog[2, (-c)*x] + (1/2)*b*c^3*d^3*PolyLog[2, c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^5, x, 4, -((b*c*d^3)/(12*x^3)) - (b*c^2*d^3)/(2*x^2) - (7*b*c^3*d^3)/(4*x) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(4*x^4) + 2*b*c^4*d^3*Log[x] - 2*b*c^4*d^3*Log[1 - c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^6, x, 4, -((b*c*d^3)/(20*x^4)) - (b*c^2*d^3)/(4*x^3) - (3*b*c^3*d^3)/(5*x^2) - (5*b*c^4*d^3)/(4*x) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(5*x^5) + (c*d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x]))/(20*x^4) + (6/5)*b*c^5*d^3*Log[x] - (6/5)*b*c^5*d^3*Log[1 - c*x]} -{((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7, x, 4, -((b*c*d^3)/(30*x^5)) - (3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3)/(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^3)/(12*x) - (d^3*(a + b*ArcTanh[c*x]))/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(5*x^5) - (3*c^2*d^3*(a + b*ArcTanh[c*x]))/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x]))/(3*x^3) + (14/15)*b*c^6*d^3*Log[x] - (37/40)*b*c^6*d^3*Log[1 - c*x] - (1/120)*b*c^6*d^3*Log[1 + c*x]} - - -{x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]), x, 7, (11*b*d^4*x)/(8*c^3) + (24*b*d^4*x^2)/(35*c^2) + (11*b*d^4*x^3)/(24*c) + (12/35)*b*d^4*x^4 + (9/40)*b*c*d^4*x^5 + (2/21)*b*c^2*d^4*x^6 + (1/56)*b*c^3*d^4*x^7 + (1/4)*d^4*x^4*(a + b*ArcTanh[c*x]) + (4/5)*c*d^4*x^5*(a + b*ArcTanh[c*x]) + c^2*d^4*x^6*(a + b*ArcTanh[c*x]) + (4/7)*c^3*d^4*x^7*(a + b*ArcTanh[c*x]) + (1/8)*c^4*d^4*x^8*(a + b*ArcTanh[c*x]) + (769*b*d^4*Log[1 - c*x])/(560*c^4) - (b*d^4*Log[1 + c*x])/(560*c^4)} -{x^2*(d + c*d*x)^4*(a + b*ArcTanh[c*x]), x, 4, (5*b*d^4*x)/(3*c^2) + (88*b*d^4*x^2)/(105*c) + (5/9)*b*d^4*x^3 + (47/140)*b*c*d^4*x^4 + (2/15)*b*c^2*d^4*x^5 + (1/42)*b*c^3*d^4*x^6 + (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*c^3) - (d^4*(1 + c*x)^6*(a + b*ArcTanh[c*x]))/(3*c^3) + (d^4*(1 + c*x)^7*(a + b*ArcTanh[c*x]))/(7*c^3) + (176*b*d^4*Log[1 - c*x])/(105*c^3)} -{x^1*(d + c*d*x)^4*(a + b*ArcTanh[c*x]), x, 4, (16*b*d^4*x)/(15*c) + (4*b*d^4*(1 + c*x)^2)/(15*c^2) + (4*b*d^4*(1 + c*x)^3)/(45*c^2) + (b*d^4*(1 + c*x)^4)/(30*c^2) + (b*d^4*(1 + c*x)^5)/(30*c^2) - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*c^2) + (d^4*(1 + c*x)^6*(a + b*ArcTanh[c*x]))/(6*c^2) + (32*b*d^4*Log[1 - c*x])/(15*c^2)} -{x^0*(d + c*d*x)^4*(a + b*ArcTanh[c*x]), x, 4, (8/5)*b*d^4*x + (2*b*d^4*(1 + c*x)^2)/(5*c) + (2*b*d^4*(1 + c*x)^3)/(15*c) + (b*d^4*(1 + c*x)^4)/(20*c) + (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*c) + (16*b*d^4*Log[1 - c*x])/(5*c)} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^1, x, 17, 4*a*c*d^4*x + (13/4)*b*c*d^4*x + (2/3)*b*c^2*d^4*x^2 + (1/12)*b*c^3*d^4*x^3 - (13/4)*b*d^4*ArcTanh[c*x] + 4*b*c*d^4*x*ArcTanh[c*x] + 3*c^2*d^4*x^2*(a + b*ArcTanh[c*x]) + (4/3)*c^3*d^4*x^3*(a + b*ArcTanh[c*x]) + (1/4)*c^4*d^4*x^4*(a + b*ArcTanh[c*x]) + a*d^4*Log[x] + (8/3)*b*d^4*Log[1 - c^2*x^2] - (1/2)*b*d^4*PolyLog[2, (-c)*x] + (1/2)*b*d^4*PolyLog[2, c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^2, x, 18, 6*a*c^2*d^4*x + 2*b*c^2*d^4*x + (1/6)*b*c^3*d^4*x^2 - 2*b*c*d^4*ArcTanh[c*x] + 6*b*c^2*d^4*x*ArcTanh[c*x] - (d^4*(a + b*ArcTanh[c*x]))/x + 2*c^3*d^4*x^2*(a + b*ArcTanh[c*x]) + (1/3)*c^4*d^4*x^3*(a + b*ArcTanh[c*x]) + 4*a*c*d^4*Log[x] + b*c*d^4*Log[x] + (8/3)*b*c*d^4*Log[1 - c^2*x^2] - 2*b*c*d^4*PolyLog[2, (-c)*x] + 2*b*c*d^4*PolyLog[2, c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^3, x, 17, -((b*c*d^4)/(2*x)) + 4*a*c^3*d^4*x + (1/2)*b*c^3*d^4*x + 4*b*c^3*d^4*x*ArcTanh[c*x] - (d^4*(a + b*ArcTanh[c*x]))/(2*x^2) - (4*c*d^4*(a + b*ArcTanh[c*x]))/x + (1/2)*c^4*d^4*x^2*(a + b*ArcTanh[c*x]) + 6*a*c^2*d^4*Log[x] + 4*b*c^2*d^4*Log[x] - 3*b*c^2*d^4*PolyLog[2, (-c)*x] + 3*b*c^2*d^4*PolyLog[2, c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^4, x, 18, -((b*c*d^4)/(6*x^2)) - (2*b*c^2*d^4)/x + a*c^4*d^4*x + 2*b*c^3*d^4*ArcTanh[c*x] + b*c^4*d^4*x*ArcTanh[c*x] - (d^4*(a + b*ArcTanh[c*x]))/(3*x^3) - (2*c*d^4*(a + b*ArcTanh[c*x]))/x^2 - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/x + 4*a*c^3*d^4*Log[x] + (19/3)*b*c^3*d^4*Log[x] - (8/3)*b*c^3*d^4*Log[1 - c^2*x^2] - 2*b*c^3*d^4*PolyLog[2, (-c)*x] + 2*b*c^3*d^4*PolyLog[2, c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^5, x, 19, -((b*c*d^4)/(12*x^3)) - (2*b*c^2*d^4)/(3*x^2) - (13*b*c^3*d^4)/(4*x) + (13/4)*b*c^4*d^4*ArcTanh[c*x] - (d^4*(a + b*ArcTanh[c*x]))/(4*x^4) - (4*c*d^4*(a + b*ArcTanh[c*x]))/(3*x^3) - (3*c^2*d^4*(a + b*ArcTanh[c*x]))/x^2 - (4*c^3*d^4*(a + b*ArcTanh[c*x]))/x + a*c^4*d^4*Log[x] + (16/3)*b*c^4*d^4*Log[x] - (8/3)*b*c^4*d^4*Log[1 - c^2*x^2] - (1/2)*b*c^4*d^4*PolyLog[2, (-c)*x] + (1/2)*b*c^4*d^4*PolyLog[2, c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^6, x, 4, -((b*c*d^4)/(20*x^4)) - (b*c^2*d^4)/(3*x^3) - (11*b*c^3*d^4)/(10*x^2) - (3*b*c^4*d^4)/x - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(5*x^5) + (16/5)*b*c^5*d^4*Log[x] - (16/5)*b*c^5*d^4*Log[1 - c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^7, x, 4, -((b*c*d^4)/(30*x^5)) - (b*c^2*d^4)/(5*x^4) - (5*b*c^3*d^4)/(9*x^3) - (16*b*c^4*d^4)/(15*x^2) - (13*b*c^5*d^4)/(6*x) - (d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(6*x^6) + (c*d^4*(1 + c*x)^5*(a + b*ArcTanh[c*x]))/(30*x^5) + (32/15)*b*c^6*d^4*Log[x] - (32/15)*b*c^6*d^4*Log[1 - c*x]} -{((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8, x, 4, -((b*c*d^4)/(42*x^6)) - (2*b*c^2*d^4)/(15*x^5) - (47*b*c^3*d^4)/(140*x^4) - (5*b*c^4*d^4)/(9*x^3) - (88*b*c^5*d^4)/(105*x^2) - (5*b*c^6*d^4)/(3*x) - (d^4*(a + b*ArcTanh[c*x]))/(7*x^7) - (2*c*d^4*(a + b*ArcTanh[c*x]))/(3*x^6) - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/(5*x^5) - (c^3*d^4*(a + b*ArcTanh[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTanh[c*x]))/(3*x^3) + (176/105)*b*c^7*d^4*Log[x] - (117/70)*b*c^7*d^4*Log[1 - c*x] - (1/210)*b*c^7*d^4*Log[1 + c*x]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x), x, 16, (a*x)/(c^3*d) - (b*x)/(2*c^3*d) + (b*x^2)/(6*c^2*d) + (b*ArcTanh[c*x])/(2*c^4*d) + (b*x*ArcTanh[c*x])/(c^3*d) - (x^2*(a + b*ArcTanh[c*x]))/(2*c^2*d) + (x^3*(a + b*ArcTanh[c*x]))/(3*c*d) + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^4*d) + (2*b*Log[1 - c^2*x^2])/(3*c^4*d) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^4*d)} -{(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x), x, 11, -((a*x)/(c^2*d)) + (b*x)/(2*c^2*d) - (b*ArcTanh[c*x])/(2*c^3*d) - (b*x*ArcTanh[c*x])/(c^2*d) + (x^2*(a + b*ArcTanh[c*x]))/(2*c*d) - ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^3*d) - (b*Log[1 - c^2*x^2])/(2*c^3*d) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^3*d)} -{(x^1*(a + b*ArcTanh[c*x]))/(d + c*d*x), x, 7, (a*x)/(c*d) + (b*x*ArcTanh[c*x])/(c*d) + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d) + (b*Log[1 - c^2*x^2])/(2*c^2*d) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^2*d)} -{x^0*(a + b*ArcTanh[c*x])/(d + c*d*x), x, 3, -(((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c*d)) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c*d)} -{(a + b*ArcTanh[c*x])/(x^1*(d + c*d*x)), x, 2, ((a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d - (b*PolyLog[2, -1 + 2/(1 + c*x)])/(2*d)} -{(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)), x, 8, -((a + b*ArcTanh[c*x])/(d*x)) + (b*c*Log[x])/d - (b*c*Log[1 - c^2*x^2])/(2*d) - (c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d + (b*c*PolyLog[2, -1 + 2/(1 + c*x)])/(2*d)} -{(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)), x, 12, -(b*c)/(2*d*x) + (b*c^2*ArcTanh[c*x])/(2*d) - (a + b*ArcTanh[c*x])/(2*d*x^2) + (c*(a + b*ArcTanh[c*x]))/(d*x) - (b*c^2*Log[x])/d + (b*c^2*Log[1 - c^2*x^2])/(2*d) + (c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d - (b*c^2*PolyLog[2, -1 + 2/(1 + c*x)])/(2*d)} -{(a + b*ArcTanh[c*x])/(x^4*(d + c*d*x)), x, 17, -(b*c)/(6*d*x^2) + (b*c^2)/(2*d*x) - (b*c^3*ArcTanh[c*x])/(2*d) - (a + b*ArcTanh[c*x])/(3*d*x^3) + (c*(a + b*ArcTanh[c*x]))/(2*d*x^2) - (c^2*(a + b*ArcTanh[c*x]))/(d*x) + (4*b*c^3*Log[x])/(3*d) - (2*b*c^3*Log[1 - c^2*x^2])/(3*d) - (c^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d + (b*c^3*PolyLog[2, -1 + 2/(1 + c*x)])/(2*d)} - - -{(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2, x, 16, -((2*a*x)/(c^3*d^2)) + (b*x)/(2*c^3*d^2) + b/(2*c^4*d^2*(1 + c*x)) - (b*ArcTanh[c*x])/(c^4*d^2) - (2*b*x*ArcTanh[c*x])/(c^3*d^2) + (x^2*(a + b*ArcTanh[c*x]))/(2*c^2*d^2) + (a + b*ArcTanh[c*x])/(c^4*d^2*(1 + c*x)) - (3*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^4*d^2) - (b*Log[1 - c^2*x^2])/(c^4*d^2) + (3*b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^4*d^2)} -{(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2, x, 13, (a*x)/(c^2*d^2) - b/(2*c^3*d^2*(1 + c*x)) + (b*ArcTanh[c*x])/(2*c^3*d^2) + (b*x*ArcTanh[c*x])/(c^2*d^2) - (a + b*ArcTanh[c*x])/(c^3*d^2*(1 + c*x)) + (2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^3*d^2) + (b*Log[1 - c^2*x^2])/(2*c^3*d^2) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(c^3*d^2)} -{(x^1*(a + b*ArcTanh[c*x]))/(d + c*d*x)^2, x, 10, b/(2*c^2*d^2*(1 + c*x)) - (b*ArcTanh[c*x])/(2*c^2*d^2) + (a + b*ArcTanh[c*x])/(c^2*d^2*(1 + c*x)) - ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^2*d^2) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^2*d^2)} -{x^0*(a + b*ArcTanh[c*x])/(d + c*d*x)^2, x, 5, -b/(2*c*d^2*(1 + c*x)) + (b*ArcTanh[c*x])/(2*c*d^2) - (a + b*ArcTanh[c*x])/(c*d^2*(1 + c*x))} -{(a + b*ArcTanh[c*x])/(x^1*(d + c*d*x)^2), x, 11, b/(2*d^2*(1 + c*x)) - (b*ArcTanh[c*x])/(2*d^2) + (a + b*ArcTanh[c*x])/(d^2*(1 + c*x)) + (a*Log[x])/d^2 + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 - (b*PolyLog[2, (-c)*x])/(2*d^2) + (b*PolyLog[2, c*x])/(2*d^2) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^2)} -{(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^2), x, 16, -((b*c)/(2*d^2*(1 + c*x))) + (b*c*ArcTanh[c*x])/(2*d^2) - (a + b*ArcTanh[c*x])/(d^2*x) - (c*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) - (2*a*c*Log[x])/d^2 + (b*c*Log[x])/d^2 - (2*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 - (b*c*Log[1 - c^2*x^2])/(2*d^2) + (b*c*PolyLog[2, (-c)*x])/d^2 - (b*c*PolyLog[2, c*x])/d^2 + (b*c*PolyLog[2, 1 - 2/(1 + c*x)])/d^2} -{(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)^2), x, 19, -((b*c)/(2*d^2*x)) + (b*c^2)/(2*d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])/(2*d^2*x^2) + (2*c*(a + b*ArcTanh[c*x]))/(d^2*x) + (c^2*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) + (3*a*c^2*Log[x])/d^2 - (2*b*c^2*Log[x])/d^2 + (3*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 + (b*c^2*Log[1 - c^2*x^2])/d^2 - (3*b*c^2*PolyLog[2, (-c)*x])/(2*d^2) + (3*b*c^2*PolyLog[2, c*x])/(2*d^2) - (3*b*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^2)} - - -{(x^4*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3, x, 21, -((3*a*x)/(c^4*d^3)) + (b*x)/(2*c^4*d^3) - b/(8*c^5*d^3*(1 + c*x)^2) + (15*b)/(8*c^5*d^3*(1 + c*x)) - (19*b*ArcTanh[c*x])/(8*c^5*d^3) - (3*b*x*ArcTanh[c*x])/(c^4*d^3) + (x^2*(a + b*ArcTanh[c*x]))/(2*c^3*d^3) - (a + b*ArcTanh[c*x])/(2*c^5*d^3*(1 + c*x)^2) + (4*(a + b*ArcTanh[c*x]))/(c^5*d^3*(1 + c*x)) - (6*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^5*d^3) - (3*b*Log[1 - c^2*x^2])/(2*c^5*d^3) + (3*b*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^3)} -{(x^3*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3, x, 18, (a*x)/(c^3*d^3) + b/(8*c^4*d^3*(1 + c*x)^2) - (11*b)/(8*c^4*d^3*(1 + c*x)) + (11*b*ArcTanh[c*x])/(8*c^4*d^3) + (b*x*ArcTanh[c*x])/(c^3*d^3) + (a + b*ArcTanh[c*x])/(2*c^4*d^3*(1 + c*x)^2) - (3*(a + b*ArcTanh[c*x]))/(c^4*d^3*(1 + c*x)) + (3*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^4*d^3) + (b*Log[1 - c^2*x^2])/(2*c^4*d^3) - (3*b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^4*d^3)} -{(x^2*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3, x, 15, -(b/(8*c^3*d^3*(1 + c*x)^2)) + (7*b)/(8*c^3*d^3*(1 + c*x)) - (7*b*ArcTanh[c*x])/(8*c^3*d^3) - (a + b*ArcTanh[c*x])/(2*c^3*d^3*(1 + c*x)^2) + (2*(a + b*ArcTanh[c*x]))/(c^3*d^3*(1 + c*x)) - ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^3*d^3) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^3*d^3)} -{(x^1*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3, x, 5, b/(8*c^2*d^3*(1 + c*x)^2) - (3*b)/(8*c^2*d^3*(1 + c*x)) - (b*ArcTanh[c*x])/(8*c^2*d^3) + (x^2*(a + b*ArcTanh[c*x]))/(2*d^3*(1 + c*x)^2)} -{x^0*(a + b*ArcTanh[c*x])/(d + c*d*x)^3, x, 5, -b/(8*c*d^3*(1 + c*x)^2) - b/(8*c*d^3*(1 + c*x)) + (b*ArcTanh[c*x])/(8*c*d^3) - (a + b*ArcTanh[c*x])/(2*c*d^3*(1 + c*x)^2)} -{(a + b*ArcTanh[c*x])/(x^1*(d + c*d*x)^3), x, 16, b/(8*d^3*(1 + c*x)^2) + (5*b)/(8*d^3*(1 + c*x)) - (5*b*ArcTanh[c*x])/(8*d^3) + (a + b*ArcTanh[c*x])/(2*d^3*(1 + c*x)^2) + (a + b*ArcTanh[c*x])/(d^3*(1 + c*x)) + (a*Log[x])/d^3 + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^3 - (b*PolyLog[2, (-c)*x])/(2*d^3) + (b*PolyLog[2, c*x])/(2*d^3) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^3)} -{(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^3), x, 21, -((b*c)/(8*d^3*(1 + c*x)^2)) - (9*b*c)/(8*d^3*(1 + c*x)) + (9*b*c*ArcTanh[c*x])/(8*d^3) - (a + b*ArcTanh[c*x])/(d^3*x) - (c*(a + b*ArcTanh[c*x]))/(2*d^3*(1 + c*x)^2) - (2*c*(a + b*ArcTanh[c*x]))/(d^3*(1 + c*x)) - (3*a*c*Log[x])/d^3 + (b*c*Log[x])/d^3 - (3*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^3 - (b*c*Log[1 - c^2*x^2])/(2*d^3) + (3*b*c*PolyLog[2, (-c)*x])/(2*d^3) - (3*b*c*PolyLog[2, c*x])/(2*d^3) + (3*b*c*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^3)} -{(a + b*ArcTanh[c*x])/(x^3*(d + c*d*x)^3), x, 24, -((b*c)/(2*d^3*x)) + (b*c^2)/(8*d^3*(1 + c*x)^2) + (13*b*c^2)/(8*d^3*(1 + c*x)) - (9*b*c^2*ArcTanh[c*x])/(8*d^3) - (a + b*ArcTanh[c*x])/(2*d^3*x^2) + (3*c*(a + b*ArcTanh[c*x]))/(d^3*x) + (c^2*(a + b*ArcTanh[c*x]))/(2*d^3*(1 + c*x)^2) + (3*c^2*(a + b*ArcTanh[c*x]))/(d^3*(1 + c*x)) + (6*a*c^2*Log[x])/d^3 - (3*b*c^2*Log[x])/d^3 + (6*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^3 + (3*b*c^2*Log[1 - c^2*x^2])/(2*d^3) - (3*b*c^2*PolyLog[2, (-c)*x])/d^3 + (3*b*c^2*PolyLog[2, c*x])/d^3 - (3*b*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/d^3} - - -{(a + b*ArcTanh[c*x])/(1 + c*x)^4, x, 5, -(b/(18*c*(1 + c*x)^3)) - b/(24*c*(1 + c*x)^2) - b/(24*c*(1 + c*x)) + (b*ArcTanh[c*x])/(24*c) - (a + b*ArcTanh[c*x])/(3*c*(1 + c*x)^3)} - - -{ArcTanh[a*x]/(c*x + a*c*x^2), x, 3, (ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])/c - PolyLog[2, -1 + 2/(1 + a*x)]/(2*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^2 when c^2 d^2 - e^2=0*) - - -(* ::Subsubsection::Closed:: *) -(*q>0*) - - -{x^3*(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 27, (a*b*d*x)/(2*c^3) + (3*b^2*d*x)/(10*c^3) + (b^2*d*x^2)/(12*c^2) + (b^2*d*x^3)/(30*c) - (3*b^2*d*ArcTanh[c*x])/(10*c^4) + (b^2*d*x*ArcTanh[c*x])/(2*c^3) + (b*d*x^2*(a + b*ArcTanh[c*x]))/(5*c^2) + (b*d*x^3*(a + b*ArcTanh[c*x]))/(6*c) + (1/10)*b*d*x^4*(a + b*ArcTanh[c*x]) - (d*(a + b*ArcTanh[c*x])^2)/(20*c^4) + (1/4)*d*x^4*(a + b*ArcTanh[c*x])^2 + (1/5)*c*d*x^5*(a + b*ArcTanh[c*x])^2 - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^4) + (b^2*d*Log[1 - c^2*x^2])/(3*c^4) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^4)} -{x^2*(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 22, (a*b*d*x)/(2*c^2) + (b^2*d*x)/(3*c^2) + (b^2*d*x^2)/(12*c) - (b^2*d*ArcTanh[c*x])/(3*c^3) + (b^2*d*x*ArcTanh[c*x])/(2*c^2) + (b*d*x^2*(a + b*ArcTanh[c*x]))/(3*c) + (1/6)*b*d*x^3*(a + b*ArcTanh[c*x]) + (d*(a + b*ArcTanh[c*x])^2)/(12*c^3) + (1/3)*d*x^3*(a + b*ArcTanh[c*x])^2 + (1/4)*c*d*x^4*(a + b*ArcTanh[c*x])^2 - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^3) + (b^2*d*Log[1 - c^2*x^2])/(3*c^3) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^3)} -{x^1*(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 17, (a*b*d*x)/c + (b^2*d*x)/(3*c) - (b^2*d*ArcTanh[c*x])/(3*c^2) + (b^2*d*x*ArcTanh[c*x])/c + (1/3)*b*d*x^2*(a + b*ArcTanh[c*x]) - (d*(a + b*ArcTanh[c*x])^2)/(6*c^2) + (1/2)*d*x^2*(a + b*ArcTanh[c*x])^2 + (1/3)*c*d*x^3*(a + b*ArcTanh[c*x])^2 - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^2) + (b^2*d*Log[1 - c^2*x^2])/(2*c^2) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^2)} -{x^0*(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 9, a*b*d*x + b^2*d*x*ArcTanh[c*x] + (d*(1 + c*x)^2*(a + b*ArcTanh[c*x])^2)/(2*c) - (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (b^2*d*Log[1 - c^2*x^2])/(2*c) - (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/c} -{((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^1, x, 13, d*(a + b*ArcTanh[c*x])^2 + c*d*x*(a + b*ArcTanh[c*x])^2 + 2*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - b^2*d*PolyLog[2, 1 - 2/(1 - c*x)] - b*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (1/2)*b^2*d*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*d*PolyLog[3, -1 + 2/(1 - c*x)]} -{((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^2, x, 12, c*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/x + 2*c*d*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + 2*b*c*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c*d*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d*PolyLog[2, -1 + 2/(1 + c*x)] + (1/2)*b^2*c*d*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*c*d*PolyLog[3, -1 + 2/(1 - c*x)]} -{((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^3, x, 14, -((b*c*d*(a + b*ArcTanh[c*x]))/x) + (3/2)*c^2*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (c*d*(a + b*ArcTanh[c*x])^2)/x + b^2*c^2*d*Log[x] - (1/2)*b^2*c^2*d*Log[1 - c^2*x^2] + 2*b*c^2*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c^2*d*PolyLog[2, -1 + 2/(1 + c*x)]} -{((d + c*d*x)*(a + b*ArcTanh[c*x])^2)/x^4, x, 18, -((b^2*c^2*d)/(3*x)) + (1/3)*b^2*c^3*d*ArcTanh[c*x] - (b*c*d*(a + b*ArcTanh[c*x]))/(3*x^2) - (b*c^2*d*(a + b*ArcTanh[c*x]))/x + (5/6)*c^3*d*(a + b*ArcTanh[c*x])^2 - (d*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (c*d*(a + b*ArcTanh[c*x])^2)/(2*x^2) + b^2*c^3*d*Log[x] - (1/2)*b^2*c^3*d*Log[1 - c^2*x^2] + (2/3)*b*c^3*d*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (1/3)*b^2*c^3*d*PolyLog[2, -1 + 2/(1 + c*x)]} - - -{x^3*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 43, (5*a*b*d^2*x)/(6*c^3) + (3*b^2*d^2*x)/(5*c^3) + (31*b^2*d^2*x^2)/(180*c^2) + (b^2*d^2*x^3)/(15*c) + (1/60)*b^2*d^2*x^4 - (3*b^2*d^2*ArcTanh[c*x])/(5*c^4) + (5*b^2*d^2*x*ArcTanh[c*x])/(6*c^3) + (2*b*d^2*x^2*(a + b*ArcTanh[c*x]))/(5*c^2) + (5*b*d^2*x^3*(a + b*ArcTanh[c*x]))/(18*c) + (1/5)*b*d^2*x^4*(a + b*ArcTanh[c*x]) + (1/15)*b*c*d^2*x^5*(a + b*ArcTanh[c*x]) - (d^2*(a + b*ArcTanh[c*x])^2)/(60*c^4) + (1/4)*d^2*x^4*(a + b*ArcTanh[c*x])^2 + (2/5)*c*d^2*x^5*(a + b*ArcTanh[c*x])^2 + (1/6)*c^2*d^2*x^6*(a + b*ArcTanh[c*x])^2 - (4*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^4) + (53*b^2*d^2*Log[1 - c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^4)} -{x^2*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 36, (a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + (b^2*d^2*x^2)/(6*c) + (1/30)*b^2*d^2*x^3 - (19*b^2*d^2*ArcTanh[c*x])/(30*c^3) + (b^2*d^2*x*ArcTanh[c*x])/c^2 + (8*b*d^2*x^2*(a + b*ArcTanh[c*x]))/(15*c) + (1/3)*b*d^2*x^3*(a + b*ArcTanh[c*x]) + (1/10)*b*c*d^2*x^4*(a + b*ArcTanh[c*x]) + (d^2*(a + b*ArcTanh[c*x])^2)/(30*c^3) + (1/3)*d^2*x^3*(a + b*ArcTanh[c*x])^2 + (1/2)*c*d^2*x^4*(a + b*ArcTanh[c*x])^2 + (1/5)*c^2*d^2*x^5*(a + b*ArcTanh[c*x])^2 - (16*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(15*c^3) + (2*b^2*d^2*Log[1 - c^2*x^2])/(3*c^3) - (8*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(15*c^3)} -{x^1*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 28, (3*a*b*d^2*x)/(2*c) + (2*b^2*d^2*x)/(3*c) + (1/12)*b^2*d^2*x^2 - (2*b^2*d^2*ArcTanh[c*x])/(3*c^2) + (3*b^2*d^2*x*ArcTanh[c*x])/(2*c) + (2/3)*b*d^2*x^2*(a + b*ArcTanh[c*x]) + (1/6)*b*c*d^2*x^3*(a + b*ArcTanh[c*x]) - (d^2*(a + b*ArcTanh[c*x])^2)/(12*c^2) + (1/2)*d^2*x^2*(a + b*ArcTanh[c*x])^2 + (2/3)*c*d^2*x^3*(a + b*ArcTanh[c*x])^2 + (1/4)*c^2*d^2*x^4*(a + b*ArcTanh[c*x])^2 - (4*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^2) + (5*b^2*d^2*Log[1 - c^2*x^2])/(6*c^2) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^2)} -{x^0*(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 12, 2*a*b*d^2*x + (1/3)*b^2*d^2*x - (b^2*d^2*ArcTanh[c*x])/(3*c) + 2*b^2*d^2*x*ArcTanh[c*x] + (1/3)*b*c*d^2*x^2*(a + b*ArcTanh[c*x]) + (d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x])^2)/(3*c) - (8*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c) + (b^2*d^2*Log[1 - c^2*x^2])/c - (4*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c)} -{(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^1, x, 19, a*b*c*d^2*x + b^2*c*d^2*x*ArcTanh[c*x] + (3/2)*d^2*(a + b*ArcTanh[c*x])^2 + 2*c*d^2*x*(a + b*ArcTanh[c*x])^2 + (1/2)*c^2*d^2*x^2*(a + b*ArcTanh[c*x])^2 + 2*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 4*b*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + (1/2)*b^2*d^2*Log[1 - c^2*x^2] - 2*b^2*d^2*PolyLog[2, 1 - 2/(1 - c*x)] - b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (1/2)*b^2*d^2*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*d^2*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^2, x, 17, 2*c*d^2*(a + b*ArcTanh[c*x])^2 - (d^2*(a + b*ArcTanh[c*x])^2)/x + c^2*d^2*x*(a + b*ArcTanh[c*x])^2 + 4*c*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 2*b*c*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + 2*b*c*d^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c*d^2*PolyLog[2, 1 - 2/(1 - c*x)] - 2*b*c*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 2*b*c*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d^2*PolyLog[2, -1 + 2/(1 + c*x)] + b^2*c*d^2*PolyLog[3, 1 - 2/(1 - c*x)] - b^2*c*d^2*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^3, x, 20, -((b*c*d^2*(a + b*ArcTanh[c*x]))/x) + (5/2)*c^2*d^2*(a + b*ArcTanh[c*x])^2 - (d^2*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (2*c*d^2*(a + b*ArcTanh[c*x])^2)/x + 2*c^2*d^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + b^2*c^2*d^2*Log[x] - (1/2)*b^2*c^2*d^2*Log[1 - c^2*x^2] + 4*b*c^2*d^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b*c^2*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c^2*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - 2*b^2*c^2*d^2*PolyLog[2, -1 + 2/(1 + c*x)] + (1/2)*b^2*c^2*d^2*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*c^2*d^2*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^4, x, 14, -((b^2*c^2*d^2)/(3*x)) + (1/3)*b^2*c^3*d^2*ArcTanh[c*x] - (b*c*d^2*(a + b*ArcTanh[c*x]))/(3*x^2) - (2*b*c^2*d^2*(a + b*ArcTanh[c*x]))/x - (d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) + (8/3)*a*b*c^3*d^2*Log[x] + 2*b^2*c^3*d^2*Log[x] + (8/3)*b*c^3*d^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - b^2*c^3*d^2*Log[1 - c^2*x^2] - (4/3)*b^2*c^3*d^2*PolyLog[2, (-c)*x] + (4/3)*b^2*c^3*d^2*PolyLog[2, c*x] + (4/3)*b^2*c^3*d^2*PolyLog[2, 1 - 2/(1 - c*x)]} - - -{x^3*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 62, (3*a*b*d^3*x)/(2*c^3) + (122*b^2*d^3*x)/(105*c^3) + (7*b^2*d^3*x^2)/(20*c^2) + (44*b^2*d^3*x^3)/(315*c) + (1/20)*b^2*d^3*x^4 + (1/105)*b^2*c*d^3*x^5 - (122*b^2*d^3*ArcTanh[c*x])/(105*c^4) + (3*b^2*d^3*x*ArcTanh[c*x])/(2*c^3) + (26*b*d^3*x^2*(a + b*ArcTanh[c*x]))/(35*c^2) + (b*d^3*x^3*(a + b*ArcTanh[c*x]))/(2*c) + (13/35)*b*d^3*x^4*(a + b*ArcTanh[c*x]) + (1/5)*b*c*d^3*x^5*(a + b*ArcTanh[c*x]) + (1/21)*b*c^2*d^3*x^6*(a + b*ArcTanh[c*x]) - (d^3*(a + b*ArcTanh[c*x])^2)/(140*c^4) + (1/4)*d^3*x^4*(a + b*ArcTanh[c*x])^2 + (3/5)*c*d^3*x^5*(a + b*ArcTanh[c*x])^2 + (1/2)*c^2*d^3*x^6*(a + b*ArcTanh[c*x])^2 + (1/7)*c^3*d^3*x^7*(a + b*ArcTanh[c*x])^2 - (52*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(35*c^4) + (11*b^2*d^3*Log[1 - c^2*x^2])/(10*c^4) - (26*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/(35*c^4)} -{x^2*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 52, (11*a*b*d^3*x)/(6*c^2) + (37*b^2*d^3*x)/(30*c^2) + (61*b^2*d^3*x^2)/(180*c) + (1/10)*b^2*d^3*x^3 + (1/60)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTanh[c*x])/(30*c^3) + (11*b^2*d^3*x*ArcTanh[c*x])/(6*c^2) + (14*b*d^3*x^2*(a + b*ArcTanh[c*x]))/(15*c) + (11/18)*b*d^3*x^3*(a + b*ArcTanh[c*x]) + (3/10)*b*c*d^3*x^4*(a + b*ArcTanh[c*x]) + (1/15)*b*c^2*d^3*x^5*(a + b*ArcTanh[c*x]) + (d^3*(a + b*ArcTanh[c*x])^2)/(60*c^3) + (1/3)*d^3*x^3*(a + b*ArcTanh[c*x])^2 + (3/4)*c*d^3*x^4*(a + b*ArcTanh[c*x])^2 + (3/5)*c^2*d^3*x^5*(a + b*ArcTanh[c*x])^2 + (1/6)*c^3*d^3*x^6*(a + b*ArcTanh[c*x])^2 - (28*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(15*c^3) + (113*b^2*d^3*Log[1 - c^2*x^2])/(90*c^3) - (14*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/(15*c^3)} -{x^1*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 38, (5*a*b*d^3*x)/(2*c) + (13*b^2*d^3*x)/(10*c) + (1/4)*b^2*d^3*x^2 + (1/30)*b^2*c*d^3*x^3 - (13*b^2*d^3*ArcTanh[c*x])/(10*c^2) + (5*b^2*d^3*x*ArcTanh[c*x])/(2*c) + (6/5)*b*d^3*x^2*(a + b*ArcTanh[c*x]) + (1/2)*b*c*d^3*x^3*(a + b*ArcTanh[c*x]) + (1/10)*b*c^2*d^3*x^4*(a + b*ArcTanh[c*x]) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(4*c^2) + (d^3*(1 + c*x)^5*(a + b*ArcTanh[c*x])^2)/(5*c^2) - (12*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(5*c^2) + (3*b^2*d^3*Log[1 - c^2*x^2])/(2*c^2) - (6*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/(5*c^2)} -{x^0*(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 16, (7/2)*a*b*d^3*x + b^2*d^3*x + (1/12)*b^2*c*d^3*x^2 - (b^2*d^3*ArcTanh[c*x])/c + (7/2)*b^2*d^3*x*ArcTanh[c*x] + b*c*d^3*x^2*(a + b*ArcTanh[c*x]) + (1/6)*b*c^2*d^3*x^3*(a + b*ArcTanh[c*x]) + (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(4*c) - (4*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c + (11*b^2*d^3*Log[1 - c^2*x^2])/(6*c) - (2*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)])/c} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^1, x, 28, 3*a*b*c*d^3*x + (1/3)*b^2*c*d^3*x - (1/3)*b^2*d^3*ArcTanh[c*x] + 3*b^2*c*d^3*x*ArcTanh[c*x] + (1/3)*b*c^2*d^3*x^2*(a + b*ArcTanh[c*x]) + (11/6)*d^3*(a + b*ArcTanh[c*x])^2 + 3*c*d^3*x*(a + b*ArcTanh[c*x])^2 + (3/2)*c^2*d^3*x^2*(a + b*ArcTanh[c*x])^2 + (1/3)*c^3*d^3*x^3*(a + b*ArcTanh[c*x])^2 + 2*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - (20/3)*b*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + (3/2)*b^2*d^3*Log[1 - c^2*x^2] - (10/3)*b^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] + (1/2)*b^2*d^3*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*d^3*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^2, x, 23, a*b*c^2*d^3*x + b^2*c^2*d^3*x*ArcTanh[c*x] + (7/2)*c*d^3*(a + b*ArcTanh[c*x])^2 - (d^3*(a + b*ArcTanh[c*x])^2)/x + 3*c^2*d^3*x*(a + b*ArcTanh[c*x])^2 + (1/2)*c^3*d^3*x^2*(a + b*ArcTanh[c*x])^2 + 6*c*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] - 6*b*c*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + (1/2)*b^2*c*d^3*Log[1 - c^2*x^2] + 2*b*c*d^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - 3*b^2*c*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - 3*b*c*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 3*b*c*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - b^2*c*d^3*PolyLog[2, -1 + 2/(1 + c*x)] + (3/2)*b^2*c*d^3*PolyLog[3, 1 - 2/(1 - c*x)] - (3/2)*b^2*c*d^3*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^3, x, 25, -((b*c*d^3*(a + b*ArcTanh[c*x]))/x) + (9/2)*c^2*d^3*(a + b*ArcTanh[c*x])^2 - (d^3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c*d^3*(a + b*ArcTanh[c*x])^2)/x + c^3*d^3*x*(a + b*ArcTanh[c*x])^2 + 6*c^2*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + b^2*c^2*d^3*Log[x] - 2*b*c^2*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - (1/2)*b^2*c^2*d^3*Log[1 - c^2*x^2] + 6*b*c^2*d^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - 3*b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + 3*b*c^2*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - 3*b^2*c^2*d^3*PolyLog[2, -1 + 2/(1 + c*x)] + (3/2)*b^2*c^2*d^3*PolyLog[3, 1 - 2/(1 - c*x)] - (3/2)*b^2*c^2*d^3*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^4, x, 28, -((b^2*c^2*d^3)/(3*x)) + (1/3)*b^2*c^3*d^3*ArcTanh[c*x] - (b*c*d^3*(a + b*ArcTanh[c*x]))/(3*x^2) - (3*b*c^2*d^3*(a + b*ArcTanh[c*x]))/x + (29/6)*c^3*d^3*(a + b*ArcTanh[c*x])^2 - (d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) - (3*c*d^3*(a + b*ArcTanh[c*x])^2)/(2*x^2) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/x + 2*c^3*d^3*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)] + 3*b^2*c^3*d^3*Log[x] - (3/2)*b^2*c^3*d^3*Log[1 - c^2*x^2] + (20/3)*b*c^3*d^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)] + b*c^3*d^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)] - (10/3)*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 + c*x)] + (1/2)*b^2*c^3*d^3*PolyLog[3, 1 - 2/(1 - c*x)] - (1/2)*b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^5, x, 18, -((b^2*c^2*d^3)/(12*x^2)) - (b^2*c^3*d^3)/x + b^2*c^4*d^3*ArcTanh[c*x] - (b*c*d^3*(a + b*ArcTanh[c*x]))/(6*x^3) - (b*c^2*d^3*(a + b*ArcTanh[c*x]))/x^2 - (7*b*c^3*d^3*(a + b*ArcTanh[c*x]))/(2*x) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(4*x^4) + 4*a*b*c^4*d^3*Log[x] + (11/3)*b^2*c^4*d^3*Log[x] + 4*b*c^4*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - (11/6)*b^2*c^4*d^3*Log[1 - c^2*x^2] - 2*b^2*c^4*d^3*PolyLog[2, (-c)*x] + 2*b^2*c^4*d^3*PolyLog[2, c*x] + 2*b^2*c^4*d^3*PolyLog[2, 1 - 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^6, x, 22, -((b^2*c^2*d^3)/(30*x^3)) - (b^2*c^3*d^3)/(4*x^2) - (13*b^2*c^4*d^3)/(10*x) + (13/10)*b^2*c^5*d^3*ArcTanh[c*x] - (b*c*d^3*(a + b*ArcTanh[c*x]))/(10*x^4) - (b*c^2*d^3*(a + b*ArcTanh[c*x]))/(2*x^3) - (6*b*c^3*d^3*(a + b*ArcTanh[c*x]))/(5*x^2) - (5*b*c^4*d^3*(a + b*ArcTanh[c*x]))/(2*x) - (d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(5*x^5) + (c*d^3*(1 + c*x)^4*(a + b*ArcTanh[c*x])^2)/(20*x^4) + (12/5)*a*b*c^5*d^3*Log[x] + 3*b^2*c^5*d^3*Log[x] + (12/5)*b*c^5*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] - (3/2)*b^2*c^5*d^3*Log[1 - c^2*x^2] - (6/5)*b^2*c^5*d^3*PolyLog[2, (-c)*x] + (6/5)*b^2*c^5*d^3*PolyLog[2, c*x] + (6/5)*b^2*c^5*d^3*PolyLog[2, 1 - 2/(1 - c*x)]} -{(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^7, x, 29, -((b^2*c^2*d^3)/(60*x^4)) - (b^2*c^3*d^3)/(10*x^3) - (61*b^2*c^4*d^3)/(180*x^2) - (37*b^2*c^5*d^3)/(30*x) + (37/30)*b^2*c^6*d^3*ArcTanh[c*x] - (b*c*d^3*(a + b*ArcTanh[c*x]))/(15*x^5) - (3*b*c^2*d^3*(a + b*ArcTanh[c*x]))/(10*x^4) - (11*b*c^3*d^3*(a + b*ArcTanh[c*x]))/(18*x^3) - (14*b*c^4*d^3*(a + b*ArcTanh[c*x]))/(15*x^2) - (11*b*c^5*d^3*(a + b*ArcTanh[c*x]))/(6*x) - (d^3*(a + b*ArcTanh[c*x])^2)/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x])^2)/(5*x^5) - (3*c^2*d^3*(a + b*ArcTanh[c*x])^2)/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x])^2)/(3*x^3) + (28/15)*a*b*c^6*d^3*Log[x] + (113/45)*b^2*c^6*d^3*Log[x] + (37/20)*b*c^6*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)] + (1/60)*b*c^6*d^3*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)] - (113/90)*b^2*c^6*d^3*Log[1 - c^2*x^2] - (14/15)*b^2*c^6*d^3*PolyLog[2, (-c)*x] + (14/15)*b^2*c^6*d^3*PolyLog[2, c*x] + (37/40)*b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 - c*x)] - (1/120)*b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 + c*x)]} - - -(* ::Subsubsection::Closed:: *) -(*q<0*) - - -{x^3/(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 26, -((a*b*x)/(c^3*d)) + (b^2*x)/(3*c^3*d) - (b^2*ArcTanh[c*x])/(3*c^4*d) - (b^2*x*ArcTanh[c*x])/(c^3*d) + (b*x^2*(a + b*ArcTanh[c*x]))/(3*c^2*d) + (11*(a + b*ArcTanh[c*x])^2)/(6*c^4*d) + (x*(a + b*ArcTanh[c*x])^2)/(c^3*d) - (x^2*(a + b*ArcTanh[c*x])^2)/(2*c^2*d) + (x^3*(a + b*ArcTanh[c*x])^2)/(3*c*d) - (8*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^4*d) + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^4*d) - (b^2*Log[1 - c^2*x^2])/(2*c^4*d) - (4*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^4*d) - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^4*d) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^4*d)} -{x^2/(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 16, (a*b*x)/(c^2*d) + (b^2*x*ArcTanh[c*x])/(c^2*d) - (3*(a + b*ArcTanh[c*x])^2)/(2*c^3*d) - (x*(a + b*ArcTanh[c*x])^2)/(c^2*d) + (x^2*(a + b*ArcTanh[c*x])^2)/(2*c*d) + (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^3*d) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d) + (b^2*Log[1 - c^2*x^2])/(2*c^3*d) + (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^3*d) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^3*d) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^3*d)} -{x^1/(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 9, (a + b*ArcTanh[c*x])^2/(c^2*d) + (x*(a + b*ArcTanh[c*x])^2)/(c*d) - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^2*d) + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^2*d) - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^2*d) - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^2*d)} -{x^0/(d + c*d*x)*(a + b*ArcTanh[c*x])^2, x, 3, -(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c*d)) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c*d) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c*d)} -{1/(d + c*d*x)*(a + b*ArcTanh[c*x])^2/x^1, x, 3, ((a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)])/d - (b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)])/d - (b^2*PolyLog[3, -1 + 2/(1 + c*x)])/(2*d)} -{1/(d + c*d*x)*(a + b*ArcTanh[c*x])^2/x^2, x, 8, (c*(a + b*ArcTanh[c*x])^2)/d - (a + b*ArcTanh[c*x])^2/(d*x) + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d - (c*(a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)])/d - (b^2*c*PolyLog[2, -1 + 2/(1 + c*x)])/d + (b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)])/d + (b^2*c*PolyLog[3, -1 + 2/(1 + c*x)])/(2*d)} -{1/(d + c*d*x)*(a + b*ArcTanh[c*x])^2/x^3, x, 17, -((b*c*(a + b*ArcTanh[c*x]))/(d*x)) - (c^2*(a + b*ArcTanh[c*x])^2)/(2*d) - (a + b*ArcTanh[c*x])^2/(2*d*x^2) + (c*(a + b*ArcTanh[c*x])^2)/(d*x) + (b^2*c^2*Log[x])/d - (b^2*c^2*Log[1 - c^2*x^2])/(2*d) - (2*b*c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d + (c^2*(a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)])/d + (b^2*c^2*PolyLog[2, -1 + 2/(1 + c*x)])/d - (b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)])/d - (b^2*c^2*PolyLog[3, -1 + 2/(1 + c*x)])/(2*d)} -{1/(d + c*d*x)*(a + b*ArcTanh[c*x])^2/x^4, x, 26, -((b^2*c^2)/(3*d*x)) + (b^2*c^3*ArcTanh[c*x])/(3*d) - (b*c*(a + b*ArcTanh[c*x]))/(3*d*x^2) + (b*c^2*(a + b*ArcTanh[c*x]))/(d*x) + (5*c^3*(a + b*ArcTanh[c*x])^2)/(6*d) - (a + b*ArcTanh[c*x])^2/(3*d*x^3) + (c*(a + b*ArcTanh[c*x])^2)/(2*d*x^2) - (c^2*(a + b*ArcTanh[c*x])^2)/(d*x) - (b^2*c^3*Log[x])/d + (b^2*c^3*Log[1 - c^2*x^2])/(2*d) + (8*b*c^3*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/(3*d) - (c^3*(a + b*ArcTanh[c*x])^2*Log[2 - 2/(1 + c*x)])/d - (4*b^2*c^3*PolyLog[2, -1 + 2/(1 + c*x)])/(3*d) + (b*c^3*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 + c*x)])/d + (b^2*c^3*PolyLog[3, -1 + 2/(1 + c*x)])/(2*d)} - - -{x^4/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 33, -((2*a*b*x)/(c^4*d^2)) + (b^2*x)/(3*c^4*d^2) - b^2/(2*c^5*d^2*(1 + c*x)) + (b^2*ArcTanh[c*x])/(6*c^5*d^2) - (2*b^2*x*ArcTanh[c*x])/(c^4*d^2) + (b*x^2*(a + b*ArcTanh[c*x]))/(3*c^3*d^2) - (b*(a + b*ArcTanh[c*x]))/(c^5*d^2*(1 + c*x)) + (29*(a + b*ArcTanh[c*x])^2)/(6*c^5*d^2) + (3*x*(a + b*ArcTanh[c*x])^2)/(c^4*d^2) - (x^2*(a + b*ArcTanh[c*x])^2)/(c^3*d^2) + (x^3*(a + b*ArcTanh[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^5*d^2*(1 + c*x)) - (20*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(3*c^5*d^2) + (4*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^5*d^2) - (b^2*Log[1 - c^2*x^2])/(c^5*d^2) - (10*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(3*c^5*d^2) - (4*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^2) - (2*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^5*d^2)} -{x^3/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 24, (a*b*x)/(c^3*d^2) + b^2/(2*c^4*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*c^4*d^2) + (b^2*x*ArcTanh[c*x])/(c^3*d^2) + (b*(a + b*ArcTanh[c*x]))/(c^4*d^2*(1 + c*x)) - (3*(a + b*ArcTanh[c*x])^2)/(c^4*d^2) - (2*x*(a + b*ArcTanh[c*x])^2)/(c^3*d^2) + (x^2*(a + b*ArcTanh[c*x])^2)/(2*c^2*d^2) + (a + b*ArcTanh[c*x])^2/(c^4*d^2*(1 + c*x)) + (4*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^4*d^2) - (3*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^4*d^2) + (b^2*Log[1 - c^2*x^2])/(2*c^4*d^2) + (2*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^4*d^2) + (3*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^4*d^2) + (3*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^4*d^2)} -{x^2/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 18, -(b^2/(2*c^3*d^2*(1 + c*x))) + (b^2*ArcTanh[c*x])/(2*c^3*d^2) - (b*(a + b*ArcTanh[c*x]))/(c^3*d^2*(1 + c*x)) + (3*(a + b*ArcTanh[c*x])^2)/(2*c^3*d^2) + (x*(a + b*ArcTanh[c*x])^2)/(c^2*d^2) - (a + b*ArcTanh[c*x])^2/(c^3*d^2*(1 + c*x)) - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^3*d^2) + (2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^2) - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^3*d^2) - (2*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^3*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^3*d^2)} -{x^1/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 13, b^2/(2*c^2*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*c^2*d^2) + (b*(a + b*ArcTanh[c*x]))/(c^2*d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])^2/(2*c^2*d^2) + (a + b*ArcTanh[c*x])^2/(c^2*d^2*(1 + c*x)) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^2*d^2) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^2*d^2)} -{x^0/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2, x, 8, -(b^2/(2*c*d^2*(1 + c*x))) + (b^2*ArcTanh[c*x])/(2*c*d^2) - (b*(a + b*ArcTanh[c*x]))/(c*d^2*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(2*c*d^2) - (a + b*ArcTanh[c*x])^2/(c*d^2*(1 + c*x))} -{1/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^1, x, 19, b^2/(2*d^2*(1 + c*x)) - (b^2*ArcTanh[c*x])/(2*d^2) + (b*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) - (a + b*ArcTanh[c*x])^2/(2*d^2) + (a + b*ArcTanh[c*x])^2/(d^2*(1 + c*x)) + (2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 + (b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 + (b^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) - (b^2*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^2)} -{1/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^2, x, 23, -((b^2*c)/(2*d^2*(1 + c*x))) + (b^2*c*ArcTanh[c*x])/(2*d^2) - (b*c*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) + (3*c*(a + b*ArcTanh[c*x])^2)/(2*d^2) - (a + b*ArcTanh[c*x])^2/(d^2*x) - (c*(a + b*ArcTanh[c*x])^2)/(d^2*(1 + c*x)) - (4*c*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 - (2*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 - (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 - (b^2*c*PolyLog[2, -1 + 2/(1 + c*x)])/d^2 - (b^2*c*PolyLog[3, 1 - 2/(1 - c*x)])/d^2 + (b^2*c*PolyLog[3, -1 + 2/(1 - c*x)])/d^2 + (b^2*c*PolyLog[3, 1 - 2/(1 + c*x)])/d^2} -{1/(d + c*d*x)^2*(a + b*ArcTanh[c*x])^2/x^3, x, 31, (b^2*c^2)/(2*d^2*(1 + c*x)) - (b^2*c^2*ArcTanh[c*x])/(2*d^2) - (b*c*(a + b*ArcTanh[c*x]))/(d^2*x) + (b*c^2*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) - (2*c^2*(a + b*ArcTanh[c*x])^2)/d^2 - (a + b*ArcTanh[c*x])^2/(2*d^2*x^2) + (2*c*(a + b*ArcTanh[c*x])^2)/(d^2*x) + (c^2*(a + b*ArcTanh[c*x])^2)/(d^2*(1 + c*x)) + (6*c^2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 + (b^2*c^2*Log[x])/d^2 + (3*c^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 - (b^2*c^2*Log[1 - c^2*x^2])/(2*d^2) - (4*b*c^2*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d^2 - (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 + (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 - (3*b*c^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 + (2*b^2*c^2*PolyLog[2, -1 + 2/(1 + c*x)])/d^2 + (3*b^2*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) - (3*b^2*c^2*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^2) - (3*b^2*c^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^2)} - - -{x^4/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 37, (a*b*x)/(c^4*d^3) - b^2/(16*c^5*d^3*(1 + c*x)^2) + (29*b^2)/(16*c^5*d^3*(1 + c*x)) - (29*b^2*ArcTanh[c*x])/(16*c^5*d^3) + (b^2*x*ArcTanh[c*x])/(c^4*d^3) - (b*(a + b*ArcTanh[c*x]))/(4*c^5*d^3*(1 + c*x)^2) + (15*b*(a + b*ArcTanh[c*x]))/(4*c^5*d^3*(1 + c*x)) - (43*(a + b*ArcTanh[c*x])^2)/(8*c^5*d^3) - (3*x*(a + b*ArcTanh[c*x])^2)/(c^4*d^3) + (x^2*(a + b*ArcTanh[c*x])^2)/(2*c^3*d^3) - (a + b*ArcTanh[c*x])^2/(2*c^5*d^3*(1 + c*x)^2) + (4*(a + b*ArcTanh[c*x])^2)/(c^5*d^3*(1 + c*x)) + (6*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^5*d^3) - (6*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^5*d^3) + (b^2*Log[1 - c^2*x^2])/(2*c^5*d^3) + (3*b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^5*d^3) + (6*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^3) + (3*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(c^5*d^3)} -{x^3/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 31, b^2/(16*c^4*d^3*(1 + c*x)^2) - (21*b^2)/(16*c^4*d^3*(1 + c*x)) + (21*b^2*ArcTanh[c*x])/(16*c^4*d^3) + (b*(a + b*ArcTanh[c*x]))/(4*c^4*d^3*(1 + c*x)^2) - (11*b*(a + b*ArcTanh[c*x]))/(4*c^4*d^3*(1 + c*x)) + (19*(a + b*ArcTanh[c*x])^2)/(8*c^4*d^3) + (x*(a + b*ArcTanh[c*x])^2)/(c^3*d^3) + (a + b*ArcTanh[c*x])^2/(2*c^4*d^3*(1 + c*x)^2) - (3*(a + b*ArcTanh[c*x])^2)/(c^4*d^3*(1 + c*x)) - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c^4*d^3) + (3*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^4*d^3) - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c^4*d^3) - (3*b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^4*d^3) - (3*b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^4*d^3)} -{x^2/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 26, -(b^2/(16*c^3*d^3*(1 + c*x)^2)) + (13*b^2)/(16*c^3*d^3*(1 + c*x)) - (13*b^2*ArcTanh[c*x])/(16*c^3*d^3) - (b*(a + b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)^2) + (7*b*(a + b*ArcTanh[c*x]))/(4*c^3*d^3*(1 + c*x)) - (7*(a + b*ArcTanh[c*x])^2)/(8*c^3*d^3) - (a + b*ArcTanh[c*x])^2/(2*c^3*d^3*(1 + c*x)^2) + (2*(a + b*ArcTanh[c*x])^2)/(c^3*d^3*(1 + c*x)) - ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(c^3*d^3) + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(c^3*d^3) + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c^3*d^3)} -{x^1/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 13, b^2/(16*c^2*d^3*(1 + c*x)^2) - (5*b^2)/(16*c^2*d^3*(1 + c*x)) + (5*b^2*ArcTanh[c*x])/(16*c^2*d^3) + (b*(a + b*ArcTanh[c*x]))/(4*c^2*d^3*(1 + c*x)^2) - (3*b*(a + b*ArcTanh[c*x]))/(4*c^2*d^3*(1 + c*x)) - (a + b*ArcTanh[c*x])^2/(8*c^2*d^3) + (x^2*(a + b*ArcTanh[c*x])^2)/(2*d^3*(1 + c*x)^2)} -{x^0/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2, x, 13, -(b^2/(16*c*d^3*(1 + c*x)^2)) - (3*b^2)/(16*c*d^3*(1 + c*x)) + (3*b^2*ArcTanh[c*x])/(16*c*d^3) - (b*(a + b*ArcTanh[c*x]))/(4*c*d^3*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x]))/(4*c*d^3*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(8*c*d^3) - (a + b*ArcTanh[c*x])^2/(2*c*d^3*(1 + c*x)^2)} -{1/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^1, x, 32, b^2/(16*d^3*(1 + c*x)^2) + (11*b^2)/(16*d^3*(1 + c*x)) - (11*b^2*ArcTanh[c*x])/(16*d^3) + (b*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)^2) + (5*b*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)) - (5*(a + b*ArcTanh[c*x])^2)/(8*d^3) + (a + b*ArcTanh[c*x])^2/(2*d^3*(1 + c*x)^2) + (a + b*ArcTanh[c*x])^2/(d^3*(1 + c*x)) + (2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^3 + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^3 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^3 + (b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^3 - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^3 + (b^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^3) - (b^2*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^3) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^3)} -{1/(d + c*d*x)^3*(a + b*ArcTanh[c*x])^2/x^2, x, 36, -((b^2*c)/(16*d^3*(1 + c*x)^2)) - (19*b^2*c)/(16*d^3*(1 + c*x)) + (19*b^2*c*ArcTanh[c*x])/(16*d^3) - (b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)^2) - (9*b*c*(a + b*ArcTanh[c*x]))/(4*d^3*(1 + c*x)) + (17*c*(a + b*ArcTanh[c*x])^2)/(8*d^3) - (a + b*ArcTanh[c*x])^2/(d^3*x) - (c*(a + b*ArcTanh[c*x])^2)/(2*d^3*(1 + c*x)^2) - (2*c*(a + b*ArcTanh[c*x])^2)/(d^3*(1 + c*x)) - (6*c*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^3 - (3*c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^3 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d^3 + (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^3 - (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^3 + (3*b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^3 - (b^2*c*PolyLog[2, -1 + 2/(1 + c*x)])/d^3 - (3*b^2*c*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^3) + (3*b^2*c*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^3) + (3*b^2*c*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^3)} - - -{(a + b*ArcTanh[c*x])^2/(1 + c*x)^4, x, 18, -(b^2/(54*c*(1 + c*x)^3)) - (5*b^2)/(144*c*(1 + c*x)^2) - (11*b^2)/(144*c*(1 + c*x)) + (11*b^2*ArcTanh[c*x])/(144*c) - (b*(a + b*ArcTanh[c*x]))/(9*c*(1 + c*x)^3) - (b*(a + b*ArcTanh[c*x]))/(12*c*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x]))/(12*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(24*c) - (a + b*ArcTanh[c*x])^2/(3*c*(1 + c*x)^3)} - - -{ArcTanh[a*x]^2/(c*x - a*c*x^2), x, 4, (ArcTanh[a*x]^2*Log[2 - 2/(1 - a*x)])/c + (ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)])/c - PolyLog[3, -1 + 2/(1 - a*x)]/(2*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^3 when c^2 d^2 - e^2=0*) - - -{(a + b*ArcTanh[c*x])^3*(1 + c*x)^3, x, 26, 3*a*b^2*x + (b^3*x)/4 - (b^3*ArcTanh[c*x])/(4*c) + 3*b^3*x*ArcTanh[c*x] + (1/4)*b^2*c*x^2*(a + b*ArcTanh[c*x]) + (4*b*(a + b*ArcTanh[c*x])^2)/c + (21/4)*b*x*(a + b*ArcTanh[c*x])^2 + (3/2)*b*c*x^2*(a + b*ArcTanh[c*x])^2 + (1/4)*b*c^2*x^3*(a + b*ArcTanh[c*x])^2 + ((1 + c*x)^4*(a + b*ArcTanh[c*x])^3)/(4*c) - (11*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (6*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c + (3*b^3*Log[1 - c^2*x^2])/(2*c) - (11*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c) - (6*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (3*b^3*PolyLog[3, 1 - 2/(1 - c*x)])/c} -{(a + b*ArcTanh[c*x])^3*(1 + c*x)^2, x, 17, a*b^2*x + b^3*x*ArcTanh[c*x] + (5*b*(a + b*ArcTanh[c*x])^2)/(2*c) + 3*b*x*(a + b*ArcTanh[c*x])^2 + (1/2)*b*c*x^2*(a + b*ArcTanh[c*x])^2 + ((1 + c*x)^3*(a + b*ArcTanh[c*x])^3)/(3*c) - (6*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (4*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c + (b^3*Log[1 - c^2*x^2])/(2*c) - (3*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/c - (4*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (2*b^3*PolyLog[3, 1 - 2/(1 - c*x)])/c} -{(a + b*ArcTanh[c*x])^3*(1 + c*x)^1, x, 11, (3*b*(a + b*ArcTanh[c*x])^2)/(2*c) + (3/2)*b*x*(a + b*ArcTanh[c*x])^2 + ((1 + c*x)^2*(a + b*ArcTanh[c*x])^3)/(2*c) - (3*b^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/c - (3*b*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c - (3*b^3*PolyLog[2, 1 - 2/(1 - c*x)])/(2*c) - (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c + (3*b^3*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c)} -{(a + b*ArcTanh[c*x])^3/(1 + c*x)^1, x, 4, -(((a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)])/c) + (3*b*(a + b*ArcTanh[c*x])^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c) + (3*b^2*(a + b*ArcTanh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)])/(2*c) + (3*b^3*PolyLog[4, 1 - 2/(1 + c*x)])/(4*c)} -{(a + b*ArcTanh[c*x])^3/(1 + c*x)^2, x, 11, -((3*b^3)/(4*c*(1 + c*x))) + (3*b^3*ArcTanh[c*x])/(4*c) - (3*b^2*(a + b*ArcTanh[c*x]))/(2*c*(1 + c*x)) + (3*b*(a + b*ArcTanh[c*x])^2)/(4*c) - (3*b*(a + b*ArcTanh[c*x])^2)/(2*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^3/(2*c) - (a + b*ArcTanh[c*x])^3/(c*(1 + c*x))} -{(a + b*ArcTanh[c*x])^3/(1 + c*x)^3, x, 24, -((3*b^3)/(64*c*(1 + c*x)^2)) - (21*b^3)/(64*c*(1 + c*x)) + (21*b^3*ArcTanh[c*x])/(64*c) - (3*b^2*(a + b*ArcTanh[c*x]))/(16*c*(1 + c*x)^2) - (9*b^2*(a + b*ArcTanh[c*x]))/(16*c*(1 + c*x)) + (9*b*(a + b*ArcTanh[c*x])^2)/(32*c) - (3*b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (3*b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^3/(8*c) - (a + b*ArcTanh[c*x])^3/(2*c*(1 + c*x)^2)} -{(a + b*ArcTanh[c*x])^3/(1 + c*x)^4, x, 42, -(b^3/(108*c*(1 + c*x)^3)) - (19*b^3)/(576*c*(1 + c*x)^2) - (85*b^3)/(576*c*(1 + c*x)) + (85*b^3*ArcTanh[c*x])/(576*c) - (b^2*(a + b*ArcTanh[c*x]))/(18*c*(1 + c*x)^3) - (5*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)^2) - (11*b^2*(a + b*ArcTanh[c*x]))/(48*c*(1 + c*x)) + (11*b*(a + b*ArcTanh[c*x])^2)/(96*c) - (b*(a + b*ArcTanh[c*x])^2)/(6*c*(1 + c*x)^3) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)^2) - (b*(a + b*ArcTanh[c*x])^2)/(8*c*(1 + c*x)) + (a + b*ArcTanh[c*x])^3/(24*c) - (a + b*ArcTanh[c*x])^3/(3*c*(1 + c*x)^3)} - - -{(x^2*ArcTanh[a*x]^3)/(c + a*c*x), x, 19, (3*ArcTanh[a*x]^2)/(2*a^3*c) + (3*x*ArcTanh[a*x]^2)/(2*a^2*c) - (3*ArcTanh[a*x]^3)/(2*a^3*c) - (x*ArcTanh[a*x]^3)/(a^2*c) + (x^2*ArcTanh[a*x]^3)/(2*a*c) - (3*ArcTanh[a*x]*Log[2/(1 - a*x)])/(a^3*c) + (3*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(a^3*c) - (ArcTanh[a*x]^3*Log[2/(1 + a*x)])/(a^3*c) - (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^3*c) + (3*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(a^3*c) + (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)])/(2*a^3*c) - (3*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^3*c) + (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/(2*a^3*c) + (3*PolyLog[4, 1 - 2/(1 + a*x)])/(4*a^3*c)} -{(x*ArcTanh[a*x]^3)/(c + a*c*x), x, 10, ArcTanh[a*x]^3/(a^2*c) + (x*ArcTanh[a*x]^3)/(a*c) - (3*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(a^2*c) + (ArcTanh[a*x]^3*Log[2/(1 + a*x)])/(a^2*c) - (3*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(a^2*c) - (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)])/(2*a^2*c) + (3*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^2*c) - (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/(2*a^2*c) - (3*PolyLog[4, 1 - 2/(1 + a*x)])/(4*a^2*c)} -{ArcTanh[a*x]^3/(c + a*c*x), x, 4, -((ArcTanh[a*x]^3*Log[2/(1 + a*x)])/(a*c)) + (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)])/(2*a*c) + (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 + a*x)])/(2*a*c) + (3*PolyLog[4, 1 - 2/(1 + a*x)])/(4*a*c)} -{ArcTanh[a*x]^3/(x*(c + a*c*x)), x, 4, (ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c - (3*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*c) - (3*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) - (3*PolyLog[4, -1 + 2/(1 + a*x)])/(4*c)} -{ArcTanh[a*x]^3/(c*x + a*c*x^2), x, 5, (ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c - (3*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*c) - (3*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) - (3*PolyLog[4, -1 + 2/(1 + a*x)])/(4*c)} -{ArcTanh[a*x]^3/(x^2*(c + a*c*x)), x, 10, (a*ArcTanh[a*x]^3)/c - ArcTanh[a*x]^3/(c*x) + (3*a*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)])/c - (a*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c - (3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/c + (3*a*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*c) - (3*a*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) + (3*a*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) + (3*a*PolyLog[4, -1 + 2/(1 + a*x)])/(4*c)} -{ArcTanh[a*x]^3/(x^3*(c + a*c*x)), x, 18, (3*a^2*ArcTanh[a*x]^2)/(2*c) - (3*a*ArcTanh[a*x]^2)/(2*c*x) - (a^2*ArcTanh[a*x]^3)/(2*c) - ArcTanh[a*x]^3/(2*c*x^2) + (a*ArcTanh[a*x]^3)/(c*x) + (3*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)])/c - (3*a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)])/c + (a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c - (3*a^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*c) + (3*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/c - (3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/(2*c) + (3*a^2*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) - (3*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/(2*c) - (3*a^2*PolyLog[4, -1 + 2/(1 + a*x)])/(4*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^4 when c^2 d^2 - e^2=0*) - - -{(x^2*ArcTanh[a*x]^4)/(c - a*c*x), x, 21, (-2*ArcTanh[a*x]^3)/(a^3*c) - (2*x*ArcTanh[a*x]^3)/(a^2*c) - ArcTanh[a*x]^4/(2*a^3*c) - (x*ArcTanh[a*x]^4)/(a^2*c) - (x^2*ArcTanh[a*x]^4)/(2*a*c) + (6*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(a^3*c) + (4*ArcTanh[a*x]^3*Log[2/(1 - a*x)])/(a^3*c) + (ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a^3*c) + (6*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(a^3*c) + (6*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/(a^3*c) + (2*ArcTanh[a*x]^3*PolyLog[2, 1 - 2/(1 - a*x)])/(a^3*c) - (3*PolyLog[3, 1 - 2/(1 - a*x)])/(a^3*c) - (6*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - a*x)])/(a^3*c) - (3*ArcTanh[a*x]^2*PolyLog[3, 1 - 2/(1 - a*x)])/(a^3*c) + (3*PolyLog[4, 1 - 2/(1 - a*x)])/(a^3*c) + (3*ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(a^3*c) - (3*PolyLog[5, 1 - 2/(1 - a*x)])/(2*a^3*c)} -{(x*ArcTanh[a*x]^4)/(c - a*c*x), x, 12, -(ArcTanh[a*x]^4/(a^2*c)) - (x*ArcTanh[a*x]^4)/(a*c) + (4*ArcTanh[a*x]^3*Log[2/(1 - a*x)])/(a^2*c) + (ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a^2*c) + (6*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/(a^2*c) + (2*ArcTanh[a*x]^3*PolyLog[2, 1 - 2/(1 - a*x)])/(a^2*c) - (6*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - a*x)])/(a^2*c) - (3*ArcTanh[a*x]^2*PolyLog[3, 1 - 2/(1 - a*x)])/(a^2*c) + (3*PolyLog[4, 1 - 2/(1 - a*x)])/(a^2*c) + (3*ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(a^2*c) - (3*PolyLog[5, 1 - 2/(1 - a*x)])/(2*a^2*c)} -{ArcTanh[a*x]^4/(c - a*c*x), x, 5, (ArcTanh[a*x]^4*Log[2/(1 - a*x)])/(a*c) + (2*ArcTanh[a*x]^3*PolyLog[2, 1 - 2/(1 - a*x)])/(a*c) - (3*ArcTanh[a*x]^2*PolyLog[3, 1 - 2/(1 - a*x)])/(a*c) + (3*ArcTanh[a*x]*PolyLog[4, 1 - 2/(1 - a*x)])/(a*c) - (3*PolyLog[5, 1 - 2/(1 - a*x)])/(2*a*c)} -{ArcTanh[a*x]^4/(x*(c - a*c*x)), x, 5, (ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*x)])/c - (3*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c + (3*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)} -{ArcTanh[a*x]^4/(c*x - a*c*x^2), x, 6, (ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*x)])/c - (3*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c + (3*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)} -{ArcTanh[a*x]^4/(x^2*(c - a*c*x)), x, 12, (a*ArcTanh[a*x]^4)/c - ArcTanh[a*x]^4/(c*x) + (a*ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (4*a*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c + (2*a*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*x)])/c - (6*a*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/c - (3*a*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c - (6*a*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/c + (3*a*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*a*PolyLog[4, -1 + 2/(1 + a*x)])/c - (3*a*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)} -{ArcTanh[a*x]^4/(x^3*(c - a*c*x)), x, 21, (2*a^2*ArcTanh[a*x]^3)/c - (2*a*ArcTanh[a*x]^3)/(c*x) + (3*a^2*ArcTanh[a*x]^4)/(2*c) - ArcTanh[a*x]^4/(2*c*x^2) - (a*ArcTanh[a*x]^4)/(c*x) + (a^2*ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c + (6*a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)])/c + (4*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)])/c + (2*a^2*ArcTanh[a*x]^3*PolyLog[2, -1 + 2/(1 - a*x)])/c - (6*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)])/c - (6*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/c - (3*a^2*ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a*x)])/c - (3*a^2*PolyLog[3, -1 + 2/(1 + a*x)])/c - (6*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)])/c + (3*a^2*ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/c - (3*a^2*PolyLog[4, -1 + 2/(1 + a*x)])/c - (3*a^2*PolyLog[5, -1 + 2/(1 - a*x)])/(2*c)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^-1 when c^2 d^2 - e^2=0*) - - -{x/(ArcTanh[a*x]*(c + a*c*x)), x, 0, Unintegrable[x/((c + a*c*x)*ArcTanh[a*x]), x]} -{1/(ArcTanh[a*x]*(c + a*c*x)), x, 0, Unintegrable[1/((c + a*c*x)*ArcTanh[a*x]), x]} -{1/(x*ArcTanh[a*x]*(c + a*c*x)), x, 0, Unintegrable[1/(x*(c + a*c*x)*ArcTanh[a*x]), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^-2 when c^2 d^2 - e^2=0*) - - -{x/(ArcTanh[a*x]^2*(c + a*c*x)), x, 0, Unintegrable[x/((c + a*c*x)*ArcTanh[a*x]^2), x]} -{1/(ArcTanh[a*x]^2*(c + a*c*x)), x, 0, Unintegrable[1/((c + a*c*x)*ArcTanh[a*x]^2), x]} -{1/(x*ArcTanh[a*x]^2*(c + a*c*x)), x, 0, Unintegrable[1/(x*(c + a*c*x)*ArcTanh[a*x]^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m / (d+e x) (a+b ArcTanh[c x])^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*(a + b*ArcTanh[c*x])/(d + e*x), x, 16, (a*d^2*x)/e^3 - (b*d*x)/(2*c*e^2) + (b*x^2)/(6*c*e) + (b*d*ArcTanh[c*x])/(2*c^2*e^2) + (b*d^2*x*ArcTanh[c*x])/e^3 - (d*x^2*(a + b*ArcTanh[c*x]))/(2*e^2) + (x^3*(a + b*ArcTanh[c*x]))/(3*e) + (d^3*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e^4 - (d^3*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^4 + (b*d^2*Log[1 - c^2*x^2])/(2*c*e^3) + (b*Log[1 - c^2*x^2])/(6*c^3*e) - (b*d^3*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e^4) + (b*d^3*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^4)} -{x^2*(a + b*ArcTanh[c*x])/(d + e*x), x, 12, -((a*d*x)/e^2) + (b*x)/(2*c*e) - (b*ArcTanh[c*x])/(2*c^2*e) - (b*d*x*ArcTanh[c*x])/e^2 + (x^2*(a + b*ArcTanh[c*x]))/(2*e) - (d^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e^3 + (d^2*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 - (b*d*Log[1 - c^2*x^2])/(2*c*e^2) + (b*d^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e^3) - (b*d^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^3)} -{x^1*(a + b*ArcTanh[c*x])/(d + e*x), x, 9, (a*x)/e + (b*x*ArcTanh[c*x])/e + (d*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e^2 - (d*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^2 + (b*Log[1 - c^2*x^2])/(2*c*e) - (b*d*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e^2) + (b*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^2)} -{x^0*(a + b*ArcTanh[c*x])/(d + e*x), x, 4, -(((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e) - (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)} -{(a + b*ArcTanh[c*x])/(x^1*(d + e*x)), x, 7, (a*Log[x])/d + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d - ((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d - (b*PolyLog[2, (-c)*x])/(2*d) + (b*PolyLog[2, c*x])/(2*d) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d) + (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d)} -{(a + b*ArcTanh[c*x])/(x^2*(d + e*x)), x, 12, -((a + b*ArcTanh[c*x])/(d*x)) + (b*c*Log[x])/d - (a*e*Log[x])/d^2 - (e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 + (e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 - (b*c*Log[1 - c^2*x^2])/(2*d) + (b*e*PolyLog[2, (-c)*x])/(2*d^2) - (b*e*PolyLog[2, c*x])/(2*d^2) + (b*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^2) - (b*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d^2)} -{(a + b*ArcTanh[c*x])/(x^3*(d + e*x)), x, 15, -((b*c)/(2*d*x)) + (b*c^2*ArcTanh[c*x])/(2*d) - (a + b*ArcTanh[c*x])/(2*d*x^2) + (e*(a + b*ArcTanh[c*x]))/(d^2*x) - (b*c*e*Log[x])/d^2 + (a*e^2*Log[x])/d^3 + (e^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^3 - (e^2*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^3 + (b*c*e*Log[1 - c^2*x^2])/(2*d^2) - (b*e^2*PolyLog[2, (-c)*x])/(2*d^3) + (b*e^2*PolyLog[2, c*x])/(2*d^3) - (b*e^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^3) + (b*e^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d^3)} - - -{x^2*(a + b*ArcTanh[c*x])^2/(d + e*x), x, 14, (a*b*x)/(c*e) + (b^2*x*ArcTanh[c*x])/(c*e) - (d*(a + b*ArcTanh[c*x])^2)/(c*e^2) - (a + b*ArcTanh[c*x])^2/(2*c^2*e) - (d*x*(a + b*ArcTanh[c*x])^2)/e^2 + (x^2*(a + b*ArcTanh[c*x])^2)/(2*e) + (2*b*d*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c*e^2) - (d^2*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e^3 + (d^2*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 + (b^2*Log[1 - c^2*x^2])/(2*c^2*e) + (b^2*d*PolyLog[2, 1 - 2/(1 - c*x)])/(c*e^2) + (b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e^3 - (b*d^2*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^3 + (b^2*d^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e^3) - (b^2*d^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^3)} -{x^1*(a + b*ArcTanh[c*x])^2/(d + e*x), x, 8, (a + b*ArcTanh[c*x])^2/(c*e) + (x*(a + b*ArcTanh[c*x])^2)/e - (2*b*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(c*e) + (d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e^2 - (d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^2 - (b^2*PolyLog[2, 1 - 2/(1 - c*x)])/(c*e) - (b*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e^2 + (b*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e^2 - (b^2*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e^2) + (b^2*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^2)} -{x^0*(a + b*ArcTanh[c*x])^2/(d + e*x), x, 1, -(((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e) - (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e)} -{(a + b*ArcTanh[c*x])^2/(x^1*(d + e*x)), x, 9, (2*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d + ((a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d - ((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d + (b*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d - (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d + (b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d + (b^2*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d) - (b^2*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d) - (b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d) + (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d)} -{(a + b*ArcTanh[c*x])^2/(x^2*(d + e*x)), x, 13, (c*(a + b*ArcTanh[c*x])^2)/d - (a + b*ArcTanh[c*x])^2/(d*x) - (2*e*(a + b*ArcTanh[c*x])^2*ArcTanh[1 - 2/(1 - c*x)])/d^2 - (e*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/d^2 + (e*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 + (2*b*c*(a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)])/d + (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/d^2 - (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, -1 + 2/(1 - c*x)])/d^2 + (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/d^2 - (b^2*c*PolyLog[2, -1 + 2/(1 + c*x)])/d - (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - c*x)])/(2*d^2) + (b^2*e*PolyLog[3, -1 + 2/(1 - c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + c*x)])/(2*d^2) - (b^2*e*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d^2)} - - -{ArcTanh[c*x]^2/(x*(d + e*x)), x, 9, (2*ArcTanh[c*x]^2*ArcTanh[1 - 2/(1 - c*x)])/d + (ArcTanh[c*x]^2*Log[2/(1 + c*x)])/d - (ArcTanh[c*x]^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d - (ArcTanh[c*x]*PolyLog[2, 1 - 2/(1 - c*x)])/d + (ArcTanh[c*x]*PolyLog[2, -1 + 2/(1 - c*x)])/d - (ArcTanh[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/d + (ArcTanh[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d + PolyLog[3, 1 - 2/(1 - c*x)]/(2*d) - PolyLog[3, -1 + 2/(1 - c*x)]/(2*d) - PolyLog[3, 1 - 2/(1 + c*x)]/(2*d) + PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(2*d)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/((d + e*x)*(a + b*ArcTan[c*x])), x, 0, Unintegrable[1/((d + e*x)*(a + b*ArcTan[c*x])), x]} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p with c^2 d+e=0*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^q (a+b ArcTanh[c x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]*(1 - a^2*x^2)*x^4, x, 9, x^2/(35*a^3) + x^4/(70*a) - (a*x^6)/42 + (1/5)*x^5*ArcTanh[a*x] - (1/7)*a^2*x^7*ArcTanh[a*x] + Log[1 - a^2*x^2]/(35*a^5)} -{ArcTanh[a*x]*(1 - a^2*x^2)*x^3, x, 9, x/(12*a^3) + x^3/(36*a) - (a*x^5)/30 - ArcTanh[a*x]/(12*a^4) + (1/4)*x^4*ArcTanh[a*x] - (1/6)*a^2*x^6*ArcTanh[a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)*x^2, x, 9, x^2/(15*a) - (a*x^4)/20 + (1/3)*x^3*ArcTanh[a*x] - (1/5)*a^2*x^5*ArcTanh[a*x] + Log[1 - a^2*x^2]/(15*a^3)} -{ArcTanh[a*x]*(1 - a^2*x^2)*x^1, x, 2, x/(4*a) - (a*x^3)/12 - ((1 - a^2*x^2)^2*ArcTanh[a*x])/(4*a^2)} -{ArcTanh[a*x]*(1 - a^2*x^2)*x^0, x, 3, (1 - a^2*x^2)/(6*a) + (2/3)*x*ArcTanh[a*x] + (1/3)*x*(1 - a^2*x^2)*ArcTanh[a*x] + Log[1 - a^2*x^2]/(3*a)} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^1, x, 5, -((a*x)/2) + (1/2)*ArcTanh[a*x] - (1/2)*a^2*x^2*ArcTanh[a*x] - (1/2)*PolyLog[2, (-a)*x] + (1/2)*PolyLog[2, a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^2, x, 8, -(ArcTanh[a*x]/x) - a^2*x*ArcTanh[a*x] + a*Log[x] - a*Log[1 - a^2*x^2]} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^3, x, 5, -(a/(2*x)) + (1/2)*a^2*ArcTanh[a*x] - ArcTanh[a*x]/(2*x^2) + (1/2)*a^2*PolyLog[2, (-a)*x] - (1/2)*a^2*PolyLog[2, a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^4, x, 10, -(a/(6*x^2)) - ArcTanh[a*x]/(3*x^3) + (a^2*ArcTanh[a*x])/x - (2/3)*a^3*Log[x] + (1/3)*a^3*Log[1 - a^2*x^2]} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^5, x, 3, -(a/(12*x^3)) + a^3/(4*x) - ((1 - a^2*x^2)^2*ArcTanh[a*x])/(4*x^4)} -{ArcTanh[a*x]*(1 - a^2*x^2)/x^6, x, 9, -(a/(20*x^4)) + a^3/(15*x^2) - ArcTanh[a*x]/(5*x^5) + (a^2*ArcTanh[a*x])/(3*x^3) - (2/15)*a^5*Log[x] + (1/15)*a^5*Log[1 - a^2*x^2]} - - -{ArcTanh[a*x]^2*(1 - a^2*x^2)*x^4, x, 34, (4*x)/(105*a^4) - (2*x^3)/(315*a^2) - x^5/105 - (4*ArcTanh[a*x])/(105*a^5) + (2*x^2*ArcTanh[a*x])/(35*a^3) + (x^4*ArcTanh[a*x])/(35*a) - (1/21)*a*x^6*ArcTanh[a*x] + (2*ArcTanh[a*x]^2)/(35*a^5) + (1/5)*x^5*ArcTanh[a*x]^2 - (1/7)*a^2*x^7*ArcTanh[a*x]^2 - (4*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a^5) - (2*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a^5)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)*x^3, x, 26, -(x^2/(180*a^2)) - x^4/60 + (x*ArcTanh[a*x])/(6*a^3) + (x^3*ArcTanh[a*x])/(18*a) - (1/15)*a*x^5*ArcTanh[a*x] - ArcTanh[a*x]^2/(12*a^4) + (1/4)*x^4*ArcTanh[a*x]^2 - (1/6)*a^2*x^6*ArcTanh[a*x]^2 + (7*Log[1 - a^2*x^2])/(90*a^4)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)*x^2, x, 24, x/(30*a^2) - x^3/30 - ArcTanh[a*x]/(30*a^3) + (2*x^2*ArcTanh[a*x])/(15*a) - (1/10)*a*x^4*ArcTanh[a*x] + (2*ArcTanh[a*x]^2)/(15*a^3) + (1/3)*x^3*ArcTanh[a*x]^2 - (1/5)*a^2*x^5*ArcTanh[a*x]^2 - (4*ArcTanh[a*x]*Log[2/(1 - a*x)])/(15*a^3) - (2*PolyLog[2, 1 - 2/(1 - a*x)])/(15*a^3)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)*x^1, x, 4, (1 - a^2*x^2)/(12*a^2) + (x*ArcTanh[a*x])/(3*a) + (x*(1 - a^2*x^2)*ArcTanh[a*x])/(6*a) - ((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/(4*a^2) + Log[1 - a^2*x^2]/(6*a^2)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)*x^0, x, 7, -(x/3) + ((1 - a^2*x^2)*ArcTanh[a*x])/(3*a) + (2*ArcTanh[a*x]^2)/(3*a) + (2/3)*x*ArcTanh[a*x]^2 + (1/3)*x*(1 - a^2*x^2)*ArcTanh[a*x]^2 - (4*ArcTanh[a*x]*Log[2/(1 - a*x)])/(3*a) - (2*PolyLog[2, 1 - 2/(1 - a*x)])/(3*a)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^1, x, 12, (-a)*x*ArcTanh[a*x] + (1/2)*ArcTanh[a*x]^2 - (1/2)*a^2*x^2*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (1/2)*Log[1 - a^2*x^2] - ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] + (1/2)*PolyLog[3, 1 - 2/(1 - a*x)] - (1/2)*PolyLog[3, -1 + 2/(1 - a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^2, x, 10, -(ArcTanh[a*x]^2/x) - a^2*x*ArcTanh[a*x]^2 + 2*a*ArcTanh[a*x]*Log[2/(1 - a*x)] + 2*a*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + a*PolyLog[2, 1 - 2/(1 - a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^3, x, 15, -((a*ArcTanh[a*x])/x) + (1/2)*a^2*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(2*x^2) - 2*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] + a^2*Log[x] - (1/2)*a^2*Log[1 - a^2*x^2] + a^2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] - a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] - (1/2)*a^2*PolyLog[3, 1 - 2/(1 - a*x)] + (1/2)*a^2*PolyLog[3, -1 + 2/(1 - a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^4, x, 13, -(a^2/(3*x)) + (1/3)*a^3*ArcTanh[a*x] - (a*ArcTanh[a*x])/(3*x^2) - (2/3)*a^3*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(3*x^3) + (a^2*ArcTanh[a*x]^2)/x - (4/3)*a^3*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + (2/3)*a^3*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^5, x, 11, -(a^2/(12*x^2)) - (a*ArcTanh[a*x])/(6*x^3) + (a^3*ArcTanh[a*x])/(2*x) - ((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/(4*x^4) - (1/3)*a^4*Log[x] + (1/6)*a^4*Log[1 - a^2*x^2]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)/x^6, x, 22, -(a^2/(30*x^3)) + a^4/(30*x) - (1/30)*a^5*ArcTanh[a*x] - (a*ArcTanh[a*x])/(10*x^4) + (2*a^3*ArcTanh[a*x])/(15*x^2) - (2/15)*a^5*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(5*x^5) + (a^2*ArcTanh[a*x]^2)/(3*x^3) - (4/15)*a^5*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + (2/15)*a^5*PolyLog[2, -1 + 2/(1 + a*x)]} - - -{ArcTanh[a*x]^3*(1 - a^2*x^2), x, 8, (-x)*ArcTanh[a*x] + ((1 - a^2*x^2)*ArcTanh[a*x]^2)/(2*a) + (2*ArcTanh[a*x]^3)/(3*a) + (2/3)*x*ArcTanh[a*x]^3 + (1/3)*x*(1 - a^2*x^2)*ArcTanh[a*x]^3 - (2*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a - Log[1 - a^2*x^2]/(2*a) - (2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + PolyLog[3, 1 - 2/(1 - a*x)]/a} - - -{x*ArcTanh[x/Sqrt[2]]/(1 - x^2), x, 10, ArcTanh[x/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + x)] - (1/2)*ArcTanh[x/Sqrt[2]]*Log[-((4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + x)))] - (1/2)*ArcTanh[x/Sqrt[2]]*Log[(4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))] - (1/2)*PolyLog[2, 1 - (2*Sqrt[2])/(Sqrt[2] + x)] + (1/4)*PolyLog[2, 1 + (4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + x))] + (1/4)*PolyLog[2, 1 - (4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/ArcTanh[a*x]*(1 - a^2*x^2)*x^1, x, 0, Unintegrable[(x*(1 - a^2*x^2))/ArcTanh[a*x], x]} -{1/ArcTanh[a*x]*(1 - a^2*x^2)*x^0, x, 0, Unintegrable[(1 - a^2*x^2)/ArcTanh[a*x], x]} -{1/ArcTanh[a*x]*(1 - a^2*x^2)/x^1, x, 0, Unintegrable[(1 - a^2*x^2)/(x*ArcTanh[a*x]), x]} - - -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)*x^1, x, 0, Unintegrable[(x*(1 - a^2*x^2))/ArcTanh[a*x]^2, x]} -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)*x^0, x, 0, Unintegrable[(1 - a^2*x^2)/ArcTanh[a*x]^2, x]} -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)/x^1, x, 0, Unintegrable[(1 - a^2*x^2)/(x*ArcTanh[a*x]^2), x]} - - -{1/ArcTanh[a*x]^3*(1 - a^2*x^2)*x^0, x, 0, Unintegrable[(1 - a^2*x^2)/ArcTanh[a*x]^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]*(1 - a^2*x^2)^2*x^4, x, 14, (4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (1/5)*x^5*ArcTanh[a*x] - (2/7)*a^2*x^7*ArcTanh[a*x] + (1/9)*a^4*x^9*ArcTanh[a*x] + (4*Log[1 - a^2*x^2])/(315*a^5)} -{ArcTanh[a*x]*(1 - a^2*x^2)^2*x^3, x, 14, x/(24*a^3) + x^3/(72*a) - (a*x^5)/24 + (a^3*x^7)/56 - ArcTanh[a*x]/(24*a^4) + (1/4)*x^4*ArcTanh[a*x] - (1/3)*a^2*x^6*ArcTanh[a*x] + (1/8)*a^4*x^8*ArcTanh[a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2*x^2, x, 14, (4*x^2)/(105*a) - (9*a*x^4)/140 + (a^3*x^6)/42 + (1/3)*x^3*ArcTanh[a*x] - (2/5)*a^2*x^5*ArcTanh[a*x] + (1/7)*a^4*x^7*ArcTanh[a*x] + (4*Log[1 - a^2*x^2])/(105*a^3)} -{ArcTanh[a*x]*(1 - a^2*x^2)^2*x^1, x, 3, x/(6*a) - (a*x^3)/9 + (a^3*x^5)/30 - ((1 - a^2*x^2)^3*ArcTanh[a*x])/(6*a^2)} -{ArcTanh[a*x]*(1 - a^2*x^2)^2*x^0, x, 4, (2*(1 - a^2*x^2))/(15*a) + (1 - a^2*x^2)^2/(20*a) + (8/15)*x*ArcTanh[a*x] + (4/15)*x*(1 - a^2*x^2)*ArcTanh[a*x] + (1/5)*x*(1 - a^2*x^2)^2*ArcTanh[a*x] + (4*Log[1 - a^2*x^2])/(15*a)} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^1, x, 10, -((3*a*x)/4) + (a^3*x^3)/12 + (3/4)*ArcTanh[a*x] - a^2*x^2*ArcTanh[a*x] + (1/4)*a^4*x^4*ArcTanh[a*x] - (1/2)*PolyLog[2, (-a)*x] + (1/2)*PolyLog[2, a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^2, x, 13, (a^3*x^2)/6 - ArcTanh[a*x]/x - 2*a^2*x*ArcTanh[a*x] + (1/3)*a^4*x^3*ArcTanh[a*x] + a*Log[x] - (4/3)*a*Log[1 - a^2*x^2]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^3, x, 9, -(a/(2*x)) + (a^3*x)/2 - ArcTanh[a*x]/(2*x^2) + (1/2)*a^4*x^2*ArcTanh[a*x] + a^2*PolyLog[2, (-a)*x] - a^2*PolyLog[2, a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^4, x, 13, -(a/(6*x^2)) - ArcTanh[a*x]/(3*x^3) + (2*a^2*ArcTanh[a*x])/x + a^4*x*ArcTanh[a*x] - (5/3)*a^3*Log[x] + (4/3)*a^3*Log[1 - a^2*x^2]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^5, x, 10, -(a/(12*x^3)) + (3*a^3)/(4*x) - (3/4)*a^4*ArcTanh[a*x] - ArcTanh[a*x]/(4*x^4) + (a^2*ArcTanh[a*x])/x^2 - (1/2)*a^4*PolyLog[2, (-a)*x] + (1/2)*a^4*PolyLog[2, a*x]} -{ArcTanh[a*x]*(1 - a^2*x^2)^2/x^6, x, 15, -(a/(20*x^4)) + (7*a^3)/(30*x^2) - ArcTanh[a*x]/(5*x^5) + (2*a^2*ArcTanh[a*x])/(3*x^3) - (a^4*ArcTanh[a*x])/x + (8/15)*a^5*Log[x] - (4/15)*a^5*Log[1 - a^2*x^2]} - - -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^4, x, 59, (29*x)/(3780*a^4) - (67*x^3)/(11340*a^2) - (23*x^5)/3780 + (a^2*x^7)/252 - (29*ArcTanh[a*x])/(3780*a^5) + (8*x^2*ArcTanh[a*x])/(315*a^3) + (4*x^4*ArcTanh[a*x])/(315*a) - (11/189)*a*x^6*ArcTanh[a*x] + (1/36)*a^3*x^8*ArcTanh[a*x] + (8*ArcTanh[a*x]^2)/(315*a^5) + (1/5)*x^5*ArcTanh[a*x]^2 - (2/7)*a^2*x^7*ArcTanh[a*x]^2 + (1/9)*a^4*x^9*ArcTanh[a*x]^2 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(315*a^5) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(315*a^5)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^3, x, 47, -((5*x^2)/(504*a^2)) - x^4/84 + (a^2*x^6)/168 + (x*ArcTanh[a*x])/(12*a^3) + (x^3*ArcTanh[a*x])/(36*a) - (1/12)*a*x^5*ArcTanh[a*x] + (1/28)*a^3*x^7*ArcTanh[a*x] - ArcTanh[a*x]^2/(24*a^4) + (1/4)*x^4*ArcTanh[a*x]^2 - (1/3)*a^2*x^6*ArcTanh[a*x]^2 + (1/8)*a^4*x^8*ArcTanh[a*x]^2 + (2*Log[1 - a^2*x^2])/(63*a^4)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^2, x, 44, -(x/(210*a^2)) - (17*x^3)/630 + (a^2*x^5)/105 + ArcTanh[a*x]/(210*a^3) + (8*x^2*ArcTanh[a*x])/(105*a) - (9/70)*a*x^4*ArcTanh[a*x] + (1/21)*a^3*x^6*ArcTanh[a*x] + (8*ArcTanh[a*x]^2)/(105*a^3) + (1/3)*x^3*ArcTanh[a*x]^2 - (2/5)*a^2*x^5*ArcTanh[a*x]^2 + (1/7)*a^4*x^7*ArcTanh[a*x]^2 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(105*a^3) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(105*a^3)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^1, x, 5, (2*(1 - a^2*x^2))/(45*a^2) + (1 - a^2*x^2)^2/(60*a^2) + (8*x*ArcTanh[a*x])/(45*a) + (4*x*(1 - a^2*x^2)*ArcTanh[a*x])/(45*a) + (x*(1 - a^2*x^2)^2*ArcTanh[a*x])/(15*a) - ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(6*a^2) + (4*Log[1 - a^2*x^2])/(45*a^2)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^0, x, 9, -((11*x)/30) + (a^2*x^3)/30 + (4*(1 - a^2*x^2)*ArcTanh[a*x])/(15*a) + ((1 - a^2*x^2)^2*ArcTanh[a*x])/(10*a) + (8*ArcTanh[a*x]^2)/(15*a) + (8/15)*x*ArcTanh[a*x]^2 + (4/15)*x*(1 - a^2*x^2)*ArcTanh[a*x]^2 + (1/5)*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(15*a) - (8*PolyLog[2, 1 - 2/(1 - a*x)])/(15*a)} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^1, x, 23, (a^2*x^2)/12 - (3/2)*a*x*ArcTanh[a*x] + (1/6)*a^3*x^3*ArcTanh[a*x] + (3/4)*ArcTanh[a*x]^2 - a^2*x^2*ArcTanh[a*x]^2 + (1/4)*a^4*x^4*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (2/3)*Log[1 - a^2*x^2] - ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] + (1/2)*PolyLog[3, 1 - 2/(1 - a*x)] - (1/2)*PolyLog[3, -1 + 2/(1 - a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^2, x, 20, (a^2*x)/3 - (1/3)*a*ArcTanh[a*x] + (1/3)*a^3*x^2*ArcTanh[a*x] - (2/3)*a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x - 2*a^2*x*ArcTanh[a*x]^2 + (1/3)*a^4*x^3*ArcTanh[a*x]^2 + (10/3)*a*ArcTanh[a*x]*Log[2/(1 - a*x)] + 2*a*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + (5/3)*a*PolyLog[2, 1 - 2/(1 - a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^3, x, 21, -((a*ArcTanh[a*x])/x) + a^3*x*ArcTanh[a*x] - ArcTanh[a*x]^2/(2*x^2) + (1/2)*a^4*x^2*ArcTanh[a*x]^2 - 4*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] + a^2*Log[x] + 2*a^2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] - 2*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] - a^2*PolyLog[3, 1 - 2/(1 - a*x)] + a^2*PolyLog[3, -1 + 2/(1 - a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^4, x, 19, -(a^2/(3*x)) + (1/3)*a^3*ArcTanh[a*x] - (a*ArcTanh[a*x])/(3*x^2) - (2/3)*a^3*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(3*x^3) + (2*a^2*ArcTanh[a*x]^2)/x + a^4*x*ArcTanh[a*x]^2 - 2*a^3*ArcTanh[a*x]*Log[2/(1 - a*x)] - (10/3)*a^3*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a^3*PolyLog[2, 1 - 2/(1 - a*x)] + (5/3)*a^3*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^5, x, 29, -(a^2/(12*x^2)) - (a*ArcTanh[a*x])/(6*x^3) + (3*a^3*ArcTanh[a*x])/(2*x) - (3/4)*a^4*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(4*x^4) + (a^2*ArcTanh[a*x]^2)/x^2 + 2*a^4*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (4/3)*a^4*Log[x] + (2/3)*a^4*Log[1 - a^2*x^2] - a^4*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + a^4*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] + (1/2)*a^4*PolyLog[3, 1 - 2/(1 - a*x)] - (1/2)*a^4*PolyLog[3, -1 + 2/(1 - a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^6, x, 27, -(a^2/(30*x^3)) + (11*a^4)/(30*x) - (11/30)*a^5*ArcTanh[a*x] - (a*ArcTanh[a*x])/(10*x^4) + (7*a^3*ArcTanh[a*x])/(15*x^2) + (8/15)*a^5*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(5*x^5) + (2*a^2*ArcTanh[a*x]^2)/(3*x^3) - (a^4*ArcTanh[a*x]^2)/x + (16/15)*a^5*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (8/15)*a^5*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^7, x, 16, -(a^2/(60*x^4)) + (7*a^4)/(90*x^2) - (a*ArcTanh[a*x])/(15*x^5) + (2*a^3*ArcTanh[a*x])/(9*x^3) - (a^5*ArcTanh[a*x])/(3*x) - ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(6*x^6) + (8/45)*a^6*Log[x] - (4/45)*a^6*Log[1 - a^2*x^2]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^8, x, 42, -(a^2/(105*x^5)) + (17*a^4)/(630*x^3) + a^6/(210*x) - (1/210)*a^7*ArcTanh[a*x] - (a*ArcTanh[a*x])/(21*x^6) + (9*a^3*ArcTanh[a*x])/(70*x^4) - (8*a^5*ArcTanh[a*x])/(105*x^2) + (8/105)*a^7*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(7*x^7) + (2*a^2*ArcTanh[a*x]^2)/(5*x^5) - (a^4*ArcTanh[a*x]^2)/(3*x^3) + (16/105)*a^7*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (8/105)*a^7*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^9, x, 56, -(a^2/(168*x^6)) + a^4/(84*x^4) + (5*a^6)/(504*x^2) - (a*ArcTanh[a*x])/(28*x^7) + (a^3*ArcTanh[a*x])/(12*x^5) - (a^5*ArcTanh[a*x])/(36*x^3) - (a^7*ArcTanh[a*x])/(12*x) + (1/24)*a^8*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(8*x^8) + (a^2*ArcTanh[a*x]^2)/(3*x^6) - (a^4*ArcTanh[a*x]^2)/(4*x^4) + (4/63)*a^8*Log[x] - (2/63)*a^8*Log[1 - a^2*x^2]} - - -{ArcTanh[a*x]^3*(1 - a^2*x^2)^2, x, 12, -((1 - a^2*x^2)/(20*a)) - x*ArcTanh[a*x] - (1/10)*x*(1 - a^2*x^2)*ArcTanh[a*x] + (2*(1 - a^2*x^2)*ArcTanh[a*x]^2)/(5*a) + (3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/(20*a) + (8*ArcTanh[a*x]^3)/(15*a) + (8/15)*x*ArcTanh[a*x]^3 + (4/15)*x*(1 - a^2*x^2)*ArcTanh[a*x]^3 + (1/5)*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3 - (8*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(5*a) - Log[1 - a^2*x^2]/(2*a) - (8*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(5*a) + (4*PolyLog[3, 1 - 2/(1 - a*x)])/(5*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/ArcTanh[a*x]*(1 - a^2*x^2)^2*x^1, x, 0, Unintegrable[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x], x]} -{1/ArcTanh[a*x]*(1 - a^2*x^2)^2*x^0, x, 0, Unintegrable[(1 - a^2*x^2)^2/ArcTanh[a*x], x]} -{1/ArcTanh[a*x]*(1 - a^2*x^2)^2/x^1, x, 0, Unintegrable[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]), x]} - - -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^1, x, 0, Unintegrable[(x*(1 - a^2*x^2)^2)/ArcTanh[a*x]^2, x]} -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)^2*x^0, x, 0, Unintegrable[(1 - a^2*x^2)^2/ArcTanh[a*x]^2, x]} -{1/ArcTanh[a*x]^2*(1 - a^2*x^2)^2/x^1, x, 0, Unintegrable[(1 - a^2*x^2)^2/(x*ArcTanh[a*x]^2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^3*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]*(1 - a^2*x^2)^3, x, 5, (4*(1 - a^2*x^2))/(35*a) + (3*(1 - a^2*x^2)^2)/(70*a) + (1 - a^2*x^2)^3/(42*a) + (16/35)*x*ArcTanh[a*x] + (8/35)*x*(1 - a^2*x^2)*ArcTanh[a*x] + (6/35)*x*(1 - a^2*x^2)^2*ArcTanh[a*x] + (1/7)*x*(1 - a^2*x^2)^3*ArcTanh[a*x] + (8*Log[1 - a^2*x^2])/(35*a)} - - -{ArcTanh[a*x]^2*(1 - a^2*x^2)^3, x, 12, -((38*x)/105) + (19*a^2*x^3)/315 - (a^4*x^5)/105 + (8*(1 - a^2*x^2)*ArcTanh[a*x])/(35*a) + (3*(1 - a^2*x^2)^2*ArcTanh[a*x])/(35*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21*a) + (16*ArcTanh[a*x]^2)/(35*a) + (16/35)*x*ArcTanh[a*x]^2 + (8/35)*x*(1 - a^2*x^2)*ArcTanh[a*x]^2 + (6/35)*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2 + (1/7)*x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2 - (32*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a) - (16*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a)} - - -{ArcTanh[a*x]^3*(1 - a^2*x^2)^3, x, 17, -((13*(1 - a^2*x^2))/(210*a)) - (1 - a^2*x^2)^2/(140*a) - (14/15)*x*ArcTanh[a*x] - (13/105)*x*(1 - a^2*x^2)*ArcTanh[a*x] - (1/35)*x*(1 - a^2*x^2)^2*ArcTanh[a*x] + (12*(1 - a^2*x^2)*ArcTanh[a*x]^2)/(35*a) + (9*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/(70*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(14*a) + (16*ArcTanh[a*x]^3)/(35*a) + (16/35)*x*ArcTanh[a*x]^3 + (8/35)*x*(1 - a^2*x^2)*ArcTanh[a*x]^3 + (6/35)*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3 + (1/7)*x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3 - (48*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(35*a) - (7*Log[1 - a^2*x^2])/(15*a) - (48*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a) + (24*PolyLog[3, 1 - 2/(1 - a*x)])/(35*a)} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*ArcTanh[a*x]/(1 - a^2*x^2), x, 8, -(x/(2*a^3)) + ArcTanh[a*x]/(2*a^4) - (x^2*ArcTanh[a*x])/(2*a^2) - ArcTanh[a*x]^2/(2*a^4) + (ArcTanh[a*x]*Log[2/(1 - a*x)])/a^4 + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a^4)} -{x^2*ArcTanh[a*x]/(1 - a^2*x^2), x, 4, -((x*ArcTanh[a*x])/a^2) + ArcTanh[a*x]^2/(2*a^3) - Log[1 - a^2*x^2]/(2*a^3)} -{x*ArcTanh[a*x]/(1 - a^2*x^2), x, 4, -(ArcTanh[a*x]^2/(2*a^2)) + (ArcTanh[a*x]*Log[2/(1 - a*x)])/a^2 + PolyLog[2, 1 - 2/(1 - a*x)]/(2*a^2)} -{ArcTanh[a*x]/(1 - a^2*x^2), x, 1, ArcTanh[a*x]^2/(2*a)} -{ArcTanh[a*x]/(x*(1 - a^2*x^2)), x, 3, (1/2)*ArcTanh[a*x]^2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (1/2)*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]/(x^2*(1 - a^2*x^2)), x, 7, -(ArcTanh[a*x]/x) + (1/2)*a*ArcTanh[a*x]^2 + a*Log[x] - (1/2)*a*Log[1 - a^2*x^2]} -{ArcTanh[a*x]/(x^3*(1 - a^2*x^2)), x, 7, -(a/(2*x)) + (1/2)*a^2*ArcTanh[a*x] - ArcTanh[a*x]/(2*x^2) + (1/2)*a^2*ArcTanh[a*x]^2 + a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (1/2)*a^2*PolyLog[2, -1 + 2/(1 + a*x)]} - - -{x^3*ArcTanh[a*x]^2/(1 - a^2*x^2), x, 10, -((x*ArcTanh[a*x])/a^3) + ArcTanh[a*x]^2/(2*a^4) - (x^2*ArcTanh[a*x]^2)/(2*a^2) - ArcTanh[a*x]^3/(3*a^4) + (ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a^4 - Log[1 - a^2*x^2]/(2*a^4) + (ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a^4 - PolyLog[3, 1 - 2/(1 - a*x)]/(2*a^4)} -{x^2*ArcTanh[a*x]^2/(1 - a^2*x^2), x, 7, -(ArcTanh[a*x]^2/a^3) - (x*ArcTanh[a*x]^2)/a^2 + ArcTanh[a*x]^3/(3*a^3) + (2*ArcTanh[a*x]*Log[2/(1 - a*x)])/a^3 + PolyLog[2, 1 - 2/(1 - a*x)]/a^3} -{x*ArcTanh[a*x]^2/(1 - a^2*x^2), x, 4, -(ArcTanh[a*x]^3/(3*a^2)) + (ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a^2 + (ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a^2 - PolyLog[3, 1 - 2/(1 - a*x)]/(2*a^2)} -{ArcTanh[a*x]^2/(1 - a^2*x^2), x, 1, ArcTanh[a*x]^3/(3*a)} -{ArcTanh[a*x]^2/(x*(1 - a^2*x^2)), x, 4, (1/3)*ArcTanh[a*x]^3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (1/2)*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)), x, 6, a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (1/3)*a*ArcTanh[a*x]^3 + 2*a*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)), x, 13, -((a*ArcTanh[a*x])/x) + (1/2)*a^2*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(2*x^2) + (1/3)*a^2*ArcTanh[a*x]^3 + a^2*Log[x] - (1/2)*a^2*Log[1 - a^2*x^2] + a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (1/2)*a^2*PolyLog[3, -1 + 2/(1 + a*x)]} - - -{x^3*ArcTanh[a*x]^3/(1 - a^2*x^2), x, 14, -((3*ArcTanh[a*x]^2)/(2*a^4)) - (3*x*ArcTanh[a*x]^2)/(2*a^3) + ArcTanh[a*x]^3/(2*a^4) - (x^2*ArcTanh[a*x]^3)/(2*a^2) - ArcTanh[a*x]^4/(4*a^4) + (3*ArcTanh[a*x]*Log[2/(1 - a*x)])/a^4 + (ArcTanh[a*x]^3*Log[2/(1 - a*x)])/a^4 + (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^4) + (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^4) - (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^4) + (3*PolyLog[4, 1 - 2/(1 - a*x)])/(4*a^4)} -{x^2*ArcTanh[a*x]^3/(1 - a^2*x^2), x, 7, -(ArcTanh[a*x]^3/a^3) - (x*ArcTanh[a*x]^3)/a^2 + ArcTanh[a*x]^4/(4*a^3) + (3*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a^3 + (3*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a^3 - (3*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^3)} -{x*ArcTanh[a*x]^3/(1 - a^2*x^2), x, 5, -(ArcTanh[a*x]^4/(4*a^2)) + (ArcTanh[a*x]^3*Log[2/(1 - a*x)])/a^2 + (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^2) - (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^2) + (3*PolyLog[4, 1 - 2/(1 - a*x)])/(4*a^2)} -{ArcTanh[a*x]^3/(1 - a^2*x^2), x, 1, ArcTanh[a*x]^4/(4*a)} -{ArcTanh[a*x]^3/(x*(1 - a^2*x^2)), x, 5, (1/4)*ArcTanh[a*x]^4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3/4)*PolyLog[4, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)), x, 7, a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (1/4)*a*ArcTanh[a*x]^4 + 3*a*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)), x, 13, (3/2)*a^2*ArcTanh[a*x]^2 - (3*a*ArcTanh[a*x]^2)/(2*x) + (1/2)*a^2*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/(2*x^2) + (1/4)*a^2*ArcTanh[a*x]^4 + 3*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3/2)*a^2*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3/4)*a^2*PolyLog[4, -1 + 2/(1 + a*x)]} - - -{ArcTanh[a*x]^(1/2)/(1 - a^2*x^2), x, 1, (2*ArcTanh[a*x]^(3/2))/(3*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x/(ArcTanh[a*x]*(1 - a^2*x^2)), x, 0, Unintegrable[x/((1 - a^2*x^2)*ArcTanh[a*x]), x]} -{1/(ArcTanh[a*x]*(1 - a^2*x^2)), x, 1, Log[ArcTanh[a*x]]/a} -{1/(x*ArcTanh[a*x]*(1 - a^2*x^2)), x, 0, Unintegrable[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]), x]} - - -{x/((1 - a^2*x^2)*ArcTanh[a*x]^2), x, 1, -(x/(a*ArcTanh[a*x])) + Unintegrable[1/ArcTanh[a*x], x]/a} -{1/((1 - a^2*x^2)*ArcTanh[a*x]^2), x, 1, -(1/(a*ArcTanh[a*x]))} -{1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^2), x, 1, -(1/(a*x*ArcTanh[a*x])) - Unintegrable[1/(x^2*ArcTanh[a*x]), x]/a} - - -{x/((1 - a^2*x^2)*ArcTanh[a*x]^3), x, 1, -(x/(2*a*ArcTanh[a*x]^2)) + Unintegrable[1/ArcTanh[a*x]^2, x]/(2*a)} -{1/((1 - a^2*x^2)*ArcTanh[a*x]^3), x, 1, -(1/(2*a*ArcTanh[a*x]^2))} -{1/(x*(1 - a^2*x^2)*ArcTanh[a*x]^3), x, 1, -(1/(2*a*x*ArcTanh[a*x]^2)) - Unintegrable[1/(x^2*ArcTanh[a*x]^2), x]/(2*a)} - - -(* ::Subsubsection::Closed:: *) -(*p symbolic*) - - -{ArcTanh[a*x]^p/(1 - a^2*x^2), x, 1, ArcTanh[a*x]^(1 + p)/(a*(1 + p))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^2*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*ArcTanh[a*x]/(1 - a^2*x^2)^2, x, 8, -(x/(4*a^3*(1 - a^2*x^2))) - ArcTanh[a*x]/(4*a^4) + ArcTanh[a*x]/(2*a^4*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(2*a^4) - (ArcTanh[a*x]*Log[2/(1 - a*x)])/a^4 - PolyLog[2, 1 - 2/(1 - a*x)]/(2*a^4)} -{x^2*ArcTanh[a*x]/(1 - a^2*x^2)^2, x, 2, -(1/(4*a^3*(1 - a^2*x^2))) + (x*ArcTanh[a*x])/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^2/(4*a^3)} -{x*ArcTanh[a*x]/(1 - a^2*x^2)^2, x, 3, -(x/(4*a*(1 - a^2*x^2))) - ArcTanh[a*x]/(4*a^2) + ArcTanh[a*x]/(2*a^2*(1 - a^2*x^2))} -{ArcTanh[a*x]/(1 - a^2*x^2)^2, x, 2, -(1/(4*a*(1 - a^2*x^2))) + (x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^2/(4*a)} -{ArcTanh[a*x]/(x*(1 - a^2*x^2)^2), x, 7, -((a*x)/(4*(1 - a^2*x^2))) - (1/4)*ArcTanh[a*x] + ArcTanh[a*x]/(2*(1 - a^2*x^2)) + (1/2)*ArcTanh[a*x]^2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (1/2)*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^2), x, 10, -(a/(4*(1 - a^2*x^2))) - ArcTanh[a*x]/x + (a^2*x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + (3/4)*a*ArcTanh[a*x]^2 + a*Log[x] - (1/2)*a*Log[1 - a^2*x^2]} -{ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^2), x, 15, -(a/(2*x)) - (a^3*x)/(4*(1 - a^2*x^2)) + (1/4)*a^2*ArcTanh[a*x] - ArcTanh[a*x]/(2*x^2) + (a^2*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + a^2*ArcTanh[a*x]^2 + 2*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a^2*PolyLog[2, -1 + 2/(1 + a*x)]} - - -{x^3*ArcTanh[a*x]^2/(1 - a^2*x^2)^2, x, 8, 1/(4*a^4*(1 - a^2*x^2)) - (x*ArcTanh[a*x])/(2*a^3*(1 - a^2*x^2)) - ArcTanh[a*x]^2/(4*a^4) + ArcTanh[a*x]^2/(2*a^4*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(3*a^4) - (ArcTanh[a*x]^2*Log[2/(1 - a*x)])/a^4 - (ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a^4 + PolyLog[3, 1 - 2/(1 - a*x)]/(2*a^4)} -{x^2*ArcTanh[a*x]^2/(1 - a^2*x^2)^2, x, 4, x/(4*a^2*(1 - a^2*x^2)) + ArcTanh[a*x]/(4*a^3) - ArcTanh[a*x]/(2*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(6*a^3)} -{x*ArcTanh[a*x]^2/(1 - a^2*x^2)^2, x, 3, 1/(4*a^2*(1 - a^2*x^2)) - (x*ArcTanh[a*x])/(2*a*(1 - a^2*x^2)) - ArcTanh[a*x]^2/(4*a^2) + ArcTanh[a*x]^2/(2*a^2*(1 - a^2*x^2))} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^2, x, 4, x/(4*(1 - a^2*x^2)) + ArcTanh[a*x]/(4*a) - ArcTanh[a*x]/(2*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(6*a)} -{ArcTanh[a*x]^2/(x*(1 - a^2*x^2)^2), x, 8, 1/(4*(1 - a^2*x^2)) - (a*x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) - (1/4)*ArcTanh[a*x]^2 + ArcTanh[a*x]^2/(2*(1 - a^2*x^2)) + (1/3)*ArcTanh[a*x]^3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (1/2)*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^2), x, 11, (a^2*x)/(4*(1 - a^2*x^2)) + (1/4)*a*ArcTanh[a*x] - (a*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (a^2*x*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (1/2)*a*ArcTanh[a*x]^3 + 2*a*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^2), x, 22, a^2/(4*(1 - a^2*x^2)) - (a*ArcTanh[a*x])/x - (a^3*x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + (1/4)*a^2*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/(2*x^2) + (a^2*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (2/3)*a^2*ArcTanh[a*x]^3 + a^2*Log[x] - (1/2)*a^2*Log[1 - a^2*x^2] + 2*a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - a^2*PolyLog[3, -1 + 2/(1 + a*x)]} - - -{x^3*ArcTanh[a*x]^3/(1 - a^2*x^2)^2, x, 11, -((3*x)/(8*a^3*(1 - a^2*x^2))) - (3*ArcTanh[a*x])/(8*a^4) + (3*ArcTanh[a*x])/(4*a^4*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(4*a^3*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(4*a^4) + ArcTanh[a*x]^3/(2*a^4*(1 - a^2*x^2)) + ArcTanh[a*x]^4/(4*a^4) - (ArcTanh[a*x]^3*Log[2/(1 - a*x)])/a^4 - (3*ArcTanh[a*x]^2*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^4) + (3*ArcTanh[a*x]*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a^4) - (3*PolyLog[4, 1 - 2/(1 - a*x)])/(4*a^4)} -{x^2*ArcTanh[a*x]^3/(1 - a^2*x^2)^2, x, 4, -(3/(8*a^3*(1 - a^2*x^2))) + (3*x*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(8*a^3) - (3*ArcTanh[a*x]^2)/(4*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(8*a^3)} -{x*ArcTanh[a*x]^3/(1 - a^2*x^2)^2, x, 5, -((3*x)/(8*a*(1 - a^2*x^2))) - (3*ArcTanh[a*x])/(8*a^2) + (3*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(4*a*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(4*a^2) + ArcTanh[a*x]^3/(2*a^2*(1 - a^2*x^2))} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^2, x, 4, -(3/(8*a*(1 - a^2*x^2))) + (3*x*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(8*a) - (3*ArcTanh[a*x]^2)/(4*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/(8*a)} -{ArcTanh[a*x]^3/(x*(1 - a^2*x^2)^2), x, 11, -((3*a*x)/(8*(1 - a^2*x^2))) - (3/8)*ArcTanh[a*x] + (3*ArcTanh[a*x])/(4*(1 - a^2*x^2)) - (3*a*x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)) - (1/4)*ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(2*(1 - a^2*x^2)) + (1/4)*ArcTanh[a*x]^4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3/4)*PolyLog[4, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^2), x, 12, -((3*a)/(8*(1 - a^2*x^2))) + (3*a^2*x*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3/8)*a*ArcTanh[a*x]^2 - (3*a*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)) + a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (a^2*x*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + (3/8)*a*ArcTanh[a*x]^4 + 3*a*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^2), x, 25, -((3*a^3*x)/(8*(1 - a^2*x^2))) - (3/8)*a^2*ArcTanh[a*x] + (3*a^2*ArcTanh[a*x])/(4*(1 - a^2*x^2)) + (3/2)*a^2*ArcTanh[a*x]^2 - (3*a*ArcTanh[a*x]^2)/(2*x) - (3*a^3*x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)) + (1/4)*a^2*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/(2*x^2) + (a^2*ArcTanh[a*x]^3)/(2*(1 - a^2*x^2)) + (1/2)*a^2*ArcTanh[a*x]^4 + 3*a^2*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] + 2*a^2*ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3/2)*a^2*PolyLog[2, -1 + 2/(1 + a*x)] - 3*a^2*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - 3*a^2*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3/2)*a^2*PolyLog[4, -1 + 2/(1 + a*x)]} - - -{Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^2, x, 9, (x*Sqrt[ArcTanh[a*x]])/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^(3/2)/(3*a) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(16*a) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(16*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4/((1 - a^2*x^2)^2*ArcTanh[a*x]), x, 0, CoshIntegral[2*ArcTanh[a*x]]/(2*a^5) - (3*Log[ArcTanh[a*x]])/(2*a^5) + Unintegrable[ArcTanh[a*x]^(-1), x]/a^4, Unintegrable[x^4/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]} -{x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]), x, 0, SinhIntegral[2*ArcTanh[a*x]]/(2*a^4) - Unintegrable[x/((1 - a^2*x^2)*ArcTanh[a*x]), x]/a^2, Unintegrable[x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]), x]} -{x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]), x, 4, CoshIntegral[2*ArcTanh[a*x]]/(2*a^3) - Log[ArcTanh[a*x]]/(2*a^3)} -{x/((1 - a^2*x^2)^2*ArcTanh[a*x]), x, 4, SinhIntegral[2*ArcTanh[a*x]]/(2*a^2)} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]), x, 4, CoshIntegral[2*ArcTanh[a*x]]/(2*a) + Log[ArcTanh[a*x]]/(2*a)} -{1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]), x, 0, SinhIntegral[2*ArcTanh[a*x]]/2 + Unintegrable[1/(x*(1 - a^2*x^2)*ArcTanh[a*x]), x], Unintegrable[1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]), x]} - - -{x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^2), x, 11, x/(a^3*ArcTanh[a*x]) - x/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a^4 - Unintegrable[1/ArcTanh[a*x], x]/a^3} -{x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]^2), x, 5, -(x^2/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[2*ArcTanh[a*x]]/a^3} -{x/((1 - a^2*x^2)^2*ArcTanh[a*x]^2), x, 9, -(x/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + CoshIntegral[2*ArcTanh[a*x]]/a^2} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^2), x, 5, -(1/(a*(1 - a^2*x^2)*ArcTanh[a*x])) + SinhIntegral[2*ArcTanh[a*x]]/a} -{1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2), x, 11, -(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]] - Unintegrable[1/(x^2*ArcTanh[a*x]), x]/a} - - -{x^3/((1 - a^2*x^2)^2*ArcTanh[a*x]^3), x, 7, x/(2*a^3*ArcTanh[a*x]^2) - x/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*a^4*(1 - a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^4 - Unintegrable[1/ArcTanh[a*x]^2, x]/(2*a^3)} -{x^2/((1 - a^2*x^2)^2*ArcTanh[a*x]^3), x, 10, -(x^2/(2*a*(1 - a^2*x^2)*ArcTanh[a*x]^2)) - x/(a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a^3} -{x/((1 - a^2*x^2)^2*ArcTanh[a*x]^3), x, 5, -(x/(2*a*(1 - a^2*x^2)*ArcTanh[a*x]^2)) - (1 + a^2*x^2)/(2*a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/a^2} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^3), x, 10, -(1/(2*a*(1 - a^2*x^2)*ArcTanh[a*x]^2)) - x/((1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a} -{1/(x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3), x, 7, -(1/(2*a*x*ArcTanh[a*x]^2)) - (a*x)/(2*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(2*(1 - a^2*x^2)*ArcTanh[a*x]) - Unintegrable[1/(x^2*ArcTanh[a*x]^2), x]/(2*a) + SinhIntegral[2*ArcTanh[a*x]]} - - -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^4), x, 6, -(1/(3*a*(1 - a^2*x^2)*ArcTanh[a*x]^3)) - x/(3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(3*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(3*a)} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^5), x, 11, -(1/(4*a*(1 - a^2*x^2)*ArcTanh[a*x]^4)) - x/(6*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (1 + a^2*x^2)/(12*a*(1 - a^2*x^2)*ArcTanh[a*x]^2) - x/(3*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/(3*a)} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^6), x, 7, -(1/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]^5)) - x/(10*(1 - a^2*x^2)*ArcTanh[a*x]^4) - (1 + a^2*x^2)/(30*a*(1 - a^2*x^2)*ArcTanh[a*x]^3) - x/(15*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (1 + a^2*x^2)/(15*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(15*a)} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^7), x, 12, -(1/(6*a*(1 - a^2*x^2)*ArcTanh[a*x]^6)) - x/(15*(1 - a^2*x^2)*ArcTanh[a*x]^5) - (1 + a^2*x^2)/(60*a*(1 - a^2*x^2)*ArcTanh[a*x]^4) - x/(45*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (1 + a^2*x^2)/(90*a*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (2*x)/(45*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*CoshIntegral[2*ArcTanh[a*x]])/(45*a)} -{1/((1 - a^2*x^2)^2*ArcTanh[a*x]^8), x, 8, -(1/(7*a*(1 - a^2*x^2)*ArcTanh[a*x]^7)) - x/(21*(1 - a^2*x^2)*ArcTanh[a*x]^6) - (1 + a^2*x^2)/(105*a*(1 - a^2*x^2)*ArcTanh[a*x]^5) - x/(105*(1 - a^2*x^2)*ArcTanh[a*x]^4) - (1 + a^2*x^2)/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (2*x)/(315*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (2*(1 + a^2*x^2))/(315*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (4*SinhIntegral[2*ArcTanh[a*x]])/(315*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^3*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^3*ArcTanh[a*x]/(1 - a^2*x^2)^3, x, 4, -(x^3/(16*a*(1 - a^2*x^2)^2)) + (3*x)/(32*a^3*(1 - a^2*x^2)) - (3*ArcTanh[a*x])/(32*a^4) + (x^4*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2)} -{x^2*ArcTanh[a*x]/(1 - a^2*x^2)^3, x, 3, -(1/(16*a^3*(1 - a^2*x^2)^2)) + 1/(16*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x])/(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^2/(16*a^3)} -{x*ArcTanh[a*x]/(1 - a^2*x^2)^3, x, 4, -(x/(16*a*(1 - a^2*x^2)^2)) - (3*x)/(32*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x])/(32*a^2) + ArcTanh[a*x]/(4*a^2*(1 - a^2*x^2)^2)} -{ArcTanh[a*x]/(1 - a^2*x^2)^3, x, 3, -(1/(16*a*(1 - a^2*x^2)^2)) - 3/(16*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^2)/(16*a)} -{ArcTanh[a*x]/(x*(1 - a^2*x^2)^3), x, 12, -((a*x)/(16*(1 - a^2*x^2)^2)) - (11*a*x)/(32*(1 - a^2*x^2)) - (11/32)*ArcTanh[a*x] + ArcTanh[a*x]/(4*(1 - a^2*x^2)^2) + ArcTanh[a*x]/(2*(1 - a^2*x^2)) + (1/2)*ArcTanh[a*x]^2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - (1/2)*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^3), x, 14, -(a/(16*(1 - a^2*x^2)^2)) - (7*a)/(16*(1 - a^2*x^2)) - ArcTanh[a*x]/x + (a^2*x*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + (15/16)*a*ArcTanh[a*x]^2 + a*Log[x] - (1/2)*a*Log[1 - a^2*x^2]} - - -{x^3*ArcTanh[a*x]^2/(1 - a^2*x^2)^3, x, 4, x^4/(32*(1 - a^2*x^2)^2) - 3/(32*a^4*(1 - a^2*x^2)) - (x^3*ArcTanh[a*x])/(8*a*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x])/(16*a^3*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^2)/(32*a^4) + (x^4*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2)} -{x^2*ArcTanh[a*x]^2/(1 - a^2*x^2)^3, x, 13, x/(32*a^2*(1 - a^2*x^2)^2) - x/(64*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]/(64*a^3) - ArcTanh[a*x]/(8*a^3*(1 - a^2*x^2)^2) + ArcTanh[a*x]/(8*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x]^2)/(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^3/(24*a^3)} -{x*ArcTanh[a*x]^2/(1 - a^2*x^2)^3, x, 4, 1/(32*a^2*(1 - a^2*x^2)^2) + 3/(32*a^2*(1 - a^2*x^2)) - (x*ArcTanh[a*x])/(8*a*(1 - a^2*x^2)^2) - (3*x*ArcTanh[a*x])/(16*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^2)/(32*a^2) + ArcTanh[a*x]^2/(4*a^2*(1 - a^2*x^2)^2)} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^3, x, 8, x/(32*(1 - a^2*x^2)^2) + (15*x)/(64*(1 - a^2*x^2)) + (15*ArcTanh[a*x])/(64*a) - ArcTanh[a*x]/(8*a*(1 - a^2*x^2)^2) - (3*ArcTanh[a*x])/(8*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x]^2)/(8*(1 - a^2*x^2)) + ArcTanh[a*x]^3/(8*a)} -{ArcTanh[a*x]^2/(x*(1 - a^2*x^2)^3), x, 13, 1/(32*(1 - a^2*x^2)^2) + 11/(32*(1 - a^2*x^2)) - (a*x*ArcTanh[a*x])/(8*(1 - a^2*x^2)^2) - (11*a*x*ArcTanh[a*x])/(16*(1 - a^2*x^2)) - (11/32)*ArcTanh[a*x]^2 + ArcTanh[a*x]^2/(4*(1 - a^2*x^2)^2) + ArcTanh[a*x]^2/(2*(1 - a^2*x^2)) + (1/3)*ArcTanh[a*x]^3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (1/2)*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^3), x, 20, (a^2*x)/(32*(1 - a^2*x^2)^2) + (31*a^2*x)/(64*(1 - a^2*x^2)) + (31/64)*a*ArcTanh[a*x] - (a*ArcTanh[a*x])/(8*(1 - a^2*x^2)^2) - (7*a*ArcTanh[a*x])/(8*(1 - a^2*x^2)) + a*ArcTanh[a*x]^2 - ArcTanh[a*x]^2/x + (a^2*x*ArcTanh[a*x]^2)/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^2)/(8*(1 - a^2*x^2)) + (5/8)*a*ArcTanh[a*x]^3 + 2*a*ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} - - -{x^3*ArcTanh[a*x]^3/(1 - a^2*x^2)^3, x, 9, -((3*x^3)/(128*a*(1 - a^2*x^2)^2)) + (45*x)/(256*a^3*(1 - a^2*x^2)) + (27*ArcTanh[a*x])/(256*a^4) + (3*x^4*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) - (9*ArcTanh[a*x])/(32*a^4*(1 - a^2*x^2)) - (3*x^3*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)^2) + (9*x*ArcTanh[a*x]^2)/(32*a^3*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^3)/(32*a^4) + (x^4*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2)} -{x^2*ArcTanh[a*x]^3/(1 - a^2*x^2)^3, x, 13, -(3/(128*a^3*(1 - a^2*x^2)^2)) + 3/(128*a^3*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)^2) - (3*x*ArcTanh[a*x])/(64*a^2*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^2)/(128*a^3) - (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)^2) + (3*ArcTanh[a*x]^2)/(16*a^3*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(4*a^2*(1 - a^2*x^2)^2) - (x*ArcTanh[a*x]^3)/(8*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]^4/(32*a^3)} -{x*ArcTanh[a*x]^3/(1 - a^2*x^2)^3, x, 9, -((3*x)/(128*a*(1 - a^2*x^2)^2)) - (45*x)/(256*a*(1 - a^2*x^2)) - (45*ArcTanh[a*x])/(256*a^2) + (3*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)^2) + (9*ArcTanh[a*x])/(32*a^2*(1 - a^2*x^2)) - (3*x*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)^2) - (9*x*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) - (3*ArcTanh[a*x]^3)/(32*a^2) + ArcTanh[a*x]^3/(4*a^2*(1 - a^2*x^2)^2)} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^3, x, 8, -(3/(128*a*(1 - a^2*x^2)^2)) - 45/(128*a*(1 - a^2*x^2)) + (3*x*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (45*x*ArcTanh[a*x])/(64*(1 - a^2*x^2)) + (45*ArcTanh[a*x]^2)/(128*a) - (3*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)^2) - (9*ArcTanh[a*x]^2)/(16*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) + (3*x*ArcTanh[a*x]^3)/(8*(1 - a^2*x^2)) + (3*ArcTanh[a*x]^4)/(32*a)} -{ArcTanh[a*x]^3/(x*(1 - a^2*x^2)^3), x, 21, -((3*a*x)/(128*(1 - a^2*x^2)^2)) - (141*a*x)/(256*(1 - a^2*x^2)) - (141/256)*ArcTanh[a*x] + (3*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (33*ArcTanh[a*x])/(32*(1 - a^2*x^2)) - (3*a*x*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2) - (33*a*x*ArcTanh[a*x]^2)/(32*(1 - a^2*x^2)) - (11/32)*ArcTanh[a*x]^3 + ArcTanh[a*x]^3/(4*(1 - a^2*x^2)^2) + ArcTanh[a*x]^3/(2*(1 - a^2*x^2)) + (1/4)*ArcTanh[a*x]^4 + ArcTanh[a*x]^3*Log[2 - 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1 + a*x)] - (3/4)*PolyLog[4, -1 + 2/(1 + a*x)]} -{ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^3), x, 21, -((3*a)/(128*(1 - a^2*x^2)^2)) - (93*a)/(128*(1 - a^2*x^2)) + (3*a^2*x*ArcTanh[a*x])/(32*(1 - a^2*x^2)^2) + (93*a^2*x*ArcTanh[a*x])/(64*(1 - a^2*x^2)) + (93/128)*a*ArcTanh[a*x]^2 - (3*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)^2) - (21*a*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + a*ArcTanh[a*x]^3 - ArcTanh[a*x]^3/x + (a^2*x*ArcTanh[a*x]^3)/(4*(1 - a^2*x^2)^2) + (7*a^2*x*ArcTanh[a*x]^3)/(8*(1 - a^2*x^2)) + (15/32)*a*ArcTanh[a*x]^4 + 3*a*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a*PolyLog[3, -1 + 2/(1 + a*x)]} - - -{Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^3, x, 15, ArcTanh[a*x]^(3/2)/(4*a) + (Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/(256*a) + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(16*a) - (Sqrt[Pi]*Erfi[2*Sqrt[ArcTanh[a*x]]])/(256*a) - (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(16*a) + (Sqrt[ArcTanh[a*x]]*Sinh[2*ArcTanh[a*x]])/(4*a) + (Sqrt[ArcTanh[a*x]]*Sinh[4*ArcTanh[a*x]])/(32*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^6/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 0, Unintegrable[x^6/((1 - a^2*x^2)^3*ArcTanh[a*x]), x]} -{x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 0, Unintegrable[x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]), x]} -{x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 5, -(CoshIntegral[2*ArcTanh[a*x]]/(2*a^5)) + CoshIntegral[4*ArcTanh[a*x]]/(8*a^5) + (3*Log[ArcTanh[a*x]])/(8*a^5)} -{x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 5, -(SinhIntegral[2*ArcTanh[a*x]]/(4*a^4)) + SinhIntegral[4*ArcTanh[a*x]]/(8*a^4)} -{x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 4, CoshIntegral[4*ArcTanh[a*x]]/(8*a^3) - Log[ArcTanh[a*x]]/(8*a^3)} -{x/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 5, SinhIntegral[2*ArcTanh[a*x]]/(4*a^2) + SinhIntegral[4*ArcTanh[a*x]]/(8*a^2)} -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]), x, 5, CoshIntegral[2*ArcTanh[a*x]]/(2*a) + CoshIntegral[4*ArcTanh[a*x]]/(8*a) + (3*Log[ArcTanh[a*x]])/(8*a)} -{1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]), x, 0, (3*SinhIntegral[2*ArcTanh[a*x]])/4 + SinhIntegral[4*ArcTanh[a*x]]/8 - Unintegrable[1/(x*(-1 + a^2*x^2)*ArcTanh[a*x]), x], Unintegrable[1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]), x]} - - -{x^5/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 32, -(x/(a^5*ArcTanh[a*x])) - x/(a^5*(1 - a^2*x^2)^2*ArcTanh[a*x]) + (2*x)/(a^5*(1 - a^2*x^2)*ArcTanh[a*x]) - (3*CoshIntegral[2*ArcTanh[a*x]])/(2*a^6) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^6) + Unintegrable[1/ArcTanh[a*x], x]/a^5} -{x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 6, -(x^4/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) - SinhIntegral[2*ArcTanh[a*x]]/a^5 + SinhIntegral[4*ArcTanh[a*x]]/(2*a^5)} -{x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 20, -(x^3/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) - CoshIntegral[2*ArcTanh[a*x]]/(2*a^4) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^4), -(x/(a^3*(1 - a^2*x^2)^2*ArcTanh[a*x])) + x/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) - CoshIntegral[2*ArcTanh[a*x]]/(2*a^4) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^4)} -{x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 12, -(x^2/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + SinhIntegral[4*ArcTanh[a*x]]/(2*a^3), -(1/(a^3*(1 - a^2*x^2)^2*ArcTanh[a*x])) + 1/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[4*ArcTanh[a*x]]/(2*a^3)} -{x/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 10, -(x/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + CoshIntegral[2*ArcTanh[a*x]]/(2*a^2) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^2)} -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 6, -(1/(a*(1 - a^2*x^2)^2*ArcTanh[a*x])) + SinhIntegral[2*ArcTanh[a*x]]/a + SinhIntegral[4*ArcTanh[a*x]]/(2*a)} -{1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^2), x, 22, -(1/(a*x*ArcTanh[a*x])) - (a*x)/((1 - a^2*x^2)^2*ArcTanh[a*x]) - (a*x)/((1 - a^2*x^2)*ArcTanh[a*x]) + (3/2)*CoshIntegral[2*ArcTanh[a*x]] + (1/2)*CoshIntegral[4*ArcTanh[a*x]] - Unintegrable[1/(x^2*ArcTanh[a*x]), x]/a} - - -{x^4/((1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 21, -(x^4/(2*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) - (2*x)/(a^4*(1 - a^2*x^2)^2*ArcTanh[a*x]) + (2*x)/(a^4*(1 - a^2*x^2)*ArcTanh[a*x]) - CoshIntegral[2*ArcTanh[a*x]]/a^5 + CoshIntegral[4*ArcTanh[a*x]]/a^5} -{x^3/((1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 25, -(x^3/(2*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) - (3*x^2)/(2*a^2*(1 - a^2*x^2)^2*ArcTanh[a*x]) - x^4/(2*(1 - a^2*x^2)^2*ArcTanh[a*x]) - SinhIntegral[2*ArcTanh[a*x]]/(2*a^4) + SinhIntegral[4*ArcTanh[a*x]]/a^4, -(x/(2*a^3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) + x/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - 2/(a^4*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 3/(2*a^4*(1 - a^2*x^2)*ArcTanh[a*x]) + (1 + a^2*x^2)/(2*a^4*(1 - a^2*x^2)*ArcTanh[a*x]) - SinhIntegral[2*ArcTanh[a*x]]/(2*a^4) + SinhIntegral[4*ArcTanh[a*x]]/a^4} -{x^2/((1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 22, -(x^2/(2*a*(-1 + a^2*x^2)^2*ArcTanh[a*x]^2)) - (2*x)/(a^2*(1 - a^2*x^2)^2*ArcTanh[a*x]) + x/(a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[4*ArcTanh[a*x]]/a^3, -(1/(2*a^3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) + 1/(2*a^3*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (2*x)/(a^2*(1 - a^2*x^2)^2*ArcTanh[a*x]) + x/(a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[4*ArcTanh[a*x]]/a^3} -{x/((1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 19, -(x/(2*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) - 2/(a^2*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 3/(2*a^2*(1 - a^2*x^2)*ArcTanh[a*x]) + SinhIntegral[2*ArcTanh[a*x]]/(2*a^2) + SinhIntegral[4*ArcTanh[a*x]]/a^2} -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 11, -(1/(2*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)) - (2*x)/((1 - a^2*x^2)^2*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a + CoshIntegral[4*ArcTanh[a*x]]/a} -{1/(x*(1 - a^2*x^2)^3*ArcTanh[a*x]^3), x, 27, -(1/(2*a*x*ArcTanh[a*x]^2)) - (a*x)/(2*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) - (a*x)/(2*(1 - a^2*x^2)*ArcTanh[a*x]^2) - 2/((1 - a^2*x^2)^2*ArcTanh[a*x]) + 3/(2*(1 - a^2*x^2)*ArcTanh[a*x]) - (1 + a^2*x^2)/(2*(1 - a^2*x^2)*ArcTanh[a*x]) - Unintegrable[1/(x^2*ArcTanh[a*x]^2), x]/(2*a) + (3/2)*SinhIntegral[2*ArcTanh[a*x]] + SinhIntegral[4*ArcTanh[a*x]]} - - -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]^4), x, 20, -(1/(3*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^3)) - (2*x)/(3*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) - 8/(3*a*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 2/(a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(3*a) + (4*SinhIntegral[4*ArcTanh[a*x]])/(3*a)} -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]^5), x, 35, -(1/(4*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^4)) - x/(3*(1 - a^2*x^2)^2*ArcTanh[a*x]^3) - 2/(3*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + 1/(2*a*(1 - a^2*x^2)*ArcTanh[a*x]^2) - (8*x)/(3*(1 - a^2*x^2)^2*ArcTanh[a*x]) + x/((1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/(3*a) + (4*CoshIntegral[4*ArcTanh[a*x]])/(3*a)} -{1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6), x, 49, -(1/(5*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^5)) - x/(5*(1 - a^2*x^2)^2*ArcTanh[a*x]^4) - 4/(15*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^3) + 1/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (8*x)/(15*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + x/(5*(1 - a^2*x^2)*ArcTanh[a*x]^2) - 32/(15*a*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 8/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (1 + a^2*x^2)/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(15*a) + (16*SinhIntegral[4*ArcTanh[a*x]])/(15*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^4*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]/(1 - a^2*x^2)^4, x, 4, -(1/(36*a*(1 - a^2*x^2)^3)) - 5/(96*a*(1 - a^2*x^2)^2) - 5/(32*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x])/(24*(1 - a^2*x^2)^2) + (5*x*ArcTanh[a*x])/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^2)/(32*a)} - - -{ArcTanh[a*x]^2/(1 - a^2*x^2)^4, x, 13, x/(108*(1 - a^2*x^2)^3) + (65*x)/(1728*(1 - a^2*x^2)^2) + (245*x)/(1152*(1 - a^2*x^2)) + (245*ArcTanh[a*x])/(1152*a) - ArcTanh[a*x]/(18*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x])/(48*a*(1 - a^2*x^2)^2) - (5*ArcTanh[a*x])/(16*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^2)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^2)/(24*(1 - a^2*x^2)^2) + (5*x*ArcTanh[a*x]^2)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^3)/(48*a)} - - -{ArcTanh[a*x]^3/(1 - a^2*x^2)^4, x, 13, -(1/(216*a*(1 - a^2*x^2)^3)) - 65/(2304*a*(1 - a^2*x^2)^2) - 245/(768*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x])/(36*(1 - a^2*x^2)^3) + (65*x*ArcTanh[a*x])/(576*(1 - a^2*x^2)^2) + (245*x*ArcTanh[a*x])/(384*(1 - a^2*x^2)) + (245*ArcTanh[a*x]^2)/(768*a) - ArcTanh[a*x]^2/(12*a*(1 - a^2*x^2)^3) - (5*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)^2) - (15*ArcTanh[a*x]^2)/(32*a*(1 - a^2*x^2)) + (x*ArcTanh[a*x]^3)/(6*(1 - a^2*x^2)^3) + (5*x*ArcTanh[a*x]^3)/(24*(1 - a^2*x^2)^2) + (5*x*ArcTanh[a*x]^3)/(16*(1 - a^2*x^2)) + (5*ArcTanh[a*x]^4)/(64*a)} - - -{Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4, x, 21, (5*ArcTanh[a*x]^(3/2))/(24*a) + (3*Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/(512*a) + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(256*a) + (Sqrt[Pi/6]*Erf[Sqrt[6]*Sqrt[ArcTanh[a*x]]])/(768*a) - (3*Sqrt[Pi]*Erfi[2*Sqrt[ArcTanh[a*x]]])/(512*a) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(256*a) - (Sqrt[Pi/6]*Erfi[Sqrt[6]*Sqrt[ArcTanh[a*x]]])/(768*a) + (15*Sqrt[ArcTanh[a*x]]*Sinh[2*ArcTanh[a*x]])/(64*a) + (3*Sqrt[ArcTanh[a*x]]*Sinh[4*ArcTanh[a*x]])/(64*a) + (Sqrt[ArcTanh[a*x]]*Sinh[6*ArcTanh[a*x]])/(192*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^8/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 0, Unintegrable[x^8/((1 - a^2*x^2)^4*ArcTanh[a*x]), x]} -{x^7/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 0, Unintegrable[x^7/((1 - a^2*x^2)^4*ArcTanh[a*x]), x]} -{x^6/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, (15*CoshIntegral[2*ArcTanh[a*x]])/(32*a^7) - (3*CoshIntegral[4*ArcTanh[a*x]])/(16*a^7) + CoshIntegral[6*ArcTanh[a*x]]/(32*a^7) - (5*Log[ArcTanh[a*x]])/(16*a^7)} -{x^5/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, (5*SinhIntegral[2*ArcTanh[a*x]])/(32*a^6) - SinhIntegral[4*ArcTanh[a*x]]/(8*a^6) + SinhIntegral[6*ArcTanh[a*x]]/(32*a^6)} -{x^4/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, -(CoshIntegral[2*ArcTanh[a*x]]/(32*a^5)) - CoshIntegral[4*ArcTanh[a*x]]/(16*a^5) + CoshIntegral[6*ArcTanh[a*x]]/(32*a^5) + Log[ArcTanh[a*x]]/(16*a^5)} -{x^3/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 5, -((3*SinhIntegral[2*ArcTanh[a*x]])/(32*a^4)) + SinhIntegral[6*ArcTanh[a*x]]/(32*a^4)} -{x^2/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, -(CoshIntegral[2*ArcTanh[a*x]]/(32*a^3)) + CoshIntegral[4*ArcTanh[a*x]]/(16*a^3) + CoshIntegral[6*ArcTanh[a*x]]/(32*a^3) - Log[ArcTanh[a*x]]/(16*a^3)} -{x^1/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, (5*SinhIntegral[2*ArcTanh[a*x]])/(32*a^2) + SinhIntegral[4*ArcTanh[a*x]]/(8*a^2) + SinhIntegral[6*ArcTanh[a*x]]/(32*a^2)} -{x^0/((1 - a^2*x^2)^4*ArcTanh[a*x]), x, 6, (15*CoshIntegral[2*ArcTanh[a*x]])/(32*a) + (3*CoshIntegral[4*ArcTanh[a*x]])/(16*a) + CoshIntegral[6*ArcTanh[a*x]]/(32*a) + (5*Log[ArcTanh[a*x]])/(16*a)} -{1/(x^1*(1 - a^2*x^2)^4*ArcTanh[a*x]), x, 0, Unintegrable[1/(x*(1 - a^2*x^2)^4*ArcTanh[a*x]), x]} -{1/(x^2*(1 - a^2*x^2)^4*ArcTanh[a*x]), x, 0, Unintegrable[1/(x^2*(1 - a^2*x^2)^4*ArcTanh[a*x]), x]} - - -{x/((1 - a^2*x^2)^4*ArcTanh[a*x]^2), x, 13, -(x/(a*(1 - a^2*x^2)^3*ArcTanh[a*x])) + (5*CoshIntegral[2*ArcTanh[a*x]])/(16*a^2) + CoshIntegral[4*ArcTanh[a*x]]/(2*a^2) + (3*CoshIntegral[6*ArcTanh[a*x]])/(16*a^2)} -{1/((1 - a^2*x^2)^4*ArcTanh[a*x]^2), x, 7, -(1/(a*(1 - a^2*x^2)^3*ArcTanh[a*x])) + (15*SinhIntegral[2*ArcTanh[a*x]])/(16*a) + (3*SinhIntegral[4*ArcTanh[a*x]])/(4*a) + (3*SinhIntegral[6*ArcTanh[a*x]])/(16*a)} - - -{x/((1 - a^2*x^2)^4*ArcTanh[a*x]^3), x, 22, -(x/(2*a*(1 - a^2*x^2)^3*ArcTanh[a*x]^2)) - 3/(a^2*(1 - a^2*x^2)^3*ArcTanh[a*x]) + 5/(2*a^2*(1 - a^2*x^2)^2*ArcTanh[a*x]) + (5*SinhIntegral[2*ArcTanh[a*x]])/(16*a^2) + SinhIntegral[4*ArcTanh[a*x]]/a^2 + (9*SinhIntegral[6*ArcTanh[a*x]])/(16*a^2)} -{1/((1 - a^2*x^2)^4*ArcTanh[a*x]^3), x, 14, -(1/(2*a*(1 - a^2*x^2)^3*ArcTanh[a*x]^2)) - (3*x)/((1 - a^2*x^2)^3*ArcTanh[a*x]) + (15*CoshIntegral[2*ArcTanh[a*x]])/(16*a) + (3*CoshIntegral[4*ArcTanh[a*x]])/(2*a) + (9*CoshIntegral[6*ArcTanh[a*x]])/(16*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^(1/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 9, -((5*x*Sqrt[1 - a^2*x^2])/(24*a^5)) - (x^3*Sqrt[1 - a^2*x^2])/(20*a^3) + (89*ArcSin[a*x])/(120*a^6) - (8*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^6) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^4) - (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(5*a^2)} -{x^4*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 7, -((5*Sqrt[1 - a^2*x^2])/(8*a^5)) + (1 - a^2*x^2)^(3/2)/(12*a^5) - (3*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(4*a^2) - (3*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(4*a^5) - (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(8*a^5) + (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(8*a^5)} -{x^3*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 5, -((x*Sqrt[1 - a^2*x^2])/(6*a^3)) + (5*ArcSin[a*x])/(6*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^2)} -{x^2*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 3, -(Sqrt[1 - a^2*x^2]/(2*a^3)) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a^2) - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a^3 - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(2*a^3) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(2*a^3)} -{x^1*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 2, ArcSin[a*x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^2} -{x^0*ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 1, -((2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a) - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a} -{ArcTanh[a*x]/(x^1*(1 - a^2*x^2)^(1/2)), x, 1, -2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^(1/2)), x, 4, -((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(1/2)), x, 3, -((a*Sqrt[1 - a^2*x^2])/(2*x)) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) - a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + (1/2)*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - (1/2)*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} - - -{x^3*ArcTanh[a*x]^2/(1 - a^2*x^2)^(1/2), x, 6, -(Sqrt[1 - a^2*x^2]/(3*a^4)) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^3) - (10*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(3*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(3*a^2) - (5*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(3*a^4) + (5*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(3*a^4)} -{x^2*ArcTanh[a*x]^2/(1 - a^2*x^2)^(1/2), x, 11, ArcSin[a*x]/a^3 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^3 - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*a^2) + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a^3 - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^3 + (I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a^3 + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^3 - (I*PolyLog[3, I*E^ArcTanh[a*x]])/a^3} -{x^1*ArcTanh[a*x]^2/(1 - a^2*x^2)^(1/2), x, 2, -((4*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a^2) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^2 - (2*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a^2 + (2*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^2} -{x^0*ArcTanh[a*x]^2/(1 - a^2*x^2)^(1/2), x, 8, (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a - (2*I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (2*I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a + (2*I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (2*I*PolyLog[3, I*E^ArcTanh[a*x]])/a} -{ArcTanh[a*x]^2/(x^1*(1 - a^2*x^2)^(1/2)), x, 8, -2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 2*PolyLog[3, -E^ArcTanh[a*x]] - 2*PolyLog[3, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^2/(x^2*(1 - a^2*x^2)^(1/2)), x, 2, -((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x) - 4*a*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 2*a*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 2*a*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^(1/2)), x, 13, -((a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x^2) - a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] - a^2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + a^2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + a^2*PolyLog[3, -E^ArcTanh[a*x]] - a^2*PolyLog[3, E^ArcTanh[a*x]]} - - -{x^3*ArcTanh[a*x]^3/(1 - a^2*x^2)^(1/2), x, 21, ArcSin[a*x]/a^4 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^4 - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*a^3) + (5*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a^4 - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(3*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(3*a^2) - (5*I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^4 + (5*I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a^4 + (5*I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^4 - (5*I*PolyLog[3, I*E^ArcTanh[a*x]])/a^4} -{x^2*ArcTanh[a*x]^3/(1 - a^2*x^2)^(1/2), x, 13, -((6*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a^3) - (3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*a^3) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(2*a^2) + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3)/a^3 - (3*I*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]])/(2*a^3) + (3*I*ArcTanh[a*x]^2*PolyLog[2, I*E^ArcTanh[a*x]])/(2*a^3) - (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a^3 + (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^3 + (3*I*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^3 - (3*I*ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a*x]])/a^3 - (3*I*PolyLog[4, (-I)*E^ArcTanh[a*x]])/a^3 + (3*I*PolyLog[4, I*E^ArcTanh[a*x]])/a^3} -{x^1*ArcTanh[a*x]^3/(1 - a^2*x^2)^(1/2), x, 9, (6*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/a^2 - (6*I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^2 + (6*I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a^2 + (6*I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^2 - (6*I*PolyLog[3, I*E^ArcTanh[a*x]])/a^2} -{x^0*ArcTanh[a*x]^3/(1 - a^2*x^2)^(1/2), x, 10, (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3)/a - (3*I*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (3*I*ArcTanh[a*x]^2*PolyLog[2, I*E^ArcTanh[a*x]])/a + (6*I*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (6*I*ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a*x]])/a - (6*I*PolyLog[4, (-I)*E^ArcTanh[a*x]])/a + (6*I*PolyLog[4, I*E^ArcTanh[a*x]])/a} -{ArcTanh[a*x]^3/(x^1*(1 - a^2*x^2)^(1/2)), x, 10, -2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 3*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]] + 3*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 6*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 6*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - 6*PolyLog[4, -E^ArcTanh[a*x]] + 6*PolyLog[4, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^3/(x^2*(1 - a^2*x^2)^(1/2)), x, 9, -6*a*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x - 6*a*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 6*a*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 6*a*PolyLog[3, -E^ArcTanh[a*x]] - 6*a*PolyLog[3, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^3/(x^3*(1 - a^2*x^2)^(1/2)), x, 13, -((3*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x)) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(2*x^2) - a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 6*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (3/2)*a^2*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]] + (3/2)*a^2*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 3*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 3*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 3*a^2*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 3*a^2*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - 3*a^2*PolyLog[4, -E^ArcTanh[a*x]] + 3*a^2*PolyLog[4, E^ArcTanh[a*x]]} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^(3/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[(x^m*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2), x]} - -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^3, x, 5, -(x/(a^3*Sqrt[1 - a^2*x^2])) - ArcSin[a*x]/a^4 + ArcTanh[a*x]/(a^4*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^4} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^2, x, 2, -(1/(a^3*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/(a^2*Sqrt[1 - a^2*x^2]) + (2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a^3 + (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a^3 - (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^3} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^1, x, 2, -(x/(a*Sqrt[1 - a^2*x^2])) + ArcTanh[a*x]/(a^2*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^0, x, 1, -(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2]} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)/x^1, x, 4, -((a*x)/Sqrt[1 - a^2*x^2]) + ArcTanh[a*x]/Sqrt[1 - a^2*x^2] - 2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)/x^2, x, 6, -(a/Sqrt[1 - a^2*x^2]) + (a^2*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x - a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)/x^3, x, 8, -((a^3*x)/Sqrt[1 - a^2*x^2]) - (a*Sqrt[1 - a^2*x^2])/(2*x) + (a^2*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) - 3*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + (3/2)*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - (3/2)*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} - - -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[(x^m*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2), x]} - -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^3, x, 5, 2/(a^4*Sqrt[1 - a^2*x^2]) - (2*x*ArcTanh[a*x])/(a^3*Sqrt[1 - a^2*x^2]) + (4*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a^4 + ArcTanh[a*x]^2/(a^4*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^4 + (2*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a^4 - (2*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^2, x, 11, (2*x)/(a^2*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(a^3*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(a^2*Sqrt[1 - a^2*x^2]) - (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a^3 + (2*I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^3 - (2*I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a^3 - (2*I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^3 + (2*I*PolyLog[3, I*E^ArcTanh[a*x]])/a^3} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^1, x, 2, 2/(a^2*Sqrt[1 - a^2*x^2]) - (2*x*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2]) + ArcTanh[a*x]^2/(a^2*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^0, x, 2, (2*x)/Sqrt[1 - a^2*x^2] - (2*ArcTanh[a*x])/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2]} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)/x^1, x, 11, 2/Sqrt[1 - a^2*x^2] - (2*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^2/Sqrt[1 - a^2*x^2] - 2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 2*PolyLog[3, -E^ArcTanh[a*x]] - 2*PolyLog[3, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)/x^2, x, 5, (2*a^2*x)/Sqrt[1 - a^2*x^2] - (2*a*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + (a^2*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x - 4*a*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 2*a*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 2*a*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)/x^3, x, 25, (2*a^2)/Sqrt[1 - a^2*x^2] - (2*a^3*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x + (a^2*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x^2) - 3*a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] - 3*a^2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 3*a^2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 3*a^2*PolyLog[3, -E^ArcTanh[a*x]] - 3*a^2*PolyLog[3, E^ArcTanh[a*x]]} - - -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[(x^m*ArcTanh[a*x]^3)/(1 - a^2*x^2)^(3/2), x]} - -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^3, x, 13, -((6*x)/(a^3*Sqrt[1 - a^2*x^2])) + (6*ArcTanh[a*x])/(a^4*Sqrt[1 - a^2*x^2]) - (3*x*ArcTanh[a*x]^2)/(a^3*Sqrt[1 - a^2*x^2]) - (6*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a^4 + ArcTanh[a*x]^3/(a^4*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/a^4 + (6*I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^4 - (6*I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a^4 - (6*I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^4 + (6*I*PolyLog[3, I*E^ArcTanh[a*x]])/a^4} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^2, x, 13, -(6/(a^3*Sqrt[1 - a^2*x^2])) + (6*x*ArcTanh[a*x])/(a^2*Sqrt[1 - a^2*x^2]) - (3*ArcTanh[a*x]^2)/(a^3*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(a^2*Sqrt[1 - a^2*x^2]) - (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3)/a^3 + (3*I*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a^3 - (3*I*ArcTanh[a*x]^2*PolyLog[2, I*E^ArcTanh[a*x]])/a^3 - (6*I*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a^3 + (6*I*ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a*x]])/a^3 + (6*I*PolyLog[4, (-I)*E^ArcTanh[a*x]])/a^3 - (6*I*PolyLog[4, I*E^ArcTanh[a*x]])/a^3} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^1, x, 3, -((6*x)/(a*Sqrt[1 - a^2*x^2])) + (6*ArcTanh[a*x])/(a^2*Sqrt[1 - a^2*x^2]) - (3*x*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + ArcTanh[a*x]^3/(a^2*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^0, x, 2, -(6/(a*Sqrt[1 - a^2*x^2])) + (6*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2]} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)/x^1, x, 14, -((6*a*x)/Sqrt[1 - a^2*x^2]) + (6*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^3/Sqrt[1 - a^2*x^2] - 2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 3*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]] + 3*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 6*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 6*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - 6*PolyLog[4, -E^ArcTanh[a*x]] + 6*PolyLog[4, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)/x^2, x, 12, -((6*a)/Sqrt[1 - a^2*x^2]) + (6*a^2*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] - 6*a*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 + (a^2*x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x - 6*a*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 6*a*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 6*a*PolyLog[3, -E^ArcTanh[a*x]] - 6*a*PolyLog[3, E^ArcTanh[a*x]]} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)/x^3, x, 28, -((6*a^3*x)/Sqrt[1 - a^2*x^2]) + (6*a^2*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a^3*x*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] - (3*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x) + (a^2*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/(2*x^2) - 3*a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^3 - 6*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (9/2)*a^2*ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]] + (9/2)*a^2*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]] + 3*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 3*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 9*a^2*ArcTanh[a*x]*PolyLog[3, -E^ArcTanh[a*x]] - 9*a^2*ArcTanh[a*x]*PolyLog[3, E^ArcTanh[a*x]] - 9*a^2*PolyLog[4, -E^ArcTanh[a*x]] + 9*a^2*PolyLog[4, E^ArcTanh[a*x]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]} - -{1/ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^2, x, 0, Unintegrable[x^2/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]} -{1/ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^1, x, 2, SinhIntegral[ArcTanh[a*x]]/a^2} -{1/ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)*x^0, x, 2, CoshIntegral[ArcTanh[a*x]]/a} -{1/ArcTanh[a*x]/(1 - a^2*x^2)^(3/2)/x^1, x, 0, Unintegrable[1/(x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]), x]} - - -{1/ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2), x]} - -{1/ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^2, x, 4, -(1/(a^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]]/a^3 - Unintegrable[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2), x]/a^2} -{1/ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^1, x, 3, -(x/(a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + CoshIntegral[ArcTanh[a*x]]/a^2} -{1/ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)*x^0, x, 3, -(1/(a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]]/a} -{1/ArcTanh[a*x]^2/(1 - a^2*x^2)^(3/2)/x^1, x, 5, -((a*x)/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) - Sqrt[1 - a^2*x^2]/(a*x*ArcTanh[a*x]) + CoshIntegral[ArcTanh[a*x]] - Unintegrable[1/(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]), x]/a} - - -{1/ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^m, x, 0, Unintegrable[x^m/((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^3), x]} - -{1/ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^2, x, 5, -(1/(2*a^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) - x/(2*a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) + CoshIntegral[ArcTanh[a*x]]/(2*a^3) - Unintegrable[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3), x]/a^2} -{1/ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^1, x, 4, -(x/(2*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) - 1/(2*a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) + SinhIntegral[ArcTanh[a*x]]/(2*a^2)} -{1/ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)*x^0, x, 4, -(1/(2*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) - x/(2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) + CoshIntegral[ArcTanh[a*x]]/(2*a)} -{1/ArcTanh[a*x]^3/(1 - a^2*x^2)^(3/2)/x^1, x, 6, -((a*x)/(2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) - Sqrt[1 - a^2*x^2]/(2*a*x*ArcTanh[a*x]^2) - 1/(2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) + (1/2)*SinhIntegral[ArcTanh[a*x]] - Unintegrable[1/(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2), x]/(2*a)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^(1/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*x^4, x, 11, Sqrt[1 - a^2*x^2]/(16*a^5) - (7*(1 - a^2*x^2)^(3/2))/(72*a^5) + (1 - a^2*x^2)^(5/2)/(30*a^5) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(24*a^2) + (1/6)*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(8*a^5) - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(16*a^5) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(16*a^5)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*x^3, x, 9, (x*Sqrt[1 - a^2*x^2])/(24*a^3) + (x^3*Sqrt[1 - a^2*x^2])/(20*a) + (11*ArcSin[a*x])/(120*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a^2) + (1/5)*x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*x^2, x, 7, Sqrt[1 - a^2*x^2]/(8*a^3) - (1 - a^2*x^2)^(3/2)/(12*a^3) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a^2) + (1/4)*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(4*a^3) - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(8*a^3) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(8*a^3)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*x^1, x, 3, (x*Sqrt[1 - a^2*x^2])/(6*a) + ArcSin[a*x]/(6*a^2) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/(3*a^2)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*x^0, x, 2, Sqrt[1 - a^2*x^2]/(2*a) + (1/2)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(2*a) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(2*a)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^1, x, 3, -ArcSin[a*x] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - 2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^2, x, 6, -((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) + 2*a*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x] - a*ArcTanh[Sqrt[1 - a^2*x^2]] + I*a*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])] - I*a*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^3, x, 5, -((a*Sqrt[1 - a^2*x^2])/(2*x)) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) + a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (1/2)*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] + (1/2)*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^4, x, 5, -((a*Sqrt[1 - a^2*x^2])/(6*x^2)) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/(3*x^3) + (1/6)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^5, x, 9, -((a*Sqrt[1 - a^2*x^2])/(12*x^3)) - (a^3*Sqrt[1 - a^2*x^2])/(24*x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(4*x^4) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*x^2) + (1/4)*a^4*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (1/8)*a^4*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] + (1/8)*a^4*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^6, x, 21, -((a*Sqrt[1 - a^2*x^2])/(20*x^4)) - (a^3*Sqrt[1 - a^2*x^2])/(24*x^2) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(5*x^5) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*x^3) + (2*a^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*x) + (11/120)*a^5*ArcTanh[Sqrt[1 - a^2*x^2]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]/x^7, x, 14, -((a*Sqrt[1 - a^2*x^2])/(30*x^5)) - (11*a^3*Sqrt[1 - a^2*x^2])/(360*x^3) + (a^5*Sqrt[1 - a^2*x^2])/(720*x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(6*x^6) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(24*x^4) + (a^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*x^2) + (1/8)*a^6*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (1/16)*a^6*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] + (1/16)*a^6*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} - - -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*x^4, x, 45, (x*Sqrt[1 - a^2*x^2])/(18*a^4) + (x^3*Sqrt[1 - a^2*x^2])/(60*a^2) - (19*ArcSin[a*x])/(360*a^5) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(360*a^5) + (11*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(180*a^3) + (x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(15*a) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(16*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(24*a^2) + (1/6)*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/(8*a^5) - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/(8*a^5) + (I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/(8*a^5) + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/(8*a^5) - (I*PolyLog[3, I*E^ArcTanh[a*x]])/(8*a^5)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*x^3, x, 21, (11*Sqrt[1 - a^2*x^2])/(60*a^4) - (1 - a^2*x^2)^(3/2)/(30*a^4) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(12*a^3) + (x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(10*a) - (11*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(30*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(15*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(15*a^2) + (1/5)*x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 - (11*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(60*a^4) + (11*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(60*a^4)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*x^2, x, 29, (x*Sqrt[1 - a^2*x^2])/(12*a^2) - ArcSin[a*x]/(6*a^3) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(12*a^3) + (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(6*a) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(8*a^2) + (1/4)*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/(4*a^3) - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/(4*a^3) + (I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/(4*a^3) + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/(4*a^3) - (I*PolyLog[3, I*E^ArcTanh[a*x]])/(4*a^3)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*x^1, x, 3, Sqrt[1 - a^2*x^2]/(3*a^2) + (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a) - (2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(3*a^2) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2)/(3*a^2) - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(3*a^2) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(3*a^2)} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2*x^0, x, 10, -(ArcSin[a*x]/a) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a + (1/2)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (I*PolyLog[3, I*E^ArcTanh[a*x]])/a} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2/x^1, x, 11, 4*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 - 2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - 2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] + 2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] + 2*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])] - 2*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]] + 2*PolyLog[3, -E^ArcTanh[a*x]] - 2*PolyLog[3, E^ArcTanh[a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2/x^2, x, 11, -((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x) - 2*a*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - 4*a*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + 2*I*a*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]] - 2*I*a*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]] + 2*a*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - 2*a*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]] - 2*I*a*PolyLog[3, (-I)*E^ArcTanh[a*x]] + 2*I*a*PolyLog[3, I*E^ArcTanh[a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2/x^3, x, 22, -((a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*x^2) + a^2*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - a^2*ArcTanh[Sqrt[1 - a^2*x^2]] + a^2*ArcTanh[a*x]*PolyLog[2, -E^ArcTanh[a*x]] - a^2*ArcTanh[a*x]*PolyLog[2, E^ArcTanh[a*x]] - a^2*PolyLog[3, -E^ArcTanh[a*x]] + a^2*PolyLog[3, E^ArcTanh[a*x]]} -{Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2/x^4, x, 6, -((a^2*Sqrt[1 - a^2*x^2])/(3*x)) - (a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*x^2) - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2)/(3*x^3) + (2/3)*a^3*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (1/3)*a^3*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] + (1/3)*a^3*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^(3/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*x^4, x, 27, (3*Sqrt[1 - a^2*x^2])/(128*a^5) + (1 - a^2*x^2)^(3/2)/(192*a^5) - (3*(1 - a^2*x^2)^(5/2))/(80*a^5) + (1 - a^2*x^2)^(7/2)/(56*a^5) - (3*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(128*a^4) - (x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(64*a^2) + (3/16)*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (1/8)*a^2*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (3*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(64*a^5) - (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(128*a^5) + (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(128*a^5)} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*x^3, x, 24, (3*x*Sqrt[1 - a^2*x^2])/(112*a^3) + (23*x^3*Sqrt[1 - a^2*x^2])/(840*a) - (1/42)*a*x^5*Sqrt[1 - a^2*x^2] + (17*ArcSin[a*x])/(560*a^4) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(35*a^4) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(35*a^2) + (8/35)*x^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (1/7)*a^2*x^6*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*x^2, x, 19, Sqrt[1 - a^2*x^2]/(16*a^3) + (1 - a^2*x^2)^(3/2)/(72*a^3) - (1 - a^2*x^2)^(5/2)/(30*a^3) - (x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*a^2) + (7/24)*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (1/6)*a^2*x^5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(8*a^3) - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(16*a^3) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(16*a^3)} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*x^1, x, 4, (3*x*Sqrt[1 - a^2*x^2])/(40*a) + (x*(1 - a^2*x^2)^(3/2))/(20*a) + (3*ArcSin[a*x])/(40*a^2) - ((1 - a^2*x^2)^(5/2)*ArcTanh[a*x])/(5*a^2)} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]*x^0, x, 3, (3*Sqrt[1 - a^2*x^2])/(8*a) + (1 - a^2*x^2)^(3/2)/(12*a) + (3/8)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (1/4)*x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x] - (3*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(4*a) - (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(8*a) + (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(8*a)} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^1, x, 7, (-(1/6))*a*x*Sqrt[1 - a^2*x^2] - (7/6)*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (1/3)*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x] - 2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^2, x, 9, (-(1/2))*a*Sqrt[1 - a^2*x^2] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x - (1/2)*a^2*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 3*a*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x] - a*ArcTanh[Sqrt[1 - a^2*x^2]] + (3/2)*I*a*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])] - (3/2)*I*a*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^3, x, 9, -((a*Sqrt[1 - a^2*x^2])/(2*x)) + a^2*ArcSin[a*x] - a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) + 3*a^2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] - (3/2)*a^2*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] + (3/2)*a^2*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^4, x, 12, -((a*Sqrt[1 - a^2*x^2])/(6*x^2)) + (a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x - ((1 - a^2*x^2)^(3/2)*ArcTanh[a*x])/(3*x^3) - 2*a^3*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x] + (7/6)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]] - I*a^3*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])] + I*a^3*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^5, x, 15, -((a*Sqrt[1 - a^2*x^2])/(12*x^3)) + (11*a^3*Sqrt[1 - a^2*x^2])/(24*x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(4*x^4) + (5*a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*x^2) - (3/4)*a^4*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + (3/8)*a^4*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - (3/8)*a^4*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^6, x, 6, (3*a^3*Sqrt[1 - a^2*x^2])/(40*x^2) - (a*(1 - a^2*x^2)^(3/2))/(20*x^4) - ((1 - a^2*x^2)^(5/2)*ArcTanh[a*x])/(5*x^5) - (3/40)*a^5*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]/x^7, x, 24, -((a*Sqrt[1 - a^2*x^2])/(30*x^5)) + (19*a^3*Sqrt[1 - a^2*x^2])/(360*x^3) + (31*a^5*Sqrt[1 - a^2*x^2])/(720*x) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(6*x^6) + (7*a^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(24*x^4) - (a^4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*x^2) - (1/8)*a^6*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + (1/16)*a^6*PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - (1/16)*a^6*PolyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]]} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcTanh[c x]^p (1-c^2 x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{ArcTanh[a*x]*(1 - a^2*x^2)^(5/2), x, 4, (5*Sqrt[1 - a^2*x^2])/(16*a) + (5*(1 - a^2*x^2)^(3/2))/(72*a) + (1 - a^2*x^2)^(5/2)/(30*a) + (5/16)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (5/24)*x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x] + (1/6)*x*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x] - (5*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(8*a) - (5*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(16*a) + (5*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(16*a)} -{ArcTanh[a*x]*(1 - a^2*x^2)^(3/2), x, 3, (3*Sqrt[1 - a^2*x^2])/(8*a) + (1 - a^2*x^2)^(3/2)/(12*a) + (3/8)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + (1/4)*x*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x] - (3*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(4*a) - (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(8*a) + (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(8*a)} -{ArcTanh[a*x]*(1 - a^2*x^2)^(1/2), x, 2, Sqrt[1 - a^2*x^2]/(2*a) + (1/2)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x] - (ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(2*a) + (I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(2*a)} -{ArcTanh[a*x]/(1 - a^2*x^2)^(5/2), x, 2, -(1/(9*a*(1 - a^2*x^2)^(3/2))) - 2/(3*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x])/(3*(1 - a^2*x^2)^(3/2)) + (2*x*ArcTanh[a*x])/(3*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]/(1 - a^2*x^2)^(7/2), x, 3, -(1/(25*a*(1 - a^2*x^2)^(5/2))) - 4/(45*a*(1 - a^2*x^2)^(3/2)) - 8/(15*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x])/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x])/(15*(1 - a^2*x^2)^(3/2)) + (8*x*ArcTanh[a*x])/(15*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]/(1 - a^2*x^2)^(9/2), x, 4, -(1/(49*a*(1 - a^2*x^2)^(7/2))) - 6/(175*a*(1 - a^2*x^2)^(5/2)) - 8/(105*a*(1 - a^2*x^2)^(3/2)) - 16/(35*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x])/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x])/(35*(1 - a^2*x^2)^(5/2)) + (8*x*ArcTanh[a*x])/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x])/(35*Sqrt[1 - a^2*x^2])} - -{ArcTanh[a*x]*(c - c*a^2*x^2)^(3/2), x, 4, (3*c*Sqrt[c - a^2*c*x^2])/(8*a) + (c - a^2*c*x^2)^(3/2)/(12*a) + (3/8)*c*x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x] + (1/4)*x*(c - a^2*c*x^2)^(3/2)*ArcTanh[a*x] - (3*c^2*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(4*a*Sqrt[c - a^2*c*x^2]) - (3*I*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(8*a*Sqrt[c - a^2*c*x^2]) + (3*I*c^2*Sqrt[1 - a^2*x^2]*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(8*a*Sqrt[c - a^2*c*x^2])} -{ArcTanh[a*x]*(c - c*a^2*x^2)^(1/2), x, 3, Sqrt[c - a^2*c*x^2]/(2*a) + (1/2)*x*Sqrt[c - a^2*c*x^2]*ArcTanh[a*x] - (c*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(a*Sqrt[c - a^2*c*x^2]) - (I*c*Sqrt[1 - a^2*x^2]*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(2*a*Sqrt[c - a^2*c*x^2]) + (I*c*Sqrt[1 - a^2*x^2]*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(2*a*Sqrt[c - a^2*c*x^2])} -{ArcTanh[a*x]/(c - c*a^2*x^2)^(1/2), x, 2, -((2*Sqrt[1 - a^2*x^2]*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/(a*Sqrt[c - a^2*c*x^2])) - (I*Sqrt[1 - a^2*x^2]*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/(a*Sqrt[c - a^2*c*x^2]) + (I*Sqrt[1 - a^2*x^2]*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(a*Sqrt[c - a^2*c*x^2])} -{ArcTanh[a*x]/(c - c*a^2*x^2)^(3/2), x, 1, -(1/(a*c*Sqrt[c - a^2*c*x^2])) + (x*ArcTanh[a*x])/(c*Sqrt[c - a^2*c*x^2])} -{ArcTanh[a*x]/(c - c*a^2*x^2)^(5/2), x, 2, -(1/(9*a*c*(c - a^2*c*x^2)^(3/2))) - 2/(3*a*c^2*Sqrt[c - a^2*c*x^2]) + (x*ArcTanh[a*x])/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcTanh[a*x])/(3*c^2*Sqrt[c - a^2*c*x^2])} -{ArcTanh[a*x]/(c - c*a^2*x^2)^(7/2), x, 3, -(1/(25*a*c*(c - a^2*c*x^2)^(5/2))) - 4/(45*a*c^2*(c - a^2*c*x^2)^(3/2)) - 8/(15*a*c^3*Sqrt[c - a^2*c*x^2]) + (x*ArcTanh[a*x])/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcTanh[a*x])/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcTanh[a*x])/(15*c^3*Sqrt[c - a^2*c*x^2])} - - -{ArcTanh[a*x]^2*(1 - a^2*x^2)^(1/2), x, 10, -(ArcSin[a*x]/a) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a + (1/2)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2 + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2)/a - (I*ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]])/a + (I*ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]])/a + (I*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (I*PolyLog[3, I*E^ArcTanh[a*x]])/a} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(5/2), x, 5, (2*x)/(27*(1 - a^2*x^2)^(3/2)) + (40*x)/(27*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(9*a*(1 - a^2*x^2)^(3/2)) - (4*ArcTanh[a*x])/(3*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(3*(1 - a^2*x^2)^(3/2)) + (2*x*ArcTanh[a*x]^2)/(3*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2), x, 9, (2*x)/(125*(1 - a^2*x^2)^(5/2)) + (272*x)/(3375*(1 - a^2*x^2)^(3/2)) + (4144*x)/(3375*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(25*a*(1 - a^2*x^2)^(5/2)) - (8*ArcTanh[a*x])/(45*a*(1 - a^2*x^2)^(3/2)) - (16*ArcTanh[a*x])/(15*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x]^2)/(15*(1 - a^2*x^2)^(3/2)) + (8*x*ArcTanh[a*x]^2)/(15*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^2/(1 - a^2*x^2)^(9/2), x, 14, (2*x)/(343*(1 - a^2*x^2)^(7/2)) + (888*x)/(42875*(1 - a^2*x^2)^(5/2)) + (30256*x)/(385875*(1 - a^2*x^2)^(3/2)) + (413312*x)/(385875*Sqrt[1 - a^2*x^2]) - (2*ArcTanh[a*x])/(49*a*(1 - a^2*x^2)^(7/2)) - (12*ArcTanh[a*x])/(175*a*(1 - a^2*x^2)^(5/2)) - (16*ArcTanh[a*x])/(105*a*(1 - a^2*x^2)^(3/2)) - (32*ArcTanh[a*x])/(35*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x]^2)/(35*(1 - a^2*x^2)^(5/2)) + (8*x*ArcTanh[a*x]^2)/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x]^2)/(35*Sqrt[1 - a^2*x^2])} - - -{ArcTanh[a*x]^3*(1 - a^2*x^2)^(1/2), x, 12, (6*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a + (3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(2*a) + (1/2)*x*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3 + (ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^3)/a - (3*I*ArcTanh[a*x]^2*PolyLog[2, (-I)*E^ArcTanh[a*x]])/(2*a) + (3*I*ArcTanh[a*x]^2*PolyLog[2, I*E^ArcTanh[a*x]])/(2*a) + (3*I*PolyLog[2, -((I*Sqrt[1 - a*x])/Sqrt[1 + a*x])])/a - (3*I*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a + (3*I*ArcTanh[a*x]*PolyLog[3, (-I)*E^ArcTanh[a*x]])/a - (3*I*ArcTanh[a*x]*PolyLog[3, I*E^ArcTanh[a*x]])/a - (3*I*PolyLog[4, (-I)*E^ArcTanh[a*x]])/a + (3*I*PolyLog[4, I*E^ArcTanh[a*x]])/a} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(5/2), x, 5, -(2/(27*a*(1 - a^2*x^2)^(3/2))) - 40/(9*a*Sqrt[1 - a^2*x^2]) + (2*x*ArcTanh[a*x])/(9*(1 - a^2*x^2)^(3/2)) + (40*x*ArcTanh[a*x])/(9*Sqrt[1 - a^2*x^2]) - ArcTanh[a*x]^2/(3*a*(1 - a^2*x^2)^(3/2)) - (2*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(3*(1 - a^2*x^2)^(3/2)) + (2*x*ArcTanh[a*x]^3)/(3*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(7/2), x, 9, -(6/(625*a*(1 - a^2*x^2)^(5/2))) - 272/(3375*a*(1 - a^2*x^2)^(3/2)) - 4144/(1125*a*Sqrt[1 - a^2*x^2]) + (6*x*ArcTanh[a*x])/(125*(1 - a^2*x^2)^(5/2)) + (272*x*ArcTanh[a*x])/(1125*(1 - a^2*x^2)^(3/2)) + (4144*x*ArcTanh[a*x])/(1125*Sqrt[1 - a^2*x^2]) - (3*ArcTanh[a*x]^2)/(25*a*(1 - a^2*x^2)^(5/2)) - (4*ArcTanh[a*x]^2)/(15*a*(1 - a^2*x^2)^(3/2)) - (8*ArcTanh[a*x]^2)/(5*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x]^3)/(15*(1 - a^2*x^2)^(3/2)) + (8*x*ArcTanh[a*x]^3)/(15*Sqrt[1 - a^2*x^2])} -{ArcTanh[a*x]^3/(1 - a^2*x^2)^(9/2), x, 14, -(6/(2401*a*(1 - a^2*x^2)^(7/2))) - 2664/(214375*a*(1 - a^2*x^2)^(5/2)) - 30256/(385875*a*(1 - a^2*x^2)^(3/2)) - 413312/(128625*a*Sqrt[1 - a^2*x^2]) + (6*x*ArcTanh[a*x])/(343*(1 - a^2*x^2)^(7/2)) + (2664*x*ArcTanh[a*x])/(42875*(1 - a^2*x^2)^(5/2)) + (30256*x*ArcTanh[a*x])/(128625*(1 - a^2*x^2)^(3/2)) + (413312*x*ArcTanh[a*x])/(128625*Sqrt[1 - a^2*x^2]) - (3*ArcTanh[a*x]^2)/(49*a*(1 - a^2*x^2)^(7/2)) - (18*ArcTanh[a*x]^2)/(175*a*(1 - a^2*x^2)^(5/2)) - (8*ArcTanh[a*x]^2)/(35*a*(1 - a^2*x^2)^(3/2)) - (48*ArcTanh[a*x]^2)/(35*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x]^3)/(35*(1 - a^2*x^2)^(5/2)) + (8*x*ArcTanh[a*x]^3)/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x]^3)/(35*Sqrt[1 - a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(1 - a^2*x^2)^(1/2)/ArcTanh[a*x], x, 0, Unintegrable[Sqrt[1 - a^2*x^2]/ArcTanh[a*x], x]} -{1/(1 - a^2*x^2)^(1/2)/ArcTanh[a*x], x, 0, Unintegrable[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]), x]} -{1/(1 - a^2*x^2)^(3/2)/ArcTanh[a*x], x, 2, CoshIntegral[ArcTanh[a*x]]/a} -{1/(1 - a^2*x^2)^(5/2)/ArcTanh[a*x], x, 5, (3*CoshIntegral[ArcTanh[a*x]])/(4*a) + CoshIntegral[3*ArcTanh[a*x]]/(4*a)} -{1/(1 - a^2*x^2)^(7/2)/ArcTanh[a*x], x, 6, (5*CoshIntegral[ArcTanh[a*x]])/(8*a) + (5*CoshIntegral[3*ArcTanh[a*x]])/(16*a) + CoshIntegral[5*ArcTanh[a*x]]/(16*a)} -{1/(1 - a^2*x^2)^(9/2)/ArcTanh[a*x], x, 7, (35*CoshIntegral[ArcTanh[a*x]])/(64*a) + (21*CoshIntegral[3*ArcTanh[a*x]])/(64*a) + (7*CoshIntegral[5*ArcTanh[a*x]])/(64*a) + CoshIntegral[7*ArcTanh[a*x]]/(64*a)} - - -{(1 - a^2*x^2)^(1/2)/ArcTanh[a*x]^2, x, 0, Unintegrable[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^2, x]} -{1/(1 - a^2*x^2)^(1/2)/ArcTanh[a*x]^2, x, 0, Unintegrable[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2), x]} -{1/(1 - a^2*x^2)^(3/2)/ArcTanh[a*x]^2, x, 3, -(1/(a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])) + SinhIntegral[ArcTanh[a*x]]/a} -{1/(1 - a^2*x^2)^(5/2)/ArcTanh[a*x]^2, x, 6, -(1/(a*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x])) + (3*SinhIntegral[ArcTanh[a*x]])/(4*a) + (3*SinhIntegral[3*ArcTanh[a*x]])/(4*a)} -{1/(1 - a^2*x^2)^(7/2)/ArcTanh[a*x]^2, x, 7, -(1/(a*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x])) + (5*SinhIntegral[ArcTanh[a*x]])/(8*a) + (15*SinhIntegral[3*ArcTanh[a*x]])/(16*a) + (5*SinhIntegral[5*ArcTanh[a*x]])/(16*a)} -{1/(1 - a^2*x^2)^(9/2)/ArcTanh[a*x]^2, x, 8, -(1/(a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x])) + (35*SinhIntegral[ArcTanh[a*x]])/(64*a) + (63*SinhIntegral[3*ArcTanh[a*x]])/(64*a) + (35*SinhIntegral[5*ArcTanh[a*x]])/(64*a) + (7*SinhIntegral[7*ArcTanh[a*x]])/(64*a)} - - -{(1 - a^2*x^2)^(1/2)/ArcTanh[a*x]^3, x, 0, Unintegrable[Sqrt[1 - a^2*x^2]/ArcTanh[a*x]^3, x]} -{1/(1 - a^2*x^2)^(1/2)/ArcTanh[a*x]^3, x, 0, Unintegrable[1/(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3), x]} -{1/(1 - a^2*x^2)^(3/2)/ArcTanh[a*x]^3, x, 4, -(1/(2*a*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)) - x/(2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]) + CoshIntegral[ArcTanh[a*x]]/(2*a)} -{1/(1 - a^2*x^2)^(5/2)/ArcTanh[a*x]^3, x, 12, -(1/(2*a*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]^2)) - (3*x)/(2*(1 - a^2*x^2)^(3/2)*ArcTanh[a*x]) + (3*CoshIntegral[ArcTanh[a*x]])/(8*a) + (9*CoshIntegral[3*ArcTanh[a*x]])/(8*a)} -{1/(1 - a^2*x^2)^(7/2)/ArcTanh[a*x]^3, x, 14, -(1/(2*a*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x]^2)) - (5*x)/(2*(1 - a^2*x^2)^(5/2)*ArcTanh[a*x]) + (5*CoshIntegral[ArcTanh[a*x]])/(16*a) + (45*CoshIntegral[3*ArcTanh[a*x]])/(32*a) + (25*CoshIntegral[5*ArcTanh[a*x]])/(32*a)} -{1/(1 - a^2*x^2)^(9/2)/ArcTanh[a*x]^3, x, 16, -(1/(2*a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^2)) - (7*x)/(2*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + (35*CoshIntegral[ArcTanh[a*x]])/(128*a) + (189*CoshIntegral[3*ArcTanh[a*x]])/(128*a) + (175*CoshIntegral[5*ArcTanh[a*x]])/(128*a) + (49*CoshIntegral[7*ArcTanh[a*x]])/(128*a)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f+g x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p with c^2 d+e=0*) - - -{(d + e*x)*(a + b*ArcTanh[c*x])^2/(1 - c^2*x^2), x, 7, (d*(a + b*ArcTanh[c*x])^3)/(3*b*c) - (e*(a + b*ArcTanh[c*x])^3)/(3*b*c^2) + (e*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c*x)])/c^2 + (b*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/c^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - c*x)])/(2*c^2)} - - -(* ::Title::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x^2)^q (a+b ArcTanh[c x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^q (a+b ArcTanh[c x])^p*) - - -{ArcTanh[a*x]*(c + d*x^2)^4, x, 4, (d*(420*a^6*c^3 + 378*a^4*c^2*d + 180*a^2*c*d^2 + 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 + 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) + (d^3*(36*a^2*c + 7*d)*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcTanh[a*x] + (4/3)*c^3*d*x^3*ArcTanh[a*x] + (6/5)*c^2*d^2*x^5*ArcTanh[a*x] + (4/7)*c*d^3*x^7*ArcTanh[a*x] + (1/9)*d^4*x^9*ArcTanh[a*x] + ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(630*a^9)} -{ArcTanh[a*x]*(c + d*x^2)^3, x, 4, (d*(35*a^4*c^2 + 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + (d^2*(21*a^2*c + 5*d)*x^4)/(140*a^3) + (d^3*x^6)/(42*a) + c^3*x*ArcTanh[a*x] + c^2*d*x^3*ArcTanh[a*x] + (3/5)*c*d^2*x^5*ArcTanh[a*x] + (1/7)*d^3*x^7*ArcTanh[a*x] + ((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Log[1 - a^2*x^2])/(70*a^7)} -{ArcTanh[a*x]*(c + d*x^2)^2, x, 5, (d*(10*a^2*c + 3*d)*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcTanh[a*x] + (2/3)*c*d*x^3*ArcTanh[a*x] + (1/5)*d^2*x^5*ArcTanh[a*x] + ((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Log[1 - a^2*x^2])/(30*a^5)} -{ArcTanh[a*x]*(c + d*x^2)^1, x, 5, (d*x^2)/(6*a) + c*x*ArcTanh[a*x] + (1/3)*d*x^3*ArcTanh[a*x] + ((3*a^2*c + d)*Log[1 - a^2*x^2])/(6*a^3)} -{ArcTanh[a*x]/(c + d*x^2)^1, x, 17, -((Log[1 - a*x]*Log[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d])) + (Log[1 + a*x]*Log[(a*(Sqrt[-c] - Sqrt[d]*x))/(a*Sqrt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] - Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 - a*x]*Log[(a*(Sqrt[-c] + Sqrt[d]*x))/(a*Sqrt[-c] + Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a*x))/(a*Sqrt[-c] + Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] - Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a*x))/(a*Sqrt[-c] + Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d])} -{ArcTanh[a*x]/(c + d*x^2)^2, x, If[$VersionNumber<11, 24, 25], (x*ArcTanh[a*x])/(2*c*(c + d*x^2)) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*ArcTanh[a*x])/(2*c^(3/2)*Sqrt[d]) + (I*Log[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) - (I*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) - (I*Log[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) + (I*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) + (a*Log[1 - a^2*x^2])/(4*c*(a^2*c + d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c + d)) + (I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) - (I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) + (I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) - (I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d])} -{ArcTanh[a*x]/(c + d*x^2)^3, x, 23, a/(8*c*(a^2*c + d)*(c + d*x^2)) + (x*ArcTanh[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcTanh[a*x])/(8*c^2*(c + d*x^2)) + (3*ArcTan[(Sqrt[d]*x)/Sqrt[c]]*ArcTanh[a*x])/(8*c^(5/2)*Sqrt[d]) + (3*I*Log[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) - (3*I*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) - (3*I*Log[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) + (3*I*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) + (a*(5*a^2*c + 3*d)*Log[1 - a^2*x^2])/(16*c^2*(a^2*c + d)^2) - (a*(5*a^2*c + 3*d)*Log[c + d*x^2])/(16*c^2*(a^2*c + d)^2) + (3*I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) - (3*I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) + (3*I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) - (3*I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d])} - - -{1/((a - a*x^2)*(b - 2*b*ArcTanh[x])), x, 1, -(Log[1 - 2*ArcTanh[x]]/(2*a*b))} - - -{ArcTanh[b*x]/(1 - x^2), x, 17, (1/4)*Log[-((b*(1 - x))/(1 - b))]*Log[1 - b*x] - (1/4)*Log[(b*(1 + x))/(1 + b)]*Log[1 - b*x] - (1/4)*Log[(b*(1 - x))/(1 + b)]*Log[1 + b*x] + (1/4)*Log[-((b*(1 + x))/(1 - b))]*Log[1 + b*x] + (1/4)*PolyLog[2, (1 - b*x)/(1 - b)] - (1/4)*PolyLog[2, (1 - b*x)/(1 + b)] + (1/4)*PolyLog[2, (1 + b*x)/(1 - b)] - (1/4)*PolyLog[2, (1 + b*x)/(1 + b)]} -{ArcTanh[a+b*x]/(1 - x^2), x, 17, (1/4)*Log[-((b*(1 - x))/(1 - a - b))]*Log[1 - a - b*x] - (1/4)*Log[(b*(1 + x))/(1 - a + b)]*Log[1 - a - b*x] - (1/4)*Log[(b*(1 - x))/(1 + a + b)]*Log[1 + a + b*x] + (1/4)*Log[-((b*(1 + x))/(1 + a - b))]*Log[1 + a + b*x] + (1/4)*PolyLog[2, (1 - a - b*x)/(1 - a - b)] - (1/4)*PolyLog[2, (1 - a - b*x)/(1 - a + b)] + (1/4)*PolyLog[2, (1 + a + b*x)/(1 + a - b)] - (1/4)*PolyLog[2, (1 + a + b*x)/(1 + a + b)]} - - -{ArcTanh[x]/(a + b*x), x, 4, -((ArcTanh[x]*Log[2/(1 + x)])/b) + (ArcTanh[x]*Log[(2*(a + b*x))/((a + b)*(1 + x))])/b + PolyLog[2, 1 - 2/(1 + x)]/(2*b) - PolyLog[2, 1 - (2*(a + b*x))/((a + b)*(1 + x))]/(2*b)} -{ArcTanh[x]/(a + b*x^2), x, 17, -((Log[1 - x]*Log[(Sqrt[-a] - Sqrt[b]*x)/(Sqrt[-a] - Sqrt[b])])/(4*Sqrt[-a]*Sqrt[b])) + (Log[1 + x]*Log[(Sqrt[-a] - Sqrt[b]*x)/(Sqrt[-a] + Sqrt[b])])/(4*Sqrt[-a]*Sqrt[b]) - (Log[1 + x]*Log[(Sqrt[-a] + Sqrt[b]*x)/(Sqrt[-a] - Sqrt[b])])/(4*Sqrt[-a]*Sqrt[b]) + (Log[1 - x]*Log[(Sqrt[-a] + Sqrt[b]*x)/(Sqrt[-a] + Sqrt[b])])/(4*Sqrt[-a]*Sqrt[b]) - PolyLog[2, -((Sqrt[b]*(1 - x))/(Sqrt[-a] - Sqrt[b]))]/(4*Sqrt[-a]*Sqrt[b]) + PolyLog[2, (Sqrt[b]*(1 - x))/(Sqrt[-a] + Sqrt[b])]/(4*Sqrt[-a]*Sqrt[b]) - PolyLog[2, -((Sqrt[b]*(1 + x))/(Sqrt[-a] - Sqrt[b]))]/(4*Sqrt[-a]*Sqrt[b]) + PolyLog[2, (Sqrt[b]*(1 + x))/(Sqrt[-a] + Sqrt[b])]/(4*Sqrt[-a]*Sqrt[b])} -{ArcTanh[x]/(a + b*x + c*x^2), x, 10, (ArcTanh[x]*Log[(2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + 2*c - Sqrt[b^2 - 4*a*c])*(1 + x))])/Sqrt[b^2 - 4*a*c] - (ArcTanh[x]*Log[(2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + 2*c + Sqrt[b^2 - 4*a*c])*(1 + x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 - (2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + 2*c - Sqrt[b^2 - 4*a*c])*(1 + x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLog[2, 1 - (2*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((b + 2*c + Sqrt[b^2 - 4*a*c])*(1 + x))]/(2*Sqrt[b^2 - 4*a*c])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x^2)^(q/2) (a+b ArcTanh[c x])^p*) - - -{ArcTanh[a*x]*(c + d*x^2)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x^2]*ArcTanh[a*x], x]} -{ArcTanh[a*x]/(c + d*x^2)^(1/2), x, 0, Unintegrable[ArcTanh[a*x]/Sqrt[c + d*x^2], x]} -{ArcTanh[a*x]/(c + d*x^2)^(3/2), x, 5, (x*ArcTanh[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]]/(c*Sqrt[a^2*c + d])} -{ArcTanh[a*x]/(c + d*x^2)^(5/2), x, 7, a/(3*c*(a^2*c + d)*Sqrt[c + d*x^2]) + (x*ArcTanh[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcTanh[a*x])/(3*c^2*Sqrt[c + d*x^2]) - ((3*a^2*c + 2*d)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(3*c^2*(a^2*c + d)^(3/2))} -{ArcTanh[a*x]/(c + d*x^2)^(7/2), x, 8, a/(15*c*(a^2*c + d)*(c + d*x^2)^(3/2)) + (a*(7*a^2*c + 4*d))/(15*c^2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcTanh[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcTanh[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcTanh[a*x])/(15*c^3*Sqrt[c + d*x^2]) - ((15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(15*c^3*(a^2*c + d)^(5/2))} -{ArcTanh[a*x]/(c + d*x^2)^(9/2), x, 8, a/(35*c*(a^2*c + d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c + 6*d))/(105*c^2*(a^2*c + d)^2*(c + d*x^2)^(3/2)) + (a*(19*a^4*c^2 + 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c + d)^3*Sqrt[c + d*x^2]) + (x*ArcTanh[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcTanh[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcTanh[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcTanh[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(35*c^4*(a^2*c + d)^(7/2))} - - -{ArcTanh[x]*(a - a*x^2)^(1/2), x, 3, (1/2)*Sqrt[a - a*x^2] + (1/2)*x*Sqrt[a - a*x^2]*ArcTanh[x] - (a*Sqrt[1 - x^2]*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]*ArcTanh[x])/Sqrt[a - a*x^2] - (I*a*Sqrt[1 - x^2]*PolyLog[2, -((I*Sqrt[1 - x])/Sqrt[1 + x])])/(2*Sqrt[a - a*x^2]) + (I*a*Sqrt[1 - x^2]*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]])/(2*Sqrt[a - a*x^2])} -{ArcTanh[x]/(a - a*x^2)^(1/2), x, 2, -((2*Sqrt[1 - x^2]*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]]*ArcTanh[x])/Sqrt[a - a*x^2]) - (I*Sqrt[1 - x^2]*PolyLog[2, -((I*Sqrt[1 - x])/Sqrt[1 + x])])/Sqrt[a - a*x^2] + (I*Sqrt[1 - x^2]*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]])/Sqrt[a - a*x^2]} -{ArcTanh[x]/(a - a*x^2)^(3/2), x, 1, -(1/(a*Sqrt[a - a*x^2])) + (x*ArcTanh[x])/(a*Sqrt[a - a*x^2])} -{ArcTanh[x]/(a - a*x^2)^(5/2), x, 2, -(1/(9*a*(a - a*x^2)^(3/2))) - 2/(3*a^2*Sqrt[a - a*x^2]) + (x*ArcTanh[x])/(3*a*(a - a*x^2)^(3/2)) + (2*x*ArcTanh[x])/(3*a^2*Sqrt[a - a*x^2])} -{ArcTanh[x]/(a - a*x^2)^(7/2), x, 3, -(1/(25*a*(a - a*x^2)^(5/2))) - 4/(45*a^2*(a - a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a - a*x^2]) + (x*ArcTanh[x])/(5*a*(a - a*x^2)^(5/2)) + (4*x*ArcTanh[x])/(15*a^2*(a - a*x^2)^(3/2)) + (8*x*ArcTanh[x])/(15*a^3*Sqrt[a - a*x^2])} - - -(* ::Title::Closed:: *) -(*Integrands of the form (h x)^m (d+e Log[f+g x^2]) (a+b ArcTanh[c x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e Log[f+g x^2]) (a+b ArcTanh[c x])*) - - -{x^4*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]), x, 26, -((2*a*e*x)/(5*c^4)) - (77*b*e*x^2)/(300*c^3) - (2*a*e*x^3)/(15*c^2) - (9*b*e*x^4)/(200*c) - (2/25)*a*e*x^5 - (2*b*e*x*ArcTanh[c*x])/(5*c^4) - (2*b*e*x^3*ArcTanh[c*x])/(15*c^2) - (2/25)*b*e*x^5*ArcTanh[c*x] + (b*e*ArcTanh[c*x]^2)/(5*c^5) - ((4*a + 3*b)*e*Log[1 - c*x])/(20*c^5) + ((4*a - 3*b)*e*Log[1 + c*x])/(20*c^5) - (23*b*e*Log[1 - c^2*x^2])/(75*c^5) - (b*e*Log[1 - c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(10*c^3) + (b*x^4*(d + e*Log[1 - c^2*x^2]))/(20*c) + (1/5)*x^5*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(10*c^5)} -{x^3*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]), x, 14, (b*(2*d - 3*e)*x)/(8*c^3) - (2*b*e*x)/(3*c^3) + (b*(2*d - e)*x^3)/(24*c) - (b*e*x^3)/(18*c) - (b*(2*d - 3*e)*ArcTanh[c*x])/(8*c^4) + (2*b*e*ArcTanh[c*x])/(3*c^4) - (e*x^2*(a + b*ArcTanh[c*x]))/(4*c^2) - (1/8)*e*x^4*(a + b*ArcTanh[c*x]) + (b*e*x*Log[1 - c^2*x^2])/(4*c^3) + (b*e*x^3*Log[1 - c^2*x^2])/(12*c) - (e*(a + b*ArcTanh[c*x])*Log[1 - c^2*x^2])/(4*c^4) + (1/4)*x^4*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])} -{x^2*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]), x, 21, -((2*a*e*x)/(3*c^2)) - (5*b*e*x^2)/(18*c) - (2/9)*a*e*x^3 - (2*b*e*x*ArcTanh[c*x])/(3*c^2) - (2/9)*b*e*x^3*ArcTanh[c*x] + (b*e*ArcTanh[c*x]^2)/(3*c^3) - ((2*a + b)*e*Log[1 - c*x])/(6*c^3) + ((2*a - b)*e*Log[1 + c*x])/(6*c^3) - (4*b*e*Log[1 - c^2*x^2])/(9*c^3) - (b*e*Log[1 - c^2*x^2]^2)/(12*c^3) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(6*c) + (1/3)*x^3*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(6*c^3)} -{x^1*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]), x, 7, (b*(d - e)*x)/(2*c) - (b*e*x)/c - (b*(d - e)*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (1/2)*d*x^2*(a + b*ArcTanh[c*x]) - (1/2)*e*x^2*(a + b*ArcTanh[c*x]) + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*ArcTanh[c*x])*Log[1 - c^2*x^2])/(2*c^2)} -{x^0*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]), x, 9, -2*a*e*x - 2*b*e*x*ArcTanh[c*x] + (e*(a + b*ArcTanh[c*x])^2)/(b*c) - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*(d + e*Log[1 - c^2*x^2])^2)/(4*c*e)} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^1, x, 14, a*d*Log[x] - (1/2)*b*e*Log[c*x]*Log[1 - c*x]^2 + (1/2)*b*e*Log[(-c)*x]*Log[1 + c*x]^2 - (1/2)*b*d*PolyLog[2, (-c)*x] + (1/2)*b*e*(Log[1 - c*x] + Log[1 + c*x] - Log[1 - c^2*x^2])*PolyLog[2, (-c)*x] + (1/2)*b*d*PolyLog[2, c*x] - (1/2)*b*e*(Log[1 - c*x] + Log[1 + c*x] - Log[1 - c^2*x^2])*PolyLog[2, c*x] - (1/2)*a*e*PolyLog[2, c^2*x^2] - b*e*Log[1 - c*x]*PolyLog[2, 1 - c*x] + b*e*Log[1 + c*x]*PolyLog[2, 1 + c*x] + b*e*PolyLog[3, 1 - c*x] - b*e*PolyLog[3, 1 + c*x]} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^2, x, 6, -((c*e*(a + b*ArcTanh[c*x])^2)/b) - ((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/x + (1/2)*b*c*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/2)*b*c*e*PolyLog[2, 1/(1 - c^2*x^2)]} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^3, x, 5, (-a)*c^2*e*Log[x] + (1/2)*(a + b)*c^2*e*Log[1 - c*x] + (1/2)*(a - b)*c^2*e*Log[1 + c*x] - (b*c*(d + e*Log[1 - c^2*x^2]))/(2*x) + (1/2)*b*c^2*ArcTanh[c*x]*(d + e*Log[1 - c^2*x^2]) - ((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/(2*x^2) + (1/2)*b*c^2*e*PolyLog[2, (-c)*x] - (1/2)*b*c^2*e*PolyLog[2, c*x]} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^4, x, 15, (2*c^2*e*(a + b*ArcTanh[c*x]))/(3*x) - (c^3*e*(a + b*ArcTanh[c*x])^2)/(3*b) - b*c^3*e*Log[x] + (1/3)*b*c^3*e*Log[1 - c^2*x^2] - (b*c*(1 - c^2*x^2)*(d + e*Log[1 - c^2*x^2]))/(6*x^2) - ((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/(3*x^3) + (1/6)*b*c^3*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/6)*b*c^3*e*PolyLog[2, 1/(1 - c^2*x^2)]} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^5, x, 10, (a*c^2*e)/(4*x^2) + (5*b*c^3*e)/(12*x) - (1/4)*b*c^4*e*ArcTanh[c*x] + (b*c^2*e*ArcTanh[c*x])/(4*x^2) - (1/2)*a*c^4*e*Log[x] + (1/12)*(3*a + 4*b)*c^4*e*Log[1 - c*x] + (1/12)*(3*a - 4*b)*c^4*e*Log[1 + c*x] - (b*c*(d + e*Log[1 - c^2*x^2]))/(12*x^3) - (b*c^3*(d + e*Log[1 - c^2*x^2]))/(4*x) + (1/4)*b*c^4*ArcTanh[c*x]*(d + e*Log[1 - c^2*x^2]) - ((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/(4*x^4) + (1/4)*b*c^4*e*PolyLog[2, (-c)*x] - (1/4)*b*c^4*e*PolyLog[2, c*x]} -{(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2])/x^6, x, 24, (7*b*c^3*e)/(60*x^2) + (2*c^2*e*(a + b*ArcTanh[c*x]))/(15*x^3) + (2*c^4*e*(a + b*ArcTanh[c*x]))/(5*x) - (c^5*e*(a + b*ArcTanh[c*x])^2)/(5*b) - (5/6)*b*c^5*e*Log[x] + (19/60)*b*c^5*e*Log[1 - c^2*x^2] - (b*c*(d + e*Log[1 - c^2*x^2]))/(20*x^4) - (b*c^3*(1 - c^2*x^2)*(d + e*Log[1 - c^2*x^2]))/(10*x^2) - ((a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/(5*x^5) + (1/10)*b*c^5*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/10)*b*c^5*e*PolyLog[2, 1/(1 - c^2*x^2)]} - - -{x^1*(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]), x, If[$VersionNumber<11, 21, 22], (b*(d - e)*x)/(2*c) - (b*e*x)/c + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c*Sqrt[g]) - (b*(d - e)*ArcTanh[c*x])/(2*c^2) + (1/2)*d*x^2*(a + b*ArcTanh[c*x]) - (1/2)*e*x^2*(a + b*ArcTanh[c*x]) - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[2/(1 + c*x)])/(c^2*g) + (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(2*c^2*g) + (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(2*c^2*g) + (b*e*x*Log[f + g*x^2])/(2*c) - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[f + g*x^2])/(2*c^2*g) + (e*(f + g*x^2)*(a + b*ArcTanh[c*x])*Log[f + g*x^2])/(2*g) + (b*e*(c^2*f + g)*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^2*g) - (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*c^2*g) - (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*c^2*g)} -{x^0*(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]), x, 28, -2*a*e*x + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - 2*b*e*x*ArcTanh[c*x] + (b*e*Sqrt[-f]*Log[1 - c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(2*Sqrt[g]) - (b*e*Sqrt[-f]*Log[1 + c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) + (b*e*Sqrt[-f]*Log[1 + c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(2*Sqrt[g]) - (b*e*Sqrt[-f]*Log[1 - c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]) + (b*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/(2*c) + (b*e*Sqrt[-f]*PolyLog[2, -((Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] - Sqrt[g]))])/(2*Sqrt[g]) - (b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) + (b*e*Sqrt[-f]*PolyLog[2, -((Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] - Sqrt[g]))])/(2*Sqrt[g]) - (b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[g]) + (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f + g)])/(2*c)} -{(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2])/x^1, x, 6, b*e*CannotIntegrate[(ArcTanh[c*x]*Log[f + g*x^2])/x, x] + a*d*Log[x] + (1/2)*a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2] - (1/2)*b*d*PolyLog[2, (-c)*x] + (1/2)*b*d*PolyLog[2, c*x] + (1/2)*a*e*PolyLog[2, 1 + (g*x^2)/f]} -{(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2])/x^2, x, 28, (2*a*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - (b*e*Sqrt[g]*Log[1 - c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(2*Sqrt[-f]) + (b*e*Sqrt[g]*Log[1 + c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) - (b*e*Sqrt[g]*Log[1 + c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - Sqrt[g])])/(2*Sqrt[-f]) + (b*e*Sqrt[g]*Log[1 - c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) - ((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/x + (1/2)*b*c*Log[-((g*x^2)/f)]*(d + e*Log[f + g*x^2]) - (1/2)*b*c*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]) - (b*e*Sqrt[g]*PolyLog[2, -((Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] - Sqrt[g]))])/(2*Sqrt[-f]) + (b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 - c*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) - (b*e*Sqrt[g]*PolyLog[2, -((Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] - Sqrt[g]))])/(2*Sqrt[-f]) + (b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 + c*x))/(c*Sqrt[-f] + Sqrt[g])])/(2*Sqrt[-f]) - (1/2)*b*c*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f + g)] + (1/2)*b*c*e*PolyLog[2, 1 + (g*x^2)/f]} -{(a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2])/x^3, x, 20, (b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f + (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[2/(1 + c*x)])/f - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(2*f) - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(2*f) - (a*e*g*Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) + (1/2)*b*c^2*ArcTanh[c*x]*(d + e*Log[f + g*x^2]) - ((a + b*ArcTanh[c*x])*(d + e*Log[f + g*x^2]))/(2*x^2) - (b*e*g*PolyLog[2, (-c)*x])/(2*f) + (b*e*g*PolyLog[2, c*x])/(2*f) - (b*e*(c^2*f + g)*PolyLog[2, 1 - 2/(1 + c*x)])/(2*f) + (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*f) + (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*f)} - - -(* ::Title::Closed:: *) -(*Integrands of the form u (a+b ArcTanh[c x])^p*) - - -{ArcTanh[c*x]*(a + b*ArcTanh[c*x])/(1 + c*x)^2, x, 16, -((a + b)/(2*c*(1 + c*x))) + ((a + b)*ArcTanh[c*x])/(2*c) - ((a + b)*ArcTanh[c*x])/(c*(1 + c*x)) - (b*(1 - c*x)*ArcTanh[c*x]^2)/(2*c*(1 + c*x)), -(a/(2*c*(1 + c*x))) - b/(2*c*(1 + c*x)) + (a*ArcTanh[c*x])/(2*c) + (b*ArcTanh[c*x])/(2*c) - (a*ArcTanh[c*x])/(c*(1 + c*x)) - (b*ArcTanh[c*x])/(c*(1 + c*x)) + (b*ArcTanh[c*x]^2)/(2*c) - (b*ArcTanh[c*x]^2)/(c*(1 + c*x))} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m deleted file mode 100644 index d012d35..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.5 u (a+b arctanh(c+d x))^p.m +++ /dev/null @@ -1,138 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcTanh[c+d x])^p*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c+d x])^p*) - - -{x^3*ArcTanh[a + b*x]^2, x, 19, -((a*x)/b^3) + (a + b*x)^2/(12*b^4) + (a*ArcTanh[a + b*x])/b^4 + ((1 + 6*a^2)*(a + b*x)*ArcTanh[a + b*x])/(2*b^4) - (a*(a + b*x)^2*ArcTanh[a + b*x])/b^4 + ((a + b*x)^3*ArcTanh[a + b*x])/(6*b^4) - (a*(1 + a^2)*ArcTanh[a + b*x]^2)/b^4 - ((1 + 6*a^2 + a^4)*ArcTanh[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcTanh[a + b*x]^2 + (2*a*(1 + a^2)*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/b^4 + Log[1 - (a + b*x)^2]/(12*b^4) + ((1 + 6*a^2)*Log[1 - (a + b*x)^2])/(4*b^4) + (a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^4} -{x^2*ArcTanh[a + b*x]^2, x, 15, x/(3*b^2) - ArcTanh[a + b*x]/(3*b^3) - (2*a*(a + b*x)*ArcTanh[a + b*x])/b^3 + ((a + b*x)^2*ArcTanh[a + b*x])/(3*b^3) + (a*(3 + a^2)*ArcTanh[a + b*x]^2)/(3*b^3) + ((1 + 3*a^2)*ArcTanh[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcTanh[a + b*x]^2 - (2*(1 + 3*a^2)*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/(3*b^3) - (a*Log[1 - (a + b*x)^2])/b^3 - ((1 + 3*a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(3*b^3)} -{x^1*ArcTanh[a + b*x]^2, x, 12, ((a + b*x)*ArcTanh[a + b*x])/b^2 - (a*ArcTanh[a + b*x]^2)/b^2 - ((1 + a^2)*ArcTanh[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcTanh[a + b*x]^2 + (2*a*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/b^2 + Log[1 - (a + b*x)^2]/(2*b^2) + (a*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^2} -{x^0*ArcTanh[a + b*x]^2, x, 6, ArcTanh[a + b*x]^2/b + ((a + b*x)*ArcTanh[a + b*x]^2)/b - (2*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/b - PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))]/b} -{ArcTanh[a + b*x]^2/x^1, x, 2, (-ArcTanh[a + b*x]^2)*Log[2/(1 + a + b*x)] + ArcTanh[a + b*x]^2*Log[(2*b*x)/((1 - a)*(1 + a + b*x))] + ArcTanh[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b*x)] - ArcTanh[a + b*x]*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))] + (1/2)*PolyLog[3, 1 - 2/(1 + a + b*x)] - (1/2)*PolyLog[3, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))]} -{ArcTanh[a + b*x]^2/x^2, x, 17, -(ArcTanh[a + b*x]^2/x) + (b*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/(1 - a) + (b*ArcTanh[a + b*x]*Log[2/(1 + a + b*x)])/(1 + a) - (2*b*ArcTanh[a + b*x]*Log[2/(1 + a + b*x)])/(1 - a^2) + (2*b*ArcTanh[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2) + (b*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(2*(1 - a)) - (b*PolyLog[2, 1 - 2/(1 + a + b*x)])/(2*(1 + a)) + (b*PolyLog[2, 1 - 2/(1 + a + b*x)])/(1 - a^2) - (b*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)} -{ArcTanh[a + b*x]^2/x^3, x, 21, -((b*ArcTanh[a + b*x])/((1 - a^2)*x)) - ArcTanh[a + b*x]^2/(2*x^2) + (b^2*Log[x])/(1 - a^2)^2 + (b^2*ArcTanh[a + b*x]*Log[2/(1 - a - b*x)])/(2*(1 - a)^2) - (b^2*Log[1 - a - b*x])/(2*(1 - a)^2*(1 + a)) - (b^2*ArcTanh[a + b*x]*Log[2/(1 + a + b*x)])/(2*(1 + a)^2) - (2*a*b^2*ArcTanh[a + b*x]*Log[2/(1 + a + b*x)])/(1 - a^2)^2 + (2*a*b^2*ArcTanh[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)^2 - (b^2*Log[1 + a + b*x])/(2*(1 - a)*(1 + a)^2) + (b^2*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(4*(1 - a)^2) + (b^2*PolyLog[2, 1 - 2/(1 + a + b*x)])/(4*(1 + a)^2) + (a*b^2*PolyLog[2, 1 - 2/(1 + a + b*x)])/(1 - a^2)^2 - (a*b^2*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)^2} - - -{ArcTanh[1 + b*x]^2/x, x, 4, (-ArcTanh[1 + b*x]^2)*Log[-(2/(b*x))] - ArcTanh[1 + b*x]*PolyLog[2, 1 + 2/(b*x)] + (1/2)*PolyLog[3, 1 + 2/(b*x)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p when d e-c f=0*) - - -{(c*e + d*e*x)^3*(a + b*ArcTanh[c + d*x]), x, 6, (b*e^3*x)/4 + (b*e^3*(c + d*x)^3)/(12*d) - (b*e^3*ArcTanh[c + d*x])/(4*d) + (e^3*(c + d*x)^4*(a + b*ArcTanh[c + d*x]))/(4*d)} -{(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x]), x, 6, (b*e^2*(c + d*x)^2)/(6*d) + (e^2*(c + d*x)^3*(a + b*ArcTanh[c + d*x]))/(3*d) + (b*e^2*Log[1 - (c + d*x)^2])/(6*d)} -{(c*e + d*e*x)^1*(a + b*ArcTanh[c + d*x]), x, 5, (b*e*x)/2 - (b*e*ArcTanh[c + d*x])/(2*d) + (e*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/(2*d)} -{(a + b*ArcTanh[c + d*x])/(c*e + d*e*x)^1, x, 3, (a*Log[c + d*x])/(d*e) - (b*PolyLog[2, -c - d*x])/(2*d*e) + (b*PolyLog[2, c + d*x])/(2*d*e)} -{(a + b*ArcTanh[c + d*x])/(c*e + d*e*x)^2, x, 7, -((a + b*ArcTanh[c + d*x])/(d*e^2*(c + d*x))) + (b*Log[c + d*x])/(d*e^2) - (b*Log[1 - (c + d*x)^2])/(2*d*e^2)} -{(a + b*ArcTanh[c + d*x])/(c*e + d*e*x)^3, x, 5, -b/(2*d*e^3*(c + d*x)) + (b*ArcTanh[c + d*x])/(2*d*e^3) - (a + b*ArcTanh[c + d*x])/(2*d*e^3*(c + d*x)^2)} - - -{(c*e + d*e*x)^3*(a + b*ArcTanh[c + d*x])^2, x, 13, (a*b*e^3*x)/2 + (b^2*e^3*(c + d*x)^2)/(12*d) + (b^2*e^3*(c + d*x)*ArcTanh[c + d*x])/(2*d) + (b*e^3*(c + d*x)^3*(a + b*ArcTanh[c + d*x]))/(6*d) - (e^3*(a + b*ArcTanh[c + d*x])^2)/(4*d) + (e^3*(c + d*x)^4*(a + b*ArcTanh[c + d*x])^2)/(4*d) + (b^2*e^3*Log[1 - (c + d*x)^2])/(3*d)} -{(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x])^2, x, 11, (1/3)*b^2*e^2*x - (b^2*e^2*ArcTanh[c + d*x])/(3*d) + (b*e^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/(3*d) + (e^2*(a + b*ArcTanh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcTanh[c + d*x])^2)/(3*d) - (2*b*e^2*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/(3*d) - (b^2*e^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(3*d)} -{(c*e + d*e*x)^1*(a + b*ArcTanh[c + d*x])^2, x, 8, a*b*e*x + (b^2*e*(c + d*x)*ArcTanh[c + d*x])/d - (e*(a + b*ArcTanh[c + d*x])^2)/(2*d) + (e*(c + d*x)^2*(a + b*ArcTanh[c + d*x])^2)/(2*d) + (b^2*e*Log[1 - (c + d*x)^2])/(2*d)} -{(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^1, x, 8, (2*(a + b*ArcTanh[c + d*x])^2*ArcTanh[1 - 2/(1 - c - d*x)])/(d*e) - (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/(d*e) + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, -1 + 2/(1 - c - d*x)])/(d*e) + (b^2*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d*e) - (b^2*PolyLog[3, -1 + 2/(1 - c - d*x)])/(2*d*e)} -{(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^2, x, 6, (a + b*ArcTanh[c + d*x])^2/(d*e^2) - (a + b*ArcTanh[c + d*x])^2/(d*e^2*(c + d*x)) + (2*b*(a + b*ArcTanh[c + d*x])*Log[2 - 2/(1 + c + d*x)])/(d*e^2) - (b^2*PolyLog[2, -1 + 2/(1 + c + d*x)])/(d*e^2)} -{(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^3, x, 10, -((b*(a + b*ArcTanh[c + d*x]))/(d*e^3*(c + d*x))) + (a + b*ArcTanh[c + d*x])^2/(2*d*e^3) - (a + b*ArcTanh[c + d*x])^2/(2*d*e^3*(c + d*x)^2) + (b^2*Log[c + d*x])/(d*e^3) - (b^2*Log[1 - (c + d*x)^2])/(2*d*e^3)} -{(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^4, x, 10, -(b^2/(3*d*e^4*(c + d*x))) + (b^2*ArcTanh[c + d*x])/(3*d*e^4) - (b*(a + b*ArcTanh[c + d*x]))/(3*d*e^4*(c + d*x)^2) + (a + b*ArcTanh[c + d*x])^2/(3*d*e^4) - (a + b*ArcTanh[c + d*x])^2/(3*d*e^4*(c + d*x)^3) + (2*b*(a + b*ArcTanh[c + d*x])*Log[2 - 2/(1 + c + d*x)])/(3*d*e^4) - (b^2*PolyLog[2, -1 + 2/(1 + c + d*x)])/(3*d*e^4)} -{(a + b*ArcTanh[c + d*x])^2/(c*e + d*e*x)^5, x, 15, -(b^2/(12*d*e^5*(c + d*x)^2)) - (b*(a + b*ArcTanh[c + d*x]))/(6*d*e^5*(c + d*x)^3) - (b*(a + b*ArcTanh[c + d*x]))/(2*d*e^5*(c + d*x)) + (a + b*ArcTanh[c + d*x])^2/(4*d*e^5) - (a + b*ArcTanh[c + d*x])^2/(4*d*e^5*(c + d*x)^4) + (2*b^2*Log[c + d*x])/(3*d*e^5) - (b^2*Log[1 - (c + d*x)^2])/(3*d*e^5)} - - -{(c*e + d*e*x)^2*(a + b*ArcTanh[c + d*x])^3, x, 14, a*b^2*e^2*x + (b^3*e^2*(c + d*x)*ArcTanh[c + d*x])/d - (b*e^2*(a + b*ArcTanh[c + d*x])^2)/(2*d) + (b*e^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x])^2)/(2*d) + (e^2*(a + b*ArcTanh[c + d*x])^3)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcTanh[c + d*x])^3)/(3*d) - (b*e^2*(a + b*ArcTanh[c + d*x])^2*Log[2/(1 - c - d*x)])/d + (b^3*e^2*Log[1 - (c + d*x)^2])/(2*d) - (b^2*e^2*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d + (b^3*e^2*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d)} -{(c*e + d*e*x)^1*(a + b*ArcTanh[c + d*x])^3, x, 10, (3*b*e*(a + b*ArcTanh[c + d*x])^2)/(2*d) + (3*b*e*(c + d*x)*(a + b*ArcTanh[c + d*x])^2)/(2*d) - (e*(a + b*ArcTanh[c + d*x])^3)/(2*d) + (e*(c + d*x)^2*(a + b*ArcTanh[c + d*x])^3)/(2*d) - (3*b^2*e*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d - (3*b^3*e*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*d)} -{(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^1, x, 10, (2*(a + b*ArcTanh[c + d*x])^3*ArcTanh[1 - 2/(1 - c - d*x)])/(d*e) - (3*b*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*d*e) + (3*b*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, -1 + 2/(1 - c - d*x)])/(2*d*e) + (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d*e) - (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[3, -1 + 2/(1 - c - d*x)])/(2*d*e) - (3*b^3*PolyLog[4, 1 - 2/(1 - c - d*x)])/(4*d*e) + (3*b^3*PolyLog[4, -1 + 2/(1 - c - d*x)])/(4*d*e)} -{(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^2, x, 7, (a + b*ArcTanh[c + d*x])^3/(d*e^2) - (a + b*ArcTanh[c + d*x])^3/(d*e^2*(c + d*x)) + (3*b*(a + b*ArcTanh[c + d*x])^2*Log[2 - 2/(1 + c + d*x)])/(d*e^2) - (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[2, -1 + 2/(1 + c + d*x)])/(d*e^2) - (3*b^3*PolyLog[3, -1 + 2/(1 + c + d*x)])/(2*d*e^2)} -{(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^3, x, 9, (3*b*(a + b*ArcTanh[c + d*x])^2)/(2*d*e^3) - (3*b*(a + b*ArcTanh[c + d*x])^2)/(2*d*e^3*(c + d*x)) + (a + b*ArcTanh[c + d*x])^3/(2*d*e^3) - (a + b*ArcTanh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcTanh[c + d*x])*Log[2 - 2/(1 + c + d*x)])/(d*e^3) - (3*b^3*PolyLog[2, -1 + 2/(1 + c + d*x)])/(2*d*e^3)} -{(a + b*ArcTanh[c + d*x])^3/(c*e + d*e*x)^4, x, 16, -((b^2*(a + b*ArcTanh[c + d*x]))/(d*e^4*(c + d*x))) + (b*(a + b*ArcTanh[c + d*x])^2)/(2*d*e^4) - (b*(a + b*ArcTanh[c + d*x])^2)/(2*d*e^4*(c + d*x)^2) + (a + b*ArcTanh[c + d*x])^3/(3*d*e^4) - (a + b*ArcTanh[c + d*x])^3/(3*d*e^4*(c + d*x)^3) + (b^3*Log[c + d*x])/(d*e^4) - (b^3*Log[1 - (c + d*x)^2])/(2*d*e^4) + (b*(a + b*ArcTanh[c + d*x])^2*Log[2 - 2/(1 + c + d*x)])/(d*e^4) - (b^2*(a + b*ArcTanh[c + d*x])*PolyLog[2, -1 + 2/(1 + c + d*x)])/(d*e^4) - (b^3*PolyLog[3, -1 + 2/(1 + c + d*x)])/(2*d*e^4)} - - -{ArcTanh[1 + x]/(2 + 2*x), x, 3, (-(1/4))*PolyLog[2, -1 - x] + (1/4)*PolyLog[2, 1 + x]} - - -{ArcTanh[a + b*x]/((a*d)/b + d*x), x, 3, -(PolyLog[2, -a - b*x]/(2*d)) + PolyLog[2, a + b*x]/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p*) - - -{(e + f*x)^3*(a + b*ArcTanh[c + d*x]), x, 7, (b*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcTanh[c + d*x]))/(4*f) + (b*(d*e + f - c*f)^4*Log[1 - c - d*x])/(8*d^4*f) - (b*(d*e - f - c*f)^4*Log[1 + c + d*x])/(8*d^4*f)} -{(e + f*x)^2*(a + b*ArcTanh[c + d*x]), x, 7, (b*f*(d*e - c*f)*x)/d^2 + (b*f^2*(c + d*x)^2)/(6*d^3) + ((e + f*x)^3*(a + b*ArcTanh[c + d*x]))/(3*f) + (b*(d*e + f - c*f)^3*Log[1 - c - d*x])/(6*d^3*f) - (b*(d*e - (1 + c)*f)^3*Log[1 + c + d*x])/(6*d^3*f)} -{(e + f*x)^1*(a + b*ArcTanh[c + d*x]), x, 7, (b*f*x)/(2*d) + ((e + f*x)^2*(a + b*ArcTanh[c + d*x]))/(2*f) + (b*(d*e + f - c*f)^2*Log[1 - c - d*x])/(4*d^2*f) - (b*(d*e - (1 + c)*f)^2*Log[1 + c + d*x])/(4*d^2*f)} -{(e + f*x)^0*(a + b*ArcTanh[c + d*x]), x, 4, a*x + (b*(c + d*x)*ArcTanh[c + d*x])/d + (b*Log[1 - (c + d*x)^2])/(2*d)} -{(a + b*ArcTanh[c + d*x])/(e + f*x)^1, x, 5, -(((a + b*ArcTanh[c + d*x])*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (b*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)} -{(a + b*ArcTanh[c + d*x])/(e + f*x)^2, x, 7, If[$VersionNumber>=8, -((a + b*ArcTanh[c + d*x])/(f*(e + f*x))) - (b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) + (b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) - (b*d*Log[e + f*x])/((d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcTanh[c + d*x])/(f*(e + f*x))) - (b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) + (b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) - (b*d*Log[e + f*x])/((d*e - f - c*f)*(d*e + f - c*f))]} -{(a + b*ArcTanh[c + d*x])/(e + f*x)^3, x, 5, If[$VersionNumber>=8, (b*d)/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)*(e + f*x)) - (a + b*ArcTanh[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*Log[1 - c - d*x])/(4*f*(d*e + f - c*f)^2) + (b*d^2*Log[1 + c + d*x])/(4*f*(d*e - f - c*f)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2), (b*d)/(2*(d*e - f - c*f)*(d*e + f - c*f)*(e + f*x)) - (a + b*ArcTanh[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*Log[1 - c - d*x])/(4*f*(d*e + f - c*f)^2) + (b*d^2*Log[1 + c + d*x])/(4*f*(d*e - f - c*f)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2)]} - - -{(e + f*x)^3*(a + b*ArcTanh[c + d*x])^2, x, 20, (b^2*f^2*(d*e - c*f)*x)/d^3 + (a*b*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*x)/(2*d^3) + (b^2*f^3*(c + d*x)^2)/(12*d^4) - (b^2*f^2*(d*e - c*f)*ArcTanh[c + d*x])/d^4 + (b^2*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*(c + d*x)*ArcTanh[c + d*x])/(2*d^4) + (b*f^2*(d*e - c*f)*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/d^4 + (b*f^3*(c + d*x)^3*(a + b*ArcTanh[c + d*x]))/(6*d^4) + ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2)/d^4 - ((d^4*e^4 - 4*c*d^3*e^3*f + 6*(1 + c^2)*d^2*e^2*f^2 - 4*c*(3 + c^2)*d*e*f^3 + (1 + 6*c^2 + c^4)*f^4)*(a + b*ArcTanh[c + d*x])^2)/(4*d^4*f) + ((e + f*x)^4*(a + b*ArcTanh[c + d*x])^2)/(4*f) - (2*b*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d^4 + (b^2*f^3*Log[1 - (c + d*x)^2])/(12*d^4) + (b^2*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*Log[1 - (c + d*x)^2])/(4*d^4) - (b^2*(d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^4} -{(e + f*x)^2*(a + b*ArcTanh[c + d*x])^2, x, 16, (b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 - (b^2*f^2*ArcTanh[c + d*x])/(3*d^3) + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcTanh[c + d*x])/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2)/(3*d^3) + ((e + f*x)^3*(a + b*ArcTanh[c + d*x])^2)/(3*f) - (2*b*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 - (c + d*x)^2])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(3*d^3)} -{(e + f*x)^1*(a + b*ArcTanh[c + d*x])^2, x, 13, (a*b*f*x)/d + (b^2*f*(c + d*x)*ArcTanh[c + d*x])/d^2 + ((d*e - c*f)*(a + b*ArcTanh[c + d*x])^2)/d^2 - ((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcTanh[c + d*x])^2)/(2*f) - (2*b*(d*e - c*f)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d^2 + (b^2*f*Log[1 - (c + d*x)^2])/(2*d^2) - (b^2*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^2} -{(e + f*x)^0*(a + b*ArcTanh[c + d*x])^2, x, 6, (a + b*ArcTanh[c + d*x])^2/d + ((c + d*x)*(a + b*ArcTanh[c + d*x])^2)/d - (2*b*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d - (b^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d} -{(a + b*ArcTanh[c + d*x])^2/(e + f*x)^1, x, 2, -(((a + b*ArcTanh[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)} -{(a + b*ArcTanh[c + d*x])^2/(e + f*x)^2, x, 24, If[$VersionNumber>=8, -((a + b*ArcTanh[c + d*x])^2/(f*(e + f*x))) + (b^2*d*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) - (a*b*d*Log[1 - c - d*x])/(f*(d*e + f - c*f)) - (b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (2*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (a*b*d*Log[1 + c + d*x])/(f*(d*e - f - c*f)) + (2*a*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (2*b^2*d*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcTanh[c + d*x])^2/(f*(e + f*x))) + (b^2*d*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) - (a*b*d*Log[1 - c - d*x])/(f*(d*e + f - c*f)) - (b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (2*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (a*b*d*Log[1 + c + d*x])/(f*(d*e - f - c*f)) + (2*a*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (2*b^2*d*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f))]} -{(a + b*ArcTanh[c + d*x])^2/(e + f*x)^3, x, 26, If[$VersionNumber>=8, -((a*b*d)/((f^2 - (d*e - c*f)^2)*(e + f*x))) + (b^2*d*ArcTanh[c + d*x])/((d*e + f - c*f)*(d*e - (1 + c)*f)*(e + f*x)) - (a + b*ArcTanh[c + d*x])^2/(2*f*(e + f*x)^2) + (b^2*d^2*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)^2) - (a*b*d^2*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)^2) + (b^2*d^2*Log[1 - c - d*x])/(2*(d*e + f - c*f)^2*(d*e - (1 + c)*f)) - (b^2*d^2*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)^2) + (2*b^2*d^2*(d*e - c*f)*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (a*b*d^2*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)^2) - (b^2*d^2*Log[1 + c + d*x])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)^2) + (b^2*d^2*f*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) - (2*a*b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) - (2*b^2*d^2*(d*e - c*f)*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (b^2*d^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(4*f*(d*e + f - c*f)^2) + (b^2*d^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)^2) - (b^2*d^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (b^2*d^2*(d*e - c*f)*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2), -((a*b*d)/((f^2 - (d*e - c*f)^2)*(e + f*x))) + (b^2*d*ArcTanh[c + d*x])/((d*e - f - c*f)*(d*e + f - c*f)*(e + f*x)) - (a + b*ArcTanh[c + d*x])^2/(2*f*(e + f*x)^2) + (b^2*d^2*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)^2) - (a*b*d^2*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)^2) + (b^2*d^2*Log[1 - c - d*x])/(2*(d*e + f - c*f)^2*(d*e - (1 + c)*f)) - (b^2*d^2*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)^2) + (2*b^2*d^2*(d*e - c*f)*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (a*b*d^2*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)^2) - (b^2*d^2*Log[1 + c + d*x])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)^2) + (b^2*d^2*f*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) - (2*a*b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) - (2*b^2*d^2*(d*e - c*f)*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (b^2*d^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(4*f*(d*e + f - c*f)^2) + (b^2*d^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)^2) - (b^2*d^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (b^2*d^2*(d*e - c*f)*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2)]} - - -{(e + f*x)^2*(a + b*ArcTanh[c + d*x])^3, x, 21, (a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcTanh[c + d*x])/d^3 - (b*f^2*(a + b*ArcTanh[c + d*x])^2)/(2*d^3) + (3*b*f*(d*e - c*f)*(a + b*ArcTanh[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcTanh[c + d*x])^2)/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x])^2)/(2*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^3)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcTanh[c + d*x])^3)/(3*d^3) + ((e + f*x)^3*(a + b*ArcTanh[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcTanh[c + d*x])^2*Log[2/(1 - c - d*x)])/d^3 + (b^3*f^2*Log[1 - (c + d*x)^2])/(2*d^3) - (3*b^3*f*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d^3 + (b^3*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^3)} -{(e + f*x)^1*(a + b*ArcTanh[c + d*x])^3, x, 15, (3*b*f*(a + b*ArcTanh[c + d*x])^2)/(2*d^2) + (3*b*f*(c + d*x)*(a + b*ArcTanh[c + d*x])^2)/(2*d^2) + ((d*e - c*f)*(a + b*ArcTanh[c + d*x])^3)/d^2 - ((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcTanh[c + d*x])^3)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcTanh[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d^2 - (3*b*(d*e - c*f)*(a + b*ArcTanh[c + d*x])^2*Log[2/(1 - c - d*x)])/d^2 - (3*b^3*f*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*d^2) - (3*b^2*(d*e - c*f)*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d^2 + (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^2)} -{(e + f*x)^0*(a + b*ArcTanh[c + d*x])^3, x, 6, (a + b*ArcTanh[c + d*x])^3/d + ((c + d*x)*(a + b*ArcTanh[c + d*x])^3)/d - (3*b*(a + b*ArcTanh[c + d*x])^2*Log[2/(1 - c - d*x)])/d - (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d + (3*b^3*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d)} -{(a + b*ArcTanh[c + d*x])^3/(e + f*x)^1, x, 2, -(((a + b*ArcTanh[c + d*x])^3*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcTanh[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (3*b*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b*(a + b*ArcTanh[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b^2*(a + b*ArcTanh[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^3*PolyLog[4, 1 - 2/(1 + c + d*x)])/(4*f) - (3*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(4*f)} -{(a + b*ArcTanh[c + d*x])^3/(e + f*x)^2, x, 33, If[$VersionNumber>=8, -((a + b*ArcTanh[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) - (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) - (3*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) + (3*a^2*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (6*a*b^2*d*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcTanh[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - c - d*x)])/(4*f*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcTanh[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) - (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) - (3*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) + (3*a^2*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (6*a*b^2*d*ArcTanh[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*ArcTanh[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*ArcTanh[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - c - d*x)])/(4*f*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*(d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e - f - c*f)*(d*e + f - c*f))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p with m symbolic*) - - -{(e + f*x)^m*(a + b*ArcTanh[c + d*x])^3, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^3, x]} -{(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcTanh[c + d*x])^2, x]} -{(e + f*x)^m*(a + b*ArcTanh[c + d*x])^1, x, 6, ((e + f*x)^(1 + m)*(a + b*ArcTanh[c + d*x]))/(f*(1 + m)) + (b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - f - c*f)])/(2*f*(d*e - (1 + c)*f)*(1 + m)*(2 + m)) - (b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + f - c*f)])/(2*f*(d*e + f - c*f)*(1 + m)*(2 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form AF[x] (a+b ArcTanh[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x^m)^q (a+b ArcTanh[c+d x])^p*) - - -{ArcTanh[a + b*x]/(c + d*x^3), x, 23, -((Log[1 - a - b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))) + (Log[1 + a + b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(2/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*(1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - PolyLog[2, (d^(1/3)*(1 - a - b*x))/(b*c^(1/3) + (1 - a)*d^(1/3))]/(6*c^(2/3)*d^(1/3)) - ((-1)^(2/3)*PolyLog[2, -(((-1)^(1/3)*d^(1/3)*(1 - a - b*x))/(b*c^(1/3) - (-1)^(1/3)*(1 - a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(1 - a - b*x))/(b*c^(1/3) + (-1)^(2/3)*(1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + PolyLog[2, -((d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^(1/3)))]/(6*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*PolyLog[2, -(((-1)^(2/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (-1)^(2/3)*(1 + a)*d^(1/3)))])/(6*c^(2/3)*d^(1/3))} -{ArcTanh[a + b*x]/(c + d*x^2), x, 17, -(Log[1 - a - b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 + a + b*x]*Log[(b*(Sqrt[-c] - Sqrt[d]*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) + (Log[1 - a - b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - (Log[1 + a + b*x]*Log[(b*(Sqrt[-c] + Sqrt[d]*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d])])/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 - a - b*x))/(b*Sqrt[-c] - (1 - a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 - a - b*x))/(b*Sqrt[-c] + (1 - a)*Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -((Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] - (1 + a)*Sqrt[d]))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, (Sqrt[d]*(1 + a + b*x))/(b*Sqrt[-c] + (1 + a)*Sqrt[d])]/(4*Sqrt[-c]*Sqrt[d])} -{ArcTanh[a + b*x]/(c + d*x^1), x, 5, -((ArcTanh[a + b*x]*Log[2/(1 + a + b*x)])/d) + (ArcTanh[a + b*x]*Log[(2*b*(c + d*x))/((b*c + d - a*d)*(1 + a + b*x))])/d + PolyLog[2, 1 - 2/(1 + a + b*x)]/(2*d) - PolyLog[2, 1 - (2*b*(c + d*x))/((b*c + d - a*d)*(1 + a + b*x))]/(2*d)} -{ArcTanh[a + b*x]/(c + d/x^1), x, 15, ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) - (d*Log[1 + a + b*x]*Log[-((b*(d + c*x))/(c + a*c - b*d))])/(2*c^2) + (d*Log[1 - a - b*x]*Log[(b*(d + c*x))/(c - a*c + b*d)])/(2*c^2) + (d*PolyLog[2, (c*(1 - a - b*x))/(c - a*c + b*d)])/(2*c^2) - (d*PolyLog[2, (c*(1 + a + b*x))/(c + a*c - b*d)])/(2*c^2)} -{ArcTanh[a + b*x]/(c + d/x^2), x, 25, ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (Sqrt[d]*Log[1 - a - b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 - a - b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/(Sqrt[-c] - a*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))} -{ArcTanh[a + b*x]/(c + d/x^3), x, 31, ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) - (d^(1/3)*Log[1 + a + b*x]*Log[-((b*(d^(1/3) + c^(1/3)*x))/((1 + a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) + (d^(1/3)*Log[1 - a - b*x]*Log[(b*(d^(1/3) + c^(1/3)*x))/((1 - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*Log[1 - a - b*x]*Log[-((b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(1 - a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*Log[1 + a + b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(1 + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*Log[1 + a + b*x]*Log[-((b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(1 + a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*Log[1 - a - b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(1 - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(1 - a - b*x))/((-1)^(1/3)*(1 - a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) + (d^(1/3)*PolyLog[2, (c^(1/3)*(1 - a - b*x))/((1 - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(1 - a - b*x))/((-1)^(2/3)*(1 - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - (d^(1/3)*PolyLog[2, (c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(1 + a + b*x))/((-1)^(2/3)*(1 + a)*c^(1/3) - b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(1 + a + b*x))/((-1)^(1/3)*(1 + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x^(m/2)) (a+b ArcTanh[c+d x])^p*) - - -(* {ArcTanh[a + b*x]/(a + b*x^(3/2)), x, 41, (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) + PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3))} *) -{ArcTanh[a + b*x]/(c + d*Sqrt[x]), x, 31, (2*Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*d) - (2*Sqrt[1 - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*d) + (c*Log[(d*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)]*Log[c + d*Sqrt[x]])/d^2 - (c*Log[(d*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]])/d^2 + (c*Log[-((d*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (c*Log[-((d*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (Sqrt[x]*Log[1 - a - b*x])/d + (c*Log[c + d*Sqrt[x]]*Log[1 - a - b*x])/d^2 + (Sqrt[x]*Log[1 + a + b*x])/d - (c*Log[c + d*Sqrt[x]]*Log[1 + a + b*x])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d)])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2} -{ArcTanh[a + b*x]/(c + d/Sqrt[x]), x, 37, (-2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2) + (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d*Sqrt[x]*Log[1 - a - b*x])/c^2 + ((1 - a - b*x)*Log[1 - a - b*x])/(2*b*c) - (d^2*Log[d + c*Sqrt[x]]*Log[1 - a - b*x])/c^3 - (d*Sqrt[x]*Log[1 + a + b*x])/c^2 + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (d^2*Log[d + c*Sqrt[x]]*Log[1 + a + b*x])/c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))])/c^3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)])/c^3} -(* {ArcTanh[a + b*x]/(a + b/x^(3/2)), x, 49, -(b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + ((1 - a - b*x)*Log[1 - a - b*x])/(2*a*b) - (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*a*b) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) - (b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) + (b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) - (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x+g x^2)^q (a+b ArcTanh[c+d x])^p*) - - -{ArcTanh[d + e*x]/(a + b*x + c*x^2), x, 12, (ArcTanh[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - (ArcTanh[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c e+d e x)^m (a+b ArcTanh[c+d x])^p / (1-(c+d x)^2)*) - - -{(c*e + d*e*x)*(a + b*ArcTanh[c + d*x])/(1 - (c + d*x)^2), x, 6, -((e*(a + b*ArcTanh[c + d*x])^2)/(2*b*d)) + (e*(a + b*ArcTanh[c + d*x])*Log[2/(1 - c - d*x)])/d + (b*e*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*d)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent functions.m deleted file mode 100644 index 176aef6..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.6 Exponentials of inverse hyperbolic tangent functions.m +++ /dev/null @@ -1,2489 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*x^4, x, 6, -((4*x^2*Sqrt[1 - a^2*x^2])/(15*a^3)) - (x^3*Sqrt[1 - a^2*x^2])/(4*a^2) - (x^4*Sqrt[1 - a^2*x^2])/(5*a) - ((64 + 45*a*x)*Sqrt[1 - a^2*x^2])/(120*a^5) + (3*ArcSin[a*x])/(8*a^5)} -{E^ArcTanh[a*x]*x^3, x, 5, -((x^2*Sqrt[1 - a^2*x^2])/(3*a^2)) - (x^3*Sqrt[1 - a^2*x^2])/(4*a) - ((16 + 9*a*x)*Sqrt[1 - a^2*x^2])/(24*a^4) + (3*ArcSin[a*x])/(8*a^4)} -{E^ArcTanh[a*x]*x^2, x, 7, -(Sqrt[1 - a^2*x^2]/a^3) - (x*Sqrt[1 - a^2*x^2])/(2*a^2) + (1 - a^2*x^2)^(3/2)/(3*a^3) + ArcSin[a*x]/(2*a^3)} -{E^ArcTanh[a*x]*x^1, x, 3, -(((2 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^2)) + ArcSin[a*x]/(2*a^2)} -{E^ArcTanh[a*x]*x^0, x, 3, -(Sqrt[1 - a^2*x^2]/a) + ArcSin[a*x]/a} -{E^ArcTanh[a*x]/x^1, x, 6, ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^ArcTanh[a*x]/x^2, x, 5, -(Sqrt[1 - a^2*x^2]/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^ArcTanh[a*x]/x^3, x, 6, -(Sqrt[1 - a^2*x^2]/(2*x^2)) - (a*Sqrt[1 - a^2*x^2])/x - (1/2)*a^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^ArcTanh[a*x]/x^4, x, 7, -(Sqrt[1 - a^2*x^2]/(3*x^3)) - (a*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x) - (1/2)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^ArcTanh[a*x]/x^5, x, 8, -(Sqrt[1 - a^2*x^2]/(4*x^4)) - (a*Sqrt[1 - a^2*x^2])/(3*x^3) - (3*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) - (2*a^3*Sqrt[1 - a^2*x^2])/(3*x) - (3/8)*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^(2*ArcTanh[a*x])*x^3, x, 3, -((2*x)/a^3) - x^2/a^2 - (2*x^3)/(3*a) - x^4/4 - (2*Log[1 - a*x])/a^4} -{E^(2*ArcTanh[a*x])*x^2, x, 3, -((2*x)/a^2) - x^2/a - x^3/3 - (2*Log[1 - a*x])/a^3} -{E^(2*ArcTanh[a*x])*x^1, x, 3, -((2*x)/a) - x^2/2 - (2*Log[1 - a*x])/a^2} -{E^(2*ArcTanh[a*x])*x^0, x, 3, -x - (2*Log[1 - a*x])/a} -{E^(2*ArcTanh[a*x])/x^1, x, 3, Log[x] - 2*Log[1 - a*x]} -{E^(2*ArcTanh[a*x])/x^2, x, 3, -(1/x) + 2*a*Log[x] - 2*a*Log[1 - a*x]} -{E^(2*ArcTanh[a*x])/x^3, x, 3, -(1/(2*x^2)) - (2*a)/x + 2*a^2*Log[x] - 2*a^2*Log[1 - a*x]} -{E^(2*ArcTanh[a*x])/x^4, x, 3, -(1/(3*x^3)) - a/x^2 - (2*a^2)/x + 2*a^3*Log[x] - 2*a^3*Log[1 - a*x]} - - -{E^(3*ArcTanh[a*x])*x^2, x, 10, (1 + a*x)^3/(a^3*Sqrt[1 - a^2*x^2]) + ((3 + a*x)^2*Sqrt[1 - a^2*x^2])/(3*a^3) + ((28 + 3*a*x)*Sqrt[1 - a^2*x^2])/(6*a^3) - (11*ArcSin[a*x])/(2*a^3)} -{E^(3*ArcTanh[a*x])*x^1, x, 9, (9*Sqrt[1 - a^2*x^2])/(2*a^2) + (3*(1 - a^2*x^2)^(3/2))/(2*a^2*(1 - a*x)) + (1 - a^2*x^2)^(5/2)/(a^2*(1 - a*x)^3) - (9*ArcSin[a*x])/(2*a^2)} -{E^(3*ArcTanh[a*x])*x^0, x, 5, (2*(1 + a*x)^2)/(a*Sqrt[1 - a^2*x^2]) + (3*Sqrt[1 - a^2*x^2])/a - (3*ArcSin[a*x])/a} -{E^(3*ArcTanh[a*x])/x^1, x, 8, (4*Sqrt[1 - a^2*x^2])/(1 - a*x) - ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])/x^2, x, 8, -(Sqrt[1 - a^2*x^2]/x) + (4*a*Sqrt[1 - a^2*x^2])/(1 - a*x) - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])/x^3, x, 12, -(Sqrt[1 - a^2*x^2]/(2*x^2)) - (3*a*Sqrt[1 - a^2*x^2])/x + (4*a^2*Sqrt[1 - a^2*x^2])/(1 - a*x) - (9/2)*a^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])/x^4, x, 14, -(Sqrt[1 - a^2*x^2]/(3*x^3)) - (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (4*a^3*Sqrt[1 - a^2*x^2])/(1 - a*x) - (11/2)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^(4*ArcTanh[a*x])*x^3, x, 3, (12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4} -{E^(4*ArcTanh[a*x])*x^2, x, 3, (8*x)/a^2 + (2*x^2)/a + x^3/3 + 4/(a^3*(1 - a*x)) + (12*Log[1 - a*x])/a^3} -{E^(4*ArcTanh[a*x])*x^1, x, 3, (4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2} -{E^(4*ArcTanh[a*x])*x^0, x, 3, x + 4/(a*(1 - a*x)) + (4*Log[1 - a*x])/a} -{E^(4*ArcTanh[a*x])/x^1, x, 3, 4/(1 - a*x) + Log[x]} -{E^(4*ArcTanh[a*x])/x^2, x, 3, -(1/x) + (4*a)/(1 - a*x) + 4*a*Log[x] - 4*a*Log[1 - a*x]} -{E^(4*ArcTanh[a*x])/x^3, x, 3, -(1/(2*x^2)) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*Log[1 - a*x]} -{E^(4*ArcTanh[a*x])/x^4, x, 3, -(1/(3*x^3)) - (2*a)/x^2 - (8*a^2)/x + (4*a^3)/(1 - a*x) + 12*a^3*Log[x] - 12*a^3*Log[1 - a*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{E^(-ArcTanh[a*x])*x^3, x, 5, -((x^2*Sqrt[1 - a^2*x^2])/(3*a^2)) + (x^3*Sqrt[1 - a^2*x^2])/(4*a) - ((16 - 9*a*x)*Sqrt[1 - a^2*x^2])/(24*a^4) - (3*ArcSin[a*x])/(8*a^4)} -{E^(-ArcTanh[a*x])*x^2, x, 7, Sqrt[1 - a^2*x^2]/a^3 - (x*Sqrt[1 - a^2*x^2])/(2*a^2) - (1 - a^2*x^2)^(3/2)/(3*a^3) + ArcSin[a*x]/(2*a^3)} -{E^(-ArcTanh[a*x])*x^1, x, 3, -(((2 - a*x)*Sqrt[1 - a^2*x^2])/(2*a^2)) - ArcSin[a*x]/(2*a^2)} -{E^(-ArcTanh[a*x])*x^0, x, 3, Sqrt[1 - a^2*x^2]/a + ArcSin[a*x]/a} -{E^(-ArcTanh[a*x])/x^1, x, 6, -ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-ArcTanh[a*x])/x^2, x, 5, -(Sqrt[1 - a^2*x^2]/x) + a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-ArcTanh[a*x])/x^3, x, 6, -(Sqrt[1 - a^2*x^2]/(2*x^2)) + (a*Sqrt[1 - a^2*x^2])/x - (1/2)*a^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-ArcTanh[a*x])/x^4, x, 7, -(Sqrt[1 - a^2*x^2]/(3*x^3)) + (a*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a^2*Sqrt[1 - a^2*x^2])/(3*x) + (1/2)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-ArcTanh[a*x])/x^5, x, 8, -(Sqrt[1 - a^2*x^2]/(4*x^4)) + (a*Sqrt[1 - a^2*x^2])/(3*x^3) - (3*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (2*a^3*Sqrt[1 - a^2*x^2])/(3*x) - (3/8)*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^(-2*ArcTanh[a*x])*x^3, x, 3, (2*x)/a^3 - x^2/a^2 + (2*x^3)/(3*a) - x^4/4 - (2*Log[1 + a*x])/a^4} -{E^(-2*ArcTanh[a*x])*x^2, x, 3, -((2*x)/a^2) + x^2/a - x^3/3 + (2*Log[1 + a*x])/a^3} -{E^(-2*ArcTanh[a*x])*x^1, x, 3, (2*x)/a - x^2/2 - (2*Log[1 + a*x])/a^2} -{E^(-2*ArcTanh[a*x])*x^0, x, 3, -x + (2*Log[1 + a*x])/a} -{E^(-2*ArcTanh[a*x])/x^1, x, 3, Log[x] - 2*Log[1 + a*x]} -{E^(-2*ArcTanh[a*x])/x^2, x, 3, -(1/x) - 2*a*Log[x] + 2*a*Log[1 + a*x]} -{E^(-2*ArcTanh[a*x])/x^3, x, 3, -(1/(2*x^2)) + (2*a)/x + 2*a^2*Log[x] - 2*a^2*Log[1 + a*x]} -{E^(-2*ArcTanh[a*x])/x^4, x, 3, -(1/(3*x^3)) + a/x^2 - (2*a^2)/x - 2*a^3*Log[x] + 2*a^3*Log[1 + a*x]} - - -{E^(-3*ArcTanh[a*x])*x^3, x, 14, (1 - a*x)^3/(a^4*Sqrt[1 - a^2*x^2]) + (27*Sqrt[1 - a^2*x^2])/(4*a^4) + (x^2*Sqrt[1 - a^2*x^2])/a^2 - (x^3*Sqrt[1 - a^2*x^2])/(4*a) + (9*(2 - 3*a*x)*Sqrt[1 - a^2*x^2])/(8*a^4) + (51*ArcSin[a*x])/(8*a^4)} -{E^(-3*ArcTanh[a*x])*x^2, x, 10, -((1 - a*x)^3/(a^3*Sqrt[1 - a^2*x^2])) - ((28 - 3*a*x)*Sqrt[1 - a^2*x^2])/(6*a^3) - ((3 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a^3) - (11*ArcSin[a*x])/(2*a^3)} -{E^(-3*ArcTanh[a*x])*x^1, x, 9, (9*Sqrt[1 - a^2*x^2])/(2*a^2) + (3*(1 - a^2*x^2)^(3/2))/(2*a^2*(1 + a*x)) + (1 - a^2*x^2)^(5/2)/(a^2*(1 + a*x)^3) + (9*ArcSin[a*x])/(2*a^2)} -{E^(-3*ArcTanh[a*x])*x^0, x, 5, -((2*(1 - a*x)^2)/(a*Sqrt[1 - a^2*x^2])) - (3*Sqrt[1 - a^2*x^2])/a - (3*ArcSin[a*x])/a} -{E^(-3*ArcTanh[a*x])/x^1, x, 8, (4*Sqrt[1 - a^2*x^2])/(1 + a*x) + ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-3*ArcTanh[a*x])/x^2, x, 8, -(Sqrt[1 - a^2*x^2]/x) - (4*a*Sqrt[1 - a^2*x^2])/(1 + a*x) + 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-3*ArcTanh[a*x])/x^3, x, 12, -(Sqrt[1 - a^2*x^2]/(2*x^2)) + (3*a*Sqrt[1 - a^2*x^2])/x + (4*a^2*Sqrt[1 - a^2*x^2])/(1 + a*x) - (9/2)*a^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-3*ArcTanh[a*x])/x^4, x, 14, -(Sqrt[1 - a^2*x^2]/(3*x^3)) + (3*a*Sqrt[1 - a^2*x^2])/(2*x^2) - (14*a^2*Sqrt[1 - a^2*x^2])/(3*x) - (4*a^3*Sqrt[1 - a^2*x^2])/(1 + a*x) + (11/2)*a^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(-3*ArcTanh[a*x])/x^5, x, 19, -(Sqrt[1 - a^2*x^2]/(4*x^4)) + (a*Sqrt[1 - a^2*x^2])/x^3 - (19*a^2*Sqrt[1 - a^2*x^2])/(8*x^2) + (6*a^3*Sqrt[1 - a^2*x^2])/x + (4*a^4*Sqrt[1 - a^2*x^2])/(1 + a*x) - (51/8)*a^4*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/2 ArcTanh[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcTanh[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/4, -(1/4), 2 + m, a*x, (-a)*x])/(1 + m)} - -{E^(ArcTanh[a*x]/2)*x^2, x, 15, -((3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(8*a^3)) - ((1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(12*a^3) - (x*(1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(3*a^2) + (3*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (3*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) + (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{E^(ArcTanh[a*x]/2)*x^1, x, 14, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*a^2)) - ((1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(2*a^2) + ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(4*Sqrt[2]*a^2) - ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(4*Sqrt[2]*a^2) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(8*Sqrt[2]*a^2) + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(8*Sqrt[2]*a^2)} -{E^(ArcTanh[a*x]/2)*x^0, x, 13, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a) + ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) - ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a) + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a)} -{E^(ArcTanh[a*x]/2)/x^1, x, 17, -2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{E^(ArcTanh[a*x]/2)/x^2, x, 6, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x) - a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^(ArcTanh[a*x]/2)/x^3, x, 7, -((a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*x)) - ((1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(2*x^2) - (1/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (1/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^(ArcTanh[a*x]/2)/x^4, x, 9, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(3*x^3)) - (5*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(12*x^2) - (11*a^2*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(24*x) - (3/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (3/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^(ArcTanh[a*x]/2)/x^5, x, 10, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*x^4)) - (7*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(24*x^3) - (29*a^2*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(96*x^2) - (83*a^3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(192*x) - (11/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (11/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^(ArcTanh[a*x]/2)/x^6, x, 11, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(5*x^5)) - (9*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(40*x^4) - (11*a^2*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(48*x^3) - (269*a^3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(960*x^2) - (611*a^4*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(1920*x) - (31/128)*a^5*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (31/128)*a^5*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -{E^(3*ArcTanh[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 3/4, -(3/4), 2 + m, a*x, (-a)*x])/(1 + m)} - -{E^((3*ArcTanh[a*x])/2)*x^3, x, 15, -((41*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*a^4)) - (x^2*(1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(4*a^2) - ((1 - a*x)^(1/4)*(1 + a*x)^(7/4)*(11 + 4*a*x))/(32*a^4) + (123*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (123*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (123*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{E^((3*ArcTanh[a*x])/2)*x^2, x, 15, (-17*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(24*a^3) - ((1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(4*a^3) - (x*(1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(3*a^2) + (17*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (17*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) + (17*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) - (17*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{E^((3*ArcTanh[a*x])/2)*x^1, x, 14, (-3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*a^2) - ((1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(2*a^2) + (9*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (9*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^((3*ArcTanh[a*x])/2)*x^0, x, 13, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a) + (3*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) - (3*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) + (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a) - (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a)} -{E^((3*ArcTanh[a*x])/2)/x^1, x, 17, 2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{E^((3*ArcTanh[a*x])/2)/x^2, x, 6, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x) + 3*a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - 3*a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((3*ArcTanh[a*x])/2)/x^3, x, 7, -((3*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*x)) - ((1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(2*x^2) + (9/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (9/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((3*ArcTanh[a*x])/2)/x^4, x, 9, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(3*x^3)) - (7*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(12*x^2) - (23*a^2*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(24*x) + (17/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (17/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((3*ArcTanh[a*x])/2)/x^5, x, 10, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*x^4)) - (3*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(8*x^3) - (15*a^2*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(32*x^2) - (63*a^3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*x) + (123/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (123/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -{E^(5*ArcTanh[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 5/4, -(5/4), 2 + m, a*x, (-a)*x])/(1 + m)} - -{E^((5*ArcTanh[a*x])/2)*x^3, x, 16, (475*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(64*a^4) + (4*x^3*(1 + a*x)^(5/4))/(a*(1 - a*x)^(1/4)) + (17*x^2*(1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(4*a^2) + ((1 - a*x)^(3/4)*(1 + a*x)^(5/4)*(521 + 452*a*x))/(96*a^4) - (475*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (475*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{E^((5*ArcTanh[a*x])/2)*x^2, x, 16, (55*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(8*a^3) + (11*(1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(4*a^3) + (2*(1 + a*x)^(9/4))/(a^3*(1 - a*x)^(1/4)) + ((1 - a*x)^(3/4)*(1 + a*x)^(9/4))/(3*a^3) - (55*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) + (55*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) + (55*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) - (55*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{E^((5*ArcTanh[a*x])/2)*x^1, x, 15, (25*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*a^2) + (5*(1 - a*x)^(3/4)*(1 + a*x)^(5/4))/(2*a^2) + (2*(1 + a*x)^(9/4))/(a^2*(1 - a*x)^(1/4)) - (25*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (25*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (25*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2) - (25*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^((5*ArcTanh[a*x])/2)*x^0, x, 14, (5*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a + (4*(1 + a*x)^(5/4))/(a*(1 - a*x)^(1/4)) - (5*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) + (5*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) + (5*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a) - (5*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a)} -{E^((5*ArcTanh[a*x])/2)/x^1, x, 19, (8*(1 + a*x)^(1/4))/(1 - a*x)^(1/4) - 2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{E^((5*ArcTanh[a*x])/2)/x^2, x, 7, (10*a*(1 + a*x)^(1/4))/(1 - a*x)^(1/4) - (1 + a*x)^(5/4)/(x*(1 - a*x)^(1/4)) - 5*a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - 5*a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((5*ArcTanh[a*x])/2)/x^3, x, 8, (25*a^2*(1 + a*x)^(1/4))/(2*(1 - a*x)^(1/4)) - (5*a*(1 + a*x)^(5/4))/(4*x*(1 - a*x)^(1/4)) - (1 + a*x)^(9/4)/(2*x^2*(1 - a*x)^(1/4)) - (25/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (25/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((5*ArcTanh[a*x])/2)/x^4, x, 10, (287*a^3*(1 + a*x)^(1/4))/(24*(1 - a*x)^(1/4)) - (1 + a*x)^(1/4)/(3*x^3*(1 - a*x)^(1/4)) - (13*a*(1 + a*x)^(1/4))/(12*x^2*(1 - a*x)^(1/4)) - (61*a^2*(1 + a*x)^(1/4))/(24*x*(1 - a*x)^(1/4)) - (55/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (55/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{E^((5*ArcTanh[a*x])/2)/x^5, x, 11, (2467*a^4*(1 + a*x)^(1/4))/(192*(1 - a*x)^(1/4)) - (1 + a*x)^(1/4)/(4*x^4*(1 - a*x)^(1/4)) - (17*a*(1 + a*x)^(1/4))/(24*x^3*(1 - a*x)^(1/4)) - (113*a^2*(1 + a*x)^(1/4))/(96*x^2*(1 - a*x)^(1/4)) - (521*a^3*(1 + a*x)^(1/4))/(192*x*(1 - a*x)^(1/4)) - (475/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (475/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^m/E^(ArcTanh[a*x]/2), x, 2, (x^(1 + m)*AppellF1[1 + m, -(1/4), 1/4, 2 + m, a*x, (-a)*x])/(1 + m)} - -{x^3/E^(ArcTanh[a*x]/2), x, 15, -((11*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*a^4)) - (x^2*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(4*a^2) - ((25 - 4*a*x)*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(96*a^4) - (11*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (11*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (11*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) + (11*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^(ArcTanh[a*x]/2), x, 15, (3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(8*a^3) + ((1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(12*a^3) - (x*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(3*a^2) + (3*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (3*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) + (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) - (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{x/E^(ArcTanh[a*x]/2), x, 14, -((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*a^2) - ((1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(2*a^2) - ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(4*Sqrt[2]*a^2) + ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(4*Sqrt[2]*a^2) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(8*Sqrt[2]*a^2) + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(8*Sqrt[2]*a^2)} -{E^(-ArcTanh[a*x]/2), x, 13, ((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a + ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) - ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a)} -{1/(E^(ArcTanh[a*x]/2)*x), x, 17, 2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{1/(E^(ArcTanh[a*x]/2)*x^2), x, 6, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/x) - a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^(ArcTanh[a*x]/2)*x^3), x, 7, (a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*x) - ((1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(2*x^2) + (1/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (1/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^(ArcTanh[a*x]/2)*x^4), x, 9, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(3*x^3)) + (5*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(12*x^2) - (11*a^2*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(24*x) - (3/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + (3/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^(ArcTanh[a*x]/2)*x^5), x, 10, -(((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*x^4)) + (7*a*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(24*x^3) - (29*a^2*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(96*x^2) + (83*a^3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(192*x) + (11/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (11/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -{x^m/E^((3*ArcTanh[a*x])/2), x, 2, (x^(1 + m)*AppellF1[1 + m, -(3/4), 3/4, 2 + m, a*x, (-a)*x])/(1 + m)} - -{x^3/E^((3*ArcTanh[a*x])/2), x, 15, -((41*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(64*a^4)) - (x^2*(1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(4*a^2) - ((11 - 4*a*x)*(1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(32*a^4) - (123*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (123*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^((3*ArcTanh[a*x])/2), x, 15, (17*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(24*a^3) + ((1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(4*a^3) - (x*(1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(3*a^2) + (17*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (17*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (17*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) + (17*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{x/E^((3*ArcTanh[a*x])/2), x, 14, (-3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*a^2) - ((1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(2*a^2) - (9*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2) - (9*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^((-3*ArcTanh[a*x])/2), x, 13, ((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a + (3*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) - (3*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) - (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a) + (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a)} -{1/(E^((3*ArcTanh[a*x])/2)*x), x, 17, -2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{1/(E^((3*ArcTanh[a*x])/2)*x^2), x, 6, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/x) + 3*a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + 3*a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((3*ArcTanh[a*x])/2)*x^3), x, 7, (3*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*x) - ((1 - a*x)^(7/4)*(1 + a*x)^(1/4))/(2*x^2) - (9/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (9/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((3*ArcTanh[a*x])/2)*x^4), x, 9, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(3*x^3)) + (7*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(12*x^2) - (23*a^2*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(24*x) + (17/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + (17/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((3*ArcTanh[a*x])/2)*x^5), x, 10, -(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(4*x^4)) + (3*a*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(8*x^3) - (15*a^2*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(32*x^2) + (63*a^3*(1 - a*x)^(3/4)*(1 + a*x)^(1/4))/(64*x) - (123/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (123/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -{x^m/E^((5*ArcTanh[a*x])/2), x, 2, (x^(1 + m)*AppellF1[1 + m, -(5/4), 5/4, 2 + m, a*x, (-a)*x])/(1 + m)} - -{x^3/E^((5*ArcTanh[a*x])/2), x, 16, -((4*x^3*(1 - a*x)^(5/4))/(a*(1 + a*x)^(1/4))) + (475*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*a^4) + (17*x^2*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(4*a^2) + ((521 - 452*a*x)*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(96*a^4) + (475*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) - (475*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (475*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (475*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)} -{x^2/E^((5*ArcTanh[a*x])/2), x, 16, (-2*(1 - a*x)^(9/4))/(a^3*(1 + a*x)^(1/4)) - (55*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(8*a^3) - (11*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(4*a^3) - ((1 - a*x)^(9/4)*(1 + a*x)^(3/4))/(3*a^3) - (55*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) + (55*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^3) - (55*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3) + (55*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(16*Sqrt[2]*a^3)} -{x/E^((5*ArcTanh[a*x])/2), x, 15, (2*(1 - a*x)^(9/4))/(a^2*(1 + a*x)^(1/4)) + (25*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*a^2) + (5*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(2*a^2) + (25*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) - (25*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a^2) + (25*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2) - (25*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a^2)} -{E^((-5*ArcTanh[a*x])/2), x, 14, (-4*(1 - a*x)^(5/4))/(a*(1 + a*x)^(1/4)) - (5*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a - (5*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) + (5*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a) - (5*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a) + (5*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(2*Sqrt[2]*a)} -{1/(E^((5*ArcTanh[a*x])/2)*x), x, 19, (8*(1 - a*x)^(1/4))/(1 + a*x)^(1/4) + 2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)] - 2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]} -{1/(E^((5*ArcTanh[a*x])/2)*x^2), x, 7, -((10*a*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)) - (1 - a*x)^(5/4)/(x*(1 + a*x)^(1/4)) - 5*a*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + 5*a*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((5*ArcTanh[a*x])/2)*x^3), x, 8, (25*a^2*(1 - a*x)^(1/4))/(2*(1 + a*x)^(1/4)) + (5*a*(1 - a*x)^(5/4))/(4*x*(1 + a*x)^(1/4)) - (1 - a*x)^(9/4)/(2*x^2*(1 + a*x)^(1/4)) + (25/4)*a^2*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (25/4)*a^2*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((5*ArcTanh[a*x])/2)*x^4), x, 10, -((287*a^3*(1 - a*x)^(1/4))/(24*(1 + a*x)^(1/4))) - (1 - a*x)^(1/4)/(3*x^3*(1 + a*x)^(1/4)) + (13*a*(1 - a*x)^(1/4))/(12*x^2*(1 + a*x)^(1/4)) - (61*a^2*(1 - a*x)^(1/4))/(24*x*(1 + a*x)^(1/4)) - (55/8)*a^3*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] + (55/8)*a^3*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} -{1/(E^((5*ArcTanh[a*x])/2)*x^5), x, 11, (2467*a^4*(1 - a*x)^(1/4))/(192*(1 + a*x)^(1/4)) - (1 - a*x)^(1/4)/(4*x^4*(1 + a*x)^(1/4)) + (17*a*(1 - a*x)^(1/4))/(24*x^3*(1 + a*x)^(1/4)) - (113*a^2*(1 - a*x)^(1/4))/(96*x^2*(1 + a*x)^(1/4)) + (521*a^3*(1 - a*x)^(1/4))/(192*x*(1 + a*x)^(1/4)) + (475/64)*a^4*ArcTan[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)] - (475/64)*a^4*ArcTanh[(1 + a*x)^(1/4)/(1 - a*x)^(1/4)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/3 ArcTanh[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcTanh[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/6, -(1/6), 2 + m, x, -x])/(1 + m)} - -{E^(ArcTanh[x]/3)*x^2, x, 16, (-(19/54))*(1 - x)^(5/6)*(1 + x)^(1/6) - (1/18)*(1 - x)^(5/6)*(1 + x)^(7/6) - (1/3)*(1 - x)^(5/6)*x*(1 + x)^(7/6) - (19/81)*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)] + (19/162)*ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - (19/162)*ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - (19*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)])/(108*Sqrt[3]) + (19*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)])/(108*Sqrt[3])} -{E^(ArcTanh[x]/3)*x^1, x, 15, (-(1/6))*(1 - x)^(5/6)*(1 + x)^(1/6) - (1/2)*(1 - x)^(5/6)*(1 + x)^(7/6) - (1/9)*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)] + (1/18)*ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - (1/18)*ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(12*Sqrt[3]) + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(12*Sqrt[3])} -{E^(ArcTanh[x]/3)*x^0, x, 14, (-(1 - x)^(5/6))*(1 + x)^(1/6) - (2/3)*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)] + (1/3)*ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - (1/3)*ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(2*Sqrt[3]) + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)]/(2*Sqrt[3])} -{E^(ArcTanh[x]/3)/x^1, x, 25, -2*ArcTan[(1 - x)^(1/6)/(1 + x)^(1/6)] + ArcTan[Sqrt[3] - (2*(1 - x)^(1/6))/(1 + x)^(1/6)] - ArcTan[Sqrt[3] + (2*(1 - x)^(1/6))/(1 + x)^(1/6)] + Sqrt[3]*ArcTan[(1 - (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]] - Sqrt[3]*ArcTan[(1 + (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]] - 2*ArcTanh[(1 + x)^(1/6)/(1 - x)^(1/6)] - (1/2)*Sqrt[3]*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) - (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)] + (1/2)*Sqrt[3]*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3) + (Sqrt[3]*(1 - x)^(1/6))/(1 + x)^(1/6)] + (1/2)*Log[1 - (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)] - (1/2)*Log[1 + (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)]} -{E^(ArcTanh[x]/3)/x^2, x, 13, -(((1 - x)^(5/6)*(1 + x)^(1/6))/x) + ArcTan[(1 - (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]]/Sqrt[3] - ArcTan[(1 + (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]]/Sqrt[3] - (2/3)*ArcTanh[(1 + x)^(1/6)/(1 - x)^(1/6)] + (1/6)*Log[1 - (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)] - (1/6)*Log[1 + (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)]} -{E^(ArcTanh[x]/3)/x^3, x, 14, -(((1 - x)^(5/6)*(1 + x)^(1/6))/(6*x)) - ((1 - x)^(5/6)*(1 + x)^(7/6))/(2*x^2) + ArcTan[(1 - (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) - ArcTan[(1 + (2*(1 + x)^(1/6))/(1 - x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) - (1/9)*ArcTanh[(1 + x)^(1/6)/(1 - x)^(1/6)] + (1/36)*Log[1 - (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)] - (1/36)*Log[1 + (1 + x)^(1/6)/(1 - x)^(1/6) + (1 + x)^(1/3)/(1 - x)^(1/3)]} - - -{E^(2*ArcTanh[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/3, -(1/3), 2 + m, x, -x])/(1 + m)} - -{E^(2*ArcTanh[x]/3)*x^2, x, 5, (-(11/27))*(1 - x)^(2/3)*(1 + x)^(1/3) - (1/9)*(1 - x)^(2/3)*(1 + x)^(4/3) - (1/3)*(1 - x)^(2/3)*x*(1 + x)^(4/3) + (22*ArcTan[1/Sqrt[3] - (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(27*Sqrt[3]) + (11/81)*Log[1 + x] + (11/27)*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)]} -{E^(2*ArcTanh[x]/3)*x^1, x, 4, (-(1/3))*(1 - x)^(2/3)*(1 + x)^(1/3) - (1/2)*(1 - x)^(2/3)*(1 + x)^(4/3) + (2*ArcTan[1/Sqrt[3] - (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(3*Sqrt[3]) + (1/9)*Log[1 + x] + (1/3)*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)]} -{E^(2*ArcTanh[x]/3)*x^0, x, 3, (-(1 - x)^(2/3))*(1 + x)^(1/3) + (2*ArcTan[1/Sqrt[3] - (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/Sqrt[3] + (1/3)*Log[1 + x] + Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)]} -{E^(2*ArcTanh[x]/3)/x^1, x, 4, Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))] + Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))] - Log[x]/2 + (1/2)*Log[1 + x] + (3/2)*Log[1 + (1 - x)^(1/3)/(1 + x)^(1/3)] + (3/2)*Log[(1 - x)^(1/3) - (1 + x)^(1/3)]} -{E^(2*ArcTanh[x]/3)/x^2, x, 3, -(((1 - x)^(2/3)*(1 + x)^(1/3))/x) + (2*ArcTan[1/Sqrt[3] + (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/Sqrt[3] - Log[x]/3 + Log[(1 - x)^(1/3) - (1 + x)^(1/3)]} -{E^(2*ArcTanh[x]/3)/x^3, x, 4, -(((1 - x)^(2/3)*(1 + x)^(1/3))/(3*x)) - ((1 - x)^(2/3)*(1 + x)^(4/3))/(2*x^2) + (2*ArcTan[1/Sqrt[3] + (2*(1 - x)^(1/3))/(Sqrt[3]*(1 + x)^(1/3))])/(3*Sqrt[3]) - Log[x]/9 + (1/3)*Log[(1 - x)^(1/3) - (1 + x)^(1/3)]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/4 ArcTanh[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcTanh[a*x]/4)*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, 1/8, -(1/8), 2 + m, a*x, (-a)*x])/(1 + m)} - -{E^(ArcTanh[a*x]/4)*x^2, x, 27, -((11*(1 - a*x)^(7/8)*(1 + a*x)^(1/8))/(32*a^3)) - ((1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(24*a^3) - (x*(1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(3*a^2) + (11*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(128*a^3) + (11*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(128*a^3) - (11*Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(128*a^3) - (11*Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(128*a^3) - (11*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(256*a^3) + (11*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(256*a^3) - (11*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(256*a^3) + (11*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(256*a^3)} -{E^(ArcTanh[a*x]/4)*x^1, x, 26, -(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/(8*a^2)) - ((1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(2*a^2) + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(32*a^2) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(32*a^2) - (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(32*a^2) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(32*a^2) - (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(64*a^2) + (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(64*a^2) - (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(64*a^2) + (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(64*a^2)} -{E^(ArcTanh[a*x]/4)*x^0, x, 25, -(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/a) + (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(4*a) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(4*a) - (Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]])/(4*a) - (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]])/(4*a) - (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(8*a) + (Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(8*a) - (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(8*a) + (Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)])/(8*a)} -{E^(ArcTanh[a*x]/4)/x^1, x, 39, -2*ArcTan[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] + Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] - Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 + Sqrt[2]]] - Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - a*x)^(1/8))/(1 + a*x)^(1/8))/Sqrt[2 - Sqrt[2]]] + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)] - 2*ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] - (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)] + (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)] - (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)] + (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - a*x)^(1/4)/(1 + a*x)^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - a*x)^(1/8))/(1 + a*x)^(1/8)] + Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/Sqrt[2] - Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)]/Sqrt[2]} -{E^(ArcTanh[a*x]/4)/x^2, x, 16, -(((1 - a*x)^(7/8)*(1 + a*x)^(1/8))/x) - (1/2)*a*ArcTan[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] + (a*ArcTan[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(2*Sqrt[2]) - (a*ArcTan[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(2*Sqrt[2]) - (1/2)*a*ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] + (a*Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(4*Sqrt[2]) - (a*Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(4*Sqrt[2])} -{E^(ArcTanh[a*x]/4)/x^3, x, 17, -((a*(1 - a*x)^(7/8)*(1 + a*x)^(1/8))/(8*x)) - ((1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(2*x^2) - (1/16)*a^2*ArcTan[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] + (a^2*ArcTan[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (a^2*ArcTan[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (1/16)*a^2*ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(1/8)] + (a^2*Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqrt[2]) - (a^2*Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqrt[2])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) with m symbolic*) - - -{E^(4*ArcTanh[a*x])*x^m, x, 4, x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 - a*x) - 4*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x]} -{E^(3*ArcTanh[a*x])*x^m, x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m)) - (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (4*a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{E^(2*ArcTanh[a*x])*x^m, x, 3, -(x^(1 + m)/(1 + m)) + (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)} -{E^(1*ArcTanh[a*x])*x^m, x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{E^(-ArcTanh[a*x])*x^m, x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) - (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{E^(-2*ArcTanh[a*x])*x^m, x, 3, -(x^(1 + m)/(1 + m)) + (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/(1 + m)} -{E^(-3*ArcTanh[a*x])*x^m, x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m)) + (a*x^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) - (4*a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) with n symbolic*) - - -{E^(n*ArcTanh[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[1 + m, n/2, -(n/2), 2 + m, a*x, (-a)*x])/(1 + m)} - -{E^(n*ArcTanh[a*x])*x^3, x, 4, -((x^2*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(4*a^2)) - ((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2)*(6 + n^2 + 2*a*n*x))/(24*a^4) - (2^(-2 + n/2)*n*(8 + n^2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(3*a^4*(2 - n))} -{E^(n*ArcTanh[a*x])*x^2, x, 4, -((n*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(6*a^3)) - (x*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(3*a^2) - (2^(n/2)*(2 + n^2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(3*a^3*(2 - n))} -{E^(n*ArcTanh[a*x])*x^1, x, 3, -(((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(2*a^2)) - (2^(n/2)*n*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(a^2*(2 - n))} -{E^(n*ArcTanh[a*x])*x^0, x, 2, -((2^(1 + n/2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(a*(2 - n)))} -{E^(n*ArcTanh[a*x])/x^1, x, 4, (2*(1 + a*x)^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (1 - a*x)/(1 + a*x)])/((1 - a*x)^(n/2)*n) - (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*n)} -{E^(n*ArcTanh[a*x])/x^2, x, 2, -((4*a*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/(2 - n))} -{E^(n*ArcTanh[a*x])/x^3, x, 3, -(((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(2*x^2)) - (2*a^2*n*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/(2 - n)} -{E^(n*ArcTanh[a*x])/x^4, x, 5, -(((1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(3*x^3)) - (a*n*(1 - a*x)^(1 - n/2)*(1 + a*x)^((2 + n)/2))/(6*x^2) - (2*a^3*(2 + n^2)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/(3*(2 - n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a x]) (c-a c x)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c a x)^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - a*c*x)^p, x, 3, -((2*Sqrt[2]*(c - a*c*x)^(1 + p)*Hypergeometric2F1[-(1/2), 1/2 + p, 3/2 + p, (1/2)*(1 - a*x)])/(a*c*(1 + 2*p)*Sqrt[1 - a*x]))} - -{E^ArcTanh[a*x]*(c - a*c*x)^4, x, 6, (7/8)*c^4*x*Sqrt[1 - a^2*x^2] + (7*c^4*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20*a) + (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^4*ArcSin[a*x])/(8*a)} -{E^ArcTanh[a*x]*(c - a*c*x)^3, x, 5, (5/8)*c^3*x*Sqrt[1 - a^2*x^2] + (5*c^3*(1 - a^2*x^2)^(3/2))/(12*a) + (c^3*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(4*a) + (5*c^3*ArcSin[a*x])/(8*a)} -{E^ArcTanh[a*x]*(c - a*c*x)^2, x, 4, (1/2)*c^2*x*Sqrt[1 - a^2*x^2] + (c^2*(1 - a^2*x^2)^(3/2))/(3*a) + (c^2*ArcSin[a*x])/(2*a)} -{E^ArcTanh[a*x]*(c - a*c*x), x, 3, (c*x*Sqrt[1 - a^2*x^2])/2 + (c*ArcSin[a*x])/(2*a)} -{E^ArcTanh[a*x]/(c - a*c*x), x, 3, (2*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x)) - ArcSin[a*x]/(a*c)} -{E^ArcTanh[a*x]/(c - a*c*x)^2, x, 2, (1 - a^2*x^2)^(3/2)/(3*a*c^2*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(c - a*c*x)^3, x, 3, (1 - a^2*x^2)^(3/2)/(5*a*c^3*(1 - a*x)^4) + (1 - a^2*x^2)^(3/2)/(15*a*c^3*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(c - a*c*x)^4, x, 4, (1 - a^2*x^2)^(3/2)/(7*a*c^4*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2))/(35*a*c^4*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(105*a*c^4*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(c - a*c*x)^5, x, 5, (1 - a^2*x^2)^(3/2)/(9*a*c^5*(1 - a*x)^6) + (1 - a^2*x^2)^(3/2)/(21*a*c^5*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2))/(105*a*c^5*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(315*a*c^5*(1 - a*x)^3)} - - -{E^(2*ArcTanh[a*x])*(c - a*c*x)^p, x, 4, -((2*(c - a*c*x)^p)/(a*p)) + (c - a*c*x)^(1 + p)/(a*c*(1 + p))} - -{E^(2*ArcTanh[a*x])*(c - a*c*x)^5, x, 3, (-2*c^5*(1 - a*x)^5)/(5*a) + (c^5*(1 - a*x)^6)/(6*a)} -{E^(2*ArcTanh[a*x])*(c - a*c*x)^4, x, 3, -((c^4*(1 - a*x)^4)/(2*a)) + (c^4*(1 - a*x)^5)/(5*a)} -{E^(2*ArcTanh[a*x])*(c - a*c*x)^3, x, 3, -((2*c^3*(1 - a*x)^3)/(3*a)) + (c^3*(1 - a*x)^4)/(4*a)} -{E^(2*ArcTanh[a*x])*(c - a*c*x)^2, x, 3, c^2*x - (a^2*c^2*x^3)/3} -{E^(2*ArcTanh[a*x])*(c - a*c*x), x, 1, c*x + (a*c*x^2)/2, (c*E^(2*ArcTanh[a*x])*(1 - a^2*x^2))/(2*a)} -{E^(2*ArcTanh[a*x])/(c - a*c*x), x, 3, 2/(a*c*(1 - a*x)) + Log[1 - a*x]/(a*c)} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^2, x, 2, x/(c^2*(1 - a*x)^2)} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^3, x, 3, 2/(3*a*c^3*(1 - a*x)^3) - 1/(2*a*c^3*(1 - a*x)^2)} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^4, x, 3, 1/(2*a*c^4*(1 - a*x)^4) - 1/(3*a*c^4*(1 - a*x)^3)} - - -{E^(3*ArcTanh[a*x])*(c - a*c*x)^p, x, 3, (4*Sqrt[2]*(c - a*c*x)^(1 + p)*Hypergeometric2F1[-(3/2), -(1/2) + p, 1/2 + p, (1/2)*(1 - a*x)])/(a*c*(1 - 2*p)*(1 - a*x)^(3/2))} - -{E^(3*ArcTanh[a*x])*(c - a*c*x)^4, x, 5, (3/8)*c^4*x*Sqrt[1 - a^2*x^2] + (1/4)*c^4*x*(1 - a^2*x^2)^(3/2) + (c^4*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^4*ArcSin[a*x])/(8*a)} -{E^(3*ArcTanh[a*x])*(c - a*c*x)^3, x, 4, (3*c^3*x*Sqrt[1 - a^2*x^2])/8 + (c^3*x*(1 - a^2*x^2)^(3/2))/4 + (3*c^3*ArcSin[a*x])/(8*a)} -{E^(3*ArcTanh[a*x])*(c - a*c*x)^2, x, 4, (1/2)*c^2*x*Sqrt[1 - a^2*x^2] - (c^2*(1 - a^2*x^2)^(3/2))/(3*a) + (c^2*ArcSin[a*x])/(2*a)} -{E^(3*ArcTanh[a*x])*(c - a*c*x), x, 4, -((3*c*Sqrt[1 - a^2*x^2])/(2*a)) - (c*(1 - a^2*x^2)^(3/2))/(2*a*(1 - a*x)) + (3*c*ArcSin[a*x])/(2*a)} -{E^(3*ArcTanh[a*x])/(c - a*c*x), x, 4, -((2*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x))) + (2*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^3) + ArcSin[a*x]/(a*c)} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^2, x, 2, (1 - a^2*x^2)^(5/2)/(5*a*c^2*(1 - a*x)^5)} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^3, x, 3, (1 - a^2*x^2)^(5/2)/(7*a*c^3*(1 - a*x)^6) + (1 - a^2*x^2)^(5/2)/(35*a*c^3*(1 - a*x)^5)} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^4, x, 4, (1 - a^2*x^2)^(5/2)/(9*a*c^4*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/2))/(63*a*c^4*(1 - a*x)^6) + (2*(1 - a^2*x^2)^(5/2))/(315*a*c^4*(1 - a*x)^5)} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^5, x, 5, (1 - a^2*x^2)^(5/2)/(11*a*c^5*(1 - a*x)^8) + (1 - a^2*x^2)^(5/2)/(33*a*c^5*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/2))/(231*a*c^5*(1 - a*x)^6) + (2*(1 - a^2*x^2)^(5/2))/(1155*a*c^5*(1 - a*x)^5)} - - -{E^(4*ArcTanh[a*x])*(c - a*c*x)^p, x, 4, (4*c*(c - a*c*x)^(-1 + p))/(a*(1 - p)) + (4*(c - a*c*x)^p)/(a*p) - (c - a*c*x)^(1 + p)/(a*c*(1 + p))} - -{E^(4*ArcTanh[a*x])*(c - a*c*x)^5, x, 3, -((c^5*(1 - a*x)^4)/a) + (4*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)} -{E^(4*ArcTanh[a*x])*(c - a*c*x)^4, x, 4, c^4*x - (2*a^2*c^4*x^3)/3 + (a^4*c^4*x^5)/5} -{E^(4*ArcTanh[a*x])*(c - a*c*x)^3, x, 3, (2*c^3*(1 + a*x)^3)/(3*a) - (c^3*(1 + a*x)^4)/(4*a)} -{E^(4*ArcTanh[a*x])*(c - a*c*x)^2, x, 2, (c^2*(1 + a*x)^3)/(3*a)} -{E^(4*ArcTanh[a*x])*(c - a*c*x), x, 3, -3*c*x - (1/2)*a*c*x^2 - (4*c*Log[1 - a*x])/a} -{E^(4*ArcTanh[a*x])/(c - a*c*x), x, 3, 2/(a*c*(1 - a*x)^2) - 4/(a*c*(1 - a*x)) - Log[1 - a*x]/(a*c)} -{E^(4*ArcTanh[a*x])/(c - a*c*x)^2, x, 2, (1 + a*x)^3/(6*a*c^2*(1 - a*x)^3)} -{E^(4*ArcTanh[a*x])/(c - a*c*x)^3, x, 3, 1/(a*c^3*(1 - a*x)^4) - 4/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)} -{E^(4*ArcTanh[a*x])/(c - a*c*x)^4, x, 3, 4/(5*a*c^4*(1 - a*x)^5) - 1/(a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a*c*x)^p/E^ArcTanh[a*x], x, 3, -((Sqrt[2]*Sqrt[1 - a*x]*(c - a*c*x)^(1 + p)*Hypergeometric2F1[1/2, 3/2 + p, 5/2 + p, (1/2)*(1 - a*x)])/(a*c*(3 + 2*p)))} - -{(c - a*c*x)^3/E^ArcTanh[a*x], x, 6, (35*c^3*Sqrt[1 - a^2*x^2])/(8*a) + (35*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/(24*a) + (7*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(12*a) + (c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) + (35*c^3*ArcSin[a*x])/(8*a)} -{(c - a*c*x)^2/E^ArcTanh[a*x], x, 5, (5*c^2*Sqrt[1 - a^2*x^2])/(2*a) + (5*c^2*(1 - a*x)*Sqrt[1 - a^2*x^2])/(6*a) + (c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) + (5*c^2*ArcSin[a*x])/(2*a)} -{(c - a*c*x)/E^ArcTanh[a*x], x, 4, (3*c*Sqrt[1 - a^2*x^2])/(2*a) + (c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(2*a) + (3*c*ArcSin[a*x])/(2*a)} -{1/(E^ArcTanh[a*x]*(c - a*c*x)), x, 2, ArcSin[a*x]/(a*c)} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^2), x, 2, Sqrt[1 - a^2*x^2]/(a*c^2*(1 - a*x))} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^3), x, 3, Sqrt[1 - a^2*x^2]/(3*a*c^3*(1 - a*x)^2) + Sqrt[1 - a^2*x^2]/(3*a*c^3*(1 - a*x))} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^4), x, 4, Sqrt[1 - a^2*x^2]/(5*a*c^4*(1 - a*x)^3) + (2*Sqrt[1 - a^2*x^2])/(15*a*c^4*(1 - a*x)^2) + (2*Sqrt[1 - a^2*x^2])/(15*a*c^4*(1 - a*x))} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^5), x, 5, Sqrt[1 - a^2*x^2]/(7*a*c^5*(1 - a*x)^4) + (3*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x)^3) + (2*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x)^2) + (2*Sqrt[1 - a^2*x^2])/(35*a*c^5*(1 - a*x))} - - -{(c - a*c*x)^p/E^(2*ArcTanh[a*x]), x, 3, -(((c - a*c*x)^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (1/2)*(1 - a*x)])/(2*a*c^2*(2 + p)))} - -{(c - a*c*x)^4/E^(2*ArcTanh[a*x]), x, 3, -16*c^4*x + (4*c^4*(1 - a*x)^2)/a + (4*c^4*(1 - a*x)^3)/(3*a) + (c^4*(1 - a*x)^4)/(2*a) + (c^4*(1 - a*x)^5)/(5*a) + (32*c^4*Log[1 + a*x])/a} -{(c - a*c*x)^3/E^(2*ArcTanh[a*x]), x, 3, -8*c^3*x + (2*c^3*(1 - a*x)^2)/a + (2*c^3*(1 - a*x)^3)/(3*a) + (c^3*(1 - a*x)^4)/(4*a) + (16*c^3*Log[1 + a*x])/a} -{(c - a*c*x)^2/E^(2*ArcTanh[a*x]), x, 3, -4*c^2*x + (c^2*(1 - a*x)^2)/a + (c^2*(1 - a*x)^3)/(3*a) + (8*c^2*Log[1 + a*x])/a} -{(c - a*c*x)/E^(2*ArcTanh[a*x]), x, 3, -3*c*x + (1/2)*a*c*x^2 + (4*c*Log[1 + a*x])/a} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)), x, 2, Log[1 + a*x]/(a*c)} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^2), x, 3, ArcTanh[a*x]/(a*c^2)} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^3), x, 4, 1/(2*a*c^3*(1 - a*x)) + ArcTanh[a*x]/(2*a*c^3)} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^4), x, 4, 1/(4*a*c^4*(1 - a*x)^2) + 1/(4*a*c^4*(1 - a*x)) + ArcTanh[a*x]/(4*a*c^4)} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^5), x, 4, 1/(6*a*c^5*(1 - a*x)^3) + 1/(8*a*c^5*(1 - a*x)^2) + 1/(8*a*c^5*(1 - a*x)) + ArcTanh[a*x]/(8*a*c^5)} - - -{(c - a*c*x)^p/E^(3*ArcTanh[a*x]), x, 3, -(((1 - a*x)^(3/2)*(c - a*c*x)^(1 + p)*Hypergeometric2F1[3/2, 5/2 + p, 7/2 + p, (1/2)*(1 - a*x)])/(Sqrt[2]*a*c*(5 + 2*p)))} - -{(c - a*c*x)^3/E^(3*ArcTanh[a*x]), x, 7, -((2*c^3*(1 - a*x)^5)/(a*Sqrt[1 - a^2*x^2])) - (315*c^3*Sqrt[1 - a^2*x^2])/(8*a) - (105*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/(8*a) - (21*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(4*a) - (9*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2])/(4*a) - (315*c^3*ArcSin[a*x])/(8*a)} -{(c - a*c*x)^2/E^(3*ArcTanh[a*x]), x, 6, -((2*c^2*(1 - a*x)^4)/(a*Sqrt[1 - a^2*x^2])) - (35*c^2*Sqrt[1 - a^2*x^2])/(2*a) - (35*c^2*(1 - a*x)*Sqrt[1 - a^2*x^2])/(6*a) - (7*c^2*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) - (35*c^2*ArcSin[a*x])/(2*a)} -{(c - a*c*x)/E^(3*ArcTanh[a*x]), x, 5, -((2*c*(1 - a*x)^3)/(a*Sqrt[1 - a^2*x^2])) - (15*c*Sqrt[1 - a^2*x^2])/(2*a) - (5*c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(2*a) - (15*c*ArcSin[a*x])/(2*a)} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)), x, 3, -((2*(1 - a*x))/(a*c*Sqrt[1 - a^2*x^2])) - ArcSin[a*x]/(a*c)} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^2), x, 2, -((1 - a*x)/(a*c^2*Sqrt[1 - a^2*x^2]))} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^3), x, 2, x/(c^3*Sqrt[1 - a^2*x^2])} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^4), x, 3, (2*x)/(3*c^4*Sqrt[1 - a^2*x^2]) + 1/(3*a*c^4*(1 - a*x)*Sqrt[1 - a^2*x^2])} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^5), x, 4, (2*x)/(5*c^5*Sqrt[1 - a^2*x^2]) + 1/(5*a*c^5*(1 - a*x)^2*Sqrt[1 - a^2*x^2]) + 1/(5*a*c^5*(1 - a*x)*Sqrt[1 - a^2*x^2])} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^6), x, 5, (8*x)/(35*c^6*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^6*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)^2*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^6*(1 - a*x)*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c a x)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - a*c*x)^(9/2), x, 6, (4096*c^6*(1 - a^2*x^2)^(3/2))/(3465*a*(c - a*c*x)^(3/2)) + (1024*c^5*(1 - a^2*x^2)^(3/2))/(1155*a*Sqrt[c - a*c*x]) + (128*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(231*a) + (32*c^3*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(99*a) + (2*c^2*(c - a*c*x)^(5/2)*(1 - a^2*x^2)^(3/2))/(11*a)} -{E^ArcTanh[a*x]*(c - a*c*x)^(7/2), x, 5, (256*c^5*(1 - a^2*x^2)^(3/2))/(315*a*(c - a*c*x)^(3/2)) + (64*c^4*(1 - a^2*x^2)^(3/2))/(105*a*Sqrt[c - a*c*x]) + (8*c^3*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(21*a) + (2*c^2*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(9*a)} -{E^ArcTanh[a*x]*(c - a*c*x)^(5/2), x, 4, (64*c^4*(1 - a^2*x^2)^(3/2))/(105*a*(c - a*c*x)^(3/2)) + (16*c^3*(1 - a^2*x^2)^(3/2))/(35*a*Sqrt[c - a*c*x]) + (2*c^2*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(7*a)} -{E^ArcTanh[a*x]*(c - a*c*x)^(3/2), x, 3, (8*c^3*(1 - a^2*x^2)^(3/2))/(15*a*(c - a*c*x)^(3/2)) + (2*c^2*(1 - a^2*x^2)^(3/2))/(5*a*Sqrt[c - a*c*x])} -{E^ArcTanh[a*x]*Sqrt[c - a*c*x], x, 2, (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2))} -{E^ArcTanh[a*x]/Sqrt[c - a*c*x], x, 4, -((2*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x])) + (2*Sqrt[2]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*Sqrt[c])} -{E^ArcTanh[a*x]/(c - a*c*x)^(3/2), x, 4, Sqrt[1 - a^2*x^2]/(a*(c - a*c*x)^(3/2)) - ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(Sqrt[2]*a*c^(3/2))} -{E^ArcTanh[a*x]/(c - a*c*x)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(2*a*(c - a*c*x)^(5/2)) - Sqrt[1 - a^2*x^2]/(8*a*c*(c - a*c*x)^(3/2)) - ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(8*Sqrt[2]*a*c^(5/2))} -{E^ArcTanh[a*x]/(c - a*c*x)^(7/2), x, 6, Sqrt[1 - a^2*x^2]/(3*a*(c - a*c*x)^(7/2)) - Sqrt[1 - a^2*x^2]/(24*a*c*(c - a*c*x)^(5/2)) - Sqrt[1 - a^2*x^2]/(32*a*c^2*(c - a*c*x)^(3/2)) - ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(32*Sqrt[2]*a*c^(7/2))} - - -{E^(2*ArcTanh[a*x])*(c - a*c*x)^(7/2), x, 4, -((4*(c - a*c*x)^(7/2))/(7*a)) + (2*(c - a*c*x)^(9/2))/(9*a*c)} -{E^(2*ArcTanh[a*x])*(c - a*c*x)^(5/2), x, 4, -((4*(c - a*c*x)^(5/2))/(5*a)) + (2*(c - a*c*x)^(7/2))/(7*a*c)} -{E^(2*ArcTanh[a*x])*(c - a*c*x)^(3/2), x, 4, -((4*(c - a*c*x)^(3/2))/(3*a)) + (2*(c - a*c*x)^(5/2))/(5*a*c)} -{E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x], x, 4, -((4*Sqrt[c - a*c*x])/a) + (2*(c - a*c*x)^(3/2))/(3*a*c)} -{E^(2*ArcTanh[a*x])/Sqrt[c - a*c*x], x, 4, 4/(a*Sqrt[c - a*c*x]) + (2*Sqrt[c - a*c*x])/(a*c)} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^(3/2), x, 4, 4/(3*a*(c - a*c*x)^(3/2)) - 2/(a*c*Sqrt[c - a*c*x])} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^(5/2), x, 4, 4/(5*a*(c - a*c*x)^(5/2)) - 2/(3*a*c*(c - a*c*x)^(3/2))} -{E^(2*ArcTanh[a*x])/(c - a*c*x)^(7/2), x, 4, 4/(7*a*(c - a*c*x)^(7/2)) - 2/(5*a*c*(c - a*c*x)^(5/2))} - - -{E^(3*ArcTanh[a*x])*(c - a*c*x)^(9/2), x, 5, (256*c^7*(1 - a^2*x^2)^(5/2))/(1155*a*(c - a*c*x)^(5/2)) + (64*c^6*(1 - a^2*x^2)^(5/2))/(231*a*(c - a*c*x)^(3/2)) + (8*c^5*(1 - a^2*x^2)^(5/2))/(33*a*Sqrt[c - a*c*x]) + (2*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(5/2))/(11*a)} -{E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2), x, 4, (64*c^6*(1 - a^2*x^2)^(5/2))/(315*a*(c - a*c*x)^(5/2)) + (16*c^5*(1 - a^2*x^2)^(5/2))/(63*a*(c - a*c*x)^(3/2)) + (2*c^4*(1 - a^2*x^2)^(5/2))/(9*a*Sqrt[c - a*c*x])} -{E^(3*ArcTanh[a*x])*(c - a*c*x)^(5/2), x, 3, (8*c^5*(1 - a^2*x^2)^(5/2))/(35*a*(c - a*c*x)^(5/2)) + (2*c^4*(1 - a^2*x^2)^(5/2))/(7*a*(c - a*c*x)^(3/2))} -{E^(3*ArcTanh[a*x])*(c - a*c*x)^(3/2), x, 2, (2*c^4*(1 - a^2*x^2)^(5/2))/(5*a*(c - a*c*x)^(5/2))} -{E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x], x, 5, -((4*c*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x])) - (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2)) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/a} -{E^(3*ArcTanh[a*x])/Sqrt[c - a*c*x], x, 5, (3*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x]) + (c^2*(1 - a^2*x^2)^(3/2))/(a*(c - a*c*x)^(5/2)) - (3*Sqrt[2]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*Sqrt[c])} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^(3/2), x, 5, -((3*Sqrt[1 - a^2*x^2])/(4*a*(c - a*c*x)^(3/2))) + (c^2*(1 - a^2*x^2)^(3/2))/(2*a*(c - a*c*x)^(7/2)) + (3*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(4*Sqrt[2]*a*c^(3/2))} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^(5/2), x, 6, -(Sqrt[1 - a^2*x^2]/(4*a*(c - a*c*x)^(5/2))) + Sqrt[1 - a^2*x^2]/(16*a*c*(c - a*c*x)^(3/2)) + (c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(9/2)) + ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(16*Sqrt[2]*a*c^(5/2))} -{E^(3*ArcTanh[a*x])/(c - a*c*x)^(7/2), x, 7, -(Sqrt[1 - a^2*x^2]/(8*a*(c - a*c*x)^(7/2))) + Sqrt[1 - a^2*x^2]/(64*a*c*(c - a*c*x)^(5/2)) + (3*Sqrt[1 - a^2*x^2])/(256*a*c^2*(c - a*c*x)^(3/2)) + (c^2*(1 - a^2*x^2)^(3/2))/(4*a*(c - a*c*x)^(11/2)) + (3*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(256*Sqrt[2]*a*c^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a*c*x)^(9/2)/E^ArcTanh[a*x], x, 7, (16384*c^5*Sqrt[1 - a^2*x^2])/(693*a*Sqrt[c - a*c*x]) + (4096*c^4*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(693*a) + (512*c^3*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(231*a) + (640*c^2*(c - a*c*x)^(5/2)*Sqrt[1 - a^2*x^2])/(693*a) + (40*c*(c - a*c*x)^(7/2)*Sqrt[1 - a^2*x^2])/(99*a) + (2*(c - a*c*x)^(9/2)*Sqrt[1 - a^2*x^2])/(11*a)} -{(c - a*c*x)^(7/2)/E^ArcTanh[a*x], x, 6, (4096*c^4*Sqrt[1 - a^2*x^2])/(315*a*Sqrt[c - a*c*x]) + (1024*c^3*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(315*a) + (128*c^2*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(105*a) + (32*c*(c - a*c*x)^(5/2)*Sqrt[1 - a^2*x^2])/(63*a) + (2*(c - a*c*x)^(7/2)*Sqrt[1 - a^2*x^2])/(9*a)} -{(c - a*c*x)^(5/2)/E^ArcTanh[a*x], x, 5, (256*c^3*Sqrt[1 - a^2*x^2])/(35*a*Sqrt[c - a*c*x]) + (64*c^2*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(35*a) + (24*c*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(35*a) + (2*(c - a*c*x)^(5/2)*Sqrt[1 - a^2*x^2])/(7*a)} -{(c - a*c*x)^(3/2)/E^ArcTanh[a*x], x, 4, (64*c^2*Sqrt[1 - a^2*x^2])/(15*a*Sqrt[c - a*c*x]) + (16*c*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(15*a) + (2*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(5*a)} -{Sqrt[c - a*c*x]/E^ArcTanh[a*x], x, 3, (8*c*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a*c*x]) + (2*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(3*a)} -{1/(E^ArcTanh[a*x]*Sqrt[c - a*c*x]), x, 2, (2*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x])} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^(3/2)), x, 3, (Sqrt[2]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(a*c^(3/2))} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^(5/2)), x, 4, Sqrt[1 - a^2*x^2]/(2*a*c*(c - a*c*x)^(3/2)) + ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(2*Sqrt[2]*a*c^(5/2))} -{1/(E^ArcTanh[a*x]*(c - a*c*x)^(7/2)), x, 5, Sqrt[1 - a^2*x^2]/(4*a*c*(c - a*c*x)^(5/2)) + (3*Sqrt[1 - a^2*x^2])/(16*a*c^2*(c - a*c*x)^(3/2)) + (3*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(16*Sqrt[2]*a*c^(7/2))} - - -{(c - a*c*x)^(7/2)/E^(2*ArcTanh[a*x]), x, 9, (32*c^3*Sqrt[c - a*c*x])/a + (16*c^2*(c - a*c*x)^(3/2))/(3*a) + (8*c*(c - a*c*x)^(5/2))/(5*a) + (4*(c - a*c*x)^(7/2))/(7*a) + (2*(c - a*c*x)^(9/2))/(9*a*c) - (32*Sqrt[2]*c^(7/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{(c - a*c*x)^(5/2)/E^(2*ArcTanh[a*x]), x, 8, (16*c^2*Sqrt[c - a*c*x])/a + (8*c*(c - a*c*x)^(3/2))/(3*a) + (4*(c - a*c*x)^(5/2))/(5*a) + (2*(c - a*c*x)^(7/2))/(7*a*c) - (16*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{(c - a*c*x)^(3/2)/E^(2*ArcTanh[a*x]), x, 7, (8*c*Sqrt[c - a*c*x])/a + (4*(c - a*c*x)^(3/2))/(3*a) + (2*(c - a*c*x)^(5/2))/(5*a*c) - (8*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - a*c*x]/E^(2*ArcTanh[a*x]), x, 6, (4*Sqrt[c - a*c*x])/a + (2*(c - a*c*x)^(3/2))/(3*a*c) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{1/(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x]), x, 5, (2*Sqrt[c - a*c*x])/(a*c) - (2*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(3/2)), x, 4, -((Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2)))} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(5/2)), x, 5, 1/(a*c^2*Sqrt[c - a*c*x]) - ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(Sqrt[2]*a*c^(5/2))} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(7/2)), x, 6, 1/(3*a*c^2*(c - a*c*x)^(3/2)) + 1/(2*a*c^3*Sqrt[c - a*c*x]) - ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(2*Sqrt[2]*a*c^(7/2))} -{1/(E^(2*ArcTanh[a*x])*(c - a*c*x)^(9/2)), x, 7, 1/(5*a*c^2*(c - a*c*x)^(5/2)) + 1/(6*a*c^3*(c - a*c*x)^(3/2)) + 1/(4*a*c^4*Sqrt[c - a*c*x]) - ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))} - - -{(c - a*c*x)^(5/2)/E^(3*ArcTanh[a*x]), x, 6, -((4096*c^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - a^2*x^2])) + (1024*c*(c - a*c*x)^(3/2))/(35*a*Sqrt[1 - a^2*x^2]) + (128*(c - a*c*x)^(5/2))/(35*a*Sqrt[1 - a^2*x^2]) + (32*(c - a*c*x)^(7/2))/(35*a*c*Sqrt[1 - a^2*x^2]) + (2*(c - a*c*x)^(9/2))/(7*a*c^2*Sqrt[1 - a^2*x^2])} -{(c - a*c*x)^(3/2)/E^(3*ArcTanh[a*x]), x, 5, -((256*c*Sqrt[c - a*c*x])/(5*a*Sqrt[1 - a^2*x^2])) + (64*(c - a*c*x)^(3/2))/(5*a*Sqrt[1 - a^2*x^2]) + (8*(c - a*c*x)^(5/2))/(5*a*c*Sqrt[1 - a^2*x^2]) + (2*(c - a*c*x)^(7/2))/(5*a*c^2*Sqrt[1 - a^2*x^2])} -{Sqrt[c - a*c*x]/E^(3*ArcTanh[a*x]), x, 4, -((64*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - a^2*x^2])) + (16*(c - a*c*x)^(3/2))/(3*a*c*Sqrt[1 - a^2*x^2]) + (2*(c - a*c*x)^(5/2))/(3*a*c^2*Sqrt[1 - a^2*x^2])} -{1/(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x]), x, 3, -((8*Sqrt[c - a*c*x])/(a*c*Sqrt[1 - a^2*x^2])) + (2*(c - a*c*x)^(3/2))/(a*c^2*Sqrt[1 - a^2*x^2])} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(3/2)), x, 2, -((2*Sqrt[c - a*c*x])/(a*c^2*Sqrt[1 - a^2*x^2]))} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(5/2)), x, 4, -(Sqrt[c - a*c*x]/(a*c^3*Sqrt[1 - a^2*x^2])) + ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])]/(Sqrt[2]*a*c^(5/2))} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(7/2)), x, 5, 1/(2*a*c^3*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2]) - (3*Sqrt[c - a*c*x])/(4*a*c^4*Sqrt[1 - a^2*x^2]) + (3*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(4*Sqrt[2]*a*c^(7/2))} -{1/(E^(3*ArcTanh[a*x])*(c - a*c*x)^(9/2)), x, 6, 1/(4*a*c^3*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2]) + 5/(16*a*c^4*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2]) - (15*Sqrt[c - a*c*x])/(32*a*c^5*Sqrt[1 - a^2*x^2]) + (15*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/(32*Sqrt[2]*a*c^(9/2))} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{E^(n*ArcTanh[a*x])*(c - a*c*x)^(7/2), x, 3, -((2^(1 + n/2)*(c - a*c*x)^(9/2)*Hypergeometric2F1[(9 - n)/2, -(n/2), (11 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(9 - n))))} -{E^(n*ArcTanh[a*x])*(c - a*c*x)^(5/2), x, 3, -((2^(1 + n/2)*(c - a*c*x)^(7/2)*Hypergeometric2F1[(7 - n)/2, -(n/2), (9 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(7 - n))))} -{E^(n*ArcTanh[a*x])*(c - a*c*x)^(3/2), x, 3, -((2^(1 + n/2)*(c - a*c*x)^(5/2)*Hypergeometric2F1[(5 - n)/2, -(n/2), (7 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(5 - n))))} -{E^(n*ArcTanh[a*x])*Sqrt[c - a*c*x], x, 3, -((2^(1 + n/2)*(c - a*c*x)^(3/2)*Hypergeometric2F1[(3 - n)/2, -(n/2), (5 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(3 - n))))} -{E^(n*ArcTanh[a*x])/Sqrt[c - a*c*x], x, 3, -((2^(1 + n/2)*Sqrt[c - a*c*x]*Hypergeometric2F1[(1 - n)/2, -(n/2), (3 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(1 - n))))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^(3/2), x, 3, (2^(1 + n/2)*Hypergeometric2F1[(1/2)*(-1 - n), -(n/2), (1 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(1 + n)*Sqrt[c - a*c*x]))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^(5/2), x, 3, (2^(1 + n/2)*Hypergeometric2F1[(1/2)*(-3 - n), -(n/2), (1/2)*(-1 - n), (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(3 + n)*(c - a*c*x)^(3/2)))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^(7/2), x, 3, (2^(1 + n/2)*Hypergeometric2F1[(1/2)*(-5 - n), -(n/2), (1/2)*(-3 - n), (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(5 + n)*(c - a*c*x)^(5/2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (c-c a x)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^ArcTanh[a*x]*x^4*(c - a*c*x), x, 5, -(c*x*Sqrt[1 - a^2*x^2])/(16*a^4) - (c*x^3*Sqrt[1 - a^2*x^2])/(24*a^2) + (c*x^5*Sqrt[1 - a^2*x^2])/6 + (c*ArcSin[a*x])/(16*a^5)} -{E^ArcTanh[a*x]*x^3*(c - a*c*x), x, 4, -(c*(1 - a^2*x^2)^(3/2))/(3*a^4) + (c*(1 - a^2*x^2)^(5/2))/(5*a^4)} -{E^ArcTanh[a*x]*x^2*(c - a*c*x), x, 4, -(c*x*Sqrt[1 - a^2*x^2])/(8*a^2) + (c*x^3*Sqrt[1 - a^2*x^2])/4 + (c*ArcSin[a*x])/(8*a^3)} -{E^ArcTanh[a*x]*x*(c - a*c*x), x, 2, -(c*(1 - a^2*x^2)^(3/2))/(3*a^2)} -{E^ArcTanh[a*x]*(c - a*c*x), x, 3, (c*x*Sqrt[1 - a^2*x^2])/2 + (c*ArcSin[a*x])/(2*a)} -{(E^ArcTanh[a*x]*(c - a*c*x))/x, x, 5, c*Sqrt[1 - a^2*x^2] - c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x))/x^2, x, 3, -((c*Sqrt[1 - a^2*x^2])/x) - a*c*ArcSin[a*x]} -{(E^ArcTanh[a*x]*(c - a*c*x))/x^3, x, 5, -(c*Sqrt[1 - a^2*x^2])/(2*x^2) + (a^2*c*ArcTanh[Sqrt[1 - a^2*x^2]])/2} -{(E^ArcTanh[a*x]*(c - a*c*x))/x^4, x, 2, -(c*(1 - a^2*x^2)^(3/2))/(3*x^3)} - - -{E^ArcTanh[a*x]*x^3*(c - a*c*x)^2, x, 6, -((c^2*x*Sqrt[1 - a^2*x^2])/(16*a^3)) - (c^2*x^2*(1 - a^2*x^2)^(3/2))/(5*a^2) + (c^2*x^3*(1 - a^2*x^2)^(3/2))/(6*a) - (c^2*(16 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^4) - (c^2*ArcSin[a*x])/(16*a^4)} -{E^ArcTanh[a*x]*x^2*(c - a*c*x)^2, x, 9, (c^2*x*Sqrt[1 - a^2*x^2])/(8*a^2) + (c^2*(1 - a^2*x^2)^(3/2))/(3*a^3) - (c^2*x*(1 - a^2*x^2)^(3/2))/(4*a^2) - (c^2*(1 - a^2*x^2)^(5/2))/(5*a^3) + (c^2*ArcSin[a*x])/(8*a^3)} -{E^ArcTanh[a*x]*x*(c - a*c*x)^2, x, 4, -((c^2*x*Sqrt[1 - a^2*x^2])/(8*a)) - (c^2*(4 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(12*a^2) - (c^2*ArcSin[a*x])/(8*a^2)} -{E^ArcTanh[a*x]*(c - a*c*x)^2, x, 4, (c^2*x*Sqrt[1 - a^2*x^2])/2 + (c^2*(1 - a^2*x^2)^(3/2))/(3*a) + (c^2*ArcSin[a*x])/(2*a)} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x, x, 7, (1/2)*c^2*(2 - a*x)*Sqrt[1 - a^2*x^2] - (1/2)*c^2*ArcSin[a*x] - c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^2, x, 7, -((c^2*(1 + a*x)*Sqrt[1 - a^2*x^2])/x) - a*c^2*ArcSin[a*x] + a*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^3, x, 7, -((c^2*(1 - 2*a*x)*Sqrt[1 - a^2*x^2])/(2*x^2)) + a^2*c^2*ArcSin[a*x] + (1/2)*a^2*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^4, x, 6, (a*c^2*Sqrt[1 - a^2*x^2])/(2*x^2) - (c^2*(1 - a^2*x^2)^(3/2))/(3*x^3) - (1/2)*a^3*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^5, x, 7, -((a^2*c^2*Sqrt[1 - a^2*x^2])/(8*x^2)) - (c^2*(1 - a^2*x^2)^(3/2))/(4*x^4) + (a*c^2*(1 - a^2*x^2)^(3/2))/(3*x^3) + (1/8)*a^4*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^6, x, 8, (a^3*c^2*Sqrt[1 - a^2*x^2])/(8*x^2) - (c^2*(1 - a^2*x^2)^(3/2))/(5*x^5) + (a*c^2*(1 - a^2*x^2)^(3/2))/(4*x^4) - (2*a^2*c^2*(1 - a^2*x^2)^(3/2))/(15*x^3) - (1/8)*a^5*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^7, x, 9, -((a^4*c^2*Sqrt[1 - a^2*x^2])/(16*x^2)) - (c^2*(1 - a^2*x^2)^(3/2))/(6*x^6) + (a*c^2*(1 - a^2*x^2)^(3/2))/(5*x^5) - (a^2*c^2*(1 - a^2*x^2)^(3/2))/(8*x^4) + (2*a^3*c^2*(1 - a^2*x^2)^(3/2))/(15*x^3) + (1/16)*a^6*c^2*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^ArcTanh[a*x]*x^3*(c - a*c*x)^3, x, 7, -((c^3*x*Sqrt[1 - a^2*x^2])/(8*a^3)) - (11*c^3*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (c^3*x^3*(1 - a^2*x^2)^(3/2))/(3*a) - (1/7)*c^3*x^4*(1 - a^2*x^2)^(3/2) - (c^3*(88 - 105*a*x)*(1 - a^2*x^2)^(3/2))/(420*a^4) - (c^3*ArcSin[a*x])/(8*a^4)} -{E^ArcTanh[a*x]*x^2*(c - a*c*x)^3, x, 6, (3*c^3*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (2*c^3*x^2*(1 - a^2*x^2)^(3/2))/(5*a) - (1/6)*c^3*x^3*(1 - a^2*x^2)^(3/2) + (c^3*(32 - 45*a*x)*(1 - a^2*x^2)^(3/2))/(120*a^3) + (3*c^3*ArcSin[a*x])/(16*a^3)} -{E^ArcTanh[a*x]*x*(c - a*c*x)^3, x, 5, -((c^3*x*Sqrt[1 - a^2*x^2])/(4*a)) - (1/5)*c^3*x^2*(1 - a^2*x^2)^(3/2) - (c^3*(14 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(30*a^2) - (c^3*ArcSin[a*x])/(4*a^2)} -{E^ArcTanh[a*x]*(c - a*c*x)^3, x, 5, (5/8)*c^3*x*Sqrt[1 - a^2*x^2] + (5*c^3*(1 - a^2*x^2)^(3/2))/(12*a) + (c^3*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(4*a) + (5*c^3*ArcSin[a*x])/(8*a)} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x, x, 8, c^3*(1 - a*x)*Sqrt[1 - a^2*x^2] - (1/3)*c^3*(1 - a^2*x^2)^(3/2) - c^3*ArcSin[a*x] - c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^2, x, 8, (-(1/2))*a*c^3*(4 + a*x)*Sqrt[1 - a^2*x^2] - (c^3*(1 - a^2*x^2)^(3/2))/x - (1/2)*a*c^3*ArcSin[a*x] + 2*a*c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^3, x, 8, (a*c^3*(4 + a*x)*Sqrt[1 - a^2*x^2])/(2*x) - (c^3*(1 - a^2*x^2)^(3/2))/(2*x^2) + 2*a^2*c^3*ArcSin[a*x] - (1/2)*a^2*c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^4, x, 8, (a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])/x^2 - (c^3*(1 - a^2*x^2)^(3/2))/(3*x^3) - a^3*c^3*ArcSin[a*x] - a^3*c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^5, x, 7, -((5*a^2*c^3*Sqrt[1 - a^2*x^2])/(8*x^2)) - (c^3*(1 - a^2*x^2)^(3/2))/(4*x^4) + (2*a*c^3*(1 - a^2*x^2)^(3/2))/(3*x^3) + (5/8)*a^4*c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^3)/x^6, x, 8, (a^3*c^3*Sqrt[1 - a^2*x^2])/(4*x^2) - (c^3*(1 - a^2*x^2)^(3/2))/(5*x^5) + (a*c^3*(1 - a^2*x^2)^(3/2))/(2*x^4) - (7*a^2*c^3*(1 - a^2*x^2)^(3/2))/(15*x^3) - (1/4)*a^5*c^3*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^ArcTanh[a*x]*x^3*(c - a*c*x)^4, x, 8, -((29*c^4*x*Sqrt[1 - a^2*x^2])/(128*a^3)) - (19*c^4*x^2*(1 - a^2*x^2)^(3/2))/(35*a^2) + (29*c^4*x^3*(1 - a^2*x^2)^(3/2))/(48*a) - (3/7)*c^4*x^4*(1 - a^2*x^2)^(3/2) + (1/8)*a*c^4*x^5*(1 - a^2*x^2)^(3/2) - (c^4*(2432 - 3045*a*x)*(1 - a^2*x^2)^(3/2))/(6720*a^4) - (29*c^4*ArcSin[a*x])/(128*a^4)} -{E^ArcTanh[a*x]*x^2*(c - a*c*x)^4, x, 7, (5*c^4*x*Sqrt[1 - a^2*x^2])/(16*a^2) + (5*c^4*x^2*(1 - a^2*x^2)^(3/2))/(7*a) - (1/2)*c^4*x^3*(1 - a^2*x^2)^(3/2) + (1/7)*a*c^4*x^4*(1 - a^2*x^2)^(3/2) + (5*c^4*(16 - 21*a*x)*(1 - a^2*x^2)^(3/2))/(168*a^3) + (5*c^4*ArcSin[a*x])/(16*a^3)} -{E^ArcTanh[a*x]*x*(c - a*c*x)^4, x, 7, -((7*c^4*x*Sqrt[1 - a^2*x^2])/(16*a)) - (7*c^4*(1 - a^2*x^2)^(3/2))/(24*a^2) - (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(40*a^2) - (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(10*a^2) - (c^4*(1 - a*x)^3*(1 - a^2*x^2)^(3/2))/(6*a^2) - (7*c^4*ArcSin[a*x])/(16*a^2)} -{E^ArcTanh[a*x]*(c - a*c*x)^4, x, 6, (7/8)*c^4*x*Sqrt[1 - a^2*x^2] + (7*c^4*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20*a) + (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^4*ArcSin[a*x])/(8*a)} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x, x, 9, (1/8)*c^4*(8 - 13*a*x)*Sqrt[1 - a^2*x^2] - c^4*(1 - a^2*x^2)^(3/2) + (1/4)*a*c^4*x*(1 - a^2*x^2)^(3/2) - (13/8)*c^4*ArcSin[a*x] - c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^2, x, 9, (-(1/2))*a*c^4*(6 - a*x)*Sqrt[1 - a^2*x^2] + (1/3)*a*c^4*(1 - a^2*x^2)^(3/2) - (c^4*(1 - a^2*x^2)^(3/2))/x + (1/2)*a*c^4*ArcSin[a*x] + 3*a*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^3, x, 9, (5/2)*a^2*c^4*(1 + a*x)*Sqrt[1 - a^2*x^2] - (c^4*(1 - a^2*x^2)^(3/2))/(2*x^2) + (3*a*c^4*(1 - a^2*x^2)^(3/2))/x + (5/2)*a^2*c^4*ArcSin[a*x] - (5/2)*a^2*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^4, x, 9, -((a^2*c^4*(6 - a*x)*Sqrt[1 - a^2*x^2])/(2*x)) - (c^4*(1 - a^2*x^2)^(3/2))/(3*x^3) + (3*a*c^4*(1 - a^2*x^2)^(3/2))/(2*x^2) - 3*a^3*c^4*ArcSin[a*x] - (1/2)*a^3*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^5, x, 9, -((c^4*Sqrt[1 - a^2*x^2])/(4*x^4)) + (a*c^4*Sqrt[1 - a^2*x^2])/x^3 - (11*a^2*c^4*Sqrt[1 - a^2*x^2])/(8*x^2) + a^4*c^4*ArcSin[a*x] + (13/8)*a^4*c^4*ArcTanh[Sqrt[1 - a^2*x^2]], -((a^2*c^4*(13 - 8*a*x)*Sqrt[1 - a^2*x^2])/(8*x^2)) - (c^4*(1 - a^2*x^2)^(3/2))/(4*x^4) + (a*c^4*(1 - a^2*x^2)^(3/2))/x^3 + a^4*c^4*ArcSin[a*x] + (13/8)*a^4*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^6, x, 8, (7*a^3*c^4*Sqrt[1 - a^2*x^2])/(8*x^2) - (c^4*(1 - a^2*x^2)^(3/2))/(5*x^5) + (3*a*c^4*(1 - a^2*x^2)^(3/2))/(4*x^4) - (17*a^2*c^4*(1 - a^2*x^2)^(3/2))/(15*x^3) - (7/8)*a^5*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} -{(E^ArcTanh[a*x]*(c - a*c*x)^4)/x^7, x, 9, -((7*a^4*c^4*Sqrt[1 - a^2*x^2])/(16*x^2)) - (c^4*(1 - a^2*x^2)^(3/2))/(6*x^6) + (3*a*c^4*(1 - a^2*x^2)^(3/2))/(5*x^5) - (7*a^2*c^4*(1 - a^2*x^2)^(3/2))/(8*x^4) + (11*a^3*c^4*(1 - a^2*x^2)^(3/2))/(15*x^3) + (7/16)*a^6*c^4*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(E^ArcTanh[a*x]*x^4)/(c - a*c*x), x, 8, (1 + a*x)^2/(a^5*c*Sqrt[1 - a^2*x^2]) + (13*Sqrt[1 - a^2*x^2])/(3*a^5*c) + (11*x*Sqrt[1 - a^2*x^2])/(8*a^4*c) + (2*x^2*Sqrt[1 - a^2*x^2])/(3*a^3*c) + (x^3*Sqrt[1 - a^2*x^2])/(4*a^2*c) - (27*ArcSin[a*x])/(8*a^5*c)} -{(E^ArcTanh[a*x]*x^3)/(c - a*c*x), x, 7, (1 + a*x)^2/(a^4*c*Sqrt[1 - a^2*x^2]) + (11*Sqrt[1 - a^2*x^2])/(3*a^4*c) + (x*Sqrt[1 - a^2*x^2])/(a^3*c) + (x^2*Sqrt[1 - a^2*x^2])/(3*a^2*c) - (3*ArcSin[a*x])/(a^4*c)} -{(E^ArcTanh[a*x]*x^2)/(c - a*c*x), x, 5, (1 + a*x)^2/(a^3*c*Sqrt[1 - a^2*x^2]) + ((6 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^3*c) - (5*ArcSin[a*x])/(2*a^3*c)} -{(E^ArcTanh[a*x]*x)/(c - a*c*x), x, 4, (2*Sqrt[1 - a^2*x^2])/(a^2*c) + (1 - a^2*x^2)^(3/2)/(a^2*c*(1 - a*x)^2) - (2*ArcSin[a*x])/(a^2*c)} -{E^ArcTanh[a*x]/(c - a*c*x), x, 3, (2*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x)) - ArcSin[a*x]/(a*c)} -{E^ArcTanh[a*x]/(x*(c - a*c*x)), x, 7, (2*(1 + a*x))/(c*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c} -{E^ArcTanh[a*x]/(x^2*(c - a*c*x)), x, 7, (2*a*(1 + a*x))/(c*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(c*x) - (2*a*ArcTanh[Sqrt[1 - a^2*x^2]])/c} -{E^ArcTanh[a*x]/(x^3*(c - a*c*x)), x, 8, (2*a^2*(1 + a*x))/(c*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(2*c*x^2) - (2*a*Sqrt[1 - a^2*x^2])/(c*x) - (5*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c)} -{E^ArcTanh[a*x]/(x^4*(c - a*c*x)), x, 9, (2*a^3*(1 + a*x))/(c*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(3*c*x^3) - (a*Sqrt[1 - a^2*x^2])/(c*x^2) - (8*a^2*Sqrt[1 - a^2*x^2])/(3*c*x) - (3*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/c} - - -{(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^2, x, 10, (1 + a*x)^3/(3*a^5*c^2*(1 - a^2*x^2)^(3/2)) - (2*(1 + a*x)^3)/(a^5*c^2*Sqrt[1 - a^2*x^2]) - (5*Sqrt[1 - a^2*x^2])/(2*a^5*c^2) - ((5 + a*x)*Sqrt[1 - a^2*x^2])/(6*a^5*c^2) - ((5 + a*x)^2*Sqrt[1 - a^2*x^2])/(3*a^5*c^2) + (17*ArcSin[a*x])/(2*a^5*c^2)} -{(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^2, x, 6, (1 + a*x)^3/(3*a^4*c^2*(1 - a^2*x^2)^(3/2)) - (3*(1 + a*x)^2)/(a^4*c^2*Sqrt[1 - a^2*x^2]) - ((12 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^4*c^2) + (11*ArcSin[a*x])/(2*a^4*c^2)} -{(E^ArcTanh[a*x]*x^2)/(c - a*c*x)^2, x, 5, -((6*Sqrt[1 - a^2*x^2])/(a^3*c^2*(1 - a*x))) + (1 - a^2*x^2)^(3/2)/(3*a^3*c^2*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a^3*c^2*(1 - a*x)^2) + (3*ArcSin[a*x])/(a^3*c^2)} -{(E^ArcTanh[a*x]*x)/(c - a*c*x)^2, x, 4, (-2*Sqrt[1 - a^2*x^2])/(a^2*c^2*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(3*a^2*c^2*(1 - a*x)^3) + ArcSin[a*x]/(a^2*c^2)} -{E^ArcTanh[a*x]/(c - a*c*x)^2, x, 2, (1 - a^2*x^2)^(3/2)/(3*a*c^2*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(x*(c - a*c*x)^2), x, 8, (4*(1 + a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) + (3 + 5*a*x)/(3*c^2*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c^2} -{E^ArcTanh[a*x]/(x^2*(c - a*c*x)^2), x, 8, (4*a*(1 + a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) + (a*(9 + 11*a*x))/(3*c^2*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(c^2*x) - (3*a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^2} -{E^ArcTanh[a*x]/(x^3*(c - a*c*x)^2), x, 9, (4*a^2*(1 + a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) + (a^2*(15 + 17*a*x))/(3*c^2*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(2*c^2*x^2) - (3*a*Sqrt[1 - a^2*x^2])/(c^2*x) - (11*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^2)} -{E^ArcTanh[a*x]/(x^4*(c - a*c*x)^2), x, 10, (4*a^3*(1 + a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) + (a^3*(21 + 23*a*x))/(3*c^2*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(3*c^2*x^3) - (3*a*Sqrt[1 - a^2*x^2])/(2*c^2*x^2) - (17*a^2*Sqrt[1 - a^2*x^2])/(3*c^2*x) - (17*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^2)} - - -{(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^3, x, 7, (1 + a*x)^4/(5*a^5*c^3*(1 - a^2*x^2)^(5/2)) - (19*(1 + a*x)^3)/(15*a^5*c^3*(1 - a^2*x^2)^(3/2)) + (6*(1 + a*x)^2)/(a^5*c^3*Sqrt[1 - a^2*x^2]) + ((20 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^5*c^3) - (19*ArcSin[a*x])/(2*a^5*c^3)} -{(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^3, x, 9, (8*Sqrt[1 - a^2*x^2])/(a^4*c^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^4*c^3*(1 - a*x)^4) - (14*(1 - a^2*x^2)^(3/2))/(15*a^4*c^3*(1 - a*x)^3) - (1 - a^2*x^2)^(3/2)/(a^4*c^3*(1 - a*x)^2) - (4*ArcSin[a*x])/(a^4*c^3)} -{(E^ArcTanh[a*x]*x^2)/(c - a*c*x)^3, x, 8, (2*Sqrt[1 - a^2*x^2])/(a^3*c^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*c^3*(1 - a*x)^4) - (3*(1 - a^2*x^2)^(3/2))/(5*a^3*c^3*(1 - a*x)^3) - ArcSin[a*x]/(a^3*c^3)} -{(E^ArcTanh[a*x]*x)/(c - a*c*x)^3, x, 3, (1 - a^2*x^2)^(3/2)/(5*a^2*c^3*(1 - a*x)^4) - (4*(1 - a^2*x^2)^(3/2))/(15*a^2*c^3*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(c - a*c*x)^3, x, 3, (1 - a^2*x^2)^(3/2)/(5*a*c^3*(1 - a*x)^4) + (1 - a^2*x^2)^(3/2)/(15*a*c^3*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(x*(c - a*c*x)^3), x, 9, (8*(1 + a*x))/(5*c^3*(1 - a^2*x^2)^(5/2)) + (4*a*x)/(5*c^3*(1 - a^2*x^2)^(3/2)) + (5 + 8*a*x)/(5*c^3*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c^3} -{E^ArcTanh[a*x]/(x^2*(c - a*c*x)^3), x, 9, (8*a*(1 + a*x))/(5*c^3*(1 - a^2*x^2)^(5/2)) + (4*a*(5 + 8*a*x))/(15*c^3*(1 - a^2*x^2)^(3/2)) + (a*(60 + 79*a*x))/(15*c^3*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(c^3*x) - (4*a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^3} -{E^ArcTanh[a*x]/(x^3*(c - a*c*x)^3), x, 10, (8*a^2*(1 + a*x))/(5*c^3*(1 - a^2*x^2)^(5/2)) + (4*a^2*(10 + 13*a*x))/(15*c^3*(1 - a^2*x^2)^(3/2)) + (a^2*(135 + 164*a*x))/(15*c^3*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(2*c^3*x^2) - (4*a*Sqrt[1 - a^2*x^2])/(c^3*x) - (19*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^3)} -{E^ArcTanh[a*x]/(x^4*(c - a*c*x)^3), x, 11, (8*a^3*(1 + a*x))/(5*c^3*(1 - a^2*x^2)^(5/2)) + (4*a^3*(5 + 6*a*x))/(5*c^3*(1 - a^2*x^2)^(3/2)) + (a^3*(80 + 93*a*x))/(5*c^3*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(3*c^3*x^3) - (2*a*Sqrt[1 - a^2*x^2])/(c^3*x^2) - (29*a^2*Sqrt[1 - a^2*x^2])/(3*c^3*x) - (18*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/c^3} - - -{(E^ArcTanh[a*x]*x^5)/(c - a*c*x)^4, x, 8, (1 + a*x)^5/(7*a^6*c^4*(1 - a^2*x^2)^(7/2)) - (33*(1 + a*x)^4)/(35*a^6*c^4*(1 - a^2*x^2)^(5/2)) + (317*(1 + a*x)^3)/(105*a^6*c^4*(1 - a^2*x^2)^(3/2)) - (10*(1 + a*x)^2)/(a^6*c^4*Sqrt[1 - a^2*x^2]) - ((30 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^6*c^4) + (29*ArcSin[a*x])/(2*a^6*c^4)} -{(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^4, x, 12, -((10*Sqrt[1 - a^2*x^2])/(a^5*c^4*(1 - a*x))) + (1 - a^2*x^2)^(3/2)/(7*a^5*c^4*(1 - a*x)^5) - (26*(1 - a^2*x^2)^(3/2))/(35*a^5*c^4*(1 - a*x)^4) + (184*(1 - a^2*x^2)^(3/2))/(105*a^5*c^4*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a^5*c^4*(1 - a*x)^2) + (5*ArcSin[a*x])/(a^5*c^4)} -{(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^4, x, 11, (-2*Sqrt[1 - a^2*x^2])/(a^4*c^4*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(7*a^4*c^4*(1 - a*x)^5) - (19*(1 - a^2*x^2)^(3/2))/(35*a^4*c^4*(1 - a*x)^4) + (86*(1 - a^2*x^2)^(3/2))/(105*a^4*c^4*(1 - a*x)^3) + ArcSin[a*x]/(a^4*c^4)} -{(E^ArcTanh[a*x]*x^2)/(c - a*c*x)^4, x, 5, (1 - a^2*x^2)^(3/2)/(7*a^3*c^4*(1 - a*x)^5) - (12*(1 - a^2*x^2)^(3/2))/(35*a^3*c^4*(1 - a*x)^4) + (23*(1 - a^2*x^2)^(3/2))/(105*a^3*c^4*(1 - a*x)^3)} -{(E^ArcTanh[a*x]*x)/(c - a*c*x)^4, x, 4, (1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^5) - (1 - a^2*x^2)^(3/2)/(7*a^2*c^4*(1 - a*x)^4) - (1 - a^2*x^2)^(3/2)/(21*a^2*c^4*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(c - a*c*x)^4, x, 4, (1 - a^2*x^2)^(3/2)/(7*a*c^4*(1 - a*x)^5) + (2*(1 - a^2*x^2)^(3/2))/(35*a*c^4*(1 - a*x)^4) + (2*(1 - a^2*x^2)^(3/2))/(105*a*c^4*(1 - a*x)^3)} -{E^ArcTanh[a*x]/(x*(c - a*c*x)^4), x, 10, (16*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) - (4*(7 - 3*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (35 + 83*a*x)/(105*c^4*(1 - a^2*x^2)^(3/2)) + (105 + 166*a*x)/(105*c^4*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c^4} -{E^ArcTanh[a*x]/(x^2*(c - a*c*x)^4), x, 10, (16*a*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) + (4*a*(7 + 17*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a*(175 + 307*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) + (a*(525 + 719*a*x))/(105*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(c^4*x) - (5*a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^4} -{E^ArcTanh[a*x]/(x^3*(c - a*c*x)^4), x, 11, (16*a^2*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) + (4*a^2*(21 + 31*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*(455 + 671*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) + (a^2*(1470 + 1867*a*x))/(105*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(2*c^4*x^2) - (5*a*Sqrt[1 - a^2*x^2])/(c^4*x) - (29*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^4)} - - -{x*E^ArcTanh[x]*(1 + x), x, 6, (-Sqrt[1 - x])*Sqrt[1 + x] - (1/3)*Sqrt[1 - x]*(1 + x)^(3/2) - (1/3)*Sqrt[1 - x]*(1 + x)^(5/2) + ArcSin[x]} -{E^ArcTanh[x]*(1 + x), x, 5, (-(3/2))*Sqrt[1 - x]*Sqrt[1 + x] - (1/2)*Sqrt[1 - x]*(1 + x)^(3/2) + (3*ArcSin[x])/2} - - -{x*E^ArcTanh[x]*(1 + x)^2, x, 7, (-(15/8))*Sqrt[1 - x]*Sqrt[1 + x] - (5/8)*Sqrt[1 - x]*(1 + x)^(3/2) - (1/4)*Sqrt[1 - x]*(1 + x)^(5/2) - (1/4)*Sqrt[1 - x]*(1 + x)^(7/2) + (15*ArcSin[x])/8} -{E^ArcTanh[x]*(1 + x)^2, x, 6, (-(5/2))*Sqrt[1 - x]*Sqrt[1 + x] - (5/6)*Sqrt[1 - x]*(1 + x)^(3/2) - (1/3)*Sqrt[1 - x]*(1 + x)^(5/2) + (5*ArcSin[x])/2} - - -{x*E^ArcTanh[x]/(1 + x), x, 2, (-Sqrt[1 - x])*Sqrt[1 + x]} -{E^ArcTanh[x]/(1 + x), x, 3, ArcSin[x]} - - -{x*E^ArcTanh[x]/(1 + x)^2, x, 4, Sqrt[1 - x]/Sqrt[1 + x] + ArcSin[x]} -{E^ArcTanh[x]/(1 + x)^2, x, 2, -(Sqrt[1 - x]/Sqrt[1 + x])} - - -(* ::Subsubsection::Closed:: *) -(*p/2>0*) - - -{x*E^ArcTanh[x]*(1 + x)^(3/2), x, 3, -8*Sqrt[1 - x] + (16/3)*(1 - x)^(3/2) - 2*(1 - x)^(5/2) + (2/7)*(1 - x)^(7/2)} -{E^ArcTanh[x]*(1 + x)^(3/2), x, 3, -8*Sqrt[1 - x] + (8/3)*(1 - x)^(3/2) - (2/5)*(1 - x)^(5/2)} - -{x*E^ArcTanh[x]*(1 - x)^(3/2), x, 5, (-(4/3))*(1 + x)^(3/2) + (6/5)*(1 + x)^(5/2) - (2/7)*(1 + x)^(7/2), (-(4/21))*(1 + x)^(3/2) + (2/35)*(1 + x)^(5/2) - (2/7)*Sqrt[1 - x]*(1 - x^2)^(3/2)} -{E^ArcTanh[x]*(1 - x)^(3/2), x, 4, (4/3)*(1 + x)^(3/2) - (2/5)*(1 + x)^(5/2)} - - -{x*E^ArcTanh[x]*(1 + x)^(1/2), x, 3, -4*Sqrt[1 - x] + 2*(1 - x)^(3/2) - (2/5)*(1 - x)^(5/2)} -{E^ArcTanh[x]*(1 + x)^(1/2), x, 3, -4*Sqrt[1 - x] + (2/3)*(1 - x)^(3/2)} - -{x*E^ArcTanh[x]*(1 - x)^(1/2), x, 4, (-(2/3))*(1 + x)^(3/2) + (2/5)*(1 + x)^(5/2)} -{E^ArcTanh[x]*(1 - x)^(1/2), x, 3, (2/3)*(1 + x)^(3/2)} - - -(* ::Subsubsection::Closed:: *) -(*p/2<0*) - - -{x*E^ArcTanh[x]/(1 + x)^(1/2), x, 3, -2*Sqrt[1 - x] + (2/3)*(1 - x)^(3/2)} -{E^ArcTanh[x]/(1 + x)^(1/2), x, 2, -2*Sqrt[1 - x]} - -{x*E^ArcTanh[x]/(1 - x)^(1/2), x, 5, -2*Sqrt[1 + x] - (2/3)*(1 + x)^(3/2) + 2*Sqrt[2]*ArcTanh[Sqrt[1 + x]/Sqrt[2]]} -{E^ArcTanh[x]/(1 - x)^(1/2), x, 5, -2*Sqrt[1 + x] + 2*Sqrt[2]*ArcTanh[Sqrt[1 + x]/Sqrt[2]]} - - -{x*E^ArcTanh[x]/(1 + x)^(3/2), x, 4, -2*Sqrt[1 - x] + Sqrt[2]*ArcTanh[Sqrt[1 - x]/Sqrt[2]]} -{E^ArcTanh[x]/(1 + x)^(3/2), x, 3, (-Sqrt[2])*ArcTanh[Sqrt[1 - x]/Sqrt[2]]} - -{x*E^ArcTanh[x]/(1 - x)^(3/2), x, 5, (5*Sqrt[1 + x])/2 + (1 + x)^(3/2)/(2*(1 - x)) - (5*ArcTanh[Sqrt[1 + x]/Sqrt[2]])/Sqrt[2]} -{E^ArcTanh[x]/(1 - x)^(3/2), x, 5, Sqrt[1 + x]/(1 - x) - ArcTanh[Sqrt[1 + x]/Sqrt[2]]/Sqrt[2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-c a x)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*x^m*Sqrt[c - a*c*x], x, 4, (2*c*x^m*(1 + a*x)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[3/2, -m, 5/2, 1 + a*x])/(((-a)*x)^m*(3*a*Sqrt[c - a*c*x]))} - - -{E^ArcTanh[a*x]*x^2*Sqrt[c - a*c*x], x, 4, -((8*c^2*(1 - a^2*x^2)^(3/2))/(105*a^3*(c - a*c*x)^(3/2))) + (2*c^2*x^2*(1 - a^2*x^2)^(3/2))/(7*a*(c - a*c*x)^(3/2)) + (8*c*(1 - a^2*x^2)^(3/2))/(35*a^3*Sqrt[c - a*c*x])} -{E^ArcTanh[a*x]*x*Sqrt[c - a*c*x], x, 3, (2*c^2*(1 - a^2*x^2)^(3/2))/(15*a^2*(c - a*c*x)^(3/2)) - (2*c*(1 - a^2*x^2)^(3/2))/(5*a^2*Sqrt[c - a*c*x])} -{E^ArcTanh[a*x]*Sqrt[c - a*c*x], x, 2, (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2))} -{(E^ArcTanh[a*x]*Sqrt[c - a*c*x])/x, x, 4, (2*c*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x] - 2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} -{(E^ArcTanh[a*x]*Sqrt[c - a*c*x])/x^2, x, 4, -((c*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a*c*x])) - a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} - - -{E^(2*ArcTanh[a*x])*x^3*Sqrt[c - a*c*x], x, 4, -((4*Sqrt[c - a*c*x])/a^4) + (14*(c - a*c*x)^(3/2))/(3*a^4*c) - (18*(c - a*c*x)^(5/2))/(5*a^4*c^2) + (10*(c - a*c*x)^(7/2))/(7*a^4*c^3) - (2*(c - a*c*x)^(9/2))/(9*a^4*c^4)} -{E^(2*ArcTanh[a*x])*x^2*Sqrt[c - a*c*x], x, 4, -((4*Sqrt[c - a*c*x])/a^3) + (10*(c - a*c*x)^(3/2))/(3*a^3*c) - (8*(c - a*c*x)^(5/2))/(5*a^3*c^2) + (2*(c - a*c*x)^(7/2))/(7*a^3*c^3)} -{E^(2*ArcTanh[a*x])*x*Sqrt[c - a*c*x], x, 4, -((4*Sqrt[c - a*c*x])/a^2) + (2*(c - a*c*x)^(3/2))/(a^2*c) - (2*(c - a*c*x)^(5/2))/(5*a^2*c^2)} -{E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x], x, 4, -((4*Sqrt[c - a*c*x])/a) + (2*(c - a*c*x)^(3/2))/(3*a*c)} -{(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x])/x, x, 5, -2*Sqrt[c - a*c*x] - 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^2, x, 5, -(Sqrt[c - a*c*x]/x) - 3*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^3, x, 6, -(Sqrt[c - a*c*x]/(2*x^2)) - (7*a*Sqrt[c - a*c*x])/(4*x) - (7/4)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4, x, 7, -(Sqrt[c - a*c*x]/(3*x^3)) - (11*a*Sqrt[c - a*c*x])/(12*x^2) - (11*a^2*Sqrt[c - a*c*x])/(8*x) - (11/8)*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^5, x, 8, -(Sqrt[c - a*c*x]/(4*x^4)) - (5*a*Sqrt[c - a*c*x])/(8*x^3) - (25*a^2*Sqrt[c - a*c*x])/(32*x^2) - (75*a^3*Sqrt[c - a*c*x])/(64*x) - (75/64)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} - - -{E^(3*ArcTanh[a*x])*x^3*Sqrt[c - a*c*x], x, 8, -((4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(a^4*c*(1 - a*x)^(3/2))) - (2*(1 + a*x)^(3/2)*(c - a*c*x)^(3/2))/(3*a^4*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(5/2)*(c - a*c*x)^(3/2))/(5*a^4*c*(1 - a*x)^(3/2)) + (2*(1 + a*x)^(7/2)*(c - a*c*x)^(3/2))/(7*a^4*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(9/2)*(c - a*c*x)^(3/2))/(9*a^4*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(a^4*c*(1 - a*x)^(3/2))} -{E^(3*ArcTanh[a*x])*x^2*Sqrt[c - a*c*x], x, 8, -((4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(a^3*c*(1 - a*x)^(3/2))) - (2*(1 + a*x)^(3/2)*(c - a*c*x)^(3/2))/(3*a^3*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(7/2)*(c - a*c*x)^(3/2))/(7*a^3*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(a^3*c*(1 - a*x)^(3/2))} -{E^(3*ArcTanh[a*x])*x*Sqrt[c - a*c*x], x, 7, -((4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(a^2*c*(1 - a*x)^(3/2))) - (2*(1 + a*x)^(3/2)*(c - a*c*x)^(3/2))/(3*a^2*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(5/2)*(c - a*c*x)^(3/2))/(5*a^2*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(a^2*c*(1 - a*x)^(3/2))} -{E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x], x, 5, -((4*c*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a*c*x])) - (2*c^2*(1 - a^2*x^2)^(3/2))/(3*a*(c - a*c*x)^(3/2)) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/(Sqrt[2]*Sqrt[c - a*c*x])])/a} -{(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x, x, 8, -((2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(c*(1 - a*x)^(3/2))) - (2*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))} -{(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^2, x, 8, -((Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(c*x*(1 - a*x)^(3/2))) - (5*a*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*a*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))} -{(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^3, x, 9, -((Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(2*c*x^2*(1 - a*x)^(3/2))) - (9*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(4*c*x*(1 - a*x)^(3/2)) - (23*a^2*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(4*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*a^2*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))} -{(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4, x, 10, -((Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(3*c*x^3*(1 - a*x)^(3/2))) - (13*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(12*c*x^2*(1 - a*x)^(3/2)) - (19*a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(8*c*x*(1 - a*x)^(3/2)) - (45*a^3*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(8*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*a^3*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))} -{(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^5, x, 11, -((Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(4*c*x^4*(1 - a*x)^(3/2))) - (17*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(24*c*x^3*(1 - a*x)^(3/2)) - (107*a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(96*c*x^2*(1 - a*x)^(3/2)) - (149*a^3*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(64*c*x*(1 - a*x)^(3/2)) - (363*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(64*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(x^m*Sqrt[c - a*c*x])/E^ArcTanh[a*x], x, 5, -((2*c*x^(1 + m)*Sqrt[1 - a^2*x^2])/((3 + 2*m)*Sqrt[c - a*c*x])) + (2*(5 + 4*m)*x^m*(1 + a*x)*Sqrt[c - a*c*x]*Hypergeometric2F1[1/2, -m, 3/2, 1 + a*x])/(((-a)*x)^m*(a*(3 + 2*m)*Sqrt[1 - a^2*x^2]))} - - -{x^2*Sqrt[c - a*c*x]/E^ArcTanh[a*x], x, 5, (104*c*Sqrt[1 - a^2*x^2])/(105*a^3*Sqrt[c - a*c*x]) + (26*c*x^2*Sqrt[1 - a^2*x^2])/(35*a*Sqrt[c - a*c*x]) - (2*c*x^3*Sqrt[1 - a^2*x^2])/(7*Sqrt[c - a*c*x]) + (104*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(105*a^3)} -{x^1*Sqrt[c - a*c*x]/E^ArcTanh[a*x], x, 4, -((8*c*Sqrt[1 - a^2*x^2])/(5*a^2*Sqrt[c - a*c*x])) - (2*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(5*a^2) - (2*(c - a*c*x)^(3/2)*Sqrt[1 - a^2*x^2])/(5*a^2*c)} -{x^0*Sqrt[c - a*c*x]/E^ArcTanh[a*x], x, 3, (8*c*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a*c*x]) + (2*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(3*a)} -{Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^1), x, 4, -((2*c*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]) - 2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} -{Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^2), x, 4, -((c*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a*c*x])) + 3*a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} -{Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^3), x, 5, -((c*Sqrt[1 - a^2*x^2])/(2*x^2*Sqrt[c - a*c*x])) + (7*a*c*Sqrt[1 - a^2*x^2])/(4*x*Sqrt[c - a*c*x]) - (7/4)*a^2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} -{Sqrt[c - a*c*x]/(E^ArcTanh[a*x]*x^4), x, 6, -((c*Sqrt[1 - a^2*x^2])/(3*x^3*Sqrt[c - a*c*x])) + (11*a*c*Sqrt[1 - a^2*x^2])/(12*x^2*Sqrt[c - a*c*x]) - (11*a^2*c*Sqrt[1 - a^2*x^2])/(8*x*Sqrt[c - a*c*x]) + (11/8)*a^3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - a^2*x^2])/Sqrt[c - a*c*x]]} - - -{(x^3*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]), x, 8, -((4*Sqrt[c - a*c*x])/a^4) - (2*(c - a*c*x)^(3/2))/(3*a^4*c) - (2*(c - a*c*x)^(5/2))/(5*a^4*c^2) + (2*(c - a*c*x)^(7/2))/(7*a^4*c^3) - (2*(c - a*c*x)^(9/2))/(9*a^4*c^4) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^4} -{(x^2*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]), x, 8, (4*Sqrt[c - a*c*x])/a^3 + (2*(c - a*c*x)^(3/2))/(3*a^3*c) + (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^3} -{(x*Sqrt[c - a*c*x])/E^(2*ArcTanh[a*x]), x, 7, -((4*Sqrt[c - a*c*x])/a^2) - (2*(c - a*c*x)^(3/2))/(3*a^2*c) - (2*(c - a*c*x)^(5/2))/(5*a^2*c^2) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^2} -{Sqrt[c - a*c*x]/E^(2*ArcTanh[a*x]), x, 6, (4*Sqrt[c - a*c*x])/a + (2*(c - a*c*x)^(3/2))/(3*a*c) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x), x, 8, -2*Sqrt[c - a*c*x] - 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^2), x, 8, -(Sqrt[c - a*c*x]/x) + 5*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^3), x, 9, -(Sqrt[c - a*c*x]/(2*x^2)) + (9*a*Sqrt[c - a*c*x])/(4*x) - (23/4)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^4), x, 10, -(Sqrt[c - a*c*x]/(3*x^3)) + (13*a*Sqrt[c - a*c*x])/(12*x^2) - (19*a^2*Sqrt[c - a*c*x])/(8*x) + (45/8)*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^5), x, 11, -(Sqrt[c - a*c*x]/(4*x^4)) + (17*a*Sqrt[c - a*c*x])/(24*x^3) - (107*a^2*Sqrt[c - a*c*x])/(96*x^2) + (149*a^3*Sqrt[c - a*c*x])/(64*x) - (363/64)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} - - -{(x^3*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]), x, 4, (8*c^2*(1 - a*x)^(3/2))/(a^4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (32*c^2*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^4*(c - a*c*x)^(3/2)) - (50*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))/(3*a^4*(c - a*c*x)^(3/2)) + (38*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(5/2))/(5*a^4*(c - a*c*x)^(3/2)) - (2*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(7/2))/(a^4*(c - a*c*x)^(3/2)) + (2*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(9/2))/(9*a^4*(c - a*c*x)^(3/2))} -{(x^2*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]), x, 4, -((8*c^2*(1 - a*x)^(3/2))/(a^3*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))) - (24*c^2*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^3*(c - a*c*x)^(3/2)) + (26*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))/(3*a^3*(c - a*c*x)^(3/2)) - (12*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(5/2))/(5*a^3*(c - a*c*x)^(3/2)) + (2*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(7/2))/(7*a^3*(c - a*c*x)^(3/2))} -{(x*Sqrt[c - a*c*x])/E^(3*ArcTanh[a*x]), x, 4, (8*c^2*(1 - a*x)^(3/2))/(a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (16*c^2*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^2*(c - a*c*x)^(3/2)) - (10*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))/(3*a^2*(c - a*c*x)^(3/2)) + (2*c^2*(1 - a*x)^(3/2)*(1 + a*x)^(5/2))/(5*a^2*(c - a*c*x)^(3/2))} -{Sqrt[c - a*c*x]/E^(3*ArcTanh[a*x]), x, 4, -((64*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - a^2*x^2])) + (16*(c - a*c*x)^(3/2))/(3*a*c*Sqrt[1 - a^2*x^2]) + (2*(c - a*c*x)^(5/2))/(3*a*c^2*Sqrt[1 - a^2*x^2])} -{Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x), x, 6, (8*c^2*(1 - a*x)^(3/2))/(Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (2*c^2*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(c - a*c*x)^(3/2) - (2*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c - a*c*x)^(3/2)} -{Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^2), x, 6, -((9*a*c^2*(1 - a*x)^(3/2))/(Sqrt[1 + a*x]*(c - a*c*x)^(3/2))) - (c^2*(1 - a*x)^(3/2))/(x*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (7*a*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(c - a*c*x)^(3/2)} -{Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^3), x, 7, (47*a^2*c^2*(1 - a*x)^(3/2))/(4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (c^2*(1 - a*x)^(3/2))/(2*x^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (13*a*c^2*(1 - a*x)^(3/2))/(4*x*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (47*a^2*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(4*(c - a*c*x)^(3/2))} -{Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^4), x, 8, -((119*a^3*c^2*(1 - a*x)^(3/2))/(8*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))) - (c^2*(1 - a*x)^(3/2))/(3*x^3*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (19*a*c^2*(1 - a*x)^(3/2))/(12*x^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (119*a^2*c^2*(1 - a*x)^(3/2))/(24*x*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (119*a^3*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(8*(c - a*c*x)^(3/2))} -{Sqrt[c - a*c*x]/(E^(3*ArcTanh[a*x])*x^5), x, 9, (1115*a^4*c^2*(1 - a*x)^(3/2))/(64*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (c^2*(1 - a*x)^(3/2))/(4*x^4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (25*a*c^2*(1 - a*x)^(3/2))/(24*x^3*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (223*a^2*c^2*(1 - a*x)^(3/2))/(96*x^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) + (1115*a^3*c^2*(1 - a*x)^(3/2))/(192*x*Sqrt[1 + a*x]*(c - a*c*x)^(3/2)) - (1115*a^4*c^2*(1 - a*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(64*(c - a*c*x)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c a x)^p with n symbolic*) - - -{(c - a*c*x)^p/E^(2*p*ArcTanh[a*x]), x, 3, -(((1 - a*x)^p*(c - a*c*x)^(1 + p)*Hypergeometric2F1[p, 1 + 2*p, 2*(1 + p), (1/2)*(1 - a*x)])/(2^p*(a*c*(1 + 2*p))))} -{E^(2*p*ArcTanh[a*x])*(c - a*c*x)^p, x, 3, ((1 + a*x)^(1 + p)*(c - a*c*x)^p)/(a*(1 + p)*(1 - a*x)^p)} - - -{E^(n*ArcTanh[a*x])*(c - a*c*x)^p, x, 3, -((2^(1 + n/2)*(c - a*c*x)^(1 + p)*Hypergeometric2F1[-(n/2), 1 - n/2 + p, 2 - n/2 + p, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*(2 - n + 2*p))))} - -{E^(n*ArcTanh[a*x])*(c - a*c*x)^3, x, 2, -((2^(1 + n/2)*c^3*(1 - a*x)^(4 - n/2)*Hypergeometric2F1[4 - n/2, -(n/2), 5 - n/2, (1/2)*(1 - a*x)])/(a*(8 - n)))} -{E^(n*ArcTanh[a*x])*(c - a*c*x)^2, x, 2, -((2^(1 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[3 - n/2, -(n/2), 4 - n/2, (1/2)*(1 - a*x)])/(a*(6 - n)))} -{E^(n*ArcTanh[a*x])*(c - a*c*x)^1, x, 2, -((2^(1 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[2 - n/2, -(n/2), 3 - n/2, (1/2)*(1 - a*x)])/(a*(4 - n)))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^1, x, 2, (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c*n))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^2, x, 2, ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^2*(2 + n))} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^3, x, 3, If[$VersionNumber>=8, ((1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^3*(4 + n)) + ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^3*(8 + 6*n + n^2)), ((1 - a*x)^((1/2)*(-2 - n))*(1 + a*x)^((2 + n)/2))/(a*c^3*(8 + 6*n + n^2)) + ((1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^3*(4 + n))]} -{E^(n*ArcTanh[a*x])/(c - a*c*x)^4, x, 4, If[$VersionNumber>=8, ((1 - a*x)^(-3 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^4*(6 + n)) + (2*(1 - a*x)^(-2 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^4*(4 + n)*(6 + n)) + (2*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^4*(6 + n)*(8 + 6*n + n^2)), (2*(1 - a*x)^((1/2)*(-4 - n))*(1 + a*x)^((2 + n)/2))/(a*c^4*(24 + 10*n + n^2)) + (2*(1 - a*x)^((1/2)*(-2 - n))*(1 + a*x)^((2 + n)/2))/(a*c^4*(48 + 44*n + 12*n^2 + n^3)) + ((1 - a*x)^(-3 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^4*(6 + n))]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a x]) (c-c/(a x))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - c/(a*x))^p, x, 3, ((c - c/(a*x))^p*x*AppellF1[1 - p, 1/2 - p, -1/2, 2 - p, a*x, -(a*x)])/((1 - p)*(1 - a*x)^p)} - -{E^ArcTanh[a*x]*(c - c/(a*x))^4, x, 10, -((c^4*(6 - a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x)) - (c^4*(1 - a^2*x^2)^(3/2))/(3*a^4*x^3) + (3*c^4*(1 - a^2*x^2)^(3/2))/(2*a^3*x^2) - (3*c^4*ArcSin[a*x])/a - (c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^ArcTanh[a*x]*(c - c/(a*x))^3, x, 9, -((c^3*(4 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x)) + (c^3*(1 - a^2*x^2)^(3/2))/(2*a^3*x^2) - (2*c^3*ArcSin[a*x])/a + (c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^ArcTanh[a*x]*(c - c/(a*x))^2, x, 8, -((c^2*(1 + a*x)*Sqrt[1 - a^2*x^2])/(a^2*x)) - (c^2*ArcSin[a*x])/a + (c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^ArcTanh[a*x]*(c - c/(a*x)), x, 6, -((c*Sqrt[1 - a^2*x^2])/a) + (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^ArcTanh[a*x]/(c - c/(a*x)), x, 5, -((2*Sqrt[1 - a^2*x^2])/(a*c)) - (1 - a^2*x^2)^(3/2)/(a*c*(1 - a*x)^2) + (2*ArcSin[a*x])/(a*c)} -{E^ArcTanh[a*x]/(c - c/(a*x))^2, x, 6, -((6*Sqrt[1 - a^2*x^2])/(a*c^2*(1 - a*x))) + (1 - a^2*x^2)^(3/2)/(3*a*c^2*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a*c^2*(1 - a*x)^2) + (3*ArcSin[a*x])/(a*c^2)} -{E^ArcTanh[a*x]/(c - c/(a*x))^3, x, 10, -((8*Sqrt[1 - a^2*x^2])/(a*c^3*(1 - a*x))) - (1 - a^2*x^2)^(3/2)/(5*a*c^3*(1 - a*x)^4) + (14*(1 - a^2*x^2)^(3/2))/(15*a*c^3*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a*c^3*(1 - a*x)^2) + (4*ArcSin[a*x])/(a*c^3)} -{E^ArcTanh[a*x]/(c - c/(a*x))^4, x, 13, -((10*Sqrt[1 - a^2*x^2])/(a*c^4*(1 - a*x))) + (1 - a^2*x^2)^(3/2)/(7*a*c^4*(1 - a*x)^5) - (26*(1 - a^2*x^2)^(3/2))/(35*a*c^4*(1 - a*x)^4) + (184*(1 - a^2*x^2)^(3/2))/(105*a*c^4*(1 - a*x)^3) + (1 - a^2*x^2)^(3/2)/(a*c^4*(1 - a*x)^2) + (5*ArcSin[a*x])/(a*c^4)} - - -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^p, x, 6, (-(c - c/(a*x))^p)*x - ((2 - p)*(c - c/(a*x))^p*Hypergeometric2F1[1, p, 1 + p, 1 - 1/(a*x)])/(a*p)} - -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^5, x, 4, c^5/(4*a^5*x^4) - c^5/(a^4*x^3) + c^5/(a^3*x^2) + (2*c^5)/(a^2*x) - c^5*x + (3*c^5*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^4, x, 4, -c^4/(3*a^4*x^3) + c^4/(a^3*x^2) - c^4*x + (2*c^4*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^3, x, 4, c^3/(2*a^3*x^2) - c^3/(a^2*x) - c^3*x + (c^3*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^2, x, 5, -(c^2/(a^2*x)) - c^2*x} -{E^(2*ArcTanh[a*x])*(c - c/(a*x)), x, 4, -(c*x) - (c*Log[x])/a} -{E^(2*ArcTanh[a*x])/(c - c/(a*x)), x, 4, -(x/c) - 2/(a*c*(1 - a*x)) - (3*Log[1 - a*x])/(a*c)} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^2, x, 4, -(x/c^2) + 1/(a*c^2*(1 - a*x)^2) - 5/(a*c^2*(1 - a*x)) - (4*Log[1 - a*x])/(a*c^2)} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^3, x, 4, -(x/c^3) - 2/(3*a*c^3*(1 - a*x)^3) + 7/(2*a*c^3*(1 - a*x)^2) - 9/(a*c^3*(1 - a*x)) - (5*Log[1 - a*x])/(a*c^3)} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^4, x, 4, -(x/c^4) + 1/(2*a*c^4*(1 - a*x)^4) - 3/(a*c^4*(1 - a*x)^3) + 8/(a*c^4*(1 - a*x)^2) - 14/(a*c^4*(1 - a*x)) - (6*Log[1 - a*x])/(a*c^4)} - - -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^4, x, 9, (c^4*(2 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x) - (c^4*(2 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(6*a^4*x^3) + (c^4*ArcSin[a*x])/a - (3*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^3, x, 7, (3*c^3*Sqrt[1 - a^2*x^2])/(2*a) + (c^3*(1 - a^2*x^2)^(3/2))/(2*a^3*x^2) - (3*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^2, x, 9, -((c^2*(1 - a*x)*Sqrt[1 - a^2*x^2])/(a^2*x)) - (c^2*ArcSin[a*x])/a - (c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^(3*ArcTanh[a*x])*(c - c/(a*x)), x, 9, (c*Sqrt[1 - a^2*x^2])/a - (2*c*ArcSin[a*x])/a + (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^(3*ArcTanh[a*x])/(c - c/(a*x)), x, 6, (4*Sqrt[1 - a^2*x^2])/(a*c) + (8*(1 - a^2*x^2)^(3/2))/(3*a*c*(1 - a*x)^2) - (1 - a^2*x^2)^(5/2)/(3*a*c*(1 - a*x)^4) - (4*ArcSin[a*x])/(a*c)} -{E^(3*ArcTanh[a*x])/(c - c/(a*x))^2, x, 9, (1 + a*x)^5/(5*a*c^2*(1 - a^2*x^2)^(5/2)) - (2*(1 + a*x)^4)/(3*a*c^2*(1 - a^2*x^2)^(3/2)) + (10*(1 + a*x)^2)/(3*a*c^2*Sqrt[1 - a^2*x^2]) + (5*Sqrt[1 - a^2*x^2])/(a*c^2) - (5*ArcSin[a*x])/(a*c^2)} -{E^(3*ArcTanh[a*x])/(c - c/(a*x))^3, x, 9, -((1 + a*x)^6/(7*a*c^3*(1 - a^2*x^2)^(7/2))) + (4*(1 + a*x)^5)/(7*a*c^3*(1 - a^2*x^2)^(5/2)) - (1 + a*x)^4/(a*c^3*(1 - a^2*x^2)^(3/2)) + (4*(1 + a*x)^2)/(a*c^3*Sqrt[1 - a^2*x^2]) + (6*Sqrt[1 - a^2*x^2])/(a*c^3) - (6*ArcSin[a*x])/(a*c^3)} -{E^(3*ArcTanh[a*x])/(c - c/(a*x))^4, x, 10, (1 + a*x)^7/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (34*(1 + a*x)^6)/(63*a*c^4*(1 - a^2*x^2)^(7/2)) + (344*(1 + a*x)^5)/(315*a*c^4*(1 - a^2*x^2)^(5/2)) - (4*(1 + a*x)^4)/(3*a*c^4*(1 - a^2*x^2)^(3/2)) + (14*(1 + a*x)^2)/(3*a*c^4*Sqrt[1 - a^2*x^2]) + (7*Sqrt[1 - a^2*x^2])/(a*c^4) - (7*ArcSin[a*x])/(a*c^4)} - - -{E^(4*ArcTanh[a*x])*(c - c/(a*x))^p, x, 7, -((c*(5 - p)*(c - c/(a*x))^(-1 + p))/(a*(1 - p))) + c*(c - c/(a*x))^(-1 + p)*x + ((4 - p)*(c - c/(a*x))^p*Hypergeometric2F1[1, p, 1 + p, 1 - 1/(a*x)])/(a*p)} - -{E^(4*ArcTanh[a*x])*(c - c/(a*x))^5, x, 4, c^5/(4*a^5*x^4) - c^5/(3*a^4*x^3) - c^5/(a^3*x^2) + (2*c^5)/(a^2*x) + c^5*x - (c^5*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a*x))^4, x, 5, -c^4/(3*a^4*x^3) + (2*c^4)/(a^2*x) + c^4*x} -{E^(4*ArcTanh[a*x])*(c - c/(a*x))^3, x, 4, c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x + (c^3*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a*x))^2, x, 4, -(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a*x)), x, 4, c*x - (c*Log[x])/a + (4*c*Log[1 - a*x])/a} -{E^(4*ArcTanh[a*x])/(c - c/(a*x)), x, 4, x/c - 2/(a*c*(1 - a*x)^2) + 8/(a*c*(1 - a*x)) + (5*Log[1 - a*x])/(a*c)} -{E^(4*ArcTanh[a*x])/(c - c/(a*x))^2, x, 4, x/c^2 + 4/(3*a*c^2*(1 - a*x)^3) - 6/(a*c^2*(1 - a*x)^2) + 13/(a*c^2*(1 - a*x)) + (6*Log[1 - a*x])/(a*c^2)} -{E^(4*ArcTanh[a*x])/(c - c/(a*x))^3, x, 4, x/c^3 - 1/(a*c^3*(1 - a*x)^4) + 16/(3*a*c^3*(1 - a*x)^3) - 25/(2*a*c^3*(1 - a*x)^2) + 19/(a*c^3*(1 - a*x)) + (7*Log[1 - a*x])/(a*c^3)} -{E^(4*ArcTanh[a*x])/(c - c/(a*x))^4, x, 4, x/c^4 + 4/(5*a*c^4*(1 - a*x)^5) - 5/(a*c^4*(1 - a*x)^4) + 41/(3*a*c^4*(1 - a*x)^3) - 22/(a*c^4*(1 - a*x)^2) + 26/(a*c^4*(1 - a*x)) + (8*Log[1 - a*x])/(a*c^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a*x))^p/E^ArcTanh[a*x], x, 3, ((c - c/(a*x))^p*x*AppellF1[1 - p, -1/2 - p, 1/2, 2 - p, a*x, -(a*x)])/((1 - p)*(1 - a*x)^p)} - -{(c - c/(a*x))^4/E^ArcTanh[a*x], x, 11, (c^4*Sqrt[1 - a^2*x^2])/a - (c^4*Sqrt[1 - a^2*x^2])/(3*a^4*x^3) + (5*c^4*Sqrt[1 - a^2*x^2])/(2*a^3*x^2) - (32*c^4*Sqrt[1 - a^2*x^2])/(3*a^2*x) + (5*c^4*ArcSin[a*x])/a + (25*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{(c - c/(a*x))^3/E^ArcTanh[a*x], x, 10, (c^3*Sqrt[1 - a^2*x^2])/a + (c^3*Sqrt[1 - a^2*x^2])/(2*a^3*x^2) - (4*c^3*Sqrt[1 - a^2*x^2])/(a^2*x) + (4*c^3*ArcSin[a*x])/a + (13*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{(c - c/(a*x))^2/E^ArcTanh[a*x], x, 9, (c^2*Sqrt[1 - a^2*x^2])/a - (c^2*Sqrt[1 - a^2*x^2])/(a^2*x) + (3*c^2*ArcSin[a*x])/a + (3*c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{(c - c/(a*x))/E^ArcTanh[a*x], x, 8, (c*Sqrt[1 - a^2*x^2])/a + (2*c*ArcSin[a*x])/a + (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))), x, 3, Sqrt[1 - a^2*x^2]/(a*c)} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^2), x, 6, Sqrt[1 - a^2*x^2]/(a*c^2) + Sqrt[1 - a^2*x^2]/(a*c^2*(1 - a*x)) - ArcSin[a*x]/(a*c^2)} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^3), x, 7, -((1 + a*x)^2/(3*a*c^3*(1 - a^2*x^2)^(3/2))) + (8*(1 + a*x))/(3*a*c^3*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^3) - (2*ArcSin[a*x])/(a*c^3)} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^4), x, 8, (1 + a*x)^3/(5*a*c^4*(1 - a^2*x^2)^(5/2)) - (6*(1 + a*x)^2)/(5*a*c^4*(1 - a^2*x^2)^(3/2)) + (24*(1 + a*x))/(5*a*c^4*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^4) - (3*ArcSin[a*x])/(a*c^4)} - - -{(c - c/(a*x))^p/E^(2*ArcTanh[a*x]), x, 8, -(((c - c/(a*x))^(2 + p)*x)/c^2) - ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (a - 1/x)/(2*a)])/(2*a*c^2*(2 + p)) + ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])/(a*c^2)} - -{(c - c/(a*x))^4/E^(2*ArcTanh[a*x]), x, 4, -c^4/(3*a^4*x^3) + (3*c^4)/(a^3*x^2) - (16*c^4)/(a^2*x) - c^4*x - (26*c^4*Log[x])/a + (32*c^4*Log[1 + a*x])/a} -{(c - c/(a*x))^3/E^(2*ArcTanh[a*x]), x, 4, c^3/(2*a^3*x^2) - (5*c^3)/(a^2*x) - c^3*x - (11*c^3*Log[x])/a + (16*c^3*Log[1 + a*x])/a} -{(c - c/(a*x))^2/E^(2*ArcTanh[a*x]), x, 4, -(c^2/(a^2*x)) - c^2*x - (4*c^2*Log[x])/a + (8*c^2*Log[1 + a*x])/a} -{(c - c/(a*x))/E^(2*ArcTanh[a*x]), x, 4, -(c*x) - (c*Log[x])/a + (4*c*Log[1 + a*x])/a} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))), x, 4, -(x/c) + Log[1 + a*x]/(a*c)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^2), x, 5, -(x/c^2) + ArcTanh[a*x]/(a*c^2)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^3), x, 4, -(x/c^3) - 1/(2*a*c^3*(1 - a*x)) - (5*Log[1 - a*x])/(4*a*c^3) + Log[1 + a*x]/(4*a*c^3)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^4), x, 4, -(x/c^4) + 1/(4*a*c^4*(1 - a*x)^2) - 7/(4*a*c^4*(1 - a*x)) - (17*Log[1 - a*x])/(8*a*c^4) + Log[1 + a*x]/(8*a*c^4)} - - -{(c - c/(a*x))^3/E^(3*ArcTanh[a*x]), x, 11, -((32*c^3*(1 - a*x))/(a*Sqrt[1 - a^2*x^2])) - (c^3*Sqrt[1 - a^2*x^2])/a + (c^3*Sqrt[1 - a^2*x^2])/(2*a^3*x^2) - (6*c^3*Sqrt[1 - a^2*x^2])/(a^2*x) - (6*c^3*ArcSin[a*x])/a + (33*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{(c - c/(a*x))^2/E^(3*ArcTanh[a*x]), x, 10, -((16*c^2*(1 - a*x))/(a*Sqrt[1 - a^2*x^2])) - (c^2*Sqrt[1 - a^2*x^2])/a - (c^2*Sqrt[1 - a^2*x^2])/(a^2*x) - (5*c^2*ArcSin[a*x])/a + (5*c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{(c - c/(a*x))/E^(3*ArcTanh[a*x]), x, 9, -((8*c*(1 - a*x))/(a*Sqrt[1 - a^2*x^2])) - (c*Sqrt[1 - a^2*x^2])/a - (4*c*ArcSin[a*x])/a + (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))), x, 5, -((1 - a*x)^2/(a*c*Sqrt[1 - a^2*x^2])) - (2*Sqrt[1 - a^2*x^2])/(a*c) - (2*ArcSin[a*x])/(a*c)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^2), x, 6, -((1 - a*x)/(a*c^2*Sqrt[1 - a^2*x^2])) - Sqrt[1 - a^2*x^2]/(a*c^2) - ArcSin[a*x]/(a*c^2)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^3), x, 5, -(1/(a*c^3*Sqrt[1 - a^2*x^2])) - Sqrt[1 - a^2*x^2]/(a*c^3)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^4), x, 7, (a^2*x^3*(1 + a*x))/(3*c^4*(1 - a^2*x^2)^(3/2)) - (x*(3 + 4*a*x))/(3*c^4*Sqrt[1 - a^2*x^2]) - (8*Sqrt[1 - a^2*x^2])/(3*a*c^4) + ArcSin[a*x]/(a*c^4)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^5), x, 8, -((1 + a*x)^2/(5*a*c^5*(1 - a^2*x^2)^(5/2))) + (22*(1 + a*x))/(15*a*c^5*(1 - a^2*x^2)^(3/2)) - (2*(30 + 23*a*x))/(15*a*c^5*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^5) + (2*ArcSin[a*x])/(a*c^5)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - c/(a*x))^(9/2), x, 8, -((a^3*(c - c/(a*x))^(9/2)*x^4*(54 - 227*a*x)*Sqrt[1 + a*x])/(105*(1 - a*x)^(9/2))) - (10*a^2*(c - c/(a*x))^(9/2)*x^3*Sqrt[1 + a*x])/(21*(1 - a*x)^(5/2)) + (2*a*(c - c/(a*x))^(9/2)*x^2*Sqrt[1 + a*x])/(5*(1 - a*x)^(3/2)) - (2*(c - c/(a*x))^(9/2)*x*Sqrt[1 + a*x])/(7*Sqrt[1 - a*x]) - (7*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)} -{E^ArcTanh[a*x]*(c - c/(a*x))^(7/2), x, 7, (2*a*(c - c/(a*x))^(7/2)*x^2*Sqrt[1 + a*x])/(3*(1 - a*x)^(3/2)) - (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x])/(5*Sqrt[1 - a*x]) - (a^2*(c - c/(a*x))^(7/2)*x^3*Sqrt[1 + a*x]*(18 + 31*a*x))/(15*(1 - a*x)^(7/2)) + (5*a^(5/2)*(c - c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)} -{E^ArcTanh[a*x]*(c - c/(a*x))^(5/2), x, 7, -((3*a^2*(c - c/(a*x))^(5/2)*x^3*Sqrt[1 + a*x])/(1 - a*x)^(5/2)) - (2*(c - c/(a*x))^(5/2)*x*(1 + a*x)^(3/2))/(3*(1 - a*x)^(5/2)) + (4*a*(c - c/(a*x))^(5/2)*x^2*(1 + a*x)^(3/2))/(1 - a*x)^(5/2) - (3*a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)} -{E^ArcTanh[a*x]*(c - c/(a*x))^(3/2), x, 7, (a*(c - c/(a*x))^(3/2)*x^2*Sqrt[1 + a*x])/(1 - a*x)^(3/2) - (2*(c - c/(a*x))^(3/2)*x*(1 - a^2*x^2)^(3/2))/(1 - a*x)^3 + (Sqrt[a]*(c - c/(a*x))^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(3/2)} -{E^ArcTanh[a*x]*Sqrt[c - c/(a*x)], x, 6, -((c*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - c/(a*x)])) + (Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x]), (Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x] + (Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{E^ArcTanh[a*x]/Sqrt[c - c/(a*x)], x, 8, -((Sqrt[1 - a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - c/(a*x)])) - (3*Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x]) + (2*Sqrt[2]*Sqrt[1 - a*x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x])} -{E^ArcTanh[a*x]/(c - c/(a*x))^(3/2), x, 9, (Sqrt[1 - a*x]*Sqrt[1 + a*x])/(a*(c - c/(a*x))^(3/2)) + (2*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^2*(c - c/(a*x))^(3/2)*x) + (5*(1 - a*x)^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2)) - (7*(1 - a*x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(Sqrt[2]*a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2))} -{E^ArcTanh[a*x]/(c - c/(a*x))^(5/2), x, 10, (Sqrt[1 - a*x]*Sqrt[1 + a*x])/(2*a*(c - c/(a*x))^(5/2)) - (11*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(8*a^2*(c - c/(a*x))^(5/2)*x) - (23*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(8*a^3*(c - c/(a*x))^(5/2)*x^2) - (7*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) + (79*(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(8*Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2))} - - -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^(9/2), x, 11, -((5*c^4*Sqrt[c - c/(a*x)])/a) - (5*c^3*(c - c/(a*x))^(3/2))/(3*a) - (c^2*(c - c/(a*x))^(5/2))/a - (5*c*(c - c/(a*x))^(7/2))/(7*a) - (c - c/(a*x))^(9/2)*x + (5*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2), x, 10, -((3*c^3*Sqrt[c - c/(a*x)])/a) - (c^2*(c - c/(a*x))^(3/2))/a - (3*c*(c - c/(a*x))^(5/2))/(5*a) - (c - c/(a*x))^(7/2)*x + (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^(5/2), x, 9, -((c^2*Sqrt[c - c/(a*x)])/a) - (c*(c - c/(a*x))^(3/2))/(3*a) - (c - c/(a*x))^(5/2)*x + (c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a*x))^(3/2), x, 8, (c*Sqrt[c - c/(a*x)])/a - (c - c/(a*x))^(3/2)*x - (c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)], x, 7, (-Sqrt[c - c/(a*x)])*x - (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcTanh[a*x])/Sqrt[c - c/(a*x)], x, 8, 5/(a*Sqrt[c - c/(a*x)]) - x/Sqrt[c - c/(a*x)] - (5*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^(3/2), x, 9, 7/(3*a*(c - c/(a*x))^(3/2)) + 7/(a*c*Sqrt[c - c/(a*x)]) - x/(c - c/(a*x))^(3/2) - (7*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(3/2))} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^(5/2), x, 10, 9/(5*a*(c - c/(a*x))^(5/2)) + 3/(a*c*(c - c/(a*x))^(3/2)) + 9/(a*c^2*Sqrt[c - c/(a*x)]) - x/(c - c/(a*x))^(5/2) - (9*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(5/2))} -{E^(2*ArcTanh[a*x])/(c - c/(a*x))^(7/2), x, 11, 11/(7*a*(c - c/(a*x))^(7/2)) + 11/(5*a*c*(c - c/(a*x))^(5/2)) + 11/(3*a*c^2*(c - c/(a*x))^(3/2)) + 11/(a*c^3*Sqrt[c - c/(a*x)]) - x/(c - c/(a*x))^(7/2) - (11*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(7/2))} - - -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^(9/2), x, 8, -((3*a^3*(c - c/(a*x))^(9/2)*x^4*Sqrt[1 + a*x])/(1 - a*x)^(9/2)) + (3*a^2*(c - c/(a*x))^(9/2)*x^3*(6 - 17*a*x)*(1 + a*x)^(3/2))/(35*(1 - a*x)^(9/2)) + (6*a*(c - c/(a*x))^(9/2)*x^2*(1 + a*x)^(3/2))/(35*(1 - a*x)^(5/2)) - (2*(c - c/(a*x))^(9/2)*x*(1 + a*x)^(3/2))/(7*(1 - a*x)^(3/2)) + (3*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2), x, 8, -((a^3*(c - c/(a*x))^(7/2)*x^4*Sqrt[1 + a*x])/(1 - a*x)^(7/2)) + (2*a^2*(c - c/(a*x))^(7/2)*x^3*(1 + a*x)^(3/2))/(3*(1 - a*x)^(7/2)) - (2*(c - c/(a*x))^(7/2)*x*(1 + a*x)^(5/2))/(5*(1 - a*x)^(7/2)) + (4*a*(c - c/(a*x))^(7/2)*x^2*(1 + a*x)^(5/2))/(3*(1 - a*x)^(7/2)) - (a^(5/2)*(c - c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2), x, 8, -((a^2*(c - c/(a*x))^(5/2)*x^3*Sqrt[1 + a*x])/(1 - a*x)^(5/2)) + (2*a*(c - c/(a*x))^(5/2)*x^2*(1 + a*x)^(3/2))/(3*(1 - a*x)^(5/2)) - (2*(c - c/(a*x))^(5/2)*x*(1 - a^2*x^2)^(5/2))/(3*(1 - a*x)^5) - (a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a*x))^(3/2), x, 7, (3*a*(c - c/(a*x))^(3/2)*x^2*Sqrt[1 + a*x])/(1 - a*x)^(3/2) - (2*(c - c/(a*x))^(3/2)*x*(1 + a*x)^(3/2))/(1 - a*x)^(3/2) + (3*Sqrt[a]*(c - c/(a*x))^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(3/2)} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)], x, 8, -((Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x]) - (5*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{E^(3*ArcTanh[a*x])/Sqrt[c - c/(a*x)], x, 9, (2*Sqrt[1 - a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - c/(a*x)]) + (1 + a*x)^(3/2)/(a*Sqrt[c - c/(a*x)]*Sqrt[1 - a*x]) + (7*Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x]) - (5*Sqrt[2]*Sqrt[1 - a*x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x])} -{E^(3*ArcTanh[a*x])/(c - c/(a*x))^(3/2), x, 10, -((21*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(8*a^2*(c - c/(a*x))^(3/2)*x)) + (1 + a*x)^(3/2)/(2*a*(c - c/(a*x))^(3/2)*Sqrt[1 - a*x]) - (9*Sqrt[1 - a*x]*(1 + a*x)^(3/2))/(8*a^2*(c - c/(a*x))^(3/2)*x) - (9*(1 - a*x)^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2)) + (51*(1 - a*x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(4*Sqrt[2]*a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2))} -{E^(3*ArcTanh[a*x])/(c - c/(a*x))^(5/2), x, 11, (103*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(32*a^3*(c - c/(a*x))^(5/2)*x^2) + (1 + a*x)^(3/2)/(3*a*(c - c/(a*x))^(5/2)*Sqrt[1 - a*x]) - (13*Sqrt[1 - a*x]*(1 + a*x)^(3/2))/(24*a^2*(c - c/(a*x))^(5/2)*x) + (43*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))/(32*a^3*(c - c/(a*x))^(5/2)*x^2) + (11*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) - (249*(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a*x))^(9/2)/E^ArcTanh[a*x], x, 8, -((94*a^2*(c - c/(a*x))^(9/2)*x^3*Sqrt[1 + a*x])/(21*(1 - a*x)^(5/2))) + (6*a*(c - c/(a*x))^(9/2)*x^2*Sqrt[1 + a*x])/(5*(1 - a*x)^(3/2)) - (2*(c - c/(a*x))^(9/2)*x*Sqrt[1 + a*x])/(7*Sqrt[1 - a*x]) + (a^3*(c - c/(a*x))^(9/2)*x^4*Sqrt[1 + a*x]*(2718 + 521*a*x))/(105*(1 - a*x)^(9/2)) + (11*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)} -{(c - c/(a*x))^(7/2)/E^ArcTanh[a*x], x, 7, (2*a*(c - c/(a*x))^(7/2)*x^2*Sqrt[1 + a*x])/(1 - a*x)^(3/2) - (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x])/(5*Sqrt[1 - a*x]) - (a^2*(c - c/(a*x))^(7/2)*x^3*Sqrt[1 + a*x]*(66 + 7*a*x))/(5*(1 - a*x)^(7/2)) - (9*a^(5/2)*(c - c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)} -{(c - c/(a*x))^(5/2)/E^ArcTanh[a*x], x, 6, -((2*(c - c/(a*x))^(5/2)*x*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x])) + (a*(c - c/(a*x))^(5/2)*x^2*(18 - a*x)*Sqrt[1 + a*x])/(3*(1 - a*x)^(5/2)) + (7*a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)} -{(c - c/(a*x))^(3/2)/E^ArcTanh[a*x], x, 6, -((2*(c - c/(a*x))^(3/2)*x*Sqrt[1 + a*x])/(1 - a*x)^(3/2)) + (a*(c - c/(a*x))^(3/2)*x^2*Sqrt[1 + a*x])/(1 - a*x)^(3/2) - (5*Sqrt[a]*(c - c/(a*x))^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(3/2)} -{Sqrt[c - c/(a*x)]/E^ArcTanh[a*x], x, 6, -((Sqrt[c - c/(a*x)]*x*Sqrt[1 - a^2*x^2])/(1 - a*x)) + (3*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{1/(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]), x, 6, (Sqrt[1 - a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - c/(a*x)]) - (Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x])} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^(3/2)), x, 9, -(((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^2*(c - c/(a*x))^(3/2)*x)) - ((1 - a*x)^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2)) + (Sqrt[2]*(1 - a*x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2))} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^(5/2)), x, 9, ((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(2*a^2*(c - c/(a*x))^(5/2)*x) + (3*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(2*a^3*(c - c/(a*x))^(5/2)*x^2) + (3*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) - (9*(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(2*Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2))} -{1/(E^ArcTanh[a*x]*(c - c/(a*x))^(7/2)), x, 10, ((1 - a*x)^(3/2)*Sqrt[1 + a*x])/(4*a^2*(c - c/(a*x))^(7/2)*x) - (15*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(16*a^3*(c - c/(a*x))^(7/2)*x^2) - (35*(1 - a*x)^(7/2)*Sqrt[1 + a*x])/(16*a^4*(c - c/(a*x))^(7/2)*x^3) - (5*(1 - a*x)^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2)) + (115*(1 - a*x)^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*Sqrt[2]*a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2))} - - -{(c - c/(a*x))^(9/2)/E^(2*ArcTanh[a*x]), x, 14, (51*c^4*Sqrt[c - c/(a*x)])/a + (19*c^3*(c - c/(a*x))^(3/2))/(3*a) + (3*c^2*(c - c/(a*x))^(5/2))/(5*a) - (5*c*(c - c/(a*x))^(7/2))/(7*a) - (c - c/(a*x))^(9/2)*x + (13*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (64*Sqrt[2]*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{(c - c/(a*x))^(7/2)/E^(2*ArcTanh[a*x]), x, 13, (21*c^3*Sqrt[c - c/(a*x)])/a + (5*c^2*(c - c/(a*x))^(3/2))/(3*a) - (3*c*(c - c/(a*x))^(5/2))/(5*a) - (c - c/(a*x))^(7/2)*x + (11*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (32*Sqrt[2]*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{(c - c/(a*x))^(5/2)/E^(2*ArcTanh[a*x]), x, 12, (7*c^2*Sqrt[c - c/(a*x)])/a - (c*(c - c/(a*x))^(3/2))/(3*a) - (c - c/(a*x))^(5/2)*x + (9*c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (16*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{(c - c/(a*x))^(3/2)/E^(2*ArcTanh[a*x]), x, 11, (c*Sqrt[c - c/(a*x)])/a - (c - c/(a*x))^(3/2)*x + (7*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (8*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - c/(a*x)]/E^(2*ArcTanh[a*x]), x, 10, (-Sqrt[c - c/(a*x)])*x + (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{1/(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)]), x, 10, -((Sqrt[c - c/(a*x)]*x)/c) + (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*Sqrt[c]) - (2*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(3/2)), x, 11, -((Sqrt[c - c/(a*x)]*x)/c^2) + ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*c^(3/2)) - (Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2))} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(5/2)), x, 11, 2/(a*c^2*Sqrt[c - c/(a*x)]) - x/(c^2*Sqrt[c - c/(a*x)]) - ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*c^(5/2)) - ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(Sqrt[2]*a*c^(5/2))} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(7/2)), x, 12, 4/(3*a*c^2*(c - c/(a*x))^(3/2)) + 7/(2*a*c^3*Sqrt[c - c/(a*x)]) - x/(c^2*(c - c/(a*x))^(3/2)) - (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(7/2)) - ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(2*Sqrt[2]*a*c^(7/2))} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(9/2)), x, 13, 6/(5*a*c^2*(c - c/(a*x))^(5/2)) + 11/(6*a*c^3*(c - c/(a*x))^(3/2)) + 21/(4*a*c^4*Sqrt[c - c/(a*x)]) - x/(c^2*(c - c/(a*x))^(5/2)) - (5*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(9/2)) - ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))} - - -{(c - c/(a*x))^(9/2)/E^(3*ArcTanh[a*x]), x, 9, (5*a^4*(c - c/(a*x))^(9/2)*x^5*(587 - 109*a*x))/(7*(1 - a*x)^(9/2)*Sqrt[1 + a*x]) + (70*a^3*(c - c/(a*x))^(9/2)*x^4)/((1 - a*x)^(5/2)*Sqrt[1 + a*x]) - (50*a^2*(c - c/(a*x))^(9/2)*x^3)/(7*(1 - a*x)^(3/2)*Sqrt[1 + a*x]) + (10*a*(c - c/(a*x))^(9/2)*x^2)/(7*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*(c - c/(a*x))^(9/2)*x*Sqrt[1 - a*x])/(7*Sqrt[1 + a*x]) - (15*a^(7/2)*(c - c/(a*x))^(9/2)*x^(9/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(9/2)} -{(c - c/(a*x))^(7/2)/E^(3*ArcTanh[a*x]), x, 8, -((a^3*(c - c/(a*x))^(7/2)*x^4*(2525 - 427*a*x))/(15*(1 - a*x)^(7/2)*Sqrt[1 + a*x])) - (398*a^2*(c - c/(a*x))^(7/2)*x^3)/(15*(1 - a*x)^(3/2)*Sqrt[1 + a*x]) + (38*a*(c - c/(a*x))^(7/2)*x^2)/(15*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*(c - c/(a*x))^(7/2)*x*Sqrt[1 - a*x])/(5*Sqrt[1 + a*x]) + (13*a^(5/2)*(c - c/(a*x))^(7/2)*x^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(7/2)} -{(c - c/(a*x))^(5/2)/E^(3*ArcTanh[a*x]), x, 7, (a^2*(c - c/(a*x))^(5/2)*x^3*(191 - 25*a*x))/(3*(1 - a*x)^(5/2)*Sqrt[1 + a*x]) + (26*a*(c - c/(a*x))^(5/2)*x^2)/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*(c - c/(a*x))^(5/2)*x*Sqrt[1 - a*x])/(3*Sqrt[1 + a*x]) - (11*a^(3/2)*(c - c/(a*x))^(5/2)*x^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(5/2)} -{(c - c/(a*x))^(3/2)/E^(3*ArcTanh[a*x]), x, 6, -((2*(c - c/(a*x))^(3/2)*x*Sqrt[1 - a*x])/Sqrt[1 + a*x]) - (a*(c - c/(a*x))^(3/2)*x^2*(23 - a*x))/((1 - a*x)^(3/2)*Sqrt[1 + a*x]) + (9*Sqrt[a]*(c - c/(a*x))^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(1 - a*x)^(3/2)} -{Sqrt[c - c/(a*x)]/E^(3*ArcTanh[a*x]), x, 6, (8*Sqrt[c - c/(a*x)]*x)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x] - (7*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]), x, 7, -((5*Sqrt[1 - a*x])/(a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])) - (x*(1 - a*x))/(Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2]) + (5*Sqrt[1 - a*x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x])} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(3/2)), x, 7, -((2*(1 - a*x)^(3/2))/(a*(c - c/(a*x))^(3/2)*Sqrt[1 + a*x])) + (3*(1 - a*x)^(3/2)*Sqrt[1 + a*x])/(a^2*(c - c/(a*x))^(3/2)*x) - (3*(1 - a*x)^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(5/2)*(c - c/(a*x))^(3/2)*x^(3/2))} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(5/2)), x, 9, (1 - a*x)^(5/2)/(a^2*(c - c/(a*x))^(5/2)*x*Sqrt[1 + a*x]) - (2*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(a^3*(c - c/(a*x))^(5/2)*x^2) + ((1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2)) + ((1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(Sqrt[2]*a^(7/2)*(c - c/(a*x))^(5/2)*x^(5/2))} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2)), x, 10, (1 - a*x)^(5/2)/(2*a^2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x]) - (1 - a*x)^(7/2)/(4*a^3*(c - c/(a*x))^(7/2)*x^2*Sqrt[1 + a*x]) + (7*(1 - a*x)^(7/2)*Sqrt[1 + a*x])/(4*a^4*(c - c/(a*x))^(7/2)*x^3) + ((1 - a*x)^(7/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2)) - (11*(1 - a*x)^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(4*Sqrt[2]*a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a x))^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(3*ArcTanh[a*x])/(x^3*(c - c/(a*x))), x, 9, -((8*a^2*(1 + a*x))/(3*c*(1 - a^2*x^2)^(3/2))) - (4*a^2*(3 + 4*a*x))/(3*c*Sqrt[1 - a^2*x^2]) + (a*Sqrt[1 - a^2*x^2])/(c*x) + (4*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/c} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a x))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]*x^m, x, 4, (2*Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[-(1/2), 1/2 + m, 3/2 + m, (-a)*x])/((1 + 2*m)*Sqrt[1 - a*x])} - -{E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]*x^2, x, 8, -(Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(12*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^3*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*Sqrt[1 - a*x])} -{E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]*x, x, 7, (Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(4*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(4*a^(3/2)*Sqrt[1 - a*x])} -{E^ArcTanh[a*x]*Sqrt[c - c/(a*x)], x, 6, (Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x] + (Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)])/x, x, 6, (-2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x] + (2*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[1 - a*x]} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)])/x^2, x, 4, (-2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(3*x*Sqrt[1 - a*x])} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)])/x^3, x, 5, -((2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(5*x^2*Sqrt[1 - a*x])) + (4*a*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(15*x*Sqrt[1 - a*x])} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)])/x^4, x, 6, -((2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(7*x^3*Sqrt[1 - a*x])) + (8*a*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(35*x^2*Sqrt[1 - a*x]) - (16*a^2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(105*x*Sqrt[1 - a*x])} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)])/x^5, x, 7, -((2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(9*x^4*Sqrt[1 - a*x])) + (4*a*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(21*x^3*Sqrt[1 - a*x]) - (16*a^2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(105*x^2*Sqrt[1 - a*x]) + (32*a^3*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(315*x*Sqrt[1 - a*x])} - - -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^3, x, 10, -((75*Sqrt[c - c/(a*x)]*x)/(64*a^3)) - (25*Sqrt[c - c/(a*x)]*x^2)/(32*a^2) - (5*Sqrt[c - c/(a*x)]*x^3)/(8*a) - (1/4)*Sqrt[c - c/(a*x)]*x^4 - (75*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4)} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^2, x, 9, -((11*Sqrt[c - c/(a*x)]*x)/(8*a^2)) - (11*Sqrt[c - c/(a*x)]*x^2)/(12*a) - (1/3)*Sqrt[c - c/(a*x)]*x^3 - (11*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(8*a^3)} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x, x, 8, -((7*Sqrt[c - c/(a*x)]*x)/(4*a)) - (1/2)*Sqrt[c - c/(a*x)]*x^2 - (7*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(4*a^2)} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)], x, 7, (-Sqrt[c - c/(a*x)])*x - (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x, x, 7, -2*Sqrt[c - c/(a*x)] - 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^2, x, 6, -4*a*Sqrt[c - c/(a*x)] + (2*a*(c - c/(a*x))^(3/2))/(3*c)} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^3, x, 6, -4*a^2*Sqrt[c - c/(a*x)] + (2*a^2*(c - c/(a*x))^(3/2))/c - (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2)} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^4, x, 6, -4*a^3*Sqrt[c - c/(a*x)] + (10*a^3*(c - c/(a*x))^(3/2))/(3*c) - (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) + (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3)} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5, x, 6, -4*a^4*Sqrt[c - c/(a*x)] + (14*a^4*(c - c/(a*x))^(3/2))/(3*c) - (18*a^4*(c - c/(a*x))^(5/2))/(5*c^2) + (10*a^4*(c - c/(a*x))^(7/2))/(7*c^3) - (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4)} - - -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^3, x, 11, -((107*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(64*a^3*Sqrt[1 - a*x])) - (21*Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(32*a^3*Sqrt[1 - a*x]) - (11*Sqrt[c - c/(a*x)]*x^2*(1 + a*x)^(3/2))/(24*a^2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x^3*(1 + a*x)^(3/2))/(4*a*Sqrt[1 - a*x]) - (363*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(7/2)*Sqrt[1 - a*x])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^2, x, 10, -((13*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[1 - a*x])) - (3*Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(4*a^2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x^2*(1 + a*x)^(3/2))/(3*a*Sqrt[1 - a*x]) - (45*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(5/2)*Sqrt[1 - a*x])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x, x, 9, -((7*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(4*a*Sqrt[1 - a*x])) - (Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(2*a*Sqrt[1 - a*x]) - (23*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(4*a^(3/2)*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(3/2)*Sqrt[1 - a*x])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)], x, 8, -((Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x]) - (5*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x, x, 8, -((2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x]) - (2*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[1 - a*x] + (4*Sqrt[2]*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a*x]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^2, x, 6, -((4*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x]) - (2*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(3*x*Sqrt[1 - a*x]) + (4*Sqrt[2]*a^(3/2)*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a*x]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^3, x, 7, -((4*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/Sqrt[1 - a*x]) - (2*a*Sqrt[c - c/(a*x)]*(1 + a*x)^(3/2))/(3*x*Sqrt[1 - a*x]) - (2*Sqrt[c - c/(a*x)]*(1 + a*x)^(5/2))/(5*x^2*Sqrt[1 - a*x]) + (4*Sqrt[2]*a^(5/2)*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a*x]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^4, x, 9, -((104*a^3*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(21*Sqrt[1 - a*x])) - (2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(7*x^3*Sqrt[1 - a*x]) - (6*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(7*x^2*Sqrt[1 - a*x]) - (32*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(21*x*Sqrt[1 - a*x]) + (4*Sqrt[2]*a^(7/2)*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a*x]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5, x, 10, -((1576*a^4*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(315*Sqrt[1 - a*x])) - (2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(9*x^4*Sqrt[1 - a*x]) - (38*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(63*x^3*Sqrt[1 - a*x]) - (92*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(105*x^2*Sqrt[1 - a*x]) - (472*a^3*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(315*x*Sqrt[1 - a*x]) + (4*Sqrt[2]*a^(9/2)*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/Sqrt[1 - a*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sqrt[c - c/(a*x)]*x^m)/E^ArcTanh[a*x], x, 5, If[$VersionNumber>=8, -((Sqrt[c - c/(a*x)]*x^(1 + m)*Sqrt[1 - a^2*x^2])/((1 + m)*(1 - a*x))) + ((3 + 4*m)*Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, (-a)*x])/((1 + m)*(1 + 2*m)*Sqrt[1 - a*x]), -((Sqrt[c - c/(a*x)]*x^(1 + m)*Sqrt[1 - a^2*x^2])/((1 + m)*(1 - a*x))) + ((3 + 4*m)*Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[1/2, 1/2 + m, 3/2 + m, (-a)*x])/((1 + 3*m + 2*m^2)*Sqrt[1 - a*x])]} - -{(Sqrt[c - c/(a*x)]*x^2)/E^ArcTanh[a*x], x, 8, -((11*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[1 - a*x])) + (11*Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(12*a*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x^3*Sqrt[1 - a^2*x^2])/(3*(1 - a*x)) + (11*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*Sqrt[1 - a*x])} -{(Sqrt[c - c/(a*x)]*x)/E^ArcTanh[a*x], x, 7, (7*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(4*a*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x^2*Sqrt[1 - a^2*x^2])/(2*(1 - a*x)) - (7*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(4*a^(3/2)*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/E^ArcTanh[a*x], x, 6, -((Sqrt[c - c/(a*x)]*x*Sqrt[1 - a^2*x^2])/(1 - a*x)) + (3*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/(E^ArcTanh[a*x]*x), x, 6, -((2*Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2])/(1 - a*x)) - (2*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[1 - a*x]} -{Sqrt[c - c/(a*x)]/(E^ArcTanh[a*x]*x^2), x, 5, (10*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x]) - (2*Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2])/(3*x*(1 - a*x))} -{Sqrt[c - c/(a*x)]/(E^ArcTanh[a*x]*x^3), x, 6, -((12*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(5*Sqrt[1 - a*x])) + (6*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(5*x*Sqrt[1 - a*x]) - (2*Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2])/(5*x^2*(1 - a*x))} -{Sqrt[c - c/(a*x)]/(E^ArcTanh[a*x]*x^4), x, 7, (208*a^3*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(105*Sqrt[1 - a*x]) + (26*a*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(35*x^2*Sqrt[1 - a*x]) - (104*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(105*x*Sqrt[1 - a*x]) - (2*Sqrt[c - c/(a*x)]*Sqrt[1 - a^2*x^2])/(7*x^3*(1 - a*x))} - - -{(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcTanh[a*x]), x, 13, (149*Sqrt[c - c/(a*x)]*x)/(64*a^3) - (107*Sqrt[c - c/(a*x)]*x^2)/(96*a^2) + (17*Sqrt[c - c/(a*x)]*x^3)/(24*a) - (1/4)*Sqrt[c - c/(a*x)]*x^4 - (363*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^4} -{(Sqrt[c - c/(a*x)]*x^2)/E^(2*ArcTanh[a*x]), x, 12, -((19*Sqrt[c - c/(a*x)]*x)/(8*a^2)) + (13*Sqrt[c - c/(a*x)]*x^2)/(12*a) - (1/3)*Sqrt[c - c/(a*x)]*x^3 + (45*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(8*a^3) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^3} -{(Sqrt[c - c/(a*x)]*x)/E^(2*ArcTanh[a*x]), x, 11, (9*Sqrt[c - c/(a*x)]*x)/(4*a) - (1/2)*Sqrt[c - c/(a*x)]*x^2 - (23*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(4*a^2) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^2} -{Sqrt[c - c/(a*x)]/E^(2*ArcTanh[a*x]), x, 10, (-Sqrt[c - c/(a*x)])*x + (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x), x, 10, -2*Sqrt[c - c/(a*x)] - 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] + 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^2), x, 8, 4*a*Sqrt[c - c/(a*x)] + (2*a*(c - c/(a*x))^(3/2))/(3*c) - 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^3), x, 9, -4*a^2*Sqrt[c - c/(a*x)] - (2*a^2*(c - c/(a*x))^(3/2))/(3*c) - (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2) + 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^4), x, 10, 4*a^3*Sqrt[c - c/(a*x)] + (2*a^3*(c - c/(a*x))^(3/2))/(3*c) + (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3) - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcTanh[a*x])*x^5), x, 10, -4*a^4*Sqrt[c - c/(a*x)] - (2*a^4*(c - c/(a*x))^(3/2))/(3*c) - (2*a^4*(c - c/(a*x))^(5/2))/(5*c^2) + (2*a^4*(c - c/(a*x))^(7/2))/(7*c^3) - (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4) + 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} - - -{(Sqrt[c - c/(a*x)]*x^3)/E^(3*ArcTanh[a*x]), x, 9, (8*Sqrt[c - c/(a*x)]*x^4)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (1115*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(64*a^3*Sqrt[1 - a*x]) + (1115*Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(96*a^2*Sqrt[1 - a*x]) - (223*Sqrt[c - c/(a*x)]*x^3*Sqrt[1 + a*x])/(24*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^4*Sqrt[1 + a*x])/(4*Sqrt[1 - a*x]) + (1115*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)*Sqrt[1 - a*x])} -{(Sqrt[c - c/(a*x)]*x^2)/E^(3*ArcTanh[a*x]), x, 8, (8*Sqrt[c - c/(a*x)]*x^3)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (119*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(8*a^2*Sqrt[1 - a*x]) - (119*Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(12*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^3*Sqrt[1 + a*x])/(3*Sqrt[1 - a*x]) - (119*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(8*a^(5/2)*Sqrt[1 - a*x])} -{(Sqrt[c - c/(a*x)]*x)/E^(3*ArcTanh[a*x]), x, 7, (8*Sqrt[c - c/(a*x)]*x^2)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (47*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(4*a*Sqrt[1 - a*x]) + (Sqrt[c - c/(a*x)]*x^2*Sqrt[1 + a*x])/(2*Sqrt[1 - a*x]) + (47*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(4*a^(3/2)*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/E^(3*ArcTanh[a*x]), x, 6, (8*Sqrt[c - c/(a*x)]*x)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/Sqrt[1 - a*x] - (7*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(Sqrt[a]*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x), x, 6, (-2*Sqrt[c - c/(a*x)])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (10*a*Sqrt[c - c/(a*x)]*x)/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (2*Sqrt[a]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[1 - a*x]} -{Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^2), x, 5, (20*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*Sqrt[c - c/(a*x)])/(3*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (46*a^2*Sqrt[c - c/(a*x)]*x)/(3*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^3), x, 6, (158*a^2*Sqrt[c - c/(a*x)])/(15*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*Sqrt[c - c/(a*x)])/(5*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (32*a*Sqrt[c - c/(a*x)])/(15*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (316*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(15*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^4), x, 7, (-2*Sqrt[c - c/(a*x)])/(7*x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (44*a*Sqrt[c - c/(a*x)])/(35*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (334*a^2*Sqrt[c - c/(a*x)])/(35*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (2672*a^3*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(105*Sqrt[1 - a*x]) - (1336*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(105*x*Sqrt[1 - a*x])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcTanh[a*x])*x^5), x, 8, (-2*Sqrt[c - c/(a*x)])/(9*x^4*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (8*a*Sqrt[c - c/(a*x)])/(9*x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (82*a^2*Sqrt[c - c/(a*x)])/(9*x^2*Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (1312*a^4*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(45*Sqrt[1 - a*x]) - (164*a^2*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(15*x^2*Sqrt[1 - a*x]) + (656*a^3*Sqrt[c - c/(a*x)]*Sqrt[1 + a*x])/(45*x*Sqrt[1 - a*x])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^p with n symbolic*) - - -{E^(n*ArcTanh[a*x])*(c - c/(a*x))^p, x, 3, ((c - c/(a*x))^p*x*AppellF1[1 - p, (n - 2*p)/2, -n/2, 2 - p, a*x, -(a*x)])/((1 - p)*(1 - a*x)^p)} - - -{(c - c/(a*x))^p/E^(2*p*ArcTanh[a*x]), x, 3, ((c - c/(a*x))^p*x*AppellF1[1 - p, -2*p, p, 2 - p, a*x, -(a*x)])/((1 - p)*(1 - a*x)^p)} -{E^(2*p*ArcTanh[a*x])*(c - c/(a*x))^p, x, 3, ((c - c/(a*x))^p*x*Hypergeometric2F1[1 - p, -p, 2 - p, (-a)*x])/((1 - a*x)^p*(1 - p))} - - -{E^(n*ArcTanh[a*x])*(c - c/(a*x))^2, x, 5, (4*c^2*(1 + a*x)^(n/2)*Hypergeometric2F1[2, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - a*x)^(n/2)*(a*n)) + (2^(n/2)*c^2*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[1 - n/2, 2 - n/2, 3 - n/2, (1/2)*(1 - a*x)])/(a*(4 - n))} -{E^(n*ArcTanh[a*x])*(c - c/(a*x))^1, x, 6, If[$VersionNumber>=8, (c*(1 - a*x)^(2 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a*(2 - n)) - (2*c*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[1, (1/2)*(-2 + n), n/2, (1 + a*x)/(1 - a*x)])/(a*(2 - n)) + (2^(n/2)*c*(1 - n)*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (1/2)*(1 - a*x)])/(a*(2 - n)*(4 - n)), (c*(1 - a*x)^(2 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a*(2 - n)) - (2*c*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[1, (1/2)*(-2 + n), n/2, (1 + a*x)/(1 - a*x)])/(a*(2 - n)) + (2^(n/2)*c*(1 - n)*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (1/2)*(1 - a*x)])/(a*(8 - 6*n + n^2))]} -{E^(n*ArcTanh[a*x])/(c - c/(a*x))^1, x, 4, -((1 + a*x)^((2 + n)/2)/((1 - a*x)^(n/2)*(a*c*n))) - (2^(1 + n/2)*(1 + n)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(a*c*(2 - n)*n)} -{E^(n*ArcTanh[a*x])/(c - c/(a*x))^2, x, 5, If[$VersionNumber>=8, ((3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^2*(2 + n)) - (x*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/c^2 - (2^(1 + n/2)*(2 + n)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c^2*n)), ((3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a*c^2*(2 + n)) - (x*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/c^2 - (2^(1 + n/2)*(2 + n)*Hypergeometric2F1[-(n/2), -(n/2), (2 - n)/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c^2*n))]} - - -{E^(n*ArcTanh[a*x])*(c - c/(a*x))^(3/2), x, 3, -((2*(c - c/(a*x))^(3/2)*x*AppellF1[-(1/2), (1/2)*(-3 + n), -(n/2), 1/2, a*x, (-a)*x])/(1 - a*x)^(3/2))} -{E^(n*ArcTanh[a*x])*Sqrt[c - c/(a*x)], x, 3, (2*Sqrt[c - c/(a*x)]*x*AppellF1[1/2, (-1 + n)/2, -n/2, 3/2, a*x, -(a*x)])/Sqrt[1 - a*x]} -{E^(n*ArcTanh[a*x])/Sqrt[c - c/(a*x)], x, 3, (2*x*Sqrt[1 - a*x]*AppellF1[3/2, (1 + n)/2, -n/2, 5/2, a*x, -(a*x)])/(3*Sqrt[c - c/(a*x)])} -{E^(n*ArcTanh[a*x])/(c - c/(a*x))^(3/2), x, 3, (2*x*(1 - a*x)^(3/2)*AppellF1[5/2, (3 + n)/2, -(n/2), 7/2, a*x, (-a)*x])/(5*(c - c/(a*x))^(3/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4, x, 11, (c^4*(16 - 35*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) - (c^4*(16 + 35*a*x)*(1 - a^2*x^2)^(3/2))/(48*a^4*x^3) + (c^4*(24 + 35*a*x)*(1 - a^2*x^2)^(5/2))/(120*a^6*x^5) - (c^4*(6 + 7*a*x)*(1 - a^2*x^2)^(7/2))/(42*a^8*x^7) + (c^4*ArcSin[a*x])/a + (35*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^3, x, 10, (c^3*(8 - 15*a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) - (c^3*(8 + 15*a*x)*(1 - a^2*x^2)^(3/2))/(24*a^4*x^3) + (c^3*(4 + 5*a*x)*(1 - a^2*x^2)^(5/2))/(20*a^6*x^5) + (c^3*ArcSin[a*x])/a + (15*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^2, x, 9, (c^2*(2 - 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x) - (c^2*(2 + 3*a*x)*(1 - a^2*x^2)^(3/2))/(6*a^4*x^3) + (c^2*ArcSin[a*x])/a + (3*c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^1, x, 8, (c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(a^2*x) + (c*ArcSin[a*x])/a + (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^1, x, 6, -((1 + a*x)/(a*c*Sqrt[1 - a^2*x^2])) - Sqrt[1 - a^2*x^2]/(a*c) + ArcSin[a*x]/(a*c)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^2, x, 6, (a^2*x^3*(1 + a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) - (x*(3 + 4*a*x))/(3*c^2*Sqrt[1 - a^2*x^2]) - (8*Sqrt[1 - a^2*x^2])/(3*a*c^2) + ArcSin[a*x]/(a*c^2)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^3, x, 7, -((a^4*x^5*(1 + a*x))/(5*c^3*(1 - a^2*x^2)^(5/2))) + (a^2*x^3*(5 + 6*a*x))/(15*c^3*(1 - a^2*x^2)^(3/2)) - (x*(5 + 8*a*x))/(5*c^3*Sqrt[1 - a^2*x^2]) - (16*Sqrt[1 - a^2*x^2])/(5*a*c^3) + ArcSin[a*x]/(a*c^3)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^4, x, 8, (a^6*x^7*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) - (a^4*x^5*(7 + 8*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*x^3*(35 + 48*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) - (x*(35 + 64*a*x))/(35*c^4*Sqrt[1 - a^2*x^2]) - (128*Sqrt[1 - a^2*x^2])/(35*a*c^4) + ArcSin[a*x]/(a*c^4)} - - -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^5, x, 4, c^5/(9*a^10*x^9) + c^5/(4*a^9*x^8) - (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) + (2*c^5)/(5*a^6*x^5) + (3*c^5)/(a^5*x^4) + (2*c^5)/(3*a^4*x^3) - (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) - c^5*x - (2*c^5*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^4, x, 4, -c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) + (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) - (2*c^4)/(a^2*x) - c^4*x - (2*c^4*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3, x, 4, c^3/(5*a^6*x^5) + c^3/(2*a^5*x^4) - c^3/(3*a^4*x^3) - (2*c^3)/(a^3*x^2) - c^3/(a^2*x) - c^3*x - (2*c^3*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^2, x, 4, -c^2/(3*a^4*x^3) - c^2/(a^3*x^2) - c^2*x - (2*c^2*Log[x])/a} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2)), x, 4, c/(a^2*x) - c*x - (2*c*Log[x])/a} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2)), x, 4, -(x/c) - 1/(a*c*(1 - a*x)) - (2*Log[1 - a*x])/(a*c)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^2, x, 4, -(x/c^2) + 1/(4*a*c^2*(1 - a*x)^2) - 7/(4*a*c^2*(1 - a*x)) - (17*Log[1 - a*x])/(8*a*c^2) + Log[1 + a*x]/(8*a*c^2)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^3, x, 4, -(x/c^3) - 1/(12*a*c^3*(1 - a*x)^3) + 5/(8*a*c^3*(1 - a*x)^2) - 39/(16*a*c^3*(1 - a*x)) + 1/(16*a*c^3*(1 + a*x)) - (9*Log[1 - a*x])/(4*a*c^3) + Log[1 + a*x]/(4*a*c^3)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^4, x, 4, -(x/c^4) + 1/(32*a*c^4*(1 - a*x)^4) - 13/(48*a*c^4*(1 - a*x)^3) + 35/(32*a*c^4*(1 - a*x)^2) - 99/(32*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)^2) + 11/(64*a*c^4*(1 + a*x)) - (303*Log[1 - a*x])/(128*a*c^4) + (47*Log[1 + a*x])/(128*a*c^4)} - - -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4, x, 12, -((3*c^4*(16 - 5*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x)) + (c^4*(16 + 5*a*x)*(1 - a^2*x^2)^(3/2))/(16*a^4*x^3) - (c^4*(24 + 5*a*x)*(1 - a^2*x^2)^(5/2))/(40*a^6*x^5) - (c^4*(1 - a^2*x^2)^(7/2))/(7*a^8*x^7) - (c^4*(1 - a^2*x^2)^(7/2))/(2*a^7*x^6) - (3*c^4*ArcSin[a*x])/a - (15*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^3, x, 11, -((3*c^3*(8 - a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x)) + (c^3*(8 + a*x)*(1 - a^2*x^2)^(3/2))/(8*a^4*x^3) + (c^3*(1 - a^2*x^2)^(5/2))/(5*a^6*x^5) + (3*c^3*(1 - a^2*x^2)^(5/2))/(4*a^5*x^4) - (3*c^3*ArcSin[a*x])/a - (3*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^2, x, 10, -((c^2*(6 + a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x)) - (c^2*(1 - a^2*x^2)^(3/2))/(3*a^4*x^3) - (3*c^2*(1 - a^2*x^2)^(3/2))/(2*a^3*x^2) - (3*c^2*ArcSin[a*x])/a + (c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^1, x, 9, (c*Sqrt[1 - a^2*x^2])/a + (c*Sqrt[1 - a^2*x^2])/(a^2*x) - (3*c*ArcSin[a*x])/a + (3*c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^1, x, 7, -((1 + a*x)^3/(3*a*c*(1 - a^2*x^2)^(3/2))) + (2*(1 + a*x)^2)/(a*c*Sqrt[1 - a^2*x^2]) + (3*Sqrt[1 - a^2*x^2])/(a*c) - (3*ArcSin[a*x])/(a*c)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^2, x, 7, (1 + a*x)^3/(5*a*c^2*(1 - a^2*x^2)^(5/2)) - (6*(1 + a*x)^2)/(5*a*c^2*(1 - a^2*x^2)^(3/2)) + (24*(1 + a*x))/(5*a*c^2*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^2) - (3*ArcSin[a*x])/(a*c^2)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^3, x, 8, -((1 + a*x)^3/(7*a*c^3*(1 - a^2*x^2)^(7/2))) + (38*(1 + a*x)^2)/(35*a*c^3*(1 - a^2*x^2)^(5/2)) - (137*(1 + a*x))/(35*a*c^3*(1 - a^2*x^2)^(3/2)) + (245 + 181*a*x)/(35*a*c^3*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^3) - (3*ArcSin[a*x])/(a*c^3)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^4, x, 9, (1 + a*x)^3/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (22*(1 + a*x)^2)/(21*a*c^4*(1 - a^2*x^2)^(7/2)) + (478*(1 + a*x))/(105*a*c^4*(1 - a^2*x^2)^(5/2)) - (2*(1155 + 829*a*x))/(315*a*c^4*(1 - a^2*x^2)^(3/2)) + (4*(630 + 431*a*x))/(315*a*c^4*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c^4) - (3*ArcSin[a*x])/(a*c^4)} - - -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^5, x, 4, c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^4, x, 4, -c^4/(7*a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^3, x, 4, c^3/(5*a^6*x^5) + c^3/(a^5*x^4) + (5*c^3)/(3*a^4*x^3) - (5*c^3)/(a^2*x) + c^3*x + (4*c^3*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^2, x, 4, -c^2/(3*a^4*x^3) - (2*c^2)/(a^3*x^2) - (6*c^2)/(a^2*x) + c^2*x + (4*c^2*Log[x])/a} -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2)), x, 4, c/(a^2*x) + c*x - (4*c*Log[x])/a + (8*c*Log[1 - a*x])/a} -{E^(4*ArcTanh[a*x])/(c - c/(a^2*x^2)), x, 4, x/c - 1/(a*c*(1 - a*x)^2) + 5/(a*c*(1 - a*x)) + (4*Log[1 - a*x])/(a*c)} -{E^(4*ArcTanh[a*x])/(c - c/(a^2*x^2))^2, x, 4, x/c^2 + 1/(3*a*c^2*(1 - a*x)^3) - 2/(a*c^2*(1 - a*x)^2) + 6/(a*c^2*(1 - a*x)) + (4*Log[1 - a*x])/(a*c^2)} -{E^(4*ArcTanh[a*x])/(c - c/(a^2*x^2))^3, x, 4, x/c^3 - 1/(8*a*c^3*(1 - a*x)^4) + 11/(12*a*c^3*(1 - a*x)^3) - 49/(16*a*c^3*(1 - a*x)^2) + 111/(16*a*c^3*(1 - a*x)) + (129*Log[1 - a*x])/(32*a*c^3) - Log[1 + a*x]/(32*a*c^3)} -{E^(4*ArcTanh[a*x])/(c - c/(a^2*x^2))^4, x, 4, x/c^4 + 1/(20*a*c^4*(1 - a*x)^5) - 7/(16*a*c^4*(1 - a*x)^4) + 83/(48*a*c^4*(1 - a*x)^3) - 67/(16*a*c^4*(1 - a*x)^2) + 501/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (261*Log[1 - a*x])/(64*a*c^4) - (5*Log[1 + a*x])/(64*a*c^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a^2*x^2))^4/E^ArcTanh[a*x], x, 11, (c^4*(16 + 35*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) - (c^4*(16 - 35*a*x)*(1 - a^2*x^2)^(3/2))/(48*a^4*x^3) + (c^4*(24 - 35*a*x)*(1 - a^2*x^2)^(5/2))/(120*a^6*x^5) - (c^4*(6 - 7*a*x)*(1 - a^2*x^2)^(7/2))/(42*a^8*x^7) + (c^4*ArcSin[a*x])/a - (35*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)} -{(c - c/(a^2*x^2))^3/E^ArcTanh[a*x], x, 10, (c^3*(8 + 15*a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) - (c^3*(8 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(24*a^4*x^3) + (c^3*(4 - 5*a*x)*(1 - a^2*x^2)^(5/2))/(20*a^6*x^5) + (c^3*ArcSin[a*x])/a - (15*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)} -{(c - c/(a^2*x^2))^2/E^ArcTanh[a*x], x, 9, (c^2*(2 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x) - (c^2*(2 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(6*a^4*x^3) + (c^2*ArcSin[a*x])/a - (3*c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{(c - c/(a^2*x^2))/E^ArcTanh[a*x], x, 8, (c*(1 + a*x)*Sqrt[1 - a^2*x^2])/(a^2*x) + (c*ArcSin[a*x])/a - (c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))), x, 6, (1 - a*x)/(a*c*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a*c) + ArcSin[a*x]/(a*c)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^2), x, 6, (a^2*x^3*(1 - a*x))/(3*c^2*(1 - a^2*x^2)^(3/2)) - (x*(3 - 4*a*x))/(3*c^2*Sqrt[1 - a^2*x^2]) + (8*Sqrt[1 - a^2*x^2])/(3*a*c^2) + ArcSin[a*x]/(a*c^2)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^3), x, 7, -((a^4*x^5*(1 - a*x))/(5*c^3*(1 - a^2*x^2)^(5/2))) + (a^2*x^3*(5 - 6*a*x))/(15*c^3*(1 - a^2*x^2)^(3/2)) - (x*(5 - 8*a*x))/(5*c^3*Sqrt[1 - a^2*x^2]) + (16*Sqrt[1 - a^2*x^2])/(5*a*c^3) + ArcSin[a*x]/(a*c^3)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^4), x, 8, (a^6*x^7*(1 - a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) - (a^4*x^5*(7 - 8*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*x^3*(35 - 48*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) - (x*(35 - 64*a*x))/(35*c^4*Sqrt[1 - a^2*x^2]) + (128*Sqrt[1 - a^2*x^2])/(35*a*c^4) + ArcSin[a*x]/(a*c^4)} - - -{(c - c/(a^2*x^2))^4/E^(2*ArcTanh[a*x]), x, 4, -c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) + (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) - (2*c^4)/(a^2*x) - c^4*x + (2*c^4*Log[x])/a} -{(c - c/(a^2*x^2))^3/E^(2*ArcTanh[a*x]), x, 4, c^3/(5*a^6*x^5) - c^3/(2*a^5*x^4) - c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) - c^3/(a^2*x) - c^3*x + (2*c^3*Log[x])/a} -{(c - c/(a^2*x^2))^2/E^(2*ArcTanh[a*x]), x, 4, -c^2/(3*a^4*x^3) + c^2/(a^3*x^2) - c^2*x + (2*c^2*Log[x])/a} -{(c - c/(a^2*x^2))/E^(2*ArcTanh[a*x]), x, 4, c/(a^2*x) - c*x + (2*c*Log[x])/a} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))), x, 4, -(x/c) + 1/(a*c*(1 + a*x)) + (2*Log[1 + a*x])/(a*c)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^2), x, 4, -(x/c^2) - 1/(4*a*c^2*(1 + a*x)^2) + 7/(4*a*c^2*(1 + a*x)) - Log[1 - a*x]/(8*a*c^2) + (17*Log[1 + a*x])/(8*a*c^2)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^3), x, 4, -(x/c^3) - 1/(16*a*c^3*(1 - a*x)) + 1/(12*a*c^3*(1 + a*x)^3) - 5/(8*a*c^3*(1 + a*x)^2) + 39/(16*a*c^3*(1 + a*x)) - Log[1 - a*x]/(4*a*c^3) + (9*Log[1 + a*x])/(4*a*c^3)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^4), x, 4, -(x/c^4) + 1/(64*a*c^4*(1 - a*x)^2) - 11/(64*a*c^4*(1 - a*x)) - 1/(32*a*c^4*(1 + a*x)^4) + 13/(48*a*c^4*(1 + a*x)^3) - 35/(32*a*c^4*(1 + a*x)^2) + 99/(32*a*c^4*(1 + a*x)) - (47*Log[1 - a*x])/(128*a*c^4) + (303*Log[1 + a*x])/(128*a*c^4)} - - -{(c - c/(a^2*x^2))^4/E^(3*ArcTanh[a*x]), x, 12, -((3*c^4*(16 + 5*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x)) + (c^4*(16 - 5*a*x)*(1 - a^2*x^2)^(3/2))/(16*a^4*x^3) - (c^4*(24 - 5*a*x)*(1 - a^2*x^2)^(5/2))/(40*a^6*x^5) - (c^4*(1 - a^2*x^2)^(7/2))/(7*a^8*x^7) + (c^4*(1 - a^2*x^2)^(7/2))/(2*a^7*x^6) - (3*c^4*ArcSin[a*x])/a + (15*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)} -{(c - c/(a^2*x^2))^3/E^(3*ArcTanh[a*x]), x, 11, -((3*c^3*(8 + a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x)) + (c^3*(8 - a*x)*(1 - a^2*x^2)^(3/2))/(8*a^4*x^3) + (c^3*(1 - a^2*x^2)^(5/2))/(5*a^6*x^5) - (3*c^3*(1 - a^2*x^2)^(5/2))/(4*a^5*x^4) - (3*c^3*ArcSin[a*x])/a + (3*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)} -{(c - c/(a^2*x^2))^2/E^(3*ArcTanh[a*x]), x, 10, -((c^2*(6 - a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x)) - (c^2*(1 - a^2*x^2)^(3/2))/(3*a^4*x^3) + (3*c^2*(1 - a^2*x^2)^(3/2))/(2*a^3*x^2) - (3*c^2*ArcSin[a*x])/a - (c^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)} -{(c - c/(a^2*x^2))/E^(3*ArcTanh[a*x]), x, 9, -((c*Sqrt[1 - a^2*x^2])/a) + (c*Sqrt[1 - a^2*x^2])/(a^2*x) - (3*c*ArcSin[a*x])/a - (3*c*ArcTanh[Sqrt[1 - a^2*x^2]])/a} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))), x, 7, (1 - a*x)^3/(3*a*c*(1 - a^2*x^2)^(3/2)) - (2*(1 - a*x)^2)/(a*c*Sqrt[1 - a^2*x^2]) - (3*Sqrt[1 - a^2*x^2])/(a*c) - (3*ArcSin[a*x])/(a*c)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^2), x, 7, -((1 - a*x)^3/(5*a*c^2*(1 - a^2*x^2)^(5/2))) + (6*(1 - a*x)^2)/(5*a*c^2*(1 - a^2*x^2)^(3/2)) - (24*(1 - a*x))/(5*a*c^2*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^2) - (3*ArcSin[a*x])/(a*c^2)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^3), x, 8, (1 - a*x)^3/(7*a*c^3*(1 - a^2*x^2)^(7/2)) - (38*(1 - a*x)^2)/(35*a*c^3*(1 - a^2*x^2)^(5/2)) + (137*(1 - a*x))/(35*a*c^3*(1 - a^2*x^2)^(3/2)) - (245 - 181*a*x)/(35*a*c^3*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^3) - (3*ArcSin[a*x])/(a*c^3)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4), x, 9, -((1 - a*x)^3/(9*a*c^4*(1 - a^2*x^2)^(9/2))) + (22*(1 - a*x)^2)/(21*a*c^4*(1 - a^2*x^2)^(7/2)) - (478*(1 - a*x))/(105*a*c^4*(1 - a^2*x^2)^(5/2)) + (2*(1155 - 829*a*x))/(315*a*c^4*(1 - a^2*x^2)^(3/2)) - (4*(630 - 431*a*x))/(315*a*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a^2*x^2]/(a*c^4) - (3*ArcSin[a*x])/(a*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(9/2), x, 4, -((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) - (a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2)) + (2*a^2*(c - c/(a^2*x^2))^(9/2)*x^3)/(3*(1 - a^2*x^2)^(9/2)) + (4*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a^2*x^2)^(9/2)) - (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) - (2*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1 - a^2*x^2)^(9/2) + (2*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) + (4*a^7*(c - c/(a^2*x^2))^(9/2)*x^8)/(1 - a^2*x^2)^(9/2) + (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) + (a^8*(c - c/(a^2*x^2))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(7/2), x, 4, -((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) - (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2)) + (3*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) + (a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(1 - a^2*x^2)^(7/2) - (3*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) - (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(1 - a^2*x^2)^(7/2) - (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) - (a^6*(c - c/(a^2*x^2))^(7/2)*x^7*Log[x])/(1 - a^2*x^2)^(7/2)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(5/2), x, 4, -((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(3*(1 - a^2*x^2)^(5/2)) + (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(5/2) + (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) + (a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a^2*x^2)^(5/2)} -{E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(3/2), x, 4, -((c - c/(a^2*x^2))^(3/2)*x)/(2*(1 - a^2*x^2)^(3/2)) - (a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a^2*x^2)^(3/2) - (a^3*(c - c/(a^2*x^2))^(3/2)*x^4)/(1 - a^2*x^2)^(3/2) - (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*Log[x])/(1 - a^2*x^2)^(3/2)} -{E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)], x, 4, (a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{E^ArcTanh[a*x]/Sqrt[c - c/(a^2*x^2)], x, 4, -(Sqrt[1 - a^2*x^2]/(a*Sqrt[c - c/(a^2*x^2)])) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^(3/2), x, 4, (1 - a^2*x^2)^(3/2)/(a^3*(c - c/(a^2*x^2))^(3/2)*x^2) + (1 - a^2*x^2)^(3/2)/(2*a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)) + (5*(1 - a^2*x^2)^(3/2)*Log[1 - a*x])/(4*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) - ((1 - a^2*x^2)^(3/2)*Log[1 + a*x])/(4*a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^(5/2), x, 4, -((1 - a^2*x^2)^(5/2)/(a^5*(c - c/(a^2*x^2))^(5/2)*x^4)) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)^2) - (1 - a^2*x^2)^(5/2)/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)) - (23*(1 - a^2*x^2)^(5/2)*Log[1 - a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) + (7*(1 - a^2*x^2)^(5/2)*Log[1 + a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{E^ArcTanh[a*x]/(c - c/(a^2*x^2))^(7/2), x, 4, (1 - a^2*x^2)^(7/2)/(a^7*(c - c/(a^2*x^2))^(7/2)*x^6) + (1 - a^2*x^2)^(7/2)/(24*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^3) - (11*(1 - a^2*x^2)^(7/2))/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^2) + (3*(1 - a^2*x^2)^(7/2))/(2*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)) + (1 - a^2*x^2)^(7/2)/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^2) - (5*(1 - a^2*x^2)^(7/2))/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)) + (51*(1 - a^2*x^2)^(7/2)*Log[1 - a*x])/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (19*(1 - a^2*x^2)^(7/2)*Log[1 + a*x])/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(9/2), x, 16, (295*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(1344*(1 - a*x)^4) - (501*a^8*(c - c/(a^2*x^2))^(9/2)*x^9)/(128*(1 - a*x)^4*(1 + a*x)^4) + (373*a^7*(c - c/(a^2*x^2))^(9/2)*x^8)/(192*(1 - a*x)^4*(1 + a*x)^3) + (501*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(640*(1 - a*x)^4*(1 + a*x)^2) + (661*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1680*(1 - a*x)^4*(1 + a*x)) - (127*a^3*(c - c/(a^2*x^2))^(9/2)*x^4*(1 + a*x))/(420*(1 - a*x)^4) + (71*a^2*(c - c/(a^2*x^2))^(9/2)*x^3*(1 + a*x))/(336*(1 - a*x)^3) - (a*(c - c/(a^2*x^2))^(9/2)*x^2*(1 + a*x))/(28*(1 - a*x)^2) - ((c - c/(a^2*x^2))^(9/2)*x*(1 + a*x))/(8*(1 - a*x)) + (2*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcSin[a*x])/((1 - a*x)^(9/2)*(1 + a*x)^(9/2)) + (245*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(128*(1 - a*x)^(9/2)*(1 + a*x)^(9/2))} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2), x, 14, -((11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3)) + (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) - (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) - (57*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) + (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) - (a*(c - c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) - ((c - c/(a^2*x^2))^(7/2)*x*(1 + a*x))/(6*(1 - a*x)) - (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) - (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2), x, 12, (5*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(8*(1 - a*x)^2) - (25*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2) + (17*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(12*(1 - a*x)^2*(1 + a*x)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2*(1 + a*x))/(6*(1 - a*x)^2) - ((c - c/(a^2*x^2))^(5/2)*x*(1 + a*x))/(4*(1 - a*x)) + (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) + (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2), x, 10, -((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a*x)) + (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(3/2)*x*(1 + a*x))/(2*(1 - a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) - (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)], x, 8, (-Sqrt[c - c/(a^2*x^2)])*x + (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcTanh[a*x])/Sqrt[c - c/(a^2*x^2)], x, 6, (2*(1 - a*x)*(1 + a*x))/(a^2*Sqrt[c - c/(a^2*x^2)]*x) + (1 + a*x)^2/(a^2*Sqrt[c - c/(a^2*x^2)]*x) - (2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcSin[a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2), x, 6, (1 + a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x) - (2*(5 - 2*a*x)*(1 - a*x)*(1 + a*x)^2)/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) + (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2), x, 8, (1 + a*x)^2/(5*a^2*(c - c/(a^2*x^2))^(5/2)*x) - (2*(1 - a*x)*(1 + a*x)^2)/(3*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) + (58*(1 - a*x)^2*(1 + a*x)^2)/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) + (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 43*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) - (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2), x, 10, (1 + a*x)^2/(7*a^2*(c - c/(a^2*x^2))^(7/2)*x) - (2*(1 - a*x)*(1 + a*x)^2)/(5*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) + (124*(1 - a*x)^2*(1 + a*x)^2)/(105*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) - (782*(1 - a*x)^3*(1 + a*x)^2)/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) - (142*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) - (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 107*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} -{E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(9/2), x, 12, (1 + a*x)^2/(9*a^2*(c - c/(a^2*x^2))^(9/2)*x) - (2*(1 - a*x)*(1 + a*x)^2)/(7*a^3*(c - c/(a^2*x^2))^(9/2)*x^2) + (214*(1 - a*x)^2*(1 + a*x)^2)/(315*a^4*(c - c/(a^2*x^2))^(9/2)*x^3) - (646*(1 - a*x)^3*(1 + a*x)^2)/(315*a^5*(c - c/(a^2*x^2))^(9/2)*x^4) + (302*(1 - a*x)^4*(1 + a*x)^2)/(21*a^6*(c - c/(a^2*x^2))^(9/2)*x^5) + (2458*(1 - a*x)^5*(1 + a*x)^2)/(315*a^7*(c - c/(a^2*x^2))^(9/2)*x^6) + (1334*(1 - a*x)^5*(1 + a*x)^3)/(315*a^8*(c - c/(a^2*x^2))^(9/2)*x^7) + (2*(1 - a*x)^5*(1 + a*x)^4*(704 + 1019*a*x))/(315*a^10*(c - c/(a^2*x^2))^(9/2)*x^9) - (2*(1 - a*x)^(9/2)*(1 + a*x)^(9/2)*ArcSin[a*x])/(a^10*(c - c/(a^2*x^2))^(9/2)*x^9)} - - -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(9/2), x, 4, -((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) - (3*a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2)) + (8*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a^2*x^2)^(9/2)) + (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) - (2*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1 - a^2*x^2)^(9/2) - (4*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) - (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) - (3*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2), x, 4, -((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) - (3*a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2)) - (a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) + (5*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a^2*x^2)^(7/2)) + (5*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) - (a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(1 - a^2*x^2)^(7/2) + (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) + (3*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*Log[x])/(1 - a^2*x^2)^(7/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2), x, 4, -((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(1 - a^2*x^2)^(5/2) - (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(5/2) - (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) - (3*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a^2*x^2)^(5/2)} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2), x, 4, -((c - c/(a^2*x^2))^(3/2)*x)/(2*(1 - a^2*x^2)^(3/2)) - (3*a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a^2*x^2)^(3/2) + (a^3*(c - c/(a^2*x^2))^(3/2)*x^4)/(1 - a^2*x^2)^(3/2) + (3*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*Log[x])/(1 - a^2*x^2)^(3/2)} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)], x, 4, -((a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{E^(3*ArcTanh[a*x])/Sqrt[c - c/(a^2*x^2)], x, 4, Sqrt[1 - a^2*x^2]/(a*Sqrt[c - c/(a^2*x^2)]) + (2*Sqrt[1 - a^2*x^2])/(a^2*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)) + (3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2), x, 4, -((1 - a^2*x^2)^(3/2)/(a^3*(c - c/(a^2*x^2))^(3/2)*x^2)) + (1 - a^2*x^2)^(3/2)/(2*a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)^2) - (3*(1 - a^2*x^2)^(3/2))/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)) - (3*(1 - a^2*x^2)^(3/2)*Log[1 - a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2), x, 4, (1 - a^2*x^2)^(5/2)/(a^5*(c - c/(a^2*x^2))^(5/2)*x^4) + (1 - a^2*x^2)^(5/2)/(6*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)^3) - (9*(1 - a^2*x^2)^(5/2))/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)^2) + (31*(1 - a^2*x^2)^(5/2))/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)) + (49*(1 - a^2*x^2)^(5/2)*Log[1 - a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) - ((1 - a^2*x^2)^(5/2)*Log[1 + a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2), x, 4, -((1 - a^2*x^2)^(7/2)/(a^7*(c - c/(a^2*x^2))^(7/2)*x^6)) + (1 - a^2*x^2)^(7/2)/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^4) - (1 - a^2*x^2)^(7/2)/(2*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^3) + (59*(1 - a^2*x^2)^(7/2))/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^2) - (75*(1 - a^2*x^2)^(7/2))/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)) + (1 - a^2*x^2)^(7/2)/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)) - (201*(1 - a^2*x^2)^(7/2)*Log[1 - a*x])/(64*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (9*(1 - a^2*x^2)^(7/2)*Log[1 + a*x])/(64*a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a^2*x^2))^(9/2)/E^ArcTanh[a*x], x, 4, -((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) + (a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2)) + (2*a^2*(c - c/(a^2*x^2))^(9/2)*x^3)/(3*(1 - a^2*x^2)^(9/2)) - (4*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a^2*x^2)^(9/2)) - (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) + (2*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1 - a^2*x^2)^(9/2) + (2*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) - (4*a^7*(c - c/(a^2*x^2))^(9/2)*x^8)/(1 - a^2*x^2)^(9/2) - (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) + (a^8*(c - c/(a^2*x^2))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)} -{(c - c/(a^2*x^2))^(7/2)/E^ArcTanh[a*x], x, 4, -((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) + (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2)) + (3*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) - (a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(1 - a^2*x^2)^(7/2) - (3*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(1 - a^2*x^2)^(7/2) + (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) - (a^6*(c - c/(a^2*x^2))^(7/2)*x^7*Log[x])/(1 - a^2*x^2)^(7/2)} -{(c - c/(a^2*x^2))^(5/2)/E^ArcTanh[a*x], x, 4, -((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(3*(1 - a^2*x^2)^(5/2)) + (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) - (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(5/2) - (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) + (a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a^2*x^2)^(5/2)} -{(c - c/(a^2*x^2))^(3/2)/E^ArcTanh[a*x], x, 4, -((c - c/(a^2*x^2))^(3/2)*x)/(2*(1 - a^2*x^2)^(3/2)) + (a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a^2*x^2)^(3/2) + (a^3*(c - c/(a^2*x^2))^(3/2)*x^4)/(1 - a^2*x^2)^(3/2) - (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*Log[x])/(1 - a^2*x^2)^(3/2)} -{Sqrt[c - c/(a^2*x^2)]/E^ArcTanh[a*x], x, 4, -((a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{1/(E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)]), x, 4, Sqrt[1 - a^2*x^2]/(a*Sqrt[c - c/(a^2*x^2)]) - (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(3/2)), x, 4, -((1 - a^2*x^2)^(3/2)/(a^3*(c - c/(a^2*x^2))^(3/2)*x^2)) + (1 - a^2*x^2)^(3/2)/(2*a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 + a*x)) - ((1 - a^2*x^2)^(3/2)*Log[1 - a*x])/(4*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) + (5*(1 - a^2*x^2)^(3/2)*Log[1 + a*x])/(4*a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(5/2)), x, 4, (1 - a^2*x^2)^(5/2)/(a^5*(c - c/(a^2*x^2))^(5/2)*x^4) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 - a*x)) + (1 - a^2*x^2)^(5/2)/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)^2) - (1 - a^2*x^2)^(5/2)/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)) + (7*(1 - a^2*x^2)^(5/2)*Log[1 - a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) - (23*(1 - a^2*x^2)^(5/2)*Log[1 + a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{1/(E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(7/2)), x, 4, -((1 - a^2*x^2)^(7/2)/(a^7*(c - c/(a^2*x^2))^(7/2)*x^6)) + (1 - a^2*x^2)^(7/2)/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)^2) - (5*(1 - a^2*x^2)^(7/2))/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)) + (1 - a^2*x^2)^(7/2)/(24*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^3) - (11*(1 - a^2*x^2)^(7/2))/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^2) + (3*(1 - a^2*x^2)^(7/2))/(2*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)) - (19*(1 - a^2*x^2)^(7/2)*Log[1 - a*x])/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (51*(1 - a^2*x^2)^(7/2)*Log[1 + a*x])/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -{(c - c/(a^2*x^2))^(9/2)/E^(2*ArcTanh[a*x]), x, 16, (11*a^8*(c - c/(a^2*x^2))^(9/2)*x^9)/(128*(1 - a*x)^4*(1 + a*x)^4) + (39*a^7*(c - c/(a^2*x^2))^(9/2)*x^8)/(64*(1 - a*x)^4*(1 + a*x)^3) - (11*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(640*(1 - a*x)^4*(1 + a*x)^2) + (a*(c - c/(a^2*x^2))^(9/2)*x^2)/(28*(1 + a*x)) - (103*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(160*(1 - a*x)^4*(1 + a*x)) + (629*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(960*(1 - a*x)^3*(1 + a*x)) - (2*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a*x)^2*(1 + a*x)) + (47*a^2*(c - c/(a^2*x^2))^(9/2)*x^3)/(336*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(9/2)*x*(1 - a*x))/(8*(1 + a*x)) - (2*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcSin[a*x])/((1 - a*x)^(9/2)*(1 + a*x)^(9/2)) + (245*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(128*(1 - a*x)^(9/2)*(1 + a*x)^(9/2))} -{(c - c/(a^2*x^2))^(7/2)/E^(2*ArcTanh[a*x]), x, 14, -((7*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3)) - (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(8*(1 - a*x)^3*(1 + a*x)^2) + (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(15*(1 + a*x)) + (19*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(16*(1 - a*x)^3*(1 + a*x)) - (2*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a*x)^2*(1 + a*x)) + (23*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(120*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(7/2)*x*(1 - a*x))/(6*(1 + a*x)) + (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) - (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))} -{(c - c/(a^2*x^2))^(5/2)/E^(2*ArcTanh[a*x]), x, 12, (7*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2) + (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(6*(1 + a*x)) - (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/((1 - a*x)^2*(1 + a*x)) + (7*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(24*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(5/2)*x*(1 - a*x))/(4*(1 + a*x)) - (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) + (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))} -{(c - c/(a^2*x^2))^(3/2)/E^(2*ArcTanh[a*x]), x, 10, (a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 + a*x) + (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) - ((c - c/(a^2*x^2))^(3/2)*x*(1 - a*x))/(2*(1 + a*x)) + (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) - (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))} -{Sqrt[c - c/(a^2*x^2)]/E^(2*ArcTanh[a*x]), x, 8, (-Sqrt[c - c/(a^2*x^2)])*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{1/(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]), x, 6, (1 - a*x)^2/(a^2*Sqrt[c - c/(a^2*x^2)]*x) + (2*(1 - a*x)*(1 + a*x))/(a^2*Sqrt[c - c/(a^2*x^2)]*x) + (2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcSin[a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2)), x, 6, (1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x) - (2*(1 - a*x)^2*(1 + a*x)*(5 + 2*a*x))/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) - (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2)), x, 8, (1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(5/2)*x) + (2*(1 - a*x)^3)/(5*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) - (2*(1 - a*x)^3*(1 + a*x))/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) + (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 13*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) + (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{1/(E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)), x, 10, (1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(7/2)*x) - (10*(1 - a*x)^3)/(3*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) - (12*(1 - a*x)^4)/(7*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) - (82*(1 - a*x)^4*(1 + a*x))/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) - (2*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) - (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 37*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -{(c - c/(a^2*x^2))^(9/2)/E^(3*ArcTanh[a*x]), x, 4, -((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) + (3*a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2)) - (8*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a^2*x^2)^(9/2)) + (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) + (2*a^5*(c - c/(a^2*x^2))^(9/2)*x^6)/(1 - a^2*x^2)^(9/2) - (4*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) + (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) - (3*a^8*(c - c/(a^2*x^2))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)} -{(c - c/(a^2*x^2))^(7/2)/E^(3*ArcTanh[a*x]), x, 4, -((c - c/(a^2*x^2))^(7/2)*x)/(6*(1 - a^2*x^2)^(7/2)) + (3*a*(c - c/(a^2*x^2))^(7/2)*x^2)/(5*(1 - a^2*x^2)^(7/2)) - (a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(4*(1 - a^2*x^2)^(7/2)) - (5*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a^2*x^2)^(7/2)) + (5*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(2*(1 - a^2*x^2)^(7/2)) + (a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(1 - a^2*x^2)^(7/2) - (a^7*(c - c/(a^2*x^2))^(7/2)*x^8)/(1 - a^2*x^2)^(7/2) + (3*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*Log[x])/(1 - a^2*x^2)^(7/2)} -{(c - c/(a^2*x^2))^(5/2)/E^(3*ArcTanh[a*x]), x, 4, -((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(1 - a^2*x^2)^(5/2) - (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) - (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(5/2) + (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) - (3*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a^2*x^2)^(5/2)} -{(c - c/(a^2*x^2))^(3/2)/E^(3*ArcTanh[a*x]), x, 4, -((c - c/(a^2*x^2))^(3/2)*x)/(2*(1 - a^2*x^2)^(3/2)) + (3*a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a^2*x^2)^(3/2) - (a^3*(c - c/(a^2*x^2))^(3/2)*x^4)/(1 - a^2*x^2)^(3/2) + (3*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*Log[x])/(1 - a^2*x^2)^(3/2)} -{Sqrt[c - c/(a^2*x^2)]/E^(3*ArcTanh[a*x]), x, 4, (a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]), x, 4, -(Sqrt[1 - a^2*x^2]/(a*Sqrt[c - c/(a^2*x^2)])) + (2*Sqrt[1 - a^2*x^2])/(a^2*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)) + (3*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2)), x, 4, (1 - a^2*x^2)^(3/2)/(a^3*(c - c/(a^2*x^2))^(3/2)*x^2) + (1 - a^2*x^2)^(3/2)/(2*a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 + a*x)^2) - (3*(1 - a^2*x^2)^(3/2))/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3*(1 + a*x)) - (3*(1 - a^2*x^2)^(3/2)*Log[1 + a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(5/2)), x, 4, -((1 - a^2*x^2)^(5/2)/(a^5*(c - c/(a^2*x^2))^(5/2)*x^4)) + (1 - a^2*x^2)^(5/2)/(6*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)^3) - (9*(1 - a^2*x^2)^(5/2))/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)^2) + (31*(1 - a^2*x^2)^(5/2))/(8*a^6*(c - c/(a^2*x^2))^(5/2)*x^5*(1 + a*x)) - ((1 - a^2*x^2)^(5/2)*Log[1 - a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) + (49*(1 - a^2*x^2)^(5/2)*Log[1 + a*x])/(16*a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)), x, 4, (1 - a^2*x^2)^(7/2)/(a^7*(c - c/(a^2*x^2))^(7/2)*x^6) + (1 - a^2*x^2)^(7/2)/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 - a*x)) + (1 - a^2*x^2)^(7/2)/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^4) - (1 - a^2*x^2)^(7/2)/(2*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^3) + (59*(1 - a^2*x^2)^(7/2))/(32*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)^2) - (75*(1 - a^2*x^2)^(7/2))/(16*a^8*(c - c/(a^2*x^2))^(7/2)*x^7*(1 + a*x)) + (9*(1 - a^2*x^2)^(7/2)*Log[1 - a*x])/(64*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (201*(1 - a^2*x^2)^(7/2)*Log[1 + a*x])/(64*a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)]*x^m, x, 4, (Sqrt[c - c/(a^2*x^2)]*x^(1 + m))/(m*Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)]*x^(2 + m))/((1 + m)*Sqrt[1 - a^2*x^2])} - -{E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)]*x^2, x, 4, (Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)]*x^4)/(3*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)]*x, x, 3, (Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] + (a*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)], x, 4, (a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)])/x, x, 4, -(Sqrt[c - c/(a^2*x^2)]/Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{(E^ArcTanh[a*x]*Sqrt[c - c/(a^2*x^2)])/x^2, x, 3, -((Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2)/(2*x*Sqrt[1 - a^2*x^2]))} - - -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3, x, 8, -((7*Sqrt[c - c/(a^2*x^2)]*x)/(8*a^3)) - (7*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(24*a^3) - (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)/(6*a^3) - (Sqrt[c - c/(a^2*x^2)]*x^2*(1 + a*x)^2)/(4*a^2) + (7*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(8*a^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2, x, 7, -((Sqrt[c - c/(a^2*x^2)]*x)/a^2) - (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(3*a^2) - (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(a^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x, x, 6, -((3*Sqrt[c - c/(a^2*x^2)]*x)/(2*a)) - (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(2*a) + (3*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(2*a*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)], x, 8, (-Sqrt[c - c/(a^2*x^2)])*x + (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x, x, 8, -Sqrt[c - c/(a^2*x^2)] + (a*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*a*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2, x, 6, (-(3/2))*a*Sqrt[c - c/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*(1 + a*x))/(2*x) - (3*a^2*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3, x, 7, (-a^2)*Sqrt[c - c/(a^2*x^2)] - (a*Sqrt[c - c/(a^2*x^2)]*(1 + a*x))/(3*x) - (Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2)/(3*x^2) - (a^3*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4, x, 9, (-(4/3))*a^3*Sqrt[c - c/(a^2*x^2)] - Sqrt[c - c/(a^2*x^2)]/(4*x^3) - (2*a*Sqrt[c - c/(a^2*x^2)])/(3*x^2) - (7*a^2*Sqrt[c - c/(a^2*x^2)])/(8*x) - (7*a^4*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5, x, 10, (-(6/5))*a^4*Sqrt[c - c/(a^2*x^2)] - Sqrt[c - c/(a^2*x^2)]/(5*x^4) - (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) - (3*a^2*Sqrt[c - c/(a^2*x^2)])/(5*x^2) - (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) - (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])} - - -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3, x, 4, (-4*Sqrt[c - c/(a^2*x^2)]*x^2)/(a^2*Sqrt[1 - a^2*x^2]) - (2*Sqrt[c - c/(a^2*x^2)]*x^3)/(a*Sqrt[1 - a^2*x^2]) - (Sqrt[c - c/(a^2*x^2)]*x^4)/Sqrt[1 - a^2*x^2] - (a*Sqrt[c - c/(a^2*x^2)]*x^5)/(4*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/(a^3*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2, x, 4, (-4*Sqrt[c - c/(a^2*x^2)]*x^2)/(a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)]*x^4)/(3*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/(a^2*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)]*x, x, 4, -((3*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/(a*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)], x, 4, -((a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x, x, 4, -(Sqrt[c - c/(a^2*x^2)]/Sqrt[1 - a^2*x^2]) + (3*a*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2, x, 4, (-3*a*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(2*x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3, x, 4, (-4*a^2*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(3*x^2*Sqrt[1 - a^2*x^2]) - (3*a*Sqrt[c - c/(a^2*x^2)])/(2*x*Sqrt[1 - a^2*x^2]) + (4*a^3*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^3*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4, x, 4, (-4*a^3*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(4*x^3*Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)])/(x^2*Sqrt[1 - a^2*x^2]) - (2*a^2*Sqrt[c - c/(a^2*x^2)])/(x*Sqrt[1 - a^2*x^2]) + (4*a^4*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5, x, 4, (-4*a^4*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(5*x^4*Sqrt[1 - a^2*x^2]) - (3*a*Sqrt[c - c/(a^2*x^2)])/(4*x^3*Sqrt[1 - a^2*x^2]) - (4*a^2*Sqrt[c - c/(a^2*x^2)])/(3*x^2*Sqrt[1 - a^2*x^2]) - (2*a^3*Sqrt[c - c/(a^2*x^2)])/(x*Sqrt[1 - a^2*x^2]) + (4*a^5*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^5*Sqrt[c - c/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sqrt[c - c/(a^2*x^2)]*x^m)/E^ArcTanh[a*x], x, 4, (Sqrt[c - c/(a^2*x^2)]*x^(1 + m))/(m*Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)]*x^(2 + m))/((1 + m)*Sqrt[1 - a^2*x^2])} - -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^ArcTanh[a*x], x, 4, (Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)]*x^4)/(3*Sqrt[1 - a^2*x^2])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^ArcTanh[a*x], x, 3, (Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] - (a*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2])} -{Sqrt[c - c/(a^2*x^2)]/E^ArcTanh[a*x], x, 4, -((a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^ArcTanh[a*x]*x), x, 4, -(Sqrt[c - c/(a^2*x^2)]/Sqrt[1 - a^2*x^2]) - (a*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^ArcTanh[a*x]*x^2), x, 3, -((Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2)/(2*x*Sqrt[1 - a^2*x^2]))} - - -{(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(2*ArcTanh[a*x]), x, 8, (7*Sqrt[c - c/(a^2*x^2)]*x)/(8*a^3) + (7*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(24*a^3) + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^2)/(6*a^3) - (Sqrt[c - c/(a^2*x^2)]*x^2*(1 - a*x)^2)/(4*a^2) + (7*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(8*a^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(2*ArcTanh[a*x]), x, 7, -((Sqrt[c - c/(a^2*x^2)]*x)/a^2) - (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(3*a^2) - (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^2)/(3*a^2) - (Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(a^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^(2*ArcTanh[a*x]), x, 6, (3*Sqrt[c - c/(a^2*x^2)]*x)/(2*a) + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(2*a) + (3*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(2*a*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/E^(2*ArcTanh[a*x]), x, 8, (-Sqrt[c - c/(a^2*x^2)])*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x), x, 8, -Sqrt[c - c/(a^2*x^2)] + (a*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (2*a*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^2), x, 6, (3/2)*a*Sqrt[c - c/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*(1 - a*x))/(2*x) - (3*a^2*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^3), x, 7, (-a^2)*Sqrt[c - c/(a^2*x^2)] + (a*Sqrt[c - c/(a^2*x^2)]*(1 - a*x))/(3*x) - (Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2)/(3*x^2) + (a^3*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^4), x, 9, (4/3)*a^3*Sqrt[c - c/(a^2*x^2)] - Sqrt[c - c/(a^2*x^2)]/(4*x^3) + (2*a*Sqrt[c - c/(a^2*x^2)])/(3*x^2) - (7*a^2*Sqrt[c - c/(a^2*x^2)])/(8*x) - (7*a^4*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcTanh[a*x])*x^5), x, 10, (-(6/5))*a^4*Sqrt[c - c/(a^2*x^2)] - Sqrt[c - c/(a^2*x^2)]/(5*x^4) + (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) - (3*a^2*Sqrt[c - c/(a^2*x^2)])/(5*x^2) + (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) + (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])} - - -{(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(3*ArcTanh[a*x]), x, 4, (-4*Sqrt[c - c/(a^2*x^2)]*x^2)/(a^2*Sqrt[1 - a^2*x^2]) + (2*Sqrt[c - c/(a^2*x^2)]*x^3)/(a*Sqrt[1 - a^2*x^2]) - (Sqrt[c - c/(a^2*x^2)]*x^4)/Sqrt[1 - a^2*x^2] + (a*Sqrt[c - c/(a^2*x^2)]*x^5)/(4*Sqrt[1 - a^2*x^2]) + (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/(a^3*Sqrt[1 - a^2*x^2])} -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(3*ArcTanh[a*x]), x, 4, (4*Sqrt[c - c/(a^2*x^2)]*x^2)/(a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)]*x^4)/(3*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/(a^2*Sqrt[1 - a^2*x^2])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^(3*ArcTanh[a*x]), x, 4, -((3*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)]*x^3)/(2*Sqrt[1 - a^2*x^2]) + (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/(a*Sqrt[1 - a^2*x^2])} -{Sqrt[c - c/(a^2*x^2)]/E^(3*ArcTanh[a*x]), x, 4, (a*Sqrt[c - c/(a^2*x^2)]*x^2)/Sqrt[1 - a^2*x^2] + (Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x), x, 4, -(Sqrt[c - c/(a^2*x^2)]/Sqrt[1 - a^2*x^2]) - (3*a*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] + (4*a*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^2), x, 4, (3*a*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(2*x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^3), x, 4, (-4*a^2*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(3*x^2*Sqrt[1 - a^2*x^2]) + (3*a*Sqrt[c - c/(a^2*x^2)])/(2*x*Sqrt[1 - a^2*x^2]) - (4*a^3*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] + (4*a^3*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^4), x, 4, (4*a^3*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(4*x^3*Sqrt[1 - a^2*x^2]) + (a*Sqrt[c - c/(a^2*x^2)])/(x^2*Sqrt[1 - a^2*x^2]) - (2*a^2*Sqrt[c - c/(a^2*x^2)])/(x*Sqrt[1 - a^2*x^2]) + (4*a^4*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^4*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcTanh[a*x])*x^5), x, 4, (-4*a^4*Sqrt[c - c/(a^2*x^2)])/Sqrt[1 - a^2*x^2] - Sqrt[c - c/(a^2*x^2)]/(5*x^4*Sqrt[1 - a^2*x^2]) + (3*a*Sqrt[c - c/(a^2*x^2)])/(4*x^3*Sqrt[1 - a^2*x^2]) - (4*a^2*Sqrt[c - c/(a^2*x^2)])/(3*x^2*Sqrt[1 - a^2*x^2]) + (2*a^3*Sqrt[c - c/(a^2*x^2)])/(x*Sqrt[1 - a^2*x^2]) - (4*a^5*Sqrt[c - c/(a^2*x^2)]*x*Log[x])/Sqrt[1 - a^2*x^2] + (4*a^5*Sqrt[c - c/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p with n symbolic*) - - -{(c - c/(a^2*x^2))^p/E^(2*p*ArcTanh[a*x]), x, 3, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[1 - 2*p, -2*p, 2 - 2*p, a*x])/((1 - a^2*x^2)^p*(1 - 2*p))} -{E^(2*p*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 3, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[1 - 2*p, -2*p, 2 - 2*p, (-a)*x])/((1 - a^2*x^2)^p*(1 - 2*p))} - - -{E^(n*ArcTanh[a*x])*(c - c/(a^2*x^2))^2, x, 10, -((4*c^2*(1 - a*x)^(3 - n/2)*(1 + a*x)^((1/2)*(-4 + n)))/(a*(4 - n))) - (c^2*(1 - a*x)^(3 - n/2)*(1 + a*x)^((1/2)*(-4 + n)))/(3*a^4*x^3) - (c^2*(10 + n)*(1 - a*x)^(3 - n/2)*(1 + a*x)^((1/2)*(-4 + n)))/(6*a^3*x^2) - (c^2*(14 + 5*n + n^2)*(1 - a*x)^(3 - n/2)*(1 + a*x)^((1/2)*(-4 + n)))/(6*a^2*x) - (c^2*n*(10 - n^2)*(1 - a*x)^(2 - n/2)*(1 + a*x)^((1/2)*(-4 + n))*Hypergeometric2F1[1, (1/2)*(-4 + n), (1/2)*(-2 + n), (1 + a*x)/(1 - a*x)])/(3*a*(4 - n)) + (2^(-1 + n/2)*c^2*n*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[(4 - n)/2, 3 - n/2, 4 - n/2, (1/2)*(1 - a*x)])/(a*(24 - 10*n + n^2))} -{E^(n*ArcTanh[a*x])*(c - c/(a^2*x^2))^1, x, 5, (4*c*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (1 - a*x)/(1 + a*x)])/(a*(2 - n)) - (2^(1 + n/2)*c*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(a*(2 - n))} -{E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^1, x, 5, -(((1 - n)*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a*c*n))) + (x*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*c) - (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a*c))} -{E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^2, x, 11, ((1 - n)*(3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a*c^2*(2 - n)) + ((3 + n)*x*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/c^2 - (a^2*x^3*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/c^2 + ((1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a*c^2*(2 - n)) - (1 + a*x)^((1/2)*(-2 + n))/((1 - a*x)^(n/2)*(a*c^2)) - ((3 + n)*(2 - n^2)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^(n/2))/(a*c^2*(4 - n^2)) - ((3 + n)*(2 - n^2)*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a*c^2*n*(4 - n^2))) - (2^(n/2)*n*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[(2 - n)/2, 1 - n/2, 2 - n/2, (1/2)*(1 - a*x)])/(a*c^2*(2 - n))} - - -{E^(n*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2), x, 9, -(((c - c/(a^2*x^2))^(3/2)*x*(1 - a*x)^((5 - n)/2)*(1 + a*x)^((1/2)*(-3 + n)))/(2*(1 - a^2*x^2)^(3/2))) - (a*(4 + n)*(c - c/(a^2*x^2))^(3/2)*x^2*(1 - a*x)^((5 - n)/2)*(1 + a*x)^((1/2)*(-3 + n)))/(2*(1 - a^2*x^2)^(3/2)) - (3*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)^((5 - n)/2)*(1 + a*x)^((1/2)*(-3 + n)))/((3 - n)*(1 - a^2*x^2)^(3/2)) - (a^2*(3 - n^2)*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Hypergeometric2F1[1, (1/2)*(-3 + n), (1/2)*(-1 + n), (1 + a*x)/(1 - a*x)])/((3 - n)*(1 - a^2*x^2)^(3/2)) + (2^((1/2)*(-1 + n))*a^2*n*(c - c/(a^2*x^2))^(3/2)*x^3*(1 - a*x)^((5 - n)/2)*Hypergeometric2F1[(3 - n)/2, (5 - n)/2, (7 - n)/2, (1/2)*(1 - a*x)])/((3 - n)*(5 - n)*(1 - a^2*x^2)^(3/2))} -{E^(n*ArcTanh[a*x])*(c - c/(a^2*x^2))^(1/2), x, 6, -((Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-1 + n)))/((1 - n)*Sqrt[1 - a^2*x^2])) + (2*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*n*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^((3 - n)/2)*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1/2)*(1 - a*x)])/((3 - 4*n + n^2)*Sqrt[1 - a^2*x^2])} -{E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^(1/2), x, 4, If[$VersionNumber>=8, -(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^2*(1 + n)*Sqrt[c - c/(a^2*x^2)]*x)) - (2^((3 + n)/2)*n*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 - n), (1 - n)/2, (3 - n)/2, (1/2)*(1 - a*x)])/(a^2*(1 - n^2)*Sqrt[c - c/(a^2*x^2)]*x), -(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^2*(1 + n)*Sqrt[c - c/(a^2*x^2)]*x)) - (2^((3 + n)/2)*n*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1/2)*(-1 - n), (1 - n)/2, (3 - n)/2, (1/2)*(1 - a*x)])/(a^2*(1 - n^2)*Sqrt[c - c/(a^2*x^2)]*x)]} -{E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^(3/2), x, 5, -(((1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(3/2))/(a^2*(c - c/(a^2*x^2))^(3/2)*x)) + ((1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(2 + 2*n + n^2 - a*n*(3 + 2*n)*x)*(1 - a^2*x^2)^(3/2))/(a^4*(1 - n^2)*(c - c/(a^2*x^2))^(3/2)*x^3) - (2^((1/2)*(-1 + n))*n*(1 - a*x)^((3 - n)/2)*(1 - a^2*x^2)^(3/2)*Hypergeometric2F1[(3 - n)/2, (3 - n)/2, (5 - n)/2, (1/2)*(1 - a*x)])/(a^4*(3 - n)*(c - c/(a^2*x^2))^(3/2)*x^3)} -{E^(n*ArcTanh[a*x])/(c - c/(a^2*x^2))^(5/2), x, 18, If[$VersionNumber>=8, ((4 + n)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^3*(3 + n)*(c - c/(a^2*x^2))^(5/2)*x^2) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^2*(c - c/(a^2*x^2))^(5/2)*x) + (n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(2 - n)*(4 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(9 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(4 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^5*(3 + n)*(c - c/(a^2*x^2))^(5/2)*x^4) - (2*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) + (2*n*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(3 - 4*n + n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*(4 + n)*(1 + 2*n - n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(3 - n)*(1 + n)*(3 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(4 + n)*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(9 - 10*n^2 + n^4)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1 + n)/2)*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) - (2^((3 + n)/2)*n*(1 - a*x)^((1/2)*(-1 - n))*(1 - a^2*x^2)^(5/2)*Hypergeometric2F1[(1/2)*(-1 - n), (1/2)*(-1 - n), (1 - n)/2, (1/2)*(1 - a*x)])/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5), ((4 + n)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^3*(3 + n)*(c - c/(a^2*x^2))^(5/2)*x^2) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^2*(c - c/(a^2*x^2))^(5/2)*x) + (n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(2 - n)*(4 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(9 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(4 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^5*(3 + n)*(c - c/(a^2*x^2))^(5/2)*x^4) - (2*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) + (2*n*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(3 - n - 3*n^2 + n^3)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*(4 + n)*(1 + 2*n - n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(9 + 9*n - n^2 - n^3)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(1 - n^2)*(c - c/(a^2*x^2))^(5/2)*x^5) - (3*(4 + n)*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*(1 - a^2*x^2)^(5/2))/(a^6*(9 - 10*n^2 + n^4)*(c - c/(a^2*x^2))^(5/2)*x^5) + (3*n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1 + n)/2)*(1 - a^2*x^2)^(5/2))/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5) - (2^((3 + n)/2)*n*(1 - a*x)^((1/2)*(-1 - n))*(1 - a^2*x^2)^(5/2)*Hypergeometric2F1[(1/2)*(-1 - n), (1/2)*(-1 - n), (1 - n)/2, (1/2)*(1 - a*x)])/(a^6*(1 + n)*(c - c/(a^2*x^2))^(5/2)*x^5)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p with p symbolic*) - - -{E^(n*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 3, ((c - c/(a^2*x^2))^p*x*AppellF1[1 - 2*p, (1/2)*(n - 2*p), -(n/2) - p, 2 - 2*p, a*x, (-a)*x])/((1 - a^2*x^2)^p*(1 - 2*p))} - - -{E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 13, (2*a*(c - c/(a^2*x^2))^p*x^2)/((1 - p)*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1/2)*(1 - 2*p), 2 - p, (1/2)*(3 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(1 - 2*p)) + (6*a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(1/2)*(3 - 2*p), 2 - p, (1/2)*(5 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(3 - 2*p)) + (a^4*(c - c/(a^2*x^2))^p*x^5*Hypergeometric2F1[(1/2)*(5 - 2*p), 2 - p, (1/2)*(7 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(5 - 2*p)) + (2*a^3*(c - c/(a^2*x^2))^p*x^4*Hypergeometric2F1[2 - p, 2 - p, 3 - p, a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(2 - p))} -{E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 7, ((c - c/(a^2*x^2))^p*x)/((1 - 2*p)*Sqrt[1 - a^2*x^2]) - (a*(c - c/(a^2*x^2))^p*x^2)/Sqrt[1 - a^2*x^2] + (3*a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(1/2)*(3 - 2*p), 3/2 - p, (1/2)*(5 - 2*p), a^2*x^2])/((1 - a^2*x^2)^p*(3 - 2*p)) + (a*(5 - 2*p)*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x^2])/((1 - a^2*x^2)^p*(2*(1 - p)))} -{E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 10, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1/2)*(1 - 2*p), 1 - p, (1/2)*(3 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(1 - 2*p)) + (a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(1/2)*(3 - 2*p), 1 - p, (1/2)*(5 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(3 - 2*p)) + (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(1 - p))} -{E^(1*ArcTanh[a*x])*(c - c/(a^2*x^2))^p, x, 5, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1/2)*(1 - 2*p), 1/2 - p, (1/2)*(3 - 2*p), a^2*x^2])/((1 - a^2*x^2)^p*(1 - 2*p)) + (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1/2 - p, 1 - p, 2 - p, a^2*x^2])/((1 - a^2*x^2)^p*(2*(1 - p)))} -{(c - c/(a^2*x^2))^p/E^ArcTanh[a*x], x, 5, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1/2)*(1 - 2*p), 1/2 - p, (1/2)*(3 - 2*p), a^2*x^2])/((1 - a^2*x^2)^p*(1 - 2*p)) - (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1/2 - p, 1 - p, 2 - p, a^2*x^2])/((1 - a^2*x^2)^p*(2*(1 - p)))} -{(c - c/(a^2*x^2))^p/E^(2*ArcTanh[a*x]), x, 10, ((c - c/(a^2*x^2))^p*x*Hypergeometric2F1[(1/2)*(1 - 2*p), 1 - p, (1/2)*(3 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(1 - 2*p)) + (a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(1/2)*(3 - 2*p), 1 - p, (1/2)*(5 - 2*p), a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(3 - 2*p)) - (a*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2])/((1 - a*x)^p*(1 + a*x)^p*(1 - p))} -{(c - c/(a^2*x^2))^p/E^(3*ArcTanh[a*x]), x, 7, ((c - c/(a^2*x^2))^p*x)/((1 - 2*p)*Sqrt[1 - a^2*x^2]) + (a*(c - c/(a^2*x^2))^p*x^2)/Sqrt[1 - a^2*x^2] + (3*a^2*(c - c/(a^2*x^2))^p*x^3*Hypergeometric2F1[(1/2)*(3 - 2*p), 3/2 - p, (1/2)*(5 - 2*p), a^2*x^2])/((1 - a^2*x^2)^p*(3 - 2*p)) - (a*(5 - 2*p)*(c - c/(a^2*x^2))^p*x^2*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x^2])/((1 - a^2*x^2)^p*(2*(1 - p)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u Sin[a x] E^(n ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sin[a x] E^ArcTanh[a x] (1+a x)^n*) - - -(* ::Subsubsection::Closed:: *) -(*n/2>0*) - - -{x*Sin[x]*E^ArcTanh[x]*(1 + x)^(1/2), x, 16, 3*Sqrt[1 - x]*Cos[x] - (1 - x)^(3/2)*Cos[x] - 3*Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] - (3/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + 2*Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + (3/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 3*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - (3/2)*Sqrt[1 - x]*Sin[x]} -{Sin[x]*E^ArcTanh[x]*(1 + x)^(1/2), x, 11, Sqrt[1 - x]*Cos[x] - Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] + 2*Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] - 2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1]} - -{x*Sin[x]*E^ArcTanh[x]*(1 - x)^(1/2), x, 13, Sqrt[1 + x]*Cos[x] - (1 + x)^(3/2)*Cos[x] - Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]] - (3/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]] + (3/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] - Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] + (3/2)*Sqrt[1 + x]*Sin[x]} -{Sin[x]*E^ArcTanh[x]*(1 - x)^(1/2), x, 7, (-Sqrt[1 + x])*Cos[x] + Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]] + Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1]} - - -{x*Sin[x]*E^ArcTanh[x]*(1 + x)^(3/2), x, 22, (17/4)*Sqrt[1 - x]*Cos[x] - 5*(1 - x)^(3/2)*Cos[x] + (1 - x)^(5/2)*Cos[x] + (15/4)*Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] - 4*Sqrt[2*Pi]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] - (15/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + 4*Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + (15/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 4*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] + (15/4)*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 4*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - (15/2)*Sqrt[1 - x]*Sin[x] + (5/2)*(1 - x)^(3/2)*Sin[x]} -{Sin[x]*E^ArcTanh[x]*(1 + x)^(3/2), x, 16, 4*Sqrt[1 - x]*Cos[x] - (1 - x)^(3/2)*Cos[x] - 2*Sqrt[2*Pi]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] - (3/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + 4*Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] + (3/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 4*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - 2*Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - (3/2)*Sqrt[1 - x]*Sin[x]} - -{x*Sin[x]*E^ArcTanh[x]*(1 - x)^(3/2), x, 19, (-(7/4))*Sqrt[1 + x]*Cos[x] - 3*(1 + x)^(3/2)*Cos[x] + (1 + x)^(5/2)*Cos[x] + (7/4)*Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]] - (9/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]] + (9/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] + (7/4)*Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] + (9/2)*Sqrt[1 + x]*Sin[x] - (5/2)*(1 + x)^(3/2)*Sin[x]} -{Sin[x]*E^ArcTanh[x]*(1 - x)^(3/2), x, 13, -2*Sqrt[1 + x]*Cos[x] + (1 + x)^(3/2)*Cos[x] + Sqrt[2*Pi]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]] + (3/2)*Sqrt[Pi/2]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]] - (3/2)*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] + Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[1 + x]]*Sin[1] - (3/2)*Sqrt[1 + x]*Sin[x]} - - -(* ::Subsubsection::Closed:: *) -(*n/2<0*) - - -{x*Sin[x]*E^ArcTanh[x]/(1 + x)^(1/2), x, 11, Sqrt[1 - x]*Cos[x] - Sqrt[Pi/2]*Cos[1]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]] + Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] - Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1] - Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1]} -{Sin[x]*E^ArcTanh[x]/(1 + x)^(1/2), x, 6, Sqrt[2*Pi]*Cos[1]*FresnelS[Sqrt[2/Pi]*Sqrt[1 - x]] - Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[1 - x]]*Sin[1]} - -(* {x*Sin[x]*E^ArcTanh[x]/(1 - x)^(1/2), x, 0, 0} *) -(* {Sin[x]*E^ArcTanh[x]/(1 - x)^(1/2), x, 0, 0} *) - - -(* {x*Sin[x]*E^ArcTanh[x]/(1 + x)^(3/2), x, 0, 0} *) -(* {Sin[x]*E^ArcTanh[x]/(1 + x)^(3/2), x, 0, 0} *) - -(* {x*Sin[x]*E^ArcTanh[x]/(1 - x)^(3/2), x, 0, 0} *) -(* {Sin[x]*E^ArcTanh[x]/(1 - x)^(3/2), x, 0, 0} *) - - -(* ::Title::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a+b x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcTanh[a+b x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a+b x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0 integer*) - - -{E^ArcTanh[a + b*x]*x^3, x, 7, -(((3 - 12*a + 12*a^2 - 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4)) - (x^2*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(4*b^2) - (Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2)*(7 - 10*a + 18*a^2 + 2*(1 - 6*a)*b*x))/(24*b^4) + ((3 - 12*a + 12*a^2 - 8*a^3)*ArcSin[a + b*x])/(8*b^4)} -{E^ArcTanh[a + b*x]*x^2, x, 7, -(((1 - 2*a + 2*a^2)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^3)) - ((1 - 4*a)*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(6*b^3) - (x*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(3*b^2) + ((1 - 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^3)} -{E^ArcTanh[a + b*x]*x^1, x, 6, -(((1 - 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^2)) - (Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(2*b^2) + ((1 - 2*a)*ArcSin[a + b*x])/(2*b^2)} -{E^ArcTanh[a + b*x]*x^0, x, 5, -((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b) + ArcSin[a + b*x]/b} -{E^ArcTanh[a + b*x]/x^1, x, 8, ArcSin[a + b*x] - (2*(1 + a)*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/Sqrt[1 - a^2]} -{E^ArcTanh[a + b*x]/x^2, x, 4, -((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 - a)*x)) - (2*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sqrt[1 - a^2])} -{E^ArcTanh[a + b*x]/x^3, x, 5, -(((1 + 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*(1 - a)^2*(1 + a)*x)) - (Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(2*(1 - a^2)*x^2) - ((1 + 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^2*(1 + a)*Sqrt[1 - a^2])} -{E^ArcTanh[a + b*x]/x^4, x, 7, -((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(3*(1 - a)*x^3)) - ((3 + 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(6*(1 - a)^2*(1 + a)*x^2) - ((4 + a)*(1 + 2*a)*b^2*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(6*(1 - a)^3*(1 + a)^2*x) - ((1 + 2*a + 2*a^2)*b^3*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*(1 - a^2)^(5/2))} - - -{E^(2*ArcTanh[a + b*x])*x^4, x, 3, -((2*(1 - a)^3*x)/b^4) - ((1 - a)^2*x^2)/b^3 - (2*(1 - a)*x^3)/(3*b^2) - x^4/(2*b) - x^5/5 - (2*(1 - a)^4*Log[1 - a - b*x])/b^5} -{E^(2*ArcTanh[a + b*x])*x^3, x, 3, -((2*(1 - a)^2*x)/b^3) - ((1 - a)*x^2)/b^2 - (2*x^3)/(3*b) - x^4/4 - (2*(1 - a)^3*Log[1 - a - b*x])/b^4} -{E^(2*ArcTanh[a + b*x])*x^2, x, 3, -((2*(1 - a)*x)/b^2) - x^2/b - x^3/3 - (2*(1 - a)^2*Log[1 - a - b*x])/b^3} -{E^(2*ArcTanh[a + b*x])*x^1, x, 3, -((2*x)/b) - x^2/2 - (2*(1 - a)*Log[1 - a - b*x])/b^2} -{E^(2*ArcTanh[a + b*x])*x^0, x, 3, -x - (2*Log[1 - a - b*x])/b} -{E^(2*ArcTanh[a + b*x])/x^1, x, 3, ((1 + a)*Log[x])/(1 - a) - (2*Log[1 - a - b*x])/(1 - a)} -{E^(2*ArcTanh[a + b*x])/x^2, x, 3, -((1 + a)/((1 - a)*x)) + (2*b*Log[x])/(1 - a)^2 - (2*b*Log[1 - a - b*x])/(1 - a)^2} -{E^(2*ArcTanh[a + b*x])/x^3, x, 3, -((1 + a)/(2*(1 - a)*x^2)) - (2*b)/((1 - a)^2*x) + (2*b^2*Log[x])/(1 - a)^3 - (2*b^2*Log[1 - a - b*x])/(1 - a)^3} -{E^(2*ArcTanh[a + b*x])/x^4, x, 3, -((1 + a)/(3*(1 - a)*x^3)) - b/((1 - a)^2*x^2) - (2*b^2)/((1 - a)^3*x) + (2*b^3*Log[x])/(1 - a)^4 - (2*b^3*Log[1 - a - b*x])/(1 - a)^4} - - -{E^(3*ArcTanh[a + b*x])*x^3, x, 8, (3*(17 - 44*a + 36*a^2 - 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4) + (2*x^3*(1 + a + b*x)^(3/2))/(b*Sqrt[1 - a - b*x]) + (9*x^2*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(4*b^2) + (Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2)*(29 - 54*a + 22*a^2 + 2*(11 - 10*a)*b*x))/(8*b^4) - (3*(17 - 44*a + 36*a^2 - 8*a^3)*ArcSin[a + b*x])/(8*b^4)} -{E^(3*ArcTanh[a + b*x])*x^2, x, 8, ((11 - 18*a + 6*a^2)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^3) + ((11 - 18*a + 6*a^2)*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(6*b^3) + ((1 - a)^2*(1 + a + b*x)^(5/2))/(b^3*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*(1 + a + b*x)^(5/2))/(3*b^3) - ((11 - 18*a + 6*a^2)*ArcSin[a + b*x])/(2*b^3)} -{E^(3*ArcTanh[a + b*x])*x^1, x, 7, (3*(3 - 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^2) + ((3 - 2*a)*Sqrt[1 - a - b*x]*(1 + a + b*x)^(3/2))/(2*b^2) + ((1 - a)*(1 + a + b*x)^(5/2))/(b^2*Sqrt[1 - a - b*x]) - (3*(3 - 2*a)*ArcSin[a + b*x])/(2*b^2)} -{E^(3*ArcTanh[a + b*x])*x^0, x, 6, (3*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b + (2*(1 + a + b*x)^(3/2))/(b*Sqrt[1 - a - b*x]) - (3*ArcSin[a + b*x])/b} -{E^(3*ArcTanh[a + b*x])/x^1, x, 8, (4*Sqrt[1 + a + b*x])/((1 - a)*Sqrt[1 - a - b*x]) - ArcSin[a + b*x] - (2*(1 + a)^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sqrt[1 - a^2])} -{E^(3*ArcTanh[a + b*x])/x^2, x, 5, (6*b*Sqrt[1 + a + b*x])/((1 - a)^2*Sqrt[1 - a - b*x]) - (1 + a + b*x)^(3/2)/((1 - a)*x*Sqrt[1 - a - b*x]) - (6*(1 + a)*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^2*Sqrt[1 - a^2])} -{E^(3*ArcTanh[a + b*x])/x^3, x, 6, (3*(3 + 2*a)*b^2*Sqrt[1 + a + b*x])/((1 - a)^3*(1 + a)*Sqrt[1 - a - b*x]) - ((3 + 2*a)*b*(1 + a + b*x)^(3/2))/(2*(1 - a)^2*(1 + a)*x*Sqrt[1 - a - b*x]) - (1 + a + b*x)^(5/2)/(2*(1 - a^2)*x^2*Sqrt[1 - a - b*x]) - (3*(3 + 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^3*Sqrt[1 - a^2])} -{E^(3*ArcTanh[a + b*x])/x^4, x, 8, ((52 + 51*a + 2*a^2)*b^3*Sqrt[1 + a + b*x])/(6*(1 - a)^4*(1 + a)*Sqrt[1 - a - b*x]) - ((1 + a)*Sqrt[1 + a + b*x])/(3*(1 - a)*x^3*Sqrt[1 - a - b*x]) - (7*b*Sqrt[1 + a + b*x])/(6*(1 - a)^2*x^2*Sqrt[1 - a - b*x]) - ((19 + 16*a)*b^2*Sqrt[1 + a + b*x])/(6*(1 - a)^3*(1 + a)*x*Sqrt[1 - a - b*x]) - ((11 + 18*a + 6*a^2)*b^3*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^4*(1 + a)*Sqrt[1 - a^2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0 integer*) - - -{E^(-ArcTanh[a + b*x])*x^3, x, 7, -(((3 + 12*a + 12*a^2 + 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4)) - (x^2*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(4*b^2) - ((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x]*(7 + 10*a + 18*a^2 - 2*(1 + 6*a)*b*x))/(24*b^4) - ((3 + 12*a + 12*a^2 + 8*a^3)*ArcSin[a + b*x])/(8*b^4)} -{E^(-ArcTanh[a + b*x])*x^2, x, 7, ((1 + 2*a + 2*a^2)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^3) + ((1 + 4*a)*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(6*b^3) - (x*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(3*b^2) + ((1 + 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^3)} -{E^(-ArcTanh[a + b*x])*x^1, x, 6, -(((1 + 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^2)) - ((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(2*b^2) - ((1 + 2*a)*ArcSin[a + b*x])/(2*b^2)} -{E^(-ArcTanh[a + b*x])*x^0, x, 5, (Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b + ArcSin[a + b*x]/b} -{E^(-ArcTanh[a + b*x])/x^1, x, 8, -ArcSin[a + b*x] - (2*(1 - a)*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/Sqrt[1 - a^2]} -{E^(-ArcTanh[a + b*x])/x^2, x, 4, -((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/((1 + a)*x)) + (2*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sqrt[1 - a^2])} -{E^(-ArcTanh[a + b*x])/x^3, x, 5, ((1 - 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*(1 - a)*(1 + a)^2*x) - ((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(2*(1 - a^2)*x^2) - ((1 - 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*(1 + a)^2*Sqrt[1 - a^2])} -{E^(-ArcTanh[a + b*x])/x^4, x, 7, -((Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(3*(1 + a)*x^3)) + ((3 - 2*a)*b*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(6*(1 - a)*(1 + a)^2*x^2) - ((1 - 2*a)*(4 - a)*b^2*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(6*(1 - a)^2*(1 + a)^3*x) + ((1 - 2*a + 2*a^2)*b^3*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*(1 - a^2)^(5/2))} - - -{E^(-2*ArcTanh[a + b*x])*x^4, x, 3, -((2*(1 + a)^3*x)/b^4) + ((1 + a)^2*x^2)/b^3 - (2*(1 + a)*x^3)/(3*b^2) + x^4/(2*b) - x^5/5 + (2*(1 + a)^4*Log[1 + a + b*x])/b^5} -{E^(-2*ArcTanh[a + b*x])*x^3, x, 3, (2*(1 + a)^2*x)/b^3 - ((1 + a)*x^2)/b^2 + (2*x^3)/(3*b) - x^4/4 - (2*(1 + a)^3*Log[1 + a + b*x])/b^4} -{E^(-2*ArcTanh[a + b*x])*x^2, x, 3, -((2*(1 + a)*x)/b^2) + x^2/b - x^3/3 + (2*(1 + a)^2*Log[1 + a + b*x])/b^3} -{E^(-2*ArcTanh[a + b*x])*x^1, x, 3, (2*x)/b - x^2/2 - (2*(1 + a)*Log[1 + a + b*x])/b^2} -{E^(-2*ArcTanh[a + b*x])*x^0, x, 3, -x + (2*Log[1 + a + b*x])/b} -{E^(-2*ArcTanh[a + b*x])/x^1, x, 3, ((1 - a)*Log[x])/(1 + a) - (2*Log[1 + a + b*x])/(1 + a)} -{E^(-2*ArcTanh[a + b*x])/x^2, x, 3, -((1 - a)/((1 + a)*x)) - (2*b*Log[x])/(1 + a)^2 + (2*b*Log[1 + a + b*x])/(1 + a)^2} -{E^(-2*ArcTanh[a + b*x])/x^3, x, 3, -((1 - a)/(2*(1 + a)*x^2)) + (2*b)/((1 + a)^2*x) + (2*b^2*Log[x])/(1 + a)^3 - (2*b^2*Log[1 + a + b*x])/(1 + a)^3} -{E^(-2*ArcTanh[a + b*x])/x^4, x, 3, -((1 - a)/(3*(1 + a)*x^3)) + b/((1 + a)^2*x^2) - (2*b^2)/((1 + a)^3*x) - (2*b^3*Log[x])/(1 + a)^4 + (2*b^3*Log[1 + a + b*x])/(1 + a)^4} - - -{E^(-3*ArcTanh[a + b*x])*x^3, x, 8, -((2*x^3*(1 - a - b*x)^(3/2))/(b*Sqrt[1 + a + b*x])) + (3*(17 + 44*a + 36*a^2 + 8*a^3)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(8*b^4) + (9*x^2*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(4*b^2) + ((1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x]*(29 + 54*a + 22*a^2 - 2*(11 + 10*a)*b*x))/(8*b^4) + (3*(17 + 44*a + 36*a^2 + 8*a^3)*ArcSin[a + b*x])/(8*b^4)} -{E^(-3*ArcTanh[a + b*x])*x^2, x, 8, -(((1 + a)^2*(1 - a - b*x)^(5/2))/(b^3*Sqrt[1 + a + b*x])) - ((11 + 18*a + 6*a^2)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^3) - ((11 + 18*a + 6*a^2)*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(6*b^3) - ((1 - a - b*x)^(5/2)*Sqrt[1 + a + b*x])/(3*b^3) - ((11 + 18*a + 6*a^2)*ArcSin[a + b*x])/(2*b^3)} -{E^(-3*ArcTanh[a + b*x])*x^1, x, 7, ((1 + a)*(1 - a - b*x)^(5/2))/(b^2*Sqrt[1 + a + b*x]) + (3*(3 + 2*a)*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/(2*b^2) + ((3 + 2*a)*(1 - a - b*x)^(3/2)*Sqrt[1 + a + b*x])/(2*b^2) + (3*(3 + 2*a)*ArcSin[a + b*x])/(2*b^2)} -{E^(-3*ArcTanh[a + b*x])*x^0, x, 6, -((2*(1 - a - b*x)^(3/2))/(b*Sqrt[1 + a + b*x])) - (3*Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b - (3*ArcSin[a + b*x])/b} -{E^(-3*ArcTanh[a + b*x])/x^1, x, 8, (4*Sqrt[1 - a - b*x])/((1 + a)*Sqrt[1 + a + b*x]) + ArcSin[a + b*x] - (2*(1 - a)^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)*Sqrt[1 - a^2])} -{E^(-3*ArcTanh[a + b*x])/x^2, x, 5, -((6*b*Sqrt[1 - a - b*x])/((1 + a)^2*Sqrt[1 + a + b*x])) - (1 - a - b*x)^(3/2)/((1 + a)*x*Sqrt[1 + a + b*x]) + (6*(1 - a)*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)^2*Sqrt[1 - a^2])} -{E^(-3*ArcTanh[a + b*x])/x^3, x, 6, (3*(3 - 2*a)*b^2*Sqrt[1 - a - b*x])/((1 - a)*(1 + a)^3*Sqrt[1 + a + b*x]) + ((3 - 2*a)*b*(1 - a - b*x)^(3/2))/(2*(1 - a)*(1 + a)^2*x*Sqrt[1 + a + b*x]) - (1 - a - b*x)^(5/2)/(2*(1 - a^2)*x^2*Sqrt[1 + a + b*x]) - (3*(3 - 2*a)*b^2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 + a)^3*Sqrt[1 - a^2])} -{E^(-3*ArcTanh[a + b*x])/x^4, x, 8, -(((52 - 51*a + 2*a^2)*b^3*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^4*Sqrt[1 + a + b*x])) - ((1 - a)*Sqrt[1 - a - b*x])/(3*(1 + a)*x^3*Sqrt[1 + a + b*x]) + (7*b*Sqrt[1 - a - b*x])/(6*(1 + a)^2*x^2*Sqrt[1 + a + b*x]) - ((19 - 16*a)*b^2*Sqrt[1 - a - b*x])/(6*(1 - a)*(1 + a)^3*x*Sqrt[1 + a + b*x]) + ((11 - 18*a + 6*a^2)*b^3*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*(1 + a)^4*Sqrt[1 - a^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a+b x]) (c+d x)^p*) - - -{E^ArcTanh[1 + b*x]/(2 + b*x), x, 4, ArcSin[1 + b*x]/b} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a+b x] / (1-(a+b x)^2)*) - - -{x^3*E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2), x, 6, ((1 - a)*x^2*Sqrt[1 + a + b*x])/(b^2*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x]*((1 - 2*a)*(4 - a) + (3 - 2*a)*b*x))/(2*b^4) - (3*(1 - 2*a + 2*a^2)*ArcSin[a + b*x])/(2*b^4)} -{x^2*E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2), x, 6, ((1 - a)^2*Sqrt[1 + a + b*x])/(b^3*Sqrt[1 - a - b*x]) + (Sqrt[1 - a - b*x]*Sqrt[1 + a + b*x])/b^3 - ((1 - 2*a)*ArcSin[a + b*x])/b^3} -{x^1*E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2), x, 5, ((1 - a)*Sqrt[1 + a + b*x])/(b^2*Sqrt[1 - a - b*x]) - ArcSin[a + b*x]/b^2} -{x^0*E^ArcTanh[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2), x, 2, Sqrt[1 + a + b*x]/(b*Sqrt[1 - a - b*x])} -{E^ArcTanh[a + b*x]/(x^1*(1 - a^2 - 2*a*b*x - b^2*x^2)), x, 4, Sqrt[1 + a + b*x]/((1 - a)*Sqrt[1 - a - b*x]) - (2*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)*Sqrt[1 - a^2])} -{E^ArcTanh[a + b*x]/(x^2*(1 - a^2 - 2*a*b*x - b^2*x^2)), x, 6, ((2 + a)*b*Sqrt[1 + a + b*x])/((1 - a)^2*(1 + a)*Sqrt[1 - a - b*x]) - Sqrt[1 + a + b*x]/((1 - a^2)*x*Sqrt[1 - a - b*x]) - (2*(1 + 2*a)*b*ArcTanh[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[1 - a - b*x])])/((1 - a)^2*(1 + a)*Sqrt[1 - a^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a+b x])*) - - -{E^(n*ArcTanh[a + b*x])*x^m, x, 4, (x^(1 + m)*(1 + a + b*x)^(n/2)*(1 - (b*x)/(1 - a))^(n/2)*AppellF1[1 + m, n/2, -(n/2), 2 + m, (b*x)/(1 - a), -((b*x)/(1 + a))])/((1 - a - b*x)^(n/2)*(1 + (b*x)/(1 + a))^(n/2)*(1 + m))} - - -{E^(n*ArcTanh[a + b*x])*x^3, x, 4, -((x^2*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(4*b^2)) - ((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2)*(6 + 18*a^2 - 10*a*n + n^2 - 2*b*(6*a - n)*x))/(24*b^4) + (2^(-2 + n/2)*(24*a^3 - 36*a^2*n + 12*a*(2 + n^2) - n*(8 + n^2))*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a - b*x)])/(3*b^4*(2 - n))} -{E^(n*ArcTanh[a + b*x])*x^2, x, 4, ((4*a - n)*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(6*b^3) - (x*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(3*b^2) - (2^(n/2)*(2 + 6*a^2 - 6*a*n + n^2)*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a - b*x)])/(3*b^3*(2 - n))} -{E^(n*ArcTanh[a + b*x])*x^1, x, 3, -(((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(2*b^2)) + (2^(n/2)*(2*a - n)*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a - b*x)])/(b^2*(2 - n))} -{E^(n*ArcTanh[a + b*x])*x^0, x, 2, -((2^(1 + n/2)*(1 - a - b*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a - b*x)])/(b*(2 - n)))} -{E^(n*ArcTanh[a + b*x])/x^1, x, 5, (2*(1 + a + b*x)^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, ((1 + a)*(1 - a - b*x))/((1 - a)*(1 + a + b*x))])/((1 - a - b*x)^(n/2)*n) - (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a - b*x)])/((1 - a - b*x)^(n/2)*n)} -{E^(n*ArcTanh[a + b*x])/x^2, x, 2, -((4*b*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, ((1 + a)*(1 - a - b*x))/((1 - a)*(1 + a + b*x))])/((1 - a)^2*(2 - n)))} -{E^(n*ArcTanh[a + b*x])/x^3, x, 3, -(((1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((2 + n)/2))/(2*(1 - a^2)*x^2)) - (2*b^2*(2*a + n)*(1 - a - b*x)^(1 - n/2)*(1 + a + b*x)^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, ((1 + a)*(1 - a - b*x))/((1 - a)*(1 + a + b*x))])/((1 - a)^3*(1 + a)*(2 - n))} - - -(* ::Title::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(n ArcTanh[a x])*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(1 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^4, x, 7, (35/128)*c^4*x*Sqrt[1 - a^2*x^2] + (35/192)*c^4*x*(1 - a^2*x^2)^(3/2) + (7/48)*c^4*x*(1 - a^2*x^2)^(5/2) + (1/8)*c^4*x*(1 - a^2*x^2)^(7/2) - (c^4*(1 - a^2*x^2)^(9/2))/(9*a) + (35*c^4*ArcSin[a*x])/(128*a)} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^3, x, 6, (5/16)*c^3*x*Sqrt[1 - a^2*x^2] + (5/24)*c^3*x*(1 - a^2*x^2)^(3/2) + (1/6)*c^3*x*(1 - a^2*x^2)^(5/2) - (c^3*(1 - a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*a)} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^2, x, 5, (3/8)*c^2*x*Sqrt[1 - a^2*x^2] + (1/4)*c^2*x*(1 - a^2*x^2)^(3/2) - (c^2*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^2*ArcSin[a*x])/(8*a)} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^1, x, 4, (1/2)*c*x*Sqrt[1 - a^2*x^2] - (c*(1 - a^2*x^2)^(3/2))/(3*a) + (c*ArcSin[a*x])/(2*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^4*E^ArcTanh[a*x]/(c - a^2*c*x^2), x, 5, (x^3*(1 + a*x))/(a^2*c*Sqrt[1 - a^2*x^2]) + (4*x^2*Sqrt[1 - a^2*x^2])/(3*a^3*c) + ((16 + 9*a*x)*Sqrt[1 - a^2*x^2])/(6*a^5*c) - (3*ArcSin[a*x])/(2*a^5*c)} -{x^3*E^ArcTanh[a*x]/(c - a^2*c*x^2), x, 4, (x^2*(1 + a*x))/(a^2*c*Sqrt[1 - a^2*x^2]) + ((4 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^4*c) - (3*ArcSin[a*x])/(2*a^4*c)} -{x^2*E^ArcTanh[a*x]/(c - a^2*c*x^2), x, 5, (1 + a*x)/(a^3*c*Sqrt[1 - a^2*x^2]) + Sqrt[1 - a^2*x^2]/(a^3*c) - ArcSin[a*x]/(a^3*c)} -{x^1*E^ArcTanh[a*x]/(c - a^2*c*x^2), x, 3, (1 + a*x)/(a^2*c*Sqrt[1 - a^2*x^2]) - ArcSin[a*x]/(a^2*c)} -{x^0*E^ArcTanh[a*x]/(c - a^2*c*x^2), x, 1, E^ArcTanh[a*x]/(a*c)} -{E^ArcTanh[a*x]/(x^1*(c - a^2*c*x^2)), x, 6, (1 + a*x)/(c*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c} -{E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)), x, 6, (1 + a*x)/(c*x*Sqrt[1 - a^2*x^2]) - (2*Sqrt[1 - a^2*x^2])/(c*x) - (a*ArcTanh[Sqrt[1 - a^2*x^2]])/c} -{E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)), x, 7, (1 + a*x)/(c*x^2*Sqrt[1 - a^2*x^2]) - (3*Sqrt[1 - a^2*x^2])/(2*c*x^2) - (2*a*Sqrt[1 - a^2*x^2])/(c*x) - (3*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c)} -{E^ArcTanh[a*x]/(x^4*(c - a^2*c*x^2)), x, 8, (1 + a*x)/(c*x^3*Sqrt[1 - a^2*x^2]) - (4*Sqrt[1 - a^2*x^2])/(3*c*x^3) - (3*a*Sqrt[1 - a^2*x^2])/(2*c*x^2) - (8*a^2*Sqrt[1 - a^2*x^2])/(3*c*x) - (3*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c)} - - -{x^6*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 6, (x^5*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (x^3*(5 + 6*a*x))/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) - (8*x^2*Sqrt[1 - a^2*x^2])/(3*a^5*c^2) - ((32 + 15*a*x)*Sqrt[1 - a^2*x^2])/(6*a^7*c^2) + (5*ArcSin[a*x])/(2*a^7*c^2)} -{x^5*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 5, (x^4*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (x^2*(4 + 5*a*x))/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) - ((16 + 15*a*x)*Sqrt[1 - a^2*x^2])/(6*a^6*c^2) + (5*ArcSin[a*x])/(2*a^6*c^2)} -{x^4*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 5, (x^3*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (x*(3 + 4*a*x))/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) - (8*Sqrt[1 - a^2*x^2])/(3*a^5*c^2) + ArcSin[a*x]/(a^5*c^2)} -{x^3*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 4, (x^2*(1 + a*x))/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - (2 + 3*a*x)/(3*a^4*c^2*Sqrt[1 - a^2*x^2]) + ArcSin[a*x]/(a^4*c^2)} -{x^2*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 4, (x^2*(1 + a*x))/(3*a*c^2*(1 - a^2*x^2)^(3/2)) - 2/(3*a^3*c^2*Sqrt[1 - a^2*x^2])} -{x^1*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 3, (1 + a*x)/(3*a^2*c^2*(1 - a^2*x^2)^(3/2)) - x/(3*a*c^2*Sqrt[1 - a^2*x^2])} -{x^0*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 3, (1 + a*x)/(3*a*c^2*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]/(x^1*(c - a^2*c*x^2)^2), x, 7, (1 + a*x)/(3*c^2*(1 - a^2*x^2)^(3/2)) + (3 + 2*a*x)/(3*c^2*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c^2} -{E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)^2), x, 7, (1 + a*x)/(3*c^2*x*(1 - a^2*x^2)^(3/2)) + (4 + 3*a*x)/(3*c^2*x*Sqrt[1 - a^2*x^2]) - (8*Sqrt[1 - a^2*x^2])/(3*c^2*x) - (a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^2} -{E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^2), x, 8, (1 + a*x)/(3*c^2*x^2*(1 - a^2*x^2)^(3/2)) + (5 + 4*a*x)/(3*c^2*x^2*Sqrt[1 - a^2*x^2]) - (5*Sqrt[1 - a^2*x^2])/(2*c^2*x^2) - (8*a*Sqrt[1 - a^2*x^2])/(3*c^2*x) - (5*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^2)} -{E^ArcTanh[a*x]/(x^4*(c - a^2*c*x^2)^2), x, 9, (1 + a*x)/(3*c^2*x^3*(1 - a^2*x^2)^(3/2)) + (6 + 5*a*x)/(3*c^2*x^3*Sqrt[1 - a^2*x^2]) - (8*Sqrt[1 - a^2*x^2])/(3*c^2*x^3) - (5*a*Sqrt[1 - a^2*x^2])/(2*c^2*x^2) - (16*a^2*Sqrt[1 - a^2*x^2])/(3*c^2*x) - (5*a^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^2)} - - -{x^7*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 6, (x^6*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^4*(6 + 7*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (x^2*(24 + 35*a*x))/(15*a^6*c^3*Sqrt[1 - a^2*x^2]) + ((32 + 35*a*x)*Sqrt[1 - a^2*x^2])/(10*a^8*c^3) - (7*ArcSin[a*x])/(2*a^8*c^3)} -{x^6*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 6, (x^5*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^3*(5 + 6*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (x*(5 + 8*a*x))/(5*a^6*c^3*Sqrt[1 - a^2*x^2]) + (16*Sqrt[1 - a^2*x^2])/(5*a^7*c^3) - ArcSin[a*x]/(a^7*c^3)} -{x^5*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 5, (x^4*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (x^2*(4 + 5*a*x))/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + (8 + 15*a*x)/(15*a^6*c^3*Sqrt[1 - a^2*x^2]) - ArcSin[a*x]/(a^6*c^3)} -{x^4*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 5, (x^4*(1 + a*x))/(5*a*c^3*(1 - a^2*x^2)^(5/2)) - 4/(15*a^5*c^3*(1 - a^2*x^2)^(3/2)) + 4/(5*a^5*c^3*Sqrt[1 - a^2*x^2])} -{x^3*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, 1/(5*a^4*c^3*(1 - a^2*x^2)^(5/2)) + (a*x^5)/(5*c^3*(1 - a^2*x^2)^(5/2)) - 1/(3*a^4*c^3*(1 - a^2*x^2)^(3/2)), (x^2*(1 + a*x))/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - (2 + 3*a*x)/(15*a^4*c^3*(1 - a^2*x^2)^(3/2)) + x/(5*a^3*c^3*Sqrt[1 - a^2*x^2])} -{x^2*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, (x^2*(1 + a*x))/(5*a*c^3*(1 - a^2*x^2)^(5/2)) - (2*(1 - a*x))/(15*a^3*c^3*(1 - a^2*x^2)^(3/2)) - (2*x)/(15*a^2*c^3*Sqrt[1 - a^2*x^2])} -{x^1*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, (1 + a*x)/(5*a^2*c^3*(1 - a^2*x^2)^(5/2)) - x/(15*a*c^3*(1 - a^2*x^2)^(3/2)) - (2*x)/(15*a*c^3*Sqrt[1 - a^2*x^2])} -{x^0*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, (1 + a*x)/(5*a*c^3*(1 - a^2*x^2)^(5/2)) + (4*x)/(15*c^3*(1 - a^2*x^2)^(3/2)) + (8*x)/(15*c^3*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]/(x^1*(c - a^2*c*x^2)^3), x, 8, (1 + a*x)/(5*c^3*(1 - a^2*x^2)^(5/2)) + (5 + 4*a*x)/(15*c^3*(1 - a^2*x^2)^(3/2)) + (15 + 8*a*x)/(15*c^3*Sqrt[1 - a^2*x^2]) - ArcTanh[Sqrt[1 - a^2*x^2]]/c^3} -{E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)^3), x, 8, (1 + a*x)/(5*c^3*x*(1 - a^2*x^2)^(5/2)) + (6 + 5*a*x)/(15*c^3*x*(1 - a^2*x^2)^(3/2)) + (8 + 5*a*x)/(5*c^3*x*Sqrt[1 - a^2*x^2]) - (16*Sqrt[1 - a^2*x^2])/(5*c^3*x) - (a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^3} -{E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^3), x, 9, (1 + a*x)/(5*c^3*x^2*(1 - a^2*x^2)^(5/2)) + (7 + 6*a*x)/(15*c^3*x^2*(1 - a^2*x^2)^(3/2)) + (35 + 24*a*x)/(15*c^3*x^2*Sqrt[1 - a^2*x^2]) - (7*Sqrt[1 - a^2*x^2])/(2*c^3*x^2) - (16*a*Sqrt[1 - a^2*x^2])/(5*c^3*x) - (7*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^3)} - - -{E^ArcTanh[a*x]/(c - a^2*c*x^2)^4, x, 5, (1 + a*x)/(7*a*c^4*(1 - a^2*x^2)^(7/2)) + (6*x)/(35*c^4*(1 - a^2*x^2)^(5/2)) + (8*x)/(35*c^4*(1 - a^2*x^2)^(3/2)) + (16*x)/(35*c^4*Sqrt[1 - a^2*x^2])} - - -{E^ArcTanh[a*x]/(c - a^2*c*x^2)^5, x, 6, (1 + a*x)/(9*a*c^5*(1 - a^2*x^2)^(9/2)) + (8*x)/(63*c^5*(1 - a^2*x^2)^(7/2)) + (16*x)/(105*c^5*(1 - a^2*x^2)^(5/2)) + (64*x)/(315*c^5*(1 - a^2*x^2)^(3/2)) + (128*x)/(315*c^5*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (1-a^2 x^2)^(p/2)*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^ArcTanh[a*x]*x^4/Sqrt[1 - a^2*x^2], x, 3, -(x/a^4) - x^2/(2*a^3) - x^3/(3*a^2) - x^4/(4*a) - Log[1 - a*x]/a^5} -{E^ArcTanh[a*x]*x^3/Sqrt[1 - a^2*x^2], x, 3, -(x/a^3) - x^2/(2*a^2) - x^3/(3*a) - Log[1 - a*x]/a^4} -{E^ArcTanh[a*x]*x^2/Sqrt[1 - a^2*x^2], x, 3, -(x/a^2) - x^2/(2*a) - Log[1 - a*x]/a^3} -{E^ArcTanh[a*x]*x^1/Sqrt[1 - a^2*x^2], x, 3, -(x/a) - Log[1 - a*x]/a^2} -{E^ArcTanh[a*x]*x^0/Sqrt[1 - a^2*x^2], x, 2, -(Log[1 - a*x]/a)} -{E^ArcTanh[a*x]/(x^1*Sqrt[1 - a^2*x^2]), x, 4, Log[x] - Log[1 - a*x]} -{E^ArcTanh[a*x]/(x^2*Sqrt[1 - a^2*x^2]), x, 3, -x^(-1) + a*Log[x] - a*Log[1 - a*x]} -{E^ArcTanh[a*x]/(x^3*Sqrt[1 - a^2*x^2]), x, 3, -1/(2*x^2) - a/x + a^2*Log[x] - a^2*Log[1 - a*x]} -{E^ArcTanh[a*x]/(x^4*Sqrt[1 - a^2*x^2]), x, 3, -1/(3*x^3) - a/(2*x^2) - a^2/x + a^3*Log[x] - a^3*Log[1 - a*x]} - - -{E^ArcTanh[a*x]*x^4/(1 - a^2*x^2)^(3/2), x, 3, x/a^4 + x^2/(2*a^3) + 1/(2*a^5*(1 - a*x)) + (7*Log[1 - a*x])/(4*a^5) + Log[1 + a*x]/(4*a^5)} -{E^ArcTanh[a*x]*x^3/(1 - a^2*x^2)^(3/2), x, 3, x/a^3 + 1/(2*a^4*(1 - a*x)) + (5*Log[1 - a*x])/(4*a^4) - Log[1 + a*x]/(4*a^4)} -{E^ArcTanh[a*x]*x^2/(1 - a^2*x^2)^(3/2), x, 3, 1/(2*a^3*(1 - a*x)) + (3*Log[1 - a*x])/(4*a^3) + Log[1 + a*x]/(4*a^3)} -{E^ArcTanh[a*x]*x^1/(1 - a^2*x^2)^(3/2), x, 4, 1/(2*a^2*(1 - a*x)) - ArcTanh[a*x]/(2*a^2)} -{E^ArcTanh[a*x]*x^0/(1 - a^2*x^2)^(3/2), x, 4, 1/(2*a*(1 - a*x)) + ArcTanh[a*x]/(2*a)} -{E^ArcTanh[a*x]/(x^1*(1 - a^2*x^2)^(3/2)), x, 3, 1/(2*(1 - a*x)) + Log[x] - (3*Log[1 - a*x])/4 - Log[1 + a*x]/4} -{E^ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^(3/2)), x, 3, -x^(-1) + a/(2*(1 - a*x)) + a*Log[x] - (5*a*Log[1 - a*x])/4 + (a*Log[1 + a*x])/4} -{E^ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(3/2)), x, 3, -1/(2*x^2) - a/x + a^2/(2*(1 - a*x)) + 2*a^2*Log[x] - (7*a^2*Log[1 - a*x])/4 - (a^2*Log[1 + a*x])/4} -{E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(3/2)), x, 3, -1/(3*x^3) - a/(2*x^2) - (2*a^2)/x + a^3/(2*(1 - a*x)) + 2*a^3*Log[x] - (9*a^3*Log[1 - a*x])/4 + (a^3*Log[1 + a*x])/4} - - -{E^ArcTanh[a*x]*x^6/(1 - a^2*x^2)^(5/2), x, 3, -(x/a^6) - x^2/(2*a^5) + 1/(8*a^7*(1 - a*x)^2) - 5/(4*a^7*(1 - a*x)) - 1/(8*a^7*(1 + a*x)) - (39*Log[1 - a*x])/(16*a^7) - (9*Log[1 + a*x])/(16*a^7)} -{E^ArcTanh[a*x]*x^5/(1 - a^2*x^2)^(5/2), x, 3, -(x/a^5) + 1/(8*a^6*(1 - a*x)^2) - 1/(a^6*(1 - a*x)) + 1/(8*a^6*(1 + a*x)) - (23*Log[1 - a*x])/(16*a^6) + (7*Log[1 + a*x])/(16*a^6)} -{E^ArcTanh[a*x]*x^4/(1 - a^2*x^2)^(5/2), x, 3, 1/(8*a^5*(1 - a*x)^2) - 3/(4*a^5*(1 - a*x)) - 1/(8*a^5*(1 + a*x)) - (11*Log[1 - a*x])/(16*a^5) - (5*Log[1 + a*x])/(16*a^5)} -{E^ArcTanh[a*x]*x^3/(1 - a^2*x^2)^(5/2), x, 4, 1/(8*a^4*(1 - a*x)^2) - 1/(2*a^4*(1 - a*x)) + 1/(8*a^4*(1 + a*x)) + (3*ArcTanh[a*x])/(8*a^4)} -{E^ArcTanh[a*x]*x^2/(1 - a^2*x^2)^(5/2), x, 4, 1/(8*a^3*(1 - a*x)^2) - 1/(4*a^3*(1 - a*x)) - 1/(8*a^3*(1 + a*x)) - ArcTanh[a*x]/(8*a^3)} -{E^ArcTanh[a*x]*x^1/(1 - a^2*x^2)^(5/2), x, 4, 1/(8*a^2*(1 - a*x)^2) + 1/(8*a^2*(1 + a*x)) - ArcTanh[a*x]/(8*a^2)} -{E^ArcTanh[a*x]*x^0/(1 - a^2*x^2)^(5/2), x, 4, 1/(8*a*(1 - a*x)^2) + 1/(4*a*(1 - a*x)) - 1/(8*a*(1 + a*x)) + (3*ArcTanh[a*x])/(8*a)} -{E^ArcTanh[a*x]/(x^1*(1 - a^2*x^2)^(5/2)), x, 3, 1/(8*(1 - a*x)^2) + 1/(2*(1 - a*x)) + 1/(8*(1 + a*x)) + Log[x] - (11*Log[1 - a*x])/16 - (5*Log[1 + a*x])/16} -{E^ArcTanh[a*x]/(x^2*(1 - a^2*x^2)^(5/2)), x, 3, -x^(-1) + a/(8*(1 - a*x)^2) + (3*a)/(4*(1 - a*x)) - a/(8*(1 + a*x)) + a*Log[x] - (23*a*Log[1 - a*x])/16 + (7*a*Log[1 + a*x])/16} -{E^ArcTanh[a*x]/(x^3*(1 - a^2*x^2)^(5/2)), x, 3, -1/(2*x^2) - a/x + a^2/(8*(1 - a*x)^2) + a^2/(1 - a*x) + a^2/(8*(1 + a*x)) + 3*a^2*Log[x] - (39*a^2*Log[1 - a*x])/16 - (9*a^2*Log[1 + a*x])/16} -{E^ArcTanh[a*x]/(x^4*(1 - a^2*x^2)^(5/2)), x, 3, -1/(3*x^3) - a/(2*x^2) - (3*a^2)/x + a^3/(8*(1 - a*x)^2) + (5*a^3)/(4*(1 - a*x)) - a^3/(8*(1 + a*x)) + 3*a^3*Log[x] - (59*a^3*Log[1 - a*x])/16 + (11*a^3*Log[1 + a*x])/16} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^2, x, 4, (x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2]) + (a*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^1, x, 4, (x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) + (a*x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^0, x, 3, (x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] + (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2])} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)/x^1, x, 4, (a*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] + (Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2]} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)/x^2, x, 4, -(Sqrt[c - a^2*c*x^2]/(x*Sqrt[1 - a^2*x^2])) + (a*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2]} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(3/2), x, 4, (2*c*(1 + a*x)^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (c*(1 + a*x)^4*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2])} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(5/2), x, 4, (c^2*(1 + a*x)^4*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) - (4*c^2*(1 + a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2]) + (c^2*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - a^2*x^2])} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^(7/2), x, 4, (8*c^3*(1 + a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2]) - (2*c^3*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) + (6*c^3*(1 + a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (c^3*(1 + a*x)^8*Sqrt[c - a^2*c*x^2])/(8*a*Sqrt[1 - a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(E^ArcTanh[a*x]*x^4)/Sqrt[c - a^2*c*x^2], x, 4, -((x*Sqrt[1 - a^2*x^2])/(a^4*Sqrt[c - a^2*c*x^2])) - (x^2*Sqrt[1 - a^2*x^2])/(2*a^3*Sqrt[c - a^2*c*x^2]) - (x^3*Sqrt[1 - a^2*x^2])/(3*a^2*Sqrt[c - a^2*c*x^2]) - (x^4*Sqrt[1 - a^2*x^2])/(4*a*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^5*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^3)/Sqrt[c - a^2*c*x^2], x, 4, -((x*Sqrt[1 - a^2*x^2])/(a^3*Sqrt[c - a^2*c*x^2])) - (x^2*Sqrt[1 - a^2*x^2])/(2*a^2*Sqrt[c - a^2*c*x^2]) - (x^3*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^4*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^2)/Sqrt[c - a^2*c*x^2], x, 4, -((x*Sqrt[1 - a^2*x^2])/(a^2*Sqrt[c - a^2*c*x^2])) - (x^2*Sqrt[1 - a^2*x^2])/(2*a*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^3*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x)/Sqrt[c - a^2*c*x^2], x, 4, -((x*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a^2*c*x^2])) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a^2*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/Sqrt[c - a^2*c*x^2], x, 3, -((Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a*Sqrt[c - a^2*c*x^2]))} -{E^ArcTanh[a*x]/(x*Sqrt[c - a^2*c*x^2]), x, 5, (Sqrt[1 - a^2*x^2]*Log[x])/Sqrt[c - a^2*c*x^2] - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} -{E^ArcTanh[a*x]/(x^2*Sqrt[c - a^2*c*x^2]), x, 4, -(Sqrt[1 - a^2*x^2]/(x*Sqrt[c - a^2*c*x^2])) + (a*Sqrt[1 - a^2*x^2]*Log[x])/Sqrt[c - a^2*c*x^2] - (a*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} -{E^ArcTanh[a*x]/(x^3*Sqrt[c - a^2*c*x^2]), x, 4, -Sqrt[1 - a^2*x^2]/(2*x^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2]*Log[x])/Sqrt[c - a^2*c*x^2] - (a^2*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} -{E^ArcTanh[a*x]/(x^4*Sqrt[c - a^2*c*x^2]), x, 4, -Sqrt[1 - a^2*x^2]/(3*x^3*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(2*x^2*Sqrt[c - a^2*c*x^2]) - (a^2*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2]*Log[x])/Sqrt[c - a^2*c*x^2] - (a^3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} - - -{(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^(3/2), x, 4, (2*x*Sqrt[1 - a^2*x^2])/(a^5*c*Sqrt[c - a^2*c*x^2]) + (x^2*Sqrt[1 - a^2*x^2])/(2*a^4*c*Sqrt[c - a^2*c*x^2]) + (x^3*Sqrt[1 - a^2*x^2])/(3*a^3*c*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(2*a^6*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (9*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^6*c*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^6*c*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2)^(3/2), x, 4, (x*Sqrt[1 - a^2*x^2])/(a^4*c*Sqrt[c - a^2*c*x^2]) + (x^2*Sqrt[1 - a^2*x^2])/(2*a^3*c*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(2*a^5*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (7*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^5*c*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^5*c*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2)^(3/2), x, 4, (x*Sqrt[1 - a^2*x^2])/(a^3*c*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(2*a^4*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^4*c*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^4*c*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^2)/(c - a^2*c*x^2)^(3/2), x, 4, Sqrt[1 - a^2*x^2]/(2*a^3*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x)/(c - a^2*c*x^2)^(3/2), x, 5, Sqrt[1 - a^2*x^2]/(2*a^2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a^2*c*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(c - a^2*c*x^2)^(3/2), x, 5, Sqrt[1 - a^2*x^2]/(2*a*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a*c*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x*(c - a^2*c*x^2)^(3/2)), x, 4, Sqrt[1 - a^2*x^2]/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)^(3/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(c*x*Sqrt[c - a^2*c*x^2])) + (a*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^(3/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(2*c*x^2*Sqrt[c - a^2*c*x^2])) - (a*Sqrt[1 - a^2*x^2])/(c*x*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (2*a^2*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (7*a^2*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c*x^2]) - (a^2*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x^4*(c - a^2*c*x^2)^(3/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(3*c*x^3*Sqrt[c - a^2*c*x^2])) - (a*Sqrt[1 - a^2*x^2])/(2*c*x^2*Sqrt[c - a^2*c*x^2]) - (2*a^2*Sqrt[1 - a^2*x^2])/(c*x*Sqrt[c - a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (2*a^3*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (9*a^3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])} - - -{(E^ArcTanh[a*x]*x^6)/(c - a^2*c*x^2)^(5/2), x, 4, -((x*Sqrt[1 - a^2*x^2])/(a^6*c^2*Sqrt[c - a^2*c*x^2])) - (x^2*Sqrt[1 - a^2*x^2])/(2*a^5*c^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a^7*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) - (5*Sqrt[1 - a^2*x^2])/(4*a^7*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a^7*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) - (39*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*a^7*c^2*Sqrt[c - a^2*c*x^2]) - (9*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*a^7*c^2*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^(5/2), x, 4, -((x*Sqrt[1 - a^2*x^2])/(a^5*c^2*Sqrt[c - a^2*c*x^2])) + Sqrt[1 - a^2*x^2]/(8*a^6*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(a^6*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a^6*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) - (23*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*a^6*c^2*Sqrt[c - a^2*c*x^2]) + (7*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*a^6*c^2*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^4)/(c - a^2*c*x^2)^(5/2), x, 4, Sqrt[1 - a^2*x^2]/(8*a^5*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2])/(4*a^5*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a^5*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) - (11*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*a^5*c^2*Sqrt[c - a^2*c*x^2]) - (5*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*a^5*c^2*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(8*a^4*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(2*a^4*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a^4*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a^4*c^2*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x^2)/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(8*a^3*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(4*a^3*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a^3*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a^3*c^2*Sqrt[c - a^2*c*x^2])} -{(E^ArcTanh[a*x]*x)/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(8*a^2*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a^2*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a^2*c^2*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(4*a*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a*c^2*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x*(c - a^2*c*x^2)^(5/2)), x, 4, Sqrt[1 - a^2*x^2]/(8*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(2*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[x])/(c^2*Sqrt[c - a^2*c*x^2]) - (11*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*c^2*Sqrt[c - a^2*c*x^2]) - (5*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*c^2*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)^(5/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(c^2*x*Sqrt[c - a^2*c*x^2])) + (a*Sqrt[1 - a^2*x^2])/(8*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + (3*a*Sqrt[1 - a^2*x^2])/(4*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(8*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (a*Sqrt[1 - a^2*x^2]*Log[x])/(c^2*Sqrt[c - a^2*c*x^2]) - (23*a*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*c^2*Sqrt[c - a^2*c*x^2]) + (7*a*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*c^2*Sqrt[c - a^2*c*x^2])} -{E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^(5/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(2*c^2*x^2*Sqrt[c - a^2*c*x^2])) - (a*Sqrt[1 - a^2*x^2])/(c^2*x*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(8*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(8*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*a^2*Sqrt[1 - a^2*x^2]*Log[x])/(c^2*Sqrt[c - a^2*c*x^2]) - (39*a^2*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*c^2*Sqrt[c - a^2*c*x^2]) - (9*a^2*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(16*c^2*Sqrt[c - a^2*c*x^2])} - - -{E^ArcTanh[a*x]/(c - a^2*c*x^2)^(7/2), x, 5, Sqrt[1 - a^2*x^2]/(24*a*c^3*(1 - a*x)^3*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(16*a*c^3*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p with m symbolic*) - - -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^2, x, 4, (c^2*x^(1 + m)*Hypergeometric2F1[-(3/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*c^2*x^(2 + m)*Hypergeometric2F1[-(3/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^1, x, 4, (c*x^(1 + m)*Hypergeometric2F1[-(1/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*c*x^(2 + m)*Hypergeometric2F1[-(1/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^1, x, 4, (x^(1 + m)*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(c*(1 + m)) + (a*x^(2 + m)*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(c*(2 + m))} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 4, (x^(1 + m)*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(c^2*(1 + m)) + (a*x^(2 + m)*Hypergeometric2F1[5/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(c^2*(2 + m))} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, (x^(1 + m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(c^3*(1 + m)) + (a*x^(2 + m)*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(c^3*(2 + m))} - - -{x^m*E^ArcTanh[a*x]*(1 - a^2*x^2)^(5/2), x, 3, x^(1 + m)/(1 + m) + (a*x^(2 + m))/(2 + m) - (2*a^2*x^(3 + m))/(3 + m) - (2*a^3*x^(4 + m))/(4 + m) + (a^4*x^(5 + m))/(5 + m) + (a^5*x^(6 + m))/(6 + m)} -{x^m*E^ArcTanh[a*x]*(1 - a^2*x^2)^(3/2), x, 3, x^(1 + m)/(1 + m) + (a*x^(2 + m))/(2 + m) - (a^2*x^(3 + m))/(3 + m) - (a^3*x^(4 + m))/(4 + m)} -{x^m*E^ArcTanh[a*x]*(1 - a^2*x^2)^(1/2), x, 3, x^(1 + m)/(1 + m) + (a*x^(2 + m))/(2 + m)} -{x^m*E^ArcTanh[a*x]/(1 - a^2*x^2)^(1/2), x, 2, (x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)} -{x^m*E^ArcTanh[a*x]/(1 - a^2*x^2)^(3/2), x, 6, (x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} -{x^m*E^ArcTanh[a*x]/(1 - a^2*x^2)^(5/2), x, 6, (x^(1 + m)*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, a^2*x^2])/(2 + m)} - - -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^(5/2), x, 4, (c^2*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) + (a*c^2*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]) - (2*a^2*c^2*x^(3 + m)*Sqrt[c - a^2*c*x^2])/((3 + m)*Sqrt[1 - a^2*x^2]) - (2*a^3*c^2*x^(4 + m)*Sqrt[c - a^2*c*x^2])/((4 + m)*Sqrt[1 - a^2*x^2]) + (a^4*c^2*x^(5 + m)*Sqrt[c - a^2*c*x^2])/((5 + m)*Sqrt[1 - a^2*x^2]) + (a^5*c^2*x^(6 + m)*Sqrt[c - a^2*c*x^2])/((6 + m)*Sqrt[1 - a^2*x^2])} -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^(3/2), x, 4, (c*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) + (a*c*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]) - (a^2*c*x^(3 + m)*Sqrt[c - a^2*c*x^2])/((3 + m)*Sqrt[1 - a^2*x^2]) - (a^3*c*x^(4 + m)*Sqrt[c - a^2*c*x^2])/((4 + m)*Sqrt[1 - a^2*x^2])} -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2), x, 4, (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) + (a*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^(1/2), x, 3, (x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/((1 + m)*Sqrt[c - a^2*c*x^2])} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^(3/2), x, 7, (x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(c*(1 + m)*Sqrt[c - a^2*c*x^2]) + (a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(c*(2 + m)*Sqrt[c - a^2*c*x^2])} -{x^m*E^ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2), x, 7, (x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[3, (1 + m)/2, (3 + m)/2, a^2*x^2])/(c^2*(1 + m)*Sqrt[c - a^2*c*x^2]) + (a*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[3, (2 + m)/2, (4 + m)/2, a^2*x^2])/(c^2*(2 + m)*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p with p symbolic*) - - -{x^m*E^ArcTanh[a*x]*(c - a^2*c*x^2)^p, x, 5, (x^(1 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(1 + m)/2, 1/2 - p, (3 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*(1 + m)) + (a*x^(2 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(2 + m)/2, 1/2 - p, (4 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*(2 + m))} - - -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p*x^3, x, 6, -((1 - a^2*x^2)^(1/2 + p)/(a^4*(1 + 2*p))) + (1 - a^2*x^2)^(3/2 + p)/(a^4*(3 + 2*p)) + (1/5)*a*x^5*Hypergeometric2F1[5/2, 1/2 - p, 7/2, a^2*x^2]} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p*x^2, x, 6, -((1 - a^2*x^2)^(1/2 + p)/(a^3*(1 + 2*p))) + (1 - a^2*x^2)^(3/2 + p)/(a^3*(3 + 2*p)) + (1/3)*x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2]} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p*x^1, x, 4, -((1 - a^2*x^2)^(1/2 + p)/(a^2*(1 + 2*p))) + (1/3)*a*x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2]} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p*x^0, x, 2, -((2^(3/2 + p)*(1 - a*x)^(1/2 + p)*Hypergeometric2F1[-(1/2) - p, 1/2 + p, 3/2 + p, (1/2)*(1 - a*x)])/(a*(1 + 2*p)))} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p/x^1, x, 5, a*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2] - ((1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p/x^2, x, 5, -(Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2]/x) - (a*(1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^ArcTanh[a*x]*(1 - a^2*x^2)^p/x^3, x, 5, -((a*Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2])/x) - (a^2*(1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[2, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} - - -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^3, x, 7, -((Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^4*(1 + 2*p))) + ((1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^4*(3 + 2*p)) + ((1/5)*a*x^5*(c - a^2*c*x^2)^p*Hypergeometric2F1[5/2, 1/2 - p, 7/2, a^2*x^2])/(1 - a^2*x^2)^p} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^2, x, 7, -((Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^3*(1 + 2*p))) + ((1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^3*(3 + 2*p)) + ((1/3)*x^3*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/(1 - a^2*x^2)^p} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^1, x, 5, -((Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^2*(1 + 2*p))) + ((1/3)*a*x^3*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/(1 - a^2*x^2)^p} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^0, x, 3, -((2^(3/2 + p)*(1 - a*x)^(1/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(1/2) - p, 1/2 + p, 3/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a*(1 + 2*p))))} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p/x^1, x, 6, (a*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2])/(1 - a^2*x^2)^p - (Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p/x^2, x, 6, -(((c - a^2*c*x^2)^p*Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2])/((1 - a^2*x^2)^p*x)) - (a*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^ArcTanh[a*x]*(c - a^2*c*x^2)^p/x^3, x, 6, -((a*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2])/((1 - a^2*x^2)^p*x)) - (a^2*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[2, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(2 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2), x, 3, (c*x^5)/5 + (a*c*x^6)/3 + (a^2*c*x^7)/7} -{E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2), x, 3, (c*x^4)/4 + (2*a*c*x^5)/5 + (a^2*c*x^6)/6} -{E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2), x, 3, (c*x^3)/3 + (a*c*x^4)/2 + (a^2*c*x^5)/5} -{E^(2*ArcTanh[a*x])*x*(c - a^2*c*x^2), x, 3, (c*x^2)/2 + (2*a*c*x^3)/3 + (a^2*c*x^4)/4} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2), x, 2, (c*(1 + a*x)^3)/(3*a)} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2))/x, x, 3, 2*a*c*x + (a^2*c*x^2)/2 + c*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2))/x^2, x, 3, -(c/x) + a^2*c*x + 2*a*c*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2))/x^3, x, 3, -c/(2*x^2) - (2*a*c)/x + a^2*c*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2))/x^4, x, 2, -(c*(1 + a*x)^3)/(3*x^3)} - - -{E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^2, x, 3, (c^2*x^5)/5 + (a*c^2*x^6)/3 - (a^3*c^2*x^8)/4 - (a^4*c^2*x^9)/9} -{E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^2, x, 3, (c^2*x^4)/4 + (2*a*c^2*x^5)/5 - (2*a^3*c^2*x^7)/7 - (a^4*c^2*x^8)/8} -{E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^2, x, 3, (c^2*x^3)/3 + (a*c^2*x^4)/2 - (a^3*c^2*x^6)/3 - (a^4*c^2*x^7)/7} -{E^(2*ArcTanh[a*x])*x*(c - a^2*c*x^2)^2, x, 3, (c^2*x^2)/2 + (2*a*c^2*x^3)/3 - (2*a^3*c^2*x^5)/5 - (a^4*c^2*x^6)/6} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 3, (c^2*(1 + a*x)^4)/(2*a) - (c^2*(1 + a*x)^5)/(5*a)} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x, x, 3, 2*a*c^2*x - (2*a^3*c^2*x^3)/3 - (a^4*c^2*x^4)/4 + c^2*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x^2, x, 3, -(c^2/x) - a^3*c^2*x^2 - (a^4*c^2*x^3)/3 + 2*a*c^2*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x^3, x, 2, -((c^2*(1 + a*x)^4)/(2*x^2))} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x^4, x, 3, -c^2/(3*x^3) - (a*c^2)/x^2 - a^4*c^2*x - 2*a^3*c^2*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x^5, x, 3, -(c^2/(4*x^4)) - (2*a*c^2)/(3*x^3) + (2*a^3*c^2)/x - a^4*c^2*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2)/x^6, x, 3, -(c^2/(5*x^5)) - (a*c^2)/(2*x^4) + (a^3*c^2)/x^2 + (a^4*c^2)/x} - - -{E^(2*ArcTanh[a*x])*x^4*(c - a^2*c*x^2)^3, x, 3, (c^3*x^5)/5 + (a*c^3*x^6)/3 - (a^2*c^3*x^7)/7 - (a^3*c^3*x^8)/2 - (a^4*c^3*x^9)/9 + (a^5*c^3*x^10)/5 + (a^6*c^3*x^11)/11} -{E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^3, x, 3, (c^3*x^4)/4 + (2*a*c^3*x^5)/5 - (a^2*c^3*x^6)/6 - (4*a^3*c^3*x^7)/7 - (a^4*c^3*x^8)/8 + (2*a^5*c^3*x^9)/9 + (a^6*c^3*x^10)/10} -{E^(2*ArcTanh[a*x])*x^2*(c - a^2*c*x^2)^3, x, 3, (4*c^3*(1 + a*x)^5)/(5*a^3) - (2*c^3*(1 + a*x)^6)/a^3 + (13*c^3*(1 + a*x)^7)/(7*a^3) - (3*c^3*(1 + a*x)^8)/(4*a^3) + (c^3*(1 + a*x)^9)/(9*a^3)} -{E^(2*ArcTanh[a*x])*x*(c - a^2*c*x^2)^3, x, 3, -((4*c^3*(1 + a*x)^5)/(5*a^2)) + (4*c^3*(1 + a*x)^6)/(3*a^2) - (5*c^3*(1 + a*x)^7)/(7*a^2) + (c^3*(1 + a*x)^8)/(8*a^2)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 3, (4*c^3*(1 + a*x)^5)/(5*a) - (2*c^3*(1 + a*x)^6)/(3*a) + (c^3*(1 + a*x)^7)/(7*a)} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x, x, 3, 2*a*c^3*x - (a^2*c^3*x^2)/2 - (4*a^3*c^3*x^3)/3 - (a^4*c^3*x^4)/4 + (2*a^5*c^3*x^5)/5 + (a^6*c^3*x^6)/6 + c^3*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^2, x, 3, -(c^3/x) - a^2*c^3*x - 2*a^3*c^3*x^2 - (a^4*c^3*x^3)/3 + (a^5*c^3*x^4)/2 + (a^6*c^3*x^5)/5 + 2*a*c^3*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^3, x, 3, -c^3/(2*x^2) - (2*a*c^3)/x - 4*a^3*c^3*x - (a^4*c^3*x^2)/2 + (2*a^5*c^3*x^3)/3 + (a^6*c^3*x^4)/4 - a^2*c^3*Log[x]} -{(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3)/x^4, x, 3, -c^3/(3*x^3) - (a*c^3)/x^2 + (a^2*c^3)/x - a^4*c^3*x + a^5*c^3*x^2 + (a^6*c^3*x^3)/3 - 4*a^3*c^3*Log[x]} - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^4, x, 3, (4*c^4*(1 + a*x)^6)/(3*a) - (12*c^4*(1 + a*x)^7)/(7*a) + (3*c^4*(1 + a*x)^8)/(4*a) - (c^4*(1 + a*x)^9)/(9*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2), x, 3, (3*x)/(a^4*c) + x^2/(a^3*c) + x^3/(3*a^2*c) + 1/(a^5*c*(1 - a*x)) + (4*Log[1 - a*x])/(a^5*c)} -{(E^(2*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2), x, 3, (2*x)/(a^3*c) + x^2/(2*a^2*c) + 1/(a^4*c*(1 - a*x)) + (3*Log[1 - a*x])/(a^4*c)} -{(E^(2*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2), x, 3, x/(a^2*c) + 1/(a^3*c*(1 - a*x)) + (2*Log[1 - a*x])/(a^3*c)} -{(E^(2*ArcTanh[a*x])*x)/(c - a^2*c*x^2), x, 3, 1/(a^2*c*(1 - a*x)) + Log[1 - a*x]/(a^2*c)} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2), x, 2, 1/(a*c*(1 - a*x))} -{E^(2*ArcTanh[a*x])/(x*(c - a^2*c*x^2)), x, 3, 1/(c*(1 - a*x)) + Log[x]/c - Log[1 - a*x]/c} -{E^(2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)), x, 3, -(1/(c*x)) + a/(c*(1 - a*x)) + (2*a*Log[x])/c - (2*a*Log[1 - a*x])/c} -{E^(2*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)), x, 3, -1/(2*c*x^2) - (2*a)/(c*x) + a^2/(c*(1 - a*x)) + (3*a^2*Log[x])/c - (3*a^2*Log[1 - a*x])/c} -{E^(2*ArcTanh[a*x])/(x^4*(c - a^2*c*x^2)), x, 3, -1/(3*c*x^3) - a/(c*x^2) - (3*a^2)/(c*x) + a^3/(c*(1 - a*x)) + (4*a^3*Log[x])/c - (4*a^3*Log[1 - a*x])/c} - - -{(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2)^2, x, 3, -(x/(a^4*c^2)) + 1/(4*a^5*c^2*(1 - a*x)^2) - 7/(4*a^5*c^2*(1 - a*x)) - (17*Log[1 - a*x])/(8*a^5*c^2) + Log[1 + a*x]/(8*a^5*c^2)} -{(E^(2*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^2, x, 3, 1/(4*a^4*c^2*(1 - a*x)^2) - 5/(4*a^4*c^2*(1 - a*x)) - (7*Log[1 - a*x])/(8*a^4*c^2) - Log[1 + a*x]/(8*a^4*c^2)} -{(E^(2*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^2, x, 4, 1/(4*a^3*c^2*(1 - a*x)^2) - 3/(4*a^3*c^2*(1 - a*x)) + ArcTanh[a*x]/(4*a^3*c^2)} -{(E^(2*ArcTanh[a*x])*x)/(c - a^2*c*x^2)^2, x, 4, 1/(4*a^2*c^2*(1 - a*x)^2) - 1/(4*a^2*c^2*(1 - a*x)) - ArcTanh[a*x]/(4*a^2*c^2)} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 4, 1/(4*a*c^2*(1 - a*x)^2) + 1/(4*a*c^2*(1 - a*x)) + ArcTanh[a*x]/(4*a*c^2)} -{E^(2*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^2), x, 3, 1/(4*c^2*(1 - a*x)^2) + 3/(4*c^2*(1 - a*x)) + Log[x]/c^2 - (7*Log[1 - a*x])/(8*c^2) - Log[1 + a*x]/(8*c^2)} -{E^(2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^2), x, 3, -(1/(c^2*x)) + a/(4*c^2*(1 - a*x)^2) + (5*a)/(4*c^2*(1 - a*x)) + (2*a*Log[x])/c^2 - (17*a*Log[1 - a*x])/(8*c^2) + (a*Log[1 + a*x])/(8*c^2)} -{E^(2*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^2), x, 3, -1/(2*c^2*x^2) - (2*a)/(c^2*x) + a^2/(4*c^2*(1 - a*x)^2) + (7*a^2)/(4*c^2*(1 - a*x)) + (4*a^2*Log[x])/c^2 - (31*a^2*Log[1 - a*x])/(8*c^2) - (a^2*Log[1 + a*x])/(8*c^2)} -{E^(2*ArcTanh[a*x])/(x^4*(c - a^2*c*x^2)^2), x, 3, -1/(3*c^2*x^3) - a/(c^2*x^2) - (4*a^2)/(c^2*x) + a^3/(4*c^2*(1 - a*x)^2) + (9*a^3)/(4*c^2*(1 - a*x)) + (6*a^3*Log[x])/c^2 - (49*a^3*Log[1 - a*x])/(8*c^2) + (a^3*Log[1 + a*x])/(8*c^2)} - - -{(E^(2*ArcTanh[a*x])*x^5)/(c - a^2*c*x^2)^3, x, 3, 1/(12*a^6*c^3*(1 - a*x)^3) - 1/(2*a^6*c^3*(1 - a*x)^2) + 23/(16*a^6*c^3*(1 - a*x)) + 1/(16*a^6*c^3*(1 + a*x)) + (13*Log[1 - a*x])/(16*a^6*c^3) + (3*Log[1 + a*x])/(16*a^6*c^3)} -{(E^(2*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2)^3, x, 4, 1/(12*a^5*c^3*(1 - a*x)^3) - 3/(8*a^5*c^3*(1 - a*x)^2) + 11/(16*a^5*c^3*(1 - a*x)) - 1/(16*a^5*c^3*(1 + a*x)) - ArcTanh[a*x]/(4*a^5*c^3)} -{(E^(2*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^3, x, 4, 1/(12*a^4*c^3*(1 - a*x)^3) - 1/(4*a^4*c^3*(1 - a*x)^2) + 3/(16*a^4*c^3*(1 - a*x)) + 1/(16*a^4*c^3*(1 + a*x)) + ArcTanh[a*x]/(8*a^4*c^3)} -{(E^(2*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^3, x, 2, -((1 - 2*a*x)/(6*a^3*c^3*(1 - a*x)^3*(1 + a*x)))} -{(E^(2*ArcTanh[a*x])*x)/(c - a^2*c*x^2)^3, x, 4, 1/(12*a^2*c^3*(1 - a*x)^3) - 1/(16*a^2*c^3*(1 - a*x)) + 1/(16*a^2*c^3*(1 + a*x)) - ArcTanh[a*x]/(8*a^2*c^3)} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 4, 1/(12*a*c^3*(1 - a*x)^3) + 1/(8*a*c^3*(1 - a*x)^2) + 3/(16*a*c^3*(1 - a*x)) - 1/(16*a*c^3*(1 + a*x)) + ArcTanh[a*x]/(4*a*c^3)} -{E^(2*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^3), x, 3, 1/(12*c^3*(1 - a*x)^3) + 1/(4*c^3*(1 - a*x)^2) + 11/(16*c^3*(1 - a*x)) + 1/(16*c^3*(1 + a*x)) + Log[x]/c^3 - (13*Log[1 - a*x])/(16*c^3) - (3*Log[1 + a*x])/(16*c^3)} -{E^(2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^3), x, 3, -(1/(c^3*x)) + a/(12*c^3*(1 - a*x)^3) + (3*a)/(8*c^3*(1 - a*x)^2) + (23*a)/(16*c^3*(1 - a*x)) - a/(16*c^3*(1 + a*x)) + (2*a*Log[x])/c^3 - (9*a*Log[1 - a*x])/(4*c^3) + (a*Log[1 + a*x])/(4*c^3)} -{E^(2*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^3), x, 3, -1/(2*c^3*x^2) - (2*a)/(c^3*x) + a^2/(12*c^3*(1 - a*x)^3) + a^2/(2*c^3*(1 - a*x)^2) + (39*a^2)/(16*c^3*(1 - a*x)) + a^2/(16*c^3*(1 + a*x)) + (5*a^2*Log[x])/c^3 - (75*a^2*Log[1 - a*x])/(16*c^3) - (5*a^2*Log[1 + a*x])/(16*c^3)} - - -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 4, 1/(32*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 3/(32*a*c^4*(1 - a*x)^2) + 5/(32*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)^2) - 5/(64*a*c^4*(1 + a*x)) + (15*ArcTanh[a*x])/(64*a*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^3, x, 7, -((3*x^2*Sqrt[c - a^2*c*x^2])/(5*a^2)) - (x^3*Sqrt[c - a^2*c*x^2])/(2*a) - (1/5)*x^4*Sqrt[c - a^2*c*x^2] - (3*(8 + 5*a*x)*Sqrt[c - a^2*c*x^2])/(20*a^4) + (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(4*a^4)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^2, x, 6, -((2*x^2*Sqrt[c - a^2*c*x^2])/(3*a)) - (1/4)*x^3*Sqrt[c - a^2*c*x^2] - ((32 + 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3) + (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^1, x, 5, (-(1/3))*x^2*Sqrt[c - a^2*c*x^2] - ((5 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(3*a^2) + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/a^2} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^0, x, 5, -((3*Sqrt[c - a^2*c*x^2])/(2*a)) - ((1 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a) + (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^1, x, 8, -Sqrt[c - a^2*c*x^2] + 2*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^2, x, 8, -(Sqrt[c - a^2*c*x^2]/x) + a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - 2*a*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^3, x, 6, -(Sqrt[c - a^2*c*x^2]/(2*x^2)) - (2*a*Sqrt[c - a^2*c*x^2])/x - (3/2)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^4, x, 7, -(Sqrt[c - a^2*c*x^2]/(3*x^3)) - (a*Sqrt[c - a^2*c*x^2])/x^2 - (5*a^2*Sqrt[c - a^2*c*x^2])/(3*x) - a^3*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^5, x, 8, -(Sqrt[c - a^2*c*x^2]/(4*x^4)) - (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) - (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) - (4*a^3*Sqrt[c - a^2*c*x^2])/(3*x) - (7/8)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)*x^3, x, 8, (c*x*Sqrt[c - a^2*c*x^2])/(8*a^3) - (11*x^2*(c - a^2*c*x^2)^(3/2))/(35*a^2) - (x^3*(c - a^2*c*x^2)^(3/2))/(3*a) - (1/7)*x^4*(c - a^2*c*x^2)^(3/2) - ((88 + 105*a*x)*(c - a^2*c*x^2)^(3/2))/(420*a^4) + (c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^4)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)*x^2, x, 7, (3*c*x*Sqrt[c - a^2*c*x^2])/(16*a^2) - (2*x^2*(c - a^2*c*x^2)^(3/2))/(5*a) - (1/6)*x^3*(c - a^2*c*x^2)^(3/2) - ((32 + 45*a*x)*(c - a^2*c*x^2)^(3/2))/(120*a^3) + (3*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a^3)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)*x^1, x, 6, (c*x*Sqrt[c - a^2*c*x^2])/(4*a) - (1/5)*x^2*(c - a^2*c*x^2)^(3/2) - ((14 + 15*a*x)*(c - a^2*c*x^2)^(3/2))/(30*a^2) + (c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(4*a^2)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)*x^0, x, 6, (5/8)*c*x*Sqrt[c - a^2*c*x^2] - (5*(c - a^2*c*x^2)^(3/2))/(12*a) - ((1 + a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) + (5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^1, x, 9, c*(1 + a*x)*Sqrt[c - a^2*c*x^2] - (1/3)*(c - a^2*c*x^2)^(3/2) + c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^2, x, 9, (1/2)*a*c*(4 - a*x)*Sqrt[c - a^2*c*x^2] - (c - a^2*c*x^2)^(3/2)/x - (1/2)*a*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - 2*a*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^3, x, 9, -((a*c*(4 - a*x)*Sqrt[c - a^2*c*x^2])/(2*x)) - (c - a^2*c*x^2)^(3/2)/(2*x^2) - 2*a^2*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - (1/2)*a^2*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^4, x, 9, -((a*c*(1 + a*x)*Sqrt[c - a^2*c*x^2])/x^2) - (c - a^2*c*x^2)^(3/2)/(3*x^3) - a^3*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + a^3*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^5, x, 7, -((5*a^2*c*Sqrt[c - a^2*c*x^2])/(8*x^2)) - (c - a^2*c*x^2)^(3/2)/(4*x^4) - (2*a*(c - a^2*c*x^2)^(3/2))/(3*x^3) + (5/8)*a^4*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^6, x, 8, -((a^3*c*Sqrt[c - a^2*c*x^2])/(4*x^2)) - (c - a^2*c*x^2)^(3/2)/(5*x^5) - (a*(c - a^2*c*x^2)^(3/2))/(2*x^4) - (7*a^2*(c - a^2*c*x^2)^(3/2))/(15*x^3) + (1/4)*a^5*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^7, x, 9, -((3*a^4*c*Sqrt[c - a^2*c*x^2])/(16*x^2)) - (c - a^2*c*x^2)^(3/2)/(6*x^6) - (2*a*(c - a^2*c*x^2)^(3/2))/(5*x^5) - (3*a^2*(c - a^2*c*x^2)^(3/2))/(8*x^4) - (4*a^3*(c - a^2*c*x^2)^(3/2))/(15*x^3) + (3/16)*a^6*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)/x^8, x, 10, -((a^5*c*Sqrt[c - a^2*c*x^2])/(8*x^2)) - (c - a^2*c*x^2)^(3/2)/(7*x^7) - (a*(c - a^2*c*x^2)^(3/2))/(3*x^6) - (11*a^2*(c - a^2*c*x^2)^(3/2))/(35*x^5) - (a^3*(c - a^2*c*x^2)^(3/2))/(4*x^4) - (22*a^4*(c - a^2*c*x^2)^(3/2))/(105*x^3) + (1/8)*a^7*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)*x^3, x, 9, (3*c^2*x*Sqrt[c - a^2*c*x^2])/(64*a^3) + (c*x*(c - a^2*c*x^2)^(3/2))/(32*a^3) - (13*x^2*(c - a^2*c*x^2)^(5/2))/(63*a^2) - (x^3*(c - a^2*c*x^2)^(5/2))/(4*a) - (1/9)*x^4*(c - a^2*c*x^2)^(5/2) - ((208 + 315*a*x)*(c - a^2*c*x^2)^(5/2))/(2520*a^4) + (3*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(64*a^4)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)*x^2, x, 8, (11*c^2*x*Sqrt[c - a^2*c*x^2])/(128*a^2) + (11*c*x*(c - a^2*c*x^2)^(3/2))/(192*a^2) - (2*x^2*(c - a^2*c*x^2)^(5/2))/(7*a) - (1/8)*x^3*(c - a^2*c*x^2)^(5/2) - ((192 + 385*a*x)*(c - a^2*c*x^2)^(5/2))/(1680*a^3) + (11*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a^3)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)*x^1, x, 7, (c^2*x*Sqrt[c - a^2*c*x^2])/(8*a) + (c*x*(c - a^2*c*x^2)^(3/2))/(12*a) - (1/7)*x^2*(c - a^2*c*x^2)^(5/2) - ((27 + 35*a*x)*(c - a^2*c*x^2)^(5/2))/(105*a^2) + (c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^2)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)*x^0, x, 7, (7/16)*c^2*x*Sqrt[c - a^2*c*x^2] + (7/24)*c*x*(c - a^2*c*x^2)^(3/2) - (7*(c - a^2*c*x^2)^(5/2))/(30*a) - ((1 + a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) + (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)/x^1, x, 10, (1/4)*c^2*(4 + 3*a*x)*Sqrt[c - a^2*c*x^2] + (1/6)*c*(2 + 3*a*x)*(c - a^2*c*x^2)^(3/2) - (1/5)*(c - a^2*c*x^2)^(5/2) + (3/4)*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - c^(5/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)/x^2, x, 10, (1/8)*a*c^2*(16 - 9*a*x)*Sqrt[c - a^2*c*x^2] + (1/12)*a*c*(8 - 9*a*x)*(c - a^2*c*x^2)^(3/2) - (c - a^2*c*x^2)^(5/2)/x - (9/8)*a*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - 2*a*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)/x^3, x, 10, (-(1/2))*a^2*c^2*(1 + 6*a*x)*Sqrt[c - a^2*c*x^2] - (a*c*(12 + a*x)*(c - a^2*c*x^2)^(3/2))/(6*x) - (c - a^2*c*x^2)^(5/2)/(2*x^2) - 3*a^2*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + (1/2)*a^2*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)/x^4, x, 10, -((a^2*c^2*(1 + 6*a*x)*Sqrt[c - a^2*c*x^2])/(2*x)) - (a*c*(6 - a*x)*(c - a^2*c*x^2)^(3/2))/(6*x^2) - (c - a^2*c*x^2)^(5/2)/(3*x^3) - (1/2)*a^3*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + 3*a^3*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)/x^5, x, 10, (a^3*c^2*(16 - 9*a*x)*Sqrt[c - a^2*c*x^2])/(8*x) - (a*c*(16 + 9*a*x)*(c - a^2*c*x^2)^(3/2))/(24*x^3) - (c - a^2*c*x^2)^(5/2)/(4*x^4) + 2*a^4*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + (9/8)*a^4*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2), x, 8, (45/128)*c^3*x*Sqrt[c - a^2*c*x^2] + (15/64)*c^2*x*(c - a^2*c*x^2)^(3/2) + (3/16)*c*x*(c - a^2*c*x^2)^(5/2) - (9*(c - a^2*c*x^2)^(7/2))/(56*a) - ((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) + (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)*x^3, x, 7, (1 + a*x)^2/(a^4*Sqrt[c - a^2*c*x^2]) + (11*Sqrt[c - a^2*c*x^2])/(3*a^4*c) + (x*Sqrt[c - a^2*c*x^2])/(a^3*c) + (x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*c) - (3*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^4*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)*x^2, x, 5, (1 + a*x)^2/(a^3*Sqrt[c - a^2*c*x^2]) + ((6 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a^3*c) - (5*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a^3*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)*x^1, x, 5, (1 + a*x)^2/(a^2*Sqrt[c - a^2*c*x^2]) + (2*Sqrt[c - a^2*c*x^2])/(a^2*c) - (2*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^2*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)*x^0, x, 4, (2*(1 + a*x))/(a*Sqrt[c - a^2*c*x^2]) - ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)/x^1, x, 5, (2*(1 + a*x))/Sqrt[c - a^2*c*x^2] - ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]/Sqrt[c]} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)/x^2, x, 6, (2*a*(1 + a*x))/Sqrt[c - a^2*c*x^2] - Sqrt[c - a^2*c*x^2]/(c*x) - (2*a*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c]} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)/x^3, x, 7, (2*a^2*(1 + a*x))/Sqrt[c - a^2*c*x^2] - Sqrt[c - a^2*c*x^2]/(2*c*x^2) - (2*a*Sqrt[c - a^2*c*x^2])/(c*x) - (5*a^2*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/(2*Sqrt[c])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)/x^4, x, 8, (2*a^3*(1 + a*x))/Sqrt[c - a^2*c*x^2] - Sqrt[c - a^2*c*x^2]/(3*c*x^3) - (a*Sqrt[c - a^2*c*x^2])/(c*x^2) - (8*a^2*Sqrt[c - a^2*c*x^2])/(3*c*x) - (3*a^3*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c]} - - -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)*x^3, x, 6, (1 + a*x)^2/(3*a^4*(c - a^2*c*x^2)^(3/2)) - (8*(1 + a*x))/(3*a^4*c*Sqrt[c - a^2*c*x^2]) - Sqrt[c - a^2*c*x^2]/(a^4*c^2) + (2*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(a^4*c^(3/2))} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)*x^2, x, 5, (1 + a*x)^2/(3*a^3*(c - a^2*c*x^2)^(3/2)) - (5*(1 + a*x))/(3*a^3*c*Sqrt[c - a^2*c*x^2]) + ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a^3*c^(3/2))} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)*x^1, x, 3, (1 + a*x)^2/(3*a^2*(c - a^2*c*x^2)^(3/2)) - (2*(1 + a*x))/(3*a^2*c*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)*x^0, x, 3, (2*(1 + a*x))/(3*a*(c - a^2*c*x^2)^(3/2)) + x/(3*c*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)/x^1, x, 7, (2*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (3 + 4*a*x)/(3*c*Sqrt[c - a^2*c*x^2]) - ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]/c^(3/2)} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)/x^2, x, 7, (2*a*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (a*(6 + 7*a*x))/(3*c*Sqrt[c - a^2*c*x^2]) - Sqrt[c - a^2*c*x^2]/(c^2*x) - (2*a*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/c^(3/2)} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)/x^3, x, 8, (2*a^2*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + (a^2*(9 + 10*a*x))/(3*c*Sqrt[c - a^2*c*x^2]) - Sqrt[c - a^2*c*x^2]/(2*c^2*x^2) - (2*a*Sqrt[c - a^2*c*x^2])/(c^2*x) - (7*a^2*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/(2*c^(3/2))} - - -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/2), x, 4, (2*(1 + a*x))/(5*a*(c - a^2*c*x^2)^(5/2)) + x/(5*c*(c - a^2*c*x^2)^(3/2)) + (2*x)/(5*c^2*Sqrt[c - a^2*c*x^2])} - - -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 5, (2*(1 + a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) + x/(7*c*(c - a^2*c*x^2)^(5/2)) + (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic*) - - -{E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^3, x, 3, (c^3*x^(1 + m))/(1 + m) + (2*a*c^3*x^(2 + m))/(2 + m) - (a^2*c^3*x^(3 + m))/(3 + m) - (4*a^3*c^3*x^(4 + m))/(4 + m) - (a^4*c^3*x^(5 + m))/(5 + m) + (2*a^5*c^3*x^(6 + m))/(6 + m) + (a^6*c^3*x^(7 + m))/(7 + m)} -{E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^2, x, 3, (c^2*x^(1 + m))/(1 + m) + (2*a*c^2*x^(2 + m))/(2 + m) - (2*a^3*c^2*x^(4 + m))/(4 + m) - (a^4*c^2*x^(5 + m))/(5 + m)} -{E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^1, x, 3, (c*x^(1 + m))/(1 + m) + (2*a*c*x^(2 + m))/(2 + m) + (a^2*c*x^(3 + m))/(3 + m)} -{(E^(2*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^1, x, 2, (x^(1 + m)*Hypergeometric2F1[2, 1 + m, 2 + m, a*x])/(c*(1 + m))} -{(E^(2*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^2, x, 6, x^(1 + m)/(4*c^2*(1 - a*x)^2) + ((2 - m)*x^(1 + m))/(4*c^2*(1 - a*x)) + (x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/(8*c^2*(1 + m)) + ((1 - 4*m + 2*m^2)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(8*c^2*(1 + m))} -{(E^(2*ArcTanh[a*x])*x^m)/(c - a^2*c*x^2)^3, x, 8, -(((2 - m)*(4 - m)*x^(1 + m))/(24*c^3*(1 + a*x))) + x^(1 + m)/(6*c^3*(1 - a*x)^3*(1 + a*x)) + ((4 - m)*x^(1 + m))/(12*c^3*(1 - a*x)^2*(1 + a*x)) + ((7 - 2*m)*(2 - m)*x^(1 + m))/(24*c^3*(1 - a*x)*(1 + a*x)) + ((2 - m)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/(16*c^3*(1 + m)) + ((2 - m)*(3 - 8*m + 2*m^2)*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(48*c^3*(1 + m))} - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2)*x^m, x, 7, If[$VersionNumber>=8, -((x^(1 + m)*(c - a^2*c*x^2)^(5/2))/(6 + m)) + (c^2*(7 + 2*m)*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(3/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(6 + m)*Sqrt[1 - a^2*x^2]) + (2*a*c^2*x^(2 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(3/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]), -((x^(1 + m)*(c - a^2*c*x^2)^(5/2))/(6 + m)) + (c^2*(7 + 2*m)*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(3/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/((6 + 7*m + m^2)*Sqrt[1 - a^2*x^2]) + (2*a*c^2*x^(2 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(3/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)*x^m, x, 7, If[$VersionNumber>=8, -((x^(1 + m)*(c - a^2*c*x^2)^(3/2))/(4 + m)) + (c*(5 + 2*m)*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(1/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(4 + m)*Sqrt[1 - a^2*x^2]) + (2*a*c*x^(2 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(1/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]), -((x^(1 + m)*(c - a^2*c*x^2)^(3/2))/(4 + m)) + (c*(5 + 2*m)*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(1/2), (1 + m)/2, (3 + m)/2, a^2*x^2])/((4 + 5*m + m^2)*Sqrt[1 - a^2*x^2]) + (2*a*c*x^(2 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-(1/2), (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])]} -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^m, x, 7, If[$VersionNumber>=8, -((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m)) + (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(2 + m)*Sqrt[c - a^2*c*x^2]) + (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2]), -((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m)) + (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[c - a^2*c*x^2]) + (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])]} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2)*x^m, x, 7, (2*x^(1 + m)*(1 + a*x))/Sqrt[c - a^2*c*x^2] - ((1 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*Sqrt[c - a^2*c*x^2]) - (2*a*(1 + m)*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2)*x^m, x, 7, (2*x^(1 + m)*(1 + a*x))/(3*(c - a^2*c*x^2)^(3/2)) + ((1 - 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/(3*c*(1 + m)*Sqrt[c - a^2*c*x^2]) + (2*a*(1 - m)*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[3/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/(3*c*(2 + m)*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, -((2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1/2)*(1 - a*x)])/((1 + a*x)^p*(a*p)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(3 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)*x^3, x, 7, -((17*c*x^2*Sqrt[1 - a^2*x^2])/(15*a^2)) - (23*c*x^3*Sqrt[1 - a^2*x^2])/(24*a) - (3/5)*c*x^4*Sqrt[1 - a^2*x^2] - (1/6)*a*c*x^5*Sqrt[1 - a^2*x^2] - (c*(544 + 345*a*x)*Sqrt[1 - a^2*x^2])/(240*a^4) + (23*c*ArcSin[a*x])/(16*a^4)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)*x^2, x, 6, -((19*c*x^2*Sqrt[1 - a^2*x^2])/(15*a)) - (3/4)*c*x^3*Sqrt[1 - a^2*x^2] - (1/5)*a*c*x^4*Sqrt[1 - a^2*x^2] - (c*(304 + 195*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3) + (13*c*ArcSin[a*x])/(8*a^3)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)*x^1, x, 6, -((15*c*Sqrt[1 - a^2*x^2])/(8*a^2)) - (5*c*(1 + a*x)*Sqrt[1 - a^2*x^2])/(8*a^2) - (c*(1 + a*x)^2*Sqrt[1 - a^2*x^2])/(4*a^2) - (c*(1 + a*x)^3*Sqrt[1 - a^2*x^2])/(4*a^2) + (15*c*ArcSin[a*x])/(8*a^2)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)*x^0, x, 5, -((5*c*Sqrt[1 - a^2*x^2])/(2*a)) - (5*c*(1 + a*x)*Sqrt[1 - a^2*x^2])/(6*a) - (c*(1 + a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) + (5*c*ArcSin[a*x])/(2*a)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^1, x, 8, -3*c*Sqrt[1 - a^2*x^2] - (1/2)*a*c*x*Sqrt[1 - a^2*x^2] + (7/2)*c*ArcSin[a*x] - c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^2, x, 8, (-a)*c*Sqrt[1 - a^2*x^2] - (c*Sqrt[1 - a^2*x^2])/x + 3*a*c*ArcSin[a*x] - 3*a*c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^3, x, 8, -((c*Sqrt[1 - a^2*x^2])/(2*x^2)) - (3*a*c*Sqrt[1 - a^2*x^2])/x + a^2*c*ArcSin[a*x] - (7/2)*a^2*c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^4, x, 7, -((c*Sqrt[1 - a^2*x^2])/(3*x^3)) - (3*a*c*Sqrt[1 - a^2*x^2])/(2*x^2) - (11*a^2*c*Sqrt[1 - a^2*x^2])/(3*x) - (5/2)*a^3*c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^5, x, 8, -((c*Sqrt[1 - a^2*x^2])/(4*x^4)) - (a*c*Sqrt[1 - a^2*x^2])/x^3 - (15*a^2*c*Sqrt[1 - a^2*x^2])/(8*x^2) - (3*a^3*c*Sqrt[1 - a^2*x^2])/x - (15/8)*a^4*c*ArcTanh[Sqrt[1 - a^2*x^2]]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)/x^6, x, 9, -((c*Sqrt[1 - a^2*x^2])/(5*x^5)) - (3*a*c*Sqrt[1 - a^2*x^2])/(4*x^4) - (19*a^2*c*Sqrt[1 - a^2*x^2])/(15*x^3) - (13*a^3*c*Sqrt[1 - a^2*x^2])/(8*x^2) - (38*a^4*c*Sqrt[1 - a^2*x^2])/(15*x) - (13/8)*a^5*c*ArcTanh[Sqrt[1 - a^2*x^2]]} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 6, (7/8)*c^2*x*Sqrt[1 - a^2*x^2] - (7*c^2*(1 - a^2*x^2)^(3/2))/(12*a) - (7*c^2*(1 + a*x)*(1 - a^2*x^2)^(3/2))/(20*a) - (c^2*(1 + a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^2*ArcSin[a*x])/(8*a)} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 7, (9/16)*c^3*x*Sqrt[1 - a^2*x^2] + (3/8)*c^3*x*(1 - a^2*x^2)^(3/2) - (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) - (3*c^3*(1 + a*x)*(1 - a^2*x^2)^(5/2))/(14*a) - (c^3*(1 + a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16*a)} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^4, x, 8, (55/128)*c^4*x*Sqrt[1 - a^2*x^2] + (55/192)*c^4*x*(1 - a^2*x^2)^(3/2) + (11/48)*c^4*x*(1 - a^2*x^2)^(5/2) - (11*c^4*(1 - a^2*x^2)^(7/2))/(56*a) - (11*c^4*(1 + a*x)*(1 - a^2*x^2)^(7/2))/(72*a) - (c^4*(1 + a*x)^2*(1 - a^2*x^2)^(7/2))/(9*a) + (55*c^4*ArcSin[a*x])/(128*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)*x^2, x, 6, (1 + a*x)^3/(3*a^3*c*(1 - a^2*x^2)^(3/2)) - (2*(1 + a*x)^2)/(a^3*c*Sqrt[1 - a^2*x^2]) - (3*Sqrt[1 - a^2*x^2])/(a^3*c) + (3*ArcSin[a*x])/(a^3*c)} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)*x^1, x, 4, (1 + a*x)^3/(3*a^2*c*(1 - a^2*x^2)^(3/2)) - (2*(1 + a*x))/(a^2*c*Sqrt[1 - a^2*x^2]) + ArcSin[a*x]/(a^2*c)} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)*x^0, x, 1, E^(3*ArcTanh[a*x])/(3*a*c)} - - -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 5, Sqrt[1 - a^2*x^2]/(5*a*c^2*(1 - a*x)^3) + (2*Sqrt[1 - a^2*x^2])/(15*a*c^2*(1 - a*x)^2) + (2*Sqrt[1 - a^2*x^2])/(15*a*c^2*(1 - a*x))} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 6, (8*x)/(35*c^3*Sqrt[1 - a^2*x^2]) + 1/(7*a*c^3*(1 - a*x)^3*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)^2*Sqrt[1 - a^2*x^2]) + 4/(35*a*c^3*(1 - a*x)*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 7, (8*x)/(63*c^4*(1 - a^2*x^2)^(3/2)) + 1/(9*a*c^4*(1 - a*x)^3*(1 - a^2*x^2)^(3/2)) + 2/(21*a*c^4*(1 - a*x)^2*(1 - a^2*x^2)^(3/2)) + 2/(21*a*c^4*(1 - a*x)*(1 - a^2*x^2)^(3/2)) + (16*x)/(63*c^4*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^3, x, 4, -((4*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - a^2*x^2])) - (2*x^2*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2]) - (4*x^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (3*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) - (a*x^5*Sqrt[c - a^2*c*x^2])/(5*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^4*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^2, x, 4, -((4*x*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2])) - (2*x^2*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) - (x^3*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] - (a*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^3*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^1, x, 4, -((4*x*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2])) - (3*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) - (a*x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^2*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^0, x, 4, -((3*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2]) - (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a*Sqrt[1 - a^2*x^2])} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^1, x, 4, -((a*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2]) + (Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^2, x, 4, -(Sqrt[c - a^2*c*x^2]/(x*Sqrt[1 - a^2*x^2])) + (3*a*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^3, x, 4, -(Sqrt[c - a^2*c*x^2]/(2*x^2*Sqrt[1 - a^2*x^2])) - (3*a*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^4, x, 4, -(Sqrt[c - a^2*c*x^2]/(3*x^3*Sqrt[1 - a^2*x^2])) - (3*a*Sqrt[c - a^2*c*x^2])/(2*x^2*Sqrt[1 - a^2*x^2]) - (4*a^2*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) + (4*a^3*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^3*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^5, x, 4, -(Sqrt[c - a^2*c*x^2]/(4*x^4*Sqrt[1 - a^2*x^2])) - (a*Sqrt[c - a^2*c*x^2])/(x^3*Sqrt[1 - a^2*x^2]) - (2*a^2*Sqrt[c - a^2*c*x^2])/(x^2*Sqrt[1 - a^2*x^2]) - (4*a^3*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) + (4*a^4*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/Sqrt[1 - a^2*x^2]} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2), x, 3, (c*(1 + a*x)^4*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2])} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2), x, 4, (2*c^2*(1 + a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2]) - (c^2*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - a^2*x^2])} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2), x, 4, (2*c^3*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (4*c^3*(1 + a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) + (c^3*(1 + a*x)^8*Sqrt[c - a^2*c*x^2])/(8*a*Sqrt[1 - a^2*x^2])} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(9/2), x, 4, (8*c^4*(1 + a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 + a*x)^8*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - a^2*x^2]) + (2*c^4*(1 + a*x)^9*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 + a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2), x, 4, (2*Sqrt[1 - a^2*x^2])/(a*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(a*Sqrt[c - a^2*c*x^2])} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2), x, 3, Sqrt[1 - a^2*x^2]/(2*a*c*(1 - a*x)^2*Sqrt[c - a^2*c*x^2])} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(6*a*c^2*(1 - a*x)^3*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a*c^2*Sqrt[c - a^2*c*x^2])} -{E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 5, Sqrt[1 - a^2*x^2]/(16*a*c^3*(1 - a*x)^4*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(12*a*c^3*(1 - a*x)^3*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^3*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(32*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic*) - - -{x^m*E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2), x, 5, -((3*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2])) - (a*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]) + (4*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/((1 + m)*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p*x^m, x, 7, -((3*x^(1 + m)*(c - a^2*c*x^2)^p)/((m + 2*p)*Sqrt[1 - a^2*x^2])) - (a*x^(2 + m)*(c - a^2*c*x^2)^p)/((1 + m + 2*p)*Sqrt[1 - a^2*x^2]) + ((3 + 4*m + 2*p)*x^(1 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(1 + m)/2, 3/2 - p, (3 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*((1 + m)*(m + 2*p))) + (a*(5 + 4*m + 6*p)*x^(2 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(2 + m)/2, 3/2 - p, (4 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*((2 + m)*(1 + m + 2*p)))} - - -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p*x^3, x, 8, (4*(c - a^2*c*x^2)^p)/(a^4*(1 - 2*p)*Sqrt[1 - a^2*x^2]) - (a*x^5*(c - a^2*c*x^2)^p)/(2*(2 + p)*Sqrt[1 - a^2*x^2]) + (7*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^4*(1 + 2*p)) - (3*(1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^4*(3 + 2*p)) + (1/(10*(2 + p)))*((a*(17 + 6*p)*x^5*(c - a^2*c*x^2)^p*Hypergeometric2F1[5/2, 3/2 - p, 7/2, a^2*x^2])/(1 - a^2*x^2)^p)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p*x^2, x, 8, (4*(c - a^2*c*x^2)^p)/(a^3*(1 - 2*p)*Sqrt[1 - a^2*x^2]) - (3*x^3*(c - a^2*c*x^2)^p)/(2*(1 + p)*Sqrt[1 - a^2*x^2]) + (5*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^3*(1 + 2*p)) - ((1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^3*(3 + 2*p)) + (1/(6*(1 + p)))*(((11 + 2*p)*x^3*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2, 3/2 - p, 5/2, a^2*x^2])/(1 - a^2*x^2)^p)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p*x^1, x, 5, -(((1 + a*x)^3*(c - a^2*c*x^2)^p)/(2*a^2*(1 + p)*Sqrt[1 - a^2*x^2])) + (3*2^(3/2 + p)*(1 - a*x)^(-(1/2) + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(3/2) - p, -(1/2) + p, 1/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a^2*(1 - p - 2*p^2)))} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p*x^0, x, 3, (2^(5/2 + p)*(1 - a*x)^(-(1/2) + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(3/2) - p, -(1/2) + p, 1/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a*(1 - 2*p)))} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p/x^1, x, 8, (4*(c - a^2*c*x^2)^p)/((1 - 2*p)*Sqrt[1 - a^2*x^2]) - (a*x*(c - a^2*c*x^2)^p)/(2*p*Sqrt[1 - a^2*x^2]) + (a*(1 + 6*p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2])/((1 - a^2*x^2)^p*(2*p)) - (Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p/x^2, x, 9, (4*a*(c - a^2*c*x^2)^p)/((1 - 2*p)*Sqrt[1 - a^2*x^2]) - (c - a^2*c*x^2)^p/(x*Sqrt[1 - a^2*x^2]) + (a^2*(5 - 2*p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2])/(1 - a^2*x^2)^p - (3*a*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p/x^3, x, 8, -((c - a^2*c*x^2)^p/(2*x^2*Sqrt[1 - a^2*x^2])) - (3*a*(c - a^2*c*x^2)^p)/(x*Sqrt[1 - a^2*x^2]) + (a^3*(7 - 6*p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2])/(1 - a^2*x^2)^p + (a^2*(9 - 2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, -(1/2) + p, 1/2 + p, 1 - a^2*x^2])/(2*(1 - 2*p)*Sqrt[1 - a^2*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(4 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^5, x, 3, (c^5*(1 + a*x)^8)/a - (4*c^5*(1 + a*x)^9)/(3*a) + (3*c^5*(1 + a*x)^10)/(5*a) - (c^5*(1 + a*x)^11)/(11*a)} -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^4, x, 3, (4*c^4*(1 + a*x)^7)/(7*a) - (c^4*(1 + a*x)^8)/(2*a) + (c^4*(1 + a*x)^9)/(9*a)} -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 3, (c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)} -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 2, (c^2*(1 + a*x)^5)/(5*a)} -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^1, x, 3, -7*c*x - 2*a*c*x^2 - (1/3)*a^2*c*x^3 - (8*c*Log[1 - a*x])/a, -4*c*x - (c*(1 + a*x)^2)/a - (c*(1 + a*x)^3)/(3*a) - (8*c*Log[1 - a*x])/a} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^1, x, 2, x/(c*(1 - a*x)^2)} -{E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 2, 1/(3*a*c^2*(1 - a*x)^3)} -{E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 4, 1/(8*a*c^3*(1 - a*x)^4) + 1/(12*a*c^3*(1 - a*x)^3) + 1/(16*a*c^3*(1 - a*x)^2) + 1/(16*a*c^3*(1 - a*x)) + ArcTanh[a*x]/(16*a*c^3)} -{E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 4, 1/(20*a*c^4*(1 - a*x)^5) + 1/(16*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 1/(16*a*c^4*(1 - a*x)^2) + 5/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (3*ArcTanh[a*x])/(32*a*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, (2^(2 + p)*c*(1 + a*x)^(1 - p)*(c - a^2*c*x^2)^(-1 + p)*Hypergeometric2F1[-2 - p, -1 + p, p, (1/2)*(1 - a*x)])/(a*(1 - p))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p / E^(1 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^4, x, 7, (35/128)*c^4*x*Sqrt[1 - a^2*x^2] + (35/192)*c^4*x*(1 - a^2*x^2)^(3/2) + (7/48)*c^4*x*(1 - a^2*x^2)^(5/2) + (1/8)*c^4*x*(1 - a^2*x^2)^(7/2) + (c^4*(1 - a^2*x^2)^(9/2))/(9*a) + (35*c^4*ArcSin[a*x])/(128*a)} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^3, x, 6, (5/16)*c^3*x*Sqrt[1 - a^2*x^2] + (5/24)*c^3*x*(1 - a^2*x^2)^(3/2) + (1/6)*c^3*x*(1 - a^2*x^2)^(5/2) + (c^3*(1 - a^2*x^2)^(7/2))/(7*a) + (5*c^3*ArcSin[a*x])/(16*a)} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^2, x, 5, (3/8)*c^2*x*Sqrt[1 - a^2*x^2] + (1/4)*c^2*x*(1 - a^2*x^2)^(3/2) + (c^2*(1 - a^2*x^2)^(5/2))/(5*a) + (3*c^2*ArcSin[a*x])/(8*a)} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^1, x, 4, (1/2)*c*x*Sqrt[1 - a^2*x^2] + (c*(1 - a^2*x^2)^(3/2))/(3*a) + (c*ArcSin[a*x])/(2*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^1, x, 1, -(1/(E^ArcTanh[a*x]*(a*c)))} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^2, x, 3, -((1 - a*x)/(3*a*c^2*(1 - a^2*x^2)^(3/2))) + (2*x)/(3*c^2*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^3, x, 4, -((1 - a*x)/(5*a*c^3*(1 - a^2*x^2)^(5/2))) + (4*x)/(15*c^3*(1 - a^2*x^2)^(3/2)) + (8*x)/(15*c^3*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^4, x, 5, -((1 - a*x)/(7*a*c^4*(1 - a^2*x^2)^(7/2))) + (6*x)/(35*c^4*(1 - a^2*x^2)^(5/2)) + (8*x)/(35*c^4*(1 - a^2*x^2)^(3/2)) + (16*x)/(35*c^4*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^5, x, 6, -((1 - a*x)/(9*a*c^5*(1 - a^2*x^2)^(9/2))) + (8*x)/(63*c^5*(1 - a^2*x^2)^(7/2)) + (16*x)/(105*c^5*(1 - a^2*x^2)^(5/2)) + (64*x)/(315*c^5*(1 - a^2*x^2)^(3/2)) + (128*x)/(315*c^5*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^m, x, 4, (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) - (a*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])} - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^2, x, 4, (x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2]) - (a*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^1, x, 4, (x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) - (a*x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)*x^0, x, 3, (x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] - (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2])} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)/x^1, x, 4, -((a*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2]) + (Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2]} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(1/2)/x^2, x, 4, -(Sqrt[c - a^2*c*x^2]/(x*Sqrt[1 - a^2*x^2])) - (a*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2]} - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(3/2), x, 4, -((2*c*(1 - a*x)^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2])) + (c*(1 - a*x)^4*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2])} - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(5/2), x, 4, -((c^2*(1 - a*x)^4*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2])) + (4*c^2*(1 - a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2]) - (c^2*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - a^2*x^2])} - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(7/2), x, 4, -((8*c^3*(1 - a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2])) + (2*c^3*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) - (6*c^3*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) + (c^3*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(8*a*Sqrt[1 - a^2*x^2])} - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^(9/2), x, 4, -((8*c^4*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2])) + (32*c^4*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) + (8*c^4*(1 - a*x)^9*Sqrt[c - a^2*c*x^2])/(9*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 - a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^(1/2), x, 3, (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(a*Sqrt[c - a^2*c*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^(3/2), x, 5, -(Sqrt[1 - a^2*x^2]/(2*a*c*(1 + a*x)*Sqrt[c - a^2*c*x^2])) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*a*c*Sqrt[c - a^2*c*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^(5/2), x, 5, Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(4*a*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a*c^2*Sqrt[c - a^2*c*x^2])} -{1/E^ArcTanh[a*x]/(c - a^2*c*x^2)^(7/2), x, 5, Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + Sqrt[1 - a^2*x^2]/(8*a*c^3*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(24*a*c^3*(1 + a*x)^3*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2])/(16*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(16*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{x^m*(c - a^2*c*x^2)^p/E^ArcTanh[a*x], x, 5, (x^(1 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(1 + m)/2, 1/2 - p, (3 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*(1 + m)) - (a*x^(2 + m)*(c - a^2*c*x^2)^p*Hypergeometric2F1[(2 + m)/2, 1/2 - p, (4 + m)/2, a^2*x^2])/((1 - a^2*x^2)^p*(2 + m))} - - -{x^3*(1 - a^2*x^2)^p/E^ArcTanh[a*x], x, 6, -((1 - a^2*x^2)^(1/2 + p)/(a^4*(1 + 2*p))) + (1 - a^2*x^2)^(3/2 + p)/(a^4*(3 + 2*p)) - (1/5)*a*x^5*Hypergeometric2F1[5/2, 1/2 - p, 7/2, a^2*x^2]} -{x^2*(1 - a^2*x^2)^p/E^ArcTanh[a*x], x, 6, (1 - a^2*x^2)^(1/2 + p)/(a^3*(1 + 2*p)) - (1 - a^2*x^2)^(3/2 + p)/(a^3*(3 + 2*p)) + (1/3)*x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2]} -{x^1*(1 - a^2*x^2)^p/E^ArcTanh[a*x], x, 4, -((1 - a^2*x^2)^(1/2 + p)/(a^2*(1 + 2*p))) - (1/3)*a*x^3*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2]} -{x^0*(1 - a^2*x^2)^p/E^ArcTanh[a*x], x, 2, -((2^(1/2 + p)*(1 - a*x)^(3/2 + p)*Hypergeometric2F1[1/2 - p, 3/2 + p, 5/2 + p, (1/2)*(1 - a*x)])/(a*(3 + 2*p)))} -{(1 - a^2*x^2)^p/(x^1*E^ArcTanh[a*x]), x, 5, (-a)*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2] - ((1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{(1 - a^2*x^2)^p/(x^2*E^ArcTanh[a*x]), x, 5, -(Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2]/x) + (a*(1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} - - -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^3, x, 7, -((Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^4*(1 + 2*p))) + ((1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^4*(3 + 2*p)) - ((1/5)*a*x^5*(c - a^2*c*x^2)^p*Hypergeometric2F1[5/2, 1/2 - p, 7/2, a^2*x^2])/(1 - a^2*x^2)^p} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^2, x, 7, (Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^3*(1 + 2*p)) - ((1 - a^2*x^2)^(3/2)*(c - a^2*c*x^2)^p)/(a^3*(3 + 2*p)) + ((1/3)*x^3*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/(1 - a^2*x^2)^p} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^1, x, 5, -((Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p)/(a^2*(1 + 2*p))) - ((1/3)*a*x^3*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/(1 - a^2*x^2)^p} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p*x^0, x, 3, -((2^(1/2 + p)*(1 - a*x)^(3/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2 - p, 3/2 + p, 5/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a*(3 + 2*p))))} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p/x^1, x, 6, ((-a)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2])/(1 - a^2*x^2)^p - (Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} -{1/E^ArcTanh[a*x]*(c - a^2*c*x^2)^p/x^2, x, 6, -(((c - a^2*c*x^2)^p*Hypergeometric2F1[-(1/2), 1/2 - p, 1/2, a^2*x^2])/((1 - a^2*x^2)^p*x)) + (a*Sqrt[1 - a^2*x^2]*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p / E^(2 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^4, x, 3, -((4*c^4*(1 - a*x)^6)/(3*a)) + (12*c^4*(1 - a*x)^7)/(7*a) - (3*c^4*(1 - a*x)^8)/(4*a) + (c^4*(1 - a*x)^9)/(9*a)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 3, -((4*c^3*(1 - a*x)^5)/(5*a)) + (2*c^3*(1 - a*x)^6)/(3*a) - (c^3*(1 - a*x)^7)/(7*a)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 3, -((c^2*(1 - a*x)^4)/(2*a)) + (c^2*(1 - a*x)^5)/(5*a)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^1, x, 2, -((c*(1 - a*x)^3)/(3*a))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^1, x, 2, -(1/(a*c*(1 + a*x)))} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 4, -(1/(4*a*c^2*(1 + a*x)^2)) - 1/(4*a*c^2*(1 + a*x)) + ArcTanh[a*x]/(4*a*c^2)} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 4, 1/(16*a*c^3*(1 - a*x)) - 1/(12*a*c^3*(1 + a*x)^3) - 1/(8*a*c^3*(1 + a*x)^2) - 3/(16*a*c^3*(1 + a*x)) + ArcTanh[a*x]/(4*a*c^3)} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 4, 1/(64*a*c^4*(1 - a*x)^2) + 5/(64*a*c^4*(1 - a*x)) - 1/(32*a*c^4*(1 + a*x)^4) - 1/(16*a*c^4*(1 + a*x)^3) - 3/(32*a*c^4*(1 + a*x)^2) - 5/(32*a*c^4*(1 + a*x)) + (15*ArcTanh[a*x])/(64*a*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^3, x, 7, -((3*x^2*Sqrt[c - a^2*c*x^2])/(5*a^2)) + (x^3*Sqrt[c - a^2*c*x^2])/(2*a) - (1/5)*x^4*Sqrt[c - a^2*c*x^2] - (3*(8 - 5*a*x)*Sqrt[c - a^2*c*x^2])/(20*a^4) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(4*a^4)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^2, x, 6, (2*x^2*Sqrt[c - a^2*c*x^2])/(3*a) - (1/4)*x^3*Sqrt[c - a^2*c*x^2] + ((32 - 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3) + (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^1, x, 5, (-(1/3))*x^2*Sqrt[c - a^2*c*x^2] - ((5 - 3*a*x)*Sqrt[c - a^2*c*x^2])/(3*a^2) - (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/a^2} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^0, x, 5, (3*Sqrt[c - a^2*c*x^2])/(2*a) + ((1 - a*x)*Sqrt[c - a^2*c*x^2])/(2*a) + (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^1, x, 8, -Sqrt[c - a^2*c*x^2] - 2*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^2, x, 8, -(Sqrt[c - a^2*c*x^2]/x) + a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + 2*a*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^3, x, 6, -(Sqrt[c - a^2*c*x^2]/(2*x^2)) + (2*a*Sqrt[c - a^2*c*x^2])/x - (3/2)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^4, x, 7, -(Sqrt[c - a^2*c*x^2]/(3*x^3)) + (a*Sqrt[c - a^2*c*x^2])/x^2 - (5*a^2*Sqrt[c - a^2*c*x^2])/(3*x) + a^3*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^5, x, 8, -(Sqrt[c - a^2*c*x^2]/(4*x^4)) + (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) - (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) + (4*a^3*Sqrt[c - a^2*c*x^2])/(3*x) - (7/8)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2), x, 6, (5/8)*c*x*Sqrt[c - a^2*c*x^2] + (5*(c - a^2*c*x^2)^(3/2))/(12*a) + ((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) + (5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)} - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2), x, 7, (7/16)*c^2*x*Sqrt[c - a^2*c*x^2] + (7/24)*c*x*(c - a^2*c*x^2)^(3/2) + (7*(c - a^2*c*x^2)^(5/2))/(30*a) + ((1 - a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) + (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)} - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2), x, 8, (45/128)*c^3*x*Sqrt[c - a^2*c*x^2] + (15/64)*c^2*x*(c - a^2*c*x^2)^(3/2) + (3/16)*c*x*(c - a^2*c*x^2)^(5/2) + (9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 - a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) + (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2), x, 4, -((2*(1 - a*x))/(a*Sqrt[c - a^2*c*x^2])) - ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*Sqrt[c])} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2), x, 3, -((2*(1 - a*x))/(3*a*(c - a^2*c*x^2)^(3/2))) + x/(3*c*Sqrt[c - a^2*c*x^2])} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/2), x, 4, -((2*(1 - a*x))/(5*a*(c - a^2*c*x^2)^(5/2))) + x/(5*c*(c - a^2*c*x^2)^(3/2)) + (2*x)/(5*c^2*Sqrt[c - a^2*c*x^2])} -{1/E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 5, -((2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2))) + x/(7*c*(c - a^2*c*x^2)^(5/2)) + (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic*) - - -{x^m*Sqrt[c - a^2*c*x^2]/E^(2*ArcTanh[a*x]), x, 7, If[$VersionNumber>=8, -((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m)) + (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(2 + m)*Sqrt[c - a^2*c*x^2]) - (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2]), -((x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m)) + (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[c - a^2*c*x^2]) - (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{1/E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, (2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1/2)*(1 + a*x)])/((1 - a*x)^p*(a*p))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p / E^(3 ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^4, x, 8, (55/128)*c^4*x*Sqrt[1 - a^2*x^2] + (55/192)*c^4*x*(1 - a^2*x^2)^(3/2) + (11/48)*c^4*x*(1 - a^2*x^2)^(5/2) + (11*c^4*(1 - a^2*x^2)^(7/2))/(56*a) + (11*c^4*(1 - a*x)*(1 - a^2*x^2)^(7/2))/(72*a) + (c^4*(1 - a*x)^2*(1 - a^2*x^2)^(7/2))/(9*a) + (55*c^4*ArcSin[a*x])/(128*a)} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 7, (9/16)*c^3*x*Sqrt[1 - a^2*x^2] + (3/8)*c^3*x*(1 - a^2*x^2)^(3/2) + (3*c^3*(1 - a^2*x^2)^(5/2))/(10*a) + (3*c^3*(1 - a*x)*(1 - a^2*x^2)^(5/2))/(14*a) + (c^3*(1 - a*x)^2*(1 - a^2*x^2)^(5/2))/(7*a) + (9*c^3*ArcSin[a*x])/(16*a)} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 6, (7/8)*c^2*x*Sqrt[1 - a^2*x^2] + (7*c^2*(1 - a^2*x^2)^(3/2))/(12*a) + (7*c^2*(1 - a*x)*(1 - a^2*x^2)^(3/2))/(20*a) + (c^2*(1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*c^2*ArcSin[a*x])/(8*a)} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^1, x, 5, (5*c*Sqrt[1 - a^2*x^2])/(2*a) + (5*c*(1 - a*x)*Sqrt[1 - a^2*x^2])/(6*a) + (c*(1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) + (5*c*ArcSin[a*x])/(2*a)} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^1, x, 1, -(1/(E^(3*ArcTanh[a*x])*(3*a*c)))} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 5, -(Sqrt[1 - a^2*x^2]/(5*a*c^2*(1 + a*x)^3)) - (2*Sqrt[1 - a^2*x^2])/(15*a*c^2*(1 + a*x)^2) - (2*Sqrt[1 - a^2*x^2])/(15*a*c^2*(1 + a*x))} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 6, (8*x)/(35*c^3*Sqrt[1 - a^2*x^2]) - 1/(7*a*c^3*(1 + a*x)^3*Sqrt[1 - a^2*x^2]) - 4/(35*a*c^3*(1 + a*x)^2*Sqrt[1 - a^2*x^2]) - 4/(35*a*c^3*(1 + a*x)*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 7, (8*x)/(63*c^4*(1 - a^2*x^2)^(3/2)) - 1/(9*a*c^4*(1 + a*x)^3*(1 - a^2*x^2)^(3/2)) - 2/(21*a*c^4*(1 + a*x)^2*(1 - a^2*x^2)^(3/2)) - 2/(21*a*c^4*(1 + a*x)*(1 - a^2*x^2)^(3/2)) + (16*x)/(63*c^4*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^3, x, 4, (4*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - a^2*x^2]) - (2*x^2*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2]) + (4*x^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (3*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) + (a*x^5*Sqrt[c - a^2*c*x^2])/(5*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^4*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^2, x, 4, -((4*x*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2])) + (2*x^2*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) - (x^3*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] + (a*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^3*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^1, x, 4, (4*x*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) - (3*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) + (a*x^3*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^2*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)*x^0, x, 4, -((3*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2]) + (a*x^2*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - a^2*x^2]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^1, x, 4, (a*x*Sqrt[c - a^2*c*x^2])/Sqrt[1 - a^2*x^2] + (Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^2, x, 4, -(Sqrt[c - a^2*c*x^2]/(x*Sqrt[1 - a^2*x^2])) - (3*a*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] + (4*a*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^3, x, 4, -(Sqrt[c - a^2*c*x^2]/(2*x^2*Sqrt[1 - a^2*x^2])) + (3*a*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) + (4*a^2*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^2*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^4, x, 4, -(Sqrt[c - a^2*c*x^2]/(3*x^3*Sqrt[1 - a^2*x^2])) + (3*a*Sqrt[c - a^2*c*x^2])/(2*x^2*Sqrt[1 - a^2*x^2]) - (4*a^2*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) - (4*a^3*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] + (4*a^3*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2)/x^5, x, 4, -(Sqrt[c - a^2*c*x^2]/(4*x^4*Sqrt[1 - a^2*x^2])) + (a*Sqrt[c - a^2*c*x^2])/(x^3*Sqrt[1 - a^2*x^2]) - (2*a^2*Sqrt[c - a^2*c*x^2])/(x^2*Sqrt[1 - a^2*x^2]) + (4*a^3*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2]) + (4*a^4*Sqrt[c - a^2*c*x^2]*Log[x])/Sqrt[1 - a^2*x^2] - (4*a^4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/Sqrt[1 - a^2*x^2]} - - -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(9/2), x, 4, -((8*c^4*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2])) + (3*c^4*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - a^2*x^2]) - (2*c^4*(1 - a*x)^9*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) + (c^4*(1 - a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2), x, 4, -((2*c^3*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2])) + (4*c^3*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (c^3*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(8*a*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2), x, 4, -((2*c^2*(1 - a*x)^5*Sqrt[c - a^2*c*x^2])/(5*a*Sqrt[1 - a^2*x^2])) + (c^2*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - a^2*x^2])} -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2), x, 3, -((c*(1 - a*x)^4*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2]))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2), x, 4, -((2*Sqrt[1 - a^2*x^2])/(a*(1 + a*x)*Sqrt[c - a^2*c*x^2])) - (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(a*Sqrt[c - a^2*c*x^2])} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2), x, 3, -(Sqrt[1 - a^2*x^2]/(2*a*c*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]))} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/2), x, 5, -(Sqrt[1 - a^2*x^2]/(6*a*c^2*(1 + a*x)^3*Sqrt[c - a^2*c*x^2])) - Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(8*a*c^2*Sqrt[c - a^2*c*x^2])} -{1/E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 5, Sqrt[1 - a^2*x^2]/(32*a*c^3*(1 - a*x)*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(16*a*c^3*(1 + a*x)^4*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(12*a*c^3*(1 + a*x)^3*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2])/(32*a*c^3*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]) - Sqrt[1 - a^2*x^2]/(8*a*c^3*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (5*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(32*a*c^3*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic*) - - -{x^m/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2), x, 5, -((3*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2])) + (a*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - a^2*x^2]) + (4*x^(1 + m)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/((1 + m)*Sqrt[1 - a^2*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{1/E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, -((2^(-(1/2) + p)*(1 - a*x)^(5/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2 - p, 5/2 + p, 7/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a*(5 + 2*p))))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(1/2 ArcTanh[a x])*) - - -(* ::Subsection:: *) -(*Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -{E^(1/2*ArcTanh[a*x])*(1 - a^2*x^2)^(5/2), x, 18, (231*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(512*a) + (231*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(1280*a) + (77*(1 - a*x)^(9/4)*(1 + a*x)^(3/4))/(960*a) - (77*(1 - a*x)^(13/4)*(1 + a*x)^(3/4))/(480*a) - (11*(1 - a*x)^(13/4)*(1 + a*x)^(7/4))/(60*a) - ((1 - a*x)^(13/4)*(1 + a*x)^(11/4))/(6*a) + (231*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(512*Sqrt[2]*a) - (231*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(512*Sqrt[2]*a) + (231*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(1024*Sqrt[2]*a) - (231*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(1024*Sqrt[2]*a)} -{E^(1/2*ArcTanh[a*x])*(1 - a^2*x^2)^(3/2), x, 16, (35*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*a) + (7*(1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(32*a) - (7*(1 - a*x)^(9/4)*(1 + a*x)^(3/4))/(24*a) - ((1 - a*x)^(9/4)*(1 + a*x)^(7/4))/(4*a) + (35*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a) - (35*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a) + (35*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a) - (35*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a)} -{E^(1/2*ArcTanh[a*x])*(1 - a^2*x^2)^(1/2), x, 14, (3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(4*a) - ((1 - a*x)^(5/4)*(1 + a*x)^(3/4))/(2*a) + (3*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a) - (3*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a) + (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a) - (3*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a)} -{E^(1/2*ArcTanh[a*x])/(1 - a^2*x^2)^(1/2), x, 12, (Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a - (Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/a + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a)} -{E^(1/2*ArcTanh[a*x])/(1 - a^2*x^2)^(3/2), x, 1, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(3*a*Sqrt[1 - a^2*x^2]))} -{E^(1/2*ArcTanh[a*x])/(1 - a^2*x^2)^(5/2), x, 2, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(35*a*(1 - a^2*x^2)^(3/2))) - (16*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(35*a*Sqrt[1 - a^2*x^2])} -{E^(1/2*ArcTanh[a*x])/(1 - a^2*x^2)^(7/2), x, 3, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 10*a*x))/(99*a*(1 - a^2*x^2)^(5/2))) - (32*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(693*a*(1 - a^2*x^2)^(3/2)) - (256*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(693*a*Sqrt[1 - a^2*x^2])} -{E^(1/2*ArcTanh[a*x])/(1 - a^2*x^2)^(9/2), x, 4, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 14*a*x))/(195*a*(1 - a^2*x^2)^(7/2))) - (112*E^((1/2)*ArcTanh[a*x])*(1 - 10*a*x))/(6435*a*(1 - a^2*x^2)^(5/2)) - (256*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(6435*a*(1 - a^2*x^2)^(3/2)) - (2048*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(6435*a*Sqrt[1 - a^2*x^2])} - - -{E^(1/2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2), x, 19, (231*c^2*(1 - a*x)^(1/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(512*a*Sqrt[1 - a^2*x^2]) + (231*c^2*(1 - a*x)^(5/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(1280*a*Sqrt[1 - a^2*x^2]) + (77*c^2*(1 - a*x)^(9/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(960*a*Sqrt[1 - a^2*x^2]) - (77*c^2*(1 - a*x)^(13/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(480*a*Sqrt[1 - a^2*x^2]) - (11*c^2*(1 - a*x)^(13/4)*(1 + a*x)^(7/4)*Sqrt[c - a^2*c*x^2])/(60*a*Sqrt[1 - a^2*x^2]) - (c^2*(1 - a*x)^(13/4)*(1 + a*x)^(11/4)*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - a^2*x^2]) + (231*c^2*Sqrt[c - a^2*c*x^2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(512*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (231*c^2*Sqrt[c - a^2*c*x^2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(512*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) + (231*c^2*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(1024*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (231*c^2*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(1024*Sqrt[2]*a*Sqrt[1 - a^2*x^2])} -{E^(1/2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2), x, 17, (35*c*(1 - a*x)^(1/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(64*a*Sqrt[1 - a^2*x^2]) + (7*c*(1 - a*x)^(5/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(32*a*Sqrt[1 - a^2*x^2]) - (7*c*(1 - a*x)^(9/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(24*a*Sqrt[1 - a^2*x^2]) - (c*(1 - a*x)^(9/4)*(1 + a*x)^(7/4)*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2]) + (35*c*Sqrt[c - a^2*c*x^2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (35*c*Sqrt[c - a^2*c*x^2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) + (35*c*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (35*c*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a*Sqrt[1 - a^2*x^2])} -{E^(1/2*ArcTanh[a*x])*(c - a^2*c*x^2)^(1/2), x, 15, (3*(1 - a*x)^(1/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - a^2*x^2]) - ((1 - a*x)^(5/4)*(1 + a*x)^(3/4)*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - a^2*x^2]) + (3*Sqrt[c - a^2*c*x^2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[c - a^2*c*x^2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(4*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) + (3*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a*Sqrt[1 - a^2*x^2]) - (3*Sqrt[c - a^2*c*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(8*Sqrt[2]*a*Sqrt[1 - a^2*x^2])} -{E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(1/2), x, 13, (Sqrt[2]*Sqrt[1 - a^2*x^2]*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(a*Sqrt[c - a^2*c*x^2]) - (Sqrt[2]*Sqrt[1 - a^2*x^2]*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(a*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a*Sqrt[c - a^2*c*x^2]) - (Sqrt[1 - a^2*x^2]*Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(Sqrt[2]*a*Sqrt[c - a^2*c*x^2])} -{E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2), x, 1, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(3*a*c*Sqrt[c - a^2*c*x^2]))} -{E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/2), x, 2, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(35*a*c*(c - a^2*c*x^2)^(3/2))) - (16*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(35*a*c^2*Sqrt[c - a^2*c*x^2])} -{E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 3, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 10*a*x))/(99*a*c*(c - a^2*c*x^2)^(5/2))) - (32*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(693*a*c^2*(c - a^2*c*x^2)^(3/2)) - (256*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(693*a*c^3*Sqrt[c - a^2*c*x^2])} -{E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(9/2), x, 4, -((2*E^((1/2)*ArcTanh[a*x])*(1 - 14*a*x))/(195*a*c*(c - a^2*c*x^2)^(7/2))) - (112*E^((1/2)*ArcTanh[a*x])*(1 - 10*a*x))/(6435*a*c^2*(c - a^2*c*x^2)^(5/2)) - (256*E^((1/2)*ArcTanh[a*x])*(1 - 6*a*x))/(6435*a*c^3*(c - a^2*c*x^2)^(3/2)) - (2048*E^((1/2)*ArcTanh[a*x])*(1 - 2*a*x))/(6435*a*c^4*Sqrt[c - a^2*c*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/4)*) - - -{x^3*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/4), x, 10, (1 - a^2*x^2)^(1/4)/(a^4*c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) + (2*Sqrt[1 - a*x]*(1 - a^2*x^2)^(1/4))/(a^4*c*(c - a^2*c*x^2)^(1/4)) - (2*(1 - a*x)^(3/2)*(1 - a^2*x^2)^(1/4))/(3*a^4*c*(c - a^2*c*x^2)^(1/4)) + ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*a^4*c*(c - a^2*c*x^2)^(1/4))} -{x^2*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/4), x, 8, (1 - a^2*x^2)^(1/4)/(a^3*c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) + (2*Sqrt[1 - a*x]*(1 - a^2*x^2)^(1/4))/(a^3*c*(c - a^2*c*x^2)^(1/4)) - ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*a^3*c*(c - a^2*c*x^2)^(1/4))} -{x^1*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/4), x, 5, (1 - a^2*x^2)^(1/4)/(a^2*c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) + ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*a^2*c*(c - a^2*c*x^2)^(1/4))} -{x^0*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(5/4), x, 5, (1 - a^2*x^2)^(1/4)/(a*c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) - ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*a*c*(c - a^2*c*x^2)^(1/4))} -{E^(1/2*ArcTanh[a*x])/(x^1*(c - a^2*c*x^2)^(5/4)), x, 8, (1 - a^2*x^2)^(1/4)/(c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) - (2*(1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]])/(c*(c - a^2*c*x^2)^(1/4)) + ((1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*c*(c - a^2*c*x^2)^(1/4))} -{E^(1/2*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(5/4)), x, 9, (2*a*(1 - a^2*x^2)^(1/4))/(c*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) - (1 - a^2*x^2)^(1/4)/(c*x*Sqrt[1 - a*x]*(c - a^2*c*x^2)^(1/4)) - (a*(1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]])/(c*(c - a^2*c*x^2)^(1/4)) - (a*(1 - a^2*x^2)^(1/4)*ArcTanh[Sqrt[1 - a*x]/Sqrt[2]])/(Sqrt[2]*c*(c - a^2*c*x^2)^(1/4))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/8)*) - - -{x^3*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(9/8), x, 5, -((4*x^2*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8))/(7*a^2*c*(1 - a*x)^(3/8)*(c - a^2*c*x^2)^(1/8))) + (8*(6 - a*x)*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8))/(21*a^4*c*(1 - a*x)^(3/8)*(c - a^2*c*x^2)^(1/8)) + (64*2^(1/8)*(1 - a*x)^(5/8)*(1 - a^2*x^2)^(1/8)*Hypergeometric2F1[5/8, 7/8, 13/8, (1/2)*(1 - a*x)])/(105*a^4*c*(c - a^2*c*x^2)^(1/8))} -{x^2*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(9/8), x, 1, (4*E^((1/2)*ArcTanh[a*x])*(2 - a*x))/(3*a^3*c*(c - a^2*c*x^2)^(1/8))} -{x^1*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(9/8), x, 4, (4*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8))/(3*a^2*c*(1 - a*x)^(3/8)*(c - a^2*c*x^2)^(1/8)) + (8*2^(1/8)*(1 - a*x)^(5/8)*(1 - a^2*x^2)^(1/8)*Hypergeometric2F1[5/8, 7/8, 13/8, (1/2)*(1 - a*x)])/(15*a^2*c*(c - a^2*c*x^2)^(1/8))} -{x^0*E^(1/2*ArcTanh[a*x])/(c - a^2*c*x^2)^(9/8), x, 3, (4*2^(1/8)*(1 - a^2*x^2)^(1/8)*Hypergeometric2F1[-(3/8), 7/8, 5/8, (1/2)*(1 - a*x)])/(3*a*c*(1 - a*x)^(3/8)*(c - a^2*c*x^2)^(1/8))} -{E^(1/2*ArcTanh[a*x])/(x^1*(c - a^2*c*x^2)^(9/8)), x, 3, -((2*2^(5/8)*(1 + a*x)^(1/8)*(1 - a^2*x^2)^(1/8)*AppellF1[1/8, 11/8, 1, 9/8, (1/2)*(1 + a*x), 1 + a*x])/(c*(c - a^2*c*x^2)^(1/8)))} - - -(* ::Subsection:: *) -(*Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (c-a^2 c x^2)^p E^(n ArcTanh[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(n*ArcTanh[a*x])*(c - a^2*c*x^2), x, 2, -((2^(2 + n/2)*c*(1 - a*x)^(2 - n/2)*Hypergeometric2F1[-1 - n/2, 2 - n/2, 3 - n/2, (1/2)*(1 - a*x)])/(a*(4 - n)))} - - -{E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 2, -((2^(3 + n/2)*c^2*(1 - a*x)^(3 - n/2)*Hypergeometric2F1[-2 - n/2, 3 - n/2, 4 - n/2, (1/2)*(1 - a*x)])/(a*(6 - n)))} - - -{E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^3, x, 2, -((2^(4 + n/2)*c^3*(1 - a*x)^(4 - n/2)*Hypergeometric2F1[-3 - n/2, 4 - n/2, 5 - n/2, (1/2)*(1 - a*x)])/(a*(8 - n)))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)*x^4, x, 5, -((n*x^2*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(6*a^3*c))) - (x^3*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(3*a^2*c)) + ((1 + a*x)^(n/2)*(6 + 8*n + n^2 + n^3 - a*n*(6 + n^2)*x))/((1 - a*x)^(n/2)*(6*a^5*c*n)) + (2^(-1 + n/2)*n*(8 + n^2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, 1 - n/2, 2 - n/2, (1/2)*(1 - a*x)])/(3*a^5*c*(2 - n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)*x^3, x, 4, -((x^2*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(2*a^2*c))) + ((1 + a*x)^(n/2)*(2 + n + n^2 - a*n^2*x))/((1 - a*x)^(n/2)*(2*a^4*c*n)) + (2^(-1 + n/2)*(2 + n^2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, 1 - n/2, 2 - n/2, (1/2)*(1 - a*x)])/(a^4*c*(2 - n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)*x^2, x, 4, E^(n*ArcTanh[a*x])/(a^3*c*n) + (2^(1 + n/2)*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (1/2)*(1 - a*x)])/(a^3*c*(2 - n)), ((1 - n)*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a^3*c*n)) - (x*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a^2*c)) + (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a^3*c))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)*x^1, x, 3, -((1 + a*x)^(n/2)/((1 - a*x)^(n/2)*(a^2*c*n))) + (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (1/2)*(1 - a*x)])/((1 - a*x)^(n/2)*(a^2*c*n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)*x^0, x, 1, E^(n*ArcTanh[a*x])/(a*c*n)} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)/x^1, x, 3, (1 + a*x)^(n/2)/((1 - a*x)^(n/2)*(c*n)) - (2*(1 + a*x)^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - a*x)^(n/2)*(c*n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)/x^2, x, 5, (a*(1 + n)*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(c*n)) - (1 + a*x)^(n/2)/((1 - a*x)^(n/2)*(c*x)) - (2*a*(1 + a*x)^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - a*x)^(n/2)*c)} - - -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2*x^4, x, 10, ((1 - n)*(3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^5*c^2*(2 - n)) + ((3 + n)*x*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^4*c^2) - (x^3*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^2*c^2) + ((1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^5*c^2*(2 - n)) - (1 + a*x)^((1/2)*(-2 + n))/((1 - a*x)^(n/2)*(a^5*c^2)) - ((3 + n)*(2 - n^2)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^(n/2))/(a^5*c^2*(4 - n^2)) - ((3 + n)*(2 - n^2)*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a^5*c^2*n*(4 - n^2))) - (2^(n/2)*n*(1 - a*x)^(1 - n/2)*Hypergeometric2F1[(2 - n)/2, 1 - n/2, 2 - n/2, (1/2)*(1 - a*x)])/(a^5*c^2*(2 - n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2*x^3, x, 10, If[$VersionNumber>=8, -(((1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^4*c^2*(2 + n))) + (2*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^4*c^2*n*(4 - n^2)) - (2*(1 + a*x)^((1/2)*(-2 + n)))/((1 - a*x)^(n/2)*(a^4*c^2*n*(2 + n))) + (3*(1 - a*x)^(-1 - n/2)*(1 + a*x)^(n/2))/(a^4*c^2*(2 + n)) + (3*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a^4*c^2*n*(2 + n))) - (3*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((2 + n)/2))/(a^4*c^2*(2 + n)) + (2^(2 + n/2)*(1 - a*x)^(-1 - n/2)*Hypergeometric2F1[-1 - n/2, -1 - n/2, -(n/2), (1/2)*(1 - a*x)])/(a^4*c^2*(2 + n)), -(((1 - a*x)^((1/2)*(-2 - n))*(1 + a*x)^((1/2)*(-2 + n)))/(a^4*c^2*(2 + n))) + (2*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(a^4*c^2*n*(4 - n^2)) - (2*(1 + a*x)^((1/2)*(-2 + n)))/((1 - a*x)^(n/2)*(a^4*c^2*n*(2 + n))) + (3*(1 - a*x)^((1/2)*(-2 - n))*(1 + a*x)^(n/2))/(a^4*c^2*(2 + n)) + (3*(1 + a*x)^(n/2))/((1 - a*x)^(n/2)*(a^4*c^2*n*(2 + n))) - (3*(1 - a*x)^((1/2)*(-2 - n))*(1 + a*x)^((2 + n)/2))/(a^4*c^2*(2 + n)) + (2^(2 + n/2)*(1 - a*x)^((1/2)*(-2 - n))*Hypergeometric2F1[(1/2)*(-2 - n), -1 - n/2, -(n/2), (1/2)*(1 - a*x)])/(a^4*c^2*(2 + n))]} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2*x^2, x, 2, -((E^(n*ArcTanh[a*x])*(2 - n^2))/(a^3*c^2*n*(4 - n^2))) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a^3*c^2*(4 - n^2)*(1 - a^2*x^2))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2*x^1, x, 3, -(E^(n*ArcTanh[a*x])/(a^2*c^2*(4 - n^2))) + (E^(n*ArcTanh[a*x])*(2 - a*n*x))/(a^2*c^2*(4 - n^2)*(1 - a^2*x^2)), -(E^(n*ArcTanh[a*x])/(a^2*c^2*(4 - n^2))) + E^(n*ArcTanh[a*x])/(2*a^2*c^2*(1 - a^2*x^2)) + (E^(n*ArcTanh[a*x])*n*(n - 2*a*x))/(2*a^2*c^2*(4 - n^2)*(1 - a^2*x^2))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2*x^0, x, 2, (2*E^(n*ArcTanh[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2/x^1, x, 6, ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(c^2*(2 + n)) - ((4 - n - n^2)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(c^2*n*(4 - n^2)) + ((4 + n)*(1 + a*x)^((1/2)*(-2 + n)))/((1 - a*x)^(n/2)*(c^2*n*(2 + n))) - (2*(1 + a*x)^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - a*x)^(n/2)*(c^2*n))} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2/x^2, x, 7, (a*(3 + n)*(1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(c^2*(2 + n)) - ((1 - a*x)^(-1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(c^2*x) - (a*(6 + 4*n - n^2 - n^3)*(1 - a*x)^(1 - n/2)*(1 + a*x)^((1/2)*(-2 + n)))/(c^2*n*(4 - n^2)) + (a*(6 + 4*n + n^2)*(1 + a*x)^((1/2)*(-2 + n)))/((1 - a*x)^(n/2)*(c^2*n*(2 + n))) - (2*a*(1 + a*x)^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - a*x)^(n/2)*c^2)} - - -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 3, If[$VersionNumber>=8, (24*E^(n*ArcTanh[a*x]))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^3*(4 - n^2)*(16 - n^2)*(1 - a^2*x^2)), (24*E^(n*ArcTanh[a*x]))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^3*(64 - 20*n^2 + n^4)*(1 - a^2*x^2))]} -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^4, x, 4, If[$VersionNumber>=8, (720*E^(n*ArcTanh[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcTanh[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*c^4*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)^2) - (360*E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^4*(4 - n^2)*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)), (720*E^(n*ArcTanh[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcTanh[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcTanh[a*x])*(n - 4*a*x))/(a*c^4*(576 - 52*n^2 + n^4)*(1 - a^2*x^2)^2) - (360*E^(n*ArcTanh[a*x])*(n - 2*a*x))/(a*c^4*(36 - n^2)*(64 - 20*n^2 + n^4)*(1 - a^2*x^2))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^(n*ArcTanh[a*x])*x^3*Sqrt[c - a^2*c*x^2], x, 5, -(x^2*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((3 + n)/2)*Sqrt[c - a^2*c*x^2])/(5*a^2*Sqrt[1 - a^2*x^2]) - ((1 - a*x)^((3 - n)/2)*(1 + a*x)^((3 + n)/2)*(8 + n^2 + 3*a*n*x)*Sqrt[c - a^2*c*x^2])/(60*a^4*Sqrt[1 - a^2*x^2]) - (2^((-1 + n)/2)*n*(11 + n^2)*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(15*a^4*(3 - n)*Sqrt[1 - a^2*x^2])} -{E^(n*ArcTanh[a*x])*x^2*Sqrt[c - a^2*c*x^2], x, 5, -(n*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((3 + n)/2)*Sqrt[c - a^2*c*x^2])/(12*a^3*Sqrt[1 - a^2*x^2]) - (x*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((3 + n)/2)*Sqrt[c - a^2*c*x^2])/(4*a^2*Sqrt[1 - a^2*x^2]) - (2^((-1 + n)/2)*(3 + n^2)*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(3*a^3*(3 - n)*Sqrt[1 - a^2*x^2])} -{E^(n*ArcTanh[a*x])*x^1*Sqrt[c - a^2*c*x^2], x, 4, -((1 - a*x)^((3 - n)/2)*(1 + a*x)^((3 + n)/2)*Sqrt[c - a^2*c*x^2])/(3*a^2*Sqrt[1 - a^2*x^2]) - (2^((3 + n)/2)*n*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(3*a^2*(3 - n)*Sqrt[1 - a^2*x^2])} -{E^(n*ArcTanh[a*x])*x^0*Sqrt[c - a^2*c*x^2], x, 3, -((2^((3 + n)/2)*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(-1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(a*(3 - n)*Sqrt[1 - a^2*x^2]))} -{(E^(n*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^1, x, 6, -(((1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[c - a^2*c*x^2])/((1 - n)*Sqrt[1 - a^2*x^2])) + (2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*n*(1 - a*x)^((3 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1/2)*(1 - a*x)])/((3 - 4*n + n^2)*Sqrt[1 - a^2*x^2])} -{(E^(n*ArcTanh[a*x])*Sqrt[c - a^2*c*x^2])/x^2, x, 6, -(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[c - a^2*c*x^2])/(x*Sqrt[1 - a^2*x^2])) - (2*a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1 + a*x)])/((1 - n)*Sqrt[1 - a^2*x^2]) + (2^((1 + n)/2)*a*(1 - a*x)^((1 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1/2)*(1 - a*x)])/((1 - n)*Sqrt[1 - a^2*x^2])} - - -{E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2), x, 3, -((2^((5 + n)/2)*c*(1 - a*x)^((5 - n)/2)*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[(1/2)*(-3 - n), (5 - n)/2, (7 - n)/2, (1/2)*(1 - a*x)])/(a*(5 - n)*Sqrt[1 - a^2*x^2]))} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(E^(n*ArcTanh[a*x])*x^3)/Sqrt[c - a^2*c*x^2], x, 5, If[$VersionNumber>=8, -(x^2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(3*a^2*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*(4 + n + n^2 + a*(1 - n)*n*x)*Sqrt[1 - a^2*x^2])/(6*a^4*(1 - n)*Sqrt[c - a^2*c*x^2]) - (2^((-1 + n)/2)*n*(5 + n^2)*(1 - a*x)^((3 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(3*a^4*(1 - n)*(3 - n)*Sqrt[c - a^2*c*x^2]), -((x^2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(3*a^2*Sqrt[c - a^2*c*x^2])) - ((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*(4 + n + n^2 + a*(1 - n)*n*x)*Sqrt[1 - a^2*x^2])/(6*a^4*(1 - n)*Sqrt[c - a^2*c*x^2]) - (2^((1/2)*(-1 + n))*n*(5 + n^2)*(1 - a*x)^((3 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1 - n)/2, (3 - n)/2, (5 - n)/2, (1/2)*(1 - a*x)])/(3*a^4*(3 - 4*n + n^2)*Sqrt[c - a^2*c*x^2])]} -{(E^(n*ArcTanh[a*x])*x^2)/Sqrt[c - a^2*c*x^2], x, 5, ((1 - n)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*a^3*(1 + n)*Sqrt[c - a^2*c*x^2]) - (x*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*a^2*Sqrt[c - a^2*c*x^2]) - (2^((1 + n)/2)*(1 + n^2)*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(-1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/(a^3*(1 - n^2)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^1)/Sqrt[c - a^2*c*x^2], x, 4, -(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^2*(1 + n)*Sqrt[c - a^2*c*x^2])) - (2^((3 + n)/2)*n*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(-1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/(a^2*(1 - n^2)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^0)/Sqrt[c - a^2*c*x^2], x, 3, -((2^((1 + n)/2)*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/(a*(1 - n)*Sqrt[c - a^2*c*x^2]))} -{E^(n*ArcTanh[a*x])/(x^1*Sqrt[c - a^2*c*x^2]), x, 3, (-2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1 + a*x)])/((1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcTanh[a*x])/(x^2*Sqrt[c - a^2*c*x^2]), x, 4, -(((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(x*Sqrt[c - a^2*c*x^2])) - (2*a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1 + a*x)])/((1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcTanh[a*x])/(x^3*Sqrt[c - a^2*c*x^2]), x, 6, -((1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*x^2*Sqrt[c - a^2*c*x^2]) - (a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1 + n)/2)*Sqrt[1 - a^2*x^2])/(2*x*Sqrt[c - a^2*c*x^2]) - (a^2*(1 + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (1 - a*x)/(1 + a*x)])/((1 - n)*Sqrt[c - a^2*c*x^2])} - - -{(E^(n*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^(3/2), x, 5, -((x^2*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^2*c*Sqrt[c - a^2*c*x^2])) + ((1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*(2 + 2*n + n^2 - a*n*(3 + 2*n)*x)*Sqrt[1 - a^2*x^2])/(a^4*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]) - (2^((-1 + n)/2)*n*(1 - a*x)^((3 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(3 - n)/2, (3 - n)/2, (5 - n)/2, (1 - a*x)/2])/(a^4*c*(3 - n)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^(3/2), x, 4, -((E^(n*ArcTanh[a*x])*(n - a*x))/(a^3*c*(1 - n^2)*Sqrt[c - a^2*c*x^2])) + (2^((1 + n)/2)*(1 - a*x)^((1 - n)/2)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (1 - a*x)/2])/(a^3*c*(1 - n)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^1)/(c - a^2*c*x^2)^(3/2), x, 1, (E^(n*ArcTanh[a*x])*(1 - a*n*x))/(a^2*c*(1 - n^2)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^0)/(c - a^2*c*x^2)^(3/2), x, 1, -((E^(n*ArcTanh[a*x])*(n - a*x))/(a*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]))} -{E^(n*ArcTanh[a*x])/(x^1*(c - a^2*c*x^2)^(3/2)), x, 6, ((1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(c*(1 + n)*Sqrt[c - a^2*c*x^2]) - ((2 + n)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(c*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + (2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c*(1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(3/2)), x, 7, (a*(2 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(c*(1 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(c*x*Sqrt[c - a^2*c*x^2]) - (a*(2 + 2*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(c*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + (2*a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c*(1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^(3/2)), x, 8, (a^2*(3 + 2*n + n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(2*c*(1 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(2*c*x^2*Sqrt[c - a^2*c*x^2]) - (a*n*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(2*c*x*Sqrt[c - a^2*c*x^2]) - (a^2*(6 + 5*n + 2*n^2 + n^3)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(2*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + (a^2*(3 + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c*(1 - n)*Sqrt[c - a^2*c*x^2])} - - -{(E^(n*ArcTanh[a*x])*x^3)/(c - a^2*c*x^2)^(5/2), x, 7, If[$VersionNumber>=8, (x^3*(1 - a*x)^((-3 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3*(2 - n)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - n^2)*Sqrt[c - a^2*c*x^2]) - (3*x*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-3 + n)/2)*Sqrt[1 - a^2*x^2])/(a^3*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + (3*(1 + 2*n - n^2)*(1 - a*x)^((-1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(3 - n)*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((-1 + n)/2)*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]), (x^3*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(a*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - (3*(2 - n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - n^2)*Sqrt[c - a^2*c*x^2]) - (3*x*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(a^3*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + (3*(1 + 2*n - n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 + 9*n - n^2 - n^3)*Sqrt[c - a^2*c*x^2]) - (3*(1 + 2*n - n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2])/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} -{(E^(n*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2)^(5/2), x, 2, -((E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a^3*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) + (E^(n*ArcTanh[a*x])*(3 - n^2)*(n - a*x))/(a^3*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^1)/(c - a^2*c*x^2)^(5/2), x, 3, E^(n*ArcTanh[a*x])/(3*a^2*c*(c - a^2*c*x^2)^(3/2)) + (E^(n*ArcTanh[a*x])*n*(n - 3*a*x))/(3*a^2*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2)) + (2*E^(n*ArcTanh[a*x])*n*(n - a*x))/(a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])} -{(E^(n*ArcTanh[a*x])*x^0)/(c - a^2*c*x^2)^(5/2), x, 2, If[$VersionNumber>=8, -((E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcTanh[a*x])*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*Sqrt[c - a^2*c*x^2]), -((E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcTanh[a*x])*(n - a*x))/(a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcTanh[a*x])/(x^1*(c - a^2*c*x^2)^(5/2)), x, 8, If[$VersionNumber>=8, ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + ((6 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - ((15 + 6*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + ((18 + 7*n - 2*n^2 - n^3)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2]), ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) + ((6 + n)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + 4*n + n^2)*Sqrt[c - a^2*c*x^2]) - ((15 + 6*n + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n - 3*n^2 - n^3)*Sqrt[c - a^2*c*x^2]) + ((18 + 7*n - 2*n^2 - n^3)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (2*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcTanh[a*x])/(x^2*(c - a^2*c*x^2)^(5/2)), x, 9, If[$VersionNumber>=8, (a*(4 + n)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*x*Sqrt[c - a^2*c*x^2]) + (a*(12 + 6*n + n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - (a*(24 + 15*n + 6*n^2 + n^3)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + (a*(24 + 18*n + 7*n^2 - 2*n^3 - n^4)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (2*a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2]), (a*(4 + n)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*x*Sqrt[c - a^2*c*x^2]) + (a*(12 + 6*n + n^2)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + 4*n + n^2)*Sqrt[c - a^2*c*x^2]) - (a*(24 + 15*n + 6*n^2 + n^3)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(3 + n - 3*n^2 - n^3)*Sqrt[c - a^2*c*x^2]) + (a*(24 + 18*n + 7*n^2 - 2*n^3 - n^4)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (2*a*n*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcTanh[a*x])/(x^3*(c - a^2*c*x^2)^(5/2)), x, 10, If[$VersionNumber>=8, (a^2*(5 + 4*n + n^2)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*x^2*Sqrt[c - a^2*c*x^2]) - (a*n*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*x*Sqrt[c - a^2*c*x^2]) + (a^2*(30 + 17*n + 6*n^2 + n^3)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(1 + n)*(3 + n)*Sqrt[c - a^2*c*x^2]) - (a^2*(75 + 54*n + 20*n^2 + 6*n^3 + n^4)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(3 + n)*(1 - n^2)*Sqrt[c - a^2*c*x^2]) + (a^2*(90 + 59*n + 8*n^2 + 2*n^3 - 2*n^4 - n^5)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (a^2*(5 + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2]), (a^2*(5 + 4*n + n^2)*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(3 + n)*Sqrt[c - a^2*c*x^2]) - ((1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*x^2*Sqrt[c - a^2*c*x^2]) - (a*n*(1 - a*x)^((1/2)*(-3 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*x*Sqrt[c - a^2*c*x^2]) + (a^2*(30 + 17*n + 6*n^2 + n^3)*(1 - a*x)^((1/2)*(-1 - n))*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(3 + 4*n + n^2)*Sqrt[c - a^2*c*x^2]) - (a^2*(75 + 54*n + 20*n^2 + 6*n^3 + n^4)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(3 + n - 3*n^2 - n^3)*Sqrt[c - a^2*c*x^2]) + (a^2*(90 + 59*n + 8*n^2 + 2*n^3 - 2*n^4 - n^5)*(1 - a*x)^((3 - n)/2)*(1 + a*x)^((1/2)*(-3 + n))*Sqrt[1 - a^2*x^2])/(2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2]) + (a^2*(5 + n^2)*(1 - a*x)^((1 - n)/2)*(1 + a*x)^((1/2)*(-1 + n))*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (1 + a*x)/(1 - a*x)])/(c^2*(1 - n)*Sqrt[c - a^2*c*x^2])]} - - -{E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2), x, 3, If[$VersionNumber>=8, -((E^(n*ArcTanh[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a*c^2*(9 - n^2)*(25 - n^2)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcTanh[a*x])*(n - a*x))/(a*c^3*(1 - n^2)*(9 - n^2)*(25 - n^2)*Sqrt[c - a^2*c*x^2]), -((E^(n*ArcTanh[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcTanh[a*x])*(n - 3*a*x))/(a*c^2*(225 - 34*n^2 + n^4)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcTanh[a*x])*(n - a*x))/(a*c^3*(25 - n^2)*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic*) - - -{x^m*E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^2, x, 2, (c^2*x^(1 + m)*AppellF1[1 + m, (1/2)*(-4 + n), -2 - n/2, 2 + m, a*x, (-a)*x])/(1 + m)} -{x^m*E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^1, x, 2, (c*x^(1 + m)*AppellF1[1 + m, (1/2)*(-2 + n), -1 - n/2, 2 + m, a*x, (-a)*x])/(1 + m)} -{x^m*E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^1, x, 2, (x^(1 + m)*AppellF1[1 + m, (2 + n)/2, 1 - n/2, 2 + m, a*x, (-a)*x])/(c*(1 + m))} -{x^m*E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^2, x, 2, (x^(1 + m)*AppellF1[1 + m, (4 + n)/2, 2 - n/2, 2 + m, a*x, (-a)*x])/(c^2*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic*) - - -{x^m*E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, (x^(1 + m)*(c - a^2*c*x^2)^p*AppellF1[1 + m, (1/2)*(n - 2*p), -(n/2) - p, 2 + m, a*x, (-a)*x])/((1 - a^2*x^2)^p*(1 + m))} - - -{x^1*E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 4, -(((1 - a*x)^(1 - n/2 + p)*(1 + a*x)^(1 + n/2 + p)*(c - a^2*c*x^2)^p)/((1 - a^2*x^2)^p*(2*a^2*(1 + p)))) - (2^(n/2 + p)*n*(1 - a*x)^(1 - n/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(n/2) - p, 1 - n/2 + p, 2 - n/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a^2*(1 + p)*(2 - n + 2*p)))} -{x^0*E^(n*ArcTanh[a*x])*(c - a^2*c*x^2)^p, x, 3, -((2^(1 + n/2 + p)*(1 - a*x)^(1 - n/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-(n/2) - p, 1 - n/2 + p, 2 - n/2 + p, (1/2)*(1 - a*x)])/((1 - a^2*x^2)^p*(a*(2 - n + 2*p))))} - - -{E^(2*(p + 1)*ArcTanh[a*x])/(1 - a^2*x^2)^p, x, 3, (1 - a*x)^(1 - 2*p)/(a*(1 - 2*p)) + 1/((1 - a*x)^(2*p)*(a*p))} -{E^(2*(p + 1)*ArcTanh[a*x])/(c - a^2*c*x^2)^p, x, 4, ((1 - a*x)^(1 - 2*p)*(1 - a^2*x^2)^p)/((c - a^2*c*x^2)^p*(a*(1 - 2*p))) + (1 - a^2*x^2)^p/((1 - a*x)^(2*p)*(c - a^2*c*x^2)^p*(a*p))} - -{(c - a^2*c*x^2)^p*E^(2*p*ArcTanh[a*x]), x, 3, ((1 + a*x)^(1 + 2*p)*(c - a^2*c*x^2)^p)/((1 - a^2*x^2)^p*(a*(1 + 2*p)))} -{(c - a^2*c*x^2)^p/E^(2*p*ArcTanh[a*x]), x, 3, -(((1 - a*x)^(1 + 2*p)*(c - a^2*c*x^2)^p)/((1 - a^2*x^2)^p*(a*(1 + 2*p))))} - - -{x^2*(E^(n*ArcTanh[a*x])/(c - a^2*c*x^2)^(n^2/2 + 1)), x, 1, (E^(n*ArcTanh[a*x])*(1 - a*n*x))/((c - a^2*c*x^2)^(n^2/2)*(a^3*c*n*(1 - n^2)))} - -{x^2*E^(6*ArcTanh[a*x])/(c - a^2*c*x^2)^19, x, 2, -((1 - 6*a*x)/(210*a^3*c^19*(1 - a*x)^21*(1 + a*x)^15))} -{x^2*E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^9, x, 2, -((1 - 4*a*x)/(60*a^3*c^9*(1 - a*x)^10*(1 + a*x)^6))} -{x^2*E^(2*ArcTanh[a*x])/(c - a^2*c*x^2)^3, x, 2, -((1 - 2*a*x)/(6*a^3*c^3*(1 - a*x)^3*(1 + a*x)))} -{x^2/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^3), x, 2, (1 + 2*a*x)/(6*a^3*c^3*(1 - a*x)*(1 + a*x)^3)} -{x^2/(E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^9), x, 2, (1 + 4*a*x)/(60*a^3*c^9*(1 - a*x)^6*(1 + a*x)^10)} - -{x^2*E^(5*ArcTanh[a*x])/(c - a^2*c*x^2)^(27/2), x, 3, -(((1 - 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(1 - a*x)^15*(1 + a*x)^10*Sqrt[c - a^2*c*x^2]))} -{x^2*E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(11/2), x, 3, -(((1 - 3*a*x)*Sqrt[1 - a^2*x^2])/(24*a^3*c^5*(1 - a*x)^6*(1 + a*x)^3*Sqrt[c - a^2*c*x^2]))} -{x^2*E^(1*ArcTanh[a*x])/(c - a^2*c*x^2)^(3/2), x, 4, Sqrt[1 - a^2*x^2]/(2*a^3*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2])} -{x^2/(E^(1*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2)), x, 4, -(Sqrt[1 - a^2*x^2]/(2*a^3*c*(1 + a*x)*Sqrt[c - a^2*c*x^2])) - (Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2]) - (3*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*a^3*c*Sqrt[c - a^2*c*x^2])} -{x^2/(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(11/2)), x, 3, ((1 + 3*a*x)*Sqrt[1 - a^2*x^2])/(24*a^3*c^5*(1 - a*x)^3*(1 + a*x)^6*Sqrt[c - a^2*c*x^2])} -{x^2/(E^(5*ArcTanh[a*x])*(c - a^2*c*x^2)^(27/2)), x, 3, ((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(1 - a*x)^10*(1 + a*x)^15*Sqrt[c - a^2*c*x^2])} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Inverse hyperbolic tangent functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Inverse hyperbolic tangent functions.m deleted file mode 100644 index db72acf..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.3 Inverse hyperbolic tangent/7.3.7 Inverse hyperbolic tangent functions.m +++ /dev/null @@ -1,621 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integrands involving inverse hyperbolic tangents of algebraic functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (a+b ArcTanh[c x/Sqrt[d+e x^2]])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcTanh[c x/Sqrt[d+e x^2]]) when e=c^2*) - - -{x^5*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 6, -((5*d^2*x*Sqrt[d + e*x^2])/(96*e^(5/2))) + (5*d*x^3*Sqrt[d + e*x^2])/(144*e^(3/2)) - (x^5*Sqrt[d + e*x^2])/(36*Sqrt[e]) + (5*d^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(96*e^3) + (1/6)*x^6*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{x^3*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 5, (3*d*x*Sqrt[d + e*x^2])/(32*e^(3/2)) - (x^3*Sqrt[d + e*x^2])/(16*Sqrt[e]) - (3*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(32*e^2) + (1/4)*x^4*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{x^1*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 4, -((x*Sqrt[d + e*x^2])/(4*Sqrt[e])) + (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(4*e) + (1/2)*x^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^1, x, 8, -((Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]^2)/(2*Sqrt[d + e*x^2])) + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[1 - E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/Sqrt[d + e*x^2] - (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*ArcSinh[(Sqrt[e]*x)/Sqrt[d]]*Log[x])/Sqrt[d + e*x^2] + ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]*Log[x] + (Sqrt[d]*Sqrt[1 + (e*x^2)/d]*PolyLog[2, E^(2*ArcSinh[(Sqrt[e]*x)/Sqrt[d]])])/(2*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^3, x, 2, -((Sqrt[e]*Sqrt[d + e*x^2])/(2*d*x)) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*x^2)} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^5, x, 3, -((Sqrt[e]*Sqrt[d + e*x^2])/(12*d*x^3)) + (e^(3/2)*Sqrt[d + e*x^2])/(6*d^2*x) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(4*x^4)} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^7, x, 4, -((Sqrt[e]*Sqrt[d + e*x^2])/(30*d*x^5)) + (2*e^(3/2)*Sqrt[d + e*x^2])/(45*d^2*x^3) - (4*e^(5/2)*Sqrt[d + e*x^2])/(45*d^3*x) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(6*x^6)} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^9, x, 5, -((Sqrt[e]*Sqrt[d + e*x^2])/(56*d*x^7)) + (3*e^(3/2)*Sqrt[d + e*x^2])/(140*d^2*x^5) - (e^(5/2)*Sqrt[d + e*x^2])/(35*d^3*x^3) + (2*e^(7/2)*Sqrt[d + e*x^2])/(35*d^4*x) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(8*x^8)} - -{x^6*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 4, (d^3*Sqrt[d + e*x^2])/(7*e^(7/2)) - (d^2*(d + e*x^2)^(3/2))/(7*e^(7/2)) + (3*d*(d + e*x^2)^(5/2))/(35*e^(7/2)) - (d + e*x^2)^(7/2)/(49*e^(7/2)) + (1/7)*x^7*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{x^4*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 4, -((d^2*Sqrt[d + e*x^2])/(5*e^(5/2))) + (2*d*(d + e*x^2)^(3/2))/(15*e^(5/2)) - (d + e*x^2)^(5/2)/(25*e^(5/2)) + (1/5)*x^5*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{x^2*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 4, (d*Sqrt[d + e*x^2])/(3*e^(3/2)) - (d + e*x^2)^(3/2)/(9*e^(3/2)) + (1/3)*x^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{x^0*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 2, -(Sqrt[d + e*x^2]/Sqrt[e]) + x*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^2, x, 4, -(ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x) - (Sqrt[e]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d]} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^4, x, 5, -((Sqrt[e]*Sqrt[d + e*x^2])/(6*d*x^2)) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(3*x^3) + (e^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(3/2))} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^6, x, 6, -((Sqrt[e]*Sqrt[d + e*x^2])/(20*d*x^4)) + (3*e^(3/2)*Sqrt[d + e*x^2])/(40*d^2*x^2) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(5*x^5) - (3*e^(5/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*d^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^(m/2) (a+b ArcTanh[c x/Sqrt[d+e x^2]]) when e=c^2*) - - -{x^(9/2)*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 6, -((60*d^2*Sqrt[x]*Sqrt[d + e*x^2])/(847*e^(5/2))) + (36*d*x^(5/2)*Sqrt[d + e*x^2])/(847*e^(3/2)) - (4*x^(9/2)*Sqrt[d + e*x^2])/(121*Sqrt[e]) + (2/11)*x^(11/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (30*d^(11/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(847*e^(11/4)*Sqrt[d + e*x^2])} -{x^(5/2)*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 5, (20*d*Sqrt[x]*Sqrt[d + e*x^2])/(147*e^(3/2)) - (4*x^(5/2)*Sqrt[d + e*x^2])/(49*Sqrt[e]) + (2/7)*x^(7/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] - (10*d^(7/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(147*e^(7/4)*Sqrt[d + e*x^2])} -{x^(1/2)*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 4, -((4*Sqrt[x]*Sqrt[d + e*x^2])/(9*Sqrt[e])) + (2/3)*x^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (2*d^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(9*e^(3/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(3/2), x, 3, -((2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[x]) + (2*e^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(d^(1/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(7/2), x, 4, -((4*Sqrt[e]*Sqrt[d + e*x^2])/(15*d*x^(3/2))) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(5*x^(5/2)) - (2*e^(5/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(15*d^(5/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(11/2), x, 5, -((4*Sqrt[e]*Sqrt[d + e*x^2])/(63*d*x^(7/2))) + (20*e^(3/2)*Sqrt[d + e*x^2])/(189*d^2*x^(3/2)) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(9*x^(9/2)) + (10*e^(9/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(189*d^(9/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(15/2), x, 6, -((4*Sqrt[e]*Sqrt[d + e*x^2])/(143*d*x^(11/2))) + (36*e^(3/2)*Sqrt[d + e*x^2])/(1001*d^2*x^(7/2)) - (60*e^(5/2)*Sqrt[d + e*x^2])/(1001*d^3*x^(3/2)) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(13*x^(13/2)) - (30*e^(13/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(1001*d^(13/4)*Sqrt[d + e*x^2])} - -{x^(7/2)*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 7, (28*d*x^(3/2)*Sqrt[d + e*x^2])/(405*e^(3/2)) - (4*x^(7/2)*Sqrt[d + e*x^2])/(81*Sqrt[e]) - (28*d^2*Sqrt[x]*Sqrt[d + e*x^2])/(135*e^2*(Sqrt[d] + Sqrt[e]*x)) + (2/9)*x^(9/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (28*d^(9/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(135*e^(9/4)*Sqrt[d + e*x^2]) - (14*d^(9/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(135*e^(9/4)*Sqrt[d + e*x^2])} -{x^(3/2)*ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]], x, 6, -((4*x^(3/2)*Sqrt[d + e*x^2])/(25*Sqrt[e])) + (12*d*Sqrt[x]*Sqrt[d + e*x^2])/(25*e*(Sqrt[d] + Sqrt[e]*x)) + (2/5)*x^(5/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] - (12*d^(5/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(25*e^(5/4)*Sqrt[d + e*x^2]) + (6*d^(5/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(25*e^(5/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(1/2), x, 5, -((4*Sqrt[x]*Sqrt[d + e*x^2])/(Sqrt[d] + Sqrt[e]*x)) + 2*Sqrt[x]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + (4*d^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(1/4)*Sqrt[d + e*x^2]) - (2*d^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(1/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(5/2), x, 6, -((4*Sqrt[e]*Sqrt[d + e*x^2])/(3*d*Sqrt[x])) + (4*e*Sqrt[x]*Sqrt[d + e*x^2])/(3*d*(Sqrt[d] + Sqrt[e]*x)) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*x^(3/2)) - (4*e^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2]) + (2*e^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2])} -{ArcTanh[Sqrt[e]*x/Sqrt[d + e*x^2]]/x^(9/2), x, 7, -((4*Sqrt[e]*Sqrt[d + e*x^2])/(35*d*x^(5/2))) + (12*e^(3/2)*Sqrt[d + e*x^2])/(35*d^2*Sqrt[x]) - (12*e^2*Sqrt[x]*Sqrt[d + e*x^2])/(35*d^2*(Sqrt[d] + Sqrt[e]*x)) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(7*x^(7/2)) + (12*e^(7/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(35*d^(7/4)*Sqrt[d + e*x^2]) - (6*e^(7/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(35*d^(7/4)*Sqrt[d + e*x^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m (a+b ArcTanh[c+d x^m])^n*) - - -{x^3*ArcTanh[a + b*x^4], x, 4, ((a + b*x^4)*ArcTanh[a + b*x^4])/(4*b) + Log[1 - (a + b*x^4)^2]/(8*b)} - - -{x^(n-1)*ArcTanh[a + b*x^n], x, 4, ((a + b*x^n)*ArcTanh[a + b*x^n])/(b*n) + Log[1 - (a + b*x^n)^2]/(2*b*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d+e x^2)^q (a+b ArcTanh[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 9, -((2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcTanh[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c) + (3*b*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (3*b*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (3*b^2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b^2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b^3*PolyLog[4, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(4*c) - (3*b^3*PolyLog[4, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(4*c)} -{(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((2*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcTanh[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c) + (b*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (b*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c - (b^2*PolyLog[3, 1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (b^2*PolyLog[3, -1 + 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c)} -{(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 2, -((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/c) + (b*PolyLog[2, -(Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) - (b*PolyLog[2, Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(2*c)} -{1/((a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse hyperbolic tangents of hyperbolic functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTanh[Tanh[a+b x]]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[Tanh[a+b x]]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcTanh[Tanh[a + b*x]]*x^m, x, 2, -((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcTanh[Tanh[a + b*x]])/(1 + m)} - -{ArcTanh[Tanh[a + b*x]]*x^2, x, 2, -((b*x^4)/12) + (1/3)*x^3*ArcTanh[Tanh[a + b*x]]} -{ArcTanh[Tanh[a + b*x]]*x^1, x, 2, -((b*x^3)/6) + (1/2)*x^2*ArcTanh[Tanh[a + b*x]]} -{ArcTanh[Tanh[a + b*x]]*x^0, x, 2, ArcTanh[Tanh[a + b*x]]^2/(2*b)} -{ArcTanh[Tanh[a + b*x]]/x^1, x, 2, b*x - (b*x - ArcTanh[Tanh[a + b*x]])*Log[x]} -{ArcTanh[Tanh[a + b*x]]/x^2, x, 2, -(ArcTanh[Tanh[a + b*x]]/x) + b*Log[x]} -{ArcTanh[Tanh[a + b*x]]/x^3, x, 2, -(b/(2*x)) - ArcTanh[Tanh[a + b*x]]/(2*x^2)} -{ArcTanh[Tanh[a + b*x]]/x^4, x, 2, -(b/(6*x^2)) - ArcTanh[Tanh[a + b*x]]/(3*x^3)} - - -{ArcTanh[Tanh[a + b*x]]^2*x^m, x, 3, (2*b^2*x^(3 + m))/(6 + 11*m + 6*m^2 + m^3) - (2*b*x^(2 + m)*ArcTanh[Tanh[a + b*x]])/(2 + 3*m + m^2) + (x^(1 + m)*ArcTanh[Tanh[a + b*x]]^2)/(1 + m)} - -{ArcTanh[Tanh[a + b*x]]^2*x^3, x, 3, (b^2*x^6)/60 - (1/10)*b*x^5*ArcTanh[Tanh[a + b*x]] + (1/4)*x^4*ArcTanh[Tanh[a + b*x]]^2} -{ArcTanh[Tanh[a + b*x]]^2*x^2, x, 3, (b^2*x^5)/30 - (1/6)*b*x^4*ArcTanh[Tanh[a + b*x]] + (1/3)*x^3*ArcTanh[Tanh[a + b*x]]^2} -{ArcTanh[Tanh[a + b*x]]^2*x^1, x, 3, (x*ArcTanh[Tanh[a + b*x]]^3)/(3*b) - ArcTanh[Tanh[a + b*x]]^4/(12*b^2)} -{ArcTanh[Tanh[a + b*x]]^2*x^0, x, 2, ArcTanh[Tanh[a + b*x]]^3/(3*b)} -{ArcTanh[Tanh[a + b*x]]^2/x^1, x, 3, (-b)*x*(b*x - ArcTanh[Tanh[a + b*x]]) + (1/2)*ArcTanh[Tanh[a + b*x]]^2 + (b*x - ArcTanh[Tanh[a + b*x]])^2*Log[x]} -{ArcTanh[Tanh[a + b*x]]^2/x^2, x, 3, 2*b^2*x - ArcTanh[Tanh[a + b*x]]^2/x - 2*b*(b*x - ArcTanh[Tanh[a + b*x]])*Log[x]} -{ArcTanh[Tanh[a + b*x]]^2/x^3, x, 3, -((b*ArcTanh[Tanh[a + b*x]])/x) - ArcTanh[Tanh[a + b*x]]^2/(2*x^2) + b^2*Log[x]} -{ArcTanh[Tanh[a + b*x]]^2/x^4, x, 1, ArcTanh[Tanh[a + b*x]]^3/(3*x^3*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^2/x^5, x, 2, -(b^2/(12*x^2)) - (b*ArcTanh[Tanh[a + b*x]])/(6*x^3) - ArcTanh[Tanh[a + b*x]]^2/(4*x^4), (b*ArcTanh[Tanh[a + b*x]]^3)/(12*x^3*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^3/(4*x^4*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{ArcTanh[Tanh[a + b*x]]^3*x^m, x, 4, -((6*b^3*x^(4 + m))/((1 + m)*(24 + 26*m + 9*m^2 + m^3))) + (6*b^2*x^(3 + m)*ArcTanh[Tanh[a + b*x]])/(6 + 11*m + 6*m^2 + m^3) - (3*b*x^(2 + m)*ArcTanh[Tanh[a + b*x]]^2)/(2 + 3*m + m^2) + (x^(1 + m)*ArcTanh[Tanh[a + b*x]]^3)/(1 + m)} - -{ArcTanh[Tanh[a + b*x]]^3*x^3, x, 4, (-(1/140))*b^3*x^7 + (1/20)*b^2*x^6*ArcTanh[Tanh[a + b*x]] - (3/20)*b*x^5*ArcTanh[Tanh[a + b*x]]^2 + (1/4)*x^4*ArcTanh[Tanh[a + b*x]]^3} -{ArcTanh[Tanh[a + b*x]]^3*x^2, x, 4, (x^2*ArcTanh[Tanh[a + b*x]]^4)/(4*b) - (x*ArcTanh[Tanh[a + b*x]]^5)/(10*b^2) + ArcTanh[Tanh[a + b*x]]^6/(60*b^3)} -{ArcTanh[Tanh[a + b*x]]^3*x^1, x, 3, (x*ArcTanh[Tanh[a + b*x]]^4)/(4*b) - ArcTanh[Tanh[a + b*x]]^5/(20*b^2)} -{ArcTanh[Tanh[a + b*x]]^3*x^0, x, 2, ArcTanh[Tanh[a + b*x]]^4/(4*b)} -{ArcTanh[Tanh[a + b*x]]^3/x^1, x, 4, b*x*(b*x - ArcTanh[Tanh[a + b*x]])^2 - (1/2)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2 + (1/3)*ArcTanh[Tanh[a + b*x]]^3 - (b*x - ArcTanh[Tanh[a + b*x]])^3*Log[x]} -{ArcTanh[Tanh[a + b*x]]^3/x^2, x, 4, -3*b^2*x*(b*x - ArcTanh[Tanh[a + b*x]]) + (3/2)*b*ArcTanh[Tanh[a + b*x]]^2 - ArcTanh[Tanh[a + b*x]]^3/x + 3*b*(b*x - ArcTanh[Tanh[a + b*x]])^2*Log[x]} -{ArcTanh[Tanh[a + b*x]]^3/x^3, x, 4, 3*b^3*x - (3*b*ArcTanh[Tanh[a + b*x]]^2)/(2*x) - ArcTanh[Tanh[a + b*x]]^3/(2*x^2) - 3*b^2*(b*x - ArcTanh[Tanh[a + b*x]])*Log[x]} -{ArcTanh[Tanh[a + b*x]]^3/x^4, x, 4, -((b^2*ArcTanh[Tanh[a + b*x]])/x) - (b*ArcTanh[Tanh[a + b*x]]^2)/(2*x^2) - ArcTanh[Tanh[a + b*x]]^3/(3*x^3) + b^3*Log[x]} -{ArcTanh[Tanh[a + b*x]]^3/x^5, x, 1, ArcTanh[Tanh[a + b*x]]^4/(4*x^4*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^3/x^6, x, 2, (b*ArcTanh[Tanh[a + b*x]]^4)/(20*x^4*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^4/(5*x^5*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{ArcTanh[Tanh[a + b*x]]^4*x^m, x, 5, (24*b^4*x^(5 + m))/((1 + m)*(2 + m)*(3 + m)*(20 + 9*m + m^2)) - (24*b^3*x^(4 + m)*ArcTanh[Tanh[a + b*x]])/((1 + m)*(24 + 26*m + 9*m^2 + m^3)) + (12*b^2*x^(3 + m)*ArcTanh[Tanh[a + b*x]]^2)/(6 + 11*m + 6*m^2 + m^3) - (4*b*x^(2 + m)*ArcTanh[Tanh[a + b*x]]^3)/(2 + 3*m + m^2) + (x^(1 + m)*ArcTanh[Tanh[a + b*x]]^4)/(1 + m)} - -{ArcTanh[Tanh[a + b*x]]^4*x^6, x, 5, (b^4*x^11)/2310 - (1/210)*b^3*x^10*ArcTanh[Tanh[a + b*x]] + (1/42)*b^2*x^9*ArcTanh[Tanh[a + b*x]]^2 - (1/14)*b*x^8*ArcTanh[Tanh[a + b*x]]^3 + (1/7)*x^7*ArcTanh[Tanh[a + b*x]]^4} -{ArcTanh[Tanh[a + b*x]]^4*x^5, x, 5, (b^4*x^10)/1260 - (1/126)*b^3*x^9*ArcTanh[Tanh[a + b*x]] + (1/28)*b^2*x^8*ArcTanh[Tanh[a + b*x]]^2 - (2/21)*b*x^7*ArcTanh[Tanh[a + b*x]]^3 + (1/6)*x^6*ArcTanh[Tanh[a + b*x]]^4} -{ArcTanh[Tanh[a + b*x]]^4*x^4, x, 5, (b^4*x^9)/630 - (1/70)*b^3*x^8*ArcTanh[Tanh[a + b*x]] + (2/35)*b^2*x^7*ArcTanh[Tanh[a + b*x]]^2 - (2/15)*b*x^6*ArcTanh[Tanh[a + b*x]]^3 + (1/5)*x^5*ArcTanh[Tanh[a + b*x]]^4} -{ArcTanh[Tanh[a + b*x]]^4*x^3, x, 5, (x^3*ArcTanh[Tanh[a + b*x]]^5)/(5*b) - (x^2*ArcTanh[Tanh[a + b*x]]^6)/(10*b^2) + (x*ArcTanh[Tanh[a + b*x]]^7)/(35*b^3) - ArcTanh[Tanh[a + b*x]]^8/(280*b^4)} -{ArcTanh[Tanh[a + b*x]]^4*x^2, x, 4, (x^2*ArcTanh[Tanh[a + b*x]]^5)/(5*b) - (x*ArcTanh[Tanh[a + b*x]]^6)/(15*b^2) + ArcTanh[Tanh[a + b*x]]^7/(105*b^3)} -{ArcTanh[Tanh[a + b*x]]^4*x^1, x, 3, (x*ArcTanh[Tanh[a + b*x]]^5)/(5*b) - ArcTanh[Tanh[a + b*x]]^6/(30*b^2)} -{ArcTanh[Tanh[a + b*x]]^4*x^0, x, 2, ArcTanh[Tanh[a + b*x]]^5/(5*b)} -{ArcTanh[Tanh[a + b*x]]^4/x^1, x, 5, (-b)*x*(b*x - ArcTanh[Tanh[a + b*x]])^3 + (1/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^2 - (1/3)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^3 + (1/4)*ArcTanh[Tanh[a + b*x]]^4 + (b*x - ArcTanh[Tanh[a + b*x]])^4*Log[x]} -{ArcTanh[Tanh[a + b*x]]^4/x^2, x, 5, 4*b^2*x*(b*x - ArcTanh[Tanh[a + b*x]])^2 - 2*b*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2 + (4/3)*b*ArcTanh[Tanh[a + b*x]]^3 - ArcTanh[Tanh[a + b*x]]^4/x - 4*b*(b*x - ArcTanh[Tanh[a + b*x]])^3*Log[x]} -{ArcTanh[Tanh[a + b*x]]^4/x^3, x, 5, -6*b^3*x*(b*x - ArcTanh[Tanh[a + b*x]]) + 3*b^2*ArcTanh[Tanh[a + b*x]]^2 - (2*b*ArcTanh[Tanh[a + b*x]]^3)/x - ArcTanh[Tanh[a + b*x]]^4/(2*x^2) + 6*b^2*(b*x - ArcTanh[Tanh[a + b*x]])^2*Log[x]} -{ArcTanh[Tanh[a + b*x]]^4/x^4, x, 5, 4*b^4*x - (2*b^2*ArcTanh[Tanh[a + b*x]]^2)/x - (2*b*ArcTanh[Tanh[a + b*x]]^3)/(3*x^2) - ArcTanh[Tanh[a + b*x]]^4/(3*x^3) - 4*b^3*(b*x - ArcTanh[Tanh[a + b*x]])*Log[x]} -{ArcTanh[Tanh[a + b*x]]^4/x^5, x, 5, -((b^3*ArcTanh[Tanh[a + b*x]])/x) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(2*x^2) - (b*ArcTanh[Tanh[a + b*x]]^3)/(3*x^3) - ArcTanh[Tanh[a + b*x]]^4/(4*x^4) + b^4*Log[x]} -{ArcTanh[Tanh[a + b*x]]^4/x^6, x, 1, ArcTanh[Tanh[a + b*x]]^5/(5*x^5*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^4/x^7, x, 2, (b*ArcTanh[Tanh[a + b*x]]^5)/(30*x^5*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^5/(6*x^6*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^4/x^8, x, 3, (b^2*ArcTanh[Tanh[a + b*x]]^5)/(105*x^5*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (b*ArcTanh[Tanh[a + b*x]]^5)/(21*x^6*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^5/(7*x^7*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^4/x^9, x, 4, -(b^4/(280*x^4)) - (b^3*ArcTanh[Tanh[a + b*x]])/(70*x^5) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(28*x^6) - (b*ArcTanh[Tanh[a + b*x]]^3)/(14*x^7) - ArcTanh[Tanh[a + b*x]]^4/(8*x^8), (b^3*ArcTanh[Tanh[a + b*x]]^5)/(280*x^5*(b*x - ArcTanh[Tanh[a + b*x]])^4) + (b^2*ArcTanh[Tanh[a + b*x]]^5)/(56*x^6*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (3*b*ArcTanh[Tanh[a + b*x]]^5)/(56*x^7*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^5/(8*x^8*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^4/x^10, x, 5, -(b^4/(630*x^5)) - (b^3*ArcTanh[Tanh[a + b*x]])/(126*x^6) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(42*x^7) - (b*ArcTanh[Tanh[a + b*x]]^3)/(18*x^8) - ArcTanh[Tanh[a + b*x]]^4/(9*x^9)} -{ArcTanh[Tanh[a + b*x]]^4/x^11, x, 5, -(b^4/(1260*x^6)) - (b^3*ArcTanh[Tanh[a + b*x]])/(210*x^7) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(60*x^8) - (2*b*ArcTanh[Tanh[a + b*x]]^3)/(45*x^9) - ArcTanh[Tanh[a + b*x]]^4/(10*x^10)} - - -{ArcTanh[Tanh[a + b*x]]^6*x^1, x, 3, (x*ArcTanh[Tanh[a + b*x]]^7)/(7*b) - ArcTanh[Tanh[a + b*x]]^8/(56*b^2)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/ArcTanh[Tanh[a + b*x]]*x^m, x, 1, -((x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/(b*x - ArcTanh[Tanh[a + b*x]])])/((1 + m)*(b*x - ArcTanh[Tanh[a + b*x]])))} - -{1/ArcTanh[Tanh[a + b*x]]*x^3, x, 5, x^3/(3*b) + (x^2*(b*x - ArcTanh[Tanh[a + b*x]]))/(2*b^2) + (x*(b*x - ArcTanh[Tanh[a + b*x]])^2)/b^3 + ((b*x - ArcTanh[Tanh[a + b*x]])^3*Log[ArcTanh[Tanh[a + b*x]]])/b^4} -{1/ArcTanh[Tanh[a + b*x]]*x^2, x, 4, x^2/(2*b) + (x*(b*x - ArcTanh[Tanh[a + b*x]]))/b^2 + ((b*x - ArcTanh[Tanh[a + b*x]])^2*Log[ArcTanh[Tanh[a + b*x]]])/b^3} -{1/ArcTanh[Tanh[a + b*x]]*x^1, x, 3, x/b + ((b*x - ArcTanh[Tanh[a + b*x]])*Log[ArcTanh[Tanh[a + b*x]]])/b^2} -{1/ArcTanh[Tanh[a + b*x]]*x^0, x, 2, Log[ArcTanh[Tanh[a + b*x]]]/b} -{1/ArcTanh[Tanh[a + b*x]]/x^1, x, 4, -(Log[x]/(b*x - ArcTanh[Tanh[a + b*x]])) + Log[ArcTanh[Tanh[a + b*x]]]/(b*x - ArcTanh[Tanh[a + b*x]])} -{1/ArcTanh[Tanh[a + b*x]]/x^2, x, 5, 1/(x*(b*x - ArcTanh[Tanh[a + b*x]])) - (b*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^2 + (b*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^2} -{1/ArcTanh[Tanh[a + b*x]]/x^3, x, 6, b/(x*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 1/(2*x^2*(b*x - ArcTanh[Tanh[a + b*x]])) - (b^2*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^3 + (b^2*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^3} - - -{1/ArcTanh[Tanh[a + b*x]]^2*x^m, x, 2, -(x^m/(b*ArcTanh[Tanh[a + b*x]])) - (x^m*Hypergeometric2F1[1, m, 1 + m, (b*x)/(b*x - ArcTanh[Tanh[a + b*x]])])/(b*(b*x - ArcTanh[Tanh[a + b*x]]))} - -{1/ArcTanh[Tanh[a + b*x]]^2*x^4, x, 6, (4*x^3)/(3*b^2) + (2*x^2*(b*x - ArcTanh[Tanh[a + b*x]]))/b^3 + (4*x*(b*x - ArcTanh[Tanh[a + b*x]])^2)/b^4 - x^4/(b*ArcTanh[Tanh[a + b*x]]) + (4*(b*x - ArcTanh[Tanh[a + b*x]])^3*Log[ArcTanh[Tanh[a + b*x]]])/b^5} -{1/ArcTanh[Tanh[a + b*x]]^2*x^3, x, 5, (3*x^2)/(2*b^2) + (3*x*(b*x - ArcTanh[Tanh[a + b*x]]))/b^3 - x^3/(b*ArcTanh[Tanh[a + b*x]]) + (3*(b*x - ArcTanh[Tanh[a + b*x]])^2*Log[ArcTanh[Tanh[a + b*x]]])/b^4} -{1/ArcTanh[Tanh[a + b*x]]^2*x^2, x, 4, (2*x)/b^2 - x^2/(b*ArcTanh[Tanh[a + b*x]]) + (2*(b*x - ArcTanh[Tanh[a + b*x]])*Log[ArcTanh[Tanh[a + b*x]]])/b^3} -{1/ArcTanh[Tanh[a + b*x]]^2*x^1, x, 3, -(x/(b*ArcTanh[Tanh[a + b*x]])) + Log[ArcTanh[Tanh[a + b*x]]]/b^2} -{1/ArcTanh[Tanh[a + b*x]]^2*x^0, x, 2, -(1/(b*ArcTanh[Tanh[a + b*x]]))} -{1/ArcTanh[Tanh[a + b*x]]^2/x^1, x, 5, -(1/((b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]])) + Log[x]/(b*x - ArcTanh[Tanh[a + b*x]])^2 - Log[ArcTanh[Tanh[a + b*x]]]/(b*x - ArcTanh[Tanh[a + b*x]])^2} -{1/ArcTanh[Tanh[a + b*x]]^2/x^2, x, 6, -((2*b)/((b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]])) + 1/(x*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]) + (2*b*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^3 - (2*b*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^3} -{1/ArcTanh[Tanh[a + b*x]]^2/x^3, x, 7, -((3*b^2)/((b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]])) + (3*b)/(2*x*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]) + 1/(2*x^2*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]) + (3*b^2*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^4 - (3*b^2*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^4} - - -{1/ArcTanh[Tanh[a + b*x]]^3*x^m, x, 3, -(x^m/(2*b*ArcTanh[Tanh[a + b*x]]^2)) - (m*x^(-1 + m))/(2*b^2*ArcTanh[Tanh[a + b*x]]) - (m*x^(-1 + m)*Hypergeometric2F1[1, -1 + m, m, (b*x)/(b*x - ArcTanh[Tanh[a + b*x]])])/(2*b^2*(b*x - ArcTanh[Tanh[a + b*x]]))} - -{1/ArcTanh[Tanh[a + b*x]]^3*x^4, x, 6, (3*x^2)/b^3 + (6*x*(b*x - ArcTanh[Tanh[a + b*x]]))/b^4 - x^4/(2*b*ArcTanh[Tanh[a + b*x]]^2) - (2*x^3)/(b^2*ArcTanh[Tanh[a + b*x]]) + (6*(b*x - ArcTanh[Tanh[a + b*x]])^2*Log[ArcTanh[Tanh[a + b*x]]])/b^5} -{1/ArcTanh[Tanh[a + b*x]]^3*x^3, x, 5, (3*x)/b^3 - x^3/(2*b*ArcTanh[Tanh[a + b*x]]^2) - (3*x^2)/(2*b^2*ArcTanh[Tanh[a + b*x]]) + (3*(b*x - ArcTanh[Tanh[a + b*x]])*Log[ArcTanh[Tanh[a + b*x]]])/b^4} -{1/ArcTanh[Tanh[a + b*x]]^3*x^2, x, 4, -(x^2/(2*b*ArcTanh[Tanh[a + b*x]]^2)) - x/(b^2*ArcTanh[Tanh[a + b*x]]) + Log[ArcTanh[Tanh[a + b*x]]]/b^3} -{1/ArcTanh[Tanh[a + b*x]]^3*x^1, x, 3, -(x/(2*b*ArcTanh[Tanh[a + b*x]]^2)) - 1/(2*b^2*ArcTanh[Tanh[a + b*x]])} -{1/ArcTanh[Tanh[a + b*x]]^3*x^0, x, 2, -(1/(2*b*ArcTanh[Tanh[a + b*x]]^2))} -{1/ArcTanh[Tanh[a + b*x]]^3/x^1, x, 6, -(1/(2*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2)) + 1/((b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]) - Log[x]/(b*x - ArcTanh[Tanh[a + b*x]])^3 + Log[ArcTanh[Tanh[a + b*x]]]/(b*x - ArcTanh[Tanh[a + b*x]])^3} -{1/ArcTanh[Tanh[a + b*x]]^3/x^2, x, 7, -((3*b)/(2*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^2)) + 1/(x*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2) + (3*b)/((b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]) - (3*b*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^4 + (3*b*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^4} -{1/ArcTanh[Tanh[a + b*x]]^3/x^3, x, 8, -((3*b^2)/((b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^2)) + (2*b)/(x*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^2) + 1/(2*x^2*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2) + (6*b^2)/((b*x - ArcTanh[Tanh[a + b*x]])^4*ArcTanh[Tanh[a + b*x]]) - (6*b^2*Log[x])/(b*x - ArcTanh[Tanh[a + b*x]])^5 + (6*b^2*Log[ArcTanh[Tanh[a + b*x]]])/(b*x - ArcTanh[Tanh[a + b*x]])^5} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[Tanh[a+b x]]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Sqrt[ArcTanh[Tanh[a + b*x]]]*x^4, x, 6, (2*x^4*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (16*x^3*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^2) + (32*x^2*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(9/2))/(315*b^4) + (256*ArcTanh[Tanh[a + b*x]]^(11/2))/(3465*b^5)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]*x^3, x, 5, (2*x^3*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (4*x^2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b^2) + (16*x*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^3) - (32*ArcTanh[Tanh[a + b*x]]^(9/2))/(315*b^4)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]*x^2, x, 4, (2*x^2*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (8*x*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^2) + (16*ArcTanh[Tanh[a + b*x]]^(7/2))/(105*b^3)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]*x^1, x, 3, (2*x*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b) - (4*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^2)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]*x^0, x, 2, (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]/x^1, x, 2, -2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] + 2*Sqrt[ArcTanh[Tanh[a + b*x]]]} -{Sqrt[ArcTanh[Tanh[a + b*x]]]/x^2, x, 2, (b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] - Sqrt[ArcTanh[Tanh[a + b*x]]]/x} -{Sqrt[ArcTanh[Tanh[a + b*x]]]/x^3, x, 4, (b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - b/(4*x*Sqrt[ArcTanh[Tanh[a + b*x]]]) + b^2/(4*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]]) - Sqrt[ArcTanh[Tanh[a + b*x]]]/(2*x^2)} -{Sqrt[ArcTanh[Tanh[a + b*x]]]/x^4, x, 6, (b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2)) + b^2/(24*x*ArcTanh[Tanh[a + b*x]]^(3/2)) - b^3/(24*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) - b/(12*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + b^3/(8*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) - Sqrt[ArcTanh[Tanh[a + b*x]]]/(3*x^3)} - - -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^4, x, 6, (2*x^4*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b) - (16*x^3*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^2) + (32*x^2*ArcTanh[Tanh[a + b*x]]^(9/2))/(105*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(11/2))/(1155*b^4) + (256*ArcTanh[Tanh[a + b*x]]^(13/2))/(15015*b^5)} -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^3, x, 5, (2*x^3*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b) - (12*x^2*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^2) + (16*x*ArcTanh[Tanh[a + b*x]]^(9/2))/(105*b^3) - (32*ArcTanh[Tanh[a + b*x]]^(11/2))/(1155*b^4)} -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^2, x, 4, (2*x^2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b) - (8*x*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^2) + (16*ArcTanh[Tanh[a + b*x]]^(9/2))/(315*b^3)} -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^1, x, 3, (2*x*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b) - (4*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^0, x, 2, (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b)} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^1, x, 3, 2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) - 2*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (2/3)*ArcTanh[Tanh[a + b*x]]^(3/2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^2, x, 3, -3*b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] + 3*b*Sqrt[ArcTanh[Tanh[a + b*x]]] - ArcTanh[Tanh[a + b*x]]^(3/2)/x} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^3, x, 3, (3*b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]) - (3*b*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*x) - ArcTanh[Tanh[a + b*x]]^(3/2)/(2*x^2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^4, x, 5, (b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - b^2/(8*x*Sqrt[ArcTanh[Tanh[a + b*x]]]) + b^3/(8*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]]) - (b*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*x^2) - ArcTanh[Tanh[a + b*x]]^(3/2)/(3*x^3)} - - -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^4, x, 6, (2*x^4*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (16*x^3*ArcTanh[Tanh[a + b*x]]^(9/2))/(63*b^2) + (32*x^2*ArcTanh[Tanh[a + b*x]]^(11/2))/(231*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(13/2))/(3003*b^4) + (256*ArcTanh[Tanh[a + b*x]]^(15/2))/(45045*b^5)} -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^3, x, 5, (2*x^3*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (4*x^2*ArcTanh[Tanh[a + b*x]]^(9/2))/(21*b^2) + (16*x*ArcTanh[Tanh[a + b*x]]^(11/2))/(231*b^3) - (32*ArcTanh[Tanh[a + b*x]]^(13/2))/(3003*b^4)} -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^2, x, 4, (2*x^2*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (8*x*ArcTanh[Tanh[a + b*x]]^(9/2))/(63*b^2) + (16*ArcTanh[Tanh[a + b*x]]^(11/2))/(693*b^3)} -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^1, x, 3, (2*x*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b) - (4*ArcTanh[Tanh[a + b*x]]^(9/2))/(63*b^2)} -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^0, x, 2, (2*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*b)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^1, x, 4, -2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2) + 2*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]] - (2/3)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2) + (2/5)*ArcTanh[Tanh[a + b*x]]^(5/2)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^2, x, 4, 5*b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) - 5*b*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (5/3)*b*ArcTanh[Tanh[a + b*x]]^(3/2) - ArcTanh[Tanh[a + b*x]]^(5/2)/x} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^3, x, 4, (-(15/4))*b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]] + (15/4)*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]] - (5*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(4*x) - ArcTanh[Tanh[a + b*x]]^(5/2)/(2*x^2)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^4, x, 4, (5*b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]) - (5*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*x) - (5*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(12*x^2) - ArcTanh[Tanh[a + b*x]]^(5/2)/(3*x^3)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^5, x, 6, (5*b^4*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(64*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - (5*b^3)/(64*x*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (5*b^4)/(64*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]]) - (5*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(32*x^2) - (5*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(24*x^3) - ArcTanh[Tanh[a + b*x]]^(5/2)/(4*x^4)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^6, x, 8, (3*b^5*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(128*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2)) + b^4/(128*x*ArcTanh[Tanh[a + b*x]]^(3/2)) - b^5/(128*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) - b^3/(64*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (3*b^5)/(128*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) - (b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(16*x^3) - (b*ArcTanh[Tanh[a + b*x]]^(3/2))/(8*x^4) - ArcTanh[Tanh[a + b*x]]^(5/2)/(5*x^5)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]*x^4, x, 6, (2*x^4*Sqrt[ArcTanh[Tanh[a + b*x]]])/b - (16*x^3*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^2) + (32*x^2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b^3) - (128*x*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^4) + (256*ArcTanh[Tanh[a + b*x]]^(9/2))/(315*b^5)} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]*x^3, x, 5, (2*x^3*Sqrt[ArcTanh[Tanh[a + b*x]]])/b - (4*x^2*ArcTanh[Tanh[a + b*x]]^(3/2))/b^2 + (16*x*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b^3) - (32*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^4)} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]*x^2, x, 4, (2*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b - (8*x*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^2) + (16*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^3)} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]*x^1, x, 3, (2*x*Sqrt[ArcTanh[Tanh[a + b*x]]])/b - (4*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^2)} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]*x^0, x, 2, (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]/x^1, x, 1, (2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]/x^2, x, 3, (b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) - 1/(x*Sqrt[ArcTanh[Tanh[a + b*x]]]) + b/((b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]/x^3, x, 5, (3*b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2)) + b/(4*x*ArcTanh[Tanh[a + b*x]]^(3/2)) - b^2/(4*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) - 1/(2*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (3*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/Sqrt[ArcTanh[Tanh[a + b*x]]]/x^4, x, 7, (5*b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*(b*x - ArcTanh[Tanh[a + b*x]])^(7/2)) - b^2/(8*x*ArcTanh[Tanh[a + b*x]]^(5/2)) + b^3/(8*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(5/2)) + b/(12*x^2*ArcTanh[Tanh[a + b*x]]^(3/2)) - (5*b^3)/(24*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2)) - 1/(3*x^3*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (5*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])} - - -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^4, x, 6, -((2*x^4)/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (16*x^3*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^2 - (32*x^2*ArcTanh[Tanh[a + b*x]]^(3/2))/b^3 + (128*x*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b^4) - (256*ArcTanh[Tanh[a + b*x]]^(7/2))/(35*b^5)} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^3, x, 5, -((2*x^3)/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (12*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^2 - (16*x*ArcTanh[Tanh[a + b*x]]^(3/2))/b^3 + (32*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*b^4)} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^2, x, 4, -((2*x^2)/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (8*x*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^2 - (16*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^3)} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^1, x, 3, -((2*x)/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (4*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^2} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^0, x, 2, -(2/(b*Sqrt[ArcTanh[Tanh[a + b*x]]]))} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^1, x, 2, -((2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - 2/((b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^2, x, 4, -((3*b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(5/2)) - 1/(x*ArcTanh[Tanh[a + b*x]]^(3/2)) + b/((b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) - (3*b)/((b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^3, x, 6, -((15*b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(7/2))) + (3*b)/(4*x*ArcTanh[Tanh[a + b*x]]^(5/2)) - (3*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(5/2)) - 1/(2*x^2*ArcTanh[Tanh[a + b*x]]^(3/2)) + (5*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2)) - (15*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^4, x, 8, -((35*b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*(b*x - ArcTanh[Tanh[a + b*x]])^(9/2))) - (5*b^2)/(8*x*ArcTanh[Tanh[a + b*x]]^(7/2)) + (5*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(7/2)) + b/(4*x^2*ArcTanh[Tanh[a + b*x]]^(5/2)) - (7*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(5/2)) - 1/(3*x^3*ArcTanh[Tanh[a + b*x]]^(3/2)) + (35*b^3)/(24*(b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^(3/2)) - (35*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^4*Sqrt[ArcTanh[Tanh[a + b*x]]])} - - -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^4, x, 6, -((2*x^4)/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2))) - (16*x^3)/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (32*x^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^3 - (128*x*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^4) + (256*ArcTanh[Tanh[a + b*x]]^(5/2))/(15*b^5)} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^3, x, 5, -((2*x^3)/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2))) - (4*x^2)/(b^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (16*x*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^3 - (32*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*b^4)} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^2, x, 4, -((2*x^2)/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2))) - (8*x)/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (16*Sqrt[ArcTanh[Tanh[a + b*x]]])/(3*b^3)} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^1, x, 3, -((2*x)/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2))) - 4/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^0, x, 2, -(2/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2)))} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^1, x, 3, (2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(5/2) - 2/(3*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) + 2/((b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^2, x, 5, (5*b*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(7/2) - 1/(x*ArcTanh[Tanh[a + b*x]]^(5/2)) + b/((b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(5/2)) - (5*b)/(3*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2)) + (5*b)/((b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^3, x, 7, (35*b^2*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(9/2)) + (5*b)/(4*x*ArcTanh[Tanh[a + b*x]]^(7/2)) - (5*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(7/2)) - 1/(2*x^2*ArcTanh[Tanh[a + b*x]]^(5/2)) + (7*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(5/2)) - (35*b^2)/(12*(b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^(3/2)) + (35*b^2)/(4*(b*x - ArcTanh[Tanh[a + b*x]])^4*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^4, x, 9, (105*b^3*ArcTan[Sqrt[ArcTanh[Tanh[a + b*x]]]/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(8*(b*x - ArcTanh[Tanh[a + b*x]])^(11/2)) - (35*b^2)/(24*x*ArcTanh[Tanh[a + b*x]]^(9/2)) + (35*b^3)/(24*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(9/2)) + (5*b)/(12*x^2*ArcTanh[Tanh[a + b*x]]^(7/2)) - (15*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(7/2)) - 1/(3*x^3*ArcTanh[Tanh[a + b*x]]^(5/2)) + (21*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^(5/2)) - (35*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^4*ArcTanh[Tanh[a + b*x]]^(3/2)) + (105*b^3)/(8*(b*x - ArcTanh[Tanh[a + b*x]])^5*Sqrt[ArcTanh[Tanh[a + b*x]]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^(m/2) ArcTanh[Tanh[a+b x]]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^(7/2)*ArcTanh[Tanh[a + b*x]], x, 2, (-4*b*x^(11/2))/99 + (2*x^(9/2)*ArcTanh[Tanh[a + b*x]])/9} -{x^(5/2)*ArcTanh[Tanh[a + b*x]], x, 2, (-4*b*x^(9/2))/63 + (2*x^(7/2)*ArcTanh[Tanh[a + b*x]])/7} -{x^(3/2)*ArcTanh[Tanh[a + b*x]], x, 2, (-4*b*x^(7/2))/35 + (2*x^(5/2)*ArcTanh[Tanh[a + b*x]])/5} -{Sqrt[x]*ArcTanh[Tanh[a + b*x]], x, 2, (-4*b*x^(5/2))/15 + (2*x^(3/2)*ArcTanh[Tanh[a + b*x]])/3} -{ArcTanh[Tanh[a + b*x]]/Sqrt[x], x, 2, (-4*b*x^(3/2))/3 + 2*Sqrt[x]*ArcTanh[Tanh[a + b*x]]} -{ArcTanh[Tanh[a + b*x]]/x^(3/2), x, 2, 4*b*Sqrt[x] - (2*ArcTanh[Tanh[a + b*x]])/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]/x^(5/2), x, 2, (-4*b)/(3*Sqrt[x]) - (2*ArcTanh[Tanh[a + b*x]])/(3*x^(3/2))} -{ArcTanh[Tanh[a + b*x]]/x^(7/2), x, 2, (-4*b)/(15*x^(3/2)) - (2*ArcTanh[Tanh[a + b*x]])/(5*x^(5/2))} - - -{x^(7/2)*ArcTanh[Tanh[a + b*x]]^2, x, 3, (16*b^2*x^(13/2))/1287 - (8*b*x^(11/2)*ArcTanh[Tanh[a + b*x]])/99 + (2*x^(9/2)*ArcTanh[Tanh[a + b*x]]^2)/9} -{x^(5/2)*ArcTanh[Tanh[a + b*x]]^2, x, 3, (16*b^2*x^(11/2))/693 - (8*b*x^(9/2)*ArcTanh[Tanh[a + b*x]])/63 + (2*x^(7/2)*ArcTanh[Tanh[a + b*x]]^2)/7} -{x^(3/2)*ArcTanh[Tanh[a + b*x]]^2, x, 3, (16*b^2*x^(9/2))/315 - (8*b*x^(7/2)*ArcTanh[Tanh[a + b*x]])/35 + (2*x^(5/2)*ArcTanh[Tanh[a + b*x]]^2)/5} -{Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2, x, 3, (16*b^2*x^(7/2))/105 - (8*b*x^(5/2)*ArcTanh[Tanh[a + b*x]])/15 + (2*x^(3/2)*ArcTanh[Tanh[a + b*x]]^2)/3} -{ArcTanh[Tanh[a + b*x]]^2/Sqrt[x], x, 3, (16*b^2*x^(5/2))/15 - (8*b*x^(3/2)*ArcTanh[Tanh[a + b*x]])/3 + 2*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2} -{ArcTanh[Tanh[a + b*x]]^2/x^(3/2), x, 3, (-16*b^2*x^(3/2))/3 + 8*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]] - (2*ArcTanh[Tanh[a + b*x]]^2)/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]^2/x^(5/2), x, 3, (16*b^2*Sqrt[x])/3 - (8*b*ArcTanh[Tanh[a + b*x]])/(3*Sqrt[x]) - (2*ArcTanh[Tanh[a + b*x]]^2)/(3*x^(3/2))} -{ArcTanh[Tanh[a + b*x]]^2/x^(7/2), x, 3, (-16*b^2)/(15*Sqrt[x]) - (8*b*ArcTanh[Tanh[a + b*x]])/(15*x^(3/2)) - (2*ArcTanh[Tanh[a + b*x]]^2)/(5*x^(5/2))} - - -{x^(7/2)*ArcTanh[Tanh[a + b*x]]^3, x, 4, (-32*b^3*x^(15/2))/6435 + (16*b^2*x^(13/2)*ArcTanh[Tanh[a + b*x]])/429 - (4*b*x^(11/2)*ArcTanh[Tanh[a + b*x]]^2)/33 + (2*x^(9/2)*ArcTanh[Tanh[a + b*x]]^3)/9} -{x^(5/2)*ArcTanh[Tanh[a + b*x]]^3, x, 4, (-32*b^3*x^(13/2))/3003 + (16*b^2*x^(11/2)*ArcTanh[Tanh[a + b*x]])/231 - (4*b*x^(9/2)*ArcTanh[Tanh[a + b*x]]^2)/21 + (2*x^(7/2)*ArcTanh[Tanh[a + b*x]]^3)/7} -{x^(3/2)*ArcTanh[Tanh[a + b*x]]^3, x, 4, (-32*b^3*x^(11/2))/1155 + (16*b^2*x^(9/2)*ArcTanh[Tanh[a + b*x]])/105 - (12*b*x^(7/2)*ArcTanh[Tanh[a + b*x]]^2)/35 + (2*x^(5/2)*ArcTanh[Tanh[a + b*x]]^3)/5} -{Sqrt[x]*ArcTanh[Tanh[a + b*x]]^3, x, 4, (-32*b^3*x^(9/2))/315 + (16*b^2*x^(7/2)*ArcTanh[Tanh[a + b*x]])/35 - (4*b*x^(5/2)*ArcTanh[Tanh[a + b*x]]^2)/5 + (2*x^(3/2)*ArcTanh[Tanh[a + b*x]]^3)/3} -{ArcTanh[Tanh[a + b*x]]^3/Sqrt[x], x, 4, (-32*b^3*x^(7/2))/35 + (16*b^2*x^(5/2)*ArcTanh[Tanh[a + b*x]])/5 - 4*b*x^(3/2)*ArcTanh[Tanh[a + b*x]]^2 + 2*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^3} -{ArcTanh[Tanh[a + b*x]]^3/x^(3/2), x, 4, (32*b^3*x^(5/2))/5 - 16*b^2*x^(3/2)*ArcTanh[Tanh[a + b*x]] + 12*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2 - (2*ArcTanh[Tanh[a + b*x]]^3)/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]^3/x^(5/2), x, 4, (-32*b^3*x^(3/2))/3 + 16*b^2*Sqrt[x]*ArcTanh[Tanh[a + b*x]] - (4*b*ArcTanh[Tanh[a + b*x]]^2)/Sqrt[x] - (2*ArcTanh[Tanh[a + b*x]]^3)/(3*x^(3/2))} -{ArcTanh[Tanh[a + b*x]]^3/x^(7/2), x, 4, (32*b^3*Sqrt[x])/5 - (16*b^2*ArcTanh[Tanh[a + b*x]])/(5*Sqrt[x]) - (4*b*ArcTanh[Tanh[a + b*x]]^2)/(5*x^(3/2)) - (2*ArcTanh[Tanh[a + b*x]]^3)/(5*x^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^(7/2)/ArcTanh[Tanh[a + b*x]], x, 5, (2*x^(7/2))/(7*b) + (2*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]]))/(5*b^2) + (2*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(3*b^3) + (2*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/b^4 - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(7/2))/b^(9/2)} -{x^(5/2)/ArcTanh[Tanh[a + b*x]], x, 4, (2*x^(5/2))/(5*b) + (2*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]))/(3*b^2) + (2*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/b^3 - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2))/b^(7/2)} -{x^(3/2)/ArcTanh[Tanh[a + b*x]], x, 3, (2*x^(3/2))/(3*b) + (2*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))/b^2 - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2))/b^(5/2)} -{Sqrt[x]/ArcTanh[Tanh[a + b*x]], x, 2, (2*Sqrt[x])/b - (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])/b^(3/2)} -{1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]), x, 1, (-2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(Sqrt[b]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])} -{1/(x^(3/2)*ArcTanh[Tanh[a + b*x]]), x, 2, (-2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(3/2) + 2/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))} -{1/(x^(5/2)*ArcTanh[Tanh[a + b*x]]), x, 3, (-2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(5/2) + (2*b)/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 2/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{1/(x^(7/2)*ArcTanh[Tanh[a + b*x]]), x, 4, (-2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(7/2) + (2*b^2)/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (2*b)/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 2/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{x^(7/2)/ArcTanh[Tanh[a + b*x]]^2, x, 5, (7*x^(5/2))/(5*b^2) + (7*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]))/(3*b^3) + (7*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/b^4 - (7*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2))/b^(9/2) - x^(7/2)/(b*ArcTanh[Tanh[a + b*x]])} -{x^(5/2)/ArcTanh[Tanh[a + b*x]]^2, x, 4, (5*x^(3/2))/(3*b^2) + (5*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))/b^3 - (5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2))/b^(7/2) - x^(5/2)/(b*ArcTanh[Tanh[a + b*x]])} -{x^(3/2)/ArcTanh[Tanh[a + b*x]]^2, x, 3, (3*Sqrt[x])/b^2 - (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])/b^(5/2) - x^(3/2)/(b*ArcTanh[Tanh[a + b*x]])} -{Sqrt[x]/ArcTanh[Tanh[a + b*x]]^2, x, 2, -(ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]/(b^(3/2)*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])) - Sqrt[x]/(b*ArcTanh[Tanh[a + b*x]])} -{1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2), x, 3, ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]/(Sqrt[b]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - 1/(b*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(b*Sqrt[x]*ArcTanh[Tanh[a + b*x]])} -{1/(x^(3/2)*ArcTanh[Tanh[a + b*x]]^2), x, 4, (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(5/2) - 3/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2) - 1/(b*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(b*x^(3/2)*ArcTanh[Tanh[a + b*x]])} -{1/(x^(5/2)*ArcTanh[Tanh[a + b*x]]^2), x, 5, (5*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(7/2) - (5*b)/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3) - 5/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) - 1/(b*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(b*x^(5/2)*ArcTanh[Tanh[a + b*x]])} -{1/(x^(7/2)*ArcTanh[Tanh[a + b*x]]^2), x, 6, (7*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(b*x - ArcTanh[Tanh[a + b*x]])^(9/2) - (7*b^2)/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^4) - (7*b)/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) - 7/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) - 1/(b*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(b*x^(7/2)*ArcTanh[Tanh[a + b*x]])} - - -{x^(7/2)/ArcTanh[Tanh[a + b*x]]^3, x, 5, (35*x^(3/2))/(12*b^3) + (35*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))/(4*b^4) - (35*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2))/(4*b^(9/2)) - x^(7/2)/(2*b*ArcTanh[Tanh[a + b*x]]^2) - (7*x^(5/2))/(4*b^2*ArcTanh[Tanh[a + b*x]])} -{x^(5/2)/ArcTanh[Tanh[a + b*x]]^3, x, 4, (15*Sqrt[x])/(4*b^3) - (15*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]])/(4*b^(7/2)) - x^(5/2)/(2*b*ArcTanh[Tanh[a + b*x]]^2) - (5*x^(3/2))/(4*b^2*ArcTanh[Tanh[a + b*x]])} -{x^(3/2)/ArcTanh[Tanh[a + b*x]]^3, x, 3, (-3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*b^(5/2)*Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]) - x^(3/2)/(2*b*ArcTanh[Tanh[a + b*x]]^2) - (3*Sqrt[x])/(4*b^2*ArcTanh[Tanh[a + b*x]])} -{Sqrt[x]/ArcTanh[Tanh[a + b*x]]^3, x, 4, ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]]/(4*b^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^(3/2)) - 1/(4*b^2*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])) - Sqrt[x]/(2*b*ArcTanh[Tanh[a + b*x]]^2) - 1/(4*b^2*Sqrt[x]*ArcTanh[Tanh[a + b*x]])} -{1/(Sqrt[x]*ArcTanh[Tanh[a + b*x]]^3), x, 5, (-3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*Sqrt[b]*(b*x - ArcTanh[Tanh[a + b*x]])^(5/2)) + 3/(4*b*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 1/(4*b^2*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(2*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^2) + 1/(4*b^2*x^(3/2)*ArcTanh[Tanh[a + b*x]])} -{1/(x^(3/2)*ArcTanh[Tanh[a + b*x]]^3), x, 6, (-15*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(7/2)) + 15/(4*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3) + 5/(4*b*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 3/(4*b^2*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(2*b*x^(3/2)*ArcTanh[Tanh[a + b*x]]^2) + 3/(4*b^2*x^(5/2)*ArcTanh[Tanh[a + b*x]])} -{1/(x^(5/2)*ArcTanh[Tanh[a + b*x]]^3), x, 7, (-35*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(9/2)) + (35*b)/(4*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^4) + 35/(12*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + 7/(4*b*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 5/(4*b^2*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(2*b*x^(5/2)*ArcTanh[Tanh[a + b*x]]^2) + 5/(4*b^2*x^(7/2)*ArcTanh[Tanh[a + b*x]])} -{1/(x^(7/2)*ArcTanh[Tanh[a + b*x]]^3), x, 8, (-63*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[b*x - ArcTanh[Tanh[a + b*x]]]])/(4*(b*x - ArcTanh[Tanh[a + b*x]])^(11/2)) + (63*b^2)/(4*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^5) + (21*b)/(4*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^4) + 63/(20*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + 9/(4*b*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + 7/(4*b^2*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]])) - 1/(2*b*x^(7/2)*ArcTanh[Tanh[a + b*x]]^2) + 7/(4*b^2*x^(9/2)*ArcTanh[Tanh[a + b*x]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^(m/2) ArcTanh[Tanh[a+b x]]^(n/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcTanh[Tanh[a + b*x]]^(1/2)*x^(3/2), x, 4, -((ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(5/2))) + (1/3)*x^(5/2)*Sqrt[ArcTanh[Tanh[a + b*x]]] - (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(12*b) - (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b^2)} -{ArcTanh[Tanh[a + b*x]]^(1/2)*x^(1/2), x, 3, -((ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(4*b^(3/2))) + (1/2)*x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]] - (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*b)} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(1/2), x, 2, -((ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]))/Sqrt[b]) + Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]]} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(3/2), x, 2, 2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]] - (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(5/2), x, 1, (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(7/2), x, 2, (4*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(15*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(9/2), x, 3, (16*b^2*ArcTanh[Tanh[a + b*x]]^(3/2))/(105*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (8*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(35*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(7*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(1/2)/x^(11/2), x, 4, (32*b^3*ArcTanh[Tanh[a + b*x]]^(3/2))/(315*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^4) + (16*b^2*ArcTanh[Tanh[a + b*x]]^(3/2))/(105*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (4*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(21*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(9*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^(3/2), x, 5, (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^4)/(64*b^(5/2)) - (1/8)*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(32*b) + (3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])/(64*b^2) + (1/4)*x^(5/2)*ArcTanh[Tanh[a + b*x]]^(3/2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)*x^(1/2), x, 4, (ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(3/2)) - (1/4)*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b) + (1/3)*x^(3/2)*ArcTanh[Tanh[a + b*x]]^(3/2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(1/2), x, 3, (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(4*Sqrt[b]) - (3/4)*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (1/2)*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^(3/2)} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(3/2), x, 3, -3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]) + 3*b*Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]] - (2*ArcTanh[Tanh[a + b*x]]^(3/2))/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(5/2), x, 3, 2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]] - (2*b*Sqrt[ArcTanh[Tanh[a + b*x]]])/Sqrt[x] - (2*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*x^(3/2))} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(7/2), x, 1, (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(9/2), x, 2, (4*b*ArcTanh[Tanh[a + b*x]]^(5/2))/(35*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(7*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(11/2), x, 3, (16*b^2*ArcTanh[Tanh[a + b*x]]^(5/2))/(315*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (8*b*ArcTanh[Tanh[a + b*x]]^(5/2))/(63*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(9*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(3/2)/x^(13/2), x, 4, (32*b^3*ArcTanh[Tanh[a + b*x]]^(5/2))/(1155*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^4) + (16*b^2*ArcTanh[Tanh[a + b*x]]^(5/2))/(231*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (4*b*ArcTanh[Tanh[a + b*x]]^(5/2))/(33*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(11*x^(11/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{ArcTanh[Tanh[a + b*x]]^(5/2)*x^(1/2), x, 5, -((5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^4)/(64*b^(3/2))) + (5/32)*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]] - (5*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])/(64*b) - (5/24)*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2) + (1/4)*x^(3/2)*ArcTanh[Tanh[a + b*x]]^(5/2)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(1/2), x, 4, -((5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*Sqrt[b])) + (5/8)*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]] - (5/12)*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2) + (1/3)*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^(5/2)} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(3/2), x, 4, (15/4)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2 - (15/4)*b*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]] + (5/2)*b*Sqrt[x]*ArcTanh[Tanh[a + b*x]]^(3/2) - (2*ArcTanh[Tanh[a + b*x]]^(5/2))/Sqrt[x]} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(5/2), x, 4, -5*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]) + 5*b^2*Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]] - (10*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*Sqrt[x]) - (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(3*x^(3/2))} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(7/2), x, 4, 2*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]] - (2*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/Sqrt[x] - (2*b*ArcTanh[Tanh[a + b*x]]^(3/2))/(3*x^(3/2)) - (2*ArcTanh[Tanh[a + b*x]]^(5/2))/(5*x^(5/2))} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(9/2), x, 1, (2*ArcTanh[Tanh[a + b*x]]^(7/2))/(7*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(11/2), x, 2, (4*b*ArcTanh[Tanh[a + b*x]]^(7/2))/(63*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(7/2))/(9*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(13/2), x, 3, (16*b^2*ArcTanh[Tanh[a + b*x]]^(7/2))/(693*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (8*b*ArcTanh[Tanh[a + b*x]]^(7/2))/(99*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(7/2))/(11*x^(11/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^(5/2)/x^(15/2), x, 4, (32*b^3*ArcTanh[Tanh[a + b*x]]^(7/2))/(3003*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]])^4) + (16*b^2*ArcTanh[Tanh[a + b*x]]^(7/2))/(429*x^(9/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (12*b*ArcTanh[Tanh[a + b*x]]^(7/2))/(143*x^(11/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*ArcTanh[Tanh[a + b*x]]^(7/2))/(13*x^(13/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/ArcTanh[Tanh[a + b*x]]^(1/2)*x^(5/2), x, 4, (5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(7/2)) + (x^(5/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/(3*b) + (5*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(12*b^2) + (5*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b^3)} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)*x^(3/2), x, 3, (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(4*b^(5/2)) + (x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/(2*b) + (3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*b^2)} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)*x^(1/2), x, 2, (ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]))/b^(3/2) + (Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]])/b} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)/x^(1/2), x, 1, (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]])/Sqrt[b]} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)/x^(3/2), x, 1, (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]]))} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)/x^(5/2), x, 2, (4*b*Sqrt[ArcTanh[Tanh[a + b*x]]])/(3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)/x^(7/2), x, 3, (16*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(15*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (8*b*Sqrt[ArcTanh[Tanh[a + b*x]]])/(15*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{1/ArcTanh[Tanh[a + b*x]]^(1/2)/x^(9/2), x, 4, (32*b^3*Sqrt[ArcTanh[Tanh[a + b*x]]])/(35*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^4) + (16*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(35*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^3) + (12*b*Sqrt[ArcTanh[Tanh[a + b*x]]])/(35*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2) + (2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(7*x^(7/2)*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^(7/2), x, 5, (35*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^3)/(8*b^(9/2)) - (2*x^(7/2))/(b*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (7*x^(5/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/(3*b^2) + (35*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(12*b^3) + (35*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])/(8*b^4)} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^(5/2), x, 4, (15*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(4*b^(7/2)) - (2*x^(5/2))/(b*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (5*x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/(2*b^2) + (15*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*b^3)} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^(3/2), x, 3, (3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]))/b^(5/2) - (2*x^(3/2))/(b*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (3*Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^2} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)*x^(1/2), x, 2, (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]])/b^(3/2) - (2*Sqrt[x])/(b*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^(1/2), x, 1, -((2*Sqrt[x])/((b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]]))} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^(3/2), x, 2, -((4*b*Sqrt[x])/((b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])) + 2/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^(5/2), x, 3, -((16*b^2*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (8*b)/(3*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + 2/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(3/2)/x^(7/2), x, 4, -((32*b^3*Sqrt[x])/(5*(b*x - ArcTanh[Tanh[a + b*x]])^4*Sqrt[ArcTanh[Tanh[a + b*x]]])) + (16*b^2)/(5*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (4*b)/(5*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + 2/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])} - - -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^(7/2), x, 5, (35*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]])^2)/(4*b^(9/2)) - (2*x^(7/2))/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2)) - (14*x^(5/2))/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (35*x^(3/2)*Sqrt[ArcTanh[Tanh[a + b*x]]])/(6*b^3) + (35*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*Sqrt[ArcTanh[Tanh[a + b*x]]])/(4*b^4)} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^(5/2), x, 4, (5*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]]*(b*x - ArcTanh[Tanh[a + b*x]]))/b^(7/2) - (2*x^(5/2))/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]]) + (5*Sqrt[x]*Sqrt[ArcTanh[Tanh[a + b*x]]])/b^3} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^(3/2), x, 3, (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[ArcTanh[Tanh[a + b*x]]]])/b^(5/2) - (2*x^(3/2))/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2)) - (2*Sqrt[x])/(b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)*x^(1/2), x, 1, -((2*x^(3/2))/(3*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)))} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^(1/2), x, 2, -((2*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2))) + (4*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^2*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^(3/2), x, 3, -((8*b*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2))) + 2/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) + (16*b*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^3*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^(5/2), x, 4, -((16*b^2*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^(3/2))) + (4*b)/(Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2)) + 2/(3*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) + (32*b^2*Sqrt[x])/(3*(b*x - ArcTanh[Tanh[a + b*x]])^4*Sqrt[ArcTanh[Tanh[a + b*x]]])} -{1/ArcTanh[Tanh[a + b*x]]^(5/2)/x^(7/2), x, 5, -((128*b^3*Sqrt[x])/(15*(b*x - ArcTanh[Tanh[a + b*x]])^4*ArcTanh[Tanh[a + b*x]]^(3/2))) + (32*b^2)/(5*Sqrt[x]*(b*x - ArcTanh[Tanh[a + b*x]])^3*ArcTanh[Tanh[a + b*x]]^(3/2)) + (16*b)/(15*x^(3/2)*(b*x - ArcTanh[Tanh[a + b*x]])^2*ArcTanh[Tanh[a + b*x]]^(3/2)) + 2/(5*x^(5/2)*(b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^(3/2)) + (256*b^3*Sqrt[x])/(15*(b*x - ArcTanh[Tanh[a + b*x]])^5*Sqrt[ArcTanh[Tanh[a + b*x]]])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[Tanh[a+b x]]^n when n symbolic*) - - -{ArcTanh[Tanh[a + b*x]]^n*x^m, x, 1, (1/(b*(1 + n)))*((x^m*ArcTanh[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, -(ArcTanh[Tanh[a + b*x]]/(b*x - ArcTanh[Tanh[a + b*x]]))])/((b*x)/(b*x - ArcTanh[Tanh[a + b*x]]))^m)} - -{ArcTanh[Tanh[a + b*x]]^n*x^4, x, 6, If[$VersionNumber>=8, (x^4*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (12*x^2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (24*x*ArcTanh[Tanh[a + b*x]]^(4 + n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)) + (24*ArcTanh[Tanh[a + b*x]]^(5 + n))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)), (x^4*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (12*x^2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2)) - (24*x*ArcTanh[Tanh[a + b*x]]^(4 + n))/(b^4*(4 + 5*n + n^2)*(6 + 5*n + n^2)) + (24*ArcTanh[Tanh[a + b*x]]^(5 + n))/(b^5*(12 + 7*n + n^2)*(10 + 17*n + 8*n^2 + n^3))]} -{ArcTanh[Tanh[a + b*x]]^n*x^3, x, 5, If[$VersionNumber>=8, (x^3*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (3*x^2*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (6*x*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (6*ArcTanh[Tanh[a + b*x]]^(4 + n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)), (x^3*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (3*x^2*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (6*x*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2)) - (6*ArcTanh[Tanh[a + b*x]]^(4 + n))/(b^4*(4 + 5*n + n^2)*(6 + 5*n + n^2))]} -{ArcTanh[Tanh[a + b*x]]^n*x^2, x, 4, If[$VersionNumber>=8, (x^2*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (2*x*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)), (x^2*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (2*x*ArcTanh[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (2*ArcTanh[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2))]} -{ArcTanh[Tanh[a + b*x]]^n*x^1, x, 3, (x*ArcTanh[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - ArcTanh[Tanh[a + b*x]]^(2 + n)/(b^2*(1 + n)*(2 + n))} -{ArcTanh[Tanh[a + b*x]]^n*x^0, x, 2, ArcTanh[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))} -{ArcTanh[Tanh[a + b*x]]^n/x^1, x, 1, (ArcTanh[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -(ArcTanh[Tanh[a + b*x]]/(b*x - ArcTanh[Tanh[a + b*x]]))])/((1 + n)*(b*x - ArcTanh[Tanh[a + b*x]]))} -{ArcTanh[Tanh[a + b*x]]^n/x^2, x, 2, -(ArcTanh[Tanh[a + b*x]]^n/x) + (b*ArcTanh[Tanh[a + b*x]]^n*Hypergeometric2F1[1, n, 1 + n, -(ArcTanh[Tanh[a + b*x]]/(b*x - ArcTanh[Tanh[a + b*x]]))])/(b*x - ArcTanh[Tanh[a + b*x]])} -{ArcTanh[Tanh[a + b*x]]^n/x^3, x, 3, -((b*n*ArcTanh[Tanh[a + b*x]]^(-1 + n))/(2*x)) - ArcTanh[Tanh[a + b*x]]^n/(2*x^2) + (b^2*n*ArcTanh[Tanh[a + b*x]]^(-1 + n)*Hypergeometric2F1[1, -1 + n, n, -(ArcTanh[Tanh[a + b*x]]/(b*x - ArcTanh[Tanh[a + b*x]]))])/(2*(b*x - ArcTanh[Tanh[a + b*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTanh[Coth[a+b x]]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[Coth[a+b x]]^n*) - - -{ArcCoth[Tanh[a + b*x]]*x^m, x, 2, -((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]])/(1 + m)} - -{ArcTanh[Coth[a + b*x]]*x^2, x, 2, -((b*x^4)/12) + (1/3)*x^3*ArcTanh[Coth[a + b*x]]} -{ArcTanh[Coth[a + b*x]]*x^1, x, 2, -((b*x^3)/6) + (1/2)*x^2*ArcTanh[Coth[a + b*x]]} -{ArcTanh[Coth[a + b*x]]*x^0, x, 2, ArcTanh[Coth[a + b*x]]^2/(2*b)} -{ArcTanh[Coth[a + b*x]]/x^1, x, 2, b*x - (b*x - ArcTanh[Coth[a + b*x]])*Log[x]} -{ArcTanh[Coth[a + b*x]]/x^2, x, 2, -(ArcTanh[Coth[a + b*x]]/x) + b*Log[x]} -{ArcTanh[Coth[a + b*x]]/x^3, x, 2, -(b/(2*x)) - ArcTanh[Coth[a + b*x]]/(2*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Hyper[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Sinh[a+b x]]*) - - -(* {ArcTanh[Sinh[x]], x, 6, 0} -{x*ArcTanh[Sinh[x]], x, 8, 0} -{x^2*ArcTanh[Sinh[x]], x, 10, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Cosh[a+b x]]*) - - -{ArcTanh[Cosh[x]], x, 6, -2*x*ArcTanh[E^x] + x*ArcTanh[Cosh[x]] - PolyLog[2, -E^x] + PolyLog[2, E^x]} -{x*ArcTanh[Cosh[x]], x, 8, (-x^2)*ArcTanh[E^x] + (1/2)*x^2*ArcTanh[Cosh[x]] - x*PolyLog[2, -E^x] + x*PolyLog[2, E^x] + PolyLog[3, -E^x] - PolyLog[3, E^x]} -{x^2*ArcTanh[Cosh[x]], x, 10, (-(2/3))*x^3*ArcTanh[E^x] + (1/3)*x^3*ArcTanh[Cosh[x]] - x^2*PolyLog[2, -E^x] + x^2*PolyLog[2, E^x] + 2*x*PolyLog[3, -E^x] - 2*x*PolyLog[3, E^x] - 2*PolyLog[4, -E^x] + 2*PolyLog[4, E^x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Tanh[a+b x]]*) - - -{ArcTanh[c + d*Tanh[a + b*x]]*x^2, x, 11, (1/3)*x^3*ArcTanh[c + d*Tanh[a + b*x]] + (1/6)*x^3*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/6)*x^3*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x^2*PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b) - (x^2*PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b) - (x*PolyLog[3, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b^2) + (x*PolyLog[3, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b^2) + PolyLog[4, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(8*b^3) - PolyLog[4, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(8*b^3)} -{ArcTanh[c + d*Tanh[a + b*x]]*x^1, x, 9, (1/2)*x^2*ArcTanh[c + d*Tanh[a + b*x]] + (1/4)*x^2*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/4)*x^2*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x*PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b) - (x*PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b) - PolyLog[3, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(8*b^2) + PolyLog[3, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(8*b^2)} -{ArcTanh[c + d*Tanh[a + b*x]]*x^0, x, 7, x*ArcTanh[c + d*Tanh[a + b*x]] + (1/2)*x*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/2)*x*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(4*b) - PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(4*b)} -{ArcTanh[c + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[c + d*Tanh[a + b*x]]/x, x]} - - -{ArcTanh[1 + d + d*Tanh[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcTanh[1 + d + d*Tanh[a + b*x]] - (1/8)*x^4*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + (3*x^2*PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))])/(8*b^2) - (3*x*PolyLog[4, -((1 + d)*E^(2*a + 2*b*x))])/(8*b^3) + (3*PolyLog[5, -((1 + d)*E^(2*a + 2*b*x))])/(16*b^4)} -{ArcTanh[1 + d + d*Tanh[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTanh[1 + d + d*Tanh[a + b*x]] - (1/6)*x^3*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + (x*PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))])/(4*b^2) - PolyLog[4, -((1 + d)*E^(2*a + 2*b*x))]/(8*b^3)} -{ArcTanh[1 + d + d*Tanh[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTanh[1 + d + d*Tanh[a + b*x]] - (1/4)*x^2*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))]/(8*b^2)} -{ArcTanh[1 + d + d*Tanh[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcTanh[1 + d + d*Tanh[a + b*x]] - (1/2)*x*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))]/(4*b)} -{ArcTanh[1 + d + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 + d + d*Tanh[a + b*x]]/x, x]} - - -{ArcTanh[1 - d - d*Tanh[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcTanh[1 - d - d*Tanh[a + b*x]] - (1/8)*x^4*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + (3*x^2*PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))])/(8*b^2) - (3*x*PolyLog[4, -((1 - d)*E^(2*a + 2*b*x))])/(8*b^3) + (3*PolyLog[5, -((1 - d)*E^(2*a + 2*b*x))])/(16*b^4)} -{ArcTanh[1 - d - d*Tanh[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTanh[1 - d - d*Tanh[a + b*x]] - (1/6)*x^3*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + (x*PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))])/(4*b^2) - PolyLog[4, -((1 - d)*E^(2*a + 2*b*x))]/(8*b^3)} -{ArcTanh[1 - d - d*Tanh[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTanh[1 - d - d*Tanh[a + b*x]] - (1/4)*x^2*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))]/(8*b^2)} -{ArcTanh[1 - d - d*Tanh[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcTanh[1 - d - d*Tanh[a + b*x]] - (1/2)*x*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))]/(4*b)} -{ArcTanh[1 - d - d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 - d - d*Tanh[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Coth[a+b x]]*) - - -{ArcTanh[c + d*Coth[a + b*x]]*x^2, x, 11, (1/3)*x^3*ArcTanh[c + d*Coth[a + b*x]] + (1/6)*x^3*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/6)*x^3*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x^2*PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b) - (x^2*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b) - (x*PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b^2) + (x*PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b^2) + PolyLog[4, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(8*b^3) - PolyLog[4, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(8*b^3)} -{ArcTanh[c + d*Coth[a + b*x]]*x^1, x, 9, (1/2)*x^2*ArcTanh[c + d*Coth[a + b*x]] + (1/4)*x^2*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/4)*x^2*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x*PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b) - (x*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b) - PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(8*b^2) + PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(8*b^2)} -{ArcTanh[c + d*Coth[a + b*x]]*x^0, x, 7, x*ArcTanh[c + d*Coth[a + b*x]] + (1/2)*x*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/2)*x*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(4*b) - PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(4*b)} -{ArcTanh[c + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[c + d*Coth[a + b*x]]/x, x]} - - -{ArcTanh[1 + d + d*Coth[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcTanh[1 + d + d*Coth[a + b*x]] - (1/8)*x^4*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + (3*x^2*PolyLog[3, (1 + d)*E^(2*a + 2*b*x)])/(8*b^2) - (3*x*PolyLog[4, (1 + d)*E^(2*a + 2*b*x)])/(8*b^3) + (3*PolyLog[5, (1 + d)*E^(2*a + 2*b*x)])/(16*b^4)} -{ArcTanh[1 + d + d*Coth[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTanh[1 + d + d*Coth[a + b*x]] - (1/6)*x^3*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + (x*PolyLog[3, (1 + d)*E^(2*a + 2*b*x)])/(4*b^2) - PolyLog[4, (1 + d)*E^(2*a + 2*b*x)]/(8*b^3)} -{ArcTanh[1 + d + d*Coth[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTanh[1 + d + d*Coth[a + b*x]] - (1/4)*x^2*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + PolyLog[3, (1 + d)*E^(2*a + 2*b*x)]/(8*b^2)} -{ArcTanh[1 + d + d*Coth[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcTanh[1 + d + d*Coth[a + b*x]] - (1/2)*x*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - PolyLog[2, (1 + d)*E^(2*a + 2*b*x)]/(4*b)} -{ArcTanh[1 + d + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 + d + d*Coth[a + b*x]]/x, x]} - - -{ArcTanh[1 - d - d*Coth[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcTanh[1 - d - d*Coth[a + b*x]] - (1/8)*x^4*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + (3*x^2*PolyLog[3, (1 - d)*E^(2*a + 2*b*x)])/(8*b^2) - (3*x*PolyLog[4, (1 - d)*E^(2*a + 2*b*x)])/(8*b^3) + (3*PolyLog[5, (1 - d)*E^(2*a + 2*b*x)])/(16*b^4)} -{ArcTanh[1 - d - d*Coth[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcTanh[1 - d - d*Coth[a + b*x]] - (1/6)*x^3*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + (x*PolyLog[3, (1 - d)*E^(2*a + 2*b*x)])/(4*b^2) - PolyLog[4, (1 - d)*E^(2*a + 2*b*x)]/(8*b^3)} -{ArcTanh[1 - d - d*Coth[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcTanh[1 - d - d*Coth[a + b*x]] - (1/4)*x^2*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + PolyLog[3, (1 - d)*E^(2*a + 2*b*x)]/(8*b^2)} -{ArcTanh[1 - d - d*Coth[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcTanh[1 - d - d*Coth[a + b*x]] - (1/2)*x*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - PolyLog[2, (1 - d)*E^(2*a + 2*b*x)]/(4*b)} -{ArcTanh[1 - d - d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 - d - d*Coth[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Trig[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Tan[a+b x]]*) - - -{(e + f*x)^3*ArcTanh[Tan[a + b*x]], x, 12, (I*(e + f*x)^4*ArcTan[E^(2*I*(a + b*x))])/(4*f) + ((e + f*x)^4*ArcTanh[Tan[a + b*x]])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (3*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (3*f*(e + f*x)^2*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3) - (3*f^3*PolyLog[5, (-I)*E^(2*I*(a + b*x))])/(16*b^4) + (3*f^3*PolyLog[5, I*E^(2*I*(a + b*x))])/(16*b^4)} -{(e + f*x)^2*ArcTanh[Tan[a + b*x]], x, 10, (I*(e + f*x)^3*ArcTan[E^(2*I*(a + b*x))])/(3*f) + ((e + f*x)^3*ArcTanh[Tan[a + b*x]])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*(e + f*x)*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(4*b^2) - (f*(e + f*x)*PolyLog[3, I*E^(2*I*(a + b*x))])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3)} -{(e + f*x)^1*ArcTanh[Tan[a + b*x]], x, 8, (I*(e + f*x)^2*ArcTan[E^(2*I*(a + b*x))])/(2*f) + ((e + f*x)^2*ArcTanh[Tan[a + b*x]])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (f*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2)} -{(e + f*x)^0*ArcTanh[Tan[a + b*x]], x, 6, I*x*ArcTan[E^(2*I*(a + b*x))] + x*ArcTanh[Tan[a + b*x]] - (I*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b)} -{ArcTanh[Tan[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcTanh[Tan[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcTanh[c + d*Tan[a + b*x]], x, 11, (1/3)*x^3*ArcTanh[c + d*Tan[a + b*x]] + (1/6)*x^3*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/6)*x^3*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*x^2*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*x^2*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b) + (x*PolyLog[3, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b^2) - (x*PolyLog[3, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b^2) + (I*PolyLog[4, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(8*b^3) - (I*PolyLog[4, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(8*b^3)} -{x^1*ArcTanh[c + d*Tan[a + b*x]], x, 9, (1/2)*x^2*ArcTanh[c + d*Tan[a + b*x]] + (1/4)*x^2*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/4)*x^2*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*x*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*x*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b) + PolyLog[3, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))]/(8*b^2) - PolyLog[3, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))]/(8*b^2)} -{x^0*ArcTanh[c + d*Tan[a + b*x]], x, 7, x*ArcTanh[c + d*Tan[a + b*x]] + (1/2)*x*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/2)*x*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b)} -{ArcTanh[c + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[c + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcTanh[1 - I*d + d*Tan[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTanh[1 - I*d + d*Tan[a + b*x]] - (1/6)*x^3*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - (x*PolyLog[3, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b^2) - (I*PolyLog[4, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(8*b^3)} -{x^1*ArcTanh[1 - I*d + d*Tan[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTanh[1 - I*d + d*Tan[a + b*x]] - (1/4)*x^2*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - PolyLog[3, -((1 - I*d)*E^(2*I*a + 2*I*b*x))]/(8*b^2)} -{x^0*ArcTanh[1 - I*d + d*Tan[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTanh[1 - I*d + d*Tan[a + b*x]] - (1/2)*x*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b)} -{ArcTanh[1 - I*d + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 - I*d + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcTanh[1 + I*d - d*Tan[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTanh[1 + I*d - d*Tan[a + b*x]] - (1/6)*x^3*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - (x*PolyLog[3, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b^2) - (I*PolyLog[4, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(8*b^3)} -{x^1*ArcTanh[1 + I*d - d*Tan[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTanh[1 + I*d - d*Tan[a + b*x]] - (1/4)*x^2*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - PolyLog[3, -((1 + I*d)*E^(2*I*a + 2*I*b*x))]/(8*b^2)} -{x^0*ArcTanh[1 + I*d - d*Tan[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTanh[1 + I*d - d*Tan[a + b*x]] - (1/2)*x*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b)} -{ArcTanh[1 + I*d - d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 + I*d - d*Tan[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcTanh[c+d Cot[a+b x]]*) - - -{(e + f*x)^3*ArcTanh[Cot[a + b*x]], x, 12, (I*(e + f*x)^4*ArcTan[E^(2*I*(a + b*x))])/(4*f) + ((e + f*x)^4*ArcTanh[Cot[a + b*x]])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (3*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (3*f*(e + f*x)^2*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3) - (3*f^3*PolyLog[5, (-I)*E^(2*I*(a + b*x))])/(16*b^4) + (3*f^3*PolyLog[5, I*E^(2*I*(a + b*x))])/(16*b^4)} -{(e + f*x)^2*ArcTanh[Cot[a + b*x]], x, 10, (I*(e + f*x)^3*ArcTan[E^(2*I*(a + b*x))])/(3*f) + ((e + f*x)^3*ArcTanh[Cot[a + b*x]])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*(e + f*x)*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(4*b^2) - (f*(e + f*x)*PolyLog[3, I*E^(2*I*(a + b*x))])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3)} -{(e + f*x)^1*ArcTanh[Cot[a + b*x]], x, 8, (I*(e + f*x)^2*ArcTan[E^(2*I*(a + b*x))])/(2*f) + ((e + f*x)^2*ArcTanh[Cot[a + b*x]])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (f*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2)} -{(e + f*x)^0*ArcTanh[Cot[a + b*x]], x, 6, I*x*ArcTan[E^(2*I*(a + b*x))] + x*ArcTanh[Cot[a + b*x]] - (I*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b)} -{ArcTanh[Cot[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcTanh[Cot[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcTanh[c + d*Cot[a + b*x]], x, 11, (1/3)*x^3*ArcTanh[c + d*Cot[a + b*x]] + (1/6)*x^3*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/6)*x^3*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*x^2*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*x^2*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b) + (x*PolyLog[3, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b^2) - (x*PolyLog[3, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b^2) + (I*PolyLog[4, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(8*b^3) - (I*PolyLog[4, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(8*b^3)} -{x^1*ArcTanh[c + d*Cot[a + b*x]], x, 9, (1/2)*x^2*ArcTanh[c + d*Cot[a + b*x]] + (1/4)*x^2*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/4)*x^2*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*x*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*x*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b) + PolyLog[3, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)]/(8*b^2) - PolyLog[3, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)]/(8*b^2)} -{x^0*ArcTanh[c + d*Cot[a + b*x]], x, 7, x*ArcTanh[c + d*Cot[a + b*x]] + (1/2)*x*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/2)*x*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b)} -{ArcTanh[c + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[c + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcTanh[1 + I*d + d*Cot[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTanh[1 + I*d + d*Cot[a + b*x]] - (1/6)*x^3*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - (x*PolyLog[3, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b^2) - (I*PolyLog[4, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(8*b^3)} -{x^1*ArcTanh[1 + I*d + d*Cot[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTanh[1 + I*d + d*Cot[a + b*x]] - (1/4)*x^2*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - PolyLog[3, (1 + I*d)*E^(2*I*a + 2*I*b*x)]/(8*b^2)} -{x^0*ArcTanh[1 + I*d + d*Cot[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTanh[1 + I*d + d*Cot[a + b*x]] - (1/2)*x*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b)} -{ArcTanh[1 + I*d + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 + I*d + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcTanh[1 - I*d - d*Cot[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcTanh[1 - I*d - d*Cot[a + b*x]] - (1/6)*x^3*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - (x*PolyLog[3, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b^2) - (I*PolyLog[4, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(8*b^3)} -{x^1*ArcTanh[1 - I*d - d*Cot[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcTanh[1 - I*d - d*Cot[a + b*x]] - (1/4)*x^2*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - PolyLog[3, (1 - I*d)*E^(2*I*a + 2*I*b*x)]/(8*b^2)} -{x^0*ArcTanh[1 - I*d - d*Cot[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcTanh[1 - I*d - d*Cot[a + b*x]] - (1/2)*x*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b)} -{ArcTanh[1 - I*d - d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcTanh[1 - I*d - d*Cot[a + b*x]]/x, x]} - - -(* ::Title::Closed:: *) -(*Integrands involving inverse hyperbolic tangents of exponentials*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcTanh[E^[a+b x]]^n*) - - -{ArcTanh[E^x], x, 2, (-(1/2))*PolyLog[2, -E^x] + (1/2)*PolyLog[2, E^x]} -{x*ArcTanh[E^x], x, 7, (-(1/2))*x*PolyLog[2, -E^x] + (1/2)*x*PolyLog[2, E^x] + (1/2)*PolyLog[3, -E^x] - (1/2)*PolyLog[3, E^x]} -{x^2*ArcTanh[E^x], x, 9, (-(1/2))*x^2*PolyLog[2, -E^x] + (1/2)*x^2*PolyLog[2, E^x] + x*PolyLog[3, -E^x] - x*PolyLog[3, E^x] - PolyLog[4, -E^x] + PolyLog[4, E^x]} - - -{ArcTanh[E^(a + b*x)], x, 2, -(PolyLog[2, -E^(a + b*x)]/(2*b)) + PolyLog[2, E^(a + b*x)]/(2*b)} -{x*ArcTanh[E^(a + b*x)], x, 7, -((x*PolyLog[2, -E^(a + b*x)])/(2*b)) + (x*PolyLog[2, E^(a + b*x)])/(2*b) + PolyLog[3, -E^(a + b*x)]/(2*b^2) - PolyLog[3, E^(a + b*x)]/(2*b^2)} -{x^2*ArcTanh[E^(a + b*x)], x, 9, -((x^2*PolyLog[2, -E^(a + b*x)])/(2*b)) + (x^2*PolyLog[2, E^(a + b*x)])/(2*b) + (x*PolyLog[3, -E^(a + b*x)])/b^2 - (x*PolyLog[3, E^(a + b*x)])/b^2 - PolyLog[4, -E^(a + b*x)]/b^3 + PolyLog[4, E^(a + b*x)]/b^3} - - -{ArcTanh[a + b*f^(c + d*x)], x, 6, -((ArcTanh[a + b*f^(c + d*x)]*Log[2/(1 + a + b*f^(c + d*x))])/(d*Log[f])) + (ArcTanh[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))])/(d*Log[f]) + PolyLog[2, 1 - 2/(1 + a + b*f^(c + d*x))]/(2*d*Log[f]) - PolyLog[2, 1 - (2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))]/(2*d*Log[f])} -{x*ArcTanh[a + b*f^(c + d*x)], x, 9, (-(1/4))*x^2*Log[1 - a - b*f^(c + d*x)] + (1/4)*x^2*Log[1 + a + b*f^(c + d*x)] + (1/4)*x^2*Log[1 - (b*f^(c + d*x))/(1 - a)] - (1/4)*x^2*Log[1 + (b*f^(c + d*x))/(1 + a)] + (x*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(2*d*Log[f]) - (x*PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f]) - PolyLog[3, (b*f^(c + d*x))/(1 - a)]/(2*d^2*Log[f]^2) + PolyLog[3, -((b*f^(c + d*x))/(1 + a))]/(2*d^2*Log[f]^2)} -{x^2*ArcTanh[a + b*f^(c + d*x)], x, 11, (-(1/6))*x^3*Log[1 - a - b*f^(c + d*x)] + (1/6)*x^3*Log[1 + a + b*f^(c + d*x)] + (1/6)*x^3*Log[1 - (b*f^(c + d*x))/(1 - a)] - (1/6)*x^3*Log[1 + (b*f^(c + d*x))/(1 + a)] + (x^2*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(2*d*Log[f]) - (x^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f]) - (x*PolyLog[3, (b*f^(c + d*x))/(1 - a)])/(d^2*Log[f]^2) + (x*PolyLog[3, -((b*f^(c + d*x))/(1 + a))])/(d^2*Log[f]^2) + PolyLog[4, (b*f^(c + d*x))/(1 - a)]/(d^3*Log[f]^3) - PolyLog[4, -((b*f^(c + d*x))/(1 + a))]/(d^3*Log[f]^3)} - - -(* ::Title::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic tangents*) - - -{E^(c*(a + b*x))*ArcTanh[Sinh[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcTanh[Sinh[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcTanh[Cosh[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcTanh[Cosh[c*(a + b*x)]])/(b*c) + Log[1 - E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcTanh[Tanh[a*c + b*c*x]], x, 3, -(E^(a*c + b*c*x)/(b*c)) + (E^(a*c + b*c*x)*ArcTanh[Tanh[c*(a + b*x)]])/(b*c)} -{E^(c*(a + b*x))*ArcTanh[Coth[a*c + b*c*x]], x, 3, -(E^(a*c + b*c*x)/(b*c)) + (E^(a*c + b*c*x)*ArcTanh[Coth[c*(a + b*x)]])/(b*c)} -{E^(c*(a + b*x))*ArcTanh[Sech[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcTanh[Sech[c*(a + b*x)]])/(b*c) + Log[1 - E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcTanh[Csch[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcTanh[Csch[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c)} - - -{((a + b*ArcTanh[c*x^n])*(d + e*Log[f*x^m]))/x, x, 11, a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) - (b*d*PolyLog[2, (-c)*x^n])/(2*n) - (b*e*Log[f*x^m]*PolyLog[2, (-c)*x^n])/(2*n) + (b*d*PolyLog[2, c*x^n])/(2*n) + (b*e*Log[f*x^m]*PolyLog[2, c*x^n])/(2*n) + (b*e*m*PolyLog[3, (-c)*x^n])/(2*n^2) - (b*e*m*PolyLog[3, c*x^n])/(2*n^2)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.1 Inverse hyperbolic cotangent functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.1 Inverse hyperbolic cotangent functions.m deleted file mode 100644 index 622f77e..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.1 Inverse hyperbolic cotangent functions.m +++ /dev/null @@ -1,612 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands involving inverse hyperbolic cotangents*) - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCoth[a x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[a x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^5*ArcCoth[a*x], x, 4, x/(6*a^5) + x^3/(18*a^3) + x^5/(30*a) + (1/6)*x^6*ArcCoth[a*x] - ArcTanh[a*x]/(6*a^6)} -{x^4*ArcCoth[a*x], x, 4, x^2/(10*a^3) + x^4/(20*a) + (1/5)*x^5*ArcCoth[a*x] + Log[1 - a^2*x^2]/(10*a^5)} -{x^3*ArcCoth[a*x], x, 4, x/(4*a^3) + x^3/(12*a) + (1/4)*x^4*ArcCoth[a*x] - ArcTanh[a*x]/(4*a^4)} -{x^2*ArcCoth[a*x], x, 4, x^2/(6*a) + (1/3)*x^3*ArcCoth[a*x] + Log[1 - a^2*x^2]/(6*a^3)} -{x^1*ArcCoth[a*x], x, 3, x/(2*a) + (1/2)*x^2*ArcCoth[a*x] - ArcTanh[a*x]/(2*a^2)} -{x^0*ArcCoth[a*x], x, 2, x*ArcCoth[a*x] + Log[1 - a^2*x^2]/(2*a)} -{ArcCoth[a*x]/x^1, x, 1, (1/2)*PolyLog[2, -(1/(a*x))] - (1/2)*PolyLog[2, 1/(a*x)]} -{ArcCoth[a*x]/x^2, x, 5, -(ArcCoth[a*x]/x) + a*Log[x] - (1/2)*a*Log[1 - a^2*x^2]} -{ArcCoth[a*x]/x^3, x, 3, -(a/(2*x)) - ArcCoth[a*x]/(2*x^2) + (1/2)*a^2*ArcTanh[a*x]} -{ArcCoth[a*x]/x^4, x, 4, -(a/(6*x^2)) - ArcCoth[a*x]/(3*x^3) + (1/3)*a^3*Log[x] - (1/6)*a^3*Log[1 - a^2*x^2]} -{ArcCoth[a*x]/x^5, x, 4, -(a/(12*x^3)) - a^3/(4*x) - ArcCoth[a*x]/(4*x^4) + (1/4)*a^4*ArcTanh[a*x]} - - -{x^5*ArcCoth[a*x]^2, x, 15, (4*x^2)/(45*a^4) + x^4/(60*a^2) + (x*ArcCoth[a*x])/(3*a^5) + (x^3*ArcCoth[a*x])/(9*a^3) + (x^5*ArcCoth[a*x])/(15*a) - ArcCoth[a*x]^2/(6*a^6) + (1/6)*x^6*ArcCoth[a*x]^2 + (23*Log[1 - a^2*x^2])/(90*a^6)} -{x^4*ArcCoth[a*x]^2, x, 14, (3*x)/(10*a^4) + x^3/(30*a^2) + (x^2*ArcCoth[a*x])/(5*a^3) + (x^4*ArcCoth[a*x])/(10*a) + ArcCoth[a*x]^2/(5*a^5) + (1/5)*x^5*ArcCoth[a*x]^2 - (3*ArcTanh[a*x])/(10*a^5) - (2*ArcCoth[a*x]*Log[2/(1 - a*x)])/(5*a^5) - PolyLog[2, 1 - 2/(1 - a*x)]/(5*a^5)} -{x^3*ArcCoth[a*x]^2, x, 10, x^2/(12*a^2) + (x*ArcCoth[a*x])/(2*a^3) + (x^3*ArcCoth[a*x])/(6*a) - ArcCoth[a*x]^2/(4*a^4) + (1/4)*x^4*ArcCoth[a*x]^2 + Log[1 - a^2*x^2]/(3*a^4)} -{x^2*ArcCoth[a*x]^2, x, 9, x/(3*a^2) + (x^2*ArcCoth[a*x])/(3*a) + ArcCoth[a*x]^2/(3*a^3) + (1/3)*x^3*ArcCoth[a*x]^2 - ArcTanh[a*x]/(3*a^3) - (2*ArcCoth[a*x]*Log[2/(1 - a*x)])/(3*a^3) - PolyLog[2, 1 - 2/(1 - a*x)]/(3*a^3)} -{x^1*ArcCoth[a*x]^2, x, 5, (x*ArcCoth[a*x])/a - ArcCoth[a*x]^2/(2*a^2) + (1/2)*x^2*ArcCoth[a*x]^2 + Log[1 - a^2*x^2]/(2*a^2)} -{x^0*ArcCoth[a*x]^2, x, 5, ArcCoth[a*x]^2/a + x*ArcCoth[a*x]^2 - (2*ArcCoth[a*x]*Log[2/(1 - a*x)])/a - PolyLog[2, 1 - 2/(1 - a*x)]/a} -{ArcCoth[a*x]^2/x^1, x, 6, 2*ArcCoth[a*x]^2*ArcCoth[1 - 2/(1 - a*x)] + ArcCoth[a*x]*PolyLog[2, 1 - 2/(1 + a*x)] - ArcCoth[a*x]*PolyLog[2, 1 - (2*a*x)/(1 + a*x)] + (1/2)*PolyLog[3, 1 - 2/(1 + a*x)] - (1/2)*PolyLog[3, 1 - (2*a*x)/(1 + a*x)]} -{ArcCoth[a*x]^2/x^2, x, 4, a*ArcCoth[a*x]^2 - ArcCoth[a*x]^2/x + 2*a*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - a*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcCoth[a*x]^2/x^3, x, 8, -((a*ArcCoth[a*x])/x) + (1/2)*a^2*ArcCoth[a*x]^2 - ArcCoth[a*x]^2/(2*x^2) + a^2*Log[x] - (1/2)*a^2*Log[1 - a^2*x^2]} -{ArcCoth[a*x]^2/x^4, x, 8, -(a^2/(3*x)) - (a*ArcCoth[a*x])/(3*x^2) + (1/3)*a^3*ArcCoth[a*x]^2 - ArcCoth[a*x]^2/(3*x^3) + (1/3)*a^3*ArcTanh[a*x] + (2/3)*a^3*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - (1/3)*a^3*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcCoth[a*x]^2/x^5, x, 13, -(a^2/(12*x^2)) - (a*ArcCoth[a*x])/(6*x^3) - (a^3*ArcCoth[a*x])/(2*x) + (1/4)*a^4*ArcCoth[a*x]^2 - ArcCoth[a*x]^2/(4*x^4) + (2/3)*a^4*Log[x] - (1/3)*a^4*Log[1 - a^2*x^2]} - - -{x^5*ArcCoth[a*x]^3, x, 33, (19*x)/(60*a^5) + x^3/(60*a^3) + (4*x^2*ArcCoth[a*x])/(15*a^4) + (x^4*ArcCoth[a*x])/(20*a^2) + (23*ArcCoth[a*x]^2)/(30*a^6) + (x*ArcCoth[a*x]^2)/(2*a^5) + (x^3*ArcCoth[a*x]^2)/(6*a^3) + (x^5*ArcCoth[a*x]^2)/(10*a) - ArcCoth[a*x]^3/(6*a^6) + (1/6)*x^6*ArcCoth[a*x]^3 - (19*ArcTanh[a*x])/(60*a^6) - (23*ArcCoth[a*x]*Log[2/(1 - a*x)])/(15*a^6) - (23*PolyLog[2, 1 - 2/(1 - a*x)])/(30*a^6)} -{x^4*ArcCoth[a*x]^3, x, 22, x^2/(20*a^3) + (9*x*ArcCoth[a*x])/(10*a^4) + (x^3*ArcCoth[a*x])/(10*a^2) - (9*ArcCoth[a*x]^2)/(20*a^5) + (3*x^2*ArcCoth[a*x]^2)/(10*a^3) + (3*x^4*ArcCoth[a*x]^2)/(20*a) + ArcCoth[a*x]^3/(5*a^5) + (1/5)*x^5*ArcCoth[a*x]^3 - (3*ArcCoth[a*x]^2*Log[2/(1 - a*x)])/(5*a^5) + Log[1 - a^2*x^2]/(2*a^5) - (3*ArcCoth[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(5*a^5) + (3*PolyLog[3, 1 - 2/(1 - a*x)])/(10*a^5)} -{x^3*ArcCoth[a*x]^3, x, 18, x/(4*a^3) + (x^2*ArcCoth[a*x])/(4*a^2) + ArcCoth[a*x]^2/a^4 + (3*x*ArcCoth[a*x]^2)/(4*a^3) + (x^3*ArcCoth[a*x]^2)/(4*a) - ArcCoth[a*x]^3/(4*a^4) + (1/4)*x^4*ArcCoth[a*x]^3 - ArcTanh[a*x]/(4*a^4) - (2*ArcCoth[a*x]*Log[2/(1 - a*x)])/a^4 - PolyLog[2, 1 - 2/(1 - a*x)]/a^4} -{x^2*ArcCoth[a*x]^3, x, 11, (x*ArcCoth[a*x])/a^2 - ArcCoth[a*x]^2/(2*a^3) + (x^2*ArcCoth[a*x]^2)/(2*a) + ArcCoth[a*x]^3/(3*a^3) + (1/3)*x^3*ArcCoth[a*x]^3 - (ArcCoth[a*x]^2*Log[2/(1 - a*x)])/a^3 + Log[1 - a^2*x^2]/(2*a^3) - (ArcCoth[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a^3 + PolyLog[3, 1 - 2/(1 - a*x)]/(2*a^3)} -{x^1*ArcCoth[a*x]^3, x, 8, (3*ArcCoth[a*x]^2)/(2*a^2) + (3*x*ArcCoth[a*x]^2)/(2*a) - ArcCoth[a*x]^3/(2*a^2) + (1/2)*x^2*ArcCoth[a*x]^3 - (3*ArcCoth[a*x]*Log[2/(1 - a*x)])/a^2 - (3*PolyLog[2, 1 - 2/(1 - a*x)])/(2*a^2)} -{x^0*ArcCoth[a*x]^3, x, 5, ArcCoth[a*x]^3/a + x*ArcCoth[a*x]^3 - (3*ArcCoth[a*x]^2*Log[2/(1 - a*x)])/a - (3*ArcCoth[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/a + (3*PolyLog[3, 1 - 2/(1 - a*x)])/(2*a)} -{ArcCoth[a*x]^3/x^1, x, 8, 2*ArcCoth[a*x]^3*ArcCoth[1 - 2/(1 - a*x)] + (3/2)*ArcCoth[a*x]^2*PolyLog[2, 1 - 2/(1 + a*x)] - (3/2)*ArcCoth[a*x]^2*PolyLog[2, 1 - (2*a*x)/(1 + a*x)] + (3/2)*ArcCoth[a*x]*PolyLog[3, 1 - 2/(1 + a*x)] - (3/2)*ArcCoth[a*x]*PolyLog[3, 1 - (2*a*x)/(1 + a*x)] + (3/4)*PolyLog[4, 1 - 2/(1 + a*x)] - (3/4)*PolyLog[4, 1 - (2*a*x)/(1 + a*x)]} -{ArcCoth[a*x]^3/x^2, x, 5, a*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/x + 3*a*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - 3*a*ArcCoth[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (3/2)*a*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcCoth[a*x]^3/x^3, x, 7, (3/2)*a^2*ArcCoth[a*x]^2 - (3*a*ArcCoth[a*x]^2)/(2*x) + (1/2)*a^2*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/(2*x^2) + 3*a^2*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - (3/2)*a^2*PolyLog[2, -1 + 2/(1 + a*x)]} -{ArcCoth[a*x]^3/x^4, x, 14, -((a^2*ArcCoth[a*x])/x) + (1/2)*a^3*ArcCoth[a*x]^2 - (a*ArcCoth[a*x]^2)/(2*x^2) + (1/3)*a^3*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/(3*x^3) + a^3*Log[x] - (1/2)*a^3*Log[1 - a^2*x^2] + a^3*ArcCoth[a*x]^2*Log[2 - 2/(1 + a*x)] - a^3*ArcCoth[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] - (1/2)*a^3*PolyLog[3, -1 + 2/(1 + a*x)]} -{ArcCoth[a*x]^3/x^5, x, 16, -(a^3/(4*x)) - (a^2*ArcCoth[a*x])/(4*x^2) + a^4*ArcCoth[a*x]^2 - (a*ArcCoth[a*x]^2)/(4*x^3) - (3*a^3*ArcCoth[a*x]^2)/(4*x) + (1/4)*a^4*ArcCoth[a*x]^3 - ArcCoth[a*x]^3/(4*x^4) + (1/4)*a^4*ArcTanh[a*x] + 2*a^4*ArcCoth[a*x]*Log[2 - 2/(1 + a*x)] - a^4*PolyLog[2, -1 + 2/(1 + a*x)]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCoth[c x])^n*) - - -{ArcCoth[c*x]^2/(d + e*x), x, 1, -((ArcCoth[c*x]^2*Log[2/(1 + c*x)])/e) + (ArcCoth[c*x]^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (ArcCoth[c*x]*PolyLog[2, 1 - 2/(1 + c*x)])/e - (ArcCoth[c*x]*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + PolyLog[3, 1 - 2/(1 + c*x)]/(2*e) - PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))]/(2*e)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcCoth[a x] (c+d x^2)^p*) - - -{ArcCoth[a*x]*(c + d*x^2)^4, x, 4, (d*(420*a^6*c^3 + 378*a^4*c^2*d + 180*a^2*c*d^2 + 35*d^3)*x^2)/(630*a^7) + (d^2*(378*a^4*c^2 + 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) + (d^3*(36*a^2*c + 7*d)*x^6)/(378*a^3) + (d^4*x^8)/(72*a) + c^4*x*ArcCoth[a*x] + (4/3)*c^3*d*x^3*ArcCoth[a*x] + (6/5)*c^2*d^2*x^5*ArcCoth[a*x] + (4/7)*c*d^3*x^7*ArcCoth[a*x] + (1/9)*d^4*x^9*ArcCoth[a*x] + ((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*Log[1 - a^2*x^2])/(630*a^9)} -{ArcCoth[a*x]*(c + d*x^2)^3, x, 4, (d*(35*a^4*c^2 + 21*a^2*c*d + 5*d^2)*x^2)/(70*a^5) + (d^2*(21*a^2*c + 5*d)*x^4)/(140*a^3) + (d^3*x^6)/(42*a) + c^3*x*ArcCoth[a*x] + c^2*d*x^3*ArcCoth[a*x] + (3/5)*c*d^2*x^5*ArcCoth[a*x] + (1/7)*d^3*x^7*ArcCoth[a*x] + ((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Log[1 - a^2*x^2])/(70*a^7)} -{ArcCoth[a*x]*(c + d*x^2)^2, x, 5, (d*(10*a^2*c + 3*d)*x^2)/(30*a^3) + (d^2*x^4)/(20*a) + c^2*x*ArcCoth[a*x] + (2/3)*c*d*x^3*ArcCoth[a*x] + (1/5)*d^2*x^5*ArcCoth[a*x] + ((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Log[1 - a^2*x^2])/(30*a^5)} -{ArcCoth[a*x]*(c + d*x^2)^1, x, 5, (d*x^2)/(6*a) + c*x*ArcCoth[a*x] + (1/3)*d*x^3*ArcCoth[a*x] + ((3*a^2*c + d)*Log[1 - a^2*x^2])/(6*a^3)} -{ArcCoth[a*x]/(c + d*x^2)^1, x, 27, -((ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 - 1/(a*x)])/(2*Sqrt[c]*Sqrt[d])) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[1 + 1/(a*x)])/(2*Sqrt[c]*Sqrt[d]) + (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[-((2*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x)))])/(2*Sqrt[c]*Sqrt[d]) - (ArcTan[(Sqrt[d]*x)/Sqrt[c]]*Log[(2*Sqrt[c]*Sqrt[d]*(1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(2*Sqrt[c]*Sqrt[d]) - (I*PolyLog[2, 1 + (2*Sqrt[c]*Sqrt[d]*(1 - a*x))/((I*a*Sqrt[c] - Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(4*Sqrt[c]*Sqrt[d]) + (I*PolyLog[2, 1 - (2*Sqrt[c]*Sqrt[d]*(1 + a*x))/((I*a*Sqrt[c] + Sqrt[d])*(Sqrt[c] - I*Sqrt[d]*x))])/(4*Sqrt[c]*Sqrt[d])} -{ArcCoth[a*x]/(c + d*x^2)^2, x, If[$VersionNumber<11, 24, 25], (x*ArcCoth[a*x])/(2*c*(c + d*x^2)) + (ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*Sqrt[d]) + (I*Log[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) - (I*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) - (I*Log[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) + (I*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/2)*Sqrt[d]) + (a*Log[1 - a^2*x^2])/(4*c*(a^2*c + d)) - (a*Log[c + d*x^2])/(4*c*(a^2*c + d)) + (I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) - (I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) + (I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d]) - (I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(8*c^(3/2)*Sqrt[d])} -{ArcCoth[a*x]/(c + d*x^2)^3, x, 23, a/(8*c*(a^2*c + d)*(c + d*x^2)) + (x*ArcCoth[a*x])/(4*c*(c + d*x^2)^2) + (3*x*ArcCoth[a*x])/(8*c^2*(c + d*x^2)) + (3*ArcCoth[a*x]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*Sqrt[d]) + (3*I*Log[(Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) - (3*I*Log[-((Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 - (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) - (3*I*Log[-((Sqrt[d]*(1 - a*x))/(I*a*Sqrt[c] - Sqrt[d]))]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) + (3*I*Log[(Sqrt[d]*(1 + a*x))/(I*a*Sqrt[c] + Sqrt[d])]*Log[1 + (I*Sqrt[d]*x)/Sqrt[c]])/(32*c^(5/2)*Sqrt[d]) + (a*(5*a^2*c + 3*d)*Log[1 - a^2*x^2])/(16*c^2*(a^2*c + d)^2) - (a*(5*a^2*c + 3*d)*Log[c + d*x^2])/(16*c^2*(a^2*c + d)^2) + (3*I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) - (3*I*PolyLog[2, (a*(Sqrt[c] - I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) + (3*I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] - I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d]) - (3*I*PolyLog[2, (a*(Sqrt[c] + I*Sqrt[d]*x))/(a*Sqrt[c] + I*Sqrt[d])])/(32*c^(5/2)*Sqrt[d])} - - -{ArcCoth[a*x]*(c + d*x^2)^(1/2), x, 0, Unintegrable[Sqrt[c + d*x^2]*ArcCoth[a*x], x]} -{ArcCoth[a*x]/(c + d*x^2)^(1/2), x, 0, Unintegrable[ArcCoth[a*x]/Sqrt[c + d*x^2], x]} -{ArcCoth[a*x]/(c + d*x^2)^(3/2), x, 5, (x*ArcCoth[a*x])/(c*Sqrt[c + d*x^2]) - ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]]/(c*Sqrt[a^2*c + d])} -{ArcCoth[a*x]/(c + d*x^2)^(5/2), x, 7, a/(3*c*(a^2*c + d)*Sqrt[c + d*x^2]) + (x*ArcCoth[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCoth[a*x])/(3*c^2*Sqrt[c + d*x^2]) - ((3*a^2*c + 2*d)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(3*c^2*(a^2*c + d)^(3/2))} -{ArcCoth[a*x]/(c + d*x^2)^(7/2), x, 8, a/(15*c*(a^2*c + d)*(c + d*x^2)^(3/2)) + (a*(7*a^2*c + 4*d))/(15*c^2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcCoth[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCoth[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCoth[a*x])/(15*c^3*Sqrt[c + d*x^2]) - ((15*a^4*c^2 + 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(15*c^3*(a^2*c + d)^(5/2))} -{ArcCoth[a*x]/(c + d*x^2)^(9/2), x, 8, a/(35*c*(a^2*c + d)*(c + d*x^2)^(5/2)) + (a*(11*a^2*c + 6*d))/(105*c^2*(a^2*c + d)^2*(c + d*x^2)^(3/2)) + (a*(19*a^4*c^2 + 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c + d)^3*Sqrt[c + d*x^2]) + (x*ArcCoth[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcCoth[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCoth[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcCoth[a*x])/(35*c^4*Sqrt[c + d*x^2]) - ((35*a^6*c^3 + 70*a^4*c^2*d + 56*a^2*c*d^2 + 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(35*c^4*(a^2*c + d)^(7/2))} - - -{ArcCoth[x]*(a - a*x^2)^(1/2), x, 3, (1/2)*Sqrt[a - a*x^2] + (1/2)*x*Sqrt[a - a*x^2]*ArcCoth[x] - (a*Sqrt[1 - x^2]*ArcCoth[x]*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/Sqrt[a - a*x^2] - (I*a*Sqrt[1 - x^2]*PolyLog[2, -((I*Sqrt[1 - x])/Sqrt[1 + x])])/(2*Sqrt[a - a*x^2]) + (I*a*Sqrt[1 - x^2]*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]])/(2*Sqrt[a - a*x^2])} -{ArcCoth[x]/(a - a*x^2)^(1/2), x, 2, -((2*Sqrt[1 - x^2]*ArcCoth[x]*ArcTan[Sqrt[1 - x]/Sqrt[1 + x]])/Sqrt[a - a*x^2]) - (I*Sqrt[1 - x^2]*PolyLog[2, -((I*Sqrt[1 - x])/Sqrt[1 + x])])/Sqrt[a - a*x^2] + (I*Sqrt[1 - x^2]*PolyLog[2, (I*Sqrt[1 - x])/Sqrt[1 + x]])/Sqrt[a - a*x^2]} -{ArcCoth[x]/(a - a*x^2)^(3/2), x, 1, -(1/(a*Sqrt[a - a*x^2])) + (x*ArcCoth[x])/(a*Sqrt[a - a*x^2])} -{ArcCoth[x]/(a - a*x^2)^(5/2), x, 2, -(1/(9*a*(a - a*x^2)^(3/2))) - 2/(3*a^2*Sqrt[a - a*x^2]) + (x*ArcCoth[x])/(3*a*(a - a*x^2)^(3/2)) + (2*x*ArcCoth[x])/(3*a^2*Sqrt[a - a*x^2])} -{ArcCoth[x]/(a - a*x^2)^(7/2), x, 3, -(1/(25*a*(a - a*x^2)^(5/2))) - 4/(45*a^2*(a - a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a - a*x^2]) + (x*ArcCoth[x])/(5*a*(a - a*x^2)^(5/2)) + (4*x*ArcCoth[x])/(15*a^2*(a - a*x^2)^(3/2)) + (8*x*ArcCoth[x])/(15*a^3*Sqrt[a - a*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcCoth[a x]^m (c+d x^2)^n*) - - -{1/((1 - x^2)*ArcCoth[x]), x, 1, Log[ArcCoth[x]]} -{ArcCoth[x]^n/(1 - x^2), x, 1, ArcCoth[x]^(1 + n)/(1 + n)} -{ArcCoth[x]^2/(1 - x^2)^2, x, 4, x/(4*(1 - x^2)) - ArcCoth[x]/(2*(1 - x^2)) + (x*ArcCoth[x]^2)/(2*(1 - x^2)) + ArcCoth[x]^3/6 + ArcTanh[x]/4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[a x] (c+d x^2)^n*) - - -{x^1*ArcCoth[x]/(1 - x^2), x, 4, (-(1/2))*ArcCoth[x]^2 + ArcCoth[x]*Log[2/(1 - x)] + (1/2)*PolyLog[2, (1 + x)/(-1 + x)]} -{x^0*ArcCoth[x]/(1 - x^2), x, 1, ArcCoth[x]^2/2} - - -{x^1*ArcCoth[x]/(1 - x^2)^2, x, 3, -(x/(4*(1 - x^2))) + ArcCoth[x]/(2*(1 - x^2)) - ArcTanh[x]/4} -{x^0*ArcCoth[x]/(1 - x^2)^2, x, 2, -(1/(4*(1 - x^2))) + (x*ArcCoth[x])/(2*(1 - x^2)) + ArcCoth[x]^2/4} - - -{x^1*ArcCoth[x]/(1 - x^2)^3, x, 4, -(x/(16*(1 - x^2)^2)) - (3*x)/(32*(1 - x^2)) + ArcCoth[x]/(4*(1 - x^2)^2) - (3*ArcTanh[x])/32} -{x^0*ArcCoth[x]/(1 - x^2)^3, x, 3, -(1/(16*(1 - x^2)^2)) - 3/(16*(1 - x^2)) + (x*ArcCoth[x])/(4*(1 - x^2)^2) + (3*x*ArcCoth[x])/(8*(1 - x^2)) + (3*ArcCoth[x]^2)/16} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCoth[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[a+b x]^n*) - - -{x^3*ArcCoth[a + b*x], x, 7, ((1 + 6*a^2)*x)/(4*b^3) - (a*(a + b*x)^2)/(2*b^4) + (a + b*x)^3/(12*b^4) + (1/4)*x^4*ArcCoth[a + b*x] + ((1 - a)^4*Log[1 - a - b*x])/(8*b^4) - ((1 + a)^4*Log[1 + a + b*x])/(8*b^4)} -{x^2*ArcCoth[a + b*x], x, 7, -((a*x)/b^2) + (a + b*x)^2/(6*b^3) + (1/3)*x^3*ArcCoth[a + b*x] + ((1 - a)^3*Log[1 - a - b*x])/(6*b^3) + ((1 + a)^3*Log[1 + a + b*x])/(6*b^3)} -{x^1*ArcCoth[a + b*x], x, 7, x/(2*b) + (1/2)*x^2*ArcCoth[a + b*x] + ((1 - a)^2*Log[1 - a - b*x])/(4*b^2) - ((1 + a)^2*Log[1 + a + b*x])/(4*b^2)} -{x^0*ArcCoth[a + b*x], x, 3, ((a + b*x)*ArcCoth[a + b*x])/b + Log[1 - (a + b*x)^2]/(2*b)} -{ArcCoth[a + b*x]/x^1, x, 5, (-ArcCoth[a + b*x])*Log[2/(1 + a + b*x)] + ArcCoth[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))] + (1/2)*PolyLog[2, 1 - 2/(1 + a + b*x)] - (1/2)*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))]} -{ArcCoth[a + b*x]/x^2, x, 7, -(ArcCoth[a + b*x]/x) + (b*Log[x])/(1 - a^2) - (b*Log[1 - a - b*x])/(2*(1 - a)) - (b*Log[1 + a + b*x])/(2*(1 + a))} -{ArcCoth[a + b*x]/x^3, x, 5, -(b/(2*(1 - a^2)*x)) - ArcCoth[a + b*x]/(2*x^2) + (a*b^2*Log[x])/(1 - a^2)^2 - (b^2*Log[1 - a - b*x])/(4*(1 - a)^2) + (b^2*Log[1 + a + b*x])/(4*(1 + a)^2)} - - -{x^3*ArcCoth[a + b*x]^2, x, 19, -((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((1 + 6*a^2)*(a + b*x)*ArcCoth[a + b*x])/(2*b^4) - (a*(a + b*x)^2*ArcCoth[a + b*x])/b^4 + ((a + b*x)^3*ArcCoth[a + b*x])/(6*b^4) - (a*(1 + a^2)*ArcCoth[a + b*x]^2)/b^4 - ((1 + 6*a^2 + a^4)*ArcCoth[a + b*x]^2)/(4*b^4) + (1/4)*x^4*ArcCoth[a + b*x]^2 + (a*ArcTanh[a + b*x])/b^4 + (2*a*(1 + a^2)*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/b^4 + Log[1 - (a + b*x)^2]/(12*b^4) + ((1 + 6*a^2)*Log[1 - (a + b*x)^2])/(4*b^4) + (a*(1 + a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^4} -{x^2*ArcCoth[a + b*x]^2, x, 15, x/(3*b^2) - (2*a*(a + b*x)*ArcCoth[a + b*x])/b^3 + ((a + b*x)^2*ArcCoth[a + b*x])/(3*b^3) + (a*(3 + a^2)*ArcCoth[a + b*x]^2)/(3*b^3) + ((1 + 3*a^2)*ArcCoth[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcCoth[a + b*x]^2 - ArcTanh[a + b*x]/(3*b^3) - (2*(1 + 3*a^2)*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/(3*b^3) - (a*Log[1 - (a + b*x)^2])/b^3 - ((1 + 3*a^2)*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(3*b^3)} -{x^1*ArcCoth[a + b*x]^2, x, 12, ((a + b*x)*ArcCoth[a + b*x])/b^2 - (a*ArcCoth[a + b*x]^2)/b^2 - ((1 + a^2)*ArcCoth[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcCoth[a + b*x]^2 + (2*a*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/b^2 + Log[1 - (a + b*x)^2]/(2*b^2) + (a*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/b^2} -{x^0*ArcCoth[a + b*x]^2, x, 6, ArcCoth[a + b*x]^2/b + ((a + b*x)*ArcCoth[a + b*x]^2)/b - (2*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/b - PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))]/b} -{ArcCoth[a + b*x]^2/x^1, x, 2, (-ArcCoth[a + b*x]^2)*Log[2/(1 + a + b*x)] + ArcCoth[a + b*x]^2*Log[(2*b*x)/((1 - a)*(1 + a + b*x))] + ArcCoth[a + b*x]*PolyLog[2, 1 - 2/(1 + a + b*x)] - ArcCoth[a + b*x]*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))] + (1/2)*PolyLog[3, 1 - 2/(1 + a + b*x)] - (1/2)*PolyLog[3, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))]} -{ArcCoth[a + b*x]^2/x^2, x, 17, -(ArcCoth[a + b*x]^2/x) + (b*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/(1 - a) + (b*ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(1 + a) - (2*b*ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(1 - a^2) + (2*b*ArcCoth[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2) + (b*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(2*(1 - a)) - (b*PolyLog[2, 1 - 2/(1 + a + b*x)])/(2*(1 + a)) + (b*PolyLog[2, 1 - 2/(1 + a + b*x)])/(1 - a^2) - (b*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)} -{ArcCoth[a + b*x]^2/x^3, x, 21, -((b*ArcCoth[a + b*x])/((1 - a^2)*x)) - ArcCoth[a + b*x]^2/(2*x^2) + (b^2*Log[x])/(1 - a^2)^2 + (b^2*ArcCoth[a + b*x]*Log[2/(1 - a - b*x)])/(2*(1 - a)^2) - (b^2*Log[1 - a - b*x])/(2*(1 - a)^2*(1 + a)) - (b^2*ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(2*(1 + a)^2) - (2*a*b^2*ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/(1 - a^2)^2 + (2*a*b^2*ArcCoth[a + b*x]*Log[(2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)^2 - (b^2*Log[1 + a + b*x])/(2*(1 - a)*(1 + a)^2) + (b^2*PolyLog[2, -((1 + a + b*x)/(1 - a - b*x))])/(4*(1 - a)^2) + (b^2*PolyLog[2, 1 - 2/(1 + a + b*x)])/(4*(1 + a)^2) + (a*b^2*PolyLog[2, 1 - 2/(1 + a + b*x)])/(1 - a^2)^2 - (a*b^2*PolyLog[2, 1 - (2*b*x)/((1 - a)*(1 + a + b*x))])/(1 - a^2)^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcCoth[a+b x] / (c+d x^n)*) - - -(* {ArcCoth[a + b*x]/(c + d*x^3), x, 51, (Log[(d^(1/3)*(1 - a - b*x))/(b*c^(1/3) + (1 - a)*d^(1/3))]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - (Log[-((d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^(1/3)))]*Log[-c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*Log[-((d^(1/3)*(1 - a - b*x))/((-1)^(1/3)*b*c^(1/3) - (1 - a)*d^(1/3)))]*Log[(-1)^(1/3)*c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/3)*Log[(d^(1/3)*(1 + a + b*x))/((-1)^(1/3)*b*c^(1/3) + (1 + a)*d^(1/3))]*Log[(-1)^(1/3)*c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*Log[(d^(1/3)*(1 - a - b*x))/((-1)^(2/3)*b*c^(1/3) + (1 - a)*d^(1/3))]*Log[(-(-1)^(2/3))*c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - ((-1)^(2/3)*Log[-((d^(1/3)*(1 + a + b*x))/((-1)^(2/3)*b*c^(1/3) - (1 + a)*d^(1/3)))]*Log[(-(-1)^(2/3))*c^(1/3) - d^(1/3)*x])/(6*c^(2/3)*d^(1/3)) - (Log[-c^(1/3) - d^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/3)*Log[(-1)^(1/3)*c^(1/3) - d^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) - ((-1)^(2/3)*Log[(-(-1)^(2/3))*c^(1/3) - d^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) + (Log[-c^(1/3) - d^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*Log[(-1)^(1/3)*c^(1/3) - d^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*Log[(-(-1)^(2/3))*c^(1/3) - d^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(2/3)*d^(1/3)) + PolyLog[2, (b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (1 - a)*d^(1/3))]/(6*c^(2/3)*d^(1/3)) - PolyLog[2, (b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (1 + a)*d^(1/3))]/(6*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*PolyLog[2, (b*((-1)^(2/3)*c^(1/3) + d^(1/3)*x))/((-1)^(2/3)*b*c^(1/3) + (1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(2/3)*PolyLog[2, (b*((-1)^(2/3)*c^(1/3) + d^(1/3)*x))/((-1)^(2/3)*b*c^(1/3) - (1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3)*b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/((-1)^(1/3)*b*c^(1/3) - (1 - a)*d^(1/3))])/(6*c^(2/3)*d^(1/3)) + ((-1)^(1/3)*PolyLog[2, ((-1)^(1/3)*b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/((-1)^(1/3)*b*c^(1/3) + (1 + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))} *) -{ArcCoth[a + b*x]/(c + d*x^2), x, 15, (Log[-((1 - a - b*x)/(a + b*x))]*Log[1 + ((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) - (Log[-((1 - a - b*x)/(a + b*x))]*Log[1 + ((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) + (Log[(1 + a + b*x)/(a + b*x)]*Log[1 - ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) - (Log[(1 + a + b*x)/(a + b*x)]*Log[1 - ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))])/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, -(((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x)))]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, -(((b^2*c + a^2*d)*(1 - a - b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] - (1 - a)*a*d)*(a + b*x)))]/(4*Sqrt[-c]*Sqrt[d]) + PolyLog[2, ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c - b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))]/(4*Sqrt[-c]*Sqrt[d]) - PolyLog[2, ((b^2*c + a^2*d)*(1 + a + b*x))/((b^2*c + b*Sqrt[-c]*Sqrt[d] + a*(1 + a)*d)*(a + b*x))]/(4*Sqrt[-c]*Sqrt[d])} -{ArcCoth[a + b*x]/(c + d*x^1), x, 5, -((ArcCoth[a + b*x]*Log[2/(1 + a + b*x)])/d) + (ArcCoth[a + b*x]*Log[(2*b*(c + d*x))/((b*c + d - a*d)*(1 + a + b*x))])/d + PolyLog[2, 1 - 2/(1 + a + b*x)]/(2*d) - PolyLog[2, 1 - (2*b*(c + d*x))/((b*c + d - a*d)*(1 + a + b*x))]/(2*d)} -{ArcCoth[a + b*x]/(c + d/x^1), x, 37, ((1 - a - b*x)*Log[-((1 - a - b*x)/(a + b*x))])/(2*b*c) + Log[a + b*x]/(2*b*c) + Log[1 + a + b*x]/(2*b*c) + ((a + b*x)*Log[(1 + a + b*x)/(a + b*x)])/(2*b*c) - (d*Log[(c*(1 - a - b*x))/(c - a*c + b*d)]*Log[d + c*x])/(2*c^2) + (d*Log[-((1 - a - b*x)/(a + b*x))]*Log[d + c*x])/(2*c^2) + (d*Log[(c*(1 + a + b*x))/(c + a*c - b*d)]*Log[d + c*x])/(2*c^2) - (d*Log[(1 + a + b*x)/(a + b*x)]*Log[d + c*x])/(2*c^2) + (d*PolyLog[2, -((b*(d + c*x))/(c + a*c - b*d))])/(2*c^2) - (d*PolyLog[2, (b*(d + c*x))/(c - a*c + b*d)])/(2*c^2), ((1 - a - b*x)*Log[-1 + a + b*x])/(2*b*c) + (x*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x]))/(2*c) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (x*(Log[a + b*x] - Log[1 + a + b*x] + Log[(1 + a + b*x)/(a + b*x)]))/(2*c) - (d*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x])*Log[d + c*x])/(2*c^2) - (d*(Log[a + b*x] - Log[1 + a + b*x] + Log[(1 + a + b*x)/(a + b*x)])*Log[d + c*x])/(2*c^2) - (d*Log[1 + a + b*x]*Log[-((b*(d + c*x))/(c + a*c - b*d))])/(2*c^2) + (d*Log[-1 + a + b*x]*Log[(b*(d + c*x))/(c - a*c + b*d)])/(2*c^2) + (d*PolyLog[2, (c*(1 - a - b*x))/(c - a*c + b*d)])/(2*c^2) - (d*PolyLog[2, (c*(1 + a + b*x))/(c + a*c - b*d)])/(2*c^2)} -{ArcCoth[a + b*x]/(c + d/x^2), x, 57, ((1 - a - b*x)*Log[-1 + a + b*x])/(2*b*c) + (x*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x]))/(2*c) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[-1 + a + b*x] - Log[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x]))/(2*c^(3/2)) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*b*c) + (x*(Log[a + b*x] - Log[1 + a + b*x] + Log[(1 + a + b*x)/(a + b*x)]))/(2*c) - (Sqrt[d]*ArcTan[(Sqrt[c]*x)/Sqrt[d]]*(Log[a + b*x] - Log[1 + a + b*x] + Log[(1 + a + b*x)/(a + b*x)]))/(2*c^(3/2)) + (Sqrt[d]*Log[-1 + a + b*x]*Log[-((b*(Sqrt[d] - Sqrt[-c]*x))/((1 - a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[1 + a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*Log[1 + a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d]))])/(4*(-c)^(3/2)) - (Sqrt[d]*Log[-1 + a + b*x]*Log[(b*(Sqrt[d] + Sqrt[-c]*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 - a - b*x))/((1 - a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] - b*Sqrt[d])])/(4*(-c)^(3/2)) - (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(1 + a + b*x))/((1 + a)*Sqrt[-c] + b*Sqrt[d])])/(4*(-c)^(3/2))} -(* {ArcCoth[a + b*x]/(c + d/x^3), x, 59, Log[1 - a - b*x]/(2*b*c) + Log[1 + a + b*x]/(2*b*c) - (d^(1/3)*Log[(c^(1/3)*(1 - a - b*x))/((1 - a)*c^(1/3) + b*d^(1/3))]*Log[-d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) + (d^(1/3)*Log[(c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) - b*d^(1/3))]*Log[-d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*Log[(c^(1/3)*(1 - a - b*x))/((1 - a)*c^(1/3) - (-1)^(1/3)*b*d^(1/3))]*Log[(-1)^(1/3)*d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*Log[(c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) + (-1)^(1/3)*b*d^(1/3))]*Log[(-1)^(1/3)*d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*Log[(c^(1/3)*(1 - a - b*x))/((1 - a)*c^(1/3) + (-1)^(2/3)*b*d^(1/3))]*Log[(-(-1)^(2/3))*d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*Log[(c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) - (-1)^(2/3)*b*d^(1/3))]*Log[(-(-1)^(2/3))*d^(1/3) - c^(1/3)*x])/(6*c^(4/3)) - ((a + b*x)*Log[1 - 1/(a + b*x)])/(2*b*c) + (d^(1/3)*Log[-d^(1/3) - c^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*Log[(-1)^(1/3)*d^(1/3) - c^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*Log[(-(-1)^(2/3))*d^(1/3) - c^(1/3)*x]*Log[1 - 1/(a + b*x)])/(6*c^(4/3)) + ((a + b*x)*Log[1 + 1/(a + b*x)])/(2*b*c) - (d^(1/3)*Log[-d^(1/3) - c^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*Log[(-1)^(1/3)*d^(1/3) - c^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*Log[(-(-1)^(2/3))*d^(1/3) - c^(1/3)*x]*Log[1 + 1/(a + b*x)])/(6*c^(4/3)) + (d^(1/3)*PolyLog[2, -((b*(d^(1/3) + c^(1/3)*x))/((1 + a)*c^(1/3) - b*d^(1/3)))])/(6*c^(4/3)) - (d^(1/3)*PolyLog[2, (b*(d^(1/3) + c^(1/3)*x))/((1 - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*PolyLog[2, -((b*((-1)^(2/3)*d^(1/3) + c^(1/3)*x))/((1 + a)*c^(1/3) - (-1)^(2/3)*b*d^(1/3)))])/(6*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*PolyLog[2, (b*((-1)^(2/3)*d^(1/3) + c^(1/3)*x))/((1 - a)*c^(1/3) + (-1)^(2/3)*b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*PolyLog[2, -(((-1)^(1/3)*b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((1 - a)*c^(1/3) - (-1)^(1/3)*b*d^(1/3)))])/(6*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((1 + a)*c^(1/3) + (-1)^(1/3)*b*d^(1/3))])/(6*c^(4/3))} *) - - -(* {ArcCoth[a + b*x]/(a + b*x^(3/2)), x, 131, ((1 - 1/Sqrt[1 + a^(-1)])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((1 + 1/Sqrt[1 + a^(-1)])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] - Sqrt[-a])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] + Sqrt[-a])*a^(2/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + a^(1/3)*b^(1/6))])/(6*(-a)^(3/2)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] - Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((1 - 1/Sqrt[1 + a^(-1)])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((1 + 1/Sqrt[1 + a^(-1)])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] - Sqrt[-a])*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] + Sqrt[-a])*a^(2/3)*Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - a^(1/3)*b^(1/6))])/(6*(-a)^(3/2)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-a] + Sqrt[b]*Sqrt[x])/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) + ((1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) - PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - a^(1/3)*b^(1/6)))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] + Sqrt[-a])*a^(2/3)*PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - a^(1/3)*b^(1/6)))])/(6*(-a)^(3/2)*b^(2/3)) + ((1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((Sqrt[1 - a] + Sqrt[-a])*a^(2/3)*PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + a^(1/3)*b^(1/6))])/(6*(-a)^(3/2)*b^(2/3)) - ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(6*Sqrt[-a]*a^(1/3)*b^(2/3))} *) -{ArcCoth[a + b*x]/(c + d*Sqrt[x]), x, 55, (2*Sqrt[1 + a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*d) - (2*Sqrt[1 - a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*d) + (c*Log[(d*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)]*Log[c + d*Sqrt[x]])/d^2 - (c*Log[(d*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)]*Log[c + d*Sqrt[x]])/d^2 + (c*Log[-((d*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (c*Log[-((d*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d))]*Log[c + d*Sqrt[x]])/d^2 - (Sqrt[x]*Log[-((1 - a - b*x)/(a + b*x))])/d + (c*Log[c + d*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/d^2 + (Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)])/d - (c*Log[c + d*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[-1 - a]*d)])/d^2 + (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[-1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c - Sqrt[1 - a]*d)])/d^2 - (c*PolyLog[2, (Sqrt[b]*(c + d*Sqrt[x]))/(Sqrt[b]*c + Sqrt[1 - a]*d)])/d^2} -{ArcCoth[a + b*x]/(c + d/Sqrt[x]), x, 65, -((2*Sqrt[1 + a]*d*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[1 + a]])/(Sqrt[b]*c^2)) + (2*Sqrt[1 - a]*d*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[1 - a]])/(Sqrt[b]*c^2) - (d^2*Log[(c*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 - (d^2*Log[(c*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + (d^2*Log[(c*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d)]*Log[d + c*Sqrt[x]])/c^3 + ((1 - a)*Log[1 - a - b*x])/(2*b*c) + (d*Sqrt[x]*Log[-((1 - a - b*x)/(a + b*x))])/c^2 - (x*Log[-((1 - a - b*x)/(a + b*x))])/(2*c) - (d^2*Log[d + c*Sqrt[x]]*Log[-((1 - a - b*x)/(a + b*x))])/c^3 + ((1 + a)*Log[1 + a + b*x])/(2*b*c) - (d*Sqrt[x]*Log[(1 + a + b*x)/(a + b*x)])/c^2 + (x*Log[(1 + a + b*x)/(a + b*x)])/(2*c) + (d^2*Log[d + c*Sqrt[x]]*Log[(1 + a + b*x)/(a + b*x)])/c^3 - (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c - Sqrt[b]*d))])/c^3 + (d^2*PolyLog[2, -((Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c - Sqrt[b]*d))])/c^3 - (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[-1 - a]*c + Sqrt[b]*d)])/c^3 + (d^2*PolyLog[2, (Sqrt[b]*(d + c*Sqrt[x]))/(Sqrt[1 - a]*c + Sqrt[b]*d)])/c^3} -(* {ArcCoth[a + b*x]/(a + b/x^(3/2)), x, 145, -((1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(6*a^(5/3)) - ((1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*(-a)^(3/2)*a^(2/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(1/3)*Sqrt[-a]*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(1/3)*Sqrt[-a]*b^(5/6)))])/(6*Sqrt[-a]*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] - Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(1/3)*Sqrt[-a]*b^(5/6)))])/(6*(-a)^(3/2)*a^(2/3)) - ((1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6))])/(6*a^(5/3)) - ((1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*(-a)^(3/2)*a^(2/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + Sqrt[-a]*b^(5/6)))])/(3*a^(5/3)) + ((Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + Sqrt[-a]*b^(5/6)))])/(6*Sqrt[-a]*a^(5/3)) - ((Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + Sqrt[-a]*b^(5/6)))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(2/3)*Sqrt[-a]*b^(5/6)))])/(3*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(2/3)*Sqrt[-a]*b^(5/6)))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[-((Sqrt[-a]*a^(1/3)*(Sqrt[-a] + Sqrt[b]*Sqrt[x]))/(a^(4/3) + (-1)^(2/3)*Sqrt[-a]*b^(5/6)))])/(6*(-a)^(3/2)*a^(2/3)) + ((1 - a)*Log[1 - a - b*x])/(2*a*b) + ((1 + a)*Log[1 + a + b*x])/(2*a*b) - (x*Log[1 - (a + b*x)^(-1)])/(2*a) - (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 - (a + b*x)^(-1)])/(3*a^(5/3)) + (x*Log[1 + (a + b*x)^(-1)])/(2*a) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 + (a + b*x)^(-1)])/(3*a^(5/3)) - ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(6*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(6*Sqrt[-a]*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(6*(-a)^(3/2)*a^(2/3)) - ((1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6)))])/(6*a^(5/3)) - ((1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6)))])/(6*a^(5/3)) + (b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) - ((1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(6*a^(5/3)) - ((1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(6*a^(5/3)) + (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) + (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + b^(5/6))])/(3*a^(5/3)) + ((Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) + ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(6*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(1/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(1/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(6*(-a)^(3/2)*a^(2/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + (-1)^(2/3)*b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) - ((-1)^(1/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(a^(4/3)/Sqrt[-a] + (-1)^(2/3)*b^(5/6))])/(6*(-a)^(3/2)*a^(2/3)) - ((-1)^(2/3)*(1 - 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) - ((-1)^(2/3)*(1 + 1/Sqrt[1 + a^(-1)])*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] - Sqrt[-a])*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*Sqrt[-a]*a^(5/3)) + ((-1)^(2/3)*(Sqrt[1 - a] + Sqrt[-a])*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(6*(-a)^(3/2)*a^(2/3))} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form ArcCoth[a+b x] / (c+d x+e x^2)*) - - -{ArcCoth[d + e*x]/(a + b*x + c*x^2), x, 12, (ArcCoth[d + e*x]*Log[(2*e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b - Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - (ArcCoth[d + e*x]*Log[(2*e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))])/Sqrt[b^2 - 4*a*c] - PolyLog[2, 1 + (2*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c - 2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c]) + PolyLog[2, 1 + (2*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e - 2*c*(d + e*x)))/((2*c*(1 - d) + (b + Sqrt[b^2 - 4*a*c])*e)*(1 + d + e*x))]/(2*Sqrt[b^2 - 4*a*c])} - - -(* ::Section::Closed:: *) -(*Integrands of the form u ArcCoth[a x^n]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*ArcCoth[Sqrt[x]], x, 6, Sqrt[x]/3 + x^(3/2)/9 + x^(5/2)/15 + (1/3)*x^3*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]/3} -{x^1*ArcCoth[Sqrt[x]], x, 5, Sqrt[x]/2 + x^(3/2)/6 + (1/2)*x^2*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]/2} -{x^0*ArcCoth[Sqrt[x]], x, 4, Sqrt[x] + x*ArcCoth[Sqrt[x]] - ArcTanh[Sqrt[x]]} -{ArcCoth[Sqrt[x]]/x^1, x, 2, PolyLog[2, -(1/Sqrt[x])] - PolyLog[2, 1/Sqrt[x]]} -{ArcCoth[Sqrt[x]]/x^2, x, 4, -(1/Sqrt[x]) - ArcCoth[Sqrt[x]]/x + ArcTanh[Sqrt[x]]} -{ArcCoth[Sqrt[x]]/x^3, x, 5, -(1/(6*x^(3/2))) - 1/(2*Sqrt[x]) - ArcCoth[Sqrt[x]]/(2*x^2) + ArcTanh[Sqrt[x]]/2} - - -{x^(3/2)*ArcCoth[Sqrt[x]], x, 3, x/5 + x^2/10 + (2/5)*x^(5/2)*ArcCoth[Sqrt[x]] + (1/5)*Log[1 - x]} -{Sqrt[x]*ArcCoth[Sqrt[x]], x, 3, x/3 + (2/3)*x^(3/2)*ArcCoth[Sqrt[x]] + (1/3)*Log[1 - x]} -{ArcCoth[Sqrt[x]]/Sqrt[x], x, 2, 2*Sqrt[x]*ArcCoth[Sqrt[x]] + Log[1 - x]} -{ArcCoth[Sqrt[x]]/x^(3/2), x, 4, -((2*ArcCoth[Sqrt[x]])/Sqrt[x]) - Log[1 - x] + Log[x]} - - -{ArcCoth[a*x^5]/x, x, 2, (1/10)*PolyLog[2, -(1/(a*x^5))] - (1/10)*PolyLog[2, 1/(a*x^5)]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{ArcCoth[1/x], x, 3, x*ArcCoth[1/x] + (1/2)*Log[1 - x^2]} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{ArcCoth[a*x^n]/x, x, 2, PolyLog[2, -(1/(x^n*a))]/(2*n) - PolyLog[2, 1/(x^n*a)]/(2*n)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCoth[c+d x])^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCoth[c+d x])^p when d e-c f=0*) - - -{(a + b*x)^1*ArcCoth[a + b*x], x, 4, x/2 + ((a + b*x)^2*ArcCoth[a + b*x])/(2*b) - ArcTanh[a + b*x]/(2*b)} -{(a + b*x)^2*ArcCoth[a + b*x], x, 5, (a + b*x)^2/(6*b) + ((a + b*x)^3*ArcCoth[a + b*x])/(3*b) + Log[1 - (a + b*x)^2]/(6*b)} -{ArcCoth[a + b*x]/(a + b*x)^1, x, 2, PolyLog[2, -(1/(a + b*x))]/(2*b) - PolyLog[2, 1/(a + b*x)]/(2*b)} -{ArcCoth[a + b*x]/(a + b*x)^2, x, 6, -(ArcCoth[a + b*x]/(b*(a + b*x))) + Log[a + b*x]/b - Log[1 - (a + b*x)^2]/(2*b)} - - -{ArcCoth[1 + x]/(2 + 2*x), x, 3, (1/4)*PolyLog[2, -(1/(1 + x))] - (1/4)*PolyLog[2, 1/(1 + x)]} - - -{ArcCoth[a + b*x]/((a*d)/b + d*x), x, 3, PolyLog[2, -(1/(a + b*x))]/(2*d) - PolyLog[2, 1/(a + b*x)]/(2*d)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCoth[c+d x])^p*) - - -{(e + f*x)^3*(a + b*ArcCoth[c + d*x]), x, 7, (b*f*(6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*x)/(4*d^3) + (b*f^2*(d*e - c*f)*(c + d*x)^2)/(2*d^4) + (b*f^3*(c + d*x)^3)/(12*d^4) + ((e + f*x)^4*(a + b*ArcCoth[c + d*x]))/(4*f) + (b*(d*e + f - c*f)^4*Log[1 - c - d*x])/(8*d^4*f) - (b*(d*e - f - c*f)^4*Log[1 + c + d*x])/(8*d^4*f)} -{(e + f*x)^2*(a + b*ArcCoth[c + d*x]), x, 7, (b*f*(d*e - c*f)*x)/d^2 + (b*f^2*(c + d*x)^2)/(6*d^3) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x]))/(3*f) + (b*(d*e + f - c*f)^3*Log[1 - c - d*x])/(6*d^3*f) - (b*(d*e - (1 + c)*f)^3*Log[1 + c + d*x])/(6*d^3*f)} -{(e + f*x)^1*(a + b*ArcCoth[c + d*x]), x, 7, (b*f*x)/(2*d) + ((e + f*x)^2*(a + b*ArcCoth[c + d*x]))/(2*f) + (b*(d*e + f - c*f)^2*Log[1 - c - d*x])/(4*d^2*f) - (b*(d*e - (1 + c)*f)^2*Log[1 + c + d*x])/(4*d^2*f)} -{(e + f*x)^0*(a + b*ArcCoth[c + d*x]), x, 4, a*x + (b*(c + d*x)*ArcCoth[c + d*x])/d + (b*Log[1 - (c + d*x)^2])/(2*d)} -{(a + b*ArcCoth[c + d*x])/(e + f*x)^1, x, 5, -(((a + b*ArcCoth[c + d*x])*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (b*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)} -{(a + b*ArcCoth[c + d*x])/(e + f*x)^2, x, 7, If[$VersionNumber>=8, -((a + b*ArcCoth[c + d*x])/(f*(e + f*x))) - (b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) + (b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) - (b*d*Log[e + f*x])/((d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcCoth[c + d*x])/(f*(e + f*x))) - (b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) + (b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) - (b*d*Log[e + f*x])/((d*e - f - c*f)*(d*e + f - c*f))]} -{(a + b*ArcCoth[c + d*x])/(e + f*x)^3, x, 5, If[$VersionNumber>=8, (b*d)/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)*(e + f*x)) - (a + b*ArcCoth[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*Log[1 - c - d*x])/(4*f*(d*e + f - c*f)^2) + (b*d^2*Log[1 + c + d*x])/(4*f*(d*e - f - c*f)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2), (b*d)/(2*(d*e - f - c*f)*(d*e + f - c*f)*(e + f*x)) - (a + b*ArcCoth[c + d*x])/(2*f*(e + f*x)^2) - (b*d^2*Log[1 - c - d*x])/(4*f*(d*e + f - c*f)^2) + (b*d^2*Log[1 + c + d*x])/(4*f*(d*e - f - c*f)^2) - (b*d^2*(d*e - c*f)*Log[e + f*x])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2)]} - - -{(e + f*x)^2*(a + b*ArcCoth[c + d*x])^2, x, 16, (b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCoth[c + d*x])/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCoth[c + d*x]))/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2)/(3*d^3) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTanh[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 - (c + d*x)^2])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(3*d^3)} -{(e + f*x)^1*(a + b*ArcCoth[c + d*x])^2, x, 13, (a*b*f*x)/d + (b^2*f*(c + d*x)*ArcCoth[c + d*x])/d^2 + ((d*e - c*f)*(a + b*ArcCoth[c + d*x])^2)/d^2 - ((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCoth[c + d*x])^2)/(2*f) - (2*b*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/d^2 + (b^2*f*Log[1 - (c + d*x)^2])/(2*d^2) - (b^2*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^2} -{(e + f*x)^0*(a + b*ArcCoth[c + d*x])^2, x, 6, (a + b*ArcCoth[c + d*x])^2/d + ((c + d*x)*(a + b*ArcCoth[c + d*x])^2)/d - (2*b*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/d - (b^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d} -{(a + b*ArcCoth[c + d*x])^2/(e + f*x)^1, x, 2, -(((a + b*ArcCoth[c + d*x])^2*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 + c + d*x)])/f - (b*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (b^2*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (b^2*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f)} -{(a + b*ArcCoth[c + d*x])^2/(e + f*x)^2, x, 24, If[$VersionNumber>=8, -((a + b*ArcCoth[c + d*x])^2/(f*(e + f*x))) + (b^2*d*ArcCoth[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) - (a*b*d*Log[1 - c - d*x])/(f*(d*e + f - c*f)) - (b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (2*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (a*b*d*Log[1 + c + d*x])/(f*(d*e - f - c*f)) + (2*a*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (2*b^2*d*ArcCoth[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcCoth[c + d*x])^2/(f*(e + f*x))) + (b^2*d*ArcCoth[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) - (a*b*d*Log[1 - c - d*x])/(f*(d*e + f - c*f)) - (b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (2*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (a*b*d*Log[1 + c + d*x])/(f*(d*e - f - c*f)) + (2*a*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (2*b^2*d*ArcCoth[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f))]} - - -{(e + f*x)^2*(a + b*ArcCoth[c + d*x])^3, x, 21, (a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCoth[c + d*x])/d^3 - (b*f^2*(a + b*ArcCoth[c + d*x])^2)/(2*d^3) + (3*b*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCoth[c + d*x])^2)/d^3 + (b*f^2*(c + d*x)^2*(a + b*ArcCoth[c + d*x])^2)/(2*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*ArcCoth[c + d*x])^3)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^3)/(3*d^3) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/d^3 - (b*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)])/d^3 + (b^3*f^2*Log[1 - (c + d*x)^2])/(2*d^3) - (3*b^3*f*(d*e - c*f)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d^3 + (b^3*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^3)} -{(e + f*x)^1*(a + b*ArcCoth[c + d*x])^3, x, 15, (3*b*f*(a + b*ArcCoth[c + d*x])^2)/(2*d^2) + (3*b*f*(c + d*x)*(a + b*ArcCoth[c + d*x])^2)/(2*d^2) + ((d*e - c*f)*(a + b*ArcCoth[c + d*x])^3)/d^2 - ((d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(a + b*ArcCoth[c + d*x])^3)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCoth[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/d^2 - (3*b*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)])/d^2 - (3*b^3*f*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*d^2) - (3*b^2*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d^2 + (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d^2)} -{(e + f*x)^0*(a + b*ArcCoth[c + d*x])^3, x, 6, (a + b*ArcCoth[c + d*x])^3/d + ((c + d*x)*(a + b*ArcCoth[c + d*x])^3)/d - (3*b*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)])/d - (3*b^2*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)])/d + (3*b^3*PolyLog[3, 1 - 2/(1 - c - d*x)])/(2*d)} -{(a + b*ArcCoth[c + d*x])^3/(e + f*x)^1, x, 2, -(((a + b*ArcCoth[c + d*x])^3*Log[2/(1 + c + d*x)])/f) + ((a + b*ArcCoth[c + d*x])^3*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/f + (3*b*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b*(a + b*ArcCoth[c + d*x])^2*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^2*(a + b*ArcCoth[c + d*x])*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*f) - (3*b^2*(a + b*ArcCoth[c + d*x])*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*f) + (3*b^3*PolyLog[4, 1 - 2/(1 + c + d*x)])/(4*f) - (3*b^3*PolyLog[4, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(4*f)} -{(a + b*ArcCoth[c + d*x])^3/(e + f*x)^2, x, 33, If[$VersionNumber>=8, -((a + b*ArcCoth[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) - (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) - (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) + (3*a^2*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (6*a*b^2*d*ArcCoth[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)*(d*e - (1 + c)*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - c - d*x)])/(4*f*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)) + (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)), -((a + b*ArcCoth[c + d*x])^3/(f*(e + f*x))) + (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 - c - d*x)])/(f*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) - (3*a^2*b*d*Log[1 - c - d*x])/(2*f*(d*e + f - c*f)) - (3*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/(f*(d*e - f - c*f)) + (6*a*b^2*d*ArcCoth[c + d*x]*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]^2*Log[2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a^2*b*d*Log[1 + c + d*x])/(2*f*(d*e - f - c*f)) + (3*a^2*b*d*Log[e + f*x])/(f^2 - (d*e - c*f)^2) - (6*a*b^2*d*ArcCoth[c + d*x]*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*ArcCoth[c + d*x]^2*Log[(2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(2*f*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 - c - d*x)])/(2*f*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*a*b^2*d*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f*(d*e - f - c*f)) - (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*a*b^2*d*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*ArcCoth[c + d*x]*PolyLog[2, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e - f - c*f)*(d*e + f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 - c - d*x)])/(4*f*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(4*f*(d*e - f - c*f)) - (3*b^3*d*PolyLog[3, 1 - 2/(1 + c + d*x)])/(2*(d*e - f - c*f)*(d*e + f - c*f)) + (3*b^3*d*PolyLog[3, 1 - (2*d*(e + f*x))/((d*e + f - c*f)*(1 + c + d*x))])/(2*(d*e - f - c*f)*(d*e + f - c*f))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e+f x)^m (a+b ArcCoth[c+d x])^p with m symbolic*) - - -{(e + f*x)^m*(a + b*ArcCoth[c + d*x])^1, x, 6, ((e + f*x)^(1 + m)*(a + b*ArcCoth[c + d*x]))/(f*(1 + m)) + (b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - f - c*f)])/(2*f*(d*e - (1 + c)*f)*(1 + m)*(2 + m)) - (b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e + f - c*f)])/(2*f*(d*e + f - c*f)*(1 + m)*(2 + m))} -{(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2, x]} -{(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3, x, 1, Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form u^m (a+b ArcCoth[Sqrt[1-c x]/Sqrt[1+c x]])^n*) - - -{(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x, 0, Unintegrable[(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]} - - -{(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x, 9, -((2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*ArcCoth[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c) - (3*b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (3*b^2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c) - (3*b^3*PolyLog[4, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(4*c) + (3*b^3*PolyLog[4, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(4*c)} -{(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x, 7, -((2*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*ArcCoth[1 - 2/(1 - Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c) - (b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/c + (b*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/c - (b^2*PolyLog[3, 1 - 2/(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x])])/(2*c) + (b^2*PolyLog[3, 1 - (2*Sqrt[1 - c*x])/(Sqrt[1 + c*x]*(1 + Sqrt[1 - c*x]/Sqrt[1 + c*x]))])/(2*c)} -{(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1/(1 - c^2*x^2), x, 2, -((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/c) - (b*PolyLog[2, -(Sqrt[1 + c*x]/Sqrt[1 - c*x])])/(2*c) + (b*PolyLog[2, Sqrt[1 + c*x]/Sqrt[1 - c*x]])/(2*c)} -{1/((a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^1*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]} -{1/((a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(1 - c^2*x^2)), x, 0, Unintegrable[1/((1 - c^2*x^2)*(a + b*ArcCoth[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCoth[Tanh[a+b x]]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[Tanh[a+b x]]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ArcCoth[Tanh[a + b*x]]*x^m, x, 2, -((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]])/(1 + m)} - -{ArcCoth[Tanh[a + b*x]]*x^2, x, 2, -((b*x^4)/12) + (1/3)*x^3*ArcCoth[Tanh[a + b*x]]} -{ArcCoth[Tanh[a + b*x]]*x^1, x, 2, -((b*x^3)/6) + (1/2)*x^2*ArcCoth[Tanh[a + b*x]]} -{ArcCoth[Tanh[a + b*x]]*x^0, x, 2, ArcCoth[Tanh[a + b*x]]^2/(2*b)} -{ArcCoth[Tanh[a + b*x]]/x^1, x, 2, b*x - (b*x - ArcCoth[Tanh[a + b*x]])*Log[x]} -{ArcCoth[Tanh[a + b*x]]/x^2, x, 2, -(ArcCoth[Tanh[a + b*x]]/x) + b*Log[x]} -{ArcCoth[Tanh[a + b*x]]/x^3, x, 2, -(b/(2*x)) - ArcCoth[Tanh[a + b*x]]/(2*x^2)} -{ArcCoth[Tanh[a + b*x]]/x^4, x, 2, -(b/(6*x^2)) - ArcCoth[Tanh[a + b*x]]/(3*x^3)} - - -{ArcCoth[Tanh[a + b*x]]^2*x^m, x, 3, (2*b^2*x^(3 + m))/(6 + 11*m + 6*m^2 + m^3) - (2*b*x^(2 + m)*ArcCoth[Tanh[a + b*x]])/(2 + 3*m + m^2) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]]^2)/(1 + m)} - -{ArcCoth[Tanh[a + b*x]]^2*x^3, x, 3, (b^2*x^6)/60 - (1/10)*b*x^5*ArcCoth[Tanh[a + b*x]] + (1/4)*x^4*ArcCoth[Tanh[a + b*x]]^2} -{ArcCoth[Tanh[a + b*x]]^2*x^2, x, 3, (b^2*x^5)/30 - (1/6)*b*x^4*ArcCoth[Tanh[a + b*x]] + (1/3)*x^3*ArcCoth[Tanh[a + b*x]]^2} -{ArcCoth[Tanh[a + b*x]]^2*x^1, x, 3, (x*ArcCoth[Tanh[a + b*x]]^3)/(3*b) - ArcCoth[Tanh[a + b*x]]^4/(12*b^2)} -{ArcCoth[Tanh[a + b*x]]^2*x^0, x, 2, ArcCoth[Tanh[a + b*x]]^3/(3*b)} -{ArcCoth[Tanh[a + b*x]]^2/x^1, x, 3, (-b)*x*(b*x - ArcCoth[Tanh[a + b*x]]) + (1/2)*ArcCoth[Tanh[a + b*x]]^2 + (b*x - ArcCoth[Tanh[a + b*x]])^2*Log[x]} -{ArcCoth[Tanh[a + b*x]]^2/x^2, x, 3, 2*b^2*x - ArcCoth[Tanh[a + b*x]]^2/x - 2*b*(b*x - ArcCoth[Tanh[a + b*x]])*Log[x]} -{ArcCoth[Tanh[a + b*x]]^2/x^3, x, 3, -((b*ArcCoth[Tanh[a + b*x]])/x) - ArcCoth[Tanh[a + b*x]]^2/(2*x^2) + b^2*Log[x]} -{ArcCoth[Tanh[a + b*x]]^2/x^4, x, 1, ArcCoth[Tanh[a + b*x]]^3/(3*x^3*(b*x - ArcCoth[Tanh[a + b*x]]))} -{ArcCoth[Tanh[a + b*x]]^2/x^5, x, 2, (b*ArcCoth[Tanh[a + b*x]]^3)/(12*x^3*(b*x - ArcCoth[Tanh[a + b*x]])^2) + ArcCoth[Tanh[a + b*x]]^3/(4*x^4*(b*x - ArcCoth[Tanh[a + b*x]]))} - - -{ArcCoth[Tanh[a + b*x]]^3*x^m, x, 4, -((6*b^3*x^(4 + m))/((1 + m)*(24 + 26*m + 9*m^2 + m^3))) + (6*b^2*x^(3 + m)*ArcCoth[Tanh[a + b*x]])/(6 + 11*m + 6*m^2 + m^3) - (3*b*x^(2 + m)*ArcCoth[Tanh[a + b*x]]^2)/(2 + 3*m + m^2) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]]^3)/(1 + m)} - -{ArcCoth[Tanh[a + b*x]]^3*x^4, x, 4, (-(1/280))*b^3*x^8 + (1/35)*b^2*x^7*ArcCoth[Tanh[a + b*x]] - (1/10)*b*x^6*ArcCoth[Tanh[a + b*x]]^2 + (1/5)*x^5*ArcCoth[Tanh[a + b*x]]^3} -{ArcCoth[Tanh[a + b*x]]^3*x^3, x, 4, (-(1/140))*b^3*x^7 + (1/20)*b^2*x^6*ArcCoth[Tanh[a + b*x]] - (3/20)*b*x^5*ArcCoth[Tanh[a + b*x]]^2 + (1/4)*x^4*ArcCoth[Tanh[a + b*x]]^3} -{ArcCoth[Tanh[a + b*x]]^3*x^2, x, 4, (x^2*ArcCoth[Tanh[a + b*x]]^4)/(4*b) - (x*ArcCoth[Tanh[a + b*x]]^5)/(10*b^2) + ArcCoth[Tanh[a + b*x]]^6/(60*b^3)} -{ArcCoth[Tanh[a + b*x]]^3*x^1, x, 3, (x*ArcCoth[Tanh[a + b*x]]^4)/(4*b) - ArcCoth[Tanh[a + b*x]]^5/(20*b^2)} -{ArcCoth[Tanh[a + b*x]]^3*x^0, x, 2, ArcCoth[Tanh[a + b*x]]^4/(4*b)} -{ArcCoth[Tanh[a + b*x]]^3/x^1, x, 4, b*x*(b*x - ArcCoth[Tanh[a + b*x]])^2 - (1/2)*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2 + (1/3)*ArcCoth[Tanh[a + b*x]]^3 - (b*x - ArcCoth[Tanh[a + b*x]])^3*Log[x]} -{ArcCoth[Tanh[a + b*x]]^3/x^2, x, 4, -3*b^2*x*(b*x - ArcCoth[Tanh[a + b*x]]) + (3/2)*b*ArcCoth[Tanh[a + b*x]]^2 - ArcCoth[Tanh[a + b*x]]^3/x + 3*b*(b*x - ArcCoth[Tanh[a + b*x]])^2*Log[x]} -{ArcCoth[Tanh[a + b*x]]^3/x^3, x, 4, 3*b^3*x - (3*b*ArcCoth[Tanh[a + b*x]]^2)/(2*x) - ArcCoth[Tanh[a + b*x]]^3/(2*x^2) - 3*b^2*(b*x - ArcCoth[Tanh[a + b*x]])*Log[x]} -{ArcCoth[Tanh[a + b*x]]^3/x^4, x, 4, -((b^2*ArcCoth[Tanh[a + b*x]])/x) - (b*ArcCoth[Tanh[a + b*x]]^2)/(2*x^2) - ArcCoth[Tanh[a + b*x]]^3/(3*x^3) + b^3*Log[x]} -{ArcCoth[Tanh[a + b*x]]^3/x^5, x, 1, ArcCoth[Tanh[a + b*x]]^4/(4*x^4*(b*x - ArcCoth[Tanh[a + b*x]]))} -{ArcCoth[Tanh[a + b*x]]^3/x^6, x, 2, (b*ArcCoth[Tanh[a + b*x]]^4)/(20*x^4*(b*x - ArcCoth[Tanh[a + b*x]])^2) + ArcCoth[Tanh[a + b*x]]^4/(5*x^5*(b*x - ArcCoth[Tanh[a + b*x]]))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{1/ArcCoth[Tanh[a + b*x]]*x^m, x, 1, -((x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/(b*x - ArcCoth[Tanh[a + b*x]])])/((1 + m)*(b*x - ArcCoth[Tanh[a + b*x]])))} - -{1/ArcCoth[Tanh[a + b*x]]*x^3, x, 5, x^3/(3*b) + (x^2*(b*x - ArcCoth[Tanh[a + b*x]]))/(2*b^2) + (x*(b*x - ArcCoth[Tanh[a + b*x]])^2)/b^3 + ((b*x - ArcCoth[Tanh[a + b*x]])^3*Log[ArcCoth[Tanh[a + b*x]]])/b^4} -{1/ArcCoth[Tanh[a + b*x]]*x^2, x, 4, x^2/(2*b) + (x*(b*x - ArcCoth[Tanh[a + b*x]]))/b^2 + ((b*x - ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^3} -{1/ArcCoth[Tanh[a + b*x]]*x^1, x, 3, x/b + ((b*x - ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^2} -{1/ArcCoth[Tanh[a + b*x]]*x^0, x, 2, Log[ArcCoth[Tanh[a + b*x]]]/b} -{1/ArcCoth[Tanh[a + b*x]]/x^1, x, 4, -(Log[x]/(b*x - ArcCoth[Tanh[a + b*x]])) + Log[ArcCoth[Tanh[a + b*x]]]/(b*x - ArcCoth[Tanh[a + b*x]])} -{1/ArcCoth[Tanh[a + b*x]]/x^2, x, 5, 1/(x*(b*x - ArcCoth[Tanh[a + b*x]])) - (b*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^2 + (b*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^2} -{1/ArcCoth[Tanh[a + b*x]]/x^3, x, 6, b/(x*(b*x - ArcCoth[Tanh[a + b*x]])^2) + 1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])) - (b^2*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^3 + (b^2*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^3} - - -{1/ArcCoth[Tanh[a + b*x]]^2*x^m, x, 2, -(x^m/(b*ArcCoth[Tanh[a + b*x]])) - (x^m*Hypergeometric2F1[1, m, 1 + m, (b*x)/(b*x - ArcCoth[Tanh[a + b*x]])])/(b*(b*x - ArcCoth[Tanh[a + b*x]]))} - -{1/ArcCoth[Tanh[a + b*x]]^2*x^4, x, 6, (4*x^3)/(3*b^2) + (2*x^2*(b*x - ArcCoth[Tanh[a + b*x]]))/b^3 + (4*x*(b*x - ArcCoth[Tanh[a + b*x]])^2)/b^4 - x^4/(b*ArcCoth[Tanh[a + b*x]]) + (4*(b*x - ArcCoth[Tanh[a + b*x]])^3*Log[ArcCoth[Tanh[a + b*x]]])/b^5} -{1/ArcCoth[Tanh[a + b*x]]^2*x^3, x, 5, (3*x^2)/(2*b^2) + (3*x*(b*x - ArcCoth[Tanh[a + b*x]]))/b^3 - x^3/(b*ArcCoth[Tanh[a + b*x]]) + (3*(b*x - ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^4} -{1/ArcCoth[Tanh[a + b*x]]^2*x^2, x, 4, (2*x)/b^2 - x^2/(b*ArcCoth[Tanh[a + b*x]]) + (2*(b*x - ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^3} -{1/ArcCoth[Tanh[a + b*x]]^2*x^1, x, 3, -(x/(b*ArcCoth[Tanh[a + b*x]])) + Log[ArcCoth[Tanh[a + b*x]]]/b^2} -{1/ArcCoth[Tanh[a + b*x]]^2*x^0, x, 2, -(1/(b*ArcCoth[Tanh[a + b*x]]))} -{1/ArcCoth[Tanh[a + b*x]]^2/x^1, x, 5, -(1/((b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]])) + Log[x]/(b*x - ArcCoth[Tanh[a + b*x]])^2 - Log[ArcCoth[Tanh[a + b*x]]]/(b*x - ArcCoth[Tanh[a + b*x]])^2} -{1/ArcCoth[Tanh[a + b*x]]^2/x^2, x, 6, -((2*b)/((b*x - ArcCoth[Tanh[a + b*x]])^2*ArcCoth[Tanh[a + b*x]])) + 1/(x*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]) + (2*b*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^3 - (2*b*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^3} -{1/ArcCoth[Tanh[a + b*x]]^2/x^3, x, 7, -((3*b^2)/((b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]])) + (3*b)/(2*x*(b*x - ArcCoth[Tanh[a + b*x]])^2*ArcCoth[Tanh[a + b*x]]) + 1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]) + (3*b^2*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^4 - (3*b^2*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^4} - - -{1/ArcCoth[Tanh[a + b*x]]^3*x^m, x, 3, -(x^m/(2*b*ArcCoth[Tanh[a + b*x]]^2)) - (m*x^(-1 + m))/(2*b^2*ArcCoth[Tanh[a + b*x]]) - (m*x^(-1 + m)*Hypergeometric2F1[1, -1 + m, m, (b*x)/(b*x - ArcCoth[Tanh[a + b*x]])])/(2*b^2*(b*x - ArcCoth[Tanh[a + b*x]]))} - -{1/ArcCoth[Tanh[a + b*x]]^3*x^4, x, 6, (3*x^2)/b^3 + (6*x*(b*x - ArcCoth[Tanh[a + b*x]]))/b^4 - x^4/(2*b*ArcCoth[Tanh[a + b*x]]^2) - (2*x^3)/(b^2*ArcCoth[Tanh[a + b*x]]) + (6*(b*x - ArcCoth[Tanh[a + b*x]])^2*Log[ArcCoth[Tanh[a + b*x]]])/b^5} -{1/ArcCoth[Tanh[a + b*x]]^3*x^3, x, 5, (3*x)/b^3 - x^3/(2*b*ArcCoth[Tanh[a + b*x]]^2) - (3*x^2)/(2*b^2*ArcCoth[Tanh[a + b*x]]) + (3*(b*x - ArcCoth[Tanh[a + b*x]])*Log[ArcCoth[Tanh[a + b*x]]])/b^4} -{1/ArcCoth[Tanh[a + b*x]]^3*x^2, x, 4, -(x^2/(2*b*ArcCoth[Tanh[a + b*x]]^2)) - x/(b^2*ArcCoth[Tanh[a + b*x]]) + Log[ArcCoth[Tanh[a + b*x]]]/b^3} -{1/ArcCoth[Tanh[a + b*x]]^3*x^1, x, 3, -(x/(2*b*ArcCoth[Tanh[a + b*x]]^2)) - 1/(2*b^2*ArcCoth[Tanh[a + b*x]])} -{1/ArcCoth[Tanh[a + b*x]]^3*x^0, x, 2, -(1/(2*b*ArcCoth[Tanh[a + b*x]]^2))} -{1/ArcCoth[Tanh[a + b*x]]^3/x^1, x, 6, -(1/(2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2)) + 1/((b*x - ArcCoth[Tanh[a + b*x]])^2*ArcCoth[Tanh[a + b*x]]) - Log[x]/(b*x - ArcCoth[Tanh[a + b*x]])^3 + Log[ArcCoth[Tanh[a + b*x]]]/(b*x - ArcCoth[Tanh[a + b*x]])^3} -{1/ArcCoth[Tanh[a + b*x]]^3/x^2, x, 7, -((3*b)/(2*(b*x - ArcCoth[Tanh[a + b*x]])^2*ArcCoth[Tanh[a + b*x]]^2)) + 1/(x*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (3*b)/((b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]]) - (3*b*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^4 + (3*b*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^4} -{1/ArcCoth[Tanh[a + b*x]]^3/x^3, x, 8, -((3*b^2)/((b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]]^2)) + (2*b)/(x*(b*x - ArcCoth[Tanh[a + b*x]])^2*ArcCoth[Tanh[a + b*x]]^2) + 1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (6*b^2)/((b*x - ArcCoth[Tanh[a + b*x]])^4*ArcCoth[Tanh[a + b*x]]) - (6*b^2*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^5 + (6*b^2*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^5} - - -(* ::Subsubsection::Closed:: *) -(*n symbolic*) - - -{ArcCoth[Tanh[a + b*x]]^n*x^m, x, 1, (1/(b*(1 + n)))*((x^m*ArcCoth[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[-m, 1 + n, 2 + n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCoth[Tanh[a + b*x]]))])/((b*x)/(b*x - ArcCoth[Tanh[a + b*x]]))^m)} - -{ArcCoth[Tanh[a + b*x]]^n*x^4, x, 6, If[$VersionNumber>=8, (x^4*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (12*x^2*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (24*x*ArcCoth[Tanh[a + b*x]]^(4 + n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)) + (24*ArcCoth[Tanh[a + b*x]]^(5 + n))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)), (x^4*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (4*x^3*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (12*x^2*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2)) - (24*x*ArcCoth[Tanh[a + b*x]]^(4 + n))/(b^4*(4 + 5*n + n^2)*(6 + 5*n + n^2)) + (24*ArcCoth[Tanh[a + b*x]]^(5 + n))/(b^5*(12 + 7*n + n^2)*(10 + 17*n + 8*n^2 + n^3))]} -{ArcCoth[Tanh[a + b*x]]^n*x^3, x, 5, If[$VersionNumber>=8, (x^3*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (3*x^2*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (6*x*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)) - (6*ArcCoth[Tanh[a + b*x]]^(4 + n))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)), (x^3*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (3*x^2*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (6*x*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2)) - (6*ArcCoth[Tanh[a + b*x]]^(4 + n))/(b^4*(4 + 5*n + n^2)*(6 + 5*n + n^2))]} -{ArcCoth[Tanh[a + b*x]]^n*x^2, x, 4, If[$VersionNumber>=8, (x^2*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (2*x*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (2*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(1 + n)*(2 + n)*(3 + n)), (x^2*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (2*x*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b^2*(1 + n)*(2 + n)) + (2*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b^3*(3 + n)*(2 + 3*n + n^2))]} -{ArcCoth[Tanh[a + b*x]]^n*x^1, x, 3, (x*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - ArcCoth[Tanh[a + b*x]]^(2 + n)/(b^2*(1 + n)*(2 + n))} -{ArcCoth[Tanh[a + b*x]]^n*x^0, x, 2, ArcCoth[Tanh[a + b*x]]^(1 + n)/(b*(1 + n))} -{ArcCoth[Tanh[a + b*x]]^n/x^1, x, 1, (ArcCoth[Tanh[a + b*x]]^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCoth[Tanh[a + b*x]]))])/((1 + n)*(b*x - ArcCoth[Tanh[a + b*x]]))} -{ArcCoth[Tanh[a + b*x]]^n/x^2, x, 2, -(ArcCoth[Tanh[a + b*x]]^n/x) + (b*ArcCoth[Tanh[a + b*x]]^n*Hypergeometric2F1[1, n, 1 + n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCoth[Tanh[a + b*x]]))])/(b*x - ArcCoth[Tanh[a + b*x]])} -{ArcCoth[Tanh[a + b*x]]^n/x^3, x, 3, -((b*n*ArcCoth[Tanh[a + b*x]]^(-1 + n))/(2*x)) - ArcCoth[Tanh[a + b*x]]^n/(2*x^2) + (b^2*n*ArcCoth[Tanh[a + b*x]]^(-1 + n)*Hypergeometric2F1[1, -1 + n, n, -(ArcCoth[Tanh[a + b*x]]/(b*x - ArcCoth[Tanh[a + b*x]]))])/(2*(b*x - ArcCoth[Tanh[a + b*x]]))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCoth[Coth[a+b x]]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[Coth[a+b x]]^n*) - - -{ArcCoth[Tanh[a + b*x]]*x^m, x, 2, -((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]])/(1 + m)} - -{ArcCoth[Coth[a + b*x]]*x^2, x, 2, -((b*x^4)/12) + (1/3)*x^3*ArcCoth[Coth[a + b*x]]} -{ArcCoth[Coth[a + b*x]]*x^1, x, 2, -((b*x^3)/6) + (1/2)*x^2*ArcCoth[Coth[a + b*x]]} -{ArcCoth[Coth[a + b*x]]*x^0, x, 2, ArcCoth[Coth[a + b*x]]^2/(2*b)} -{ArcCoth[Coth[a + b*x]]/x^1, x, 2, b*x - (b*x - ArcCoth[Coth[a + b*x]])*Log[x]} -{ArcCoth[Coth[a + b*x]]/x^2, x, 2, -(ArcCoth[Coth[a + b*x]]/x) + b*Log[x]} -{ArcCoth[Coth[a + b*x]]/x^3, x, 2, -(b/(2*x)) - ArcCoth[Coth[a + b*x]]/(2*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Hyper[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Sinh[a+b x]]*) - - -(* {ArcCoth[Sinh[x]], x, 6, 0} -{x*ArcCoth[Sinh[x]], x, 8, 0} -{x^2*ArcCoth[Sinh[x]], x, 10, 0} *) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Cosh[a+b x]]*) - - -{ArcCoth[Cosh[x]], x, 6, x*ArcCoth[Cosh[x]] - 2*x*ArcTanh[E^x] - PolyLog[2, -E^x] + PolyLog[2, E^x]} -{x*ArcCoth[Cosh[x]], x, 8, (1/2)*x^2*ArcCoth[Cosh[x]] - x^2*ArcTanh[E^x] - x*PolyLog[2, -E^x] + x*PolyLog[2, E^x] + PolyLog[3, -E^x] - PolyLog[3, E^x]} -{x^2*ArcCoth[Cosh[x]], x, 10, (1/3)*x^3*ArcCoth[Cosh[x]] - (2/3)*x^3*ArcTanh[E^x] - x^2*PolyLog[2, -E^x] + x^2*PolyLog[2, E^x] + 2*x*PolyLog[3, -E^x] - 2*x*PolyLog[3, E^x] - 2*PolyLog[4, -E^x] + 2*PolyLog[4, E^x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Tanh[a+b x]]*) - - -{ArcCoth[c + d*Tanh[a + b*x]]*x^2, x, 11, (1/3)*x^3*ArcCoth[c + d*Tanh[a + b*x]] + (1/6)*x^3*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/6)*x^3*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x^2*PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b) - (x^2*PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b) - (x*PolyLog[3, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b^2) + (x*PolyLog[3, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b^2) + PolyLog[4, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(8*b^3) - PolyLog[4, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(8*b^3)} -{ArcCoth[c + d*Tanh[a + b*x]]*x^1, x, 9, (1/2)*x^2*ArcCoth[c + d*Tanh[a + b*x]] + (1/4)*x^2*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/4)*x^2*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x*PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))])/(4*b) - (x*PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))])/(4*b) - PolyLog[3, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(8*b^2) + PolyLog[3, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(8*b^2)} -{ArcCoth[c + d*Tanh[a + b*x]]*x^0, x, 7, x*ArcCoth[c + d*Tanh[a + b*x]] + (1/2)*x*Log[1 + ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/2)*x*Log[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + PolyLog[2, -(((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d))]/(4*b) - PolyLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(4*b)} -{ArcCoth[c + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[c + d*Tanh[a + b*x]]/x, x]} - - -{ArcCoth[1 + d + d*Tanh[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcCoth[1 + d + d*Tanh[a + b*x]] - (1/8)*x^4*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + (3*x^2*PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))])/(8*b^2) - (3*x*PolyLog[4, -((1 + d)*E^(2*a + 2*b*x))])/(8*b^3) + (3*PolyLog[5, -((1 + d)*E^(2*a + 2*b*x))])/(16*b^4)} -{ArcCoth[1 + d + d*Tanh[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCoth[1 + d + d*Tanh[a + b*x]] - (1/6)*x^3*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + (x*PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))])/(4*b^2) - PolyLog[4, -((1 + d)*E^(2*a + 2*b*x))]/(8*b^3)} -{ArcCoth[1 + d + d*Tanh[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCoth[1 + d + d*Tanh[a + b*x]] - (1/4)*x^2*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))])/(4*b) + PolyLog[3, -((1 + d)*E^(2*a + 2*b*x))]/(8*b^2)} -{ArcCoth[1 + d + d*Tanh[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcCoth[1 + d + d*Tanh[a + b*x]] - (1/2)*x*Log[1 + (1 + d)*E^(2*a + 2*b*x)] - PolyLog[2, -((1 + d)*E^(2*a + 2*b*x))]/(4*b)} -{ArcCoth[1 + d + d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 + d + d*Tanh[a + b*x]]/x, x]} - - -{ArcCoth[1 - d - d*Tanh[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcCoth[1 - d - d*Tanh[a + b*x]] - (1/8)*x^4*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + (3*x^2*PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))])/(8*b^2) - (3*x*PolyLog[4, -((1 - d)*E^(2*a + 2*b*x))])/(8*b^3) + (3*PolyLog[5, -((1 - d)*E^(2*a + 2*b*x))])/(16*b^4)} -{ArcCoth[1 - d - d*Tanh[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCoth[1 - d - d*Tanh[a + b*x]] - (1/6)*x^3*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + (x*PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))])/(4*b^2) - PolyLog[4, -((1 - d)*E^(2*a + 2*b*x))]/(8*b^3)} -{ArcCoth[1 - d - d*Tanh[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCoth[1 - d - d*Tanh[a + b*x]] - (1/4)*x^2*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))])/(4*b) + PolyLog[3, -((1 - d)*E^(2*a + 2*b*x))]/(8*b^2)} -{ArcCoth[1 - d - d*Tanh[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcCoth[1 - d - d*Tanh[a + b*x]] - (1/2)*x*Log[1 + (1 - d)*E^(2*a + 2*b*x)] - PolyLog[2, -((1 - d)*E^(2*a + 2*b*x))]/(4*b)} -{ArcCoth[1 - d - d*Tanh[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 - d - d*Tanh[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Coth[a+b x]]*) - - -{ArcCoth[c + d*Coth[a + b*x]]*x^2, x, 11, (1/3)*x^3*ArcCoth[c + d*Coth[a + b*x]] + (1/6)*x^3*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/6)*x^3*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x^2*PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b) - (x^2*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b) - (x*PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b^2) + (x*PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b^2) + PolyLog[4, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(8*b^3) - PolyLog[4, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(8*b^3)} -{ArcCoth[c + d*Coth[a + b*x]]*x^1, x, 9, (1/2)*x^2*ArcCoth[c + d*Coth[a + b*x]] + (1/4)*x^2*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/4)*x^2*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + (x*PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)])/(4*b) - (x*PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(4*b) - PolyLog[3, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(8*b^2) + PolyLog[3, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(8*b^2)} -{ArcCoth[c + d*Coth[a + b*x]]*x^0, x, 7, x*ArcCoth[c + d*Coth[a + b*x]] + (1/2)*x*Log[1 - ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)] - (1/2)*x*Log[1 - ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)] + PolyLog[2, ((1 - c - d)*E^(2*a + 2*b*x))/(1 - c + d)]/(4*b) - PolyLog[2, ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)]/(4*b)} -{ArcCoth[c + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[c + d*Coth[a + b*x]]/x, x]} - - -{ArcCoth[1 + d + d*Coth[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcCoth[1 + d + d*Coth[a + b*x]] - (1/8)*x^4*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + (3*x^2*PolyLog[3, (1 + d)*E^(2*a + 2*b*x)])/(8*b^2) - (3*x*PolyLog[4, (1 + d)*E^(2*a + 2*b*x)])/(8*b^3) + (3*PolyLog[5, (1 + d)*E^(2*a + 2*b*x)])/(16*b^4)} -{ArcCoth[1 + d + d*Coth[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCoth[1 + d + d*Coth[a + b*x]] - (1/6)*x^3*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + (x*PolyLog[3, (1 + d)*E^(2*a + 2*b*x)])/(4*b^2) - PolyLog[4, (1 + d)*E^(2*a + 2*b*x)]/(8*b^3)} -{ArcCoth[1 + d + d*Coth[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCoth[1 + d + d*Coth[a + b*x]] - (1/4)*x^2*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, (1 + d)*E^(2*a + 2*b*x)])/(4*b) + PolyLog[3, (1 + d)*E^(2*a + 2*b*x)]/(8*b^2)} -{ArcCoth[1 + d + d*Coth[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcCoth[1 + d + d*Coth[a + b*x]] - (1/2)*x*Log[1 - (1 + d)*E^(2*a + 2*b*x)] - PolyLog[2, (1 + d)*E^(2*a + 2*b*x)]/(4*b)} -{ArcCoth[1 + d + d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 + d + d*Coth[a + b*x]]/x, x]} - - -{ArcCoth[1 - d - d*Coth[a + b*x]]*x^3, x, 8, (b*x^5)/20 + (1/4)*x^4*ArcCoth[1 - d - d*Coth[a + b*x]] - (1/8)*x^4*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x^3*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + (3*x^2*PolyLog[3, (1 - d)*E^(2*a + 2*b*x)])/(8*b^2) - (3*x*PolyLog[4, (1 - d)*E^(2*a + 2*b*x)])/(8*b^3) + (3*PolyLog[5, (1 - d)*E^(2*a + 2*b*x)])/(16*b^4)} -{ArcCoth[1 - d - d*Coth[a + b*x]]*x^2, x, 7, (b*x^4)/12 + (1/3)*x^3*ArcCoth[1 - d - d*Coth[a + b*x]] - (1/6)*x^3*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x^2*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + (x*PolyLog[3, (1 - d)*E^(2*a + 2*b*x)])/(4*b^2) - PolyLog[4, (1 - d)*E^(2*a + 2*b*x)]/(8*b^3)} -{ArcCoth[1 - d - d*Coth[a + b*x]]*x^1, x, 6, (b*x^3)/6 + (1/2)*x^2*ArcCoth[1 - d - d*Coth[a + b*x]] - (1/4)*x^2*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - (x*PolyLog[2, (1 - d)*E^(2*a + 2*b*x)])/(4*b) + PolyLog[3, (1 - d)*E^(2*a + 2*b*x)]/(8*b^2)} -{ArcCoth[1 - d - d*Coth[a + b*x]]*x^0, x, 5, (b*x^2)/2 + x*ArcCoth[1 - d - d*Coth[a + b*x]] - (1/2)*x*Log[1 - (1 - d)*E^(2*a + 2*b*x)] - PolyLog[2, (1 - d)*E^(2*a + 2*b*x)]/(4*b)} -{ArcCoth[1 - d - d*Coth[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 - d - d*Coth[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Trig[a+b x]]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Tan[a+b x]]*) - - -{(e + f*x)^3*ArcCoth[Tan[a + b*x]], x, 12, ((e + f*x)^4*ArcCoth[Tan[a + b*x]])/(4*f) + (I*(e + f*x)^4*ArcTan[E^(2*I*(a + b*x))])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (3*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (3*f*(e + f*x)^2*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3) - (3*f^3*PolyLog[5, (-I)*E^(2*I*(a + b*x))])/(16*b^4) + (3*f^3*PolyLog[5, I*E^(2*I*(a + b*x))])/(16*b^4)} -{(e + f*x)^2*ArcCoth[Tan[a + b*x]], x, 10, ((e + f*x)^3*ArcCoth[Tan[a + b*x]])/(3*f) + (I*(e + f*x)^3*ArcTan[E^(2*I*(a + b*x))])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*(e + f*x)*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(4*b^2) - (f*(e + f*x)*PolyLog[3, I*E^(2*I*(a + b*x))])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3)} -{(e + f*x)^1*ArcCoth[Tan[a + b*x]], x, 8, ((e + f*x)^2*ArcCoth[Tan[a + b*x]])/(2*f) + (I*(e + f*x)^2*ArcTan[E^(2*I*(a + b*x))])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (f*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2)} -{(e + f*x)^0*ArcCoth[Tan[a + b*x]], x, 6, x*ArcCoth[Tan[a + b*x]] + I*x*ArcTan[E^(2*I*(a + b*x))] - (I*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b)} -{ArcCoth[Tan[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcCoth[Tan[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcCoth[c + d*Tan[a + b*x]], x, 11, (1/3)*x^3*ArcCoth[c + d*Tan[a + b*x]] + (1/6)*x^3*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/6)*x^3*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*x^2*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*x^2*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b) + (x*PolyLog[3, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b^2) - (x*PolyLog[3, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b^2) + (I*PolyLog[4, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(8*b^3) - (I*PolyLog[4, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(8*b^3)} -{x^1*ArcCoth[c + d*Tan[a + b*x]], x, 9, (1/2)*x^2*ArcCoth[c + d*Tan[a + b*x]] + (1/4)*x^2*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/4)*x^2*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*x*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*x*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b) + PolyLog[3, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))]/(8*b^2) - PolyLog[3, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))]/(8*b^2)} -{x^0*ArcCoth[c + d*Tan[a + b*x]], x, 7, x*ArcCoth[c + d*Tan[a + b*x]] + (1/2)*x*Log[1 + ((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d)] - (1/2)*x*Log[1 + ((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d)] - (I*PolyLog[2, -(((1 - c + I*d)*E^(2*I*a + 2*I*b*x))/(1 - c - I*d))])/(4*b) + (I*PolyLog[2, -(((1 + c - I*d)*E^(2*I*a + 2*I*b*x))/(1 + c + I*d))])/(4*b)} -{ArcCoth[c + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[c + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCoth[1 - I*d + d*Tan[a + b*x]] - (1/6)*x^3*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - (x*PolyLog[3, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b^2) - (I*PolyLog[4, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(8*b^3)} -{x^1*ArcCoth[1 - I*d + d*Tan[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCoth[1 - I*d + d*Tan[a + b*x]] - (1/4)*x^2*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - PolyLog[3, -((1 - I*d)*E^(2*I*a + 2*I*b*x))]/(8*b^2)} -{x^0*ArcCoth[1 - I*d + d*Tan[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCoth[1 - I*d + d*Tan[a + b*x]] - (1/2)*x*Log[1 + (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, -((1 - I*d)*E^(2*I*a + 2*I*b*x))])/(4*b)} -{ArcCoth[1 - I*d + d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 - I*d + d*Tan[a + b*x]]/x, x]} - - -{x^2*ArcCoth[1 + I*d - d*Tan[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCoth[1 + I*d - d*Tan[a + b*x]] - (1/6)*x^3*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - (x*PolyLog[3, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b^2) - (I*PolyLog[4, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(8*b^3)} -{x^1*ArcCoth[1 + I*d - d*Tan[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCoth[1 + I*d - d*Tan[a + b*x]] - (1/4)*x^2*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b) - PolyLog[3, -((1 + I*d)*E^(2*I*a + 2*I*b*x))]/(8*b^2)} -{x^0*ArcCoth[1 + I*d - d*Tan[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCoth[1 + I*d - d*Tan[a + b*x]] - (1/2)*x*Log[1 + (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, -((1 + I*d)*E^(2*I*a + 2*I*b*x))])/(4*b)} -{ArcCoth[1 + I*d - d*Tan[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 + I*d - d*Tan[a + b*x]]/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcCoth[c+d Cot[a+b x]]*) - - -{(e + f*x)^3*ArcCoth[Cot[a + b*x]], x, 12, ((e + f*x)^4*ArcCoth[Cot[a + b*x]])/(4*f) + (I*(e + f*x)^4*ArcTan[E^(2*I*(a + b*x))])/(4*f) - (I*(e + f*x)^3*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^3*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (3*f*(e + f*x)^2*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (3*f*(e + f*x)^2*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2) + (3*I*f^2*(e + f*x)*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (3*I*f^2*(e + f*x)*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3) - (3*f^3*PolyLog[5, (-I)*E^(2*I*(a + b*x))])/(16*b^4) + (3*f^3*PolyLog[5, I*E^(2*I*(a + b*x))])/(16*b^4)} -{(e + f*x)^2*ArcCoth[Cot[a + b*x]], x, 10, ((e + f*x)^3*ArcCoth[Cot[a + b*x]])/(3*f) + (I*(e + f*x)^3*ArcTan[E^(2*I*(a + b*x))])/(3*f) - (I*(e + f*x)^2*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)^2*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*(e + f*x)*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(4*b^2) - (f*(e + f*x)*PolyLog[3, I*E^(2*I*(a + b*x))])/(4*b^2) + (I*f^2*PolyLog[4, (-I)*E^(2*I*(a + b*x))])/(8*b^3) - (I*f^2*PolyLog[4, I*E^(2*I*(a + b*x))])/(8*b^3)} -{(e + f*x)^1*ArcCoth[Cot[a + b*x]], x, 8, ((e + f*x)^2*ArcCoth[Cot[a + b*x]])/(2*f) + (I*(e + f*x)^2*ArcTan[E^(2*I*(a + b*x))])/(2*f) - (I*(e + f*x)*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*(e + f*x)*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b) + (f*PolyLog[3, (-I)*E^(2*I*(a + b*x))])/(8*b^2) - (f*PolyLog[3, I*E^(2*I*(a + b*x))])/(8*b^2)} -{(e + f*x)^0*ArcCoth[Cot[a + b*x]], x, 6, x*ArcCoth[Cot[a + b*x]] + I*x*ArcTan[E^(2*I*(a + b*x))] - (I*PolyLog[2, (-I)*E^(2*I*(a + b*x))])/(4*b) + (I*PolyLog[2, I*E^(2*I*(a + b*x))])/(4*b)} -{ArcCoth[Cot[a + b*x]]/(e + f*x)^1, x, 0, CannotIntegrate[ArcCoth[Cot[a + b*x]]/(e + f*x), x]} - - -{x^2*ArcCoth[c + d*Cot[a + b*x]], x, 11, (1/3)*x^3*ArcCoth[c + d*Cot[a + b*x]] + (1/6)*x^3*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/6)*x^3*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*x^2*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*x^2*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b) + (x*PolyLog[3, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b^2) - (x*PolyLog[3, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b^2) + (I*PolyLog[4, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(8*b^3) - (I*PolyLog[4, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(8*b^3)} -{x^1*ArcCoth[c + d*Cot[a + b*x]], x, 9, (1/2)*x^2*ArcCoth[c + d*Cot[a + b*x]] + (1/4)*x^2*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/4)*x^2*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*x*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*x*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b) + PolyLog[3, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)]/(8*b^2) - PolyLog[3, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)]/(8*b^2)} -{x^0*ArcCoth[c + d*Cot[a + b*x]], x, 7, x*ArcCoth[c + d*Cot[a + b*x]] + (1/2)*x*Log[1 - ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)] - (1/2)*x*Log[1 - ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)] - (I*PolyLog[2, ((1 - c - I*d)*E^(2*I*a + 2*I*b*x))/(1 - c + I*d)])/(4*b) + (I*PolyLog[2, ((1 + c + I*d)*E^(2*I*a + 2*I*b*x))/(1 + c - I*d)])/(4*b)} -{ArcCoth[c + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[c + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCoth[1 + I*d + d*Cot[a + b*x]] - (1/6)*x^3*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - (x*PolyLog[3, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b^2) - (I*PolyLog[4, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(8*b^3)} -{x^1*ArcCoth[1 + I*d + d*Cot[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCoth[1 + I*d + d*Cot[a + b*x]] - (1/4)*x^2*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - PolyLog[3, (1 + I*d)*E^(2*I*a + 2*I*b*x)]/(8*b^2)} -{x^0*ArcCoth[1 + I*d + d*Cot[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCoth[1 + I*d + d*Cot[a + b*x]] - (1/2)*x*Log[1 - (1 + I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, (1 + I*d)*E^(2*I*a + 2*I*b*x)])/(4*b)} -{ArcCoth[1 + I*d + d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 + I*d + d*Cot[a + b*x]]/x, x]} - - -{x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]], x, 7, (1/12)*I*b*x^4 + (1/3)*x^3*ArcCoth[1 - I*d - d*Cot[a + b*x]] - (1/6)*x^3*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x^2*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - (x*PolyLog[3, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b^2) - (I*PolyLog[4, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(8*b^3)} -{x^1*ArcCoth[1 - I*d - d*Cot[a + b*x]], x, 6, (1/6)*I*b*x^3 + (1/2)*x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]] - (1/4)*x^2*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*x*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b) - PolyLog[3, (1 - I*d)*E^(2*I*a + 2*I*b*x)]/(8*b^2)} -{x^0*ArcCoth[1 - I*d - d*Cot[a + b*x]], x, 5, (1/2)*I*b*x^2 + x*ArcCoth[1 - I*d - d*Cot[a + b*x]] - (1/2)*x*Log[1 - (1 - I*d)*E^(2*I*a + 2*I*b*x)] + (I*PolyLog[2, (1 - I*d)*E^(2*I*a + 2*I*b*x)])/(4*b)} -{ArcCoth[1 - I*d - d*Cot[a + b*x]]/x^1, x, 0, CannotIntegrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e Log[f x^m]) (a+b ArcCoth[c x^n])*) - - -{((a + b*ArcCoth[c*x^n])*(d + e*Log[f*x^m]))/x, x, 11, a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) + (b*d*PolyLog[2, -(1/(x^n*c))])/(2*n) + (b*e*Log[f*x^m]*PolyLog[2, -(1/(x^n*c))])/(2*n) - (b*d*PolyLog[2, 1/(x^n*c)])/(2*n) - (b*e*Log[f*x^m]*PolyLog[2, 1/(x^n*c)])/(2*n) + (b*e*m*PolyLog[3, -(1/(x^n*c))])/(2*n^2) - (b*e*m*PolyLog[3, 1/(x^n*c)])/(2*n^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m (d+e Log[f+g x^2]) (a+b ArcCoth[c x])*) - - -{x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, If[$VersionNumber<9, 18, 23], If[$VersionNumber<9, (b*(6*d - 11*e)*x)/(36*c^5) - (23*b*e*x)/(45*c^5) + (b*(6*d - 5*e)*x^3)/(108*c^3) - (8*b*e*x^3)/(135*c^3) + (b*(3*d - e)*x^5)/(90*c) - (b*e*x^5)/(75*c) - (e*x^2*(a + b*ArcCoth[c*x]))/(6*c^4) - (e*x^4*(a + b*ArcCoth[c*x]))/(12*c^2) - (1/18)*e*x^6*(a + b*ArcCoth[c*x]) - (b*(6*d - 11*e)*ArcTanh[c*x])/(36*c^6) + (23*b*e*ArcTanh[c*x])/(45*c^6) + (b*e*x*Log[1 - c^2*x^2])/(6*c^5) + (b*e*x^3*Log[1 - c^2*x^2])/(18*c^3) + (b*e*x^5*Log[1 - c^2*x^2])/(30*c) - (e*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(6*c^6) + (1/6)*x^6*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), (b*(3*d - e)*x)/(18*c^5) - (137*b*e*x)/(180*c^5) + (b*(3*d - e)*x^3)/(54*c^3) - (47*b*e*x^3)/(540*c^3) + (b*(3*d - e)*x^5)/(90*c) - (b*e*x^5)/(75*c) - (e*x^2*(a + b*ArcCoth[c*x]))/(6*c^4) - (e*x^4*(a + b*ArcCoth[c*x]))/(12*c^2) - (1/18)*e*x^6*(a + b*ArcCoth[c*x]) - (b*(3*d - e)*ArcTanh[c*x])/(18*c^6) + (137*b*e*ArcTanh[c*x])/(180*c^6) + (b*e*x*Log[1 - c^2*x^2])/(6*c^5) + (b*e*x^3*Log[1 - c^2*x^2])/(18*c^3) + (b*e*x^5*Log[1 - c^2*x^2])/(30*c) - (e*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(6*c^6) + (1/6)*x^6*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])]} -{x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, 14, (b*(2*d - 3*e)*x)/(8*c^3) - (2*b*e*x)/(3*c^3) + (b*(2*d - e)*x^3)/(24*c) - (b*e*x^3)/(18*c) - (e*x^2*(a + b*ArcCoth[c*x]))/(4*c^2) - (1/8)*e*x^4*(a + b*ArcCoth[c*x]) - (b*(2*d - 3*e)*ArcTanh[c*x])/(8*c^4) + (2*b*e*ArcTanh[c*x])/(3*c^4) + (b*e*x*Log[1 - c^2*x^2])/(4*c^3) + (b*e*x^3*Log[1 - c^2*x^2])/(12*c) - (e*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(4*c^4) + (1/4)*x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])} -{x^1*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, 7, (b*(d - e)*x)/(2*c) - (b*e*x)/c + (1/2)*d*x^2*(a + b*ArcCoth[c*x]) - (1/2)*e*x^2*(a + b*ArcCoth[c*x]) - (b*(d - e)*ArcTanh[c*x])/(2*c^2) + (b*e*ArcTanh[c*x])/c^2 + (b*e*x*Log[1 - c^2*x^2])/(2*c) - (e*(1 - c^2*x^2)*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(2*c^2)} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^1, x, 21, (-(1/2))*b*e*Log[1 + 1/(c*x)]^2*Log[-(1/(c*x))] + (1/2)*b*e*Log[1 - 1/(c*x)]^2*Log[1/(c*x)] + a*d*Log[x] - b*e*Log[(c + 1/x)/c]*PolyLog[2, (c + 1/x)/c] + b*e*Log[1 - 1/(c*x)]*PolyLog[2, 1 - 1/(c*x)] + (1/2)*b*d*PolyLog[2, -(1/(c*x))] + (1/2)*b*e*Log[(-c^2)*x^2]*PolyLog[2, -(1/(c*x))] - (1/2)*b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log[(-c^2)*x^2] - Log[1 - c^2*x^2])*PolyLog[2, -(1/(c*x))] - (1/2)*b*d*PolyLog[2, 1/(c*x)] - (1/2)*b*e*Log[(-c^2)*x^2]*PolyLog[2, 1/(c*x)] + (1/2)*b*e*(Log[1 - 1/(c*x)] + Log[1 + 1/(c*x)] + Log[(-c^2)*x^2] - Log[1 - c^2*x^2])*PolyLog[2, 1/(c*x)] - (1/2)*a*e*PolyLog[2, c^2*x^2] + b*e*PolyLog[3, (c + 1/x)/c] - b*e*PolyLog[3, 1 - 1/(c*x)] + b*e*PolyLog[3, -(1/(c*x))] - b*e*PolyLog[3, 1/(c*x)]} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^3, x, 13, (-(1/2))*b*c^2*e*ArcCoth[c*x]^2 - (1/2)*b*c^2*e*ArcTanh[c*x]^2 - a*c^2*e*Log[x] + b*c^2*e*ArcTanh[c*x]*Log[2/(1 - c*x)] + (1/2)*(a + b)*c^2*e*Log[1 - c*x] + (1/2)*(a - b)*c^2*e*Log[1 + c*x] - (b*c*(d + e*Log[1 - c^2*x^2]))/(2*x) - ((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/(2*x^2) + (1/2)*b*c^2*ArcTanh[c*x]*(d + e*Log[1 - c^2*x^2]) - b*c^2*e*ArcCoth[c*x]*Log[2 - 2/(1 + c*x)] + (1/2)*b*c^2*e*PolyLog[2, 1 - 2/(1 - c*x)] + (1/2)*b*c^2*e*PolyLog[2, -1 + 2/(1 + c*x)]} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^5, x, 17, (a*c^2*e)/(4*x^2) + (5*b*c^3*e)/(12*x) + (b*c^2*e*ArcCoth[c*x])/(4*x^2) - (1/4)*b*c^4*e*ArcCoth[c*x]^2 - (1/4)*b*c^4*e*ArcTanh[c*x] - (1/4)*b*c^4*e*ArcTanh[c*x]^2 - (1/2)*a*c^4*e*Log[x] + (1/2)*b*c^4*e*ArcTanh[c*x]*Log[2/(1 - c*x)] + (1/12)*(3*a + 4*b)*c^4*e*Log[1 - c*x] + (1/12)*(3*a - 4*b)*c^4*e*Log[1 + c*x] - (b*c*(d + e*Log[1 - c^2*x^2]))/(12*x^3) - (b*c^3*(d + e*Log[1 - c^2*x^2]))/(4*x) - ((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/(4*x^4) + (1/4)*b*c^4*ArcTanh[c*x]*(d + e*Log[1 - c^2*x^2]) - (1/2)*b*c^4*e*ArcCoth[c*x]*Log[2 - 2/(1 + c*x)] + (1/4)*b*c^4*e*PolyLog[2, 1 - 2/(1 - c*x)] + (1/4)*b*c^4*e*PolyLog[2, -1 + 2/(1 + c*x)]} - -{x^4*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, 26, -((2*a*e*x)/(5*c^4)) - (77*b*e*x^2)/(300*c^3) - (2*a*e*x^3)/(15*c^2) - (9*b*e*x^4)/(200*c) - (2/25)*a*e*x^5 - (2*b*e*x*ArcCoth[c*x])/(5*c^4) - (2*b*e*x^3*ArcCoth[c*x])/(15*c^2) - (2/25)*b*e*x^5*ArcCoth[c*x] + (b*e*ArcCoth[c*x]^2)/(5*c^5) - ((4*a + 3*b)*e*Log[1 - c*x])/(20*c^5) + ((4*a - 3*b)*e*Log[1 + c*x])/(20*c^5) - (23*b*e*Log[1 - c^2*x^2])/(75*c^5) - (b*e*Log[1 - c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(10*c^3) + (b*x^4*(d + e*Log[1 - c^2*x^2]))/(20*c) + (1/5)*x^5*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(10*c^5)} -{x^2*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, 21, -((2*a*e*x)/(3*c^2)) - (5*b*e*x^2)/(18*c) - (2/9)*a*e*x^3 - (2*b*e*x*ArcCoth[c*x])/(3*c^2) - (2/9)*b*e*x^3*ArcCoth[c*x] + (b*e*ArcCoth[c*x]^2)/(3*c^3) - ((2*a + b)*e*Log[1 - c*x])/(6*c^3) + ((2*a - b)*e*Log[1 + c*x])/(6*c^3) - (4*b*e*Log[1 - c^2*x^2])/(9*c^3) - (b*e*Log[1 - c^2*x^2]^2)/(12*c^3) + (b*x^2*(d + e*Log[1 - c^2*x^2]))/(6*c) + (1/3)*x^3*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*Log[1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(6*c^3)} -{x^0*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]), x, 9, -2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (e*(a + b*ArcCoth[c*x])^2)/(b*c) - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]) + (b*(d + e*Log[1 - c^2*x^2])^2)/(4*c*e)} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^2, x, 6, -((c*e*(a + b*ArcCoth[c*x])^2)/b) - ((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/x + (1/2)*b*c*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/2)*b*c*e*PolyLog[2, 1/(1 - c^2*x^2)]} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^4, x, 15, (2*c^2*e*(a + b*ArcCoth[c*x]))/(3*x) - (c^3*e*(a + b*ArcCoth[c*x])^2)/(3*b) - b*c^3*e*Log[x] + (1/3)*b*c^3*e*Log[1 - c^2*x^2] - (b*c*(1 - c^2*x^2)*(d + e*Log[1 - c^2*x^2]))/(6*x^2) - ((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/(3*x^3) + (1/6)*b*c^3*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/6)*b*c^3*e*PolyLog[2, 1/(1 - c^2*x^2)]} -{(a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2])/x^6, x, 24, (7*b*c^3*e)/(60*x^2) + (2*c^2*e*(a + b*ArcCoth[c*x]))/(15*x^3) + (2*c^4*e*(a + b*ArcCoth[c*x]))/(5*x) - (c^5*e*(a + b*ArcCoth[c*x])^2)/(5*b) - (5/6)*b*c^5*e*Log[x] + (19/60)*b*c^5*e*Log[1 - c^2*x^2] - (b*c*(d + e*Log[1 - c^2*x^2]))/(20*x^4) - (b*c^3*(1 - c^2*x^2)*(d + e*Log[1 - c^2*x^2]))/(10*x^2) - ((a + b*ArcCoth[c*x])*(d + e*Log[1 - c^2*x^2]))/(5*x^5) + (1/10)*b*c^5*(d + e*Log[1 - c^2*x^2])*Log[1 - 1/(1 - c^2*x^2)] - (1/10)*b*c^5*e*PolyLog[2, 1/(1 - c^2*x^2)]} - - -{x^1*(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]), x, If[$VersionNumber<11, 21, 22], (b*(d - e)*x)/(2*c) - (b*e*x)/c + (1/2)*d*x^2*(a + b*ArcCoth[c*x]) - (1/2)*e*x^2*(a + b*ArcCoth[c*x]) + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c*Sqrt[g]) - (b*(d - e)*ArcTanh[c*x])/(2*c^2) - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[2/(1 + c*x)])/(c^2*g) + (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(2*c^2*g) + (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(2*c^2*g) + (b*e*x*Log[f + g*x^2])/(2*c) + (e*(f + g*x^2)*(a + b*ArcCoth[c*x])*Log[f + g*x^2])/(2*g) - (b*e*(c^2*f + g)*ArcTanh[c*x]*Log[f + g*x^2])/(2*c^2*g) + (b*e*(c^2*f + g)*PolyLog[2, 1 - 2/(1 + c*x)])/(2*c^2*g) - (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*c^2*g) - (b*e*(c^2*f + g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*c^2*g)} -{x^0*(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]), x, 38, -2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 - 1/(c*x)])/Sqrt[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + 1/(c*x)])/Sqrt[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g] - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]) + (b*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/(2*c) + (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f + g)])/(2*c) - (I*b*e*Sqrt[f]*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[g]) + (I*b*e*Sqrt[f]*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[g])} -{(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2])/x^1, x, 6, b*e*CannotIntegrate[(ArcCoth[c*x]*Log[f + g*x^2])/x, x] + a*d*Log[x] + (1/2)*a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2] + (1/2)*b*d*PolyLog[2, -(1/(c*x))] - (1/2)*b*d*PolyLog[2, 1/(c*x)] + (1/2)*a*e*PolyLog[2, 1 + (g*x^2)/f]} -{(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2])/x^2, x, 38, (2*a*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - (b*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 - 1/(c*x)])/Sqrt[f] + (b*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + 1/(c*x)])/Sqrt[f] + (b*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[-((2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)))])/Sqrt[f] - (b*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[f] - ((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x + (1/2)*b*c*Log[-((g*x^2)/f)]*(d + e*Log[f + g*x^2]) - (1/2)*b*c*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]) - (1/2)*b*c*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f + g)] + (1/2)*b*c*e*PolyLog[2, 1 + (g*x^2)/f] - (I*b*e*Sqrt[g]*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]) + (I*b*e*Sqrt[g]*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f])} -{(a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2])/x^3, x, 32, (b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f + (b*e*g*ArcCoth[c*x]*Log[2/(1 + c*x)])/f + b*c^2*e*ArcTanh[c*x]*Log[2/(1 + c*x)] - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(2*f) - (1/2)*b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))] - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(2*f) - (1/2)*b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))] - (a*e*g*Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) - ((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/(2*x^2) + (1/2)*b*c^2*ArcTanh[c*x]*(d + e*Log[f + g*x^2]) + (b*e*g*PolyLog[2, -(1/(c*x))])/(2*f) - (b*e*g*PolyLog[2, 1/(c*x)])/(2*f) - (1/2)*b*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)] - (b*e*g*PolyLog[2, 1 - 2/(1 + c*x)])/(2*f) + (1/4)*b*c^2*e*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))] + (b*e*g*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*f) + (1/4)*b*c^2*e*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))] + (b*e*g*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*f)} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse hyperbolic cotangents of exponentials*) - - -{ArcCoth[E^x], x, 2, (1/2)*PolyLog[2, -E^(-x)] - (1/2)*PolyLog[2, E^(-x)]} -{x*ArcCoth[E^x], x, 7, (1/2)*x*PolyLog[2, -E^(-x)] - (1/2)*x*PolyLog[2, E^(-x)] + (1/2)*PolyLog[3, -E^(-x)] - (1/2)*PolyLog[3, E^(-x)]} -{x^2*ArcCoth[E^x], x, 9, (1/2)*x^2*PolyLog[2, -E^(-x)] - (1/2)*x^2*PolyLog[2, E^(-x)] + x*PolyLog[3, -E^(-x)] - x*PolyLog[3, E^(-x)] + PolyLog[4, -E^(-x)] - PolyLog[4, E^(-x)]} - - -{ArcCoth[E^(a + b*x)], x, 2, PolyLog[2, -E^(-a - b*x)]/(2*b) - PolyLog[2, E^(-a - b*x)]/(2*b)} -{x*ArcCoth[E^(a + b*x)], x, 7, (x*PolyLog[2, -E^(-a - b*x)])/(2*b) - (x*PolyLog[2, E^(-a - b*x)])/(2*b) + PolyLog[3, -E^(-a - b*x)]/(2*b^2) - PolyLog[3, E^(-a - b*x)]/(2*b^2)} -{x^2*ArcCoth[E^(a + b*x)], x, 9, (x^2*PolyLog[2, -E^(-a - b*x)])/(2*b) - (x^2*PolyLog[2, E^(-a - b*x)])/(2*b) + (x*PolyLog[3, -E^(-a - b*x)])/b^2 - (x*PolyLog[3, E^(-a - b*x)])/b^2 + PolyLog[4, -E^(-a - b*x)]/b^3 - PolyLog[4, E^(-a - b*x)]/b^3} - - -{ArcCoth[a + b*f^(c + d*x)], x, 6, -((ArcCoth[a + b*f^(c + d*x)]*Log[2/(1 + a + b*f^(c + d*x))])/(d*Log[f])) + (ArcCoth[a + b*f^(c + d*x)]*Log[(2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))])/(d*Log[f]) + PolyLog[2, 1 - 2/(1 + a + b*f^(c + d*x))]/(2*d*Log[f]) - PolyLog[2, 1 - (2*b*f^(c + d*x))/((1 - a)*(1 + a + b*f^(c + d*x)))]/(2*d*Log[f])} -{x*ArcCoth[a + b*f^(c + d*x)], x, 25, (1/4)*x^2*Log[1 - (b*f^(c + d*x))/(1 - a)] - (1/4)*x^2*Log[1 + (b*f^(c + d*x))/(1 + a)] - (1/4)*x^2*Log[1 - 1/(a + b*f^(c + d*x))] + (1/4)*x^2*Log[1 + 1/(a + b*f^(c + d*x))] + (x*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(2*d*Log[f]) - (x*PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f]) - PolyLog[3, (b*f^(c + d*x))/(1 - a)]/(2*d^2*Log[f]^2) + PolyLog[3, -((b*f^(c + d*x))/(1 + a))]/(2*d^2*Log[f]^2)} -{x^2*ArcCoth[a + b*f^(c + d*x)], x, 29, (1/6)*x^3*Log[1 - (b*f^(c + d*x))/(1 - a)] - (1/6)*x^3*Log[1 + (b*f^(c + d*x))/(1 + a)] - (1/6)*x^3*Log[1 - 1/(a + b*f^(c + d*x))] + (1/6)*x^3*Log[1 + 1/(a + b*f^(c + d*x))] + (x^2*PolyLog[2, (b*f^(c + d*x))/(1 - a)])/(2*d*Log[f]) - (x^2*PolyLog[2, -((b*f^(c + d*x))/(1 + a))])/(2*d*Log[f]) - (x*PolyLog[3, (b*f^(c + d*x))/(1 - a)])/(d^2*Log[f]^2) + (x*PolyLog[3, -((b*f^(c + d*x))/(1 + a))])/(d^2*Log[f]^2) + PolyLog[4, (b*f^(c + d*x))/(1 - a)]/(d^3*Log[f]^3) - PolyLog[4, -((b*f^(c + d*x))/(1 + a))]/(d^3*Log[f]^3)} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic cotangents*) - - -{1/((a - a*x^2)*(b - 2*b*ArcCoth[x])), x, 1, -(Log[1 - 2*ArcCoth[x]]/(2*a*b))} - - -{x^3*ArcCoth[a + b*x^4], x, 4, ((a + b*x^4)*ArcCoth[a + b*x^4])/(4*b) + Log[1 - (a + b*x^4)^2]/(8*b)} - - -{x^(n-1)*ArcCoth[a + b*x^n], x, 4, ((a + b*x^n)*ArcCoth[a + b*x^n])/(b*n) + Log[1 - (a + b*x^n)^2]/(2*b*n)} - - -{E^(c*(a + b*x))*ArcCoth[Sinh[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcCoth[Sinh[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c)} -{E^(c*(a + b*x))*ArcCoth[Cosh[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcCoth[Cosh[c*(a + b*x)]])/(b*c) + Log[1 - E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcCoth[Tanh[a*c + b*c*x]], x, 3, -(E^(a*c + b*c*x)/(b*c)) + (E^(a*c + b*c*x)*ArcCoth[Tanh[c*(a + b*x)]])/(b*c)} -{E^(c*(a + b*x))*ArcCoth[Coth[a*c + b*c*x]], x, 3, -(E^(a*c + b*c*x)/(b*c)) + (E^(a*c + b*c*x)*ArcCoth[Coth[c*(a + b*x)]])/(b*c)} -{E^(c*(a + b*x))*ArcCoth[Sech[a*c + b*c*x]], x, 5, (E^(a*c + b*c*x)*ArcCoth[Sech[c*(a + b*x)]])/(b*c) + Log[1 - E^(2*c*(a + b*x))]/(b*c)} -{E^(c*(a + b*x))*ArcCoth[Csch[a*c + b*c*x]], x, 8, (E^(a*c + b*c*x)*ArcCoth[Csch[c*(a + b*x)]])/(b*c) + ((1 - Sqrt[2])*Log[3 - 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c) + ((1 + Sqrt[2])*Log[3 + 2*Sqrt[2] - E^(2*c*(a + b*x))])/(2*b*c)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.2 Exponentials of inverse hyperbolic cotangent functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.2 Exponentials of inverse hyperbolic cotangent functions.m deleted file mode 100644 index 021cafb..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.2 Exponentials of inverse hyperbolic cotangent functions.m +++ /dev/null @@ -1,1480 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands involving exponentials of inverse hyperbolic cotangents*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x])*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*x^3, x, 8, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^3) + (3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(8*a^2) + (Sqrt[1 - 1/(a^2*x^2)]*x^3)/(3*a) + (1/4)*Sqrt[1 - 1/(a^2*x^2)]*x^4 + (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a^4)} -{E^ArcCoth[a*x]*x^2, x, 7, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) + (Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*a) + (1/3)*Sqrt[1 - 1/(a^2*x^2)]*x^3 + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(2*a^3)} -{E^ArcCoth[a*x]*x^1, x, 6, (Sqrt[1 - 1/(a^2*x^2)]*x)/a + (1/2)*Sqrt[1 - 1/(a^2*x^2)]*x^2 + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(2*a^2)} -{E^ArcCoth[a*x], x, 5, Sqrt[1 - 1/(a^2*x^2)]*x + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/a} -{E^ArcCoth[a*x]/x^1, x, 6, -ArcCsc[a*x] + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]} -{E^ArcCoth[a*x]/x^2, x, 3, a*Sqrt[1 - 1/(a^2*x^2)] - a*ArcCsc[a*x]} -{E^ArcCoth[a*x]/x^3, x, 3, (1/2)*a*Sqrt[1 - 1/(a^2*x^2)]*(2*a + 1/x) - (1/2)*a^2*ArcCsc[a*x]} -{E^ArcCoth[a*x]/x^4, x, 7, a^3*Sqrt[1 - 1/(a^2*x^2)] - (1/3)*a^3*(1 - 1/(a^2*x^2))^(3/2) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (1/2)*a^3*ArcCsc[a*x]} -{E^ArcCoth[a*x]/x^5, x, 5, (1/24)*a^3*Sqrt[1 - 1/(a^2*x^2)]*(16*a + 9/x) + (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(3*x^2) - (3/8)*a^4*ArcCsc[a*x]} - - -{E^(2*ArcCoth[a*x])*x^3, x, 4, (2*x)/a^3 + x^2/a^2 + (2*x^3)/(3*a) + x^4/4 + (2*Log[1 - a*x])/a^4} -{E^(2*ArcCoth[a*x])*x^2, x, 4, (2*x)/a^2 + x^2/a + x^3/3 + (2*Log[1 - a*x])/a^3} -{E^(2*ArcCoth[a*x])*x^1, x, 4, (2*x)/a + x^2/2 + (2*Log[1 - a*x])/a^2} -{E^(2*ArcCoth[a*x]), x, 4, x + (2*Log[1 - a*x])/a} -{E^(2*ArcCoth[a*x])/x^1, x, 4, -Log[x] + 2*Log[1 - a*x]} -{E^(2*ArcCoth[a*x])/x^2, x, 4, x^(-1) - 2*a*Log[x] + 2*a*Log[1 - a*x]} -{E^(2*ArcCoth[a*x])/x^3, x, 4, 1/(2*x^2) + (2*a)/x - 2*a^2*Log[x] + 2*a^2*Log[1 - a*x]} -{E^(2*ArcCoth[a*x])/x^4, x, 4, 1/(3*x^3) + a/x^2 + (2*a^2)/x - 2*a^3*Log[x] + 2*a^3*Log[1 - a*x]} - - -{E^(3*ArcCoth[a*x])*x^2, x, 14, -((4*Sqrt[1 - 1/(a^2*x^2)])/(a^2*(a - 1/x))) + (14*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) + (3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*a) + (1/3)*Sqrt[1 - 1/(a^2*x^2)]*x^3 + (11*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^3)} -{E^(3*ArcCoth[a*x])*x^1, x, 12, -((4*Sqrt[1 - 1/(a^2*x^2)])/(a*(a - 1/x))) + (3*Sqrt[1 - 1/(a^2*x^2)]*x)/a + (1/2)*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (9*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^2)} -{E^(3*ArcCoth[a*x]), x, 8, -((4*Sqrt[1 - 1/(a^2*x^2)])/(a - 1/x)) + Sqrt[1 - 1/(a^2*x^2)]*x + (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^(3*ArcCoth[a*x])/x^1, x, 8, -((4*a*Sqrt[1 - 1/(a^2*x^2)])/(a - 1/x)) + ArcCsc[a*x] + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]} -{E^(3*ArcCoth[a*x])/x^2, x, 5, -3*a*Sqrt[1 - 1/(a^2*x^2)] - (2*(a + 1/x)^2)/(a*Sqrt[1 - 1/(a^2*x^2)]) + 3*a*ArcCsc[a*x]} -{E^(3*ArcCoth[a*x])/x^3, x, 9, (-(9/2))*a^2*Sqrt[1 - 1/(a^2*x^2)] - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a - 1/x)^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/2))/(2*(a - 1/x)) + (9/2)*a^2*ArcCsc[a*x]} -{E^(3*ArcCoth[a*x])/x^4, x, 10, -((a + 1/x)^3/Sqrt[1 - 1/(a^2*x^2)]) - (1/3)*a*Sqrt[1 - 1/(a^2*x^2)]*(3*a + 1/x)^2 - (1/6)*a^2*Sqrt[1 - 1/(a^2*x^2)]*(28*a + 3/x) + (11/2)*a^3*ArcCsc[a*x]} - - -{E^(4*ArcCoth[a*x])*x^3, x, 4, (12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4} -{E^(4*ArcCoth[a*x])*x^2, x, 4, (8*x)/a^2 + (2*x^2)/a + x^3/3 + 4/(a^3*(1 - a*x)) + (12*Log[1 - a*x])/a^3} -{E^(4*ArcCoth[a*x])*x^1, x, 4, (4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2} -{E^(4*ArcCoth[a*x]), x, 4, x + 4/(a*(1 - a*x)) + (4*Log[1 - a*x])/a} -{E^(4*ArcCoth[a*x])/x^1, x, 4, 4/(1 - a*x) + Log[x]} -{E^(4*ArcCoth[a*x])/x^2, x, 4, -x^(-1) + (4*a)/(1 - a*x) + 4*a*Log[x] - 4*a*Log[1 - a*x]} -{E^(4*ArcCoth[a*x])/x^3, x, 4, -1/(2*x^2) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*Log[1 - a*x]} -{E^(4*ArcCoth[a*x])/x^4, x, 4, -1/(3*x^3) - (2*a)/x^2 - (8*a^2)/x + (4*a^3)/(1 - a*x) + 12*a^3*Log[x] - 12*a^3*Log[1 - a*x]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3/E^ArcCoth[a*x], x, 8, -((2*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^3)) + (3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(8*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^3)/(3*a) + (1/4)*Sqrt[1 - 1/(a^2*x^2)]*x^4 + (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a^4)} -{x^2/E^ArcCoth[a*x], x, 7, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*a) + (1/3)*Sqrt[1 - 1/(a^2*x^2)]*x^3 - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(2*a^3)} -{x^1/E^ArcCoth[a*x], x, 6, -((Sqrt[1 - 1/(a^2*x^2)]*x)/a) + (1/2)*Sqrt[1 - 1/(a^2*x^2)]*x^2 + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(2*a^2)} -{E^(-ArcCoth[a*x]), x, 5, Sqrt[1 - 1/(a^2*x^2)]*x - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/a} -{1/(E^ArcCoth[a*x]*x^1), x, 6, ArcCsc[a*x] + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]} -{1/(E^ArcCoth[a*x]*x^2), x, 3, (-a)*Sqrt[1 - 1/(a^2*x^2)] - a*ArcCsc[a*x]} -{1/(E^ArcCoth[a*x]*x^3), x, 3, (1/2)*a*Sqrt[1 - 1/(a^2*x^2)]*(2*a - 1/x) + (1/2)*a^2*ArcCsc[a*x]} -{1/(E^ArcCoth[a*x]*x^4), x, 7, (-a^3)*Sqrt[1 - 1/(a^2*x^2)] + (1/3)*a^3*(1 - 1/(a^2*x^2))^(3/2) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(2*x) - (1/2)*a^3*ArcCsc[a*x]} -{1/(E^ArcCoth[a*x]*x^5), x, 5, (1/24)*a^3*Sqrt[1 - 1/(a^2*x^2)]*(16*a - 9/x) - (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) + (a^2*Sqrt[1 - 1/(a^2*x^2)])/(3*x^2) + (3/8)*a^4*ArcCsc[a*x]} - - -{x^3/E^(2*ArcCoth[a*x]), x, 4, (-2*x)/a^3 + x^2/a^2 - (2*x^3)/(3*a) + x^4/4 + (2*Log[1 + a*x])/a^4} -{x^2/E^(2*ArcCoth[a*x]), x, 4, (2*x)/a^2 - x^2/a + x^3/3 - (2*Log[1 + a*x])/a^3} -{x^1/E^(2*ArcCoth[a*x]), x, 4, (-2*x)/a + x^2/2 + (2*Log[1 + a*x])/a^2} -{E^(-2*ArcCoth[a*x]), x, 4, x - (2*Log[1 + a*x])/a} -{1/(E^(2*ArcCoth[a*x])*x^1), x, 4, -Log[x] + 2*Log[1 + a*x]} -{1/(E^(2*ArcCoth[a*x])*x^2), x, 4, x^(-1) + 2*a*Log[x] - 2*a*Log[1 + a*x]} -{1/(E^(2*ArcCoth[a*x])*x^3), x, 4, 1/(2*x^2) - (2*a)/x - 2*a^2*Log[x] + 2*a^2*Log[1 + a*x]} -{1/(E^(2*ArcCoth[a*x])*x^4), x, 4, 1/(3*x^3) - a/x^2 + (2*a^2)/x + 2*a^3*Log[x] - 2*a^3*Log[1 + a*x]} - - -{x^3/E^(3*ArcCoth[a*x]), x, 19, -((4*Sqrt[1 - 1/(a^2*x^2)])/(a^3*(a + 1/x))) - (6*Sqrt[1 - 1/(a^2*x^2)]*x)/a^3 + (19*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(8*a^2) - (Sqrt[1 - 1/(a^2*x^2)]*x^3)/a + (1/4)*Sqrt[1 - 1/(a^2*x^2)]*x^4 + (51*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a^4)} -{x^2/E^(3*ArcCoth[a*x]), x, 14, (4*Sqrt[1 - 1/(a^2*x^2)])/(a^2*(a + 1/x)) + (14*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*a^2) - (3*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*a) + (1/3)*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (11*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^3)} -{x^1/E^(3*ArcCoth[a*x]), x, 12, -((4*Sqrt[1 - 1/(a^2*x^2)])/(a*(a + 1/x))) - (3*Sqrt[1 - 1/(a^2*x^2)]*x)/a + (1/2)*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (9*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a^2)} -{E^(-3*ArcCoth[a*x]), x, 8, (4*Sqrt[1 - 1/(a^2*x^2)])/(a + 1/x) + Sqrt[1 - 1/(a^2*x^2)]*x - (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{1/(E^(3*ArcCoth[a*x])*x^1), x, 8, -((4*a*Sqrt[1 - 1/(a^2*x^2)])/(a + 1/x)) - ArcCsc[a*x] + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]} -{1/(E^(3*ArcCoth[a*x])*x^2), x, 5, 3*a*Sqrt[1 - 1/(a^2*x^2)] + (2*(a - 1/x)^2)/(a*Sqrt[1 - 1/(a^2*x^2)]) + 3*a*ArcCsc[a*x]} -{1/(E^(3*ArcCoth[a*x])*x^3), x, 9, (-(9/2))*a^2*Sqrt[1 - 1/(a^2*x^2)] - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a + 1/x)^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/2))/(2*(a + 1/x)) - (9/2)*a^2*ArcCsc[a*x]} -{1/(E^(3*ArcCoth[a*x])*x^4), x, 10, (1/6)*a^2*Sqrt[1 - 1/(a^2*x^2)]*(28*a - 3/x) + (a - 1/x)^3/Sqrt[1 - 1/(a^2*x^2)] + (1/3)*a*Sqrt[1 - 1/(a^2*x^2)]*(3*a - 1/x)^2 + (11/2)*a^3*ArcCsc[a*x]} -{1/(E^(3*ArcCoth[a*x])*x^5), x, 14, (-(27/4))*a^4*Sqrt[1 - 1/(a^2*x^2)] - (9/8)*a^3*Sqrt[1 - 1/(a^2*x^2)]*(2*a - 3/x) - (a*(a - 1/x)^3)/Sqrt[1 - 1/(a^2*x^2)] + (a*Sqrt[1 - 1/(a^2*x^2)])/(4*x^3) - (a^2*Sqrt[1 - 1/(a^2*x^2)])/x^2 - (51/8)*a^4*ArcCsc[a*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/2 ArcCoth[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcCoth[a*x]/2)*x^4, x, 11, (611*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4) + (269*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3) + (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(48*a^2) + (9*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^5 + (31*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (31*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{E^(ArcCoth[a*x]/2)*x^3, x, 10, (83*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(192*a^3) + (29*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2) + (7*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(24*a) + (1/4)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4 + (11*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (11*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{E^(ArcCoth[a*x]/2)*x^2, x, 9, (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(24*a^2) + (5*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3 + (3*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) + (3*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{E^(ArcCoth[a*x]/2)*x, x, 7, ((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(4*a) + (1/2)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4)*x^2 + ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*a^2) + ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*a^2)} -{E^(ArcCoth[a*x]/2), x, 6, (1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x + ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/a + ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/a} -{E^(ArcCoth[a*x]/2)/x, x, 17, (-Sqrt[2])*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{E^(ArcCoth[a*x]/2)/x^2, x, 13, a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4) - (a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) - (a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{E^(ArcCoth[a*x]/2)/x^3, x, 14, (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/2 - (a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) + (a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) + (a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{E^(ArcCoth[a*x]/2)/x^4, x, 15, (3*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/8 + (a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/12 + (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/(3*x) - (3*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (3*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (3*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) - (3*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -{E^((3*ArcCoth[a*x])/2)*x^4, x, 11, (557*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(640*a^4) + (157*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(320*a^3) + (5*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(16*a^2) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5 - (237*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (237*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{E^((3*ArcCoth[a*x])/2)*x^3, x, 10, (63*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(64*a^3) + (15*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(32*a^2) + (3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(8*a) + (1/4)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4 - (123*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (123*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{E^((3*ArcCoth[a*x])/2)*x^2, x, 9, (23*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) + (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3 - (17*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) + (17*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{E^((3*ArcCoth[a*x])/2)*x, x, 7, (3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(4*a) + (1/2)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4)*x^2 - (9*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2) + (9*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2)} -{E^((3*ArcCoth[a*x])/2), x, 6, (1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x - (3*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a + (3*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a} -{E^((3*ArcCoth[a*x])/2)/x, x, 17, (-Sqrt[2])*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{E^((3*ArcCoth[a*x])/2)/x^2, x, 13, a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) - (3*a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (3*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (3*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) + (3*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{E^((3*ArcCoth[a*x])/2)/x^3, x, 14, (3*a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 + (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/2 - (9*a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) + (9*a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (9*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (9*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{E^((3*ArcCoth[a*x])/2)/x^4, x, 15, (17*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/24 + (a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/4 + (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/(3*x) - (17*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (17*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (17*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) + (17*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -{E^((5*ArcCoth[a*x])/2)*x^4, x, 12, -((26111*(1 + 1/(a*x))^(1/4))/(1920*a^5*(1 - 1/(a*x))^(1/4))) + (5533*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4*(1 - 1/(a*x))^(1/4)) + (1189*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 - 1/(a*x))^(1/4)) + (181*(1 + 1/(a*x))^(1/4)*x^3)/(240*a^2*(1 - 1/(a*x))^(1/4)) + (21*(1 + 1/(a*x))^(1/4)*x^4)/(40*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^5)/(5*(1 - 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) + (1003*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{E^((5*ArcCoth[a*x])/2)*x^3, x, 11, -((2467*(1 + 1/(a*x))^(1/4))/(192*a^4*(1 - 1/(a*x))^(1/4))) + (521*(1 + 1/(a*x))^(1/4)*x)/(192*a^3*(1 - 1/(a*x))^(1/4)) + (113*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 - 1/(a*x))^(1/4)) + (17*(1 + 1/(a*x))^(1/4)*x^3)/(24*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) + (475*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (475*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{E^((5*ArcCoth[a*x])/2)*x^2, x, 10, -((287*(1 + 1/(a*x))^(1/4))/(24*a^3*(1 - 1/(a*x))^(1/4))) + (61*(1 + 1/(a*x))^(1/4)*x)/(24*a^2*(1 - 1/(a*x))^(1/4)) + (13*(1 + 1/(a*x))^(1/4)*x^2)/(12*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^3)/(3*(1 - 1/(a*x))^(1/4)) + (55*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) + (55*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{E^((5*ArcCoth[a*x])/2)*x, x, 8, -((25*(1 + 1/(a*x))^(1/4))/(2*a^2*(1 - 1/(a*x))^(1/4))) + (5*(1 + 1/(a*x))^(5/4)*x)/(4*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(9/4)*x^2)/(2*(1 - 1/(a*x))^(1/4)) + (25*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2) + (25*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2)} -{E^((5*ArcCoth[a*x])/2), x, 7, -((10*(1 + 1/(a*x))^(1/4))/(a*(1 - 1/(a*x))^(1/4))) + ((1 + 1/(a*x))^(5/4)*x)/(1 - 1/(a*x))^(1/4) + (5*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a + (5*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a} -{E^((5*ArcCoth[a*x])/2)/x, x, 19, -((8*(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4)) + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{E^((5*ArcCoth[a*x])/2)/x^2, x, 14, -5*a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4) - (4*a*(1 + 1/(a*x))^(5/4))/(1 - 1/(a*x))^(1/4) + (5*a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (5*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (5*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) + (5*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{E^((5*ArcCoth[a*x])/2)/x^3, x, 15, (-25*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 - (5*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/2 - (2*a^2*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) + (25*a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (25*a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (25*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (25*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{E^((5*ArcCoth[a*x])/2)/x^4, x, 16, (-55*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/8 - (11*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/4 - (2*a^3*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) - (a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(9/4))/3 + (55*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (55*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) + (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^4/E^(ArcCoth[a*x]/2), x, 11, (611*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(1920*a^4) - (269*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(960*a^3) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(48*a^2) - (9*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5 + (31*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) - (31*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{x^3/E^(ArcCoth[a*x]/2), x, 10, -((83*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(192*a^3)) + (29*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(96*a^2) - (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(24*a) + (1/4)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4 - (11*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (11*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{x^2/E^(ArcCoth[a*x]/2), x, 9, (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) - (5*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3 + (3*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) - (3*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{x/E^(ArcCoth[a*x]/2), x, 7, -(((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(4*a)) + (1/2)*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4)*x^2 - ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*a^2) + ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*a^2)} -{E^(-ArcCoth[a*x]/2), x, 6, (1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x + ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/a - ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/a} -{1/(E^(ArcCoth[a*x]/2)*x), x, 17, Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{1/(E^(ArcCoth[a*x]/2)*x^2), x, 13, -(a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)) - (a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) + (a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{1/(E^(ArcCoth[a*x]/2)*x^3), x, 14, (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 + (a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/2 + (a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) + (a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{1/(E^(ArcCoth[a*x]/2)*x^4), x, 15, (-3*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 - (a^3*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/12 + (a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/(3*x) - (3*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (3*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (3*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) + (3*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -{x^4/E^((3*ArcCoth[a*x])/2), x, 11, (557*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(640*a^4) - (157*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(320*a^3) + (5*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(16*a^2) - (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^5 - (237*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) - (237*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{x^3/E^((3*ArcCoth[a*x])/2), x, 10, -((63*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(64*a^3)) + (15*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(32*a^2) - (3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(8*a) + (1/4)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4 + (123*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (123*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{x^2/E^((3*ArcCoth[a*x])/2), x, 9, (23*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(24*a^2) - (7*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3 - (17*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) - (17*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{x/E^((3*ArcCoth[a*x])/2), x, 7, -((3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(4*a)) + (1/2)*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4)*x^2 + (9*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2) + (9*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2)} -{E^((-3*ArcCoth[a*x])/2), x, 6, (1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x - (3*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a - (3*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a} -{1/(E^((3*ArcCoth[a*x])/2)*x), x, 17, Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{1/(E^((3*ArcCoth[a*x])/2)*x^2), x, 13, -(a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)) - (3*a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (3*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (3*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) - (3*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{1/(E^((3*ArcCoth[a*x])/2)*x^3), x, 14, (3*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/2 + (9*a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (9*a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (9*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (9*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{1/(E^((3*ArcCoth[a*x])/2)*x^4), x, 15, (-17*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/24 - (a^3*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/(3*x) - (17*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (17*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (17*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) - (17*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -{x^4/E^((5*ArcCoth[a*x])/2), x, 12, (26111*(1 - 1/(a*x))^(1/4))/(1920*a^5*(1 + 1/(a*x))^(1/4)) + (5533*(1 - 1/(a*x))^(1/4)*x)/(1920*a^4*(1 + 1/(a*x))^(1/4)) - (1189*(1 - 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 + 1/(a*x))^(1/4)) + (181*(1 - 1/(a*x))^(1/4)*x^3)/(240*a^2*(1 + 1/(a*x))^(1/4)) - (21*(1 - 1/(a*x))^(1/4)*x^4)/(40*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^5)/(5*(1 + 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) - (1003*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)} -{x^3/E^((5*ArcCoth[a*x])/2), x, 11, -((2467*(1 - 1/(a*x))^(1/4))/(192*a^4*(1 + 1/(a*x))^(1/4))) - (521*(1 - 1/(a*x))^(1/4)*x)/(192*a^3*(1 + 1/(a*x))^(1/4)) + (113*(1 - 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 + 1/(a*x))^(1/4)) - (17*(1 - 1/(a*x))^(1/4)*x^3)/(24*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^4)/(4*(1 + 1/(a*x))^(1/4)) - (475*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4) + (475*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)} -{x^2/E^((5*ArcCoth[a*x])/2), x, 10, (287*(1 - 1/(a*x))^(1/4))/(24*a^3*(1 + 1/(a*x))^(1/4)) + (61*(1 - 1/(a*x))^(1/4)*x)/(24*a^2*(1 + 1/(a*x))^(1/4)) - (13*(1 - 1/(a*x))^(1/4)*x^2)/(12*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^3)/(3*(1 + 1/(a*x))^(1/4)) + (55*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3) - (55*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)} -{x/E^((5*ArcCoth[a*x])/2), x, 8, -((25*(1 - 1/(a*x))^(1/4))/(2*a^2*(1 + 1/(a*x))^(1/4))) - (5*(1 - 1/(a*x))^(5/4)*x)/(4*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(9/4)*x^2)/(2*(1 + 1/(a*x))^(1/4)) - (25*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2) + (25*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(4*a^2)} -{E^((-5*ArcCoth[a*x])/2), x, 7, (10*(1 - 1/(a*x))^(1/4))/(a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(5/4)*x)/(1 + 1/(a*x))^(1/4) + (5*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a - (5*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/a} -{1/(E^((5*ArcCoth[a*x])/2)*x), x, 19, -((8*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] - 2*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)]/Sqrt[2]} -{1/(E^((5*ArcCoth[a*x])/2)*x^2), x, 14, (4*a*(1 - 1/(a*x))^(5/4))/(1 + 1/(a*x))^(1/4) + 5*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) + (5*a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (5*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (5*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) - (5*a*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])} -{1/(E^((5*ArcCoth[a*x])/2)*x^3), x, 15, (-2*a^2*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) - (25*a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 - (5*a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/2 - (25*a^2*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) + (25*a^2*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(4*Sqrt[2]) - (25*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (25*a^2*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2])} -{1/(E^((5*ArcCoth[a*x])/2)*x^4), x, 16, (2*a^3*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) + (55*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 + (11*a^3*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/4 + (a^3*(1 - 1/(a*x))^(9/4)*(1 + 1/(a*x))^(3/4))/3 + (55*a^3*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) - (55*a^3*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(8*Sqrt[2]) + (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2]) - (55*a^3*Log[1 + Sqrt[1 - 1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(16*Sqrt[2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/3 ArcCoth[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcCoth[x]/3)*x^2, x, 16, (11/27)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x + (7/18)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x^2 + (1/3)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x^3 - (19*ArcTan[(1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]])/(54*Sqrt[3]) + (19*ArcTan[(1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]])/(54*Sqrt[3]) + (19/81)*ArcTanh[(1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] - (19/324)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] + (19/324)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)]} -{E^(ArcCoth[x]/3)*x, x, 14, (1/6)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x + (1/2)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6)*x^2 - ArcTan[(1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) + ArcTan[(1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/(6*Sqrt[3]) + (1/9)*ArcTanh[(1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] - (1/36)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] + (1/36)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)]} -{E^(ArcCoth[x]/3), x, 13, (1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x - ArcTan[(1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/Sqrt[3] + ArcTan[(1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]]/Sqrt[3] + (2/3)*ArcTanh[(1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] - (1/6)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] + (1/6)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)]} -{E^(ArcCoth[x]/3)/x, x, 25, (-Sqrt[3])*ArcTan[(1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]] + Sqrt[3]*ArcTan[(1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/Sqrt[3]] - ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + ArcTan[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + 2*ArcTan[((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)] + 2*ArcTanh[(1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] - (1/2)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] + (1/2)*Log[1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)] + (1/2)*Sqrt[3]*Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)] - (1/2)*Sqrt[3]*Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)]} -{E^(ArcCoth[x]/3)/x^2, x, 14, (1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) - (1/3)*ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (1/3)*ArcTan[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (2/3)*ArcTan[((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)] + Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)]/(2*Sqrt[3]) - Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)]/(2*Sqrt[3])} -{E^(ArcCoth[x]/3)/x^3, x, 15, (1/6)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) + (1/2)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6) - (1/18)*ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (1/18)*ArcTan[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (1/9)*ArcTan[((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)] + Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)]/(12*Sqrt[3]) - Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)]/(12*Sqrt[3])} -{E^(ArcCoth[x]/3)/x^4, x, 16, (19/54)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) + (1/18)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6) + ((1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6))/(3*x) - (19/162)*ArcTan[Sqrt[3] - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (19/162)*ArcTan[Sqrt[3] + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)] + (19/81)*ArcTan[((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)] + (19*Log[1 - (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)])/(108*Sqrt[3]) - (19*Log[1 + (Sqrt[3]*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)])/(108*Sqrt[3])} - - -{E^((2*ArcCoth[x])/3)*x^2, x, 6, (14/27)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x + (4/9)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x^2 + (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x^3 - (22*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))])/(27*Sqrt[3]) - (11/27)*Log[(1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)] - (11*Log[x])/81} -{E^((2*ArcCoth[x])/3)*x, x, 4, (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x + (1/2)*(1 + 1/x)^(4/3)*((-1 + x)/x)^(2/3)*x^2 - (2*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))])/(3*Sqrt[3]) - (1/3)*Log[(1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)] - Log[x]/9} -{E^((2*ArcCoth[x])/3), x, 3, (1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x - (2*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))])/Sqrt[3] - Log[(1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)] - Log[x]/3} -{E^((2*ArcCoth[x])/3)/x, x, 4, (-Sqrt[3])*ArcTan[1/Sqrt[3] - (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))] - Sqrt[3]*ArcTan[1/Sqrt[3] + (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))] - (3/2)*Log[(1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)] - (3/2)*Log[1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)] - (1/2)*Log[1 + 1/x] - Log[x]/2} -{E^((2*ArcCoth[x])/3)/x^2, x, 3, (1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3) - (2*ArcTan[1/Sqrt[3] - (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))])/Sqrt[3] - Log[1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)] - (1/3)*Log[1 + 1/x]} -{E^((2*ArcCoth[x])/3)/x^3, x, 4, (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3) + (1/2)*(1 + 1/x)^(4/3)*((-1 + x)/x)^(2/3) - (2*ArcTan[1/Sqrt[3] - (2*((-1 + x)/x)^(1/3))/(Sqrt[3]*(1 + 1/x)^(1/3))])/(3*Sqrt[3]) - (1/3)*Log[1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)] - (1/9)*Log[1 + 1/x]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n/4 ArcCoth[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^(ArcCoth[a*x]/4)*x^2, x, 19, (37*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(96*a^2) + (3*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^2)/(8*a) + (1/3)*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^3 - (11*ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)])/(64*Sqrt[2]*a^3) + (11*ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)])/(64*Sqrt[2]*a^3) + (11*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)])/(64*a^3) + (11*ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)])/(64*a^3) - (11*Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3) + (11*Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*Sqrt[2]*a^3)} -{E^(ArcCoth[a*x]/4)*x^1, x, 17, ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(8*a) + (1/2)*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8)*x^2 - ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(16*Sqrt[2]*a^2) + ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) + ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(16*a^2) - Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(32*Sqrt[2]*a^2) + Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(32*Sqrt[2]*a^2)} -{E^(ArcCoth[a*x]/4)*x^0, x, 16, (1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x - ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(2*Sqrt[2]*a) + ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)]/(2*Sqrt[2]*a) + ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(2*a) + ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)]/(2*a) - Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*Sqrt[2]*a) + Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/(4*Sqrt[2]*a)} -{E^(ArcCoth[a*x]/4)/x^1, x, 39, (-Sqrt[2 + Sqrt[2]])*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] - Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] + Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] + Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] - Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)] + Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)] + 2*ArcTan[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)] + 2*ArcTanh[(1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)] + (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/2)*Sqrt[2 - Sqrt[2]]*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] + (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/2)*Sqrt[2 + Sqrt[2]]*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - Log[1 - (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/Sqrt[2] + Log[1 + (Sqrt[2]*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)]/Sqrt[2]} -{E^(ArcCoth[a*x]/4)/x^2, x, 25, a*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) - (1/4)*Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] - (1/4)*Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] + (1/4)*Sqrt[2 + Sqrt[2]]*a*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] + (1/4)*Sqrt[2 - Sqrt[2]]*a*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] + (1/8)*Sqrt[2 - Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/8)*Sqrt[2 - Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] + (1/8)*Sqrt[2 + Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/8)*Sqrt[2 + Sqrt[2]]*a*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)]} -{E^(ArcCoth[a*x]/4)/x^3, x, 26, (1/8)*a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) + (1/2)*a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8) - (1/32)*Sqrt[2 + Sqrt[2]]*a^2*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] - (1/32)*Sqrt[2 - Sqrt[2]]*a^2*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] + (1/32)*Sqrt[2 + Sqrt[2]]*a^2*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]] + (1/32)*Sqrt[2 - Sqrt[2]]*a^2*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt[2]]] + (1/64)*Sqrt[2 - Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/64)*Sqrt[2 - Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] + (1/64)*Sqrt[2 + Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)] - (1/64)*Sqrt[2 + Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) with m symbolic*) - - -{E^(4*ArcCoth[a*x])*x^m, x, 5, x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 - a*x) - 4*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x]} -{E^(3*ArcCoth[a*x])*x^m, x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m)) - (x^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m) + (4*x^m*Hypergeometric2F1[3/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m)} -{E^(2*ArcCoth[a*x])*x^m, x, 4, x^(1 + m)/(1 + m) - (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(1 + m)} -{E^(1*ArcCoth[a*x])*x^m, x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m) + (x^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m)} -{x^m/E^(1*ArcCoth[a*x]), x, 4, (x^(1 + m)*Hypergeometric2F1[1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m) - (x^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m)} -{x^m/E^(2*ArcCoth[a*x]), x, 4, x^(1 + m)/(1 + m) - (2*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/(1 + m)} -{x^m/E^(3*ArcCoth[a*x]), x, 9, -((3*x^(1 + m)*Hypergeometric2F1[1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m)) + (x^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m) + (4*x^(1 + m)*Hypergeometric2F1[3/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)])/(1 + m) - (4*x^m*Hypergeometric2F1[3/2, -(m/2), 1 - m/2, 1/(a^2*x^2)])/(a*m)} - - -{E^(5*ArcCoth[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 5/4, -(5/4), -m, 1/(a*x), -(1/(a*x))])/(1 + m)} -{E^(3*ArcCoth[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 3/4, -(3/4), -m, 1/(a*x), -(1/(a*x))])/(1 + m)} -{E^(1*ArcCoth[a*x]/2)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 1/4, -(1/4), -m, 1/(a*x), -(1/(a*x))])/(1 + m)} -{x^m/E^(1*ArcCoth[a*x]/2), x, 2, (x^(1 + m)*AppellF1[-1 - m, -(1/4), 1/4, -m, 1/(a*x), -(1/(a*x))])/(1 + m)} -{x^m/E^(3*ArcCoth[a*x]/2), x, 2, (x^(1 + m)*AppellF1[-1 - m, -(3/4), 3/4, -m, 1/(a*x), -(1/(a*x))])/(1 + m)} -{x^m/E^(5*ArcCoth[a*x]/2), x, 2, (x^(1 + m)*AppellF1[-1 - m, -(5/4), 5/4, -m, 1/(a*x), -(1/(a*x))])/(1 + m)} - - -{E^(2*ArcCoth[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 1/3, -(1/3), -m, 1/x, -(1/x)])/(1 + m)} -{E^(1*ArcCoth[x]/3)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 1/6, -(1/6), -m, 1/x, -(1/x)])/(1 + m)} - - -{E^(ArcCoth[a*x]/4)*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, 1/8, -(1/8), -m, 1/(a*x), -(1/(a*x))])/(1 + m)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) with n symbolic*) - - -{E^(n*ArcCoth[a*x])*x^m, x, 2, (x^(1 + m)*AppellF1[-1 - m, n/2, -(n/2), -m, 1/(a*x), -(1/(a*x))])/(1 + m)} - - -{E^(n*ArcCoth[a*x])*x^2, x, 5, (n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/(6*a) + (1/3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3 + (2*(2 + n^2)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)])/(3*a^3*(2 - n))} -{E^(n*ArcCoth[a*x])*x^1, x, 3, (1/2)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2 + (2*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)])/(a^2*(2 - n))} -{E^(n*ArcCoth[a*x])*x^0, x, 2, (4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)])/(a*(2 - n))} -{E^(n*ArcCoth[a*x])/x^1, x, 4, -((2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)])/((1 - 1/(a*x))^(n/2)*n)) + (2^(1 + n/2)*Hypergeometric2F1[-(n/2), -(n/2), 1 - n/2, (a - 1/x)/(2*a)])/((1 - 1/(a*x))^(n/2)*n)} -{E^(n*ArcCoth[a*x])/x^2, x, 2, (2^(1 + n/2)*a*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)])/(2 - n)} -{E^(n*ArcCoth[a*x])/x^3, x, 3, (1/2)*a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2) + (2^(n/2)*a^2*n*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)])/(2 - n)} -{E^(n*ArcCoth[a*x])/x^4, x, 4, (1/6)*a^3*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2) + (a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(3*x) + (2^(n/2)*a^3*(2 + n^2)*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)])/(3*(2 - n))} -{E^(n*ArcCoth[a*x])/x^5, x, 4, (1/24)*a^3*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(a*(6 + n^2) + (2*n)/x) + (a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(4*x^2) + (2^(-2 + n/2)*a^4*n*(8 + n^2)*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)])/(3*(2 - n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcCoth[a x]) (c-c a x)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a x)^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - a*c*x)^p, x, 4, (Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^p)/(1 + p) + (((a - 1/x)/(a + 1/x))^(1/2 - p)*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p*Hypergeometric2F1[1/2 - p, -p, 1 - p, 2/((a + 1/x)*x)])/(a*p*(1 + p)*Sqrt[1 - 1/(a*x)])} - -{E^ArcCoth[a*x]*(c - a*c*x)^4, x, 9, (-(7/8))*a*c^4*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (17/15)*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (3/4)*a^3*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (1/5)*a^4*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^5 + (7*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{E^ArcCoth[a*x]*(c - a*c*x)^3, x, 8, (-(5/8))*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (2/3)*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (1/4)*a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (5*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{E^ArcCoth[a*x]*(c - a*c*x)^2, x, 7, (-(1/2))*a*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (1/3)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3 + (c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{E^ArcCoth[a*x]*(c - a*c*x), x, 6, (-(1/2))*a*c*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{E^ArcCoth[a*x]/(c - a*c*x), x, 7, (2*(a + 1/x))/(a^2*c*Sqrt[1 - 1/(a^2*x^2)]) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c)} -{E^ArcCoth[a*x]/(c - a*c*x)^2, x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2))/(3*c^2*(a - 1/x)^3))} -{E^ArcCoth[a*x]/(c - a*c*x)^3, x, 4, (a^3*(1 - 1/(a^2*x^2))^(3/2))/(5*c^3*(a - 1/x)^4) - (4*a^2*(1 - 1/(a^2*x^2))^(3/2))/(15*c^3*(a - 1/x)^3)} -{E^ArcCoth[a*x]/(c - a*c*x)^4, x, 6, -((a^4*(1 - 1/(a^2*x^2))^(3/2))/(7*c^4*(a - 1/x)^5)) + (12*a^3*(1 - 1/(a^2*x^2))^(3/2))/(35*c^4*(a - 1/x)^4) - (23*a^2*(1 - 1/(a^2*x^2))^(3/2))/(105*c^4*(a - 1/x)^3)} -{E^ArcCoth[a*x]/(c - a*c*x)^5, x, 8, (a^5*(1 - 1/(a^2*x^2))^(3/2))/(9*c^5*(a - 1/x)^6) - (8*a^4*(1 - 1/(a^2*x^2))^(3/2))/(21*c^5*(a - 1/x)^5) + (47*a^3*(1 - 1/(a^2*x^2))^(3/2))/(105*c^5*(a - 1/x)^4) - (58*a^2*(1 - 1/(a^2*x^2))^(3/2))/(315*c^5*(a - 1/x)^3)} - - -{E^(2*ArcCoth[a*x])*(c - a*c*x)^p, x, 5, (2*(c - a*c*x)^p)/(a*p) - (c - a*c*x)^(1 + p)/(a*c*(1 + p))} - -{E^(2*ArcCoth[a*x])*(c - a*c*x)^5, x, 4, (2*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^4, x, 4, (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*a)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^3, x, 4, (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^2, x, 4, -(c^2*x) + (a^2*c^2*x^3)/3} -{E^(2*ArcCoth[a*x])*(c - a*c*x), x, 1, -(c*x) - (a*c*x^2)/2, (c*E^(2*ArcCoth[a*x])*(1 - a^2*x^2))/(2*a)} -{E^(2*ArcCoth[a*x])/(c - a*c*x), x, 4, -2/(a*c*(1 - a*x)) - Log[1 - a*x]/(a*c)} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^2, x, 3, -(x/(c^2*(1 - a*x)^2))} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^3, x, 4, -2/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^4, x, 4, -1/(2*a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^3)} - - -{E^(3*ArcCoth[a*x])*(c - a*c*x)^p, x, 5, (3*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p)/(a*p*(1 + p)*Sqrt[1 - 1/(a*x)]) + ((1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^p)/((1 + p)*Sqrt[1 - 1/(a*x)]) - (3*((a - 1/x)/(a + 1/x))^(3/2 - p)*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, 2/((a + 1/x)*x)])/(a^2*p*(1 - p^2)*(1 - 1/(a*x))^(3/2)*x)} - -{E^(3*ArcCoth[a*x])*(c - a*c*x)^4, x, 8, (3/8)*a*c^4*Sqrt[1 - 1/(a^2*x^2)]*x^2 - (1/4)*a^3*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (1/5)*a^4*c^4*(1 - 1/(a^2*x^2))^(5/2)*x^5 - (3*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^3, x, 7, (3/8)*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2 - (1/4)*a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4 - (3*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^2, x, 8, (1/2)*a*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (1/3)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{E^(3*ArcCoth[a*x])*(c - a*c*x), x, 8, -2*c*Sqrt[1 - 1/(a^2*x^2)]*x - (1/2)*a*c*Sqrt[1 - 1/(a^2*x^2)]*x^2 - (3*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{E^(3*ArcCoth[a*x])/(c - a*c*x), x, 9, (8*(a + 1/x))/(3*a^2*c*(1 - 1/(a^2*x^2))^(3/2)) + 4/(3*a^2*c*Sqrt[1 - 1/(a^2*x^2)]*x) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c)} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^2, x, 3, -((a^4*(1 - 1/(a^2*x^2))^(5/2))/(5*c^2*(a - 1/x)^5))} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^3, x, 4, (a^5*(1 - 1/(a^2*x^2))^(5/2))/(7*c^3*(a - 1/x)^6) - (6*a^4*(1 - 1/(a^2*x^2))^(5/2))/(35*c^3*(a - 1/x)^5)} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^4, x, 6, -((47*(a + 1/x)^5)/(315*a^6*c^4*(1 - 1/(a^2*x^2))^(5/2))) + (16*(a + 1/x)^6)/(63*a^7*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (a + 1/x)^7/(9*a^8*c^4*(1 - 1/(a^2*x^2))^(9/2))} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^5, x, 7, -((152*(a + 1/x)^5)/(1155*a^6*c^5*(1 - 1/(a^2*x^2))^(5/2))) + (79*(a + 1/x)^6)/(231*a^7*c^5*(1 - 1/(a^2*x^2))^(7/2)) - (10*(a + 1/x)^7)/(33*a^8*c^5*(1 - 1/(a^2*x^2))^(9/2)) + (a + 1/x)^8/(11*a^9*c^5*(1 - 1/(a^2*x^2))^(11/2))} - - -{E^(4*ArcCoth[a*x])*(c - a*c*x)^p, x, 5, (4*c*(c - a*c*x)^(-1 + p))/(a*(1 - p)) + (4*(c - a*c*x)^p)/(a*p) - (c - a*c*x)^(1 + p)/(a*c*(1 + p))} - -{E^(4*ArcCoth[a*x])*(c - a*c*x)^5, x, 4, -((c^5*(1 - a*x)^4)/a) + (4*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)} -{E^(4*ArcCoth[a*x])*(c - a*c*x)^4, x, 5, c^4*x - (2*a^2*c^4*x^3)/3 + (a^4*c^4*x^5)/5} -{E^(4*ArcCoth[a*x])*(c - a*c*x)^3, x, 4, (2*c^3*(1 + a*x)^3)/(3*a) - (c^3*(1 + a*x)^4)/(4*a)} -{E^(4*ArcCoth[a*x])*(c - a*c*x)^2, x, 3, (c^2*(1 + a*x)^3)/(3*a)} -{E^(4*ArcCoth[a*x])*(c - a*c*x), x, 4, -3*c*x - (1/2)*a*c*x^2 - (4*c*Log[1 - a*x])/a} -{E^(4*ArcCoth[a*x])/(c - a*c*x), x, 4, 2/(a*c*(1 - a*x)^2) - 4/(a*c*(1 - a*x)) - Log[1 - a*x]/(a*c)} -{E^(4*ArcCoth[a*x])/(c - a*c*x)^2, x, 3, (1 + a*x)^3/(6*a*c^2*(1 - a*x)^3)} -{E^(4*ArcCoth[a*x])/(c - a*c*x)^3, x, 4, 1/(a*c^3*(1 - a*x)^4) - 4/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)} -{E^(4*ArcCoth[a*x])/(c - a*c*x)^4, x, 4, 4/(5*a*c^4*(1 - a*x)^5) - 1/(a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a*c*x)^p/E^ArcCoth[a*x], x, 3, (((a - 1/x)/(a + 1/x))^(-(1/2) - p)*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^p*Hypergeometric2F1[-1 - p, -(1/2) - p, -p, 2/((a + 1/x)*x)])/(1 + p)} - -{(c - a*c*x)^3/E^ArcCoth[a*x], x, 9, (20/3)*c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (27/8)*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (4/3)*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (1/4)*a^3*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^4 - (35*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{(c - a*c*x)^2/E^ArcCoth[a*x], x, 8, (11/3)*c^2*Sqrt[1 - 1/(a^2*x^2)]*x - (3/2)*a*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (1/3)*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (5*c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{(c - a*c*x)/E^ArcCoth[a*x], x, 7, 2*c*Sqrt[1 - 1/(a^2*x^2)]*x - (1/2)*a*c*Sqrt[1 - 1/(a^2*x^2)]*x^2 - (3*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{1/(E^ArcCoth[a*x]*(c - a*c*x)), x, 5, -(ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^2), x, 3, -(Sqrt[1 - 1/(a^2*x^2)]/(c^2*(a - 1/x)))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^3), x, 4, (a*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - 1/x)^2) - (2*Sqrt[1 - 1/(a^2*x^2)])/(3*c^3*(a - 1/x))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^4), x, 6, -((a^2*Sqrt[1 - 1/(a^2*x^2)])/(5*c^4*(a - 1/x)^3)) + (8*a*Sqrt[1 - 1/(a^2*x^2)])/(15*c^4*(a - 1/x)^2) - (7*Sqrt[1 - 1/(a^2*x^2)])/(15*c^4*(a - 1/x))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^5), x, 7, (a^3*Sqrt[1 - 1/(a^2*x^2)])/(7*c^5*(a - 1/x)^4) - (18*a^2*Sqrt[1 - 1/(a^2*x^2)])/(35*c^5*(a - 1/x)^3) + (23*a*Sqrt[1 - 1/(a^2*x^2)])/(35*c^5*(a - 1/x)^2) - (12*Sqrt[1 - 1/(a^2*x^2)])/(35*c^5*(a - 1/x))} - - -{(c - a*c*x)^p/E^(2*ArcCoth[a*x]), x, 4, ((c - a*c*x)^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (1/2)*(1 - a*x)])/(2*a*c^2*(2 + p))} - -{(c - a*c*x)^4/E^(2*ArcCoth[a*x]), x, 4, 16*c^4*x - (4*c^4*(1 - a*x)^2)/a - (4*c^4*(1 - a*x)^3)/(3*a) - (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*a) - (32*c^4*Log[1 + a*x])/a} -{(c - a*c*x)^3/E^(2*ArcCoth[a*x]), x, 4, 8*c^3*x - (2*c^3*(1 - a*x)^2)/a - (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a) - (16*c^3*Log[1 + a*x])/a} -{(c - a*c*x)^2/E^(2*ArcCoth[a*x]), x, 4, 4*c^2*x - (c^2*(1 - a*x)^2)/a - (c^2*(1 - a*x)^3)/(3*a) - (8*c^2*Log[1 + a*x])/a} -{(c - a*c*x)/E^(2*ArcCoth[a*x]), x, 4, 3*c*x - (1/2)*a*c*x^2 - (4*c*Log[1 + a*x])/a} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)), x, 3, -(Log[1 + a*x]/(a*c))} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^2), x, 4, -(ArcTanh[a*x]/(a*c^2))} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^3), x, 5, -(1/(2*a*c^3*(1 - a*x))) - ArcTanh[a*x]/(2*a*c^3)} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^4), x, 5, -(1/(4*a*c^4*(1 - a*x)^2)) - 1/(4*a*c^4*(1 - a*x)) - ArcTanh[a*x]/(4*a*c^4)} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^5), x, 5, -(1/(6*a*c^5*(1 - a*x)^3)) - 1/(8*a*c^5*(1 - a*x)^2) - 1/(8*a*c^5*(1 - a*x)) - ArcTanh[a*x]/(8*a*c^5)} - - -{(c - a*c*x)^p/E^(3*ArcCoth[a*x]), x, 3, (((a - 1/x)/(a + 1/x))^(-(3/2) - p)*(1 - 1/(a*x))^(3/2)*x*(c - a*c*x)^p*Hypergeometric2F1[-(3/2) - p, -1 - p, -p, 2/((a + 1/x)*x)])/((1 + p)*Sqrt[1 + 1/(a*x)])} - -{(c - a*c*x)^3/E^(3*ArcCoth[a*x]), x, 10, (32*c^3*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + 30*c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (67/8)*a*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^2 + 2*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (1/4)*a^3*c^3*Sqrt[1 - 1/(a^2*x^2)]*x^4 - (315*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(8*a)} -{(c - a*c*x)^2/E^(3*ArcCoth[a*x]), x, 9, (16*c^2*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (35/3)*c^2*Sqrt[1 - 1/(a^2*x^2)]*x - (5/2)*a*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^2 + (1/3)*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x^3 - (35*c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{(c - a*c*x)/E^(3*ArcCoth[a*x]), x, 8, (8*c*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + 4*c*Sqrt[1 - 1/(a^2*x^2)]*x - (1/2)*a*c*Sqrt[1 - 1/(a^2*x^2)]*x^2 - (15*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(2*a)} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)), x, 6, (2*(a - 1/x))/(a^2*c*Sqrt[1 - 1/(a^2*x^2)]) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c)} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2), x, 3, (a - 1/x)/(a^2*c^2*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^3), x, 3, 1/(a*c^3*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^4), x, 5, 2/(3*a*c^4*Sqrt[1 - 1/(a^2*x^2)]) - 1/(3*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*(a - 1/x)*x^2)} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^5), x, 6, -((4*(a + 1/x))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2))) + (a + 1/x)^2/(5*a^3*c^5*(1 - 1/(a^2*x^2))^(5/2)) + (5*a + 2/x)/(5*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6), x, 7, -((46*(a + 1/x))/(35*a^2*c^6*(1 - 1/(a^2*x^2))^(3/2))) + (24*(a + 1/x)^2)/(35*a^3*c^6*(1 - 1/(a^2*x^2))^(5/2)) - (a + 1/x)^3/(7*a^4*c^6*(1 - 1/(a^2*x^2))^(7/2)) + (35*a + 13/x)/(35*a^2*c^6*Sqrt[1 - 1/(a^2*x^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a x)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - a*c*x)^(9/2), x, 7, -((32*(a - 1/x)^3*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(99*a^4*(1 - 1/(a*x))^(9/2))) + (9088*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(3465*a^4*(1 - 1/(a*x))^(9/2)*x^3) - (768*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(385*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (128*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(231*a^2*(1 - 1/(a*x))^(9/2)*x) + (2*(a - 1/x)^4*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(9/2))/(11*a^4*(1 - 1/(a*x))^(9/2))} -{E^ArcCoth[a*x]*(c - a*c*x)^(7/2), x, 6, -((8*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(21*a*(1 - 1/(a*x))^(7/2))) - (568*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(315*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (48*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(35*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(a - 1/x)^3*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(7/2))/(9*a^3*(1 - 1/(a*x))^(7/2))} -{E^ArcCoth[a*x]*(c - a*c*x)^(5/2), x, 5, (64*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(105*(c - a*c*x)^(3/2)) + (16*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(35*Sqrt[c - a*c*x]) + (2/7)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*Sqrt[c - a*c*x], -((36*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2))) + (142*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(5/2))/(105*a^2*(1 - 1/(a*x))^(5/2)*x) + (2*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(5/2))/(7*(1 - 1/(a*x))^(5/2))} -{E^ArcCoth[a*x]*(c - a*c*x)^(3/2), x, 4, (8*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(15*(c - a*c*x)^(3/2)) + (2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(5*Sqrt[c - a*c*x]), -((14*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(3/2))/(15*a*(1 - 1/(a*x))^(3/2))) + (2*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(3/2))/(5*(1 - 1/(a*x))^(3/2))} -{E^ArcCoth[a*x]*(c - a*c*x)^(1/2), x, 1, (2*E^ArcCoth[a*x]*(1 + a*x)*Sqrt[c - a*c*x])/(3*a)} -{E^ArcCoth[a*x]/(c - a*c*x)^(1/2), x, 5, (2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/Sqrt[c - a*c*x] - (2*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[1/x]*Sqrt[c - a*c*x])} -{E^ArcCoth[a*x]/(c - a*c*x)^(3/2), x, 5, -((a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/((a - 1/x)*(c - a*c*x)^(3/2))) - (Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[2]*(1/x)^(3/2)*(c - a*c*x)^(3/2))} -{E^ArcCoth[a*x]/(c - a*c*x)^(5/2), x, 6, (a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(8*(a - 1/x)*(c - a*c*x)^(5/2)) - (a^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^2)/(4*(a - 1/x)^2*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(8*Sqrt[2]*(1/x)^(5/2)*(c - a*c*x)^(5/2))} -{E^ArcCoth[a*x]/(c - a*c*x)^(7/2), x, 7, -((a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^2)/(6*(a - 1/x)^3*(c - a*c*x)^(7/2))) - (a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^3)/(32*(a - 1/x)*(c - a*c*x)^(7/2)) + (a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^3)/(16*(a - 1/x)^2*(c - a*c*x)^(7/2)) - (a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(32*Sqrt[2]*(1/x)^(7/2)*(c - a*c*x)^(7/2))} - - -{E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2), x, 5, (4*(c - a*c*x)^(7/2))/(7*a) - (2*(c - a*c*x)^(9/2))/(9*a*c)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^(5/2), x, 5, (4*(c - a*c*x)^(5/2))/(5*a) - (2*(c - a*c*x)^(7/2))/(7*a*c)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^(3/2), x, 5, (4*(c - a*c*x)^(3/2))/(3*a) - (2*(c - a*c*x)^(5/2))/(5*a*c)} -{E^(2*ArcCoth[a*x])*(c - a*c*x)^(1/2), x, 5, (4*Sqrt[c - a*c*x])/a - (2*(c - a*c*x)^(3/2))/(3*a*c)} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^(1/2), x, 5, -(4/(a*Sqrt[c - a*c*x])) - (2*Sqrt[c - a*c*x])/(a*c)} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^(3/2), x, 5, -(4/(3*a*(c - a*c*x)^(3/2))) + 2/(a*c*Sqrt[c - a*c*x])} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^(5/2), x, 5, -(4/(5*a*(c - a*c*x)^(5/2))) + 2/(3*a*c*(c - a*c*x)^(3/2))} -{E^(2*ArcCoth[a*x])/(c - a*c*x)^(7/2), x, 5, -(4/(7*a*(c - a*c*x)^(7/2))) + 2/(5*a*c*(c - a*c*x)^(5/2))} - - -{E^(3*ArcCoth[a*x])*(c - a*c*x)^(9/2), x, 6, (-8*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(33*a*(1 - 1/(a*x))^(9/2)) - (856*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(1155*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (16*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(21*a^2*(1 - 1/(a*x))^(9/2)*x) + (2*(a - x^(-1))^3*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(9/2))/(11*a^3*(1 - 1/(a*x))^(9/2))} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2), x, 5, (-44*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(7/2))/(63*a*(1 - 1/(a*x))^(7/2)) + (214*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(7/2))/(315*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(7/2))/(9*(1 - 1/(a*x))^(7/2))} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2), x, 4, (-18*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2)) + (2*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(5/2))/(7*(1 - 1/(a*x))^(5/2))} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2), x, 1, (2*E^(3*ArcCoth[a*x])*(1 + a*x)*(c - a*c*x)^(3/2))/(5*a)} -{E^(3*ArcCoth[a*x])*(c - a*c*x)^(1/2), x, 6, (4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(a*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(3/2)*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^(1/2), x, 6, -((6*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/((a - 1/x)*Sqrt[c - a*c*x])) + (2*a*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x)/((a - 1/x)*Sqrt[c - a*c*x]) - (3*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[1/x]*Sqrt[c - a*c*x])} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^(3/2), x, 6, -((3*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/(4*(a - 1/x)*(c - a*c*x)^(3/2))) - (a^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x)/(2*(a - 1/x)^2*(c - a*c*x)^(3/2)) - (3*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(4*Sqrt[2]*(1/x)^(3/2)*(c - a*c*x)^(3/2))} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^(5/2), x, 7, (a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(16*(a - 1/x)*(c - a*c*x)^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^2)/(24*(a - 1/x)^2*(c - a*c*x)^(5/2)) - (a^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^2)/(6*(a - 1/x)^3*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(16*Sqrt[2]*(1/x)^(5/2)*(c - a*c*x)^(5/2))} -{E^(3*ArcCoth[a*x])/(c - a*c*x)^(7/2), x, 8, -((a^5*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^2)/(8*(a - 1/x)^4*(c - a*c*x)^(7/2))) - (3*a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^3)/(256*(a - 1/x)*(c - a*c*x)^(7/2)) - (a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^3)/(128*(a - 1/x)^2*(c - a*c*x)^(7/2)) + (a^5*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^3)/(32*(a - 1/x)^3*(c - a*c*x)^(7/2)) - (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(256*Sqrt[2]*(1/x)^(7/2)*(c - a*c*x)^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a*c*x)^(9/2)/E^ArcCoth[a*x], x, 8, (16384*c^5*Sqrt[1 - 1/(a^2*x^2)]*x)/(693*Sqrt[c - a*c*x]) + (4096/693)*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x] + (512/231)*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2) + (640/693)*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(5/2) + (40/99)*c*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(7/2) + (2/11)*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(9/2), -((40*(a - 1/x)^4*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(99*a^5*(1 - 1/(a*x))^(9/2))) - (22016*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x^4) + (1024*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(99*a^4*(1 - 1/(a*x))^(9/2)*x^3) - (512*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(231*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (640*(a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x) + (2*(a - 1/x)^5*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^(9/2))/(11*a^5*(1 - 1/(a*x))^(9/2))} -{(c - a*c*x)^(7/2)/E^ArcCoth[a*x], x, 7, (4096*c^4*Sqrt[1 - 1/(a^2*x^2)]*x)/(315*Sqrt[c - a*c*x]) + (1024/315)*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x] + (128/105)*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2) + (32/63)*c*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(5/2) + (2/9)*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(7/2), -((32*(a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2))/(63*a^4*(1 - 1/(a*x))^(7/2))) + (5504*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2))/(315*a^4*(1 - 1/(a*x))^(7/2)*x^3) - (256*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2))/(45*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (128*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2))/(105*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(a - 1/x)^4*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^(7/2))/(9*a^4*(1 - 1/(a*x))^(7/2))} -{(c - a*c*x)^(5/2)/E^ArcCoth[a*x], x, 6, (256*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)/(35*Sqrt[c - a*c*x]) + (64/35)*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x] + (24/35)*c*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2) + (2/7)*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(5/2), -((24*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2))) - (344*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2))/(35*a^3*(1 - 1/(a*x))^(5/2)*x^2) + (16*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2))/(5*a^2*(1 - 1/(a*x))^(5/2)*x) + (2*(a - 1/x)^3*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2))} -{(c - a*c*x)^(3/2)/E^ArcCoth[a*x], x, 5, (64*c^2*Sqrt[1 - 1/(a^2*x^2)]*x)/(15*Sqrt[c - a*c*x]) + (16/15)*c*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x] + (2/5)*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2), -((28*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(3/2))/(15*a*(1 - 1/(a*x))^(3/2))) + (86*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(3/2))/(15*a^2*(1 - 1/(a*x))^(3/2)*x) + (2*Sqrt[1 + 1/(a*x)]*x*(c - a*c*x)^(3/2))/(5*(1 - 1/(a*x))^(3/2))} -{(c - a*c*x)^(1/2)/E^ArcCoth[a*x], x, 4, (8*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*Sqrt[c - a*c*x]) + (2/3)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x], -((10*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])) + (2*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)])} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^(1/2)), x, 1, (2*(1 + a*x))/(E^ArcCoth[a*x]*(a*Sqrt[c - a*c*x]))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)), x, 4, -((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((1/x)^(3/2)*(c - a*c*x)^(3/2)))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)), x, 5, -((a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(2*(a - 1/x)*(c - a*c*x)^(5/2))) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(2*Sqrt[2]*(1/x)^(5/2)*(c - a*c*x)^(5/2))} -{1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)), x, 6, -((a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^2)/(4*(a - 1/x)^2*(c - a*c*x)^(7/2))) + (3*a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^3)/(16*(a - 1/x)*(c - a*c*x)^(7/2)) - (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(16*Sqrt[2]*(1/x)^(7/2)*(c - a*c*x)^(7/2))} - - -{(c - a*c*x)^(7/2)/E^(2*ArcCoth[a*x]), x, 10, -((32*c^3*Sqrt[c - a*c*x])/a) - (16*c^2*(c - a*c*x)^(3/2))/(3*a) - (8*c*(c - a*c*x)^(5/2))/(5*a) - (4*(c - a*c*x)^(7/2))/(7*a) - (2*(c - a*c*x)^(9/2))/(9*a*c) + (32*Sqrt[2]*c^(7/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{(c - a*c*x)^(5/2)/E^(2*ArcCoth[a*x]), x, 9, -((16*c^2*Sqrt[c - a*c*x])/a) - (8*c*(c - a*c*x)^(3/2))/(3*a) - (4*(c - a*c*x)^(5/2))/(5*a) - (2*(c - a*c*x)^(7/2))/(7*a*c) + (16*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{(c - a*c*x)^(3/2)/E^(2*ArcCoth[a*x]), x, 8, -((8*c*Sqrt[c - a*c*x])/a) - (4*(c - a*c*x)^(3/2))/(3*a) - (2*(c - a*c*x)^(5/2))/(5*a*c) + (8*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{(c - a*c*x)^(1/2)/E^(2*ArcCoth[a*x]), x, 7, -((4*Sqrt[c - a*c*x])/a) - (2*(c - a*c*x)^(3/2))/(3*a*c) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(1/2)), x, 6, -((2*Sqrt[c - a*c*x])/(a*c)) + (2*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(3/2)), x, 5, (Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2))} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(5/2)), x, 6, -(1/(a*c^2*Sqrt[c - a*c*x])) + ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(Sqrt[2]*a*c^(5/2))} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2)), x, 7, -(1/(3*a*c^2*(c - a*c*x)^(3/2))) - 1/(2*a*c^3*Sqrt[c - a*c*x]) + ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(2*Sqrt[2]*a*c^(7/2))} -{1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(9/2)), x, 8, -(1/(5*a*c^2*(c - a*c*x)^(5/2))) - 1/(6*a*c^3*(c - a*c*x)^(3/2)) - 1/(4*a*c^4*Sqrt[c - a*c*x]) + ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))} - - -{(c - a*c*x)^(9/2)/E^(3*ArcCoth[a*x]), x, 9, -((16*(a - 1/x)^5*(c - a*c*x)^(9/2))/(33*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)])) - (94208*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^5) - (40960*(c - a*c*x)^(9/2))/(231*a^5*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^4) + (4096*(c - a*c*x)^(9/2))/(231*a^4*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^3) - (1024*(a - 1/x)^3*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^2) + (320*(a - 1/x)^4*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - 1/x)^6*x*(c - a*c*x)^(9/2))/(11*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)])} -{(c - a*c*x)^(7/2)/E^(3*ArcCoth[a*x]), x, 8, -((40*(a - 1/x)^4*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])) + (11776*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^4) + (5120*(c - a*c*x)^(7/2))/(63*a^4*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^3) - (512*(c - a*c*x)^(7/2))/(63*a^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x^2) + (128*(a - 1/x)^3*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - 1/x)^5*x*(c - a*c*x)^(7/2))/(9*a^5*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])} -{(c - a*c*x)^(5/2)/E^(3*ArcCoth[a*x]), x, 7, -((32*(a - 1/x)^3*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])) - (2944*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3) - (256*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2) + (128*(c - a*c*x)^(5/2))/(35*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - 1/x)^4*x*(c - a*c*x)^(5/2))/(7*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])} -{(c - a*c*x)^(3/2)/E^(3*ArcCoth[a*x]), x, 6, -((8*(c - a*c*x)^(3/2))/(5*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])) + (184*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2) + (16*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - 1/x)^3*x*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])} -{(c - a*c*x)^(1/2)/E^(3*ArcCoth[a*x]), x, 5, -((20*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])) - (46*Sqrt[c - a*c*x])/(3*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) + (2*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(1/2)), x, 4, (6*Sqrt[1 - 1/(a*x)])/(a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]) + (2*Sqrt[1 - 1/(a*x)]*x)/(Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2)), x, 1, -((2*(1 + a*x))/(E^(3*ArcCoth[a*x])*(a*(c - a*c*x)^(3/2))))} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2)), x, 5, (a*(1 - 1/(a*x))^(5/2)*x^2)/(Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(5/2)) - (a^(3/2)*(1 - 1/(a*x))^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[2]*(1/x)^(5/2)*(c - a*c*x)^(5/2))} -{1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2)), x, 6, -((a^2*(1 - 1/(a*x))^(7/2)*x^2)/(2*(a - 1/x)*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2))) - (3*a^2*(1 - 1/(a*x))^(7/2)*x^3)/(4*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(7/2)) + (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(4*Sqrt[2]*(1/x)^(7/2)*(c - a*c*x)^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a x)^p*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^ArcCoth[x]*x*(1 + x), x, 7, Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]*x + (1/3)*(1 + 1/x)^(3/2)*Sqrt[(-1 + x)/x]*x^2 + (1/3)*(1 + 1/x)^(5/2)*Sqrt[(-1 + x)/x]*x^3 + ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} -{E^ArcCoth[x]*(1 + x), x, 6, (3/2)*Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]*x + (1/2)*(1 + 1/x)^(3/2)*Sqrt[(-1 + x)/x]*x^2 + (3/2)*ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} - -{E^ArcCoth[x]*(1 - x)*x, x, 3, (-(1/3))*(1 - 1/x^2)^(3/2)*x^3} -{E^ArcCoth[x]*(1 - x), x, 6, (-(1/2))*Sqrt[1 - 1/x^2]*x^2 + (1/2)*ArcTanh[Sqrt[1 - 1/x^2]]} - - -{E^ArcCoth[x]*x*(1 + x)^2, x, 8, (15/8)*Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]*x + (5/8)*(1 + 1/x)^(3/2)*Sqrt[(-1 + x)/x]*x^2 + (1/4)*(1 + 1/x)^(5/2)*Sqrt[(-1 + x)/x]*x^3 + (1/4)*(1 + 1/x)^(7/2)*Sqrt[(-1 + x)/x]*x^4 + (15/8)*ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} -{E^ArcCoth[x]*(1 + x)^2, x, 7, (5/2)*Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]*x + (5/6)*(1 + 1/x)^(3/2)*Sqrt[(-1 + x)/x]*x^2 + (1/3)*(1 + 1/x)^(5/2)*Sqrt[(-1 + x)/x]*x^3 + (5/2)*ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} - -{E^ArcCoth[x]*(1 - x)^2*x, x, 8, (1/8)*Sqrt[1 - 1/x^2]*x^2 - (1/3)*(1 - 1/x^2)^(3/2)*x^3 + (1/4)*(1 - 1/x^2)^(3/2)*x^4 - (1/8)*ArcTanh[Sqrt[1 - 1/x^2]]} -{E^ArcCoth[x]*(1 - x)^2, x, 7, (-(1/2))*Sqrt[1 - 1/x^2]*x^2 + (1/3)*(1 - 1/x^2)^(3/2)*x^3 + (1/2)*ArcTanh[Sqrt[1 - 1/x^2]]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(E^ArcCoth[x]*x)/(1 + x), x, 3, Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]*x} -{E^ArcCoth[x]/(1 + x), x, 4, ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} - -{(E^ArcCoth[x]*x)/(1 - x), x, 8, (2*(1 + 1/x))/Sqrt[1 - 1/x^2] - Sqrt[1 - 1/x^2]*x - 2*ArcTanh[Sqrt[1 - 1/x^2]]} -{E^ArcCoth[x]/(1 - x), x, 7, (2*(1 + 1/x))/Sqrt[1 - 1/x^2] - ArcTanh[Sqrt[1 - 1/x^2]]} - - -{(E^ArcCoth[x]*x)/(1 + x)^2, x, 5, -(Sqrt[(-1 + x)/x]/Sqrt[1 + 1/x]) + ArcTanh[Sqrt[1 + 1/x]*Sqrt[(-1 + x)/x]]} -{E^ArcCoth[x]/(1 + x)^2, x, 3, Sqrt[(-1 + x)/x]/Sqrt[1 + 1/x]} - -{(E^ArcCoth[x]*x)/(1 - x)^2, x, 9, -((4*(1 + 1/x))/(3*(1 - 1/x^2)^(3/2))) - (3 + 5/x)/(3*Sqrt[1 - 1/x^2]) + ArcTanh[Sqrt[1 - 1/x^2]]} -{E^ArcCoth[x]/(1 - x)^2, x, 3, -((1 - 1/x^2)^(3/2)/(3*(1 - 1/x)^3))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a x)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*x^m*Sqrt[c - a*c*x], x, 3, (2*x^(1 + m)*Sqrt[c - a*c*x]*Hypergeometric2F1[-(1/2), -(3/2) - m, -(1/2) - m, -(1/(a*x))])/((3 + 2*m)*Sqrt[1 - 1/(a*x)])} - -{E^ArcCoth[a*x]*x^2*Sqrt[c - a*c*x], x, 5, (16*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) - (8*(1 + 1/(a*x))^(3/2)*x^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x^3*Sqrt[c - a*c*x])/(7*Sqrt[1 - 1/(a*x)])} -{E^ArcCoth[a*x]*x*Sqrt[c - a*c*x], x, 4, -((4*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(15*a*Sqrt[1 - 1/(a*x)])) + (2*(1 + 1/(a*x))^(3/2)*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)])} -{E^ArcCoth[a*x]*Sqrt[c - a*c*x], x, 1, (2*E^ArcCoth[a*x]*(1 + a*x)*Sqrt[c - a*c*x])/(3*a)} -{(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x, x, 5, (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/Sqrt[1 - 1/(a*x)] - (2*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(Sqrt[a]*Sqrt[1 - 1/(a*x)])} -{(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x^2, x, 5, -((Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x)) - (Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)]} - - -{E^(2*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x], x, 5, (4*Sqrt[c - a*c*x])/a^4 - (14*(c - a*c*x)^(3/2))/(3*a^4*c) + (18*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (10*(c - a*c*x)^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4)} -{E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x], x, 5, (4*Sqrt[c - a*c*x])/a^3 - (10*(c - a*c*x)^(3/2))/(3*a^3*c) + (8*(c - a*c*x)^(5/2))/(5*a^3*c^2) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3)} -{E^(2*ArcCoth[a*x])*x*Sqrt[c - a*c*x], x, 5, (4*Sqrt[c - a*c*x])/a^2 - (2*(c - a*c*x)^(3/2))/(a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2)} -{E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x], x, 5, (4*Sqrt[c - a*c*x])/a - (2*(c - a*c*x)^(3/2))/(3*a*c)} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x, x, 6, 2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^2, x, 6, Sqrt[c - a*c*x]/x + 3*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^3, x, 7, Sqrt[c - a*c*x]/(2*x^2) + (7*a*Sqrt[c - a*c*x])/(4*x) + (7/4)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4, x, 8, Sqrt[c - a*c*x]/(3*x^3) + (11*a*Sqrt[c - a*c*x])/(12*x^2) + (11*a^2*Sqrt[c - a*c*x])/(8*x) + (11/8)*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5, x, 9, Sqrt[c - a*c*x]/(4*x^4) + (5*a*Sqrt[c - a*c*x])/(8*x^3) + (25*a^2*Sqrt[c - a*c*x])/(32*x^2) + (75*a^3*Sqrt[c - a*c*x])/(64*x) + (75/64)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]} - - -{E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x], x, 10, (1576*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(315*a^4*Sqrt[1 - 1/(a*x)]) + (472*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(315*a^3*Sqrt[1 - 1/(a*x)]) + (92*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) + (38*Sqrt[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(63*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^4*Sqrt[c - a*c*x])/(9*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(9/2)*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x], x, 9, (104*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(21*a^3*Sqrt[1 - 1/(a*x)]) + (32*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(21*a^2*Sqrt[1 - 1/(a*x)]) + (6*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(7*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(7*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(7/2)*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*x*Sqrt[c - a*c*x], x, 7, (4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(a^2*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(5/2)*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(5/2)*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x], x, 6, (4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(a*Sqrt[1 - 1/(a*x)]) + (2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(a^(3/2)*Sqrt[1 - 1/(a*x)])} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x, x, 8, (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[1/x]/Sqrt[a]])/(Sqrt[a]*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(Sqrt[a]*Sqrt[1 - 1/(a*x)])} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^2, x, 8, (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x) + (5*Sqrt[a]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[1/x]/Sqrt[a]])/Sqrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[a]*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^3, x, 9, (7*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(2*Sqrt[1 - 1/(a*x)]*x) + (23*a^(3/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[1/x]/Sqrt[a]])/(4*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(3/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4, x, 10, (a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*x^2) + (13*a^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(8*Sqrt[1 - 1/(a*x)]*x) + (3*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (45*a^(5/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[1/x]/Sqrt[a]])/(8*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(5/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5, x, 11, (a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x^3) + (11*a^2*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(24*Sqrt[1 - 1/(a*x)]*x^2) + (107*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1 - 1/(a*x)]*x) + (21*a^3*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(32*Sqrt[1 - 1/(a*x)]*x) + (363*a^(7/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[1/x]/Sqrt[a]])/(64*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^(7/2)*Sqrt[1/x]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[1 - 1/(a*x)]} - - -{E^ArcCoth[x]*x*(1 + x)^(3/2), x, 6, (46*Sqrt[-((1 - x)/x)]*(1 + x)^(3/2))/(21*(1 + 1/x)^(3/2)) + (92*Sqrt[-((1 - x)/x)]*(1 + x)^(3/2))/(21*(1 + 1/x)^(3/2)*x) + (8*Sqrt[-((1 - x)/x)]*x*(1 + x)^(3/2))/(7*(1 + 1/x)^(3/2)) + (2*Sqrt[-((1 - x)/x)]*x^2*(1 + x)^(3/2))/(7*(1 + 1/x)^(3/2))} -{E^ArcCoth[x]*(1 + x)^(3/2), x, 5, (28*Sqrt[-((1 - x)/x)]*(1 + x)^(3/2))/(15*(1 + 1/x)^(3/2)) + (86*Sqrt[-((1 - x)/x)]*(1 + x)^(3/2))/(15*(1 + 1/x)^(3/2)*x) + (2*Sqrt[-((1 - x)/x)]*x*(1 + x)^(3/2))/(5*(1 + 1/x)^(3/2))} - -{E^ArcCoth[x]*(1 - x)^(3/2)*x, x, 5, (44*(1 + 1/x)^(3/2)*(1 - x)^(3/2))/(105*(1 - 1/x)^(3/2)) - (22*(1 + 1/x)^(3/2)*(1 - x)^(3/2)*x)/(35*(1 - 1/x)^(3/2)) + (2*(1 + 1/x)^(3/2)*(1 - x)^(3/2)*x^2)/(7*(1 - 1/x)^(3/2))} -{E^ArcCoth[x]*(1 - x)^(3/2), x, 4, (-14*(1 + x^(-1))^(3/2)*(1 - x)^(3/2))/(15*(1 - x^(-1))^(3/2)) + (2*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x)/(5*(1 - x^(-1))^(3/2))} - - -{E^ArcCoth[x]*x*Sqrt[1 + x], x, 5, (12*Sqrt[-((1 - x)/x)]*Sqrt[1 + x])/(5*Sqrt[1 + 1/x]) + (6*Sqrt[-((1 - x)/x)]*x*Sqrt[1 + x])/(5*Sqrt[1 + 1/x]) + (2*Sqrt[-((1 - x)/x)]*x^2*Sqrt[1 + x])/(5*Sqrt[1 + 1/x])} -{E^ArcCoth[x]*Sqrt[1 + x], x, 4, (10*Sqrt[-((1 - x)/x)]*Sqrt[1 + x])/(3*Sqrt[1 + 1/x]) + (2*Sqrt[-((1 - x)/x)]*x*Sqrt[1 + x])/(3*Sqrt[1 + 1/x])} - -{E^ArcCoth[x]*Sqrt[1 - x]*x, x, 4, -((4*(1 + 1/x)^(3/2)*Sqrt[1 - x]*x)/(15*Sqrt[1 - 1/x])) + (2*(1 + 1/x)^(3/2)*Sqrt[1 - x]*x^2)/(5*Sqrt[1 - 1/x])} -{E^ArcCoth[x]*Sqrt[1 - x], x, 1, (2/3)*E^ArcCoth[x]*Sqrt[1 - x]*(1 + x)} - - -{(E^ArcCoth[x]*x)/Sqrt[1 + x], x, 4, (4*Sqrt[1 + 1/x]*Sqrt[-((1 - x)/x)]*x)/(3*Sqrt[1 + x]) + (2*Sqrt[1 + 1/x]*Sqrt[-((1 - x)/x)]*x^2)/(3*Sqrt[1 + x])} -{E^ArcCoth[x]/Sqrt[1 + x], x, 3, (2*Sqrt[1 + 1/x]*Sqrt[-((1 - x)/x)]*x)/Sqrt[1 + x]} - -{(E^ArcCoth[x]*x)/Sqrt[1 - x], x, 6, (2*Sqrt[1 - 1/x]*Sqrt[1 + 1/x]*x)/Sqrt[1 - x] + (2*Sqrt[1 - 1/x]*(1 + 1/x)^(3/2)*x^2)/(3*Sqrt[1 - x]) - (2*Sqrt[2]*Sqrt[1 - 1/x]*ArcTanh[(Sqrt[2]*Sqrt[1/x])/Sqrt[1 + 1/x]])/(Sqrt[1 - x]*Sqrt[1/x])} -{E^ArcCoth[x]/Sqrt[1 - x], x, 5, (2*Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x)/Sqrt[1 - x] - (2*Sqrt[2]*Sqrt[1 - x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[1 + x^(-1)]])/(Sqrt[1 - x]*Sqrt[x^(-1)])} - - -{(E^ArcCoth[x]*x)/(1 + x)^(3/2), x, 5, (2*(1 + 1/x)^(3/2)*Sqrt[-((1 - x)/x)]*x^2)/(1 + x)^(3/2) + (Sqrt[2]*(1 + 1/x)^(3/2)*ArcTan[(Sqrt[2]*Sqrt[1/x])/Sqrt[-((1 - x)/x)]])/((1/x)^(3/2)*(1 + x)^(3/2))} -{E^ArcCoth[x]/(1 + x)^(3/2), x, 4, -((Sqrt[2]*(1 + 1/x)^(3/2)*ArcTan[(Sqrt[2]*Sqrt[1/x])/Sqrt[-((1 - x)/x)]])/((1/x)^(3/2)*(1 + x)^(3/2)))} - -{(E^ArcCoth[x]*x)/(1 - x)^(3/2), x, 6, (5*(1 - 1/x)^(3/2)*Sqrt[1 + 1/x]*x^2)/(2*(1 - x)^(3/2)) - (Sqrt[1 - 1/x]*(1 + 1/x)^(3/2)*x^2)/(2*(1 - x)^(3/2)) - (5*(1 - 1/x)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[1/x])/Sqrt[1 + 1/x]])/(Sqrt[2]*(1 - x)^(3/2)*(1/x)^(3/2))} -{E^ArcCoth[x]/(1 - x)^(3/2), x, 5, -((Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x)/(1 - x)^(3/2)) - ((1 - x^(-1))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[1 + x^(-1)]])/(Sqrt[2]*(1 - x)^(3/2)*(x^(-1))^(3/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(x^m*Sqrt[c - a*c*x])/E^ArcCoth[a*x], x, 4, If[$VersionNumber>=8, (2*Sqrt[1 + 1/(a*x)]*x^(1 + m)*Sqrt[c - a*c*x])/((3 + 2*m)*Sqrt[1 - 1/(a*x)]) - (2*(5 + 4*m)*x^m*Sqrt[c - a*c*x]*Hypergeometric2F1[1/2, -(1/2) - m, 1/2 - m, -(1/(a*x))])/(a*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - 1/(a*x)]), (2*Sqrt[1 + 1/(a*x)]*x^(1 + m)*Sqrt[c - a*c*x])/((3 + 2*m)*Sqrt[1 - 1/(a*x)]) - (2*(5 + 4*m)*x^m*Sqrt[c - a*c*x]*Hypergeometric2F1[1/2, -(1/2) - m, 1/2 - m, -(1/(a*x))])/(a*(3 + 8*m + 4*m^2)*Sqrt[1 - 1/(a*x)])]} - -{(x^2*Sqrt[c - a*c*x])/E^ArcCoth[a*x], x, 6, (152*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(105*a^2*Sqrt[c - a*c*x]) + (38*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x])/(105*a^2) + (6*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2))/(35*a^2*c) - (2*Sqrt[1 - 1/(a^2*x^2)]*x^2*(c - a*c*x)^(3/2))/(7*a*c), -((208*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(105*a^3*Sqrt[1 - 1/(a*x)])) + (104*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) - (26*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(7*Sqrt[1 - 1/(a*x)])} -{(x*Sqrt[c - a*c*x])/E^ArcCoth[a*x], x, 5, -((8*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(5*a*Sqrt[c - a*c*x])) - (2*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x])/(5*a) - (2*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2))/(5*a*c), (12*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(5*a^2*Sqrt[1 - 1/(a*x)]) - (6*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(5*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/E^ArcCoth[a*x], x, 4, (8*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*Sqrt[c - a*c*x]) + (2/3)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x], -((10*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])) + (2*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^ArcCoth[a*x]*x), x, 5, (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(Sqrt[a]*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^ArcCoth[a*x]*x^2), x, 5, (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x) - (3*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)]} - - -{(x^3*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]), x, 9, (4*Sqrt[c - a*c*x])/a^4 + (2*(c - a*c*x)^(3/2))/(3*a^4*c) + (2*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (2*(c - a*c*x)^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^4} -{(x^2*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]), x, 9, -((4*Sqrt[c - a*c*x])/a^3) - (2*(c - a*c*x)^(3/2))/(3*a^3*c) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^3} -{(x*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]), x, 8, (4*Sqrt[c - a*c*x])/a^2 + (2*(c - a*c*x)^(3/2))/(3*a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a^2} -{Sqrt[c - a*c*x]/E^(2*ArcCoth[a*x]), x, 7, -((4*Sqrt[c - a*c*x])/a) - (2*(c - a*c*x)^(3/2))/(3*a*c) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x), x, 9, 2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^2), x, 9, Sqrt[c - a*c*x]/x - 5*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^3), x, 10, Sqrt[c - a*c*x]/(2*x^2) - (9*a*Sqrt[c - a*c*x])/(4*x) + (23/4)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^4), x, 11, Sqrt[c - a*c*x]/(3*x^3) - (13*a*Sqrt[c - a*c*x])/(12*x^2) + (19*a^2*Sqrt[c - a*c*x])/(8*x) - (45/8)*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^5), x, 12, Sqrt[c - a*c*x]/(4*x^4) - (17*a*Sqrt[c - a*c*x])/(24*x^3) + (107*a^2*Sqrt[c - a*c*x])/(96*x^2) - (149*a^3*Sqrt[c - a*c*x])/(64*x) + (363/64)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]} - - -{(x^3*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]), x, 8, (1312*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(45*a^4*Sqrt[1 - 1/(a*x)]) - (656*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(45*a^3*Sqrt[1 - 1/(a*x)]) - (82*x^2*Sqrt[c - a*c*x])/(9*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (164*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]) - (8*x^3*Sqrt[c - a*c*x])/(9*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (2*x^4*Sqrt[c - a*c*x])/(9*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])} -{(x^2*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]), x, 7, -((2672*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(105*a^3*Sqrt[1 - 1/(a*x)])) - (334*x*Sqrt[c - a*c*x])/(35*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (1336*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) - (44*x^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (2*x^3*Sqrt[c - a*c*x])/(7*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])} -{(x^1*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]), x, 6, -((158*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])) + (316*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]) - (32*x*Sqrt[c - a*c*x])/(15*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (2*x^2*Sqrt[c - a*c*x])/(5*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])} -{Sqrt[c - a*c*x]/E^(3*ArcCoth[a*x]), x, 5, -((20*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])) - (46*Sqrt[c - a*c*x])/(3*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) + (2*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^1), x, 6, (2*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (10*Sqrt[c - a*c*x])/(a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) - (2*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(Sqrt[a]*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^2), x, 6, (-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*x) + (7*Sqrt[a]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/Sqrt[1 - 1/(a*x)]} -{Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^3), x, 7, (-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^2) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(2*Sqrt[1 - 1/(a*x)]*x^2) + (47*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) - (47*a^(3/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(4*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4), x, 8, (-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^3) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)]*x^3) + (119*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(12*Sqrt[1 - 1/(a*x)]*x^2) - (119*a^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(8*Sqrt[1 - 1/(a*x)]*x) + (119*a^(5/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(8*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^5), x, 9, (-8*Sqrt[c - a*c*x])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x^4) - (Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x^4) + (223*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(24*Sqrt[1 - 1/(a*x)]*x^3) - (1115*a^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(96*Sqrt[1 - 1/(a*x)]*x^2) + (1115*a^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(64*Sqrt[1 - 1/(a*x)]*x) - (1115*a^(7/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(64*Sqrt[1 - 1/(a*x)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a x)^p with n symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2 + 2), x, 6, -(((56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a*(4 + n)*(6 + n))) + (2*(56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a^2*(6 + n)*(8 + 6*n + n^2)*x) + ((8 + n)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/2))/(6 + n) - ((a - 1/x)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/2))/a} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2 + 1), x, 4, If[$VersionNumber>=8, -((2*(6 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((2 + n)/2))/(a*(2 + n)*(4 + n))) + (2*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((2 + n)/2))/(4 + n), -((2*(6 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((2 + n)/2))/(a*(8 + 6*n + n^2))) + (2*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((2 + n)/2))/(4 + n)]} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2 + 0), x, 1, (2*E^(n*ArcCoth[a*x])*(1 + a*x)*(c - a*c*x)^(n/2))/(a*(2 + n))} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2 - 1), x, 3, (2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^(n/2)*x*(c - a*c*x)^((1/2)*(-2 + n))*Hypergeometric2F1[1, -(n/2), 1 - n/2, 2/((a + 1/x)*x)])/n} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2 - 2), x, 3, -((2*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x*(c - a*c*x)^((1/2)*(-4 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, 2/((a + 1/x)*x)])/(2 - n))} - - -{E^(n*ArcCoth[a*x])*(c - a*c*x)^p, x, 3, (((a - 1/x)/(a + 1/x))^((1/2)*(n - 2*p))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^p*Hypergeometric2F1[(1/2)*(n - 2*p), -1 - p, -p, 2/((a + 1/x)*x)])/((1 - 1/(a*x))^(n/2)*(1 + p))} - -{E^(n*ArcCoth[a*x])*(c - a*c*x)^3, x, 3, -((32*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((1/2)*(-8 + n))*Hypergeometric2F1[5, 4 - n/2, 5 - n/2, (a - 1/x)/(a + 1/x)])/(a*(8 - n)))} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^2, x, 3, (16*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((1/2)*(-6 + n))*Hypergeometric2F1[4, 3 - n/2, 4 - n/2, (a - 1/x)/(a + 1/x)])/(a*(6 - n))} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^1, x, 3, -((8*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-4 + n))*Hypergeometric2F1[3, 2 - n/2, 3 - n/2, (a - 1/x)/(a + 1/x)])/(a*(4 - n)))} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^1, x, 3, (2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c*n))} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^2, x, 3, -(((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^2*(2 + n)))} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^3, x, 4, If[$VersionNumber>=8, ((1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(4 + n)) - ((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(2 + n)*(4 + n)), -(((3 + n)*(1 - 1/(a*x))^((1/2)*(-2 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(8 + 6*n + n^2))) + ((1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(4 + n))]} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^4, x, 6, If[$VersionNumber>=8, ((5 + n)*(1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(4 + n)*(6 + n)) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)*(8 + 6*n + n^2)) - ((1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a^2*c^4*x), -(((14 + 8*n + n^2)*(1 - 1/(a*x))^((1/2)*(-4 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(24 + 10*n + n^2))) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^((1/2)*(-2 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(48 + 44*n + 12*n^2 + n^3)) + ((5 + n)*(1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)) - ((1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a^2*c^4*x)]} - - -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(5/2), x, 3, ((2/7)*((a - 1/x)/(a + 1/x))^((1/2)*(-5 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^(5/2)*Hypergeometric2F1[-(7/2), (1/2)*(-5 + n), -(5/2), 2/((a + 1/x)*x)])/(1 - 1/(a*x))^(n/2)} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(3/2), x, 3, ((2/5)*((a - 1/x)/(a + 1/x))^((1/2)*(-3 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^(3/2)*Hypergeometric2F1[-(5/2), (1/2)*(-3 + n), -(3/2), 2/((a + 1/x)*x)])/(1 - 1/(a*x))^(n/2)} -{E^(n*ArcCoth[a*x])*(c - a*c*x)^(1/2), x, 3, ((2/3)*((a - 1/x)/(a + 1/x))^((1/2)*(-1 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*Sqrt[c - a*c*x]*Hypergeometric2F1[-(3/2), (1/2)*(-1 + n), -(1/2), 2/((a + 1/x)*x)])/(1 - 1/(a*x))^(n/2)} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^(1/2), x, 3, (2*((a - 1/x)/(a + 1/x))^((1 + n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x*Hypergeometric2F1[-(1/2), (1 + n)/2, 1/2, 2/((a + 1/x)*x)])/((1 - 1/(a*x))^(n/2)*Sqrt[c - a*c*x])} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^(3/2), x, 3, -((2*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)])/((1 - 1/(a*x))^(n/2)*(c - a*c*x)^(3/2)))} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^(5/2), x, 4, -((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((3 + n)*(c - a*c*x)^(5/2))) + (a*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)])/((3 + n)*(c - a*c*x)^(5/2))} -{E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2), x, 5, -((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((5 + n)*(c - a*c*x)^(7/2))) + (3*a^2*(1 - 1/(a*x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3)/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2)) - (3*a^2*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 - 1/(a*x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)])/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcCoth[a x]) (c-c/(a x))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a x))^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - c/(a*x))^4, x, 9, -((c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a)) + (c^4*Sqrt[1 - 1/(a^2*x^2)]*(6*a - 1/x))/(2*a^2) + c^4*(1 - 1/(a^2*x^2))^(3/2)*x - (c^4*ArcCsc[a*x])/(2*a) - (3*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^ArcCoth[a*x]*(c - c/(a*x))^3, x, 8, (c^3*Sqrt[1 - 1/(a^2*x^2)]*(4*a + 1/x))/(2*a^2) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (c^3*ArcCsc[a*x])/(2*a) - (2*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^ArcCoth[a*x]*(c - c/(a*x))^2, x, 7, (c^2*Sqrt[1 - 1/(a^2*x^2)]*(a + 1/x)*x)/a + (c^2*ArcCsc[a*x])/a - (c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^ArcCoth[a*x]*(c - c/(a*x)), x, 3, c*Sqrt[1 - 1/(a^2*x^2)]*x + (c*ArcCsc[a*x])/a} -{E^ArcCoth[a*x]/(c - c/(a*x)), x, 7, -((2*(a + 1/x))/(a^2*c*Sqrt[1 - 1/(a^2*x^2)])) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c + (2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c)} -{E^ArcCoth[a*x]/(c - c/(a*x))^2, x, 8, -((4*(a + 1/x))/(3*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))) - (9*a + 11/x)/(3*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^2 + (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^2)} -{E^ArcCoth[a*x]/(c - c/(a*x))^3, x, 9, -((8*(a + 1/x))/(5*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2))) - (4*(5*a + 8/x))/(15*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) - (60*a + 79/x)/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)} -{E^ArcCoth[a*x]/(c - c/(a*x))^4, x, 10, -((16*(a + 1/x))/(7*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2))) - (4*(7*a + 17/x))/(35*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (175*a + 307/x)/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (525*a + 719/x)/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)} - - -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^5, x, 5, -c^5/(4*a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^4, x, 5, c^4/(3*a^4*x^3) - c^4/(a^3*x^2) + c^4*x - (2*c^4*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^3, x, 5, -c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x - (c^3*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^2, x, 6, c^2/(a^2*x) + c^2*x} -{E^(2*ArcCoth[a*x])*(c - c/(a*x)), x, 5, c*x + (c*Log[x])/a} -{E^(2*ArcCoth[a*x])/(c - c/(a*x)), x, 5, x/c + 2/(a*c*(1 - a*x)) + (3*Log[1 - a*x])/(a*c)} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^2, x, 5, x/c^2 - 1/(a*c^2*(1 - a*x)^2) + 5/(a*c^2*(1 - a*x)) + (4*Log[1 - a*x])/(a*c^2)} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^3, x, 5, x/c^3 + 2/(3*a*c^3*(1 - a*x)^3) - 7/(2*a*c^3*(1 - a*x)^2) + 9/(a*c^3*(1 - a*x)) + (5*Log[1 - a*x])/(a*c^3)} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^4, x, 5, x/c^4 - 1/(2*a*c^4*(1 - a*x)^4) + 3/(a*c^4*(1 - a*x)^3) - 8/(a*c^4*(1 - a*x)^2) + 14/(a*c^4*(1 - a*x)) + (6*Log[1 - a*x])/(a*c^4)} - - -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^4, x, 8, (c^4*Sqrt[1 - 1/(a^2*x^2)]*(2*a + 3/x))/(2*a^2) + (c^4*(1 - 1/(a^2*x^2))^(3/2)*(3*a + 1/x)*x)/(3*a) + (3*c^4*ArcCsc[a*x])/(2*a) - (c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^3, x, 4, (3*c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (3*c^3*ArcCsc[a*x])/(2*a)} -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^2, x, 8, (c^2*Sqrt[1 - 1/(a^2*x^2)]*(a - 1/x)*x)/a + (c^2*ArcCsc[a*x])/a + (c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a*x)), x, 8, c*Sqrt[1 - 1/(a^2*x^2)]*x - (c*ArcCsc[a*x])/a + (2*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{E^(3*ArcCoth[a*x])/(c - c/(a*x)), x, 8, -((8*(a + 1/x))/(3*a^2*c*(1 - 1/(a^2*x^2))^(3/2))) - (4*(3*a + 4/x))/(3*a^2*c*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c + (4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c)} -{E^(3*ArcCoth[a*x])/(c - c/(a*x))^2, x, 9, -((16*(a + 1/x))/(5*a^2*c^2*(1 - 1/(a^2*x^2))^(5/2))) - (4*(5*a + 11/x))/(15*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)) - (75*a + 103/x)/(15*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^2 + (5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^2)} -{E^(3*ArcCoth[a*x])/(c - c/(a*x))^3, x, 10, -((32*(a + 1/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(7/2))) - (2*(7*a + 13/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) - (42*a + 59/x)/(7*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) - 16/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)*x) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)} -{E^(3*ArcCoth[a*x])/(c - c/(a*x))^4, x, 11, (16*(9*a - 5/x))/(63*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (64*(a + 1/x))/(9*a^2*c^4*(1 - 1/(a^2*x^2))^(9/2)) - (8*(21*a + 41/x))/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (735*a + 1417/x)/(315*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (2205*a + 3149/x)/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (7*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)} - - -{E^(4*ArcCoth[a*x])*(c - c/(a*x))^5, x, 5, c^5/(4*a^5*x^4) - c^5/(3*a^4*x^3) - c^5/(a^3*x^2) + (2*c^5)/(a^2*x) + c^5*x - (c^5*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a*x))^4, x, 6, -c^4/(3*a^4*x^3) + (2*c^4)/(a^2*x) + c^4*x} -{E^(4*ArcCoth[a*x])*(c - c/(a*x))^3, x, 5, c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x + (c^3*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a*x))^2, x, 5, -(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a*x)), x, 5, c*x - (c*Log[x])/a + (4*c*Log[1 - a*x])/a} -{E^(4*ArcCoth[a*x])/(c - c/(a*x)), x, 5, x/c - 2/(a*c*(1 - a*x)^2) + 8/(a*c*(1 - a*x)) + (5*Log[1 - a*x])/(a*c)} -{E^(4*ArcCoth[a*x])/(c - c/(a*x))^2, x, 5, x/c^2 + 4/(3*a*c^2*(1 - a*x)^3) - 6/(a*c^2*(1 - a*x)^2) + 13/(a*c^2*(1 - a*x)) + (6*Log[1 - a*x])/(a*c^2)} -{E^(4*ArcCoth[a*x])/(c - c/(a*x))^3, x, 5, x/c^3 - 1/(a*c^3*(1 - a*x)^4) + 16/(3*a*c^3*(1 - a*x)^3) - 25/(2*a*c^3*(1 - a*x)^2) + 19/(a*c^3*(1 - a*x)) + (7*Log[1 - a*x])/(a*c^3)} -{E^(4*ArcCoth[a*x])/(c - c/(a*x))^4, x, 5, x/c^4 + 4/(5*a*c^4*(1 - a*x)^5) - 5/(a*c^4*(1 - a*x)^4) + 41/(3*a*c^4*(1 - a*x)^3) - 22/(a*c^4*(1 - a*x)^2) + 26/(a*c^4*(1 - a*x)) + (8*Log[1 - a*x])/(a*c^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a*x))^4/E^ArcCoth[a*x], x, 10, -((32*c^4*Sqrt[1 - 1/(a^2*x^2)])/(3*a)) - (c^4*Sqrt[1 - 1/(a^2*x^2)])/(3*a^3*x^2) + (5*c^4*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^4*Sqrt[1 - 1/(a^2*x^2)]*x - (25*c^4*ArcCsc[a*x])/(2*a) - (5*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^3/E^ArcCoth[a*x], x, 9, -((4*c^3*Sqrt[1 - 1/(a^2*x^2)])/a) + (c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x - (13*c^3*ArcCsc[a*x])/(2*a) - (4*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^2/E^ArcCoth[a*x], x, 8, -((c^2*Sqrt[1 - 1/(a^2*x^2)])/a) + c^2*Sqrt[1 - 1/(a^2*x^2)]*x - (3*c^2*ArcCsc[a*x])/a - (3*c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^1/E^ArcCoth[a*x], x, 7, c*Sqrt[1 - 1/(a^2*x^2)]*x - (c*ArcCsc[a*x])/a - (2*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^1), x, 2, (Sqrt[1 - 1/(a^2*x^2)]*x)/c} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^2), x, 6, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/c^2 - (a*Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*(a - 1/x)) + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c^2)} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^3), x, 8, -((2*(a + 1/x))/(3*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2))) - (6*a + 7/x)/(3*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^3)} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^4), x, 9, -((4*(a + 1/x))/(5*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2))) - (5*a + 7/x)/(5*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (15*a + 19/x)/(5*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^4)} - - -{(c - c/(a*x))^4/E^(2*ArcCoth[a*x]), x, 5, c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*Log[x])/a - (32*c^4*Log[1 + a*x])/a} -{(c - c/(a*x))^3/E^(2*ArcCoth[a*x]), x, 5, -c^3/(2*a^3*x^2) + (5*c^3)/(a^2*x) + c^3*x + (11*c^3*Log[x])/a - (16*c^3*Log[1 + a*x])/a} -{(c - c/(a*x))^2/E^(2*ArcCoth[a*x]), x, 5, c^2/(a^2*x) + c^2*x + (4*c^2*Log[x])/a - (8*c^2*Log[1 + a*x])/a} -{(c - c/(a*x))^1/E^(2*ArcCoth[a*x]), x, 5, c*x + (c*Log[x])/a - (4*c*Log[1 + a*x])/a} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^1), x, 5, x/c - Log[1 + a*x]/(a*c)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^2), x, 6, x/c^2 - ArcTanh[a*x]/(a*c^2)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^3), x, 5, x/c^3 + 1/(2*a*c^3*(1 - a*x)) + (5*Log[1 - a*x])/(4*a*c^3) - Log[1 + a*x]/(4*a*c^3)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^4), x, 5, x/c^4 - 1/(4*a*c^4*(1 - a*x)^2) + 7/(4*a*c^4*(1 - a*x)) + (17*Log[1 - a*x])/(8*a*c^4) - Log[1 + a*x]/(8*a*c^4)} - - -{(c - c/(a*x))^4/E^(3*ArcCoth[a*x]), x, 11, (68*c^4*Sqrt[1 - 1/(a^2*x^2)])/(3*a) + (64*c^4*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (c^4*Sqrt[1 - 1/(a^2*x^2)])/(3*a^3*x^2) - (7*c^4*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^4*Sqrt[1 - 1/(a^2*x^2)]*x + (91*c^4*ArcCsc[a*x])/(2*a) - (7*c^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^3/E^(3*ArcCoth[a*x]), x, 10, (6*c^3*Sqrt[1 - 1/(a^2*x^2)])/a + (32*c^3*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (c^3*Sqrt[1 - 1/(a^2*x^2)])/(2*a^2*x) + c^3*Sqrt[1 - 1/(a^2*x^2)]*x + (33*c^3*ArcCsc[a*x])/(2*a) - (6*c^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^2/E^(3*ArcCoth[a*x]), x, 9, (c^2*Sqrt[1 - 1/(a^2*x^2)])/a + (16*c^2*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + c^2*Sqrt[1 - 1/(a^2*x^2)]*x + (5*c^2*ArcCsc[a*x])/a - (5*c^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{(c - c/(a*x))^1/E^(3*ArcCoth[a*x]), x, 8, (8*c*(a - 1/x))/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + c*Sqrt[1 - 1/(a^2*x^2)]*x + (c*ArcCsc[a*x])/a - (4*c*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/a} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^1), x, 6, (2*(a - 1/x))/(a^2*c*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c - (2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^2), x, 6, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/c^2 - ((a - 1/x)*x)/(a*c^2*Sqrt[1 - 1/(a^2*x^2)]) - ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c^2)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^3), x, 3, -(2/(a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)) + x/(c^3*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^4), x, 7, (8*Sqrt[1 - 1/(a^2*x^2)]*x)/(3*c^4) - (a*x)/(3*c^4*Sqrt[1 - 1/(a^2*x^2)]*(a - 1/x)) - ((4*a + 3/x)*x)/(3*a*c^4*Sqrt[1 - 1/(a^2*x^2)]) + ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]/(a*c^4)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5), x, 9, -((2*(a + 1/x))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(5/2))) - (10*a + 13/x)/(15*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2)) - (30*a + 41/x)/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^5 + (2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a*c^5)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a x))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - c/(a*x))^(9/2), x, 8, (173*c^5*Sqrt[1 - 1/(a^2*x^2)])/(105*a*Sqrt[c - c/(a*x)]) + (227*c^4*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/(105*a) + (59*c^3*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(35*a) + (9*c^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(5/2))/(7*a) + c*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(7/2)*x - (7*c^(9/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a, ((400*a - 227/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(105*a^2*(1 - 1/(a*x))^(9/2)) + (59*(a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(35*a^3*(1 - 1/(a*x))^(9/2)) + (9*(a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(7*a^4*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^4*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2)*x)/(a^4*(1 - 1/(a*x))^(9/2)) - (7*(c - c/(a*x))^(9/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(9/2))} -{E^ArcCoth[a*x]*(c - c/(a*x))^(7/2), x, 7, (49*c^4*Sqrt[1 - 1/(a^2*x^2)])/(15*a*Sqrt[c - c/(a*x)]) + (31*c^3*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)])/(15*a) + (7*c^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*a) + c*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(5/2)*x - (5*c^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a, ((80*a - 31/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(15*a^2*(1 - 1/(a*x))^(7/2)) + (7*(a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)*x)/(a^3*(1 - 1/(a*x))^(7/2)) - (5*(c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(7/2))} -{E^ArcCoth[a*x]*(c - c/(a*x))^(5/2), x, 7, -((2*c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a*(c - c/(a*x))^(3/2))) + (3*c^3*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a*x)]) + (c^4*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) - (3*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a, (3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2))/(a*(1 - 1/(a*x))^(5/2)) - (2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2))/(3*a*(1 - 1/(a*x))^(5/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2)*x)/(1 - 1/(a*x))^(5/2) - (3*(c - c/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(5/2))} -{E^ArcCoth[a*x]*(c - c/(a*x))^(3/2), x, 5, (c^2*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a*x)]) + (c^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)], x, 4, (c*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)] + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{E^ArcCoth[a*x]/Sqrt[c - c/(a*x)], x, 8, (Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/Sqrt[c - c/(a*x)] + (3*Sqrt[1 - 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[c - c/(a*x)]) - (2*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*Sqrt[c - c/(a*x)])} -{E^ArcCoth[a*x]/(c - c/(a*x))^(3/2), x, 9, (-2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])/((a - x^(-1))*(c - c/(a*x))^(3/2)) + (a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))*(c - c/(a*x))^(3/2)) + (5*(1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(3/2)) - (7*(1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(Sqrt[2]*a*(c - c/(a*x))^(3/2))} -{E^ArcCoth[a*x]/(c - c/(a*x))^(5/2), x, 10, (-3*a*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(2*(a - x^(-1))^2*(c - c/(a*x))^(5/2)) - (23*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(8*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))^2*(c - c/(a*x))^(5/2)) + (7*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - (79*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(8*Sqrt[2]*a*(c - c/(a*x))^(5/2))} - - -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2), x, 12, (5*c^4*Sqrt[c - c/(a*x)])/a + (5*c^3*(c - c/(a*x))^(3/2))/(3*a) + (c^2*(c - c/(a*x))^(5/2))/a + (5*c*(c - c/(a*x))^(7/2))/(7*a) + (c - c/(a*x))^(9/2)*x - (5*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2), x, 11, (3*c^3*Sqrt[c - c/(a*x)])/a + (c^2*(c - c/(a*x))^(3/2))/a + (3*c*(c - c/(a*x))^(5/2))/(5*a) + (c - c/(a*x))^(7/2)*x - (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2), x, 10, (c^2*Sqrt[c - c/(a*x)])/a + (c*(c - c/(a*x))^(3/2))/(3*a) + (c - c/(a*x))^(5/2)*x - (c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a*x))^(3/2), x, 9, -((c*Sqrt[c - c/(a*x)])/a) + (c - c/(a*x))^(3/2)*x + (c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)], x, 8, Sqrt[c - c/(a*x)]*x + (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{E^(2*ArcCoth[a*x])/Sqrt[c - c/(a*x)], x, 9, -5/(a*Sqrt[c - c/(a*x)]) + x/Sqrt[c - c/(a*x)] + (5*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*Sqrt[c])} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^(3/2), x, 10, -7/(3*a*(c - c/(a*x))^(3/2)) - 7/(a*c*Sqrt[c - c/(a*x)]) + x/(c - c/(a*x))^(3/2) + (7*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(3/2))} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^(5/2), x, 11, -9/(5*a*(c - c/(a*x))^(5/2)) - 3/(a*c*(c - c/(a*x))^(3/2)) - 9/(a*c^2*Sqrt[c - c/(a*x)]) + x/(c - c/(a*x))^(5/2) + (9*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(5/2))} -{E^(2*ArcCoth[a*x])/(c - c/(a*x))^(7/2), x, 12, -11/(7*a*(c - c/(a*x))^(7/2)) - 11/(5*a*c*(c - c/(a*x))^(5/2)) - 11/(3*a*c^2*(c - c/(a*x))^(3/2)) - 11/(a*c^3*Sqrt[c - c/(a*x)]) + x/(c - c/(a*x))^(7/2) + (11*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(7/2))} - - -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^(9/2), x, 8, (3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(a*(1 - 1/(a*x))^(9/2)) + (3*(28*a - 17/x)*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(35*a^2*(1 - 1/(a*x))^(9/2)) + (9*(a - 1/x)^2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(7*a^3*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^3*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2)*x)/(a^3*(1 - 1/(a*x))^(9/2)) - (3*(c - c/(a*x))^(9/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(9/2))} -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2), x, 8, (Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(7/2))/(3*a*(1 - 1/(a*x))^(7/2)) - (2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2))/(5*a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2)*x)/(1 - 1/(a*x))^(7/2) - ((c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(7/2))} -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2), x, 6, -((c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a*(c - c/(a*x))^(3/2))) - (c^3*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a*x)]) + (c^5*(1 - 1/(a^2*x^2))^(5/2)*x)/(c - c/(a*x))^(5/2) + (c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2), x, 5, -((3*c^2*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a*x)])) + (c^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) + (3*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)], x, 8, (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/Sqrt[1 - 1/(a*x)] + (5*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])/Sqrt[c - c/(a*x)], x, 9, -((3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/((a - 1/x)*Sqrt[c - c/(a*x)])) + (a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x)/((a - 1/x)*Sqrt[c - c/(a*x)]) + (7*Sqrt[1 - 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[c - c/(a*x)]) - (5*Sqrt[2]*Sqrt[1 - 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*Sqrt[c - c/(a*x)])} -{E^(3*ArcCoth[a*x])/(c - c/(a*x))^(3/2), x, 10, -((2*a*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])/((a - 1/x)^2*(c - c/(a*x))^(3/2))) - (15*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])/(4*(a - 1/x)*(c - c/(a*x))^(3/2)) + (a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/((a - 1/x)^2*(c - c/(a*x))^(3/2)) + (9*(1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(3/2)) - (51*(1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(4*Sqrt[2]*a*(c - c/(a*x))^(3/2))} -{E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2), x, 11, -((5*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(3*(a - 1/x)^3*(c - c/(a*x))^(5/2))) - (29*a*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(12*(a - 1/x)^2*(c - c/(a*x))^(5/2)) - (73*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(16*(a - 1/x)*(c - c/(a*x))^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x)/((a - 1/x)^3*(c - c/(a*x))^(5/2)) + (11*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - (249*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*Sqrt[2]*a*(c - c/(a*x))^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a*x))^(7/2)/E^ArcCoth[a*x], x, 7, -(((80*a - 7/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(5*a^2*(1 - 1/(a*x))^(7/2))) + (3*(a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)*x)/(a^3*(1 - 1/(a*x))^(7/2)) - (9*(c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(7/2))} -{(c - c/(a*x))^(5/2)/E^ArcCoth[a*x], x, 6, -(((16*a + 1/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2))/(3*a^2*(1 - 1/(a*x))^(5/2))) + ((a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2)*x)/(a^2*(1 - 1/(a*x))^(5/2)) - (7*(c - c/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(5/2))} -{(c - c/(a*x))^(3/2)/E^ArcCoth[a*x], x, 6, (-2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(3/2))/(a*(1 - 1/(a*x))^(3/2)) + (Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(3/2)*x)/(1 - 1/(a*x))^(3/2) - (5*(c - c/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(3/2))} -{Sqrt[c - c/(a*x)]/E^ArcCoth[a*x], x, 4, (c*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)] - (3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{1/(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)] - ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]]/(a*Sqrt[c])} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^(3/2)), x, 9, ((1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x)/(c - c/(a*x))^(3/2) + ((1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(3/2)) - (Sqrt[2]*(1 - 1/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*(c - c/(a*x))^(3/2))} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)), x, 10, (-3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(2*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))*(c - c/(a*x))^(5/2)) + (3*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - (9*(1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(2*Sqrt[2]*a*(c - c/(a*x))^(5/2))} -{1/(E^ArcCoth[a*x]*(c - c/(a*x))^(7/2)), x, 11, (-5*a*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])/(4*(a - x^(-1))^2*(c - c/(a*x))^(7/2)) - (35*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])/(16*(a - x^(-1))*(c - c/(a*x))^(7/2)) + (a^2*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x)/((a - x^(-1))^2*(c - c/(a*x))^(7/2)) + (5*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) - (115*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(16*Sqrt[2]*a*(c - c/(a*x))^(7/2))} - - -{(c - c/(a*x))^(7/2)/E^(2*ArcCoth[a*x]), x, 14, -((21*c^3*Sqrt[c - c/(a*x)])/a) - (5*c^2*(c - c/(a*x))^(3/2))/(3*a) + (3*c*(c - c/(a*x))^(5/2))/(5*a) + (c - c/(a*x))^(7/2)*x - (11*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (32*Sqrt[2]*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{(c - c/(a*x))^(5/2)/E^(2*ArcCoth[a*x]), x, 13, -((7*c^2*Sqrt[c - c/(a*x)])/a) + (c*(c - c/(a*x))^(3/2))/(3*a) + (c - c/(a*x))^(5/2)*x - (9*c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (16*Sqrt[2]*c^(5/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{(c - c/(a*x))^(3/2)/E^(2*ArcCoth[a*x]), x, 12, -((c*Sqrt[c - c/(a*x)])/a) + (c - c/(a*x))^(3/2)*x - (7*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (8*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]), x, 11, Sqrt[c - c/(a*x)]*x - (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]), x, 11, (Sqrt[c - c/(a*x)]*x)/c - (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*Sqrt[c]) + (2*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(3/2)), x, 12, (Sqrt[c - c/(a*x)]*x)/c^2 - ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*c^(3/2)) + (Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2))} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2)), x, 12, -(2/(a*c^2*Sqrt[c - c/(a*x)])) + x/(c^2*Sqrt[c - c/(a*x)]) + ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*c^(5/2)) + ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(Sqrt[2]*a*c^(5/2))} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2)), x, 13, -(4/(3*a*c^2*(c - c/(a*x))^(3/2))) - 7/(2*a*c^3*Sqrt[c - c/(a*x)]) + x/(c^2*(c - c/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(7/2)) + ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(2*Sqrt[2]*a*c^(7/2))} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2)), x, 14, -(6/(5*a*c^2*(c - c/(a*x))^(5/2))) - 11/(6*a*c^3*(c - c/(a*x))^(3/2)) - 21/(4*a*c^4*Sqrt[c - c/(a*x)]) + x/(c^2*(c - c/(a*x))^(5/2)) + (5*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(9/2)) + ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))} - - -{(c - c/(a*x))^(9/2)/E^(3*ArcCoth[a*x]), x, 9, (10*(a - 1/x)^4*(c - c/(a*x))^(9/2))/(a^5*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]) + (5*(304*a - 65/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(7*a^2*(1 - 1/(a*x))^(9/2)) + (135*(a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(7*a^3*(1 - 1/(a*x))^(9/2)) + (65*(a - 1/x)^3*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(9/2))/(7*a^4*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^5*(c - c/(a*x))^(9/2)*x)/(a^5*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]) - (15*(c - c/(a*x))^(9/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(9/2))} -{(c - c/(a*x))^(7/2)/E^(3*ArcCoth[a*x]), x, 8, (10*(a - 1/x)^3*(c - c/(a*x))^(7/2))/(a^4*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]) + ((1360*a - 311/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(15*a^2*(1 - 1/(a*x))^(7/2)) + (47*(a - 1/x)^2*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^4*(c - c/(a*x))^(7/2)*x)/(a^4*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]) - (13*(c - c/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(7/2))} -{(c - c/(a*x))^(5/2)/E^(3*ArcCoth[a*x]), x, 7, (10*(a - 1/x)^2*(c - c/(a*x))^(5/2))/(a^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) + ((112*a - 29/x)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2))/(3*a^2*(1 - 1/(a*x))^(5/2)) + ((a - 1/x)^3*(c - c/(a*x))^(5/2)*x)/(a^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - (11*(c - c/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(5/2))} -{(c - c/(a*x))^(3/2)/E^(3*ArcCoth[a*x]), x, 6, ((21*a + 1/x)*(c - c/(a*x))^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + ((a - 1/x)^2*(c - c/(a*x))^(3/2)*x)/(a^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) - (9*(c - c/(a*x))^(3/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(1 - 1/(a*x))^(3/2))} -{Sqrt[c - c/(a*x)]/E^(3*ArcCoth[a*x]), x, 6, (9*Sqrt[c - c/(a*x)])/(a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt[c - c/(a*x)]*x)/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (7*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[1 - 1/(a*x)])} -{1/(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]), x, 5, (5*Sqrt[c - c/(a*x)])/(a*c*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a*x)]*x)/(c*Sqrt[1 - 1/(a^2*x^2)]) - (5*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(a*Sqrt[c])} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2)), x, 5, (3*Sqrt[1 - 1/(a^2*x^2)]*x)/(c*Sqrt[c - c/(a*x)]) - (2*Sqrt[c - c/(a*x)]*x)/(c^2*Sqrt[1 - 1/(a^2*x^2)]) - (3*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(a*c^(3/2))} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2)), x, 9, (2*(1 - 1/(a*x))^(5/2))/(a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2)) + ((1 - 1/(a*x))^(5/2)*x)/(Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*x))^(5/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(Sqrt[2]*a*(c - c/(a*x))^(5/2))} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)), x, 10, (7*(1 - 1/(a*x))^(7/2))/(4*a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) - (3*(1 - 1/(a*x))^(7/2))/(2*(a - x^(-1))*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) + (a*(1 - 1/(a*x))^(7/2)*x)/((a - x^(-1))*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(7/2)) + ((1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*(c - c/(a*x))^(7/2)) - (11*(1 - 1/(a*x))^(7/2)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(4*Sqrt[2]*a*(c - c/(a*x))^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c/(a x))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^m, x, 3, (Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[-(1/2), -1 - m, -m, -(1/(a*x))])/((1 + m)*Sqrt[1 - 1/(a*x)])} - -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^2, x, 6, -((c*Sqrt[1 - 1/(a^2*x^2)]*x)/(8*a^2*Sqrt[c - c/(a*x)])) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(12*a*Sqrt[c - c/(a*x)]) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^3)/(3*Sqrt[c - c/(a*x)]) + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(8*a^3)} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^1, x, 5, (c*Sqrt[1 - 1/(a^2*x^2)]*x)/(4*a*Sqrt[c - c/(a*x)]) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*Sqrt[c - c/(a*x)]) - (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(4*a^2)} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^0, x, 4, (c*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)] + (Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]/x^1, x, 4, -((2*c*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]) + 2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]]} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]/x^2, x, 2, -((2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)))} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]/x^3, x, 3, -((2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(15*(c - c/(a*x))^(3/2))) + (2*a^2*c*(1 - 1/(a^2*x^2))^(3/2))/(5*Sqrt[c - c/(a*x)])} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]/x^4, x, 4, (8*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(105*(c - c/(a*x))^(3/2)) - (8*a^3*c*(1 - 1/(a^2*x^2))^(3/2))/(35*Sqrt[c - c/(a*x)]) - (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(7*(c - c/(a*x))^(3/2)*x^2)} -{E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]/x^5, x, 5, -((16*a^4*c^2*(1 - 1/(a^2*x^2))^(3/2))/(315*(c - c/(a*x))^(3/2))) + (16*a^4*c*(1 - 1/(a^2*x^2))^(3/2))/(105*Sqrt[c - c/(a*x)]) - (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(9*(c - c/(a*x))^(3/2)*x^3) + (4*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(21*(c - c/(a*x))^(3/2)*x^2)} - - -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3, x, 11, (75*Sqrt[c - c/(a*x)]*x)/(64*a^3) + (25*Sqrt[c - c/(a*x)]*x^2)/(32*a^2) + (5*Sqrt[c - c/(a*x)]*x^3)/(8*a) + (Sqrt[c - c/(a*x)]*x^4)/4 + (75*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4)} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^2, x, 10, (11*Sqrt[c - c/(a*x)]*x)/(8*a^2) + (11*Sqrt[c - c/(a*x)]*x^2)/(12*a) + (Sqrt[c - c/(a*x)]*x^3)/3 + (11*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(8*a^3)} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x, x, 9, (7*Sqrt[c - c/(a*x)]*x)/(4*a) + (Sqrt[c - c/(a*x)]*x^2)/2 + (7*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(4*a^2)} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)], x, 8, Sqrt[c - c/(a*x)]*x + (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x, x, 8, 2*Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^2, x, 7, 4*a*Sqrt[c - c/(a*x)] - (2*a*(c - c/(a*x))^(3/2))/(3*c)} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^3, x, 7, 4*a^2*Sqrt[c - c/(a*x)] - (2*a^2*(c - c/(a*x))^(3/2))/c + (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2)} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4, x, 7, 4*a^3*Sqrt[c - c/(a*x)] - (10*a^3*(c - c/(a*x))^(3/2))/(3*c) + (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3)} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5, x, 7, 4*a^4*Sqrt[c - c/(a*x)] - (14*a^4*(c - c/(a*x))^(3/2))/(3*c) + (18*a^4*(c - c/(a*x))^(5/2))/(5*c^2) - (10*a^4*(c - c/(a*x))^(7/2))/(7*c^3) + (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4)} - - -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3, x, 11, (149*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(64*a^3*Sqrt[1 - 1/(a*x)]) + (107*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^2)/(96*a^2*Sqrt[1 - 1/(a*x)]) + (17*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^3)/(24*a*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^4)/(4*Sqrt[1 - 1/(a*x)]) + (363*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(64*a^4*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^4*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^2, x, 10, (19*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(8*a^2*Sqrt[1 - 1/(a*x)]) + (13*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^2)/(12*a*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^3)/(3*Sqrt[1 - 1/(a*x)]) + (45*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(8*a^3*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^3*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x, x, 9, (9*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(4*a*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^2)/(2*Sqrt[1 - 1/(a*x)]) + (23*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(4*a^2*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^2*Sqrt[1 - 1/(a*x)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)], x, 8, (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/Sqrt[1 - 1/(a*x)] + (5*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a*Sqrt[1 - 1/(a*x)])} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x, x, 8, (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/Sqrt[1 - 1/(a*x)] - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^2, x, 5, (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) + (4*a*c*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] - 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^3, x, 6, (2*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2))/(5*(c - c/(a*x))^(5/2)) + (2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) + (4*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4, x, 7, (4*a^3*c^3*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(5/2)) + (2*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) - (2*a^3*c^2*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(3/2)) + (4*a^3*c*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/(Sqrt[2]*Sqrt[c - c/(a*x)])]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5, x, 8, (4*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(5/2)*Sqrt[c - c/(a*x)])/(5*Sqrt[1 - 1/(a*x)]) - (2*a^4*(1 + 1/(a*x))^(7/2)*Sqrt[c - c/(a*x)])/(7*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(9/2)*Sqrt[c - c/(a*x)])/(9*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*a^4*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sqrt[c - c/(a*x)]*x^m)/E^ArcCoth[a*x], x, 4, (Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^(1 + m))/((1 + m)*Sqrt[1 - 1/(a*x)]) - ((3 + 4*m)*Sqrt[c - c/(a*x)]*x^m*Hypergeometric2F1[1/2, -m, 1 - m, -(1/(a*x))])/(2*a*m*(1 + m)*Sqrt[1 - 1/(a*x)])} - -{(Sqrt[c - c/(a*x)]*x^2)/E^ArcCoth[a*x], x, 6, (11*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(8*a^2*Sqrt[c - c/(a*x)]) - (11*c*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(12*a*Sqrt[c - c/(a*x)]) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^3)/(3*Sqrt[c - c/(a*x)]) - (11*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(8*a^3)} -{(Sqrt[c - c/(a*x)]*x)/E^ArcCoth[a*x], x, 5, -((7*c*Sqrt[1 - 1/(a^2*x^2)]*x)/(4*a*Sqrt[c - c/(a*x)])) + (c*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(2*Sqrt[c - c/(a*x)]) + (7*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/(4*a^2)} -{Sqrt[c - c/(a*x)]/E^ArcCoth[a*x], x, 4, (c*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a*x)] - (3*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]])/a} -{Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x), x, 4, (2*c*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)])/Sqrt[c - c/(a*x)]]} -{Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^2), x, 3, -((8*a*c*Sqrt[1 - 1/(a^2*x^2)])/(3*Sqrt[c - c/(a*x)])) - (2/3)*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]} -{Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^3), x, 4, (8*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/(5*Sqrt[c - c/(a*x)]) + (2/5)*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)] + (2*a^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*c)} -{Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^4), x, 5, -((104*a^3*c*Sqrt[1 - 1/(a^2*x^2)])/(105*Sqrt[c - c/(a*x)])) - (104/105)*a^3*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)] + (2*c*Sqrt[1 - 1/(a^2*x^2)])/(7*Sqrt[c - c/(a*x)]*x^3) - (26*a*c*Sqrt[1 - 1/(a^2*x^2)])/(35*Sqrt[c - c/(a*x)]*x^2)} - - -{(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[a*x]), x, 14, -((149*Sqrt[c - c/(a*x)]*x)/(64*a^3)) + (107*Sqrt[c - c/(a*x)]*x^2)/(96*a^2) - (17*Sqrt[c - c/(a*x)]*x^3)/(24*a) + (1/4)*Sqrt[c - c/(a*x)]*x^4 + (363*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(64*a^4) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^4} -{(Sqrt[c - c/(a*x)]*x^2)/E^(2*ArcCoth[a*x]), x, 13, (19*Sqrt[c - c/(a*x)]*x)/(8*a^2) - (13*Sqrt[c - c/(a*x)]*x^2)/(12*a) + (1/3)*Sqrt[c - c/(a*x)]*x^3 - (45*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(8*a^3) + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^3} -{(Sqrt[c - c/(a*x)]*x)/E^(2*ArcCoth[a*x]), x, 12, -((9*Sqrt[c - c/(a*x)]*x)/(4*a)) + (1/2)*Sqrt[c - c/(a*x)]*x^2 + (23*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(4*a^2) - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a^2} -{Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]), x, 11, Sqrt[c - c/(a*x)]*x - (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a} -{Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x), x, 11, 2*Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^2), x, 9, -4*a*Sqrt[c - c/(a*x)] - (2*a*(c - c/(a*x))^(3/2))/(3*c) + 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^3), x, 10, 4*a^2*Sqrt[c - c/(a*x)] + (2*a^2*(c - c/(a*x))^(3/2))/(3*c) + (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2) - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^4), x, 11, -4*a^3*Sqrt[c - c/(a*x)] - (2*a^3*(c - c/(a*x))^(3/2))/(3*c) - (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3) + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} -{Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^5), x, 11, 4*a^4*Sqrt[c - c/(a*x)] + (2*a^4*(c - c/(a*x))^(3/2))/(3*c) + (2*a^4*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^4*(c - c/(a*x))^(7/2))/(7*c^3) + (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4) - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]} - - -{(Sqrt[c - c/(a*x)]*x^3)/E^(3*ArcCoth[a*x]), x, 9, -((1115*Sqrt[c - c/(a*x)])/(64*a^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])) - (1115*Sqrt[c - c/(a*x)]*x)/(192*a^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (223*Sqrt[c - c/(a*x)]*x^2)/(96*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (25*Sqrt[c - c/(a*x)]*x^3)/(24*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt[c - c/(a*x)]*x^4)/(4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (1115*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(64*a^4*Sqrt[1 - 1/(a*x)])} -{(Sqrt[c - c/(a*x)]*x^2)/E^(3*ArcCoth[a*x]), x, 8, (119*Sqrt[c - c/(a*x)])/(8*a^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (119*Sqrt[c - c/(a*x)]*x)/(24*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (19*Sqrt[c - c/(a*x)]*x^2)/(12*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt[c - c/(a*x)]*x^3)/(3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (119*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(8*a^3*Sqrt[1 - 1/(a*x)])} -{(Sqrt[c - c/(a*x)]*x)/E^(3*ArcCoth[a*x]), x, 7, (-47*Sqrt[c - c/(a*x)])/(4*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (13*Sqrt[c - c/(a*x)]*x)/(4*a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt[c - c/(a*x)]*x^2)/(2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (47*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(4*a^2*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - c/(a*x)]/E^(3*ArcCoth[a*x]), x, 6, (9*Sqrt[c - c/(a*x)])/(a*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (Sqrt[c - c/(a*x)]*x)/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (7*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/(a*Sqrt[1 - 1/(a*x)])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x), x, 6, (-8*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]])/Sqrt[1 - 1/(a*x)]} -{Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^2), x, 4, (64*a*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a^2*x^2)]) - (16*a*(c - c/(a*x))^(3/2))/(3*c*Sqrt[1 - 1/(a^2*x^2)]) - (2*a*(c - c/(a*x))^(5/2))/(3*c^2*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^3), x, 5, -((224*a^2*c*Sqrt[1 - 1/(a^2*x^2)])/(15*Sqrt[c - c/(a*x)])) - (56/15)*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)] - (7*a^2*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(5*c) - (a^2*(c - c/(a*x))^(7/2))/(c^3*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^4), x, 6, (1888*a^3*c*Sqrt[1 - 1/(a^2*x^2)])/(105*Sqrt[c - c/(a*x)]) + (472/105)*a^3*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)] + (59*a^3*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(3/2))/(35*c) + (2*a^3*Sqrt[1 - 1/(a^2*x^2)]*(c - c/(a*x))^(5/2))/(7*c^2) + (a^3*(c - c/(a*x))^(7/2))/(c^3*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5), x, 4, -((8*a^4*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])) - (32*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (50*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a*x)]) - (38*a^4*(1 + 1/(a*x))^(5/2)*Sqrt[c - c/(a*x)])/(5*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(7/2)*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] - (2*a^4*(1 + 1/(a*x))^(9/2)*Sqrt[c - c/(a*x)])/(9*Sqrt[1 - 1/(a*x)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a x))^p with n symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c/(a*x)), x, 5, c*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^(n/2)*x - (2*c*(1 - n)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - 1/(a*x))^(n/2)*(a*n)) - (2^(n/2)*c*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, 1 - n/2, 2 - n/2, (a - 1/x)/(2*a)])/(a*(2 - n))} -{E^(n*ArcCoth[a*x])/(c - c/(a*x)), x, 3, ((1 + 1/(a*x))^((2 + n)/2)*x)/((1 - 1/(a*x))^(n/2)*c) - (2*(1 + n)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c*n))} -{E^(n*ArcCoth[a*x])/(c - c/(a*x))^2, x, 5, -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^2*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x)/c^2 - (2*(2 + n)*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c^2*n))} - - -{E^(n*ArcCoth[a*x])*(c - c/(a*x))^(3/2), x, 3, -((2^(5/2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^(3/2)*AppellF1[(2 + n)/2, (1/2)*(-3 + n), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/(a*(2 + n)*(1 - 1/(a*x))^(3/2)))} -{E^(n*ArcCoth[a*x])*Sqrt[c - c/(a*x)], x, 3, -((2^(3/2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*Sqrt[c - c/(a*x)]*AppellF1[(2 + n)/2, (1/2)*(-1 + n), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/(a*(2 + n)*Sqrt[1 - 1/(a*x)]))} -{E^(n*ArcCoth[a*x])/Sqrt[c - c/(a*x)], x, 3, -((2^(1/2 - n/2)*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^((2 + n)/2)*AppellF1[(2 + n)/2, (1 + n)/2, 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/(a*(2 + n)*Sqrt[c - c/(a*x)]))} -{E^(n*ArcCoth[a*x])/(c - c/(a*x))^(3/2), x, 3, -((2^(-(1/2) - n/2)*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^((2 + n)/2)*AppellF1[(2 + n)/2, (3 + n)/2, 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/(a*(2 + n)*(c - c/(a*x))^(3/2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a x))^p with p symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c/(a*x))^p, x, 3, -((2^(1 - n/2 + p)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^p*AppellF1[(2 + n)/2, (1/2)*(n - 2*p), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/((1 - 1/(a*x))^p*(a*(2 + n))))} - - -{(c - c/(a*x))^p*E^(2*p*ArcCoth[a*x]), x, 3, -(((1 + 1/(a*x))^(1 + p)*(c - c/(a*x))^p*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + 1/(a*x)])/((1 - 1/(a*x))^p*(a*(1 + p))))} -{(c - c/(a*x))^p/E^(2*p*ArcCoth[a*x]), x, 3, -((4^p*(1 + 1/(a*x))^(1 - p)*(c - c/(a*x))^p*AppellF1[1 - p, -2*p, 2, 2 - p, (a + 1/x)/(2*a), 1 + 1/(a*x)])/((1 - 1/(a*x))^p*(a*(1 - p))))} - - -{(c - c/(a*x))^p*E^(2*ArcCoth[a*x]), x, 7, (c - c/(a*x))^p*x + ((2 - p)*(c - c/(a*x))^p*Hypergeometric2F1[1, p, 1 + p, 1 - 1/(a*x)])/(a*p)} -{(c - c/(a*x))^p*E^ArcCoth[a*x], x, 3, -((2^(1/2 + p)*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^p*AppellF1[3/2, 1/2 - p, 2, 5/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/((1 - 1/(a*x))^p*(3*a)))} -{(c - c/(a*x))^p/E^ArcCoth[a*x], x, 3, -((2^(3/2 + p)*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^p*AppellF1[1/2, -(1/2) - p, 2, 3/2, (a + 1/x)/(2*a), 1 + 1/(a*x)])/((1 - 1/(a*x))^p*a))} -{(c - c/(a*x))^p/E^(2*ArcCoth[a*x]), x, 9, ((c - c/(a*x))^(2 + p)*x)/c^2 + ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (a - 1/x)/(2*a)])/(2*a*c^2*(2 + p)) - ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])/(a*c^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcCoth[a x]) (c-c a^2 x^2)^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a^2 x^2)^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^4, x, 13, (35/128)*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (35/384)*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (7/192)*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 + (1/64)*a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 + (1/144)*a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5 - (5/144)*a^5*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(11/2)*x^6 + (5/72)*a^6*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7 - (7/72)*a^7*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(11/2)*x^9 + (35*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^3, x, 11, (5/16)*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (5/48)*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (1/24)*a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 + (1/56)*a^3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 - (1/14)*a^4*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5 + (5/42)*a^5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7 + (5*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(16*a)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^2, x, 9, (3/8)*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (1/8)*a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (1/20)*a^2*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (3/20)*a^3*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 + (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(8*a)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2), x, 7, (1/2)*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (1/6)*a*c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 + (c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(2*a)} -{E^ArcCoth[a*x]/(c - a^2*c*x^2), x, 1, E^ArcCoth[a*x]/(a*c)} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^2, x, 2, (2*E^ArcCoth[a*x])/(3*a*c^2) - (E^ArcCoth[a*x]*(1 - 2*a*x))/(3*a*c^2*(1 - a^2*x^2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^3, x, 3, (8*E^ArcCoth[a*x])/(15*a*c^3) - (E^ArcCoth[a*x]*(1 - 4*a*x))/(15*a*c^3*(1 - a^2*x^2)^2) - (4*E^ArcCoth[a*x]*(1 - 2*a*x))/(15*a*c^3*(1 - a^2*x^2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^4, x, 4, (16*E^ArcCoth[a*x])/(35*a*c^4) - (E^ArcCoth[a*x]*(1 - 6*a*x))/(35*a*c^4*(1 - a^2*x^2)^3) - (2*E^ArcCoth[a*x]*(1 - 4*a*x))/(35*a*c^4*(1 - a^2*x^2)^2) - (8*E^ArcCoth[a*x]*(1 - 2*a*x))/(35*a*c^4*(1 - a^2*x^2))} - - -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5, x, 4, -((16*c^5*(1 + a*x)^7)/(7*a)) + (4*c^5*(1 + a*x)^8)/a - (8*c^5*(1 + a*x)^9)/(3*a) + (4*c^5*(1 + a*x)^10)/(5*a) - (c^5*(1 + a*x)^11)/(11*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^4, x, 4, -((4*c^4*(1 + a*x)^6)/(3*a)) + (12*c^4*(1 + a*x)^7)/(7*a) - (3*c^4*(1 + a*x)^8)/(4*a) + (c^4*(1 + a*x)^9)/(9*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^3, x, 4, -((4*c^3*(1 + a*x)^5)/(5*a)) + (2*c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^2, x, 4, -((c^2*(1 + a*x)^4)/(2*a)) + (c^2*(1 + a*x)^5)/(5*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^1, x, 3, -((c*(1 + a*x)^3)/(3*a))} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^1, x, 3, -(1/(a*c*(1 - a*x)))} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^2, x, 5, -(1/(4*a*c^2*(1 - a*x)^2)) - 1/(4*a*c^2*(1 - a*x)) - ArcTanh[a*x]/(4*a*c^2)} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^3, x, 5, -(1/(12*a*c^3*(1 - a*x)^3)) - 1/(8*a*c^3*(1 - a*x)^2) - 3/(16*a*c^3*(1 - a*x)) + 1/(16*a*c^3*(1 + a*x)) - ArcTanh[a*x]/(4*a*c^3)} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^4, x, 5, -(1/(32*a*c^4*(1 - a*x)^4)) - 1/(16*a*c^4*(1 - a*x)^3) - 3/(32*a*c^4*(1 - a*x)^2) - 5/(32*a*c^4*(1 - a*x)) + 1/(64*a*c^4*(1 + a*x)^2) + 5/(64*a*c^4*(1 + a*x)) - (15*ArcTanh[a*x])/(64*a*c^4)} - - -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4, x, 13, (-(55/128))*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (55/384)*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (11/192)*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (11/448)*a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 - (11*a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5)/1008 - (5*a^5*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(11/2)*x^6)/1008 + (5/168)*a^6*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(13/2)*x^7 - (5/72)*a^7*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(13/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(13/2)*x^9 - (55*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^3, x, 11, (-(9/16))*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (3/16)*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (3/40)*a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (9/280)*a^3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 - (1/70)*a^4*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5 + (1/14)*a^5*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(11/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7 - (9*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(16*a)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^2, x, 9, (-(7/8))*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (7/24)*a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (7/60)*a^2*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (1/20)*a^3*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 + (1/5)*a^4*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5 - (7*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(8*a)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2), x, 7, (-(5/2))*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (5/6)*a*c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (5*c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(2*a)} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2), x, 1, E^(3*ArcCoth[a*x])/(3*a*c)} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^2, x, 2, -((2*E^(3*ArcCoth[a*x]))/(15*a*c^2)) + (E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(5*a*c^2*(1 - a^2*x^2))} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^3, x, 3, -((8*E^(3*ArcCoth[a*x]))/(35*a*c^3)) - (E^(3*ArcCoth[a*x])*(3 - 4*a*x))/(7*a*c^3*(1 - a^2*x^2)^2) + (12*E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(35*a*c^3*(1 - a^2*x^2))} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^4, x, 4, -((16*E^(3*ArcCoth[a*x]))/(63*a*c^4)) - (E^(3*ArcCoth[a*x])*(1 - 2*a*x))/(9*a*c^4*(1 - a^2*x^2)^3) - (10*E^(3*ArcCoth[a*x])*(3 - 4*a*x))/(63*a*c^4*(1 - a^2*x^2)^2) + (8*E^(3*ArcCoth[a*x])*(3 - 2*a*x))/(21*a*c^4*(1 - a^2*x^2))} - - -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^5, x, 4, (c^5*(1 + a*x)^8)/a - (4*c^5*(1 + a*x)^9)/(3*a) + (3*c^5*(1 + a*x)^10)/(5*a) - (c^5*(1 + a*x)^11)/(11*a)} -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^4, x, 4, (4*c^4*(1 + a*x)^7)/(7*a) - (c^4*(1 + a*x)^8)/(2*a) + (c^4*(1 + a*x)^9)/(9*a)} -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3, x, 4, (c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)} -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^2, x, 3, (c^2*(1 + a*x)^5)/(5*a)} -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^1, x, 4, -4*c*x - (c*(1 + a*x)^2)/a - (c*(1 + a*x)^3)/(3*a) - (8*c*Log[1 - a*x])/a} -{E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^1, x, 3, x/(c*(1 - a*x)^2)} -{E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^2, x, 3, 1/(3*a*c^2*(1 - a*x)^3)} -{E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^3, x, 5, 1/(8*a*c^3*(1 - a*x)^4) + 1/(12*a*c^3*(1 - a*x)^3) + 1/(16*a*c^3*(1 - a*x)^2) + 1/(16*a*c^3*(1 - a*x)) + ArcTanh[a*x]/(16*a*c^3)} -{E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^4, x, 5, 1/(20*a*c^4*(1 - a*x)^5) + 1/(16*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 1/(16*a*c^4*(1 - a*x)^2) + 5/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (3*ArcTanh[a*x])/(32*a*c^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a^2*c*x^2)^4/E^ArcCoth[a*x], x, 13, (-(35/128))*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (35/384)*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (7/192)*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (1/64)*a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 + (1/16)*a^4*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)*x^5 - (5/48)*a^5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x^6 + (1/8)*a^6*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7 - (1/8)*a^7*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(9/2)*x^9 - (35*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)} -{(c - a^2*c*x^2)^3/E^ArcCoth[a*x], x, 11, (-(5/16))*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (5/48)*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (1/24)*a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 + (1/8)*a^3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 - (1/6)*a^4*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 + (1/6)*a^5*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7 - (5*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(16*a)} -{(c - a^2*c*x^2)^2/E^ArcCoth[a*x], x, 9, (-(3/8))*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (1/8)*a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (1/4)*a^2*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (1/4)*a^3*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5 - (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(8*a)} -{(c - a^2*c*x^2)/E^ArcCoth[a*x], x, 7, (-(1/2))*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (1/2)*a*c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x^3 - (c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(2*a)} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)), x, 1, -(1/(a*c*E^ArcCoth[a*x]))} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^2), x, 2, -(2/(E^ArcCoth[a*x]*(3*a*c^2))) + (1 + 2*a*x)/(E^ArcCoth[a*x]*(3*a*c^2*(1 - a^2*x^2)))} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^3), x, 3, -(8/(E^ArcCoth[a*x]*(15*a*c^3))) + (1 + 4*a*x)/(E^ArcCoth[a*x]*(15*a*c^3*(1 - a^2*x^2)^2)) + (4*(1 + 2*a*x))/(E^ArcCoth[a*x]*(15*a*c^3*(1 - a^2*x^2)))} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^4), x, 4, -(16/(E^ArcCoth[a*x]*(35*a*c^4))) + (1 + 6*a*x)/(E^ArcCoth[a*x]*(35*a*c^4*(1 - a^2*x^2)^3)) + (2*(1 + 4*a*x))/(E^ArcCoth[a*x]*(35*a*c^4*(1 - a^2*x^2)^2)) + (8*(1 + 2*a*x))/(E^ArcCoth[a*x]*(35*a*c^4*(1 - a^2*x^2)))} - - -{(c - a^2*c*x^2)^4/E^(2*ArcCoth[a*x]), x, 4, (4*c^4*(1 - a*x)^6)/(3*a) - (12*c^4*(1 - a*x)^7)/(7*a) + (3*c^4*(1 - a*x)^8)/(4*a) - (c^4*(1 - a*x)^9)/(9*a)} -{(c - a^2*c*x^2)^3/E^(2*ArcCoth[a*x]), x, 4, (4*c^3*(1 - a*x)^5)/(5*a) - (2*c^3*(1 - a*x)^6)/(3*a) + (c^3*(1 - a*x)^7)/(7*a)} -{(c - a^2*c*x^2)^2/E^(2*ArcCoth[a*x]), x, 4, (c^2*(1 - a*x)^4)/(2*a) - (c^2*(1 - a*x)^5)/(5*a)} -{(c - a^2*c*x^2)/E^(2*ArcCoth[a*x]), x, 3, (c*(1 - a*x)^3)/(3*a)} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)), x, 3, 1/(a*c*(1 + a*x))} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^2), x, 5, 1/(4*a*c^2*(1 + a*x)^2) + 1/(4*a*c^2*(1 + a*x)) - ArcTanh[a*x]/(4*a*c^2)} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^3), x, 5, -(1/(16*a*c^3*(1 - a*x))) + 1/(12*a*c^3*(1 + a*x)^3) + 1/(8*a*c^3*(1 + a*x)^2) + 3/(16*a*c^3*(1 + a*x)) - ArcTanh[a*x]/(4*a*c^3)} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^4), x, 5, -(1/(64*a*c^4*(1 - a*x)^2)) - 5/(64*a*c^4*(1 - a*x)) + 1/(32*a*c^4*(1 + a*x)^4) + 1/(16*a*c^4*(1 + a*x)^3) + 3/(32*a*c^4*(1 + a*x)^2) + 5/(32*a*c^4*(1 + a*x)) - (15*ArcTanh[a*x])/(64*a*c^4)} - - -{(c - a^2*c*x^2)^4/E^(3*ArcCoth[a*x]), x, 13, (55/128)*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (55/384)*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (11/192)*a^2*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 - (11/64)*a^3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x^4 + (11/48)*a^4*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 - (11/48)*a^5*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x^6 + (11/56)*a^6*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7 - (11/72)*a^7*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(7/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(11/2)*(1 + 1/(a*x))^(7/2)*x^9 + (55*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(128*a)} -{(c - a^2*c*x^2)^3/E^(3*ArcCoth[a*x]), x, 11, (9/16)*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (3/16)*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 - (3/8)*a^2*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)*x^3 + (3/8)*a^3*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4 - (3/10)*a^4*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5 + (3/14)*a^5*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(5/2)*x^7 + (9*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(16*a)} -{(c - a^2*c*x^2)^2/E^(3*ArcCoth[a*x]), x, 9, (7/8)*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x - (7/8)*a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x^2 + (7/12)*a^2*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x^3 - (7/20)*a^3*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^5 + (7*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(8*a)} -{(c - a^2*c*x^2)/E^(3*ArcCoth[a*x]), x, 7, (-(5/2))*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]*x + (5/6)*a*c*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x^2 - (1/3)*a^2*c*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3 + (5*c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(2*a)} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)), x, 1, -1/(3*a*c*E^(3*ArcCoth[a*x]))} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^2), x, 2, 2/(E^(3*ArcCoth[a*x])*(15*a*c^2)) - (3 + 2*a*x)/(E^(3*ArcCoth[a*x])*(5*a*c^2*(1 - a^2*x^2)))} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^3), x, 3, 8/(E^(3*ArcCoth[a*x])*(35*a*c^3)) + (3 + 4*a*x)/(E^(3*ArcCoth[a*x])*(7*a*c^3*(1 - a^2*x^2)^2)) - (12*(3 + 2*a*x))/(E^(3*ArcCoth[a*x])*(35*a*c^3*(1 - a^2*x^2)))} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4), x, 4, 16/(E^(3*ArcCoth[a*x])*(63*a*c^4)) + (1 + 2*a*x)/(E^(3*ArcCoth[a*x])*(9*a*c^4*(1 - a^2*x^2)^3)) + (10*(3 + 4*a*x))/(E^(3*ArcCoth[a*x])*(63*a*c^4*(1 - a^2*x^2)^2)) - (8*(3 + 2*a*x))/(E^(3*ArcCoth[a*x])*(21*a*c^4*(1 - a^2*x^2)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a^2 x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^(9/2), x, 4, (8*(1 + a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 + a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (8*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2), x, 4, -((8*(1 + a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)) + (2*(1 + a*x)^6*(c - a^2*c*x^2)^(7/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 + a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 + a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^(5/2), x, 4, ((1 + a*x)^4*(c - a^2*c*x^2)^(5/2))/(a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) - (4*(1 + a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) + ((1 + a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^(3/2), x, 4, -((2*(1 + a*x)^3*(c - a^2*c*x^2)^(3/2))/(3*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)) + ((1 + a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)} -{E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2], x, 3, Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]/Sqrt[c - a^2*c*x^2], x, 3, (Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2), x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*ArcTanh[a*x])/(2*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2), x, 5, -((a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(7/2), x, 5, (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 - a*x)^3*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) + (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(16*(c - a^2*c*x^2)^(7/2))} - - -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2), x, 10, (-(77/256))*c^4*x*Sqrt[c - a^2*c*x^2] - (77/384)*c^3*x*(c - a^2*c*x^2)^(3/2) - (77/480)*c^2*x*(c - a^2*c*x^2)^(5/2) - (11/80)*c*x*(c - a^2*c*x^2)^(7/2) + (11*(c - a^2*c*x^2)^(9/2))/(90*a) + ((1 + a*x)*(c - a^2*c*x^2)^(9/2))/(10*a) - (77*c^(9/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(256*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2), x, 9, (-(45/128))*c^3*x*Sqrt[c - a^2*c*x^2] - (15/64)*c^2*x*(c - a^2*c*x^2)^(3/2) - (3/16)*c*x*(c - a^2*c*x^2)^(5/2) + (9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) - (45*c^(7/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(128*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2), x, 8, (-(7/16))*c^2*x*Sqrt[c - a^2*c*x^2] - (7/24)*c*x*(c - a^2*c*x^2)^(3/2) + (7*(c - a^2*c*x^2)^(5/2))/(30*a) + ((1 + a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2), x, 7, (-(5/8))*c*x*Sqrt[c - a^2*c*x^2] + (5*(c - a^2*c*x^2)^(3/2))/(12*a) + ((1 + a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) - (5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)} -{E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2], x, 6, (3*Sqrt[c - a^2*c*x^2])/(2*a) + ((1 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{E^(2*ArcCoth[a*x])/Sqrt[c - a^2*c*x^2], x, 5, -((2*(1 + a*x))/(a*Sqrt[c - a^2*c*x^2])) + ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*Sqrt[c])} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2), x, 4, -((2*(1 + a*x))/(3*a*(c - a^2*c*x^2)^(3/2))) - x/(3*c*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2), x, 5, -((2*(1 + a*x))/(5*a*(c - a^2*c*x^2)^(5/2))) - x/(5*c*(c - a^2*c*x^2)^(3/2)) - (2*x)/(5*c^2*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2), x, 6, -((2*(1 + a*x))/(7*a*(c - a^2*c*x^2)^(7/2))) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) - (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])} -{E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2), x, 7, -((2*(1 + a*x))/(9*a*(c - a^2*c*x^2)^(9/2))) - x/(9*c*(c - a^2*c*x^2)^(7/2)) - (2*x)/(15*c^2*(c - a^2*c*x^2)^(5/2)) - (8*x)/(45*c^3*(c - a^2*c*x^2)^(3/2)) - (16*x)/(45*c^4*Sqrt[c - a^2*c*x^2])} - - -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2), x, 4, -((8*(1 + a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(2*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (2*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2), x, 4, (2*(1 + a*x)^6*(c - a^2*c*x^2)^(7/2))/(3*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (4*(1 + a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 + a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2), x, 4, -((2*(1 + a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)) + ((1 + a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2), x, 3, ((1 + a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)} -{E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2], x, 4, (3*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)} -{E^(3*ArcCoth[a*x])/Sqrt[c - a^2*c*x^2], x, 4, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/((1 - a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2), x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)^2*(c - a^2*c*x^2)^(3/2)))} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(6*(1 - a*x)^3*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2), x, 5, -((a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^4*(c - a^2*c*x^2)^(7/2))) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(12*(1 - a*x)^3*(c - a^2*c*x^2)^(7/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(32*(c - a^2*c*x^2)^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - a^2*c*x^2)^(9/2)/E^ArcCoth[a*x], x, 4, (8*(1 - a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 - a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 - a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (8*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 - a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)} -{(c - a^2*c*x^2)^(7/2)/E^ArcCoth[a*x], x, 4, -((8*(1 - a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)) + (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)} -{(c - a^2*c*x^2)^(5/2)/E^ArcCoth[a*x], x, 4, ((1 - a*x)^4*(c - a^2*c*x^2)^(5/2))/(a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) - (4*(1 - a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) + ((1 - a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)} -{(c - a^2*c*x^2)^(3/2)/E^ArcCoth[a*x], x, 4, -((2*(1 - a*x)^3*(c - a^2*c*x^2)^(3/2))/(3*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)) + ((1 - a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)} -{Sqrt[c - a^2*c*x^2]/E^ArcCoth[a*x], x, 3, -(Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)])) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2]), x, 3, (Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[c - a^2*c*x^2]} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(3/2)), x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 + a*x)*(c - a^2*c*x^2)^(3/2)) - (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*ArcTanh[a*x])/(2*(c - a^2*c*x^2)^(3/2))} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(5/2)), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)^2*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2)), x, 5, -((a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2))) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(16*(c - a^2*c*x^2)^(7/2))} - - -{(c - a^2*c*x^2)^(5/2)/E^(2*ArcCoth[a*x]), x, 8, (-(7/16))*c^2*x*Sqrt[c - a^2*c*x^2] - (7/24)*c*x*(c - a^2*c*x^2)^(3/2) - (7*(c - a^2*c*x^2)^(5/2))/(30*a) - ((1 - a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(16*a)} -{(c - a^2*c*x^2)^(3/2)/E^(2*ArcCoth[a*x]), x, 7, (-(5/8))*c*x*Sqrt[c - a^2*c*x^2] - (5*(c - a^2*c*x^2)^(3/2))/(12*a) - ((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) - (5*c^(3/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a)} -{Sqrt[c - a^2*c*x^2]/E^(2*ArcCoth[a*x]), x, 6, -((3*Sqrt[c - a^2*c*x^2])/(2*a)) - ((1 - a*x)*Sqrt[c - a^2*c*x^2])/(2*a) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{1/(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]), x, 5, (2*(1 - a*x))/(a*Sqrt[c - a^2*c*x^2]) + ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/(a*Sqrt[c])} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2)), x, 4, (2*(1 - a*x))/(3*a*(c - a^2*c*x^2)^(3/2)) - x/(3*c*Sqrt[c - a^2*c*x^2])} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2)), x, 5, (2*(1 - a*x))/(5*a*(c - a^2*c*x^2)^(5/2)) - x/(5*c*(c - a^2*c*x^2)^(3/2)) - (2*x)/(5*c^2*Sqrt[c - a^2*c*x^2])} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)), x, 6, (2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) - (8*x)/(21*c^3*Sqrt[c - a^2*c*x^2])} -{1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2)), x, 7, (2*(1 - a*x))/(9*a*(c - a^2*c*x^2)^(9/2)) - x/(9*c*(c - a^2*c*x^2)^(7/2)) - (2*x)/(15*c^2*(c - a^2*c*x^2)^(5/2)) - (8*x)/(45*c^3*(c - a^2*c*x^2)^(3/2)) - (16*x)/(45*c^4*Sqrt[c - a^2*c*x^2])} - - -{(c - a^2*c*x^2)^(9/2)/E^(3*ArcCoth[a*x]), x, 4, -((8*(1 - a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)) + (3*(1 - a*x)^8*(c - a^2*c*x^2)^(9/2))/(2*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (2*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 - a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)} -{(c - a^2*c*x^2)^(7/2)/E^(3*ArcCoth[a*x]), x, 4, (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7/2))/(3*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (4*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)} -{(c - a^2*c*x^2)^(5/2)/E^(3*ArcCoth[a*x]), x, 4, -((2*(1 - a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)) + ((1 - a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)} -{(c - a^2*c*x^2)^(3/2)/E^(3*ArcCoth[a*x]), x, 3, ((1 - a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)} -{Sqrt[c - a^2*c*x^2]/E^(3*ArcCoth[a*x]), x, 4, -((3*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)])) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)} -{1/(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]), x, 4, (2*Sqrt[1 - 1/(a^2*x^2)]*x)/((1 + a*x)*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[c - a^2*c*x^2]} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2)), x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 + a*x)^2*(c - a^2*c*x^2)^(3/2)))} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2)), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(6*(1 + a*x)^3*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)^2*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)), x, 5, (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 + a*x)^4*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(12*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) + (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*ArcTanh[a*x])/(32*(c - a^2*c*x^2)^(7/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a^2 x^2)^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*x^2*Sqrt[c - a^2*c*x^2], x, 4, (x^2*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^3*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*x*Sqrt[c - a^2*c*x^2], x, 4, (x*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^2*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2], x, 3, Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)])} -{(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2])/x, x, 4, Sqrt[c - a^2*c*x^2]/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - a^2*c*x^2]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2])/x^2, x, 4, -(Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)) + (Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} - - -{E^(2*ArcCoth[a*x])*x^3*Sqrt[c - a^2*c*x^2], x, 8, (3*x^2*Sqrt[c - a^2*c*x^2])/(5*a^2) + (x^3*Sqrt[c - a^2*c*x^2])/(2*a) + (1/5)*x^4*Sqrt[c - a^2*c*x^2] + (3*(8 + 5*a*x)*Sqrt[c - a^2*c*x^2])/(20*a^4) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(4*a^4)} -{E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2], x, 7, (2*x^2*Sqrt[c - a^2*c*x^2])/(3*a) + (1/4)*x^3*Sqrt[c - a^2*c*x^2] + ((32 + 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3) - (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)} -{E^(2*ArcCoth[a*x])*x*Sqrt[c - a^2*c*x^2], x, 6, (1/3)*x^2*Sqrt[c - a^2*c*x^2] + ((5 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(3*a^2) - (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/a^2} -{E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2], x, 6, (3*Sqrt[c - a^2*c*x^2])/(2*a) + ((1 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x, x, 9, Sqrt[c - a^2*c*x^2] - 2*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^2, x, 9, Sqrt[c - a^2*c*x^2]/x - a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + 2*a*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^3, x, 7, Sqrt[c - a^2*c*x^2]/(2*x^2) + (2*a*Sqrt[c - a^2*c*x^2])/x + (3/2)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^4, x, 8, Sqrt[c - a^2*c*x^2]/(3*x^3) + (a*Sqrt[c - a^2*c*x^2])/x^2 + (5*a^2*Sqrt[c - a^2*c*x^2])/(3*x) + a^3*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^5, x, 9, Sqrt[c - a^2*c*x^2]/(4*x^4) + (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) + (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) + (4*a^3*Sqrt[c - a^2*c*x^2])/(3*x) + (7/8)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a^2*c*x^2], x, 4, (4*Sqrt[c - a^2*c*x^2])/(a^4*Sqrt[1 - 1/(a^2*x^2)]) + (2*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (4*x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]) + (3*x^3*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^4*Sqrt[c - a^2*c*x^2])/(5*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^5*Sqrt[1 - 1/(a^2*x^2)]*x)} -{E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2], x, 4, (4*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (2*x*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (x^2*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x^3*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x)} -{E^(3*ArcCoth[a*x])*x*Sqrt[c - a^2*c*x^2], x, 4, (4*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (3*x*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^2*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x)} -{E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2], x, 4, (3*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x, x, 4, Sqrt[c - a^2*c*x^2]/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - a^2*c*x^2]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^2, x, 4, Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (3*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^3, x, 4, Sqrt[c - a^2*c*x^2]/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (3*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^4, x, 4, Sqrt[c - a^2*c*x^2]/(3*a*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (3*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (4*a*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a^2*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a^2*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^5, x, 4, Sqrt[c - a^2*c*x^2]/(4*a*Sqrt[1 - 1/(a^2*x^2)]*x^5) + Sqrt[c - a^2*c*x^2]/(Sqrt[1 - 1/(a^2*x^2)]*x^4) + (2*a*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^3) + (4*a^2*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a^3*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a^3*Sqrt[c - a^2*c*x^2]*Log[1 - a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} - - -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)*x^4, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^4)/(a*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^5)/(2*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*a^2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (7*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*a^2*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*a^2*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)*x^3, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^4)/(c - a^2*c*x^2)^(3/2) + ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*a*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (5*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*a*(c - a^2*c*x^2)^(3/2)) - ((1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*a*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)*x^2, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (3*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)*x^1, x, 5, (a*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) - (a*(1 - 1/(a^2*x^2))^(3/2)*x^3*ArcTanh[a*x])/(2*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)*x^0, x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*ArcTanh[a*x])/(2*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)/x^1, x, 4, (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[x])/(c - a^2*c*x^2)^(3/2) - (3*a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*(c - a^2*c*x^2)^(3/2)) - (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)/x^2, x, 4, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*x^2)/(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[x])/(c - a^2*c*x^2)^(3/2) - (5*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*(c - a^2*c*x^2)^(3/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2)/x^3, x, 4, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(2*(c - a^2*c*x^2)^(3/2))) - (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^2)/(c - a^2*c*x^2)^(3/2) + (a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (2*a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[x])/(c - a^2*c*x^2)^(3/2) - (7*a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 - a*x])/(4*(c - a^2*c*x^2)^(3/2)) - (a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*Log[1 + a*x])/(4*(c - a^2*c*x^2)^(3/2))} - - -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^5, x, 4, ((1 - 1/(a^2*x^2))^(5/2)*x^6)/(c - a^2*c*x^2)^(5/2) - ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*a*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) + ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(a*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*a*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (23*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 - a*x])/(16*a*(c - a^2*c*x^2)^(5/2)) - (7*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 + a*x])/(16*a*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^4, x, 4, -(((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (11*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 - a*x])/(16*(c - a^2*c*x^2)^(5/2)) + (5*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 + a*x])/(16*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^3, x, 5, -((a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(2*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^2, x, 5, -((a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^1, x, 5, -((a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)*x^0, x, 5, -((a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*ArcTanh[a*x])/(8*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)/x^1, x, 4, -((a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(2*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[x])/(c - a^2*c*x^2)^(5/2) + (11*a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 - a*x])/(16*(c - a^2*c*x^2)^(5/2)) + (5*a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 + a*x])/(16*(c - a^2*c*x^2)^(5/2))} -{E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2)/x^2, x, 4, (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^4)/(c - a^2*c*x^2)^(5/2) - (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[x])/(c - a^2*c*x^2)^(5/2) + (23*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 - a*x])/(16*(c - a^2*c*x^2)^(5/2)) - (7*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*Log[1 + a*x])/(16*(c - a^2*c*x^2)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(x^2*Sqrt[c - a^2*c*x^2])/E^ArcCoth[a*x], x, 4, -(x^2*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^3*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - 1/(a^2*x^2)])} -{(x*Sqrt[c - a^2*c*x^2])/E^ArcCoth[a*x], x, 4, -(x*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^2*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - a^2*c*x^2]/E^ArcCoth[a*x], x, 3, -(Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)])) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - a^2*c*x^2]/(E^ArcCoth[a*x]*x), x, 4, Sqrt[c - a^2*c*x^2]/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - a^2*c*x^2]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^ArcCoth[a*x]*x^2), x, 4, Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} - - -{(x^3*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]), x, 8, (3*x^2*Sqrt[c - a^2*c*x^2])/(5*a^2) - (x^3*Sqrt[c - a^2*c*x^2])/(2*a) + (1/5)*x^4*Sqrt[c - a^2*c*x^2] + (3*(8 - 5*a*x)*Sqrt[c - a^2*c*x^2])/(20*a^4) + (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(4*a^4)} -{(x^2*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]), x, 7, -((2*x^2*Sqrt[c - a^2*c*x^2])/(3*a)) + (1/4)*x^3*Sqrt[c - a^2*c*x^2] - ((32 - 21*a*x)*Sqrt[c - a^2*c*x^2])/(24*a^3) - (7*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(8*a^3)} -{(x^1*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]), x, 6, (1/3)*x^2*Sqrt[c - a^2*c*x^2] + ((5 - 3*a*x)*Sqrt[c - a^2*c*x^2])/(3*a^2) + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/a^2} -{Sqrt[c - a^2*c*x^2]/E^(2*ArcCoth[a*x]), x, 6, -((3*Sqrt[c - a^2*c*x^2])/(2*a)) - ((1 - a*x)*Sqrt[c - a^2*c*x^2])/(2*a) - (3*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)} -{Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^1), x, 9, Sqrt[c - a^2*c*x^2] + 2*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^2), x, 9, Sqrt[c - a^2*c*x^2]/x - a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] - 2*a*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^3), x, 7, Sqrt[c - a^2*c*x^2]/(2*x^2) - (2*a*Sqrt[c - a^2*c*x^2])/x + (3/2)*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^4), x, 8, Sqrt[c - a^2*c*x^2]/(3*x^3) - (a*Sqrt[c - a^2*c*x^2])/x^2 + (5*a^2*Sqrt[c - a^2*c*x^2])/(3*x) - a^3*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} -{Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^5), x, 9, Sqrt[c - a^2*c*x^2]/(4*x^4) - (2*a*Sqrt[c - a^2*c*x^2])/(3*x^3) + (7*a^2*Sqrt[c - a^2*c*x^2])/(8*x^2) - (4*a^3*Sqrt[c - a^2*c*x^2])/(3*x) + (7/8)*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]]} - - -{(x^3*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]), x, 4, (4*Sqrt[c - a^2*c*x^2])/(a^4*Sqrt[1 - 1/(a^2*x^2)]) - (2*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (4*x^2*Sqrt[c - a^2*c*x^2])/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]) - (3*x^3*Sqrt[c - a^2*c*x^2])/(4*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^4*Sqrt[c - a^2*c*x^2])/(5*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^5*Sqrt[1 - 1/(a^2*x^2)]*x)} -{(x^2*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]), x, 4, (-4*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (2*x*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (x^2*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (x^3*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x)} -{(x*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]), x, 4, (4*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (3*x*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (x^2*Sqrt[c - a^2*c*x^2])/(3*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/E^(3*ArcCoth[a*x]), x, 4, -((3*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - 1/(a^2*x^2)])) + (x*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x), x, 4, Sqrt[c - a^2*c*x^2]/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - a^2*c*x^2]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^2), x, 4, -(Sqrt[c - a^2*c*x^2]/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)) - (3*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^3), x, 4, -(Sqrt[c - a^2*c*x^2]/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^3)) + (3*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^4), x, 4, -(Sqrt[c - a^2*c*x^2]/(3*a*Sqrt[1 - 1/(a^2*x^2)]*x^4)) + (3*Sqrt[c - a^2*c*x^2])/(2*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (4*a*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a^2*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a^2*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} -{Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^5), x, 4, -(Sqrt[c - a^2*c*x^2]/(4*a*Sqrt[1 - 1/(a^2*x^2)]*x^5)) + Sqrt[c - a^2*c*x^2]/(Sqrt[1 - 1/(a^2*x^2)]*x^4) - (2*a*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^3) + (4*a^2*Sqrt[c - a^2*c*x^2])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a^3*Sqrt[c - a^2*c*x^2]*Log[x])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^3*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(Sqrt[1 - 1/(a^2*x^2)]*x)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a^2 x^2)^p with m symbolic*) - - -{E^(3*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2], x, 5, (3*x^m*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]) + (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - 1/(a^2*x^2)]) - (4*x^m*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, 1 + m, 2 + m, a*x])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)])} -{E^(2*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2], x, 8, If[$VersionNumber>=8, (x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(2 + m)*Sqrt[c - a^2*c*x^2]) - (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2]), (x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[c - a^2*c*x^2]) - (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])]} -{E^(1*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2], x, 4, (x^m*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]) + (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - 1/(a^2*x^2)])} -{(x^m*Sqrt[c - a^2*c*x^2])/E^(1*ArcCoth[a*x]), x, 4, -((x^m*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)])) + (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - 1/(a^2*x^2)])} -{(x^m*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]), x, 8, If[$VersionNumber>=8, (x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((1 + m)*(2 + m)*Sqrt[c - a^2*c*x^2]) + (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2]), (x^(1 + m)*Sqrt[c - a^2*c*x^2])/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/((2 + 3*m + m^2)*Sqrt[c - a^2*c*x^2]) + (2*a*c*x^(2 + m)*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^2*c*x^2])]} -{(x^m*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]), x, 5, -((3*x^m*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)])) + (x^(1 + m)*Sqrt[c - a^2*c*x^2])/((2 + m)*Sqrt[1 - 1/(a^2*x^2)]) + (4*x^m*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, 1 + m, 2 + m, (-a)*x])/(a*(1 + m)*Sqrt[1 - 1/(a^2*x^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a^2 x^2)^p with n symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c*a^2*x^2)^3, x, 3, -((256*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((1/2)*(-8 + n))*Hypergeometric2F1[8, 4 - n/2, 5 - n/2, (a - 1/x)/(a + 1/x)])/(a*(8 - n)))} -{E^(n*ArcCoth[a*x])*(c - c*a^2*x^2)^2, x, 3, (64*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((1/2)*(-6 + n))*Hypergeometric2F1[6, 3 - n/2, 4 - n/2, (a - 1/x)/(a + 1/x)])/(a*(6 - n))} -{E^(n*ArcCoth[a*x])*(c - c*a^2*x^2)^1, x, 3, -((16*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-4 + n))*Hypergeometric2F1[4, 2 - n/2, 3 - n/2, (a - 1/x)/(a + 1/x)])/(a*(4 - n)))} -{E^(n*ArcCoth[a*x])*(c - c*a^2*x^2)^0, x, 2, (4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)])/(a*(2 - n))} -{E^(n*ArcCoth[a*x])/(c - c*a^2*x^2)^1, x, 1, E^(n*ArcCoth[a*x])/(a*c*n)} -{E^(n*ArcCoth[a*x])/(c - c*a^2*x^2)^2, x, 2, (2*E^(n*ArcCoth[a*x]))/(a*c^2*n*(4 - n^2)) - (E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))} -{E^(n*ArcCoth[a*x])/(c - c*a^2*x^2)^3, x, 3, If[$VersionNumber>=8, (24*E^(n*ArcCoth[a*x]))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^3*(4 - n^2)*(16 - n^2)*(1 - a^2*x^2)), (24*E^(n*ArcCoth[a*x]))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^3*(64 - 20*n^2 + n^4)*(1 - a^2*x^2))]} -{E^(n*ArcCoth[a*x])/(c - c*a^2*x^2)^4, x, 4, If[$VersionNumber>=8, (720*E^(n*ArcCoth[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^4*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)^2) - (360*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^4*(4 - n^2)*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)), (720*E^(n*ArcCoth[a*x]))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (E^(n*ArcCoth[a*x])*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*E^(n*ArcCoth[a*x])*(n - 4*a*x))/(a*c^4*(576 - 52*n^2 + n^4)*(1 - a^2*x^2)^2) - (360*E^(n*ArcCoth[a*x])*(n - 2*a*x))/(a*c^4*(36 - n^2)*(64 - 20*n^2 + n^4)*(1 - a^2*x^2))]} - - -{E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2), x, 3, (32*(1 - 1/(a*x))^((5 - n)/2)*(1 + 1/(a*x))^((1/2)*(-5 + n))*(c - a^2*c*x^2)^(3/2)*Hypergeometric2F1[5, (5 - n)/2, (7 - n)/2, (a - 1/x)/(a + 1/x)])/(a^4*(5 - n)*(1 - 1/(a^2*x^2))^(3/2)*x^3)} -{E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^(1/2), x, 3, (8*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[3, (3 - n)/2, (5 - n)/2, (a - 1/x)/(a + 1/x)])/(a^2*(3 - n)*Sqrt[1 - 1/(a^2*x^2)]*x)} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(1/2), x, 3, (2*Sqrt[1 - 1/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)])/((1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2), x, 1, -((E^(n*ArcCoth[a*x])*(n - a*x))/(a*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]))} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2), x, 2, If[$VersionNumber>=8, -((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*Sqrt[c - a^2*c*x^2]), -((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2), x, 3, If[$VersionNumber>=8, -((E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^2*(9 - n^2)*(25 - n^2)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^3*(1 - n^2)*(9 - n^2)*(25 - n^2)*Sqrt[c - a^2*c*x^2]), -((E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^2*(225 - 34*n^2 + n^4)*(c - a^2*c*x^2)^(3/2)) - (120*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^3*(25 - n^2)*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2), x, 4, If[$VersionNumber>=8, -((E^(n*ArcCoth[a*x])*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c^2*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(5/2)) - (840*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^3*(9 - n^2)*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(3/2)) - (5040*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^4*(1 - n^2)*(9 - n^2)*(25 - n^2)*(49 - n^2)*Sqrt[c - a^2*c*x^2]),-((E^(n*ArcCoth[a*x])*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*E^(n*ArcCoth[a*x])*(n - 5*a*x))/(a*c^2*(1225 - 74*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (840*E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c^3*(49 - n^2)*(225 - 34*n^2 + n^4)*(c - a^2*c*x^2)^(3/2)) - (5040*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^4*(1225 - 74*n^2 + n^4)*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} - - -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2)*x^3, x, 7, If[$VersionNumber>=8, -(((2 + n)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 + n)*(c - a^2*c*x^2)^(3/2))) + ((2 + 2*n + n^2)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 - n)*(1 + n)*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^4)/(c - a^2*c*x^2)^(3/2) - (2*n*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)])/(a*(1 - n)*(c - a^2*c*x^2)^(3/2)), -(((2 + n)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 + n)*(c - a^2*c*x^2)^(3/2))) + ((2 + 2*n + n^2)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 - n^2)*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^4)/(c - a^2*c*x^2)^(3/2) - (2*n*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)])/(a*(1 - n)*(c - a^2*c*x^2)^(3/2))]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2)*x^2, x, 4, -((E^(n*ArcCoth[a*x])*(n - a*x))/(a^3*c*(1 - n^2)*Sqrt[c - a^2*c*x^2])) - (2*Sqrt[1 - 1/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)])/(a^2*c*(1 - n)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2)*x^1, x, 1, (E^(n*ArcCoth[a*x])*(1 - a*n*x))/(a^2*c*(1 - n^2)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2)*x^0, x, 1, -((E^(n*ArcCoth[a*x])*(n - a*x))/(a*c*(1 - n^2)*Sqrt[c - a^2*c*x^2]))} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2)/x^1, x, 5, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/((1 + n)*(c - a^2*c*x^2)^(3/2))) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/((1 - n^2)*(c - a^2*c*x^2)^(3/2)) - (2^((1 + n)/2)*a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*x^3*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (a - 1/x)/(2*a)])/((1 - n)*(c - a^2*c*x^2)^(3/2))} - - -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)*x^4, x, 8, If[$VersionNumber>=8, -(((1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((1 - n)*(1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - n)*(c - a^2*c*x^2)^(5/2)), -(((1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5*Hypergeometric2F1[1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - n)*(c - a^2*c*x^2)^(5/2))]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)*x^3, x, 6, If[$VersionNumber>=8, -((a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)), -((a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2))]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)*x^2, x, 2, -((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a^3*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) + (E^(n*ArcCoth[a*x])*(3 - n^2)*(n - a*x))/(a^3*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)*x^1, x, 2, (E^(n*ArcCoth[a*x])*(3 - a*n*x))/(a^2*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2)) + (2*E^(n*ArcCoth[a*x])*n*(n - a*x))/(a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)*x^0, x, 2, If[$VersionNumber>=8, -((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*Sqrt[c - a^2*c*x^2]), -((E^(n*ArcCoth[a*x])*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*E^(n*ArcCoth[a*x])*(n - a*x))/(a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])]} -{E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2)/x^1, x, 15, If[$VersionNumber>=8, -((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*x^5*Hypergeometric2F1[(1/2)*(-3 - n), (1/2)*(-3 - n), (1/2)*(-1 - n), (a - 1/x)/(2*a)])/((3 + n)*(c - a^2*c*x^2)^(5/2)), -((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*x^5*Hypergeometric2F1[(1/2)*(-3 - n), (1/2)*(-3 - n), (1/2)*(-1 - n), (a - 1/x)/(2*a)])/((3 + n)*(c - a^2*c*x^2)^(5/2))]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c a^2 x^2)^p with p symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c*a^2*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^((1/2)*(n - 2*p))*(1 - 1/(a*x))^(-(n/2) + p)*(1 + 1/(a*x))^(1 + n/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, (1/2)*(n - 2*p), -2*p, 2/((a + 1/x)*x)])/((1 - 1/(a^2*x^2))^p*(1 + 2*p))} - - -{(c - a^2*c*x^2)^p*E^(2*p*ArcCoth[a*x]), x, 3, ((1 + 1/(a*x))^(1 + 2*p)*x*(c - a^2*c*x^2)^p)/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)} -{(c - a^2*c*x^2)^p/E^(2*p*ArcCoth[a*x]), x, 3, ((1 - 1/(a*x))^(1 + 2*p)*x*(c - a^2*c*x^2)^p)/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)} - - -{E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^p, x, 4, (2^(2 + p)*c*(1 + a*x)^(1 - p)*(c - a^2*c*x^2)^(-1 + p)*Hypergeometric2F1[-2 - p, -1 + p, p, (1/2)*(1 - a*x)])/(a*(1 - p))} -{E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^(3/2 - p)*(1 - 1/(a*x))^(-(3/2) + p)*(1 + 1/(a*x))^(5/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, 3/2 - p, -2*p, 2/((a + 1/x)*x)])/((1 - 1/(a^2*x^2))^p*(1 + 2*p))} -{E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^p, x, 4, (2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1/2)*(1 - a*x)])/((1 + a*x)^p*(a*p))} -{E^ArcCoth[a*x]*(c - a^2*c*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^(1/2 - p)*(1 - 1/(a*x))^(-(1/2) + p)*(1 + 1/(a*x))^(3/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, 1/2 - p, -2*p, 2/((a + 1/x)*x)])/((1 - 1/(a^2*x^2))^p*(1 + 2*p))} -{(c - a^2*c*x^2)^p/E^ArcCoth[a*x], x, 3, (((a - 1/x)/(a + 1/x))^(-(1/2) - p)*(1 - 1/(a*x))^(1/2 + p)*(1 + 1/(a*x))^(1/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -(1/2) - p, -2*p, 2/((a + 1/x)*x)])/((1 - 1/(a^2*x^2))^p*(1 + 2*p))} -{(c - a^2*c*x^2)^p/E^(2*ArcCoth[a*x]), x, 4, -((2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1/2)*(1 + a*x)])/((1 - a*x)^p*(a*p)))} -{(c - a^2*c*x^2)^p/E^(3*ArcCoth[a*x]), x, 3, (((a - 1/x)/(a + 1/x))^(-(3/2) - p)*(1 - 1/(a*x))^(3/2 + p)*(1 + 1/(a*x))^(-(1/2) + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -(3/2) - p, -2*p, 2/((a + 1/x)*x)])/((1 - 1/(a^2*x^2))^p*(1 + 2*p))} - - -(* ::Section::Closed:: *) -(*Integrands of the form u E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^p*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^p*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^4, x, 14, -((51*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a)) - (67*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(48*a) - (91*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(120*a) - (131*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280*a) + (61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (47*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))/(42*a) + (8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2))/(7*a) + c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*x + (35*c^4*ArcCsc[a*x])/(16*a) + (c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^3, x, 12, -((23*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(8*a)) - (31*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(24*a) - (43*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(60*a) + (23*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(20*a) + (6*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2))/(5*a) + c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x + (15*c^3*ArcCsc[a*x])/(8*a) + (c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^2, x, 10, -((5*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a)) - (7*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(6*a) + (4*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x + (3*c^2*ArcCsc[a*x])/(2*a) + (c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^ArcCoth[a*x]*(c - c/(a^2*x^2)), x, 9, -((2*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/a) + c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x + (c*ArcCsc[a*x])/a + (c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2)), x, 6, -((2*Sqrt[1 + 1/(a*x)])/(a*c*Sqrt[1 - 1/(a*x)])) + (Sqrt[1 + 1/(a*x)]*x)/(c*Sqrt[1 - 1/(a*x)]) + ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c)} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^2, x, 8, -(4/(3*a*c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])) - 11/(3*a*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (8*Sqrt[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) + ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^3, x, 10, -(6/(5*a*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2))) - 29/(15*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) - 34/(5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (21*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (16*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^3)} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^4, x, 12, -(8/(7*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2))) - 11/(7*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)) - 62/(21*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - 269/(21*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)) + (262*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (163*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(3/2)) + (128*Sqrt[1 - 1/(a*x)])/(35*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)) + ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^4)} - - -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^5, x, 5, -(c^5/(9*a^10*x^9)) - c^5/(4*a^9*x^8) + (3*c^5)/(7*a^8*x^7) + (4*c^5)/(3*a^7*x^6) - (2*c^5)/(5*a^6*x^5) - (3*c^5)/(a^5*x^4) - (2*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) + (3*c^5)/(a^2*x) + c^5*x + (2*c^5*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^4, x, 5, c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x + (2*c^4*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^3, x, 5, -(c^3/(5*a^6*x^5)) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^2, x, 5, c^2/(3*a^4*x^3) + c^2/(a^3*x^2) + c^2*x + (2*c^2*Log[x])/a} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2)), x, 5, -(c/(a^2*x)) + c*x + (2*c*Log[x])/a} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2)), x, 5, x/c + 1/(a*c*(1 - a*x)) + (2*Log[1 - a*x])/(a*c)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^2, x, 5, x/c^2 - 1/(4*a*c^2*(1 - a*x)^2) + 7/(4*a*c^2*(1 - a*x)) + (17*Log[1 - a*x])/(8*a*c^2) - Log[1 + a*x]/(8*a*c^2)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^3, x, 5, x/c^3 + 1/(12*a*c^3*(1 - a*x)^3) - 5/(8*a*c^3*(1 - a*x)^2) + 39/(16*a*c^3*(1 - a*x)) - 1/(16*a*c^3*(1 + a*x)) + (9*Log[1 - a*x])/(4*a*c^3) - Log[1 + a*x]/(4*a*c^3)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^4, x, 5, x/c^4 - 1/(32*a*c^4*(1 - a*x)^4) + 13/(48*a*c^4*(1 - a*x)^3) - 35/(32*a*c^4*(1 - a*x)^2) + 99/(32*a*c^4*(1 - a*x)) + 1/(64*a*c^4*(1 + a*x)^2) - 11/(64*a*c^4*(1 + a*x)) + (303*Log[1 - a*x])/(128*a*c^4) - (47*Log[1 + a*x])/(128*a*c^4)} - - -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4, x, 14, -((63*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a)) - (37*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(16*a) - (61*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(40*a) - (303*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(280*a) - (57*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(70*a) + (15*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(11/2))/(14*a) + (8*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2))/(7*a) + c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*x + (15*c^4*ArcCsc[a*x])/(16*a) + (3*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^3, x, 12, -((27*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(8*a)) - (17*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(8*a) - (29*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(20*a) - (21*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(20*a) + (6*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(5*a) + c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x + (3*c^3*ArcCsc[a*x])/(8*a) + (3*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2, x, 10, -((5*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a)) - (11*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(6*a) - (4*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)*x - (c^2*ArcCsc[a*x])/(2*a) + (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2)), x, 8, c*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)*x - (3*c*ArcCsc[a*x])/a + (3*c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)), x, 7, -((5*Sqrt[1 + 1/(a*x)])/(3*a*c*(1 - 1/(a*x))^(3/2))) - (14*Sqrt[1 + 1/(a*x)])/(3*a*c*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*x)/(c*(1 - 1/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^2, x, 9, -((6*Sqrt[1 + 1/(a*x)])/(5*a*c^2*(1 - 1/(a*x))^(5/2))) - (9*Sqrt[1 + 1/(a*x)])/(5*a*c^2*(1 - 1/(a*x))^(3/2)) - (24*Sqrt[1 + 1/(a*x)])/(5*a*c^2*Sqrt[1 - 1/(a*x)]) + (Sqrt[1 + 1/(a*x)]*x)/(c^2*(1 - 1/(a*x))^(5/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^3, x, 10, -(8/(7*a*c^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)])) - 53/(35*a*c^3*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - 88/(35*a*c^3*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]) - 281/(35*a*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (176*Sqrt[1 - 1/(a*x)])/(35*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^3)} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^4, x, 12, -(10/(9*a*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2))) - 29/(21*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)) - 208/(105*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 1147/(315*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) - 1462/(105*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (2609*Sqrt[1 - 1/(a*x)])/(315*a*c^4*(1 + 1/(a*x))^(3/2)) + (1664*Sqrt[1 - 1/(a*x)])/(315*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)} - - -{E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^5, x, 5, c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^4, x, 5, -(c^4/(7*a^8*x^7)) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^3, x, 5, c^3/(5*a^6*x^5) + c^3/(a^5*x^4) + (5*c^3)/(3*a^4*x^3) - (5*c^3)/(a^2*x) + c^3*x + (4*c^3*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^2, x, 5, -(c^2/(3*a^4*x^3)) - (2*c^2)/(a^3*x^2) - (6*c^2)/(a^2*x) + c^2*x + (4*c^2*Log[x])/a} -{E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2)), x, 5, c/(a^2*x) + c*x - (4*c*Log[x])/a + (8*c*Log[1 - a*x])/a} -{E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2)), x, 5, x/c - 1/(a*c*(1 - a*x)^2) + 5/(a*c*(1 - a*x)) + (4*Log[1 - a*x])/(a*c)} -{E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^2, x, 5, x/c^2 + 1/(3*a*c^2*(1 - a*x)^3) - 2/(a*c^2*(1 - a*x)^2) + 6/(a*c^2*(1 - a*x)) + (4*Log[1 - a*x])/(a*c^2)} -{E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^3, x, 5, x/c^3 - 1/(8*a*c^3*(1 - a*x)^4) + 11/(12*a*c^3*(1 - a*x)^3) - 49/(16*a*c^3*(1 - a*x)^2) + 111/(16*a*c^3*(1 - a*x)) + (129*Log[1 - a*x])/(32*a*c^3) - Log[1 + a*x]/(32*a*c^3)} -{E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^4, x, 5, x/c^4 + 1/(20*a*c^4*(1 - a*x)^5) - 7/(16*a*c^4*(1 - a*x)^4) + 83/(48*a*c^4*(1 - a*x)^3) - 67/(16*a*c^4*(1 - a*x)^2) + 501/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (261*Log[1 - a*x])/(64*a*c^4) - (5*Log[1 + a*x])/(64*a*c^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a^2*x^2))^4/E^ArcCoth[a*x], x, 14, -((19*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a)) - (c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(16*a) + (7*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(40*a) + (19*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(40*a) + (29*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2))/(30*a) + (7*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2))/(6*a) + (8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2))/(7*a) + c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(7/2)*x + (35*c^4*ArcCsc[a*x])/(16*a) - (c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))^3/E^ArcCoth[a*x], x, 12, -((7*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(8*a)) + (c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(24*a) + (11*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(12*a) + (5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))/(4*a) + (6*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2))/(5*a) + c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x + (15*c^3*ArcCsc[a*x])/(8*a) - (c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))^2/E^ArcCoth[a*x], x, 10, -((c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a)) + (3*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(2*a) + (4*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2))/(3*a) + c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x + (3*c^2*ArcCsc[a*x])/(2*a) - (c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))/E^ArcCoth[a*x], x, 9, (2*c*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/a + c*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x + (c*ArcCsc[a*x])/a - (c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))), x, 6, (2*Sqrt[1 - 1/(a*x)])/(a*c*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c*Sqrt[1 + 1/(a*x)]) - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c)} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^2), x, 8, -(2/(a*c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))) + (5*Sqrt[1 - 1/(a*x)])/(3*a*c^2*(1 + 1/(a*x))^(3/2)) + (8*Sqrt[1 - 1/(a*x)])/(3*a*c^2*Sqrt[1 + 1/(a*x)]) + x/(c^2*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^2)} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^3), x, 10, -(4/(3*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))) - 13/(3*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)) + (14*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(5/2)) + (11*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (16*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^3)} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^4), x, 12, -(6/(5*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2))) - 31/(15*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)) - 28/(3*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) + (115*Sqrt[1 - 1/(a*x)])/(21*a*c^4*(1 + 1/(a*x))^(7/2)) + (122*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (93*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(3/2)) + (128*Sqrt[1 - 1/(a*x)])/(35*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^4)} - - -{(c - c/(a^2*x^2))^4/E^(2*ArcCoth[a*x]), x, 5, c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x - (2*c^4*Log[x])/a} -{(c - c/(a^2*x^2))^3/E^(2*ArcCoth[a*x]), x, 5, -(c^3/(5*a^6*x^5)) + c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) - (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x - (2*c^3*Log[x])/a} -{(c - c/(a^2*x^2))^2/E^(2*ArcCoth[a*x]), x, 5, c^2/(3*a^4*x^3) - c^2/(a^3*x^2) + c^2*x - (2*c^2*Log[x])/a} -{(c - c/(a^2*x^2))/E^(2*ArcCoth[a*x]), x, 5, -(c/(a^2*x)) + c*x - (2*c*Log[x])/a} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))), x, 5, x/c - 1/(a*c*(1 + a*x)) - (2*Log[1 + a*x])/(a*c)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^2), x, 5, x/c^2 + 1/(4*a*c^2*(1 + a*x)^2) - 7/(4*a*c^2*(1 + a*x)) + Log[1 - a*x]/(8*a*c^2) - (17*Log[1 + a*x])/(8*a*c^2)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^3), x, 5, x/c^3 + 1/(16*a*c^3*(1 - a*x)) - 1/(12*a*c^3*(1 + a*x)^3) + 5/(8*a*c^3*(1 + a*x)^2) - 39/(16*a*c^3*(1 + a*x)) + Log[1 - a*x]/(4*a*c^3) - (9*Log[1 + a*x])/(4*a*c^3)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^4), x, 5, x/c^4 - 1/(64*a*c^4*(1 - a*x)^2) + 11/(64*a*c^4*(1 - a*x)) + 1/(32*a*c^4*(1 + a*x)^4) - 13/(48*a*c^4*(1 + a*x)^3) + 35/(32*a*c^4*(1 + a*x)^2) - 99/(32*a*c^4*(1 + a*x)) + (47*Log[1 - a*x])/(128*a*c^4) - (303*Log[1 + a*x])/(128*a*c^4)} - - -{(c - c/(a^2*x^2))^4/E^(3*ArcCoth[a*x]), x, 14, (33*c^4*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(16*a) + (27*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(16*a) - (3*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(8*a) + (5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))/(8*a) + (11*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2))/(10*a) + (17*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2))/(14*a) + (8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(5/2))/(7*a) + c^4*(1 - 1/(a*x))^(11/2)*(1 + 1/(a*x))^(5/2)*x + (15*c^4*ArcCsc[a*x])/(16*a) - (3*c^4*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))^3/E^(3*ArcCoth[a*x]), x, 12, (21*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(8*a) + (3*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(8*a) + (5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2))/(4*a) + (27*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2))/(20*a) + (6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2))/(5*a) + c^3*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)*x + (3*c^3*ArcCsc[a*x])/(8*a) - (3*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))^2/E^(3*ArcCoth[a*x]), x, 10, (5*c^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(2*a) + (11*c^2*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)])/(6*a) + (4*c^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])/(3*a) + c^2*(1 - 1/(a*x))^(7/2)*Sqrt[1 + 1/(a*x)]*x - (c^2*ArcCsc[a*x])/(2*a) - (3*c^2*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{(c - c/(a^2*x^2))/E^(3*ArcCoth[a*x]), x, 8, c*(1 - 1/(a*x))^(3/2)*Sqrt[1 + 1/(a*x)]*x - (3*c*ArcCsc[a*x])/a - (3*c*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))), x, 7, (5*Sqrt[1 - 1/(a*x)])/(3*a*c*(1 + 1/(a*x))^(3/2)) + (14*Sqrt[1 - 1/(a*x)])/(3*a*c*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c*(1 + 1/(a*x))^(3/2)) - (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2), x, 9, (6*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(5/2)) + (9*Sqrt[1 - 1/(a*x)])/(5*a*c^2*(1 + 1/(a*x))^(3/2)) + (24*Sqrt[1 - 1/(a*x)])/(5*a*c^2*Sqrt[1 + 1/(a*x)]) + (Sqrt[1 - 1/(a*x)]*x)/(c^2*(1 + 1/(a*x))^(5/2)) - (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^2)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^3), x, 10, -(2/(a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))) + (11*Sqrt[1 - 1/(a*x)])/(7*a*c^3*(1 + 1/(a*x))^(7/2)) + (54*Sqrt[1 - 1/(a*x)])/(35*a*c^3*(1 + 1/(a*x))^(5/2)) + (71*Sqrt[1 - 1/(a*x)])/(35*a*c^3*(1 + 1/(a*x))^(3/2)) + (176*Sqrt[1 - 1/(a*x)])/(35*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) - (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^3)} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4), x, 12, -(4/(3*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))) - 5/(a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)) + (28*Sqrt[1 - 1/(a*x)])/(9*a*c^4*(1 + 1/(a*x))^(9/2)) + (139*Sqrt[1 - 1/(a*x)])/(63*a*c^4*(1 + 1/(a*x))^(7/2)) + (202*Sqrt[1 - 1/(a*x)])/(105*a*c^4*(1 + 1/(a*x))^(5/2)) + (719*Sqrt[1 - 1/(a*x)])/(315*a*c^4*(1 + 1/(a*x))^(3/2)) + (1664*Sqrt[1 - 1/(a*x)])/(315*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)) - (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(7/2), x, 4, (c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6) + (c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^3*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2), x, 4, -((c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)) - (c^2*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (2*c^2*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (c^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(3/2), x, 4, (c*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (c*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (c*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]/Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a^2*x^2)] + (Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(a*Sqrt[c - c/(a^2*x^2)])} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(3/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(2*a*c*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (5*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(4*a*c*Sqrt[c - c/(a^2*x^2)]) - (Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(4*a*c*Sqrt[c - c/(a^2*x^2)])} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(5/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + Sqrt[1 - 1/(a^2*x^2)]/(a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) - Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (23*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (7*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])} -{E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(7/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(24*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^3) - (11*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (3*Sqrt[1 - 1/(a^2*x^2)])/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - (5*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)])} - - -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2), x, 15, (11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3) - (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) + (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) + (57*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) - (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) + (a*(c - c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(7/2)*x*(1 + a*x))/(6*(1 - a*x)) + (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) + (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2), x, 13, -((5*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(8*(1 - a*x)^2)) + (25*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2) - (17*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(12*(1 - a*x)^2*(1 + a*x)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2*(1 + a*x))/(6*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(5/2)*x*(1 + a*x))/(4*(1 - a*x)) - (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))} -{E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2), x, 11, (a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a*x) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(3/2)*x*(1 + a*x))/(2*(1 - a*x)) + (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)], x, 9, Sqrt[c - c/(a^2*x^2)]*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)], x, 7, -((2*(1 - a*x)*(1 + a*x))/(a^2*Sqrt[c - c/(a^2*x^2)]*x)) - (1 + a*x)^2/(a^2*Sqrt[c - c/(a^2*x^2)]*x) + (2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcSin[a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(3/2), x, 7, -((1 + a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x)) + (2*(5 - 2*a*x)*(1 - a*x)*(1 + a*x)^2)/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) - (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2), x, 9, -((1 + a*x)^2/(5*a^2*(c - c/(a^2*x^2))^(5/2)*x)) + (2*(1 - a*x)*(1 + a*x)^2)/(3*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) - (58*(1 - a*x)^2*(1 + a*x)^2)/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 43*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) + (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(7/2), x, 11, -((1 + a*x)^2/(7*a^2*(c - c/(a^2*x^2))^(7/2)*x)) + (2*(1 - a*x)*(1 + a*x)^2)/(5*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) - (124*(1 - a*x)^2*(1 + a*x)^2)/(105*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) + (782*(1 - a*x)^3*(1 + a*x)^2)/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) + (142*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) + (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 107*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(9/2), x, 4, (c^4*Sqrt[c - c/(a^2*x^2)])/(8*a^9*Sqrt[1 - 1/(a^2*x^2)]*x^8) + (3*c^4*Sqrt[c - c/(a^2*x^2)])/(7*a^8*Sqrt[1 - 1/(a^2*x^2)]*x^7) - (8*c^4*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) - (3*c^4*Sqrt[c - c/(a^2*x^2)])/(2*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (2*c^4*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (4*c^4*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (c^4*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (3*c^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2), x, 4, -((c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6)) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) - (c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (5*c^3*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (5*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (c^3*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (3*c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2), x, 4, (c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (c^2*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (2*c^2*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (3*c^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2), x, 4, -((c*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)) - (3*c*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (3*c*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a^2*x^2)] + (2*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(a*Sqrt[c - c/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(3/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(2*a*c*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (3*Sqrt[1 - 1/(a^2*x^2)])/(a*c*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(a*c*Sqrt[c - c/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(6*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^3) - (9*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (31*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + (49*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(7/2), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^4) + Sqrt[1 - 1/(a^2*x^2)]/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^3) - (59*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (75*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) - Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (9*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)])} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c - c/(a^2*x^2))^(7/2)/E^ArcCoth[a*x], x, 4, -((c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6)) + (c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^3*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (3*c^3*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(c - c/(a^2*x^2))^(5/2)/E^ArcCoth[a*x], x, 4, (c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^2*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (2*c^2*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (c^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(c - c/(a^2*x^2))^(3/2)/E^ArcCoth[a*x], x, 4, -((c*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)) + (c*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (c*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/E^ArcCoth[a*x], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a^2*x^2)] - (Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(a*Sqrt[c - c/(a^2*x^2)])} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(3/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(2*a*c*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(4*a*c*Sqrt[c - c/(a^2*x^2)]) - (5*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(4*a*c*Sqrt[c - c/(a^2*x^2)])} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - Sqrt[1 - 1/(a^2*x^2)]/(a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (7*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (23*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])} -{1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(7/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2) + (5*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) - Sqrt[1 - 1/(a^2*x^2)]/(24*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^3) + (11*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - (3*Sqrt[1 - 1/(a^2*x^2)])/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (19*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (51*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)])} - - -{(c - c/(a^2*x^2))^(7/2)/E^(2*ArcCoth[a*x]), x, 15, (7*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(8*(1 - a*x)^3*(1 + a*x)^2) - (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(15*(1 + a*x)) - (19*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(16*(1 - a*x)^3*(1 + a*x)) + (2*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a*x)^2*(1 + a*x)) - (23*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(120*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(7/2)*x*(1 - a*x))/(6*(1 + a*x)) - (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcSin[a*x])/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) + (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))} -{(c - c/(a^2*x^2))^(5/2)/E^(2*ArcCoth[a*x]), x, 13, -((7*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(6*(1 + a*x)) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/((1 - a*x)^2*(1 + a*x)) - (7*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(24*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(5/2)*x*(1 - a*x))/(4*(1 + a*x)) + (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcSin[a*x])/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))} -{(c - c/(a^2*x^2))^(3/2)/E^(2*ArcCoth[a*x]), x, 11, -((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 + a*x)) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(3/2)*x*(1 - a*x))/(2*(1 + a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))} -{Sqrt[c - c/(a^2*x^2)]/E^(2*ArcCoth[a*x]), x, 9, Sqrt[c - c/(a^2*x^2)]*x + (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]), x, 7, -((1 - a*x)^2/(a^2*Sqrt[c - c/(a^2*x^2)]*x)) - (2*(1 - a*x)*(1 + a*x))/(a^2*Sqrt[c - c/(a^2*x^2)]*x) - (2*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcSin[a*x])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2)), x, 7, -((1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x)) + (2*(1 - a*x)^2*(1 + a*x)*(5 + 2*a*x))/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) + (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*ArcSin[a*x])/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)), x, 9, -((1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(5/2)*x)) - (2*(1 - a*x)^3)/(5*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) + (2*(1 - a*x)^3*(1 + a*x))/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 13*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) - (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*ArcSin[a*x])/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)} -{1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)), x, 11, -((1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(7/2)*x)) + (10*(1 - a*x)^3)/(3*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) + (12*(1 - a*x)^4)/(7*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) + (82*(1 - a*x)^4*(1 + a*x))/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) + (2*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) + (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 37*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*ArcSin[a*x])/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)} - - -{(c - c/(a^2*x^2))^(9/2)/E^(3*ArcCoth[a*x]), x, 4, -((c^4*Sqrt[c - c/(a^2*x^2)])/(8*a^9*Sqrt[1 - 1/(a^2*x^2)]*x^8)) + (3*c^4*Sqrt[c - c/(a^2*x^2)])/(7*a^8*Sqrt[1 - 1/(a^2*x^2)]*x^7) - (8*c^4*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*c^4*Sqrt[c - c/(a^2*x^2)])/(2*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (2*c^4*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (4*c^4*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (c^4*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (3*c^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(c - c/(a^2*x^2))^(7/2)/E^(3*ArcCoth[a*x]), x, 4, (c^3*Sqrt[c - c/(a^2*x^2)])/(6*a^7*Sqrt[1 - 1/(a^2*x^2)]*x^6) - (3*c^3*Sqrt[c - c/(a^2*x^2)])/(5*a^6*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (c^3*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (5*c^3*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (5*c^3*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (c^3*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^3*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (3*c^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(c - c/(a^2*x^2))^(5/2)/E^(3*ArcCoth[a*x]), x, 4, -((c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)) + (c^2*Sqrt[c - c/(a^2*x^2)])/(a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3) - (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (2*c^2*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (3*c^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(c - c/(a^2*x^2))^(3/2)/E^(3*ArcCoth[a*x]), x, 4, (c*Sqrt[c - c/(a^2*x^2)])/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (3*c*Sqrt[c - c/(a^2*x^2)])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (3*c*Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/E^(3*ArcCoth[a*x]), x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - c/(a^2*x^2)] - (2*Sqrt[1 - 1/(a^2*x^2)])/(a*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) - (3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(a*Sqrt[c - c/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(2*a*c*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - (3*Sqrt[1 - 1/(a^2*x^2)])/(a*c*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) - (3*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(a*c*Sqrt[c - c/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*Sqrt[c - c/(a^2*x^2)]) - Sqrt[1 - 1/(a^2*x^2)]/(6*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^3) + (9*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - (31*Sqrt[1 - 1/(a^2*x^2)])/(8*a*c^2*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)]) - (49*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(16*a*c^2*Sqrt[c - c/(a^2*x^2)])} -{1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)), x, 4, (Sqrt[1 - 1/(a^2*x^2)]*x)/(c^3*Sqrt[c - c/(a^2*x^2)]) + Sqrt[1 - 1/(a^2*x^2)]/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 - a*x)) + Sqrt[1 - 1/(a^2*x^2)]/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^4) - Sqrt[1 - 1/(a^2*x^2)]/(2*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^3) + (59*Sqrt[1 - 1/(a^2*x^2)])/(32*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2) - (75*Sqrt[1 - 1/(a^2*x^2)])/(16*a*c^3*Sqrt[c - c/(a^2*x^2)]*(1 + a*x)) + (9*Sqrt[1 - 1/(a^2*x^2)]*Log[1 - a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)]) - (201*Sqrt[1 - 1/(a^2*x^2)]*Log[1 + a*x])/(64*a*c^3*Sqrt[c - c/(a^2*x^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^(p/2)*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x^m, x, 4, (Sqrt[c - c/(a^2*x^2)]*x^m)/(a*m*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^(1 + m))/((1 + m)*Sqrt[1 - 1/(a^2*x^2)])} - -{E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x^2, x, 4, (Sqrt[c - c/(a^2*x^2)]*x^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^3)/(3*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x, x, 3, (Sqrt[c - c/(a^2*x^2)]*x)/(a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^2)/(2*Sqrt[1 - 1/(a^2*x^2)])} -{E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)])/x, x, 4, -(Sqrt[c - c/(a^2*x^2)]/(a*Sqrt[1 - 1/(a^2*x^2)]*x)) + (Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)]} -{(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)])/x^2, x, 3, -((Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^2))} - - -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3, x, 9, (7*Sqrt[c - c/(a^2*x^2)]*x)/(8*a^3) + (7*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(24*a^3) + (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)/(6*a^3) + (Sqrt[c - c/(a^2*x^2)]*x^2*(1 + a*x)^2)/(4*a^2) - (7*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(8*a^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2, x, 8, (Sqrt[c - c/(a^2*x^2)]*x)/a^2 + (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)/(3*a^2) - (Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(a^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x, x, 7, (3*Sqrt[c - c/(a^2*x^2)]*x)/(2*a) + (Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))/(2*a) - (3*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(2*a*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)], x, 9, Sqrt[c - c/(a^2*x^2)]*x - (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x, x, 9, Sqrt[c - c/(a^2*x^2)] - (a*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (2*a*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2, x, 7, (3/2)*a*Sqrt[c - c/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*(1 + a*x))/(2*x) + (3*a^2*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3, x, 8, a^2*Sqrt[c - c/(a^2*x^2)] + (a*Sqrt[c - c/(a^2*x^2)]*(1 + a*x))/(3*x) + (Sqrt[c - c/(a^2*x^2)]*(1 + a*x)^2)/(3*x^2) + (a^3*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4, x, 10, (4/3)*a^3*Sqrt[c - c/(a^2*x^2)] + Sqrt[c - c/(a^2*x^2)]/(4*x^3) + (2*a*Sqrt[c - c/(a^2*x^2)])/(3*x^2) + (7*a^2*Sqrt[c - c/(a^2*x^2)])/(8*x) + (7*a^4*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5, x, 11, (6/5)*a^4*Sqrt[c - c/(a^2*x^2)] + Sqrt[c - c/(a^2*x^2)]/(5*x^4) + (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) + (3*a^2*Sqrt[c - c/(a^2*x^2)])/(5*x^2) + (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) + (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])} - - -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3, x, 4, (4*Sqrt[c - c/(a^2*x^2)]*x)/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (2*Sqrt[c - c/(a^2*x^2)]*x^2)/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^3)/(a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^4)/(4*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/(a^4*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2, x, 4, (4*Sqrt[c - c/(a^2*x^2)]*x)/(a^2*Sqrt[1 - 1/(a^2*x^2)]) + (3*Sqrt[c - c/(a^2*x^2)]*x^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^3)/(3*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/(a^3*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x, x, 4, (3*Sqrt[c - c/(a^2*x^2)]*x)/(a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^2)/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)])} -{E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x, x, 4, Sqrt[c - c/(a^2*x^2)]/(a*Sqrt[1 - 1/(a^2*x^2)]*x) - (3*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2, x, 4, Sqrt[c - c/(a^2*x^2)]/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (3*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3, x, 4, Sqrt[c - c/(a^2*x^2)]/(3*a*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (3*Sqrt[c - c/(a^2*x^2)])/(2*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a^2*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4, x, 4, Sqrt[c - c/(a^2*x^2)]/(4*a*Sqrt[1 - 1/(a^2*x^2)]*x^4) + Sqrt[c - c/(a^2*x^2)]/(Sqrt[1 - 1/(a^2*x^2)]*x^3) + (2*a*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a^2*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a^3*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5, x, 4, Sqrt[c - c/(a^2*x^2)]/(5*a*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*Sqrt[c - c/(a^2*x^2)])/(4*Sqrt[1 - 1/(a^2*x^2)]*x^4) + (4*a*Sqrt[c - c/(a^2*x^2)])/(3*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (2*a^2*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a^3*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/Sqrt[1 - 1/(a^2*x^2)]} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(Sqrt[c - c/(a^2*x^2)]*x^m)/E^ArcCoth[a*x], x, 4, -((Sqrt[c - c/(a^2*x^2)]*x^m)/(a*m*Sqrt[1 - 1/(a^2*x^2)])) + (Sqrt[c - c/(a^2*x^2)]*x^(1 + m))/((1 + m)*Sqrt[1 - 1/(a^2*x^2)])} -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^ArcCoth[a*x], x, 4, -(Sqrt[c - c/(a^2*x^2)]*x^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^3)/(3*Sqrt[1 - 1/(a^2*x^2)])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^ArcCoth[a*x], x, 3, -((Sqrt[c - c/(a^2*x^2)]*x)/(a*Sqrt[1 - 1/(a^2*x^2)])) + (Sqrt[c - c/(a^2*x^2)]*x^2)/(2*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/E^ArcCoth[a*x], x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/(E^ArcCoth[a*x]*x), x, 4, Sqrt[c - c/(a^2*x^2)]/(a*Sqrt[1 - 1/(a^2*x^2)]*x) + (Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)]} -{Sqrt[c - c/(a^2*x^2)]/(E^ArcCoth[a*x]*x^2), x, 3, (Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^2)} - - -{(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(2*ArcCoth[a*x]), x, 9, -((7*Sqrt[c - c/(a^2*x^2)]*x)/(8*a^3)) - (7*Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(24*a^3) - (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^2)/(6*a^3) + (Sqrt[c - c/(a^2*x^2)]*x^2*(1 - a*x)^2)/(4*a^2) - (7*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(8*a^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(2*ArcCoth[a*x]), x, 8, (Sqrt[c - c/(a^2*x^2)]*x)/a^2 + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x)^2)/(3*a^2) + (Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(a^2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^(2*ArcCoth[a*x]), x, 7, -((3*Sqrt[c - c/(a^2*x^2)]*x)/(2*a)) - (Sqrt[c - c/(a^2*x^2)]*x*(1 - a*x))/(2*a) - (3*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(2*a*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/E^(2*ArcCoth[a*x]), x, 9, Sqrt[c - c/(a^2*x^2)]*x + (2*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) + (Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x), x, 9, Sqrt[c - c/(a^2*x^2)] - (a*Sqrt[c - c/(a^2*x^2)]*x*ArcSin[a*x])/(Sqrt[1 - a*x]*Sqrt[1 + a*x]) - (2*a*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^2), x, 7, (-(3/2))*a*Sqrt[c - c/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*(1 - a*x))/(2*x) + (3*a^2*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^3), x, 8, a^2*Sqrt[c - c/(a^2*x^2)] - (a*Sqrt[c - c/(a^2*x^2)]*(1 - a*x))/(3*x) + (Sqrt[c - c/(a^2*x^2)]*(1 - a*x)^2)/(3*x^2) - (a^3*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^4), x, 10, (-(4/3))*a^3*Sqrt[c - c/(a^2*x^2)] + Sqrt[c - c/(a^2*x^2)]/(4*x^3) - (2*a*Sqrt[c - c/(a^2*x^2)])/(3*x^2) + (7*a^2*Sqrt[c - c/(a^2*x^2)])/(8*x) + (7*a^4*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*Sqrt[1 - a*x]*Sqrt[1 + a*x])} -{Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^5), x, 11, (6/5)*a^4*Sqrt[c - c/(a^2*x^2)] + Sqrt[c - c/(a^2*x^2)]/(5*x^4) - (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) + (3*a^2*Sqrt[c - c/(a^2*x^2)])/(5*x^2) - (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) - (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])} - - -{(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(3*ArcCoth[a*x]), x, 4, (-4*Sqrt[c - c/(a^2*x^2)]*x)/(a^3*Sqrt[1 - 1/(a^2*x^2)]) + (2*Sqrt[c - c/(a^2*x^2)]*x^2)/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (Sqrt[c - c/(a^2*x^2)]*x^3)/(a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^4)/(4*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/(a^4*Sqrt[1 - 1/(a^2*x^2)])} -{(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(3*ArcCoth[a*x]), x, 4, (4*Sqrt[c - c/(a^2*x^2)]*x)/(a^2*Sqrt[1 - 1/(a^2*x^2)]) - (3*Sqrt[c - c/(a^2*x^2)]*x^2)/(2*a*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[c - c/(a^2*x^2)]*x^3)/(3*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/(a^3*Sqrt[1 - 1/(a^2*x^2)])} -{(Sqrt[c - c/(a^2*x^2)]*x)/E^(3*ArcCoth[a*x]), x, 4, -((3*Sqrt[c - c/(a^2*x^2)]*x)/(a*Sqrt[1 - 1/(a^2*x^2)])) + (Sqrt[c - c/(a^2*x^2)]*x^2)/(2*Sqrt[1 - 1/(a^2*x^2)]) + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/(a^2*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/E^(3*ArcCoth[a*x]), x, 4, (Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] + (Sqrt[c - c/(a^2*x^2)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)]) - (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/(a*Sqrt[1 - 1/(a^2*x^2)])} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x), x, 4, -(Sqrt[c - c/(a^2*x^2)]/(a*Sqrt[1 - 1/(a^2*x^2)]*x)) - (3*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^2), x, 4, -(Sqrt[c - c/(a^2*x^2)]/(2*a*Sqrt[1 - 1/(a^2*x^2)]*x^2)) + (3*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] - (4*a*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^3), x, 4, -(Sqrt[c - c/(a^2*x^2)]/(3*a*Sqrt[1 - 1/(a^2*x^2)]*x^3)) + (3*Sqrt[c - c/(a^2*x^2)])/(2*Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^2*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a^2*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^4), x, 4, -(Sqrt[c - c/(a^2*x^2)]/(4*a*Sqrt[1 - 1/(a^2*x^2)]*x^4)) + Sqrt[c - c/(a^2*x^2)]/(Sqrt[1 - 1/(a^2*x^2)]*x^3) - (2*a*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) + (4*a^2*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) + (4*a^3*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] - (4*a^3*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/Sqrt[1 - 1/(a^2*x^2)]} -{Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^5), x, 4, -(Sqrt[c - c/(a^2*x^2)]/(5*a*Sqrt[1 - 1/(a^2*x^2)]*x^5)) + (3*Sqrt[c - c/(a^2*x^2)])/(4*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (4*a*Sqrt[c - c/(a^2*x^2)])/(3*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (2*a^2*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x^2) - (4*a^3*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/Sqrt[1 - 1/(a^2*x^2)] + (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[1 + a*x])/Sqrt[1 - 1/(a^2*x^2)]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^p with n symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c/(a^2*x^2)), x, 4, (4*c*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)])/(a*(2 - n)) - (2^(1 + n/2)*c*(1 - 1/(a*x))^(1 - n/2)*Hypergeometric2F1[1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)])/(a*(2 - n))} -{E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2)), x, 5, -(((1 + n)*(1 + 1/(a*x))^(n/2))/((1 - 1/(a*x))^(n/2)*(a*c*n))) + ((1 + 1/(a*x))^(n/2)*x)/((1 - 1/(a*x))^(n/2)*c) + (2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c))} -{E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2))^2, x, 7, If[$VersionNumber>=8, -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 - n)*n*(2 + n)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/((1 - 1/(a*x))^(n/2)*(a*c^2*n*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x)/c^2 + (2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c^2)), -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*n*(4 - n^2)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/((1 - 1/(a*x))^(n/2)*(a*c^2*n*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x)/c^2 + (2*(1 + 1/(a*x))^(n/2)*Hypergeometric2F1[1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)])/((1 - 1/(a*x))^(n/2)*(a*c^2))]} - - -{E^(n*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)], x, 6, (Sqrt[c - c/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2)*x)/Sqrt[1 - 1/(a^2*x^2)] + (2*n*Sqrt[c - c/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)])/(a*(1 - n)*Sqrt[1 - 1/(a^2*x^2)]) - (2^((1 + n)/2)*Sqrt[c - c/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*Hypergeometric2F1[(1 - n)/2, (1 - n)/2, (3 - n)/2, (a - 1/x)/(2*a)])/(a*(1 - n)*Sqrt[1 - 1/(a^2*x^2)])} -{E^(n*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)], x, 4, (Sqrt[1 - 1/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1 + n)/2)*x)/Sqrt[c - c/(a^2*x^2)] + (2*n*Sqrt[1 - 1/(a^2*x^2)]*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*Hypergeometric2F1[1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)])/(a*(1 - n)*Sqrt[c - c/(a^2*x^2)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^p with p symbolic*) - - -{E^(n*ArcCoth[a*x])*(c - c/(a^2*x^2))^p, x, 3, -((2^(1 - n/2 + p)*(c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + n/2 + p)*AppellF1[1 + n/2 + p, (1/2)*(n - 2*p), 2, 2 + n/2 + p, (a + 1/x)/(2*a), 1 + 1/(a*x)])/((1 - 1/(a^2*x^2))^p*(a*(2 + n + 2*p))))} - - -{(c - c/(a^2*x^2))^p/E^(2*p*ArcCoth[a*x]), x, 3, ((c - c/(a^2*x^2))^p*(1 - 1/(a*x))^(1 + 2*p)*Hypergeometric2F1[2, 1 + 2*p, 2*(1 + p), 1 - 1/(a*x)])/((1 - 1/(a^2*x^2))^p*(a*(1 + 2*p)))} -{E^(2*p*ArcCoth[a*x])*(c - c/(a^2*x^2))^p, x, 3, -(((c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + 2*p)*Hypergeometric2F1[2, 1 + 2*p, 2*(1 + p), 1 + 1/(a*x)])/((1 - 1/(a^2*x^2))^p*(a*(1 + 2*p))))} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m deleted file mode 100644 index b7d7b9b..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.1 u (a+b arcsech(c x))^n.m +++ /dev/null @@ -1,361 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcSech[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSech[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ArcSech[a x]^n*) - - -{x^4*ArcSech[a*x]^2, x, 9, -((3*x)/(20*a^4)) - x^3/(30*a^2) - (3*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(20*a^4) - (x^3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(10*a^2) + (1/5)*x^5*ArcSech[a*x]^2 - (3*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]])/(10*a^5) + (3*I*PolyLog[2, (-I)*E^ArcSech[a*x]])/(20*a^5) - (3*I*PolyLog[2, I*E^ArcSech[a*x]])/(20*a^5)} -{x^3*ArcSech[a*x]^2, x, 5, -(x^2/(12*a^2)) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(3*a^4) - (x^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(6*a^2) + (1/4)*x^4*ArcSech[a*x]^2 - Log[x]/(3*a^4)} -{x^2*ArcSech[a*x]^2, x, 8, -(x/(3*a^2)) - (x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(3*a^2) + (1/3)*x^3*ArcSech[a*x]^2 - (2*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]])/(3*a^3) + (I*PolyLog[2, (-I)*E^ArcSech[a*x]])/(3*a^3) - (I*PolyLog[2, I*E^ArcSech[a*x]])/(3*a^3)} -{x^1*ArcSech[a*x]^2, x, 4, -((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/a^2) + (1/2)*x^2*ArcSech[a*x]^2 - Log[x]/a^2} -{x^0*ArcSech[a*x]^2, x, 7, x*ArcSech[a*x]^2 - (4*ArcSech[a*x]*ArcTan[E^ArcSech[a*x]])/a + (2*I*PolyLog[2, (-I)*E^ArcSech[a*x]])/a - (2*I*PolyLog[2, I*E^ArcSech[a*x]])/a} -{ArcSech[a*x]^2/x^1, x, 6, (1/3)*ArcSech[a*x]^3 - ArcSech[a*x]^2*Log[1 + E^(2*ArcSech[a*x])] - ArcSech[a*x]*PolyLog[2, -E^(2*ArcSech[a*x])] + (1/2)*PolyLog[3, -E^(2*ArcSech[a*x])]} -{ArcSech[a*x]^2/x^2, x, 4, -(2/x) + (2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/x - ArcSech[a*x]^2/x} -{ArcSech[a*x]^2/x^3, x, 4, -(((1 - a*x)*(1 + a*x))/(4*x^2)) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(2*x^2) - (1/4)*a^2*ArcSech[a*x]^2 - ((1 - a*x)*(1 + a*x)*ArcSech[a*x]^2)/(2*x^2)} -{ArcSech[a*x]^2/x^4, x, 5, -(2/(27*x^3)) - (4*a^2)/(9*x) + (2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(9*x^3) + (4*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x])/(9*x) - ArcSech[a*x]^2/(3*x^3)} - - -{x^4*ArcSech[a*x]^3, x, 14, (x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(20*a^4) - (9*x*ArcSech[a*x])/(20*a^4) - (x^3*ArcSech[a*x])/(10*a^2) - (9*x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(40*a^4) - (3*x^3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(20*a^2) + (1/5)*x^5*ArcSech[a*x]^3 - (9*ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]])/(20*a^5) + ArcTan[(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a*x)]/(2*a^5) + (9*I*ArcSech[a*x]*PolyLog[2, (-I)*E^ArcSech[a*x]])/(20*a^5) - (9*I*ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]])/(20*a^5) - (9*I*PolyLog[3, (-I)*E^ArcSech[a*x]])/(20*a^5) + (9*I*PolyLog[3, I*E^ArcSech[a*x]])/(20*a^5)} -{x^3*ArcSech[a*x]^3, x, 10, (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(4*a^4) - (x^2*ArcSech[a*x])/(4*a^2) - ArcSech[a*x]^2/(2*a^4) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(2*a^4) - (x^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(4*a^2) + (1/4)*x^4*ArcSech[a*x]^3 + (ArcSech[a*x]*Log[1 + E^(2*ArcSech[a*x])])/a^4 + PolyLog[2, -E^(2*ArcSech[a*x])]/(2*a^4)} -{x^2*ArcSech[a*x]^3, x, 11, -((x*ArcSech[a*x])/a^2) - (x*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(2*a^2) + (1/3)*x^3*ArcSech[a*x]^3 - (ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]])/a^3 + ArcTan[(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(a*x)]/a^3 + (I*ArcSech[a*x]*PolyLog[2, (-I)*E^ArcSech[a*x]])/a^3 - (I*ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]])/a^3 - (I*PolyLog[3, (-I)*E^ArcSech[a*x]])/a^3 + (I*PolyLog[3, I*E^ArcSech[a*x]])/a^3} -{x^1*ArcSech[a*x]^3, x, 7, -((3*ArcSech[a*x]^2)/(2*a^2)) - (3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(2*a^2) + (1/2)*x^2*ArcSech[a*x]^3 + (3*ArcSech[a*x]*Log[1 + E^(2*ArcSech[a*x])])/a^2 + (3*PolyLog[2, -E^(2*ArcSech[a*x])])/(2*a^2)} -{x^0*ArcSech[a*x]^3, x, 9, x*ArcSech[a*x]^3 - (6*ArcSech[a*x]^2*ArcTan[E^ArcSech[a*x]])/a + (6*I*ArcSech[a*x]*PolyLog[2, (-I)*E^ArcSech[a*x]])/a - (6*I*ArcSech[a*x]*PolyLog[2, I*E^ArcSech[a*x]])/a - (6*I*PolyLog[3, (-I)*E^ArcSech[a*x]])/a + (6*I*PolyLog[3, I*E^ArcSech[a*x]])/a} -{ArcSech[a*x]^3/x^1, x, 7, (1/4)*ArcSech[a*x]^4 - ArcSech[a*x]^3*Log[1 + E^(2*ArcSech[a*x])] - (3/2)*ArcSech[a*x]^2*PolyLog[2, -E^(2*ArcSech[a*x])] + (3/2)*ArcSech[a*x]*PolyLog[3, -E^(2*ArcSech[a*x])] - (3/4)*PolyLog[4, -E^(2*ArcSech[a*x])]} -{ArcSech[a*x]^3/x^2, x, 5, (6*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/x - (6*ArcSech[a*x])/x + (3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/x - ArcSech[a*x]^3/x} -{ArcSech[a*x]^3/x^3, x, 6, (3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(8*x^2) - (3/8)*a^2*ArcSech[a*x] - (3*(1 - a*x)*(1 + a*x)*ArcSech[a*x])/(4*x^2) + (3*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(4*x^2) - (1/4)*a^2*ArcSech[a*x]^3 - ((1 - a*x)*(1 + a*x)*ArcSech[a*x]^3)/(2*x^2)} -{ArcSech[a*x]^3/x^4, x, 8, (14*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/(9*x) + (2*((1 - a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3)/(27*x^3) - (2*ArcSech[a*x])/(9*x^3) - (4*a^2*ArcSech[a*x])/(3*x) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*x^3) + (2*a^2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)*ArcSech[a*x]^2)/(3*x) - ArcSech[a*x]^3/(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcSech[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^6*(a + b*ArcSech[c*x]), x, 8, -((5*b*x*Sqrt[1 - c*x])/(112*c^6*Sqrt[1/(1 + c*x)])) - (5*b*x^3*Sqrt[1 - c*x])/(168*c^4*Sqrt[1/(1 + c*x)]) - (b*x^5*Sqrt[1 - c*x])/(42*c^2*Sqrt[1/(1 + c*x)]) + (1/7)*x^7*(a + b*ArcSech[c*x]) + (5*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(112*c^7)} -{x^5*(a + b*ArcSech[c*x]), x, 6, -((4*b*Sqrt[1 - c*x])/(45*c^6*Sqrt[1/(1 + c*x)])) - (2*b*x^2*Sqrt[1 - c*x])/(45*c^4*Sqrt[1/(1 + c*x)]) - (b*x^4*Sqrt[1 - c*x])/(30*c^2*Sqrt[1/(1 + c*x)]) + (1/6)*x^6*(a + b*ArcSech[c*x])} -{x^4*(a + b*ArcSech[c*x]), x, 6, -((3*b*x*Sqrt[1 - c*x])/(40*c^4*Sqrt[1/(1 + c*x)])) - (b*x^3*Sqrt[1 - c*x])/(20*c^2*Sqrt[1/(1 + c*x)]) + (1/5)*x^5*(a + b*ArcSech[c*x]) + (3*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(40*c^5)} -{x^3*(a + b*ArcSech[c*x]), x, 4, -((b*Sqrt[1 - c*x])/(6*c^4*Sqrt[1/(1 + c*x)])) - (b*x^2*Sqrt[1 - c*x])/(12*c^2*Sqrt[1/(1 + c*x)]) + (1/4)*x^4*(a + b*ArcSech[c*x])} -{x^2*(a + b*ArcSech[c*x]), x, 4, -((b*x*Sqrt[1 - c*x])/(6*c^2*Sqrt[1/(1 + c*x)])) + (1/3)*x^3*(a + b*ArcSech[c*x]) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3)} -{x^1*(a + b*ArcSech[c*x]), x, 2, -((b*Sqrt[1 - c*x])/(2*c^2*Sqrt[1/(1 + c*x)])) + (1/2)*x^2*(a + b*ArcSech[c*x])} -{x^0*(a + b*ArcSech[c*x]), x, 3, a*x + b*x*ArcSech[c*x] + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c} -{(a + b*ArcSech[c*x])/x^1, x, 6, -((a + b*ArcSech[c*x])^2/(2*b)) - (a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])] + (1/2)*b*PolyLog[2, -E^(-2*ArcSech[c*x])]} -{(a + b*ArcSech[c*x])/x^2, x, 2, (b*Sqrt[1 - c*x])/(x*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/x} -{(a + b*ArcSech[c*x])/x^3, x, 5, (b*Sqrt[1 - c*x])/(4*x^2*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/(2*x^2) + (1/4)*b*c^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]]} -{(a + b*ArcSech[c*x])/x^4, x, 4, (b*Sqrt[1 - c*x])/(9*x^3*Sqrt[1/(1 + c*x)]) + (2*b*c^2*Sqrt[1 - c*x])/(9*x*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/(3*x^3)} -{(a + b*ArcSech[c*x])/x^5, x, 7, (b*Sqrt[1 - c*x])/(16*x^4*Sqrt[1/(1 + c*x)]) + (3*b*c^2*Sqrt[1 - c*x])/(32*x^2*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/(4*x^4) + (3/32)*b*c^4*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]]} -{(a + b*ArcSech[c*x])/x^6, x, 6, (b*Sqrt[1 - c*x])/(25*x^5*Sqrt[1/(1 + c*x)]) + (4*b*c^2*Sqrt[1 - c*x])/(75*x^3*Sqrt[1/(1 + c*x)]) + (8*b*c^4*Sqrt[1 - c*x])/(75*x*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/(5*x^5)} -{(a + b*ArcSech[c*x])/x^7, x, 9, (b*Sqrt[1 - c*x])/(36*x^6*Sqrt[1/(1 + c*x)]) + (5*b*c^2*Sqrt[1 - c*x])/(144*x^4*Sqrt[1/(1 + c*x)]) + (5*b*c^4*Sqrt[1 - c*x])/(96*x^2*Sqrt[1/(1 + c*x)]) - (a + b*ArcSech[c*x])/(6*x^6) + (5/96)*b*c^6*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]]} - - -{x^3*(a + b*ArcSech[c*x])^2, x, 5, -((b^2*x^2)/(12*c^2)) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(3*c^4) - (b*x^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(6*c^2) + (1/4)*x^4*(a + b*ArcSech[c*x])^2 - (b^2*Log[x])/(3*c^4)} -{x^2*(a + b*ArcSech[c*x])^2, x, 8, -((b^2*x)/(3*c^2)) - (b*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(3*c^2) + (1/3)*x^3*(a + b*ArcSech[c*x])^2 - (2*b*(a + b*ArcSech[c*x])*ArcTan[E^ArcSech[c*x]])/(3*c^3) + (I*b^2*PolyLog[2, (-I)*E^ArcSech[c*x]])/(3*c^3) - (I*b^2*PolyLog[2, I*E^ArcSech[c*x]])/(3*c^3)} -{x^1*(a + b*ArcSech[c*x])^2, x, 4, -((b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/c^2) + (1/2)*x^2*(a + b*ArcSech[c*x])^2 - (b^2*Log[x])/c^2} -{x^0*(a + b*ArcSech[c*x])^2, x, 7, x*(a + b*ArcSech[c*x])^2 - (4*b*(a + b*ArcSech[c*x])*ArcTan[E^ArcSech[c*x]])/c + (2*I*b^2*PolyLog[2, (-I)*E^ArcSech[c*x]])/c - (2*I*b^2*PolyLog[2, I*E^ArcSech[c*x]])/c} -{(a + b*ArcSech[c*x])^2/x^1, x, 6, (a + b*ArcSech[c*x])^3/(3*b) - (a + b*ArcSech[c*x])^2*Log[1 + E^(2*ArcSech[c*x])] - b*(a + b*ArcSech[c*x])*PolyLog[2, -E^(2*ArcSech[c*x])] + (1/2)*b^2*PolyLog[3, -E^(2*ArcSech[c*x])]} -{(a + b*ArcSech[c*x])^2/x^2, x, 4, -((2*b^2)/x) + (2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/x - (a + b*ArcSech[c*x])^2/x} -{(a + b*ArcSech[c*x])^2/x^3, x, 4, -((b^2*(1 - c*x)*(1 + c*x))/(4*x^2)) - (1/2)*a*b*c^2*ArcSech[c*x] - (1/4)*b^2*c^2*ArcSech[c*x]^2 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(2*x^2) - ((1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*x^2)} -{(a + b*ArcSech[c*x])^2/x^4, x, 5, -((2*b^2)/(27*x^3)) - (4*b^2*c^2)/(9*x) + (2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(9*x^3) + (4*b*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(9*x) - (a + b*ArcSech[c*x])^2/(3*x^3)} -{(a + b*ArcSech[c*x])^2/x^5, x, 5, -(b^2/(32*x^4)) - (3*b^2*c^2)/(32*x^2) + (3/16)*a*b*c^4*ArcSech[c*x] + (3/32)*b^2*c^4*ArcSech[c*x]^2 + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(8*x^4) + (3*b*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x]))/(16*x^2) - (a + b*ArcSech[c*x])^2/(4*x^4)} - - -{x^3*(a + b*ArcSech[c*x])^3, x, 10, (b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(4*c^4) - (b^2*x^2*(a + b*ArcSech[c*x]))/(4*c^2) - (b*(a + b*ArcSech[c*x])^2)/(2*c^4) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^4) - (b*x^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(4*c^2) + (1/4)*x^4*(a + b*ArcSech[c*x])^3 + (b^2*(a + b*ArcSech[c*x])*Log[1 + E^(2*ArcSech[c*x])])/c^4 + (b^3*PolyLog[2, -E^(2*ArcSech[c*x])])/(2*c^4)} -{x^2*(a + b*ArcSech[c*x])^3, x, 11, -((b^2*x*(a + b*ArcSech[c*x]))/c^2) - (b*x*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2) + (1/3)*x^3*(a + b*ArcSech[c*x])^3 - (b*(a + b*ArcSech[c*x])^2*ArcTan[E^ArcSech[c*x]])/c^3 + (b^3*ArcTan[(Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(c*x)])/c^3 + (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, (-I)*E^ArcSech[c*x]])/c^3 - (I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, I*E^ArcSech[c*x]])/c^3 - (I*b^3*PolyLog[3, (-I)*E^ArcSech[c*x]])/c^3 + (I*b^3*PolyLog[3, I*E^ArcSech[c*x]])/c^3} -{x^1*(a + b*ArcSech[c*x])^3, x, 7, -((3*b*(a + b*ArcSech[c*x])^2)/(2*c^2)) - (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(2*c^2) + (1/2)*x^2*(a + b*ArcSech[c*x])^3 + (3*b^2*(a + b*ArcSech[c*x])*Log[1 + E^(2*ArcSech[c*x])])/c^2 + (3*b^3*PolyLog[2, -E^(2*ArcSech[c*x])])/(2*c^2)} -{x^0*(a + b*ArcSech[c*x])^3, x, 9, x*(a + b*ArcSech[c*x])^3 - (6*b*(a + b*ArcSech[c*x])^2*ArcTan[E^ArcSech[c*x]])/c + (6*I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, (-I)*E^ArcSech[c*x]])/c - (6*I*b^2*(a + b*ArcSech[c*x])*PolyLog[2, I*E^ArcSech[c*x]])/c - (6*I*b^3*PolyLog[3, (-I)*E^ArcSech[c*x]])/c + (6*I*b^3*PolyLog[3, I*E^ArcSech[c*x]])/c} -{(a + b*ArcSech[c*x])^3/x^1, x, 7, (a + b*ArcSech[c*x])^4/(4*b) - (a + b*ArcSech[c*x])^3*Log[1 + E^(2*ArcSech[c*x])] - (3/2)*b*(a + b*ArcSech[c*x])^2*PolyLog[2, -E^(2*ArcSech[c*x])] + (3/2)*b^2*(a + b*ArcSech[c*x])*PolyLog[3, -E^(2*ArcSech[c*x])] - (3/4)*b^3*PolyLog[4, -E^(2*ArcSech[c*x])]} -{(a + b*ArcSech[c*x])^3/x^2, x, 5, (6*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/x - (6*b^2*(a + b*ArcSech[c*x]))/x + (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/x - (a + b*ArcSech[c*x])^3/x} -{(a + b*ArcSech[c*x])^3/x^3, x, 6, (3*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(8*x^2) - (3/8)*b^3*c^2*ArcSech[c*x] - (3*b^2*(1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x]))/(4*x^2) + (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(4*x^2) - (1/4)*c^2*(a + b*ArcSech[c*x])^3 - ((1 - c*x)*(1 + c*x)*(a + b*ArcSech[c*x])^3)/(2*x^2)} -{(a + b*ArcSech[c*x])^3/x^4, x, 8, (14*b^3*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(9*x) + (2*b^3*((1 - c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)/(27*x^3) - (2*b^2*(a + b*ArcSech[c*x]))/(9*x^3) - (4*b^2*c^2*(a + b*ArcSech[c*x]))/(3*x) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(3*x^3) + (2*b*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(3*x) - (a + b*ArcSech[c*x])^3/(3*x^3)} -{(a + b*ArcSech[c*x])^3/x^5, x, 10, (3*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(128*x^4) + (45*b^3*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(256*x^2) + (45/256)*b^3*c^4*ArcSech[c*x] - (3*b^2*(a + b*ArcSech[c*x]))/(32*x^4) - (9*b^2*c^2*(a + b*ArcSech[c*x]))/(32*x^2) + (3*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(16*x^4) + (9*b*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(32*x^2) + (3/32)*c^4*(a + b*ArcSech[c*x])^3 - (a + b*ArcSech[c*x])^3/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/(a + b*ArcSech[c*x]), x, 0, Unintegrable[x/(a + b*ArcSech[c*x]), x]} -{x^0/(a + b*ArcSech[c*x]), x, 0, Unintegrable[1/(a + b*ArcSech[c*x]), x]} -{1/(x^1*(a + b*ArcSech[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcSech[c*x])), x]} -{1/(x^2*(a + b*ArcSech[c*x])), x, 4, (c*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/b - (c*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/b} -{1/(x^3*(a + b*ArcSech[c*x])), x, 6, (c^2*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b])/(2*b) - (c^2*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSech[c*x]])/(2*b)} -{1/(x^4*(a + b*ArcSech[c*x])), x, 9, (c^3*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/(4*b) + (c^3*CoshIntegral[(3*a)/b + 3*ArcSech[c*x]]*Sinh[(3*a)/b])/(4*b) - (c^3*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/(4*b) - (c^3*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSech[c*x]])/(4*b)} - - -{x^1/(a + b*ArcSech[c*x])^2, x, 0, Unintegrable[x/(a + b*ArcSech[c*x])^2, x]} -{x^0/(a + b*ArcSech[c*x])^2, x, 0, Unintegrable[1/(a + b*ArcSech[c*x])^2, x]} -{1/(x^1*(a + b*ArcSech[c*x])^2), x, 0, Unintegrable[1/(x*(a + b*ArcSech[c*x])^2), x]} -{1/(x^2*(a + b*ArcSech[c*x])^2), x, 5, (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(b*x*(a + b*ArcSech[c*x])) - (c*Cosh[a/b]*CoshIntegral[a/b + ArcSech[c*x]])/b^2 + (c*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/b^2} -{1/(x^3*(a + b*ArcSech[c*x])^2), x, 7, -((c^2*Cosh[(2*a)/b]*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]])/b^2) + (c^2*Sinh[2*ArcSech[c*x]])/(2*b*(a + b*ArcSech[c*x])) + (c^2*Sinh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSech[c*x]])/b^2} -{1/(x^4*(a + b*ArcSech[c*x])^2), x, 11, (c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(4*b*x*(a + b*ArcSech[c*x])) - (c^3*Cosh[a/b]*CoshIntegral[a/b + ArcSech[c*x]])/(4*b^2) - (3*c^3*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSech[c*x]])/(4*b^2) + (c^3*Sinh[3*ArcSech[c*x]])/(4*b*(a + b*ArcSech[c*x])) + (c^3*Sinh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/(4*b^2) + (3*c^3*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSech[c*x]])/(4*b^2)} - - -{x^1/(a + b*ArcSech[c*x])^3, x, 0, Unintegrable[x/(a + b*ArcSech[c*x])^3, x]} -{x^0/(a + b*ArcSech[c*x])^3, x, 0, Unintegrable[1/(a + b*ArcSech[c*x])^3, x]} -{1/(x^1*(a + b*ArcSech[c*x])^3), x, 0, Unintegrable[1/(x*(a + b*ArcSech[c*x])^3), x]} -{1/(x^2*(a + b*ArcSech[c*x])^3), x, 6, (Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(2*b*x*(a + b*ArcSech[c*x])^2) + 1/(2*b^2*x*(a + b*ArcSech[c*x])) + (c*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/(2*b^3) - (c*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/(2*b^3)} -{1/(x^3*(a + b*ArcSech[c*x])^3), x, 8, (c^2*Cosh[2*ArcSech[c*x]])/(2*b^2*(a + b*ArcSech[c*x])) + (c^2*CoshIntegral[(2*a)/b + 2*ArcSech[c*x]]*Sinh[(2*a)/b])/b^3 + (c^2*Sinh[2*ArcSech[c*x]])/(4*b*(a + b*ArcSech[c*x])^2) - (c^2*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSech[c*x]])/b^3} -{1/(x^4*(a + b*ArcSech[c*x])^3), x, 13, (c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(8*b*x*(a + b*ArcSech[c*x])^2) + c^2/(8*b^2*x*(a + b*ArcSech[c*x])) + (3*c^3*Cosh[3*ArcSech[c*x]])/(8*b^2*(a + b*ArcSech[c*x])) + (c^3*CoshIntegral[a/b + ArcSech[c*x]]*Sinh[a/b])/(8*b^3) + (9*c^3*CoshIntegral[(3*a)/b + 3*ArcSech[c*x]]*Sinh[(3*a)/b])/(8*b^3) + (c^3*Sinh[3*ArcSech[c*x]])/(8*b*(a + b*ArcSech[c*x])^2) - (c^3*Cosh[a/b]*SinhIntegral[a/b + ArcSech[c*x]])/(8*b^3) - (9*c^3*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSech[c*x]])/(8*b^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcSech[c x])^n when m symbolic*) - - -{(d*x)^m*(a + b*ArcSech[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcSech[c*x])^3, x]} -{(d*x)^m*(a + b*ArcSech[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcSech[c*x])^2, x]} -{(d*x)^m*(a + b*ArcSech[c*x]), x, 3, ((d*x)^(1 + m)*(a + b*ArcSech[c*x]))/(d*(1 + m)) + (b*(d*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(d*(1 + m)^2)} -{(d*x)^m/(a + b*ArcSech[c*x]), x, 0, Unintegrable[(d*x)^m/(a + b*ArcSech[c*x]), x]} -{(d*x)^m/(a + b*ArcSech[c*x])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcSech[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^m (a+b ArcSech[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSech[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcSech[c*x]), x, 9, -((b*e*(9*c^2*d^2 + e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^4)) - (b*d*e^2*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*c^2) - (b*e^3*x^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(12*c^2) + ((d + e*x)^4*(a + b*ArcSech[c*x]))/(4*e) + (b*d*(2*c^2*d^2 + e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(2*c^3) - (b*d^4*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*e)} -{(d + e*x)^2*(a + b*ArcSech[c*x]), x, 8, -((b*d*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/c^2) - (b*e^2*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^2) + ((d + e*x)^3*(a + b*ArcSech[c*x]))/(3*e) + (b*(6*c^2*d^2 + e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3) - (b*d^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(3*e)} -{(d + e*x)^1*(a + b*ArcSech[c*x]), x, 7, -((b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*c^2)) + ((d + e*x)^2*(a + b*ArcSech[c*x]))/(2*e) + (b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c - (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(2*e)} -{(d + e*x)^0*(a + b*ArcSech[c*x]), x, 3, a*x + b*x*ArcSech[c*x] + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c} -{(a + b*ArcSech[c*x])/(d + e*x)^1, x, 4, -(((a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e) + ((a + b*ArcSech[c*x])*Log[1 + (e - Sqrt[(-c^2)*d^2 + e^2])/(E^ArcSech[c*x]*(c*d))])/e + ((a + b*ArcSech[c*x])*Log[1 + (e + Sqrt[(-c^2)*d^2 + e^2])/(E^ArcSech[c*x]*(c*d))])/e + (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e) - (b*PolyLog[2, -((e - Sqrt[(-c^2)*d^2 + e^2])/(E^ArcSech[c*x]*(c*d)))])/e - (b*PolyLog[2, -((e + Sqrt[(-c^2)*d^2 + e^2])/(E^ArcSech[c*x]*(c*d)))])/e} -{(a + b*ArcSech[c*x])/(d + e*x)^2, x, 8, -((a + b*ArcSech[c*x])/(e*(d + e*x))) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(d*Sqrt[c^2*d^2 - e^2]) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(d*e)} -{(a + b*ArcSech[c*x])/(d + e*x)^3, x, 11, (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(2*d*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSech[c*x])/(2*e*(d + e*x)^2) + (b*c^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*(c^2*d^2 - e^2)^(3/2)) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(2*d^2*Sqrt[c^2*d^2 - e^2]) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(2*d^2*e)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^(m/2) (a+b ArcSech[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^(3/2)*(a + b*ArcSech[c*x]), x, 21, -((4*b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/(15*c^2)) + (2*(d + e*x)^(5/2)*(a + b*ArcSech[c*x]))/(5*e) - (28*b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^3*Sqrt[d + e*x]) - (4*b*d^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*e*Sqrt[d + e*x])} -{(d + e*x)^(1/2)*(a + b*ArcSech[c*x]), x, 14, (2*(d + e*x)^(3/2)*(a + b*ArcSech[c*x]))/(3*e) - (4*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*Sqrt[d + e*x]) - (4*b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*e*Sqrt[d + e*x])} -{(a + b*ArcSech[c*x])/(d + e*x)^(1/2), x, 8, (2*Sqrt[d + e*x]*(a + b*ArcSech[c*x]))/e - (4*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) - (4*b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*Sqrt[d + e*x])} -{(a + b*ArcSech[c*x])/(d + e*x)^(3/2), x, 5, -((2*(a + b*ArcSech[c*x]))/(e*Sqrt[d + e*x])) + (4*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*Sqrt[d + e*x])} -{(a + b*ArcSech[c*x])/(d + e*x)^(5/2), x, 11, (4*b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*(a + b*ArcSech[c*x]))/(3*e*(d + e*x)^(3/2)) - (4*b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*e*Sqrt[d + e*x])} -{(a + b*ArcSech[c*x])/(d + e*x)^(7/2), x, 18, (4*b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(15*d*(c^2*d^2 - e^2)*(d + e*x)^(3/2)) + (16*b*c^2*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(15*(c^2*d^2 - e^2)^2*Sqrt[d + e*x]) + (4*b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(5*d^2*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*(a + b*ArcSech[c*x]))/(5*e*(d + e*x)^(5/2)) - (16*b*c^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*(c^2*d^2 - e^2)^2*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*d^2*(c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) + (4*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*d^2*e*Sqrt[d + e*x])} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcSech[c x])^n when m symbolic*) - - -{(d + e*x)^m*(a + b*ArcSech[c*x]), x, 1, ((d + e*x)^(1 + m)*(a + b*ArcSech[c*x]))/(e*(1 + m)) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Unintegrable[(d + e*x)^(1 + m)/(x*Sqrt[1 - c^2*x^2]), x])/(e*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSech[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcSech[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 6, -((b*(42*c^2*d + 25*e)*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(560*c^6)) - (b*(42*c^2*d + 25*e)*x^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(840*c^4) - (b*e*x^5*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(42*c^2) + (1/5)*d*x^5*(a + b*ArcSech[c*x]) + (1/7)*e*x^7*(a + b*ArcSech[c*x]) + (b*(42*c^2*d + 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(560*c^7)} -{x^2*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 5, -((b*(20*c^2*d + 9*e)*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(120*c^4)) - (b*e*x^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(20*c^2) + (1/3)*d*x^3*(a + b*ArcSech[c*x]) + (1/5)*e*x^5*(a + b*ArcSech[c*x]) + (b*(20*c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(120*c^5)} -{x^0*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 4, -((b*e*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^2)) + d*x*(a + b*ArcSech[c*x]) + (1/3)*e*x^3*(a + b*ArcSech[c*x]) + (b*(6*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3)} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x^2, x, 3, (b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/x - (d*(a + b*ArcSech[c*x]))/x + e*x*(a + b*ArcSech[c*x]) + (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x^4, x, 4, (b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(9*x^3) + (b*(2*c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(9*x) - (d*(a + b*ArcSech[c*x]))/(3*x^3) - (e*(a + b*ArcSech[c*x]))/x} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x^6, x, 5, (b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(25*x^5) + (b*(12*c^2*d + 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x^3) + (2*b*c^2*(12*c^2*d + 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x) - (d*(a + b*ArcSech[c*x]))/(5*x^5) - (e*(a + b*ArcSech[c*x]))/(3*x^3)} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x^8, x, 6, (b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(49*x^7) + (b*(30*c^2*d + 49*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(1225*x^5) + (4*b*c^2*(30*c^2*d + 49*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3675*x^3) + (8*b*c^4*(30*c^2*d + 49*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3675*x) - (d*(a + b*ArcSech[c*x]))/(7*x^7) - (e*(a + b*ArcSech[c*x]))/(5*x^5)} - -{x^5*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 5, -((b*(4*c^2*d + 3*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(24*c^8)) + (b*(8*c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(72*c^8) - (b*(4*c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(120*c^8) + (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(7/2))/(56*c^8) + (1/6)*d*x^6*(a + b*ArcSech[c*x]) + (1/8)*e*x^8*(a + b*ArcSech[c*x])} -{x^3*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 5, -((b*(3*c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(12*c^6)) + (b*(3*c^2*d + 4*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(36*c^6) - (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(30*c^6) + (1/4)*d*x^4*(a + b*ArcSech[c*x]) + (1/6)*e*x^6*(a + b*ArcSech[c*x])} -{x*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 7, -((b*(2*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(4*c^4)) + (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(12*c^4) + ((d + e*x^2)^2*(a + b*ArcSech[c*x]))/(4*e) - (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*e)} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x, x, 12, -((b*e*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(2*c)) + (I*b*d*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (1/2)*e*x^2*(a + b*ArcSech[c*x]) - (b*d*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*d*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1/x])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - d*(a + b*ArcSech[c*x])*Log[1/x] + (I*b*d*Sqrt[1 - 1/(c^2*x^2)]*PolyLog[2, E^(2*I*ArcCsc[c*x])])/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])} -{((d + e*x^2)*(a + b*ArcSech[c*x]))/x^3, x, 14, (b*c*d*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(4*x) + (I*b*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (1/4)*b*c^2*d*ArcSech[c*x] - (d*(a + b*ArcSech[c*x]))/(2*x^2) - (b*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1/x])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - e*(a + b*ArcSech[c*x])*Log[1/x] + (I*b*e*Sqrt[1 - 1/(c^2*x^2)]*PolyLog[2, E^(2*I*ArcCsc[c*x])])/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])} - - -{x^2*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x, 6, -((b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(1680*c^6)) - (b*e*(84*c^2*d + 25*e)*x^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(840*c^4) - (b*e^2*x^5*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(42*c^2) + (1/3)*d^2*x^3*(a + b*ArcSech[c*x]) + (2/5)*d*e*x^5*(a + b*ArcSech[c*x]) + (1/7)*e^2*x^7*(a + b*ArcSech[c*x]) + (b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(1680*c^7)} -{x^0*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x, 5, -((b*e*(40*c^2*d + 9*e)*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(120*c^4)) - (b*e^2*x^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(20*c^2) + d^2*x*(a + b*ArcSech[c*x]) + (2/3)*d*e*x^3*(a + b*ArcSech[c*x]) + (1/5)*e^2*x^5*(a + b*ArcSech[c*x]) + (b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(120*c^5)} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^2, x, 5, (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/x - (b*e^2*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^2) - (d^2*(a + b*ArcSech[c*x]))/x + 2*d*e*x*(a + b*ArcSech[c*x]) + (1/3)*e^2*x^3*(a + b*ArcSech[c*x]) + (b*e*(12*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(6*c^3)} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^4, x, 5, (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(9*x^3) + (2*b*d*(c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(9*x) - (d^2*(a + b*ArcSech[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcSech[c*x]))/x + e^2*x*(a + b*ArcSech[c*x]) + (b*e^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^6, x, 5, (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(25*x^5) + (2*b*d*(6*c^2*d + 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x^3) + (b*(24*c^4*d^2 + 100*c^2*d*e + 225*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(225*x) - (d^2*(a + b*ArcSech[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcSech[c*x]))/(3*x^3) - (e^2*(a + b*ArcSech[c*x]))/x} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^8, x, 6, (b*d^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(49*x^7) + (2*b*d*(15*c^2*d + 49*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(1225*x^5) + (b*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(11025*x^3) + (2*b*c^2*(360*c^4*d^2 + 1176*c^2*d*e + 1225*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(11025*x) - (d^2*(a + b*ArcSech[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcSech[c*x]))/(5*x^5) - (e^2*(a + b*ArcSech[c*x]))/(3*x^3)} - -{x^3*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x, 5, -((b*(6*c^4*d^2 + 8*c^2*d*e + 3*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(24*c^8)) + (b*(6*c^4*d^2 + 16*c^2*d*e + 9*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(72*c^8) - (b*e*(8*c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(120*c^8) + (b*e^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(7/2))/(56*c^8) + (1/4)*d^2*x^4*(a + b*ArcSech[c*x]) + (1/3)*d*e*x^6*(a + b*ArcSech[c*x]) + (1/8)*e^2*x^8*(a + b*ArcSech[c*x])} -{x^1*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x, 7, -((b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^6)) + (b*e*(3*c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(18*c^6) - (b*e^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(30*c^6) + ((d + e*x^2)^3*(a + b*ArcSech[c*x]))/(6*e) - (b*d^3*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*e)} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^1, x, 13, -((b*e*(6*c^2*d + e)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(6*c^3)) - (b*e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x^3)/(12*c) + (I*b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + d*e*x^2*(a + b*ArcSech[c*x]) + (1/4)*e^2*x^4*(a + b*ArcSech[c*x]) - (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*d^2*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1/x])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - d^2*(a + b*ArcSech[c*x])*Log[1/x] + (I*b*d^2*Sqrt[1 - 1/(c^2*x^2)]*PolyLog[2, E^(2*I*ArcCsc[c*x])])/(2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])} -{((d + e*x^2)^2*(a + b*ArcSech[c*x]))/x^3, x, 15, (b*c*d^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(4*x) - (b*e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(2*c) + (I*b*d*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]^2)/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (1/4)*b*c^2*d^2*ArcSech[c*x] - (d^2*(a + b*ArcSech[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcSech[c*x]) - (2*b*d*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1 - E^(2*I*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (2*b*d*e*Sqrt[1 - 1/(c^2*x^2)]*ArcCsc[c*x]*Log[1/x])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - 2*d*e*(a + b*ArcSech[c*x])*Log[1/x] + (I*b*d*e*Sqrt[1 - 1/(c^2*x^2)]*PolyLog[2, E^(2*I*ArcCsc[c*x])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2), x, 24, (x*(a + b*ArcSech[c*x]))/e - (b*ArcTan[Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]])/(c*e) + (Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^(3/2))} -{(x*(a + b*ArcSech[c*x]))/(d + e*x^2), x, 26, -((a + b*ArcSech[c*x])^2/(b*e)) - ((a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e) + (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e)} -{(a + b*ArcSech[c*x])/(d + e*x^2), x, 19, ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcSech[c*x])/(x*(d + e*x^2)), x, 19, (a + b*ArcSech[c*x])^2/(2*b*d) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d)} -{(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)), x, 24, (b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d - a/(d*x) - (b*ArcSech[c*x])/(d*x) + (Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2))} - - -{(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 32, -((b*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)/(2*c*e^2)) + (d*(a + b*ArcSech[c*x]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcSech[c*x]))/(2*e^2) + (2*d*(a + b*ArcSech[c*x])^2)/(b*e^3) - (b*d*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (2*d*(a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e^3 - (d*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (d*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - (b*d*PolyLog[2, -E^(-2*ArcSech[c*x])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3} -{(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 30, -((a + b*ArcSech[c*x])/(2*e*(e + d/x^2))) - (a + b*ArcSech[c*x])^2/(b*e^2) + (b*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(3/2)*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e^2 + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e^2) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2)} -{(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 8, -((a + b*ArcSech[c*x])/(2*e*(d + e*x^2))) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(2*d*e) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(2*d*Sqrt[e]*Sqrt[c^2*d + e])} -{(a + b*ArcSech[c*x])/(x*(d + e*x^2)^2), x, 25, -((e*(a + b*ArcSech[c*x]))/(2*d^2*(e + d/x^2))) + (a + b*ArcSech[c*x])^2/(2*b*d^2) + (b*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^2)} - -{(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 50, -((d*(a + b*ArcSech[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] - d/x))) + (d*(a + b*ArcSech[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcSech[c*x]))/e^2 + (b*d*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) + (b*d*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) - (b*ArcTan[Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]])/(c*e^2) + (3*Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*e^(5/2))} -{(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 27, (a + b*ArcSech[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcSech[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) - (b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2))} -{(a + b*ArcSech[c*x])/(d + e*x^2)^2, x, 47, -((a + b*ArcSech[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x))) + (a + b*ArcSech[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) + (b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)^2), x, 50, (b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/d^2 - a/(d^2*x) - (b*ArcSech[c*x])/(d^2*x) + (e*(a + b*ArcSech[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcSech[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(2*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(4*(-d)^(5/2))} - - -{(x^5*(a + b*ArcSech[c*x]))/(d + e*x^2)^3, x, 35, (b*d*(c^2 - 1/x^2))/(8*c*e^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (a + b*ArcSech[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcSech[c*x])/(2*e^2*(e + d/x^2)) - (a + b*ArcSech[c*x])^2/(b*e^3) + (b*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(2*e^(5/2)*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (b*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcSech[c*x])*Log[1 + E^(-2*ArcSech[c*x])])/e^3 + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3) + (b*PolyLog[2, -E^(-2*ArcSech[c*x])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^3)} -{(x^3*(a + b*ArcSech[c*x]))/(d + e*x^2)^3, x, 6, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(8*e*(c^2*d + e)*(d + e*x^2)) + (x^4*(a + b*ArcSech[c*x]))/(4*d*(d + e*x^2)^2) - (b*(c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(8*d*e^(3/2)*(c^2*d + e)^(3/2))} -{(x*(a + b*ArcSech[c*x]))/(d + e*x^2)^3, x, 9, -((b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2))) - (a + b*ArcSech[c*x])/(4*e*(d + e*x^2)^2) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(4*d^2*e) - (b*(3*c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[(Sqrt[e]*Sqrt[1 - c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt[e]*(c^2*d + e)^(3/2))} -{(a + b*ArcSech[c*x])/(x*(d + e*x^2)^3), x, 30, -((b*e*(c^2 - 1/x^2))/(8*c*d^2*(c^2*d + e)*(e + d/x^2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)) + (e^2*(a + b*ArcSech[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSech[c*x]))/(d^3*(e + d/x^2)) + (a + b*ArcSech[c*x])^2/(2*b*d^3) + (b*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(d^3*Sqrt[c^2*d + e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (b*Sqrt[e]*(c^2*d + 2*e)*Sqrt[-1 + 1/(c^2*x^2)]*ArcTanh[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[-1 + 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3)} - -{(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3, x, 35, (b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcSech[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcSech[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*ArcSech[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcSech[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (3*b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) - (b*d*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)*e) - (3*b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) - (b*d*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)*e) + (3*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2))} -{(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^3, x, 63, (b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*Sqrt[-d]*Sqrt[e]*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcSech[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcSech[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcSech[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcSech[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) + (b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) - (b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2))} -{(a + b*ArcSech[c*x])/(d + e*x^2)^3, x, 81, (b*c*Sqrt[e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[e]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcSech[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcSech[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(a + b*ArcSech[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcSech[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (5*b*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) + (5*b*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d^2*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]) - (b*e*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*d*(c*d - Sqrt[-d]*Sqrt[e])^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)) + (3*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcSech[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSech[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcSech[c*x])/(Sqrt[e] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcSech[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, If[$VersionNumber>=8, 12, 13], (b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(1680*c^6*e^2) + (b*(29*c^2*d - 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^4*e^2) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c^2*e^2) + (d^2*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcSech[c*x]))/(7*e^3) - (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^7*e^(5/2)) - (8*b*d^(7/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(105*e^3)} -{x^3*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, If[$VersionNumber>=8, 11, 12], If[$VersionNumber>=8, -((b*(c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^4*e)) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c^2*e) - (d*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^2) + (b*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^5*e^(3/2)) + (2*b*d^(5/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(15*e^2), -((b*(c^2*d + 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^4*e)) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c^2*e) - (d*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^2) + (b*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^5*e^(3/2)) + (2*b*d^(5/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(15*e^2)]} -{x*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, 10, -((b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c^2)) + ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e) - (b*(3*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^3*Sqrt[e]) - (b*d^(3/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*e)} -{Sqrt[d + e*x^2]*(a + b*ArcSech[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x, x]} -{Sqrt[d + e*x^2]*(a + b*ArcSech[c*x])/x^3, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^3, x]} - -{x^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, 0, Unintegrable[x^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]} -{x^0*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]} -{Sqrt[d + e*x^2]*(a + b*ArcSech[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/x^2, x]} -{Sqrt[d + e*x^2]*(a + b*ArcSech[c*x])/x^4, x, 9, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*x^3) + (2*b*(c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*x) - ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*d*x^3) + (2*b*c*(c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(2*c^2*d + 3*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*c*d*Sqrt[d + e*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcSech[c*x])/x^6, x, If[$VersionNumber>=8, 10, 26], If[$VersionNumber>=8, (b*(12*c^2*d - e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(225*d*x^3) + (b*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(225*d^2*x) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(25*d*x^5) - ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(15*d^2*x^3) + (b*c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(225*d^2*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 30*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(225*c*d^2*Sqrt[d + e*x^2]), (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(25*x^5) + (b*e*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(45*d*x^3) + (b*(4*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d*x^3) - (2*b*e^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(15*d^2*x) + (b*e*(2*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*x) + (b*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d^2*x) - ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(15*d^2*x^3) - (2*b*c*e^2*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(15*d^2*Sqrt[1 + (e*x^2)/d]) + (b*c*e*(2*c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(45*d^2*Sqrt[1 + (e*x^2)/d]) + (b*c*(8*c^4*d^2 + 3*c^2*d*e - 2*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d^2*Sqrt[1 + (e*x^2)/d]) - (b*c*(8*c^2*d - e)*(c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[d + e*x^2]) - (2*b*c*e*(c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(45*d*Sqrt[d + e*x^2]) + (2*b*e^2*(c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(15*c*d^2*Sqrt[d + e*x^2])]} - - -{x^3*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x, 12, (b*(3*c^4*d^2 - 38*c^2*d*e - 25*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(560*c^6*e) - (b*(13*c^2*d + 25*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^4*e) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c^2*e) - (d*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcSech[c*x]))/(7*e^2) + (b*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 - 25*e^3)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^7*e^(3/2)) + (2*b*d^(7/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(35*e^2)} -{x^1*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x, 11, -((b*(7*c^2*d + 3*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(40*c^4)) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c^2) + ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e) - (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^5*Sqrt[e]) - (b*d^(5/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(5*e)} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^1, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^3, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^3, x]} - -{x^2*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x, 0, Unintegrable[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]} -{x^0*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x, 0, Unintegrable[(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^2, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^2, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^4, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/x^4, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^6, x, 10, (4*b*(c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*x^3) + (b*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d*x) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(25*x^5) - ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*d*x^5) + (b*c*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(75*d*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*(8*c^4*d^2 + 19*c^2*d*e + 15*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(75*c*d*Sqrt[d + e*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x])/x^8, x, 11, (b*(120*c^4*d^2 + 159*c^2*d*e - 37*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(3675*d*x^3) + (b*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(3675*d^2*x) + (b*(30*c^2*d + 11*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*x^5) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^7) - ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(35*d^2*x^5) + (b*c*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3675*d^2*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d + e)*(120*c^6*d^3 + 204*c^4*d^2*e + 17*c^2*d*e^2 - 105*e^3)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3675*c*d^2*Sqrt[d + e*x^2])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x, 11, (b*(19*c^2*d - 9*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^4*e^2) - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c^2*e^2) + (d^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^3) - (b*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^5*e^(5/2)) - (8*b*d^(5/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(15*e^3)} -{x^3*(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x, 10, -((b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c^2*e)) - (d*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^2) + (b*(3*c^2*d - e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^3*e^(3/2)) + (2*b*d^(3/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*e^2)} -{x^1*(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x, 10, (Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(c*Sqrt[e]) - (b*Sqrt[d]*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/e} -{(a + b*ArcSech[c*x])/(x^1*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x*Sqrt[d + e*x^2]), x]} -{(a + b*ArcSech[c*x])/(x^3*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x^3*Sqrt[d + e*x^2]), x]} - -{x^2*(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(x^2*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x]} -{x^0*(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcSech[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcSech[c*x])/(x^2*Sqrt[d + e*x^2]), x, 9, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(d*x) - (Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/(d*x) + (b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d*Sqrt[d + e*x^2])} -{(a + b*ArcSech[c*x])/(x^4*Sqrt[d + e*x^2]), x, 9, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*x^3) + (b*(2*c^2*d - 5*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*x) - (Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/(3*d^2*x) + (b*c*(2*c^2*d - 5*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d^2*Sqrt[1 + (e*x^2)/d]) - (2*b*(c^2*d - 3*e)*(c^2*d + e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(9*c*d^2*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 10, -((b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c^2*e^2)) - (d^2*(a + b*ArcSech[c*x]))/(e^3*Sqrt[d + e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) + (b*(9*c^2*d - e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^3*e^(5/2)) + (8*b*d^(3/2)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*e^3)} -{x^3*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 9, (d*(a + b*ArcSech[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^2 - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(c*e^(3/2)) - (2*b*Sqrt[d]*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/e^2} -{x^1*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 5, -((a + b*ArcSech[c*x])/(e*Sqrt[d + e*x^2])) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(Sqrt[d]*e)} -{(a + b*ArcSech[c*x])/(x^1*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(3/2)), x]} - -{x^4*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]} -{x^2*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^2*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]} -{x^0*(a + b*ArcSech[c*x])/(d + e*x^2)^(3/2), x, 4, (x*(a + b*ArcSech[c*x]))/(d*Sqrt[d + e*x^2]) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d*Sqrt[d + e*x^2])} -{(a + b*ArcSech[c*x])/(x^2*(d + e*x^2)^(3/2)), x, 8, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(d^2*x) - (a + b*ArcSech[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcSech[c*x]))/(d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(d^2*Sqrt[1 + (e*x^2)/d]) - (b*(c^2*d + 2*e)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d^2*Sqrt[d + e*x^2])} - - -{x^5*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 10, -((b*d*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*e^2*(c^2*d + e)*Sqrt[d + e*x^2])) - (d^2*(a + b*ArcSech[c*x]))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcSech[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^3 - (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(c*e^(5/2)) - (8*b*Sqrt[d]*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*e^3)} -{x^3*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 7, (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*e*(c^2*d + e)*Sqrt[d + e*x^2]) + (d*(a + b*ArcSech[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcSech[c*x])/(e^2*Sqrt[d + e*x^2]) + (2*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*Sqrt[d]*e^2)} -{x^1*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 6, -((b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2])) - (a + b*ArcSech[c*x])/(3*e*(d + e*x^2)^(3/2)) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(3*d^(3/2)*e)} -{(a + b*ArcSech[c*x])/(x^1*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x^3*(d + e*x^2)^(5/2)), x]} - -{x^6*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^6*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2), x]} -{x^4*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^(5/2), x]} -{x^2*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 8, -((b*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2])) + (x^3*(a + b*ArcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d*e*(c^2*d + e)*Sqrt[1 + (e*x^2)/d]) + (b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*c*d*e*Sqrt[d + e*x^2])} -{x^0*(a + b*ArcSech[c*x])/(d + e*x^2)^(5/2), x, 8, (b*e*x*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d^2*(c^2*d + e)*Sqrt[d + e*x^2]) + (x*(a + b*ArcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSech[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*(c^2*d + e)*Sqrt[1 + (e*x^2)/d]) + (2*b*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*c*d^2*Sqrt[d + e*x^2])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSech[c x]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[c*x]), x, 5, If[$VersionNumber>=8, -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^6*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*(f*x)^(3 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^4*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)) - (b*e^3*(f*x)^(5 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f^5*(6 + m)*(7 + m)) + (d^3*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSech[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSech[c*x]))/(f^7*(7 + m)) + (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^6*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)), -((b*e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^6*f*(6 + m)*(8 + 6*m + m^2)*(105 + 71*m + 15*m^2 + m^3))) - (b*e^2*(e*(5 + m)^2 + 3*c^2*d*(42 + 13*m + m^2))*(f*x)^(3 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^4*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)) - (b*e^3*(f*x)^(5 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f^5*(6 + m)*(7 + m)) + (d^3*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcSech[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSech[c*x]))/(f^7*(7 + m)) + (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^6*f*(1 + m)*(2 + m)*(4 + m)*(6 + m))]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcSech[c*x]), x, 5, If[$VersionNumber>=8, -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^4*f*(2 + m)*(3 + m)*(4 + m)*(5 + m))) - (b*e^2*(f*x)^(3 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f^3*(4 + m)*(5 + m)) + (d^2*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSech[c*x]))/(f^5*(5 + m)) + (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^4*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)), -((b*e*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^4*f*(4 + m)*(5 + m)*(6 + 5*m + m^2))) - (b*e^2*(f*x)^(3 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f^3*(4 + m)*(5 + m)) + (d^2*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcSech[c*x]))/(f^5*(5 + m)) + (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 + 2*c^2*d*(20 + 9*m + m^2)))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^4*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m))]} -{(f*x)^m*(d + e*x^2)*(a + b*ArcSech[c*x]), x, 4, -((b*e*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(c^2*f*(2 + m)*(3 + m))) + (d*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (b*(e*(1 + m)^2 + c^2*d*(2 + m)*(3 + m))*(f*x)^(1 + m)*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c^2*f*(1 + m)^2*(2 + m)*(3 + m))} -{((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2), x]} -{((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^2, x]} - - -{(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x, 0, Unintegrable[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]), x]} -{(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x, 0, Unintegrable[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]), x]} -{((f*x)^m*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x, 0, Unintegrable[((f*x)^m*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x]} -{((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcSech[c*x]))/(d + e*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^4)^p (a+b ArcSech[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^4)^(p/2) (a+b ArcSech[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^11*(a + b*ArcSech[c*x])/Sqrt[1 - c^4*x^4], x, 15, -((4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)) + (7*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(90*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (13*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(150*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (3*b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(7/2))/(70*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(9/2))/(90*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcSech[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcSech[c*x]))/(10*c^12) + (4*b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(15*c^13*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)} -{x^7*(a + b*ArcSech[c*x])/Sqrt[1 - c^4*x^4], x, 12, -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)) + (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(3/2))/(18*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/2))/(30*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^8) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcSech[c*x]))/(6*c^8) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(3*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)} -{x^3*(a + b*ArcSech[c*x])/Sqrt[1 - c^4*x^4], x, 7, -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(2*c^5*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^4) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(2*c^5*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)} -{(a + b*ArcSech[c*x])/(x^1*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x*Sqrt[1 - c^4*x^4]), x]} -{(a + b*ArcSech[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcSech[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]} - - -(* ::Section:: *) -(*Integrands of the form u (a+b ArcSech[c x])^n*) diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Inverse hyperbolic secant functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Inverse hyperbolic secant functions.m deleted file mode 100644 index 97a8860..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.5 Inverse hyperbolic secant/7.5.2 Inverse hyperbolic secant functions.m +++ /dev/null @@ -1,204 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Inverse Hyperbolic Secants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcSech[a+b x]^n*) - - -{x^3*ArcSech[a + b*x], x, 8, -(((2 + 17*a^2)*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(12*b^4)) - (x^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(12*b^2) + (a*(a + b*x)*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(3*b^4) - (a^4*ArcSech[a + b*x])/(4*b^4) + (1/4)*x^4*ArcSech[a + b*x] + (a*(1 + 2*a^2)*ArcTan[(Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(a + b*x)])/(2*b^4)} -{x^2*ArcSech[a + b*x], x, 7, (5*a*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*b^3) - (x*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*b^2) + (a^3*ArcSech[a + b*x])/(3*b^3) + (1/3)*x^3*ArcSech[a + b*x] - ((1 + 6*a^2)*ArcTan[(Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(a + b*x)])/(6*b^3)} -{x^1*ArcSech[a + b*x], x, 6, -((Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(2*b^2)) - (a^2*ArcSech[a + b*x])/(2*b^2) + (1/2)*x^2*ArcSech[a + b*x] + (a*ArcTan[(Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(a + b*x)])/b^2} -{x^0*ArcSech[a + b*x], x, 4, ((a + b*x)*ArcSech[a + b*x])/b - (2*ArcTan[Sqrt[(1 - a - b*x)/(1 + a + b*x)]])/b} -{ArcSech[a + b*x]/x^1, x, 14, ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - ArcSech[a + b*x]*Log[1 + E^(2*ArcSech[a + b*x])] + PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - (1/2)*PolyLog[2, -E^(2*ArcSech[a + b*x])]} -{ArcSech[a + b*x]/x^2, x, 5, -((b*ArcSech[a + b*x])/a) - ArcSech[a + b*x]/x + (2*b*ArcTanh[(Sqrt[1 + a]*Tanh[(1/2)*ArcSech[a + b*x]])/Sqrt[1 - a]])/(a*Sqrt[1 - a^2])} -{ArcSech[a + b*x]/x^3, x, 7, (b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(2*a*(1 - a^2)*x) + (b^2*ArcSech[a + b*x])/(2*a^2) - ArcSech[a + b*x]/(2*x^2) - ((1 - 2*a^2)*b^2*ArcTanh[(Sqrt[1 + a]*Tanh[(1/2)*ArcSech[a + b*x]])/Sqrt[1 - a]])/(a^2*(1 - a^2)^(3/2))} -{ArcSech[a + b*x]/x^4, x, 8, (b*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*a*(1 - a^2)*x^2) - ((2 - 5*a^2)*b^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x))/(6*a^2*(1 - a^2)^2*x) - (b^3*ArcSech[a + b*x])/(3*a^3) - ArcSech[a + b*x]/(3*x^3) + ((2 - 5*a^2 + 6*a^4)*b^3*ArcTanh[(Sqrt[1 + a]*Tanh[(1/2)*ArcSech[a + b*x]])/Sqrt[1 - a]])/(3*a^3*(1 - a^2)^(5/2))} - - -{x^2*ArcSech[a + b*x]^2, x, 17, -(x/(3*b^2)) + (2*a*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x])/b^3 - ((a + b*x)*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x])/(3*b^3) + (a^3*ArcSech[a + b*x]^2)/(3*b^3) + (1/3)*x^3*ArcSech[a + b*x]^2 - (2*ArcSech[a + b*x]*ArcTan[E^ArcSech[a + b*x]])/(3*b^3) - (4*a^2*ArcSech[a + b*x]*ArcTan[E^ArcSech[a + b*x]])/b^3 + (2*a*Log[a + b*x])/b^3 + (I*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/(3*b^3) + (2*I*a^2*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/b^3 - (I*PolyLog[2, I*E^ArcSech[a + b*x]])/(3*b^3) - (2*I*a^2*PolyLog[2, I*E^ArcSech[a + b*x]])/b^3} -{x^1*ArcSech[a + b*x]^2, x, 11, -((Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x])/b^2) - (a^2*ArcSech[a + b*x]^2)/(2*b^2) + (1/2)*x^2*ArcSech[a + b*x]^2 + (4*a*ArcSech[a + b*x]*ArcTan[E^ArcSech[a + b*x]])/b^2 - Log[a + b*x]/b^2 - (2*I*a*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/b^2 + (2*I*a*PolyLog[2, I*E^ArcSech[a + b*x]])/b^2} -{x^0*ArcSech[a + b*x]^2, x, 8, ((a + b*x)*ArcSech[a + b*x]^2)/b - (4*ArcSech[a + b*x]*ArcTan[E^ArcSech[a + b*x]])/b + (2*I*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/b - (2*I*PolyLog[2, I*E^ArcSech[a + b*x]])/b} -{ArcSech[a + b*x]^2/x^1, x, 17, ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - ArcSech[a + b*x]^2*Log[1 + E^(2*ArcSech[a + b*x])] + 2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + 2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - ArcSech[a + b*x]*PolyLog[2, -E^(2*ArcSech[a + b*x])] - 2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] - 2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] + (1/2)*PolyLog[3, -E^(2*ArcSech[a + b*x])]} -{ArcSech[a + b*x]^2/x^2, x, 12, -((b*ArcSech[a + b*x]^2)/a) - ArcSech[a + b*x]^2/x + (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (2*b*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*b*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])} -{ArcSech[a + b*x]^2/x^3, x, 23, (b^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x])/(a*(1 - a^2)*(a + b*x)*(1 - a/(a + b*x))) + (b^2*ArcSech[a + b*x]^2)/(2*a^2) - ArcSech[a + b*x]^2/(2*x^2) + (b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) - (2*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - (b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (2*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (b^2*Log[x/(a + b*x)])/(a^2*(1 - a^2)) + (b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) - (2*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - (b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (2*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2])} - - -{x^1*ArcSech[a + b*x]^3, x, 16, -((3*ArcSech[a + b*x]^2)/(2*b^2)) - (3*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x]^2)/(2*b^2) - (a^2*ArcSech[a + b*x]^3)/(2*b^2) + (1/2)*x^2*ArcSech[a + b*x]^3 + (6*a*ArcSech[a + b*x]^2*ArcTan[E^ArcSech[a + b*x]])/b^2 + (3*ArcSech[a + b*x]*Log[1 + E^(2*ArcSech[a + b*x])])/b^2 - (6*I*a*ArcSech[a + b*x]*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/b^2 + (6*I*a*ArcSech[a + b*x]*PolyLog[2, I*E^ArcSech[a + b*x]])/b^2 + (3*PolyLog[2, -E^(2*ArcSech[a + b*x])])/(2*b^2) + (6*I*a*PolyLog[3, (-I)*E^ArcSech[a + b*x]])/b^2 - (6*I*a*PolyLog[3, I*E^ArcSech[a + b*x]])/b^2} -{x^0*ArcSech[a + b*x]^3, x, 10, ((a + b*x)*ArcSech[a + b*x]^3)/b - (6*ArcSech[a + b*x]^2*ArcTan[E^ArcSech[a + b*x]])/b + (6*I*ArcSech[a + b*x]*PolyLog[2, (-I)*E^ArcSech[a + b*x]])/b - (6*I*ArcSech[a + b*x]*PolyLog[2, I*E^ArcSech[a + b*x]])/b - (6*I*PolyLog[3, (-I)*E^ArcSech[a + b*x]])/b + (6*I*PolyLog[3, I*E^ArcSech[a + b*x]])/b} -{ArcSech[a + b*x]^3/x^1, x, 20, ArcSech[a + b*x]^3*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + ArcSech[a + b*x]^3*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - ArcSech[a + b*x]^3*Log[1 + E^(2*ArcSech[a + b*x])] + 3*ArcSech[a + b*x]^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + 3*ArcSech[a + b*x]^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - (3/2)*ArcSech[a + b*x]^2*PolyLog[2, -E^(2*ArcSech[a + b*x])] - 6*ArcSech[a + b*x]*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] - 6*ArcSech[a + b*x]*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] + (3/2)*ArcSech[a + b*x]*PolyLog[3, -E^(2*ArcSech[a + b*x])] + 6*PolyLog[4, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])] + 6*PolyLog[4, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])] - (3/4)*PolyLog[4, -E^(2*ArcSech[a + b*x])]} -{ArcSech[a + b*x]^3/x^2, x, 14, -((b*ArcSech[a + b*x]^3)/a) - ArcSech[a + b*x]^3/x + (3*b*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (3*b*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (6*b*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (6*b*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (6*b*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (6*b*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2])} -{ArcSech[a + b*x]^3/x^3, x, 32, -((3*b^2*ArcSech[a + b*x]^2)/(2*a^2*(1 - a^2))) + (3*b^2*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a + b*x)*ArcSech[a + b*x]^2)/(2*a*(1 - a^2)*(a + b*x)*(1 - a/(a + b*x))) + (b^2*ArcSech[a + b*x]^3)/(2*a^2) - ArcSech[a + b*x]^3/(2*x^2) + (3*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) - (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*ArcSech[a + b*x]*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(2*a^2*(1 - a^2)^(3/2)) + (3*b^2*ArcSech[a + b*x]^2*Log[1 - (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)) + (3*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) - (6*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)) - (3*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*ArcSech[a + b*x]*PolyLog[2, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) - (3*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) + (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 - Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2]) + (3*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*(1 - a^2)^(3/2)) - (6*b^2*PolyLog[3, (a*E^ArcSech[a + b*x])/(1 + Sqrt[1 - a^2])])/(a^2*Sqrt[1 - a^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcSech[a x^n]*) - - -{x^3*ArcSech[Sqrt[x]], x, 4, -((1 - x)/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])) + (1 - x)^2/(4*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) - (3*(1 - x)^3)/(20*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1 - x)^4/(28*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1/4)*x^4*ArcSech[Sqrt[x]]} -{x^2*ArcSech[Sqrt[x]], x, 4, -((1 - x)/(3*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])) + (2*(1 - x)^2)/(9*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) - (1 - x)^3/(15*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1/3)*x^3*ArcSech[Sqrt[x]]} -{x^1*ArcSech[Sqrt[x]], x, 4, -((1 - x)/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])) + (1 - x)^2/(6*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x]) + (1/2)*x^2*ArcSech[Sqrt[x]]} -{x^0*ArcSech[Sqrt[x]], x, 3, -((1 - x)/(Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])) + x*ArcSech[Sqrt[x]]} -{ArcSech[Sqrt[x]]/x^1, x, 7, ArcSech[Sqrt[x]]^2 - 2*ArcSech[Sqrt[x]]*Log[1 + E^(2*ArcSech[Sqrt[x]])] - PolyLog[2, -E^(2*ArcSech[Sqrt[x]])]} -{ArcSech[Sqrt[x]]/x^2, x, 5, (1 - x)/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/x + (Sqrt[1 - x]*ArcTanh[Sqrt[1 - x]])/(2*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])} -{ArcSech[Sqrt[x]]/x^3, x, 6, (1 - x)/(8*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(5/2)) + (3*(1 - x))/(16*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/(2*x^2) + (3*Sqrt[1 - x]*ArcTanh[Sqrt[1 - x]])/(16*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])} -{ArcSech[Sqrt[x]]/x^4, x, 7, (1 - x)/(18*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(7/2)) + (5*(1 - x))/(72*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(5/2)) + (5*(1 - x))/(48*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*x^(3/2)) - ArcSech[Sqrt[x]]/(3*x^3) + (5*Sqrt[1 - x]*ArcTanh[Sqrt[1 - x]])/(48*Sqrt[-1 + 1/Sqrt[x]]*Sqrt[1 + 1/Sqrt[x]]*Sqrt[x])} - - -{ArcSech[1/x], x, 3, -(Sqrt[-1 + x]*Sqrt[1 + x]) + x*ArcCosh[x]} - - -{ArcSech[a*x^n]/x, x, 7, ArcSech[a*x^n]^2/(2*n) - (ArcSech[a*x^n]*Log[1 + E^(2*ArcSech[a*x^n])])/n - PolyLog[2, -E^(2*ArcSech[a*x^n])]/(2*n)} -{ArcSech[a*x^5]/x, x, 7, (1/10)*ArcSech[a*x^5]^2 - (1/5)*ArcSech[a*x^5]*Log[1 + E^(2*ArcSech[a*x^5])] - (1/10)*PolyLog[2, -E^(2*ArcSech[a*x^5])]} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse hyperbolic secants of exponentials*) - - -{ArcSech[c*E^(a + b*x)], x, 7, ArcSech[c*E^(a + b*x)]^2/(2*b) - (ArcSech[c*E^(a + b*x)]*Log[1 + E^(2*ArcSech[c*E^(a + b*x)])])/b - PolyLog[2, -E^(2*ArcSech[c*E^(a + b*x)])]/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse hyperbolic secants*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcSech[a x^p]*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{E^ArcSech[a*x]*x^4, x, 5, -((2*E^ArcSech[a*x]*x)/(15*a^4)) + x^2/(15*a^3) - (E^ArcSech[a*x]*x^3)/(15*a^2) + x^4/(20*a) + (1/5)*E^ArcSech[a*x]*x^5, x^4/(20*a) + (1/5)*E^ArcSech[a*x]*x^5 - (2*Sqrt[1 - a*x])/(15*a^5*Sqrt[1/(1 + a*x)]) - (x^2*Sqrt[1 - a*x])/(15*a^3*Sqrt[1/(1 + a*x)])} -{E^ArcSech[a*x]*x^3, x, 5, x^3/(12*a) + (1/4)*E^ArcSech[a*x]*x^4 - (x*Sqrt[1 - a*x])/(8*a^3*Sqrt[1/(1 + a*x)]) + (Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcSin[a*x])/(8*a^4)} -{E^ArcSech[a*x]*x^2, x, 3, -((E^ArcSech[a*x]*x)/(3*a^2)) + x^2/(6*a) + (1/3)*E^ArcSech[a*x]*x^3, x^2/(6*a) + (1/3)*E^ArcSech[a*x]*x^3 - Sqrt[1 - a*x]/(3*a^3*Sqrt[1/(1 + a*x)])} -{E^ArcSech[a*x]*x^1, x, 4, x/(2*a) + (1/2)*E^ArcSech[a*x]*x^2 + (Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcSin[a*x])/(2*a^2)} -{E^ArcSech[a*x]*x^0, x, 3, E^ArcSech[a*x]*x - ArcSech[a*x]/a + Log[x]/a, E^ArcSech[a*x]*x - (2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/a + Log[x]/a} -{E^ArcSech[a*x]/x^1, x, 5, -(2/(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + 2*ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]], -(1/(a*x)) - Sqrt[1 - a*x]/(a*x*Sqrt[1/(1 + a*x)]) - Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcSin[a*x]} -{E^ArcSech[a*x]/x^2, x, 6, -(E^ArcSech[a*x]/(2*x)) + a*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]], 1/(2*a*x^2) - E^ArcSech[a*x]/x + Sqrt[1 - a*x]/(2*a*x^2*Sqrt[1/(1 + a*x)]) + (1/2)*a*Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]]} -{E^ArcSech[a*x]/x^3, x, 5, -(1/(3*a*x^3)) - (8*a^2*((1 - a*x)/(1 + a*x))^(3/2))/(3*(1 - (1 - a*x)/(1 + a*x))^3), 1/(6*a*x^3) - E^ArcSech[a*x]/(2*x^2) + Sqrt[1 - a*x]/(6*a*x^3*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(3*x*Sqrt[1/(1 + a*x)])} -{E^ArcSech[a*x]/x^4, x, 8, 1/(12*a*x^4) - E^ArcSech[a*x]/(3*x^3) + Sqrt[1 - a*x]/(12*a*x^4*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(8*x^2*Sqrt[1/(1 + a*x)]) + (1/8)*a^3*Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]]} -{E^ArcSech[a*x]/x^5, x, 7, 1/(20*a*x^5) - E^ArcSech[a*x]/(4*x^4) + Sqrt[1 - a*x]/(20*a*x^5*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(15*x^3*Sqrt[1/(1 + a*x)]) + (2*a^3*Sqrt[1 - a*x])/(15*x*Sqrt[1/(1 + a*x)])} -{E^ArcSech[a*x]/x^6, x, 10, 1/(30*a*x^6) - E^ArcSech[a*x]/(5*x^5) + Sqrt[1 - a*x]/(30*a*x^6*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(24*x^4*Sqrt[1/(1 + a*x)]) + (a^3*Sqrt[1 - a*x])/(16*x^2*Sqrt[1/(1 + a*x)]) + (1/16)*a^5*Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]]} -{E^ArcSech[a*x]/x^7, x, 9, 1/(42*a*x^7) - E^ArcSech[a*x]/(6*x^6) + Sqrt[1 - a*x]/(42*a*x^7*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(35*x^5*Sqrt[1/(1 + a*x)]) + (4*a^3*Sqrt[1 - a*x])/(105*x^3*Sqrt[1/(1 + a*x)]) + (8*a^5*Sqrt[1 - a*x])/(105*x*Sqrt[1/(1 + a*x)])} -{E^ArcSech[a*x]/x^8, x, 12, 1/(56*a*x^8) - E^ArcSech[a*x]/(7*x^7) + Sqrt[1 - a*x]/(56*a*x^8*Sqrt[1/(1 + a*x)]) + (a*Sqrt[1 - a*x])/(48*x^6*Sqrt[1/(1 + a*x)]) + (5*a^3*Sqrt[1 - a*x])/(192*x^4*Sqrt[1/(1 + a*x)]) + (5*a^5*Sqrt[1 - a*x])/(128*x^2*Sqrt[1/(1 + a*x)]) + (5/128)*a^7*Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]]} - - -(* Mathematica 8 is unable to validate some of the following antiderivatives. *) -{E^ArcSech[a*x^2]*x^7, x, 6, x^6/(24*a) + (1/8)*E^ArcSech[a*x^2]*x^8 - (x^2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(16*a^3) + (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2])/(16*a^4)} -{E^ArcSech[a*x^2]*x^6, x, 5, (2*x^5)/(35*a) + (1/7)*E^ArcSech[a*x^2]*x^7 - (2*x*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(21*a^3) + (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/(21*a^(7/2))} -{E^ArcSech[a*x^2]*x^5, x, 4, x^4/(12*a) + (1/6)*E^ArcSech[a*x^2]*x^6 - Sqrt[1 - a*x^2]/(6*a^3*Sqrt[1/(1 + a*x^2)]), x^4/(12*a) + (1/6)*E^ArcSech[a*x^2]*x^6 - (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(6*a^3)} -{E^ArcSech[a*x^2]*x^4, x, 7, (2*x^3)/(15*a) + (1/5)*E^ArcSech[a*x^2]*x^5 + (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]*x], -1])/(5*a^(5/2)) - (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/(5*a^(5/2))} -{E^ArcSech[a*x^2]*x^3, x, 5, x^2/(4*a) + (1/4)*E^ArcSech[a*x^2]*x^4 + (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2])/(4*a^2)} -{E^ArcSech[a*x^2]*x^2, x, 4, (2*x)/(3*a) + (1/3)*E^ArcSech[a*x^2]*x^3 + (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/(3*a^(3/2))} -{E^ArcSech[a*x^2]*x^1, x, 6, (1/2)*E^ArcSech[a*x^2]*x^2 - (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]])/(2*a) + Log[x]/a} -{E^ArcSech[a*x^2]*x^0, x, 8, -(2/(a*x)) + E^ArcSech[a*x^2]*x - (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(a*x) - (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticE[ArcSin[Sqrt[a]*x], -1])/Sqrt[a] + (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1])/Sqrt[a]} -{E^ArcSech[a*x^2]/x^1, x, 5, -(1/(2*a*x^2)) - Sqrt[1 - a*x^2]/(2*a*x^2*Sqrt[1/(1 + a*x^2)]) - (1/2)*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2], -(1/(2*a*x^2)) - (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(2*a*x^2) - (1/2)*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2]} -{E^ArcSech[a*x^2]/x^2, x, 5, 2/(3*a*x^3) - E^ArcSech[a*x^2]/x + (2*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(3*a*x^3) - (2/3)*Sqrt[a]*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x], -1]} -{E^ArcSech[a*x^2]/x^3, x, 7, 1/(4*a*x^4) - E^ArcSech[a*x^2]/(2*x^2) + (Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(4*a*x^4) + (1/4)*a*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*ArcTanh[Sqrt[1 - a^2*x^4]]} - - -(* ::Subsubsection:: *) -(*p<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcSech[a x^p] with m symbolic*) - - -{E^ArcSech[a*x^3]*x^m, x, 4, -((3*x^(-2 + m))/(a*(2 + m - m^2))) + (E^ArcSech[a*x^3]*x^(1 + m))/(1 + m) - (3*x^(-2 + m)*Sqrt[1/(1 + a*x^3)]*Sqrt[1 + a*x^3]*Hypergeometric2F1[1/2, (1/6)*(-2 + m), (4 + m)/6, a^2*x^6])/(a*(2 + m - m^2))} -{E^ArcSech[a*x^2]*x^m, x, 4, -((2*x^(-1 + m))/(a*(1 - m^2))) + (E^ArcSech[a*x^2]*x^(1 + m))/(1 + m) - (2*x^(-1 + m)*Sqrt[1/(1 + a*x^2)]*Sqrt[1 + a*x^2]*Hypergeometric2F1[1/2, (1/4)*(-1 + m), (3 + m)/4, a^2*x^4])/(a*(1 - m^2))} -{E^ArcSech[a*x^1]*x^m, x, 4, x^m/(a*m*(1 + m)) + (E^ArcSech[a*x]*x^(1 + m))/(1 + m) + (x^m*Sqrt[1/(1 + a*x)]*Sqrt[1 + a*x]*Hypergeometric2F1[1/2, m/2, (2 + m)/2, a^2*x^2])/(a*m*(1 + m))} -{E^ArcSech[a/x^1]*x^m, x, 5, (E^ArcSech[a/x]*x^(1 + m))/(1 + m) - x^(2 + m)/(a*(2 + 3*m + m^2)) - (Sqrt[1/(1 + a/x)]*Sqrt[1 + a/x]*x^(2 + m)*Hypergeometric2F1[1/2, (1/2)*(-2 - m), -(m/2), a^2/x^2])/(a*(2 + 3*m + m^2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcSech[a x^p] with p symbolic*) - - -{E^ArcSech[a*x^p]*x^m, x, 4, (E^ArcSech[a*x^p]*x^(1 + m))/(1 + m) + (p*x^(1 + m - p))/(a*(1 + m)*(1 + m - p)) + (p*x^(1 + m - p)*Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*Hypergeometric2F1[1/2, (1 + m - p)/(2*p), (1 + m + p)/(2*p), a^2*x^(2*p)])/(a*(1 + m)*(1 + m - p))} - - -{E^ArcSech[a*x^p]*x^1, x, 4, (1/2)*E^ArcSech[a*x^p]*x^2 + (p*x^(2 - p))/(2*a*(2 - p)) + (p*x^(2 - p)*Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*Hypergeometric2F1[1/2, (1/2)*(-1 + 2/p), (1/2)*(1 + 2/p), a^2*x^(2*p)])/(2*a*(2 - p))} -{E^ArcSech[a*x^p]*x^0, x, 4, E^ArcSech[a*x^p]*x + (p*x^(1 - p))/(a*(1 - p)) + (p*x^(1 - p)*Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*Hypergeometric2F1[1/2, (1/2)*(-1 + 1/p), (1 + p)/(2*p), a^2*x^(2*p)])/(a*(1 - p))} -{E^ArcSech[a*x^p]/x^1, x, 6, -(1/(x^p*(a*p))) - Sqrt[1 - a*x^p]/(x^p*(a*p*Sqrt[1/(1 + a*x^p)])) - (Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*ArcSin[a*x^p])/p, -(1/(x^p*(a*p))) - (Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*Sqrt[1 - a^2*x^(2*p)])/(x^p*(a*p)) - (Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*ArcCsc[1/(x^p*a)])/p} -{E^ArcSech[a*x^p]/x^2, x, 4, -(E^ArcSech[a*x^p]/x) + (p*x^(-1 - p))/(a*(1 + p)) + (1/(a*(1 + p)))*(p*x^(-1 - p)*Sqrt[1/(1 + a*x^p)]*Sqrt[1 + a*x^p]*Hypergeometric2F1[1/2, -((1 + p)/(2*p)), -((1 - p)/(2*p)), a^2*x^(2*p)])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcSech[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -(* {E^(2*ArcSech[a*x])*x^m, x, 14, 0} *) - -{E^(2*ArcSech[a*x])*x^4, x, 9, (5*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2)/(4*a^5) + ((1 - a*x)*(1 + a*x)^4)/(5*a^5) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^4*(5 - 6*Sqrt[(1 - a*x)/(1 + a*x)]))/(10*a^5) + ((1 + a*x)*(4 - Sqrt[(1 - a*x)/(1 + a*x)]))/(4*a^5) - ((1 + a*x)^3*(4 + 45*Sqrt[(1 - a*x)/(1 + a*x)]))/(30*a^5) - ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/(2*a^5)} -{E^(2*ArcSech[a*x])*x^3, x, 8, -(x/a^3) + ((1 - a*x)*(1 + a*x)^3)/(4*a^4) + ((1 + a*x)^2*(3 - 8*Sqrt[(1 - a*x)/(1 + a*x)]))/(6*a^4) + (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^3*(4 - 3*Sqrt[(1 - a*x)/(1 + a*x)]))/(6*a^4)} -{E^(2*ArcSech[a*x])*x^2, x, 7, ((1 + a*x)*(1 - Sqrt[(1 - a*x)/(1 + a*x)])*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))/(2*a^3) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^2*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3)/(6*a^3) + ((1 + a*x)^3*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4)/(12*a^3) - (2*ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]])/a^3} -{E^(2*ArcSech[a*x])*x^1, x, 8, -((1 + a*x)^2/(2*a^2)) + ((1 + a*x)*(1 + 2*Sqrt[(1 - a*x)/(1 + a*x)]))/a^2 + (2*Log[1 + a*x])/a^2 + (4*Log[1 - Sqrt[(1 - a*x)/(1 + a*x)]])/a^2} -{E^(2*ArcSech[a*x])*x^0, x, 7, -x - 4/(a*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + (4*ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]])/a} -{E^(2*ArcSech[a*x])/x^1, x, 5, -(2/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) + 2/(1 - Sqrt[(1 - a*x)/(1 + a*x)]) - Log[1 + a*x] - 2*Log[1 - Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(2*ArcSech[a*x])/x^2, x, 4, -((4*a)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3)) + (2*a)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2} -{E^(2*ArcSech[a*x])/x^3, x, 5, -(a^2/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) + (2*a^2)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3 - (3*a^2)/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^2/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) + (1/2)*a^2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(2*ArcSech[a*x])/x^4, x, 4, -((4*a^3)/(5*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5)) + (2*a^3)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4 - (7*a^3)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) + (3*a^3)/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^3/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))} -{E^(2*ArcSech[a*x])/x^5, x, 5, -((2*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6)) + (2*a^4)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5 - (3*a^4)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4 + (8*a^4)/(3*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (11*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) + (3*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^4/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^4/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) + (1/4)*a^4*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(2*ArcSech[a*x])/x^6, x, 4, -((4*a^5)/(7*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^7)) + (2*a^5)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^6 - (18*a^5)/(5*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5) + (4*a^5)/(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4 - (35*a^5)/(12*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) + (11*a^5)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - a^5/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^5/(12*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) + a^5/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) - a^5/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -(* {E^(-ArcSech[a*x])*x^m, x, 7, 0} *) - -{E^(-ArcSech[a*x])*x^4, x, 8, -(x/a^4) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^5)/(5*a^5) + ((1 + a*x)^2*(9 + 4*Sqrt[(1 - a*x)/(1 + a*x)]))/(6*a^5) + ((1 + a*x)^4*(5 + 16*Sqrt[(1 - a*x)/(1 + a*x)]))/(20*a^5) - ((1 + a*x)^3*(15 + 17*Sqrt[(1 - a*x)/(1 + a*x)]))/(15*a^5)} -{E^(-ArcSech[a*x])*x^3, x, 7, -((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^4)/(4*a^4)) + ((1 + a*x)*(8 + Sqrt[(1 - a*x)/(1 + a*x)]))/(8*a^4) - ((1 + a*x)^2*(8 + 5*Sqrt[(1 - a*x)/(1 + a*x)]))/(8*a^4) + ((1 + a*x)^3*(4 + 9*Sqrt[(1 - a*x)/(1 + a*x)]))/(12*a^4) + ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/(4*a^4)} -{E^(-ArcSech[a*x])*x^2, x, 6, -(x/a^2) - (Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)^3)/(3*a^3) + ((1 + a*x)^2*(3 + 4*Sqrt[(1 - a*x)/(1 + a*x)]))/(6*a^3)} -{E^(-ArcSech[a*x])*x^1, x, 5, ((1 + a*x)^2*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2)/(4*a^2) + ((1 + a*x)*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))/(2*a^2) + ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]/a^2} -{E^(-ArcSech[a*x])*x^0, x, 6, -((Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x))/a) + Log[1 + a*x]/a + (2*Log[1 + Sqrt[(1 - a*x)/(1 + a*x)]])/a} -{E^(-ArcSech[a*x])/x^1, x, 5, -(2/(1 + Sqrt[(1 - a*x)/(1 + a*x)])) - 2*ArcTan[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(-ArcSech[a*x])/x^2, x, 5, -(a/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2) + a/(1 + Sqrt[(1 - a*x)/(1 + a*x)]) - a*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(-ArcSech[a*x])/x^3, x, 4, -(a^2/(2*(1 - Sqrt[(1 - a*x)/(1 + a*x)]))) - (2*a^2)/(3*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) + a^2/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - a^2/(2*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))} -{E^(-ArcSech[a*x])/x^4, x, 5, -(a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2)) + a^3/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^3/(2*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) + a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3 - a^3/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 + a^3/(2*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) - (1/4)*a^3*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(-ArcSech[a*x])/x^5, x, 4, -(a^4/(6*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3)) + a^4/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - (3*a^4)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - (2*a^4)/(5*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5) + a^4/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4 - (4*a^4)/(3*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) + a^4/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - (3*a^4)/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))} -{E^(-ArcSech[a*x])/x^6, x, 5, -(a^5/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4)) + a^5/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) - (3*a^5)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) + a^5/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - a^5/(3*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6) + a^5/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5 - (13*a^5)/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) + (19*a^5)/(12*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) - a^5/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 + (3*a^5)/(8*(1 + Sqrt[(1 - a*x)/(1 + a*x)])) - (1/8)*a^5*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]]} -{E^(-ArcSech[a*x])/x^7, x, 4, -(a^6/(10*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^5)) + a^6/(4*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^4) - (5*a^6)/(12*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^3) + (3*a^6)/(8*(1 - Sqrt[(1 - a*x)/(1 + a*x)])^2) - (5*a^6)/(16*(1 - Sqrt[(1 - a*x)/(1 + a*x)])) - (2*a^6)/(7*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^7) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^6 - (19*a^6)/(10*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^5) + (9*a^6)/(4*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^4) - (11*a^6)/(6*(1 + Sqrt[(1 - a*x)/(1 + a*x)])^3) + a^6/(1 + Sqrt[(1 - a*x)/(1 + a*x)])^2 - (5*a^6)/(16*(1 + Sqrt[(1 - a*x)/(1 + a*x)]))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcSech[a x]) / (1-a^2 x^2)*) - - -{(d*x)^m*E^ArcSech[c*x]/(1 - c^2*x^2), x, 5, ((d*x)^m*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*Hypergeometric2F1[1/2, m/2, (2 + m)/2, c^2*x^2])/(c*m) + ((d*x)^m*Hypergeometric2F1[1, m/2, (2 + m)/2, c^2*x^2])/(c*m)} - - -{x^4*E^ArcSech[c*x]/(1 - c^2*x^2), x, 8, -(x^2/(2*c^3)) - (2*Sqrt[1 - c*x])/(3*c^5*Sqrt[1/(1 + c*x)]) - (x^2*Sqrt[1 - c*x])/(3*c^3*Sqrt[1/(1 + c*x)]) - Log[1 - c^2*x^2]/(2*c^5)} -{x^3*E^ArcSech[c*x]/(1 - c^2*x^2), x, 7, -(x/c^3) - (x*Sqrt[1 - c*x])/(2*c^3*Sqrt[1/(1 + c*x)]) + (Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/(2*c^4) + ArcTanh[c*x]/c^4} -{x^2*E^ArcSech[c*x]/(1 - c^2*x^2), x, 4, -(Sqrt[1 - c*x]/(c^3*Sqrt[1/(1 + c*x)])) - Log[1 - c^2*x^2]/(2*c^3)} -{x^1*E^ArcSech[c*x]/(1 - c^2*x^2), x, 5, (Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcSin[c*x])/c^2 + ArcTanh[c*x]/c^2} -{x^0*E^ArcSech[c*x]/(1 - c^2*x^2), x, 8, -((Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]])/c) + Log[x]/c - Log[1 - c^2*x^2]/(2*c)} -{E^ArcSech[c*x]/(x^1*(1 - c^2*x^2)), x, 5, -(1/(c*x)) - Sqrt[1 - c*x]/(c*x*Sqrt[1/(1 + c*x)]) + ArcTanh[c*x]} -{E^ArcSech[c*x]/(x^2*(1 - c^2*x^2)), x, 9, -(1/(2*c*x^2)) - Sqrt[1 - c*x]/(2*c*x^2*Sqrt[1/(1 + c*x)]) - (1/2)*c*Sqrt[1/(1 + c*x)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c*x]*Sqrt[1 + c*x]] + c*Log[x] - (1/2)*c*Log[1 - c^2*x^2]} -{E^ArcSech[c*x]/(x^3*(1 - c^2*x^2)), x, 8, -(1/(3*c*x^3)) - c/x - Sqrt[1 - c*x]/(3*c*x^3*Sqrt[1/(1 + c*x)]) - (2*c*Sqrt[1 - c*x])/(3*x*Sqrt[1/(1 + c*x)]) + c^2*ArcTanh[c*x]} - - -{x*(a*x*E^ArcSech[a*x] - 1)/(1 - a^2*x^2), x, 7, -(E^ArcSech[a*x]*x)/a, -(Sqrt[1 - a*x]/(a^2*Sqrt[1/(1 + a*x)]))} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic secants*) - - -{ArcSech[a + b*x]/((a*d)/b + d*x), x, 8, ArcSech[a + b*x]^2/(2*d) - (ArcSech[a + b*x]*Log[1 + E^(2*ArcSech[a + b*x])])/d - PolyLog[2, -E^(2*ArcSech[a + b*x])]/(2*d)} - - -{x^3*ArcSech[a + b*x^4], x, 5, ((a + b*x^4)*ArcSech[a + b*x^4])/(4*b) - ArcTan[Sqrt[(1 - a - b*x^4)/(1 + a + b*x^4)]]/(2*b)} - -{x^(n-1)*ArcSech[a + b*x^n], x, 5, ((a + b*x^n)*ArcSech[a + b*x^n])/(b*n) - (2*ArcTan[Sqrt[(1 - a - b*x^n)/(1 + a + b*x^n)]])/(b*n)} diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.1 u (a+b arccsch(c x))^n.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.1 u (a+b arccsch(c x))^n.m deleted file mode 100644 index fa0e54c..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.1 u (a+b arccsch(c x))^n.m +++ /dev/null @@ -1,342 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integrands of the form u (a+b ArcCsch[c x])^n*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCsch[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b ArcCsch[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^6*(a + b*ArcCsch[c*x]), x, 7, (5*b*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(112*c^5) - (5*b*Sqrt[1 + 1/(c^2*x^2)]*x^4)/(168*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^6)/(42*c) + (x^7*(a + b*ArcCsch[c*x]))/7 - (5*b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(112*c^7)} -{x^5*(a + b*ArcCsch[c*x]), x, 4, (4*b*Sqrt[1 + 1/(c^2*x^2)]*x)/(45*c^5) - (2*b*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(45*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^5)/(30*c) + (x^6*(a + b*ArcCsch[c*x]))/6} -{x^4*(a + b*ArcCsch[c*x]), x, 6, (-3*b*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(40*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^4)/(20*c) + (x^5*(a + b*ArcCsch[c*x]))/5 + (3*b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(40*c^5)} -{x^3*(a + b*ArcCsch[c*x]), x, 3, -(b*Sqrt[1 + 1/(c^2*x^2)]*x)/(6*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(12*c) + (x^4*(a + b*ArcCsch[c*x]))/4} -{x^2*(a + b*ArcCsch[c*x]), x, 5, (b*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(6*c) + (x^3*(a + b*ArcCsch[c*x]))/3 - (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(6*c^3)} -{x^1*(a + b*ArcCsch[c*x]), x, 2, (b*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c) + (x^2*(a + b*ArcCsch[c*x]))/2} -{x^0*(a + b*ArcCsch[c*x]), x, 5, a*x + b*x*ArcCsch[c*x] + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c} -{(a + b*ArcCsch[c*x])/x^1, x, 6, -((a + b*ArcCsch[c*x])^2/(2*b)) - (a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])] + (1/2)*b*PolyLog[2, E^(-2*ArcCsch[c*x])]} -{(a + b*ArcCsch[c*x])/x^2, x, 2, b*c*Sqrt[1 + 1/(c^2*x^2)] - (a + b*ArcCsch[c*x])/x} -{(a + b*ArcCsch[c*x])/x^3, x, 4, (b*c*Sqrt[1 + 1/(c^2*x^2)])/(4*x) - (b*c^2*ArcCsch[c*x])/4 - (a + b*ArcCsch[c*x])/(2*x^2)} -{(a + b*ArcCsch[c*x])/x^4, x, 4, -(b*c^3*Sqrt[1 + 1/(c^2*x^2)])/3 + (b*c^3*(1 + 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsch[c*x])/(3*x^3)} -{(a + b*ArcCsch[c*x])/x^5, x, 5, (b*c*Sqrt[1 + 1/(c^2*x^2)])/(16*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(32*x) + (3*b*c^4*ArcCsch[c*x])/32 - (a + b*ArcCsch[c*x])/(4*x^4)} -{(a + b*ArcCsch[c*x])/x^6, x, 4, (b*c^5*Sqrt[1 + 1/(c^2*x^2)])/5 - (2*b*c^5*(1 + 1/(c^2*x^2))^(3/2))/15 + (b*c^5*(1 + 1/(c^2*x^2))^(5/2))/25 - (a + b*ArcCsch[c*x])/(5*x^5)} -{(a + b*ArcCsch[c*x])/x^7, x, 6, (b*c*Sqrt[1 + 1/(c^2*x^2)])/(36*x^5) - (5*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(144*x^3) + (5*b*c^5*Sqrt[1 + 1/(c^2*x^2)])/(96*x) - (5*b*c^6*ArcCsch[c*x])/96 - (a + b*ArcCsch[c*x])/(6*x^6)} - - -{x^3*(a + b*ArcCsch[c*x])^2, x, 5, (b^2*x^2)/(12*c^2) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x]))/(3*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3*(a + b*ArcCsch[c*x]))/(6*c) + (x^4*(a + b*ArcCsch[c*x])^2)/4 - (b^2*Log[x])/(3*c^4)} -{x^2*(a + b*ArcCsch[c*x])^2, x, 8, (b^2*x)/(3*c^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x]))/(3*c) + (x^3*(a + b*ArcCsch[c*x])^2)/3 - (2*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/(3*c^3) - (b^2*PolyLog[2, -E^ArcCsch[c*x]])/(3*c^3) + (b^2*PolyLog[2, E^ArcCsch[c*x]])/(3*c^3)} -{x^1*(a + b*ArcCsch[c*x])^2, x, 4, (b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x]))/c + (x^2*(a + b*ArcCsch[c*x])^2)/2 + (b^2*Log[x])/c^2} -{x^0*(a + b*ArcCsch[c*x])^2, x, 7, x*(a + b*ArcCsch[c*x])^2 + (4*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/c + (2*b^2*PolyLog[2, -E^ArcCsch[c*x]])/c - (2*b^2*PolyLog[2, E^ArcCsch[c*x]])/c} -{(a + b*ArcCsch[c*x])^2/x^1, x, 6, (a + b*ArcCsch[c*x])^3/(3*b) - (a + b*ArcCsch[c*x])^2*Log[1 - E^(2*ArcCsch[c*x])] - b*(a + b*ArcCsch[c*x])*PolyLog[2, E^(2*ArcCsch[c*x])] + (b^2*PolyLog[3, E^(2*ArcCsch[c*x])])/2} -{(a + b*ArcCsch[c*x])^2/x^2, x, 4, (-2*b^2)/x + 2*b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]) - (a + b*ArcCsch[c*x])^2/x} -{(a + b*ArcCsch[c*x])^2/x^3, x, 4, -b^2/(4*x^2) - (a*b*c^2*ArcCsch[c*x])/2 - (b^2*c^2*ArcCsch[c*x]^2)/4 + (b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(2*x) - (a + b*ArcCsch[c*x])^2/(2*x^2)} -{(a + b*ArcCsch[c*x])^2/x^4, x, 5, (-2*b^2)/(27*x^3) + (4*b^2*c^2)/(9*x) - (4*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/9 + (2*b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(9*x^2) - (a + b*ArcCsch[c*x])^2/(3*x^3)} -{(a + b*ArcCsch[c*x])^2/x^5, x, 5, -b^2/(32*x^4) + (3*b^2*c^2)/(32*x^2) + (3*a*b*c^4*ArcCsch[c*x])/16 + (3*b^2*c^4*ArcCsch[c*x]^2)/32 + (b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(8*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x]))/(16*x) - (a + b*ArcCsch[c*x])^2/(4*x^4)} - - -{x^3*(a + b*ArcCsch[c*x])^3, x, 10, (b^3*Sqrt[1 + 1/(c^2*x^2)]*x)/(4*c^3) + (b^2*x^2*(a + b*ArcCsch[c*x]))/(4*c^2) - (b*(a + b*ArcCsch[c*x])^2)/(2*c^4) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x])^2)/(2*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3*(a + b*ArcCsch[c*x])^2)/(4*c) + (x^4*(a + b*ArcCsch[c*x])^3)/4 + (b^2*(a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/c^4 + (b^3*PolyLog[2, E^(2*ArcCsch[c*x])])/(2*c^4)} -{x^2*(a + b*ArcCsch[c*x])^3, x, 11, (b^2*x*(a + b*ArcCsch[c*x]))/c^2 + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x])^2)/(2*c) + (x^3*(a + b*ArcCsch[c*x])^3)/3 - (b*(a + b*ArcCsch[c*x])^2*ArcTanh[E^ArcCsch[c*x]])/c^3 + (b^3*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c^3 - (b^2*(a + b*ArcCsch[c*x])*PolyLog[2, -E^ArcCsch[c*x]])/c^3 + (b^2*(a + b*ArcCsch[c*x])*PolyLog[2, E^ArcCsch[c*x]])/c^3 + (b^3*PolyLog[3, -E^ArcCsch[c*x]])/c^3 - (b^3*PolyLog[3, E^ArcCsch[c*x]])/c^3} -{x^1*(a + b*ArcCsch[c*x])^3, x, 7, (3*b*(a + b*ArcCsch[c*x])^2)/(2*c^2) + (3*b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x])^2)/(2*c) + (x^2*(a + b*ArcCsch[c*x])^3)/2 - (3*b^2*(a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/c^2 - (3*b^3*PolyLog[2, E^(2*ArcCsch[c*x])])/(2*c^2)} -{x^0*(a + b*ArcCsch[c*x])^3, x, 9, x*(a + b*ArcCsch[c*x])^3 + (6*b*(a + b*ArcCsch[c*x])^2*ArcTanh[E^ArcCsch[c*x]])/c + (6*b^2*(a + b*ArcCsch[c*x])*PolyLog[2, -E^ArcCsch[c*x]])/c - (6*b^2*(a + b*ArcCsch[c*x])*PolyLog[2, E^ArcCsch[c*x]])/c - (6*b^3*PolyLog[3, -E^ArcCsch[c*x]])/c + (6*b^3*PolyLog[3, E^ArcCsch[c*x]])/c} -{(a + b*ArcCsch[c*x])^3/x^1, x, 7, (a + b*ArcCsch[c*x])^4/(4*b) - (a + b*ArcCsch[c*x])^3*Log[1 - E^(2*ArcCsch[c*x])] - (3*b*(a + b*ArcCsch[c*x])^2*PolyLog[2, E^(2*ArcCsch[c*x])])/2 + (3*b^2*(a + b*ArcCsch[c*x])*PolyLog[3, E^(2*ArcCsch[c*x])])/2 - (3*b^3*PolyLog[4, E^(2*ArcCsch[c*x])])/4} -{(a + b*ArcCsch[c*x])^3/x^2, x, 5, 6*b^3*c*Sqrt[1 + 1/(c^2*x^2)] - (6*b^2*(a + b*ArcCsch[c*x]))/x + 3*b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2 - (a + b*ArcCsch[c*x])^3/x} -{(a + b*ArcCsch[c*x])^3/x^3, x, 6, (3*b^3*c*Sqrt[1 + 1/(c^2*x^2)])/(8*x) - (3*b^3*c^2*ArcCsch[c*x])/8 - (3*b^2*(a + b*ArcCsch[c*x]))/(4*x^2) + (3*b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(4*x) - (c^2*(a + b*ArcCsch[c*x])^3)/4 - (a + b*ArcCsch[c*x])^3/(2*x^2)} -{(a + b*ArcCsch[c*x])^3/x^4, x, 8, (-14*b^3*c^3*Sqrt[1 + 1/(c^2*x^2)])/9 + (2*b^3*c^3*(1 + 1/(c^2*x^2))^(3/2))/27 - (2*b^2*(a + b*ArcCsch[c*x]))/(9*x^3) + (4*b^2*c^2*(a + b*ArcCsch[c*x]))/(3*x) - (2*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/3 + (b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(3*x^2) - (a + b*ArcCsch[c*x])^3/(3*x^3)} -{(a + b*ArcCsch[c*x])^3/x^5, x, 10, (3*b^3*c*Sqrt[1 + 1/(c^2*x^2)])/(128*x^3) - (45*b^3*c^3*Sqrt[1 + 1/(c^2*x^2)])/(256*x) + (45*b^3*c^4*ArcCsch[c*x])/256 - (3*b^2*(a + b*ArcCsch[c*x]))/(32*x^4) + (9*b^2*c^2*(a + b*ArcCsch[c*x]))/(32*x^2) + (3*b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(16*x^3) - (9*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(32*x) + (3*c^4*(a + b*ArcCsch[c*x])^3)/32 - (a + b*ArcCsch[c*x])^3/(4*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^1/(a + b*ArcCsch[c*x]), x, 0, Unintegrable[x/(a + b*ArcCsch[c*x]), x]} -{x^0/(a + b*ArcCsch[c*x]), x, 0, Unintegrable[1/(a + b*ArcCsch[c*x]), x]} -{1/(x^1*(a + b*ArcCsch[c*x])), x, 0, Unintegrable[1/(x*(a + b*ArcCsch[c*x])), x]} -{1/(x^2*(a + b*ArcCsch[c*x])), x, 4, -((c*Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]])/b) + (c*Sinh[a/b]*SinhIntegral[a/b + ArcCsch[c*x]])/b} -{1/(x^3*(a + b*ArcCsch[c*x])), x, 6, (c^2*CoshIntegral[(2*a)/b + 2*ArcCsch[c*x]]*Sinh[(2*a)/b])/(2*b) - (c^2*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCsch[c*x]])/(2*b)} -{1/(x^4*(a + b*ArcCsch[c*x])), x, 9, (c^3*Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]])/(4*b) - (c^3*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcCsch[c*x]])/(4*b) - (c^3*Sinh[a/b]*SinhIntegral[a/b + ArcCsch[c*x]])/(4*b) + (c^3*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCsch[c*x]])/(4*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m (a+b ArcCsch[c x])^n with m symbolic*) - - -{(d*x)^m*(a + b*ArcCsch[c*x])^3, x, 0, Unintegrable[(d*x)^m*(a + b*ArcCsch[c*x])^3, x]} -{(d*x)^m*(a + b*ArcCsch[c*x])^2, x, 0, Unintegrable[(d*x)^m*(a + b*ArcCsch[c*x])^2, x]} -{(d*x)^m*(a + b*ArcCsch[c*x]), x, 3, ((d*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(d*(1 + m)) + (b*(d*x)^m*Hypergeometric2F1[1/2, -(m/2), 1 - m/2, -(1/(c^2*x^2))])/(c*m*(1 + m))} -{(d*x)^m/(a + b*ArcCsch[c*x]), x, 0, Unintegrable[(d*x)^m/(a + b*ArcCsch[c*x]), x]} -{(d*x)^m/(a + b*ArcCsch[c*x])^2, x, 0, Unintegrable[(d*x)^m/(a + b*ArcCsch[c*x])^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x)^m (a+b ArcCsch[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m (a+b ArcCsch[c x])^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(d + e*x)^3*(a + b*ArcCsch[c*x]), x, 11, (b*e*(9*c^2*d^2 - e^2)*Sqrt[1 + 1/(c^2*x^2)]*x)/(6*c^3) + (b*d*e^2*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(2*c) + (b*e^3*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(12*c) - (b*d^4*ArcCsch[c*x])/(4*e) + ((d + e*x)^4*(a + b*ArcCsch[c*x]))/(4*e) + (b*d*(2*c^2*d^2 - e^2)*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(2*c^3)} -{(d + e*x)^2*(a + b*ArcCsch[c*x]), x, 10, (b*d*e*Sqrt[1 + 1/(c^2*x^2)]*x)/c + (b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(6*c) - (b*d^3*ArcCsch[c*x])/(3*e) + ((d + e*x)^3*(a + b*ArcCsch[c*x]))/(3*e) + (b*(6*c^2*d^2 - e^2)*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(6*c^3)} -{(d + e*x)*(a + b*ArcCsch[c*x]), x, 9, (b*e*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c) - (b*d^2*ArcCsch[c*x])/(2*e) + ((d + e*x)^2*(a + b*ArcCsch[c*x]))/(2*e) + (b*d*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c} -{a + b*ArcCsch[c*x], x, 5, a*x + b*x*ArcCsch[c*x] + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c} -{(a + b*ArcCsch[c*x])/(d + e*x), x, 4, ((a + b*ArcCsch[c*x])*Log[1 - ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e + ((a + b*ArcCsch[c*x])*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e - ((a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/e + (b*PolyLog[2, ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e + (b*PolyLog[2, ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)])/e - (b*PolyLog[2, E^(2*ArcCsch[c*x])])/(2*e)} -{(a + b*ArcCsch[c*x])/(d + e*x)^2, x, 7, (b*ArcCsch[c*x])/(d*e) - (a + b*ArcCsch[c*x])/(e*(d + e*x)) + (b*ArcTanh[(c^2*d - e/x)/(c*Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])])/(d*Sqrt[c^2*d^2 + e^2])} -{(a + b*ArcCsch[c*x])/(d + e*x)^3, x, 8, -(b*c*e*Sqrt[1 + 1/(c^2*x^2)])/(2*d*(c^2*d^2 + e^2)*(e + d/x)) + (b*ArcCsch[c*x])/(2*d^2*e) - (a + b*ArcCsch[c*x])/(2*e*(d + e*x)^2) + (b*(2*c^2*d^2 + e^2)*ArcTanh[(c^2*d - e/x)/(c*Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])])/(2*d^2*(c^2*d^2 + e^2)^(3/2))} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x)^(p/2) (a+b ArcCsch[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -(* {x^3*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]), x, 39, -((4*b*(25/c^2 + (6*d^2)/e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/(945*c)) + (8*b*d^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/(315*c*e^2) - (32*b*d*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x)^(3/2))/(315*c*e^2) + (4*b*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x)^(5/2))/(63*c*e^2) - (2*d^3*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^4) + (6*d^2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^4) - (6*d*(d + e*x)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^4) + (2*(d + e*x)^(9/2)*(a + b*ArcCsch[c*x]))/(9*e^4) + (64*b*Sqrt[-c^2]*d^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(315*c*e^3*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]) - (8*b*c*d*(c^2*d^2 - 3*e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(3/2)*e^3*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]) - (8*b*c*d*(3*c^2*d^2 + 41*e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(945*(-c^2)^(3/2)*e^3*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]) - (64*b*Sqrt[-c^2]*d^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(315*c*e^3*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]) - (8*b*c*d^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(315*(-c^2)^(3/2)*e^3*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]) - (4*b*c*(6*c^4*d^4 + 31*c^2*d^2*e^2 + 25*e^4)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(945*(-c^2)^(5/2)*e^3*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]) + (64*b*c*d^5*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(315*e^4*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2])} *) -{x^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]), x, If[$VersionNumber>=8, 31, 27], If[$VersionNumber>=8, -((4*b*d*Sqrt[d + e*x]*(1 + c^2*x^2))/(105*c^3*e*Sqrt[1 + 1/(c^2*x^2)]*x)) + (4*b*(d + e*x)^(3/2)*(1 + c^2*x^2))/(35*c^3*e*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^3) - (32*b*c*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (4*b*c*(c^2*d^2 - 3*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(35*(-c^2)^(5/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (32*b*c*d^3*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*c*d*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(5/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^4*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(105*c*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]), (4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(35*c^3*Sqrt[1 + 1/(c^2*x^2)]) + (8*b*d*Sqrt[d + e*x]*(1 + c^2*x^2))/(105*c^3*e*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^3) - (4*b*c*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(35*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*(2*c^2*d^2 + 9*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(5/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (32*b*c*d^3*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*c*d*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(5/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^4*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(105*c*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])]} -{x^1*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]), x, If[$VersionNumber>=8, 24, 20], (4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*Sqrt[1 + 1/(c^2*x^2)]*x) - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) + (8*b*c*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (8*b*c*d^2*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*c*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(5/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(15*c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]), x, 15, (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e) + (4*b*c*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*c*d*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{Sqrt[d + e*x]*(a + b*ArcCsch[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x, x]} -{Sqrt[d + e*x]*(a + b*ArcCsch[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2, x]} - - -{(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]), x, 22, (4*b*e*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e) + (28*b*c*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) - (4*b*c*(2*c^2*d^2 - e^2)*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(5/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^3*(a + b*ArcCsch[c*x])/Sqrt[d + e*x], x, 27, (4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(35*c^3*e*Sqrt[1 + 1/(c^2*x^2)]) - (4*b*d*Sqrt[d + e*x]*(1 + c^2*x^2))/(21*c^3*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) - (2*d^3*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^4) + (24*b*c*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*(2*c^2*d^2 + 9*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (64*b*c*d^3*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*c*d*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^4*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(35*c*e^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsch[c*x])/Sqrt[d + e*x], x, 20, (4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*e*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*d^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3) - (4*b*c*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(5*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (32*b*c*d^2*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*c*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(5/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(15*c*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsch[c*x])/Sqrt[d + e*x], x, 14, -((2*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2) + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + (4*b*c*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (8*b*c*d*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (8*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsch[c*x])/Sqrt[d + e*x], x, 9, (2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e + (4*b*c*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/((-c^2)^(3/2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsch[c*x])/(x^1*Sqrt[d + e*x]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x]), x]} -{(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x]), x]} - - -{x^3*(a + b*ArcCsch[c*x])/(d + e*x)^(3/2), x, 23, (4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*d^3*(a + b*ArcCsch[c*x]))/(e^4*Sqrt[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/e^4 + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^4) - (32*b*c*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (8*b*c*d^2*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/((-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*c*(2*c^2*d^2 - e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (64*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*e^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsch[c*x])/(d + e*x)^(3/2), x, 16, -((2*d^2*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x])) - (4*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + (4*b*c*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (20*b*c*d*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (32*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsch[c*x])/(d + e*x)^(3/2), x, 11, (2*d*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (4*b*c*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/((-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (8*b*d*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsch[c*x])/(d + e*x)^(3/2), x, 6, -((2*(a + b*ArcCsch[c*x]))/(e*Sqrt[d + e*x])) + (4*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsch[c*x])/(x^1*(d + e*x)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(3/2)), x]} -{(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)), x]} - - -{x^3*(a + b*ArcCsch[c*x])/(d + e*x)^(5/2), x, 31, (4*b*d^2*(1 + c^2*x^2))/(3*c*e^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (2*d^3*(a + b*ArcCsch[c*x]))/(3*e^4*(d + e*x)^(3/2)) - (6*d^2*(a + b*ArcCsch[c*x]))/(e^4*Sqrt[d + e*x]) - (6*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 + (2*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^4) - (8*b*Sqrt[-c^2]*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*c*e^3*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*(2*c^2*d^2 + e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e^3*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (32*b*c*d*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^2*(a + b*ArcCsch[c*x])/(d + e*x)^(5/2), x, 25, -((4*b*d*(1 + c^2*x^2))/(3*c*e*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])) - (2*d^2*(a + b*ArcCsch[c*x]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 + (4*b*Sqrt[-c^2]*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*c*e^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/((-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^1*(a + b*ArcCsch[c*x])/(d + e*x)^(5/2), x, 19, (4*b*(1 + c^2*x^2))/(3*c*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (2*d*(a + b*ArcCsch[c*x]))/(3*e^2*(d + e*x)^(3/2)) - (2*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) - (4*b*Sqrt[-c^2]*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*c*e*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (8*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{x^0*(a + b*ArcCsch[c*x])/(d + e*x)^(5/2), x, 12, -((4*b*e*(1 + c^2*x^2))/(3*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])) - (2*(a + b*ArcCsch[c*x]))/(3*e*(d + e*x)^(3/2)) + (4*b*Sqrt[-c^2]*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(3*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*d*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} -{(a + b*ArcCsch[c*x])/(x^1*(d + e*x)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(5/2)), x]} -{(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(5/2)), x]} - - -{(a + b*ArcCsch[c*x])/(d + e*x)^(7/2), x, 19, -((4*b*e*(1 + c^2*x^2))/(15*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 + c^2*x^2))/(15*(c^2*d^2 + e^2)^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*e*(1 + c^2*x^2))/(5*c*d^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsch[c*x]))/(5*e*(d + e*x)^(5/2)) - (4*b*c*(7*c^2*d^2 + 3*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*Sqrt[-c^2]*e)/((-c^2)*d + Sqrt[-c^2]*e)])/(15*Sqrt[-c^2]*d^2*(c^2*d^2 + e^2)^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) - (4*b*Sqrt[-c^2]*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*d^2*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]), -((4*b*e*(1 + c^2*x^2))/(15*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x)^(3/2))) - (16*b*c*e*(1 + c^2*x^2))/(15*(c^2*d^2 + e^2)^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*e*(1 + c^2*x^2))/(5*c*d^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsch[c*x]))/(5*e*(d + e*x)^(5/2)) + (16*b*c*Sqrt[-c^2]*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*(c^2*d^2 + e^2)^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*Sqrt[-c^2]*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(5*c*d^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]) - (4*b*Sqrt[-c^2]*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], -((2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e))])/(15*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (4*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(5*c*d^2*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCsch[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^p (a+b ArcCsch[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^4*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 7, -((b*(42*c^2*d - 25*e)*x^2*Sqrt[-1 - c^2*x^2])/(560*c^5*Sqrt[(-c^2)*x^2])) + (b*(42*c^2*d - 25*e)*x^4*Sqrt[-1 - c^2*x^2])/(840*c^3*Sqrt[(-c^2)*x^2]) + (b*e*x^6*Sqrt[-1 - c^2*x^2])/(42*c*Sqrt[(-c^2)*x^2]) + (1/5)*d*x^5*(a + b*ArcCsch[c*x]) + (1/7)*e*x^7*(a + b*ArcCsch[c*x]) - (b*(42*c^2*d - 25*e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(560*c^6*Sqrt[(-c^2)*x^2])} -{x^2*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 6, (b*(20*c^2*d - 9*e)*x^2*Sqrt[-1 - c^2*x^2])/(120*c^3*Sqrt[(-c^2)*x^2]) + (b*e*x^4*Sqrt[-1 - c^2*x^2])/(20*c*Sqrt[(-c^2)*x^2]) + (1/3)*d*x^3*(a + b*ArcCsch[c*x]) + (1/5)*e*x^5*(a + b*ArcCsch[c*x]) + (b*(20*c^2*d - 9*e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(120*c^4*Sqrt[(-c^2)*x^2])} -{(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 5, (b*e*x^2*Sqrt[-1 - c^2*x^2])/(6*c*Sqrt[(-c^2)*x^2]) + d*x*(a + b*ArcCsch[c*x]) + (1/3)*e*x^3*(a + b*ArcCsch[c*x]) - (b*(6*c^2*d - e)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(6*c^2*Sqrt[(-c^2)*x^2])} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^2, x, 4, (b*c*d*Sqrt[-1 - c^2*x^2])/Sqrt[(-c^2)*x^2] - (d*(a + b*ArcCsch[c*x]))/x + e*x*(a + b*ArcCsch[c*x]) - (b*e*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/Sqrt[(-c^2)*x^2]} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^4, x, 4, -((b*c*(2*c^2*d - 9*e)*Sqrt[-1 - c^2*x^2])/(9*Sqrt[(-c^2)*x^2])) + (b*c*d*Sqrt[-1 - c^2*x^2])/(9*x^2*Sqrt[(-c^2)*x^2]) - (d*(a + b*ArcCsch[c*x]))/(3*x^3) - (e*(a + b*ArcCsch[c*x]))/x} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6, x, 5, (2*b*c^3*(12*c^2*d - 25*e)*Sqrt[-1 - c^2*x^2])/(225*Sqrt[(-c^2)*x^2]) + (b*c*d*Sqrt[-1 - c^2*x^2])/(25*x^4*Sqrt[(-c^2)*x^2]) - (b*c*(12*c^2*d - 25*e)*Sqrt[-1 - c^2*x^2])/(225*x^2*Sqrt[(-c^2)*x^2]) - (d*(a + b*ArcCsch[c*x]))/(5*x^5) - (e*(a + b*ArcCsch[c*x]))/(3*x^3)} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8, x, 6, -((8*b*c^5*(30*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(3675*Sqrt[(-c^2)*x^2])) + (b*c*d*Sqrt[-1 - c^2*x^2])/(49*x^6*Sqrt[(-c^2)*x^2]) - (b*c*(30*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(1225*x^4*Sqrt[(-c^2)*x^2]) + (4*b*c^3*(30*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(3675*x^2*Sqrt[(-c^2)*x^2]) - (d*(a + b*ArcCsch[c*x]))/(7*x^7) - (e*(a + b*ArcCsch[c*x]))/(5*x^5)} - -{x^5*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 5, (b*(4*c^2*d - 3*e)*x*Sqrt[-1 - c^2*x^2])/(24*c^7*Sqrt[(-c^2)*x^2]) + (b*(8*c^2*d - 9*e)*x*(-1 - c^2*x^2)^(3/2))/(72*c^7*Sqrt[(-c^2)*x^2]) + (b*(4*c^2*d - 9*e)*x*(-1 - c^2*x^2)^(5/2))/(120*c^7*Sqrt[(-c^2)*x^2]) - (b*e*x*(-1 - c^2*x^2)^(7/2))/(56*c^7*Sqrt[(-c^2)*x^2]) + (1/6)*d*x^6*(a + b*ArcCsch[c*x]) + (1/8)*e*x^8*(a + b*ArcCsch[c*x])} -{x^3*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 5, -((b*(3*c^2*d - 2*e)*x*Sqrt[-1 - c^2*x^2])/(12*c^5*Sqrt[(-c^2)*x^2])) - (b*(3*c^2*d - 4*e)*x*(-1 - c^2*x^2)^(3/2))/(36*c^5*Sqrt[(-c^2)*x^2]) + (b*e*x*(-1 - c^2*x^2)^(5/2))/(30*c^5*Sqrt[(-c^2)*x^2]) + (1/4)*d*x^4*(a + b*ArcCsch[c*x]) + (1/6)*e*x^6*(a + b*ArcCsch[c*x])} -{x*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 6, (b*(2*c^2*d - e)*x*Sqrt[-1 - c^2*x^2])/(4*c^3*Sqrt[(-c^2)*x^2]) - (b*e*x*(-1 - c^2*x^2)^(3/2))/(12*c^3*Sqrt[(-c^2)*x^2]) + ((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/(4*e) - (b*c*d^2*x*ArcTan[Sqrt[-1 - c^2*x^2]])/(4*e*Sqrt[(-c^2)*x^2])} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x, x, 11, (b*e*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c) + (1/2)*b*d*ArcCsch[c*x]^2 + (1/2)*e*x^2*(a + b*ArcCsch[c*x]) - b*d*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] + b*d*ArcCsch[c*x]*Log[1/x] - d*(a + b*ArcCsch[c*x])*Log[1/x] - (1/2)*b*d*PolyLog[2, E^(2*ArcCsch[c*x])]} -{((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^3, x, 13, (b*c*d*Sqrt[1 + 1/(c^2*x^2)])/(4*x) - (1/4)*b*c^2*d*ArcCsch[c*x] + (1/2)*b*e*ArcCsch[c*x]^2 - (d*(a + b*ArcCsch[c*x]))/(2*x^2) - b*e*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] + b*e*ArcCsch[c*x]*Log[1/x] - e*(a + b*ArcCsch[c*x])*Log[1/x] - (1/2)*b*e*PolyLog[2, E^(2*ArcCsch[c*x])]} - - -{x^2*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x, 7, (b*(280*c^4*d^2 - 252*c^2*d*e + 75*e^2)*x^2*Sqrt[-1 - c^2*x^2])/(1680*c^5*Sqrt[(-c^2)*x^2]) + (b*(84*c^2*d - 25*e)*e*x^4*Sqrt[-1 - c^2*x^2])/(840*c^3*Sqrt[(-c^2)*x^2]) + (b*e^2*x^6*Sqrt[-1 - c^2*x^2])/(42*c*Sqrt[(-c^2)*x^2]) + (1/3)*d^2*x^3*(a + b*ArcCsch[c*x]) + (2/5)*d*e*x^5*(a + b*ArcCsch[c*x]) + (1/7)*e^2*x^7*(a + b*ArcCsch[c*x]) + (b*(280*c^4*d^2 - 252*c^2*d*e + 75*e^2)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(1680*c^6*Sqrt[(-c^2)*x^2])} -{x^0*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x, 6, (b*(40*c^2*d - 9*e)*e*x^2*Sqrt[-1 - c^2*x^2])/(120*c^3*Sqrt[(-c^2)*x^2]) + (b*e^2*x^4*Sqrt[-1 - c^2*x^2])/(20*c*Sqrt[(-c^2)*x^2]) + d^2*x*(a + b*ArcCsch[c*x]) + (2/3)*d*e*x^3*(a + b*ArcCsch[c*x]) + (1/5)*e^2*x^5*(a + b*ArcCsch[c*x]) - (b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(120*c^4*Sqrt[(-c^2)*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^2, x, 6, (b*c*d^2*Sqrt[-1 - c^2*x^2])/Sqrt[(-c^2)*x^2] + (b*e^2*x^2*Sqrt[-1 - c^2*x^2])/(6*c*Sqrt[(-c^2)*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/x + 2*d*e*x*(a + b*ArcCsch[c*x]) + (1/3)*e^2*x^3*(a + b*ArcCsch[c*x]) - (b*(12*c^2*d - e)*e*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/(6*c^2*Sqrt[(-c^2)*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4, x, 6, -((2*b*c*d*(c^2*d - 9*e)*Sqrt[-1 - c^2*x^2])/(9*Sqrt[(-c^2)*x^2])) + (b*c*d^2*Sqrt[-1 - c^2*x^2])/(9*x^2*Sqrt[(-c^2)*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcCsch[c*x]))/x + e^2*x*(a + b*ArcCsch[c*x]) - (b*e^2*x*ArcTan[(c*x)/Sqrt[-1 - c^2*x^2]])/Sqrt[(-c^2)*x^2]} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^6, x, 5, (b*c*(24*c^4*d^2 - 100*c^2*d*e + 225*e^2)*Sqrt[-1 - c^2*x^2])/(225*Sqrt[(-c^2)*x^2]) + (b*c*d^2*Sqrt[-1 - c^2*x^2])/(25*x^4*Sqrt[(-c^2)*x^2]) - (2*b*c*d*(6*c^2*d - 25*e)*Sqrt[-1 - c^2*x^2])/(225*x^2*Sqrt[(-c^2)*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcCsch[c*x]))/(3*x^3) - (e^2*(a + b*ArcCsch[c*x]))/x} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^8, x, 6, -((2*b*c^3*(360*c^4*d^2 - 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 - c^2*x^2])/(11025*Sqrt[(-c^2)*x^2])) + (b*c*d^2*Sqrt[-1 - c^2*x^2])/(49*x^6*Sqrt[(-c^2)*x^2]) - (2*b*c*d*(15*c^2*d - 49*e)*Sqrt[-1 - c^2*x^2])/(1225*x^4*Sqrt[(-c^2)*x^2]) + (b*c*(360*c^4*d^2 - 1176*c^2*d*e + 1225*e^2)*Sqrt[-1 - c^2*x^2])/(11025*x^2*Sqrt[(-c^2)*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcCsch[c*x]))/(5*x^5) - (e^2*(a + b*ArcCsch[c*x]))/(3*x^3)} - -{x^3*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x, 5, -((b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x*Sqrt[-1 - c^2*x^2])/(24*c^7*Sqrt[(-c^2)*x^2])) - (b*(6*c^4*d^2 - 16*c^2*d*e + 9*e^2)*x*(-1 - c^2*x^2)^(3/2))/(72*c^7*Sqrt[(-c^2)*x^2]) + (b*(8*c^2*d - 9*e)*e*x*(-1 - c^2*x^2)^(5/2))/(120*c^7*Sqrt[(-c^2)*x^2]) - (b*e^2*x*(-1 - c^2*x^2)^(7/2))/(56*c^7*Sqrt[(-c^2)*x^2]) + (1/4)*d^2*x^4*(a + b*ArcCsch[c*x]) + (1/3)*d*e*x^6*(a + b*ArcCsch[c*x]) + (1/8)*e^2*x^8*(a + b*ArcCsch[c*x])} -{x^1*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x, 6, (b*(3*c^4*d^2 - 3*c^2*d*e + e^2)*x*Sqrt[-1 - c^2*x^2])/(6*c^5*Sqrt[(-c^2)*x^2]) - (b*(3*c^2*d - 2*e)*e*x*(-1 - c^2*x^2)^(3/2))/(18*c^5*Sqrt[(-c^2)*x^2]) + (b*e^2*x*(-1 - c^2*x^2)^(5/2))/(30*c^5*Sqrt[(-c^2)*x^2]) + ((d + e*x^2)^3*(a + b*ArcCsch[c*x]))/(6*e) - (b*c*d^3*x*ArcTan[Sqrt[-1 - c^2*x^2]])/(6*e*Sqrt[(-c^2)*x^2])} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^1, x, 12, (b*(6*c^2*d - e)*e*Sqrt[1 + 1/(c^2*x^2)]*x)/(6*c^3) + (b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x^3)/(12*c) + (1/2)*b*d^2*ArcCsch[c*x]^2 + d*e*x^2*(a + b*ArcCsch[c*x]) + (1/4)*e^2*x^4*(a + b*ArcCsch[c*x]) - b*d^2*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] + b*d^2*ArcCsch[c*x]*Log[1/x] - d^2*(a + b*ArcCsch[c*x])*Log[1/x] - (1/2)*b*d^2*PolyLog[2, E^(2*ArcCsch[c*x])]} -{((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^3, x, 14, (b*c*d^2*Sqrt[1 + 1/(c^2*x^2)])/(4*x) + (b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c) - (1/4)*b*c^2*d^2*ArcCsch[c*x] + b*d*e*ArcCsch[c*x]^2 - (d^2*(a + b*ArcCsch[c*x]))/(2*x^2) + (1/2)*e^2*x^2*(a + b*ArcCsch[c*x]) - 2*b*d*e*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] + 2*b*d*e*ArcCsch[c*x]*Log[1/x] - 2*d*e*(a + b*ArcCsch[c*x])*Log[1/x] - b*d*e*PolyLog[2, E^(2*ArcCsch[c*x])]} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2), x, 25, (x*(a + b*ArcCsch[c*x]))/e + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(c*e) + (Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^(3/2)) + (Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^(3/2)) - (b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*e^(3/2)) + (b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^(3/2))} -{(x*(a + b*ArcCsch[c*x]))/(d + e*x^2), x, 26, -((a + b*ArcCsch[c*x])^2/(b*e)) - ((a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e) + (b*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*e) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e)} -{(a + b*ArcCsch[c*x])/(d + e*x^2), x, 19, ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*Sqrt[-d]*Sqrt[e]) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*Sqrt[-d]*Sqrt[e])} -{(a + b*ArcCsch[c*x])/(x*(d + e*x^2)), x, 19, (a + b*ArcCsch[c*x])^2/(2*b*d) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*d) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d)} -{(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)), x, 24, (b*c*Sqrt[1 + 1/(c^2*x^2)])/d - a/(d*x) - (b*ArcCsch[c*x])/(d*x) + (Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*(-d)^(3/2))} - - -{(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 31, (b*Sqrt[1 + 1/(c^2*x^2)]*x)/(2*c*e^2) + (d*(a + b*ArcCsch[c*x]))/(2*e^2*(e + d/x^2)) + (x^2*(a + b*ArcCsch[c*x]))/(2*e^2) + (2*d*(a + b*ArcCsch[c*x])^2)/(b*e^3) - (b*d*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(2*Sqrt[c^2*d - e]*e^(5/2)) + (2*d*(a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/e^3 - (d*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/e^3 - (b*d*PolyLog[2, E^(-2*ArcCsch[c*x])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/e^3 - (b*d*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/e^3 - (b*d*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/e^3} -{(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 29, -((a + b*ArcCsch[c*x])/(2*e*(e + d/x^2))) - (a + b*ArcCsch[c*x])^2/(b*e^2) + (b*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(2*Sqrt[c^2*d - e]*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e^2 + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^2) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^2) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^2) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^2) + (b*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*e^2) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^2) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*e^2) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^2)} -{(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 7, -((a + b*ArcCsch[c*x])/(2*e*(d + e*x^2))) + (b*c*x*ArcTan[Sqrt[-1 - c^2*x^2]])/(2*d*e*Sqrt[(-c^2)*x^2]) + (b*c*x*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - e]])/(2*d*Sqrt[c^2*d - e]*Sqrt[e]*Sqrt[(-c^2)*x^2])} -{(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^2), x, 24, -((e*(a + b*ArcCsch[c*x]))/(2*d^2*(e + d/x^2))) + (a + b*ArcCsch[c*x])^2/(2*b*d^2) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d - e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^2) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^2) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^2) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^2) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*d^2) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^2)} - -{(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 51, -((d*(a + b*ArcCsch[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] - d/x))) + (d*(a + b*ArcCsch[c*x]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcCsch[c*x]))/e^2 + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(c*e^2) + (b*Sqrt[d]*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[c^2*d - e]*e^2) + (b*Sqrt[d]*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[c^2*d - e]*e^2) + (3*Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*e^(5/2))} -{(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 27, (a + b*ArcCsch[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCsch[c*x])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[d]*Sqrt[c^2*d - e]*e) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[d]*Sqrt[c^2*d - e]*e) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*Sqrt[-d]*e^(3/2))} -{(a + b*ArcCsch[c*x])/(d + e*x^2)^2, x, 47, -((a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] - d/x))) + (a + b*ArcCsch[c*x])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d - e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d - e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(3/2)*Sqrt[e])} -{(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)^2), x, 50, (b*c*Sqrt[1 + 1/(c^2*x^2)])/d^2 - a/(d^2*x) - (b*ArcCsch[c*x])/(d^2*x) + (e*(a + b*ArcCsch[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (e*(a + b*ArcCsch[c*x]))/(4*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d - e]) - (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(5/2)*Sqrt[c^2*d - e]) - (3*Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(4*(-d)^(5/2))} - - -{(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3, x, 33, (b*c*d*Sqrt[1 + 1/(c^2*x^2)])/(8*(c^2*d - e)*e^2*(e + d/x^2)*x) - (a + b*ArcCsch[c*x])/(4*e*(e + d/x^2)^2) - (a + b*ArcCsch[c*x])/(2*e^2*(e + d/x^2)) - (a + b*ArcCsch[c*x])^2/(b*e^3) + (b*(c^2*d - 2*e)*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(8*(c^2*d - e)^(3/2)*e^(5/2)) + (b*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(2*Sqrt[c^2*d - e]*e^(5/2)) - ((a + b*ArcCsch[c*x])*Log[1 - E^(-2*ArcCsch[c*x])])/e^3 + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^3) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^3) + ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^3) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^3) + (b*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*e^3) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*e^3) + (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*e^3)} -{(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3, x, 6, -((b*c*x*Sqrt[-1 - c^2*x^2])/(8*(c^2*d - e)*e*Sqrt[(-c^2)*x^2]*(d + e*x^2))) + (x^4*(a + b*ArcCsch[c*x]))/(4*d*(d + e*x^2)^2) + (b*c*(c^2*d - 2*e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - e]])/(8*d*(c^2*d - e)^(3/2)*e^(3/2)*Sqrt[(-c^2)*x^2])} -{(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3, x, 8, (b*c*x*Sqrt[-1 - c^2*x^2])/(8*d*(c^2*d - e)*Sqrt[(-c^2)*x^2]*(d + e*x^2)) - (a + b*ArcCsch[c*x])/(4*e*(d + e*x^2)^2) + (b*c*x*ArcTan[Sqrt[-1 - c^2*x^2]])/(4*d^2*e*Sqrt[(-c^2)*x^2]) + (b*c*(3*c^2*d - 2*e)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/Sqrt[c^2*d - e]])/(8*d^2*(c^2*d - e)^(3/2)*Sqrt[e]*Sqrt[(-c^2)*x^2])} -{(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^3), x, 28, -((b*c*e*Sqrt[1 + 1/(c^2*x^2)])/(8*d^2*(c^2*d - e)*(e + d/x^2)*x)) + (e^2*(a + b*ArcCsch[c*x]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcCsch[c*x]))/(d^3*(e + d/x^2)) + (a + b*ArcCsch[c*x])^2/(2*b*d^3) - (b*(c^2*d - 2*e)*Sqrt[e]*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d - e)^(3/2)) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)])/(d^3*Sqrt[c^2*d - e]) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^3) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^3) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^3) - ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(2*d^3) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(2*d^3)} - -{(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3, x, 35, -((b*c*Sqrt[-d]*Sqrt[1 + 1/(c^2*x^2)])/(16*(c^2*d - e)*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x))) - (b*c*Sqrt[-d]*Sqrt[1 + 1/(c^2*x^2)])/(16*(c^2*d - e)*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + b*ArcCsch[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b*ArcCsch[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*ArcCsch[c*x]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcCsch[c*x]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (3*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*Sqrt[d]*Sqrt[c^2*d - e]*e^2) + (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*Sqrt[d]*(c^2*d - e)^(3/2)*e) - (3*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*Sqrt[d]*Sqrt[c^2*d - e]*e^2) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*Sqrt[d]*(c^2*d - e)^(3/2)*e) + (3*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2)) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(16*Sqrt[-d]*e^(5/2)) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*Sqrt[-d]*e^(5/2))} -{(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3, x, 63, -((b*c*Sqrt[1 + 1/(c^2*x^2)])/(16*Sqrt[-d]*(c^2*d - e)*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x))) - (b*c*Sqrt[1 + 1/(c^2*x^2)])/(16*Sqrt[-d]*(c^2*d - e)*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)) + (a + b*ArcCsch[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (a + b*ArcCsch[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcCsch[c*x])/(16*Sqrt[-d]*Sqrt[e]*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (a + b*ArcCsch[c*x])/(16*d*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d - e)^(3/2)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(3/2)*Sqrt[c^2*d - e]*e) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(3/2)*(c^2*d - e)^(3/2)) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(3/2)*Sqrt[c^2*d - e]*e) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2)) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(16*(-d)^(3/2)*e^(3/2)) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(3/2)*e^(3/2))} -{(a + b*ArcCsch[c*x])/(d + e*x^2)^3, x, 81, -((b*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d - e)*(Sqrt[-d]*Sqrt[e] - d/x))) - (b*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)])/(16*(-d)^(3/2)*(c^2*d - e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[e]*(a + b*ArcCsch[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) - (5*(a + b*ArcCsch[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[e]*(a + b*ArcCsch[c*x]))/(16*(-d)^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) + (5*(a + b*ArcCsch[c*x]))/(16*d^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (5*b*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d - e]) + (b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d - e)^(3/2)) + (5*b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(5/2)*Sqrt[c^2*d - e]) + (b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(16*d^(5/2)*(c^2*d - e)^(3/2)) + (3*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCsch[c*x])*Log[1 - (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCsch[c*x])*Log[1 + (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] - Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (c*Sqrt[-d]*E^ArcCsch[c*x])/(Sqrt[e] + Sqrt[(-c^2)*d + e])])/(16*(-d)^(5/2)*Sqrt[e])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcCsch[c x])*) - - -(* ::Subsubsection::Closed:: *) -(*p>0*) - - -{x^5*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, If[$VersionNumber>=8, 12, 13], -((b*(23*c^4*d^2 - 12*c^2*d*e - 75*e^2)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(1680*c^5*e^2*Sqrt[(-c^2)*x^2])) - (b*(29*c^2*d + 25*e)*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e^2*Sqrt[(-c^2)*x^2]) + (b*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e^2*Sqrt[(-c^2)*x^2]) + (d^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^3) + (b*(105*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 - 75*e^3)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^6*e^(5/2)*Sqrt[(-c^2)*x^2]) + (8*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(105*e^3*Sqrt[(-c^2)*x^2])} -{x^3*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, If[$VersionNumber>=8, 11, 12], (b*(c^2*d - 9*e)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e*Sqrt[(-c^2)*x^2]) + (b*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e*Sqrt[(-c^2)*x^2]) - (d*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) - (b*(15*c^4*d^2 + 10*c^2*d*e - 9*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(3/2)*Sqrt[(-c^2)*x^2]) - (2*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(15*e^2*Sqrt[(-c^2)*x^2])} -{x*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, 9, (b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*Sqrt[(-c^2)*x^2]) + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e) + (b*(3*c^2*d - e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*Sqrt[e]*Sqrt[(-c^2)*x^2]) + (b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e*Sqrt[(-c^2)*x^2])} -{Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x])/x^1, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x, x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x])/x^3, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^3, x]} - -{x^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]} -{x^0*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x])/x^2, x, 0, Unintegrable[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^2, x]} -{Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x])/x^4, x, 8, -((2*b*c^3*(c^2*d - 2*e)*x^2*Sqrt[d + e*x^2])/(9*d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])) - (2*b*c*(c^2*d - 2*e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*Sqrt[(-c^2)*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*b*c^2*(c^2*d - 2*e)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*(c^2*d - 3*e)*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} -{Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x])/x^6, x, If[$VersionNumber>=8, 9, 23], If[$VersionNumber>=8, (b*c^3*(24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*x^2*Sqrt[d + e*x^2])/(225*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (b*c*(24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(225*d^2*Sqrt[(-c^2)*x^2]) - (b*c*(12*c^2*d + e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(225*d*x^2*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(25*d*x^4*Sqrt[(-c^2)*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(15*d^2*x^3) - (b*c^2*(24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(225*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (2*b*e*(6*c^4*d^2 - 4*c^2*d*e - 15*e^2)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(225*d^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]), -((b*c^3*(2*c^2*d - e)*e*x^2*Sqrt[d + e*x^2])/(45*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])) - (2*b*c^3*e^2*x^2*Sqrt[d + e*x^2])/(15*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (b*c^3*(8*c^4*d^2 - 3*c^2*d*e - 2*e^2)*x^2*Sqrt[d + e*x^2])/(75*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) - (b*c*(2*c^2*d - e)*e*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(45*d^2*Sqrt[(-c^2)*x^2]) - (2*b*c*e^2*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(15*d^2*Sqrt[(-c^2)*x^2]) + (b*c*(8*c^4*d^2 - 3*c^2*d*e - 2*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d^2*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(25*x^4*Sqrt[(-c^2)*x^2]) - (b*c*(4*c^2*d - e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d*x^2*Sqrt[(-c^2)*x^2]) + (b*c*e*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(45*d*x^2*Sqrt[(-c^2)*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(15*d^2*x^3) + (b*c^2*(2*c^2*d - e)*e*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(45*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (2*b*c^2*e^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(15*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*c^2*(8*c^4*d^2 - 3*c^2*d*e - 2*e^2)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(75*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*c^2*(4*c^2*d - e)*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(75*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*c^2*e^2*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(45*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (2*b*e^3*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(15*d^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])]} - - -{x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x, 12, -((b*(3*c^4*d^2 + 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(560*c^5*e*Sqrt[(-c^2)*x^2])) + (b*(13*c^2*d - 25*e)*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[(-c^2)*x^2]) + (b*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[(-c^2)*x^2]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^2) - (b*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[(-c^2)*x^2]) - (2*b*c*d^(7/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(35*e^2*Sqrt[(-c^2)*x^2])} -{x^1*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x, 10, (b*(7*c^2*d - 3*e)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(40*c^3*Sqrt[(-c^2)*x^2]) + (b*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*Sqrt[(-c^2)*x^2]) + ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e) + (b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[(-c^2)*x^2]) + (b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(5*e*Sqrt[(-c^2)*x^2])} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^1, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^3, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^3, x]} - -{x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]} -{x^0*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^2, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^2, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^4, x, 0, Unintegrable[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^4, x]} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^6, x, 9, (b*c^3*(8*c^4*d^2 - 23*c^2*d*e + 23*e^2)*x^2*Sqrt[d + e*x^2])/(75*d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (b*c*(8*c^4*d^2 - 23*c^2*d*e + 23*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*d*Sqrt[(-c^2)*x^2]) - (4*b*c*(c^2*d - 2*e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(75*x^2*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(25*x^4*Sqrt[(-c^2)*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*d*x^5) - (b*c^2*(8*c^4*d^2 - 23*c^2*d*e + 23*e^2)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(75*d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*e*(4*c^4*d^2 - 11*c^2*d*e + 15*e^2)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(75*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} -{(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x])/x^8, x, 10, -((b*c^3*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x^2*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])) - (b*c*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(3675*d^2*Sqrt[(-c^2)*x^2]) + (b*c*(120*c^4*d^2 - 159*c^2*d*e - 37*e^2)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(3675*d*x^2*Sqrt[(-c^2)*x^2]) - (b*c*(30*c^2*d - 11*e)*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(1225*d*x^4*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(5/2))/(49*d*x^6*Sqrt[(-c^2)*x^2]) - ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(7*d*x^7) + (2*e*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(35*d^2*x^5) + (b*c^2*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(3675*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*e*(120*c^6*d^3 - 249*c^4*d^2*e + 71*c^2*d*e^2 + 210*e^3)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(3675*d^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^5*(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x, 11, -((b*(19*c^2*d + 9*e)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e^2*Sqrt[(-c^2)*x^2])) + (b*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(20*c*e^2*Sqrt[(-c^2)*x^2]) + (d^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3) + (b*(45*c^4*d^2 + 10*c^2*d*e + 9*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(5/2)*Sqrt[(-c^2)*x^2]) + (8*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(15*e^3*Sqrt[(-c^2)*x^2])} -{x^3*(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x, 10, (b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e*Sqrt[(-c^2)*x^2]) - (d*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) - (b*(3*c^2*d + e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(3/2)*Sqrt[(-c^2)*x^2]) - (2*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e^2*Sqrt[(-c^2)*x^2])} -{x^1*(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x, 9, (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(e*Sqrt[(-c^2)*x^2])} -{(a + b*ArcCsch[c*x])/(x^1*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x^2]), x]} -{(a + b*ArcCsch[c*x])/(x^3*Sqrt[d + e*x^2]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^3*Sqrt[d + e*x^2]), x]} - -{x^2*(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]} -{x^0*(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x, 0, Unintegrable[(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x]} -{(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x^2]), x, 8, (b*c^3*x^2*Sqrt[d + e*x^2])/(d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(d*Sqrt[(-c^2)*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/(d*x) - (b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(d*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} -{(a + b*ArcCsch[c*x])/(x^4*Sqrt[d + e*x^2]), x, 8, -((b*c^3*(2*c^2*d + 5*e)*x^2*Sqrt[d + e*x^2])/(9*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])) - (b*c*(2*c^2*d + 5*e)*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d^2*Sqrt[(-c^2)*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*x^2*Sqrt[(-c^2)*x^2]) - (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/(3*d^2*x) + (b*c^2*(2*c^2*d + 5*e)*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) - (b*e*(c^2*d + 6*e)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} - - -{x^5*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 10, (b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e^2*Sqrt[(-c^2)*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) - (b*(9*c^2*d + e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(5/2)*Sqrt[(-c^2)*x^2]) - (8*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e^3*Sqrt[(-c^2)*x^2])} -{x^3*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 9, (d*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^2 + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(3/2)*Sqrt[(-c^2)*x^2]) + (2*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(e^2*Sqrt[(-c^2)*x^2])} -{x^1*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 4, -((a + b*ArcCsch[c*x])/(e*Sqrt[d + e*x^2])) - (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(Sqrt[d]*e*Sqrt[(-c^2)*x^2])} -{(a + b*ArcCsch[c*x])/(x^1*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(3/2)), x]} -{(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(3/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(3/2)), x]} - -{x^4*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]} -{x^2*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 0, Unintegrable[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]} -{x^0*(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2), x, 3, (x*(a + b*ArcCsch[c*x]))/(d*Sqrt[d + e*x^2]) - (b*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} -{(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)^(3/2)), x, 7, (b*c^3*x^2*Sqrt[d + e*x^2])/(d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(d^2*Sqrt[(-c^2)*x^2]) - (a + b*ArcCsch[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcCsch[c*x]))/(d^2*Sqrt[d + e*x^2]) - (b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(d^2*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (2*b*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(d^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} - - -{x^5*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 10, (b*c*d*x*Sqrt[-1 - c^2*x^2])/(3*(c^2*d - e)*e^2*Sqrt[(-c^2)*x^2]*Sqrt[d + e*x^2]) - (d^2*(a + b*ArcCsch[c*x]))/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^3 + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[(-c^2)*x^2]) + (8*b*c*Sqrt[d]*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e^3*Sqrt[(-c^2)*x^2])} -{x^3*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 7, -((b*c*x*Sqrt[-1 - c^2*x^2])/(3*(c^2*d - e)*e*Sqrt[(-c^2)*x^2]*Sqrt[d + e*x^2])) + (d*(a + b*ArcCsch[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcCsch[c*x])/(e^2*Sqrt[d + e*x^2]) - (2*b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*Sqrt[d]*e^2*Sqrt[(-c^2)*x^2])} -{x^1*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 5, (b*c*x*Sqrt[-1 - c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[(-c^2)*x^2]*Sqrt[d + e*x^2]) - (a + b*ArcCsch[c*x])/(3*e*(d + e*x^2)^(3/2)) - (b*c*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*d^(3/2)*e*Sqrt[(-c^2)*x^2])} -{(a + b*ArcCsch[c*x])/(x^1*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(5/2)), x]} -{(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)), x]} - -{x^6*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^6*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2), x]} -{x^4*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 0, Unintegrable[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2), x]} -{x^2*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 7, (b*c*x^2*Sqrt[-1 - c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[(-c^2)*x^2]*Sqrt[d + e*x^2]) + (b*c^3*x^2*Sqrt[d + e*x^2])/(3*d*(c^2*d - e)*e*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) + (x^3*(a + b*ArcCsch[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(3*d*(c^2*d - e)*e*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(3*d^2*(c^2*d - e)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} -{x^0*(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2), x, 5, (x*(a + b*ArcCsch[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcCsch[c*x]))/(3*d^2*Sqrt[d + e*x^2]) - (b*c*Sqrt[e]*x*Sqrt[-1 - c^2*x^2]*EllipticE[ArcTan[(Sqrt[e]*x)/Sqrt[d]], 1 - (c^2*d)/e])/(3*d^(3/2)*(c^2*d - e)*Sqrt[(-c^2)*x^2]*Sqrt[(d*(1 + c^2*x^2))/(d + e*x^2)]*Sqrt[d + e*x^2]) - (b*(3*c^2*d - 2*e)*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(3*d^3*(c^2*d - e)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcCsch[c x]) when m symbolic*) - - -{(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsch[c*x]), x, 6, If[$VersionNumber>=8, (b*e*(e^2*(15 + 8*m + m^2)^2 - 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 - c^2*x^2])/(c^5*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[(-c^2)*x^2]) - (b*e^2*(e*(5 + m)^2 - 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 - c^2*x^2])/(c^3*f^3*(4 + m)*(5 + m)*(6 + m)*(7 + m)*Sqrt[(-c^2)*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 - c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[(-c^2)*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsch[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsch[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCsch[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) - (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 - 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/((3 + m)*(5 + m)*(7 + m)))*x*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]), (b*e*(e^2*(15 + 8*m + m^2)^2 - 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*x*(f*x)^(1 + m)*Sqrt[-1 - c^2*x^2])/(c^5*f*(6 + m)*(8 + 6*m + m^2)*(105 + 71*m + 15*m^2 + m^3)*Sqrt[(-c^2)*x^2]) - (b*e^2*(e*(5 + m)^2 - 3*c^2*d*(42 + 13*m + m^2))*x*(f*x)^(3 + m)*Sqrt[-1 - c^2*x^2])/(c^3*f^3*(4 + m)*(6 + m)*(35 + 12*m + m^2)*Sqrt[(-c^2)*x^2]) + (b*e^3*x*(f*x)^(5 + m)*Sqrt[-1 - c^2*x^2])/(c*f^5*(6 + m)*(7 + m)*Sqrt[(-c^2)*x^2]) + (d^3*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + m)*(a + b*ArcCsch[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*ArcCsch[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCsch[c*x]))/(f^7*(7 + m)) - (b*((c^6*d^3*(2 + m)*(4 + m)*(6 + m))/(1 + m) - (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 - 3*c^2*d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)))/(105 + 71*m + 15*m^2 + m^3))*x*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(c^5*f*(1 + m)*(2 + m)*(4 + m)*(6 + m)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])]} -{(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x, 6, If[$VersionNumber>=8, -((b*e*(e*(3 + m)^2 - 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 - c^2*x^2])/(c^3*f*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[(-c^2)*x^2])) + (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 - c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[(-c^2)*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcCsch[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCsch[c*x]))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 - 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]), -((b*e*(e*(3 + m)^2 - 2*c^2*d*(20 + 9*m + m^2))*x*(f*x)^(1 + m)*Sqrt[-1 - c^2*x^2])/(c^3*f*(4 + m)*(5 + m)*(6 + 5*m + m^2)*Sqrt[(-c^2)*x^2])) + (b*e^2*x*(f*x)^(3 + m)*Sqrt[-1 - c^2*x^2])/(c*f^3*(4 + m)*(5 + m)*Sqrt[(-c^2)*x^2]) + (d^2*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m)*(a + b*ArcCsch[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCsch[c*x]))/(f^5*(5 + m)) - (b*(c^4*d^2*(2 + m)*(3 + m)*(4 + m)*(5 + m) + e*(1 + m)^2*(e*(3 + m)^2 - 2*c^2*d*(20 + 9*m + m^2)))*x*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(c^3*f*(1 + m)^2*(2 + m)*(3 + m)*(4 + m)*(5 + m)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])]} -{(f*x)^m*(d + e*x^2)*(a + b*ArcCsch[c*x]), x, 5, (b*e*x*(f*x)^(1 + m)*Sqrt[-1 - c^2*x^2])/(c*f*(6 + 5*m + m^2)*Sqrt[(-c^2)*x^2]) + (d*(f*x)^(1 + m)*(a + b*ArcCsch[c*x]))/(f*(1 + m)) + (e*(f*x)^(3 + m)*(a + b*ArcCsch[c*x]))/(f^3*(3 + m)) + (b*(e*(1 + m)^2 - c^2*d*(2 + m)*(3 + m))*x*(f*x)^(1 + m)*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, (-c^2)*x^2])/(c*f*(1 + m)^2*(2 + m)*(3 + m)*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2])} -{((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2), x]} -{((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x]} - - -{(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]} -{(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x, 0, Unintegrable[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]} -{((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]} -{((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x, 0, Unintegrable[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (f x)^m (d+e x^4)^p (a+b ArcCsch[c x])^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (d+e x^4)^(p/2) (a+b ArcCsch[c x])*) - - -(* ::Subsubsection:: *) -(*p>0*) - - -(* ::Subsubsection::Closed:: *) -(*p<0*) - - -{x^11*(a + b*ArcCsch[c*x])/Sqrt[1 - c^4*x^4], x, 16, -((4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(15*c^13*Sqrt[1 + 1/(c^2*x^2)]*x)) + (7*b*(1 - c^2*x^2)^(3/2)*Sqrt[1 + c^2*x^2])/(90*c^13*Sqrt[1 + 1/(c^2*x^2)]*x) - (13*b*(1 - c^2*x^2)^(5/2)*Sqrt[1 + c^2*x^2])/(150*c^13*Sqrt[1 + 1/(c^2*x^2)]*x) + (3*b*(1 - c^2*x^2)^(7/2)*Sqrt[1 + c^2*x^2])/(70*c^13*Sqrt[1 + 1/(c^2*x^2)]*x) - (b*(1 - c^2*x^2)^(9/2)*Sqrt[1 + c^2*x^2])/(90*c^13*Sqrt[1 + 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsch[c*x]))/(2*c^12) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcCsch[c*x]))/(3*c^12) - ((1 - c^4*x^4)^(5/2)*(a + b*ArcCsch[c*x]))/(10*c^12) + (4*b*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(15*c^13*Sqrt[1 + 1/(c^2*x^2)]*x)} -{x^7*(a + b*ArcCsch[c*x])/Sqrt[1 - c^4*x^4], x, 13, -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[1 + 1/(c^2*x^2)]*x)) + (b*(1 - c^2*x^2)^(3/2)*Sqrt[1 + c^2*x^2])/(18*c^9*Sqrt[1 + 1/(c^2*x^2)]*x) - (b*(1 - c^2*x^2)^(5/2)*Sqrt[1 + c^2*x^2])/(30*c^9*Sqrt[1 + 1/(c^2*x^2)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsch[c*x]))/(2*c^8) + ((1 - c^4*x^4)^(3/2)*(a + b*ArcCsch[c*x]))/(6*c^8) + (b*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(3*c^9*Sqrt[1 + 1/(c^2*x^2)]*x)} -{x^3*(a + b*ArcCsch[c*x])/Sqrt[1 - c^4*x^4], x, 8, (b*x*Sqrt[1 - c^4*x^4])/(2*c^3*Sqrt[(-c^2)*x^2]*Sqrt[-1 - c^2*x^2]) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsch[c*x]))/(2*c^4) - (b*x*ArcTan[Sqrt[1 - c^4*x^4]/Sqrt[-1 - c^2*x^2]])/(2*c^3*Sqrt[(-c^2)*x^2]), -((b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(2*c^5*Sqrt[1 + 1/(c^2*x^2)]*x)) - (Sqrt[1 - c^4*x^4]*(a + b*ArcCsch[c*x]))/(2*c^4) + (b*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 - c^2*x^2]])/(2*c^5*Sqrt[1 + 1/(c^2*x^2)]*x)} -{(a + b*ArcCsch[c*x])/(x^1*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x*Sqrt[1 - c^4*x^4]), x]} -{(a + b*ArcCsch[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x, 0, Unintegrable[(a + b*ArcCsch[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]} - - -(* ::Section:: *) -(*Integrands of the form u (a+b ArcCsch[c x])^n*) diff --git a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.2 Inverse hyperbolic cosecant functions.m b/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.2 Inverse hyperbolic cosecant functions.m deleted file mode 100644 index 04dc6b0..0000000 --- a/test/methods/rule_based/test_files/7 Inverse hyperbolic functions/7.6 Inverse hyperbolic cosecant/7.6.2 Inverse hyperbolic cosecant functions.m +++ /dev/null @@ -1,145 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Inverse Hyperbolic Cosecants*) - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCsch[a+b x]^n*) - - -{x^3*ArcCsch[a + b*x], x, 8, -(((2 - 17*a^2)*(a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(12*b^4)) + (x^2*(a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(12*b^2) - (a*(a + b*x)^2*Sqrt[1 + 1/(a + b*x)^2])/(3*b^4) - (a^4*ArcCsch[a + b*x])/(4*b^4) + (1/4)*x^4*ArcCsch[a + b*x] + (a*(1 - 2*a^2)*ArcTanh[Sqrt[1 + 1/(a + b*x)^2]])/(2*b^4)} -{x^2*ArcCsch[a + b*x], x, 7, -((5*a*(a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(6*b^3)) + (x*(a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(6*b^2) + (a^3*ArcCsch[a + b*x])/(3*b^3) + (1/3)*x^3*ArcCsch[a + b*x] - ((1 - 6*a^2)*ArcTanh[Sqrt[1 + 1/(a + b*x)^2]])/(6*b^3)} -{x^1*ArcCsch[a + b*x], x, 6, ((a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(2*b^2) - (a^2*ArcCsch[a + b*x])/(2*b^2) + (1/2)*x^2*ArcCsch[a + b*x] - (a*ArcTanh[Sqrt[1 + 1/(a + b*x)^2]])/b^2} -{ArcCsch[a + b*x]/x^1, x, 14, ArcCsch[a + b*x]*Log[1 - (a*E^ArcCsch[a + b*x])/(1 - Sqrt[1 + a^2])] + ArcCsch[a + b*x]*Log[1 - (a*E^ArcCsch[a + b*x])/(1 + Sqrt[1 + a^2])] - ArcCsch[a + b*x]*Log[1 - E^(2*ArcCsch[a + b*x])] + PolyLog[2, (a*E^ArcCsch[a + b*x])/(1 - Sqrt[1 + a^2])] + PolyLog[2, (a*E^ArcCsch[a + b*x])/(1 + Sqrt[1 + a^2])] - (1/2)*PolyLog[2, E^(2*ArcCsch[a + b*x])]} -{ArcCsch[a + b*x]/x^2, x, 6, -((b*ArcCsch[a + b*x])/a) - ArcCsch[a + b*x]/x + (2*b*ArcTanh[(a + Tanh[(1/2)*ArcCsch[a + b*x]])/Sqrt[1 + a^2]])/(a*Sqrt[1 + a^2])} -{ArcCsch[a + b*x]/x^3, x, 8, (b*(a + b*x)*Sqrt[1 + 1/(a + b*x)^2])/(2*a*(1 + a^2)*x) + (b^2*ArcCsch[a + b*x])/(2*a^2) - ArcCsch[a + b*x]/(2*x^2) - ((1 + 2*a^2)*b^2*ArcTanh[(a + Tanh[(1/2)*ArcCsch[a + b*x]])/Sqrt[1 + a^2]])/(a^2*(1 + a^2)^(3/2))} - - -{(e + f*x)^3*(a + b*ArcCsch[c + d*x])^2, x, 20, (b^2*f^2*(d*e - c*f)*x)/d^3 + (b^2*f^3*(c + d*x)^2)/(12*d^4) - (b*f^3*(c + d*x)*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/(3*d^4) + (3*b*f*(d*e - c*f)^2*(c + d*x)*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/d^4 + (b*f^2*(d*e - c*f)*(c + d*x)^2*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/d^4 + (b*f^3*(c + d*x)^3*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/(6*d^4) - ((d*e - c*f)^4*(a + b*ArcCsch[c + d*x])^2)/(4*d^4*f) + ((e + f*x)^4*(a + b*ArcCsch[c + d*x])^2)/(4*f) - (2*b*f^2*(d*e - c*f)*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^4 + (4*b*(d*e - c*f)^3*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^4 - (b^2*f^3*Log[c + d*x])/(3*d^4) + (3*b^2*f*(d*e - c*f)^2*Log[c + d*x])/d^4 - (b^2*f^2*(d*e - c*f)*PolyLog[2, -E^ArcCsch[c + d*x]])/d^4 + (2*b^2*(d*e - c*f)^3*PolyLog[2, -E^ArcCsch[c + d*x]])/d^4 + (b^2*f^2*(d*e - c*f)*PolyLog[2, E^ArcCsch[c + d*x]])/d^4 - (2*b^2*(d*e - c*f)^3*PolyLog[2, E^ArcCsch[c + d*x]])/d^4} -{(e + f*x)^2*(a + b*ArcCsch[c + d*x])^2, x, 17, (b^2*f^2*x)/(3*d^2) + (2*b*f*(d*e - c*f)*(c + d*x)*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/d^3 + (b*f^2*(c + d*x)^2*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/(3*d^3) - ((d*e - c*f)^3*(a + b*ArcCsch[c + d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCsch[c + d*x])^2)/(3*f) - (2*b*f^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/(3*d^3) + (4*b*(d*e - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^3 + (2*b^2*f*(d*e - c*f)*Log[c + d*x])/d^3 - (b^2*f^2*PolyLog[2, -E^ArcCsch[c + d*x]])/(3*d^3) + (2*b^2*(d*e - c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d^3 + (b^2*f^2*PolyLog[2, E^ArcCsch[c + d*x]])/(3*d^3) - (2*b^2*(d*e - c*f)^2*PolyLog[2, E^ArcCsch[c + d*x]])/d^3} -{(e + f*x)^1*(a + b*ArcCsch[c + d*x])^2, x, 11, (b*f*(c + d*x)*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/d^2 - ((d*e - c*f)^2*(a + b*ArcCsch[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCsch[c + d*x])^2)/(2*f) + (4*b*(d*e - c*f)*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^2 + (b^2*f*Log[c + d*x])/d^2 + (2*b^2*(d*e - c*f)*PolyLog[2, -E^ArcCsch[c + d*x]])/d^2 - (2*b^2*(d*e - c*f)*PolyLog[2, E^ArcCsch[c + d*x]])/d^2} -{(e + f*x)^0*(a + b*ArcCsch[c + d*x])^2, x, 8, ((c + d*x)*(a + b*ArcCsch[c + d*x])^2)/d + (4*b*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d + (2*b^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d - (2*b^2*PolyLog[2, E^ArcCsch[c + d*x]])/d} -{(a + b*ArcCsch[c + d*x])^2/(e + f*x)^1, x, 17, -(((a + b*ArcCsch[c + d*x])^2*Log[1 - E^(2*ArcCsch[c + d*x])])/f) + ((a + b*ArcCsch[c + d*x])^2*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f + ((a + b*ArcCsch[c + d*x])^2*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/f - (b*(a + b*ArcCsch[c + d*x])*PolyLog[2, E^(2*ArcCsch[c + d*x])])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (2*b*(a + b*ArcCsch[c + d*x])*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f + (b^2*PolyLog[3, E^(2*ArcCsch[c + d*x])])/(2*f) - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f - (2*b^2*PolyLog[3, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/f} -{(a + b*ArcCsch[c + d*x])^2/(e + f*x)^2, x, 12, (d*(a + b*ArcCsch[c + d*x])^2)/(f*(d*e - c*f)) - (a + b*ArcCsch[c + d*x])^2/(f*(e + f*x)) - (2*b*d*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b*d*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) - (2*b^2*d*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (2*b^2*d*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])} -{(a + b*ArcCsch[c + d*x])^2/(e + f*x)^3, x, 23, -((b*d^2*f*Sqrt[1 + 1/(c + d*x)^2]*(a + b*ArcCsch[c + d*x]))/((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)*(f + (d*e - c*f)/(c + d*x)))) + (d^2*(a + b*ArcCsch[c + d*x])^2)/(2*f*(d*e - c*f)^2) - (a + b*ArcCsch[c + d*x])^2/(2*f*(e + f*x)^2) + (b*d^2*f^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b*d^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) - (b*d^2*f^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b*d^2*(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (b^2*d^2*f*Log[f + (d*e - c*f)/(c + d*x)])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (b^2*d^2*f^2*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) - (2*b^2*d^2*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) - (b^2*d^2*f^2*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)^2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)^(3/2)) + (2*b^2*d^2*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)^2*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m ArcCsch[a x^n]*) - - -{x^3*ArcCsch[Sqrt[x]], x, 4, -((Sqrt[-1 - x]*Sqrt[x])/(4*Sqrt[-x])) - ((-1 - x)^(3/2)*Sqrt[x])/(4*Sqrt[-x]) - (3*(-1 - x)^(5/2)*Sqrt[x])/(20*Sqrt[-x]) - ((-1 - x)^(7/2)*Sqrt[x])/(28*Sqrt[-x]) + (1/4)*x^4*ArcCsch[Sqrt[x]]} -{x^2*ArcCsch[Sqrt[x]], x, 4, (Sqrt[-1 - x]*Sqrt[x])/(3*Sqrt[-x]) + (2*(-1 - x)^(3/2)*Sqrt[x])/(9*Sqrt[-x]) + ((-1 - x)^(5/2)*Sqrt[x])/(15*Sqrt[-x]) + (1/3)*x^3*ArcCsch[Sqrt[x]]} -{x^1*ArcCsch[Sqrt[x]], x, 4, -((Sqrt[-1 - x]*Sqrt[x])/(2*Sqrt[-x])) - ((-1 - x)^(3/2)*Sqrt[x])/(6*Sqrt[-x]) + (1/2)*x^2*ArcCsch[Sqrt[x]]} -{x^0*ArcCsch[Sqrt[x]], x, 3, (Sqrt[-1 - x]*Sqrt[x])/Sqrt[-x] + x*ArcCsch[Sqrt[x]]} -{ArcCsch[Sqrt[x]]/x^1, x, 7, ArcCsch[Sqrt[x]]^2 - 2*ArcCsch[Sqrt[x]]*Log[1 - E^(2*ArcCsch[Sqrt[x]])] - PolyLog[2, E^(2*ArcCsch[Sqrt[x]])]} -{ArcCsch[Sqrt[x]]/x^2, x, 5, Sqrt[-1 - x]/(2*Sqrt[-x]*Sqrt[x]) - ArcCsch[Sqrt[x]]/x - (Sqrt[x]*ArcTan[Sqrt[-1 - x]])/(2*Sqrt[-x])} -{ArcCsch[Sqrt[x]]/x^3, x, 6, Sqrt[-1 - x]/(8*Sqrt[-x]*x^(3/2)) - (3*Sqrt[-1 - x])/(16*Sqrt[-x]*Sqrt[x]) - ArcCsch[Sqrt[x]]/(2*x^2) + (3*Sqrt[x]*ArcTan[Sqrt[-1 - x]])/(16*Sqrt[-x])} -{ArcCsch[Sqrt[x]]/x^4, x, 7, Sqrt[-1 - x]/(18*Sqrt[-x]*x^(5/2)) - (5*Sqrt[-1 - x])/(72*Sqrt[-x]*x^(3/2)) + (5*Sqrt[-1 - x])/(48*Sqrt[-x]*Sqrt[x]) - ArcCsch[Sqrt[x]]/(3*x^3) - (5*Sqrt[x]*ArcTan[Sqrt[-1 - x]])/(48*Sqrt[-x])} - - -{ArcCsch[1/x], x, 3, -Sqrt[1 + x^2] + x*ArcSinh[x]} - - -{ArcCsch[a*x^n]/x, x, 7, ArcCsch[a*x^n]^2/(2*n) - (ArcCsch[a*x^n]*Log[1 - E^(2*ArcCsch[a*x^n])])/n - PolyLog[2, E^(2*ArcCsch[a*x^n])]/(2*n)} -{ArcCsch[a*x^5]/x, x, 7, (1/10)*ArcCsch[a*x^5]^2 - (1/5)*ArcCsch[a*x^5]*Log[1 - E^(2*ArcCsch[a*x^5])] - (1/10)*PolyLog[2, E^(2*ArcCsch[a*x^5])]} - - -(* ::Section::Closed:: *) -(*Integrands involving inverse hyperbolic cosecants of exponentials*) - - -{ArcCsch[c*E^(a + b*x)], x, 7, ArcCsch[c*E^(a + b*x)]^2/(2*b) - (ArcCsch[c*E^(a + b*x)]*Log[1 - E^(2*ArcCsch[c*E^(a + b*x)])])/b - PolyLog[2, E^(2*ArcCsch[c*E^(a + b*x)])]/(2*b)} - - -(* ::Section::Closed:: *) -(*Integrands involving exponentials of inverse hyperbolic cosecants*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^ArcCsch[a x^p]*) - - -{x^m*E^ArcCsch[a*x], x, 4, x^m/(a*m) + (x^(1 + m)*Hypergeometric2F1[-(1/2), (1/2)*(-1 - m), (1 - m)/2, -(1/(a^2*x^2))])/(1 + m)} - -{x^4*E^ArcCsch[a*x], x, 4, -((2*(1 + 1/(a^2*x^2))^(3/2)*x^3)/(15*a^2)) + x^4/(4*a) + (1/5)*(1 + 1/(a^2*x^2))^(3/2)*x^5} -{x^3*E^ArcCsch[a*x], x, 7, (Sqrt[1 + 1/(a^2*x^2)]*x^2)/(8*a^2) + x^3/(3*a) + (1/4)*Sqrt[1 + 1/(a^2*x^2)]*x^4 - ArcTanh[Sqrt[1 + 1/(a^2*x^2)]]/(8*a^4)} -{x^2*E^ArcCsch[a*x], x, 3, x^2/(2*a) + (1/3)*(1 + 1/(a^2*x^2))^(3/2)*x^3} -{x^1*E^ArcCsch[a*x], x, 6, x/a + (1/2)*Sqrt[1 + 1/(a^2*x^2)]*x^2 + ArcTanh[Sqrt[1 + 1/(a^2*x^2)]]/(2*a^2)} -{x^0*E^ArcCsch[a*x], x, 5, E^ArcCsch[a*x]*x - ArcCsch[a*x]/a + Log[x]/a, Sqrt[1 + 1/(a^2*x^2)]*x - ArcCsch[a*x]/a + Log[x]/a} -{E^ArcCsch[a*x]/x^1, x, 6, -Sqrt[1 + 1/(a^2*x^2)] - 1/(a*x) + ArcTanh[Sqrt[1 + 1/(a^2*x^2)]]} -{E^ArcCsch[a*x]/x^2, x, 5, -(1/(2*a*x^2)) - Sqrt[1 + 1/(a^2*x^2)]/(2*x) - (1/2)*a*ArcCsch[a*x]} -{E^ArcCsch[a*x]/x^3, x, 3, (-(1/3))*a^2*(1 + 1/(a^2*x^2))^(3/2) - 1/(3*a*x^3)} -{E^ArcCsch[a*x]/x^4, x, 6, -(1/(4*a*x^4)) - Sqrt[1 + 1/(a^2*x^2)]/(4*x^3) - (a^2*Sqrt[1 + 1/(a^2*x^2)])/(8*x) + (1/8)*a^3*ArcCsch[a*x]} -{E^ArcCsch[a*x]/x^5, x, 5, (1/3)*a^4*(1 + 1/(a^2*x^2))^(3/2) - (1/5)*a^4*(1 + 1/(a^2*x^2))^(5/2) - 1/(5*a*x^5)} - - -{x^m*E^ArcCsch[a*x^2], x, 4, -(x^(-1 + m)/(a*(1 - m))) + (x^(1 + m)*Hypergeometric2F1[-(1/2), (1/4)*(-1 - m), (3 - m)/4, -(1/(a^2*x^4))])/(1 + m)} - -{x^4*E^ArcCsch[a*x^2], x, 8, -((2*Sqrt[1 + 1/(a^2*x^4)])/(5*a^2*(a + 1/x^2)*x)) + (2*Sqrt[1 + 1/(a^2*x^4)]*x)/(5*a^2) + x^3/(3*a) + (1/5)*Sqrt[1 + 1/(a^2*x^4)]*x^5 + (2*Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(5*a^(7/2)*Sqrt[1 + 1/(a^2*x^4)])} -{x^3*E^ArcCsch[a*x^2], x, 6, x^2/(2*a) + (1/4)*Sqrt[1 + 1/(a^2*x^4)]*x^4 + ArcTanh[Sqrt[1 + 1/(a^2*x^4)]]/(4*a^2)} -{x^2*E^ArcCsch[a*x^2], x, 5, x/a + (1/3)*Sqrt[1 + 1/(a^2*x^4)]*x^3 - (Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(3*a^(5/2)*Sqrt[1 + 1/(a^2*x^4)])} -{x^1*E^ArcCsch[a*x^2], x, 6, (1/2)*Sqrt[1 + 1/(a^2*x^4)]*x^2 - ArcCsch[a*x^2]/(2*a) + Log[x]/a} -{x^0*E^ArcCsch[a*x^2], x, 7, -(1/(a*x)) - (2*Sqrt[1 + 1/(a^2*x^4)])/((a + 1/x^2)*x) + Sqrt[1 + 1/(a^2*x^4)]*x + (2*Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(a^(3/2)*Sqrt[1 + 1/(a^2*x^4)])} -{E^ArcCsch[a*x^2]/x^1, x, 6, (-(1/2))*Sqrt[1 + 1/(a^2*x^4)] - 1/(2*a*x^2) + (1/2)*ArcTanh[Sqrt[1 + 1/(a^2*x^4)]]} -{E^ArcCsch[a*x^2]/x^2, x, 5, -(1/(3*a*x^3)) - Sqrt[1 + 1/(a^2*x^4)]/(3*x) - (Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(3*Sqrt[a]*Sqrt[1 + 1/(a^2*x^4)])} -{E^ArcCsch[a*x^2]/x^3, x, 6, -(1/(4*a*x^4)) - Sqrt[1 + 1/(a^2*x^4)]/(4*x^2) - (1/4)*a*ArcCsch[a*x^2]} -{E^ArcCsch[a*x^2]/x^4, x, 7, -(1/(5*a*x^5)) - Sqrt[1 + 1/(a^2*x^4)]/(5*x^3) - (2*a^2*Sqrt[1 + 1/(a^2*x^4)])/(5*(a + 1/x^2)*x) + (2*Sqrt[a]*Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticE[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 + 1/(a^2*x^4)]) - (Sqrt[a]*Sqrt[(a^2 + 1/x^4)/(a + 1/x^2)^2]*(a + 1/x^2)*EllipticF[2*ArcCot[Sqrt[a]*x], 1/2])/(5*Sqrt[1 + 1/(a^2*x^4)])} -{E^ArcCsch[a*x^2]/x^5, x, 3, (-(1/6))*a^2*(1 + 1/(a^2*x^4))^(3/2) - 1/(6*a*x^6)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCsch[a x])*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^m*E^(2*ArcCsch[a*x]), x, 5, -((2*x^(-1 + m))/(a^2*(1 - m))) + x^(1 + m)/(1 + m) + (2*x^m*Hypergeometric2F1[-(1/2), -(m/2), 1 - m/2, -(1/(a^2*x^2))])/(a*m)} - -{x^4*E^(2*ArcCsch[a*x]), x, 8, (Sqrt[1 + 1/(a^2*x^2)]*x^2)/(4*a^3) + (2*x^3)/(3*a^2) + (Sqrt[1 + 1/(a^2*x^2)]*x^4)/(2*a) + x^5/5 - ArcTanh[Sqrt[1 + 1/(a^2*x^2)]]/(4*a^5)} -{x^3*E^(2*ArcCsch[a*x]), x, 4, x^2/a^2 + (2*(1 + 1/(a^2*x^2))^(3/2)*x^3)/(3*a) + x^4/4} -{x^2*E^(2*ArcCsch[a*x]), x, 7, (2*x)/a^2 + (Sqrt[1 + 1/(a^2*x^2)]*x^2)/a + x^3/3 + ArcTanh[Sqrt[1 + 1/(a^2*x^2)]]/a^3} -{x^1*E^(2*ArcCsch[a*x]), x, 6, (2*Sqrt[1 + 1/(a^2*x^2)]*x)/a + x^2/2 - (2*ArcCsch[a*x])/a^2 + (2*Log[x])/a^2} -{x^0*E^(2*ArcCsch[a*x]), x, 7, -((2*Sqrt[1 + 1/(a^2*x^2)])/a) - 2/(a^2*x) + x + (2*ArcTanh[Sqrt[1 + 1/(a^2*x^2)]])/a} -{E^(2*ArcCsch[a*x])/x^1, x, 6, -(1/(a^2*x^2)) - Sqrt[1 + 1/(a^2*x^2)]/(a*x) - ArcCsch[a*x] + Log[x]} -{E^(2*ArcCsch[a*x])/x^2, x, 4, (-(2/3))*a*(1 + 1/(a^2*x^2))^(3/2) - 2/(3*a^2*x^3) - 1/x, (-(1/2))*a*Sqrt[1 + 1/(a^2*x^2)] - (1/6)*a*(Sqrt[1 + 1/(a^2*x^2)] + 1/(a*x))^3 - 1/(2*x)} -{E^(2*ArcCsch[a*x])/x^3, x, 7, -(1/(2*a^2*x^4)) - Sqrt[1 + 1/(a^2*x^2)]/(2*a*x^3) - 1/(2*x^2) - (a*Sqrt[1 + 1/(a^2*x^2)])/(4*x) + (1/4)*a^2*ArcCsch[a*x]} -{E^(2*ArcCsch[a*x])/x^4, x, 6, (2/3)*a^3*(1 + 1/(a^2*x^2))^(3/2) - (2/5)*a^3*(1 + 1/(a^2*x^2))^(5/2) - 2/(5*a^2*x^5) - 1/(3*x^3)} -{E^(2*ArcCsch[a*x])/x^5, x, 8, -(1/(3*a^2*x^6)) - Sqrt[1 + 1/(a^2*x^2)]/(3*a*x^5) - 1/(4*x^4) - (a*Sqrt[1 + 1/(a^2*x^2)])/(12*x^3) + (a^3*Sqrt[1 + 1/(a^2*x^2)])/(8*x) - (1/8)*a^4*ArcCsch[a*x]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(n ArcCsch[a x]) / (1-a^2 x^2)*) - - -{(d*x)^m*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 4, -((d*(d*x)^(-1 + m)*Hypergeometric2F1[1/2, (1 - m)/2, (3 - m)/2, -(1/(c^2*x^2))])/(c^2*(1 - m))) + ((d*x)^m*Hypergeometric2F1[1, m/2, (2 + m)/2, (-c^2)*x^2])/(c*m)} - - -{x^5*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 9, -(x/c^5) - (3*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(8*c^4) + x^3/(3*c^3) + (Sqrt[1 + 1/(c^2*x^2)]*x^4)/(4*c^2) + ArcTan[c*x]/c^6 + (3*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(8*c^6)} -{x^4*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 6, -((2*Sqrt[1 + 1/(c^2*x^2)]*x)/(3*c^4)) + x^2/(2*c^3) + (Sqrt[1 + 1/(c^2*x^2)]*x^3)/(3*c^2) - Log[1 + c^2*x^2]/(2*c^5)} -{x^3*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 7, x/c^3 + (Sqrt[1 + 1/(c^2*x^2)]*x^2)/(2*c^2) - ArcTan[c*x]/c^4 - ArcTanh[Sqrt[1 + 1/(c^2*x^2)]]/(2*c^4)} -{x^2*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 3, (Sqrt[1 + 1/(c^2*x^2)]*x)/c^2 + Log[1 + c^2*x^2]/(2*c^3)} -{x^1*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 5, ArcTan[c*x]/c^2 + ArcTanh[Sqrt[1 + 1/(c^2*x^2)]]/c^2} -{x^0*E^ArcCsch[c*x]/(1 + c^2*x^2), x, 7, -(ArcCsch[c*x]/c) + Log[x]/c - Log[1 + c^2*x^2]/(2*c)} -{E^ArcCsch[c*x]/(x^1*(1 + c^2*x^2)), x, 4, -Sqrt[1 + 1/(c^2*x^2)] - 1/(c*x) - ArcTan[c*x]} -{E^ArcCsch[c*x]/(x^2*(1 + c^2*x^2)), x, 7, -(1/(2*c*x^2)) - Sqrt[1 + 1/(c^2*x^2)]/(2*x) + (1/2)*c*ArcCsch[c*x] - c*Log[x] + (1/2)*c*Log[1 + c^2*x^2]} -{E^ArcCsch[c*x]/(x^3*(1 + c^2*x^2)), x, 7, c^2*Sqrt[1 + 1/(c^2*x^2)] - (1/3)*c^2*(1 + 1/(c^2*x^2))^(3/2) - 1/(3*c*x^3) + c/x + c^2*ArcTan[c*x]} - - -(* ::Section::Closed:: *) -(*Miscellaneous integrands involving inverse hyperbolic cosecants*) - - -{ArcCsch[a + b*x]/((a*d)/b + d*x), x, 8, ArcCsch[a + b*x]^2/(2*d) - (ArcCsch[a + b*x]*Log[1 - E^(2*ArcCsch[a + b*x])])/d - PolyLog[2, E^(2*ArcCsch[a + b*x])]/(2*d)} - - -{x^3*ArcCsch[a + b*x^4], x, 6, ((a + b*x^4)*ArcCsch[a + b*x^4])/(4*b) + ArcTanh[Sqrt[1 + 1/(a + b*x^4)^2]]/(4*b)} - -{x^(n-1)*ArcCsch[a + b*x^n], x, 6, ((a + b*x^n)*ArcCsch[a + b*x^n])/(b*n) + ArcTanh[Sqrt[1 + 1/(a + b*x^n)^2]]/(b*n)} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.1 Error functions.m b/test/methods/rule_based/test_files/8 Special functions/8.1 Error functions.m deleted file mode 100644 index bfb5b2e..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.1 Error functions.m +++ /dev/null @@ -1,587 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integration Problems Involving The Error Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erf[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erf[b x]*) - - -{x^5*Erf[b*x], x, 5, (5*x)/(E^(b^2*x^2)*(8*b^5*Sqrt[Pi])) + (5*x^3)/(E^(b^2*x^2)*(12*b^3*Sqrt[Pi])) + x^5/(E^(b^2*x^2)*(6*b*Sqrt[Pi])) - (5*Erf[b*x])/(16*b^6) + (1/6)*x^6*Erf[b*x]} -{x^3*Erf[b*x], x, 4, (3*x)/(E^(b^2*x^2)*(8*b^3*Sqrt[Pi])) + x^3/(E^(b^2*x^2)*(4*b*Sqrt[Pi])) - (3*Erf[b*x])/(16*b^4) + (1/4)*x^4*Erf[b*x]} -{x^1*Erf[b*x], x, 3, x/(E^(b^2*x^2)*(2*b*Sqrt[Pi])) - Erf[b*x]/(4*b^2) + (1/2)*x^2*Erf[b*x]} -{Erf[b*x]/x^1, x, 1, (2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-b^2)*x^2])/Sqrt[Pi]} -{Erf[b*x]/x^3, x, 3, -(b/(E^(b^2*x^2)*(Sqrt[Pi]*x))) - b^2*Erf[b*x] - Erf[b*x]/(2*x^2)} -{Erf[b*x]/x^5, x, 4, -(b/(E^(b^2*x^2)*(6*Sqrt[Pi]*x^3))) + b^3/(E^(b^2*x^2)*(3*Sqrt[Pi]*x)) + (1/3)*b^4*Erf[b*x] - Erf[b*x]/(4*x^4)} -{Erf[b*x]/x^7, x, 5, -(b/(E^(b^2*x^2)*(15*Sqrt[Pi]*x^5))) + (2*b^3)/(E^(b^2*x^2)*(45*Sqrt[Pi]*x^3)) - (4*b^5)/(E^(b^2*x^2)*(45*Sqrt[Pi]*x)) - (4/45)*b^6*Erf[b*x] - Erf[b*x]/(6*x^6)} - -{x^6*Erf[b*x], x, 5, 6/(E^(b^2*x^2)*(7*b^7*Sqrt[Pi])) + (6*x^2)/(E^(b^2*x^2)*(7*b^5*Sqrt[Pi])) + (3*x^4)/(E^(b^2*x^2)*(7*b^3*Sqrt[Pi])) + x^6/(E^(b^2*x^2)*(7*b*Sqrt[Pi])) + (1/7)*x^7*Erf[b*x]} -{x^4*Erf[b*x], x, 4, 2/(E^(b^2*x^2)*(5*b^5*Sqrt[Pi])) + (2*x^2)/(E^(b^2*x^2)*(5*b^3*Sqrt[Pi])) + x^4/(E^(b^2*x^2)*(5*b*Sqrt[Pi])) + (1/5)*x^5*Erf[b*x]} -{x^2*Erf[b*x], x, 3, 1/(E^(b^2*x^2)*(3*b^3*Sqrt[Pi])) + x^2/(E^(b^2*x^2)*(3*b*Sqrt[Pi])) + (1/3)*x^3*Erf[b*x]} -{x^0*Erf[b*x], x, 1, 1/(E^(b^2*x^2)*(b*Sqrt[Pi])) + x*Erf[b*x]} -{Erf[b*x]/x^2, x, 2, -(Erf[b*x]/x) + (b*ExpIntegralEi[(-b^2)*x^2])/Sqrt[Pi]} -{Erf[b*x]/x^4, x, 3, -(b/(E^(b^2*x^2)*(3*Sqrt[Pi]*x^2))) - Erf[b*x]/(3*x^3) - (b^3*ExpIntegralEi[(-b^2)*x^2])/(3*Sqrt[Pi])} -{Erf[b*x]/x^6, x, 4, -(b/(E^(b^2*x^2)*(10*Sqrt[Pi]*x^4))) + b^3/(E^(b^2*x^2)*(10*Sqrt[Pi]*x^2)) - Erf[b*x]/(5*x^5) + (b^5*ExpIntegralEi[(-b^2)*x^2])/(10*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erf[a+b x]*) - - -{(c + d*x)^3*Erf[a + b*x], x, 12, (d^2*(b*c - a*d))/(E^(a + b*x)^2*(b^4*Sqrt[Pi])) + (b*c - a*d)^3/(E^(a + b*x)^2*(b^4*Sqrt[Pi])) + (3*d^3*(a + b*x))/(E^(a + b*x)^2*(8*b^4*Sqrt[Pi])) + (3*d*(b*c - a*d)^2*(a + b*x))/(E^(a + b*x)^2*(2*b^4*Sqrt[Pi])) + (d^2*(b*c - a*d)*(a + b*x)^2)/(E^(a + b*x)^2*(b^4*Sqrt[Pi])) + (d^3*(a + b*x)^3)/(E^(a + b*x)^2*(4*b^4*Sqrt[Pi])) - (3*d^3*Erf[a + b*x])/(16*b^4) - (3*d*(b*c - a*d)^2*Erf[a + b*x])/(4*b^4) - ((b*c - a*d)^4*Erf[a + b*x])/(4*b^4*d) + ((c + d*x)^4*Erf[a + b*x])/(4*d)} -{(c + d*x)^2*Erf[a + b*x], x, 9, d^2/(E^(a + b*x)^2*(3*b^3*Sqrt[Pi])) + (b*c - a*d)^2/(E^(a + b*x)^2*(b^3*Sqrt[Pi])) + (d*(b*c - a*d)*(a + b*x))/(E^(a + b*x)^2*(b^3*Sqrt[Pi])) + (d^2*(a + b*x)^2)/(E^(a + b*x)^2*(3*b^3*Sqrt[Pi])) - (d*(b*c - a*d)*Erf[a + b*x])/(2*b^3) - ((b*c - a*d)^3*Erf[a + b*x])/(3*b^3*d) + ((c + d*x)^3*Erf[a + b*x])/(3*d)} -{(c + d*x)^1*Erf[a + b*x], x, 7, (b*c - a*d)/(E^(a + b*x)^2*(b^2*Sqrt[Pi])) + (d*(a + b*x))/(E^(a + b*x)^2*(2*b^2*Sqrt[Pi])) - (d*Erf[a + b*x])/(4*b^2) - ((b*c - a*d)^2*Erf[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erf[a + b*x])/(2*d)} -{(c + d*x)^0*Erf[a + b*x], x, 1, 1/(E^(a + b*x)^2*(b*Sqrt[Pi])) + ((a + b*x)*Erf[a + b*x])/b} -{Erf[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Erf[a + b*x]/(c + d*x), x]} -{Erf[a + b*x]/(c + d*x)^2, x, 1, -(Erf[a + b*x]/(d*(c + d*x))) + (2*b*Unintegrable[1/(E^(a + b*x)^2*(c + d*x)), x])/(d*Sqrt[Pi])} -{Erf[a + b*x]/(c + d*x)^3, x, 3, -(b/(E^(a + b*x)^2*(d^2*Sqrt[Pi]*(c + d*x)))) - (b^2*Erf[a + b*x])/d^3 - Erf[a + b*x]/(2*d*(c + d*x)^2) + (2*b^2*(b*c - a*d)*Unintegrable[1/(E^(a + b*x)^2*(c + d*x)), x])/(d^3*Sqrt[Pi])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erf[a+b x]^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erf[b x]^2*) - - -{x^5*Erf[b*x]^2, x, 12, 11/(E^(2*b^2*x^2)*(12*b^6*Pi)) + (7*x^2)/(E^(2*b^2*x^2)*(12*b^4*Pi)) + x^4/(E^(2*b^2*x^2)*(6*b^2*Pi)) + (5*x*Erf[b*x])/(E^(b^2*x^2)*(4*b^5*Sqrt[Pi])) + (5*x^3*Erf[b*x])/(E^(b^2*x^2)*(6*b^3*Sqrt[Pi])) + (x^5*Erf[b*x])/(E^(b^2*x^2)*(3*b*Sqrt[Pi])) - (5*Erf[b*x]^2)/(16*b^6) + (1/6)*x^6*Erf[b*x]^2} -{x^3*Erf[b*x]^2, x, 8, 1/(E^(2*b^2*x^2)*(2*b^4*Pi)) + x^2/(E^(2*b^2*x^2)*(4*b^2*Pi)) + (3*x*Erf[b*x])/(E^(b^2*x^2)*(4*b^3*Sqrt[Pi])) + (x^3*Erf[b*x])/(E^(b^2*x^2)*(2*b*Sqrt[Pi])) - (3*Erf[b*x]^2)/(16*b^4) + (1/4)*x^4*Erf[b*x]^2} -{x^1*Erf[b*x]^2, x, 5, 1/(E^(2*b^2*x^2)*(2*b^2*Pi)) + (x*Erf[b*x])/(E^(b^2*x^2)*(b*Sqrt[Pi])) - Erf[b*x]^2/(4*b^2) + (1/2)*x^2*Erf[b*x]^2} -{Erf[b*x]^2/x^1, x, 0, Unintegrable[Erf[b*x]^2/x, x]} -{Erf[b*x]^2/x^3, x, 5, -((2*b*Erf[b*x])/(E^(b^2*x^2)*(Sqrt[Pi]*x))) - b^2*Erf[b*x]^2 - Erf[b*x]^2/(2*x^2) + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/Pi} -{Erf[b*x]^2/x^5, x, 8, -(b^2/(E^(2*b^2*x^2)*(3*Pi*x^2))) - (b*Erf[b*x])/(E^(b^2*x^2)*(3*Sqrt[Pi]*x^3)) + (2*b^3*Erf[b*x])/(E^(b^2*x^2)*(3*Sqrt[Pi]*x)) + (1/3)*b^4*Erf[b*x]^2 - Erf[b*x]^2/(4*x^4) - (4*b^4*ExpIntegralEi[-2*b^2*x^2])/(3*Pi)} -{Erf[b*x]^2/x^7, x, 12, -(b^2/(E^(2*b^2*x^2)*(15*Pi*x^4))) + (2*b^4)/(E^(2*b^2*x^2)*(9*Pi*x^2)) - (2*b*Erf[b*x])/(E^(b^2*x^2)*(15*Sqrt[Pi]*x^5)) + (4*b^3*Erf[b*x])/(E^(b^2*x^2)*(45*Sqrt[Pi]*x^3)) - (8*b^5*Erf[b*x])/(E^(b^2*x^2)*(45*Sqrt[Pi]*x)) - (4/45)*b^6*Erf[b*x]^2 - Erf[b*x]^2/(6*x^6) + (28*b^6*ExpIntegralEi[-2*b^2*x^2])/(45*Pi)} - -{x^4*Erf[b*x]^2, x, 10, (11*x)/(E^(2*b^2*x^2)*(20*b^4*Pi)) + x^3/(E^(2*b^2*x^2)*(5*b^2*Pi)) + (4*Erf[b*x])/(E^(b^2*x^2)*(5*b^5*Sqrt[Pi])) + (4*x^2*Erf[b*x])/(E^(b^2*x^2)*(5*b^3*Sqrt[Pi])) + (2*x^4*Erf[b*x])/(E^(b^2*x^2)*(5*b*Sqrt[Pi])) + (1/5)*x^5*Erf[b*x]^2 - (43*Erf[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])} -{x^2*Erf[b*x]^2, x, 6, x/(E^(2*b^2*x^2)*(3*b^2*Pi)) + (2*Erf[b*x])/(E^(b^2*x^2)*(3*b^3*Sqrt[Pi])) + (2*x^2*Erf[b*x])/(E^(b^2*x^2)*(3*b*Sqrt[Pi])) + (1/3)*x^3*Erf[b*x]^2 - (5*Erf[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])} -{x^0*Erf[b*x]^2, x, 4, (2*Erf[b*x])/(E^(b^2*x^2)*(b*Sqrt[Pi])) + x*Erf[b*x]^2 - (Sqrt[2/Pi]*Erf[Sqrt[2]*b*x])/b} -{Erf[b*x]^2/x^2, x, 0, Unintegrable[Erf[b*x]^2/x^2, x]} -{Erf[b*x]^2/x^4, x, 0, Unintegrable[Erf[b*x]^2/x^4, x]} -{Erf[b*x]^2/x^6, x, 0, Unintegrable[Erf[b*x]^2/x^6, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erf[a+b x]^2*) - - -{(c + d*x)^2*Erf[a + b*x]^2, x, 16, (d*(b*c - a*d))/(E^(2*(a + b*x)^2)*(b^3*Pi)) + (d^2*(a + b*x))/(E^(2*(a + b*x)^2)*(3*b^3*Pi)) + (2*d^2*Erf[a + b*x])/(E^(a + b*x)^2*(3*b^3*Sqrt[Pi])) + (2*(b*c - a*d)^2*Erf[a + b*x])/(E^(a + b*x)^2*(b^3*Sqrt[Pi])) + (2*d*(b*c - a*d)*(a + b*x)*Erf[a + b*x])/(E^(a + b*x)^2*(b^3*Sqrt[Pi])) + (2*d^2*(a + b*x)^2*Erf[a + b*x])/(E^(a + b*x)^2*(3*b^3*Sqrt[Pi])) - (d*(b*c - a*d)*Erf[a + b*x]^2)/(2*b^3) + ((b*c - a*d)^2*(a + b*x)*Erf[a + b*x]^2)/b^3 + (d*(b*c - a*d)*(a + b*x)^2*Erf[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*Erf[a + b*x]^2)/(3*b^3) - ((b*c - a*d)^2*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b^3 - (5*d^2*Erf[Sqrt[2]*(a + b*x)])/(6*b^3*Sqrt[2*Pi])} -{(c + d*x)^1*Erf[a + b*x]^2, x, 10, d/(E^(2*(a + b*x)^2)*(2*b^2*Pi)) + (2*(b*c - a*d)*Erf[a + b*x])/(E^(a + b*x)^2*(b^2*Sqrt[Pi])) + (d*(a + b*x)*Erf[a + b*x])/(E^(a + b*x)^2*(b^2*Sqrt[Pi])) - (d*Erf[a + b*x]^2)/(4*b^2) + ((b*c - a*d)*(a + b*x)*Erf[a + b*x]^2)/b^2 + (d*(a + b*x)^2*Erf[a + b*x]^2)/(2*b^2) - ((b*c - a*d)*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b^2} -{(c + d*x)^0*Erf[a + b*x]^2, x, 4, (2*Erf[a + b*x])/(E^(a + b*x)^2*(b*Sqrt[Pi])) + ((a + b*x)*Erf[a + b*x]^2)/b - (Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b} -{Erf[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Erf[a + b*x]^2/(c + d*x), x]} -{Erf[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Erf[a + b*x]^2/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Erf[d (a+b Log[c x^n])]*) - - -{x^2*Erf[d*(a + b*Log[c*x^n])], x, 5, (x^3*Erf[d*(a + b*Log[c*x^n])])/3 - (E^((9 - 12*a*b*d^2*n)/(4*b^2*d^2*n^2))*x^3*Erf[(2*a*b*d^2 - 3/n + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(3*(c*x^n)^(3/n))} -{x^1*Erf[d*(a + b*Log[c*x^n])], x, 5, (x^2*Erf[d*(a + b*Log[c*x^n])])/2 - (E^((1 - 2*a*b*d^2*n)/(b^2*d^2*n^2))*x^2*Erf[(a*b*d^2 - n^(-1) + b^2*d^2*Log[c*x^n])/(b*d)])/(2*(c*x^n)^(2/n))} -{x^0*Erf[d*(a + b*Log[c*x^n])], x, 5, x*Erf[d*(a + b*Log[c*x^n])] - (E^((1 - 4*a*b*d^2*n)/(4*b^2*d^2*n^2))*x*Erf[(2*a*b*d^2 - n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(c*x^n)^n^(-1)} -{Erf[d*(a + b*Log[c*x^n])]/x^1, x, 3, 1/(b*d*E^(d^2*(a + b*Log[c*x^n])^2)*n*Sqrt[Pi]) + (Erf[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)} -{Erf[d*(a + b*Log[c*x^n])]/x^2, x, 5, -(Erf[d*(a + b*Log[c*x^n])]/x) + (E^(1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erf[(2*a*b*d^2 + n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/x} -{Erf[d*(a + b*Log[c*x^n])]/x^3, x, 5, -Erf[d*(a + b*Log[c*x^n])]/(2*x^2) + (E^((1 + 2*a*b*d^2*n)/(b^2*d^2*n^2))*(c*x^n)^(2/n)*Erf[(1 + a*b*d^2*n + b^2*d^2*n*Log[c*x^n])/(b*d*n)])/(2*x^2)} - - -{(e*x)^m*Erf[d*(a + b*Log[c*x^n])], x, 5, ((e*x)^(1 + m)*Erf[d*(a + b*Log[c*x^n])])/(e*(1 + m)) + (E^(((1 + m)*(1 + m - 4*a*b*d^2*n))/(4*b^2*d^2*n^2))*x*(e*x)^m*Erf[(1 + m - 2*a*b*d^2*n - 2*b^2*d^2*n*Log[c*x^n])/(2*b*d*n)])/((1 + m)*(c*x^n)^((1 + m)/n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m E^(c+d x^2) Erf[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c-b^2 x^2) Erf[b x]^n*) - - -{E^(c - b^2*x^2)*Erf[b*x]^2, x, 2, (E^c*Sqrt[Pi]*Erf[b*x]^3)/(6*b)} -{E^(c - b^2*x^2)*Erf[b*x]^1, x, 2, (E^c*Sqrt[Pi]*Erf[b*x]^2)/(4*b)} -{E^(c - b^2*x^2)/Erf[b*x]^1, x, 2, (E^c*Sqrt[Pi]*Log[Erf[b*x]])/(2*b)} -{E^(c - b^2*x^2)/Erf[b*x]^2, x, 2, -((E^c*Sqrt[Pi])/(2*b*Erf[b*x]))} -{E^(c - b^2*x^2)/Erf[b*x]^3, x, 2, -((E^c*Sqrt[Pi])/(4*b*Erf[b*x]^2))} - - -{E^(c - b^2*x^2)*Erf[b*x]^n, x, 2, (E^c*Sqrt[Pi]*Erf[b*x]^(1 + n))/(2*b*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erf[b x]*) - - -{x^5*E^(c + d*x^2)*Erf[b*x], x, 9, -((b*E^(c - (b^2 - d)*x^2)*x)/((b^2 - d)*d^2*Sqrt[Pi])) + (3*b*E^(c - (b^2 - d)*x^2)*x)/(4*(b^2 - d)^2*d*Sqrt[Pi]) + (b*E^(c - (b^2 - d)*x^2)*x^3)/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*Erf[b*x])/d^3 - (E^(c + d*x^2)*x^2*Erf[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erf[b*x])/(2*d) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(Sqrt[b^2 - d]*d^3) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*(b^2 - d)^(3/2)*d^2) - (3*b*E^c*Erf[Sqrt[b^2 - d]*x])/(8*(b^2 - d)^(5/2)*d)} -{x^3*E^(c + d*x^2)*Erf[b*x], x, 5, (b*E^(c - (b^2 - d)*x^2)*x)/(2*(b^2 - d)*d*Sqrt[Pi]) - (E^(c + d*x^2)*Erf[b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erf[b*x])/(2*d) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d^2) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(4*(b^2 - d)^(3/2)*d)} -{x^1*E^(c + d*x^2)*Erf[b*x], x, 2, (E^(c + d*x^2)*Erf[b*x])/(2*d) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d)} -{E^(c + d*x^2)*Erf[b*x]/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erf[b*x])/x, x]} -{E^(c + d*x^2)*Erf[b*x]/x^3, x, 3, -((b*E^(c - (b^2 - d)*x^2))/(Sqrt[Pi]*x)) - (E^(c + d*x^2)*Erf[b*x])/(2*x^2) - b*Sqrt[b^2 - d]*E^c*Erf[Sqrt[b^2 - d]*x] + d*Unintegrable[(E^(c + d*x^2)*Erf[b*x])/x, x]} -{E^(c + d*x^2)*Erf[b*x]/x^5, x, 7, -((b*E^(c - (b^2 - d)*x^2))/(6*Sqrt[Pi]*x^3)) + (b*(b^2 - d)*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x) - (b*d*E^(c - (b^2 - d)*x^2))/(2*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erf[b*x])/(4*x^4) - (d*E^(c + d*x^2)*Erf[b*x])/(4*x^2) + (1/3)*b*(b^2 - d)^(3/2)*E^c*Erf[Sqrt[b^2 - d]*x] - (1/2)*b*Sqrt[b^2 - d]*d*E^c*Erf[Sqrt[b^2 - d]*x] + (1/2)*d^2*Unintegrable[(E^(c + d*x^2)*Erf[b*x])/x, x]} - -{x^4*E^(c + d*x^2)*Erf[b*x], x, 5, -((3*b*E^(c - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi])) + (b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]) + (b*E^(c - (b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erf[b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erf[b*x])/(2*d) + (3*Unintegrable[E^(c + d*x^2)*Erf[b*x], x])/(4*d^2)} -{x^2*E^(c + d*x^2)*Erf[b*x], x, 2, (b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erf[b*x])/(2*d) - Unintegrable[E^(c + d*x^2)*Erf[b*x], x]/(2*d)} -{x^0*E^(c + d*x^2)*Erf[b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erf[b*x], x]} -{E^(c + d*x^2)*Erf[b*x]/x^2, x, 2, -((E^(c + d*x^2)*Erf[b*x])/x) + (b*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erf[b*x], x]} -{E^(c + d*x^2)*Erf[b*x]/x^4, x, 5, -((b*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2)) - (E^(c + d*x^2)*Erf[b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erf[b*x])/(3*x) - (b*(b^2 - d)*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) + (2*b*d*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) + (4/3)*d^2*Unintegrable[E^(c + d*x^2)*Erf[b*x], x]} - - -{x^5*E^(c + b^2*x^2)*Erf[b*x], x, 8, -((2*E^c*x)/(b^5*Sqrt[Pi])) + (2*E^c*x^3)/(3*b^3*Sqrt[Pi]) - (E^c*x^5)/(5*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erf[b*x])/b^6 - (E^(c + b^2*x^2)*x^2*Erf[b*x])/b^4 + (E^(c + b^2*x^2)*x^4*Erf[b*x])/(2*b^2)} -{x^3*E^(c + b^2*x^2)*Erf[b*x], x, 5, (E^c*x)/(b^3*Sqrt[Pi]) - (E^c*x^3)/(3*b*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erf[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erf[b*x])/(2*b^2)} -{x^1*E^(c + b^2*x^2)*Erf[b*x], x, 2, -((E^c*x)/(b*Sqrt[Pi])) + (E^(c + b^2*x^2)*Erf[b*x])/(2*b^2)} -{E^(c + b^2*x^2)*Erf[b*x]/x^1, x, 1, (2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} -{E^(c + b^2*x^2)*Erf[b*x]/x^3, x, 4, -((b*E^c)/(Sqrt[Pi]*x)) - (E^(c + b^2*x^2)*Erf[b*x])/(2*x^2) + (2*b^3*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} -{E^(c + b^2*x^2)*Erf[b*x]/x^5, x, 7, -((b*E^c)/(6*Sqrt[Pi]*x^3)) - (b^3*E^c)/(2*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erf[b*x])/(4*x^4) - (b^2*E^(c + b^2*x^2)*Erf[b*x])/(4*x^2) + (b^5*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} - -{x^4*E^(c + b^2*x^2)*Erf[b*x], x, 7, (3*E^c*x^2)/(4*b^3*Sqrt[Pi]) - (E^c*x^4)/(4*b*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erf[b*x])/(4*b^4) + (E^(c + b^2*x^2)*x^3*Erf[b*x])/(2*b^2) + (3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(4*b^3*Sqrt[Pi])} -{x^2*E^(c + b^2*x^2)*Erf[b*x], x, 4, -((E^c*x^2)/(2*b*Sqrt[Pi])) + (E^(c + b^2*x^2)*x*Erf[b*x])/(2*b^2) - (E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*b*Sqrt[Pi])} -{x^0*E^(c + b^2*x^2)*Erf[b*x], x, 1, (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]} -{E^(c + b^2*x^2)*Erf[b*x]/x^2, x, 4, -((E^(c + b^2*x^2)*Erf[b*x])/x) + (2*b^3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi] + (2*b*E^c*Log[x])/Sqrt[Pi]} -{E^(c + b^2*x^2)*Erf[b*x]/x^4, x, 7, -((b*E^c)/(3*Sqrt[Pi]*x^2)) - (E^(c + b^2*x^2)*Erf[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erf[b*x])/(3*x) + (4*b^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) + (4*b^3*E^c*Log[x])/(3*Sqrt[Pi])} - - -{x^5*E^(-b^2*x^2)*Erf[b*x], x, 9, -((11*x)/(E^(2*b^2*x^2)*(16*b^5*Sqrt[Pi]))) - x^3/(E^(2*b^2*x^2)*(4*b^3*Sqrt[Pi])) - Erf[b*x]/(E^(b^2*x^2)*b^6) - (x^2*Erf[b*x])/(E^(b^2*x^2)*b^4) - (x^4*Erf[b*x])/(E^(b^2*x^2)*(2*b^2)) + (43*Erf[Sqrt[2]*b*x])/(32*Sqrt[2]*b^6)} -{x^3*E^(-b^2*x^2)*Erf[b*x], x, 5, -(x/(E^(2*b^2*x^2)*(4*b^3*Sqrt[Pi]))) - Erf[b*x]/(E^(b^2*x^2)*(2*b^4)) - (x^2*Erf[b*x])/(E^(b^2*x^2)*(2*b^2)) + (5*Erf[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)} -{x^1*E^(-b^2*x^2)*Erf[b*x], x, 2, -(Erf[b*x]/(E^(b^2*x^2)*(2*b^2))) + Erf[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2)} -{E^(-b^2*x^2)*Erf[b*x]/x^1, x, 0, Unintegrable[Erf[b*x]/(E^(b^2*x^2)*x), x]} -{E^(-b^2*x^2)*Erf[b*x]/x^3, x, 3, -(b/(E^(2*b^2*x^2)*(Sqrt[Pi]*x))) - Erf[b*x]/(E^(b^2*x^2)*(2*x^2)) - Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - b^2*Unintegrable[Erf[b*x]/(E^(b^2*x^2)*x), x]} -{E^(-b^2*x^2)*Erf[b*x]/x^5, x, 7, -(b/(E^(2*b^2*x^2)*(6*Sqrt[Pi]*x^3))) + (7*b^3)/(E^(2*b^2*x^2)*(6*Sqrt[Pi]*x)) - Erf[b*x]/(E^(b^2*x^2)*(4*x^4)) + (b^2*Erf[b*x])/(E^(b^2*x^2)*(4*x^2)) + (b^4*Erf[Sqrt[2]*b*x])/Sqrt[2] + (2/3)*Sqrt[2]*b^4*Erf[Sqrt[2]*b*x] + (1/2)*b^4*Unintegrable[Erf[b*x]/(E^(b^2*x^2)*x), x]} - -{x^4*E^(-b^2*x^2)*Erf[b*x], x, 7, -(1/(E^(2*b^2*x^2)*(2*b^5*Sqrt[Pi]))) - x^2/(E^(2*b^2*x^2)*(4*b^3*Sqrt[Pi])) - (3*x*Erf[b*x])/(E^(b^2*x^2)*(4*b^4)) - (x^3*Erf[b*x])/(E^(b^2*x^2)*(2*b^2)) + (3*Sqrt[Pi]*Erf[b*x]^2)/(16*b^5)} -{x^2*E^(-b^2*x^2)*Erf[b*x], x, 4, -(1/(E^(2*b^2*x^2)*(4*b^3*Sqrt[Pi]))) - (x*Erf[b*x])/(E^(b^2*x^2)*(2*b^2)) + (Sqrt[Pi]*Erf[b*x]^2)/(8*b^3)} -{x^0*E^(-b^2*x^2)*Erf[b*x], x, 2, (Sqrt[Pi]*Erf[b*x]^2)/(4*b)} -{E^(-b^2*x^2)*Erf[b*x]/x^2, x, 4, -(Erf[b*x]/(E^(b^2*x^2)*x)) - (1/2)*b*Sqrt[Pi]*Erf[b*x]^2 + (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]} -{E^(-b^2*x^2)*Erf[b*x]/x^4, x, 7, -(b/(E^(2*b^2*x^2)*(3*Sqrt[Pi]*x^2))) - Erf[b*x]/(E^(b^2*x^2)*(3*x^3)) + (2*b^2*Erf[b*x])/(E^(b^2*x^2)*(3*x)) + (1/3)*b^3*Sqrt[Pi]*Erf[b*x]^2 - (4*b^3*ExpIntegralEi[-2*b^2*x^2])/(3*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erf[a+b x]*) - - -{x^3*E^(c + d*x^2)*Erf[a + b*x], x, 10, -((a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi])) + (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)*d*Sqrt[Pi]) - (E^(c + d*x^2)*Erf[a + b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erf[a + b*x])/(2*d) + (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*Sqrt[b^2 - d]*d^2) - (a^2*b^3*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(5/2)*d) - (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(3/2)*d)} -{x^1*E^(c + d*x^2)*Erf[a + b*x], x, 3, (E^(c + d*x^2)*Erf[a + b*x])/(2*d) - (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*Sqrt[b^2 - d]*d)} -{E^(c + d*x^2)*Erf[a + b*x]/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erf[a + b*x])/x, x]} -{E^(c + d*x^2)*Erf[a + b*x]/x^3, x, 4, -((b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(Sqrt[Pi]*x)) - (E^(c + d*x^2)*Erf[a + b*x])/(2*x^2) - b*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]] - (2*a*b^2*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi] + d*Unintegrable[(E^(c + d*x^2)*Erf[a + b*x])/x, x]} - -{x^4*E^(c + d*x^2)*Erf[a + b*x], x, 15, -((3*b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi])) + (a^2*b^3*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^3*d*Sqrt[Pi]) + (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)^2*d*Sqrt[Pi]) + (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erf[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erf[a + b*x])/(2*d) - (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(3/2)*d^2) + (a^3*b^4*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(7/2)*d) + (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(5/2)*d) + (3*Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x])/(4*d^2)} -{x^2*E^(c + d*x^2)*Erf[a + b*x], x, 4, (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erf[a + b*x])/(2*d) + (a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(3/2)*d) - Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x]/(2*d)} -{x^0*E^(c + d*x^2)*Erf[a + b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x]} -{E^(c + d*x^2)*Erf[a + b*x]/x^2, x, 1, -((E^(c + d*x^2)*Erf[a + b*x])/x) + (2*b*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x]} -{E^(c + d*x^2)*Erf[a + b*x]/x^4, x, 6, -((b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2)) + (2*a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erf[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erf[a + b*x])/(3*x) + (2/3)*a*b^2*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]] + (4*a^2*b^3*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) - (2*b*(b^2 - d)*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4*b*d*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4/3)*d^2*Unintegrable[E^(c + d*x^2)*Erf[a + b*x], x]} - - -{Erf[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erf[b*x])/(E^(b^2*x^2)*x), x, 4, -(b/(E^(2*b^2*x^2)*(Sqrt[Pi]*x))) - Erf[b*x]/(E^(b^2*x^2)*(2*x^2)) - Sqrt[2]*b^2*Erf[Sqrt[2]*b*x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Trig[c+d x^2] Erf[a+b x]^n*) - - -{Sin[c + I*b^2*x^2]*Erf[b*x], x, 4, -((I*E^(I*c)*Sqrt[Pi]*Erf[b*x]^2)/(8*b)) + (I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} -{Sin[c - I*b^2*x^2]*Erf[b*x], x, 4, (I*Sqrt[Pi]*Erf[b*x]^2)/(E^(I*c)*(8*b)) - (I*b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} - - -{Cos[c + I*b^2*x^2]*Erf[b*x], x, 4, (E^(I*c)*Sqrt[Pi]*Erf[b*x]^2)/(8*b) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} -{Cos[c - I*b^2*x^2]*Erf[b*x], x, 4, (Sqrt[Pi]*Erf[b*x]^2)/(E^(I*c)*(8*b)) + (b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} - - -{Sinh[c + b^2*x^2]*Erf[b*x], x, 4, -((Sqrt[Pi]*Erf[b*x]^2)/(E^c*(8*b))) + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} -{Sinh[c - b^2*x^2]*Erf[b*x], x, 4, (E^c*Sqrt[Pi]*Erf[b*x]^2)/(8*b) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^c*(2*Sqrt[Pi]))} - - -{Cosh[c + b^2*x^2]*Erf[b*x], x, 4, (Sqrt[Pi]*Erf[b*x]^2)/(E^c*(8*b)) + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} -{Cosh[c - b^2*x^2]*Erf[b*x], x, 4, (E^c*Sqrt[Pi]*Erf[b*x]^2)/(8*b) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^c*(2*Sqrt[Pi]))} - - -(* ::Title::Closed:: *) -(*Integration Problems Involving The Complementary Error Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erfc[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erfc[b x]*) - - -{x^5*Erfc[b*x], x, 5, (-5*x)/(8*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3)/(12*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^5/(6*b*E^(b^2*x^2)*Sqrt[Pi]) + (5*Erf[b*x])/(16*b^6) + (x^6*Erfc[b*x])/6} -{x^3*Erfc[b*x], x, 4, (-3*x)/(8*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^3/(4*b*E^(b^2*x^2)*Sqrt[Pi]) + (3*Erf[b*x])/(16*b^4) + (x^4*Erfc[b*x])/4} -{x^1*Erfc[b*x], x, 3, -x/(2*b*E^(b^2*x^2)*Sqrt[Pi]) + Erf[b*x]/(4*b^2) + (x^2*Erfc[b*x])/2} -{Erfc[b*x]/x^1, x, 2, (-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + Log[x]} -{Erfc[b*x]/x^3, x, 3, b/(E^(b^2*x^2)*Sqrt[Pi]*x) + b^2*Erf[b*x] - Erfc[b*x]/(2*x^2)} -{Erfc[b*x]/x^5, x, 4, b/(6*E^(b^2*x^2)*Sqrt[Pi]*x^3) - b^3/(3*E^(b^2*x^2)*Sqrt[Pi]*x) - (b^4*Erf[b*x])/3 - Erfc[b*x]/(4*x^4)} -{Erfc[b*x]/x^7, x, 5, b/(15*E^(b^2*x^2)*Sqrt[Pi]*x^5) - (2*b^3)/(45*E^(b^2*x^2)*Sqrt[Pi]*x^3) + (4*b^5)/(45*E^(b^2*x^2)*Sqrt[Pi]*x) + (4*b^6*Erf[b*x])/45 - Erfc[b*x]/(6*x^6)} - -{x^6*Erfc[b*x], x, 5, -6/(7*b^7*E^(b^2*x^2)*Sqrt[Pi]) - (6*x^2)/(7*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (3*x^4)/(7*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^6/(7*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^7*Erfc[b*x])/7} -{x^4*Erfc[b*x], x, 4, -2/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (2*x^2)/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^4/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erfc[b*x])/5} -{x^2*Erfc[b*x], x, 3, -1/(3*b^3*E^(b^2*x^2)*Sqrt[Pi]) - x^2/(3*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^3*Erfc[b*x])/3} -{x^0*Erfc[b*x], x, 1, -(1/(b*E^(b^2*x^2)*Sqrt[Pi])) + x*Erfc[b*x]} -{Erfc[b*x]/x^2, x, 2, -(Erfc[b*x]/x) - (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]} -{Erfc[b*x]/x^4, x, 3, b/(3*E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(3*x^3) + (b^3*ExpIntegralEi[-(b^2*x^2)])/(3*Sqrt[Pi])} -{Erfc[b*x]/x^6, x, 4, b/(10*E^(b^2*x^2)*Sqrt[Pi]*x^4) - b^3/(10*E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(5*x^5) - (b^5*ExpIntegralEi[-(b^2*x^2)])/(10*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erfc[a+b x]*) - - -{(c + d*x)^3*Erfc[a + b*x], x, 12, -((d^2*(b*c - a*d))/(b^4*E^(a + b*x)^2*Sqrt[Pi])) - (b*c - a*d)^3/(b^4*E^(a + b*x)^2*Sqrt[Pi]) - (3*d^3*(a + b*x))/(8*b^4*E^(a + b*x)^2*Sqrt[Pi]) - (3*d*(b*c - a*d)^2*(a + b*x))/(2*b^4*E^(a + b*x)^2*Sqrt[Pi]) - (d^2*(b*c - a*d)*(a + b*x)^2)/(b^4*E^(a + b*x)^2*Sqrt[Pi]) - (d^3*(a + b*x)^3)/(4*b^4*E^(a + b*x)^2*Sqrt[Pi]) + (3*d^3*Erf[a + b*x])/(16*b^4) + (3*d*(b*c - a*d)^2*Erf[a + b*x])/(4*b^4) + ((b*c - a*d)^4*Erf[a + b*x])/(4*b^4*d) + ((c + d*x)^4*Erfc[a + b*x])/(4*d)} -{(c + d*x)^2*Erfc[a + b*x], x, 9, -d^2/(3*b^3*E^(a + b*x)^2*Sqrt[Pi]) - (b*c - a*d)^2/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (d*(b*c - a*d)*(a + b*x))/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (d^2*(a + b*x)^2)/(3*b^3*E^(a + b*x)^2*Sqrt[Pi]) + (d*(b*c - a*d)*Erf[a + b*x])/(2*b^3) + ((b*c - a*d)^3*Erf[a + b*x])/(3*b^3*d) + ((c + d*x)^3*Erfc[a + b*x])/(3*d)} -{(c + d*x)^1*Erfc[a + b*x], x, 7, -((b*c - a*d)/(b^2*E^(a + b*x)^2*Sqrt[Pi])) - (d*(a + b*x))/(2*b^2*E^(a + b*x)^2*Sqrt[Pi]) + (d*Erf[a + b*x])/(4*b^2) + ((b*c - a*d)^2*Erf[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erfc[a + b*x])/(2*d)} -{(c + d*x)^0*Erfc[a + b*x], x, 1, -(1/(b*E^(a + b*x)^2*Sqrt[Pi])) + ((a + b*x)*Erfc[a + b*x])/b} -{Erfc[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Erfc[a + b*x]/(c + d*x), x]} -{Erfc[a + b*x]/(c + d*x)^2, x, 1, -(Erfc[a + b*x]/(d*(c + d*x))) - (2*b*Unintegrable[1/(E^(a + b*x)^2*(c + d*x)), x])/(d*Sqrt[Pi])} -{Erfc[a + b*x]/(c + d*x)^3, x, 3, b/(d^2*E^(a + b*x)^2*Sqrt[Pi]*(c + d*x)) + (b^2*Erf[a + b*x])/d^3 - Erfc[a + b*x]/(2*d*(c + d*x)^2) - (2*b^2*(b*c - a*d)*Unintegrable[1/(E^(a + b*x)^2*(c + d*x)), x])/(d^3*Sqrt[Pi])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erfc[a+b x]^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erfc[b x]^2*) - - -{x^5*Erfc[b*x]^2, x, 12, 11/(12*b^6*E^(2*b^2*x^2)*Pi) + (7*x^2)/(12*b^4*E^(2*b^2*x^2)*Pi) + x^4/(6*b^2*E^(2*b^2*x^2)*Pi) - (5*x*Erfc[b*x])/(4*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (5*x^3*Erfc[b*x])/(6*b^3*E^(b^2*x^2)*Sqrt[Pi]) - (x^5*Erfc[b*x])/(3*b*E^(b^2*x^2)*Sqrt[Pi]) - (5*Erfc[b*x]^2)/(16*b^6) + (x^6*Erfc[b*x]^2)/6} -{x^3*Erfc[b*x]^2, x, 8, 1/(2*b^4*E^(2*b^2*x^2)*Pi) + x^2/(4*b^2*E^(2*b^2*x^2)*Pi) - (3*x*Erfc[b*x])/(4*b^3*E^(b^2*x^2)*Sqrt[Pi]) - (x^3*Erfc[b*x])/(2*b*E^(b^2*x^2)*Sqrt[Pi]) - (3*Erfc[b*x]^2)/(16*b^4) + (x^4*Erfc[b*x]^2)/4} -{x^1*Erfc[b*x]^2, x, 5, 1/(2*b^2*E^(2*b^2*x^2)*Pi) - (x*Erfc[b*x])/(b*E^(b^2*x^2)*Sqrt[Pi]) - Erfc[b*x]^2/(4*b^2) + (x^2*Erfc[b*x]^2)/2} -{Erfc[b*x]^2/x^1, x, 0, Unintegrable[Erfc[b*x]^2/x, x]} -{Erfc[b*x]^2/x^3, x, 5, (2*b*Erfc[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) - b^2*Erfc[b*x]^2 - Erfc[b*x]^2/(2*x^2) + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/Pi} -{Erfc[b*x]^2/x^5, x, 8, -b^2/(3*E^(2*b^2*x^2)*Pi*x^2) + (b*Erfc[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x^3) - (2*b^3*Erfc[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x) + (b^4*Erfc[b*x]^2)/3 - Erfc[b*x]^2/(4*x^4) - (4*b^4*ExpIntegralEi[-2*b^2*x^2])/(3*Pi)} -{Erfc[b*x]^2/x^7, x, 12, -b^2/(15*E^(2*b^2*x^2)*Pi*x^4) + (2*b^4)/(9*E^(2*b^2*x^2)*Pi*x^2) + (2*b*Erfc[b*x])/(15*E^(b^2*x^2)*Sqrt[Pi]*x^5) - (4*b^3*Erfc[b*x])/(45*E^(b^2*x^2)*Sqrt[Pi]*x^3) + (8*b^5*Erfc[b*x])/(45*E^(b^2*x^2)*Sqrt[Pi]*x) - (4*b^6*Erfc[b*x]^2)/45 - Erfc[b*x]^2/(6*x^6) + (28*b^6*ExpIntegralEi[-2*b^2*x^2])/(45*Pi)} - -{x^4*Erfc[b*x]^2, x, 10, (11*x)/(20*b^4*E^(2*b^2*x^2)*Pi) + x^3/(5*b^2*E^(2*b^2*x^2)*Pi) - (43*Erf[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi]) - (4*Erfc[b*x])/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) - (4*x^2*Erfc[b*x])/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) - (2*x^4*Erfc[b*x])/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erfc[b*x]^2)/5} -{x^2*Erfc[b*x]^2, x, 6, x/(3*b^2*E^(2*b^2*x^2)*Pi) - (5*Erf[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi]) - (2*Erfc[b*x])/(3*b^3*E^(b^2*x^2)*Sqrt[Pi]) - (2*x^2*Erfc[b*x])/(3*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^3*Erfc[b*x]^2)/3} -{x^0*Erfc[b*x]^2, x, 4, -((Sqrt[2/Pi]*Erf[Sqrt[2]*b*x])/b) - (2*Erfc[b*x])/(b*E^(b^2*x^2)*Sqrt[Pi]) + x*Erfc[b*x]^2} -{Erfc[b*x]^2/x^2, x, 0, Unintegrable[Erfc[b*x]^2/x^2, x]} -{Erfc[b*x]^2/x^4, x, 0, Unintegrable[Erfc[b*x]^2/x^4, x]} -{Erfc[b*x]^2/x^6, x, 0, Unintegrable[Erfc[b*x]^2/x^6, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erfc[a+b x]^2*) - - -{(c + d*x)^2*Erfc[a + b*x]^2, x, 16, (d*(b*c - a*d))/(b^3*E^(2*(a + b*x)^2)*Pi) + (d^2*(a + b*x))/(3*b^3*E^(2*(a + b*x)^2)*Pi) - ((b*c - a*d)^2*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b^3 - (5*d^2*Erf[Sqrt[2]*(a + b*x)])/(6*b^3*Sqrt[2*Pi]) - (2*d^2*Erfc[a + b*x])/(3*b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*(b*c - a*d)^2*Erfc[a + b*x])/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*d*(b*c - a*d)*(a + b*x)*Erfc[a + b*x])/(b^3*E^(a + b*x)^2*Sqrt[Pi]) - (2*d^2*(a + b*x)^2*Erfc[a + b*x])/(3*b^3*E^(a + b*x)^2*Sqrt[Pi]) - (d*(b*c - a*d)*Erfc[a + b*x]^2)/(2*b^3) + ((b*c - a*d)^2*(a + b*x)*Erfc[a + b*x]^2)/b^3 + (d*(b*c - a*d)*(a + b*x)^2*Erfc[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*Erfc[a + b*x]^2)/(3*b^3)} -{(c + d*x)^1*Erfc[a + b*x]^2, x, 10, d/(2*b^2*E^(2*(a + b*x)^2)*Pi) - ((b*c - a*d)*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b^2 - (2*(b*c - a*d)*Erfc[a + b*x])/(b^2*E^(a + b*x)^2*Sqrt[Pi]) - (d*(a + b*x)*Erfc[a + b*x])/(b^2*E^(a + b*x)^2*Sqrt[Pi]) - (d*Erfc[a + b*x]^2)/(4*b^2) + ((b*c - a*d)*(a + b*x)*Erfc[a + b*x]^2)/b^2 + (d*(a + b*x)^2*Erfc[a + b*x]^2)/(2*b^2)} -{(c + d*x)^0*Erfc[a + b*x]^2, x, 4, -((Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)])/b) - (2*Erfc[a + b*x])/(b*E^(a + b*x)^2*Sqrt[Pi]) + ((a + b*x)*Erfc[a + b*x]^2)/b} -{Erfc[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Erfc[a + b*x]^2/(c + d*x), x]} -{Erfc[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Erfc[a + b*x]^2/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Erfc[d (a+b Log[c x^n])]*) - - -{x^2*Erfc[d*(a + b*Log[c*x^n])], x, 5, (E^((9 - 12*a*b*d^2*n)/(4*b^2*d^2*n^2))*x^3*Erf[(2*a*b*d^2 - 3/n + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(3*(c*x^n)^(3/n)) + (x^3*Erfc[d*(a + b*Log[c*x^n])])/3} -{x^1*Erfc[d*(a + b*Log[c*x^n])], x, 5, (E^((1 - 2*a*b*d^2*n)/(b^2*d^2*n^2))*x^2*Erf[(a*b*d^2 - n^(-1) + b^2*d^2*Log[c*x^n])/(b*d)])/(2*(c*x^n)^(2/n)) + (x^2*Erfc[d*(a + b*Log[c*x^n])])/2} -{x^0*Erfc[d*(a + b*Log[c*x^n])], x, 5, (E^((1 - 4*a*b*d^2*n)/(4*b^2*d^2*n^2))*x*Erf[(2*a*b*d^2 - n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(c*x^n)^n^(-1) + x*Erfc[d*(a + b*Log[c*x^n])]} -{Erfc[d*(a + b*Log[c*x^n])]/x^1, x, 3, -(1/(b*d*E^(d^2*(a + b*Log[c*x^n])^2)*n*Sqrt[Pi])) + (Erfc[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)} -{Erfc[d*(a + b*Log[c*x^n])]/x^2, x, 5, -((E^(1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erf[(2*a*b*d^2 + n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/x) - Erfc[d*(a + b*Log[c*x^n])]/x} -{Erfc[d*(a + b*Log[c*x^n])]/x^3, x, 5, -(E^((1 + 2*a*b*d^2*n)/(b^2*d^2*n^2))*(c*x^n)^(2/n)*Erf[(1 + a*b*d^2*n + b^2*d^2*n*Log[c*x^n])/(b*d*n)])/(2*x^2) - Erfc[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*Erfc[d*(a + b*Log[c*x^n])], x, 5, -((E^(((1 + m)*(1 + m - 4*a*b*d^2*n))/(4*b^2*d^2*n^2))*x*(e*x)^m*Erf[(1 + m - 2*a*b*d^2*n - 2*b^2*d^2*n*Log[c*x^n])/(2*b*d*n)])/((1 + m)*(c*x^n)^((1 + m)/n))) + ((e*x)^(1 + m)*Erfc[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m E^(c+d x^2) Erfc[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c-b^2 x^2) Erfc[b x]^n*) - - -{E^(c - b^2*x^2)*Erfc[b*x]^2, x, 2, -((E^c*Sqrt[Pi]*Erfc[b*x]^3)/(6*b))} -{E^(c - b^2*x^2)*Erfc[b*x]^1, x, 2, -((E^c*Sqrt[Pi]*Erfc[b*x]^2)/(4*b))} -{E^(c - b^2*x^2)/Erfc[b*x]^1, x, 2, -((E^c*Sqrt[Pi]*Log[Erfc[b*x]])/(2*b))} -{E^(c - b^2*x^2)/Erfc[b*x]^2, x, 2, (E^c*Sqrt[Pi])/(2*b*Erfc[b*x])} -{E^(c - b^2*x^2)/Erfc[b*x]^3, x, 2, (E^c*Sqrt[Pi])/(4*b*Erfc[b*x]^2)} - - -{E^(c - b^2*x^2)*Erfc[b*x]^n, x, 2, -((E^c*Sqrt[Pi]*Erfc[b*x]^(1 + n))/(2*b*(1 + n)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erfc[b x]*) - - -{E^(c + d*x^2)*x^5*Erfc[b*x], x, 9, (b*E^(c - (b^2 - d)*x^2)*x)/((b^2 - d)*d^2*Sqrt[Pi]) - (3*b*E^(c - (b^2 - d)*x^2)*x)/(4*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(c - (b^2 - d)*x^2)*x^3)/(2*(b^2 - d)*d*Sqrt[Pi]) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(Sqrt[b^2 - d]*d^3) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*(b^2 - d)^(3/2)*d^2) + (3*b*E^c*Erf[Sqrt[b^2 - d]*x])/(8*(b^2 - d)^(5/2)*d) + (E^(c + d*x^2)*Erfc[b*x])/d^3 - (E^(c + d*x^2)*x^2*Erfc[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erfc[b*x])/(2*d)} -{E^(c + d*x^2)*x^3*Erfc[b*x], x, 5, -(b*E^(c - (b^2 - d)*x^2)*x)/(2*(b^2 - d)*d*Sqrt[Pi]) - (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d^2) + (b*E^c*Erf[Sqrt[b^2 - d]*x])/(4*(b^2 - d)^(3/2)*d) - (E^(c + d*x^2)*Erfc[b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfc[b*x])/(2*d)} -{E^(c + d*x^2)*x^1*Erfc[b*x], x, 2, (b*E^c*Erf[Sqrt[b^2 - d]*x])/(2*Sqrt[b^2 - d]*d) + (E^(c + d*x^2)*Erfc[b*x])/(2*d)} -{(E^(c + d*x^2)*Erfc[b*x])/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erfc[b*x])/x, x]} -{(E^(c + d*x^2)*Erfc[b*x])/x^3, x, 3, (b*E^(c - (b^2 - d)*x^2))/(Sqrt[Pi]*x) + b*Sqrt[b^2 - d]*E^c*Erf[Sqrt[b^2 - d]*x] - (E^(c + d*x^2)*Erfc[b*x])/(2*x^2) + d*Unintegrable[(E^(c + d*x^2)*Erfc[b*x])/x, x]} -{(E^(c + d*x^2)*Erfc[b*x])/x^5, x, 7, (b*E^(c - (b^2 - d)*x^2))/(6*Sqrt[Pi]*x^3) - (b*(b^2 - d)*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x) + (b*d*E^(c - (b^2 - d)*x^2))/(2*Sqrt[Pi]*x) - (b*(b^2 - d)^(3/2)*E^c*Erf[Sqrt[b^2 - d]*x])/3 + (b*Sqrt[b^2 - d]*d*E^c*Erf[Sqrt[b^2 - d]*x])/2 - (E^(c + d*x^2)*Erfc[b*x])/(4*x^4) - (d*E^(c + d*x^2)*Erfc[b*x])/(4*x^2) + (d^2*Unintegrable[(E^(c + d*x^2)*Erfc[b*x])/x, x])/2} - -{E^(c + d*x^2)*x^4*Erfc[b*x], x, 5, (3*b*E^(c - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) - (b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(c - (b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfc[b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfc[b*x])/(2*d) + (3*Unintegrable[E^(c + d*x^2)*Erfc[b*x], x])/(4*d^2)} -{E^(c + d*x^2)*x^2*Erfc[b*x], x, 2, -(b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erfc[b*x])/(2*d) - Unintegrable[E^(c + d*x^2)*Erfc[b*x], x]/(2*d)} -{E^(c + d*x^2)*x^0*Erfc[b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erfc[b*x], x]} -{(E^(c + d*x^2)*Erfc[b*x])/x^2, x, 2, -((E^(c + d*x^2)*Erfc[b*x])/x) - (b*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erfc[b*x], x]} -{(E^(c + d*x^2)*Erfc[b*x])/x^4, x, 5, (b*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (E^(c + d*x^2)*Erfc[b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[b*x])/(3*x) + (b*(b^2 - d)*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) - (2*b*d*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfc[b*x], x])/3} - - -{E^(c + b^2*x^2)*x^5*Erfc[b*x], x, 8, (2*E^c*x)/(b^5*Sqrt[Pi]) - (2*E^c*x^3)/(3*b^3*Sqrt[Pi]) + (E^c*x^5)/(5*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfc[b*x])/b^6 - (E^(c + b^2*x^2)*x^2*Erfc[b*x])/b^4 + (E^(c + b^2*x^2)*x^4*Erfc[b*x])/(2*b^2)} -{E^(c + b^2*x^2)*x^3*Erfc[b*x], x, 5, -((E^c*x)/(b^3*Sqrt[Pi])) + (E^c*x^3)/(3*b*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erfc[b*x])/(2*b^2)} -{E^(c + b^2*x^2)*x^1*Erfc[b*x], x, 2, (E^c*x)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfc[b*x])/(2*b^2)} -{(E^(c + b^2*x^2)*Erfc[b*x])/x^1, x, 3, (E^c*ExpIntegralEi[b^2*x^2])/2 - (2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} -{(E^(c + b^2*x^2)*Erfc[b*x])/x^3, x, 6, (b*E^c)/(Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*x^2) + (b^2*E^c*ExpIntegralEi[b^2*x^2])/2 - (2*b^3*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} -{(E^(c + b^2*x^2)*Erfc[b*x])/x^5, x, 9, (b*E^c)/(6*Sqrt[Pi]*x^3) + (b^3*E^c)/(2*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(4*x^4) - (b^2*E^(c + b^2*x^2)*Erfc[b*x])/(4*x^2) + (b^4*E^c*ExpIntegralEi[b^2*x^2])/4 - (b^5*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} - -{E^(c + b^2*x^2)*x^4*Erfc[b*x], x, 9, -((3*E^c*x^2)/(4*b^3*Sqrt[Pi])) + (E^c*x^4)/(4*b*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erfc[b*x])/(4*b^4) + (E^(c + b^2*x^2)*x^3*Erfc[b*x])/(2*b^2) + (3*E^c*Sqrt[Pi]*Erfi[b*x])/(8*b^5) - (3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(4*b^3*Sqrt[Pi])} -{E^(c + b^2*x^2)*x^2*Erfc[b*x], x, 6, (E^c*x^2)/(2*b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfc[b*x])/(2*b^2) - (E^c*Sqrt[Pi]*Erfi[b*x])/(4*b^3) + (E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*b*Sqrt[Pi])} -{E^(c + b^2*x^2)*x^0*Erfc[b*x], x, 3, (E^c*Sqrt[Pi]*Erfi[b*x])/(2*b) - (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]} -{(E^(c + b^2*x^2)*Erfc[b*x])/x^2, x, 6, -((E^(c + b^2*x^2)*Erfc[b*x])/x) + b*E^c*Sqrt[Pi]*Erfi[b*x] - (2*b^3*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi] - (2*b*E^c*Log[x])/Sqrt[Pi]} -{(E^(c + b^2*x^2)*Erfc[b*x])/x^4, x, 9, (b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erfc[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erfc[b*x])/(3*x) + (2/3)*b^3*E^c*Sqrt[Pi]*Erfi[b*x] - (4*b^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) - (4*b^3*E^c*Log[x])/(3*Sqrt[Pi])} - - -{x^5*Erfc[b*x]/E^(b^2*x^2), x, 9, (11*x)/(16*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^3/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (43*Erf[Sqrt[2]*b*x])/(32*Sqrt[2]*b^6) - Erfc[b*x]/(b^6*E^(b^2*x^2)) - (x^2*Erfc[b*x])/(b^4*E^(b^2*x^2)) - (x^4*Erfc[b*x])/(2*b^2*E^(b^2*x^2))} -{x^3*Erfc[b*x]/E^(b^2*x^2), x, 5, x/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (5*Erf[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4) - Erfc[b*x]/(2*b^4*E^(b^2*x^2)) - (x^2*Erfc[b*x])/(2*b^2*E^(b^2*x^2))} -{x^1*Erfc[b*x]/E^(b^2*x^2), x, 2, -Erf[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2) - Erfc[b*x]/(2*b^2*E^(b^2*x^2))} -{Erfc[b*x]/(E^(b^2*x^2)*x^1), x, 0, Unintegrable[Erfc[b*x]/(E^(b^2*x^2)*x), x]} -{Erfc[b*x]/(E^(b^2*x^2)*x^3), x, 3, b/(E^(2*b^2*x^2)*Sqrt[Pi]*x) + Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - Erfc[b*x]/(2*E^(b^2*x^2)*x^2) - b^2*Unintegrable[Erfc[b*x]/(E^(b^2*x^2)*x), x]} -{Erfc[b*x]/(E^(b^2*x^2)*x^5), x, 7, b/(6*E^(2*b^2*x^2)*Sqrt[Pi]*x^3) - (7*b^3)/(6*E^(2*b^2*x^2)*Sqrt[Pi]*x) - (b^4*Erf[Sqrt[2]*b*x])/Sqrt[2] - (2*Sqrt[2]*b^4*Erf[Sqrt[2]*b*x])/3 - Erfc[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfc[b*x])/(4*E^(b^2*x^2)*x^2) + (b^4*Unintegrable[Erfc[b*x]/(E^(b^2*x^2)*x), x])/2} - -{x^4*Erfc[b*x]/E^(b^2*x^2), x, 7, 1/(2*b^5*E^(2*b^2*x^2)*Sqrt[Pi]) + x^2/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (3*x*Erfc[b*x])/(4*b^4*E^(b^2*x^2)) - (x^3*Erfc[b*x])/(2*b^2*E^(b^2*x^2)) - (3*Sqrt[Pi]*Erfc[b*x]^2)/(16*b^5)} -{x^2*Erfc[b*x]/E^(b^2*x^2), x, 4, 1/(4*b^3*E^(2*b^2*x^2)*Sqrt[Pi]) - (x*Erfc[b*x])/(2*b^2*E^(b^2*x^2)) - (Sqrt[Pi]*Erfc[b*x]^2)/(8*b^3)} -{x^0*Erfc[b*x]/E^(b^2*x^2), x, 2, -(Sqrt[Pi]*Erfc[b*x]^2)/(4*b)} -{Erfc[b*x]/(E^(b^2*x^2)*x^2), x, 4, -(Erfc[b*x]/(E^(b^2*x^2)*x)) + (b*Sqrt[Pi]*Erfc[b*x]^2)/2 - (b*ExpIntegralEi[-2*b^2*x^2])/Sqrt[Pi]} -{Erfc[b*x]/(E^(b^2*x^2)*x^4), x, 7, b/(3*E^(2*b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(3*E^(b^2*x^2)*x^3) + (2*b^2*Erfc[b*x])/(3*E^(b^2*x^2)*x) - (b^3*Sqrt[Pi]*Erfc[b*x]^2)/3 + (4*b^3*ExpIntegralEi[-2*b^2*x^2])/(3*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erfc[a+b x]*) - - -{E^(c + d*x^2)*x^3*Erfc[a + b*x], x, 10, (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)*d*Sqrt[Pi]) - (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*Sqrt[b^2 - d]*d^2) + (a^2*b^3*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(5/2)*d) + (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(3/2)*d) - (E^(c + d*x^2)*Erfc[a + b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfc[a + b*x])/(2*d)} -{E^(c + d*x^2)*x^1*Erfc[a + b*x], x, 3, (b*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*Sqrt[b^2 - d]*d) + (E^(c + d*x^2)*Erfc[a + b*x])/(2*d)} -{E^(c + d*x^2)*Erfc[a + b*x]/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erfc[a + b*x])/x, x]} -{E^(c + d*x^2)*Erfc[a + b*x]/x^3, x, 4, (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(Sqrt[Pi]*x) + b*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]] - (E^(c + d*x^2)*Erfc[a + b*x])/(2*x^2) + (2*a*b^2*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi] + d*Unintegrable[(E^(c + d*x^2)*Erfc[a + b*x])/x, x]} - -{E^(c + d*x^2)*x^4*Erfc[a + b*x], x, 15, (3*b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) - (a^2*b^3*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^3*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]) + (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) + (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(3/2)*d^2) - (a^3*b^4*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(7/2)*d) - (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(5/2)*d) - (3*E^(c + d*x^2)*x*Erfc[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfc[a + b*x])/(2*d) + (3*Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x])/(4*d^2)} -{E^(c + d*x^2)*x^2*Erfc[a + b*x], x, 4, -(b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) - (a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(3/2)*d) + (E^(c + d*x^2)*x*Erfc[a + b*x])/(2*d) - Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x]/(2*d)} -{E^(c + d*x^2)*x^0*Erfc[a + b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x]} -{E^(c + d*x^2)*Erfc[a + b*x]/x^2, x, 1, -((E^(c + d*x^2)*Erfc[a + b*x])/x) - (2*b*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x]} -{E^(c + d*x^2)*Erfc[a + b*x]/x^4, x, 6, (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (2*a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x) - (2*a*b^2*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/3 - (E^(c + d*x^2)*Erfc[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[a + b*x])/(3*x) - (4*a^2*b^3*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (2*b*(b^2 - d)*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) - (4*b*d*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x])/3} - - -{Erfc[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfc[b*x])/(E^(b^2*x^2)*x), x, 4, b/(E^(2*b^2*x^2)*Sqrt[Pi]*x) + Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - Erfc[b*x]/(2*E^(b^2*x^2)*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Trig[c+d x^2] Erfc[a+b x]^n*) - - -{Sin[c + I*b^2*x^2]*Erfc[b*x], x, 6, (I*E^(I*c)*Sqrt[Pi]*Erfc[b*x]^2)/(8*b) + (I*Sqrt[Pi]*Erfi[b*x])/(E^(I*c)*(4*b)) - (I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} -{Sin[c - I*b^2*x^2]*Erfc[b*x], x, 6, -((I*Sqrt[Pi]*Erfc[b*x]^2)/(E^(I*c)*(8*b))) - (I*E^(I*c)*Sqrt[Pi]*Erfi[b*x])/(4*b) + (I*b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} - - -{Cos[c + I*b^2*x^2]*Erfc[b*x], x, 6, -((E^(I*c)*Sqrt[Pi]*Erfc[b*x]^2)/(8*b)) + (Sqrt[Pi]*Erfi[b*x])/(E^(I*c)*(4*b)) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} -{Cos[c - I*b^2*x^2]*Erfc[b*x], x, 6, -((Sqrt[Pi]*Erfc[b*x]^2)/(E^(I*c)*(8*b))) + (E^(I*c)*Sqrt[Pi]*Erfi[b*x])/(4*b) - (b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} - - -{Sinh[c + b^2*x^2]*Erfc[b*x], x, 6, (Sqrt[Pi]*Erfc[b*x]^2)/(E^c*(8*b)) + (E^c*Sqrt[Pi]*Erfi[b*x])/(4*b) - (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} -{Sinh[c - b^2*x^2]*Erfc[b*x], x, 6, -((E^c*Sqrt[Pi]*Erfc[b*x]^2)/(8*b)) - (Sqrt[Pi]*Erfi[b*x])/(E^c*(4*b)) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^c*(2*Sqrt[Pi]))} - - -{Cosh[c + b^2*x^2]*Erfc[b*x], x, 6, -((Sqrt[Pi]*Erfc[b*x]^2)/(E^c*(8*b))) + (E^c*Sqrt[Pi]*Erfi[b*x])/(4*b) - (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(2*Sqrt[Pi])} -{Cosh[c - b^2*x^2]*Erfc[b*x], x, 6, -((E^c*Sqrt[Pi]*Erfc[b*x]^2)/(8*b)) + (Sqrt[Pi]*Erfi[b*x])/(E^c*(4*b)) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(E^c*(2*Sqrt[Pi]))} - - -(* ::Title::Closed:: *) -(*Integration Problems Involving The Imaginary Error Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erfi[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erfi[b x]*) - - -{x^5*Erfi[b*x], x, 5, (-5*E^(b^2*x^2)*x)/(8*b^5*Sqrt[Pi]) + (5*E^(b^2*x^2)*x^3)/(12*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^5)/(6*b*Sqrt[Pi]) + (5*Erfi[b*x])/(16*b^6) + (x^6*Erfi[b*x])/6} -{x^3*Erfi[b*x], x, 4, (3*E^(b^2*x^2)*x)/(8*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^3)/(4*b*Sqrt[Pi]) - (3*Erfi[b*x])/(16*b^4) + (x^4*Erfi[b*x])/4} -{x^1*Erfi[b*x], x, 3, -(E^(b^2*x^2)*x)/(2*b*Sqrt[Pi]) + Erfi[b*x]/(4*b^2) + (x^2*Erfi[b*x])/2} -{Erfi[b*x]/x^1, x, 1, (2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]} -{Erfi[b*x]/x^3, x, 3, -((b*E^(b^2*x^2))/(Sqrt[Pi]*x)) + b^2*Erfi[b*x] - Erfi[b*x]/(2*x^2)} -{Erfi[b*x]/x^5, x, 4, -(b*E^(b^2*x^2))/(6*Sqrt[Pi]*x^3) - (b^3*E^(b^2*x^2))/(3*Sqrt[Pi]*x) + (b^4*Erfi[b*x])/3 - Erfi[b*x]/(4*x^4)} -{Erfi[b*x]/x^7, x, 5, -(b*E^(b^2*x^2))/(15*Sqrt[Pi]*x^5) - (2*b^3*E^(b^2*x^2))/(45*Sqrt[Pi]*x^3) - (4*b^5*E^(b^2*x^2))/(45*Sqrt[Pi]*x) + (4*b^6*Erfi[b*x])/45 - Erfi[b*x]/(6*x^6)} - -{x^6*Erfi[b*x], x, 5, (6*E^(b^2*x^2))/(7*b^7*Sqrt[Pi]) - (6*E^(b^2*x^2)*x^2)/(7*b^5*Sqrt[Pi]) + (3*E^(b^2*x^2)*x^4)/(7*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^6)/(7*b*Sqrt[Pi]) + (x^7*Erfi[b*x])/7} -{x^4*Erfi[b*x], x, 4, (-2*E^(b^2*x^2))/(5*b^5*Sqrt[Pi]) + (2*E^(b^2*x^2)*x^2)/(5*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^4)/(5*b*Sqrt[Pi]) + (x^5*Erfi[b*x])/5} -{x^2*Erfi[b*x], x, 3, E^(b^2*x^2)/(3*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^2)/(3*b*Sqrt[Pi]) + (x^3*Erfi[b*x])/3} -{x^0*Erfi[b*x], x, 1, -(E^(b^2*x^2)/(b*Sqrt[Pi])) + x*Erfi[b*x]} -{Erfi[b*x]/x^2, x, 2, -(Erfi[b*x]/x) + (b*ExpIntegralEi[b^2*x^2])/Sqrt[Pi]} -{Erfi[b*x]/x^4, x, 3, -(b*E^(b^2*x^2))/(3*Sqrt[Pi]*x^2) - Erfi[b*x]/(3*x^3) + (b^3*ExpIntegralEi[b^2*x^2])/(3*Sqrt[Pi])} -{Erfi[b*x]/x^6, x, 4, -(b*E^(b^2*x^2))/(10*Sqrt[Pi]*x^4) - (b^3*E^(b^2*x^2))/(10*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*x^5) + (b^5*ExpIntegralEi[b^2*x^2])/(10*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erfi[a+b x]*) - - -{(c + d*x)^3*Erfi[a + b*x], x, 12, (d^2*(b*c - a*d)*E^(a + b*x)^2)/(b^4*Sqrt[Pi]) - ((b*c - a*d)^3*E^(a + b*x)^2)/(b^4*Sqrt[Pi]) + (3*d^3*E^(a + b*x)^2*(a + b*x))/(8*b^4*Sqrt[Pi]) - (3*d*(b*c - a*d)^2*E^(a + b*x)^2*(a + b*x))/(2*b^4*Sqrt[Pi]) - (d^2*(b*c - a*d)*E^(a + b*x)^2*(a + b*x)^2)/(b^4*Sqrt[Pi]) - (d^3*E^(a + b*x)^2*(a + b*x)^3)/(4*b^4*Sqrt[Pi]) - (3*d^3*Erfi[a + b*x])/(16*b^4) + (3*d*(b*c - a*d)^2*Erfi[a + b*x])/(4*b^4) - ((b*c - a*d)^4*Erfi[a + b*x])/(4*b^4*d) + ((c + d*x)^4*Erfi[a + b*x])/(4*d)} -{(c + d*x)^2*Erfi[a + b*x], x, 9, (d^2*E^(a + b*x)^2)/(3*b^3*Sqrt[Pi]) - ((b*c - a*d)^2*E^(a + b*x)^2)/(b^3*Sqrt[Pi]) - (d*(b*c - a*d)*E^(a + b*x)^2*(a + b*x))/(b^3*Sqrt[Pi]) - (d^2*E^(a + b*x)^2*(a + b*x)^2)/(3*b^3*Sqrt[Pi]) + (d*(b*c - a*d)*Erfi[a + b*x])/(2*b^3) - ((b*c - a*d)^3*Erfi[a + b*x])/(3*b^3*d) + ((c + d*x)^3*Erfi[a + b*x])/(3*d)} -{(c + d*x)^1*Erfi[a + b*x], x, 7, -(((b*c - a*d)*E^(a + b*x)^2)/(b^2*Sqrt[Pi])) - (d*E^(a + b*x)^2*(a + b*x))/(2*b^2*Sqrt[Pi]) + (d*Erfi[a + b*x])/(4*b^2) - ((b*c - a*d)^2*Erfi[a + b*x])/(2*b^2*d) + ((c + d*x)^2*Erfi[a + b*x])/(2*d)} -{(c + d*x)^0*Erfi[a + b*x], x, 1, -(E^(a + b*x)^2/(b*Sqrt[Pi])) + ((a + b*x)*Erfi[a + b*x])/b} -{Erfi[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Erfi[a + b*x]/(c + d*x), x]} -{Erfi[a + b*x]/(c + d*x)^2, x, 1, -(Erfi[a + b*x]/(d*(c + d*x))) + (2*b*Unintegrable[E^(a + b*x)^2/(c + d*x), x])/(d*Sqrt[Pi])} -{Erfi[a + b*x]/(c + d*x)^3, x, 3, -((b*E^(a + b*x)^2)/(d^2*Sqrt[Pi]*(c + d*x))) + (b^2*Erfi[a + b*x])/d^3 - Erfi[a + b*x]/(2*d*(c + d*x)^2) - (2*b^2*(b*c - a*d)*Unintegrable[E^(a + b*x)^2/(c + d*x), x])/(d^3*Sqrt[Pi])} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Erfi[a+b x]^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Erfi[b x]^2*) - - -{x^5*Erfi[b*x]^2, x, 12, (11*E^(2*b^2*x^2))/(12*b^6*Pi) - (7*E^(2*b^2*x^2)*x^2)/(12*b^4*Pi) + (E^(2*b^2*x^2)*x^4)/(6*b^2*Pi) - (5*E^(b^2*x^2)*x*Erfi[b*x])/(4*b^5*Sqrt[Pi]) + (5*E^(b^2*x^2)*x^3*Erfi[b*x])/(6*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^5*Erfi[b*x])/(3*b*Sqrt[Pi]) + (5*Erfi[b*x]^2)/(16*b^6) + (x^6*Erfi[b*x]^2)/6} -{x^3*Erfi[b*x]^2, x, 8, -E^(2*b^2*x^2)/(2*b^4*Pi) + (E^(2*b^2*x^2)*x^2)/(4*b^2*Pi) + (3*E^(b^2*x^2)*x*Erfi[b*x])/(4*b^3*Sqrt[Pi]) - (E^(b^2*x^2)*x^3*Erfi[b*x])/(2*b*Sqrt[Pi]) - (3*Erfi[b*x]^2)/(16*b^4) + (x^4*Erfi[b*x]^2)/4} -{x^1*Erfi[b*x]^2, x, 5, E^(2*b^2*x^2)/(2*b^2*Pi) - (E^(b^2*x^2)*x*Erfi[b*x])/(b*Sqrt[Pi]) + Erfi[b*x]^2/(4*b^2) + (x^2*Erfi[b*x]^2)/2} -{Erfi[b*x]^2/x^1, x, 0, Unintegrable[Erfi[b*x]^2/x, x]} -{Erfi[b*x]^2/x^3, x, 5, (-2*b*E^(b^2*x^2)*Erfi[b*x])/(Sqrt[Pi]*x) + b^2*Erfi[b*x]^2 - Erfi[b*x]^2/(2*x^2) + (2*b^2*ExpIntegralEi[2*b^2*x^2])/Pi} -{Erfi[b*x]^2/x^5, x, 8, -(b^2*E^(2*b^2*x^2))/(3*Pi*x^2) - (b*E^(b^2*x^2)*Erfi[b*x])/(3*Sqrt[Pi]*x^3) - (2*b^3*E^(b^2*x^2)*Erfi[b*x])/(3*Sqrt[Pi]*x) + (b^4*Erfi[b*x]^2)/3 - Erfi[b*x]^2/(4*x^4) + (4*b^4*ExpIntegralEi[2*b^2*x^2])/(3*Pi)} -{Erfi[b*x]^2/x^7, x, 12, -(b^2*E^(2*b^2*x^2))/(15*Pi*x^4) - (2*b^4*E^(2*b^2*x^2))/(9*Pi*x^2) - (2*b*E^(b^2*x^2)*Erfi[b*x])/(15*Sqrt[Pi]*x^5) - (4*b^3*E^(b^2*x^2)*Erfi[b*x])/(45*Sqrt[Pi]*x^3) - (8*b^5*E^(b^2*x^2)*Erfi[b*x])/(45*Sqrt[Pi]*x) + (4*b^6*Erfi[b*x]^2)/45 - Erfi[b*x]^2/(6*x^6) + (28*b^6*ExpIntegralEi[2*b^2*x^2])/(45*Pi)} - -{x^4*Erfi[b*x]^2, x, 10, (-11*E^(2*b^2*x^2)*x)/(20*b^4*Pi) + (E^(2*b^2*x^2)*x^3)/(5*b^2*Pi) - (4*E^(b^2*x^2)*Erfi[b*x])/(5*b^5*Sqrt[Pi]) + (4*E^(b^2*x^2)*x^2*Erfi[b*x])/(5*b^3*Sqrt[Pi]) - (2*E^(b^2*x^2)*x^4*Erfi[b*x])/(5*b*Sqrt[Pi]) + (x^5*Erfi[b*x]^2)/5 + (43*Erfi[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])} -{x^2*Erfi[b*x]^2, x, 6, (E^(2*b^2*x^2)*x)/(3*b^2*Pi) + (2*E^(b^2*x^2)*Erfi[b*x])/(3*b^3*Sqrt[Pi]) - (2*E^(b^2*x^2)*x^2*Erfi[b*x])/(3*b*Sqrt[Pi]) + (x^3*Erfi[b*x]^2)/3 - (5*Erfi[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])} -{x^0*Erfi[b*x]^2, x, 4, (-2*E^(b^2*x^2)*Erfi[b*x])/(b*Sqrt[Pi]) + x*Erfi[b*x]^2 + (Sqrt[2/Pi]*Erfi[Sqrt[2]*b*x])/b} -{Erfi[b*x]^2/x^2, x, 0, Unintegrable[Erfi[b*x]^2/x^2, x]} -{Erfi[b*x]^2/x^4, x, 0, Unintegrable[Erfi[b*x]^2/x^4, x]} -{Erfi[b*x]^2/x^6, x, 0, Unintegrable[Erfi[b*x]^2/x^6, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Erfi[a+b x]^2*) - - -{(c + d*x)^2*Erfi[a + b*x]^2, x, 16, (d*(b*c - a*d)*E^(2*(a + b*x)^2))/(b^3*Pi) + (d^2*E^(2*(a + b*x)^2)*(a + b*x))/(3*b^3*Pi) + (2*d^2*E^(a + b*x)^2*Erfi[a + b*x])/(3*b^3*Sqrt[Pi]) - (2*(b*c - a*d)^2*E^(a + b*x)^2*Erfi[a + b*x])/(b^3*Sqrt[Pi]) - (2*d*(b*c - a*d)*E^(a + b*x)^2*(a + b*x)*Erfi[a + b*x])/(b^3*Sqrt[Pi]) - (2*d^2*E^(a + b*x)^2*(a + b*x)^2*Erfi[a + b*x])/(3*b^3*Sqrt[Pi]) + (d*(b*c - a*d)*Erfi[a + b*x]^2)/(2*b^3) + ((b*c - a*d)^2*(a + b*x)*Erfi[a + b*x]^2)/b^3 + (d*(b*c - a*d)*(a + b*x)^2*Erfi[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*Erfi[a + b*x]^2)/(3*b^3) + ((b*c - a*d)^2*Sqrt[2/Pi]*Erfi[Sqrt[2]*(a + b*x)])/b^3 - (5*d^2*Erfi[Sqrt[2]*(a + b*x)])/(6*b^3*Sqrt[2*Pi])} -{(c + d*x)^1*Erfi[a + b*x]^2, x, 10, (d*E^(2*(a + b*x)^2))/(2*b^2*Pi) - (2*(b*c - a*d)*E^(a + b*x)^2*Erfi[a + b*x])/(b^2*Sqrt[Pi]) - (d*E^(a + b*x)^2*(a + b*x)*Erfi[a + b*x])/(b^2*Sqrt[Pi]) + (d*Erfi[a + b*x]^2)/(4*b^2) + ((b*c - a*d)*(a + b*x)*Erfi[a + b*x]^2)/b^2 + (d*(a + b*x)^2*Erfi[a + b*x]^2)/(2*b^2) + ((b*c - a*d)*Sqrt[2/Pi]*Erfi[Sqrt[2]*(a + b*x)])/b^2} -{(c + d*x)^0*Erfi[a + b*x]^2, x, 4, (-2*E^(a + b*x)^2*Erfi[a + b*x])/(b*Sqrt[Pi]) + ((a + b*x)*Erfi[a + b*x]^2)/b + (Sqrt[2/Pi]*Erfi[Sqrt[2]*(a + b*x)])/b} -{Erfi[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[Erfi[a + b*x]^2/(c + d*x), x]} -{Erfi[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[Erfi[a + b*x]^2/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Erfi[d (a+b Log[c x^n])]*) - - -{x^2*Erfi[d*(a + b*Log[c*x^n])], x, 5, (x^3*Erfi[d*(a + b*Log[c*x^n])])/3 - (x^3*Erfi[(2*a*b*d^2 + 3/n + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(3*E^((3*(3 + 4*a*b*d^2*n))/(4*b^2*d^2*n^2))*(c*x^n)^(3/n))} -{x^1*Erfi[d*(a + b*Log[c*x^n])], x, 5, (x^2*Erfi[d*(a + b*Log[c*x^n])])/2 - (x^2*Erfi[(a*b*d^2 + n^(-1) + b^2*d^2*Log[c*x^n])/(b*d)])/(2*E^((1 + 2*a*b*d^2*n)/(b^2*d^2*n^2))*(c*x^n)^(2/n))} -{x^0*Erfi[d*(a + b*Log[c*x^n])], x, 5, x*Erfi[d*(a + b*Log[c*x^n])] - (x*Erfi[(2*a*b*d^2 + n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/(E^((1 + 4*a*b*d^2*n)/(4*b^2*d^2*n^2))*(c*x^n)^n^(-1))} -{Erfi[d*(a + b*Log[c*x^n])]/x^1, x, 3, -(E^(a*d + b*d*Log[c*x^n])^2/(b*d*n*Sqrt[Pi])) + (Erfi[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)} -{Erfi[d*(a + b*Log[c*x^n])]/x^2, x, 5, -(Erfi[d*(a + b*Log[c*x^n])]/x) + (E^(-1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erfi[(2*a*b*d^2 - n^(-1) + 2*b^2*d^2*Log[c*x^n])/(2*b*d)])/x} -{Erfi[d*(a + b*Log[c*x^n])]/x^3, x, 5, -Erfi[d*(a + b*Log[c*x^n])]/(2*x^2) + ((c*x^n)^(2/n)*Erfi[(a*b*d^2 - n^(-1) + b^2*d^2*Log[c*x^n])/(b*d)])/(2*E^((1 - 2*a*b*d^2*n)/(b^2*d^2*n^2))*x^2)} - - -{(e*x)^m*Erfi[d*(a + b*Log[c*x^n])], x, 5, ((e*x)^(1 + m)*Erfi[d*(a + b*Log[c*x^n])])/(e*(1 + m)) - (x*(e*x)^m*Erfi[(1 + m + 2*a*b*d^2*n + 2*b^2*d^2*n*Log[c*x^n])/(2*b*d*n)])/(E^(((1 + m)*(1 + m + 4*a*b*d^2*n))/(4*b^2*d^2*n^2))*(1 + m)*(c*x^n)^((1 + m)/n))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m E^(c+d x^2) Erfi[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c+b^2 x^2) Erfi[b x]^n*) - - -{E^(c + b^2*x^2)*Erfi[b*x]^2, x, 2, (E^c*Sqrt[Pi]*Erfi[b*x]^3)/(6*b)} -{E^(c + b^2*x^2)*Erfi[b*x]^1, x, 2, (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(4*b)} -{E^(c + b^2*x^2)/Erfi[b*x]^1, x, 2, (E^c*Sqrt[Pi]*Log[Erfi[b*x]])/(2*b)} -{E^(c + b^2*x^2)/Erfi[b*x]^2, x, 2, -((E^c*Sqrt[Pi])/(2*b*Erfi[b*x]))} -{E^(c + b^2*x^2)/Erfi[b*x]^3, x, 2, -((E^c*Sqrt[Pi])/(4*b*Erfi[b*x]^2))} - - -{E^(c + b^2*x^2)*Erfi[b*x]^n, x, 2, (E^c*Sqrt[Pi]*Erfi[b*x]^(1 + n))/(2*b*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erfi[b x]*) - - -{E^(c + d*x^2)*x^5*Erfi[b*x], x, 9, (3*b*E^(c + (b^2 + d)*x^2)*x)/(4*d*(b^2 + d)^2*Sqrt[Pi]) + (b*E^(c + (b^2 + d)*x^2)*x)/(d^2*(b^2 + d)*Sqrt[Pi]) - (b*E^(c + (b^2 + d)*x^2)*x^3)/(2*d*(b^2 + d)*Sqrt[Pi]) + (E^(c + d*x^2)*Erfi[b*x])/d^3 - (E^(c + d*x^2)*x^2*Erfi[b*x])/d^2 + (E^(c + d*x^2)*x^4*Erfi[b*x])/(2*d) - (3*b*E^c*Erfi[Sqrt[b^2 + d]*x])/(8*d*(b^2 + d)^(5/2)) - (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(2*d^2*(b^2 + d)^(3/2)) - (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(d^3*Sqrt[b^2 + d])} -{E^(c + d*x^2)*x^3*Erfi[b*x], x, 5, -(b*E^(c + (b^2 + d)*x^2)*x)/(2*d*(b^2 + d)*Sqrt[Pi]) - (E^(c + d*x^2)*Erfi[b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfi[b*x])/(2*d) + (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(4*d*(b^2 + d)^(3/2)) + (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(2*d^2*Sqrt[b^2 + d])} -{E^(c + d*x^2)*x^1*Erfi[b*x], x, 2, (E^(c + d*x^2)*Erfi[b*x])/(2*d) - (b*E^c*Erfi[Sqrt[b^2 + d]*x])/(2*d*Sqrt[b^2 + d])} -{(E^(c + d*x^2)*Erfi[b*x])/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erfi[b*x])/x, x]} -{(E^(c + d*x^2)*Erfi[b*x])/x^3, x, 3, -((b*E^(c + (b^2 + d)*x^2))/(Sqrt[Pi]*x)) - (E^(c + d*x^2)*Erfi[b*x])/(2*x^2) + b*Sqrt[b^2 + d]*E^c*Erfi[Sqrt[b^2 + d]*x] + d*Unintegrable[(E^(c + d*x^2)*Erfi[b*x])/x, x]} -{(E^(c + d*x^2)*Erfi[b*x])/x^5, x, 7, -(b*E^(c + (b^2 + d)*x^2))/(6*Sqrt[Pi]*x^3) - (b*d*E^(c + (b^2 + d)*x^2))/(2*Sqrt[Pi]*x) - (b*(b^2 + d)*E^(c + (b^2 + d)*x^2))/(3*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erfi[b*x])/(4*x^4) - (d*E^(c + d*x^2)*Erfi[b*x])/(4*x^2) + (b*d*Sqrt[b^2 + d]*E^c*Erfi[Sqrt[b^2 + d]*x])/2 + (b*(b^2 + d)^(3/2)*E^c*Erfi[Sqrt[b^2 + d]*x])/3 + (d^2*Unintegrable[(E^(c + d*x^2)*Erfi[b*x])/x, x])/2} - -{E^(c + d*x^2)*x^4*Erfi[b*x], x, 5, (b*E^(c + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) + (3*b*E^(c + (b^2 + d)*x^2))/(4*d^2*(b^2 + d)*Sqrt[Pi]) - (b*E^(c + (b^2 + d)*x^2)*x^2)/(2*d*(b^2 + d)*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfi[b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfi[b*x])/(2*d) + (3*Unintegrable[E^(c + d*x^2)*Erfi[b*x], x])/(4*d^2)} -{E^(c + d*x^2)*x^2*Erfi[b*x], x, 2, -(b*E^(c + (b^2 + d)*x^2))/(2*d*(b^2 + d)*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erfi[b*x])/(2*d) - Unintegrable[E^(c + d*x^2)*Erfi[b*x], x]/(2*d)} -{E^(c + d*x^2)*x^0*Erfi[b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erfi[b*x], x]} -{(E^(c + d*x^2)*Erfi[b*x])/x^2, x, 2, -((E^(c + d*x^2)*Erfi[b*x])/x) + (b*E^c*ExpIntegralEi[(b^2 + d)*x^2])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erfi[b*x], x]} -{(E^(c + d*x^2)*Erfi[b*x])/x^4, x, 5, -(b*E^(c + (b^2 + d)*x^2))/(3*Sqrt[Pi]*x^2) - (E^(c + d*x^2)*Erfi[b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfi[b*x])/(3*x) + (2*b*d*E^c*ExpIntegralEi[(b^2 + d)*x^2])/(3*Sqrt[Pi]) + (b*(b^2 + d)*E^c*ExpIntegralEi[(b^2 + d)*x^2])/(3*Sqrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfi[b*x], x])/3} - - -{x^5*Erfi[b*x]/E^(b^2*x^2), x, 6, (2*x)/(b^5*Sqrt[Pi]) + (2*x^3)/(3*b^3*Sqrt[Pi]) + x^5/(5*b*Sqrt[Pi]) - Erfi[b*x]/(b^6*E^(b^2*x^2)) - (x^2*Erfi[b*x])/(b^4*E^(b^2*x^2)) - (x^4*Erfi[b*x])/(2*b^2*E^(b^2*x^2))} -{x^3*Erfi[b*x]/E^(b^2*x^2), x, 4, x/(b^3*Sqrt[Pi]) + x^3/(3*b*Sqrt[Pi]) - Erfi[b*x]/(2*b^4*E^(b^2*x^2)) - (x^2*Erfi[b*x])/(2*b^2*E^(b^2*x^2))} -{x^1*Erfi[b*x]/E^(b^2*x^2), x, 2, x/(b*Sqrt[Pi]) - Erfi[b*x]/(2*b^2*E^(b^2*x^2))} -{Erfi[b*x]/(E^(b^2*x^2)*x^1), x, 1, (2*b*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]} -{Erfi[b*x]/(E^(b^2*x^2)*x^3), x, 3, -(b/(Sqrt[Pi]*x)) - Erfi[b*x]/(2*E^(b^2*x^2)*x^2) - (2*b^3*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]} -{Erfi[b*x]/(E^(b^2*x^2)*x^5), x, 5, -b/(6*Sqrt[Pi]*x^3) + b^3/(2*Sqrt[Pi]*x) - Erfi[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfi[b*x])/(4*E^(b^2*x^2)*x^2) + (b^5*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]} - -{x^6*Erfi[b*x]/E^(b^2*x^2), x, 7, (15*x^2)/(8*b^5*Sqrt[Pi]) + (5*x^4)/(8*b^3*Sqrt[Pi]) + x^6/(6*b*Sqrt[Pi]) - (15*x*Erfi[b*x])/(E^(b^2*x^2)*(8*b^6)) - (5*x^3*Erfi[b*x])/(E^(b^2*x^2)*(4*b^4)) - (x^5*Erfi[b*x])/(E^(b^2*x^2)*(2*b^2)) + (15*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(8*b^5*Sqrt[Pi])} -{x^4*Erfi[b*x]/E^(b^2*x^2), x, 5, (3*x^2)/(4*b^3*Sqrt[Pi]) + x^4/(4*b*Sqrt[Pi]) - (3*x*Erfi[b*x])/(E^(b^2*x^2)*(4*b^4)) - (x^3*Erfi[b*x])/(E^(b^2*x^2)*(2*b^2)) + (3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(4*b^3*Sqrt[Pi])} -{x^2*Erfi[b*x]/E^(b^2*x^2), x, 3, x^2/(2*b*Sqrt[Pi]) - (x*Erfi[b*x])/(E^(b^2*x^2)*(2*b^2)) + (x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(2*b*Sqrt[Pi])} -{x^0*Erfi[b*x]/E^(b^2*x^2), x, 1, (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/Sqrt[Pi]} -{Erfi[b*x]/(E^(b^2*x^2)*x^2), x, 3, -(Erfi[b*x]/(E^(b^2*x^2)*x)) - (2*b^3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/Sqrt[Pi] + (2*b*Log[x])/Sqrt[Pi]} -{Erfi[b*x]/(E^(b^2*x^2)*x^4), x, 5, -(b/(3*Sqrt[Pi]*x^2)) - Erfi[b*x]/(E^(b^2*x^2)*(3*x^3)) + (2*b^2*Erfi[b*x])/(E^(b^2*x^2)*(3*x)) + (4*b^5*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(3*Sqrt[Pi]) - (4*b^3*Log[x])/(3*Sqrt[Pi])} -{Erfi[b*x]/(E^(b^2*x^2)*x^6), x, 7, -(b/(10*Sqrt[Pi]*x^4)) + (2*b^3)/(15*Sqrt[Pi]*x^2) - Erfi[b*x]/(E^(b^2*x^2)*(5*x^5)) + (2*b^2*Erfi[b*x])/(E^(b^2*x^2)*(15*x^3)) - (4*b^4*Erfi[b*x])/(E^(b^2*x^2)*(15*x)) - (8*b^7*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(15*Sqrt[Pi]) + (8*b^5*Log[x])/(15*Sqrt[Pi])} - - -{E^(c + b^2*x^2)*x^5*Erfi[b*x], x, 9, (11*E^(c + 2*b^2*x^2)*x)/(16*b^5*Sqrt[Pi]) - (E^(c + 2*b^2*x^2)*x^3)/(4*b^3*Sqrt[Pi]) + (E^(c + b^2*x^2)*Erfi[b*x])/b^6 - (E^(c + b^2*x^2)*x^2*Erfi[b*x])/b^4 + (E^(c + b^2*x^2)*x^4*Erfi[b*x])/(2*b^2) - (43*E^c*Erfi[Sqrt[2]*b*x])/(32*Sqrt[2]*b^6)} -{E^(c + b^2*x^2)*x^3*Erfi[b*x], x, 5, -(E^(c + 2*b^2*x^2)*x)/(4*b^3*Sqrt[Pi]) - (E^(c + b^2*x^2)*Erfi[b*x])/(2*b^4) + (E^(c + b^2*x^2)*x^2*Erfi[b*x])/(2*b^2) + (5*E^c*Erfi[Sqrt[2]*b*x])/(8*Sqrt[2]*b^4)} -{E^(c + b^2*x^2)*x^1*Erfi[b*x], x, 2, (E^(c + b^2*x^2)*Erfi[b*x])/(2*b^2) - (E^c*Erfi[Sqrt[2]*b*x])/(2*Sqrt[2]*b^2)} -{(E^(c + b^2*x^2)*Erfi[b*x])/x^1, x, 0, Unintegrable[(E^(c + b^2*x^2)*Erfi[b*x])/x, x]} -{(E^(c + b^2*x^2)*Erfi[b*x])/x^3, x, 3, -((b*E^(c + 2*b^2*x^2))/(Sqrt[Pi]*x)) - (E^(c + b^2*x^2)*Erfi[b*x])/(2*x^2) + Sqrt[2]*b^2*E^c*Erfi[Sqrt[2]*b*x] + b^2*Unintegrable[(E^(c + b^2*x^2)*Erfi[b*x])/x, x]} -{(E^(c + b^2*x^2)*Erfi[b*x])/x^5, x, 7, -(b*E^(c + 2*b^2*x^2))/(6*Sqrt[Pi]*x^3) - (7*b^3*E^(c + 2*b^2*x^2))/(6*Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfi[b*x])/(4*x^4) - (b^2*E^(c + b^2*x^2)*Erfi[b*x])/(4*x^2) + (b^4*E^c*Erfi[Sqrt[2]*b*x])/Sqrt[2] + (2*Sqrt[2]*b^4*E^c*Erfi[Sqrt[2]*b*x])/3 + (b^4*Unintegrable[(E^(c + b^2*x^2)*Erfi[b*x])/x, x])/2} - -{E^(c + b^2*x^2)*x^4*Erfi[b*x], x, 7, E^(c + 2*b^2*x^2)/(2*b^5*Sqrt[Pi]) - (E^(c + 2*b^2*x^2)*x^2)/(4*b^3*Sqrt[Pi]) - (3*E^(c + b^2*x^2)*x*Erfi[b*x])/(4*b^4) + (E^(c + b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) + (3*E^c*Sqrt[Pi]*Erfi[b*x]^2)/(16*b^5)} -{E^(c + b^2*x^2)*x^2*Erfi[b*x], x, 4, -E^(c + 2*b^2*x^2)/(4*b^3*Sqrt[Pi]) + (E^(c + b^2*x^2)*x*Erfi[b*x])/(2*b^2) - (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*b^3)} -{E^(c + b^2*x^2)*x^0*Erfi[b*x], x, 2, (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(4*b)} -{(E^(c + b^2*x^2)*Erfi[b*x])/x^2, x, 4, -((E^(c + b^2*x^2)*Erfi[b*x])/x) + (b*E^c*Sqrt[Pi]*Erfi[b*x]^2)/2 + (b*E^c*ExpIntegralEi[2*b^2*x^2])/Sqrt[Pi]} -{(E^(c + b^2*x^2)*Erfi[b*x])/x^4, x, 7, -(b*E^(c + 2*b^2*x^2))/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erfi[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erfi[b*x])/(3*x) + (b^3*E^c*Sqrt[Pi]*Erfi[b*x]^2)/3 + (4*b^3*E^c*ExpIntegralEi[2*b^2*x^2])/(3*Sqrt[Pi])} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(c+d x^2) Erfi[a+b x]*) - - -{E^(c + d*x^2)*x^3*Erfi[a + b*x], x, 10, (a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) - (b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)*x)/(2*d*(b^2 + d)*Sqrt[Pi]) - (E^(c + d*x^2)*Erfi[a + b*x])/(2*d^2) + (E^(c + d*x^2)*x^2*Erfi[a + b*x])/(2*d) - (a^2*b^3*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)^(5/2)) + (b*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d*(b^2 + d)^(3/2)) + (b*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d^2*Sqrt[b^2 + d])} -{E^(c + d*x^2)*x^1*Erfi[a + b*x], x, 3, (E^(c + d*x^2)*Erfi[a + b*x])/(2*d) - (b*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*Sqrt[b^2 + d])} -{E^(c + d*x^2)*Erfi[a + b*x]/x^1, x, 0, Unintegrable[(E^(c + d*x^2)*Erfi[a + b*x])/x, x]} -{E^(c + d*x^2)*Erfi[a + b*x]/x^3, x, 4, -((b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(Sqrt[Pi]*x)) - (E^(c + d*x^2)*Erfi[a + b*x])/(2*x^2) + b*Sqrt[b^2 + d]*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]] + (2*a*b^2*Unintegrable[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/Sqrt[Pi] + d*Unintegrable[(E^(c + d*x^2)*Erfi[a + b*x])/x, x]} - -{E^(c + d*x^2)*x^4*Erfi[a + b*x], x, 15, -(a^2*b^3*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)^3*Sqrt[Pi]) + (b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) + (3*b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(4*d^2*(b^2 + d)*Sqrt[Pi]) + (a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)*x)/(2*d*(b^2 + d)^2*Sqrt[Pi]) - (b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)*x^2)/(2*d*(b^2 + d)*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfi[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfi[a + b*x])/(2*d) + (a^3*b^4*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)^(7/2)) - (3*a*b^2*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d*(b^2 + d)^(5/2)) - (3*a*b^2*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d^2*(b^2 + d)^(3/2)) + (3*Unintegrable[E^(c + d*x^2)*Erfi[a + b*x], x])/(4*d^2)} -{E^(c + d*x^2)*x^2*Erfi[a + b*x], x, 4, -(b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(2*d*(b^2 + d)*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erfi[a + b*x])/(2*d) + (a*b^2*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)^(3/2)) - Unintegrable[E^(c + d*x^2)*Erfi[a + b*x], x]/(2*d)} -{E^(c + d*x^2)*x^0*Erfi[a + b*x], x, 0, Unintegrable[E^(c + d*x^2)*Erfi[a + b*x], x]} -{E^(c + d*x^2)*Erfi[a + b*x]/x^2, x, 1, -((E^(c + d*x^2)*Erfi[a + b*x])/x) + (2*b*Unintegrable[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/Sqrt[Pi] + 2*d*Unintegrable[E^(c + d*x^2)*Erfi[a + b*x], x]} -{E^(c + d*x^2)*Erfi[a + b*x]/x^4, x, 6, -(b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(3*Sqrt[Pi]*x^2) - (2*a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(3*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erfi[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfi[a + b*x])/(3*x) + (2*a*b^2*Sqrt[b^2 + d]*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/3 + (4*a^2*b^3*Unintegrable[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4*b*d*Unintegrable[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (2*b*(b^2 + d)*Unintegrable[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfi[a + b*x], x])/3} - - -{Erfi[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfi[b*x])/(E^(b^2*x^2)*x), x, 5, -(b/(Sqrt[Pi]*x)) - Erfi[b*x]/(2*E^(b^2*x^2)*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Trig[c+d x^2] Erfi[a+b x]^n*) - - -{Sin[c + I*b^2*x^2]*Erfi[b*x], x, 4, (I*Sqrt[Pi]*Erfi[b*x]^2)/(E^(I*c)*(8*b)) - (I*b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(2*Sqrt[Pi])} -{Sin[c - I*b^2*x^2]*Erfi[b*x], x, 4, -((I*E^(I*c)*Sqrt[Pi]*Erfi[b*x]^2)/(8*b)) + (I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} - - -{Cos[c + I*b^2*x^2]*Erfi[b*x], x, 4, (Sqrt[Pi]*Erfi[b*x]^2)/(E^(I*c)*(8*b)) + (b*E^(I*c)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(2*Sqrt[Pi])} -{Cos[c - I*b^2*x^2]*Erfi[b*x], x, 4, (E^(I*c)*Sqrt[Pi]*Erfi[b*x]^2)/(8*b) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(E^(I*c)*(2*Sqrt[Pi]))} - - -{Sinh[c + b^2*x^2]*Erfi[b*x], x, 4, (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*b) - (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(E^c*(2*Sqrt[Pi]))} -{Sinh[c - b^2*x^2]*Erfi[b*x], x, 4, -((Sqrt[Pi]*Erfi[b*x]^2)/(E^c*(8*b))) + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(2*Sqrt[Pi])} - - -{Cosh[c + b^2*x^2]*Erfi[b*x], x, 4, (E^c*Sqrt[Pi]*Erfi[b*x]^2)/(8*b) + (b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(E^c*(2*Sqrt[Pi]))} -{Cosh[c - b^2*x^2]*Erfi[b*x], x, 4, (Sqrt[Pi]*Erfi[b*x]^2)/(E^c*(8*b)) + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-b^2)*x^2])/(2*Sqrt[Pi])} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.10 Formal derivatives.m b/test/methods/rule_based/test_files/8 Special functions/8.10 Formal derivatives.m deleted file mode 100644 index da8ba60..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.10 Formal derivatives.m +++ /dev/null @@ -1,156 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Differentiation Function Integration Problems*) - - -{f'[x], x, 1, f[x]} -{f''[x], x, 1, f'[x]} -{f'''[x], x, 1, f''[x]} -{Derivative[n][f][x], x, 1, Derivative[n-1][f][x]} -{Derivative[1][u][x]*Derivative[2][u][x], x, 2, Derivative[1][u][x]^2/2} - - -{f'[x]/f[x], x, 2, Log[f[x]]} -{f'[x]/(a+b*f[x]), x, 2, Log[a+b*f[x]]/b} - -{f[x]*f'[x], x, 2, f[x]^2/2} -{(a+b*f[x])*f'[x], x, 2, a*f[x] + (1/2)*b*f[x]^2} - -{f'[x]/Sqrt[f[x]], x, 2, 2*Sqrt[f[x]]} -{f'[x]/Sqrt[a+b*f[x]], x, 2, (2*Sqrt[a+b*f[x]])/b} - -{f[x]^n*f'[x], x, 2, f[x]^(n+1)/(n+1)} -{(a+b*f[x])^n*f'[x], x, 2, (a+b*f[x])^(n+1)/(b*(n+1))} - - -{f''[x]/f'[x], x, 2, Log[f'[x]]} -{f''[x]/(a+b*f'[x]), x, 2, Log[a+b*f'[x]]/b} - -{f'[x]*f''[x], x, 2, f'[x]^2/2} -{(a+b*f'[x])*f''[x], x, 2, a*f'[x] + (1/2)*b*f'[x]^2} - -{f''[x]/Sqrt[f'[x]], x, 2, 2*Sqrt[f'[x]]} -{f''[x]/Sqrt[a+b*f'[x]], x, 2, (2*Sqrt[a+b*f'[x]])/b} - -{f'[x]^n*f''[x], x, 2, f'[x]^(n+1)/(n+1)} -{(a+b*f'[x])^n*f''[x], x, 2, (a+b*f'[x])^(n+1)/(b*(n+1))} - - -{f[g[x]]*g'[x], x, 1, CannotIntegrate[f[g[x]]*g'[x], x]} -{f[g'[x]]*g''[x], x, 1, CannotIntegrate[f[g'[x]]*g''[x], x]} - - -{f'[x]*g[x] + f[x]*g'[x], x, -1, f[x]*g[x]} -{(f'[x]*g[x] - f[x]*g'[x])/g[x]^2, x, 1, f[x]/g[x]} -{(f'[x]*g[x] - f[x]*g'[x])/(f[x]*g[x]), x, 1, Log[f[x]/g[x]]} - - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(1 + f[x]^2*g[x]^2), x, 2, ArcTan[f[x]*g[x]]} -{(g[x]*Derivative[1][f][x] - f[x]*g'[x])/(f[x]^2 + g[x]^2), x, 2, ArcTan[f[x]/g[x]]} -{-((g[x]*Derivative[1][f][x] + f[x]*g'[x])/(1 + f[x]^2*g[x]^2)), x, 2, -ArcTan[f[x]*g[x]]} - - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(1 - f[x]^2*g[x]^2), x, 2, ArcTanh[f[x]*g[x]]} -{((-g[x])*Derivative[1][f][x] + f[x]*g'[x])/(f[x]^2 - g[x]^2), x, 2, ArcTanh[f[x]/g[x]]} - - -{(f[x]^(-1 + m)*g[x]^(-1 + n)*(m*g[x]*Derivative[1][f][x] + n*f[x]*g'[x]))/(1 - f[x]^(2*m)*g[x]^(2*n)), x, 2, ArcTanh[f[x]^m*g[x]^n]} -{(f[x]^(-1 + m)*g[x]^(-1 + n)*((-m)*g[x]*Derivative[1][f][x] + n*f[x]*g'[x]))/(f[x]^(2*m) - g[x]^(2*n)), x, 3, ArcTanh[g[x]^n/f[x]^m]} -{(f[x]^(-1 + m)*g[x]^(-1 - n)*((-m)*g[x]*Derivative[1][f][x] - n*f[x]*g'[x]))/(f[x]^(2*m) - g[x]^(-2*n)), x, 3, ArcTanh[1/(f[x]^m*g[x]^n)]} - - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*f[x]^1*g[x]^1), x, 2, Log[a + b*f[x]*g[x]]/b} -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*f[x]^2*g[x]^2), x, 2, ArcTan[(Sqrt[b]*f[x]*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*f[x]^3*g[x]^3), x, 7, -(ArcTan[(a^(1/3) - 2*b^(1/3)*f[x]*g[x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3))) + Log[a^(1/3) + b^(1/3)*f[x]*g[x]]/(3*a^(2/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*f[x]*g[x] + b^(2/3)*f[x]^2*g[x]^2]/(6*a^(2/3)*b^(1/3))} - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*(f[x]*g[x])^(1/2)), x, 4, (2*Sqrt[f[x]*g[x]])/b - (2*a*Log[a + b*Sqrt[f[x]*g[x]]])/b^2} -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*(f[x]*g[x])^(3/2)), x, 8, -((2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sqrt[f[x]*g[x]])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(2/3))) - (2*Log[a^(1/3) + b^(1/3)*Sqrt[f[x]*g[x]]])/(3*a^(1/3)*b^(2/3)) + Log[a^(2/3) + b^(2/3)*f[x]*g[x] - a^(1/3)*b^(1/3)*Sqrt[f[x]*g[x]]]/(3*a^(1/3)*b^(2/3))} -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*(f[x]*g[x])^(5/2)), x, 8, -((Sqrt[2*(5 - Sqrt[5])]*ArcTan[Sqrt[(1/5)*(5 - 2*Sqrt[5])] + (2*Sqrt[2/(5 + Sqrt[5])]*b^(1/5)*Sqrt[f[x]*g[x]])/a^(1/5)])/(5*a^(3/5)*b^(2/5))) - (Sqrt[2*(5 + Sqrt[5])]*ArcTan[Sqrt[(1/5)*(5 + 2*Sqrt[5])] - (Sqrt[(2/5)*(5 + Sqrt[5])]*b^(1/5)*Sqrt[f[x]*g[x]])/a^(1/5)])/(5*a^(3/5)*b^(2/5)) - (2*Log[a^(1/5) + b^(1/5)*Sqrt[f[x]*g[x]]])/(5*a^(3/5)*b^(2/5)) + ((1 - Sqrt[5])*Log[2*a^(2/5) + 2*b^(2/5)*f[x]*g[x] - a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]] - Sqrt[5]*a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]]])/(10*a^(3/5)*b^(2/5)) + ((1 + Sqrt[5])*Log[2*a^(2/5) + 2*b^(2/5)*f[x]*g[x] - a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]] + Sqrt[5]*a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]]])/(10*a^(3/5)*b^(2/5))} - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*(f[x]*g[x])^n), x, 2, (f[x]*g[x]*Hypergeometric2F1[1, 1/n, 1 + 1/n, -((b*(f[x]*g[x])^n)/a)])/a} -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*f[x]^n*g[x]^n), x, 2, CannotIntegrate[(g[x]*Derivative[1][f][x])/(a + b*f[x]^n*g[x]^n), x] + CannotIntegrate[(f[x]*g'[x])/(a + b*f[x]^n*g[x]^n), x]} - - -{Cos[x]*g[E^x]*Derivative[1][f][Sin[x]] + E^x*f[Sin[x]]*Derivative[1][g][E^x], x, -1, f[Sin[x]]*g[E^x]} - - -{F^(a + b*x)*Derivative[3][f][x], x, 3, b^2*F^(a + b*x)*f[x]*Log[F]^2 - b^3*CannotIntegrate[F^(a + b*x)*f[x], x]*Log[F]^3 - b*F^(a + b*x)*Log[F]*Derivative[1][f][x] + F^(a + b*x)*Derivative[2][f][x]} -{F^(a + b*x)*Derivative[2][f][x], x, 2, (-b)*F^(a + b*x)*f[x]*Log[F] + b^2*CannotIntegrate[F^(a + b*x)*f[x], x]*Log[F]^2 + F^(a + b*x)*Derivative[1][f][x]} -{F^(a + b*x)*Derivative[1][f][x], x, 1, F^(a + b*x)*f[x] - b*CannotIntegrate[F^(a + b*x)*f[x], x]*Log[F]} -{F^(a + b*x)*Derivative[0][f][x], x, 0, CannotIntegrate[F^(a + b*x)*f[x], x]} -{F^(a + b*x)*Derivative[-1][f][x], x, 1, -(CannotIntegrate[F^(a + b*x)*f[x], x]/(b*Log[F])) + (F^(a + b*x)*Derivative[-1][f][x])/(b*Log[F])} -{F^(a + b*x)*Derivative[-2][f][x], x, 2, CannotIntegrate[F^(a + b*x)*f[x], x]/(b^2*Log[F]^2) + (F^(a + b*x)*Derivative[-2][f][x])/(b*Log[F]) - (F^(a + b*x)*Derivative[-1][f][x])/(b^2*Log[F]^2)} -{F^(a + b*x)*Derivative[-3][f][x], x, 3, -(CannotIntegrate[F^(a + b*x)*f[x], x]/(b^3*Log[F]^3)) + (F^(a + b*x)*Derivative[-3][f][x])/(b*Log[F]) - (F^(a + b*x)*Derivative[-2][f][x])/(b^2*Log[F]^2) + (F^(a + b*x)*Derivative[-1][f][x])/(b^3*Log[F]^3)} - -{F^(a + b*x)*Derivative[3][f][x] + b^3*F^(a + b*x)*f[x]*Log[F]^3, x, 4, b^2*F^(a + b*x)*f[x]*Log[F]^2 - b*F^(a + b*x)*Log[F]*Derivative[1][f][x] + F^(a + b*x)*Derivative[2][f][x]} - - -{Sin[a + b*x]*Derivative[3][f][x], x, 3, b^3*CannotIntegrate[Cos[a + b*x]*f[x], x] - b^2*f[x]*Sin[a + b*x] - b*Cos[a + b*x]*Derivative[1][f][x] + Sin[a + b*x]*Derivative[2][f][x]} -{Sin[a + b*x]*Derivative[2][f][x], x, 2, (-b)*Cos[a + b*x]*f[x] - b^2*CannotIntegrate[f[x]*Sin[a + b*x], x] + Sin[a + b*x]*Derivative[1][f][x]} -{Sin[a + b*x]*Derivative[1][f][x], x, 1, (-b)*CannotIntegrate[Cos[a + b*x]*f[x], x] + f[x]*Sin[a + b*x]} -{Sin[a + b*x]*Derivative[0][f][x], x, 0, CannotIntegrate[f[x]*Sin[a + b*x], x]} -{Sin[a + b*x]*Derivative[-1][f][x], x, 1, CannotIntegrate[Cos[a + b*x]*f[x], x]/b - (Cos[a + b*x]*Derivative[-1][f][x])/b} -{Sin[a + b*x]*Derivative[-2][f][x], x, 2, -(CannotIntegrate[f[x]*Sin[a + b*x], x]/b^2) - (Cos[a + b*x]*Derivative[-2][f][x])/b + (Sin[a + b*x]*Derivative[-1][f][x])/b^2} -{Sin[a + b*x]*Derivative[-3][f][x], x, 3, -(CannotIntegrate[Cos[a + b*x]*f[x], x]/b^3) - (Cos[a + b*x]*Derivative[-3][f][x])/b + (Sin[a + b*x]*Derivative[-2][f][x])/b^2 + (Cos[a + b*x]*Derivative[-1][f][x])/b^3} - -{Sin[a + b*x]*Derivative[3][f][x] - b^3*Cos[a + b*x]*f[x], x, 4, (-b^2)*f[x]*Sin[a + b*x] - b*Cos[a + b*x]*Derivative[1][f][x] + Sin[a + b*x]*Derivative[2][f][x]} - - -{Cos[a + b*x]*Derivative[3][f][x], x, 3, (-b^2)*Cos[a + b*x]*f[x] - b^3*CannotIntegrate[f[x]*Sin[a + b*x], x] + b*Sin[a + b*x]*Derivative[1][f][x] + Cos[a + b*x]*Derivative[2][f][x]} -{Cos[a + b*x]*Derivative[2][f][x], x, 2, (-b^2)*CannotIntegrate[Cos[a + b*x]*f[x], x] + b*f[x]*Sin[a + b*x] + Cos[a + b*x]*Derivative[1][f][x]} -{Cos[a + b*x]*Derivative[1][f][x], x, 1, Cos[a + b*x]*f[x] + b*CannotIntegrate[f[x]*Sin[a + b*x], x]} -{Cos[a + b*x]*Derivative[0][f][x], x, 0, CannotIntegrate[Cos[a + b*x]*f[x], x]} -{Cos[a + b*x]*Derivative[-1][f][x], x, 1, -(CannotIntegrate[f[x]*Sin[a + b*x], x]/b) + (Sin[a + b*x]*Derivative[-1][f][x])/b} -{Cos[a + b*x]*Derivative[-2][f][x], x, 2, -(CannotIntegrate[Cos[a + b*x]*f[x], x]/b^2) + (Sin[a + b*x]*Derivative[-2][f][x])/b + (Cos[a + b*x]*Derivative[-1][f][x])/b^2} -{Cos[a + b*x]*Derivative[-3][f][x], x, 3, CannotIntegrate[f[x]*Sin[a + b*x], x]/b^3 + (Sin[a + b*x]*Derivative[-3][f][x])/b + (Cos[a + b*x]*Derivative[-2][f][x])/b^2 - (Sin[a + b*x]*Derivative[-1][f][x])/b^3} - -{Cos[a + b*x]*Derivative[-2][f][x] + Cos[a + b*x]*(f[x]/b^2), x, 3, (Sin[a + b*x]*Derivative[-2][f][x])/b + (Cos[a + b*x]*Derivative[-1][f][x])/b^2} - - -{Cos[f[x]*g[x]]*(g[x]*Derivative[1][f][x] + f[x]*g'[x]), x, 2, Sin[f[x]*g[x]]} -{Cos[g[x]*Derivative[m][f][x]]*(g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]), x, 2, Sin[g[x]*Derivative[m][f][x]]} -{Cos[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]*(Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]), x, 2, Sin[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]} - - -{(g[x]*Derivative[1][f][x] + f[x]*g'[x])/(a + b*f[x]^2*g[x]^2), x, 2, ArcTan[(Sqrt[b]*f[x]*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x])/(a + b*g[x]^2*Derivative[m][f][x]^2), x, 2, ArcTan[(Sqrt[b]*g[x]*Derivative[m][f][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(Derivative[1 + m][f][x]*Derivative[n][g][x] + Derivative[m][f][x]*Derivative[1 + n][g][x])/(a + b*Derivative[m][f][x]^2*Derivative[n][g][x]^2), x, 2, ArcTan[(Sqrt[b]*Derivative[m][f][x]*Derivative[n][g][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} - - -{Derivative[1][F][f[x]*g[x]]*(g[x]*Derivative[1][f][x] + f[x]*g'[x]), x, 2, F[f[x]*g[x]]} -{Derivative[1][F][g[x]*Derivative[m][f][x]]*(g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]), x, 2, F[g[x]*Derivative[m][f][x]]} -{Derivative[1][F][Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]*(Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]), x, 2, F[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]} - - -{Cos[f[x]^2*g[x]]*f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*g'[x]), x, 2, Sin[f[x]^2*g[x]]} -{Cos[g[x]^2*Derivative[m][f][x]]*g[x]*(2*g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]), x, 2, Sin[g[x]^2*Derivative[m][f][x]]} -{Cos[g[x]*Derivative[m][f][x]^2]*Derivative[m][f][x]*(g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]), x, 2, Sin[g[x]*Derivative[m][f][x]^2]} -{Cos[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]*Derivative[-1 + m][f][x]*(2*Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]), x, 2, Sin[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]} - - -{(f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*g'[x]))/(a + b*f[x]^4*g[x]^2), x, 2, ArcTan[(Sqrt[b]*f[x]^2*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(g[x]*(2*g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^4*Derivative[m][f][x]^2), x, 2, ArcTan[(Sqrt[b]*g[x]^2*Derivative[m][f][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(Derivative[m][f][x]*(g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^2*Derivative[m][f][x]^4), x, 2, ArcTan[(Sqrt[b]*g[x]*Derivative[m][f][x]^2)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(Derivative[m][f][x]*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + Derivative[m][f][x]*Derivative[1 + n][g][x]))/(a + b*Derivative[m][f][x]^4*Derivative[n][g][x]^2), x, 2, ArcTan[(Sqrt[b]*Derivative[m][f][x]^2*Derivative[n][g][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} - - -{f[x]*Derivative[1][F][f[x]^2*g[x]]*(2*g[x]*Derivative[1][f][x] + f[x]*g'[x]), x, 2, F[f[x]^2*g[x]]} -{g[x]*Derivative[1][F][g[x]^2*Derivative[m][f][x]]*(2*g'[x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]), x, 2, F[g[x]^2*Derivative[m][f][x]]} -{Derivative[1][F][g[x]*Derivative[m][f][x]^2]*Derivative[m][f][x]*(g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]), x, 2, F[g[x]*Derivative[m][f][x]^2]} -{Derivative[1][F][Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]*Derivative[-1 + m][f][x]*(2*Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]), x, 2, F[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]} - - -{Cos[f[x]^2*g[x]^3]*f[x]*g[x]^2*(2*g[x]*Derivative[1][f][x] + 3*f[x]*g'[x]), x, 2, Sin[f[x]^2*g[x]^3]} -{Cos[g[x]^3*Derivative[m][f][x]^2]*g[x]^2*Derivative[m][f][x]*(3*g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]), x, 2, Sin[g[x]^3*Derivative[m][f][x]^2]} -{Cos[Derivative[m][f][x]^2*Derivative[n][g][x]^3]*Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x]), x, 2, Sin[Derivative[m][f][x]^2*Derivative[n][g][x]^3]} - - -{(f[x]*g[x]^2*(2*g[x]*Derivative[1][f][x] + 3*f[x]*g'[x]))/(a + b*f[x]^4*g[x]^6), x, 2, ArcTan[(Sqrt[b]*f[x]^2*g[x]^3)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{(g[x]^2*Derivative[m][f][x]*(3*g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^6*Derivative[m][f][x]^4), x, 2, ArcTan[(Sqrt[b]*g[x]^3*Derivative[m][f][x]^2)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} -{Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x])/(a + b*Derivative[m][f][x]^4*Derivative[n][g][x]^6), x, 2, ArcTan[(Sqrt[b]*Derivative[m][f][x]^2*Derivative[n][g][x]^3)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])} - - -{f[x]*g[x]^2*Derivative[1][F][f[x]^2*g[x]^3]*(2*g[x]*Derivative[1][f][x] + 3*f[x]*g'[x]), x, 2, F[f[x]^2*g[x]^3]} -{g[x]^2*Derivative[1][F][g[x]^3*Derivative[m][f][x]^2]*Derivative[m][f][x]*(3*g'[x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]), x, 2, F[g[x]^3*Derivative[m][f][x]^2]} -{Derivative[1][F][Derivative[m][f][x]^2*Derivative[n][g][x]^3]*Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x]), x, 2, F[Derivative[m][f][x]^2*Derivative[n][g][x]^3]} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.2 Fresnel integral functions.m b/test/methods/rule_based/test_files/8 Special functions/8.2 Fresnel integral functions.m deleted file mode 100644 index e884fe1..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.2 Fresnel integral functions.m +++ /dev/null @@ -1,418 +0,0 @@ -(* ::Package:: *) - -(* ::Title::Closed:: *) -(*Integration Problems Involving The Fresnel S Integral Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelS[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m FresnelS[b x]*) - - -{x^7*FresnelS[b*x], x, 6, -((35*x^3*Cos[(1/2)*b^2*Pi*x^2])/(8*b^5*Pi^3)) + (x^7*Cos[(1/2)*b^2*Pi*x^2])/(8*b*Pi) - (105*FresnelS[b*x])/(8*b^8*Pi^4) + (1/8)*x^8*FresnelS[b*x] + (105*x*Sin[(1/2)*b^2*Pi*x^2])/(8*b^7*Pi^4) - (7*x^5*Sin[(1/2)*b^2*Pi*x^2])/(8*b^3*Pi^2)} -{x^6*FresnelS[b*x], x, 6, -((24*x^2*Cos[(1/2)*b^2*Pi*x^2])/(7*b^5*Pi^3)) + (x^6*Cos[(1/2)*b^2*Pi*x^2])/(7*b*Pi) + (1/7)*x^7*FresnelS[b*x] + (48*Sin[(1/2)*b^2*Pi*x^2])/(7*b^7*Pi^4) - (6*x^4*Sin[(1/2)*b^2*Pi*x^2])/(7*b^3*Pi^2)} -{x^5*FresnelS[b*x], x, 5, -((5*x*Cos[(1/2)*b^2*Pi*x^2])/(2*b^5*Pi^3)) + (x^5*Cos[(1/2)*b^2*Pi*x^2])/(6*b*Pi) + (5*FresnelC[b*x])/(2*b^6*Pi^3) + (1/6)*x^6*FresnelS[b*x] - (5*x^3*Sin[(1/2)*b^2*Pi*x^2])/(6*b^3*Pi^2)} -{x^4*FresnelS[b*x], x, 5, -((8*Cos[(1/2)*b^2*Pi*x^2])/(5*b^5*Pi^3)) + (x^4*Cos[(1/2)*b^2*Pi*x^2])/(5*b*Pi) + (1/5)*x^5*FresnelS[b*x] - (4*x^2*Sin[(1/2)*b^2*Pi*x^2])/(5*b^3*Pi^2)} -{x^3*FresnelS[b*x], x, 4, (x^3*Cos[(1/2)*b^2*Pi*x^2])/(4*b*Pi) + (3*FresnelS[b*x])/(4*b^4*Pi^2) + (1/4)*x^4*FresnelS[b*x] - (3*x*Sin[(1/2)*b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^2*FresnelS[b*x], x, 4, (x^2*Cos[(1/2)*b^2*Pi*x^2])/(3*b*Pi) + (1/3)*x^3*FresnelS[b*x] - (2*Sin[(1/2)*b^2*Pi*x^2])/(3*b^3*Pi^2)} -{x^1*FresnelS[b*x], x, 3, (x*Cos[(1/2)*b^2*Pi*x^2])/(2*b*Pi) - FresnelC[b*x]/(2*b^2*Pi) + (1/2)*x^2*FresnelS[b*x]} -{x^0*FresnelS[b*x], x, 1, Cos[(1/2)*b^2*Pi*x^2]/(b*Pi) + x*FresnelS[b*x]} -{FresnelS[b*x]/x^1, x, 3, (1/2)*I*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-(1/2))*I*b^2*Pi*x^2] - (1/2)*I*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (1/2)*I*b^2*Pi*x^2]} -{FresnelS[b*x]/x^2, x, 2, -(FresnelS[b*x]/x) + (1/2)*b*SinIntegral[(1/2)*b^2*Pi*x^2]} -{FresnelS[b*x]/x^3, x, 3, (1/2)*b^2*Pi*FresnelC[b*x] - FresnelS[b*x]/(2*x^2) - (b*Sin[(1/2)*b^2*Pi*x^2])/(2*x)} -{FresnelS[b*x]/x^4, x, 4, (1/12)*b^3*Pi*CosIntegral[(1/2)*b^2*Pi*x^2] - FresnelS[b*x]/(3*x^3) - (b*Sin[(1/2)*b^2*Pi*x^2])/(6*x^2)} -{FresnelS[b*x]/x^5, x, 4, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(12*x)) - (1/12)*b^4*Pi^2*FresnelS[b*x] - FresnelS[b*x]/(4*x^4) - (b*Sin[(1/2)*b^2*Pi*x^2])/(12*x^3)} -{FresnelS[b*x]/x^6, x, 5, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(40*x^2)) - FresnelS[b*x]/(5*x^5) - (b*Sin[(1/2)*b^2*Pi*x^2])/(20*x^4) - (1/80)*b^5*Pi^2*SinIntegral[(1/2)*b^2*Pi*x^2]} -{FresnelS[b*x]/x^7, x, 5, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(90*x^3)) - (1/90)*b^6*Pi^3*FresnelC[b*x] - FresnelS[b*x]/(6*x^6) - (b*Sin[(1/2)*b^2*Pi*x^2])/(30*x^5) + (b^5*Pi^2*Sin[(1/2)*b^2*Pi*x^2])/(90*x)} -{FresnelS[b*x]/x^8, x, 6, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(168*x^4)) - (1/672)*b^7*Pi^3*CosIntegral[(1/2)*b^2*Pi*x^2] - FresnelS[b*x]/(7*x^7) - (b*Sin[(1/2)*b^2*Pi*x^2])/(42*x^6) + (b^5*Pi^2*Sin[(1/2)*b^2*Pi*x^2])/(336*x^2)} -{FresnelS[b*x]/x^9, x, 6, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(280*x^5)) + (b^7*Pi^3*Cos[(1/2)*b^2*Pi*x^2])/(840*x) + (1/840)*b^8*Pi^4*FresnelS[b*x] - FresnelS[b*x]/(8*x^8) - (b*Sin[(1/2)*b^2*Pi*x^2])/(56*x^7) + (b^5*Pi^2*Sin[(1/2)*b^2*Pi*x^2])/(840*x^3)} -{FresnelS[b*x]/x^10, x, 7, -((b^3*Pi*Cos[(1/2)*b^2*Pi*x^2])/(432*x^6)) + (b^7*Pi^3*Cos[(1/2)*b^2*Pi*x^2])/(3456*x^2) - FresnelS[b*x]/(9*x^9) - (b*Sin[(1/2)*b^2*Pi*x^2])/(72*x^8) + (b^5*Pi^2*Sin[(1/2)*b^2*Pi*x^2])/(1728*x^4) + (b^9*Pi^4*SinIntegral[(1/2)*b^2*Pi*x^2])/6912} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelS[a+b x]*) - - -{(c + d*x)^3*FresnelS[a + b*x], x, 14, ((b*c - a*d)^3*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) + (3*d*(b*c - a*d)^2*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^4*Pi) + (d^2*(b*c - a*d)*(a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) + (d^3*(a + b*x)^3*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi) - (3*d*(b*c - a*d)^2*FresnelC[a + b*x])/(2*b^4*Pi) - ((b*c - a*d)^4*FresnelS[a + b*x])/(4*b^4*d) + (3*d^3*FresnelS[a + b*x])/(4*b^4*Pi^2) + ((c + d*x)^4*FresnelS[a + b*x])/(4*d) - (2*d^2*(b*c - a*d)*Sin[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi^2) - (3*d^3*(a + b*x)*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi^2)} -{(c + d*x)^2*FresnelS[a + b*x], x, 11, ((b*c - a*d)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) + (d*(b*c - a*d)*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) + (d^2*(a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi) - (d*(b*c - a*d)*FresnelC[a + b*x])/(b^3*Pi) - ((b*c - a*d)^3*FresnelS[a + b*x])/(3*b^3*d) + ((c + d*x)^3*FresnelS[a + b*x])/(3*d) - (2*d^2*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)} -{(c + d*x)^1*FresnelS[a + b*x], x, 8, ((b*c - a*d)*Cos[(1/2)*Pi*(a + b*x)^2])/(b^2*Pi) + (d*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^2*Pi) - (d*FresnelC[a + b*x])/(2*b^2*Pi) - ((b*c - a*d)^2*FresnelS[a + b*x])/(2*b^2*d) + ((c + d*x)^2*FresnelS[a + b*x])/(2*d)} -{(c + d*x)^0*FresnelS[a + b*x], x, 1, Cos[(1/2)*Pi*(a + b*x)^2]/(b*Pi) + ((a + b*x)*FresnelS[a + b*x])/b} -{FresnelS[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[FresnelS[a + b*x]/(c + d*x), x]} -{FresnelS[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[FresnelS[a + b*x]/(c + d*x)^2, x]} - - -{x^3*FresnelS[a + b*x], x, 14, -((a^3*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi)) + (3*a^2*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^4*Pi) - (a*(a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) + ((a + b*x)^3*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi) - (3*a^2*FresnelC[a + b*x])/(2*b^4*Pi) - (a^4*FresnelS[a + b*x])/(4*b^4) + (3*FresnelS[a + b*x])/(4*b^4*Pi^2) + (1/4)*x^4*FresnelS[a + b*x] + (2*a*Sin[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi^2) - (3*(a + b*x)*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi^2)} -{x^2*FresnelS[a + b*x], x, 11, (a^2*Cos[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) - (a*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) + ((a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi) + (a*FresnelC[a + b*x])/(b^3*Pi) + (a^3*FresnelS[a + b*x])/(3*b^3) + (1/3)*x^3*FresnelS[a + b*x] - (2*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)} -{x^1*FresnelS[a + b*x], x, 8, -((a*Cos[(1/2)*Pi*(a + b*x)^2])/(b^2*Pi)) + ((a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(2*b^2*Pi) - FresnelC[a + b*x]/(2*b^2*Pi) - (a^2*FresnelS[a + b*x])/(2*b^2) + (1/2)*x^2*FresnelS[a + b*x]} -{x^0*FresnelS[a + b*x], x, 1, Cos[(1/2)*Pi*(a + b*x)^2]/(b*Pi) + ((a + b*x)*FresnelS[a + b*x])/b} -{FresnelS[a + b*x]/x^1, x, 0, Unintegrable[FresnelS[a + b*x]/x, x]} -{FresnelS[a + b*x]/x^2, x, 0, Unintegrable[FresnelS[a + b*x]/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelS[a+b x]^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m FresnelS[b x]^2*) - - -{x^7*FresnelS[b*x]^2, x, 23, -((105*x^2)/(16*b^6*Pi^4)) + (7*x^6)/(48*b^2*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) + (x^6*Cos[b^2*Pi*x^2])/(16*b^2*Pi^2) - (35*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(4*b^5*Pi^3) + (x^7*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(4*b*Pi) - (105*FresnelS[b*x]^2)/(8*b^8*Pi^4) + (1/8)*x^8*FresnelS[b*x]^2 + (105*x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(4*b^7*Pi^4) - (7*x^5*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(4*b^3*Pi^2) + (10*Sin[b^2*Pi*x^2])/(b^8*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)} -{x^6*FresnelS[b*x]^2, x, 19, -((48*x)/(7*b^6*Pi^4)) + (6*x^5)/(35*b^2*Pi^2) - (21*x*Cos[b^2*Pi*x^2])/(8*b^6*Pi^4) + (x^5*Cos[b^2*Pi*x^2])/(14*b^2*Pi^2) + (531*FresnelC[Sqrt[2]*b*x])/(56*Sqrt[2]*b^7*Pi^4) - (48*x^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(7*b^5*Pi^3) + (2*x^6*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(7*b*Pi) + (1/7)*x^7*FresnelS[b*x]^2 + (96*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(7*b^7*Pi^4) - (12*x^4*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(7*b^3*Pi^2) - (17*x^3*Sin[b^2*Pi*x^2])/(28*b^4*Pi^3)} -{x^5*FresnelS[b*x]^2, x, 16, (5*x^4)/(24*b^2*Pi^2) - (11*Cos[b^2*Pi*x^2])/(6*b^6*Pi^4) + (x^4*Cos[b^2*Pi*x^2])/(12*b^2*Pi^2) - (5*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^5*Pi^3) + (x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(3*b*Pi) + (5*FresnelC[b*x]*FresnelS[b*x])/(2*b^6*Pi^3) + (1/6)*x^6*FresnelS[b*x]^2 - (5*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^4*Pi^3) + (5*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^4*Pi^3) - (5*x^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(3*b^3*Pi^2) - (7*x^2*Sin[b^2*Pi*x^2])/(12*b^4*Pi^3)} -{x^4*FresnelS[b*x]^2, x, 12, (4*x^3)/(15*b^2*Pi^2) + (x^3*Cos[b^2*Pi*x^2])/(10*b^2*Pi^2) - (16*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(5*b^5*Pi^3) + (2*x^4*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(5*b*Pi) + (1/5)*x^5*FresnelS[b*x]^2 + (43*FresnelS[Sqrt[2]*b*x])/(20*Sqrt[2]*b^5*Pi^3) - (8*x^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(5*b^3*Pi^2) - (11*x*Sin[b^2*Pi*x^2])/(20*b^4*Pi^3)} -{x^3*FresnelS[b*x]^2, x, 10, (3*x^2)/(8*b^2*Pi^2) + (x^2*Cos[b^2*Pi*x^2])/(8*b^2*Pi^2) + (x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(2*b*Pi) + (3*FresnelS[b*x]^2)/(4*b^4*Pi^2) + (1/4)*x^4*FresnelS[b*x]^2 - (3*x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(2*b^3*Pi^2) - Sin[b^2*Pi*x^2]/(2*b^4*Pi^3)} -{x^2*FresnelS[b*x]^2, x, 8, (2*x)/(3*b^2*Pi^2) + (x*Cos[b^2*Pi*x^2])/(6*b^2*Pi^2) - (5*FresnelC[Sqrt[2]*b*x])/(6*Sqrt[2]*b^3*Pi^2) + (2*x^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(3*b*Pi) + (1/3)*x^3*FresnelS[b*x]^2 - (4*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(3*b^3*Pi^2)} -{x^1*FresnelS[b*x]^2, x, 5, Cos[b^2*Pi*x^2]/(4*b^2*Pi^2) + (x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b*Pi) - (FresnelC[b*x]*FresnelS[b*x])/(2*b^2*Pi) + (1/2)*x^2*FresnelS[b*x]^2 + (I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*Pi) - (I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*Pi)} -{x^0*FresnelS[b*x]^2, x, 4, (2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b*Pi) + x*FresnelS[b*x]^2 - FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b*Pi)} -{FresnelS[b*x]^2/x^1, x, 0, Unintegrable[FresnelS[b*x]^2/x, x]} -{FresnelS[b*x]^2/x^2, x, 1, -(FresnelS[b*x]^2/x) + 2*b*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{FresnelS[b*x]^2/x^3, x, 1, -(FresnelS[b*x]^2/(2*x^2)) + b*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{FresnelS[b*x]^2/x^4, x, 4, -(b^2/(6*x)) + (b^2*Cos[b^2*Pi*x^2])/(6*x) - FresnelS[b*x]^2/(3*x^3) + (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(3*x^2) + (1/3)*b^3*Pi*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{FresnelS[b*x]^2/x^5, x, 9, -(b^2/(24*x^2)) + (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(6*x) - (1/12)*b^4*Pi^2*FresnelS[b*x]^2 - FresnelS[b*x]^2/(4*x^4) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(6*x^3) + (1/12)*b^4*Pi*SinIntegral[b^2*Pi*x^2]} -{FresnelS[b*x]^2/x^6, x, 8, -(b^2/(60*x^3)) + (b^2*Cos[b^2*Pi*x^2])/(60*x^3) + (7*b^5*Pi^2*FresnelC[Sqrt[2]*b*x])/(60*Sqrt[2]) - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(20*x^2) - FresnelS[b*x]^2/(5*x^5) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(10*x^4) - (7*b^4*Pi*Sin[b^2*Pi*x^2])/(120*x) - (1/20)*b^5*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{FresnelS[b*x]^2/x^7, x, 10, -(b^2/(120*x^4)) + (b^2*Cos[b^2*Pi*x^2])/(120*x^4) + (1/72)*b^6*Pi^2*CosIntegral[b^2*Pi*x^2] - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(45*x^3) - FresnelS[b*x]^2/(6*x^6) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(15*x^5) - (b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (1/45)*b^5*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{FresnelS[b*x]^2/x^8, x, 13, -(b^2/(210*x^5)) + (b^6*Pi^2)/(336*x) + (b^2*Cos[b^2*Pi*x^2])/(210*x^5) - (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x) - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(84*x^4) - FresnelS[b*x]^2/(7*x^7) - (b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/(72*Sqrt[2]) - (2/315)*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*b*x] - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(21*x^6) + (b^5*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(168*x^2) - (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) - (1/168)*b^7*Pi^3*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{FresnelS[b*x]^2/x^9, x, 20, -(b^2/(336*x^6)) + (b^6*Pi^2)/(1680*x^2) + (b^2*Cos[b^2*Pi*x^2])/(336*x^6) - (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2) - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(140*x^5) + (b^7*Pi^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(420*x) + (1/840)*b^8*Pi^4*FresnelS[b*x]^2 - FresnelS[b*x]^2/(8*x^8) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(28*x^7) + (b^5*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(420*x^3) - (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) - (1/280)*b^8*Pi^3*SinIntegral[b^2*Pi*x^2]} -{FresnelS[b*x]^2/x^10, x, 19, -(b^2/(504*x^7)) + (b^6*Pi^2)/(5184*x^3) + (b^2*Cos[b^2*Pi*x^2])/(504*x^7) - (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181440*x^3) - (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) - (b^3*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(216*x^6) + (b^7*Pi^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(1728*x^2) - FresnelS[b*x]^2/(9*x^9) - (b*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(36*x^8) + (b^5*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(864*x^4) - (19*b^4*Pi*Sin[b^2*Pi*x^2])/(15120*x^5) + (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x])/1728} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelS[a+b x]^2*) - - -{(c + d*x)^2*FresnelS[a + b*x]^2, x, 18, (2*d^2*x)/(3*b^2*Pi^2) + (d*(b*c - a*d)*Cos[Pi*(a + b*x)^2])/(2*b^3*Pi^2) + (d^2*(a + b*x)*Cos[Pi*(a + b*x)^2])/(6*b^3*Pi^2) - (5*d^2*FresnelC[Sqrt[2]*(a + b*x)])/(6*Sqrt[2]*b^3*Pi^2) + (2*(b*c - a*d)^2*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b^3*Pi) + (2*d*(b*c - a*d)*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b^3*Pi) + (2*d^2*(a + b*x)^2*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(3*b^3*Pi) - (d*(b*c - a*d)*FresnelC[a + b*x]*FresnelS[a + b*x])/(b^3*Pi) + ((b*c - a*d)^2*(a + b*x)*FresnelS[a + b*x]^2)/b^3 + (d*(b*c - a*d)*(a + b*x)^2*FresnelS[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*FresnelS[a + b*x]^2)/(3*b^3) - ((b*c - a*d)^2*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^3*Pi) + (I*d*(b*c - a*d)*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*Pi*(a + b*x)^2])/(4*b^3*Pi) - (I*d*(b*c - a*d)*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*Pi*(a + b*x)^2])/(4*b^3*Pi) - (4*d^2*FresnelS[a + b*x]*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)} -{(c + d*x)^1*FresnelS[a + b*x]^2, x, 10, (d*Cos[Pi*(a + b*x)^2])/(4*b^2*Pi^2) + (2*(b*c - a*d)*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b^2*Pi) + (d*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b^2*Pi) - (d*FresnelC[a + b*x]*FresnelS[a + b*x])/(2*b^2*Pi) + ((b*c - a*d)*(a + b*x)*FresnelS[a + b*x]^2)/b^2 + (d*(a + b*x)^2*FresnelS[a + b*x]^2)/(2*b^2) - ((b*c - a*d)*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^2*Pi) + (I*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*Pi*(a + b*x)^2])/(8*b^2*Pi) - (I*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*Pi*(a + b*x)^2])/(8*b^2*Pi)} -{(c + d*x)^0*FresnelS[a + b*x]^2, x, 4, (2*Cos[(1/2)*Pi*(a + b*x)^2]*FresnelS[a + b*x])/(b*Pi) + ((a + b*x)*FresnelS[a + b*x]^2)/b - FresnelS[Sqrt[2]*(a + b*x)]/(Sqrt[2]*b*Pi)} -{FresnelS[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[FresnelS[a + b*x]^2/(c + d*x), x]} -{FresnelS[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[FresnelS[a + b*x]^2/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m FresnelS[d (a+b Log[c x^n])]*) - - -{x^2*FresnelS[d*(a + b*Log[c*x^n])], x, 10, ((1/12 - I/12)*E^((-3*a)/(b*n) + ((9*I)/2)/(b^2*d^2*n^2*Pi))*x^3*Erf[((1/2 + I/2)*(3/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) + ((1/12 - I/12)*E^((-3*a)/(b*n) - ((9*I)/2)/(b^2*d^2*n^2*Pi))*x^3*Erfi[((1/2 + I/2)*(3/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) + (x^3*FresnelS[d*(a + b*Log[c*x^n])])/3} -{x^1*FresnelS[d*(a + b*Log[c*x^n])], x, 10, ((1/8 - I/8)*E^((2*I - 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*x^2*Erf[((1/2 + I/2)*(2/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(2/n) + ((1/8 - I/8)*x^2*Erfi[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*(I + a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*(c*x^n)^(2/n)) + (x^2*FresnelS[d*(a + b*Log[c*x^n])])/2} -{x^0*FresnelS[d*(a + b*Log[c*x^n])], x, 10, ((1/4 - I/4)*x*Erf[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + ((1/4 - I/4)*x*Erfi[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + x*FresnelS[d*(a + b*Log[c*x^n])]} -{FresnelS[d*(a + b*Log[c*x^n])]/x^1, x, 3, Cos[(d^2*Pi*(a + b*Log[c*x^n])^2)/2]/(b*d*n*Pi) + (FresnelS[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)} -{FresnelS[d*(a + b*Log[c*x^n])]/x^2, x, 10, ((1/4 - I/4)*E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*Erf[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x + ((1/4 - I/4)*E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*Erfi[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x - FresnelS[d*(a + b*Log[c*x^n])]/x} -{FresnelS[d*(a + b*Log[c*x^n])]/x^3, x, 10, ((1/8 - I/8)*E^((2*I + 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*(c*x^n)^(2/n)*Erf[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x^2 + ((1/8 - I/8)*(c*x^n)^(2/n)*Erfi[((1/2 + I/2)*(2/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*(I - a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x^2) - FresnelS[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*FresnelS[d*(a + b*Log[c*x^n])], x, 10, ((1/4 - I/4)*E^(((I/2)*(1 + m)*(1 + m + (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x*(e*x)^m*Erf[((1/2 + I/2)*(1 + m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/((1 + m)*(c*x^n)^((1 + m)/n)) + ((1/4 - I/4)*x*(e*x)^m*Erfi[((1/2 + I/2)*(1 + m - I*a*b*d^2*n*Pi - I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/(E^(((I/2)*(1 + m)*(1 + m - (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*(1 + m)*(c*x^n)^((1 + m)/n)) + ((e*x)^(1 + m)*FresnelS[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(c+d x^2) FresnelS[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c+d x^2) FresnelS[b x] when d^2=Pi^2/4 b^4*) - - -{E^(c + Pi/2*I*b^2*x^2)*FresnelS[b*x], x, 4, If[$VersionNumber>=8, -((E^c*Erfi[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/(8*b)) + (1/4)*I*b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2], (E^c*Erf[(1/2 - I/2)*b*Sqrt[Pi]*x]^2)/(8*b) + (1/4)*I*b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2]]} -{E^(c - Pi/2*I*b^2*x^2)*FresnelS[b*x], x, 4, (E^c*Erf[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/(8*b) - (1/4)*I*b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sin[c+d x^2] FresnelS[a+b x]^n*) - - -{Sin[c + Pi/2*b^2*x^2]*FresnelS[b*x], x, 4, (Cos[c]*FresnelS[b*x]^2)/(2*b) + (FresnelC[b*x]*FresnelS[b*x]*Sin[c])/(2*b) - (1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2]*Sin[c] + (1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2]*Sin[c]} -{Cos[c + Pi/2*b^2*x^2]*FresnelS[b*x], x, 4, (Cos[c]*FresnelC[b*x]*FresnelS[b*x])/(2*b) - (1/8)*I*b*x^2*Cos[c]*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2] + (1/8)*I*b*x^2*Cos[c]*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2] - (FresnelS[b*x]^2*Sin[c])/(2*b)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[Pi/2 b^2 x^2] FresnelS[b x]^n*) - - -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]^2, x, 2, FresnelS[b*x]^3/(3*b)} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]^1, x, 2, FresnelS[b*x]^2/(2*b)} -{Sin[Pi/2*b^2*x^2]/FresnelS[b*x]^1, x, 2, Log[FresnelS[b*x]]/b} -{Sin[Pi/2*b^2*x^2]/FresnelS[b*x]^2, x, 2, -(1/(b*FresnelS[b*x]))} -{Sin[Pi/2*b^2*x^2]/FresnelS[b*x]^3, x, 2, -(1/(2*b*FresnelS[b*x]^2))} - - -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]^n, x, 2, FresnelS[b*x]^(1 + n)/(b*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sin[Pi/2 b^2 x^2] FresnelS[b x]*) - - -{x^8*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 22, (105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) - (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (35*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^6*Pi^3) - (x^7*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) + (105*FresnelS[b*x]^2)/(2*b^9*Pi^4) - (105*x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^8*Pi^4) + (7*x^5*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) - (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5*Pi^3)} -{x^7*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 18, (24*x)/(b^7*Pi^4) - (3*x^5)/(5*b^3*Pi^2) + (147*x*Cos[b^2*Pi*x^2])/(16*b^7*Pi^4) - (x^5*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (531*FresnelC[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) + (24*x^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^6*Pi^3) - (x^6*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) - (48*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^8*Pi^4) + (6*x^4*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) + (17*x^3*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)} -{x^6*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 15, -((5*x^4)/(8*b^3*Pi^2)) + (11*Cos[b^2*Pi*x^2])/(2*b^7*Pi^4) - (x^4*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (15*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^6*Pi^3) - (x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) - (15*FresnelC[b*x]*FresnelS[b*x])/(2*b^7*Pi^3) + (15*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^5*Pi^3) - (15*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^5*Pi^3) + (5*x^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) + (7*x^2*Sin[b^2*Pi*x^2])/(4*b^5*Pi^3)} -{x^5*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 11, -((2*x^3)/(3*b^3*Pi^2)) - (x^3*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (8*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^6*Pi^3) - (x^4*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) - (43*FresnelS[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (4*x^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) + (11*x*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)} -{x^4*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 9, -((3*x^2)/(4*b^3*Pi^2)) - (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) - (3*FresnelS[b*x]^2)/(2*b^5*Pi^2) + (3*x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) + Sin[b^2*Pi*x^2]/(b^5*Pi^3)} -{x^3*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 7, -(x/(b^3*Pi^2)) - (x*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (5*FresnelC[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) - (x^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) + (2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2)} -{x^2*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 4, -(Cos[b^2*Pi*x^2]/(4*b^3*Pi^2)) - (x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi) + (FresnelC[b*x]*FresnelS[b*x])/(2*b^3*Pi) - (I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b*Pi) + (I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b*Pi)} -{x^1*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 2, -((Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^2*Pi)) + FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)} -{x^0*Sin[Pi/2*b^2*x^2]*FresnelS[b*x], x, 2, FresnelS[b*x]^2/(2*b)} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^1, x, 0, Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^2, x, 0, Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^3, x, 3, -(b/(4*x)) + (b*Cos[b^2*Pi*x^2])/(4*x) + (b^2*Pi*FresnelS[Sqrt[2]*b*x])/(2*Sqrt[2]) - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(2*x^2) + (1/2)*b^2*Pi*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^4, x, 8, -(b/(12*x^2)) + (b*Cos[b^2*Pi*x^2])/(12*x^2) - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(3*x) - (1/6)*b^3*Pi^2*FresnelS[b*x]^2 - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(3*x^3) + (1/6)*b^3*Pi*SinIntegral[b^2*Pi*x^2]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^5, x, 7, -(b/(24*x^3)) + (b*Cos[b^2*Pi*x^2])/(24*x^3) + (7*b^4*Pi^2*FresnelC[Sqrt[2]*b*x])/(24*Sqrt[2]) - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(8*x^2) - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(4*x^4) - (7*b^3*Pi*Sin[b^2*Pi*x^2])/(48*x) - (1/8)*b^4*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^6, x, 9, -(b/(40*x^4)) + (b*Cos[b^2*Pi*x^2])/(40*x^4) + (1/24)*b^5*Pi^2*CosIntegral[b^2*Pi*x^2] - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(15*x^3) - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(5*x^5) - (b^3*Pi*Sin[b^2*Pi*x^2])/(24*x^2) - (1/15)*b^4*Pi^2*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^7, x, 12, -(b/(60*x^5)) + (b^5*Pi^2)/(96*x) + (b*Cos[b^2*Pi*x^2])/(60*x^5) - (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(1440*x) - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(24*x^4) - (7*b^6*Pi^3*FresnelS[Sqrt[2]*b*x])/(144*Sqrt[2]) - (1/45)*Sqrt[2]*b^6*Pi^3*FresnelS[Sqrt[2]*b*x] - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(6*x^6) + (b^4*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(48*x^2) - (13*b^3*Pi*Sin[b^2*Pi*x^2])/(720*x^3) - (1/48)*b^6*Pi^3*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^8, x, 19, -(b/(84*x^6)) + (b^5*Pi^2)/(420*x^2) + (b*Cos[b^2*Pi*x^2])/(84*x^6) - (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(35*x^5) + (b^6*Pi^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(105*x) + (1/210)*b^7*Pi^4*FresnelS[b*x]^2 - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(7*x^7) + (b^4*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(105*x^3) - (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) - (1/70)*b^7*Pi^3*SinIntegral[b^2*Pi*x^2]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^9, x, 18, -(b/(112*x^7)) + (b^5*Pi^2)/(1152*x^3) + (b*Cos[b^2*Pi*x^2])/(112*x^7) - (187*b^5*Pi^2*Cos[b^2*Pi*x^2])/(40320*x^3) - (853*b^8*Pi^4*FresnelC[Sqrt[2]*b*x])/(40320*Sqrt[2]) - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(48*x^6) + (b^6*Pi^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(384*x^2) - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(8*x^8) + (b^4*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(192*x^4) - (19*b^3*Pi*Sin[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*Sin[b^2*Pi*x^2])/(80640*x) + (1/384)*b^8*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{Sin[Pi/2*b^2*x^2]*FresnelS[b*x]/x^10, x, 22, -(b/(144*x^8)) + (b^5*Pi^2)/(2520*x^4) + (b*Cos[b^2*Pi*x^2])/(144*x^8) - (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(30240*x^4) - (5*b^9*Pi^4*CosIntegral[b^2*Pi*x^2])/2016 - (b^2*Pi*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(63*x^7) + (b^6*Pi^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(945*x^3) - (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(9*x^9) + (b^4*Pi^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(315*x^5) - (11*b^3*Pi*Sin[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Sin[b^2*Pi*x^2])/(2016*x^2) + (1/945)*b^8*Pi^4*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Sin[c+d x^2] FresnelS[b x]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[c+d x^2] FresnelS[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[Pi/2 b^2 x^2] FresnelS[b x]^n*) - - -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]^n, x, 0, Unintegrable[Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x]^n, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[Pi/2 b^2 x^2] FresnelS[b x]*) - - -{x^8*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 23, (35*x^4)/(8*b^5*Pi^3) - x^8/(16*b*Pi) - (40*Cos[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Cos[b^2*Pi*x^2])/(2*b^5*Pi^3) - (105*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) + (105*FresnelC[b*x]*FresnelS[b*x])/(2*b^9*Pi^4) - (105*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^7*Pi^4) + (105*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^7*Pi^4) - (35*x^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^7*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) - (55*x^2*Sin[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^7*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 18, (4*x^3)/(b^5*Pi^3) - x^7/(14*b*Pi) + (17*x^3*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) - (48*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^8*Pi^4) + (6*x^4*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) + (531*FresnelS[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) - (24*x^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^6*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) - (147*x*Sin[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^6*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 16, (15*x^2)/(4*b^5*Pi^3) - x^6/(12*b*Pi) + (7*x^2*Cos[b^2*Pi*x^2])/(4*b^5*Pi^3) + (5*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) + (15*FresnelS[b*x]^2)/(2*b^7*Pi^3) - (15*x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^5*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) - (11*Sin[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^5*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 13, (4*x)/(b^5*Pi^3) - x^5/(10*b*Pi) + (11*x*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) - (43*FresnelC[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (4*x^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) - (8*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^4*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + (x^3*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^4*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 10, -(x^4/(8*b*Pi)) + Cos[b^2*Pi*x^2]/(b^5*Pi^3) + (3*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) - (3*FresnelC[b*x]*FresnelS[b*x])/(2*b^5*Pi^2) + (3*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^3*Pi^2) - (3*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^3*Pi^2) + (x^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + (x^2*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^3*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 7, -(x^3/(6*b*Pi)) + (2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(b^4*Pi^2) - (5*FresnelS[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) + (x^2*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + (x*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^2*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 5, -(x^2/(4*b*Pi)) - FresnelS[b*x]^2/(2*b^3*Pi) + (x*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + Sin[b^2*Pi*x^2]/(4*b^3*Pi^2)} -{x^1*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 4, -(x/(2*b*Pi)) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi) + (FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi)} -{x^0*Cos[Pi/2*b^2*x^2]*FresnelS[b*x], x, 1, (FresnelC[b*x]*FresnelS[b*x])/(2*b) - (1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2] + (1/8)*I*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^1, x, 0, Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^2, x, 4, -((Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x) - (1/2)*b*Pi*FresnelS[b*x]^2 + (1/4)*b*SinIntegral[b^2*Pi*x^2]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^3, x, 3, (b^2*Pi*FresnelC[Sqrt[2]*b*x])/(2*Sqrt[2]) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(2*x^2) - (b*Sin[b^2*Pi*x^2])/(4*x) - (1/2)*b^2*Pi*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^4, x, 4, (1/12)*b^3*Pi*CosIntegral[b^2*Pi*x^2] - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(3*x^3) - (b*Sin[b^2*Pi*x^2])/(12*x^2) - (1/3)*b^2*Pi*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^5, x, 7, (b^3*Pi)/(16*x) - (7*b^3*Pi*Cos[b^2*Pi*x^2])/(48*x) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(4*x^4) - (7*b^4*Pi^2*FresnelS[Sqrt[2]*b*x])/(24*Sqrt[2]) + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(8*x^2) - (b*Sin[b^2*Pi*x^2])/(24*x^3) - (1/8)*b^4*Pi^2*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^6, x, 13, (b^3*Pi)/(60*x^2) - (b^3*Pi*Cos[b^2*Pi*x^2])/(24*x^2) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(5*x^5) + (b^4*Pi^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(15*x) + (1/30)*b^5*Pi^3*FresnelS[b*x]^2 + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(15*x^3) - (b*Sin[b^2*Pi*x^2])/(40*x^4) - (7/120)*b^5*Pi^2*SinIntegral[b^2*Pi*x^2]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^7, x, 12, (b^3*Pi)/(144*x^3) - (13*b^3*Pi*Cos[b^2*Pi*x^2])/(720*x^3) - (7*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/(144*Sqrt[2]) - (1/45)*Sqrt[2]*b^6*Pi^3*FresnelC[Sqrt[2]*b*x] - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(6*x^6) + (b^4*Pi^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(48*x^2) + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(24*x^4) - (b*Sin[b^2*Pi*x^2])/(60*x^5) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(1440*x) + (1/48)*b^6*Pi^3*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^8, x, 15, (b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (1/84)*b^7*Pi^3*CosIntegral[b^2*Pi*x^2] - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(7*x^7) + (b^4*Pi^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(105*x^3) + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(35*x^5) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) + (1/105)*b^6*Pi^3*Unintegrable[(FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/x^2, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^9, x, 18, (b^3*Pi)/(480*x^5) - (b^7*Pi^3)/(768*x) - (19*b^3*Pi*Cos[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*Cos[b^2*Pi*x^2])/(80640*x) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(8*x^8) + (b^4*Pi^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(192*x^4) + (853*b^8*Pi^4*FresnelS[Sqrt[2]*b*x])/(40320*Sqrt[2]) + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(48*x^6) - (b^6*Pi^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(384*x^2) - (b*Sin[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^2*Sin[b^2*Pi*x^2])/(40320*x^3) + (1/384)*b^8*Pi^4*Unintegrable[(Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/x, x]} -{Cos[Pi/2*b^2*x^2]*FresnelS[b*x]/x^10, x, 26, (b^3*Pi)/(756*x^6) - (b^7*Pi^3)/(3780*x^2) - (11*b^3*Pi*Cos[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Cos[b^2*Pi*x^2])/(2016*x^2) - (Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(9*x^9) + (b^4*Pi^2*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(315*x^5) - (b^8*Pi^4*Cos[(1/2)*b^2*Pi*x^2]*FresnelS[b*x])/(945*x) - (b^9*Pi^5*FresnelS[b*x]^2)/1890 + (b^2*Pi*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(63*x^7) - (b^6*Pi^3*FresnelS[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(945*x^3) - (b*Sin[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(30240*x^4) + (83*b^9*Pi^4*SinIntegral[b^2*Pi*x^2])/30240} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Cos[c+d x^2] FresnelS[b x]*) - - -(* ::Title::Closed:: *) -(*Integration Problems Involving The Fresnel C Integral Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelC[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m FresnelC[b x]*) - - -{x^7*FresnelC[b*x], x, 6, (105*x*Cos[(b^2*Pi*x^2)/2])/(8*b^7*Pi^4) - (7*x^5*Cos[(b^2*Pi*x^2)/2])/(8*b^3*Pi^2) - (105*FresnelC[b*x])/(8*b^8*Pi^4) + (x^8*FresnelC[b*x])/8 + (35*x^3*Sin[(b^2*Pi*x^2)/2])/(8*b^5*Pi^3) - (x^7*Sin[(b^2*Pi*x^2)/2])/(8*b*Pi)} -{x^6*FresnelC[b*x], x, 6, (48*Cos[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) - (6*x^4*Cos[(b^2*Pi*x^2)/2])/(7*b^3*Pi^2) + (x^7*FresnelC[b*x])/7 + (24*x^2*Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3) - (x^6*Sin[(b^2*Pi*x^2)/2])/(7*b*Pi)} -{x^5*FresnelC[b*x], x, 5, (-5*x^3*Cos[(b^2*Pi*x^2)/2])/(6*b^3*Pi^2) + (x^6*FresnelC[b*x])/6 - (5*FresnelS[b*x])/(2*b^6*Pi^3) + (5*x*Sin[(b^2*Pi*x^2)/2])/(2*b^5*Pi^3) - (x^5*Sin[(b^2*Pi*x^2)/2])/(6*b*Pi)} -{x^4*FresnelC[b*x], x, 5, (-4*x^2*Cos[(b^2*Pi*x^2)/2])/(5*b^3*Pi^2) + (x^5*FresnelC[b*x])/5 + (8*Sin[(b^2*Pi*x^2)/2])/(5*b^5*Pi^3) - (x^4*Sin[(b^2*Pi*x^2)/2])/(5*b*Pi)} -{x^3*FresnelC[b*x], x, 4, (-3*x*Cos[(b^2*Pi*x^2)/2])/(4*b^3*Pi^2) + (3*FresnelC[b*x])/(4*b^4*Pi^2) + (x^4*FresnelC[b*x])/4 - (x^3*Sin[(b^2*Pi*x^2)/2])/(4*b*Pi)} -{x^2*FresnelC[b*x], x, 4, (-2*Cos[(b^2*Pi*x^2)/2])/(3*b^3*Pi^2) + (x^3*FresnelC[b*x])/3 - (x^2*Sin[(b^2*Pi*x^2)/2])/(3*b*Pi)} -{x^1*FresnelC[b*x], x, 3, (x^2*FresnelC[b*x])/2 + FresnelS[b*x]/(2*b^2*Pi) - (x*Sin[(b^2*Pi*x^2)/2])/(2*b*Pi)} -{x^0*FresnelC[b*x], x, 1, x*FresnelC[b*x] - Sin[(b^2*Pi*x^2)/2]/(b*Pi)} -{FresnelC[b*x]/x^1, x, 3, (b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-I/2)*b^2*Pi*x^2])/2 + (b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (I/2)*b^2*Pi*x^2])/2} -{FresnelC[b*x]/x^2, x, 2, (b*CosIntegral[(b^2*Pi*x^2)/2])/2 - FresnelC[b*x]/x} -{FresnelC[b*x]/x^3, x, 3, -(b*Cos[(b^2*Pi*x^2)/2])/(2*x) - FresnelC[b*x]/(2*x^2) - (b^2*Pi*FresnelS[b*x])/2} -{FresnelC[b*x]/x^4, x, 4, -(b*Cos[(b^2*Pi*x^2)/2])/(6*x^2) - FresnelC[b*x]/(3*x^3) - (b^3*Pi*SinIntegral[(b^2*Pi*x^2)/2])/12} -{FresnelC[b*x]/x^5, x, 4, -(b*Cos[(b^2*Pi*x^2)/2])/(12*x^3) - (b^4*Pi^2*FresnelC[b*x])/12 - FresnelC[b*x]/(4*x^4) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(12*x)} -{FresnelC[b*x]/x^6, x, 5, -(b*Cos[(b^2*Pi*x^2)/2])/(20*x^4) - (b^5*Pi^2*CosIntegral[(b^2*Pi*x^2)/2])/80 - FresnelC[b*x]/(5*x^5) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(40*x^2)} -{FresnelC[b*x]/x^7, x, 5, -(b*Cos[(b^2*Pi*x^2)/2])/(30*x^5) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2])/(90*x) - FresnelC[b*x]/(6*x^6) + (b^6*Pi^3*FresnelS[b*x])/90 + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(90*x^3)} -{FresnelC[b*x]/x^8, x, 6, -(b*Cos[(b^2*Pi*x^2)/2])/(42*x^6) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2])/(336*x^2) - FresnelC[b*x]/(7*x^7) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(168*x^4) + (b^7*Pi^3*SinIntegral[(b^2*Pi*x^2)/2])/672} -{FresnelC[b*x]/x^9, x, 6, -(b*Cos[(b^2*Pi*x^2)/2])/(56*x^7) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2])/(840*x^3) + (b^8*Pi^4*FresnelC[b*x])/840 - FresnelC[b*x]/(8*x^8) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(280*x^5) - (b^7*Pi^3*Sin[(b^2*Pi*x^2)/2])/(840*x)} -{FresnelC[b*x]/x^10, x, 7, -(b*Cos[(b^2*Pi*x^2)/2])/(72*x^8) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2])/(1728*x^4) + (b^9*Pi^4*CosIntegral[(b^2*Pi*x^2)/2])/6912 - FresnelC[b*x]/(9*x^9) + (b^3*Pi*Sin[(b^2*Pi*x^2)/2])/(432*x^6) - (b^7*Pi^3*Sin[(b^2*Pi*x^2)/2])/(3456*x^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelC[a+b x]*) - - -{(c + d*x)^3*FresnelC[a + b*x], x, 14, -((2*d^2*(b*c - a*d)*Cos[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi^2)) - (3*d^3*(a + b*x)*Cos[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi^2) - ((b*c - a*d)^4*FresnelC[a + b*x])/(4*b^4*d) + (3*d^3*FresnelC[a + b*x])/(4*b^4*Pi^2) + ((c + d*x)^4*FresnelC[a + b*x])/(4*d) + (3*d*(b*c - a*d)^2*FresnelS[a + b*x])/(2*b^4*Pi) - ((b*c - a*d)^3*Sin[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) - (3*d*(b*c - a*d)^2*(a + b*x)*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b^4*Pi) - (d^2*(b*c - a*d)*(a + b*x)^2*Sin[(1/2)*Pi*(a + b*x)^2])/(b^4*Pi) - (d^3*(a + b*x)^3*Sin[(1/2)*Pi*(a + b*x)^2])/(4*b^4*Pi)} -{(c + d*x)^2*FresnelC[a + b*x], x, 11, -((2*d^2*Cos[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi^2)) - ((b*c - a*d)^3*FresnelC[a + b*x])/(3*b^3*d) + ((c + d*x)^3*FresnelC[a + b*x])/(3*d) + (d*(b*c - a*d)*FresnelS[a + b*x])/(b^3*Pi) - ((b*c - a*d)^2*Sin[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) - (d*(b*c - a*d)*(a + b*x)*Sin[(1/2)*Pi*(a + b*x)^2])/(b^3*Pi) - (d^2*(a + b*x)^2*Sin[(1/2)*Pi*(a + b*x)^2])/(3*b^3*Pi)} -{(c + d*x)^1*FresnelC[a + b*x], x, 8, -(((b*c - a*d)^2*FresnelC[a + b*x])/(2*b^2*d)) + ((c + d*x)^2*FresnelC[a + b*x])/(2*d) + (d*FresnelS[a + b*x])/(2*b^2*Pi) - ((b*c - a*d)*Sin[(1/2)*Pi*(a + b*x)^2])/(b^2*Pi) - (d*(a + b*x)*Sin[(1/2)*Pi*(a + b*x)^2])/(2*b^2*Pi)} -{(c + d*x)^0*FresnelC[a + b*x], x, 1, ((a + b*x)*FresnelC[a + b*x])/b - Sin[(Pi*(a + b*x)^2)/2]/(b*Pi)} -{FresnelC[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[FresnelC[a + b*x]/(c + d*x), x]} -{FresnelC[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[FresnelC[a + b*x]/(c + d*x)^2, x]} - - -{x^3*FresnelC[a + b*x], x, 14, (2*a*Cos[(Pi*(a + b*x)^2)/2])/(b^4*Pi^2) - (3*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi^2) - (a^4*FresnelC[a + b*x])/(4*b^4) + (3*FresnelC[a + b*x])/(4*b^4*Pi^2) + (x^4*FresnelC[a + b*x])/4 + (3*a^2*FresnelS[a + b*x])/(2*b^4*Pi) + (a^3*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - (3*a^2*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/(2*b^4*Pi) + (a*(a + b*x)^2*Sin[(Pi*(a + b*x)^2)/2])/(b^4*Pi) - ((a + b*x)^3*Sin[(Pi*(a + b*x)^2)/2])/(4*b^4*Pi)} -{x^2*FresnelC[a + b*x], x, 11, (-2*Cos[(Pi*(a + b*x)^2)/2])/(3*b^3*Pi^2) + (a^3*FresnelC[a + b*x])/(3*b^3) + (x^3*FresnelC[a + b*x])/3 - (a*FresnelS[a + b*x])/(b^3*Pi) - (a^2*Sin[(Pi*(a + b*x)^2)/2])/(b^3*Pi) + (a*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/(b^3*Pi) - ((a + b*x)^2*Sin[(Pi*(a + b*x)^2)/2])/(3*b^3*Pi)} -{x^1*FresnelC[a + b*x], x, 8, -(a^2*FresnelC[a + b*x])/(2*b^2) + (x^2*FresnelC[a + b*x])/2 + FresnelS[a + b*x]/(2*b^2*Pi) + (a*Sin[(Pi*(a + b*x)^2)/2])/(b^2*Pi) - ((a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/(2*b^2*Pi)} -{x^0*FresnelC[a + b*x], x, 1, ((a + b*x)*FresnelC[a + b*x])/b - Sin[(Pi*(a + b*x)^2)/2]/(b*Pi)} -{FresnelC[a + b*x]/x^1, x, 0, Unintegrable[FresnelC[a + b*x]/x, x]} -{FresnelC[a + b*x]/x^2, x, 0, Unintegrable[FresnelC[a + b*x]/x^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelC[a+b x]^2*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m FresnelC[b x]^2*) - - -{x^7*FresnelC[b*x]^2, x, 23, -((105*x^2)/(16*b^6*Pi^4)) + (7*x^6)/(48*b^2*Pi^2) + (55*x^2*Cos[b^2*Pi*x^2])/(16*b^6*Pi^4) - (x^6*Cos[b^2*Pi*x^2])/(16*b^2*Pi^2) + (105*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(4*b^7*Pi^4) - (7*x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(4*b^3*Pi^2) - (105*FresnelC[b*x]^2)/(8*b^8*Pi^4) + (1/8)*x^8*FresnelC[b*x]^2 + (35*x^3*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(4*b^5*Pi^3) - (x^7*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(4*b*Pi) - (10*Sin[b^2*Pi*x^2])/(b^8*Pi^5) + (5*x^4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)} -{x^6*FresnelC[b*x]^2, x, 19, (-48*x)/(7*b^6*Pi^4) + (6*x^5)/(35*b^2*Pi^2) + (21*x*Cos[b^2*Pi*x^2])/(8*b^6*Pi^4) - (x^5*Cos[b^2*Pi*x^2])/(14*b^2*Pi^2) + (96*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(7*b^7*Pi^4) - (12*x^4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(7*b^3*Pi^2) + (x^7*FresnelC[b*x]^2)/7 - (531*FresnelC[Sqrt[2]*b*x])/(56*Sqrt[2]*b^7*Pi^4) + (48*x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3) - (2*x^6*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*b*Pi) + (17*x^3*Sin[b^2*Pi*x^2])/(28*b^4*Pi^3)} -{x^5*FresnelC[b*x]^2, x, 16, (5*x^4)/(24*b^2*Pi^2) + (11*Cos[b^2*Pi*x^2])/(6*b^6*Pi^4) - (x^4*Cos[b^2*Pi*x^2])/(12*b^2*Pi^2) - (5*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(3*b^3*Pi^2) + (1/6)*x^6*FresnelC[b*x]^2 - (5*FresnelC[b*x]*FresnelS[b*x])/(2*b^6*Pi^3) - (5*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^4*Pi^3) + (5*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^4*Pi^3) + (5*x*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^5*Pi^3) - (x^5*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(3*b*Pi) + (7*x^2*Sin[b^2*Pi*x^2])/(12*b^4*Pi^3)} -{x^4*FresnelC[b*x]^2, x, 12, (4*x^3)/(15*b^2*Pi^2) - (x^3*Cos[b^2*Pi*x^2])/(10*b^2*Pi^2) - (8*x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(5*b^3*Pi^2) + (x^5*FresnelC[b*x]^2)/5 - (43*FresnelS[Sqrt[2]*b*x])/(20*Sqrt[2]*b^5*Pi^3) + (16*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(5*b^5*Pi^3) - (2*x^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(5*b*Pi) + (11*x*Sin[b^2*Pi*x^2])/(20*b^4*Pi^3)} -{x^3*FresnelC[b*x]^2, x, 10, (3*x^2)/(8*b^2*Pi^2) - (x^2*Cos[b^2*Pi*x^2])/(8*b^2*Pi^2) - (3*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(2*b^3*Pi^2) + (3*FresnelC[b*x]^2)/(4*b^4*Pi^2) + (1/4)*x^4*FresnelC[b*x]^2 - (x^3*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(2*b*Pi) + Sin[b^2*Pi*x^2]/(2*b^4*Pi^3)} -{x^2*FresnelC[b*x]^2, x, 8, (2*x)/(3*b^2*Pi^2) - (x*Cos[b^2*Pi*x^2])/(6*b^2*Pi^2) - (4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(3*b^3*Pi^2) + (x^3*FresnelC[b*x]^2)/3 + (5*FresnelC[Sqrt[2]*b*x])/(6*Sqrt[2]*b^3*Pi^2) - (2*x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*b*Pi)} -{x^1*FresnelC[b*x]^2, x, 5, -Cos[b^2*Pi*x^2]/(4*b^2*Pi^2) + (x^2*FresnelC[b*x]^2)/2 + (FresnelC[b*x]*FresnelS[b*x])/(2*b^2*Pi) + ((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2])/Pi - ((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/Pi - (x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b*Pi)} -{x^0*FresnelC[b*x]^2, x, 4, x*FresnelC[b*x]^2 + FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b*Pi) - (2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b*Pi)} -{FresnelC[b*x]^2/x^1, x, 0, Unintegrable[FresnelC[b*x]^2/x, x]} -{FresnelC[b*x]^2/x^2, x, 1, -(FresnelC[b*x]^2/x) + 2*b*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x]} -{FresnelC[b*x]^2/x^3, x, 1, -FresnelC[b*x]^2/(2*x^2) + b*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x]} -{FresnelC[b*x]^2/x^4, x, 4, -b^2/(6*x) - (b^2*Cos[b^2*Pi*x^2])/(6*x) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(3*x^2) - FresnelC[b*x]^2/(3*x^3) - (b^3*Pi*FresnelS[Sqrt[2]*b*x])/(3*Sqrt[2]) - (b^3*Pi*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/3} -{FresnelC[b*x]^2/x^5, x, 9, -b^2/(24*x^2) - (b^2*Cos[b^2*Pi*x^2])/(24*x^2) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(6*x^3) - (b^4*Pi^2*FresnelC[b*x]^2)/12 - FresnelC[b*x]^2/(4*x^4) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x) - (b^4*Pi*SinIntegral[b^2*Pi*x^2])/12} -{FresnelC[b*x]^2/x^6, x, 8, -b^2/(60*x^3) - (b^2*Cos[b^2*Pi*x^2])/(60*x^3) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(10*x^4) - FresnelC[b*x]^2/(5*x^5) - (7*b^5*Pi^2*FresnelC[Sqrt[2]*b*x])/(60*Sqrt[2]) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(20*x^2) + (7*b^4*Pi*Sin[b^2*Pi*x^2])/(120*x) - (b^5*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/20} -{FresnelC[b*x]^2/x^7, x, 10, -b^2/(120*x^4) - (b^2*Cos[b^2*Pi*x^2])/(120*x^4) - (b^6*Pi^2*CosIntegral[b^2*Pi*x^2])/72 - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(15*x^5) - FresnelC[b*x]^2/(6*x^6) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(45*x^3) + (b^4*Pi*Sin[b^2*Pi*x^2])/(72*x^2) - (b^5*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/45} -{FresnelC[b*x]^2/x^8, x, 13, -b^2/(210*x^5) + (b^6*Pi^2)/(336*x) - (b^2*Cos[b^2*Pi*x^2])/(210*x^5) + (67*b^6*Pi^2*Cos[b^2*Pi*x^2])/(5040*x) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(21*x^6) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(168*x^2) - FresnelC[b*x]^2/(7*x^7) + (b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/(72*Sqrt[2]) + (2*Sqrt[2]*b^7*Pi^3*FresnelS[Sqrt[2]*b*x])/315 + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(84*x^4) + (13*b^4*Pi*Sin[b^2*Pi*x^2])/(2520*x^3) + (b^7*Pi^3*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/168} -{FresnelC[b*x]^2/x^9, x, 20, -b^2/(336*x^6) + (b^6*Pi^2)/(1680*x^2) - (b^2*Cos[b^2*Pi*x^2])/(336*x^6) + (b^6*Pi^2*Cos[b^2*Pi*x^2])/(336*x^2) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(28*x^7) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(420*x^3) + (b^8*Pi^4*FresnelC[b*x]^2)/840 - FresnelC[b*x]^2/(8*x^8) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(140*x^5) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(420*x) + (b^4*Pi*Sin[b^2*Pi*x^2])/(420*x^4) + (b^8*Pi^3*SinIntegral[b^2*Pi*x^2])/280} -{FresnelC[b*x]^2/x^10, x, 19, -b^2/(504*x^7) + (b^6*Pi^2)/(5184*x^3) - (b^2*Cos[b^2*Pi*x^2])/(504*x^7) + (187*b^6*Pi^2*Cos[b^2*Pi*x^2])/(181440*x^3) - (b*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(36*x^8) + (b^5*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(864*x^4) - FresnelC[b*x]^2/(9*x^9) + (853*b^9*Pi^4*FresnelC[Sqrt[2]*b*x])/(181440*Sqrt[2]) + (b^3*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(216*x^6) - (b^7*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(1728*x^2) + (19*b^4*Pi*Sin[b^2*Pi*x^2])/(15120*x^5) - (853*b^8*Pi^3*Sin[b^2*Pi*x^2])/(362880*x) + (b^9*Pi^4*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/1728} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m FresnelC[a+b x]^2*) - - -{(c + d*x)^2*FresnelC[a + b*x]^2, x, 18, (2*d^2*x)/(3*b^2*Pi^2) - (d*(b*c - a*d)*Cos[Pi*(a + b*x)^2])/(2*b^3*Pi^2) - (d^2*(a + b*x)*Cos[Pi*(a + b*x)^2])/(6*b^3*Pi^2) - (4*d^2*Cos[(Pi*(a + b*x)^2)/2]*FresnelC[a + b*x])/(3*b^3*Pi^2) + ((b*c - a*d)^2*(a + b*x)*FresnelC[a + b*x]^2)/b^3 + (d*(b*c - a*d)*(a + b*x)^2*FresnelC[a + b*x]^2)/b^3 + (d^2*(a + b*x)^3*FresnelC[a + b*x]^2)/(3*b^3) + (5*d^2*FresnelC[Sqrt[2]*(a + b*x)])/(6*Sqrt[2]*b^3*Pi^2) + (d*(b*c - a*d)*FresnelC[a + b*x]*FresnelS[a + b*x])/(b^3*Pi) + ((b*c - a*d)^2*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^3*Pi) + ((I/4)*d*(b*c - a*d)*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*Pi*(a + b*x)^2])/(b^3*Pi) - ((I/4)*d*(b*c - a*d)*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*Pi*(a + b*x)^2])/(b^3*Pi) - (2*(b*c - a*d)^2*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(b^3*Pi) - (2*d*(b*c - a*d)*(a + b*x)*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(b^3*Pi) - (2*d^2*(a + b*x)^2*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(3*b^3*Pi)} -{(c + d*x)^1*FresnelC[a + b*x]^2, x, 10, -(d*Cos[Pi*(a + b*x)^2])/(4*b^2*Pi^2) + ((b*c - a*d)*(a + b*x)*FresnelC[a + b*x]^2)/b^2 + (d*(a + b*x)^2*FresnelC[a + b*x]^2)/(2*b^2) + (d*FresnelC[a + b*x]*FresnelS[a + b*x])/(2*b^2*Pi) + ((b*c - a*d)*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*b^2*Pi) + ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*Pi*(a + b*x)^2])/(b^2*Pi) - ((I/8)*d*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*Pi*(a + b*x)^2])/(b^2*Pi) - (2*(b*c - a*d)*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(b^2*Pi) - (d*(a + b*x)*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(b^2*Pi)} -{(c + d*x)^0*FresnelC[a + b*x]^2, x, 4, ((a + b*x)*FresnelC[a + b*x]^2)/b + FresnelS[Sqrt[2]*(a + b*x)]/(Sqrt[2]*b*Pi) - (2*FresnelC[a + b*x]*Sin[(Pi*(a + b*x)^2)/2])/(b*Pi)} -{FresnelC[a + b*x]^2/(c + d*x)^1, x, 0, Unintegrable[FresnelC[a + b*x]^2/(c + d*x), x]} -{FresnelC[a + b*x]^2/(c + d*x)^2, x, 0, Unintegrable[FresnelC[a + b*x]^2/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m FresnelC[d (a+b Log[c x^n])]*) - - -{x^2*FresnelC[d*(a + b*Log[c*x^n])], x, 10, ((1/12 + I/12)*E^((-3*a)/(b*n) + ((9*I)/2)/(b^2*d^2*n^2*Pi))*x^3*Erf[((1/2 + I/2)*(3/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) - ((1/12 + I/12)*E^((-3*a)/(b*n) - ((9*I)/2)/(b^2*d^2*n^2*Pi))*x^3*Erfi[((1/2 + I/2)*(3/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(3/n) + (x^3*FresnelC[d*(a + b*Log[c*x^n])])/3} -{x^1*FresnelC[d*(a + b*Log[c*x^n])], x, 10, ((1/8 + I/8)*E^((2*I - 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*x^2*Erf[((1/2 + I/2)*(2/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(c*x^n)^(2/n) - ((1/8 + I/8)*x^2*Erfi[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*(I + a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*(c*x^n)^(2/n)) + (x^2*FresnelC[d*(a + b*Log[c*x^n])])/2} -{x^0*FresnelC[d*(a + b*Log[c*x^n])], x, 10, ((1/4 + I/4)*x*Erf[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) - ((1/4 + I/4)*x*Erfi[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)) + x*FresnelC[d*(a + b*Log[c*x^n])]} -{FresnelC[d*(a + b*Log[c*x^n])]/x^1, x, 3, (FresnelC[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sin[(d^2*Pi*(a + b*Log[c*x^n])^2)/2]/(b*d*n*Pi)} -{FresnelC[d*(a + b*Log[c*x^n])]/x^2, x, 10, ((1/4 + I/4)*E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*Erf[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x - ((1/4 + I/4)*E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*Erfi[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x - FresnelC[d*(a + b*Log[c*x^n])]/x} -{FresnelC[d*(a + b*Log[c*x^n])]/x^3, x, 10, ((1/8 + I/8)*E^((2*I + 2*a*b*d^2*n*Pi)/(b^2*d^2*n^2*Pi))*(c*x^n)^(2/n)*Erf[((1/2 + I/2)*(2/n - I*a*b*d^2*Pi - I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x^2 - ((1/8 + I/8)*(c*x^n)^(2/n)*Erfi[((1/2 + I/2)*(2/n + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/(E^((2*(I - a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x^2) - FresnelC[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*FresnelC[d*(a + b*Log[c*x^n])], x, 10, ((1/4 + I/4)*E^(((I/2)*(1 + m)*(1 + m + (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*x*(e*x)^m*Erf[((1/2 + I/2)*(1 + m + I*a*b*d^2*n*Pi + I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/((1 + m)*(c*x^n)^((1 + m)/n)) - ((1/4 + I/4)*x*(e*x)^m*Erfi[((1/2 + I/2)*(1 + m - I*a*b*d^2*n*Pi - I*b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])])/(E^(((I/2)*(1 + m)*(1 + m - (2*I)*a*b*d^2*n*Pi))/(b^2*d^2*n^2*Pi))*(1 + m)*(c*x^n)^((1 + m)/n)) + ((e*x)^(1 + m)*FresnelC[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form E^(c+d x^2) FresnelC[a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form E^(c+d x^2) FresnelC[b x] when d^2=Pi^2/4 b^4*) - - -{E^(c + (I/2)*b^2*Pi*x^2)*FresnelC[b*x], x, 4, If[$VersionNumber>=8, ((-I/8)*E^c*Erfi[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/b + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/4, (I*E^c*Erf[(1/2 - I/2)*b*Sqrt[Pi]*x]^2)/(8*b) + (1/4)*b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2]]} -{E^(c - (I/2)*b^2*Pi*x^2)*FresnelC[b*x], x, 4, ((-I/8)*E^c*Erf[(1/2 + I/2)*b*Sqrt[Pi]*x]^2)/b + (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2])/4} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Cos[c+d x^2] FresnelC[a+b x]^n*) - - -{Sin[c + (b^2*Pi*x^2)/2]*FresnelC[b*x], x, 4, (Cos[c]*FresnelC[b*x]*FresnelS[b*x])/(2*b) + (I/8)*b*x^2*Cos[c]*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2] - (I/8)*b*x^2*Cos[c]*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2] + (FresnelC[b*x]^2*Sin[c])/(2*b)} -{Cos[c + (b^2*Pi*x^2)/2]*FresnelC[b*x], x, 4, (Cos[c]*FresnelC[b*x]^2)/(2*b) - (FresnelC[b*x]*FresnelS[b*x]*Sin[c])/(2*b) - (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2]*Sin[c] + (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]*Sin[c]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Cos[Pi/2 b^2 x^2] FresnelC[b x]^n*) - - -{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]^2, x, 2, FresnelC[b*x]^3/(3*b)} -{Cos[Pi/2*b^2*x^2]*FresnelC[b*x]^1, x, 2, FresnelC[b*x]^2/(2*b)} -{Cos[Pi/2*b^2*x^2]/FresnelC[b*x]^1, x, 2, Log[FresnelC[b*x]]/b} -{Cos[Pi/2*b^2*x^2]/FresnelC[b*x]^2, x, 2, -(1/(b*FresnelC[b*x]))} -{Cos[Pi/2*b^2*x^2]/FresnelC[b*x]^3, x, 2, -(1/(2*b*FresnelC[b*x]^2))} - - -{Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x]^n, x, 2, FresnelC[b*x]^(1 + n)/(b*(1 + n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Cos[Pi/2 b^2 x^2] FresnelC[b x]*) - - -{x^8*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 22, (105*x^2)/(4*b^7*Pi^4) - (7*x^6)/(12*b^3*Pi^2) - (55*x^2*Cos[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (105*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^8*Pi^4) + (7*x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^4*Pi^2) + (105*FresnelC[b*x]^2)/(2*b^9*Pi^4) - (35*x^3*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^7*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) + (40*Sin[b^2*Pi*x^2])/(b^9*Pi^5) - (5*x^4*Sin[b^2*Pi*x^2])/(2*b^5*Pi^3)} -{x^7*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 18, (24*x)/(b^7*Pi^4) - (3*x^5)/(5*b^3*Pi^2) - (147*x*Cos[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) - (48*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^8*Pi^4) + (6*x^4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^4*Pi^2) + (531*FresnelC[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) - (24*x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^6*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (17*x^3*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)} -{x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 15, -((5*x^4)/(8*b^3*Pi^2)) - (11*Cos[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (5*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^4*Pi^2) + (15*FresnelC[b*x]*FresnelS[b*x])/(2*b^7*Pi^3) + (15*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^5*Pi^3) - (15*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^5*Pi^3) - (15*x*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^6*Pi^3) + (x^5*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) - (7*x^2*Sin[b^2*Pi*x^2])/(4*b^5*Pi^3)} -{x^5*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 11, (-2*x^3)/(3*b^3*Pi^2) + (x^3*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (4*x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^4*Pi^2) + (43*FresnelS[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) - (8*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^6*Pi^3) + (x^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) - (11*x*Sin[b^2*Pi*x^2])/(8*b^5*Pi^3)} -{x^4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 9, -((3*x^2)/(4*b^3*Pi^2)) + (x^2*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (3*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^4*Pi^2) - (3*FresnelC[b*x]^2)/(2*b^5*Pi^2) + (x^3*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^2*Pi) - Sin[b^2*Pi*x^2]/(b^5*Pi^3)} -{x^3*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 7, -(x/(b^3*Pi^2)) + (x*Cos[b^2*Pi*x^2])/(4*b^3*Pi^2) + (2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^4*Pi^2) - (5*FresnelC[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) + (x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)} -{x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 4, Cos[b^2*Pi*x^2]/(4*b^3*Pi^2) - (FresnelC[b*x]*FresnelS[b*x])/(2*b^3*Pi) - ((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2])/(b*Pi) + ((I/8)*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2])/(b*Pi) + (x*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)} -{x^1*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 2, -FresnelS[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi) + (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)} -{x^0*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x], x, 2, FresnelC[b*x]^2/(2*b)} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^1, x, 0, Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x]} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x, 0, Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x]} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^3, x, 3, -b/(4*x) - (b*Cos[b^2*Pi*x^2])/(4*x) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(2*x^2) - (b^2*Pi*FresnelS[Sqrt[2]*b*x])/(2*Sqrt[2]) - (b^2*Pi*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/2} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^4, x, 8, -b/(12*x^2) - (b*Cos[b^2*Pi*x^2])/(12*x^2) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(3*x^3) - (b^3*Pi^2*FresnelC[b*x]^2)/6 + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x) - (b^3*Pi*SinIntegral[b^2*Pi*x^2])/6} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^5, x, 7, -b/(24*x^3) - (b*Cos[b^2*Pi*x^2])/(24*x^3) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(4*x^4) - (7*b^4*Pi^2*FresnelC[Sqrt[2]*b*x])/(24*Sqrt[2]) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^2) + (7*b^3*Pi*Sin[b^2*Pi*x^2])/(48*x) - (b^4*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/8} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^6, x, 9, -b/(40*x^4) - (b*Cos[b^2*Pi*x^2])/(40*x^4) - (b^5*Pi^2*CosIntegral[b^2*Pi*x^2])/24 - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(5*x^5) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(15*x^3) + (b^3*Pi*Sin[b^2*Pi*x^2])/(24*x^2) - (b^4*Pi^2*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/15} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^7, x, 12, -b/(60*x^5) + (b^5*Pi^2)/(96*x) - (b*Cos[b^2*Pi*x^2])/(60*x^5) + (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(1440*x) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(6*x^6) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(48*x^2) + (7*b^6*Pi^3*FresnelS[Sqrt[2]*b*x])/(144*Sqrt[2]) + (Sqrt[2]*b^6*Pi^3*FresnelS[Sqrt[2]*b*x])/45 + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(24*x^4) + (13*b^3*Pi*Sin[b^2*Pi*x^2])/(720*x^3) + (b^6*Pi^3*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/48} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^8, x, 19, -b/(84*x^6) + (b^5*Pi^2)/(420*x^2) - (b*Cos[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Cos[b^2*Pi*x^2])/(84*x^2) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(7*x^7) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(105*x^3) + (b^7*Pi^4*FresnelC[b*x]^2)/210 + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(35*x^5) - (b^6*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(105*x) + (b^3*Pi*Sin[b^2*Pi*x^2])/(105*x^4) + (b^7*Pi^3*SinIntegral[b^2*Pi*x^2])/70} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^9, x, 18, -b/(112*x^7) + (b^5*Pi^2)/(1152*x^3) - (b*Cos[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^2*Cos[b^2*Pi*x^2])/(40320*x^3) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(8*x^8) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(192*x^4) + (853*b^8*Pi^4*FresnelC[Sqrt[2]*b*x])/(40320*Sqrt[2]) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(48*x^6) - (b^6*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(384*x^2) + (19*b^3*Pi*Sin[b^2*Pi*x^2])/(3360*x^5) - (853*b^7*Pi^3*Sin[b^2*Pi*x^2])/(80640*x) + (b^8*Pi^4*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/384} -{(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^10, x, 22, -b/(144*x^8) + (b^5*Pi^2)/(2520*x^4) - (b*Cos[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Cos[b^2*Pi*x^2])/(30240*x^4) + (5*b^9*Pi^4*CosIntegral[b^2*Pi*x^2])/2016 - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(9*x^9) + (b^4*Pi^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(315*x^5) + (b^2*Pi*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(63*x^7) - (b^6*Pi^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(945*x^3) + (11*b^3*Pi*Sin[b^2*Pi*x^2])/(3024*x^6) - (5*b^7*Pi^3*Sin[b^2*Pi*x^2])/(2016*x^2) + (b^8*Pi^4*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/945} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Cos[c+d x^2] FresnelC[b x]*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Sin[c+d x^2] FresnelC[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form Sin[Pi/2 b^2 x^2] FresnelC[b x]^n*) - - -{FresnelC[b*x]^n*Sin[(b^2*Pi*x^2)/2], x, 0, Unintegrable[FresnelC[b*x]^n*Sin[(b^2*Pi*x^2)/2], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Sin[Pi/2 b^2 x^2] FresnelC[b x]*) - - -{x^8*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 23, -((35*x^4)/(8*b^5*Pi^3)) + x^8/(16*b*Pi) - (40*Cos[b^2*Pi*x^2])/(b^9*Pi^5) + (5*x^4*Cos[b^2*Pi*x^2])/(2*b^5*Pi^3) + (35*x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^6*Pi^3) - (x^7*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) + (105*FresnelC[b*x]*FresnelS[b*x])/(2*b^9*Pi^4) + (105*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^7*Pi^4) - (105*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^7*Pi^4) - (105*x*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^8*Pi^4) + (7*x^5*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) - (55*x^2*Sin[b^2*Pi*x^2])/(4*b^7*Pi^4) + (x^6*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^7*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 18, (-4*x^3)/(b^5*Pi^3) + x^7/(14*b*Pi) + (17*x^3*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) + (24*x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^6*Pi^3) - (x^6*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi) + (531*FresnelS[Sqrt[2]*b*x])/(16*Sqrt[2]*b^8*Pi^4) - (48*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^8*Pi^4) + (6*x^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) - (147*x*Sin[b^2*Pi*x^2])/(16*b^7*Pi^4) + (x^5*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^6*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 16, -((15*x^2)/(4*b^5*Pi^3)) + x^6/(12*b*Pi) + (7*x^2*Cos[b^2*Pi*x^2])/(4*b^5*Pi^3) + (15*x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^6*Pi^3) - (x^5*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) - (15*FresnelC[b*x]^2)/(2*b^7*Pi^3) + (5*x^3*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) - (11*Sin[b^2*Pi*x^2])/(2*b^7*Pi^4) + (x^4*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^5*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 13, (-4*x)/(b^5*Pi^3) + x^5/(10*b*Pi) + (11*x*Cos[b^2*Pi*x^2])/(8*b^5*Pi^3) + (8*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^6*Pi^3) - (x^4*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi) - (43*FresnelC[Sqrt[2]*b*x])/(8*Sqrt[2]*b^6*Pi^3) + (4*x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) + (x^3*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 10, x^4/(8*b*Pi) + Cos[b^2*Pi*x^2]/(b^5*Pi^3) - (x^3*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) - (3*FresnelC[b*x]*FresnelS[b*x])/(2*b^5*Pi^2) - (3*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-(1/2))*I*b^2*Pi*x^2])/(8*b^3*Pi^2) + (3*I*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (1/2)*I*b^2*Pi*x^2])/(8*b^3*Pi^2) + (3*x*FresnelC[b*x]*Sin[(1/2)*b^2*Pi*x^2])/(b^4*Pi^2) + (x^2*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 7, x^3/(6*b*Pi) - (x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi) - (5*FresnelS[Sqrt[2]*b*x])/(4*Sqrt[2]*b^4*Pi^2) + (2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^4*Pi^2) + (x*Sin[b^2*Pi*x^2])/(4*b^3*Pi^2)} -{x^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 5, x^2/(4*b*Pi) - (x*Cos[(1/2)*b^2*Pi*x^2]*FresnelC[b*x])/(b^2*Pi) + FresnelC[b*x]^2/(2*b^3*Pi) + Sin[b^2*Pi*x^2]/(4*b^3*Pi^2)} -{x^1*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 4, x/(2*b*Pi) - (Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi) + FresnelC[Sqrt[2]*b*x]/(2*Sqrt[2]*b^2*Pi)} -{x^0*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2], x, 1, (FresnelC[b*x]*FresnelS[b*x])/(2*b) + (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-I/2)*b^2*Pi*x^2] - (I/8)*b*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/2)*b^2*Pi*x^2]} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^1, x, 0, Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x]} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^2, x, 4, (b*Pi*FresnelC[b*x]^2)/2 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x + (b*SinIntegral[b^2*Pi*x^2])/4} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^3, x, 3, (b^2*Pi*FresnelC[Sqrt[2]*b*x])/(2*Sqrt[2]) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(2*x^2) - (b*Sin[b^2*Pi*x^2])/(4*x) + (b^2*Pi*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/2} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^4, x, 4, (b^3*Pi*CosIntegral[b^2*Pi*x^2])/12 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(3*x^3) - (b*Sin[b^2*Pi*x^2])/(12*x^2) + (b^2*Pi*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/3} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^5, x, 7, -(b^3*Pi)/(16*x) - (7*b^3*Pi*Cos[b^2*Pi*x^2])/(48*x) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(8*x^2) - (7*b^4*Pi^2*FresnelS[Sqrt[2]*b*x])/(24*Sqrt[2]) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(4*x^4) - (b*Sin[b^2*Pi*x^2])/(24*x^3) - (b^4*Pi^2*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/8} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^6, x, 13, -(b^3*Pi)/(60*x^2) - (b^3*Pi*Cos[b^2*Pi*x^2])/(24*x^2) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(15*x^3) - (b^5*Pi^3*FresnelC[b*x]^2)/30 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(5*x^5) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(15*x) - (b*Sin[b^2*Pi*x^2])/(40*x^4) - (7*b^5*Pi^2*SinIntegral[b^2*Pi*x^2])/120} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^7, x, 12, -(b^3*Pi)/(144*x^3) - (13*b^3*Pi*Cos[b^2*Pi*x^2])/(720*x^3) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(24*x^4) - (7*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/(144*Sqrt[2]) - (Sqrt[2]*b^6*Pi^3*FresnelC[Sqrt[2]*b*x])/45 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(6*x^6) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(48*x^2) - (b*Sin[b^2*Pi*x^2])/(60*x^5) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(1440*x) - (b^6*Pi^3*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x, x])/48} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^8, x, 15, -(b^3*Pi)/(280*x^4) - (b^3*Pi*Cos[b^2*Pi*x^2])/(105*x^4) - (b^7*Pi^3*CosIntegral[b^2*Pi*x^2])/84 - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(35*x^5) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(7*x^7) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(105*x^3) - (b*Sin[b^2*Pi*x^2])/(84*x^6) + (b^5*Pi^2*Sin[b^2*Pi*x^2])/(84*x^2) - (b^6*Pi^3*Unintegrable[(Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/x^2, x])/105} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^9, x, 18, -(b^3*Pi)/(480*x^5) + (b^7*Pi^3)/(768*x) - (19*b^3*Pi*Cos[b^2*Pi*x^2])/(3360*x^5) + (853*b^7*Pi^3*Cos[b^2*Pi*x^2])/(80640*x) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(48*x^6) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(384*x^2) + (853*b^8*Pi^4*FresnelS[Sqrt[2]*b*x])/(40320*Sqrt[2]) - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(8*x^8) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(192*x^4) - (b*Sin[b^2*Pi*x^2])/(112*x^7) + (187*b^5*Pi^2*Sin[b^2*Pi*x^2])/(40320*x^3) + (b^8*Pi^4*Unintegrable[(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x, x])/384} -{(FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/x^10, x, 26, -(b^3*Pi)/(756*x^6) + (b^7*Pi^3)/(3780*x^2) - (11*b^3*Pi*Cos[b^2*Pi*x^2])/(3024*x^6) + (5*b^7*Pi^3*Cos[b^2*Pi*x^2])/(2016*x^2) - (b^2*Pi*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(63*x^7) + (b^6*Pi^3*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(945*x^3) + (b^9*Pi^5*FresnelC[b*x]^2)/1890 - (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(9*x^9) + (b^4*Pi^2*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(315*x^5) - (b^8*Pi^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(945*x) - (b*Sin[b^2*Pi*x^2])/(144*x^8) + (67*b^5*Pi^2*Sin[b^2*Pi*x^2])/(30240*x^4) + (83*b^9*Pi^4*SinIntegral[b^2*Pi*x^2])/30240} - - -(* ::Subsection:: *) -(*Integrands of the form x^m Sin[c+d x^2] FresnelC[b x]*) diff --git a/test/methods/rule_based/test_files/8 Special functions/8.3 Exponential integral functions.m b/test/methods/rule_based/test_files/8 Special functions/8.3 Exponential integral functions.m deleted file mode 100644 index 31c3960..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.3 Exponential integral functions.m +++ /dev/null @@ -1,414 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Exponential Integral Functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m ExpIntegralE[n, b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m ExpIntegralE[n, b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^2*ExpIntegralE[1, b*x], x, 1, -(x^3*ExpIntegralE[-2, b*x])/3 + (x^3*ExpIntegralE[1, b*x])/3} -{x^1*ExpIntegralE[1, b*x], x, 1, -(x^2*ExpIntegralE[-1, b*x])/2 + (x^2*ExpIntegralE[1, b*x])/2} -{x^0*ExpIntegralE[1, b*x], x, 1, -(ExpIntegralE[2, b*x]/b)} -{ExpIntegralE[1, b*x]/x^1, x, 1, b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)] - EulerGamma*Log[x] - Log[b*x]^2/2} -{ExpIntegralE[1, b*x]/x^2, x, 1, -(ExpIntegralE[1, b*x]/x) + ExpIntegralE[2, b*x]/x} -{ExpIntegralE[1, b*x]/x^3, x, 1, -ExpIntegralE[1, b*x]/(2*x^2) + ExpIntegralE[3, b*x]/(2*x^2)} -{ExpIntegralE[1, b*x]/x^4, x, 1, -(ExpIntegralE[1, b*x]/(3*x^3)) + ExpIntegralE[4, b*x]/(3*x^3)} - - -{x^2*ExpIntegralE[2, b*x], x, 1, -(x^3*ExpIntegralE[-2, b*x])/4 + (x^3*ExpIntegralE[2, b*x])/4} -{x^1*ExpIntegralE[2, b*x], x, 1, -(x^2*ExpIntegralE[-1, b*x])/3 + (x^2*ExpIntegralE[2, b*x])/3} -{ExpIntegralE[2, b*x], x, 1, -(ExpIntegralE[3, b*x]/b)} -{ExpIntegralE[2, b*x]/x^1, x, 1, -ExpIntegralE[1, b*x] + ExpIntegralE[2, b*x]} -{ExpIntegralE[2, b*x]/x^2, x, 2, -(ExpIntegralE[2, b*x]/x) - b^2*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)] + b*EulerGamma*Log[x] + (b*Log[b*x]^2)/2} -{ExpIntegralE[2, b*x]/x^3, x, 1, -(ExpIntegralE[2, b*x]/x^2) + ExpIntegralE[3, b*x]/x^2} -{ExpIntegralE[2, b*x]/x^4, x, 1, -(ExpIntegralE[2, b*x]/(2*x^3)) + ExpIntegralE[4, b*x]/(2*x^3)} -{ExpIntegralE[2, b*x]/x^5, x, 1, -(ExpIntegralE[2, b*x]/(3*x^4)) + ExpIntegralE[5, b*x]/(3*x^4)} - - -{x^2*ExpIntegralE[3, b*x], x, 1, -(x^3*ExpIntegralE[-2, b*x])/5 + (x^3*ExpIntegralE[3, b*x])/5} -{x^1*ExpIntegralE[3, b*x], x, 1, -(x^2*ExpIntegralE[-1, b*x])/4 + (x^2*ExpIntegralE[3, b*x])/4} -{x^0*ExpIntegralE[3, b*x], x, 1, -(ExpIntegralE[4, b*x]/b)} -{ExpIntegralE[3, b*x]/x^1, x, 1, -ExpIntegralE[1, b*x]/2 + ExpIntegralE[3, b*x]/2} -{ExpIntegralE[3, b*x]/x^2, x, 1, -(ExpIntegralE[2, b*x]/x) + ExpIntegralE[3, b*x]/x} -{ExpIntegralE[3, b*x]/x^3, x, 3, (b*ExpIntegralE[2, b*x])/(2*x) - ExpIntegralE[3, b*x]/(2*x^2) + (b^3*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -(b*x)])/2 - (b^2*EulerGamma*Log[x])/2 - (b^2*Log[b*x]^2)/4} -{ExpIntegralE[3, b*x]/x^4, x, 1, -(ExpIntegralE[3, b*x]/x^3) + ExpIntegralE[4, b*x]/x^3} -{ExpIntegralE[3, b*x]/x^5, x, 1, -(ExpIntegralE[3, b*x]/(2*x^4)) + ExpIntegralE[5, b*x]/(2*x^4)} -{ExpIntegralE[3, b*x]/x^6, x, 1, -(ExpIntegralE[3, b*x]/(3*x^5)) + ExpIntegralE[6, b*x]/(3*x^5)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^3*ExpIntegralE[-1, b*x], x, 1, -(x^4*ExpIntegralE[-3, b*x])/2 + (x^4*ExpIntegralE[-1, b*x])/2} -{x^2*ExpIntegralE[-1, b*x], x, 1, -(x^3*ExpIntegralE[-2, b*x]) + x^3*ExpIntegralE[-1, b*x]} -{x^1*ExpIntegralE[-1, b*x], x, 3, -(1/(b^2*E^(b*x))) + ExpIntegralEi[-(b*x)]/b^2} -{x^0*ExpIntegralE[-1, b*x], x, 1, -(1/(b^2*E^(b*x)*x))} -{ExpIntegralE[-1, b*x]/x^1, x, 1, -ExpIntegralE[-1, b*x]/2 + ExpIntegralE[1, b*x]/2} -{ExpIntegralE[-1, b*x]/x^2, x, 1, -ExpIntegralE[-1, b*x]/(3*x) + ExpIntegralE[2, b*x]/(3*x)} -{ExpIntegralE[-1, b*x]/x^3, x, 1, -ExpIntegralE[-1, b*x]/(4*x^2) + ExpIntegralE[3, b*x]/(4*x^2)} - - -{x^4*ExpIntegralE[-2, b*x], x, 1, (-(1/2))*x^5*ExpIntegralE[-4, b*x] + (1/2)*x^5*ExpIntegralE[-2, b*x]} -{x^3*ExpIntegralE[-2, b*x], x, 1, -(x^4*ExpIntegralE[-3, b*x]) + x^4*ExpIntegralE[-2, b*x]} -{x^2*ExpIntegralE[-2, b*x], x, 4, -2/(b^3*E^(b*x)) - (x^2*ExpIntegralE[-1, b*x])/b + (2*ExpIntegralEi[-(b*x)])/b^3} -{x^1*ExpIntegralE[-2, b*x], x, 1, -(x^2*ExpIntegralE[-2, b*x]) + x^2*ExpIntegralE[-1, b*x]} -{x^0*ExpIntegralE[-1, b*x], x, 1, -(1/(b^2*E^(b*x)*x))} -{ExpIntegralE[-2, b*x]/x^1, x, 1, -ExpIntegralE[-2, b*x]/3 + ExpIntegralE[1, b*x]/3} -{ExpIntegralE[-2, b*x]/x^2, x, 1, -ExpIntegralE[-2, b*x]/(4*x) + ExpIntegralE[2, b*x]/(4*x)} -{ExpIntegralE[-2, b*x]/x^3, x, 1, -ExpIntegralE[-2, b*x]/(5*x^2) + ExpIntegralE[3, b*x]/(5*x^2)} - - -{x^5*ExpIntegralE[-3, b*x], x, 1, (-(1/2))*x^6*ExpIntegralE[-5, b*x] + (1/2)*x^6*ExpIntegralE[-3, b*x]} -{x^4*ExpIntegralE[-3, b*x], x, 1, (-x^5)*ExpIntegralE[-4, b*x] + x^5*ExpIntegralE[-3, b*x]} -{x^3*ExpIntegralE[-3, b*x], x, 5, -6/(b^4*E^(b*x)) - (x^3*ExpIntegralE[-2, b*x])/b - (3*x^2*ExpIntegralE[-1, b*x])/b^2 + (6*ExpIntegralEi[-(b*x)])/b^4} -{x^2*ExpIntegralE[-3, b*x], x, 1, -(x^3*ExpIntegralE[-3, b*x]) + x^3*ExpIntegralE[-2, b*x]} -{x^1*ExpIntegralE[-3, b*x], x, 1, -(x^2*ExpIntegralE[-3, b*x])/2 + (x^2*ExpIntegralE[-1, b*x])/2} -{x^0*ExpIntegralE[-1, b*x], x, 1, -(1/(b^2*E^(b*x)*x))} -{ExpIntegralE[-3, b*x]/x^1, x, 1, -ExpIntegralE[-3, b*x]/4 + ExpIntegralE[1, b*x]/4} -{ExpIntegralE[-3, b*x]/x^2, x, 1, -ExpIntegralE[-3, b*x]/(5*x) + ExpIntegralE[2, b*x]/(5*x)} -{ExpIntegralE[-3, b*x]/x^3, x, 1, -ExpIntegralE[-3, b*x]/(6*x^2) + ExpIntegralE[3, b*x]/(6*x^2)} - - -{x^3*ExpIntegralE[-3, b*x], x, 5, -(6/(E^(b*x)*b^4)) - (x^3*ExpIntegralE[-2, b*x])/b - (3*x^2*ExpIntegralE[-1, b*x])/b^2 + (6*ExpIntegralEi[(-b)*x])/b^4} -{x^2*ExpIntegralE[-2, b*x], x, 4, -(2/(E^(b*x)*b^3)) - (x^2*ExpIntegralE[-1, b*x])/b + (2*ExpIntegralEi[(-b)*x])/b^3} -{x^1*ExpIntegralE[-1, b*x], x, 3, -(1/(E^(b*x)*b^2)) + ExpIntegralEi[(-b)*x]/b^2} -{x^0*ExpIntegralE[0, b*x], x, 2, ExpIntegralEi[(-b)*x]/b} -{ExpIntegralE[1, b*x]/x^1, x, 1, b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x] - EulerGamma*Log[x] - (1/2)*Log[b*x]^2} -{ExpIntegralE[2, b*x]/x^2, x, 2, -(ExpIntegralE[2, b*x]/x) - b^2*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x] + b*EulerGamma*Log[x] + (1/2)*b*Log[b*x]^2} -{ExpIntegralE[3, b*x]/x^3, x, 3, (b*ExpIntegralE[2, b*x])/(2*x) - ExpIntegralE[3, b*x]/(2*x^2) + (1/2)*b^3*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x] - (1/2)*b^2*EulerGamma*Log[x] - (1/4)*b^2*Log[b*x]^2} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) ExpIntegralE[n/2, b x]*) - - -{(d*x)^(3/2)*ExpIntegralE[-3/2, b*x], x, 1, -((4*(d*x)^(5/2)*HypergeometricPFQ[{5/2, 5/2}, {7/2, 7/2}, (-b)*x])/(25*d)) + (3*Sqrt[Pi]*(d*x)^(3/2)*Log[x])/(4*b*(b*x)^(3/2))} -{(d*x)^(1/2)*ExpIntegralE[-1/2, b*x], x, 1, -((4*(d*x)^(3/2)*HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, (-b)*x])/(9*d)) + (Sqrt[Pi]*Sqrt[d*x]*Log[x])/(2*b*Sqrt[b*x])} -{ExpIntegralE[1/2, b*x]/(d*x)^(1/2), x, 1, -((4*Sqrt[d*x]*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-b)*x])/d) + (Sqrt[Pi]*Sqrt[b*x]*Log[x])/(b*Sqrt[d*x])} -{ExpIntegralE[3/2, b*x]/(d*x)^(3/2), x, 1, -((4*HypergeometricPFQ[{-(1/2), -(1/2)}, {1/2, 1/2}, (-b)*x])/(d*Sqrt[d*x])) - (2*Sqrt[Pi]*(b*x)^(3/2)*Log[x])/(b*(d*x)^(3/2))} -{ExpIntegralE[5/2, b*x]/(d*x)^(5/2), x, 1, -((4*HypergeometricPFQ[{-(3/2), -(3/2)}, {-(1/2), -(1/2)}, (-b)*x])/(9*d*(d*x)^(3/2))) + (4*Sqrt[Pi]*(b*x)^(5/2)*Log[x])/(3*b*(d*x)^(5/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m ExpIntegralE[n, b x] with n symbolic*) - - -{x^m*ExpIntegralE[n, x], x, 1, -((x^(1 + m)*ExpIntegralE[-m, x])/(m + n)) + (x^(1 + m)*ExpIntegralE[n, x])/(m + n)} -{x^m*ExpIntegralE[n, b*x], x, 1, -((x^(1 + m)*ExpIntegralE[-m, b*x])/(m + n)) + (x^(1 + m)*ExpIntegralE[n, b*x])/(m + n)} -{(d*x)^m*ExpIntegralE[n, x], x, 1, -(((d*x)^(1 + m)*ExpIntegralE[-m, x])/(d*(m + n))) + ((d*x)^(1 + m)*ExpIntegralE[n, x])/(d*(m + n))} -{(d*x)^m*ExpIntegralE[n, b*x], x, 1, -(((d*x)^(1 + m)*ExpIntegralE[-m, b*x])/(d*(m + n))) + ((d*x)^(1 + m)*ExpIntegralE[n, b*x])/(d*(m + n))} - - -{ExpIntegralE[n, x]/x^n, x, 1, -((x^(1 - n)*HypergeometricPFQ[{1 - n, 1 - n}, {2 - n, 2 - n}, -x])/(1 - n)^2) + Gamma[1 - n]*Log[x]} -{ExpIntegralE[n, b*x]/x^n, x, 1, -((x^(1 - n)*HypergeometricPFQ[{1 - n, 1 - n}, {2 - n, 2 - n}, (-b)*x])/(1 - n)^2) + ((b*x)^n*Gamma[1 - n]*Log[x])/(x^n*b)} -{ExpIntegralE[n, x]/(d*x)^n, x, 1, -(((d*x)^(1 - n)*HypergeometricPFQ[{1 - n, 1 - n}, {2 - n, 2 - n}, -x])/(d*(1 - n)^2)) + (x^n*Gamma[1 - n]*Log[x])/(d*x)^n} -{ExpIntegralE[n, b*x]/(d*x)^n, x, 1, -(((d*x)^(1 - n)*HypergeometricPFQ[{1 - n, 1 - n}, {2 - n, 2 - n}, (-b)*x])/(d*(1 - n)^2)) + ((b*x)^n*Gamma[1 - n]*Log[x])/((d*x)^n*b)} - - -{x^2*ExpIntegralE[n, b*x], x, 1, -((x^3*ExpIntegralE[-2, b*x])/(2 + n)) + (x^3*ExpIntegralE[n, b*x])/(2 + n)} -{x^1*ExpIntegralE[n, b*x], x, 1, -((x^2*ExpIntegralE[-1, b*x])/(1 + n)) + (x^2*ExpIntegralE[n, b*x])/(1 + n)} -{x^0*ExpIntegralE[n, b*x], x, 1, -(ExpIntegralE[1 + n, b*x]/b)} -{ExpIntegralE[n, b*x]/x^1, x, 1, ExpIntegralE[1, b*x]/(1 - n) - ExpIntegralE[n, b*x]/(1 - n)} -{ExpIntegralE[n, b*x]/x^2, x, 1, ExpIntegralE[2, b*x]/((2 - n)*x) - ExpIntegralE[n, b*x]/((2 - n)*x)} -{ExpIntegralE[n, b*x]/x^3, x, 1, ExpIntegralE[3, b*x]/((3 - n)*x^2) - ExpIntegralE[n, b*x]/((3 - n)*x^2)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x]*) -(**) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{(c + d*x)^3*ExpIntegralE[1, a + b*x], x, 4, -(((c + d*x)^3*ExpIntegralE[2, a + b*x])/b) - (3*d*(c + d*x)^2*ExpIntegralE[3, a + b*x])/b^2 - (6*d^2*(c + d*x)*ExpIntegralE[4, a + b*x])/b^3 - (6*d^3*ExpIntegralE[5, a + b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[1, a + b*x], x, 3, -(((c + d*x)^2*ExpIntegralE[2, a + b*x])/b) - (2*d*(c + d*x)*ExpIntegralE[3, a + b*x])/b^2 - (2*d^2*ExpIntegralE[4, a + b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[1, a + b*x], x, 2, -(((c + d*x)*ExpIntegralE[2, a + b*x])/b) - (d*ExpIntegralE[3, a + b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[1, a + b*x], x, 1, -(ExpIntegralE[2, a + b*x]/b)} -{ExpIntegralE[1, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[ExpIntegralE[1, a + b*x]/(c + d*x), x]} -{ExpIntegralE[1, a + b*x]/(c + d*x)^2, x, 5, -(ExpIntegralE[1, a + b*x]/(d*(c + d*x))) - (b*ExpIntegralEi[-a - b*x])/(d*(b*c - a*d)) + (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d*(b*c - a*d))} -{ExpIntegralE[1, a + b*x]/(c + d*x)^3, x, 7, -(b*E^(-a - b*x))/(2*d*(b*c - a*d)*(c + d*x)) - ExpIntegralE[1, a + b*x]/(2*d*(c + d*x)^2) - (b^2*ExpIntegralEi[-a - b*x])/(2*d*(b*c - a*d)^2) + (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d*(b*c - a*d)^2) - (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^2*(b*c - a*d))} -{ExpIntegralE[1, a + b*x]/(c + d*x)^4, x, 10, -(b*E^(-a - b*x))/(6*d*(b*c - a*d)*(c + d*x)^2) - (b^2*E^(-a - b*x))/(3*d*(b*c - a*d)^2*(c + d*x)) + (b^2*E^(-a - b*x))/(6*d^2*(b*c - a*d)*(c + d*x)) - ExpIntegralE[1, a + b*x]/(3*d*(c + d*x)^3) - (b^3*ExpIntegralEi[-a - b*x])/(3*d*(b*c - a*d)^3) + (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(3*d*(b*c - a*d)^3) - (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(3*d^2*(b*c - a*d)^2) + (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^3*(b*c - a*d))} - - -{(c + d*x)^3*ExpIntegralE[2, a + b*x], x, 4, -(((c + d*x)^3*ExpIntegralE[3, a + b*x])/b) - (3*d*(c + d*x)^2*ExpIntegralE[4, a + b*x])/b^2 - (6*d^2*(c + d*x)*ExpIntegralE[5, a + b*x])/b^3 - (6*d^3*ExpIntegralE[6, a + b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[2, a + b*x], x, 3, -(((c + d*x)^2*ExpIntegralE[3, a + b*x])/b) - (2*d*(c + d*x)*ExpIntegralE[4, a + b*x])/b^2 - (2*d^2*ExpIntegralE[5, a + b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[2, a + b*x], x, 2, -(((c + d*x)*ExpIntegralE[3, a + b*x])/b) - (d*ExpIntegralE[4, a + b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[2, a + b*x], x, 1, -(ExpIntegralE[3, a + b*x]/b)} -{ExpIntegralE[2, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[ExpIntegralE[2, a + b*x]/(c + d*x), x]} -{ExpIntegralE[2, a + b*x]/(c + d*x)^2, x, 1, -(ExpIntegralE[2, a + b*x]/(d*(c + d*x))) - (b*Unintegrable[ExpIntegralE[1, a + b*x]/(c + d*x), x])/d} -{ExpIntegralE[2, a + b*x]/(c + d*x)^3, x, 6, (b*ExpIntegralE[1, a + b*x])/(2*d^2*(c + d*x)) - ExpIntegralE[2, a + b*x]/(2*d*(c + d*x)^2) + (b^2*ExpIntegralEi[-a - b*x])/(2*d^2*(b*c - a*d)) - (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^2*(b*c - a*d))} -{ExpIntegralE[2, a + b*x]/(c + d*x)^4, x, 8, (b^2*E^(-a - b*x))/(6*d^2*(b*c - a*d)*(c + d*x)) + (b*ExpIntegralE[1, a + b*x])/(6*d^2*(c + d*x)^2) - ExpIntegralE[2, a + b*x]/(3*d*(c + d*x)^3) + (b^3*ExpIntegralEi[-a - b*x])/(6*d^2*(b*c - a*d)^2) - (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^2*(b*c - a*d)^2) + (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^3*(b*c - a*d))} - - -{(c + d*x)^3*ExpIntegralE[3, a + b*x], x, 4, -(((c + d*x)^3*ExpIntegralE[4, a + b*x])/b) - (3*d*(c + d*x)^2*ExpIntegralE[5, a + b*x])/b^2 - (6*d^2*(c + d*x)*ExpIntegralE[6, a + b*x])/b^3 - (6*d^3*ExpIntegralE[7, a + b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[3, a + b*x], x, 3, -(((c + d*x)^2*ExpIntegralE[4, a + b*x])/b) - (2*d*(c + d*x)*ExpIntegralE[5, a + b*x])/b^2 - (2*d^2*ExpIntegralE[6, a + b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[3, a + b*x], x, 2, -(((c + d*x)*ExpIntegralE[4, a + b*x])/b) - (d*ExpIntegralE[5, a + b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[3, a + b*x], x, 1, -(ExpIntegralE[4, a + b*x]/b)} -{ExpIntegralE[3, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[ExpIntegralE[3, a + b*x]/(c + d*x), x]} -{ExpIntegralE[3, a + b*x]/(c + d*x)^2, x, 1, -(ExpIntegralE[3, a + b*x]/(d*(c + d*x))) - (b*Unintegrable[ExpIntegralE[2, a + b*x]/(c + d*x), x])/d} -{ExpIntegralE[3, a + b*x]/(c + d*x)^3, x, 2, (b*ExpIntegralE[2, a + b*x])/(2*d^2*(c + d*x)) - ExpIntegralE[3, a + b*x]/(2*d*(c + d*x)^2) + (b^2*Unintegrable[ExpIntegralE[1, a + b*x]/(c + d*x), x])/(2*d^2)} -{ExpIntegralE[3, a + b*x]/(c + d*x)^4, x, 7, -(b^2*ExpIntegralE[1, a + b*x])/(6*d^3*(c + d*x)) + (b*ExpIntegralE[2, a + b*x])/(6*d^2*(c + d*x)^2) - ExpIntegralE[3, a + b*x]/(3*d*(c + d*x)^3) - (b^3*ExpIntegralEi[-a - b*x])/(6*d^3*(b*c - a*d)) + (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^3*(b*c - a*d))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c + d*x)^4*ExpIntegralE[-1, a + b*x], x, 10, (-8*d^4*E^(-a - b*x))/b^5 - (4*d^3*(b*c - a*d)*E^(-a - b*x))/b^5 - (4*d^2*(b*c - a*d)^2*E^(-a - b*x))/b^5 - (8*d^3*E^(-a - b*x)*(c + d*x))/b^4 - (4*d^2*(b*c - a*d)*E^(-a - b*x)*(c + d*x))/b^4 - (4*d^2*E^(-a - b*x)*(c + d*x)^2)/b^3 - (E^(-a - b*x)*(c + d*x)^4)/(b*(a + b*x)) + (4*d*(b*c - a*d)^3*ExpIntegralEi[-a - b*x])/b^5} -{(c + d*x)^3*ExpIntegralE[-1, a + b*x], x, 7, (-3*d^3*E^(-a - b*x))/b^4 - (3*d^2*(b*c - a*d)*E^(-a - b*x))/b^4 - (3*d^2*E^(-a - b*x)*(c + d*x))/b^3 - (E^(-a - b*x)*(c + d*x)^3)/(b*(a + b*x)) + (3*d*(b*c - a*d)^2*ExpIntegralEi[-a - b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[-1, a + b*x], x, 5, (-2*d^2*E^(-a - b*x))/b^3 - (E^(-a - b*x)*(c + d*x)^2)/(b*(a + b*x)) + (2*d*(b*c - a*d)*ExpIntegralEi[-a - b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[-1, a + b*x], x, 2, -((E^(-a - b*x)*(c + d*x))/(b*(a + b*x))) + (d*ExpIntegralEi[-a - b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[-1, a + b*x], x, 1, -(E^(-a - b*x)/(b*(a + b*x)))} -{ExpIntegralE[-1, a + b*x]/(c + d*x)^1, x, 7, -((d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x))) - E^(-a - b*x)/(b*(a + b*x)*(c + d*x)) - (d*ExpIntegralEi[-a - b*x])/(b*c - a*d)^2 + (d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 - (E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)} -{ExpIntegralE[-1, a + b*x]/(c + d*x)^2, x, 10, -((d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)^2)) - E^(-a - b*x)/(b*(a + b*x)*(c + d*x)^2) - (2*d*E^(-a - b*x))/((b*c - a*d)^2*(c + d*x)) + E^(-a - b*x)/((b*c - a*d)*(c + d*x)) - (2*b*d*ExpIntegralEi[-a - b*x])/(b*c - a*d)^3 + (2*b*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 - (2*b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 + (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d*(b*c - a*d))} -{ExpIntegralE[-1, a + b*x]/(c + d*x)^3, x, 14, -((d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)^3)) - E^(-a - b*x)/(b*(a + b*x)*(c + d*x)^3) - (3*d*E^(-a - b*x))/(2*(b*c - a*d)^2*(c + d*x)^2) + E^(-a - b*x)/(2*(b*c - a*d)*(c + d*x)^2) - (3*b*d*E^(-a - b*x))/((b*c - a*d)^3*(c + d*x)) + (3*b*E^(-a - b*x))/(2*(b*c - a*d)^2*(c + d*x)) - (b*E^(-a - b*x))/(2*d*(b*c - a*d)*(c + d*x)) - (3*b^2*d*ExpIntegralEi[-a - b*x])/(b*c - a*d)^4 + (3*b^2*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^4 - (3*b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 + (3*b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d*(b*c - a*d)^2) - (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^2*(b*c - a*d))} - - -{(c + d*x)^4*ExpIntegralE[-2, a + b*x], x, 8, (-12*d^4*E^(-a - b*x))/b^5 - (12*d^3*(b*c - a*d)*E^(-a - b*x))/b^5 - (12*d^3*E^(-a - b*x)*(c + d*x))/b^4 - (4*d*E^(-a - b*x)*(c + d*x)^3)/(b^2*(a + b*x)) - ((c + d*x)^4*ExpIntegralE[-1, a + b*x])/b + (12*d^2*(b*c - a*d)^2*ExpIntegralEi[-a - b*x])/b^5} -{(c + d*x)^3*ExpIntegralE[-2, a + b*x], x, 6, (-6*d^3*E^(-a - b*x))/b^4 - (3*d*E^(-a - b*x)*(c + d*x)^2)/(b^2*(a + b*x)) - ((c + d*x)^3*ExpIntegralE[-1, a + b*x])/b + (6*d^2*(b*c - a*d)*ExpIntegralEi[-a - b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[-2, a + b*x], x, 3, (-2*d*E^(-a - b*x)*(c + d*x))/(b^2*(a + b*x)) - ((c + d*x)^2*ExpIntegralE[-1, a + b*x])/b + (2*d^2*ExpIntegralEi[-a - b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[-2, a + b*x], x, 2, -((d*E^(-a - b*x))/(b^2*(a + b*x))) - ((c + d*x)*ExpIntegralE[-1, a + b*x])/b} -{(c + d*x)^0*ExpIntegralE[-2, a + b*x], x, 1, -(ExpIntegralE[-1, a + b*x]/b)} -{ExpIntegralE[-2, a + b*x]/(c + d*x)^1, x, 11, (d^2*E^(-a - b*x))/(b^2*(b*c - a*d)*(c + d*x)^2) + (d*E^(-a - b*x))/(b^2*(a + b*x)*(c + d*x)^2) + (2*d^2*E^(-a - b*x))/(b*(b*c - a*d)^2*(c + d*x)) - (d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)) - ExpIntegralE[-1, a + b*x]/(b*(c + d*x)) + (2*d^2*ExpIntegralEi[-a - b*x])/(b*c - a*d)^3 - (2*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 + (2*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 - (E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)} -{ExpIntegralE[-2, a + b*x]/(c + d*x)^2, x, 15, (2*d^2*E^(-a - b*x))/(b^2*(b*c - a*d)*(c + d*x)^3) + (2*d*E^(-a - b*x))/(b^2*(a + b*x)*(c + d*x)^3) + (3*d^2*E^(-a - b*x))/(b*(b*c - a*d)^2*(c + d*x)^2) - (d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)^2) + (6*d^2*E^(-a - b*x))/((b*c - a*d)^3*(c + d*x)) - (3*d*E^(-a - b*x))/((b*c - a*d)^2*(c + d*x)) + E^(-a - b*x)/((b*c - a*d)*(c + d*x)) - ExpIntegralE[-1, a + b*x]/(b*(c + d*x)^2) + (6*b*d^2*ExpIntegralEi[-a - b*x])/(b*c - a*d)^4 - (6*b*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^4 + (6*b*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 - (3*b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 + (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d*(b*c - a*d))} -{ExpIntegralE[-2, a + b*x]/(c + d*x)^3, x, 20, (3*d^2*E^(-a - b*x))/(b^2*(b*c - a*d)*(c + d*x)^4) + (3*d*E^(-a - b*x))/(b^2*(a + b*x)*(c + d*x)^4) + (4*d^2*E^(-a - b*x))/(b*(b*c - a*d)^2*(c + d*x)^3) - (d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)^3) + (6*d^2*E^(-a - b*x))/((b*c - a*d)^3*(c + d*x)^2) - (2*d*E^(-a - b*x))/((b*c - a*d)^2*(c + d*x)^2) + E^(-a - b*x)/(2*(b*c - a*d)*(c + d*x)^2) + (12*b*d^2*E^(-a - b*x))/((b*c - a*d)^4*(c + d*x)) - (6*b*d*E^(-a - b*x))/((b*c - a*d)^3*(c + d*x)) + (2*b*E^(-a - b*x))/((b*c - a*d)^2*(c + d*x)) - (b*E^(-a - b*x))/(2*d*(b*c - a*d)*(c + d*x)) - ExpIntegralE[-1, a + b*x]/(b*(c + d*x)^3) + (12*b^2*d^2*ExpIntegralEi[-a - b*x])/(b*c - a*d)^5 - (12*b^2*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^5 + (12*b^2*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^4 - (6*b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 + (2*b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d*(b*c - a*d)^2) - (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^2*(b*c - a*d))} - - -{(c + d*x)^4*ExpIntegralE[-3, a + b*x], x, 7, (-24*d^4*E^(-a - b*x))/b^5 - (12*d^2*E^(-a - b*x)*(c + d*x)^2)/(b^3*(a + b*x)) - ((c + d*x)^4*ExpIntegralE[-2, a + b*x])/b - (4*d*(c + d*x)^3*ExpIntegralE[-1, a + b*x])/b^2 + (24*d^3*(b*c - a*d)*ExpIntegralEi[-a - b*x])/b^5} -{(c + d*x)^3*ExpIntegralE[-3, a + b*x], x, 4, (-6*d^2*E^(-a - b*x)*(c + d*x))/(b^3*(a + b*x)) - ((c + d*x)^3*ExpIntegralE[-2, a + b*x])/b - (3*d*(c + d*x)^2*ExpIntegralE[-1, a + b*x])/b^2 + (6*d^3*ExpIntegralEi[-a - b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[-3, a + b*x], x, 3, (-2*d^2*E^(-a - b*x))/(b^3*(a + b*x)) - ((c + d*x)^2*ExpIntegralE[-2, a + b*x])/b - (2*d*(c + d*x)*ExpIntegralE[-1, a + b*x])/b^2} -{(c + d*x)^1*ExpIntegralE[-3, a + b*x], x, 2, -(((c + d*x)*ExpIntegralE[-2, a + b*x])/b) - (d*ExpIntegralE[-1, a + b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[-3, a + b*x], x, 1, -(ExpIntegralE[-2, a + b*x]/b)} -{ExpIntegralE[-3, a + b*x]/(c + d*x)^1, x, 16, (-2*d^3*E^(-a - b*x))/(b^3*(b*c - a*d)*(c + d*x)^3) - (2*d^2*E^(-a - b*x))/(b^3*(a + b*x)*(c + d*x)^3) - (3*d^3*E^(-a - b*x))/(b^2*(b*c - a*d)^2*(c + d*x)^2) + (d^2*E^(-a - b*x))/(b^2*(b*c - a*d)*(c + d*x)^2) - (6*d^3*E^(-a - b*x))/(b*(b*c - a*d)^3*(c + d*x)) + (3*d^2*E^(-a - b*x))/(b*(b*c - a*d)^2*(c + d*x)) - (d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)) - ExpIntegralE[-2, a + b*x]/(b*(c + d*x)) + (d*ExpIntegralE[-1, a + b*x])/(b^2*(c + d*x)^2) - (6*d^3*ExpIntegralEi[-a - b*x])/(b*c - a*d)^4 + (6*d^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^4 - (6*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 + (3*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 - (E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)} -{ExpIntegralE[-3, a + b*x]/(c + d*x)^2, x, 21, (-6*d^3*E^(-a - b*x))/(b^3*(b*c - a*d)*(c + d*x)^4) - (6*d^2*E^(-a - b*x))/(b^3*(a + b*x)*(c + d*x)^4) - (8*d^3*E^(-a - b*x))/(b^2*(b*c - a*d)^2*(c + d*x)^3) + (2*d^2*E^(-a - b*x))/(b^2*(b*c - a*d)*(c + d*x)^3) - (12*d^3*E^(-a - b*x))/(b*(b*c - a*d)^3*(c + d*x)^2) + (4*d^2*E^(-a - b*x))/(b*(b*c - a*d)^2*(c + d*x)^2) - (d*E^(-a - b*x))/(b*(b*c - a*d)*(c + d*x)^2) - (24*d^3*E^(-a - b*x))/((b*c - a*d)^4*(c + d*x)) + (12*d^2*E^(-a - b*x))/((b*c - a*d)^3*(c + d*x)) - (4*d*E^(-a - b*x))/((b*c - a*d)^2*(c + d*x)) + E^(-a - b*x)/((b*c - a*d)*(c + d*x)) - ExpIntegralE[-2, a + b*x]/(b*(c + d*x)^2) + (2*d*ExpIntegralE[-1, a + b*x])/(b^2*(c + d*x)^3) - (24*b*d^3*ExpIntegralEi[-a - b*x])/(b*c - a*d)^5 + (24*b*d^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^5 - (24*b*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^4 + (12*b*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3 - (4*b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 + (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d*(b*c - a*d))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x] when m symbolic*) - - -{(c + d*x)^m*ExpIntegralE[3, a + b*x], x, 3, If[$VersionNumber>=8, (b^2*(c + d*x)^(3 + m)*ExpIntegralE[1, a + b*x])/(d^3*(1 + m)*(2 + m)*(3 + m)) + (b*(c + d*x)^(2 + m)*ExpIntegralE[2, a + b*x])/(d^2*(1 + m)*(2 + m)) + ((c + d*x)^(1 + m)*ExpIntegralE[3, a + b*x])/(d*(1 + m)) + (b^3*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(3 + m))/(a + b*x), x])/(d^3*(1 + m)*(2 + m)*(3 + m)), (b^2*(c + d*x)^(3 + m)*ExpIntegralE[1, a + b*x])/(d^3*(3 + m)*(2 + 3*m + m^2)) + (b*(c + d*x)^(2 + m)*ExpIntegralE[2, a + b*x])/(d^2*(1 + m)*(2 + m)) + ((c + d*x)^(1 + m)*ExpIntegralE[3, a + b*x])/(d*(1 + m)) + (b^3*Int[(E^(-a - b*x)*(c + d*x)^(3 + m))/(a + b*x), x])/(d^3*(3 + m)*(2 + 3*m + m^2))]} -{(c + d*x)^m*ExpIntegralE[2, a + b*x], x, 2, (b*(c + d*x)^(2 + m)*ExpIntegralE[1, a + b*x])/(d^2*(1 + m)*(2 + m)) + ((c + d*x)^(1 + m)*ExpIntegralE[2, a + b*x])/(d*(1 + m)) + (b^2*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(2 + m))/(a + b*x), x])/(d^2*(1 + m)*(2 + m))} -{(c + d*x)^m*ExpIntegralE[1, a + b*x], x, 1, ((c + d*x)^(1 + m)*ExpIntegralE[1, a + b*x])/(d*(1 + m)) + (b*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(1 + m))/(a + b*x), x])/(d*(1 + m))} -{(c + d*x)^m*ExpIntegralE[-1, a + b*x], x, 1, -((E^(-a - b*x)*(c + d*x)^m)/(b*(a + b*x))) + (d*m*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(-1 + m))/(a + b*x), x])/b} -{(c + d*x)^m*ExpIntegralE[-2, a + b*x], x, 2, -((d*E^(-a - b*x)*m*(c + d*x)^(-1 + m))/(b^2*(a + b*x))) - ((c + d*x)^m*ExpIntegralE[-1, a + b*x])/b - (d^2*(1 - m)*m*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(-2 + m))/(a + b*x), x])/b^2} -{(c + d*x)^m*ExpIntegralE[-3, a + b*x], x, 3, (d^2*E^(-a - b*x)*(1 - m)*m*(c + d*x)^(-2 + m))/(b^3*(a + b*x)) - ((c + d*x)^m*ExpIntegralE[-2, a + b*x])/b - (d*m*(c + d*x)^(-1 + m)*ExpIntegralE[-1, a + b*x])/b^2 + (d^3*(1 - m)*(2 - m)*m*CannotIntegrate[(E^(-a - b*x)*(c + d*x)^(-3 + m))/(a + b*x), x])/b^3} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralE[n, a+b x] when n symbolic*) - - -{(c + d*x)^m*ExpIntegralE[n, a + b*x], x, 0, Unintegrable[(c + d*x)^m*ExpIntegralE[n, a + b*x], x]} - - -{(c + d*x)^3*ExpIntegralE[n, a + b*x], x, 4, -(((c + d*x)^3*ExpIntegralE[1 + n, a + b*x])/b) - (3*d*(c + d*x)^2*ExpIntegralE[2 + n, a + b*x])/b^2 - (6*d^2*(c + d*x)*ExpIntegralE[3 + n, a + b*x])/b^3 - (6*d^3*ExpIntegralE[4 + n, a + b*x])/b^4} -{(c + d*x)^2*ExpIntegralE[n, a + b*x], x, 3, -(((c + d*x)^2*ExpIntegralE[1 + n, a + b*x])/b) - (2*d*(c + d*x)*ExpIntegralE[2 + n, a + b*x])/b^2 - (2*d^2*ExpIntegralE[3 + n, a + b*x])/b^3} -{(c + d*x)^1*ExpIntegralE[n, a + b*x], x, 2, -(((c + d*x)*ExpIntegralE[1 + n, a + b*x])/b) - (d*ExpIntegralE[2 + n, a + b*x])/b^2} -{(c + d*x)^0*ExpIntegralE[n, a + b*x], x, 1, -(ExpIntegralE[1 + n, a + b*x]/b)} -{ExpIntegralE[n, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[ExpIntegralE[n, a + b*x]/(c + d*x), x]} -{ExpIntegralE[n, a + b*x]/(c + d*x)^2, x, 0, Unintegrable[ExpIntegralE[n, a + b*x]/(c + d*x)^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m ExpIntegralEi[b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m ExpIntegralEi[b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{ExpIntegralEi[b*x]*x^3, x, 6, (3*E^(b*x))/(2*b^4) - (3*E^(b*x)*x)/(2*b^3) + (3*E^(b*x)*x^2)/(4*b^2) - (E^(b*x)*x^3)/(4*b) + (1/4)*x^4*ExpIntegralEi[b*x]} -{ExpIntegralEi[b*x]*x^2, x, 5, -((2*E^(b*x))/(3*b^3)) + (2*E^(b*x)*x)/(3*b^2) - (E^(b*x)*x^2)/(3*b) + (1/3)*x^3*ExpIntegralEi[b*x]} -{ExpIntegralEi[b*x]*x^1, x, 4, E^(b*x)/(2*b^2) - (E^(b*x)*x)/(2*b) + (1/2)*x^2*ExpIntegralEi[b*x]} -{ExpIntegralEi[b*x]*x^0, x, 1, -(E^(b*x)/b) + ((b*x)*ExpIntegralEi[b*x])/b} -{ExpIntegralEi[b*x]/x^1, x, 2, b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + EulerGamma*Log[x] + (ExpIntegralE[1, (-b)*x] + ExpIntegralEi[b*x])*Log[x] + (1/2)*Log[(-b)*x]^2} -{ExpIntegralEi[b*x]/x^2, x, 4, -(E^(b*x)/x) + b*ExpIntegralEi[b*x] - ExpIntegralEi[b*x]/x} -{ExpIntegralEi[b*x]/x^3, x, 5, -(E^(b*x)/(4*x^2)) - (b*E^(b*x))/(4*x) + (1/4)*b^2*ExpIntegralEi[b*x] - ExpIntegralEi[b*x]/(2*x^2)} -{ExpIntegralEi[b*x]/x^4, x, 6, -(E^(b*x)/(9*x^3)) - (b*E^(b*x))/(18*x^2) - (b^2*E^(b*x))/(18*x) + (1/18)*b^3*ExpIntegralEi[b*x] - ExpIntegralEi[b*x]/(3*x^3)} - - -{ExpIntegralEi[b*x]^2*x^2, x, 11, -((5*E^(2*b*x))/(6*b^3)) + (E^(2*b*x)*x)/(3*b^2) - (4*E^(b*x)*ExpIntegralEi[b*x])/(3*b^3) + (4*E^(b*x)*x*ExpIntegralEi[b*x])/(3*b^2) - (2*E^(b*x)*x^2*ExpIntegralEi[b*x])/(3*b) + (1/3)*x^3*ExpIntegralEi[b*x]^2 + (4*ExpIntegralEi[2*b*x])/(3*b^3)} -{ExpIntegralEi[b*x]^2*x^1, x, 7, E^(2*b*x)/(2*b^2) + (E^(b*x)*ExpIntegralEi[b*x])/b^2 - (E^(b*x)*x*ExpIntegralEi[b*x])/b + (1/2)*x^2*ExpIntegralEi[b*x]^2 - ExpIntegralEi[2*b*x]/b^2} -{ExpIntegralEi[b*x]^2*x^0, x, 4, -((2*E^(b*x)*ExpIntegralEi[b*x])/b) + x*ExpIntegralEi[b*x]^2 + (2*ExpIntegralEi[2*b*x])/b} -{ExpIntegralEi[b*x]^2/x^1, x, 0, CannotIntegrate[ExpIntegralEi[b*x]^2/x, x]} -{ExpIntegralEi[b*x]^2/x^2, x, 0, CannotIntegrate[ExpIntegralEi[b*x]^2/x^2, x]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m ExpIntegralEi[b x]^n with m symbolic*) - - -{(d*x)^m*ExpIntegralEi[b*x]^3, x, 0, CannotIntegrate[(d*x)^m*ExpIntegralEi[b*x]^3, x]} -{(d*x)^m*ExpIntegralEi[b*x]^2, x, 0, CannotIntegrate[(d*x)^m*ExpIntegralEi[b*x]^2, x]} -{(d*x)^m*ExpIntegralEi[b*x]^1, x, 4, ((d*x)^(1 + m)*ExpIntegralEi[b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(b*(1 + m)))} -{(d*x)^m/ExpIntegralEi[b*x]^1, x, 0, CannotIntegrate[(d*x)^m/ExpIntegralEi[b*x], x]} -{(d*x)^m/ExpIntegralEi[b*x]^2, x, 0, CannotIntegrate[(d*x)^m/ExpIntegralEi[b*x]^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{x^3*ExpIntegralEi[a + b*x], x, 14, (3*E^(a + b*x))/(2*b^4) + (a*E^(a + b*x))/(2*b^4) + (a^2*E^(a + b*x))/(4*b^4) + (a^3*E^(a + b*x))/(4*b^4) - (3*E^(a + b*x)*x)/(2*b^3) - (a*E^(a + b*x)*x)/(2*b^3) - (a^2*E^(a + b*x)*x)/(4*b^3) + (3*E^(a + b*x)*x^2)/(4*b^2) + (a*E^(a + b*x)*x^2)/(4*b^2) - (E^(a + b*x)*x^3)/(4*b) - (a^4*ExpIntegralEi[a + b*x])/(4*b^4) + (1/4)*x^4*ExpIntegralEi[a + b*x]} -{x^2*ExpIntegralEi[a + b*x], x, 10, -((2*E^(a + b*x))/(3*b^3)) - (a*E^(a + b*x))/(3*b^3) - (a^2*E^(a + b*x))/(3*b^3) + (2*E^(a + b*x)*x)/(3*b^2) + (a*E^(a + b*x)*x)/(3*b^2) - (E^(a + b*x)*x^2)/(3*b) + (a^3*ExpIntegralEi[a + b*x])/(3*b^3) + (1/3)*x^3*ExpIntegralEi[a + b*x]} -{x^1*ExpIntegralEi[a + b*x], x, 7, E^(a + b*x)/(2*b^2) + (a*E^(a + b*x))/(2*b^2) - (E^(a + b*x)*x)/(2*b) - (a^2*ExpIntegralEi[a + b*x])/(2*b^2) + (1/2)*x^2*ExpIntegralEi[a + b*x]} -{x^0*ExpIntegralEi[a + b*x], x, 1, -(E^(a + b*x)/b) + ((a + b*x)*ExpIntegralEi[a + b*x])/b} -{ExpIntegralEi[a + b*x]/x^1, x, 0, Unintegrable[ExpIntegralEi[a + b*x]/x, x]} -{ExpIntegralEi[a + b*x]/x^2, x, 5, (b*E^a*ExpIntegralEi[b*x])/a - (b*ExpIntegralEi[a + b*x])/a - ExpIntegralEi[a + b*x]/x} -{ExpIntegralEi[a + b*x]/x^3, x, 7, -((b*E^(a + b*x))/(2*a*x)) - (b^2*E^a*ExpIntegralEi[b*x])/(2*a^2) + (b^2*E^a*ExpIntegralEi[b*x])/(2*a) + (b^2*ExpIntegralEi[a + b*x])/(2*a^2) - ExpIntegralEi[a + b*x]/(2*x^2)} - - -(* {x^3*ExpIntegralEi[a + b*x]^2, x, 51, (2*E^(2*a + 2*b*x))/b^4 + (3*a*E^(2*a + 2*b*x))/(2*b^4) + (3*a^2*E^(2*a + 2*b*x))/(4*b^4) - (E^(2*a + 2*b*x)*x)/b^3 - (a*E^(2*a + 2*b*x)*x)/(2*b^3) + (E^(2*a + 2*b*x)*x^2)/(4*b^2) + (3*E^(a + b*x)*ExpIntegralEi[a + b*x])/b^4 + (a*E^(a + b*x)*ExpIntegralEi[a + b*x])/b^4 + (a^2*E^(a + b*x)*ExpIntegralEi[a + b*x])/(2*b^4) + (a^3*E^(a + b*x)*ExpIntegralEi[a + b*x])/(2*b^4) - (3*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b^3 - (a*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b^3 - (a^2*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/(2*b^3) + (3*E^(a + b*x)*x^2*ExpIntegralEi[a + b*x])/(2*b^2) + (a*E^(a + b*x)*x^2*ExpIntegralEi[a + b*x])/(2*b^2) - (E^(a + b*x)*x^3*ExpIntegralEi[a + b*x])/(2*b) + (a^3*x*ExpIntegralEi[a + b*x]^2)/(4*b^3) + (1/4)*x^4*ExpIntegralEi[a + b*x]^2 - (a^3*(a + b*x)*ExpIntegralEi[a + b*x]^2)/(4*b^4) - (3*ExpIntegralEi[2*a + 2*b*x])/b^4 - (4*a*ExpIntegralEi[2*a + 2*b*x])/b^4 - (3*a^2*ExpIntegralEi[2*a + 2*b*x])/b^4 - (2*a^3*ExpIntegralEi[2*a + 2*b*x])/b^4} *) -{x^2*ExpIntegralEi[a + b*x]^2, x, 26, -((5*E^(2*a + 2*b*x))/(6*b^3)) - (2*a*E^(2*a + 2*b*x))/(3*b^3) + (E^(2*a + 2*b*x)*x)/(3*b^2) - (4*E^(a + b*x)*ExpIntegralEi[a + b*x])/(3*b^3) - (2*a*E^(a + b*x)*ExpIntegralEi[a + b*x])/(3*b^3) - (2*a^2*E^(a + b*x)*ExpIntegralEi[a + b*x])/(3*b^3) + (4*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/(3*b^2) + (2*a*E^(a + b*x)*x*ExpIntegralEi[a + b*x])/(3*b^2) - (2*E^(a + b*x)*x^2*ExpIntegralEi[a + b*x])/(3*b) - (a^2*x*ExpIntegralEi[a + b*x]^2)/(3*b^2) + (1/3)*x^3*ExpIntegralEi[a + b*x]^2 + (a^2*(a + b*x)*ExpIntegralEi[a + b*x]^2)/(3*b^3) + (4*ExpIntegralEi[2*(a + b*x)])/(3*b^3) + (2*a*ExpIntegralEi[2*(a + b*x)])/b^3 + (2*a^2*ExpIntegralEi[2*(a + b*x)])/b^3} -{x^1*ExpIntegralEi[a + b*x]^2, x, 11, E^(2*a + 2*b*x)/(2*b^2) + (E^(a + b*x)*ExpIntegralEi[a + b*x])/b^2 + (a*E^(a + b*x)*ExpIntegralEi[a + b*x])/b^2 - (E^(a + b*x)*x*ExpIntegralEi[a + b*x])/b + (a*x*ExpIntegralEi[a + b*x]^2)/(2*b) + (1/2)*x^2*ExpIntegralEi[a + b*x]^2 - (a*(a + b*x)*ExpIntegralEi[a + b*x]^2)/(2*b^2) - ExpIntegralEi[2*(a + b*x)]/b^2 - (2*a*ExpIntegralEi[2*(a + b*x)])/b^2} -{x^0*ExpIntegralEi[a + b*x]^2, x, 3, -((2*E^(a + b*x)*ExpIntegralEi[a + b*x])/b) + ((a + b*x)*ExpIntegralEi[a + b*x]^2)/b + (2*ExpIntegralEi[2*(a + b*x)])/b} -{ExpIntegralEi[a + b*x]^2/x^1, x, 0, CannotIntegrate[ExpIntegralEi[a + b*x]^2/x, x]} -{ExpIntegralEi[a + b*x]^2/x^2, x, 0, CannotIntegrate[ExpIntegralEi[a + b*x]^2/x^2, x]} - - -{x^2*ExpIntegralEi[a + b*x]^3, x, 0, CannotIntegrate[x^2*ExpIntegralEi[a + b*x]^3, x]} -{x^1*ExpIntegralEi[a + b*x]^3, x, 0, CannotIntegrate[x*ExpIntegralEi[a + b*x]^3, x]} -{x^0*ExpIntegralEi[a + b*x]^3, x, 1, CannotIntegrate[ExpIntegralEi[a + b*x]^3,x]} -{ExpIntegralEi[a + b*x]^3/x^1, x, 0, CannotIntegrate[ExpIntegralEi[a + b*x]^3/x, x]} -{ExpIntegralEi[a + b*x]^3/x^2, x, 0, CannotIntegrate[ExpIntegralEi[a + b*x]^3/x^2, x]} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m ExpIntegralEi[a+b x]^n with m symbolic*) - - -{(c + d*x)^m*ExpIntegralEi[a + b*x]^3, x, 0, CannotIntegrate[(c + d*x)^m*ExpIntegralEi[a + b*x]^3, x]} -{(c + d*x)^m*ExpIntegralEi[a + b*x]^2, x, 0, CannotIntegrate[(c + d*x)^m*ExpIntegralEi[a + b*x]^2, x]} -{(c + d*x)^m*ExpIntegralEi[a + b*x]^1, x, 1, ((c + d*x)^(1 + m)*ExpIntegralEi[a + b*x])/(d*(1 + m)) - (b*CannotIntegrate[(E^(a + b*x)*(c + d*x)^(1 + m))/(a + b*x), x])/(d*(1 + m))} -{(c + d*x)^m/ExpIntegralEi[a + b*x]^1, x, 0, CannotIntegrate[(c + d*x)^m/ExpIntegralEi[a + b*x], x]} -{(c + d*x)^m/ExpIntegralEi[a + b*x]^2, x, 0, CannotIntegrate[(c + d*x)^m/ExpIntegralEi[a + b*x]^2, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m ExpIntegralEi[d (a+b Log[c x^n])]*) - - -{x^2*ExpIntegralEi[d*(a + b*Log[c*x^n])], x, 3, -(x^3*ExpIntegralEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(3*E^((3*a)/(b*n))*(c*x^n)^(3/n)) + (x^3*ExpIntegralEi[d*(a + b*Log[c*x^n])])/3} -{x^1*ExpIntegralEi[d*(a + b*Log[c*x^n])], x, 3, -(x^2*ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((2*a)/(b*n))*(c*x^n)^(2/n)) + (x^2*ExpIntegralEi[d*(a + b*Log[c*x^n])])/2} -{x^0*ExpIntegralEi[d*(a + b*Log[c*x^n])], x, 4, -((x*ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^(a/(b*n))*(c*x^n)^n^(-1))) + x*ExpIntegralEi[d*(a + b*Log[c*x^n])]} -{ExpIntegralEi[d*(a + b*Log[c*x^n])]/x^1, x, 3, -((E^(a*d)*(c*x^n)^(b*d))/(b*d*n)) + (ExpIntegralEi[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)} -{ExpIntegralEi[d*(a + b*Log[c*x^n])]/x^2, x, 3, (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/x - ExpIntegralEi[d*(a + b*Log[c*x^n])]/x} -{ExpIntegralEi[d*(a + b*Log[c*x^n])]/x^3, x, 3, (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x^2) - ExpIntegralEi[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*ExpIntegralEi[d*(a + b*Log[c*x^n])], x, 3, -(((e*x)^(1 + m)*ExpIntegralEi[((1 + m + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(e*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n))) + ((e*x)^(1 + m)*ExpIntegralEi[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form x^m E^(a+b x) ExpIntegralEi[c+d x]^n*) -(**) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(b x) ExpIntegralEi[b x]^n*) - - -(* Integrands of x^m*E^(b*x)*ExpIntegralEi[b*x] where m is an integer *) -{E^(b*x)*ExpIntegralEi[b*x]/x^3, x, 10, -(E^(2*b*x)/(4*x^2)) - (b*E^(2*b*x))/x - (E^(b*x)*ExpIntegralEi[b*x])/(2*x^2) - (b*E^(b*x)*ExpIntegralEi[b*x])/(2*x) + (1/4)*b^2*ExpIntegralEi[b*x]^2 + 2*b^2*ExpIntegralEi[2*b*x]} -{E^(b*x)*ExpIntegralEi[b*x]/x^2, x, 5, -(E^(2*b*x)/x) - (E^(b*x)*ExpIntegralEi[b*x])/x + (1/2)*b*ExpIntegralEi[b*x]^2 + 2*b*ExpIntegralEi[2*b*x]} -{E^(b*x)*ExpIntegralEi[b*x]/x, x, 1, (1/2)*ExpIntegralEi[b*x]^2} -{E^(b*x)*ExpIntegralEi[b*x], x, 3, (E^(b*x)*ExpIntegralEi[b*x])/b - ExpIntegralEi[2*b*x]/b} -{x*E^(b*x)*ExpIntegralEi[b*x], x, 6, -(E^(2*b*x)/(2*b^2)) - (E^(b*x)*ExpIntegralEi[b*x])/b^2 + (E^(b*x)*x*ExpIntegralEi[b*x])/b + ExpIntegralEi[2*b*x]/b^2} -{x^2*E^(b*x)*ExpIntegralEi[b*x], x, 10, (5*E^(2*b*x))/(4*b^3) - (E^(2*b*x)*x)/(2*b^2) + (2*E^(b*x)*ExpIntegralEi[b*x])/b^3 - (2*E^(b*x)*x*ExpIntegralEi[b*x])/b^2 + (E^(b*x)*x^2*ExpIntegralEi[b*x])/b - (2*ExpIntegralEi[2*b*x])/b^3} -{x^3*E^(b*x)*ExpIntegralEi[b*x], x, 15, -((4*E^(2*b*x))/b^4) + (2*E^(2*b*x)*x)/b^3 - (E^(2*b*x)*x^2)/(2*b^2) - (6*E^(b*x)*ExpIntegralEi[b*x])/b^4 + (6*E^(b*x)*x*ExpIntegralEi[b*x])/b^3 - (3*E^(b*x)*x^2*ExpIntegralEi[b*x])/b^2 + (E^(b*x)*x^3*ExpIntegralEi[b*x])/b + (6*ExpIntegralEi[2*b*x])/b^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m E^(a+b x) ExpIntegralEi[c+d x]^n*) - - -{x^3*E^(a + b*x)*ExpIntegralEi[c + d*x], x, 24, -((2*E^(a + c + (b + d)*x))/(b*(b + d)^3)) - (3*E^(a + c + (b + d)*x))/(b^2*(b + d)^2) - (c*E^(a + c + (b + d)*x))/(b*d*(b + d)^2) - (6*E^(a + c + (b + d)*x))/(b^3*(b + d)) - (c^2*E^(a + c + (b + d)*x))/(b*d^2*(b + d)) - (3*c*E^(a + c + (b + d)*x))/(b^2*d*(b + d)) + (2*E^(a + c + (b + d)*x)*x)/(b*(b + d)^2) + (3*E^(a + c + (b + d)*x)*x)/(b^2*(b + d)) + (c*E^(a + c + (b + d)*x)*x)/(b*d*(b + d)) - (E^(a + c + (b + d)*x)*x^2)/(b*(b + d)) - (6*E^(a + b*x)*ExpIntegralEi[c + d*x])/b^4 + (6*E^(a + b*x)*x*ExpIntegralEi[c + d*x])/b^3 - (3*E^(a + b*x)*x^2*ExpIntegralEi[c + d*x])/b^2 + (E^(a + b*x)*x^3*ExpIntegralEi[c + d*x])/b + (6*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/b^4 + (c^3*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b*d^3) + (3*c^2*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b^2*d^2) + (6*c*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b^3*d)} -{x^2*E^(a + b*x)*ExpIntegralEi[c + d*x], x, 14, E^(a + c + (b + d)*x)/(b*(b + d)^2) + (2*E^(a + c + (b + d)*x))/(b^2*(b + d)) + (c*E^(a + c + (b + d)*x))/(b*d*(b + d)) - (E^(a + c + (b + d)*x)*x)/(b*(b + d)) + (2*E^(a + b*x)*ExpIntegralEi[c + d*x])/b^3 - (2*E^(a + b*x)*x*ExpIntegralEi[c + d*x])/b^2 + (E^(a + b*x)*x^2*ExpIntegralEi[c + d*x])/b - (2*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/b^3 - (c^2*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b*d^2) - (2*c*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b^2*d)} -{x^1*E^(a + b*x)*ExpIntegralEi[c + d*x], x, 7, -(E^(a + c + (b + d)*x)/(b*(b + d))) - (E^(a + b*x)*ExpIntegralEi[c + d*x])/b^2 + (E^(a + b*x)*x*ExpIntegralEi[c + d*x])/b + (E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/b^2 + (c*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(b*d)} -{x^0*E^(a + b*x)*ExpIntegralEi[c + d*x], x, 2, (E^(a + b*x)*ExpIntegralEi[c + d*x])/b - (E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/b} -{E^(a + b*x)*ExpIntegralEi[c + d*x]/x^1, x, 0, CannotIntegrate[(E^(a + b*x)*ExpIntegralEi[c + d*x])/x, x]} -{E^(a + b*x)*ExpIntegralEi[c + d*x]/x^2, x, 5, (d*E^(a + c)*ExpIntegralEi[(b + d)*x])/c - (E^(a + b*x)*ExpIntegralEi[c + d*x])/x - (d*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/c + b*CannotIntegrate[(E^(a + b*x)*ExpIntegralEi[c + d*x])/x, x]} -{E^(a + b*x)*ExpIntegralEi[c + d*x]/x^3, x, 12, -((d*E^(a + c + (b + d)*x))/(2*c*x)) + (b*d*E^(a + c)*ExpIntegralEi[(b + d)*x])/(2*c) - (d^2*E^(a + c)*ExpIntegralEi[(b + d)*x])/(2*c^2) + (d*(b + d)*E^(a + c)*ExpIntegralEi[(b + d)*x])/(2*c) - (E^(a + b*x)*ExpIntegralEi[c + d*x])/(2*x^2) - (b*E^(a + b*x)*ExpIntegralEi[c + d*x])/(2*x) - (b*d*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(2*c) + (d^2*E^(a - (b*c)/d)*ExpIntegralEi[((b + d)*(c + d*x))/d])/(2*c^2) + (1/2)*b^2*CannotIntegrate[(E^(a + b*x)*ExpIntegralEi[c + d*x])/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m LogIntegral[b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m LogIntegral[b x]^n*) - - -{x^2*LogIntegral[b*x], x, 3, -(ExpIntegralEi[4*Log[b*x]]/(3*b^3)) + (1/3)*x^3*LogIntegral[b*x]} -{x^1*LogIntegral[b*x], x, 3, -(ExpIntegralEi[3*Log[b*x]]/(2*b^2)) + (1/2)*x^2*LogIntegral[b*x]} -{x^0*LogIntegral[b*x], x, 1, -(ExpIntegralEi[2*Log[b*x]]/b) + x*LogIntegral[b*x]} -{LogIntegral[b*x]/x^1, x, 1, (-b)*x + Log[b*x]*LogIntegral[b*x]} -{LogIntegral[b*x]/x^2, x, 3, b*Log[Log[b*x]] - LogIntegral[b*x]/x} -{LogIntegral[b*x]/x^3, x, 3, (1/2)*b^2*ExpIntegralEi[-Log[b*x]] - LogIntegral[b*x]/(2*x^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m LogIntegral[b x]^n with m symbolic*) - - -{(d*x)^m*LogIntegral[b*x], x, 3, -((b*(b*x)^(-2 - m)*(d*x)^(2 + m)*ExpIntegralEi[(2 + m)*Log[b*x]])/(d^2*(1 + m))) + ((d*x)^(1 + m)*LogIntegral[b*x])/(d*(1 + m))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m LogIntegral[a+b x]^n*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m LogIntegral[a+b x]^n*) - - -{x^2*LogIntegral[a + b*x], x, 14, -((a^2*ExpIntegralEi[2*Log[a + b*x]])/b^3) + (a*ExpIntegralEi[3*Log[a + b*x]])/b^3 - ExpIntegralEi[4*Log[a + b*x]]/(3*b^3) + (a^3*LogIntegral[a + b*x])/(3*b^3) + (1/3)*x^3*LogIntegral[a + b*x]} -{x^1*LogIntegral[a + b*x], x, 11, (a*ExpIntegralEi[2*Log[a + b*x]])/b^2 - ExpIntegralEi[3*Log[a + b*x]]/(2*b^2) - (a^2*LogIntegral[a + b*x])/(2*b^2) + (1/2)*x^2*LogIntegral[a + b*x]} -{x^0*LogIntegral[a + b*x], x, 1, -(ExpIntegralEi[2*Log[a + b*x]]/b) + ((a + b*x)*LogIntegral[a + b*x])/b} -{LogIntegral[a + b*x]/x^1, x, 0, Unintegrable[LogIntegral[a + b*x]/x, x]} -{LogIntegral[a + b*x]/x^2, x, 1, b*Unintegrable[1/(x*Log[a + b*x]), x] - LogIntegral[a + b*x]/x} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m LogIntegral[a+b x]^n with m symbolic*) - - -{(d*x)^m*LogIntegral[a + b*x], x, 1, ((d*x)^(1 + m)*LogIntegral[a + b*x])/(d*(1 + m)) - (b*Unintegrable[(d*x)^(1 + m)/Log[a + b*x], x])/(d*(1 + m))} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.4 Trig integral functions.m b/test/methods/rule_based/test_files/8 Special functions/8.4 Trig integral functions.m deleted file mode 100644 index 4ee1427..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.4 Trig integral functions.m +++ /dev/null @@ -1,276 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Trig Integral Functions*) - - -(* ::Section::Closed:: *) -(*Sine integral function*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m SinIntegral[b x]^n*) - - -{x^m*SinIntegral[b*x], x, 5, (x^m*Gamma[1 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b*(1 + m))) + (x^m*Gamma[1 + m, I*b*x])/((I*b*x)^m*(2*b*(1 + m))) + (x^(1 + m)*SinIntegral[b*x])/(1 + m)} - -{x^3*SinIntegral[b*x], x, 6, -((3*x*Cos[b*x])/(2*b^3)) + (x^3*Cos[b*x])/(4*b) + (3*Sin[b*x])/(2*b^4) - (3*x^2*Sin[b*x])/(4*b^2) + (1/4)*x^4*SinIntegral[b*x]} -{x^2*SinIntegral[b*x], x, 5, -((2*Cos[b*x])/(3*b^3)) + (x^2*Cos[b*x])/(3*b) - (2*x*Sin[b*x])/(3*b^2) + (1/3)*x^3*SinIntegral[b*x]} -{x^1*SinIntegral[b*x], x, 4, (x*Cos[b*x])/(2*b) - Sin[b*x]/(2*b^2) + (1/2)*x^2*SinIntegral[b*x]} -{x^0*SinIntegral[b*x], x, 1, Cos[b*x]/b + x*SinIntegral[b*x]} -{SinIntegral[b*x]/x^1, x, 1, (1/2)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x] + (1/2)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x]} -{SinIntegral[b*x]/x^2, x, 4, b*CosIntegral[b*x] - Sin[b*x]/x - SinIntegral[b*x]/x} -{SinIntegral[b*x]/x^3, x, 5, -((b*Cos[b*x])/(4*x)) - Sin[b*x]/(4*x^2) - (1/4)*b^2*SinIntegral[b*x] - SinIntegral[b*x]/(2*x^2)} - - -{x^m*SinIntegral[b*x]^2, x, 0, CannotIntegrate[x^m*SinIntegral[b*x]^2, x]} - -{x^3*SinIntegral[b*x]^2, x, 19, x^2/(2*b^2) + (3*CosIntegral[2*b*x])/(2*b^4) - (3*Log[x])/(2*b^4) - (x*Cos[b*x]*Sin[b*x])/b^3 + (2*Sin[b*x]^2)/b^4 - (x^2*Sin[b*x]^2)/(4*b^2) - (3*x*Cos[b*x]*SinIntegral[b*x])/b^3 + (x^3*Cos[b*x]*SinIntegral[b*x])/(2*b) + (3*Sin[b*x]*SinIntegral[b*x])/b^4 - (3*x^2*Sin[b*x]*SinIntegral[b*x])/(2*b^2) + (1/4)*x^4*SinIntegral[b*x]^2} -{x^2*SinIntegral[b*x]^2, x, 15, (5*x)/(6*b^2) - (5*Cos[b*x]*Sin[b*x])/(6*b^3) - (x*Sin[b*x]^2)/(3*b^2) - (4*Cos[b*x]*SinIntegral[b*x])/(3*b^3) + (2*x^2*Cos[b*x]*SinIntegral[b*x])/(3*b) - (4*x*Sin[b*x]*SinIntegral[b*x])/(3*b^2) + (1/3)*x^3*SinIntegral[b*x]^2 + (2*SinIntegral[2*b*x])/(3*b^3)} -{x^1*SinIntegral[b*x]^2, x, 10, -(CosIntegral[2*b*x]/(2*b^2)) + Log[x]/(2*b^2) - Sin[b*x]^2/(2*b^2) + (x*Cos[b*x]*SinIntegral[b*x])/b - (Sin[b*x]*SinIntegral[b*x])/b^2 + (1/2)*x^2*SinIntegral[b*x]^2} -{x^0*SinIntegral[b*x]^2, x, 6, (2*Cos[b*x]*SinIntegral[b*x])/b + x*SinIntegral[b*x]^2 - SinIntegral[2*b*x]/b} -{SinIntegral[b*x]^2/x^1, x, 0, CannotIntegrate[SinIntegral[b*x]^2/x, x]} -{SinIntegral[b*x]^2/x^2, x, 0, CannotIntegrate[SinIntegral[b*x]^2/x^2, x]} -{SinIntegral[b*x]^2/x^3, x, 0, CannotIntegrate[SinIntegral[b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m SinIntegral[a+b x]^n*) - - -{x^m*SinIntegral[a + b*x], x, 1, -((b*CannotIntegrate[(x^(1 + m)*Sin[a + b*x])/(a + b*x), x])/(1 + m)) + (x^(1 + m)*SinIntegral[a + b*x])/(1 + m)} - -{x^3*SinIntegral[a + b*x], x, 14, (a*Cos[a + b*x])/(2*b^4) - (a^3*Cos[a + b*x])/(4*b^4) - (3*x*Cos[a + b*x])/(2*b^3) + (a^2*x*Cos[a + b*x])/(4*b^3) - (a*x^2*Cos[a + b*x])/(4*b^2) + (x^3*Cos[a + b*x])/(4*b) + (3*Sin[a + b*x])/(2*b^4) - (a^2*Sin[a + b*x])/(4*b^4) + (a*x*Sin[a + b*x])/(2*b^3) - (3*x^2*Sin[a + b*x])/(4*b^2) - (a^4*SinIntegral[a + b*x])/(4*b^4) + (1/4)*x^4*SinIntegral[a + b*x]} -{x^2*SinIntegral[a + b*x], x, 10, -((2*Cos[a + b*x])/(3*b^3)) + (a^2*Cos[a + b*x])/(3*b^3) - (a*x*Cos[a + b*x])/(3*b^2) + (x^2*Cos[a + b*x])/(3*b) + (a*Sin[a + b*x])/(3*b^3) - (2*x*Sin[a + b*x])/(3*b^2) + (a^3*SinIntegral[a + b*x])/(3*b^3) + (1/3)*x^3*SinIntegral[a + b*x]} -{x^1*SinIntegral[a + b*x], x, 7, -((a*Cos[a + b*x])/(2*b^2)) + (x*Cos[a + b*x])/(2*b) - Sin[a + b*x]/(2*b^2) - (a^2*SinIntegral[a + b*x])/(2*b^2) + (1/2)*x^2*SinIntegral[a + b*x]} -{x^0*SinIntegral[a + b*x], x, 1, Cos[a + b*x]/b + ((a + b*x)*SinIntegral[a + b*x])/b} -{SinIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[SinIntegral[a + b*x]/x, x]} -{SinIntegral[a + b*x]/x^2, x, 7, (b*CosIntegral[b*x]*Sin[a])/a + (b*Cos[a]*SinIntegral[b*x])/a - (b*SinIntegral[a + b*x])/a - SinIntegral[a + b*x]/x} -{SinIntegral[a + b*x]/x^3, x, 11, (b^2*Cos[a]*CosIntegral[b*x])/(2*a) - (b^2*CosIntegral[b*x]*Sin[a])/(2*a^2) - (b*Sin[a + b*x])/(2*a*x) - (b^2*Cos[a]*SinIntegral[b*x])/(2*a^2) - (b^2*Sin[a]*SinIntegral[b*x])/(2*a) + (b^2*SinIntegral[a + b*x])/(2*a^2) - SinIntegral[a + b*x]/(2*x^2)} - - -{x^m*SinIntegral[a + b*x]^2, x, 0, CannotIntegrate[x^m*SinIntegral[a + b*x]^2, x]} - -{x^2*SinIntegral[a + b*x]^2, x, 39, (2*x)/(3*b^2) - (a*Cos[2*a + 2*b*x])/(3*b^3) + (x*Cos[2*a + 2*b*x])/(6*b^2) + (a*CosIntegral[2*a + 2*b*x])/b^3 - (a*Log[a + b*x])/b^3 - (2*Cos[a + b*x]*Sin[a + b*x])/(3*b^3) - Sin[2*a + 2*b*x]/(12*b^3) - (4*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b^3) + (2*a^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b^3) - (2*a*x*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b^2) + (2*x^2*Cos[a + b*x]*SinIntegral[a + b*x])/(3*b) + (2*a*Sin[a + b*x]*SinIntegral[a + b*x])/(3*b^3) - (4*x*Sin[a + b*x]*SinIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^3) - (a*x*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*SinIntegral[a + b*x]^2)/(3*b) + (2*SinIntegral[2*a + 2*b*x])/(3*b^3) - (a^2*SinIntegral[2*a + 2*b*x])/b^3} -{x^1*SinIntegral[a + b*x]^2, x, 17, Cos[2*a + 2*b*x]/(4*b^2) - CosIntegral[2*a + 2*b*x]/(2*b^2) + Log[a + b*x]/(2*b^2) - (a*Cos[a + b*x]*SinIntegral[a + b*x])/b^2 + (x*Cos[a + b*x]*SinIntegral[a + b*x])/b - (Sin[a + b*x]*SinIntegral[a + b*x])/b^2 - (a*(a + b*x)*SinIntegral[a + b*x]^2)/(2*b^2) + (x*(a + b*x)*SinIntegral[a + b*x]^2)/(2*b) + (a*SinIntegral[2*a + 2*b*x])/b^2} -{x^0*SinIntegral[a + b*x]^2, x, 5, (2*Cos[a + b*x]*SinIntegral[a + b*x])/b + ((a + b*x)*SinIntegral[a + b*x]^2)/b - SinIntegral[2*a + 2*b*x]/b} -{SinIntegral[a + b*x]^2/x^1, x, 0, CannotIntegrate[SinIntegral[a + b*x]^2/x, x]} -{SinIntegral[a + b*x]^2/x^2, x, 0, CannotIntegrate[SinIntegral[a + b*x]^2/x^2, x]} -{SinIntegral[a + b*x]^2/x^3, x, 0, CannotIntegrate[SinIntegral[a + b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m SinIntegral[d (a+b Log[c x^n])]*) - - -{x^2*SinIntegral[d*(a + b*Log[c*x^n])], x, 7, ((-(1/6))*I*x^3*ExpIntegralEi[((3 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) + ((1/6)*I*x^3*ExpIntegralEi[((3 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*x^n)^(3/n)) + (1/3)*x^3*SinIntegral[d*(a + b*Log[c*x^n])]} -{x^1*SinIntegral[d*(a + b*Log[c*x^n])], x, 7, ((-(1/4))*I*x^2*ExpIntegralEi[((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) + ((1/4)*I*x^2*ExpIntegralEi[((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n)) + (1/2)*x^2*SinIntegral[d*(a + b*Log[c*x^n])]} -{x^0*SinIntegral[d*(a + b*Log[c*x^n])], x, 7, ((-(1/2))*I*x*ExpIntegralEi[((1 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^(a/(b*n))*(c*x^n)^n^(-1)) + ((1/2)*I*x*ExpIntegralEi[((1 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^(a/(b*n))*(c*x^n)^n^(-1)) + x*SinIntegral[d*(a + b*Log[c*x^n])]} -{SinIntegral[d*(a + b*Log[c*x^n])]/x^1, x, 3, Cos[d*(a + b*Log[c*x^n])]/(b*d*n) + ((a + b*Log[c*x^n])*SinIntegral[d*(a + b*Log[c*x^n])])/(b*n)} -{SinIntegral[d*(a + b*Log[c*x^n])]/x^2, x, 7, -((I*E^(a/(b*n))*(c*x^n)^(1/n)*ExpIntegralEi[-(((1 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x)) + (I*E^(a/(b*n))*(c*x^n)^(1/n)*ExpIntegralEi[-(((1 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x) - SinIntegral[d*(a + b*Log[c*x^n])]/x} -{SinIntegral[d*(a + b*Log[c*x^n])]/x^3, x, 7, -((I*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2)) + (I*E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2) - SinIntegral[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*SinIntegral[d*(a + b*Log[c*x^n])], x, 7, -((I*x*(e*x)^m*ExpIntegralEi[((1 + m - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(2*(1 + m)))) + (I*x*(e*x)^m*ExpIntegralEi[((1 + m + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(2*(1 + m))) + ((e*x)^(1 + m)*SinIntegral[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[b x] SinIntegral[b x]^n*) - - -{Sin[b*x]*SinIntegral[b*x]/x^3, x, 14, b^2*CosIntegral[2*b*x] - (b*Cos[b*x]*Sin[b*x])/(2*x) - Sin[b*x]^2/(4*x^2) - (b*Sin[2*b*x])/(4*x) - (b*Cos[b*x]*SinIntegral[b*x])/(2*x) - (Sin[b*x]*SinIntegral[b*x])/(2*x^2) - (1/4)*b^2*SinIntegral[b*x]^2} -{Sin[b*x]*SinIntegral[b*x]/x^2, x, 5, b*CannotIntegrate[(Cos[b*x]*SinIntegral[b*x])/x, x] - Sin[b*x]^2/x - (Sin[b*x]*SinIntegral[b*x])/x + b*SinIntegral[2*b*x]} -{Sin[b*x]*SinIntegral[b*x]/x, x, 1, (1/2)*SinIntegral[b*x]^2} -{Sin[b*x]*SinIntegral[b*x], x, 5, -((Cos[b*x]*SinIntegral[b*x])/b) + SinIntegral[2*b*x]/(2*b)} -{x*Sin[b*x]*SinIntegral[b*x], x, 9, CosIntegral[2*b*x]/(2*b^2) - Log[x]/(2*b^2) + Sin[b*x]^2/(2*b^2) - (x*Cos[b*x]*SinIntegral[b*x])/b + (Sin[b*x]*SinIntegral[b*x])/b^2} -{x^2*Sin[b*x]*SinIntegral[b*x], x, 14, -((5*x)/(4*b^2)) + (5*Cos[b*x]*Sin[b*x])/(4*b^3) + (x*Sin[b*x]^2)/(2*b^2) + (2*Cos[b*x]*SinIntegral[b*x])/b^3 - (x^2*Cos[b*x]*SinIntegral[b*x])/b + (2*x*Sin[b*x]*SinIntegral[b*x])/b^2 - SinIntegral[2*b*x]/b^3} -{x^3*Sin[b*x]*SinIntegral[b*x], x, 18, -(x^2/b^2) - (3*CosIntegral[2*b*x])/b^4 + (3*Log[x])/b^4 + (2*x*Cos[b*x]*Sin[b*x])/b^3 - (4*Sin[b*x]^2)/b^4 + (x^2*Sin[b*x]^2)/(2*b^2) + (6*x*Cos[b*x]*SinIntegral[b*x])/b^3 - (x^3*Cos[b*x]*SinIntegral[b*x])/b - (6*Sin[b*x]*SinIntegral[b*x])/b^4 + (3*x^2*Sin[b*x]*SinIntegral[b*x])/b^2} - - -{Cos[b*x]*SinIntegral[b*x]/x^3, x, 12, -((b*Cos[2*b*x])/(4*x)) - (1/2)*b^2*CannotIntegrate[(Cos[b*x]*SinIntegral[b*x])/x, x] + (b*Sin[b*x]^2)/(2*x) - Sin[2*b*x]/(8*x^2) - (Cos[b*x]*SinIntegral[b*x])/(2*x^2) + (b*Sin[b*x]*SinIntegral[b*x])/(2*x) - b^2*SinIntegral[2*b*x]} -{Cos[b*x]*SinIntegral[b*x]/x^2, x, 7, b*CosIntegral[2*b*x] - Sin[2*b*x]/(2*x) - (Cos[b*x]*SinIntegral[b*x])/x - (1/2)*b*SinIntegral[b*x]^2} -{Cos[b*x]*SinIntegral[b*x]/x, x, 0, CannotIntegrate[(Cos[b*x]*SinIntegral[b*x])/x, x]} -{Cos[b*x]*SinIntegral[b*x], x, 5, CosIntegral[2*b*x]/(2*b) - Log[x]/(2*b) + (Sin[b*x]*SinIntegral[b*x])/b} -{x*Cos[b*x]*SinIntegral[b*x], x, 9, -(x/(2*b)) + (Cos[b*x]*Sin[b*x])/(2*b^2) + (Cos[b*x]*SinIntegral[b*x])/b^2 + (x*Sin[b*x]*SinIntegral[b*x])/b - SinIntegral[2*b*x]/(2*b^2)} -{x^2*Cos[b*x]*SinIntegral[b*x], x, 13, -(x^2/(4*b)) - CosIntegral[2*b*x]/b^3 + Log[x]/b^3 + (x*Cos[b*x]*Sin[b*x])/(2*b^2) - (5*Sin[b*x]^2)/(4*b^3) + (2*x*Cos[b*x]*SinIntegral[b*x])/b^2 - (2*Sin[b*x]*SinIntegral[b*x])/b^3 + (x^2*Sin[b*x]*SinIntegral[b*x])/b} -{x^3*Cos[b*x]*SinIntegral[b*x], x, 20, (4*x)/b^3 - x^3/(6*b) - (4*Cos[b*x]*Sin[b*x])/b^4 + (x^2*Cos[b*x]*Sin[b*x])/(2*b^2) - (2*x*Sin[b*x]^2)/b^3 - (6*Cos[b*x]*SinIntegral[b*x])/b^4 + (3*x^2*Cos[b*x]*SinIntegral[b*x])/b^2 - (6*x*Sin[b*x]*SinIntegral[b*x])/b^3 + (x^3*Sin[b*x]*SinIntegral[b*x])/b + (3*SinIntegral[2*b*x])/b^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[b x] SinIntegral[d x]^n*) - - -{Sin[5*x]*SinIntegral[2*x], x, 6, (-(1/5))*Cos[5*x]*SinIntegral[2*x] - (1/10)*SinIntegral[3*x] + (1/10)*SinIntegral[7*x]} - - -{Cos[5*x]*SinIntegral[2*x], x, 6, (-(1/10))*CosIntegral[3*x] + (1/10)*CosIntegral[7*x] + (1/5)*Sin[5*x]*SinIntegral[2*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[a+b x] SinIntegral[a+b x]^n*) - - -(* {x^3*Sin[a + b*x]*SinIntegral[a + b*x], x, 32, (7*a*x)/(4*b^3) - x^2/b^2 + (3*Cos[a + b*x]^2)/b^4 - (a^2*Cos[a + b*x]^2)/(2*b^4) - (3*CosIntegral[2*a + 2*b*x])/b^4 + (3*a^2*CosIntegral[2*a + 2*b*x])/(2*b^4) + (3*Log[a + b*x])/b^4 - (3*a^2*Log[a + b*x])/(2*b^4) - (7*a*Cos[a + b*x]*Sin[a + b*x])/(4*b^4) + (2*x*Cos[a + b*x]*Sin[a + b*x])/b^3 - Sin[a + b*x]^2/b^4 - (a*x*Sin[a + b*x]^2)/(2*b^3) + (x^2*Sin[a + b*x]^2)/(2*b^2) + (6*x*Cos[a + b*x]*SinIntegral[a + b*x])/b^3 - (x^3*Cos[a + b*x]*SinIntegral[a + b*x])/b - (6*Sin[a + b*x]*SinIntegral[a + b*x])/b^4 + (3*x^2*Sin[a + b*x]*SinIntegral[a + b*x])/b^2 + (3*a*SinIntegral[2*a + 2*b*x])/b^4 - (a^3*SinIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Sin[a + b*x]*SinIntegral[a + b*x], x, 21, -(x/b^2) + (a*Cos[2*a + 2*b*x])/(4*b^3) - (x*Cos[2*a + 2*b*x])/(4*b^2) - (a*CosIntegral[2*a + 2*b*x])/b^3 + (a*Log[a + b*x])/b^3 + (Cos[a + b*x]*Sin[a + b*x])/b^3 + Sin[2*a + 2*b*x]/(8*b^3) + (2*Cos[a + b*x]*SinIntegral[a + b*x])/b^3 - (x^2*Cos[a + b*x]*SinIntegral[a + b*x])/b + (2*x*Sin[a + b*x]*SinIntegral[a + b*x])/b^2 - SinIntegral[2*a + 2*b*x]/b^3 + (a^2*SinIntegral[2*a + 2*b*x])/(2*b^3)} -{x^1*Sin[a + b*x]*SinIntegral[a + b*x], x, 11, -(Cos[2*a + 2*b*x]/(4*b^2)) + CosIntegral[2*a + 2*b*x]/(2*b^2) - Log[a + b*x]/(2*b^2) - (x*Cos[a + b*x]*SinIntegral[a + b*x])/b + (Sin[a + b*x]*SinIntegral[a + b*x])/b^2 - (a*SinIntegral[2*a + 2*b*x])/(2*b^2)} -{x^0*Sin[a + b*x]*SinIntegral[a + b*x], x, 4, -((Cos[a + b*x]*SinIntegral[a + b*x])/b) + SinIntegral[2*a + 2*b*x]/(2*b)} -{Sin[a + b*x]*SinIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Sin[a + b*x]*SinIntegral[a + b*x])/x, x]} - - -(* {x^3*Cos[a + b*x]*SinIntegral[a + b*x], x, 32, (4*x)/b^3 - (a^2*x)/(2*b^3) + (a*x^2)/(4*b^2) - x^3/(6*b) - (3*a*Cos[a + b*x]^2)/(2*b^4) + (3*a*CosIntegral[2*a + 2*b*x])/b^4 - (a^3*CosIntegral[2*a + 2*b*x])/(2*b^4) - (3*a*Log[a + b*x])/b^4 + (a^3*Log[a + b*x])/(2*b^4) - (4*Cos[a + b*x]*Sin[a + b*x])/b^4 + (a^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^4) - (a*x*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (a*Sin[a + b*x]^2)/(4*b^4) - (2*x*Sin[a + b*x]^2)/b^3 - (6*Cos[a + b*x]*SinIntegral[a + b*x])/b^4 + (3*x^2*Cos[a + b*x]*SinIntegral[a + b*x])/b^2 - (6*x*Sin[a + b*x]*SinIntegral[a + b*x])/b^3 + (x^3*Sin[a + b*x]*SinIntegral[a + b*x])/b + (3*SinIntegral[2*a + 2*b*x])/b^4 - (3*a^2*SinIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Cos[a + b*x]*SinIntegral[a + b*x], x, 21, (a*x)/(2*b^2) - x^2/(4*b) + Cos[2*a + 2*b*x]/(2*b^3) - CosIntegral[2*a + 2*b*x]/b^3 + (a^2*CosIntegral[2*a + 2*b*x])/(2*b^3) + Log[a + b*x]/b^3 - (a^2*Log[a + b*x])/(2*b^3) - (a*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) - Sin[a + b*x]^2/(4*b^3) + (2*x*Cos[a + b*x]*SinIntegral[a + b*x])/b^2 - (2*Sin[a + b*x]*SinIntegral[a + b*x])/b^3 + (x^2*Sin[a + b*x]*SinIntegral[a + b*x])/b + (a*SinIntegral[2*a + 2*b*x])/b^3} -{x^1*Cos[a + b*x]*SinIntegral[a + b*x], x, 12, -(x/(2*b)) - (a*CosIntegral[2*a + 2*b*x])/(2*b^2) + (a*Log[a + b*x])/(2*b^2) + (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (Cos[a + b*x]*SinIntegral[a + b*x])/b^2 + (x*Sin[a + b*x]*SinIntegral[a + b*x])/b - SinIntegral[2*a + 2*b*x]/(2*b^2)} -{x^0*Cos[a + b*x]*SinIntegral[a + b*x], x, 4, CosIntegral[2*a + 2*b*x]/(2*b) - Log[a + b*x]/(2*b) + (Sin[a + b*x]*SinIntegral[a + b*x])/b} -{Cos[a + b*x]*SinIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Cos[a + b*x]*SinIntegral[a + b*x])/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[a+b x] SinIntegral[c+d x]^n*) - - -(* {x^2*Sin[a + b*x]*SinIntegral[c + d*x], x, 46, -((c*Cos[a - c + (b - d)*x])/(2*b*(b - d)*d)) + (x*Cos[a - c + (b - d)*x])/(2*b*(b - d)) + (c*Cos[a + c + (b + d)*x])/(2*b*d*(b + d)) - (x*Cos[a + c + (b + d)*x])/(2*b*(b + d)) + (CosIntegral[((b - d)*(c + d*x))/d]*(2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d]))/(2*b^3*d^2) - (CosIntegral[((b + d)*(c + d*x))/d]*(2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d]))/(2*b^3*d^2) - Sin[a - c + (b - d)*x]/(2*b*(b - d)^2) - Sin[a - c + (b - d)*x]/(b^2*(b - d)) + Sin[a + c + (b + d)*x]/(2*b*(b + d)^2) + Sin[a + c + (b + d)*x]/(b^2*(b + d)) + (((2 - b^2*x^2)*Cos[a + b*x] + 2*b*x*Sin[a + b*x])*SinIntegral[c + d*x])/b^3 - (((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d])*SinIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) + (((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d])*SinIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Sin[a + b*x]*SinIntegral[c + d*x], x, 24, Cos[a - c + (b - d)*x]/(2*b*(b - d)) - Cos[a + c + (b + d)*x]/(2*b*(b + d)) - (Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b*d) - (c*CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (x*Cos[a + b*x]*SinIntegral[c + d*x])/b + (Sin[a + b*x]*SinIntegral[c + d*x])/b^2 - (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)} -{x^0*Sin[a + b*x]*SinIntegral[c + d*x], x, 9, -((CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b)) + (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cos[a + b*x]*SinIntegral[c + d*x])/b + (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Sin[a + b*x]*SinIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Sin[a + b*x]*SinIntegral[c + d*x])/x, x]} - - -(* {x^2*Cos[a + b*x]*SinIntegral[c + d*x], x, 46, -(Cos[a - c + (b - d)*x]/(2*b*(b - d)^2)) - Cos[a - c + (b - d)*x]/(b^2*(b - d)) + Cos[a + c + (b + d)*x]/(2*b*(b + d)^2) + Cos[a + c + (b + d)*x]/(b^2*(b + d)) - (CosIntegral[((b - d)*(c + d*x))/d]*((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (CosIntegral[((b + d)*(c + d*x))/d]*((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (c*Sin[a - c + (b - d)*x])/(2*b*(b - d)*d) - (x*Sin[a - c + (b - d)*x])/(2*b*(b - d)) - (c*Sin[a + c + (b + d)*x])/(2*b*d*(b + d)) + (x*Sin[a + c + (b + d)*x])/(2*b*(b + d)) + ((2*b*x*Cos[a + b*x] - (2 - b^2*x^2)*Sin[a + b*x])*SinIntegral[c + d*x])/b^3 - ((2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d])*SinIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) + ((2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d])*SinIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Cos[a + b*x]*SinIntegral[c + d*x], x, 24, (c*Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (c*Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) + (CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b^2) - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b^2) - Sin[a - c + (b - d)*x]/(2*b*(b - d)) + Sin[a + c + (b + d)*x]/(2*b*(b + d)) + (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Cos[a + b*x]*SinIntegral[c + d*x])/b^2 + (x*Sin[a + b*x]*SinIntegral[c + d*x])/b - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)} -{x^0*Cos[a + b*x]*SinIntegral[c + d*x], x, 9, -((Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b)) + (Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b) + (Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) + (Sin[a + b*x]*SinIntegral[c + d*x])/b - (Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Cos[a + b*x]*SinIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Cos[a + b*x]*SinIntegral[c + d*x])/x, x]} - - -(* ::Section::Closed:: *) -(*Cosine integral function*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m CosIntegral[b x]^n*) - - -{x^m*CosIntegral[b*x], x, 5, (x^(1 + m)*CosIntegral[b*x])/(1 + m) + (I*x^m*Gamma[1 + m, (-I)*b*x])/(((-I)*b*x)^m*(2*b*(1 + m))) - (I*x^m*Gamma[1 + m, I*b*x])/((I*b*x)^m*(2*b*(1 + m)))} - -{x^3*CosIntegral[b*x], x, 6, (3*Cos[b*x])/(2*b^4) - (3*x^2*Cos[b*x])/(4*b^2) + (1/4)*x^4*CosIntegral[b*x] + (3*x*Sin[b*x])/(2*b^3) - (x^3*Sin[b*x])/(4*b)} -{x^2*CosIntegral[b*x], x, 5, -((2*x*Cos[b*x])/(3*b^2)) + (1/3)*x^3*CosIntegral[b*x] + (2*Sin[b*x])/(3*b^3) - (x^2*Sin[b*x])/(3*b)} -{x^1*CosIntegral[b*x], x, 4, -(Cos[b*x]/(2*b^2)) + (1/2)*x^2*CosIntegral[b*x] - (x*Sin[b*x])/(2*b)} -{x^0*CosIntegral[b*x], x, 1, x*CosIntegral[b*x] - Sin[b*x]/b} -{CosIntegral[b*x]/x^1, x, 1, (-(1/2))*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-I)*b*x] + (1/2)*I*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, I*b*x] + EulerGamma*Log[x] + (1/2)*Log[b*x]^2} -{CosIntegral[b*x]/x^2, x, 4, -(Cos[b*x]/x) - CosIntegral[b*x]/x - b*SinIntegral[b*x]} -{CosIntegral[b*x]/x^3, x, 5, -(Cos[b*x]/(4*x^2)) - (1/4)*b^2*CosIntegral[b*x] - CosIntegral[b*x]/(2*x^2) + (b*Sin[b*x])/(4*x)} - - -{x^m*CosIntegral[b*x]^2, x, 0, CannotIntegrate[x^m*CosIntegral[b*x]^2, x]} - -{x^3*CosIntegral[b*x]^2, x, 19, x^2/(4*b^2) + (3*Cos[b*x]^2)/(8*b^4) + (3*Cos[b*x]*CosIntegral[b*x])/b^4 - (3*x^2*Cos[b*x]*CosIntegral[b*x])/(2*b^2) + (1/4)*x^4*CosIntegral[b*x]^2 - (3*CosIntegral[2*b*x])/(2*b^4) - (3*Log[x])/(2*b^4) + (x*Cos[b*x]*Sin[b*x])/b^3 + (3*x*CosIntegral[b*x]*Sin[b*x])/b^3 - (x^3*CosIntegral[b*x]*Sin[b*x])/(2*b) - (13*Sin[b*x]^2)/(8*b^4) + (x^2*Sin[b*x]^2)/(4*b^2)} -{x^2*CosIntegral[b*x]^2, x, 15, x/(2*b^2) - (4*x*Cos[b*x]*CosIntegral[b*x])/(3*b^2) + (1/3)*x^3*CosIntegral[b*x]^2 + (5*Cos[b*x]*Sin[b*x])/(6*b^3) + (4*CosIntegral[b*x]*Sin[b*x])/(3*b^3) - (2*x^2*CosIntegral[b*x]*Sin[b*x])/(3*b) + (x*Sin[b*x]^2)/(3*b^2) - (2*SinIntegral[2*b*x])/(3*b^3)} -{x^1*CosIntegral[b*x]^2, x, 10, -((Cos[b*x]*CosIntegral[b*x])/b^2) + (1/2)*x^2*CosIntegral[b*x]^2 + CosIntegral[2*b*x]/(2*b^2) + Log[x]/(2*b^2) - (x*CosIntegral[b*x]*Sin[b*x])/b + Sin[b*x]^2/(2*b^2)} -{x^0*CosIntegral[b*x]^2, x, 6, x*CosIntegral[b*x]^2 - (2*CosIntegral[b*x]*Sin[b*x])/b + SinIntegral[2*b*x]/b} -{CosIntegral[b*x]^2/x^1, x, 0, CannotIntegrate[CosIntegral[b*x]^2/x, x]} -{CosIntegral[b*x]^2/x^2, x, 0, CannotIntegrate[CosIntegral[b*x]^2/x^2, x]} -{CosIntegral[b*x]^2/x^3, x, 0, CannotIntegrate[CosIntegral[b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m CosIntegral[a+b x]^n*) - - -{x^m*CosIntegral[a + b*x], x, 1, (x^(1 + m)*CosIntegral[a + b*x])/(1 + m) - (b*CannotIntegrate[(x^(1 + m)*Cos[a + b*x])/(a + b*x), x])/(1 + m)} - -{x^3*CosIntegral[a + b*x], x, 14, (3*Cos[a + b*x])/(2*b^4) - (a^2*Cos[a + b*x])/(4*b^4) + (a*x*Cos[a + b*x])/(2*b^3) - (3*x^2*Cos[a + b*x])/(4*b^2) - (a^4*CosIntegral[a + b*x])/(4*b^4) + (1/4)*x^4*CosIntegral[a + b*x] - (a*Sin[a + b*x])/(2*b^4) + (a^3*Sin[a + b*x])/(4*b^4) + (3*x*Sin[a + b*x])/(2*b^3) - (a^2*x*Sin[a + b*x])/(4*b^3) + (a*x^2*Sin[a + b*x])/(4*b^2) - (x^3*Sin[a + b*x])/(4*b)} -{x^2*CosIntegral[a + b*x], x, 10, (a*Cos[a + b*x])/(3*b^3) - (2*x*Cos[a + b*x])/(3*b^2) + (a^3*CosIntegral[a + b*x])/(3*b^3) + (1/3)*x^3*CosIntegral[a + b*x] + (2*Sin[a + b*x])/(3*b^3) - (a^2*Sin[a + b*x])/(3*b^3) + (a*x*Sin[a + b*x])/(3*b^2) - (x^2*Sin[a + b*x])/(3*b)} -{x^1*CosIntegral[a + b*x], x, 7, -(Cos[a + b*x]/(2*b^2)) - (a^2*CosIntegral[a + b*x])/(2*b^2) + (1/2)*x^2*CosIntegral[a + b*x] + (a*Sin[a + b*x])/(2*b^2) - (x*Sin[a + b*x])/(2*b)} -{x^0*CosIntegral[a + b*x], x, 1, ((a + b*x)*CosIntegral[a + b*x])/b - Sin[a + b*x]/b} -{CosIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[CosIntegral[a + b*x]/x, x]} -{CosIntegral[a + b*x]/x^2, x, 7, (b*Cos[a]*CosIntegral[b*x])/a - (b*CosIntegral[a + b*x])/a - CosIntegral[a + b*x]/x - (b*Sin[a]*SinIntegral[b*x])/a} -{CosIntegral[a + b*x]/x^3, x, 11, -((b*Cos[a + b*x])/(2*a*x)) - (b^2*Cos[a]*CosIntegral[b*x])/(2*a^2) + (b^2*CosIntegral[a + b*x])/(2*a^2) - CosIntegral[a + b*x]/(2*x^2) - (b^2*CosIntegral[b*x]*Sin[a])/(2*a) - (b^2*Cos[a]*SinIntegral[b*x])/(2*a) + (b^2*Sin[a]*SinIntegral[b*x])/(2*a^2)} - - -{x^m*CosIntegral[a + b*x]^2, x, 0, CannotIntegrate[x^m*CosIntegral[a + b*x]^2, x]} - -{x^2*CosIntegral[a + b*x]^2, x, 39, (2*x)/(3*b^2) + (a*Cos[2*a + 2*b*x])/(3*b^3) - (x*Cos[2*a + 2*b*x])/(6*b^2) + (2*a*Cos[a + b*x]*CosIntegral[a + b*x])/(3*b^3) - (4*x*Cos[a + b*x]*CosIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*CosIntegral[a + b*x]^2)/(3*b^3) - (a*x*(a + b*x)*CosIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*CosIntegral[a + b*x]^2)/(3*b) - (a*CosIntegral[2*a + 2*b*x])/b^3 - (a*Log[a + b*x])/b^3 + (2*Cos[a + b*x]*Sin[a + b*x])/(3*b^3) + (4*CosIntegral[a + b*x]*Sin[a + b*x])/(3*b^3) - (2*a^2*CosIntegral[a + b*x]*Sin[a + b*x])/(3*b^3) + (2*a*x*CosIntegral[a + b*x]*Sin[a + b*x])/(3*b^2) - (2*x^2*CosIntegral[a + b*x]*Sin[a + b*x])/(3*b) + Sin[2*a + 2*b*x]/(12*b^3) - (2*SinIntegral[2*a + 2*b*x])/(3*b^3) + (a^2*SinIntegral[2*a + 2*b*x])/b^3} -{x^1*CosIntegral[a + b*x]^2, x, 17, -(Cos[2*a + 2*b*x]/(4*b^2)) - (Cos[a + b*x]*CosIntegral[a + b*x])/b^2 - (a*(a + b*x)*CosIntegral[a + b*x]^2)/(2*b^2) + (x*(a + b*x)*CosIntegral[a + b*x]^2)/(2*b) + CosIntegral[2*a + 2*b*x]/(2*b^2) + Log[a + b*x]/(2*b^2) + (a*CosIntegral[a + b*x]*Sin[a + b*x])/b^2 - (x*CosIntegral[a + b*x]*Sin[a + b*x])/b - (a*SinIntegral[2*a + 2*b*x])/b^2} -{x^0*CosIntegral[a + b*x]^2, x, 5, ((a + b*x)*CosIntegral[a + b*x]^2)/b - (2*CosIntegral[a + b*x]*Sin[a + b*x])/b + SinIntegral[2*a + 2*b*x]/b} -{CosIntegral[a + b*x]^2/x^1, x, 0, CannotIntegrate[CosIntegral[a + b*x]^2/x, x]} -{CosIntegral[a + b*x]^2/x^2, x, 0, CannotIntegrate[CosIntegral[a + b*x]^2/x^2, x]} -{CosIntegral[a + b*x]^2/x^3, x, 0, CannotIntegrate[CosIntegral[a + b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m CosIntegral[d (a+b Log[c x^n])]*) - - -{x^2*CosIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^3*CosIntegral[d*(a + b*Log[c*x^n])])/3 - (x^3*ExpIntegralEi[((3 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) - (x^3*ExpIntegralEi[((3 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n))} -{x^1*CosIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^2*CosIntegral[d*(a + b*Log[c*x^n])])/2 - (x^2*ExpIntegralEi[((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n)) - (x^2*ExpIntegralEi[((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n))} -{x^0*CosIntegral[d*(a + b*Log[c*x^n])], x, 7, x*CosIntegral[d*(a + b*Log[c*x^n])] - (x*ExpIntegralEi[((1 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) - (x*ExpIntegralEi[((1 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1))} -{CosIntegral[d*(a + b*Log[c*x^n])]/x^1, x, 3, (CosIntegral[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sin[d*(a + b*Log[c*x^n])]/(b*d*n)} -{CosIntegral[d*(a + b*Log[c*x^n])]/x^2, x, 7, -(CosIntegral[d*(a + b*Log[c*x^n])]/x) + (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x) + (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x)} -{CosIntegral[d*(a + b*Log[c*x^n])]/x^3, x, 7, -CosIntegral[d*(a + b*Log[c*x^n])]/(2*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2)} - - -{(e*x)^m*CosIntegral[d*(a + b*Log[c*x^n])], x, 7, ((e*x)^(1 + m)*CosIntegral[d*(a + b*Log[c*x^n])])/(e*(1 + m)) - (x*(e*x)^m*ExpIntegralEi[((1 + m - I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n)) - (x*(e*x)^m*ExpIntegralEi[((1 + m + I*b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[b x] CosIntegral[b x]^n*) - - -{Sin[b*x]*CosIntegral[b*x]/x^3, x, 12, -((b*Cos[b*x]^2)/(2*x)) - (b*Cos[2*b*x])/(4*x) - (b*Cos[b*x]*CosIntegral[b*x])/(2*x) - (1/2)*b^2*CannotIntegrate[(CosIntegral[b*x]*Sin[b*x])/x, x] - (CosIntegral[b*x]*Sin[b*x])/(2*x^2) - Sin[2*b*x]/(8*x^2) - b^2*SinIntegral[2*b*x]} -{Sin[b*x]*CosIntegral[b*x]/x^2, x, 7, (1/2)*b*CosIntegral[b*x]^2 + b*CosIntegral[2*b*x] - (CosIntegral[b*x]*Sin[b*x])/x - Sin[2*b*x]/(2*x)} -{Sin[b*x]*CosIntegral[b*x]/x, x, 0, CannotIntegrate[(CosIntegral[b*x]*Sin[b*x])/x, x]} -{Sin[b*x]*CosIntegral[b*x], x, 5, -((Cos[b*x]*CosIntegral[b*x])/b) + CosIntegral[2*b*x]/(2*b) + Log[x]/(2*b)} -{x*Sin[b*x]*CosIntegral[b*x], x, 9, x/(2*b) - (x*Cos[b*x]*CosIntegral[b*x])/b + (Cos[b*x]*Sin[b*x])/(2*b^2) + (CosIntegral[b*x]*Sin[b*x])/b^2 - SinIntegral[2*b*x]/(2*b^2)} -{x^2*Sin[b*x]*CosIntegral[b*x], x, 13, x^2/(4*b) + Cos[b*x]^2/(4*b^3) + (2*Cos[b*x]*CosIntegral[b*x])/b^3 - (x^2*Cos[b*x]*CosIntegral[b*x])/b - CosIntegral[2*b*x]/b^3 - Log[x]/b^3 + (x*Cos[b*x]*Sin[b*x])/(2*b^2) + (2*x*CosIntegral[b*x]*Sin[b*x])/b^2 - Sin[b*x]^2/b^3} -{x^3*Sin[b*x]*CosIntegral[b*x], x, 20, -((5*x)/(2*b^3)) + x^3/(6*b) + (x*Cos[b*x]^2)/(2*b^3) + (6*x*Cos[b*x]*CosIntegral[b*x])/b^3 - (x^3*Cos[b*x]*CosIntegral[b*x])/b - (4*Cos[b*x]*Sin[b*x])/b^4 + (x^2*Cos[b*x]*Sin[b*x])/(2*b^2) - (6*CosIntegral[b*x]*Sin[b*x])/b^4 + (3*x^2*CosIntegral[b*x]*Sin[b*x])/b^2 - (3*x*Sin[b*x]^2)/(2*b^3) + (3*SinIntegral[2*b*x])/b^4} - - -{Cos[b*x]*CosIntegral[b*x]/x^3, x, 14, -(Cos[b*x]^2/(4*x^2)) - (Cos[b*x]*CosIntegral[b*x])/(2*x^2) - (1/4)*b^2*CosIntegral[b*x]^2 - b^2*CosIntegral[2*b*x] + (b*Cos[b*x]*Sin[b*x])/(2*x) + (b*CosIntegral[b*x]*Sin[b*x])/(2*x) + (b*Sin[2*b*x])/(4*x)} -{Cos[b*x]*CosIntegral[b*x]/x^2, x, 5, -(Cos[b*x]^2/x) - (Cos[b*x]*CosIntegral[b*x])/x - b*CannotIntegrate[(CosIntegral[b*x]*Sin[b*x])/x, x] - b*SinIntegral[2*b*x]} -{Cos[b*x]*CosIntegral[b*x]/x, x, 1, (1/2)*CosIntegral[b*x]^2} -{Cos[b*x]*CosIntegral[b*x], x, 5, (CosIntegral[b*x]*Sin[b*x])/b - SinIntegral[2*b*x]/(2*b)} -{x*Cos[b*x]*CosIntegral[b*x], x, 9, (Cos[b*x]*CosIntegral[b*x])/b^2 - CosIntegral[2*b*x]/(2*b^2) - Log[x]/(2*b^2) + (x*CosIntegral[b*x]*Sin[b*x])/b - Sin[b*x]^2/(2*b^2)} -{x^2*Cos[b*x]*CosIntegral[b*x], x, 14, -((3*x)/(4*b^2)) + (2*x*Cos[b*x]*CosIntegral[b*x])/b^2 - (5*Cos[b*x]*Sin[b*x])/(4*b^3) - (2*CosIntegral[b*x]*Sin[b*x])/b^3 + (x^2*CosIntegral[b*x]*Sin[b*x])/b - (x*Sin[b*x]^2)/(2*b^2) + SinIntegral[2*b*x]/b^3} -{x^3*Cos[b*x]*CosIntegral[b*x], x, 18, -(x^2/(2*b^2)) - (3*Cos[b*x]^2)/(4*b^4) - (6*Cos[b*x]*CosIntegral[b*x])/b^4 + (3*x^2*Cos[b*x]*CosIntegral[b*x])/b^2 + (3*CosIntegral[2*b*x])/b^4 + (3*Log[x])/b^4 - (2*x*Cos[b*x]*Sin[b*x])/b^3 - (6*x*CosIntegral[b*x]*Sin[b*x])/b^3 + (x^3*CosIntegral[b*x]*Sin[b*x])/b + (13*Sin[b*x]^2)/(4*b^4) - (x^2*Sin[b*x]^2)/(2*b^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[b x] CosIntegral[d x]^n*) - - -{Sin[5*x]*CosIntegral[2*x], x, 6, (-(1/5))*Cos[5*x]*CosIntegral[2*x] + (1/10)*CosIntegral[3*x] + (1/10)*CosIntegral[7*x]} - - -{Cos[5*x]*CosIntegral[2*x], x, 6, (1/5)*CosIntegral[2*x]*Sin[5*x] - (1/10)*SinIntegral[3*x] - (1/10)*SinIntegral[7*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[a+b x] CosIntegral[a+b x]^n*) - - -(* {x^3*Sin[a + b*x]*CosIntegral[a + b*x], x, 32, -((5*x)/(2*b^3)) + (a^2*x)/(2*b^3) - (a*x^2)/(4*b^2) + x^3/(6*b) - (7*a*Cos[a + b*x]^2)/(4*b^4) + (x*Cos[a + b*x]^2)/(2*b^3) + (6*x*Cos[a + b*x]*CosIntegral[a + b*x])/b^3 - (x^3*Cos[a + b*x]*CosIntegral[a + b*x])/b + (3*a*CosIntegral[2*a + 2*b*x])/b^4 - (a^3*CosIntegral[2*a + 2*b*x])/(2*b^4) + (3*a*Log[a + b*x])/b^4 - (a^3*Log[a + b*x])/(2*b^4) - (4*Cos[a + b*x]*Sin[a + b*x])/b^4 + (a^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^4) - (a*x*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x^2*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) - (6*CosIntegral[a + b*x]*Sin[a + b*x])/b^4 + (3*x^2*CosIntegral[a + b*x]*Sin[a + b*x])/b^2 - (3*x*Sin[a + b*x]^2)/(2*b^3) + (3*SinIntegral[2*a + 2*b*x])/b^4 - (3*a^2*SinIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Sin[a + b*x]*CosIntegral[a + b*x], x, 21, -((a*x)/(2*b^2)) + x^2/(4*b) + Cos[a + b*x]^2/(4*b^3) + Cos[2*a + 2*b*x]/(2*b^3) + (2*Cos[a + b*x]*CosIntegral[a + b*x])/b^3 - (x^2*Cos[a + b*x]*CosIntegral[a + b*x])/b - CosIntegral[2*a + 2*b*x]/b^3 + (a^2*CosIntegral[2*a + 2*b*x])/(2*b^3) - Log[a + b*x]/b^3 + (a^2*Log[a + b*x])/(2*b^3) - (a*Cos[a + b*x]*Sin[a + b*x])/(2*b^3) + (x*Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (2*x*CosIntegral[a + b*x]*Sin[a + b*x])/b^2 + (a*SinIntegral[2*a + 2*b*x])/b^3} -{x^1*Sin[a + b*x]*CosIntegral[a + b*x], x, 12, x/(2*b) - (x*Cos[a + b*x]*CosIntegral[a + b*x])/b - (a*CosIntegral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^2) + (Cos[a + b*x]*Sin[a + b*x])/(2*b^2) + (CosIntegral[a + b*x]*Sin[a + b*x])/b^2 - SinIntegral[2*a + 2*b*x]/(2*b^2)} -{x^0*Sin[a + b*x]*CosIntegral[a + b*x], x, 4, -((Cos[a + b*x]*CosIntegral[a + b*x])/b) + CosIntegral[2*a + 2*b*x]/(2*b) + Log[a + b*x]/(2*b)} -{Sin[a + b*x]*CosIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Sin[a + b*x]*CosIntegral[a + b*x])/x, x]} - - -(* {x^3*Cos[a + b*x]*CosIntegral[a + b*x], x, 32, (5*a*x)/(4*b^3) - x^2/(2*b^2) - (15*Cos[a + b*x]^2)/(4*b^4) + (a^2*Cos[a + b*x]^2)/(2*b^4) - (6*Cos[a + b*x]*CosIntegral[a + b*x])/b^4 + (3*x^2*Cos[a + b*x]*CosIntegral[a + b*x])/b^2 + (3*CosIntegral[2*a + 2*b*x])/b^4 - (3*a^2*CosIntegral[2*a + 2*b*x])/(2*b^4) + (3*Log[a + b*x])/b^4 - (3*a^2*Log[a + b*x])/(2*b^4) + (7*a*Cos[a + b*x]*Sin[a + b*x])/(4*b^4) - (2*x*Cos[a + b*x]*Sin[a + b*x])/b^3 - (6*x*CosIntegral[a + b*x]*Sin[a + b*x])/b^3 + (x^3*CosIntegral[a + b*x]*Sin[a + b*x])/b + Sin[a + b*x]^2/(4*b^4) + (a*x*Sin[a + b*x]^2)/(2*b^3) - (x^2*Sin[a + b*x]^2)/(2*b^2) - (3*a*SinIntegral[2*a + 2*b*x])/b^4 + (a^3*SinIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Cos[a + b*x]*CosIntegral[a + b*x], x, 21, -(x/b^2) - (a*Cos[2*a + 2*b*x])/(4*b^3) + (x*Cos[2*a + 2*b*x])/(4*b^2) + (2*x*Cos[a + b*x]*CosIntegral[a + b*x])/b^2 + (a*CosIntegral[2*a + 2*b*x])/b^3 + (a*Log[a + b*x])/b^3 - (Cos[a + b*x]*Sin[a + b*x])/b^3 - (2*CosIntegral[a + b*x]*Sin[a + b*x])/b^3 + (x^2*CosIntegral[a + b*x]*Sin[a + b*x])/b - Sin[2*a + 2*b*x]/(8*b^3) + SinIntegral[2*a + 2*b*x]/b^3 - (a^2*SinIntegral[2*a + 2*b*x])/(2*b^3)} -{x^1*Cos[a + b*x]*CosIntegral[a + b*x], x, 11, Cos[2*a + 2*b*x]/(4*b^2) + (Cos[a + b*x]*CosIntegral[a + b*x])/b^2 - CosIntegral[2*a + 2*b*x]/(2*b^2) - Log[a + b*x]/(2*b^2) + (x*CosIntegral[a + b*x]*Sin[a + b*x])/b + (a*SinIntegral[2*a + 2*b*x])/(2*b^2)} -{x^0*Cos[a + b*x]*CosIntegral[a + b*x], x, 4, (CosIntegral[a + b*x]*Sin[a + b*x])/b - SinIntegral[2*a + 2*b*x]/(2*b)} -{Cos[a + b*x]*CosIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Cos[a + b*x]*CosIntegral[a + b*x])/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Trig[a+b x] CosIntegral[c+d x]^n*) - - -(* {x^2*Sin[a + b*x]*CosIntegral[c + d*x], x, 46, Cos[a - c + (b - d)*x]/(2*b*(b - d)^2) + Cos[a - c + (b - d)*x]/(b^2*(b - d)) + Cos[a + c + (b + d)*x]/(2*b*(b + d)^2) + Cos[a + c + (b + d)*x]/(b^2*(b + d)) + (CosIntegral[((b - d)*(c + d*x))/d]*((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (CosIntegral[((b + d)*(c + d*x))/d]*((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (CosIntegral[c + d*x]*((2 - b^2*x^2)*Cos[a + b*x] + 2*b*x*Sin[a + b*x]))/b^3 - (c*Sin[a - c + (b - d)*x])/(2*b*(b - d)*d) + (x*Sin[a - c + (b - d)*x])/(2*b*(b - d)) - (c*Sin[a + c + (b + d)*x])/(2*b*d*(b + d)) + (x*Sin[a + c + (b + d)*x])/(2*b*(b + d)) + ((2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d])*SinIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) + ((2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d])*SinIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Sin[a + b*x]*CosIntegral[c + d*x], x, 24, -((c*Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d)) - (x*Cos[a + b*x]*CosIntegral[c + d*x])/b - (c*Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b^2) - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b^2) + (CosIntegral[c + d*x]*Sin[a + b*x])/b^2 + Sin[a - c + (b - d)*x]/(2*b*(b - d)) + Sin[a + c + (b + d)*x]/(2*b*(b + d)) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)} -{x^0*Sin[a + b*x]*CosIntegral[c + d*x], x, 9, (Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cos[a + b*x]*CosIntegral[c + d*x])/b + (Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b) - (Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Sin[a + b*x]*CosIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(CosIntegral[c + d*x]*Sin[a + b*x])/x, x]} - - -(* {x^2*Cos[a + b*x]*CosIntegral[c + d*x], x, 46, -((c*Cos[a - c + (b - d)*x])/(2*b*(b - d)*d)) + (x*Cos[a - c + (b - d)*x])/(2*b*(b - d)) - (c*Cos[a + c + (b + d)*x])/(2*b*d*(b + d)) + (x*Cos[a + c + (b + d)*x])/(2*b*(b + d)) + (CosIntegral[((b - d)*(c + d*x))/d]*(2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (CosIntegral[((b + d)*(c + d*x))/d]*(2*b*c*d*Cos[a - (b*c)/d] - (b^2*c^2 - 2*d^2)*Sin[a - (b*c)/d]))/(2*b^3*d^2) + (CosIntegral[c + d*x]*(2*b*x*Cos[a + b*x] - (2 - b^2*x^2)*Sin[a + b*x]))/b^3 - Sin[a - c + (b - d)*x]/(2*b*(b - d)^2) - Sin[a - c + (b - d)*x]/(b^2*(b - d)) - Sin[a + c + (b + d)*x]/(2*b*(b + d)^2) - Sin[a + c + (b + d)*x]/(b^2*(b + d)) - (((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d])*SinIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) - (((b^2*c^2 - 2*d^2)*Cos[a - (b*c)/d] + 2*b*c*d*Sin[a - (b*c)/d])*SinIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Cos[a + b*x]*CosIntegral[c + d*x], x, 24, Cos[a - c + (b - d)*x]/(2*b*(b - d)) + Cos[a + c + (b + d)*x]/(2*b*(b + d)) - (Cos[a - (b*c)/d]*CosIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (Cos[a + b*x]*CosIntegral[c + d*x])/b^2 - (Cos[a - (b*c)/d]*CosIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (c*CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b*d) + (x*CosIntegral[c + d*x]*Sin[a + b*x])/b + (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Sin[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) + (Sin[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)} -{x^0*Cos[a + b*x]*CosIntegral[c + d*x], x, 9, -((CosIntegral[(c*(b - d))/d + (b - d)*x]*Sin[a - (b*c)/d])/(2*b)) - (CosIntegral[(c*(b + d))/d + (b + d)*x]*Sin[a - (b*c)/d])/(2*b) + (CosIntegral[c + d*x]*Sin[a + b*x])/b - (Cos[a - (b*c)/d]*SinIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cos[a - (b*c)/d]*SinIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Cos[a + b*x]*CosIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Cos[a + b*x]*CosIntegral[c + d*x])/x, x]} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.5 Hyperbolic integral functions.m b/test/methods/rule_based/test_files/8 Special functions/8.5 Hyperbolic integral functions.m deleted file mode 100644 index 6a99ab5..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.5 Hyperbolic integral functions.m +++ /dev/null @@ -1,276 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Hyperbolic Integral Functions*) - - -(* ::Section::Closed:: *) -(*Hyperbolic sine integral function*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m SinhIntegral[b x]^n*) - - -{x^m*SinhIntegral[b*x], x, 5, -((x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b*(1 + m)))) - (x^m*Gamma[1 + m, b*x])/((b*x)^m*(2*b*(1 + m))) + (x^(1 + m)*SinhIntegral[b*x])/(1 + m)} - -{x^3*SinhIntegral[b*x], x, 6, -((3*x*Cosh[b*x])/(2*b^3)) - (x^3*Cosh[b*x])/(4*b) + (3*Sinh[b*x])/(2*b^4) + (3*x^2*Sinh[b*x])/(4*b^2) + (1/4)*x^4*SinhIntegral[b*x]} -{x^2*SinhIntegral[b*x], x, 5, -((2*Cosh[b*x])/(3*b^3)) - (x^2*Cosh[b*x])/(3*b) + (2*x*Sinh[b*x])/(3*b^2) + (1/3)*x^3*SinhIntegral[b*x]} -{x^1*SinhIntegral[b*x], x, 4, -((x*Cosh[b*x])/(2*b)) + Sinh[b*x]/(2*b^2) + (1/2)*x^2*SinhIntegral[b*x]} -{x^0*SinhIntegral[b*x], x, 1, -(Cosh[b*x]/b) + x*SinhIntegral[b*x]} -{SinhIntegral[b*x]/x^1, x, 1, (1/2)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x] + (1/2)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x]} -{SinhIntegral[b*x]/x^2, x, 4, b*CoshIntegral[b*x] - Sinh[b*x]/x - SinhIntegral[b*x]/x} -{SinhIntegral[b*x]/x^3, x, 5, -((b*Cosh[b*x])/(4*x)) - Sinh[b*x]/(4*x^2) + (1/4)*b^2*SinhIntegral[b*x] - SinhIntegral[b*x]/(2*x^2)} - - -{x^m*SinhIntegral[b*x]^2, x, 0, CannotIntegrate[x^m*SinhIntegral[b*x]^2, x]} - -{x^3*SinhIntegral[b*x]^2, x, 19, x^2/(2*b^2) - (3*CoshIntegral[2*b*x])/(2*b^4) + (3*Log[x])/(2*b^4) - (x*Cosh[b*x]*Sinh[b*x])/b^3 + (2*Sinh[b*x]^2)/b^4 + (x^2*Sinh[b*x]^2)/(4*b^2) - (3*x*Cosh[b*x]*SinhIntegral[b*x])/b^3 - (x^3*Cosh[b*x]*SinhIntegral[b*x])/(2*b) + (3*Sinh[b*x]*SinhIntegral[b*x])/b^4 + (3*x^2*Sinh[b*x]*SinhIntegral[b*x])/(2*b^2) + (1/4)*x^4*SinhIntegral[b*x]^2} -{x^2*SinhIntegral[b*x]^2, x, 15, (5*x)/(6*b^2) - (5*Cosh[b*x]*Sinh[b*x])/(6*b^3) + (x*Sinh[b*x]^2)/(3*b^2) - (4*Cosh[b*x]*SinhIntegral[b*x])/(3*b^3) - (2*x^2*Cosh[b*x]*SinhIntegral[b*x])/(3*b) + (4*x*Sinh[b*x]*SinhIntegral[b*x])/(3*b^2) + (1/3)*x^3*SinhIntegral[b*x]^2 + (2*SinhIntegral[2*b*x])/(3*b^3)} -{x^1*SinhIntegral[b*x]^2, x, 10, -(CoshIntegral[2*b*x]/(2*b^2)) + Log[x]/(2*b^2) + Sinh[b*x]^2/(2*b^2) - (x*Cosh[b*x]*SinhIntegral[b*x])/b + (Sinh[b*x]*SinhIntegral[b*x])/b^2 + (1/2)*x^2*SinhIntegral[b*x]^2} -{x^0*SinhIntegral[b*x]^2, x, 6, -((2*Cosh[b*x]*SinhIntegral[b*x])/b) + x*SinhIntegral[b*x]^2 + SinhIntegral[2*b*x]/b} -{SinhIntegral[b*x]^2/x^1, x, 0, CannotIntegrate[SinhIntegral[b*x]^2/x, x]} -{SinhIntegral[b*x]^2/x^2, x, 0, CannotIntegrate[SinhIntegral[b*x]^2/x^2, x]} -{SinhIntegral[b*x]^2/x^3, x, 0, CannotIntegrate[SinhIntegral[b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m SinhIntegral[a+b x]^n*) - - -{x^m*SinhIntegral[a + b*x], x, 1, -((b*CannotIntegrate[(x^(1 + m)*Sinh[a + b*x])/(a + b*x), x])/(1 + m)) + (x^(1 + m)*SinhIntegral[a + b*x])/(1 + m)} - -{x^3*SinhIntegral[a + b*x], x, 14, (a*Cosh[a + b*x])/(2*b^4) + (a^3*Cosh[a + b*x])/(4*b^4) - (3*x*Cosh[a + b*x])/(2*b^3) - (a^2*x*Cosh[a + b*x])/(4*b^3) + (a*x^2*Cosh[a + b*x])/(4*b^2) - (x^3*Cosh[a + b*x])/(4*b) + (3*Sinh[a + b*x])/(2*b^4) + (a^2*Sinh[a + b*x])/(4*b^4) - (a*x*Sinh[a + b*x])/(2*b^3) + (3*x^2*Sinh[a + b*x])/(4*b^2) - (a^4*SinhIntegral[a + b*x])/(4*b^4) + (1/4)*x^4*SinhIntegral[a + b*x]} -{x^2*SinhIntegral[a + b*x], x, 10, -((2*Cosh[a + b*x])/(3*b^3)) - (a^2*Cosh[a + b*x])/(3*b^3) + (a*x*Cosh[a + b*x])/(3*b^2) - (x^2*Cosh[a + b*x])/(3*b) - (a*Sinh[a + b*x])/(3*b^3) + (2*x*Sinh[a + b*x])/(3*b^2) + (a^3*SinhIntegral[a + b*x])/(3*b^3) + (1/3)*x^3*SinhIntegral[a + b*x]} -{x^1*SinhIntegral[a + b*x], x, 7, (a*Cosh[a + b*x])/(2*b^2) - (x*Cosh[a + b*x])/(2*b) + Sinh[a + b*x]/(2*b^2) - (a^2*SinhIntegral[a + b*x])/(2*b^2) + (1/2)*x^2*SinhIntegral[a + b*x]} -{x^0*SinhIntegral[a + b*x], x, 1, -(Cosh[a + b*x]/b) + ((a + b*x)*SinhIntegral[a + b*x])/b} -{SinhIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[SinhIntegral[a + b*x]/x, x]} -{SinhIntegral[a + b*x]/x^2, x, 7, (b*CoshIntegral[b*x]*Sinh[a])/a + (b*Cosh[a]*SinhIntegral[b*x])/a - (b*SinhIntegral[a + b*x])/a - SinhIntegral[a + b*x]/x} -{SinhIntegral[a + b*x]/x^3, x, 11, (b^2*Cosh[a]*CoshIntegral[b*x])/(2*a) - (b^2*CoshIntegral[b*x]*Sinh[a])/(2*a^2) - (b*Sinh[a + b*x])/(2*a*x) - (b^2*Cosh[a]*SinhIntegral[b*x])/(2*a^2) + (b^2*Sinh[a]*SinhIntegral[b*x])/(2*a) + (b^2*SinhIntegral[a + b*x])/(2*a^2) - SinhIntegral[a + b*x]/(2*x^2)} - - -{x^m*SinhIntegral[a + b*x]^2, x, 0, CannotIntegrate[x^m*SinhIntegral[a + b*x]^2, x]} - -{x^2*SinhIntegral[a + b*x]^2, x, 39, (2*x)/(3*b^2) - (a*Cosh[2*a + 2*b*x])/(3*b^3) + (x*Cosh[2*a + 2*b*x])/(6*b^2) + (a*CoshIntegral[2*a + 2*b*x])/b^3 - (a*Log[a + b*x])/b^3 - (2*Cosh[a + b*x]*Sinh[a + b*x])/(3*b^3) - Sinh[2*a + 2*b*x]/(12*b^3) - (4*Cosh[a + b*x]*SinhIntegral[a + b*x])/(3*b^3) - (2*a^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/(3*b^3) + (2*a*x*Cosh[a + b*x]*SinhIntegral[a + b*x])/(3*b^2) - (2*x^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/(3*b) - (2*a*Sinh[a + b*x]*SinhIntegral[a + b*x])/(3*b^3) + (4*x*Sinh[a + b*x]*SinhIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*SinhIntegral[a + b*x]^2)/(3*b^3) - (a*x*(a + b*x)*SinhIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*SinhIntegral[a + b*x]^2)/(3*b) + (2*SinhIntegral[2*a + 2*b*x])/(3*b^3) + (a^2*SinhIntegral[2*a + 2*b*x])/b^3} -{x^1*SinhIntegral[a + b*x]^2, x, 17, Cosh[2*a + 2*b*x]/(4*b^2) - CoshIntegral[2*a + 2*b*x]/(2*b^2) + Log[a + b*x]/(2*b^2) + (a*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^2 - (x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b^2 - (a*(a + b*x)*SinhIntegral[a + b*x]^2)/(2*b^2) + (x*(a + b*x)*SinhIntegral[a + b*x]^2)/(2*b) - (a*SinhIntegral[2*a + 2*b*x])/b^2} -{x^0*SinhIntegral[a + b*x]^2, x, 5, -((2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b) + ((a + b*x)*SinhIntegral[a + b*x]^2)/b + SinhIntegral[2*a + 2*b*x]/b} -{SinhIntegral[a + b*x]^2/x^1, x, 0, CannotIntegrate[SinhIntegral[a + b*x]^2/x, x]} -{SinhIntegral[a + b*x]^2/x^2, x, 0, CannotIntegrate[SinhIntegral[a + b*x]^2/x^2, x]} -{SinhIntegral[a + b*x]^2/x^3, x, 0, CannotIntegrate[SinhIntegral[a + b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m SinhIntegral[d (a+b Log[c x^n])]*) - - -{x^2*SinhIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^3*ExpIntegralEi[((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) - (x^3*ExpIntegralEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) + (x^3*SinhIntegral[d*(a + b*Log[c*x^n])])/3} -{x^1*SinhIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^2*ExpIntegralEi[((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n)) - (x^2*ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n)) + (x^2*SinhIntegral[d*(a + b*Log[c*x^n])])/2} -{x^0*SinhIntegral[d*(a + b*Log[c*x^n])], x, 7, (x*ExpIntegralEi[((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) - (x*ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) + x*SinhIntegral[d*(a + b*Log[c*x^n])]} -{SinhIntegral[d*(a + b*Log[c*x^n])]/x^1, x, 3, -(Cosh[d*(a + b*Log[c*x^n])]/(b*d*n)) + ((a + b*Log[c*x^n])*SinhIntegral[d*(a + b*Log[c*x^n])])/(b*n)} -{SinhIntegral[d*(a + b*Log[c*x^n])]/x^2, x, 7, (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x) - (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x) - SinhIntegral[d*(a + b*Log[c*x^n])]/x} -{SinhIntegral[d*(a + b*Log[c*x^n])]/x^3, x, 7, (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2) - (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2) - SinhIntegral[d*(a + b*Log[c*x^n])]/(2*x^2)} - - -{(e*x)^m*SinhIntegral[d*(a + b*Log[c*x^n])], x, 7, (x*(e*x)^m*ExpIntegralEi[((1 + m - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n)) - (x*(e*x)^m*ExpIntegralEi[((1 + m + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n)) + ((e*x)^(1 + m)*SinhIntegral[d*(a + b*Log[c*x^n])])/(e*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[b x] SinhIntegral[b x]^n*) - - -{Sinh[b*x]*SinhIntegral[b*x]/x^3, x, 14, b^2*CoshIntegral[2*b*x] - (b*Cosh[b*x]*Sinh[b*x])/(2*x) - Sinh[b*x]^2/(4*x^2) - (b*Sinh[2*b*x])/(4*x) - (b*Cosh[b*x]*SinhIntegral[b*x])/(2*x) - (Sinh[b*x]*SinhIntegral[b*x])/(2*x^2) + (1/4)*b^2*SinhIntegral[b*x]^2} -{Sinh[b*x]*SinhIntegral[b*x]/x^2, x, 5, b*CannotIntegrate[(Cosh[b*x]*SinhIntegral[b*x])/x, x] - Sinh[b*x]^2/x - (Sinh[b*x]*SinhIntegral[b*x])/x + b*SinhIntegral[2*b*x]} -{Sinh[b*x]*SinhIntegral[b*x]/x, x, 1, (1/2)*SinhIntegral[b*x]^2} -{Sinh[b*x]*SinhIntegral[b*x], x, 5, (Cosh[b*x]*SinhIntegral[b*x])/b - SinhIntegral[2*b*x]/(2*b)} -{x*Sinh[b*x]*SinhIntegral[b*x], x, 9, CoshIntegral[2*b*x]/(2*b^2) - Log[x]/(2*b^2) - Sinh[b*x]^2/(2*b^2) + (x*Cosh[b*x]*SinhIntegral[b*x])/b - (Sinh[b*x]*SinhIntegral[b*x])/b^2} -{x^2*Sinh[b*x]*SinhIntegral[b*x], x, 14, -((5*x)/(4*b^2)) + (5*Cosh[b*x]*Sinh[b*x])/(4*b^3) - (x*Sinh[b*x]^2)/(2*b^2) + (2*Cosh[b*x]*SinhIntegral[b*x])/b^3 + (x^2*Cosh[b*x]*SinhIntegral[b*x])/b - (2*x*Sinh[b*x]*SinhIntegral[b*x])/b^2 - SinhIntegral[2*b*x]/b^3} -{x^3*Sinh[b*x]*SinhIntegral[b*x], x, 18, -(x^2/b^2) + (3*CoshIntegral[2*b*x])/b^4 - (3*Log[x])/b^4 + (2*x*Cosh[b*x]*Sinh[b*x])/b^3 - (4*Sinh[b*x]^2)/b^4 - (x^2*Sinh[b*x]^2)/(2*b^2) + (6*x*Cosh[b*x]*SinhIntegral[b*x])/b^3 + (x^3*Cosh[b*x]*SinhIntegral[b*x])/b - (6*Sinh[b*x]*SinhIntegral[b*x])/b^4 - (3*x^2*Sinh[b*x]*SinhIntegral[b*x])/b^2} - - -{Cosh[b*x]*SinhIntegral[b*x]/x^3, x, 12, -((b*Cosh[2*b*x])/(4*x)) + (1/2)*b^2*CannotIntegrate[(Cosh[b*x]*SinhIntegral[b*x])/x, x] - (b*Sinh[b*x]^2)/(2*x) - Sinh[2*b*x]/(8*x^2) - (Cosh[b*x]*SinhIntegral[b*x])/(2*x^2) - (b*Sinh[b*x]*SinhIntegral[b*x])/(2*x) + b^2*SinhIntegral[2*b*x]} -{Cosh[b*x]*SinhIntegral[b*x]/x^2, x, 7, b*CoshIntegral[2*b*x] - Sinh[2*b*x]/(2*x) - (Cosh[b*x]*SinhIntegral[b*x])/x + (1/2)*b*SinhIntegral[b*x]^2} -{Cosh[b*x]*SinhIntegral[b*x]/x, x, 0, CannotIntegrate[(Cosh[b*x]*SinhIntegral[b*x])/x, x]} -{Cosh[b*x]*SinhIntegral[b*x], x, 5, -(CoshIntegral[2*b*x]/(2*b)) + Log[x]/(2*b) + (Sinh[b*x]*SinhIntegral[b*x])/b} -{x*Cosh[b*x]*SinhIntegral[b*x], x, 9, x/(2*b) - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - (Cosh[b*x]*SinhIntegral[b*x])/b^2 + (x*Sinh[b*x]*SinhIntegral[b*x])/b + SinhIntegral[2*b*x]/(2*b^2)} -{x^2*Cosh[b*x]*SinhIntegral[b*x], x, 13, x^2/(4*b) - CoshIntegral[2*b*x]/b^3 + Log[x]/b^3 - (x*Cosh[b*x]*Sinh[b*x])/(2*b^2) + (5*Sinh[b*x]^2)/(4*b^3) - (2*x*Cosh[b*x]*SinhIntegral[b*x])/b^2 + (2*Sinh[b*x]*SinhIntegral[b*x])/b^3 + (x^2*Sinh[b*x]*SinhIntegral[b*x])/b} -{x^3*Cosh[b*x]*SinhIntegral[b*x], x, 20, (4*x)/b^3 + x^3/(6*b) - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cosh[b*x]*Sinh[b*x])/(2*b^2) + (2*x*Sinh[b*x]^2)/b^3 - (6*Cosh[b*x]*SinhIntegral[b*x])/b^4 - (3*x^2*Cosh[b*x]*SinhIntegral[b*x])/b^2 + (6*x*Sinh[b*x]*SinhIntegral[b*x])/b^3 + (x^3*Sinh[b*x]*SinhIntegral[b*x])/b + (3*SinhIntegral[2*b*x])/b^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[b x] SinhIntegral[d x]^n*) - - -{Sinh[5*x]*SinhIntegral[2*x], x, 6, (1/5)*Cosh[5*x]*SinhIntegral[2*x] + (1/10)*SinhIntegral[3*x] - (1/10)*SinhIntegral[7*x]} - - -{Cosh[5*x]*SinhIntegral[2*x], x, 6, (1/10)*CoshIntegral[3*x] - (1/10)*CoshIntegral[7*x] + (1/5)*Sinh[5*x]*SinhIntegral[2*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[a+b x] SinhIntegral[a+b x]^n*) - - -(* {x^3*Sinh[a + b*x]*SinhIntegral[a + b*x], x, 32, (7*a*x)/(4*b^3) - x^2/b^2 + (3*Cosh[a + b*x]^2)/b^4 - (a^2*Cosh[a + b*x]^2)/(2*b^4) - (3*CosIntegral[2*a + 2*b*x])/b^4 + (3*a^2*CosIntegral[2*a + 2*b*x])/(2*b^4) + (3*Log[a + b*x])/b^4 - (3*a^2*Log[a + b*x])/(2*b^4) - (7*a*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^4) + (2*x*Cosh[a + b*x]*Sinh[a + b*x])/b^3 - Sinh[a + b*x]^2/b^4 - (a*x*Sinh[a + b*x]^2)/(2*b^3) + (x^2*Sinh[a + b*x]^2)/(2*b^2) + (6*x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^3 - (x^3*Cosh[a + b*x]*SinhIntegral[a + b*x])/b - (6*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^4 + (3*x^2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^2 + (3*a*SinhIntegral[2*a + 2*b*x])/b^4 - (a^3*SinhIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Sinh[a + b*x]*SinhIntegral[a + b*x], x, 21, -(x/b^2) + (a*Cosh[2*a + 2*b*x])/(4*b^3) - (x*Cosh[2*a + 2*b*x])/(4*b^2) - (a*CoshIntegral[2*a + 2*b*x])/b^3 + (a*Log[a + b*x])/b^3 + (Cosh[a + b*x]*Sinh[a + b*x])/b^3 + Sinh[2*a + 2*b*x]/(8*b^3) + (2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^3 + (x^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b - (2*x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^2 - SinhIntegral[2*a + 2*b*x]/b^3 - (a^2*SinhIntegral[2*a + 2*b*x])/(2*b^3)} -{x^1*Sinh[a + b*x]*SinhIntegral[a + b*x], x, 11, -(Cosh[2*a + 2*b*x]/(4*b^2)) + CoshIntegral[2*a + 2*b*x]/(2*b^2) - Log[a + b*x]/(2*b^2) + (x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b - (Sinh[a + b*x]*SinhIntegral[a + b*x])/b^2 + (a*SinhIntegral[2*a + 2*b*x])/(2*b^2)} -{x^0*Sinh[a + b*x]*SinhIntegral[a + b*x], x, 4, (Cosh[a + b*x]*SinhIntegral[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b)} -{Sinh[a + b*x]*SinhIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Sinh[a + b*x]*SinhIntegral[a + b*x])/x, x]} - - -(* {x^3*Cosh[a + b*x]*SinhIntegral[a + b*x], x, 32, (4*x)/b^3 - (a^2*x)/(2*b^3) + (a*x^2)/(4*b^2) - x^3/(6*b) - (3*a*Cosh[a + b*x]^2)/(2*b^4) + (3*a*CosIntegral[2*a + 2*b*x])/b^4 - (a^3*CosIntegral[2*a + 2*b*x])/(2*b^4) - (3*a*Log[a + b*x])/b^4 + (a^3*Log[a + b*x])/(2*b^4) - (4*Cosh[a + b*x]*Sinh[a + b*x])/b^4 + (a^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^4) - (a*x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) + (x^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + (a*Sinh[a + b*x]^2)/(4*b^4) - (2*x*Sinh[a + b*x]^2)/b^3 - (6*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^4 + (3*x^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^2 - (6*x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^3 + (x^3*Sinh[a + b*x]*SinhIntegral[a + b*x])/b + (3*SinhIntegral[2*a + 2*b*x])/b^4 - (3*a^2*SinhIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Cosh[a + b*x]*SinhIntegral[a + b*x], x, 21, -((a*x)/(2*b^2)) + x^2/(4*b) + Cosh[2*a + 2*b*x]/(2*b^3) - CoshIntegral[2*a + 2*b*x]/b^3 - (a^2*CoshIntegral[2*a + 2*b*x])/(2*b^3) + Log[a + b*x]/b^3 + (a^2*Log[a + b*x])/(2*b^3) + (a*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) - (2*x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b^2 + (2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b^3 + (x^2*Sinh[a + b*x]*SinhIntegral[a + b*x])/b - (a*SinhIntegral[2*a + 2*b*x])/b^3} -{x^1*Cosh[a + b*x]*SinhIntegral[a + b*x], x, 12, x/(2*b) + (a*CoshIntegral[2*a + 2*b*x])/(2*b^2) - (a*Log[a + b*x])/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) - (Cosh[a + b*x]*SinhIntegral[a + b*x])/b^2 + (x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b + SinhIntegral[2*a + 2*b*x]/(2*b^2)} -{x^0*Cosh[a + b*x]*SinhIntegral[a + b*x], x, 4, -(CoshIntegral[2*a + 2*b*x]/(2*b)) + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b} -{Cosh[a + b*x]*SinhIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Cosh[a + b*x]*SinhIntegral[a + b*x])/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[a+b x] SinhIntegral[c+d x]^n*) - - -(* {x^2*Sinh[a + b*x]*SinhIntegral[c + d*x], x, 46, -((c*Cosh[a - c + (b - d)*x])/(2*b*(b - d)*d)) + (x*Cosh[a - c + (b - d)*x])/(2*b*(b - d)) + (c*Cosh[a + c + (b + d)*x])/(2*b*d*(b + d)) - (x*Cosh[a + c + (b + d)*x])/(2*b*(b + d)) + (CoshIntegral[((b - d)*(c + d*x))/d]*(2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (CoshIntegral[((b + d)*(c + d*x))/d]*(2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - Sinh[a - c + (b - d)*x]/(2*b*(b - d)^2) - Sinh[a - c + (b - d)*x]/(b^2*(b - d)) + Sinh[a + c + (b + d)*x]/(2*b*(b + d)^2) + Sinh[a + c + (b + d)*x]/(b^2*(b + d)) + (((2 + b^2*x^2)*Cosh[a + b*x] - 2*b*x*Sinh[a + b*x])*SinhIntegral[c + d*x])/b^3 + (((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d])*SinhIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) - (((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d])*SinhIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Sinh[a + b*x]*SinhIntegral[c + d*x], x, 24, Cosh[a - c + (b - d)*x]/(2*b*(b - d)) - Cosh[a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) - (c*CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b*d) + (c*CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b*d) - (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (x*Cosh[a + b*x]*SinhIntegral[c + d*x])/b - (Sinh[a + b*x]*SinhIntegral[c + d*x])/b^2 + (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)} -{x^0*Sinh[a + b*x]*SinhIntegral[c + d*x], x, 9, (CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b) - (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b) + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) + (Cosh[a + b*x]*SinhIntegral[c + d*x])/b - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Sinh[a + b*x]*SinhIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Sinh[a + b*x]*SinhIntegral[c + d*x])/x, x]} - - -(* {x^2*Cosh[a + b*x]*SinhIntegral[c + d*x], x, 46, -(Cosh[a - c + (b - d)*x]/(2*b*(b - d)^2)) - Cosh[a - c + (b - d)*x]/(b^2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*b*(b + d)^2) + Cosh[a + c + (b + d)*x]/(b^2*(b + d)) + (CoshIntegral[((b - d)*(c + d*x))/d]*((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (CoshIntegral[((b + d)*(c + d*x))/d]*((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (c*Sinh[a - c + (b - d)*x])/(2*b*(b - d)*d) + (x*Sinh[a - c + (b - d)*x])/(2*b*(b - d)) + (c*Sinh[a + c + (b + d)*x])/(2*b*d*(b + d)) - (x*Sinh[a + c + (b + d)*x])/(2*b*(b + d)) - ((2*b*x*Cosh[a + b*x] - (2 + b^2*x^2)*Sinh[a + b*x])*SinhIntegral[c + d*x])/b^3 + ((2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d])*SinhIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) - ((2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d])*SinhIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Cosh[a + b*x]*SinhIntegral[c + d*x], x, 24, -((c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d)) + (c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) - (CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + Sinh[a - c + (b - d)*x]/(2*b*(b - d)) - Sinh[a + c + (b + d)*x]/(2*b*(b + d)) - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) - (Cosh[a + b*x]*SinhIntegral[c + d*x])/b^2 + (x*Sinh[a + b*x]*SinhIntegral[c + d*x])/b + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)} -{x^0*Cosh[a + b*x]*SinhIntegral[c + d*x], x, 9, (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) + (Sinh[a + b*x]*SinhIntegral[c + d*x])/b - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Cosh[a + b*x]*SinhIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Cosh[a + b*x]*SinhIntegral[c + d*x])/x, x]} - - -(* ::Section::Closed:: *) -(*Hyperbolic cosine integral function*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m CoshIntegral[b x]^n*) - - -{x^m*CoshIntegral[b*x], x, 5, (x^(1 + m)*CoshIntegral[b*x])/(1 + m) - (x^m*Gamma[1 + m, (-b)*x])/(((-b)*x)^m*(2*b*(1 + m))) + (x^m*Gamma[1 + m, b*x])/((b*x)^m*(2*b*(1 + m)))} - -{x^3*CoshIntegral[b*x], x, 6, (3*Cosh[b*x])/(2*b^4) + (3*x^2*Cosh[b*x])/(4*b^2) + (1/4)*x^4*CoshIntegral[b*x] - (3*x*Sinh[b*x])/(2*b^3) - (x^3*Sinh[b*x])/(4*b)} -{x^2*CoshIntegral[b*x], x, 5, (2*x*Cosh[b*x])/(3*b^2) + (1/3)*x^3*CoshIntegral[b*x] - (2*Sinh[b*x])/(3*b^3) - (x^2*Sinh[b*x])/(3*b)} -{x^1*CoshIntegral[b*x], x, 4, Cosh[b*x]/(2*b^2) + (1/2)*x^2*CoshIntegral[b*x] - (x*Sinh[b*x])/(2*b)} -{x^0*CoshIntegral[b*x], x, 1, x*CoshIntegral[b*x] - Sinh[b*x]/b} -{CoshIntegral[b*x]/x^1, x, 1, (-(1/2))*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-b)*x] + (1/2)*b*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, b*x] + EulerGamma*Log[x] + (1/2)*Log[b*x]^2} -{CoshIntegral[b*x]/x^2, x, 4, -(Cosh[b*x]/x) - CoshIntegral[b*x]/x + b*SinhIntegral[b*x]} -{CoshIntegral[b*x]/x^3, x, 5, -(Cosh[b*x]/(4*x^2)) + (1/4)*b^2*CoshIntegral[b*x] - CoshIntegral[b*x]/(2*x^2) - (b*Sinh[b*x])/(4*x)} - - -{x^m*CoshIntegral[b*x]^2, x, 0, CannotIntegrate[x^m*CoshIntegral[b*x]^2, x]} - -{x^3*CoshIntegral[b*x]^2, x, 19, -(x^2/(4*b^2)) + (3*Cosh[b*x]^2)/(8*b^4) + (3*Cosh[b*x]*CoshIntegral[b*x])/b^4 + (3*x^2*Cosh[b*x]*CoshIntegral[b*x])/(2*b^2) + (1/4)*x^4*CoshIntegral[b*x]^2 - (3*CoshIntegral[2*b*x])/(2*b^4) - (3*Log[x])/(2*b^4) - (x*Cosh[b*x]*Sinh[b*x])/b^3 - (3*x*CoshIntegral[b*x]*Sinh[b*x])/b^3 - (x^3*CoshIntegral[b*x]*Sinh[b*x])/(2*b) + (13*Sinh[b*x]^2)/(8*b^4) + (x^2*Sinh[b*x]^2)/(4*b^2)} -{x^2*CoshIntegral[b*x]^2, x, 15, -(x/(2*b^2)) + (4*x*Cosh[b*x]*CoshIntegral[b*x])/(3*b^2) + (1/3)*x^3*CoshIntegral[b*x]^2 - (5*Cosh[b*x]*Sinh[b*x])/(6*b^3) - (4*CoshIntegral[b*x]*Sinh[b*x])/(3*b^3) - (2*x^2*CoshIntegral[b*x]*Sinh[b*x])/(3*b) + (x*Sinh[b*x]^2)/(3*b^2) + (2*SinhIntegral[2*b*x])/(3*b^3)} -{x^1*CoshIntegral[b*x]^2, x, 10, (Cosh[b*x]*CoshIntegral[b*x])/b^2 + (1/2)*x^2*CoshIntegral[b*x]^2 - CoshIntegral[2*b*x]/(2*b^2) - Log[x]/(2*b^2) - (x*CoshIntegral[b*x]*Sinh[b*x])/b + Sinh[b*x]^2/(2*b^2)} -{x^0*CoshIntegral[b*x]^2, x, 6, x*CoshIntegral[b*x]^2 - (2*CoshIntegral[b*x]*Sinh[b*x])/b + SinhIntegral[2*b*x]/b} -{CoshIntegral[b*x]^2/x^1, x, 0, CannotIntegrate[CoshIntegral[b*x]^2/x, x]} -{CoshIntegral[b*x]^2/x^2, x, 0, CannotIntegrate[CoshIntegral[b*x]^2/x^2, x]} -{CoshIntegral[b*x]^2/x^3, x, 0, CannotIntegrate[CoshIntegral[b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m CoshIntegral[a+b x]^n*) - - -{x^m*CoshIntegral[a + b*x], x, 1, (x^(1 + m)*CoshIntegral[a + b*x])/(1 + m) - (b*CannotIntegrate[(x^(1 + m)*Cosh[a + b*x])/(a + b*x), x])/(1 + m)} - -{x^3*CoshIntegral[a + b*x], x, 14, (3*Cosh[a + b*x])/(2*b^4) + (a^2*Cosh[a + b*x])/(4*b^4) - (a*x*Cosh[a + b*x])/(2*b^3) + (3*x^2*Cosh[a + b*x])/(4*b^2) - (a^4*CoshIntegral[a + b*x])/(4*b^4) + (1/4)*x^4*CoshIntegral[a + b*x] + (a*Sinh[a + b*x])/(2*b^4) + (a^3*Sinh[a + b*x])/(4*b^4) - (3*x*Sinh[a + b*x])/(2*b^3) - (a^2*x*Sinh[a + b*x])/(4*b^3) + (a*x^2*Sinh[a + b*x])/(4*b^2) - (x^3*Sinh[a + b*x])/(4*b)} -{x^2*CoshIntegral[a + b*x], x, 10, -((a*Cosh[a + b*x])/(3*b^3)) + (2*x*Cosh[a + b*x])/(3*b^2) + (a^3*CoshIntegral[a + b*x])/(3*b^3) + (1/3)*x^3*CoshIntegral[a + b*x] - (2*Sinh[a + b*x])/(3*b^3) - (a^2*Sinh[a + b*x])/(3*b^3) + (a*x*Sinh[a + b*x])/(3*b^2) - (x^2*Sinh[a + b*x])/(3*b)} -{x^1*CoshIntegral[a + b*x], x, 7, Cosh[a + b*x]/(2*b^2) - (a^2*CoshIntegral[a + b*x])/(2*b^2) + (1/2)*x^2*CoshIntegral[a + b*x] + (a*Sinh[a + b*x])/(2*b^2) - (x*Sinh[a + b*x])/(2*b)} -{x^0*CoshIntegral[a + b*x], x, 1, ((a + b*x)*CoshIntegral[a + b*x])/b - Sinh[a + b*x]/b} -{CoshIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[CoshIntegral[a + b*x]/x, x]} -{CoshIntegral[a + b*x]/x^2, x, 7, (b*Cosh[a]*CoshIntegral[b*x])/a - (b*CoshIntegral[a + b*x])/a - CoshIntegral[a + b*x]/x + (b*Sinh[a]*SinhIntegral[b*x])/a} -{CoshIntegral[a + b*x]/x^3, x, 11, -((b*Cosh[a + b*x])/(2*a*x)) - (b^2*Cosh[a]*CoshIntegral[b*x])/(2*a^2) + (b^2*CoshIntegral[a + b*x])/(2*a^2) - CoshIntegral[a + b*x]/(2*x^2) + (b^2*CoshIntegral[b*x]*Sinh[a])/(2*a) + (b^2*Cosh[a]*SinhIntegral[b*x])/(2*a) - (b^2*Sinh[a]*SinhIntegral[b*x])/(2*a^2)} - - -{x^m*CoshIntegral[a + b*x]^2, x, 0, CannotIntegrate[x^m*CoshIntegral[a + b*x]^2, x]} - -{x^2*CoshIntegral[a + b*x]^2, x, 39, -((2*x)/(3*b^2)) - (a*Cosh[2*a + 2*b*x])/(3*b^3) + (x*Cosh[2*a + 2*b*x])/(6*b^2) - (2*a*Cosh[a + b*x]*CoshIntegral[a + b*x])/(3*b^3) + (4*x*Cosh[a + b*x]*CoshIntegral[a + b*x])/(3*b^2) + (a^2*(a + b*x)*CoshIntegral[a + b*x]^2)/(3*b^3) - (a*x*(a + b*x)*CoshIntegral[a + b*x]^2)/(3*b^2) + (x^2*(a + b*x)*CoshIntegral[a + b*x]^2)/(3*b) + (a*CoshIntegral[2*a + 2*b*x])/b^3 + (a*Log[a + b*x])/b^3 - (2*Cosh[a + b*x]*Sinh[a + b*x])/(3*b^3) - (4*CoshIntegral[a + b*x]*Sinh[a + b*x])/(3*b^3) - (2*a^2*CoshIntegral[a + b*x]*Sinh[a + b*x])/(3*b^3) + (2*a*x*CoshIntegral[a + b*x]*Sinh[a + b*x])/(3*b^2) - (2*x^2*CoshIntegral[a + b*x]*Sinh[a + b*x])/(3*b) - Sinh[2*a + 2*b*x]/(12*b^3) + (2*SinhIntegral[2*a + 2*b*x])/(3*b^3) + (a^2*SinhIntegral[2*a + 2*b*x])/b^3} -{x^1*CoshIntegral[a + b*x]^2, x, 17, Cosh[2*a + 2*b*x]/(4*b^2) + (Cosh[a + b*x]*CoshIntegral[a + b*x])/b^2 - (a*(a + b*x)*CoshIntegral[a + b*x]^2)/(2*b^2) + (x*(a + b*x)*CoshIntegral[a + b*x]^2)/(2*b) - CoshIntegral[2*a + 2*b*x]/(2*b^2) - Log[a + b*x]/(2*b^2) + (a*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^2 - (x*CoshIntegral[a + b*x]*Sinh[a + b*x])/b - (a*SinhIntegral[2*a + 2*b*x])/b^2} -{x^0*CoshIntegral[a + b*x]^2, x, 5, ((a + b*x)*CoshIntegral[a + b*x]^2)/b - (2*CoshIntegral[a + b*x]*Sinh[a + b*x])/b + SinhIntegral[2*a + 2*b*x]/b} -{CoshIntegral[a + b*x]^2/x^1, x, 0, CannotIntegrate[CoshIntegral[a + b*x]^2/x, x]} -{CoshIntegral[a + b*x]^2/x^2, x, 0, CannotIntegrate[CoshIntegral[a + b*x]^2/x^2, x]} -{CoshIntegral[a + b*x]^2/x^3, x, 0, CannotIntegrate[CoshIntegral[a + b*x]^2/x^3, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (e x)^m CoshIntegral[d (a+b Log[c x^n])]*) - - -{x^2*CoshIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^3*CoshIntegral[d*(a + b*Log[c*x^n])])/3 - (x^3*ExpIntegralEi[((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) - (x^3*ExpIntegralEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n))} -{x^1*CoshIntegral[d*(a + b*Log[c*x^n])], x, 7, (x^2*CoshIntegral[d*(a + b*Log[c*x^n])])/2 - (x^2*ExpIntegralEi[((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n)) - (x^2*ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*(c*x^n)^(2/n))} -{x^0*CoshIntegral[d*(a + b*Log[c*x^n])], x, 7, x*CoshIntegral[d*(a + b*Log[c*x^n])] - (x*ExpIntegralEi[((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1)) - (x*ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^(a/(b*n))*(c*x^n)^n^(-1))} -{CoshIntegral[d*(a + b*Log[c*x^n])]/x^1, x, 3, (CoshIntegral[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n) - Sinh[d*(a + b*Log[c*x^n])]/(b*d*n)} -{CoshIntegral[d*(a + b*Log[c*x^n])]/x^2, x, 7, -(CoshIntegral[d*(a + b*Log[c*x^n])]/x) + (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x) + (E^(a/(b*n))*(c*x^n)^n^(-1)*ExpIntegralEi[-(((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*x)} -{CoshIntegral[d*(a + b*Log[c*x^n])]/x^3, x, 7, -CoshIntegral[d*(a + b*Log[c*x^n])]/(2*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(4*x^2)} - - -{(e*x)^m*CoshIntegral[d*(a + b*Log[c*x^n])], x, 7, ((e*x)^(1 + m)*CoshIntegral[d*(a + b*Log[c*x^n])])/(e*(1 + m)) - (x*(e*x)^m*ExpIntegralEi[((1 + m - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n)) - (x*(e*x)^m*ExpIntegralEi[((1 + m + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*E^((a*(1 + m))/(b*n))*(1 + m)*(c*x^n)^((1 + m)/n))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[b x] CoshIntegral[b x]^n*) - - -{Cosh[b*x]*CoshIntegral[b*x]/x^3, x, 14, -(Cosh[b*x]^2/(4*x^2)) - (Cosh[b*x]*CoshIntegral[b*x])/(2*x^2) + (1/4)*b^2*CoshIntegral[b*x]^2 + b^2*CoshIntegral[2*b*x] - (b*Cosh[b*x]*Sinh[b*x])/(2*x) - (b*CoshIntegral[b*x]*Sinh[b*x])/(2*x) - (b*Sinh[2*b*x])/(4*x)} -{Cosh[b*x]*CoshIntegral[b*x]/x^2, x, 5, -(Cosh[b*x]^2/x) - (Cosh[b*x]*CoshIntegral[b*x])/x + b*CannotIntegrate[(CoshIntegral[b*x]*Sinh[b*x])/x, x] + b*SinhIntegral[2*b*x]} -{Cosh[b*x]*CoshIntegral[b*x]/x, x, 1, (1/2)*CoshIntegral[b*x]^2} -{Cosh[b*x]*CoshIntegral[b*x], x, 5, (CoshIntegral[b*x]*Sinh[b*x])/b - SinhIntegral[2*b*x]/(2*b)} -{x*Cosh[b*x]*CoshIntegral[b*x], x, 9, -((Cosh[b*x]*CoshIntegral[b*x])/b^2) + CoshIntegral[2*b*x]/(2*b^2) + Log[x]/(2*b^2) + (x*CoshIntegral[b*x]*Sinh[b*x])/b - Sinh[b*x]^2/(2*b^2)} -{x^2*Cosh[b*x]*CoshIntegral[b*x], x, 14, (3*x)/(4*b^2) - (2*x*Cosh[b*x]*CoshIntegral[b*x])/b^2 + (5*Cosh[b*x]*Sinh[b*x])/(4*b^3) + (2*CoshIntegral[b*x]*Sinh[b*x])/b^3 + (x^2*CoshIntegral[b*x]*Sinh[b*x])/b - (x*Sinh[b*x]^2)/(2*b^2) - SinhIntegral[2*b*x]/b^3} -{x^3*Cosh[b*x]*CoshIntegral[b*x], x, 18, x^2/(2*b^2) - (3*Cosh[b*x]^2)/(4*b^4) - (6*Cosh[b*x]*CoshIntegral[b*x])/b^4 - (3*x^2*Cosh[b*x]*CoshIntegral[b*x])/b^2 + (3*CoshIntegral[2*b*x])/b^4 + (3*Log[x])/b^4 + (2*x*Cosh[b*x]*Sinh[b*x])/b^3 + (6*x*CoshIntegral[b*x]*Sinh[b*x])/b^3 + (x^3*CoshIntegral[b*x]*Sinh[b*x])/b - (13*Sinh[b*x]^2)/(4*b^4) - (x^2*Sinh[b*x]^2)/(2*b^2)} - - -{Sinh[b*x]*CoshIntegral[b*x]/x^3, x, 12, -((b*Cosh[b*x]^2)/(2*x)) - (b*Cosh[2*b*x])/(4*x) - (b*Cosh[b*x]*CoshIntegral[b*x])/(2*x) + (1/2)*b^2*CannotIntegrate[(CoshIntegral[b*x]*Sinh[b*x])/x, x] - (CoshIntegral[b*x]*Sinh[b*x])/(2*x^2) - Sinh[2*b*x]/(8*x^2) + b^2*SinhIntegral[2*b*x]} -{Sinh[b*x]*CoshIntegral[b*x]/x^2, x, 7, (1/2)*b*CoshIntegral[b*x]^2 + b*CoshIntegral[2*b*x] - (CoshIntegral[b*x]*Sinh[b*x])/x - Sinh[2*b*x]/(2*x)} -{Sinh[b*x]*CoshIntegral[b*x]/x, x, 0, CannotIntegrate[(CoshIntegral[b*x]*Sinh[b*x])/x, x]} -{Sinh[b*x]*CoshIntegral[b*x], x, 5, (Cosh[b*x]*CoshIntegral[b*x])/b - CoshIntegral[2*b*x]/(2*b) - Log[x]/(2*b)} -{x*Sinh[b*x]*CoshIntegral[b*x], x, 9, -(x/(2*b)) + (x*Cosh[b*x]*CoshIntegral[b*x])/b - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - (CoshIntegral[b*x]*Sinh[b*x])/b^2 + SinhIntegral[2*b*x]/(2*b^2)} -{x^2*Sinh[b*x]*CoshIntegral[b*x], x, 13, -(x^2/(4*b)) + Cosh[b*x]^2/(4*b^3) + (2*Cosh[b*x]*CoshIntegral[b*x])/b^3 + (x^2*Cosh[b*x]*CoshIntegral[b*x])/b - CoshIntegral[2*b*x]/b^3 - Log[x]/b^3 - (x*Cosh[b*x]*Sinh[b*x])/(2*b^2) - (2*x*CoshIntegral[b*x]*Sinh[b*x])/b^2 + Sinh[b*x]^2/b^3} -{x^3*Sinh[b*x]*CoshIntegral[b*x], x, 20, -((5*x)/(2*b^3)) - x^3/(6*b) + (x*Cosh[b*x]^2)/(2*b^3) + (6*x*Cosh[b*x]*CoshIntegral[b*x])/b^3 + (x^3*Cosh[b*x]*CoshIntegral[b*x])/b - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cosh[b*x]*Sinh[b*x])/(2*b^2) - (6*CoshIntegral[b*x]*Sinh[b*x])/b^4 - (3*x^2*CoshIntegral[b*x]*Sinh[b*x])/b^2 + (3*x*Sinh[b*x]^2)/(2*b^3) + (3*SinhIntegral[2*b*x])/b^4} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[b x] CoshIntegral[d x]^n*) - - -{Sinh[5*x]*CoshIntegral[2*x], x, 6, (1/5)*Cosh[5*x]*CoshIntegral[2*x] - (1/10)*CoshIntegral[3*x] - (1/10)*CoshIntegral[7*x]} - - -{Cosh[5*x]*CoshIntegral[2*x], x, 6, (1/5)*CoshIntegral[2*x]*Sinh[5*x] - (1/10)*SinhIntegral[3*x] - (1/10)*SinhIntegral[7*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[a+b x] CoshIntegral[a+b x]^n*) - - -(* {x^3*Sinh[a + b*x]*CoshIntegral[a + b*x], x, 32, -((5*x)/(2*b^3)) - (a^2*x)/(2*b^3) + (a*x^2)/(4*b^2) - x^3/(6*b) - (7*a*Cosh[a + b*x]^2)/(4*b^4) + (x*Cosh[a + b*x]^2)/(2*b^3) + (6*x*Cosh[a + b*x]*CoshIntegral[a + b*x])/b^3 + (x^3*Cosh[a + b*x]*CoshIntegral[a + b*x])/b + (3*a*CoshIntegral[2*a + 2*b*x])/b^4 + (a^3*CoshIntegral[2*a + 2*b*x])/(2*b^4) + (3*a*Log[a + b*x])/b^4 + (a^3*Log[a + b*x])/(2*b^4) - (4*Cosh[a + b*x]*Sinh[a + b*x])/b^4 - (a^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^4) + (a*x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) - (x^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) - (6*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^4 - (3*x^2*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^2 + (3*x*Sinh[a + b*x]^2)/(2*b^3) + (3*SinhIntegral[2*a + 2*b*x])/b^4 + (3*a^2*SinhIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Sinh[a + b*x]*CoshIntegral[a + b*x], x, 21, (a*x)/(2*b^2) - x^2/(4*b) + Cosh[a + b*x]^2/(4*b^3) + Cosh[2*a + 2*b*x]/(2*b^3) + (2*Cosh[a + b*x]*CoshIntegral[a + b*x])/b^3 + (x^2*Cosh[a + b*x]*CoshIntegral[a + b*x])/b - CoshIntegral[2*a + 2*b*x]/b^3 - (a^2*CoshIntegral[2*a + 2*b*x])/(2*b^3) - Log[a + b*x]/b^3 - (a^2*Log[a + b*x])/(2*b^3) + (a*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) - (2*x*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^2 - (a*SinhIntegral[2*a + 2*b*x])/b^3} -{x^1*Sinh[a + b*x]*CoshIntegral[a + b*x], x, 12, -(x/(2*b)) + (x*Cosh[a + b*x]*CoshIntegral[a + b*x])/b + (a*CoshIntegral[2*a + 2*b*x])/(2*b^2) + (a*Log[a + b*x])/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) - (CoshIntegral[a + b*x]*Sinh[a + b*x])/b^2 + SinhIntegral[2*a + 2*b*x]/(2*b^2)} -{x^0*Sinh[a + b*x]*CoshIntegral[a + b*x], x, 4, (Cosh[a + b*x]*CoshIntegral[a + b*x])/b - CoshIntegral[2*a + 2*b*x]/(2*b) - Log[a + b*x]/(2*b)} -{Sinh[a + b*x]*CoshIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Sinh[a + b*x]*CoshIntegral[a + b*x])/x, x]} - - -(* {x^3*Cosh[a + b*x]*CoshIntegral[a + b*x], x, 32, -((5*a*x)/(4*b^3)) + x^2/(2*b^2) - (15*Cosh[a + b*x]^2)/(4*b^4) - (a^2*Cosh[a + b*x]^2)/(2*b^4) - (6*Cosh[a + b*x]*CoshIntegral[a + b*x])/b^4 - (3*x^2*Cosh[a + b*x]*CoshIntegral[a + b*x])/b^2 + (3*CoshIntegral[2*a + 2*b*x])/b^4 + (3*a^2*CoshIntegral[2*a + 2*b*x])/(2*b^4) + (3*Log[a + b*x])/b^4 + (3*a^2*Log[a + b*x])/(2*b^4) - (7*a*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^4) + (2*x*Cosh[a + b*x]*Sinh[a + b*x])/b^3 + (6*x*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^3 + (x^3*CoshIntegral[a + b*x]*Sinh[a + b*x])/b - Sinh[a + b*x]^2/(4*b^4) + (a*x*Sinh[a + b*x]^2)/(2*b^3) - (x^2*Sinh[a + b*x]^2)/(2*b^2) + (3*a*SinhIntegral[2*a + 2*b*x])/b^4 + (a^3*SinhIntegral[2*a + 2*b*x])/(2*b^4)} *) -{x^2*Cosh[a + b*x]*CoshIntegral[a + b*x], x, 21, x/b^2 + (a*Cosh[2*a + 2*b*x])/(4*b^3) - (x*Cosh[2*a + 2*b*x])/(4*b^2) - (2*x*Cosh[a + b*x]*CoshIntegral[a + b*x])/b^2 - (a*CoshIntegral[2*a + 2*b*x])/b^3 - (a*Log[a + b*x])/b^3 + (Cosh[a + b*x]*Sinh[a + b*x])/b^3 + (2*CoshIntegral[a + b*x]*Sinh[a + b*x])/b^3 + (x^2*CoshIntegral[a + b*x]*Sinh[a + b*x])/b + Sinh[2*a + 2*b*x]/(8*b^3) - SinhIntegral[2*a + 2*b*x]/b^3 - (a^2*SinhIntegral[2*a + 2*b*x])/(2*b^3)} -{x^1*Cosh[a + b*x]*CoshIntegral[a + b*x], x, 11, -(Cosh[2*a + 2*b*x]/(4*b^2)) - (Cosh[a + b*x]*CoshIntegral[a + b*x])/b^2 + CoshIntegral[2*a + 2*b*x]/(2*b^2) + Log[a + b*x]/(2*b^2) + (x*CoshIntegral[a + b*x]*Sinh[a + b*x])/b + (a*SinhIntegral[2*a + 2*b*x])/(2*b^2)} -{x^0*Cosh[a + b*x]*CoshIntegral[a + b*x], x, 4, (CoshIntegral[a + b*x]*Sinh[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b)} -{Cosh[a + b*x]*CoshIntegral[a + b*x]/x^1, x, 0, CannotIntegrate[(Cosh[a + b*x]*CoshIntegral[a + b*x])/x, x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Hyper[a+b x] CoshIntegral[c+d x]^n*) - - -(* {x^2*Sinh[a + b*x]*CoshIntegral[c + d*x], x, 46, Cosh[a - c + (b - d)*x]/(2*b*(b - d)^2) + Cosh[a - c + (b - d)*x]/(b^2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*b*(b + d)^2) + Cosh[a + c + (b + d)*x]/(b^2*(b + d)) - (CoshIntegral[((b - d)*(c + d*x))/d]*((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (CoshIntegral[((b + d)*(c + d*x))/d]*((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d]))/(2*b^3*d^2) + (CoshIntegral[c + d*x]*((2 + b^2*x^2)*Cosh[a + b*x] - 2*b*x*Sinh[a + b*x]))/b^3 + (c*Sinh[a - c + (b - d)*x])/(2*b*(b - d)*d) - (x*Sinh[a - c + (b - d)*x])/(2*b*(b - d)) + (c*Sinh[a + c + (b + d)*x])/(2*b*d*(b + d)) - (x*Sinh[a + c + (b + d)*x])/(2*b*(b + d)) - ((2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d])*SinhIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) - ((2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d])*SinhIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Sinh[a + b*x]*CoshIntegral[c + d*x], x, 24, (c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (x*Cosh[a + b*x]*CoshIntegral[c + d*x])/b + (c*Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) + (CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b^2) + (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b^2) - (CoshIntegral[c + d*x]*Sinh[a + b*x])/b^2 - Sinh[a - c + (b - d)*x]/(2*b*(b - d)) - Sinh[a + c + (b + d)*x]/(2*b*(b + d)) + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d)} -{x^0*Sinh[a + b*x]*CoshIntegral[c + d*x], x, 9, -((Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b)) + (Cosh[a + b*x]*CoshIntegral[c + d*x])/b - (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b) - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Sinh[a + b*x]*CoshIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(CoshIntegral[c + d*x]*Sinh[a + b*x])/x, x]} - - -(* {x^2*Cosh[a + b*x]*CoshIntegral[c + d*x], x, 46, (c*Cosh[a - c + (b - d)*x])/(2*b*(b - d)*d) - (x*Cosh[a - c + (b - d)*x])/(2*b*(b - d)) + (c*Cosh[a + c + (b + d)*x])/(2*b*d*(b + d)) - (x*Cosh[a + c + (b + d)*x])/(2*b*(b + d)) - (CoshIntegral[((b - d)*(c + d*x))/d]*(2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (CoshIntegral[((b + d)*(c + d*x))/d]*(2*b*c*d*Cosh[a - (b*c)/d] + (b^2*c^2 + 2*d^2)*Sinh[a - (b*c)/d]))/(2*b^3*d^2) - (CoshIntegral[c + d*x]*(2*b*x*Cosh[a + b*x] - (2 + b^2*x^2)*Sinh[a + b*x]))/b^3 + Sinh[a - c + (b - d)*x]/(2*b*(b - d)^2) + Sinh[a - c + (b - d)*x]/(b^2*(b - d)) + Sinh[a + c + (b + d)*x]/(2*b*(b + d)^2) + Sinh[a + c + (b + d)*x]/(b^2*(b + d)) - (((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d])*SinhIntegral[((b - d)*(c + d*x))/d])/(2*b^3*d^2) - (((b^2*c^2 + 2*d^2)*Cosh[a - (b*c)/d] + 2*b*c*d*Sinh[a - (b*c)/d])*SinhIntegral[((b + d)*(c + d*x))/d])/(2*b^3*d^2)} *) -{x^1*Cosh[a + b*x]*CoshIntegral[c + d*x], x, 24, -(Cosh[a - c + (b - d)*x]/(2*b*(b - d))) - Cosh[a + c + (b + d)*x]/(2*b*(b + d)) + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) - (Cosh[a + b*x]*CoshIntegral[c + d*x])/b^2 + (Cosh[a - (b*c)/d]*CoshIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2) + (c*CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b*d) + (c*CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b*d) + (x*CoshIntegral[c + d*x]*Sinh[a + b*x])/b + (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b*d) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b^2) + (c*Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b*d) + (Sinh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b^2)} -{x^0*Cosh[a + b*x]*CoshIntegral[c + d*x], x, 9, -((CoshIntegral[(c*(b - d))/d + (b - d)*x]*Sinh[a - (b*c)/d])/(2*b)) - (CoshIntegral[(c*(b + d))/d + (b + d)*x]*Sinh[a - (b*c)/d])/(2*b) + (CoshIntegral[c + d*x]*Sinh[a + b*x])/b - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b - d))/d + (b - d)*x])/(2*b) - (Cosh[a - (b*c)/d]*SinhIntegral[(c*(b + d))/d + (b + d)*x])/(2*b)} -{Cosh[a + b*x]*CoshIntegral[c + d*x]/x^1, x, 0, CannotIntegrate[(Cosh[a + b*x]*CoshIntegral[c + d*x])/x, x]} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.6 Gamma functions.m b/test/methods/rule_based/test_files/8 Special functions/8.6 Gamma functions.m deleted file mode 100644 index 0c01201..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.6 Gamma functions.m +++ /dev/null @@ -1,417 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Gamma Functions*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m Gamma[n, b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Gamma[n, b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{Gamma[0, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[0, a*x] - Gamma[101, a*x]/(101*a^101)} - -{Gamma[0, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[0, a*x] - Gamma[3, a*x]/(3*a^3)} -{Gamma[0, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[0, a*x] - Gamma[2, a*x]/(2*a^2)} -{Gamma[0, a*x]*x^0, x, 1, -(1/(E^(a*x)*a)) + x*Gamma[0, a*x]} -{Gamma[0, a*x]/x^1, x, 1, a*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-a)*x] - EulerGamma*Log[x] - (1/2)*Log[a*x]^2} -{Gamma[0, a*x]/x^2, x, 1, a*Gamma[-1, a*x] - Gamma[0, a*x]/x} -{Gamma[0, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-2, a*x] - Gamma[0, a*x]/(2*x^2)} -{Gamma[0, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-3, a*x] - Gamma[0, a*x]/(3*x^3)} - - -{Gamma[1, a*x]*x^3, x, 4, -(6/(E^(a*x)*a^4)) - (6*x)/(E^(a*x)*a^3) - (3*x^2)/(E^(a*x)*a^2) - x^3/(E^(a*x)*a)} -{Gamma[1, a*x]*x^2, x, 3, -(2/(E^(a*x)*a^3)) - (2*x)/(E^(a*x)*a^2) - x^2/(E^(a*x)*a)} -{Gamma[1, a*x]*x^1, x, 2, -(1/(E^(a*x)*a^2)) - x/(E^(a*x)*a)} -{Gamma[1, a*x]*x^0, x, 1, -(1/(E^(a*x)*a))} -{Gamma[1, a*x]/x^1, x, 1, ExpIntegralEi[(-a)*x]} -{Gamma[1, a*x]/x^2, x, 2, -(1/(E^(a*x)*x)) - a*ExpIntegralEi[(-a)*x]} -{Gamma[1, a*x]/x^3, x, 3, -(1/(E^(a*x)*(2*x^2))) + a/(E^(a*x)*(2*x)) + (1/2)*a^2*ExpIntegralEi[(-a)*x]} -{Gamma[1, a*x]/x^4, x, 4, -(1/(E^(a*x)*(3*x^3))) + a/(E^(a*x)*(6*x^2)) - a^2/(E^(a*x)*(6*x)) - (1/6)*a^3*ExpIntegralEi[(-a)*x]} - - -{Gamma[2, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[2, a*x] - Gamma[103, a*x]/(101*a^101)} - -{Gamma[2, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[2, a*x] - Gamma[5, a*x]/(3*a^3)} -{Gamma[2, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[2, a*x] - Gamma[4, a*x]/(2*a^2)} -{Gamma[2, a*x]*x^0, x, 1, x*Gamma[2, a*x] - Gamma[3, a*x]/a} -{Gamma[2, a*x]/x^1, x, 2, -E^(-a*x) + ExpIntegralEi[-a*x]} -{Gamma[2, a*x]/x^2, x, 1, a/E^(a*x) - Gamma[2, a*x]/x} -{Gamma[2, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[0, a*x] - Gamma[2, a*x]/(2*x^2)} -{Gamma[2, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-1, a*x] - Gamma[2, a*x]/(3*x^3)} - - -{Gamma[3, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[3, a*x] - Gamma[104, a*x]/(101*a^101)} - -{Gamma[3, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[3, a*x] - Gamma[6, a*x]/(3*a^3)} -{Gamma[3, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[3, a*x] - Gamma[5, a*x]/(2*a^2)} -{Gamma[3, a*x]*x^0, x, 1, x*Gamma[3, a*x] - Gamma[4, a*x]/a} -{Gamma[3, a*x]/x^1, x, 3, -2/E^(a*x) + 2*ExpIntegralEi[(-a)*x] - Gamma[2, a*x]} -{Gamma[3, a*x]/x^2, x, 1, a*Gamma[2, a*x] - Gamma[3, a*x]/x} -{Gamma[3, a*x]/x^3, x, 1, ((1/2)*a^2)/E^(a*x) - Gamma[3, a*x]/(2*x^2)} -{Gamma[3, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[0, a*x] - Gamma[3, a*x]/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{Gamma[-1, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[-1, a*x] - Gamma[100, a*x]/(101*a^101)} - -{Gamma[-1, a*x]*x^3, x, 1, (1/4)*x^4*Gamma[-1, a*x] - Gamma[3, a*x]/(4*a^4)} -{Gamma[-1, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[-1, a*x] - Gamma[2, a*x]/(3*a^3)} -{Gamma[-1, a*x]*x^1, x, 1, -(1/(E^(a*x)*(2*a^2))) + (1/2)*x^2*Gamma[-1, a*x]} -{Gamma[-1, a*x]*x^0, x, 1, x*Gamma[-1, a*x] - Gamma[0, a*x]/a} -{Gamma[-1, a*x]/x^1, x, 2, -Gamma[-1, a*x] - a*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-a)*x] + EulerGamma*Log[x] + (1/2)*Log[a*x]^2} -{Gamma[-1, a*x]/x^2, x, 1, a*Gamma[-2, a*x] - Gamma[-1, a*x]/x} -{Gamma[-1, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-3, a*x] - Gamma[-1, a*x]/(2*x^2)} -{Gamma[-1, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-4, a*x] - Gamma[-1, a*x]/(3*x^3)} - - -{Gamma[-2, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[-2, a*x] - Gamma[99, a*x]/(101*a^101)} - -{Gamma[-2, a*x]*x^3, x, 1, (1/4)*x^4*Gamma[-2, a*x] - Gamma[2, a*x]/(4*a^4)} -{Gamma[-2, a*x]*x^2, x, 1, -(1/(E^(a*x)*(3*a^3))) + (1/3)*x^3*Gamma[-2, a*x]} -{Gamma[-2, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[-2, a*x] - Gamma[0, a*x]/(2*a^2)} -{Gamma[-2, a*x]*x^0, x, 1, x*Gamma[-2, a*x] - Gamma[-1, a*x]/a} -{Gamma[-2, a*x]/x^1, x, 3, (-(1/2))*Gamma[-2, a*x] + (1/2)*Gamma[-1, a*x] + (1/2)*a*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-a)*x] - (1/2)*EulerGamma*Log[x] - (1/4)*Log[a*x]^2} -{Gamma[-2, a*x]/x^2, x, 1, a*Gamma[-3, a*x] - Gamma[-2, a*x]/x} -{Gamma[-2, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-4, a*x] - Gamma[-2, a*x]/(2*x^2)} -{Gamma[-2, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-5, a*x] - Gamma[-2, a*x]/(3*x^3)} - - -{Gamma[-3, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[-3, a*x] - Gamma[98, a*x]/(101*a^101)} - -{Gamma[-3, a*x]*x^3, x, 1, -(1/(E^(a*x)*(4*a^4))) + (1/4)*x^4*Gamma[-3, a*x]} -{Gamma[-3, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[-3, a*x] - Gamma[0, a*x]/(3*a^3)} -{Gamma[-3, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[-3, a*x] - Gamma[-1, a*x]/(2*a^2)} -{Gamma[-3, a*x]*x^0, x, 1, x*Gamma[-3, a*x] - Gamma[-2, a*x]/a} -{Gamma[-3, a*x]/x^1, x, 4, (-(1/3))*Gamma[-3, a*x] + (1/6)*Gamma[-2, a*x] - (1/6)*Gamma[-1, a*x] - (1/6)*a*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, (-a)*x] + (1/6)*EulerGamma*Log[x] + (1/12)*Log[a*x]^2} -{Gamma[-3, a*x]/x^2, x, 1, a*Gamma[-4, a*x] - Gamma[-3, a*x]/x} -{Gamma[-3, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-5, a*x] - Gamma[-3, a*x]/(2*x^2)} -{Gamma[-3, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-6, a*x] - Gamma[-3, a*x]/(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Gamma[n/2, b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>0*) - - -{Gamma[1/2, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[1/2, a*x] - Gamma[203/2, a*x]/(101*a^101)} - -{Gamma[1/2, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[1/2, a*x] - Gamma[7/2, a*x]/(3*a^3)} -{Gamma[1/2, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[1/2, a*x] - Gamma[5/2, a*x]/(2*a^2)} -{Gamma[1/2, a*x]*x^0, x, 1, x*Gamma[1/2, a*x] - Gamma[3/2, a*x]/a} -{Gamma[1/2, a*x]/x^1, x, 1, -4*Sqrt[a*x]*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-a)*x] + Sqrt[Pi]*Log[x]} -{Gamma[1/2, a*x]/x^2, x, 1, a*Gamma[-(1/2), a*x] - Gamma[1/2, a*x]/x} -{Gamma[1/2, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-(3/2), a*x] - Gamma[1/2, a*x]/(2*x^2)} -{Gamma[1/2, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-(5/2), a*x] - Gamma[1/2, a*x]/(3*x^3)} - - -{Gamma[3/2, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[3/2, a*x] - Gamma[205/2, a*x]/(101*a^101)} - -{Gamma[3/2, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[3/2, a*x] - Gamma[9/2, a*x]/(3*a^3)} -{Gamma[3/2, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[3/2, a*x] - Gamma[7/2, a*x]/(2*a^2)} -{Gamma[3/2, a*x]*x^0, x, 1, x*Gamma[3/2, a*x] - Gamma[5/2, a*x]/a} -{Gamma[3/2, a*x]/x^1, x, 1, (-(4/9))*(a*x)^(3/2)*HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, (-a)*x] + (1/2)*Sqrt[Pi]*Log[x]} -{Gamma[3/2, a*x]/x^2, x, 1, a*Gamma[1/2, a*x] - Gamma[3/2, a*x]/x} -{Gamma[3/2, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-(1/2), a*x] - Gamma[3/2, a*x]/(2*x^2)} -{Gamma[3/2, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-(3/2), a*x] - Gamma[3/2, a*x]/(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m Gamma[n, b x] when m symbolic*) - - -{(d*x)^m*Gamma[3, b*x], x, 1, ((d*x)^(1 + m)*Gamma[3, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[4 + m, b*x])/((b*x)^m*(b*(1 + m)))} -{(d*x)^m*Gamma[2, b*x], x, 1, ((d*x)^(1 + m)*Gamma[2, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[3 + m, b*x])/((b*x)^m*(b*(1 + m)))} -{(d*x)^m*Gamma[1, b*x], x, 1, -(((d*x)^m*Gamma[1 + m, b*x])/((b*x)^m*b))} -{(d*x)^m*Gamma[0, b*x], x, 1, ((d*x)^(1 + m)*Gamma[0, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[1 + m, b*x])/((b*x)^m*(b*(1 + m)))} -{(d*x)^m*Gamma[-1, b*x], x, 1, ((d*x)^(1 + m)*Gamma[-1, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[m, b*x])/((b*x)^m*(b*(1 + m)))} -{(d*x)^m*Gamma[-2, b*x], x, 1, ((d*x)^(1 + m)*Gamma[-2, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[-1 + m, b*x])/((b*x)^m*(b*(1 + m)))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m Gamma[n, b x] when n symbolic*) - - -{x^m*Gamma[n, x], x, 1, (x^(1 + m)*Gamma[n, x])/(1 + m) - Gamma[1 + m + n, x]/(1 + m)} -{x^m*Gamma[n, b*x], x, 1, (x^(1 + m)*Gamma[n, b*x])/(1 + m) - (x^m*Gamma[1 + m + n, b*x])/((b*x)^m*(b*(1 + m)))} -{(d*x)^m*Gamma[n, x], x, 1, ((d*x)^(1 + m)*Gamma[n, x])/(d*(1 + m)) - ((d*x)^m*Gamma[1 + m + n, x])/(x^m*(1 + m))} -{(d*x)^m*Gamma[n, b*x], x, 1, ((d*x)^(1 + m)*Gamma[n, b*x])/(d*(1 + m)) - ((d*x)^m*Gamma[1 + m + n, b*x])/((b*x)^m*(b*(1 + m)))} - - -{Gamma[n, a*x]*x^100, x, 1, (1/101)*x^101*Gamma[n, a*x] - Gamma[101 + n, a*x]/(101*a^101)} - -{Gamma[n, a*x]*x^2, x, 1, (1/3)*x^3*Gamma[n, a*x] - Gamma[3 + n, a*x]/(3*a^3)} -{Gamma[n, a*x]*x^1, x, 1, (1/2)*x^2*Gamma[n, a*x] - Gamma[2 + n, a*x]/(2*a^2)} -{Gamma[n, a*x]*x^0, x, 1, x*Gamma[n, a*x] - Gamma[1 + n, a*x]/a} -{Gamma[n, a*x]/x^1, x, 1, -(((a*x)^n*HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, (-a)*x])/n^2) + Gamma[n]*Log[x]} -{Gamma[n, a*x]/x^2, x, 1, a*Gamma[-1 + n, a*x] - Gamma[n, a*x]/x} -{Gamma[n, a*x]/x^3, x, 1, (1/2)*a^2*Gamma[-2 + n, a*x] - Gamma[n, a*x]/(2*x^2)} -{Gamma[n, a*x]/x^4, x, 1, (1/3)*a^3*Gamma[-3 + n, a*x] - Gamma[n, a*x]/(3*x^3)} - - -{Gamma[n, 2*x]*x^100, x, 1, (1/101)*x^101*Gamma[n, 2*x] - Gamma[101 + n, 2*x]/256065421246102339102334047485952} - -{Gamma[n, 2*x]*x^2, x, 1, (1/3)*x^3*Gamma[n, 2*x] - (1/24)*Gamma[3 + n, 2*x]} -{Gamma[n, 2*x]*x^1, x, 1, (1/2)*x^2*Gamma[n, 2*x] - (1/8)*Gamma[2 + n, 2*x]} -{Gamma[n, 2*x]*x^0, x, 1, x*Gamma[n, 2*x] - (1/2)*Gamma[1 + n, 2*x]} -{Gamma[n, 2*x]/x^1, x, 1, -((2^n*x^n*HypergeometricPFQ[{n, n}, {1 + n, 1 + n}, -2*x])/n^2) + Gamma[n]*Log[x]} -{Gamma[n, 2*x]/x^2, x, 1, 2*Gamma[-1 + n, 2*x] - Gamma[n, 2*x]/x} -{Gamma[n, 2*x]/x^3, x, 1, 2*Gamma[-2 + n, 2*x] - Gamma[n, 2*x]/(2*x^2)} -{Gamma[n, 2*x]/x^4, x, 1, (8/3)*Gamma[-3 + n, 2*x] - Gamma[n, 2*x]/(3*x^3)} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m Gamma[n, a+b x]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Gamma[n, a+b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{(c+ d*x)^3*Gamma[0, a + b*x], x, 8, -(((b*c - a*d)^3*E^(-a - b*x))/(4*b^4)) - ((b*c - a*d)^4*Gamma[0, a + b*x])/(4*b^4*d) + ((c + d*x)^4*Gamma[0, a + b*x])/(4*d) - (d*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[2, (b*(c + d*x))/d])/(4*b^4) - (d^2*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[3, (b*(c + d*x))/d])/(4*b^4) - (d^3*E^(-a + (b*c)/d)*Gamma[4, (b*(c + d*x))/d])/(4*b^4)} -{(c+ d*x)^2*Gamma[0, a + b*x], x, 7, -(((b*c - a*d)^2*E^(-a - b*x))/(3*b^3)) - ((b*c - a*d)^3*Gamma[0, a + b*x])/(3*b^3*d) + ((c + d*x)^3*Gamma[0, a + b*x])/(3*d) - (d*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[2, (b*(c + d*x))/d])/(3*b^3) - (d^2*E^(-a + (b*c)/d)*Gamma[3, (b*(c + d*x))/d])/(3*b^3)} -{(c+ d*x)^1*Gamma[0, a + b*x], x, 6, -(((b*c - a*d)*E^(-a - b*x))/(2*b^2)) - ((b*c - a*d)^2*Gamma[0, a + b*x])/(2*b^2*d) + ((c + d*x)^2*Gamma[0, a + b*x])/(2*d) - (d*E^(-a + (b*c)/d)*Gamma[2, (b*(c + d*x))/d])/(2*b^2)} -{(c+ d*x)^0*Gamma[0, a + b*x], x, 1, -(E^(-a - b*x)/b) + ((a + b*x)*Gamma[0, a + b*x])/b} -{Gamma[0, a + b*x]/(c+ d*x)^1, x, 0, Unintegrable[Gamma[0, a + b*x]/(c + d*x), x]} -{Gamma[0, a + b*x]/(c+ d*x)^2, x, 5, (b*Gamma[0, a + b*x])/(d*(b*c - a*d)) - Gamma[0, a + b*x]/(d*(c + d*x)) - (b*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(d*(b*c - a*d))} -{Gamma[0, a + b*x]/(c+ d*x)^3, x, 6, -((b^2*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*d^2*(b*c - a*d))) + (b^2*Gamma[0, a + b*x])/(2*d*(b*c - a*d)^2) - Gamma[0, a + b*x]/(2*d*(c + d*x)^2) - (b^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(2*d*(b*c - a*d)^2)} -{Gamma[0, a + b*x]/(c+ d*x)^4, x, 7, -((b^3*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*d^3*(b*c - a*d))) - (b^3*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(3*d^2*(b*c - a*d)^2) + (b^3*Gamma[0, a + b*x])/(3*d*(b*c - a*d)^3) - Gamma[0, a + b*x]/(3*d*(c + d*x)^3) - (b^3*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(3*d*(b*c - a*d)^3)} - - -{(c+ d*x)^4*Gamma[1, a + b*x], x, 5, -((24*d^4*E^(-a - b*x))/b^5) - (24*d^3*E^(-a - b*x)*(c + d*x))/b^4 - (12*d^2*E^(-a - b*x)*(c + d*x)^2)/b^3 - (4*d*E^(-a - b*x)*(c + d*x)^3)/b^2 - (E^(-a - b*x)*(c + d*x)^4)/b} -{(c+ d*x)^3*Gamma[1, a + b*x], x, 4, -((6*d^3*E^(-a - b*x))/b^4) - (6*d^2*E^(-a - b*x)*(c + d*x))/b^3 - (3*d*E^(-a - b*x)*(c + d*x)^2)/b^2 - (E^(-a - b*x)*(c + d*x)^3)/b} -{(c+ d*x)^2*Gamma[1, a + b*x], x, 3, -((2*d^2*E^(-a - b*x))/b^3) - (2*d*E^(-a - b*x)*(c + d*x))/b^2 - (E^(-a - b*x)*(c + d*x)^2)/b} -{(c+ d*x)^1*Gamma[1, a + b*x], x, 2, -((d*E^(-a - b*x))/b^2) - (E^(-a - b*x)*(c + d*x))/b} -{(c+ d*x)^0*Gamma[1, a + b*x], x, 1, -(E^(-a - b*x)/b)} -{Gamma[1, a + b*x]/(c+ d*x)^1, x, 1, (E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d} -{Gamma[1, a + b*x]/(c+ d*x)^2, x, 2, -(E^(-a - b*x)/(d*(c + d*x))) - (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^2} -{Gamma[1, a + b*x]/(c+ d*x)^3, x, 3, -(E^(-a - b*x)/(2*d*(c + d*x)^2)) + (b*E^(-a - b*x))/(2*d^2*(c + d*x)) + (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^3)} -{Gamma[1, a + b*x]/(c+ d*x)^4, x, 4, -(E^(-a - b*x)/(3*d*(c + d*x)^3)) + (b*E^(-a - b*x))/(6*d^2*(c + d*x)^2) - (b^2*E^(-a - b*x))/(6*d^3*(c + d*x)) - (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^4)} - - -{(c+ d*x)^3*Gamma[2, a + b*x], x, 5, ((c + d*x)^4*Gamma[2, a + b*x])/(4*d) + (d^2*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[5, (b*(c + d*x))/d])/(4*b^4) - (d^3*E^(-a + (b*c)/d)*Gamma[6, (b*(c + d*x))/d])/(4*b^4)} -{(c+ d*x)^2*Gamma[2, a + b*x], x, 5, ((c + d*x)^3*Gamma[2, a + b*x])/(3*d) + (d*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[4, (b*(c + d*x))/d])/(3*b^3) - (d^2*E^(-a + (b*c)/d)*Gamma[5, (b*(c + d*x))/d])/(3*b^3)} -{(c+ d*x)^1*Gamma[2, a + b*x], x, 5, ((c + d*x)^2*Gamma[2, a + b*x])/(2*d) + ((b*c - a*d)*E^(-a + (b*c)/d)*Gamma[3, (b*(c + d*x))/d])/(2*b^2) - (d*E^(-a + (b*c)/d)*Gamma[4, (b*(c + d*x))/d])/(2*b^2)} -{(c+ d*x)^0*Gamma[2, a + b*x], x, 1, ((a + b*x)*Gamma[2, a + b*x])/b - Gamma[3, a + b*x]/b} -{Gamma[2, a + b*x]/(c+ d*x)^1, x, 6, -(E^(-a - b*x)/d) + (E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d - ((b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^2} -{Gamma[2, a + b*x]/(c+ d*x)^2, x, 5, (b*E^(-a - b*x))/d^2 - (b*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/d^3 - Gamma[2, a + b*x]/(d*(c + d*x))} -{Gamma[2, a + b*x]/(c+ d*x)^3, x, 5, -((b^2*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*d^4)) + (b^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(2*d^3) - Gamma[2, a + b*x]/(2*d*(c + d*x)^2)} -{Gamma[2, a + b*x]/(c+ d*x)^4, x, 5, -((b^3*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*d^5)) + (b^3*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(3*d^4) - Gamma[2, a + b*x]/(3*d*(c + d*x)^3)} -{Gamma[2, a + b*x]/(c+ d*x)^5, x, 5, -((b^4*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[-3, (b*(c + d*x))/d])/(4*d^6)) + (b^4*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(4*d^5) - Gamma[2, a + b*x]/(4*d*(c + d*x)^4)} - - -{(c+ d*x)^3*Gamma[3, a + b*x], x, 6, ((c + d*x)^4*Gamma[3, a + b*x])/(4*d) - (d*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[5, (b*(c + d*x))/d])/(4*b^4) + (d^2*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[6, (b*(c + d*x))/d])/(2*b^4) - (d^3*E^(-a + (b*c)/d)*Gamma[7, (b*(c + d*x))/d])/(4*b^4)} -{(c+ d*x)^2*Gamma[3, a + b*x], x, 6, ((c + d*x)^3*Gamma[3, a + b*x])/(3*d) - ((b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[4, (b*(c + d*x))/d])/(3*b^3) + (2*d*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[5, (b*(c + d*x))/d])/(3*b^3) - (d^2*E^(-a + (b*c)/d)*Gamma[6, (b*(c + d*x))/d])/(3*b^3)} -{(c+ d*x)^1*Gamma[3, a + b*x], x, 6, -(((b*c - a*d)^2*Gamma[3, a + b*x])/(2*b^2*d)) + ((c + d*x)^2*Gamma[3, a + b*x])/(2*d) - ((b*c - a*d)*Gamma[4, a + b*x])/b^2 - (d*Gamma[5, a + b*x])/(2*b^2)} -{(c+ d*x)^0*Gamma[3, a + b*x], x, 1, ((a + b*x)*Gamma[3, a + b*x])/b - Gamma[4, a + b*x]/b} -{Gamma[3, a + b*x]/(c+ d*x)^1, x, 13, -((3*E^(-a - b*x))/d) + ((b*c - a*d)*E^(-a - b*x))/d^2 - (E^(-a - b*x)*(a + b*x))/d + (2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d - (2*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^2 + ((b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^3} -{Gamma[3, a + b*x]/(c+ d*x)^2, x, 6, -((b*(b*c - a*d)*E^(-a - b*x))/d^3) + (b*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/d^4 + (b*Gamma[2, a + b*x])/d^2 - Gamma[3, a + b*x]/(d*(c + d*x))} -{Gamma[3, a + b*x]/(c+ d*x)^3, x, 6, (b^2*E^(-a - b*x))/(2*d^3) + (b^2*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*d^5) - (b^2*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/d^4 - Gamma[3, a + b*x]/(2*d*(c + d*x)^2)} -{Gamma[3, a + b*x]/(c+ d*x)^4, x, 6, (b^3*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*d^6) - (2*b^3*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(3*d^5) + (b^3*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(3*d^4) - Gamma[3, a + b*x]/(3*d*(c + d*x)^3)} -{Gamma[3, a + b*x]/(c+ d*x)^5, x, 6, (b^4*(b*c - a*d)^2*E^(-a + (b*c)/d)*Gamma[-3, (b*(c + d*x))/d])/(4*d^7) - (b^4*(b*c - a*d)*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(2*d^6) + (b^4*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(4*d^5) - Gamma[3, a + b*x]/(4*d*(c + d*x)^4)} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{(c+ d*x)^3*Gamma[-1, a + b*x], x, 8, -((3*d*(b*c - a*d)^2*E^(-a - b*x))/(2*b^4)) - ((b*c - a*d)^4*Gamma[-1, a + b*x])/(4*b^4*d) + ((c + d*x)^4*Gamma[-1, a + b*x])/(4*d) - ((b*c - a*d)^3*Gamma[0, a + b*x])/b^4 - (d^2*(b*c - a*d)*Gamma[2, a + b*x])/b^4 - (d^3*Gamma[3, a + b*x])/(4*b^4)} -{(c+ d*x)^2*Gamma[-1, a + b*x], x, 7, -((d*(3*b*c - 2*a*d)*E^(-a - b*x))/(3*b^3)) - ((b*c - a*d)^3*Gamma[-1, a + b*x])/(3*b^3*d) + ((c + d*x)^3*Gamma[-1, a + b*x])/(3*d) - ((b*c - a*d)^2*Gamma[0, a + b*x])/b^3 - (d^2*Gamma[2, b*x])/(E^a*(3*b^3))} -{(c+ d*x)^1*Gamma[-1, a + b*x], x, 6, -((d*E^(-a - b*x))/(2*b^2)) - ((b*c - a*d)^2*Gamma[-1, a + b*x])/(2*b^2*d) + ((c + d*x)^2*Gamma[-1, a + b*x])/(2*d) - ((b*c - a*d)*Gamma[0, a + b*x])/b^2} -{(c+ d*x)^0*Gamma[-1, a + b*x], x, 1, ((a + b*x)*Gamma[-1, a + b*x])/b - Gamma[0, a + b*x]/b} -{Gamma[-1, a + b*x]/(c+ d*x)^1, x, 0, Unintegrable[Gamma[-1, a + b*x]/(c + d*x), x]} -{Gamma[-1, a + b*x]/(c+ d*x)^2, x, 6, (b*Gamma[-1, a + b*x])/(d*(b*c - a*d)) - Gamma[-1, a + b*x]/(d*(c + d*x)) - (b*Gamma[0, a + b*x])/(b*c - a*d)^2 + (b*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^2} -{Gamma[-1, a + b*x]/(c+ d*x)^3, x, 7, (b^2*Gamma[-1, a + b*x])/(2*d*(b*c - a*d)^2) - Gamma[-1, a + b*x]/(2*d*(c + d*x)^2) + (b^2*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*d*(b*c - a*d)^2) - (b^2*Gamma[0, a + b*x])/(b*c - a*d)^3 + (b^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^3} -{Gamma[-1, a + b*x]/(c+ d*x)^4, x, 8, (b^3*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*d^2*(b*c - a*d)^2) + (b^3*Gamma[-1, a + b*x])/(3*d*(b*c - a*d)^3) - Gamma[-1, a + b*x]/(3*d*(c + d*x)^3) + (2*b^3*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(3*d*(b*c - a*d)^3) - (b^3*Gamma[0, a + b*x])/(b*c - a*d)^4 + (b^3*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^4} - - -{(c+ d*x)^3*Gamma[-2, a + b*x], x, 8, -((d^2*(4*b*c - 3*a*d)*E^(-a - b*x))/(4*b^4)) - ((b*c - a*d)^4*Gamma[-2, a + b*x])/(4*b^4*d) + ((c + d*x)^4*Gamma[-2, a + b*x])/(4*d) - ((b*c - a*d)^3*Gamma[-1, a + b*x])/b^4 - (3*d*(b*c - a*d)^2*Gamma[0, a + b*x])/(2*b^4) - (d^3*Gamma[2, b*x])/(E^a*(4*b^4))} -{(c+ d*x)^2*Gamma[-2, a + b*x], x, 7, -((d^2*E^(-a - b*x))/(3*b^3)) - ((b*c - a*d)^3*Gamma[-2, a + b*x])/(3*b^3*d) + ((c + d*x)^3*Gamma[-2, a + b*x])/(3*d) - ((b*c - a*d)^2*Gamma[-1, a + b*x])/b^3 - (d*(b*c - a*d)*Gamma[0, a + b*x])/b^3} -{(c+ d*x)^1*Gamma[-2, a + b*x], x, 6, -(((b*c - a*d)^2*Gamma[-2, a + b*x])/(2*b^2*d)) + ((c + d*x)^2*Gamma[-2, a + b*x])/(2*d) - ((b*c - a*d)*Gamma[-1, a + b*x])/b^2 - (d*Gamma[0, a + b*x])/(2*b^2)} -{(c+ d*x)^0*Gamma[-2, a + b*x], x, 1, ((a + b*x)*Gamma[-2, a + b*x])/b - Gamma[-1, a + b*x]/b} -{Gamma[-2, a + b*x]/(c+ d*x)^1, x, 0, Unintegrable[Gamma[-2, a + b*x]/(c + d*x), x]} -{Gamma[-2, a + b*x]/(c+ d*x)^2, x, 7, (b*Gamma[-2, a + b*x])/(d*(b*c - a*d)) - Gamma[-2, a + b*x]/(d*(c + d*x)) - (b*Gamma[-1, a + b*x])/(b*c - a*d)^2 + (b*d*Gamma[0, a + b*x])/(b*c - a*d)^3 - (b*d*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^3} -{Gamma[-2, a + b*x]/(c+ d*x)^3, x, 8, (b^2*Gamma[-2, a + b*x])/(2*d*(b*c - a*d)^2) - Gamma[-2, a + b*x]/(2*d*(c + d*x)^2) - (b^2*Gamma[-1, a + b*x])/(b*c - a*d)^3 - (b^2*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*(b*c - a*d)^3) + (3*b^2*d*Gamma[0, a + b*x])/(2*(b*c - a*d)^4) - (3*b^2*d*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(2*(b*c - a*d)^4)} -{Gamma[-2, a + b*x]/(c+ d*x)^4, x, 9, (b^3*Gamma[-2, a + b*x])/(3*d*(b*c - a*d)^3) - Gamma[-2, a + b*x]/(3*d*(c + d*x)^3) - (b^3*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*d*(b*c - a*d)^3) - (b^3*Gamma[-1, a + b*x])/(b*c - a*d)^4 - (b^3*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(b*c - a*d)^4 + (2*b^3*d*Gamma[0, a + b*x])/(b*c - a*d)^5 - (2*b^3*d*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^5} - - -{(c+ d*x)^3*Gamma[-3, a + b*x], x, 8, -((d^3*E^(-a - b*x))/(4*b^4)) - ((b*c - a*d)^4*Gamma[-3, a + b*x])/(4*b^4*d) + ((c + d*x)^4*Gamma[-3, a + b*x])/(4*d) - ((b*c - a*d)^3*Gamma[-2, a + b*x])/b^4 - (3*d*(b*c - a*d)^2*Gamma[-1, a + b*x])/(2*b^4) - (d^2*(b*c - a*d)*Gamma[0, a + b*x])/b^4} -{(c+ d*x)^2*Gamma[-3, a + b*x], x, 7, -(((b*c - a*d)^3*Gamma[-3, a + b*x])/(3*b^3*d)) + ((c + d*x)^3*Gamma[-3, a + b*x])/(3*d) - ((b*c - a*d)^2*Gamma[-2, a + b*x])/b^3 - (d*(b*c - a*d)*Gamma[-1, a + b*x])/b^3 - (d^2*Gamma[0, a + b*x])/(3*b^3)} -{(c+ d*x)^1*Gamma[-3, a + b*x], x, 6, -(((b*c - a*d)^2*Gamma[-3, a + b*x])/(2*b^2*d)) + ((c + d*x)^2*Gamma[-3, a + b*x])/(2*d) - ((b*c - a*d)*Gamma[-2, a + b*x])/b^2 - (d*Gamma[-1, a + b*x])/(2*b^2)} -{(c+ d*x)^0*Gamma[-3, a + b*x], x, 1, ((a + b*x)*Gamma[-3, a + b*x])/b - Gamma[-2, a + b*x]/b} -{Gamma[-3, a + b*x]/(c+ d*x)^1, x, 0, Unintegrable[Gamma[-3, a + b*x]/(c + d*x), x]} -{Gamma[-3, a + b*x]/(c+ d*x)^2, x, 8, (b*Gamma[-3, a + b*x])/(d*(b*c - a*d)) - Gamma[-3, a + b*x]/(d*(c + d*x)) - (b*Gamma[-2, a + b*x])/(b*c - a*d)^2 + (b*d*Gamma[-1, a + b*x])/(b*c - a*d)^3 - (b*d^2*Gamma[0, a + b*x])/(b*c - a*d)^4 + (b*d^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^4} -{Gamma[-3, a + b*x]/(c+ d*x)^3, x, 9, (b^2*Gamma[-3, a + b*x])/(2*d*(b*c - a*d)^2) - Gamma[-3, a + b*x]/(2*d*(c + d*x)^2) - (b^2*Gamma[-2, a + b*x])/(b*c - a*d)^3 + (3*b^2*d*Gamma[-1, a + b*x])/(2*(b*c - a*d)^4) + (b^2*d*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(2*(b*c - a*d)^4) - (2*b^2*d^2*Gamma[0, a + b*x])/(b*c - a*d)^5 + (2*b^2*d^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(b*c - a*d)^5} -{Gamma[-3, a + b*x]/(c+ d*x)^4, x, 10, (b^3*Gamma[-3, a + b*x])/(3*d*(b*c - a*d)^3) - Gamma[-3, a + b*x]/(3*d*(c + d*x)^3) - (b^3*Gamma[-2, a + b*x])/(b*c - a*d)^4 + (b^3*E^(-a + (b*c)/d)*Gamma[-2, (b*(c + d*x))/d])/(3*(b*c - a*d)^4) + (2*b^3*d*Gamma[-1, a + b*x])/(b*c - a*d)^5 + (4*b^3*d*E^(-a + (b*c)/d)*Gamma[-1, (b*(c + d*x))/d])/(3*(b*c - a*d)^5) - (10*b^3*d^2*Gamma[0, a + b*x])/(3*(b*c - a*d)^6) + (10*b^3*d^2*E^(-a + (b*c)/d)*Gamma[0, (b*(c + d*x))/d])/(3*(b*c - a*d)^6)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/2) Gamma[n, a+b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{x^(5/2)*Gamma[2, a + b*x], x, 5, (2/7)*x^(7/2)*Gamma[2, a + b*x] - (2*a*Sqrt[x]*Gamma[9/2, b*x])/(E^a*(7*b^3*Sqrt[b*x])) - (2*Sqrt[x]*Gamma[11/2, b*x])/(E^a*(7*b^3*Sqrt[b*x]))} -{x^(3/2)*Gamma[2, a + b*x], x, 5, (2/5)*x^(5/2)*Gamma[2, a + b*x] - (2*a*Sqrt[x]*Gamma[7/2, b*x])/(E^a*(5*b^2*Sqrt[b*x])) - (2*Sqrt[x]*Gamma[9/2, b*x])/(E^a*(5*b^2*Sqrt[b*x]))} -{x^(1/2)*Gamma[2, a + b*x], x, 5, (2/3)*x^(3/2)*Gamma[2, a + b*x] - (2*a*Sqrt[x]*Gamma[5/2, b*x])/(E^a*(3*b*Sqrt[b*x])) - (2*Sqrt[x]*Gamma[7/2, b*x])/(E^a*(3*b*Sqrt[b*x]))} -{Gamma[2, a + b*x]/x^(1/2), x, 5, -((2*a*Sqrt[x]*Gamma[3/2, b*x])/(E^a*Sqrt[b*x])) + 2*Sqrt[x]*Gamma[2, a + b*x] - (2*Sqrt[x]*Gamma[5/2, b*x])/(E^a*Sqrt[b*x])} -{Gamma[2, a + b*x]/x^(3/2), x, 5, (2*a*Sqrt[b*x]*Gamma[1/2, b*x])/(E^a*Sqrt[x]) + (2*b*Sqrt[x]*Gamma[3/2, b*x])/(E^a*Sqrt[b*x]) - (2*Gamma[2, a + b*x])/Sqrt[x]} -{Gamma[2, a + b*x]/x^(5/2), x, 5, (2*a*b*Sqrt[b*x]*Gamma[-(1/2), b*x])/(E^a*(3*Sqrt[x])) + (2*b*Sqrt[b*x]*Gamma[1/2, b*x])/(E^a*(3*Sqrt[x])) - (2*Gamma[2, a + b*x])/(3*x^(3/2))} -{Gamma[2, a + b*x]/x^(7/2), x, 5, (2*a*b^2*Sqrt[b*x]*Gamma[-(3/2), b*x])/(E^a*(5*Sqrt[x])) + (2*b^2*Sqrt[b*x]*Gamma[-(1/2), b*x])/(E^a*(5*Sqrt[x])) - (2*Gamma[2, a + b*x])/(5*x^(5/2))} -{Gamma[2, a + b*x]/x^(9/2), x, 5, (2*a*b^3*Sqrt[b*x]*Gamma[-(5/2), b*x])/(E^a*(7*Sqrt[x])) + (2*b^3*Sqrt[b*x]*Gamma[-(3/2), b*x])/(E^a*(7*Sqrt[x])) - (2*Gamma[2, a + b*x])/(7*x^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*n<0*) - - -{x^(3/2)*Gamma[-2, a + b*x], x, 0, Unintegrable[x^(3/2)*Gamma[-2, a + b*x], x]} -{x^(1/2)*Gamma[-2, a + b*x], x, 0, Unintegrable[Sqrt[x]*Gamma[-2, a + b*x], x]} -{Gamma[-2, a + b*x]/x^(1/2), x, 0, Unintegrable[Gamma[-2, a + b*x]/Sqrt[x], x]} -{Gamma[-2, a + b*x]/x^(3/2), x, 0, Unintegrable[Gamma[-2, a + b*x]/x^(3/2), x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^(m/3) Gamma[n, a+b x]*) - - -(* ::Subsubsection::Closed:: *) -(*n>=0*) - - -{x^(4/3)*Gamma[2, a + b*x], x, 5, (3/7)*x^(7/3)*Gamma[2, a + b*x] - (3*a*x^(1/3)*Gamma[10/3, b*x])/(E^a*(7*b^2*(b*x)^(1/3))) - (3*x^(1/3)*Gamma[13/3, b*x])/(E^a*(7*b^2*(b*x)^(1/3)))} -{x^(2/3)*Gamma[2, a + b*x], x, 5, (3/5)*x^(5/3)*Gamma[2, a + b*x] - (3*a*x^(2/3)*Gamma[8/3, b*x])/(E^a*(5*b*(b*x)^(2/3))) - (3*x^(2/3)*Gamma[11/3, b*x])/(E^a*(5*b*(b*x)^(2/3)))} -{x^(1/3)*Gamma[2, a + b*x], x, 5, (3/4)*x^(4/3)*Gamma[2, a + b*x] - (3*a*x^(1/3)*Gamma[7/3, b*x])/(E^a*(4*b*(b*x)^(1/3))) - (3*x^(1/3)*Gamma[10/3, b*x])/(E^a*(4*b*(b*x)^(1/3)))} -{Gamma[2, a + b*x]/x^(1/3), x, 5, -((3*a*x^(2/3)*Gamma[5/3, b*x])/(E^a*(2*(b*x)^(2/3)))) + (3/2)*x^(2/3)*Gamma[2, a + b*x] - (3*x^(2/3)*Gamma[8/3, b*x])/(E^a*(2*(b*x)^(2/3)))} -{Gamma[2, a + b*x]/x^(2/3), x, 5, -((3*a*x^(1/3)*Gamma[4/3, b*x])/(E^a*(b*x)^(1/3))) + 3*x^(1/3)*Gamma[2, a + b*x] - (3*x^(1/3)*Gamma[7/3, b*x])/(E^a*(b*x)^(1/3))} -{Gamma[2, a + b*x]/x^(4/3), x, 5, (3*a*(b*x)^(1/3)*Gamma[2/3, b*x])/(E^a*x^(1/3)) + (3*b*x^(2/3)*Gamma[5/3, b*x])/(E^a*(b*x)^(2/3)) - (3*Gamma[2, a + b*x])/x^(1/3)} - - -(* ::Subsubsection:: *) -(*n<0*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Gamma[n, a+b x] when m symbolic*) - - -{(c + d*x)^m*Gamma[3, a + b*x], x, 6, ((c + d*x)^(1 + m)*Gamma[3, a + b*x])/(d*(1 + m)) - ((b*c - a*d)^2*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[2 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*(b*d^2*(1 + m))) + (2*(b*c - a*d)*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[3 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*(b*d*(1 + m))) - (E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[4 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*(b*(1 + m)))} -{(c + d*x)^m*Gamma[2, a + b*x], x, 5, ((c + d*x)^(1 + m)*Gamma[2, a + b*x])/(d*(1 + m)) + ((b*c - a*d)*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[2 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*(b*d*(1 + m))) - (E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[3 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*(b*(1 + m)))} -{(c + d*x)^m*Gamma[1, a + b*x], x, 1, -((E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(((b*(c + d*x))/d)^m*b))} -{(c + d*x)^m*Gamma[0, a + b*x], x, 0, Unintegrable[(c + d*x)^m*Gamma[0, a + b*x], x]} -{(c + d*x)^m*Gamma[-1, a + b*x], x, 0, Unintegrable[(c + d*x)^m*Gamma[-1, a + b*x], x]} -{(c + d*x)^m*Gamma[-2, a + b*x], x, 0, Unintegrable[(c + d*x)^m*Gamma[-2, a + b*x], x]} -{(c + d*x)^m*Gamma[-3, a + b*x], x, 0, Unintegrable[(c + d*x)^m*Gamma[-3, a + b*x], x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (c+d x)^m Gamma[n, a+b x] when n symbolic*) - - -{x^m*Gamma[n, a + b*x], x, 0, Unintegrable[x^m*Gamma[n, a + b*x], x]} -{(c + d*x)^m*Gamma[n, a + b*x], x, 0, Unintegrable[(c + d*x)^m*Gamma[n, a + b*x], x]} - - -{(c + d*x)^4*Gamma[n, a + b*x], x, 9, -(((b*c - a*d)^5*Gamma[n, a + b*x])/(5*b^5*d)) + ((c + d*x)^5*Gamma[n, a + b*x])/(5*d) - ((b*c - a*d)^4*Gamma[1 + n, a + b*x])/b^5 - (2*d*(b*c - a*d)^3*Gamma[2 + n, a + b*x])/b^5 - (2*d^2*(b*c - a*d)^2*Gamma[3 + n, a + b*x])/b^5 - (d^3*(b*c - a*d)*Gamma[4 + n, a + b*x])/b^5 - (d^4*Gamma[5 + n, a + b*x])/(5*b^5)} -{(c + d*x)^3*Gamma[n, a + b*x], x, 8, -(((b*c - a*d)^4*Gamma[n, a + b*x])/(4*b^4*d)) + ((c + d*x)^4*Gamma[n, a + b*x])/(4*d) - ((b*c - a*d)^3*Gamma[1 + n, a + b*x])/b^4 - (3*d*(b*c - a*d)^2*Gamma[2 + n, a + b*x])/(2*b^4) - (d^2*(b*c - a*d)*Gamma[3 + n, a + b*x])/b^4 - (d^3*Gamma[4 + n, a + b*x])/(4*b^4)} -{(c + d*x)^2*Gamma[n, a + b*x], x, 7, -(((b*c - a*d)^3*Gamma[n, a + b*x])/(3*b^3*d)) + ((c + d*x)^3*Gamma[n, a + b*x])/(3*d) - ((b*c - a*d)^2*Gamma[1 + n, a + b*x])/b^3 - (d*(b*c - a*d)*Gamma[2 + n, a + b*x])/b^3 - (d^2*Gamma[3 + n, a + b*x])/(3*b^3)} -{(c + d*x)^1*Gamma[n, a + b*x], x, 6, -(((b*c - a*d)^2*Gamma[n, a + b*x])/(2*b^2*d)) + ((c + d*x)^2*Gamma[n, a + b*x])/(2*d) - ((b*c - a*d)*Gamma[1 + n, a + b*x])/b^2 - (d*Gamma[2 + n, a + b*x])/(2*b^2)} -{(c + d*x)^0*Gamma[n, a + b*x], x, 1, ((a + b*x)*Gamma[n, a + b*x])/b - Gamma[1 + n, a + b*x]/b} -{Gamma[n, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[Gamma[n, a + b*x]/(c + d*x), x]} -{Gamma[n, a + b*x]/(c + d*x)^2, x, 0, Unintegrable[Gamma[n, a + b*x]/(c + d*x)^2, x]} -{Gamma[n, a + b*x]/(c + d*x)^3, x, 0, Unintegrable[Gamma[n, a + b*x]/(c + d*x)^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (e x)^m Gamma[p, d (a+b Log[c x^n])]*) - - -{x^2*Gamma[p, d*(a + b*Log[c*x^n])], x, 4, (1/3)*x^3*Gamma[p, d*(a + b*Log[c*x^n])] - ((1/3)*x^3*Gamma[p, -(((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))]*(d*(a + b*Log[c*x^n]))^p)/(E^((3*a)/(b*n))*(c*x^n)^(3/n)*(-(((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)))^p)} -{x^1*Gamma[p, d*(a + b*Log[c*x^n])], x, 4, (1/2)*x^2*Gamma[p, d*(a + b*Log[c*x^n])] - ((1/2)*x^2*Gamma[p, -(((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))]*(d*(a + b*Log[c*x^n]))^p)/(E^((2*a)/(b*n))*(c*x^n)^(2/n)*(-(((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)))^p)} -{x^0*Gamma[p, d*(a + b*Log[c*x^n])], x, 5, x*Gamma[p, d*(a + b*Log[c*x^n])] - (x*Gamma[p, -(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))]*(d*(a + b*Log[c*x^n]))^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)))^p)} -{Gamma[p, d*(a + b*Log[c*x^n])]/x^1, x, 3, -(Gamma[1 + p, a*d + b*d*Log[c*x^n]]/(b*d*n)) + (Gamma[p, a*d + b*d*Log[c*x^n]]*(a + b*Log[c*x^n]))/(b*n)} -{Gamma[p, d*(a + b*Log[c*x^n])]/x^2, x, 4, -(Gamma[p, d*(a + b*Log[c*x^n])]/x) + (E^(a/(b*n))*(c*x^n)^(1/n)*Gamma[p, ((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]*(d*(a + b*Log[c*x^n]))^p)/((((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))^p*x)} -{Gamma[p, d*(a + b*Log[c*x^n])]/x^3, x, 4, -(Gamma[p, d*(a + b*Log[c*x^n])]/(2*x^2)) + (E^((2*a)/(b*n))*(c*x^n)^(2/n)*Gamma[p, ((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]*(d*(a + b*Log[c*x^n]))^p)/((((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))^p*(2*x^2))} - - -{(e*x)^m*Gamma[p, d*(a + b*Log[c*x^n])], x, 4, ((e*x)^(1 + m)*Gamma[p, d*(a + b*Log[c*x^n])])/(e*(1 + m)) - ((e*x)^(1 + m)*Gamma[p, -(((1 + m - b*d*n)*(a + b*Log[c*x^n]))/(b*n))]*(d*(a + b*Log[c*x^n]))^p)/(E^((a*(1 + m))/(b*n))*(c*x^n)^((1 + m)/n)*(-(((1 + m - b*d*n)*(a + b*Log[c*x^n]))/(b*n)))^p*(e*(1 + m)))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m LogGamma[a+b x]*) - - -{(c + d*x)^3*LogGamma[a + b*x], x, 4, -((6*d^3*PolyGamma[-5, a + b*x])/b^4) + (6*d^2*(c + d*x)*PolyGamma[-4, a + b*x])/b^3 - (3*d*(c + d*x)^2*PolyGamma[-3, a + b*x])/b^2 + ((c + d*x)^3*PolyGamma[-2, a + b*x])/b} -{(c + d*x)^2*LogGamma[a + b*x], x, 3, (2*d^2*PolyGamma[-4, a + b*x])/b^3 - (2*d*(c + d*x)*PolyGamma[-3, a + b*x])/b^2 + ((c + d*x)^2*PolyGamma[-2, a + b*x])/b} -{(c + d*x)^1*LogGamma[a + b*x], x, 2, -((d*PolyGamma[-3, a + b*x])/b^2) + ((c + d*x)*PolyGamma[-2, a + b*x])/b} -{(c + d*x)^0*LogGamma[a + b*x], x, 1, PolyGamma[-2, a + b*x]/b} -{LogGamma[a + b*x]/(c + d*x)^1, x, 0, Unintegrable[LogGamma[a + b*x]/(c + d*x), x]} -{LogGamma[a + b*x]/(c + d*x)^2, x, 0, Unintegrable[LogGamma[a + b*x]/(c + d*x)^2, x]} - - -{(c + d*x)^(3/2)*LogGamma[a + b*x], x, 0, Unintegrable[(c + d*x)^(3/2)*LogGamma[a + b*x], x]} -{(c + d*x)^(1/2)*LogGamma[a + b*x], x, 0, Unintegrable[Sqrt[c + d*x]*LogGamma[a + b*x], x]} -{LogGamma[a + b*x]/(c + d*x)^(1/2), x, 0, Unintegrable[LogGamma[a + b*x]/Sqrt[c + d*x], x]} - - -{(c + d*x)^2*Log[Gamma[a + b*x]], x, 6, ((c + d*x)^3*Log[Gamma[a + b*x]])/(3*d) - ((c + d*x)^3*LogGamma[a + b*x])/(3*d) + (2*d^2*PolyGamma[-4, a + b*x])/b^3 - (2*d*(c + d*x)*PolyGamma[-3, a + b*x])/b^2 + ((c + d*x)^2*PolyGamma[-2, a + b*x])/b} -{(c + d*x)^1*Log[Gamma[a + b*x]], x, 5, ((c + d*x)^2*Log[Gamma[a + b*x]])/(2*d) - ((c + d*x)^2*LogGamma[a + b*x])/(2*d) - (d*PolyGamma[-3, a + b*x])/b^2 + ((c + d*x)*PolyGamma[-2, a + b*x])/b} -{(c + d*x)^0*Log[Gamma[a + b*x]], x, 4, x*Log[Gamma[a + b*x]] - x*LogGamma[a + b*x] + PolyGamma[-2, a + b*x]/b} -{Log[Gamma[a + b*x]]/(c + d*x)^1, x, 2, (Log[c + d*x]*(Log[Gamma[a + b*x]] - LogGamma[a + b*x]))/d + Unintegrable[LogGamma[a + b*x]/(c + d*x), x]} -{Log[Gamma[a + b*x]]/(c + d*x)^2, x, 2, -(Log[Gamma[a + b*x]]/(d*(c + d*x))) + (b*Unintegrable[PolyGamma[0, a + b*x]/(c + d*x), x])/d} - - -(* ::Section::Closed:: *) -(*Integrands of the form (c+d x)^m PolyGamma[n, a+b x]*) - - -{(c + d*x)^m*PolyGamma[n, a + b*x], x, 0, Unintegrable[(c + d*x)^m*PolyGamma[n, a + b*x], x]} - - -{(c + d*x)^3*PolyGamma[n, a + b*x], x, 4, -((6*d^3*PolyGamma[-4 + n, a + b*x])/b^4) + (6*d^2*(c + d*x)*PolyGamma[-3 + n, a + b*x])/b^3 - (3*d*(c + d*x)^2*PolyGamma[-2 + n, a + b*x])/b^2 + ((c + d*x)^3*PolyGamma[-1 + n, a + b*x])/b} -{(c + d*x)^2*PolyGamma[n, a + b*x], x, 3, (2*d^2*PolyGamma[-3 + n, a + b*x])/b^3 - (2*d*(c + d*x)*PolyGamma[-2 + n, a + b*x])/b^2 + ((c + d*x)^2*PolyGamma[-1 + n, a + b*x])/b} -{(c + d*x)^1*PolyGamma[n, a + b*x], x, 2, -((d*PolyGamma[-2 + n, a + b*x])/b^2) + ((c + d*x)*PolyGamma[-1 + n, a + b*x])/b} -{(c + d*x)^0*PolyGamma[n, a + b*x], x, 1, PolyGamma[-1 + n, a + b*x]/b} -{PolyGamma[n, a + b*x]/(c + d*x)^1, x, 0, Unintegrable[PolyGamma[n, a + b*x]/(c + d*x), x]} -{PolyGamma[n, a + b*x]/(c + d*x)^2, x, 1, -(PolyGamma[n, a + b*x]/(d*(c + d*x))) + (b*Unintegrable[PolyGamma[1 + n, a + b*x]/(c + d*x), x])/d} -{PolyGamma[n, a + b*x]/(c + d*x)^3, x, 2, -(PolyGamma[n, a + b*x]/(2*d*(c + d*x)^2)) - (b*PolyGamma[1 + n, a + b*x])/(2*d^2*(c + d*x)) + (b^2*Unintegrable[PolyGamma[2 + n, a + b*x]/(c + d*x), x])/(2*d^2)} - - -{(c + d*x)^(3/2)*PolyGamma[n, a + b*x], x, 2, -((3*d*Sqrt[c + d*x]*PolyGamma[-2 + n, a + b*x])/(2*b^2)) + ((c + d*x)^(3/2)*PolyGamma[-1 + n, a + b*x])/b + (3*d^2*Unintegrable[PolyGamma[-2 + n, a + b*x]/Sqrt[c + d*x], x])/(4*b^2)} -{(c + d*x)^(1/2)*PolyGamma[n, a + b*x], x, 1, (Sqrt[c + d*x]*PolyGamma[-1 + n, a + b*x])/b - (d*Unintegrable[PolyGamma[-1 + n, a + b*x]/Sqrt[c + d*x], x])/(2*b)} -{PolyGamma[n, a + b*x]/(c + d*x)^(1/2), x, 0, Unintegrable[PolyGamma[n, a + b*x]/Sqrt[c + d*x], x]} -{PolyGamma[n, a + b*x]/(c + d*x)^(3/2), x, 1, -((2*PolyGamma[n, a + b*x])/(d*Sqrt[c + d*x])) + (2*b*Unintegrable[PolyGamma[1 + n, a + b*x]/Sqrt[c + d*x], x])/d} - - -{x^2*PolyGamma[1, a + b*x], x, 3, -((2*x*LogGamma[a + b*x])/b^2) + (2*PolyGamma[-2, a + b*x])/b^3 + (x^2*PolyGamma[0, a + b*x])/b} - - -{PolyGamma[1, a + b*x]/x^2 - (b*PolyGamma[2, a + b*x])/x, x, 2, -(PolyGamma[1, a + b*x]/x)} -{PolyGamma[n, a + b*x]/x^2 - (b*PolyGamma[1 + n, a + b*x])/x, x, 2, -(PolyGamma[n, a + b*x]/x)} - - -(* ::Section::Closed:: *) -(*Integrands of the form Gamma[c+dx]^m PolyGamma[n, a+b x]*) - - -{Gamma[a + b*x]^n*PolyGamma[0, a + b*x], x, 1, Gamma[a + b*x]^n/(b*n)} -{(a + b*x)!^n*PolyGamma[0, 1 + a + b*x], x, 1, (a + b*x)!^n/(b*n)} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.7 Zeta function.m b/test/methods/rule_based/test_files/8 Special functions/8.7 Zeta function.m deleted file mode 100644 index 740252b..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.7 Zeta function.m +++ /dev/null @@ -1,32 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving Zeta Functions*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Zeta[2, a+b x]*) - - -{x^2*Zeta[2, a + b*x], x, 4, -((2*x*LogGamma[a + b*x])/b^2) + (2*PolyGamma[-2, a + b*x])/b^3 + (x^2*PolyGamma[0, a + b*x])/b} -{x^1*Zeta[2, a + b*x], x, 3, -(LogGamma[a + b*x]/b^2) + (x*PolyGamma[0, a + b*x])/b} -{x^0*Zeta[2, a + b*x], x, 2, PolyGamma[0, a + b*x]/b} -{Zeta[2, a + b*x]/x^1, x, 1, Unintegrable[PolyGamma[1, a + b*x]/x, x]} -{Zeta[2, a + b*x]/x^2, x, 2, b*Unintegrable[PolyGamma[2, a + b*x]/x, x] - PolyGamma[1, a + b*x]/x} -{Zeta[2, a + b*x]/x^3, x, 3, (1/2)*b^2*Unintegrable[PolyGamma[3, a + b*x]/x, x] - PolyGamma[1, a + b*x]/(2*x^2) - (b*PolyGamma[2, a + b*x])/(2*x)} - -{Zeta[2, a + b*x]/x^2 - b*(PolyGamma[2, a + b*x]/x), x, 3, -(PolyGamma[1, a + b*x]/x)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Zeta[s, a+b x]*) - - -{x^2*Zeta[s, a + b*x], x, 3, If[$VersionNumber>=8, (2*Zeta[-3 + s, a + b*x])/(b^3*(1 - s)*(2 - s)*(3 - s)) - (2*x*Zeta[-2 + s, a + b*x])/(b^2*(1 - s)*(2 - s)) + (x^2*Zeta[-1 + s, a + b*x])/(b*(1 - s)), (2*Zeta[-3 + s, a + b*x])/(b^3*(1 - s)*(6 - 5*s + s^2)) - (2*x*Zeta[-2 + s, a + b*x])/(b^2*(1 - s)*(2 - s)) + (x^2*Zeta[-1 + s, a + b*x])/(b*(1 - s))]} -{x^1*Zeta[s, a + b*x], x, 2, -(Zeta[-2 + s, a + b*x]/(b^2*(1 - s)*(2 - s))) + (x*Zeta[-1 + s, a + b*x])/(b*(1 - s))} -{x^0*Zeta[s, a + b*x], x, 1, Zeta[-1 + s, a + b*x]/(b*(1 - s))} -{Zeta[s, a + b*x]/x^1, x, 0, CannotIntegrate[Zeta[s, a + b*x]/x, x]} -{Zeta[s, a + b*x]/x^2, x, 1, (-b)*s*CannotIntegrate[Zeta[1 + s, a + b*x]/x, x] - Zeta[s, a + b*x]/x} -{Zeta[s, a + b*x]/x^3, x, 2, (1/2)*b^2*s*(1 + s)*CannotIntegrate[Zeta[2 + s, a + b*x]/x, x] - Zeta[s, a + b*x]/(2*x^2) + (b*s*Zeta[1 + s, a + b*x])/(2*x)} - -{Zeta[s, a + b*x]/x^2 + b*s*(Zeta[1 + s, a + b*x]/x), x, 2, -(Zeta[s, a + b*x]/x)} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.8 Polylogarithm function.m b/test/methods/rule_based/test_files/8 Special functions/8.8 Polylogarithm function.m deleted file mode 100644 index 810fd71..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.8 Polylogarithm function.m +++ /dev/null @@ -1,362 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving the Polylogarithm Function*) - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[n, a x^q]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m PolyLog[n, a x^q]*) - - -(* ::Subsubsection::Closed:: *) -(*q=1*) - - -{x^4*PolyLog[2, a*x], x, 4, -(x/(25*a^4)) - x^2/(50*a^3) - x^3/(75*a^2) - x^4/(100*a) - x^5/125 - Log[1 - a*x]/(25*a^5) + (1/25)*x^5*Log[1 - a*x] + (1/5)*x^5*PolyLog[2, a*x]} -{x^3*PolyLog[2, a*x], x, 4, -(x/(16*a^3)) - x^2/(32*a^2) - x^3/(48*a) - x^4/64 - Log[1 - a*x]/(16*a^4) + (1/16)*x^4*Log[1 - a*x] + (1/4)*x^4*PolyLog[2, a*x]} -{x^2*PolyLog[2, a*x], x, 4, -(x/(9*a^2)) - x^2/(18*a) - x^3/27 - Log[1 - a*x]/(9*a^3) + (1/9)*x^3*Log[1 - a*x] + (1/3)*x^3*PolyLog[2, a*x]} -{x^1*PolyLog[2, a*x], x, 4, -(x/(4*a)) - x^2/8 - Log[1 - a*x]/(4*a^2) + (1/4)*x^2*Log[1 - a*x] + (1/2)*x^2*PolyLog[2, a*x]} -{x^0*PolyLog[2, a*x], x, 3, -x - ((1 - a*x)*Log[1 - a*x])/a + x*PolyLog[2, a*x]} -{PolyLog[2, a*x]/x^1, x, 1, PolyLog[3, a*x]} -{PolyLog[2, a*x]/x^2, x, 5, a*Log[x] - a*Log[1 - a*x] + Log[1 - a*x]/x - PolyLog[2, a*x]/x} -{PolyLog[2, a*x]/x^3, x, 4, -(a/(4*x)) + (1/4)*a^2*Log[x] - (1/4)*a^2*Log[1 - a*x] + Log[1 - a*x]/(4*x^2) - PolyLog[2, a*x]/(2*x^2)} -{PolyLog[2, a*x]/x^4, x, 4, -(a/(18*x^2)) - a^2/(9*x) + (1/9)*a^3*Log[x] - (1/9)*a^3*Log[1 - a*x] + Log[1 - a*x]/(9*x^3) - PolyLog[2, a*x]/(3*x^3)} -{PolyLog[2, a*x]/x^5, x, 4, -(a/(48*x^3)) - a^2/(32*x^2) - a^3/(16*x) + (1/16)*a^4*Log[x] - (1/16)*a^4*Log[1 - a*x] + Log[1 - a*x]/(16*x^4) - PolyLog[2, a*x]/(4*x^4)} - - -{x^3*PolyLog[3, a*x], x, 5, x/(64*a^3) + x^2/(128*a^2) + x^3/(192*a) + x^4/256 + Log[1 - a*x]/(64*a^4) - (1/64)*x^4*Log[1 - a*x] - (1/16)*x^4*PolyLog[2, a*x] + (1/4)*x^4*PolyLog[3, a*x]} -{x^2*PolyLog[3, a*x], x, 5, x/(27*a^2) + x^2/(54*a) + x^3/81 + Log[1 - a*x]/(27*a^3) - (1/27)*x^3*Log[1 - a*x] - (1/9)*x^3*PolyLog[2, a*x] + (1/3)*x^3*PolyLog[3, a*x]} -{x^1*PolyLog[3, a*x], x, 5, x/(8*a) + x^2/16 + Log[1 - a*x]/(8*a^2) - (1/8)*x^2*Log[1 - a*x] - (1/4)*x^2*PolyLog[2, a*x] + (1/2)*x^2*PolyLog[3, a*x]} -{x^0*PolyLog[3, a*x], x, 4, x + ((1 - a*x)*Log[1 - a*x])/a - x*PolyLog[2, a*x] + x*PolyLog[3, a*x]} -{PolyLog[3, a*x]/x^1, x, 1, PolyLog[4, a*x]} -{PolyLog[3, a*x]/x^2, x, 6, a*Log[x] - a*Log[1 - a*x] + Log[1 - a*x]/x - PolyLog[2, a*x]/x - PolyLog[3, a*x]/x} -{PolyLog[3, a*x]/x^3, x, 5, -(a/(8*x)) + (1/8)*a^2*Log[x] - (1/8)*a^2*Log[1 - a*x] + Log[1 - a*x]/(8*x^2) - PolyLog[2, a*x]/(4*x^2) - PolyLog[3, a*x]/(2*x^2)} -{PolyLog[3, a*x]/x^4, x, 5, -(a/(54*x^2)) - a^2/(27*x) + (1/27)*a^3*Log[x] - (1/27)*a^3*Log[1 - a*x] + Log[1 - a*x]/(27*x^3) - PolyLog[2, a*x]/(9*x^3) - PolyLog[3, a*x]/(3*x^3)} - - -(* ::Subsubsection::Closed:: *) -(*q=2*) - - -{x^5*PolyLog[2, a*x^2], x, 5, -(x^2/(18*a^2)) - x^4/(36*a) - x^6/54 - Log[1 - a*x^2]/(18*a^3) + (1/18)*x^6*Log[1 - a*x^2] + (1/6)*x^6*PolyLog[2, a*x^2]} -{x^3*PolyLog[2, a*x^2], x, 5, -(x^2/(8*a)) - x^4/16 - Log[1 - a*x^2]/(8*a^2) + (1/8)*x^4*Log[1 - a*x^2] + (1/4)*x^4*PolyLog[2, a*x^2]} -{x^1*PolyLog[2, a*x^2], x, 4, -(x^2/2) - ((1 - a*x^2)*Log[1 - a*x^2])/(2*a) + (1/2)*x^2*PolyLog[2, a*x^2]} -{PolyLog[2, a*x^2]/x^1, x, 1, (1/2)*PolyLog[3, a*x^2]} -{PolyLog[2, a*x^2]/x^3, x, 6, a*Log[x] - (1/2)*a*Log[1 - a*x^2] + Log[1 - a*x^2]/(2*x^2) - PolyLog[2, a*x^2]/(2*x^2)} -{PolyLog[2, a*x^2]/x^5, x, 5, -(a/(8*x^2)) + (1/4)*a^2*Log[x] - (1/8)*a^2*Log[1 - a*x^2] + Log[1 - a*x^2]/(8*x^4) - PolyLog[2, a*x^2]/(4*x^4)} -{PolyLog[2, a*x^2]/x^7, x, 5, -(a/(36*x^4)) - a^2/(18*x^2) + (1/9)*a^3*Log[x] - (1/18)*a^3*Log[1 - a*x^2] + Log[1 - a*x^2]/(18*x^6) - PolyLog[2, a*x^2]/(6*x^6)} - -{x^4*PolyLog[2, a*x^2], x, 5, -((4*x)/(25*a^2)) - (4*x^3)/(75*a) - (4*x^5)/125 + (4*ArcTanh[Sqrt[a]*x])/(25*a^(5/2)) + (2/25)*x^5*Log[1 - a*x^2] + (1/5)*x^5*PolyLog[2, a*x^2]} -{x^2*PolyLog[2, a*x^2], x, 5, -((4*x)/(9*a)) - (4*x^3)/27 + (4*ArcTanh[Sqrt[a]*x])/(9*a^(3/2)) + (2/9)*x^3*Log[1 - a*x^2] + (1/3)*x^3*PolyLog[2, a*x^2]} -{x^0*PolyLog[2, a*x^2], x, 4, -4*x + (4*ArcTanh[Sqrt[a]*x])/Sqrt[a] + 2*x*Log[1 - a*x^2] + x*PolyLog[2, a*x^2]} -{PolyLog[2, a*x^2]/x^2, x, 3, 4*Sqrt[a]*ArcTanh[Sqrt[a]*x] + (2*Log[1 - a*x^2])/x - PolyLog[2, a*x^2]/x} -{PolyLog[2, a*x^2]/x^4, x, 4, -((4*a)/(9*x)) + (4/9)*a^(3/2)*ArcTanh[Sqrt[a]*x] + (2*Log[1 - a*x^2])/(9*x^3) - PolyLog[2, a*x^2]/(3*x^3)} -{PolyLog[2, a*x^2]/x^6, x, 5, -((4*a)/(75*x^3)) - (4*a^2)/(25*x) + (4/25)*a^(5/2)*ArcTanh[Sqrt[a]*x] + (2*Log[1 - a*x^2])/(25*x^5) - PolyLog[2, a*x^2]/(5*x^5)} - - -{x^5*PolyLog[3, a*x^2], x, 6, x^2/(54*a^2) + x^4/(108*a) + x^6/162 + Log[1 - a*x^2]/(54*a^3) - (1/54)*x^6*Log[1 - a*x^2] - (1/18)*x^6*PolyLog[2, a*x^2] + (1/6)*x^6*PolyLog[3, a*x^2]} -{x^3*PolyLog[3, a*x^2], x, 6, x^2/(16*a) + x^4/32 + Log[1 - a*x^2]/(16*a^2) - (1/16)*x^4*Log[1 - a*x^2] - (1/8)*x^4*PolyLog[2, a*x^2] + (1/4)*x^4*PolyLog[3, a*x^2]} -{x^1*PolyLog[3, a*x^2], x, 5, x^2/2 + ((1 - a*x^2)*Log[1 - a*x^2])/(2*a) - (1/2)*x^2*PolyLog[2, a*x^2] + (1/2)*x^2*PolyLog[3, a*x^2]} -{PolyLog[3, a*x^2]/x^1, x, 1, (1/2)*PolyLog[4, a*x^2]} -{PolyLog[3, a*x^2]/x^3, x, 7, a*Log[x] - (1/2)*a*Log[1 - a*x^2] + Log[1 - a*x^2]/(2*x^2) - PolyLog[2, a*x^2]/(2*x^2) - PolyLog[3, a*x^2]/(2*x^2)} -{PolyLog[3, a*x^2]/x^5, x, 6, -(a/(16*x^2)) + (1/8)*a^2*Log[x] - (1/16)*a^2*Log[1 - a*x^2] + Log[1 - a*x^2]/(16*x^4) - PolyLog[2, a*x^2]/(8*x^4) - PolyLog[3, a*x^2]/(4*x^4)} -{PolyLog[3, a*x^2]/x^7, x, 6, -(a/(108*x^4)) - a^2/(54*x^2) + (1/27)*a^3*Log[x] - (1/54)*a^3*Log[1 - a*x^2] + Log[1 - a*x^2]/(54*x^6) - PolyLog[2, a*x^2]/(18*x^6) - PolyLog[3, a*x^2]/(6*x^6)} - -{x^4*PolyLog[3, a*x^2], x, 6, (8*x)/(125*a^2) + (8*x^3)/(375*a) + (8*x^5)/625 - (8*ArcTanh[Sqrt[a]*x])/(125*a^(5/2)) - (4/125)*x^5*Log[1 - a*x^2] - (2/25)*x^5*PolyLog[2, a*x^2] + (1/5)*x^5*PolyLog[3, a*x^2]} -{x^2*PolyLog[3, a*x^2], x, 6, (8*x)/(27*a) + (8*x^3)/81 - (8*ArcTanh[Sqrt[a]*x])/(27*a^(3/2)) - (4/27)*x^3*Log[1 - a*x^2] - (2/9)*x^3*PolyLog[2, a*x^2] + (1/3)*x^3*PolyLog[3, a*x^2]} -{x^0*PolyLog[3, a*x^2], x, 5, 8*x - (8*ArcTanh[Sqrt[a]*x])/Sqrt[a] - 4*x*Log[1 - a*x^2] - 2*x*PolyLog[2, a*x^2] + x*PolyLog[3, a*x^2]} -{PolyLog[3, a*x^2]/x^2, x, 4, 8*Sqrt[a]*ArcTanh[Sqrt[a]*x] + (4*Log[1 - a*x^2])/x - (2*PolyLog[2, a*x^2])/x - PolyLog[3, a*x^2]/x} -{PolyLog[3, a*x^2]/x^4, x, 5, -((8*a)/(27*x)) + (8/27)*a^(3/2)*ArcTanh[Sqrt[a]*x] + (4*Log[1 - a*x^2])/(27*x^3) - (2*PolyLog[2, a*x^2])/(9*x^3) - PolyLog[3, a*x^2]/(3*x^3)} -{PolyLog[3, a*x^2]/x^6, x, 6, -((8*a)/(375*x^3)) - (8*a^2)/(125*x) + (8/125)*a^(5/2)*ArcTanh[Sqrt[a]*x] + (4*Log[1 - a*x^2])/(125*x^5) - (2*PolyLog[2, a*x^2])/(25*x^5) - PolyLog[3, a*x^2]/(5*x^5)} - - -(* ::Subsubsection::Closed:: *) -(*q symbolic*) - - -{x^2*PolyLog[2, a*x^q], x, 3, (a*q^2*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(9*(3 + q)) + (1/9)*q*x^3*Log[1 - a*x^q] + (1/3)*x^3*PolyLog[2, a*x^q]} -{x^1*PolyLog[2, a*x^q], x, 3, (a*q^2*x^(2 + q)*Hypergeometric2F1[1, (2 + q)/q, 2*(1 + 1/q), a*x^q])/(4*(2 + q)) + (1/4)*q*x^2*Log[1 - a*x^q] + (1/2)*x^2*PolyLog[2, a*x^q]} -{x^0*PolyLog[2, a*x^q], x, 3, (a*q^2*x^(1 + q)*Hypergeometric2F1[1, 1 + 1/q, 2 + 1/q, a*x^q])/(1 + q) + q*x*Log[1 - a*x^q] + x*PolyLog[2, a*x^q]} -{PolyLog[2, a*x^q]/x^1, x, 1, PolyLog[3, a*x^q]/q} -{PolyLog[2, a*x^q]/x^2, x, 3, -((a*q^2*x^(-1 + q)*Hypergeometric2F1[1, -((1 - q)/q), 2 - 1/q, a*x^q])/(1 - q)) + (q*Log[1 - a*x^q])/x - PolyLog[2, a*x^q]/x} -{PolyLog[2, a*x^q]/x^3, x, 3, -((a*q^2*x^(-2 + q)*Hypergeometric2F1[1, -((2 - q)/q), 2*(1 - 1/q), a*x^q])/(4*(2 - q))) + (q*Log[1 - a*x^q])/(4*x^2) - PolyLog[2, a*x^q]/(2*x^2)} -{PolyLog[2, a*x^q]/x^4, x, 3, -((a*q^2*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(9*(3 - q))) + (q*Log[1 - a*x^q])/(9*x^3) - PolyLog[2, a*x^q]/(3*x^3)} - - -{x^2*PolyLog[3, a*x^q], x, 4, -((a*q^3*x^(3 + q)*Hypergeometric2F1[1, (3 + q)/q, 2 + 3/q, a*x^q])/(27*(3 + q))) - (1/27)*q^2*x^3*Log[1 - a*x^q] - (1/9)*q*x^3*PolyLog[2, a*x^q] + (1/3)*x^3*PolyLog[3, a*x^q]} -{x^1*PolyLog[3, a*x^q], x, 4, -((a*q^3*x^(2 + q)*Hypergeometric2F1[1, (2 + q)/q, 2*(1 + 1/q), a*x^q])/(8*(2 + q))) - (1/8)*q^2*x^2*Log[1 - a*x^q] - (1/4)*q*x^2*PolyLog[2, a*x^q] + (1/2)*x^2*PolyLog[3, a*x^q]} -{x^0*PolyLog[3, a*x^q], x, 4, -((a*q^3*x^(1 + q)*Hypergeometric2F1[1, 1 + 1/q, 2 + 1/q, a*x^q])/(1 + q)) - q^2*x*Log[1 - a*x^q] - q*x*PolyLog[2, a*x^q] + x*PolyLog[3, a*x^q]} -{PolyLog[3, a*x^q]/x^1, x, 1, PolyLog[4, a*x^q]/q} -{PolyLog[3, a*x^q]/x^2, x, 4, -((a*q^3*x^(-1 + q)*Hypergeometric2F1[1, -((1 - q)/q), 2 - 1/q, a*x^q])/(1 - q)) + (q^2*Log[1 - a*x^q])/x - (q*PolyLog[2, a*x^q])/x - PolyLog[3, a*x^q]/x} -{PolyLog[3, a*x^q]/x^3, x, 4, -((a*q^3*x^(-2 + q)*Hypergeometric2F1[1, -((2 - q)/q), 2*(1 - 1/q), a*x^q])/(8*(2 - q))) + (q^2*Log[1 - a*x^q])/(8*x^2) - (q*PolyLog[2, a*x^q])/(4*x^2) - PolyLog[3, a*x^q]/(2*x^2)} -{PolyLog[3, a*x^q]/x^4, x, 4, -((a*q^3*x^(-3 + q)*Hypergeometric2F1[1, -((3 - q)/q), 2 - 3/q, a*x^q])/(27*(3 - q))) + (q^2*Log[1 - a*x^q])/(27*x^3) - (q*PolyLog[2, a*x^q])/(9*x^3) - PolyLog[3, a*x^q]/(3*x^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^(m/2) PolyLog[n, a x^q]*) - - -(* ::Subsubsection::Closed:: *) -(*q=1*) - - -{(d*x)^(3/2)*PolyLog[2, a*x], x, 7, -((8*d*Sqrt[d*x])/(25*a^2)) - (8*(d*x)^(3/2))/(75*a) - (8*(d*x)^(5/2))/(125*d) + (8*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(25*a^(5/2)) + (4*(d*x)^(5/2)*Log[1 - a*x])/(25*d) + (2*(d*x)^(5/2)*PolyLog[2, a*x])/(5*d)} -{(d*x)^(1/2)*PolyLog[2, a*x], x, 6, -((8*Sqrt[d*x])/(9*a)) - (8*(d*x)^(3/2))/(27*d) + (8*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/2)) + (4*(d*x)^(3/2)*Log[1 - a*x])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a*x])/(3*d)} -{PolyLog[2, a*x]/(d*x)^(1/2), x, 5, -((8*Sqrt[d*x])/d) + (8*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[a]*Sqrt[d]) + (4*Sqrt[d*x]*Log[1 - a*x])/d + (2*Sqrt[d*x]*PolyLog[2, a*x])/d} -{PolyLog[2, a*x]/(d*x)^(3/2), x, 4, (8*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (4*Log[1 - a*x])/(d*Sqrt[d*x]) - (2*PolyLog[2, a*x])/(d*Sqrt[d*x])} -{PolyLog[2, a*x]/(d*x)^(5/2), x, 5, -((8*a)/(9*d^2*Sqrt[d*x])) + (8*a^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(9*d^(5/2)) + (4*Log[1 - a*x])/(9*d*(d*x)^(3/2)) - (2*PolyLog[2, a*x])/(3*d*(d*x)^(3/2))} -{PolyLog[2, a*x]/(d*x)^(7/2), x, 6, -((8*a)/(75*d^2*(d*x)^(3/2))) - (8*a^2)/(25*d^3*Sqrt[d*x]) + (8*a^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(25*d^(7/2)) + (4*Log[1 - a*x])/(25*d*(d*x)^(5/2)) - (2*PolyLog[2, a*x])/(5*d*(d*x)^(5/2))} - - -{(d*x)^(5/2)*PolyLog[3, a*x], x, 9, (16*d^2*Sqrt[d*x])/(343*a^3) + (16*d*(d*x)^(3/2))/(1029*a^2) + (16*(d*x)^(5/2))/(1715*a) + (16*(d*x)^(7/2))/(2401*d) - (16*d^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/2)) - (8*(d*x)^(7/2)*Log[1 - a*x])/(343*d) - (4*(d*x)^(7/2)*PolyLog[2, a*x])/(49*d) + (2*(d*x)^(7/2)*PolyLog[3, a*x])/(7*d)} -{(d*x)^(3/2)*PolyLog[3, a*x], x, 8, (16*d*Sqrt[d*x])/(125*a^2) + (16*(d*x)^(3/2))/(375*a) + (16*(d*x)^(5/2))/(625*d) - (16*d^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(125*a^(5/2)) - (8*(d*x)^(5/2)*Log[1 - a*x])/(125*d) - (4*(d*x)^(5/2)*PolyLog[2, a*x])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3, a*x])/(5*d)} -{(d*x)^(1/2)*PolyLog[3, a*x], x, 7, (16*Sqrt[d*x])/(27*a) + (16*(d*x)^(3/2))/(81*d) - (16*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/2)) - (8*(d*x)^(3/2)*Log[1 - a*x])/(27*d) - (4*(d*x)^(3/2)*PolyLog[2, a*x])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x])/(3*d)} -{PolyLog[3, a*x]/(d*x)^(1/2), x, 6, (16*Sqrt[d*x])/d - (16*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(Sqrt[a]*Sqrt[d]) - (8*Sqrt[d*x]*Log[1 - a*x])/d - (4*Sqrt[d*x]*PolyLog[2, a*x])/d + (2*Sqrt[d*x]*PolyLog[3, a*x])/d} -{PolyLog[3, a*x]/(d*x)^(3/2), x, 5, (16*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (8*Log[1 - a*x])/(d*Sqrt[d*x]) - (4*PolyLog[2, a*x])/(d*Sqrt[d*x]) - (2*PolyLog[3, a*x])/(d*Sqrt[d*x])} -{PolyLog[3, a*x]/(d*x)^(5/2), x, 6, -((16*a)/(27*d^2*Sqrt[d*x])) + (16*a^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(27*d^(5/2)) + (8*Log[1 - a*x])/(27*d*(d*x)^(3/2)) - (4*PolyLog[2, a*x])/(9*d*(d*x)^(3/2)) - (2*PolyLog[3, a*x])/(3*d*(d*x)^(3/2))} -{PolyLog[3, a*x]/(d*x)^(7/2), x, 7, -((16*a)/(375*d^2*(d*x)^(3/2))) - (16*a^2)/(125*d^3*Sqrt[d*x]) + (16*a^(5/2)*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/(125*d^(7/2)) + (8*Log[1 - a*x])/(125*d*(d*x)^(5/2)) - (4*PolyLog[2, a*x])/(25*d*(d*x)^(5/2)) - (2*PolyLog[3, a*x])/(5*d*(d*x)^(5/2))} - - -(* ::Subsubsection::Closed:: *) -(*q=2*) - - -{(d*x)^(3/2)*PolyLog[2, a*x^2], x, 9, -((32*d*Sqrt[d*x])/(25*a)) - (32*(d*x)^(5/2))/(125*d) + (16*d^(3/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(25*a^(5/4)) + (16*d^(3/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(25*a^(5/4)) + (8*(d*x)^(5/2)*Log[1 - a*x^2])/(25*d) + (2*(d*x)^(5/2)*PolyLog[2, a*x^2])/(5*d)} -{(d*x)^(1/2)*PolyLog[2, a*x^2], x, 8, -((32*(d*x)^(3/2))/(27*d)) - (16*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (16*Sqrt[d]*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*a^(3/4)) + (8*(d*x)^(3/2)*Log[1 - a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a*x^2])/(3*d)} -{PolyLog[2, a*x^2]/(d*x)^(1/2), x, 8, -((32*Sqrt[d*x])/d) + (16*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(a^(1/4)*Sqrt[d]) + (16*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(a^(1/4)*Sqrt[d]) + (8*Sqrt[d*x]*Log[1 - a*x^2])/d + (2*Sqrt[d*x]*PolyLog[2, a*x^2])/d} -{PolyLog[2, a*x^2]/(d*x)^(3/2), x, 7, -((16*a^(1/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2)) + (16*a^(1/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (8*Log[1 - a*x^2])/(d*Sqrt[d*x]) - (2*PolyLog[2, a*x^2])/(d*Sqrt[d*x])} -{PolyLog[2, a*x^2]/(d*x)^(5/2), x, 7, (16*a^(3/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*d^(5/2)) + (16*a^(3/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(9*d^(5/2)) + (8*Log[1 - a*x^2])/(9*d*(d*x)^(3/2)) - (2*PolyLog[2, a*x^2])/(3*d*(d*x)^(3/2))} -{PolyLog[2, a*x^2]/(d*x)^(7/2), x, 8, -((32*a)/(25*d^3*Sqrt[d*x])) - (16*a^(5/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(25*d^(7/2)) + (16*a^(5/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(25*d^(7/2)) + (8*Log[1 - a*x^2])/(25*d*(d*x)^(5/2)) - (2*PolyLog[2, a*x^2])/(5*d*(d*x)^(5/2))} - - -{(d*x)^(5/2)*PolyLog[3, a*x^2], x, 10, (128*d*(d*x)^(3/2))/(1029*a) + (128*(d*x)^(7/2))/(2401*d) + (64*d^(5/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/4)) - (64*d^(5/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*a^(7/4)) - (32*(d*x)^(7/2)*Log[1 - a*x^2])/(343*d) - (8*(d*x)^(7/2)*PolyLog[2, a*x^2])/(49*d) + (2*(d*x)^(7/2)*PolyLog[3, a*x^2])/(7*d)} -{(d*x)^(3/2)*PolyLog[3, a*x^2], x, 10, (128*d*Sqrt[d*x])/(125*a) + (128*(d*x)^(5/2))/(625*d) - (64*d^(3/2)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*a^(5/4)) - (64*d^(3/2)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*a^(5/4)) - (32*(d*x)^(5/2)*Log[1 - a*x^2])/(125*d) - (8*(d*x)^(5/2)*PolyLog[2, a*x^2])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3, a*x^2])/(5*d)} -{(d*x)^(1/2)*PolyLog[3, a*x^2], x, 9, (128*(d*x)^(3/2))/(81*d) + (64*Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (64*Sqrt[d]*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*a^(3/4)) - (32*(d*x)^(3/2)*Log[1 - a*x^2])/(27*d) - (8*(d*x)^(3/2)*PolyLog[2, a*x^2])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x^2])/(3*d)} -{PolyLog[3, a*x^2]/(d*x)^(1/2), x, 9, (128*Sqrt[d*x])/d - (64*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(a^(1/4)*Sqrt[d]) - (64*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(a^(1/4)*Sqrt[d]) - (32*Sqrt[d*x]*Log[1 - a*x^2])/d - (8*Sqrt[d*x]*PolyLog[2, a*x^2])/d + (2*Sqrt[d*x]*PolyLog[3, a*x^2])/d} -{PolyLog[3, a*x^2]/(d*x)^(3/2), x, 8, -((64*a^(1/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2)) + (64*a^(1/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/d^(3/2) + (32*Log[1 - a*x^2])/(d*Sqrt[d*x]) - (8*PolyLog[2, a*x^2])/(d*Sqrt[d*x]) - (2*PolyLog[3, a*x^2])/(d*Sqrt[d*x])} -{PolyLog[3, a*x^2]/(d*x)^(5/2), x, 8, (64*a^(3/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*d^(5/2)) + (64*a^(3/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(27*d^(5/2)) + (32*Log[1 - a*x^2])/(27*d*(d*x)^(3/2)) - (8*PolyLog[2, a*x^2])/(9*d*(d*x)^(3/2)) - (2*PolyLog[3, a*x^2])/(3*d*(d*x)^(3/2))} -{PolyLog[3, a*x^2]/(d*x)^(7/2), x, 9, -((128*a)/(125*d^3*Sqrt[d*x])) - (64*a^(5/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*d^(7/2)) + (64*a^(5/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(125*d^(7/2)) + (32*Log[1 - a*x^2])/(125*d*(d*x)^(5/2)) - (8*PolyLog[2, a*x^2])/(25*d*(d*x)^(5/2)) - (2*PolyLog[3, a*x^2])/(5*d*(d*x)^(5/2))} -{PolyLog[3, a*x^2]/(d*x)^(9/2), x, 9, -((128*a)/(1029*d^3*(d*x)^(3/2))) + (64*a^(7/4)*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*d^(9/2)) + (64*a^(7/4)*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(343*d^(9/2)) + (32*Log[1 - a*x^2])/(343*d*(d*x)^(7/2)) - (8*PolyLog[2, a*x^2])/(49*d*(d*x)^(7/2)) - (2*PolyLog[3, a*x^2])/(7*d*(d*x)^(7/2))} - - -(* ::Subsubsection::Closed:: *) -(*q symbolic*) - - -{(d*x)^(3/2)*PolyLog[2, a*x^q], x, 4, (8*a*d*q^2*x^(2 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (5/2 + q)/q, (1/2)*(4 + 5/q), a*x^q])/(25*(5 + 2*q)) + (4*q*(d*x)^(5/2)*Log[1 - a*x^q])/(25*d) + (2*(d*x)^(5/2)*PolyLog[2, a*x^q])/(5*d)} -{(d*x)^(1/2)*PolyLog[2, a*x^q], x, 4, (8*a*q^2*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (1/2)*(4 + 3/q), a*x^q])/(9*(3 + 2*q)) + (4*q*(d*x)^(3/2)*Log[1 - a*x^q])/(9*d) + (2*(d*x)^(3/2)*PolyLog[2, a*x^q])/(3*d)} -{PolyLog[2, a*x^q]/(d*x)^(1/2), x, 4, (8*a*q^2*x^q*Sqrt[d*x]*Hypergeometric2F1[1, (1/2 + q)/q, (1/2)*(4 + 1/q), a*x^q])/(d*(1 + 2*q)) + (4*q*Sqrt[d*x]*Log[1 - a*x^q])/d + (2*Sqrt[d*x]*PolyLog[2, a*x^q])/d} -{PolyLog[2, a*x^q]/(d*x)^(3/2), x, 4, -((8*a*q^2*x^q*Hypergeometric2F1[1, (1/2)*(2 - 1/q), (1/2)*(4 - 1/q), a*x^q])/(d*(1 - 2*q)*Sqrt[d*x])) + (4*q*Log[1 - a*x^q])/(d*Sqrt[d*x]) - (2*PolyLog[2, a*x^q])/(d*Sqrt[d*x])} -{PolyLog[2, a*x^q]/(d*x)^(5/2), x, 4, -((8*a*q^2*x^(-1 + q)*Hypergeometric2F1[1, (1/2)*(2 - 3/q), (1/2)*(4 - 3/q), a*x^q])/(9*d^2*(3 - 2*q)*Sqrt[d*x])) + (4*q*Log[1 - a*x^q])/(9*d*(d*x)^(3/2)) - (2*PolyLog[2, a*x^q])/(3*d*(d*x)^(3/2))} - - -{(d*x)^(3/2)*PolyLog[3, a*x^q], x, 5, -((16*a*d*q^3*x^(2 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (5/2 + q)/q, (1/2)*(4 + 5/q), a*x^q])/(125*(5 + 2*q))) - (8*q^2*(d*x)^(5/2)*Log[1 - a*x^q])/(125*d) - (4*q*(d*x)^(5/2)*PolyLog[2, a*x^q])/(25*d) + (2*(d*x)^(5/2)*PolyLog[3, a*x^q])/(5*d)} -{(d*x)^(1/2)*PolyLog[3, a*x^q], x, 5, -((16*a*q^3*x^(1 + q)*Sqrt[d*x]*Hypergeometric2F1[1, (3/2 + q)/q, (1/2)*(4 + 3/q), a*x^q])/(27*(3 + 2*q))) - (8*q^2*(d*x)^(3/2)*Log[1 - a*x^q])/(27*d) - (4*q*(d*x)^(3/2)*PolyLog[2, a*x^q])/(9*d) + (2*(d*x)^(3/2)*PolyLog[3, a*x^q])/(3*d)} -{PolyLog[3, a*x^q]/(d*x)^(1/2), x, 5, -((16*a*q^3*x^q*Sqrt[d*x]*Hypergeometric2F1[1, (1/2 + q)/q, (1/2)*(4 + 1/q), a*x^q])/(d*(1 + 2*q))) - (8*q^2*Sqrt[d*x]*Log[1 - a*x^q])/d - (4*q*Sqrt[d*x]*PolyLog[2, a*x^q])/d + (2*Sqrt[d*x]*PolyLog[3, a*x^q])/d} -{PolyLog[3, a*x^q]/(d*x)^(3/2), x, 5, -((16*a*q^3*x^q*Hypergeometric2F1[1, (1/2)*(2 - 1/q), (1/2)*(4 - 1/q), a*x^q])/(d*(1 - 2*q)*Sqrt[d*x])) + (8*q^2*Log[1 - a*x^q])/(d*Sqrt[d*x]) - (4*q*PolyLog[2, a*x^q])/(d*Sqrt[d*x]) - (2*PolyLog[3, a*x^q])/(d*Sqrt[d*x])} -{PolyLog[3, a*x^q]/(d*x)^(5/2), x, 5, -((16*a*q^3*x^(-1 + q)*Hypergeometric2F1[1, (1/2)*(2 - 3/q), (1/2)*(4 - 3/q), a*x^q])/(27*d^2*(3 - 2*q)*Sqrt[d*x])) + (8*q^2*Log[1 - a*x^q])/(27*d*(d*x)^(3/2)) - (4*q*PolyLog[2, a*x^q])/(9*d*(d*x)^(3/2)) - (2*PolyLog[3, a*x^q])/(3*d*(d*x)^(3/2))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m PolyLog[n/2, a x^q]*) - - -{PolyLog[3/2, a*x], x, 2, (-x)*PolyLog[1/2, a*x] + x*PolyLog[3/2, a*x] + Unintegrable[PolyLog[-(1/2), a*x], x]} -{PolyLog[1/2, a*x], x, 1, x*PolyLog[1/2, a*x] - Unintegrable[PolyLog[-(1/2), a*x], x]} -{PolyLog[-1/2, a*x], x, 0, Unintegrable[PolyLog[-(1/2), a*x], x]} -{PolyLog[-3/2, a*x], x, 1, x*PolyLog[-(1/2), a*x] - Unintegrable[PolyLog[-(1/2), a*x], x]} -{PolyLog[-5/2, a*x], x, 2, x*PolyLog[-(3/2), a*x] - x*PolyLog[-(1/2), a*x] + Unintegrable[PolyLog[-(1/2), a*x], x]} - - -{PolyLog[-3/2, a*x] + PolyLog[-1/2, a*x], x, 2, x*PolyLog[-1/2, a*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[n, a x^q] with m symbolic*) - - -{(d*x)^m*PolyLog[2, a*x], x, 3, (a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^2*(2 + m)) + ((d*x)^(1 + m)*Log[1 - a*x])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m))} -{(d*x)^m*PolyLog[3, a*x], x, 4, -((a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^3*(2 + m))) - ((d*x)^(1 + m)*Log[1 - a*x])/(d*(1 + m)^3) - ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x])/(d*(1 + m))} -{(d*x)^m*PolyLog[4, a*x], x, 5, (a*(d*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, a*x])/(d^2*(1 + m)^4*(2 + m)) + ((d*x)^(1 + m)*Log[1 - a*x])/(d*(1 + m)^4) + ((d*x)^(1 + m)*PolyLog[2, a*x])/(d*(1 + m)^3) - ((d*x)^(1 + m)*PolyLog[3, a*x])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[4, a*x])/(d*(1 + m))} - - -{(d*x)^m*PolyLog[2, a*x^2], x, 4, (4*a*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, a*x^2])/(d^3*(1 + m)^2*(3 + m)) + (2*(d*x)^(1 + m)*Log[1 - a*x^2])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[2, a*x^2])/(d*(1 + m))} -{(d*x)^m*PolyLog[3, a*x^2], x, 5, -((8*a*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, a*x^2])/(d^3*(1 + m)^3*(3 + m))) - (4*(d*x)^(1 + m)*Log[1 - a*x^2])/(d*(1 + m)^3) - (2*(d*x)^(1 + m)*PolyLog[2, a*x^2])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x^2])/(d*(1 + m))} -{(d*x)^m*PolyLog[4, a*x^2], x, 6, (16*a*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, a*x^2])/(d^3*(1 + m)^4*(3 + m)) + (8*(d*x)^(1 + m)*Log[1 - a*x^2])/(d*(1 + m)^4) + (4*(d*x)^(1 + m)*PolyLog[2, a*x^2])/(d*(1 + m)^3) - (2*(d*x)^(1 + m)*PolyLog[3, a*x^2])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[4, a*x^2])/(d*(1 + m))} - - -{(d*x)^m*PolyLog[2, a*x^3], x, 4, (9*a*(d*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/3, (7 + m)/3, a*x^3])/(d^4*(1 + m)^2*(4 + m)) + (3*(d*x)^(1 + m)*Log[1 - a*x^3])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[2, a*x^3])/(d*(1 + m))} -{(d*x)^m*PolyLog[3, a*x^3], x, 5, -((27*a*(d*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/3, (7 + m)/3, a*x^3])/(d^4*(1 + m)^3*(4 + m))) - (9*(d*x)^(1 + m)*Log[1 - a*x^3])/(d*(1 + m)^3) - (3*(d*x)^(1 + m)*PolyLog[2, a*x^3])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x^3])/(d*(1 + m))} -{(d*x)^m*PolyLog[4, a*x^3], x, 6, (81*a*(d*x)^(4 + m)*Hypergeometric2F1[1, (4 + m)/3, (7 + m)/3, a*x^3])/(d^4*(1 + m)^4*(4 + m)) + (27*(d*x)^(1 + m)*Log[1 - a*x^3])/(d*(1 + m)^4) + (9*(d*x)^(1 + m)*PolyLog[2, a*x^3])/(d*(1 + m)^3) - (3*(d*x)^(1 + m)*PolyLog[3, a*x^3])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[4, a*x^3])/(d*(1 + m))} - - -{(d*x)^m*PolyLog[2, a*x^q], x, 4, (a*q^2*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, a*x^q])/((1 + m)^2*(1 + m + q)) + (q*(d*x)^(1 + m)*Log[1 - a*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[2, a*x^q])/(d*(1 + m))} -{(d*x)^m*PolyLog[3, a*x^q], x, 5, -((a*q^3*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, a*x^q])/((1 + m)^3*(1 + m + q))) - (q^2*(d*x)^(1 + m)*Log[1 - a*x^q])/(d*(1 + m)^3) - (q*(d*x)^(1 + m)*PolyLog[2, a*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[3, a*x^q])/(d*(1 + m))} -{(d*x)^m*PolyLog[4, a*x^q], x, 6, (a*q^4*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, a*x^q])/((1 + m)^4*(1 + m + q)) + (q^3*(d*x)^(1 + m)*Log[1 - a*x^q])/(d*(1 + m)^4) + (q^2*(d*x)^(1 + m)*PolyLog[2, a*x^q])/(d*(1 + m)^3) - (q*(d*x)^(1 + m)*PolyLog[3, a*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog[4, a*x^q])/(d*(1 + m))} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[n, a x^q] with n symbolic*) - - -{x^1*PolyLog[n, a*x], x, 0, Unintegrable[x*PolyLog[n, a*x], x]} -{x^0*PolyLog[n, a*x], x, 0, Unintegrable[PolyLog[n, a*x], x]} -{PolyLog[n, a*x]/x^1, x, 1, PolyLog[1 + n, a*x]} -{PolyLog[n, a*x]/x^2, x, 0, Unintegrable[PolyLog[n, a*x]/x^2, x]} -{PolyLog[n, a*x]/x^3, x, 0, Unintegrable[PolyLog[n, a*x]/x^3, x]} - - -{x^1*PolyLog[n, a*x^q], x, 0, Unintegrable[x*PolyLog[n, a*x^q], x]} -{x^0*PolyLog[n, a*x^q], x, 0, Unintegrable[PolyLog[n, a*x^q], x]} -{PolyLog[n, a*x^q]/x^1, x, 1, PolyLog[1 + n, a*x^q]/q} -{PolyLog[n, a*x^q]/x^2, x, 0, Unintegrable[PolyLog[n, a*x^q]/x^2, x]} -{PolyLog[n, a*x^q]/x^3, x, 0, Unintegrable[PolyLog[n, a*x^q]/x^3, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[n, c (a+b x)]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m PolyLog[n, c (a+b x)]*) - - -{x^2*PolyLog[2, c*(a + b*x)], x, 13, -((a^2*x)/(3*b^2)) + (a*(1 - a*c)*x)/(6*b^2*c) - ((1 - a*c)^2*x)/(9*b^2*c^2) + (a*x^2)/(12*b) - ((1 - a*c)*x^2)/(18*b*c) - x^3/27 + (a*(1 - a*c)^2*Log[1 - a*c - b*c*x])/(6*b^3*c^2) - ((1 - a*c)^3*Log[1 - a*c - b*c*x])/(9*b^3*c^3) - (a*x^2*Log[1 - a*c - b*c*x])/(6*b) + (1/9)*x^3*Log[1 - a*c - b*c*x] - (a^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^3*c) + (a^3*PolyLog[2, c*(a + b*x)])/(3*b^3) + (1/3)*x^3*PolyLog[2, c*(a + b*x)]} -{x^1*PolyLog[2, c*(a + b*x)], x, 10, (a*x)/(2*b) - ((1 - a*c)*x)/(4*b*c) - x^2/8 - ((1 - a*c)^2*Log[1 - a*c - b*c*x])/(4*b^2*c^2) + (1/4)*x^2*Log[1 - a*c - b*c*x] + (a*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(2*b^2*c) - (a^2*PolyLog[2, c*(a + b*x)])/(2*b^2) + (1/2)*x^2*PolyLog[2, c*(a + b*x)]} -{x^0*PolyLog[2, c*(a + b*x)], x, 7, -x - ((1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c) + (a*PolyLog[2, c*(a + b*x)])/b + x*PolyLog[2, c*(a + b*x)]} -{PolyLog[2, c*(a + b*x)]/x^1, x, 3, Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)] + (1/2)*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2 + (1/2)*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2 + (Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)] + Log[x]*PolyLog[2, c*(a + b*x)] + Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))] - Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))] + (Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)] - PolyLog[3, -((b*x)/a)] + PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))] - PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))] - PolyLog[3, 1 - c*(a + b*x)]} -{PolyLog[2, c*(a + b*x)]/x^2, x, 7, -((b*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/a) - (b*PolyLog[2, c*(a + b*x)])/a - PolyLog[2, c*(a + b*x)]/x - (b*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a} -{PolyLog[2, c*(a + b*x)]/x^3, x, 11, (b^2*c*Log[x])/(2*a*(1 - a*c)) - (b^2*c*Log[1 - a*c - b*c*x])/(2*a*(1 - a*c)) + (b*Log[1 - a*c - b*c*x])/(2*a*x) + (b^2*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a^2) + (b^2*PolyLog[2, c*(a + b*x)])/(2*a^2) - PolyLog[2, c*(a + b*x)]/(2*x^2) + (b^2*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2)} -{PolyLog[2, c*(a + b*x)]/x^4, x, 14, -((b^2*c)/(6*a*(1 - a*c)*x)) + (b^3*c^2*Log[x])/(6*a*(1 - a*c)^2) - (b^3*c*Log[x])/(3*a^2*(1 - a*c)) - (b^3*c^2*Log[1 - a*c - b*c*x])/(6*a*(1 - a*c)^2) + (b^3*c*Log[1 - a*c - b*c*x])/(3*a^2*(1 - a*c)) + (b*Log[1 - a*c - b*c*x])/(6*a*x^2) - (b^2*Log[1 - a*c - b*c*x])/(3*a^2*x) - (b^3*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(3*a^3) - (b^3*PolyLog[2, c*(a + b*x)])/(3*a^3) - PolyLog[2, c*(a + b*x)]/(3*x^3) - (b^3*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(3*a^3)} - - -{x^2*PolyLog[3, c*(a + b*x)], x, 33, (11*a^2*x)/(18*b^2) - (5*a*(1 - a*c)*x)/(36*b^2*c) + ((1 - a*c)^2*x)/(27*b^2*c^2) - (5*a*x^2)/(72*b) + ((1 - a*c)*x^2)/(54*b*c) + x^3/81 - (5*a*(1 - a*c)^2*Log[1 - a*c - b*c*x])/(36*b^3*c^2) + ((1 - a*c)^3*Log[1 - a*c - b*c*x])/(27*b^3*c^3) + (5*a*x^2*Log[1 - a*c - b*c*x])/(36*b) - (1/27)*x^3*Log[1 - a*c - b*c*x] + (11*a^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(18*b^3*c) - (11*a^3*PolyLog[2, c*(a + b*x)])/(18*b^3) - (a^2*x*PolyLog[2, c*(a + b*x)])/(3*b^2) + (a*x^2*PolyLog[2, c*(a + b*x)])/(6*b) - (1/9)*x^3*PolyLog[2, c*(a + b*x)] + (2*a^3*PolyLog[3, c*(a + b*x)])/(3*b^3) - ((a^3 - b^3*x^3)*PolyLog[3, c*(a + b*x)])/(3*b^3)} -{x^1*PolyLog[3, c*(a + b*x)], x, 19, -((3*a*x)/(4*b)) + ((1 - a*c)*x)/(8*b*c) + x^2/16 + ((1 - a*c)^2*Log[1 - a*c - b*c*x])/(8*b^2*c^2) - (1/8)*x^2*Log[1 - a*c - b*c*x] - (3*a*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(4*b^2*c) + (3*a^2*PolyLog[2, c*(a + b*x)])/(4*b^2) + (a*x*PolyLog[2, c*(a + b*x)])/(2*b) - (1/4)*x^2*PolyLog[2, c*(a + b*x)] - ((a^2 - b^2*x^2)*PolyLog[3, c*(a + b*x)])/(2*b^2)} -{x^0*PolyLog[3, c*(a + b*x)], x, 9, x + ((1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c) - (a*PolyLog[2, c*(a + b*x)])/b - x*PolyLog[2, c*(a + b*x)] + (a*PolyLog[3, c*(a + b*x)])/b + x*PolyLog[3, c*(a + b*x)]} -{PolyLog[3, c*(a + b*x)]/x^1, x, 1, Int[PolyLog[3, a*c + b*c*x]/x, x]} -{PolyLog[3, c*(a + b*x)]/x^2, x, 6, (b*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/a + (b*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(2*a) + (b*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(2*a) + (b*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/a + (b*Log[x]*PolyLog[2, c*(a + b*x)])/a + (b*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/a - (b*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/a + (b*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)])/a - (b*PolyLog[3, -((b*x)/a)])/a - (2*b*PolyLog[3, c*(a + b*x)])/a + ((b - a/x)*PolyLog[3, c*(a + b*x)])/a + (b*PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))])/a - (b*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/a - (b*PolyLog[3, 1 - c*(a + b*x)])/a} -{PolyLog[3, c*(a + b*x)]/x^3, x, 12, -((b^2*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a^2)) - (b^2*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/(2*a^2) - (b^2*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(4*a^2) - (b^2*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(4*a^2) - (b^2*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/(2*a^2) - (b^2*PolyLog[2, c*(a + b*x)])/(2*a^2) - (b*PolyLog[2, c*(a + b*x)])/(2*a*x) - (b^2*Log[x]*PolyLog[2, c*(a + b*x)])/(2*a^2) - (b^2*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2) - (b^2*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/(2*a^2) + (b^2*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/(2*a^2) - (b^2*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)])/(2*a^2) + (b^2*PolyLog[3, -((b*x)/a)])/(2*a^2) + ((b^2 - a^2/x^2)*PolyLog[3, c*(a + b*x)])/(2*a^2) - (b^2*PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))])/(2*a^2) + (b^2*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/(2*a^2) + (b^2*PolyLog[3, 1 - c*(a + b*x)])/(2*a^2)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form (d+e x)^m PolyLog[n,c (a+b x)]*) - - -{PolyLog[2, c*(a + b*x)]*(d + e*x)^3, x, 16, -(((b*d - a*e)^3*x)/(4*b^3)) - ((b*d - a*e)^2*(b*c*d + e - a*c*e)*x)/(8*b^3*c) - ((b*d - a*e)*(b*c*d + e - a*c*e)^2*x)/(12*b^3*c^2) - ((b*c*d + e - a*c*e)^3*x)/(16*b^3*c^3) - ((b*d - a*e)^2*(d + e*x)^2)/(16*b^2*e) - ((b*d - a*e)*(b*c*d + e - a*c*e)*(d + e*x)^2)/(24*b^2*c*e) - ((b*c*d + e - a*c*e)^2*(d + e*x)^2)/(32*b^2*c^2*e) - ((b*d - a*e)*(d + e*x)^3)/(36*b*e) - ((b*c*d + e - a*c*e)*(d + e*x)^3)/(48*b*c*e) - (d + e*x)^4/(64*e) - ((b*d - a*e)^2*(b*c*d + e - a*c*e)^2*Log[1 - a*c - b*c*x])/(8*b^4*c^2*e) - ((b*d - a*e)*(b*c*d + e - a*c*e)^3*Log[1 - a*c - b*c*x])/(12*b^4*c^3*e) - ((b*c*d + e - a*c*e)^4*Log[1 - a*c - b*c*x])/(16*b^4*c^4*e) - ((b*d - a*e)^3*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(4*b^4*c) + ((b*d - a*e)^2*(d + e*x)^2*Log[1 - a*c - b*c*x])/(8*b^2*e) + ((b*d - a*e)*(d + e*x)^3*Log[1 - a*c - b*c*x])/(12*b*e) + ((d + e*x)^4*Log[1 - a*c - b*c*x])/(16*e) - ((b*d - a*e)^4*PolyLog[2, c*(a + b*x)])/(4*b^4*e) + ((d + e*x)^4*PolyLog[2, c*(a + b*x)])/(4*e)} -{PolyLog[2, c*(a + b*x)]*(d + e*x)^2, x, 13, -(((b*d - a*e)^2*x)/(3*b^2)) - ((b*d - a*e)*(b*c*d + e - a*c*e)*x)/(6*b^2*c) - ((b*c*d + e - a*c*e)^2*x)/(9*b^2*c^2) - ((b*d - a*e)*(d + e*x)^2)/(12*b*e) - ((b*c*d + e - a*c*e)*(d + e*x)^2)/(18*b*c*e) - (d + e*x)^3/(27*e) - ((b*d - a*e)*(b*c*d + e - a*c*e)^2*Log[1 - a*c - b*c*x])/(6*b^3*c^2*e) - ((b*c*d + e - a*c*e)^3*Log[1 - a*c - b*c*x])/(9*b^3*c^3*e) - ((b*d - a*e)^2*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^3*c) + ((b*d - a*e)*(d + e*x)^2*Log[1 - a*c - b*c*x])/(6*b*e) + ((d + e*x)^3*Log[1 - a*c - b*c*x])/(9*e) - ((b*d - a*e)^3*PolyLog[2, c*(a + b*x)])/(3*b^3*e) + ((d + e*x)^3*PolyLog[2, c*(a + b*x)])/(3*e)} -{PolyLog[2, c*(a + b*x)]*(d + e*x)^1, x, 10, -(((b*d - a*e)*x)/(2*b)) - ((b*c*d + e - a*c*e)*x)/(4*b*c) - (d + e*x)^2/(8*e) - ((b*c*d + e - a*c*e)^2*Log[1 - a*c - b*c*x])/(4*b^2*c^2*e) - ((b*d - a*e)*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(2*b^2*c) + ((d + e*x)^2*Log[1 - a*c - b*c*x])/(4*e) - ((b*d - a*e)^2*PolyLog[2, c*(a + b*x)])/(2*b^2*e) + ((d + e*x)^2*PolyLog[2, c*(a + b*x)])/(2*e)} -{PolyLog[2, c*(a + b*x)]*(d + e*x)^0, x, 7, -x - ((1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c) + (a*PolyLog[2, c*(a + b*x)])/b + x*PolyLog[2, c*(a + b*x)]} -{PolyLog[2, c*(a + b*x)]/(d + e*x)^1, x, 3, ((Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(2*e) + (Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/e - ((Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*e) + (Log[d + e*x]*PolyLog[2, c*(a + b*x)])/e + ((Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/e + ((Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/e - (Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/e + (Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/e - PolyLog[3, (b*(d + e*x))/(b*d - a*e)]/e - PolyLog[3, 1 - c*(a + b*x)]/e - PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))]/e + PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))]/e} -{PolyLog[2, c*(a + b*x)]/(d + e*x)^2, x, 8, (b*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(e*(b*d - a*e)) + (b*PolyLog[2, c*(a + b*x)])/(e*(b*d - a*e)) - PolyLog[2, c*(a + b*x)]/(e*(d + e*x)) + (b*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(e*(b*d - a*e))} -{PolyLog[2, c*(a + b*x)]/(d + e*x)^3, x, 12, (b^2*c*Log[1 - a*c - b*c*x])/(2*e*(b*d - a*e)*(b*c*d + e - a*c*e)) - (b*Log[1 - a*c - b*c*x])/(2*e*(b*d - a*e)*(d + e*x)) - (b^2*c*Log[d + e*x])/(2*e*(b*d - a*e)*(b*c*d + e - a*c*e)) + (b^2*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(2*e*(b*d - a*e)^2) + (b^2*PolyLog[2, c*(a + b*x)])/(2*e*(b*d - a*e)^2) - PolyLog[2, c*(a + b*x)]/(2*e*(d + e*x)^2) + (b^2*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(2*e*(b*d - a*e)^2)} -{PolyLog[2, c*(a + b*x)]/(d + e*x)^4, x, 15, (b^2*c)/(6*e*(b*d - a*e)*(b*c*d + e - a*c*e)*(d + e*x)) + (b^3*c^2*Log[1 - a*c - b*c*x])/(6*e*(b*d - a*e)*(b*c*d + e - a*c*e)^2) + (b^3*c*Log[1 - a*c - b*c*x])/(3*e*(b*d - a*e)^2*(b*c*d + e - a*c*e)) - (b*Log[1 - a*c - b*c*x])/(6*e*(b*d - a*e)*(d + e*x)^2) - (b^2*Log[1 - a*c - b*c*x])/(3*e*(b*d - a*e)^2*(d + e*x)) - (b^3*c^2*Log[d + e*x])/(6*e*(b*d - a*e)*(b*c*d + e - a*c*e)^2) - (b^3*c*Log[d + e*x])/(3*e*(b*d - a*e)^2*(b*c*d + e - a*c*e)) + (b^3*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*e*(b*d - a*e)^3) + (b^3*PolyLog[2, c*(a + b*x)])/(3*e*(b*d - a*e)^3) - PolyLog[2, c*(a + b*x)]/(3*e*(d + e*x)^3) + (b^3*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(3*e*(b*d - a*e)^3)} - - -(* Following integrands are equal. *) -{PolyLog[2, x]/(-1 + x), x, 5, Log[1-x]^2 Log[x]+2 Log[1-x] PolyLog[2,1-x]+Log[1-x] PolyLog[2,x]-2 PolyLog[3,1-x]} -{-PolyLog[2, x]/(1 - x), x, 5, Log[1-x]^2 Log[x]+2 Log[1-x] PolyLog[2,1-x]+Log[1-x] PolyLog[2,x]-2 PolyLog[3,1-x]} - -{PolyLog[2, x]/((-1 + x)*x), x, 8, Log[1-x]^2 Log[x]+2 Log[1-x] PolyLog[2,1-x]+Log[1-x] PolyLog[2,x]-2 PolyLog[3,1-x]-PolyLog[3,x]} -{-PolyLog[2, x]/((1 - x)*x), x, 8, Log[1-x]^2 Log[x]+2 Log[1-x] PolyLog[2,1-x]+Log[1-x] PolyLog[2,x]-2 PolyLog[3,1-x]-PolyLog[3,x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form PolyLog[n, e ((a + b x) / (c + d x))^n] / ((a + b x) (c + d x))*) - - -{PolyLog[n, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 1, PolyLog[1 + n, e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n)} - - -{PolyLog[3, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 1, PolyLog[4, e*((a + b*x)/(c + d*x))^n]/(n*(b*c - a*d))} -{PolyLog[2, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 1, PolyLog[3, e*((a + b*x)/(c + d*x))^n]/(n*(b*c - a*d))} -{PolyLog[1, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 1, PolyLog[2, e*((a + b*x)/(c + d*x))^n]/(n*(b*c - a*d))} -{PolyLog[0, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 2, PolyLog[1, e*((a + b*x)/(c + d*x))^n]/(n*(b*c - a*d))} -{PolyLog[-1, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 2, 1/((b*c - a*d)*n*(1 - e*((a + b*x)/(c + d*x))^n))} -{PolyLog[-2, e*((a + b*x)/(c + d*x))^n]/((a + b*x)*(c + d*x)), x, 4, PolyLog[-1, e*((a + b*x)/(c + d*x))^n]/(n*(b*c - a*d))} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m PolyLog[n, d (F^(c (a + b x)))^p]*) - - -{x^3*PolyLog[n, d*(F^(c*(a + b*x)))^p], x, 5, (x^3*PolyLog[1 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - (3*x^2*PolyLog[2 + n, d*(F^(c*(a + b*x)))^p])/(b^2*c^2*p^2*Log[F]^2) + (6*x*PolyLog[3 + n, d*(F^(c*(a + b*x)))^p])/(b^3*c^3*p^3*Log[F]^3) - (6*PolyLog[4 + n, d*(F^(c*(a + b*x)))^p])/(b^4*c^4*p^4*Log[F]^4)} -{x^2*PolyLog[n, d*(F^(c*(a + b*x)))^p], x, 4, (x^2*PolyLog[1 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - (2*x*PolyLog[2 + n, d*(F^(c*(a + b*x)))^p])/(b^2*c^2*p^2*Log[F]^2) + (2*PolyLog[3 + n, d*(F^(c*(a + b*x)))^p])/(b^3*c^3*p^3*Log[F]^3)} -{x^1*PolyLog[n, d*(F^(c*(a + b*x)))^p], x, 3, (x*PolyLog[1 + n, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]) - PolyLog[2 + n, d*(F^(c*(a + b*x)))^p]/(b^2*c^2*p^2*Log[F]^2)} -{x^0*PolyLog[n, d*(F^(c*(a + b*x)))^p], x, 2, PolyLog[1 + n, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])} -{PolyLog[n, d*(F^(c*(a + b*x)))^p]/x^1, x, 1, CannotIntegrate[PolyLog[n, d*(F^(a*c + b*c*x))^p]/x, x]} - - -(* ::Section::Closed:: *) -(*Integrands of the form (d x)^m P[x] (g+h Log[f (d+e x)^n]) PolyLog[2, c (a+b x)]*) - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m Log[1-c x] PolyLog[2, c x]*) - - -{x^3*Log[1 - c*x]*PolyLog[2, c*x], x, 38, (355*x)/(576*c^3) + (139*x^2)/(1152*c^2) + (67*x^3)/(1728*c) + (3*x^4)/256 + (139*Log[1 - c*x])/(576*c^4) - (x^2*Log[1 - c*x])/(8*c^2) - (5*x^3*Log[1 - c*x])/(72*c) - (3/64)*x^4*Log[1 - c*x] + (3*(1 - c*x)*Log[1 - c*x])/(8*c^4) - Log[1 - c*x]^2/(16*c^4) + (1/16)*x^4*Log[1 - c*x]^2 - (Log[c*x]*Log[1 - c*x]^2)/(4*c^4) - (x*PolyLog[2, c*x])/(4*c^3) - (x^2*PolyLog[2, c*x])/(8*c^2) - (x^3*PolyLog[2, c*x])/(12*c) - (1/16)*x^4*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(4*c^4) + (1/4)*x^4*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, 1 - c*x])/(2*c^4) + PolyLog[3, 1 - c*x]/(2*c^4)} -{x^2*Log[1 - c*x]*PolyLog[2, c*x], x, 31, (31*x)/(36*c^2) + (11*x^2)/(72*c) + x^3/27 + (11*Log[1 - c*x])/(36*c^3) - (7*x^2*Log[1 - c*x])/(36*c) - (1/9)*x^3*Log[1 - c*x] + (5*(1 - c*x)*Log[1 - c*x])/(9*c^3) - Log[1 - c*x]^2/(9*c^3) + (1/9)*x^3*Log[1 - c*x]^2 - (Log[c*x]*Log[1 - c*x]^2)/(3*c^3) - (x*PolyLog[2, c*x])/(3*c^2) - (x^2*PolyLog[2, c*x])/(6*c) - (1/9)*x^3*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(3*c^3) + (1/3)*x^3*Log[1 - c*x]*PolyLog[2, c*x] - (2*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(3*c^3) + (2*PolyLog[3, 1 - c*x])/(3*c^3)} -{x^1*Log[1 - c*x]*PolyLog[2, c*x], x, 22, (13*x)/(8*c) + x^2/16 + (1 - c*x)^2/(8*c^2) + Log[1 - c*x]/(8*c^2) - (1/8)*x^2*Log[1 - c*x] + (3*(1 - c*x)*Log[1 - c*x])/(2*c^2) - ((1 - c*x)^2*Log[1 - c*x])/(4*c^2) - ((1 - c*x)*Log[1 - c*x]^2)/(2*c^2) + ((1 - c*x)^2*Log[1 - c*x]^2)/(4*c^2) - (Log[c*x]*Log[1 - c*x]^2)/(2*c^2) - (x*PolyLog[2, c*x])/(2*c) - (1/4)*x^2*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(2*c^2) + (1/2)*x^2*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, 1 - c*x])/c^2 + PolyLog[3, 1 - c*x]/c^2} -{x^0*Log[1 - c*x]*PolyLog[2, c*x], x, 15, 3*x + (3*(1 - c*x)*Log[1 - c*x])/c - ((1 - c*x)*Log[1 - c*x]^2)/c - (Log[c*x]*Log[1 - c*x]^2)/c - x*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/c + x*Log[1 - c*x]*PolyLog[2, c*x] - (2*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c + (2*PolyLog[3, 1 - c*x])/c} -{Log[1 - c*x]*PolyLog[2, c*x]/x^1, x, 1, (-(1/2))*PolyLog[2, c*x]^2} -{Log[1 - c*x]*PolyLog[2, c*x]/x^2, x, 10, ((1 - c*x)*Log[1 - c*x]^2)/x + c*Log[c*x]*Log[1 - c*x]^2 - 2*c*PolyLog[2, c*x] + c*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/x + 2*c*Log[1 - c*x]*PolyLog[2, 1 - c*x] - c*PolyLog[3, c*x] - 2*c*PolyLog[3, 1 - c*x]} -{Log[1 - c*x]*PolyLog[2, c*x]/x^3, x, 23, (-c^2)*Log[x] + c^2*Log[1 - c*x] - (c*Log[1 - c*x])/x - (1/4)*c^2*Log[1 - c*x]^2 + Log[1 - c*x]^2/(4*x^2) + (1/2)*c^2*Log[c*x]*Log[1 - c*x]^2 - (1/2)*c^2*PolyLog[2, c*x] + (c*PolyLog[2, c*x])/(2*x) + (1/2)*c^2*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(2*x^2) + c^2*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/2)*c^2*PolyLog[3, c*x] - c^2*PolyLog[3, 1 - c*x]} -{Log[1 - c*x]*PolyLog[2, c*x]/x^4, x, 30, (7*c^2)/(36*x) - (3/4)*c^3*Log[x] + (3/4)*c^3*Log[1 - c*x] - (7*c*Log[1 - c*x])/(36*x^2) - (5*c^2*Log[1 - c*x])/(9*x) - (1/9)*c^3*Log[1 - c*x]^2 + Log[1 - c*x]^2/(9*x^3) + (1/3)*c^3*Log[c*x]*Log[1 - c*x]^2 - (2/9)*c^3*PolyLog[2, c*x] + (c*PolyLog[2, c*x])/(6*x^2) + (c^2*PolyLog[2, c*x])/(3*x) + (1/3)*c^3*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(3*x^3) + (2/3)*c^3*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/3)*c^3*PolyLog[3, c*x] - (2/3)*c^3*PolyLog[3, 1 - c*x]} -{Log[1 - c*x]*PolyLog[2, c*x]/x^5, x, 37, (5*c^2)/(144*x^2) + (7*c^3)/(36*x) - (41/72)*c^4*Log[x] + (41/72)*c^4*Log[1 - c*x] - (5*c*Log[1 - c*x])/(72*x^3) - (c^2*Log[1 - c*x])/(8*x^2) - (3*c^3*Log[1 - c*x])/(8*x) - (1/16)*c^4*Log[1 - c*x]^2 + Log[1 - c*x]^2/(16*x^4) + (1/4)*c^4*Log[c*x]*Log[1 - c*x]^2 - (1/8)*c^4*PolyLog[2, c*x] + (c*PolyLog[2, c*x])/(12*x^3) + (c^2*PolyLog[2, c*x])/(8*x^2) + (c^3*PolyLog[2, c*x])/(4*x) + (1/4)*c^4*Log[1 - c*x]*PolyLog[2, c*x] - (Log[1 - c*x]*PolyLog[2, c*x])/(4*x^4) + (1/2)*c^4*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/4)*c^4*PolyLog[3, c*x] - (1/2)*c^4*PolyLog[3, 1 - c*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (g+h Log[1-c x]) PolyLog[2, c x]*) - - -{x^2*(g + h*Log[1 - c*x])*PolyLog[2, c*x], x, 25, (121*h*x)/(108*c^2) + (13*h*x^2)/(216*c) + (h*x^3)/81 + (h*(1 - c*x)^2)/(6*c^3) - (2*h*(1 - c*x)^3)/(81*c^3) + (13*h*Log[1 - c*x])/(108*c^3) - (h*x^2*Log[1 - c*x])/(12*c) - (1/27)*h*x^3*Log[1 - c*x] + (h*(1 - c*x)*Log[1 - c*x])/(3*c^3) + (h*Log[1 - c*x]^2)/(9*c^3) - (h*Log[c*x]*Log[1 - c*x]^2)/(3*c^3) + (1/9)*x^3*Log[1 - c*x]*(g + h*Log[1 - c*x]) + ((1 - c*x)*(g + 2*h*Log[1 - c*x]))/(3*c^3) - ((1 - c*x)^2*(g + 2*h*Log[1 - c*x]))/(6*c^3) + ((1 - c*x)^3*(g + 2*h*Log[1 - c*x]))/(27*c^3) - (Log[1 - c*x]*(g + 2*h*Log[1 - c*x]))/(9*c^3) - (h*x*PolyLog[2, c*x])/(3*c^2) - (h*x^2*PolyLog[2, c*x])/(6*c) - (1/9)*h*x^3*PolyLog[2, c*x] - (h*Log[1 - c*x]*PolyLog[2, c*x])/(3*c^3) + (1/3)*x^3*(g + h*Log[1 - c*x])*PolyLog[2, c*x] - (2*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(3*c^3) + (2*h*PolyLog[3, 1 - c*x])/(3*c^3)} -{x^1*(g + h*Log[1 - c*x])*PolyLog[2, c*x], x, 21, (13*h*x)/(8*c) + (h*x^2)/16 + (h*(1 - c*x)^2)/(8*c^2) + (h*Log[1 - c*x])/(8*c^2) - (1/8)*h*x^2*Log[1 - c*x] + (h*(1 - c*x)*Log[1 - c*x])/(2*c^2) + (h*Log[1 - c*x]^2)/(4*c^2) - (h*Log[c*x]*Log[1 - c*x]^2)/(2*c^2) + (1/4)*x^2*Log[1 - c*x]*(g + h*Log[1 - c*x]) + ((1 - c*x)*(g + 2*h*Log[1 - c*x]))/(2*c^2) - ((1 - c*x)^2*(g + 2*h*Log[1 - c*x]))/(8*c^2) - (Log[1 - c*x]*(g + 2*h*Log[1 - c*x]))/(4*c^2) - (h*x*PolyLog[2, c*x])/(2*c) - (1/4)*h*x^2*PolyLog[2, c*x] - (h*Log[1 - c*x]*PolyLog[2, c*x])/(2*c^2) + (1/2)*x^2*(g + h*Log[1 - c*x])*PolyLog[2, c*x] - (h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c^2 + (h*PolyLog[3, 1 - c*x])/c^2} -{x^0*(g + h*Log[1 - c*x])*PolyLog[2, c*x], x, 18, (-g)*x + 3*h*x - (g*(1 - c*x)*Log[1 - c*x])/c + (3*h*(1 - c*x)*Log[1 - c*x])/c - (h*(1 - c*x)*Log[1 - c*x]^2)/c - (h*Log[c*x]*Log[1 - c*x]^2)/c - h*x*PolyLog[2, c*x] - (h*Log[1 - c*x]*PolyLog[2, c*x])/c + x*(g + h*Log[1 - c*x])*PolyLog[2, c*x] - (2*h*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c + (2*h*PolyLog[3, 1 - c*x])/c} -{(g + h*Log[1 - c*x])*PolyLog[2, c*x]/x^1, x, 3, (-(1/2))*h*PolyLog[2, c*x]^2 + g*PolyLog[3, c*x]} -{(g + h*Log[1 - c*x])*PolyLog[2, c*x]/x^2, x, 12, c*h*Log[c*x]*Log[1 - c*x]^2 + (Log[1 - c*x]*(g + h*Log[1 - c*x]))/x + c*(g + 2*h*Log[1 - c*x])*Log[1 - 1/(1 - c*x)] + c*h*Log[1 - c*x]*PolyLog[2, c*x] - ((g + h*Log[1 - c*x])*PolyLog[2, c*x])/x - 2*c*h*PolyLog[2, 1/(1 - c*x)] + 2*c*h*Log[1 - c*x]*PolyLog[2, 1 - c*x] - c*h*PolyLog[3, c*x] - 2*c*h*PolyLog[3, 1 - c*x]} -{(g + h*Log[1 - c*x])*PolyLog[2, c*x]/x^3, x, 20, (-c^2)*h*Log[x] + (1/2)*c^2*h*Log[1 - c*x] - (c*h*Log[1 - c*x])/(2*x) + (1/2)*c^2*h*Log[c*x]*Log[1 - c*x]^2 + (Log[1 - c*x]*(g + h*Log[1 - c*x]))/(4*x^2) - (c*(1 - c*x)*(g + 2*h*Log[1 - c*x]))/(4*x) + (1/4)*c^2*(g + 2*h*Log[1 - c*x])*Log[1 - 1/(1 - c*x)] + (c*h*PolyLog[2, c*x])/(2*x) + (1/2)*c^2*h*Log[1 - c*x]*PolyLog[2, c*x] - ((g + h*Log[1 - c*x])*PolyLog[2, c*x])/(2*x^2) - (1/2)*c^2*h*PolyLog[2, 1/(1 - c*x)] + c^2*h*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/2)*c^2*h*PolyLog[3, c*x] - c^2*h*PolyLog[3, 1 - c*x]} -{(g + h*Log[1 - c*x])*PolyLog[2, c*x]/x^4, x, 28, (7*c^2*h)/(36*x) - (3/4)*c^3*h*Log[x] + (19/36)*c^3*h*Log[1 - c*x] - (c*h*Log[1 - c*x])/(12*x^2) - (c^2*h*Log[1 - c*x])/(3*x) + (1/3)*c^3*h*Log[c*x]*Log[1 - c*x]^2 + (Log[1 - c*x]*(g + h*Log[1 - c*x]))/(9*x^3) - (c*(g + 2*h*Log[1 - c*x]))/(18*x^2) - (c^2*(1 - c*x)*(g + 2*h*Log[1 - c*x]))/(9*x) + (1/9)*c^3*(g + 2*h*Log[1 - c*x])*Log[1 - 1/(1 - c*x)] + (c*h*PolyLog[2, c*x])/(6*x^2) + (c^2*h*PolyLog[2, c*x])/(3*x) + (1/3)*c^3*h*Log[1 - c*x]*PolyLog[2, c*x] - ((g + h*Log[1 - c*x])*PolyLog[2, c*x])/(3*x^3) - (2/9)*c^3*h*PolyLog[2, 1/(1 - c*x)] + (2/3)*c^3*h*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/3)*c^3*h*PolyLog[3, c*x] - (2/3)*c^3*h*PolyLog[3, 1 - c*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (g+h Log[f (d+e x)^n]) PolyLog[2, c (a+b x)]*) - - -{x^2*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x, 108, -((a^2*g*x)/(3*b^2)) + (a*(1 - a*c)*g*x)/(6*b^2*c) - ((1 - a*c)^2*g*x)/(9*b^2*c^2) + (7*a^2*h*n*x)/(9*b^2) - (11*a*(1 - a*c)*h*n*x)/(36*b^2*c) + (5*(1 - a*c)^2*h*n*x)/(27*b^2*c^2) + (13*d^2*h*n*x)/(27*e^2) + (5*a*d*h*n*x)/(12*b*e) - (7*(1 - a*c)*d*h*n*x)/(36*b*c*e) - (a*h*n*x^2)/(9*b) + (7*(1 - a*c)*h*n*x^2)/(108*b*c) - (19*d*h*n*x^2)/(216*e) + (1/27)*h*n*x^3 - (5*a*(1 - a*c)^2*h*n*Log[1 - a*c - b*c*x])/(36*b^3*c^2) + (2*(1 - a*c)^3*h*n*Log[1 - a*c - b*c*x])/(27*b^3*c^3) - (5*(1 - a*c)^2*d*h*n*Log[1 - a*c - b*c*x])/(36*b^2*c^2*e) + (5*a*h*n*x^2*Log[1 - a*c - b*c*x])/(36*b) + (5*d*h*n*x^2*Log[1 - a*c - b*c*x])/(36*e) - (2/27)*h*n*x^3*Log[1 - a*c - b*c*x] + (4*a^2*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(9*b^3*c) + (4*d^2*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(9*b*c*e^2) + (a*d*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(3*b^2*c*e) - (d^3*h*n*Log[d + e*x])/(27*e^3) - (a*d^2*h*n*Log[d + e*x])/(12*b*e^2) + ((1 - a*c)*d^2*h*n*Log[d + e*x])/(18*b*c*e^2) + (d^3*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(9*e^3) + (a*d^2*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*b*e^2) + (a^2*d*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*b^2*e) - (a^2*h*(d + e*x)*Log[f*(d + e*x)^n])/(3*b^2*e) + (a*(1 - a*c)*h*(d + e*x)*Log[f*(d + e*x)^n])/(6*b^2*c*e) - ((1 - a*c)^2*h*(d + e*x)*Log[f*(d + e*x)^n])/(9*b^2*c^2*e) + (a*x^2*(g + h*Log[f*(d + e*x)^n]))/(12*b) - ((1 - a*c)*x^2*(g + h*Log[f*(d + e*x)^n]))/(18*b*c) - (1/27)*x^3*(g + h*Log[f*(d + e*x)^n]) + (a^2*x*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(3*b^2) - (a*x^2*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(6*b) + (1/9)*x^3*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]) - (a^2*(1 - a*c)*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(3*b^3*c) + (a*(1 - a*c)^2*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(6*b^3*c^2) - ((1 - a*c)^3*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(9*b^3*c^3) - (a^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(6*b^3) + (d^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(6*e^3) - (a^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*b^3) + (d^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*e^3) + (a^3*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(6*b^3) - (d^3*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(6*e^3) + (a^3*g*PolyLog[2, c*(a + b*x)])/(3*b^3) - (a^3*h*n*PolyLog[2, c*(a + b*x)])/(9*b^3) - (a*d^2*h*n*PolyLog[2, c*(a + b*x)])/(3*b*e^2) - (a^2*d*h*n*PolyLog[2, c*(a + b*x)])/(6*b^2*e) - (d^2*h*n*x*PolyLog[2, c*(a + b*x)])/(3*e^2) + (d*h*n*x^2*PolyLog[2, c*(a + b*x)])/(6*e) - (1/9)*h*n*x^3*PolyLog[2, c*(a + b*x)] + (d^3*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/(3*e^3) - (a^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(3*b^3) + (1/3)*x^3*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + (d^3*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(9*e^3) + (a*d^2*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(6*b*e^2) + (a^2*d*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(3*b^2*e) - (a^3*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*b^3) + (d^3*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*e^3) - (a^2*(1 - a*c)*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*b^3*c) + (a*(1 - a*c)^2*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*b^3*c^2) - ((1 - a*c)^3*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(9*b^3*c^3) - (a^3*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(3*b^3) + (d^3*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(3*e^3) + (a^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*b^3) - (d^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*e^3) - (a^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*b^3) + (d^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*e^3) + (a^3*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*b^3) - (d^3*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*e^3) + (a^3*h*n*PolyLog[3, 1 - c*(a + b*x)])/(3*b^3) - (d^3*h*n*PolyLog[3, 1 - c*(a + b*x)])/(3*e^3) + (a^3*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*b^3) - (d^3*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*e^3) - (a^3*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*b^3) + (d^3*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*e^3)} -{x^1*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x, 67, (a*g*x)/(2*b) - ((1 - a*c)*g*x)/(4*b*c) - (5*a*h*n*x)/(4*b) + ((1 - a*c)*h*n*x)/(2*b*c) - (7*d*h*n*x)/(8*e) + (3/16)*h*n*x^2 + ((1 - a*c)^2*h*n*Log[1 - a*c - b*c*x])/(4*b^2*c^2) - (1/4)*h*n*x^2*Log[1 - a*c - b*c*x] - (3*a*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(4*b^2*c) - (3*d*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(4*b*c*e) + (d^2*h*n*Log[d + e*x])/(8*e^2) - (d^2*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(4*e^2) - (a*d*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(2*b*e) + (a*h*(d + e*x)*Log[f*(d + e*x)^n])/(2*b*e) - ((1 - a*c)*h*(d + e*x)*Log[f*(d + e*x)^n])/(4*b*c*e) - (1/8)*x^2*(g + h*Log[f*(d + e*x)^n]) - (a*x*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(2*b) + (1/4)*x^2*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]) + (a*(1 - a*c)*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(2*b^2*c) - ((1 - a*c)^2*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(4*b^2*c^2) + (a^2*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(4*b^2) - (d^2*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(4*e^2) + (a^2*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(2*b^2) - (d^2*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(2*e^2) - (a^2*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(4*b^2) + (d^2*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(4*e^2) - (a^2*g*PolyLog[2, c*(a + b*x)])/(2*b^2) + (a^2*h*n*PolyLog[2, c*(a + b*x)])/(4*b^2) + (a*d*h*n*PolyLog[2, c*(a + b*x)])/(2*b*e) + (d*h*n*x*PolyLog[2, c*(a + b*x)])/(2*e) - (1/4)*h*n*x^2*PolyLog[2, c*(a + b*x)] - (d^2*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/(2*e^2) + (a^2*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(2*b^2) + (1/2)*x^2*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] - (d^2*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(4*e^2) - (a*d*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(2*b*e) + (a^2*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(2*b^2) - (d^2*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(2*e^2) + (a*(1 - a*c)*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(2*b^2*c) - ((1 - a*c)^2*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(4*b^2*c^2) + (a^2*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(2*b^2) - (d^2*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(2*e^2) - (a^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*b^2) + (d^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*e^2) + (a^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*b^2) - (d^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*e^2) - (a^2*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(2*b^2) + (d^2*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(2*e^2) - (a^2*h*n*PolyLog[3, 1 - c*(a + b*x)])/(2*b^2) + (d^2*h*n*PolyLog[3, 1 - c*(a + b*x)])/(2*e^2) - (a^2*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*b^2) + (d^2*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*e^2) + (a^2*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*b^2) - (d^2*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*e^2)} -{x^0*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)], x, 42, (-g)*x + 3*h*n*x - (g*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c) + (2*h*n*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c) + (d*h*n*Log[c*(a + b*x)]*Log[1 - a*c - b*c*x]*Log[-d - e*x])/e + (d*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/e + (d*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))]^2)/(2*e) - (d*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[1 - a*c - b*c*x] + Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))])^2)/(2*e) - (h*(d + e*x)*Log[f*(d + e*x)^n])/e + h*x*Log[1 - a*c - b*c*x]*Log[f*(d + e*x)^n] - ((1 - a*c)*h*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*Log[f*(d + e*x)^n])/(b*c) - (a*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(2*b) - (a*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/b + (a*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*b) + (a*g*PolyLog[2, c*(a + b*x)])/b - (a*h*n*PolyLog[2, c*(a + b*x)])/b - (a*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/b + x*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)] + (d*h*n*(Log[-d - e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))])*PolyLog[2, 1 - a*c - b*c*x])/e + (d*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/e - h*n*x*PolyLog[2, a*c + b*c*x] + (d*h*n*Log[-d - e*x]*PolyLog[2, a*c + b*c*x])/e - (d*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))]*PolyLog[2, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/e + (d*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))]*PolyLog[2, ((b*d - a*e)*(1 - a*c - b*c*x))/(b*(d + e*x))])/e + (d*h*n*(Log[1 - a*c - b*c*x] + Log[(b*(d + e*x))/((b*d - a*e)*(1 - a*c - b*c*x))])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/e - (a*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/b - ((1 - a*c)*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(b*c) - (a*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/b + (a*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/b - (a*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/b - (d*h*n*PolyLog[3, 1 - a*c - b*c*x])/e - (d*h*n*PolyLog[3, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/e + (d*h*n*PolyLog[3, ((b*d - a*e)*(1 - a*c - b*c*x))/(b*(d + e*x))])/e + (a*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/b - (d*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/e + (a*h*n*PolyLog[3, 1 - c*(a + b*x)])/b + (a*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/b - (a*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/b} -{(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)]/x^1, x, 0, Unintegrable[((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x, x]} -{(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)]/x^2, x, 22, -((b*g*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/a) - (b*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Log[d + e*x])/a - (b*h*n*(Log[(b*c*x)/(1 - a*c)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*x)/((1 - a*c)*(d + e*x))])*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]^2)/(2*a) + (b*h*n*(Log[(b*c*x)/(1 - a*c)] - Log[-((e*x)/d)])*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])^2)/(2*a) + (b*h*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*(n*Log[d + e*x] - Log[f*(d + e*x)^n]))/a + (b*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(2*a) - (e*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(2*d) + (e*h*n*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/d + (b*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/a - (e*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/d - (b*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*a) + (e*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*d) + (e*h*n*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(2*d) + (e*h*n*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(2*d) + (e*h*n*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/d - (b*g*PolyLog[2, c*(a + b*x)])/a + (e*h*n*Log[x]*PolyLog[2, c*(a + b*x)])/d - (e*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/d + (b*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/a - ((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x - (b*g*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a - (b*h*n*(Log[d + e*x] - Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a + (b*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/a - (b*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/a + (b*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/a + (b*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/a - (e*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/d - (b*h*n*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 + (e*x)/d])/a + (e*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/d - (e*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/d + (b*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/a - (e*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/d + (e*h*n*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)])/d - (b*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/a + (e*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/d + (b*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/a - (e*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/d - (e*h*n*PolyLog[3, -((b*x)/a)])/d + (b*h*n*PolyLog[3, 1 - (b*c*x)/(1 - a*c)])/a - (b*h*n*PolyLog[3, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/a + (b*h*n*PolyLog[3, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/a - (b*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/a + (e*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/d + (b*h*n*PolyLog[3, 1 + (e*x)/d])/a + (e*h*n*PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))])/d - (e*h*n*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/d - (b*h*n*PolyLog[3, 1 - c*(a + b*x)])/a - (b*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/a + (e*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/d + (b*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/a - (e*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/d} -{(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)]/x^3, x, 44, (b^2*g*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a^2) - (b*e*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(a*d) + (b^2*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Log[d + e*x])/(2*a^2) + (b*e*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(2*a*d) + (b^2*h*n*(Log[(b*c*x)/(1 - a*c)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*x)/((1 - a*c)*(d + e*x))])*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]^2)/(4*a^2) - (b^2*h*n*(Log[(b*c*x)/(1 - a*c)] - Log[-((e*x)/d)])*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])^2)/(4*a^2) - (b^2*h*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*(n*Log[d + e*x] - Log[f*(d + e*x)^n]))/(2*a^2) + (b^2*c*Log[-((e*x)/d)]*(g + h*Log[f*(d + e*x)^n]))/(2*a*(1 - a*c)) + (b*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(2*a*x) - (b^2*c*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(2*a*(1 - a*c)) - (b^2*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(4*a^2) + (e^2*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(4*d^2) - (e^2*h*n*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/(2*d^2) - (b^2*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(2*a^2) + (e^2*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(2*d^2) + (b^2*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(4*a^2) - (e^2*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(4*d^2) - (e^2*h*n*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(4*d^2) - (e^2*h*n*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(4*d^2) - (e^2*h*n*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/(2*d^2) + (b^2*g*PolyLog[2, c*(a + b*x)])/(2*a^2) - (b*e*h*n*PolyLog[2, c*(a + b*x)])/(2*a*d) - (e*h*n*PolyLog[2, c*(a + b*x)])/(2*d*x) - (e^2*h*n*Log[x]*PolyLog[2, c*(a + b*x)])/(2*d^2) + (e^2*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/(2*d^2) - (b^2*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(2*a^2) - ((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(2*x^2) + (b*e*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(2*a*d) + (b^2*g*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2) - (b*e*h*n*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(a*d) + (b^2*h*n*(Log[d + e*x] - Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2) - (b^2*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2) + (b^2*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/(2*a^2) - (b^2*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/(2*a^2) - (b^2*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(2*a^2) + (e^2*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(2*d^2) - (b^2*c*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(2*a*(1 - a*c)) + (b^2*c*h*n*PolyLog[2, 1 + (e*x)/d])/(2*a*(1 - a*c)) + (b^2*h*n*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 + (e*x)/d])/(2*a^2) - (e^2*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/(2*d^2) + (e^2*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/(2*d^2) - (b^2*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(2*a^2) + (e^2*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(2*d^2) - (e^2*h*n*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)])/(2*d^2) + (b^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*a^2) - (e^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*d^2) - (b^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*a^2) + (e^2*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*d^2) + (e^2*h*n*PolyLog[3, -((b*x)/a)])/(2*d^2) - (b^2*h*n*PolyLog[3, 1 - (b*c*x)/(1 - a*c)])/(2*a^2) + (b^2*h*n*PolyLog[3, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/(2*a^2) - (b^2*h*n*PolyLog[3, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/(2*a^2) + (b^2*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(2*a^2) - (e^2*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(2*d^2) - (b^2*h*n*PolyLog[3, 1 + (e*x)/d])/(2*a^2) - (e^2*h*n*PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))])/(2*d^2) + (e^2*h*n*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/(2*d^2) + (b^2*h*n*PolyLog[3, 1 - c*(a + b*x)])/(2*a^2) + (b^2*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*a^2) - (e^2*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(2*d^2) - (b^2*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*a^2) + (e^2*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(2*d^2)} -{(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)]/x^4, x, 78, (b^2*c*e*h*n*Log[x])/(2*a*(1 - a*c)*d) - (b^2*c*e*h*n*Log[1 - a*c - b*c*x])/(3*a*(1 - a*c)*d) + (b*e*h*n*Log[1 - a*c - b*c*x])/(3*a*d*x) - (b^3*g*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(3*a^3) + (b^2*e*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a^2*d) + (b*e^2*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(2*a*d^2) - (b^2*c*e*h*n*Log[d + e*x])/(6*a*(1 - a*c)*d) - (b^3*h*n*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*Log[d + e*x])/(3*a^3) - (b^2*e*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*a^2*d) - (b*e^2*h*n*Log[1 - a*c - b*c*x]*Log[(b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*a*d^2) - (b^3*h*n*(Log[(b*c*x)/(1 - a*c)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*x)/((1 - a*c)*(d + e*x))])*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]^2)/(6*a^3) + (b^3*h*n*(Log[(b*c*x)/(1 - a*c)] - Log[-((e*x)/d)])*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])^2)/(6*a^3) + (b^3*h*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x]*(n*Log[d + e*x] - Log[f*(d + e*x)^n]))/(3*a^3) - (b^2*c*(g + h*Log[f*(d + e*x)^n]))/(6*a*(1 - a*c)*x) + (b^3*c^2*Log[-((e*x)/d)]*(g + h*Log[f*(d + e*x)^n]))/(6*a*(1 - a*c)^2) - (b^3*c*Log[-((e*x)/d)]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*(1 - a*c)) + (b*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(6*a*x^2) - (b^2*Log[1 - a*c - b*c*x]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*x) - (b^3*c^2*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(6*a*(1 - a*c)^2) + (b^3*c*Log[(e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)]*(g + h*Log[f*(d + e*x)^n]))/(3*a^2*(1 - a*c)) + (b^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(6*a^3) - (e^3*h*n*(Log[c*(a + b*x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)*(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]^2)/(6*d^3) + (e^3*h*n*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/(3*d^3) + (b^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*a^3) - (e^3*h*n*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/(3*d^3) - (b^3*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(6*a^3) + (e^3*h*n*(Log[c*(a + b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(6*d^3) + (e^3*h*n*(Log[1 + (b*x)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 - c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(6*d^3) + (e^3*h*n*(Log[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])^2)/(6*d^3) + (e^3*h*n*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, -((b*x)/a)])/(3*d^3) - (b^3*g*PolyLog[2, c*(a + b*x)])/(3*a^3) + (b^2*e*h*n*PolyLog[2, c*(a + b*x)])/(6*a^2*d) + (b*e^2*h*n*PolyLog[2, c*(a + b*x)])/(3*a*d^2) - (e*h*n*PolyLog[2, c*(a + b*x)])/(6*d*x^2) + (e^2*h*n*PolyLog[2, c*(a + b*x)])/(3*d^2*x) + (e^3*h*n*Log[x]*PolyLog[2, c*(a + b*x)])/(3*d^3) - (e^3*h*n*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/(3*d^3) + (b^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(3*a^3) - ((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/(3*x^3) - (b^2*e*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(3*a^2*d) - (b*e^2*h*n*PolyLog[2, (e*(1 - a*c - b*c*x))/(b*c*d + e - a*c*e)])/(6*a*d^2) - (b^3*g*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(3*a^3) + (b^2*e*h*n*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a^2*d) + (b*e^2*h*n*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(2*a*d^2) - (b^3*h*n*(Log[d + e*x] - Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(3*a^3) + (b^3*h*(n*Log[d + e*x] - Log[f*(d + e*x)^n])*PolyLog[2, 1 - (b*c*x)/(1 - a*c)])/(3*a^3) - (b^3*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/(3*a^3) + (b^3*h*n*Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))]*PolyLog[2, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/(3*a^3) + (b^3*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*a^3) - (e^3*h*n*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/(3*d^3) - (b^3*c^2*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(6*a*(1 - a*c)^2) + (b^3*c*h*n*PolyLog[2, (b*c*(d + e*x))/(b*c*d + e - a*c*e)])/(3*a^2*(1 - a*c)) + (b^3*c^2*h*n*PolyLog[2, 1 + (e*x)/d])/(6*a*(1 - a*c)^2) - (b^3*c*h*n*PolyLog[2, 1 + (e*x)/d])/(3*a^2*(1 - a*c)) - (b^3*h*n*(Log[1 - a*c - b*c*x] + Log[((1 - a*c)*(d + e*x))/(d*(1 - a*c - b*c*x))])*PolyLog[2, 1 + (e*x)/d])/(3*a^3) + (e^3*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x))))])/(3*d^3) - (e^3*h*n*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 - c*(a + b*x)))])/(3*d^3) + (b^3*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(3*a^3) - (e^3*h*n*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/(3*d^3) + (e^3*h*n*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b*x))])*PolyLog[2, 1 - c*(a + b*x)])/(3*d^3) - (b^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*a^3) + (e^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*d^3) + (b^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*a^3) - (e^3*h*n*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*d^3) - (e^3*h*n*PolyLog[3, -((b*x)/a)])/(3*d^3) + (b^3*h*n*PolyLog[3, 1 - (b*c*x)/(1 - a*c)])/(3*a^3) - (b^3*h*n*PolyLog[3, (d*(1 - a*c - b*c*x))/((1 - a*c)*(d + e*x))])/(3*a^3) + (b^3*h*n*PolyLog[3, -((e*(1 - a*c - b*c*x))/(b*c*(d + e*x)))])/(3*a^3) - (b^3*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*a^3) + (e^3*h*n*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/(3*d^3) + (b^3*h*n*PolyLog[3, 1 + (e*x)/d])/(3*a^3) + (e^3*h*n*PolyLog[3, -((b*x)/(a*(1 - c*(a + b*x))))])/(3*d^3) - (e^3*h*n*PolyLog[3, -((b*c*x)/(1 - c*(a + b*x)))])/(3*d^3) - (b^3*h*n*PolyLog[3, 1 - c*(a + b*x)])/(3*a^3) - (b^3*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*a^3) + (e^3*h*n*PolyLog[3, -((e*(1 - c*(a + b*x)))/(b*c*(d + e*x)))])/(3*d^3) + (b^3*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*a^3) - (e^3*h*n*PolyLog[3, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/(3*d^3)} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b x) Log[1-c x] PolyLog[2, c x]*) - - -{x^2*(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x], x, 52, (53*b*x)/(192*c^3) + (11*a*x)/(27*c^2) + (49*(3*b + 4*a*c)*x)/(432*c^3) + (29*b*x^2)/(384*c^2) + (5*a*x^2)/(54*c) + (13*(3*b + 4*a*c)*x^2)/(864*c^2) + (2*a*x^3)/81 + (17*b*x^3)/(576*c) + ((3*b + 4*a*c)*x^3)/(324*c) + (3*b*x^4)/256 + (29*b*Log[1 - c*x])/(192*c^4) + (5*a*Log[1 - c*x])/(27*c^3) + (13*(3*b + 4*a*c)*Log[1 - c*x])/(432*c^4) - (b*x^2*Log[1 - c*x])/(16*c^2) - (a*x^2*Log[1 - c*x])/(9*c) - ((3*b + 4*a*c)*x^2*Log[1 - c*x])/(48*c^2) - (2/27)*a*x^3*Log[1 - c*x] - (b*x^3*Log[1 - c*x])/(24*c) - ((3*b + 4*a*c)*x^3*Log[1 - c*x])/(108*c) - (3/64)*b*x^4*Log[1 - c*x] + (b*(1 - c*x)*Log[1 - c*x])/(8*c^4) + (2*a*(1 - c*x)*Log[1 - c*x])/(9*c^3) + ((3*b + 4*a*c)*(1 - c*x)*Log[1 - c*x])/(12*c^4) - (b*Log[1 - c*x]^2)/(16*c^4) - (a*Log[1 - c*x]^2)/(9*c^3) + (1/9)*a*x^3*Log[1 - c*x]^2 + (1/16)*b*x^4*Log[1 - c*x]^2 - ((3*b + 4*a*c)*Log[c*x]*Log[1 - c*x]^2)/(12*c^4) - ((3*b + 4*a*c)*x*PolyLog[2, c*x])/(12*c^3) - ((3*b + 4*a*c)*x^2*PolyLog[2, c*x])/(24*c^2) - ((3*b + 4*a*c)*x^3*PolyLog[2, c*x])/(36*c) - (1/16)*b*x^4*PolyLog[2, c*x] - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, c*x])/(12*c^4) + (1/12)*(4*a*x^3 + 3*b*x^4)*Log[1 - c*x]*PolyLog[2, c*x] - ((3*b + 4*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(6*c^4) + ((3*b + 4*a*c)*PolyLog[3, 1 - c*x])/(6*c^4)} -{x^1*(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x], x, 40, (4*b*x)/(9*c^2) + (a*x)/c + (5*(2*b + 3*a*c)*x)/(24*c^2) + (b*x^2)/(9*c) + ((2*b + 3*a*c)*x^2)/(48*c) + (b*x^3)/27 + (a*(1 - c*x)^2)/(8*c^2) + (2*b*Log[1 - c*x])/(9*c^3) + ((2*b + 3*a*c)*Log[1 - c*x])/(24*c^3) - (b*x^2*Log[1 - c*x])/(9*c) - ((2*b + 3*a*c)*x^2*Log[1 - c*x])/(24*c) - (1/9)*b*x^3*Log[1 - c*x] + (2*b*(1 - c*x)*Log[1 - c*x])/(9*c^3) + (a*(1 - c*x)*Log[1 - c*x])/c^2 + ((2*b + 3*a*c)*(1 - c*x)*Log[1 - c*x])/(6*c^3) - (a*(1 - c*x)^2*Log[1 - c*x])/(4*c^2) - (b*Log[1 - c*x]^2)/(9*c^3) + (1/9)*b*x^3*Log[1 - c*x]^2 - (a*(1 - c*x)*Log[1 - c*x]^2)/(2*c^2) + (a*(1 - c*x)^2*Log[1 - c*x]^2)/(4*c^2) - ((2*b + 3*a*c)*Log[c*x]*Log[1 - c*x]^2)/(6*c^3) - ((2*b + 3*a*c)*x*PolyLog[2, c*x])/(6*c^2) - ((2*b + 3*a*c)*x^2*PolyLog[2, c*x])/(12*c) - (1/9)*b*x^3*PolyLog[2, c*x] - ((2*b + 3*a*c)*Log[1 - c*x]*PolyLog[2, c*x])/(6*c^3) + (1/6)*(3*a*x^2 + 2*b*x^3)*Log[1 - c*x]*PolyLog[2, c*x] - ((2*b + 3*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/(3*c^3) + ((2*b + 3*a*c)*PolyLog[3, 1 - c*x])/(3*c^3)} -{x^0*(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x], x, 26, 2*a*x + (9*b*x)/(8*c) + ((b + 2*a*c)*x)/(2*c) + (b*x^2)/16 + (b*(1 - c*x)^2)/(8*c^2) + (b*Log[1 - c*x])/(8*c^2) - (1/8)*b*x^2*Log[1 - c*x] + (b*(1 - c*x)*Log[1 - c*x])/c^2 + (2*a*(1 - c*x)*Log[1 - c*x])/c + ((b + 2*a*c)*(1 - c*x)*Log[1 - c*x])/(2*c^2) - (b*(1 - c*x)^2*Log[1 - c*x])/(4*c^2) - (b*(1 - c*x)*Log[1 - c*x]^2)/(2*c^2) - (a*(1 - c*x)*Log[1 - c*x]^2)/c + (b*(1 - c*x)^2*Log[1 - c*x]^2)/(4*c^2) - ((b + 2*a*c)*Log[c*x]*Log[1 - c*x]^2)/(2*c^2) - ((b + 2*a*c)*x*PolyLog[2, c*x])/(2*c) - (1/4)*b*x^2*PolyLog[2, c*x] - ((b + 2*a*c)*Log[1 - c*x]*PolyLog[2, c*x])/(2*c^2) + (1/2)*(2*a*x + b*x^2)*Log[1 - c*x]*PolyLog[2, c*x] - ((b + 2*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c^2 + ((b + 2*a*c)*PolyLog[3, 1 - c*x])/c^2} -{(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x]/x^1, x, 18, 3*b*x + (3*b*(1 - c*x)*Log[1 - c*x])/c - (b*(1 - c*x)*Log[1 - c*x]^2)/c - (b*Log[c*x]*Log[1 - c*x]^2)/c - b*x*PolyLog[2, c*x] - (b*Log[1 - c*x]*PolyLog[2, c*x])/c + b*x*Log[1 - c*x]*PolyLog[2, c*x] - (1/2)*a*PolyLog[2, c*x]^2 - (2*b*Log[1 - c*x]*PolyLog[2, 1 - c*x])/c + (2*b*PolyLog[3, 1 - c*x])/c} -{(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x]/x^2, x, 13, (a*(1 - c*x)*Log[1 - c*x]^2)/x + a*c*Log[c*x]*Log[1 - c*x]^2 - 2*a*c*PolyLog[2, c*x] + a*c*Log[1 - c*x]*PolyLog[2, c*x] - (a*Log[1 - c*x]*PolyLog[2, c*x])/x - (1/2)*b*PolyLog[2, c*x]^2 + 2*a*c*Log[1 - c*x]*PolyLog[2, 1 - c*x] - a*c*PolyLog[3, c*x] - 2*a*c*PolyLog[3, 1 - c*x]} -{(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x]/x^3, x, 30, (-a)*c^2*Log[x] + a*c^2*Log[1 - c*x] - (a*c*Log[1 - c*x])/x - (1/4)*a*c^2*Log[1 - c*x]^2 + (a*Log[1 - c*x]^2)/(4*x^2) + (b*(1 - c*x)*Log[1 - c*x]^2)/x - (b^2*Log[c*x]*Log[1 - c*x]^2)/(2*a) + ((b + a*c)^2*Log[c*x]*Log[1 - c*x]^2)/(2*a) - 2*b*c*PolyLog[2, c*x] - (1/2)*a*c^2*PolyLog[2, c*x] + (a*c*PolyLog[2, c*x])/(2*x) + ((b + a*c)^2*Log[1 - c*x]*PolyLog[2, c*x])/(2*a) - ((a + b*x)^2*Log[1 - c*x]*PolyLog[2, c*x])/(2*a*x^2) - (b^2*Log[1 - c*x]*PolyLog[2, 1 - c*x])/a + ((b + a*c)^2*Log[1 - c*x]*PolyLog[2, 1 - c*x])/a - (1/2)*c*(2*b + a*c)*PolyLog[3, c*x] + (b^2*PolyLog[3, 1 - c*x])/a - ((b + a*c)^2*PolyLog[3, 1 - c*x])/a} -{(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x]/x^4, x, 41, (7*a*c^2)/(36*x) - (1/2)*b*c^2*Log[x] - (5/12)*a*c^3*Log[x] - (1/6)*c^2*(3*b + 2*a*c)*Log[x] + (1/2)*b*c^2*Log[1 - c*x] + (5/12)*a*c^3*Log[1 - c*x] + (1/6)*c^2*(3*b + 2*a*c)*Log[1 - c*x] - (7*a*c*Log[1 - c*x])/(36*x^2) - (b*c*Log[1 - c*x])/(2*x) - (2*a*c^2*Log[1 - c*x])/(9*x) - (c*(3*b + 2*a*c)*Log[1 - c*x])/(6*x) - (1/4)*b*c^2*Log[1 - c*x]^2 - (1/9)*a*c^3*Log[1 - c*x]^2 + (a*Log[1 - c*x]^2)/(9*x^3) + (b*Log[1 - c*x]^2)/(4*x^2) + (1/6)*c^2*(3*b + 2*a*c)*Log[c*x]*Log[1 - c*x]^2 - (1/2)*b*c^2*PolyLog[2, c*x] - (2/9)*a*c^3*PolyLog[2, c*x] + (a*c*PolyLog[2, c*x])/(6*x^2) + (c*(3*b + 2*a*c)*PolyLog[2, c*x])/(6*x) + (1/6)*c^2*(3*b + 2*a*c)*Log[1 - c*x]*PolyLog[2, c*x] - (1/6)*((2*a)/x^3 + (3*b)/x^2)*Log[1 - c*x]*PolyLog[2, c*x] + (1/3)*c^2*(3*b + 2*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/6)*c^2*(3*b + 2*a*c)*PolyLog[3, c*x] - (1/3)*c^2*(3*b + 2*a*c)*PolyLog[3, 1 - c*x]} -{(a + b*x)*Log[1 - c*x]*PolyLog[2, c*x]/x^5, x, 51, (5*a*c^2)/(144*x^2) + (b*c^2)/(9*x) + (19*a*c^3)/(144*x) + (c^2*(4*b + 3*a*c))/(48*x) - (1/3)*b*c^3*Log[x] - (37/144)*a*c^4*Log[x] - (5/48)*c^3*(4*b + 3*a*c)*Log[x] + (1/3)*b*c^3*Log[1 - c*x] + (37/144)*a*c^4*Log[1 - c*x] + (5/48)*c^3*(4*b + 3*a*c)*Log[1 - c*x] - (5*a*c*Log[1 - c*x])/(72*x^3) - (b*c*Log[1 - c*x])/(9*x^2) - (a*c^2*Log[1 - c*x])/(16*x^2) - (c*(4*b + 3*a*c)*Log[1 - c*x])/(48*x^2) - (2*b*c^2*Log[1 - c*x])/(9*x) - (a*c^3*Log[1 - c*x])/(8*x) - (c^2*(4*b + 3*a*c)*Log[1 - c*x])/(12*x) - (1/9)*b*c^3*Log[1 - c*x]^2 - (1/16)*a*c^4*Log[1 - c*x]^2 + (a*Log[1 - c*x]^2)/(16*x^4) + (b*Log[1 - c*x]^2)/(9*x^3) + (1/12)*c^3*(4*b + 3*a*c)*Log[c*x]*Log[1 - c*x]^2 - (2/9)*b*c^3*PolyLog[2, c*x] - (1/8)*a*c^4*PolyLog[2, c*x] + (a*c*PolyLog[2, c*x])/(12*x^3) + (c*(4*b + 3*a*c)*PolyLog[2, c*x])/(24*x^2) + (c^2*(4*b + 3*a*c)*PolyLog[2, c*x])/(12*x) + (1/12)*c^3*(4*b + 3*a*c)*Log[1 - c*x]*PolyLog[2, c*x] - (1/12)*((3*a)/x^4 + (4*b)/x^3)*Log[1 - c*x]*PolyLog[2, c*x] + (1/6)*c^3*(4*b + 3*a*c)*Log[1 - c*x]*PolyLog[2, 1 - c*x] - (1/12)*c^3*(4*b + 3*a*c)*PolyLog[3, c*x] - (1/6)*c^3*(4*b + 3*a*c)*PolyLog[3, 1 - c*x]} - - -(* ::Subsection::Closed:: *) -(*Integrands of the form x^m (a+b x+c x^2) Log[1-d x] PolyLog[2, d x]*) - - -{x^1*(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x], x, 60, (53*c*x)/(192*d^3) + (11*b*x)/(27*d^2) + (a*x)/d + ((3*c + 4*b*d)*x)/(108*d^3) + (5*(3*c + 4*b*d + 6*a*d^2)*x)/(48*d^3) + (29*c*x^2)/(384*d^2) + (5*b*x^2)/(54*d) + ((3*c + 4*b*d)*x^2)/(216*d^2) + ((3*c + 4*b*d + 6*a*d^2)*x^2)/(96*d^2) + (2*b*x^3)/81 + (17*c*x^3)/(576*d) + ((3*c + 4*b*d)*x^3)/(324*d) + (3*c*x^4)/256 + (a*(1 - d*x)^2)/(8*d^2) + (29*c*Log[1 - d*x])/(192*d^4) + (5*b*Log[1 - d*x])/(27*d^3) + ((3*c + 4*b*d)*Log[1 - d*x])/(108*d^4) + ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x])/(48*d^4) - (c*x^2*Log[1 - d*x])/(16*d^2) - (b*x^2*Log[1 - d*x])/(9*d) - ((3*c + 4*b*d + 6*a*d^2)*x^2*Log[1 - d*x])/(48*d^2) - (2/27)*b*x^3*Log[1 - d*x] - (c*x^3*Log[1 - d*x])/(24*d) - ((3*c + 4*b*d)*x^3*Log[1 - d*x])/(108*d) - (3/64)*c*x^4*Log[1 - d*x] + (c*(1 - d*x)*Log[1 - d*x])/(8*d^4) + (2*b*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (a*(1 - d*x)*Log[1 - d*x])/d^2 + ((3*c + 4*b*d + 6*a*d^2)*(1 - d*x)*Log[1 - d*x])/(12*d^4) - (a*(1 - d*x)^2*Log[1 - d*x])/(4*d^2) - (c*Log[1 - d*x]^2)/(16*d^4) - (b*Log[1 - d*x]^2)/(9*d^3) + (1/9)*b*x^3*Log[1 - d*x]^2 + (1/16)*c*x^4*Log[1 - d*x]^2 - (a*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2) + (a*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((3*c + 4*b*d + 6*a*d^2)*Log[d*x]*Log[1 - d*x]^2)/(12*d^4) - ((3*c + 4*b*d + 6*a*d^2)*x*PolyLog[2, d*x])/(12*d^3) - ((3*c + 4*b*d + 6*a*d^2)*x^2*PolyLog[2, d*x])/(24*d^2) - ((3*c + 4*b*d)*x^3*PolyLog[2, d*x])/(36*d) - (1/16)*c*x^4*PolyLog[2, d*x] - ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x]*PolyLog[2, d*x])/(12*d^4) + (1/12)*(6*a*x^2 + 4*b*x^3 + 3*c*x^4)*Log[1 - d*x]*PolyLog[2, d*x] - ((3*c + 4*b*d + 6*a*d^2)*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(6*d^4) + ((3*c + 4*b*d + 6*a*d^2)*PolyLog[3, 1 - d*x])/(6*d^4)} -{x^0*(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x], x, 43, 2*a*x + (4*c*x)/(9*d^2) + (b*x)/d + ((2*c + 3*b*d)*x)/(24*d^2) + ((2*c + 3*d*(b + 2*a*d))*x)/(6*d^2) + (c*x^2)/(9*d) + ((2*c + 3*b*d)*x^2)/(48*d) + (c*x^3)/27 + (b*(1 - d*x)^2)/(8*d^2) + (2*c*Log[1 - d*x])/(9*d^3) + ((2*c + 3*b*d)*Log[1 - d*x])/(24*d^3) - (c*x^2*Log[1 - d*x])/(9*d) - ((2*c + 3*b*d)*x^2*Log[1 - d*x])/(24*d) - (1/9)*c*x^3*Log[1 - d*x] + (2*c*(1 - d*x)*Log[1 - d*x])/(9*d^3) + (b*(1 - d*x)*Log[1 - d*x])/d^2 + (2*a*(1 - d*x)*Log[1 - d*x])/d + ((2*c + 3*d*(b + 2*a*d))*(1 - d*x)*Log[1 - d*x])/(6*d^3) - (b*(1 - d*x)^2*Log[1 - d*x])/(4*d^2) - (c*Log[1 - d*x]^2)/(9*d^3) + (1/9)*c*x^3*Log[1 - d*x]^2 - (b*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2) - (a*(1 - d*x)*Log[1 - d*x]^2)/d + (b*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((2*c + 3*d*(b + 2*a*d))*Log[d*x]*Log[1 - d*x]^2)/(6*d^3) - ((2*c + 3*d*(b + 2*a*d))*x*PolyLog[2, d*x])/(6*d^2) - ((2*c + 3*b*d)*x^2*PolyLog[2, d*x])/(12*d) - (1/9)*c*x^3*PolyLog[2, d*x] - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, d*x])/(6*d^3) + (1/6)*(6*a*x + 3*b*x^2 + 2*c*x^3)*Log[1 - d*x]*PolyLog[2, d*x] - ((2*c + 3*d*(b + 2*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x])/(3*d^3) + ((2*c + 3*d*(b + 2*a*d))*PolyLog[3, 1 - d*x])/(3*d^3)} -{(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x]/x^1, x, 29, 2*b*x + (9*c*x)/(8*d) + ((c + 2*b*d)*x)/(2*d) + (c*x^2)/16 + (c*(1 - d*x)^2)/(8*d^2) + (c*Log[1 - d*x])/(8*d^2) - (1/8)*c*x^2*Log[1 - d*x] + (c*(1 - d*x)*Log[1 - d*x])/d^2 + (2*b*(1 - d*x)*Log[1 - d*x])/d + ((c + 2*b*d)*(1 - d*x)*Log[1 - d*x])/(2*d^2) - (c*(1 - d*x)^2*Log[1 - d*x])/(4*d^2) - (c*(1 - d*x)*Log[1 - d*x]^2)/(2*d^2) - (b*(1 - d*x)*Log[1 - d*x]^2)/d + (c*(1 - d*x)^2*Log[1 - d*x]^2)/(4*d^2) - ((c + 2*b*d)*Log[d*x]*Log[1 - d*x]^2)/(2*d^2) - ((c + 2*b*d)*x*PolyLog[2, d*x])/(2*d) - (1/4)*c*x^2*PolyLog[2, d*x] - ((c + 2*b*d)*Log[1 - d*x]*PolyLog[2, d*x])/(2*d^2) + (1/2)*(2*b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x] - (1/2)*a*PolyLog[2, d*x]^2 - ((c + 2*b*d)*Log[1 - d*x]*PolyLog[2, 1 - d*x])/d^2 + ((c + 2*b*d)*PolyLog[3, 1 - d*x])/d^2} -{(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x]/x^2, x, 19, 3*c*x + (3*c*(1 - d*x)*Log[1 - d*x])/d - (c*(1 - d*x)*Log[1 - d*x]^2)/d + (a*(1 - d*x)*Log[1 - d*x]^2)/x + (a - c/d^2)*d*Log[d*x]*Log[1 - d*x]^2 - 2*a*d*PolyLog[2, d*x] - c*x*PolyLog[2, d*x] + (a - c/d^2)*d*Log[1 - d*x]*PolyLog[2, d*x] - (a/x - c*x)*Log[1 - d*x]*PolyLog[2, d*x] - (1/2)*b*PolyLog[2, d*x]^2 + 2*(a - c/d^2)*d*Log[1 - d*x]*PolyLog[2, 1 - d*x] - a*d*PolyLog[3, d*x] - 2*(a - c/d^2)*d*PolyLog[3, 1 - d*x]} -{(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x]/x^3, x, 32, (-a)*d^2*Log[x] + a*d^2*Log[1 - d*x] - (a*d*Log[1 - d*x])/x - (1/4)*a*d^2*Log[1 - d*x]^2 + (a*Log[1 - d*x]^2)/(4*x^2) + (b*(1 - d*x)*Log[1 - d*x]^2)/x - (b^2*Log[d*x]*Log[1 - d*x]^2)/(2*a) + ((b + a*d)^2*Log[d*x]*Log[1 - d*x]^2)/(2*a) - 2*b*d*PolyLog[2, d*x] - (1/2)*a*d^2*PolyLog[2, d*x] + (a*d*PolyLog[2, d*x])/(2*x) + ((b + a*d)^2*Log[1 - d*x]*PolyLog[2, d*x])/(2*a) - ((a + b*x)^2*Log[1 - d*x]*PolyLog[2, d*x])/(2*a*x^2) - (1/2)*c*PolyLog[2, d*x]^2 - (b^2*Log[1 - d*x]*PolyLog[2, 1 - d*x])/a + ((b + a*d)^2*Log[1 - d*x]*PolyLog[2, 1 - d*x])/a - (1/2)*d*(2*b + a*d)*PolyLog[3, d*x] + (b^2*PolyLog[3, 1 - d*x])/a - ((b + a*d)^2*PolyLog[3, 1 - d*x])/a} -{(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x]/x^4, x, 43, (7*a*d^2)/(36*x) - (1/2)*b*d^2*Log[x] - (5/12)*a*d^3*Log[x] - (1/6)*d^2*(3*b + 2*a*d)*Log[x] + (1/2)*b*d^2*Log[1 - d*x] + (5/12)*a*d^3*Log[1 - d*x] + (1/6)*d^2*(3*b + 2*a*d)*Log[1 - d*x] - (7*a*d*Log[1 - d*x])/(36*x^2) - (b*d*Log[1 - d*x])/(2*x) - (2*a*d^2*Log[1 - d*x])/(9*x) - (d*(3*b + 2*a*d)*Log[1 - d*x])/(6*x) - (1/4)*b*d^2*Log[1 - d*x]^2 - (1/9)*a*d^3*Log[1 - d*x]^2 + (a*Log[1 - d*x]^2)/(9*x^3) + (b*Log[1 - d*x]^2)/(4*x^2) + (c*(1 - d*x)*Log[1 - d*x]^2)/x + (1/6)*d*(6*c + d*(3*b + 2*a*d))*Log[d*x]*Log[1 - d*x]^2 - 2*c*d*PolyLog[2, d*x] - (1/2)*b*d^2*PolyLog[2, d*x] - (2/9)*a*d^3*PolyLog[2, d*x] + (a*d*PolyLog[2, d*x])/(6*x^2) + (d*(3*b + 2*a*d)*PolyLog[2, d*x])/(6*x) + (1/6)*d*(6*c + d*(3*b + 2*a*d))*Log[1 - d*x]*PolyLog[2, d*x] - (1/6)*((2*a)/x^3 + (3*b)/x^2 + (6*c)/x)*Log[1 - d*x]*PolyLog[2, d*x] + (1/3)*d*(6*c + d*(3*b + 2*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x] - (1/6)*d*(6*c + d*(3*b + 2*a*d))*PolyLog[3, d*x] - (1/3)*d*(6*c + d*(3*b + 2*a*d))*PolyLog[3, 1 - d*x]} -{(a + b*x + c*x^2)*Log[1 - d*x]*PolyLog[2, d*x]/x^5, x, 61, (5*a*d^2)/(144*x^2) + (b*d^2)/(9*x) + (19*a*d^3)/(144*x) + (d^2*(4*b + 3*a*d))/(48*x) - (1/2)*c*d^2*Log[x] - (1/3)*b*d^3*Log[x] - (37/144)*a*d^4*Log[x] - (1/48)*d^3*(4*b + 3*a*d)*Log[x] - (1/12)*d^2*(6*c + d*(4*b + 3*a*d))*Log[x] + (1/2)*c*d^2*Log[1 - d*x] + (1/3)*b*d^3*Log[1 - d*x] + (37/144)*a*d^4*Log[1 - d*x] + (1/48)*d^3*(4*b + 3*a*d)*Log[1 - d*x] + (1/12)*d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x] - (5*a*d*Log[1 - d*x])/(72*x^3) - (b*d*Log[1 - d*x])/(9*x^2) - (a*d^2*Log[1 - d*x])/(16*x^2) - (d*(4*b + 3*a*d)*Log[1 - d*x])/(48*x^2) - (c*d*Log[1 - d*x])/(2*x) - (2*b*d^2*Log[1 - d*x])/(9*x) - (a*d^3*Log[1 - d*x])/(8*x) - (d*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x])/(12*x) - (1/4)*c*d^2*Log[1 - d*x]^2 - (1/9)*b*d^3*Log[1 - d*x]^2 - (1/16)*a*d^4*Log[1 - d*x]^2 + (a*Log[1 - d*x]^2)/(16*x^4) + (b*Log[1 - d*x]^2)/(9*x^3) + (c*Log[1 - d*x]^2)/(4*x^2) + (1/12)*d^2*(6*c + d*(4*b + 3*a*d))*Log[d*x]*Log[1 - d*x]^2 - (1/2)*c*d^2*PolyLog[2, d*x] - (2/9)*b*d^3*PolyLog[2, d*x] - (1/8)*a*d^4*PolyLog[2, d*x] + (a*d*PolyLog[2, d*x])/(12*x^3) + (d*(4*b + 3*a*d)*PolyLog[2, d*x])/(24*x^2) + (d*(6*c + d*(4*b + 3*a*d))*PolyLog[2, d*x])/(12*x) + (1/12)*d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, d*x] - (1/12)*((3*a)/x^4 + (4*b)/x^3 + (6*c)/x^2)*Log[1 - d*x]*PolyLog[2, d*x] + (1/6)*d^2*(6*c + d*(4*b + 3*a*d))*Log[1 - d*x]*PolyLog[2, 1 - d*x] - (1/12)*d^2*(6*c + d*(4*b + 3*a*d))*PolyLog[3, d*x] - (1/6)*d^2*(6*c + d*(4*b + 3*a*d))*PolyLog[3, 1 - d*x]} diff --git a/test/methods/rule_based/test_files/8 Special functions/8.9 Product logarithm function.m b/test/methods/rule_based/test_files/8 Special functions/8.9 Product logarithm function.m deleted file mode 100644 index 5ee5952..0000000 --- a/test/methods/rule_based/test_files/8 Special functions/8.9 Product logarithm function.m +++ /dev/null @@ -1,554 +0,0 @@ -(* ::Package:: *) - -(* ::Title:: *) -(*Integration Problems Involving the Lambert W (ProductLogarithm) Function*) - - -(* ::Subsection::Closed:: *) -(*Integrands involving ProductLog[a+b x]*) - - -(* ::Subsubsection::Closed:: *) -(*ProductLog[a+b x]^n*) - - -{ProductLog[a + b*x]^4, x, 6, 96*x - (96*(a + b*x))/(b*ProductLog[a + b*x]) - (48*(a + b*x)*ProductLog[a + b*x])/b + (16*(a + b*x)*ProductLog[a + b*x]^2)/b - (4*(a + b*x)*ProductLog[a + b*x]^3)/b + ((a + b*x)*ProductLog[a + b*x]^4)/b} -{ProductLog[a + b*x]^3, x, 5, -18*x + (18*(a + b*x))/(b*ProductLog[a + b*x]) + (9*(a + b*x)*ProductLog[a + b*x])/b - (3*(a + b*x)*ProductLog[a + b*x]^2)/b + ((a + b*x)*ProductLog[a + b*x]^3)/b} -{ProductLog[a + b*x]^2, x, 4, 4*x - (4*(a + b*x))/(b*ProductLog[a + b*x]) - (2*(a + b*x)*ProductLog[a + b*x])/b + ((a + b*x)*ProductLog[a + b*x]^2)/b} -{ProductLog[a + b*x], x, 3, -x + (a + b*x)/(b*ProductLog[a + b*x]) + ((a + b*x)*ProductLog[a + b*x])/b} -{1/ProductLog[a + b*x], x, 2, ExpIntegralEi[ProductLog[a + b*x]]/b + (a + b*x)/(b*ProductLog[a + b*x])} -{1/ProductLog[a + b*x]^2, x, 2, (2*ExpIntegralEi[ProductLog[a + b*x]])/b - (a + b*x)/(b*ProductLog[a + b*x]^2)} -{1/ProductLog[a + b*x]^3, x, 3, (3*ExpIntegralEi[ProductLog[a + b*x]])/(2*b) - (a + b*x)/(2*b*ProductLog[a + b*x]^3) - (3*(a + b*x))/(2*b*ProductLog[a + b*x]^2)} -{1/ProductLog[a + b*x]^4, x, 4, (2*ExpIntegralEi[ProductLog[a + b*x]])/(3*b) - (a + b*x)/(3*b*ProductLog[a + b*x]^4) - (2*(a + b*x))/(3*b*ProductLog[a + b*x]^3) - (2*(a + b*x))/(3*b*ProductLog[a + b*x]^2)} -{1/ProductLog[a + b*x]^5, x, 5, (5*ExpIntegralEi[ProductLog[a + b*x]])/(24*b) - (a + b*x)/(4*b*ProductLog[a + b*x]^5) - (5*(a + b*x))/(12*b*ProductLog[a + b*x]^4) - (5*(a + b*x))/(24*b*ProductLog[a + b*x]^3) - (5*(a + b*x))/(24*b*ProductLog[a + b*x]^2)} - - -{(c*ProductLog[a+b*x])^(5/2), x, 5, (75*c^(5/2)*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(16*b) - (75*c^3*(a + b*x))/(8*b*Sqrt[c*ProductLog[a + b*x]]) + (25*c^2*(a + b*x)*Sqrt[c*ProductLog[a + b*x]])/(4*b) - (5*c*(a + b*x)*(c*ProductLog[a + b*x])^(3/2))/(2*b) + ((a + b*x)*(c*ProductLog[a + b*x])^(5/2))/b} -{(c*ProductLog[a+b*x])^(3/2), x, 4, -((9*c^(3/2)*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(8*b)) + (9*c^2*(a + b*x))/(4*b*Sqrt[c*ProductLog[a + b*x]]) - (3*c*(a + b*x)*Sqrt[c*ProductLog[a + b*x]])/(2*b) + ((a + b*x)*(c*ProductLog[a + b*x])^(3/2))/b} -{Sqrt[c*ProductLog[a+b*x]], x, 3, (Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(4*b) - (c*(a + b*x))/(2*b*Sqrt[c*ProductLog[a + b*x]]) + ((a + b*x)*Sqrt[c*ProductLog[a + b*x]])/b} -{1/Sqrt[c*ProductLog[a+b*x]], x, 2, (Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(2*b*Sqrt[c]) + (a + b*x)/(b*Sqrt[c*ProductLog[a + b*x]])} -{1/(c*ProductLog[a+b*x])^(3/2), x, 2, (3*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(b*c^(3/2)) - (2*(a + b*x))/(b*(c*ProductLog[a + b*x])^(3/2))} -{1/(c*ProductLog[a+b*x])^(5/2), x, 3, (10*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(3*b*c^(5/2)) - (2*(a + b*x))/(3*b*(c*ProductLog[a + b*x])^(5/2)) - (10*(a + b*x))/(3*b*c*(c*ProductLog[a + b*x])^(3/2))} -{1/(c*ProductLog[a+b*x])^(7/2), x, 4, (28*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(15*b*c^(7/2)) - (2*(a + b*x))/(5*b*(c*ProductLog[a + b*x])^(7/2)) - (14*(a + b*x))/(15*b*c*(c*ProductLog[a + b*x])^(5/2)) - (28*(a + b*x))/(15*b*c^2*(c*ProductLog[a + b*x])^(3/2))} - - -{(-c*ProductLog[a+b*x])^(5/2), x, 5, (75*c^(5/2)*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(16*b) + (75*c^3*(a + b*x))/(8*b*Sqrt[(-c)*ProductLog[a + b*x]]) + (25*c^2*(a + b*x)*Sqrt[(-c)*ProductLog[a + b*x]])/(4*b) + (5*c*(a + b*x)*((-c)*ProductLog[a + b*x])^(3/2))/(2*b) + ((a + b*x)*((-c)*ProductLog[a + b*x])^(5/2))/b} -{(-c*ProductLog[a+b*x])^(3/2), x, 4, (9*c^(3/2)*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(8*b) + (9*c^2*(a + b*x))/(4*b*Sqrt[(-c)*ProductLog[a + b*x]]) + (3*c*(a + b*x)*Sqrt[(-c)*ProductLog[a + b*x]])/(2*b) + ((a + b*x)*((-c)*ProductLog[a + b*x])^(3/2))/b} -{Sqrt[-c*ProductLog[a+b*x]], x, 3, (Sqrt[c]*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(4*b) + (c*(a + b*x))/(2*b*Sqrt[(-c)*ProductLog[a + b*x]]) + ((a + b*x)*Sqrt[(-c)*ProductLog[a + b*x]])/b} -{1/Sqrt[-c*ProductLog[a+b*x]], x, 2, -((Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(2*b*Sqrt[c])) + (a + b*x)/(b*Sqrt[(-c)*ProductLog[a + b*x]])} -{1/(-c*ProductLog[a+b*x])^(3/2), x, 2, (3*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(b*c^(3/2)) - (2*(a + b*x))/(b*((-c)*ProductLog[a + b*x])^(3/2))} -{1/(-c*ProductLog[a+b*x])^(5/2), x, 3, -((10*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(3*b*c^(5/2))) - (2*(a + b*x))/(3*b*((-c)*ProductLog[a + b*x])^(5/2)) + (10*(a + b*x))/(3*b*c*((-c)*ProductLog[a + b*x])^(3/2))} -{1/(-c*ProductLog[a+b*x])^(7/2), x, 4, (28*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(15*b*c^(7/2)) - (2*(a + b*x))/(5*b*((-c)*ProductLog[a + b*x])^(7/2)) + (14*(a + b*x))/(15*b*c*((-c)*ProductLog[a + b*x])^(5/2)) - (28*(a + b*x))/(15*b*c^2*((-c)*ProductLog[a + b*x])^(3/2))} - - -{(c*ProductLog[a + b*x])^n, x, 2, ((a + b*x)*(c*ProductLog[a + b*x])^n)/b - (n*Gamma[1 + n, -ProductLog[a + b*x]]*(c*ProductLog[a + b*x])^n)/((-ProductLog[a + b*x])^n*b)} - - -(* ::Subsubsection::Closed:: *) -(*x^m ProductLog[a+b x]^n*) - - -{x^3*ProductLog[a + b*x], x, 20, (a^3*x)/b^3 - (3*a^2*(a + b*x)^2)/(4*b^4) + (a*(a + b*x)^3)/(3*b^4) - (a + b*x)^4/(16*b^4) - (3*(a + b*x)^4)/(512*b^4*ProductLog[a + b*x]^4) - (2*a*(a + b*x)^3)/(27*b^4*ProductLog[a + b*x]^3) + (3*(a + b*x)^4)/(128*b^4*ProductLog[a + b*x]^3) - (3*a^2*(a + b*x)^2)/(8*b^4*ProductLog[a + b*x]^2) + (2*a*(a + b*x)^3)/(9*b^4*ProductLog[a + b*x]^2) - (3*(a + b*x)^4)/(64*b^4*ProductLog[a + b*x]^2) - (a^3*(a + b*x))/(b^4*ProductLog[a + b*x]) + (3*a^2*(a + b*x)^2)/(4*b^4*ProductLog[a + b*x]) - (a*(a + b*x)^3)/(3*b^4*ProductLog[a + b*x]) + (a + b*x)^4/(16*b^4*ProductLog[a + b*x]) - (a^3*(a + b*x)*ProductLog[a + b*x])/b^4 + (3*a^2*(a + b*x)^2*ProductLog[a + b*x])/(2*b^4) - (a*(a + b*x)^3*ProductLog[a + b*x])/b^4 + ((a + b*x)^4*ProductLog[a + b*x])/(4*b^4)} -{x^2*ProductLog[a + b*x], x, 14, -((a^2*x)/b^2) + (a*(a + b*x)^2)/(2*b^3) - (a + b*x)^3/(9*b^3) + (2*(a + b*x)^3)/(81*b^3*ProductLog[a + b*x]^3) + (a*(a + b*x)^2)/(4*b^3*ProductLog[a + b*x]^2) - (2*(a + b*x)^3)/(27*b^3*ProductLog[a + b*x]^2) + (a^2*(a + b*x))/(b^3*ProductLog[a + b*x]) - (a*(a + b*x)^2)/(2*b^3*ProductLog[a + b*x]) + (a + b*x)^3/(9*b^3*ProductLog[a + b*x]) + (a^2*(a + b*x)*ProductLog[a + b*x])/b^3 - (a*(a + b*x)^2*ProductLog[a + b*x])/b^3 + ((a + b*x)^3*ProductLog[a + b*x])/(3*b^3)} -{x*ProductLog[a + b*x], x, 9, (a*x)/b - (a + b*x)^2/(4*b^2) - (a + b*x)^2/(8*b^2*ProductLog[a + b*x]^2) - (a*(a + b*x))/(b^2*ProductLog[a + b*x]) + (a + b*x)^2/(4*b^2*ProductLog[a + b*x]) - (a*(a + b*x)*ProductLog[a + b*x])/b^2 + ((a + b*x)^2*ProductLog[a + b*x])/(2*b^2)} -{ProductLog[a + b*x], x, 3, -x + (a + b*x)/(b*ProductLog[a + b*x]) + ((a + b*x)*ProductLog[a + b*x])/b} -{ProductLog[a + b*x]/x, x, 0, CannotIntegrate[ProductLog[a + b*x]/x, x]} -{ProductLog[a + b*x]/x^2, x, 0, CannotIntegrate[ProductLog[a + b*x]/x^2, x]} - - -{x^3*ProductLog[a + b*x]^2, x, 24, -((4*a^3*x)/b^3) + (9*a^2*(a + b*x)^2)/(4*b^4) - (8*a*(a + b*x)^3)/(9*b^4) + (5*(a + b*x)^4)/(32*b^4) + (15*(a + b*x)^4)/(1024*b^4*ProductLog[a + b*x]^4) + (16*a*(a + b*x)^3)/(81*b^4*ProductLog[a + b*x]^3) - (15*(a + b*x)^4)/(256*b^4*ProductLog[a + b*x]^3) + (9*a^2*(a + b*x)^2)/(8*b^4*ProductLog[a + b*x]^2) - (16*a*(a + b*x)^3)/(27*b^4*ProductLog[a + b*x]^2) + (15*(a + b*x)^4)/(128*b^4*ProductLog[a + b*x]^2) + (4*a^3*(a + b*x))/(b^4*ProductLog[a + b*x]) - (9*a^2*(a + b*x)^2)/(4*b^4*ProductLog[a + b*x]) + (8*a*(a + b*x)^3)/(9*b^4*ProductLog[a + b*x]) - (5*(a + b*x)^4)/(32*b^4*ProductLog[a + b*x]) + (2*a^3*(a + b*x)*ProductLog[a + b*x])/b^4 - (3*a^2*(a + b*x)^2*ProductLog[a + b*x])/(2*b^4) + (2*a*(a + b*x)^3*ProductLog[a + b*x])/(3*b^4) - ((a + b*x)^4*ProductLog[a + b*x])/(8*b^4) - (a^3*(a + b*x)*ProductLog[a + b*x]^2)/b^4 + (3*a^2*(a + b*x)^2*ProductLog[a + b*x]^2)/(2*b^4) - (a*(a + b*x)^3*ProductLog[a + b*x]^2)/b^4 + ((a + b*x)^4*ProductLog[a + b*x]^2)/(4*b^4)} -{x^2*ProductLog[a + b*x]^2, x, 17, (4*a^2*x)/b^2 - (3*a*(a + b*x)^2)/(2*b^3) + (8*(a + b*x)^3)/(27*b^3) - (16*(a + b*x)^3)/(243*b^3*ProductLog[a + b*x]^3) - (3*a*(a + b*x)^2)/(4*b^3*ProductLog[a + b*x]^2) + (16*(a + b*x)^3)/(81*b^3*ProductLog[a + b*x]^2) - (4*a^2*(a + b*x))/(b^3*ProductLog[a + b*x]) + (3*a*(a + b*x)^2)/(2*b^3*ProductLog[a + b*x]) - (8*(a + b*x)^3)/(27*b^3*ProductLog[a + b*x]) - (2*a^2*(a + b*x)*ProductLog[a + b*x])/b^3 + (a*(a + b*x)^2*ProductLog[a + b*x])/b^3 - (2*(a + b*x)^3*ProductLog[a + b*x])/(9*b^3) + (a^2*(a + b*x)*ProductLog[a + b*x]^2)/b^3 - (a*(a + b*x)^2*ProductLog[a + b*x]^2)/b^3 + ((a + b*x)^3*ProductLog[a + b*x]^2)/(3*b^3)} -{x*ProductLog[a + b*x]^2, x, 11, -((4*a*x)/b) + (3*(a + b*x)^2)/(4*b^2) + (3*(a + b*x)^2)/(8*b^2*ProductLog[a + b*x]^2) + (4*a*(a + b*x))/(b^2*ProductLog[a + b*x]) - (3*(a + b*x)^2)/(4*b^2*ProductLog[a + b*x]) + (2*a*(a + b*x)*ProductLog[a + b*x])/b^2 - ((a + b*x)^2*ProductLog[a + b*x])/(2*b^2) - (a*(a + b*x)*ProductLog[a + b*x]^2)/b^2 + ((a + b*x)^2*ProductLog[a + b*x]^2)/(2*b^2)} -{ProductLog[a + b*x]^2, x, 4, 4*x - (4*(a + b*x))/(b*ProductLog[a + b*x]) - (2*(a + b*x)*ProductLog[a + b*x])/b + ((a + b*x)*ProductLog[a + b*x]^2)/b} -{ProductLog[a + b*x]^2/x, x, 0, CannotIntegrate[ProductLog[a + b*x]^2/x, x]} -{ProductLog[a + b*x]^2/x^2, x, 0, CannotIntegrate[ProductLog[a + b*x]^2/x^2, x]} - - -{x^3/Sqrt[c*ProductLog[a + b*x]], x, 16, -((a^3*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^4*Sqrt[c])) - (15*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(8192*b^4*Sqrt[c]) - (3*a^2*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(16*b^4*Sqrt[c]) - (a*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(24*b^4*Sqrt[c]) + (15*c^3*(a + b*x)^4)/(2048*b^4*(c*ProductLog[a + b*x])^(7/2)) + (a*c^2*(a + b*x)^3)/(12*b^4*(c*ProductLog[a + b*x])^(5/2)) - (5*c^2*(a + b*x)^4)/(256*b^4*(c*ProductLog[a + b*x])^(5/2)) + (3*a^2*c*(a + b*x)^2)/(8*b^4*(c*ProductLog[a + b*x])^(3/2)) - (a*c*(a + b*x)^3)/(6*b^4*(c*ProductLog[a + b*x])^(3/2)) + (c*(a + b*x)^4)/(32*b^4*(c*ProductLog[a + b*x])^(3/2)) - (a^3*(a + b*x))/(b^4*Sqrt[c*ProductLog[a + b*x]]) + (3*a^2*(a + b*x)^2)/(2*b^4*Sqrt[c*ProductLog[a + b*x]]) - (a*(a + b*x)^3)/(b^4*Sqrt[c*ProductLog[a + b*x]]) + (a + b*x)^4/(4*b^4*Sqrt[c*ProductLog[a + b*x]])} -{x^2/Sqrt[c*ProductLog[a + b*x]], x, 11, (a^2*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^3*Sqrt[c]) + (a*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(8*b^3*Sqrt[c]) + (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(72*b^3*Sqrt[c]) - (c^2*(a + b*x)^3)/(36*b^3*(c*ProductLog[a + b*x])^(5/2)) - (a*c*(a + b*x)^2)/(4*b^3*(c*ProductLog[a + b*x])^(3/2)) + (c*(a + b*x)^3)/(18*b^3*(c*ProductLog[a + b*x])^(3/2)) + (a^2*(a + b*x))/(b^3*Sqrt[c*ProductLog[a + b*x]]) - (a*(a + b*x)^2)/(b^3*Sqrt[c*ProductLog[a + b*x]]) + (a + b*x)^3/(3*b^3*Sqrt[c*ProductLog[a + b*x]])} -{x/Sqrt[c*ProductLog[a + b*x]], x, 7, -((a*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^2*Sqrt[c])) - (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(16*b^2*Sqrt[c]) + (c*(a + b*x)^2)/(8*b^2*(c*ProductLog[a + b*x])^(3/2)) - (a*(a + b*x))/(b^2*Sqrt[c*ProductLog[a + b*x]]) + (a + b*x)^2/(2*b^2*Sqrt[c*ProductLog[a + b*x]])} -{1/Sqrt[c*ProductLog[a + b*x]], x, 2, (Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(2*b*Sqrt[c]) + (a + b*x)/(b*Sqrt[c*ProductLog[a + b*x]])} -{1/Sqrt[c*ProductLog[a + b*x]]/x, x, 1, (CannotIntegrate[1/(x*Sqrt[ProductLog[a + b*x]]), x]*Sqrt[ProductLog[a + b*x]])/Sqrt[c*ProductLog[a + b*x]]} -{1/Sqrt[c*ProductLog[a + b*x]]/x^2, x, 1, (CannotIntegrate[1/(x^2*Sqrt[ProductLog[a + b*x]]), x]*Sqrt[ProductLog[a + b*x]])/Sqrt[c*ProductLog[a + b*x]]} - - -{x^3/Sqrt[-c*ProductLog[a + b*x]], x, 16, (a^3*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^4*Sqrt[c]) + (15*Sqrt[Pi]*Erf[(2*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(8192*b^4*Sqrt[c]) + (3*a^2*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(16*b^4*Sqrt[c]) + (a*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(24*b^4*Sqrt[c]) - (15*c^3*(a + b*x)^4)/(2048*b^4*((-c)*ProductLog[a + b*x])^(7/2)) + (a*c^2*(a + b*x)^3)/(12*b^4*((-c)*ProductLog[a + b*x])^(5/2)) - (5*c^2*(a + b*x)^4)/(256*b^4*((-c)*ProductLog[a + b*x])^(5/2)) - (3*a^2*c*(a + b*x)^2)/(8*b^4*((-c)*ProductLog[a + b*x])^(3/2)) + (a*c*(a + b*x)^3)/(6*b^4*((-c)*ProductLog[a + b*x])^(3/2)) - (c*(a + b*x)^4)/(32*b^4*((-c)*ProductLog[a + b*x])^(3/2)) - (a^3*(a + b*x))/(b^4*Sqrt[(-c)*ProductLog[a + b*x]]) + (3*a^2*(a + b*x)^2)/(2*b^4*Sqrt[(-c)*ProductLog[a + b*x]]) - (a*(a + b*x)^3)/(b^4*Sqrt[(-c)*ProductLog[a + b*x]]) + (a + b*x)^4/(4*b^4*Sqrt[(-c)*ProductLog[a + b*x]])} -{x^2/Sqrt[-c*ProductLog[a + b*x]], x, 11, -((a^2*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^3*Sqrt[c])) - (a*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(8*b^3*Sqrt[c]) - (Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(72*b^3*Sqrt[c]) - (c^2*(a + b*x)^3)/(36*b^3*((-c)*ProductLog[a + b*x])^(5/2)) + (a*c*(a + b*x)^2)/(4*b^3*((-c)*ProductLog[a + b*x])^(3/2)) - (c*(a + b*x)^3)/(18*b^3*((-c)*ProductLog[a + b*x])^(3/2)) + (a^2*(a + b*x))/(b^3*Sqrt[(-c)*ProductLog[a + b*x]]) - (a*(a + b*x)^2)/(b^3*Sqrt[(-c)*ProductLog[a + b*x]]) + (a + b*x)^3/(3*b^3*Sqrt[(-c)*ProductLog[a + b*x]])} -{x/Sqrt[-c*ProductLog[a + b*x]], x, 7, (a*Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(2*b^2*Sqrt[c]) + (Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[(-c)*ProductLog[a + b*x]])/Sqrt[c]])/(16*b^2*Sqrt[c]) - (c*(a + b*x)^2)/(8*b^2*((-c)*ProductLog[a + b*x])^(3/2)) - (a*(a + b*x))/(b^2*Sqrt[(-c)*ProductLog[a + b*x]]) + (a + b*x)^2/(2*b^2*Sqrt[(-c)*ProductLog[a + b*x]])} -{1/Sqrt[-c*ProductLog[a + b*x]], x, 2, -((Sqrt[Pi]*Erf[Sqrt[(-c)*ProductLog[a + b*x]]/Sqrt[c]])/(2*b*Sqrt[c])) + (a + b*x)/(b*Sqrt[(-c)*ProductLog[a + b*x]])} -{1/Sqrt[-c*ProductLog[a + b*x]]/x, x, 1, (CannotIntegrate[1/(x*Sqrt[ProductLog[a + b*x]]), x]*Sqrt[ProductLog[a + b*x]])/Sqrt[(-c)*ProductLog[a + b*x]]} -{1/Sqrt[-c*ProductLog[a + b*x]]/x^2, x, 1, (CannotIntegrate[1/(x^2*Sqrt[ProductLog[a + b*x]]), x]*Sqrt[ProductLog[a + b*x]])/Sqrt[(-c)*ProductLog[a + b*x]]} - - -{x^3*Sqrt[c*ProductLog[a + b*x]], x, 20, -((a^3*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(4*b^4)) - (105*Sqrt[c]*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(65536*b^4) - (9*a^2*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(64*b^4) - (5*a*Sqrt[c]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(144*b^4) + (105*c^4*(a + b*x)^4)/(16384*b^4*(c*ProductLog[a + b*x])^(7/2)) + (5*a*c^3*(a + b*x)^3)/(72*b^4*(c*ProductLog[a + b*x])^(5/2)) - (35*c^3*(a + b*x)^4)/(2048*b^4*(c*ProductLog[a + b*x])^(5/2)) + (9*a^2*c^2*(a + b*x)^2)/(32*b^4*(c*ProductLog[a + b*x])^(3/2)) - (5*a*c^2*(a + b*x)^3)/(36*b^4*(c*ProductLog[a + b*x])^(3/2)) + (7*c^2*(a + b*x)^4)/(256*b^4*(c*ProductLog[a + b*x])^(3/2)) + (a^3*c*(a + b*x))/(2*b^4*Sqrt[c*ProductLog[a + b*x]]) - (3*a^2*c*(a + b*x)^2)/(8*b^4*Sqrt[c*ProductLog[a + b*x]]) + (a*c*(a + b*x)^3)/(6*b^4*Sqrt[c*ProductLog[a + b*x]]) - (c*(a + b*x)^4)/(32*b^4*Sqrt[c*ProductLog[a + b*x]]) - (a^3*(a + b*x)*Sqrt[c*ProductLog[a + b*x]])/b^4 + (3*a^2*(a + b*x)^2*Sqrt[c*ProductLog[a + b*x]])/(2*b^4) - (a*(a + b*x)^3*Sqrt[c*ProductLog[a + b*x]])/b^4 + ((a + b*x)^4*Sqrt[c*ProductLog[a + b*x]])/(4*b^4)} -{x^2*Sqrt[c*ProductLog[a + b*x]], x, 14, (a^2*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(4*b^3) + (3*a*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(32*b^3) + (5*Sqrt[c]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(432*b^3) - (5*c^3*(a + b*x)^3)/(216*b^3*(c*ProductLog[a + b*x])^(5/2)) - (3*a*c^2*(a + b*x)^2)/(16*b^3*(c*ProductLog[a + b*x])^(3/2)) + (5*c^2*(a + b*x)^3)/(108*b^3*(c*ProductLog[a + b*x])^(3/2)) - (a^2*c*(a + b*x))/(2*b^3*Sqrt[c*ProductLog[a + b*x]]) + (a*c*(a + b*x)^2)/(4*b^3*Sqrt[c*ProductLog[a + b*x]]) - (c*(a + b*x)^3)/(18*b^3*Sqrt[c*ProductLog[a + b*x]]) + (a^2*(a + b*x)*Sqrt[c*ProductLog[a + b*x]])/b^3 - (a*(a + b*x)^2*Sqrt[c*ProductLog[a + b*x]])/b^3 + ((a + b*x)^3*Sqrt[c*ProductLog[a + b*x]])/(3*b^3)} -{x*Sqrt[c*ProductLog[a + b*x]], x, 9, -((a*Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(4*b^2)) - (3*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[c]])/(64*b^2) + (3*c^2*(a + b*x)^2)/(32*b^2*(c*ProductLog[a + b*x])^(3/2)) + (a*c*(a + b*x))/(2*b^2*Sqrt[c*ProductLog[a + b*x]]) - (c*(a + b*x)^2)/(8*b^2*Sqrt[c*ProductLog[a + b*x]]) - (a*(a + b*x)*Sqrt[c*ProductLog[a + b*x]])/b^2 + ((a + b*x)^2*Sqrt[c*ProductLog[a + b*x]])/(2*b^2)} -{Sqrt[c*ProductLog[a + b*x]], x, 3, (Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a + b*x]]/Sqrt[c]])/(4*b) - (c*(a + b*x))/(2*b*Sqrt[c*ProductLog[a + b*x]]) + ((a + b*x)*Sqrt[c*ProductLog[a + b*x]])/b} -{Sqrt[c*ProductLog[a + b*x]]/x, x, 1, (CannotIntegrate[Sqrt[ProductLog[a + b*x]]/x, x]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[ProductLog[a + b*x]]} -{Sqrt[c*ProductLog[a + b*x]]/x^2, x, 1, (CannotIntegrate[Sqrt[ProductLog[a + b*x]]/x^2, x]*Sqrt[c*ProductLog[a + b*x]])/Sqrt[ProductLog[a + b*x]]} - - -(* ::Subsubsection::Closed:: *) -(*x^m / (d+d ProductLog[a+b x])*) -(**) - - -{x^3/(d + d*ProductLog[a + b*x]), x, 12, -((3*(a + b*x)^4)/(128*b^4*d*ProductLog[a + b*x]^4)) - (2*a*(a + b*x)^3)/(9*b^4*d*ProductLog[a + b*x]^3) + (3*(a + b*x)^4)/(32*b^4*d*ProductLog[a + b*x]^3) - (3*a^2*(a + b*x)^2)/(4*b^4*d*ProductLog[a + b*x]^2) + (2*a*(a + b*x)^3)/(3*b^4*d*ProductLog[a + b*x]^2) - (3*(a + b*x)^4)/(16*b^4*d*ProductLog[a + b*x]^2) - (a^3*(a + b*x))/(b^4*d*ProductLog[a + b*x]) + (3*a^2*(a + b*x)^2)/(2*b^4*d*ProductLog[a + b*x]) - (a*(a + b*x)^3)/(b^4*d*ProductLog[a + b*x]) + (a + b*x)^4/(4*b^4*d*ProductLog[a + b*x])} -{x^2/(d + d*ProductLog[a + b*x]), x, 8, (2*(a + b*x)^3)/(27*b^3*d*ProductLog[a + b*x]^3) + (a*(a + b*x)^2)/(2*b^3*d*ProductLog[a + b*x]^2) - (2*(a + b*x)^3)/(9*b^3*d*ProductLog[a + b*x]^2) + (a^2*(a + b*x))/(b^3*d*ProductLog[a + b*x]) - (a*(a + b*x)^2)/(b^3*d*ProductLog[a + b*x]) + (a + b*x)^3/(3*b^3*d*ProductLog[a + b*x])} -{x/(d + d*ProductLog[a + b*x]), x, 5, -((a + b*x)^2/(4*b^2*d*ProductLog[a + b*x]^2)) - (a*(a + b*x))/(b^2*d*ProductLog[a + b*x]) + (a + b*x)^2/(2*b^2*d*ProductLog[a + b*x])} -{1/(d + d*ProductLog[a + b*x]), x, 1, (a + b*x)/(b*d*ProductLog[a + b*x])} -{1/(x*(d + d*ProductLog[a + b*x])), x, 2, CannotIntegrate[1/(x*(1 + ProductLog[a + b*x])), x]/d} -{1/(x^2*(d + d*ProductLog[a + b*x])), x, 2, CannotIntegrate[1/(x^2*(1 + ProductLog[a + b*x])), x]/d} - - -(* ::Subsection::Closed:: *) -(*Integrands involving ProductLog[a x^n]*) - - -(* ::Subsubsection::Closed:: *) -(*x^m ProductLog[a x]^p*) - - -{x^3*ProductLog[a*x], x, 6, -(x^4/16) - (3*x^4)/(512*ProductLog[a*x]^4) + (3*x^4)/(128*ProductLog[a*x]^3) - (3*x^4)/(64*ProductLog[a*x]^2) + x^4/(16*ProductLog[a*x]) + (1/4)*x^4*ProductLog[a*x]} -{x^2*ProductLog[a*x], x, 5, -(x^3/9) + (2*x^3)/(81*ProductLog[a*x]^3) - (2*x^3)/(27*ProductLog[a*x]^2) + x^3/(9*ProductLog[a*x]) + (1/3)*x^3*ProductLog[a*x]} -{x*ProductLog[a*x], x, 4, -(x^2/4) - x^2/(8*ProductLog[a*x]^2) + x^2/(4*ProductLog[a*x]) + (1/2)*x^2*ProductLog[a*x]} -{ProductLog[a*x], x, 3, -x + x/ProductLog[a*x] + x*ProductLog[a*x]} -{ProductLog[a*x]/x, x, 2, ProductLog[a*x] + (1/2)*ProductLog[a*x]^2} -{ProductLog[a*x]/x^2, x, 2, a*ExpIntegralEi[-ProductLog[a*x]] - ProductLog[a*x]/x} -{ProductLog[a*x]/x^3, x, 2, (-a^2)*ExpIntegralEi[-2*ProductLog[a*x]] - ProductLog[a*x]/x^2} -{ProductLog[a*x]/x^4, x, 3, (3/2)*a^3*ExpIntegralEi[-3*ProductLog[a*x]] - ProductLog[a*x]/(2*x^3) + ProductLog[a*x]^2/(2*x^3)} -{ProductLog[a*x]/x^5, x, 4, (-(8/3))*a^4*ExpIntegralEi[-4*ProductLog[a*x]] - ProductLog[a*x]/(3*x^4) + ProductLog[a*x]^2/(6*x^4) - (2*ProductLog[a*x]^3)/(3*x^4)} -{ProductLog[a*x]/x^6, x, 5, (125/24)*a^5*ExpIntegralEi[-5*ProductLog[a*x]] - ProductLog[a*x]/(4*x^5) + ProductLog[a*x]^2/(12*x^5) - (5*ProductLog[a*x]^3)/(24*x^5) + (25*ProductLog[a*x]^4)/(24*x^5)} - - -{x^2*ProductLog[a*x]^2, x, 6, (8*x^3)/27 - (16*x^3)/(243*ProductLog[a*x]^3) + (16*x^3)/(81*ProductLog[a*x]^2) - (8*x^3)/(27*ProductLog[a*x]) - (2/9)*x^3*ProductLog[a*x] + (1/3)*x^3*ProductLog[a*x]^2} -{x*ProductLog[a*x]^2, x, 5, (3*x^2)/4 + (3*x^2)/(8*ProductLog[a*x]^2) - (3*x^2)/(4*ProductLog[a*x]) - (1/2)*x^2*ProductLog[a*x] + (1/2)*x^2*ProductLog[a*x]^2} -{ProductLog[a*x]^2, x, 4, 4*x - (4*x)/ProductLog[a*x] - 2*x*ProductLog[a*x] + x*ProductLog[a*x]^2} -{ProductLog[a*x]^2/x, x, 2, (1/2)*ProductLog[a*x]^2 + (1/3)*ProductLog[a*x]^3} -{ProductLog[a*x]^2/x^2, x, 2, -((2*ProductLog[a*x])/x) - ProductLog[a*x]^2/x} -{ProductLog[a*x]^2/x^3, x, 2, a^2*ExpIntegralEi[-2*ProductLog[a*x]] - ProductLog[a*x]^2/(2*x^2)} -{ProductLog[a*x]^2/x^4, x, 2, -2*a^3*ExpIntegralEi[-3*ProductLog[a*x]] - ProductLog[a*x]^2/x^3} -{ProductLog[a*x]^2/x^5, x, 3, 4*a^4*ExpIntegralEi[-4*ProductLog[a*x]] - ProductLog[a*x]^2/(2*x^4) + ProductLog[a*x]^3/x^4} -{ProductLog[a*x]^2/x^6, x, 4, (-(25/3))*a^5*ExpIntegralEi[-5*ProductLog[a*x]] - ProductLog[a*x]^2/(3*x^5) + ProductLog[a*x]^3/(3*x^5) - (5*ProductLog[a*x]^4)/(3*x^5)} -{ProductLog[a*x]^2/x^7, x, 5, 18*a^6*ExpIntegralEi[-6*ProductLog[a*x]] - ProductLog[a*x]^2/(4*x^6) + ProductLog[a*x]^3/(6*x^6) - ProductLog[a*x]^4/(2*x^6) + (3*ProductLog[a*x]^5)/x^6} - - -{x^2*ProductLog[a*x]^3, x, 7, -((20*x^3)/27) + (40*x^3)/(243*ProductLog[a*x]^3) - (40*x^3)/(81*ProductLog[a*x]^2) + (20*x^3)/(27*ProductLog[a*x]) + (5/9)*x^3*ProductLog[a*x] - (1/3)*x^3*ProductLog[a*x]^2 + (1/3)*x^3*ProductLog[a*x]^3} -{x*ProductLog[a*x]^3, x, 6, -((9*x^2)/4) - (9*x^2)/(8*ProductLog[a*x]^2) + (9*x^2)/(4*ProductLog[a*x]) + (3/2)*x^2*ProductLog[a*x] - (3/4)*x^2*ProductLog[a*x]^2 + (1/2)*x^2*ProductLog[a*x]^3} -{ProductLog[a*x]^3, x, 5, -18*x + (18*x)/ProductLog[a*x] + 9*x*ProductLog[a*x] - 3*x*ProductLog[a*x]^2 + x*ProductLog[a*x]^3} -{ProductLog[a*x]^3/x, x, 2, (1/3)*ProductLog[a*x]^3 + (1/4)*ProductLog[a*x]^4} -{ProductLog[a*x]^3/x^2, x, 3, -((3*ProductLog[a*x])/x) - (3*ProductLog[a*x]^2)/x - ProductLog[a*x]^3/x} -{ProductLog[a*x]^3/x^3, x, 2, -((3*ProductLog[a*x]^2)/(4*x^2)) - ProductLog[a*x]^3/(2*x^2)} -{ProductLog[a*x]^3/x^4, x, 2, a^3*ExpIntegralEi[-3*ProductLog[a*x]] - ProductLog[a*x]^3/(3*x^3)} -{ProductLog[a*x]^3/x^5, x, 2, -3*a^4*ExpIntegralEi[-4*ProductLog[a*x]] - ProductLog[a*x]^3/x^4} -{ProductLog[a*x]^3/x^6, x, 3, (15/2)*a^5*ExpIntegralEi[-5*ProductLog[a*x]] - ProductLog[a*x]^3/(2*x^5) + (3*ProductLog[a*x]^4)/(2*x^5)} -{ProductLog[a*x]^3/x^7, x, 4, -18*a^6*ExpIntegralEi[-6*ProductLog[a*x]] - ProductLog[a*x]^3/(3*x^6) + ProductLog[a*x]^4/(2*x^6) - (3*ProductLog[a*x]^5)/x^6} -{ProductLog[a*x]^3/x^8, x, 5, (343/8)*a^7*ExpIntegralEi[-7*ProductLog[a*x]] - ProductLog[a*x]^3/(4*x^7) + ProductLog[a*x]^4/(4*x^7) - (7*ProductLog[a*x]^5)/(8*x^7) + (49*ProductLog[a*x]^6)/(8*x^7)} - - -{x^4/ProductLog[a*x], x, 5, -((6*x^5)/(3125*ProductLog[a*x]^5)) + (6*x^5)/(625*ProductLog[a*x]^4) - (3*x^5)/(125*ProductLog[a*x]^3) + x^5/(25*ProductLog[a*x]^2) + x^5/(5*ProductLog[a*x])} -{x^3/ProductLog[a*x], x, 4, x^4/(128*ProductLog[a*x]^4) - x^4/(32*ProductLog[a*x]^3) + x^4/(16*ProductLog[a*x]^2) + x^4/(4*ProductLog[a*x])} -{x^2/ProductLog[a*x], x, 3, -(x^3/(27*ProductLog[a*x]^3)) + x^3/(9*ProductLog[a*x]^2) + x^3/(3*ProductLog[a*x])} -{x/ProductLog[a*x], x, 2, x^2/(4*ProductLog[a*x]^2) + x^2/(2*ProductLog[a*x])} -{1/ProductLog[a*x], x, 2, ExpIntegralEi[ProductLog[a*x]]/a + x/ProductLog[a*x]} -{1/(x*ProductLog[a*x]), x, 2, Log[ProductLog[a*x]] - 1/ProductLog[a*x]} -{1/(x^2*ProductLog[a*x]), x, 3, -(1/(2*x)) - (1/2)*a*ExpIntegralEi[-ProductLog[a*x]] - 1/(2*x*ProductLog[a*x])} -{1/(x^3*ProductLog[a*x]), x, 4, -(1/(6*x^2)) + (2/3)*a^2*ExpIntegralEi[-2*ProductLog[a*x]] - 1/(3*x^2*ProductLog[a*x]) + ProductLog[a*x]/(3*x^2)} -{1/(x^4*ProductLog[a*x]), x, 5, -(1/(12*x^3)) - (9/8)*a^3*ExpIntegralEi[-3*ProductLog[a*x]] - 1/(4*x^3*ProductLog[a*x]) + ProductLog[a*x]/(8*x^3) - (3*ProductLog[a*x]^2)/(8*x^3)} - - -{x^5/ProductLog[a*x]^2, x, 5, -(x^6/(648*ProductLog[a*x]^6)) + x^6/(108*ProductLog[a*x]^5) - x^6/(36*ProductLog[a*x]^4) + x^6/(18*ProductLog[a*x]^3) + x^6/(6*ProductLog[a*x]^2)} -{x^4/ProductLog[a*x]^2, x, 4, (4*x^5)/(625*ProductLog[a*x]^5) - (4*x^5)/(125*ProductLog[a*x]^4) + (2*x^5)/(25*ProductLog[a*x]^3) + x^5/(5*ProductLog[a*x]^2)} -{x^3/ProductLog[a*x]^2, x, 3, -(x^4/(32*ProductLog[a*x]^4)) + x^4/(8*ProductLog[a*x]^3) + x^4/(4*ProductLog[a*x]^2)} -{x^2/ProductLog[a*x]^2, x, 2, (2*x^3)/(9*ProductLog[a*x]^3) + x^3/(3*ProductLog[a*x]^2)} -{x/ProductLog[a*x]^2, x, 2, ExpIntegralEi[2*ProductLog[a*x]]/a^2 + x^2/(2*ProductLog[a*x]^2)} -{1/ProductLog[a*x]^2, x, 2, (2*ExpIntegralEi[ProductLog[a*x]])/a - x/ProductLog[a*x]^2} -{1/(x*ProductLog[a*x]^2), x, 2, -(1/(2*ProductLog[a*x]^2)) - 1/ProductLog[a*x]} -{1/(x^2*ProductLog[a*x]^2), x, 4, 1/(3*x) + (1/3)*a*ExpIntegralEi[-ProductLog[a*x]] - 1/(3*x*ProductLog[a*x]^2) - 1/(3*x*ProductLog[a*x])} -{1/(x^3*ProductLog[a*x]^2), x, 5, 1/(6*x^2) - (2/3)*a^2*ExpIntegralEi[-2*ProductLog[a*x]] - 1/(4*x^2*ProductLog[a*x]^2) - 1/(6*x^2*ProductLog[a*x]) - ProductLog[a*x]/(3*x^2)} - - -{x^6/ProductLog[a*x]^3, x, 5, -((18*x^7)/(16807*ProductLog[a*x]^7)) + (18*x^7)/(2401*ProductLog[a*x]^6) - (9*x^7)/(343*ProductLog[a*x]^5) + (3*x^7)/(49*ProductLog[a*x]^4) + x^7/(7*ProductLog[a*x]^3)} -{x^5/ProductLog[a*x]^3, x, 4, x^6/(216*ProductLog[a*x]^6) - x^6/(36*ProductLog[a*x]^5) + x^6/(12*ProductLog[a*x]^4) + x^6/(6*ProductLog[a*x]^3)} -{x^4/ProductLog[a*x]^3, x, 3, -((3*x^5)/(125*ProductLog[a*x]^5)) + (3*x^5)/(25*ProductLog[a*x]^4) + x^5/(5*ProductLog[a*x]^3)} -{x^3/ProductLog[a*x]^3, x, 2, (3*x^4)/(16*ProductLog[a*x]^4) + x^4/(4*ProductLog[a*x]^3)} -{x^2/ProductLog[a*x]^3, x, 2, ExpIntegralEi[3*ProductLog[a*x]]/a^3 + x^3/(3*ProductLog[a*x]^3)} -{x/ProductLog[a*x]^3, x, 2, (3*ExpIntegralEi[2*ProductLog[a*x]])/a^2 - x^2/ProductLog[a*x]^3} -{1/ProductLog[a*x]^3, x, 3, (3*ExpIntegralEi[ProductLog[a*x]])/(2*a) - x/(2*ProductLog[a*x]^3) - (3*x)/(2*ProductLog[a*x]^2)} -{1/(x*ProductLog[a*x]^3), x, 2, -(1/(3*ProductLog[a*x]^3)) - 1/(2*ProductLog[a*x]^2)} -{1/(x^2*ProductLog[a*x]^3), x, 5, -(1/(8*x)) - (1/8)*a*ExpIntegralEi[-ProductLog[a*x]] - 1/(4*x*ProductLog[a*x]^3) - 1/(4*x*ProductLog[a*x]^2) + 1/(8*x*ProductLog[a*x])} -{1/(x^3*ProductLog[a*x]^3), x, 6, -(1/(10*x^2)) + (2/5)*a^2*ExpIntegralEi[-2*ProductLog[a*x]] - 1/(5*x^2*ProductLog[a*x]^3) - 3/(20*x^2*ProductLog[a*x]^2) + 1/(10*x^2*ProductLog[a*x]) + ProductLog[a*x]/(5*x^2)} - - -{x^3*Sqrt[c*ProductLog[a*x]], x, 6, -((105*Sqrt[c]*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(65536*a^4)) + (105*c^4*x^4)/(16384*(c*ProductLog[a*x])^(7/2)) - (35*c^3*x^4)/(2048*(c*ProductLog[a*x])^(5/2)) + (7*c^2*x^4)/(256*(c*ProductLog[a*x])^(3/2)) - (c*x^4)/(32*Sqrt[c*ProductLog[a*x]]) + (1/4)*x^4*Sqrt[c*ProductLog[a*x]]} -{x^2*Sqrt[c*ProductLog[a*x]], x, 5, (5*Sqrt[c]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(432*a^3) - (5*c^3*x^3)/(216*(c*ProductLog[a*x])^(5/2)) + (5*c^2*x^3)/(108*(c*ProductLog[a*x])^(3/2)) - (c*x^3)/(18*Sqrt[c*ProductLog[a*x]]) + (1/3)*x^3*Sqrt[c*ProductLog[a*x]]} -{x*Sqrt[c*ProductLog[a*x]], x, 4, -((3*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(64*a^2)) + (3*c^2*x^2)/(32*(c*ProductLog[a*x])^(3/2)) - (c*x^2)/(8*Sqrt[c*ProductLog[a*x]]) + (1/2)*x^2*Sqrt[c*ProductLog[a*x]]} -{Sqrt[c*ProductLog[a*x]], x, 3, (Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x]]/Sqrt[c]])/(4*a) - (c*x)/(2*Sqrt[c*ProductLog[a*x]]) + x*Sqrt[c*ProductLog[a*x]]} -{Sqrt[c*ProductLog[a*x]]/x, x, 2, 2*Sqrt[c*ProductLog[a*x]] + (2*(c*ProductLog[a*x])^(3/2))/(3*c)} -{Sqrt[c*ProductLog[a*x]]/x^2, x, 2, (-a)*Sqrt[c]*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x]]/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/x} -{Sqrt[c*ProductLog[a*x]]/x^3, x, 3, (2/3)*a^2*Sqrt[c]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/(3*x^2) + (2*(c*ProductLog[a*x])^(3/2))/(3*c*x^2)} -{Sqrt[c*ProductLog[a*x]]/x^4, x, 4, (-(4/5))*a^3*Sqrt[c]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/(5*x^3) + (2*(c*ProductLog[a*x])^(3/2))/(15*c*x^3) - (4*(c*ProductLog[a*x])^(5/2))/(5*c^2*x^3)} -{Sqrt[c*ProductLog[a*x]]/x^5, x, 5, (256/105)*a^4*Sqrt[c]*Sqrt[Pi]*Erf[(2*Sqrt[c*ProductLog[a*x]])/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/(7*x^4) + (2*(c*ProductLog[a*x])^(3/2))/(35*c*x^4) - (16*(c*ProductLog[a*x])^(5/2))/(105*c^2*x^4) + (128*(c*ProductLog[a*x])^(7/2))/(105*c^3*x^4)} -{Sqrt[c*ProductLog[a*x]]/x^6, x, 6, (-(400/189))*a^5*Sqrt[c]*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/(9*x^5) + (2*(c*ProductLog[a*x])^(3/2))/(63*c*x^5) - (4*(c*ProductLog[a*x])^(5/2))/(63*c^2*x^5) + (40*(c*ProductLog[a*x])^(7/2))/(189*c^3*x^5) - (400*(c*ProductLog[a*x])^(9/2))/(189*c^4*x^5)} - - -{x^4/Sqrt[c*ProductLog[a*x]], x, 6, (21*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(20000*a^5*Sqrt[c]) - (21*c^4*x^5)/(10000*(c*ProductLog[a*x])^(9/2)) + (7*c^3*x^5)/(1000*(c*ProductLog[a*x])^(7/2)) - (7*c^2*x^5)/(500*(c*ProductLog[a*x])^(5/2)) + (c*x^5)/(50*(c*ProductLog[a*x])^(3/2)) + x^5/(5*Sqrt[c*ProductLog[a*x]])} -{x^3/Sqrt[c*ProductLog[a*x]], x, 5, -((15*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(8192*a^4*Sqrt[c])) + (15*c^3*x^4)/(2048*(c*ProductLog[a*x])^(7/2)) - (5*c^2*x^4)/(256*(c*ProductLog[a*x])^(5/2)) + (c*x^4)/(32*(c*ProductLog[a*x])^(3/2)) + x^4/(4*Sqrt[c*ProductLog[a*x]])} -{x^2/Sqrt[c*ProductLog[a*x]], x, 4, (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(72*a^3*Sqrt[c]) - (c^2*x^3)/(36*(c*ProductLog[a*x])^(5/2)) + (c*x^3)/(18*(c*ProductLog[a*x])^(3/2)) + x^3/(3*Sqrt[c*ProductLog[a*x]])} -{x/Sqrt[c*ProductLog[a*x]], x, 3, -((Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(16*a^2*Sqrt[c])) + (c*x^2)/(8*(c*ProductLog[a*x])^(3/2)) + x^2/(2*Sqrt[c*ProductLog[a*x]])} -{1/Sqrt[c*ProductLog[a*x]], x, 2, (Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x]]/Sqrt[c]])/(2*a*Sqrt[c]) + x/Sqrt[c*ProductLog[a*x]]} -{1/(x*Sqrt[c*ProductLog[a*x]]), x, 2, -(2/Sqrt[c*ProductLog[a*x]]) + (2*Sqrt[c*ProductLog[a*x]])/c} -{1/(x^2*Sqrt[c*ProductLog[a*x]]), x, 3, -((2*a*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x]]/Sqrt[c]])/(3*Sqrt[c])) - 2/(3*x*Sqrt[c*ProductLog[a*x]]) - (2*Sqrt[c*ProductLog[a*x]])/(3*c*x)} -{1/(x^3*Sqrt[c*ProductLog[a*x]]), x, 4, (8*a^2*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(15*Sqrt[c]) - 2/(5*x^2*Sqrt[c*ProductLog[a*x]]) - (2*Sqrt[c*ProductLog[a*x]])/(15*c*x^2) + (8*(c*ProductLog[a*x])^(3/2))/(15*c^2*x^2)} -{1/(x^4*Sqrt[c*ProductLog[a*x]]), x, 5, -((24*a^3*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(35*Sqrt[c])) - 2/(7*x^3*Sqrt[c*ProductLog[a*x]]) - (2*Sqrt[c*ProductLog[a*x]])/(35*c*x^3) + (4*(c*ProductLog[a*x])^(3/2))/(35*c^2*x^3) - (24*(c*ProductLog[a*x])^(5/2))/(35*c^3*x^3)} -{1/(x^5*Sqrt[c*ProductLog[a*x]]), x, 6, (2048*a^4*Sqrt[Pi]*Erf[(2*Sqrt[c*ProductLog[a*x]])/Sqrt[c]])/(945*Sqrt[c]) - 2/(9*x^4*Sqrt[c*ProductLog[a*x]]) - (2*Sqrt[c*ProductLog[a*x]])/(63*c*x^4) + (16*(c*ProductLog[a*x])^(3/2))/(315*c^2*x^4) - (128*(c*ProductLog[a*x])^(5/2))/(945*c^3*x^4) + (1024*(c*ProductLog[a*x])^(7/2))/(945*c^4*x^4)} - - -{x^2*(c*ProductLog[a*x])^p, x, 3, (3^(-3 - p)*x^2*Gamma[3 + p, -3*ProductLog[a*x]]*(-ProductLog[a*x])^(-2 - p)*(c*ProductLog[a*x])^p)/(E^(2*ProductLog[a*x])*a) + (3^(-4 - p)*x^2*Gamma[4 + p, -3*ProductLog[a*x]]*(-ProductLog[a*x])^(-3 - p)*(c*ProductLog[a*x])^(1 + p))/(E^(2*ProductLog[a*x])*(a*c))} -{x*(c*ProductLog[a*x])^p, x, 3, (2^(-2 - p)*x*Gamma[2 + p, -2*ProductLog[a*x]]*(-ProductLog[a*x])^(-1 - p)*(c*ProductLog[a*x])^p)/(E^ProductLog[a*x]*a) + (2^(-3 - p)*x*Gamma[3 + p, -2*ProductLog[a*x]]*(-ProductLog[a*x])^(-2 - p)*(c*ProductLog[a*x])^(1 + p))/(E^ProductLog[a*x]*(a*c))} -{(c*ProductLog[a*x])^p/x, x, 2, (c*ProductLog[a*x])^p/p + (c*ProductLog[a*x])^(1 + p)/(c*(1 + p))} -{(c*ProductLog[a*x])^p/x^2, x, 3, -((E^(2*ProductLog[a*x])*Gamma[-1 + p, ProductLog[a*x]]*ProductLog[a*x]^(2 - p)*(c*ProductLog[a*x])^p)/(a*x^2)) - (E^(2*ProductLog[a*x])*Gamma[p, ProductLog[a*x]]*ProductLog[a*x]^(1 - p)*(c*ProductLog[a*x])^(1 + p))/(a*c*x^2)} -{(c*ProductLog[a*x])^p/x^3, x, 3, -((2^(2 - p)*E^(3*ProductLog[a*x])*Gamma[-2 + p, 2*ProductLog[a*x]]*ProductLog[a*x]^(3 - p)*(c*ProductLog[a*x])^p)/(a*x^3)) - (2^(1 - p)*E^(3*ProductLog[a*x])*Gamma[-1 + p, 2*ProductLog[a*x]]*ProductLog[a*x]^(2 - p)*(c*ProductLog[a*x])^(1 + p))/(a*c*x^3)} - - -{x^m*ProductLog[a*x], x, 3, (x^m*Gamma[3 + m, -(1 + m)*ProductLog[a*x]]*ProductLog[a*x]^2*(-(1 + m)*ProductLog[a*x])^(-2 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m))) + (x^m*Gamma[2 + m, -(1 + m)*ProductLog[a*x]]*ProductLog[a*x]*(-(1 + m)*ProductLog[a*x])^(-1 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)))} -{x^m*ProductLog[a*x]^2, x, 3, (x^m*Gamma[4 + m, -(1 + m)*ProductLog[a*x]]*ProductLog[a*x]^3*(-(1 + m)*ProductLog[a*x])^(-3 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m))) + (x^m*Gamma[3 + m, -(1 + m)*ProductLog[a*x]]*ProductLog[a*x]^2*(-(1 + m)*ProductLog[a*x])^(-2 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)))} -{x^m/ProductLog[a*x], x, 3, (x^m*Gamma[m, -(1 + m)*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(1 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)*ProductLog[a*x])) + (x^m*Gamma[1 + m, -(1 + m)*ProductLog[a*x]])/(E^(m*ProductLog[a*x])*(-(1 + m)*ProductLog[a*x])^m*(a*(1 + m)))} -{x^m/ProductLog[a*x]^2, x, 3, (x^m*Gamma[m, -(1 + m)*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(1 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)*ProductLog[a*x])) + (x^m*Gamma[-1 + m, -(1 + m)*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(2 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)*ProductLog[a*x]^2))} -{x^m*Sqrt[c*ProductLog[a*x]], x, 3, (x^m*Gamma[5/2 + m, -(1 + m)*ProductLog[a*x]]*(c*ProductLog[a*x])^(3/2)*(-(1 + m)*ProductLog[a*x])^(-(3/2) - m))/(E^(m*ProductLog[a*x])*(a*c*(1 + m))) + (x^m*Gamma[3/2 + m, -(1 + m)*ProductLog[a*x]]*Sqrt[c*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(-(1/2) - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)))} -{x^m/Sqrt[c*ProductLog[a*x]], x, 3, (x^m*Gamma[3/2 + m, -(1 + m)*ProductLog[a*x]]*Sqrt[c*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(-(1/2) - m))/(E^(m*ProductLog[a*x])*(a*c*(1 + m))) + (x^m*Gamma[1/2 + m, -(1 + m)*ProductLog[a*x]]*(-(1 + m)*ProductLog[a*x])^(1/2 - m))/(E^(m*ProductLog[a*x])*(a*(1 + m)*Sqrt[c*ProductLog[a*x]]))} - - -{x^m*(c*ProductLog[a*x])^p, x, 3, (1/(a*c*(1 + m)))*((x^m*Gamma[2 + m + p, -(1 + m)*ProductLog[a*x]]*(c*ProductLog[a*x])^(1 + p)*(-(1 + m)*ProductLog[a*x])^(-1 - m - p))/E^(m*ProductLog[a*x])) + (1/(a*(1 + m)))*((x^m*Gamma[1 + m + p, -(1 + m)*ProductLog[a*x]]*(c*ProductLog[a*x])^p*(-(1 + m)*ProductLog[a*x])^(-m - p))/E^(m*ProductLog[a*x]))} - - -(* ::Subsubsection::Closed:: *) -(*x^m ProductLog[a x^2]^p*) - - -{x^4*ProductLog[a*x^2], x, 0, CannotIntegrate[x^4*ProductLog[a*x^2], x]} -{x^3*ProductLog[a*x^2], x, 5, -(x^4/8) - x^4/(16*ProductLog[a*x^2]^2) + x^4/(8*ProductLog[a*x^2]) + (1/4)*x^4*ProductLog[a*x^2]} -{x^2*ProductLog[a*x^2], x, 0, CannotIntegrate[x^2*ProductLog[a*x^2], x]} -{x*ProductLog[a*x^2], x, 4, -(x^2/2) + x^2/(2*ProductLog[a*x^2]) + (1/2)*x^2*ProductLog[a*x^2]} -{ProductLog[a*x^2], x, 0, CannotIntegrate[ProductLog[a*x^2], x]} -{ProductLog[a*x^2]/x, x, 2, (1/2)*ProductLog[a*x^2] + (1/4)*ProductLog[a*x^2]^2} -{ProductLog[a*x^2]/x^2, x, 0, CannotIntegrate[ProductLog[a*x^2]/x^2, x]} -{ProductLog[a*x^2]/x^3, x, 2, (1/2)*a*ExpIntegralEi[-ProductLog[a*x^2]] - ProductLog[a*x^2]/(2*x^2)} -{ProductLog[a*x^2]/x^4, x, 0, CannotIntegrate[ProductLog[a*x^2]/x^4, x]} -{ProductLog[a*x^2]/x^5, x, 2, (-(1/2))*a^2*ExpIntegralEi[-2*ProductLog[a*x^2]] - ProductLog[a*x^2]/(2*x^4)} -{ProductLog[a*x^2]/x^6, x, 0, CannotIntegrate[ProductLog[a*x^2]/x^6, x]} -{ProductLog[a*x^2]/x^7, x, 3, (3/4)*a^3*ExpIntegralEi[-3*ProductLog[a*x^2]] - ProductLog[a*x^2]/(4*x^6) + ProductLog[a*x^2]^2/(4*x^6)} - - -{x^3*ProductLog[a*x^2]^2, x, 6, (3*x^4)/8 + (3*x^4)/(16*ProductLog[a*x^2]^2) - (3*x^4)/(8*ProductLog[a*x^2]) - (1/4)*x^4*ProductLog[a*x^2] + (1/4)*x^4*ProductLog[a*x^2]^2} -{x^2*ProductLog[a*x^2]^2, x, 0, CannotIntegrate[x^2*ProductLog[a*x^2]^2, x]} -{x*ProductLog[a*x^2]^2, x, 5, 2*x^2 - (2*x^2)/ProductLog[a*x^2] - x^2*ProductLog[a*x^2] + (1/2)*x^2*ProductLog[a*x^2]^2} -{ProductLog[a*x^2]^2, x, 0, CannotIntegrate[ProductLog[a*x^2]^2, x]} -{ProductLog[a*x^2]^2/x, x, 2, (1/4)*ProductLog[a*x^2]^2 + (1/6)*ProductLog[a*x^2]^3} -{ProductLog[a*x^2]^2/x^2, x, 0, CannotIntegrate[ProductLog[a*x^2]^2/x^2, x]} -{ProductLog[a*x^2]^2/x^3, x, 2, -(ProductLog[a*x^2]/x^2) - ProductLog[a*x^2]^2/(2*x^2)} -{ProductLog[a*x^2]^2/x^4, x, 0, CannotIntegrate[ProductLog[a*x^2]^2/x^4, x]} -{ProductLog[a*x^2]^2/x^5, x, 2, (1/2)*a^2*ExpIntegralEi[-2*ProductLog[a*x^2]] - ProductLog[a*x^2]^2/(4*x^4)} -{ProductLog[a*x^2]^2/x^6, x, 0, CannotIntegrate[ProductLog[a*x^2]^2/x^6, x]} -{ProductLog[a*x^2]^2/x^7, x, 2, (-a^3)*ExpIntegralEi[-3*ProductLog[a*x^2]] - ProductLog[a*x^2]^2/(2*x^6)} -{ProductLog[a*x^2]^2/x^8, x, 0, CannotIntegrate[ProductLog[a*x^2]^2/x^8, x]} -{ProductLog[a*x^2]^2/x^9, x, 3, 2*a^4*ExpIntegralEi[-4*ProductLog[a*x^2]] - ProductLog[a*x^2]^2/(4*x^8) + ProductLog[a*x^2]^3/(2*x^8)} - - -{x^2*ProductLog[a*x^2]^3, x, 0, CannotIntegrate[x^2*ProductLog[a*x^2]^3, x]} -{x*ProductLog[a*x^2]^3, x, 6, -9*x^2 + (9*x^2)/ProductLog[a*x^2] + (9/2)*x^2*ProductLog[a*x^2] - (3/2)*x^2*ProductLog[a*x^2]^2 + (1/2)*x^2*ProductLog[a*x^2]^3} -{ProductLog[a*x^2]^3, x, 0, CannotIntegrate[ProductLog[a*x^2]^3, x]} -{ProductLog[a*x^2]^3/x, x, 2, (1/6)*ProductLog[a*x^2]^3 + (1/8)*ProductLog[a*x^2]^4} -{ProductLog[a*x^2]^3/x^2, x, 0, CannotIntegrate[ProductLog[a*x^2]^3/x^2, x]} -{ProductLog[a*x^2]^3/x^3, x, 3, -((3*ProductLog[a*x^2])/(2*x^2)) - (3*ProductLog[a*x^2]^2)/(2*x^2) - ProductLog[a*x^2]^3/(2*x^2)} -{ProductLog[a*x^2]^3/x^4, x, 0, CannotIntegrate[ProductLog[a*x^2]^3/x^4, x]} -{ProductLog[a*x^2]^3/x^5, x, 2, -((3*ProductLog[a*x^2]^2)/(8*x^4)) - ProductLog[a*x^2]^3/(4*x^4)} -{ProductLog[a*x^2]^3/x^6, x, 0, CannotIntegrate[ProductLog[a*x^2]^3/x^6, x]} -{ProductLog[a*x^2]^3/x^7, x, 2, (1/2)*a^3*ExpIntegralEi[-3*ProductLog[a*x^2]] - ProductLog[a*x^2]^3/(6*x^6)} -{ProductLog[a*x^2]^3/x^8, x, 0, CannotIntegrate[ProductLog[a*x^2]^3/x^8, x]} -{ProductLog[a*x^2]^3/x^9, x, 2, (-(3/2))*a^4*ExpIntegralEi[-4*ProductLog[a*x^2]] - ProductLog[a*x^2]^3/(2*x^8)} - - -{x^5/ProductLog[a*x^2], x, 3, -(x^6/(54*ProductLog[a*x^2]^3)) + x^6/(18*ProductLog[a*x^2]^2) + x^6/(6*ProductLog[a*x^2])} -{x^4/ProductLog[a*x^2], x, 0, CannotIntegrate[x^4/ProductLog[a*x^2], x]} -{x^3/ProductLog[a*x^2], x, 2, x^4/(8*ProductLog[a*x^2]^2) + x^4/(4*ProductLog[a*x^2])} -{x^2/ProductLog[a*x^2], x, 0, CannotIntegrate[x^2/ProductLog[a*x^2], x]} -{x/ProductLog[a*x^2], x, 2, ExpIntegralEi[ProductLog[a*x^2]]/(2*a) + x^2/(2*ProductLog[a*x^2])} -{1/ProductLog[a*x^2], x, 0, CannotIntegrate[1/ProductLog[a*x^2], x]} -{1/(x*ProductLog[a*x^2]), x, 2, (1/2)*Log[ProductLog[a*x^2]] - 1/(2*ProductLog[a*x^2])} -{1/(x^2*ProductLog[a*x^2]), x, 0, CannotIntegrate[1/(x^2*ProductLog[a*x^2]), x]} -{1/(x^3*ProductLog[a*x^2]), x, 4, -(1/(4*x^2)) - (1/4)*a*ExpIntegralEi[-ProductLog[a*x^2]] - 1/(4*x^2*ProductLog[a*x^2])} -{1/(x^4*ProductLog[a*x^2]), x, 0, CannotIntegrate[1/(x^4*ProductLog[a*x^2]), x]} -{1/(x^5*ProductLog[a*x^2]), x, 5, -(1/(12*x^4)) + (1/3)*a^2*ExpIntegralEi[-2*ProductLog[a*x^2]] - 1/(6*x^4*ProductLog[a*x^2]) + ProductLog[a*x^2]/(6*x^4)} - - -{x^7/ProductLog[a*x^2]^2, x, 3, -(x^8/(64*ProductLog[a*x^2]^4)) + x^8/(16*ProductLog[a*x^2]^3) + x^8/(8*ProductLog[a*x^2]^2)} -{x^6/ProductLog[a*x^2]^2, x, 0, CannotIntegrate[x^6/ProductLog[a*x^2]^2, x]} -{x^5/ProductLog[a*x^2]^2, x, 2, x^6/(9*ProductLog[a*x^2]^3) + x^6/(6*ProductLog[a*x^2]^2)} -{x^4/ProductLog[a*x^2]^2, x, 0, CannotIntegrate[x^4/ProductLog[a*x^2]^2, x]} -{x^3/ProductLog[a*x^2]^2, x, 2, ExpIntegralEi[2*ProductLog[a*x^2]]/(2*a^2) + x^4/(4*ProductLog[a*x^2]^2)} -{x^2/ProductLog[a*x^2]^2, x, 0, CannotIntegrate[x^2/ProductLog[a*x^2]^2, x]} -{x/ProductLog[a*x^2]^2, x, 2, ExpIntegralEi[ProductLog[a*x^2]]/a - x^2/(2*ProductLog[a*x^2]^2)} -{1/ProductLog[a*x^2]^2, x, 0, CannotIntegrate[1/ProductLog[a*x^2]^2, x]} -{1/(x*ProductLog[a*x^2]^2), x, 2, -(1/(4*ProductLog[a*x^2]^2)) - 1/(2*ProductLog[a*x^2])} -{1/(x^2*ProductLog[a*x^2]^2), x, 0, CannotIntegrate[1/(x^2*ProductLog[a*x^2]^2), x]} -{1/(x^3*ProductLog[a*x^2]^2), x, 5, 1/(6*x^2) + (1/6)*a*ExpIntegralEi[-ProductLog[a*x^2]] - 1/(6*x^2*ProductLog[a*x^2]^2) - 1/(6*x^2*ProductLog[a*x^2])} - - -{x^7*Sqrt[c*ProductLog[a*x^2]], x, 6, -((105*Sqrt[c]*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(131072*a^4)) + (105*c^4*x^8)/(32768*(c*ProductLog[a*x^2])^(7/2)) - (35*c^3*x^8)/(4096*(c*ProductLog[a*x^2])^(5/2)) + (7*c^2*x^8)/(512*(c*ProductLog[a*x^2])^(3/2)) - (c*x^8)/(64*Sqrt[c*ProductLog[a*x^2]]) + (1/8)*x^8*Sqrt[c*ProductLog[a*x^2]]} -{x^6*Sqrt[c*ProductLog[a*x^2]], x, 5, (48*c^4*x^7)/(16807*(c*ProductLog[a*x^2])^(7/2)) - (24*c^3*x^7)/(2401*(c*ProductLog[a*x^2])^(5/2)) + (6*c^2*x^7)/(343*(c*ProductLog[a*x^2])^(3/2)) - (c*x^7)/(49*Sqrt[c*ProductLog[a*x^2]]) + (1/7)*x^7*Sqrt[c*ProductLog[a*x^2]]} -{x^5*Sqrt[c*ProductLog[a*x^2]], x, 5, (5*Sqrt[c]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(864*a^3) - (5*c^3*x^6)/(432*(c*ProductLog[a*x^2])^(5/2)) + (5*c^2*x^6)/(216*(c*ProductLog[a*x^2])^(3/2)) - (c*x^6)/(36*Sqrt[c*ProductLog[a*x^2]]) + (1/6)*x^6*Sqrt[c*ProductLog[a*x^2]]} -{x^4*Sqrt[c*ProductLog[a*x^2]], x, 4, -((8*c^3*x^5)/(625*(c*ProductLog[a*x^2])^(5/2))) + (4*c^2*x^5)/(125*(c*ProductLog[a*x^2])^(3/2)) - (c*x^5)/(25*Sqrt[c*ProductLog[a*x^2]]) + (1/5)*x^5*Sqrt[c*ProductLog[a*x^2]]} -{x^3*Sqrt[c*ProductLog[a*x^2]], x, 4, -((3*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(128*a^2)) + (3*c^2*x^4)/(64*(c*ProductLog[a*x^2])^(3/2)) - (c*x^4)/(16*Sqrt[c*ProductLog[a*x^2]]) + (1/4)*x^4*Sqrt[c*ProductLog[a*x^2]]} -{x^2*Sqrt[c*ProductLog[a*x^2]], x, 3, (2*c^2*x^3)/(27*(c*ProductLog[a*x^2])^(3/2)) - (c*x^3)/(9*Sqrt[c*ProductLog[a*x^2]]) + (1/3)*x^3*Sqrt[c*ProductLog[a*x^2]]} -{x*Sqrt[c*ProductLog[a*x^2]], x, 3, (Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^2]]/Sqrt[c]])/(8*a) - (c*x^2)/(4*Sqrt[c*ProductLog[a*x^2]]) + (1/2)*x^2*Sqrt[c*ProductLog[a*x^2]]} -{Sqrt[c*ProductLog[a*x^2]], x, 2, -((c*x)/Sqrt[c*ProductLog[a*x^2]]) + x*Sqrt[c*ProductLog[a*x^2]]} -{Sqrt[c*ProductLog[a*x^2]]/x, x, 2, Sqrt[c*ProductLog[a*x^2]] + (c*ProductLog[a*x^2])^(3/2)/(3*c)} -{Sqrt[c*ProductLog[a*x^2]]/x^2, x, 1, (CannotIntegrate[Sqrt[ProductLog[a*x^2]]/x^2, x]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[ProductLog[a*x^2]]} -{Sqrt[c*ProductLog[a*x^2]]/x^3, x, 2, (-(1/2))*a*Sqrt[c]*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^2]]/Sqrt[c]] - Sqrt[c*ProductLog[a*x^2]]/x^2} -{Sqrt[c*ProductLog[a*x^2]]/x^4, x, 1, (CannotIntegrate[Sqrt[ProductLog[a*x^2]]/x^4, x]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[ProductLog[a*x^2]]} -{Sqrt[c*ProductLog[a*x^2]]/x^5, x, 3, (1/3)*a^2*Sqrt[c]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]] - Sqrt[c*ProductLog[a*x^2]]/(3*x^4) + (c*ProductLog[a*x^2])^(3/2)/(3*c*x^4)} -{Sqrt[c*ProductLog[a*x^2]]/x^6, x, 1, (CannotIntegrate[Sqrt[ProductLog[a*x^2]]/x^6, x]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[ProductLog[a*x^2]]} -{Sqrt[c*ProductLog[a*x^2]]/x^7, x, 4, (-(2/5))*a^3*Sqrt[c]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]] - Sqrt[c*ProductLog[a*x^2]]/(5*x^6) + (c*ProductLog[a*x^2])^(3/2)/(15*c*x^6) - (2*(c*ProductLog[a*x^2])^(5/2))/(5*c^2*x^6)} - - -{x^7/Sqrt[c*ProductLog[a*x^2]], x, 5, -((15*Sqrt[Pi]*Erfi[(2*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(16384*a^4*Sqrt[c])) + (15*c^3*x^8)/(4096*(c*ProductLog[a*x^2])^(7/2)) - (5*c^2*x^8)/(512*(c*ProductLog[a*x^2])^(5/2)) + (c*x^8)/(64*(c*ProductLog[a*x^2])^(3/2)) + x^8/(8*Sqrt[c*ProductLog[a*x^2]])} -{x^6/Sqrt[c*ProductLog[a*x^2]], x, 4, (8*c^3*x^7)/(2401*(c*ProductLog[a*x^2])^(7/2)) - (4*c^2*x^7)/(343*(c*ProductLog[a*x^2])^(5/2)) + (c*x^7)/(49*(c*ProductLog[a*x^2])^(3/2)) + x^7/(7*Sqrt[c*ProductLog[a*x^2]])} -{x^5/Sqrt[c*ProductLog[a*x^2]], x, 4, (Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(144*a^3*Sqrt[c]) - (c^2*x^6)/(72*(c*ProductLog[a*x^2])^(5/2)) + (c*x^6)/(36*(c*ProductLog[a*x^2])^(3/2)) + x^6/(6*Sqrt[c*ProductLog[a*x^2]])} -{x^4/Sqrt[c*ProductLog[a*x^2]], x, 3, -((2*c^2*x^5)/(125*(c*ProductLog[a*x^2])^(5/2))) + (c*x^5)/(25*(c*ProductLog[a*x^2])^(3/2)) + x^5/(5*Sqrt[c*ProductLog[a*x^2]])} -{x^3/Sqrt[c*ProductLog[a*x^2]], x, 3, -((Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(32*a^2*Sqrt[c])) + (c*x^4)/(16*(c*ProductLog[a*x^2])^(3/2)) + x^4/(4*Sqrt[c*ProductLog[a*x^2]])} -{x^2/Sqrt[c*ProductLog[a*x^2]], x, 2, (c*x^3)/(9*(c*ProductLog[a*x^2])^(3/2)) + x^3/(3*Sqrt[c*ProductLog[a*x^2]])} -{x/Sqrt[c*ProductLog[a*x^2]], x, 2, (Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^2]]/Sqrt[c]])/(4*a*Sqrt[c]) + x^2/(2*Sqrt[c*ProductLog[a*x^2]])} -{1/Sqrt[c*ProductLog[a*x^2]], x, 1, (CannotIntegrate[1/Sqrt[ProductLog[a*x^2]], x]*Sqrt[ProductLog[a*x^2]])/Sqrt[c*ProductLog[a*x^2]]} -{1/(x*Sqrt[c*ProductLog[a*x^2]]), x, 2, -(1/Sqrt[c*ProductLog[a*x^2]]) + Sqrt[c*ProductLog[a*x^2]]/c} -{1/(x^2*Sqrt[c*ProductLog[a*x^2]]), x, 1, (CannotIntegrate[1/(x^2*Sqrt[ProductLog[a*x^2]]), x]*Sqrt[ProductLog[a*x^2]])/Sqrt[c*ProductLog[a*x^2]]} -{1/(x^3*Sqrt[c*ProductLog[a*x^2]]), x, 3, -((a*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^2]]/Sqrt[c]])/(3*Sqrt[c])) - 1/(3*x^2*Sqrt[c*ProductLog[a*x^2]]) - Sqrt[c*ProductLog[a*x^2]]/(3*c*x^2)} -{1/(x^4*Sqrt[c*ProductLog[a*x^2]]), x, 1, (CannotIntegrate[1/(x^4*Sqrt[ProductLog[a*x^2]]), x]*Sqrt[ProductLog[a*x^2]])/Sqrt[c*ProductLog[a*x^2]]} -{1/(x^5*Sqrt[c*ProductLog[a*x^2]]), x, 4, (4*a^2*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(15*Sqrt[c]) - 1/(5*x^4*Sqrt[c*ProductLog[a*x^2]]) - Sqrt[c*ProductLog[a*x^2]]/(15*c*x^4) + (4*(c*ProductLog[a*x^2])^(3/2))/(15*c^2*x^4)} -{1/(x^6*Sqrt[c*ProductLog[a*x^2]]), x, 1, (CannotIntegrate[1/(x^6*Sqrt[ProductLog[a*x^2]]), x]*Sqrt[ProductLog[a*x^2]])/Sqrt[c*ProductLog[a*x^2]]} -{1/(x^7*Sqrt[c*ProductLog[a*x^2]]), x, 5, -((12*a^3*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[c*ProductLog[a*x^2]])/Sqrt[c]])/(35*Sqrt[c])) - 1/(7*x^6*Sqrt[c*ProductLog[a*x^2]]) - Sqrt[c*ProductLog[a*x^2]]/(35*c*x^6) + (2*(c*ProductLog[a*x^2])^(3/2))/(35*c^2*x^6) - (12*(c*ProductLog[a*x^2])^(5/2))/(35*c^3*x^6)} - - -{x^2*(c*ProductLog[a*x^2])^p, x, 1, (CannotIntegrate[x^2*ProductLog[a*x^2]^p, x]*(c*ProductLog[a*x^2])^p)/ProductLog[a*x^2]^p} -{x*(c*ProductLog[a*x^2])^p, x, 3, (1/2)*x^2*(c*ProductLog[a*x^2])^p - (p*Gamma[1 + p, -ProductLog[a*x^2]]*(c*ProductLog[a*x^2])^p)/((-ProductLog[a*x^2])^p*(2*a))} -{(c*ProductLog[a*x^2])^p/x, x, 2, (c*ProductLog[a*x^2])^p/(2*p) + (c*ProductLog[a*x^2])^(1 + p)/(2*c*(1 + p))} -{(c*ProductLog[a*x^2])^p/x^2, x, 1, (CannotIntegrate[ProductLog[a*x^2]^p/x^2, x]*(c*ProductLog[a*x^2])^p)/ProductLog[a*x^2]^p} -{(c*ProductLog[a*x^2])^p/x^3, x, 5, -((E^(2*ProductLog[a*x^2])*Gamma[-1 + p, ProductLog[a*x^2]]*ProductLog[a*x^2]^(2 - p)*(c*ProductLog[a*x^2])^p)/(2*a*x^4)) - (E^(2*ProductLog[a*x^2])*Gamma[p, ProductLog[a*x^2]]*ProductLog[a*x^2]^(2 - p)*(c*ProductLog[a*x^2])^p)/(2*a*x^4)} - - -(* ::Subsubsection::Closed:: *) -(*x^m ProductLog[a/x]^p*) - - -{x^4*ProductLog[a/x], x, 5, (-(125/24))*a^5*ExpIntegralEi[-5*ProductLog[a/x]] + (1/4)*x^5*ProductLog[a/x] - (1/12)*x^5*ProductLog[a/x]^2 + (5/24)*x^5*ProductLog[a/x]^3 - (25/24)*x^5*ProductLog[a/x]^4} -{x^3*ProductLog[a/x], x, 4, (8/3)*a^4*ExpIntegralEi[-4*ProductLog[a/x]] + (1/3)*x^4*ProductLog[a/x] - (1/6)*x^4*ProductLog[a/x]^2 + (2/3)*x^4*ProductLog[a/x]^3} -{x^2*ProductLog[a/x], x, 3, (-(3/2))*a^3*ExpIntegralEi[-3*ProductLog[a/x]] + (1/2)*x^3*ProductLog[a/x] - (1/2)*x^3*ProductLog[a/x]^2} -{x*ProductLog[a/x], x, 2, a^2*ExpIntegralEi[-2*ProductLog[a/x]] + x^2*ProductLog[a/x]} -{ProductLog[a/x], x, 3, (-a)*ExpIntegralEi[-ProductLog[a/x]] + x*ProductLog[a/x]} -{ProductLog[a/x]/x, x, 2, -ProductLog[a/x] - (1/2)*ProductLog[a/x]^2} -{ProductLog[a/x]/x^2, x, 4, 1/x - 1/(x*ProductLog[a/x]) - ProductLog[a/x]/x} -{ProductLog[a/x]/x^3, x, 5, 1/(4*x^2) + 1/(8*x^2*ProductLog[a/x]^2) - 1/(4*x^2*ProductLog[a/x]) - ProductLog[a/x]/(2*x^2)} -{ProductLog[a/x]/x^4, x, 6, 1/(9*x^3) - 2/(81*x^3*ProductLog[a/x]^3) + 2/(27*x^3*ProductLog[a/x]^2) - 1/(9*x^3*ProductLog[a/x]) - ProductLog[a/x]/(3*x^3)} -{ProductLog[a/x]/x^5, x, 7, 1/(16*x^4) + 3/(512*x^4*ProductLog[a/x]^4) - 3/(128*x^4*ProductLog[a/x]^3) + 3/(64*x^4*ProductLog[a/x]^2) - 1/(16*x^4*ProductLog[a/x]) - ProductLog[a/x]/(4*x^4)} - - -{x^4*ProductLog[a/x]^2, x, 4, (25/3)*a^5*ExpIntegralEi[-5*ProductLog[a/x]] + (1/3)*x^5*ProductLog[a/x]^2 - (1/3)*x^5*ProductLog[a/x]^3 + (5/3)*x^5*ProductLog[a/x]^4} -{x^3*ProductLog[a/x]^2, x, 3, -4*a^4*ExpIntegralEi[-4*ProductLog[a/x]] + (1/2)*x^4*ProductLog[a/x]^2 - x^4*ProductLog[a/x]^3} -{x^2*ProductLog[a/x]^2, x, 2, 2*a^3*ExpIntegralEi[-3*ProductLog[a/x]] + x^3*ProductLog[a/x]^2} -{x*ProductLog[a/x]^2, x, 2, (-a^2)*ExpIntegralEi[-2*ProductLog[a/x]] + (1/2)*x^2*ProductLog[a/x]^2} -{ProductLog[a/x]^2, x, 2, 2*x*ProductLog[a/x] + x*ProductLog[a/x]^2} -{ProductLog[a/x]^2/x, x, 2, (-(1/2))*ProductLog[a/x]^2 - (1/3)*ProductLog[a/x]^3} -{ProductLog[a/x]^2/x^2, x, 5, -(4/x) + 4/(x*ProductLog[a/x]) + (2*ProductLog[a/x])/x - ProductLog[a/x]^2/x} -{ProductLog[a/x]^2/x^3, x, 6, -(3/(4*x^2)) - 3/(8*x^2*ProductLog[a/x]^2) + 3/(4*x^2*ProductLog[a/x]) + ProductLog[a/x]/(2*x^2) - ProductLog[a/x]^2/(2*x^2)} -{ProductLog[a/x]^2/x^4, x, 7, -(8/(27*x^3)) + 16/(243*x^3*ProductLog[a/x]^3) - 16/(81*x^3*ProductLog[a/x]^2) + 8/(27*x^3*ProductLog[a/x]) + (2*ProductLog[a/x])/(9*x^3) - ProductLog[a/x]^2/(3*x^3)} -{ProductLog[a/x]^2/x^5, x, 8, -(5/(32*x^4)) - 15/(1024*x^4*ProductLog[a/x]^4) + 15/(256*x^4*ProductLog[a/x]^3) - 15/(128*x^4*ProductLog[a/x]^2) + 5/(32*x^4*ProductLog[a/x]) + ProductLog[a/x]/(8*x^4) - ProductLog[a/x]^2/(4*x^4)} - - -{x^3*Sqrt[ProductLog[a/x]], x, 5, (-(256/105))*a^4*Sqrt[Pi]*Erf[2*Sqrt[ProductLog[a/x]]] + (2/7)*x^4*Sqrt[ProductLog[a/x]] - (2/35)*x^4*ProductLog[a/x]^(3/2) + (16/105)*x^4*ProductLog[a/x]^(5/2) - (128/105)*x^4*ProductLog[a/x]^(7/2)} -{x^2*Sqrt[ProductLog[a/x]], x, 4, (4/5)*a^3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ProductLog[a/x]]] + (2/5)*x^3*Sqrt[ProductLog[a/x]] - (2/15)*x^3*ProductLog[a/x]^(3/2) + (4/5)*x^3*ProductLog[a/x]^(5/2)} -{x*Sqrt[ProductLog[a/x]], x, 3, (-(2/3))*a^2*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ProductLog[a/x]]] + (2/3)*x^2*Sqrt[ProductLog[a/x]] - (2/3)*x^2*ProductLog[a/x]^(3/2)} -{Sqrt[ProductLog[a/x]], x, 2, a*Sqrt[Pi]*Erf[Sqrt[ProductLog[a/x]]] + 2*x*Sqrt[ProductLog[a/x]]} -{Sqrt[ProductLog[a/x]]/x, x, 2, -2*Sqrt[ProductLog[a/x]] - (2/3)*ProductLog[a/x]^(3/2)} -{Sqrt[ProductLog[a/x]]/x^2, x, 3, -((Sqrt[Pi]*Erfi[Sqrt[ProductLog[a/x]]])/(4*a)) + 1/(2*x*Sqrt[ProductLog[a/x]]) - Sqrt[ProductLog[a/x]]/x} -{Sqrt[ProductLog[a/x]]/x^3, x, 4, (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ProductLog[a/x]]])/(64*a^2) - 3/(32*x^2*ProductLog[a/x]^(3/2)) + 1/(8*x^2*Sqrt[ProductLog[a/x]]) - Sqrt[ProductLog[a/x]]/(2*x^2)} -{Sqrt[ProductLog[a/x]]/x^4, x, 5, -((5*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ProductLog[a/x]]])/(432*a^3)) + 5/(216*x^3*ProductLog[a/x]^(5/2)) - 5/(108*x^3*ProductLog[a/x]^(3/2)) + 1/(18*x^3*Sqrt[ProductLog[a/x]]) - Sqrt[ProductLog[a/x]]/(3*x^3)} - - -{x^3/Sqrt[ProductLog[a/x]], x, 6, (-(2048/945))*a^4*Sqrt[Pi]*Erf[2*Sqrt[ProductLog[a/x]]] + (2*x^4)/(9*Sqrt[ProductLog[a/x]]) + (2/63)*x^4*Sqrt[ProductLog[a/x]] - (16/315)*x^4*ProductLog[a/x]^(3/2) + (128/945)*x^4*ProductLog[a/x]^(5/2) - (1024/945)*x^4*ProductLog[a/x]^(7/2)} -{x^2/Sqrt[ProductLog[a/x]], x, 5, (24/35)*a^3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ProductLog[a/x]]] + (2*x^3)/(7*Sqrt[ProductLog[a/x]]) + (2/35)*x^3*Sqrt[ProductLog[a/x]] - (4/35)*x^3*ProductLog[a/x]^(3/2) + (24/35)*x^3*ProductLog[a/x]^(5/2)} -{x/Sqrt[ProductLog[a/x]], x, 4, (-(8/15))*a^2*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ProductLog[a/x]]] + (2*x^2)/(5*Sqrt[ProductLog[a/x]]) + (2/15)*x^2*Sqrt[ProductLog[a/x]] - (8/15)*x^2*ProductLog[a/x]^(3/2)} -{1/Sqrt[ProductLog[a/x]], x, 4, (2/3)*a*Sqrt[Pi]*Erf[Sqrt[ProductLog[a/x]]] + (2*x)/(3*Sqrt[ProductLog[a/x]]) + (2/3)*x*Sqrt[ProductLog[a/x]]} -{1/(x*Sqrt[ProductLog[a/x]]), x, 2, 2/Sqrt[ProductLog[a/x]] - 2*Sqrt[ProductLog[a/x]]} -{1/(x^2*Sqrt[ProductLog[a/x]]), x, 2, -((Sqrt[Pi]*Erfi[Sqrt[ProductLog[a/x]]])/(2*a)) - 1/(x*Sqrt[ProductLog[a/x]])} -{1/(x^3*Sqrt[ProductLog[a/x]]), x, 3, (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ProductLog[a/x]]])/(16*a^2) - 1/(8*x^2*ProductLog[a/x]^(3/2)) - 1/(2*x^2*Sqrt[ProductLog[a/x]])} -{1/(x^4*Sqrt[ProductLog[a/x]]), x, 4, -((Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ProductLog[a/x]]])/(72*a^3)) + 1/(36*x^3*ProductLog[a/x]^(5/2)) - 1/(18*x^3*ProductLog[a/x]^(3/2)) - 1/(3*x^3*Sqrt[ProductLog[a/x]])} - - -{x^2*(c*ProductLog[a/x])^p, x, 4, (3^(3 - p)*E^(4*ProductLog[a/x])*x^4*Gamma[-3 + p, 3*ProductLog[a/x]]*ProductLog[a/x]^(4 - p)*(c*ProductLog[a/x])^p)/a + (3^(2 - p)*E^(4*ProductLog[a/x])*x^4*Gamma[-2 + p, 3*ProductLog[a/x]]*ProductLog[a/x]^(3 - p)*(c*ProductLog[a/x])^(1 + p))/(a*c)} -{x*(c*ProductLog[a/x])^p, x, 4, (2^(2 - p)*E^(3*ProductLog[a/x])*x^3*Gamma[-2 + p, 2*ProductLog[a/x]]*ProductLog[a/x]^(3 - p)*(c*ProductLog[a/x])^p)/a + (2^(1 - p)*E^(3*ProductLog[a/x])*x^3*Gamma[-1 + p, 2*ProductLog[a/x]]*ProductLog[a/x]^(2 - p)*(c*ProductLog[a/x])^(1 + p))/(a*c)} -{(c*ProductLog[a/x])^p/x, x, 2, -((c*ProductLog[a/x])^p/p) - (c*ProductLog[a/x])^(1 + p)/(c*(1 + p))} -{(c*ProductLog[a/x])^p/x^2, x, 3, -((c*ProductLog[a/x])^p/x) + (p*Gamma[1 + p, -ProductLog[a/x]]*(c*ProductLog[a/x])^p)/((-ProductLog[a/x])^p*a)} -{(c*ProductLog[a/x])^p/x^3, x, 4, -((2^(-2 - p)*Gamma[2 + p, -2*ProductLog[a/x]]*(-ProductLog[a/x])^(-1 - p)*(c*ProductLog[a/x])^p)/(E^ProductLog[a/x]*(a*x))) - (2^(-3 - p)*Gamma[3 + p, -2*ProductLog[a/x]]*(-ProductLog[a/x])^(-2 - p)*(c*ProductLog[a/x])^(1 + p))/(E^ProductLog[a/x]*(a*c*x))} - - -(* ::Subsubsection::Closed:: *) -(*ProductLog[a x^n]^p*) - - -{ProductLog[a/x^(1/4)]^5, x, 2, (5/4)*x*ProductLog[a/x^(1/4)]^4 + x*ProductLog[a/x^(1/4)]^5} -{ProductLog[a/x^(1/3)]^4, x, 2, (4/3)*x*ProductLog[a/x^(1/3)]^3 + x*ProductLog[a/x^(1/3)]^4} -{ProductLog[a/Sqrt[x]]^3, x, 2, (3/2)*x*ProductLog[a/Sqrt[x]]^2 + x*ProductLog[a/Sqrt[x]]^3} -{ProductLog[a/x]^2, x, 2, 2*x*ProductLog[a/x] + x*ProductLog[a/x]^2} -{1/ProductLog[a*Sqrt[x]], x, 2, x/(2*ProductLog[a*Sqrt[x]]^2) + x/ProductLog[a*Sqrt[x]]} -{1/ProductLog[a*x^(1/3)]^2, x, 2, (2*x)/(3*ProductLog[a*x^(1/3)]^3) + x/ProductLog[a*x^(1/3)]^2} -{1/ProductLog[a*x^(1/4)]^3, x, 2, (3*x)/(4*ProductLog[a*x^(1/4)]^4) + x/ProductLog[a*x^(1/4)]^3} - - -{ProductLog[a/x^(1/5)]^4, x, 2, 20*a^5*ExpIntegralEi[-5*ProductLog[a/x^(1/5)]] + 5*x*ProductLog[a/x^(1/5)]^4} -{ProductLog[a/x^(1/4)]^3, x, 2, 12*a^4*ExpIntegralEi[-4*ProductLog[a/x^(1/4)]] + 4*x*ProductLog[a/x^(1/4)]^3} -{ProductLog[a/x^(1/3)]^2, x, 2, 6*a^3*ExpIntegralEi[-3*ProductLog[a/x^(1/3)]] + 3*x*ProductLog[a/x^(1/3)]^2} -{ProductLog[a/Sqrt[x]], x, 2, 2*a^2*ExpIntegralEi[-2*ProductLog[a/Sqrt[x]]] + 2*x*ProductLog[a/Sqrt[x]]} -{1/ProductLog[a*x]^2, x, 2, (2*ExpIntegralEi[ProductLog[a*x]])/a - x/ProductLog[a*x]^2} -{1/ProductLog[a*Sqrt[x]]^3, x, 2, (6*ExpIntegralEi[2*ProductLog[a*Sqrt[x]]])/a^2 - (2*x)/ProductLog[a*Sqrt[x]]^3} -{1/ProductLog[a*x^(1/3)]^4, x, 2, (12*ExpIntegralEi[3*ProductLog[a*x^(1/3)]])/a^3 - (3*x)/ProductLog[a*x^(1/3)]^4} -{1/ProductLog[a*x^(1/4)]^5, x, 2, (20*ExpIntegralEi[4*ProductLog[a*x^(1/4)]])/a^4 - (4*x)/ProductLog[a*x^(1/4)]^5} - - -{ProductLog[a*x^n]^((-1 + n)/n), x, 2, ((1 - n)*x)/ProductLog[a*x^n]^n^(-1) + x/ProductLog[a*x^n]^((1 - n)/n)} -{ProductLog[a*x^(1/(1 - p))]^p, x, 2, x*ProductLog[a*x^(1/(1 - p))]^p - (p/(1 - p))*x*ProductLog[a*x^(1/(1 - p))]^(p - 1)} - - -(* ::Subsubsection::Closed:: *) -(*x^m ProductLog[a x^n]^p*) - - -{x^(-1 - n)*(c*ProductLog[a*x^n])^(9/2), x, 5, (135*a*c^(9/2)*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(16*n) - (135*c^3*(c*ProductLog[a*x^n])^(3/2))/(x^n*(8*n)) - (45*c^2*(c*ProductLog[a*x^n])^(5/2))/(x^n*(4*n)) - (9*c*(c*ProductLog[a*x^n])^(7/2))/(x^n*(2*n)) - (c*ProductLog[a*x^n])^(9/2)/(x^n*n)} -{x^(-1 - n)*(c*ProductLog[a*x^n])^(7/2), x, 4, (21*a*c^(7/2)*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(8*n) - (21*c^2*(c*ProductLog[a*x^n])^(3/2))/(x^n*(4*n)) - (7*c*(c*ProductLog[a*x^n])^(5/2))/(x^n*(2*n)) - (c*ProductLog[a*x^n])^(7/2)/(x^n*n)} -{x^(-1 - n)*(c*ProductLog[a*x^n])^(5/2), x, 3, (5*a*c^(5/2)*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(4*n) - (5*c*(c*ProductLog[a*x^n])^(3/2))/(x^n*(2*n)) - (c*ProductLog[a*x^n])^(5/2)/(x^n*n)} -{x^(-1 - n)*(c*ProductLog[a*x^n])^(3/2), x, 2, (3*a*c^(3/2)*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(2*n) - (c*ProductLog[a*x^n])^(3/2)/(x^n*n)} -{x^(-1 - n)*(c*ProductLog[a*x^n])^(1/2), x, 2, -((a*Sqrt[c]*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/n) - (2*Sqrt[c*ProductLog[a*x^n]])/(x^n*n)} -{x^(-1 - n)/(c*ProductLog[a*x^n])^(1/2), x, 3, -((2*a*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(3*Sqrt[c]*n)) - 2/(x^n*(3*n*Sqrt[c*ProductLog[a*x^n]])) - (2*Sqrt[c*ProductLog[a*x^n]])/(x^n*(3*c*n))} -{x^(-1 - n)/(c*ProductLog[a*x^n])^(3/2), x, 4, (4*a*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(5*c^(3/2)*n) - 2/(x^n*(5*n*(c*ProductLog[a*x^n])^(3/2))) - 2/(x^n*(5*c*n*Sqrt[c*ProductLog[a*x^n]])) + (4*Sqrt[c*ProductLog[a*x^n]])/(x^n*(5*c^2*n))} -{x^(-1 - n)/(c*ProductLog[a*x^n])^(5/2), x, 5, -((8*a*Sqrt[Pi]*Erf[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(21*c^(5/2)*n)) - 2/(x^n*(7*n*(c*ProductLog[a*x^n])^(5/2))) - 2/(x^n*(7*c*n*(c*ProductLog[a*x^n])^(3/2))) + 4/(x^n*(21*c^2*n*Sqrt[c*ProductLog[a*x^n]])) - (8*Sqrt[c*ProductLog[a*x^n]])/(x^n*(21*c^3*n))} - - -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(11/2), x, 5, (165*a^2*c^(11/2)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(256*n) - (165*c^3*(c*ProductLog[a*x^n])^(5/2))/(x^(2*n)*(128*n)) - (55*c^2*(c*ProductLog[a*x^n])^(7/2))/(x^(2*n)*(32*n)) - (11*c*(c*ProductLog[a*x^n])^(9/2))/(x^(2*n)*(8*n)) - (c*ProductLog[a*x^n])^(11/2)/(x^(2*n)*(2*n))} -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(9/2), x, 4, (27*a^2*c^(9/2)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(64*n) - (27*c^2*(c*ProductLog[a*x^n])^(5/2))/(x^(2*n)*(32*n)) - (9*c*(c*ProductLog[a*x^n])^(7/2))/(x^(2*n)*(8*n)) - (c*ProductLog[a*x^n])^(9/2)/(x^(2*n)*(2*n))} -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(7/2), x, 3, (7*a^2*c^(7/2)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(16*n) - (7*c*(c*ProductLog[a*x^n])^(5/2))/(x^(2*n)*(8*n)) - (c*ProductLog[a*x^n])^(7/2)/(x^(2*n)*(2*n))} -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(5/2), x, 2, (5*a^2*c^(5/2)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(4*n) - (c*ProductLog[a*x^n])^(5/2)/(x^(2*n)*(2*n))} -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(3/2), x, 2, -((3*a^2*c^(3/2)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/n) - (2*(c*ProductLog[a*x^n])^(3/2))/(x^(2*n)*n)} -{x^(-1 - 2*n)*(c*ProductLog[a*x^n])^(1/2), x, 3, (2*a^2*Sqrt[c]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(3*n) - (2*Sqrt[c*ProductLog[a*x^n]])/(x^(2*n)*(3*n)) + (2*(c*ProductLog[a*x^n])^(3/2))/(x^(2*n)*(3*c*n))} -{x^(-1 - 2*n)/(c*ProductLog[a*x^n])^(1/2), x, 4, (8*a^2*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(15*Sqrt[c]*n) - 2/(x^(2*n)*(5*n*Sqrt[c*ProductLog[a*x^n]])) - (2*Sqrt[c*ProductLog[a*x^n]])/(x^(2*n)*(15*c*n)) + (8*(c*ProductLog[a*x^n])^(3/2))/(x^(2*n)*(15*c^2*n))} -{x^(-1 - 2*n)/(c*ProductLog[a*x^n])^(3/2), x, 5, -((32*a^2*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(35*c^(3/2)*n)) - 2/(x^(2*n)*(7*n*(c*ProductLog[a*x^n])^(3/2))) - 6/(x^(2*n)*(35*c*n*Sqrt[c*ProductLog[a*x^n]])) + (8*Sqrt[c*ProductLog[a*x^n]])/(x^(2*n)*(35*c^2*n)) - (32*(c*ProductLog[a*x^n])^(3/2))/(x^(2*n)*(35*c^3*n))} - - -{x^(-1 + n)*(c*ProductLog[a*x^n])^(5/2), x, 5, (75*c^(5/2)*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(16*a*n) - (75*c^3*x^n)/(8*n*Sqrt[c*ProductLog[a*x^n]]) + (25*c^2*x^n*Sqrt[c*ProductLog[a*x^n]])/(4*n) - (5*c*x^n*(c*ProductLog[a*x^n])^(3/2))/(2*n) + (x^n*(c*ProductLog[a*x^n])^(5/2))/n} -{x^(-1 + n)*(c*ProductLog[a*x^n])^(3/2), x, 4, -((9*c^(3/2)*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(8*a*n)) + (9*c^2*x^n)/(4*n*Sqrt[c*ProductLog[a*x^n]]) - (3*c*x^n*Sqrt[c*ProductLog[a*x^n]])/(2*n) + (x^n*(c*ProductLog[a*x^n])^(3/2))/n} -{x^(-1 + n)*(c*ProductLog[a*x^n])^(1/2), x, 3, (Sqrt[c]*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(4*a*n) - (c*x^n)/(2*n*Sqrt[c*ProductLog[a*x^n]]) + (x^n*Sqrt[c*ProductLog[a*x^n]])/n} -{x^(-1 + n)/(c*ProductLog[a*x^n])^(1/2), x, 2, (Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(2*a*Sqrt[c]*n) + x^n/(n*Sqrt[c*ProductLog[a*x^n]])} -{x^(-1 + n)/(c*ProductLog[a*x^n])^(3/2), x, 2, (3*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(a*c^(3/2)*n) - (2*x^n)/(n*(c*ProductLog[a*x^n])^(3/2))} -{x^(-1 + n)/(c*ProductLog[a*x^n])^(5/2), x, 3, (10*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(3*a*c^(5/2)*n) - (2*x^n)/(3*n*(c*ProductLog[a*x^n])^(5/2)) - (10*x^n)/(3*c*n*(c*ProductLog[a*x^n])^(3/2))} -{x^(-1 + n)/(c*ProductLog[a*x^n])^(7/2), x, 4, (28*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(15*a*c^(7/2)*n) - (2*x^n)/(5*n*(c*ProductLog[a*x^n])^(7/2)) - (14*x^n)/(15*c*n*(c*ProductLog[a*x^n])^(5/2)) - (28*x^n)/(15*c^2*n*(c*ProductLog[a*x^n])^(3/2))} -{x^(-1 + n)/(c*ProductLog[a*x^n])^(9/2), x, 5, (24*Sqrt[Pi]*Erfi[Sqrt[c*ProductLog[a*x^n]]/Sqrt[c]])/(35*a*c^(9/2)*n) - (2*x^n)/(7*n*(c*ProductLog[a*x^n])^(9/2)) - (18*x^n)/(35*c*n*(c*ProductLog[a*x^n])^(7/2)) - (12*x^n)/(35*c^2*n*(c*ProductLog[a*x^n])^(5/2)) - (24*x^n)/(35*c^3*n*(c*ProductLog[a*x^n])^(3/2))} - - -{x^(-1 + 2*n)*(c*ProductLog[a*x^n])^(3/2), x, 5, (45*c^(3/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(256*a^2*n) - (45*c^3*x^(2*n))/(128*n*(c*ProductLog[a*x^n])^(3/2)) + (15*c^2*x^(2*n))/(32*n*Sqrt[c*ProductLog[a*x^n]]) - (3*c*x^(2*n)*Sqrt[c*ProductLog[a*x^n]])/(8*n) + (x^(2*n)*(c*ProductLog[a*x^n])^(3/2))/(2*n)} -{x^(-1 + 2*n)*(c*ProductLog[a*x^n])^(1/2), x, 4, -((3*Sqrt[c]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(64*a^2*n)) + (3*c^2*x^(2*n))/(32*n*(c*ProductLog[a*x^n])^(3/2)) - (c*x^(2*n))/(8*n*Sqrt[c*ProductLog[a*x^n]]) + (x^(2*n)*Sqrt[c*ProductLog[a*x^n]])/(2*n)} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(1/2), x, 3, -((Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(16*a^2*Sqrt[c]*n)) + (c*x^(2*n))/(8*n*(c*ProductLog[a*x^n])^(3/2)) + x^(2*n)/(2*n*Sqrt[c*ProductLog[a*x^n]])} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(3/2), x, 2, (3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(4*a^2*c^(3/2)*n) + x^(2*n)/(2*n*(c*ProductLog[a*x^n])^(3/2))} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(5/2), x, 2, (5*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(a^2*c^(5/2)*n) - (2*x^(2*n))/(n*(c*ProductLog[a*x^n])^(5/2))} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(7/2), x, 3, (14*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(3*a^2*c^(7/2)*n) - (2*x^(2*n))/(3*n*(c*ProductLog[a*x^n])^(7/2)) - (14*x^(2*n))/(3*c*n*(c*ProductLog[a*x^n])^(5/2))} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(9/2), x, 4, (24*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(5*a^2*c^(9/2)*n) - (2*x^(2*n))/(5*n*(c*ProductLog[a*x^n])^(9/2)) - (6*x^(2*n))/(5*c*n*(c*ProductLog[a*x^n])^(7/2)) - (24*x^(2*n))/(5*c^2*n*(c*ProductLog[a*x^n])^(5/2))} -{x^(-1 + 2*n)/(c*ProductLog[a*x^n])^(11/2), x, 5, (352*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[c*ProductLog[a*x^n]])/Sqrt[c]])/(105*a^2*c^(11/2)*n) - (2*x^(2*n))/(7*n*(c*ProductLog[a*x^n])^(11/2)) - (22*x^(2*n))/(35*c*n*(c*ProductLog[a*x^n])^(9/2)) - (88*x^(2*n))/(105*c^2*n*(c*ProductLog[a*x^n])^(7/2)) - (352*x^(2*n))/(105*c^3*n*(c*ProductLog[a*x^n])^(5/2))} - - -{ProductLog[a*x^n]^4/x^(3*n + 1), x, 2, -((4*ProductLog[a*x^n]^3)/(x^(3*n)*(9*n))) - ProductLog[a*x^n]^4/(x^(3*n)*(3*n))} -{ProductLog[a*x^n]^3/x^(2*n + 1), x, 2, -((3*ProductLog[a*x^n]^2)/(x^(2*n)*(4*n))) - ProductLog[a*x^n]^3/(x^(2*n)*(2*n))} -{ProductLog[a*x^n]^2/x^(n + 1), x, 2, -((2*ProductLog[a*x^n])/(x^n*n)) - ProductLog[a*x^n]^2/(x^n*n)} -{x^(2*n - 1)/ProductLog[a*x^n], x, 2, x^(2*n)/(4*n*ProductLog[a*x^n]^2) + x^(2*n)/(2*n*ProductLog[a*x^n])} -{x^(3*n - 1)/ProductLog[a*x^n]^2, x, 2, (2*x^(3*n))/(9*n*ProductLog[a*x^n]^3) + x^(3*n)/(3*n*ProductLog[a*x^n]^2)} -{x^(4*n - 1)/ProductLog[a*x^n]^3, x, 2, (3*x^(4*n))/(16*n*ProductLog[a*x^n]^4) + x^(4*n)/(4*n*ProductLog[a*x^n]^3)} - - -{x^(-1 - n*(1 + p))*(c*ProductLog[a*x^n])^p, x, 2, -((c*ProductLog[a*x^n])^p/(x^(n*(1 + p))*n)) - (p*(c*ProductLog[a*x^n])^p*CannotIntegrate[(x^(-1 - n*(1 + p))*ProductLog[a*x^n]^(1 + p))/(1 + ProductLog[a*x^n]), x])/ProductLog[a*x^n]^p} -{x^(-1 + n*(0 - p))*(c*ProductLog[a*x^n])^p, x, 2, -((c*ProductLog[a*x^n])^p/(x^(n*p)*(n*p))) + (CannotIntegrate[(x^(-1 - n*p)*ProductLog[a*x^n]^p)/(1 + ProductLog[a*x^n]), x]*(c*ProductLog[a*x^n])^p)/ProductLog[a*x^n]^p} -{x^(-1 + n*(1 - p))*(c*ProductLog[a*x^n])^p, x, 2, -((c*p*x^(n*(1 - p))*(c*ProductLog[a*x^n])^(-1 + p))/(n*(1 - p)^2)) + (x^(n*(1 - p))*(c*ProductLog[a*x^n])^p)/(n*(1 - p))} -{x^(-1 + n*(2 - p))*(c*ProductLog[a*x^n])^p, x, 3, (c^2*p*x^(n*(2 - p))*(c*ProductLog[a*x^n])^(-2 + p))/(n*(2 - p)^3) - (c*p*x^(n*(2 - p))*(c*ProductLog[a*x^n])^(-1 + p))/(n*(2 - p)^2) + (x^(n*(2 - p))*(c*ProductLog[a*x^n])^p)/(n*(2 - p))} -{x^(-1 + n*(3 - p))*(c*ProductLog[a*x^n])^p, x, 4, -((2*c^3*p*x^(n*(3 - p))*(c*ProductLog[a*x^n])^(-3 + p))/(n*(3 - p)^4)) + (2*c^2*p*x^(n*(3 - p))*(c*ProductLog[a*x^n])^(-2 + p))/(n*(3 - p)^3) - (c*p*x^(n*(3 - p))*(c*ProductLog[a*x^n])^(-1 + p))/(n*(3 - p)^2) + (x^(n*(3 - p))*(c*ProductLog[a*x^n])^p)/(n*(3 - p))} - - -(* ::Subsubsection::Closed:: *) -(*x^m / (1+ProductLog[a x^n])*) - - -{x^3/(1 + ProductLog[a*x]), x, 4, -((3*x^4)/(128*ProductLog[a*x]^4)) + (3*x^4)/(32*ProductLog[a*x]^3) - (3*x^4)/(16*ProductLog[a*x]^2) + x^4/(4*ProductLog[a*x])} -{x^2/(1 + ProductLog[a*x]), x, 3, (2*x^3)/(27*ProductLog[a*x]^3) - (2*x^3)/(9*ProductLog[a*x]^2) + x^3/(3*ProductLog[a*x])} -{x/(1 + ProductLog[a*x]), x, 2, -(x^2/(4*ProductLog[a*x]^2)) + x^2/(2*ProductLog[a*x])} -{1/(1 + ProductLog[a*x]), x, 1, a*x/(a*ProductLog[a*x])} -{1/(x*(1 + ProductLog[a*x])), x, 1, Log[ProductLog[a*x]]} -{1/(x^2*(1 + ProductLog[a*x])), x, 2, -(1/x) - a*ExpIntegralEi[-ProductLog[a*x]]} -{1/(x^3*(1 + ProductLog[a*x])), x, 3, -(1/(2*x^2)) + 2*a^2*ExpIntegralEi[-2*ProductLog[a*x]] + ProductLog[a*x]/x^2} -{1/(x^4*(1 + ProductLog[a*x])), x, 4, -(1/(3*x^3)) - (9/2)*a^3*ExpIntegralEi[-3*ProductLog[a*x]] + ProductLog[a*x]/(2*x^3) - (3*ProductLog[a*x]^2)/(2*x^3)} - - -{x^3/(1 + ProductLog[a*x^2]), x, 3, -(x^4/(8*ProductLog[a*x^2]^2)) + x^4/(4*ProductLog[a*x^2])} -{x^2/(1 + ProductLog[a*x^2]), x, 0, CannotIntegrate[x^2/(1 + ProductLog[a*x^2]), x]} -{x/(1 + ProductLog[a*x^2]), x, 2, x^2/(2*ProductLog[a*x^2])} -{1/(1 + ProductLog[a*x^2]), x, 0, CannotIntegrate[1/(1 + ProductLog[a*x^2]), x]} -{1/(x*(1 + ProductLog[a*x^2])), x, 1, (1/2)*Log[ProductLog[a*x^2]]} -{1/(x^2*(1 + ProductLog[a*x^2])), x, 0, CannotIntegrate[1/(x^2*(1 + ProductLog[a*x^2])), x]} -{1/(x^3*(1 + ProductLog[a*x^2])), x, 3, -(1/(2*x^2)) - (1/2)*a*ExpIntegralEi[-ProductLog[a*x^2]]} -{1/(x^4*(1 + ProductLog[a*x^2])), x, 0, CannotIntegrate[1/(x^4*(1 + ProductLog[a*x^2])), x]} - - -{x^3/(1 + ProductLog[a/x]), x, 6, x^4/4 - (32/3)*a^4*ExpIntegralEi[-4*ProductLog[a/x]] - (1/3)*x^4*ProductLog[a/x] + (2/3)*x^4*ProductLog[a/x]^2 - (8/3)*x^4*ProductLog[a/x]^3} -{x^2/(1 + ProductLog[a/x]), x, 5, x^3/3 + (9/2)*a^3*ExpIntegralEi[-3*ProductLog[a/x]] - (1/2)*x^3*ProductLog[a/x] + (3/2)*x^3*ProductLog[a/x]^2} -{x/(1 + ProductLog[a/x]), x, 4, x^2/2 - 2*a^2*ExpIntegralEi[-2*ProductLog[a/x]] - x^2*ProductLog[a/x]} -{1/(1 + ProductLog[a/x]), x, 3, x + a*ExpIntegralEi[-ProductLog[a/x]]} -{1/(x*(1 + ProductLog[a/x])), x, 1, -Log[ProductLog[a/x]]} -{1/(x^2*(1 + ProductLog[a/x])), x, 2, -(1/(x*ProductLog[a/x]))} -{1/(x^3*(1 + ProductLog[a/x])), x, 3, 1/(4*x^2*ProductLog[a/x]^2) - 1/(2*x^2*ProductLog[a/x])} -{1/(x^4*(1 + ProductLog[a/x])), x, 4, -(2/(27*x^3*ProductLog[a/x]^3)) + 2/(9*x^3*ProductLog[a/x]^2) - 1/(3*x^3*ProductLog[a/x])} - - -{x^5/(1 + ProductLog[a/x^2]), x, 6, x^6/6 + (9/4)*a^3*ExpIntegralEi[-3*ProductLog[a/x^2]] - (1/4)*x^6*ProductLog[a/x^2] + (3/4)*x^6*ProductLog[a/x^2]^2} -{x^3/(1 + ProductLog[a/x^2]), x, 5, x^4/4 - a^2*ExpIntegralEi[-2*ProductLog[a/x^2]] - (1/2)*x^4*ProductLog[a/x^2]} -{x^1/(1 + ProductLog[a/x^2]), x, 4, x^2/2 + (1/2)*a*ExpIntegralEi[-ProductLog[a/x^2]]} -{1/(x^1*(1 + ProductLog[a/x^2])), x, 1, (-(1/2))*Log[ProductLog[a/x^2]]} -{1/(x^3*(1 + ProductLog[a/x^2])), x, 3, -(1/(2*x^2*ProductLog[a/x^2]))} - -{x^4/(1 + ProductLog[a/x^2]), x, 1, CannotIntegrate[x^4/(1 + ProductLog[a/x^2]), x]} -{x^2/(1 + ProductLog[a/x^2]), x, 1, CannotIntegrate[x^2/(1 + ProductLog[a/x^2]), x]} -{x^0/(1 + ProductLog[a/x^2]), x, 1, CannotIntegrate[1/(1 + ProductLog[a/x^2]), x]} -{1/(x^2*(1 + ProductLog[a/x^2])), x, 1, CannotIntegrate[1/(x^2*(1 + ProductLog[a/x^2])), x]} -{1/(x^4*(1 + ProductLog[a/x^2])), x, 1, CannotIntegrate[1/(x^4*(1 + ProductLog[a/x^2])), x]} - - -{x^m/(d + d*ProductLog[a*x]), x, 1, (x^m*Gamma[1 + m, (-(1 + m)*ProductLog[a*x])])/(E^(m*ProductLog[a*x])*(-(1 + m)*ProductLog[a*x])^m*(a*d*(1 + m)))} - - -(* ::Subsubsection::Closed:: *) -(*ProductLog[a x^n]^p / (1+ProductLog[a x^n])*) - - -{ProductLog[a/x^(1/4)]^5/(1 + ProductLog[a/x^(1/4)]), x, 1, x*ProductLog[a/x^(1/4)]^4} -{ProductLog[a/x^(1/3)]^4/(1 + ProductLog[a/x^(1/3)]), x, 1, x*ProductLog[a/x^(1/3)]^3} -{ProductLog[a/Sqrt[x]]^3/(1 + ProductLog[a/Sqrt[x]]), x, 1, x*ProductLog[a/Sqrt[x]]^2} -{ProductLog[a/x]^2/(1 + ProductLog[a/x]), x, 1, x*ProductLog[a/x]} -{1/(ProductLog[a*Sqrt[x]]*(1 + ProductLog[a*Sqrt[x]])), x, 1, x/ProductLog[a*Sqrt[x]]^2} -{1/(ProductLog[a*x^(1/3)]^2*(1 + ProductLog[a*x^(1/3)])), x, 1, x/ProductLog[a*x^(1/3)]^3} -{1/(ProductLog[a*x^(1/4)]^3*(1 + ProductLog[a*x^(1/4)])), x, 1, x/ProductLog[a*x^(1/4)]^4} - - -{ProductLog[a/x^(1/4)]^4/(1 + ProductLog[a/x^(1/4)]), x, 1, -4*a^4*ExpIntegralEi[-4*ProductLog[a/x^(1/4)]]} -{ProductLog[a/x^(1/3)]^3/(1 + ProductLog[a/x^(1/3)]), x, 1, -3*a^3*ExpIntegralEi[-3*ProductLog[a/x^(1/3)]]} -{ProductLog[a/Sqrt[x]]^2/(1 + ProductLog[a/Sqrt[x]]), x, 1, -2*a^2*ExpIntegralEi[-2*ProductLog[a/Sqrt[x]]]} -{ProductLog[a/x]/(1 + ProductLog[a/x]), x, 1, (-a)*ExpIntegralEi[-ProductLog[a/x]]} -{1/(ProductLog[a*x]*(1 + ProductLog[a*x])), x, 1, ExpIntegralEi[ProductLog[a*x]]/a} -{1/(ProductLog[a*Sqrt[x]]^2*(1 + ProductLog[a*Sqrt[x]])), x, 1, (2*ExpIntegralEi[2*ProductLog[a*Sqrt[x]]])/a^2} -{1/(ProductLog[a*x^(1/3)]^3*(1 + ProductLog[a*x^(1/3)])), x, 1, (3*ExpIntegralEi[3*ProductLog[a*x^(1/3)]])/a^3} -{1/(ProductLog[a*x^(1/4)]^4*(1 + ProductLog[a*x^(1/4)])), x, 1, (4*ExpIntegralEi[4*ProductLog[a*x^(1/4)]])/a^4} - - -{ProductLog[a*x^n]^(1 - 1/n)/(1 + ProductLog[a*x^n]), x, 1, x/ProductLog[a*x^n]^(n^(-1))} -{ProductLog[a*x^(1/(1 - p))]^p/(1 + ProductLog[a*x^(1/(1 - p))]), x, 1, x*ProductLog[a*x^(1/(1 - p))]^(p - 1)}